Properties

Label 315.2.bo
Level 315
Weight 2
Character orbit bo
Rep. character \(\chi_{315}(4,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 88
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.bo (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 315 \)
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 104 104 0
Cusp forms 88 88 0
Eisenstein series 16 16 0

Trace form

\( 88q + 42q^{4} - 2q^{5} - 2q^{9} + O(q^{10}) \) \( 88q + 42q^{4} - 2q^{5} - 2q^{9} + 2q^{10} - 8q^{14} + 4q^{15} - 34q^{16} - 4q^{19} - 12q^{20} - 8q^{21} + 18q^{24} - 2q^{25} - 32q^{26} - 14q^{29} - 34q^{30} + 2q^{31} - 8q^{34} + 24q^{35} - 12q^{36} - 20q^{39} + 16q^{40} - 34q^{41} - 16q^{44} - 18q^{45} + 10q^{46} - 14q^{49} + 34q^{50} + 2q^{51} - 30q^{55} + 18q^{56} + 50q^{59} + 54q^{60} + 8q^{61} - 56q^{64} - 8q^{65} - 74q^{66} - 32q^{69} - 24q^{70} + 12q^{71} - 76q^{74} + 6q^{75} + 12q^{76} - 4q^{79} - 7q^{80} - 70q^{81} - 52q^{84} - 7q^{85} - 88q^{86} + 40q^{89} + 5q^{90} - 36q^{91} - 2q^{94} - 37q^{95} + 84q^{96} - 30q^{99} + O(q^{100}) \)

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.bo.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.bo.b \(84\) \(2.515\) None \(0\) \(0\) \(-6\) \(0\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 3 T^{2} + 5 T^{4} + 12 T^{6} + 16 T^{8} \))
$3$ (\( ( 1 - 3 T^{2} )^{2} \))
$5$ (\( ( 1 - 2 T + 5 T^{2} )^{2} \))
$7$ (\( 1 + 11 T^{2} + 49 T^{4} \))
$11$ (\( ( 1 - 6 T + 11 T^{2} )^{4} \))
$13$ (\( ( 1 - 6 T + 23 T^{2} - 78 T^{3} + 169 T^{4} )( 1 + 6 T + 23 T^{2} + 78 T^{3} + 169 T^{4} ) \))
$17$ (\( ( 1 - 8 T + 47 T^{2} - 136 T^{3} + 289 T^{4} )( 1 + 8 T + 47 T^{2} + 136 T^{3} + 289 T^{4} ) \))
$19$ (\( ( 1 + 6 T + 17 T^{2} + 114 T^{3} + 361 T^{4} )^{2} \))
$23$ (\( ( 1 - 37 T^{2} + 529 T^{4} )^{2} \))
$29$ (\( ( 1 + 2 T - 25 T^{2} + 58 T^{3} + 841 T^{4} )^{2} \))
$31$ (\( ( 1 - 11 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2} \))
$37$ (\( 1 + 10 T^{2} - 1269 T^{4} + 13690 T^{6} + 1874161 T^{8} \))
$41$ (\( ( 1 + 2 T - 37 T^{2} + 82 T^{3} + 1681 T^{4} )^{2} \))
$43$ (\( 1 + 85 T^{2} + 5376 T^{4} + 157165 T^{6} + 3418801 T^{8} \))
$47$ (\( 1 + 85 T^{2} + 5016 T^{4} + 187765 T^{6} + 4879681 T^{8} \))
$53$ (\( 1 - 38 T^{2} - 1365 T^{4} - 106742 T^{6} + 7890481 T^{8} \))
$59$ (\( ( 1 - 4 T - 43 T^{2} - 236 T^{3} + 3481 T^{4} )^{2} \))
$61$ (\( ( 1 + 7 T - 12 T^{2} + 427 T^{3} + 3721 T^{4} )^{2} \))
$67$ (\( 1 + 85 T^{2} + 2736 T^{4} + 381565 T^{6} + 20151121 T^{8} \))
$71$ (\( ( 1 - 4 T + 71 T^{2} )^{4} \))
$73$ (\( ( 1 + 73 T^{2} + 5329 T^{4} )^{2} \))
$79$ (\( ( 1 + 14 T + 117 T^{2} + 1106 T^{3} + 6241 T^{4} )^{2} \))
$83$ (\( 1 + 150 T^{2} + 15611 T^{4} + 1033350 T^{6} + 47458321 T^{8} \))
$89$ (\( ( 1 + 3 T - 80 T^{2} + 267 T^{3} + 7921 T^{4} )^{2} \))
$97$ (\( 1 + 190 T^{2} + 26691 T^{4} + 1787710 T^{6} + 88529281 T^{8} \))
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