LMFDB - Embedded newform 1638.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1638,2,Mod(1,1638)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1638, 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("1638.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1638.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:	$13.0794958511$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 182)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1638.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} -4.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{4} -4.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} -4.00000 q^{10} +1.00000 q^{11} +1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -4.00000 q^{17} +2.00000 q^{19} -4.00000 q^{20} +1.00000 q^{22} +7.00000 q^{23} +11.0000 q^{25} +1.00000 q^{26} -1.00000 q^{28} +8.00000 q^{29} +3.00000 q^{31} +1.00000 q^{32} -4.00000 q^{34} +4.00000 q^{35} +7.00000 q^{37} +2.00000 q^{38} -4.00000 q^{40} +7.00000 q^{41} -8.00000 q^{43} +1.00000 q^{44} +7.00000 q^{46} -3.00000 q^{47} +1.00000 q^{49} +11.0000 q^{50} +1.00000 q^{52} -4.00000 q^{55} -1.00000 q^{56} +8.00000 q^{58} +6.00000 q^{59} -13.0000 q^{61} +3.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} +7.00000 q^{67} -4.00000 q^{68} +4.00000 q^{70} -4.00000 q^{71} +9.00000 q^{73} +7.00000 q^{74} +2.00000 q^{76} -1.00000 q^{77} -13.0000 q^{79} -4.00000 q^{80} +7.00000 q^{82} +16.0000 q^{83} +16.0000 q^{85} -8.00000 q^{86} +1.00000 q^{88} +6.00000 q^{89} -1.00000 q^{91} +7.00000 q^{92} -3.00000 q^{94} -8.00000 q^{95} +11.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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
$3$	0	0
$4$	1.00000	0.500000
$5$	−4.00000	−1.78885	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	−4.00000	−1.78885	−0.894427	+	0.447214i	$0.852416\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	−4.00000	−1.26491
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	−4.00000	−0.894427
$21$	0	0
$22$	1.00000	0.213201
$23$	7.00000	1.45960	0.729800	−	0.683660i	$-0.239613\pi$
$23$	7.00000	1.45960	0.729800	+	0.683660i	$0.239613\pi$
$24$	0	0
$25$	11.0000	2.20000
$26$	1.00000	0.196116
$27$	0	0
$28$	−1.00000	−0.188982
$29$	8.00000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	0	0
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−4.00000	−0.685994
$35$	4.00000	0.676123
$36$	0	0
$37$	7.00000	1.15079	0.575396	−	0.817875i	$-0.304848\pi$
$37$	7.00000	1.15079	0.575396	+	0.817875i	$0.304848\pi$
$38$	2.00000	0.324443
$39$	0	0
$40$	−4.00000	−0.632456
$41$	7.00000	1.09322	0.546608	−	0.837389i	$-0.315919\pi$
$41$	7.00000	1.09322	0.546608	+	0.837389i	$0.315919\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	7.00000	1.03209
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	11.0000	1.55563
$51$	0	0
$52$	1.00000	0.138675
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	−1.00000	−0.133631
$57$	0	0
$58$	8.00000	1.05045
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	−13.0000	−1.66448	−0.832240	−	0.554416i	$-0.812942\pi$
$61$	−13.0000	−1.66448	−0.832240	+	0.554416i	$0.812942\pi$
$62$	3.00000	0.381000
$63$	0	0
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	0	0
$67$	7.00000	0.855186	0.427593	−	0.903971i	$-0.359362\pi$
$67$	7.00000	0.855186	0.427593	+	0.903971i	$0.359362\pi$
$68$	−4.00000	−0.485071
$69$	0	0
$70$	4.00000	0.478091
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	0	0
$73$	9.00000	1.05337	0.526685	−	0.850060i	$-0.323435\pi$
$73$	9.00000	1.05337	0.526685	+	0.850060i	$0.323435\pi$
$74$	7.00000	0.813733
$75$	0	0
$76$	2.00000	0.229416
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−13.0000	−1.46261	−0.731307	−	0.682048i	$-0.761089\pi$
$79$	−13.0000	−1.46261	−0.731307	+	0.682048i	$0.761089\pi$
$80$	−4.00000	−0.447214
$81$	0	0
$82$	7.00000	0.773021
$83$	16.0000	1.75623	0.878114	−	0.478451i	$-0.158802\pi$
$83$	16.0000	1.75623	0.878114	+	0.478451i	$0.158802\pi$
$84$	0	0
$85$	16.0000	1.73544
$86$	−8.00000	−0.862662
$87$	0	0
$88$	1.00000	0.106600
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	7.00000	0.729800
$93$	0	0
$94$	−3.00000	−0.309426
