LMFDB - Embedded newform 8020.2.a.d.1.22

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8020,2,Mod(1,8020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8020, 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("8020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8020 = 2^{2} \cdot 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8020.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:	$64.0400224211$
Analytic rank:	$1$
Dimension:	$29$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.22
Character	$\chi$	$=$	8020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.72093 q^{3} +1.00000 q^{5} -0.760805 q^{7} -0.0384167 q^{9} +O(q^{10})$	$q+1.72093 q^{3} +1.00000 q^{5} -0.760805 q^{7} -0.0384167 q^{9} -0.652862 q^{11} +3.86396 q^{13} +1.72093 q^{15} -4.93483 q^{17} -1.98637 q^{19} -1.30929 q^{21} +1.36607 q^{23} +1.00000 q^{25} -5.22889 q^{27} -10.3212 q^{29} -3.31825 q^{31} -1.12353 q^{33} -0.760805 q^{35} +9.99229 q^{37} +6.64958 q^{39} -9.93098 q^{41} -5.32276 q^{43} -0.0384167 q^{45} +10.0149 q^{47} -6.42118 q^{49} -8.49247 q^{51} +14.4137 q^{53} -0.652862 q^{55} -3.41840 q^{57} -9.17463 q^{59} -7.13964 q^{61} +0.0292276 q^{63} +3.86396 q^{65} +7.04691 q^{67} +2.35090 q^{69} -13.9255 q^{71} -10.7148 q^{73} +1.72093 q^{75} +0.496701 q^{77} +8.03161 q^{79} -8.88327 q^{81} -13.5096 q^{83} -4.93483 q^{85} -17.7621 q^{87} -7.74030 q^{89} -2.93972 q^{91} -5.71046 q^{93} -1.98637 q^{95} +10.7522 q^{97} +0.0250808 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9}+O(q^{10}) $ 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9	$ 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9} + 2 q^{11} - 23 q^{13} - 3 q^{15} - 30 q^{17} - 6 q^{19} - 16 q^{21} - 21 q^{23} + 29 q^{25} - 15 q^{27} - 35 q^{29} - 7 q^{31} - 36 q^{33} - 8 q^{35} - 31 q^{37} - 11 q^{39} - 24 q^{41} - 17 q^{43} + 10 q^{45} - 17 q^{47} + q^{49} + 8 q^{51} - 57 q^{53} + 2 q^{55} - 46 q^{57} - 9 q^{59} - 27 q^{61} - 34 q^{63} - 23 q^{65} - 21 q^{67} - 28 q^{69} - 19 q^{71} - 81 q^{73} - 3 q^{75} - 66 q^{77} - 17 q^{79} - 39 q^{81} - 30 q^{83} - 30 q^{85} - 20 q^{87} - 38 q^{89} + q^{91} - 75 q^{93} - 6 q^{95} - 48 q^{97} - 7 q^{99}+O(q^{100}) $ 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9 + 2 * q^11 - 23 * q^13 - 3 * q^15 - 30 * q^17 - 6 * q^19 - 16 * q^21 - 21 * q^23 + 29 * q^25 - 15 * q^27 - 35 * q^29 - 7 * q^31 - 36 * q^33 - 8 * q^35 - 31 * q^37 - 11 * q^39 - 24 * q^41 - 17 * q^43 + 10 * q^45 - 17 * q^47 + q^49 + 8 * q^51 - 57 * q^53 + 2 * q^55 - 46 * q^57 - 9 * q^59 - 27 * q^61 - 34 * q^63 - 23 * q^65 - 21 * q^67 - 28 * q^69 - 19 * q^71 - 81 * q^73 - 3 * q^75 - 66 * q^77 - 17 * q^79 - 39 * q^81 - 30 * q^83 - 30 * q^85 - 20 * q^87 - 38 * q^89 + q^91 - 75 * q^93 - 6 * q^95 - 48 * q^97 - 7 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	1.72093	0.993577	0.496788	−	0.867872i	$-0.334513\pi$
$3$	1.72093	0.993577	0.496788	+	0.867872i	$0.334513\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.760805	−0.287557	−0.143779	−	0.989610i	$-0.545925\pi$
$7$	−0.760805	−0.287557	−0.143779	+	0.989610i	$0.545925\pi$
$8$	0	0
$9$	−0.0384167	−0.0128056
$10$	0	0
$11$	−0.652862	−0.196845	−0.0984227	−	0.995145i	$-0.531380\pi$
$11$	−0.652862	−0.196845	−0.0984227	+	0.995145i	$0.531380\pi$
$12$	0	0
$13$	3.86396	1.07167	0.535835	−	0.844323i	$-0.319997\pi$
$13$	3.86396	1.07167	0.535835	+	0.844323i	$0.319997\pi$
$14$	0	0
$15$	1.72093	0.444341
$16$	0	0
$17$	−4.93483	−1.19687	−0.598435	−	0.801171i	$-0.704211\pi$
$17$	−4.93483	−1.19687	−0.598435	+	0.801171i	$0.704211\pi$
$18$	0	0
$19$	−1.98637	−0.455705	−0.227852	−	0.973696i	$-0.573170\pi$
$19$	−1.98637	−0.455705	−0.227852	+	0.973696i	$0.573170\pi$
$20$	0	0
$21$	−1.30929	−0.285710
$22$	0	0
$23$	1.36607	0.284845	0.142423	−	0.989806i	$-0.454511\pi$
$23$	1.36607	0.284845	0.142423	+	0.989806i	$0.454511\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.22889	−1.00630
$28$	0	0
$29$	−10.3212	−1.91660	−0.958302	−	0.285759i	$-0.907754\pi$
$29$	−10.3212	−1.91660	−0.958302	+	0.285759i	$0.907754\pi$
$30$	0	0
$31$	−3.31825	−0.595976	−0.297988	−	0.954570i	$-0.596316\pi$
$31$	−3.31825	−0.595976	−0.297988	+	0.954570i	$0.596316\pi$
$32$	0	0
$33$	−1.12353	−0.195581
$34$	0	0
$35$	−0.760805	−0.128600
$36$	0	0
$37$	9.99229	1.64272	0.821361	−	0.570409i	$-0.193215\pi$
$37$	9.99229	1.64272	0.821361	+	0.570409i	$0.193215\pi$
$38$	0	0
$39$	6.64958	1.06479
$40$	0	0
$41$	−9.93098	−1.55096	−0.775479	−	0.631373i	$-0.782492\pi$
$41$	−9.93098	−1.55096	−0.775479	+	0.631373i	$0.782492\pi$
$42$	0	0
$43$	−5.32276	−0.811713	−0.405857	−	0.913937i	$-0.633027\pi$
$43$	−5.32276	−0.811713	−0.405857	+	0.913937i	$0.633027\pi$
$44$	0	0
$45$	−0.0384167	−0.00572683
$46$	0	0
$47$	10.0149	1.46082	0.730410	−	0.683009i	$-0.239329\pi$
$47$	10.0149	1.46082	0.730410	+	0.683009i	$0.239329\pi$
