Space of modular forms of level 75, weight 2, and Character 75.g

• Label: 75.2.g
• Level: $75$
• Weight: $2$
• Character orbit: 75.g
• Rep. character: $\chi_{75}(16,\cdot)$
• Character field: $\Q(\zeta_{5})$
• Dimension: $24$
• Newform subspaces: $3$
• Sturm bound: $20$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	75.g (of order \(5\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 25 \)
Character field:			\(\Q(\zeta_{5})\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(20\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(75, [\chi])\).

	Total	New	Old
Modular forms	48	24	24
Cusp forms	32	24	8
Eisenstein series	16	0	16

Trace form

24 * q - 2 * q^2 - 8 * q^4 - 6 * q^5 - 2 * q^6 - 8 * q^7 + 12 * q^8 - 6 * q^9 - 14 * q^10 + 6 * q^11 - 8 * q^12 - 12 * q^13 - 12 * q^14 - 4 * q^15 - 10 * q^17 + 8 * q^18 - 2 * q^19 + 16 * q^20 - 4 * q^21 + 30 * q^22 + 36 * q^23 + 24 * q^24 - 16 * q^25 - 32 * q^26 - 26 * q^28 - 4 * q^29 + 44 * q^30 - 6 * q^31 + 12 * q^32 - 6 * q^33 - 6 * q^34 - 30 * q^35 - 8 * q^36 - 8 * q^37 - 26 * q^38 - 8 * q^39 + 12 * q^40 - 6 * q^41 - 14 * q^42 + 32 * q^43 + 26 * q^44 - 6 * q^45 - 16 * q^46 - 16 * q^47 - 32 * q^48 + 40 * q^49 + 86 * q^50 + 32 * q^51 + 40 * q^52 + 36 * q^53 - 2 * q^54 + 14 * q^55 - 16 * q^57 + 38 * q^58 + 12 * q^59 - 6 * q^60 - 20 * q^61 + 14 * q^62 + 2 * q^63 - 2 * q^64 + 12 * q^65 - 16 * q^66 - 124 * q^68 - 12 * q^69 - 70 * q^70 + 8 * q^71 + 12 * q^72 - 24 * q^73 - 72 * q^74 - 24 * q^75 + 32 * q^76 - 56 * q^77 - 40 * q^78 - 20 * q^79 - 154 * q^80 - 6 * q^81 + 4 * q^82 - 26 * q^83 + 12 * q^84 - 2 * q^85 + 36 * q^86 + 8 * q^87 + 72 * q^88 + 48 * q^89 + 16 * q^90 - 26 * q^91 + 22 * q^92 + 88 * q^93 - 58 * q^94 + 96 * q^95 + 6 * q^96 + 42 * q^98 - 4 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(75, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
75.2.g.a	$75$	$2$	75.g	25.d	$5$	$4$	$1$	$0.599$	\(\Q(\zeta_{10})\)	None		✓			75.2.g.a	$2$	$0$	\(-1\)	\(-1\)	\(5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+(-1+\zeta_{10}-\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
75.2.g.b	$75$	$2$	75.g	25.d	$5$	$8$	$2$	$0.599$	8.0.26265625.1	None		✓			75.2.g.b	$2$	$0$	\(-1\)	\(-2\)	\(-5\)	\(4\)	$5$	$\mathrm{SU}(2)[C_{5}]$	\(q+(-\beta _{3}-\beta _{7})q^{2}+(-1+\beta _{1}+\beta _{3}+\cdots)q^{3}+\cdots\)
75.2.g.c	$75$	$2$	75.g	25.d	$5$	$12$	$3$	$0.599$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓	✓		75.2.g.c	$2$	$0$	\(0\)	\(3\)	\(-6\)	\(-12\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q-\beta _{2}q^{2}+\beta _{8}q^{3}+(-1+\beta _{1}-\beta _{5}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(75, [\chi])\) into lower level spaces

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