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	Dim.
\(+\)	\(+\)	\(+\)	\(+\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)	\(1\)
\(+\)	\(-\)	\(+\)	\(-\)	\(1\)
\(+\)	\(-\)	\(-\)	\(+\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)	\(2\)
Plus space			\(+\)	\(2\)
Minus space			\(-\)	\(6\)

Trace form

\( 8 q + 8 q^{4} + O(q^{10}) \)

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	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	✓	$1$	$0$	\(-1\)	\(0\)	\(-4\)	\(-1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-4q^{5}-q^{7}-q^{8}+4q^{10}+\cdots\)
378.2.a.b	$378$	$2$	378.a	1.a	$1$	$1$	$1$	$3.018$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(-3\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-3q^{5}+q^{7}-q^{8}+3q^{10}+\cdots\)
378.2.a.c	$378$	$2$	378.a	1.a	$1$	$1$	$1$	$3.018$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(-1\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{5}-q^{7}-q^{8}+q^{10}+\cdots\)
378.2.a.d	$378$	$2$	378.a	1.a	$1$	$1$	$1$	$3.018$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{7}-q^{8}+5q^{13}-q^{14}+\cdots\)
378.2.a.e	$378$	$2$	378.a	1.a	$1$	$1$	$1$	$3.018$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{7}+q^{8}+5q^{13}+q^{14}+\cdots\)
378.2.a.f	$378$	$2$	378.a	1.a	$1$	$1$	$1$	$3.018$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(1\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{5}-q^{7}+q^{8}+q^{10}+\cdots\)
378.2.a.g	$378$	$2$	378.a	1.a	$1$	$1$	$1$	$3.018$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(3\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+3q^{5}+q^{7}+q^{8}+3q^{10}+\cdots\)
378.2.a.h	$378$	$2$	378.a	1.a	$1$	$1$	$1$	$3.018$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(4\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+4q^{5}-q^{7}+q^{8}+4q^{10}+\cdots\)

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

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