$48$	0	0
$49$	−6.42118	−0.917311
$50$	0	0
$51$	−8.49247	−1.18918
$52$	0	0
$53$	14.4137	1.97987	0.989937	−	0.141512i	$-0.0451965\pi$
$53$	14.4137	1.97987	0.989937	+	0.141512i	$0.0451965\pi$
$54$	0	0
$55$	−0.652862	−0.0880319
$56$	0	0
$57$	−3.41840	−0.452778
$58$	0	0
$59$	−9.17463	−1.19443	−0.597217	−	0.802079i	$-0.703727\pi$
$59$	−9.17463	−1.19443	−0.597217	+	0.802079i	$0.703727\pi$
$60$	0	0
$61$	−7.13964	−0.914137	−0.457069	−	0.889431i	$-0.651101\pi$
$61$	−7.13964	−0.914137	−0.457069	+	0.889431i	$0.651101\pi$
$62$	0	0
$63$	0.0292276	0.00368234
$64$	0	0
$65$	3.86396	0.479265
$66$	0	0
$67$	7.04691	0.860917	0.430458	−	0.902610i	$-0.358352\pi$
$67$	7.04691	0.860917	0.430458	+	0.902610i	$0.358352\pi$
$68$	0	0
$69$	2.35090	0.283016
$70$	0	0
$71$	−13.9255	−1.65266	−0.826329	−	0.563188i	$-0.809575\pi$
$71$	−13.9255	−1.65266	−0.826329	+	0.563188i	$0.809575\pi$
$72$	0	0
$73$	−10.7148	−1.25407	−0.627035	−	0.778991i	$-0.715732\pi$
$73$	−10.7148	−1.25407	−0.627035	+	0.778991i	$0.715732\pi$
$74$	0	0
$75$	1.72093	0.198715
$76$	0	0
$77$	0.496701	0.0566043
$78$	0	0
$79$	8.03161	0.903627	0.451814	−	0.892112i	$-0.350777\pi$
$79$	8.03161	0.903627	0.451814	+	0.892112i	$0.350777\pi$
$80$	0	0
$81$	−8.88327	−0.987030
$82$	0	0
$83$	−13.5096	−1.48287	−0.741435	−	0.671025i	$-0.765854\pi$
$83$	−13.5096	−1.48287	−0.741435	+	0.671025i	$0.765854\pi$
$84$	0	0
$85$	−4.93483	−0.535257
$86$	0	0
$87$	−17.7621	−1.90429
$88$	0	0
$89$	−7.74030	−0.820471	−0.410235	−	0.911980i	$-0.634553\pi$
$89$	−7.74030	−0.820471	−0.410235	+	0.911980i	$0.634553\pi$
$90$	0	0
$91$	−2.93972	−0.308166
$92$	0	0
$93$	−5.71046	−0.592148
$94$	0	0
$95$	−1.98637	−0.203797
$96$	0	0
$97$	10.7522	1.09172	0.545862	−	0.837875i	$-0.316202\pi$
$97$	10.7522	1.09172	0.545862	+	0.837875i	$0.316202\pi$
$98$	0	0
$99$	0.0250808	0.00252072
$100$	0	0
$101$	−7.65730	−0.761930	−0.380965	−	0.924589i	$-0.624408\pi$
$101$	−7.65730	−0.761930	−0.380965	+	0.924589i	$0.624408\pi$
$102$	0	0
$103$	−15.6735	−1.54436	−0.772179	−	0.635405i	$-0.780833\pi$
$103$	−15.6735	−1.54436	−0.772179	+	0.635405i	$0.780833\pi$
$104$	0	0
$105$	−1.30929	−0.127773
$106$	0	0
$107$	7.08240	0.684681	0.342341	−	0.939576i	$-0.388780\pi$
$107$	7.08240	0.684681	0.342341	+	0.939576i	$0.388780\pi$
$108$	0	0
$109$	7.04970	0.675238	0.337619	−	0.941283i	$-0.390378\pi$
$109$	7.04970	0.675238	0.337619	+	0.941283i	$0.390378\pi$
$110$	0	0
$111$	17.1960	1.63217
$112$	0	0
$113$	3.17792	0.298954	0.149477	−	0.988765i	$-0.452241\pi$
$113$	3.17792	0.298954	0.149477	+	0.988765i	$0.452241\pi$
$114$	0	0
$115$	1.36607	0.127387
$116$	0	0
$117$	−0.148441	−0.0137233
$118$	0	0
$119$	3.75444	0.344169
$120$	0	0
$121$	−10.5738	−0.961252
$122$	0	0
$123$	−17.0905	−1.54100
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	11.8480	1.05134	0.525672	−	0.850688i	$-0.323814\pi$
$127$	11.8480	1.05134	0.525672	+	0.850688i	$0.323814\pi$
$128$	0	0
$129$	−9.16007	−0.806499
$130$	0	0
$131$	−11.9375	−1.04298	−0.521491	−	0.853257i	$-0.674624\pi$
$131$	−11.9375	−1.04298	−0.521491	+	0.853257i	$0.674624\pi$
$132$	0	0
$133$	1.51124	0.131041
$134$	0	0
$135$	−5.22889	−0.450031
$136$	0	0
$137$	6.25459	0.534365	0.267183	−	0.963646i	$-0.413907\pi$
$137$	6.25459	0.534365	0.267183	+	0.963646i	$0.413907\pi$
$138$	0	0
$139$	21.9129	1.85863	0.929315	−	0.369289i	$-0.120399\pi$
$139$	21.9129	1.85863	0.929315	+	0.369289i	$0.120399\pi$
$140$	0	0
$141$	17.2349	1.45144
$142$	0	0
$143$	−2.52263	−0.210953
$144$	0	0
$145$	−10.3212	−0.857131
$146$	0	0
$147$	−11.0504	−0.911419
$148$	0	0
$149$	6.14550	0.503459	0.251730	−	0.967798i	$-0.419001\pi$
$149$	6.14550	0.503459	0.251730	+	0.967798i	$0.419001\pi$
$150$	0	0
$151$	19.3041	1.57095	0.785474	−	0.618895i	$-0.212419\pi$
$151$	19.3041	1.57095	0.785474	+	0.618895i	$0.212419\pi$
$152$	0	0
$153$	0.189580	0.0153266
$154$	0	0
$155$	−3.31825	−0.266528
$156$	0	0
$157$	−5.14615	−0.410707	−0.205354	−	0.978688i	$-0.565834\pi$
$157$	−5.14615	−0.410707	−0.205354	+	0.978688i	$0.565834\pi$
$158$	0	0
$159$	24.8049	1.96716
$160$	0	0
$161$	−1.03931	−0.0819094
$162$	0	0
$163$	−22.3057	−1.74712	−0.873560	−	0.486717i	$-0.838194\pi$
$163$	−22.3057	−1.74712	−0.873560	+	0.486717i	$0.838194\pi$
$164$	0	0
$165$	−1.12353	−0.0874665
$166$	0	0
$167$	6.61970	0.512248	0.256124	−	0.966644i	$-0.417554\pi$
$167$	6.61970	0.512248	0.256124	+	0.966644i	$0.417554\pi$
$168$	0	0
$169$	1.93017	0.148475
$170$	0	0
$171$	0.0763099	0.00583556
$172$	0	0
$173$	−1.81687	−0.138134	−0.0690669	−	0.997612i	$-0.522002\pi$
$173$	−1.81687	−0.138134	−0.0690669	+	0.997612i	$0.522002\pi$
$174$	0	0
$175$	−0.760805	−0.0575115
$176$	0	0
$177$	−15.7888	−1.18676
$178$	0	0
$179$	−16.1383	−1.20623	−0.603116	−	0.797653i	$-0.706074\pi$
$179$	−16.1383	−1.20623	−0.603116	+	0.797653i	$0.706074\pi$
$180$	0	0
$181$	−12.5680	−0.934171	−0.467086	−	0.884212i	$-0.654696\pi$
