LMFDB - Embedded newform 4576.2.a.p.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4576,2,Mod(1,4576)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4576.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4576, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4576 = 2^{5} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4576.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-1,0,-2,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$36.5395439649$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.106740016.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 9x^{4} + 7x^{3} + 15x^{2} + x - 2 $ x^6 - x^5 - 9x^4 + 7x^3 + 15*x^2 + x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.320976$ of defining polynomial
Character	$\chi$	$=$	4576.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.320976 q^{3} +0.406820 q^{5} -1.49015 q^{7} -2.89697 q^{9} -1.00000 q^{11} -1.00000 q^{13} -0.130579 q^{15} +2.52708 q^{17} +6.31684 q^{19} +0.478304 q^{21} -0.262686 q^{23} -4.83450 q^{25} +1.89279 q^{27} -0.276240 q^{29} -1.20763 q^{31} +0.320976 q^{33} -0.606224 q^{35} +8.52043 q^{37} +0.320976 q^{39} -6.54180 q^{41} -7.30248 q^{43} -1.17855 q^{45} +7.69440 q^{47} -4.77944 q^{49} -0.811131 q^{51} -9.76992 q^{53} -0.406820 q^{55} -2.02755 q^{57} -6.07501 q^{59} -5.77525 q^{61} +4.31694 q^{63} -0.406820 q^{65} +8.91405 q^{67} +0.0843160 q^{69} +9.70930 q^{71} +2.70239 q^{73} +1.55176 q^{75} +1.49015 q^{77} +8.16918 q^{79} +8.08338 q^{81} +15.7556 q^{83} +1.02806 q^{85} +0.0886665 q^{87} +16.3781 q^{89} +1.49015 q^{91} +0.387619 q^{93} +2.56981 q^{95} -3.20763 q^{97} +2.89697 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{3} - 2 q^{5} + 5 q^{7} + q^{9} - 6 q^{11} - 6 q^{13} + 5 q^{15} + 6 q^{17} - 2 q^{19} + 20 q^{23} + 4 q^{25} - q^{27} - 3 q^{29} + 11 q^{31} + q^{33} + 9 q^{35} - 3 q^{37} + q^{39} - q^{41}+ \cdots - q^{99}+O(q^{100}) $ 6 * q - q^3 - 2 * q^5 + 5 * q^7 + q^9 - 6 * q^11 - 6 * q^13 + 5 * q^15 + 6 * q^17 - 2 * q^19 + 20 * q^23 + 4 * q^25 - q^27 - 3 * q^29 + 11 * q^31 + q^33 + 9 * q^35 - 3 * q^37 + q^39 - q^41 - 9 * q^43 - 23 * q^45 - 8 * q^47 + q^49 + 10 * q^51 - 2 * q^53 + 2 * q^55 + 12 * q^57 + 6 * q^59 - 11 * q^61 + 29 * q^63 + 2 * q^65 + 14 * q^67 + 7 * q^69 + 19 * q^71 - q^73 - 6 * q^75 - 5 * q^77 + 36 * q^79 - 6 * q^81 - 2 * q^83 + 22 * q^85 + 24 * q^87 - 5 * q^89 - 5 * q^91 + 35 * q^93 + 38 * q^95 - q^97 - q^99

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
$3$	−0.320976	−0.185316	−0.0926578	−	0.995698i	$-0.529536\pi$
$3$	−0.320976	−0.185316	−0.0926578	+	0.995698i	$0.529536\pi$
$4$	0	0
$5$	0.406820	0.181935	0.0909676	−	0.995854i	$-0.471004\pi$
$5$	0.406820	0.181935	0.0909676	+	0.995854i	$0.471004\pi$
$6$	0	0
$7$	−1.49015	−0.563226	−0.281613	−	0.959528i	$-0.590869\pi$
$7$	−1.49015	−0.563226	−0.281613	+	0.959528i	$0.590869\pi$
$8$	0	0
$9$	−2.89697	−0.965658
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−0.130579	−0.0337154
$16$	0	0
$17$	2.52708	0.612906	0.306453	−	0.951886i	$-0.400858\pi$
$17$	2.52708	0.612906	0.306453	+	0.951886i	$0.400858\pi$
$18$	0	0
$19$	6.31684	1.44918	0.724591	−	0.689179i	$-0.242029\pi$
$19$	6.31684	1.44918	0.724591	+	0.689179i	$0.242029\pi$
$20$	0	0
$21$	0.478304	0.104374
$22$	0	0
$23$	−0.262686	−0.0547739	−0.0273869	−	0.999625i	$-0.508719\pi$
$23$	−0.262686	−0.0547739	−0.0273869	+	0.999625i	$0.508719\pi$
$24$	0	0
$25$	−4.83450	−0.966900
$26$	0	0
$27$	1.89279	0.364267
$28$	0	0
$29$	−0.276240	−0.0512965	−0.0256483	−	0.999671i	$-0.508165\pi$
$29$	−0.276240	−0.0512965	−0.0256483	+	0.999671i	$0.508165\pi$
$30$	0	0
$31$	−1.20763	−0.216896	−0.108448	−	0.994102i	$-0.534588\pi$
$31$	−1.20763	−0.216896	−0.108448	+	0.994102i	$0.534588\pi$
$32$	0	0
$33$	0.320976	0.0558748
$34$	0	0
$35$	−0.606224	−0.102471
$36$	0	0
$37$	8.52043	1.40075	0.700375	−	0.713775i	$-0.253016\pi$
$37$	8.52043	1.40075	0.700375	+	0.713775i	$0.253016\pi$
$38$	0	0
$39$	0.320976	0.0513973
$40$	0	0
$41$	−6.54180	−1.02166	−0.510829	−	0.859683i	$-0.670661\pi$
$41$	−6.54180	−1.02166	−0.510829	+	0.859683i	$0.670661\pi$
$42$	0	0
$43$	−7.30248	−1.11362	−0.556809	−	0.830641i	$-0.687974\pi$
$43$	−7.30248	−1.11362	−0.556809	+	0.830641i	$0.687974\pi$
$44$	0	0
$45$	−1.17855	−0.175687
$46$	0	0
$47$	7.69440	1.12234	0.561172	−	0.827699i	$-0.310351\pi$
$47$	7.69440	1.12234	0.561172	+	0.827699i	$0.310351\pi$
$48$	0	0
$49$	−4.77944	−0.682777
$50$	0	0
$51$	−0.811131	−0.113581
$52$	0	0
$53$	−9.76992	−1.34200	−0.671001	−	0.741457i	$-0.734135\pi$
$53$	−9.76992	−1.34200	−0.671001	+	0.741457i	$0.734135\pi$
$54$	0	0
$55$	−0.406820	−0.0548555
$56$	0	0
$57$	−2.02755	−0.268556
$58$	0	0
$59$	−6.07501	−0.790899	−0.395449	−	0.918488i	$-0.629411\pi$
