LMFDB - Embedded newform 920.2.a.j.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [920,2,Mod(1,920)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("920.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(920, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 920 = 2^{3} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	920.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$7.34623698596$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.13955077.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 14x^{3} - x^{2} + 32x + 16 $ x^5 - 14x^3 - x^2 + 32x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.31091$ of defining polynomial
Character	$\chi$	$=$	920.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.31091 q^{3} -1.00000 q^{5} -4.66212 q^{7} -1.28151 q^{9} +2.23020 q^{11} -2.80072 q^{13} +1.31091 q^{15} +7.63271 q^{17} -1.36222 q^{19} +6.11163 q^{21} -1.00000 q^{23} +1.00000 q^{25} +5.61268 q^{27} +8.94362 q^{29} -1.58140 q^{31} -2.92360 q^{33} +4.66212 q^{35} -1.40251 q^{37} +3.67149 q^{39} +10.7134 q^{41} +1.28151 q^{45} -7.26391 q^{47} +14.7353 q^{49} -10.0058 q^{51} -8.38212 q^{53} -2.23020 q^{55} +1.78575 q^{57} -4.88331 q^{59} +4.33282 q^{61} +5.97454 q^{63} +2.80072 q^{65} +8.54355 q^{67} +1.31091 q^{69} +8.81604 q^{71} -5.26391 q^{73} -1.31091 q^{75} -10.3974 q^{77} +7.08222 q^{79} -3.51322 q^{81} -4.59749 q^{83} -7.63271 q^{85} -11.7243 q^{87} -4.70241 q^{89} +13.0573 q^{91} +2.07308 q^{93} +1.36222 q^{95} +16.5160 q^{97} -2.85802 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{5} - 2 q^{7} + 13 q^{9} - q^{11} + 4 q^{13} + 4 q^{17} + 7 q^{19} + 6 q^{21} - 5 q^{23} + 5 q^{25} + 3 q^{27} + 4 q^{29} + 19 q^{31} + 17 q^{33} + 2 q^{35} + 15 q^{37} + 19 q^{39} + 25 q^{41}+ \cdots - 65 q^{99}+O(q^{100}) $ 5 * q - 5 * q^5 - 2 * q^7 + 13 * q^9 - q^11 + 4 * q^13 + 4 * q^17 + 7 * q^19 + 6 * q^21 - 5 * q^23 + 5 * q^25 + 3 * q^27 + 4 * q^29 + 19 * q^31 + 17 * q^33 + 2 * q^35 + 15 * q^37 + 19 * q^39 + 25 * q^41 - 13 * q^45 - 11 * q^47 + 25 * q^49 + 19 * q^51 + 3 * q^53 + q^55 + 48 * q^57 - q^59 - 5 * q^61 - 41 * q^63 - 4 * q^65 + 9 * q^67 + q^71 - q^73 + 18 * q^77 - 2 * q^79 + 57 * q^81 - 45 * q^83 - 4 * q^85 - 9 * q^87 + 6 * q^89 + 11 * q^91 - 39 * q^93 - 7 * q^95 + 25 * q^97 - 65 * q^99

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
$3$	−1.31091	−0.756856	−0.378428	−	0.925631i	$-0.623535\pi$
$3$	−1.31091	−0.756856	−0.378428	+	0.925631i	$0.623535\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.66212	−1.76211	−0.881057	−	0.473010i	$-0.843168\pi$
$7$	−4.66212	−1.76211	−0.881057	+	0.473010i	$0.843168\pi$
$8$	0	0
$9$	−1.28151	−0.427169
$10$	0	0
$11$	2.23020	0.672430	0.336215	−	0.941785i	$-0.390853\pi$
$11$	2.23020	0.672430	0.336215	+	0.941785i	$0.390853\pi$
$12$	0	0
$13$	−2.80072	−0.776779	−0.388389	−	0.921495i	$-0.626968\pi$
$13$	−2.80072	−0.776779	−0.388389	+	0.921495i	$0.626968\pi$
$14$	0	0
$15$	1.31091	0.338476
$16$	0	0
$17$	7.63271	1.85120	0.925602	−	0.378498i	$-0.123559\pi$
$17$	7.63271	1.85120	0.925602	+	0.378498i	$0.123559\pi$
$18$	0	0
$19$	−1.36222	−0.312515	−0.156258	−	0.987716i	$-0.549943\pi$
$19$	−1.36222	−0.312515	−0.156258	+	0.987716i	$0.549943\pi$
$20$	0	0
$21$	6.11163	1.33367
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.61268	1.08016
$28$	0	0
$29$	8.94362	1.66079	0.830395	−	0.557176i	$-0.188115\pi$
$29$	8.94362	1.66079	0.830395	+	0.557176i	$0.188115\pi$
$30$	0	0
$31$	−1.58140	−0.284028	−0.142014	−	0.989865i	$-0.545358\pi$
$31$	−1.58140	−0.284028	−0.142014	+	0.989865i	$0.545358\pi$
$32$	0	0
$33$	−2.92360	−0.508933
$34$	0	0
$35$	4.66212	0.788042
$36$	0	0
$37$	−1.40251	−0.230572	−0.115286	−	0.993332i	$-0.536778\pi$
$37$	−1.40251	−0.230572	−0.115286	+	0.993332i	$0.536778\pi$
$38$	0	0
$39$	3.67149	0.587909
$40$	0	0
$41$	10.7134	1.67316	0.836578	−	0.547848i	$-0.184553\pi$
$41$	10.7134	1.67316	0.836578	+	0.547848i	$0.184553\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	1.28151	0.191036
$46$	0	0
$47$	−7.26391	−1.05955	−0.529775	−	0.848138i	$-0.677724\pi$
$47$	−7.26391	−1.05955	−0.529775	+	0.848138i	$0.677724\pi$
$48$	0	0
$49$	14.7353	2.10505
$50$	0	0
$51$	−10.0058	−1.40109
$52$	0	0
$53$	−8.38212	−1.15137	−0.575686	−	0.817671i	$-0.695265\pi$
$53$	−8.38212	−1.15137	−0.575686	+	0.817671i	$0.695265\pi$
$54$	0	0
$55$	−2.23020	−0.300720
$56$	0	0
$57$	1.78575	0.236529
$58$	0	0
$59$	−4.88331	−0.635752	−0.317876	−	0.948132i	$-0.602970\pi$
$59$	−4.88331	−0.635752	−0.317876	+	0.948132i	$0.602970\pi$
$60$	0	0
$61$	4.33282	0.554760	0.277380	−	0.960760i	$-0.410534\pi$
$61$	4.33282	0.554760	0.277380	+	0.960760i	$0.410534\pi$
$62$	0	0
$63$	5.97454	0.752721
$64$	0	0
$65$	2.80072	0.347386
$66$	0	0
