LMFDB - Embedded newform 8000.2.a.bz.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8000,2,Mod(1,8000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8000, 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("8000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8000 = 2^{6} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8000.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:	$63.8803216170$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.26208800000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 11x^{6} + 34x^{4} - 30x^{2} + 5 $ x^8 - 11x^6 + 34x^4 - 30*x^2 + 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 4000)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-1.81062$ of defining polynomial
Character	$\chi$	$=$	8000.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.74204 q^{3} +1.42258 q^{7} +4.51880 q^{9} +O(q^{10})$	$q+2.74204 q^{3} +1.42258 q^{7} +4.51880 q^{9} +0.375210 q^{11} +6.69354 q^{13} -1.44329 q^{17} -6.63539 q^{19} +3.90077 q^{21} +6.67477 q^{23} +4.16462 q^{27} -0.174739 q^{29} -7.61770 q^{31} +1.02884 q^{33} +8.13684 q^{37} +18.3540 q^{39} +7.97627 q^{41} +4.98010 q^{43} +1.11109 q^{47} -4.97627 q^{49} -3.95758 q^{51} +1.52786 q^{53} -18.1945 q^{57} -10.1292 q^{59} +8.96751 q^{61} +6.42834 q^{63} -8.03444 q^{67} +18.3025 q^{69} +7.84959 q^{71} +8.05227 q^{73} +0.533765 q^{77} +0.231892 q^{79} -2.13684 q^{81} +7.21897 q^{83} -0.479142 q^{87} +5.85410 q^{89} +9.52208 q^{91} -20.8881 q^{93} +8.49508 q^{97} +1.69550 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^9	$ 8 q + 8 q^{9} + 8 q^{13} - 20 q^{17} + 12 q^{21} + 16 q^{29} - 36 q^{33} + 28 q^{37} + 8 q^{41} + 16 q^{49} + 48 q^{53} - 20 q^{57} + 28 q^{61} + 28 q^{69} + 44 q^{77} + 20 q^{81} + 20 q^{89} + 4 q^{93} - 16 q^{97}+O(q^{100}) $ 8 * q + 8 * q^9 + 8 * q^13 - 20 * q^17 + 12 * q^21 + 16 * q^29 - 36 * q^33 + 28 * q^37 + 8 * q^41 + 16 * q^49 + 48 * q^53 - 20 * q^57 + 28 * q^61 + 28 * q^69 + 44 * q^77 + 20 * q^81 + 20 * q^89 + 4 * q^93 - 16 * 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
$3$	2.74204	1.58312	0.791560	−	0.611092i	$-0.209269\pi$
$3$	2.74204	1.58312	0.791560	+	0.611092i	$0.209269\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.42258	0.537684	0.268842	−	0.963184i	$-0.413359\pi$
$7$	1.42258	0.537684	0.268842	+	0.963184i	$0.413359\pi$
$8$	0	0
$9$	4.51880	1.50627
$10$	0	0
$11$	0.375210	0.113130	0.0565650	−	0.998399i	$-0.481985\pi$
$11$	0.375210	0.113130	0.0565650	+	0.998399i	$0.481985\pi$
$12$	0	0
$13$	6.69354	1.85645	0.928227	−	0.372014i	$-0.121333\pi$
$13$	6.69354	1.85645	0.928227	+	0.372014i	$0.121333\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.44329	−0.350050	−0.175025	−	0.984564i	$-0.556001\pi$
$17$	−1.44329	−0.350050	−0.175025	+	0.984564i	$0.556001\pi$
$18$	0	0
$19$	−6.63539	−1.52226	−0.761131	−	0.648598i	$-0.775356\pi$
$19$	−6.63539	−1.52226	−0.761131	+	0.648598i	$0.775356\pi$
$20$	0	0
$21$	3.90077	0.851217
$22$	0	0
$23$	6.67477	1.39179	0.695893	−	0.718145i	$-0.255009\pi$
$23$	6.67477	1.39179	0.695893	+	0.718145i	$0.255009\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.16462	0.801482
$28$	0	0
$29$	−0.174739	−0.0324482	−0.0162241	−	0.999868i	$-0.505165\pi$
$29$	−0.174739	−0.0324482	−0.0162241	+	0.999868i	$0.505165\pi$
$30$	0	0
$31$	−7.61770	−1.36818	−0.684089	−	0.729398i	$-0.739800\pi$
$31$	−7.61770	−1.36818	−0.684089	+	0.729398i	$0.739800\pi$
$32$	0	0
$33$	1.02884	0.179098
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.13684	1.33769	0.668844	−	0.743403i	$-0.266790\pi$
$37$	8.13684	1.33769	0.668844	+	0.743403i	$0.266790\pi$
$38$	0	0
$39$	18.3540	2.93899
$40$	0	0
$41$	7.97627	1.24568	0.622842	−	0.782347i	$-0.285978\pi$
$41$	7.97627	1.24568	0.622842	+	0.782347i	$0.285978\pi$
$42$	0	0
$43$	4.98010	0.759457	0.379729	−	0.925098i	$-0.376017\pi$
$43$	4.98010	0.759457	0.379729	+	0.925098i	$0.376017\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.11109	0.162070	0.0810348	−	0.996711i	$-0.474178\pi$
$47$	1.11109	0.162070	0.0810348	+	0.996711i	$0.474178\pi$
$48$	0	0
$49$	−4.97627	−0.710896
$50$	0	0
$51$	−3.95758	−0.554172
$52$	0	0
$53$	1.52786	0.209868	0.104934	−	0.994479i	$-0.466537\pi$
$53$	1.52786	0.209868	0.104934	+	0.994479i	$0.466537\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−18.1945	−2.40992
$58$	0	0
$59$	−10.1292	−1.31871	−0.659353	−	0.751833i	$-0.729170\pi$
$59$	−10.1292	−1.31871	−0.659353	+	0.751833i	$0.729170\pi$
$60$	0	0
$61$	8.96751	1.14817	0.574086	−	0.818795i	$-0.305357\pi$
$61$	8.96751	1.14817	0.574086	+	0.818795i	$0.305357\pi$
$62$	0	0
$63$	6.42834	0.809895
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−8.03444	−0.981564	−0.490782	−	0.871283i	$-0.663289\pi$
