Space of modular forms of level 9904, weight 2, and trivial character

• Label: 9904.2.a
• Level: $9904$
• Weight: $2$
• Character orbit: 9904.a
• Rep. character: $\chi_{9904}(1,\cdot)$
• Character field: $\Q$
• Dimension: $309$
• Newform subspaces: $17$
• Sturm bound: $2480$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 9904 = 2^{4} \cdot 619 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9904.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 17 \)
Sturm bound:			\(2480\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(9904))\).

	Total	New	Old
Modular forms	1246	309	937
Cusp forms	1235	309	926
Eisenstein series	11	0	11

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

\(2\)	\(619\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(70\)
\(+\)	\(-\)	$-$	\(85\)
\(-\)	\(+\)	$-$	\(77\)
\(-\)	\(-\)	$+$	\(77\)
Plus space		\(+\)	\(147\)
Minus space		\(-\)	\(162\)

Trace form

\( 309 q - 2 q^{5} - 2 q^{7} + 309 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(9904))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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	619
9904.2.a.a	$9904$	$2$	9904.a	1.a	$1$	$1$	$1$	$79.084$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(3\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+3q^{5}+4q^{7}+q^{9}+6q^{11}+\cdots\)
9904.2.a.b	$9904$	$2$	9904.a	1.a	$1$	$1$	$1$	$79.084$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}-3q^{9}-q^{11}+q^{13}-4q^{17}+\cdots\)
9904.2.a.c	$9904$	$2$	9904.a	1.a	$1$	$1$	$1$	$79.084$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}-3q^{9}+q^{11}-q^{13}+4q^{17}+\cdots\)
9904.2.a.d	$9904$	$2$	9904.a	1.a	$1$	$2$	$2$	$79.084$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(-3\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2\beta q^{3}+(-2+\beta )q^{5}+(1+\beta )q^{7}+\cdots\)
9904.2.a.e	$9904$	$2$	9904.a	1.a	$1$	$2$	$2$	$79.084$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(0\)	\(4\)	\(1\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+(1-\beta )q^{5}-\beta q^{7}+q^{9}+(4+\cdots)q^{11}+\cdots\)
9904.2.a.f	$9904$	$2$	9904.a	1.a	$1$	$3$	$3$	$79.084$	3.3.469.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(8\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{5}+(3-\beta _{1})q^{7}-3q^{9}+(1-\beta _{1}+\cdots)q^{11}+\cdots\)
9904.2.a.g	$9904$	$2$	9904.a	1.a	$1$	$8$	$8$	$79.084$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(2\)	\(-4\)	\(10\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{5}q^{3}-\beta _{7}q^{5}+(1-\beta _{4})q^{7}+(-\beta _{1}+\cdots)q^{9}+\cdots\)
9904.2.a.h	$9904$	$2$	9904.a	1.a	$1$	$17$	$17$	$79.084$	\(\mathbb{Q}[x]/(x^{17} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-5\)	\(7\)	\(-9\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-\beta _{6}q^{5}+(-1-\beta _{4})q^{7}+(1+\cdots)q^{9}+\cdots\)
9904.2.a.i	$9904$	$2$	9904.a	1.a	$1$	$19$	$19$	$79.084$	\(\mathbb{Q}[x]/(x^{19} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-7\)	\(2\)	\(-14\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-\beta _{14}q^{5}+(-1+\beta _{7})q^{7}+\cdots\)
9904.2.a.j	$9904$	$2$	9904.a	1.a	$1$	$21$	$21$	$79.084$		None	✓	$1$	$1$	\(0\)	\(5\)	\(-21\)	\(4\)		$-$	$-$	$+$		$\mathrm{SU}(2)$
9904.2.a.k	$9904$	$2$	9904.a	1.a	$1$	$23$	$23$	$79.084$		None	✓	$1$	$0$	\(0\)	\(5\)	\(-2\)	\(9\)		$-$	$+$	$-$		$\mathrm{SU}(2)$
9904.2.a.l	$9904$	$2$	9904.a	1.a	$1$	$28$	$28$	$79.084$		None	✓	$1$	$1$	\(0\)	\(-5\)	\(-2\)	\(-5\)		$-$	$-$	$+$		$\mathrm{SU}(2)$
9904.2.a.m	$9904$	$2$	9904.a	1.a	$1$	$29$	$29$	$79.084$		None	✓	$1$	$1$	\(0\)	\(3\)	\(-5\)	\(8\)		$+$	$+$	$+$		$\mathrm{SU}(2)$
9904.2.a.n	$9904$	$2$	9904.a	1.a	$1$	$30$	$30$	$79.084$		None	✓	$1$	$0$	\(0\)	\(-1\)	\(21\)	\(-2\)		$-$	$+$	$-$		$\mathrm{SU}(2)$
9904.2.a.o	$9904$	$2$	9904.a	1.a	$1$	$36$	$36$	$79.084$		None	✓	$1$	$0$	\(0\)	\(11\)	\(-9\)	\(12\)		$+$	$-$	$-$		$\mathrm{SU}(2)$
9904.2.a.p	$9904$	$2$	9904.a	1.a	$1$	$40$	$40$	$79.084$		None	✓	$1$	$1$	\(0\)	\(-8\)	\(5\)	\(-16\)		$+$	$+$	$+$		$\mathrm{SU}(2)$
9904.2.a.q	$9904$	$2$	9904.a	1.a	$1$	$48$	$48$	$79.084$		None	✓	$1$	$0$	\(0\)	\(-4\)	\(6\)	\(-11\)		$+$	$-$	$-$		$\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(9904))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(9904)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(619))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1238))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2476))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(4952))\)\(^{\oplus 2}\)