LMFDB - Embedded newform 10000.2.a.be.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [10000,2,Mod(1,10000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(10000, 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("10000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 10000 = 2^{4} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	10000.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:	$79.8504020213$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.6152203125.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 8x^{6} + 20x^{5} + 26x^{4} - 35x^{3} - 27x^{2} + 16x + 1 $ x^8 - 3x^7 - 8x^6 + 20x^5 + 26x^4 - 35x^3 - 27x^2 + 16*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 625)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.32610$ of defining polynomial
Character	$\chi$	$=$	10000.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.09326 q^{3} -0.0237879 q^{7} +6.56824 q^{9} +O(q^{10})$	$q-3.09326 q^{3} -0.0237879 q^{7} +6.56824 q^{9} -3.58329 q^{11} +3.77587 q^{13} -3.62303 q^{17} +2.43084 q^{19} +0.0735820 q^{21} -1.71538 q^{23} -11.0375 q^{27} +3.85734 q^{29} +6.00979 q^{31} +11.0840 q^{33} -0.369309 q^{37} -11.6797 q^{39} -7.80900 q^{41} -0.174574 q^{43} -7.81082 q^{47} -6.99943 q^{49} +11.2070 q^{51} -8.97184 q^{53} -7.51920 q^{57} +4.45536 q^{59} +9.21403 q^{61} -0.156244 q^{63} +4.47385 q^{67} +5.30612 q^{69} +9.69458 q^{71} -3.95387 q^{73} +0.0852388 q^{77} +9.68349 q^{79} +14.4370 q^{81} +8.95717 q^{83} -11.9317 q^{87} -17.0151 q^{89} -0.0898200 q^{91} -18.5898 q^{93} +2.76438 q^{97} -23.5359 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 5 q^{3} - 10 q^{7} + 9 q^{9}+O(q^{10}) $ 8 * q - 5 * q^3 - 10 * q^7 + 9 * q^9	$ 8 q - 5 q^{3} - 10 q^{7} + 9 q^{9} - q^{11} + 10 q^{13} + 15 q^{17} + 10 q^{19} - 14 q^{21} - 30 q^{23} - 20 q^{27} + 10 q^{29} + 9 q^{31} + 5 q^{33} - 10 q^{37} - 8 q^{39} - 4 q^{41} - 30 q^{47} - 4 q^{49} + 14 q^{51} + 10 q^{53} - 10 q^{57} + 5 q^{59} + 6 q^{61} - 10 q^{67} + 3 q^{69} + 9 q^{71} + 5 q^{77} + 20 q^{79} + 8 q^{81} - 40 q^{83} - 40 q^{87} - 5 q^{89} - 6 q^{91} - 40 q^{93} + 22 q^{99}+O(q^{100}) $ 8 * q - 5 * q^3 - 10 * q^7 + 9 * q^9 - q^11 + 10 * q^13 + 15 * q^17 + 10 * q^19 - 14 * q^21 - 30 * q^23 - 20 * q^27 + 10 * q^29 + 9 * q^31 + 5 * q^33 - 10 * q^37 - 8 * q^39 - 4 * q^41 - 30 * q^47 - 4 * q^49 + 14 * q^51 + 10 * q^53 - 10 * q^57 + 5 * q^59 + 6 * q^61 - 10 * q^67 + 3 * q^69 + 9 * q^71 + 5 * q^77 + 20 * q^79 + 8 * q^81 - 40 * q^83 - 40 * q^87 - 5 * q^89 - 6 * q^91 - 40 * q^93 + 22 * 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
$3$	−3.09326	−1.78589	−0.892946	−	0.450163i	$-0.851366\pi$
$3$	−3.09326	−1.78589	−0.892946	+	0.450163i	$0.851366\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.0237879	−0.00899097	−0.00449549	−	0.999990i	$-0.501431\pi$
$7$	−0.0237879	−0.00899097	−0.00449549	+	0.999990i	$0.501431\pi$
$8$	0	0
$9$	6.56824	2.18941
$10$	0	0
$11$	−3.58329	−1.08040	−0.540201	−	0.841536i	$-0.681652\pi$
$11$	−3.58329	−1.08040	−0.540201	+	0.841536i	$0.681652\pi$
$12$	0	0
$13$	3.77587	1.04724	0.523619	−	0.851952i	$-0.324582\pi$
$13$	3.77587	1.04724	0.523619	+	0.851952i	$0.324582\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.62303	−0.878713	−0.439357	−	0.898313i	$-0.644794\pi$
$17$	−3.62303	−0.878713	−0.439357	+	0.898313i	$0.644794\pi$
$18$	0	0
$19$	2.43084	0.557672	0.278836	−	0.960339i	$-0.410051\pi$
$19$	2.43084	0.557672	0.278836	+	0.960339i	$0.410051\pi$
$20$	0	0
$21$	0.0735820	0.0160569
$22$	0	0
$23$	−1.71538	−0.357682	−0.178841	−	0.983878i	$-0.557235\pi$
$23$	−1.71538	−0.357682	−0.178841	+	0.983878i	$0.557235\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−11.0375	−2.12416
$28$	0	0
$29$	3.85734	0.716290	0.358145	−	0.933666i	$-0.383409\pi$
$29$	3.85734	0.716290	0.358145	+	0.933666i	$0.383409\pi$
$30$	0	0
$31$	6.00979	1.07939	0.539695	−	0.841861i	$-0.318540\pi$
$31$	6.00979	1.07939	0.539695	+	0.841861i	$0.318540\pi$
$32$	0	0
$33$	11.0840	1.92948
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.369309	−0.0607139	−0.0303570	−	0.999539i	$-0.509664\pi$
$37$	−0.369309	−0.0607139	−0.0303570	+	0.999539i	$0.509664\pi$
$38$	0	0
$39$	−11.6797	−1.87025
$40$	0	0
$41$	−7.80900	−1.21956	−0.609780	−	0.792570i	$-0.708742\pi$
$41$	−7.80900	−1.21956	−0.609780	+	0.792570i	$0.708742\pi$
$42$	0	0
$43$	−0.174574	−0.0266224	−0.0133112	−	0.999911i	$-0.504237\pi$
$43$	−0.174574	−0.0266224	−0.0133112	+	0.999911i	$0.504237\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−7.81082	−1.13932	−0.569662	−	0.821879i	$-0.692926\pi$
$47$	−7.81082	−1.13932	−0.569662	+	0.821879i	$0.692926\pi$
$48$	0	0
$49$	−6.99943	−0.999919
$50$	0	0
$51$	11.2070	1.56929
$52$	0	0
$53$	−8.97184	−1.23238	−0.616189	−	0.787599i	$-0.711324\pi$
$53$	−8.97184	−1.23238	−0.616189	+	0.787599i	$0.711324\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−7.51920	−0.995943
$58$	0	0
