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

• Label: 378.3.s
• Level: $378$
• Weight: $3$
• Character orbit: 378.s
• Rep. character: $\chi_{378}(53,\cdot)$
• 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.s (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 21 \)
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

\( 44 q + 44 q^{4} + 6 q^{7} + O(q^{10}) \)

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.s.a	$378$	$3$	378.s	21.h	$6$	$4$	$2$	$10.300$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-28\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{1}q^{2}+2\beta _{2}q^{4}+2\beta _{1}q^{5}-7q^{7}+\cdots\)
378.3.s.b	$378$	$3$	378.s	21.h	$6$	$4$	$2$	$10.300$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(22\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{2}+2\beta _{2}q^{4}+3\beta _{1}q^{5}+(8-5\beta _{2}+\cdots)q^{7}+\cdots\)
378.3.s.c	$378$	$3$	378.s	21.h	$6$	$4$	$2$	$10.300$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(28\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{1}q^{2}+2\beta _{2}q^{4}+6\beta _{1}q^{5}+7q^{7}+\cdots\)
378.3.s.d	$378$	$3$	378.s	21.h	$6$	$8$	$4$	$10.300$	8.0.\(\cdots\).1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-24\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{5}q^{2}+(2+2\beta _{1})q^{4}-\beta _{2}q^{5}+(-2+\cdots)q^{7}+\cdots\)
378.3.s.e	$378$	$3$	378.s	21.h	$6$	$24$	$12$	$10.300$		None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(8\)		$\mathrm{SU}(2)[C_{6}]$

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

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