Space of modular forms of level 325, weight 6, and Character 325.b

• Label: 325.6.b
• Level: $325$
• Weight: $6$
• Character orbit: 325.b
• Rep. character: $\chi_{325}(274,\cdot)$
• Character field: $\Q$
• Dimension: $90$
• Newform subspaces: $9$
• Sturm bound: $210$
• Trace bound: $6$

Defining parameters

Level:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	325.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(210\)
Trace bound:			\(6\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(325, [\chi])\).

	Total	New	Old
Modular forms	182	90	92
Cusp forms	170	90	80
Eisenstein series	12	0	12

Trace form

\( 90 q - 1400 q^{4} + 664 q^{6} - 8174 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(325, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
325.6.b.a	$325$	$6$	325.b	5.b	$2$	$2$	$2$	$52.125$	\(\Q(\sqrt{-1}) \)	None					65.6.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+5iq^{2}-6iq^{3}+7q^{4}+30q^{6}-244iq^{7}+\cdots\)
325.6.b.b	$325$	$6$	325.b	5.b	$2$	$4$	$4$	$52.125$	\(\Q(i, \sqrt{17})\)	None					13.6.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}+3\beta _{2})q^{2}+(6\beta _{1}-11\beta _{2})q^{3}+\cdots\)
325.6.b.c	$325$	$6$	325.b	5.b	$2$	$6$	$6$	$52.125$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None					13.6.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}+2\beta _{2})q^{2}+(\beta _{1}-3\beta _{2}-\beta _{5})q^{3}+\cdots\)
325.6.b.d	$325$	$6$	325.b	5.b	$2$	$6$	$6$	$52.125$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None					65.6.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}+\beta _{2})q^{2}+(-5\beta _{2}-\beta _{5})q^{3}+(4+\cdots)q^{4}+\cdots\)
325.6.b.e	$325$	$6$	325.b	5.b	$2$	$8$	$8$	$52.125$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None					65.6.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{7}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(2\beta _{1}+\beta _{2}-\beta _{3})q^{2}+(-\beta _{1}-\beta _{3}+\cdots)q^{3}+\cdots\)
325.6.b.f	$325$	$6$	325.b	5.b	$2$	$12$	$12$	$52.125$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None					65.6.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{11}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-\beta _{7}-3\beta _{8})q^{3}+(-23+\cdots)q^{4}+\cdots\)
325.6.b.g	$325$	$6$	325.b	5.b	$2$	$12$	$12$	$52.125$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None					65.6.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{14}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-\beta _{1}+6\beta _{3}-\beta _{7})q^{3}+(-22+\cdots)q^{4}+\cdots\)
325.6.b.h	$325$	$6$	325.b	5.b	$2$	$18$	$18$	$52.125$	\(\mathbb{Q}[x]/(x^{18} + \cdots)\)	None		✓			325.6.a.h	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{7}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}-\beta _{10})q^{2}+(\beta _{10}-\beta _{12})q^{3}+(-11+\cdots)q^{4}+\cdots\)
325.6.b.i	$325$	$6$	325.b	5.b	$2$	$22$	$22$	$52.125$		None		✓	✓		325.6.a.j	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{6}^{\mathrm{old}}(325, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(325, [\chi]) \simeq \) \(S_{6}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 2}\)