Space of modular forms of level 4949, weight 2, and trivial character

• Label: 4949.2.a
• Level: $4949$
• Weight: $2$
• Character orbit: 4949.a
• Rep. character: $\chi_{4949}(1,\cdot)$
• Character field: $\Q$
• Dimension: $341$
• Newform subspaces: $27$
• Sturm bound: $952$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 4949 = 7^{2} \cdot 101 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4949.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 27 \)
Sturm bound:			\(952\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4949))\).

	Total	New	Old
Modular forms	484	341	143
Cusp forms	469	341	128
Eisenstein series	15	0	15

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

\(7\)	\(101\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(76\)
\(+\)	\(-\)	\(-\)	\(90\)
\(-\)	\(+\)	\(-\)	\(98\)
\(-\)	\(-\)	\(+\)	\(77\)
Plus space		\(+\)	\(153\)
Minus space		\(-\)	\(188\)

Trace form

\( 341 q - q^{2} + 2 q^{3} + 341 q^{4} + 2 q^{5} + 6 q^{6} - 3 q^{8} + 343 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4949))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		7	101
4949.2.a.a	$4949$	$2$	4949.a	1.a	$1$	$1$	$1$	$39.518$	\(\Q\)	None	✓				707.2.a.a	$1$	$0$	\(-2\)	\(2\)	\(3\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{3}+2q^{4}+3q^{5}-4q^{6}+\cdots\)
4949.2.a.b	$4949$	$2$	4949.a	1.a	$1$	$1$	$1$	$39.518$	\(\Q\)	None	✓				707.2.e.a	$1$	$1$	\(-1\)	\(-1\)	\(3\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}-q^{4}+3q^{5}+q^{6}+3q^{8}+\cdots\)
4949.2.a.c	$4949$	$2$	4949.a	1.a	$1$	$1$	$1$	$39.518$	\(\Q\)	None	✓				707.2.e.a	$1$	$1$	\(-1\)	\(1\)	\(-3\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}-q^{4}-3q^{5}-q^{6}+3q^{8}+\cdots\)
4949.2.a.d	$4949$	$2$	4949.a	1.a	$1$	$1$	$1$	$39.518$	\(\Q\)	None	✓				101.2.a.a	$1$	$1$	\(0\)	\(2\)	\(1\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-2q^{4}+q^{5}+q^{9}-2q^{11}+\cdots\)
4949.2.a.e	$4949$	$2$	4949.a	1.a	$1$	$1$	$1$	$39.518$	\(\Q\)	None	✓	✓			4949.2.a.e	$1$	$0$	\(1\)	\(-3\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-3q^{3}-q^{4}-3q^{6}-3q^{8}+6q^{9}+\cdots\)
4949.2.a.f	$4949$	$2$	4949.a	1.a	$1$	$1$	$1$	$39.518$	\(\Q\)	None	✓	✓			4949.2.a.e	$1$	$1$	\(1\)	\(3\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+3q^{3}-q^{4}+3q^{6}-3q^{8}+6q^{9}+\cdots\)
4949.2.a.g	$4949$	$2$	4949.a	1.a	$1$	$2$	$2$	$39.518$	\(\Q(\sqrt{5}) \)	None	✓	✓			4949.2.a.g	$1$	$2$	\(-3\)	\(-2\)	\(-3\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}-q^{3}+3\beta q^{4}+(-2+\cdots)q^{5}+\cdots\)
4949.2.a.h	$4949$	$2$	4949.a	1.a	$1$	$2$	$2$	$39.518$	\(\Q(\sqrt{5}) \)	None	✓	✓			4949.2.a.g	$1$	$1$	\(-3\)	\(2\)	\(3\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}+q^{3}+3\beta q^{4}+(2-\beta )q^{5}+\cdots\)
4949.2.a.i	$4949$	$2$	4949.a	1.a	$1$	$2$	$2$	$39.518$	\(\Q(\sqrt{13}) \)	None	✓				707.2.a.b	$1$	$1$	\(-1\)	\(-2\)	\(1\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}-q^{3}+(1+\beta )q^{4}+(1-\beta )q^{5}+\cdots\)
4949.2.a.j	$4949$	$2$	4949.a	1.a	$1$	$2$	$2$	$39.518$	\(\Q(\sqrt{2}) \)	None	✓				707.2.e.b	$1$	$0$	\(2\)	\(-2\)	\(2\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(-1+\beta )q^{3}-q^{4}+(1-2\beta )q^{5}+\cdots\)
4949.2.a.k	$4949$	$2$	4949.a	1.a	$1$	$2$	$2$	$39.518$	\(\Q(\sqrt{2}) \)	None	✓				707.2.e.b	$1$	$0$	\(2\)	\(2\)	\(-2\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(1-\beta )q^{3}-q^{4}+(-1+2\beta )q^{5}+\cdots\)
4949.2.a.l	$4949$	$2$	4949.a	1.a	$1$	$4$	$4$	$39.518$	4.4.16609.1	None	✓				707.2.a.c	$1$	$1$	\(0\)	\(-4\)	\(-1\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{3}-2q^{4}-\beta _{3}q^{5}+(2+\cdots)q^{9}+\cdots\)
