Properties

Label 3.44.a.a
Level 3
Weight 44
Character orbit 3.a
Self dual Yes
Analytic conductor 35.133
Analytic rank 1
Dimension 3
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 44 \)
Character orbit: \([\chi]\) = 3.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(35.1331186037\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{5}\cdot 5\cdot 11 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 1619008 - \beta_{1} ) q^{2} \) \( + 10460353203 q^{3} \) \( + ( -593686009856 - 3392488 \beta_{1} + 4 \beta_{2} ) q^{4} \) \( + ( -169251107035090 + 87483970 \beta_{1} - 1317 \beta_{2} ) q^{5} \) \( + ( 16935395518482624 - 10460353203 \beta_{1} ) q^{6} \) \( + ( -544389767902363096 + 407863957886 \beta_{1} + 1184197 \beta_{2} ) q^{7} \) \( + ( 3732086898157617152 + 606520958464 \beta_{1} + 19428096 \beta_{2} ) q^{8} \) \( + 109418989131512359209 q^{9} \) \(+O(q^{10})\) \( q\) \(+(1619008 - \beta_{1}) q^{2}\) \(+10460353203 q^{3}\) \(+(-593686009856 - 3392488 \beta_{1} + 4 \beta_{2}) q^{4}\) \(+(-169251107035090 + 87483970 \beta_{1} - 1317 \beta_{2}) q^{5}\) \(+(16935395518482624 - 10460353203 \beta_{1}) q^{6}\) \(+(-544389767902363096 + 407863957886 \beta_{1} + 1184197 \beta_{2}) q^{7}\) \(+(3732086898157617152 + 606520958464 \beta_{1} + 19428096 \beta_{2}) q^{8}\) \(+\)\(10\!\cdots\!09\)\( q^{9}\) \(+(-\)\(76\!\cdots\!40\)\( + 1235354107077970 \beta_{1} - 2278729792 \beta_{2}) q^{10}\) \(+(-\)\(92\!\cdots\!40\)\( + 7617306526252796 \beta_{1} - 14933373558 \beta_{2}) q^{11}\) \(+(-\)\(62\!\cdots\!68\)\( - 35486622716939064 \beta_{1} + 41841412812 \beta_{2}) q^{12}\) \(+(-\)\(33\!\cdots\!50\)\( - 202202396925505444 \beta_{1} + 641923722778 \beta_{2}) q^{13}\) \(+(-\)\(31\!\cdots\!76\)\( + 448634534786039896 \beta_{1} + 102843306048 \beta_{2}) q^{14}\) \(+(-\)\(17\!\cdots\!70\)\( + 915113225800655910 \beta_{1} - 13776285168351 \beta_{2}) q^{15}\) \(+(\)\(78\!\cdots\!12\)\( + 13746042266337042432 \beta_{1} - 9157309919232 \beta_{2}) q^{16}\) \(+(-\)\(53\!\cdots\!94\)\( + 90683027194690706820 \beta_{1} + 201871787631414 \beta_{2}) q^{17}\) \(+(\)\(17\!\cdots\!72\)\( - \)\(10\!\cdots\!09\)\( \beta_{1}) q^{18}\) \(+(-\)\(10\!\cdots\!88\)\( + \)\(40\!\cdots\!04\)\( \beta_{1} - 2147995640621886 \beta_{2}) q^{19}\) \(+(-\)\(66\!\cdots\!20\)\( + \)\(37\!\cdots\!60\)\( \beta_{1} + 3305756267279544 \beta_{2}) q^{20}\) \(+(-\)\(56\!\cdots\!88\)\( + \)\(42\!\cdots\!58\)\( \beta_{1} + 12387118881932991 \beta_{2}) q^{21}\) \(+(-\)\(57\!\cdots\!08\)\( + \)\(33\!\cdots\!40\)\( \beta_{1} - 52339689282150272 \beta_{2}) q^{22}\) \(+(\)\(38\!\cdots\!68\)\( - \)\(60\!\cdots\!28\)\( \beta_{1} + 32453105180134026 \beta_{2}) q^{23}\) \(+(\)\(39\!\cdots\!56\)\( + \)\(63\!\cdots\!92\)\( \beta_{1} + 203224746221791488 \beta_{2}) q^{24}\) \(+(\)\(60\!\cdots\!75\)\( - \)\(14\!\cdots\!00\)\( \beta_{1} - 758897906364611660 \beta_{2}) q^{25}\) \(+(\)\(59\!\cdots\!12\)\( - \)\(47\!\cdots\!90\)\( \beta_{1} + 1748929988964422784 \beta_{2}) q^{26}\) \(+\)\(11\!\cdots\!27\)\( q^{27}\) \(+(-\)\(28\!\cdots\!48\)\( + \)\(29\!