Space of modular forms of level 4900, weight 2, and trivial character

• Label: 4900.2.a
• Level: $4900$
• Weight: $2$
• Character orbit: 4900.a
• Rep. character: $\chi_{4900}(1,\cdot)$
• Character field: $\Q$
• Dimension: $65$
• Newform subspaces: $36$
• Sturm bound: $1680$
• Trace bound: $23$

Defining parameters

Level:	\( N \)	\(=\)	\( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4900.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 36 \)
Sturm bound:			\(1680\)
Trace bound:			\(23\)
Distinguishing \(T_p\):			\(3\), \(11\), \(13\), \(19\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4900))\).

	Total	New	Old
Modular forms	912	65	847
Cusp forms	769	65	704
Eisenstein series	143	0	143

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
\(+\)	\(+\)	\(+\)	\(+\)		\(114\)	\(0\)	\(114\)		\(91\)	\(0\)	\(91\)		\(23\)	\(0\)	\(23\)
\(+\)	\(+\)	\(-\)	\(-\)		\(120\)	\(0\)	\(120\)		\(96\)	\(0\)	\(96\)		\(24\)	\(0\)	\(24\)
\(+\)	\(-\)	\(+\)	\(-\)		\(118\)	\(0\)	\(118\)		\(94\)	\(0\)	\(94\)		\(24\)	\(0\)	\(24\)
\(+\)	\(-\)	\(-\)	\(+\)		\(116\)	\(0\)	\(116\)		\(92\)	\(0\)	\(92\)		\(24\)	\(0\)	\(24\)
\(-\)	\(+\)	\(+\)	\(-\)		\(114\)	\(16\)	\(98\)		\(102\)	\(16\)	\(86\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(-\)	\(+\)		\(108\)	\(15\)	\(93\)		\(96\)	\(15\)	\(81\)		\(12\)	\(0\)	\(12\)
\(-\)	\(-\)	\(+\)	\(+\)		\(110\)	\(15\)	\(95\)		\(98\)	\(15\)	\(83\)		\(12\)	\(0\)	\(12\)
\(-\)	\(-\)	\(-\)	\(-\)		\(112\)	\(19\)	\(93\)		\(100\)	\(19\)	\(81\)		\(12\)	\(0\)	\(12\)
Plus space			\(+\)		\(448\)	\(30\)	\(418\)		\(377\)	\(30\)	\(347\)		\(71\)	\(0\)	\(71\)
Minus space			\(-\)		\(464\)	\(35\)	\(429\)		\(392\)	\(35\)	\(357\)		\(72\)	\(0\)	\(72\)

