LMFDB - Embedded newform 6480.2.a.bo.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6480,2,Mod(1,6480)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6480, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6480.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6480 = 2^{4} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6480.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:	$51.7430605098$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3240)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.37228$ of defining polynomial
Character	$\chi$	$=$	6480.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -3.37228 q^{7} -2.37228 q^{11} -3.37228 q^{13} -6.74456 q^{17} -1.00000 q^{19} -5.37228 q^{23} +1.00000 q^{25} +2.37228 q^{29} -11.1168 q^{31} -3.37228 q^{35} +6.00000 q^{37} +0.255437 q^{41} +4.74456 q^{43} +9.37228 q^{47} +4.37228 q^{49} +10.1168 q^{53} -2.37228 q^{55} -5.00000 q^{59} +12.7446 q^{61} -3.37228 q^{65} +0.744563 q^{67} +4.37228 q^{71} +14.7446 q^{73} +8.00000 q^{77} +2.74456 q^{79} +10.0000 q^{83} -6.74456 q^{85} -4.37228 q^{89} +11.3723 q^{91} -1.00000 q^{95} -4.74456 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - q^{7} + q^{11} - q^{13} - 2 q^{17} - 2 q^{19} - 5 q^{23} + 2 q^{25} - q^{29} - 5 q^{31} - q^{35} + 12 q^{37} + 12 q^{41} - 2 q^{43} + 13 q^{47} + 3 q^{49} + 3 q^{53} + q^{55} - 10 q^{59}+ \cdots + 2 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 - q^7 + q^11 - q^13 - 2 * q^17 - 2 * q^19 - 5 * q^23 + 2 * q^25 - q^29 - 5 * q^31 - q^35 + 12 * q^37 + 12 * q^41 - 2 * q^43 + 13 * q^47 + 3 * q^49 + 3 * q^53 + q^55 - 10 * q^59 + 14 * q^61 - q^65 - 10 * q^67 + 3 * q^71 + 18 * q^73 + 16 * q^77 - 6 * q^79 + 20 * q^83 - 2 * q^85 - 3 * q^89 + 17 * q^91 - 2 * q^95 + 2 * q^97

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	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.37228	−1.27460	−0.637301	−	0.770615i	$-0.719949\pi$
$7$	−3.37228	−1.27460	−0.637301	+	0.770615i	$0.719949\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.37228	−0.715270	−0.357635	−	0.933862i	$-0.616417\pi$
$11$	−2.37228	−0.715270	−0.357635	+	0.933862i	$0.616417\pi$
$12$	0	0
$13$	−3.37228	−0.935303	−0.467651	−	0.883913i	$-0.654900\pi$
$13$	−3.37228	−0.935303	−0.467651	+	0.883913i	$0.654900\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.74456	−1.63580	−0.817898	−	0.575363i	$-0.804861\pi$
$17$	−6.74456	−1.63580	−0.817898	+	0.575363i	$0.804861\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.37228	−1.12020	−0.560099	−	0.828426i	$-0.689237\pi$
$23$	−5.37228	−1.12020	−0.560099	+	0.828426i	$0.689237\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.37228	0.440522	0.220261	−	0.975441i	$-0.429309\pi$
$29$	2.37228	0.440522	0.220261	+	0.975441i	$0.429309\pi$
$30$	0	0
$31$	−11.1168	−1.99664	−0.998322	−	0.0579057i	$-0.981558\pi$
$31$	−11.1168	−1.99664	−0.998322	+	0.0579057i	$0.981558\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.37228	−0.570020
$36$	0	0
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.255437	0.0398926	0.0199463	−	0.999801i	$-0.493650\pi$
$41$	0.255437	0.0398926	0.0199463	+	0.999801i	$0.493650\pi$
$42$	0	0
$43$	4.74456	0.723539	0.361770	−	0.932268i	$-0.382173\pi$
$43$	4.74456	0.723539	0.361770	+	0.932268i	$0.382173\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.37228	1.36709	0.683544	−	0.729909i	$-0.260438\pi$
$47$	9.37228	1.36709	0.683544	+	0.729909i	$0.260438\pi$
$48$	0	0
$49$	4.37228	0.624612
$50$	0	0
$51$	0	0
$52$	0	0
$53$	10.1168	1.38966	0.694828	−	0.719176i	$-0.255481\pi$
$53$	10.1168	1.38966	0.694828	+	0.719176i	$0.255481\pi$
$54$	0	0
$55$	−2.37228	−0.319878
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−5.00000	−0.650945	−0.325472	−	0.945552i	$-0.605523\pi$
$59$	−5.00000	−0.650945	−0.325472	+	0.945552i	$0.605523\pi$
$60$	0	0
$61$	12.7446	1.63177	0.815887	−	0.578211i	$-0.196249\pi$
$61$	12.7446	1.63177	0.815887	+	0.578211i	$0.196249\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.37228	−0.418280
$66$	0	0
$67$	0.744563	0.0909628	0.0454814	−	0.998965i	$-0.485518\pi$
$67$	0.744563	0.0909628	0.0454814	+	0.998965i	$0.485518\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4.37228	0.518894	0.259447	−	0.965757i	$-0.416460\pi$
$71$	4.37228	0.518894	0.259447	+	0.965757i	$0.416460\pi$
$72$	0	0
$73$	14.7446	1.72572	0.862860	−	0.505443i	$-0.168671\pi$
$73$	14.7446	1.72572	0.862860	+	0.505443i	$0.168671\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8.00000	0.911685
