Space of modular forms of level 475, weight 4, and Character 475.b

• Label: 475.4.b
• Level: $475$
• Weight: $4$
• Character orbit: 475.b
• Rep. character: $\chi_{475}(324,\cdot)$
• Character field: $\Q$
• Dimension: $80$
• Newform subspaces: $11$
• Sturm bound: $200$
• Trace bound: $6$

Defining parameters

Level:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	475.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(200\)
Trace bound:			\(6\)
Distinguishing \(T_p\):			\(2\), \(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(475, [\chi])\).

	Total	New	Old
Modular forms	156	80	76
Cusp forms	144	80	64
Eisenstein series	12	0	12

Trace form

\( 80 q - 312 q^{4} - 8 q^{6} - 756 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(475, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
475.4.b.a	$475$	$4$	475.b	5.b	$2$	$2$	$2$	$28.026$	\(\Q(\sqrt{-1}) \)	None		$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+5iq^{2}-4iq^{3}-17q^{4}+20q^{6}+\cdots\)
475.4.b.b	$475$	$4$	475.b	5.b	$2$	$2$	$2$	$28.026$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3iq^{2}+5iq^{3}-q^{4}-15q^{6}-iq^{7}+\cdots\)
475.4.b.c	$475$	$4$	475.b	5.b	$2$	$2$	$2$	$28.026$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3iq^{2}-5iq^{3}-q^{4}+15q^{6}-11iq^{7}+\cdots\)
475.4.b.d	$475$	$4$	475.b	5.b	$2$	$2$	$2$	$28.026$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3iq^{2}-7iq^{3}-q^{4}+21q^{6}+11iq^{7}+\cdots\)
475.4.b.e	$475$	$4$	475.b	5.b	$2$	$2$	$2$	$28.026$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-2iq^{3}+8q^{4}-11iq^{7}+11q^{9}+\cdots\)
475.4.b.f	$475$	$4$	475.b	5.b	$2$	$6$	$6$	$28.026$	6.0.158155776.1	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{2}+\beta _{4}-\beta _{5})q^{2}+(2\beta _{4}-\beta _{5})q^{3}+\cdots\)
475.4.b.g	$475$	$4$	475.b	5.b	$2$	$6$	$6$	$28.026$	6.0.27206656.1	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{5}q^{2}+(\beta _{2}-\beta _{4}-\beta _{5})q^{3}+(-2\beta _{1}+\cdots)q^{4}+\cdots\)
475.4.b.h	$475$	$4$	475.b	5.b	$2$	$10$	$10$	$28.026$	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{7}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{5}q^{2}+(\beta _{5}+\beta _{7}+\beta _{8})q^{3}+(-3+\cdots)q^{4}+\cdots\)
475.4.b.i	$475$	$4$	475.b	5.b	$2$	$12$	$12$	$28.026$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{7}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-\beta _{1}-\beta _{7}-\beta _{8})q^{3}+(-4+\cdots)q^{4}+\cdots\)
475.4.b.j	$475$	$4$	475.b	5.b	$2$	$18$	$18$	$28.026$	\(\mathbb{Q}[x]/(x^{18} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$5^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{10}q^{2}+(\beta _{10}-\beta _{12})q^{3}+(-5+\beta _{2}+\cdots)q^{4}+\cdots\)
475.4.b.k	$475$	$4$	475.b	5.b	$2$	$18$	$18$	$28.026$	\(\mathbb{Q}[x]/(x^{18} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$5^{6}\cdot 23^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{10}q^{2}+\beta _{12}q^{3}+(-5-\beta _{4})q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(475, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(475, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 2}\)