LMFDB - Embedded newform 3192.2.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3192,2,Mod(1,3192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3192, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3192.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3192.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:	$25.4882483252$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	3192.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -0.732051 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -0.732051 q^{5} +1.00000 q^{7} +1.00000 q^{9} +1.46410 q^{13} +0.732051 q^{15} -4.73205 q^{17} -1.00000 q^{19} -1.00000 q^{21} -4.46410 q^{25} -1.00000 q^{27} +7.66025 q^{29} +2.00000 q^{31} -0.732051 q^{35} -10.0000 q^{37} -1.46410 q^{39} +4.92820 q^{41} -8.92820 q^{43} -0.732051 q^{45} +5.66025 q^{47} +1.00000 q^{49} +4.73205 q^{51} -3.26795 q^{53} +1.00000 q^{57} +8.00000 q^{59} -2.00000 q^{61} +1.00000 q^{63} -1.07180 q^{65} -6.53590 q^{67} +1.26795 q^{71} -10.3923 q^{73} +4.46410 q^{75} +4.00000 q^{79} +1.00000 q^{81} -0.196152 q^{83} +3.46410 q^{85} -7.66025 q^{87} -8.92820 q^{89} +1.46410 q^{91} -2.00000 q^{93} +0.732051 q^{95} +2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} - 4 q^{13} - 2 q^{15} - 6 q^{17} - 2 q^{19} - 2 q^{21} - 2 q^{25} - 2 q^{27} - 2 q^{29} + 4 q^{31} + 2 q^{35} - 20 q^{37} + 4 q^{39} - 4 q^{41} - 4 q^{43} + 2 q^{45} - 6 q^{47} + 2 q^{49} + 6 q^{51} - 10 q^{53} + 2 q^{57} + 16 q^{59} - 4 q^{61} + 2 q^{63} - 16 q^{65} - 20 q^{67} + 6 q^{71} + 2 q^{75} + 8 q^{79} + 2 q^{81} + 10 q^{83} + 2 q^{87} - 4 q^{89} - 4 q^{91} - 4 q^{93} - 2 q^{95} + 4 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 - 4 * q^13 - 2 * q^15 - 6 * q^17 - 2 * q^19 - 2 * q^21 - 2 * q^25 - 2 * q^27 - 2 * q^29 + 4 * q^31 + 2 * q^35 - 20 * q^37 + 4 * q^39 - 4 * q^41 - 4 * q^43 + 2 * q^45 - 6 * q^47 + 2 * q^49 + 6 * q^51 - 10 * q^53 + 2 * q^57 + 16 * q^59 - 4 * q^61 + 2 * q^63 - 16 * q^65 - 20 * q^67 + 6 * q^71 + 2 * q^75 + 8 * q^79 + 2 * q^81 + 10 * q^83 + 2 * q^87 - 4 * q^89 - 4 * q^91 - 4 * q^93 - 2 * q^95 + 4 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−0.732051	−0.327383	−0.163692	−	0.986512i	$-0.552340\pi$
$5$	−0.732051	−0.327383	−0.163692	+	0.986512i	$0.552340\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	1.46410	0.406069	0.203034	−	0.979172i	$-0.434920\pi$
$13$	1.46410	0.406069	0.203034	+	0.979172i	$0.434920\pi$
$14$	0	0
$15$	0.732051	0.189015
$16$	0	0
$17$	−4.73205	−1.14769	−0.573845	−	0.818964i	$-0.694549\pi$
$17$	−4.73205	−1.14769	−0.573845	+	0.818964i	$0.694549\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−4.46410	−0.892820
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	7.66025	1.42247	0.711237	−	0.702953i	$-0.248135\pi$
$29$	7.66025	1.42247	0.711237	+	0.702953i	$0.248135\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.732051	−0.123739
$36$	0	0
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	0	0
$39$	−1.46410	−0.234444
$40$	0	0
$41$	4.92820	0.769656	0.384828	−	0.922988i	$-0.374261\pi$
$41$	4.92820	0.769656	0.384828	+	0.922988i	$0.374261\pi$
$42$	0	0
$43$	−8.92820	−1.36154	−0.680769	−	0.732498i	$-0.738354\pi$
