Space of modular forms of level 325, weight 2, and Character 325.b

• Label: 325.2.b
• Level: $325$
• Weight: $2$
• Character orbit: 325.b
• Rep. character: $\chi_{325}(274,\cdot)$
• Character field: $\Q$
• Dimension: $18$
• Newform subspaces: $6$
• Sturm bound: $70$
• Trace bound: $6$

Defining parameters

Level:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	325.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(70\)
Trace bound:			\(6\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(325, [\chi])\).

	Total	New	Old
Modular forms	42	18	24
Cusp forms	30	18	12
Eisenstein series	12	0	12

Trace form

18 * q - 10 * q^4 - 12 * q^6 - 14 * q^9 + 12 * q^11 - 4 * q^14 + 2 * q^16 + 8 * q^19 - 12 * q^21 + 8 * q^24 + 6 * q^26 + 28 * q^29 + 8 * q^31 + 18 * q^36 - 8 * q^39 - 76 * q^44 - 28 * q^46 - 22 * q^49 + 92 * q^54 - 44 * q^56 - 48 * q^59 - 44 * q^61 + 70 * q^64 - 8 * q^66 + 28 * q^69 + 52 * q^71 + 4 * q^74 - 40 * q^76 - 12 * q^79 - 30 * q^81 + 80 * q^84 + 84 * q^86 + 16 * q^89 + 4 * q^94 + 32 * q^96 - 76 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(325, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
325.2.b.a	$325$	$2$	325.b	5.b	$2$	$2$	$2$	$2.595$	\(\Q(\sqrt{-1}) \)	None		✓			325.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{2}-i q^{3}-2 q^{4}+2 q^{6}+2 i q^{7}+\cdots\)
325.2.b.b	$325$	$2$	325.b	5.b	$2$	$2$	$2$	$2.595$	\(\Q(\sqrt{-1}) \)	None					65.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}-2 i q^{3}+q^{4}+2 q^{6}+4 i q^{7}+\cdots\)
325.2.b.c	$325$	$2$	325.b	5.b	$2$	$2$	$2$	$2.595$	\(\Q(\sqrt{-1}) \)	None		✓			325.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}+2 q^{4}+4 i q^{7}+2 q^{9}-6 q^{11}+\cdots\)
325.2.b.d	$325$	$2$	325.b	5.b	$2$	$4$	$4$	$2.595$	\(\Q(\zeta_{8})\)	None		✓			325.2.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta_{2}+\beta_1)q^{2}+2\beta_{2} q^{3}+(-2\beta_{3}-1)q^{4}+\cdots\)
325.2.b.e	$325$	$2$	325.b	5.b	$2$	$4$	$4$	$2.595$	\(\Q(\zeta_{12})\)	None					65.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta_{2} q^{2}+(-\beta_{2}-\beta_1)q^{3}-q^{4}+\cdots\)
325.2.b.f	$325$	$2$	325.b	5.b	$2$	$4$	$4$	$2.595$	\(\Q(\zeta_{8})\)	None					65.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta_{2}+\beta_1)q^{2}-\beta_{2} q^{3}+(-2\beta_{3}-1)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(325, [\chi])\) into lower level spaces

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