Properties

Label 315.2.t
Level 315
Weight 2
Character orbit t
Rep. character \(\chi_{315}(101,\cdot)\)
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.t (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

\( 64q - 64q^{4} + 2q^{7} + 2q^{9} + O(q^{10}) \) \( 64q - 64q^{4} + 2q^{7} + 2q^{9} + 6q^{11} + 30q^{12} + 6q^{13} - 36q^{14} - 2q^{15} + 64q^{16} - 26q^{18} - 6q^{21} - 18q^{23} - 32q^{25} + 24q^{26} + 18q^{27} - 8q^{28} + 18q^{29} + 2q^{30} + 36q^{36} - 2q^{37} - 60q^{38} + 6q^{39} - 6q^{41} - 8q^{43} + 42q^{44} - 18q^{45} + 6q^{46} - 72q^{47} - 60q^{48} - 8q^{49} - 12q^{51} - 24q^{52} - 48q^{53} - 6q^{54} + 78q^{56} + 6q^{57} + 60q^{59} + 36q^{60} - 64q^{63} - 64q^{64} + 108q^{66} - 28q^{67} - 30q^{68} - 12q^{70} + 62q^{72} + 120q^{74} - 54q^{77} - 24q^{78} + 8q^{79} - 46q^{81} - 60q^{83} - 72q^{84} - 6q^{85} - 6q^{86} + 18q^{87} + 42q^{89} + 54q^{90} - 6q^{91} + 12q^{92} + 42q^{93} - 42q^{96} + 6q^{97} + 54q^{98} - 2q^{99} + O(q^{100}) \)

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.t.a \(2\) \(2.515\) \(\Q(\sqrt{-3}) \) None \(0\) \(-3\) \(1\) \(4\) \(q+(1-2\zeta_{6})q^{2}+(-2+\zeta_{6})q^{3}-q^{4}+\cdots\)
315.2.t.b \(30\) \(2.515\) None \(0\) \(4\) \(15\) \(-3\)
315.2.t.c \(32\) \(2.515\) None \(0\) \(-1\) \(-16\) \(1\)

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

$p$ $F_p(T)$
$2$ (\( 1 - T^{2} + 4 T^{4} \))
$3$ (\( 1 + 3 T + 3 T^{2} \))
$5$ (\( 1 - T + T^{2} \))
$7$ (\( 1 - 4 T + 7 T^{2} \))
$11$ (\( 1 + 6 T + 23 T^{2} + 66 T^{3} + 121 T^{4} \))
$13$ (\( ( 1 - 7 T + 13 T^{2} )( 1 - 5 T + 13 T^{2} ) \))
$17$ (\( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \))
$19$ (\( 1 + 19 T^{2} + 361 T^{4} \))
$23$ (\( 1 + 9 T + 50 T^{2} + 207 T^{3} + 529 T^{4} \))
$29$ (\( 1 + 29 T^{2} + 841 T^{4} \))
$31$ (\( 1 - 50 T^{2} + 961 T^{4} \))
$37$ (\( 1 - 2 T - 33 T^{2} - 74 T^{3} + 1369 T^{4} \))
$41$ (\( 1 - 6 T - 5 T^{2} - 246 T^{3} + 1681 T^{4} \))
$43$ (\( 1 + T - 42 T^{2} + 43 T^{3} + 1849 T^{4} \))
$47$ (\( ( 1 - 9 T + 47 T^{2} )^{2} \))
$53$ (\( 1 - 6 T + 65 T^{2} - 318 T^{3} + 2809 T^{4} \))
$59$ (\( ( 1 + 59 T^{2} )^{2} \))
$61$ (\( 1 + 25 T^{2} + 3721 T^{4} \))
$67$ (\( ( 1 - 5 T + 67 T^{2} )^{2} \))
$71$ (\( 1 - 94 T^{2} + 5041 T^{4} \))
$73$ (\( 1 + 18 T + 181 T^{2} + 1314 T^{3} + 5329 T^{4} \))
$79$ (\( ( 1 + 10 T + 79 T^{2} )^{2} \))
$83$ (\( 1 + 12 T + 61 T^{2} + 996 T^{3} + 6889 T^{4} \))
$89$ (\( 1 - 15 T + 136 T^{2} - 1335 T^{3} + 7921 T^{4} \))
$97$ (\( 1 - 12 T + 145 T^{2} - 1164 T^{3} + 9409 T^{4} \))
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