$95$	−8.00000	−0.820783
$96$	0	0
$97$	11.0000	1.11688	0.558440	−	0.829545i	$-0.311400\pi$
$97$	11.0000	1.11688	0.558440	+	0.829545i	$0.311400\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	11.0000	1.10000
$101$	−9.00000	−0.895533	−0.447767	−	0.894150i	$-0.647781\pi$
$101$	−9.00000	−0.895533	−0.447767	+	0.894150i	$0.647781\pi$
$102$	0	0
$103$	10.0000	0.985329	0.492665	−	0.870219i	$-0.336023\pi$
$103$	10.0000	0.985329	0.492665	+	0.870219i	$0.336023\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	0	0
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	−4.00000	−0.381385
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	−1.00000	−0.0940721	−0.0470360	−	0.998893i	$-0.514978\pi$
$113$	−1.00000	−0.0940721	−0.0470360	+	0.998893i	$0.514978\pi$
$114$	0	0
$115$	−28.0000	−2.61101
$116$	8.00000	0.742781
$117$	0	0
$118$	6.00000	0.552345
$119$	4.00000	0.366679
$120$	0	0
$121$	−10.0000	−0.909091
$122$	−13.0000	−1.17696
$123$	0	0
$124$	3.00000	0.269408
$125$	−24.0000	−2.14663
$126$	0	0
$127$	13.0000	1.15356	0.576782	−	0.816898i	$-0.304308\pi$
$127$	13.0000	1.15356	0.576782	+	0.816898i	$0.304308\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−4.00000	−0.350823
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−2.00000	−0.173422
$134$	7.00000	0.604708
$135$	0	0
$136$	−4.00000	−0.342997
$137$	14.0000	1.19610	0.598050	−	0.801459i	$-0.295942\pi$
$137$	14.0000	1.19610	0.598050	+	0.801459i	$0.295942\pi$
$138$	0	0
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	4.00000	0.338062
$141$	0	0
$142$	−4.00000	−0.335673
$143$	1.00000	0.0836242
$144$	0	0
$145$	−32.0000	−2.65746
$146$	9.00000	0.744845
$147$	0	0
$148$	7.00000	0.575396
$149$	−15.0000	−1.22885	−0.614424	−	0.788976i	$-0.710612\pi$
$149$	−15.0000	−1.22885	−0.614424	+	0.788976i	$0.710612\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	2.00000	0.162221
$153$	0	0
$154$	−1.00000	−0.0805823
$155$	−12.0000	−0.963863
$156$	0	0
$157$	7.00000	0.558661	0.279330	−	0.960195i	$-0.409888\pi$
$157$	7.00000	0.558661	0.279330	+	0.960195i	$0.409888\pi$
$158$	−13.0000	−1.03422
$159$	0	0
$160$	−4.00000	−0.316228
$161$	−7.00000	−0.551677
$162$	0	0
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	7.00000	0.546608
$165$	0	0
$166$	16.0000	1.24184
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	16.0000	1.22714
$171$	0	0
$172$	−8.00000	−0.609994
$173$	−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
$175$	−11.0000	−0.831522
$176$	1.00000	0.0753778
$177$	0	0
$178$	6.00000	0.449719
$179$	18.0000	1.34538	0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.0000	1.34538	0.672692	+	0.739923i	$0.265138\pi$
$180$	0	0
$181$	13.0000	0.966282	0.483141	−	0.875542i	$-0.339496\pi$
$181$	13.0000	0.966282	0.483141	+	0.875542i	$0.339496\pi$
$182$	−1.00000	−0.0741249
$183$	0	0
$184$	7.00000	0.516047
$185$	−28.0000	−2.05860
$186$	0	0
$187$	−4.00000	−0.292509
$188$	−3.00000	−0.218797
$189$	0	0
$190$	−8.00000	−0.580381
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	11.0000	0.789754
$195$	0	0
$196$	1.00000	0.0714286
$197$	−3.00000	−0.213741	−0.106871	−	0.994273i	$-0.534083\pi$
$197$	−3.00000	−0.213741	−0.106871	+	0.994273i	$0.534083\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	11.0000	0.777817
$201$	0	0
$202$	−9.00000	−0.633238
$203$	−8.00000	−0.561490
$204$	0	0
$205$	−28.0000	−1.95560
$206$	10.0000	0.696733
$207$	0	0
$208$	1.00000	0.0693375
$209$	2.00000	0.138343
$210$	0	0
$211$	22.0000	1.51454	0.757271	−	0.653101i	$-0.226532\pi$
$211$	22.0000	1.51454	0.757271	+	0.653101i	$0.226532\pi$
$212$	0	0
$213$	0	0
$214$	−12.0000	−0.820303
$215$	32.0000	2.18238
$216$	0	0
$217$	−3.00000	−0.203653
$218$	14.0000	0.948200
$219$	0	0
$220$	−4.00000	−0.269680
$221$	−4.00000	−0.269069
$222$	0	0
$223$	9.00000	0.602685	0.301342	−	0.953516i	$-0.402565\pi$
$223$	9.00000	0.602685	0.301342	+	0.953516i	$0.402565\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−1.00000	−0.0665190
$227$	−28.0000	−1.85843	−0.929213	−	0.369546i	$-0.879513\pi$
$227$	−28.0000	−1.85843	−0.929213	+	0.369546i	$0.879513\pi$
$228$	0	0
$229$	−20.0000	−1.32164	−0.660819	−	0.750546i	$-0.729791\pi$
