Space of modular forms of level 408, weight 2, and Character 408.bc

• Label: 408.2.bc
• Level: $408$
• Weight: $2$
• Character orbit: 408.bc
• Rep. character: $\chi_{408}(229,\cdot)$
• Character field: $\Q(\zeta_{8})$
• Dimension: $144$
• Newform subspaces: $1$
• Sturm bound: $144$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 408 = 2^{3} \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	408.bc (of order \(8\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 136 \)
Character field:			\(\Q(\zeta_{8})\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(144\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(408, [\chi])\).

	Total	New	Old
Modular forms	304	144	160
Cusp forms	272	144	128
Eisenstein series	32	0	32

Trace form

144 * q + 8 * q^6 - 32 * q^14 - 24 * q^22 + 8 * q^24 - 16 * q^25 - 32 * q^26 + 24 * q^28 + 80 * q^32 - 8 * q^34 + 8 * q^36 + 88 * q^40 - 16 * q^41 - 24 * q^42 + 32 * q^44 - 16 * q^46 - 8 * q^54 - 96 * q^56 - 24 * q^58 + 8 * q^60 - 16 * q^62 + 80 * q^65 - 128 * q^68 - 56 * q^70 - 128 * q^71 - 80 * q^73 - 48 * q^74 - 88 * q^76 - 24 * q^78 - 144 * q^80 + 8 * q^82 + 48 * q^84 - 64 * q^86 - 120 * q^88 + 80 * q^92

Decomposition of \(S_{2}^{\mathrm{new}}(408, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
408.2.bc.a	$408$	$2$	408.bc	136.o	$8$	$144$	$36$	$3.258$		None		✓	✓	✓	408.2.bc.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{8}]$

Decomposition of \(S_{2}^{\mathrm{old}}(408, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(408, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(136, [\chi])\)\(^{\oplus 2}\)