$78$	0	0
$79$	2.74456	0.308787	0.154394	−	0.988009i	$-0.450658\pi$
$79$	2.74456	0.308787	0.154394	+	0.988009i	$0.450658\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	10.0000	1.09764	0.548821	−	0.835940i	$-0.315077\pi$
$83$	10.0000	1.09764	0.548821	+	0.835940i	$0.315077\pi$
$84$	0	0
$85$	−6.74456	−0.731551
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−4.37228	−0.463461	−0.231730	−	0.972780i	$-0.574439\pi$
$89$	−4.37228	−0.463461	−0.231730	+	0.972780i	$0.574439\pi$
$90$	0	0
$91$	11.3723	1.19214
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−4.74456	−0.481737	−0.240869	−	0.970558i	$-0.577432\pi$
$97$	−4.74456	−0.481737	−0.240869	+	0.970558i	$0.577432\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−15.8614	−1.57827	−0.789134	−	0.614220i	$-0.789471\pi$
$101$	−15.8614	−1.57827	−0.789134	+	0.614220i	$0.789471\pi$
$102$	0	0
$103$	0.116844	0.0115130	0.00575649	−	0.999983i	$-0.498168\pi$
$103$	0.116844	0.0115130	0.00575649	+	0.999983i	$0.498168\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	0	0
$109$	0.883156	0.0845910	0.0422955	−	0.999105i	$-0.486533\pi$
$109$	0.883156	0.0845910	0.0422955	+	0.999105i	$0.486533\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	14.7446	1.38705	0.693526	−	0.720432i	$-0.256056\pi$
$113$	14.7446	1.38705	0.693526	+	0.720432i	$0.256056\pi$
$114$	0	0
$115$	−5.37228	−0.500968
$116$	0	0
$117$	0	0
$118$	0	0
$119$	22.7446	2.08499
$120$	0	0
$121$	−5.37228	−0.488389
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	12.8614	1.14127	0.570633	−	0.821205i	$-0.306698\pi$
$127$	12.8614	1.14127	0.570633	+	0.821205i	$0.306698\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−6.25544	−0.546540	−0.273270	−	0.961937i	$-0.588105\pi$
$131$	−6.25544	−0.546540	−0.273270	+	0.961937i	$0.588105\pi$
$132$	0	0
$133$	3.37228	0.292414
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.25544	−0.449002	−0.224501	−	0.974474i	$-0.572075\pi$
$137$	−5.25544	−0.449002	−0.224501	+	0.974474i	$0.572075\pi$
$138$	0	0
$139$	−15.0000	−1.27228	−0.636142	−	0.771572i	$-0.719471\pi$
$139$	−15.0000	−1.27228	−0.636142	+	0.771572i	$0.719471\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	8.00000	0.668994
$144$	0	0
$145$	2.37228	0.197007
$146$	0	0
$147$	0	0
$148$	0	0
$149$	19.4891	1.59661	0.798306	−	0.602252i	$-0.205730\pi$
$149$	19.4891	1.59661	0.798306	+	0.602252i	$0.205730\pi$
$150$	0	0
$151$	−24.3723	−1.98339	−0.991694	−	0.128619i	$-0.958946\pi$
$151$	−24.3723	−1.98339	−0.991694	+	0.128619i	$0.958946\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−11.1168	−0.892926
$156$	0	0
$157$	22.8614	1.82454	0.912269	−	0.409591i	$-0.134328\pi$
$157$	22.8614	1.82454	0.912269	+	0.409591i	$0.134328\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	18.1168	1.42781
$162$	0	0
$163$	−9.48913	−0.743246	−0.371623	−	0.928384i	$-0.621199\pi$
$163$	−9.48913	−0.743246	−0.371623	+	0.928384i	$0.621199\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	−1.62772	−0.125209
$170$	0	0
$171$	0	0
$172$	0	0
$173$	14.8614	1.12989	0.564946	−	0.825128i	$-0.308897\pi$
$173$	14.8614	1.12989	0.564946	+	0.825128i	$0.308897\pi$
$174$	0	0
$175$	−3.37228	−0.254921
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4.25544	0.318066	0.159033	−	0.987273i	$-0.449162\pi$
$179$	4.25544	0.318066	0.159033	+	0.987273i	$0.449162\pi$
$180$	0	0
$181$	−13.8614	−1.03031	−0.515155	−	0.857097i	$-0.672266\pi$
$181$	−13.8614	−1.03031	−0.515155	+	0.857097i	$0.672266\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	6.00000	0.441129
$186$	0	0
$187$	16.0000	1.17004
$188$	0	0
$189$	0	0
$190$	0	0
$191$	5.62772	0.407207	0.203604	−	0.979053i	$-0.434735\pi$
$191$	5.62772	0.407207	0.203604	+	0.979053i	$0.434735\pi$
$192$	0	0
$193$	0.744563	0.0535948	0.0267974	−	0.999641i	$-0.491469\pi$
$193$	0.744563	0.0535948	0.0267974	+	0.999641i	$0.491469\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−0.627719	−0.0447231	−0.0223616	−	0.999750i	$-0.507118\pi$
$197$	−0.627719	−0.0447231	−0.0223616	+	0.999750i	$0.507118\pi$
$198$	0	0
$199$	−13.4891	−0.956219	−0.478109	−	0.878300i	$-0.658678\pi$
$199$	−13.4891	−0.956219	−0.478109	+	0.878300i	$0.658678\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−8.00000	−0.561490
$204$	0	0
$205$	0.255437	0.0178405
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.37228	0.164094
$210$	0	0
$211$	−17.0000	−1.17033	−0.585164	−	0.810915i	$-0.698970\pi$
