LMFDB - Embedded newform 4840.2.a.bg.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4840,2,Mod(1,4840)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4840 = 2^{3} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4840.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$38.6475945783$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 21x^{6} + 15x^{5} + 140x^{4} - 60x^{3} - 295x^{2} + 50x + 100 $ x^8 - x^7 - 21x^6 + 15x^5 + 140x^4 - 60x^3 - 295x^2 + 50x + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 440)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.77190$ of defining polynomial
Character	$\chi$	$=$	4840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.77190 q^{3} +1.00000 q^{5} -4.55506 q^{7} +4.68342 q^{9} +2.90026 q^{13} -2.77190 q^{15} -7.08802 q^{17} -7.14347 q^{19} +12.6262 q^{21} -1.86699 q^{23} +1.00000 q^{25} -4.66626 q^{27} -1.01070 q^{29} +4.41034 q^{31} -4.55506 q^{35} -5.39094 q^{37} -8.03922 q^{39} -4.48790 q^{41} -6.73002 q^{43} +4.68342 q^{45} +7.85017 q^{47} +13.7486 q^{49} +19.6473 q^{51} +0.714458 q^{53} +19.8010 q^{57} -3.80330 q^{59} +2.11595 q^{61} -21.3332 q^{63} +2.90026 q^{65} -12.8384 q^{67} +5.17511 q^{69} +4.21095 q^{71} +1.36576 q^{73} -2.77190 q^{75} -12.2375 q^{79} -1.11585 q^{81} -16.3351 q^{83} -7.08802 q^{85} +2.80156 q^{87} -11.8165 q^{89} -13.2108 q^{91} -12.2250 q^{93} -7.14347 q^{95} -10.8544 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{3} + 8 q^{5} - 6 q^{7} + 19 q^{9} + 12 q^{13} + q^{15} + 2 q^{17} - 6 q^{19} + 6 q^{21} + 10 q^{23} + 8 q^{25} + 13 q^{27} + 8 q^{29} + 19 q^{31} - 6 q^{35} + 12 q^{37} - 21 q^{39} - 3 q^{41}+ \cdots + 21 q^{97}+O(q^{100}) $ 8 * q + q^3 + 8 * q^5 - 6 * q^7 + 19 * q^9 + 12 * q^13 + q^15 + 2 * q^17 - 6 * q^19 + 6 * q^21 + 10 * q^23 + 8 * q^25 + 13 * q^27 + 8 * q^29 + 19 * q^31 - 6 * q^35 + 12 * q^37 - 21 * q^39 - 3 * q^41 - 8 * q^43 + 19 * q^45 + 10 * q^47 + 22 * q^49 - 7 * q^51 + 28 * q^53 + 25 * q^57 + 25 * q^59 + 10 * q^61 - 64 * q^63 + 12 * q^65 - 2 * q^67 + 18 * q^69 + 25 * q^71 + 38 * q^73 + q^75 - 38 * q^79 + 32 * q^81 - 28 * q^83 + 2 * q^85 + 15 * q^87 + 12 * q^89 + 8 * q^91 - 15 * q^93 - 6 * q^95 + 21 * q^97

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 $

$2$	0	0
$2$	0	0
$3$	−2.77190	−1.60036	−0.800178	−	0.599763i	$-0.795262\pi$
$3$	−2.77190	−1.60036	−0.800178	+	0.599763i	$0.795262\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.55506	−1.72165	−0.860825	−	0.508901i	$-0.830052\pi$
$7$	−4.55506	−1.72165	−0.860825	+	0.508901i	$0.830052\pi$
$8$	0	0
$9$	4.68342	1.56114
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	2.90026	0.804387	0.402193	−	0.915555i	$-0.368248\pi$
$13$	2.90026	0.804387	0.402193	+	0.915555i	$0.368248\pi$
$14$	0	0
$15$	−2.77190	−0.715701
$16$	0	0
$17$	−7.08802	−1.71910	−0.859548	−	0.511054i	$-0.829255\pi$
$17$	−7.08802	−1.71910	−0.859548	+	0.511054i	$0.829255\pi$
$18$	0	0
$19$	−7.14347	−1.63882	−0.819412	−	0.573205i	$-0.805700\pi$
$19$	−7.14347	−1.63882	−0.819412	+	0.573205i	$0.805700\pi$
$20$	0	0
$21$	12.6262	2.75525
$22$	0	0
$23$	−1.86699	−0.389295	−0.194647	−	0.980873i	$-0.562356\pi$
$23$	−1.86699	−0.389295	−0.194647	+	0.980873i	$0.562356\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.66626	−0.898023
$28$	0	0
$29$	−1.01070	−0.187682	−0.0938412	−	0.995587i	$-0.529915\pi$
$29$	−1.01070	−0.187682	−0.0938412	+	0.995587i	$0.529915\pi$
$30$	0	0
$31$	4.41034	0.792121	0.396061	−	0.918224i	$-0.370377\pi$
$31$	4.41034	0.792121	0.396061	+	0.918224i	$0.370377\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.55506	−0.769945
$36$	0	0
$37$	−5.39094	−0.886265	−0.443132	−	0.896456i	$-0.646133\pi$
$37$	−5.39094	−0.886265	−0.443132	+	0.896456i	$0.646133\pi$
$38$	0	0
$39$	−8.03922	−1.28730
$40$	0	0
$41$	−4.48790	−0.700892	−0.350446	−	0.936583i	$-0.613970\pi$
$41$	−4.48790	−0.700892	−0.350446	+	0.936583i	$0.613970\pi$
$42$	0	0
$43$	−6.73002	−1.02632	−0.513159	−	0.858293i	$-0.671525\pi$
$43$	−6.73002	−1.02632	−0.513159	+	0.858293i	$0.671525\pi$
$44$	0	0
$45$	4.68342	0.698163
$46$	0	0
$47$	7.85017	1.14507	0.572533	−	0.819882i	$-0.305961\pi$
$47$	7.85017	1.14507	0.572533	+	0.819882i	$0.305961\pi$
$48$	0	0
$49$	13.7486	1.96408
$50$	0	0
$51$	19.6473	2.75117
$52$	0	0
$53$	0.714458	0.0981383	0.0490692	−	0.998795i	$-0.484375\pi$
$53$	0.714458	0.0981383	0.0490692	+	0.998795i	$0.484375\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	19.8010	2.62270
$58$	0	0
$59$	−3.80330	−0.495147	−0.247574	−	0.968869i	$-0.579633\pi$
$59$	−3.80330	−0.495147	−0.247574	+	0.968869i	$0.579633\pi$
$60$	0	0
$61$	2.11595	0.270920	0.135460	−	0.990783i	$-0.456749\pi$
$61$	2.11595	0.270920	0.135460	+	0.990783i	$0.456749\pi$
