Properties

Label 315.2.r
Level 315
Weight 2
Character orbit r
Rep. character \(\chi_{315}(184,\cdot)\)
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.r (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

\( 88q - 84q^{4} + q^{5} - 8q^{9} + O(q^{10}) \) \( 88q - 84q^{4} + q^{5} - 8q^{9} + 2q^{10} + 16q^{14} + 4q^{15} + 68q^{16} - 4q^{19} - 12q^{20} - 8q^{21} + q^{25} - 32q^{26} - 14q^{29} + 5q^{30} - 4q^{31} - 8q^{34} + 24q^{35} - 12q^{36} - 32q^{39} - 8q^{40} - 34q^{41} - 16q^{44} + 27q^{45} + 10q^{46} + 4q^{49} + 34q^{50} - 52q^{51} + 24q^{54} - 30q^{55} + 6q^{56} - 100q^{59} - 21q^{60} - 16q^{61} - 56q^{64} + 16q^{65} + 40q^{66} - 32q^{69} - 9q^{70} + 12q^{71} + 38q^{74} + 42q^{75} + 12q^{76} + 8q^{79} - 7q^{80} + 20q^{81} + 74q^{84} - 7q^{85} + 44q^{86} + 40q^{89} + 5q^{90} - 36q^{91} + 4q^{94} + 74q^{95} + 48q^{96} - 30q^{99} + O(q^{100}) \)

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.r.a \(4\) \(2.515\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(-2\) \(0\) \(q+\zeta_{12}^{3}q^{2}+(\zeta_{12}+\zeta_{12}^{3})q^{3}+q^{4}+\cdots\)
315.2.r.b \(84\) \(2.515\) None \(0\) \(0\) \(3\) \(0\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 - 3 T^{2} + 4 T^{4} )^{2} \))
$3$ (\( 1 + 3 T^{2} + 9 T^{4} \))
$5$ (\( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} \))
$7$ (\( 1 + 2 T^{2} + 49 T^{4} \))
$11$ (\( ( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} )^{2} \))
$13$ (\( ( 1 - 6 T + 23 T^{2} - 78 T^{3} + 169 T^{4} )( 1 + 6 T + 23 T^{2} + 78 T^{3} + 169 T^{4} ) \))
$17$ (\( ( 1 - 8 T + 47 T^{2} - 136 T^{3} + 289 T^{4} )( 1 + 8 T + 47 T^{2} + 136 T^{3} + 289 T^{4} ) \))
$19$ (\( ( 1 + 6 T + 17 T^{2} + 114 T^{3} + 361 T^{4} )^{2} \))
$23$ (\( 1 + 37 T^{2} + 840 T^{4} + 19573 T^{6} + 279841 T^{8} \))
$29$ (\( ( 1 + 2 T - 25 T^{2} + 58 T^{3} + 841 T^{4} )^{2} \))
$31$ (\( ( 1 + 4 T + 31 T^{2} )^{4} \))
$37$ (\( 1 + 10 T^{2} - 1269 T^{4} + 13690 T^{6} + 1874161 T^{8} \))
$41$ (\( ( 1 + 2 T - 37 T^{2} + 82 T^{3} + 1681 T^{4} )^{2} \))
$43$ (\( 1 + 85 T^{2} + 5376 T^{4} + 157165 T^{6} + 3418801 T^{8} \))
$47$ (\( ( 1 - 85 T^{2} + 2209 T^{4} )^{2} \))
$53$ (\( 1 - 38 T^{2} - 1365 T^{4} - 106742 T^{6} + 7890481 T^{8} \))
$59$ (\( ( 1 + 4 T + 59 T^{2} )^{4} \))
$61$ (\( ( 1 - 7 T + 61 T^{2} )^{4} \))
$67$ (\( ( 1 - 85 T^{2} + 4489 T^{4} )^{2} \))
$71$ (\( ( 1 - 4 T + 71 T^{2} )^{4} \))
$73$ (\( ( 1 + 73 T^{2} + 5329 T^{4} )^{2} \))
$79$ (\( ( 1 - 14 T + 79 T^{2} )^{4} \))
$83$ (\( 1 + 150 T^{2} + 15611 T^{4} + 1033350 T^{6} + 47458321 T^{8} \))
$89$ (\( ( 1 + 3 T - 80 T^{2} + 267 T^{3} + 7921 T^{4} )^{2} \))
$97$ (\( 1 + 190 T^{2} + 26691 T^{4} + 1787710 T^{6} + 88529281 T^{8} \))
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