Space of modular forms of level 5030, weight 2, and trivial character

• Label: 5030.2.a
• Level: $5030$
• Weight: $2$
• Character orbit: 5030.a
• Rep. character: $\chi_{5030}(1,\cdot)$
• Character field: $\Q$
• Dimension: $169$
• Newform subspaces: $20$
• Sturm bound: $1512$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 5030 = 2 \cdot 5 \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5030.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 20 \)
Sturm bound:			\(1512\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(5030))\).

	Total	New	Old
Modular forms	760	169	591
Cusp forms	753	169	584
Eisenstein series	7	0	7

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

\(2\)	\(5\)	\(503\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(86\)	\(18\)	\(68\)		\(86\)	\(18\)	\(68\)		\(0\)	\(0\)	\(0\)
\(+\)	\(+\)	\(-\)	\(-\)		\(103\)	\(24\)	\(79\)		\(102\)	\(24\)	\(78\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(+\)	\(-\)		\(99\)	\(24\)	\(75\)		\(98\)	\(24\)	\(74\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)	\(+\)		\(92\)	\(18\)	\(74\)		\(91\)	\(18\)	\(73\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)		\(99\)	\(26\)	\(73\)		\(98\)	\(26\)	\(72\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(91\)	\(17\)	\(74\)		\(90\)	\(17\)	\(73\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(96\)	\(17\)	\(79\)		\(95\)	\(17\)	\(78\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(94\)	\(25\)	\(69\)		\(93\)	\(25\)	\(68\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(365\)	\(70\)	\(295\)		\(362\)	\(70\)	\(292\)		\(3\)	\(0\)	\(3\)
Minus space			\(-\)		\(395\)	\(99\)	\(296\)		\(391\)	\(99\)	\(292\)		\(4\)	\(0\)	\(4\)

