Space of modular forms of level 98, weight 2, and trivial character

• Label: 98.2.a
• Level: $98$
• Weight: $2$
• Character orbit: 98.a
• Rep. character: $\chi_{98}(1,\cdot)$
• Character field: $\Q$
• Dimension: $3$
• Newform subspaces: $2$
• Sturm bound: $28$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 98 = 2 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	98.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(28\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(98))\).

	Total	New	Old
Modular forms	22	3	19
Cusp forms	7	3	4
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.

\(2\)	\(7\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(4\)	\(0\)	\(4\)		\(1\)	\(0\)	\(1\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)		\(7\)	\(1\)	\(6\)		\(3\)	\(1\)	\(2\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)		\(6\)	\(2\)	\(4\)		\(2\)	\(2\)	\(0\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)		\(5\)	\(0\)	\(5\)		\(1\)	\(0\)	\(1\)		\(4\)	\(0\)	\(4\)
Plus space		\(+\)		\(9\)	\(0\)	\(9\)		\(2\)	\(0\)	\(2\)		\(7\)	\(0\)	\(7\)
Minus space		\(-\)		\(13\)	\(3\)	\(10\)		\(5\)	\(3\)	\(2\)		\(8\)	\(0\)	\(8\)

Trace form

3 * q + q^2 + 2 * q^3 + 3 * q^4 - 2 * q^6 + q^8 - q^9 - 4 * q^11 + 2 * q^12 + 4 * q^13 - 8 * q^15 + 3 * q^16 - 6 * q^17 - 3 * q^18 - 2 * q^19 - 4 * q^22 - 8 * q^23 - 2 * q^24 + q^25 - 4 * q^26 - 4 * q^27 - 2 * q^29 - 8 * q^30 + 4 * q^31 + q^32 + 6 * q^34 - q^36 + 22 * q^37 + 2 * q^38 + 8 * q^39 - 6 * q^41 + 12 * q^43 - 4 * q^44 - 8 * q^46 + 12 * q^47 + 2 * q^48 + 11 * q^50 - 8 * q^51 + 4 * q^52 + 2 * q^53 + 4 * q^54 + 16 * q^57 + 10 * q^58 + 6 * q^59 - 8 * q^60 - 8 * q^61 - 4 * q^62 + 3 * q^64 + 20 * q^67 - 6 * q^68 - 24 * q^71 - 3 * q^72 - 2 * q^73 + 18 * q^74 - 10 * q^75 - 2 * q^76 - 8 * q^78 - 21 * q^81 + 6 * q^82 + 6 * q^83 - 8 * q^85 - 4 * q^86 - 12 * q^87 - 4 * q^88 + 6 * q^89 - 8 * q^92 - 16 * q^93 - 12 * q^94 - 40 * q^95 - 2 * q^96 + 10 * q^97 + 4 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(98))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	7
98.2.a.a	$98$	$2$	98.a	1.a	$1$	$1$	$1$	$0.783$	\(\Q\)	None	✓		✓	✓	14.2.a.a	$1$	$0$	\(-1\)	\(2\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+2q^{3}+q^{4}-2q^{6}-q^{8}+q^{9}+\cdots\)
98.2.a.b	$98$	$2$	98.a	1.a	$1$	$2$	$2$	$0.783$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓	✓	98.2.a.b	$2$	$0$	\(2\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+\beta q^{3}+q^{4}-2\beta q^{5}+\beta q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(98))\) into lower level spaces

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