$229$	−20.0000	−1.32164	−0.660819	+	0.750546i	$0.729791\pi$
$230$	−28.0000	−1.84627
$231$	0	0
$232$	8.00000	0.525226
$233$	13.0000	0.851658	0.425829	−	0.904804i	$-0.359982\pi$
$233$	13.0000	0.851658	0.425829	+	0.904804i	$0.359982\pi$
$234$	0	0
$235$	12.0000	0.782794
$236$	6.00000	0.390567
$237$	0	0
$238$	4.00000	0.259281
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	−10.0000	−0.642824
$243$	0	0
$244$	−13.0000	−0.832240
$245$	−4.00000	−0.255551
$246$	0	0
$247$	2.00000	0.127257
$248$	3.00000	0.190500
$249$	0	0
$250$	−24.0000	−1.51789
$251$	−17.0000	−1.07303	−0.536515	−	0.843891i	$-0.680260\pi$
$251$	−17.0000	−1.07303	−0.536515	+	0.843891i	$0.680260\pi$
$252$	0	0
$253$	7.00000	0.440086
$254$	13.0000	0.815693
$255$	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
$259$	−7.00000	−0.434959
$260$	−4.00000	−0.248069
$261$	0	0
$262$	0	0
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	0	0
$265$	0	0
$266$	−2.00000	−0.122628
$267$	0	0
$268$	7.00000	0.427593
$269$	−3.00000	−0.182913	−0.0914566	−	0.995809i	$-0.529152\pi$
$269$	−3.00000	−0.182913	−0.0914566	+	0.995809i	$0.529152\pi$
$270$	0	0
$271$	−3.00000	−0.182237	−0.0911185	−	0.995840i	$-0.529044\pi$
$271$	−3.00000	−0.182237	−0.0911185	+	0.995840i	$0.529044\pi$
$272$	−4.00000	−0.242536
$273$	0	0
$274$	14.0000	0.845771
$275$	11.0000	0.663325
$276$	0	0
$277$	−14.0000	−0.841178	−0.420589	−	0.907251i	$-0.638177\pi$
$277$	−14.0000	−0.841178	−0.420589	+	0.907251i	$0.638177\pi$
$278$	−20.0000	−1.19952
$279$	0	0
$280$	4.00000	0.239046
$281$	20.0000	1.19310	0.596550	−	0.802576i	$-0.296538\pi$
$281$	20.0000	1.19310	0.596550	+	0.802576i	$0.296538\pi$
$282$	0	0
$283$	17.0000	1.01055	0.505273	−	0.862960i	$-0.331392\pi$
$283$	17.0000	1.01055	0.505273	+	0.862960i	$0.331392\pi$
$284$	−4.00000	−0.237356
$285$	0	0
$286$	1.00000	0.0591312
$287$	−7.00000	−0.413197
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	−32.0000	−1.87910
$291$	0	0
$292$	9.00000	0.526685
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	0	0
$295$	−24.0000	−1.39733
$296$	7.00000	0.406867
$297$	0	0
$298$	−15.0000	−0.868927
$299$	7.00000	0.404820
$300$	0	0
$301$	8.00000	0.461112
$302$	−4.00000	−0.230174
$303$	0	0
$304$	2.00000	0.114708
$305$	52.0000	2.97751
$306$	0	0
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	−1.00000	−0.0569803
$309$	0	0
$310$	−12.0000	−0.681554
$311$	30.0000	1.70114	0.850572	−	0.525859i	$-0.176256\pi$
$311$	30.0000	1.70114	0.850572	+	0.525859i	$0.176256\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	7.00000	0.395033
$315$	0	0
$316$	−13.0000	−0.731307
$317$	13.0000	0.730153	0.365076	−	0.930978i	$-0.381043\pi$
$317$	13.0000	0.730153	0.365076	+	0.930978i	$0.381043\pi$
$318$	0	0
$319$	8.00000	0.447914
$320$	−4.00000	−0.223607
$321$	0	0
$322$	−7.00000	−0.390095
$323$	−8.00000	−0.445132
$324$	0	0
$325$	11.0000	0.610170
$326$	12.0000	0.664619
$327$	0	0
$328$	7.00000	0.386510
$329$	3.00000	0.165395
$330$	0	0
$331$	15.0000	0.824475	0.412237	−	0.911077i	$-0.364747\pi$
$331$	15.0000	0.824475	0.412237	+	0.911077i	$0.364747\pi$
$332$	16.0000	0.878114
$333$	0	0
$334$	0	0
$335$	−28.0000	−1.52980
$336$	0	0
$337$	9.00000	0.490261	0.245131	−	0.969490i	$-0.421169\pi$
$337$	9.00000	0.490261	0.245131	+	0.969490i	$0.421169\pi$
$338$	1.00000	0.0543928
$339$	0	0
$340$	16.0000	0.867722
$341$	3.00000	0.162459
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−8.00000	−0.431331
$345$	0	0
$346$	−14.0000	−0.752645
$347$	32.0000	1.71785	0.858925	−	0.512101i	$-0.171133\pi$
$347$	32.0000	1.71785	0.858925	+	0.512101i	$0.171133\pi$
$348$	0	0
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	−11.0000	−0.587975
$351$	0	0
$352$	1.00000	0.0533002
$353$	11.0000	0.585471	0.292735	−	0.956193i	$-0.405434\pi$
$353$	11.0000	0.585471	0.292735	+	0.956193i	$0.405434\pi$
$354$	0	0
$355$	16.0000	0.849192
$356$	6.00000	0.317999
$357$	0	0
$358$	18.0000	0.951330
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	13.0000	0.683265
$363$	0	0
$364$	−1.00000	−0.0524142
$365$	−36.0000	−1.88433
$366$	0	0
$367$	12.0000	0.626395	0.313197	−	0.949688i	$-0.398600\pi$
$367$	12.0000	0.626395	0.313197	+	0.949688i	$0.398600\pi$
$368$	7.00000	0.364900
$369$	0	0
