Properties

Label 1.34.a
Level 1
Weight 34
Character orbit a
Rep. character \(\chi_{1}(1,\cdot)\)
Character field \(\Q\)
Dimension 2
Newform subspaces 1
Sturm bound 2
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 34 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(2\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{34}(\Gamma_0(1))\).

Total New Old
Modular forms 3 3 0
Cusp forms 2 2 0
Eisenstein series 1 1 0

Trace form

\( 2q - 121680q^{2} + 37919880q^{3} + 14652233984q^{4} - 181061536500q^{5} - 9928922193216q^{6} - 67153080066800q^{7} - 2818750585098240q^{8} - 8021136954970854q^{9} + O(q^{10}) \) \( 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} + 195605979111894351872q^{16} - 79361149261175525340q^{17} + 198985322728543319280q^{18} - 1360696443041697171800q^{19} - 4310900037459183168000q^{20} + 4836438843082801141824q^{21} + 24751741840613395122240q^{22} + 26163854053674631579440q^{23} - 100235518588319710003200q^{24} - 191814088084783724781250q^{25} + 272731683634642092710304q^{26} - 272704742315639906405040q^{27} + 1890794991723644827125760q^{28} - 1674519971630399929109700q^{29} + 1829469606462141559632000q^{30} - 6238301225723632337095616q^{31} - 5708163006744585940500480q^{32} - 7725507376265263259943840q^{33} + 69860790536166916073075808q^{34} - 13581143663546216842308000q^{35} - 23595755096197135608576768q^{36} - 104802741516758180098173620q^{37} - 36544090911856460663722560q^{38} - 85026272438019668836783248q^{39} + 405758480738092701742080000q^{40} + 277566613022002182732188244q^{41} - 409610685367579027895216640q^{42} + 1567661189367072113768291800q^{43} - 3022085914387766085808346112q^{44} + 435982958561646097146265500q^{45} - 5613550634763385681470869376q^{46} + 5421061461431251260476972640q^{47} + 9331035686759324258712944640q^{48} + 2489799365888888933444851314q^{49} + 7229107516344249428363250000q^{50} - 21794809591439435388668624496q^{51} - 32956722688372988680703091200q^{52} - 26867414785542903092975171220q^{53} + 84050074928235529084016073600q^{54} + 20908570805489145209222682000q^{55} - 25575237795171758759822131200q^{56} + 11431882183748158902465090720q^{57} + 250532370716581201796821222560q^{58} - 305786409635298433026663995400q^{59} - 221757467882938689123571968000q^{60} - 5745893983413264859067362436q^{61} - 424581887479352364037004290560q^{62} + 500999494400792409792776544720q^{63} + 864365322360380859973709594624q^{64} + 361623311364964256200313721000q^{65} + 583558358556830038058269663488q^{66} + 1598906337224495341750209565960q^{67} - 8494558579722382470601861086720q^{68} + 1750848530317427976275742443712q^{69} + 1775546165763388432770973344000q^{70} - 2669761937956104715736810084976q^{71} + 9530438520123487645446231060480q^{72} + 946314321191998870965166536340q^{73} + 1457936998080019494211346192928q^{74} - 2251234781189208709982677125000q^{75} + 4551314657470802548563162035200q^{76} - 30864620292716415864215760033600q^{77} + 18267352963262685465124679337600q^{78} - 8535199972765636469730106191200q^{79} - 35800819471604723635001769984000q^{80} + 18372400450705101408694430349042q^{81} + 62171822095602398237444019107040q^{82} + 29072605803747945444403983322920q^{83} + 49469534061147727842067908599808q^{84} + 72477213859794605306963311287000q^{85} - 312696885737578501917214826406336q^{86} - 78129025791401437906177732680720q^{87} + 13282372192867492753038509752320q^{88} + 132719467655604316264938597726900q^{89} - 98726386114587723068546425404000q^{90} + 26902102147452358410548887211744q^{91} + 681044992402323869436382557788160q^{92} + 132607664882695458799488635592960q^{93} - 450506433888867898835049561530112q^{94} + 3378716387067841884487537110000q^{95} - 793791282556560211499025092837376q^{96} - 367787330930535727150839877856060q^{97} + 1163527978168280943548587223169840q^{98} - 926100695332764780030656940070728q^{99} + O(q^{100}) \)

