Space of modular forms of level 507, weight 4, and Character 507.b

• Label: 507.4.b
• Level: $507$
• Weight: $4$
• Character orbit: 507.b
• Rep. character: $\chi_{507}(337,\cdot)$
• Character field: $\Q$
• Dimension: $76$
• Newform subspaces: $11$
• Sturm bound: $242$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	507.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(242\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(507, [\chi])\).

	Total	New	Old
Modular forms	196	76	120
Cusp forms	168	76	92
Eisenstein series	28	0	28

Trace form

76 * q - 296 * q^4 + 684 * q^9 + 32 * q^10 - 48 * q^12 + 304 * q^14 + 936 * q^16 - 316 * q^17 - 408 * q^22 + 80 * q^23 - 2100 * q^25 + 308 * q^29 + 528 * q^30 - 1040 * q^35 - 2664 * q^36 + 224 * q^38 + 280 * q^40 - 1296 * q^42 + 160 * q^43 + 1104 * q^48 - 3024 * q^49 + 192 * q^51 + 196 * q^53 - 208 * q^55 - 2208 * q^56 + 2340 * q^61 - 2788 * q^62 - 6012 * q^64 - 2184 * q^66 + 6348 * q^68 + 1272 * q^69 - 540 * q^74 + 480 * q^75 + 392 * q^77 - 5060 * q^79 + 6156 * q^81 + 3668 * q^82 + 1212 * q^87 - 2240 * q^88 + 288 * q^90 + 8620 * q^92 - 5520 * q^94 - 2000 * q^95

Decomposition of \(S_{4}^{\mathrm{new}}(507, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
507.4.b.a	$507$	$4$	507.b	13.b	$2$	$2$	$2$	$29.914$	\(\Q(\sqrt{-3}) \)	None					39.4.j.a	$2$	$0$	\(0\)	\(-6\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-3 q^{3}+8 q^{4}+3\beta q^{5}-6\beta q^{7}+\cdots\)
507.4.b.b	$507$	$4$	507.b	13.b	$2$	$2$	$2$	$29.914$	\(\Q(\sqrt{-1}) \)	None					39.4.a.a	$2$	$0$	\(0\)	\(-6\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-3 q^{3}+8 q^{4}-6\beta q^{5}-\beta q^{7}+9 q^{9}+\cdots\)
507.4.b.c	$507$	$4$	507.b	13.b	$2$	$2$	$2$	$29.914$	\(\Q(\sqrt{-1}) \)	None					39.4.e.a	$2$	$0$	\(0\)	\(6\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{2}+3 q^{3}-q^{4}-9 i q^{5}+9 i q^{6}+\cdots\)
507.4.b.d	$507$	$4$	507.b	13.b	$2$	$2$	$2$	$29.914$	\(\Q(\sqrt{-1}) \)	None					39.4.e.b	$2$	$0$	\(0\)	\(6\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}+3 q^{3}+7 q^{4}-7 i q^{5}+3 i q^{6}+\cdots\)
507.4.b.e	$507$	$4$	507.b	13.b	$2$	$4$	$4$	$29.914$	\(\Q(\sqrt{-3}, \sqrt{-17})\)	None					39.4.j.b	$2$	$0$	\(0\)	\(-12\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}-3q^{3}-9q^{4}+(-2\beta _{1}+3\beta _{2}+\cdots)q^{5}+\cdots\)
507.4.b.f	$507$	$4$	507.b	13.b	$2$	$4$	$4$	$29.914$	\(\Q(i, \sqrt{14})\)	None					39.4.a.b	$2$	$0$	\(0\)	\(-12\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}-3q^{3}+(-7-\beta _{3})q^{4}+(7\beta _{1}+\cdots)q^{5}+\cdots\)
507.4.b.g	$507$	$4$	507.b	13.b	$2$	$6$	$6$	$29.914$	6.0.158155776.1	None					39.4.a.c	$2$	$0$	\(0\)	\(18\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{4}q^{2}+3q^{3}+(-3+\beta _{1})q^{4}+(\beta _{2}+\cdots)q^{5}+\cdots\)
507.4.b.h	$507$	$4$	507.b	13.b	$2$	$8$	$8$	$29.914$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None					39.4.e.c	$2$	$0$	\(0\)	\(-24\)	\(0\)	\(0\)	$2^{6}\cdot 13^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{2}-3q^{3}+(-6+\beta _{1})q^{4}+(\beta _{2}+\cdots)q^{5}+\cdots\)
507.4.b.i	$507$	$4$	507.b	13.b	$2$	$10$	$10$	$29.914$	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)	None					39.4.j.c	$2$	$0$	\(0\)	\(30\)	\(0\)	\(0\)	$2^{5}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+3q^{3}+(-6+\beta _{4})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots\)
507.4.b.j	$507$	$4$	507.b	13.b	$2$	$18$	$18$	$29.914$	\(\mathbb{Q}[x]/(x^{18} + \cdots)\)	None		✓			507.4.a.n	$2$	$0$	\(0\)	\(-54\)	\(0\)	\(0\)	$13^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{10}q^{2}-3q^{3}+(-4+\beta _{2})q^{4}+(2\beta _{10}+\cdots)q^{5}+\cdots\)
507.4.b.k	$507$	$4$	507.b	13.b	$2$	$18$	$18$	$29.914$	\(\mathbb{Q}[x]/(x^{18} + \cdots)\)	None		✓			507.4.a.o	$2$	$0$	\(0\)	\(54\)	\(0\)	\(0\)	$13^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{9}q^{2}+3q^{3}+(-5+\beta _{2})q^{4}-\beta _{15}q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(507, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(507, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(169, [\chi])\)\(^{\oplus 2}\)