LMFDB - Embedded newform 800.4.a.bb.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,4,Mod(1,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("800.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	800.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:	$47.2015280046$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.2106005.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 24x^{2} + 25x - 6 $ x^4 - 2x^3 - 24x^2 + 25*x - 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.610728$ of defining polynomial
Character	$\chi$	$=$	800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.221457 q^{3} +22.7992 q^{7} -26.9510 q^{9} +O(q^{10})$	$q+0.221457 q^{3} +22.7992 q^{7} -26.9510 q^{9} +66.8474 q^{11} -32.9510 q^{13} +35.9510 q^{17} -44.0482 q^{19} +5.04904 q^{21} +139.010 q^{23} -11.9478 q^{27} +134.853 q^{29} -229.321 q^{31} +14.8038 q^{33} +79.1471 q^{37} -7.29722 q^{39} +384.804 q^{41} -251.234 q^{43} -241.490 q^{47} +176.804 q^{49} +7.96159 q^{51} -222.000 q^{53} -9.75478 q^{57} +552.707 q^{59} +494.951 q^{61} -614.460 q^{63} +574.631 q^{67} +30.7847 q^{69} +654.533 q^{71} -1131.17 q^{73} +1524.07 q^{77} -179.504 q^{79} +725.030 q^{81} +810.131 q^{83} +29.8641 q^{87} -29.7548 q^{89} -751.256 q^{91} -50.7847 q^{93} +383.510 q^{97} -1801.60 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 96 q^{9}+O(q^{10}) $ 4 * q + 96 * q^9	$ 4 q + 96 q^{9} + 72 q^{13} - 60 q^{17} + 224 q^{21} - 72 q^{29} - 756 q^{33} + 928 q^{37} + 724 q^{41} - 108 q^{49} - 888 q^{53} + 980 q^{57} + 1776 q^{61} + 3384 q^{69} - 1060 q^{73} + 2224 q^{77} + 7180 q^{81} + 900 q^{89} - 3464 q^{93} - 504 q^{97}+O(q^{100}) $ 4 * q + 96 * q^9 + 72 * q^13 - 60 * q^17 + 224 * q^21 - 72 * q^29 - 756 * q^33 + 928 * q^37 + 724 * q^41 - 108 * q^49 - 888 * q^53 + 980 * q^57 + 1776 * q^61 + 3384 * q^69 - 1060 * q^73 + 2224 * q^77 + 7180 * q^81 + 900 * q^89 - 3464 * q^93 - 504 * q^97

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
$3$	0.221457	0.0426194	0.0213097	−	0.999773i	$-0.493216\pi$
$3$	0.221457	0.0426194	0.0213097	+	0.999773i	$0.493216\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	22.7992	1.23104	0.615521	−	0.788121i	$-0.288946\pi$
$7$	22.7992	1.23104	0.615521	+	0.788121i	$0.288946\pi$
$8$	0	0
$9$	−26.9510	−0.998184
$10$	0	0
$11$	66.8474	1.83230	0.916148	−	0.400840i	$-0.131282\pi$
$11$	66.8474	1.83230	0.916148	+	0.400840i	$0.131282\pi$
$12$	0	0
$13$	−32.9510	−0.702996	−0.351498	−	0.936189i	$-0.614328\pi$
$13$	−32.9510	−0.702996	−0.351498	+	0.936189i	$0.614328\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	35.9510	0.512905	0.256453	−	0.966557i	$-0.417446\pi$
$17$	35.9510	0.512905	0.256453	+	0.966557i	$0.417446\pi$
$18$	0	0
$19$	−44.0482	−0.531861	−0.265930	−	0.963992i	$-0.585679\pi$
$19$	−44.0482	−0.531861	−0.265930	+	0.963992i	$0.585679\pi$
$20$	0	0
$21$	5.04904	0.0524663
$22$	0	0
$23$	139.010	1.26024	0.630121	−	0.776497i	$-0.283005\pi$
$23$	139.010	1.26024	0.630121	+	0.776497i	$0.283005\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−11.9478	−0.0851614
$28$	0	0
$29$	134.853	0.863502	0.431751	−	0.901993i	$-0.357896\pi$
$29$	134.853	0.863502	0.431751	+	0.901993i	$0.357896\pi$
$30$	0	0
$31$	−229.321	−1.32862	−0.664310	−	0.747457i	$-0.731275\pi$
$31$	−229.321	−1.32862	−0.664310	+	0.747457i	$0.731275\pi$
$32$	0	0
$33$	14.8038	0.0780914
$34$	0	0
$35$	0	0
$36$	0	0
$37$	79.1471	0.351668	0.175834	−	0.984420i	$-0.443738\pi$
$37$	79.1471	0.351668	0.175834	+	0.984420i	$0.443738\pi$
$38$	0	0
$39$	−7.29722	−0.0299613
$40$	0	0
$41$	384.804	1.46576	0.732881	−	0.680357i	$-0.238175\pi$
$41$	384.804	1.46576	0.732881	+	0.680357i	$0.238175\pi$
$42$	0	0
$43$	−251.234	−0.890997	−0.445498	−	0.895283i	$-0.646973\pi$
$43$	−251.234	−0.890997	−0.445498	+	0.895283i	$0.646973\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−241.490	−0.749467	−0.374733	−	0.927133i	$-0.622266\pi$
$47$	−241.490	−0.749467	−0.374733	+	0.927133i	$0.622266\pi$
$48$	0	0
$49$	176.804	0.515463
$50$	0	0
$51$	7.96159	0.0218597
$52$	0	0
$53$	−222.000	−0.575359	−0.287680	−	0.957727i	$-0.592884\pi$
$53$	−222.000	−0.575359	−0.287680	+	0.957727i	$0.592884\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−9.75478	−0.0226676
$58$	0	0
$59$	552.707	1.21960	0.609799	−	0.792556i	$-0.291250\pi$
$59$	552.707	1.21960	0.609799	+	0.792556i	$0.291250\pi$
$60$	0	0
$61$	494.951	1.03888	0.519442	−	0.854505i	$-0.326140\pi$
$61$	494.951	1.03888	0.519442	+	0.854505i	$0.326140\pi$
$62$	0	0
$63$	−614.460	−1.22881
$64$	0	0
$65$	0	0
$66$	0	0
$67$	574.631	1.04780	0.523898	−	0.851781i	$-0.324477\pi$
$67$	574.631	1.04780	0.523898	+	0.851781i	$0.324477\pi$
$68$	0	0