$181$	−12.5680	−0.934171	−0.467086	+	0.884212i	$0.654696\pi$
$182$	0	0
$183$	−12.2868	−0.908265
$184$	0	0
$185$	9.99229	0.734648
$186$	0	0
$187$	3.22176	0.235598
$188$	0	0
$189$	3.97816	0.289369
$190$	0	0
$191$	3.42750	0.248005	0.124003	−	0.992282i	$-0.460427\pi$
$191$	3.42750	0.248005	0.124003	+	0.992282i	$0.460427\pi$
$192$	0	0
$193$	8.37515	0.602856	0.301428	−	0.953489i	$-0.402537\pi$
$193$	8.37515	0.602856	0.301428	+	0.953489i	$0.402537\pi$
$194$	0	0
$195$	6.64958	0.476187
$196$	0	0
$197$	−24.8102	−1.76765	−0.883826	−	0.467816i	$-0.845041\pi$
$197$	−24.8102	−1.76765	−0.883826	+	0.467816i	$0.845041\pi$
$198$	0	0
$199$	9.28487	0.658187	0.329094	−	0.944297i	$-0.393257\pi$
$199$	9.28487	0.658187	0.329094	+	0.944297i	$0.393257\pi$
$200$	0	0
$201$	12.1272	0.855387
$202$	0	0
$203$	7.85244	0.551133
$204$	0	0
$205$	−9.93098	−0.693610
$206$	0	0
$207$	−0.0524800	−0.00364761
$208$	0	0
$209$	1.29683	0.0897034
$210$	0	0
$211$	−8.18778	−0.563670	−0.281835	−	0.959463i	$-0.590943\pi$
$211$	−8.18778	−0.563670	−0.281835	+	0.959463i	$0.590943\pi$
$212$	0	0
$213$	−23.9648	−1.64204
$214$	0	0
$215$	−5.32276	−0.363009
$216$	0	0
$217$	2.52454	0.171377
$218$	0	0
$219$	−18.4393	−1.24602
$220$	0	0
$221$	−19.0680	−1.28265
$222$	0	0
$223$	9.79248	0.655753	0.327877	−	0.944721i	$-0.393667\pi$
$223$	9.79248	0.655753	0.327877	+	0.944721i	$0.393667\pi$
$224$	0	0
$225$	−0.0384167	−0.00256112
$226$	0	0
$227$	16.7291	1.11035	0.555173	−	0.831735i	$-0.312652\pi$
$227$	16.7291	1.11035	0.555173	+	0.831735i	$0.312652\pi$
$228$	0	0
$229$	22.5673	1.49129	0.745645	−	0.666343i	$-0.232142\pi$
$229$	22.5673	1.49129	0.745645	+	0.666343i	$0.232142\pi$
$230$	0	0
$231$	0.854785	0.0562407
$232$	0	0
$233$	−20.7163	−1.35717	−0.678584	−	0.734523i	$-0.737406\pi$
$233$	−20.7163	−1.35717	−0.678584	+	0.734523i	$0.737406\pi$
$234$	0	0
$235$	10.0149	0.653299
$236$	0	0
$237$	13.8218	0.897823
$238$	0	0
$239$	−23.5335	−1.52225	−0.761127	−	0.648603i	$-0.775353\pi$
$239$	−23.5335	−1.52225	−0.761127	+	0.648603i	$0.775353\pi$
$240$	0	0
$241$	−10.0016	−0.644258	−0.322129	−	0.946696i	$-0.604398\pi$
$241$	−10.0016	−0.644258	−0.322129	+	0.946696i	$0.604398\pi$
$242$	0	0
$243$	0.399214	0.0256096
$244$	0	0
$245$	−6.42118	−0.410234
$246$	0	0
$247$	−7.67526	−0.488365
$248$	0	0
$249$	−23.2490	−1.47335
$250$	0	0
$251$	8.35994	0.527675	0.263837	−	0.964567i	$-0.415012\pi$
$251$	8.35994	0.527675	0.263837	+	0.964567i	$0.415012\pi$
$252$	0	0
$253$	−0.891856	−0.0560705
$254$	0	0
$255$	−8.49247	−0.531819
$256$	0	0
$257$	8.39120	0.523428	0.261714	−	0.965145i	$-0.415712\pi$
$257$	8.39120	0.523428	0.261714	+	0.965145i	$0.415712\pi$
$258$	0	0
$259$	−7.60218	−0.472377
$260$	0	0
$261$	0.396508	0.0245432
$262$	0	0
$263$	5.57234	0.343605	0.171803	−	0.985131i	$-0.445041\pi$
$263$	5.57234	0.343605	0.171803	+	0.985131i	$0.445041\pi$
$264$	0	0
$265$	14.4137	0.885426
$266$	0	0
$267$	−13.3205	−0.815200
$268$	0	0
$269$	27.9623	1.70489	0.852445	−	0.522817i	$-0.175119\pi$
$269$	27.9623	1.70489	0.852445	+	0.522817i	$0.175119\pi$
$270$	0	0
$271$	−31.0852	−1.88829	−0.944147	−	0.329525i	$-0.893111\pi$
$271$	−31.0852	−1.88829	−0.944147	+	0.329525i	$0.893111\pi$
$272$	0	0
$273$	−5.05904	−0.306187
$274$	0	0
$275$	−0.652862	−0.0393691
$276$	0	0
$277$	−3.44790	−0.207164	−0.103582	−	0.994621i	$-0.533030\pi$
$277$	−3.44790	−0.207164	−0.103582	+	0.994621i	$0.533030\pi$
$278$	0	0
$279$	0.127476	0.00763181
$280$	0	0
$281$	−27.1536	−1.61985	−0.809924	−	0.586535i	$-0.800492\pi$
$281$	−27.1536	−1.61985	−0.809924	+	0.586535i	$0.800492\pi$
$282$	0	0
$283$	11.6485	0.692429	0.346214	−	0.938155i	$-0.387467\pi$
$283$	11.6485	0.692429	0.346214	+	0.938155i	$0.387467\pi$
$284$	0	0
$285$	−3.41840	−0.202488
$286$	0	0
$287$	7.55554	0.445989
$288$	0	0
$289$	7.35250	0.432500
$290$	0	0
$291$	18.5038	1.08471
$292$	0	0
$293$	−8.70418	−0.508504	−0.254252	−	0.967138i	$-0.581829\pi$
$293$	−8.70418	−0.508504	−0.254252	+	0.967138i	$0.581829\pi$
$294$	0	0
$295$	−9.17463	−0.534167
$296$	0	0
$297$	3.41374	0.198085
$298$	0	0
$299$	5.27844	0.305260
$300$	0	0
$301$	4.04958	0.233414
$302$	0	0
$303$	−13.1776	−0.757036
$304$	0	0
$305$	−7.13964	−0.408815
$306$	0	0
$307$	−17.0599	−0.973658	−0.486829	−	0.873497i	$-0.661846\pi$
$307$	−17.0599	−0.973658	−0.486829	+	0.873497i	$0.661846\pi$
$308$	0	0
$309$	−26.9730	−1.53444
$310$	0	0
$311$	−2.65203	−0.150383	−0.0751914	−	0.997169i	$-0.523957\pi$
$311$	−2.65203	−0.150383	−0.0751914	+	0.997169i	$0.523957\pi$
$312$	0	0
$313$	9.65157	0.545539	0.272770	−	0.962079i	$-0.412060\pi$
$313$	9.65157	0.545539	0.272770	+	0.962079i	$0.412060\pi$
$314$	0	0
$315$	0.0292276	0.00164679
$316$	0	0
$317$	−16.0490	−0.901401	−0.450700	−	0.892675i	$-0.648826\pi$
$317$	−16.0490	−0.901401	−0.450700	+	0.892675i	$0.648826\pi$
$318$	0	0
$319$	6.73834	0.377274
$320$	0	0
$321$	12.1883	0.680283
$322$	0	0