$67$	8.54355	1.04376	0.521880	−	0.853019i	$-0.325231\pi$
$67$	8.54355	1.04376	0.521880	+	0.853019i	$0.325231\pi$
$68$	0	0
$69$	1.31091	0.157815
$70$	0	0
$71$	8.81604	1.04627	0.523136	−	0.852249i	$-0.324762\pi$
$71$	8.81604	1.04627	0.523136	+	0.852249i	$0.324762\pi$
$72$	0	0
$73$	−5.26391	−0.616095	−0.308047	−	0.951371i	$-0.599675\pi$
$73$	−5.26391	−0.616095	−0.308047	+	0.951371i	$0.599675\pi$
$74$	0	0
$75$	−1.31091	−0.151371
$76$	0	0
$77$	−10.3974	−1.18490
$78$	0	0
$79$	7.08222	0.796812	0.398406	−	0.917209i	$-0.369563\pi$
$79$	7.08222	0.796812	0.398406	+	0.917209i	$0.369563\pi$
$80$	0	0
$81$	−3.51322	−0.390357
$82$	0	0
$83$	−4.59749	−0.504640	−0.252320	−	0.967644i	$-0.581194\pi$
$83$	−4.59749	−0.504640	−0.252320	+	0.967644i	$0.581194\pi$
$84$	0	0
$85$	−7.63271	−0.827884
$86$	0	0
$87$	−11.7243	−1.25698
$88$	0	0
$89$	−4.70241	−0.498454	−0.249227	−	0.968445i	$-0.580177\pi$
$89$	−4.70241	−0.498454	−0.249227	+	0.968445i	$0.580177\pi$
$90$	0	0
$91$	13.0573	1.36877
$92$	0	0
$93$	2.07308	0.214968
$94$	0	0
$95$	1.36222	0.139761
$96$	0	0
$97$	16.5160	1.67695	0.838474	−	0.544942i	$-0.183448\pi$
$97$	16.5160	1.67695	0.838474	+	0.544942i	$0.183448\pi$
$98$	0	0
$99$	−2.85802	−0.287241
$100$	0	0
$101$	10.7267	1.06735	0.533676	−	0.845689i	$-0.320810\pi$
$101$	10.7267	1.06735	0.533676	+	0.845689i	$0.320810\pi$
$102$	0	0
$103$	1.95300	0.192435	0.0962175	−	0.995360i	$-0.469326\pi$
$103$	1.95300	0.192435	0.0962175	+	0.995360i	$0.469326\pi$
$104$	0	0
$105$	−6.11163	−0.596434
$106$	0	0
$107$	−1.97566	−0.190994	−0.0954972	−	0.995430i	$-0.530444\pi$
$107$	−1.97566	−0.190994	−0.0954972	+	0.995430i	$0.530444\pi$
$108$	0	0
$109$	−8.92360	−0.854725	−0.427363	−	0.904080i	$-0.640557\pi$
$109$	−8.92360	−0.854725	−0.427363	+	0.904080i	$0.640557\pi$
$110$	0	0
$111$	1.83857	0.174510
$112$	0	0
$113$	17.3486	1.63202	0.816008	−	0.578040i	$-0.196182\pi$
$113$	17.3486	1.63202	0.816008	+	0.578040i	$0.196182\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	3.58914	0.331816
$118$	0	0
$119$	−35.5846	−3.26203
$120$	0	0
$121$	−6.02621	−0.547838
$122$	0	0
$123$	−14.0444	−1.26634
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	19.4872	1.72921	0.864603	−	0.502455i	$-0.167570\pi$
$127$	19.4872	1.72921	0.864603	+	0.502455i	$0.167570\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	5.68050	0.496308	0.248154	−	0.968721i	$-0.420176\pi$
$131$	5.68050	0.496308	0.248154	+	0.968721i	$0.420176\pi$
$132$	0	0
$133$	6.35084	0.550687
$134$	0	0
$135$	−5.61268	−0.483063
$136$	0	0
$137$	−8.68645	−0.742134	−0.371067	−	0.928606i	$-0.621008\pi$
$137$	−8.68645	−0.742134	−0.371067	+	0.928606i	$0.621008\pi$
$138$	0	0
$139$	−9.22569	−0.782513	−0.391256	−	0.920282i	$-0.627959\pi$
$139$	−9.22569	−0.782513	−0.391256	+	0.920282i	$0.627959\pi$
$140$	0	0
$141$	9.52236	0.801927
$142$	0	0
$143$	−6.24615	−0.522329
$144$	0	0
$145$	−8.94362	−0.742728
$146$	0	0
$147$	−19.3167	−1.59322
$148$	0	0
$149$	18.6056	1.52423	0.762115	−	0.647441i	$-0.224161\pi$
$149$	18.6056	1.52423	0.762115	+	0.647441i	$0.224161\pi$
$150$	0	0
$151$	17.4316	1.41856	0.709280	−	0.704927i	$-0.249020\pi$
$151$	17.4316	1.41856	0.709280	+	0.704927i	$0.249020\pi$
$152$	0	0
$153$	−9.78138	−0.790778
$154$	0	0
$155$	1.58140	0.127021
$156$	0	0
$157$	0.839497	0.0669992	0.0334996	−	0.999439i	$-0.489335\pi$
$157$	0.839497	0.0669992	0.0334996	+	0.999439i	$0.489335\pi$
$158$	0	0
$159$	10.9882	0.871423
$160$	0	0
$161$	4.66212	0.367426
$162$	0	0
$163$	−14.5673	−1.14100	−0.570500	−	0.821297i	$-0.693251\pi$
$163$	−14.5673	−1.14100	−0.570500	+	0.821297i	$0.693251\pi$
$164$	0	0
$165$	2.92360	0.227602
$166$	0	0
$167$	−5.03842	−0.389884	−0.194942	−	0.980815i	$-0.562452\pi$
$167$	−5.03842	−0.389884	−0.194942	+	0.980815i	$0.562452\pi$
$168$	0	0
$169$	−5.15599	−0.396615
$170$	0	0
$171$	1.74570	0.133497
$172$	0	0
$173$	−11.3124	−0.860067	−0.430034	−	0.902813i	$-0.641498\pi$
$173$	−11.3124	−0.860067	−0.430034	+	0.902813i	$0.641498\pi$
$174$	0	0
$175$	−4.66212	−0.352423
$176$	0	0
$177$	6.40159	0.481173
$178$	0	0
$179$	−24.3053	−1.81667	−0.908333	−	0.418247i	$-0.862645\pi$
$179$	−24.3053	−1.81667	−0.908333	+	0.418247i	$0.862645\pi$
$180$	0	0
$181$	−19.4829	−1.44815	−0.724075	−	0.689721i	$-0.757733\pi$
$181$	−19.4829	−1.44815	−0.724075	+	0.689721i	$0.757733\pi$
$182$	0	0
$183$	−5.67994	−0.419874
$184$	0	0
$185$	1.40251	0.103115
$186$	0	0
$187$	17.0225	1.24481
$188$	0	0
$189$	−26.1670	−1.90337
$190$	0	0
$191$	2.62183	0.189709	0.0948543	−	0.995491i	$-0.469761\pi$
$191$	2.62183	0.189709	0.0948543	+	0.995491i	$0.469761\pi$
$192$	0	0
$193$	17.4332	1.25487	0.627436	−	0.778668i	$-0.284105\pi$
$193$	17.4332	1.25487	0.627436	+	0.778668i	$0.284105\pi$
$194$	0	0
$195$	−3.67149	−0.262921
