Space of modular forms of level 507, weight 4, and trivial character

• Label: 507.4.a
• Level: $507$
• Weight: $4$
• Character orbit: 507.a
• Rep. character: $\chi_{507}(1,\cdot)$
• Character field: $\Q$
• Dimension: $78$
• Newform subspaces: $18$
• Sturm bound: $242$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	507.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 18 \)
Sturm bound:			\(242\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(507))\).

	Total	New	Old
Modular forms	196	78	118
Cusp forms	168	78	90
Eisenstein series	28	0	28

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

\(3\)	\(13\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(51\)	\(20\)	\(31\)		\(44\)	\(20\)	\(24\)		\(7\)	\(0\)	\(7\)
\(+\)	\(-\)	\(-\)		\(47\)	\(19\)	\(28\)		\(40\)	\(19\)	\(21\)		\(7\)	\(0\)	\(7\)
\(-\)	\(+\)	\(-\)		\(47\)	\(16\)	\(31\)		\(40\)	\(16\)	\(24\)		\(7\)	\(0\)	\(7\)
\(-\)	\(-\)	\(+\)		\(51\)	\(23\)	\(28\)		\(44\)	\(23\)	\(21\)		\(7\)	\(0\)	\(7\)
Plus space		\(+\)		\(102\)	\(43\)	\(59\)		\(88\)	\(43\)	\(45\)		\(14\)	\(0\)	\(14\)
Minus space		\(-\)		\(94\)	\(35\)	\(59\)		\(80\)	\(35\)	\(45\)		\(14\)	\(0\)	\(14\)

