Space of modular forms of level 1359, weight 2, and trivial character

• Label: 1359.2.a
• Level: $1359$
• Weight: $2$
• Character orbit: 1359.a
• Rep. character: $\chi_{1359}(1,\cdot)$
• Character field: $\Q$
• Dimension: $63$
• Newform subspaces: $13$
• Sturm bound: $304$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 1359 = 3^{2} \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1359.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(304\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1359))\).

	Total	New	Old
Modular forms	156	63	93
Cusp forms	149	63	86
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.

\(3\)	\(151\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(6\)
\(+\)	\(-\)	$-$	\(20\)
\(-\)	\(+\)	$-$	\(22\)
\(-\)	\(-\)	$+$	\(15\)
Plus space		\(+\)	\(21\)
Minus space		\(-\)	\(42\)

Trace form

\( 63 q + 64 q^{4} + 2 q^{5} + 2 q^{7} + 6 q^{8} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1359))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Minimal twist	Inner twists	Rank*	Traces					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	151
1359.2.a.a	$1359$	$2$	1359.a	1.a	$1$	$2$	$2$	$10.852$	\(\Q(\sqrt{5}) \)	None	✓		453.2.a.d	$1$	$1$	\(-3\)	\(0\)	\(3\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}+3\beta q^{4}+(2-\beta )q^{5}+\cdots\)
1359.2.a.b	$1359$	$2$	1359.a	1.a	$1$	$2$	$2$	$10.852$	\(\Q(\sqrt{3}) \)	None	✓		453.2.a.c	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+q^{4}-2q^{5}+q^{7}-\beta q^{8}-2\beta q^{10}+\cdots\)
1359.2.a.c	$1359$	$2$	1359.a	1.a	$1$	$2$	$2$	$10.852$	\(\Q(\sqrt{5}) \)	None	✓		453.2.a.b	$1$	$1$	\(1\)	\(0\)	\(1\)	\(-6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1+\beta )q^{4}+(1-\beta )q^{5}-3q^{7}+\cdots\)
1359.2.a.d	$1359$	$2$	1359.a	1.a	$1$	$2$	$2$	$10.852$	\(\Q(\sqrt{5}) \)	None	✓		453.2.a.a	$1$	$0$	\(3\)	\(0\)	\(-3\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+3\beta q^{4}+(-2+\beta )q^{5}+\cdots\)
1359.2.a.e	$1359$	$2$	1359.a	1.a	$1$	$3$	$3$	$10.852$	3.3.257.1	None	✓		151.2.a.b	$1$	$1$	\(0\)	\(0\)	\(-5\)	\(-6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+(1+\beta _{1}-\beta _{2})q^{4}+(-2+\beta _{1}+\cdots)q^{5}+\cdots\)
1359.2.a.f	$1359$	$2$	1359.a	1.a	$1$	$3$	$3$	$10.852$	\(\Q(\zeta_{14})^+\)	None	✓		453.2.a.e	$1$	$0$	\(1\)	\(0\)	\(3\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{2}q^{4}+(2-\beta _{1}+2\beta _{2})q^{5}+\cdots\)
1359.2.a.g	$1359$	$2$	1359.a	1.a	$1$	$3$	$3$	$10.852$	\(\Q(\zeta_{14})^+\)	None	✓		151.2.a.a	$1$	$0$	\(2\)	\(0\)	\(7\)	\(-3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(1-2\beta _{1}+\beta _{2})q^{4}+(3+\cdots)q^{5}+\cdots\)
1359.2.a.h	$1359$	$2$	1359.a	1.a	$1$	$5$	$5$	$10.852$	5.5.1190005.1	None	✓		453.2.a.f	$1$	$0$	\(3\)	\(0\)	\(4\)	\(-6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(3-\beta _{1}+\beta _{2})q^{4}+(1+\cdots)q^{5}+\cdots\)
1359.2.a.i	$1359$	$2$	1359.a	1.a	$1$	$6$	$6$	$10.852$	6.6.4838537.1	None	✓		151.2.a.c	$1$	$1$	\(-1\)	\(0\)	\(-6\)	\(3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{4}+(-1-\beta _{4}+\cdots)q^{5}+\cdots\)
1359.2.a.j	$1359$	$2$	1359.a	1.a	$1$	$6$	$6$	$10.852$	6.6.3356224.1	None	✓	✓	1359.2.a.j	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-6\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{2}q^{4}+(-\beta _{4}+\beta _{5})q^{5}+\cdots\)
1359.2.a.k	$1359$	$2$	1359.a	1.a	$1$	$6$	$6$	$10.852$	6.6.19783872.1	None	✓	✓	1359.2.a.k	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{2}q^{4}+\beta _{1}q^{5}+2\beta _{3}q^{7}+\cdots\)
1359.2.a.l	$1359$	$2$	1359.a	1.a	$1$	$9$	$9$	$10.852$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓		453.2.a.g	$1$	$0$	\(-6\)	\(0\)	\(2\)	\(14\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(1+\beta _{2}+\beta _{3})q^{4}+\cdots\)
1359.2.a.m	$1359$	$2$	1359.a	1.a	$1$	$14$	$14$	$10.852$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓	✓	1359.2.a.m	$2$	$0$	\(0\)	\(0\)	\(0\)	\(14\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{5}-\beta _{6})q^{4}+\beta _{12}q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1359))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1359)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(151))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(453))\)\(^{\oplus 2}\)