Properties

Label 3.40.a.a
Level 3
Weight 40
Character orbit 3.a
Self dual Yes
Analytic conductor 28.902
Analytic rank 1
Dimension 3
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 40 \)
Character orbit: \([\chi]\) = 3.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(28.9018654169\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{6}\cdot 5 \)
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.

\(f(q)\) \(=\) \( q\) \( + ( -369000 - \beta_{1} ) q^{2} \) \( + 1162261467 q^{3} \) \( + ( 335300075200 + 785144 \beta_{1} + 16 \beta_{2} ) q^{4} \) \( + ( 3119016476430 + 32303812 \beta_{1} - 513 \beta_{2} ) q^{5} \) \( + ( -428874481323000 - 1162261467 \beta_{1} ) q^{6} \) \( + ( 4613961459153368 - 30823389604 \beta_{1} + 348985 \beta_{2} ) q^{7} \) \( + ( -508852326783711744 - 484502113856 \beta_{1} - 17712000 \beta_{2} ) q^{8} \) \( + 1350851717672992089 q^{9} \) \(+O(q^{10})\) \( q\) \(+(-369000 - \beta_{1}) q^{2}\) \(+1162261467 q^{3}\) \(+(335300075200 + 785144 \beta_{1} + 16 \beta_{2}) q^{4}\) \(+(3119016476430 + 32303812 \beta_{1} - 513 \beta_{2}) q^{5}\) \(+(-428874481323000 - 1162261467 \beta_{1}) q^{6}\) \(+(4613961459153368 - 30823389604 \beta_{1} + 348985 \beta_{2}) q^{7}\) \(+(-508852326783711744 - 484502113856 \beta_{1} - 17712000 \beta_{2}) q^{8}\) \(+1350851717672992089 q^{9}\) \(+(-25343199726347478960 - 4627593552014 \beta_{1} - 351748864 \beta_{2}) q^{10}\) \(+(-13350109203715747932 + 87947509241336 \beta_{1} + 4631921442 \beta_{2}) q^{11}\) \(+(\)\(38\!\cdots\!00\)\( + 912542617246248 \beta_{1} + 18596183472 \beta_{2}) q^{12}\) \(+(-\)\(46\!\cdots\!98\)\( - 569679887088520 \beta_{1} - 206263539326 \beta_{2}) q^{13}\) \(+(\)\(21\!\cdots\!32\)\( + 94200779733928 \beta_{1} + 380851317504 \beta_{2}) q^{14}\) \(+(\)\(36\!\cdots\!10\)\( + 37545475924812204 \beta_{1} - 596240132571 \beta_{2}) q^{15}\) \(+(\)\(36\!\cdots\!28\)\( + 690890446516379136 \beta_{1} + 4656654271488 \beta_{2}) q^{16}\) \(+(-\)\(56\!\cdots\!06\)\( + 926992962792897096 \beta_{1} - 6009628140594 \beta_{2}) q^{17}\) \(+(-\)\(49\!\cdots\!00\)\( - 1350851717672992089 \beta_{1}) q^{18}\) \(+(-\)\(23\!\cdots\!96\)\( + 2957557631558772120 \beta_{1} - 160826348093766 \beta_{2}) q^{19}\) \(+(\)\(11\!\cdots\!80\)\( + 17692841649648817552 \beta_{1} + 469278711728352 \beta_{2}) q^{20}\) \(+(\)\(53\!\cdots\!56\)\( - 35824838019057589068 \beta_{1} + 405611818060995 \beta_{2}) q^{21}\) \(+(-\)\(60\!\cdots\!32\)\( - \)\(13\!\cdots\!48\)\( \beta_{1} - 2897971855497728 \beta_{2}) q^{22}\) \(+(-\)\(85\!\cdots\!24\)\( - \)\(24\!\cdots\!96\)\( \beta_{1} + 7342398742559922 \beta_{2}) q^{23}\) \(+(-\)\(59\!\cdots\!48\)\( - \)\(56\!\cdots\!52\)\( \beta_{1} - 20585975103504000 \beta_{2}) q^{24}\) \(+(-\)\(74\!\cdots\!25\)\( - \)\(53\!\cdots\!20\)\( \beta_{1} + 8936618308826980 \beta_{2}) q^{25}\) \(+(\)\(21\!\cdots\!52\)\( + \)\(96\!\cdots\!66\)\( \beta_{1} + 75502035906725376 \beta_{2}) q^{26}\) \(+\)\(15\!\cdots\!63\)\( q^{27}\) \(+(-\)\(10\!\cdots\!16\)\( - \)\(13\!