$62$	0	0
$63$	−21.3332	−2.68774
$64$	0	0
$65$	2.90026	0.359733
$66$	0	0
$67$	−12.8384	−1.56847	−0.784233	−	0.620467i	$-0.786943\pi$
$67$	−12.8384	−1.56847	−0.784233	+	0.620467i	$0.786943\pi$
$68$	0	0
$69$	5.17511	0.623010
$70$	0	0
$71$	4.21095	0.499747	0.249874	−	0.968278i	$-0.419611\pi$
$71$	4.21095	0.499747	0.249874	+	0.968278i	$0.419611\pi$
$72$	0	0
$73$	1.36576	0.159850	0.0799249	−	0.996801i	$-0.474532\pi$
$73$	1.36576	0.159850	0.0799249	+	0.996801i	$0.474532\pi$
$74$	0	0
$75$	−2.77190	−0.320071
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−12.2375	−1.37682	−0.688411	−	0.725320i	$-0.741692\pi$
$79$	−12.2375	−1.37682	−0.688411	+	0.725320i	$0.741692\pi$
$80$	0	0
$81$	−1.11585	−0.123983
$82$	0	0
$83$	−16.3351	−1.79301	−0.896503	−	0.443037i	$-0.853901\pi$
$83$	−16.3351	−1.79301	−0.896503	+	0.443037i	$0.853901\pi$
$84$	0	0
$85$	−7.08802	−0.768803
$86$	0	0
$87$	2.80156	0.300359
$88$	0	0
$89$	−11.8165	−1.25255	−0.626273	−	0.779604i	$-0.715420\pi$
$89$	−11.8165	−1.25255	−0.626273	+	0.779604i	$0.715420\pi$
$90$	0	0
$91$	−13.2108	−1.38487
$92$	0	0
$93$	−12.2250	−1.26768
$94$	0	0
$95$	−7.14347	−0.732904
$96$	0	0
$97$	−10.8544	−1.10210	−0.551048	−	0.834473i	$-0.685772\pi$
$97$	−10.8544	−1.10210	−0.551048	+	0.834473i	$0.685772\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	12.6665	1.26037	0.630184	−	0.776446i	$-0.282979\pi$
$101$	12.6665	1.26037	0.630184	+	0.776446i	$0.282979\pi$
$102$	0	0
$103$	−6.18162	−0.609093	−0.304547	−	0.952497i	$-0.598505\pi$
$103$	−6.18162	−0.609093	−0.304547	+	0.952497i	$0.598505\pi$
$104$	0	0
$105$	12.6262	1.23219
$106$	0	0
$107$	1.00774	0.0974223	0.0487111	−	0.998813i	$-0.484489\pi$
$107$	1.00774	0.0974223	0.0487111	+	0.998813i	$0.484489\pi$
$108$	0	0
$109$	6.60399	0.632548	0.316274	−	0.948668i	$-0.397568\pi$
$109$	6.60399	0.632548	0.316274	+	0.948668i	$0.397568\pi$
$110$	0	0
$111$	14.9431	1.41834
$112$	0	0
$113$	−10.2611	−0.965279	−0.482640	−	0.875819i	$-0.660322\pi$
$113$	−10.2611	−0.965279	−0.482640	+	0.875819i	$0.660322\pi$
$114$	0	0
$115$	−1.86699	−0.174098
$116$	0	0
$117$	13.5831	1.25576
$118$	0	0
$119$	32.2863	2.95968
$120$	0	0
$121$	0	0
$122$	0	0
$123$	12.4400	1.12168
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	17.7211	1.57249	0.786245	−	0.617914i	$-0.212022\pi$
$127$	17.7211	1.57249	0.786245	+	0.617914i	$0.212022\pi$
$128$	0	0
$129$	18.6549	1.64247
$130$	0	0
$131$	14.3999	1.25812	0.629061	−	0.777356i	$-0.283440\pi$
$131$	14.3999	1.25812	0.629061	+	0.777356i	$0.283440\pi$
$132$	0	0
$133$	32.5389	2.82148
$134$	0	0
$135$	−4.66626	−0.401608
$136$	0	0
$137$	4.46970	0.381872	0.190936	−	0.981603i	$-0.438848\pi$
$137$	4.46970	0.381872	0.190936	+	0.981603i	$0.438848\pi$
$138$	0	0
$139$	4.84523	0.410967	0.205483	−	0.978661i	$-0.434123\pi$
$139$	4.84523	0.410967	0.205483	+	0.978661i	$0.434123\pi$
$140$	0	0
$141$	−21.7599	−1.83251
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−1.01070	−0.0839341
$146$	0	0
$147$	−38.1096	−3.14323
$148$	0	0
$149$	−10.7263	−0.878734	−0.439367	−	0.898308i	$-0.644797\pi$
$149$	−10.7263	−0.878734	−0.439367	+	0.898308i	$0.644797\pi$
$150$	0	0
$151$	−0.201814	−0.0164234	−0.00821172	−	0.999966i	$-0.502614\pi$
$151$	−0.201814	−0.0164234	−0.00821172	+	0.999966i	$0.502614\pi$
$152$	0	0
$153$	−33.1961	−2.68375
$154$	0	0
$155$	4.41034	0.354247
$156$	0	0
$157$	−2.92198	−0.233199	−0.116600	−	0.993179i	$-0.537199\pi$
$157$	−2.92198	−0.233199	−0.116600	+	0.993179i	$0.537199\pi$
$158$	0	0
$159$	−1.98040	−0.157056
$160$	0	0
$161$	8.50425	0.670229
$162$	0	0
$163$	8.48158	0.664329	0.332164	−	0.943222i	$-0.392221\pi$
$163$	8.48158	0.664329	0.332164	+	0.943222i	$0.392221\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−10.0642	−0.778790	−0.389395	−	0.921071i	$-0.627316\pi$
$167$	−10.0642	−0.778790	−0.389395	+	0.921071i	$0.627316\pi$
$168$	0	0
$169$	−4.58851	−0.352962
$170$	0	0
$171$	−33.4558	−2.55843
$172$	0	0
$173$	5.77262	0.438884	0.219442	−	0.975626i	$-0.429576\pi$
$173$	5.77262	0.438884	0.219442	+	0.975626i	$0.429576\pi$
$174$	0	0
$175$	−4.55506	−0.344330
$176$	0	0
$177$	10.5424	0.792412
$178$	0	0
$179$	16.8671	1.26071	0.630353	−	0.776309i	$-0.282910\pi$
$179$	16.8671	1.26071	0.630353	+	0.776309i	$0.282910\pi$
$180$	0	0
$181$	15.1054	1.12277	0.561387	−	0.827553i	$-0.310268\pi$
$181$	15.1054	1.12277	0.561387	+	0.827553i	$0.310268\pi$
$182$	0	0
$183$	−5.86520	−0.433568
$184$	0	0
$185$	−5.39094	−0.396350
$186$	0	0
$187$	0	0
$188$	0	0
$189$	21.2551	1.54608
$190$	0	0
$191$	10.4338	0.754966	0.377483	−	0.926016i	$-0.376790\pi$
$191$	10.4338	0.754966	0.377483	+	0.926016i	$0.376790\pi$
$192$	0	0
$193$	13.9250	1.00234	0.501172	−	0.865348i	$-0.332902\pi$
