Space of modular forms of level 490, weight 4, and trivial character

• Label: 490.4.a
• Level: $490$
• Weight: $4$
• Character orbit: 490.a
• Rep. character: $\chi_{490}(1,\cdot)$
• Character field: $\Q$
• Dimension: $41$
• Newform subspaces: $25$
• Sturm bound: $336$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	490.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 25 \)
Sturm bound:			\(336\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(490))\).

	Total	New	Old
Modular forms	268	41	227
Cusp forms	236	41	195
Eisenstein series	32	0	32

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

\(2\)	\(5\)	\(7\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(36\)	\(4\)	\(32\)		\(32\)	\(4\)	\(28\)		\(4\)	\(0\)	\(4\)
\(+\)	\(+\)	\(-\)	\(-\)		\(32\)	\(6\)	\(26\)		\(28\)	\(6\)	\(22\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)		\(32\)	\(4\)	\(28\)		\(28\)	\(4\)	\(24\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)		\(34\)	\(6\)	\(28\)		\(30\)	\(6\)	\(24\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)		\(34\)	\(5\)	\(29\)		\(30\)	\(5\)	\(25\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)		\(33\)	\(6\)	\(27\)		\(29\)	\(6\)	\(23\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)		\(34\)	\(7\)	\(27\)		\(30\)	\(7\)	\(23\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)		\(33\)	\(3\)	\(30\)		\(29\)	\(3\)	\(26\)		\(4\)	\(0\)	\(4\)
Plus space			\(+\)		\(137\)	\(23\)	\(114\)		\(121\)	\(23\)	\(98\)		\(16\)	\(0\)	\(16\)
Minus space			\(-\)		\(131\)	\(18\)	\(113\)		\(115\)	\(18\)	\(97\)		\(16\)	\(0\)	\(16\)

