Space of modular forms of level 475, weight 4, and trivial character

• Label: 475.4.a
• Level: $475$
• Weight: $4$
• Character orbit: 475.a
• Rep. character: $\chi_{475}(1,\cdot)$
• Character field: $\Q$
• Dimension: $86$
• Newform subspaces: $15$
• Sturm bound: $200$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	475.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 15 \)
Sturm bound:			\(200\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(475))\).

	Total	New	Old
Modular forms	156	86	70
Cusp forms	144	86	58
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.

\(5\)	\(19\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(42\)	\(21\)	\(21\)		\(39\)	\(21\)	\(18\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)		\(36\)	\(19\)	\(17\)		\(33\)	\(19\)	\(14\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)		\(38\)	\(21\)	\(17\)		\(35\)	\(21\)	\(14\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)		\(40\)	\(25\)	\(15\)		\(37\)	\(25\)	\(12\)		\(3\)	\(0\)	\(3\)
Plus space		\(+\)		\(82\)	\(46\)	\(36\)		\(76\)	\(46\)	\(30\)		\(6\)	\(0\)	\(6\)
Minus space		\(-\)		\(74\)	\(40\)	\(34\)		\(68\)	\(40\)	\(28\)		\(6\)	\(0\)	\(6\)

Trace form

86 * q - 4 * q^2 + 4 * q^3 + 358 * q^4 - 10 * q^6 - 4 * q^7 + 764 * q^9 + 38 * q^11 + 56 * q^12 - 92 * q^13 + 36 * q^14 + 1338 * q^16 - 40 * q^17 - 36 * q^18 + 38 * q^19 - 240 * q^21 - 488 * q^22 - 94 * q^23 + 254 * q^24 + 114 * q^26 - 56 * q^27 + 606 * q^28 - 788 * q^29 + 44 * q^31 + 1096 * q^32 + 356 * q^33 - 208 * q^34 + 4088 * q^36 + 428 * q^37 - 114 * q^38 + 606 * q^39 - 96 * q^41 + 1246 * q^42 + 518 * q^43 - 1596 * q^44 + 328 * q^46 - 1922 * q^47 + 1996 * q^48 + 4410 * q^49 - 1212 * q^51 - 1144 * q^52 + 1180 * q^53 + 2182 * q^54 + 724 * q^56 - 114 * q^57 - 262 * q^58 + 1140 * q^59 + 1230 * q^61 + 1468 * q^62 - 66 * q^63 + 6366 * q^64 - 2056 * q^66 - 376 * q^67 + 58 * q^68 - 6212 * q^69 - 3360 * q^71 - 1876 * q^72 - 2020 * q^73 + 4540 * q^74 + 380 * q^76 - 262 * q^77 + 4444 * q^78 - 148 * q^79 + 7262 * q^81 - 3896 * q^82 + 20 * q^83 - 5152 * q^84 - 5560 * q^86 - 450 * q^87 - 2404 * q^88 + 2000 * q^89 - 2584 * q^91 - 2110 * q^92 - 5172 * q^93 + 712 * q^94 + 1122 * q^96 - 820 * q^97 + 2164 * q^98 - 2158 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(475))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		5	19
475.4.a.a	$475$	$4$	475.a	1.a	$1$	$1$	$1$	$28.026$	\(\Q\)	None	✓				95.4.a.d	$1$	$1$	\(-5\)	\(-4\)	\(0\)	\(32\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{2}-4q^{3}+17q^{4}+20q^{6}+2^{5}q^{7}+\cdots\)
475.4.a.b	$475$	$4$	475.a	1.a	$1$	$1$	$1$	$28.026$	\(\Q\)	None	✓				95.4.a.c	$1$	$1$	\(-3\)	\(-7\)	\(0\)	\(-11\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}-7q^{3}+q^{4}+21q^{6}-11q^{7}+\cdots\)
475.4.a.c	$475$	$4$	475.a	1.a	$1$	$1$	$1$	$28.026$	\(\Q\)	None	✓				95.4.a.b	$1$	$1$	\(-3\)	\(5\)	\(0\)	\(1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}+5q^{3}+q^{4}-15q^{6}+q^{7}+\cdots\)
475.4.a.d	$475$	$4$	475.a	1.a	$1$	$1$	$1$	$28.026$	\(\Q\)	None	✓				95.4.a.a	$1$	$1$	\(0\)	\(-4\)	\(0\)	\(22\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{3}-8q^{4}+22q^{7}-11q^{9}-12q^{11}+\cdots\)