$67$	−8.03444	−0.981564	−0.490782	+	0.871283i	$0.663289\pi$
$68$	0	0
$69$	18.3025	2.20336
$70$	0	0
$71$	7.84959	0.931575	0.465788	−	0.884897i	$-0.345771\pi$
$71$	7.84959	0.931575	0.465788	+	0.884897i	$0.345771\pi$
$72$	0	0
$73$	8.05227	0.942447	0.471223	−	0.882014i	$-0.343813\pi$
$73$	8.05227	0.942447	0.471223	+	0.882014i	$0.343813\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.533765	0.0608281
$78$	0	0
$79$	0.231892	0.0260899	0.0130450	−	0.999915i	$-0.495848\pi$
$79$	0.231892	0.0260899	0.0130450	+	0.999915i	$0.495848\pi$
$80$	0	0
$81$	−2.13684	−0.237426
$82$	0	0
$83$	7.21897	0.792385	0.396192	−	0.918168i	$-0.370331\pi$
$83$	7.21897	0.792385	0.396192	+	0.918168i	$0.370331\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−0.479142	−0.0513694
$88$	0	0
$89$	5.85410	0.620534	0.310267	−	0.950650i	$-0.399582\pi$
$89$	5.85410	0.620534	0.310267	+	0.950650i	$0.399582\pi$
$90$	0	0
$91$	9.52208	0.998185
$92$	0	0
$93$	−20.8881	−2.16599
$94$	0	0
$95$	0	0
$96$	0	0
$97$	8.49508	0.862544	0.431272	−	0.902222i	$-0.358065\pi$
$97$	8.49508	0.862544	0.431272	+	0.902222i	$0.358065\pi$
$98$	0	0
$99$	1.69550	0.170404
$100$	0	0
$101$	−14.8126	−1.47390	−0.736952	−	0.675945i	$-0.763736\pi$
$101$	−14.8126	−1.47390	−0.736952	+	0.675945i	$0.763736\pi$
$102$	0	0
$103$	10.4629	1.03094	0.515468	−	0.856909i	$-0.327618\pi$
$103$	10.4629	1.03094	0.515468	+	0.856909i	$0.327618\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	17.7067	1.71177	0.855884	−	0.517167i	$-0.173014\pi$
$107$	17.7067	1.71177	0.855884	+	0.517167i	$0.173014\pi$
$108$	0	0
$109$	−4.65564	−0.445929	−0.222965	−	0.974827i	$-0.571573\pi$
$109$	−4.65564	−0.445929	−0.222965	+	0.974827i	$0.571573\pi$
$110$	0	0
$111$	22.3116	2.11772
$112$	0	0
$113$	−8.89201	−0.836489	−0.418245	−	0.908334i	$-0.637355\pi$
$113$	−8.89201	−0.836489	−0.418245	+	0.908334i	$0.637355\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	30.2468	2.79632
$118$	0	0
$119$	−2.05320	−0.188216
$120$	0	0
$121$	−10.8592	−0.987202
$122$	0	0
$123$	21.8713	1.97207
$124$	0	0
$125$	0	0
$126$	0	0
$127$	4.37299	0.388040	0.194020	−	0.980998i	$-0.437847\pi$
$127$	4.37299	0.388040	0.194020	+	0.980998i	$0.437847\pi$
$128$	0	0
$129$	13.6556	1.20231
$130$	0	0
$131$	11.2001	0.978554	0.489277	−	0.872128i	$-0.337261\pi$
$131$	11.2001	0.978554	0.489277	+	0.872128i	$0.337261\pi$
$132$	0	0
$133$	−9.43935	−0.818495
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−16.3582	−1.39758	−0.698789	−	0.715327i	$-0.746278\pi$
$137$	−16.3582	−1.39758	−0.698789	+	0.715327i	$0.746278\pi$
$138$	0	0
$139$	−19.4796	−1.65224	−0.826119	−	0.563495i	$-0.809456\pi$
$139$	−19.4796	−1.65224	−0.826119	+	0.563495i	$0.809456\pi$
$140$	0	0
$141$	3.04667	0.256576
$142$	0	0
$143$	2.51148	0.210021
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−13.6452	−1.12543
$148$	0	0
$149$	5.76077	0.471941	0.235970	−	0.971760i	$-0.424173\pi$
$149$	5.76077	0.471941	0.235970	+	0.971760i	$0.424173\pi$
$150$	0	0
$151$	−13.0389	−1.06109	−0.530545	−	0.847657i	$-0.678013\pi$
$151$	−13.0389	−1.06109	−0.530545	+	0.847657i	$0.678013\pi$
$152$	0	0
$153$	−6.52196	−0.527269
$154$	0	0
$155$	0	0
$156$	0	0
$157$	10.3876	0.829018	0.414509	−	0.910045i	$-0.363953\pi$
$157$	10.3876	0.829018	0.414509	+	0.910045i	$0.363953\pi$
$158$	0	0
$159$	4.18947	0.332247
$160$	0	0
$161$	9.49538	0.748340
$162$	0	0
$163$	10.0898	0.790294	0.395147	−	0.918618i	$-0.370694\pi$
$163$	10.0898	0.790294	0.395147	+	0.918618i	$0.370694\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−14.3977	−1.11413	−0.557065	−	0.830469i	$-0.688073\pi$
$167$	−14.3977	−1.11413	−0.557065	+	0.830469i	$0.688073\pi$
$168$	0	0
$169$	31.8035	2.44642
$170$	0	0
$171$	−29.9840	−2.29293
$172$	0	0
$173$	5.88610	0.447512	0.223756	−	0.974645i	$-0.428168\pi$
$173$	5.88610	0.447512	0.223756	+	0.974645i	$0.428168\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−27.7746	−2.08767
$178$	0	0
$179$	22.5982	1.68907	0.844534	−	0.535502i	$-0.179878\pi$
$179$	22.5982	1.68907	0.844534	+	0.535502i	$0.179878\pi$
$180$	0	0
$181$	−11.6845	−0.868500	−0.434250	−	0.900792i	$-0.642987\pi$
$181$	−11.6845	−0.868500	−0.434250	+	0.900792i	$0.642987\pi$
$182$	0	0
$183$	24.5893	1.81769
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−0.541538	−0.0396012
$188$	0	0
$189$	5.92449	0.430943
$190$	0	0
$191$	−25.8053	−1.86721	−0.933605	−	0.358305i	$-0.883355\pi$
$191$	−25.8053	−1.86721	−0.933605	+	0.358305i	$0.883355\pi$
$192$	0	0
$193$	−3.46624	−0.249505	−0.124753	−	0.992188i	$-0.539814\pi$
$193$	−3.46624	−0.249505	−0.124753	+	0.992188i	$0.539814\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3.50098	−0.249434	−0.124717	−	0.992192i	$-0.539802\pi$
