Space of modular forms of level 325, weight 3, and Character 325.g

• Label: 325.3.g
• Level: $325$
• Weight: $3$
• Character orbit: 325.g
• Rep. character: $\chi_{325}(99,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $80$
• Newform subspaces: $6$
• Sturm bound: $105$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	325.g (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 65 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(105\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(325, [\chi])\).

	Total	New	Old
Modular forms	152	88	64
Cusp forms	128	80	48
Eisenstein series	24	8	16

Trace form

80 * q - 20 * q^6 - 208 * q^9 + 4 * q^11 + 40 * q^14 - 360 * q^16 + 96 * q^19 + 28 * q^21 - 8 * q^24 + 112 * q^26 + 272 * q^29 - 8 * q^31 - 52 * q^34 + 28 * q^39 + 212 * q^41 - 604 * q^44 + 244 * q^46 + 52 * q^54 - 160 * q^59 - 768 * q^61 + 128 * q^66 - 668 * q^71 + 432 * q^74 + 1112 * q^76 + 800 * q^79 + 208 * q^81 - 296 * q^84 - 492 * q^86 - 172 * q^89 + 48 * q^91 + 584 * q^94 - 456 * q^96 + 1708 * q^99

Decomposition of \(S_{3}^{\mathrm{new}}(325, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
325.3.g.a	$325$	$3$	325.g	65.g	$4$	$4$	$2$	$8.856$	\(\Q(i, \sqrt{10})\)	None					13.3.d.a	$2$	$0$	\(-4\)	\(0\)	\(0\)	\(12\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+\beta _{1}-\beta _{2})q^{2}+(\beta _{1}+\beta _{2}+\beta _{3})q^{3}+\cdots\)
325.3.g.b	$325$	$3$	325.g	65.g	$4$	$4$	$2$	$8.856$	\(\Q(i, \sqrt{10})\)	None					13.3.d.a	$2$	$0$	\(4\)	\(0\)	\(0\)	\(-12\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1+\beta _{1}+\beta _{2})q^{2}+(\beta _{1}-\beta _{2}+\beta _{3})q^{3}+\cdots\)
325.3.g.c	$325$	$3$	325.g	65.g	$4$	$16$	$8$	$8.856$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None					65.3.j.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-20\)	$2^{3}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{2}q^{2}+\beta _{12}q^{3}+(\beta _{1}-\beta _{2}+\beta _{8}+\cdots)q^{4}+\cdots\)
325.3.g.d	$325$	$3$	325.g	65.g	$4$	$16$	$8$	$8.856$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None					65.3.j.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(20\)	$2^{3}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{2}+\beta _{12}q^{3}+(-\beta _{1}+\beta _{2}-\beta _{8}+\cdots)q^{4}+\cdots\)
325.3.g.e	$325$	$3$	325.g	65.g	$4$	$20$	$10$	$8.856$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None		✓			325.3.j.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-20\)	$2^{3}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{6}q^{2}+\beta _{4}q^{3}+(-2\beta _{8}-\beta _{13})q^{4}+\cdots\)
325.3.g.f	$325$	$3$	325.g	65.g	$4$	$20$	$10$	$8.856$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None		✓			325.3.j.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(20\)	$2^{3}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{6}q^{2}-\beta _{4}q^{3}+(-2\beta _{8}-\beta _{13})q^{4}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(325, [\chi])\) into lower level spaces

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