LMFDB - Embedded newform 3380.2.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,2,Mod(1,3380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3380, 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("3380.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3380 = 2^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3380.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:	$26.9894358832$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.4752.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 3x^{2} + 4x + 1 $ x^4 - 2x^3 - 3x^2 + 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 260)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.49551$ of defining polynomial
Character	$\chi$	$=$	3380.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.0947876 q^{3} -1.00000 q^{5} +0.826838 q^{7} -2.99102 q^{9} +O(q^{10})$	$q-0.0947876 q^{3} -1.00000 q^{5} +0.826838 q^{7} -2.99102 q^{9} -1.73205 q^{11} +0.0947876 q^{15} +1.43213 q^{17} +1.06939 q^{19} -0.0783740 q^{21} +3.08580 q^{23} +1.00000 q^{25} +0.567874 q^{27} +7.45512 q^{29} +5.84325 q^{31} +0.164177 q^{33} -0.826838 q^{35} -0.983586 q^{37} -4.26795 q^{41} -9.54092 q^{43} +2.99102 q^{45} -3.46410 q^{47} -6.31634 q^{49} -0.135748 q^{51} +0.334308 q^{53} +1.73205 q^{55} -0.101365 q^{57} -11.5245 q^{59} +2.71649 q^{61} -2.47309 q^{63} -13.7550 q^{67} -0.292496 q^{69} -9.77689 q^{71} -11.1806 q^{73} -0.0947876 q^{75} -1.43213 q^{77} -0.252387 q^{79} +8.91922 q^{81} +5.67165 q^{83} -1.43213 q^{85} -0.706653 q^{87} -4.59630 q^{89} -0.553868 q^{93} -1.06939 q^{95} -9.53434 q^{97} +5.18059 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} - 4 q^{5} - 6 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 2 * q^3 - 4 * q^5 - 6 * q^7 + 4 * q^9	$ 4 q + 2 q^{3} - 4 q^{5} - 6 q^{7} + 4 q^{9} - 2 q^{15} + 6 q^{17} - 12 q^{21} - 6 q^{23} + 4 q^{25} + 2 q^{27} - 6 q^{33} + 6 q^{35} - 18 q^{37} - 24 q^{41} + 10 q^{43} - 4 q^{45} - 4 q^{49} + 12 q^{53} + 18 q^{57} - 12 q^{59} + 4 q^{61} - 12 q^{63} - 18 q^{67} - 24 q^{69} - 12 q^{71} - 24 q^{73} + 2 q^{75} - 6 q^{77} - 8 q^{79} - 8 q^{81} + 36 q^{83} - 6 q^{85} + 6 q^{87} - 12 q^{89} - 48 q^{93} - 6 q^{97}+O(q^{100}) $ 4 * q + 2 * q^3 - 4 * q^5 - 6 * q^7 + 4 * q^9 - 2 * q^15 + 6 * q^17 - 12 * q^21 - 6 * q^23 + 4 * q^25 + 2 * q^27 - 6 * q^33 + 6 * q^35 - 18 * q^37 - 24 * q^41 + 10 * q^43 - 4 * q^45 - 4 * q^49 + 12 * q^53 + 18 * q^57 - 12 * q^59 + 4 * q^61 - 12 * q^63 - 18 * q^67 - 24 * q^69 - 12 * q^71 - 24 * q^73 + 2 * q^75 - 6 * q^77 - 8 * q^79 - 8 * q^81 + 36 * q^83 - 6 * q^85 + 6 * q^87 - 12 * q^89 - 48 * q^93 - 6 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−0.0947876	−0.0547256	−0.0273628	−	0.999626i	$-0.508711\pi$
$3$	−0.0947876	−0.0547256	−0.0273628	+	0.999626i	$0.508711\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0.826838	0.312516	0.156258	−	0.987716i	$-0.450057\pi$
$7$	0.826838	0.312516	0.156258	+	0.987716i	$0.450057\pi$
$8$	0	0
$9$	−2.99102	−0.997005
$10$	0	0
$11$	−1.73205	−0.522233	−0.261116	−	0.965307i	$-0.584091\pi$
$11$	−1.73205	−0.522233	−0.261116	+	0.965307i	$0.584091\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0.0947876	0.0244740
$16$	0	0
$17$	1.43213	0.347342	0.173671	−	0.984804i	$-0.444437\pi$
$17$	1.43213	0.347342	0.173671	+	0.984804i	$0.444437\pi$
$18$	0	0
$19$	1.06939	0.245335	0.122667	−	0.992448i	$-0.460855\pi$
$19$	1.06939	0.245335	0.122667	+	0.992448i	$0.460855\pi$
$20$	0	0
$21$	−0.0783740	−0.0171026
$22$	0	0
$23$	3.08580	0.643434	0.321717	−	0.946836i	$-0.395740\pi$
$23$	3.08580	0.643434	0.321717	+	0.946836i	$0.395740\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0.567874	0.109287
$28$	0	0
$29$	7.45512	1.38438	0.692190	−	0.721715i	$-0.256646\pi$
$29$	7.45512	1.38438	0.692190	+	0.721715i	$0.256646\pi$
$30$	0	0
$31$	5.84325	1.04948	0.524740	−	0.851263i	$-0.324163\pi$
$31$	5.84325	1.04948	0.524740	+	0.851263i	$0.324163\pi$
$32$	0	0
$33$	0.164177	0.0285795
$34$	0	0
$35$	−0.826838	−0.139761
$36$	0	0
$37$	−0.983586	−0.161701	−0.0808503	−	0.996726i	$-0.525764\pi$
$37$	−0.983586	−0.161701	−0.0808503	+	0.996726i	$0.525764\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.26795	−0.666542	−0.333271	−	0.942831i	$-0.608152\pi$
$41$	−4.26795	−0.666542	−0.333271	+	0.942831i	$0.608152\pi$
$42$	0	0
$43$	−9.54092	−1.45498	−0.727488	−	0.686120i	$-0.759312\pi$
$43$	−9.54092	−1.45498	−0.727488	+	0.686120i	$0.759312\pi$
$44$	0	0
$45$	2.99102	0.445874
$46$	0	0
$47$	−3.46410	−0.505291	−0.252646	−	0.967559i	$-0.581301\pi$
$47$	−3.46410	−0.505291	−0.252646	+	0.967559i	$0.581301\pi$
$48$	0	0
$49$	−6.31634	−0.902334
$50$	0	0
$51$	−0.135748	−0.0190085
$52$	0	0
$53$	0.334308	0.0459207	0.0229603	−	0.999736i	$-0.492691\pi$
$53$	0.334308	0.0459207	0.0229603	+	0.999736i	$0.492691\pi$
$54$	0	0
$55$	1.73205	0.233550
$56$	0	0
$57$	−0.101365	−0.0134261
$58$	0	0
$59$	−11.5245	−1.50036	−0.750181	−	0.661232i	$-0.770034\pi$