$196$	0	0
$197$	−23.4402	−1.67004	−0.835022	−	0.550217i	$-0.814545\pi$
$197$	−23.4402	−1.67004	−0.835022	+	0.550217i	$0.814545\pi$
$198$	0	0
$199$	−1.29759	−0.0919839	−0.0459920	−	0.998942i	$-0.514645\pi$
$199$	−1.29759	−0.0919839	−0.0459920	+	0.998942i	$0.514645\pi$
$200$	0	0
$201$	−11.1998	−0.789976
$202$	0	0
$203$	−41.6962	−2.92650
$204$	0	0
$205$	−10.7134	−0.748258
$206$	0	0
$207$	1.28151	0.0890709
$208$	0	0
$209$	−3.03803	−0.210145
$210$	0	0
$211$	14.7619	1.01625	0.508127	−	0.861282i	$-0.330338\pi$
$211$	14.7619	1.01625	0.508127	+	0.861282i	$0.330338\pi$
$212$	0	0
$213$	−11.5571	−0.791877
$214$	0	0
$215$	0	0
$216$	0	0
$217$	7.37268	0.500490
$218$	0	0
$219$	6.90053	0.466295
$220$	0	0
$221$	−21.3771	−1.43798
$222$	0	0
$223$	11.8434	0.793095	0.396548	−	0.918014i	$-0.370208\pi$
$223$	11.8434	0.793095	0.396548	+	0.918014i	$0.370208\pi$
$224$	0	0
$225$	−1.28151	−0.0854338
$226$	0	0
$227$	−5.27556	−0.350151	−0.175075	−	0.984555i	$-0.556017\pi$
$227$	−5.27556	−0.350151	−0.175075	+	0.984555i	$0.556017\pi$
$228$	0	0
$229$	−1.23878	−0.0818611	−0.0409305	−	0.999162i	$-0.513032\pi$
$229$	−1.23878	−0.0818611	−0.0409305	+	0.999162i	$0.513032\pi$
$230$	0	0
$231$	13.6301	0.896798
$232$	0	0
$233$	2.66412	0.174533	0.0872663	−	0.996185i	$-0.472187\pi$
$233$	2.66412	0.174533	0.0872663	+	0.996185i	$0.472187\pi$
$234$	0	0
$235$	7.26391	0.473845
$236$	0	0
$237$	−9.28418	−0.603072
$238$	0	0
$239$	26.2577	1.69847	0.849235	−	0.528014i	$-0.177063\pi$
$239$	26.2577	1.69847	0.849235	+	0.528014i	$0.177063\pi$
$240$	0	0
$241$	−2.22326	−0.143213	−0.0716063	−	0.997433i	$-0.522813\pi$
$241$	−2.22326	−0.143213	−0.0716063	+	0.997433i	$0.522813\pi$
$242$	0	0
$243$	−12.2325	−0.784717
$244$	0	0
$245$	−14.7353	−0.941406
$246$	0	0
$247$	3.81520	0.242755
$248$	0	0
$249$	6.02690	0.381940
$250$	0	0
$251$	13.4829	0.851031	0.425515	−	0.904951i	$-0.360093\pi$
$251$	13.4829	0.851031	0.425515	+	0.904951i	$0.360093\pi$
$252$	0	0
$253$	−2.23020	−0.140211
$254$	0	0
$255$	10.0058	0.626589
$256$	0	0
$257$	22.0281	1.37408	0.687039	−	0.726620i	$-0.258910\pi$
$257$	22.0281	1.37408	0.687039	+	0.726620i	$0.258910\pi$
$258$	0	0
$259$	6.53868	0.406294
$260$	0	0
$261$	−11.4613	−0.709438
$262$	0	0
$263$	22.3164	1.37609	0.688043	−	0.725670i	$-0.258470\pi$
$263$	22.3164	1.37609	0.688043	+	0.725670i	$0.258470\pi$
$264$	0	0
$265$	8.38212	0.514909
$266$	0	0
$267$	6.16445	0.377258
$268$	0	0
$269$	10.5050	0.640501	0.320251	−	0.947333i	$-0.396233\pi$
$269$	10.5050	0.640501	0.320251	+	0.947333i	$0.396233\pi$
$270$	0	0
$271$	−3.84696	−0.233686	−0.116843	−	0.993150i	$-0.537277\pi$
$271$	−3.84696	−0.233686	−0.116843	+	0.993150i	$0.537277\pi$
$272$	0	0
$273$	−17.1169	−1.03596
$274$	0	0
$275$	2.23020	0.134486
$276$	0	0
$277$	14.8653	0.893172	0.446586	−	0.894741i	$-0.352640\pi$
$277$	14.8653	0.893172	0.446586	+	0.894741i	$0.352640\pi$
$278$	0	0
$279$	2.02658	0.121328
$280$	0	0
$281$	4.41659	0.263472	0.131736	−	0.991285i	$-0.457945\pi$
$281$	4.41659	0.263472	0.131736	+	0.991285i	$0.457945\pi$
$282$	0	0
$283$	3.88495	0.230936	0.115468	−	0.993311i	$-0.463163\pi$
$283$	3.88495	0.230936	0.115468	+	0.993311i	$0.463163\pi$
$284$	0	0
$285$	−1.78575	−0.105779
$286$	0	0
$287$	−49.9472	−2.94829
$288$	0	0
$289$	41.2583	2.42696
$290$	0	0
$291$	−21.6511	−1.26921
$292$	0	0
$293$	−25.0257	−1.46202	−0.731009	−	0.682368i	$-0.760950\pi$
$293$	−25.0257	−1.46202	−0.731009	+	0.682368i	$0.760950\pi$
$294$	0	0
$295$	4.88331	0.284317
$296$	0	0
$297$	12.5174	0.726333
$298$	0	0
$299$	2.80072	0.161970
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−14.0618	−0.807831
$304$	0	0
$305$	−4.33282	−0.248096
$306$	0	0
$307$	22.0197	1.25673	0.628365	−	0.777918i	$-0.283724\pi$
$307$	22.0197	1.25673	0.628365	+	0.777918i	$0.283724\pi$
$308$	0	0
$309$	−2.56021	−0.145645
$310$	0	0
$311$	19.6217	1.11264	0.556322	−	0.830967i	$-0.312212\pi$
$311$	19.6217	1.11264	0.556322	+	0.830967i	$0.312212\pi$
$312$	0	0
$313$	−17.9496	−1.01457	−0.507285	−	0.861778i	$-0.669351\pi$
$313$	−17.9496	−1.01457	−0.507285	+	0.861778i	$0.669351\pi$
$314$	0	0
$315$	−5.97454	−0.336627
$316$	0	0
$317$	34.3185	1.92752	0.963761	−	0.266768i	$-0.0859559\pi$
$317$	34.3185	1.92752	0.963761	+	0.266768i	$0.0859559\pi$
$318$	0	0
$319$	19.9461	1.11676
$320$	0	0
$321$	2.58992	0.144555
$322$	0	0
$323$	−10.3974	−0.578529
$324$	0	0
$325$	−2.80072	−0.155356
$326$	0	0
$327$	11.6981	0.646904
$328$	0	0
$329$	33.8652	1.86705
$330$	0	0
$331$	−0.299762	−0.0164764	−0.00823821	−	0.999966i	$-0.502622\pi$
$331$	−0.299762	−0.0164764	−0.00823821	+	0.999966i	$0.502622\pi$
$332$	0	0
$333$	1.79733	0.0984931
$334$	0	0
$335$	−8.54355	−0.466784
$336$	0	0
$337$	−22.2965	−1.21457	−0.607284	−	0.794485i	$-0.707741\pi$
