Space of modular forms of level 75, weight 22, and Character 75.b

• Label: 75.22.b
• Level: $75$
• Weight: $22$
• Character orbit: 75.b
• Rep. character: $\chi_{75}(49,\cdot)$
• Character field: $\Q$
• Dimension: $64$
• Newform subspaces: $10$
• Sturm bound: $220$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	75.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(220\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{22}(75, [\chi])\).

	Total	New	Old
Modular forms	216	64	152
Cusp forms	204	64	140
Eisenstein series	12	0	12

Trace form

\( 64 q - 66781932 q^{4} + 31414068 q^{6} - 223154201664 q^{9} + O(q^{10}) \)

Decomposition of \(S_{22}^{\mathrm{new}}(75, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
75.22.b.a	$75$	$22$	75.b	5.b	$2$	$2$	$2$	$209.608$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2844iq^{2}-3^{10}iq^{3}-5991184q^{4}+\cdots\)
75.22.b.b	$75$	$22$	75.b	5.b	$2$	$2$	$2$	$209.608$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+12^{3}iq^{2}+3^{10}iq^{3}-888832q^{4}+\cdots\)
75.22.b.c	$75$	$22$	75.b	5.b	$2$	$2$	$2$	$209.608$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+544iq^{2}-3^{10}iq^{3}+1801216q^{4}+\cdots\)
75.22.b.d	$75$	$22$	75.b	5.b	$2$	$4$	$4$	$209.608$	\(\Q(i, \sqrt{649})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{4}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(333\beta _{1}+\beta _{2})q^{2}-3^{10}\beta _{1}q^{3}+(-589618+\cdots)q^{4}+\cdots\)
75.22.b.e	$75$	$22$	75.b	5.b	$2$	$6$	$6$	$209.608$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{2}\cdot 5^{2}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}+268\beta _{2})q^{2}-3^{10}\beta _{2}q^{3}+(-2238318+\cdots)q^{4}+\cdots\)
75.22.b.f	$75$	$22$	75.b	5.b	$2$	$6$	$6$	$209.608$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}\cdot 3^{8}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-234\beta _{1}+\beta _{2})q^{2}-3^{10}\beta _{1}q^{3}+\cdots\)
75.22.b.g	$75$	$22$	75.b	5.b	$2$	$6$	$6$	$209.608$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}\cdot 3^{4}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(767\beta _{1}+\beta _{4})q^{2}+3^{10}\beta _{1}q^{3}+(-421908+\cdots)q^{4}+\cdots\)
75.22.b.h	$75$	$22$	75.b	5.b	$2$	$8$	$8$	$209.608$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}\cdot 3^{8}\cdot 5^{4}\cdot 7^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}+224\beta _{5})q^{2}-3^{10}\beta _{5}q^{3}+(290854+\cdots)q^{4}+\cdots\)
75.22.b.i	$75$	$22$	75.b	5.b	$2$	$12$	$12$	$209.608$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{39}\cdot 3^{12}\cdot 5^{14}\cdot 7^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-15\beta _{5}+\beta _{6})q^{2}-3^{10}\beta _{5}q^{3}+(-731822+\cdots)q^{4}+\cdots\)
75.22.b.j	$75$	$22$	75.b	5.b	$2$	$16$	$16$	$209.608$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{46}\cdot 3^{20}\cdot 5^{28}\cdot 7^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(83\beta _{8}+\beta _{9})q^{2}+3^{10}\beta _{8}q^{3}+(-1335201+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{22}^{\mathrm{old}}(75, [\chi])\) into lower level spaces

\( S_{22}^{\mathrm{old}}(75, [\chi]) \cong \) \(S_{22}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 2}\)