Properties

Label 1.34.a.a
Level 1
Weight 34
Character orbit 1.a
Self dual Yes
Analytic conductor 6.898
Analytic rank 1
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 34 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(6.8982828881\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 72\sqrt{2356201}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -60840 - \beta ) q^{2} \) \( + ( 18959940 + 312 \beta ) q^{3} \) \( + ( 7326116992 + 121680 \beta ) q^{4} \) \( + ( -90530768250 - 1004000 \beta ) q^{5} \) \( + ( -4964461096608 - 37942020 \beta ) q^{6} \) \( + ( -33576540033400 + 801594864 \beta ) q^{7} \) \( + ( -1409375292549120 - 6139193600 \beta ) q^{8} \) \( + ( -4010568477485427 + 11831002560 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \(+(-60840 - \beta) q^{2}\) \(+(18959940 + 312 \beta) q^{3}\) \(+(7326116992 + 121680 \beta) q^{4}\) \(+(-90530768250 - 1004000 \beta) q^{5}\) \(+(-4964461096608 - 37942020 \beta) q^{6}\) \(+(-33576540033400 + 801594864 \beta) q^{7}\) \(+(-1409375292549120 - 6139193600 \beta) q^{8}\) \(+(-4010568477485427 + 11831002560 \beta) q^{9}\) \(+(17771296108266000 + 151614128250 \beta) q^{10}\) \(+(66935907720957132 - 1346611783000 \beta) q^{11}\) \(+(602617716665233920 + 4592794000704 \beta) q^{12}\) \(+(-1490805239129721970 - 3738595861728 \beta) q^{13}\) \(+(-7748320631234170176 - 15192491492360 \beta) q^{14}\) \(+(-5542640034569937000 - 47281379454000 \beta) q^{15}\) \(+(97802989555947175936 + 737660590018560 \beta) q^{16}\) \(+(-39680574630587762670 - 2662090686805056 \beta) q^{17}\) \(+(99492661364271659640 + 3290770281735027 \beta) q^{18}\) \(+(-\)\(68\!\cdots\!00\)\( + 4884703150768920 \beta) q^{19}\) \(+(-\)\(21\!\cdots\!00\)\( - 18371205340628000 \beta) q^{20}\) \(+(\)\(24\!\cdots\!12\)\( + 4722310035327360 \beta) q^{21}\) \(+(\)\(12\!\cdots\!20\)\( + 14991953156762868 \beta) q^{22}\) \(+(\)\(13\!\cdots\!20\)\( + 164629195362887632 \beta) q^{23}\) \(+(-\)\(50\!\cdots\!00\)\( - 556123833579709440 \beta) q^{24}\) \(+(-\)\(95\!\cdots\!25\)\( + 181785782646000000 \beta) q^{25}\) \(+(\)\(13\!\cdots\!52\)\( + 1718261411357253490 \beta) q^{26}\) \(+(-\)\(13\!\cdots\!20\)\( - 2761409163063330000 \beta) q^{27}\) \(+(\)\(94\!\cdots\!80\)\( + 1786984362586217088 \beta) q^{28}\) \(+(-\)\(83\!\cdots\!50\)\( - 6085145360184173920 \beta) q^{29}\) \(+(\)\(91\!\cdots\!00\)\( + 8419239160551297000 \beta) q^{30}\) \(+(-\)\(31\!\cdots\!08\)\( + 32916497064471576000 \beta) q^{31}\) \(+(-\)\(28\!\cdots\!40\)\( - 89946988381051355136 \beta) q^{32}\) \(+(-\)\(38\!\cdots\!20\)\( - 4647675400034394816 \beta) q^{33}\) \(+(\)\(34\!\cdots\!04\)\( + \)\(20\!\cdots\!10\)\( \beta) q^{34}\) \(+(-\)\(67\!\cdots\!00\)\( - 38858152669640668000 \beta) q^{35}\) \(+(-\)\(11\!\cdots\!84\)\( - \)\(40\!\cdots\!40\)\( \beta) q^{36}\) \(+(-\)\(52\!\cdots\!10\)\( + \)\(20\!\cdots\!04\)\( \beta) q^{37}\) \(+(-\)\(18\!\cdots\!80\)\( + \)\(38\!\cdots\!00\)\( \beta) q^{38}\) \(+(-\)\(42\!\cdots\!24\)\( - \)\(53\!\cdots\!60\)\( \beta) q^{39}\) \(+(\)\(20\!\cdots\!00\)\( + \)\(19\!\cdots\!00\)\( \beta) q^{40}\) \(+(\)\(13\!\cdots\!22\)\( - \)\(32\!\cdots\!00\)\( \beta) q^{41}\) \(+(-\)\(20\!\cdots\!20\)\( - \)\(27\!\cdots\!12\)\( \beta) q^{42}\) \(+(\)\(78\!\cdots\!00\)\( + \)\(88\!\cdots\!52\)\( \beta) q^{43}\) \(+(-\)\(15\!\cdots\!56\)\( - \)\(17\!\cdots\!40\)\( \beta) q^{44}\) \(+(\)\(21\!\cdots\!50\)\( + \)\(29\!\cdots\!00\)\( \beta) q^{45}\) \(+(-\)\(28\!\cdots\!88\)\( - \)\(23\!