Properties

Label 364.2.g
Level 364
Weight 2
Character orbit g
Rep. character \(\chi_{364}(337,\cdot)\)
Character field \(\Q\)
Dimension 8
Newform subspaces 1
Sturm bound 112
Trace bound 0

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Defining parameters

Level: \( N \) = \( 364 = 2^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 364.g (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 13 \)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(112\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(364, [\chi])\).

Total New Old
Modular forms 62 8 54
Cusp forms 50 8 42
Eisenstein series 12 0 12

Trace form

\( 8q + 8q^{9} + O(q^{10}) \) \( 8q + 8q^{9} + 2q^{13} + 4q^{17} - 6q^{23} + 2q^{25} + 12q^{27} - 2q^{29} + 6q^{35} - 6q^{43} - 8q^{49} - 8q^{51} + 22q^{53} - 20q^{55} + 8q^{61} - 6q^{65} - 20q^{69} - 20q^{75} - 26q^{79} - 24q^{81} - 32q^{87} - 10q^{91} - 18q^{95} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(364, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
364.2.g.a \(8\) \(2.907\) \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{2}q^{3}+(-\beta _{3}-\beta _{4})q^{5}+\beta _{3}q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(364, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(364, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(91, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(182, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( ( 1 + 4 T^{2} - 2 T^{3} + 14 T^{4} - 6 T^{5} + 36 T^{6} + 81 T^{8} )^{2} \)
$5$ \( 1 - 21 T^{2} + 249 T^{4} - 2006 T^{6} + 11626 T^{8} - 50150 T^{10} + 155625 T^{12} - 328125 T^{14} + 390625 T^{16} \)
$7$ \( ( 1 + T^{2} )^{4} \)
$11$ \( 1 - 52 T^{2} + 1368 T^{4} - 24280 T^{6} + 312502 T^{8} - 2937880 T^{10} + 20028888 T^{12} - 92121172 T^{14} + 214358881 T^{16} \)
$13$ \( 1 - 2 T + 20 T^{2} - 94 T^{3} + 278 T^{4} - 1222 T^{5} + 3380 T^{6} - 4394 T^{7} + 28561 T^{8} \)
$17$ \( ( 1 - 2 T + 16 T^{2} - 24 T^{3} + 10 T^{4} - 408 T^{5} + 4624 T^{6} - 9826 T^{7} + 83521 T^{8} )^{2} \)
$19$ \( 1 - 81 T^{2} + 3529 T^{4} - 102890 T^{6} + 2243374 T^{8} - 37143290 T^{10} + 459902809 T^{12} - 3810716361 T^{14} + 16983563041 T^{16} \)
$23$ \( ( 1 + 3 T + 79 T^{2} + 160 T^{3} + 2540 T^{4} + 3680 T^{5} + 41791 T^{6} + 36501 T^{7} + 279841 T^{8} )^{2} \)
$29$ \( ( 1 + T + 33 T^{2} + 2 T^{3} + 1414 T^{4} + 58 T^{5} + 27753 T^{6} + 24389 T^{7} + 707281 T^{8} )^{2} \)
$31$ \( 1 - 125 T^{2} + 8493 T^{4} - 384510 T^{6} + 13439406 T^{8} - 369514110 T^{10} + 7843463853 T^{12} - 110937960125 T^{14} + 852891037441 T^{16} \)
$37$ \( 1 - 84 T^{2} + 2216 T^{4} + 61384 T^{6} - 5827706 T^{8} + 84034696 T^{10} + 4153140776 T^{12} - 215521018356 T^{14} + 3512479453921 T^{16} \)
$41$ \( 1 - 172 T^{2} + 16612 T^{4} - 1090964 T^{6} + 51790710 T^{8} - 1833910484 T^{10} + 46941541732 T^{12} - 817017929452 T^{14} + 7984925229121 T^{16} \)
$43$ \( ( 1 + 3 T + 87 T^{2} + 20 T^{3} + 4016 T^{4} + 860 T^{5} + 160863 T^{6} + 238521 T^{7} + 3418801 T^{8} )^{2} \)
$47$ \( 1 - 137 T^{2} + 13681 T^{4} - 938098 T^{6} + 50458758 T^{8} - 2072258482 T^{10} + 66758915761 T^{12} - 1476752500073 T^{14} + 23811286661761 T^{16} \)
$53$ \( ( 1 - 11 T + 209 T^{2} - 1686 T^{3} + 16482 T^{4} - 89358 T^{5} + 587081 T^{6} - 1637647 T^{7} + 7890481 T^{8} )^{2} \)
$59$ \( 1 - 292 T^{2} + 39492 T^{4} - 3405100 T^{6} + 222712566 T^{8} - 11853153100 T^{10} + 478538820612 T^{12} - 12316715823172 T^{14} + 146830437604321 T^{16} \)
$61$ \( ( 1 - 4 T + 132 T^{2} - 1068 T^{3} + 8646 T^{4} - 65148 T^{5} + 491172 T^{6} - 907924 T^{7} + 13845841 T^{8} )^{2} \)
$67$ \( 1 - 168 T^{2} + 18332 T^{4} - 1592024 T^{6} + 121444582 T^{8} - 7146595736 T^{10} + 369410350172 T^{12} - 15197008204392 T^{14} + 406067677556641 T^{16} \)
$71$ \( 1 - 160 T^{2} + 10920 T^{4} - 867532 T^{6} + 79016406 T^{8} - 4373228812 T^{10} + 277495556520 T^{12} - 20496045427360 T^{14} + 645753531245761 T^{16} \)
$73$ \( 1 - 305 T^{2} + 45549 T^{4} - 4497786 T^{6} + 354140538 T^{8} - 23968701594 T^{10} + 1293511479309 T^{12} - 46156939018145 T^{14} + 806460091894081 T^{16} \)
$79$ \( ( 1 + 13 T + 219 T^{2} + 2704 T^{3} + 22084 T^{4} + 213616 T^{5} + 1366779 T^{6} + 6409507 T^{7} + 38950081 T^{8} )^{2} \)
$83$ \( 1 - 117 T^{2} + 15309 T^{4} - 1167406 T^{6} + 97224854 T^{8} - 8042259934 T^{10} + 726539436189 T^{12} - 38252023684173 T^{14} + 2252292232139041 T^{16} \)
$89$ \( 1 - 273 T^{2} + 54405 T^{4} - 7250626 T^{6} + 745491194 T^{8} - 57432208546 T^{10} + 3413491621605 T^{12} - 135675892432353 T^{14} + 3936588805702081 T^{16} \)
$97$ \( 1 - 329 T^{2} + 69397 T^{4} - 9804594 T^{6} + 1094792986 T^{8} - 92251424946 T^{10} + 6143666513557 T^{12} - 274047789621641 T^{14} + 7837433594376961 T^{16} \)
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