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 + 652 * q^11 + 2588 * q^14 + 16768 * q^16 - 7608 * q^19 - 11540 * q^21 + 1100 * q^24 + 4056 * q^26 - 15004 * q^29 + 28632 * q^31 + 76760 * q^34 + 116936 * q^36 - 12168 * q^39 + 4816 * q^41 - 19464 * q^44 - 23476 * q^46 - 146134 * q^49 - 39648 * q^51 + 21784 * q^54 - 208076 * q^56 - 80384 * q^59 + 69100 * q^61 - 203848 * q^64 - 159340 * q^66 + 286500 * q^69 + 149876 * q^71 - 433092 * q^74 + 399380 * q^76 + 308868 * q^79 + 528618 * q^81 - 250904 * q^84 - 242380 * q^86 - 29296 * q^89 + 140608 * q^91 - 644140 * q^94 + 1068020 * q^96 + 450724 * q^99

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	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+5 i q^{2}-6 i q^{3}+7 q^{4}+30 q^{6}+\cdots\)
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

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