$337$	−22.2965	−1.21457	−0.607284	+	0.794485i	$0.707741\pi$
$338$	0	0
$339$	−22.7425	−1.23520
$340$	0	0
$341$	−3.52684	−0.190989
$342$	0	0
$343$	−36.0630	−1.94722
$344$	0	0
$345$	−1.31091	−0.0705772
$346$	0	0
$347$	33.1240	1.77819	0.889094	−	0.457725i	$-0.151336\pi$
$347$	33.1240	1.77819	0.889094	+	0.457725i	$0.151336\pi$
$348$	0	0
$349$	−22.0041	−1.17785	−0.588926	−	0.808187i	$-0.700449\pi$
$349$	−22.0041	−1.17785	−0.588926	+	0.808187i	$0.700449\pi$
$350$	0	0
$351$	−15.7195	−0.839046
$352$	0	0
$353$	−15.6085	−0.830759	−0.415379	−	0.909648i	$-0.636351\pi$
$353$	−15.6085	−0.830759	−0.415379	+	0.909648i	$0.636351\pi$
$354$	0	0
$355$	−8.81604	−0.467907
$356$	0	0
$357$	46.6483	2.46889
$358$	0	0
$359$	0.263781	0.0139218	0.00696092	−	0.999976i	$-0.497784\pi$
$359$	0.263781	0.0139218	0.00696092	+	0.999976i	$0.497784\pi$
$360$	0	0
$361$	−17.1444	−0.902334
$362$	0	0
$363$	7.89984	0.414634
$364$	0	0
$365$	5.26391	0.275526
$366$	0	0
$367$	10.8615	0.566967	0.283484	−	0.958977i	$-0.408510\pi$
$367$	10.8615	0.566967	0.283484	+	0.958977i	$0.408510\pi$
$368$	0	0
$369$	−13.7293	−0.714721
$370$	0	0
$371$	39.0784	2.02885
$372$	0	0
$373$	26.8889	1.39225	0.696127	−	0.717919i	$-0.254905\pi$
$373$	26.8889	1.39225	0.696127	+	0.717919i	$0.254905\pi$
$374$	0	0
$375$	1.31091	0.0676952
$376$	0	0
$377$	−25.0485	−1.29007
$378$	0	0
$379$	−2.15824	−0.110861	−0.0554306	−	0.998463i	$-0.517653\pi$
$379$	−2.15824	−0.110861	−0.0554306	+	0.998463i	$0.517653\pi$
$380$	0	0
$381$	−25.5460	−1.30876
$382$	0	0
$383$	−4.62814	−0.236487	−0.118244	−	0.992985i	$-0.537726\pi$
$383$	−4.62814	−0.236487	−0.118244	+	0.992985i	$0.537726\pi$
$384$	0	0
$385$	10.3974	0.529903
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−18.4988	−0.937929	−0.468964	−	0.883217i	$-0.655373\pi$
$389$	−18.4988	−0.937929	−0.468964	+	0.883217i	$0.655373\pi$
$390$	0	0
$391$	−7.63271	−0.386003
$392$	0	0
$393$	−7.44664	−0.375634
$394$	0	0
$395$	−7.08222	−0.356345
$396$	0	0
$397$	−10.2685	−0.515360	−0.257680	−	0.966230i	$-0.582958\pi$
$397$	−10.2685	−0.515360	−0.257680	+	0.966230i	$0.582958\pi$
$398$	0	0
$399$	−8.32539	−0.416791
$400$	0	0
$401$	−0.885607	−0.0442251	−0.0221125	−	0.999755i	$-0.507039\pi$
$401$	−0.885607	−0.0442251	−0.0221125	+	0.999755i	$0.507039\pi$
$402$	0	0
$403$	4.42906	0.220627
$404$	0	0
$405$	3.51322	0.174573
$406$	0	0
$407$	−3.12788	−0.155043
$408$	0	0
$409$	24.3497	1.20401	0.602006	−	0.798491i	$-0.294368\pi$
$409$	24.3497	1.20401	0.602006	+	0.798491i	$0.294368\pi$
$410$	0	0
$411$	11.3872	0.561688
$412$	0	0
$413$	22.7665	1.12027
$414$	0	0
$415$	4.59749	0.225682
$416$	0	0
$417$	12.0941	0.592249
$418$	0	0
$419$	25.4535	1.24348	0.621742	−	0.783222i	$-0.286425\pi$
$419$	25.4535	1.24348	0.621742	+	0.783222i	$0.286425\pi$
$420$	0	0
$421$	15.2376	0.742634	0.371317	−	0.928506i	$-0.378906\pi$
$421$	15.2376	0.742634	0.371317	+	0.928506i	$0.378906\pi$
$422$	0	0
$423$	9.30876	0.452607
$424$	0	0
$425$	7.63271	0.370241
$426$	0	0
$427$	−20.2001	−0.977551
$428$	0	0
$429$	8.18816	0.395328
$430$	0	0
$431$	−34.6166	−1.66742	−0.833711	−	0.552202i	$-0.813788\pi$
$431$	−34.6166	−1.66742	−0.833711	+	0.552202i	$0.813788\pi$
$432$	0	0
$433$	28.4774	1.36854	0.684268	−	0.729230i	$-0.260122\pi$
$433$	28.4774	1.36854	0.684268	+	0.729230i	$0.260122\pi$
$434$	0	0
$435$	11.7243	0.562138
$436$	0	0
$437$	1.36222	0.0651639
$438$	0	0
$439$	21.6009	1.03095	0.515477	−	0.856904i	$-0.327615\pi$
$439$	21.6009	1.03095	0.515477	+	0.856904i	$0.327615\pi$
$440$	0	0
$441$	−18.8834	−0.899211
$442$	0	0
$443$	40.3700	1.91804	0.959018	−	0.283346i	$-0.0914445\pi$
$443$	40.3700	1.91804	0.959018	+	0.283346i	$0.0914445\pi$
$444$	0	0
$445$	4.70241	0.222915
$446$	0	0
$447$	−24.3903	−1.15362
$448$	0	0
$449$	5.95484	0.281026	0.140513	−	0.990079i	$-0.455125\pi$
$449$	5.95484	0.281026	0.140513	+	0.990079i	$0.455125\pi$
$450$	0	0
$451$	23.8931	1.12508
$452$	0	0
$453$	−22.8512	−1.07365
$454$	0	0
$455$	−13.0573	−0.612134
$456$	0	0
$457$	−5.66169	−0.264843	−0.132421	−	0.991194i	$-0.542275\pi$
$457$	−5.66169	−0.264843	−0.132421	+	0.991194i	$0.542275\pi$
$458$	0	0
$459$	42.8400	1.99960
$460$	0	0
$461$	−9.26391	−0.431463	−0.215732	−	0.976453i	$-0.569214\pi$
$461$	−9.26391	−0.431463	−0.215732	+	0.976453i	$0.569214\pi$
$462$	0	0
$463$	37.9899	1.76554	0.882769	−	0.469807i	$-0.155676\pi$
$463$	37.9899	1.76554	0.882769	+	0.469807i	$0.155676\pi$
$464$	0	0
$465$	−2.07308	−0.0961368
$466$	0	0
$467$	−26.2865	−1.21640	−0.608198	−	0.793785i	$-0.708107\pi$
$467$	−26.2865	−1.21640	−0.608198	+	0.793785i	$0.708107\pi$
$468$	0	0
$469$	−39.8310	−1.83922
$470$	0	0
$471$	−1.10051	−0.0507087
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−1.36222	−0.0625030
$476$	0	0
$477$	10.7417	0.491831
