Space of modular forms of level 975, weight 2, and Character 975.t

• Label: 975.2.t
• Level: $975$
• Weight: $2$
• Character orbit: 975.t
• Rep. character: $\chi_{975}(268,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $84$
• Newform subspaces: $5$
• Sturm bound: $280$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	975.t (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 65 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(280\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(975, [\chi])\).

	Total	New	Old
Modular forms	304	84	220
Cusp forms	256	84	172
Eisenstein series	48	0	48

Trace form

84 * q + 4 * q^2 + 84 * q^4 + 12 * q^8 + 16 * q^11 + 8 * q^12 + 84 * q^16 - 28 * q^17 - 20 * q^19 + 4 * q^21 + 32 * q^22 + 8 * q^23 + 12 * q^31 + 68 * q^32 + 8 * q^33 - 4 * q^34 + 16 * q^39 - 68 * q^41 + 80 * q^44 + 32 * q^46 + 16 * q^48 - 196 * q^49 - 48 * q^52 - 20 * q^53 + 64 * q^59 - 16 * q^61 + 72 * q^62 + 84 * q^64 - 32 * q^66 - 32 * q^67 - 60 * q^68 + 16 * q^69 - 80 * q^71 - 56 * q^73 - 48 * q^77 - 40 * q^78 - 84 * q^81 + 4 * q^82 + 16 * q^84 - 32 * q^86 + 24 * q^87 + 72 * q^88 - 12 * q^89 + 92 * q^91 + 32 * q^92 - 48 * q^97 + 12 * q^98 + 16 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(975, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
975.2.t.a	$975$	$2$	975.t	65.k	$4$	$4$	$2$	$7.785$	\(\Q(\zeta_{8})\)	None		✓			975.2.k.a	$2$	$0$	\(-4\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1-\zeta_{8}+\zeta_{8}^{3})q^{2}-\zeta_{8}q^{3}+(1+\cdots)q^{4}+\cdots\)
975.2.t.b	$975$	$2$	975.t	65.k	$4$	$4$	$2$	$7.785$	\(\Q(\zeta_{8})\)	None		✓			975.2.k.a	$2$	$0$	\(4\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1-\zeta_{8}+\zeta_{8}^{3})q^{2}+\zeta_{8}^{3}q^{3}+(1-2\zeta_{8}+\cdots)q^{4}+\cdots\)
975.2.t.c	$975$	$2$	975.t	65.k	$4$	$8$	$4$	$7.785$	8.0.959512576.1	None		✓			975.2.k.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(\beta _{1}-\beta _{5})q^{2}+\beta _{5}q^{3}+(-1-\beta _{2}+\cdots)q^{6}+\cdots\)
975.2.t.d	$975$	$2$	975.t	65.k	$4$	$28$	$14$	$7.785$		None					195.2.k.a	$2$	$0$	\(4\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{4}]$
975.2.t.e	$975$	$2$	975.t	65.k	$4$	$40$	$20$	$7.785$		None		✓	✓		975.2.k.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{4}]$

Decomposition of \(S_{2}^{\mathrm{old}}(975, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(975, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(195, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(325, [\chi])\)\(^{\oplus 2}\)