$59$	−11.5245	−1.50036	−0.750181	+	0.661232i	$0.770034\pi$
$60$	0	0
$61$	2.71649	0.347811	0.173905	−	0.984762i	$-0.444361\pi$
$61$	2.71649	0.347811	0.173905	+	0.984762i	$0.444361\pi$
$62$	0	0
$63$	−2.47309	−0.311580
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−13.7550	−1.68045	−0.840223	−	0.542241i	$-0.817576\pi$
$67$	−13.7550	−1.68045	−0.840223	+	0.542241i	$0.817576\pi$
$68$	0	0
$69$	−0.292496	−0.0352124
$70$	0	0
$71$	−9.77689	−1.16030	−0.580152	−	0.814508i	$-0.697007\pi$
$71$	−9.77689	−1.16030	−0.580152	+	0.814508i	$0.697007\pi$
$72$	0	0
$73$	−11.1806	−1.30859	−0.654295	−	0.756240i	$-0.727034\pi$
$73$	−11.1806	−1.30859	−0.654295	+	0.756240i	$0.727034\pi$
$74$	0	0
$75$	−0.0947876	−0.0109451
$76$	0	0
$77$	−1.43213	−0.163206
$78$	0	0
$79$	−0.252387	−0.0283958	−0.0141979	−	0.999899i	$-0.504519\pi$
$79$	−0.252387	−0.0283958	−0.0141979	+	0.999899i	$0.504519\pi$
$80$	0	0
$81$	8.91922	0.991024
$82$	0	0
$83$	5.67165	0.622544	0.311272	−	0.950321i	$-0.399245\pi$
$83$	5.67165	0.622544	0.311272	+	0.950321i	$0.399245\pi$
$84$	0	0
$85$	−1.43213	−0.155336
$86$	0	0
$87$	−0.706653	−0.0757611
$88$	0	0
$89$	−4.59630	−0.487207	−0.243604	−	0.969875i	$-0.578330\pi$
$89$	−4.59630	−0.487207	−0.243604	+	0.969875i	$0.578330\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−0.553868	−0.0574334
$94$	0	0
$95$	−1.06939	−0.109717
$96$	0	0
$97$	−9.53434	−0.968066	−0.484033	−	0.875050i	$-0.660829\pi$
$97$	−9.53434	−0.968066	−0.484033	+	0.875050i	$0.660829\pi$
$98$	0	0
$99$	5.18059	0.520669
$100$	0	0
$101$	−5.80144	−0.577265	−0.288632	−	0.957440i	$-0.593201\pi$
$101$	−5.80144	−0.577265	−0.288632	+	0.957440i	$0.593201\pi$
$102$	0	0
$103$	−10.0760	−0.992814	−0.496407	−	0.868090i	$-0.665348\pi$
$103$	−10.0760	−0.992814	−0.496407	+	0.868090i	$0.665348\pi$
$104$	0	0
$105$	0.0783740	0.00764852
$106$	0	0
$107$	16.2795	1.57380	0.786902	−	0.617078i	$-0.211684\pi$
$107$	16.2795	1.57380	0.786902	+	0.617078i	$0.211684\pi$
$108$	0	0
$109$	3.12979	0.299780	0.149890	−	0.988703i	$-0.452108\pi$
$109$	3.12979	0.299780	0.149890	+	0.988703i	$0.452108\pi$
$110$	0	0
$111$	0.0932318	0.00884917
$112$	0	0
$113$	−10.1708	−0.956784	−0.478392	−	0.878146i	$-0.658780\pi$
$113$	−10.1708	−0.956784	−0.478392	+	0.878146i	$0.658780\pi$
$114$	0	0
$115$	−3.08580	−0.287753
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.18414	0.108550
$120$	0	0
$121$	−8.00000	−0.727273
$122$	0	0
$123$	0.404549	0.0364769
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	5.96802	0.529577	0.264788	−	0.964307i	$-0.414698\pi$
$127$	5.96802	0.529577	0.264788	+	0.964307i	$0.414698\pi$
$128$	0	0
$129$	0.904361	0.0796245
$130$	0	0
$131$	16.6267	1.45268	0.726342	−	0.687334i	$-0.241219\pi$
$131$	16.6267	1.45268	0.726342	+	0.687334i	$0.241219\pi$
$132$	0	0
$133$	0.884212	0.0766709
$134$	0	0
$135$	−0.567874	−0.0488748
$136$	0	0
$137$	−0.404549	−0.0345629	−0.0172815	−	0.999851i	$-0.505501\pi$
$137$	−0.404549	−0.0345629	−0.0172815	+	0.999851i	$0.505501\pi$
$138$	0	0
$139$	−9.31634	−0.790201	−0.395101	−	0.918638i	$-0.629290\pi$
$139$	−9.31634	−0.790201	−0.395101	+	0.918638i	$0.629290\pi$
$140$	0	0
$141$	0.328354	0.0276524
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−7.45512	−0.619114
$146$	0	0
$147$	0.598710	0.0493808
$148$	0	0
$149$	−10.8678	−0.890325	−0.445162	−	0.895450i	$-0.646854\pi$
$149$	−10.8678	−0.890325	−0.445162	+	0.895450i	$0.646854\pi$
$150$	0	0
$151$	0.991015	0.0806477	0.0403238	−	0.999187i	$-0.487161\pi$
$151$	0.991015	0.0806477	0.0403238	+	0.999187i	$0.487161\pi$
$152$	0	0
$153$	−4.28351	−0.346301
$154$	0	0
$155$	−5.84325	−0.469341
$156$	0	0
$157$	17.5729	1.40247	0.701235	−	0.712930i	$-0.252632\pi$
$157$	17.5729	1.40247	0.701235	+	0.712930i	$0.252632\pi$
$158$	0	0
$159$	−0.0316882	−0.00251304
$160$	0	0
$161$	2.55146	0.201083
$162$	0	0
$163$	−17.2820	−1.35363	−0.676814	−	0.736154i	$-0.736640\pi$
$163$	−17.2820	−1.35363	−0.676814	+	0.736154i	$0.736640\pi$
$164$	0	0
$165$	−0.164177	−0.0127812
$166$	0	0
$167$	3.05955	0.236755	0.118378	−	0.992969i	$-0.462231\pi$
$167$	3.05955	0.236755	0.118378	+	0.992969i	$0.462231\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−3.19856	−0.244600
$172$	0	0
$173$	−3.42011	−0.260026	−0.130013	−	0.991512i	$-0.541502\pi$
$173$	−3.42011	−0.260026	−0.130013	+	0.991512i	$0.541502\pi$
$174$	0	0
$175$	0.826838	0.0625031
$176$	0	0
$177$	1.09238	0.0821083
$178$	0	0
$179$	−10.3822	−0.776001	−0.388000	−	0.921659i	$-0.626834\pi$
$179$	−10.3822	−0.776001	−0.388000	+	0.921659i	$0.626834\pi$
$180$	0	0
$181$	10.3492	0.769247	0.384624	−	0.923073i	$-0.374331\pi$
$181$	10.3492	0.769247	0.384624	+	0.923073i	$0.374331\pi$
$182$	0	0
$183$	−0.257489	−0.0190342
$184$	0	0
$185$	0.983586	0.0723147
$186$	0	0
$187$	−2.48052	−0.181393
$188$	0	0
$189$	0.469540	0.0341540
$190$	0	0
$191$	−15.5059	−1.12197	−0.560984	−	0.827826i	$-0.689577\pi$
$191$	−15.5059	−1.12197	−0.560984	+	0.827826i	$0.689577\pi$