\cdots\!48\)\( \beta_{1} - 12060227383697492832 \beta_{2}) q^{28}\) \(+(-\)\(94\!\cdots\!86\)\( - \)\(12\!\cdots\!42\)\( \beta_{1} + 20892595961060032401 \beta_{2}) q^{29}\) \(+(-\)\(79\!\cdots\!20\)\( + \)\(12\!\cdots\!10\)\( \beta_{1} - 23836318478518723776 \beta_{2}) q^{30}\) \(+(-\)\(18\!\cdots\!12\)\( + \)\(35\!\cdots\!42\)\( \beta_{1} + \)\(13\!\cdots\!53\)\( \beta_{2}) q^{31}\) \(+(-\)\(96\!\cdots\!96\)\( + \)\(17\!\cdots\!16\)\( \beta_{1} - \)\(23\!\cdots\!48\)\( \beta_{2}) q^{32}\) \(+(-\)\(96\!\cdots\!20\)\( + \)\(79\!\cdots\!88\)\( \beta_{1} - \)\(15\!\cdots\!74\)\( \beta_{2}) q^{33}\) \(+(-\)\(59\!\cdots\!92\)\( + \)\(74\!\cdots\!34\)\( \beta_{1} - 67083608408202293376 \beta_{2}) q^{34}\) \(+(-\)\(12\!\cdots\!40\)\( - \)\(46\!\cdots\!80\)\( \beta_{1} + \)\(16\!\cdots\!58\)\( \beta_{2}) q^{35}\) \(+(-\)\(64\!\cdots\!04\)\( - \)\(37\!\cdots\!92\)\( \beta_{1} + \)\(43\!\cdots\!36\)\( \beta_{2}) q^{36}\) \(+(-\)\(25\!\cdots\!26\)\( - \)\(15\!\cdots\!64\)\( \beta_{1} - \)\(37\!\cdots\!36\)\( \beta_{2}) q^{37}\) \(+(-\)\(39\!\cdots\!36\)\( + \)\(32\!\cdots\!48\)\( \beta_{1} - \)\(47\!\cdots\!12\)\( \beta_{2}) q^{38}\) \(+(-\)\(34\!\cdots\!50\)\( - \)\(21\!\cdots\!32\)\( \beta_{1} + \)\(67\!\cdots\!34\)\( \beta_{2}) q^{39}\) \(+(-\)\(25\!\cdots\!00\)\( + \)\(15\!\cdots\!00\)\( \beta_{1} + \)\(98\!\cdots\!80\)\( \beta_{2}) q^{40}\) \(+(-\)\(50\!\cdots\!22\)\( - \)\(21\!\cdots\!68\)\( \beta_{1} - \)\(13\!\cdots\!78\)\( \beta_{2}) q^{41}\) \(+(-\)\(33\!\cdots\!28\)\( + \)\(46\!\cdots\!88\)\( \beta_{1} + \)\(10\!\cdots\!44\)\( \beta_{2}) q^{42}\) \(+(-\)\(16\!\cdots\!96\)\( + \)\(14\!\cdots\!44\)\( \beta_{1} + \)\(35\!\cdots\!70\)\( \beta_{2}) q^{43}\) \(+(-\)\(19\!\cdots\!24\)\( + \)\(85\!\cdots\!20\)\( \beta_{1} - \)\(77\!\cdots\!88\)\( \beta_{2}) q^{44}\) \(+(-\)\(18\!\cdots\!10\)\( + \)\(95\!\cdots\!30\)\( \beta_{1} - \)\(14\!\cdots\!53\)\( \beta_{2}) q^{45}\) \(+(\)\(40\!\cdots\!88\)\( - \)\(16\!\cdots\!48\)\( \beta_{1} + \)\(29\!\cdots\!48\)\( \beta_{2}) q^{46}\) \(+(\)\(17\!\cdots\!12\)\( - \)\(41\!\cdots\!68\)\( \beta_{1} + \)\(50\!\cdots\!22\)\( \beta_{2}) q^{47}\) \(+(\)\(82\!\cdots\!36\)\( + \)\(14\!\cdots\!96\)\( \beta_{1} - \)\(95\!\cdots\!96\)\( \beta_{2}) q^{48}\) \(+(\)\(39\!\cdots\!01\)\( + \)\(23\!\cdots\!36\)\( \beta_{1} - \)\(14\!\cdots\!52\)\( \beta_{2}) q^{49}\) \(+(\)\(17\!\cdots\!00\)\( - \)\(33\!\cdots\!75\)\( \beta_{1} - \)\(53\!\cdots\!60\)\( \beta_{2}) q^{50}\) \(+(-\)\(55\!\cdots\!82\)\( + \)\(94\!\cdots\!60\)\( \beta_{1} + \)\(21\!\cdots\!42\)\( \beta_{2}) q^{51}\) \(+(\)\(65\!\cdots\!36\)\( - \)\(85\!\cdots\!20\)\( \beta_{1} - \)\(12\!\cdots\!40\)\( \beta_{2}) q^{52}\) \(+(\)\(10\!\cdots\!78\)\( + \)\(57\!\cdots\!02\)\( \beta_{1} + \)\(38\!\cdots\!57\)\( \beta_{2}) q^{53}\) \(+(\)\(18\!\cdots\!16\)\( - \)\(11\!\cdots\!27\)\( \beta_{1}) q^{54}\) \(+(\)\(24\!\cdots\!20\)\( - \)\(95\!\cdots\!60\)\( \beta_{1} + \)\(20\!\cdots\!16\)\( \beta_{2}) q^{55}\) \(+(\)\(21\!\cdots\!40\)\( + \)\(77\!\cdots\!00\)\( \beta_{1} - \)\(19\!