Space of modular forms of level 75, weight 5, and Character 75.f

• Label: 75.5.f
• Level: $75$
• Weight: $5$
• Character orbit: 75.f
• Rep. character: $\chi_{75}(7,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $24$
• Newform subspaces: $5$
• Sturm bound: $50$
• Trace bound: $6$

Defining parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	75.f (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(50\)
Trace bound:			\(6\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(75, [\chi])\).

	Total	New	Old
Modular forms	92	24	68
Cusp forms	68	24	44
Eisenstein series	24	0	24

Trace form

24 * q - 72 * q^6 - 20 * q^7 - 180 * q^8 + 576 * q^11 + 360 * q^12 + 340 * q^13 - 3800 * q^16 - 900 * q^17 + 396 * q^21 + 1100 * q^22 + 1560 * q^23 + 288 * q^26 - 3580 * q^28 - 796 * q^31 - 4980 * q^32 - 2700 * q^33 + 3672 * q^36 + 3820 * q^37 + 7680 * q^38 + 14064 * q^41 + 7380 * q^42 + 1240 * q^43 + 2824 * q^46 - 4800 * q^47 - 3600 * q^48 - 12528 * q^51 + 1240 * q^52 - 1020 * q^53 + 8520 * q^56 + 5400 * q^57 - 2340 * q^58 + 13140 * q^61 - 28680 * q^62 - 540 * q^63 - 20016 * q^66 + 8920 * q^67 + 1920 * q^68 - 15072 * q^71 + 4860 * q^72 - 11600 * q^73 - 32368 * q^76 + 360 * q^77 + 4680 * q^78 - 17496 * q^81 + 27200 * q^82 + 32400 * q^83 - 24504 * q^86 - 10620 * q^87 + 14340 * q^88 + 78844 * q^91 + 31800 * q^92 - 14040 * q^93 + 74016 * q^96 - 58640 * q^97 - 46440 * q^98

Decomposition of \(S_{5}^{\mathrm{new}}(75, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
75.5.f.a	$75$	$5$	75.f	5.c	$4$	$4$	$2$	$7.753$	\(\Q(i, \sqrt{6})\)	None		✓			75.5.f.a	$2$	$0$	\(-12\)	\(0\)	\(0\)	\(72\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-3+3\beta _{2}-2\beta _{3})q^{2}-3\beta _{1}q^{3}+\cdots\)
75.5.f.b	$75$	$5$	75.f	5.c	$4$	$4$	$2$	$7.753$	\(\Q(i, \sqrt{6})\)	None		✓			75.5.f.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{2}+3\beta _{3}q^{3}-13\beta _{2}q^{4}-9q^{6}+\cdots\)
75.5.f.c	$75$	$5$	75.f	5.c	$4$	$4$	$2$	$7.753$	\(\Q(i, \sqrt{6})\)	None		✓			75.5.f.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+2\beta _{1}q^{2}-3\beta _{3}q^{3}-4\beta _{2}q^{4}+18q^{6}+\cdots\)
75.5.f.d	$75$	$5$	75.f	5.c	$4$	$4$	$2$	$7.753$	\(\Q(i, \sqrt{6})\)	None		✓			75.5.f.a	$2$	$0$	\(12\)	\(0\)	\(0\)	\(-72\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(3-3\beta _{2}+2\beta _{3})q^{2}+3\beta _{1}q^{3}+(12\beta _{1}+\cdots)q^{4}+\cdots\)
75.5.f.e	$75$	$5$	75.f	5.c	$4$	$8$	$4$	$7.753$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None			✓		15.5.f.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-20\)	$3^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{4}q^{2}-\beta _{3}q^{3}+(\beta _{1}+12\beta _{2}+2\beta _{3}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{5}^{\mathrm{old}}(75, [\chi])\) into lower level spaces

\( S_{5}^{\mathrm{old}}(75, [\chi]) \simeq \) \(S_{5}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 2}\)