Space of modular forms of level 325, weight 4, and Character 325.c

• Label: 325.4.c
• Level: $325$
• Weight: $4$
• Character orbit: 325.c
• Rep. character: $\chi_{325}(51,\cdot)$
• Character field: $\Q$
• Dimension: $64$
• Newform subspaces: $7$
• Sturm bound: $140$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	325.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(140\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(325, [\chi])\).

	Total	New	Old
Modular forms	110	70	40
Cusp forms	98	64	34
Eisenstein series	12	6	6

Trace form

\( 64 q + 2 q^{3} - 246 q^{4} + 574 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(325, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
325.4.c.a	$325$	$4$	325.c	13.b	$2$	$2$	$2$	$19.176$	\(\Q(\sqrt{-1}) \)	None		$2$	$1$	\(0\)	\(-4\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{2}-2q^{3}+7q^{4}-2iq^{6}-24iq^{7}+\cdots\)
325.4.c.b	$325$	$4$	325.c	13.b	$2$	$2$	$2$	$19.176$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{2}+q^{3}-q^{4}+iq^{6}-5iq^{7}+\cdots\)
325.4.c.c	$325$	$4$	325.c	13.b	$2$	$2$	$2$	$19.176$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{2}+2q^{3}+7q^{4}+2iq^{6}-24iq^{7}+\cdots\)
325.4.c.d	$325$	$4$	325.c	13.b	$2$	$14$	$14$	$19.176$	\(\mathbb{Q}[x]/(x^{14} + \cdots)\)	None		$2$	$0$	\(0\)	\(-12\)	\(0\)	\(0\)	$2^{9}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-1-\beta _{3})q^{3}+(-4+\beta _{2}+\cdots)q^{4}+\cdots\)
325.4.c.e	$325$	$4$	325.c	13.b	$2$	$14$	$14$	$19.176$	\(\mathbb{Q}[x]/(x^{14} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{2}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+\beta _{4}q^{3}+(-4+\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{6}+\cdots\)
325.4.c.f	$325$	$4$	325.c	13.b	$2$	$14$	$14$	$19.176$	\(\mathbb{Q}[x]/(x^{14} + \cdots)\)	None		$2$	$0$	\(0\)	\(12\)	\(0\)	\(0\)	$2^{9}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(1+\beta _{3})q^{3}+(-4+\beta _{2})q^{4}+\cdots\)
325.4.c.g	$325$	$4$	325.c	13.b	$2$	$16$	$16$	$19.176$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{14}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{9}q^{2}+\beta _{10}q^{3}+(-6-\beta _{3})q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(325, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(325, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 2}\)