\cdots\!28\)\( \beta_{2}) q^{56}\) \(+(-\)\(11\!\cdots\!64\)\( + \)\(42\!\cdots\!12\)\( \beta_{1} - \)\(22\!\cdots\!58\)\( \beta_{2}) q^{57}\) \(+(\)\(55\!\cdots\!88\)\( - \)\(27\!\cdots\!14\)\( \beta_{1} + \)\(81\!\cdots\!04\)\( \beta_{2}) q^{58}\) \(+(-\)\(59\!\cdots\!92\)\( + \)\(17\!\cdots\!36\)\( \beta_{1} - \)\(22\!\cdots\!36\)\( \beta_{2}) q^{59}\) \(+(-\)\(69\!\cdots\!60\)\( + \)\(39\!\cdots\!80\)\( \beta_{1} + \)\(34\!\cdots\!32\)\( \beta_{2}) q^{60}\) \(+(-\)\(41\!\cdots\!06\)\( - \)\(87\!\cdots\!96\)\( \beta_{1} - \)\(17\!\cdots\!68\)\( \beta_{2}) q^{61}\) \(+(-\)\(22\!\cdots\!52\)\( - \)\(14\!\cdots\!48\)\( \beta_{1} + \)\(62\!\cdots\!40\)\( \beta_{2}) q^{62}\) \(+(-\)\(59\!\cdots\!64\)\( + \)\(44\!\cdots\!74\)\( \beta_{1} + \)\(12\!\cdots\!73\)\( \beta_{2}) q^{63}\) \(+(-\)\(32\!\cdots\!12\)\( + \)\(17\!\cdots\!40\)\( \beta_{1} - \)\(33\!\cdots\!36\)\( \beta_{2}) q^{64}\) \(+(-\)\(85\!\cdots\!80\)\( + \)\(20\!\cdots\!40\)\( \beta_{1} + \)\(89\!\cdots\!26\)\( \beta_{2}) q^{65}\) \(+(-\)\(60\!\cdots\!24\)\( + \)\(34\!\cdots\!20\)\( \beta_{1} - \)\(54\!\cdots\!16\)\( \beta_{2}) q^{66}\) \(+(\)\(43\!\cdots\!16\)\( - \)\(72\!\cdots\!40\)\( \beta_{1} + \)\(22\!\cdots\!08\)\( \beta_{2}) q^{67}\) \(+(-\)\(90\!\cdots\!56\)\( - \)\(26\!\cdots\!08\)\( \beta_{1} - \)\(21\!\cdots\!84\)\( \beta_{2}) q^{68}\) \(+(\)\(40\!\cdots\!04\)\( - \)\(63\!\cdots\!84\)\( \beta_{1} + \)\(33\!\cdots\!78\)\( \beta_{2}) q^{69}\) \(+(\)\(62\!\cdots\!60\)\( - \)\(76\!\cdots\!80\)\( \beta_{1} + \)\(43\!\cdots\!08\)\( \beta_{2}) q^{70}\) \(+(\)\(74\!\cdots\!72\)\( + \)\(47\!\cdots\!60\)\( \beta_{1} - \)\(13\!\cdots\!02\)\( \beta_{2}) q^{71}\) \(+(\)\(40\!\cdots\!68\)\( + \)\(66\!\cdots\!76\)\( \beta_{1} + \)\(21\!\cdots\!64\)\( \beta_{2}) q^{72}\) \(+(\)\(10\!\cdots\!18\)\( + \)\(89\!\cdots\!92\)\( \beta_{1} - \)\(57\!\cdots\!40\)\( \beta_{2}) q^{73}\) \(+(\)\(42\!\cdots\!44\)\( + \)\(25\!\cdots\!46\)\( \beta_{1} + \)\(52\!\cdots\!60\)\( \beta_{2}) q^{74}\) \(+(\)\(63\!\cdots\!25\)\( - \)\(15\!\cdots\!00\)\( \beta_{1} - \)\(79\!\cdots\!80\)\( \beta_{2}) q^{75}\) \(+(-\)\(15\!\cdots\!68\)\( + \)\(95\!\cdots\!24\)\( \beta_{1} - \)\(11\!\cdots\!36\)\( \beta_{2}) q^{76}\) \(+(\)\(53\!\cdots\!48\)\( - \)\(68\!\cdots\!28\)\( \beta_{1} + \)\(22\!\cdots\!80\)\( \beta_{2}) q^{77}\) \(+(\)\(61\!\cdots\!36\)\( - \)\(49\!\cdots\!70\)\( \beta_{1} + \)\(18\!\cdots\!52\)\( \beta_{2}) q^{78}\) \(+(-\)\(14\!\cdots\!60\)\( + \)\(10\!\cdots\!50\)\( \beta_{1} - \)\(24\!\cdots\!83\)\( \beta_{2}) q^{79}\) \(+(\)\(17\!\cdots\!60\)\( - \)\(14\!\cdots\!80\)\( \beta_{1} - \)\(15\!\cdots\!72\)\( \beta_{2}) q^{80}\) \(+\)\(11\!\cdots\!81\)\( q^{81}\) \(+(-\)\(69\!\cdots\!32\)\( + \)\(47\!\cdots\!02\)\( \beta_{1} + \)\(66\!\cdots\!64\)\( \beta_{2}) q^{82}\) \(+(\)\(29\!\cdots\!56\)\( - \)\(90\!\cdots\!92\)\( \beta_{1} - \)\(21\!\cdots\!90\)\( \beta_{2}) q^{83}\) \(+(-\)\(29\!\cdots\!44\)\( + \)\(30\!\cdots\!44\)\( \beta_{1} - \)\(12\!\cdots\!96\)\( \beta_{2}) q^{84}\) \(+(-\)\(20\!\cdots\!40\)\( - \)\(98\!\cdots\!80\)\( \beta_{1} + \)\(23\!