$323$	9.80240	0.545420
$324$	0	0
$325$	3.86396	0.214334
$326$	0	0
$327$	12.1320	0.670901
$328$	0	0
$329$	−7.61937	−0.420069
$330$	0	0
$331$	−8.95995	−0.492484	−0.246242	−	0.969208i	$-0.579196\pi$
$331$	−8.95995	−0.492484	−0.246242	+	0.969208i	$0.579196\pi$
$332$	0	0
$333$	−0.383871	−0.0210360
$334$	0	0
$335$	7.04691	0.385014
$336$	0	0
$337$	−15.8951	−0.865861	−0.432931	−	0.901427i	$-0.642520\pi$
$337$	−15.8951	−0.865861	−0.432931	+	0.901427i	$0.642520\pi$
$338$	0	0
$339$	5.46897	0.297034
$340$	0	0
$341$	2.16636	0.117315
$342$	0	0
$343$	10.2109	0.551337
$344$	0	0
$345$	2.35090	0.126568
$346$	0	0
$347$	2.59972	0.139560	0.0697801	−	0.997562i	$-0.477770\pi$
$347$	2.59972	0.139560	0.0697801	+	0.997562i	$0.477770\pi$
$348$	0	0
$349$	24.5046	1.31170	0.655851	−	0.754891i	$-0.272310\pi$
$349$	24.5046	1.31170	0.655851	+	0.754891i	$0.272310\pi$
$350$	0	0
$351$	−20.2042	−1.07842
$352$	0	0
$353$	−7.40740	−0.394256	−0.197128	−	0.980378i	$-0.563161\pi$
$353$	−7.40740	−0.394256	−0.197128	+	0.980378i	$0.563161\pi$
$354$	0	0
$355$	−13.9255	−0.739091
$356$	0	0
$357$	6.46111	0.341958
$358$	0	0
$359$	−21.7628	−1.14860	−0.574299	−	0.818646i	$-0.694725\pi$
$359$	−21.7628	−1.14860	−0.574299	+	0.818646i	$0.694725\pi$
$360$	0	0
$361$	−15.0543	−0.792333
$362$	0	0
$363$	−18.1967	−0.955077
$364$	0	0
$365$	−10.7148	−0.560837
$366$	0	0
$367$	22.8734	1.19398	0.596992	−	0.802247i	$-0.296363\pi$
$367$	22.8734	1.19398	0.596992	+	0.802247i	$0.296363\pi$
$368$	0	0
$369$	0.381516	0.0198609
$370$	0	0
$371$	−10.9660	−0.569327
$372$	0	0
$373$	−27.7839	−1.43860	−0.719299	−	0.694701i	$-0.755537\pi$
$373$	−27.7839	−1.43860	−0.719299	+	0.694701i	$0.755537\pi$
$374$	0	0
$375$	1.72093	0.0888682
$376$	0	0
$377$	−39.8808	−2.05396
$378$	0	0
$379$	17.4528	0.896489	0.448244	−	0.893911i	$-0.352049\pi$
$379$	17.4528	0.896489	0.448244	+	0.893911i	$0.352049\pi$
$380$	0	0
$381$	20.3896	1.04459
$382$	0	0
$383$	34.2663	1.75093	0.875463	−	0.483286i	$-0.160557\pi$
$383$	34.2663	1.75093	0.875463	+	0.483286i	$0.160557\pi$
$384$	0	0
$385$	0.496701	0.0253142
$386$	0	0
$387$	0.204483	0.0103945
$388$	0	0
$389$	10.1379	0.514013	0.257006	−	0.966410i	$-0.417264\pi$
$389$	10.1379	0.514013	0.257006	+	0.966410i	$0.417264\pi$
$390$	0	0
$391$	−6.74132	−0.340923
$392$	0	0
$393$	−20.5435	−1.03628
$394$	0	0
$395$	8.03161	0.404114
$396$	0	0
$397$	20.4215	1.02493	0.512463	−	0.858709i	$-0.328733\pi$
$397$	20.4215	1.02493	0.512463	+	0.858709i	$0.328733\pi$
$398$	0	0
$399$	2.60073	0.130199
$400$	0	0
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	0	0
$403$	−12.8216	−0.638689
$404$	0	0
$405$	−8.88327	−0.441413
$406$	0	0
$407$	−6.52359	−0.323362
$408$	0	0
$409$	8.49301	0.419952	0.209976	−	0.977707i	$-0.432661\pi$
$409$	8.49301	0.419952	0.209976	+	0.977707i	$0.432661\pi$
$410$	0	0
$411$	10.7637	0.530933
$412$	0	0
$413$	6.98010	0.343468
$414$	0	0
$415$	−13.5096	−0.663160
$416$	0	0
$417$	37.7105	1.84669
$418$	0	0
$419$	−17.3374	−0.846989	−0.423495	−	0.905899i	$-0.639197\pi$
$419$	−17.3374	−0.846989	−0.423495	+	0.905899i	$0.639197\pi$
$420$	0	0
$421$	−13.7305	−0.669182	−0.334591	−	0.942363i	$-0.608598\pi$
$421$	−13.7305	−0.669182	−0.334591	+	0.942363i	$0.608598\pi$
$422$	0	0
$423$	−0.384739	−0.0187066
$424$	0	0
$425$	−4.93483	−0.239374
$426$	0	0
$427$	5.43187	0.262867
$428$	0	0
$429$	−4.34126	−0.209598
$430$	0	0
$431$	11.2374	0.541285	0.270643	−	0.962680i	$-0.412764\pi$
$431$	11.2374	0.541285	0.270643	+	0.962680i	$0.412764\pi$
$432$	0	0
$433$	30.5533	1.46830	0.734148	−	0.678990i	$-0.237582\pi$
$433$	30.5533	1.46830	0.734148	+	0.678990i	$0.237582\pi$
$434$	0	0
$435$	−17.7621	−0.851625
$436$	0	0
$437$	−2.71352	−0.129805
$438$	0	0
$439$	−29.2502	−1.39604	−0.698018	−	0.716080i	$-0.745935\pi$
$439$	−29.2502	−1.39604	−0.698018	+	0.716080i	$0.745935\pi$
$440$	0	0
$441$	0.246681	0.0117467
$442$	0	0
$443$	35.5372	1.68842	0.844211	−	0.536010i	$-0.180069\pi$
$443$	35.5372	1.68842	0.844211	+	0.536010i	$0.180069\pi$
$444$	0	0
$445$	−7.74030	−0.366926
$446$	0	0
$447$	10.5759	0.500225
$448$	0	0
$449$	4.98400	0.235209	0.117605	−	0.993060i	$-0.462478\pi$
$449$	4.98400	0.235209	0.117605	+	0.993060i	$0.462478\pi$
$450$	0	0
$451$	6.48356	0.305299
$452$	0	0
$453$	33.2210	1.56086
$454$	0	0
$455$	−2.93972	−0.137816
$456$	0	0
$457$	23.9253	1.11918	0.559589	−	0.828770i	$-0.310959\pi$
$457$	23.9253	1.11918	0.559589	+	0.828770i	$0.310959\pi$
$458$	0	0
$459$	25.8036	1.20441
$460$	0	0
$461$	−0.692805	−0.0322671	−0.0161336	−	0.999870i	$-0.505136\pi$
$461$	−0.692805	−0.0322671	−0.0161336	+	0.999870i	$0.505136\pi$
$462$	0	0
$463$	−17.7380	−0.824357	−0.412178	−	0.911103i	$-0.635232\pi$
$463$	−17.7380	−0.824357	−0.412178	+	0.911103i	$0.635232\pi$
$464$	0	0
$465$	−5.71046	−0.264816
$466$	0	0
$467$	14.1937	0.656805	0.328402	−	0.944538i	$-0.393490\pi$
$467$	14.1937	0.656805	0.328402	+	0.944538i	$0.393490\pi$