$192$	0	0
$193$	−5.57028	−0.400958	−0.200479	−	0.979698i	$-0.564250\pi$
$193$	−5.57028	−0.400958	−0.200479	+	0.979698i	$0.564250\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−24.8728	−1.77212	−0.886058	−	0.463575i	$-0.846566\pi$
$197$	−24.8728	−1.77212	−0.886058	+	0.463575i	$0.846566\pi$
$198$	0	0
$199$	−18.6489	−1.32198	−0.660991	−	0.750393i	$-0.729864\pi$
$199$	−18.6489	−1.32198	−0.660991	+	0.750393i	$0.729864\pi$
$200$	0	0
$201$	1.30381	0.0919635
$202$	0	0
$203$	6.16418	0.432640
$204$	0	0
$205$	4.26795	0.298087
$206$	0	0
$207$	−9.22968	−0.641507
$208$	0	0
$209$	−1.85224	−0.128122
$210$	0	0
$211$	9.64469	0.663968	0.331984	−	0.943285i	$-0.392282\pi$
$211$	9.64469	0.663968	0.331984	+	0.943285i	$0.392282\pi$
$212$	0	0
$213$	0.926728	0.0634984
$214$	0	0
$215$	9.54092	0.650685
$216$	0	0
$217$	4.83143	0.327979
$218$	0	0
$219$	1.05978	0.0716134
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−1.16115	−0.0777561	−0.0388780	−	0.999244i	$-0.512378\pi$
$223$	−1.16115	−0.0777561	−0.0388780	+	0.999244i	$0.512378\pi$
$224$	0	0
$225$	−2.99102	−0.199401
$226$	0	0
$227$	27.2460	1.80838	0.904191	−	0.427129i	$-0.140475\pi$
$227$	27.2460	1.80838	0.904191	+	0.427129i	$0.140475\pi$
$228$	0	0
$229$	−24.3432	−1.60864	−0.804322	−	0.594193i	$-0.797471\pi$
$229$	−24.3432	−1.60864	−0.804322	+	0.594193i	$0.797471\pi$
$230$	0	0
$231$	0.135748	0.00893155
$232$	0	0
$233$	23.0238	1.50834	0.754171	−	0.656678i	$-0.228039\pi$
$233$	23.0238	1.50834	0.754171	+	0.656678i	$0.228039\pi$
$234$	0	0
$235$	3.46410	0.225973
$236$	0	0
$237$	0.0239232	0.00155398
$238$	0	0
$239$	23.7057	1.53340	0.766698	−	0.642008i	$-0.221898\pi$
$239$	23.7057	1.53340	0.766698	+	0.642008i	$0.221898\pi$
$240$	0	0
$241$	−10.8307	−0.697668	−0.348834	−	0.937185i	$-0.613422\pi$
$241$	−10.8307	−0.697668	−0.348834	+	0.937185i	$0.613422\pi$
$242$	0	0
$243$	−2.54905	−0.163522
$244$	0	0
$245$	6.31634	0.403536
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−0.537602	−0.0340691
$250$	0	0
$251$	−1.12081	−0.0707449	−0.0353724	−	0.999374i	$-0.511262\pi$
$251$	−1.12081	−0.0707449	−0.0353724	+	0.999374i	$0.511262\pi$
$252$	0	0
$253$	−5.34477	−0.336023
$254$	0	0
$255$	0.135748	0.00850086
$256$	0	0
$257$	16.6307	1.03739	0.518697	−	0.854958i	$-0.326417\pi$
$257$	16.6307	1.03739	0.518697	+	0.854958i	$0.326417\pi$
$258$	0	0
$259$	−0.813267	−0.0505340
$260$	0	0
$261$	−22.2984	−1.38023
$262$	0	0
$263$	24.5020	1.51085	0.755427	−	0.655232i	$-0.227429\pi$
$263$	24.5020	1.51085	0.755427	+	0.655232i	$0.227429\pi$
$264$	0	0
$265$	−0.334308	−0.0205364
$266$	0	0
$267$	0.435672	0.0266627
$268$	0	0
$269$	−6.53287	−0.398316	−0.199158	−	0.979967i	$-0.563821\pi$
$269$	−6.53287	−0.398316	−0.199158	+	0.979967i	$0.563821\pi$
$270$	0	0
$271$	−5.65608	−0.343583	−0.171791	−	0.985133i	$-0.554955\pi$
$271$	−5.65608	−0.343583	−0.171791	+	0.985133i	$0.554955\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.73205	−0.104447
$276$	0	0
$277$	3.71564	0.223251	0.111626	−	0.993750i	$-0.464394\pi$
$277$	3.71564	0.223251	0.111626	+	0.993750i	$0.464394\pi$
$278$	0	0
$279$	−17.4773	−1.04634
$280$	0	0
$281$	−9.70447	−0.578920	−0.289460	−	0.957190i	$-0.593476\pi$
$281$	−9.70447	−0.578920	−0.289460	+	0.957190i	$0.593476\pi$
$282$	0	0
$283$	−24.1976	−1.43840	−0.719200	−	0.694803i	$-0.755491\pi$
$283$	−24.1976	−1.43840	−0.719200	+	0.694803i	$0.755491\pi$
$284$	0	0
$285$	0.101365	0.00600433
$286$	0	0
$287$	−3.52890	−0.208305
$288$	0	0
$289$	−14.9490	−0.879354
$290$	0	0
$291$	0.903737	0.0529780
$292$	0	0
$293$	−3.62828	−0.211966	−0.105983	−	0.994368i	$-0.533799\pi$
$293$	−3.62828	−0.211966	−0.105983	+	0.994368i	$0.533799\pi$
$294$	0	0
$295$	11.5245	0.670983
$296$	0	0
$297$	−0.983586	−0.0570735
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−7.88880	−0.454703
$302$	0	0
$303$	0.549905	0.0315912
$304$	0	0
$305$	−2.71649	−0.155546
$306$	0	0
$307$	9.40129	0.536560	0.268280	−	0.963341i	$-0.413545\pi$
$307$	9.40129	0.536560	0.268280	+	0.963341i	$0.413545\pi$
$308$	0	0
$309$	0.955077	0.0543324
$310$	0	0
$311$	−25.5370	−1.44807	−0.724034	−	0.689764i	$-0.757714\pi$
$311$	−25.5370	−1.44807	−0.724034	+	0.689764i	$0.757714\pi$
$312$	0	0
$313$	5.25656	0.297118	0.148559	−	0.988904i	$-0.452536\pi$
$313$	5.25656	0.297118	0.148559	+	0.988904i	$0.452536\pi$
$314$	0	0
$315$	2.47309	0.139343
$316$	0	0
$317$	−14.1536	−0.794947	−0.397474	−	0.917614i	$-0.630113\pi$
$317$	−14.1536	−0.794947	−0.397474	+	0.917614i	$0.630113\pi$
$318$	0	0
$319$	−12.9126	−0.722969
$320$	0	0
$321$	−1.54310	−0.0861274
$322$	0	0
$323$	1.53150	0.0852150
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−0.296666	−0.0164056
$328$	0	0
$329$	−2.86425	−0.157911
$330$	0	0
$331$	18.5843	1.02148	0.510742	−	0.859734i	$-0.329371\pi$
$331$	18.5843	1.02148	0.510742	+	0.859734i	$0.329371\pi$
$332$	0	0
$333$	2.94192	0.161216
$334$	0	0
$335$	13.7550	0.751518
$336$	0	0