\cdots\!64\)\( \beta_{1} - 315943026832013952 \beta_{2}) q^{28}\) \(+(-\)\(15\!\cdots\!62\)\( + \)\(18\!\cdots\!44\)\( \beta_{1} + 119886169875921837 \beta_{2}) q^{29}\) \(+(-\)\(29\!\cdots\!20\)\( - \)\(53\!\cdots\!38\)\( \beta_{1} - 408824150688223488 \beta_{2}) q^{30}\) \(+(-\)\(94\!\cdots\!96\)\( + \)\(90\!\cdots\!60\)\( \beta_{1} + 2713599485454965797 \beta_{2}) q^{31}\) \(+(-\)\(37\!\cdots\!92\)\( - \)\(49\!\cdots\!28\)\( \beta_{1} - 2815744285881851904 \beta_{2}) q^{32}\) \(+(-\)\(15\!\cdots\!44\)\( + \)\(10\!\cdots\!12\)\( \beta_{1} + 5383503810207675414 \beta_{2}) q^{33}\) \(+(-\)\(67\!\cdots\!00\)\( - \)\(18\!\cdots\!46\)\( \beta_{1} - 12897652529867331072 \beta_{2}) q^{34}\) \(+(-\)\(92\!\cdots\!00\)\( + \)\(64\!\cdots\!20\)\( \beta_{1} - 14321872082373654330 \beta_{2}) q^{35}\) \(+(\)\(45\!\cdots\!00\)\( + \)\(10\!\cdots\!16\)\( \beta_{1} + 21613627482767873424 \beta_{2}) q^{36}\) \(+(-\)\(17\!\cdots\!62\)\( + \)\(19\!\cdots\!00\)\( \beta_{1} + 85573013785692403224 \beta_{2}) q^{37}\) \(+(-\)\(13\!\cdots\!88\)\( + \)\(48\!\cdots\!24\)\( \beta_{1} + 4442002987126795776 \beta_{2}) q^{38}\) \(+(-\)\(53\!\cdots\!66\)\( - \)\(66\!\cdots\!40\)\( \beta_{1} - \)\(23\!\cdots\!42\)\( \beta_{2}) q^{39}\) \(+(-\)\(34\!\cdots\!80\)\( - \)\(26\!\cdots\!12\)\( \beta_{1} - \)\(24\!\cdots\!12\)\( \beta_{2}) q^{40}\) \(+(\)\(15\!\cdots\!94\)\( - \)\(38\!\cdots\!24\)\( \beta_{1} + \)\(53\!\cdots\!38\)\( \beta_{2}) q^{41}\) \(+(\)\(24\!\cdots\!44\)\( + \)\(10\!\cdots\!76\)\( \beta_{1} + \)\(44\!\cdots\!68\)\( \beta_{2}) q^{42}\) \(+(-\)\(49\!\cdots\!96\)\( - \)\(68\!\cdots\!24\)\( \beta_{1} - \)\(61\!\cdots\!18\)\( \beta_{2}) q^{43}\) \(+(\)\(12\!\cdots\!16\)\( + \)\(13\!\cdots\!40\)\( \beta_{1} + \)\(48\!\cdots\!40\)\( \beta_{2}) q^{44}\) \(+(\)\(42\!\cdots\!70\)\( + \)\(43\!\cdots\!68\)\( \beta_{1} - \)\(69\!\cdots\!57\)\( \beta_{2}) q^{45}\) \(+(\)\(21\!\cdots\!24\)\( + \)\(17\!\cdots\!12\)\( \beta_{1} + \)\(15\!\cdots\!04\)\( \beta_{2}) q^{46}\) \(+(-\)\(43\!\cdots\!48\)\( - \)\(20\!\cdots\!40\)\( \beta_{1} - \)\(99\!\cdots\!42\)\( \beta_{2}) q^{47}\) \(+(\)\(42\!\cdots\!76\)\( + \)\(80\!\cdots\!12\)\( \beta_{1} + \)\(54\!\cdots\!96\)\( \beta_{2}) q^{48}\) \(+(-\)\(44\!\cdots\!71\)\( - \)\(74\!\cdots\!80\)\( \beta_{1} + \)\(16\!\cdots\!36\)\( \beta_{2}) q^{49}\) \(+(\)\(67\!\cdots\!00\)\( + \)\(75\!\cdots\!65\)\( \beta_{1} + \)\(57\!\cdots\!40\)\( \beta_{2}) q^{50}\) \(+(-\)\(65\!\cdots\!02\)\( + \)\(10\!\cdots\!32\)\( \beta_{1} - \)\(69\!\cdots\!98\)\( \beta_{2}) q^{51}\) \(+(-\)\(54\!\cdots\!76\)\( - \)\(76\!\cdots\!84\)\( \beta_{1} - \)\(65\!\cdots\!24\)\( \beta_{2}) q^{52}\) \(+(-\)\(36\!\cdots\!14\)\( + \)\(10\!\cdots\!24\)\( \beta_{1} + \)\(18\!\cdots\!37\)\( \beta_{2}) q^{53}\) \(+(-\)\(57\!\cdots\!00\)\( - \)\(15\!\cdots\!63\)\( \beta_{1}) q^{54}\) \(+(-\)\(50\!\cdots\!20\)\( + \)\(25\!\cdots\!52\)\( \beta_{1} + \)\(11\!\cdots\!52\)\( \beta_{2}) q^{55}\) \(+(\)\(21\!\cdots\!00\)\( + \)\(23\!\cdots\!44\)\( \beta_{1} + \)\(10\!\cdots\!84\)\( \beta_{2}) q^{56}\) \(+(-\)\(26\!\cdots\!32\)\( + \)\(34\!