Trace form

65 * q + 2 * q^3 + 67 * q^9 - 2 * q^11 - 2 * q^13 - 10 * q^17 + 12 * q^19 + 8 * q^23 + 8 * q^27 - 6 * q^29 - 12 * q^33 - 16 * q^37 - 16 * q^39 + 18 * q^41 - 18 * q^43 - 10 * q^47 + 26 * q^51 - 20 * q^53 + 26 * q^57 + 40 * q^59 + 22 * q^61 - 24 * q^69 - 16 * q^71 + 18 * q^73 - 10 * q^79 + 37 * q^81 - 10 * q^83 - 48 * q^87 - 2 * q^89 - 18 * q^93 + 6 * q^97 - 8 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4900))\) 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
4900.2.a.a	$4900$	$2$	4900.a	1.a	$1$	$1$	$1$	$39.127$	\(\Q\)	None	✓				140.2.i.b	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+6q^{9}-2q^{11}-6q^{13}+2q^{17}+\cdots\)
4900.2.a.b	$4900$	$2$	4900.a	1.a	$1$	$1$	$1$	$39.127$	\(\Q\)	None	✓				140.2.e.a	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+6q^{9}+3q^{11}-q^{13}+5q^{17}+\cdots\)
4900.2.a.c	$4900$	$2$	4900.a	1.a	$1$	$1$	$1$	$39.127$	\(\Q\)	None	✓	✓			4900.2.a.c	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{9}-q^{11}-2q^{13}+4q^{17}+\cdots\)
4900.2.a.d	$4900$	$2$	4900.a	1.a	$1$	$1$	$1$	$39.127$	\(\Q\)	None	✓	✓			4900.2.a.c	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{9}-q^{11}-2q^{13}+4q^{17}+\cdots\)
4900.2.a.e	$4900$	$2$	4900.a	1.a	$1$	$1$	$1$	$39.127$	\(\Q\)	None	✓				20.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{9}+2q^{13}-6q^{17}+4q^{19}+\cdots\)
4900.2.a.f	$4900$	$2$	4900.a	1.a	$1$	$1$	$1$	$39.127$	\(\Q\)	None	✓				700.2.a.c	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{9}+3q^{11}-4q^{13}-2q^{19}+\cdots\)
4900.2.a.g	$4900$	$2$	4900.a	1.a	$1$	$1$	$1$	$39.127$	\(\Q\)	None	✓				28.2.e.a	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{9}-3q^{11}-2q^{13}-3q^{17}+\cdots\)
4900.2.a.h	$4900$	$2$	4900.a	1.a	$1$	$1$	$1$	$39.127$	\(\Q\)	None	✓				980.2.a.d	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{9}-q^{11}+5q^{13}-q^{17}+\cdots\)
4900.2.a.i	$4900$	$2$	4900.a	1.a	$1$	$1$	$1$	$39.127$	\(\Q\)	None	✓				140.2.i.a	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{9}+6q^{11}-2q^{13}+6q^{17}+\cdots\)
4900.2.a.j	$4900$	$2$	4900.a	1.a	$1$	$1$	$1$	$39.127$	\(\Q\)	None	✓				700.2.a.e	$1$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{9}-5q^{11}-6q^{13}+4q^{17}+6q^{19}+\cdots\)
4900.2.a.k	$4900$	$2$	4900.a	1.a	$1$	$1$	$1$	$39.127$	\(\Q\)	None	✓				700.2.a.e	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{9}-5q^{11}+6q^{13}-4q^{17}+6q^{19}+\cdots\)
4900.2.a.l	$4900$	$2$	4900.a	1.a	$1$	$1$	$1$	$39.127$	\(\Q\)	None	✓				140.2.e.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{9}-4q^{13}-4q^{17}-4q^{19}+8q^{23}+\cdots\)
4900.2.a.m	$4900$	$2$	4900.a	1.a	$1$	$1$	$1$	$39.127$	\(\Q\)	None	✓				140.2.e.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{9}+4q^{13}+4q^{17}-4q^{19}-8q^{23}+\cdots\)
4900.2.a.n	$4900$	$2$	4900.a	1.a	$1$	$1$	$1$	$39.127$	\(\Q\)	None	✓				28.2.e.a	$1$	$1$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{9}-3q^{11}+2q^{13}+3q^{17}+\cdots\)
4900.2.a.o	$4900$	$2$	4900.a	1.a	$1$	$1$	$1$	$39.127$	\(\Q\)	None	✓				980.2.a.d	$1$	$1$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{9}-q^{11}-5q^{13}+q^{17}+\cdots\)
4900.2.a.p	$4900$	$2$	4900.a	1.a	$1$	$1$	$1$	$39.127$	\(\Q\)	None	✓				140.2.a.a	$1$	$1$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{9}+3q^{11}-q^{13}-3q^{17}+\cdots\)
4900.2.a.q	$4900$	$2$	4900.a	1.a	$1$	$1$	$1$	$39.127$	\(\Q\)	None	✓				140.2.i.a	$1$	$1$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{9}+6q^{11}+2q^{13}-6q^{17}+\cdots\)
4900.2.a.r	$4900$	$2$	4900.a	1.a	$1$	$1$	$1$	$39.127$	\(\Q\)	None	✓	✓			4900.2.a.c	$1$	$1$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{9}-q^{11}+2q^{13}-4q^{17}+\cdots\)
4900.2.a.s	$4900$	$2$	4900.a	1.a	$1$	$1$	$1$	$39.127$	\(\Q\)	None	✓	✓			4900.2.a.c	$1$	$0$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{9}-q^{11}+2q^{13}-4q^{17}+\cdots\)
4900.2.a.t	$4900$	$2$	4900.a	1.a	$1$	$1$	$1$	$39.127$	\(\Q\)	None	✓				700.2.a.c	$1$	$0$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{9}+3q^{11}+4q^{13}-2q^{19}+\cdots\)