$59$	4.45536	0.580038	0.290019	−	0.957021i	$-0.406338\pi$
$59$	4.45536	0.580038	0.290019	+	0.957021i	$0.406338\pi$
$60$	0	0
$61$	9.21403	1.17974	0.589868	−	0.807500i	$-0.299180\pi$
$61$	9.21403	1.17974	0.589868	+	0.807500i	$0.299180\pi$
$62$	0	0
$63$	−0.156244	−0.0196849
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.47385	0.546568	0.273284	−	0.961933i	$-0.411890\pi$
$67$	4.47385	0.546568	0.273284	+	0.961933i	$0.411890\pi$
$68$	0	0
$69$	5.30612	0.638782
$70$	0	0
$71$	9.69458	1.15054	0.575268	−	0.817965i	$-0.304898\pi$
$71$	9.69458	1.15054	0.575268	+	0.817965i	$0.304898\pi$
$72$	0	0
$73$	−3.95387	−0.462765	−0.231383	−	0.972863i	$-0.574325\pi$
$73$	−3.95387	−0.462765	−0.231383	+	0.972863i	$0.574325\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.0852388	0.00971386
$78$	0	0
$79$	9.68349	1.08948	0.544739	−	0.838606i	$-0.316629\pi$
$79$	9.68349	1.08948	0.544739	+	0.838606i	$0.316629\pi$
$80$	0	0
$81$	14.4370	1.60411
$82$	0	0
$83$	8.95717	0.983177	0.491589	−	0.870828i	$-0.336416\pi$
$83$	8.95717	0.983177	0.491589	+	0.870828i	$0.336416\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−11.9317	−1.27922
$88$	0	0
$89$	−17.0151	−1.80359	−0.901797	−	0.432161i	$-0.857751\pi$
$89$	−17.0151	−1.80359	−0.901797	+	0.432161i	$0.857751\pi$
$90$	0	0
$91$	−0.0898200	−0.00941569
$92$	0	0
$93$	−18.5898	−1.92767
$94$	0	0
$95$	0	0
$96$	0	0
$97$	2.76438	0.280680	0.140340	−	0.990103i	$-0.455180\pi$
$97$	2.76438	0.280680	0.140340	+	0.990103i	$0.455180\pi$
$98$	0	0
$99$	−23.5359	−2.36545
$100$	0	0
$101$	10.3526	1.03012	0.515062	−	0.857153i	$-0.327769\pi$
$101$	10.3526	1.03012	0.515062	+	0.857153i	$0.327769\pi$
$102$	0	0
$103$	−18.2913	−1.80230	−0.901150	−	0.433508i	$-0.857276\pi$
$103$	−18.2913	−1.80230	−0.901150	+	0.433508i	$0.857276\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−15.8786	−1.53504	−0.767522	−	0.641023i	$-0.778510\pi$
$107$	−15.8786	−1.53504	−0.767522	+	0.641023i	$0.778510\pi$
$108$	0	0
$109$	5.96109	0.570969	0.285484	−	0.958383i	$-0.407845\pi$
$109$	5.96109	0.570969	0.285484	+	0.958383i	$0.407845\pi$
$110$	0	0
$111$	1.14237	0.108429
$112$	0	0
$113$	2.42830	0.228435	0.114218	−	0.993456i	$-0.463564\pi$
$113$	2.42830	0.228435	0.114218	+	0.993456i	$0.463564\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	24.8008	2.29284
$118$	0	0
$119$	0.0861842	0.00790049
$120$	0	0
$121$	1.83995	0.167268
$122$	0	0
$123$	24.1552	2.17800
$124$	0	0
$125$	0	0
$126$	0	0
$127$	17.1073	1.51803	0.759013	−	0.651075i	$-0.225682\pi$
$127$	17.1073	1.51803	0.759013	+	0.651075i	$0.225682\pi$
$128$	0	0
$129$	0.540004	0.0475447
$130$	0	0
$131$	−4.78955	−0.418464	−0.209232	−	0.977866i	$-0.567096\pi$
$131$	−4.78955	−0.418464	−0.209232	+	0.977866i	$0.567096\pi$
$132$	0	0
$133$	−0.0578245	−0.00501402
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.8104	1.09446	0.547231	−	0.836982i	$-0.315682\pi$
$137$	12.8104	1.09446	0.547231	+	0.836982i	$0.315682\pi$
$138$	0	0
$139$	7.94020	0.673479	0.336739	−	0.941598i	$-0.390676\pi$
$139$	7.94020	0.673479	0.336739	+	0.941598i	$0.390676\pi$
$140$	0	0
$141$	24.1609	2.03471
$142$	0	0
$143$	−13.5300	−1.13144
$144$	0	0
$145$	0	0
$146$	0	0
$147$	21.6510	1.78575
$148$	0	0
$149$	−5.62724	−0.461002	−0.230501	−	0.973072i	$-0.574036\pi$
$149$	−5.62724	−0.461002	−0.230501	+	0.973072i	$0.574036\pi$
$150$	0	0
$151$	7.36960	0.599730	0.299865	−	0.953982i	$-0.403058\pi$
$151$	7.36960	0.599730	0.299865	+	0.953982i	$0.403058\pi$
$152$	0	0
$153$	−23.7969	−1.92387
$154$	0	0
$155$	0	0
$156$	0	0
$157$	8.63091	0.688822	0.344411	−	0.938819i	$-0.388079\pi$
$157$	8.63091	0.688822	0.344411	+	0.938819i	$0.388079\pi$
$158$	0	0
$159$	27.7522	2.20089
$160$	0	0
$161$	0.0408053	0.00321591
$162$	0	0
$163$	−4.74964	−0.372020	−0.186010	−	0.982548i	$-0.559556\pi$
$163$	−4.74964	−0.372020	−0.186010	+	0.982548i	$0.559556\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−18.5967	−1.43906	−0.719528	−	0.694463i	$-0.755642\pi$
$167$	−18.5967	−1.43906	−0.719528	+	0.694463i	$0.755642\pi$
$168$	0	0
$169$	1.25720	0.0967079
$170$	0	0
$171$	15.9663	1.22097
$172$	0	0
$173$	15.7957	1.20092	0.600462	−	0.799653i	$-0.294983\pi$
$173$	15.7957	1.20092	0.600462	+	0.799653i	$0.294983\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−13.7816	−1.03589
$178$	0	0
$179$	−9.36946	−0.700306	−0.350153	−	0.936692i	$-0.613870\pi$
$179$	−9.36946	−0.700306	−0.350153	+	0.936692i	$0.613870\pi$
$180$	0	0
$181$	−5.84630	−0.434552	−0.217276	−	0.976110i	$-0.569717\pi$
$181$	−5.84630	−0.434552	−0.217276	+	0.976110i	$0.569717\pi$
$182$	0	0
$183$	−28.5014	−2.10688
$184$	0	0
$185$	0	0
$186$	0	0
$187$	12.9824	0.949364
$188$	0	0
$189$	0.262558	0.0190983
$190$	0	0
$191$	21.8947	1.58425	0.792123	−	0.610362i	$-0.208976\pi$
$191$	21.8947	1.58425	0.792123	+	0.610362i	$0.208976\pi$
$192$	0	0
$193$	−25.2541	−1.81783	−0.908916	−	0.416980i	$-0.863089\pi$
