Space of modular forms of level 378, weight 2, and Character 378.k

• Label: 378.2.k
• Level: $378$
• Weight: $2$
• Character orbit: 378.k
• Rep. character: $\chi_{378}(215,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $20$
• Newform subspaces: $4$
• Sturm bound: $144$
• Trace bound: $10$

Defining parameters

Level:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
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:			\(144\)
Trace bound:			\(10\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(378, [\chi])\).

	Total	New	Old
Modular forms	168	20	148
Cusp forms	120	20	100
Eisenstein series	48	0	48

Trace form

20 * q + 10 * q^4 + 10 * q^7 - 6 * q^10 - 10 * q^16 + 12 * q^19 + 12 * q^22 - 4 * q^25 + 8 * q^28 + 24 * q^31 + 14 * q^37 - 6 * q^40 + 44 * q^43 - 58 * q^49 - 18 * q^52 - 24 * q^58 - 30 * q^61 - 20 * q^64 - 26 * q^67 - 30 * q^70 - 120 * q^73 - 2 * q^79 - 24 * q^82 - 72 * q^85 + 6 * q^88 + 48 * q^91 + 120 * q^94

Decomposition of \(S_{2}^{\mathrm{new}}(378, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
378.2.k.a	$378$	$2$	378.k	21.g	$6$	$4$	$2$	$3.018$	\(\Q(\zeta_{12})\)	None		✓			378.2.k.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}+(-1+3\zeta_{12}^{2}+\cdots)q^{7}+\cdots\)
378.2.k.b	$378$	$2$	378.k	21.g	$6$	$4$	$2$	$3.018$	\(\Q(\zeta_{12})\)	None		✓			378.2.k.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}+(-2\zeta_{12}+4\zeta_{12}^{3})q^{5}+\cdots\)
378.2.k.c	$378$	$2$	378.k	21.g	$6$	$4$	$2$	$3.018$	\(\Q(\zeta_{12})\)	None		✓			378.2.k.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}+(\zeta_{12}-2\zeta_{12}^{3})q^{5}+\cdots\)
378.2.k.d	$378$	$2$	378.k	21.g	$6$	$8$	$4$	$3.018$	\(\Q(\zeta_{24})\)	None		✓	✓		378.2.k.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta_1 q^{2}+\beta_{2} q^{4}+(-\beta_{6}+\beta_{5})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(378, [\chi])\) into lower level spaces

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