Space of modular forms of level 98, weight 8, and trivial character

• Label: 98.8.a
• Level: $98$
• Weight: $8$
• Character orbit: 98.a
• Rep. character: $\chi_{98}(1,\cdot)$
• Character field: $\Q$
• Dimension: $23$
• Newform subspaces: $12$
• Sturm bound: $112$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 98 = 2 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	98.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(112\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(98))\).

	Total	New	Old
Modular forms	106	23	83
Cusp forms	90	23	67
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
\(+\)	\(+\)	\(+\)		\(28\)	\(6\)	\(22\)		\(24\)	\(6\)	\(18\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)		\(25\)	\(6\)	\(19\)		\(21\)	\(6\)	\(15\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)		\(26\)	\(4\)	\(22\)		\(22\)	\(4\)	\(18\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)		\(27\)	\(7\)	\(20\)		\(23\)	\(7\)	\(16\)		\(4\)	\(0\)	\(4\)
Plus space		\(+\)		\(55\)	\(13\)	\(42\)		\(47\)	\(13\)	\(34\)		\(8\)	\(0\)	\(8\)
Minus space		\(-\)		\(51\)	\(10\)	\(41\)		\(43\)	\(10\)	\(33\)		\(8\)	\(0\)	\(8\)

Trace form

23 * q - 8 * q^2 + 66 * q^3 + 1472 * q^4 + 36 * q^5 - 592 * q^6 - 512 * q^8 + 14883 * q^9 + 4096 * q^10 - 6028 * q^11 + 4224 * q^12 + 2352 * q^13 + 43832 * q^15 + 94208 * q^16 - 15222 * q^17 - 38824 * q^18 + 93038 * q^19 + 2304 * q^20 + 57952 * q^22 - 2984 * q^23 - 37888 * q^24 - 4395 * q^25 + 154784 * q^26 + 49212 * q^27 + 192646 * q^29 - 220992 * q^30 - 195612 * q^31 - 32768 * q^32 + 802880 * q^33 + 153296 * q^34 + 952512 * q^36 - 806266 * q^37 + 688976 * q^38 + 347128 * q^39 + 262144 * q^40 - 253302 * q^41 + 1597476 * q^43 - 385792 * q^44 + 760052 * q^45 + 875712 * q^46 - 2378676 * q^47 + 270336 * q^48 - 2038552 * q^50 + 3965224 * q^51 + 150528 * q^52 + 2249066 * q^53 - 3501664 * q^54 - 5083696 * q^55 + 3068232 * q^57 + 1571440 * q^58 + 4864758 * q^59 + 2805248 * q^60 - 1562532 * q^61 + 6598112 * q^62 + 6029312 * q^64 + 9257976 * q^65 + 3473408 * q^66 - 5907300 * q^67 - 974208 * q^68 - 10979008 * q^69 + 698328 * q^71 - 2484736 * q^72 - 2937386 * q^73 + 5063088 * q^74 + 14355910 * q^75 + 5954432 * q^76 - 7449664 * q^78 - 12865584 * q^79 + 147456 * q^80 - 1507889 * q^81 - 4106928 * q^82 - 4771242 * q^83 + 8239320 * q^85 - 24026272 * q^86 - 29343644 * q^87 + 3708928 * q^88 + 22097454 * q^89 + 45466112 * q^90 - 190976 * q^92 - 12335128 * q^93 + 2976 * q^94 - 32614984 * q^95 - 2424832 * q^96 - 26400710 * q^97 - 21405796 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(98))\) 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
98.8.a.a	$98$	$8$	98.a	1.a	$1$	$1$	$1$	$30.614$	\(\Q\)	None	✓				2.8.a.a	$1$	$1$	\(-8\)	\(-12\)	\(210\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}-12q^{3}+2^{6}q^{4}+210q^{5}+\cdots\)