$193$	−25.2541	−1.81783	−0.908916	+	0.416980i	$0.863089\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.4538	1.02979	0.514897	−	0.857252i	$-0.327830\pi$
$197$	14.4538	1.02979	0.514897	+	0.857252i	$0.327830\pi$
$198$	0	0
$199$	3.77734	0.267768	0.133884	−	0.990997i	$-0.457255\pi$
$199$	3.77734	0.267768	0.133884	+	0.990997i	$0.457255\pi$
$200$	0	0
$201$	−13.8388	−0.976112
$202$	0	0
$203$	−0.0917579	−0.00644014
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−11.2670	−0.783113
$208$	0	0
$209$	−8.71039	−0.602510
$210$	0	0
$211$	18.9006	1.30117	0.650586	−	0.759432i	$-0.274523\pi$
$211$	18.9006	1.30117	0.650586	+	0.759432i	$0.274523\pi$
$212$	0	0
$213$	−29.9878	−2.05473
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−0.142960	−0.00970477
$218$	0	0
$219$	12.2303	0.826449
$220$	0	0
$221$	−13.6801	−0.920222
$222$	0	0
$223$	11.3556	0.760426	0.380213	−	0.924899i	$-0.375851\pi$
$223$	11.3556	0.760426	0.380213	+	0.924899i	$0.375851\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−11.1977	−0.743216	−0.371608	−	0.928390i	$-0.621194\pi$
$227$	−11.1977	−0.743216	−0.371608	+	0.928390i	$0.621194\pi$
$228$	0	0
$229$	9.64969	0.637669	0.318835	−	0.947810i	$-0.396709\pi$
$229$	9.64969	0.637669	0.318835	+	0.947810i	$0.396709\pi$
$230$	0	0
$231$	−0.263665	−0.0173479
$232$	0	0
$233$	−15.5261	−1.01715	−0.508573	−	0.861019i	$-0.669827\pi$
$233$	−15.5261	−1.01715	−0.508573	+	0.861019i	$0.669827\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−29.9535	−1.94569
$238$	0	0
$239$	−14.3342	−0.927203	−0.463601	−	0.886044i	$-0.653443\pi$
$239$	−14.3342	−0.927203	−0.463601	+	0.886044i	$0.653443\pi$
$240$	0	0
$241$	25.3114	1.63045	0.815224	−	0.579145i	$-0.196614\pi$
$241$	25.3114	1.63045	0.815224	+	0.579145i	$0.196614\pi$
$242$	0	0
$243$	−11.5450	−0.740613
$244$	0	0
$245$	0	0
$246$	0	0
$247$	9.17853	0.584016
$248$	0	0
$249$	−27.7068	−1.75585
$250$	0	0
$251$	−4.23698	−0.267436	−0.133718	−	0.991019i	$-0.542692\pi$
$251$	−4.23698	−0.267436	−0.133718	+	0.991019i	$0.542692\pi$
$252$	0	0
$253$	6.14671	0.386440
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−20.4007	−1.27256	−0.636281	−	0.771458i	$-0.719528\pi$
$257$	−20.4007	−1.27256	−0.636281	+	0.771458i	$0.719528\pi$
$258$	0	0
$259$	0.00878507	0.000545877 0
$260$	0	0
$261$	25.3359	1.56825
$262$	0	0
$263$	−28.6678	−1.76773	−0.883865	−	0.467741i	$-0.845068\pi$
$263$	−28.6678	−1.76773	−0.883865	+	0.467741i	$0.845068\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	52.6320	3.22102
$268$	0	0
$269$	7.94652	0.484508	0.242254	−	0.970213i	$-0.422113\pi$
$269$	7.94652	0.484508	0.242254	+	0.970213i	$0.422113\pi$
$270$	0	0
$271$	−9.55487	−0.580417	−0.290208	−	0.956963i	$-0.593725\pi$
$271$	−9.55487	−0.580417	−0.290208	+	0.956963i	$0.593725\pi$
$272$	0	0
$273$	0.277836	0.0168154
$274$	0	0
$275$	0	0
$276$	0	0
$277$	17.1904	1.03287	0.516436	−	0.856326i	$-0.327258\pi$
$277$	17.1904	1.03287	0.516436	+	0.856326i	$0.327258\pi$
$278$	0	0
$279$	39.4737	2.36323
$280$	0	0
$281$	27.2182	1.62370	0.811851	−	0.583864i	$-0.198460\pi$
$281$	27.2182	1.62370	0.811851	+	0.583864i	$0.198460\pi$
$282$	0	0
$283$	3.47901	0.206806	0.103403	−	0.994640i	$-0.467027\pi$
$283$	3.47901	0.206806	0.103403	+	0.994640i	$0.467027\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.185760	0.0109650
$288$	0	0
$289$	−3.87366	−0.227863
$290$	0	0
$291$	−8.55092	−0.501264
$292$	0	0
$293$	23.4941	1.37254	0.686271	−	0.727346i	$-0.259246\pi$
$293$	23.4941	1.37254	0.686271	+	0.727346i	$0.259246\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	39.5504	2.29495
$298$	0	0
$299$	−6.47706	−0.374578
$300$	0	0
$301$	0.00415276	0.000239361 0
$302$	0	0
$303$	−32.0233	−1.83969
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−1.11253	−0.0634952	−0.0317476	−	0.999496i	$-0.510107\pi$
$307$	−1.11253	−0.0634952	−0.0317476	+	0.999496i	$0.510107\pi$
$308$	0	0
$309$	56.5798	3.21871
$310$	0	0
$311$	14.7529	0.836561	0.418280	−	0.908318i	$-0.362633\pi$
$311$	14.7529	0.836561	0.418280	+	0.908318i	$0.362633\pi$
$312$	0	0
$313$	4.94834	0.279697	0.139848	−	0.990173i	$-0.455339\pi$
$313$	4.94834	0.279697	0.139848	+	0.990173i	$0.455339\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	22.6855	1.27414	0.637072	−	0.770804i	$-0.280145\pi$
$317$	22.6855	1.27414	0.637072	+	0.770804i	$0.280145\pi$
$318$	0	0
$319$	−13.8220	−0.773881
$320$	0	0
$321$	49.1166	2.74142
$322$	0	0
$323$	−8.80699	−0.490034
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−18.4392	−1.01969
$328$	0	0
$329$	0.185803	0.0102436
$330$	0	0
$331$	−1.73258	−0.0952312	−0.0476156	−	0.998866i	$-0.515162\pi$
$331$	−1.73258	−0.0952312	−0.0476156	+	0.998866i	$0.515162\pi$
$332$	0	0
$333$	−2.42571	−0.132928
$334$	0	0
$335$	0	0
$336$	0	0
$337$	13.9811	0.761596	0.380798	−	0.924658i	$-0.375649\pi$
$337$	13.9811	0.761596	0.380798	+	0.924658i	$0.375649\pi$
$338$	0	0
$339$	−7.51136	−0.407961
$340$	0	0
$341$	−21.5348	−1.16617
$342$	0	0
$343$	0.333017	0.0179812