$43$	−8.92820	−1.36154	−0.680769	+	0.732498i	$0.738354\pi$
$44$	0	0
$45$	−0.732051	−0.109128
$46$	0	0
$47$	5.66025	0.825633	0.412816	−	0.910814i	$-0.364545\pi$
$47$	5.66025	0.825633	0.412816	+	0.910814i	$0.364545\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	4.73205	0.662620
$52$	0	0
$53$	−3.26795	−0.448887	−0.224444	−	0.974487i	$-0.572056\pi$
$53$	−3.26795	−0.448887	−0.224444	+	0.974487i	$0.572056\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−1.07180	−0.132940
$66$	0	0
$67$	−6.53590	−0.798487	−0.399244	−	0.916845i	$-0.630727\pi$
$67$	−6.53590	−0.798487	−0.399244	+	0.916845i	$0.630727\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.26795	0.150478	0.0752389	−	0.997166i	$-0.476028\pi$
$71$	1.26795	0.150478	0.0752389	+	0.997166i	$0.476028\pi$
$72$	0	0
$73$	−10.3923	−1.21633	−0.608164	−	0.793812i	$-0.708094\pi$
$73$	−10.3923	−1.21633	−0.608164	+	0.793812i	$0.708094\pi$
$74$	0	0
$75$	4.46410	0.515470
$76$	0	0
$77$	0	0
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−0.196152	−0.0215305	−0.0107653	−	0.999942i	$-0.503427\pi$
$83$	−0.196152	−0.0215305	−0.0107653	+	0.999942i	$0.503427\pi$
$84$	0	0
$85$	3.46410	0.375735
$86$	0	0
$87$	−7.66025	−0.821265
$88$	0	0
$89$	−8.92820	−0.946388	−0.473194	−	0.880958i	$-0.656899\pi$
$89$	−8.92820	−0.946388	−0.473194	+	0.880958i	$0.656899\pi$
$90$	0	0
$91$	1.46410	0.153480
$92$	0	0
$93$	−2.00000	−0.207390
$94$	0	0
$95$	0.732051	0.0751068
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	4.73205	0.470857	0.235428	−	0.971892i	$-0.424351\pi$
$101$	4.73205	0.470857	0.235428	+	0.971892i	$0.424351\pi$
$102$	0	0
$103$	10.9282	1.07679	0.538394	−	0.842693i	$-0.319031\pi$
$103$	10.9282	1.07679	0.538394	+	0.842693i	$0.319031\pi$
$104$	0	0
$105$	0.732051	0.0714408
$106$	0	0
$107$	−1.26795	−0.122577	−0.0612886	−	0.998120i	$-0.519521\pi$
$107$	−1.26795	−0.122577	−0.0612886	+	0.998120i	$0.519521\pi$
$108$	0	0
$109$	−18.3923	−1.76166	−0.880832	−	0.473430i	$-0.843016\pi$
$109$	−18.3923	−1.76166	−0.880832	+	0.473430i	$0.843016\pi$
$110$	0	0
$111$	10.0000	0.949158
$112$	0	0
$113$	−10.1962	−0.959173	−0.479587	−	0.877495i	$-0.659213\pi$
$113$	−10.1962	−0.959173	−0.479587	+	0.877495i	$0.659213\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.46410	0.135356
$118$	0	0
$119$	−4.73205	−0.433786
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	−4.92820	−0.444361
$124$	0	0
$125$	6.92820	0.619677
$126$	0	0
$127$	−0.392305	−0.0348114	−0.0174057	−	0.999849i	$-0.505541\pi$
$127$	−0.392305	−0.0348114	−0.0174057	+	0.999849i	$0.505541\pi$
$128$	0	0
$129$	8.92820	0.786084
$130$	0	0
$131$	−16.5885	−1.44934	−0.724670	−	0.689096i	$-0.758008\pi$
$131$	−16.5885	−1.44934	−0.724670	+	0.689096i	$0.758008\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	0.732051	0.0630049
$136$	0	0
$137$	−0.928203	−0.0793018	−0.0396509	−	0.999214i	$-0.512625\pi$
$137$	−0.928203	−0.0793018	−0.0396509	+	0.999214i	$0.512625\pi$
$138$	0	0
$139$	19.3205	1.63874	0.819372	−	0.573262i	$-0.194322\pi$
$139$	19.3205	1.63874	0.819372	+	0.573262i	$0.194322\pi$
$140$	0	0
$141$	−5.66025	−0.476679
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−5.60770	−0.465694
