Space of modular forms of level 9999, weight 2, and trivial character

• Label: 9999.2.a
• Level: $9999$
• Weight: $2$
• Character orbit: 9999.a
• Rep. character: $\chi_{9999}(1,\cdot)$
• Character field: $\Q$
• Dimension: $418$
• Newform subspaces: $34$
• Sturm bound: $2448$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 9999 = 3^{2} \cdot 11 \cdot 101 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9999.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 34 \)
Sturm bound:			\(2448\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(9999))\).

	Total	New	Old
Modular forms	1232	418	814
Cusp forms	1217	418	799
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.

\(3\)	\(11\)	\(101\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(31\)
\(+\)	\(+\)	\(-\)	\(-\)	\(53\)
\(+\)	\(-\)	\(+\)	\(-\)	\(53\)
\(+\)	\(-\)	\(-\)	\(+\)	\(31\)
\(-\)	\(+\)	\(+\)	\(-\)	\(67\)
\(-\)	\(+\)	\(-\)	\(+\)	\(56\)
\(-\)	\(-\)	\(+\)	\(+\)	\(58\)
\(-\)	\(-\)	\(-\)	\(-\)	\(69\)
Plus space			\(+\)	\(176\)
Minus space			\(-\)	\(242\)

Trace form

\( 418 q + 430 q^{4} + 2 q^{5} + 4 q^{7} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(9999))\) 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	11	101
9999.2.a.a	$9999$	$2$	9999.a	1.a	$1$	$1$	$1$	$79.842$	\(\Q\)	None	✓	✓			9999.2.a.a	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(-1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}-q^{7}+q^{11}+4q^{13}+\cdots\)
9999.2.a.b	$9999$	$2$	9999.a	1.a	$1$	$1$	$1$	$79.842$	\(\Q\)	None	✓				3333.2.a.g	$1$	$0$	\(-1\)	\(0\)	\(-2\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}-2q^{5}+4q^{7}+3q^{8}+2q^{10}+\cdots\)
9999.2.a.c	$9999$	$2$	9999.a	1.a	$1$	$1$	$1$	$79.842$	\(\Q\)	None	✓	✓			9999.2.a.c	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+2q^{7}+3q^{8}-q^{11}-2q^{13}+\cdots\)
9999.2.a.d	$9999$	$2$	9999.a	1.a	$1$	$1$	$1$	$79.842$	\(\Q\)	None	✓				3333.2.a.d	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}-4q^{5}+3q^{7}-q^{11}+4q^{16}+\cdots\)
9999.2.a.e	$9999$	$2$	9999.a	1.a	$1$	$1$	$1$	$79.842$	\(\Q\)	None	✓				1111.2.a.b	$1$	$2$	\(0\)	\(0\)	\(-3\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}-3q^{5}-q^{11}-5q^{13}+4q^{16}+\cdots\)
9999.2.a.f	$9999$	$2$	9999.a	1.a	$1$	$1$	$1$	$79.842$	\(\Q\)	None	✓				3333.2.a.f	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-5\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}-2q^{5}-5q^{7}+q^{11}+2q^{13}+\cdots\)
9999.2.a.g	$9999$	$2$	9999.a	1.a	$1$	$1$	$1$	$79.842$	\(\Q\)	None	✓				3333.2.a.c	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+q^{5}-4q^{7}+q^{11}-q^{13}+\cdots\)
9999.2.a.h	$9999$	$2$	9999.a	1.a	$1$	$1$	$1$	$79.842$	\(\Q\)	None	✓				3333.2.a.e	$1$	$1$	\(0\)	\(0\)	\(1\)	\(4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+q^{5}+4q^{7}+q^{11}-q^{13}+\cdots\)
