Space of modular forms of level 408, weight 4, and Character 408.c

• Label: 408.4.c
• Level: $408$
• Weight: $4$
• Character orbit: 408.c
• Rep. character: $\chi_{408}(169,\cdot)$
• Character field: $\Q$
• Dimension: $26$
• Newform subspaces: $4$
• Sturm bound: $288$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 408 = 2^{3} \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	408.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 17 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(288\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(408, [\chi])\).

	Total	New	Old
Modular forms	224	26	198
Cusp forms	208	26	182
Eisenstein series	16	0	16

Trace form

\( 26 q - 234 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(408, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
408.4.c.a	$408$	$4$	408.c	17.b	$2$	$2$	$2$	$24.073$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3iq^{3}+18iq^{5}+14iq^{7}-9q^{9}+\cdots\)
408.4.c.b	$408$	$4$	408.c	17.b	$2$	$6$	$6$	$24.073$	6.0.\(\cdots\).1	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}+(-2\beta _{1}+\beta _{4})q^{5}+(-2\beta _{1}+\cdots)q^{7}+\cdots\)
408.4.c.c	$408$	$4$	408.c	17.b	$2$	$6$	$6$	$24.073$	6.0.\(\cdots\).1	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+3\beta _{1}q^{3}+(-11\beta _{1}-\beta _{5})q^{5}+(-11\beta _{1}+\cdots)q^{7}+\cdots\)
408.4.c.d	$408$	$4$	408.c	17.b	$2$	$12$	$12$	$24.073$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{15}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{5}q^{3}+\beta _{6}q^{5}+(-\beta _{5}-\beta _{6}-\beta _{7}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(408, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(408, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(17, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(34, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(51, [\chi])\)\(^{\oplus 4}\)