Space of modular forms of level 1275, weight 2, and Character 1275.d

• Label: 1275.2.d
• Level: $1275$
• Weight: $2$
• Character orbit: 1275.d
• Rep. character: $\chi_{1275}(424,\cdot)$
• Character field: $\Q$
• Dimension: $56$
• Newform subspaces: $10$
• Sturm bound: $360$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 1275 = 3 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1275.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 85 \)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(360\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1275, [\chi])\).

	Total	New	Old
Modular forms	192	56	136
Cusp forms	168	56	112
Eisenstein series	24	0	24

Trace form

56 * q - 68 * q^4 + 56 * q^9 + 108 * q^16 + 36 * q^19 + 8 * q^21 - 8 * q^26 + 52 * q^34 - 68 * q^36 + 64 * q^49 + 16 * q^51 - 32 * q^59 - 148 * q^64 - 8 * q^66 - 4 * q^69 - 48 * q^76 + 56 * q^81 + 32 * q^84 - 96 * q^86 + 40 * q^89 - 72 * q^94

Decomposition of \(S_{2}^{\mathrm{new}}(1275, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1275.2.d.a	$1275$	$2$	1275.d	85.c	$2$	$2$	$2$	$10.181$	\(\Q(\sqrt{-1}) \)	None					51.2.d.a	$2$	$0$	\(0\)	\(-2\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{2}-q^{3}-2 q^{4}-2 i q^{6}+2 q^{7}+\cdots\)
1275.2.d.b	$1275$	$2$	1275.d	85.c	$2$	$2$	$2$	$10.181$	\(\Q(\sqrt{-1}) \)	None					51.2.d.b	$2$	$0$	\(0\)	\(-2\)	\(0\)	\(-8\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}-q^{3}+q^{4}-i q^{6}-4 q^{7}+\cdots\)
1275.2.d.c	$1275$	$2$	1275.d	85.c	$2$	$2$	$2$	$10.181$	\(\Q(\sqrt{-1}) \)	None					51.2.d.a	$2$	$0$	\(0\)	\(2\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{2}+q^{3}-2 q^{4}+2 i q^{6}-2 q^{7}+\cdots\)
1275.2.d.d	$1275$	$2$	1275.d	85.c	$2$	$2$	$2$	$10.181$	\(\Q(\sqrt{-1}) \)	None					51.2.d.b	$2$	$0$	\(0\)	\(2\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}+q^{3}+q^{4}+i q^{6}+4 q^{7}+\cdots\)
1275.2.d.e	$1275$	$2$	1275.d	85.c	$2$	$4$	$4$	$10.181$	\(\Q(i, \sqrt{13})\)	None					255.2.g.a	$2$	$0$	\(0\)	\(-4\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}-q^{3}+(-2+\beta _{3})q^{4}-\beta _{1}q^{6}+\cdots\)
1275.2.d.f	$1275$	$2$	1275.d	85.c	$2$	$4$	$4$	$10.181$	\(\Q(i, \sqrt{13})\)	None					255.2.g.a	$2$	$0$	\(0\)	\(4\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+q^{3}+(-2+\beta _{3})q^{4}+\beta _{1}q^{6}+\cdots\)
1275.2.d.g	$1275$	$2$	1275.d	85.c	$2$	$8$	$8$	$10.181$	8.0.\(\cdots\).1	None					255.2.g.b	$2$	$0$	\(0\)	\(-8\)	\(0\)	\(4\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}-q^{3}+(-1+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
1275.2.d.h	$1275$	$2$	1275.d	85.c	$2$	$8$	$8$	$10.181$	8.0.\(\cdots\).1	None					255.2.g.b	$2$	$0$	\(0\)	\(8\)	\(0\)	\(-4\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+q^{3}+(-1+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
1275.2.d.i	$1275$	$2$	1275.d	85.c	$2$	$12$	$12$	$10.181$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓			1275.2.g.e	$2$	$0$	\(0\)	\(-12\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}-q^{3}+(-1+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
1275.2.d.j	$1275$	$2$	1275.d	85.c	$2$	$12$	$12$	$10.181$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓			1275.2.g.e	$2$	$0$	\(0\)	\(12\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+q^{3}+(-1+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1275, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1275, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(85, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(255, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(425, [\chi])\)\(^{\oplus 2}\)