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Properties

Label 315.2.bs
Level $315$
Weight $2$
Character orbit 315.bs
Rep. character $\chi_{315}(52,\cdot)$
Character field $\Q(\zeta_{12})$
Dimension $176$
Newform subspaces $5$
Sturm bound $96$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 315.bs (of order \(12\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 315 \)
Character field: \(\Q(\zeta_{12})\)
Newform subspaces: \( 5 \)
Sturm bound: \(96\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(2\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(315, [\chi])\).

Total New Old
Modular forms 208 208 0
Cusp forms 176 176 0
Eisenstein series 32 32 0

Trace form

\( 176 q - 4 q^{2} - 6 q^{3} - 6 q^{5} - 2 q^{7} - 8 q^{8} + O(q^{10}) \) \( 176 q - 4 q^{2} - 6 q^{3} - 6 q^{5} - 2 q^{7} - 8 q^{8} - 12 q^{10} - 18 q^{12} - 2 q^{15} - 136 q^{16} - 30 q^{17} + 22 q^{18} + 48 q^{20} - 12 q^{22} - 6 q^{23} + 2 q^{25} - 24 q^{26} - 36 q^{27} + 4 q^{28} - 14 q^{30} - 60 q^{32} - 6 q^{33} - 44 q^{35} + 40 q^{36} - 4 q^{37} - 6 q^{38} - 6 q^{40} + 12 q^{41} - 40 q^{42} - 4 q^{43} - 54 q^{45} - 20 q^{46} + 18 q^{48} - 28 q^{50} + 32 q^{51} + 42 q^{52} + 16 q^{53} + 128 q^{56} + 8 q^{57} + 22 q^{58} + 86 q^{60} - 6 q^{63} - 52 q^{65} - 12 q^{66} - 4 q^{67} + 90 q^{68} + 18 q^{70} - 64 q^{71} + 106 q^{72} - 12 q^{73} - 54 q^{75} + 48 q^{76} + 74 q^{77} - 62 q^{78} + 12 q^{80} - 64 q^{81} - 24 q^{82} - 84 q^{83} - 4 q^{85} - 40 q^{86} + 12 q^{87} + 42 q^{88} + 36 q^{90} - 16 q^{91} - 8 q^{92} + 58 q^{93} + 100 q^{95} - 168 q^{96} + 120 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(315, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
315.2.bs.a 315.bs 315.as $4$ $2.515$ \(\Q(\zeta_{12})\) None \(-2\) \(0\) \(2\) \(-10\) $\mathrm{SU}(2)[C_{12}]$ \(q+(-\zeta_{12}-\zeta_{12}^{2})q^{2}+(\zeta_{12}-2\zeta_{12}^{3})q^{3}+\cdots\)
315.2.bs.b 315.bs 315.as $4$ $2.515$ \(\Q(\zeta_{12})\) None \(-2\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{12}]$ \(q+(-1+\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+\cdots\)
315.2.bs.c 315.bs 315.as $4$ $2.515$ \(\Q(\zeta_{12})\) None \(-2\) \(6\) \(-4\) \(8\) $\mathrm{SU}(2)[C_{12}]$ \(q+(-\zeta_{12}-\zeta_{12}^{2})q^{2}+(2-\zeta_{12}^{2})q^{3}+\cdots\)
315.2.bs.d 315.bs 315.as $4$ $2.515$ \(\Q(\zeta_{12})\) None \(-2\) \(6\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{12}]$ \(q+(-1+\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+\cdots\)
315.2.bs.e 315.bs 315.as $160$ $2.515$ None \(4\) \(-18\) \(-6\) \(0\) $\mathrm{SU}(2)[C_{12}]$