$211$	−17.0000	−1.17033	−0.585164	+	0.810915i	$0.698970\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.74456	0.323576
$216$	0	0
$217$	37.4891	2.54493
$218$	0	0
$219$	0	0
$220$	0	0
$221$	22.7446	1.52996
$222$	0	0
$223$	−25.4891	−1.70688	−0.853439	−	0.521193i	$-0.825487\pi$
$223$	−25.4891	−1.70688	−0.853439	+	0.521193i	$0.825487\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	17.4891	1.16079	0.580397	−	0.814334i	$-0.302897\pi$
$227$	17.4891	1.16079	0.580397	+	0.814334i	$0.302897\pi$
$228$	0	0
$229$	−16.7446	−1.10651	−0.553256	−	0.833011i	$-0.686615\pi$
$229$	−16.7446	−1.10651	−0.553256	+	0.833011i	$0.686615\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−14.2337	−0.932480	−0.466240	−	0.884658i	$-0.654392\pi$
$233$	−14.2337	−0.932480	−0.466240	+	0.884658i	$0.654392\pi$
$234$	0	0
$235$	9.37228	0.611380
$236$	0	0
$237$	0	0
$238$	0	0
$239$	20.2337	1.30881	0.654404	−	0.756145i	$-0.272919\pi$
$239$	20.2337	1.30881	0.654404	+	0.756145i	$0.272919\pi$
$240$	0	0
$241$	−10.6060	−0.683191	−0.341595	−	0.939847i	$-0.610967\pi$
$241$	−10.6060	−0.683191	−0.341595	+	0.939847i	$0.610967\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	4.37228	0.279335
$246$	0	0
$247$	3.37228	0.214573
$248$	0	0
$249$	0	0
$250$	0	0
$251$	12.8614	0.811805	0.405902	−	0.913916i	$-0.366957\pi$
$251$	12.8614	0.811805	0.405902	+	0.913916i	$0.366957\pi$
$252$	0	0
$253$	12.7446	0.801244
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0.510875	0.0318675	0.0159337	−	0.999873i	$-0.494928\pi$
$257$	0.510875	0.0318675	0.0159337	+	0.999873i	$0.494928\pi$
$258$	0	0
$259$	−20.2337	−1.25726
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4.11684	−0.253855	−0.126928	−	0.991912i	$-0.540512\pi$
$263$	−4.11684	−0.253855	−0.126928	+	0.991912i	$0.540512\pi$
$264$	0	0
$265$	10.1168	0.621473
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−5.11684	−0.311979	−0.155990	−	0.987759i	$-0.549857\pi$
$269$	−5.11684	−0.311979	−0.155990	+	0.987759i	$0.549857\pi$
$270$	0	0
$271$	9.25544	0.562228	0.281114	−	0.959674i	$-0.409296\pi$
$271$	9.25544	0.562228	0.281114	+	0.959674i	$0.409296\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.37228	−0.143054
$276$	0	0
$277$	0.116844	0.00702047	0.00351024	−	0.999994i	$-0.498883\pi$
$277$	0.116844	0.00702047	0.00351024	+	0.999994i	$0.498883\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	17.3723	1.03634	0.518172	−	0.855277i	$-0.326613\pi$
$281$	17.3723	1.03634	0.518172	+	0.855277i	$0.326613\pi$
$282$	0	0
$283$	−16.7446	−0.995361	−0.497680	−	0.867360i	$-0.665815\pi$
$283$	−16.7446	−0.995361	−0.497680	+	0.867360i	$0.665815\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.861407	−0.0508472
$288$	0	0
$289$	28.4891	1.67583
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−14.8614	−0.868213	−0.434106	−	0.900862i	$-0.642936\pi$
$293$	−14.8614	−0.868213	−0.434106	+	0.900862i	$0.642936\pi$
$294$	0	0
$295$	−5.00000	−0.291111
$296$	0	0
$297$	0	0
$298$	0	0
$299$	18.1168	1.04772
$300$	0	0
$301$	−16.0000	−0.922225
$302$	0	0
$303$	0	0
$304$	0	0
$305$	12.7446	0.729752
$306$	0	0
$307$	−0.744563	−0.0424944	−0.0212472	−	0.999774i	$-0.506764\pi$
$307$	−0.744563	−0.0424944	−0.0212472	+	0.999774i	$0.506764\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−17.8614	−1.01283	−0.506414	−	0.862291i	$-0.669029\pi$
$311$	−17.8614	−1.01283	−0.506414	+	0.862291i	$0.669029\pi$
$312$	0	0
$313$	−4.00000	−0.226093	−0.113047	−	0.993590i	$-0.536061\pi$
$313$	−4.00000	−0.226093	−0.113047	+	0.993590i	$0.536061\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−10.6277	−0.596912	−0.298456	−	0.954423i	$-0.596472\pi$
$317$	−10.6277	−0.596912	−0.298456	+	0.954423i	$0.596472\pi$
$318$	0	0
$319$	−5.62772	−0.315092
$320$	0	0
$321$	0	0
$322$	0	0
$323$	6.74456	0.375278
$324$	0	0
$325$	−3.37228	−0.187061
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−31.6060	−1.74249
$330$	0	0
$331$	−13.6277	−0.749047	−0.374524	−	0.927217i	$-0.622194\pi$
$331$	−13.6277	−0.749047	−0.374524	+	0.927217i	$0.622194\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0.744563	0.0406798
$336$	0	0
$337$	−10.7446	−0.585294	−0.292647	−	0.956221i	$-0.594536\pi$
$337$	−10.7446	−0.585294	−0.292647	+	0.956221i	$0.594536\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	26.3723	1.42814
$342$	0	0
$343$	8.86141	0.478471
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−4.23369	−0.227276	−0.113638	−	0.993522i	$-0.536250\pi$
$347$	−4.23369	−0.227276	−0.113638	+	0.993522i	$0.536250\pi$
$348$	0	0
$349$	−19.6277	−1.05065	−0.525324	−	0.850902i	$-0.676056\pi$
