{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T19:38:09Z","timestamp":1776800289959,"version":"3.51.2"},"reference-count":18,"publisher":"American Mathematical Society (AMS)","issue":"300","license":[{"start":{"date-parts":[[2016,10,21]],"date-time":"2016-10-21T00:00:00Z","timestamp":1477008000000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    This article focuses on the theta series on the 6-fold cover of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper G normal upper L Subscript 2\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"normal\">G<\/mml:mi>\n                              <mml:mi mathvariant=\"normal\">L<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathrm {GL}_2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We investigate the Fourier coefficients\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"tau left-parenthesis r right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03c4\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>r<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\tau (r)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of the theta series, and give partially proven, partially conjectured values for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"tau left-parenthesis pi right-parenthesis squared\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03c4\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>\n                              \u03c0\n                              \n                            <\/mml:mi>\n                            <mml:msup>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\tau (\\pi )^2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"tau left-parenthesis pi squared right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03c4\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>\n                                \u03c0\n                                \n                              <\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\tau (\\pi ^2)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"tau left-parenthesis pi Superscript 4 Baseline right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03c4\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>\n                                \u03c0\n                                \n                              <\/mml:mi>\n                              <mml:mn>4<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\tau (\\pi ^4)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for prime values\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"pi\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03c0\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\pi<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/mcom3044","type":"journal-article","created":{"date-parts":[[2015,2,11]],"date-time":"2015-02-11T13:10:27Z","timestamp":1423660227000},"page":"1901-1927","source":"Crossref","is-referenced-by-count":2,"title":["Fourier coefficients of sextic theta series"],"prefix":"10.1090","volume":"85","author":[{"given":"Reinier","family":"Br\u00f6ker","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jeff","family":"Hoffstein","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2015,10,21]]},"reference":[{"issue":"78","key":"1","doi-asserted-by":"publisher","first-page":"4211","DOI":"10.1155\/S1073792804133473","article-title":"Nonvanishing twists of GL(2) automorphic \ud835\udc3f-functions","author":"Brubaker, Ben","year":"2004","journal-title":"Int. Math. Res. Not.","ISSN":"https:\/\/id.crossref.org\/issn\/1073-7928","issn-type":"print"},{"key":"2","volume-title":"Algebraic number theory","year":"1967"},{"key":"3","isbn-type":"print","doi-asserted-by":"publisher","first-page":"149","DOI":"10.1007\/978-1-4614-1219-9_6","article-title":"Double Dirichlet series and theta functions","author":"Chinta, Gautam","year":"2012","ISBN":"https:\/\/id.crossref.org\/isbn\/9781461412182"},{"key":"4","series-title":"Graduate Texts in Mathematics","isbn-type":"print","volume-title":"Number theory. Vol. II. Analytic and modern tools","volume":"240","author":"Cohen, Henri","year":"2007","ISBN":"https:\/\/id.crossref.org\/isbn\/9780387498935"},{"key":"5","unstructured":"C. Eckhardt, Eine Vermuting \u00fcber biquadratische Thetareihen und ihre numerische Untersuchung, PhD thesis, University of G\u00f6ttingen, 1989."},{"issue":"2","key":"6","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1112\/plms\/s3-64.2.225","article-title":"On the Fourier coefficients of biquadratic theta series","volume":"64","author":"Eckhardt, C.","year":"1992","journal-title":"Proc. London Math. Soc. (3)","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6115","issn-type":"print"},{"key":"7","unstructured":"S. Friedberg, D. Ginzburg, Descent and theta functions for metaplectic groups, preprint, 2014."},{"key":"8","isbn-type":"print","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1090\/crmp\/001\/02","article-title":"Eisenstein series and theta functions on the metaplectic group","author":"Hoffstein, Jeff","year":"1993","ISBN":"https:\/\/id.crossref.org\/isbn\/0821869973"},{"issue":"59","key":"9","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1007\/BF02698770","article-title":"Metaplectic forms","author":"Kazhdan, D. A.","year":"1984","journal-title":"Inst. Hautes \\'{E}tudes Sci. Publ. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0073-8301","issn-type":"print"},{"key":"10","unstructured":"T. Kubota, On Automorphic Forms and the Reciprocity Law in a Number Field, Springer, Lecture Notes Math., (1973), p. 348."},{"key":"11","series-title":"Grundlehren der Mathematischen Wissenschaften","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-03983-0","volume-title":"Algebraic Number Theory","volume":"322","author":"Neukirch, J\u00fcrgen","year":"1999"},{"key":"12","doi-asserted-by":"publisher","first-page":"125","DOI":"10.1515\/crll.1977.296.125","article-title":"A cubic analogue of the theta series","volume":"296","author":"Patterson, S. J.","year":"1977","journal-title":"J. Reine Angew. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0075-4102","issn-type":"print"},{"key":"13","first-page":"1","article-title":"A heuristic principle and applications to Gauss sums","volume":"52","author":"Patterson, S. J.","year":"1987","journal-title":"J. Indian Math. Soc. (N.S.)","ISSN":"https:\/\/id.crossref.org\/issn\/0019-5839","issn-type":"print"},{"issue":"2","key":"14","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1112\/plms\/s3-54.2.193","article-title":"The distribution of general Gauss sums and similar arithmetic functions at prime arguments","volume":"54","author":"Patterson, S. J.","year":"1987","journal-title":"Proc. London Math. Soc. (3)","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6115","issn-type":"print"},{"key":"15","doi-asserted-by":"publisher","first-page":"111","DOI":"10.1515\/crll.1979.310.111","article-title":"The distribution of Kummer sums at prime arguments","volume":"310","author":"Heath-Brown, D. R.","year":"1979","journal-title":"J. Reine Angew. 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Wellhausen, Fourierkoeffizienten von Thetafunktionen sechster Ordnung, PhD thesis, University of G\u00f6ttingen, 1996."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2016-85-300\/S0025-5718-2015-03044-0\/S0025-5718-2015-03044-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2016-85-300\/S0025-5718-2015-03044-0\/S0025-5718-2015-03044-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:49:44Z","timestamp":1776797384000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2016-85-300\/S0025-5718-2015-03044-0\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,10,21]]},"references-count":18,"journal-issue":{"issue":"300","published-print":{"date-parts":[[2016,7]]}},"alternative-id":["S0025-5718-2015-03044-0"],"URL":"https:\/\/doi.org\/10.1090\/mcom3044","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2015,10,21]]}}}