$197$	−3.50098	−0.249434	−0.124717	+	0.992192i	$0.539802\pi$
$198$	0	0
$199$	−23.6318	−1.67522	−0.837609	−	0.546271i	$-0.816047\pi$
$199$	−23.6318	−1.67522	−0.837609	+	0.546271i	$0.816047\pi$
$200$	0	0
$201$	−22.0308	−1.55393
$202$	0	0
$203$	−0.248580	−0.0174469
$204$	0	0
$205$	0	0
$206$	0	0
$207$	30.1620	2.09640
$208$	0	0
$209$	−2.48966	−0.172213
$210$	0	0
$211$	−16.7646	−1.15412	−0.577060	−	0.816702i	$-0.695800\pi$
$211$	−16.7646	−1.15412	−0.577060	+	0.816702i	$0.695800\pi$
$212$	0	0
$213$	21.5239	1.47479
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−10.8368	−0.735647
$218$	0	0
$219$	22.0797	1.49201
$220$	0	0
$221$	−9.66075	−0.649853
$222$	0	0
$223$	22.8544	1.53045	0.765223	−	0.643766i	$-0.222629\pi$
$223$	22.8544	1.53045	0.765223	+	0.643766i	$0.222629\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−13.3731	−0.887601	−0.443801	−	0.896126i	$-0.646370\pi$
$227$	−13.3731	−0.887601	−0.443801	+	0.896126i	$0.646370\pi$
$228$	0	0
$229$	24.6141	1.62654	0.813272	−	0.581883i	$-0.197684\pi$
$229$	24.6141	1.62654	0.813272	+	0.581883i	$0.197684\pi$
$230$	0	0
$231$	1.46361	0.0962982
$232$	0	0
$233$	−25.0288	−1.63969	−0.819847	−	0.572583i	$-0.805942\pi$
$233$	−25.0288	−1.63969	−0.819847	+	0.572583i	$0.805942\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0.635859	0.0413035
$238$	0	0
$239$	−5.17178	−0.334535	−0.167267	−	0.985912i	$-0.553494\pi$
$239$	−5.17178	−0.334535	−0.167267	+	0.985912i	$0.553494\pi$
$240$	0	0
$241$	−9.79247	−0.630789	−0.315394	−	0.948961i	$-0.602137\pi$
$241$	−9.79247	−0.630789	−0.315394	+	0.948961i	$0.602137\pi$
$242$	0	0
$243$	−18.3532	−1.17736
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−44.4142	−2.82601
$248$	0	0
$249$	19.7947	1.25444
$250$	0	0
$251$	12.4690	0.787037	0.393519	−	0.919317i	$-0.371258\pi$
$251$	12.4690	0.787037	0.393519	+	0.919317i	$0.371258\pi$
$252$	0	0
$253$	2.50444	0.157453
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−9.35873	−0.583781	−0.291891	−	0.956452i	$-0.594284\pi$
$257$	−9.35873	−0.583781	−0.291891	+	0.956452i	$0.594284\pi$
$258$	0	0
$259$	11.5753	0.719253
$260$	0	0
$261$	−0.789611	−0.0488757
$262$	0	0
$263$	−18.7934	−1.15885	−0.579426	−	0.815025i	$-0.696723\pi$
$263$	−18.7934	−1.15885	−0.579426	+	0.815025i	$0.696723\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	16.0522	0.982379
$268$	0	0
$269$	22.5298	1.37367	0.686834	−	0.726815i	$-0.259000\pi$
$269$	22.5298	1.37367	0.686834	+	0.726815i	$0.259000\pi$
$270$	0	0
$271$	5.56450	0.338019	0.169010	−	0.985614i	$-0.445943\pi$
$271$	5.56450	0.338019	0.169010	+	0.985614i	$0.445943\pi$
$272$	0	0
$273$	26.1099	1.58025
$274$	0	0
$275$	0	0
$276$	0	0
$277$	21.3025	1.27994	0.639972	−	0.768398i	$-0.278946\pi$
$277$	21.3025	1.27994	0.639972	+	0.768398i	$0.278946\pi$
$278$	0	0
$279$	−34.4229	−2.06084
$280$	0	0
$281$	−19.5765	−1.16784	−0.583918	−	0.811813i	$-0.698481\pi$
$281$	−19.5765	−1.16784	−0.583918	+	0.811813i	$0.698481\pi$
$282$	0	0
$283$	20.0079	1.18935	0.594674	−	0.803967i	$-0.297281\pi$
$283$	20.0079	1.18935	0.594674	+	0.803967i	$0.297281\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	11.3469	0.669784
$288$	0	0
$289$	−14.9169	−0.877465
$290$	0	0
$291$	23.2939	1.36551
$292$	0	0
$293$	8.75122	0.511252	0.255626	−	0.966776i	$-0.417718\pi$
$293$	8.75122	0.511252	0.255626	+	0.966776i	$0.417718\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.56261	0.0906716
$298$	0	0
$299$	44.6779	2.58379
$300$	0	0
$301$	7.08457	0.408348
$302$	0	0
$303$	−40.6167	−2.33337
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−12.2633	−0.699903	−0.349952	−	0.936768i	$-0.613802\pi$
$307$	−12.2633	−0.699903	−0.349952	+	0.936768i	$0.613802\pi$
$308$	0	0
$309$	28.6896	1.63209
$310$	0	0
$311$	−0.838994	−0.0475750	−0.0237875	−	0.999717i	$-0.507573\pi$
$311$	−0.838994	−0.0475750	−0.0237875	+	0.999717i	$0.507573\pi$
$312$	0	0
$313$	14.5005	0.819616	0.409808	−	0.912172i	$-0.365596\pi$
$313$	14.5005	0.819616	0.409808	+	0.912172i	$0.365596\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	15.2796	0.858186	0.429093	−	0.903260i	$-0.358833\pi$
$317$	15.2796	0.858186	0.429093	+	0.903260i	$0.358833\pi$
$318$	0	0
$319$	−0.0655638	−0.00367087
$320$	0	0
$321$	48.5525	2.70993
$322$	0	0
$323$	9.57682	0.532868
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−12.7660	−0.705959
$328$	0	0
$329$	1.58062	0.0871422
$330$	0	0
$331$	29.4107	1.61656	0.808280	−	0.588798i	$-0.200399\pi$
$331$	29.4107	1.61656	0.808280	+	0.588798i	$0.200399\pi$
$332$	0	0
$333$	36.7687	2.01491
$334$	0	0
$335$	0	0
$336$	0	0
$337$	25.3548	1.38116	0.690581	−	0.723255i	$-0.257355\pi$
