Space of modular forms of level 98, weight 4, and trivial character

• Label: 98.4.a
• Level: $98$
• Weight: $4$
• Character orbit: 98.a
• Rep. character: $\chi_{98}(1,\cdot)$
• Character field: $\Q$
• Dimension: $10$
• Newform subspaces: $8$
• Sturm bound: $56$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 98 = 2 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	98.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(56\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(98))\).

	Total	New	Old
Modular forms	50	10	40
Cusp forms	34	10	24
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
\(+\)	\(+\)	$+$	\(3\)
\(+\)	\(-\)	$-$	\(2\)
\(-\)	\(+\)	$-$	\(1\)
\(-\)	\(-\)	$+$	\(4\)
Plus space		\(+\)	\(7\)
Minus space		\(-\)	\(3\)

Trace form

\( 10 q - 6 q^{3} + 40 q^{4} + 26 q^{5} + 20 q^{6} + 126 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(98))\) 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
98.4.a.a	$98$	$4$	98.a	1.a	$1$	$1$	$1$	$5.782$	\(\Q\)	None	✓	$1$	$1$	\(-2\)	\(-8\)	\(14\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-8q^{3}+4q^{4}+14q^{5}+2^{4}q^{6}+\cdots\)
98.4.a.b	$98$	$4$	98.a	1.a	$1$	$1$	$1$	$5.782$	\(\Q\)	None	✓	$1$	$0$	\(-2\)	\(-1\)	\(7\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-q^{3}+4q^{4}+7q^{5}+2q^{6}+\cdots\)
98.4.a.c	$98$	$4$	98.a	1.a	$1$	$1$	$1$	$5.782$	\(\Q\)	None	✓	$1$	$1$	\(-2\)	\(1\)	\(-7\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+q^{3}+4q^{4}-7q^{5}-2q^{6}+\cdots\)
98.4.a.d	$98$	$4$	98.a	1.a	$1$	$1$	$1$	$5.782$	\(\Q\)	None	✓	$1$	$1$	\(2\)	\(-5\)	\(-9\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-5q^{3}+4q^{4}-9q^{5}-10q^{6}+\cdots\)
98.4.a.e	$98$	$4$	98.a	1.a	$1$	$1$	$1$	$5.782$	\(\Q\)	None	✓	$1$	$0$	\(2\)	\(2\)	\(12\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{3}+4q^{4}+12q^{5}+4q^{6}+\cdots\)
98.4.a.f	$98$	$4$	98.a	1.a	$1$	$1$	$1$	$5.782$	\(\Q\)	None	✓	$1$	$0$	\(2\)	\(5\)	\(9\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+5q^{3}+4q^{4}+9q^{5}+10q^{6}+\cdots\)
98.4.a.g	$98$	$4$	98.a	1.a	$1$	$2$	$2$	$5.782$	\(\Q(\sqrt{2}) \)	None	✓	$2$	$0$	\(-4\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+5\beta q^{3}+4q^{4}+14\beta q^{5}-10\beta q^{6}+\cdots\)
98.4.a.h	$98$	$4$	98.a	1.a	$1$	$2$	$2$	$5.782$	\(\Q(\sqrt{22}) \)	None	✓	$2$	$0$	\(4\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+2q^{2}+\beta q^{3}+4q^{4}-\beta q^{5}+2\beta q^{6}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(98))\) into lower level spaces

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