$370$	−28.0000	−1.45565
$371$	0	0
$372$	0	0
$373$	−32.0000	−1.65690	−0.828449	−	0.560065i	$-0.810776\pi$
$373$	−32.0000	−1.65690	−0.828449	+	0.560065i	$0.810776\pi$
$374$	−4.00000	−0.206835
$375$	0	0
$376$	−3.00000	−0.154713
$377$	8.00000	0.412021
$378$	0	0
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	−8.00000	−0.410391
$381$	0	0
$382$	−24.0000	−1.22795
$383$	−21.0000	−1.07305	−0.536525	−	0.843884i	$-0.680263\pi$
$383$	−21.0000	−1.07305	−0.536525	+	0.843884i	$0.680263\pi$
$384$	0	0
$385$	4.00000	0.203859
$386$	−4.00000	−0.203595
$387$	0	0
$388$	11.0000	0.558440
$389$	−8.00000	−0.405616	−0.202808	−	0.979219i	$-0.565007\pi$
$389$	−8.00000	−0.405616	−0.202808	+	0.979219i	$0.565007\pi$
$390$	0	0
$391$	−28.0000	−1.41602
$392$	1.00000	0.0505076
$393$	0	0
$394$	−3.00000	−0.151138
$395$	52.0000	2.61640
$396$	0	0
$397$	8.00000	0.401508	0.200754	−	0.979642i	$-0.435661\pi$
$397$	8.00000	0.401508	0.200754	+	0.979642i	$0.435661\pi$
$398$	16.0000	0.802008
$399$	0	0
$400$	11.0000	0.550000
$401$	−8.00000	−0.399501	−0.199750	−	0.979847i	$-0.564013\pi$
$401$	−8.00000	−0.399501	−0.199750	+	0.979847i	$0.564013\pi$
$402$	0	0
$403$	3.00000	0.149441
$404$	−9.00000	−0.447767
$405$	0	0
$406$	−8.00000	−0.397033
$407$	7.00000	0.346977
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	−28.0000	−1.38282
$411$	0	0
$412$	10.0000	0.492665
$413$	−6.00000	−0.295241
$414$	0	0
$415$	−64.0000	−3.14164
$416$	1.00000	0.0490290
$417$	0	0
$418$	2.00000	0.0978232
$419$	19.0000	0.928211	0.464105	−	0.885780i	$-0.346376\pi$
$419$	19.0000	0.928211	0.464105	+	0.885780i	$0.346376\pi$
$420$	0	0
$421$	−11.0000	−0.536107	−0.268054	−	0.963404i	$-0.586380\pi$
$421$	−11.0000	−0.536107	−0.268054	+	0.963404i	$0.586380\pi$
$422$	22.0000	1.07094
$423$	0	0
$424$	0	0
$425$	−44.0000	−2.13431
$426$	0	0
$427$	13.0000	0.629114
$428$	−12.0000	−0.580042
$429$	0	0
$430$	32.0000	1.54318
$431$	−2.00000	−0.0963366	−0.0481683	−	0.998839i	$-0.515338\pi$
$431$	−2.00000	−0.0963366	−0.0481683	+	0.998839i	$0.515338\pi$
$432$	0	0
$433$	−18.0000	−0.865025	−0.432512	−	0.901628i	$-0.642373\pi$
$433$	−18.0000	−0.865025	−0.432512	+	0.901628i	$0.642373\pi$
$434$	−3.00000	−0.144005
$435$	0	0
$436$	14.0000	0.670478
$437$	14.0000	0.669711
$438$	0	0
$439$	−2.00000	−0.0954548	−0.0477274	−	0.998860i	$-0.515198\pi$
$439$	−2.00000	−0.0954548	−0.0477274	+	0.998860i	$0.515198\pi$
$440$	−4.00000	−0.190693
$441$	0	0
$442$	−4.00000	−0.190261
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	−24.0000	−1.13771
$446$	9.00000	0.426162
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	7.00000	0.329617
$452$	−1.00000	−0.0470360
$453$	0	0
$454$	−28.0000	−1.31411
$455$	4.00000	0.187523
$456$	0	0
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	−20.0000	−0.934539
$459$	0	0
$460$	−28.0000	−1.30551
$461$	−16.0000	−0.745194	−0.372597	−	0.927993i	$-0.621533\pi$
$461$	−16.0000	−0.745194	−0.372597	+	0.927993i	$0.621533\pi$
$462$	0	0
$463$	−12.0000	−0.557687	−0.278844	−	0.960337i	$-0.589951\pi$
$463$	−12.0000	−0.557687	−0.278844	+	0.960337i	$0.589951\pi$
$464$	8.00000	0.371391
$465$	0	0
$466$	13.0000	0.602213
$467$	20.0000	0.925490	0.462745	−	0.886492i	$-0.346865\pi$
$467$	20.0000	0.925490	0.462745	+	0.886492i	$0.346865\pi$
$468$	0	0
$469$	−7.00000	−0.323230
$470$	12.0000	0.553519
$471$	0	0
$472$	6.00000	0.276172
$473$	−8.00000	−0.367840
$474$	0	0
$475$	22.0000	1.00943
$476$	4.00000	0.183340
$477$	0	0
$478$	6.00000	0.274434
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	7.00000	0.319173
$482$	−10.0000	−0.455488
$483$	0	0
$484$	−10.0000	−0.454545
$485$	−44.0000	−1.99794
$486$	0	0
$487$	26.0000	1.17817	0.589086	−	0.808070i	$-0.299488\pi$
$487$	26.0000	1.17817	0.589086	+	0.808070i	$0.299488\pi$
$488$	−13.0000	−0.588482
$489$	0	0
$490$	−4.00000	−0.180702
$491$	−40.0000	−1.80517	−0.902587	−	0.430507i	$-0.858335\pi$
$491$	−40.0000	−1.80517	−0.902587	+	0.430507i	$0.858335\pi$
$492$	0	0
$493$	−32.0000	−1.44121
$494$	2.00000	0.0899843
$495$	0	0
$496$	3.00000	0.134704
$497$	4.00000	0.179425
$498$	0	0
$499$	37.0000	1.65635	0.828174	−	0.560471i	$-0.189380\pi$
$499$	37.0000	1.65635	0.828174	+	0.560471i	$0.189380\pi$