$193$	13.9250	1.00234	0.501172	+	0.865348i	$0.332902\pi$
$194$	0	0
$195$	−8.03922	−0.575700
$196$	0	0
$197$	−2.25234	−0.160473	−0.0802363	−	0.996776i	$-0.525567\pi$
$197$	−2.25234	−0.160473	−0.0802363	+	0.996776i	$0.525567\pi$
$198$	0	0
$199$	22.1246	1.56837	0.784184	−	0.620528i	$-0.213082\pi$
$199$	22.1246	1.56837	0.784184	+	0.620528i	$0.213082\pi$
$200$	0	0
$201$	35.5869	2.51010
$202$	0	0
$203$	4.60380	0.323123
$204$	0	0
$205$	−4.48790	−0.313448
$206$	0	0
$207$	−8.74390	−0.607743
$208$	0	0
$209$	0	0
$210$	0	0
$211$	7.77119	0.534991	0.267495	−	0.963559i	$-0.413804\pi$
$211$	7.77119	0.534991	0.267495	+	0.963559i	$0.413804\pi$
$212$	0	0
$213$	−11.6723	−0.799774
$214$	0	0
$215$	−6.73002	−0.458984
$216$	0	0
$217$	−20.0894	−1.36376
$218$	0	0
$219$	−3.78574	−0.255817
$220$	0	0
$221$	−20.5571	−1.38282
$222$	0	0
$223$	−7.92084	−0.530419	−0.265209	−	0.964191i	$-0.585441\pi$
$223$	−7.92084	−0.530419	−0.265209	+	0.964191i	$0.585441\pi$
$224$	0	0
$225$	4.68342	0.312228
$226$	0	0
$227$	−20.2672	−1.34518	−0.672590	−	0.740015i	$-0.734818\pi$
$227$	−20.2672	−1.34518	−0.672590	+	0.740015i	$0.734818\pi$
$228$	0	0
$229$	9.88050	0.652922	0.326461	−	0.945211i	$-0.394144\pi$
$229$	9.88050	0.652922	0.326461	+	0.945211i	$0.394144\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	9.79865	0.641931	0.320966	−	0.947091i	$-0.395993\pi$
$233$	9.79865	0.641931	0.320966	+	0.947091i	$0.395993\pi$
$234$	0	0
$235$	7.85017	0.512089
$236$	0	0
$237$	33.9210	2.20341
$238$	0	0
$239$	−11.0149	−0.712493	−0.356247	−	0.934392i	$-0.615944\pi$
$239$	−11.0149	−0.712493	−0.356247	+	0.934392i	$0.615944\pi$
$240$	0	0
$241$	−13.3108	−0.857421	−0.428711	−	0.903442i	$-0.641032\pi$
$241$	−13.3108	−0.857421	−0.428711	+	0.903442i	$0.641032\pi$
$242$	0	0
$243$	17.0918	1.09644
$244$	0	0
$245$	13.7486	0.878363
$246$	0	0
$247$	−20.7179	−1.31825
$248$	0	0
$249$	45.2791	2.86945
$250$	0	0
$251$	−14.5916	−0.921011	−0.460506	−	0.887657i	$-0.652332\pi$
$251$	−14.5916	−0.921011	−0.460506	+	0.887657i	$0.652332\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	19.6473	1.23036
$256$	0	0
$257$	0.965528	0.0602280	0.0301140	−	0.999546i	$-0.490413\pi$
$257$	0.965528	0.0602280	0.0301140	+	0.999546i	$0.490413\pi$
$258$	0	0
$259$	24.5560	1.52584
$260$	0	0
$261$	−4.73353	−0.292998
$262$	0	0
$263$	20.7753	1.28106	0.640529	−	0.767934i	$-0.278715\pi$
$263$	20.7753	1.28106	0.640529	+	0.767934i	$0.278715\pi$
$264$	0	0
$265$	0.714458	0.0438888
$266$	0	0
$267$	32.7541	2.00452
$268$	0	0
$269$	−3.91387	−0.238633	−0.119317	−	0.992856i	$-0.538070\pi$
$269$	−3.91387	−0.238633	−0.119317	+	0.992856i	$0.538070\pi$
$270$	0	0
$271$	−14.2782	−0.867338	−0.433669	−	0.901072i	$-0.642781\pi$
$271$	−14.2782	−0.867338	−0.433669	+	0.901072i	$0.642781\pi$
$272$	0	0
$273$	36.6191	2.21629
$274$	0	0
$275$	0	0
$276$	0	0
$277$	3.76298	0.226095	0.113048	−	0.993590i	$-0.463939\pi$
$277$	3.76298	0.226095	0.113048	+	0.993590i	$0.463939\pi$
$278$	0	0
$279$	20.6555	1.23661
$280$	0	0
$281$	−10.3189	−0.615576	−0.307788	−	0.951455i	$-0.599589\pi$
$281$	−10.3189	−0.615576	−0.307788	+	0.951455i	$0.599589\pi$
$282$	0	0
$283$	30.4565	1.81045	0.905224	−	0.424934i	$-0.139703\pi$
$283$	30.4565	1.81045	0.905224	+	0.424934i	$0.139703\pi$
$284$	0	0
$285$	19.8010	1.17291
$286$	0	0
$287$	20.4426	1.20669
$288$	0	0
$289$	33.2400	1.95529
$290$	0	0
$291$	30.0873	1.76375
$292$	0	0
$293$	−7.90548	−0.461843	−0.230922	−	0.972972i	$-0.574174\pi$
$293$	−7.90548	−0.461843	−0.230922	+	0.972972i	$0.574174\pi$
$294$	0	0
$295$	−3.80330	−0.221437
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−5.41475	−0.313143
$300$	0	0
$301$	30.6556	1.76696
$302$	0	0
$303$	−35.1104	−2.01704
$304$	0	0
$305$	2.11595	0.121159
$306$	0	0
$307$	14.2585	0.813777	0.406889	−	0.913478i	$-0.366614\pi$
$307$	14.2585	0.813777	0.406889	+	0.913478i	$0.366614\pi$
$308$	0	0
$309$	17.1348	0.974766
$310$	0	0
$311$	17.2118	0.975991	0.487995	−	0.872846i	$-0.337728\pi$
$311$	17.2118	0.975991	0.487995	+	0.872846i	$0.337728\pi$
$312$	0	0
$313$	−7.96400	−0.450152	−0.225076	−	0.974341i	$-0.572263\pi$
$313$	−7.96400	−0.450152	−0.225076	+	0.974341i	$0.572263\pi$
$314$	0	0
$315$	−21.3332	−1.20199
$316$	0	0
$317$	−13.7307	−0.771191	−0.385595	−	0.922668i	$-0.626004\pi$
$317$	−13.7307	−0.771191	−0.385595	+	0.922668i	$0.626004\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−2.79336	−0.155910
$322$	0	0
$323$	50.6330	2.81730
$324$	0	0
$325$	2.90026	0.160877
$326$	0	0
$327$	−18.3056	−1.01230
$328$	0	0
$329$	−35.7580	−1.97140
$330$	0	0
$331$	10.6971	0.587967	0.293983	−	0.955811i	$-0.405019\pi$
$331$	10.6971	0.587967	0.293983	+	0.955811i	$0.405019\pi$
$332$	0	0
$333$	−25.2480	−1.38358
