Space of modular forms of level 378, weight 5, and Character 378.b

• Label: 378.5.b
• Level: $378$
• Weight: $5$
• Character orbit: 378.b
• Rep. character: $\chi_{378}(323,\cdot)$
• Character field: $\Q$
• Dimension: $32$
• Newform subspaces: $3$
• Sturm bound: $360$
• Trace bound: $10$

Defining parameters

Level:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	378.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 3 \)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(360\)
Trace bound:			\(10\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(378, [\chi])\).

	Total	New	Old
Modular forms	300	32	268
Cusp forms	276	32	244
Eisenstein series	24	0	24

Trace form

\( 32 q - 256 q^{4} + O(q^{10}) \)

Decomposition of \(S_{5}^{\mathrm{new}}(378, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
378.5.b.a	$378$	$5$	378.b	3.b	$2$	$8$	$8$	$39.074$	8.0.\(\cdots\).29	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{9}\cdot 3^{4}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}-8q^{4}+(-4\beta _{1}+\beta _{3}+\beta _{7})q^{5}+\cdots\)
378.5.b.b	$378$	$5$	378.b	3.b	$2$	$8$	$8$	$39.074$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}\cdot 3^{4}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}-8q^{4}+(-\beta _{1}-\beta _{2})q^{5}+\beta _{3}q^{7}+\cdots\)
378.5.b.c	$378$	$5$	378.b	3.b	$2$	$16$	$16$	$39.074$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{20}\cdot 3^{24}\cdot 7^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{5}q^{2}-8q^{4}+(2\beta _{5}-\beta _{11})q^{5}-\beta _{1}q^{7}+\cdots\)

Decomposition of \(S_{5}^{\mathrm{old}}(378, [\chi])\) into lower level spaces

\( S_{5}^{\mathrm{old}}(378, [\chi]) \cong \) \(S_{5}^{\mathrm{new}}(6, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(189, [\chi])\)\(^{\oplus 2}\)