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

• Label: 378.4.k
• Level: $378$
• Weight: $4$
• Character orbit: 378.k
• Rep. character: $\chi_{378}(215,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $64$
• 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.k (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 21 \)
Character field:			\(\Q(\zeta_{6})\)
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	456	64	392
Cusp forms	408	64	344
Eisenstein series	48	0	48

Trace form

\( 64 q + 128 q^{4} - 40 q^{7} + O(q^{10}) \)

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.k.a	$378$	$4$	378.k	21.g	$6$	$12$	$6$	$22.303$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-42\)	$2^{8}\cdot 3^{6}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\beta _{2}+\beta _{7})q^{2}+(4-4\beta _{1})q^{4}+(\beta _{2}+\cdots)q^{5}+\cdots\)
378.4.k.b	$378$	$4$	378.k	21.g	$6$	$16$	$8$	$22.303$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-52\)	$3^{12}$	$\mathrm{SU}(2)[C_{6}]$	\(q-2\beta _{2}q^{2}-4\beta _{1}q^{4}+(-\beta _{2}-2\beta _{9}+\cdots)q^{5}+\cdots\)
378.4.k.c	$378$	$4$	378.k	21.g	$6$	$16$	$8$	$22.303$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(50\)	$3^{12}$	$\mathrm{SU}(2)[C_{6}]$	\(q-2\beta _{5}q^{2}+(4-4\beta _{2})q^{4}+(-\beta _{5}-2\beta _{9}+\cdots)q^{5}+\cdots\)
378.4.k.d	$378$	$4$	378.k	21.g	$6$	$20$	$10$	$22.303$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$2^{16}\cdot 3^{18}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{1}q^{2}-4\beta _{5}q^{4}+\beta _{6}q^{5}+(1+\beta _{5}+\cdots)q^{7}+\cdots\)

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

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