Space of modular forms of level 425, weight 4, and trivial character

• Label: 425.4.a
• Level: $425$
• Weight: $4$
• Character orbit: 425.a
• Rep. character: $\chi_{425}(1,\cdot)$
• Character field: $\Q$
• Dimension: $76$
• Newform subspaces: $15$
• Sturm bound: $180$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 425 = 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	425.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 15 \)
Sturm bound:			\(180\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(425))\).

	Total	New	Old
Modular forms	142	76	66
Cusp forms	130	76	54
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.

\(5\)	\(17\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(39\)	\(21\)	\(18\)		\(36\)	\(21\)	\(15\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)		\(33\)	\(15\)	\(18\)		\(30\)	\(15\)	\(15\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)		\(32\)	\(18\)	\(14\)		\(29\)	\(18\)	\(11\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)		\(38\)	\(22\)	\(16\)		\(35\)	\(22\)	\(13\)		\(3\)	\(0\)	\(3\)
Plus space		\(+\)		\(77\)	\(43\)	\(34\)		\(71\)	\(43\)	\(28\)		\(6\)	\(0\)	\(6\)
Minus space		\(-\)		\(65\)	\(33\)	\(32\)		\(59\)	\(33\)	\(26\)		\(6\)	\(0\)	\(6\)

Trace form

76 * q - 2 * q^2 + 4 * q^3 + 314 * q^4 - 10 * q^6 + 18 * q^7 + 66 * q^8 + 624 * q^9 + 92 * q^11 + 190 * q^12 + 20 * q^13 - 36 * q^14 + 1074 * q^16 - 34 * q^17 + 294 * q^18 - 36 * q^19 + 48 * q^21 - 150 * q^22 - 182 * q^23 + 354 * q^24 + 460 * q^26 - 476 * q^27 + 100 * q^28 - 214 * q^29 + 22 * q^31 + 766 * q^32 + 292 * q^33 + 68 * q^34 + 2050 * q^36 + 490 * q^37 - 1600 * q^38 + 828 * q^39 + 276 * q^41 - 220 * q^42 + 712 * q^43 - 1294 * q^44 + 464 * q^46 - 368 * q^47 + 962 * q^48 + 3868 * q^49 + 204 * q^51 - 268 * q^52 + 820 * q^53 - 980 * q^54 + 1744 * q^56 + 672 * q^57 + 2686 * q^58 - 328 * q^59 + 1534 * q^61 + 712 * q^62 - 2338 * q^63 + 4626 * q^64 - 732 * q^66 - 2500 * q^67 - 408 * q^68 - 5016 * q^69 - 650 * q^71 + 5882 * q^72 - 1848 * q^73 + 530 * q^74 - 1088 * q^76 - 592 * q^77 + 2424 * q^78 - 2362 * q^79 + 6824 * q^81 - 4420 * q^82 + 1560 * q^83 + 5556 * q^84 + 2540 * q^86 + 884 * q^87 - 90 * q^88 - 3096 * q^89 + 2432 * q^91 - 5268 * q^92 - 4152 * q^93 + 2440 * q^94 - 1578 * q^96 + 2180 * q^97 + 1122 * q^98 + 9168 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(425))\) 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	17
425.4.a.a	$425$	$4$	425.a	1.a	$1$	$1$	$1$	$25.076$	\(\Q\)	None	✓				85.4.a.c	$1$	$0$	\(-3\)	\(-10\)	\(0\)	\(22\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}-10q^{3}+q^{4}+30q^{6}+22q^{7}+\cdots\)
425.4.a.b	$425$	$4$	425.a	1.a	$1$	$1$	$1$	$25.076$	\(\Q\)	None	✓				85.4.a.b	$1$	$0$	\(-3\)	\(5\)	\(0\)	\(22\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}+5q^{3}+q^{4}-15q^{6}+22q^{7}+\cdots\)
425.4.a.c	$425$	$4$	425.a	1.a	$1$	$1$	$1$	$25.076$	\(\Q\)	None	✓				85.4.a.a	$1$	$1$	\(-3\)	\(7\)	\(0\)	\(22\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}+7q^{3}+q^{4}-21q^{6}+22q^{7}+\cdots\)
