Properties

Label 315.2.l
Level 315
Weight 2
Character orbit l
Rep. character \(\chi_{315}(121,\cdot)\)
Character field \(\Q(\zeta_{3})\)
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.l (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 63 \)
Character field: \(\Q(\zeta_{3})\)
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

\( 64q + 64q^{4} - 4q^{5} - 4q^{6} - 2q^{7} - 6q^{9} + O(q^{10}) \) \( 64q + 64q^{4} - 4q^{5} - 4q^{6} - 2q^{7} - 6q^{9} + 2q^{11} + 10q^{12} + 2q^{13} + 12q^{14} - 2q^{15} + 64q^{16} - 16q^{17} - 6q^{18} - 4q^{19} - 12q^{20} - 22q^{21} - 6q^{23} - 32q^{24} - 32q^{25} - 8q^{26} + 6q^{27} - 8q^{28} - 10q^{29} + 2q^{30} - 16q^{31} - 40q^{32} - 28q^{33} - 60q^{36} + 2q^{37} - 44q^{38} + 2q^{39} + 10q^{41} + 12q^{42} + 8q^{43} - 14q^{44} + 6q^{45} - 6q^{46} + 104q^{47} - 64q^{48} + 4q^{49} + 56q^{51} + 8q^{52} - 30q^{54} - 42q^{56} + 26q^{57} + 20q^{59} - 36q^{60} - 16q^{61} - 24q^{62} + 44q^{63} + 64q^{64} - 4q^{65} - 8q^{66} - 28q^{67} - 58q^{68} - 24q^{69} - 12q^{70} + 48q^{71} - 34q^{72} - 28q^{73} + 44q^{74} - 16q^{76} + 70q^{77} + 40q^{78} - 16q^{79} - 28q^{80} - 46q^{81} - 68q^{83} - 24q^{84} + 6q^{85} + 2q^{86} + 26q^{87} - 14q^{89} + 18q^{90} - 22q^{91} - 100q^{92} + 6q^{93} - 24q^{94} - 158q^{96} + 2q^{97} - 10q^{98} - 90q^{99} + O(q^{100}) \)

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.l.a \(4\) \(2.515\) \(\Q(\sqrt{-3}, \sqrt{13})\) None \(-2\) \(6\) \(2\) \(10\) \(q+(-1+\beta _{3})q^{2}+(1-\beta _{2})q^{3}+(2-\beta _{3})q^{4}+\cdots\)
315.2.l.b \(24\) \(2.515\) None \(2\) \(-5\) \(12\) \(-11\)
315.2.l.c \(36\) \(2.515\) None \(0\) \(-1\) \(-18\) \(-1\)

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

$p$ $F_p(T)$
$2$ (\( ( 1 + T + T^{2} + 2 T^{3} + 4 T^{4} )^{2} \))
$3$ (\( ( 1 - 3 T + 3 T^{2} )^{2} \))
$5$ (\( ( 1 - T + T^{2} )^{2} \))
$7$ (\( ( 1 - 5 T + 7 T^{2} )^{2} \))
$11$ (\( ( 1 - 11 T^{2} + 121 T^{4} )^{2} \))
$13$ (\( 1 - 4 T - T^{2} + 36 T^{3} - 88 T^{4} + 468 T^{5} - 169 T^{6} - 8788 T^{7} + 28561 T^{8} \))
$17$ (\( 1 + 4 T - 9 T^{2} - 36 T^{3} + 64 T^{4} - 612 T^{5} - 2601 T^{6} + 19652 T^{7} + 83521 T^{8} \))
$19$ (\( 1 - 25 T^{2} + 264 T^{4} - 9025 T^{6} + 130321 T^{8} \))
$23$ (\( 1 + 4 T + 18 T^{2} - 192 T^{3} - 893 T^{4} - 4416 T^{5} + 9522 T^{6} + 48668 T^{7} + 279841 T^{8} \))
$29$ (\( 1 + 2 T - 3 T^{2} - 102 T^{3} - 908 T^{4} - 2958 T^{5} - 2523 T^{6} + 48778 T^{7} + 707281 T^{8} \))
$31$ (\( ( 1 + 49 T^{2} + 961 T^{4} )^{2} \))
$37$ (\( 1 - 61 T^{2} + 2352 T^{4} - 83509 T^{6} + 1874161 T^{8} \))
$41$ (\( ( 1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4} )^{2} \))
$43$ (\( 1 - 6 T - 7 T^{2} + 258 T^{3} - 1548 T^{4} + 11094 T^{5} - 12943 T^{6} - 477042 T^{7} + 3418801 T^{8} \))
$47$ (\( ( 1 + 2 T + 43 T^{2} + 94 T^{3} + 2209 T^{4} )^{2} \))
$53$ (\( 1 + 8 T - 45 T^{2} + 24 T^{3} + 5680 T^{4} + 1272 T^{5} - 126405 T^{6} + 1191016 T^{7} + 7890481 T^{8} \))
$59$ (\( ( 1 + 16 T + 169 T^{2} + 944 T^{3} + 3481 T^{4} )^{2} \))
$61$ (\( ( 1 + 6 T + 79 T^{2} + 366 T^{3} + 3721 T^{4} )^{2} \))
$67$ (\( ( 1 + 6 T + 91 T^{2} + 402 T^{3} + 4489 T^{4} )^{2} \))
$71$ (\( ( 1 + 71 T^{2} )^{4} \))
$73$ (\( 1 - 133 T^{2} + 12360 T^{4} - 708757 T^{6} + 28398241 T^{8} \))
$79$ (\( ( 1 - 4 T + 45 T^{2} - 316 T^{3} + 6241 T^{4} )^{2} \))
$83$ (\( 1 - 14 T + 33 T^{2} + 42 T^{3} + 3412 T^{4} + 3486 T^{5} + 227337 T^{6} - 8005018 T^{7} + 47458321 T^{8} \))
$89$ (\( ( 1 + 3 T - 80 T^{2} + 267 T^{3} + 7921 T^{4} )^{2} \))
$97$ (\( 1 + 20 T + 119 T^{2} + 1740 T^{3} + 30752 T^{4} + 168780 T^{5} + 1119671 T^{6} + 18253460 T^{7} + 88529281 T^{8} \))
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