Properties

Label 630.2.k
Level $630$
Weight $2$
Character orbit 630.k
Rep. character $\chi_{630}(361,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $24$
Newform subspaces $10$
Sturm bound $288$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 630.k (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 10 \)
Sturm bound: \(288\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(630, [\chi])\).

Total New Old
Modular forms 320 24 296
Cusp forms 256 24 232
Eisenstein series 64 0 64

Trace form

\( 24 q - 12 q^{4} - 2 q^{5} - 8 q^{7} + O(q^{10}) \) \( 24 q - 12 q^{4} - 2 q^{5} - 8 q^{7} + 2 q^{10} - 6 q^{11} + 10 q^{14} - 12 q^{16} - 12 q^{17} + 6 q^{19} + 4 q^{20} + 8 q^{22} + 12 q^{23} - 12 q^{25} + 6 q^{26} + 4 q^{28} + 20 q^{29} - 8 q^{31} - 16 q^{34} + 6 q^{35} - 12 q^{37} - 16 q^{38} + 2 q^{40} - 40 q^{41} + 56 q^{43} - 6 q^{44} - 16 q^{46} + 4 q^{47} + 42 q^{49} - 20 q^{53} - 16 q^{55} - 2 q^{56} + 36 q^{59} - 42 q^{61} + 16 q^{62} + 24 q^{64} - 6 q^{65} - 48 q^{67} - 12 q^{68} + 6 q^{70} + 32 q^{71} + 28 q^{73} + 6 q^{74} - 12 q^{76} + 32 q^{77} - 8 q^{79} - 2 q^{80} - 16 q^{82} + 16 q^{83} - 16 q^{85} - 18 q^{86} - 4 q^{88} + 6 q^{89} + 76 q^{91} - 24 q^{92} + 6 q^{94} - 4 q^{95} - 32 q^{97} - 24 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(630, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
630.2.k.a 630.k 7.c $2$ $5.031$ \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(-1\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-\zeta_{6}q^{5}+(-3+\cdots)q^{7}+\cdots\)
630.2.k.b 630.k 7.c $2$ $5.031$ \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(-1\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-\zeta_{6}q^{5}+(1+\cdots)q^{7}+\cdots\)
630.2.k.c 630.k 7.c $2$ $5.031$ \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(-1\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-\zeta_{6}q^{5}+(1+\cdots)q^{7}+\cdots\)
630.2.k.d 630.k 7.c $2$ $5.031$ \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(1\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+\zeta_{6}q^{5}+(-3+\cdots)q^{7}+\cdots\)
630.2.k.e 630.k 7.c $2$ $5.031$ \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(-1\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-\zeta_{6}q^{5}+(-1+\cdots)q^{7}+\cdots\)
630.2.k.f 630.k 7.c $2$ $5.031$ \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(-1\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-\zeta_{6}q^{5}+(-1+\cdots)q^{7}+\cdots\)
630.2.k.g 630.k 7.c $2$ $5.031$ \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(1\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+\zeta_{6}q^{5}+(-1+\cdots)q^{7}+\cdots\)
630.2.k.h 630.k 7.c $2$ $5.031$ \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(1\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+\zeta_{6}q^{5}+(3+\cdots)q^{7}+\cdots\)
630.2.k.i 630.k 7.c $4$ $5.031$ \(\Q(\sqrt{-3}, \sqrt{7})\) None \(-2\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1-\beta _{2})q^{2}+\beta _{2}q^{4}+(1+\beta _{2})q^{5}+\cdots\)
630.2.k.j 630.k 7.c $4$ $5.031$ \(\Q(\sqrt{-3}, \sqrt{7})\) None \(2\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\beta _{2})q^{2}+\beta _{2}q^{4}+(-1-\beta _{2})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(630, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(630, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(315, [\chi])\)\(^{\oplus 2}\)