LMFDB - Embedded newform 8470.2.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8470,2,Mod(1,8470)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8470, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8470.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8470.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:	$67.6332905120$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8470.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -3.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -3.00000 q^{9} -1.00000 q^{10} +6.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -6.00000 q^{17} -3.00000 q^{18} -6.00000 q^{19} -1.00000 q^{20} +8.00000 q^{23} +1.00000 q^{25} +6.00000 q^{26} +1.00000 q^{28} +6.00000 q^{29} -8.00000 q^{31} +1.00000 q^{32} -6.00000 q^{34} -1.00000 q^{35} -3.00000 q^{36} -4.00000 q^{37} -6.00000 q^{38} -1.00000 q^{40} -10.0000 q^{41} -4.00000 q^{43} +3.00000 q^{45} +8.00000 q^{46} -10.0000 q^{47} +1.00000 q^{49} +1.00000 q^{50} +6.00000 q^{52} +1.00000 q^{56} +6.00000 q^{58} -4.00000 q^{59} +8.00000 q^{61} -8.00000 q^{62} -3.00000 q^{63} +1.00000 q^{64} -6.00000 q^{65} -6.00000 q^{67} -6.00000 q^{68} -1.00000 q^{70} +8.00000 q^{71} -3.00000 q^{72} -10.0000 q^{73} -4.00000 q^{74} -6.00000 q^{76} -2.00000 q^{79} -1.00000 q^{80} +9.00000 q^{81} -10.0000 q^{82} +4.00000 q^{83} +6.00000 q^{85} -4.00000 q^{86} +6.00000 q^{89} +3.00000 q^{90} +6.00000 q^{91} +8.00000 q^{92} -10.0000 q^{94} +6.00000 q^{95} +16.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	−3.00000	−1.00000
$10$	−1.00000	−0.316228
$11$	0	0
$11$	0	0
$12$	0	0
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	−3.00000	−0.707107
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	0	0
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	6.00000	1.17670
$27$	0	0
$28$	1.00000	0.188982
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−6.00000	−1.02899
$35$	−1.00000	−0.169031
$36$	−3.00000	−0.500000
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	−6.00000	−0.973329
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	3.00000	0.447214
$46$	8.00000	1.17954
$47$	−10.0000	−1.45865	−0.729325	−	0.684167i	$-0.760166\pi$
$47$	−10.0000	−1.45865	−0.729325	+	0.684167i	$0.760166\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	0	0
$52$	6.00000	0.832050
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	1.00000	0.133631
$57$	0	0
$58$	6.00000	0.787839
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	−8.00000	−1.01600
$63$	−3.00000	−0.377964
$64$	1.00000	0.125000
$65$	−6.00000	−0.744208
$66$	0	0
$67$	−6.00000	−0.733017	−0.366508	−	0.930415i	$-0.619447\pi$
$67$	−6.00000	−0.733017	−0.366508	+	0.930415i	$0.619447\pi$
$68$	−6.00000	−0.727607
$69$	0	0
$70$	−1.00000	−0.119523
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	−3.00000	−0.353553
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	−4.00000	−0.464991
$75$	0	0
$76$	−6.00000	−0.688247
$77$	0	0
$78$	0	0
$79$	−2.00000	−0.225018	−0.112509	−	0.993651i	$-0.535889\pi$
$79$	−2.00000	−0.225018	−0.112509	+	0.993651i	$0.535889\pi$
$80$	−1.00000	−0.111803
$81$	9.00000	1.00000
$82$	−10.0000	−1.10432
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	6.00000	0.650791
$86$	−4.00000	−0.431331
$87$	0	0
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	3.00000	0.316228
$91$	6.00000	0.628971
$92$	8.00000	0.834058
$93$	0	0
$94$	−10.0000	−1.03142
$95$	6.00000	0.615587
$96$	0	0
$97$	16.0000	1.62455	0.812277	−	0.583272i	$-0.198228\pi$
$97$	16.0000	1.62455	0.812277	+	0.583272i	$0.198228\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−4.00000	−0.398015	−0.199007	−	0.979998i	$-0.563772\pi$
$101$	−4.00000	−0.398015	−0.199007	+	0.979998i	$0.563772\pi$
$102$	0	0
$103$	10.0000	0.985329	0.492665	−	0.870219i	$-0.336023\pi$
