Properties

Label 315.2.z
Level 315
Weight 2
Character orbit z
Rep. character \(\chi_{315}(104,\cdot)\)
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.z (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

\( 88q - 44q^{4} + O(q^{10}) \) \( 88q - 44q^{4} - 6q^{11} - 12q^{14} - 24q^{15} - 44q^{16} + 6q^{21} - 2q^{25} - 48q^{29} + 36q^{30} - 36q^{36} + 54q^{39} - 24q^{46} - 8q^{49} - 42q^{50} - 18q^{51} + 24q^{56} + 96q^{60} + 64q^{64} - 6q^{70} + 12q^{74} - 34q^{79} - 60q^{84} + 8q^{85} + 156q^{86} + 20q^{91} - 24q^{95} - 72q^{99} + O(q^{100}) \)

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.z.a \(8\) \(2.515\) 8.0.\(\cdots\).2 \(\Q(\sqrt{-35}) \) \(0\) \(0\) \(0\) \(0\) \(q+\beta _{5}q^{3}+(2+2\beta _{4})q^{4}+(\beta _{1}-\beta _{6})q^{5}+\cdots\)
315.2.z.b \(80\) \(2.515\) None \(0\) \(0\) \(0\) \(0\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 - 2 T^{2} + 4 T^{4} )^{4} \))
$3$ (\( ( 1 - T^{2} + 9 T^{4} )^{2} \))
$5$ (\( ( 1 - 5 T^{2} + 25 T^{4} )^{2} \))
$7$ (\( ( 1 + 7 T^{2} + 49 T^{4} )^{2} \))
$11$ (\( ( 1 - 3 T + 11 T^{2} )^{4}( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} )^{2} \))
$13$ (\( ( 1 + 19 T^{2} + 169 T^{4} )^{2}( 1 - 19 T^{2} + 192 T^{4} - 3211 T^{6} + 28561 T^{8} ) \))
$17$ (\( ( 1 + 29 T^{2} + 552 T^{4} + 8381 T^{6} + 83521 T^{8} )^{2} \))
$19$ (\( ( 1 - 19 T^{2} )^{8} \))
$23$ (\( ( 1 - 23 T^{2} + 529 T^{4} )^{4} \))
$29$ (\( ( 1 - 9 T + 52 T^{2} - 261 T^{3} + 841 T^{4} )^{2}( 1 + 9 T + 52 T^{2} + 261 T^{3} + 841 T^{4} )^{2} \))
$31$ (\( ( 1 + 31 T^{2} + 961 T^{4} )^{4} \))
$37$ (\( ( 1 - 37 T^{2} )^{8} \))
$41$ (\( ( 1 - 41 T^{2} + 1681 T^{4} )^{4} \))
$43$ (\( ( 1 + 43 T^{2} + 1849 T^{4} )^{4} \))
$47$ (\( ( 1 + 31 T^{2} + 2209 T^{4} )^{2}( 1 - 31 T^{2} - 1248 T^{4} - 68479 T^{6} + 4879681 T^{8} ) \))
$53$ (\( ( 1 + 53 T^{2} )^{8} \))
$59$ (\( ( 1 - 59 T^{2} + 3481 T^{4} )^{4} \))
$61$ (\( ( 1 + 61 T^{2} + 3721 T^{4} )^{4} \))
$67$ (\( ( 1 + 67 T^{2} + 4489 T^{4} )^{4} \))
$71$ (\( ( 1 - 12 T + 73 T^{2} - 852 T^{3} + 5041 T^{4} )^{2}( 1 + 12 T + 73 T^{2} + 852 T^{3} + 5041 T^{4} )^{2} \))
$73$ (\( ( 1 - 34 T^{2} - 4173 T^{4} - 181186 T^{6} + 28398241 T^{8} )^{2} \))
$79$ (\( ( 1 + T + 79 T^{2} )^{4}( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} )^{2} \))
$83$ (\( ( 1 - 86 T^{2} + 6889 T^{4} )^{2}( 1 + 86 T^{2} + 507 T^{4} + 592454 T^{6} + 47458321 T^{8} ) \))
$89$ (\( ( 1 + 89 T^{2} )^{8} \))
$97$ (\( ( 1 - 149 T^{2} + 9409 T^{4} )^{2}( 1 + 149 T^{2} + 12792 T^{4} + 1401941 T^{6} + 88529281 T^{8} ) \))
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