$349$	−19.6277	−1.05065	−0.525324	+	0.850902i	$0.676056\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−20.9783	−1.11656	−0.558280	−	0.829653i	$-0.688538\pi$
$353$	−20.9783	−1.11656	−0.558280	+	0.829653i	$0.688538\pi$
$354$	0	0
$355$	4.37228	0.232057
$356$	0	0
$357$	0	0
$358$	0	0
$359$	16.3723	0.864096	0.432048	−	0.901851i	$-0.357791\pi$
$359$	16.3723	0.864096	0.432048	+	0.901851i	$0.357791\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	0	0
$364$	0	0
$365$	14.7446	0.771766
$366$	0	0
$367$	30.2337	1.57819	0.789093	−	0.614274i	$-0.210551\pi$
$367$	30.2337	1.57819	0.789093	+	0.614274i	$0.210551\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−34.1168	−1.77126
$372$	0	0
$373$	−10.7446	−0.556332	−0.278166	−	0.960533i	$-0.589727\pi$
$373$	−10.7446	−0.556332	−0.278166	+	0.960533i	$0.589727\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−8.00000	−0.412021
$378$	0	0
$379$	26.3505	1.35354	0.676768	−	0.736196i	$-0.263380\pi$
$379$	26.3505	1.35354	0.676768	+	0.736196i	$0.263380\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	19.3723	0.989877	0.494939	−	0.868928i	$-0.335191\pi$
$383$	19.3723	0.989877	0.494939	+	0.868928i	$0.335191\pi$
$384$	0	0
$385$	8.00000	0.407718
$386$	0	0
$387$	0	0
$388$	0	0
$389$	20.7446	1.05179	0.525896	−	0.850549i	$-0.323730\pi$
$389$	20.7446	1.05179	0.525896	+	0.850549i	$0.323730\pi$
$390$	0	0
$391$	36.2337	1.83242
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2.74456	0.138094
$396$	0	0
$397$	34.0000	1.70641	0.853206	−	0.521575i	$-0.174655\pi$
$397$	34.0000	1.70641	0.853206	+	0.521575i	$0.174655\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	36.3505	1.81526	0.907629	−	0.419772i	$-0.137890\pi$
$401$	36.3505	1.81526	0.907629	+	0.419772i	$0.137890\pi$
$402$	0	0
$403$	37.4891	1.86747
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−14.2337	−0.705538
$408$	0	0
$409$	−21.3723	−1.05679	−0.528396	−	0.848998i	$-0.677206\pi$
$409$	−21.3723	−1.05679	−0.528396	+	0.848998i	$0.677206\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	16.8614	0.829696
$414$	0	0
$415$	10.0000	0.490881
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−34.9783	−1.70880	−0.854400	−	0.519616i	$-0.826075\pi$
$419$	−34.9783	−1.70880	−0.854400	+	0.519616i	$0.826075\pi$
$420$	0	0
$421$	28.3723	1.38278	0.691390	−	0.722482i	$-0.256999\pi$
$421$	28.3723	1.38278	0.691390	+	0.722482i	$0.256999\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−6.74456	−0.327159
$426$	0	0
$427$	−42.9783	−2.07986
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−9.62772	−0.463751	−0.231875	−	0.972745i	$-0.574486\pi$
$431$	−9.62772	−0.463751	−0.231875	+	0.972745i	$0.574486\pi$
$432$	0	0
$433$	−41.2119	−1.98052	−0.990260	−	0.139233i	$-0.955536\pi$
$433$	−41.2119	−1.98052	−0.990260	+	0.139233i	$0.955536\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.37228	0.256991
$438$	0	0
$439$	32.0951	1.53182	0.765908	−	0.642951i	$-0.222290\pi$
$439$	32.0951	1.53182	0.765908	+	0.642951i	$0.222290\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	28.9783	1.37680	0.688399	−	0.725332i	$-0.258314\pi$
$443$	28.9783	1.37680	0.688399	+	0.725332i	$0.258314\pi$
$444$	0	0
$445$	−4.37228	−0.207266
$446$	0	0
$447$	0	0
$448$	0	0
$449$	34.7228	1.63867	0.819335	−	0.573314i	$-0.194343\pi$
$449$	34.7228	1.63867	0.819335	+	0.573314i	$0.194343\pi$
$450$	0	0
$451$	−0.605969	−0.0285340
$452$	0	0
$453$	0	0
$454$	0	0
$455$	11.3723	0.533141
$456$	0	0
$457$	−5.76631	−0.269737	−0.134868	−	0.990864i	$-0.543061\pi$
$457$	−5.76631	−0.269737	−0.134868	+	0.990864i	$0.543061\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	31.1168	1.44926	0.724628	−	0.689140i	$-0.242012\pi$
$461$	31.1168	1.44926	0.724628	+	0.689140i	$0.242012\pi$
$462$	0	0
$463$	−6.86141	−0.318877	−0.159438	−	0.987208i	$-0.550968\pi$
$463$	−6.86141	−0.318877	−0.159438	+	0.987208i	$0.550968\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−16.2337	−0.751205	−0.375603	−	0.926781i	$-0.622564\pi$
$467$	−16.2337	−0.751205	−0.375603	+	0.926781i	$0.622564\pi$
$468$	0	0
$469$	−2.51087	−0.115941
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−11.2554	−0.517526
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	6.88316	0.314499	0.157250	−	0.987559i	$-0.449737\pi$
$479$	6.88316	0.314499	0.157250	+	0.987559i	$0.449737\pi$
$480$	0	0
$481$	−20.2337	−0.922577
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.74456	−0.215439
$486$	0	0
$487$	−19.6060	−0.888431	−0.444216	−	0.895920i	$-0.646518\pi$