$69$	30.7847	0.0537107
$70$	0	0
$71$	654.533	1.09407	0.547034	−	0.837110i	$-0.315757\pi$
$71$	654.533	1.09407	0.547034	+	0.837110i	$0.315757\pi$
$72$	0	0
$73$	−1131.17	−1.81360	−0.906801	−	0.421558i	$-0.861483\pi$
$73$	−1131.17	−1.81360	−0.906801	+	0.421558i	$0.861483\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1524.07	2.25563
$78$	0	0
$79$	−179.504	−0.255643	−0.127821	−	0.991797i	$-0.540798\pi$
$79$	−179.504	−0.255643	−0.127821	+	0.991797i	$0.540798\pi$
$80$	0	0
$81$	725.030	0.994554
$82$	0	0
$83$	810.131	1.07137	0.535683	−	0.844419i	$-0.320054\pi$
$83$	810.131	1.07137	0.535683	+	0.844419i	$0.320054\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	29.8641	0.0368019
$88$	0	0
$89$	−29.7548	−0.0354382	−0.0177191	−	0.999843i	$-0.505640\pi$
$89$	−29.7548	−0.0354382	−0.0177191	+	0.999843i	$0.505640\pi$
$90$	0	0
$91$	−751.256	−0.865418
$92$	0	0
$93$	−50.7847	−0.0566250
$94$	0	0
$95$	0	0
$96$	0	0
$97$	383.510	0.401438	0.200719	−	0.979649i	$-0.435672\pi$
$97$	383.510	0.401438	0.200719	+	0.979649i	$0.435672\pi$
$98$	0	0
$99$	−1801.60	−1.82897
$100$	0	0
$101$	1272.26	1.25342	0.626708	−	0.779254i	$-0.284402\pi$
$101$	1272.26	1.25342	0.626708	+	0.779254i	$0.284402\pi$
$102$	0	0
$103$	1423.51	1.36177	0.680884	−	0.732391i	$-0.261596\pi$
$103$	1423.51	1.36177	0.680884	+	0.732391i	$0.261596\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1663.68	−1.50312	−0.751560	−	0.659665i	$-0.770698\pi$
$107$	−1663.68	−1.50312	−0.751560	+	0.659665i	$0.770698\pi$
$108$	0	0
$109$	1847.87	1.62380	0.811899	−	0.583798i	$-0.198434\pi$
$109$	1847.87	1.62380	0.811899	+	0.583798i	$0.198434\pi$
$110$	0	0
$111$	17.5277	0.0149879
$112$	0	0
$113$	−1063.39	−0.885270	−0.442635	−	0.896702i	$-0.645956\pi$
$113$	−1063.39	−0.885270	−0.442635	+	0.896702i	$0.645956\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	888.060	0.701719
$118$	0	0
$119$	819.653	0.631408
$120$	0	0
$121$	3137.58	2.35731
$122$	0	0
$123$	85.2175	0.0624699
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2120.07	1.48131	0.740653	−	0.671887i	$-0.234516\pi$
$127$	2120.07	1.48131	0.740653	+	0.671887i	$0.234516\pi$
$128$	0	0
$129$	−55.6376	−0.0379738
$130$	0	0
$131$	675.329	0.450410	0.225205	−	0.974311i	$-0.427695\pi$
$131$	675.329	0.450410	0.225205	+	0.974311i	$0.427695\pi$
$132$	0	0
$133$	−1004.26	−0.654743
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1089.92	−0.679695	−0.339848	−	0.940481i	$-0.610375\pi$
$137$	−1089.92	−0.679695	−0.339848	+	0.940481i	$0.610375\pi$
$138$	0	0
$139$	1429.48	0.872283	0.436141	−	0.899878i	$-0.356345\pi$
$139$	1429.48	0.872283	0.436141	+	0.899878i	$0.356345\pi$
$140$	0	0
$141$	−53.4797	−0.0319418
$142$	0	0
$143$	−2202.69	−1.28810
$144$	0	0
$145$	0	0
$146$	0	0
$147$	39.1544	0.0219687
$148$	0	0
$149$	2040.36	1.12183	0.560916	−	0.827873i	$-0.310449\pi$
$149$	2040.36	1.12183	0.560916	+	0.827873i	$0.310449\pi$
$150$	0	0
$151$	−174.232	−0.0938995	−0.0469498	−	0.998897i	$-0.514950\pi$
$151$	−174.232	−0.0938995	−0.0469498	+	0.998897i	$0.514950\pi$
$152$	0	0
$153$	−968.913	−0.511974
$154$	0	0
$155$	0	0
$156$	0	0
$157$	3494.72	1.77649	0.888247	−	0.459367i	$-0.151924\pi$
$157$	3494.72	1.77649	0.888247	+	0.459367i	$0.151924\pi$
$158$	0	0
$159$	−49.1634	−0.0245215
$160$	0	0
$161$	3169.31	1.55141
$162$	0	0
$163$	−2989.55	−1.43656	−0.718282	−	0.695752i	$-0.755071\pi$
$163$	−2989.55	−1.43656	−0.718282	+	0.695752i	$0.755071\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2215.51	−1.02659	−0.513297	−	0.858211i	$-0.671576\pi$
$167$	−2215.51	−1.02659	−0.513297	+	0.858211i	$0.671576\pi$
$168$	0	0
$169$	−1111.23	−0.505796
$170$	0	0
$171$	1187.14	0.530895
$172$	0	0
$173$	−829.646	−0.364606	−0.182303	−	0.983242i	$-0.558355\pi$
$173$	−829.646	−0.364606	−0.182303	+	0.983242i	$0.558355\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	122.401	0.0519785
$178$	0	0
$179$	700.164	0.292362	0.146181	−	0.989258i	$-0.453302\pi$
$179$	700.164	0.292362	0.146181	+	0.989258i	$0.453302\pi$
$180$	0	0
$181$	−1157.22	−0.475222	−0.237611	−	0.971360i	$-0.576364\pi$
$181$	−1157.22	−0.475222	−0.237611	+	0.971360i	$0.576364\pi$
$182$	0	0
$183$	109.610	0.0442767
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2403.23	0.939794
$188$	0	0
$189$	−272.401	−0.104837
$190$	0	0
$191$	−2306.05	−0.873613	−0.436807	−	0.899555i	$-0.643891\pi$
$191$	−2306.05	−0.873613	−0.436807	+	0.899555i	$0.643891\pi$
$192$	0	0
$193$	−567.461	−0.211641	−0.105820	−	0.994385i	$-0.533747\pi$
$193$	−567.461	−0.211641	−0.105820	+	0.994385i	$0.533747\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4301.74	1.55577	0.777885	−	0.628407i	$-0.216293\pi$
$197$	4301.74	1.55577	0.777885	+	0.628407i	$0.216293\pi$