Trace form

41 * q + 2 * q^2 + 4 * q^3 + 164 * q^4 - 5 * q^5 - 24 * q^6 + 8 * q^8 + 361 * q^9 - 10 * q^10 - 32 * q^11 + 16 * q^12 + 90 * q^13 + 40 * q^15 + 656 * q^16 - 78 * q^17 - 150 * q^18 + 360 * q^19 - 20 * q^20 + 136 * q^22 - 204 * q^23 - 96 * q^24 + 1025 * q^25 + 196 * q^26 + 304 * q^27 + 410 * q^29 + 200 * q^30 + 104 * q^31 + 32 * q^32 + 520 * q^33 - 348 * q^34 + 1444 * q^36 + 2038 * q^37 - 160 * q^38 + 1608 * q^39 - 40 * q^40 + 850 * q^41 + 640 * q^43 - 128 * q^44 + 135 * q^45 - 328 * q^46 - 1204 * q^47 + 64 * q^48 + 50 * q^50 - 312 * q^51 + 360 * q^52 - 570 * q^53 + 432 * q^54 - 420 * q^55 - 2968 * q^57 + 588 * q^58 - 696 * q^59 + 160 * q^60 - 886 * q^61 + 384 * q^62 + 2624 * q^64 + 1130 * q^65 + 288 * q^66 - 3240 * q^67 - 312 * q^68 - 248 * q^69 - 1216 * q^71 - 600 * q^72 - 1942 * q^73 + 756 * q^74 + 100 * q^75 + 1440 * q^76 - 832 * q^78 - 4672 * q^79 - 80 * q^80 + 5265 * q^81 - 444 * q^82 - 4396 * q^83 + 10 * q^85 - 1880 * q^86 + 3408 * q^87 + 544 * q^88 - 462 * q^89 - 530 * q^90 - 816 * q^92 - 2120 * q^93 + 1096 * q^94 - 1580 * q^95 - 384 * q^96 + 3074 * q^97 + 5488 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(490))\) 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	7
490.4.a.a	$490$	$4$	490.a	1.a	$1$	$1$	$1$	$28.911$	\(\Q\)	None	✓				70.4.e.b	$1$	$0$	\(-2\)	\(-10\)	\(5\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-10q^{3}+4q^{4}+5q^{5}+20q^{6}+\cdots\)
490.4.a.b	$490$	$4$	490.a	1.a	$1$	$1$	$1$	$28.911$	\(\Q\)	None	✓				70.4.a.d	$1$	$1$	\(-2\)	\(-4\)	\(-5\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-4q^{3}+4q^{4}-5q^{5}+8q^{6}+\cdots\)
490.4.a.c	$490$	$4$	490.a	1.a	$1$	$1$	$1$	$28.911$	\(\Q\)	None	✓				70.4.e.c	$1$	$0$	\(-2\)	\(-1\)	\(-5\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-q^{3}+4q^{4}-5q^{5}+2q^{6}+\cdots\)
490.4.a.d	$490$	$4$	490.a	1.a	$1$	$1$	$1$	$28.911$	\(\Q\)	None	✓				70.4.a.c	$1$	$0$	\(-2\)	\(1\)	\(5\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+q^{3}+4q^{4}+5q^{5}-2q^{6}+\cdots\)
490.4.a.e	$490$	$4$	490.a	1.a	$1$	$1$	$1$	$28.911$	\(\Q\)	None	✓				70.4.e.c	$1$	$0$	\(-2\)	\(1\)	\(5\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+q^{3}+4q^{4}+5q^{5}-2q^{6}+\cdots\)
490.4.a.f	$490$	$4$	490.a	1.a	$1$	$1$	$1$	$28.911$	\(\Q\)	None	✓				70.4.a.b	$1$	$1$	\(-2\)	\(3\)	\(-5\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+3q^{3}+4q^{4}-5q^{5}-6q^{6}+\cdots\)
490.4.a.g	$490$	$4$	490.a	1.a	$1$	$1$	$1$	$28.911$	\(\Q\)	None	✓				70.4.a.a	$1$	$0$	\(-2\)	\(8\)	\(5\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+8q^{3}+4q^{4}+5q^{5}-2^{4}q^{6}+\cdots\)
490.4.a.h	$490$	$4$	490.a	1.a	$1$	$1$	$1$	$28.911$	\(\Q\)	None	✓				70.4.e.b	$1$	$0$	\(-2\)	\(10\)	\(-5\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+10q^{3}+4q^{4}-5q^{5}-20q^{6}+\cdots\)
490.4.a.i	$490$	$4$	490.a	1.a	$1$	$1$	$1$	$28.911$	\(\Q\)	None	✓				70.4.a.f	$1$	$1$	\(2\)	\(-7\)	\(5\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-7q^{3}+4q^{4}+5q^{5}-14q^{6}+\cdots\)
490.4.a.j	$490$	$4$	490.a	1.a	$1$	$1$	$1$	$28.911$	\(\Q\)	None	✓				70.4.a.e	$1$	$0$	\(2\)	\(-5\)	\(-5\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-5q^{3}+4q^{4}-5q^{5}-10q^{6}+\cdots\)
490.4.a.k	$490$	$4$	490.a	1.a	$1$	$1$	$1$	$28.911$	\(\Q\)	None	✓				70.4.e.a	$1$	$1$	\(2\)	\(-1\)	\(5\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-q^{3}+4q^{4}+5q^{5}-2q^{6}+\cdots\)
