Space of modular forms of level 75, weight 8, and Character 75.e

• Label: 75.8.e
• Level: $75$
• Weight: $8$
• Character orbit: 75.e
• Rep. character: $\chi_{75}(32,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $80$
• Newform subspaces: $5$
• Sturm bound: $80$
• Trace bound: $6$

Defining parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	75.e (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 15 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(80\)
Trace bound:			\(6\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(75, [\chi])\).

	Total	New	Old
Modular forms	152	88	64
Cusp forms	128	80	48
Eisenstein series	24	8	16

Trace form

80 * q - 24 * q^3 - 1860 * q^6 - 1344 * q^7 + 2028 * q^12 - 16848 * q^13 - 239120 * q^16 + 106560 * q^18 - 168300 * q^21 - 166020 * q^22 + 264168 * q^27 - 602028 * q^28 + 382700 * q^31 + 877800 * q^33 + 709020 * q^36 + 635616 * q^37 - 1219380 * q^42 - 1397328 * q^43 - 5574360 * q^46 + 476796 * q^48 + 3601440 * q^51 + 6426336 * q^52 - 8457264 * q^57 - 986340 * q^58 - 17552300 * q^61 + 1973952 * q^63 + 36430980 * q^66 + 16720176 * q^67 - 36245880 * q^72 - 4059528 * q^73 - 51035000 * q^76 + 26156280 * q^78 + 30015720 * q^81 + 53633520 * q^82 - 44480160 * q^87 - 74717340 * q^88 - 64039500 * q^91 + 58742592 * q^93 + 203047980 * q^96 + 75148056 * q^97

Decomposition of \(S_{8}^{\mathrm{new}}(75, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
75.8.e.a	$75$	$8$	75.e	15.e	$4$	$4$	$2$	$23.429$	\(\Q(i, \sqrt{6})\)	\(\Q(\sqrt{-15}) \)		✓			75.8.e.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+13\beta _{1}q^{2}+3^{3}\beta _{3}q^{3}+379\beta _{2}q^{4}+\cdots\)
75.8.e.b	$75$	$8$	75.e	15.e	$4$	$4$	$2$	$23.429$	\(\Q(i, \sqrt{6})\)	\(\Q(\sqrt{-3}) \)		✓			75.8.e.b	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q-3^{3}\beta _{3}q^{3}+2^{7}\beta _{2}q^{4}+757\beta _{1}q^{7}+\cdots\)
75.8.e.c	$75$	$8$	75.e	15.e	$4$	$16$	$8$	$23.429$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		✓			75.8.e.c	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{28}\cdot 3^{12}\cdot 5^{8}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{4}q^{2}+(\beta _{5}-\beta _{8})q^{3}+(-2\beta _{6}-\beta _{13}+\cdots)q^{4}+\cdots\)
75.8.e.d	$75$	$8$	75.e	15.e	$4$	$24$	$12$	$23.429$		None					15.8.e.a	$4$	$0$	\(0\)	\(-24\)	\(0\)	\(-1344\)		$\mathrm{SU}(2)[C_{4}]$
75.8.e.e	$75$	$8$	75.e	15.e	$4$	$32$	$16$	$23.429$		None		✓	✓		75.8.e.e	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{4}]$

Decomposition of \(S_{8}^{\mathrm{old}}(75, [\chi])\) into lower level spaces

\( S_{8}^{\mathrm{old}}(75, [\chi]) \simeq \) \(S_{8}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 2}\)