Properties

Label 315.2.j
Level $315$
Weight $2$
Character orbit 315.j
Rep. character $\chi_{315}(46,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $28$
Newform subspaces $7$
Sturm bound $96$
Trace bound $2$

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Defining parameters

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

Dimensions

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

Total New Old
Modular forms 112 28 84
Cusp forms 80 28 52
Eisenstein series 32 0 32

Trace form

\( 28 q - 2 q^{2} - 18 q^{4} + 2 q^{5} + 6 q^{7} + 12 q^{8} + O(q^{10}) \) \( 28 q - 2 q^{2} - 18 q^{4} + 2 q^{5} + 6 q^{7} + 12 q^{8} - 2 q^{10} + 16 q^{13} + 8 q^{14} - 22 q^{16} + 12 q^{17} - 4 q^{19} - 20 q^{20} + 24 q^{22} - 10 q^{23} - 14 q^{25} - 4 q^{26} - 14 q^{28} + 12 q^{29} + 8 q^{31} - 14 q^{32} - 40 q^{34} - 4 q^{35} - 12 q^{37} - 24 q^{38} - 6 q^{40} + 12 q^{41} - 36 q^{43} - 16 q^{44} + 22 q^{46} + 12 q^{47} - 10 q^{49} + 4 q^{50} - 28 q^{52} + 8 q^{53} + 24 q^{55} + 18 q^{56} + 10 q^{58} - 12 q^{59} + 46 q^{61} + 116 q^{64} + 8 q^{65} + 22 q^{67} + 8 q^{68} + 14 q^{70} - 16 q^{71} - 32 q^{73} + 4 q^{74} - 48 q^{76} - 72 q^{77} - 4 q^{79} + 22 q^{80} + 2 q^{82} - 12 q^{83} + 8 q^{85} - 14 q^{86} - 76 q^{88} - 22 q^{89} - 24 q^{91} + 76 q^{92} + 36 q^{94} - 24 q^{97} + 14 q^{98} + O(q^{100}) \)

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.j.a 315.j 7.c $2$ $2.515$ \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(1\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\zeta_{6}q^{2}+(-2+2\zeta_{6})q^{4}+\zeta_{6}q^{5}+\cdots\)
315.2.j.b 315.j 7.c $2$ $2.515$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-1\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{4}-\zeta_{6}q^{5}+(2+\zeta_{6})q^{7}+\cdots\)
315.2.j.c 315.j 7.c $4$ $2.515$ \(\Q(\zeta_{12})\) None \(-2\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+(-2+2\zeta_{12}+\cdots)q^{4}+\cdots\)
315.2.j.d 315.j 7.c $4$ $2.515$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(0\) \(-2\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{2}+(-1-\beta _{2})q^{5}+(-1-\beta _{1}+\cdots)q^{7}+\cdots\)
315.2.j.e 315.j 7.c $4$ $2.515$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(2\) \(0\) \(2\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\beta _{1}+\beta _{2})q^{2}+(2\beta _{1}+\beta _{2}+2\beta _{3})q^{4}+\cdots\)
315.2.j.f 315.j 7.c $6$ $2.515$ 6.0.4406832.1 None \(-2\) \(0\) \(-3\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{1}+\beta _{2}-\beta _{4})q^{2}+(-2+\beta _{1}+\cdots)q^{4}+\cdots\)
315.2.j.g 315.j 7.c $6$ $2.515$ 6.0.4406832.1 None \(2\) \(0\) \(3\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{1}-\beta _{2}+\beta _{4})q^{2}+(-2+\beta _{1}+\beta _{3}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(315, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(315, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)