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−6.39230	−0.523678	−0.261839	−	0.965112i	$-0.584329\pi$
$149$	−6.39230	−0.523678	−0.261839	+	0.965112i	$0.584329\pi$
$150$	0	0
$151$	−9.46410	−0.770178	−0.385089	−	0.922880i	$-0.625829\pi$
$151$	−9.46410	−0.770178	−0.385089	+	0.922880i	$0.625829\pi$
$152$	0	0
$153$	−4.73205	−0.382564
$154$	0	0
$155$	−1.46410	−0.117599
$156$	0	0
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	0	0
$159$	3.26795	0.259165
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−20.9282	−1.63922	−0.819612	−	0.572919i	$-0.805811\pi$
$163$	−20.9282	−1.63922	−0.819612	+	0.572919i	$0.805811\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−23.3205	−1.80460	−0.902298	−	0.431114i	$-0.858121\pi$
$167$	−23.3205	−1.80460	−0.902298	+	0.431114i	$0.858121\pi$
$168$	0	0
$169$	−10.8564	−0.835108
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	−3.07180	−0.233544	−0.116772	−	0.993159i	$-0.537255\pi$
$173$	−3.07180	−0.233544	−0.116772	+	0.993159i	$0.537255\pi$
$174$	0	0
$175$	−4.46410	−0.337454
$176$	0	0
$177$	−8.00000	−0.601317
$178$	0	0
$179$	−3.80385	−0.284313	−0.142156	−	0.989844i	$-0.545404\pi$
$179$	−3.80385	−0.284313	−0.142156	+	0.989844i	$0.545404\pi$
$180$	0	0
$181$	−8.92820	−0.663628	−0.331814	−	0.943345i	$-0.607661\pi$
$181$	−8.92820	−0.663628	−0.331814	+	0.943345i	$0.607661\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	0	0
$185$	7.32051	0.538214
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−1.07180	−0.0775525	−0.0387762	−	0.999248i	$-0.512346\pi$
$191$	−1.07180	−0.0775525	−0.0387762	+	0.999248i	$0.512346\pi$
$192$	0	0
$193$	3.85641	0.277590	0.138795	−	0.990321i	$-0.455677\pi$
$193$	3.85641	0.277590	0.138795	+	0.990321i	$0.455677\pi$
$194$	0	0
$195$	1.07180	0.0767530
$196$	0	0
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	0	0
$199$	8.39230	0.594915	0.297457	−	0.954735i	$-0.403861\pi$
$199$	8.39230	0.594915	0.297457	+	0.954735i	$0.403861\pi$
$200$	0	0
$201$	6.53590	0.461007
$202$	0	0
$203$	7.66025	0.537644
$204$	0	0
$205$	−3.60770	−0.251972
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	0	0
$213$	−1.26795	−0.0868784
$214$	0	0
$215$	6.53590	0.445745
$216$	0	0
$217$	2.00000	0.135769
$218$	0	0
$219$	10.3923	0.702247
$220$	0	0
$221$	−6.92820	−0.466041
$222$	0	0
$223$	−7.07180	−0.473563	−0.236781	−	0.971563i	$-0.576092\pi$
$223$	−7.07180	−0.473563	−0.236781	+	0.971563i	$0.576092\pi$
$224$	0	0
$225$	−4.46410	−0.297607
$226$	0	0
$227$	−4.39230	−0.291528	−0.145764	−	0.989319i	$-0.546564\pi$
$227$	−4.39230	−0.291528	−0.145764	+	0.989319i	$0.546564\pi$
$228$	0	0
$229$	−20.2487	−1.33807	−0.669036	−	0.743230i	$-0.733293\pi$
$229$	−20.2487	−1.33807	−0.669036	+	0.743230i	$0.733293\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	18.3923	1.20492	0.602460	−	0.798149i	$-0.294187\pi$
$233$	18.3923	1.20492	0.602460	+	0.798149i	$0.294187\pi$
$234$	0	0
$235$	−4.14359	−0.270298
$236$	0	0
$237$	−4.00000	−0.259828
$238$	0	0
$239$	16.3923	1.06033	0.530165	−	0.847894i	$-0.322130\pi$
$239$	16.3923	1.06033	0.530165	+	0.847894i	$0.322130\pi$
$240$	0	0
$241$	0.928203	0.0597908	0.0298954	−	0.999553i	$-0.490483\pi$
$241$	0.928203	0.0597908	0.0298954	+	0.999553i	$0.490483\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−0.732051	−0.0467690