9999.2.a.i	$9999$	$2$	9999.a	1.a	$1$	$1$	$1$	$79.842$	\(\Q\)	None	✓				1111.2.a.a	$1$	$0$	\(0\)	\(0\)	\(3\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+3q^{5}-q^{11}+4q^{13}+4q^{16}+\cdots\)
9999.2.a.j	$9999$	$2$	9999.a	1.a	$1$	$1$	$1$	$79.842$	\(\Q\)	None	✓	✓			9999.2.a.c	$1$	$1$	\(1\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}+2q^{7}-3q^{8}+q^{11}-2q^{13}+\cdots\)
9999.2.a.k	$9999$	$2$	9999.a	1.a	$1$	$1$	$1$	$79.842$	\(\Q\)	None	✓				3333.2.a.b	$1$	$0$	\(2\)	\(0\)	\(-1\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}-q^{5}-2q^{7}-2q^{10}+\cdots\)
9999.2.a.l	$9999$	$2$	9999.a	1.a	$1$	$1$	$1$	$79.842$	\(\Q\)	None	✓	✓			9999.2.a.a	$1$	$1$	\(2\)	\(0\)	\(0\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}-q^{7}-q^{11}+4q^{13}+\cdots\)
9999.2.a.m	$9999$	$2$	9999.a	1.a	$1$	$1$	$1$	$79.842$	\(\Q\)	None	✓				3333.2.a.a	$1$	$1$	\(2\)	\(0\)	\(3\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}+3q^{5}-2q^{7}+6q^{10}+\cdots\)
9999.2.a.n	$9999$	$2$	9999.a	1.a	$1$	$2$	$2$	$79.842$	\(\Q(\sqrt{5}) \)	None	✓				1111.2.a.d	$1$	$0$	\(-1\)	\(0\)	\(3\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-1+\beta )q^{4}+(2-\beta )q^{5}-2q^{7}+\cdots\)
9999.2.a.o	$9999$	$2$	9999.a	1.a	$1$	$2$	$2$	$79.842$	\(\Q(\sqrt{13}) \)	None	✓				3333.2.a.i	$1$	$1$	\(-1\)	\(0\)	\(1\)	\(-3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(1+\beta )q^{4}+\beta q^{5}+(-1-\beta )q^{7}+\cdots\)
9999.2.a.p	$9999$	$2$	9999.a	1.a	$1$	$2$	$2$	$79.842$	\(\Q(\sqrt{6}) \)	None	✓				3333.2.a.h	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}-\beta q^{5}-q^{7}+q^{11}+(-2-\beta )q^{13}+\cdots\)
9999.2.a.q	$9999$	$2$	9999.a	1.a	$1$	$2$	$2$	$79.842$	\(\Q(\sqrt{5}) \)	None	✓				1111.2.a.c	$1$	$0$	\(0\)	\(0\)	\(1\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-2\beta )q^{2}+3q^{4}+(1-\beta )q^{5}+(2+\cdots)q^{7}+\cdots\)
9999.2.a.r	$9999$	$2$	9999.a	1.a	$1$	$5$	$5$	$79.842$	5.5.1060708.1	None	✓				3333.2.a.j	$1$	$0$	\(-2\)	\(0\)	\(-3\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}+(1+\beta _{1}+\beta _{4})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
9999.2.a.s	$9999$	$2$	9999.a	1.a	$1$	$8$	$8$	$79.842$	8.8.\(\cdots\).1	None	✓				3333.2.a.k	$1$	$1$	\(2\)	\(0\)	\(6\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-\beta _{4}-\beta _{6})q^{2}+(1-\beta _{6})q^{4}+(1-\beta _{1}+\cdots)q^{5}+\cdots\)
9999.2.a.t	$9999$	$2$	9999.a	1.a	$1$	$9$	$9$	$79.842$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓				1111.2.a.e	$1$	$0$	\(6\)	\(0\)	\(6\)	\(-3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(1-\beta _{1}+\beta _{2})q^{4}+(1+\cdots)q^{5}+\cdots\)
9999.2.a.u	$9999$	$2$	9999.a	1.a	$1$	$14$	$14$	$79.842$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓				3333.2.a.l	$1$	$1$	\(5\)	\(0\)	\(2\)	\(-12\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{4}-\beta _{12}q^{5}+(-1+\cdots)q^{7}+\cdots\)