$500$	−24.0000	−1.07331
$501$	0	0
$502$	−17.0000	−0.758747
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	0	0
$505$	36.0000	1.60198
$506$	7.00000	0.311188
$507$	0	0
$508$	13.0000	0.576782
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	0	0
$511$	−9.00000	−0.398137
$512$	1.00000	0.0441942
$513$	0	0
$514$	12.0000	0.529297
$515$	−40.0000	−1.76261
$516$	0	0
$517$	−3.00000	−0.131940
$518$	−7.00000	−0.307562
$519$	0	0
$520$	−4.00000	−0.175412
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	0	0
$523$	−15.0000	−0.655904	−0.327952	−	0.944694i	$-0.606358\pi$
$523$	−15.0000	−0.655904	−0.327952	+	0.944694i	$0.606358\pi$
$524$	0	0
$525$	0	0
$526$	−16.0000	−0.697633
$527$	−12.0000	−0.522728
$528$	0	0
$529$	26.0000	1.13043
$530$	0	0
$531$	0	0
$532$	−2.00000	−0.0867110
$533$	7.00000	0.303204
$534$	0	0
$535$	48.0000	2.07522
$536$	7.00000	0.302354
$537$	0	0
$538$	−3.00000	−0.129339
$539$	1.00000	0.0430730
$540$	0	0
$541$	−38.0000	−1.63375	−0.816874	−	0.576816i	$-0.804295\pi$
$541$	−38.0000	−1.63375	−0.816874	+	0.576816i	$0.804295\pi$
$542$	−3.00000	−0.128861
$543$	0	0
$544$	−4.00000	−0.171499
$545$	−56.0000	−2.39878
$546$	0	0
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	14.0000	0.598050
$549$	0	0
$550$	11.0000	0.469042
$551$	16.0000	0.681623
$552$	0	0
$553$	13.0000	0.552816
$554$	−14.0000	−0.594803
$555$	0	0
$556$	−20.0000	−0.848189
$557$	−31.0000	−1.31351	−0.656756	−	0.754103i	$-0.728072\pi$
$557$	−31.0000	−1.31351	−0.656756	+	0.754103i	$0.728072\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	4.00000	0.169031
$561$	0	0
$562$	20.0000	0.843649
$563$	31.0000	1.30649	0.653247	−	0.757145i	$-0.273406\pi$
$563$	31.0000	1.30649	0.653247	+	0.757145i	$0.273406\pi$
$564$	0	0
$565$	4.00000	0.168281
$566$	17.0000	0.714563
$567$	0	0
$568$	−4.00000	−0.167836
$569$	−29.0000	−1.21574	−0.607872	−	0.794035i	$-0.707976\pi$
$569$	−29.0000	−1.21574	−0.607872	+	0.794035i	$0.707976\pi$
$570$	0	0
$571$	26.0000	1.08807	0.544033	−	0.839064i	$-0.316897\pi$
$571$	26.0000	1.08807	0.544033	+	0.839064i	$0.316897\pi$
$572$	1.00000	0.0418121
$573$	0	0
$574$	−7.00000	−0.292174
$575$	77.0000	3.21112
$576$	0	0
$577$	−18.0000	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$578$	−1.00000	−0.0415945
$579$	0	0
$580$	−32.0000	−1.32873
$581$	−16.0000	−0.663792
$582$	0	0
$583$	0	0
$584$	9.00000	0.372423
$585$	0	0
$586$	−14.0000	−0.578335
$587$	18.0000	0.742940	0.371470	−	0.928445i	$-0.378854\pi$
$587$	18.0000	0.742940	0.371470	+	0.928445i	$0.378854\pi$
$588$	0	0
$589$	6.00000	0.247226
$590$	−24.0000	−0.988064
$591$	0	0
$592$	7.00000	0.287698
$593$	10.0000	0.410651	0.205325	−	0.978694i	$-0.434175\pi$
$593$	10.0000	0.410651	0.205325	+	0.978694i	$0.434175\pi$
$594$	0	0
$595$	−16.0000	−0.655936
$596$	−15.0000	−0.614424
$597$	0	0
$598$	7.00000	0.286251
$599$	−27.0000	−1.10319	−0.551595	−	0.834112i	$-0.685981\pi$
$599$	−27.0000	−1.10319	−0.551595	+	0.834112i	$0.685981\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	8.00000	0.326056
$603$	0	0
$604$	−4.00000	−0.162758
$605$	40.0000	1.62623
$606$	0	0
$607$	14.0000	0.568242	0.284121	−	0.958788i	$-0.408298\pi$
$607$	14.0000	0.568242	0.284121	+	0.958788i	$0.408298\pi$
$608$	2.00000	0.0811107
$609$	0	0
$610$	52.0000	2.10542
$611$	−3.00000	−0.121367
$612$	0	0
$613$	17.0000	0.686624	0.343312	−	0.939222i	$-0.388451\pi$
$613$	17.0000	0.686624	0.343312	+	0.939222i	$0.388451\pi$
$614$	−12.0000	−0.484281
$615$	0	0
$616$	−1.00000	−0.0402911
$617$	38.0000	1.52982	0.764911	−	0.644136i	$-0.222783\pi$
$617$	38.0000	1.52982	0.764911	+	0.644136i	$0.222783\pi$
$618$	0	0
$619$	14.0000	0.562708	0.281354	−	0.959604i	$-0.409217\pi$
$619$	14.0000	0.562708	0.281354	+	0.959604i	$0.409217\pi$
$620$	−12.0000	−0.481932
$621$	0	0
$622$	30.0000	1.20289
$623$	−6.00000	−0.240385
$624$	0	0
$625$	41.0000	1.64000
$626$	6.00000	0.239808
$627$	0	0
$628$	7.00000	0.279330
$629$	−28.0000	−1.11643
$630$	0	0
$631$	22.0000	0.875806	0.437903	−	0.899022i	$-0.355721\pi$
$631$	22.0000	0.875806	0.437903	+	0.899022i	$0.355721\pi$
$632$	−13.0000	−0.517112
$633$	0	0
$634$	13.0000	0.516296
$635$	−52.0000	−2.06356
$636$	0	0
$637$	1.00000	0.0396214
$638$	8.00000	0.316723
