Space of modular forms of level 98, weight 10, and trivial character

• Label: 98.10.a
• Level: $98$
• Weight: $10$
• Character orbit: 98.a
• Rep. character: $\chi_{98}(1,\cdot)$
• Character field: $\Q$
• Dimension: $31$
• Newform subspaces: $12$
• Sturm bound: $140$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 98 = 2 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	98.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(140\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(98))\).

	Total	New	Old
Modular forms	134	31	103
Cusp forms	118	31	87
Eisenstein series	16	0	16

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

\(2\)	\(7\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(7\)
\(+\)	\(-\)	$-$	\(8\)
\(-\)	\(+\)	$-$	\(9\)
\(-\)	\(-\)	$+$	\(7\)
Plus space		\(+\)	\(14\)
Minus space		\(-\)	\(17\)

Trace form

\( 31 q + 16 q^{2} + 6 q^{3} + 7936 q^{4} + 756 q^{5} - 544 q^{6} + 4096 q^{8} + 164295 q^{9} + O(q^{10}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(98))\) 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	7
98.10.a.a	$98$	$10$	98.a	1.a	$1$	$1$	$1$	$50.474$	\(\Q\)	None	✓	$1$	$0$	\(-16\)	\(6\)	\(-560\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+6q^{3}+2^{8}q^{4}-560q^{5}+\cdots\)
98.10.a.b	$98$	$10$	98.a	1.a	$1$	$1$	$1$	$50.474$	\(\Q\)	None	✓	$1$	$1$	\(16\)	\(-170\)	\(-544\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}-170q^{3}+2^{8}q^{4}-544q^{5}+\cdots\)
98.10.a.c	$98$	$10$	98.a	1.a	$1$	$1$	$1$	$50.474$	\(\Q\)	None	✓	$1$	$1$	\(16\)	\(156\)	\(-870\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+156q^{3}+2^{8}q^{4}-870q^{5}+\cdots\)
98.10.a.d	$98$	$10$	98.a	1.a	$1$	$2$	$2$	$50.474$	\(\Q(\sqrt{43}) \)	None	✓	$2$	$0$	\(-32\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+\beta q^{3}+2^{8}q^{4}+35\beta q^{5}+\cdots\)
98.10.a.e	$98$	$10$	98.a	1.a	$1$	$2$	$2$	$50.474$	\(\Q(\sqrt{2305}) \)	None	✓	$1$	$0$	\(-32\)	\(14\)	\(2730\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+(7+5\beta )q^{3}+2^{8}q^{4}+(1365+\cdots)q^{5}+\cdots\)
98.10.a.f	$98$	$10$	98.a	1.a	$1$	$2$	$2$	$50.474$	\(\Q(\sqrt{106}) \)	None	✓	$2$	$1$	\(32\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+3\beta q^{3}+2^{8}q^{4}-31\beta q^{5}+\cdots\)
98.10.a.g	$98$	$10$	98.a	1.a	$1$	$3$	$3$	$50.474$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$1$	\(-48\)	\(-71\)	\(1085\)	\(0\)		$+$	$+$	$+$	$2^{3}\cdot 7$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+(-24-\beta _{1})q^{3}+2^{8}q^{4}+\cdots\)
98.10.a.h	$98$	$10$	98.a	1.a	$1$	$3$	$3$	$50.474$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$0$	\(-48\)	\(71\)	\(-1085\)	\(0\)		$+$	$-$	$-$	$2^{3}\cdot 7$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+(24+\beta _{1})q^{3}+2^{8}q^{4}+(-363+\cdots)q^{5}+\cdots\)
98.10.a.i	$98$	$10$	98.a	1.a	$1$	$3$	$3$	$50.474$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$1$	\(48\)	\(-233\)	\(-733\)	\(0\)		$-$	$-$	$+$	$2^{3}\cdot 3\cdot 7$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+(-78-\beta _{1})q^{3}+2^{8}q^{4}+\cdots\)
98.10.a.j	$98$	$10$	98.a	1.a	$1$	$3$	$3$	$50.474$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$0$	\(48\)	\(233\)	\(733\)	\(0\)		$-$	$+$	$-$	$2^{3}\cdot 3\cdot 7$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+(78+\beta _{1})q^{3}+2^{8}q^{4}+(246+\cdots)q^{5}+\cdots\)
98.10.a.k	$98$	$10$	98.a	1.a	$1$	$4$	$4$	$50.474$	\(\Q(\sqrt{2}, \sqrt{4817})\)	None	✓	$2$	$1$	\(-64\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2\cdot 3^{2}\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+(5\beta _{1}+\beta _{3})q^{3}+2^{8}q^{4}+(104\beta _{1}+\cdots)q^{5}+\cdots\)
98.10.a.l	$98$	$10$	98.a	1.a	$1$	$6$	$6$	$50.474$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$2$	$0$	\(96\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{4}\cdot 3^{2}\cdot 7^{6}$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+(\beta _{2}+\beta _{3})q^{3}+2^{8}q^{4}+(-2\beta _{2}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(98))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(98)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 2}\)