$468$	0	0
$469$	−5.36132	−0.247563
$470$	0	0
$471$	−8.85613	−0.408069
$472$	0	0
$473$	3.47503	0.159782
$474$	0	0
$475$	−1.98637	−0.0911410
$476$	0	0
$477$	−0.553727	−0.0253534
$478$	0	0
$479$	−7.19514	−0.328754	−0.164377	−	0.986398i	$-0.552561\pi$
$479$	−7.19514	−0.328754	−0.164377	+	0.986398i	$0.552561\pi$
$480$	0	0
$481$	38.6098	1.76045
$482$	0	0
$483$	−1.78858	−0.0813832
$484$	0	0
$485$	10.7522	0.488234
$486$	0	0
$487$	−1.50836	−0.0683503	−0.0341752	−	0.999416i	$-0.510880\pi$
$487$	−1.50836	−0.0683503	−0.0341752	+	0.999416i	$0.510880\pi$
$488$	0	0
$489$	−38.3865	−1.73590
$490$	0	0
$491$	27.2395	1.22930	0.614650	−	0.788800i	$-0.289297\pi$
$491$	27.2395	1.22930	0.614650	+	0.788800i	$0.289297\pi$
$492$	0	0
$493$	50.9334	2.29393
$494$	0	0
$495$	0.0250808	0.00112730
$496$	0	0
$497$	10.5946	0.475234
$498$	0	0
$499$	2.23977	0.100266	0.0501329	−	0.998743i	$-0.484036\pi$
$499$	2.23977	0.100266	0.0501329	+	0.998743i	$0.484036\pi$
$500$	0	0
$501$	11.3920	0.508958
$502$	0	0
$503$	21.6349	0.964653	0.482326	−	0.875992i	$-0.339792\pi$
$503$	21.6349	0.964653	0.482326	+	0.875992i	$0.339792\pi$
$504$	0	0
$505$	−7.65730	−0.340745
$506$	0	0
$507$	3.32168	0.147521
$508$	0	0
$509$	−27.0233	−1.19779	−0.598894	−	0.800828i	$-0.704393\pi$
$509$	−27.0233	−1.19779	−0.598894	+	0.800828i	$0.704393\pi$
$510$	0	0
$511$	8.15186	0.360617
$512$	0	0
$513$	10.3865	0.458576
$514$	0	0
$515$	−15.6735	−0.690658
$516$	0	0
$517$	−6.53834	−0.287556
$518$	0	0
$519$	−3.12669	−0.137247
$520$	0	0
$521$	−14.2520	−0.624389	−0.312195	−	0.950018i	$-0.601064\pi$
$521$	−14.2520	−0.624389	−0.312195	+	0.950018i	$0.601064\pi$
$522$	0	0
$523$	−12.5054	−0.546824	−0.273412	−	0.961897i	$-0.588152\pi$
$523$	−12.5054	−0.546824	−0.273412	+	0.961897i	$0.588152\pi$
$524$	0	0
$525$	−1.30929	−0.0571420
$526$	0	0
$527$	16.3750	0.713306
$528$	0	0
$529$	−21.1339	−0.918863
$530$	0	0
$531$	0.352459	0.0152954
$532$	0	0
$533$	−38.3729	−1.66211
$534$	0	0
$535$	7.08240	0.306199
$536$	0	0
$537$	−27.7728	−1.19848
$538$	0	0
$539$	4.19214	0.180568
$540$	0	0
$541$	−5.81739	−0.250109	−0.125055	−	0.992150i	$-0.539911\pi$
$541$	−5.81739	−0.250109	−0.125055	+	0.992150i	$0.539911\pi$
$542$	0	0
$543$	−21.6286	−0.928171
$544$	0	0
$545$	7.04970	0.301976
$546$	0	0
$547$	−29.8910	−1.27805	−0.639024	−	0.769187i	$-0.720662\pi$
$547$	−29.8910	−1.27805	−0.639024	+	0.769187i	$0.720662\pi$
$548$	0	0
$549$	0.274282	0.0117061
$550$	0	0
$551$	20.5018	0.873405
$552$	0	0
$553$	−6.11049	−0.259845
$554$	0	0
$555$	17.1960	0.729929
$556$	0	0
$557$	−15.6552	−0.663330	−0.331665	−	0.943397i	$-0.607610\pi$
$557$	−15.6552	−0.663330	−0.331665	+	0.943397i	$0.607610\pi$
$558$	0	0
$559$	−20.5669	−0.869888
$560$	0	0
$561$	5.54441	0.234085
$562$	0	0
$563$	−29.1913	−1.23027	−0.615134	−	0.788423i	$-0.710898\pi$
$563$	−29.1913	−1.23027	−0.615134	+	0.788423i	$0.710898\pi$
$564$	0	0
$565$	3.17792	0.133696
$566$	0	0
$567$	6.75844	0.283828
$568$	0	0
$569$	10.2164	0.428296	0.214148	−	0.976801i	$-0.431303\pi$
$569$	10.2164	0.428296	0.214148	+	0.976801i	$0.431303\pi$
$570$	0	0
$571$	−13.6675	−0.571967	−0.285984	−	0.958234i	$-0.592320\pi$
$571$	−13.6675	−0.571967	−0.285984	+	0.958234i	$0.592320\pi$
$572$	0	0
$573$	5.89847	0.246412
$574$	0	0
$575$	1.36607	0.0569691
$576$	0	0
$577$	14.1970	0.591027	0.295514	−	0.955339i	$-0.404509\pi$
$577$	14.1970	0.591027	0.295514	+	0.955339i	$0.404509\pi$
$578$	0	0
$579$	14.4130	0.598984
$580$	0	0
$581$	10.2782	0.426410
$582$	0	0
$583$	−9.41015	−0.389729
$584$	0	0
$585$	−0.148441	−0.00613727
$586$	0	0
$587$	−28.8087	−1.18906	−0.594532	−	0.804072i	$-0.702663\pi$
$587$	−28.8087	−1.18906	−0.594532	+	0.804072i	$0.702663\pi$
$588$	0	0
$589$	6.59128	0.271589
$590$	0	0
$591$	−42.6965	−1.75630
$592$	0	0
$593$	3.27914	0.134658	0.0673290	−	0.997731i	$-0.478552\pi$
$593$	3.27914	0.134658	0.0673290	+	0.997731i	$0.478552\pi$
$594$	0	0
$595$	3.75444	0.153917
$596$	0	0
$597$	15.9786	0.653959
$598$	0	0
$599$	22.7592	0.929917	0.464959	−	0.885332i	$-0.346069\pi$
$599$	22.7592	0.929917	0.464959	+	0.885332i	$0.346069\pi$
$600$	0	0
$601$	−21.5210	−0.877861	−0.438930	−	0.898521i	$-0.644643\pi$
$601$	−21.5210	−0.877861	−0.438930	+	0.898521i	$0.644643\pi$
$602$	0	0
$603$	−0.270719	−0.0110245
$604$	0	0
$605$	−10.5738	−0.429885
$606$	0	0
$607$	25.5316	1.03629	0.518147	−	0.855291i	$-0.326622\pi$
$607$	25.5316	1.03629	0.518147	+	0.855291i	$0.326622\pi$
$608$	0	0
$609$	13.5135	0.547593
$610$	0	0
$611$	38.6971	1.56552
$612$	0	0
$613$	2.44813	0.0988792	0.0494396	−	0.998777i	$-0.484256\pi$
$613$	2.44813	0.0988792	0.0494396	+	0.998777i	$0.484256\pi$
$614$	0	0
$615$	−17.0905	−0.689154
$616$	0	0
$617$	16.4489	0.662209	0.331104	−	0.943594i	$-0.392579\pi$
$617$	16.4489	0.662209	0.331104	+	0.943594i	$0.392579\pi$
$618$	0	0
$619$	−21.6324	−0.869480	−0.434740	−	0.900556i	$-0.643160\pi$
