Space of modular forms of level 1369, weight 2, and trivial character

• Label: 1369.2.a
• Level: $1369$
• Weight: $2$
• Character orbit: 1369.a
• Rep. character: $\chi_{1369}(1,\cdot)$
• Character field: $\Q$
• Dimension: $94$
• Newform subspaces: $15$
• Sturm bound: $234$
• Trace bound: $6$

Defining parameters

Level:	\( N \)	\(=\)	\( 1369 = 37^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1369.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 15 \)
Sturm bound:			\(234\)
Trace bound:			\(6\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1369))\).

	Total	New	Old
Modular forms	135	129	6
Cusp forms	98	94	4
Eisenstein series	37	35	2

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

\(37\)	Dim
\(+\)	\(43\)
\(-\)	\(51\)

Trace form

\( 94 q + 2 q^{2} + 2 q^{3} + 78 q^{4} + 2 q^{5} - 6 q^{6} + 2 q^{7} + 56 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1369))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		37
1369.2.a.a	$1369$	$2$	1369.a	1.a	$1$	$1$	$1$	$10.932$	\(\Q\)	None	✓	$1$	$0$	\(-2\)	\(1\)	\(-2\)	\(3\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+q^{3}+2q^{4}-2q^{5}-2q^{6}+\cdots\)
1369.2.a.b	$1369$	$2$	1369.a	1.a	$1$	$1$	$1$	$10.932$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(-1\)	\(2\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}-q^{5}+2q^{7}+3q^{8}-3q^{9}+\cdots\)
1369.2.a.c	$1369$	$2$	1369.a	1.a	$1$	$1$	$1$	$10.932$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{4}-q^{7}-2q^{9}+3q^{11}+\cdots\)
1369.2.a.d	$1369$	$2$	1369.a	1.a	$1$	$1$	$1$	$10.932$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(0\)	\(1\)	\(2\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}+q^{5}+2q^{7}-3q^{8}-3q^{9}+\cdots\)
1369.2.a.e	$1369$	$2$	1369.a	1.a	$1$	$1$	$1$	$10.932$	\(\Q\)	None	✓	$1$	$1$	\(2\)	\(-3\)	\(2\)	\(-1\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}+2q^{4}+2q^{5}-6q^{6}+\cdots\)
1369.2.a.f	$1369$	$2$	1369.a	1.a	$1$	$1$	$1$	$10.932$	\(\Q\)	None	✓	$1$	$0$	\(2\)	\(1\)	\(2\)	\(3\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+q^{3}+2q^{4}+2q^{5}+2q^{6}+\cdots\)
1369.2.a.g	$1369$	$2$	1369.a	1.a	$1$	$2$	$2$	$10.932$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(-2\)	\(2\)	\(4\)	\(0\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(1-\beta )q^{3}-q^{4}+(2+\beta )q^{5}+\cdots\)
1369.2.a.h	$1369$	$2$	1369.a	1.a	$1$	$2$	$2$	$10.932$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(2\)	\(2\)	\(-4\)	\(0\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(1+\beta )q^{3}-q^{4}+(-2+\beta )q^{5}+\cdots\)
1369.2.a.i	$1369$	$2$	1369.a	1.a	$1$	$3$	$3$	$10.932$	\(\Q(\zeta_{18})^+\)	None	✓	$1$	$1$	\(-3\)	\(0\)	\(3\)	\(0\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}-\beta _{2}q^{3}+(1-2\beta _{1}+\cdots)q^{4}+\cdots\)
1369.2.a.j	$1369$	$2$	1369.a	1.a	$1$	$3$	$3$	$10.932$	\(\Q(\zeta_{18})^+\)	None	✓	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-6\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-1-\beta _{1}+\beta _{2})q^{3}+\beta _{2}q^{4}+\cdots\)
1369.2.a.k	$1369$	$2$	1369.a	1.a	$1$	$3$	$3$	$10.932$	\(\Q(\zeta_{18})^+\)	None	✓	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-6\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-1-\beta _{1}+\beta _{2})q^{3}+\beta _{2}q^{4}+\cdots\)
1369.2.a.l	$1369$	$2$	1369.a	1.a	$1$	$3$	$3$	$10.932$	\(\Q(\zeta_{18})^+\)	None	✓	$1$	$1$	\(3\)	\(0\)	\(-3\)	\(0\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}-\beta _{2}q^{3}+(1-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
1369.2.a.m	$1369$	$2$	1369.a	1.a	$1$	$18$	$18$	$10.932$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(6\)	\(0\)	\(12\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(\beta _{1}-\beta _{13})q^{2}+(-\beta _{7}+\beta _{14})q^{3}+\cdots\)
1369.2.a.n	$1369$	$2$	1369.a	1.a	$1$	$27$	$27$	$10.932$		None	✓	$1$	$1$	\(-9\)	\(-1\)	\(-17\)	\(-3\)		$+$	$+$		$\mathrm{SU}(2)$
1369.2.a.o	$1369$	$2$	1369.a	1.a	$1$	$27$	$27$	$10.932$		None	✓	$1$	$0$	\(9\)	\(-1\)	\(17\)	\(-3\)		$-$	$-$		$\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1369))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1369)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(37))\)\(^{\oplus 2}\)