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−5.08173	−0.272802	−0.136401	−	0.990654i	$-0.543553\pi$
$347$	−5.08173	−0.272802	−0.136401	+	0.990654i	$0.543553\pi$
$348$	0	0
$349$	−8.88643	−0.475680	−0.237840	−	0.971304i	$-0.576439\pi$
$349$	−8.88643	−0.475680	−0.237840	+	0.971304i	$0.576439\pi$
$350$	0	0
$351$	−41.6761	−2.22450
$352$	0	0
$353$	9.41440	0.501078	0.250539	−	0.968107i	$-0.419392\pi$
$353$	9.41440	0.501078	0.250539	+	0.968107i	$0.419392\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−0.266590	−0.0141094
$358$	0	0
$359$	−9.11498	−0.481070	−0.240535	−	0.970640i	$-0.577323\pi$
$359$	−9.11498	−0.481070	−0.240535	+	0.970640i	$0.577323\pi$
$360$	0	0
$361$	−13.0910	−0.689002
$362$	0	0
$363$	−5.69143	−0.298723
$364$	0	0
$365$	0	0
$366$	0	0
$367$	17.8784	0.933246	0.466623	−	0.884456i	$-0.345471\pi$
$367$	17.8784	0.933246	0.466623	+	0.884456i	$0.345471\pi$
$368$	0	0
$369$	−51.2914	−2.67012
$370$	0	0
$371$	0.213421	0.0110803
$372$	0	0
$373$	−3.07395	−0.159163	−0.0795814	−	0.996828i	$-0.525358\pi$
$373$	−3.07395	−0.159163	−0.0795814	+	0.996828i	$0.525358\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	14.5648	0.750126
$378$	0	0
$379$	−25.5866	−1.31430	−0.657149	−	0.753761i	$-0.728238\pi$
$379$	−25.5866	−1.31430	−0.657149	+	0.753761i	$0.728238\pi$
$380$	0	0
$381$	−52.9173	−2.71103
$382$	0	0
$383$	2.22529	0.113707	0.0568535	−	0.998383i	$-0.481893\pi$
$383$	2.22529	0.113707	0.0568535	+	0.998383i	$0.481893\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.14665	−0.0582873
$388$	0	0
$389$	4.64413	0.235467	0.117733	−	0.993045i	$-0.462437\pi$
$389$	4.64413	0.235467	0.117733	+	0.993045i	$0.462437\pi$
$390$	0	0
$391$	6.21488	0.314300
$392$	0	0
$393$	14.8153	0.747333
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−37.9452	−1.90442	−0.952208	−	0.305452i	$-0.901193\pi$
$397$	−37.9452	−1.90442	−0.952208	+	0.305452i	$0.901193\pi$
$398$	0	0
$399$	0.178866	0.00895449
$400$	0	0
$401$	−16.7187	−0.834890	−0.417445	−	0.908702i	$-0.637074\pi$
$401$	−16.7187	−0.834890	−0.417445	+	0.908702i	$0.637074\pi$
$402$	0	0
$403$	22.6922	1.13038
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.32334	0.0655955
$408$	0	0
$409$	0.473132	0.0233949	0.0116974	−	0.999932i	$-0.496277\pi$
$409$	0.473132	0.0233949	0.0116974	+	0.999932i	$0.496277\pi$
$410$	0	0
$411$	−39.6257	−1.95459
$412$	0	0
$413$	−0.105983	−0.00521511
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−24.5611	−1.20276
$418$	0	0
$419$	−24.2348	−1.18395	−0.591975	−	0.805957i	$-0.701651\pi$
$419$	−24.2348	−1.18395	−0.591975	+	0.805957i	$0.701651\pi$
$420$	0	0
$421$	−31.3482	−1.52782	−0.763908	−	0.645325i	$-0.776722\pi$
$421$	−31.3482	−1.52782	−0.763908	+	0.645325i	$0.776722\pi$
$422$	0	0
$423$	−51.3033	−2.49445
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−0.219182	−0.0106070
$428$	0	0
$429$	41.8519	2.02063
$430$	0	0
$431$	2.00066	0.0963686	0.0481843	−	0.998838i	$-0.484657\pi$
$431$	2.00066	0.0963686	0.0481843	+	0.998838i	$0.484657\pi$
$432$	0	0
$433$	7.07253	0.339884	0.169942	−	0.985454i	$-0.445642\pi$
$433$	7.07253	0.339884	0.169942	+	0.985454i	$0.445642\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.16981	−0.199469
$438$	0	0
$439$	−12.9620	−0.618644	−0.309322	−	0.950957i	$-0.600102\pi$
$439$	−12.9620	−0.618644	−0.309322	+	0.950957i	$0.600102\pi$
$440$	0	0
$441$	−45.9739	−2.18924
$442$	0	0
$443$	−21.8687	−1.03901	−0.519506	−	0.854467i	$-0.673884\pi$
$443$	−21.8687	−1.03901	−0.519506	+	0.854467i	$0.673884\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	17.4065	0.823299
$448$	0	0
$449$	0.399626	0.0188595	0.00942976	−	0.999956i	$-0.496998\pi$
$449$	0.399626	0.0188595	0.00942976	+	0.999956i	$0.496998\pi$
$450$	0	0
$451$	27.9819	1.31762
$452$	0	0
$453$	−22.7961	−1.07105
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−35.2247	−1.64774	−0.823872	−	0.566777i	$-0.808190\pi$
$457$	−35.2247	−1.64774	−0.823872	+	0.566777i	$0.808190\pi$
$458$	0	0
$459$	39.9891	1.86653
$460$	0	0
$461$	−15.9615	−0.743402	−0.371701	−	0.928353i	$-0.621225\pi$
$461$	−15.9615	−0.743402	−0.371701	+	0.928353i	$0.621225\pi$
$462$	0	0
$463$	30.4831	1.41667	0.708336	−	0.705875i	$-0.249446\pi$
$463$	30.4831	1.41667	0.708336	+	0.705875i	$0.249446\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6.64657	−0.307567	−0.153783	−	0.988105i	$-0.549146\pi$
$467$	−6.64657	−0.307567	−0.153783	+	0.988105i	$0.549146\pi$
$468$	0	0
$469$	−0.106424	−0.00491418
$470$	0	0
$471$	−26.6976	−1.23016
$472$	0	0
$473$	0.625551	0.0287628
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−58.9292	−2.69818
$478$	0	0
$479$	−22.2408	−1.01621	−0.508104	−	0.861296i	$-0.669653\pi$
$479$	−22.2408	−1.01621	−0.508104	+	0.861296i	$0.669653\pi$
$480$	0	0
$481$	−1.39446	−0.0635820
$482$	0	0
$483$	−0.126221	−0.00574327
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−10.1851	−0.461531	−0.230765	−	0.973009i	$-0.574123\pi$
$487$	−10.1851	−0.461531	−0.230765	+	0.973009i	$0.574123\pi$