$337$	25.3548	1.38116	0.690581	+	0.723255i	$0.257355\pi$
$338$	0	0
$339$	−24.3823	−1.32426
$340$	0	0
$341$	−2.85823	−0.154782
$342$	0	0
$343$	−17.0372	−0.919921
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−19.5058	−1.04713	−0.523563	−	0.851987i	$-0.675397\pi$
$347$	−19.5058	−1.04713	−0.523563	+	0.851987i	$0.675397\pi$
$348$	0	0
$349$	1.56625	0.0838396	0.0419198	−	0.999121i	$-0.486653\pi$
$349$	1.56625	0.0838396	0.0419198	+	0.999121i	$0.486653\pi$
$350$	0	0
$351$	27.8761	1.48791
$352$	0	0
$353$	11.2409	0.598292	0.299146	−	0.954207i	$-0.403298\pi$
$353$	11.2409	0.598292	0.299146	+	0.954207i	$0.403298\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−5.62996	−0.297969
$358$	0	0
$359$	2.67781	0.141329	0.0706647	−	0.997500i	$-0.477488\pi$
$359$	2.67781	0.141329	0.0706647	+	0.997500i	$0.477488\pi$
$360$	0	0
$361$	25.0284	1.31728
$362$	0	0
$363$	−29.7764	−1.56286
$364$	0	0
$365$	0	0
$366$	0	0
$367$	13.5005	0.704721	0.352361	−	0.935864i	$-0.385379\pi$
$367$	13.5005	0.704721	0.352361	+	0.935864i	$0.385379\pi$
$368$	0	0
$369$	36.0432	1.87633
$370$	0	0
$371$	2.17350	0.112843
$372$	0	0
$373$	−33.3373	−1.72614	−0.863069	−	0.505086i	$-0.831461\pi$
$373$	−33.3373	−1.72614	−0.863069	+	0.505086i	$0.831461\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.16962	−0.0602387
$378$	0	0
$379$	16.7876	0.862320	0.431160	−	0.902276i	$-0.358104\pi$
$379$	16.7876	0.862320	0.431160	+	0.902276i	$0.358104\pi$
$380$	0	0
$381$	11.9909	0.614314
$382$	0	0
$383$	7.68358	0.392612	0.196306	−	0.980543i	$-0.437105\pi$
$383$	7.68358	0.392612	0.196306	+	0.980543i	$0.437105\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	22.5041	1.14395
$388$	0	0
$389$	−10.8825	−0.551763	−0.275881	−	0.961192i	$-0.588970\pi$
$389$	−10.8825	−0.551763	−0.275881	+	0.961192i	$0.588970\pi$
$390$	0	0
$391$	−9.63366	−0.487195
$392$	0	0
$393$	30.7111	1.54917
$394$	0	0
$395$	0	0
$396$	0	0
$397$	13.1940	0.662190	0.331095	−	0.943598i	$-0.392582\pi$
$397$	13.1940	0.662190	0.331095	+	0.943598i	$0.392582\pi$
$398$	0	0
$399$	−25.8831	−1.29578
$400$	0	0
$401$	−23.7885	−1.18794	−0.593971	−	0.804486i	$-0.702441\pi$
$401$	−23.7885	−1.18794	−0.593971	+	0.804486i	$0.702441\pi$
$402$	0	0
$403$	−50.9894	−2.53996
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.05302	0.151333
$408$	0	0
$409$	−7.42547	−0.367166	−0.183583	−	0.983004i	$-0.558770\pi$
$409$	−7.42547	−0.367166	−0.183583	+	0.983004i	$0.558770\pi$
$410$	0	0
$411$	−44.8550	−2.21253
$412$	0	0
$413$	−14.4095	−0.709047
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−53.4139	−2.61569
$418$	0	0
$419$	−26.6991	−1.30434	−0.652168	−	0.758075i	$-0.726140\pi$
$419$	−26.6991	−1.30434	−0.652168	+	0.758075i	$0.726140\pi$
$420$	0	0
$421$	26.4572	1.28944	0.644722	−	0.764417i	$-0.276973\pi$
$421$	26.4572	1.28944	0.644722	+	0.764417i	$0.276973\pi$
$422$	0	0
$423$	5.02081	0.244120
$424$	0	0
$425$	0	0
$426$	0	0
$427$	12.7570	0.617353
$428$	0	0
$429$	6.88659	0.332488
$430$	0	0
$431$	−30.7080	−1.47915	−0.739576	−	0.673073i	$-0.764974\pi$
$431$	−30.7080	−1.47915	−0.739576	+	0.673073i	$0.764974\pi$
$432$	0	0
$433$	−10.7781	−0.517963	−0.258981	−	0.965882i	$-0.583387\pi$
$433$	−10.7781	−0.517963	−0.258981	+	0.965882i	$0.583387\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−44.2897	−2.11866
$438$	0	0
$439$	18.3540	0.875988	0.437994	−	0.898978i	$-0.355689\pi$
$439$	18.3540	0.875988	0.437994	+	0.898978i	$0.355689\pi$
$440$	0	0
$441$	−22.4868	−1.07080
$442$	0	0
$443$	−25.2995	−1.20202	−0.601008	−	0.799243i	$-0.705234\pi$
$443$	−25.2995	−1.20202	−0.601008	+	0.799243i	$0.705234\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	15.7963	0.747138
$448$	0	0
$449$	31.9061	1.50574	0.752870	−	0.658169i	$-0.228669\pi$
$449$	31.9061	1.50574	0.752870	+	0.658169i	$0.228669\pi$
$450$	0	0
$451$	2.99278	0.140924
$452$	0	0
$453$	−35.7532	−1.67983
$454$	0	0
$455$	0	0
$456$	0	0
$457$	17.5279	0.819919	0.409959	−	0.912104i	$-0.365543\pi$
$457$	17.5279	0.819919	0.409959	+	0.912104i	$0.365543\pi$
$458$	0	0
$459$	−6.01078	−0.280559
$460$	0	0
$461$	21.7173	1.01147	0.505737	−	0.862688i	$-0.331221\pi$
$461$	21.7173	1.01147	0.505737	+	0.862688i	$0.331221\pi$
$462$	0	0
$463$	−8.89860	−0.413553	−0.206777	−	0.978388i	$-0.566297\pi$
$463$	−8.89860	−0.413553	−0.206777	+	0.978388i	$0.566297\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−38.5182	−1.78241	−0.891205	−	0.453601i	$-0.850139\pi$
$467$	−38.5182	−1.78241	−0.891205	+	0.453601i	$0.850139\pi$
$468$	0	0
$469$	−11.4296	−0.527771
$470$	0	0
$471$	28.4832	1.31243
$472$	0	0
$473$	1.86858	0.0859174
$474$	0	0
$475$	0	0
$476$	0	0
$477$	6.90411	0.316118
$478$	0	0
$479$	−9.02656	−0.412434	−0.206217	−	0.978506i	$-0.566115\pi$