$487$	−19.6060	−0.888431	−0.444216	+	0.895920i	$0.646518\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	25.9783	1.17238	0.586191	−	0.810173i	$-0.300627\pi$
$491$	25.9783	1.17238	0.586191	+	0.810173i	$0.300627\pi$
$492$	0	0
$493$	−16.0000	−0.720604
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−14.7446	−0.661384
$498$	0	0
$499$	−19.7446	−0.883888	−0.441944	−	0.897043i	$-0.645711\pi$
$499$	−19.7446	−0.883888	−0.441944	+	0.897043i	$0.645711\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−10.9783	−0.489496	−0.244748	−	0.969587i	$-0.578705\pi$
$503$	−10.9783	−0.489496	−0.244748	+	0.969587i	$0.578705\pi$
$504$	0	0
$505$	−15.8614	−0.705823
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−38.0000	−1.68432	−0.842160	−	0.539227i	$-0.818716\pi$
$509$	−38.0000	−1.68432	−0.842160	+	0.539227i	$0.818716\pi$
$510$	0	0
$511$	−49.7228	−2.19961
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0.116844	0.00514876
$516$	0	0
$517$	−22.2337	−0.977836
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−0.116844	−0.00511903	−0.00255951	−	0.999997i	$-0.500815\pi$
$521$	−0.116844	−0.00511903	−0.00255951	+	0.999997i	$0.500815\pi$
$522$	0	0
$523$	40.2337	1.75930	0.879648	−	0.475625i	$-0.157778\pi$
$523$	40.2337	1.75930	0.879648	+	0.475625i	$0.157778\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	74.9783	3.26610
$528$	0	0
$529$	5.86141	0.254844
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−0.861407	−0.0373117
$534$	0	0
$535$	6.00000	0.259403
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−10.3723	−0.446766
$540$	0	0
$541$	13.3505	0.573984	0.286992	−	0.957933i	$-0.407345\pi$
$541$	13.3505	0.573984	0.286992	+	0.957933i	$0.407345\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0.883156	0.0378302
$546$	0	0
$547$	40.0000	1.71028	0.855138	−	0.518400i	$-0.173472\pi$
$547$	40.0000	1.71028	0.855138	+	0.518400i	$0.173472\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2.37228	−0.101063
$552$	0	0
$553$	−9.25544	−0.393581
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−31.6060	−1.33919	−0.669594	−	0.742727i	$-0.733532\pi$
$557$	−31.6060	−1.33919	−0.669594	+	0.742727i	$0.733532\pi$
$558$	0	0
$559$	−16.0000	−0.676728
$560$	0	0
$561$	0	0
$562$	0	0
$563$	10.2337	0.431299	0.215649	−	0.976471i	$-0.430813\pi$
$563$	10.2337	0.431299	0.215649	+	0.976471i	$0.430813\pi$
$564$	0	0
$565$	14.7446	0.620308
$566$	0	0
$567$	0	0
$568$	0	0
$569$	8.76631	0.367503	0.183751	−	0.982973i	$-0.441176\pi$
$569$	8.76631	0.367503	0.183751	+	0.982973i	$0.441176\pi$
$570$	0	0
$571$	3.11684	0.130436	0.0652179	−	0.997871i	$-0.479226\pi$
$571$	3.11684	0.130436	0.0652179	+	0.997871i	$0.479226\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−5.37228	−0.224040
$576$	0	0
$577$	−9.48913	−0.395037	−0.197519	−	0.980299i	$-0.563288\pi$
$577$	−9.48913	−0.395037	−0.197519	+	0.980299i	$0.563288\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−33.7228	−1.39906
$582$	0	0
$583$	−24.0000	−0.993978
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−30.0000	−1.23823	−0.619116	−	0.785299i	$-0.712509\pi$
$587$	−30.0000	−1.23823	−0.619116	+	0.785299i	$0.712509\pi$
$588$	0	0
$589$	11.1168	0.458062
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−28.9783	−1.18999	−0.594997	−	0.803728i	$-0.702847\pi$
$593$	−28.9783	−1.18999	−0.594997	+	0.803728i	$0.702847\pi$
$594$	0	0
$595$	22.7446	0.932436
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−15.3505	−0.627206	−0.313603	−	0.949554i	$-0.601536\pi$
$599$	−15.3505	−0.627206	−0.313603	+	0.949554i	$0.601536\pi$
$600$	0	0
$601$	13.0000	0.530281	0.265141	−	0.964210i	$-0.414582\pi$
$601$	13.0000	0.530281	0.265141	+	0.964210i	$0.414582\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−5.37228	−0.218414
$606$	0	0
$607$	44.7446	1.81613	0.908063	−	0.418834i	$-0.137561\pi$
$607$	44.7446	1.81613	0.908063	+	0.418834i	$0.137561\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−31.6060	−1.27864
$612$	0	0
$613$	18.3505	0.741171	0.370586	−	0.928798i	$-0.379157\pi$
$613$	18.3505	0.741171	0.370586	+	0.928798i	$0.379157\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	13.7228	0.552460	0.276230	−	0.961092i	$-0.410915\pi$
$617$	13.7228	0.552460	0.276230	+	0.961092i	$0.410915\pi$
$618$	0	0
$619$	32.6277	1.31142	0.655709	−	0.755013i	$-0.272370\pi$
$619$	32.6277	1.31142	0.655709	+	0.755013i	$0.272370\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	14.7446	0.590728