$198$	0	0
$199$	−258.531	−0.0920945	−0.0460473	−	0.998939i	$-0.514662\pi$
$199$	−258.531	−0.0920945	−0.0460473	+	0.998939i	$0.514662\pi$
$200$	0	0
$201$	127.256	0.0446564
$202$	0	0
$203$	3074.54	1.06301
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−3746.45	−1.25795
$208$	0	0
$209$	−2944.51	−0.974526
$210$	0	0
$211$	−3083.01	−1.00589	−0.502946	−	0.864318i	$-0.667750\pi$
$211$	−3083.01	−1.00589	−0.502946	+	0.864318i	$0.667750\pi$
$212$	0	0
$213$	144.951	0.0466285
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−5228.33	−1.63559
$218$	0	0
$219$	−250.505	−0.0772947
$220$	0	0
$221$	−1184.62	−0.360570
$222$	0	0
$223$	5701.49	1.71211	0.856053	−	0.516888i	$-0.172909\pi$
$223$	5701.49	1.71211	0.856053	+	0.516888i	$0.172909\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1893.20	0.553551	0.276775	−	0.960935i	$-0.410734\pi$
$227$	1893.20	0.553551	0.276775	+	0.960935i	$0.410734\pi$
$228$	0	0
$229$	3968.92	1.14530	0.572650	−	0.819800i	$-0.305915\pi$
$229$	3968.92	1.14530	0.572650	+	0.819800i	$0.305915\pi$
$230$	0	0
$231$	337.516	0.0961337
$232$	0	0
$233$	−3215.84	−0.904192	−0.452096	−	0.891969i	$-0.649324\pi$
$233$	−3215.84	−0.904192	−0.452096	+	0.891969i	$0.649324\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−39.7524	−0.0108953
$238$	0	0
$239$	−6465.86	−1.74997	−0.874983	−	0.484154i	$-0.839128\pi$
$239$	−6465.86	−1.74997	−0.874983	+	0.484154i	$0.839128\pi$
$240$	0	0
$241$	2418.68	0.646476	0.323238	−	0.946318i	$-0.395229\pi$
$241$	2418.68	0.646476	0.323238	+	0.946318i	$0.395229\pi$
$242$	0	0
$243$	483.154	0.127549
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1451.43	0.373896
$248$	0	0
$249$	179.409	0.0456610
$250$	0	0
$251$	5331.27	1.34066	0.670332	−	0.742061i	$-0.266152\pi$
$251$	5331.27	1.34066	0.670332	+	0.742061i	$0.266152\pi$
$252$	0	0
$253$	9292.45	2.30914
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6579.19	−1.59688	−0.798441	−	0.602073i	$-0.794342\pi$
$257$	−6579.19	−1.59688	−0.798441	+	0.602073i	$0.794342\pi$
$258$	0	0
$259$	1804.49	0.432918
$260$	0	0
$261$	−3634.41	−0.861933
$262$	0	0
$263$	−5228.27	−1.22581	−0.612907	−	0.790155i	$-0.710000\pi$
$263$	−5228.27	−1.22581	−0.612907	+	0.790155i	$0.710000\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−6.58940	−0.00151036
$268$	0	0
$269$	−335.667	−0.0760818	−0.0380409	−	0.999276i	$-0.512112\pi$
$269$	−335.667	−0.0760818	−0.0380409	+	0.999276i	$0.512112\pi$
$270$	0	0
$271$	5998.13	1.34450	0.672252	−	0.740323i	$-0.265327\pi$
$271$	5998.13	1.34450	0.672252	+	0.740323i	$0.265327\pi$
$272$	0	0
$273$	−166.371	−0.0368836
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1386.93	0.300839	0.150420	−	0.988622i	$-0.451938\pi$
$277$	1386.93	0.300839	0.150420	+	0.988622i	$0.451938\pi$
$278$	0	0
$279$	6180.42	1.32621
$280$	0	0
$281$	−5596.57	−1.18813	−0.594063	−	0.804419i	$-0.702477\pi$
$281$	−5596.57	−1.18813	−0.594063	+	0.804419i	$0.702477\pi$
$282$	0	0
$283$	−1305.37	−0.274191	−0.137095	−	0.990558i	$-0.543777\pi$
$283$	−1305.37	−0.274191	−0.137095	+	0.990558i	$0.543777\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8773.22	1.80441
$288$	0	0
$289$	−3620.53	−0.736928
$290$	0	0
$291$	84.9309	0.0171091
$292$	0	0
$293$	3603.03	0.718400	0.359200	−	0.933261i	$-0.383050\pi$
$293$	3603.03	0.718400	0.359200	+	0.933261i	$0.383050\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−798.681	−0.156041
$298$	0	0
$299$	−4580.51	−0.885945
$300$	0	0
$301$	−5727.94	−1.09685
$302$	0	0
$303$	281.752	0.0534199
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−9456.83	−1.75808	−0.879039	−	0.476750i	$-0.841815\pi$
$307$	−9456.83	−1.75808	−0.879039	+	0.476750i	$0.841815\pi$
$308$	0	0
$309$	315.245	0.0580378
$310$	0	0
$311$	−6805.49	−1.24085	−0.620424	−	0.784266i	$-0.713040\pi$
$311$	−6805.49	−1.24085	−0.620424	+	0.784266i	$0.713040\pi$
$312$	0	0
$313$	3229.55	0.583210	0.291605	−	0.956539i	$-0.405811\pi$
$313$	3229.55	0.583210	0.291605	+	0.956539i	$0.405811\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1536.59	0.272251	0.136125	−	0.990692i	$-0.456535\pi$
$317$	1536.59	0.272251	0.136125	+	0.990692i	$0.456535\pi$
$318$	0	0
$319$	9014.57	1.58219
$320$	0	0
$321$	−368.433	−0.0640621
$322$	0	0
$323$	−1583.58	−0.272794
$324$	0	0
$325$	0	0
$326$	0	0
$327$	409.224	0.0692053
$328$	0	0
$329$	−5505.78	−0.922625
$330$	0	0
$331$	−7492.66	−1.24421	−0.622105	−	0.782934i	$-0.713722\pi$
$331$	−7492.66	−1.24421	−0.622105	+	0.782934i	$0.713722\pi$
$332$	0	0
$333$	−2133.09	−0.351029
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−10345.3	−1.67224	−0.836120	−	0.548546i	$-0.815181\pi$
$337$	−10345.3	−1.67224	−0.836120	+	0.548546i	$0.815181\pi$
$338$	0	0