$478$	0	0
$479$	−31.0770	−1.41994	−0.709971	−	0.704231i	$-0.751292\pi$
$479$	−31.0770	−1.41994	−0.709971	+	0.704231i	$0.751292\pi$
$480$	0	0
$481$	3.92804	0.179103
$482$	0	0
$483$	−6.11163	−0.278089
$484$	0	0
$485$	−16.5160	−0.749954
$486$	0	0
$487$	14.9591	0.677860	0.338930	−	0.940812i	$-0.389935\pi$
$487$	14.9591	0.677860	0.338930	+	0.940812i	$0.389935\pi$
$488$	0	0
$489$	19.0965	0.863573
$490$	0	0
$491$	24.1333	1.08912	0.544561	−	0.838721i	$-0.316696\pi$
$491$	24.1333	1.08912	0.544561	+	0.838721i	$0.316696\pi$
$492$	0	0
$493$	68.2641	3.07446
$494$	0	0
$495$	2.85802	0.128458
$496$	0	0
$497$	−41.1014	−1.84365
$498$	0	0
$499$	−17.7266	−0.793552	−0.396776	−	0.917915i	$-0.629871\pi$
$499$	−17.7266	−0.793552	−0.396776	+	0.917915i	$0.629871\pi$
$500$	0	0
$501$	6.60492	0.295086
$502$	0	0
$503$	−7.69566	−0.343133	−0.171566	−	0.985173i	$-0.554883\pi$
$503$	−7.69566	−0.343133	−0.171566	+	0.985173i	$0.554883\pi$
$504$	0	0
$505$	−10.7267	−0.477334
$506$	0	0
$507$	6.75906	0.300180
$508$	0	0
$509$	−33.0564	−1.46520	−0.732600	−	0.680659i	$-0.761693\pi$
$509$	−33.0564	−1.46520	−0.732600	+	0.680659i	$0.761693\pi$
$510$	0	0
$511$	24.5410	1.08563
$512$	0	0
$513$	−7.64572	−0.337567
$514$	0	0
$515$	−1.95300	−0.0860595
$516$	0	0
$517$	−16.2000	−0.712474
$518$	0	0
$519$	14.8296	0.650947
$520$	0	0
$521$	10.6047	0.464598	0.232299	−	0.972644i	$-0.425375\pi$
$521$	10.6047	0.464598	0.232299	+	0.972644i	$0.425375\pi$
$522$	0	0
$523$	−34.5128	−1.50914	−0.754570	−	0.656219i	$-0.772155\pi$
$523$	−34.5128	−1.50914	−0.754570	+	0.656219i	$0.772155\pi$
$524$	0	0
$525$	6.11163	0.266733
$526$	0	0
$527$	−12.0704	−0.525794
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	6.25799	0.271574
$532$	0	0
$533$	−30.0053	−1.29967
$534$	0	0
$535$	1.97566	0.0854153
$536$	0	0
$537$	31.8622	1.37495
$538$	0	0
$539$	32.8627	1.41550
$540$	0	0
$541$	−27.3344	−1.17520	−0.587598	−	0.809153i	$-0.699926\pi$
$541$	−27.3344	−1.17520	−0.587598	+	0.809153i	$0.699926\pi$
$542$	0	0
$543$	25.5403	1.09604
$544$	0	0
$545$	8.92360	0.382245
$546$	0	0
$547$	34.6190	1.48020	0.740101	−	0.672496i	$-0.234778\pi$
$547$	34.6190	1.48020	0.740101	+	0.672496i	$0.234778\pi$
$548$	0	0
$549$	−5.55254	−0.236977
$550$	0	0
$551$	−12.1832	−0.519022
$552$	0	0
$553$	−33.0181	−1.40407
$554$	0	0
$555$	−1.83857	−0.0780430
$556$	0	0
$557$	25.2490	1.06983	0.534917	−	0.844905i	$-0.320343\pi$
$557$	25.2490	1.06983	0.534917	+	0.844905i	$0.320343\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−22.3150	−0.942139
$562$	0	0
$563$	41.0793	1.73128	0.865642	−	0.500663i	$-0.166910\pi$
$563$	41.0793	1.73128	0.865642	+	0.500663i	$0.166910\pi$
$564$	0	0
$565$	−17.3486	−0.729860
$566$	0	0
$567$	16.3790	0.687854
$568$	0	0
$569$	−28.4230	−1.19155	−0.595776	−	0.803150i	$-0.703156\pi$
$569$	−28.4230	−1.19155	−0.595776	+	0.803150i	$0.703156\pi$
$570$	0	0
$571$	13.4690	0.563659	0.281830	−	0.959464i	$-0.409059\pi$
$571$	13.4690	0.563659	0.281830	+	0.959464i	$0.409059\pi$
$572$	0	0
$573$	−3.43698	−0.143582
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	3.69392	0.153780	0.0768899	−	0.997040i	$-0.475501\pi$
$577$	3.69392	0.153780	0.0768899	+	0.997040i	$0.475501\pi$
$578$	0	0
$579$	−22.8534	−0.949757
$580$	0	0
$581$	21.4340	0.889233
$582$	0	0
$583$	−18.6938	−0.774218
$584$	0	0
$585$	−3.58914	−0.148393
$586$	0	0
$587$	19.1042	0.788513	0.394257	−	0.919000i	$-0.371002\pi$
$587$	19.1042	0.788513	0.394257	+	0.919000i	$0.371002\pi$
$588$	0	0
$589$	2.15422	0.0887631
$590$	0	0
$591$	30.7280	1.26398
$592$	0	0
$593$	13.9194	0.571602	0.285801	−	0.958289i	$-0.407740\pi$
$593$	13.9194	0.571602	0.285801	+	0.958289i	$0.407740\pi$
$594$	0	0
$595$	35.5846	1.45883
$596$	0	0
$597$	1.70103	0.0696186
$598$	0	0
$599$	10.9498	0.447396	0.223698	−	0.974659i	$-0.428187\pi$
$599$	10.9498	0.447396	0.223698	+	0.974659i	$0.428187\pi$
$600$	0	0
$601$	28.7034	1.17084	0.585418	−	0.810731i	$-0.300930\pi$
$601$	28.7034	1.17084	0.585418	+	0.810731i	$0.300930\pi$
$602$	0	0
$603$	−10.9486	−0.445862
$604$	0	0
$605$	6.02621	0.245000
$606$	0	0
$607$	−43.7745	−1.77675	−0.888376	−	0.459117i	$-0.848166\pi$
$607$	−43.7745	−1.77675	−0.888376	+	0.459117i	$0.848166\pi$
$608$	0	0
$609$	54.6601	2.21494
$610$	0	0
$611$	20.3442	0.823036
$612$	0	0
$613$	6.59492	0.266366	0.133183	−	0.991091i	$-0.457480\pi$
$613$	6.59492	0.266366	0.133183	+	0.991091i	$0.457480\pi$
$614$	0	0
$615$	14.0444	0.566324
$616$	0	0
$617$	20.3939	0.821029	0.410514	−	0.911854i	$-0.365349\pi$
$617$	20.3939	0.821029	0.410514	+	0.911854i	$0.365349\pi$
$618$	0	0
$619$	0.513389	0.0206348	0.0103174	−	0.999947i	$-0.496716\pi$
$619$	0.513389	0.0206348	0.0103174	+	0.999947i	$0.496716\pi$
$620$	0	0
$621$	−5.61268	−0.225229
$622$	0	0