4900.2.a.u	$4900$	$2$	4900.a	1.a	$1$	$1$	$1$	$39.127$	\(\Q\)	None	✓				140.2.a.b	$1$	$1$	\(0\)	\(3\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+6q^{9}-5q^{11}-3q^{13}-q^{17}+\cdots\)
4900.2.a.v	$4900$	$2$	4900.a	1.a	$1$	$1$	$1$	$39.127$	\(\Q\)	None	✓				140.2.i.b	$1$	$0$	\(0\)	\(3\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+6q^{9}-2q^{11}+6q^{13}-2q^{17}+\cdots\)
4900.2.a.w	$4900$	$2$	4900.a	1.a	$1$	$1$	$1$	$39.127$	\(\Q\)	None	✓				140.2.e.a	$1$	$0$	\(0\)	\(3\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+6q^{9}+3q^{11}+q^{13}-5q^{17}+\cdots\)
4900.2.a.x	$4900$	$2$	4900.a	1.a	$1$	$2$	$2$	$39.127$	\(\Q(\sqrt{2}) \)	None	✓				980.2.a.j	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}-2\beta q^{9}+(-1-2\beta )q^{11}+\cdots\)
4900.2.a.y	$4900$	$2$	4900.a	1.a	$1$	$2$	$2$	$39.127$	\(\Q(\sqrt{2}) \)	None	✓				196.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2\beta q^{3}+5q^{9}+4q^{11}-3\beta q^{13}+\cdots\)
4900.2.a.z	$4900$	$2$	4900.a	1.a	$1$	$2$	$2$	$39.127$	\(\Q(\sqrt{2}) \)	None	✓				980.2.a.j	$1$	$0$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+2\beta q^{9}+(-1+2\beta )q^{11}+\cdots\)
4900.2.a.ba	$4900$	$2$	4900.a	1.a	$1$	$3$	$3$	$39.127$	3.3.257.1	None	✓				700.2.i.d	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$3$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+(2-\beta _{1}-\beta _{2})q^{9}+(1-\beta _{2})q^{11}+\cdots\)
4900.2.a.bb	$4900$	$2$	4900.a	1.a	$1$	$3$	$3$	$39.127$	3.3.257.1	None	✓		✓		700.2.i.d	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+(2-\beta _{1}-\beta _{2})q^{9}+(1-\beta _{2})q^{11}+\cdots\)
4900.2.a.bc	$4900$	$2$	4900.a	1.a	$1$	$3$	$3$	$39.127$	3.3.257.1	None	✓				700.2.i.d	$1$	$0$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+(2-\beta _{1}-\beta _{2})q^{9}+(1-\beta _{2})q^{11}+\cdots\)
4900.2.a.bd	$4900$	$2$	4900.a	1.a	$1$	$3$	$3$	$39.127$	3.3.257.1	None	✓				700.2.i.d	$1$	$0$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$3$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+(2-\beta _{1}-\beta _{2})q^{9}+(1-\beta _{2})q^{11}+\cdots\)
4900.2.a.be	$4900$	$2$	4900.a	1.a	$1$	$4$	$4$	$39.127$	\(\Q(\sqrt{3}, \sqrt{19})\)	None	✓				140.2.q.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+(-2-\beta _{3})q^{11}+(2\beta _{1}-2\beta _{2}+\cdots)q^{13}+\cdots\)
4900.2.a.bf	$4900$	$2$	4900.a	1.a	$1$	$4$	$4$	$39.127$	\(\Q(\sqrt{3}, \sqrt{19})\)	None	✓				140.2.q.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+(-2-\beta _{3})q^{11}+(2\beta _{1}-2\beta _{2}+\cdots)q^{13}+\cdots\)
4900.2.a.bg	$4900$	$2$	4900.a	1.a	$1$	$4$	$4$	$39.127$	\(\Q(\sqrt{2}, \sqrt{5})\)	None	✓	✓			4900.2.a.bg	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{3}q^{9}+(-1-2\beta _{3})q^{11}+\cdots\)
4900.2.a.bh	$4900$	$2$	4900.a	1.a	$1$	$4$	$4$	$39.127$	\(\Q(\zeta_{24})^+\)	None	✓				980.2.e.e	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-q^{11}+\beta _{1}q^{13}-3\beta _{1}q^{17}+\cdots\)
4900.2.a.bi	$4900$	$2$	4900.a	1.a	$1$	$4$	$4$	$39.127$	\(\Q(\sqrt{2}, \sqrt{5})\)	None	✓	✓	✓		4900.2.a.bg	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{3}q^{9}+(-1-2\beta _{3})q^{11}+\cdots\)
4900.2.a.bj	$4900$	$2$	4900.a	1.a	$1$	$4$	$4$	$39.127$	\(\Q(\sqrt{5}, \sqrt{21})\)	\(\Q(\sqrt{-35}) \)	✓				980.2.e.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$2^{3}$	$N(\mathrm{U}(1))$	\(q+\beta _{1}q^{3}+(4+\beta _{3})q^{9}+(1-\beta _{3})q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4900))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4900)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(140))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(245))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(350))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(490))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(700))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(980))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1225))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2450))\)\(^{\oplus 2}\)