$334$	0	0
$335$	−12.8384	−0.701439
$336$	0	0
$337$	11.4797	0.625338	0.312669	−	0.949862i	$-0.398777\pi$
$337$	11.4797	0.625338	0.312669	+	0.949862i	$0.398777\pi$
$338$	0	0
$339$	28.4426	1.54479
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−30.7401	−1.65981
$344$	0	0
$345$	5.17511	0.278618
$346$	0	0
$347$	−0.423181	−0.0227176	−0.0113588	−	0.999935i	$-0.503616\pi$
$347$	−0.423181	−0.0227176	−0.0113588	+	0.999935i	$0.503616\pi$
$348$	0	0
$349$	0.847568	0.0453693	0.0226846	−	0.999743i	$-0.492779\pi$
$349$	0.847568	0.0453693	0.0226846	+	0.999743i	$0.492779\pi$
$350$	0	0
$351$	−13.5334	−0.722357
$352$	0	0
$353$	9.79855	0.521524	0.260762	−	0.965403i	$-0.416026\pi$
$353$	9.79855	0.521524	0.260762	+	0.965403i	$0.416026\pi$
$354$	0	0
$355$	4.21095	0.223494
$356$	0	0
$357$	−89.4944	−4.73655
$358$	0	0
$359$	7.60668	0.401465	0.200733	−	0.979646i	$-0.435668\pi$
$359$	7.60668	0.401465	0.200733	+	0.979646i	$0.435668\pi$
$360$	0	0
$361$	32.0291	1.68574
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.36576	0.0714870
$366$	0	0
$367$	−1.86780	−0.0974981	−0.0487491	−	0.998811i	$-0.515523\pi$
$367$	−1.86780	−0.0974981	−0.0487491	+	0.998811i	$0.515523\pi$
$368$	0	0
$369$	−21.0187	−1.09419
$370$	0	0
$371$	−3.25440	−0.168960
$372$	0	0
$373$	20.0214	1.03667	0.518335	−	0.855178i	$-0.326552\pi$
$373$	20.0214	1.03667	0.518335	+	0.855178i	$0.326552\pi$
$374$	0	0
$375$	−2.77190	−0.143140
$376$	0	0
$377$	−2.93129	−0.150969
$378$	0	0
$379$	24.0052	1.23307	0.616533	−	0.787329i	$-0.288537\pi$
$379$	24.0052	1.23307	0.616533	+	0.787329i	$0.288537\pi$
$380$	0	0
$381$	−49.1210	−2.51655
$382$	0	0
$383$	19.6767	1.00543	0.502716	−	0.864452i	$-0.332334\pi$
$383$	19.6767	1.00543	0.502716	+	0.864452i	$0.332334\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−31.5195	−1.60223
$388$	0	0
$389$	25.0051	1.26781	0.633904	−	0.773412i	$-0.281451\pi$
$389$	25.0051	1.26781	0.633904	+	0.773412i	$0.281451\pi$
$390$	0	0
$391$	13.2333	0.669235
$392$	0	0
$393$	−39.9149	−2.01344
$394$	0	0
$395$	−12.2375	−0.615734
$396$	0	0
$397$	−19.3120	−0.969240	−0.484620	−	0.874725i	$-0.661042\pi$
$397$	−19.3120	−0.969240	−0.484620	+	0.874725i	$0.661042\pi$
$398$	0	0
$399$	−90.1946	−4.51538
$400$	0	0
$401$	23.2469	1.16090	0.580448	−	0.814297i	$-0.302877\pi$
$401$	23.2469	1.16090	0.580448	+	0.814297i	$0.302877\pi$
$402$	0	0
$403$	12.7911	0.637172
$404$	0	0
$405$	−1.11585	−0.0554470
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−5.92254	−0.292851	−0.146425	−	0.989222i	$-0.546777\pi$
$409$	−5.92254	−0.292851	−0.146425	+	0.989222i	$0.546777\pi$
$410$	0	0
$411$	−12.3895	−0.611131
$412$	0	0
$413$	17.3242	0.852470
$414$	0	0
$415$	−16.3351	−0.801857
$416$	0	0
$417$	−13.4305	−0.657693
$418$	0	0
$419$	−1.11739	−0.0545881	−0.0272940	−	0.999627i	$-0.508689\pi$
$419$	−1.11739	−0.0545881	−0.0272940	+	0.999627i	$0.508689\pi$
$420$	0	0
$421$	17.2344	0.839956	0.419978	−	0.907534i	$-0.362038\pi$
$421$	17.2344	0.839956	0.419978	+	0.907534i	$0.362038\pi$
$422$	0	0
$423$	36.7656	1.78761
$424$	0	0
$425$	−7.08802	−0.343819
$426$	0	0
$427$	−9.63828	−0.466429
$428$	0	0
$429$	0	0
$430$	0	0
$431$	7.74895	0.373254	0.186627	−	0.982431i	$-0.440244\pi$
$431$	7.74895	0.373254	0.186627	+	0.982431i	$0.440244\pi$
$432$	0	0
$433$	3.93864	0.189279	0.0946395	−	0.995512i	$-0.469830\pi$
$433$	3.93864	0.189279	0.0946395	+	0.995512i	$0.469830\pi$
$434$	0	0
$435$	2.80156	0.134324
$436$	0	0
$437$	13.3368	0.637985
$438$	0	0
$439$	−34.7505	−1.65855	−0.829275	−	0.558841i	$-0.811246\pi$
$439$	−34.7505	−1.65855	−0.829275	+	0.558841i	$0.811246\pi$
$440$	0	0
$441$	64.3903	3.06620
$442$	0	0
$443$	24.3227	1.15560	0.577802	−	0.816177i	$-0.303910\pi$
$443$	24.3227	1.15560	0.577802	+	0.816177i	$0.303910\pi$
$444$	0	0
$445$	−11.8165	−0.560155
$446$	0	0
$447$	29.7322	1.40629
$448$	0	0
$449$	−18.9235	−0.893057	−0.446528	−	0.894770i	$-0.647340\pi$
$449$	−18.9235	−0.893057	−0.446528	+	0.894770i	$0.647340\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0.559409	0.0262833
$454$	0	0
$455$	−13.2108	−0.619334
$456$	0	0
$457$	−16.2210	−0.758787	−0.379393	−	0.925235i	$-0.623867\pi$
$457$	−16.2210	−0.758787	−0.379393	+	0.925235i	$0.623867\pi$
$458$	0	0
$459$	33.0745	1.54379
$460$	0	0
$461$	−11.3851	−0.530258	−0.265129	−	0.964213i	$-0.585415\pi$
$461$	−11.3851	−0.530258	−0.265129	+	0.964213i	$0.585415\pi$
$462$	0	0
$463$	−36.4523	−1.69408	−0.847040	−	0.531529i	$-0.821618\pi$
$463$	−36.4523	−1.69408	−0.847040	+	0.531529i	$0.821618\pi$
$464$	0	0
$465$	−12.2250	−0.566922
$466$	0	0
$467$	28.0455	1.29779	0.648895	−	0.760878i	$-0.275232\pi$
$467$	28.0455	1.29779	0.648895	+	0.760878i	$0.275232\pi$
$468$	0	0
$469$	58.4799	2.70035
$470$	0	0
