Properties

Label 1.62.a.a
Level 1
Weight 62
Character orbit 1.a
Self dual Yes
Analytic conductor 23.566
Analytic rank 1
Dimension 4
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 62 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(23.5656183265\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{28}\cdot 3^{8}\cdot 5^{2}\cdot 7 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(+(286578000 - \beta_{1}) q^{2}\) \(+(-143430899755500 + 48580 \beta_{1} + \beta_{2}) q^{3}\) \(+(1100749418957151232 - 838426474 \beta_{1} + 343 \beta_{2} + \beta_{3}) q^{4}\) \(+(-\)\(13\!\cdots\!50\)\( + 100252966240 \beta_{1} - 48540 \beta_{2} - 120 \beta_{3}) q^{5}\) \(+(-\)\(20\!\cdots\!88\)\( + 157027925895036 \beta_{1} - 109667352 \beta_{2} - 189864 \beta_{3}) q^{6}\) \(+(-\)\(15\!\cdots\!00\)\( + 14513866966047176 \beta_{1} - 46993043006 \beta_{2} - 19628000 \beta_{3}) q^{7}\) \(+(\)\(24\!\cdots\!00\)\( - 1373007096556182656 \beta_{1} - 7496719202112 \beta_{2} + 1190856000 \beta_{3}) q^{8}\) \(+(\)\(29\!\cdots\!13\)\( + 1326939277665425472 \beta_{1} - 362685230082504 \beta_{2} - 19284989328 \beta_{3}) q^{9}\) \(+O(q^{10})\) \( q\) \(+(286578000 - \beta_{1}) q^{2}\) \(+(-143430899755500 + 48580 \beta_{1} + \beta_{2}) q^{3}\) \(+(1100749418957151232 - 838426474 \beta_{1} + 343 \beta_{2} + \beta_{3}) q^{4}\) \(+(-\)\(13\!\cdots\!50\)\( + 100252966240 \beta_{1} - 48540 \beta_{2} - 120 \beta_{3}) q^{5}\) \(+(-\)\(20\!\cdots\!88\)\( + 157027925895036 \beta_{1} - 109667352 \beta_{2} - 189864 \beta_{3}) q^{6}\) \(+(-\)\(15\!\cdots\!00\)\( + 14513866966047176 \beta_{1} - 46993043006 \beta_{2} - 19628000 \beta_{3}) q^{7}\) \(+(\)\(24\!\cdots\!00\)\( - 1373007096556182656 \beta_{1} - 7496719202112 \beta_{2} + 1190856000 \beta_{3}) q^{8}\) \(+(\)\(29\!\cdots\!13\)\( + 1326939277665425472 \beta_{1} - 362685230082504 \beta_{2} - 19284989328 \beta_{3}) q^{9}\) \(+(-\)\(37\!\cdots\!00\)\( + \)\(44\!\cdots\!30\)\( \beta_{1} + 900415543063520 \beta_{2} - 141501833440 \beta_{3}) q^{10}\) \(+(-\)\(10\!\cdots\!88\)\( + \)\(12\!\cdots\!80\)\( \beta_{1} + 130667557581888915 \beta_{2} + 11035029161280 \beta_{3}) q^{11}\) \(+(-\)\(24\!\cdots\!00\)\( + \)\(57\!\cdots\!92\)\( \beta_{1} - 877602562285349684 \beta_{2} - 217648250923500 \beta_{3}) q^{12}\) \(+(\)\(26\!\cdots\!50\)\( + \)\(39\!\cdots\!32\)\( \beta_{1} - 12482522028380172860 \beta_{2} + 2435574015901000 \beta_{3}) q^{13}\) \(+(-\)\(52\!\cdots\!76\)\( + \)\(65\!\cdots\!28\)\( \beta_{1} + \)\(15\!\cdots\!04\)\( \beta_{2} - 15743169581051472 \beta_{3}) q^{14}\) \(+(\)\(28\!\cdots\!00\)\( - \)\(69\!\cdots\!60\)\( \beta_{1} + \)\(10\!\cdots\!10\)\( \beta_{2} + 26260172144333280 \beta_{3}) q^{15}\) \(+(\)\(27\!\cdots\!56\)\( - \)\(36\!\cdots\!88\)\( \beta_{1} - \)\(88\!\cdots\!84\)\( \beta_{2} + 603732751100608512 \beta_{3}) q^{16}\) \(+(-\)\(10\!\cdots\!50\)\( - \)\(38\!