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

• Label: 378.2.d
• Level: $378$
• Weight: $2$
• Character orbit: 378.d
• Rep. character: $\chi_{378}(377,\cdot)$
• Character field: $\Q$
• Dimension: $12$
• Newform subspaces: $3$
• Sturm bound: $144$
• Trace bound: $22$

Defining parameters

Level:	$N$	$=$	$378 = 2 \cdot 3^{3} \cdot 7$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	378.d (of order $2$ and degree $1$)
Character conductor:	$\operatorname{cond}(\chi)$	$=$	$21$
Character field:			$\Q$
Newform subspaces:			$3$
Sturm bound:			$144$
Trace bound:			$22$
Distinguishing $T_p$:			$5$, $13$

Dimensions

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

	Total	New	Old
Modular forms	84	12	72
Cusp forms	60	12	48
Eisenstein series	24	0	24

Trace form

$12 q - 12 q^{4} + 6 q^{7} + 12 q^{16} + 12 q^{25} - 6 q^{28} + 12 q^{37} - 60 q^{43} + 30 q^{49} - 36 q^{58} - 12 q^{64} - 12 q^{67} - 18 q^{70} + 24 q^{79} + 72 q^{85} - 36 q^{91}+O(q^{100})$

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.d.a	$378$	$2$	378.d	21.c	$2$	$4$	$4$	$3.018$	$\Q(\zeta_{12})$	None		✓			378.2.d.a	$4$	$0$	$0$	$0$	$0$	$-10$	$1$	$\mathrm{SU}(2)[C_{2}]$	$q-\zeta_{12}^{3}q^{2}-q^{4}+(-2\zeta_{12}+\zeta_{12}^{3})q^{5}+\cdots$
378.2.d.b	$378$	$2$	378.d	21.c	$2$	$4$	$4$	$3.018$	$\Q(\zeta_{12})$	None		✓			378.2.d.b	$4$	$0$	$0$	$0$	$0$	$8$	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta_1 q^{2}-q^{4}-\beta_{3} q^{5}+(-\beta_{2}+2)q^{7}+\cdots$
378.2.d.c	$378$	$2$	378.d	21.c	$2$	$4$	$4$	$3.018$	$\Q(\zeta_{12})$	None		✓			378.2.d.c	$4$	$0$	$0$	$0$	$0$	$8$	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta_1 q^{2}-q^{4}-2\beta_{3} q^{5}+(-\beta_{2}+2)q^{7}+\cdots$

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

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