LMFDB - Embedded newform 867.2.d.f.577.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [867,2,Mod(577,867)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(867, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("867.577"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 867 = 3 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	867.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$6.92302985525$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.836829184.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 14x^{6} + 61x^{4} + 84x^{2} + 4 $ x^8 + 14x^6 + 61x^4 + 84*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 51)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			577.7
Root			$-0.222191i$ of defining polynomial
Character	$\chi$	$=$	867.577
Dual form			867.2.d.f.577.8

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.63640 q^{2} -1.00000i q^{3} +4.95063 q^{4} -2.22219i q^{5} -2.63640i q^{6} +2.31423i q^{7} +7.77906 q^{8} -1.00000 q^{9} -5.85860i q^{10} -1.95063i q^{11} -4.95063i q^{12} +1.32218 q^{13} +6.10124i q^{14} -2.22219 q^{15} +10.6075 q^{16} -2.63640 q^{18} -5.95063 q^{19} -11.0012i q^{20} +2.31423 q^{21} -5.14265i q^{22} +4.77906i q^{23} -7.77906i q^{24} +0.0618655 q^{25} +3.48580 q^{26} +1.00000i q^{27} +11.4569i q^{28} +0.172822i q^{29} -5.85860 q^{30} +0.444383i q^{31} +12.4075 q^{32} -1.95063 q^{33} +5.14265 q^{35} -4.95063 q^{36} +5.31548i q^{37} -15.6883 q^{38} -1.32218i q^{39} -17.2866i q^{40} -4.02222i q^{41} +6.10124 q^{42} -8.33468 q^{43} -9.65685i q^{44} +2.22219i q^{45} +12.5995i q^{46} +2.38404 q^{47} -10.6075i q^{48} +1.64436 q^{49} +0.163102 q^{50} +6.54562 q^{52} +0.727190 q^{53} +2.63640i q^{54} -4.33468 q^{55} +18.0025i q^{56} +5.95063i q^{57} +0.455630i q^{58} +11.7870 q^{59} -11.0012 q^{60} +8.68702i q^{61} +1.17157i q^{62} -2.31423i q^{63} +11.4963 q^{64} -2.93813i q^{65} -5.14265 q^{66} -14.1866 q^{67} +4.77906 q^{69} +13.5581 q^{70} +6.10124i q^{71} -7.77906 q^{72} -10.5569i q^{73} +14.0137i q^{74} -0.0618655i q^{75} -29.4594 q^{76} +4.51420 q^{77} -3.48580i q^{78} -0.698270i q^{79} -23.5719i q^{80} +1.00000 q^{81} -10.6042i q^{82} -8.52971 q^{83} +11.4569 q^{84} -21.9736 q^{86} +0.172822 q^{87} -15.1741i q^{88} +5.40297 q^{89} +5.85860i q^{90} +3.05982i q^{91} +23.6594i q^{92} +0.444383 q^{93} +6.28531 q^{94} +13.2234i q^{95} -12.4075i q^{96} -4.71594i q^{97} +4.33519 q^{98} +1.95063i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{2} + 12 q^{4} + 12 q^{8} - 8 q^{9} + 4 q^{13} - 12 q^{15} + 12 q^{16} - 4 q^{18} - 20 q^{19} + 8 q^{21} - 4 q^{25} + 40 q^{26} - 24 q^{30} + 28 q^{32} + 12 q^{33} + 8 q^{35} - 12 q^{36} - 24 q^{38}+ \cdots + 76 q^{98}+O(q^{100}) $ 8 * q + 4 * q^2 + 12 * q^4 + 12 * q^8 - 8 * q^9 + 4 * q^13 - 12 * q^15 + 12 * q^16 - 4 * q^18 - 20 * q^19 + 8 * q^21 - 4 * q^25 + 40 * q^26 - 24 * q^30 + 28 * q^32 + 12 * q^33 + 8 * q^35 - 12 * q^36 - 24 * q^38 - 8 * q^42 - 28 * q^43 + 8 * q^47 - 60 * q^50 - 16 * q^52 + 40 * q^53 + 4 * q^55 + 48 * q^59 - 32 * q^60 - 4 * q^64 - 8 * q^66 + 8 * q^67 - 12 * q^69 + 8 * q^70 - 12 * q^72 - 80 * q^76 + 24 * q^77 + 8 * q^81 + 8 * q^83 + 48 * q^84 - 8 * q^86 - 32 * q^87 + 8 * q^89 - 8 * q^93 - 16 * q^94 + 76 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/867\mathbb{Z}\right)^\times$.

$n$	$290$	$292$
$\chi(n)$	$1$	$-1$

Coefficient data

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

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.63640	1.86422	0.932110	−	0.362176i	$-0.117966\pi$
$2$	2.63640	1.86422	0.932110	+	0.362176i	$0.117966\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	4.95063	2.47532
$5$	− 2.22219i	− 0.993794i	−0.867809	−	0.496897i	$-0.834473\pi$
$5$	− 2.22219i	− 0.993794i	0.867809	−	0.496897i	$-0.165527\pi$
$6$	− 2.63640i	− 1.07631i
$7$	2.31423i	0.874695i	0.899293	+	0.437348i	$0.144082\pi$
$7$	2.31423i	0.874695i	−0.899293	+	0.437348i	$0.855918\pi$
$8$	7.77906	2.75031
$9$	−1.00000	−0.333333
$10$	− 5.85860i	− 1.85265i
$11$	− 1.95063i	− 0.588137i	−0.955784	−	0.294069i	$-0.904991\pi$
$11$	− 1.95063i	− 0.588137i	0.955784	−	0.294069i	$-0.0950094\pi$
$12$	− 4.95063i	− 1.42912i
$13$	1.32218	0.366706	0.183353	−	0.983047i	$-0.441305\pi$
$13$	1.32218	0.366706	0.183353	+	0.983047i	$0.441305\pi$
$14$	6.10124i	1.63062i
$15$	−2.22219	−0.573767
$16$	10.6075	2.65187
$17$	0	0
$17$	0	0
$18$	−2.63640	−0.621407
$19$	−5.95063	−1.36517	−0.682584	−	0.730807i	$-0.739144\pi$
$19$	−5.95063	−1.36517	−0.682584	+	0.730807i	$0.739144\pi$
$20$	− 11.0012i	− 2.45995i
$21$	2.31423	0.505006
$22$	− 5.14265i	− 1.09642i
$23$	4.77906i	0.996503i	0.867033	+	0.498251i	$0.166024\pi$
$23$	4.77906i	0.996503i	−0.867033	+	0.498251i	$0.833976\pi$
$24$	− 7.77906i	− 1.58789i
$25$	0.0618655	0.0123731
$26$	3.48580	0.683621
$27$	1.00000i	0.192450i
$28$	11.4569i	2.16515i
$29$	0.172822i	0.0320923i	0.999871	+	0.0160462i	$0.00510787\pi$
$29$	0.172822i	0.0320923i	−0.999871	+	0.0160462i	$0.994892\pi$
$30$	−5.85860	−1.06963
$31$	0.444383i	0.0798135i	0.999203	+	0.0399067i	$0.0127061\pi$
$31$	0.444383i	0.0798135i	−0.999203	+	0.0399067i	$0.987294\pi$
$32$	12.4075	2.19336
$33$	−1.95063	−0.339561
$34$	0	0
$35$	5.14265	0.869267
$36$	−4.95063	−0.825105
$37$	5.31548i	0.873859i	0.899496	+	0.436929i	$0.143934\pi$
$37$	5.31548i	0.873859i	−0.899496	+	0.436929i	$0.856066\pi$
$38$	−15.6883	−2.54497
$39$	− 1.32218i	− 0.211718i
$40$	− 17.2866i	− 2.73324i
$41$	− 4.02222i	− 0.628165i	−0.949396	−	0.314082i	$-0.898303\pi$
$41$	− 4.02222i	− 0.628165i	0.949396	−	0.314082i	$-0.101697\pi$
$42$	6.10124	0.941441
$43$	−8.33468	−1.27103	−0.635513	−	0.772090i	$-0.719211\pi$
$43$	−8.33468	−1.27103	−0.635513	+	0.772090i	$0.719211\pi$
$44$	− 9.65685i	− 1.45583i
$45$	2.22219i	0.331265i
$46$	12.5995i	1.85770i
$47$	2.38404	0.347749	0.173874	−	0.984768i	$-0.444371\pi$
$47$	2.38404	0.347749	0.173874	+	0.984768i	$0.444371\pi$
$48$	− 10.6075i	− 1.53106i
$49$	1.64436	0.234908
$50$	0.163102	0.0230662
$51$	0	0
$52$	6.54562	0.907714
$53$	0.727190	0.0998872	0.0499436	−	0.998752i	$-0.484096\pi$
$53$	0.727190	0.0998872	0.0499436	+	0.998752i	$0.484096\pi$
$54$	2.63640i	0.358769i