\cdots\!60\)\( \beta_{1} + \)\(33\!\cdots\!52\)\( \beta_{2} - 7263830520990282000 \beta_{3}) q^{17}\) \(+(\)\(39\!\cdots\!00\)\( + \)\(17\!\cdots\!15\)\( \beta_{1} + \)\(18\!\cdots\!36\)\( \beta_{2} + 42183522593278488000 \beta_{3}) q^{18}\) \(+(-\)\(89\!\cdots\!80\)\( + \)\(48\!\cdots\!64\)\( \beta_{1} - \)\(16\!\cdots\!23\)\( \beta_{2} - \)\(10\!\cdots\!36\)\( \beta_{3}) q^{19}\) \(+(-\)\(12\!\cdots\!00\)\( + \)\(66\!\cdots\!80\)\( \beta_{1} + \)\(97\!\cdots\!70\)\( \beta_{2} - \)\(34\!\cdots\!90\)\( \beta_{3}) q^{20}\) \(+(-\)\(13\!\cdots\!68\)\( - \)\(11\!\cdots\!48\)\( \beta_{1} + \)\(31\!\cdots\!36\)\( \beta_{2} + \)\(50\!\cdots\!52\)\( \beta_{3}) q^{21}\) \(+(-\)\(43\!\cdots\!00\)\( - \)\(79\!\cdots\!92\)\( \beta_{1} - \)\(10\!\cdots\!60\)\( \beta_{2} - \)\(26\!\cdots\!00\)\( \beta_{3}) q^{22}\) \(+(-\)\(10\!\cdots\!00\)\( + \)\(24\!\cdots\!24\)\( \beta_{1} - \)\(24\!\cdots\!66\)\( \beta_{2} + \)\(75\!\cdots\!00\)\( \beta_{3}) q^{23}\) \(+(-\)\(15\!\cdots\!40\)\( + \)\(67\!\cdots\!92\)\( \beta_{1} + \)\(18\!\cdots\!56\)\( \beta_{2} - \)\(10\!\cdots\!08\)\( \beta_{3}) q^{24}\) \(+(-\)\(41\!\cdots\!25\)\( - \)\(79\!\cdots\!00\)\( \beta_{1} - \)\(11\!\cdots\!00\)\( \beta_{2} + \)\(41\!\cdots\!00\)\( \beta_{3}) q^{25}\) \(+(-\)\(12\!\cdots\!08\)\( - \)\(54\!\cdots\!42\)\( \beta_{1} - \)\(19\!\cdots\!56\)\( \beta_{2} - \)\(11\!\cdots\!92\)\( \beta_{3}) q^{26}\) \(+(-\)\(32\!\cdots\!00\)\( - \)\(18\!\cdots\!96\)\( \beta_{1} + \)\(19\!\cdots\!58\)\( \beta_{2} + \)\(11\!\cdots\!00\)\( \beta_{3}) q^{27}\) \(+(-\)\(19\!\cdots\!00\)\( + \)\(86\!\cdots\!96\)\( \beta_{1} + \)\(19\!\cdots\!76\)\( \beta_{2} - \)\(48\!\cdots\!00\)\( \beta_{3}) q^{28}\) \(+(\)\(16\!\cdots\!30\)\( + \)\(32\!\cdots\!76\)\( \beta_{1} - \)\(54\!\cdots\!32\)\( \beta_{2} + \)\(93\!\cdots\!76\)\( \beta_{3}) q^{29}\) \(+(\)\(23\!\cdots\!00\)\( - \)\(12\!\cdots\!20\)\( \beta_{1} - \)\(18\!\cdots\!80\)\( \beta_{2} + \)\(64\!\cdots\!60\)\( \beta_{3}) q^{30}\) \(+(\)\(81\!\cdots\!32\)\( - \)\(32\!\cdots\!60\)\( \beta_{1} + \)\(47\!\cdots\!20\)\( \beta_{2} - \)\(92\!\cdots\!60\)\( \beta_{3}) q^{31}\) \(+(\)\(73\!\cdots\!00\)\( - \)\(27\!\cdots\!56\)\( \beta_{1} + \)\(14\!\cdots\!80\)\( \beta_{2} + \)\(24\!\cdots\!00\)\( \beta_{3}) q^{32}\) \(+(\)\(20\!\cdots\!00\)\( + \)\(41\!\cdots\!20\)\( \beta_{1} - \)\(54\!\cdots\!68\)\( \beta_{2} - \)\(19\!\cdots\!00\)\( \beta_{3}) q^{33}\) \(+(\)\(99\!\cdots\!04\)\( + \)\(22\!\cdots\!42\)\( \beta_{1} + \)\(54\!\cdots\!56\)\( \beta_{2} - \)\(38\!\cdots\!08\)\( \beta_{3}) q^{34}\) \(+(\)\(23\!\cdots\!00\)\( - \)\(10\!\cdots\!20\)\( \beta_{1} - \)\(23\!\cdots\!80\)\( \beta_{2} + \)\(57\!\cdots\!60\)\( \beta_{3}) q^{35}\) \(+(-\)\(12\!\cdots\!84\)\( - \)\(89\!\cdots\!58\)\( \beta_{1} + \)\(48\!\cdots\!31\)\( \beta_{2} + \)\(18\!\cdots\!17\)\( \beta_{3}) q^{36}\) \(+(-\)\(17\!\cdots\!50\)\( - \)\(22\!