$337$	−22.4060	−1.22053	−0.610267	−	0.792196i	$-0.708938\pi$
$337$	−22.4060	−1.22053	−0.610267	+	0.792196i	$0.708938\pi$
$338$	0	0
$339$	0.964061	0.0523606
$340$	0	0
$341$	−10.1208	−0.548073
$342$	0	0
$343$	−11.0105	−0.594509
$344$	0	0
$345$	0.292496	0.0157474
$346$	0	0
$347$	20.1725	1.08291	0.541457	−	0.840728i	$-0.317873\pi$
$347$	20.1725	1.08291	0.541457	+	0.840728i	$0.317873\pi$
$348$	0	0
$349$	−28.5943	−1.53062	−0.765310	−	0.643662i	$-0.777414\pi$
$349$	−28.5943	−1.53062	−0.765310	+	0.643662i	$0.777414\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	11.1563	0.593792	0.296896	−	0.954910i	$-0.404049\pi$
$353$	11.1563	0.593792	0.296896	+	0.954910i	$0.404049\pi$
$354$	0	0
$355$	9.77689	0.518904
$356$	0	0
$357$	−0.112241	−0.00594045
$358$	0	0
$359$	−11.0490	−0.583145	−0.291572	−	0.956549i	$-0.594178\pi$
$359$	−11.0490	−0.583145	−0.291572	+	0.956549i	$0.594178\pi$
$360$	0	0
$361$	−17.8564	−0.939811
$362$	0	0
$363$	0.758301	0.0398005
$364$	0	0
$365$	11.1806	0.585219
$366$	0	0
$367$	24.4052	1.27394	0.636970	−	0.770889i	$-0.280188\pi$
$367$	24.4052	1.27394	0.636970	+	0.770889i	$0.280188\pi$
$368$	0	0
$369$	12.7655	0.664545
$370$	0	0
$371$	0.276418	0.0143509
$372$	0	0
$373$	−5.31132	−0.275010	−0.137505	−	0.990501i	$-0.543908\pi$
$373$	−5.31132	−0.275010	−0.137505	+	0.990501i	$0.543908\pi$
$374$	0	0
$375$	0.0947876	0.00489481
$376$	0	0
$377$	0	0
$378$	0	0
$379$	33.5405	1.72286	0.861430	−	0.507876i	$-0.169569\pi$
$379$	33.5405	1.72286	0.861430	+	0.507876i	$0.169569\pi$
$380$	0	0
$381$	−0.565695	−0.0289814
$382$	0	0
$383$	−23.8027	−1.21626	−0.608131	−	0.793836i	$-0.708081\pi$
$383$	−23.8027	−1.21626	−0.608131	+	0.793836i	$0.708081\pi$
$384$	0	0
$385$	1.43213	0.0729879
$386$	0	0
$387$	28.5370	1.45062
$388$	0	0
$389$	26.2787	1.33238	0.666191	−	0.745781i	$-0.267923\pi$
$389$	26.2787	1.33238	0.666191	+	0.745781i	$0.267923\pi$
$390$	0	0
$391$	4.41926	0.223492
$392$	0	0
$393$	−1.57601	−0.0794990
$394$	0	0
$395$	0.252387	0.0126990
$396$	0	0
$397$	33.2920	1.67088	0.835439	−	0.549584i	$-0.185214\pi$
$397$	33.2920	1.67088	0.835439	+	0.549584i	$0.185214\pi$
$398$	0	0
$399$	−0.0838123	−0.00419586
$400$	0	0
$401$	−14.1692	−0.707576	−0.353788	−	0.935326i	$-0.615107\pi$
$401$	−14.1692	−0.707576	−0.353788	+	0.935326i	$0.615107\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−8.91922	−0.443200
$406$	0	0
$407$	1.70362	0.0844454
$408$	0	0
$409$	−6.84983	−0.338702	−0.169351	−	0.985556i	$-0.554167\pi$
$409$	−6.84983	−0.338702	−0.169351	+	0.985556i	$0.554167\pi$
$410$	0	0
$411$	0.0383462	0.00189148
$412$	0	0
$413$	−9.52890	−0.468887
$414$	0	0
$415$	−5.67165	−0.278410
$416$	0	0
$417$	0.883073	0.0432443
$418$	0	0
$419$	−16.3822	−0.800322	−0.400161	−	0.916445i	$-0.631046\pi$
$419$	−16.3822	−0.800322	−0.400161	+	0.916445i	$0.631046\pi$
$420$	0	0
$421$	−21.7045	−1.05781	−0.528906	−	0.848681i	$-0.677397\pi$
$421$	−21.7045	−1.05781	−0.528906	+	0.848681i	$0.677397\pi$
$422$	0	0
$423$	10.3612	0.503778
$424$	0	0
$425$	1.43213	0.0694683
$426$	0	0
$427$	2.24610	0.108696
$428$	0	0
$429$	0	0
$430$	0	0
$431$	32.3888	1.56011	0.780056	−	0.625710i	$-0.215191\pi$
$431$	32.3888	1.56011	0.780056	+	0.625710i	$0.215191\pi$
$432$	0	0
$433$	28.7975	1.38392	0.691959	−	0.721937i	$-0.256748\pi$
$433$	28.7975	1.38392	0.691959	+	0.721937i	$0.256748\pi$
$434$	0	0
$435$	0.706653	0.0338814
$436$	0	0
$437$	3.29992	0.157857
$438$	0	0
$439$	−17.5998	−0.839995	−0.419997	−	0.907525i	$-0.637969\pi$
$439$	−17.5998	−0.839995	−0.419997	+	0.907525i	$0.637969\pi$
$440$	0	0
$441$	18.8923	0.899632
$442$	0	0
$443$	−14.4043	−0.684370	−0.342185	−	0.939633i	$-0.611167\pi$
$443$	−14.4043	−0.684370	−0.342185	+	0.939633i	$0.611167\pi$
$444$	0	0
$445$	4.59630	0.217886
$446$	0	0
$447$	1.03013	0.0487236
$448$	0	0
$449$	−2.98861	−0.141041	−0.0705206	−	0.997510i	$-0.522466\pi$
$449$	−2.98861	−0.141041	−0.0705206	+	0.997510i	$0.522466\pi$
$450$	0	0
$451$	7.39230	0.348090
$452$	0	0
$453$	−0.0939360	−0.00441350
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−26.6204	−1.24525	−0.622626	−	0.782520i	$-0.713934\pi$
$457$	−26.6204	−1.24525	−0.622626	+	0.782520i	$0.713934\pi$
$458$	0	0
$459$	0.813267	0.0379601
$460$	0	0
$461$	−8.38876	−0.390703	−0.195352	−	0.980733i	$-0.562585\pi$
$461$	−8.38876	−0.390703	−0.195352	+	0.980733i	$0.562585\pi$
$462$	0	0
$463$	21.3014	0.989960	0.494980	−	0.868904i	$-0.335175\pi$
$463$	21.3014	0.989960	0.494980	+	0.868904i	$0.335175\pi$
$464$	0	0
$465$	0.553868	0.0256850
$466$	0	0
$467$	−2.12392	−0.0982833	−0.0491417	−	0.998792i	$-0.515649\pi$
$467$	−2.12392	−0.0982833	−0.0491417	+	0.998792i	$0.515649\pi$
$468$	0	0
$469$	−11.3732	−0.525165
$470$	0	0
$471$	−1.66569	−0.0767511
$472$	0	0
$473$	16.5254	0.759837
$474$	0	0
$475$	1.06939	0.0490669
$476$	0	0
$477$	−0.999919	−0.0457832
$478$	0	0
$479$	29.0543	1.32752	0.663762	−	0.747944i	$-0.268959\pi$