$471$	8.09943	0.373202
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−7.14347	−0.327765
$476$	0	0
$477$	3.34610	0.153208
$478$	0	0
$479$	−36.3433	−1.66057	−0.830284	−	0.557340i	$-0.811822\pi$
$479$	−36.3433	−1.66057	−0.830284	+	0.557340i	$0.811822\pi$
$480$	0	0
$481$	−15.6351	−0.712899
$482$	0	0
$483$	−23.5729	−1.07261
$484$	0	0
$485$	−10.8544	−0.492873
$486$	0	0
$487$	41.8132	1.89474	0.947368	−	0.320146i	$-0.103732\pi$
$487$	41.8132	1.89474	0.947368	+	0.320146i	$0.103732\pi$
$488$	0	0
$489$	−23.5101	−1.06316
$490$	0	0
$491$	19.3555	0.873501	0.436751	−	0.899583i	$-0.356129\pi$
$491$	19.3555	0.873501	0.436751	+	0.899583i	$0.356129\pi$
$492$	0	0
$493$	7.16386	0.322644
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−19.1811	−0.860390
$498$	0	0
$499$	13.3514	0.597691	0.298846	−	0.954301i	$-0.403398\pi$
$499$	13.3514	0.597691	0.298846	+	0.954301i	$0.403398\pi$
$500$	0	0
$501$	27.8969	1.24634
$502$	0	0
$503$	35.8888	1.60020	0.800100	−	0.599866i	$-0.204780\pi$
$503$	35.8888	1.60020	0.800100	+	0.599866i	$0.204780\pi$
$504$	0	0
$505$	12.6665	0.563654
$506$	0	0
$507$	12.7189	0.564865
$508$	0	0
$509$	17.9153	0.794080	0.397040	−	0.917801i	$-0.370037\pi$
$509$	17.9153	0.794080	0.397040	+	0.917801i	$0.370037\pi$
$510$	0	0
$511$	−6.22111	−0.275206
$512$	0	0
$513$	33.3333	1.47170
$514$	0	0
$515$	−6.18162	−0.272395
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−16.0011	−0.702370
$520$	0	0
$521$	7.25941	0.318040	0.159020	−	0.987275i	$-0.449167\pi$
$521$	7.25941	0.318040	0.159020	+	0.987275i	$0.449167\pi$
$522$	0	0
$523$	34.0176	1.48748	0.743742	−	0.668467i	$-0.233049\pi$
$523$	34.0176	1.48748	0.743742	+	0.668467i	$0.233049\pi$
$524$	0	0
$525$	12.6262	0.551051
$526$	0	0
$527$	−31.2606	−1.36173
$528$	0	0
$529$	−19.5143	−0.848450
$530$	0	0
$531$	−17.8124	−0.772994
$532$	0	0
$533$	−13.0161	−0.563788
$534$	0	0
$535$	1.00774	0.0435686
$536$	0	0
$537$	−46.7539	−2.01758
$538$	0	0
$539$	0	0
$540$	0	0
$541$	19.0875	0.820637	0.410319	−	0.911942i	$-0.365417\pi$
$541$	19.0875	0.820637	0.410319	+	0.911942i	$0.365417\pi$
$542$	0	0
$543$	−41.8706	−1.79684
$544$	0	0
$545$	6.60399	0.282884
$546$	0	0
$547$	−23.3090	−0.996621	−0.498311	−	0.866999i	$-0.666046\pi$
$547$	−23.3090	−0.996621	−0.498311	+	0.866999i	$0.666046\pi$
$548$	0	0
$549$	9.90988	0.422944
$550$	0	0
$551$	7.21991	0.307578
$552$	0	0
$553$	55.7424	2.37041
$554$	0	0
$555$	14.9431	0.634300
$556$	0	0
$557$	−28.3838	−1.20266	−0.601329	−	0.799001i	$-0.705362\pi$
$557$	−28.3838	−1.20266	−0.601329	+	0.799001i	$0.705362\pi$
$558$	0	0
$559$	−19.5188	−0.825557
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−8.13215	−0.342729	−0.171365	−	0.985208i	$-0.554818\pi$
$563$	−8.13215	−0.342729	−0.171365	+	0.985208i	$0.554818\pi$
$564$	0	0
$565$	−10.2611	−0.431686
$566$	0	0
$567$	5.08276	0.213456
$568$	0	0
$569$	16.3414	0.685067	0.342534	−	0.939506i	$-0.388715\pi$
$569$	16.3414	0.685067	0.342534	+	0.939506i	$0.388715\pi$
$570$	0	0
$571$	33.1999	1.38937	0.694687	−	0.719312i	$-0.255543\pi$
$571$	33.1999	1.38937	0.694687	+	0.719312i	$0.255543\pi$
$572$	0	0
$573$	−28.9215	−1.20822
$574$	0	0
$575$	−1.86699	−0.0778589
$576$	0	0
$577$	−45.0285	−1.87456	−0.937281	−	0.348576i	$-0.886665\pi$
$577$	−45.0285	−1.87456	−0.937281	+	0.348576i	$0.886665\pi$
$578$	0	0
$579$	−38.5987	−1.60411
$580$	0	0
$581$	74.4072	3.08693
$582$	0	0
$583$	0	0
$584$	0	0
$585$	13.5831	0.561593
$586$	0	0
$587$	−44.6981	−1.84489	−0.922444	−	0.386131i	$-0.873811\pi$
$587$	−44.6981	−1.84489	−0.922444	+	0.386131i	$0.873811\pi$
$588$	0	0
$589$	−31.5052	−1.29815
$590$	0	0
$591$	6.24326	0.256813
$592$	0	0
$593$	−18.7395	−0.769538	−0.384769	−	0.923013i	$-0.625719\pi$
$593$	−18.7395	−0.769538	−0.384769	+	0.923013i	$0.625719\pi$
$594$	0	0
$595$	32.2863	1.32361
$596$	0	0
$597$	−61.3270	−2.50995
$598$	0	0
$599$	−19.5241	−0.797735	−0.398867	−	0.917009i	$-0.630597\pi$
$599$	−19.5241	−0.797735	−0.398867	+	0.917009i	$0.630597\pi$
$600$	0	0
$601$	−31.3146	−1.27735	−0.638674	−	0.769477i	$-0.720517\pi$
$601$	−31.3146	−1.27735	−0.638674	+	0.769477i	$0.720517\pi$
$602$	0	0
$603$	−60.1278	−2.44859
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−22.0334	−0.894309	−0.447154	−	0.894457i	$-0.647563\pi$
$607$	−22.0334	−0.894309	−0.447154	+	0.894457i	$0.647563\pi$
$608$	0	0
$609$	−12.7613	−0.517112
$610$	0	0
$611$	22.7675	0.921075
$612$	0	0
$613$	37.8937	1.53051	0.765256	−	0.643726i	$-0.222612\pi$
$613$	37.8937	1.53051	0.765256	+	0.643726i	$0.222612\pi$
$614$	0	0
$615$	12.4400	0.501629
$616$	0	0
$617$	−34.0287	−1.36995	−0.684973	−	0.728569i	$-0.740186\pi$
$617$	−34.0287	−1.36995	−0.684973	+	0.728569i	$0.740186\pi$
$618$	0	0
$619$	−26.4162	−1.06176	−0.530878	−	0.847448i	$-0.678138\pi$
