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

• Label: 378.2.h
• Level: $378$
• Weight: $2$
• Character orbit: 378.h
• Rep. character: $\chi_{378}(289,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $16$
• Newform subspaces: $4$
• Sturm bound: $144$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	378.h (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 63 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(144\)
Trace bound:			\(2\)
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	16	152
Cusp forms	120	16	104
Eisenstein series	48	0	48

Trace form

16 * q - 8 * q^4 - 8 * q^5 - 2 * q^7 + 8 * q^11 + 2 * q^13 - 2 * q^14 - 8 * q^16 + 14 * q^17 - 4 * q^19 + 4 * q^20 - 4 * q^23 + 16 * q^25 + 16 * q^26 - 2 * q^28 + 10 * q^29 + 2 * q^31 + 14 * q^35 + 2 * q^37 - 24 * q^38 + 6 * q^41 + 2 * q^43 - 4 * q^44 - 6 * q^46 + 6 * q^47 - 14 * q^49 + 4 * q^50 - 4 * q^52 - 24 * q^53 - 12 * q^55 + 4 * q^56 - 12 * q^58 + 22 * q^59 + 8 * q^61 - 44 * q^62 + 16 * q^64 + 6 * q^65 + 14 * q^67 - 28 * q^68 - 52 * q^71 - 28 * q^73 - 12 * q^74 - 4 * q^76 + 34 * q^77 + 20 * q^79 + 4 * q^80 - 16 * q^83 + 12 * q^85 + 24 * q^86 + 36 * q^89 - 16 * q^91 + 2 * q^92 + 12 * q^94 - 34 * q^95 + 2 * q^97 + 24 * q^98

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.h.a	$378$	$2$	378.h	63.g	$3$	$2$	$1$	$3.018$	\(\Q(\sqrt{-3}) \)	None					126.2.e.a	$2$	$0$	\(-1\)	\(0\)	\(-6\)	\(5\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}-3q^{5}+(3+\cdots)q^{7}+\cdots\)
378.2.h.b	$378$	$2$	378.h	63.g	$3$	$2$	$1$	$3.018$	\(\Q(\sqrt{-3}) \)	None					126.2.e.b	$2$	$0$	\(1\)	\(0\)	\(6\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}+3q^{5}+(1-3\zeta_{6})q^{7}+\cdots\)
378.2.h.c	$378$	$2$	378.h	63.g	$3$	$6$	$3$	$3.018$	6.0.309123.1	None					126.2.e.c	$2$	$0$	\(-3\)	\(0\)	\(2\)	\(-4\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\beta _{4})q^{2}-\beta _{4}q^{4}-\beta _{3}q^{5}+(-1+\cdots)q^{7}+\cdots\)
378.2.h.d	$378$	$2$	378.h	63.g	$3$	$6$	$3$	$3.018$	6.0.309123.1	None					126.2.e.d	$2$	$0$	\(3\)	\(0\)	\(-10\)	\(-2\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{4}q^{2}+(-1+\beta _{4})q^{4}+(-2-\beta _{1}+\cdots)q^{5}+\cdots\)

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

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