$639$	0	0
$640$	−4.00000	−0.158114
$641$	25.0000	0.987441	0.493720	−	0.869621i	$-0.335637\pi$
$641$	25.0000	0.987441	0.493720	+	0.869621i	$0.335637\pi$
$642$	0	0
$643$	−28.0000	−1.10421	−0.552106	−	0.833774i	$-0.686176\pi$
$643$	−28.0000	−1.10421	−0.552106	+	0.833774i	$0.686176\pi$
$644$	−7.00000	−0.275839
$645$	0	0
$646$	−8.00000	−0.314756
$647$	48.0000	1.88707	0.943537	−	0.331266i	$-0.107476\pi$
$647$	48.0000	1.88707	0.943537	+	0.331266i	$0.107476\pi$
$648$	0	0
$649$	6.00000	0.235521
$650$	11.0000	0.431455
$651$	0	0
$652$	12.0000	0.469956
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	7.00000	0.273304
$657$	0	0
$658$	3.00000	0.116952
$659$	−6.00000	−0.233727	−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000	−0.233727	−0.116863	+	0.993148i	$0.537284\pi$
$660$	0	0
$661$	−8.00000	−0.311164	−0.155582	−	0.987823i	$-0.549725\pi$
$661$	−8.00000	−0.311164	−0.155582	+	0.987823i	$0.549725\pi$
$662$	15.0000	0.582992
$663$	0	0
$664$	16.0000	0.620920
$665$	8.00000	0.310227
$666$	0	0
$667$	56.0000	2.16833
$668$	0	0
$669$	0	0
$670$	−28.0000	−1.08173
$671$	−13.0000	−0.501859
$672$	0	0
$673$	−33.0000	−1.27206	−0.636028	−	0.771666i	$-0.719424\pi$
$673$	−33.0000	−1.27206	−0.636028	+	0.771666i	$0.719424\pi$
$674$	9.00000	0.346667
$675$	0	0
$676$	1.00000	0.0384615
$677$	−13.0000	−0.499631	−0.249815	−	0.968294i	$-0.580370\pi$
$677$	−13.0000	−0.499631	−0.249815	+	0.968294i	$0.580370\pi$
$678$	0	0
$679$	−11.0000	−0.422141
$680$	16.0000	0.613572
$681$	0	0
$682$	3.00000	0.114876
$683$	−31.0000	−1.18618	−0.593091	−	0.805135i	$-0.702093\pi$
$683$	−31.0000	−1.18618	−0.593091	+	0.805135i	$0.702093\pi$
$684$	0	0
$685$	−56.0000	−2.13965
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	−8.00000	−0.304997
$689$	0	0
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	−14.0000	−0.532200
$693$	0	0
$694$	32.0000	1.21470
$695$	80.0000	3.03457
$696$	0	0
$697$	−28.0000	−1.06058
$698$	2.00000	0.0757011
$699$	0	0
$700$	−11.0000	−0.415761
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	14.0000	0.528020
$704$	1.00000	0.0376889
$705$	0	0
$706$	11.0000	0.413990
$707$	9.00000	0.338480
$708$	0	0
$709$	−51.0000	−1.91535	−0.957673	−	0.287860i	$-0.907056\pi$
$709$	−51.0000	−1.91535	−0.957673	+	0.287860i	$0.907056\pi$
$710$	16.0000	0.600469
$711$	0	0
$712$	6.00000	0.224860
$713$	21.0000	0.786456
$714$	0	0
$715$	−4.00000	−0.149592
$716$	18.0000	0.672692
$717$	0	0
$718$	0	0
$719$	22.0000	0.820462	0.410231	−	0.911982i	$-0.365448\pi$
$719$	22.0000	0.820462	0.410231	+	0.911982i	$0.365448\pi$
$720$	0	0
$721$	−10.0000	−0.372419
$722$	−15.0000	−0.558242
$723$	0	0
$724$	13.0000	0.483141
$725$	88.0000	3.26824
$726$	0	0
$727$	18.0000	0.667583	0.333792	−	0.942647i	$-0.391672\pi$
$727$	18.0000	0.667583	0.333792	+	0.942647i	$0.391672\pi$
$728$	−1.00000	−0.0370625
$729$	0	0
$730$	−36.0000	−1.33242
$731$	32.0000	1.18356
$732$	0	0
$733$	−12.0000	−0.443230	−0.221615	−	0.975134i	$-0.571133\pi$
$733$	−12.0000	−0.443230	−0.221615	+	0.975134i	$0.571133\pi$
$734$	12.0000	0.442928
$735$	0	0
$736$	7.00000	0.258023
$737$	7.00000	0.257848
$738$	0	0
$739$	−8.00000	−0.294285	−0.147142	−	0.989115i	$-0.547008\pi$
$739$	−8.00000	−0.294285	−0.147142	+	0.989115i	$0.547008\pi$
$740$	−28.0000	−1.02930
$741$	0	0
$742$	0	0
$743$	44.0000	1.61420	0.807102	−	0.590412i	$-0.201035\pi$
$743$	44.0000	1.61420	0.807102	+	0.590412i	$0.201035\pi$
$744$	0	0
$745$	60.0000	2.19823
$746$	−32.0000	−1.17160
$747$	0	0
$748$	−4.00000	−0.146254
$749$	12.0000	0.438470
$750$	0	0
$751$	37.0000	1.35015	0.675075	−	0.737749i	$-0.264111\pi$
$751$	37.0000	1.35015	0.675075	+	0.737749i	$0.264111\pi$
$752$	−3.00000	−0.109399
$753$	0	0
$754$	8.00000	0.291343
$755$	16.0000	0.582300
$756$	0	0
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	−8.00000	−0.290573
$759$	0	0
$760$	−8.00000	−0.290191
$761$	−1.00000	−0.0362500	−0.0181250	−	0.999836i	$-0.505770\pi$
$761$	−1.00000	−0.0362500	−0.0181250	+	0.999836i	$0.505770\pi$
$762$	0	0
$763$	−14.0000	−0.506834
$764$	−24.0000	−0.868290
$765$	0	0
$766$	−21.0000	−0.758761
$767$	6.00000	0.216647
$768$	0	0
$769$	25.0000	0.901523	0.450762	−	0.892644i	$-0.351152\pi$