$619$	−21.6324	−0.869480	−0.434740	+	0.900556i	$0.643160\pi$
$620$	0	0
$621$	−7.14303	−0.286640
$622$	0	0
$623$	5.88886	0.235932
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	2.23174	0.0891272
$628$	0	0
$629$	−49.3102	−1.96613
$630$	0	0
$631$	31.7671	1.26463	0.632314	−	0.774712i	$-0.282105\pi$
$631$	31.7671	1.26463	0.632314	+	0.774712i	$0.282105\pi$
$632$	0	0
$633$	−14.0906	−0.560049
$634$	0	0
$635$	11.8480	0.470175
$636$	0	0
$637$	−24.8112	−0.983054
$638$	0	0
$639$	0.534974	0.0211632
$640$	0	0
$641$	43.3716	1.71308	0.856538	−	0.516084i	$-0.172611\pi$
$641$	43.3716	1.71308	0.856538	+	0.516084i	$0.172611\pi$
$642$	0	0
$643$	33.3120	1.31370	0.656848	−	0.754023i	$-0.271889\pi$
$643$	33.3120	1.31370	0.656848	+	0.754023i	$0.271889\pi$
$644$	0	0
$645$	−9.16007	−0.360677
$646$	0	0
$647$	41.9564	1.64948	0.824738	−	0.565516i	$-0.191323\pi$
$647$	41.9564	1.64948	0.824738	+	0.565516i	$0.191323\pi$
$648$	0	0
$649$	5.98977	0.235119
$650$	0	0
$651$	4.34455	0.170276
$652$	0	0
$653$	−8.93044	−0.349475	−0.174738	−	0.984615i	$-0.555908\pi$
$653$	−8.93044	−0.349475	−0.174738	+	0.984615i	$0.555908\pi$
$654$	0	0
$655$	−11.9375	−0.466436
$656$	0	0
$657$	0.411627	0.0160591
$658$	0	0
$659$	7.71756	0.300634	0.150317	−	0.988638i	$-0.451971\pi$
$659$	7.71756	0.300634	0.150317	+	0.988638i	$0.451971\pi$
$660$	0	0
$661$	24.7869	0.964098	0.482049	−	0.876144i	$-0.339893\pi$
$661$	24.7869	0.964098	0.482049	+	0.876144i	$0.339893\pi$
$662$	0	0
$663$	−32.8145	−1.27441
$664$	0	0
$665$	1.51124	0.0586034
$666$	0	0
$667$	−14.0995	−0.545936
$668$	0	0
$669$	16.8521	0.651541
$670$	0	0
$671$	4.66120	0.179944
$672$	0	0
$673$	−21.3106	−0.821462	−0.410731	−	0.911757i	$-0.634726\pi$
$673$	−21.3106	−0.821462	−0.410731	+	0.911757i	$0.634726\pi$
$674$	0	0
$675$	−5.22889	−0.201260
$676$	0	0
$677$	−8.55536	−0.328809	−0.164405	−	0.986393i	$-0.552570\pi$
$677$	−8.55536	−0.328809	−0.164405	+	0.986393i	$0.552570\pi$
$678$	0	0
$679$	−8.18036	−0.313933
$680$	0	0
$681$	28.7894	1.10321
$682$	0	0
$683$	−1.77696	−0.0679936	−0.0339968	−	0.999422i	$-0.510824\pi$
$683$	−1.77696	−0.0679936	−0.0339968	+	0.999422i	$0.510824\pi$
$684$	0	0
$685$	6.25459	0.238975
$686$	0	0
$687$	38.8367	1.48171
$688$	0	0
$689$	55.6939	2.12177
$690$	0	0
$691$	−31.4584	−1.19673	−0.598366	−	0.801223i	$-0.704183\pi$
$691$	−31.4584	−1.19673	−0.598366	+	0.801223i	$0.704183\pi$
$692$	0	0
$693$	−0.0190816	−0.000724851 0
$694$	0	0
$695$	21.9129	0.831204
$696$	0	0
$697$	49.0077	1.85630
$698$	0	0
$699$	−35.6511	−1.34845
$700$	0	0
$701$	−32.0155	−1.20921	−0.604604	−	0.796526i	$-0.706669\pi$
$701$	−32.0155	−1.20921	−0.604604	+	0.796526i	$0.706669\pi$
$702$	0	0
$703$	−19.8484	−0.748596
$704$	0	0
$705$	17.2349	0.649102
$706$	0	0
$707$	5.82571	0.219099
$708$	0	0
$709$	26.6454	1.00069	0.500345	−	0.865826i	$-0.333207\pi$
$709$	26.6454	1.00069	0.500345	+	0.865826i	$0.333207\pi$
$710$	0	0
$711$	−0.308548	−0.0115715
$712$	0	0
$713$	−4.53297	−0.169761
$714$	0	0
$715$	−2.52263	−0.0943411
$716$	0	0
$717$	−40.4993	−1.51247
$718$	0	0
$719$	18.1215	0.675818	0.337909	−	0.941179i	$-0.390280\pi$
$719$	18.1215	0.675818	0.337909	+	0.941179i	$0.390280\pi$
$720$	0	0
$721$	11.9245	0.444092
$722$	0	0
$723$	−17.2120	−0.640120
$724$	0	0
$725$	−10.3212	−0.383321
$726$	0	0
$727$	−47.1575	−1.74897	−0.874487	−	0.485049i	$-0.838802\pi$
$727$	−47.1575	−1.74897	−0.874487	+	0.485049i	$0.838802\pi$
$728$	0	0
$729$	27.3368	1.01248
$730$	0	0
$731$	26.2669	0.971516
$732$	0	0
$733$	−2.01104	−0.0742796	−0.0371398	−	0.999310i	$-0.511825\pi$
$733$	−2.01104	−0.0742796	−0.0371398	+	0.999310i	$0.511825\pi$
$734$	0	0
$735$	−11.0504	−0.407599
$736$	0	0
$737$	−4.60066	−0.169467
$738$	0	0
$739$	28.5604	1.05061	0.525305	−	0.850914i	$-0.323951\pi$
$739$	28.5604	1.05061	0.525305	+	0.850914i	$0.323951\pi$
$740$	0	0
$741$	−13.2085	−0.485228
$742$	0	0
$743$	−0.505179	−0.0185332	−0.00926660	−	0.999957i	$-0.502950\pi$
$743$	−0.505179	−0.0185332	−0.00926660	+	0.999957i	$0.502950\pi$
$744$	0	0
$745$	6.14550	0.225154
$746$	0	0
$747$	0.518994	0.0189890
$748$	0	0
$749$	−5.38832	−0.196885
$750$	0	0
$751$	−9.74255	−0.355511	−0.177755	−	0.984075i	$-0.556884\pi$
$751$	−9.74255	−0.355511	−0.177755	+	0.984075i	$0.556884\pi$
$752$	0	0
$753$	14.3868	0.524285
$754$	0	0
$755$	19.3041	0.702549
$756$	0	0
$757$	−48.9606	−1.77950	−0.889752	−	0.456443i	$-0.849123\pi$
$757$	−48.9606	−1.77950	−0.889752	+	0.456443i	$0.849123\pi$
$758$	0	0
$759$	−1.53482	−0.0557103
$760$	0	0
$761$	−1.72941	−0.0626909	−0.0313454	−	0.999509i	$-0.509979\pi$
$761$	−1.72941	−0.0626909	−0.0313454	+	0.999509i	$0.509979\pi$
$762$	0	0
$763$	−5.36344	−0.194170
$764$	0	0
$765$	0.189580	0.00685427
$766$	0	0
$767$	−35.4504	−1.28004
$768$	0	0
$769$	−17.7885	−0.641471	−0.320735	−	0.947169i	$-0.603930\pi$
$769$	−17.7885	−0.641471	−0.320735	+	0.947169i	$0.603930\pi$
