Space of modular forms of level 75, weight 4, and Character 75.e

• Label: 75.4.e
• Level: $75$
• Weight: $4$
• Character orbit: 75.e
• Rep. character: $\chi_{75}(32,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $32$
• Newform subspaces: $4$
• Sturm bound: $40$
• Trace bound: $6$

Defining parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	75.e (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 15 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(40\)
Trace bound:			\(6\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(75, [\chi])\).

	Total	New	Old
Modular forms	72	40	32
Cusp forms	48	32	16
Eisenstein series	24	8	16

Trace form

32 * q + 6 * q^3 + 12 * q^6 + 16 * q^7 - 132 * q^12 - 68 * q^13 - 784 * q^16 + 240 * q^18 + 972 * q^21 + 500 * q^22 - 702 * q^27 - 508 * q^28 - 896 * q^31 + 240 * q^33 + 2364 * q^36 + 1156 * q^37 - 540 * q^42 - 548 * q^43 - 1496 * q^46 + 1116 * q^48 - 1128 * q^51 - 224 * q^52 - 684 * q^57 - 60 * q^58 - 2216 * q^61 - 1428 * q^63 - 1380 * q^66 - 404 * q^67 + 1800 * q^72 + 2512 * q^73 + 10248 * q^76 + 360 * q^78 + 1332 * q^81 - 2800 * q^82 + 1680 * q^87 - 2460 * q^88 - 3536 * q^91 - 3408 * q^93 - 10164 * q^96 - 1904 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(75, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
75.4.e.a	$75$	$4$	75.e	15.e	$4$	$4$	$2$	$4.425$	\(\Q(i, \sqrt{6})\)	\(\Q(\sqrt{-3}) \)		✓			75.4.e.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{2}$	$\mathrm{U}(1)[D_{4}]$	\(q+\beta _{1}q^{3}-8\beta _{2}q^{4}-6\beta _{3}q^{7}+3^{3}\beta _{2}q^{9}+\cdots\)
75.4.e.b	$75$	$4$	75.e	15.e	$4$	$4$	$2$	$4.425$	\(\Q(i, \sqrt{6})\)	\(\Q(\sqrt{-15}) \)		✓			75.4.e.b	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{2}$	$\mathrm{U}(1)[D_{4}]$	\(q+\beta _{1}q^{2}-\beta _{3}q^{3}+19\beta _{2}q^{4}+3^{3}q^{6}+\cdots\)
75.4.e.c	$75$	$4$	75.e	15.e	$4$	$8$	$4$	$4.425$	8.0.\(\cdots\).8	None					15.4.e.a	$4$	$0$	\(0\)	\(6\)	\(0\)	\(16\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{3}q^{2}+(1+\beta _{2}-\beta _{5}+\beta _{6}-\beta _{7})q^{3}+\cdots\)
75.4.e.d	$75$	$4$	75.e	15.e	$4$	$16$	$8$	$4.425$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		✓	✓		75.4.e.d	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3^{8}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{3}q^{2}-\beta _{8}q^{3}+(6\beta _{1}+\beta _{5})q^{4}+(-6+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(75, [\chi])\) into lower level spaces

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