$339$	−235.496	−0.0377297
$340$	0	0
$341$	−15329.5	−2.43443
$342$	0	0
$343$	−3789.14	−0.596485
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3077.69	−0.476136	−0.238068	−	0.971248i	$-0.576514\pi$
$347$	−3077.69	−0.476136	−0.238068	+	0.971248i	$0.576514\pi$
$348$	0	0
$349$	−2568.28	−0.393917	−0.196958	−	0.980412i	$-0.563106\pi$
$349$	−2568.28	−0.393917	−0.196958	+	0.980412i	$0.563106\pi$
$350$	0	0
$351$	393.692	0.0598682
$352$	0	0
$353$	6843.39	1.03183	0.515916	−	0.856639i	$-0.327452\pi$
$353$	6843.39	1.03183	0.515916	+	0.856639i	$0.327452\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	181.518	0.0269102
$358$	0	0
$359$	−7296.40	−1.07267	−0.536336	−	0.844005i	$-0.680192\pi$
$359$	−7296.40	−1.07267	−0.536336	+	0.844005i	$0.680192\pi$
$360$	0	0
$361$	−4918.75	−0.717124
$362$	0	0
$363$	694.838	0.100467
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−11304.9	−1.60793	−0.803966	−	0.594675i	$-0.797281\pi$
$367$	−11304.9	−1.60793	−0.803966	+	0.594675i	$0.797281\pi$
$368$	0	0
$369$	−10370.8	−1.46310
$370$	0	0
$371$	−5061.42	−0.708291
$372$	0	0
$373$	−1924.45	−0.267142	−0.133571	−	0.991039i	$-0.542644\pi$
$373$	−1924.45	−0.267142	−0.133571	+	0.991039i	$0.542644\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4443.53	−0.607038
$378$	0	0
$379$	−4477.79	−0.606883	−0.303441	−	0.952850i	$-0.598136\pi$
$379$	−4477.79	−0.606883	−0.303441	+	0.952850i	$0.598136\pi$
$380$	0	0
$381$	469.505	0.0631324
$382$	0	0
$383$	175.452	0.0234078	0.0117039	−	0.999932i	$-0.496274\pi$
$383$	175.452	0.0234078	0.0117039	+	0.999932i	$0.496274\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6771.00	0.889378
$388$	0	0
$389$	−1208.94	−0.157573	−0.0787864	−	0.996892i	$-0.525105\pi$
$389$	−1208.94	−0.157573	−0.0787864	+	0.996892i	$0.525105\pi$
$390$	0	0
$391$	4997.54	0.646384
$392$	0	0
$393$	149.556	0.0191962
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−7449.92	−0.941815	−0.470908	−	0.882182i	$-0.656074\pi$
$397$	−7449.92	−0.941815	−0.470908	+	0.882182i	$0.656074\pi$
$398$	0	0
$399$	−222.401	−0.0279047
$400$	0	0
$401$	−2421.19	−0.301517	−0.150759	−	0.988571i	$-0.548172\pi$
$401$	−2421.19	−0.301517	−0.150759	+	0.988571i	$0.548172\pi$
$402$	0	0
$403$	7556.34	0.934015
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5290.78	0.644359
$408$	0	0
$409$	9305.41	1.12499	0.562497	−	0.826799i	$-0.309841\pi$
$409$	9305.41	1.12499	0.562497	+	0.826799i	$0.309841\pi$
$410$	0	0
$411$	−241.371	−0.0289682
$412$	0	0
$413$	12601.3	1.50138
$414$	0	0
$415$	0	0
$416$	0	0
$417$	316.569	0.0371762
$418$	0	0
$419$	15744.4	1.83572	0.917859	−	0.396907i	$-0.129916\pi$
$419$	15744.4	1.83572	0.917859	+	0.396907i	$0.129916\pi$
$420$	0	0
$421$	−7645.39	−0.885068	−0.442534	−	0.896752i	$-0.645920\pi$
$421$	−7645.39	−0.885068	−0.442534	+	0.896752i	$0.645920\pi$
$422$	0	0
$423$	6508.39	0.748106
$424$	0	0
$425$	0	0
$426$	0	0
$427$	11284.5	1.27891
$428$	0	0
$429$	−487.800	−0.0548979
$430$	0	0
$431$	6915.12	0.772829	0.386415	−	0.922325i	$-0.373713\pi$
$431$	6915.12	0.772829	0.386415	+	0.922325i	$0.373713\pi$
$432$	0	0
$433$	14836.3	1.64662	0.823310	−	0.567592i	$-0.192125\pi$
$433$	14836.3	1.64662	0.823310	+	0.567592i	$0.192125\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6123.13	−0.670273
$438$	0	0
$439$	−15194.2	−1.65189	−0.825944	−	0.563752i	$-0.809357\pi$
$439$	−15194.2	−1.65189	−0.825944	+	0.563752i	$0.809357\pi$
$440$	0	0
$441$	−4765.03	−0.514527
$442$	0	0
$443$	−4707.62	−0.504889	−0.252444	−	0.967611i	$-0.581234\pi$
$443$	−4707.62	−0.504889	−0.252444	+	0.967611i	$0.581234\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	451.852	0.0478118
$448$	0	0
$449$	−2980.66	−0.313287	−0.156644	−	0.987655i	$-0.550067\pi$
$449$	−2980.66	−0.313287	−0.156644	+	0.987655i	$0.550067\pi$
$450$	0	0
$451$	25723.1	2.68571
$452$	0	0
$453$	−38.5850	−0.00400194
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−4677.53	−0.478787	−0.239393	−	0.970923i	$-0.576949\pi$
$457$	−4677.53	−0.478787	−0.239393	+	0.970923i	$0.576949\pi$
$458$	0	0
$459$	−429.535	−0.0436797
$460$	0	0
$461$	2410.16	0.243497	0.121749	−	0.992561i	$-0.461150\pi$
$461$	2410.16	0.243497	0.121749	+	0.992561i	$0.461150\pi$
$462$	0	0
$463$	−4926.69	−0.494520	−0.247260	−	0.968949i	$-0.579530\pi$
$463$	−4926.69	−0.494520	−0.247260	+	0.968949i	$0.579530\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1492.64	0.147904	0.0739521	−	0.997262i	$-0.476439\pi$
$467$	1492.64	0.147904	0.0739521	+	0.997262i	$0.476439\pi$
$468$	0	0
$469$	13101.1	1.28988
$470$	0	0
$471$	773.931	0.0757131
$472$	0	0
$473$	−16794.4	−1.63257
$474$	0	0
$475$	0	0
$476$	0	0
$477$	5983.11	0.574314
$478$	0	0