$623$	21.9232	0.878333
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	3.98259	0.159049
$628$	0	0
$629$	−10.7050	−0.426835
$630$	0	0
$631$	16.3428	0.650596	0.325298	−	0.945612i	$-0.394535\pi$
$631$	16.3428	0.650596	0.325298	+	0.945612i	$0.394535\pi$
$632$	0	0
$633$	−19.3516	−0.769157
$634$	0	0
$635$	−19.4872	−0.773325
$636$	0	0
$637$	−41.2695	−1.63516
$638$	0	0
$639$	−11.2978	−0.446935
$640$	0	0
$641$	−33.7798	−1.33422	−0.667110	−	0.744959i	$-0.732469\pi$
$641$	−33.7798	−1.33422	−0.667110	+	0.744959i	$0.732469\pi$
$642$	0	0
$643$	−2.18090	−0.0860062	−0.0430031	−	0.999075i	$-0.513693\pi$
$643$	−2.18090	−0.0860062	−0.0430031	+	0.999075i	$0.513693\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−34.7346	−1.36556	−0.682778	−	0.730625i	$-0.739229\pi$
$647$	−34.7346	−1.36556	−0.682778	+	0.730625i	$0.739229\pi$
$648$	0	0
$649$	−10.8907	−0.427499
$650$	0	0
$651$	−9.66494	−0.378799
$652$	0	0
$653$	−6.68309	−0.261530	−0.130765	−	0.991413i	$-0.541743\pi$
$653$	−6.68309	−0.261530	−0.130765	+	0.991413i	$0.541743\pi$
$654$	0	0
$655$	−5.68050	−0.221956
$656$	0	0
$657$	6.74575	0.263177
$658$	0	0
$659$	−45.8903	−1.78763	−0.893815	−	0.448435i	$-0.851982\pi$
$659$	−45.8903	−1.78763	−0.893815	+	0.448435i	$0.851982\pi$
$660$	0	0
$661$	13.6251	0.529957	0.264978	−	0.964254i	$-0.414635\pi$
$661$	13.6251	0.529957	0.264978	+	0.964254i	$0.414635\pi$
$662$	0	0
$663$	28.0234	1.08834
$664$	0	0
$665$	−6.35084	−0.246275
$666$	0	0
$667$	−8.94362	−0.346299
$668$	0	0
$669$	−15.5257	−0.600259
$670$	0	0
$671$	9.66304	0.373038
$672$	0	0
$673$	−25.3475	−0.977075	−0.488537	−	0.872543i	$-0.662469\pi$
$673$	−25.3475	−0.977075	−0.488537	+	0.872543i	$0.662469\pi$
$674$	0	0
$675$	5.61268	0.216032
$676$	0	0
$677$	−17.2279	−0.662123	−0.331062	−	0.943609i	$-0.607407\pi$
$677$	−17.2279	−0.662123	−0.331062	+	0.943609i	$0.607407\pi$
$678$	0	0
$679$	−76.9996	−2.95497
$680$	0	0
$681$	6.91579	0.265014
$682$	0	0
$683$	39.5454	1.51316	0.756582	−	0.653899i	$-0.226868\pi$
$683$	39.5454	1.51316	0.756582	+	0.653899i	$0.226868\pi$
$684$	0	0
$685$	8.68645	0.331892
$686$	0	0
$687$	1.62394	0.0619570
$688$	0	0
$689$	23.4759	0.894361
$690$	0	0
$691$	−29.0501	−1.10512	−0.552558	−	0.833474i	$-0.686348\pi$
$691$	−29.0501	−1.10512	−0.552558	+	0.833474i	$0.686348\pi$
$692$	0	0
$693$	13.3244	0.506152
$694$	0	0
$695$	9.22569	0.349950
$696$	0	0
$697$	81.7725	3.09735
$698$	0	0
$699$	−3.49244	−0.132096
$700$	0	0
$701$	−17.7735	−0.671298	−0.335649	−	0.941987i	$-0.608956\pi$
$701$	−17.7735	−0.671298	−0.335649	+	0.941987i	$0.608956\pi$
$702$	0	0
$703$	1.91053	0.0720571
$704$	0	0
$705$	−9.52236	−0.358633
$706$	0	0
$707$	−50.0093	−1.88079
$708$	0	0
$709$	−14.8606	−0.558103	−0.279052	−	0.960276i	$-0.590020\pi$
$709$	−14.8606	−0.558103	−0.279052	+	0.960276i	$0.590020\pi$
$710$	0	0
$711$	−9.07592	−0.340374
$712$	0	0
$713$	1.58140	0.0592240
$714$	0	0
$715$	6.24615	0.233593
$716$	0	0
$717$	−34.4216	−1.28550
$718$	0	0
$719$	27.4365	1.02321	0.511604	−	0.859221i	$-0.329051\pi$
$719$	27.4365	1.02321	0.511604	+	0.859221i	$0.329051\pi$
$720$	0	0
$721$	−9.10512	−0.339092
$722$	0	0
$723$	2.91449	0.108391
$724$	0	0
$725$	8.94362	0.332158
$726$	0	0
$727$	−23.3674	−0.866649	−0.433325	−	0.901238i	$-0.642660\pi$
$727$	−23.3674	−0.866649	−0.433325	+	0.901238i	$0.642660\pi$
$728$	0	0
$729$	26.5754	0.984275
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−2.53381	−0.0935884	−0.0467942	−	0.998905i	$-0.514900\pi$
$733$	−2.53381	−0.0935884	−0.0467942	+	0.998905i	$0.514900\pi$
$734$	0	0
$735$	19.3167	0.712508
$736$	0	0
$737$	19.0538	0.701856
$738$	0	0
$739$	25.9318	0.953918	0.476959	−	0.878925i	$-0.341739\pi$
$739$	25.9318	0.953918	0.476959	+	0.878925i	$0.341739\pi$
$740$	0	0
$741$	−5.00139	−0.183731
$742$	0	0
$743$	−2.11370	−0.0775442	−0.0387721	−	0.999248i	$-0.512345\pi$
$743$	−2.11370	−0.0775442	−0.0387721	+	0.999248i	$0.512345\pi$
$744$	0	0
$745$	−18.6056	−0.681657
$746$	0	0
$747$	5.89172	0.215567
$748$	0	0
$749$	9.21077	0.336554
$750$	0	0
$751$	2.54986	0.0930459	0.0465229	−	0.998917i	$-0.485186\pi$
$751$	2.54986	0.0930459	0.0465229	+	0.998917i	$0.485186\pi$
$752$	0	0
$753$	−17.6749	−0.644107
$754$	0	0
$755$	−17.4316	−0.634399
$756$	0	0
$757$	30.6897	1.11544	0.557718	−	0.830030i	$-0.311677\pi$
$757$	30.6897	1.11544	0.557718	+	0.830030i	$0.311677\pi$
$758$	0	0
$759$	2.92360	0.106120
$760$	0	0
$761$	19.3112	0.700032	0.350016	−	0.936744i	$-0.386176\pi$
$761$	19.3112	0.700032	0.350016	+	0.936744i	$0.386176\pi$
$762$	0	0
$763$	41.6028	1.50612
$764$	0	0
$765$	9.78138	0.353646
$766$	0	0
$767$	13.6767	0.493839
$768$	0	0
$769$	−8.52624	−0.307464	−0.153732	−	0.988113i	$-0.549129\pi$
$769$	−8.52624	−0.307464	−0.153732	+	0.988113i	$0.549129\pi$
