Space of modular forms of level 98, weight 8, and Character 98.c

• Label: 98.8.c
• Level: $98$
• Weight: $8$
• Character orbit: 98.c
• Rep. character: $\chi_{98}(67,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $48$
• Newform subspaces: $14$
• Sturm bound: $112$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 98 = 2 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	98.c (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(112\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(98, [\chi])\).

	Total	New	Old
Modular forms	212	48	164
Cusp forms	180	48	132
Eisenstein series	32	0	32

Trace form

48 * q - 1536 * q^4 - 252 * q^5 + 1792 * q^6 - 20496 * q^9 - 1792 * q^10 + 14164 * q^11 - 24080 * q^13 - 46688 * q^15 - 98304 * q^16 - 82740 * q^17 + 48352 * q^18 - 43456 * q^19 + 32256 * q^20 - 49216 * q^22 - 148576 * q^23 - 57344 * q^24 - 229660 * q^25 - 75264 * q^26 + 483840 * q^27 - 83872 * q^29 - 288576 * q^30 - 236264 * q^31 + 282100 * q^33 + 200704 * q^34 + 2623488 * q^36 + 463172 * q^37 - 309120 * q^38 - 1183936 * q^39 - 114688 * q^40 + 700560 * q^41 + 4566760 * q^43 + 906496 * q^44 + 1794856 * q^45 - 207936 * q^46 + 357000 * q^47 + 3741440 * q^50 + 2411060 * q^51 + 770560 * q^52 - 2199644 * q^53 - 1269632 * q^54 - 10240496 * q^55 + 2775168 * q^57 - 1214848 * q^58 + 2502864 * q^59 + 1494016 * q^60 + 2741284 * q^61 - 1532160 * q^62 + 12582912 * q^64 - 4024440 * q^65 - 5411840 * q^66 + 484528 * q^67 - 5295360 * q^68 + 3498712 * q^69 - 10567152 * q^71 + 3094528 * q^72 + 2914380 * q^73 + 285056 * q^74 + 10993136 * q^75 + 5562368 * q^76 + 28912384 * q^78 - 7130528 * q^79 - 1032192 * q^80 - 8928028 * q^81 - 15697920 * q^82 + 21245280 * q^83 - 40069944 * q^85 + 4742880 * q^86 - 234640 * q^87 + 1574912 * q^88 + 10716300 * q^89 + 19102720 * q^90 + 19017728 * q^92 + 7475404 * q^93 - 25452672 * q^94 - 18284768 * q^95 - 3670016 * q^96 - 8250480 * q^97 - 160824728 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(98, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
98.8.c.a	$98$	$8$	98.c	7.c	$3$	$2$	$1$	$30.614$	\(\Q(\sqrt{-3}) \)	None					14.8.a.b	$2$	$0$	\(-8\)	\(-66\)	\(-400\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-8\zeta_{6}q^{2}+(-66+66\zeta_{6})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots\)
98.8.c.b	$98$	$8$	98.c	7.c	$3$	$2$	$1$	$30.614$	\(\Q(\sqrt{-3}) \)	None					14.8.a.b	$2$	$0$	\(-8\)	\(66\)	\(400\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-8\zeta_{6}q^{2}+(66-66\zeta_{6})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots\)
98.8.c.c	$98$	$8$	98.c	7.c	$3$	$2$	$1$	$30.614$	\(\Q(\sqrt{-3}) \)	None					14.8.a.a	$2$	$0$	\(8\)	\(-82\)	\(448\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+8\zeta_{6}q^{2}+(-82+82\zeta_{6})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots\)