\cdots\!72\)\( \beta_{1} - \)\(16\!\cdots\!12\)\( \beta_{2} - \)\(11\!\cdots\!00\)\( \beta_{3}) q^{37}\) \(+(-\)\(16\!\cdots\!00\)\( + \)\(61\!\cdots\!16\)\( \beta_{1} + \)\(83\!\cdots\!32\)\( \beta_{2} - \)\(29\!\cdots\!00\)\( \beta_{3}) q^{38}\) \(+(-\)\(13\!\cdots\!44\)\( + \)\(35\!\cdots\!28\)\( \beta_{1} + \)\(34\!\cdots\!54\)\( \beta_{2} + \)\(98\!\cdots\!28\)\( \beta_{3}) q^{39}\) \(+(-\)\(17\!\cdots\!00\)\( + \)\(13\!\cdots\!00\)\( \beta_{1} + \)\(29\!\cdots\!00\)\( \beta_{2} - \)\(62\!\cdots\!00\)\( \beta_{3}) q^{40}\) \(+(\)\(40\!\cdots\!42\)\( - \)\(27\!\cdots\!60\)\( \beta_{1} - \)\(10\!\cdots\!80\)\( \beta_{2} - \)\(34\!\cdots\!60\)\( \beta_{3}) q^{41}\) \(+(\)\(37\!\cdots\!00\)\( - \)\(15\!\cdots\!84\)\( \beta_{1} - \)\(38\!\cdots\!24\)\( \beta_{2} + \)\(89\!\cdots\!00\)\( \beta_{3}) q^{42}\) \(+(\)\(18\!\cdots\!00\)\( + \)\(70\!\cdots\!68\)\( \beta_{1} + \)\(15\!\cdots\!67\)\( \beta_{2} - \)\(33\!\cdots\!00\)\( \beta_{3}) q^{43}\) \(+(\)\(38\!\cdots\!84\)\( + \)\(67\!\cdots\!72\)\( \beta_{1} - \)\(86\!\cdots\!04\)\( \beta_{2} - \)\(13\!\cdots\!28\)\( \beta_{3}) q^{44}\) \(+(\)\(15\!\cdots\!50\)\( + \)\(10\!\cdots\!20\)\( \beta_{1} - \)\(59\!\cdots\!20\)\( \beta_{2} - \)\(23\!\cdots\!60\)\( \beta_{3}) q^{45}\) \(+(-\)\(37\!\cdots\!28\)\( - \)\(52\!\cdots\!80\)\( \beta_{1} - \)\(56\!\cdots\!40\)\( \beta_{2} + \)\(62\!\cdots\!20\)\( \beta_{3}) q^{46}\) \(+(-\)\(54\!\cdots\!00\)\( - \)\(53\!\cdots\!48\)\( \beta_{1} + \)\(90\!\cdots\!16\)\( \beta_{2} + \)\(28\!\cdots\!00\)\( \beta_{3}) q^{47}\) \(+(-\)\(21\!\cdots\!00\)\( + \)\(76\!\cdots\!64\)\( \beta_{1} + \)\(24\!\cdots\!64\)\( \beta_{2} - \)\(47\!\cdots\!00\)\( \beta_{3}) q^{48}\) \(+(\)\(38\!\cdots\!57\)\( - \)\(12\!\cdots\!80\)\( \beta_{1} - \)\(50\!\cdots\!40\)\( \beta_{2} + \)\(74\!\cdots\!20\)\( \beta_{3}) q^{49}\) \(+(-\)\(93\!\cdots\!00\)\( + \)\(40\!\cdots\!25\)\( \beta_{1} - \)\(28\!\cdots\!00\)\( \beta_{2} + \)\(11\!\cdots\!00\)\( \beta_{3}) q^{50}\) \(+(\)\(51\!\cdots\!72\)\( - \)\(23\!\cdots\!36\)\( \beta_{1} - \)\(84\!\cdots\!98\)\( \beta_{2} - \)\(13\!\cdots\!36\)\( \beta_{3}) q^{51}\) \(+(\)\(85\!\cdots\!00\)\( + \)\(29\!\cdots\!36\)\( \beta_{1} + \)\(41\!\cdots\!74\)\( \beta_{2} + \)\(20\!\cdots\!50\)\( \beta_{3}) q^{52}\) \(+(\)\(21\!\cdots\!50\)\( - \)\(18\!\cdots\!80\)\( \beta_{1} - \)\(98\!\cdots\!44\)\( \beta_{2} - \)\(81\!\cdots\!00\)\( \beta_{3}) q^{53}\) \(+(\)\(51\!\cdots\!20\)\( - \)\(25\!\cdots\!76\)\( \beta_{1} - \)\(84\!\cdots\!68\)\( \beta_{2} + \)\(20\!\cdots\!24\)\( \beta_{3}) q^{54}\) \(+(-\)\(47\!\cdots\!00\)\( - \)\(80\!\cdots\!20\)\( \beta_{1} + \)\(10\!\cdots\!70\)\( \beta_{2} + \)\(16\!\cdots\!60\)\( \beta_{3}) q^{55}\) \(+(-\)\(22\!\cdots\!80\)\( + \)\(19\!\cdots\!64\)\( \beta_{1} - \)\(22\!\cdots\!48\)\( \beta_{2} - \)\(97\!\cdots\!36\)\( \beta_{3}) q^{56}\) \(+(-\)\(12\!\cdots\!00\)\( - \)\(14\!