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−40.4674	−1.61354
$630$	0	0
$631$	42.3723	1.68681	0.843407	−	0.537275i	$-0.180546\pi$
$631$	42.3723	1.68681	0.843407	+	0.537275i	$0.180546\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	12.8614	0.510389
$636$	0	0
$637$	−14.7446	−0.584201
$638$	0	0
$639$	0	0
$640$	0	0
$641$	42.6060	1.68283	0.841417	−	0.540386i	$-0.181722\pi$
$641$	42.6060	1.68283	0.841417	+	0.540386i	$0.181722\pi$
$642$	0	0
$643$	−32.7446	−1.29132	−0.645660	−	0.763625i	$-0.723418\pi$
$643$	−32.7446	−1.29132	−0.645660	+	0.763625i	$0.723418\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	31.7228	1.24715	0.623576	−	0.781763i	$-0.285679\pi$
$647$	31.7228	1.24715	0.623576	+	0.781763i	$0.285679\pi$
$648$	0	0
$649$	11.8614	0.465601
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−13.2554	−0.518725	−0.259363	−	0.965780i	$-0.583512\pi$
$653$	−13.2554	−0.518725	−0.259363	+	0.965780i	$0.583512\pi$
$654$	0	0
$655$	−6.25544	−0.244420
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−38.5842	−1.50303	−0.751514	−	0.659717i	$-0.770676\pi$
$659$	−38.5842	−1.50303	−0.751514	+	0.659717i	$0.770676\pi$
$660$	0	0
$661$	18.0951	0.703818	0.351909	−	0.936034i	$-0.385533\pi$
$661$	18.0951	0.703818	0.351909	+	0.936034i	$0.385533\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.37228	0.130771
$666$	0	0
$667$	−12.7446	−0.493471
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−30.2337	−1.16716
$672$	0	0
$673$	−23.4891	−0.905439	−0.452720	−	0.891653i	$-0.649546\pi$
$673$	−23.4891	−0.905439	−0.452720	+	0.891653i	$0.649546\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−3.88316	−0.149242	−0.0746209	−	0.997212i	$-0.523775\pi$
$677$	−3.88316	−0.149242	−0.0746209	+	0.997212i	$0.523775\pi$
$678$	0	0
$679$	16.0000	0.614024
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−3.76631	−0.144114	−0.0720570	−	0.997401i	$-0.522956\pi$
$683$	−3.76631	−0.144114	−0.0720570	+	0.997401i	$0.522956\pi$
$684$	0	0
$685$	−5.25544	−0.200800
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−34.1168	−1.29975
$690$	0	0
$691$	10.3505	0.393753	0.196876	−	0.980428i	$-0.436920\pi$
$691$	10.3505	0.393753	0.196876	+	0.980428i	$0.436920\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−15.0000	−0.568982
$696$	0	0
$697$	−1.72281	−0.0652562
$698$	0	0
$699$	0	0
$700$	0	0
$701$	9.11684	0.344338	0.172169	−	0.985067i	$-0.444922\pi$
$701$	9.11684	0.344338	0.172169	+	0.985067i	$0.444922\pi$
$702$	0	0
$703$	−6.00000	−0.226294
$704$	0	0
$705$	0	0
$706$	0	0
$707$	53.4891	2.01167
$708$	0	0
$709$	−50.4674	−1.89534	−0.947671	−	0.319248i	$-0.896570\pi$
$709$	−50.4674	−1.89534	−0.947671	+	0.319248i	$0.896570\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	59.7228	2.23664
$714$	0	0
$715$	8.00000	0.299183
$716$	0	0
$717$	0	0
$718$	0	0
$719$	50.6060	1.88728	0.943642	−	0.330968i	$-0.107375\pi$
$719$	50.6060	1.88728	0.943642	+	0.330968i	$0.107375\pi$
$720$	0	0
$721$	−0.394031	−0.0146745
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.37228	0.0881043
$726$	0	0
$727$	29.8832	1.10830	0.554152	−	0.832415i	$-0.313042\pi$
$727$	29.8832	1.10830	0.554152	+	0.832415i	$0.313042\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−32.0000	−1.18356
$732$	0	0
$733$	16.9783	0.627106	0.313553	−	0.949571i	$-0.398481\pi$
$733$	16.9783	0.627106	0.313553	+	0.949571i	$0.398481\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.76631	−0.0650629
$738$	0	0
$739$	−15.8614	−0.583471	−0.291736	−	0.956499i	$-0.594233\pi$
$739$	−15.8614	−0.583471	−0.291736	+	0.956499i	$0.594233\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	6.51087	0.238861	0.119430	−	0.992843i	$-0.461893\pi$
$743$	6.51087	0.238861	0.119430	+	0.992843i	$0.461893\pi$
$744$	0	0
$745$	19.4891	0.714026
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−20.2337	−0.739323
$750$	0	0
$751$	3.76631	0.137435	0.0687173	−	0.997636i	$-0.478109\pi$
$751$	3.76631	0.137435	0.0687173	+	0.997636i	$0.478109\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−24.3723	−0.886998
$756$	0	0
$757$	9.88316	0.359209	0.179605	−	0.983739i	$-0.442518\pi$
$757$	9.88316	0.359209	0.179605	+	0.983739i	$0.442518\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	17.2337	0.624721	0.312360	−	0.949964i	$-0.398880\pi$
$761$	17.2337	0.624721	0.312360	+	0.949964i	$0.398880\pi$
$762$	0	0
$763$	−2.97825	−0.107820
$764$	0	0
$765$	0	0
$766$	0	0
$767$	16.8614	0.608830
$768$	0	0
$769$	19.3505	0.697798	0.348899	−	0.937160i	$-0.386556\pi$
