Space of modular forms of level 1337, weight 2, and trivial character

• Label: 1337.2.a
• Level: $1337$
• Weight: $2$
• Character orbit: 1337.a
• Rep. character: $\chi_{1337}(1,\cdot)$
• Character field: $\Q$
• Dimension: $95$
• Newform subspaces: $5$
• Sturm bound: $256$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 1337 = 7 \cdot 191 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1337.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(256\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1337))\).

	Total	New	Old
Modular forms	130	95	35
Cusp forms	127	95	32
Eisenstein series	3	0	3

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

\(7\)	\(191\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(27\)
\(+\)	\(-\)	$-$	\(20\)
\(-\)	\(+\)	$-$	\(33\)
\(-\)	\(-\)	$+$	\(15\)
Plus space		\(+\)	\(42\)
Minus space		\(-\)	\(53\)

Trace form

\( 95 q - q^{2} + 91 q^{4} - 2 q^{5} - 12 q^{6} + q^{7} - 9 q^{8} + 91 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1337))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		7	191
1337.2.a.a	$1337$	$2$	1337.a	1.a	$1$	$1$	$1$	$10.676$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(-2\)	\(-2\)	\(-1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-2q^{3}-q^{4}-2q^{5}-2q^{6}-q^{7}+\cdots\)
1337.2.a.b	$1337$	$2$	1337.a	1.a	$1$	$15$	$15$	$10.676$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	$1$	$1$	\(-4\)	\(-7\)	\(-3\)	\(15\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+\beta _{4}q^{3}+\beta _{2}q^{4}+(-1-\beta _{1}+\cdots)q^{5}+\cdots\)
1337.2.a.c	$1337$	$2$	1337.a	1.a	$1$	$19$	$19$	$10.676$	\(\mathbb{Q}[x]/(x^{19} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(11\)	\(7\)	\(-19\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{6})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
1337.2.a.d	$1337$	$2$	1337.a	1.a	$1$	$27$	$27$	$10.676$		None	✓	$1$	$1$	\(-3\)	\(-11\)	\(-9\)	\(-27\)		$+$	$+$	$+$		$\mathrm{SU}(2)$
1337.2.a.e	$1337$	$2$	1337.a	1.a	$1$	$33$	$33$	$10.676$		None	✓	$1$	$0$	\(5\)	\(9\)	\(5\)	\(33\)		$-$	$+$	$-$		$\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1337))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1337)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(191))\)\(^{\oplus 2}\)