Space of modular forms of level 765, weight 2, and Character 765.g

• Label: 765.2.g
• Level: $765$
• Weight: $2$
• Character orbit: 765.g
• Rep. character: $\chi_{765}(271,\cdot)$
• Character field: $\Q$
• Dimension: $30$
• Newform subspaces: $4$
• Sturm bound: $216$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 765 = 3^{2} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	765.g (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 17 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(216\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(765, [\chi])\).

	Total	New	Old
Modular forms	116	30	86
Cusp forms	100	30	70
Eisenstein series	16	0	16

Trace form

30 * q + 2 * q^2 + 34 * q^4 + 6 * q^8 + 8 * q^13 + 26 * q^16 + 10 * q^17 - 16 * q^19 - 30 * q^25 - 4 * q^26 + 74 * q^32 + 6 * q^34 - 12 * q^35 - 24 * q^38 - 20 * q^43 - 12 * q^47 - 22 * q^49 - 2 * q^50 + 16 * q^52 - 20 * q^53 + 8 * q^55 - 16 * q^59 + 50 * q^64 - 20 * q^67 + 26 * q^68 - 32 * q^76 + 28 * q^77 - 20 * q^83 + 4 * q^85 + 56 * q^86 - 20 * q^89 - 32 * q^94 - 102 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(765, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
765.2.g.a	$765$	$2$	765.g	17.b	$2$	$4$	$4$	$6.109$	\(\Q(i, \sqrt{13})\)	None					255.2.g.a	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1+\beta _{3})q^{2}+(2-\beta _{3})q^{4}-\beta _{2}q^{5}+\cdots\)
765.2.g.b	$765$	$2$	765.g	17.b	$2$	$6$	$6$	$6.109$	6.0.350464.1	None					85.2.d.a	$2$	$0$	\(2\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{2}+(\beta _{1}-\beta _{2})q^{4}+\beta _{3}q^{5}+(2\beta _{4}+\cdots)q^{7}+\cdots\)
765.2.g.c	$765$	$2$	765.g	17.b	$2$	$8$	$8$	$6.109$	8.0.\(\cdots\).1	None					255.2.g.b	$2$	$0$	\(2\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{2}+(1-\beta _{2}-\beta _{3})q^{4}-\beta _{5}q^{5}+\cdots\)
765.2.g.d	$765$	$2$	765.g	17.b	$2$	$12$	$12$	$6.109$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		✓	✓		765.2.g.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{2}+(1+\beta _{1})q^{4}+\beta _{6}q^{5}+\beta _{10}q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(765, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(765, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(51, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(85, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(153, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(255, [\chi])\)\(^{\oplus 2}\)