Properties

Label 315.2.bb
Level 315
Weight 2
Character orbit bb
Rep. character \(\chi_{315}(89,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 32
Newform subspaces 2
Sturm bound 96
Trace bound 1

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Defining parameters

Level: \( N \) = \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 315.bb (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 105 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 2 \)
Sturm bound: \(96\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

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

Total New Old
Modular forms 112 32 80
Cusp forms 80 32 48
Eisenstein series 32 0 32

Trace form

\( 32q - 16q^{4} + O(q^{10}) \) \( 32q - 16q^{4} - 12q^{10} - 16q^{16} - 24q^{19} + 20q^{25} - 24q^{31} + 96q^{40} - 24q^{46} + 8q^{49} - 24q^{61} - 16q^{64} - 48q^{70} - 32q^{79} - 128q^{85} + 88q^{91} + 48q^{94} + O(q^{100}) \)

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.bb.a \(8\) \(2.515\) 8.0.\(\cdots\).5 None \(0\) \(0\) \(0\) \(0\) \(q+2\beta _{4}q^{4}+(\beta _{1}-\beta _{2}+\beta _{5})q^{5}+(\beta _{3}+\cdots)q^{7}+\cdots\)
315.2.bb.b \(24\) \(2.515\) None \(0\) \(0\) \(0\) \(0\)

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

$p$ $F_p(T)$
$2$ (\( ( 1 - 2 T^{2} + 4 T^{4} )^{4} \))
$3$ 1
$5$ (\( 1 - 4 T^{2} - 9 T^{4} - 100 T^{6} + 625 T^{8} \))
$7$ (\( ( 1 + 7 T^{2} + 49 T^{4} )^{2} \))
$11$ (\( ( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} )^{2}( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} )^{2} \))
$13$ (\( ( 1 + 19 T^{2} + 169 T^{4} )^{4} \))
$17$ (\( ( 1 + 20 T^{2} + 111 T^{4} + 5780 T^{6} + 83521 T^{8} )^{2} \))
$19$ (\( ( 1 - 3 T + 22 T^{2} - 57 T^{3} + 361 T^{4} )^{4} \))
$23$ (\( ( 1 - 4 T^{2} - 513 T^{4} - 2116 T^{6} + 279841 T^{8} )^{2} \))
$29$ (\( ( 1 - 56 T^{2} + 841 T^{4} )^{4} \))
$31$ (\( ( 1 - 9 T + 58 T^{2} - 279 T^{3} + 961 T^{4} )^{4} \))
$37$ (\( ( 1 + 53 T^{2} + 1440 T^{4} + 72557 T^{6} + 1874161 T^{8} )^{2} \))
$41$ (\( ( 1 + 58 T^{2} + 1681 T^{4} )^{4} \))
$43$ (\( ( 1 - 65 T^{2} + 1849 T^{4} )^{4} \))
$47$ (\( ( 1 + 80 T^{2} + 4191 T^{4} + 176720 T^{6} + 4879681 T^{8} )^{2} \))
$53$ (\( ( 1 + 62 T^{2} + 1035 T^{4} + 174158 T^{6} + 7890481 T^{8} )^{2} \))
$59$ (\( ( 1 - 64 T^{2} + 615 T^{4} - 222784 T^{6} + 12117361 T^{8} )^{2} \))
$61$ (\( ( 1 + 18 T + 169 T^{2} + 1098 T^{3} + 3721 T^{4} )^{4} \))
$67$ (\( ( 1 - 55 T^{2} - 1464 T^{4} - 246895 T^{6} + 20151121 T^{8} )^{2} \))
$71$ (\( ( 1 - 14 T^{2} + 5041 T^{4} )^{4} \))
$73$ (\( ( 1 + 29 T^{2} - 4488 T^{4} + 154541 T^{6} + 28398241 T^{8} )^{2} \))
$79$ (\( ( 1 - 7 T - 30 T^{2} - 553 T^{3} + 6241 T^{4} )^{4} \))
$83$ (\( ( 1 + 58 T^{2} + 6889 T^{4} )^{4} \))
$89$ (\( ( 1 + 116 T^{2} + 5535 T^{4} + 918836 T^{6} + 62742241 T^{8} )^{2} \))
$97$ (\( ( 1 + 166 T^{2} + 9409 T^{4} )^{4} \))
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