$770$	0	0
$771$	14.4406	0.520066
$772$	0	0
$773$	−40.0519	−1.44057	−0.720283	−	0.693681i	$-0.755988\pi$
$773$	−40.0519	−1.44057	−0.720283	+	0.693681i	$0.755988\pi$
$774$	0	0
$775$	−3.31825	−0.119195
$776$	0	0
$777$	−13.0828	−0.469342
$778$	0	0
$779$	19.7266	0.706779
$780$	0	0
$781$	9.09146	0.325318
$782$	0	0
$783$	53.9685	1.92868
$784$	0	0
$785$	−5.14615	−0.183674
$786$	0	0
$787$	−38.0521	−1.35641	−0.678206	−	0.734872i	$-0.737242\pi$
$787$	−38.0521	−1.35641	−0.678206	+	0.734872i	$0.737242\pi$
$788$	0	0
$789$	9.58958	0.341398
$790$	0	0
$791$	−2.41778	−0.0859663
$792$	0	0
$793$	−27.5873	−0.979653
$794$	0	0
$795$	24.8049	0.879739
$796$	0	0
$797$	33.2602	1.17814	0.589068	−	0.808083i	$-0.299495\pi$
$797$	33.2602	1.17814	0.589068	+	0.808083i	$0.299495\pi$
$798$	0	0
$799$	−49.4217	−1.74841
$800$	0	0
$801$	0.297357	0.0105066
$802$	0	0
$803$	6.99528	0.246858
$804$	0	0
$805$	−1.03931	−0.0366310
$806$	0	0
$807$	48.1210	1.69394
$808$	0	0
$809$	33.4576	1.17631	0.588153	−	0.808750i	$-0.299855\pi$
$809$	33.4576	1.17631	0.588153	+	0.808750i	$0.299855\pi$
$810$	0	0
$811$	23.1382	0.812492	0.406246	−	0.913764i	$-0.366838\pi$
$811$	23.1382	0.812492	0.406246	+	0.913764i	$0.366838\pi$
$812$	0	0
$813$	−53.4954	−1.87616
$814$	0	0
$815$	−22.3057	−0.781335
$816$	0	0
$817$	10.5730	0.369902
$818$	0	0
$819$	0.112934	0.00394625
$820$	0	0
$821$	−35.8563	−1.25139	−0.625697	−	0.780067i	$-0.715185\pi$
$821$	−35.8563	−1.25139	−0.625697	+	0.780067i	$0.715185\pi$
$822$	0	0
$823$	−22.2014	−0.773894	−0.386947	−	0.922102i	$-0.626470\pi$
$823$	−22.2014	−0.773894	−0.386947	+	0.922102i	$0.626470\pi$
$824$	0	0
$825$	−1.12353	−0.0391162
$826$	0	0
$827$	3.90990	0.135961	0.0679803	−	0.997687i	$-0.478345\pi$
$827$	3.90990	0.135961	0.0679803	+	0.997687i	$0.478345\pi$
$828$	0	0
$829$	−41.4720	−1.44038	−0.720191	−	0.693776i	$-0.755946\pi$
$829$	−41.4720	−1.44038	−0.720191	+	0.693776i	$0.755946\pi$
$830$	0	0
$831$	−5.93357	−0.205833
$832$	0	0
$833$	31.6874	1.09790
$834$	0	0
$835$	6.61970	0.229084
$836$	0	0
$837$	17.3508	0.599730
$838$	0	0
$839$	−1.84404	−0.0636633	−0.0318316	−	0.999493i	$-0.510134\pi$
$839$	−1.84404	−0.0636633	−0.0318316	+	0.999493i	$0.510134\pi$
$840$	0	0
$841$	77.5277	2.67337
$842$	0	0
$843$	−46.7293	−1.60944
$844$	0	0
$845$	1.93017	0.0664000
$846$	0	0
$847$	8.04458	0.276415
$848$	0	0
$849$	20.0461	0.687981
$850$	0	0
$851$	13.6502	0.467922
$852$	0	0
$853$	24.3690	0.834380	0.417190	−	0.908819i	$-0.363015\pi$
$853$	24.3690	0.834380	0.417190	+	0.908819i	$0.363015\pi$
$854$	0	0
$855$	0.0763099	0.00260974
$856$	0	0
$857$	37.5108	1.28135	0.640673	−	0.767814i	$-0.278656\pi$
$857$	37.5108	1.28135	0.640673	+	0.767814i	$0.278656\pi$
$858$	0	0
$859$	3.40008	0.116009	0.0580046	−	0.998316i	$-0.481526\pi$
$859$	3.40008	0.116009	0.0580046	+	0.998316i	$0.481526\pi$
$860$	0	0
$861$	13.0025	0.443125
$862$	0	0
$863$	9.60453	0.326942	0.163471	−	0.986548i	$-0.447731\pi$
$863$	9.60453	0.326942	0.163471	+	0.986548i	$0.447731\pi$
$864$	0	0
$865$	−1.81687	−0.0617753
$866$	0	0
$867$	12.6531	0.429722
$868$	0	0
$869$	−5.24354	−0.177875
$870$	0	0
$871$	27.2290	0.922618
$872$	0	0
$873$	−0.413066	−0.0139802
$874$	0	0
$875$	−0.760805	−0.0257199
$876$	0	0
$877$	14.9669	0.505397	0.252698	−	0.967545i	$-0.418682\pi$
$877$	14.9669	0.505397	0.252698	+	0.967545i	$0.418682\pi$
$878$	0	0
$879$	−14.9792	−0.505237
$880$	0	0
$881$	−27.7123	−0.933652	−0.466826	−	0.884349i	$-0.654603\pi$
$881$	−27.7123	−0.933652	−0.466826	+	0.884349i	$0.654603\pi$
$882$	0	0
$883$	−6.06031	−0.203946	−0.101973	−	0.994787i	$-0.532515\pi$
$883$	−6.06031	−0.203946	−0.101973	+	0.994787i	$0.532515\pi$
$884$	0	0
$885$	−15.7888	−0.530736
$886$	0	0
$887$	29.7934	1.00036	0.500181	−	0.865921i	$-0.333267\pi$
$887$	29.7934	1.00036	0.500181	+	0.865921i	$0.333267\pi$
$888$	0	0
$889$	−9.01405	−0.302321
$890$	0	0
$891$	5.79955	0.194292
$892$	0	0
$893$	−19.8933	−0.665703
$894$	0	0
$895$	−16.1383	−0.539443
$896$	0	0
$897$	9.08380	0.303299
$898$	0	0
$899$	34.2484	1.14225
$900$	0	0
$901$	−71.1291	−2.36965
$902$	0	0
$903$	6.96903	0.231915
$904$	0	0
$905$	−12.5680	−0.417774
$906$	0	0
$907$	38.1062	1.26530	0.632648	−	0.774439i	$-0.281968\pi$
$907$	38.1062	1.26530	0.632648	+	0.774439i	$0.281968\pi$
$908$	0	0
$909$	0.294169	0.00975695
$910$	0	0
$911$	−35.3926	−1.17261	−0.586304	−	0.810091i	$-0.699418\pi$
$911$	−35.3926	−1.17261	−0.586304	+	0.810091i	$0.699418\pi$
$912$	0	0
$913$	8.81990	0.291896
$914$	0	0
$915$	−12.2868	−0.406189
$916$	0	0
$917$	9.08209	0.299917
$918$	0	0
$919$	14.7532	0.486665	0.243332	−	0.969943i	$-0.421759\pi$
$919$	14.7532	0.486665	0.243332	+	0.969943i	$0.421759\pi$
$920$	0	0
$921$	−29.3587	−0.967403
$922$	0	0
$923$	−53.8077	−1.77110
$924$	0	0
$925$	9.99229	0.328544
$926$	0	0
$927$	0.602126	0.0197764
$928$	0	0
$929$	14.2017	0.465942	0.232971	−	0.972484i	$-0.425155\pi$