$103$	10.0000	0.985329	0.492665	+	0.870219i	$0.336023\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	0	0
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	0	0
$112$	1.00000	0.0944911
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	−8.00000	−0.746004
$116$	6.00000	0.557086
$117$	−18.0000	−1.66410
$118$	−4.00000	−0.368230
$119$	−6.00000	−0.550019
$120$	0	0
$121$	0	0
$122$	8.00000	0.724286
$123$	0	0
$124$	−8.00000	−0.718421
$125$	−1.00000	−0.0894427
$126$	−3.00000	−0.267261
$127$	−20.0000	−1.77471	−0.887357	−	0.461084i	$-0.847461\pi$
$127$	−20.0000	−1.77471	−0.887357	+	0.461084i	$0.847461\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−6.00000	−0.526235
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	−6.00000	−0.520266
$134$	−6.00000	−0.518321
$135$	0	0
$136$	−6.00000	−0.514496
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	2.00000	0.169638	0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000	0.169638	0.0848189	+	0.996396i	$0.472969\pi$
$140$	−1.00000	−0.0845154
$141$	0	0
$142$	8.00000	0.671345
$143$	0	0
$144$	−3.00000	−0.250000
$145$	−6.00000	−0.498273
$146$	−10.0000	−0.827606
$147$	0	0
$148$	−4.00000	−0.328798
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	10.0000	0.813788	0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000	0.813788	0.406894	+	0.913475i	$0.366612\pi$
$152$	−6.00000	−0.486664
$153$	18.0000	1.45521
$154$	0	0
$155$	8.00000	0.642575
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	−2.00000	−0.159111
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	8.00000	0.630488
$162$	9.00000	0.707107
$163$	−22.0000	−1.72317	−0.861586	−	0.507611i	$-0.830529\pi$
$163$	−22.0000	−1.72317	−0.861586	+	0.507611i	$0.830529\pi$
$164$	−10.0000	−0.780869
$165$	0	0
$166$	4.00000	0.310460
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	6.00000	0.460179
$171$	18.0000	1.37649
$172$	−4.00000	−0.304997
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	0	0
$178$	6.00000	0.449719
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	3.00000	0.223607
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	6.00000	0.444750
$183$	0	0
$184$	8.00000	0.589768
$185$	4.00000	0.294086
$186$	0	0
$187$	0	0
$188$	−10.0000	−0.729325
$189$	0	0
$190$	6.00000	0.435286
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	16.0000	1.14873
$195$	0	0
$196$	1.00000	0.0714286
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	0	0
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−4.00000	−0.281439
$203$	6.00000	0.421117
$204$	0	0
$205$	10.0000	0.698430
$206$	10.0000	0.696733
$207$	−24.0000	−1.66812
$208$	6.00000	0.416025
$209$	0	0
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	0	0
$213$	0	0
$214$	−4.00000	−0.273434
$215$	4.00000	0.272798
$216$	0	0
$217$	−8.00000	−0.543075
$218$	2.00000	0.135457
$219$	0	0
$220$	0	0
$221$	−36.0000	−2.42162
$222$	0	0
$223$	−10.0000	−0.669650	−0.334825	−	0.942280i	$-0.608677\pi$
$223$	−10.0000	−0.669650	−0.334825	+	0.942280i	$0.608677\pi$
$224$	1.00000	0.0668153
$225$	−3.00000	−0.200000
$226$	−2.00000	−0.133038
$227$	8.00000	0.530979	0.265489	−	0.964114i	$-0.414466\pi$
$227$	8.00000	0.530979	0.265489	+	0.964114i	$0.414466\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	−8.00000	−0.527504
$231$	0	0
$232$	6.00000	0.393919
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	−18.0000	−1.17670
$235$	10.0000	0.652328
$236$	−4.00000	−0.260378
$237$	0	0
$238$	−6.00000	−0.388922
$239$	10.0000	0.646846	0.323423	−	0.946254i	$-0.395166\pi$
$239$	10.0000	0.646846	0.323423	+	0.946254i	$0.395166\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	0	0
$244$	8.00000	0.512148
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−36.0000	−2.29063
$248$	−8.00000	−0.508001
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	−3.00000	−0.188982
$253$	0	0
$254$	−20.0000	−1.25491
