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

• Label: 75.2.e
• Level: $75$
• Weight: $2$
• Character orbit: 75.e
• Rep. character: $\chi_{75}(32,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $8$
• Newform subspaces: $2$
• Sturm bound: $20$
• Trace bound: $6$

Defining parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	75.e (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 15 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(20\)
Trace bound:			\(6\)
Distinguishing \(T_p\):			\(2\)

Dimensions

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

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

Trace form

\( 8 q - 12 q^{6} + 4 q^{16} - 12 q^{21} - 4 q^{31} + 36 q^{36} - 24 q^{46} + 48 q^{51} - 44 q^{61} - 8 q^{76} - 72 q^{81} + 36 q^{91} - 36 q^{96}+O(q^{100}) \)

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.e.a	$75$	$2$	75.e	15.e	$4$	$4$	$2$	$0.599$	\(\Q(i, \sqrt{6})\)	\(\Q(\sqrt{-15}) \)		✓			75.2.e.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+\beta _{1}q^{2}+\beta _{3}q^{3}+\beta _{2}q^{4}-3q^{6}-\beta _{3}q^{8}+\cdots\)
75.2.e.b	$75$	$2$	75.e	15.e	$4$	$4$	$2$	$0.599$	\(\Q(i, \sqrt{6})\)	\(\Q(\sqrt{-3}) \)		✓			75.2.e.b	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+\beta _{1}q^{3}-2\beta _{2}q^{4}+\beta _{3}q^{7}+3\beta _{2}q^{9}+\cdots\)