Properties

Label 315.2.bh
Level 315
Weight 2
Character orbit bh
Rep. character \(\chi_{315}(169,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 72
Newform subspaces 3
Sturm bound 96
Trace bound 4

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Defining parameters

Level: \( N \) = \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 315.bh (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 45 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 3 \)
Sturm bound: \(96\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(2\)

Dimensions

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

Total New Old
Modular forms 104 72 32
Cusp forms 88 72 16
Eisenstein series 16 0 16

Trace form

\( 72q + 36q^{4} - 2q^{5} - 8q^{9} + O(q^{10}) \) \( 72q + 36q^{4} - 2q^{5} - 8q^{9} + 18q^{11} - 8q^{14} - 20q^{15} - 36q^{16} - 6q^{20} - 2q^{21} - 12q^{24} + 6q^{25} - 32q^{26} - 20q^{29} + 20q^{30} - 12q^{31} + 12q^{34} - 36q^{36} - 14q^{39} + 8q^{41} + 32q^{44} + 18q^{45} + 24q^{46} + 36q^{49} - 50q^{50} + 74q^{51} + 36q^{54} - 12q^{55} + 24q^{56} - 4q^{59} - 48q^{60} - 96q^{64} - 20q^{65} - 128q^{66} - 20q^{69} - 168q^{71} - 28q^{74} + 60q^{75} - 12q^{76} - 6q^{79} + 104q^{80} - 40q^{81} - 16q^{84} - 24q^{85} + 44q^{86} - 80q^{89} - 16q^{90} - 12q^{91} - 48q^{94} + 20q^{95} + 180q^{96} + 144q^{99} + O(q^{100}) \)

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.bh.a \(4\) \(2.515\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(4\) \(0\) \(q+\zeta_{12}q^{2}+(2\zeta_{12}-\zeta_{12}^{3})q^{3}-\zeta_{12}^{2}q^{4}+\cdots\)
315.2.bh.b \(4\) \(2.515\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(4\) \(0\) \(q+2\zeta_{12}q^{2}+(\zeta_{12}-2\zeta_{12}^{3})q^{3}+2\zeta_{12}^{2}q^{4}+\cdots\)
315.2.bh.c \(64\) \(2.515\) None \(0\) \(0\) \(-10\) \(0\)

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

\( S_{2}^{\mathrm{old}}(315, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 3 T^{2} + 5 T^{4} + 12 T^{6} + 16 T^{8} \))(\( ( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} )( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} ) \))
$3$ (\( ( 1 - 3 T^{2} )^{2} \))(\( 1 + 3 T^{2} + 9 T^{4} \))
$5$ (\( 1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4} \))(\( 1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4} \))
$7$ (\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))
$11$ (\( ( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} )^{2} \))
$13$ (\( 1 + 25 T^{2} + 456 T^{4} + 4225 T^{6} + 28561 T^{8} \))(\( ( 1 - T^{2} + 169 T^{4} )( 1 + 23 T^{2} + 169 T^{4} ) \))
$17$ (\( ( 1 + 15 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 8 T + 17 T^{2} )^{2}( 1 + 8 T + 17 T^{2} )^{2} \))
$19$ (\( ( 1 + 6 T + 19 T^{2} )^{4} \))(\( ( 1 + 6 T + 19 T^{2} )^{4} \))
$23$ (\( ( 1 + 23 T^{2} + 529 T^{4} )^{2} \))(\( 1 + 10 T^{2} - 429 T^{4} + 5290 T^{6} + 279841 T^{8} \))
$29$ (\( ( 1 - 10 T + 71 T^{2} - 290 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 + 5 T - 4 T^{2} + 145 T^{3} + 841 T^{4} )^{2} \))
$31$ (\( ( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 - 11 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2} \))
$37$ (\( ( 1 - 10 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2} \))
$41$ (\( ( 1 + 2 T - 37 T^{2} + 82 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 8 T + 23 T^{2} + 328 T^{3} + 1681 T^{4} )^{2} \))
$43$ (\( 1 + 70 T^{2} + 3051 T^{4} + 129430 T^{6} + 3418801 T^{8} \))(\( 1 + 82 T^{2} + 4875 T^{4} + 151618 T^{6} + 3418801 T^{8} \))
$47$ (\( 1 + 13 T^{2} - 2040 T^{4} + 28717 T^{6} + 4879681 T^{8} \))(\( 1 + 85 T^{2} + 5016 T^{4} + 187765 T^{6} + 4879681 T^{8} \))
$53$ (\( ( 1 - 70 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 53 T^{2} )^{4} \))
$59$ (\( ( 1 - 4 T - 43 T^{2} - 236 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 + 14 T + 137 T^{2} + 826 T^{3} + 3481 T^{4} )^{2} \))
$61$ (\( ( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} )^{2} \))
$67$ (\( 1 + 118 T^{2} + 9435 T^{4} + 529702 T^{6} + 20151121 T^{8} \))(\( 1 + 34 T^{2} - 3333 T^{4} + 152626 T^{6} + 20151121 T^{8} \))
$71$ (\( ( 1 + 5 T + 71 T^{2} )^{4} \))(\( ( 1 - 7 T + 71 T^{2} )^{4} \))
$73$ (\( ( 1 - 137 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 - 65 T^{2} + 5329 T^{4} )^{2} \))
$79$ (\( ( 1 - 7 T - 30 T^{2} - 553 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} )^{2} \))
$83$ (\( 1 + 141 T^{2} + 12992 T^{4} + 971349 T^{6} + 47458321 T^{8} \))(\( 1 - 123 T^{2} + 8240 T^{4} - 847347 T^{6} + 47458321 T^{8} \))
$89$ (\( ( 1 - 18 T + 89 T^{2} )^{4} \))(\( ( 1 - 12 T + 89 T^{2} )^{4} \))
$97$ (\( ( 1 + 2 T^{2} + 9409 T^{4} )( 1 + 167 T^{2} + 9409 T^{4} ) \))(\( 1 + 145 T^{2} + 11616 T^{4} + 1364305 T^{6} + 88529281 T^{8} \))
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