Properties

Label 315.2.b
Level 315
Weight 2
Character orbit b
Rep. character \(\chi_{315}(251,\cdot)\)
Character field \(\Q\)
Dimension 8
Newform subspaces 2
Sturm bound 96
Trace bound 5

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Defining parameters

Level: \( N \) = \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 315.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 21 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(96\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(47\)

Dimensions

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

Total New Old
Modular forms 56 8 48
Cusp forms 40 8 32
Eisenstein series 16 0 16

Trace form

\( 8q + 8q^{7} + O(q^{10}) \) \( 8q + 8q^{7} - 8q^{16} - 40q^{22} + 8q^{25} + 16q^{37} + 16q^{43} + 8q^{46} - 40q^{49} + 8q^{58} + 32q^{64} - 32q^{67} + 24q^{70} - 32q^{79} - 8q^{88} + 48q^{91} + O(q^{100}) \)

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.b.a \(4\) \(2.515\) \(\Q(\sqrt{-2}, \sqrt{3})\) None \(0\) \(0\) \(-4\) \(4\) \(q+\beta _{1}q^{2}+\beta _{2}q^{4}-q^{5}+(1+\beta _{1}+\beta _{3})q^{7}+\cdots\)
315.2.b.b \(4\) \(2.515\) \(\Q(\sqrt{-2}, \sqrt{3})\) None \(0\) \(0\) \(4\) \(4\) \(q+\beta _{1}q^{2}+\beta _{2}q^{4}+q^{5}+(1-\beta _{1}-\beta _{3})q^{7}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(315, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 4 T^{2} + 9 T^{4} - 16 T^{6} + 16 T^{8} \))(\( 1 - 4 T^{2} + 9 T^{4} - 16 T^{6} + 16 T^{8} \))
$3$ (\( \))(\( \))
$5$ (\( ( 1 + T )^{4} \))(\( ( 1 - T )^{4} \))
$7$ (\( ( 1 - 2 T + 7 T^{2} )^{2} \))(\( ( 1 - 2 T + 7 T^{2} )^{2} \))
$11$ (\( 1 - 16 T^{2} + 114 T^{4} - 1936 T^{6} + 14641 T^{8} \))(\( 1 - 16 T^{2} + 114 T^{4} - 1936 T^{6} + 14641 T^{8} \))
$13$ (\( 1 - 4 T^{2} - 90 T^{4} - 676 T^{6} + 28561 T^{8} \))(\( 1 - 4 T^{2} - 90 T^{4} - 676 T^{6} + 28561 T^{8} \))
$17$ (\( ( 1 + 22 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 + 22 T^{2} + 289 T^{4} )^{2} \))
$19$ (\( 1 - 28 T^{2} + 486 T^{4} - 10108 T^{6} + 130321 T^{8} \))(\( 1 - 28 T^{2} + 486 T^{4} - 10108 T^{6} + 130321 T^{8} \))
$23$ (\( ( 1 - 44 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 - 44 T^{2} + 529 T^{4} )^{2} \))
$29$ (\( ( 1 - 56 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 56 T^{2} + 841 T^{4} )^{2} \))
$31$ (\( 1 - 76 T^{2} + 2934 T^{4} - 73036 T^{6} + 923521 T^{8} \))(\( 1 - 76 T^{2} + 2934 T^{4} - 73036 T^{6} + 923521 T^{8} \))
$37$ (\( ( 1 - 4 T + 66 T^{2} - 148 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 - 4 T + 66 T^{2} - 148 T^{3} + 1369 T^{4} )^{2} \))
$41$ (\( ( 1 - 26 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 26 T^{2} + 1681 T^{4} )^{2} \))
$43$ (\( ( 1 - 4 T + 42 T^{2} - 172 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 - 4 T + 42 T^{2} - 172 T^{3} + 1849 T^{4} )^{2} \))
$47$ (\( ( 1 - 12 T + 118 T^{2} - 564 T^{3} + 2209 T^{4} )^{2} \))(\( ( 1 + 12 T + 118 T^{2} + 564 T^{3} + 2209 T^{4} )^{2} \))
$53$ (\( 1 - 88 T^{2} + 5826 T^{4} - 247192 T^{6} + 7890481 T^{8} \))(\( 1 - 88 T^{2} + 5826 T^{4} - 247192 T^{6} + 7890481 T^{8} \))
$59$ (\( ( 1 - 12 T + 142 T^{2} - 708 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 + 12 T + 142 T^{2} + 708 T^{3} + 3481 T^{4} )^{2} \))
$61$ (\( 1 - 52 T^{2} + 1206 T^{4} - 193492 T^{6} + 13845841 T^{8} \))(\( 1 - 52 T^{2} + 1206 T^{4} - 193492 T^{6} + 13845841 T^{8} \))
$67$ (\( ( 1 + 8 T + 102 T^{2} + 536 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 + 8 T + 102 T^{2} + 536 T^{3} + 4489 T^{4} )^{2} \))
$71$ (\( 1 + 32 T^{2} + 5538 T^{4} + 161312 T^{6} + 25411681 T^{8} \))(\( 1 + 32 T^{2} + 5538 T^{4} + 161312 T^{6} + 25411681 T^{8} \))
$73$ (\( 1 - 244 T^{2} + 25110 T^{4} - 1300276 T^{6} + 28398241 T^{8} \))(\( 1 - 244 T^{2} + 25110 T^{4} - 1300276 T^{6} + 28398241 T^{8} \))
$79$ (\( ( 1 + 8 T + 126 T^{2} + 632 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 + 8 T + 126 T^{2} + 632 T^{3} + 6241 T^{4} )^{2} \))
$83$ (\( ( 1 + 12 T + 190 T^{2} + 996 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 - 12 T + 190 T^{2} - 996 T^{3} + 6889 T^{4} )^{2} \))
$89$ (\( ( 1 - 12 T + 166 T^{2} - 1068 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 12 T + 166 T^{2} + 1068 T^{3} + 7921 T^{4} )^{2} \))
$97$ (\( 1 - 52 T^{2} + 15606 T^{4} - 489268 T^{6} + 88529281 T^{8} \))(\( 1 - 52 T^{2} + 15606 T^{4} - 489268 T^{6} + 88529281 T^{8} \))
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