$769$	25.0000	0.901523	0.450762	+	0.892644i	$0.351152\pi$
$770$	4.00000	0.144150
$771$	0	0
$772$	−4.00000	−0.143963
$773$	−38.0000	−1.36677	−0.683383	−	0.730061i	$-0.739492\pi$
$773$	−38.0000	−1.36677	−0.683383	+	0.730061i	$0.739492\pi$
$774$	0	0
$775$	33.0000	1.18539
$776$	11.0000	0.394877
$777$	0	0
$778$	−8.00000	−0.286814
$779$	14.0000	0.501602
$780$	0	0
$781$	−4.00000	−0.143131
$782$	−28.0000	−1.00128
$783$	0	0
$784$	1.00000	0.0357143
$785$	−28.0000	−0.999363
$786$	0	0
$787$	−40.0000	−1.42585	−0.712923	−	0.701242i	$-0.752629\pi$
$787$	−40.0000	−1.42585	−0.712923	+	0.701242i	$0.752629\pi$
$788$	−3.00000	−0.106871
$789$	0	0
$790$	52.0000	1.85008
$791$	1.00000	0.0355559
$792$	0	0
$793$	−13.0000	−0.461644
$794$	8.00000	0.283909
$795$	0	0
$796$	16.0000	0.567105
$797$	−21.0000	−0.743858	−0.371929	−	0.928261i	$-0.621304\pi$
$797$	−21.0000	−0.743858	−0.371929	+	0.928261i	$0.621304\pi$
$798$	0	0
$799$	12.0000	0.424529
$800$	11.0000	0.388909
$801$	0	0
$802$	−8.00000	−0.282490
$803$	9.00000	0.317603
$804$	0	0
$805$	28.0000	0.986870
$806$	3.00000	0.105670
$807$	0	0
$808$	−9.00000	−0.316619
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	−8.00000	−0.280745
$813$	0	0
$814$	7.00000	0.245350
$815$	−48.0000	−1.68137
$816$	0	0
$817$	−16.0000	−0.559769
$818$	−10.0000	−0.349642
$819$	0	0
$820$	−28.0000	−0.977802
$821$	−46.0000	−1.60541	−0.802706	−	0.596376i	$-0.796607\pi$
$821$	−46.0000	−1.60541	−0.802706	+	0.596376i	$0.796607\pi$
$822$	0	0
$823$	−35.0000	−1.22002	−0.610012	−	0.792392i	$-0.708835\pi$
$823$	−35.0000	−1.22002	−0.610012	+	0.792392i	$0.708835\pi$
$824$	10.0000	0.348367
$825$	0	0
$826$	−6.00000	−0.208767
$827$	−4.00000	−0.139094	−0.0695468	−	0.997579i	$-0.522155\pi$
$827$	−4.00000	−0.139094	−0.0695468	+	0.997579i	$0.522155\pi$
$828$	0	0
$829$	−18.0000	−0.625166	−0.312583	−	0.949890i	$-0.601194\pi$
$829$	−18.0000	−0.625166	−0.312583	+	0.949890i	$0.601194\pi$
$830$	−64.0000	−2.22147
$831$	0	0
$832$	1.00000	0.0346688
$833$	−4.00000	−0.138592
$834$	0	0
$835$	0	0
$836$	2.00000	0.0691714
$837$	0	0
$838$	19.0000	0.656344
$839$	−21.0000	−0.725001	−0.362500	−	0.931984i	$-0.618077\pi$
$839$	−21.0000	−0.725001	−0.362500	+	0.931984i	$0.618077\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	−11.0000	−0.379085
$843$	0	0
$844$	22.0000	0.757271
$845$	−4.00000	−0.137604
$846$	0	0
$847$	10.0000	0.343604
$848$	0	0
$849$	0	0
$850$	−44.0000	−1.50919
$851$	49.0000	1.67970
$852$	0	0
$853$	−44.0000	−1.50653	−0.753266	−	0.657716i	$-0.771523\pi$
$853$	−44.0000	−1.50653	−0.753266	+	0.657716i	$0.771523\pi$
$854$	13.0000	0.444851
$855$	0	0
$856$	−12.0000	−0.410152
$857$	−42.0000	−1.43469	−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000	−1.43469	−0.717346	+	0.696717i	$0.754643\pi$
$858$	0	0
$859$	31.0000	1.05771	0.528853	−	0.848713i	$-0.322622\pi$
$859$	31.0000	1.05771	0.528853	+	0.848713i	$0.322622\pi$
$860$	32.0000	1.09119
$861$	0	0
$862$	−2.00000	−0.0681203
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	0	0
$865$	56.0000	1.90406
$866$	−18.0000	−0.611665
$867$	0	0
$868$	−3.00000	−0.101827
$869$	−13.0000	−0.440995
$870$	0	0
$871$	7.00000	0.237186
$872$	14.0000	0.474100
$873$	0	0
$874$	14.0000	0.473557
$875$	24.0000	0.811348
$876$	0	0
$877$	29.0000	0.979260	0.489630	−	0.871930i	$-0.337132\pi$
$877$	29.0000	0.979260	0.489630	+	0.871930i	$0.337132\pi$
$878$	−2.00000	−0.0674967
$879$	0	0
$880$	−4.00000	−0.134840
$881$	14.0000	0.471672	0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000	0.471672	0.235836	+	0.971793i	$0.424217\pi$
$882$	0	0
$883$	−34.0000	−1.14419	−0.572096	−	0.820187i	$-0.693869\pi$
$883$	−34.0000	−1.14419	−0.572096	+	0.820187i	$0.693869\pi$
$884$	−4.00000	−0.134535
$885$	0	0
$886$	0	0
$887$	−14.0000	−0.470074	−0.235037	−	0.971986i	$-0.575521\pi$
$887$	−14.0000	−0.470074	−0.235037	+	0.971986i	$0.575521\pi$
$888$	0	0
$889$	−13.0000	−0.436006
$890$	−24.0000	−0.804482
$891$	0	0
$892$	9.00000	0.301342
$893$	−6.00000	−0.200782
$894$	0	0
$895$	−72.0000	−2.40669
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−6.00000	−0.200223
$899$	24.0000	0.800445
$900$	0	0
$901$	0	0
$902$	7.00000	0.233075
$903$	0	0
$904$	−1.00000	−0.0332595
