Space of modular forms of level 507, weight 2, and trivial character

• Label: 507.2.a
• Level: $507$
• Weight: $2$
• Character orbit: 507.a
• Rep. character: $\chi_{507}(1,\cdot)$
• Character field: $\Q$
• Dimension: $25$
• Newform subspaces: $12$
• Sturm bound: $121$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(121\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(507))\).

	Total	New	Old
Modular forms	74	25	49
Cusp forms	47	25	22
Eisenstein series	27	0	27

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	Dim
\(+\)	\(+\)	\(+\)	\(6\)
\(+\)	\(-\)	\(-\)	\(7\)
\(-\)	\(+\)	\(-\)	\(9\)
\(-\)	\(-\)	\(+\)	\(3\)
Plus space		\(+\)	\(9\)
Minus space		\(-\)	\(16\)

Trace form

25 * q + q^2 - q^3 + 27 * q^4 - 2 * q^5 + 3 * q^6 + 4 * q^7 + 9 * q^8 + 25 * q^9 + 2 * q^10 - 7 * q^12 - 8 * q^14 + 2 * q^15 + 19 * q^16 - 10 * q^17 + q^18 - 14 * q^20 - 4 * q^21 - 16 * q^22 + 4 * q^23 + 3 * q^24 + 19 * q^25 - q^27 + 12 * q^28 - 2 * q^29 + 6 * q^30 + 4 * q^31 - 11 * q^32 + 8 * q^33 - 14 * q^34 + 20 * q^35 + 27 * q^36 + 6 * q^37 + 4 * q^38 + 2 * q^40 - 22 * q^41 - 12 * q^42 - 8 * q^43 + 8 * q^44 - 2 * q^45 - 8 * q^46 + 12 * q^47 - 15 * q^48 + 9 * q^49 + 7 * q^50 - 6 * q^51 - 18 * q^53 + 3 * q^54 - 20 * q^55 - 32 * q^56 + 14 * q^58 - 16 * q^59 - 18 * q^60 - 18 * q^61 - 36 * q^62 + 4 * q^63 + 15 * q^64 - 4 * q^66 + 6 * q^68 + 4 * q^69 - 8 * q^70 - 4 * q^71 + 9 * q^72 - 14 * q^73 + 2 * q^74 - 15 * q^75 - 16 * q^76 - 20 * q^79 + 2 * q^80 + 25 * q^81 - 2 * q^82 + 20 * q^84 + 28 * q^85 + 36 * q^86 - 18 * q^87 - 24 * q^88 - 22 * q^89 + 2 * q^90 - 20 * q^92 + 12 * q^93 - 8 * q^94 - 36 * q^95 - q^96 - 6 * q^97 - 7 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(507))\) 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
507.2.a.a	$507$	$2$	507.a	1.a	$1$	$1$	$1$	$4.048$	\(\Q\)	None	✓				39.2.a.a	$1$	$1$	\(-1\)	\(-1\)	\(-2\)	\(4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}-q^{4}-2q^{5}+q^{6}+4q^{7}+\cdots\)
507.2.a.b	$507$	$2$	507.a	1.a	$1$	$1$	$1$	$4.048$	\(\Q\)	None	✓				39.2.e.a	$1$	$1$	\(-1\)	\(-1\)	\(1\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}-q^{4}+q^{5}+q^{6}-2q^{7}+\cdots\)
507.2.a.c	$507$	$2$	507.a	1.a	$1$	$1$	$1$	$4.048$	\(\Q\)	None	✓				39.2.e.a	$1$	$1$	\(1\)	\(-1\)	\(-1\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}-q^{4}-q^{5}-q^{6}+2q^{7}+\cdots\)
507.2.a.d	$507$	$2$	507.a	1.a	$1$	$2$	$2$	$4.048$	\(\Q(\sqrt{17}) \)	None	✓				39.2.e.b	$1$	$0$	\(-1\)	\(2\)	\(3\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+q^{3}+(2+\beta )q^{4}+(2-\beta )q^{5}+\cdots\)
507.2.a.e	$507$	$2$	507.a	1.a	$1$	$2$	$2$	$4.048$	\(\Q(\sqrt{3}) \)	None	✓				39.2.j.a	$2$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{4}+2\beta q^{5}+\beta q^{7}+q^{9}+\cdots\)
507.2.a.f	$507$	$2$	507.a	1.a	$1$	$2$	$2$	$4.048$	\(\Q(\sqrt{3}) \)	None	✓				39.2.b.a	$2$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-q^{3}+q^{4}-\beta q^{6}+2\beta q^{7}+\cdots\)
507.2.a.g	$507$	$2$	507.a	1.a	$1$	$2$	$2$	$4.048$	\(\Q(\sqrt{17}) \)	None	✓				39.2.e.b	$1$	$0$	\(1\)	\(2\)	\(-3\)	\(-3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+q^{3}+(2+\beta )q^{4}+(-2+\beta )q^{5}+\cdots\)
507.2.a.h	$507$	$2$	507.a	1.a	$1$	$2$	$2$	$4.048$	\(\Q(\sqrt{2}) \)	None	✓				39.2.a.b	$1$	$0$	\(2\)	\(2\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+q^{3}+(1+2\beta )q^{4}-2\beta q^{5}+\cdots\)
507.2.a.i	$507$	$2$	507.a	1.a	$1$	$3$	$3$	$4.048$	\(\Q(\zeta_{14})^+\)	None	✓	✓	✓		507.2.a.i	$1$	$1$	\(-3\)	\(-3\)	\(-6\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1}+\beta _{2})q^{2}-q^{3}+(4-\beta _{1}+\cdots)q^{4}+\cdots\)
507.2.a.j	$507$	$2$	507.a	1.a	$1$	$3$	$3$	$4.048$	\(\Q(\zeta_{14})^+\)	None	✓	✓	✓	✓	507.2.a.j	$1$	$1$	\(-1\)	\(3\)	\(-4\)	\(-10\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+q^{3}+\beta _{2}q^{4}+(-2+\beta _{1}+\cdots)q^{5}+\cdots\)
507.2.a.k	$507$	$2$	507.a	1.a	$1$	$3$	$3$	$4.048$	\(\Q(\zeta_{14})^+\)	None	✓	✓	✓		507.2.a.j	$1$	$0$	\(1\)	\(3\)	\(4\)	\(10\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+q^{3}+\beta _{2}q^{4}+(2-\beta _{1}+\beta _{2})q^{5}+\cdots\)
507.2.a.l	$507$	$2$	507.a	1.a	$1$	$3$	$3$	$4.048$	\(\Q(\zeta_{14})^+\)	None	✓	✓	✓		507.2.a.i	$1$	$0$	\(3\)	\(-3\)	\(6\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1}-\beta _{2})q^{2}-q^{3}+(4-\beta _{1})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(507))\) into lower level spaces

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