$55$	−4.33468	−0.584488
$56$	18.0025i	2.40569i
$57$	5.95063i	0.788180i
$58$	0.455630i	0.0598271i
$59$	11.7870	1.53454	0.767269	−	0.641325i	$-0.221615\pi$
$59$	11.7870	1.53454	0.767269	+	0.641325i	$0.221615\pi$
$60$	−11.0012	−1.42026
$61$	8.68702i	1.11226i	0.831096	+	0.556130i	$0.187714\pi$
$61$	8.68702i	1.11226i	−0.831096	+	0.556130i	$0.812286\pi$
$62$	1.17157i	0.148790i
$63$	− 2.31423i	− 0.291565i
$64$	11.4963	1.43703
$65$	− 2.93813i	− 0.364431i
$66$	−5.14265	−0.633017
$67$	−14.1866	−1.73317	−0.866583	−	0.499034i	$-0.833688\pi$
$67$	−14.1866	−1.73317	−0.866583	+	0.499034i	$0.833688\pi$
$68$	0	0
$69$	4.77906	0.575331
$70$	13.5581	1.62050
$71$	6.10124i	0.724084i	0.932162	+	0.362042i	$0.117920\pi$
$71$	6.10124i	0.724084i	−0.932162	+	0.362042i	$0.882080\pi$
$72$	−7.77906	−0.916771
$73$	− 10.5569i	− 1.23559i	−0.786340	−	0.617794i	$-0.788027\pi$
$73$	− 10.5569i	− 1.23559i	0.786340	−	0.617794i	$-0.211973\pi$
$74$	14.0137i	1.62906i
$75$	− 0.0618655i	− 0.00714361i
$76$	−29.4594	−3.37922
$77$	4.51420	0.514441
$78$	− 3.48580i	− 0.394689i
$79$	− 0.698270i	− 0.0785616i	−0.999228	−	0.0392808i	$-0.987493\pi$
$79$	− 0.698270i	− 0.0785616i	0.999228	−	0.0392808i	$-0.0125067\pi$
$80$	− 23.5719i	− 2.63541i
$81$	1.00000	0.111111
$82$	− 10.6042i	− 1.17104i
$83$	−8.52971	−0.936258	−0.468129	−	0.883660i	$-0.655072\pi$
$83$	−8.52971	−0.936258	−0.468129	+	0.883660i	$0.655072\pi$
$84$	11.4569	1.25005
$85$	0	0
$86$	−21.9736	−2.36947
$87$	0.172822	0.0185285
$88$	− 15.1741i	− 1.61756i
$89$	5.40297	0.572713	0.286357	−	0.958123i	$-0.407556\pi$
$89$	5.40297	0.572713	0.286357	+	0.958123i	$0.407556\pi$
$90$	5.85860i	0.617550i
$91$	3.05982i	0.320756i
$92$	23.6594i	2.46666i
$93$	0.444383	0.0460803
$94$	6.28531	0.648280
$95$	13.2234i	1.35670i
$96$	− 12.4075i	− 1.26634i
$97$	− 4.71594i	− 0.478831i	−0.970917	−	0.239416i	$-0.923044\pi$
$97$	− 4.71594i	− 0.478831i	0.970917	−	0.239416i	$-0.0769559\pi$
$98$	4.33519	0.437921
$99$	1.95063i	0.196046i
$100$	0.306273	0.0306273
$101$	−7.11123	−0.707594	−0.353797	−	0.935322i	$-0.615110\pi$
$101$	−7.11123	−0.707594	−0.353797	+	0.935322i	$0.615110\pi$
$102$	0	0
$103$	−9.22344	−0.908813	−0.454406	−	0.890795i	$-0.650149\pi$
$103$	−9.22344	−0.908813	−0.454406	+	0.890795i	$0.650149\pi$
$104$	10.2853	1.00856
$105$	− 5.14265i	− 0.501872i
$106$	1.91717	0.186212
$107$	− 5.18254i	− 0.501015i	−0.968115	−	0.250508i	$-0.919402\pi$
$107$	− 5.18254i	− 0.501015i	0.968115	−	0.250508i	$-0.0805975\pi$
$108$	4.95063i	0.476375i
$109$	19.9753i	1.91329i	0.291255	+	0.956646i	$0.405927\pi$
$109$	19.9753i	1.91329i	−0.291255	+	0.956646i	$0.594073\pi$
$110$	−11.4280	−1.08961
$111$	5.31548	0.504523
$112$	24.5481i	2.31958i
$113$	17.4372i	1.64035i	0.572113	+	0.820175i	$0.306124\pi$
$113$	17.4372i	1.64035i	−0.572113	+	0.820175i	$0.693876\pi$
$114$	15.6883i	1.46934i
$115$	10.6200	0.990318
$116$	0.855580i	0.0794386i
$117$	−1.32218	−0.122235
$118$	31.0753	2.86072
$119$	0	0
$120$	−17.2866	−1.57804
$121$	7.19504	0.654094
$122$	22.9025i	2.07350i
$123$	−4.02222	−0.362671
$124$	2.19998i	0.197564i
$125$	− 11.2484i	− 1.00609i
$126$	− 6.10124i	− 0.543541i
$127$	9.85189	0.874214	0.437107	−	0.899409i	$-0.356003\pi$
$127$	9.85189	0.874214	0.437107	+	0.899409i	$0.356003\pi$
$128$	5.49375	0.485584
$129$	8.33468i	0.733827i
$130$	− 7.74611i	− 0.679379i
$131$	− 7.60749i	− 0.664669i	−0.943162	−	0.332335i	$-0.892164\pi$
$131$	− 7.60749i	− 0.664669i	0.943162	−	0.332335i	$-0.107836\pi$
$132$	−9.65685	−0.840521
$133$	− 13.7711i	− 1.19411i
$134$	−37.4015	−3.23100
$135$	2.22219	0.191256
$136$	0	0
$137$	22.0723	1.88577	0.942883	−	0.333125i	$-0.108103\pi$
$137$	22.0723	1.88577	0.942883	+	0.333125i	$0.108103\pi$
$138$	12.5995	1.07254
$139$	3.58454i	0.304036i	0.988378	+	0.152018i	$0.0485772\pi$
$139$	3.58454i	0.304036i	−0.988378	+	0.152018i	$0.951423\pi$
$140$	25.4594	2.15171
$141$	− 2.38404i	− 0.200773i
$142$	16.0853i	1.34985i
$143$	− 2.57908i	− 0.215674i
$144$	−10.6075	−0.883957
$145$	0.384044	0.0318931
$146$	− 27.8322i	− 2.30341i
$147$	− 1.64436i	− 0.135624i
$148$	26.3150i	2.16308i
$149$	−11.7711	−0.964327	−0.482163	−	0.876081i	$-0.660149\pi$
$149$	−11.7711	−0.964327	−0.482163	+	0.876081i	$0.660149\pi$
$150$	− 0.163102i	− 0.0133173i
$151$	16.1037	1.31050	0.655252	−	0.755411i	$-0.272563\pi$
$151$	16.1037	1.31050	0.655252	+	0.755411i	$0.272563\pi$
$152$	−46.2903	−3.75464
$153$	0	0
$154$	11.9013	0.959031
$155$	0.987504	0.0793182
$156$	− 6.54562i	− 0.524069i
$157$	−6.45535	−0.515193	−0.257597	−	0.966253i	$-0.582931\pi$
$157$	−6.45535	−0.515193	−0.257597	+	0.966253i	$0.582931\pi$
$158$	− 1.84092i	− 0.146456i
$159$	− 0.727190i	− 0.0576699i
$160$	− 27.5719i	− 2.17975i
$161$	−11.0598	−0.871636
$162$	2.63640	0.207136
$163$	− 16.7322i	− 1.31057i	−0.755383	−	0.655283i	$-0.772549\pi$
$163$	− 16.7322i	− 1.31057i	0.755383	−	0.655283i	$-0.227451\pi$
$164$	− 19.9125i	− 1.55491i
$165$	4.33468i	0.337454i
$166$	−22.4878	−1.74539
$167$	1.60499i	0.124198i	0.998070	+	0.0620988i	$0.0197794\pi$
$167$	1.60499i	0.124198i	−0.998070	+	0.0620988i	$0.980221\pi$
$168$	18.0025	1.38892
$169$	−11.2518	−0.865526
$170$	0	0
$171$	5.95063	0.455056
$172$	−41.2619	−3.14619
$173$	− 17.1359i	− 1.30282i	−0.758725	−	0.651411i	$-0.774177\pi$
$173$	− 17.1359i	− 1.30282i	0.758725	−	0.651411i	$-0.225823\pi$
$174$	0.455630	0.0345412
$175$	0.143171i	0.0108227i
$176$	− 20.6913i	− 1.55966i
$177$	− 11.7870i	− 0.885966i
$178$	14.2444	1.06766
$179$	−11.0753	−0.827809	−0.413905	−	0.910320i	$-0.635835\pi$
$179$	−11.0753	−0.827809	−0.413905	+	0.910320i	$0.635835\pi$
$180$	11.0012i	0.819985i
$181$	− 19.6995i	− 1.46425i	−0.681168	−	0.732127i	$-0.738528\pi$
$181$	− 19.6995i	− 1.46425i	0.681168	−	0.732127i	$-0.261472\pi$
$182$	8.06693i	0.597960i
$183$	8.68702	0.642163
$184$	37.1766i	2.74069i
$185$	11.8120	0.868436
$186$	1.17157	0.0859039
$187$	0	0
$188$	11.8025	0.860788
$189$	−2.31423	−0.168335
$190$	34.8623i	2.52918i
$191$	1.60953	0.116461	0.0582307	−	0.998303i	$-0.481454\pi$
$191$	1.60953	0.116461	0.0582307	+	0.998303i	$0.481454\pi$
$192$	− 11.4963i	− 0.829670i
$193$	− 4.47403i	− 0.322048i	−0.986950	−	0.161024i	$-0.948520\pi$