$59$	−6.07501	−0.790899	−0.395449	+	0.918488i	$0.629411\pi$
$60$	0	0
$61$	−5.77525	−0.739445	−0.369723	−	0.929142i	$-0.620547\pi$
$61$	−5.77525	−0.739445	−0.369723	+	0.929142i	$0.620547\pi$
$62$	0	0
$63$	4.31694	0.543883
$64$	0	0
$65$	−0.406820	−0.0504598
$66$	0	0
$67$	8.91405	1.08903	0.544513	−	0.838753i	$-0.316715\pi$
$67$	8.91405	1.08903	0.544513	+	0.838753i	$0.316715\pi$
$68$	0	0
$69$	0.0843160	0.0101505
$70$	0	0
$71$	9.70930	1.15228	0.576141	−	0.817350i	$-0.304558\pi$
$71$	9.70930	1.15228	0.576141	+	0.817350i	$0.304558\pi$
$72$	0	0
$73$	2.70239	0.316291	0.158145	−	0.987416i	$-0.449449\pi$
$73$	2.70239	0.316291	0.158145	+	0.987416i	$0.449449\pi$
$74$	0	0
$75$	1.55176	0.179182
$76$	0	0
$77$	1.49015	0.169819
$78$	0	0
$79$	8.16918	0.919104	0.459552	−	0.888151i	$-0.348010\pi$
$79$	8.16918	0.919104	0.459552	+	0.888151i	$0.348010\pi$
$80$	0	0
$81$	8.08338	0.898154
$82$	0	0
$83$	15.7556	1.72940	0.864701	−	0.502286i	$-0.167508\pi$
$83$	15.7556	1.72940	0.864701	+	0.502286i	$0.167508\pi$
$84$	0	0
$85$	1.02806	0.111509
$86$	0	0
$87$	0.0886665	0.00950605
$88$	0	0
$89$	16.3781	1.73608	0.868040	−	0.496495i	$-0.165380\pi$
$89$	16.3781	1.73608	0.868040	+	0.496495i	$0.165380\pi$
$90$	0	0
$91$	1.49015	0.156211
$92$	0	0
$93$	0.387619	0.0401943
$94$	0	0
$95$	2.56981	0.263657
$96$	0	0
$97$	−3.20763	−0.325685	−0.162843	−	0.986652i	$-0.552066\pi$
$97$	−3.20763	−0.325685	−0.162843	+	0.986652i	$0.552066\pi$
$98$	0	0
$99$	2.89697	0.291157
$100$	0	0
$101$	−2.26672	−0.225547	−0.112774	−	0.993621i	$-0.535973\pi$
$101$	−2.26672	−0.225547	−0.112774	+	0.993621i	$0.535973\pi$
$102$	0	0
$103$	−2.98917	−0.294531	−0.147266	−	0.989097i	$-0.547047\pi$
$103$	−2.98917	−0.294531	−0.147266	+	0.989097i	$0.547047\pi$
$104$	0	0
$105$	0.194583	0.0189894
$106$	0	0
$107$	−5.61525	−0.542847	−0.271423	−	0.962460i	$-0.587494\pi$
$107$	−5.61525	−0.542847	−0.271423	+	0.962460i	$0.587494\pi$
$108$	0	0
$109$	3.76912	0.361016	0.180508	−	0.983574i	$-0.442226\pi$
$109$	3.76912	0.361016	0.180508	+	0.983574i	$0.442226\pi$
$110$	0	0
$111$	−2.73485	−0.259581
$112$	0	0
$113$	10.3944	0.977819	0.488909	−	0.872335i	$-0.337395\pi$
$113$	10.3944	0.977819	0.488909	+	0.872335i	$0.337395\pi$
$114$	0	0
$115$	−0.106866	−0.00996530
$116$	0	0
$117$	2.89697	0.267825
$118$	0	0
$119$	−3.76573	−0.345204
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	2.09976	0.189329
$124$	0	0
$125$	−4.00087	−0.357848
$126$	0	0
$127$	19.8997	1.76582	0.882908	−	0.469545i	$-0.155582\pi$
$127$	19.8997	1.76582	0.882908	+	0.469545i	$0.155582\pi$
$128$	0	0
$129$	2.34392	0.206371
$130$	0	0
$131$	−1.95531	−0.170836	−0.0854182	−	0.996345i	$-0.527223\pi$
$131$	−1.95531	−0.170836	−0.0854182	+	0.996345i	$0.527223\pi$
$132$	0	0
$133$	−9.41307	−0.816216
$134$	0	0
$135$	0.770023	0.0662730
$136$	0	0
$137$	−3.89832	−0.333056	−0.166528	−	0.986037i	$-0.553256\pi$
$137$	−3.89832	−0.333056	−0.166528	+	0.986037i	$0.553256\pi$
$138$	0	0
$139$	13.4538	1.14113	0.570567	−	0.821251i	$-0.306724\pi$
$139$	13.4538	1.14113	0.570567	+	0.821251i	$0.306724\pi$
$140$	0	0
$141$	−2.46972	−0.207988
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−0.112380	−0.00933265
$146$	0	0
$147$	1.53409	0.126529
$148$	0	0
$149$	11.1315	0.911928	0.455964	−	0.889998i	$-0.349295\pi$
$149$	11.1315	0.911928	0.455964	+	0.889998i	$0.349295\pi$
$150$	0	0
$151$	−8.98575	−0.731250	−0.365625	−	0.930762i	$-0.619145\pi$
$151$	−8.98575	−0.731250	−0.365625	+	0.930762i	$0.619145\pi$
$152$	0	0
$153$	−7.32087	−0.591858
$154$	0	0
$155$	−0.491286	−0.0394611
$156$	0	0
$157$	6.18728	0.493798	0.246899	−	0.969041i	$-0.420588\pi$
$157$	6.18728	0.493798	0.246899	+	0.969041i	$0.420588\pi$
$158$	0	0
$159$	3.13591	0.248694
$160$	0	0
$161$	0.391443	0.0308500
$162$	0	0
$163$	16.7850	1.31470	0.657350	−	0.753586i	$-0.271677\pi$
$163$	16.7850	1.31470	0.657350	+	0.753586i	$0.271677\pi$
$164$	0	0
$165$	0.130579	0.0101656
$166$	0	0
$167$	22.0963	1.70986	0.854930	−	0.518744i	$-0.173600\pi$
$167$	22.0963	1.70986	0.854930	+	0.518744i	$0.173600\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−18.2997	−1.39941
$172$	0	0
$173$	17.2915	1.31465	0.657325	−	0.753607i	$-0.271688\pi$
$173$	17.2915	1.31465	0.657325	+	0.753607i	$0.271688\pi$
$174$	0	0
$175$	7.20415	0.544583
$176$	0	0
$177$	1.94993	0.146566
$178$	0	0
$179$	13.0247	0.973514	0.486757	−	0.873537i	$-0.338180\pi$
$179$	13.0247	0.973514	0.486757	+	0.873537i	$0.338180\pi$
$180$	0	0
$181$	11.5936	0.861745	0.430872	−	0.902413i	$-0.358206\pi$
$181$	11.5936	0.861745	0.430872	+	0.902413i	$0.358206\pi$
$182$	0	0
$183$	1.85372	0.137031
$184$	0	0
$185$	3.46628	0.254846
$186$	0	0
$187$	−2.52708	−0.184798
$188$	0	0
$189$	−2.82055	−0.205165
$190$	0	0
$191$	8.68418	0.628365	0.314183	−	0.949363i	$-0.398270\pi$