$479$	−9.02656	−0.412434	−0.206217	+	0.978506i	$0.566115\pi$
$480$	0	0
$481$	54.4642	2.48336
$482$	0	0
$483$	26.0367	1.18471
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−8.70742	−0.394571	−0.197286	−	0.980346i	$-0.563213\pi$
$487$	−8.70742	−0.394571	−0.197286	+	0.980346i	$0.563213\pi$
$488$	0	0
$489$	27.6667	1.25113
$490$	0	0
$491$	13.8779	0.626300	0.313150	−	0.949704i	$-0.398616\pi$
$491$	13.8779	0.626300	0.313150	+	0.949704i	$0.398616\pi$
$492$	0	0
$493$	0.252200	0.0113585
$494$	0	0
$495$	0	0
$496$	0	0
$497$	11.1666	0.500893
$498$	0	0
$499$	16.5152	0.739320	0.369660	−	0.929167i	$-0.379474\pi$
$499$	16.5152	0.739320	0.369660	+	0.929167i	$0.379474\pi$
$500$	0	0
$501$	−39.4792	−1.76380
$502$	0	0
$503$	−22.1447	−0.987384	−0.493692	−	0.869637i	$-0.664353\pi$
$503$	−22.1447	−0.987384	−0.493692	+	0.869637i	$0.664353\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	87.2065	3.87298
$508$	0	0
$509$	−10.0870	−0.447099	−0.223549	−	0.974693i	$-0.571764\pi$
$509$	−10.0870	−0.447099	−0.223549	+	0.974693i	$0.571764\pi$
$510$	0	0
$511$	11.4550	0.506738
$512$	0	0
$513$	−27.6339	−1.22007
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0.416893	0.0183349
$518$	0	0
$519$	16.1400	0.708465
$520$	0	0
$521$	−1.44890	−0.0634773	−0.0317386	−	0.999496i	$-0.510104\pi$
$521$	−1.44890	−0.0634773	−0.0317386	+	0.999496i	$0.510104\pi$
$522$	0	0
$523$	14.6510	0.640643	0.320322	−	0.947309i	$-0.396209\pi$
$523$	14.6510	0.640643	0.320322	+	0.947309i	$0.396209\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10.9946	0.478932
$528$	0	0
$529$	21.5526	0.937068
$530$	0	0
$531$	−45.7718	−1.98632
$532$	0	0
$533$	53.3895	2.31256
$534$	0	0
$535$	0	0
$536$	0	0
$537$	61.9652	2.67400
$538$	0	0
$539$	−1.86715	−0.0804237
$540$	0	0
$541$	18.2216	0.783407	0.391704	−	0.920091i	$-0.371886\pi$
$541$	18.2216	0.783407	0.391704	+	0.920091i	$0.371886\pi$
$542$	0	0
$543$	−32.0393	−1.37494
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−41.5712	−1.77746	−0.888729	−	0.458433i	$-0.848411\pi$
$547$	−41.5712	−1.77746	−0.888729	+	0.458433i	$0.848411\pi$
$548$	0	0
$549$	40.5224	1.72945
$550$	0	0
$551$	1.15946	0.0493947
$552$	0	0
$553$	0.329885	0.0140281
$554$	0	0
$555$	0	0
$556$	0	0
$557$	18.8479	0.798611	0.399306	−	0.916818i	$-0.369251\pi$
$557$	18.8479	0.798611	0.399306	+	0.916818i	$0.369251\pi$
$558$	0	0
$559$	33.3345	1.40990
$560$	0	0
$561$	−1.48492	−0.0626934
$562$	0	0
$563$	−31.4716	−1.32637	−0.663186	−	0.748455i	$-0.730796\pi$
$563$	−31.4716	−1.32637	−0.663186	+	0.748455i	$0.730796\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−3.03981	−0.127660
$568$	0	0
$569$	−8.55987	−0.358848	−0.179424	−	0.983772i	$-0.557423\pi$
$569$	−8.55987	−0.358848	−0.179424	+	0.983772i	$0.557423\pi$
$570$	0	0
$571$	45.4513	1.90208	0.951039	−	0.309072i	$-0.100018\pi$
$571$	45.4513	1.90208	0.951039	+	0.309072i	$0.100018\pi$
$572$	0	0
$573$	−70.7594	−2.95602
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−0.414454	−0.0172539	−0.00862697	−	0.999963i	$-0.502746\pi$
$577$	−0.414454	−0.0172539	−0.00862697	+	0.999963i	$0.502746\pi$
$578$	0	0
$579$	−9.50457	−0.394996
$580$	0	0
$581$	10.2695	0.426052
$582$	0	0
$583$	0.573269	0.0237424
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−6.48226	−0.267552	−0.133776	−	0.991012i	$-0.542710\pi$
$587$	−6.48226	−0.267552	−0.133776	+	0.991012i	$0.542710\pi$
$588$	0	0
$589$	50.5464	2.08273
$590$	0	0
$591$	−9.59983	−0.394884
$592$	0	0
$593$	−27.6861	−1.13693	−0.568467	−	0.822706i	$-0.692463\pi$
$593$	−27.6861	−1.13693	−0.568467	+	0.822706i	$0.692463\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−64.7995	−2.65207
$598$	0	0
$599$	24.7575	1.01156	0.505781	−	0.862662i	$-0.331204\pi$
$599$	24.7575	1.01156	0.505781	+	0.862662i	$0.331204\pi$
$600$	0	0
$601$	12.0046	0.489679	0.244840	−	0.969564i	$-0.421265\pi$
$601$	12.0046	0.489679	0.244840	+	0.969564i	$0.421265\pi$
$602$	0	0
$603$	−36.3061	−1.47850
$604$	0	0
$605$	0	0
$606$	0	0
$607$	9.92131	0.402694	0.201347	−	0.979520i	$-0.435468\pi$
$607$	9.92131	0.402694	0.201347	+	0.979520i	$0.435468\pi$
$608$	0	0
$609$	−0.681617	−0.0276205
$610$	0	0
$611$	7.43715	0.300875
$612$	0	0
$613$	−11.0841	−0.447682	−0.223841	−	0.974626i	$-0.571860\pi$
$613$	−11.0841	−0.447682	−0.223841	+	0.974626i	$0.571860\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	8.63191	0.347508	0.173754	−	0.984789i	$-0.444410\pi$
$617$	8.63191	0.347508	0.173754	+	0.984789i	$0.444410\pi$
$618$	0	0
$619$	−28.3115	−1.13794	−0.568968	−	0.822360i	$-0.692657\pi$
$619$	−28.3115	−1.13794	−0.568968	+	0.822360i	$0.692657\pi$
$620$	0	0
$621$	27.7979	1.11549
$622$	0	0
$623$	8.32791	0.333651
