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

• Label: 765.2.t
• Level: $765$
• Weight: $2$
• Character orbit: 765.t
• Rep. character: $\chi_{765}(64,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $88$
• Newform subspaces: $7$
• Sturm bound: $216$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 765 = 3^{2} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	765.t (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 85 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(216\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(2\)

Dimensions

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

	Total	New	Old
Modular forms	232	96	136
Cusp forms	200	88	112
Eisenstein series	32	8	24

Trace form

88 * q + 80 * q^4 - 2 * q^5 - 6 * q^10 + 4 * q^11 + 20 * q^14 + 56 * q^16 + 2 * q^20 + 8 * q^29 - 20 * q^34 - 8 * q^35 - 26 * q^40 - 4 * q^41 + 64 * q^44 - 40 * q^46 + 12 * q^50 + 80 * q^56 + 28 * q^61 + 40 * q^64 - 28 * q^71 + 40 * q^74 + 16 * q^79 + 38 * q^80 + 58 * q^85 - 80 * q^86 - 72 * q^89 + 4 * q^91 + 36 * q^95

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.t.a	$765$	$2$	765.t	85.j	$4$	$2$	$1$	$6.109$	\(\Q(\sqrt{-1}) \)	None					85.2.j.a	$2$	$0$	\(-2\)	\(0\)	\(2\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-q^{2}-q^{4}+(2 i+1)q^{5}+(3 i+3)q^{7}+\cdots\)
765.2.t.b	$765$	$2$	765.t	85.j	$4$	$2$	$1$	$6.109$	\(\Q(\sqrt{-1}) \)	None					85.2.j.a	$2$	$0$	\(2\)	\(0\)	\(4\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+q^{2}-q^{4}+(i+2)q^{5}+(-3 i-3)q^{7}+\cdots\)
765.2.t.c	$765$	$2$	765.t	85.j	$4$	$8$	$4$	$6.109$	8.0.959512576.1	None		✓			765.2.t.c	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(\beta _{3}-\beta _{5})q^{2}+(2\beta _{5}-\beta _{7})q^{5}+(-1+\cdots)q^{7}+\cdots\)
765.2.t.d	$765$	$2$	765.t	85.j	$4$	$8$	$4$	$6.109$	8.0.3317760000.5	\(\Q(\sqrt{-15}) \)		✓			765.2.t.d	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{4}]$	\(q+\beta _{3}q^{2}+(2+\beta _{6}+\beta _{7})q^{4}+(-\beta _{3}+\cdots)q^{5}+\cdots\)
765.2.t.e	$765$	$2$	765.t	85.j	$4$	$12$	$6$	$6.109$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None					85.2.j.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{2}q^{2}+(1+\beta _{1}-\beta _{6})q^{4}+\beta _{11}q^{5}+\cdots\)
765.2.t.f	$765$	$2$	765.t	85.j	$4$	$24$	$12$	$6.109$		None		✓			765.2.t.f	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{4}]$
765.2.t.g	$765$	$2$	765.t	85.j	$4$	$32$	$16$	$6.109$		None			✓		255.2.s.a	$4$	$0$	\(0\)	\(0\)	\(-8\)	\(0\)		$\mathrm{SU}(2)[C_{4}]$

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

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