$246$	0	0
$247$	−1.46410	−0.0931586
$248$	0	0
$249$	0.196152	0.0124307
$250$	0	0
$251$	17.6603	1.11471	0.557353	−	0.830276i	$-0.311817\pi$
$251$	17.6603	1.11471	0.557353	+	0.830276i	$0.311817\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−3.46410	−0.216930
$256$	0	0
$257$	4.53590	0.282942	0.141471	−	0.989942i	$-0.454817\pi$
$257$	4.53590	0.282942	0.141471	+	0.989942i	$0.454817\pi$
$258$	0	0
$259$	−10.0000	−0.621370
$260$	0	0
$261$	7.66025	0.474158
$262$	0	0
$263$	15.3205	0.944703	0.472351	−	0.881410i	$-0.343405\pi$
$263$	15.3205	0.944703	0.472351	+	0.881410i	$0.343405\pi$
$264$	0	0
$265$	2.39230	0.146958
$266$	0	0
$267$	8.92820	0.546397
$268$	0	0
$269$	−31.1769	−1.90089	−0.950445	−	0.310893i	$-0.899372\pi$
$269$	−31.1769	−1.90089	−0.950445	+	0.310893i	$0.899372\pi$
$270$	0	0
$271$	12.7846	0.776610	0.388305	−	0.921531i	$-0.373061\pi$
$271$	12.7846	0.776610	0.388305	+	0.921531i	$0.373061\pi$
$272$	0	0
$273$	−1.46410	−0.0886115
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−8.39230	−0.504245	−0.252122	−	0.967695i	$-0.581129\pi$
$277$	−8.39230	−0.504245	−0.252122	+	0.967695i	$0.581129\pi$
$278$	0	0
$279$	2.00000	0.119737
$280$	0	0
$281$	6.19615	0.369631	0.184816	−	0.982773i	$-0.440831\pi$
$281$	6.19615	0.369631	0.184816	+	0.982773i	$0.440831\pi$
$282$	0	0
$283$	0.392305	0.0233201	0.0116601	−	0.999932i	$-0.496288\pi$
$283$	0.392305	0.0233201	0.0116601	+	0.999932i	$0.496288\pi$
$284$	0	0
$285$	−0.732051	−0.0433629
$286$	0	0
$287$	4.92820	0.290903
$288$	0	0
$289$	5.39230	0.317194
$290$	0	0
$291$	−2.00000	−0.117242
$292$	0	0
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	0	0
$295$	−5.85641	−0.340973
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−8.92820	−0.514613
$302$	0	0
$303$	−4.73205	−0.271849
$304$	0	0
$305$	1.46410	0.0838342
$306$	0	0
$307$	−22.7846	−1.30039	−0.650193	−	0.759769i	$-0.725312\pi$
$307$	−22.7846	−1.30039	−0.650193	+	0.759769i	$0.725312\pi$
$308$	0	0
$309$	−10.9282	−0.621684
$310$	0	0
$311$	−12.5885	−0.713826	−0.356913	−	0.934138i	$-0.616171\pi$
$311$	−12.5885	−0.713826	−0.356913	+	0.934138i	$0.616171\pi$
$312$	0	0
$313$	−2.00000	−0.113047	−0.0565233	−	0.998401i	$-0.518002\pi$
$313$	−2.00000	−0.113047	−0.0565233	+	0.998401i	$0.518002\pi$
$314$	0	0
$315$	−0.732051	−0.0412464
$316$	0	0
$317$	1.80385	0.101314	0.0506571	−	0.998716i	$-0.483868\pi$
$317$	1.80385	0.101314	0.0506571	+	0.998716i	$0.483868\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	1.26795	0.0707700
$322$	0	0
$323$	4.73205	0.263298
$324$	0	0
$325$	−6.53590	−0.362546
$326$	0	0
$327$	18.3923	1.01710
$328$	0	0
$329$	5.66025	0.312060
$330$	0	0
$331$	−26.2487	−1.44276	−0.721380	−	0.692540i	$-0.756492\pi$
$331$	−26.2487	−1.44276	−0.721380	+	0.692540i	$0.756492\pi$
$332$	0	0
$333$	−10.0000	−0.547997
$334$	0	0
$335$	4.78461	0.261411
$336$	0	0
$337$	9.60770	0.523365	0.261682	−	0.965154i	$-0.415723\pi$
$337$	9.60770	0.523365	0.261682	+	0.965154i	$0.415723\pi$
$338$	0	0
$339$	10.1962	0.553779
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.39230	0.235791	0.117896	−	0.993026i	$-0.462385\pi$
$347$	4.39230	0.235791	0.117896	+	0.993026i	$0.462385\pi$