\cdots\!00\)\( \beta) q^{46}\) \(+(\)\(27\!\cdots\!20\)\( + \)\(49\!\cdots\!84\)\( \beta) q^{47}\) \(+(\)\(46\!\cdots\!20\)\( + \)\(44\!\cdots\!32\)\( \beta) q^{48}\) \(+(\)\(12\!\cdots\!57\)\( - \)\(53\!\cdots\!00\)\( \beta) q^{49}\) \(+(\)\(36\!\cdots\!00\)\( + \)\(84\!\cdots\!25\)\( \beta) q^{50}\) \(+(-\)\(10\!\cdots\!48\)\( - \)\(62\!\cdots\!80\)\( \beta) q^{51}\) \(+(-\)\(16\!\cdots\!00\)\( - \)\(20\!\cdots\!76\)\( \beta) q^{52}\) \(+(-\)\(13\!\cdots\!10\)\( + \)\(20\!\cdots\!12\)\( \beta) q^{53}\) \(+(\)\(42\!\cdots\!00\)\( + \)\(30\!\cdots\!20\)\( \beta) q^{54}\) \(+(\)\(10\!\cdots\!00\)\( + \)\(54\!\cdots\!00\)\( \beta) q^{55}\) \(+(-\)\(12\!\cdots\!00\)\( - \)\(92\!\cdots\!80\)\( \beta) q^{56}\) \(+(\)\(57\!\cdots\!60\)\( - \)\(11\!\cdots\!00\)\( \beta) q^{57}\) \(+(\)\(12\!\cdots\!80\)\( + \)\(12\!\cdots\!50\)\( \beta) q^{58}\) \(+(-\)\(15\!\cdots\!00\)\( + \)\(31\!\cdots\!60\)\( \beta) q^{59}\) \(+(-\)\(11\!\cdots\!00\)\( - \)\(10\!\cdots\!00\)\( \beta) q^{60}\) \(+(-\)\(28\!\cdots\!18\)\( - \)\(63\!\cdots\!00\)\( \beta) q^{61}\) \(+(-\)\(21\!\cdots\!80\)\( + \)\(11\!\cdots\!08\)\( \beta) q^{62}\) \(+(\)\(25\!\cdots\!60\)\( - \)\(36\!\cdots\!28\)\( \beta) q^{63}\) \(+(\)\(43\!\cdots\!12\)\( + \)\(19\!\cdots\!60\)\( \beta) q^{64}\) \(+(\)\(18\!\cdots\!00\)\( + \)\(18\!\cdots\!00\)\( \beta) q^{65}\) \(+(\)\(29\!\cdots\!44\)\( + \)\(41\!\cdots\!60\)\( \beta) q^{66}\) \(+(\)\(79\!\cdots\!80\)\( + \)\(28\!\cdots\!44\)\( \beta) q^{67}\) \(+(-\)\(42\!\cdots\!60\)\( - \)\(24\!\cdots\!52\)\( \beta) q^{68}\) \(+(\)\(87\!\cdots\!56\)\( + \)\(72\!\cdots\!20\)\( \beta) q^{69}\) \(+(\)\(88\!\cdots\!00\)\( + \)\(91\!\cdots\!00\)\( \beta) q^{70}\) \(+(-\)\(13\!\cdots\!88\)\( + \)\(18\!\cdots\!00\)\( \beta) q^{71}\) \(+(\)\(47\!\cdots\!40\)\( + \)\(79\!\cdots\!00\)\( \beta) q^{72}\) \(+(\)\(47\!\cdots\!70\)\( - \)\(59\!\cdots\!68\)\( \beta) q^{73}\) \(+(\)\(72\!\cdots\!64\)\( + \)\(40\!\cdots\!50\)\( \beta) q^{74}\) \(+(-\)\(11\!\cdots\!00\)\( - \)\(26\!\cdots\!00\)\( \beta) q^{75}\) \(+(\)\(22\!\cdots\!00\)\( - \)\(46\!\cdots\!60\)\( \beta) q^{76}\) \(+(-\)\(15\!\cdots\!00\)\( + \)\(98\!\cdots\!48\)\( \beta) q^{77}\) \(+(\)\(91\!\cdots\!00\)\( + \)\(75\!\cdots\!24\)\( \beta) q^{78}\) \(+(-\)\(42\!\cdots\!00\)\( - \)\(11\!\cdots\!20\)\( \beta) q^{79}\) \(+(-\)\(17\!\cdots\!00\)\( - \)\(16\!\cdots\!00\)\( \beta) q^{80}\) \(+(\)\(91\!\cdots\!21\)\( - \)\(16\!\cdots\!20\)\( \beta) q^{81}\) \(+(\)\(31\!\cdots\!20\)\( + \)\(58\!\cdots\!78\)\( \beta) q^{82}\) \(+(\)\(14\!\cdots\!60\)\( + \)\(24\!\cdots\!92\)\( \beta) q^{83}\) \(+(\)\(24\!\cdots\!04\)\( + \)\(32\!\cdots\!80\)\( \beta) q^{84}\) \(+(\)\(36\!\cdots\!00\)\( + \)\(28\!\cdots\!00\)\( \beta) q^{85}\) \(+(-\)\(15\!\cdots\!68\)\( - \)\(13\!\cdots\!80\)\( \beta) q^{86}\) \(+(-\)\(39\!\cdots\!60\)\( - \)\(37\!\cdots\!00\)\( \beta) q^{87}\) \(+(\)\(66\!\cdots\!60\)\( + \)\(14\!\cdots\!00\)\( \beta) q^{88}\) \(+(\)\(66\!\cdots\!50\)\( - \)\(38\!\cdots\!60\)\( \beta) q^{89}\) \(+(-\)\(49\!\cdots\!00\)\( - \)\(39\!\cdots\!50\)\( \beta) q^{90}\) \(+(\)\(13\!\cdots\!72\)\( - \)\(10\!\cdots\!80\)\( \beta) q^{91}\) \(+(\)\(34\!\cdots\!80\)\( + \)\(27\!\cdots\!44\)\( \beta) q^{92}\) \(+(\)\(66\!\cdots\!80\)\( - \)\(34\!\cdots\!96\)\( \beta) q^{93}\) \(+(-\)\(22\!\cdots\!56\)\( - \)\(30\!\cdots\!80\)\( \beta) q^{94}\) \(+(\)\(16\!\cdots\!00\)\( + \)\(24\!\cdots\!00\)\( \beta) q^{95}\) \(+(-\)\(39\!\cdots\!88\)\( - \)\(25\!\cdots\!20\)\( \beta) q^{96}\) \(+(-\)\(18\!