$193$	− 4.47403i	− 0.322048i	0.986950	−	0.161024i	$-0.0514797\pi$
$194$	− 12.4331i	− 0.892647i
$195$	−2.93813	−0.210404
$196$	8.14061	0.581472
$197$	7.39376i	0.526784i	0.964689	+	0.263392i	$0.0848412\pi$
$197$	7.39376i	0.526784i	−0.964689	+	0.263392i	$0.915159\pi$
$198$	5.14265i	0.365472i
$199$	− 7.19657i	− 0.510151i	−0.966921	−	0.255076i	$-0.917900\pi$
$199$	− 7.19657i	− 0.510151i	0.966921	−	0.255076i	$-0.0821003\pi$
$200$	0.481255	0.0340299
$201$	14.1866i	1.00064i
$202$	−18.7481	−1.31911
$203$	−0.399950	−0.0280710
$204$	0	0
$205$	−8.93813	−0.624266
$206$	−24.3167	−1.69423
$207$	− 4.77906i	− 0.332168i
$208$	14.0250	0.972458
$209$	11.6075i	0.802907i
$210$	− 13.5581i	− 0.935599i
$211$	− 13.4185i	− 0.923766i	−0.886941	−	0.461883i	$-0.847174\pi$
$211$	− 13.4185i	− 0.923766i	0.886941	−	0.461883i	$-0.152826\pi$
$212$	3.60005	0.247252
$213$	6.10124	0.418050
$214$	− 13.6633i	− 0.934003i
$215$	18.5212i	1.26314i
$216$	7.77906i	0.529298i
$217$	−1.02840	−0.0698125
$218$	52.6631i	3.56680i
$219$	−10.5569	−0.713367
$220$	−21.4594	−1.44679
$221$	0	0
$222$	14.0137	0.940541
$223$	−21.2643	−1.42397	−0.711983	−	0.702197i	$-0.752203\pi$
$223$	−21.2643	−1.42397	−0.711983	+	0.702197i	$0.752203\pi$
$224$	28.7138i	1.91852i
$225$	−0.0618655	−0.00412437
$226$	45.9714i	3.05797i
$227$	− 4.95404i	− 0.328811i	−0.986393	−	0.164406i	$-0.947429\pi$
$227$	− 4.95404i	− 0.328811i	0.986393	−	0.164406i	$-0.0525706\pi$
$228$	29.4594i	1.95099i
$229$	−1.83842	−0.121487	−0.0607433	−	0.998153i	$-0.519347\pi$
$229$	−1.83842	−0.121487	−0.0607433	+	0.998153i	$0.519347\pi$
$230$	27.9986	1.84617
$231$	− 4.51420i	− 0.297013i
$232$	1.34440i	0.0882639i
$233$	1.49500i	0.0979408i	0.998800	+	0.0489704i	$0.0155940\pi$
$233$	1.49500i	0.0979408i	−0.998800	+	0.0489704i	$0.984406\pi$
$234$	−3.48580	−0.227874
$235$	− 5.29780i	− 0.345591i
$236$	58.3531	3.79847
$237$	−0.698270	−0.0453575
$238$	0	0
$239$	−10.0250	−0.648463	−0.324231	−	0.945978i	$-0.605106\pi$
$239$	−10.0250	−0.648463	−0.324231	+	0.945978i	$0.605106\pi$
$240$	−23.5719	−1.52156
$241$	− 11.5155i	− 0.741775i	−0.928678	−	0.370888i	$-0.879053\pi$
$241$	− 11.5155i	− 0.741775i	0.928678	−	0.370888i	$-0.120947\pi$
$242$	18.9690	1.21938
$243$	− 1.00000i	− 0.0641500i
$244$	43.0062i	2.75319i
$245$	− 3.65408i	− 0.233450i
$246$	−10.6042	−0.676099
$247$	−7.86780	−0.500616
$248$	3.45688i	0.219512i
$249$	8.52971i	0.540549i
$250$	− 29.6554i	− 1.87557i
$251$	−0.940179	−0.0593436	−0.0296718	−	0.999560i	$-0.509446\pi$
$251$	−0.940179	−0.0593436	−0.0296718	+	0.999560i	$0.509446\pi$
$252$	− 11.4569i	− 0.721716i
$253$	9.32218	0.586080
$254$	25.9736	1.62973
$255$	0	0
$256$	−8.50875	−0.531797
$257$	−2.15515	−0.134435	−0.0672173	−	0.997738i	$-0.521412\pi$
$257$	−2.15515	−0.134435	−0.0672173	+	0.997738i	$0.521412\pi$
$258$	21.9736i	1.36802i
$259$	−12.3012	−0.764360
$260$	− 14.5456i	− 0.902081i
$261$	− 0.172822i	− 0.0106974i
$262$	− 20.0564i	− 1.23909i
$263$	13.1427	0.810411	0.405205	−	0.914226i	$-0.367200\pi$
$263$	13.1427	0.810411	0.405205	+	0.914226i	$0.367200\pi$
$264$	−15.1741	−0.933900
$265$	− 1.61596i	− 0.0992674i
$266$	− 36.3062i	− 2.22608i
$267$	− 5.40297i	− 0.330656i
$268$	−70.2325	−4.29013
$269$	− 15.1743i	− 0.925196i	−0.886568	−	0.462598i	$-0.846917\pi$
$269$	− 15.1743i	− 0.925196i	0.886568	−	0.462598i	$-0.153083\pi$
$270$	5.85860	0.356543
$271$	11.8360	0.718985	0.359493	−	0.933148i	$-0.382950\pi$
$271$	11.8360	0.718985	0.359493	+	0.933148i	$0.382950\pi$
$272$	0	0
$273$	3.05982	0.185189
$274$	58.1916	3.51548
$275$	− 0.120677i	− 0.00727708i
$276$	23.6594	1.42413
$277$	− 26.9025i	− 1.61642i	−0.588898	−	0.808208i	$-0.700438\pi$
$277$	− 26.9025i	− 1.61642i	0.588898	−	0.808208i	$-0.299562\pi$
$278$	9.45029i	0.566791i
$279$	− 0.444383i	− 0.0266045i
$280$	40.0050	2.39076
$281$	22.2339	1.32636	0.663181	−	0.748459i	$-0.269206\pi$
$281$	22.2339	1.32636	0.663181	+	0.748459i	$0.269206\pi$
$282$	− 6.28531i	− 0.374285i
$283$	− 23.9577i	− 1.42414i	−0.702111	−	0.712068i	$-0.747759\pi$
$283$	− 23.9577i	− 1.42414i	0.702111	−	0.712068i	$-0.252241\pi$
$284$	30.2050i	1.79234i
$285$	13.2234	0.783289
$286$	− 6.79951i	− 0.402063i
$287$	9.30832	0.549453
$288$	−12.4075	−0.731120
$289$	0	0
$290$	1.01250	0.0594558
$291$	−4.71594	−0.276453
$292$	− 52.2632i	− 3.05847i
$293$	7.44387	0.434875	0.217438	−	0.976074i	$-0.430230\pi$
$293$	7.44387	0.434875	0.217438	+	0.976074i	$0.430230\pi$
$294$	− 4.33519i	− 0.252834i
$295$	− 26.1930i	− 1.52502i
$296$	41.3494i	2.40338i
$297$	1.95063	0.113187
$298$	−31.0334	−1.79772
$299$	6.31877i	0.365424i
$300$	− 0.306273i	− 0.0176827i
$301$	− 19.2883i	− 1.11176i
$302$	42.4560	2.44307
$303$	7.11123i	0.408530i
$304$	−63.1212	−3.62025
$305$	19.3042	1.10536
$306$	0	0
$307$	−14.2184	−0.811486	−0.405743	−	0.913987i	$-0.632987\pi$
$307$	−14.2184	−0.811486	−0.405743	+	0.913987i	$0.632987\pi$
$308$	22.3481	1.27340
$309$	9.22344i	0.524703i
$310$	2.60346	0.147867
$311$	10.3616i	0.587550i	0.955875	+	0.293775i	$0.0949116\pi$
$311$	10.3616i	0.587550i	−0.955875	+	0.293775i	$0.905088\pi$
$312$	− 10.2853i	− 0.582291i
$313$	− 17.4262i	− 0.984987i	−0.870316	−	0.492493i	$-0.836086\pi$
$313$	− 17.4262i	− 0.984987i	0.870316	−	0.492493i	$-0.163914\pi$
$314$	−17.0189	−0.960433
$315$	−5.14265	−0.289756
$316$	− 3.45688i	− 0.194465i
$317$	− 14.1753i	− 0.796165i	−0.917350	−	0.398083i	$-0.869676\pi$
$317$	− 14.1753i	− 0.796165i	0.917350	−	0.398083i	$-0.130324\pi$
$318$	− 1.91717i	− 0.107509i
$319$	0.337113	0.0188747
$320$	− 25.5469i	− 1.42811i
$321$	−5.18254	−0.289261
$322$	−29.1582	−1.62492
$323$	0	0
$324$	4.95063	0.275035
$325$	0.0817972	0.00453730
$326$	− 44.1128i	− 2.44318i
$327$	19.9753	1.10464
$328$	− 31.2891i	− 1.72765i
$329$	5.51722i	0.304174i
$330$	11.4280i	0.629088i
$331$	−10.8485	−0.596287	−0.298143	−	0.954521i	$-0.596367\pi$
$331$	−10.8485	−0.596287	−0.298143	+	0.954521i	$0.596367\pi$
$332$	−42.2275	−2.31753
$333$	− 5.31548i	− 0.291286i
$334$	4.23139i	0.231532i
$335$	31.5253i	1.72241i
$336$	24.5481	1.33921
$337$	− 9.81770i	− 0.534804i	−0.963585	−	0.267402i	$-0.913835\pi$
$337$	− 9.81770i	− 0.534804i	0.963585	−	0.267402i	$-0.0861652\pi$
$338$	−29.6644	−1.61353
