Space of modular forms of level 368, weight 7, and Character 368.f

• Label: 368.7.f
• Level: $368$
• Weight: $7$
• Character orbit: 368.f
• Rep. character: $\chi_{368}(321,\cdot)$
• Character field: $\Q$
• Dimension: $71$
• Newform subspaces: $6$
• Sturm bound: $336$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 368 = 2^{4} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	368.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 23 \)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(336\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(368, [\chi])\).

	Total	New	Old
Modular forms	294	73	221
Cusp forms	282	71	211
Eisenstein series	12	2	10

Trace form

\( 71 q + 2 q^{3} + 16765 q^{9} + O(q^{10}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(368, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
368.7.f.a	$368$	$7$	368.f	23.b	$2$	$1$	$1$	$84.660$	\(\Q\)	\(\Q(\sqrt{-23}) \)	✓				23.7.b.a	$2$	$0$	\(0\)	\(38\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+38q^{3}+715q^{9}+1082q^{13}+23^{3}q^{23}+\cdots\)
368.7.f.b	$368$	$7$	368.f	23.b	$2$	$2$	$2$	$84.660$	\(\Q(\sqrt{69}) \)	\(\Q(\sqrt{-23}) \)	✓				23.7.b.b	$2$	$0$	\(0\)	\(-38\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{U}(1)[D_{2}]$	\(q+(-19-\beta )q^{3}+(736+38\beta )q^{9}+(-541+\cdots)q^{13}+\cdots\)
368.7.f.c	$368$	$7$	368.f	23.b	$2$	$8$	$8$	$84.660$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None					23.7.b.c	$2$	$0$	\(0\)	\(-30\)	\(0\)	\(0\)	$2^{6}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-4-\beta _{3})q^{3}-\beta _{4}q^{5}+(\beta _{2}-\beta _{4}+\cdots)q^{7}+\cdots\)
368.7.f.d	$368$	$7$	368.f	23.b	$2$	$12$	$12$	$84.660$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None					92.7.d.a	$2$	$0$	\(0\)	\(-32\)	\(0\)	\(0\)	$2^{20}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-3-\beta _{2})q^{3}+\beta _{1}q^{5}+\beta _{7}q^{7}+(320+\cdots)q^{9}+\cdots\)
368.7.f.e	$368$	$7$	368.f	23.b	$2$	$12$	$12$	$84.660$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None					46.7.b.a	$2$	$0$	\(0\)	\(64\)	\(0\)	\(0\)	$2^{20}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(5-\beta _{2})q^{3}+\beta _{6}q^{5}+\beta _{8}q^{7}+(396+\cdots)q^{9}+\cdots\)
368.7.f.f	$368$	$7$	368.f	23.b	$2$	$36$	$36$	$84.660$		None			✓		184.7.f.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{7}^{\mathrm{old}}(368, [\chi])\) into lower level spaces

\( S_{7}^{\mathrm{old}}(368, [\chi]) \simeq \) \(S_{7}^{\mathrm{new}}(23, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(46, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(92, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(184, [\chi])\)\(^{\oplus 2}\)