$488$	0	0
$489$	14.6918	0.664388
$490$	0	0
$491$	−6.64004	−0.299661	−0.149830	−	0.988712i	$-0.547873\pi$
$491$	−6.64004	−0.299661	−0.149830	+	0.988712i	$0.547873\pi$
$492$	0	0
$493$	−13.9752	−0.629413
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−0.230614	−0.0103444
$498$	0	0
$499$	−1.08397	−0.0485253	−0.0242626	−	0.999706i	$-0.507724\pi$
$499$	−1.08397	−0.0485253	−0.0242626	+	0.999706i	$0.507724\pi$
$500$	0	0
$501$	57.5244	2.57000
$502$	0	0
$503$	1.98603	0.0885525	0.0442762	−	0.999019i	$-0.485902\pi$
$503$	1.98603	0.0885525	0.0442762	+	0.999019i	$0.485902\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−3.88885	−0.172710
$508$	0	0
$509$	2.30915	0.102351	0.0511757	−	0.998690i	$-0.483703\pi$
$509$	2.30915	0.102351	0.0511757	+	0.998690i	$0.483703\pi$
$510$	0	0
$511$	0.0940541	0.00416071
$512$	0	0
$513$	−26.8303	−1.18459
$514$	0	0
$515$	0	0
$516$	0	0
$517$	27.9884	1.23093
$518$	0	0
$519$	−48.8601	−2.14472
$520$	0	0
$521$	−11.0447	−0.483879	−0.241940	−	0.970291i	$-0.577784\pi$
$521$	−11.0447	−0.483879	−0.241940	+	0.970291i	$0.577784\pi$
$522$	0	0
$523$	−26.0025	−1.13701	−0.568505	−	0.822680i	$-0.692478\pi$
$523$	−26.0025	−1.13701	−0.568505	+	0.822680i	$0.692478\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−21.7736	−0.948475
$528$	0	0
$529$	−20.0575	−0.872064
$530$	0	0
$531$	29.2638	1.26994
$532$	0	0
$533$	−29.4858	−1.27717
$534$	0	0
$535$	0	0
$536$	0	0
$537$	28.9821	1.25067
$538$	0	0
$539$	25.0810	1.08031
$540$	0	0
$541$	−0.0182391	−0.000784162 0	−0.000392081	−	1.00000i	$-0.500125\pi$
$541$	−0.0182391	−0.000784162 0	−0.000392081		1.00000i	$0.500125\pi$
$542$	0	0
$543$	18.0841	0.776063
$544$	0	0
$545$	0	0
$546$	0	0
$547$	9.96048	0.425879	0.212940	−	0.977065i	$-0.431696\pi$
$547$	9.96048	0.425879	0.212940	+	0.977065i	$0.431696\pi$
$548$	0	0
$549$	60.5199	2.58293
$550$	0	0
$551$	9.37656	0.399455
$552$	0	0
$553$	−0.230350	−0.00979547
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−34.9291	−1.47999	−0.739996	−	0.672611i	$-0.765173\pi$
$557$	−34.9291	−1.47999	−0.739996	+	0.672611i	$0.765173\pi$
$558$	0	0
$559$	−0.659171	−0.0278800
$560$	0	0
$561$	−40.1577	−1.69546
$562$	0	0
$563$	−15.7939	−0.665634	−0.332817	−	0.942991i	$-0.607999\pi$
$563$	−15.7939	−0.665634	−0.332817	+	0.942991i	$0.607999\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−0.343426	−0.0144225
$568$	0	0
$569$	−14.5605	−0.610408	−0.305204	−	0.952287i	$-0.598725\pi$
$569$	−14.5605	−0.610408	−0.305204	+	0.952287i	$0.598725\pi$
$570$	0	0
$571$	−27.6138	−1.15560	−0.577802	−	0.816177i	$-0.696089\pi$
$571$	−27.6138	−1.15560	−0.577802	+	0.816177i	$0.696089\pi$
$572$	0	0
$573$	−67.7259	−2.82929
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−27.1520	−1.13035	−0.565176	−	0.824970i	$-0.691192\pi$
$577$	−27.1520	−1.13035	−0.565176	+	0.824970i	$0.691192\pi$
$578$	0	0
$579$	78.1175	3.24645
$580$	0	0
$581$	−0.213072	−0.00883972
$582$	0	0
$583$	32.1487	1.33146
$584$	0	0
$585$	0	0
$586$	0	0
$587$	10.7037	0.441787	0.220894	−	0.975298i	$-0.429103\pi$
$587$	10.7037	0.441787	0.220894	+	0.975298i	$0.429103\pi$
$588$	0	0
$589$	14.6088	0.601946
$590$	0	0
$591$	−44.7094	−1.83910
$592$	0	0
$593$	−3.84629	−0.157948	−0.0789740	−	0.996877i	$-0.525164\pi$
$593$	−3.84629	−0.157948	−0.0789740	+	0.996877i	$0.525164\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−11.6843	−0.478206
$598$	0	0
$599$	−46.1423	−1.88532	−0.942662	−	0.333750i	$-0.891686\pi$
$599$	−46.1423	−1.88532	−0.942662	+	0.333750i	$0.891686\pi$
$600$	0	0
$601$	−13.3119	−0.543005	−0.271503	−	0.962438i	$-0.587521\pi$
$601$	−13.3119	−0.543005	−0.271503	+	0.962438i	$0.587521\pi$
$602$	0	0
$603$	29.3853	1.19666
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−34.0838	−1.38342	−0.691709	−	0.722176i	$-0.743142\pi$
$607$	−34.0838	−1.38342	−0.691709	+	0.722176i	$0.743142\pi$
$608$	0	0
$609$	0.283831	0.0115014
$610$	0	0
$611$	−29.4926	−1.19314
$612$	0	0
$613$	25.4809	1.02916	0.514582	−	0.857441i	$-0.327947\pi$
$613$	25.4809	1.02916	0.514582	+	0.857441i	$0.327947\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−26.9453	−1.08478	−0.542389	−	0.840127i	$-0.682480\pi$
$617$	−26.9453	−1.08478	−0.542389	+	0.840127i	$0.682480\pi$
$618$	0	0
$619$	−39.3289	−1.58076	−0.790381	−	0.612615i	$-0.790118\pi$
$619$	−39.3289	−1.58076	−0.790381	+	0.612615i	$0.790118\pi$
$620$	0	0
$621$	18.9335	0.759775
$622$	0	0
$623$	0.404752	0.0162161
$624$	0	0
$625$	0	0
$626$	0	0
$627$	26.9435	1.07602
$628$	0	0
$629$	1.33802	0.0533502
$630$	0	0
$631$	−11.2443	−0.447630	−0.223815	−	0.974632i	$-0.571851\pi$
$631$	−11.2443	−0.447630	−0.223815	+	0.974632i	$0.571851\pi$
$632$	0	0
$633$	−58.4645	−2.32375
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−26.4290	−1.04715
$638$	0	0
$639$	63.6763	2.51900
$640$	0	0
$641$	17.4773	0.690311	0.345155	−	0.938546i	$-0.387826\pi$
$641$	17.4773	0.690311	0.345155	+	0.938546i	$0.387826\pi$
$642$	0	0
