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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(28\)	\(0\)	\(28\)		\(15\)	\(0\)	\(15\)		\(13\)	\(0\)	\(13\)
\(+\)	\(-\)	\(-\)		\(31\)	\(0\)	\(31\)		\(17\)	\(0\)	\(17\)		\(14\)	\(0\)	\(14\)
\(-\)	\(+\)	\(-\)		\(28\)	\(8\)	\(20\)		\(21\)	\(8\)	\(13\)		\(7\)	\(0\)	\(7\)
\(-\)	\(-\)	\(+\)		\(25\)	\(5\)	\(20\)		\(18\)	\(5\)	\(13\)		\(7\)	\(0\)	\(7\)
Plus space		\(+\)		\(53\)	\(5\)	\(48\)		\(33\)	\(5\)	\(28\)		\(20\)	\(0\)	\(20\)
Minus space		\(-\)		\(59\)	\(8\)	\(51\)		\(38\)	\(8\)	\(30\)		\(21\)	\(0\)	\(21\)

Trace form

13 * q - 2 * q^5 + 2 * q^7 + 17 * q^9 + 2 * q^11 - 8 * q^17 + 6 * q^19 - 6 * q^23 + 17 * q^25 - 6 * q^27 - 2 * q^29 - 10 * q^31 + 6 * q^37 + 6 * q^41 - 2 * q^43 + 6 * q^45 + 2 * q^47 + 19 * q^49 - 2 * q^51 - 6 * q^53 + 8 * q^55 + 10 * q^59 + 4 * q^61 - 6 * q^63 - 10 * q^67 - 4 * q^69 - 10 * q^71 - 2 * q^73 + 2 * q^75 + 6 * q^79 + 5 * q^81 + 6 * q^83 - 12 * q^85 + 4 * q^87 + 6 * q^89 + 18 * q^95 - 2 * q^97 - 6 * q^99

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	Twist minimal	Largest	Maximal	Minimal twist	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	✓				52.2.e.b	$1$	$0$	\(0\)	\(-2\)	\(-3\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-3q^{5}-4q^{7}+q^{9}+6q^{15}+\cdots\)
676.2.a.b	$676$	$2$	676.a	1.a	$1$	$1$	$1$	$5.398$	\(\Q\)	None	✓				52.2.e.b	$1$	$0$	\(0\)	\(-2\)	\(3\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+3q^{5}+4q^{7}+q^{9}-6q^{15}+\cdots\)
676.2.a.c	$676$	$2$	676.a	1.a	$1$	$1$	$1$	$5.398$	\(\Q\)	None	✓				52.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+2q^{7}-3q^{9}+2q^{11}+6q^{17}+\cdots\)
676.2.a.d	$676$	$2$	676.a	1.a	$1$	$1$	$1$	$5.398$	\(\Q\)	None	✓				52.2.e.a	$1$	$0$	\(0\)	\(3\)	\(-2\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-2q^{5}-q^{7}+6q^{9}+5q^{11}+\cdots\)
676.2.a.e	$676$	$2$	676.a	1.a	$1$	$1$	$1$	$5.398$	\(\Q\)	None	✓				52.2.e.a	$1$	$0$	\(0\)	\(3\)	\(2\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+2q^{5}+q^{7}+6q^{9}-5q^{11}+\cdots\)
676.2.a.f	$676$	$2$	676.a	1.a	$1$	$2$	$2$	$5.398$	\(\Q(\sqrt{3}) \)	None	✓				52.2.h.a	$2$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta q^{7}-2q^{9}-3\beta q^{11}-3q^{17}+\cdots\)
676.2.a.g	$676$	$2$	676.a	1.a	$1$	$3$	$3$	$5.398$	\(\Q(\zeta_{14})^+\)	None	✓	✓	✓		676.2.a.g	$1$	$1$	\(0\)	\(0\)	\(-8\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-\beta _{1}-\beta _{2})q^{3}+(-2-\beta _{1}+\beta _{2})q^{5}+\cdots\)
676.2.a.h	$676$	$2$	676.a	1.a	$1$	$3$	$3$	$5.398$	\(\Q(\zeta_{14})^+\)	None	✓	✓	✓		676.2.a.g	$1$	$0$	\(0\)	\(0\)	\(8\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-\beta _{1}-\beta _{2})q^{3}+(2+\beta _{1}-\beta _{2})q^{5}+\cdots\)

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

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