Space of modular forms of level 750, weight 3, and Character 750.f

• Label: 750.3.f
• Level: $750$
• Weight: $3$
• Character orbit: 750.f
• Rep. character: $\chi_{750}(193,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $64$
• Newform subspaces: $4$
• Sturm bound: $450$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 750 = 2 \cdot 3 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	750.f (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(450\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(750, [\chi])\).

	Total	New	Old
Modular forms	640	64	576
Cusp forms	560	64	496
Eisenstein series	80	0	80

Trace form

64 * q - 16 * q^11 - 256 * q^16 - 192 * q^21 + 64 * q^26 + 480 * q^31 + 384 * q^36 - 128 * q^41 - 32 * q^46 + 48 * q^51 + 64 * q^56 - 896 * q^61 - 96 * q^66 + 64 * q^71 + 32 * q^76 - 576 * q^81 + 96 * q^86 + 1008 * q^91

Decomposition of \(S_{3}^{\mathrm{new}}(750, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
750.3.f.a	$750$	$3$	750.f	5.c	$4$	$16$	$8$	$20.436$	16.0.\(\cdots\).9	None		✓			750.3.f.a	$2$	$0$	\(-16\)	\(0\)	\(0\)	\(-16\)	$2^{8}\cdot 5^{8}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+\beta _{2})q^{2}-\beta _{1}q^{3}-2\beta _{2}q^{4}+\cdots\)
750.3.f.b	$750$	$3$	750.f	5.c	$4$	$16$	$8$	$20.436$	16.0.\(\cdots\).2	None		✓			750.3.f.b	$2$	$0$	\(-16\)	\(0\)	\(0\)	\(24\)	$2^{8}\cdot 5^{8}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1-\beta _{1})q^{2}-\beta _{2}q^{3}+2\beta _{1}q^{4}+\cdots\)
750.3.f.c	$750$	$3$	750.f	5.c	$4$	$16$	$8$	$20.436$	16.0.\(\cdots\).2	None		✓			750.3.f.b	$2$	$0$	\(16\)	\(0\)	\(0\)	\(-24\)	$2^{8}\cdot 5^{8}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1-\beta _{1})q^{2}-\beta _{7}q^{3}-2\beta _{1}q^{4}+(-\beta _{2}+\cdots)q^{6}+\cdots\)
750.3.f.d	$750$	$3$	750.f	5.c	$4$	$16$	$8$	$20.436$	16.0.\(\cdots\).9	None		✓			750.3.f.a	$2$	$0$	\(16\)	\(0\)	\(0\)	\(16\)	$2^{8}\cdot 5^{8}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1+\beta _{2})q^{2}-\beta _{3}q^{3}+2\beta _{2}q^{4}+(\beta _{1}+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(750, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(750, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(125, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(250, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(375, [\chi])\)\(^{\oplus 2}\)