$191$	8.68418	0.628365	0.314183	+	0.949363i	$0.398270\pi$
$192$	0	0
$193$	−19.6041	−1.41114	−0.705569	−	0.708642i	$-0.749308\pi$
$193$	−19.6041	−1.41114	−0.705569	+	0.708642i	$0.749308\pi$
$194$	0	0
$195$	0.130579	0.00935098
$196$	0	0
$197$	−17.9622	−1.27975	−0.639876	−	0.768478i	$-0.721014\pi$
$197$	−17.9622	−1.27975	−0.639876	+	0.768478i	$0.721014\pi$
$198$	0	0
$199$	21.3333	1.51227	0.756137	−	0.654413i	$-0.227084\pi$
$199$	21.3333	1.51227	0.756137	+	0.654413i	$0.227084\pi$
$200$	0	0
$201$	−2.86120	−0.201813
$202$	0	0
$203$	0.411641	0.0288915
$204$	0	0
$205$	−2.66133	−0.185875
$206$	0	0
$207$	0.760995	0.0528928
$208$	0	0
$209$	−6.31684	−0.436945
$210$	0	0
$211$	−18.0179	−1.24041	−0.620203	−	0.784441i	$-0.712950\pi$
$211$	−18.0179	−1.24041	−0.620203	+	0.784441i	$0.712950\pi$
$212$	0	0
$213$	−3.11645	−0.213536
$214$	0	0
$215$	−2.97079	−0.202606
$216$	0	0
$217$	1.79955	0.122162
$218$	0	0
$219$	−0.867403	−0.0586136
$220$	0	0
$221$	−2.52708	−0.169990
$222$	0	0
$223$	19.7929	1.32543	0.662714	−	0.748872i	$-0.269404\pi$
$223$	19.7929	1.32543	0.662714	+	0.748872i	$0.269404\pi$
$224$	0	0
$225$	14.0054	0.933694
$226$	0	0
$227$	−20.9990	−1.39375	−0.696876	−	0.717192i	$-0.745427\pi$
$227$	−20.9990	−1.39375	−0.696876	+	0.717192i	$0.745427\pi$
$228$	0	0
$229$	−5.45778	−0.360660	−0.180330	−	0.983606i	$-0.557717\pi$
$229$	−5.45778	−0.360660	−0.180330	+	0.983606i	$0.557717\pi$
$230$	0	0
$231$	−0.478304	−0.0314701
$232$	0	0
$233$	−17.5712	−1.15113	−0.575565	−	0.817756i	$-0.695218\pi$
$233$	−17.5712	−1.15113	−0.575565	+	0.817756i	$0.695218\pi$
$234$	0	0
$235$	3.13023	0.204194
$236$	0	0
$237$	−2.62211	−0.170324
$238$	0	0
$239$	−9.00408	−0.582425	−0.291213	−	0.956658i	$-0.594059\pi$
$239$	−9.00408	−0.582425	−0.291213	+	0.956658i	$0.594059\pi$
$240$	0	0
$241$	−5.22730	−0.336720	−0.168360	−	0.985726i	$-0.553847\pi$
$241$	−5.22730	−0.336720	−0.168360	+	0.985726i	$0.553847\pi$
$242$	0	0
$243$	−8.27294	−0.530709
$244$	0	0
$245$	−1.94437	−0.124221
$246$	0	0
$247$	−6.31684	−0.401931
$248$	0	0
$249$	−5.05717	−0.320485
$250$	0	0
$251$	−18.6998	−1.18032	−0.590161	−	0.807285i	$-0.700936\pi$
$251$	−18.6998	−1.18032	−0.590161	+	0.807285i	$0.700936\pi$
$252$	0	0
$253$	0.262686	0.0165149
$254$	0	0
$255$	−0.329984	−0.0206644
$256$	0	0
$257$	18.5746	1.15865	0.579324	−	0.815097i	$-0.303316\pi$
$257$	18.5746	1.15865	0.579324	+	0.815097i	$0.303316\pi$
$258$	0	0
$259$	−12.6968	−0.788938
$260$	0	0
$261$	0.800261	0.0495349
$262$	0	0
$263$	0.423790	0.0261320	0.0130660	−	0.999915i	$-0.495841\pi$
$263$	0.423790	0.0261320	0.0130660	+	0.999915i	$0.495841\pi$
$264$	0	0
$265$	−3.97460	−0.244157
$266$	0	0
$267$	−5.25699	−0.321723
$268$	0	0
$269$	−4.99053	−0.304278	−0.152139	−	0.988359i	$-0.548616\pi$
$269$	−4.99053	−0.304278	−0.152139	+	0.988359i	$0.548616\pi$
$270$	0	0
$271$	18.5550	1.12713	0.563567	−	0.826070i	$-0.309429\pi$
$271$	18.5550	1.12713	0.563567	+	0.826070i	$0.309429\pi$
$272$	0	0
$273$	−0.478304	−0.0289483
$274$	0	0
$275$	4.83450	0.291531
$276$	0	0
$277$	4.24246	0.254905	0.127452	−	0.991845i	$-0.459320\pi$
$277$	4.24246	0.254905	0.127452	+	0.991845i	$0.459320\pi$
$278$	0	0
$279$	3.49847	0.209448
$280$	0	0
$281$	25.2069	1.50372	0.751859	−	0.659324i	$-0.229157\pi$
$281$	25.2069	1.50372	0.751859	+	0.659324i	$0.229157\pi$
$282$	0	0
$283$	9.72942	0.578354	0.289177	−	0.957276i	$-0.406618\pi$
$283$	9.72942	0.578354	0.289177	+	0.957276i	$0.406618\pi$
$284$	0	0
$285$	−0.824849	−0.0488598
$286$	0	0
$287$	9.74829	0.575423
$288$	0	0
$289$	−10.6139	−0.624346
$290$	0	0
$291$	1.02957	0.0603546
$292$	0	0
$293$	12.3558	0.721832	0.360916	−	0.932598i	$-0.382464\pi$
$293$	12.3558	0.721832	0.360916	+	0.932598i	$0.382464\pi$
$294$	0	0
$295$	−2.47143	−0.143892
$296$	0	0
$297$	−1.89279	−0.109831
$298$	0	0
$299$	0.262686	0.0151915
$300$	0	0
$301$	10.8818	0.627218
$302$	0	0
$303$	0.727564	0.0417974
$304$	0	0
$305$	−2.34949	−0.134531
$306$	0	0
$307$	−21.9045	−1.25016	−0.625078	−	0.780563i	$-0.714933\pi$
$307$	−21.9045	−1.25016	−0.625078	+	0.780563i	$0.714933\pi$
$308$	0	0
$309$	0.959451	0.0545813
$310$	0	0
$311$	11.5441	0.654606	0.327303	−	0.944919i	$-0.393860\pi$
$311$	11.5441	0.654606	0.327303	+	0.944919i	$0.393860\pi$
$312$	0	0
$313$	−1.14635	−0.0647956	−0.0323978	−	0.999475i	$-0.510314\pi$
$313$	−1.14635	−0.0647956	−0.0323978	+	0.999475i	$0.510314\pi$
$314$	0	0
$315$	1.75622	0.0989516
$316$	0	0
$317$	−24.3371	−1.36691	−0.683454	−	0.729993i	$-0.739523\pi$
$317$	−24.3371	−1.36691	−0.683454	+	0.729993i	$0.739523\pi$
$318$	0	0
$319$	0.276240	0.0154665
$320$	0	0
$321$	1.80236	0.100598
$322$	0	0
$323$	15.9631	0.888212
$324$	0	0
$325$	4.83450	0.268170
$326$	0	0
$327$	−1.20980	−0.0669019
$328$	0	0
$329$	−11.4658	−0.632133
$330$	0	0
$331$	−32.0581	−1.76207	−0.881036	−	0.473050i	$-0.843153\pi$