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−6.82676	−0.272634
$628$	0	0
$629$	−11.7439	−0.468258
$630$	0	0
$631$	−1.67249	−0.0665806	−0.0332903	−	0.999446i	$-0.510599\pi$
$631$	−1.67249	−0.0665806	−0.0332903	+	0.999446i	$0.510599\pi$
$632$	0	0
$633$	−45.9692	−1.82711
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−33.3089	−1.31975
$638$	0	0
$639$	35.4707	1.40320
$640$	0	0
$641$	−5.13714	−0.202905	−0.101452	−	0.994840i	$-0.532349\pi$
$641$	−5.13714	−0.202905	−0.101452	+	0.994840i	$0.532349\pi$
$642$	0	0
$643$	21.6526	0.853895	0.426948	−	0.904276i	$-0.359589\pi$
$643$	21.6526	0.853895	0.426948	+	0.904276i	$0.359589\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−37.4222	−1.47122	−0.735608	−	0.677407i	$-0.763104\pi$
$647$	−37.4222	−1.47122	−0.735608	+	0.677407i	$0.763104\pi$
$648$	0	0
$649$	−3.80057	−0.149185
$650$	0	0
$651$	−29.7149	−1.16462
$652$	0	0
$653$	47.5991	1.86270	0.931349	−	0.364128i	$-0.118633\pi$
$653$	47.5991	1.86270	0.931349	+	0.364128i	$0.118633\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	36.3866	1.41958
$658$	0	0
$659$	−34.3398	−1.33769	−0.668844	−	0.743403i	$-0.733211\pi$
$659$	−34.3398	−1.33769	−0.668844	+	0.743403i	$0.733211\pi$
$660$	0	0
$661$	−17.1096	−0.665488	−0.332744	−	0.943017i	$-0.607975\pi$
$661$	−17.1096	−0.665488	−0.332744	+	0.943017i	$0.607975\pi$
$662$	0	0
$663$	−26.4902	−1.02879
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.16634	−0.0451610
$668$	0	0
$669$	62.6678	2.42288
$670$	0	0
$671$	3.36470	0.129893
$672$	0	0
$673$	5.70339	0.219849	0.109925	−	0.993940i	$-0.464939\pi$
$673$	5.70339	0.219849	0.109925	+	0.993940i	$0.464939\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	21.8474	0.839664	0.419832	−	0.907602i	$-0.362089\pi$
$677$	21.8474	0.839664	0.419832	+	0.907602i	$0.362089\pi$
$678$	0	0
$679$	12.0849	0.463776
$680$	0	0
$681$	−36.6695	−1.40518
$682$	0	0
$683$	−29.3290	−1.12224	−0.561121	−	0.827734i	$-0.689630\pi$
$683$	−29.3290	−1.12224	−0.561121	+	0.827734i	$0.689630\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	67.4929	2.57501
$688$	0	0
$689$	10.2268	0.389611
$690$	0	0
$691$	−10.9452	−0.416374	−0.208187	−	0.978089i	$-0.566756\pi$
$691$	−10.9452	−0.416374	−0.208187	+	0.978089i	$0.566756\pi$
$692$	0	0
$693$	2.41198	0.0916234
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−11.5121	−0.436053
$698$	0	0
$699$	−68.6302	−2.59583
$700$	0	0
$701$	15.1593	0.572559	0.286279	−	0.958146i	$-0.407581\pi$
$701$	15.1593	0.572559	0.286279	+	0.958146i	$0.407581\pi$
$702$	0	0
$703$	−53.9911	−2.03631
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−21.0720	−0.792494
$708$	0	0
$709$	−44.1990	−1.65993	−0.829965	−	0.557816i	$-0.811640\pi$
$709$	−44.1990	−1.65993	−0.829965	+	0.557816i	$0.811640\pi$
$710$	0	0
$711$	1.04788	0.0392984
$712$	0	0
$713$	−50.8464	−1.90421
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−14.1812	−0.529608
$718$	0	0
$719$	17.6408	0.657891	0.328945	−	0.944349i	$-0.393307\pi$
$719$	17.6408	0.657891	0.328945	+	0.944349i	$0.393307\pi$
$720$	0	0
$721$	14.8842	0.554317
$722$	0	0
$723$	−26.8514	−0.998614
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−12.9714	−0.481081	−0.240541	−	0.970639i	$-0.577325\pi$
$727$	−12.9714	−0.481081	−0.240541	+	0.970639i	$0.577325\pi$
$728$	0	0
$729$	−43.9146	−1.62647
$730$	0	0
$731$	−7.18775	−0.265848
$732$	0	0
$733$	16.4951	0.609260	0.304630	−	0.952471i	$-0.401467\pi$
$733$	16.4951	0.609260	0.304630	+	0.952471i	$0.401467\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3.01460	−0.111044
$738$	0	0
$739$	8.43368	0.310238	0.155119	−	0.987896i	$-0.450424\pi$
$739$	8.43368	0.310238	0.155119	+	0.987896i	$0.450424\pi$
$740$	0	0
$741$	−121.786	−4.47391
$742$	0	0
$743$	−33.9514	−1.24556	−0.622778	−	0.782399i	$-0.713996\pi$
$743$	−33.9514	−1.24556	−0.622778	+	0.782399i	$0.713996\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	32.6211	1.19354
$748$	0	0
$749$	25.1891	0.920390
$750$	0	0
$751$	−45.4283	−1.65770	−0.828851	−	0.559470i	$-0.811005\pi$
$751$	−45.4283	−1.65770	−0.828851	+	0.559470i	$0.811005\pi$
$752$	0	0
$753$	34.1906	1.24597
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−16.8920	−0.613950	−0.306975	−	0.951718i	$-0.599317\pi$
$757$	−16.8920	−0.613950	−0.306975	+	0.951718i	$0.599317\pi$
$758$	0	0
$759$	6.86728	0.249266
$760$	0	0
$761$	−47.1179	−1.70802	−0.854012	−	0.520254i	$-0.825837\pi$
$761$	−47.1179	−1.70802	−0.854012	+	0.520254i	$0.825837\pi$
$762$	0	0
$763$	−6.62300	−0.239769
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−67.8001	−2.44812
$768$	0	0
$769$	−18.3741	−0.662586	−0.331293	−	0.943528i	$-0.607485\pi$
$769$	−18.3741	−0.662586	−0.331293	+	0.943528i	$0.607485\pi$