\cdots\!40\)\( \beta_{1} - \)\(18\!\cdots\!22\)\( \beta_{2}) q^{57}\) \(+(-\)\(79\!\cdots\!76\)\( + \)\(50\!\cdots\!70\)\( \beta_{1} - \)\(32\!\cdots\!76\)\( \beta_{2}) q^{58}\) \(+(\)\(11\!\cdots\!16\)\( - \)\(40\!\cdots\!52\)\( \beta_{1} + \)\(26\!\cdots\!44\)\( \beta_{2}) q^{59}\) \(+(\)\(12\!\cdots\!60\)\( + \)\(20\!\cdots\!84\)\( \beta_{1} + \)\(54\!\cdots\!84\)\( \beta_{2}) q^{60}\) \(+(-\)\(87\!\cdots\!34\)\( - \)\(59\!\cdots\!00\)\( \beta_{1} - \)\(18\!\cdots\!56\)\( \beta_{2}) q^{61}\) \(+(-\)\(32\!\cdots\!04\)\( - \)\(64\!\cdots\!80\)\( \beta_{1} - \)\(23\!\cdots\!92\)\( \beta_{2}) q^{62}\) \(+(\)\(62\!\cdots\!52\)\( - \)\(41\!\cdots\!56\)\( \beta_{1} + \)\(47\!\cdots\!65\)\( \beta_{2}) q^{63}\) \(+(\)\(30\!\cdots\!68\)\( + \)\(26\!\cdots\!08\)\( \beta_{1} + \)\(62\!\cdots\!28\)\( \beta_{2}) q^{64}\) \(+(\)\(86\!\cdots\!20\)\( - \)\(30\!\cdots\!52\)\( \beta_{1} - \)\(71\!\cdots\!02\)\( \beta_{2}) q^{65}\) \(+(-\)\(70\!\cdots\!44\)\( - \)\(15\!\cdots\!16\)\( \beta_{1} - \)\(33\!\cdots\!76\)\( \beta_{2}) q^{66}\) \(+(-\)\(66\!\cdots\!96\)\( + \)\(26\!\cdots\!20\)\( \beta_{1} - \)\(14\!\cdots\!60\)\( \beta_{2}) q^{67}\) \(+(\)\(42\!\cdots\!00\)\( + \)\(54\!\cdots\!12\)\( \beta_{1} + \)\(10\!\cdots\!40\)\( \beta_{2}) q^{68}\) \(+(-\)\(99\!\cdots\!08\)\( - \)\(28\!\cdots\!32\)\( \beta_{1} + \)\(85\!\cdots\!74\)\( \beta_{2}) q^{69}\) \(+(-\)\(14\!\cdots\!00\)\( + \)\(98\!\cdots\!60\)\( \beta_{1} - \)\(57\!\cdots\!40\)\( \beta_{2}) q^{70}\) \(+(-\)\(10\!\cdots\!48\)\( - \)\(25\!\cdots\!16\)\( \beta_{1} + \)\(16\!\cdots\!74\)\( \beta_{2}) q^{71}\) \(+(-\)\(68\!\cdots\!16\)\( - \)\(65\!\cdots\!84\)\( \beta_{1} - \)\(23\!\cdots\!00\)\( \beta_{2}) q^{72}\) \(+(-\)\(12\!\cdots\!74\)\( - \)\(16\!\cdots\!88\)\( \beta_{1} + \)\(37\!\cdots\!20\)\( \beta_{2}) q^{73}\) \(+(-\)\(84\!\cdots\!08\)\( - \)\(10\!\cdots\!50\)\( \beta_{1} - \)\(59\!\cdots\!44\)\( \beta_{2}) q^{74}\) \(+(-\)\(86\!\cdots\!75\)\( - \)\(62\!\cdots\!40\)\( \beta_{1} + \)\(10\!\cdots\!60\)\( \beta_{2}) q^{75}\) \(+(-\)\(18\!\cdots\!56\)\( - \)\(23\!\cdots\!16\)\( \beta_{1} + \)\(97\!\cdots\!68\)\( \beta_{2}) q^{76}\) \(+(-\)\(33\!\cdots\!24\)\( - \)\(20\!\cdots\!64\)\( \beta_{1} - \)\(58\!\cdots\!76\)\( \beta_{2}) q^{77}\) \(+(\)\(24\!\cdots\!84\)\( + \)\(11\!\cdots\!22\)\( \beta_{1} + \)\(87\!\cdots\!92\)\( \beta_{2}) q^{78}\) \(+(-\)\(34\!\cdots\!00\)\( + \)\(73\!\cdots\!44\)\( \beta_{1} - \)\(36\!\cdots\!91\)\( \beta_{2}) q^{79}\) \(+(\)\(15\!\cdots\!20\)\( + \)\(10\!\cdots\!88\)\( \beta_{1} + \)\(24\!\cdots\!88\)\( \beta_{2}) q^{80}\) \(+\)\(18\!\cdots\!21\)\( q^{81}\) \(+(-\)\(55\!\cdots\!84\)\( - \)\(28\!\cdots\!82\)\( \beta_{1} - \)\(16\!\cdots\!44\)\( \beta_{2}) q^{82}\) \(+(\)\(23\!\cdots\!80\)\( + \)\(12\!\cdots\!08\)\( \beta_{1} - \)\(53\!\cdots\!30\)\( \beta_{2}) q^{83}\) \(+(-\)\(12\!\cdots\!72\)\( - \)\(15\!\cdots\!88\)\( \beta_{1} - \)\(36\!\cdots\!84\)\( \beta_{2}) q^{84}\) \(+(\)\(25\!\cdots\!20\)\( - \)\(13\!\cdots\!52\)\( \beta_{1} + \)\(44\!\cdots\!98\)\( \beta_{2}) q^{85}\) \(+(\)\(53\!\cdots\!68\)\( + \)\(47\!