\cdots\!52\)\( \beta_{1} + \)\(67\!\cdots\!96\)\( \beta_{2} + \)\(14\!\cdots\!00\)\( \beta_{3}) q^{57}\) \(+(-\)\(10\!\cdots\!00\)\( - \)\(17\!\cdots\!06\)\( \beta_{1} - \)\(68\!\cdots\!12\)\( \beta_{2} + \)\(82\!\cdots\!00\)\( \beta_{3}) q^{58}\) \(+(-\)\(53\!\cdots\!40\)\( - \)\(31\!\cdots\!88\)\( \beta_{1} - \)\(66\!\cdots\!09\)\( \beta_{2} - \)\(38\!\cdots\!88\)\( \beta_{3}) q^{59}\) \(+(\)\(41\!\cdots\!00\)\( - \)\(28\!\cdots\!20\)\( \beta_{1} - \)\(69\!\cdots\!80\)\( \beta_{2} + \)\(11\!\cdots\!60\)\( \beta_{3}) q^{60}\) \(+(\)\(10\!\cdots\!62\)\( + \)\(63\!\cdots\!00\)\( \beta_{1} + \)\(25\!\cdots\!00\)\( \beta_{2} + \)\(50\!\cdots\!00\)\( \beta_{3}) q^{61}\) \(+(\)\(12\!\cdots\!00\)\( + \)\(94\!\cdots\!28\)\( \beta_{1} + \)\(72\!\cdots\!20\)\( \beta_{2} - \)\(11\!\cdots\!00\)\( \beta_{3}) q^{62}\) \(+(\)\(44\!\cdots\!00\)\( + \)\(22\!\cdots\!96\)\( \beta_{1} - \)\(15\!\cdots\!82\)\( \beta_{2} - \)\(82\!\cdots\!00\)\( \beta_{3}) q^{63}\) \(+(\)\(49\!\cdots\!52\)\( - \)\(57\!\cdots\!88\)\( \beta_{1} + \)\(11\!\cdots\!16\)\( \beta_{2} + \)\(25\!\cdots\!12\)\( \beta_{3}) q^{64}\) \(+(-\)\(10\!\cdots\!00\)\( - \)\(36\!\cdots\!60\)\( \beta_{1} - \)\(50\!\cdots\!40\)\( \beta_{2} - \)\(25\!\cdots\!20\)\( \beta_{3}) q^{65}\) \(+(-\)\(81\!\cdots\!56\)\( - \)\(12\!\cdots\!08\)\( \beta_{1} + \)\(18\!\cdots\!56\)\( \beta_{2} + \)\(26\!\cdots\!92\)\( \beta_{3}) q^{66}\) \(+(-\)\(37\!\cdots\!00\)\( + \)\(19\!\cdots\!60\)\( \beta_{1} + \)\(59\!\cdots\!17\)\( \beta_{2} + \)\(26\!\cdots\!00\)\( \beta_{3}) q^{67}\) \(+(-\)\(49\!\cdots\!00\)\( + \)\(18\!\cdots\!24\)\( \beta_{1} - \)\(61\!\cdots\!58\)\( \beta_{2} - \)\(15\!\cdots\!50\)\( \beta_{3}) q^{68}\) \(+(-\)\(16\!\cdots\!04\)\( + \)\(21\!\cdots\!04\)\( \beta_{1} - \)\(16\!\cdots\!28\)\( \beta_{2} + \)\(37\!\cdots\!04\)\( \beta_{3}) q^{69}\) \(+(\)\(41\!\cdots\!00\)\( - \)\(41\!\cdots\!40\)\( \beta_{1} - \)\(38\!\cdots\!60\)\( \beta_{2} + \)\(15\!\cdots\!20\)\( \beta_{3}) q^{70}\) \(+(\)\(66\!\cdots\!72\)\( - \)\(92\!\cdots\!00\)\( \beta_{1} + \)\(71\!\cdots\!50\)\( \beta_{2} + \)\(40\!\cdots\!00\)\( \beta_{3}) q^{71}\) \(+(\)\(25\!\cdots\!00\)\( - \)\(11\!\cdots\!88\)\( \beta_{1} - \)\(57\!\cdots\!76\)\( \beta_{2} - \)\(69\!\cdots\!00\)\( \beta_{3}) q^{72}\) \(+(\)\(10\!\cdots\!50\)\( + \)\(79\!\cdots\!44\)\( \beta_{1} + \)\(17\!\cdots\!24\)\( \beta_{2} - \)\(19\!\cdots\!00\)\( \beta_{3}) q^{73}\) \(+(\)\(68\!\cdots\!64\)\( + \)\(10\!\cdots\!30\)\( \beta_{1} + \)\(32\!\cdots\!40\)\( \beta_{2} + \)\(45\!\cdots\!80\)\( \beta_{3}) q^{74}\) \(+(\)\(57\!\cdots\!00\)\( - \)\(17\!\cdots\!00\)\( \beta_{1} - \)\(42\!\cdots\!25\)\( \beta_{2} - \)\(13\!\cdots\!00\)\( \beta_{3}) q^{75}\) \(+(-\)\(22\!\cdots\!60\)\( + \)\(14\!\cdots\!68\)\( \beta_{1} + \)\(58\!\cdots\!24\)\( \beta_{2} - \)\(59\!\cdots\!32\)\( \beta_{3}) q^{76}\) \(+(-\)\(15\!\cdots\!00\)\( - \)\(14\!