$770$	0	0
$771$	−25.6620	−0.924195
$772$	0	0
$773$	15.0020	0.539583	0.269791	−	0.962919i	$-0.413045\pi$
$773$	15.0020	0.539583	0.269791	+	0.962919i	$0.413045\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	31.7399	1.13866
$778$	0	0
$779$	−52.9257	−1.89626
$780$	0	0
$781$	2.94524	0.105389
$782$	0	0
$783$	−0.727722	−0.0260067
$784$	0	0
$785$	0	0
$786$	0	0
$787$	33.4508	1.19239	0.596196	−	0.802839i	$-0.296678\pi$
$787$	33.4508	1.19239	0.596196	+	0.802839i	$0.296678\pi$
$788$	0	0
$789$	−51.5324	−1.83460
$790$	0	0
$791$	−12.6496	−0.449767
$792$	0	0
$793$	60.0244	2.13153
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−38.1970	−1.35301	−0.676503	−	0.736440i	$-0.736505\pi$
$797$	−38.1970	−1.35301	−0.676503	+	0.736440i	$0.736505\pi$
$798$	0	0
$799$	−1.60364	−0.0567325
$800$	0	0
$801$	26.4535	0.934689
$802$	0	0
$803$	3.02129	0.106619
$804$	0	0
$805$	0	0
$806$	0	0
$807$	61.7777	2.17468
$808$	0	0
$809$	−20.4597	−0.719326	−0.359663	−	0.933082i	$-0.617108\pi$
$809$	−20.4597	−0.719326	−0.359663	+	0.933082i	$0.617108\pi$
$810$	0	0
$811$	18.7948	0.659973	0.329987	−	0.943986i	$-0.392956\pi$
$811$	18.7948	0.659973	0.329987	+	0.943986i	$0.392956\pi$
$812$	0	0
$813$	15.2581	0.535125
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−33.0449	−1.15609
$818$	0	0
$819$	43.0284	1.50353
$820$	0	0
$821$	19.2705	0.672545	0.336273	−	0.941765i	$-0.390834\pi$
$821$	19.2705	0.672545	0.336273	+	0.941765i	$0.390834\pi$
$822$	0	0
$823$	21.6432	0.754434	0.377217	−	0.926125i	$-0.376881\pi$
$823$	21.6432	0.754434	0.377217	+	0.926125i	$0.376881\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	19.3833	0.674024	0.337012	−	0.941500i	$-0.390584\pi$
$827$	19.3833	0.674024	0.337012	+	0.941500i	$0.390584\pi$
$828$	0	0
$829$	−35.9313	−1.24794	−0.623972	−	0.781447i	$-0.714482\pi$
$829$	−35.9313	−1.24794	−0.623972	+	0.781447i	$0.714482\pi$
$830$	0	0
$831$	58.4124	2.02630
$832$	0	0
$833$	7.18223	0.248850
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−31.7248	−1.09657
$838$	0	0
$839$	−50.7061	−1.75057	−0.875285	−	0.483607i	$-0.839326\pi$
$839$	−50.7061	−1.75057	−0.875285	+	0.483607i	$0.839326\pi$
$840$	0	0
$841$	−28.9695	−0.998947
$842$	0	0
$843$	−53.6796	−1.84882
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−15.4481	−0.530802
$848$	0	0
$849$	54.8626	1.88288
$850$	0	0
$851$	54.3115	1.86177
$852$	0	0
$853$	−17.1309	−0.586552	−0.293276	−	0.956028i	$-0.594745\pi$
$853$	−17.1309	−0.586552	−0.293276	+	0.956028i	$0.594745\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	29.7130	1.01498	0.507489	−	0.861658i	$-0.330574\pi$
$857$	29.7130	1.01498	0.507489	+	0.861658i	$0.330574\pi$
$858$	0	0
$859$	4.76274	0.162503	0.0812513	−	0.996694i	$-0.474108\pi$
$859$	4.76274	0.162503	0.0812513	+	0.996694i	$0.474108\pi$
$860$	0	0
$861$	31.1136	1.06035
$862$	0	0
$863$	30.5440	1.03973	0.519865	−	0.854248i	$-0.325982\pi$
$863$	30.5440	1.03973	0.519865	+	0.854248i	$0.325982\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−40.9028	−1.38913
$868$	0	0
$869$	0.0870082	0.00295155
$870$	0	0
$871$	−53.7789	−1.82223
$872$	0	0
$873$	38.3876	1.29922
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−26.2296	−0.885710	−0.442855	−	0.896593i	$-0.646034\pi$
$877$	−26.2296	−0.885710	−0.442855	+	0.896593i	$0.646034\pi$
$878$	0	0
$879$	23.9962	0.809373
$880$	0	0
$881$	−3.04667	−0.102645	−0.0513224	−	0.998682i	$-0.516344\pi$
$881$	−3.04667	−0.102645	−0.0513224	+	0.998682i	$0.516344\pi$
$882$	0	0
$883$	−29.5403	−0.994111	−0.497056	−	0.867719i	$-0.665586\pi$
$883$	−29.5403	−0.994111	−0.497056	+	0.867719i	$0.665586\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−34.4623	−1.15713	−0.578565	−	0.815637i	$-0.696387\pi$
$887$	−34.4623	−1.15713	−0.578565	+	0.815637i	$0.696387\pi$
$888$	0	0
$889$	6.22092	0.208643
$890$	0	0
$891$	−0.801761	−0.0268600
$892$	0	0
$893$	−7.37253	−0.246712
$894$	0	0
$895$	0	0
$896$	0	0
$897$	122.509	4.09044
$898$	0	0
$899$	1.33111	0.0443950
$900$	0	0
$901$	−2.20516	−0.0734645
$902$	0	0
$903$	19.4262	0.646463
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−14.0918	−0.467909	−0.233954	−	0.972248i	$-0.575167\pi$
$907$	−14.0918	−0.467909	−0.233954	+	0.972248i	$0.575167\pi$
$908$	0	0
$909$	−66.9350	−2.22009
$910$	0	0
$911$	−1.09390	−0.0362424	−0.0181212	−	0.999836i	$-0.505768\pi$
$911$	−1.09390	−0.0362424	−0.0181212	+	0.999836i	$0.505768\pi$
$912$	0	0
$913$	2.70863	0.0896425
$914$	0	0
$915$	0	0
$916$	0	0
$917$	15.9330	0.526153
$918$	0	0
$919$	24.0443	0.793148	0.396574	−	0.918003i	$-0.370199\pi$
$919$	24.0443	0.793148	0.396574	+	0.918003i	$0.370199\pi$
$920$	0	0
$921$	−33.6265	−1.10803
$922$	0	0
$923$	52.5416	1.72943
$924$	0	0
$925$	0	0