$331$	−32.0581	−1.76207	−0.881036	+	0.473050i	$0.843153\pi$
$332$	0	0
$333$	−24.6835	−1.35265
$334$	0	0
$335$	3.62641	0.198132
$336$	0	0
$337$	31.0725	1.69263	0.846314	−	0.532685i	$-0.178817\pi$
$337$	31.0725	1.69263	0.846314	+	0.532685i	$0.178817\pi$
$338$	0	0
$339$	−3.33634	−0.181205
$340$	0	0
$341$	1.20763	0.0653967
$342$	0	0
$343$	17.5532	0.947783
$344$	0	0
$345$	0.0343014	0.00184673
$346$	0	0
$347$	18.7285	1.00540	0.502700	−	0.864461i	$-0.332340\pi$
$347$	18.7285	1.00540	0.502700	+	0.864461i	$0.332340\pi$
$348$	0	0
$349$	33.9345	1.81647	0.908235	−	0.418460i	$-0.137430\pi$
$349$	33.9345	1.81647	0.908235	+	0.418460i	$0.137430\pi$
$350$	0	0
$351$	−1.89279	−0.101030
$352$	0	0
$353$	−23.2673	−1.23839	−0.619196	−	0.785236i	$-0.712541\pi$
$353$	−23.2673	−1.23839	−0.619196	+	0.785236i	$0.712541\pi$
$354$	0	0
$355$	3.94993	0.209641
$356$	0	0
$357$	1.20871	0.0639717
$358$	0	0
$359$	22.4412	1.18440	0.592202	−	0.805790i	$-0.298259\pi$
$359$	22.4412	1.18440	0.592202	+	0.805790i	$0.298259\pi$
$360$	0	0
$361$	20.9024	1.10013
$362$	0	0
$363$	−0.320976	−0.0168469
$364$	0	0
$365$	1.09939	0.0575445
$366$	0	0
$367$	3.03677	0.158518	0.0792591	−	0.996854i	$-0.474745\pi$
$367$	3.03677	0.158518	0.0792591	+	0.996854i	$0.474745\pi$
$368$	0	0
$369$	18.9514	0.986572
$370$	0	0
$371$	14.5587	0.755850
$372$	0	0
$373$	−27.8349	−1.44124	−0.720618	−	0.693332i	$-0.756142\pi$
$373$	−27.8349	−1.44124	−0.720618	+	0.693332i	$0.756142\pi$
$374$	0	0
$375$	1.28418	0.0663149
$376$	0	0
$377$	0.276240	0.0142271
$378$	0	0
$379$	−1.84707	−0.0948777	−0.0474388	−	0.998874i	$-0.515106\pi$
$379$	−1.84707	−0.0948777	−0.0474388	+	0.998874i	$0.515106\pi$
$380$	0	0
$381$	−6.38734	−0.327233
$382$	0	0
$383$	24.0057	1.22664	0.613318	−	0.789836i	$-0.289835\pi$
$383$	24.0057	1.22664	0.613318	+	0.789836i	$0.289835\pi$
$384$	0	0
$385$	0.606224	0.0308960
$386$	0	0
$387$	21.1551	1.07537
$388$	0	0
$389$	−17.9711	−0.911172	−0.455586	−	0.890192i	$-0.650570\pi$
$389$	−17.9711	−0.911172	−0.455586	+	0.890192i	$0.650570\pi$
$390$	0	0
$391$	−0.663828	−0.0335712
$392$	0	0
$393$	0.627609	0.0316587
$394$	0	0
$395$	3.32338	0.167217
$396$	0	0
$397$	26.1878	1.31433	0.657164	−	0.753748i	$-0.271756\pi$
$397$	26.1878	1.31433	0.657164	+	0.753748i	$0.271756\pi$
$398$	0	0
$399$	3.02137	0.151258
$400$	0	0
$401$	4.07164	0.203328	0.101664	−	0.994819i	$-0.467583\pi$
$401$	4.07164	0.203328	0.101664	+	0.994819i	$0.467583\pi$
$402$	0	0
$403$	1.20763	0.0601562
$404$	0	0
$405$	3.28848	0.163406
$406$	0	0
$407$	−8.52043	−0.422342
$408$	0	0
$409$	−5.12715	−0.253521	−0.126761	−	0.991933i	$-0.540458\pi$
$409$	−5.12715	−0.253521	−0.126761	+	0.991933i	$0.540458\pi$
$410$	0	0
$411$	1.25127	0.0617204
$412$	0	0
$413$	9.05271	0.445455
$414$	0	0
$415$	6.40969	0.314639
$416$	0	0
$417$	−4.31834	−0.211470
$418$	0	0
$419$	6.05740	0.295924	0.147962	−	0.988993i	$-0.452729\pi$
$419$	6.05740	0.295924	0.147962	+	0.988993i	$0.452729\pi$
$420$	0	0
$421$	−22.4873	−1.09597	−0.547983	−	0.836490i	$-0.684604\pi$
$421$	−22.4873	−1.09597	−0.547983	+	0.836490i	$0.684604\pi$
$422$	0	0
$423$	−22.2905	−1.08380
$424$	0	0
$425$	−12.2171	−0.592618
$426$	0	0
$427$	8.60602	0.416474
$428$	0	0
$429$	−0.320976	−0.0154969
$430$	0	0
$431$	25.2957	1.21845	0.609224	−	0.792998i	$-0.291481\pi$
$431$	25.2957	1.21845	0.609224	+	0.792998i	$0.291481\pi$
$432$	0	0
$433$	−6.46157	−0.310523	−0.155262	−	0.987873i	$-0.549622\pi$
$433$	−6.46157	−0.310523	−0.155262	+	0.987873i	$0.549622\pi$
$434$	0	0
$435$	0.0360713	0.00172949
$436$	0	0
$437$	−1.65935	−0.0793773
$438$	0	0
$439$	21.7620	1.03864	0.519321	−	0.854579i	$-0.326185\pi$
$439$	21.7620	1.03864	0.519321	+	0.854579i	$0.326185\pi$
$440$	0	0
$441$	13.8459	0.659329
$442$	0	0
$443$	−15.6715	−0.744574	−0.372287	−	0.928118i	$-0.621426\pi$
$443$	−15.6715	−0.744574	−0.372287	+	0.928118i	$0.621426\pi$
$444$	0	0
$445$	6.66295	0.315854
$446$	0	0
$447$	−3.57294	−0.168994
$448$	0	0
$449$	−6.33091	−0.298774	−0.149387	−	0.988779i	$-0.547730\pi$
$449$	−6.33091	−0.298774	−0.149387	+	0.988779i	$0.547730\pi$
$450$	0	0
$451$	6.54180	0.308041
$452$	0	0
$453$	2.88421	0.135512
$454$	0	0
$455$	0.606224	0.0284202
$456$	0	0
$457$	11.6052	0.542868	0.271434	−	0.962457i	$-0.412502\pi$
$457$	11.6052	0.542868	0.271434	+	0.962457i	$0.412502\pi$
$458$	0	0
$459$	4.78322	0.223261
$460$	0	0
$461$	14.3894	0.670181	0.335091	−	0.942186i	$-0.391233\pi$
$461$	14.3894	0.670181	0.335091	+	0.942186i	$0.391233\pi$
$462$	0	0
$463$	−4.19439	−0.194930	−0.0974649	−	0.995239i	$-0.531073\pi$
$463$	−4.19439	−0.194930	−0.0974649	+	0.995239i	$0.531073\pi$
$464$	0	0
$465$	0.157691	0.00731275
$466$	0	0
$467$	−0.727501	−0.0336647	−0.0168324	−	0.999858i	$-0.505358\pi$
$467$	−0.727501	−0.0336647	−0.0168324	+	0.999858i	$0.505358\pi$
