Space of modular forms of level 676, weight 4, and Character 676.d

• Label: 676.4.d
• Level: $676$
• Weight: $4$
• Character orbit: 676.d
• Rep. character: $\chi_{676}(337,\cdot)$
• Character field: $\Q$
• Dimension: $38$
• Newform subspaces: $5$
• Sturm bound: $364$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 676 = 2^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	676.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(364\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(676, [\chi])\).

	Total	New	Old
Modular forms	294	38	256
Cusp forms	252	38	214
Eisenstein series	42	0	42

Trace form

\( 38 q - 6 q^{3} + 356 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(676, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
676.4.d.a	$676$	$4$	676.d	13.b	$2$	$2$	$2$	$39.885$	\(\Q(\sqrt{-1}) \)	None					52.4.a.a	$2$	$0$	\(0\)	\(-6\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3q^{3}-13iq^{5}+11iq^{7}-18q^{9}+\cdots\)
676.4.d.b	$676$	$4$	676.d	13.b	$2$	$4$	$4$	$39.885$	\(\Q(i, \sqrt{217})\)	None					52.4.a.b	$2$	$0$	\(0\)	\(-6\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-\beta _{3})q^{3}+(\beta _{1}+6\beta _{2})q^{5}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
676.4.d.c	$676$	$4$	676.d	13.b	$2$	$6$	$6$	$39.885$	6.0.\(\cdots\).1	None					52.4.e.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{3}+(\beta _{1}+2\beta _{2}-\beta _{4})q^{5}+(2\beta _{1}+\cdots)q^{7}+\cdots\)
676.4.d.d	$676$	$4$	676.d	13.b	$2$	$8$	$8$	$39.885$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None					52.4.h.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{4}q^{3}+(-\beta _{1}-2\beta _{2})q^{5}+(-3\beta _{2}+\cdots)q^{7}+\cdots\)
676.4.d.e	$676$	$4$	676.d	13.b	$2$	$18$	$18$	$39.885$	\(\mathbb{Q}[x]/(x^{18} + \cdots)\)	None		✓	✓		676.4.a.h	$2$	$0$	\(0\)	\(6\)	\(0\)	\(0\)	$2^{6}\cdot 13^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}-\beta _{12}q^{5}-\beta _{10}q^{7}+(7+\beta _{1}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(676, [\chi])\) into lower level spaces

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