$479$	11609.5	1.10742	0.553708	−	0.832711i	$-0.313212\pi$
$479$	11609.5	1.10742	0.553708	+	0.832711i	$0.313212\pi$
$480$	0	0
$481$	−2607.97	−0.247221
$482$	0	0
$483$	701.867	0.0661202
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−8199.43	−0.762940	−0.381470	−	0.924381i	$-0.624582\pi$
$487$	−8199.43	−0.762940	−0.381470	+	0.924381i	$0.624582\pi$
$488$	0	0
$489$	−662.057	−0.0612255
$490$	0	0
$491$	−6788.22	−0.623927	−0.311963	−	0.950094i	$-0.600987\pi$
$491$	−6788.22	−0.623927	−0.311963	+	0.950094i	$0.600987\pi$
$492$	0	0
$493$	4848.09	0.442894
$494$	0	0
$495$	0	0
$496$	0	0
$497$	14922.8	1.34684
$498$	0	0
$499$	−18758.2	−1.68283	−0.841415	−	0.540390i	$-0.818277\pi$
$499$	−18758.2	−1.68283	−0.841415	+	0.540390i	$0.818277\pi$
$500$	0	0
$501$	−490.640	−0.0437528
$502$	0	0
$503$	13192.9	1.16947	0.584733	−	0.811226i	$-0.301199\pi$
$503$	13192.9	1.16947	0.584733	+	0.811226i	$0.301199\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−246.091	−0.0215567
$508$	0	0
$509$	−22232.9	−1.93607	−0.968033	−	0.250823i	$-0.919299\pi$
$509$	−22232.9	−1.93607	−0.968033	+	0.250823i	$0.919299\pi$
$510$	0	0
$511$	−25789.7	−2.23262
$512$	0	0
$513$	526.280	0.0452940
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−16143.0	−1.37325
$518$	0	0
$519$	−183.731	−0.0155393
$520$	0	0
$521$	−6609.16	−0.555763	−0.277881	−	0.960615i	$-0.589632\pi$
$521$	−6609.16	−0.555763	−0.277881	+	0.960615i	$0.589632\pi$
$522$	0	0
$523$	−13834.2	−1.15665	−0.578323	−	0.815808i	$-0.696292\pi$
$523$	−13834.2	−1.15665	−0.578323	+	0.815808i	$0.696292\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8244.30	−0.681456
$528$	0	0
$529$	7156.73	0.588208
$530$	0	0
$531$	−14896.0	−1.21738
$532$	0	0
$533$	−12679.7	−1.03043
$534$	0	0
$535$	0	0
$536$	0	0
$537$	155.056	0.0124603
$538$	0	0
$539$	11818.9	0.944481
$540$	0	0
$541$	20781.9	1.65154	0.825771	−	0.564005i	$-0.190740\pi$
$541$	20781.9	1.65154	0.825771	+	0.564005i	$0.190740\pi$
$542$	0	0
$543$	−256.273	−0.0202537
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−9805.41	−0.766452	−0.383226	−	0.923655i	$-0.625187\pi$
$547$	−9805.41	−0.766452	−0.383226	+	0.923655i	$0.625187\pi$
$548$	0	0
$549$	−13339.4	−1.03700
$550$	0	0
$551$	−5940.03	−0.459263
$552$	0	0
$553$	−4092.55	−0.314707
$554$	0	0
$555$	0	0
$556$	0	0
$557$	11660.5	0.887023	0.443512	−	0.896269i	$-0.353733\pi$
$557$	11660.5	0.887023	0.443512	+	0.896269i	$0.353733\pi$
$558$	0	0
$559$	8278.41	0.626367
$560$	0	0
$561$	532.212	0.0400535
$562$	0	0
$563$	9368.59	0.701313	0.350656	−	0.936504i	$-0.385958\pi$
$563$	9368.59	0.701313	0.350656	+	0.936504i	$0.385958\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	16530.1	1.22434
$568$	0	0
$569$	9830.77	0.724301	0.362150	−	0.932120i	$-0.382043\pi$
$569$	9830.77	0.724301	0.362150	+	0.932120i	$0.382043\pi$
$570$	0	0
$571$	6529.64	0.478559	0.239279	−	0.970951i	$-0.423089\pi$
$571$	6529.64	0.478559	0.239279	+	0.970951i	$0.423089\pi$
$572$	0	0
$573$	−510.691	−0.0372329
$574$	0	0
$575$	0	0
$576$	0	0
$577$	14167.7	1.02220	0.511098	−	0.859522i	$-0.329239\pi$
$577$	14167.7	1.02220	0.511098	+	0.859522i	$0.329239\pi$
$578$	0	0
$579$	−125.668	−0.00902001
$580$	0	0
$581$	18470.3	1.31890
$582$	0	0
$583$	−14840.1	−1.05423
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−18531.5	−1.30303	−0.651513	−	0.758638i	$-0.725865\pi$
$587$	−18531.5	−1.30303	−0.651513	+	0.758638i	$0.725865\pi$
$588$	0	0
$589$	10101.2	0.706641
$590$	0	0
$591$	952.651	0.0663060
$592$	0	0
$593$	1492.11	0.103328	0.0516642	−	0.998665i	$-0.483547\pi$
$593$	1492.11	0.103328	0.0516642	+	0.998665i	$0.483547\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−57.2536	−0.00392501
$598$	0	0
$599$	9668.23	0.659488	0.329744	−	0.944070i	$-0.393038\pi$
$599$	9668.23	0.659488	0.329744	+	0.944070i	$0.393038\pi$
$600$	0	0
$601$	6895.64	0.468018	0.234009	−	0.972234i	$-0.424815\pi$
$601$	6895.64	0.468018	0.234009	+	0.972234i	$0.424815\pi$
$602$	0	0
$603$	−15486.8	−1.04589
$604$	0	0
$605$	0	0
$606$	0	0
$607$	11614.6	0.776640	0.388320	−	0.921525i	$-0.373056\pi$
$607$	11614.6	0.776640	0.388320	+	0.921525i	$0.373056\pi$
$608$	0	0
$609$	680.878	0.0453047
$610$	0	0
$611$	7957.33	0.526872
$612$	0	0
$613$	−22009.2	−1.45015	−0.725075	−	0.688670i	$-0.758195\pi$
$613$	−22009.2	−1.45015	−0.725075	+	0.688670i	$0.758195\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	10913.5	0.712089	0.356045	−	0.934469i	$-0.384125\pi$
$617$	10913.5	0.712089	0.356045	+	0.934469i	$0.384125\pi$
$618$	0	0
$619$	1100.19	0.0714385	0.0357193	−	0.999362i	$-0.488628\pi$
$619$	1100.19	0.0714385	0.0357193	+	0.999362i	$0.488628\pi$
$620$	0	0
$621$	−1660.86	−0.107324
$622$	0	0
$623$	−678.385	−0.0436259