$770$	0	0
$771$	−28.8770	−1.03998
$772$	0	0
$773$	29.5107	1.06143	0.530713	−	0.847551i	$-0.321924\pi$
$773$	29.5107	1.06143	0.530713	+	0.847551i	$0.321924\pi$
$774$	0	0
$775$	−1.58140	−0.0568056
$776$	0	0
$777$	−8.57164	−0.307506
$778$	0	0
$779$	−14.5941	−0.522887
$780$	0	0
$781$	19.6615	0.703545
$782$	0	0
$783$	50.1977	1.79392
$784$	0	0
$785$	−0.839497	−0.0299629
$786$	0	0
$787$	−37.1733	−1.32508	−0.662542	−	0.749025i	$-0.730522\pi$
$787$	−37.1733	−1.32508	−0.662542	+	0.749025i	$0.730522\pi$
$788$	0	0
$789$	−29.2548	−1.04150
$790$	0	0
$791$	−80.8811	−2.87580
$792$	0	0
$793$	−12.1350	−0.430926
$794$	0	0
$795$	−10.9882	−0.389712
$796$	0	0
$797$	16.9954	0.602008	0.301004	−	0.953623i	$-0.402678\pi$
$797$	16.9954	0.602008	0.301004	+	0.953623i	$0.402678\pi$
$798$	0	0
$799$	−55.4434	−1.96144
$800$	0	0
$801$	6.02617	0.212924
$802$	0	0
$803$	−11.7396	−0.414281
$804$	0	0
$805$	−4.66212	−0.164318
$806$	0	0
$807$	−13.7711	−0.484767
$808$	0	0
$809$	16.8409	0.592095	0.296047	−	0.955173i	$-0.404331\pi$
$809$	16.8409	0.592095	0.296047	+	0.955173i	$0.404331\pi$
$810$	0	0
$811$	4.44804	0.156192	0.0780959	−	0.996946i	$-0.475116\pi$
$811$	4.44804	0.156192	0.0780959	+	0.996946i	$0.475116\pi$
$812$	0	0
$813$	5.04303	0.176867
$814$	0	0
$815$	14.5673	0.510271
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−16.7330	−0.584698
$820$	0	0
$821$	−16.7361	−0.584093	−0.292047	−	0.956404i	$-0.594336\pi$
$821$	−16.7361	−0.584093	−0.292047	+	0.956404i	$0.594336\pi$
$822$	0	0
$823$	−43.6605	−1.52191	−0.760955	−	0.648805i	$-0.775269\pi$
$823$	−43.6605	−1.52191	−0.760955	+	0.648805i	$0.775269\pi$
$824$	0	0
$825$	−2.92360	−0.101787
$826$	0	0
$827$	−37.9750	−1.32052	−0.660260	−	0.751037i	$-0.729554\pi$
$827$	−37.9750	−1.32052	−0.660260	+	0.751037i	$0.729554\pi$
$828$	0	0
$829$	17.9046	0.621851	0.310925	−	0.950434i	$-0.399361\pi$
$829$	17.9046	0.621851	0.310925	+	0.950434i	$0.399361\pi$
$830$	0	0
$831$	−19.4872	−0.676002
$832$	0	0
$833$	112.471	3.89687
$834$	0	0
$835$	5.03842	0.174362
$836$	0	0
$837$	−8.87591	−0.306796
$838$	0	0
$839$	−42.4503	−1.46555	−0.732773	−	0.680473i	$-0.761774\pi$
$839$	−42.4503	−1.46555	−0.732773	+	0.680473i	$0.761774\pi$
$840$	0	0
$841$	50.9884	1.75822
$842$	0	0
$843$	−5.78976	−0.199410
$844$	0	0
$845$	5.15599	0.177372
$846$	0	0
$847$	28.0949	0.965353
$848$	0	0
$849$	−5.09283	−0.174785
$850$	0	0
$851$	1.40251	0.0480775
$852$	0	0
$853$	−22.9952	−0.787339	−0.393670	−	0.919252i	$-0.628795\pi$
$853$	−22.9952	−0.787339	−0.393670	+	0.919252i	$0.628795\pi$
$854$	0	0
$855$	−1.74570	−0.0597016
$856$	0	0
$857$	−9.46053	−0.323166	−0.161583	−	0.986859i	$-0.551660\pi$
$857$	−9.46053	−0.323166	−0.161583	+	0.986859i	$0.551660\pi$
$858$	0	0
$859$	−6.20043	−0.211556	−0.105778	−	0.994390i	$-0.533733\pi$
$859$	−6.20043	−0.211556	−0.105778	+	0.994390i	$0.533733\pi$
$860$	0	0
$861$	65.4765	2.23143
$862$	0	0
$863$	−9.68754	−0.329768	−0.164884	−	0.986313i	$-0.552725\pi$
$863$	−9.68754	−0.329768	−0.164884	+	0.986313i	$0.552725\pi$
$864$	0	0
$865$	11.3124	0.384634
$866$	0	0
$867$	−54.0860	−1.83686
$868$	0	0
$869$	15.7948	0.535801
$870$	0	0
$871$	−23.9280	−0.810771
$872$	0	0
$873$	−21.1654	−0.716340
$874$	0	0
$875$	4.66212	0.157608
$876$	0	0
$877$	51.5663	1.74127	0.870636	−	0.491928i	$-0.163708\pi$
$877$	51.5663	1.74127	0.870636	+	0.491928i	$0.163708\pi$
$878$	0	0
$879$	32.8065	1.10654
$880$	0	0
$881$	−33.5969	−1.13191	−0.565954	−	0.824437i	$-0.691492\pi$
$881$	−33.5969	−1.13191	−0.565954	+	0.824437i	$0.691492\pi$
$882$	0	0
$883$	11.7914	0.396814	0.198407	−	0.980120i	$-0.436423\pi$
$883$	11.7914	0.396814	0.198407	+	0.980120i	$0.436423\pi$
$884$	0	0
$885$	−6.40159	−0.215187
$886$	0	0
$887$	−54.9578	−1.84530	−0.922652	−	0.385634i	$-0.873983\pi$
$887$	−54.9578	−1.84530	−0.922652	+	0.385634i	$0.873983\pi$
$888$	0	0
$889$	−90.8515	−3.04706
$890$	0	0
$891$	−7.83517	−0.262488
$892$	0	0
$893$	9.89506	0.331126
$894$	0	0
$895$	24.3053	0.812438
$896$	0	0
$897$	−3.67149	−0.122588
$898$	0	0
$899$	−14.1435	−0.471711
$900$	0	0
$901$	−63.9783	−2.13143
$902$	0	0
$903$	0	0
$904$	0	0
$905$	19.4829	0.647632
$906$	0	0
$907$	8.41429	0.279392	0.139696	−	0.990194i	$-0.455387\pi$
$907$	8.41429	0.279392	0.139696	+	0.990194i	$0.455387\pi$
$908$	0	0
$909$	−13.7464	−0.455940
$910$	0	0
$911$	−26.4143	−0.875146	−0.437573	−	0.899183i	$-0.644162\pi$
$911$	−26.4143	−0.875146	−0.437573	+	0.899183i	$0.644162\pi$
$912$	0	0
$913$	−10.2533	−0.339335
$914$	0	0
$915$	5.67994	0.187773
$916$	0	0
$917$	−26.4832	−0.874551
$918$	0	0
$919$	59.2734	1.95525	0.977624	−	0.210359i	$-0.0674633\pi$
$919$	59.2734	1.95525	0.977624	+	0.210359i	$0.0674633\pi$
$920$	0	0
$921$	−28.8659	−0.951164
$922$	0	0
$923$	−24.6912	−0.812722
$924$	0	0