$255$	0	0
$256$	1.00000	0.0625000
$257$	12.0000	0.748539	0.374270	−	0.927320i	$-0.377893\pi$
$257$	12.0000	0.748539	0.374270	+	0.927320i	$0.377893\pi$
$258$	0	0
$259$	−4.00000	−0.248548
$260$	−6.00000	−0.372104
$261$	−18.0000	−1.11417
$262$	−6.00000	−0.370681
$263$	−4.00000	−0.246651	−0.123325	−	0.992366i	$-0.539356\pi$
$263$	−4.00000	−0.246651	−0.123325	+	0.992366i	$0.539356\pi$
$264$	0	0
$265$	0	0
$266$	−6.00000	−0.367884
$267$	0	0
$268$	−6.00000	−0.366508
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	−6.00000	−0.363803
$273$	0	0
$274$	−18.0000	−1.08742
$275$	0	0
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	2.00000	0.119952
$279$	24.0000	1.43684
$280$	−1.00000	−0.0597614
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	−24.0000	−1.42665	−0.713326	−	0.700832i	$-0.752812\pi$
$283$	−24.0000	−1.42665	−0.713326	+	0.700832i	$0.752812\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	0	0
$287$	−10.0000	−0.590281
$288$	−3.00000	−0.176777
$289$	19.0000	1.11765
$290$	−6.00000	−0.352332
$291$	0	0
$292$	−10.0000	−0.585206
$293$	22.0000	1.28525	0.642627	−	0.766179i	$-0.277845\pi$
$293$	22.0000	1.28525	0.642627	+	0.766179i	$0.277845\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	−4.00000	−0.232495
$297$	0	0
$298$	−6.00000	−0.347571
$299$	48.0000	2.77591
$300$	0	0
$301$	−4.00000	−0.230556
$302$	10.0000	0.575435
$303$	0	0
$304$	−6.00000	−0.344124
$305$	−8.00000	−0.458079
$306$	18.0000	1.02899
$307$	16.0000	0.913168	0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000	0.913168	0.456584	+	0.889680i	$0.349073\pi$
$308$	0	0
$309$	0	0
$310$	8.00000	0.454369
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	28.0000	1.58265	0.791327	−	0.611393i	$-0.209391\pi$
$313$	28.0000	1.58265	0.791327	+	0.611393i	$0.209391\pi$
$314$	−10.0000	−0.564333
$315$	3.00000	0.169031
$316$	−2.00000	−0.112509
$317$	−8.00000	−0.449325	−0.224662	−	0.974437i	$-0.572128\pi$
$317$	−8.00000	−0.449325	−0.224662	+	0.974437i	$0.572128\pi$
$318$	0	0
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	8.00000	0.445823
$323$	36.0000	2.00309
$324$	9.00000	0.500000
$325$	6.00000	0.332820
$326$	−22.0000	−1.21847
$327$	0	0
$328$	−10.0000	−0.552158
$329$	−10.0000	−0.551318
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	4.00000	0.219529
$333$	12.0000	0.657596
$334$	−8.00000	−0.437741
$335$	6.00000	0.327815
$336$	0	0
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	23.0000	1.25104
$339$	0	0
$340$	6.00000	0.325396
$341$	0	0
$342$	18.0000	0.973329
$343$	1.00000	0.0539949
$344$	−4.00000	−0.215666
$345$	0	0
$346$	−14.0000	−0.752645
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	12.0000	0.642345	0.321173	−	0.947021i	$-0.395923\pi$
$349$	12.0000	0.642345	0.321173	+	0.947021i	$0.395923\pi$
$350$	1.00000	0.0534522
$351$	0	0
$352$	0	0
$353$	−4.00000	−0.212899	−0.106449	−	0.994318i	$-0.533948\pi$
$353$	−4.00000	−0.212899	−0.106449	+	0.994318i	$0.533948\pi$
$354$	0	0
$355$	−8.00000	−0.424596
$356$	6.00000	0.317999
$357$	0	0
$358$	0	0
$359$	14.0000	0.738892	0.369446	−	0.929252i	$-0.379548\pi$
$359$	14.0000	0.738892	0.369446	+	0.929252i	$0.379548\pi$
$360$	3.00000	0.158114
$361$	17.0000	0.894737
$362$	−22.0000	−1.15629
$363$	0	0
$364$	6.00000	0.314485
$365$	10.0000	0.523424
$366$	0	0
$367$	2.00000	0.104399	0.0521996	−	0.998637i	$-0.483377\pi$
$367$	2.00000	0.104399	0.0521996	+	0.998637i	$0.483377\pi$
$368$	8.00000	0.417029
$369$	30.0000	1.56174
$370$	4.00000	0.207950
$371$	0	0
$372$	0	0
$373$	−18.0000	−0.932005	−0.466002	−	0.884783i	$-0.654306\pi$
$373$	−18.0000	−0.932005	−0.466002	+	0.884783i	$0.654306\pi$
$374$	0	0
$375$	0	0
$376$	−10.0000	−0.515711
$377$	36.0000	1.85409
$378$	0	0
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	6.00000	0.307794
$381$	0	0
$382$	−8.00000	−0.409316