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−652.082	−0.0415337
$628$	0	0
$629$	2845.42	0.180372
$630$	0	0
$631$	19436.0	1.22621	0.613103	−	0.790003i	$-0.289921\pi$
$631$	19436.0	1.22621	0.613103	+	0.790003i	$0.289921\pi$
$632$	0	0
$633$	−682.754	−0.0428705
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−5825.86	−0.362369
$638$	0	0
$639$	−17640.3	−1.09208
$640$	0	0
$641$	16001.9	0.986014	0.493007	−	0.870025i	$-0.335898\pi$
$641$	16001.9	0.986014	0.493007	+	0.870025i	$0.335898\pi$
$642$	0	0
$643$	7965.67	0.488547	0.244273	−	0.969706i	$-0.421451\pi$
$643$	7965.67	0.488547	0.244273	+	0.969706i	$0.421451\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4065.35	0.247025	0.123513	−	0.992343i	$-0.460584\pi$
$647$	4065.35	0.247025	0.123513	+	0.992343i	$0.460584\pi$
$648$	0	0
$649$	36947.0	2.23466
$650$	0	0
$651$	−1157.85	−0.0697078
$652$	0	0
$653$	−2625.87	−0.157363	−0.0786816	−	0.996900i	$-0.525071\pi$
$653$	−2625.87	−0.157363	−0.0786816	+	0.996900i	$0.525071\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	30486.0	1.81031
$658$	0	0
$659$	−6099.81	−0.360569	−0.180284	−	0.983615i	$-0.557702\pi$
$659$	−6099.81	−0.360569	−0.180284	+	0.983615i	$0.557702\pi$
$660$	0	0
$661$	21701.3	1.27698	0.638490	−	0.769630i	$-0.279559\pi$
$661$	21701.3	1.27698	0.638490	+	0.769630i	$0.279559\pi$
$662$	0	0
$663$	−262.342	−0.0153673
$664$	0	0
$665$	0	0
$666$	0	0
$667$	18745.9	1.08822
$668$	0	0
$669$	1262.63	0.0729690
$670$	0	0
$671$	33086.2	1.90354
$672$	0	0
$673$	27560.5	1.57857	0.789287	−	0.614024i	$-0.210450\pi$
$673$	27560.5	1.57857	0.789287	+	0.614024i	$0.210450\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−22643.1	−1.28545	−0.642723	−	0.766099i	$-0.722195\pi$
$677$	−22643.1	−1.28545	−0.642723	+	0.766099i	$0.722195\pi$
$678$	0	0
$679$	8743.71	0.494187
$680$	0	0
$681$	419.262	0.0235920
$682$	0	0
$683$	27407.8	1.53548	0.767739	−	0.640763i	$-0.221382\pi$
$683$	27407.8	1.53548	0.767739	+	0.640763i	$0.221382\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	878.945	0.0488120
$688$	0	0
$689$	7315.11	0.404475
$690$	0	0
$691$	−24189.2	−1.33170	−0.665848	−	0.746088i	$-0.731930\pi$
$691$	−24189.2	−1.33170	−0.665848	+	0.746088i	$0.731930\pi$
$692$	0	0
$693$	−41075.1	−2.25154
$694$	0	0
$695$	0	0
$696$	0	0
$697$	13834.1	0.751797
$698$	0	0
$699$	−712.171	−0.0385361
$700$	0	0
$701$	28135.4	1.51592	0.757961	−	0.652300i	$-0.226196\pi$
$701$	28135.4	1.51592	0.757961	+	0.652300i	$0.226196\pi$
$702$	0	0
$703$	−3486.29	−0.187038
$704$	0	0
$705$	0	0
$706$	0	0
$707$	29006.6	1.54301
$708$	0	0
$709$	20075.8	1.06342	0.531708	−	0.846928i	$-0.321550\pi$
$709$	20075.8	1.06342	0.531708	+	0.846928i	$0.321550\pi$
$710$	0	0
$711$	4837.80	0.255178
$712$	0	0
$713$	−31877.8	−1.67438
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−1431.91	−0.0745825
$718$	0	0
$719$	−17789.9	−0.922742	−0.461371	−	0.887207i	$-0.652642\pi$
$719$	−17789.9	−0.922742	−0.461371	+	0.887207i	$0.652642\pi$
$720$	0	0
$721$	32454.8	1.67639
$722$	0	0
$723$	535.633	0.0275524
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−28884.8	−1.47356	−0.736781	−	0.676132i	$-0.763655\pi$
$727$	−28884.8	−1.47356	−0.736781	+	0.676132i	$0.763655\pi$
$728$	0	0
$729$	−19468.8	−0.989118
$730$	0	0
$731$	−9032.11	−0.456997
$732$	0	0
$733$	22393.4	1.12840	0.564202	−	0.825637i	$-0.309184\pi$
$733$	22393.4	1.12840	0.564202	+	0.825637i	$0.309184\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	38412.6	1.91987
$738$	0	0
$739$	3465.65	0.172511	0.0862557	−	0.996273i	$-0.472510\pi$
$739$	3465.65	0.172511	0.0862557	+	0.996273i	$0.472510\pi$
$740$	0	0
$741$	321.429	0.0159352
$742$	0	0
$743$	12870.3	0.635485	0.317742	−	0.948177i	$-0.397075\pi$
$743$	12870.3	0.635485	0.317742	+	0.948177i	$0.397075\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−21833.8	−1.06942
$748$	0	0
$749$	−37930.5	−1.85040
$750$	0	0
$751$	−19763.7	−0.960304	−0.480152	−	0.877185i	$-0.659419\pi$
$751$	−19763.7	−0.960304	−0.480152	+	0.877185i	$0.659419\pi$
$752$	0	0
$753$	1180.65	0.0571383
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−10045.4	−0.482306	−0.241153	−	0.970487i	$-0.577525\pi$
$757$	−10045.4	−0.482306	−0.241153	+	0.970487i	$0.577525\pi$
$758$	0	0
$759$	2057.88	0.0984140
$760$	0	0
$761$	−28857.9	−1.37464	−0.687318	−	0.726357i	$-0.741212\pi$
$761$	−28857.9	−1.37464	−0.687318	+	0.726357i	$0.741212\pi$
$762$	0	0
$763$	42130.0	1.99896
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−18212.2	−0.857373
$768$	0	0
$769$	7056.72	0.330913	0.165456	−	0.986217i	$-0.447090\pi$
$769$	7056.72	0.330913	0.165456	+	0.986217i	$0.447090\pi$
$770$	0	0
$771$	−1457.01	−0.0680582
$772$	0	0
$773$	−33312.1	−1.55000	−0.775001	−	0.631960i	$-0.782251\pi$