$468$	0	0
$469$	−13.2833	−0.613367
$470$	0	0
$471$	−1.98597	−0.0915085
$472$	0	0
$473$	7.30248	0.335768
$474$	0	0
$475$	−30.5387	−1.40121
$476$	0	0
$477$	28.3032	1.29591
$478$	0	0
$479$	4.73550	0.216371	0.108185	−	0.994131i	$-0.465496\pi$
$479$	4.73550	0.216371	0.108185	+	0.994131i	$0.465496\pi$
$480$	0	0
$481$	−8.52043	−0.388498
$482$	0	0
$483$	−0.125644	−0.00571699
$484$	0	0
$485$	−1.30493	−0.0592536
$486$	0	0
$487$	−29.6760	−1.34475	−0.672375	−	0.740211i	$-0.734726\pi$
$487$	−29.6760	−1.34475	−0.672375	+	0.740211i	$0.734726\pi$
$488$	0	0
$489$	−5.38757	−0.243634
$490$	0	0
$491$	−9.11125	−0.411185	−0.205593	−	0.978638i	$-0.565912\pi$
$491$	−9.11125	−0.411185	−0.205593	+	0.978638i	$0.565912\pi$
$492$	0	0
$493$	−0.698080	−0.0314399
$494$	0	0
$495$	1.17855	0.0529717
$496$	0	0
$497$	−14.4684	−0.648995
$498$	0	0
$499$	−18.5863	−0.832035	−0.416018	−	0.909357i	$-0.636575\pi$
$499$	−18.5863	−0.832035	−0.416018	+	0.909357i	$0.636575\pi$
$500$	0	0
$501$	−7.09237	−0.316864
$502$	0	0
$503$	34.2704	1.52804	0.764022	−	0.645191i	$-0.223222\pi$
$503$	34.2704	1.52804	0.764022	+	0.645191i	$0.223222\pi$
$504$	0	0
$505$	−0.922147	−0.0410350
$506$	0	0
$507$	−0.320976	−0.0142550
$508$	0	0
$509$	−3.57365	−0.158399	−0.0791997	−	0.996859i	$-0.525236\pi$
$509$	−3.57365	−0.158399	−0.0791997	+	0.996859i	$0.525236\pi$
$510$	0	0
$511$	−4.02698	−0.178143
$512$	0	0
$513$	11.9564	0.527889
$514$	0	0
$515$	−1.21605	−0.0535856
$516$	0	0
$517$	−7.69440	−0.338399
$518$	0	0
$519$	−5.55016	−0.243625
$520$	0	0
$521$	25.8691	1.13335	0.566674	−	0.823942i	$-0.308230\pi$
$521$	25.8691	1.13335	0.566674	+	0.823942i	$0.308230\pi$
$522$	0	0
$523$	−14.6243	−0.639477	−0.319738	−	0.947506i	$-0.603595\pi$
$523$	−14.6243	−0.639477	−0.319738	+	0.947506i	$0.603595\pi$
$524$	0	0
$525$	−2.31236	−0.100920
$526$	0	0
$527$	−3.05177	−0.132937
$528$	0	0
$529$	−22.9310	−0.997000
$530$	0	0
$531$	17.5991	0.763738
$532$	0	0
$533$	6.54180	0.283357
$534$	0	0
$535$	−2.28439	−0.0987630
$536$	0	0
$537$	−4.18063	−0.180407
$538$	0	0
$539$	4.77944	0.205865
$540$	0	0
$541$	9.07699	0.390250	0.195125	−	0.980778i	$-0.437489\pi$
$541$	9.07699	0.390250	0.195125	+	0.980778i	$0.437489\pi$
$542$	0	0
$543$	−3.72127	−0.159695
$544$	0	0
$545$	1.53335	0.0656815
$546$	0	0
$547$	−43.7355	−1.87000	−0.934998	−	0.354652i	$-0.884599\pi$
$547$	−43.7355	−1.87000	−0.934998	+	0.354652i	$0.884599\pi$
$548$	0	0
$549$	16.7308	0.714051
$550$	0	0
$551$	−1.74496	−0.0743380
$552$	0	0
$553$	−12.1733	−0.517663
$554$	0	0
$555$	−1.11259	−0.0472269
$556$	0	0
$557$	−45.8315	−1.94194	−0.970971	−	0.239199i	$-0.923115\pi$
$557$	−45.8315	−1.94194	−0.970971	+	0.239199i	$0.923115\pi$
$558$	0	0
$559$	7.30248	0.308862
$560$	0	0
$561$	0.811131	0.0342460
$562$	0	0
$563$	24.6000	1.03677	0.518384	−	0.855148i	$-0.326534\pi$
$563$	24.6000	1.03677	0.518384	+	0.855148i	$0.326534\pi$
$564$	0	0
$565$	4.22863	0.177900
$566$	0	0
$567$	−12.0455	−0.505863
$568$	0	0
$569$	−11.7250	−0.491538	−0.245769	−	0.969328i	$-0.579040\pi$
$569$	−11.7250	−0.491538	−0.245769	+	0.969328i	$0.579040\pi$
$570$	0	0
$571$	10.3184	0.431813	0.215907	−	0.976414i	$-0.430729\pi$
$571$	10.3184	0.431813	0.215907	+	0.976414i	$0.430729\pi$
$572$	0	0
$573$	−2.78741	−0.116446
$574$	0	0
$575$	1.26996	0.0529608
$576$	0	0
$577$	−12.7317	−0.530029	−0.265015	−	0.964244i	$-0.585377\pi$
$577$	−12.7317	−0.530029	−0.265015	+	0.964244i	$0.585377\pi$
$578$	0	0
$579$	6.29246	0.261506
$580$	0	0
$581$	−23.4783	−0.974044
$582$	0	0
$583$	9.76992	0.404629
$584$	0	0
$585$	1.17855	0.0487269
$586$	0	0
$587$	31.1975	1.28766	0.643829	−	0.765169i	$-0.277345\pi$
$587$	31.1975	1.28766	0.643829	+	0.765169i	$0.277345\pi$
$588$	0	0
$589$	−7.62839	−0.314322
$590$	0	0
$591$	5.76543	0.237158
$592$	0	0
$593$	−9.83653	−0.403938	−0.201969	−	0.979392i	$-0.564734\pi$
$593$	−9.83653	−0.403938	−0.201969	+	0.979392i	$0.564734\pi$
$594$	0	0
$595$	−1.53197	−0.0628048
$596$	0	0
$597$	−6.84746	−0.280248
$598$	0	0
$599$	5.25105	0.214552	0.107276	−	0.994229i	$-0.465787\pi$
$599$	5.25105	0.214552	0.107276	+	0.994229i	$0.465787\pi$
$600$	0	0
$601$	40.9551	1.67059	0.835297	−	0.549800i	$-0.185296\pi$
$601$	40.9551	1.67059	0.835297	+	0.549800i	$0.185296\pi$
$602$	0	0
$603$	−25.8238	−1.05163
$604$	0	0
$605$	0.406820	0.0165396
$606$	0	0
$607$	−35.5744	−1.44392	−0.721960	−	0.691935i	$-0.756759\pi$
$607$	−35.5744	−1.44392	−0.721960	+	0.691935i	$0.756759\pi$
$608$	0	0
$609$	−0.132127	−0.00535405
$610$	0	0
$611$	−7.69440	−0.311282
$612$	0	0
$613$	32.9365	1.33029	0.665147	−	0.746712i	$-0.268369\pi$
$613$	32.9365	1.33029	0.665147	+	0.746712i	$0.268369\pi$
$614$	0	0
$615$	0.854224	0.0344456
$616$	0	0
$617$	−38.6317	−1.55525	−0.777627	−	0.628726i	$-0.783577\pi$