Decomposition of \(S_{34}^{\mathrm{new}}(\Gamma_0(1))\) into newform subspaces

Label Dim. \(A\) Field CM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1.34.a.a \(2\) \(6.898\) \(\mathbb{Q}[x]/(x^{2} - \cdots)\) None \(-121680\) \(37919880\) \(-181061536500\) \(-6\!\cdots\!00\) \(+\) \(q+(-60840-\beta )q^{2}+(18959940+312\beta )q^{3}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 121680 T + 8666828800 T^{2} + 1045223241154560 T^{3} + 73786976294838206464 T^{4} \)
$3$ \( 1 - 37919880 T + 10288587693648150 T^{2} - \)\(21\!\cdots\!40\)\( T^{3} + \)\(30\!\cdots\!29\)\( T^{4} \)
$5$ \( 1 + 181061536500 T + \)\(22\!\cdots\!50\)\( T^{2} + \)\(21\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!25\)\( T^{4} \)
$7$ \( 1 + 67153080066800 T + \)\(87\!\cdots\!50\)\( T^{2} + \)\(51\!\cdots\!00\)\( T^{3} + \)\(59\!\cdots\!49\)\( T^{4} \)
$11$ \( 1 - 133871815441914264 T + \)\(28\!\cdots\!86\)\( T^{2} - \)\(31\!\cdots\!84\)\( T^{3} + \)\(53\!\cdots\!61\)\( T^{4} \)
$13$ \( 1 + 2981610478259443940 T + \)\(13\!\cdots\!50\)\( T^{2} + \)\(17\!\cdots\!20\)\( T^{3} + \)\(33\!\cdots\!09\)\( T^{4} \)
$17$ \( 1 + 79361149261175525340 T - \)\(44\!\cdots\!50\)\( T^{2} + \)\(31\!\cdots\!80\)\( T^{3} + \)\(16\!\cdots\!69\)\( T^{4} \)
$19$ \( 1 + \)\(13\!\cdots\!00\)\( T + \)\(33\!\cdots\!18\)\( T^{2} + \)\(21\!\cdots\!00\)\( T^{3} + \)\(24\!\cdots\!81\)\( T^{4} \)
$23$ \( 1 - \)\(26\!\cdots\!40\)\( T + \)\(15\!\cdots\!50\)\( T^{2} - \)\(22\!\cdots\!20\)\( T^{3} + \)\(74\!\cdots\!89\)\( T^{4} \)
$29$ \( 1 + \)\(16\!\cdots\!00\)\( T + \)\(38\!\cdots\!78\)\( T^{2} + \)\(30\!\cdots\!00\)\( T^{3} + \)\(32\!\cdots\!21\)\( T^{4} \)
$31$ \( 1 + \)\(62\!\cdots\!16\)\( T + \)\(29\!\cdots\!46\)\( T^{2} + \)\(10\!\cdots\!56\)\( T^{3} + \)\(26\!\cdots\!81\)\( T^{4} \)
$37$ \( 1 + \)\(10\!\cdots\!20\)\( T + \)\(13\!\cdots\!50\)\( T^{2} + \)\(59\!\cdots\!40\)\( T^{3} + \)\(31\!\cdots\!09\)\( T^{4} \)
$41$ \( 1 - \)\(27\!\cdots\!44\)\( T + \)\(22\!\cdots\!26\)\( T^{2} - \)\(46\!\cdots\!24\)\( T^{3} + \)\(27\!\cdots\!41\)\( T^{4} \)
$43$ \( 1 - \)\(15\!\cdots\!00\)\( T + \)\(12\!\cdots\!50\)\( T^{2} - \)\(12\!\cdots\!00\)\( T^{3} + \)\(64\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 - \)\(54\!\cdots\!40\)\( T + \)\(37\!\cdots\!50\)\( T^{2} - \)\(81\!\cdots\!80\)\( T^{3} + \)\(22\!\cdots\!29\)\( T^{4} \)
$53$ \( 1 + \)\(26\!\cdots\!20\)\( T + \)\(12\!\cdots\!50\)\( T^{2} + \)\(21\!\cdots\!60\)\( T^{3} + \)\(63\!\cdots\!29\)\( T^{4} \)
$59$ \( 1 + \)\(30\!\cdots\!00\)\( T + \)\(77\!\cdots\!58\)\( T^{2} + \)\(83\!\cdots\!00\)\( T^{3} + \)\(75\!\cdots\!41\)\( T^{4} \)
$61$ \( 1 + \)\(57\!\cdots\!36\)\( T + \)\(16\!\cdots\!86\)\( T^{2} + \)\(47\!\cdots\!16\)\( T^{3} + \)\(67\!\cdots\!61\)\( T^{4} \)
$67$ \( 1 - \)\(15\!\cdots\!60\)\( T + \)\(41\!\cdots\!50\)\( T^{2} - \)\(29\!\cdots\!20\)\( T^{3} + \)\(33\!\cdots\!69\)\( T^{4} \)
$71$ \( 1 + \)\(26\!\cdots\!76\)\( T + \)\(22\!\cdots\!66\)\( T^{2} + \)\(32\!\cdots\!36\)\( T^{3} + \)\(15\!\cdots\!21\)\( T^{4} \)
$73$ \( 1 - \)\(94\!\cdots\!40\)\( T + \)\(19\!\cdots\!50\)\( T^{2} - \)\(29\!\cdots\!20\)\( T^{3} + \)\(95\!\cdots\!89\)\( T^{4} \)
$79$ \( 1 + \)\(85\!\cdots\!00\)\( T + \)\(85\!\cdots\!78\)\( T^{2} + \)\(35\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!21\)\( T^{4} \)
$83$ \( 1 - \)\(29\!\cdots\!20\)\( T + \)\(37\!\cdots\!50\)\( T^{2} - \)\(62\!\cdots\!60\)\( T^{3} + \)\(45\!\cdots\!69\)\( T^{4} \)
$89$ \( 1 - \)\(13\!\cdots\!00\)\( T + \)\(47\!\cdots\!38\)\( T^{2} - \)\(28\!\cdots\!00\)\( T^{3} + \)\(45\!\cdots\!61\)\( T^{4} \)
$97$ \( 1 + \)\(36\!\cdots\!60\)\( T + \)\(42\!\cdots\!50\)\( T^{2} + \)\(13\!\cdots\!20\)\( T^{3} + \)\(13\!\cdots\!29\)\( T^{4} \)
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