$383$	−38.0000	−1.94171	−0.970855	−	0.239669i	$-0.922961\pi$
$383$	−38.0000	−1.94171	−0.970855	+	0.239669i	$0.922961\pi$
$384$	0	0
$385$	0	0
$386$	−10.0000	−0.508987
$387$	12.0000	0.609994
$388$	16.0000	0.812277
$389$	−38.0000	−1.92668	−0.963338	−	0.268290i	$-0.913542\pi$
$389$	−38.0000	−1.92668	−0.963338	+	0.268290i	$0.913542\pi$
$390$	0	0
$391$	−48.0000	−2.42746
$392$	1.00000	0.0505076
$393$	0	0
$394$	−18.0000	−0.906827
$395$	2.00000	0.100631
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	−20.0000	−1.00251
$399$	0	0
$400$	1.00000	0.0500000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	−48.0000	−2.39105
$404$	−4.00000	−0.199007
$405$	−9.00000	−0.447214
$406$	6.00000	0.297775
$407$	0	0
$408$	0	0
$409$	−30.0000	−1.48340	−0.741702	−	0.670729i	$-0.765981\pi$
$409$	−30.0000	−1.48340	−0.741702	+	0.670729i	$0.765981\pi$
$410$	10.0000	0.493865
$411$	0	0
$412$	10.0000	0.492665
$413$	−4.00000	−0.196827
$414$	−24.0000	−1.17954
$415$	−4.00000	−0.196352
$416$	6.00000	0.294174
$417$	0	0
$418$	0	0
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	0	0
$421$	−34.0000	−1.65706	−0.828529	−	0.559946i	$-0.810822\pi$
$421$	−34.0000	−1.65706	−0.828529	+	0.559946i	$0.810822\pi$
$422$	20.0000	0.973585
$423$	30.0000	1.45865
$424$	0	0
$425$	−6.00000	−0.291043
$426$	0	0
$427$	8.00000	0.387147
$428$	−4.00000	−0.193347
$429$	0	0
$430$	4.00000	0.192897
$431$	−18.0000	−0.867029	−0.433515	−	0.901146i	$-0.642727\pi$
$431$	−18.0000	−0.867029	−0.433515	+	0.901146i	$0.642727\pi$
$432$	0	0
$433$	4.00000	0.192228	0.0961139	−	0.995370i	$-0.469359\pi$
$433$	4.00000	0.192228	0.0961139	+	0.995370i	$0.469359\pi$
$434$	−8.00000	−0.384012
$435$	0	0
$436$	2.00000	0.0957826
$437$	−48.0000	−2.29615
$438$	0	0
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	−36.0000	−1.71235
$443$	−6.00000	−0.285069	−0.142534	−	0.989790i	$-0.545525\pi$
$443$	−6.00000	−0.285069	−0.142534	+	0.989790i	$0.545525\pi$
$444$	0	0
$445$	−6.00000	−0.284427
$446$	−10.0000	−0.473514
$447$	0	0
$448$	1.00000	0.0472456
$449$	−34.0000	−1.60456	−0.802280	−	0.596948i	$-0.796380\pi$
$449$	−34.0000	−1.60456	−0.802280	+	0.596948i	$0.796380\pi$
$450$	−3.00000	−0.141421
$451$	0	0
$452$	−2.00000	−0.0940721
$453$	0	0
$454$	8.00000	0.375459
$455$	−6.00000	−0.281284
$456$	0	0
$457$	34.0000	1.59045	0.795226	−	0.606313i	$-0.207352\pi$
$457$	34.0000	1.59045	0.795226	+	0.606313i	$0.207352\pi$
$458$	−6.00000	−0.280362
$459$	0	0
$460$	−8.00000	−0.373002
$461$	16.0000	0.745194	0.372597	−	0.927993i	$-0.378467\pi$
$461$	16.0000	0.745194	0.372597	+	0.927993i	$0.378467\pi$
$462$	0	0
$463$	32.0000	1.48717	0.743583	−	0.668644i	$-0.233125\pi$
$463$	32.0000	1.48717	0.743583	+	0.668644i	$0.233125\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	18.0000	0.833834
$467$	−36.0000	−1.66588	−0.832941	−	0.553362i	$-0.813345\pi$
$467$	−36.0000	−1.66588	−0.832941	+	0.553362i	$0.813345\pi$
$468$	−18.0000	−0.832050
$469$	−6.00000	−0.277054
$470$	10.0000	0.461266
$471$	0	0
$472$	−4.00000	−0.184115
$473$	0	0
$474$	0	0
$475$	−6.00000	−0.275299
$476$	−6.00000	−0.275010
$477$	0	0
$478$	10.0000	0.457389
$479$	20.0000	0.913823	0.456912	−	0.889512i	$-0.348956\pi$
$479$	20.0000	0.913823	0.456912	+	0.889512i	$0.348956\pi$
$480$	0	0
$481$	−24.0000	−1.09431
$482$	−10.0000	−0.455488
$483$	0	0
$484$	0	0
$485$	−16.0000	−0.726523
$486$	0	0
$487$	−20.0000	−0.906287	−0.453143	−	0.891438i	$-0.649697\pi$
$487$	−20.0000	−0.906287	−0.453143	+	0.891438i	$0.649697\pi$
$488$	8.00000	0.362143
$489$	0	0
$490$	−1.00000	−0.0451754
$491$	−32.0000	−1.44414	−0.722070	−	0.691820i	$-0.756809\pi$
$491$	−32.0000	−1.44414	−0.722070	+	0.691820i	$0.756809\pi$
$492$	0	0
$493$	−36.0000	−1.62136
$494$	−36.0000	−1.61972
$495$	0	0
$496$	−8.00000	−0.359211
$497$	8.00000	0.358849
$498$	0	0
$499$	36.0000	1.61158	0.805791	−	0.592200i	$-0.201741\pi$