\cdots\!36\)\( \beta_{1} + \)\(12\!\cdots\!92\)\( \beta_{2}) q^{86}\) \(+(-\)\(17\!\cdots\!54\)\( + \)\(21\!\cdots\!48\)\( \beta_{1} + \)\(13\!\cdots\!79\)\( \beta_{2}) q^{87}\) \(+(-\)\(11\!\cdots\!84\)\( - \)\(12\!\cdots\!72\)\( \beta_{1} - \)\(71\!\cdots\!16\)\( \beta_{2}) q^{88}\) \(+(\)\(45\!\cdots\!14\)\( - \)\(22\!\cdots\!72\)\( \beta_{1} + \)\(34\!\cdots\!12\)\( \beta_{2}) q^{89}\) \(+(-\)\(34\!\cdots\!40\)\( - \)\(62\!\cdots\!46\)\( \beta_{1} - \)\(47\!\cdots\!96\)\( \beta_{2}) q^{90}\) \(+(-\)\(86\!\cdots\!36\)\( + \)\(28\!\cdots\!04\)\( \beta_{1} - \)\(90\!\cdots\!14\)\( \beta_{2}) q^{91}\) \(+(-\)\(45\!\cdots\!12\)\( - \)\(12\!\cdots\!56\)\( \beta_{1} - \)\(48\!\cdots\!52\)\( \beta_{2}) q^{92}\) \(+(-\)\(10\!\cdots\!32\)\( + \)\(10\!\cdots\!20\)\( \beta_{1} + \)\(31\!\cdots\!99\)\( \beta_{2}) q^{93}\) \(+(\)\(15\!\cdots\!84\)\( + \)\(31\!\cdots\!04\)\( \beta_{1} + \)\(64\!\cdots\!92\)\( \beta_{2}) q^{94}\) \(+(\)\(15\!\cdots\!80\)\( - \)\(22\!\cdots\!08\)\( \beta_{1} + \)\(61\!\cdots\!92\)\( \beta_{2}) q^{95}\) \(+(-\)\(43\!\cdots\!64\)\( - \)\(57\!\cdots\!76\)\( \beta_{1} - \)\(32\!\cdots\!68\)\( \beta_{2}) q^{96}\) \(+(-\)\(44\!\cdots\!06\)\( + \)\(35\!\cdots\!36\)\( \beta_{1} - \)\(27\!\cdots\!24\)\( \beta_{2}) q^{97}\) \(+(\)\(57\!\cdots\!28\)\( - \)\(30\!\cdots\!77\)\( \beta_{1} + \)\(66\!\cdots\!64\)\( \beta_{2}) q^{98}\) \(+(-\)\(18\!\cdots\!48\)\( + \)\(11\!\cdots\!04\)\( \beta_{1} + \)\(62\!\cdots\!38\)\( \beta_{2}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(3q \) \(\mathstrut -\mathstrut 1107000q^{2} \) \(\mathstrut +\mathstrut 3486784401q^{3} \) \(\mathstrut +\mathstrut 1005900225600q^{4} \) \(\mathstrut +\mathstrut 9357049429290q^{5} \) \(\mathstrut -\mathstrut 1286623443969000q^{6} \) \(\mathstrut +\mathstrut 13841884377460104q^{7} \) \(\mathstrut -\mathstrut 1526556980351135232q^{8} \) \(\mathstrut +\mathstrut 4052555153018976267q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut -\mathstrut 1107000q^{2} \) \(\mathstrut +\mathstrut 3486784401q^{3} \) \(\mathstrut +\mathstrut 1005900225600q^{4} \) \(\mathstrut +\mathstrut 9357049429290q^{5} \) \(\mathstrut -\mathstrut 1286623443969000q^{6} \) \(\mathstrut +\mathstrut 13841884377460104q^{7} \) \(\mathstrut -\mathstrut 1526556980351135232q^{8} \) \(\mathstrut +\mathstrut 4052555153018976267q^{9} \) \(\mathstrut -\mathstrut 76029599179042436880q^{10} \) \(\mathstrut -\mathstrut 40050327611147243796q^{11} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!00\)\(q^{12} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!94\)\(q^{13} \) \(\mathstrut +\mathstrut \)\(64\!\cdots\!96\)\(q^{14} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!30\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!84\)\(q^{16} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!18\)\(q^{17} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!00\)\(q^{18} \) \(\mathstrut -\mathstrut \)\(69\!