$479$	29.0543	1.32752	0.663762	+	0.747944i	$0.268959\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−0.241847	−0.0110044
$484$	0	0
$485$	9.53434	0.432932
$486$	0	0
$487$	30.3190	1.37388	0.686941	−	0.726713i	$-0.258953\pi$
$487$	30.3190	1.37388	0.686941	+	0.726713i	$0.258953\pi$
$488$	0	0
$489$	1.63811	0.0740781
$490$	0	0
$491$	38.1519	1.72177	0.860884	−	0.508800i	$-0.169911\pi$
$491$	38.1519	1.72177	0.860884	+	0.508800i	$0.169911\pi$
$492$	0	0
$493$	10.6767	0.480853
$494$	0	0
$495$	−5.18059	−0.232850
$496$	0	0
$497$	−8.08391	−0.362613
$498$	0	0
$499$	−16.5179	−0.739444	−0.369722	−	0.929142i	$-0.620547\pi$
$499$	−16.5179	−0.739444	−0.369722	+	0.929142i	$0.620547\pi$
$500$	0	0
$501$	−0.290008	−0.0129566
$502$	0	0
$503$	11.7616	0.524425	0.262212	−	0.965010i	$-0.415548\pi$
$503$	11.7616	0.524425	0.262212	+	0.965010i	$0.415548\pi$
$504$	0	0
$505$	5.80144	0.258161
$506$	0	0
$507$	0	0
$508$	0	0
$509$	31.8617	1.41224	0.706122	−	0.708091i	$-0.250443\pi$
$509$	31.8617	1.41224	0.706122	+	0.708091i	$0.250443\pi$
$510$	0	0
$511$	−9.24454	−0.408954
$512$	0	0
$513$	0.607278	0.0268120
$514$	0	0
$515$	10.0760	0.444000
$516$	0	0
$517$	6.00000	0.263880
$518$	0	0
$519$	0.324184	0.0142301
$520$	0	0
$521$	−19.5013	−0.854367	−0.427183	−	0.904165i	$-0.640494\pi$
$521$	−19.5013	−0.854367	−0.427183	+	0.904165i	$0.640494\pi$
$522$	0	0
$523$	−44.5659	−1.94873	−0.974365	−	0.224971i	$-0.927771\pi$
$523$	−44.5659	−1.94873	−0.974365	+	0.224971i	$0.927771\pi$
$524$	0	0
$525$	−0.0783740	−0.00342052
$526$	0	0
$527$	8.36827	0.364528
$528$	0	0
$529$	−13.4778	−0.585992
$530$	0	0
$531$	34.4700	1.49587
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−16.2795	−0.703826
$536$	0	0
$537$	0.984102	0.0424671
$538$	0	0
$539$	10.9402	0.471229
$540$	0	0
$541$	3.74450	0.160989	0.0804943	−	0.996755i	$-0.474350\pi$
$541$	3.74450	0.160989	0.0804943	+	0.996755i	$0.474350\pi$
$542$	0	0
$543$	−0.980972	−0.0420976
$544$	0	0
$545$	−3.12979	−0.134066
$546$	0	0
$547$	38.3803	1.64102	0.820511	−	0.571630i	$-0.193689\pi$
$547$	38.3803	1.64102	0.820511	+	0.571630i	$0.193689\pi$
$548$	0	0
$549$	−8.12506	−0.346769
$550$	0	0
$551$	7.97242	0.339637
$552$	0	0
$553$	−0.208683	−0.00887412
$554$	0	0
$555$	−0.0932318	−0.00395747
$556$	0	0
$557$	26.6462	1.12903	0.564517	−	0.825421i	$-0.309062\pi$
$557$	26.6462	1.12903	0.564517	+	0.825421i	$0.309062\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0.235122	0.00992686
$562$	0	0
$563$	−16.6935	−0.703547	−0.351774	−	0.936085i	$-0.614421\pi$
$563$	−16.6935	−0.703547	−0.351774	+	0.936085i	$0.614421\pi$
$564$	0	0
$565$	10.1708	0.427887
$566$	0	0
$567$	7.37475	0.309710
$568$	0	0
$569$	−43.9240	−1.84139	−0.920694	−	0.390285i	$-0.872376\pi$
$569$	−43.9240	−1.84139	−0.920694	+	0.390285i	$0.872376\pi$
$570$	0	0
$571$	11.4641	0.479758	0.239879	−	0.970803i	$-0.422892\pi$
$571$	11.4641	0.479758	0.239879	+	0.970803i	$0.422892\pi$
$572$	0	0
$573$	1.46977	0.0614004
$574$	0	0
$575$	3.08580	0.128687
$576$	0	0
$577$	−44.9354	−1.87069	−0.935343	−	0.353743i	$-0.884909\pi$
$577$	−44.9354	−1.87069	−0.935343	+	0.353743i	$0.884909\pi$
$578$	0	0
$579$	0.527994	0.0219427
$580$	0	0
$581$	4.68953	0.194555
$582$	0	0
$583$	−0.579038	−0.0239813
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−24.5217	−1.01212	−0.506059	−	0.862499i	$-0.668898\pi$
$587$	−24.5217	−1.01212	−0.506059	+	0.862499i	$0.668898\pi$
$588$	0	0
$589$	6.24871	0.257474
$590$	0	0
$591$	2.35763	0.0969801
$592$	0	0
$593$	−18.7655	−0.770607	−0.385303	−	0.922790i	$-0.625903\pi$
$593$	−18.7655	−0.770607	−0.385303	+	0.922790i	$0.625903\pi$
$594$	0	0
$595$	−1.18414	−0.0485449
$596$	0	0
$597$	1.76768	0.0723464
$598$	0	0
$599$	−35.9293	−1.46803	−0.734015	−	0.679133i	$-0.762356\pi$
$599$	−35.9293	−1.46803	−0.734015	+	0.679133i	$0.762356\pi$
$600$	0	0
$601$	−39.7726	−1.62236	−0.811179	−	0.584798i	$-0.801174\pi$
$601$	−39.7726	−1.62236	−0.811179	+	0.584798i	$0.801174\pi$
$602$	0	0
$603$	41.1415	1.67541
$604$	0	0
$605$	8.00000	0.325246
$606$	0	0
$607$	33.4613	1.35815	0.679076	−	0.734068i	$-0.262381\pi$
$607$	33.4613	1.35815	0.679076	+	0.734068i	$0.262381\pi$
$608$	0	0
$609$	−0.584287	−0.0236765
$610$	0	0
$611$	0	0
$612$	0	0
$613$	19.7699	0.798499	0.399249	−	0.916842i	$-0.369271\pi$
$613$	19.7699	0.798499	0.399249	+	0.916842i	$0.369271\pi$
$614$	0	0
$615$	−0.404549	−0.0163130
$616$	0	0
$617$	19.5087	0.785391	0.392696	−	0.919668i	$-0.371543\pi$
$617$	19.5087	0.785391	0.392696	+	0.919668i	$0.371543\pi$
$618$	0	0
$619$	40.4640	1.62639	0.813193	−	0.581994i	$-0.197727\pi$
$619$	40.4640	1.62639	0.813193	+	0.581994i	$0.197727\pi$
$620$	0	0
$621$	1.75235	0.0703193
$622$	0	0
$623$	−3.80040	−0.152260
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0.175569	0.00701155
$628$	0	0
$629$	−1.40862	−0.0561653
$630$	0	0
$631$	−19.5961	−0.780109	−0.390054	−	0.920792i	$-0.627544\pi$
$631$	−19.5961	−0.780109	−0.390054	+	0.920792i	$0.627544\pi$
$632$	0	0
$633$	−0.914197	−0.0363361
$634$	0	0