$499$	36.0000	1.61158	0.805791	+	0.592200i	$0.201741\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−12.0000	−0.535586
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	−3.00000	−0.133631
$505$	4.00000	0.177998
$506$	0	0
$507$	0	0
$508$	−20.0000	−0.887357
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	−10.0000	−0.442374
$512$	1.00000	0.0441942
$513$	0	0
$514$	12.0000	0.529297
$515$	−10.0000	−0.440653
$516$	0	0
$517$	0	0
$518$	−4.00000	−0.175750
$519$	0	0
$520$	−6.00000	−0.263117
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	−18.0000	−0.787839
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	−6.00000	−0.262111
$525$	0	0
$526$	−4.00000	−0.174408
$527$	48.0000	2.09091
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	12.0000	0.520756
$532$	−6.00000	−0.260133
$533$	−60.0000	−2.59889
$534$	0	0
$535$	4.00000	0.172935
$536$	−6.00000	−0.259161
$537$	0	0
$538$	−18.0000	−0.776035
$539$	0	0
$540$	0	0
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	20.0000	0.859074
$543$	0	0
$544$	−6.00000	−0.257248
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	−18.0000	−0.768922
$549$	−24.0000	−1.02430
$550$	0	0
$551$	−36.0000	−1.53365
$552$	0	0
$553$	−2.00000	−0.0850487
$554$	10.0000	0.424859
$555$	0	0
$556$	2.00000	0.0848189
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	24.0000	1.01600
$559$	−24.0000	−1.01509
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	0	0
$563$	−36.0000	−1.51722	−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000	−1.51722	−0.758610	+	0.651546i	$0.774121\pi$
$564$	0	0
$565$	2.00000	0.0841406
$566$	−24.0000	−1.00880
$567$	9.00000	0.377964
$568$	8.00000	0.335673
$569$	40.0000	1.67689	0.838444	−	0.544988i	$-0.183466\pi$
$569$	40.0000	1.67689	0.838444	+	0.544988i	$0.183466\pi$
$570$	0	0
$571$	36.0000	1.50655	0.753277	−	0.657704i	$-0.228472\pi$
$571$	36.0000	1.50655	0.753277	+	0.657704i	$0.228472\pi$
$572$	0	0
$573$	0	0
$574$	−10.0000	−0.417392
$575$	8.00000	0.333623
$576$	−3.00000	−0.125000
$577$	32.0000	1.33218	0.666089	−	0.745873i	$-0.267967\pi$
$577$	32.0000	1.33218	0.666089	+	0.745873i	$0.267967\pi$
$578$	19.0000	0.790296
$579$	0	0
$580$	−6.00000	−0.249136
$581$	4.00000	0.165948
$582$	0	0
$583$	0	0
$584$	−10.0000	−0.413803
$585$	18.0000	0.744208
$586$	22.0000	0.908812
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	48.0000	1.97781
$590$	4.00000	0.164677
$591$	0	0
$592$	−4.00000	−0.164399
$593$	14.0000	0.574911	0.287456	−	0.957794i	$-0.407191\pi$
$593$	14.0000	0.574911	0.287456	+	0.957794i	$0.407191\pi$
$594$	0	0
$595$	6.00000	0.245976
$596$	−6.00000	−0.245770
$597$	0	0
$598$	48.0000	1.96287
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	42.0000	1.71322	0.856608	−	0.515968i	$-0.172568\pi$
$601$	42.0000	1.71322	0.856608	+	0.515968i	$0.172568\pi$
$602$	−4.00000	−0.163028
$603$	18.0000	0.733017
$604$	10.0000	0.406894
$605$	0	0
$606$	0	0
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	−6.00000	−0.243332
$609$	0	0
$610$	−8.00000	−0.323911
$611$	−60.0000	−2.42734
$612$	18.0000	0.727607
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	16.0000	0.645707
$615$	0	0
$616$	0	0
$617$	−38.0000	−1.52982	−0.764911	−	0.644136i	$-0.777217\pi$
$617$	−38.0000	−1.52982	−0.764911	+	0.644136i	$0.777217\pi$
$618$	0	0
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	8.00000	0.321288
$621$	0	0
$622$	−12.0000	−0.481156
$623$	6.00000	0.240385
$624$	0	0
$625$	1.00000	0.0400000
$626$	28.0000	1.11911
$627$	0	0
$628$	−10.0000	−0.399043
$629$	24.0000	0.956943
$630$	3.00000	0.119523
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	−2.00000	−0.0795557
$633$	0	0
$634$	−8.00000	−0.317721
$635$	20.0000	0.793676
$636$	0	0
$637$	6.00000	0.237729
$638$	0	0
$639$	−24.0000	−0.949425
$640$	−1.00000	−0.0395285
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	0	0
