Space of modular forms of level 678, weight 2, and trivial character

• Label: 678.2.a
• Level: $678$
• Weight: $2$
• Character orbit: 678.a
• Rep. character: $\chi_{678}(1,\cdot)$
• Character field: $\Q$
• Dimension: $19$
• Newform subspaces: $11$
• Sturm bound: $228$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 678 = 2 \cdot 3 \cdot 113 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	678.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(228\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(678))\).

	Total	New	Old
Modular forms	118	19	99
Cusp forms	111	19	92
Eisenstein series	7	0	7

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

\(2\)	\(3\)	\(113\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(3\)
\(+\)	\(+\)	\(-\)	$-$	\(2\)
\(+\)	\(-\)	\(+\)	$-$	\(4\)
\(+\)	\(-\)	\(-\)	$+$	\(1\)
\(-\)	\(+\)	\(+\)	$-$	\(3\)
\(-\)	\(+\)	\(-\)	$+$	\(1\)
\(-\)	\(-\)	\(-\)	$-$	\(5\)
Plus space			\(+\)	\(5\)
Minus space			\(-\)	\(14\)

Trace form

\( 19 q - q^{2} + q^{3} + 19 q^{4} + 2 q^{5} + q^{6} + 8 q^{7} - q^{8} + 19 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(678))\) 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	113
678.2.a.a	$678$	$2$	678.a	1.a	$1$	$1$	$1$	$5.414$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(-1\)	\(1\)	\(1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}+q^{7}+\cdots\)
678.2.a.b	$678$	$2$	678.a	1.a	$1$	$1$	$1$	$5.414$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(1\)	\(-1\)	\(-3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}-3q^{7}+\cdots\)
678.2.a.c	$678$	$2$	678.a	1.a	$1$	$1$	$1$	$5.414$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(-1\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-q^{6}-4q^{7}+q^{8}+\cdots\)
678.2.a.d	$678$	$2$	678.a	1.a	$1$	$1$	$1$	$5.414$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(1\)	\(-4\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}-4q^{5}+q^{6}+4q^{7}+\cdots\)
678.2.a.e	$678$	$2$	678.a	1.a	$1$	$1$	$1$	$5.414$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(1\)	\(-1\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+q^{7}+\cdots\)
678.2.a.f	$678$	$2$	678.a	1.a	$1$	$1$	$1$	$5.414$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(1\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+2q^{5}+q^{6}+q^{8}+\cdots\)
678.2.a.g	$678$	$2$	678.a	1.a	$1$	$2$	$2$	$5.414$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(-2\)	\(-2\)	\(-4\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+(-2+\beta )q^{5}+q^{6}+\cdots\)
678.2.a.h	$678$	$2$	678.a	1.a	$1$	$2$	$2$	$5.414$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(-2\)	\(-2\)	\(4\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+2q^{5}+q^{6}+(-1+\cdots)q^{7}+\cdots\)
678.2.a.i	$678$	$2$	678.a	1.a	$1$	$2$	$2$	$5.414$	\(\Q(\sqrt{61}) \)	None	✓	$1$	$0$	\(2\)	\(2\)	\(4\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+2q^{5}+q^{6}+(1-\beta )q^{7}+\cdots\)
678.2.a.j	$678$	$2$	678.a	1.a	$1$	$3$	$3$	$5.414$	3.3.469.1	None	✓	$1$	$0$	\(3\)	\(-3\)	\(1\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+(1-\beta _{1}+\beta _{2})q^{5}+\cdots\)
678.2.a.k	$678$	$2$	678.a	1.a	$1$	$4$	$4$	$5.414$	4.4.77976.1	None	✓	$1$	$0$	\(-4\)	\(4\)	\(0\)	\(5\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}+\beta _{2}q^{5}-q^{6}+(1+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(678))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(678)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(113))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(226))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(339))\)\(^{\oplus 2}\)