425.4.a.d	$425$	$4$	425.a	1.a	$1$	$1$	$1$	$25.076$	\(\Q\)	None	✓				17.4.a.a	$1$	$0$	\(3\)	\(8\)	\(0\)	\(28\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}+8q^{3}+q^{4}+24q^{6}+28q^{7}+\cdots\)
425.4.a.e	$425$	$4$	425.a	1.a	$1$	$2$	$2$	$25.076$	\(\Q(\sqrt{3}) \)	None	✓				85.4.a.d	$1$	$1$	\(4\)	\(2\)	\(0\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{2}+(1+\beta )q^{3}+(-1+4\beta )q^{4}+\cdots\)
425.4.a.f	$425$	$4$	425.a	1.a	$1$	$3$	$3$	$25.076$	3.3.568.1	None	✓				85.4.a.f	$1$	$1$	\(-3\)	\(-9\)	\(0\)	\(-34\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1}+\beta _{2})q^{2}+(-4+3\beta _{1}+\cdots)q^{3}+\cdots\)
425.4.a.g	$425$	$4$	425.a	1.a	$1$	$3$	$3$	$25.076$	3.3.2636.1	None	✓				17.4.a.b	$1$	$1$	\(-1\)	\(-4\)	\(0\)	\(-22\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-\beta _{1}+\beta _{2})q^{2}+(-2+\beta _{1}-2\beta _{2})q^{3}+\cdots\)
425.4.a.h	$425$	$4$	425.a	1.a	$1$	$3$	$3$	$25.076$	3.3.1304.1	None	✓				85.4.a.e	$1$	$0$	\(6\)	\(4\)	\(0\)	\(-8\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(2+\beta _{1})q^{2}+(1+\beta _{1}+\beta _{2})q^{3}+(3+\cdots)q^{4}+\cdots\)
425.4.a.i	$425$	$4$	425.a	1.a	$1$	$5$	$5$	$25.076$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				85.4.a.g	$1$	$0$	\(-2\)	\(1\)	\(0\)	\(-10\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{2}+\beta _{2}q^{3}+(6-2\beta _{1}+\beta _{3})q^{4}+\cdots\)
425.4.a.j	$425$	$4$	425.a	1.a	$1$	$6$	$6$	$25.076$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓			425.4.a.j	$1$	$1$	\(-1\)	\(6\)	\(0\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{3})q^{3}+(2+\beta _{2})q^{4}+\cdots\)
425.4.a.k	$425$	$4$	425.a	1.a	$1$	$6$	$6$	$25.076$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓		425.4.a.j	$1$	$1$	\(1\)	\(-6\)	\(0\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-1-\beta _{3})q^{3}+(2+\beta _{2})q^{4}+\cdots\)
425.4.a.l	$425$	$4$	425.a	1.a	$1$	$10$	$10$	$25.076$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓			425.4.a.l	$1$	$0$	\(-1\)	\(6\)	\(0\)	\(16\)		$-$	$-$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1-\beta _{6})q^{3}+(6+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
425.4.a.m	$425$	$4$	425.a	1.a	$1$	$10$	$10$	$25.076$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓	✓		425.4.a.l	$1$	$0$	\(1\)	\(-6\)	\(0\)	\(-16\)		$+$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-1+\beta _{6})q^{3}+(6+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
425.4.a.n	$425$	$4$	425.a	1.a	$1$	$12$	$12$	$25.076$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓		✓		85.4.b.a	$1$	$1$	\(-4\)	\(-12\)	\(0\)	\(-70\)		$-$	$+$	$-$	$2^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-1+\beta _{4})q^{3}+(4+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
425.4.a.o	$425$	$4$	425.a	1.a	$1$	$12$	$12$	$25.076$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓		✓		85.4.b.a	$1$	$0$	\(4\)	\(12\)	\(0\)	\(70\)		$-$	$-$	$+$	$2^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1-\beta _{4})q^{3}+(4+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(425))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(425)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(85))\)\(^{\oplus 2}\)