$643$	−16.0000	−0.630978	−0.315489	−	0.948929i	$-0.602169\pi$
$643$	−16.0000	−0.630978	−0.315489	+	0.948929i	$0.602169\pi$
$644$	8.00000	0.315244
$645$	0	0
$646$	36.0000	1.41640
$647$	46.0000	1.80845	0.904223	−	0.427060i	$-0.140451\pi$
$647$	46.0000	1.80845	0.904223	+	0.427060i	$0.140451\pi$
$648$	9.00000	0.353553
$649$	0	0
$650$	6.00000	0.235339
$651$	0	0
$652$	−22.0000	−0.861586
$653$	−24.0000	−0.939193	−0.469596	−	0.882881i	$-0.655601\pi$
$653$	−24.0000	−0.939193	−0.469596	+	0.882881i	$0.655601\pi$
$654$	0	0
$655$	6.00000	0.234439
$656$	−10.0000	−0.390434
$657$	30.0000	1.17041
$658$	−10.0000	−0.389841
$659$	24.0000	0.934907	0.467454	−	0.884018i	$-0.345171\pi$
$659$	24.0000	0.934907	0.467454	+	0.884018i	$0.345171\pi$
$660$	0	0
$661$	−26.0000	−1.01128	−0.505641	−	0.862744i	$-0.668744\pi$
$661$	−26.0000	−1.01128	−0.505641	+	0.862744i	$0.668744\pi$
$662$	−4.00000	−0.155464
$663$	0	0
$664$	4.00000	0.155230
$665$	6.00000	0.232670
$666$	12.0000	0.464991
$667$	48.0000	1.85857
$668$	−8.00000	−0.309529
$669$	0	0
$670$	6.00000	0.231800
$671$	0	0
$672$	0	0
$673$	−34.0000	−1.31060	−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000	−1.31060	−0.655302	+	0.755367i	$0.727459\pi$
$674$	−2.00000	−0.0770371
$675$	0	0
$676$	23.0000	0.884615
$677$	30.0000	1.15299	0.576497	−	0.817099i	$-0.304419\pi$
$677$	30.0000	1.15299	0.576497	+	0.817099i	$0.304419\pi$
$678$	0	0
$679$	16.0000	0.614024
$680$	6.00000	0.230089
$681$	0	0
$682$	0	0
$683$	42.0000	1.60709	0.803543	−	0.595247i	$-0.202946\pi$
$683$	42.0000	1.60709	0.803543	+	0.595247i	$0.202946\pi$
$684$	18.0000	0.688247
$685$	18.0000	0.687745
$686$	1.00000	0.0381802
$687$	0	0
$688$	−4.00000	−0.152499
$689$	0	0
$690$	0	0
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	−14.0000	−0.532200
$693$	0	0
$694$	12.0000	0.455514
$695$	−2.00000	−0.0758643
$696$	0	0
$697$	60.0000	2.27266
$698$	12.0000	0.454207
$699$	0	0
$700$	1.00000	0.0377964
$701$	−22.0000	−0.830929	−0.415464	−	0.909610i	$-0.636381\pi$
$701$	−22.0000	−0.830929	−0.415464	+	0.909610i	$0.636381\pi$
$702$	0	0
$703$	24.0000	0.905177
$704$	0	0
$705$	0	0
$706$	−4.00000	−0.150542
$707$	−4.00000	−0.150435
$708$	0	0
$709$	34.0000	1.27690	0.638448	−	0.769665i	$-0.279577\pi$
$709$	34.0000	1.27690	0.638448	+	0.769665i	$0.279577\pi$
$710$	−8.00000	−0.300235
$711$	6.00000	0.225018
$712$	6.00000	0.224860
$713$	−64.0000	−2.39682
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	14.0000	0.522475
$719$	−40.0000	−1.49175	−0.745874	−	0.666087i	$-0.767968\pi$
$719$	−40.0000	−1.49175	−0.745874	+	0.666087i	$0.767968\pi$
$720$	3.00000	0.111803
$721$	10.0000	0.372419
$722$	17.0000	0.632674
$723$	0	0
$724$	−22.0000	−0.817624
$725$	6.00000	0.222834
$726$	0	0
$727$	−6.00000	−0.222528	−0.111264	−	0.993791i	$-0.535490\pi$
$727$	−6.00000	−0.222528	−0.111264	+	0.993791i	$0.535490\pi$
$728$	6.00000	0.222375
$729$	−27.0000	−1.00000
$730$	10.0000	0.370117
$731$	24.0000	0.887672
$732$	0	0
$733$	34.0000	1.25582	0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000	1.25582	0.627909	+	0.778287i	$0.283911\pi$
$734$	2.00000	0.0738213
$735$	0	0
$736$	8.00000	0.294884
$737$	0	0
$738$	30.0000	1.10432
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	4.00000	0.147043
$741$	0	0
$742$	0	0
$743$	−28.0000	−1.02722	−0.513610	−	0.858024i	$-0.671692\pi$
$743$	−28.0000	−1.02722	−0.513610	+	0.858024i	$0.671692\pi$
$744$	0	0
$745$	6.00000	0.219823
$746$	−18.0000	−0.659027
$747$	−12.0000	−0.439057
$748$	0	0
$749$	−4.00000	−0.146157
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	−10.0000	−0.364662
$753$	0	0
$754$	36.0000	1.31104
$755$	−10.0000	−0.363937
$756$	0	0
$757$	44.0000	1.59921	0.799604	−	0.600528i	$-0.205043\pi$
$757$	44.0000	1.59921	0.799604	+	0.600528i	$0.205043\pi$
$758$	16.0000	0.581146
$759$	0	0
$760$	6.00000	0.217643
$761$	−46.0000	−1.66750	−0.833749	−	0.552143i	$-0.813810\pi$
