Space of modular forms of level 75, weight 12, and Character 75.e

• Label: 75.12.e
• Level: $75$
• Weight: $12$
• Character orbit: 75.e
• Rep. character: $\chi_{75}(32,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $128$
• Newform subspaces: $5$
• Sturm bound: $120$
• Trace bound: $6$

Defining parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
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:			\(120\)
Trace bound:			\(6\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(75, [\chi])\).

	Total	New	Old
Modular forms	232	136	96
Cusp forms	208	128	80
Eisenstein series	24	8	16

Trace form

128 * q - 504 * q^3 - 5892 * q^6 - 31504 * q^7 + 1714188 * q^12 + 3448472 * q^13 - 93950736 * q^16 - 3729120 * q^18 - 67141692 * q^21 + 21743260 * q^22 - 113282712 * q^27 - 253355948 * q^28 - 406638564 * q^31 + 563443560 * q^33 - 2181876324 * q^36 - 1292573224 * q^37 + 5132304780 * q^42 - 4012128208 * q^43 + 16341659816 * q^46 + 13847034876 * q^48 - 11380265232 * q^51 - 10287332704 * q^52 + 10576855656 * q^57 - 56277468420 * q^58 + 44502748396 * q^61 + 50371112832 * q^63 - 38023124700 * q^66 - 37314988144 * q^67 + 92983114440 * q^72 - 51361482568 * q^73 + 303664299272 * q^76 + 104085843000 * q^78 - 25282289952 * q^81 + 56747065840 * q^82 - 84564024000 * q^87 - 148338843420 * q^88 - 242604946924 * q^91 - 38932525008 * q^93 + 871417041324 * q^96 + 294950050616 * q^97

Decomposition of \(S_{12}^{\mathrm{new}}(75, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
75.12.e.a	$75$	$12$	75.e	15.e	$4$	$4$	$2$	$57.626$	\(\Q(i, \sqrt{6})\)	\(\Q(\sqrt{-3}) \)		✓			75.12.e.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q-3^{5}\beta _{3}q^{3}+2^{11}\beta _{2}q^{4}-25807\beta _{1}q^{7}+\cdots\)
75.12.e.b	$75$	$12$	75.e	15.e	$4$	$4$	$2$	$57.626$	\(\Q(i, \sqrt{6})\)	\(\Q(\sqrt{-15}) \)		✓			75.12.e.b	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+43\beta _{1}q^{2}-3^{5}\beta _{3}q^{3}+3499\beta _{2}q^{4}+\cdots\)
75.12.e.c	$75$	$12$	75.e	15.e	$4$	$24$	$12$	$57.626$		None		✓			75.12.e.c	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{4}]$
75.12.e.d	$75$	$12$	75.e	15.e	$4$	$40$	$20$	$57.626$		None					15.12.e.a	$4$	$0$	\(0\)	\(-504\)	\(0\)	\(-31504\)		$\mathrm{SU}(2)[C_{4}]$
75.12.e.e	$75$	$12$	75.e	15.e	$4$	$56$	$28$	$57.626$		None		✓	✓		75.12.e.e	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{4}]$

Decomposition of \(S_{12}^{\mathrm{old}}(75, [\chi])\) into lower level spaces

\( S_{12}^{\mathrm{old}}(75, [\chi]) \simeq \) \(S_{12}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 2}\)