\cdots\!68\)\( \beta_{1} + \)\(38\!\cdots\!68\)\( \beta_{2} + \)\(34\!\cdots\!00\)\( \beta_{3}) q^{77}\) \(+(-\)\(15\!\cdots\!00\)\( - \)\(58\!\cdots\!84\)\( \beta_{1} - \)\(80\!\cdots\!36\)\( \beta_{2} - \)\(44\!\cdots\!00\)\( \beta_{3}) q^{78}\) \(+(-\)\(54\!\cdots\!20\)\( - \)\(32\!\cdots\!24\)\( \beta_{1} - \)\(14\!\cdots\!32\)\( \beta_{2} + \)\(16\!\cdots\!76\)\( \beta_{3}) q^{79}\) \(+(-\)\(20\!\cdots\!00\)\( + \)\(22\!\cdots\!40\)\( \beta_{1} + \)\(20\!\cdots\!60\)\( \beta_{2} - \)\(83\!\cdots\!20\)\( \beta_{3}) q^{80}\) \(+(\)\(41\!\cdots\!21\)\( + \)\(57\!\cdots\!56\)\( \beta_{1} - \)\(15\!\cdots\!92\)\( \beta_{2} - \)\(21\!\cdots\!44\)\( \beta_{3}) q^{81}\) \(+(\)\(10\!\cdots\!00\)\( + \)\(18\!\cdots\!18\)\( \beta_{1} + \)\(29\!\cdots\!20\)\( \beta_{2} + \)\(29\!\cdots\!00\)\( \beta_{3}) q^{82}\) \(+(-\)\(48\!\cdots\!00\)\( + \)\(14\!\cdots\!56\)\( \beta_{1} + \)\(64\!\cdots\!33\)\( \beta_{2} - \)\(66\!\cdots\!00\)\( \beta_{3}) q^{83}\) \(+(\)\(67\!\cdots\!24\)\( - \)\(38\!\cdots\!04\)\( \beta_{1} - \)\(13\!\cdots\!72\)\( \beta_{2} + \)\(13\!\cdots\!96\)\( \beta_{3}) q^{84}\) \(+(\)\(59\!\cdots\!00\)\( - \)\(22\!\cdots\!20\)\( \beta_{1} + \)\(74\!\cdots\!20\)\( \beta_{2} + \)\(18\!\cdots\!60\)\( \beta_{3}) q^{85}\) \(+(-\)\(17\!\cdots\!68\)\( - \)\(99\!\cdots\!96\)\( \beta_{1} + \)\(95\!\cdots\!72\)\( \beta_{2} - \)\(29\!\cdots\!96\)\( \beta_{3}) q^{86}\) \(+(-\)\(64\!\cdots\!00\)\( + \)\(24\!\cdots\!32\)\( \beta_{1} - \)\(16\!\cdots\!86\)\( \beta_{2} + \)\(17\!\cdots\!00\)\( \beta_{3}) q^{87}\) \(+(-\)\(11\!\cdots\!00\)\( + \)\(47\!\cdots\!28\)\( \beta_{1} + \)\(32\!\cdots\!56\)\( \beta_{2} - \)\(24\!\cdots\!00\)\( \beta_{3}) q^{88}\) \(+(-\)\(15\!\cdots\!10\)\( + \)\(39\!\cdots\!28\)\( \beta_{1} - \)\(20\!\cdots\!96\)\( \beta_{2} + \)\(35\!\cdots\!28\)\( \beta_{3}) q^{89}\) \(+(-\)\(31\!\cdots\!00\)\( - \)\(35\!\cdots\!10\)\( \beta_{1} + \)\(19\!\cdots\!60\)\( \beta_{2} - \)\(32\!\cdots\!20\)\( \beta_{3}) q^{90}\) \(+(-\)\(11\!\cdots\!88\)\( - \)\(72\!\cdots\!16\)\( \beta_{1} - \)\(95\!\cdots\!88\)\( \beta_{2} - \)\(80\!\cdots\!16\)\( \beta_{3}) q^{91}\) \(+(\)\(40\!\cdots\!00\)\( - \)\(12\!\cdots\!40\)\( \beta_{1} + \)\(15\!\cdots\!92\)\( \beta_{2} - \)\(16\!\cdots\!00\)\( \beta_{3}) q^{92}\) \(+(-\)\(10\!\cdots\!00\)\( - \)\(27\!\cdots\!60\)\( \beta_{1} + \)\(20\!\cdots\!92\)\( \beta_{2} - \)\(29\!\cdots\!00\)\( \beta_{3}) q^{93}\) \(+(\)\(16\!\cdots\!44\)\( + \)\(17\!\cdots\!04\)\( \beta_{1} - \)\(12\!\cdots\!28\)\( \beta_{2} + \)\(41\!\cdots\!04\)\( \beta_{3}) q^{94}\) \(+(\)\(27\!\cdots\!00\)\( - \)\(17\!\cdots\!00\)\( \beta_{1} - \)\(70\!\cdots\!50\)\( \beta_{2} + \)\(70\!\cdots\!00\)\( \beta_{3}) q^{95}\) \(+(\)\(37\!\cdots\!52\)\( + \)\(19\!\cdots\!76\)\( \beta_{1} - \)\(10\!\cdots\!32\)\( \beta_{2} - \)\(10\!\cdots\!24\)\( \beta_{3}) q^{96}\) \(+(-\)\(20\!