Space of modular forms of level 676, weight 2, and trivial character

• Label: 676.2.a
• Level: $676$
• Weight: $2$
• Character orbit: 676.a
• Rep. character: $\chi_{676}(1,\cdot)$
• Character field: $\Q$
• Dimension: $13$
• Newform subspaces: $8$
• Sturm bound: $182$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 676 = 2^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	676.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(182\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(676))\).

	Total	New	Old
Modular forms	112	13	99
Cusp forms	71	13	58
Eisenstein series	41	0	41

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
\(-\)	\(+\)	$-$	\(8\)
\(-\)	\(-\)	$+$	\(5\)
Plus space		\(+\)	\(5\)
Minus space		\(-\)	\(8\)

Trace form

\( 13 q - 2 q^{5} + 2 q^{7} + 17 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(676))\) 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
676.2.a.a	$676$	$2$	676.a	1.a	$1$	$1$	$1$	$5.398$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(-3\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-3q^{5}-4q^{7}+q^{9}+6q^{15}+\cdots\)
676.2.a.b	$676$	$2$	676.a	1.a	$1$	$1$	$1$	$5.398$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(3\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+3q^{5}+4q^{7}+q^{9}-6q^{15}+\cdots\)
676.2.a.c	$676$	$2$	676.a	1.a	$1$	$1$	$1$	$5.398$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+2q^{7}-3q^{9}+2q^{11}+6q^{17}+\cdots\)
676.2.a.d	$676$	$2$	676.a	1.a	$1$	$1$	$1$	$5.398$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(3\)	\(-2\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-2q^{5}-q^{7}+6q^{9}+5q^{11}+\cdots\)
676.2.a.e	$676$	$2$	676.a	1.a	$1$	$1$	$1$	$5.398$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(3\)	\(2\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+2q^{5}+q^{7}+6q^{9}-5q^{11}+\cdots\)
676.2.a.f	$676$	$2$	676.a	1.a	$1$	$2$	$2$	$5.398$	\(\Q(\sqrt{3}) \)	None	✓	$2$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta q^{7}-2q^{9}-3\beta q^{11}-3q^{17}+\cdots\)
676.2.a.g	$676$	$2$	676.a	1.a	$1$	$3$	$3$	$5.398$	\(\Q(\zeta_{14})^+\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-8\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-\beta _{1}-\beta _{2})q^{3}+(-2-\beta _{1}+\beta _{2})q^{5}+\cdots\)
676.2.a.h	$676$	$2$	676.a	1.a	$1$	$3$	$3$	$5.398$	\(\Q(\zeta_{14})^+\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(8\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-\beta _{1}-\beta _{2})q^{3}+(2+\beta _{1}-\beta _{2})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(676))\) into lower level spaces

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