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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(14\)	\(3\)	\(11\)		\(10\)	\(3\)	\(7\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)		\(11\)	\(2\)	\(9\)		\(7\)	\(2\)	\(5\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)		\(12\)	\(1\)	\(11\)		\(8\)	\(1\)	\(7\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)		\(13\)	\(4\)	\(9\)		\(9\)	\(4\)	\(5\)		\(4\)	\(0\)	\(4\)
Plus space		\(+\)		\(27\)	\(7\)	\(20\)		\(19\)	\(7\)	\(12\)		\(8\)	\(0\)	\(8\)
Minus space		\(-\)		\(23\)	\(3\)	\(20\)		\(15\)	\(3\)	\(12\)		\(8\)	\(0\)	\(8\)

Trace form

10 * q - 6 * q^3 + 40 * q^4 + 26 * q^5 + 20 * q^6 + 126 * q^9 - 4 * q^10 - 12 * q^11 - 24 * q^12 - 74 * q^13 + 92 * q^15 + 160 * q^16 + 40 * q^17 + 128 * q^18 - 82 * q^19 + 104 * q^20 - 80 * q^22 + 268 * q^23 + 80 * q^24 + 310 * q^25 - 76 * q^26 - 180 * q^27 - 328 * q^29 - 432 * q^30 - 308 * q^31 + 320 * q^33 + 376 * q^34 + 504 * q^36 - 916 * q^37 + 156 * q^38 - 1352 * q^39 - 16 * q^40 - 288 * q^41 + 288 * q^43 - 48 * q^44 + 242 * q^45 - 80 * q^46 - 12 * q^47 - 96 * q^48 - 1192 * q^50 - 884 * q^51 - 296 * q^52 - 1152 * q^53 - 40 * q^54 + 184 * q^55 + 156 * q^57 + 24 * q^58 + 62 * q^59 + 368 * q^60 - 182 * q^61 - 328 * q^62 + 640 * q^64 + 476 * q^65 - 256 * q^66 + 3044 * q^67 + 160 * q^68 + 656 * q^69 - 704 * q^71 + 512 * q^72 + 612 * q^73 - 232 * q^74 - 530 * q^75 - 328 * q^76 + 128 * q^78 + 2028 * q^79 + 416 * q^80 + 1066 * q^81 + 72 * q^82 + 694 * q^83 + 1240 * q^85 + 416 * q^86 - 1628 * q^87 - 320 * q^88 - 420 * q^89 - 1588 * q^90 + 1072 * q^92 + 4916 * q^93 + 72 * q^94 + 916 * q^95 + 320 * q^96 - 24 * q^97 - 1936 * q^99

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	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
98.4.a.a	$98$	$4$	98.a	1.a	$1$	$1$	$1$	$5.782$	\(\Q\)	None	✓				14.4.a.a	$1$	$1$	\(-2\)	\(-8\)	\(14\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-8q^{3}+4q^{4}+14q^{5}+2^{4}q^{6}+\cdots\)
98.4.a.b	$98$	$4$	98.a	1.a	$1$	$1$	$1$	$5.782$	\(\Q\)	None	✓				14.4.c.b	$1$	$0$	\(-2\)	\(-1\)	\(7\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-q^{3}+4q^{4}+7q^{5}+2q^{6}+\cdots\)
98.4.a.c	$98$	$4$	98.a	1.a	$1$	$1$	$1$	$5.782$	\(\Q\)	None	✓				14.4.c.b	$1$	$1$	\(-2\)	\(1\)	\(-7\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+q^{3}+4q^{4}-7q^{5}-2q^{6}+\cdots\)
98.4.a.d	$98$	$4$	98.a	1.a	$1$	$1$	$1$	$5.782$	\(\Q\)	None	✓		✓	✓	14.4.c.a	$1$	$1$	\(2\)	\(-5\)	\(-9\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-5q^{3}+4q^{4}-9q^{5}-10q^{6}+\cdots\)
98.4.a.e	$98$	$4$	98.a	1.a	$1$	$1$	$1$	$5.782$	\(\Q\)	None	✓				14.4.a.b	$1$	$0$	\(2\)	\(2\)	\(12\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{3}+4q^{4}+12q^{5}+4q^{6}+\cdots\)
98.4.a.f	$98$	$4$	98.a	1.a	$1$	$1$	$1$	$5.782$	\(\Q\)	None	✓				14.4.c.a	$1$	$0$	\(2\)	\(5\)	\(9\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+5q^{3}+4q^{4}+9q^{5}+10q^{6}+\cdots\)
98.4.a.g	$98$	$4$	98.a	1.a	$1$	$2$	$2$	$5.782$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓		98.4.a.g	$2$	$0$	\(-4\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+5\beta q^{3}+4q^{4}+14\beta q^{5}-10\beta q^{6}+\cdots\)
98.4.a.h	$98$	$4$	98.a	1.a	$1$	$2$	$2$	$5.782$	\(\Q(\sqrt{22}) \)	None	✓	✓	✓		98.4.a.h	$2$	$0$	\(4\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+2q^{2}+\beta q^{3}+4q^{4}-\beta q^{5}+2\beta q^{6}+\cdots\)

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

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