$769$	19.3505	0.697798	0.348899	+	0.937160i	$0.386556\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	18.0000	0.647415	0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000	0.647415	0.323708	+	0.946157i	$0.395071\pi$
$774$	0	0
$775$	−11.1168	−0.399329
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−0.255437	−0.00915199
$780$	0	0
$781$	−10.3723	−0.371149
$782$	0	0
$783$	0	0
$784$	0	0
$785$	22.8614	0.815959
$786$	0	0
$787$	41.7228	1.48726	0.743629	−	0.668593i	$-0.233103\pi$
$787$	41.7228	1.48726	0.743629	+	0.668593i	$0.233103\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−49.7228	−1.76794
$792$	0	0
$793$	−42.9783	−1.52620
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−20.2337	−0.716714	−0.358357	−	0.933585i	$-0.616663\pi$
$797$	−20.2337	−0.716714	−0.358357	+	0.933585i	$0.616663\pi$
$798$	0	0
$799$	−63.2119	−2.23628
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−34.9783	−1.23436
$804$	0	0
$805$	18.1168	0.638535
$806$	0	0
$807$	0	0
$808$	0	0
$809$	46.2119	1.62473	0.812363	−	0.583153i	$-0.198181\pi$
$809$	46.2119	1.62473	0.812363	+	0.583153i	$0.198181\pi$
$810$	0	0
$811$	37.3505	1.31155	0.655777	−	0.754954i	$-0.272341\pi$
$811$	37.3505	1.31155	0.655777	+	0.754954i	$0.272341\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−9.48913	−0.332390
$816$	0	0
$817$	−4.74456	−0.165991
$818$	0	0
$819$	0	0
$820$	0	0
$821$	35.6277	1.24342	0.621708	−	0.783249i	$-0.286439\pi$
$821$	35.6277	1.24342	0.621708	+	0.783249i	$0.286439\pi$
$822$	0	0
$823$	23.2554	0.810634	0.405317	−	0.914176i	$-0.367161\pi$
$823$	23.2554	0.810634	0.405317	+	0.914176i	$0.367161\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	45.9565	1.59806	0.799032	−	0.601288i	$-0.205346\pi$
$827$	45.9565	1.59806	0.799032	+	0.601288i	$0.205346\pi$
$828$	0	0
$829$	−43.5842	−1.51374	−0.756871	−	0.653564i	$-0.773273\pi$
$829$	−43.5842	−1.51374	−0.756871	+	0.653564i	$0.773273\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−29.4891	−1.02174
$834$	0	0
$835$	12.0000	0.415277
$836$	0	0
$837$	0	0
$838$	0	0
$839$	28.6060	0.987588	0.493794	−	0.869579i	$-0.335610\pi$
$839$	28.6060	0.987588	0.493794	+	0.869579i	$0.335610\pi$
$840$	0	0
$841$	−23.3723	−0.805941
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.62772	−0.0559952
$846$	0	0
$847$	18.1168	0.622502
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−32.2337	−1.10496
$852$	0	0
$853$	10.7446	0.367887	0.183943	−	0.982937i	$-0.441114\pi$
$853$	10.7446	0.367887	0.183943	+	0.982937i	$0.441114\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−33.2554	−1.13598	−0.567992	−	0.823034i	$-0.692280\pi$
$857$	−33.2554	−1.13598	−0.567992	+	0.823034i	$0.692280\pi$
$858$	0	0
$859$	1.35053	0.0460796	0.0230398	−	0.999735i	$-0.492666\pi$
$859$	1.35053	0.0460796	0.0230398	+	0.999735i	$0.492666\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	51.6060	1.75669	0.878344	−	0.478029i	$-0.158649\pi$
$863$	51.6060	1.75669	0.878344	+	0.478029i	$0.158649\pi$
$864$	0	0
$865$	14.8614	0.505303
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−6.51087	−0.220866
$870$	0	0
$871$	−2.51087	−0.0850777
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3.37228	−0.114004
$876$	0	0
$877$	26.1168	0.881903	0.440952	−	0.897531i	$-0.354641\pi$
$877$	26.1168	0.881903	0.440952	+	0.897531i	$0.354641\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	19.6277	0.661275	0.330637	−	0.943758i	$-0.392736\pi$
$881$	19.6277	0.661275	0.330637	+	0.943758i	$0.392736\pi$
$882$	0	0
$883$	−40.2337	−1.35397	−0.676986	−	0.735996i	$-0.736714\pi$
$883$	−40.2337	−1.35397	−0.676986	+	0.735996i	$0.736714\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	13.1386	0.441151	0.220575	−	0.975370i	$-0.429206\pi$
$887$	13.1386	0.441151	0.220575	+	0.975370i	$0.429206\pi$
$888$	0	0
$889$	−43.3723	−1.45466
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−9.37228	−0.313631
$894$	0	0
$895$	4.25544	0.142244
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−26.3723	−0.879565
$900$	0	0
$901$	−68.2337	−2.27319
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−13.8614	−0.460769
$906$	0	0
$907$	12.5109	0.415417	0.207708	−	0.978191i	$-0.433399\pi$
$907$	12.5109	0.415417	0.207708	+	0.978191i	$0.433399\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	55.0733	1.82466	0.912331	−	0.409454i	$-0.134281\pi$
$911$	55.0733	1.82466	0.912331	+	0.409454i	$0.134281\pi$
$912$	0	0
$913$	−23.7228	−0.785111
$914$	0	0
$915$	0	0
$916$	0	0
$917$	21.0951	0.696621
$918$	0	0
