Properties

Label 315.2.k
Level 315
Weight 2
Character orbit k
Rep. character \(\chi_{315}(16,\cdot)\)
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.k (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

\( 64q - 32q^{4} + 8q^{5} - 4q^{6} - 2q^{7} + O(q^{10}) \) \( 64q - 32q^{4} + 8q^{5} - 4q^{6} - 2q^{7} - 4q^{11} - 20q^{12} + 2q^{13} - 18q^{14} - 2q^{15} - 32q^{16} - 16q^{17} - 12q^{18} - 4q^{19} - 12q^{20} - 4q^{21} + 12q^{23} - 14q^{24} + 64q^{25} - 8q^{26} + 6q^{27} - 8q^{28} - 10q^{29} - 4q^{30} + 8q^{31} + 20q^{32} - 4q^{33} - 60q^{36} + 2q^{37} + 88q^{38} - 28q^{39} + 10q^{41} - 36q^{42} + 8q^{43} - 14q^{44} + 6q^{45} - 6q^{46} - 52q^{47} - 64q^{48} - 14q^{49} - 46q^{51} - 16q^{52} + 120q^{54} + 102q^{56} + 26q^{57} - 10q^{59} + 30q^{60} + 8q^{61} - 24q^{62} + 26q^{63} + 64q^{64} + 2q^{65} + 16q^{66} + 14q^{67} + 116q^{68} - 24q^{69} + 6q^{70} + 48q^{71} - 28q^{72} - 28q^{73} - 88q^{74} - 16q^{76} + 10q^{77} + 40q^{78} + 8q^{79} - 28q^{80} + 44q^{81} - 68q^{83} + 54q^{84} + 6q^{85} - 4q^{86} - 22q^{87} - 14q^{89} + 18q^{90} - 22q^{91} - 100q^{92} - 36q^{93} + 12q^{94} + 64q^{96} + 2q^{97} - 10q^{98} - 90q^{99} + O(q^{100}) \)

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.k.a \(4\) \(2.515\) \(\Q(\sqrt{-3}, \sqrt{13})\) None \(1\) \(0\) \(-4\) \(-8\) \(q+\beta _{1}q^{2}+(1+2\beta _{2})q^{3}+(-1+\beta _{1}+\cdots)q^{4}+\cdots\)
315.2.k.b \(24\) \(2.515\) None \(-1\) \(1\) \(-24\) \(7\)
315.2.k.c \(36\) \(2.515\) None \(0\) \(-1\) \(36\) \(-1\)

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

$p$ $F_p(T)$
$2$ (\( 1 - T + 3 T^{3} - 5 T^{4} + 6 T^{5} - 8 T^{7} + 16 T^{8} \))
$3$ (\( ( 1 + 3 T^{2} )^{2} \))
$5$ (\( ( 1 + T )^{4} \))
$7$ (\( ( 1 + 4 T + 7 T^{2} )^{2} \))
$11$ (\( ( 1 + 11 T^{2} )^{4} \))
$13$ (\( 1 - 4 T - T^{2} + 36 T^{3} - 88 T^{4} + 468 T^{5} - 169 T^{6} - 8788 T^{7} + 28561 T^{8} \))
$17$ (\( 1 + 4 T - 9 T^{2} - 36 T^{3} + 64 T^{4} - 612 T^{5} - 2601 T^{6} + 19652 T^{7} + 83521 T^{8} \))
$19$ (\( 1 - 25 T^{2} + 264 T^{4} - 9025 T^{6} + 130321 T^{8} \))
$23$ (\( ( 1 - 4 T - 2 T^{2} - 92 T^{3} + 529 T^{4} )^{2} \))
$29$ (\( 1 + 2 T - 3 T^{2} - 102 T^{3} - 908 T^{4} - 2958 T^{5} - 2523 T^{6} + 48778 T^{7} + 707281 T^{8} \))
$31$ (\( 1 - 49 T^{2} + 1440 T^{4} - 47089 T^{6} + 923521 T^{8} \))
$37$ (\( 1 - 61 T^{2} + 2352 T^{4} - 83509 T^{6} + 1874161 T^{8} \))
$41$ (\( ( 1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4} )^{2} \))
$43$ (\( 1 - 6 T - 7 T^{2} + 258 T^{3} - 1548 T^{4} + 11094 T^{5} - 12943 T^{6} - 477042 T^{7} + 3418801 T^{8} \))
$47$ (\( 1 - 2 T - 39 T^{2} + 102 T^{3} - 548 T^{4} + 4794 T^{5} - 86151 T^{6} - 207646 T^{7} + 4879681 T^{8} \))
$53$ (\( 1 + 8 T - 45 T^{2} + 24 T^{3} + 5680 T^{4} + 1272 T^{5} - 126405 T^{6} + 1191016 T^{7} + 7890481 T^{8} \))
$59$ (\( 1 - 16 T + 87 T^{2} - 816 T^{3} + 9976 T^{4} - 48144 T^{5} + 302847 T^{6} - 3286064 T^{7} + 12117361 T^{8} \))
$61$ (\( 1 - 6 T - 43 T^{2} + 258 T^{3} + 324 T^{4} + 15738 T^{5} - 160003 T^{6} - 1361886 T^{7} + 13845841 T^{8} \))
$67$ (\( 1 - 6 T - 55 T^{2} + 258 T^{3} + 1380 T^{4} + 17286 T^{5} - 246895 T^{6} - 1804578 T^{7} + 20151121 T^{8} \))
$71$ (\( ( 1 + 71 T^{2} )^{4} \))
$73$ (\( 1 - 133 T^{2} + 12360 T^{4} - 708757 T^{6} + 28398241 T^{8} \))
$79$ (\( 1 + 4 T - 29 T^{2} - 452 T^{3} - 5480 T^{4} - 35708 T^{5} - 180989 T^{6} + 1972156 T^{7} + 38950081 T^{8} \))
$83$ (\( 1 - 14 T + 33 T^{2} + 42 T^{3} + 3412 T^{4} + 3486 T^{5} + 227337 T^{6} - 8005018 T^{7} + 47458321 T^{8} \))
$89$ (\( ( 1 + 3 T - 80 T^{2} + 267 T^{3} + 7921 T^{4} )^{2} \))
$97$ (\( 1 + 20 T + 119 T^{2} + 1740 T^{3} + 30752 T^{4} + 168780 T^{5} + 1119671 T^{6} + 18253460 T^{7} + 88529281 T^{8} \))
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