\cdots\!50\)\( + \)\(47\!\cdots\!52\)\( \beta_{1} - \)\(30\!\cdots\!84\)\( \beta_{2} - \)\(42\!\cdots\!00\)\( \beta_{3}) q^{97}\) \(+(\)\(42\!\cdots\!00\)\( - \)\(25\!\cdots\!77\)\( \beta_{1} - \)\(48\!\cdots\!40\)\( \beta_{2} + \)\(22\!\cdots\!00\)\( \beta_{3}) q^{98}\) \(+(-\)\(78\!\cdots\!44\)\( - \)\(27\!\cdots\!96\)\( \beta_{1} + \)\(21\!\cdots\!47\)\( \beta_{2} + \)\(10\!\cdots\!04\)\( \beta_{3}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut 1146312000q^{2} \) \(\mathstrut -\mathstrut 573723599022000q^{3} \) \(\mathstrut +\mathstrut 4402997675828604928q^{4} \) \(\mathstrut -\mathstrut \)\(52\!\cdots\!00\)\(q^{5} \) \(\mathstrut -\mathstrut \)\(81\!\cdots\!52\)\(q^{6} \) \(\mathstrut -\mathstrut \)\(63\!\cdots\!00\)\(q^{7} \) \(\mathstrut +\mathstrut \)\(97\!\cdots\!00\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!52\)\(q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut 1146312000q^{2} \) \(\mathstrut -\mathstrut 573723599022000q^{3} \) \(\mathstrut +\mathstrut 4402997675828604928q^{4} \) \(\mathstrut -\mathstrut \)\(52\!\cdots\!00\)\(q^{5} \) \(\mathstrut -\mathstrut \)\(81\!\cdots\!52\)\(q^{6} \) \(\mathstrut -\mathstrut \)\(63\!\cdots\!00\)\(q^{7} \) \(\mathstrut +\mathstrut \)\(97\!\cdots\!00\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!52\)\(q^{9} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!00\)\(q^{10} \) \(\mathstrut -\mathstrut \)\(41\!\cdots\!52\)\(q^{11} \) \(\mathstrut -\mathstrut \)\(99\!\cdots\!00\)\(q^{12} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!00\)\(q^{13} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!04\)\(q^{14} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!00\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!24\)\(q^{16} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!00\)\(q^{17} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!00\)\(q^{18} \) \(\mathstrut -\mathstrut \)\(35\!\cdots\!20\)\(q^{19} \) \(\mathstrut -\mathstrut \)\(50\!\cdots\!00\)\(q^{20} \) \(\mathstrut -\mathstrut \)\(55\!\cdots\!72\)\(q^{21} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!00\)\(q^{22} \) \(\mathstrut -\mathstrut \)\(41\!\cdots\!00\)\(q^{23} \) \(\mathstrut -\mathstrut \)\(61\!\cdots\!60\)\(q^{24} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!00\)\(q^{25} \) \(\mathstrut -\mathstrut \)\(49\!\cdots\!32\)\(q^{26} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!00\)\(q^{27} \) \(\mathstrut -\mathstrut \)\(79\!\cdots\!00\)\(q^{28} \) \(\mathstrut +\mathstrut \)\(66\!\cdots\!20\)\(q^{29} \) \(\mathstrut +\mathstrut \)\(95\!\cdots\!00\)\(q^{30} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!28\)\(q^{31} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!00\)\(q^{32} \) \(\mathstrut +\mathstrut \)\(80\!\cdots\!00\)\(q^{33} \) \(\mathstrut +\mathstrut \)\(39\!\cdots\!16\)\(q^{34} \) \(\mathstrut +\mathstrut \)\(94\!