Embedded newform 99.2.p.a.2.1

• Label: 99.2.p.a.2.1
• Level: $99$
• Weight: $2$
• Character: 99.2
• Analytic conductor: $0.791$
• Analytic rank: $0$
• Dimension: $80$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 99 = 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	99.p (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.790518980011\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(10\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			2.1
Character	\(\chi\)	\(=\)	99.2
Dual form			99.2.p.a.50.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.66735 - 1.85178i) q^{2} +(1.13023 + 1.31247i) q^{3} +(-0.439972 + 4.18606i) q^{4} +(1.05074 + 0.946093i) q^{5} +(0.545914 - 4.28128i) q^{6} +(1.27299 + 2.85917i) q^{7} +(4.45339 - 3.23558i) q^{8} +(-0.445156 + 2.96679i) q^{9} -3.52320i q^{10} +(-0.715855 - 3.23845i) q^{11} +(-5.99134 + 4.15376i) q^{12} +(-1.11949 - 5.26678i) q^{13} +(3.17204 - 7.12451i) q^{14} +(-0.0541368 + 2.44837i) q^{15} +(-5.18263 - 1.10160i) q^{16} +(0.663043 + 2.04064i) q^{17} +(6.23605 - 4.12234i) q^{18} +(1.17531 + 1.61768i) q^{19} +(-4.42270 + 3.98221i) q^{20} +(-2.31381 + 4.90228i) q^{21} +(-4.80330 + 6.72522i) q^{22} +(-1.15277 + 0.665550i) q^{23} +(9.27995 + 2.18799i) q^{24} +(-0.313674 - 2.98441i) q^{25} +(-7.88632 + 10.8546i) q^{26} +(-4.39695 + 2.76890i) q^{27} +(-12.5287 + 4.07083i) q^{28} +(-0.937918 + 0.417588i) q^{29} +(4.62410 - 3.98203i) q^{30} +(9.20270 - 1.95610i) q^{31} +(1.09663 + 1.89941i) q^{32} +(3.44129 - 4.59973i) q^{33} +(2.67328 - 4.63025i) q^{34} +(-1.36746 + 4.20862i) q^{35} +(-12.2233 - 3.16875i) q^{36} +(-5.64362 - 4.10033i) q^{37} +(1.03592 - 4.87364i) q^{38} +(5.64721 - 7.42198i) q^{39} +(7.74052 + 0.813562i) q^{40} +(-2.92640 - 1.30292i) q^{41} +(12.9358 - 3.88914i) q^{42} +(-4.42393 - 2.55416i) q^{43} +(13.8713 - 1.57178i) q^{44} +(-3.27460 + 2.69617i) q^{45} +(3.15451 + 1.02496i) q^{46} +(-4.64300 + 0.487999i) q^{47} +(-4.41175 - 8.04711i) q^{48} +(-1.87046 + 2.07736i) q^{49} +(-5.00345 + 5.55689i) q^{50} +(-1.92888 + 3.17662i) q^{51} +(22.5396 - 2.36901i) q^{52} +(-4.78123 - 1.55352i) q^{53} +(12.4586 + 3.52544i) q^{54} +(2.31170 - 4.08004i) q^{55} +(14.9202 + 8.61416i) q^{56} +(-0.794780 + 3.37091i) q^{57} +(2.33711 + 1.04055i) q^{58} +(10.5570 + 1.10959i) q^{59} +(-10.2252 - 1.30384i) q^{60} +(0.610725 - 2.87323i) q^{61} +(-18.9663 - 13.7799i) q^{62} +(-9.04924 + 2.50390i) q^{63} +(-1.58577 + 4.88050i) q^{64} +(3.80657 - 6.59317i) q^{65} +(-14.2555 + 1.29686i) q^{66} +(-0.745757 - 1.29169i) q^{67} +(-8.83394 + 1.87771i) q^{68} +(-2.17641 - 0.760747i) q^{69} +(10.0734 - 4.48499i) q^{70} +(10.6126 - 3.44824i) q^{71} +(7.61682 + 14.6526i) q^{72} +(-5.45572 + 7.50916i) q^{73} +(1.81698 + 17.2874i) q^{74} +(3.56242 - 3.78476i) q^{75} +(-7.28879 + 4.20819i) q^{76} +(8.34801 - 6.16925i) q^{77} +(-23.1597 + 1.91763i) q^{78} +(-7.85688 + 7.07437i) q^{79} +(-4.40339 - 6.06075i) q^{80} +(-8.60367 - 2.64137i) q^{81} +(2.46661 + 7.59144i) q^{82} +(6.66232 + 1.41612i) q^{83} +(-19.5032 - 11.8426i) q^{84} +(-1.23394 + 2.77149i) q^{85} +(2.64649 + 12.4508i) q^{86} +(-1.60814 - 0.759018i) q^{87} +(-13.6662 - 12.1059i) q^{88} +8.06536i q^{89} +(10.4526 + 1.56837i) q^{90} +(13.6335 - 9.90535i) q^{91} +(-2.27885 - 5.11837i) q^{92} +(12.9685 + 9.86743i) q^{93} +(8.64515 + 7.78413i) q^{94} +(-0.295523 + 2.81172i) q^{95} +(-1.25348 + 3.58606i) q^{96} +(-1.58246 - 1.75751i) q^{97} +6.96550 q^{98} +(9.92646 - 0.682176i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	80 * q - 15 * q^2 - 3 * q^3 + 5 * q^4 - 6 * q^5 - 15 * q^6 - 5 * q^7 - q^9 - 3 * q^11 - 54 * q^12 - 5 * q^13 - 9 * q^14 + 5 * q^16 - 50 * q^19 - 3 * q^20 - 11 * q^22 - 42 * q^23 - 5 * q^24 - 2 * q^25 + 3 * q^27 - 20 * q^28 + 30 * q^29 + 50 * q^30 - 6 * q^31 + 4 * q^33 - 10 * q^34 - 17 * q^36 - 6 * q^37 + 9 * q^38 + 85 * q^39 + 15 * q^40 - 15 * q^41 + 19 * q^42 - 12 * q^45 - 40 * q^46 - 21 * q^47 + 70 * q^48 - q^49 + 60 * q^50 - 45 * q^51 - 5 * q^52 - 18 * q^55 + 90 * q^56 + 60 * q^57 - 29 * q^58 + 81 * q^59 + 43 * q^60 - 5 * q^61 + 15 * q^63 - 8 * q^64 - 39 * q^66 + 10 * q^67 + 180 * q^68 - 20 * q^69 + 30 * q^70 + 5 * q^72 - 20 * q^73 - 15 * q^74 - 30 * q^75 + 33 * q^77 + 152 * q^78 - 5 * q^79 - 73 * q^81 - 2 * q^82 - 60 * q^83 - 135 * q^84 - 5 * q^85 - 48 * q^86 - 59 * q^88 - 70 * q^90 + 52 * q^91 - 213 * q^92 - 34 * q^93 - 5 * q^94 - 135 * q^95 - 145 * q^96 + 27 * q^97 + 35 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/99\mathbb{Z}\right)^\times\).

