Space of modular forms of level 576, weight 7, and Character 576.g

• Label: 576.7.g
• Level: $576$
• Weight: $7$
• Character orbit: 576.g
• Rep. character: $\chi_{576}(127,\cdot)$
• Character field: $\Q$
• Dimension: $59$
• Newform subspaces: $18$
• Sturm bound: $672$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	576.g (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 4 \)
Character field:			\(\Q\)
Newform subspaces:			\( 18 \)
Sturm bound:			\(672\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(576, [\chi])\).

	Total	New	Old
Modular forms	600	61	539
Cusp forms	552	59	493
Eisenstein series	48	2	46

Trace form

59 * q - 2 * q^5 - 5038 * q^13 + 4890 * q^17 + 182169 * q^25 - 33202 * q^29 + 59042 * q^37 - 33398 * q^41 - 890773 * q^49 + 522942 * q^53 + 523058 * q^61 - 69292 * q^65 + 162502 * q^73 + 564224 * q^77 - 1886012 * q^85 + 1248810 * q^89 - 314762 * q^97

Decomposition of \(S_{7}^{\mathrm{new}}(576, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
576.7.g.a	$576$	$7$	576.g	4.b	$2$	$1$	$1$	$132.511$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				36.7.d.a	$2$	$0$	\(0\)	\(0\)	\(-88\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-88q^{5}-4070q^{13}-9776q^{17}+\cdots\)
576.7.g.b	$576$	$7$	576.g	4.b	$2$	$1$	$1$	$132.511$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				36.7.d.a	$2$	$0$	\(0\)	\(0\)	\(88\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+88q^{5}-4070q^{13}+9776q^{17}+\cdots\)
576.7.g.c	$576$	$7$	576.g	4.b	$2$	$1$	$1$	$132.511$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				16.7.c.a	$2$	$0$	\(0\)	\(0\)	\(234\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+234q^{5}+4070q^{13}+990q^{17}+\cdots\)
576.7.g.d	$576$	$7$	576.g	4.b	$2$	$2$	$2$	$132.511$	\(\Q(\sqrt{-3}) \)	None					16.7.c.b	$2$	$0$	\(0\)	\(0\)	\(-300\)	\(0\)	$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q-150 q^{5}-22\beta q^{7}+33\beta q^{11}+\cdots\)
576.7.g.e	$576$	$7$	576.g	4.b	$2$	$2$	$2$	$132.511$	\(\Q(\sqrt{-3}) \)	None					48.7.g.a	$2$	$0$	\(0\)	\(0\)	\(-180\)	\(0\)	$2^{3}\cdot 3^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-90 q^{5}+\beta q^{7}+9\beta q^{11}-1762 q^{13}+\cdots\)
576.7.g.f	$576$	$7$	576.g	4.b	$2$	$2$	$2$	$132.511$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)					144.7.g.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{2}\cdot 5$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta q^{7}+506 q^{13}+14\beta q^{19}-15625 q^{25}+\cdots\)
576.7.g.g	$576$	$7$	576.g	4.b	$2$	$2$	$2$	$132.511$	\(\Q(\sqrt{-3}) \)	None					48.7.g.b	$2$	$0$	\(0\)	\(0\)	\(12\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+6 q^{5}+29\beta q^{7}+93\beta q^{11}+2654 q^{13}+\cdots\)
576.7.g.h	$576$	$7$	576.g	4.b	$2$	$2$	$2$	$132.511$	\(\Q(\sqrt{-15}) \)	None					4.7.b.a	$2$	$0$	\(0\)	\(0\)	\(20\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+10q^{5}+10\beta q^{7}-31\beta q^{11}-1466q^{13}+\cdots\)