$617$	−38.6317	−1.55525	−0.777627	+	0.628726i	$0.783577\pi$
$618$	0	0
$619$	−33.9917	−1.36624	−0.683120	−	0.730306i	$-0.739377\pi$
$619$	−33.9917	−1.36624	−0.683120	+	0.730306i	$0.739377\pi$
$620$	0	0
$621$	−0.497209	−0.0199523
$622$	0	0
$623$	−24.4060	−0.977804
$624$	0	0
$625$	22.5449	0.901794
$626$	0	0
$627$	2.02755	0.0809727
$628$	0	0
$629$	21.5318	0.858528
$630$	0	0
$631$	2.35121	0.0936001	0.0468000	−	0.998904i	$-0.485098\pi$
$631$	2.35121	0.0936001	0.0468000	+	0.998904i	$0.485098\pi$
$632$	0	0
$633$	5.78333	0.229867
$634$	0	0
$635$	8.09561	0.321264
$636$	0	0
$637$	4.77944	0.189368
$638$	0	0
$639$	−28.1276	−1.11271
$640$	0	0
$641$	−11.7314	−0.463364	−0.231682	−	0.972792i	$-0.574423\pi$
$641$	−11.7314	−0.463364	−0.231682	+	0.972792i	$0.574423\pi$
$642$	0	0
$643$	−6.20535	−0.244715	−0.122358	−	0.992486i	$-0.539045\pi$
$643$	−6.20535	−0.244715	−0.122358	+	0.992486i	$0.539045\pi$
$644$	0	0
$645$	0.953553	0.0375461
$646$	0	0
$647$	−34.6388	−1.36179	−0.680895	−	0.732381i	$-0.738409\pi$
$647$	−34.6388	−1.36179	−0.680895	+	0.732381i	$0.738409\pi$
$648$	0	0
$649$	6.07501	0.238465
$650$	0	0
$651$	−0.577613	−0.0226384
$652$	0	0
$653$	11.0014	0.430518	0.215259	−	0.976557i	$-0.430940\pi$
$653$	11.0014	0.430518	0.215259	+	0.976557i	$0.430940\pi$
$654$	0	0
$655$	−0.795459	−0.0310812
$656$	0	0
$657$	−7.82876	−0.305429
$658$	0	0
$659$	18.7085	0.728781	0.364390	−	0.931246i	$-0.381277\pi$
$659$	18.7085	0.728781	0.364390	+	0.931246i	$0.381277\pi$
$660$	0	0
$661$	27.2734	1.06081	0.530407	−	0.847743i	$-0.322039\pi$
$661$	27.2734	1.06081	0.530407	+	0.847743i	$0.322039\pi$
$662$	0	0
$663$	0.811131	0.0315017
$664$	0	0
$665$	−3.82942	−0.148499
$666$	0	0
$667$	0.0725645	0.00280971
$668$	0	0
$669$	−6.35304	−0.245623
$670$	0	0
$671$	5.77525	0.222951
$672$	0	0
$673$	1.93812	0.0747089	0.0373545	−	0.999302i	$-0.488107\pi$
$673$	1.93812	0.0747089	0.0373545	+	0.999302i	$0.488107\pi$
$674$	0	0
$675$	−9.15068	−0.352210
$676$	0	0
$677$	12.5740	0.483258	0.241629	−	0.970369i	$-0.422318\pi$
$677$	12.5740	0.483258	0.241629	+	0.970369i	$0.422318\pi$
$678$	0	0
$679$	4.77986	0.183434
$680$	0	0
$681$	6.74017	0.258284
$682$	0	0
$683$	12.2609	0.469152	0.234576	−	0.972098i	$-0.424630\pi$
$683$	12.2609	0.469152	0.234576	+	0.972098i	$0.424630\pi$
$684$	0	0
$685$	−1.58591	−0.0605946
$686$	0	0
$687$	1.75182	0.0668360
$688$	0	0
$689$	9.76992	0.372204
$690$	0	0
$691$	10.0782	0.383393	0.191696	−	0.981454i	$-0.438601\pi$
$691$	10.0782	0.383393	0.191696	+	0.981454i	$0.438601\pi$
$692$	0	0
$693$	−4.31694	−0.163987
$694$	0	0
$695$	5.47326	0.207612
$696$	0	0
$697$	−16.5316	−0.626180
$698$	0	0
$699$	5.63995	0.213322
$700$	0	0
$701$	8.47088	0.319941	0.159970	−	0.987122i	$-0.448860\pi$
$701$	8.47088	0.319941	0.159970	+	0.987122i	$0.448860\pi$
$702$	0	0
$703$	53.8222	2.02994
$704$	0	0
$705$	−1.00473	−0.0378403
$706$	0	0
$707$	3.37777	0.127034
$708$	0	0
$709$	−17.2510	−0.647876	−0.323938	−	0.946078i	$-0.605007\pi$
$709$	−17.2510	−0.647876	−0.323938	+	0.946078i	$0.605007\pi$
$710$	0	0
$711$	−23.6659	−0.887541
$712$	0	0
$713$	0.317227	0.0118802
$714$	0	0
$715$	0.406820	0.0152142
$716$	0	0
$717$	2.89009	0.107933
$718$	0	0
$719$	−10.7487	−0.400858	−0.200429	−	0.979708i	$-0.564234\pi$
$719$	−10.7487	−0.400858	−0.200429	+	0.979708i	$0.564234\pi$
$720$	0	0
$721$	4.45432	0.165888
$722$	0	0
$723$	1.67784	0.0623995
$724$	0	0
$725$	1.33548	0.0495986
$726$	0	0
$727$	−29.6911	−1.10118	−0.550591	−	0.834775i	$-0.685598\pi$
$727$	−29.6911	−1.10118	−0.550591	+	0.834775i	$0.685598\pi$
$728$	0	0
$729$	−21.5947	−0.799805
$730$	0	0
$731$	−18.4539	−0.682543
$732$	0	0
$733$	43.1043	1.59209	0.796047	−	0.605235i	$-0.206921\pi$
$733$	43.1043	1.59209	0.796047	+	0.605235i	$0.206921\pi$
$734$	0	0
$735$	0.624096	0.0230201
$736$	0	0
$737$	−8.91405	−0.328353
$738$	0	0
$739$	−8.59678	−0.316238	−0.158119	−	0.987420i	$-0.550543\pi$
$739$	−8.59678	−0.316238	−0.158119	+	0.987420i	$0.550543\pi$
$740$	0	0
$741$	2.02755	0.0744840
$742$	0	0
$743$	−2.74932	−0.100863	−0.0504313	−	0.998728i	$-0.516060\pi$
$743$	−2.74932	−0.100863	−0.0504313	+	0.998728i	$0.516060\pi$
$744$	0	0
$745$	4.52851	0.165912
$746$	0	0
$747$	−45.6436	−1.67001
$748$	0	0
$749$	8.36759	0.305745
$750$	0	0
$751$	7.81172	0.285054	0.142527	−	0.989791i	$-0.454477\pi$
$751$	7.81172	0.285054	0.142527	+	0.989791i	$0.454477\pi$
$752$	0	0
$753$	6.00220	0.218732
$754$	0	0
$755$	−3.65558	−0.133040
$756$	0	0
$757$	26.1209	0.949379	0.474689	−	0.880153i	$-0.342560\pi$
$757$	26.1209	0.949379	0.474689	+	0.880153i	$0.342560\pi$
$758$	0	0
$759$	−0.0843160	−0.00306048
$760$	0	0
$761$	45.4058	1.64596	0.822980	−	0.568071i	$-0.192310\pi$
$761$	45.4058	1.64596	0.822980	+	0.568071i	$0.192310\pi$
$762$	0	0
$763$	−5.61657	−0.203333
$764$	0	0
$765$	−2.97827	−0.107680
