Space of modular forms of level 759, weight 2, and trivial character

• Label: 759.2.a
• Level: $759$
• Weight: $2$
• Character orbit: 759.a
• Rep. character: $\chi_{759}(1,\cdot)$
• Character field: $\Q$
• Dimension: $39$
• Newform subspaces: $10$
• Sturm bound: $192$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 759 = 3 \cdot 11 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	759.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(192\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(759))\).

	Total	New	Old
Modular forms	100	39	61
Cusp forms	93	39	54
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.

\(3\)	\(11\)	\(23\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(3\)
\(+\)	\(+\)	\(-\)	$-$	\(6\)
\(+\)	\(-\)	\(+\)	$-$	\(8\)
\(+\)	\(-\)	\(-\)	$+$	\(1\)
\(-\)	\(+\)	\(+\)	$-$	\(8\)
\(-\)	\(+\)	\(-\)	$+$	\(3\)
\(-\)	\(-\)	\(+\)	$+$	\(1\)
\(-\)	\(-\)	\(-\)	$-$	\(9\)
Plus space			\(+\)	\(8\)
Minus space			\(-\)	\(31\)

Trace form

\( 39 q + q^{2} + 3 q^{3} + 41 q^{4} + 10 q^{5} - 3 q^{6} + 21 q^{8} + 39 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(759))\) 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	11	23
759.2.a.a	$759$	$2$	759.a	1.a	$1$	$1$	$1$	$6.061$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(-1\)	\(0\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}-q^{4}+q^{6}-2q^{7}+3q^{8}+\cdots\)
759.2.a.b	$759$	$2$	759.a	1.a	$1$	$1$	$1$	$6.061$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(1\)	\(-2\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}-q^{4}-2q^{5}-q^{6}+3q^{8}+\cdots\)
759.2.a.c	$759$	$2$	759.a	1.a	$1$	$2$	$2$	$6.061$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(-1\)	\(2\)	\(-2\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+q^{3}+(-1+\beta )q^{4}-q^{5}-\beta q^{6}+\cdots\)
759.2.a.d	$759$	$2$	759.a	1.a	$1$	$2$	$2$	$6.061$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(2\)	\(2\)	\(4\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+q^{3}+(1+2\beta )q^{4}+2q^{5}+\cdots\)
759.2.a.e	$759$	$2$	759.a	1.a	$1$	$3$	$3$	$6.061$	3.3.148.1	None	✓	$1$	$1$	\(-1\)	\(-3\)	\(-4\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-q^{3}+(\beta _{1}+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
759.2.a.f	$759$	$2$	759.a	1.a	$1$	$3$	$3$	$6.061$	3.3.148.1	None	✓	$1$	$1$	\(-1\)	\(3\)	\(-2\)	\(-6\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+q^{3}+(\beta _{1}+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
759.2.a.g	$759$	$2$	759.a	1.a	$1$	$5$	$5$	$6.061$	5.5.1563364.1	None	✓	$1$	$0$	\(-2\)	\(5\)	\(4\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}+(1+\beta _{1}+\cdots)q^{5}+\cdots\)
759.2.a.h	$759$	$2$	759.a	1.a	$1$	$6$	$6$	$6.061$	6.6.4222000.1	None	✓	$1$	$0$	\(2\)	\(-6\)	\(6\)	\(0\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-q^{3}+(\beta _{1}+\beta _{2})q^{4}+(1-\beta _{3}+\cdots)q^{5}+\cdots\)
759.2.a.i	$759$	$2$	759.a	1.a	$1$	$8$	$8$	$6.061$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(2\)	\(-8\)	\(6\)	\(-6\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-q^{3}+(2+\beta _{2})q^{4}+(1+\beta _{4}+\cdots)q^{5}+\cdots\)
759.2.a.j	$759$	$2$	759.a	1.a	$1$	$8$	$8$	$6.061$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(2\)	\(8\)	\(0\)	\(6\)		$-$	$+$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}-\beta _{6}q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(759))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(759)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(253))\)\(^{\oplus 2}\)