Space of modular forms of level 75, weight 9, and Character 75.c

• Label: 75.9.c
• Level: $75$
• Weight: $9$
• Character orbit: 75.c
• Rep. character: $\chi_{75}(26,\cdot)$
• Character field: $\Q$
• Dimension: $48$
• Newform subspaces: $8$
• Sturm bound: $90$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	75.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 3 \)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(90\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(75, [\chi])\).

	Total	New	Old
Modular forms	86	54	32
Cusp forms	74	48	26
Eisenstein series	12	6	6

Trace form

\( 48 q + 22 q^{3} - 5648 q^{4} + 1998 q^{6} - 3656 q^{7} - 690 q^{9} + O(q^{10}) \)

Decomposition of \(S_{9}^{\mathrm{new}}(75, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
75.9.c.a	$75$	$9$	75.c	3.b	$2$	$1$	$1$	$30.553$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$0$	\(0\)	\(-81\)	\(0\)	\(4273\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-3^{4}q^{3}+2^{8}q^{4}+4273q^{7}+3^{8}q^{9}+\cdots\)
75.9.c.b	$75$	$9$	75.c	3.b	$2$	$1$	$1$	$30.553$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$0$	\(0\)	\(81\)	\(0\)	\(-4273\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+3^{4}q^{3}+2^{8}q^{4}-4273q^{7}+3^{8}q^{9}+\cdots\)
75.9.c.c	$75$	$9$	75.c	3.b	$2$	$2$	$2$	$30.553$	\(\Q(\sqrt{-14}) \)	None		$2$	$0$	\(0\)	\(-90\)	\(0\)	\(3500\)	$2\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{2}+(-45-3\beta )q^{3}-248q^{4}+\cdots\)
75.9.c.d	$75$	$9$	75.c	3.b	$2$	$2$	$2$	$30.553$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-15}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+17iq^{2}-3^{4}iq^{3}-33q^{4}+1377q^{6}+\cdots\)
75.9.c.e	$75$	$9$	75.c	3.b	$2$	$10$	$10$	$30.553$	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)	None		$2$	$0$	\(0\)	\(-25\)	\(0\)	\(1960\)	$2^{8}\cdot 3^{11}\cdot 5^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}+(-2-\beta _{2}+\beta _{4})q^{3}+(-155+\cdots)q^{4}+\cdots\)
75.9.c.f	$75$	$9$	75.c	3.b	$2$	$10$	$10$	$30.553$	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)	None		$2$	$0$	\(0\)	\(25\)	\(0\)	\(-1960\)	$2^{8}\cdot 3^{11}\cdot 5^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}+(2-\beta _{2}-\beta _{4})q^{3}+(-155+\cdots)q^{4}+\cdots\)
75.9.c.g	$75$	$9$	75.c	3.b	$2$	$10$	$10$	$30.553$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None		$2$	$0$	\(0\)	\(112\)	\(0\)	\(-7156\)	$2^{5}\cdot 3^{11}\cdot 5^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(11+2\beta _{1}+\beta _{2})q^{3}+(-79+\cdots)q^{4}+\cdots\)
75.9.c.h	$75$	$9$	75.c	3.b	$2$	$12$	$12$	$30.553$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{16}\cdot 3^{18}\cdot 5^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{2}+(\beta _{2}-\beta _{3})q^{3}+(-142-\beta _{5}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{9}^{\mathrm{old}}(75, [\chi])\) into lower level spaces

\( S_{9}^{\mathrm{old}}(75, [\chi]) \cong \) \(S_{9}^{\mathrm{new}}(3, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 2}\)