Space of modular forms of level 378, weight 2, and trivial character

• Label: 378.2.a
• Level: $378$
• Weight: $2$
• Character orbit: 378.a
• Rep. character: $\chi_{378}(1,\cdot)$
• Character field: $\Q$
• Dimension: $8$
• Newform subspaces: $8$
• Sturm bound: $144$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	378.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(144\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(378))\).

	Total	New	Old
Modular forms	84	8	76
Cusp forms	61	8	53
Eisenstein series	23	0	23

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(3\)	\(7\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(7\)	\(1\)	\(6\)		\(5\)	\(1\)	\(4\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)		\(11\)	\(1\)	\(10\)		\(8\)	\(1\)	\(7\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(+\)	\(-\)		\(13\)	\(1\)	\(12\)		\(10\)	\(1\)	\(9\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)		\(11\)	\(1\)	\(10\)		\(8\)	\(1\)	\(7\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)		\(11\)	\(2\)	\(9\)		\(8\)	\(2\)	\(6\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)		\(10\)	\(0\)	\(10\)		\(7\)	\(0\)	\(7\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)		\(11\)	\(0\)	\(11\)		\(8\)	\(0\)	\(8\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)		\(10\)	\(2\)	\(8\)		\(7\)	\(2\)	\(5\)		\(3\)	\(0\)	\(3\)
Plus space			\(+\)		\(39\)	\(2\)	\(37\)		\(28\)	\(2\)	\(26\)		\(11\)	\(0\)	\(11\)
Minus space			\(-\)		\(45\)	\(6\)	\(39\)		\(33\)	\(6\)	\(27\)		\(12\)	\(0\)	\(12\)

Trace form

8 * q + 8 * q^4 + 16 * q^10 + 8 * q^13 + 8 * q^16 - 8 * q^19 - 4 * q^22 + 12 * q^25 - 16 * q^31 + 8 * q^34 + 4 * q^37 + 16 * q^40 - 20 * q^43 - 16 * q^46 + 8 * q^49 + 8 * q^52 - 40 * q^55 + 4 * q^58 - 16 * q^61 + 8 * q^64 - 36 * q^67 - 4 * q^70 - 24 * q^73 - 8 * q^76 - 40 * q^79 - 16 * q^85 - 4 * q^88 - 4 * q^91 - 24 * q^94 + 48 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(378))\) 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					A-L signs			Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3	7
378.2.a.a	$378$	$2$	378.a	1.a	$1$	$1$	$1$	$3.018$	\(\Q\)	None	✓	✓	✓	✓	378.2.a.a	$1$	$0$	\(-1\)	\(0\)	\(-4\)	\(-1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-4q^{5}-q^{7}-q^{8}+4q^{10}+\cdots\)
378.2.a.b	$378$	$2$	378.a	1.a	$1$	$1$	$1$	$3.018$	\(\Q\)	None	✓	✓	✓	✓	378.2.a.b	$1$	$1$	\(-1\)	\(0\)	\(-3\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-3q^{5}+q^{7}-q^{8}+3q^{10}+\cdots\)
378.2.a.c	$378$	$2$	378.a	1.a	$1$	$1$	$1$	$3.018$	\(\Q\)	None	✓	✓	✓	✓	378.2.a.c	$1$	$1$	\(-1\)	\(0\)	\(-1\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{5}-q^{7}-q^{8}+q^{10}+\cdots\)
378.2.a.d	$378$	$2$	378.a	1.a	$1$	$1$	$1$	$3.018$	\(\Q\)	None	✓	✓	✓	✓	378.2.a.d	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{7}-q^{8}+5q^{13}-q^{14}+\cdots\)
378.2.a.e	$378$	$2$	378.a	1.a	$1$	$1$	$1$	$3.018$	\(\Q\)	None	✓	✓			378.2.a.d	$1$	$0$	\(1\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{7}+q^{8}+5q^{13}+q^{14}+\cdots\)
378.2.a.f	$378$	$2$	378.a	1.a	$1$	$1$	$1$	$3.018$	\(\Q\)	None	✓	✓			378.2.a.c	$1$	$0$	\(1\)	\(0\)	\(1\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{5}-q^{7}+q^{8}+q^{10}+\cdots\)
378.2.a.g	$378$	$2$	378.a	1.a	$1$	$1$	$1$	$3.018$	\(\Q\)	None	✓	✓			378.2.a.b	$1$	$0$	\(1\)	\(0\)	\(3\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+3q^{5}+q^{7}+q^{8}+3q^{10}+\cdots\)
378.2.a.h	$378$	$2$	378.a	1.a	$1$	$1$	$1$	$3.018$	\(\Q\)	None	✓	✓			378.2.a.a	$1$	$0$	\(1\)	\(0\)	\(4\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+4q^{5}-q^{7}+q^{8}+4q^{10}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(378))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(378)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(189))\)\(^{\oplus 2}\)