490.4.a.l	$490$	$4$	490.a	1.a	$1$	$1$	$1$	$28.911$	\(\Q\)	None	✓	✓			490.4.a.l	$1$	$1$	\(2\)	\(-1\)	\(5\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-q^{3}+4q^{4}+5q^{5}-2q^{6}+\cdots\)
490.4.a.m	$490$	$4$	490.a	1.a	$1$	$1$	$1$	$28.911$	\(\Q\)	None	✓				70.4.e.a	$1$	$1$	\(2\)	\(1\)	\(-5\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+q^{3}+4q^{4}-5q^{5}+2q^{6}+\cdots\)
490.4.a.n	$490$	$4$	490.a	1.a	$1$	$1$	$1$	$28.911$	\(\Q\)	None	✓	✓			490.4.a.l	$1$	$0$	\(2\)	\(1\)	\(-5\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+q^{3}+4q^{4}-5q^{5}+2q^{6}+\cdots\)
490.4.a.o	$490$	$4$	490.a	1.a	$1$	$1$	$1$	$28.911$	\(\Q\)	None	✓				10.4.a.a	$1$	$0$	\(2\)	\(8\)	\(-5\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+8q^{3}+4q^{4}-5q^{5}+2^{4}q^{6}+\cdots\)
490.4.a.p	$490$	$4$	490.a	1.a	$1$	$2$	$2$	$28.911$	\(\Q(\sqrt{177}) \)	None	✓	✓			490.4.a.p	$1$	$1$	\(-4\)	\(-5\)	\(-10\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+(-2-\beta )q^{3}+4q^{4}-5q^{5}+\cdots\)
490.4.a.q	$490$	$4$	490.a	1.a	$1$	$2$	$2$	$28.911$	\(\Q(\sqrt{2}) \)	None	✓	✓			490.4.a.q	$1$	$1$	\(-4\)	\(-2\)	\(10\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+(-1+3\beta )q^{3}+4q^{4}+5q^{5}+\cdots\)
490.4.a.r	$490$	$4$	490.a	1.a	$1$	$2$	$2$	$28.911$	\(\Q(\sqrt{46}) \)	None	✓				70.4.e.d	$1$	$1$	\(-4\)	\(-2\)	\(10\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+(-1+\beta )q^{3}+4q^{4}+5q^{5}+\cdots\)
490.4.a.s	$490$	$4$	490.a	1.a	$1$	$2$	$2$	$28.911$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓		490.4.a.q	$1$	$0$	\(-4\)	\(2\)	\(-10\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+(1+3\beta )q^{3}+4q^{4}-5q^{5}+\cdots\)
490.4.a.t	$490$	$4$	490.a	1.a	$1$	$2$	$2$	$28.911$	\(\Q(\sqrt{46}) \)	None	✓				70.4.e.d	$1$	$1$	\(-4\)	\(2\)	\(-10\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+(1+\beta )q^{3}+4q^{4}-5q^{5}+(-2+\cdots)q^{6}+\cdots\)
490.4.a.u	$490$	$4$	490.a	1.a	$1$	$2$	$2$	$28.911$	\(\Q(\sqrt{177}) \)	None	✓	✓	✓		490.4.a.p	$1$	$0$	\(-4\)	\(5\)	\(10\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+(3-\beta )q^{3}+4q^{4}+5q^{5}+(-6+\cdots)q^{6}+\cdots\)
490.4.a.v	$490$	$4$	490.a	1.a	$1$	$3$	$3$	$28.911$	3.3.115880.1	None	✓		✓		70.4.e.e	$1$	$0$	\(6\)	\(-4\)	\(-15\)	\(0\)		$-$	$+$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q+2q^{2}+(-1-\beta _{1})q^{3}+4q^{4}-5q^{5}+\cdots\)
490.4.a.w	$490$	$4$	490.a	1.a	$1$	$3$	$3$	$28.911$	3.3.115880.1	None	✓				70.4.e.e	$1$	$0$	\(6\)	\(4\)	\(15\)	\(0\)		$-$	$-$	$+$	$+$	$3$	$\mathrm{SU}(2)$	\(q+2q^{2}+(1+\beta _{1})q^{3}+4q^{4}+5q^{5}+\cdots\)
490.4.a.x	$490$	$4$	490.a	1.a	$1$	$4$	$4$	$28.911$	\(\Q(\sqrt{2}, \sqrt{113})\)	None	✓	✓	✓		490.4.a.x	$1$	$1$	\(8\)	\(-10\)	\(-20\)	\(0\)		$-$	$+$	$+$	$-$	$7$	$\mathrm{SU}(2)$	\(q+2q^{2}+(-3-\beta _{1})q^{3}+4q^{4}-5q^{5}+\cdots\)
490.4.a.y	$490$	$4$	490.a	1.a	$1$	$4$	$4$	$28.911$	\(\Q(\sqrt{2}, \sqrt{113})\)	None	✓	✓	✓		490.4.a.x	$1$	$0$	\(8\)	\(10\)	\(20\)	\(0\)		$-$	$-$	$+$	$+$	$7$	$\mathrm{SU}(2)$	\(q+2q^{2}+(2-\beta _{1})q^{3}+4q^{4}+5q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(490))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(490)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(245))\)\(^{\oplus 2}\)