Space of modular forms of level 378, weight 4, and Character 378.d

• Label: 378.4.d
• Level: $378$
• Weight: $4$
• Character orbit: 378.d
• Rep. character: $\chi_{378}(377,\cdot)$
• Character field: $\Q$
• Dimension: $32$
• Newform subspaces: $4$
• Sturm bound: $288$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	378.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 21 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(288\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(378, [\chi])\).

	Total	New	Old
Modular forms	228	32	196
Cusp forms	204	32	172
Eisenstein series	24	0	24

Trace form

\( 32 q - 128 q^{4} - 56 q^{7} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(378, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
378.4.d.a	$378$	$4$	378.d	21.c	$2$	$4$	$4$	$22.303$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(14\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\zeta_{12}q^{2}-4q^{4}+(-2\zeta_{12}+4\zeta_{12}^{3})q^{5}+\cdots\)
378.4.d.b	$378$	$4$	378.d	21.c	$2$	$4$	$4$	$22.303$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(56\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\zeta_{12}q^{2}-4q^{4}+2\zeta_{12}^{3}q^{5}+(14+\cdots)q^{7}+\cdots\)
378.4.d.c	$378$	$4$	378.d	21.c	$2$	$12$	$12$	$22.303$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-102\)	$2^{10}\cdot 3^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}-4q^{4}+(\beta _{5}-\beta _{6})q^{5}+(-9+\cdots)q^{7}+\cdots\)
378.4.d.d	$378$	$4$	378.d	21.c	$2$	$12$	$12$	$22.303$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-24\)	$2^{16}\cdot 3^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{2}-4q^{4}-\beta _{6}q^{5}+(-2+\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(378, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(378, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(189, [\chi])\)\(^{\oplus 2}\)