$635$	−5.96802	−0.236834
$636$	0	0
$637$	0	0
$638$	0	0
$639$	29.2428	1.15683
$640$	0	0
$641$	31.0477	1.22631	0.613155	−	0.789963i	$-0.289900\pi$
$641$	31.0477	1.22631	0.613155	+	0.789963i	$0.289900\pi$
$642$	0	0
$643$	5.94659	0.234511	0.117255	−	0.993102i	$-0.462590\pi$
$643$	5.94659	0.234511	0.117255	+	0.993102i	$0.462590\pi$
$644$	0	0
$645$	−0.904361	−0.0356092
$646$	0	0
$647$	42.5020	1.67092	0.835462	−	0.549548i	$-0.185200\pi$
$647$	42.5020	1.67092	0.835462	+	0.549548i	$0.185200\pi$
$648$	0	0
$649$	19.9610	0.783539
$650$	0	0
$651$	−0.457959	−0.0179488
$652$	0	0
$653$	−30.6107	−1.19789	−0.598945	−	0.800790i	$-0.704413\pi$
$653$	−30.6107	−1.19789	−0.598945	+	0.800790i	$0.704413\pi$
$654$	0	0
$655$	−16.6267	−0.649660
$656$	0	0
$657$	33.4413	1.30467
$658$	0	0
$659$	−25.4383	−0.990935	−0.495467	−	0.868626i	$-0.665003\pi$
$659$	−25.4383	−0.990935	−0.495467	+	0.868626i	$0.665003\pi$
$660$	0	0
$661$	0.333603	0.0129757	0.00648784	−	0.999979i	$-0.497935\pi$
$661$	0.333603	0.0129757	0.00648784	+	0.999979i	$0.497935\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−0.884212	−0.0342883
$666$	0	0
$667$	23.0050	0.890758
$668$	0	0
$669$	0.110062	0.00425525
$670$	0	0
$671$	−4.70510	−0.181638
$672$	0	0
$673$	9.81412	0.378306	0.189153	−	0.981948i	$-0.439426\pi$
$673$	9.81412	0.378306	0.189153	+	0.981948i	$0.439426\pi$
$674$	0	0
$675$	0.567874	0.0218575
$676$	0	0
$677$	−23.2414	−0.893241	−0.446620	−	0.894724i	$-0.647373\pi$
$677$	−23.2414	−0.893241	−0.446620	+	0.894724i	$0.647373\pi$
$678$	0	0
$679$	−7.88336	−0.302536
$680$	0	0
$681$	−2.58258	−0.0989648
$682$	0	0
$683$	5.26710	0.201540	0.100770	−	0.994910i	$-0.467869\pi$
$683$	5.26710	0.201540	0.100770	+	0.994910i	$0.467869\pi$
$684$	0	0
$685$	0.404549	0.0154570
$686$	0	0
$687$	2.30743	0.0880341
$688$	0	0
$689$	0	0
$690$	0	0
$691$	19.7840	0.752618	0.376309	−	0.926494i	$-0.377193\pi$
$691$	19.7840	0.752618	0.376309	+	0.926494i	$0.377193\pi$
$692$	0	0
$693$	4.28351	0.162717
$694$	0	0
$695$	9.31634	0.353389
$696$	0	0
$697$	−6.11224	−0.231518
$698$	0	0
$699$	−2.18237	−0.0825450
$700$	0	0
$701$	−18.1256	−0.684595	−0.342298	−	0.939592i	$-0.611205\pi$
$701$	−18.1256	−0.684595	−0.342298	+	0.939592i	$0.611205\pi$
$702$	0	0
$703$	−1.05184	−0.0396708
$704$	0	0
$705$	−0.328354	−0.0123665
$706$	0	0
$707$	−4.79685	−0.180404
$708$	0	0
$709$	−25.8092	−0.969283	−0.484642	−	0.874713i	$-0.661050\pi$
$709$	−25.8092	−0.969283	−0.484642	+	0.874713i	$0.661050\pi$
$710$	0	0
$711$	0.754894	0.0283107
$712$	0	0
$713$	18.0311	0.675271
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−2.24701	−0.0839161
$718$	0	0
$719$	−9.02677	−0.336642	−0.168321	−	0.985732i	$-0.553834\pi$
$719$	−9.02677	−0.336642	−0.168321	+	0.985732i	$0.553834\pi$
$720$	0	0
$721$	−8.33120	−0.310270
$722$	0	0
$723$	1.02662	0.0381803
$724$	0	0
$725$	7.45512	0.276876
$726$	0	0
$727$	−24.8934	−0.923245	−0.461623	−	0.887076i	$-0.652733\pi$
$727$	−24.8934	−0.923245	−0.461623	+	0.887076i	$0.652733\pi$
$728$	0	0
$729$	−26.5160	−0.982075
$730$	0	0
$731$	−13.6638	−0.505374
$732$	0	0
$733$	−13.2793	−0.490484	−0.245242	−	0.969462i	$-0.578867\pi$
$733$	−13.2793	−0.490484	−0.245242	+	0.969462i	$0.578867\pi$
$734$	0	0
$735$	−0.598710	−0.0220838
$736$	0	0
$737$	23.8244	0.877584
$738$	0	0
$739$	−19.5902	−0.720636	−0.360318	−	0.932830i	$-0.617332\pi$
$739$	−19.5902	−0.720636	−0.360318	+	0.932830i	$0.617332\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	51.2460	1.88003	0.940017	−	0.341128i	$-0.110809\pi$
$743$	51.2460	1.88003	0.940017	+	0.341128i	$0.110809\pi$
$744$	0	0
$745$	10.8678	0.398165
$746$	0	0
$747$	−16.9640	−0.620680
$748$	0	0
$749$	13.4606	0.491838
$750$	0	0
$751$	−21.1985	−0.773543	−0.386772	−	0.922175i	$-0.626410\pi$
$751$	−21.1985	−0.773543	−0.386772	+	0.922175i	$0.626410\pi$
$752$	0	0
$753$	0.106239	0.00387156
$754$	0	0
$755$	−0.991015	−0.0360667
$756$	0	0
$757$	−32.9494	−1.19757	−0.598783	−	0.800911i	$-0.704349\pi$
$757$	−32.9494	−1.19757	−0.598783	+	0.800911i	$0.704349\pi$
$758$	0	0
$759$	0.506618	0.0183891
$760$	0	0
$761$	52.2349	1.89351	0.946756	−	0.321952i	$-0.104339\pi$
$761$	52.2349	1.89351	0.946756	+	0.321952i	$0.104339\pi$
$762$	0	0
$763$	2.58783	0.0936859
$764$	0	0
$765$	4.28351	0.154871
$766$	0	0
$767$	0	0
$768$	0	0
$769$	26.8719	0.969025	0.484513	−	0.874784i	$-0.338997\pi$
$769$	26.8719	0.969025	0.484513	+	0.874784i	$0.338997\pi$
$770$	0	0
$771$	−1.57638	−0.0567720
$772$	0	0
$773$	−11.9549	−0.429989	−0.214994	−	0.976615i	$-0.568973\pi$
$773$	−11.9549	−0.429989	−0.214994	+	0.976615i	$0.568973\pi$
$774$	0	0
$775$	5.84325	0.209896
$776$	0	0
$777$	0.0770876	0.00276550
$778$	0	0
$779$	−4.56410	−0.163526
$780$	0	0
$781$	16.9341	0.605949
$782$	0	0
$783$	4.23357	0.151295
$784$	0	0
$785$	−17.5729	−0.627204
$786$	0	0
$787$	34.4937	1.22957	0.614783	−	0.788696i	$-0.289243\pi$
$787$	34.4937	1.22957	0.614783	+	0.788696i	$0.289243\pi$
$788$	0	0
$789$	−2.32248	−0.0826825