$643$	8.72320	0.344009	0.172005	−	0.985096i	$-0.444976\pi$
$643$	8.72320	0.344009	0.172005	+	0.985096i	$0.444976\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−25.0654	−0.985421	−0.492710	−	0.870193i	$-0.663994\pi$
$647$	−25.0654	−0.985421	−0.492710	+	0.870193i	$0.663994\pi$
$648$	0	0
$649$	−15.9648	−0.626674
$650$	0	0
$651$	0.442212	0.0173317
$652$	0	0
$653$	29.9777	1.17312	0.586559	−	0.809906i	$-0.300482\pi$
$653$	29.9777	1.17312	0.586559	+	0.809906i	$0.300482\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−25.9699	−1.01318
$658$	0	0
$659$	4.36601	0.170076	0.0850379	−	0.996378i	$-0.472899\pi$
$659$	4.36601	0.170076	0.0850379	+	0.996378i	$0.472899\pi$
$660$	0	0
$661$	−19.4529	−0.756628	−0.378314	−	0.925677i	$-0.623496\pi$
$661$	−19.4529	−0.756628	−0.378314	+	0.925677i	$0.623496\pi$
$662$	0	0
$663$	42.3160	1.64342
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−6.61681	−0.256204
$668$	0	0
$669$	−35.1257	−1.35804
$670$	0	0
$671$	−33.0165	−1.27459
$672$	0	0
$673$	−36.2275	−1.39647	−0.698234	−	0.715870i	$-0.746030\pi$
$673$	−36.2275	−1.39647	−0.698234	+	0.715870i	$0.746030\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2.04046	0.0784214	0.0392107	−	0.999231i	$-0.487516\pi$
$677$	2.04046	0.0784214	0.0392107	+	0.999231i	$0.487516\pi$
$678$	0	0
$679$	−0.0657586	−0.00252359
$680$	0	0
$681$	34.6373	1.32730
$682$	0	0
$683$	−15.9105	−0.608799	−0.304400	−	0.952544i	$-0.598456\pi$
$683$	−15.9105	−0.608799	−0.304400	+	0.952544i	$0.598456\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−29.8490	−1.13881
$688$	0	0
$689$	−33.8765	−1.29059
$690$	0	0
$691$	3.90166	0.148426	0.0742130	−	0.997242i	$-0.476356\pi$
$691$	3.90166	0.148426	0.0742130	+	0.997242i	$0.476356\pi$
$692$	0	0
$693$	0.559869	0.0212677
$694$	0	0
$695$	0	0
$696$	0	0
$697$	28.2922	1.07164
$698$	0	0
$699$	48.0261	1.81651
$700$	0	0
$701$	30.3587	1.14663	0.573316	−	0.819335i	$-0.305657\pi$
$701$	30.3587	1.14663	0.573316	+	0.819335i	$0.305657\pi$
$702$	0	0
$703$	−0.897729	−0.0338585
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−0.246267	−0.00926181
$708$	0	0
$709$	−48.5677	−1.82400	−0.911999	−	0.410193i	$-0.865461\pi$
$709$	−48.5677	−1.82400	−0.911999	+	0.410193i	$0.865461\pi$
$710$	0	0
$711$	63.6035	2.38532
$712$	0	0
$713$	−10.3091	−0.386078
$714$	0	0
$715$	0	0
$716$	0	0
$717$	44.3394	1.65588
$718$	0	0
$719$	−4.76238	−0.177607	−0.0888033	−	0.996049i	$-0.528304\pi$
$719$	−4.76238	−0.177607	−0.0888033	+	0.996049i	$0.528304\pi$
$720$	0	0
$721$	0.435112	0.0162044
$722$	0	0
$723$	−78.2946	−2.91181
$724$	0	0
$725$	0	0
$726$	0	0
$727$	3.38691	0.125614	0.0628068	−	0.998026i	$-0.479995\pi$
$727$	3.38691	0.125614	0.0628068	+	0.998026i	$0.479995\pi$
$728$	0	0
$729$	−7.59939	−0.281459
$730$	0	0
$731$	0.632488	0.0233934
$732$	0	0
$733$	15.4009	0.568847	0.284423	−	0.958699i	$-0.408198\pi$
$733$	15.4009	0.568847	0.284423	+	0.958699i	$0.408198\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−16.0311	−0.590513
$738$	0	0
$739$	−9.98465	−0.367291	−0.183646	−	0.982993i	$-0.558790\pi$
$739$	−9.98465	−0.367291	−0.183646	+	0.982993i	$0.558790\pi$
$740$	0	0
$741$	−28.3915	−1.04299
$742$	0	0
$743$	41.4419	1.52036	0.760178	−	0.649715i	$-0.225112\pi$
$743$	41.4419	1.52036	0.760178	+	0.649715i	$0.225112\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	58.8328	2.15258
$748$	0	0
$749$	0.377719	0.0138015
$750$	0	0
$751$	−21.1036	−0.770082	−0.385041	−	0.922900i	$-0.625813\pi$
$751$	−21.1036	−0.770082	−0.385041	+	0.922900i	$0.625813\pi$
$752$	0	0
$753$	13.1061	0.477612
$754$	0	0
$755$	0	0
$756$	0	0
$757$	40.7168	1.47988	0.739938	−	0.672675i	$-0.234855\pi$
$757$	40.7168	1.47988	0.739938	+	0.672675i	$0.234855\pi$
$758$	0	0
$759$	−19.0133	−0.690141
$760$	0	0
$761$	−5.07664	−0.184028	−0.0920140	−	0.995758i	$-0.529330\pi$
$761$	−5.07664	−0.184028	−0.0920140	+	0.995758i	$0.529330\pi$
$762$	0	0
$763$	−0.141802	−0.00513357
$764$	0	0
$765$	0	0
$766$	0	0
$767$	16.8229	0.607438
$768$	0	0
$769$	−46.7687	−1.68652	−0.843261	−	0.537505i	$-0.819367\pi$
$769$	−46.7687	−1.68652	−0.843261	+	0.537505i	$0.819367\pi$
$770$	0	0
$771$	63.1046	2.27266
$772$	0	0
$773$	19.0938	0.686756	0.343378	−	0.939197i	$-0.388429\pi$
$773$	19.0938	0.686756	0.343378	+	0.939197i	$0.388429\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−0.0271745	−0.000974878 0
$778$	0	0
$779$	−18.9824	−0.680115
$780$	0	0
$781$	−34.7385	−1.24304
$782$	0	0
$783$	−42.5753	−1.52152
$784$	0	0
$785$	0	0
$786$	0	0
$787$	4.02513	0.143481	0.0717403	−	0.997423i	$-0.477145\pi$
$787$	4.02513	0.143481	0.0717403	+	0.997423i	$0.477145\pi$
$788$	0	0
$789$	88.6768	3.15698
$790$	0	0
$791$	−0.0577642	−0.00205386
$792$	0	0
$793$	34.7910	1.23546
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−15.9847	−0.566205	−0.283103	−	0.959090i	$-0.591364\pi$
$797$	−15.9847	−0.566205	−0.283103	+	0.959090i	$0.591364\pi$
$798$	0	0
$799$	28.2988	1.00114
$800$	0	0
$801$	−111.759	−3.94881