Trace form

169 * q + q^2 + 4 * q^3 + 169 * q^4 - q^5 + 8 * q^7 + q^8 + 169 * q^9 - q^10 + 4 * q^12 + 14 * q^13 - 4 * q^15 + 169 * q^16 + 10 * q^17 - 3 * q^18 + 16 * q^19 - q^20 + 12 * q^22 + 24 * q^23 + 169 * q^25 - 6 * q^26 + 16 * q^27 + 8 * q^28 - 14 * q^29 - 4 * q^30 + q^32 - 8 * q^33 + 10 * q^34 - 8 * q^35 + 169 * q^36 + 18 * q^37 - 16 * q^39 - q^40 + 10 * q^41 - 8 * q^42 + 20 * q^43 + 3 * q^45 - 8 * q^46 - 24 * q^47 + 4 * q^48 + 169 * q^49 + q^50 - 8 * q^51 + 14 * q^52 - 38 * q^53 - 24 * q^54 - 12 * q^55 - 48 * q^57 + 10 * q^58 - 24 * q^59 - 4 * q^60 - 14 * q^61 - 24 * q^62 - 24 * q^63 + 169 * q^64 - 14 * q^65 - 40 * q^66 + 28 * q^67 + 10 * q^68 - 64 * q^69 - 8 * q^70 - 32 * q^71 - 3 * q^72 + 18 * q^73 - 22 * q^74 + 4 * q^75 + 16 * q^76 - 24 * q^79 - q^80 + 177 * q^81 + 18 * q^82 + 4 * q^83 - 14 * q^85 + 8 * q^86 - 48 * q^87 + 12 * q^88 - 30 * q^89 - 13 * q^90 + 32 * q^91 + 24 * q^92 + 16 * q^93 - 16 * q^94 + 12 * q^95 + 34 * q^97 - 7 * q^98 - 40 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(5030))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	5	503
5030.2.a.a	$5030$	$2$	5030.a	1.a	$1$	$1$	$1$	$40.165$	\(\Q\)	None	✓	✓			5030.2.a.a	$1$	$1$	\(-1\)	\(-1\)	\(1\)	\(-5\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-5q^{7}+\cdots\)
5030.2.a.b	$5030$	$2$	5030.a	1.a	$1$	$1$	$1$	$40.165$	\(\Q\)	None	✓	✓			5030.2.a.b	$1$	$1$	\(-1\)	\(-1\)	\(1\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}+q^{7}+\cdots\)
5030.2.a.c	$5030$	$2$	5030.a	1.a	$1$	$1$	$1$	$40.165$	\(\Q\)	None	✓	✓			5030.2.a.c	$1$	$1$	\(-1\)	\(3\)	\(1\)	\(-3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+3q^{3}+q^{4}+q^{5}-3q^{6}-3q^{7}+\cdots\)
5030.2.a.d	$5030$	$2$	5030.a	1.a	$1$	$1$	$1$	$40.165$	\(\Q\)	None	✓	✓			5030.2.a.d	$1$	$1$	\(1\)	\(-3\)	\(1\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-3q^{3}+q^{4}+q^{5}-3q^{6}-q^{7}+\cdots\)
5030.2.a.e	$5030$	$2$	5030.a	1.a	$1$	$1$	$1$	$40.165$	\(\Q\)	None	✓	✓			5030.2.a.e	$1$	$0$	\(1\)	\(-1\)	\(-1\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+4q^{7}+\cdots\)
5030.2.a.f	$5030$	$2$	5030.a	1.a	$1$	$1$	$1$	$40.165$	\(\Q\)	None	✓	✓			5030.2.a.f	$1$	$1$	\(1\)	\(0\)	\(-1\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{5}-2q^{7}+q^{8}-3q^{9}+\cdots\)
5030.2.a.g	$5030$	$2$	5030.a	1.a	$1$	$1$	$1$	$40.165$	\(\Q\)	None	✓	✓			5030.2.a.g	$1$	$1$	\(1\)	\(1\)	\(1\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}-q^{7}+\cdots\)
5030.2.a.h	$5030$	$2$	5030.a	1.a	$1$	$1$	$1$	$40.165$	\(\Q\)	None	✓	✓			5030.2.a.h	$1$	$1$	\(1\)	\(2\)	\(-1\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+2q^{3}+q^{4}-q^{5}+2q^{6}-2q^{7}+\cdots\)
5030.2.a.i	$5030$	$2$	5030.a	1.a	$1$	$1$	$1$	$40.165$	\(\Q\)	None	✓	✓			5030.2.a.i	$1$	$0$	\(1\)	\(3\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+3q^{3}+q^{4}+q^{5}+3q^{6}+q^{8}+\cdots\)
5030.2.a.j	$5030$	$2$	5030.a	1.a	$1$	$2$	$2$	$40.165$	\(\Q(\sqrt{3}) \)	None	✓	✓			5030.2.a.j	$1$	$0$	\(-2\)	\(2\)	\(2\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(1+\beta )q^{3}+q^{4}+q^{5}+(-1+\cdots)q^{6}+\cdots\)
5030.2.a.k	$5030$	$2$	5030.a	1.a	$1$	$2$	$2$	$40.165$	\(\Q(\sqrt{2}) \)	None	✓	✓			5030.2.a.k	$1$	$1$	\(2\)	\(0\)	\(2\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+\beta q^{3}+q^{4}+q^{5}+\beta q^{6}-2q^{7}+\cdots\)
5030.2.a.l	$5030$	$2$	5030.a	1.a	$1$	$3$	$3$	$40.165$	3.3.316.1	None	✓	✓			5030.2.a.l	$1$	$0$	\(-3\)	\(3\)	\(3\)	\(2\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}+(1+\beta _{2})q^{7}+\cdots\)
5030.2.a.m	$5030$	$2$	5030.a	1.a	$1$	$13$	$13$	$40.165$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓	✓	✓		5030.2.a.m	$1$	$1$	\(13\)	\(-8\)	\(13\)	\(-13\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(-1+\beta _{1})q^{3}+q^{4}+q^{5}+(-1+\cdots)q^{6}+\cdots\)
5030.2.a.n	$5030$	$2$	5030.a	1.a	$1$	$15$	$15$	$40.165$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	✓	✓		5030.2.a.n	$1$	$1$	\(-15\)	\(-3\)	\(15\)	\(-6\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-\beta _{1}q^{3}+q^{4}+q^{5}+\beta _{1}q^{6}+\cdots\)
5030.2.a.o	$5030$	$2$	5030.a	1.a	$1$	$15$	$15$	$40.165$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	✓	✓		5030.2.a.o	$1$	$1$	\(15\)	\(-6\)	\(-15\)	\(-3\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+q^{2}-\beta _{1}q^{3}+q^{4}-q^{5}-\beta _{1}q^{6}+\cdots\)
5030.2.a.p	$5030$	$2$	5030.a	1.a	$1$	$18$	$18$	$40.165$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	✓	✓	✓	✓	5030.2.a.p	$1$	$1$	\(-18\)	\(4\)	\(-18\)	\(-1\)		$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{2}+\beta _{1}q^{3}+q^{4}-q^{5}-\beta _{1}q^{6}+\cdots\)
5030.2.a.q	$5030$	$2$	5030.a	1.a	$1$	$19$	$19$	$40.165$	\(\mathbb{Q}[x]/(x^{19} - \cdots)\)	None	✓	✓	✓		5030.2.a.q	$1$	$0$	\(-19\)	\(-2\)	\(19\)	\(9\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-\beta _{1}q^{3}+q^{4}+q^{5}+\beta _{1}q^{6}+\cdots\)
5030.2.a.r	$5030$	$2$	5030.a	1.a	$1$	$24$	$24$	$40.165$		None	✓	✓	✓	✓	5030.2.a.r	$1$	$0$	\(-24\)	\(-3\)	\(-24\)	\(3\)		$+$	$+$	$-$	$-$		$\mathrm{SU}(2)$
5030.2.a.s	$5030$	$2$	5030.a	1.a	$1$	$24$	$24$	$40.165$		None	✓	✓	✓		5030.2.a.s	$1$	$0$	\(24\)	\(6\)	\(24\)	\(17\)		$-$	$-$	$-$	$-$		$\mathrm{SU}(2)$
5030.2.a.t	$5030$	$2$	5030.a	1.a	$1$	$25$	$25$	$40.165$		None	✓	✓	✓		5030.2.a.t	$1$	$0$	\(25\)	\(8\)	\(-25\)	\(9\)		$-$	$+$	$+$	$-$		$\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(5030))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(5030)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(503))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1006))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2515))\)\(^{\oplus 2}\)