\cdots\!88\)\(q^{19} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!40\)\(q^{20} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!68\)\(q^{21} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!96\)\(q^{22} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!72\)\(q^{23} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!44\)\(q^{24} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!75\)\(q^{25} \) \(\mathstrut +\mathstrut \)\(64\!\cdots\!56\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(47\!\cdots\!89\)\(q^{27} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!48\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(46\!\cdots\!86\)\(q^{29} \) \(\mathstrut -\mathstrut \)\(88\!\cdots\!60\)\(q^{30} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!88\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!76\)\(q^{32} \) \(\mathstrut -\mathstrut \)\(46\!\cdots\!32\)\(q^{33} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!00\)\(q^{34} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!00\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!00\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(52\!\cdots\!86\)\(q^{37} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!64\)\(q^{38} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!98\)\(q^{39} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!40\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(47\!\cdots\!82\)\(q^{41} \) \(\mathstrut +\mathstrut \)\(74\!\cdots\!32\)\(q^{42} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!88\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!48\)\(q^{44} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!10\)\(q^{45} \) \(\mathstrut +\mathstrut \)\(64\!\cdots\!72\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!44\)\(q^{47} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!28\)\(q^{48} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!13\)\(q^{49} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!00\)\(q^{50} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!06\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!28\)\(q^{52} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!42\)\(q^{53} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!00\)\(q^{54} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!60\)\(q^{55} \) \(\mathstrut +\mathstrut \)\(63\!\cdots\!00\)\(q^{56} \) \(\mathstrut -\mathstrut \)\(80\!\cdots\!96\)\(q^{57} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!28\)\(q^{58} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!48\)\(q^{59} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!80\)\(q^{60} \) \(\mathstrut -\mathstrut \)\(26\!\cdots\!02\)\(q^{61} \) \(\mathstrut -\mathstrut \)\(98\!\cdots\!12\)\(q^{62} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!56\)\(q^{63} \) \(\mathstrut +\mathstrut \)\(92\!\cdots\!04\)\(q^{64} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!60\)\(q^{65} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!32\)\(q^{66} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!88\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!00\)\(q^{68} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!24\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!00\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!44\)\(q^{71} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!48\)\(q^{72} \) \(\mathstrut -\mathstrut \)\(38\!