$348$	0	0
$349$	−32.9282	−1.76261	−0.881303	−	0.472551i	$-0.843333\pi$
$349$	−32.9282	−1.76261	−0.881303	+	0.472551i	$0.843333\pi$
$350$	0	0
$351$	−1.46410	−0.0781480
$352$	0	0
$353$	5.80385	0.308908	0.154454	−	0.988000i	$-0.450638\pi$
$353$	5.80385	0.308908	0.154454	+	0.988000i	$0.450638\pi$
$354$	0	0
$355$	−0.928203	−0.0492639
$356$	0	0
$357$	4.73205	0.250447
$358$	0	0
$359$	17.0718	0.901015	0.450507	−	0.892773i	$-0.351243\pi$
$359$	17.0718	0.901015	0.450507	+	0.892773i	$0.351243\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	11.0000	0.577350
$364$	0	0
$365$	7.60770	0.398205
$366$	0	0
$367$	25.4641	1.32922	0.664608	−	0.747193i	$-0.268599\pi$
$367$	25.4641	1.32922	0.664608	+	0.747193i	$0.268599\pi$
$368$	0	0
$369$	4.92820	0.256552
$370$	0	0
$371$	−3.26795	−0.169663
$372$	0	0
$373$	−12.9282	−0.669397	−0.334698	−	0.942325i	$-0.608634\pi$
$373$	−12.9282	−0.669397	−0.334698	+	0.942325i	$0.608634\pi$
$374$	0	0
$375$	−6.92820	−0.357771
$376$	0	0
$377$	11.2154	0.577622
$378$	0	0
$379$	19.3205	0.992428	0.496214	−	0.868200i	$-0.334723\pi$
$379$	19.3205	0.992428	0.496214	+	0.868200i	$0.334723\pi$
$380$	0	0
$381$	0.392305	0.0200984
$382$	0	0
$383$	23.3205	1.19162	0.595811	−	0.803125i	$-0.296831\pi$
$383$	23.3205	1.19162	0.595811	+	0.803125i	$0.296831\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−8.92820	−0.453846
$388$	0	0
$389$	−27.4641	−1.39249	−0.696243	−	0.717807i	$-0.745146\pi$
$389$	−27.4641	−1.39249	−0.696243	+	0.717807i	$0.745146\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	16.5885	0.836777
$394$	0	0
$395$	−2.92820	−0.147334
$396$	0	0
$397$	33.7128	1.69200	0.845999	−	0.533185i	$-0.179005\pi$
$397$	33.7128	1.69200	0.845999	+	0.533185i	$0.179005\pi$
$398$	0	0
$399$	1.00000	0.0500626
$400$	0	0
$401$	−10.1962	−0.509172	−0.254586	−	0.967050i	$-0.581939\pi$
$401$	−10.1962	−0.509172	−0.254586	+	0.967050i	$0.581939\pi$
$402$	0	0
$403$	2.92820	0.145864
$404$	0	0
$405$	−0.732051	−0.0363759
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−7.60770	−0.376176	−0.188088	−	0.982152i	$-0.560229\pi$
$409$	−7.60770	−0.376176	−0.188088	+	0.982152i	$0.560229\pi$
$410$	0	0
$411$	0.928203	0.0457849
$412$	0	0
$413$	8.00000	0.393654
$414$	0	0
$415$	0.143594	0.00704873
$416$	0	0
$417$	−19.3205	−0.946129
$418$	0	0
$419$	17.2679	0.843595	0.421797	−	0.906690i	$-0.361399\pi$
$419$	17.2679	0.843595	0.421797	+	0.906690i	$0.361399\pi$
$420$	0	0
$421$	16.2487	0.791914	0.395957	−	0.918269i	$-0.370413\pi$
$421$	16.2487	0.791914	0.395957	+	0.918269i	$0.370413\pi$
$422$	0	0
$423$	5.66025	0.275211
$424$	0	0
$425$	21.1244	1.02468
$426$	0	0
$427$	−2.00000	−0.0967868
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−38.0526	−1.83293	−0.916464	−	0.400118i	$-0.868969\pi$
$431$	−38.0526	−1.83293	−0.916464	+	0.400118i	$0.868969\pi$
$432$	0	0
$433$	11.8564	0.569783	0.284891	−	0.958560i	$-0.408043\pi$
$433$	11.8564	0.569783	0.284891	+	0.958560i	$0.408043\pi$
$434$	0	0
$435$	5.60770	0.268868
$436$	0	0
$437$	0	0
$438$	0	0
$439$	8.14359	0.388673	0.194336	−	0.980935i	$-0.437745\pi$
$439$	8.14359	0.388673	0.194336	+	0.980935i	$0.437745\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−41.4641	−1.97002	−0.985009	−	0.172500i	$-0.944815\pi$
