Space of modular forms of level 1336, weight 2, and trivial character

• Label: 1336.2.a
• Level: $1336$
• Weight: $2$
• Character orbit: 1336.a
• Rep. character: $\chi_{1336}(1,\cdot)$
• Character field: $\Q$
• Dimension: $42$
• Newform subspaces: $5$
• Sturm bound: $336$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 1336 = 2^{3} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1336.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(336\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1336))\).

	Total	New	Old
Modular forms	172	42	130
Cusp forms	165	42	123
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.

\(2\)	\(167\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(9\)
\(+\)	\(-\)	$-$	\(12\)
\(-\)	\(+\)	$-$	\(12\)
\(-\)	\(-\)	$+$	\(9\)
Plus space		\(+\)	\(18\)
Minus space		\(-\)	\(24\)

Trace form

\( 42 q - 2 q^{5} + 38 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1336))\) 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	167
1336.2.a.a	$1336$	$2$	1336.a	1.a	$1$	$2$	$2$	$10.668$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(0\)	\(3\)	\(0\)	\(-7\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(1-2\beta )q^{5}+(-4+\beta )q^{7}+\cdots\)
1336.2.a.b	$1336$	$2$	1336.a	1.a	$1$	$7$	$7$	$10.668$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-6\)	\(0\)	\(-1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-\beta _{4}+\beta _{5})q^{5}+(-1+\cdots)q^{7}+\cdots\)
1336.2.a.c	$1336$	$2$	1336.a	1.a	$1$	$9$	$9$	$10.668$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(-8\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1-\beta _{2}-\beta _{8})q^{5}+(\beta _{1}+\cdots)q^{7}+\cdots\)
1336.2.a.d	$1336$	$2$	1336.a	1.a	$1$	$12$	$12$	$10.668$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(8\)	\(-4\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(1-\beta _{9})q^{5}+\beta _{11}q^{7}+(2+\cdots)q^{9}+\cdots\)
1336.2.a.e	$1336$	$2$	1336.a	1.a	$1$	$12$	$12$	$10.668$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(5\)	\(-2\)	\(10\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{9}q^{5}+(1+\beta _{11})q^{7}+(2+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1336))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1336)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(167))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(334))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(668))\)\(^{\oplus 2}\)