$802$	0	0
$803$	14.1678	0.499972
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−24.5806	−0.865279
$808$	0	0
$809$	−29.2015	−1.02667	−0.513335	−	0.858188i	$-0.671590\pi$
$809$	−29.2015	−1.02667	−0.513335	+	0.858188i	$0.671590\pi$
$810$	0	0
$811$	−7.11719	−0.249918	−0.124959	−	0.992162i	$-0.539880\pi$
$811$	−7.11719	−0.249918	−0.124959	+	0.992162i	$0.539880\pi$
$812$	0	0
$813$	29.5557	1.03656
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−0.424362	−0.0148466
$818$	0	0
$819$	−0.589959	−0.0206148
$820$	0	0
$821$	−32.6032	−1.13786	−0.568929	−	0.822387i	$-0.692642\pi$
$821$	−32.6032	−1.13786	−0.568929	+	0.822387i	$0.692642\pi$
$822$	0	0
$823$	−27.1361	−0.945905	−0.472953	−	0.881088i	$-0.656812\pi$
$823$	−27.1361	−0.945905	−0.472953	+	0.881088i	$0.656812\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	54.8232	1.90639	0.953195	−	0.302357i	$-0.0977736\pi$
$827$	54.8232	1.90639	0.953195	+	0.302357i	$0.0977736\pi$
$828$	0	0
$829$	0.471969	0.0163922	0.00819608	−	0.999966i	$-0.497391\pi$
$829$	0.471969	0.0163922	0.00819608	+	0.999966i	$0.497391\pi$
$830$	0	0
$831$	−53.1744	−1.84460
$832$	0	0
$833$	25.3591	0.878642
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−66.3329	−2.29280
$838$	0	0
$839$	38.4737	1.32826	0.664129	−	0.747618i	$-0.268802\pi$
$839$	38.4737	1.32826	0.664129	+	0.747618i	$0.268802\pi$
$840$	0	0
$841$	−14.1209	−0.486929
$842$	0	0
$843$	−84.1929	−2.89976
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−0.0437684	−0.00150390
$848$	0	0
$849$	−10.7615	−0.369333
$850$	0	0
$851$	0.633505	0.0217163
$852$	0	0
$853$	−35.8541	−1.22762	−0.613811	−	0.789453i	$-0.710364\pi$
$853$	−35.8541	−1.22762	−0.613811	+	0.789453i	$0.710364\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0.386299	0.0131957	0.00659786	−	0.999978i	$-0.497900\pi$
$857$	0.386299	0.0131957	0.00659786	+	0.999978i	$0.497900\pi$
$858$	0	0
$859$	−22.2889	−0.760487	−0.380244	−	0.924886i	$-0.624160\pi$
$859$	−22.2889	−0.760487	−0.380244	+	0.924886i	$0.624160\pi$
$860$	0	0
$861$	−0.574602	−0.0195824
$862$	0	0
$863$	−9.29808	−0.316510	−0.158255	−	0.987398i	$-0.550587\pi$
$863$	−9.29808	−0.316510	−0.158255	+	0.987398i	$0.550587\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	11.9822	0.406938
$868$	0	0
$869$	−34.6987	−1.17707
$870$	0	0
$871$	16.8927	0.572387
$872$	0	0
$873$	18.1571	0.614524
$874$	0	0
$875$	0	0
$876$	0	0
$877$	29.1786	0.985292	0.492646	−	0.870230i	$-0.336030\pi$
$877$	29.1786	0.985292	0.492646	+	0.870230i	$0.336030\pi$
$878$	0	0
$879$	−72.6734	−2.45121
$880$	0	0
$881$	−40.0737	−1.35012	−0.675059	−	0.737764i	$-0.735882\pi$
$881$	−40.0737	−1.35012	−0.675059	+	0.737764i	$0.735882\pi$
$882$	0	0
$883$	−30.8339	−1.03764	−0.518822	−	0.854882i	$-0.673629\pi$
$883$	−30.8339	−1.03764	−0.518822	+	0.854882i	$0.673629\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	38.5528	1.29448	0.647239	−	0.762287i	$-0.275924\pi$
$887$	38.5528	1.29448	0.647239	+	0.762287i	$0.275924\pi$
$888$	0	0
$889$	−0.406946	−0.0136485
$890$	0	0
$891$	−51.7320	−1.73309
$892$	0	0
$893$	−18.9868	−0.635370
$894$	0	0
$895$	0	0
$896$	0	0
$897$	20.0352	0.668956
$898$	0	0
$899$	23.1818	0.773156
$900$	0	0
$901$	32.5052	1.08291
$902$	0	0
$903$	−0.0128455	−0.000427473 0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	4.77670	0.158608	0.0793038	−	0.996850i	$-0.474730\pi$
$907$	4.77670	0.158608	0.0793038	+	0.996850i	$0.474730\pi$
$908$	0	0
$909$	67.9984	2.25537
$910$	0	0
$911$	11.5332	0.382111	0.191056	−	0.981579i	$-0.438809\pi$
$911$	11.5332	0.382111	0.191056	+	0.981579i	$0.438809\pi$
$912$	0	0
$913$	−32.0961	−1.06223
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0.113933	0.00376240
$918$	0	0
$919$	41.3268	1.36325	0.681623	−	0.731704i	$-0.261274\pi$
$919$	41.3268	1.36325	0.681623	+	0.731704i	$0.261274\pi$
$920$	0	0
$921$	3.44133	0.113396
$922$	0	0
$923$	36.6055	1.20488
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−120.142	−3.94598
$928$	0	0
$929$	−13.4174	−0.440212	−0.220106	−	0.975476i	$-0.570640\pi$
$929$	−13.4174	−0.440212	−0.220106	+	0.975476i	$0.570640\pi$
$930$	0	0
$931$	−17.0145	−0.557627
$932$	0	0
$933$	−45.6345	−1.49401
$934$	0	0
$935$	0	0
$936$	0	0
$937$	41.9929	1.37185	0.685923	−	0.727674i	$-0.259398\pi$
$937$	41.9929	1.37185	0.685923	+	0.727674i	$0.259398\pi$
$938$	0	0
$939$	−15.3065	−0.499508
$940$	0	0
$941$	40.0565	1.30580	0.652902	−	0.757443i	$-0.273551\pi$
$941$	40.0565	1.30580	0.652902	+	0.757443i	$0.273551\pi$
$942$	0	0
$943$	13.3954	0.436215
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−43.9751	−1.42900	−0.714499	−	0.699637i	$-0.753345\pi$
$947$	−43.9751	−1.42900	−0.714499	+	0.699637i	$0.753345\pi$
$948$	0	0
$949$	−14.9293	−0.484625
$950$	0	0
$951$	−70.1721	−2.27549
$952$	0	0
$953$	40.6171	1.31572	0.657859	−	0.753141i	$-0.271462\pi$
$953$	40.6171	1.31572	0.657859	+	0.753141i	$0.271462\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	42.7548	1.38207