\(n\)	\(46\)	\(56\)
\(\chi(n)\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{1}{6}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.66735	−	1.85178i	−1.17899	−	1.30940i	−0.941107	−	0.338109i	\(-0.890213\pi\)
							−0.237885	−	0.971293i	\(-0.576454\pi\)
\(3\)	1.13023	+	1.31247i	0.652539	+	0.757755i
							\(5\)	1.05074	+	0.946093i	0.469906	+	0.423106i	0.869761	−	0.493473i	\(-0.164273\pi\)
−0.399855	+	0.916579i	\(0.630939\pi\)
\(7\)	1.27299	+	2.85917i	0.481143	+	1.08067i	0.977189	+	0.212369i	\(0.0681180\pi\)
							−0.496046	+	0.868296i	\(0.665215\pi\)
\(11\)	−0.715855	−	3.23845i	−0.215838	−	0.976429i
							\(13\)	−1.11949	−	5.26678i	−0.310490	−	1.46074i	−0.805884	−	0.592074i	\(-0.798309\pi\)
0.495393	−	0.868669i	\(-0.335024\pi\)
\(17\)	0.663043	+	2.04064i	0.160812	+	0.494927i	0.998703	−	0.0509094i	\(-0.0162120\pi\)
							−0.837892	+	0.545836i	\(0.816212\pi\)
\(19\)	1.17531	+	1.61768i	0.269635	+	0.371121i	0.922266	−	0.386555i	\(-0.126335\pi\)
							−0.652632	+	0.757675i	\(0.726335\pi\)
\(23\)	−1.15277	+	0.665550i	−0.240369	+	0.138777i	−0.615346	−	0.788257i	\(-0.710984\pi\)
							0.374978	+	0.927034i	\(0.377650\pi\)
\(29\)	−0.937918	+	0.417588i	−0.174167	+	0.0775441i	−0.491968	−	0.870613i	\(-0.663722\pi\)
							0.317801	+	0.948157i	\(0.397056\pi\)
\(31\)	9.20270	−	1.95610i	1.65285	−	0.351325i	0.715207	−	0.698913i	\(-0.246332\pi\)
							0.937647	+	0.347588i	\(0.112999\pi\)
\(37\)	−5.64362	−	4.10033i	−0.927805	−	0.674090i	0.0176490	−	0.999844i	\(-0.494382\pi\)
							−0.945455	+	0.325754i	\(0.894382\pi\)
\(41\)	−2.92640	−	1.30292i	−0.457026	−	0.203481i	0.165293	−	0.986244i	\(-0.447143\pi\)
							−0.622320	+	0.782763i	\(0.713810\pi\)
\(43\)	−4.42393	−	2.55416i	−0.674642	−	0.389505i	0.123191	−	0.992383i	\(-0.460687\pi\)
							−0.797833	+	0.602878i	\(0.794021\pi\)
\(47\)	−4.64300	+	0.487999i	−0.677251	+	0.0711819i	−0.436908	−	0.899506i	\(-0.643926\pi\)
							−0.240343	+	0.970688i	\(0.577260\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	99.2.p.a.2.1	✓	80
3.2	odd	2		297.2.t.a.35.10		80
9.2	odd	6		891.2.k.a.728.20		80
9.4	even	3		297.2.t.a.233.10		80
9.5	odd	6	inner	99.2.p.a.68.1	yes	80
9.7	even	3		891.2.k.a.728.1		80
11.6	odd	10	inner	99.2.p.a.83.1	yes	80
33.17	even	10		297.2.t.a.116.10		80
99.50	even	30	inner	99.2.p.a.50.1	yes	80
99.61	odd	30		891.2.k.a.809.20		80
99.83	even	30		891.2.k.a.809.1		80
99.94	odd	30		297.2.t.a.17.10		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.2.p.a.2.1	✓	80	1.1	even	1	trivial
99.2.p.a.50.1	yes	80	99.50	even	30	inner
99.2.p.a.68.1	yes	80	9.5	odd	6	inner
99.2.p.a.83.1	yes	80	11.6	odd	10	inner
297.2.t.a.17.10		80	99.94	odd	30
297.2.t.a.35.10		80	3.2	odd	2
297.2.t.a.116.10		80	33.17	even	10
297.2.t.a.233.10		80	9.4	even	3
891.2.k.a.728.1		80	9.7	even	3
891.2.k.a.728.20		80	9.2	odd	6
891.2.k.a.809.1		80	99.83	even	30
891.2.k.a.809.20		80	99.61	odd	30