$766$	0	0
$767$	6.07501	0.219356
$768$	0	0
$769$	−10.6495	−0.384032	−0.192016	−	0.981392i	$-0.561503\pi$
$769$	−10.6495	−0.384032	−0.192016	+	0.981392i	$0.561503\pi$
$770$	0	0
$771$	−5.96199	−0.214716
$772$	0	0
$773$	−13.3587	−0.480480	−0.240240	−	0.970714i	$-0.577226\pi$
$773$	−13.3587	−0.480480	−0.240240	+	0.970714i	$0.577226\pi$
$774$	0	0
$775$	5.83827	0.209717
$776$	0	0
$777$	4.07536	0.146203
$778$	0	0
$779$	−41.3235	−1.48057
$780$	0	0
$781$	−9.70930	−0.347426
$782$	0	0
$783$	−0.522864	−0.0186856
$784$	0	0
$785$	2.51711	0.0898393
$786$	0	0
$787$	−28.3908	−1.01202	−0.506011	−	0.862527i	$-0.668881\pi$
$787$	−28.3908	−1.01202	−0.506011	+	0.862527i	$0.668881\pi$
$788$	0	0
$789$	−0.136026	−0.00484266
$790$	0	0
$791$	−15.4892	−0.550733
$792$	0	0
$793$	5.77525	0.205085
$794$	0	0
$795$	1.27575	0.0452462
$796$	0	0
$797$	−24.0049	−0.850297	−0.425148	−	0.905124i	$-0.639778\pi$
$797$	−24.0049	−0.850297	−0.425148	+	0.905124i	$0.639778\pi$
$798$	0	0
$799$	19.4443	0.687891
$800$	0	0
$801$	−47.4471	−1.67646
$802$	0	0
$803$	−2.70239	−0.0953653
$804$	0	0
$805$	0.159247	0.00561271
$806$	0	0
$807$	1.60184	0.0563875
$808$	0	0
$809$	6.87470	0.241702	0.120851	−	0.992671i	$-0.461438\pi$
$809$	6.87470	0.241702	0.120851	+	0.992671i	$0.461438\pi$
$810$	0	0
$811$	−52.8647	−1.85633	−0.928165	−	0.372170i	$-0.878614\pi$
$811$	−52.8647	−1.85633	−0.928165	+	0.372170i	$0.878614\pi$
$812$	0	0
$813$	−5.95570	−0.208875
$814$	0	0
$815$	6.82845	0.239190
$816$	0	0
$817$	−46.1286	−1.61383
$818$	0	0
$819$	−4.31694	−0.150846
$820$	0	0
$821$	45.0563	1.57248	0.786238	−	0.617924i	$-0.212026\pi$
$821$	45.0563	1.57248	0.786238	+	0.617924i	$0.212026\pi$
$822$	0	0
$823$	−1.85920	−0.0648076	−0.0324038	−	0.999475i	$-0.510316\pi$
$823$	−1.85920	−0.0648076	−0.0324038	+	0.999475i	$0.510316\pi$
$824$	0	0
$825$	−1.55176	−0.0540253
$826$	0	0
$827$	−11.0273	−0.383456	−0.191728	−	0.981448i	$-0.561409\pi$
$827$	−11.0273	−0.383456	−0.191728	+	0.981448i	$0.561409\pi$
$828$	0	0
$829$	8.23844	0.286133	0.143066	−	0.989713i	$-0.454304\pi$
$829$	8.23844	0.286133	0.143066	+	0.989713i	$0.454304\pi$
$830$	0	0
$831$	−1.36173	−0.0472378
$832$	0	0
$833$	−12.0780	−0.418478
$834$	0	0
$835$	8.98919	0.311084
$836$	0	0
$837$	−2.28578	−0.0790082
$838$	0	0
$839$	−10.9649	−0.378551	−0.189275	−	0.981924i	$-0.560614\pi$
$839$	−10.9649	−0.378551	−0.189275	+	0.981924i	$0.560614\pi$
$840$	0	0
$841$	−28.9237	−0.997369
$842$	0	0
$843$	−8.09082	−0.278662
$844$	0	0
$845$	0.406820	0.0139950
$846$	0	0
$847$	−1.49015	−0.0512023
$848$	0	0
$849$	−3.12291	−0.107178
$850$	0	0
$851$	−2.23820	−0.0767245
$852$	0	0
$853$	−43.5163	−1.48997	−0.744985	−	0.667081i	$-0.767543\pi$
$853$	−43.5163	−1.48997	−0.744985	+	0.667081i	$0.767543\pi$
$854$	0	0
$855$	−7.44468	−0.254603
$856$	0	0
$857$	−36.5293	−1.24782	−0.623908	−	0.781498i	$-0.714456\pi$
$857$	−36.5293	−1.24782	−0.623908	+	0.781498i	$0.714456\pi$
$858$	0	0
$859$	38.9062	1.32746	0.663732	−	0.747971i	$-0.268972\pi$
$859$	38.9062	1.32746	0.663732	+	0.747971i	$0.268972\pi$
$860$	0	0
$861$	−3.12897	−0.106635
$862$	0	0
$863$	−47.7942	−1.62693	−0.813467	−	0.581611i	$-0.802423\pi$
$863$	−47.7942	−1.62693	−0.813467	+	0.581611i	$0.802423\pi$
$864$	0	0
$865$	7.03453	0.239181
$866$	0	0
$867$	3.40680	0.115701
$868$	0	0
$869$	−8.16918	−0.277120
$870$	0	0
$871$	−8.91405	−0.302041
$872$	0	0
$873$	9.29241	0.314501
$874$	0	0
$875$	5.96191	0.201549
$876$	0	0
$877$	22.5633	0.761909	0.380955	−	0.924594i	$-0.375595\pi$
$877$	22.5633	0.761909	0.380955	+	0.924594i	$0.375595\pi$
$878$	0	0
$879$	−3.96591	−0.133767
$880$	0	0
$881$	33.7545	1.13722	0.568610	−	0.822608i	$-0.307482\pi$
$881$	33.7545	1.13722	0.568610	+	0.822608i	$0.307482\pi$
$882$	0	0
$883$	−44.9557	−1.51288	−0.756440	−	0.654063i	$-0.773063\pi$
$883$	−44.9557	−1.51288	−0.756440	+	0.654063i	$0.773063\pi$
$884$	0	0
$885$	0.793271	0.0266655
$886$	0	0
$887$	20.8902	0.701424	0.350712	−	0.936483i	$-0.385940\pi$
$887$	20.8902	0.701424	0.350712	+	0.936483i	$0.385940\pi$
$888$	0	0
$889$	−29.6537	−0.994553
$890$	0	0
$891$	−8.08338	−0.270804
$892$	0	0
$893$	48.6043	1.62648
$894$	0	0
$895$	5.29872	0.177117
$896$	0	0
$897$	−0.0843160	−0.00281523
$898$	0	0
$899$	0.333595	0.0111260
$900$	0	0
$901$	−24.6893	−0.822521
$902$	0	0
$903$	−3.49280	−0.116233
$904$	0	0
$905$	4.71650	0.156782
$906$	0	0
$907$	−46.0531	−1.52917	−0.764584	−	0.644524i	$-0.777056\pi$
$907$	−46.0531	−1.52917	−0.764584	+	0.644524i	$0.777056\pi$
$908$	0	0
$909$	6.56664	0.217802
$910$	0	0
$911$	51.2590	1.69829	0.849143	−	0.528164i	$-0.177119\pi$
$911$	51.2590	1.69829	0.849143	+	0.528164i	$0.177119\pi$
$912$	0	0
$913$	−15.7556	−0.521435
$914$	0	0
$915$	0.754129	0.0249307
$916$	0	0
$917$	2.91372	0.0962195
$918$	0	0