$790$	0	0
$791$	−8.40957	−0.299010
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0.0316882	0.00112387
$796$	0	0
$797$	33.2049	1.17618	0.588089	−	0.808796i	$-0.299880\pi$
$797$	33.2049	1.17618	0.588089	+	0.808796i	$0.299880\pi$
$798$	0	0
$799$	−4.96103	−0.175509
$800$	0	0
$801$	13.7476	0.485748
$802$	0	0
$803$	19.3654	0.683388
$804$	0	0
$805$	−2.55146	−0.0899272
$806$	0	0
$807$	0.619235	0.0217981
$808$	0	0
$809$	−8.70277	−0.305973	−0.152987	−	0.988228i	$-0.548889\pi$
$809$	−8.70277	−0.305973	−0.152987	+	0.988228i	$0.548889\pi$
$810$	0	0
$811$	−7.69132	−0.270079	−0.135039	−	0.990840i	$-0.543116\pi$
$811$	−7.69132	−0.270079	−0.135039	+	0.990840i	$0.543116\pi$
$812$	0	0
$813$	0.536127	0.0188028
$814$	0	0
$815$	17.2820	0.605360
$816$	0	0
$817$	−10.2030	−0.356956
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−0.615411	−0.0214780	−0.0107390	−	0.999942i	$-0.503418\pi$
$821$	−0.615411	−0.0214780	−0.0107390	+	0.999942i	$0.503418\pi$
$822$	0	0
$823$	−22.3375	−0.778638	−0.389319	−	0.921103i	$-0.627290\pi$
$823$	−22.3375	−0.778638	−0.389319	+	0.921103i	$0.627290\pi$
$824$	0	0
$825$	0.164177	0.00571591
$826$	0	0
$827$	33.8701	1.17778	0.588890	−	0.808213i	$-0.299565\pi$
$827$	33.8701	1.17778	0.588890	+	0.808213i	$0.299565\pi$
$828$	0	0
$829$	−34.5293	−1.19925	−0.599626	−	0.800280i	$-0.704684\pi$
$829$	−34.5293	−1.19925	−0.599626	+	0.800280i	$0.704684\pi$
$830$	0	0
$831$	−0.352196	−0.0122176
$832$	0	0
$833$	−9.04579	−0.313418
$834$	0	0
$835$	−3.05955	−0.105880
$836$	0	0
$837$	3.31823	0.114695
$838$	0	0
$839$	−29.9513	−1.03404	−0.517018	−	0.855975i	$-0.672958\pi$
$839$	−29.9513	−1.03404	−0.517018	+	0.855975i	$0.672958\pi$
$840$	0	0
$841$	26.5788	0.916509
$842$	0	0
$843$	0.919864	0.0316818
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−6.61471	−0.227284
$848$	0	0
$849$	2.29363	0.0787173
$850$	0	0
$851$	−3.03515	−0.104044
$852$	0	0
$853$	−54.1009	−1.85238	−0.926189	−	0.377059i	$-0.876935\pi$
$853$	−54.1009	−1.85238	−0.926189	+	0.377059i	$0.876935\pi$
$854$	0	0
$855$	3.19856	0.109388
$856$	0	0
$857$	16.4383	0.561521	0.280761	−	0.959778i	$-0.409413\pi$
$857$	16.4383	0.561521	0.280761	+	0.959778i	$0.409413\pi$
$858$	0	0
$859$	32.7187	1.11635	0.558174	−	0.829724i	$-0.311502\pi$
$859$	32.7187	1.11635	0.558174	+	0.829724i	$0.311502\pi$
$860$	0	0
$861$	0.334496	0.0113996
$862$	0	0
$863$	−55.7922	−1.89919	−0.949594	−	0.313482i	$-0.898504\pi$
$863$	−55.7922	−1.89919	−0.949594	+	0.313482i	$0.898504\pi$
$864$	0	0
$865$	3.42011	0.116287
$866$	0	0
$867$	1.41698	0.0481232
$868$	0	0
$869$	0.437148	0.0148292
$870$	0	0
$871$	0	0
$872$	0	0
$873$	28.5174	0.965167
$874$	0	0
$875$	−0.826838	−0.0279522
$876$	0	0
$877$	38.6921	1.30654	0.653271	−	0.757125i	$-0.273396\pi$
$877$	38.6921	1.30654	0.653271	+	0.757125i	$0.273396\pi$
$878$	0	0
$879$	0.343916	0.0116000
$880$	0	0
$881$	−5.42115	−0.182643	−0.0913216	−	0.995821i	$-0.529109\pi$
$881$	−5.42115	−0.182643	−0.0913216	+	0.995821i	$0.529109\pi$
$882$	0	0
$883$	−21.2583	−0.715397	−0.357699	−	0.933837i	$-0.616439\pi$
$883$	−21.2583	−0.715397	−0.357699	+	0.933837i	$0.616439\pi$
$884$	0	0
$885$	−1.09238	−0.0367200
$886$	0	0
$887$	0.566171	0.0190101	0.00950507	−	0.999955i	$-0.496974\pi$
$887$	0.566171	0.0190101	0.00950507	+	0.999955i	$0.496974\pi$
$888$	0	0
$889$	4.93459	0.165501
$890$	0	0
$891$	−15.4485	−0.517546
$892$	0	0
$893$	−3.70447	−0.123965
$894$	0	0
$895$	10.3822	0.347038
$896$	0	0
$897$	0	0
$898$	0	0
$899$	43.5621	1.45288
$900$	0	0
$901$	0.478771	0.0159502
$902$	0	0
$903$	0.747760	0.0248839
$904$	0	0
$905$	−10.3492	−0.344018
$906$	0	0
$907$	37.0516	1.23028	0.615139	−	0.788419i	$-0.289100\pi$
$907$	37.0516	1.23028	0.615139	+	0.788419i	$0.289100\pi$
$908$	0	0
$909$	17.3522	0.575536
$910$	0	0
$911$	−27.9952	−0.927522	−0.463761	−	0.885960i	$-0.653500\pi$
$911$	−27.9952	−0.927522	−0.463761	+	0.885960i	$0.653500\pi$
$912$	0	0
$913$	−9.82358	−0.325113
$914$	0	0
$915$	0.257489	0.00851234
$916$	0	0
$917$	13.7476	0.453986
$918$	0	0
$919$	36.9441	1.21867	0.609337	−	0.792911i	$-0.291436\pi$
$919$	36.9441	1.21867	0.609337	+	0.792911i	$0.291436\pi$
$920$	0	0
$921$	−0.891126	−0.0293636
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−0.983586	−0.0323401
$926$	0	0
$927$	30.1374	0.989841
$928$	0	0
$929$	−13.3290	−0.437310	−0.218655	−	0.975802i	$-0.570167\pi$
$929$	−13.3290	−0.437310	−0.218655	+	0.975802i	$0.570167\pi$
$930$	0	0
$931$	−6.75462	−0.221374
$932$	0	0
$933$	2.42059	0.0792464
$934$	0	0
$935$	2.48052	0.0811215
$936$	0	0
$937$	13.0922	0.427702	0.213851	−	0.976866i	$-0.431399\pi$
$937$	13.0922	0.427702	0.213851	+	0.976866i	$0.431399\pi$
$938$	0	0
$939$	−0.498256	−0.0162600
$940$	0	0
$941$	43.0399	1.40306	0.701531	−	0.712639i	$-0.252500\pi$
$941$	43.0399	1.40306	0.701531	+	0.712639i	$0.252500\pi$
$942$	0	0
$943$	−13.1700	−0.428876
$944$	0	0
$945$	−0.469540	−0.0152741
$946$	0	0
$947$	30.4045	0.988015	0.494008	−	0.869458i	$-0.335532\pi$