\cdots\!30\)\( + \)\(53\!\cdots\!84\)\( \beta) q^{97}\) \(+(\)\(58\!\cdots\!20\)\( + \)\(20\!\cdots\!43\)\( \beta) q^{98}\) \(+(-\)\(46\!\cdots\!64\)\( + \)\(61\!\cdots\!20\)\( \beta) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 121680q^{2} \) \(\mathstrut +\mathstrut 37919880q^{3} \) \(\mathstrut +\mathstrut 14652233984q^{4} \) \(\mathstrut -\mathstrut 181061536500q^{5} \) \(\mathstrut -\mathstrut 9928922193216q^{6} \) \(\mathstrut -\mathstrut 67153080066800q^{7} \) \(\mathstrut -\mathstrut 2818750585098240q^{8} \) \(\mathstrut -\mathstrut 8021136954970854q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 121680q^{2} \) \(\mathstrut +\mathstrut 37919880q^{3} \) \(\mathstrut +\mathstrut 14652233984q^{4} \) \(\mathstrut -\mathstrut 181061536500q^{5} \) \(\mathstrut -\mathstrut 9928922193216q^{6} \) \(\mathstrut -\mathstrut 67153080066800q^{7} \) \(\mathstrut -\mathstrut 2818750585098240q^{8} \) \(\mathstrut -\mathstrut 8021136954970854q^{9} \) \(\mathstrut +\mathstrut 35542592216532000q^{10} \) \(\mathstrut +\mathstrut 133871815441914264q^{11} \) \(\mathstrut +\mathstrut 1205235433330467840q^{12} \) \(\mathstrut -\mathstrut 2981610478259443940q^{13} \) \(\mathstrut -\mathstrut 15496641262468340352q^{14} \) \(\mathstrut -\mathstrut 11085280069139874000q^{15} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!72\)\(q^{16} \) \(\mathstrut -\mathstrut 79361149261175525340q^{17} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!80\)\(q^{18} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!00\)\(q^{19} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!00\)\(q^{20} \) \(\mathstrut +\mathstrut \)\(48\!\cdots\!24\)\(q^{21} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!40\)\(q^{22} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!40\)\(q^{23} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!00\)\(q^{24} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!50\)\(q^{25} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!04\)\(q^{26} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!40\)\(q^{27} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!60\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!00\)\(q^{29} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!00\)\(q^{30} \) \(\mathstrut -\mathstrut \)\(62\!\cdots\!16\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(57\!\cdots\!80\)\(q^{32} \) \(\mathstrut -\mathstrut \)\(77\!\cdots\!40\)\(q^{33} \) \(\mathstrut +\mathstrut \)\(69\!\cdots\!08\)\(q^{34} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!00\)\(q^{35} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!68\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!20\)\(q^{37} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!60\)\(q^{38} \) \(\mathstrut -\mathstrut \)\(85\!\cdots\!48\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(40\!\cdots\!00\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!44\)\(q^{41} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!40\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!00\)\(q^{43} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!12\)\(q^{44} \) \(\mathstrut +\mathstrut \)\(43\!\cdots\!00\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(56\!\cdots\!76\)\(q^{46} \) \(\mathstrut +\mathstrut \)\(54\!\cdots\!40\)\(q^{47} \) \(\mathstrut +\mathstrut \)\(93\!\cdots\!40\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!14\)\(q^{49} \) \(\mathstrut +\mathstrut \)\(72\!