Space of modular forms of level 75, weight 4, and trivial character

• Label: 75.4.a
• Level: $75$
• Weight: $4$
• Character orbit: 75.a
• Rep. character: $\chi_{75}(1,\cdot)$
• Character field: $\Q$
• Dimension: $10$
• Newform subspaces: $6$
• Sturm bound: $40$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(40\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(75))\).

	Total	New	Old
Modular forms	36	10	26
Cusp forms	24	10	14
Eisenstein series	12	0	12

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)	\(5\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(10\)	\(3\)	\(7\)		\(7\)	\(3\)	\(4\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)		\(8\)	\(2\)	\(6\)		\(5\)	\(2\)	\(3\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)		\(8\)	\(1\)	\(7\)		\(5\)	\(1\)	\(4\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)		\(10\)	\(4\)	\(6\)		\(7\)	\(4\)	\(3\)		\(3\)	\(0\)	\(3\)
Plus space		\(+\)		\(20\)	\(7\)	\(13\)		\(14\)	\(7\)	\(7\)		\(6\)	\(0\)	\(6\)
Minus space		\(-\)		\(16\)	\(3\)	\(13\)		\(10\)	\(3\)	\(7\)		\(6\)	\(0\)	\(6\)

Trace form

10 * q - 4 * q^2 + 60 * q^4 + 4 * q^7 + 36 * q^8 + 90 * q^9 + 24 * q^12 - 96 * q^13 + 60 * q^14 + 260 * q^16 - 40 * q^17 - 36 * q^18 - 20 * q^19 + 60 * q^21 + 20 * q^22 + 288 * q^23 - 180 * q^24 - 1020 * q^26 - 188 * q^28 - 300 * q^29 + 260 * q^31 - 116 * q^32 - 228 * q^33 - 520 * q^34 + 540 * q^36 + 104 * q^37 + 392 * q^38 + 420 * q^39 + 420 * q^41 + 252 * q^42 + 96 * q^43 - 1440 * q^44 - 1520 * q^46 - 232 * q^47 - 336 * q^48 + 950 * q^49 + 80 * q^52 - 112 * q^53 - 360 * q^56 - 312 * q^57 + 4 * q^58 + 2600 * q^61 - 312 * q^62 + 36 * q^63 + 2060 * q^64 + 280 * q^67 - 152 * q^68 + 360 * q^69 + 960 * q^71 + 324 * q^72 + 468 * q^73 + 5280 * q^74 - 1680 * q^76 + 1728 * q^77 + 600 * q^78 - 1760 * q^79 + 810 * q^81 - 1112 * q^82 - 1368 * q^83 - 1560 * q^84 + 2700 * q^86 - 924 * q^87 + 276 * q^88 + 1860 * q^89 - 1500 * q^91 - 1056 * q^92 + 1464 * q^93 - 4240 * q^94 - 5220 * q^96 - 580 * q^97 - 404 * q^98

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(75))\) 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					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	5
75.4.a.a	$75$	$4$	75.a	1.a	$1$	$1$	$1$	$4.425$	\(\Q\)	None	✓		✓	✓	15.4.a.b	$1$	$1$	\(-3\)	\(3\)	\(0\)	\(-20\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}+3q^{3}+q^{4}-9q^{6}-20q^{7}+\cdots\)
75.4.a.b	$75$	$4$	75.a	1.a	$1$	$1$	$1$	$4.425$	\(\Q\)	None	✓				15.4.a.a	$1$	$0$	\(-1\)	\(-3\)	\(0\)	\(24\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-3q^{3}-7q^{4}+3q^{6}+24q^{7}+\cdots\)
75.4.a.c	$75$	$4$	75.a	1.a	$1$	$2$	$2$	$4.425$	\(\Q(\sqrt{41}) \)	None	✓		✓	✓	15.4.b.a	$1$	$1$	\(-3\)	\(-6\)	\(0\)	\(-6\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}-3q^{3}+(3+3\beta )q^{4}+\cdots\)
75.4.a.d	$75$	$4$	75.a	1.a	$1$	$2$	$2$	$4.425$	\(\Q(\sqrt{19}) \)	None	✓	✓			75.4.a.d	$1$	$0$	\(-2\)	\(6\)	\(0\)	\(26\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+3q^{3}+(12-2\beta )q^{4}+\cdots\)
75.4.a.e	$75$	$4$	75.a	1.a	$1$	$2$	$2$	$4.425$	\(\Q(\sqrt{19}) \)	None	✓	✓	✓		75.4.a.d	$1$	$0$	\(2\)	\(-6\)	\(0\)	\(-26\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}-3q^{3}+(12+2\beta )q^{4}+(-3+\cdots)q^{6}+\cdots\)
75.4.a.f	$75$	$4$	75.a	1.a	$1$	$2$	$2$	$4.425$	\(\Q(\sqrt{41}) \)	None	✓				15.4.b.a	$1$	$0$	\(3\)	\(6\)	\(0\)	\(6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+3q^{3}+(3+3\beta )q^{4}+(3+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(75))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(75)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)