$339$	17.4372	0.947057
$340$	0	0
$341$	0.866827	0.0469413
$342$	15.6883	0.848325
$343$	20.0050i	1.08017i
$344$	−64.8359	−3.49572
$345$	− 10.6200i	− 0.571761i
$346$	− 45.1773i	− 2.42875i
$347$	26.0469i	1.39827i	0.714988	+	0.699136i	$0.246432\pi$
$347$	26.0469i	1.39827i	−0.714988	+	0.699136i	$0.753568\pi$
$348$	0.855580	0.0458639
$349$	18.6609	0.998894	0.499447	−	0.866344i	$-0.333537\pi$
$349$	18.6609	0.998894	0.499447	+	0.866344i	$0.333537\pi$
$350$	0.377456i	0.0201759i
$351$	1.32218i	0.0705727i
$352$	− 24.2025i	− 1.29000i
$353$	11.8339	0.629857	0.314929	−	0.949115i	$-0.398019\pi$
$353$	11.8339	0.629857	0.314929	+	0.949115i	$0.398019\pi$
$354$	− 31.0753i	− 1.65164i
$355$	13.5581	0.719590
$356$	26.7481	1.41765
$357$	0	0
$358$	−29.1991	−1.54322
$359$	14.1392	0.746241	0.373120	−	0.927783i	$-0.378288\pi$
$359$	14.1392	0.746241	0.373120	+	0.927783i	$0.378288\pi$
$360$	17.2866i	0.911081i
$361$	16.4100	0.863685
$362$	− 51.9359i	− 2.72969i
$363$	− 7.19504i	− 0.377642i
$364$	15.1480i	0.793973i
$365$	−23.4594	−1.22792
$366$	22.9025	1.19713
$367$	3.72362i	0.194371i	0.995266	+	0.0971856i	$0.0309841\pi$
$367$	3.72362i	0.194371i	−0.995266	+	0.0971856i	$0.969016\pi$
$368$	50.6938i	2.64260i
$369$	4.02222i	0.209388i
$370$	31.1412	1.61896
$371$	1.68288i	0.0873709i
$372$	2.19998	0.114063
$373$	1.53618	0.0795403	0.0397701	−	0.999209i	$-0.487337\pi$
$373$	1.53618	0.0795403	0.0397701	+	0.999209i	$0.487337\pi$
$374$	0	0
$375$	−11.2484	−0.580867
$376$	18.5456	0.956417
$377$	0.228502i	0.0117685i
$378$	−6.10124	−0.313814
$379$	36.1511i	1.85696i	0.371388	+	0.928478i	$0.378882\pi$
$379$	36.1511i	1.85696i	−0.371388	+	0.928478i	$0.621118\pi$
$380$	65.4644i	3.35825i
$381$	− 9.85189i	− 0.504728i
$382$	4.24337	0.217110
$383$	16.3492	0.835404	0.417702	−	0.908584i	$-0.362836\pi$
$383$	16.3492	0.835404	0.417702	+	0.908584i	$0.362836\pi$
$384$	− 5.49375i	− 0.280352i
$385$	− 10.0314i	− 0.511248i
$386$	− 11.7954i	− 0.600368i
$387$	8.33468	0.423675
$388$	− 23.3469i	− 1.18526i
$389$	16.4533	0.834216	0.417108	−	0.908857i	$-0.363044\pi$
$389$	16.4533	0.834216	0.417108	+	0.908857i	$0.363044\pi$
$390$	−7.74611	−0.392240
$391$	0	0
$392$	12.7916	0.646071
$393$	−7.60749	−0.383747
$394$	19.4930i	0.982041i
$395$	−1.55169	−0.0780740
$396$	9.65685i	0.485275i
$397$	36.4606i	1.82991i	0.403560	+	0.914953i	$0.367773\pi$
$397$	36.4606i	1.82991i	−0.403560	+	0.914953i	$0.632227\pi$
$398$	− 18.9731i	− 0.951034i
$399$	−13.7711	−0.689418
$400$	0.656237	0.0328119
$401$	− 9.67907i	− 0.483350i	−0.970357	−	0.241675i	$-0.922303\pi$
$401$	− 9.67907i	− 0.483350i	0.970357	−	0.241675i	$-0.0776967\pi$
$402$	37.4015i	1.86542i
$403$	0.587553i	0.0292681i
$404$	−35.2051	−1.75152
$405$	− 2.22219i	− 0.110422i
$406$	−1.05443	−0.0523305
$407$	10.3685	0.513949
$408$	0	0
$409$	−2.52228	−0.124719	−0.0623593	−	0.998054i	$-0.519862\pi$
$409$	−2.52228	−0.124719	−0.0623593	+	0.998054i	$0.519862\pi$
$410$	−23.5645	−1.16377
$411$	− 22.0723i	− 1.08875i
$412$	−45.6619	−2.24960
$413$	27.2778i	1.34225i
$414$	− 12.5995i	− 0.619233i
$415$	18.9547i	0.930448i
$416$	16.4049	0.804319
$417$	3.58454	0.175536
$418$	30.6020i	1.49679i
$419$	26.3262i	1.28612i	0.765816	+	0.643060i	$0.222335\pi$
$419$	26.3262i	1.28612i	−0.765816	+	0.643060i	$0.777665\pi$
$420$	− 25.4594i	− 1.24229i
$421$	−6.61089	−0.322195	−0.161098	−	0.986938i	$-0.551503\pi$
$421$	−6.61089	−0.322195	−0.161098	+	0.986938i	$0.551503\pi$
$422$	− 35.3765i	− 1.72210i
$423$	−2.38404	−0.115916
$424$	5.65685	0.274721
$425$	0	0
$426$	16.0853	0.779337
$427$	−20.1037	−0.972888
$428$	− 25.6569i	− 1.24017i
$429$	−2.57908	−0.124519
$430$	48.8295i	2.35477i
$431$	− 28.1262i	− 1.35479i	−0.735618	−	0.677396i	$-0.763108\pi$
$431$	− 28.1262i	− 1.35479i	0.735618	−	0.677396i	$-0.236892\pi$
$432$	10.6075i	0.510353i
$433$	−17.1497	−0.824162	−0.412081	−	0.911147i	$-0.635198\pi$
$433$	−17.1497	−0.824162	−0.412081	+	0.911147i	$0.635198\pi$
$434$	−2.71128	−0.130146
$435$	− 0.384044i	− 0.0184135i
$436$	98.8906i	4.73600i
$437$	− 28.4384i	− 1.36039i
$438$	−27.8322	−1.32987
$439$	8.22497i	0.392556i	0.980548	+	0.196278i	$0.0628856\pi$
$439$	8.22497i	0.392556i	−0.980548	+	0.196278i	$0.937114\pi$
$440$	−33.7197	−1.60752
$441$	−1.64436	−0.0783028
$442$	0	0
$443$	−18.4658	−0.877339	−0.438669	−	0.898649i	$-0.644550\pi$
$443$	−18.4658	−0.877339	−0.438669	+	0.898649i	$0.644550\pi$
$444$	26.3150	1.24885
$445$	− 12.0064i	− 0.569159i
$446$	−56.0614	−2.65458
$447$	11.7711i	0.556754i
$448$	26.6049i	1.25696i
$449$	− 30.9250i	− 1.45944i	−0.683745	−	0.729721i	$-0.739650\pi$
$449$	− 30.9250i	− 1.45944i	0.683745	−	0.729721i	$-0.260350\pi$
$450$	−0.163102	−0.00768873
$451$	−7.84586	−0.369447
$452$	86.3250i	4.06038i
$453$	− 16.1037i	− 0.756620i
$454$	− 13.0609i	− 0.612976i
$455$	6.79951	0.318766
$456$	46.2903i	2.16774i
$457$	15.5337	0.726638	0.363319	−	0.931665i	$-0.381644\pi$
$457$	15.5337	0.726638	0.363319	+	0.931665i	$0.381644\pi$
$458$	−4.84683	−0.226478
$459$	0	0
$460$	52.5756	2.45135
$461$	38.9032	1.81190	0.905952	−	0.423380i	$-0.139156\pi$
$461$	38.9032	1.81190	0.905952	+	0.423380i	$0.139156\pi$
$462$	− 11.9013i	− 0.553697i
$463$	−22.7572	−1.05762	−0.528808	−	0.848742i	$-0.677361\pi$
$463$	−22.7572	−1.05762	−0.528808	+	0.848742i	$0.677361\pi$
$464$	1.83321i	0.0851047i
$465$	− 0.987504i	− 0.0457944i
$466$	3.94143i	0.182583i
$467$	12.5551	0.580981	0.290490	−	0.956878i	$-0.406182\pi$
$467$	12.5551	0.580981	0.290490	+	0.956878i	$0.406182\pi$
$468$	−6.54562	−0.302571
$469$	− 32.8309i	− 1.51599i
$470$	− 13.9672i	− 0.644257i
$471$	6.45535i	0.297447i
$472$	91.6918	4.22046
$473$	16.2579i	0.747538i
$474$	−1.84092	−0.0845564
$475$	−0.368139	−0.0168914
$476$	0	0
$477$	−0.727190	−0.0332957
$478$	−26.4299	−1.20888
$479$	37.2418i	1.70162i	0.525471	+	0.850812i	$0.323889\pi$
$479$	37.2418i	1.70162i	−0.525471	+	0.850812i	$0.676111\pi$
$480$	−27.5719	−1.25848
$481$	7.02801i	0.320450i
$482$	− 30.3594i	− 1.38283i
$483$	11.0598i	0.503239i
$484$	35.6200	1.61909
$485$	−10.4797	−0.475860
$486$	− 2.63640i	− 0.119590i
$487$	1.64488i	0.0745364i	0.999305	+	0.0372682i	$0.0118656\pi$
$487$	1.64488i	0.0745364i	−0.999305	+	0.0372682i	$0.988134\pi$