475.4.a.e	$475$	$4$	475.a	1.a	$1$	$1$	$1$	$28.026$	\(\Q\)	None	✓				19.4.a.a	$1$	$1$	\(3\)	\(5\)	\(0\)	\(-11\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}+5q^{3}+q^{4}+15q^{6}-11q^{7}+\cdots\)
475.4.a.f	$475$	$4$	475.a	1.a	$1$	$3$	$3$	$28.026$	3.3.3144.1	None	✓				19.4.a.b	$1$	$0$	\(-3\)	\(-1\)	\(0\)	\(35\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1}+\beta _{2})q^{2}+(\beta _{1}-2\beta _{2})q^{3}+\cdots\)
475.4.a.g	$475$	$4$	475.a	1.a	$1$	$3$	$3$	$28.026$	3.3.1304.1	None	✓				95.4.a.e	$1$	$0$	\(3\)	\(11\)	\(0\)	\(5\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}+(4+\beta _{1}+\beta _{2})q^{3}+(1+\cdots)q^{4}+\cdots\)
475.4.a.h	$475$	$4$	475.a	1.a	$1$	$5$	$5$	$28.026$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				95.4.a.f	$1$	$1$	\(3\)	\(4\)	\(0\)	\(-72\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(\beta _{1}+\beta _{3})q^{3}+(4-\beta _{1}+\cdots)q^{4}+\cdots\)
475.4.a.i	$475$	$4$	475.a	1.a	$1$	$6$	$6$	$28.026$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				95.4.a.g	$1$	$0$	\(1\)	\(-5\)	\(0\)	\(-5\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-1+\beta _{1}+\beta _{5})q^{3}+(4+\beta _{2}+\cdots)q^{4}+\cdots\)
475.4.a.j	$475$	$4$	475.a	1.a	$1$	$9$	$9$	$28.026$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓			475.4.a.j	$1$	$1$	\(-7\)	\(-6\)	\(0\)	\(0\)		$-$	$+$	$-$	$23$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(-1-\beta _{4})q^{3}+(5+\cdots)q^{4}+\cdots\)
475.4.a.k	$475$	$4$	475.a	1.a	$1$	$9$	$9$	$28.026$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓		475.4.a.k	$1$	$1$	\(-5\)	\(-6\)	\(0\)	\(-28\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(-\beta _{1}+\beta _{4})q^{3}+(5+\cdots)q^{4}+\cdots\)
475.4.a.l	$475$	$4$	475.a	1.a	$1$	$9$	$9$	$28.026$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓			475.4.a.k	$1$	$0$	\(5\)	\(6\)	\(0\)	\(28\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(\beta _{1}-\beta _{4})q^{3}+(5-\beta _{1}+\cdots)q^{4}+\cdots\)
475.4.a.m	$475$	$4$	475.a	1.a	$1$	$9$	$9$	$28.026$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓		475.4.a.j	$1$	$0$	\(7\)	\(6\)	\(0\)	\(0\)		$+$	$+$	$+$	$23$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(1+\beta _{4})q^{3}+(5-2\beta _{1}+\cdots)q^{4}+\cdots\)
475.4.a.n	$475$	$4$	475.a	1.a	$1$	$12$	$12$	$28.026$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓		✓		95.4.b.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{7}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-\beta _{5}q^{3}+(3+\beta _{2})q^{4}+(-5+\cdots)q^{6}+\cdots\)
475.4.a.o	$475$	$4$	475.a	1.a	$1$	$16$	$16$	$28.026$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓		✓		95.4.b.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{5}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-\beta _{11}q^{3}+(5+\beta _{2})q^{4}+(5+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(475))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(475)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(95))\)\(^{\oplus 2}\)