$919$	−24.8832	−0.820820	−0.410410	−	0.911901i	$-0.634614\pi$
$919$	−24.8832	−0.820820	−0.410410	+	0.911901i	$0.634614\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−14.7446	−0.485323
$924$	0	0
$925$	6.00000	0.197279
$926$	0	0
$927$	0	0
$928$	0	0
$929$	25.8614	0.848485	0.424243	−	0.905549i	$-0.360540\pi$
$929$	25.8614	0.848485	0.424243	+	0.905549i	$0.360540\pi$
$930$	0	0
$931$	−4.37228	−0.143296
$932$	0	0
$933$	0	0
$934$	0	0
$935$	16.0000	0.523256
$936$	0	0
$937$	31.2554	1.02107	0.510535	−	0.859857i	$-0.329447\pi$
$937$	31.2554	1.02107	0.510535	+	0.859857i	$0.329447\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−42.4674	−1.38440	−0.692198	−	0.721707i	$-0.743358\pi$
$941$	−42.4674	−1.38440	−0.692198	+	0.721707i	$0.743358\pi$
$942$	0	0
$943$	−1.37228	−0.0446876
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−48.7446	−1.58399	−0.791993	−	0.610531i	$-0.790956\pi$
$947$	−48.7446	−1.58399	−0.791993	+	0.610531i	$0.790956\pi$
$948$	0	0
$949$	−49.7228	−1.61407
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−12.0000	−0.388718	−0.194359	−	0.980930i	$-0.562263\pi$
$953$	−12.0000	−0.388718	−0.194359	+	0.980930i	$0.562263\pi$
$954$	0	0
$955$	5.62772	0.182109
$956$	0	0
$957$	0	0
$958$	0	0
$959$	17.7228	0.572299
$960$	0	0
$961$	92.5842	2.98659
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0.744563	0.0239683
$966$	0	0
$967$	41.4891	1.33420	0.667100	−	0.744968i	$-0.267535\pi$
$967$	41.4891	1.33420	0.667100	+	0.744968i	$0.267535\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−48.7228	−1.56359	−0.781795	−	0.623536i	$-0.785696\pi$
$971$	−48.7228	−1.56359	−0.781795	+	0.623536i	$0.785696\pi$
$972$	0	0
$973$	50.5842	1.62166
$974$	0	0
$975$	0	0
$976$	0	0
$977$	52.2337	1.67110	0.835552	−	0.549412i	$-0.185148\pi$
$977$	52.2337	1.67110	0.835552	+	0.549412i	$0.185148\pi$
$978$	0	0
$979$	10.3723	0.331500
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−17.4891	−0.557816	−0.278908	−	0.960318i	$-0.589973\pi$
$983$	−17.4891	−0.557816	−0.278908	+	0.960318i	$0.589973\pi$
$984$	0	0
$985$	−0.627719	−0.0200008
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−25.4891	−0.810507
$990$	0	0
$991$	2.37228	0.0753580	0.0376790	−	0.999290i	$-0.488004\pi$
$991$	2.37228	0.0753580	0.0376790	+	0.999290i	$0.488004\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−13.4891	−0.427634
$996$	0	0
$997$	−42.3505	−1.34125	−0.670627	−	0.741794i	$-0.733975\pi$
$997$	−42.3505	−1.34125	−0.670627	+	0.741794i	$0.733975\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6480.2.a.bo.1.1		2
3.2	odd	2		6480.2.a.bd.1.1		2
4.3	odd	2		3240.2.a.m.1.2	yes	2
12.11	even	2		3240.2.a.i.1.2	✓	2
36.7	odd	6		3240.2.q.ba.1081.1		4
36.11	even	6		3240.2.q.bd.1081.1		4
36.23	even	6		3240.2.q.bd.2161.1		4
36.31	odd	6		3240.2.q.ba.2161.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3240.2.a.i.1.2	✓	2	12.11	even	2
3240.2.a.m.1.2	yes	2	4.3	odd	2
3240.2.q.ba.1081.1		4	36.7	odd	6
3240.2.q.ba.2161.1		4	36.31	odd	6
3240.2.q.bd.1081.1		4	36.11	even	6
3240.2.q.bd.2161.1		4	36.23	even	6
6480.2.a.bd.1.1		2	3.2	odd	2
6480.2.a.bo.1.1		2	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:	\( N \)	\(=\)	\( 6480 = 2^{4} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6480.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -3.37228 q^{7} -2.37228 q^{11} -3.37228 q^{13} -6.74456 q^{17} -1.00000 q^{19} -5.37228 q^{23} +1.00000 q^{25} +2.37228 q^{29} -11.1168 q^{31} -3.37228 q^{35} +6.00000 q^{37} +0.255437 q^{41} +4.74456 q^{43} +9.37228 q^{47} +4.37228 q^{49} +10.1168 q^{53} -2.37228 q^{55} -5.00000 q^{59} +12.7446 q^{61} -3.37228 q^{65} +0.744563 q^{67} +4.37228 q^{71} +14.7446 q^{73} +8.00000 q^{77} +2.74456 q^{79} +10.0000 q^{83} -6.74456 q^{85} -4.37228 q^{89} +11.3723 q^{91} -1.00000 q^{95} -4.74456 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - q^{7} + q^{11} - q^{13} - 2 q^{17} - 2 q^{19} - 5 q^{23} + 2 q^{25} - q^{29} - 5 q^{31} - q^{35} + 12 q^{37} + 12 q^{41} - 2 q^{43} + 13 q^{47} + 3 q^{49} + 3 q^{53} + q^{55} - 10 q^{59}+ \cdots + 2 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 - q^7 + q^11 - q^13 - 2 * q^17 - 2 * q^19 - 5 * q^23 + 2 * q^25 - q^29 - 5 * q^31 - q^35 + 12 * q^37 + 12 * q^41 - 2 * q^43 + 13 * q^47 + 3 * q^49 + 3 * q^53 + q^55 - 10 * q^59 + 14 * q^61 - q^65 - 10 * q^67 + 3 * q^71 + 18 * q^73 + 16 * q^77 - 6 * q^79 + 20 * q^83 - 2 * q^85 - 3 * q^89 + 17 * q^91 - 2 * q^95 + 2 * q^97

Introduction

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