Space of modular forms of level 75, weight 7, and Character 75.c

• Label: 75.7.c
• Level: $75$
• Weight: $7$
• Character orbit: 75.c
• Rep. character: $\chi_{75}(26,\cdot)$
• Character field: $\Q$
• Dimension: $35$
• Newform subspaces: $6$
• Sturm bound: $70$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	75.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 3 \)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(70\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(75, [\chi])\).

	Total	New	Old
Modular forms	66	41	25
Cusp forms	54	35	19
Eisenstein series	12	6	6

Trace form

\( 35 q + 7 q^{3} - 1088 q^{4} - 354 q^{6} + 126 q^{7} + 651 q^{9} + O(q^{10}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(75, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
75.7.c.a	$75$	$7$	75.c	3.b	$2$	$1$	$1$	$17.254$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$0$	\(0\)	\(27\)	\(0\)	\(286\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+3^{3}q^{3}+2^{6}q^{4}+286q^{7}+3^{6}q^{9}+\cdots\)
75.7.c.b	$75$	$7$	75.c	3.b	$2$	$2$	$2$	$17.254$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-15}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+11iq^{2}-3^{3}iq^{3}-57q^{4}+297q^{6}+\cdots\)
75.7.c.c	$75$	$7$	75.c	3.b	$2$	$8$	$8$	$17.254$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(0\)	\(-20\)	\(0\)	\(-160\)	$2^{2}\cdot 3^{5}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}+(-3+\beta _{3})q^{3}+(-40-\beta _{3}+\cdots)q^{4}+\cdots\)
75.7.c.d	$75$	$7$	75.c	3.b	$2$	$8$	$8$	$17.254$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		$2$	$0$	\(0\)	\(-10\)	\(0\)	\(280\)	$2^{3}\cdot 3^{5}\cdot 5^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-1-\beta _{3})q^{3}+(-40+\beta _{2}+\cdots)q^{4}+\cdots\)
75.7.c.e	$75$	$7$	75.c	3.b	$2$	$8$	$8$	$17.254$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}\cdot 3^{6}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+(-\beta _{1}+\beta _{2})q^{3}+(-9-\beta _{3}+\cdots)q^{4}+\cdots\)
75.7.c.f	$75$	$7$	75.c	3.b	$2$	$8$	$8$	$17.254$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		$2$	$0$	\(0\)	\(10\)	\(0\)	\(-280\)	$2^{3}\cdot 3^{5}\cdot 5^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(1-\beta _{3})q^{3}+(-40+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{7}^{\mathrm{old}}(75, [\chi])\) into lower level spaces

\( S_{7}^{\mathrm{old}}(75, [\chi]) \cong \) \(S_{7}^{\mathrm{new}}(3, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 2}\)