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

• Label: 576.7.e
• Level: $576$
• Weight: $7$
• Character orbit: 576.e
• Rep. character: $\chi_{576}(449,\cdot)$
• Character field: $\Q$
• Dimension: $48$
• 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.e (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 3 \)
Character field:			\(\Q\)
Newform subspaces:			\( 18 \)
Sturm bound:			\(672\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

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

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

Trace form

\( 48 q + O(q^{10}) \)

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	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
576.7.e.a	$576$	$7$	576.e	3.b	$2$	$2$	$2$	$132.511$	\(\Q(\sqrt{-2}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-1048\)	$3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+5\beta q^{5}-524q^{7}+68\beta q^{11}-344q^{13}+\cdots\)
576.7.e.b	$576$	$7$	576.e	3.b	$2$	$2$	$2$	$132.511$	\(\Q(\sqrt{-2}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-968\)	$3$	$\mathrm{SU}(2)[C_{2}]$	\(q+41\beta q^{5}-22^{2}q^{7}+316\beta q^{11}-3368q^{13}+\cdots\)
576.7.e.c	$576$	$7$	576.e	3.b	$2$	$2$	$2$	$132.511$	\(\Q(\sqrt{-2}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-488\)	$3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+5\beta q^{5}-244q^{7}+188\beta q^{11}+2728q^{13}+\cdots\)
576.7.e.d	$576$	$7$	576.e	3.b	$2$	$2$	$2$	$132.511$	\(\Q(\sqrt{-2}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-176\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+119\beta q^{5}-88q^{7}-872\beta q^{11}-472q^{13}+\cdots\)
576.7.e.e	$576$	$7$	576.e	3.b	$2$	$2$	$2$	$132.511$	\(\Q(\sqrt{-2}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-120\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+5\beta q^{5}-60q^{7}-236\beta q^{11}-1192q^{13}+\cdots\)
576.7.e.f	$576$	$7$	576.e	3.b	$2$	$2$	$2$	$132.511$	\(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+73\beta q^{5}-1656q^{13}+4393\beta q^{17}+\cdots\)
576.7.e.g	$576$	$7$	576.e	3.b	$2$	$2$	$2$	$132.511$	\(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+161\beta q^{5}+1656q^{13}-5383\beta q^{17}+\cdots\)
576.7.e.h	$576$	$7$	576.e	3.b	$2$	$2$	$2$	$132.511$	\(\Q(\sqrt{-2}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(120\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+5\beta q^{5}+60q^{7}+236\beta q^{11}-1192q^{13}+\cdots\)
576.7.e.i	$576$	$7$	576.e	3.b	$2$	$2$	$2$	$132.511$	\(\Q(\sqrt{-2}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(176\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+119\beta q^{5}+88q^{7}+872\beta q^{11}-472q^{13}+\cdots\)
576.7.e.j	$576$	$7$	576.e	3.b	$2$	$2$	$2$	$132.511$	\(\Q(\sqrt{-2}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(488\)	$3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+5\beta q^{5}+244q^{7}-188\beta q^{11}+2728q^{13}+\cdots\)
576.7.e.k	$576$	$7$	576.e	3.b	$2$	$2$	$2$	$132.511$	\(\Q(\sqrt{-2}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(968\)	$3$	$\mathrm{SU}(2)[C_{2}]$	\(q+41\beta q^{5}+22^{2}q^{7}-316\beta q^{11}-3368q^{13}+\cdots\)
576.7.e.l	$576$	$7$	576.e	3.b	$2$	$2$	$2$	$132.511$	\(\Q(\sqrt{-2}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(1048\)	$3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+5\beta q^{5}+524q^{7}-68\beta q^{11}-344q^{13}+\cdots\)
576.7.e.m	$576$	$7$	576.e	3.b	$2$	$4$	$4$	$132.511$	\(\Q(\sqrt{-2}, \sqrt{145})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-432\)	$2^{7}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-5^{2}\beta _{1}-\beta _{3})q^{5}+(-108-\beta _{2}+\cdots)q^{7}+\cdots\)
576.7.e.n	$576$	$7$	576.e	3.b	$2$	$4$	$4$	$132.511$	\(\Q(\sqrt{-2}, \sqrt{-41})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-32\)	$2^{10}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+65\beta _{1}q^{5}+(-8+\beta _{2})q^{7}+(-520\beta _{1}+\cdots)q^{11}+\cdots\)
576.7.e.o	$576$	$7$	576.e	3.b	$2$	$4$	$4$	$132.511$	\(\Q(\sqrt{-2}, \sqrt{3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{9}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-55\beta _{1}q^{5}+\beta _{3}q^{7}-7\beta _{2}q^{11}-504q^{13}+\cdots\)
576.7.e.p	$576$	$7$	576.e	3.b	$2$	$4$	$4$	$132.511$	\(\Q(\sqrt{-2}, \sqrt{7})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{9}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+95\beta _{1}q^{5}+\beta _{3}q^{7}-\beta _{2}q^{11}+2040q^{13}+\cdots\)
576.7.e.q	$576$	$7$	576.e	3.b	$2$	$4$	$4$	$132.511$	\(\Q(\sqrt{-2}, \sqrt{-41})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(32\)	$2^{10}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+65\beta _{1}q^{5}+(8-\beta _{2})q^{7}+(520\beta _{1}-\beta _{3})q^{11}+\cdots\)
576.7.e.r	$576$	$7$	576.e	3.b	$2$	$4$	$4$	$132.511$	\(\Q(\sqrt{-2}, \sqrt{145})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(432\)	$2^{7}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-5^{2}\beta _{1}-\beta _{3})q^{5}+(108+\beta _{2})q^{7}+\cdots\)

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

\( S_{7}^{\mathrm{old}}(576, [\chi]) \cong \) \(S_{7}^{\mathrm{new}}(3, [\chi])\)\(^{\oplus 14}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(6, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 7}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(144, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(288, [\chi])\)\(^{\oplus 2}\)