576.7.g.i	$576$	$7$	576.g	4.b	$2$	$2$	$2$	$132.511$	\(\Q(\sqrt{-1}) \)	None					32.7.c.a	$2$	$0$	\(0\)	\(0\)	\(100\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+50 q^{5}-46\beta q^{7}-527\beta q^{11}+\cdots\)
576.7.g.j	$576$	$7$	576.g	4.b	$2$	$2$	$2$	$132.511$	\(\Q(\sqrt{-3}) \)	None					48.7.g.c	$2$	$0$	\(0\)	\(0\)	\(300\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+150 q^{5}-47\beta q^{7}+213\beta q^{11}+\cdots\)
576.7.g.k	$576$	$7$	576.g	4.b	$2$	$4$	$4$	$132.511$	\(\Q(\zeta_{12})\)	None					96.7.g.a	$2$	$0$	\(0\)	\(0\)	\(-200\)	\(0\)	$2^{8}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-5\beta_{2}-50)q^{5}+(5\beta_{3}+73\beta_1)q^{7}+\cdots\)
576.7.g.l	$576$	$7$	576.g	4.b	$2$	$4$	$4$	$132.511$	\(\Q(i, \sqrt{6})\)	None					32.7.c.b	$2$	$0$	\(0\)	\(0\)	\(-56\)	\(0\)	$2^{17}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-14-\beta _{3})q^{5}+(-2\beta _{1}+5\beta _{2})q^{7}+\cdots\)
576.7.g.m	$576$	$7$	576.g	4.b	$2$	$4$	$4$	$132.511$	\(\Q(\sqrt{-3}, \sqrt{10})\)	None					144.7.g.g	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{16}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{5}-\beta _{2}q^{7}+\beta _{3}q^{11}-262q^{13}+\cdots\)
576.7.g.n	$576$	$7$	576.g	4.b	$2$	$4$	$4$	$132.511$	\(\Q(\sqrt{13}, \sqrt{-51})\)	None					36.7.d.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{18}$	$\mathrm{SU}(2)[C_{2}]$	\(q-5\beta _{1}q^{5}-\beta _{3}q^{7}-\beta _{2}q^{11}+2458q^{13}+\cdots\)
576.7.g.o	$576$	$7$	576.g	4.b	$2$	$6$	$6$	$132.511$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None					288.7.g.d	$2$	$0$	\(0\)	\(0\)	\(-312\)	\(0\)	$2^{25}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-52-\beta _{1})q^{5}+(\beta _{3}-\beta _{5})q^{7}+(\beta _{3}+\cdots)q^{11}+\cdots\)
576.7.g.p	$576$	$7$	576.g	4.b	$2$	$6$	$6$	$132.511$	6.0.50898483.1	None					12.7.d.a	$2$	$0$	\(0\)	\(0\)	\(-44\)	\(0\)	$2^{30}\cdot 3^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-7-\beta _{1})q^{5}+(2\beta _{2}-\beta _{3})q^{7}+(-2\beta _{3}+\cdots)q^{11}+\cdots\)
576.7.g.q	$576$	$7$	576.g	4.b	$2$	$6$	$6$	$132.511$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None					288.7.g.d	$2$	$0$	\(0\)	\(0\)	\(312\)	\(0\)	$2^{25}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(52+\beta _{1})q^{5}+(-\beta _{3}+\beta _{5})q^{7}+(\beta _{3}+\cdots)q^{11}+\cdots\)
576.7.g.r	$576$	$7$	576.g	4.b	$2$	$8$	$8$	$132.511$	8.0.\(\cdots\).5	None			✓		96.7.g.b	$2$	$0$	\(0\)	\(0\)	\(112\)	\(0\)	$2^{41}\cdot 3^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(14-\beta _{1})q^{5}+(2\beta _{2}-\beta _{3}-\beta _{5})q^{7}+\cdots\)

Decomposition of \(S_{7}^{\mathrm{old}}(576, [\chi])\) into lower level spaces

\( S_{7}^{\mathrm{old}}(576, [\chi]) \simeq \) \(S_{7}^{\mathrm{new}}(4, [\chi])\)\(^{\oplus 15}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(144, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(288, [\chi])\)\(^{\oplus 2}\)