$488$	67.5769i	3.05906i
$489$	−16.7322	−0.756656
$490$	− 9.63363i	− 0.435203i
$491$	−4.30121	−0.194111	−0.0970555	−	0.995279i	$-0.530942\pi$
$491$	−4.30121	−0.194111	−0.0970555	+	0.995279i	$0.530942\pi$
$492$	−19.9125	−0.897725
$493$	0	0
$494$	−20.7427	−0.933258
$495$	4.33468	0.194829
$496$	4.71378i	0.211655i
$497$	−14.1196	−0.633352
$498$	22.4878i	1.00770i
$499$	− 1.49528i	− 0.0669379i	−0.999440	−	0.0334689i	$-0.989345\pi$
$499$	− 1.49528i	− 0.0669379i	0.999440	−	0.0334689i	$-0.0106555\pi$
$500$	− 55.6868i	− 2.49039i
$501$	1.60499	0.0717055
$502$	−2.47869	−0.110629
$503$	1.68678i	0.0752100i	0.999293	+	0.0376050i	$0.0119729\pi$
$503$	1.68678i	0.0752100i	−0.999293	+	0.0376050i	$0.988027\pi$
$504$	− 18.0025i	− 0.801895i
$505$	15.8025i	0.703203i
$506$	24.5770	1.09258
$507$	11.2518i	0.499712i
$508$	48.7731	2.16396
$509$	−6.91070	−0.306312	−0.153156	−	0.988202i	$-0.548944\pi$
$509$	−6.91070	−0.306312	−0.153156	+	0.988202i	$0.548944\pi$
$510$	0	0
$511$	24.4310	1.08076
$512$	−33.4200	−1.47697
$513$	− 5.95063i	− 0.262727i
$514$	−5.68185	−0.250616
$515$	20.4963i	0.903173i
$516$	41.2619i	1.81645i
$517$	− 4.65039i	− 0.204524i
$518$	−32.4310	−1.42494
$519$	−17.1359	−0.752185
$520$	− 22.8559i	− 1.00230i
$521$	− 13.3963i	− 0.586901i	−0.955974	−	0.293450i	$-0.905196\pi$
$521$	− 13.3963i	− 0.586901i	0.955974	−	0.293450i	$-0.0948036\pi$
$522$	− 0.455630i	− 0.0199424i
$523$	−4.40904	−0.192794	−0.0963969	−	0.995343i	$-0.530732\pi$
$523$	−4.40904	−0.192794	−0.0963969	+	0.995343i	$0.530732\pi$
$524$	− 37.6619i	− 1.64527i
$525$	0.143171	0.00624848
$526$	34.6494	1.51078
$527$	0	0
$528$	−20.6913	−0.900473
$529$	0.160604	0.00698276
$530$	− 4.26031i	− 0.185056i
$531$	−11.7870	−0.511513
$532$	− 68.1757i	− 2.95579i
$533$	− 5.31809i	− 0.230352i
$534$	− 14.2444i	− 0.616416i
$535$	−11.5166	−0.497906
$536$	−110.358	−4.76675
$537$	11.0753i	0.477936i
$538$	− 40.0057i	− 1.72477i
$539$	− 3.20754i	− 0.138158i
$540$	11.0012	0.473418
$541$	− 23.1155i	− 0.993813i	−0.867804	−	0.496906i	$-0.834469\pi$
$541$	− 23.1155i	− 0.993813i	0.867804	−	0.496906i	$-0.165531\pi$
$542$	31.2045	1.34035
$543$	−19.6995	−0.845387
$544$	0	0
$545$	44.3890	1.90142
$546$	8.06693	0.345233
$547$	− 9.48476i	− 0.405539i	−0.979226	−	0.202770i	$-0.935006\pi$
$547$	− 9.48476i	− 0.405539i	0.979226	−	0.202770i	$-0.0649943\pi$
$548$	109.272	4.66786
$549$	− 8.68702i	− 0.370753i
$550$	− 0.318153i	− 0.0135661i
$551$	− 1.02840i	− 0.0438114i
$552$	37.1766	1.58234
$553$	1.61596	0.0687174
$554$	− 70.9259i	− 3.01335i
$555$	− 11.8120i	− 0.501392i
$556$	17.7457i	0.752586i
$557$	−38.1670	−1.61719	−0.808593	−	0.588368i	$-0.799771\pi$
$557$	−38.1670	−1.61719	−0.808593	+	0.588368i	$0.799771\pi$
$558$	− 1.17157i	− 0.0495966i
$559$	−11.0199	−0.466093
$560$	54.5506	2.30518
$561$	0	0
$562$	58.6175	2.47263
$563$	13.8184	0.582377	0.291189	−	0.956666i	$-0.405949\pi$
$563$	13.8184	0.582377	0.291189	+	0.956666i	$0.405949\pi$
$564$	− 11.8025i	− 0.496976i
$565$	38.7487	1.63017
$566$	− 63.1621i	− 2.65490i
$567$	2.31423i	0.0971884i
$568$	47.4619i	1.99146i
$569$	12.9516	0.542959	0.271480	−	0.962444i	$-0.412487\pi$
$569$	12.9516	0.542959	0.271480	+	0.962444i	$0.412487\pi$
$570$	34.8623	1.46022
$571$	6.51381i	0.272594i	0.990668	+	0.136297i	$0.0435202\pi$
$571$	6.51381i	0.272594i	−0.990668	+	0.136297i	$0.956480\pi$
$572$	− 12.7681i	− 0.533861i
$573$	− 1.60953i	− 0.0672391i
$574$	24.5405	1.02430
$575$	0.295659i	0.0123298i
$576$	−11.4963	−0.479010
$577$	15.6703	0.652364	0.326182	−	0.945307i	$-0.394238\pi$
$577$	15.6703	0.652364	0.326182	+	0.945307i	$0.394238\pi$
$578$	0	0
$579$	−4.47403	−0.185935
$580$	1.90126	0.0789456
$581$	− 19.7397i	− 0.818940i
$582$	−12.4331	−0.515370
$583$	− 1.41848i	− 0.0587474i
$584$	− 82.1225i	− 3.39825i
$585$	2.93813i	0.121477i
$586$	19.6250	0.810703
$587$	14.9766	0.618150	0.309075	−	0.951038i	$-0.399981\pi$
$587$	14.9766	0.618150	0.309075	+	0.951038i	$0.399981\pi$
$588$	− 8.14061i	− 0.335713i
$589$	− 2.64436i	− 0.108959i
$590$	− 69.0553i	− 2.84296i
$591$	7.39376	0.304139
$592$	56.3838i	2.31736i
$593$	−46.3701	−1.90419	−0.952096	−	0.305801i	$-0.901076\pi$
$593$	−46.3701	−1.90419	−0.952096	+	0.305801i	$0.901076\pi$
$594$	5.14265	0.211006
$595$	0	0
$596$	−58.2744	−2.38701
$597$	−7.19657	−0.294536
$598$	16.6588i	0.681230i
$599$	28.7068	1.17293	0.586464	−	0.809975i	$-0.300520\pi$
$599$	28.7068	1.17293	0.586464	+	0.809975i	$0.300520\pi$
$600$	− 0.481255i	− 0.0196472i
$601$	20.7887i	0.847987i	0.905665	+	0.423993i	$0.139372\pi$
$601$	20.7887i	0.847987i	−0.905665	+	0.423993i	$0.860628\pi$
$602$	− 50.8518i	− 2.07257i
$603$	14.1866	0.577722
$604$	79.7237	3.24391
$605$	− 15.9888i	− 0.650035i
$606$	18.7481i	0.761589i
$607$	17.2060i	0.698372i	0.937053	+	0.349186i	$0.113542\pi$
$607$	17.2060i	0.698372i	−0.937053	+	0.349186i	$0.886458\pi$
$608$	−73.8325	−2.99430
$609$	0.399950i	0.0162068i
$610$	50.8938	2.06063
$611$	3.15213	0.127522
$612$	0	0
$613$	−42.7657	−1.72729	−0.863644	−	0.504102i	$-0.831824\pi$
$613$	−42.7657	−1.72729	−0.863644	+	0.504102i	$0.831824\pi$
$614$	−37.4854	−1.51279
$615$	8.93813i	0.360420i
$616$	35.1162	1.41487
$617$	38.2796i	1.54108i	0.637392	+	0.770540i	$0.280013\pi$
$617$	38.2796i	1.54108i	−0.637392	+	0.770540i	$0.719987\pi$
$618$	24.3167i	0.978162i
$619$	9.27242i	0.372690i	0.982484	+	0.186345i	$0.0596642\pi$
$619$	9.27242i	0.372690i	−0.982484	+	0.186345i	$0.940336\pi$
$620$	4.88877	0.196338
$621$	−4.77906	−0.191777
$622$	27.3172i	1.09532i
$623$	12.5037i	0.500950i
$624$	− 14.0250i	− 0.561449i
$625$	−24.6868	−0.987474
$626$	− 45.9425i	− 1.83623i
$627$	11.6075	0.463558
$628$	−31.9581	−1.27527
$629$	0	0
$630$	−13.5581	−0.540168
$631$	33.8478	1.34746	0.673730	−	0.738978i	$-0.264691\pi$
$631$	33.8478	1.34746	0.673730	+	0.738978i	$0.264691\pi$
$632$	− 5.43189i	− 0.216069i
$633$	−13.4185	−0.533337
$634$	− 37.3719i	− 1.48423i
$635$	− 21.8928i	− 0.868789i
$636$	− 3.60005i	− 0.142751i
$637$	2.17414	0.0861424
$638$	0.888765	0.0351866
$639$	− 6.10124i	− 0.241361i
$640$	− 12.2082i	− 0.482570i
$641$	− 10.6716i	− 0.421502i	−0.977540	−	0.210751i	$-0.932409\pi$
