Space of modular forms of level 75, weight 10, and Character 75.e

• Label: 75.10.e
• Level: $75$
• Weight: $10$
• Character orbit: 75.e
• Rep. character: $\chi_{75}(32,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $104$
• Newform subspaces: $7$
• Sturm bound: $100$
• Trace bound: $6$

Defining parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	75.e (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 15 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(100\)
Trace bound:			\(6\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(75, [\chi])\).

	Total	New	Old
Modular forms	192	112	80
Cusp forms	168	104	64
Eisenstein series	24	8	16

Trace form

104 * q + 150 * q^3 + 9084 * q^6 + 9760 * q^7 - 191940 * q^12 - 114260 * q^13 - 5661328 * q^16 - 994800 * q^18 - 1184436 * q^21 - 308260 * q^22 - 4786830 * q^27 + 12953300 * q^28 - 8821952 * q^31 - 19638120 * q^33 + 101033628 * q^36 + 35578660 * q^37 - 46648260 * q^42 + 118721020 * q^43 - 6290072 * q^46 - 275395140 * q^48 + 252102264 * q^51 + 157103680 * q^52 - 48163500 * q^57 + 209634300 * q^58 + 291515608 * q^61 - 89858580 * q^63 - 493527900 * q^66 - 153311540 * q^67 + 1534046760 * q^72 - 628070360 * q^73 + 3547891656 * q^76 + 403337880 * q^78 - 1980657756 * q^81 - 4138031440 * q^82 + 2878139760 * q^87 + 595851300 * q^88 + 7018496848 * q^91 + 293672640 * q^93 - 14700037908 * q^96 - 3987642200 * q^97

Decomposition of \(S_{10}^{\mathrm{new}}(75, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
75.10.e.a	$75$	$10$	75.e	15.e	$4$	$4$	$2$	$38.628$	\(\Q(i, \sqrt{86})\)	None		✓			75.10.e.a	$4$	$0$	\(0\)	\(-360\)	\(0\)	\(25200\)	$3^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+2\beta _{1}q^{2}+(-90-90\beta _{2}-3\beta _{3})q^{3}+\cdots\)
75.10.e.b	$75$	$10$	75.e	15.e	$4$	$4$	$2$	$38.628$	\(\Q(i, \sqrt{6})\)	\(\Q(\sqrt{-3}) \)		✓			75.10.e.b	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{6}$	$\mathrm{U}(1)[D_{4}]$	\(q-3\beta _{3}q^{3}+2^{9}\beta _{2}q^{4}-38\beta _{1}q^{7}-3^{9}\beta _{2}q^{9}+\cdots\)
75.10.e.c	$75$	$10$	75.e	15.e	$4$	$4$	$2$	$38.628$	\(\Q(i, \sqrt{6})\)	\(\Q(\sqrt{-15}) \)		✓			75.10.e.c	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{4}$	$\mathrm{U}(1)[D_{4}]$	\(q+\beta _{1}q^{2}-9\beta _{3}q^{3}-269\beta _{2}q^{4}+3^{7}q^{6}+\cdots\)
75.10.e.d	$75$	$10$	75.e	15.e	$4$	$4$	$2$	$38.628$	\(\Q(i, \sqrt{86})\)	None		✓			75.10.e.a	$4$	$0$	\(0\)	\(360\)	\(0\)	\(-25200\)	$3^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+2\beta _{1}q^{2}+(90+90\beta _{2}-3\beta _{3})q^{3}+\cdots\)
75.10.e.e	$75$	$10$	75.e	15.e	$4$	$8$	$4$	$38.628$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		✓			75.10.e.e	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}\cdot 3^{8}$	$\mathrm{SU}(2)[C_{4}]$	\(q+2\beta _{3}q^{2}+(-9\beta _{2}-\beta _{5})q^{3}+208\beta _{1}q^{4}+\cdots\)
75.10.e.f	$75$	$10$	75.e	15.e	$4$	$32$	$16$	$38.628$		None					15.10.e.a	$4$	$0$	\(0\)	\(150\)	\(0\)	\(9760\)		$\mathrm{SU}(2)[C_{4}]$
75.10.e.g	$75$	$10$	75.e	15.e	$4$	$48$	$24$	$38.628$		None		✓	✓		75.10.e.g	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{4}]$

Decomposition of \(S_{10}^{\mathrm{old}}(75, [\chi])\) into lower level spaces

\( S_{10}^{\mathrm{old}}(75, [\chi]) \simeq \) \(S_{10}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 2}\)