$926$	0	0
$927$	47.2796	1.55286
$928$	0	0
$929$	−6.72651	−0.220690	−0.110345	−	0.993893i	$-0.535196\pi$
$929$	−6.72651	−0.220690	−0.110345	+	0.993893i	$0.535196\pi$
$930$	0	0
$931$	33.0195	1.08217
$932$	0	0
$933$	−2.30056	−0.0753169
$934$	0	0
$935$	0	0
$936$	0	0
$937$	14.8582	0.485397	0.242699	−	0.970102i	$-0.421967\pi$
$937$	14.8582	0.485397	0.242699	+	0.970102i	$0.421967\pi$
$938$	0	0
$939$	39.7610	1.29755
$940$	0	0
$941$	−30.2845	−0.987247	−0.493623	−	0.869676i	$-0.664328\pi$
$941$	−30.2845	−0.987247	−0.493623	+	0.869676i	$0.664328\pi$
$942$	0	0
$943$	53.2398	1.73373
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−26.3050	−0.854799	−0.427399	−	0.904063i	$-0.640570\pi$
$947$	−26.3050	−0.854799	−0.427399	+	0.904063i	$0.640570\pi$
$948$	0	0
$949$	53.8982	1.74961
$950$	0	0
$951$	41.8972	1.35861
$952$	0	0
$953$	−31.4448	−1.01860	−0.509298	−	0.860590i	$-0.670095\pi$
$953$	−31.4448	−1.01860	−0.509298	+	0.860590i	$0.670095\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−0.179779	−0.00581142
$958$	0	0
$959$	−23.2709	−0.751455
$960$	0	0
$961$	27.0293	0.871914
$962$	0	0
$963$	80.0129	2.57838
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−7.87393	−0.253209	−0.126604	−	0.991953i	$-0.540408\pi$
$967$	−7.87393	−0.253209	−0.126604	+	0.991953i	$0.540408\pi$
$968$	0	0
$969$	26.2601	0.843594
$970$	0	0
$971$	−15.6762	−0.503072	−0.251536	−	0.967848i	$-0.580936\pi$
$971$	−15.6762	−0.503072	−0.251536	+	0.967848i	$0.580936\pi$
$972$	0	0
$973$	−27.7112	−0.888382
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−8.11293	−0.259555	−0.129778	−	0.991543i	$-0.541426\pi$
$977$	−8.11293	−0.259555	−0.129778	+	0.991543i	$0.541426\pi$
$978$	0	0
$979$	2.19652	0.0702009
$980$	0	0
$981$	−21.0379	−0.671689
$982$	0	0
$983$	−19.3833	−0.618232	−0.309116	−	0.951024i	$-0.600033\pi$
$983$	−19.3833	−0.618232	−0.309116	+	0.951024i	$0.600033\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	4.33412	0.137956
$988$	0	0
$989$	33.2410	1.05700
$990$	0	0
$991$	37.8708	1.20301	0.601503	−	0.798870i	$-0.294569\pi$
$991$	37.8708	1.20301	0.601503	+	0.798870i	$0.294569\pi$
$992$	0	0
$993$	80.6455	2.55921
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−55.4647	−1.75659	−0.878293	−	0.478123i	$-0.841317\pi$
$997$	−55.4647	−1.75659	−0.878293	+	0.478123i	$0.841317\pi$
$998$	0	0
$999$	33.8868	1.07213

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8000.2.a.bz.1.7		8
4.3	odd	2	inner	8000.2.a.bz.1.2		8
5.4	even	2		8000.2.a.by.1.2		8
8.3	odd	2		4000.2.a.o.1.7	yes	8
8.5	even	2		4000.2.a.o.1.2	✓	8
20.19	odd	2		8000.2.a.by.1.7		8
40.3	even	4		4000.2.c.i.1249.14		16
40.13	odd	4		4000.2.c.i.1249.3		16
40.19	odd	2		4000.2.a.p.1.2	yes	8
40.27	even	4		4000.2.c.i.1249.4		16
40.29	even	2		4000.2.a.p.1.7	yes	8
40.37	odd	4		4000.2.c.i.1249.13		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4000.2.a.o.1.2	✓	8	8.5	even	2
4000.2.a.o.1.7	yes	8	8.3	odd	2
4000.2.a.p.1.2	yes	8	40.19	odd	2
4000.2.a.p.1.7	yes	8	40.29	even	2
4000.2.c.i.1249.3		16	40.13	odd	4
4000.2.c.i.1249.4		16	40.27	even	4
4000.2.c.i.1249.13		16	40.37	odd	4
4000.2.c.i.1249.14		16	40.3	even	4
8000.2.a.by.1.2		8	5.4	even	2
8000.2.a.by.1.7		8	20.19	odd	2
8000.2.a.bz.1.2		8	4.3	odd	2	inner
8000.2.a.bz.1.7		8	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:	\( N \)	\(=\)	\( 8000 = 2^{6} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8000.a (trivial)

\(f(q)\)	\(=\)	\(q+2.74204 q^{3} +1.42258 q^{7} +4.51880 q^{9} +O(q^{10})\)	\(q+2.74204 q^{3} +1.42258 q^{7} +4.51880 q^{9} +0.375210 q^{11} +6.69354 q^{13} -1.44329 q^{17} -6.63539 q^{19} +3.90077 q^{21} +6.67477 q^{23} +4.16462 q^{27} -0.174739 q^{29} -7.61770 q^{31} +1.02884 q^{33} +8.13684 q^{37} +18.3540 q^{39} +7.97627 q^{41} +4.98010 q^{43} +1.11109 q^{47} -4.97627 q^{49} -3.95758 q^{51} +1.52786 q^{53} -18.1945 q^{57} -10.1292 q^{59} +8.96751 q^{61} +6.42834 q^{63} -8.03444 q^{67} +18.3025 q^{69} +7.84959 q^{71} +8.05227 q^{73} +0.533765 q^{77} +0.231892 q^{79} -2.13684 q^{81} +7.21897 q^{83} -0.479142 q^{87} +5.85410 q^{89} +9.52208 q^{91} -20.8881 q^{93} +8.49508 q^{97} +1.69550 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^9	\( 8 q + 8 q^{9} + 8 q^{13} - 20 q^{17} + 12 q^{21} + 16 q^{29} - 36 q^{33} + 28 q^{37} + 8 q^{41} + 16 q^{49} + 48 q^{53} - 20 q^{57} + 28 q^{61} + 28 q^{69} + 44 q^{77} + 20 q^{81} + 20 q^{89} + 4 q^{93} - 16 q^{97}+O(q^{100}) \) 8 * q + 8 * q^9 + 8 * q^13 - 20 * q^17 + 12 * q^21 + 16 * q^29 - 36 * q^33 + 28 * q^37 + 8 * q^41 + 16 * q^49 + 48 * q^53 - 20 * q^57 + 28 * q^61 + 28 * q^69 + 44 * q^77 + 20 * q^81 + 20 * q^89 + 4 * q^93 - 16 * q^97

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