$641$	− 10.6716i	− 0.421502i	0.977540	−	0.210751i	$-0.0675909\pi$
$642$	−13.6633	−0.539247
$643$	39.3137i	1.55038i	0.631727	+	0.775191i	$0.282346\pi$
$643$	39.3137i	1.55038i	−0.631727	+	0.775191i	$0.717654\pi$
$644$	−54.7531	−2.15757
$645$	18.5212	0.729273
$646$	0	0
$647$	−27.9895	−1.10038	−0.550190	−	0.835039i	$-0.685445\pi$
$647$	−27.9895	−1.10038	−0.550190	+	0.835039i	$0.685445\pi$
$648$	7.77906	0.305590
$649$	− 22.9921i	− 0.902519i
$650$	0.215651	0.00845852
$651$	1.02840i	0.0403063i
$652$	− 82.8349i	− 3.24406i
$653$	40.9519i	1.60257i	0.598282	+	0.801285i	$0.295850\pi$
$653$	40.9519i	1.60257i	−0.598282	+	0.801285i	$0.704150\pi$
$654$	52.6631	2.05929
$655$	−16.9053	−0.660544
$656$	− 42.6656i	− 1.66581i
$657$	10.5569i	0.411863i
$658$	14.5456i	0.567047i
$659$	−10.8164	−0.421349	−0.210674	−	0.977556i	$-0.567566\pi$
$659$	−10.8164	−0.421349	−0.210674	+	0.977556i	$0.567566\pi$
$660$	21.4594i	0.835305i
$661$	−20.0794	−0.780997	−0.390498	−	0.920604i	$-0.627697\pi$
$661$	−20.0794	−0.780997	−0.390498	+	0.920604i	$0.627697\pi$
$662$	−28.6010	−1.11161
$663$	0	0
$664$	−66.3531	−2.57500
$665$	−30.6020	−1.18670
$666$	− 14.0137i	− 0.543022i
$667$	−0.825928	−0.0319801
$668$	7.94569i	0.307428i
$669$	21.2643i	0.822127i
$670$	83.1134i	3.21095i
$671$	16.9452	0.654161
$672$	28.7138	1.10766
$673$	27.4103i	1.05659i	0.849061	+	0.528294i	$0.177168\pi$
$673$	27.4103i	1.05659i	−0.849061	+	0.528294i	$0.822832\pi$
$674$	− 25.8834i	− 0.996992i
$675$	0.0618655i	0.00238120i
$676$	−55.7037	−2.14245
$677$	− 22.7259i	− 0.873426i	−0.899601	−	0.436713i	$-0.856142\pi$
$677$	− 22.7259i	− 0.873426i	0.899601	−	0.436713i	$-0.143858\pi$
$678$	45.9714	1.76552
$679$	10.9138	0.418832
$680$	0	0
$681$	−4.95404	−0.189839
$682$	2.28531	0.0875089
$683$	37.2065i	1.42367i	0.702348	+	0.711834i	$0.252135\pi$
$683$	37.2065i	1.42367i	−0.702348	+	0.711834i	$0.747865\pi$
$684$	29.4594	1.12641
$685$	− 49.0489i	− 1.87406i
$686$	52.7413i	2.01367i
$687$	1.83842i	0.0701403i
$688$	−88.4099	−3.37060
$689$	0.961475	0.0366293
$690$	− 27.9986i	− 1.06589i
$691$	18.0560i	0.686883i	0.939174	+	0.343442i	$0.111593\pi$
$691$	18.0560i	0.686883i	−0.939174	+	0.343442i	$0.888407\pi$
$692$	− 84.8338i	− 3.22490i
$693$	−4.51420	−0.171480
$694$	68.6703i	2.60669i
$695$	7.96553	0.302150
$696$	1.34440	0.0509592
$697$	0	0
$698$	49.1976	1.86216
$699$	1.49500	0.0565461
$700$	0.708785i	0.0267896i
$701$	9.99896	0.377656	0.188828	−	0.982010i	$-0.439531\pi$
$701$	9.99896	0.377656	0.188828	+	0.982010i	$0.439531\pi$
$702$	3.48580i	0.131563i
$703$	− 31.6304i	− 1.19296i
$704$	− 22.4249i	− 0.845172i
$705$	−5.29780	−0.199527
$706$	31.1991	1.17419
$707$	− 16.4570i	− 0.618929i
$708$	− 58.3531i	− 2.19305i
$709$	− 1.09395i	− 0.0410843i	−0.999789	−	0.0205422i	$-0.993461\pi$
$709$	− 1.09395i	− 0.0410843i	0.999789	−	0.0205422i	$-0.00653923\pi$
$710$	35.7447	1.34147
$711$	0.698270i	0.0261872i
$712$	42.0300	1.57514
$713$	−2.12373	−0.0795344
$714$	0	0
$715$	−5.73122	−0.214335
$716$	−54.8299	−2.04909
$717$	10.0250i	0.374390i
$718$	37.2768	1.39116
$719$	− 42.4818i	− 1.58430i	−0.610324	−	0.792152i	$-0.708961\pi$
$719$	− 42.4818i	− 1.58430i	0.610324	−	0.792152i	$-0.291039\pi$
$720$	23.5719i	0.878471i
$721$	− 21.3451i	− 0.794934i
$722$	43.2634	1.61010
$723$	−11.5155	−0.428264
$724$	− 97.5251i	− 3.62449i
$725$	0.0106917i	0 0.000397081i
$726$	− 18.9690i	− 0.704007i
$727$	45.2169	1.67700	0.838501	−	0.544900i	$-0.183432\pi$
$727$	45.2169	1.67700	0.838501	+	0.544900i	$0.183432\pi$
$728$	23.8025i	0.882180i
$729$	−1.00000	−0.0370370
$730$	−61.8484	−2.28911
$731$	0	0
$732$	43.0062	1.58956
$733$	21.5990	0.797778	0.398889	−	0.916999i	$-0.369396\pi$
$733$	21.5990	0.797778	0.398889	+	0.916999i	$0.369396\pi$
$734$	9.81697i	0.362351i
$735$	−3.65408	−0.134783
$736$	59.2962i	2.18569i
$737$	27.6728i	1.01934i
$738$	10.6042i	0.390346i
$739$	−19.0419	−0.700467	−0.350233	−	0.936663i	$-0.613898\pi$
$739$	−19.0419	−0.700467	−0.350233	+	0.936663i	$0.613898\pi$
$740$	58.4769	2.14965
$741$	7.86780i	0.289031i
$742$	4.43676i	0.162879i
$743$	26.4343i	0.969779i	0.874575	+	0.484889i	$0.161140\pi$
$743$	26.4343i	0.969779i	−0.874575	+	0.484889i	$0.838860\pi$
$744$	3.45688	0.126735
$745$	26.1576i	0.958342i
$746$	4.04999	0.148281
$747$	8.52971	0.312086
$748$	0	0
$749$	11.9936	0.438236
$750$	−29.6554	−1.08286
$751$	− 4.64686i	− 0.169566i	−0.996399	−	0.0847831i	$-0.972980\pi$
$751$	− 4.64686i	− 0.169566i	0.996399	−	0.0847831i	$-0.0270197\pi$
$752$	25.2887	0.922185
$753$	0.940179i	0.0342620i
$754$	0.602424i	0.0219390i
$755$	− 35.7856i	− 1.30237i
$756$	−11.4569	−0.416683
$757$	10.4074	0.378263	0.189131	−	0.981952i	$-0.439433\pi$
$757$	10.4074	0.378263	0.189131	+	0.981952i	$0.439433\pi$
$758$	95.3088i	3.46177i
$759$	− 9.32218i	− 0.338374i
$760$	102.866i	3.73134i
$761$	−38.6365	−1.40057	−0.700286	−	0.713862i	$-0.746944\pi$
$761$	−38.6365	−1.40057	−0.700286	+	0.713862i	$0.746944\pi$
$762$	− 25.9736i	− 0.940924i
$763$	−46.2275	−1.67355
$764$	7.96819	0.288279
$765$	0	0
$766$	43.1031	1.55738
$767$	15.5845	0.562725
$768$	8.50875i	0.307033i
$769$	−0.178098	−0.00642238	−0.00321119	−	0.999995i	$-0.501022\pi$
$769$	−0.178098	−0.00642238	−0.00321119	+	0.999995i	$0.501022\pi$
$770$	− 26.4469i	− 0.953079i
$771$	2.15515i	0.0776158i
$772$	− 22.1493i	− 0.797171i
$773$	−4.15554	−0.149464	−0.0747322	−	0.997204i	$-0.523810\pi$
$773$	−4.15554	−0.149464	−0.0747322	+	0.997204i	$0.523810\pi$
$774$	21.9736	0.789824
$775$	0.0274920i	0 0.000987540i
$776$	− 36.6856i	− 1.31694i
$777$	12.3012i	0.441304i
$778$	43.3776	1.55516
$779$	23.9347i	0.857551i
$780$	−14.5456	−0.520817
$781$	11.9013	0.425861
$782$	0	0
$783$	−0.172822	−0.00617617
$784$	17.4425	0.622946
$785$	14.3450i	0.511996i
$786$	−20.0564	−0.715389
$787$	− 1.73865i	− 0.0619762i	−0.999520	−	0.0309881i	$-0.990135\pi$
$787$	− 1.73865i	− 0.0619762i	0.999520	−	0.0309881i	$-0.00986540\pi$
$788$	36.6038i	1.30396i
$789$	− 13.1427i	− 0.467891i
$790$	−4.09088	−0.145547
$791$	−40.3535	−1.43481
$792$	15.1741i	0.539187i
$793$	11.4858i	0.407873i
$794$	96.1250i	3.41135i
$795$	−1.61596	−0.0573120
$796$	− 35.6275i	− 1.26278i
$797$	22.2085	0.786666	0.393333	−	0.919396i	$-0.371322\pi$