$919$	−24.5960	−0.811346	−0.405673	−	0.914018i	$-0.632963\pi$
$919$	−24.5960	−0.811346	−0.405673	+	0.914018i	$0.632963\pi$
$920$	0	0
$921$	7.03082	0.231673
$922$	0	0
$923$	−9.70930	−0.319585
$924$	0	0
$925$	−41.1920	−1.35438
$926$	0	0
$927$	8.65954	0.284417
$928$	0	0
$929$	22.3197	0.732286	0.366143	−	0.930559i	$-0.380678\pi$
$929$	22.3197	0.732286	0.366143	+	0.930559i	$0.380678\pi$
$930$	0	0
$931$	−30.1909	−0.989468
$932$	0	0
$933$	−3.70538	−0.121309
$934$	0	0
$935$	−1.02806	−0.0336213
$936$	0	0
$937$	40.3010	1.31658	0.658288	−	0.752766i	$-0.271281\pi$
$937$	40.3010	1.31658	0.658288	+	0.752766i	$0.271281\pi$
$938$	0	0
$939$	0.367951	0.0120076
$940$	0	0
$941$	−41.4004	−1.34962	−0.674808	−	0.737993i	$-0.735773\pi$
$941$	−41.4004	−1.34962	−0.674808	+	0.737993i	$0.735773\pi$
$942$	0	0
$943$	1.71844	0.0559601
$944$	0	0
$945$	−1.14745	−0.0373267
$946$	0	0
$947$	−4.04040	−0.131295	−0.0656477	−	0.997843i	$-0.520911\pi$
$947$	−4.04040	−0.131295	−0.0656477	+	0.997843i	$0.520911\pi$
$948$	0	0
$949$	−2.70239	−0.0877233
$950$	0	0
$951$	7.81163	0.253309
$952$	0	0
$953$	26.6692	0.863900	0.431950	−	0.901898i	$-0.357826\pi$
$953$	26.6692	0.863900	0.431950	+	0.901898i	$0.357826\pi$
$954$	0	0
$955$	3.53289	0.114322
$956$	0	0
$957$	−0.0886665	−0.00286618
$958$	0	0
$959$	5.80910	0.187586
$960$	0	0
$961$	−29.5416	−0.952956
$962$	0	0
$963$	16.2672	0.524205
$964$	0	0
$965$	−7.97535	−0.256736
$966$	0	0
$967$	−26.2328	−0.843588	−0.421794	−	0.906692i	$-0.638600\pi$
$967$	−26.2328	−0.843588	−0.421794	+	0.906692i	$0.638600\pi$
$968$	0	0
$969$	−5.12378	−0.164600
$970$	0	0
$971$	28.3624	0.910192	0.455096	−	0.890442i	$-0.349605\pi$
$971$	28.3624	0.910192	0.455096	+	0.890442i	$0.349605\pi$
$972$	0	0
$973$	−20.0482	−0.642716
$974$	0	0
$975$	−1.55176	−0.0496960
$976$	0	0
$977$	43.5330	1.39275	0.696373	−	0.717680i	$-0.254796\pi$
$977$	43.5330	1.39275	0.696373	+	0.717680i	$0.254796\pi$
$978$	0	0
$979$	−16.3781	−0.523448
$980$	0	0
$981$	−10.9190	−0.348618
$982$	0	0
$983$	−10.9898	−0.350520	−0.175260	−	0.984522i	$-0.556077\pi$
$983$	−10.9898	−0.350520	−0.175260	+	0.984522i	$0.556077\pi$
$984$	0	0
$985$	−7.30736	−0.232832
$986$	0	0
$987$	3.68026	0.117144
$988$	0	0
$989$	1.91826	0.0609971
$990$	0	0
$991$	20.3784	0.647340	0.323670	−	0.946170i	$-0.395083\pi$
$991$	20.3784	0.647340	0.323670	+	0.946170i	$0.395083\pi$
$992$	0	0
$993$	10.2899	0.326539
$994$	0	0
$995$	8.67879	0.275136
$996$	0	0
$997$	23.1387	0.732810	0.366405	−	0.930455i	$-0.380588\pi$
$997$	23.1387	0.732810	0.366405	+	0.930455i	$0.380588\pi$
$998$	0	0
$999$	16.1274	0.510247

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4576.2.a.p.1.3	✓	6
4.3	odd	2		4576.2.a.q.1.4	yes	6
8.3	odd	2		9152.2.a.cp.1.3		6
8.5	even	2		9152.2.a.cr.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4576.2.a.p.1.3	✓	6	1.1	even	1	trivial
4576.2.a.q.1.4	yes	6	4.3	odd	2
9152.2.a.cp.1.3		6	8.3	odd	2
9152.2.a.cr.1.4		6	8.5	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4576 = 2^{5} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4576.a (trivial)

\(f(q)\)	\(=\)	\(q-0.320976 q^{3} +0.406820 q^{5} -1.49015 q^{7} -2.89697 q^{9} -1.00000 q^{11} -1.00000 q^{13} -0.130579 q^{15} +2.52708 q^{17} +6.31684 q^{19} +0.478304 q^{21} -0.262686 q^{23} -4.83450 q^{25} +1.89279 q^{27} -0.276240 q^{29} -1.20763 q^{31} +0.320976 q^{33} -0.606224 q^{35} +8.52043 q^{37} +0.320976 q^{39} -6.54180 q^{41} -7.30248 q^{43} -1.17855 q^{45} +7.69440 q^{47} -4.77944 q^{49} -0.811131 q^{51} -9.76992 q^{53} -0.406820 q^{55} -2.02755 q^{57} -6.07501 q^{59} -5.77525 q^{61} +4.31694 q^{63} -0.406820 q^{65} +8.91405 q^{67} +0.0843160 q^{69} +9.70930 q^{71} +2.70239 q^{73} +1.55176 q^{75} +1.49015 q^{77} +8.16918 q^{79} +8.08338 q^{81} +15.7556 q^{83} +1.02806 q^{85} +0.0886665 q^{87} +16.3781 q^{89} +1.49015 q^{91} +0.387619 q^{93} +2.56981 q^{95} -3.20763 q^{97} +2.89697 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{3} - 2 q^{5} + 5 q^{7} + q^{9} - 6 q^{11} - 6 q^{13} + 5 q^{15} + 6 q^{17} - 2 q^{19} + 20 q^{23} + 4 q^{25} - q^{27} - 3 q^{29} + 11 q^{31} + q^{33} + 9 q^{35} - 3 q^{37} + q^{39} - q^{41}+ \cdots - q^{99}+O(q^{100}) \) 6 * q - q^3 - 2 * q^5 + 5 * q^7 + q^9 - 6 * q^11 - 6 * q^13 + 5 * q^15 + 6 * q^17 - 2 * q^19 + 20 * q^23 + 4 * q^25 - q^27 - 3 * q^29 + 11 * q^31 + q^33 + 9 * q^35 - 3 * q^37 + q^39 - q^41 - 9 * q^43 - 23 * q^45 - 8 * q^47 + q^49 + 10 * q^51 - 2 * q^53 + 2 * q^55 + 12 * q^57 + 6 * q^59 - 11 * q^61 + 29 * q^63 + 2 * q^65 + 14 * q^67 + 7 * q^69 + 19 * q^71 - q^73 - 6 * q^75 - 5 * q^77 + 36 * q^79 - 6 * q^81 - 2 * q^83 + 22 * q^85 + 24 * q^87 - 5 * q^89 - 5 * q^91 + 35 * q^93 + 38 * q^95 - q^97 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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