Trace form

78 * q - 4 * q^2 + 320 * q^4 - 16 * q^5 - 32 * q^7 - 48 * q^8 + 702 * q^9 + 72 * q^10 + 96 * q^11 + 272 * q^14 + 24 * q^15 + 1384 * q^16 + 64 * q^17 - 36 * q^18 - 216 * q^19 + 92 * q^20 - 84 * q^21 + 272 * q^22 + 224 * q^23 + 180 * q^24 + 1850 * q^25 + 48 * q^28 - 544 * q^29 - 288 * q^30 - 128 * q^31 - 28 * q^32 - 192 * q^33 - 440 * q^34 - 224 * q^35 + 2880 * q^36 + 12 * q^37 - 664 * q^38 - 112 * q^40 - 296 * q^41 + 360 * q^42 + 416 * q^43 - 20 * q^44 - 144 * q^45 - 248 * q^46 + 48 * q^47 + 528 * q^48 + 4394 * q^49 + 380 * q^50 + 864 * q^51 + 600 * q^53 + 1024 * q^55 + 440 * q^56 + 84 * q^57 + 896 * q^58 + 1072 * q^59 + 1188 * q^60 - 584 * q^61 - 1068 * q^62 - 288 * q^63 + 7204 * q^64 - 1608 * q^66 + 360 * q^67 - 972 * q^68 - 168 * q^69 - 656 * q^70 + 976 * q^71 - 432 * q^72 + 1804 * q^73 - 2108 * q^74 - 480 * q^75 - 2248 * q^76 - 1704 * q^77 - 700 * q^79 + 700 * q^80 + 6318 * q^81 - 3948 * q^82 + 3072 * q^83 - 624 * q^84 - 2432 * q^85 - 2640 * q^86 + 1476 * q^87 - 1272 * q^88 - 1376 * q^89 + 648 * q^90 - 1156 * q^92 - 1428 * q^93 + 824 * q^94 + 4944 * q^95 - 840 * q^96 - 2820 * q^97 + 5644 * q^98 + 864 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(507))\) 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	13
507.4.a.a	$507$	$4$	507.a	1.a	$1$	$1$	$1$	$29.914$	\(\Q\)	None	✓				39.4.e.a	$1$	$1$	\(-3\)	\(3\)	\(9\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}+3q^{3}+q^{4}+9q^{5}-9q^{6}+\cdots\)
507.4.a.b	$507$	$4$	507.a	1.a	$1$	$1$	$1$	$29.914$	\(\Q\)	None	✓				39.4.e.b	$1$	$1$	\(-1\)	\(3\)	\(7\)	\(-10\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+3q^{3}-7q^{4}+7q^{5}-3q^{6}+\cdots\)
507.4.a.c	$507$	$4$	507.a	1.a	$1$	$1$	$1$	$29.914$	\(\Q\)	None	✓				39.4.a.a	$1$	$0$	\(0\)	\(-3\)	\(12\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-8q^{4}+12q^{5}-2q^{7}+9q^{9}+\cdots\)
507.4.a.d	$507$	$4$	507.a	1.a	$1$	$1$	$1$	$29.914$	\(\Q\)	None	✓				39.4.e.b	$1$	$1$	\(1\)	\(3\)	\(-7\)	\(10\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+3q^{3}-7q^{4}-7q^{5}+3q^{6}+\cdots\)
507.4.a.e	$507$	$4$	507.a	1.a	$1$	$1$	$1$	$29.914$	\(\Q\)	None	✓				39.4.e.a	$1$	$1$	\(3\)	\(3\)	\(-9\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}+3q^{3}+q^{4}-9q^{5}+9q^{6}+\cdots\)
507.4.a.f	$507$	$4$	507.a	1.a	$1$	$2$	$2$	$29.914$	\(\Q(\sqrt{14}) \)	None	✓				39.4.a.b	$1$	$0$	\(-2\)	\(-6\)	\(-24\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}-3q^{3}+(7-2\beta )q^{4}+\cdots\)
507.4.a.g	$507$	$4$	507.a	1.a	$1$	$2$	$2$	$29.914$	\(\Q(\sqrt{3}) \)	None	✓				39.4.j.a	$2$	$1$	\(0\)	\(-6\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-8q^{4}+3\beta q^{5}+6\beta q^{7}+9q^{9}+\cdots\)
507.4.a.h	$507$	$4$	507.a	1.a	$1$	$3$	$3$	$29.914$	3.3.3144.1	None	✓				39.4.a.c	$1$	$1$	\(-2\)	\(9\)	\(-4\)	\(-30\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+3q^{3}+(3+\beta _{2})q^{4}+\cdots\)
507.4.a.i	$507$	$4$	507.a	1.a	$1$	$4$	$4$	$29.914$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				39.4.e.c	$1$	$0$	\(-2\)	\(-12\)	\(6\)	\(14\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-3q^{3}+(5+\beta _{1}+\beta _{2})q^{4}+\cdots\)
507.4.a.j	$507$	$4$	507.a	1.a	$1$	$4$	$4$	$29.914$	4.4.5054412.1	None	✓				39.4.b.a	$2$	$1$	\(0\)	\(-12\)	\(0\)	\(0\)		$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-3q^{3}+(7+\beta _{3})q^{4}-\beta _{2}q^{5}+\cdots\)
507.4.a.k	$507$	$4$	507.a	1.a	$1$	$4$	$4$	$29.914$	\(\Q(\sqrt{3}, \sqrt{17})\)	None	✓				39.4.j.b	$2$	$1$	\(0\)	\(-12\)	\(0\)	\(0\)		$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{2}-3q^{3}+9q^{4}+(3\beta _{1}-2\beta _{3})q^{5}+\cdots\)
507.4.a.l	$507$	$4$	507.a	1.a	$1$	$4$	$4$	$29.914$	4.4.1362828.1	None	✓				39.4.b.b	$2$	$0$	\(0\)	\(12\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+3q^{3}+(4+\beta _{3})q^{4}+(2\beta _{1}+\cdots)q^{5}+\cdots\)
507.4.a.m	$507$	$4$	507.a	1.a	$1$	$4$	$4$	$29.914$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				39.4.e.c	$1$	$0$	\(2\)	\(-12\)	\(-6\)	\(-14\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-3q^{3}+(5+\beta _{1}+\beta _{2})q^{4}+\cdots\)
507.4.a.n	$507$	$4$	507.a	1.a	$1$	$9$	$9$	$29.914$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓		507.4.a.n	$1$	$1$	\(-8\)	\(-27\)	\(-41\)	\(-1\)		$+$	$-$	$-$	$13^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}-3q^{3}+(3-2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
507.4.a.o	$507$	$4$	507.a	1.a	$1$	$9$	$9$	$29.914$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓		507.4.a.o	$1$	$1$	\(-6\)	\(27\)	\(-33\)	\(-83\)		$-$	$+$	$-$	$13^{2}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{3})q^{2}+3q^{3}+(5+\beta _{3}+\beta _{5}+\cdots)q^{4}+\cdots\)
507.4.a.p	$507$	$4$	507.a	1.a	$1$	$9$	$9$	$29.914$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓			507.4.a.o	$1$	$0$	\(6\)	\(27\)	\(33\)	\(83\)		$-$	$-$	$+$	$13^{2}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{3})q^{2}+3q^{3}+(5+\beta _{3}+\beta _{5}+\cdots)q^{4}+\cdots\)
507.4.a.q	$507$	$4$	507.a	1.a	$1$	$9$	$9$	$29.914$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓		507.4.a.n	$1$	$0$	\(8\)	\(-27\)	\(41\)	\(1\)		$+$	$+$	$+$	$13^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}-3q^{3}+(3-2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
507.4.a.r	$507$	$4$	507.a	1.a	$1$	$10$	$10$	$29.914$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓		✓		39.4.j.c	$2$	$0$	\(0\)	\(30\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+3q^{3}+(6+\beta _{4})q^{4}+(\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(507))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(507)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 2}\)