Space of modular forms of level 676, weight 6, and Character 676.d

• Label: 676.6.d
• Level: $676$
• Weight: $6$
• Character orbit: 676.d
• Rep. character: $\chi_{676}(337,\cdot)$
• Character field: $\Q$
• Dimension: $64$
• Newform subspaces: $7$
• Sturm bound: $546$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 676 = 2^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	676.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(546\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(676, [\chi])\).

	Total	New	Old
Modular forms	476	64	412
Cusp forms	434	64	370
Eisenstein series	42	0	42

Trace form

\( 64 q + 4964 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(676, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
676.6.d.a	$676$	$6$	676.d	13.b	$2$	$2$	$2$	$108.419$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(-24\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-12q^{3}+3^{3}iq^{5}+44iq^{7}-99q^{9}+\cdots\)
676.6.d.b	$676$	$6$	676.d	13.b	$2$	$2$	$2$	$108.419$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(-10\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-5q^{3}+3iq^{5}+53iq^{7}-218q^{9}+\cdots\)
676.6.d.c	$676$	$6$	676.d	13.b	$2$	$2$	$2$	$108.419$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(34\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+17q^{3}-91iq^{5}+233iq^{7}+46q^{9}+\cdots\)
676.6.d.d	$676$	$6$	676.d	13.b	$2$	$6$	$6$	$108.419$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None		$2$	$0$	\(0\)	\(24\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(5-2\beta _{3}-\beta _{5})q^{3}+(5\beta _{1}+2\beta _{2}+5\beta _{4}+\cdots)q^{5}+\cdots\)
676.6.d.e	$676$	$6$	676.d	13.b	$2$	$10$	$10$	$108.419$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None		$2$	$0$	\(0\)	\(-18\)	\(0\)	\(0\)	$2^{15}\cdot 3\cdot 13^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-2-\beta _{5})q^{3}+(-6\beta _{1}-\beta _{8})q^{5}+\cdots\)
676.6.d.f	$676$	$6$	676.d	13.b	$2$	$12$	$12$	$108.419$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		$2$	$0$	\(0\)	\(18\)	\(0\)	\(0\)	$2^{12}\cdot 13^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(2+\beta _{2})q^{3}+\beta _{9}q^{5}+\beta _{7}q^{7}+(115+\cdots)q^{9}+\cdots\)
676.6.d.g	$676$	$6$	676.d	13.b	$2$	$30$	$30$	$108.419$		None		$2$	$0$	\(0\)	\(-24\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{6}^{\mathrm{old}}(676, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(676, [\chi]) \cong \) \(S_{6}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(52, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(169, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(338, [\chi])\)\(^{\oplus 2}\)