$619$	−26.4162	−1.06176	−0.530878	+	0.847448i	$0.678138\pi$
$620$	0	0
$621$	8.71187	0.349595
$622$	0	0
$623$	53.8248	2.15645
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	38.2111	1.52357
$630$	0	0
$631$	−38.5953	−1.53645	−0.768226	−	0.640178i	$-0.778861\pi$
$631$	−38.5953	−1.53645	−0.768226	+	0.640178i	$0.778861\pi$
$632$	0	0
$633$	−21.5410	−0.856176
$634$	0	0
$635$	17.7211	0.703239
$636$	0	0
$637$	39.8744	1.57988
$638$	0	0
$639$	19.7216	0.780175
$640$	0	0
$641$	−18.2250	−0.719843	−0.359921	−	0.932983i	$-0.617196\pi$
$641$	−18.2250	−0.719843	−0.359921	+	0.932983i	$0.617196\pi$
$642$	0	0
$643$	10.5456	0.415879	0.207940	−	0.978142i	$-0.433324\pi$
$643$	10.5456	0.415879	0.207940	+	0.978142i	$0.433324\pi$
$644$	0	0
$645$	18.6549	0.734537
$646$	0	0
$647$	16.9165	0.665057	0.332529	−	0.943093i	$-0.392098\pi$
$647$	16.9165	0.665057	0.332529	+	0.943093i	$0.392098\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	55.6857	2.18249
$652$	0	0
$653$	47.9383	1.87597	0.937986	−	0.346673i	$-0.112689\pi$
$653$	47.9383	1.87597	0.937986	+	0.346673i	$0.112689\pi$
$654$	0	0
$655$	14.3999	0.562649
$656$	0	0
$657$	6.39641	0.249548
$658$	0	0
$659$	−5.96793	−0.232477	−0.116239	−	0.993221i	$-0.537084\pi$
$659$	−5.96793	−0.232477	−0.116239	+	0.993221i	$0.537084\pi$
$660$	0	0
$661$	−22.5701	−0.877875	−0.438937	−	0.898518i	$-0.644645\pi$
$661$	−22.5701	−0.877875	−0.438937	+	0.898518i	$0.644645\pi$
$662$	0	0
$663$	56.9821	2.21300
$664$	0	0
$665$	32.5389	1.26181
$666$	0	0
$667$	1.88697	0.0730637
$668$	0	0
$669$	21.9558	0.848858
$670$	0	0
$671$	0	0
$672$	0	0
$673$	6.69924	0.258237	0.129118	−	0.991629i	$-0.458785\pi$
$673$	6.69924	0.258237	0.129118	+	0.991629i	$0.458785\pi$
$674$	0	0
$675$	−4.66626	−0.179605
$676$	0	0
$677$	41.5199	1.59574	0.797870	−	0.602830i	$-0.205960\pi$
$677$	41.5199	1.59574	0.797870	+	0.602830i	$0.205960\pi$
$678$	0	0
$679$	49.4424	1.89743
$680$	0	0
$681$	56.1786	2.15277
$682$	0	0
$683$	−13.4576	−0.514940	−0.257470	−	0.966286i	$-0.582889\pi$
$683$	−13.4576	−0.514940	−0.257470	+	0.966286i	$0.582889\pi$
$684$	0	0
$685$	4.46970	0.170778
$686$	0	0
$687$	−27.3877	−1.04491
$688$	0	0
$689$	2.07211	0.0789411
$690$	0	0
$691$	−29.4815	−1.12153	−0.560765	−	0.827975i	$-0.689493\pi$
$691$	−29.4815	−1.12153	−0.560765	+	0.827975i	$0.689493\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.84523	0.183790
$696$	0	0
$697$	31.8103	1.20490
$698$	0	0
$699$	−27.1609	−1.02732
$700$	0	0
$701$	−9.03034	−0.341071	−0.170536	−	0.985352i	$-0.554550\pi$
$701$	−9.03034	−0.341071	−0.170536	+	0.985352i	$0.554550\pi$
$702$	0	0
$703$	38.5100	1.45243
$704$	0	0
$705$	−21.7599	−0.819524
$706$	0	0
$707$	−57.6969	−2.16991
$708$	0	0
$709$	49.0834	1.84336	0.921682	−	0.387945i	$-0.126815\pi$
$709$	49.0834	1.84336	0.921682	+	0.387945i	$0.126815\pi$
$710$	0	0
$711$	−57.3132	−2.14941
$712$	0	0
$713$	−8.23407	−0.308368
$714$	0	0
$715$	0	0
$716$	0	0
$717$	30.5321	1.14024
$718$	0	0
$719$	−48.7090	−1.81654	−0.908269	−	0.418387i	$-0.862596\pi$
$719$	−48.7090	−1.81654	−0.908269	+	0.418387i	$0.862596\pi$
$720$	0	0
$721$	28.1576	1.04865
$722$	0	0
$723$	36.8961	1.37218
$724$	0	0
$725$	−1.01070	−0.0375365
$726$	0	0
$727$	−50.6964	−1.88023	−0.940113	−	0.340864i	$-0.889281\pi$
$727$	−50.6964	−1.88023	−0.940113	+	0.340864i	$0.889281\pi$
$728$	0	0
$729$	−44.0292	−1.63071
$730$	0	0
$731$	47.7025	1.76434
$732$	0	0
$733$	−47.1585	−1.74184	−0.870920	−	0.491425i	$-0.836476\pi$
$733$	−47.1585	−1.74184	−0.870920	+	0.491425i	$0.836476\pi$
$734$	0	0
$735$	−38.1096	−1.40569
$736$	0	0
$737$	0	0
$738$	0	0
$739$	18.1569	0.667912	0.333956	−	0.942589i	$-0.391616\pi$
$739$	18.1569	0.667912	0.333956	+	0.942589i	$0.391616\pi$
$740$	0	0
$741$	57.4279	2.10967
$742$	0	0
$743$	−3.73625	−0.137070	−0.0685349	−	0.997649i	$-0.521832\pi$
$743$	−3.73625	−0.137070	−0.0685349	+	0.997649i	$0.521832\pi$
$744$	0	0
$745$	−10.7263	−0.392982
$746$	0	0
$747$	−76.5040	−2.79913
$748$	0	0
$749$	−4.59033	−0.167727
$750$	0	0
$751$	−27.8430	−1.01600	−0.508002	−	0.861356i	$-0.669616\pi$
$751$	−27.8430	−1.01600	−0.508002	+	0.861356i	$0.669616\pi$
$752$	0	0
$753$	40.4463	1.47395
$754$	0	0
$755$	−0.201814	−0.00734478
$756$	0	0
$757$	45.0313	1.63669	0.818346	−	0.574726i	$-0.194891\pi$
$757$	45.0313	1.63669	0.818346	+	0.574726i	$0.194891\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−18.7068	−0.678120	−0.339060	−	0.940765i	$-0.610109\pi$
$761$	−18.7068	−0.678120	−0.339060	+	0.940765i	$0.610109\pi$
$762$	0	0
$763$	−30.0816	−1.08903
$764$	0	0
$765$	−33.1961	−1.20021
$766$	0	0
$767$	−11.0305	−0.398290
$768$	0	0
$769$	−13.5701	−0.489351	−0.244675	−	0.969605i	$-0.578681\pi$
