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

Related objects

Downloads

Learn more about

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.

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

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 \(S_{34}^{\mathrm{new}}(\Gamma_0(1))\).