98.8.a.b	$98$	$8$	98.a	1.a	$1$	$1$	$1$	$30.614$	\(\Q\)	None	✓				14.8.a.a	$1$	$1$	\(-8\)	\(82\)	\(-448\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}+82q^{3}+2^{6}q^{4}-448q^{5}+\cdots\)
98.8.a.c	$98$	$8$	98.a	1.a	$1$	$1$	$1$	$30.614$	\(\Q\)	None	✓				14.8.a.b	$1$	$0$	\(8\)	\(66\)	\(400\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+66q^{3}+2^{6}q^{4}+20^{2}q^{5}+\cdots\)
98.8.a.d	$98$	$8$	98.a	1.a	$1$	$2$	$2$	$30.614$	\(\Q(\sqrt{949}) \)	None	✓				14.8.c.b	$1$	$1$	\(-16\)	\(-56\)	\(14\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-8q^{2}+(-28-\beta )q^{3}+2^{6}q^{4}+(7+\cdots)q^{5}+\cdots\)
98.8.a.e	$98$	$8$	98.a	1.a	$1$	$2$	$2$	$30.614$	\(\Q(\sqrt{22}) \)	None	✓	✓			98.8.a.e	$2$	$1$	\(-16\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-8q^{2}+3\beta q^{3}+2^{6}q^{4}+7\beta q^{5}-24\beta q^{6}+\cdots\)
98.8.a.f	$98$	$8$	98.a	1.a	$1$	$2$	$2$	$30.614$	\(\Q(\sqrt{949}) \)	None	✓				14.8.c.b	$1$	$0$	\(-16\)	\(56\)	\(-14\)	\(0\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-8q^{2}+(28-\beta )q^{3}+2^{6}q^{4}+(-7+\cdots)q^{5}+\cdots\)
98.8.a.g	$98$	$8$	98.a	1.a	$1$	$2$	$2$	$30.614$	\(\Q(\sqrt{1969}) \)	None	✓				14.8.a.c	$1$	$0$	\(16\)	\(-70\)	\(-126\)	\(0\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+8q^{2}+(-35-\beta )q^{3}+2^{6}q^{4}+(-63+\cdots)q^{5}+\cdots\)
98.8.a.h	$98$	$8$	98.a	1.a	$1$	$2$	$2$	$30.614$	\(\Q(\sqrt{2389}) \)	None	✓				14.8.c.a	$1$	$1$	\(16\)	\(-56\)	\(-238\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+8q^{2}+(-28-\beta )q^{3}+2^{6}q^{4}+(-119+\cdots)q^{5}+\cdots\)
98.8.a.i	$98$	$8$	98.a	1.a	$1$	$2$	$2$	$30.614$	\(\Q(\sqrt{2}) \)	None	✓	✓			98.8.a.i	$2$	$1$	\(16\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+15\beta q^{3}+2^{6}q^{4}-86\beta q^{5}+\cdots\)
98.8.a.j	$98$	$8$	98.a	1.a	$1$	$2$	$2$	$30.614$	\(\Q(\sqrt{37}) \)	None	✓	✓			98.8.a.j	$2$	$0$	\(16\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+8q^{2}-\beta q^{3}+2^{6}q^{4}-5\beta q^{5}-8\beta q^{6}+\cdots\)
98.8.a.k	$98$	$8$	98.a	1.a	$1$	$2$	$2$	$30.614$	\(\Q(\sqrt{2389}) \)	None	✓				14.8.c.a	$1$	$0$	\(16\)	\(56\)	\(238\)	\(0\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+8q^{2}+(28-\beta )q^{3}+2^{6}q^{4}+(119+\cdots)q^{5}+\cdots\)
98.8.a.l	$98$	$8$	98.a	1.a	$1$	$4$	$4$	$30.614$	\(\Q(\sqrt{2}, \sqrt{1801})\)	None	✓	✓	✓		98.8.a.l	$2$	$0$	\(-32\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2^{3}\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q-8q^{2}+(2\beta _{1}+\beta _{3})q^{3}+2^{6}q^{4}+10\beta _{1}q^{5}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(98))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(98)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 2}\)