\cdots\!22\)\(q^{73} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!24\)\(q^{74} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!25\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(55\!\cdots\!68\)\(q^{76} \) \(\mathstrut -\mathstrut \)\(99\!\cdots\!72\)\(q^{77} \) \(\mathstrut +\mathstrut \)\(74\!\cdots\!52\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!00\)\(q^{79} \) \(\mathstrut +\mathstrut \)\(45\!\cdots\!60\)\(q^{80} \) \(\mathstrut +\mathstrut \)\(54\!\cdots\!63\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!52\)\(q^{82} \) \(\mathstrut +\mathstrut \)\(69\!\cdots\!40\)\(q^{83} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!16\)\(q^{84} \) \(\mathstrut +\mathstrut \)\(76\!\cdots\!60\)\(q^{85} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!04\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(53\!\cdots\!62\)\(q^{87} \) \(\mathstrut -\mathstrut \)\(34\!\cdots\!52\)\(q^{88} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!42\)\(q^{89} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!20\)\(q^{90} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!08\)\(q^{91} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!36\)\(q^{92} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!96\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(45\!\cdots\!52\)\(q^{94} \) \(\mathstrut +\mathstrut \)\(46\!\cdots\!40\)\(q^{95} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!92\)\(q^{96} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!18\)\(q^{97} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!84\)\(q^{98} \) \(\mathstrut -\mathstrut \)\(54\!\cdots\!44\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3}\mathstrut -\mathstrut \) \(216694123\) \(x\mathstrut -\mathstrut \) \(94580724378\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 72 \nu \)
\(\beta_{2}\)\(=\)\( 324 \nu^{2} - 212148 \nu - 46805930568 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/72\)
\(\nu^{2}\)\(=\)\((\)\(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(5893\) \(\beta_{1}\mathstrut +\mathstrut \) \(93611861136\)\()/648\)

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
14934.1
−436.856
−14497.2
−1.44426e6 1.16226e9 1.53612e12 2.64208e13 −1.67860e15 −2.07513e16 −1.42456e18 1.35085e18 −3.81584e19
1.2 −337546. 1.16226e9 −4.35818e11 2.60351e13 −3.92317e14 −1.06972e16 3.32677e17 1.35085e18 −8.78806e18
1.3 674802. 1.16226e9 −9.43987e10 −4.30989e13 7.84296e14 4.52904e16 −4.34676e17 1.35085e18 −2.90832e19
\(n\): 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 \(T_{2}^{3} \) \(\mathstrut +\mathstrut 1107000 T_{2}^{2} \) \(\mathstrut -\mathstrut 714859333632 T_{2} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!56\)\( \) acting on \(S_{40}^{\mathrm{new}}(\Gamma_0(3))\).