$773$	−33312.1	−1.55000	−0.775001	+	0.631960i	$0.782251\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	399.617	0.0184507
$778$	0	0
$779$	−16949.9	−0.779581
$780$	0	0
$781$	43753.9	2.00466
$782$	0	0
$783$	−1611.20	−0.0735370
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−15240.9	−0.690318	−0.345159	−	0.938544i	$-0.612175\pi$
$787$	−15240.9	−0.690318	−0.345159	+	0.938544i	$0.612175\pi$
$788$	0	0
$789$	−1157.84	−0.0522435
$790$	0	0
$791$	−24244.5	−1.08980
$792$	0	0
$793$	−16309.1	−0.730332
$794$	0	0
$795$	0	0
$796$	0	0
$797$	20742.7	0.921886	0.460943	−	0.887430i	$-0.347511\pi$
$797$	20742.7	0.921886	0.460943	+	0.887430i	$0.347511\pi$
$798$	0	0
$799$	−8681.80	−0.384405
$800$	0	0
$801$	801.920	0.0353738
$802$	0	0
$803$	−75615.5	−3.32306
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−74.3359	−0.00324256
$808$	0	0
$809$	24441.1	1.06218	0.531089	−	0.847316i	$-0.321783\pi$
$809$	24441.1	1.06218	0.531089	+	0.847316i	$0.321783\pi$
$810$	0	0
$811$	18571.3	0.804104	0.402052	−	0.915617i	$-0.368297\pi$
$811$	18571.3	0.804104	0.402052	+	0.915617i	$0.368297\pi$
$812$	0	0
$813$	1328.33	0.0573020
$814$	0	0
$815$	0	0
$816$	0	0
$817$	11066.4	0.473886
$818$	0	0
$819$	20247.1	0.863846
$820$	0	0
$821$	1293.62	0.0549912	0.0274956	−	0.999622i	$-0.491247\pi$
$821$	1293.62	0.0549912	0.0274956	+	0.999622i	$0.491247\pi$
$822$	0	0
$823$	−10159.6	−0.430305	−0.215152	−	0.976580i	$-0.569025\pi$
$823$	−10159.6	−0.430305	−0.215152	+	0.976580i	$0.569025\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−304.104	−0.0127869	−0.00639343	−	0.999980i	$-0.502035\pi$
$827$	−304.104	−0.0127869	−0.00639343	+	0.999980i	$0.502035\pi$
$828$	0	0
$829$	9680.47	0.405569	0.202784	−	0.979223i	$-0.435001\pi$
$829$	9680.47	0.405569	0.202784	+	0.979223i	$0.435001\pi$
$830$	0	0
$831$	307.145	0.0128216
$832$	0	0
$833$	6356.27	0.264384
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2739.88	0.113147
$838$	0	0
$839$	28033.4	1.15354	0.576770	−	0.816907i	$-0.304313\pi$
$839$	28033.4	1.15354	0.576770	+	0.816907i	$0.304313\pi$
$840$	0	0
$841$	−6203.70	−0.254365
$842$	0	0
$843$	−1239.40	−0.0506372
$844$	0	0
$845$	0	0
$846$	0	0
$847$	71534.3	2.90194
$848$	0	0
$849$	−289.083	−0.0116859
$850$	0	0
$851$	11002.2	0.443186
$852$	0	0
$853$	1303.27	0.0523132	0.0261566	−	0.999658i	$-0.491673\pi$
$853$	1303.27	0.0523132	0.0261566	+	0.999658i	$0.491673\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	17269.4	0.688344	0.344172	−	0.938907i	$-0.388160\pi$
$857$	17269.4	0.688344	0.344172	+	0.938907i	$0.388160\pi$
$858$	0	0
$859$	34492.8	1.37006	0.685029	−	0.728515i	$-0.259789\pi$
$859$	34492.8	1.37006	0.685029	+	0.728515i	$0.259789\pi$
$860$	0	0
$861$	1942.89	0.0769031
$862$	0	0
$863$	−9348.29	−0.368737	−0.184368	−	0.982857i	$-0.559024\pi$
$863$	−9348.29	−0.368737	−0.184368	+	0.982857i	$0.559024\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−801.791	−0.0314075
$868$	0	0
$869$	−11999.4	−0.468413
$870$	0	0
$871$	−18934.6	−0.736597
$872$	0	0
$873$	−10335.9	−0.400709
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−44386.3	−1.70903	−0.854516	−	0.519426i	$-0.826146\pi$
$877$	−44386.3	−1.70903	−0.854516	+	0.519426i	$0.826146\pi$
$878$	0	0
$879$	797.916	0.0306178
$880$	0	0
$881$	−17267.9	−0.660354	−0.330177	−	0.943919i	$-0.607108\pi$
$881$	−17267.9	−0.660354	−0.330177	+	0.943919i	$0.607108\pi$
$882$	0	0
$883$	−21218.0	−0.808655	−0.404328	−	0.914614i	$-0.632494\pi$
$883$	−21218.0	−0.808655	−0.404328	+	0.914614i	$0.632494\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−19106.0	−0.723243	−0.361621	−	0.932325i	$-0.617777\pi$
$887$	−19106.0	−0.723243	−0.361621	+	0.932325i	$0.617777\pi$
$888$	0	0
$889$	48336.0	1.82355
$890$	0	0
$891$	48466.4	1.82232
$892$	0	0
$893$	10637.2	0.398612
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−1014.39	−0.0377585
$898$	0	0
$899$	−30924.6	−1.14727
$900$	0	0
$901$	−7981.11	−0.295105
$902$	0	0
$903$	−1268.49	−0.0467473
$904$	0	0
$905$	0	0
$906$	0	0
$907$	7041.23	0.257773	0.128887	−	0.991659i	$-0.458860\pi$
$907$	7041.23	0.257773	0.128887	+	0.991659i	$0.458860\pi$
$908$	0	0
$909$	−34288.7	−1.25114
$910$	0	0
$911$	−4329.40	−0.157453	−0.0787263	−	0.996896i	$-0.525085\pi$
$911$	−4329.40	−0.157453	−0.0787263	+	0.996896i	$0.525085\pi$
$912$	0	0
$913$	54155.1	1.96306
$914$	0	0
$915$	0	0
$916$	0	0
$917$	15397.0	0.554474
$918$	0	0
$919$	6812.79	0.244541	0.122270	−	0.992497i	$-0.460982\pi$
$919$	6812.79	0.244541	0.122270	+	0.992497i	$0.460982\pi$
$920$	0	0
$921$	−2094.28	−0.0749282
$922$	0	0
$923$	−21567.5	−0.769126
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−38364.8	−1.35930
$928$	0	0
