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Properties

Label 315.2.cg
Level $315$
Weight $2$
Character orbit 315.cg
Rep. character $\chi_{315}(157,\cdot)$
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.cg (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 \(T_p\): \(2\), \(11\)

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

\( 176 q + 2 q^{2} - 6 q^{3} - 2 q^{7} - 8 q^{8} + O(q^{10}) \) \( 176 q + 2 q^{2} - 6 q^{3} - 2 q^{7} - 8 q^{8} - 12 q^{10} - 6 q^{12} - 2 q^{15} + 68 q^{16} - 30 q^{17} - 26 q^{18} - 48 q^{20} - 36 q^{21} - 12 q^{22} + 12 q^{23} - 4 q^{25} - 24 q^{26} + 36 q^{27} + 4 q^{28} - 44 q^{30} - 12 q^{31} + 30 q^{32} + 24 q^{33} - 44 q^{35} + 40 q^{36} - 4 q^{37} - 12 q^{41} - 10 q^{42} - 4 q^{43} + 12 q^{45} - 20 q^{46} - 6 q^{47} - 18 q^{48} - 28 q^{50} - 28 q^{51} + 16 q^{53} - 88 q^{56} + 8 q^{57} - 44 q^{58} - 10 q^{60} - 48 q^{61} + 54 q^{63} + 26 q^{65} - 84 q^{66} + 2 q^{67} - 42 q^{70} - 64 q^{71} + 10 q^{72} - 12 q^{73} - 6 q^{75} - 48 q^{76} - 106 q^{77} - 62 q^{78} + 12 q^{80} + 20 q^{81} - 24 q^{82} + 84 q^{83} - 4 q^{85} + 80 q^{86} - 24 q^{87} - 84 q^{88} - 36 q^{90} - 16 q^{91} - 8 q^{92} + 28 q^{93} - 50 q^{95} + 72 q^{96} + 120 q^{98} + O(q^{100}) \)

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.cg.a 315.cg 315.bg $4$ $2.515$ \(\Q(\zeta_{12})\) None \(-2\) \(0\) \(-8\) \(0\) $\mathrm{SU}(2)[C_{12}]$ \(q+(-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+(-1+2\zeta_{12}^{2}+\cdots)q^{3}+\cdots\)
315.2.cg.b 315.cg 315.bg $4$ $2.515$ \(\Q(\zeta_{12})\) None \(-2\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{12}]$ \(q+(-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+(\zeta_{12}-2\zeta_{12}^{3})q^{3}+\cdots\)
315.2.cg.c 315.cg 315.bg $4$ $2.515$ \(\Q(\zeta_{12})\) None \(4\) \(0\) \(8\) \(-10\) $\mathrm{SU}(2)[C_{12}]$ \(q+(1+\zeta_{12}-\zeta_{12}^{3})q^{2}+(2\zeta_{12}-\zeta_{12}^{3})q^{3}+\cdots\)
315.2.cg.d 315.cg 315.bg $4$ $2.515$ \(\Q(\zeta_{12})\) None \(4\) \(6\) \(-4\) \(2\) $\mathrm{SU}(2)[C_{12}]$ \(q+(1+\zeta_{12}-\zeta_{12}^{3})q^{2}+(1+\zeta_{12}^{2}+\cdots)q^{3}+\cdots\)
315.2.cg.e 315.cg 315.bg $160$ $2.515$ None \(-2\) \(-12\) \(0\) \(6\) $\mathrm{SU}(2)[C_{12}]$