\cdots\!38\)\( \beta_{2}) q^{85}\) \(+(-\)\(35\!\cdots\!40\)\( + \)\(19\!\cdots\!16\)\( \beta_{1} - \)\(51\!\cdots\!56\)\( \beta_{2}) q^{86}\) \(+(-\)\(98\!\cdots\!58\)\( - \)\(13\!\cdots\!26\)\( \beta_{1} + \)\(21\!\cdots\!03\)\( \beta_{2}) q^{87}\) \(+(-\)\(28\!\cdots\!28\)\( + \)\(11\!\cdots\!24\)\( \beta_{1} + \)\(51\!\cdots\!28\)\( \beta_{2}) q^{88}\) \(+(\)\(21\!\cdots\!02\)\( + \)\(24\!\cdots\!04\)\( \beta_{1} - \)\(14\!\cdots\!20\)\( \beta_{2}) q^{89}\) \(+(-\)\(83\!\cdots\!60\)\( + \)\(13\!\cdots\!30\)\( \beta_{1} - \)\(24\!\cdots\!28\)\( \beta_{2}) q^{90}\) \(+(\)\(45\!\cdots\!08\)\( + \)\(23\!\cdots\!72\)\( \beta_{1} - \)\(15\!\cdots\!34\)\( \beta_{2}) q^{91}\) \(+(\)\(12\!\cdots\!24\)\( - \)\(36\!\cdots\!24\)\( \beta_{1} + \)\(81\!\cdots\!12\)\( \beta_{2}) q^{92}\) \(+(-\)\(19\!\cdots\!36\)\( + \)\(37\!\cdots\!26\)\( \beta_{1} + \)\(14\!\cdots\!59\)\( \beta_{2}) q^{93}\) \(+(\)\(26\!\cdots\!40\)\( - \)\(12\!\cdots\!32\)\( \beta_{1} + \)\(24\!\cdots\!64\)\( \beta_{2}) q^{94}\) \(+(\)\(31\!\cdots\!00\)\( - \)\(59\!\cdots\!00\)\( \beta_{1} - \)\(14\!\cdots\!60\)\( \beta_{2}) q^{95}\) \(+(-\)\(10\!\cdots\!88\)\( + \)\(18\!\cdots\!48\)\( \beta_{1} - \)\(25\!\cdots\!44\)\( \beta_{2}) q^{96}\) \(+(\)\(20\!\cdots\!26\)\( + \)\(21\!\cdots\!80\)\( \beta_{1} - \)\(24\!\cdots\!92\)\( \beta_{2}) q^{97}\) \(+(-\)\(67\!\cdots\!20\)\( + \)\(10\!\cdots\!59\)\( \beta_{1} - \)\(30\!\cdots\!16\)\( \beta_{2}) q^{98}\) \(+(-\)\(10\!\cdots\!60\)\( + \)\(83\!\cdots\!64\)\( \beta_{1} - \)\(16\!\cdots\!22\)\( \beta_{2}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(3q \) \(\mathstrut +\mathstrut 4857024q^{2} \) \(\mathstrut +\mathstrut 31381059609q^{3} \) \(\mathstrut -\mathstrut 1781058029568q^{4} \) \(\mathstrut -\mathstrut 507753321105270q^{5} \) \(\mathstrut +\mathstrut 50806186555447872q^{6} \) \(\mathstrut -\mathstrut 1633169303707089288q^{7} \) \(\mathstrut +\mathstrut 11196260694472851456q^{8} \) \(\mathstrut +\mathstrut 328256967394537077627q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut +\mathstrut 4857024q^{2} \) \(\mathstrut +\mathstrut 31381059609q^{3} \) \(\mathstrut -\mathstrut 1781058029568q^{4} \) \(\mathstrut -\mathstrut 507753321105270q^{5} \) \(\mathstrut +\mathstrut 50806186555447872q^{6} \) \(\mathstrut -\mathstrut 1633169303707089288q^{7} \) \(\mathstrut +\mathstrut 11196260694472851456q^{8} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!27\)\(q^{9} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!20\)\(q^{10} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!20\)\(q^{11} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!04\)\(q^{12} \) \(\mathstrut -\mathstrut \)\(99\!\cdots\!50\)\(q^{13} \) \(\mathstrut -\mathstrut \)\(94\!\cdots\!28\)\(q^{14} \) \(\mathstrut -\mathstrut \)\(53\!\cdots\!10\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!36\)\(q^{16} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!82\)\(q^{17} \) \(\mathstrut +\mathstrut \)\(53\!