Space of modular forms of level 1335, weight 2, and trivial character

• Label: 1335.2.a
• Level: $1335$
• Weight: $2$
• Character orbit: 1335.a
• Rep. character: $\chi_{1335}(1,\cdot)$
• Character field: $\Q$
• Dimension: $59$
• Newform subspaces: $11$
• Sturm bound: $360$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 1335 = 3 \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1335.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(360\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1335))\).

	Total	New	Old
Modular forms	184	59	125
Cusp forms	177	59	118
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.

\(3\)	\(5\)	\(89\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(6\)
\(+\)	\(+\)	\(-\)	$-$	\(9\)
\(+\)	\(-\)	\(+\)	$-$	\(10\)
\(+\)	\(-\)	\(-\)	$+$	\(3\)
\(-\)	\(+\)	\(+\)	$-$	\(11\)
\(-\)	\(+\)	\(-\)	$+$	\(4\)
\(-\)	\(-\)	\(+\)	$+$	\(3\)
\(-\)	\(-\)	\(-\)	$-$	\(13\)
Plus space			\(+\)	\(16\)
Minus space			\(-\)	\(43\)

Trace form

\( 59 q + 5 q^{2} + 3 q^{3} + 69 q^{4} - q^{5} + q^{6} + 8 q^{7} + 9 q^{8} + 59 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1335))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	5	89
1335.2.a.a	$1335$	$2$	1335.a	1.a	$1$	$1$	$1$	$10.660$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(1\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}-q^{4}+q^{5}-q^{6}+3q^{8}+\cdots\)
1335.2.a.b	$1335$	$2$	1335.a	1.a	$1$	$1$	$1$	$10.660$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(1\)	\(-1\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}-q^{4}-q^{5}+q^{6}+4q^{7}+\cdots\)
1335.2.a.c	$1335$	$2$	1335.a	1.a	$1$	$2$	$2$	$10.660$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(-1\)	\(2\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+q^{3}+(2+\beta )q^{4}+q^{5}-\beta q^{6}+\cdots\)
1335.2.a.d	$1335$	$2$	1335.a	1.a	$1$	$3$	$3$	$10.660$	\(\Q(\zeta_{14})^+\)	None	✓	$1$	$1$	\(-2\)	\(3\)	\(3\)	\(-5\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{2}+q^{3}+(\beta _{1}+\beta _{2})q^{4}+\cdots\)
1335.2.a.e	$1335$	$2$	1335.a	1.a	$1$	$3$	$3$	$10.660$	\(\Q(\zeta_{18})^+\)	None	✓	$1$	$1$	\(0\)	\(-3\)	\(3\)	\(3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-q^{3}+\beta _{2}q^{4}+q^{5}+\beta _{1}q^{6}+\cdots\)
1335.2.a.f	$1335$	$2$	1335.a	1.a	$1$	$4$	$4$	$10.660$	4.4.2777.1	None	✓	$1$	$1$	\(0\)	\(4\)	\(-4\)	\(-7\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}+q^{3}+(1-\beta _{3})q^{4}-q^{5}+\beta _{2}q^{6}+\cdots\)
1335.2.a.g	$1335$	$2$	1335.a	1.a	$1$	$6$	$6$	$10.660$	6.6.10407557.1	None	✓	$1$	$1$	\(-4\)	\(-6\)	\(-6\)	\(1\)		$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{2}-q^{3}+(2+\beta _{2}+\beta _{3}+\cdots)q^{4}+\cdots\)
1335.2.a.h	$1335$	$2$	1335.a	1.a	$1$	$9$	$9$	$10.660$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$0$	\(5\)	\(-9\)	\(-9\)	\(-3\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}-q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
1335.2.a.i	$1335$	$2$	1335.a	1.a	$1$	$10$	$10$	$10.660$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(10\)	\(-10\)	\(9\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}-q^{5}+\beta _{1}q^{6}+\cdots\)
1335.2.a.j	$1335$	$2$	1335.a	1.a	$1$	$10$	$10$	$10.660$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$0$	\(1\)	\(-10\)	\(10\)	\(-1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+q^{5}-\beta _{1}q^{6}+\cdots\)
1335.2.a.k	$1335$	$2$	1335.a	1.a	$1$	$10$	$10$	$10.660$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$0$	\(6\)	\(10\)	\(10\)	\(7\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1335))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1335)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(89))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(267))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(445))\)\(^{\oplus 2}\)