$443$	−41.4641	−1.97002	−0.985009	+	0.172500i	$0.944815\pi$
$444$	0	0
$445$	6.53590	0.309831
$446$	0	0
$447$	6.39230	0.302346
$448$	0	0
$449$	8.73205	0.412091	0.206045	−	0.978542i	$-0.433941\pi$
$449$	8.73205	0.412091	0.206045	+	0.978542i	$0.433941\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	9.46410	0.444662
$454$	0	0
$455$	−1.07180	−0.0502466
$456$	0	0
$457$	0.392305	0.0183512	0.00917562	−	0.999958i	$-0.497079\pi$
$457$	0.392305	0.0183512	0.00917562	+	0.999958i	$0.497079\pi$
$458$	0	0
$459$	4.73205	0.220873
$460$	0	0
$461$	31.6603	1.47457	0.737283	−	0.675585i	$-0.236109\pi$
$461$	31.6603	1.47457	0.737283	+	0.675585i	$0.236109\pi$
$462$	0	0
$463$	−5.85641	−0.272170	−0.136085	−	0.990697i	$-0.543452\pi$
$463$	−5.85641	−0.272170	−0.136085	+	0.990697i	$0.543452\pi$
$464$	0	0
$465$	1.46410	0.0678961
$466$	0	0
$467$	38.0526	1.76086	0.880431	−	0.474174i	$-0.157253\pi$
$467$	38.0526	1.76086	0.880431	+	0.474174i	$0.157253\pi$
$468$	0	0
$469$	−6.53590	−0.301800
$470$	0	0
$471$	14.0000	0.645086
$472$	0	0
$473$	0	0
$474$	0	0
$475$	4.46410	0.204827
$476$	0	0
$477$	−3.26795	−0.149629
$478$	0	0
$479$	10.3397	0.472435	0.236218	−	0.971700i	$-0.424092\pi$
$479$	10.3397	0.472435	0.236218	+	0.971700i	$0.424092\pi$
$480$	0	0
$481$	−14.6410	−0.667573
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.46410	−0.0664814
$486$	0	0
$487$	−1.85641	−0.0841218	−0.0420609	−	0.999115i	$-0.513392\pi$
$487$	−1.85641	−0.0841218	−0.0420609	+	0.999115i	$0.513392\pi$
$488$	0	0
$489$	20.9282	0.946406
$490$	0	0
$491$	−15.3205	−0.691405	−0.345702	−	0.938344i	$-0.612359\pi$
$491$	−15.3205	−0.691405	−0.345702	+	0.938344i	$0.612359\pi$
$492$	0	0
$493$	−36.2487	−1.63256
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.26795	0.0568753
$498$	0	0
$499$	17.8564	0.799363	0.399681	−	0.916654i	$-0.369121\pi$
$499$	17.8564	0.799363	0.399681	+	0.916654i	$0.369121\pi$
$500$	0	0
$501$	23.3205	1.04188
$502$	0	0
$503$	1.26795	0.0565351	0.0282675	−	0.999600i	$-0.491001\pi$
$503$	1.26795	0.0565351	0.0282675	+	0.999600i	$0.491001\pi$
$504$	0	0
$505$	−3.46410	−0.154150
$506$	0	0
$507$	10.8564	0.482150
$508$	0	0
$509$	40.2487	1.78399	0.891996	−	0.452043i	$-0.149304\pi$
$509$	40.2487	1.78399	0.891996	+	0.452043i	$0.149304\pi$
$510$	0	0
$511$	−10.3923	−0.459728
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	−8.00000	−0.352522
$516$	0	0
$517$	0	0
$518$	0	0
$519$	3.07180	0.134837
$520$	0	0
$521$	−12.2487	−0.536626	−0.268313	−	0.963332i	$-0.586466\pi$
$521$	−12.2487	−0.536626	−0.268313	+	0.963332i	$0.586466\pi$
$522$	0	0
$523$	−22.0000	−0.961993	−0.480996	−	0.876723i	$-0.659725\pi$
$523$	−22.0000	−0.961993	−0.480996	+	0.876723i	$0.659725\pi$
$524$	0	0
$525$	4.46410	0.194829
$526$	0	0
$527$	−9.46410	−0.412263
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	8.00000	0.347170
$532$	0	0
$533$	7.21539	0.312533
$534$	0	0
$535$	0.928203	0.0401297
$536$	0	0
$537$	3.80385	0.164148
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	0	0
$543$	8.92820	0.383146
$544$	0	0
$545$	13.4641	0.576739
$546$	0	0
$547$	43.7128	1.86902	0.934512	−	0.355930i	$-0.115836\pi$
$547$	43.7128	1.86902	0.934512	+	0.355930i	$0.115836\pi$