\cdots\!00\)\(q^{35} \) \(\mathstrut -\mathstrut \)\(49\!\cdots\!36\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(71\!\cdots\!00\)\(q^{37} \) \(\mathstrut -\mathstrut \)\(65\!\cdots\!00\)\(q^{38} \) \(\mathstrut -\mathstrut \)\(53\!\cdots\!76\)\(q^{39} \) \(\mathstrut -\mathstrut \)\(69\!\cdots\!00\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!68\)\(q^{41} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!00\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(75\!\cdots\!00\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!36\)\(q^{44} \) \(\mathstrut +\mathstrut \)\(60\!\cdots\!00\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!12\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!00\)\(q^{47} \) \(\mathstrut -\mathstrut \)\(84\!\cdots\!00\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!28\)\(q^{49} \) \(\mathstrut -\mathstrut \)\(37\!\cdots\!00\)\(q^{50} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!88\)\(q^{51} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!00\)\(q^{52} \) \(\mathstrut +\mathstrut \)\(84\!\cdots\!00\)\(q^{53} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!80\)\(q^{54} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!00\)\(q^{55} \) \(\mathstrut -\mathstrut \)\(89\!\cdots\!20\)\(q^{56} \) \(\mathstrut -\mathstrut \)\(48\!\cdots\!00\)\(q^{57} \) \(\mathstrut -\mathstrut \)\(42\!\cdots\!00\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!60\)\(q^{59} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!00\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(42\!\cdots\!48\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(51\!\cdots\!00\)\(q^{62} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!00\)\(q^{63} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!08\)\(q^{64} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!00\)\(q^{65} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!24\)\(q^{66} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!00\)\(q^{67} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!00\)\(q^{68} \) \(\mathstrut -\mathstrut \)\(65\!\cdots\!16\)\(q^{69} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!00\)\(q^{70} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!88\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!00\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(43\!\cdots\!00\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!56\)\(q^{74} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!00\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(91\!\cdots\!40\)\(q^{76} \) \(\mathstrut -\mathstrut \)\(62\!\cdots\!00\)\(q^{77} \) \(\mathstrut -\mathstrut \)\(62\!\cdots\!00\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!80\)\(q^{79} \) \(\mathstrut -\mathstrut \)\(81\!