Space of modular forms of level 1339, weight 2, and trivial character

• Label: 1339.2.a
• Level: $1339$
• Weight: $2$
• Character orbit: 1339.a
• Rep. character: $\chi_{1339}(1,\cdot)$
• Character field: $\Q$
• Dimension: $103$
• Newform subspaces: $7$
• Sturm bound: $242$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 1339 = 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1339.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(242\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1339))\).

	Total	New	Old
Modular forms	122	103	19
Cusp forms	119	103	16
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.

\(13\)	\(103\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(22\)
\(+\)	\(-\)	$-$	\(30\)
\(-\)	\(+\)	$-$	\(29\)
\(-\)	\(-\)	$+$	\(22\)
Plus space		\(+\)	\(44\)
Minus space		\(-\)	\(59\)

Trace form

\( 103 q - 3 q^{2} + 97 q^{4} + 2 q^{5} - 3 q^{8} + 103 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1339))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		13	103
1339.2.a.a	$1339$	$2$	1339.a	1.a	$1$	$1$	$1$	$10.692$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(-1\)	\(1\)	\(4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}-q^{4}+q^{5}-q^{6}+4q^{7}+\cdots\)
1339.2.a.b	$1339$	$2$	1339.a	1.a	$1$	$1$	$1$	$10.692$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-4q^{7}-3q^{8}-3q^{9}+6q^{11}+\cdots\)
1339.2.a.c	$1339$	$2$	1339.a	1.a	$1$	$3$	$3$	$10.692$	3.3.148.1	None	✓	$1$	$1$	\(-1\)	\(-1\)	\(-7\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-\beta _{1}-\beta _{2})q^{3}+(\beta _{1}+\beta _{2})q^{4}+\cdots\)
1339.2.a.d	$1339$	$2$	1339.a	1.a	$1$	$19$	$19$	$10.692$	\(\mathbb{Q}[x]/(x^{19} - \cdots)\)	None	✓	$1$	$1$	\(-9\)	\(-2\)	\(-18\)	\(-8\)		$-$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-\beta _{14}q^{3}+(\beta _{1}+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
1339.2.a.e	$1339$	$2$	1339.a	1.a	$1$	$21$	$21$	$10.692$		None	✓	$1$	$1$	\(-1\)	\(-6\)	\(-18\)	\(-2\)		$+$	$+$	$+$		$\mathrm{SU}(2)$
1339.2.a.f	$1339$	$2$	1339.a	1.a	$1$	$28$	$28$	$10.692$		None	✓	$1$	$0$	\(6\)	\(5\)	\(27\)	\(6\)		$-$	$+$	$-$		$\mathrm{SU}(2)$
1339.2.a.g	$1339$	$2$	1339.a	1.a	$1$	$30$	$30$	$10.692$		None	✓	$1$	$0$	\(0\)	\(5\)	\(17\)	\(2\)		$+$	$-$	$-$		$\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1339))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1339)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(103))\)\(^{\oplus 2}\)