$548$	0	0
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	−7.66025	−0.326338
$552$	0	0
$553$	4.00000	0.170097
$554$	0	0
$555$	−7.32051	−0.310738
$556$	0	0
$557$	27.4641	1.16369	0.581846	−	0.813299i	$-0.302331\pi$
$557$	27.4641	1.16369	0.581846	+	0.813299i	$0.302331\pi$
$558$	0	0
$559$	−13.0718	−0.552878
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−18.5359	−0.781195	−0.390597	−	0.920562i	$-0.627732\pi$
$563$	−18.5359	−0.781195	−0.390597	+	0.920562i	$0.627732\pi$
$564$	0	0
$565$	7.46410	0.314017
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−2.58846	−0.108514	−0.0542569	−	0.998527i	$-0.517279\pi$
$569$	−2.58846	−0.108514	−0.0542569	+	0.998527i	$0.517279\pi$
$570$	0	0
$571$	1.85641	0.0776882	0.0388441	−	0.999245i	$-0.487632\pi$
$571$	1.85641	0.0776882	0.0388441	+	0.999245i	$0.487632\pi$
$572$	0	0
$573$	1.07180	0.0447750
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−10.0000	−0.416305	−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000	−0.416305	−0.208153	+	0.978096i	$0.566745\pi$
$578$	0	0
$579$	−3.85641	−0.160267
$580$	0	0
$581$	−0.196152	−0.00813777
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−1.07180	−0.0443133
$586$	0	0
$587$	−39.9090	−1.64722	−0.823610	−	0.567157i	$-0.808043\pi$
$587$	−39.9090	−1.64722	−0.823610	+	0.567157i	$0.808043\pi$
$588$	0	0
$589$	−2.00000	−0.0824086
$590$	0	0
$591$	18.0000	0.740421
$592$	0	0
$593$	38.1962	1.56853	0.784264	−	0.620427i	$-0.213041\pi$
$593$	38.1962	1.56853	0.784264	+	0.620427i	$0.213041\pi$
$594$	0	0
$595$	3.46410	0.142014
$596$	0	0
$597$	−8.39230	−0.343474
$598$	0	0
$599$	−10.7321	−0.438500	−0.219250	−	0.975669i	$-0.570361\pi$
$599$	−10.7321	−0.438500	−0.219250	+	0.975669i	$0.570361\pi$
$600$	0	0
$601$	34.0000	1.38689	0.693444	−	0.720510i	$-0.256092\pi$
$601$	34.0000	1.38689	0.693444	+	0.720510i	$0.256092\pi$
$602$	0	0
$603$	−6.53590	−0.266162
$604$	0	0
$605$	8.05256	0.327383
$606$	0	0
$607$	−26.9282	−1.09298	−0.546491	−	0.837465i	$-0.684037\pi$
$607$	−26.9282	−1.09298	−0.546491	+	0.837465i	$0.684037\pi$
$608$	0	0
$609$	−7.66025	−0.310409
$610$	0	0
$611$	8.28719	0.335264
$612$	0	0
$613$	30.2487	1.22173	0.610867	−	0.791733i	$-0.290821\pi$
$613$	30.2487	1.22173	0.610867	+	0.791733i	$0.290821\pi$
$614$	0	0
$615$	3.60770	0.145476
$616$	0	0
$617$	44.6410	1.79718	0.898590	−	0.438790i	$-0.144593\pi$
$617$	44.6410	1.79718	0.898590	+	0.438790i	$0.144593\pi$
$618$	0	0
$619$	49.1769	1.97659	0.988294	−	0.152564i	$-0.0487531\pi$
$619$	49.1769	1.97659	0.988294	+	0.152564i	$0.0487531\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−8.92820	−0.357701
$624$	0	0
$625$	17.2487	0.689948
$626$	0	0
$627$	0	0
$628$	0	0
$629$	47.3205	1.88679
$630$	0	0
$631$	27.0718	1.07771	0.538856	−	0.842398i	$-0.318857\pi$
$631$	27.0718	1.07771	0.538856	+	0.842398i	$0.318857\pi$
$632$	0	0
$633$	−4.00000	−0.158986
$634$	0	0
$635$	0.287187	0.0113967
$636$	0	0
$637$	1.46410	0.0580098
$638$	0	0
$639$	1.26795	0.0501593
$640$	0	0
$641$	−23.6603	−0.934524	−0.467262	−	0.884119i	$-0.654759\pi$
$641$	−23.6603	−0.934524	−0.467262	+	0.884119i	$0.654759\pi$
$642$	0	0
$643$	−0.392305	−0.0154710	−0.00773550	−	0.999970i	$-0.502462\pi$