98.8.c.d	$98$	$8$	98.c	7.c	$3$	$2$	$1$	$30.614$	\(\Q(\sqrt{-3}) \)	None					2.8.a.a	$2$	$0$	\(8\)	\(-12\)	\(210\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+8\zeta_{6}q^{2}+(-12+12\zeta_{6})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots\)
98.8.c.e	$98$	$8$	98.c	7.c	$3$	$2$	$1$	$30.614$	\(\Q(\sqrt{-3}) \)	None					2.8.a.a	$2$	$0$	\(8\)	\(12\)	\(-210\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+8\zeta_{6}q^{2}+(12-12\zeta_{6})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots\)
98.8.c.f	$98$	$8$	98.c	7.c	$3$	$2$	$1$	$30.614$	\(\Q(\sqrt{-3}) \)	None					14.8.a.a	$2$	$0$	\(8\)	\(82\)	\(-448\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+8\zeta_{6}q^{2}+(82-82\zeta_{6})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots\)
98.8.c.g	$98$	$8$	98.c	7.c	$3$	$4$	$2$	$30.614$	\(\Q(\sqrt{-3}, \sqrt{1969})\)	None					14.8.a.c	$2$	$0$	\(-16\)	\(-70\)	\(-126\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-8+8\beta _{1})q^{2}+(-35\beta _{1}-\beta _{2})q^{3}+\cdots\)
98.8.c.h	$98$	$8$	98.c	7.c	$3$	$4$	$2$	$30.614$	\(\Q(\sqrt{-3}, \sqrt{2389})\)	None					14.8.c.a	$2$	$0$	\(-16\)	\(-56\)	\(-238\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-8+8\beta _{1})q^{2}+(-28\beta _{1}-\beta _{2})q^{3}+\cdots\)
98.8.c.i	$98$	$8$	98.c	7.c	$3$	$4$	$2$	$30.614$	\(\Q(\sqrt{-3}, \sqrt{37})\)	None		✓			98.8.a.j	$4$	$0$	\(-16\)	\(0\)	\(0\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-8+8\beta _{1})q^{2}-\beta _{2}q^{3}-2^{6}\beta _{1}q^{4}+\cdots\)
98.8.c.j	$98$	$8$	98.c	7.c	$3$	$4$	$2$	$30.614$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		✓			98.8.a.i	$4$	$0$	\(-16\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+8\beta _{2}q^{2}+15\beta _{1}q^{3}+(-2^{6}-2^{6}\beta _{2}+\cdots)q^{4}+\cdots\)
98.8.c.k	$98$	$8$	98.c	7.c	$3$	$4$	$2$	$30.614$	\(\Q(\sqrt{-3}, \sqrt{1969})\)	None					14.8.a.c	$2$	$0$	\(-16\)	\(70\)	\(126\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-8+8\beta _{1})q^{2}+(35\beta _{1}-\beta _{2})q^{3}+\cdots\)
98.8.c.l	$98$	$8$	98.c	7.c	$3$	$4$	$2$	$30.614$	\(\Q(\sqrt{-3}, \sqrt{22})\)	None		✓			98.8.a.e	$4$	$0$	\(16\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{3}]$	\(q-8\beta _{2}q^{2}+3\beta _{1}q^{3}+(-2^{6}-2^{6}\beta _{2}+\cdots)q^{4}+\cdots\)
98.8.c.m	$98$	$8$	98.c	7.c	$3$	$4$	$2$	$30.614$	\(\Q(\sqrt{-3}, \sqrt{949})\)	None					14.8.c.b	$2$	$0$	\(16\)	\(56\)	\(-14\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(8-8\beta _{1})q^{2}+(28\beta _{1}+\beta _{2})q^{3}-2^{6}\beta _{1}q^{4}+\cdots\)
98.8.c.n	$98$	$8$	98.c	7.c	$3$	$8$	$4$	$30.614$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		✓	✓		98.8.a.l	$4$	$0$	\(32\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 7^{4}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(8+8\beta _{1})q^{2}+(\beta _{2}+2\beta _{3})q^{3}+2^{6}\beta _{1}q^{4}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(98, [\chi])\) into lower level spaces

\( S_{8}^{\mathrm{old}}(98, [\chi]) \simeq \) \(S_{8}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 2}\)