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Properties

Label 8.6.b
Level 8
Weight 6
Character orbit b
Rep. character \(\chi_{8}(5,\cdot)\)
Character field \(\Q\)
Dimension 4
Newform subspaces 1
Sturm bound 6
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 8.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 8 \)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(6\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(8, [\chi])\).

Total New Old
Modular forms 6 6 0
Cusp forms 4 4 0
Eisenstein series 2 2 0

Trace form

\( 4q - 2q^{2} + 20q^{4} - 116q^{6} + 96q^{7} - 248q^{8} - 164q^{9} + O(q^{10}) \) \( 4q - 2q^{2} + 20q^{4} - 116q^{6} + 96q^{7} - 248q^{8} - 164q^{9} + 632q^{10} + 1576q^{12} - 2384q^{14} - 416q^{15} - 3312q^{16} + 200q^{17} + 4754q^{18} + 4624q^{20} - 5636q^{22} + 2336q^{23} - 7792q^{24} + 1556q^{25} + 5608q^{26} + 5152q^{28} - 2128q^{30} - 12928q^{31} + 5408q^{32} - 2352q^{33} - 4772q^{34} - 10164q^{36} + 15980q^{38} + 35104q^{39} + 16032q^{40} - 4568q^{41} - 26144q^{42} - 29112q^{44} + 29200q^{46} - 54720q^{47} + 35616q^{48} + 9828q^{49} - 47498q^{50} - 36560q^{52} + 23288q^{54} + 85472q^{55} + 40768q^{56} - 2032q^{57} + 3784q^{58} + 2592q^{60} + 34496q^{62} - 153440q^{63} - 41920q^{64} - 19520q^{65} + 43224q^{66} + 10344q^{68} - 68928q^{70} + 206688q^{71} - 83272q^{72} + 39976q^{73} + 17464q^{74} + 99944q^{76} - 174064q^{78} - 247872q^{79} - 35520q^{80} + 29684q^{81} + 161132q^{82} + 196672q^{84} - 18500q^{86} + 307872q^{87} - 167216q^{88} - 84632q^{89} + 142280q^{90} - 49056q^{92} - 98784q^{94} - 259744q^{95} - 115648q^{96} - 99576q^{97} - 117042q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(8, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
8.6.b.a \(4\) \(1.283\) 4.0.218489.1 None \(-2\) \(0\) \(0\) \(96\) \(q+(-1-\beta _{1})q^{2}+(-\beta _{1}+\beta _{3})q^{3}+(5+\cdots)q^{4}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T - 8 T^{2} + 64 T^{3} + 1024 T^{4} \)
$3$ \( 1 - 404 T^{2} + 84150 T^{4} - 23855796 T^{6} + 3486784401 T^{8} \)
$5$ \( 1 - 7028 T^{2} + 24404246 T^{4} - 68632812500 T^{6} + 95367431640625 T^{8} \)
$7$ \( ( 1 - 48 T + 15502 T^{2} - 806736 T^{3} + 282475249 T^{4} )^{2} \)
$11$ \( 1 - 296436 T^{2} + 49128544726 T^{4} - 7688786399022036 T^{6} + \)\(67\!\cdots\!01\)\( T^{8} \)
$13$ \( 1 - 894228 T^{2} + 396323515894 T^{4} - 123276923449147572 T^{6} + \)\(19\!\cdots\!01\)\( T^{8} \)
$17$ \( ( 1 - 100 T + 2767462 T^{2} - 141985700 T^{3} + 2015993900449 T^{4} )^{2} \)
$19$ \( 1 - 6794580 T^{2} + 21506967947254 T^{4} - 41658020173929518580 T^{6} + \)\(37\!\cdots\!01\)\( T^{8} \)
$23$ \( ( 1 - 1168 T + 10055470 T^{2} - 7517648624 T^{3} + 41426511213649 T^{4} )^{2} \)
$29$ \( 1 - 31255380 T^{2} + 976386653995702 T^{4} - \)\(13\!\cdots\!80\)\( T^{6} + \)\(17\!\cdots\!01\)\( T^{8} \)
$31$ \( ( 1 + 6464 T + 65013054 T^{2} + 185058832064 T^{3} + 819628286980801 T^{4} )^{2} \)
$37$ \( 1 - 241262580 T^{2} + 24149916431784598 T^{4} - \)\(11\!\cdots\!20\)\( T^{6} + \)\(23\!\cdots\!01\)\( T^{8} \)
$41$ \( ( 1 + 2284 T + 146603254 T^{2} + 264615563084 T^{3} + 13422659310152401 T^{4} )^{2} \)
$43$ \( 1 - 466346868 T^{2} + 96250708269010006 T^{4} - \)\(10\!\cdots\!32\)\( T^{6} + \)\(46\!\cdots\!01\)\( T^{8} \)
$47$ \( ( 1 + 27360 T + 591338206 T^{2} + 6274879391520 T^{3} + 52599132235830049 T^{4} )^{2} \)
$53$ \( 1 - 1039152180 T^{2} + 595616955270391126 T^{4} - \)\(18\!\cdots\!20\)\( T^{6} + \)\(30\!\cdots\!01\)\( T^{8} \)
$59$ \( 1 - 1537424180 T^{2} + 1399694789142612374 T^{4} - \)\(78\!\cdots\!80\)\( T^{6} + \)\(26\!\cdots\!01\)\( T^{8} \)
$61$ \( 1 + 741098540 T^{2} + 1478044222094100534 T^{4} + \)\(52\!\cdots\!40\)\( T^{6} + \)\(50\!\cdots\!01\)\( T^{8} \)
$67$ \( 1 - 1366835860 T^{2} + 3274116308996825526 T^{4} - \)\(24\!\cdots\!40\)\( T^{6} + \)\(33\!\cdots\!01\)\( T^{8} \)
$71$ \( ( 1 - 103344 T + 6217736974 T^{2} - 186456278049744 T^{3} + 3255243551009881201 T^{4} )^{2} \)
$73$ \( ( 1 - 19988 T + 2541602870 T^{2} - 41436555000884 T^{3} + 4297625829703557649 T^{4} )^{2} \)
$79$ \( ( 1 + 123936 T + 9855929374 T^{2} + 381358061866464 T^{3} + 9468276082626847201 T^{4} )^{2} \)
$83$ \( 1 - 10047855188 T^{2} + 55381071937674414326 T^{4} - \)\(15\!\cdots\!12\)\( T^{6} + \)\(24\!\cdots\!01\)\( T^{8} \)
$89$ \( ( 1 + 42316 T + 4292401174 T^{2} + 236295059643884 T^{3} + 31181719929966183601 T^{4} )^{2} \)
$97$ \( ( 1 + 49788 T + 16391371462 T^{2} + 427546496715516 T^{3} + 73742412689492826049 T^{4} )^{2} \)
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