Space of modular forms of level 676, weight 6, and trivial character

• Label: 676.6.a
• Level: $676$
• Weight: $6$
• Character orbit: 676.a
• Rep. character: $\chi_{676}(1,\cdot)$
• Character field: $\Q$
• Dimension: $64$
• Newform subspaces: $10$
• Sturm bound: $546$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 676 = 2^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	676.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(546\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(676))\).

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

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(13\)	Fricke	Dim
\(-\)	\(+\)	$-$	\(33\)
\(-\)	\(-\)	$+$	\(31\)
Plus space		\(+\)	\(31\)
Minus space		\(-\)	\(33\)

Trace form

\( 64 q - 12 q^{3} + 32 q^{5} + 130 q^{7} + 4900 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(676))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	13
676.6.a.a	$676$	$6$	676.a	1.a	$1$	$1$	$1$	$108.419$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-12\)	\(-54\)	\(88\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-12q^{3}-54q^{5}+88q^{7}-99q^{9}+\cdots\)
676.6.a.b	$676$	$6$	676.a	1.a	$1$	$1$	$1$	$108.419$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-5\)	\(3\)	\(-53\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{3}+3q^{5}-53q^{7}-218q^{9}+702q^{11}+\cdots\)
676.6.a.c	$676$	$6$	676.a	1.a	$1$	$1$	$1$	$108.419$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(17\)	\(91\)	\(233\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+17q^{3}+91q^{5}+233q^{7}+46q^{9}+\cdots\)
676.6.a.d	$676$	$6$	676.a	1.a	$1$	$3$	$3$	$108.419$	3.3.203961.1	None	✓	$1$	$0$	\(0\)	\(12\)	\(-8\)	\(-138\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(5+2\beta _{1}-\beta _{2})q^{3}+(-6-5\beta _{1}+5\beta _{2})q^{5}+\cdots\)
676.6.a.e	$676$	$6$	676.a	1.a	$1$	$6$	$6$	$108.419$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{5}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{2}q^{5}-\beta _{3}q^{7}+(118-\beta _{1}+\cdots)q^{9}+\cdots\)
676.6.a.f	$676$	$6$	676.a	1.a	$1$	$6$	$6$	$108.419$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(9\)	\(-11\)	\(-7\)		$-$	$+$	$-$	$2^{6}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{3}+(-2-\beta _{3})q^{5}+(-2\beta _{1}+\cdots)q^{7}+\cdots\)
676.6.a.g	$676$	$6$	676.a	1.a	$1$	$6$	$6$	$108.419$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(9\)	\(11\)	\(7\)		$-$	$+$	$-$	$2^{6}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{3}+(2+\beta _{3})q^{5}+(2\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
676.6.a.h	$676$	$6$	676.a	1.a	$1$	$10$	$10$	$108.419$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$2$	$1$	\(0\)	\(-18\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{10}\cdot 3\cdot 13^{2}$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{3})q^{3}+(-6\beta _{2}-\beta _{4})q^{5}+\cdots\)
676.6.a.i	$676$	$6$	676.a	1.a	$1$	$15$	$15$	$108.419$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-12\)	\(-132\)	\(21\)		$-$	$-$	$+$	$2^{12}\cdot 13^{7}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{3}+(-9-\beta _{2}-\beta _{3})q^{5}+\cdots\)
676.6.a.j	$676$	$6$	676.a	1.a	$1$	$15$	$15$	$108.419$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-12\)	\(132\)	\(-21\)		$-$	$+$	$-$	$2^{12}\cdot 13^{7}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{3}+(9+\beta _{2}+\beta _{3})q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(676))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(676)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(338))\)\(^{\oplus 2}\)