9999.2.a.v	$9999$	$2$	9999.a	1.a	$1$	$15$	$15$	$79.842$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓				3333.2.a.m	$1$	$1$	\(-6\)	\(0\)	\(-6\)	\(6\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}-\beta _{9}q^{5}-\beta _{6}q^{7}+\cdots\)
9999.2.a.w	$9999$	$2$	9999.a	1.a	$1$	$16$	$16$	$79.842$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓				3333.2.a.n	$1$	$1$	\(-2\)	\(0\)	\(-2\)	\(3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}-\beta _{14}q^{5}-\beta _{5}q^{7}+\cdots\)
9999.2.a.x	$9999$	$2$	9999.a	1.a	$1$	$16$	$16$	$79.842$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓				1111.2.a.f	$1$	$0$	\(5\)	\(0\)	\(9\)	\(-1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(1+\beta _{9})q^{5}+\cdots\)
9999.2.a.y	$9999$	$2$	9999.a	1.a	$1$	$21$	$21$	$79.842$		None	✓				3333.2.a.o	$1$	$0$	\(-5\)	\(0\)	\(4\)	\(19\)		$-$	$+$	$+$	$-$		$\mathrm{SU}(2)$
9999.2.a.z	$9999$	$2$	9999.a	1.a	$1$	$24$	$24$	$79.842$		None	✓				3333.2.a.q	$1$	$0$	\(4\)	\(0\)	\(8\)	\(0\)		$-$	$-$	$-$	$-$		$\mathrm{SU}(2)$
9999.2.a.ba	$9999$	$2$	9999.a	1.a	$1$	$24$	$24$	$79.842$		None	✓		✓		3333.2.a.p	$1$	$0$	\(6\)	\(0\)	\(4\)	\(-14\)		$-$	$+$	$+$	$-$		$\mathrm{SU}(2)$
9999.2.a.bb	$9999$	$2$	9999.a	1.a	$1$	$25$	$25$	$79.842$		None	✓		✓		1111.2.a.g	$1$	$1$	\(-8\)	\(0\)	\(-8\)	\(7\)		$-$	$+$	$-$	$+$		$\mathrm{SU}(2)$
9999.2.a.bc	$9999$	$2$	9999.a	1.a	$1$	$27$	$27$	$79.842$		None	✓		✓		1111.2.a.h	$1$	$1$	\(-5\)	\(0\)	\(-15\)	\(-3\)		$-$	$-$	$+$	$+$		$\mathrm{SU}(2)$
9999.2.a.bd	$9999$	$2$	9999.a	1.a	$1$	$29$	$29$	$79.842$		None	✓	✓	✓		9999.2.a.bd	$1$	$1$	\(-2\)	\(0\)	\(-4\)	\(-9\)		$+$	$-$	$-$	$+$		$\mathrm{SU}(2)$
9999.2.a.be	$9999$	$2$	9999.a	1.a	$1$	$29$	$29$	$79.842$		None	✓		✓		3333.2.a.r	$1$	$0$	\(-1\)	\(0\)	\(-4\)	\(10\)		$-$	$-$	$-$	$-$		$\mathrm{SU}(2)$
9999.2.a.bf	$9999$	$2$	9999.a	1.a	$1$	$29$	$29$	$79.842$		None	✓	✓	✓		9999.2.a.bd	$1$	$1$	\(2\)	\(0\)	\(4\)	\(-9\)		$+$	$+$	$+$	$+$		$\mathrm{SU}(2)$
9999.2.a.bg	$9999$	$2$	9999.a	1.a	$1$	$53$	$53$	$79.842$		None	✓	✓	✓	✓	9999.2.a.bg	$1$	$0$	\(-3\)	\(0\)	\(4\)	\(12\)		$+$	$+$	$-$	$-$		$\mathrm{SU}(2)$
9999.2.a.bh	$9999$	$2$	9999.a	1.a	$1$	$53$	$53$	$79.842$		None	✓	✓	✓	✓	9999.2.a.bg	$1$	$0$	\(3\)	\(0\)	\(-4\)	\(12\)		$+$	$-$	$+$	$-$		$\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(9999))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(9999)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(101))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(303))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(909))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1111))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3333))\)\(^{\oplus 2}\)