4949.2.a.m	$4949$	$2$	4949.a	1.a	$1$	$4$	$4$	$39.518$	4.4.23724.1	None	✓	✓			4949.2.a.m	$1$	$1$	\(1\)	\(-2\)	\(3\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-\beta _{1}+\beta _{2})q^{3}+(2+\beta _{2}+\cdots)q^{4}+\cdots\)
4949.2.a.n	$4949$	$2$	4949.a	1.a	$1$	$4$	$4$	$39.518$	4.4.23724.1	None	✓	✓			4949.2.a.m	$1$	$0$	\(1\)	\(2\)	\(-3\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(\beta _{1}-\beta _{2})q^{3}+(2+\beta _{2})q^{4}+\cdots\)
4949.2.a.o	$4949$	$2$	4949.a	1.a	$1$	$7$	$7$	$39.518$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓				101.2.a.b	$1$	$0$	\(0\)	\(-4\)	\(3\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-1-\beta _{4})q^{3}+(1+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
4949.2.a.p	$4949$	$2$	4949.a	1.a	$1$	$8$	$8$	$39.518$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓				707.2.a.d	$1$	$1$	\(-3\)	\(3\)	\(2\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2}+\beta _{3}+\beta _{7})q^{3}+(1+\cdots)q^{4}+\cdots\)
4949.2.a.q	$4949$	$2$	4949.a	1.a	$1$	$10$	$10$	$39.518$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓				707.2.a.e	$1$	$0$	\(-2\)	\(9\)	\(7\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{3})q^{3}+(-\beta _{7}-\beta _{8}+\cdots)q^{4}+\cdots\)
4949.2.a.r	$4949$	$2$	4949.a	1.a	$1$	$11$	$11$	$39.518$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓				707.2.a.f	$1$	$1$	\(2\)	\(-3\)	\(-4\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}-\beta _{5}q^{3}+(2+\beta _{1})q^{4}+\beta _{7}q^{5}+\cdots\)
4949.2.a.s	$4949$	$2$	4949.a	1.a	$1$	$15$	$15$	$39.518$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓				707.2.a.g	$1$	$0$	\(5\)	\(-1\)	\(-10\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-\beta _{6}q^{3}+(1+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
4949.2.a.t	$4949$	$2$	4949.a	1.a	$1$	$18$	$18$	$39.518$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	✓	✓			4949.2.a.t	$1$	$1$	\(-1\)	\(-13\)	\(-10\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-1+\beta _{10})q^{3}+(1+\beta _{2}+\cdots)q^{4}+\cdots\)
4949.2.a.u	$4949$	$2$	4949.a	1.a	$1$	$18$	$18$	$39.518$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	✓	✓			4949.2.a.t	$1$	$0$	\(-1\)	\(13\)	\(10\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1-\beta _{10})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
4949.2.a.v	$4949$	$2$	4949.a	1.a	$1$	$25$	$25$	$39.518$		None	✓		✓		707.2.e.c	$1$	$1$	\(-1\)	\(0\)	\(-3\)	\(0\)		$-$	$-$	$+$		$\mathrm{SU}(2)$
4949.2.a.w	$4949$	$2$	4949.a	1.a	$1$	$25$	$25$	$39.518$		None	✓				707.2.e.c	$1$	$1$	\(-1\)	\(0\)	\(3\)	\(0\)		$+$	$+$	$+$		$\mathrm{SU}(2)$
4949.2.a.x	$4949$	$2$	4949.a	1.a	$1$	$38$	$38$	$39.518$		None	✓		✓		707.2.e.d	$1$	$0$	\(2\)	\(-3\)	\(2\)	\(0\)		$-$	$+$	$-$		$\mathrm{SU}(2)$
4949.2.a.y	$4949$	$2$	4949.a	1.a	$1$	$38$	$38$	$39.518$		None	✓				707.2.e.d	$1$	$0$	\(2\)	\(3\)	\(-2\)	\(0\)		$+$	$-$	$-$		$\mathrm{SU}(2)$
4949.2.a.z	$4949$	$2$	4949.a	1.a	$1$	$50$	$50$	$39.518$		None	✓	✓	✓		4949.2.a.z	$1$	$1$	\(0\)	\(-20\)	\(-8\)	\(0\)		$+$	$+$	$+$		$\mathrm{SU}(2)$
4949.2.a.ba	$4949$	$2$	4949.a	1.a	$1$	$50$	$50$	$39.518$		None	✓	✓	✓		4949.2.a.z	$1$	$0$	\(0\)	\(20\)	\(8\)	\(0\)		$+$	$-$	$-$		$\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4949))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4949)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(101))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(707))\)\(^{\oplus 2}\)