$797$	22.2085	0.786666	0.393333	+	0.919396i	$0.371322\pi$
$798$	−36.3062	−1.28523
$799$	0	0
$800$	0.767597	0.0271386
$801$	−5.40297	−0.190904
$802$	− 25.5179i	− 0.901070i
$803$	−20.5926	−0.726695
$804$	70.2325i	2.47691i
$805$	24.5770i	0.866227i
$806$	1.54903i	0.0545622i
$807$	−15.1743	−0.534162
$808$	−55.3187	−1.94611
$809$	− 3.22865i	− 0.113513i	−0.998388	−	0.0567567i	$-0.981924\pi$
$809$	− 3.22865i	− 0.113513i	0.998388	−	0.0567567i	$-0.0180759\pi$
$810$	− 5.85860i	− 0.205850i
$811$	25.2160i	0.885454i	0.896657	+	0.442727i	$0.145989\pi$
$811$	25.2160i	0.885454i	−0.896657	+	0.442727i	$0.854011\pi$
$812$	−1.98000	−0.0694846
$813$	− 11.8360i	− 0.415106i
$814$	27.3356	0.958114
$815$	−37.1821	−1.30243
$816$	0	0
$817$	49.5966	1.73516
$818$	−6.64975	−0.232503
$819$	− 3.05982i	− 0.106919i
$820$	−44.2494	−1.54526
$821$	− 34.4282i	− 1.20155i	−0.799417	−	0.600776i	$-0.794858\pi$
$821$	− 34.4282i	− 1.20155i	0.799417	−	0.600776i	$-0.205142\pi$
$822$	− 58.1916i	− 2.02966i
$823$	17.7994i	0.620447i	0.950664	+	0.310224i	$0.100404\pi$
$823$	17.7994i	0.620447i	−0.950664	+	0.310224i	$0.899596\pi$
$824$	−71.7497	−2.49952
$825$	−0.120677	−0.00420143
$826$	71.9153i	2.50225i
$827$	1.95063i	0.0678301i	0.999425	+	0.0339150i	$0.0107976\pi$
$827$	1.95063i	0.0678301i	−0.999425	+	0.0339150i	$0.989202\pi$
$828$	− 23.6594i	− 0.822219i
$829$	0.904671	0.0314205	0.0157103	−	0.999877i	$-0.494999\pi$
$829$	0.904671	0.0314205	0.0157103	+	0.999877i	$0.494999\pi$
$830$	49.9722i	1.73456i
$831$	−26.9025	−0.933238
$832$	15.2001	0.526969
$833$	0	0
$834$	9.45029	0.327237
$835$	3.56659	0.123427
$836$	57.4644i	1.98745i
$837$	−0.444383	−0.0153601
$838$	69.4065i	2.39761i
$839$	24.2763i	0.838110i	0.907961	+	0.419055i	$0.137639\pi$
$839$	24.2763i	0.838110i	−0.907961	+	0.419055i	$0.862361\pi$
$840$	− 40.0050i	− 1.38030i
$841$	28.9701	0.998970
$842$	−17.4290	−0.600643
$843$	− 22.2339i	− 0.765776i
$844$	− 66.4299i	− 2.28661i
$845$	25.0037i	0.860155i
$846$	−6.28531	−0.216093
$847$	16.6509i	0.572133i
$848$	7.71366	0.264888
$849$	−23.9577	−0.822225
$850$	0	0
$851$	−25.4030	−0.870802
$852$	30.2050	1.03481
$853$	38.1105i	1.30488i	0.757841	+	0.652439i	$0.226254\pi$
$853$	38.1105i	1.30488i	−0.757841	+	0.652439i	$0.773746\pi$
$854$	−53.0016	−1.81368
$855$	− 13.2234i	− 0.452232i
$856$	− 40.3153i	− 1.37795i
$857$	6.80024i	0.232292i	0.993232	+	0.116146i	$0.0370540\pi$
$857$	6.80024i	0.232292i	−0.993232	+	0.116146i	$0.962946\pi$
$858$	−6.79951	−0.232131
$859$	−22.3072	−0.761113	−0.380557	−	0.924758i	$-0.624268\pi$
$859$	−22.3072	−0.761113	−0.380557	+	0.924758i	$0.624268\pi$
$860$	91.6918i	3.12667i
$861$	− 9.30832i	− 0.317227i
$862$	− 74.1521i	− 2.52563i
$863$	0.400593	0.0136363	0.00681816	−	0.999977i	$-0.497830\pi$
$863$	0.400593	0.0136363	0.00681816	+	0.999977i	$0.497830\pi$
$864$	12.4075i	0.422112i
$865$	−38.0794	−1.29474
$866$	−45.2135	−1.53642
$867$	0	0
$868$	−5.09124	−0.172808
$869$	−1.36207	−0.0462050
$870$	− 1.01250i	− 0.0343268i
$871$	−18.7572	−0.635563
$872$	155.389i	5.26215i
$873$	4.71594i	0.159610i
$874$	− 74.9752i	− 2.53607i
$875$	26.0314	0.880023
$876$	−52.2632	−1.76581
$877$	− 0.210829i	− 0.00711920i	−0.999994	−	0.00355960i	$-0.998867\pi$
$877$	− 0.210829i	− 0.00711920i	0.999994	−	0.00355960i	$-0.00113306\pi$
$878$	21.6843i	0.731811i
$879$	− 7.44387i	− 0.251075i
$880$	−45.9800	−1.54999
$881$	− 51.7963i	− 1.74506i	−0.488560	−	0.872530i	$-0.662478\pi$
$881$	− 51.7963i	− 1.74506i	0.488560	−	0.872530i	$-0.337522\pi$
$882$	−4.33519	−0.145974
$883$	−23.5757	−0.793385	−0.396692	−	0.917952i	$-0.629842\pi$
$883$	−23.5757	−0.793385	−0.396692	+	0.917952i	$0.629842\pi$
$884$	0	0
$885$	−26.1930	−0.880468
$886$	−48.6834	−1.63555
$887$	− 16.3341i	− 0.548446i	−0.961666	−	0.274223i	$-0.911579\pi$
$887$	− 16.3341i	− 0.548446i	0.961666	−	0.274223i	$-0.0884207\pi$
$888$	41.3494	1.38759
$889$	22.7995i	0.764671i
$890$	− 31.6538i	− 1.06104i
$891$	− 1.95063i	− 0.0653486i
$892$	−105.272	−3.52476
$893$	−14.1866	−0.474735
$894$	31.0334i	1.03791i
$895$	24.6115i	0.822672i
$896$	12.7138i	0.424738i
$897$	6.31877	0.210978
$898$	− 81.5308i	− 2.72072i
$899$	−0.0767993	−0.00256140
$900$	−0.306273	−0.0102091
$901$	0	0
$902$	−20.6849	−0.688731
$903$	−19.2883	−0.641875
$904$	135.645i	4.51148i
$905$	−43.7761	−1.45517
$906$	− 42.4560i	− 1.41051i
$907$	30.7826i	1.02212i	0.859546	+	0.511059i	$0.170747\pi$
$907$	30.7826i	1.02212i	−0.859546	+	0.511059i	$0.829253\pi$
$908$	− 24.5256i	− 0.813911i
$909$	7.11123	0.235865
$910$	17.9263	0.594250
$911$	7.19491i	0.238378i	0.992872	+	0.119189i	$0.0380295\pi$
$911$	7.19491i	0.238378i	−0.992872	+	0.119189i	$0.961971\pi$
$912$	63.1212i	2.09015i
$913$	16.6383i	0.550648i
$914$	40.9532	1.35461
$915$	− 19.3042i	− 0.638178i
$916$	−9.10136	−0.300717
$917$	17.6054	0.581383
$918$	0	0
$919$	11.6873	0.385527	0.192764	−	0.981245i	$-0.438255\pi$
$919$	11.6873	0.385527	0.192764	+	0.981245i	$0.438255\pi$
$920$	82.6135	2.72368
$921$	14.2184i	0.468512i
$922$	102.565	3.37779
$923$	8.06693i	0.265526i
$924$	− 22.3481i	− 0.735200i
$925$	0.328845i	0.0108123i
$926$	−59.9971	−1.97163
$927$	9.22344	0.302938
$928$	2.14430i	0.0703899i
$929$	14.1828i	0.465321i	0.972558	+	0.232661i	$0.0747431\pi$
$929$	14.1828i	0.465321i	−0.972558	+	0.232661i	$0.925257\pi$
$930$	− 2.60346i	− 0.0853708i
$931$	−9.78497	−0.320689
$932$	7.40120i	0.242434i
$933$	10.3616	0.339222
$934$	33.1003	1.08308
$935$	0	0
$936$	−10.2853	−0.336186
$937$	−25.4693	−0.832044	−0.416022	−	0.909355i	$-0.636576\pi$
$937$	−25.4693	−0.832044	−0.416022	+	0.909355i	$0.636576\pi$
$938$	− 86.5556i	− 2.82614i
$939$	−17.4262	−0.568682
$940$	− 26.2275i	− 0.855446i
$941$	58.5838i	1.90978i	0.296964	+	0.954889i	$0.404026\pi$
$941$	58.5838i	1.90978i	−0.296964	+	0.954889i	$0.595974\pi$
$942$	17.0189i	0.554506i
$943$	19.2224	0.625968
$944$	125.031	4.06940
$945$	5.14265i	0.167291i
$946$	42.8623i	1.39358i
$947$	− 19.0443i	− 0.618857i	−0.950923	−	0.309428i	$-0.899862\pi$
$947$	− 19.0443i	− 0.618857i	0.950923	−	0.309428i	$-0.100138\pi$
$948$	−3.45688	−0.112274
$949$	− 13.9581i	− 0.453098i
$950$	−0.970563	−0.0314892
$951$	−14.1753	−0.459666
$952$	0	0