\cdots\!16\)\(q^{18} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!64\)\(q^{19} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!60\)\(q^{20} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!64\)\(q^{21} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!24\)\(q^{22} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!04\)\(q^{23} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!68\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!25\)\(q^{25} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!36\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!81\)\(q^{27} \) \(\mathstrut -\mathstrut \)\(84\!\cdots\!44\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!58\)\(q^{29} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!60\)\(q^{30} \) \(\mathstrut -\mathstrut \)\(56\!\cdots\!36\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!88\)\(q^{32} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!60\)\(q^{33} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!76\)\(q^{34} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!20\)\(q^{35} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!12\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(77\!\cdots\!78\)\(q^{37} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!08\)\(q^{38} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!50\)\(q^{39} \) \(\mathstrut -\mathstrut \)\(75\!\cdots\!00\)\(q^{40} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!66\)\(q^{41} \) \(\mathstrut -\mathstrut \)\(99\!\cdots\!84\)\(q^{42} \) \(\mathstrut -\mathstrut \)\(50\!\cdots\!88\)\(q^{43} \) \(\mathstrut -\mathstrut \)\(58\!\cdots\!72\)\(q^{44} \) \(\mathstrut -\mathstrut \)\(55\!\cdots\!30\)\(q^{45} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!64\)\(q^{46} \) \(\mathstrut +\mathstrut \)\(51\!\cdots\!36\)\(q^{47} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!08\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!03\)\(q^{49} \) \(\mathstrut +\mathstrut \)\(53\!\cdots\!00\)\(q^{50} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!46\)\(q^{51} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!08\)\(q^{52} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!34\)\(q^{53} \) \(\mathstrut +\mathstrut \)\(55\!\cdots\!48\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(72\!\cdots\!60\)\(q^{55} \) \(\mathstrut +\mathstrut \)\(64\!\cdots\!20\)\(q^{56} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!92\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!64\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!76\)\(q^{59} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!80\)\(q^{60} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!18\)\(q^{61} \) \(\mathstrut -\mathstrut \)\(68\!\cdots\!56\)\(q^{62} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!92\)\(q^{63} \) \(\mathstrut -\mathstrut \)\(97\!