$929$	−17406.0	−0.614716	−0.307358	−	0.951594i	$-0.599445\pi$
$929$	−17406.0	−0.614716	−0.307358	+	0.951594i	$0.599445\pi$
$930$	0	0
$931$	−7787.89	−0.274155
$932$	0	0
$933$	−1507.12	−0.0528842
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−24803.5	−0.864775	−0.432387	−	0.901688i	$-0.642329\pi$
$937$	−24803.5	−0.864775	−0.432387	+	0.901688i	$0.642329\pi$
$938$	0	0
$939$	715.206	0.0248561
$940$	0	0
$941$	31855.3	1.10356	0.551781	−	0.833989i	$-0.313948\pi$
$941$	31855.3	1.10356	0.551781	+	0.833989i	$0.313948\pi$
$942$	0	0
$943$	53491.5	1.84721
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1411.03	0.0484185	0.0242093	−	0.999707i	$-0.492293\pi$
$947$	1411.03	0.0484185	0.0242093	+	0.999707i	$0.492293\pi$
$948$	0	0
$949$	37273.0	1.27496
$950$	0	0
$951$	340.288	0.0116032
$952$	0	0
$953$	−30090.1	−1.02278	−0.511392	−	0.859348i	$-0.670870\pi$
$953$	−30090.1	−1.02278	−0.511392	+	0.859348i	$0.670870\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1996.34	0.0674320
$958$	0	0
$959$	−24849.3	−0.836733
$960$	0	0
$961$	22797.0	0.765232
$962$	0	0
$963$	44837.7	1.50039
$964$	0	0
$965$	0	0
$966$	0	0
$967$	5840.40	0.194224	0.0971120	−	0.995273i	$-0.469039\pi$
$967$	5840.40	0.194224	0.0971120	+	0.995273i	$0.469039\pi$
$968$	0	0
$969$	−350.694	−0.0116263
$970$	0	0
$971$	−20428.7	−0.675166	−0.337583	−	0.941296i	$-0.609609\pi$
$971$	−20428.7	−0.675166	−0.337583	+	0.941296i	$0.609609\pi$
$972$	0	0
$973$	32591.1	1.07382
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−39986.7	−1.30940	−0.654702	−	0.755887i	$-0.727206\pi$
$977$	−39986.7	−1.30940	−0.654702	+	0.755887i	$0.727206\pi$
$978$	0	0
$979$	−1989.03	−0.0649333
$980$	0	0
$981$	−49801.9	−1.62085
$982$	0	0
$983$	1096.54	0.0355790	0.0177895	−	0.999842i	$-0.494337\pi$
$983$	1096.54	0.0355790	0.0177895	+	0.999842i	$0.494337\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−1219.29	−0.0393217
$988$	0	0
$989$	−34924.0	−1.12287
$990$	0	0
$991$	12697.6	0.407016	0.203508	−	0.979073i	$-0.434766\pi$
$991$	12697.6	0.407016	0.203508	+	0.979073i	$0.434766\pi$
$992$	0	0
$993$	−1659.30	−0.0530275
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−54735.3	−1.73870	−0.869350	−	0.494197i	$-0.835462\pi$
$997$	−54735.3	−1.73870	−0.869350	+	0.494197i	$0.835462\pi$
$998$	0	0
$999$	−945.635	−0.0299485

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.4.a.bb.1.3	yes	4
4.3	odd	2	inner	800.4.a.bb.1.2	yes	4
5.2	odd	4		800.4.c.o.449.4		8
5.3	odd	4		800.4.c.o.449.6		8
5.4	even	2		800.4.a.ba.1.2	✓	4
8.3	odd	2		1600.4.a.cw.1.3		4
8.5	even	2		1600.4.a.cw.1.2		4
20.3	even	4		800.4.c.o.449.3		8
20.7	even	4		800.4.c.o.449.5		8
20.19	odd	2		800.4.a.ba.1.3	yes	4
40.19	odd	2		1600.4.a.cx.1.2		4
40.29	even	2		1600.4.a.cx.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
800.4.a.ba.1.2	✓	4	5.4	even	2
800.4.a.ba.1.3	yes	4	20.19	odd	2
800.4.a.bb.1.2	yes	4	4.3	odd	2	inner
800.4.a.bb.1.3	yes	4	1.1	even	1	trivial
800.4.c.o.449.3		8	20.3	even	4
800.4.c.o.449.4		8	5.2	odd	4
800.4.c.o.449.5		8	20.7	even	4
800.4.c.o.449.6		8	5.3	odd	4
1600.4.a.cw.1.2		4	8.5	even	2
1600.4.a.cw.1.3		4	8.3	odd	2
1600.4.a.cx.1.2		4	40.19	odd	2
1600.4.a.cx.1.3		4	40.29	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:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	800.a (trivial)

\(f(q)\)	\(=\)	\(q+0.221457 q^{3} +22.7992 q^{7} -26.9510 q^{9} +O(q^{10})\)	\(q+0.221457 q^{3} +22.7992 q^{7} -26.9510 q^{9} +66.8474 q^{11} -32.9510 q^{13} +35.9510 q^{17} -44.0482 q^{19} +5.04904 q^{21} +139.010 q^{23} -11.9478 q^{27} +134.853 q^{29} -229.321 q^{31} +14.8038 q^{33} +79.1471 q^{37} -7.29722 q^{39} +384.804 q^{41} -251.234 q^{43} -241.490 q^{47} +176.804 q^{49} +7.96159 q^{51} -222.000 q^{53} -9.75478 q^{57} +552.707 q^{59} +494.951 q^{61} -614.460 q^{63} +574.631 q^{67} +30.7847 q^{69} +654.533 q^{71} -1131.17 q^{73} +1524.07 q^{77} -179.504 q^{79} +725.030 q^{81} +810.131 q^{83} +29.8641 q^{87} -29.7548 q^{89} -751.256 q^{91} -50.7847 q^{93} +383.510 q^{97} -1801.60 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 96 q^{9}+O(q^{10}) \) 4 * q + 96 * q^9	\( 4 q + 96 q^{9} + 72 q^{13} - 60 q^{17} + 224 q^{21} - 72 q^{29} - 756 q^{33} + 928 q^{37} + 724 q^{41} - 108 q^{49} - 888 q^{53} + 980 q^{57} + 1776 q^{61} + 3384 q^{69} - 1060 q^{73} + 2224 q^{77} + 7180 q^{81} + 900 q^{89} - 3464 q^{93} - 504 q^{97}+O(q^{100}) \) 4 * q + 96 * q^9 + 72 * q^13 - 60 * q^17 + 224 * q^21 - 72 * q^29 - 756 * q^33 + 928 * q^37 + 724 * q^41 - 108 * q^49 - 888 * q^53 + 980 * q^57 + 1776 * q^61 + 3384 * q^69 - 1060 * q^73 + 2224 * q^77 + 7180 * q^81 + 900 * q^89 - 3464 * q^93 - 504 * q^97

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