$769$	−13.5701	−0.489351	−0.244675	+	0.969605i	$0.578681\pi$
$770$	0	0
$771$	−2.67635	−0.0963862
$772$	0	0
$773$	−3.27790	−0.117898	−0.0589489	−	0.998261i	$-0.518775\pi$
$773$	−3.27790	−0.117898	−0.0589489	+	0.998261i	$0.518775\pi$
$774$	0	0
$775$	4.41034	0.158424
$776$	0	0
$777$	−68.0668	−2.44188
$778$	0	0
$779$	32.0592	1.14864
$780$	0	0
$781$	0	0
$782$	0	0
$783$	4.71619	0.168543
$784$	0	0
$785$	−2.92198	−0.104290
$786$	0	0
$787$	2.54160	0.0905981	0.0452991	−	0.998973i	$-0.485576\pi$
$787$	2.54160	0.0905981	0.0452991	+	0.998973i	$0.485576\pi$
$788$	0	0
$789$	−57.5870	−2.05015
$790$	0	0
$791$	46.7397	1.66187
$792$	0	0
$793$	6.13680	0.217924
$794$	0	0
$795$	−1.98040	−0.0702377
$796$	0	0
$797$	26.2914	0.931290	0.465645	−	0.884972i	$-0.345822\pi$
$797$	26.2914	0.931290	0.465645	+	0.884972i	$0.345822\pi$
$798$	0	0
$799$	−55.6422	−1.96848
$800$	0	0
$801$	−55.3416	−1.95540
$802$	0	0
$803$	0	0
$804$	0	0
$805$	8.50425	0.299736
$806$	0	0
$807$	10.8489	0.381898
$808$	0	0
$809$	−7.94284	−0.279255	−0.139628	−	0.990204i	$-0.544591\pi$
$809$	−7.94284	−0.279255	−0.139628	+	0.990204i	$0.544591\pi$
$810$	0	0
$811$	19.4167	0.681811	0.340906	−	0.940098i	$-0.389266\pi$
$811$	19.4167	0.681811	0.340906	+	0.940098i	$0.389266\pi$
$812$	0	0
$813$	39.5777	1.38805
$814$	0	0
$815$	8.48158	0.297097
$816$	0	0
$817$	48.0757	1.68196
$818$	0	0
$819$	−61.8719	−2.16198
$820$	0	0
$821$	−13.0819	−0.456561	−0.228281	−	0.973595i	$-0.573310\pi$
$821$	−13.0819	−0.456561	−0.228281	+	0.973595i	$0.573310\pi$
$822$	0	0
$823$	31.4670	1.09687	0.548435	−	0.836193i	$-0.315224\pi$
$823$	31.4670	1.09687	0.548435	+	0.836193i	$0.315224\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−6.47448	−0.225140	−0.112570	−	0.993644i	$-0.535908\pi$
$827$	−6.47448	−0.225140	−0.112570	+	0.993644i	$0.535908\pi$
$828$	0	0
$829$	−50.0421	−1.73804	−0.869018	−	0.494781i	$-0.835248\pi$
$829$	−50.0421	−1.73804	−0.869018	+	0.494781i	$0.835248\pi$
$830$	0	0
$831$	−10.4306	−0.361833
$832$	0	0
$833$	−97.4500	−3.37644
$834$	0	0
$835$	−10.0642	−0.348286
$836$	0	0
$837$	−20.5798	−0.711343
$838$	0	0
$839$	10.0716	0.347709	0.173855	−	0.984771i	$-0.444378\pi$
$839$	10.0716	0.347709	0.173855	+	0.984771i	$0.444378\pi$
$840$	0	0
$841$	−27.9785	−0.964775
$842$	0	0
$843$	28.6030	0.985141
$844$	0	0
$845$	−4.58851	−0.157850
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−84.4222	−2.89736
$850$	0	0
$851$	10.0648	0.345018
$852$	0	0
$853$	−24.2805	−0.831348	−0.415674	−	0.909514i	$-0.636454\pi$
$853$	−24.2805	−0.831348	−0.415674	+	0.909514i	$0.636454\pi$
$854$	0	0
$855$	−33.4558	−1.14417
$856$	0	0
$857$	23.8353	0.814200	0.407100	−	0.913384i	$-0.366540\pi$
$857$	23.8353	0.814200	0.407100	+	0.913384i	$0.366540\pi$
$858$	0	0
$859$	27.1693	0.927004	0.463502	−	0.886096i	$-0.346593\pi$
$859$	27.1693	0.927004	0.463502	+	0.886096i	$0.346593\pi$
$860$	0	0
$861$	−56.6649	−1.93114
$862$	0	0
$863$	33.5003	1.14036	0.570182	−	0.821518i	$-0.306873\pi$
$863$	33.5003	1.14036	0.570182	+	0.821518i	$0.306873\pi$
$864$	0	0
$865$	5.77262	0.196275
$866$	0	0
$867$	−92.1378	−3.12917
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−37.2348	−1.26165
$872$	0	0
$873$	−50.8357	−1.72053
$874$	0	0
$875$	−4.55506	−0.153989
$876$	0	0
$877$	−4.10027	−0.138456	−0.0692281	−	0.997601i	$-0.522054\pi$
$877$	−4.10027	−0.138456	−0.0692281	+	0.997601i	$0.522054\pi$
$878$	0	0
$879$	21.9132	0.739113
$880$	0	0
$881$	34.8285	1.17340	0.586701	−	0.809804i	$-0.300426\pi$
$881$	34.8285	1.17340	0.586701	+	0.809804i	$0.300426\pi$
$882$	0	0
$883$	−26.6934	−0.898304	−0.449152	−	0.893455i	$-0.648274\pi$
$883$	−26.6934	−0.898304	−0.449152	+	0.893455i	$0.648274\pi$
$884$	0	0
$885$	10.5424	0.354377
$886$	0	0
$887$	51.4083	1.72612	0.863060	−	0.505101i	$-0.168545\pi$
$887$	51.4083	1.72612	0.863060	+	0.505101i	$0.168545\pi$
$888$	0	0
$889$	−80.7205	−2.70728
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−56.0775	−1.87656
$894$	0	0
$895$	16.8671	0.563805
$896$	0	0
$897$	15.0091	0.501141
$898$	0	0
$899$	−4.45754	−0.148667
$900$	0	0
$901$	−5.06409	−0.168709
$902$	0	0
$903$	−84.9743	−2.82777
$904$	0	0
$905$	15.1054	0.502120
$906$	0	0
$907$	−6.75322	−0.224237	−0.112118	−	0.993695i	$-0.535764\pi$
$907$	−6.75322	−0.224237	−0.112118	+	0.993695i	$0.535764\pi$
$908$	0	0
$909$	59.3227	1.96761
$910$	0	0
$911$	−13.0900	−0.433692	−0.216846	−	0.976206i	$-0.569577\pi$
$911$	−13.0900	−0.433692	−0.216846	+	0.976206i	$0.569577\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−5.86520	−0.193898
$916$	0	0
$917$	−65.5922	−2.16605
$918$	0	0
$919$	0.533564	0.0176007	0.00880033	−	0.999961i	$-0.497199\pi$
$919$	0.533564	0.0176007	0.00880033	+	0.999961i	$0.497199\pi$
$920$	0	0
$921$	−39.5232	−1.30233
