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

• Label: 75.4.b
• Level: $75$
• Weight: $4$
• Character orbit: 75.b
• Rep. character: $\chi_{75}(49,\cdot)$
• Character field: $\Q$
• Dimension: $8$
• Newform subspaces: $3$
• Sturm bound: $40$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	75.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(40\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(2\)

Dimensions

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

	Total	New	Old
Modular forms	36	8	28
Cusp forms	24	8	16
Eisenstein series	12	0	12

Trace form

8 * q - 36 * q^4 - 24 * q^6 - 72 * q^9 + 112 * q^11 - 324 * q^14 + 180 * q^16 + 276 * q^19 - 108 * q^21 + 468 * q^24 - 764 * q^26 - 488 * q^29 + 68 * q^31 + 1088 * q^34 + 324 * q^36 + 204 * q^39 + 1688 * q^41 + 712 * q^44 - 2232 * q^46 - 1100 * q^49 - 816 * q^51 + 216 * q^54 - 480 * q^56 - 496 * q^59 + 2340 * q^61 - 3284 * q^64 + 1488 * q^66 - 504 * q^69 + 416 * q^71 - 1904 * q^74 + 784 * q^76 + 3280 * q^79 + 648 * q^81 - 936 * q^84 - 1508 * q^86 - 984 * q^89 - 2492 * q^91 + 824 * q^94 - 2748 * q^96 - 1008 * q^99

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.b.a	$75$	$4$	75.b	5.b	$2$	$2$	$2$	$4.425$	\(\Q(\sqrt{-1}) \)	None					15.4.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{2}+3 i q^{3}-q^{4}-9 q^{6}+20 i q^{7}+\cdots\)
75.4.b.b	$75$	$4$	75.b	5.b	$2$	$2$	$2$	$4.425$	\(\Q(\sqrt{-1}) \)	None					15.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}-3 i q^{3}+7 q^{4}+3 q^{6}-24 i q^{7}+\cdots\)
75.4.b.c	$75$	$4$	75.b	5.b	$2$	$4$	$4$	$4.425$	\(\Q(i, \sqrt{19})\)	None		✓	✓		75.4.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{1}+\beta _{3})q^{2}-3\beta _{1}q^{3}+(-12+\cdots)q^{4}+\cdots\)

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

\( S_{4}^{\mathrm{old}}(75, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 2}\)