$958$	0	0
$959$	−0.304731	−0.00984028
$960$	0	0
$961$	5.11756	0.165083
$962$	0	0
$963$	−104.295	−3.36084
$964$	0	0
$965$	0	0
$966$	0	0
$967$	16.0002	0.514531	0.257265	−	0.966341i	$-0.417179\pi$
$967$	16.0002	0.514531	0.257265	+	0.966341i	$0.417179\pi$
$968$	0	0
$969$	27.2423	0.875148
$970$	0	0
$971$	−55.5201	−1.78173	−0.890863	−	0.454272i	$-0.849899\pi$
$971$	−55.5201	−1.78173	−0.890863	+	0.454272i	$0.849899\pi$
$972$	0	0
$973$	−0.188880	−0.00605523
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6.32539	0.202367	0.101184	−	0.994868i	$-0.467737\pi$
$977$	6.32539	0.202367	0.101184	+	0.994868i	$0.467737\pi$
$978$	0	0
$979$	60.9699	1.94861
$980$	0	0
$981$	39.1539	1.25009
$982$	0	0
$983$	−48.6422	−1.55144	−0.775722	−	0.631074i	$-0.782614\pi$
$983$	−48.6422	−1.55144	−0.775722	+	0.631074i	$0.782614\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−0.574736	−0.0182940
$988$	0	0
$989$	0.299462	0.00952234
$990$	0	0
$991$	44.1763	1.40331	0.701653	−	0.712519i	$-0.252446\pi$
$991$	44.1763	1.40331	0.701653	+	0.712519i	$0.252446\pi$
$992$	0	0
$993$	5.35931	0.170073
$994$	0	0
$995$	0	0
$996$	0	0
$997$	40.2262	1.27398	0.636988	−	0.770873i	$-0.280180\pi$
$997$	40.2262	1.27398	0.636988	+	0.770873i	$0.280180\pi$
$998$	0	0
$999$	4.07623	0.128966

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.be.1.1		8
4.3	odd	2		625.2.a.g.1.4	yes	8
5.4	even	2		10000.2.a.bn.1.8		8
12.11	even	2		5625.2.a.s.1.5		8
20.3	even	4		625.2.b.d.624.7		16
20.7	even	4		625.2.b.d.624.10		16
20.19	odd	2		625.2.a.e.1.5	✓	8
60.59	even	2		5625.2.a.be.1.4		8
100.3	even	20		625.2.e.j.249.4		32
100.11	odd	10		625.2.d.n.501.2		16
100.19	odd	10		625.2.d.q.251.2		16
100.23	even	20		625.2.e.k.124.4		32
100.27	even	20		625.2.e.k.124.5		32
100.31	odd	10		625.2.d.m.251.3		16
100.39	odd	10		625.2.d.p.501.3		16
100.47	even	20		625.2.e.j.249.5		32
100.59	odd	10		625.2.d.p.126.3		16
100.63	even	20		625.2.e.k.499.5		32
100.67	even	20		625.2.e.j.374.4		32
100.71	odd	10		625.2.d.m.376.3		16
100.79	odd	10		625.2.d.q.376.2		16
100.83	even	20		625.2.e.j.374.5		32
100.87	even	20		625.2.e.k.499.4		32
100.91	odd	10		625.2.d.n.126.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
625.2.a.e.1.5	✓	8	20.19	odd	2
625.2.a.g.1.4	yes	8	4.3	odd	2
625.2.b.d.624.7		16	20.3	even	4
625.2.b.d.624.10		16	20.7	even	4
625.2.d.m.251.3		16	100.31	odd	10
625.2.d.m.376.3		16	100.71	odd	10
625.2.d.n.126.2		16	100.91	odd	10
625.2.d.n.501.2		16	100.11	odd	10
625.2.d.p.126.3		16	100.59	odd	10
625.2.d.p.501.3		16	100.39	odd	10
625.2.d.q.251.2		16	100.19	odd	10
625.2.d.q.376.2		16	100.79	odd	10
625.2.e.j.249.4		32	100.3	even	20
625.2.e.j.249.5		32	100.47	even	20
625.2.e.j.374.4		32	100.67	even	20
625.2.e.j.374.5		32	100.83	even	20
625.2.e.k.124.4		32	100.23	even	20
625.2.e.k.124.5		32	100.27	even	20
625.2.e.k.499.4		32	100.87	even	20
625.2.e.k.499.5		32	100.63	even	20
5625.2.a.s.1.5		8	12.11	even	2
5625.2.a.be.1.4		8	60.59	even	2
10000.2.a.be.1.1		8	1.1	even	1	trivial
10000.2.a.bn.1.8		8	5.4	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:	\( N \)	\(=\)	\( 10000 = 2^{4} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	10000.a (trivial)

\(f(q)\)	\(=\)	\(q-3.09326 q^{3} -0.0237879 q^{7} +6.56824 q^{9} +O(q^{10})\)	\(q-3.09326 q^{3} -0.0237879 q^{7} +6.56824 q^{9} -3.58329 q^{11} +3.77587 q^{13} -3.62303 q^{17} +2.43084 q^{19} +0.0735820 q^{21} -1.71538 q^{23} -11.0375 q^{27} +3.85734 q^{29} +6.00979 q^{31} +11.0840 q^{33} -0.369309 q^{37} -11.6797 q^{39} -7.80900 q^{41} -0.174574 q^{43} -7.81082 q^{47} -6.99943 q^{49} +11.2070 q^{51} -8.97184 q^{53} -7.51920 q^{57} +4.45536 q^{59} +9.21403 q^{61} -0.156244 q^{63} +4.47385 q^{67} +5.30612 q^{69} +9.69458 q^{71} -3.95387 q^{73} +0.0852388 q^{77} +9.68349 q^{79} +14.4370 q^{81} +8.95717 q^{83} -11.9317 q^{87} -17.0151 q^{89} -0.0898200 q^{91} -18.5898 q^{93} +2.76438 q^{97} -23.5359 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 5 q^{3} - 10 q^{7} + 9 q^{9}+O(q^{10}) \) 8 * q - 5 * q^3 - 10 * q^7 + 9 * q^9	\( 8 q - 5 q^{3} - 10 q^{7} + 9 q^{9} - q^{11} + 10 q^{13} + 15 q^{17} + 10 q^{19} - 14 q^{21} - 30 q^{23} - 20 q^{27} + 10 q^{29} + 9 q^{31} + 5 q^{33} - 10 q^{37} - 8 q^{39} - 4 q^{41} - 30 q^{47} - 4 q^{49} + 14 q^{51} + 10 q^{53} - 10 q^{57} + 5 q^{59} + 6 q^{61} - 10 q^{67} + 3 q^{69} + 9 q^{71} + 5 q^{77} + 20 q^{79} + 8 q^{81} - 40 q^{83} - 40 q^{87} - 5 q^{89} - 6 q^{91} - 40 q^{93} + 22 q^{99}+O(q^{100}) \) 8 * q - 5 * q^3 - 10 * q^7 + 9 * q^9 - q^11 + 10 * q^13 + 15 * q^17 + 10 * q^19 - 14 * q^21 - 30 * q^23 - 20 * q^27 + 10 * q^29 + 9 * q^31 + 5 * q^33 - 10 * q^37 - 8 * q^39 - 4 * q^41 - 30 * q^47 - 4 * q^49 + 14 * q^51 + 10 * q^53 - 10 * q^57 + 5 * q^59 + 6 * q^61 - 10 * q^67 + 3 * q^69 + 9 * q^71 + 5 * q^77 + 20 * q^79 + 8 * q^81 - 40 * q^83 - 40 * q^87 - 5 * q^89 - 6 * q^91 - 40 * q^93 + 22 * q^99

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