Space of modular forms of level 684, weight 2, and trivial character

• Label: 684.2.a
• Level: $684$
• Weight: $2$
• Character orbit: 684.a
• Rep. character: $\chi_{684}(1,\cdot)$
• Character field: $\Q$
• Dimension: $7$
• Newform subspaces: $5$
• Sturm bound: $240$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	684.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(240\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(684))\).

	Total	New	Old
Modular forms	132	7	125
Cusp forms	109	7	102
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.

\(2\)	\(3\)	\(19\)	Fricke	Dim
\(-\)	\(+\)	\(+\)	$-$	\(2\)
\(-\)	\(-\)	\(+\)	$+$	\(2\)
\(-\)	\(-\)	\(-\)	$-$	\(3\)
Plus space			\(+\)	\(2\)
Minus space			\(-\)	\(5\)

Trace form

\( 7 q - q^{5} + 5 q^{7} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(684))\) 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	19
684.2.a.a	$684$	$2$	684.a	1.a	$1$	$1$	$1$	$5.462$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-2q^{11}+2q^{13}-6q^{17}-q^{19}+\cdots\)
684.2.a.b	$684$	$2$	684.a	1.a	$1$	$1$	$1$	$5.462$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-3q^{7}-5q^{11}-4q^{13}+3q^{17}+\cdots\)
684.2.a.c	$684$	$2$	684.a	1.a	$1$	$1$	$1$	$5.462$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(3\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}+q^{7}+5q^{11}-6q^{13}+5q^{17}+\cdots\)
684.2.a.d	$684$	$2$	684.a	1.a	$1$	$2$	$2$	$5.462$	\(\Q(\sqrt{33}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{5}+(1-\beta )q^{7}+(1+\beta )q^{11}+\cdots\)
684.2.a.e	$684$	$2$	684.a	1.a	$1$	$2$	$2$	$5.462$	\(\Q(\sqrt{7}) \)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(6\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+3q^{7}+\beta q^{11}+2q^{13}-3\beta q^{17}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(684))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(684)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(114))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(171))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(228))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(342))\)\(^{\oplus 2}\)