Properties

Label 315.2.g
Level 315
Weight 2
Character orbit g
Rep. character \(\chi_{315}(314,\cdot)\)
Character field \(\Q\)
Dimension 16
Newform subspaces 1
Sturm bound 96
Trace bound 0

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Defining parameters

Level: \( N \) = \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 315.g (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 105 \)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(96\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(315, [\chi])\).

Total New Old
Modular forms 56 16 40
Cusp forms 40 16 24
Eisenstein series 16 0 16

Trace form

\( 16q + 16q^{4} + O(q^{10}) \) \( 16q + 16q^{4} + 16q^{16} + 16q^{25} - 96q^{46} + 16q^{49} - 80q^{64} - 48q^{70} - 64q^{79} + 32q^{85} + 32q^{91} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(315, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
315.2.g.a \(16\) \(2.515\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{3}q^{2}+(1+\beta _{14})q^{4}-\beta _{4}q^{5}-\beta _{12}q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(315, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(315, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 2 T^{2} + 3 T^{4} + 8 T^{6} + 16 T^{8} )^{4} \)
$3$ \( \)
$5$ \( ( 1 - 4 T^{2} + 30 T^{4} - 100 T^{6} + 625 T^{8} )^{2} \)
$7$ \( ( 1 - 4 T^{2} + 6 T^{4} - 196 T^{6} + 2401 T^{8} )^{2} \)
$11$ \( ( 1 - 20 T^{2} + 121 T^{4} )^{8} \)
$13$ \( ( 1 + 20 T^{2} + 222 T^{4} + 3380 T^{6} + 28561 T^{8} )^{4} \)
$17$ \( ( 1 - 4 T^{2} + 558 T^{4} - 1156 T^{6} + 83521 T^{8} )^{4} \)
$19$ \( ( 1 - 4 T^{2} + 510 T^{4} - 1444 T^{6} + 130321 T^{8} )^{4} \)
$23$ \( ( 1 + 56 T^{2} + 1818 T^{4} + 29624 T^{6} + 279841 T^{8} )^{4} \)
$29$ \( ( 1 - 16 T^{2} + 1362 T^{4} - 13456 T^{6} + 707281 T^{8} )^{4} \)
$31$ \( ( 1 - 52 T^{2} + 1422 T^{4} - 49972 T^{6} + 923521 T^{8} )^{4} \)
$37$ \( ( 1 - 76 T^{2} + 3318 T^{4} - 104044 T^{6} + 1874161 T^{8} )^{4} \)
$41$ \( ( 1 + 44 T^{2} + 3246 T^{4} + 73964 T^{6} + 2825761 T^{8} )^{4} \)
$43$ \( ( 1 - 100 T^{2} + 6102 T^{4} - 184900 T^{6} + 3418801 T^{8} )^{4} \)
$47$ \( ( 1 - 124 T^{2} + 7398 T^{4} - 273916 T^{6} + 4879681 T^{8} )^{4} \)
$53$ \( ( 1 + 104 T^{2} + 6378 T^{4} + 292136 T^{6} + 7890481 T^{8} )^{4} \)
$59$ \( ( 1 + 92 T^{2} + 4374 T^{4} + 320252 T^{6} + 12117361 T^{8} )^{4} \)
$61$ \( ( 1 - 52 T^{2} + 6582 T^{4} - 193492 T^{6} + 13845841 T^{8} )^{4} \)
$67$ \( ( 1 - 124 T^{2} + 12438 T^{4} - 556636 T^{6} + 20151121 T^{8} )^{4} \)
$71$ \( ( 1 - 184 T^{2} + 18162 T^{4} - 927544 T^{6} + 25411681 T^{8} )^{4} \)
$73$ \( ( 1 + 164 T^{2} + 13326 T^{4} + 873956 T^{6} + 28398241 T^{8} )^{4} \)
$79$ \( ( 1 + 4 T + 79 T^{2} )^{16} \)
$83$ \( ( 1 - 268 T^{2} + 30870 T^{4} - 1846252 T^{6} + 47458321 T^{8} )^{4} \)
$89$ \( ( 1 + 140 T^{2} + 18798 T^{4} + 1108940 T^{6} + 62742241 T^{8} )^{4} \)
$97$ \( ( 1 + 356 T^{2} + 50286 T^{4} + 3349604 T^{6} + 88529281 T^{8} )^{4} \)
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