\cdots\!36\)\(q^{64} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!40\)\(q^{65} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!72\)\(q^{66} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!48\)\(q^{67} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!68\)\(q^{68} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!12\)\(q^{69} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!80\)\(q^{70} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!16\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!04\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!54\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!32\)\(q^{74} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!75\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!04\)\(q^{76} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!44\)\(q^{77} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!08\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!80\)\(q^{79} \) \(\mathstrut +\mathstrut \)\(51\!\cdots\!80\)\(q^{80} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!43\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!96\)\(q^{82} \) \(\mathstrut +\mathstrut \)\(89\!\cdots\!68\)\(q^{83} \) \(\mathstrut -\mathstrut \)\(88\!\cdots\!32\)\(q^{84} \) \(\mathstrut -\mathstrut \)\(60\!\cdots\!20\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!20\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!74\)\(q^{87} \) \(\mathstrut -\mathstrut \)\(86\!\cdots\!84\)\(q^{88} \) \(\mathstrut +\mathstrut \)\(65\!\cdots\!06\)\(q^{89} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!80\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!24\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!72\)\(q^{92} \) \(\mathstrut -\mathstrut \)\(59\!\cdots\!08\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(78\!\cdots\!20\)\(q^{94} \) \(\mathstrut +\mathstrut \)\(93\!\cdots\!00\)\(q^{95} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!64\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(61\!\cdots\!78\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!60\)\(q^{98} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!80\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3}\mathstrut -\mathstrut \) \(x^{2}\mathstrut -\mathstrut \) \(908401710\) \(x\mathstrut +\mathstrut \) \(974756489742\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 96 \nu - 32 \)
\(\beta_{2}\)\(=\)\( 2304 \nu^{2} + 3705792 \nu - 1395306262592 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(32\)\()/96\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2}\mathstrut -\mathstrut \) \(38602\) \(\beta_{1}\mathstrut +\mathstrut \) \(1395305027328\)\()/2304\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
29588.6
1074.41
−30662.1
−1.22147e6 1.04604e10 −7.30411e12 −8.84097e14 −1.27770e16 1.48034e18 1.96659e19 1.09419e20 1.07990e21
1.2 1.51590e6 1.04604e10 −6.49815e12 1.66864e15 1.58568e16 −2.14679e18 −2.31845e19 1.09419e20 2.52949e21
1.3 4.56260e6 1.04604e10 1.20212e13 −1.29230e15 4.77264e16 −9.66721e17 1.47149e19 1.09419e20 −5.89624e21
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{3} \) \(\mathstrut -\mathstrut 4857024 T_{2}^{2} \) \(\mathstrut -\mathstrut 508269450240 T_{2} \) \(\mathstrut +\mathstrut \)\(84\!\cdots\!88\)\( \) acting on \(S_{44}^{\mathrm{new}}(\Gamma_0(3))\).