↑

Properties

Label 630.4.k
Level $630$
Weight $4$
Character orbit 630.k
Rep. character $\chi_{630}(361,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $80$
Newform subspaces $20$
Sturm bound $576$
Trace bound $7$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 630.k (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 20 \)
Sturm bound: \(576\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(630, [\chi])\).

Total New Old
Modular forms 896 80 816
Cusp forms 832 80 752
Eisenstein series 64 0 64

Trace form

\( 80 q - 160 q^{4} + 10 q^{5} - 4 q^{7} + O(q^{10}) \) \( 80 q - 160 q^{4} + 10 q^{5} - 4 q^{7} - 20 q^{10} + 22 q^{11} + 384 q^{13} + 52 q^{14} - 640 q^{16} + 140 q^{17} + 162 q^{19} - 80 q^{20} + 112 q^{22} - 168 q^{23} - 1000 q^{25} + 4 q^{26} - 64 q^{28} + 444 q^{29} + 200 q^{31} + 480 q^{34} + 90 q^{35} + 364 q^{37} + 208 q^{38} - 80 q^{40} - 1648 q^{41} - 1520 q^{43} + 88 q^{44} + 296 q^{46} + 236 q^{47} - 1162 q^{49} - 768 q^{52} + 1852 q^{53} + 480 q^{55} - 368 q^{56} + 288 q^{58} - 860 q^{59} + 938 q^{61} - 4128 q^{62} + 5120 q^{64} - 710 q^{65} - 516 q^{67} + 560 q^{68} + 180 q^{70} + 1664 q^{71} - 196 q^{73} + 804 q^{74} - 1296 q^{76} + 4688 q^{77} - 2256 q^{79} + 160 q^{80} - 2112 q^{82} - 3016 q^{83} + 2560 q^{85} + 1748 q^{86} - 224 q^{88} - 374 q^{89} - 1292 q^{91} + 1344 q^{92} + 460 q^{94} + 580 q^{95} + 3056 q^{97} - 2736 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(630, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
630.4.k.a 630.k 7.c $2$ $37.171$ \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(-5\) \(-35\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\zeta_{6}q^{2}+(-4+4\zeta_{6})q^{4}-5\zeta_{6}q^{5}+\cdots\)
630.4.k.b 630.k 7.c $2$ $37.171$ \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(-5\) \(17\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\zeta_{6}q^{2}+(-4+4\zeta_{6})q^{4}-5\zeta_{6}q^{5}+\cdots\)
630.4.k.c 630.k 7.c $2$ $37.171$ \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(-5\) \(28\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\zeta_{6}q^{2}+(-4+4\zeta_{6})q^{4}-5\zeta_{6}q^{5}+\cdots\)
630.4.k.d 630.k 7.c $2$ $37.171$ \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(5\) \(7\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\zeta_{6}q^{2}+(-4+4\zeta_{6})q^{4}+5\zeta_{6}q^{5}+\cdots\)
630.4.k.e 630.k 7.c $2$ $37.171$ \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(5\) \(20\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\zeta_{6}q^{2}+(-4+4\zeta_{6})q^{4}+5\zeta_{6}q^{5}+\cdots\)
630.4.k.f 630.k 7.c $2$ $37.171$ \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(-5\) \(-28\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\zeta_{6}q^{2}+(-4+4\zeta_{6})q^{4}-5\zeta_{6}q^{5}+\cdots\)
630.4.k.g 630.k 7.c $2$ $37.171$ \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(-5\) \(-20\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\zeta_{6}q^{2}+(-4+4\zeta_{6})q^{4}-5\zeta_{6}q^{5}+\cdots\)
630.4.k.h 630.k 7.c $2$ $37.171$ \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(-5\) \(-7\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\zeta_{6}q^{2}+(-4+4\zeta_{6})q^{4}-5\zeta_{6}q^{5}+\cdots\)
630.4.k.i 630.k 7.c $2$ $37.171$ \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(-5\) \(35\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\zeta_{6}q^{2}+(-4+4\zeta_{6})q^{4}-5\zeta_{6}q^{5}+\cdots\)
630.4.k.j 630.k 7.c $2$ $37.171$ \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(5\) \(35\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\zeta_{6}q^{2}+(-4+4\zeta_{6})q^{4}+5\zeta_{6}q^{5}+\cdots\)
630.4.k.k 630.k 7.c $4$ $37.171$ \(\Q(\sqrt{-3}, \sqrt{-5})\) None \(-4\) \(0\) \(-10\) \(-20\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\beta _{2})q^{2}-4\beta _{2}q^{4}+(-5+5\beta _{2}+\cdots)q^{5}+\cdots\)
630.4.k.l 630.k 7.c $4$ $37.171$ \(\Q(\sqrt{-3}, \sqrt{46})\) None \(-4\) \(0\) \(10\) \(6\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2-2\beta _{2})q^{2}+4\beta _{2}q^{4}+(5+5\beta _{2}+\cdots)q^{5}+\cdots\)
630.4.k.m 630.k 7.c $4$ $37.171$ \(\Q(\sqrt{-3}, \sqrt{295})\) None \(4\) \(0\) \(-10\) \(24\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2+2\beta _{2})q^{2}+4\beta _{2}q^{4}+(-5-5\beta _{2}+\cdots)q^{5}+\cdots\)
630.4.k.n 630.k 7.c $4$ $37.171$ \(\Q(\sqrt{-3}, \sqrt{46})\) None \(4\) \(0\) \(10\) \(-6\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\beta _{2}q^{2}+(-4-4\beta _{2})q^{4}-5\beta _{2}q^{5}+\cdots\)
630.4.k.o 630.k 7.c $6$ $37.171$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-6\) \(0\) \(15\) \(-22\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\beta _{2})q^{2}-4\beta _{2}q^{4}+(5-5\beta _{2}+\cdots)q^{5}+\cdots\)
630.4.k.p 630.k 7.c $6$ $37.171$ 6.0.\(\cdots\).1 None \(-6\) \(0\) \(15\) \(-13\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2-2\beta _{3})q^{2}+4\beta _{3}q^{4}+(5+5\beta _{3}+\cdots)q^{5}+\cdots\)
630.4.k.q 630.k 7.c $6$ $37.171$ 6.0.\(\cdots\).1 None \(6\) \(0\) \(-15\) \(-13\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\beta _{3}q^{2}+(-4-4\beta _{3})q^{4}+5\beta _{3}q^{5}+\cdots\)
630.4.k.r 630.k 7.c $6$ $37.171$ 6.0.\(\cdots\).2 None \(6\) \(0\) \(15\) \(14\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\beta _{3})q^{2}-4\beta _{3}q^{4}+(5-5\beta _{3}+\cdots)q^{5}+\cdots\)
630.4.k.s 630.k 7.c $10$ $37.171$ \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None \(-10\) \(0\) \(-25\) \(-13\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\beta _{3})q^{2}-4\beta _{3}q^{4}+(-5+5\beta _{3}+\cdots)q^{5}+\cdots\)
630.4.k.t 630.k 7.c $10$ $37.171$ \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None \(10\) \(0\) \(25\) \(-13\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\beta _{3})q^{2}-4\beta _{3}q^{4}+(5-5\beta _{3}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(630, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(630, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 2}\)