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Properties

Label 560.4.q
Level $560$
Weight $4$
Character orbit 560.q
Rep. character $\chi_{560}(81,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $96$
Newform subspaces $18$
Sturm bound $384$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 560.q (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 18 \)
Sturm bound: \(384\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(560, [\chi])\).

Total New Old
Modular forms 600 96 504
Cusp forms 552 96 456
Eisenstein series 48 0 48

Trace form

\( 96 q + 12 q^{3} - 36 q^{7} - 432 q^{9} + O(q^{10}) \) \( 96 q + 12 q^{3} - 36 q^{7} - 432 q^{9} + 20 q^{11} + 180 q^{19} - 68 q^{21} + 164 q^{23} - 1200 q^{25} - 864 q^{27} - 344 q^{29} - 240 q^{31} + 8 q^{37} + 696 q^{39} - 296 q^{41} - 1480 q^{43} - 220 q^{45} - 940 q^{47} + 464 q^{49} + 544 q^{51} + 16 q^{53} + 880 q^{55} + 336 q^{57} + 344 q^{59} + 200 q^{61} - 1336 q^{63} - 140 q^{65} - 264 q^{67} - 4112 q^{69} + 160 q^{71} + 216 q^{73} + 300 q^{75} + 1496 q^{77} - 552 q^{79} - 4196 q^{81} - 3536 q^{83} - 4644 q^{87} + 644 q^{89} - 4232 q^{91} - 112 q^{93} + 3168 q^{97} + 4488 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(560, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
560.4.q.a 560.q 7.c $2$ $33.041$ \(\Q(\sqrt{-3}) \) None 140.4.i.a \(0\) \(-2\) \(-5\) \(-20\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\zeta_{6})q^{3}-5\zeta_{6}q^{5}+(-1-18\zeta_{6})q^{7}+\cdots\)
560.4.q.b 560.q 7.c $2$ $33.041$ \(\Q(\sqrt{-3}) \) None 35.4.e.a \(0\) \(-2\) \(-5\) \(28\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\zeta_{6})q^{3}-5\zeta_{6}q^{5}+(7+14\zeta_{6})q^{7}+\cdots\)
560.4.q.c 560.q 7.c $2$ $33.041$ \(\Q(\sqrt{-3}) \) None 140.4.i.b \(0\) \(-2\) \(5\) \(28\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\zeta_{6})q^{3}+5\zeta_{6}q^{5}+(21-14\zeta_{6})q^{7}+\cdots\)
560.4.q.d 560.q 7.c $2$ $33.041$ \(\Q(\sqrt{-3}) \) None 70.4.e.c \(0\) \(-1\) \(5\) \(-17\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{3}+5\zeta_{6}q^{5}+(1-19\zeta_{6})q^{7}+\cdots\)
560.4.q.e 560.q 7.c $2$ $33.041$ \(\Q(\sqrt{-3}) \) None 70.4.e.a \(0\) \(1\) \(5\) \(-35\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}+5\zeta_{6}q^{5}+(-21+7\zeta_{6})q^{7}+\cdots\)
560.4.q.f 560.q 7.c $2$ $33.041$ \(\Q(\sqrt{-3}) \) None 280.4.q.a \(0\) \(7\) \(5\) \(-7\) $\mathrm{SU}(2)[C_{3}]$ \(q+(7-7\zeta_{6})q^{3}+5\zeta_{6}q^{5}+(7-21\zeta_{6})q^{7}+\cdots\)
560.4.q.g 560.q 7.c $2$ $33.041$ \(\Q(\sqrt{-3}) \) None 70.4.e.b \(0\) \(10\) \(5\) \(-28\) $\mathrm{SU}(2)[C_{3}]$ \(q+(10-10\zeta_{6})q^{3}+5\zeta_{6}q^{5}+(-21+\cdots)q^{7}+\cdots\)
560.4.q.h 560.q 7.c $4$ $33.041$ \(\Q(\sqrt{-3}, \sqrt{46})\) None 140.4.i.e \(0\) \(-6\) \(-10\) \(6\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-3+\beta _{1}-3\beta _{2})q^{3}+5\beta _{2}q^{5}+(-1+\cdots)q^{7}+\cdots\)
560.4.q.i 560.q 7.c $4$ $33.041$ \(\Q(\sqrt{2}, \sqrt{-3})\) None 35.4.e.b \(0\) \(-2\) \(-10\) \(-22\) $\mathrm{SU}(2)[C_{3}]$ \(q+(3\beta _{1}+\beta _{2}+3\beta _{3})q^{3}+(-5-5\beta _{2}+\cdots)q^{5}+\cdots\)
560.4.q.j 560.q 7.c $4$ $33.041$ \(\Q(\sqrt{-3}, \sqrt{46})\) None 70.4.e.d \(0\) \(-2\) \(-10\) \(-6\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\beta _{1}-\beta _{2})q^{3}+5\beta _{2}q^{5}+(1+\cdots)q^{7}+\cdots\)
560.4.q.k 560.q 7.c $4$ $33.041$ \(\Q(\sqrt{-3}, \sqrt{37})\) None 140.4.i.d \(0\) \(0\) \(10\) \(-36\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{2}q^{3}+(5-5\beta _{1})q^{5}+(-8-2\beta _{1}+\cdots)q^{7}+\cdots\)
560.4.q.l 560.q 7.c $4$ $33.041$ \(\Q(\sqrt{-3}, \sqrt{22})\) None 140.4.i.c \(0\) \(10\) \(-10\) \(54\) $\mathrm{SU}(2)[C_{3}]$ \(q+(5+\beta _{1}+5\beta _{2})q^{3}+5\beta _{2}q^{5}+(15+\cdots)q^{7}+\cdots\)
560.4.q.m 560.q 7.c $6$ $33.041$ 6.0.\(\cdots\).2 None 70.4.e.e \(0\) \(4\) \(-15\) \(-14\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{1}+\beta _{3})q^{3}+(-5+5\beta _{3})q^{5}+(1+\cdots)q^{7}+\cdots\)
560.4.q.n 560.q 7.c $10$ $33.041$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None 35.4.e.c \(0\) \(-8\) \(25\) \(62\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{3}+\beta _{4}-2\beta _{5})q^{3}+(5-5\beta _{5})q^{5}+\cdots\)
560.4.q.o 560.q 7.c $10$ $33.041$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None 280.4.q.b \(0\) \(6\) \(-25\) \(-28\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\beta _{3}-\beta _{7})q^{3}-5\beta _{3}q^{5}+(-3+\cdots)q^{7}+\cdots\)
560.4.q.p 560.q 7.c $12$ $33.041$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 280.4.q.e \(0\) \(-8\) \(30\) \(14\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1-\beta _{1}+\beta _{3})q^{3}+5\beta _{3}q^{5}+(\beta _{1}+\cdots)q^{7}+\cdots\)
560.4.q.q 560.q 7.c $12$ $33.041$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None 280.4.q.d \(0\) \(0\) \(-30\) \(-16\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{5}q^{3}+(-5+5\beta _{2})q^{5}+(-3+4\beta _{2}+\cdots)q^{7}+\cdots\)
560.4.q.r 560.q 7.c $12$ $33.041$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 280.4.q.c \(0\) \(7\) \(30\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\beta _{3}+\beta _{5})q^{3}+5\beta _{3}q^{5}+(1-\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(560, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(560, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(140, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(280, [\chi])\)\(^{\oplus 2}\)