Space of modular forms of level 765, weight 2, and trivial character

• Label: 765.2.a
• Level: $765$
• Weight: $2$
• Character orbit: 765.a
• Rep. character: $\chi_{765}(1,\cdot)$
• Character field: $\Q$
• Dimension: $28$
• Newform subspaces: $13$
• Sturm bound: $216$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 765 = 3^{2} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	765.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(216\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(765))\).

	Total	New	Old
Modular forms	116	28	88
Cusp forms	101	28	73
Eisenstein series	15	0	15

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

\(3\)	\(5\)	\(17\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(2\)
\(+\)	\(+\)	\(-\)	$-$	\(4\)
\(+\)	\(-\)	\(+\)	$-$	\(4\)
\(+\)	\(-\)	\(-\)	$+$	\(2\)
\(-\)	\(+\)	\(+\)	$-$	\(3\)
\(-\)	\(+\)	\(-\)	$+$	\(4\)
\(-\)	\(-\)	\(+\)	$+$	\(3\)
\(-\)	\(-\)	\(-\)	$-$	\(6\)
Plus space			\(+\)	\(11\)
Minus space			\(-\)	\(17\)

Trace form

\( 28 q - 4 q^{2} + 28 q^{4} + 2 q^{5} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(765))\) 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	17
765.2.a.a	$765$	$2$	765.a	1.a	$1$	$1$	$1$	$6.109$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(1\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+q^{5}-2q^{7}+3q^{8}-q^{10}+\cdots\)
765.2.a.b	$765$	$2$	765.a	1.a	$1$	$1$	$1$	$6.109$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(1\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+q^{5}+4q^{7}+3q^{8}-q^{10}+\cdots\)
765.2.a.c	$765$	$2$	765.a	1.a	$1$	$1$	$1$	$6.109$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(-1\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-q^{5}+4q^{7}-3q^{8}-q^{10}+\cdots\)
765.2.a.d	$765$	$2$	765.a	1.a	$1$	$2$	$2$	$6.109$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(-3\)	\(0\)	\(-2\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}+3\beta q^{4}-q^{5}+(1-2\beta )q^{7}+\cdots\)
765.2.a.e	$765$	$2$	765.a	1.a	$1$	$2$	$2$	$6.109$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(2\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-1+\beta )q^{4}+q^{5}+(-1-2\beta )q^{7}+\cdots\)
765.2.a.f	$765$	$2$	765.a	1.a	$1$	$2$	$2$	$6.109$	\(\Q(\sqrt{13}) \)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(2\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(1+\beta )q^{4}+q^{5}+(-1+2\beta )q^{7}+\cdots\)
765.2.a.g	$765$	$2$	765.a	1.a	$1$	$2$	$2$	$6.109$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+q^{4}-q^{5}+(-1-\beta )q^{7}-\beta q^{8}+\cdots\)
765.2.a.h	$765$	$2$	765.a	1.a	$1$	$2$	$2$	$6.109$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(1\)	\(0\)	\(-2\)	\(-4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1+\beta )q^{4}-q^{5}+(-1-2\beta )q^{7}+\cdots\)
765.2.a.i	$765$	$2$	765.a	1.a	$1$	$2$	$2$	$6.109$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(2\)	\(0\)	\(2\)	\(-4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(1+2\beta )q^{4}+q^{5}+(-2+\cdots)q^{7}+\cdots\)
765.2.a.j	$765$	$2$	765.a	1.a	$1$	$3$	$3$	$6.109$	3.3.229.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}-q^{5}+(1+\beta _{1}+\cdots)q^{7}+\cdots\)
765.2.a.k	$765$	$2$	765.a	1.a	$1$	$3$	$3$	$6.109$	3.3.621.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{1}+\beta _{2})q^{4}-q^{5}+\beta _{2}q^{7}+\cdots\)
765.2.a.l	$765$	$2$	765.a	1.a	$1$	$3$	$3$	$6.109$	3.3.621.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(3\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(2+\beta _{1}+\beta _{2})q^{4}+q^{5}+\beta _{2}q^{7}+\cdots\)
765.2.a.m	$765$	$2$	765.a	1.a	$1$	$4$	$4$	$6.109$	4.4.13768.1	None	✓	$1$	$0$	\(-1\)	\(0\)	\(4\)	\(4\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{2}+(2-\beta _{1})q^{4}+q^{5}+(1-\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(765))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(765)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(51))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(85))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(153))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(255))\)\(^{\oplus 2}\)