Space of modular forms of level 99, weight 8, and trivial character

• Label: 99.8.a
• Level: $99$
• Weight: $8$
• Character orbit: 99.a
• Rep. character: $\chi_{99}(1,\cdot)$
• Character field: $\Q$
• Dimension: $28$
• Newform subspaces: $9$
• Sturm bound: $96$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 99 = 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	99.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(96\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(99))\).

	Total	New	Old
Modular forms	88	28	60
Cusp forms	80	28	52
Eisenstein series	8	0	8

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

\(3\)	\(11\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(5\)
\(+\)	\(-\)	$-$	\(5\)
\(-\)	\(+\)	$-$	\(8\)
\(-\)	\(-\)	$+$	\(10\)
Plus space		\(+\)	\(15\)
Minus space		\(-\)	\(13\)

Trace form

\( 28 q - 8 q^{2} + 1548 q^{4} + 709 q^{5} + 726 q^{7} - 5136 q^{8} + O(q^{10}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(99))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	11
99.8.a.a	$99$	$8$	99.a	1.a	$1$	$1$	$1$	$30.926$	\(\Q\)	None	✓	$1$	$0$	\(-10\)	\(0\)	\(410\)	\(-1028\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-10q^{2}-28q^{4}+410q^{5}-1028q^{7}+\cdots\)
99.8.a.b	$99$	$8$	99.a	1.a	$1$	$2$	$2$	$30.926$	\(\Q(\sqrt{97}) \)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(194\)	\(-418\)		$-$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(90+\beta )q^{4}+(90+14\beta )q^{5}+\cdots\)
99.8.a.c	$99$	$8$	99.a	1.a	$1$	$2$	$2$	$30.926$	\(\Q(\sqrt{15}) \)	None	✓	$1$	$1$	\(8\)	\(0\)	\(470\)	\(-1228\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(4+\beta )q^{2}+(-52+8\beta )q^{4}+(235+\cdots)q^{5}+\cdots\)
99.8.a.d	$99$	$8$	99.a	1.a	$1$	$2$	$2$	$30.926$	\(\Q(\sqrt{177}) \)	None	✓	$1$	$0$	\(19\)	\(0\)	\(34\)	\(-166\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(10-\beta )q^{2}+(2^{4}-19\beta )q^{4}+(6^{2}-38\beta )q^{5}+\cdots\)
99.8.a.e	$99$	$8$	99.a	1.a	$1$	$3$	$3$	$30.926$	3.3.115512.1	None	✓	$1$	$0$	\(-9\)	\(0\)	\(444\)	\(1614\)		$-$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{1})q^{2}+(-5+13\beta _{1}-\beta _{2})q^{4}+\cdots\)
99.8.a.f	$99$	$8$	99.a	1.a	$1$	$4$	$4$	$30.926$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(-15\)	\(0\)	\(-306\)	\(890\)		$-$	$+$	$-$	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-4+\beta _{1})q^{2}+(142-2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
99.8.a.g	$99$	$8$	99.a	1.a	$1$	$4$	$4$	$30.926$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-537\)	\(170\)		$-$	$-$	$+$	$2\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(151+\beta _{1}+\beta _{2}+\beta _{3})q^{4}+\cdots\)
99.8.a.h	$99$	$8$	99.a	1.a	$1$	$5$	$5$	$30.926$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(-8\)	\(0\)	\(-500\)	\(446\)		$+$	$-$	$-$	$2^{4}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{1})q^{2}+(35+4\beta _{1}-\beta _{3})q^{4}+\cdots\)
99.8.a.i	$99$	$8$	99.a	1.a	$1$	$5$	$5$	$30.926$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(8\)	\(0\)	\(500\)	\(446\)		$+$	$+$	$+$	$2^{4}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+(2+\beta _{1})q^{2}+(35+4\beta _{1}-\beta _{3})q^{4}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(99))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(99)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 2}\)