$761$	−46.0000	−1.66750	−0.833749	+	0.552143i	$0.813810\pi$
$762$	0	0
$763$	2.00000	0.0724049
$764$	−8.00000	−0.289430
$765$	−18.0000	−0.650791
$766$	−38.0000	−1.37300
$767$	−24.0000	−0.866590
$768$	0	0
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	0	0
$771$	0	0
$772$	−10.0000	−0.359908
$773$	46.0000	1.65451	0.827253	−	0.561830i	$-0.189903\pi$
$773$	46.0000	1.65451	0.827253	+	0.561830i	$0.189903\pi$
$774$	12.0000	0.431331
$775$	−8.00000	−0.287368
$776$	16.0000	0.574367
$777$	0	0
$778$	−38.0000	−1.36237
$779$	60.0000	2.14972
$780$	0	0
$781$	0	0
$782$	−48.0000	−1.71648
$783$	0	0
$784$	1.00000	0.0357143
$785$	10.0000	0.356915
$786$	0	0
$787$	4.00000	0.142585	0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000	0.142585	0.0712923	+	0.997455i	$0.477288\pi$
$788$	−18.0000	−0.641223
$789$	0	0
$790$	2.00000	0.0711568
$791$	−2.00000	−0.0711118
$792$	0	0
$793$	48.0000	1.70453
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	−20.0000	−0.708881
$797$	22.0000	0.779280	0.389640	−	0.920967i	$-0.372599\pi$
$797$	22.0000	0.779280	0.389640	+	0.920967i	$0.372599\pi$
$798$	0	0
$799$	60.0000	2.12265
$800$	1.00000	0.0353553
$801$	−18.0000	−0.635999
$802$	−18.0000	−0.635602
$803$	0	0
$804$	0	0
$805$	−8.00000	−0.281963
$806$	−48.0000	−1.69073
$807$	0	0
$808$	−4.00000	−0.140720
$809$	16.0000	0.562530	0.281265	−	0.959630i	$-0.409246\pi$
$809$	16.0000	0.562530	0.281265	+	0.959630i	$0.409246\pi$
$810$	−9.00000	−0.316228
$811$	−26.0000	−0.912983	−0.456492	−	0.889728i	$-0.650894\pi$
$811$	−26.0000	−0.912983	−0.456492	+	0.889728i	$0.650894\pi$
$812$	6.00000	0.210559
$813$	0	0
$814$	0	0
$815$	22.0000	0.770626
$816$	0	0
$817$	24.0000	0.839654
$818$	−30.0000	−1.04893
$819$	−18.0000	−0.628971
$820$	10.0000	0.349215
$821$	−22.0000	−0.767805	−0.383903	−	0.923374i	$-0.625420\pi$
$821$	−22.0000	−0.767805	−0.383903	+	0.923374i	$0.625420\pi$
$822$	0	0
$823$	44.0000	1.53374	0.766872	−	0.641800i	$-0.221812\pi$
$823$	44.0000	1.53374	0.766872	+	0.641800i	$0.221812\pi$
$824$	10.0000	0.348367
$825$	0	0
$826$	−4.00000	−0.139178
$827$	20.0000	0.695468	0.347734	−	0.937593i	$-0.386951\pi$
$827$	20.0000	0.695468	0.347734	+	0.937593i	$0.386951\pi$
$828$	−24.0000	−0.834058
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	−4.00000	−0.138842
$831$	0	0
$832$	6.00000	0.208013
$833$	−6.00000	−0.207888
$834$	0	0
$835$	8.00000	0.276851
$836$	0	0
$837$	0	0
$838$	−20.0000	−0.690889
$839$	−24.0000	−0.828572	−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000	−0.828572	−0.414286	+	0.910147i	$0.635969\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−34.0000	−1.17172
$843$	0	0
$844$	20.0000	0.688428
$845$	−23.0000	−0.791224
$846$	30.0000	1.03142
$847$	0	0
$848$	0	0
$849$	0	0
$850$	−6.00000	−0.205798
$851$	−32.0000	−1.09695
$852$	0	0
$853$	−22.0000	−0.753266	−0.376633	−	0.926363i	$-0.622918\pi$
$853$	−22.0000	−0.753266	−0.376633	+	0.926363i	$0.622918\pi$
$854$	8.00000	0.273754
$855$	−18.0000	−0.615587
$856$	−4.00000	−0.136717
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	0	0
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	4.00000	0.136399
$861$	0	0
$862$	−18.0000	−0.613082
$863$	−4.00000	−0.136162	−0.0680808	−	0.997680i	$-0.521688\pi$
$863$	−4.00000	−0.136162	−0.0680808	+	0.997680i	$0.521688\pi$
$864$	0	0
$865$	14.0000	0.476014
$866$	4.00000	0.135926
$867$	0	0
$868$	−8.00000	−0.271538
$869$	0	0
$870$	0	0
$871$	−36.0000	−1.21981
$872$	2.00000	0.0677285
$873$	−48.0000	−1.62455
$874$	−48.0000	−1.62362
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	46.0000	1.55331	0.776655	−	0.629926i	$-0.216915\pi$
$877$	46.0000	1.55331	0.776655	+	0.629926i	$0.216915\pi$
$878$	28.0000	0.944954
$879$	0	0
$880$	0	0
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	−3.00000	−0.101015
$883$	54.0000	1.81724	0.908622	−	0.417619i	$-0.137135\pi$
