Properties

Label 315.2.be
Level 315
Weight 2
Character orbit be
Rep. character \(\chi_{315}(236,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 64
Newform subspaces 3
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.be (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 63 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 3 \)
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 64 40
Cusp forms 88 64 24
Eisenstein series 16 0 16

Trace form

\( 64q + 32q^{4} + 2q^{7} - 4q^{9} + O(q^{10}) \) \( 64q + 32q^{4} + 2q^{7} - 4q^{9} - 6q^{13} - 6q^{14} - 2q^{15} - 32q^{16} + 28q^{18} - 24q^{21} - 18q^{24} + 64q^{25} - 24q^{26} - 18q^{27} - 8q^{28} + 18q^{29} - 4q^{30} + 24q^{31} - 24q^{33} + 36q^{36} - 2q^{37} - 120q^{38} - 36q^{39} + 6q^{41} - 8q^{43} - 42q^{44} - 18q^{45} + 6q^{46} - 36q^{47} + 60q^{48} + 10q^{49} + 42q^{51} + 48q^{53} - 36q^{54} + 102q^{56} + 6q^{57} + 30q^{59} - 30q^{60} - 60q^{61} + 14q^{63} - 64q^{64} + 6q^{65} + 48q^{66} + 14q^{67} - 60q^{68} + 6q^{70} - 76q^{72} - 54q^{77} - 24q^{78} - 4q^{79} + 44q^{81} + 60q^{83} - 54q^{84} - 6q^{85} - 102q^{87} - 42q^{89} - 54q^{90} - 6q^{91} + 12q^{92} - 60q^{93} + 60q^{96} - 6q^{97} - 54q^{98} - 2q^{99} + O(q^{100}) \)

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.be.a \(2\) \(2.515\) \(\Q(\sqrt{-3}) \) None \(-3\) \(0\) \(2\) \(-5\) \(q+(-1-\zeta_{6})q^{2}+(1-2\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
315.2.be.b \(30\) \(2.515\) None \(3\) \(-1\) \(30\) \(6\)
315.2.be.c \(32\) \(2.515\) None \(0\) \(1\) \(-32\) \(1\)

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

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4} \))
$3$ (\( 1 + 3 T^{2} \))
$5$ (\( ( 1 - T )^{2} \))
$7$ (\( 1 + 5 T + 7 T^{2} \))
$11$ (\( 1 - 10 T^{2} + 121 T^{4} \))
$13$ (\( ( 1 + 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))
$17$ (\( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} \))
$19$ (\( 1 + 19 T^{2} + 361 T^{4} \))
$23$ (\( 1 - 19 T^{2} + 529 T^{4} \))
$29$ (\( 1 + 29 T^{2} + 841 T^{4} \))
$31$ (\( 1 + 6 T + 43 T^{2} + 186 T^{3} + 961 T^{4} \))
$37$ (\( 1 - 2 T - 33 T^{2} - 74 T^{3} + 1369 T^{4} \))
$41$ (\( 1 + 6 T - 5 T^{2} + 246 T^{3} + 1681 T^{4} \))
$43$ (\( 1 + T - 42 T^{2} + 43 T^{3} + 1849 T^{4} \))
$47$ (\( 1 - 9 T + 34 T^{2} - 423 T^{3} + 2209 T^{4} \))
$53$ (\( 1 + 6 T + 65 T^{2} + 318 T^{3} + 2809 T^{4} \))
$59$ (\( 1 - 59 T^{2} + 3481 T^{4} \))
$61$ (\( 1 + 21 T + 208 T^{2} + 1281 T^{3} + 3721 T^{4} \))
$67$ (\( ( 1 - 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))
$71$ (\( 1 - 94 T^{2} + 5041 T^{4} \))
$73$ (\( 1 + 18 T + 181 T^{2} + 1314 T^{3} + 5329 T^{4} \))
$79$ (\( 1 - 10 T + 21 T^{2} - 790 T^{3} + 6241 T^{4} \))
$83$ (\( 1 - 12 T + 61 T^{2} - 996 T^{3} + 6889 T^{4} \))
$89$ (\( 1 + 15 T + 136 T^{2} + 1335 T^{3} + 7921 T^{4} \))
$97$ (\( 1 + 12 T + 145 T^{2} + 1164 T^{3} + 9409 T^{4} \))
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