Space of modular forms of level 765, weight 4, and Character 765.g

• Label: 765.4.g
• Level: $765$
• Weight: $4$
• Character orbit: 765.g
• Rep. character: $\chi_{765}(271,\cdot)$
• Character field: $\Q$
• Dimension: $90$
• Newform subspaces: $5$
• Sturm bound: $432$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 765 = 3^{2} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	765.g (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 17 \)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(432\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(765, [\chi])\).

	Total	New	Old
Modular forms	332	90	242
Cusp forms	316	90	226
Eisenstein series	16	0	16

Trace form

90 * q - 4 * q^2 + 348 * q^4 - 48 * q^8 - 68 * q^13 + 1268 * q^16 + 62 * q^17 + 164 * q^19 - 2250 * q^25 + 764 * q^26 - 228 * q^32 + 1152 * q^34 + 320 * q^35 - 944 * q^38 + 1604 * q^43 + 692 * q^47 - 2994 * q^49 + 100 * q^50 - 3980 * q^52 + 2548 * q^53 + 160 * q^55 + 532 * q^59 + 5884 * q^64 - 3188 * q^67 - 1064 * q^68 - 1500 * q^70 + 196 * q^76 - 4136 * q^77 - 7204 * q^83 + 760 * q^85 + 4024 * q^86 + 3512 * q^89 + 6852 * q^94 + 7840 * q^98

Decomposition of \(S_{4}^{\mathrm{new}}(765, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
765.4.g.a	$765$	$4$	765.g	17.b	$2$	$2$	$2$	$45.136$	\(\Q(\sqrt{-1}) \)	None					85.4.d.a	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-q^{2}-7 q^{4}-5 i q^{5}-14 i q^{7}+\cdots\)
765.4.g.b	$765$	$4$	765.g	17.b	$2$	$16$	$16$	$45.136$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None					255.4.g.a	$2$	$0$	\(-4\)	\(0\)	\(0\)	\(0\)	$2^{9}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{4}q^{2}+(2-\beta _{2})q^{4}-5\beta _{7}q^{5}+(2\beta _{1}+\cdots)q^{7}+\cdots\)
765.4.g.c	$765$	$4$	765.g	17.b	$2$	$16$	$16$	$45.136$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None					85.4.d.b	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{2}+(6-\beta _{2})q^{4}-\beta _{7}q^{5}+(\beta _{7}+\cdots)q^{7}+\cdots\)
765.4.g.d	$765$	$4$	765.g	17.b	$2$	$20$	$20$	$45.136$	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)	None					255.4.g.b	$2$	$0$	\(4\)	\(0\)	\(0\)	\(0\)	$2^{13}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{2}+(5-\beta _{2})q^{4}-5\beta _{9}q^{5}+\beta _{14}q^{7}+\cdots\)
765.4.g.e	$765$	$4$	765.g	17.b	$2$	$36$	$36$	$45.136$		None		✓	✓		765.4.g.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{4}^{\mathrm{old}}(765, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(765, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(17, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(51, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(85, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(153, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(255, [\chi])\)\(^{\oplus 2}\)