$953$	19.8922	0.644371	0.322185	−	0.946677i	$-0.395583\pi$
$953$	19.8922	0.644371	0.322185	+	0.946677i	$0.395583\pi$
$954$	−1.91717	−0.0620706
$955$	− 3.57668i	− 0.115739i
$956$	−49.6300	−1.60515
$957$	− 0.337113i	− 0.0108973i
$958$	98.1846i	3.17220i
$959$	51.0803i	1.64947i
$960$	−25.5469	−0.824522
$961$	30.8025	0.993630
$962$	18.5287i	0.597389i
$963$	5.18254i	0.167005i
$964$	− 57.0087i	− 1.83613i
$965$	−9.94216	−0.320049
$966$	29.1582i	0.938149i
$967$	−53.0310	−1.70536	−0.852680	−	0.522433i	$-0.825025\pi$
$967$	−53.0310	−1.70536	−0.852680	+	0.522433i	$0.825025\pi$
$968$	55.9706	1.79896
$969$	0	0
$970$	−27.6288	−0.887108
$971$	−3.02535	−0.0970880	−0.0485440	−	0.998821i	$-0.515458\pi$
$971$	−3.02535	−0.0970880	−0.0485440	+	0.998821i	$0.515458\pi$
$972$	− 4.95063i	− 0.158792i
$973$	−8.29543	−0.265939
$974$	4.33656i	0.138952i
$975$	− 0.0817972i	− 0.00261961i
$976$	92.1475i	2.94957i
$977$	−11.2664	−0.360444	−0.180222	−	0.983626i	$-0.557682\pi$
$977$	−11.2664	−0.360444	−0.180222	+	0.983626i	$0.557682\pi$
$978$	−44.1128	−1.41057
$979$	− 10.5392i	− 0.336834i
$980$	− 18.0900i	− 0.577864i
$981$	− 19.9753i	− 0.637764i
$982$	−11.3397	−0.361866
$983$	48.3581i	1.54238i	0.636603	+	0.771192i	$0.280339\pi$
$983$	48.3581i	1.54238i	−0.636603	+	0.771192i	$0.719661\pi$
$984$	−31.2891	−0.997459
$985$	16.4304	0.523515
$986$	0	0
$987$	5.51722	0.175615
$988$	−38.9506	−1.23918
$989$	− 39.8319i	− 1.26658i
$990$	11.4280	0.363204
$991$	− 24.0403i	− 0.763667i	−0.924231	−	0.381833i	$-0.875293\pi$
$991$	− 24.0403i	− 0.763667i	0.924231	−	0.381833i	$-0.124707\pi$
$992$	5.51368i	0.175060i
$993$	10.8485i	0.344266i
$994$	−37.2251	−1.18071
$995$	−15.9921	−0.506985
$996$	42.2275i	1.33803i
$997$	26.3398i	0.834189i	0.908863	+	0.417095i	$0.136952\pi$
$997$	26.3398i	0.834189i	−0.908863	+	0.417095i	$0.863048\pi$
$998$	− 3.94216i	− 0.124787i
$999$	−5.31548	−0.168174

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	867.2.d.f.577.7		8
17.2	even	8		51.2.e.a.13.4	yes	8
17.3	odd	16		867.2.h.k.688.4		16
17.4	even	4		867.2.a.k.1.1		4
17.5	odd	16		867.2.h.i.757.2		16
17.6	odd	16		867.2.h.k.712.3		16
17.7	odd	16		867.2.h.i.733.1		16
17.8	even	8		51.2.e.a.4.1	✓	8
17.9	even	8		867.2.e.g.616.1		8
17.10	odd	16		867.2.h.i.733.2		16
17.11	odd	16		867.2.h.k.712.4		16
17.12	odd	16		867.2.h.i.757.1		16
17.13	even	4		867.2.a.l.1.1		4
17.14	odd	16		867.2.h.k.688.3		16
17.15	even	8		867.2.e.g.829.4		8
17.16	even	2	inner	867.2.d.f.577.8		8
51.2	odd	8		153.2.f.b.64.1		8
51.8	odd	8		153.2.f.b.55.4		8
51.38	odd	4		2601.2.a.bf.1.4		4
51.47	odd	4		2601.2.a.be.1.4		4
68.19	odd	8		816.2.bd.e.625.4		8
68.59	odd	8		816.2.bd.e.769.4		8
204.59	even	8		2448.2.be.x.1585.1		8
204.155	even	8		2448.2.be.x.1441.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.2.e.a.4.1	✓	8	17.8	even	8
51.2.e.a.13.4	yes	8	17.2	even	8
153.2.f.b.55.4		8	51.8	odd	8
153.2.f.b.64.1		8	51.2	odd	8
816.2.bd.e.625.4		8	68.19	odd	8
816.2.bd.e.769.4		8	68.59	odd	8
867.2.a.k.1.1		4	17.4	even	4
867.2.a.l.1.1		4	17.13	even	4
867.2.d.f.577.7		8	1.1	even	1	trivial
867.2.d.f.577.8		8	17.16	even	2	inner
867.2.e.g.616.1		8	17.9	even	8
867.2.e.g.829.4		8	17.15	even	8
867.2.h.i.733.1		16	17.7	odd	16
867.2.h.i.733.2		16	17.10	odd	16
867.2.h.i.757.1		16	17.12	odd	16
867.2.h.i.757.2		16	17.5	odd	16
867.2.h.k.688.3		16	17.14	odd	16
867.2.h.k.688.4		16	17.3	odd	16
867.2.h.k.712.3		16	17.6	odd	16
867.2.h.k.712.4		16	17.11	odd	16
2448.2.be.x.1441.1		8	204.155	even	8
2448.2.be.x.1585.1		8	204.59	even	8
2601.2.a.be.1.4		4	51.47	odd	4
2601.2.a.bf.1.4		4	51.38	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 867 = 3 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	867.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.63640 q^{2} -1.00000i q^{3} +4.95063 q^{4} -2.22219i q^{5} -2.63640i q^{6} +2.31423i q^{7} +7.77906 q^{8} -1.00000 q^{9} -5.85860i q^{10} -1.95063i q^{11} -4.95063i q^{12} +1.32218 q^{13} +6.10124i q^{14} -2.22219 q^{15} +10.6075 q^{16} -2.63640 q^{18} -5.95063 q^{19} -11.0012i q^{20} +2.31423 q^{21} -5.14265i q^{22} +4.77906i q^{23} -7.77906i q^{24} +0.0618655 q^{25} +3.48580 q^{26} +1.00000i q^{27} +11.4569i q^{28} +0.172822i q^{29} -5.85860 q^{30} +0.444383i q^{31} +12.4075 q^{32} -1.95063 q^{33} +5.14265 q^{35} -4.95063 q^{36} +5.31548i q^{37} -15.6883 q^{38} -1.32218i q^{39} -17.2866i q^{40} -4.02222i q^{41} +6.10124 q^{42} -8.33468 q^{43} -9.65685i q^{44} +2.22219i q^{45} +12.5995i q^{46} +2.38404 q^{47} -10.6075i q^{48} +1.64436 q^{49} +0.163102 q^{50} +6.54562 q^{52} +0.727190 q^{53} +2.63640i q^{54} -4.33468 q^{55} +18.0025i q^{56} +5.95063i q^{57} +0.455630i q^{58} +11.7870 q^{59} -11.0012 q^{60} +8.68702i q^{61} +1.17157i q^{62} -2.31423i q^{63} +11.4963 q^{64} -2.93813i q^{65} -5.14265 q^{66} -14.1866 q^{67} +4.77906 q^{69} +13.5581 q^{70} +6.10124i q^{71} -7.77906 q^{72} -10.5569i q^{73} +14.0137i q^{74} -0.0618655i q^{75} -29.4594 q^{76} +4.51420 q^{77} -3.48580i q^{78} -0.698270i q^{79} -23.5719i q^{80} +1.00000 q^{81} -10.6042i q^{82} -8.52971 q^{83} +11.4569 q^{84} -21.9736 q^{86} +0.172822 q^{87} -15.1741i q^{88} +5.40297 q^{89} +5.85860i q^{90} +3.05982i q^{91} +23.6594i q^{92} +0.444383 q^{93} +6.28531 q^{94} +13.2234i q^{95} -12.4075i q^{96} -4.71594i q^{97} +4.33519 q^{98} +1.95063i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{2} + 12 q^{4} + 12 q^{8} - 8 q^{9} + 4 q^{13} - 12 q^{15} + 12 q^{16} - 4 q^{18} - 20 q^{19} + 8 q^{21} - 4 q^{25} + 40 q^{26} - 24 q^{30} + 28 q^{32} + 12 q^{33} + 8 q^{35} - 12 q^{36} - 24 q^{38}+ \cdots + 76 q^{98}+O(q^{100}) \) 8 * q + 4 * q^2 + 12 * q^4 + 12 * q^8 - 8 * q^9 + 4 * q^13 - 12 * q^15 + 12 * q^16 - 4 * q^18 - 20 * q^19 + 8 * q^21 - 4 * q^25 + 40 * q^26 - 24 * q^30 + 28 * q^32 + 12 * q^33 + 8 * q^35 - 12 * q^36 - 24 * q^38 - 8 * q^42 - 28 * q^43 + 8 * q^47 - 60 * q^50 - 16 * q^52 + 40 * q^53 + 4 * q^55 + 48 * q^59 - 32 * q^60 - 4 * q^64 - 8 * q^66 + 8 * q^67 - 12 * q^69 + 8 * q^70 - 12 * q^72 - 80 * q^76 + 24 * q^77 + 8 * q^81 + 8 * q^83 + 48 * q^84 - 8 * q^86 - 32 * q^87 + 8 * q^89 - 8 * q^93 - 16 * q^94 + 76 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more