$947$	30.4045	0.988015	0.494008	+	0.869458i	$0.335532\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	1.34159	0.0435040
$952$	0	0
$953$	−21.4022	−0.693287	−0.346643	−	0.937997i	$-0.612679\pi$
$953$	−21.4022	−0.693287	−0.346643	+	0.937997i	$0.612679\pi$
$954$	0	0
$955$	15.5059	0.501760
$956$	0	0
$957$	1.22396	0.0395649
$958$	0	0
$959$	−0.334496	−0.0108014
$960$	0	0
$961$	3.14359	0.101406
$962$	0	0
$963$	−48.6924	−1.56909
$964$	0	0
$965$	5.57028	0.179314
$966$	0	0
$967$	−41.8892	−1.34707	−0.673533	−	0.739157i	$-0.735224\pi$
$967$	−41.8892	−1.34707	−0.673533	+	0.739157i	$0.735224\pi$
$968$	0	0
$969$	−0.145167	−0.00466344
$970$	0	0
$971$	−28.8253	−0.925047	−0.462524	−	0.886607i	$-0.653056\pi$
$971$	−28.8253	−0.925047	−0.462524	+	0.886607i	$0.653056\pi$
$972$	0	0
$973$	−7.70311	−0.246950
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−54.4678	−1.74258	−0.871289	−	0.490770i	$-0.836716\pi$
$977$	−54.4678	−1.74258	−0.871289	+	0.490770i	$0.836716\pi$
$978$	0	0
$979$	7.96103	0.254436
$980$	0	0
$981$	−9.36126	−0.298882
$982$	0	0
$983$	56.5991	1.80523	0.902616	−	0.430447i	$-0.141644\pi$
$983$	56.5991	1.80523	0.902616	+	0.430447i	$0.141644\pi$
$984$	0	0
$985$	24.8728	0.792514
$986$	0	0
$987$	0.271496	0.00864180
$988$	0	0
$989$	−29.4414	−0.936182
$990$	0	0
$991$	22.6925	0.720850	0.360425	−	0.932788i	$-0.382632\pi$
$991$	22.6925	0.720850	0.360425	+	0.932788i	$0.382632\pi$
$992$	0	0
$993$	−1.76156	−0.0559014
$994$	0	0
$995$	18.6489	0.591209
$996$	0	0
$997$	46.7228	1.47972	0.739862	−	0.672758i	$-0.234891\pi$
$997$	46.7228	1.47972	0.739862	+	0.672758i	$0.234891\pi$
$998$	0	0
$999$	−0.558553	−0.0176718

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.2.a.p.1.2		4
13.2	odd	12		260.2.x.a.121.3	yes	8
13.5	odd	4		3380.2.f.i.3041.4		8
13.7	odd	12		260.2.x.a.101.3	✓	8
13.8	odd	4		3380.2.f.i.3041.3		8
13.12	even	2		3380.2.a.q.1.2		4
39.2	even	12		2340.2.dj.d.901.2		8
39.20	even	12		2340.2.dj.d.361.4		8
52.7	even	12		1040.2.da.c.881.2		8
52.15	even	12		1040.2.da.c.641.2		8
65.2	even	12		1300.2.ba.c.849.2		8
65.7	even	12		1300.2.ba.b.49.3		8
65.28	even	12		1300.2.ba.b.849.3		8
65.33	even	12		1300.2.ba.c.49.2		8
65.54	odd	12		1300.2.y.b.901.2		8
65.59	odd	12		1300.2.y.b.101.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.x.a.101.3	✓	8	13.7	odd	12
260.2.x.a.121.3	yes	8	13.2	odd	12
1040.2.da.c.641.2		8	52.15	even	12
1040.2.da.c.881.2		8	52.7	even	12
1300.2.y.b.101.2		8	65.59	odd	12
1300.2.y.b.901.2		8	65.54	odd	12
1300.2.ba.b.49.3		8	65.7	even	12
1300.2.ba.b.849.3		8	65.28	even	12
1300.2.ba.c.49.2		8	65.33	even	12
1300.2.ba.c.849.2		8	65.2	even	12
2340.2.dj.d.361.4		8	39.20	even	12
2340.2.dj.d.901.2		8	39.2	even	12
3380.2.a.p.1.2		4	1.1	even	1	trivial
3380.2.a.q.1.2		4	13.12	even	2
3380.2.f.i.3041.3		8	13.8	odd	4
3380.2.f.i.3041.4		8	13.5	odd	4

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:	\( N \)	\(=\)	\( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3380.a (trivial)

\(f(q)\)	\(=\)	\(q-0.0947876 q^{3} -1.00000 q^{5} +0.826838 q^{7} -2.99102 q^{9} +O(q^{10})\)	\(q-0.0947876 q^{3} -1.00000 q^{5} +0.826838 q^{7} -2.99102 q^{9} -1.73205 q^{11} +0.0947876 q^{15} +1.43213 q^{17} +1.06939 q^{19} -0.0783740 q^{21} +3.08580 q^{23} +1.00000 q^{25} +0.567874 q^{27} +7.45512 q^{29} +5.84325 q^{31} +0.164177 q^{33} -0.826838 q^{35} -0.983586 q^{37} -4.26795 q^{41} -9.54092 q^{43} +2.99102 q^{45} -3.46410 q^{47} -6.31634 q^{49} -0.135748 q^{51} +0.334308 q^{53} +1.73205 q^{55} -0.101365 q^{57} -11.5245 q^{59} +2.71649 q^{61} -2.47309 q^{63} -13.7550 q^{67} -0.292496 q^{69} -9.77689 q^{71} -11.1806 q^{73} -0.0947876 q^{75} -1.43213 q^{77} -0.252387 q^{79} +8.91922 q^{81} +5.67165 q^{83} -1.43213 q^{85} -0.706653 q^{87} -4.59630 q^{89} -0.553868 q^{93} -1.06939 q^{95} -9.53434 q^{97} +5.18059 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} - 4 q^{5} - 6 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 2 * q^3 - 4 * q^5 - 6 * q^7 + 4 * q^9	\( 4 q + 2 q^{3} - 4 q^{5} - 6 q^{7} + 4 q^{9} - 2 q^{15} + 6 q^{17} - 12 q^{21} - 6 q^{23} + 4 q^{25} + 2 q^{27} - 6 q^{33} + 6 q^{35} - 18 q^{37} - 24 q^{41} + 10 q^{43} - 4 q^{45} - 4 q^{49} + 12 q^{53} + 18 q^{57} - 12 q^{59} + 4 q^{61} - 12 q^{63} - 18 q^{67} - 24 q^{69} - 12 q^{71} - 24 q^{73} + 2 q^{75} - 6 q^{77} - 8 q^{79} - 8 q^{81} + 36 q^{83} - 6 q^{85} + 6 q^{87} - 12 q^{89} - 48 q^{93} - 6 q^{97}+O(q^{100}) \) 4 * q + 2 * q^3 - 4 * q^5 - 6 * q^7 + 4 * q^9 - 2 * q^15 + 6 * q^17 - 12 * q^21 - 6 * q^23 + 4 * q^25 + 2 * q^27 - 6 * q^33 + 6 * q^35 - 18 * q^37 - 24 * q^41 + 10 * q^43 - 4 * q^45 - 4 * q^49 + 12 * q^53 + 18 * q^57 - 12 * q^59 + 4 * q^61 - 12 * q^63 - 18 * q^67 - 24 * q^69 - 12 * q^71 - 24 * q^73 + 2 * q^75 - 6 * q^77 - 8 * q^79 - 8 * q^81 + 36 * q^83 - 6 * q^85 + 6 * q^87 - 12 * q^89 - 48 * q^93 - 6 * q^97

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