$925$	−1.40251	−0.0461143
$926$	0	0
$927$	−2.50279	−0.0822023
$928$	0	0
$929$	−17.2475	−0.565871	−0.282935	−	0.959139i	$-0.591308\pi$
$929$	−17.2475	−0.565871	−0.282935	+	0.959139i	$0.591308\pi$
$930$	0	0
$931$	−20.0728	−0.657859
$932$	0	0
$933$	−25.7223	−0.842111
$934$	0	0
$935$	−17.0225	−0.556694
$936$	0	0
$937$	−21.6918	−0.708642	−0.354321	−	0.935124i	$-0.615288\pi$
$937$	−21.6918	−0.708642	−0.354321	+	0.935124i	$0.615288\pi$
$938$	0	0
$939$	23.5303	0.767883
$940$	0	0
$941$	−11.1158	−0.362365	−0.181182	−	0.983449i	$-0.557992\pi$
$941$	−11.1158	−0.362365	−0.181182	+	0.983449i	$0.557992\pi$
$942$	0	0
$943$	−10.7134	−0.348877
$944$	0	0
$945$	26.1670	0.851212
$946$	0	0
$947$	−40.8093	−1.32613	−0.663063	−	0.748564i	$-0.730744\pi$
$947$	−40.8093	−1.32613	−0.663063	+	0.748564i	$0.730744\pi$
$948$	0	0
$949$	14.7427	0.478569
$950$	0	0
$951$	−44.9886	−1.45886
$952$	0	0
$953$	11.8237	0.383009	0.191504	−	0.981492i	$-0.438663\pi$
$953$	11.8237	0.383009	0.191504	+	0.981492i	$0.438663\pi$
$954$	0	0
$955$	−2.62183	−0.0848403
$956$	0	0
$957$	−26.1475	−0.845230
$958$	0	0
$959$	40.4973	1.30772
$960$	0	0
$961$	−28.4992	−0.919328
$962$	0	0
$963$	2.53183	0.0815869
$964$	0	0
$965$	−17.4332	−0.561195
$966$	0	0
$967$	27.8536	0.895710	0.447855	−	0.894106i	$-0.352188\pi$
$967$	27.8536	0.895710	0.447855	+	0.894106i	$0.352188\pi$
$968$	0	0
$969$	13.6301	0.437863
$970$	0	0
$971$	−16.5249	−0.530309	−0.265155	−	0.964206i	$-0.585423\pi$
$971$	−16.5249	−0.530309	−0.265155	+	0.964206i	$0.585423\pi$
$972$	0	0
$973$	43.0112	1.37888
$974$	0	0
$975$	3.67149	0.117582
$976$	0	0
$977$	−50.0059	−1.59983	−0.799915	−	0.600114i	$-0.795122\pi$
$977$	−50.0059	−1.59983	−0.799915	+	0.600114i	$0.795122\pi$
$978$	0	0
$979$	−10.4873	−0.335176
$980$	0	0
$981$	11.4357	0.365112
$982$	0	0
$983$	22.7894	0.726869	0.363434	−	0.931620i	$-0.381604\pi$
$983$	22.7894	0.726869	0.363434	+	0.931620i	$0.381604\pi$
$984$	0	0
$985$	23.4402	0.746866
$986$	0	0
$987$	−44.3943	−1.41309
$988$	0	0
$989$	0	0
$990$	0	0
$991$	33.9822	1.07948	0.539740	−	0.841832i	$-0.318522\pi$
$991$	33.9822	1.07948	0.539740	+	0.841832i	$0.318522\pi$
$992$	0	0
$993$	0.392962	0.0124703
$994$	0	0
$995$	1.29759	0.0411365
$996$	0	0
$997$	−17.7258	−0.561382	−0.280691	−	0.959798i	$-0.590564\pi$
$997$	−17.7258	−0.561382	−0.280691	+	0.959798i	$0.590564\pi$
$998$	0	0
$999$	−7.87186	−0.249055

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	920.2.a.j.1.2	✓	5
3.2	odd	2		8280.2.a.bs.1.1		5
4.3	odd	2		1840.2.a.v.1.4		5
5.2	odd	4		4600.2.e.u.4049.7		10
5.3	odd	4		4600.2.e.u.4049.4		10
5.4	even	2		4600.2.a.be.1.4		5
8.3	odd	2		7360.2.a.cp.1.2		5
8.5	even	2		7360.2.a.co.1.4		5
20.19	odd	2		9200.2.a.cu.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.j.1.2	✓	5	1.1	even	1	trivial
1840.2.a.v.1.4		5	4.3	odd	2
4600.2.a.be.1.4		5	5.4	even	2
4600.2.e.u.4049.4		10	5.3	odd	4
4600.2.e.u.4049.7		10	5.2	odd	4
7360.2.a.co.1.4		5	8.5	even	2
7360.2.a.cp.1.2		5	8.3	odd	2
8280.2.a.bs.1.1		5	3.2	odd	2
9200.2.a.cu.1.2		5	20.19	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	920.a (trivial)

\(f(q)\)	\(=\)	\(q-1.31091 q^{3} -1.00000 q^{5} -4.66212 q^{7} -1.28151 q^{9} +2.23020 q^{11} -2.80072 q^{13} +1.31091 q^{15} +7.63271 q^{17} -1.36222 q^{19} +6.11163 q^{21} -1.00000 q^{23} +1.00000 q^{25} +5.61268 q^{27} +8.94362 q^{29} -1.58140 q^{31} -2.92360 q^{33} +4.66212 q^{35} -1.40251 q^{37} +3.67149 q^{39} +10.7134 q^{41} +1.28151 q^{45} -7.26391 q^{47} +14.7353 q^{49} -10.0058 q^{51} -8.38212 q^{53} -2.23020 q^{55} +1.78575 q^{57} -4.88331 q^{59} +4.33282 q^{61} +5.97454 q^{63} +2.80072 q^{65} +8.54355 q^{67} +1.31091 q^{69} +8.81604 q^{71} -5.26391 q^{73} -1.31091 q^{75} -10.3974 q^{77} +7.08222 q^{79} -3.51322 q^{81} -4.59749 q^{83} -7.63271 q^{85} -11.7243 q^{87} -4.70241 q^{89} +13.0573 q^{91} +2.07308 q^{93} +1.36222 q^{95} +16.5160 q^{97} -2.85802 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{5} - 2 q^{7} + 13 q^{9} - q^{11} + 4 q^{13} + 4 q^{17} + 7 q^{19} + 6 q^{21} - 5 q^{23} + 5 q^{25} + 3 q^{27} + 4 q^{29} + 19 q^{31} + 17 q^{33} + 2 q^{35} + 15 q^{37} + 19 q^{39} + 25 q^{41}+ \cdots - 65 q^{99}+O(q^{100}) \) 5 * q - 5 * q^5 - 2 * q^7 + 13 * q^9 - q^11 + 4 * q^13 + 4 * q^17 + 7 * q^19 + 6 * q^21 - 5 * q^23 + 5 * q^25 + 3 * q^27 + 4 * q^29 + 19 * q^31 + 17 * q^33 + 2 * q^35 + 15 * q^37 + 19 * q^39 + 25 * q^41 - 13 * q^45 - 11 * q^47 + 25 * q^49 + 19 * q^51 + 3 * q^53 + q^55 + 48 * q^57 - q^59 - 5 * q^61 - 41 * q^63 - 4 * q^65 + 9 * q^67 + q^71 - q^73 + 18 * q^77 - 2 * q^79 + 57 * q^81 - 45 * q^83 - 4 * q^85 - 9 * q^87 + 6 * q^89 + 11 * q^91 - 39 * q^93 - 7 * q^95 + 25 * q^97 - 65 * q^99

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