$905$	−52.0000	−1.72854
$906$	0	0
$907$	54.0000	1.79304	0.896520	−	0.443003i	$-0.146087\pi$
$907$	54.0000	1.79304	0.896520	+	0.443003i	$0.146087\pi$
$908$	−28.0000	−0.929213
$909$	0	0
$910$	4.00000	0.132599
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	0	0
$913$	16.0000	0.529523
$914$	18.0000	0.595387
$915$	0	0
$916$	−20.0000	−0.660819
$917$	0	0
$918$	0	0
$919$	−37.0000	−1.22052	−0.610259	−	0.792202i	$-0.708935\pi$
$919$	−37.0000	−1.22052	−0.610259	+	0.792202i	$0.708935\pi$
$920$	−28.0000	−0.923133
$921$	0	0
$922$	−16.0000	−0.526932
$923$	−4.00000	−0.131662
$924$	0	0
$925$	77.0000	2.53174
$926$	−12.0000	−0.394344
$927$	0	0
$928$	8.00000	0.262613
$929$	27.0000	0.885841	0.442921	−	0.896561i	$-0.353942\pi$
$929$	27.0000	0.885841	0.442921	+	0.896561i	$0.353942\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	13.0000	0.425829
$933$	0	0
$934$	20.0000	0.654420
$935$	16.0000	0.523256
$936$	0	0
$937$	−48.0000	−1.56809	−0.784046	−	0.620703i	$-0.786847\pi$
$937$	−48.0000	−1.56809	−0.784046	+	0.620703i	$0.786847\pi$
$938$	−7.00000	−0.228558
$939$	0	0
$940$	12.0000	0.391397
$941$	−14.0000	−0.456387	−0.228193	−	0.973616i	$-0.573282\pi$
$941$	−14.0000	−0.456387	−0.228193	+	0.973616i	$0.573282\pi$
$942$	0	0
$943$	49.0000	1.59566
$944$	6.00000	0.195283
$945$	0	0
$946$	−8.00000	−0.260102
$947$	8.00000	0.259965	0.129983	−	0.991516i	$-0.458508\pi$
$947$	8.00000	0.259965	0.129983	+	0.991516i	$0.458508\pi$
$948$	0	0
$949$	9.00000	0.292152
$950$	22.0000	0.713774
$951$	0	0
$952$	4.00000	0.129641
$953$	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
$955$	96.0000	3.10649
$956$	6.00000	0.194054
$957$	0	0
$958$	0	0
$959$	−14.0000	−0.452084
$960$	0	0
$961$	−22.0000	−0.709677
$962$	7.00000	0.225689
$963$	0	0
$964$	−10.0000	−0.322078
$965$	16.0000	0.515058
$966$	0	0
$967$	18.0000	0.578841	0.289420	−	0.957202i	$-0.406537\pi$
$967$	18.0000	0.578841	0.289420	+	0.957202i	$0.406537\pi$
$968$	−10.0000	−0.321412
$969$	0	0
$970$	−44.0000	−1.41275
$971$	45.0000	1.44412	0.722059	−	0.691831i	$-0.243196\pi$
$971$	45.0000	1.44412	0.722059	+	0.691831i	$0.243196\pi$
$972$	0	0
$973$	20.0000	0.641171
$974$	26.0000	0.833094
$975$	0	0
$976$	−13.0000	−0.416120
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	0	0
$979$	6.00000	0.191761
$980$	−4.00000	−0.127775
$981$	0	0
$982$	−40.0000	−1.27645
$983$	−32.0000	−1.02064	−0.510321	−	0.859984i	$-0.670473\pi$
$983$	−32.0000	−1.02064	−0.510321	+	0.859984i	$0.670473\pi$
$984$	0	0
$985$	12.0000	0.382352
$986$	−32.0000	−1.01909
$987$	0	0
$988$	2.00000	0.0636285
$989$	−56.0000	−1.78070
$990$	0	0
$991$	−33.0000	−1.04828	−0.524140	−	0.851632i	$-0.675613\pi$
$991$	−33.0000	−1.04828	−0.524140	+	0.851632i	$0.675613\pi$
$992$	3.00000	0.0952501
$993$	0	0
$994$	4.00000	0.126872
$995$	−64.0000	−2.02894
$996$	0	0
$997$	31.0000	0.981780	0.490890	−	0.871222i	$-0.336672\pi$
$997$	31.0000	0.981780	0.490890	+	0.871222i	$0.336672\pi$
$998$	37.0000	1.17121
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1638.2.a.k.1.1		1
3.2	odd	2		182.2.a.a.1.1	✓	1
12.11	even	2		1456.2.a.e.1.1		1
15.14	odd	2		4550.2.a.t.1.1		1
21.2	odd	6		1274.2.f.n.1145.1		2
21.5	even	6		1274.2.f.t.1145.1		2
21.11	odd	6		1274.2.f.n.79.1		2
21.17	even	6		1274.2.f.t.79.1		2
21.20	even	2		1274.2.a.b.1.1		1
24.5	odd	2		5824.2.a.g.1.1		1
24.11	even	2		5824.2.a.w.1.1		1
39.5	even	4		2366.2.d.g.337.2		2
39.8	even	4		2366.2.d.g.337.1		2
39.38	odd	2		2366.2.a.m.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
182.2.a.a.1.1	✓	1	3.2	odd	2
1274.2.a.b.1.1		1	21.20	even	2
1274.2.f.n.79.1		2	21.11	odd	6
1274.2.f.n.1145.1		2	21.2	odd	6
1274.2.f.t.79.1		2	21.17	even	6
1274.2.f.t.1145.1		2	21.5	even	6
1456.2.a.e.1.1		1	12.11	even	2
1638.2.a.k.1.1		1	1.1	even	1	trivial
2366.2.a.m.1.1		1	39.38	odd	2
2366.2.d.g.337.1		2	39.8	even	4
2366.2.d.g.337.2		2	39.5	even	4
4550.2.a.t.1.1		1	15.14	odd	2
5824.2.a.g.1.1		1	24.5	odd	2
5824.2.a.w.1.1		1	24.11	even	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:	\( N \)	\(=\)	\( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1638.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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