\cdots\!00\)\(q^{80} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!84\)\(q^{81} \) \(\mathstrut +\mathstrut \)\(41\!\cdots\!00\)\(q^{82} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!00\)\(q^{83} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!96\)\(q^{84} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!00\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(71\!\cdots\!72\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!00\)\(q^{87} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!00\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(60\!\cdots\!40\)\(q^{89} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!00\)\(q^{90} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!52\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!00\)\(q^{92} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!00\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(64\!\cdots\!76\)\(q^{94} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!00\)\(q^{95} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!08\)\(q^{96} \) \(\mathstrut -\mathstrut \)\(80\!\cdots\!00\)\(q^{97} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!00\)\(q^{98} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!76\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut -\mathstrut \) \(x^{3}\mathstrut -\mathstrut \) \(180363795469121\) \(x^{2}\mathstrut +\mathstrut \) \(166129321978984507920\) \(x\mathstrut +\mathstrut \) \(2785609847439483545242446300\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 192 \nu - 48 \)
\(\beta_{2}\)\(=\)\((\)\( 12 \nu^{3} + 20695116 \nu^{2} - 1765108515928884 \nu - 371162118903041598888 \)\()/13402613\)
\(\beta_{3}\)\(=\)\((\)\( -588 \nu^{3} + 69567928692 \nu^{2} + 184007562081366900 \nu - 6347030828361347486393976 \)\()/1914659\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(48\)\()/192\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(343\) \(\beta_{2}\mathstrut -\mathstrut \) \(265270378\) \(\beta_{1}\mathstrut +\mathstrut \) \(3324465478086847488\)\()/36864\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(1724593\) \(\beta_{3}\mathstrut +\mathstrut \) \(40581291737\) \(\beta_{2}\mathstrut +\mathstrut \) \(28699219691868298\) \(\beta_{1}\mathstrut -\mathstrut \) \(4593138507376746544705536\)\()/36864\)

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
1.21599e7
4.76281e6
−3.61406e6
−1.33086e7
−2.04812e9 1.78996e14 1.88894e18 −2.27994e20 −3.66606e23 −4.42559e25 8.53860e26 −9.51338e28 4.66958e29
1.2 −6.27881e8 −6.22194e14 −1.91161e18 2.33985e20 3.90664e23 6.25792e25 2.64806e27 2.59952e29 −1.46915e29
1.3 9.80478e8 2.49037e14 −1.34451e18 1.59503e20 2.44176e23 1.63507e25 −3.57909e27 −6.51539e28 1.56389e29
1.4 2.84183e9 −3.79563e14 5.77017e18 −6.89620e20 −1.07865e24 −9.80127e25 9.84504e27 1.68945e28 −1.95979e30
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{62}^{\mathrm{new}}(\Gamma_0(1))\).