$922$	0	0
$923$	12.2128	0.401990
$924$	0	0
$925$	−5.39094	−0.177253
$926$	0	0
$927$	−28.9511	−0.950879
$928$	0	0
$929$	−9.92150	−0.325514	−0.162757	−	0.986666i	$-0.552039\pi$
$929$	−9.92150	−0.325514	−0.162757	+	0.986666i	$0.552039\pi$
$930$	0	0
$931$	−98.2124	−3.21878
$932$	0	0
$933$	−47.7093	−1.56193
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−3.57002	−0.116627	−0.0583137	−	0.998298i	$-0.518572\pi$
$937$	−3.57002	−0.116627	−0.0583137	+	0.998298i	$0.518572\pi$
$938$	0	0
$939$	22.0754	0.720403
$940$	0	0
$941$	−48.8240	−1.59162	−0.795809	−	0.605548i	$-0.792954\pi$
$941$	−48.8240	−1.59162	−0.795809	+	0.605548i	$0.792954\pi$
$942$	0	0
$943$	8.37887	0.272853
$944$	0	0
$945$	21.2551	0.691428
$946$	0	0
$947$	15.1463	0.492190	0.246095	−	0.969246i	$-0.420852\pi$
$947$	15.1463	0.492190	0.246095	+	0.969246i	$0.420852\pi$
$948$	0	0
$949$	3.96105	0.128581
$950$	0	0
$951$	38.0600	1.23418
$952$	0	0
$953$	25.6087	0.829547	0.414773	−	0.909925i	$-0.363861\pi$
$953$	25.6087	0.829547	0.414773	+	0.909925i	$0.363861\pi$
$954$	0	0
$955$	10.4338	0.337631
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−20.3597	−0.657450
$960$	0	0
$961$	−11.5489	−0.372544
$962$	0	0
$963$	4.71969	0.152090
$964$	0	0
$965$	13.9250	0.448262
$966$	0	0
$967$	19.9187	0.640544	0.320272	−	0.947326i	$-0.396226\pi$
$967$	19.9187	0.640544	0.320272	+	0.947326i	$0.396226\pi$
$968$	0	0
$969$	−140.350	−4.50868
$970$	0	0
$971$	−14.3646	−0.460980	−0.230490	−	0.973075i	$-0.574033\pi$
$971$	−14.3646	−0.460980	−0.230490	+	0.973075i	$0.574033\pi$
$972$	0	0
$973$	−22.0703	−0.707541
$974$	0	0
$975$	−8.03922	−0.257461
$976$	0	0
$977$	48.4441	1.54987	0.774933	−	0.632044i	$-0.217784\pi$
$977$	48.4441	1.54987	0.774933	+	0.632044i	$0.217784\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	30.9293	0.987496
$982$	0	0
$983$	10.0332	0.320010	0.160005	−	0.987116i	$-0.448849\pi$
$983$	10.0332	0.320010	0.160005	+	0.987116i	$0.448849\pi$
$984$	0	0
$985$	−2.25234	−0.0717655
$986$	0	0
$987$	99.1175	3.15495
$988$	0	0
$989$	12.5649	0.399540
$990$	0	0
$991$	−4.44920	−0.141333	−0.0706667	−	0.997500i	$-0.522513\pi$
$991$	−4.44920	−0.141333	−0.0706667	+	0.997500i	$0.522513\pi$
$992$	0	0
$993$	−29.6513	−0.940956
$994$	0	0
$995$	22.1246	0.701396
$996$	0	0
$997$	42.4383	1.34404	0.672018	−	0.740535i	$-0.265428\pi$
$997$	42.4383	1.34404	0.672018	+	0.740535i	$0.265428\pi$
$998$	0	0
$999$	25.1555	0.795886

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4840.2.a.bg.1.2		8
4.3	odd	2		9680.2.a.df.1.7		8
11.5	even	5		440.2.y.d.201.1	yes	16
11.9	even	5		440.2.y.d.81.1	✓	16
11.10	odd	2		4840.2.a.bh.1.2		8
44.27	odd	10		880.2.bo.k.641.4		16
44.31	odd	10		880.2.bo.k.81.4		16
44.43	even	2		9680.2.a.de.1.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.y.d.81.1	✓	16	11.9	even	5
440.2.y.d.201.1	yes	16	11.5	even	5
880.2.bo.k.81.4		16	44.31	odd	10
880.2.bo.k.641.4		16	44.27	odd	10
4840.2.a.bg.1.2		8	1.1	even	1	trivial
4840.2.a.bh.1.2		8	11.10	odd	2
9680.2.a.de.1.7		8	44.43	even	2
9680.2.a.df.1.7		8	4.3	odd	2

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4840.a (trivial)

\(f(q)\)	\(=\)	\(q-2.77190 q^{3} +1.00000 q^{5} -4.55506 q^{7} +4.68342 q^{9} +2.90026 q^{13} -2.77190 q^{15} -7.08802 q^{17} -7.14347 q^{19} +12.6262 q^{21} -1.86699 q^{23} +1.00000 q^{25} -4.66626 q^{27} -1.01070 q^{29} +4.41034 q^{31} -4.55506 q^{35} -5.39094 q^{37} -8.03922 q^{39} -4.48790 q^{41} -6.73002 q^{43} +4.68342 q^{45} +7.85017 q^{47} +13.7486 q^{49} +19.6473 q^{51} +0.714458 q^{53} +19.8010 q^{57} -3.80330 q^{59} +2.11595 q^{61} -21.3332 q^{63} +2.90026 q^{65} -12.8384 q^{67} +5.17511 q^{69} +4.21095 q^{71} +1.36576 q^{73} -2.77190 q^{75} -12.2375 q^{79} -1.11585 q^{81} -16.3351 q^{83} -7.08802 q^{85} +2.80156 q^{87} -11.8165 q^{89} -13.2108 q^{91} -12.2250 q^{93} -7.14347 q^{95} -10.8544 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{3} + 8 q^{5} - 6 q^{7} + 19 q^{9} + 12 q^{13} + q^{15} + 2 q^{17} - 6 q^{19} + 6 q^{21} + 10 q^{23} + 8 q^{25} + 13 q^{27} + 8 q^{29} + 19 q^{31} - 6 q^{35} + 12 q^{37} - 21 q^{39} - 3 q^{41}+ \cdots + 21 q^{97}+O(q^{100}) \) 8 * q + q^3 + 8 * q^5 - 6 * q^7 + 19 * q^9 + 12 * q^13 + q^15 + 2 * q^17 - 6 * q^19 + 6 * q^21 + 10 * q^23 + 8 * q^25 + 13 * q^27 + 8 * q^29 + 19 * q^31 - 6 * q^35 + 12 * q^37 - 21 * q^39 - 3 * q^41 - 8 * q^43 + 19 * q^45 + 10 * q^47 + 22 * q^49 - 7 * q^51 + 28 * q^53 + 25 * q^57 + 25 * q^59 + 10 * q^61 - 64 * q^63 + 12 * q^65 - 2 * q^67 + 18 * q^69 + 25 * q^71 + 38 * q^73 + q^75 - 38 * q^79 + 32 * q^81 - 28 * q^83 + 2 * q^85 + 15 * q^87 + 12 * q^89 + 8 * q^91 - 15 * q^93 - 6 * q^95 + 21 * q^97

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