Space of modular forms of level 378, weight 3, and Character 378.n

• Label: 378.3.n
• Level: $378$
• Weight: $3$
• Character orbit: 378.n
• Rep. character: $\chi_{378}(271,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $44$
• Newform subspaces: $5$
• Sturm bound: $216$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	378.n (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(216\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(378, [\chi])\).

	Total	New	Old
Modular forms	312	44	268
Cusp forms	264	44	220
Eisenstein series	48	0	48

Trace form

\( 44 q - 44 q^{4} - 14 q^{7} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(378, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
378.3.n.a	$378$	$3$	378.n	7.d	$6$	$4$	$2$	$10.300$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-28\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\beta _{1}-\beta _{3})q^{2}+(-2-2\beta _{2})q^{4}+\cdots\)
378.3.n.b	$378$	$3$	378.n	7.d	$6$	$4$	$2$	$10.300$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(26\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\beta _{1}-\beta _{3})q^{2}+(-2-2\beta _{2})q^{4}+\cdots\)
378.3.n.c	$378$	$3$	378.n	7.d	$6$	$12$	$6$	$10.300$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(-6\)	\(0\)	$3^{6}\cdot 7$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{3}+\beta _{7})q^{2}+2\beta _{2}q^{4}+(-1-\beta _{1}+\cdots)q^{5}+\cdots\)
378.3.n.d	$378$	$3$	378.n	7.d	$6$	$12$	$6$	$10.300$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-12\)	$2^{2}\cdot 3^{6}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{3}-\beta _{4})q^{2}+2\beta _{1}q^{4}+\beta _{2}q^{5}+(\beta _{1}+\cdots)q^{7}+\cdots\)
378.3.n.e	$378$	$3$	378.n	7.d	$6$	$12$	$6$	$10.300$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(6\)	\(0\)	$3^{6}\cdot 7$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\beta _{3}-\beta _{7})q^{2}+2\beta _{2}q^{4}+(1+\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(378, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(378, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(189, [\chi])\)\(^{\oplus 2}\)