$929$	14.2017	0.465942	0.232971	+	0.972484i	$0.425155\pi$
$930$	0	0
$931$	12.7548	0.418023
$932$	0	0
$933$	−4.56395	−0.149417
$934$	0	0
$935$	3.22176	0.105363
$936$	0	0
$937$	24.1590	0.789240	0.394620	−	0.918844i	$-0.370876\pi$
$937$	24.1590	0.789240	0.394620	+	0.918844i	$0.370876\pi$
$938$	0	0
$939$	16.6096	0.542035
$940$	0	0
$941$	4.67803	0.152499	0.0762497	−	0.997089i	$-0.475705\pi$
$941$	4.67803	0.152499	0.0762497	+	0.997089i	$0.475705\pi$
$942$	0	0
$943$	−13.5664	−0.441783
$944$	0	0
$945$	3.97816	0.129410
$946$	0	0
$947$	−59.3083	−1.92726	−0.963631	−	0.267237i	$-0.913889\pi$
$947$	−59.3083	−1.92726	−0.963631	+	0.267237i	$0.913889\pi$
$948$	0	0
$949$	−41.4015	−1.34395
$950$	0	0
$951$	−27.6191	−0.895611
$952$	0	0
$953$	−27.9313	−0.904783	−0.452392	−	0.891819i	$-0.649429\pi$
$953$	−27.9313	−0.904783	−0.452392	+	0.891819i	$0.649429\pi$
$954$	0	0
$955$	3.42750	0.110911
$956$	0	0
$957$	11.5962	0.374851
$958$	0	0
$959$	−4.75852	−0.153661
$960$	0	0
$961$	−19.9892	−0.644813
$962$	0	0
$963$	−0.272083	−0.00876774
$964$	0	0
$965$	8.37515	0.269605
$966$	0	0
$967$	31.0464	0.998383	0.499192	−	0.866492i	$-0.333630\pi$
$967$	31.0464	0.998383	0.499192	+	0.866492i	$0.333630\pi$
$968$	0	0
$969$	16.8692	0.541916
$970$	0	0
$971$	17.2101	0.552297	0.276149	−	0.961115i	$-0.410942\pi$
$971$	17.2101	0.552297	0.276149	+	0.961115i	$0.410942\pi$
$972$	0	0
$973$	−16.6715	−0.534462
$974$	0	0
$975$	6.64958	0.212957
$976$	0	0
$977$	−42.5682	−1.36188	−0.680938	−	0.732341i	$-0.738428\pi$
$977$	−42.5682	−1.36188	−0.680938	+	0.732341i	$0.738428\pi$
$978$	0	0
$979$	5.05335	0.161506
$980$	0	0
$981$	−0.270826	−0.00864682
$982$	0	0
$983$	41.5662	1.32576	0.662878	−	0.748728i	$-0.269335\pi$
$983$	41.5662	1.32576	0.662878	+	0.748728i	$0.269335\pi$
$984$	0	0
$985$	−24.8102	−0.790518
$986$	0	0
$987$	−13.1124	−0.417371
$988$	0	0
$989$	−7.27127	−0.231213
$990$	0	0
$991$	−53.7367	−1.70700	−0.853501	−	0.521091i	$-0.825525\pi$
$991$	−53.7367	−1.70700	−0.853501	+	0.521091i	$0.825525\pi$
$992$	0	0
$993$	−15.4194	−0.489320
$994$	0	0
$995$	9.28487	0.294350
$996$	0	0
$997$	24.4742	0.775106	0.387553	−	0.921847i	$-0.373320\pi$
$997$	24.4742	0.775106	0.387553	+	0.921847i	$0.373320\pi$
$998$	0	0
$999$	−52.2486	−1.65307

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.d.1.22	✓	29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.d.1.22	✓	29	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:	\( N \)	\(=\)	\( 8020 = 2^{2} \cdot 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8020.a (trivial)

\(f(q)\)	\(=\)	\(q+1.72093 q^{3} +1.00000 q^{5} -0.760805 q^{7} -0.0384167 q^{9} +O(q^{10})\)	\(q+1.72093 q^{3} +1.00000 q^{5} -0.760805 q^{7} -0.0384167 q^{9} -0.652862 q^{11} +3.86396 q^{13} +1.72093 q^{15} -4.93483 q^{17} -1.98637 q^{19} -1.30929 q^{21} +1.36607 q^{23} +1.00000 q^{25} -5.22889 q^{27} -10.3212 q^{29} -3.31825 q^{31} -1.12353 q^{33} -0.760805 q^{35} +9.99229 q^{37} +6.64958 q^{39} -9.93098 q^{41} -5.32276 q^{43} -0.0384167 q^{45} +10.0149 q^{47} -6.42118 q^{49} -8.49247 q^{51} +14.4137 q^{53} -0.652862 q^{55} -3.41840 q^{57} -9.17463 q^{59} -7.13964 q^{61} +0.0292276 q^{63} +3.86396 q^{65} +7.04691 q^{67} +2.35090 q^{69} -13.9255 q^{71} -10.7148 q^{73} +1.72093 q^{75} +0.496701 q^{77} +8.03161 q^{79} -8.88327 q^{81} -13.5096 q^{83} -4.93483 q^{85} -17.7621 q^{87} -7.74030 q^{89} -2.93972 q^{91} -5.71046 q^{93} -1.98637 q^{95} +10.7522 q^{97} +0.0250808 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9}+O(q^{10}) \) 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9	\( 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9} + 2 q^{11} - 23 q^{13} - 3 q^{15} - 30 q^{17} - 6 q^{19} - 16 q^{21} - 21 q^{23} + 29 q^{25} - 15 q^{27} - 35 q^{29} - 7 q^{31} - 36 q^{33} - 8 q^{35} - 31 q^{37} - 11 q^{39} - 24 q^{41} - 17 q^{43} + 10 q^{45} - 17 q^{47} + q^{49} + 8 q^{51} - 57 q^{53} + 2 q^{55} - 46 q^{57} - 9 q^{59} - 27 q^{61} - 34 q^{63} - 23 q^{65} - 21 q^{67} - 28 q^{69} - 19 q^{71} - 81 q^{73} - 3 q^{75} - 66 q^{77} - 17 q^{79} - 39 q^{81} - 30 q^{83} - 30 q^{85} - 20 q^{87} - 38 q^{89} + q^{91} - 75 q^{93} - 6 q^{95} - 48 q^{97} - 7 q^{99}+O(q^{100}) \) 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9 + 2 * q^11 - 23 * q^13 - 3 * q^15 - 30 * q^17 - 6 * q^19 - 16 * q^21 - 21 * q^23 + 29 * q^25 - 15 * q^27 - 35 * q^29 - 7 * q^31 - 36 * q^33 - 8 * q^35 - 31 * q^37 - 11 * q^39 - 24 * q^41 - 17 * q^43 + 10 * q^45 - 17 * q^47 + q^49 + 8 * q^51 - 57 * q^53 + 2 * q^55 - 46 * q^57 - 9 * q^59 - 27 * q^61 - 34 * q^63 - 23 * q^65 - 21 * q^67 - 28 * q^69 - 19 * q^71 - 81 * q^73 - 3 * q^75 - 66 * q^77 - 17 * q^79 - 39 * q^81 - 30 * q^83 - 30 * q^85 - 20 * q^87 - 38 * q^89 + q^91 - 75 * q^93 - 6 * q^95 - 48 * q^97 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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