\cdots\!00\)\(q^{50} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!96\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!00\)\(q^{52} \) \(\mathstrut -\mathstrut \)\(26\!\cdots\!20\)\(q^{53} \) \(\mathstrut +\mathstrut \)\(84\!\cdots\!00\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!00\)\(q^{55} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!00\)\(q^{56} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!20\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!60\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!00\)\(q^{59} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!00\)\(q^{60} \) \(\mathstrut -\mathstrut \)\(57\!\cdots\!36\)\(q^{61} \) \(\mathstrut -\mathstrut \)\(42\!\cdots\!60\)\(q^{62} \) \(\mathstrut +\mathstrut \)\(50\!\cdots\!20\)\(q^{63} \) \(\mathstrut +\mathstrut \)\(86\!\cdots\!24\)\(q^{64} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!00\)\(q^{65} \) \(\mathstrut +\mathstrut \)\(58\!\cdots\!88\)\(q^{66} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!60\)\(q^{67} \) \(\mathstrut -\mathstrut \)\(84\!\cdots\!20\)\(q^{68} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!12\)\(q^{69} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!00\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(26\!\cdots\!76\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(95\!\cdots\!80\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(94\!\cdots\!40\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!28\)\(q^{74} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!00\)\(q^{75} \) \(\mathstrut +\mathstrut \)\(45\!\cdots\!00\)\(q^{76} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!00\)\(q^{77} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!00\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(85\!\cdots\!00\)\(q^{79} \) \(\mathstrut -\mathstrut \)\(35\!\cdots\!00\)\(q^{80} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!42\)\(q^{81} \) \(\mathstrut +\mathstrut \)\(62\!\cdots\!40\)\(q^{82} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!20\)\(q^{83} \) \(\mathstrut +\mathstrut \)\(49\!\cdots\!08\)\(q^{84} \) \(\mathstrut +\mathstrut \)\(72\!\cdots\!00\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!36\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(78\!\cdots\!20\)\(q^{87} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!20\)\(q^{88} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!00\)\(q^{89} \) \(\mathstrut -\mathstrut \)\(98\!\cdots\!00\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!44\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(68\!\cdots\!60\)\(q^{92} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!60\)\(q^{93} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!12\)\(q^{94} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!00\)\(q^{95} \) \(\mathstrut -\mathstrut \)\(79\!\cdots\!76\)\(q^{96} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!60\)\(q^{97} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!40\)\(q^{98} \) \(\mathstrut -\mathstrut \)\(92\!\cdots\!28\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
767.996
−766.996
−171359. 5.34420e7 2.07741e10 −2.01492e11 −9.15779e12 5.50153e13 −2.08788e15 −2.70301e15 3.45276e16
1.2 49679.4 −1.55221e7 −6.12189e9 2.04307e10 −7.71130e11 −1.22168e14 −7.30875e14 −5.31812e15 1.01499e15
\(n\): 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 \(S_{34}^{\mathrm{new}}(\Gamma_0(1))\).