Space of modular forms of level 75, weight 3, and Character 75.f

• Label: 75.3.f
• Level: $75$
• Weight: $3$
• Character orbit: 75.f
• Rep. character: $\chi_{75}(7,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $12$
• Newform subspaces: $3$
• Sturm bound: $30$
• Trace bound: $6$

Defining parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	75.f (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(30\)
Trace bound:			\(6\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(75, [\chi])\).

	Total	New	Old
Modular forms	52	12	40
Cusp forms	28	12	16
Eisenstein series	24	0	24

Trace form

12 * q + 4 * q^2 + 24 * q^6 - 4 * q^7 - 12 * q^8 - 32 * q^11 - 24 * q^12 + 32 * q^13 - 40 * q^16 + 40 * q^17 + 12 * q^18 - 12 * q^21 - 20 * q^22 - 56 * q^23 - 56 * q^26 - 44 * q^28 + 172 * q^31 + 76 * q^32 + 36 * q^33 + 96 * q^36 - 64 * q^37 + 96 * q^38 - 368 * q^41 - 12 * q^42 + 8 * q^43 - 88 * q^46 - 128 * q^47 - 48 * q^48 - 144 * q^51 + 80 * q^52 - 56 * q^53 + 840 * q^56 + 72 * q^57 + 12 * q^58 + 60 * q^61 - 88 * q^62 - 12 * q^63 - 48 * q^66 + 200 * q^67 + 104 * q^68 + 544 * q^71 + 36 * q^72 - 76 * q^73 - 464 * q^76 - 88 * q^77 - 120 * q^78 - 108 * q^81 - 128 * q^82 + 16 * q^83 - 1112 * q^86 + 84 * q^87 - 12 * q^88 - 268 * q^91 - 104 * q^92 + 72 * q^93 - 192 * q^96 + 20 * q^97 + 188 * q^98

Decomposition of \(S_{3}^{\mathrm{new}}(75, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
75.3.f.a	$75$	$3$	75.f	5.c	$4$	$4$	$2$	$2.044$	\(\Q(i, \sqrt{6})\)	None		✓			75.3.f.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{2}-\beta _{3}q^{3}-\beta _{2}q^{4}+3q^{6}+6\beta _{1}q^{7}+\cdots\)
75.3.f.b	$75$	$3$	75.f	5.c	$4$	$4$	$2$	$2.044$	\(\Q(i, \sqrt{6})\)	None		✓			75.3.f.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+2\beta _{1}q^{2}-\beta _{3}q^{3}+8\beta _{2}q^{4}+6q^{6}+\cdots\)
75.3.f.c	$75$	$3$	75.f	5.c	$4$	$4$	$2$	$2.044$	\(\Q(i, \sqrt{6})\)	None					15.3.f.a	$2$	$0$	\(4\)	\(0\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1+\beta _{1}+\beta _{2})q^{2}+\beta _{3}q^{3}+(2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(75, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(75, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 2}\)