Space of modular forms of level 650, weight 2, and Character 650.j

• Label: 650.2.j
• Level: $650$
• Weight: $2$
• Character orbit: 650.j
• Rep. character: $\chi_{650}(307,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $42$
• Newform subspaces: $9$
• Sturm bound: $210$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	650.j (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 65 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(210\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(650, [\chi])\).

	Total	New	Old
Modular forms	234	42	192
Cusp forms	186	42	144
Eisenstein series	48	0	48

Trace form

42 * q - 42 * q^4 + 24 * q^11 + 14 * q^13 + 42 * q^16 + 2 * q^17 + 26 * q^18 + 16 * q^19 - 8 * q^21 - 24 * q^23 + 12 * q^27 + 8 * q^31 + 26 * q^34 - 32 * q^37 + 14 * q^41 + 12 * q^42 + 24 * q^43 - 24 * q^44 + 24 * q^46 + 64 * q^47 + 90 * q^49 - 14 * q^52 - 30 * q^53 - 28 * q^58 + 40 * q^59 - 12 * q^62 - 42 * q^64 - 64 * q^66 - 2 * q^68 - 64 * q^69 - 26 * q^72 - 16 * q^76 - 8 * q^77 + 32 * q^78 - 106 * q^81 + 34 * q^82 + 8 * q^83 + 8 * q^84 + 16 * q^87 - 18 * q^89 + 24 * q^91 + 24 * q^92 - 32 * q^93 - 88 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(650, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
650.2.j.a	$650$	$2$	650.j	65.f	$4$	$2$	$1$	$5.190$	\(\Q(\sqrt{-1}) \)	None					130.2.g.b	$2$	$0$	\(0\)	\(-2\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+i q^{2}+(i-1)q^{3}-q^{4}+(-i-1)q^{6}+\cdots\)
650.2.j.b	$650$	$2$	650.j	65.f	$4$	$2$	$1$	$5.190$	\(\Q(\sqrt{-1}) \)	None		✓			650.2.g.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-i q^{2}-q^{4}-2 q^{7}+i q^{8}+3 i q^{9}+\cdots\)
650.2.j.c	$650$	$2$	650.j	65.f	$4$	$2$	$1$	$5.190$	\(\Q(\sqrt{-1}) \)	None		✓			650.2.g.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+i q^{2}-q^{4}+2 q^{7}-i q^{8}+3 i q^{9}+\cdots\)
650.2.j.d	$650$	$2$	650.j	65.f	$4$	$2$	$1$	$5.190$	\(\Q(\sqrt{-1}) \)	None					130.2.g.c	$2$	$0$	\(0\)	\(2\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-i q^{2}+(-i+1)q^{3}-q^{4}+(-i-1)q^{6}+\cdots\)
650.2.j.e	$650$	$2$	650.j	65.f	$4$	$2$	$1$	$5.190$	\(\Q(\sqrt{-1}) \)	None					130.2.g.a	$2$	$0$	\(0\)	\(4\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+i q^{2}+(-2 i+2)q^{3}-q^{4}+(2 i+2)q^{6}+\cdots\)
650.2.j.f	$650$	$2$	650.j	65.f	$4$	$4$	$2$	$5.190$	\(\Q(i, \sqrt{11})\)	None					130.2.g.d	$2$	$0$	\(0\)	\(-2\)	\(0\)	\(-12\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{1}q^{2}+(-\beta _{1}+\beta _{2})q^{3}-q^{4}+(-1+\cdots)q^{6}+\cdots\)
650.2.j.g	$650$	$2$	650.j	65.f	$4$	$4$	$2$	$5.190$	\(\Q(\zeta_{12})\)	None					130.2.g.e	$2$	$0$	\(0\)	\(-2\)	\(0\)	\(-4\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta_{2} q^{2}-\beta_1 q^{3}-q^{4}+(\beta_{3}-\beta_{2}+1)q^{6}+\cdots\)
650.2.j.h	$650$	$2$	650.j	65.f	$4$	$12$	$6$	$5.190$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓			650.2.g.h	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{4}q^{2}-\beta _{1}q^{3}-q^{4}+\beta _{5}q^{6}-\beta _{10}q^{7}+\cdots\)
650.2.j.i	$650$	$2$	650.j	65.f	$4$	$12$	$6$	$5.190$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓			650.2.g.h	$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{4}q^{2}-\beta _{5}q^{3}-q^{4}-\beta _{1}q^{6}+\beta _{10}q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(650, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(650, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(130, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(325, [\chi])\)\(^{\oplus 2}\)