$883$	54.0000	1.81724	0.908622	+	0.417619i	$0.137135\pi$
$884$	−36.0000	−1.21081
$885$	0	0
$886$	−6.00000	−0.201574
$887$	−32.0000	−1.07445	−0.537227	−	0.843437i	$-0.680528\pi$
$887$	−32.0000	−1.07445	−0.537227	+	0.843437i	$0.680528\pi$
$888$	0	0
$889$	−20.0000	−0.670778
$890$	−6.00000	−0.201120
$891$	0	0
$892$	−10.0000	−0.334825
$893$	60.0000	2.00782
$894$	0	0
$895$	0	0
$896$	1.00000	0.0334077
$897$	0	0
$898$	−34.0000	−1.13459
$899$	−48.0000	−1.60089
$900$	−3.00000	−0.100000
$901$	0	0
$902$	0	0
$903$	0	0
$904$	−2.00000	−0.0665190
$905$	22.0000	0.731305
$906$	0	0
$907$	−26.0000	−0.863316	−0.431658	−	0.902037i	$-0.642071\pi$
$907$	−26.0000	−0.863316	−0.431658	+	0.902037i	$0.642071\pi$
$908$	8.00000	0.265489
$909$	12.0000	0.398015
$910$	−6.00000	−0.198898
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	34.0000	1.12462
$915$	0	0
$916$	−6.00000	−0.198246
$917$	−6.00000	−0.198137
$918$	0	0
$919$	10.0000	0.329870	0.164935	−	0.986304i	$-0.447259\pi$
$919$	10.0000	0.329870	0.164935	+	0.986304i	$0.447259\pi$
$920$	−8.00000	−0.263752
$921$	0	0
$922$	16.0000	0.526932
$923$	48.0000	1.57994
$924$	0	0
$925$	−4.00000	−0.131519
$926$	32.0000	1.05159
$927$	−30.0000	−0.985329
$928$	6.00000	0.196960
$929$	−34.0000	−1.11550	−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000	−1.11550	−0.557752	+	0.830008i	$0.688336\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	18.0000	0.589610
$933$	0	0
$934$	−36.0000	−1.17796
$935$	0	0
$936$	−18.0000	−0.588348
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	−6.00000	−0.195907
$939$	0	0
$940$	10.0000	0.326164
$941$	−12.0000	−0.391189	−0.195594	−	0.980685i	$-0.562664\pi$
$941$	−12.0000	−0.391189	−0.195594	+	0.980685i	$0.562664\pi$
$942$	0	0
$943$	−80.0000	−2.60516
$944$	−4.00000	−0.130189
$945$	0	0
$946$	0	0
$947$	−58.0000	−1.88475	−0.942373	−	0.334563i	$-0.891411\pi$
$947$	−58.0000	−1.88475	−0.942373	+	0.334563i	$0.891411\pi$
$948$	0	0
$949$	−60.0000	−1.94768
$950$	−6.00000	−0.194666
$951$	0	0
$952$	−6.00000	−0.194461
$953$	34.0000	1.10137	0.550684	−	0.834714i	$-0.314367\pi$
$953$	34.0000	1.10137	0.550684	+	0.834714i	$0.314367\pi$
$954$	0	0
$955$	8.00000	0.258874
$956$	10.0000	0.323423
$957$	0	0
$958$	20.0000	0.646171
$959$	−18.0000	−0.581250
$960$	0	0
$961$	33.0000	1.06452
$962$	−24.0000	−0.773791
$963$	12.0000	0.386695
$964$	−10.0000	−0.322078
$965$	10.0000	0.321911
$966$	0	0
$967$	−20.0000	−0.643157	−0.321578	−	0.946883i	$-0.604213\pi$
$967$	−20.0000	−0.643157	−0.321578	+	0.946883i	$0.604213\pi$
$968$	0	0
$969$	0	0
$970$	−16.0000	−0.513729
$971$	28.0000	0.898563	0.449281	−	0.893390i	$-0.351680\pi$
$971$	28.0000	0.898563	0.449281	+	0.893390i	$0.351680\pi$
$972$	0	0
$973$	2.00000	0.0641171
$974$	−20.0000	−0.640841
$975$	0	0
$976$	8.00000	0.256074
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	0	0
$979$	0	0
$980$	−1.00000	−0.0319438
$981$	−6.00000	−0.191565
$982$	−32.0000	−1.02116
$983$	42.0000	1.33959	0.669796	−	0.742545i	$-0.266382\pi$
$983$	42.0000	1.33959	0.669796	+	0.742545i	$0.266382\pi$
$984$	0	0
$985$	18.0000	0.573528
$986$	−36.0000	−1.14647
$987$	0	0
$988$	−36.0000	−1.14531
$989$	−32.0000	−1.01754
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	−8.00000	−0.254000
$993$	0	0
$994$	8.00000	0.253745
$995$	20.0000	0.634043
$996$	0	0
$997$	−22.0000	−0.696747	−0.348373	−	0.937356i	$-0.613266\pi$
$997$	−22.0000	−0.696747	−0.348373	+	0.937356i	$0.613266\pi$
$998$	36.0000	1.13956
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.y.1.1	yes	1
11.10	odd	2		8470.2.a.h.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8470.2.a.h.1.1	✓	1	11.10	odd	2
8470.2.a.y.1.1	yes	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8470.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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