$643$	−0.392305	−0.0154710	−0.00773550	+	0.999970i	$0.502462\pi$
$644$	0	0
$645$	−6.53590	−0.257351
$646$	0	0
$647$	13.6603	0.537040	0.268520	−	0.963274i	$-0.413465\pi$
$647$	13.6603	0.537040	0.268520	+	0.963274i	$0.413465\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−2.00000	−0.0783862
$652$	0	0
$653$	−1.21539	−0.0475619	−0.0237809	−	0.999717i	$-0.507570\pi$
$653$	−1.21539	−0.0475619	−0.0237809	+	0.999717i	$0.507570\pi$
$654$	0	0
$655$	12.1436	0.474489
$656$	0	0
$657$	−10.3923	−0.405442
$658$	0	0
$659$	−6.05256	−0.235774	−0.117887	−	0.993027i	$-0.537612\pi$
$659$	−6.05256	−0.235774	−0.117887	+	0.993027i	$0.537612\pi$
$660$	0	0
$661$	−13.4641	−0.523693	−0.261846	−	0.965110i	$-0.584331\pi$
$661$	−13.4641	−0.523693	−0.261846	+	0.965110i	$0.584331\pi$
$662$	0	0
$663$	6.92820	0.269069
$664$	0	0
$665$	0.732051	0.0283877
$666$	0	0
$667$	0	0
$668$	0	0
$669$	7.07180	0.273411
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−20.9282	−0.806723	−0.403361	−	0.915041i	$-0.632158\pi$
$673$	−20.9282	−0.806723	−0.403361	+	0.915041i	$0.632158\pi$
$674$	0	0
$675$	4.46410	0.171823
$676$	0	0
$677$	−5.32051	−0.204484	−0.102242	−	0.994760i	$-0.532602\pi$
$677$	−5.32051	−0.204484	−0.102242	+	0.994760i	$0.532602\pi$
$678$	0	0
$679$	2.00000	0.0767530
$680$	0	0
$681$	4.39230	0.168313
$682$	0	0
$683$	1.26795	0.0485167	0.0242584	−	0.999706i	$-0.492278\pi$
$683$	1.26795	0.0485167	0.0242584	+	0.999706i	$0.492278\pi$
$684$	0	0
$685$	0.679492	0.0259621
$686$	0	0
$687$	20.2487	0.772537
$688$	0	0
$689$	−4.78461	−0.182279
$690$	0	0
$691$	32.7846	1.24719	0.623593	−	0.781749i	$-0.285672\pi$
$691$	32.7846	1.24719	0.623593	+	0.781749i	$0.285672\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−14.1436	−0.536497
$696$	0	0
$697$	−23.3205	−0.883327
$698$	0	0
$699$	−18.3923	−0.695661
$700$	0	0
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	0	0
$703$	10.0000	0.377157
$704$	0	0
$705$	4.14359	0.156057
$706$	0	0
$707$	4.73205	0.177967
$708$	0	0
$709$	−20.3923	−0.765849	−0.382925	−	0.923780i	$-0.625083\pi$
$709$	−20.3923	−0.765849	−0.382925	+	0.923780i	$0.625083\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−16.3923	−0.612182
$718$	0	0
$719$	−4.98076	−0.185751	−0.0928755	−	0.995678i	$-0.529606\pi$
$719$	−4.98076	−0.185751	−0.0928755	+	0.995678i	$0.529606\pi$
$720$	0	0
$721$	10.9282	0.406988
$722$	0	0
$723$	−0.928203	−0.0345202
$724$	0	0
$725$	−34.1962	−1.27001
$726$	0	0
$727$	5.07180	0.188103	0.0940513	−	0.995567i	$-0.470018\pi$
$727$	5.07180	0.188103	0.0940513	+	0.995567i	$0.470018\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	42.2487	1.56263
$732$	0	0
$733$	−44.2487	−1.63436	−0.817182	−	0.576380i	$-0.804465\pi$
$733$	−44.2487	−1.63436	−0.817182	+	0.576380i	$0.804465\pi$
$734$	0	0
$735$	0.732051	0.0270021
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−47.8564	−1.76043	−0.880213	−	0.474578i	$-0.842601\pi$
$739$	−47.8564	−1.76043	−0.880213	+	0.474578i	$0.842601\pi$
$740$	0	0
$741$	1.46410	0.0537851
$742$	0	0
$743$	−5.26795	−0.193262	−0.0966312	−	0.995320i	$-0.530807\pi$
$743$	−5.26795	−0.193262	−0.0966312	+	0.995320i	$0.530807\pi$
$744$	0	0
$745$	4.67949	0.171443
$746$	0	0
$747$	−0.196152	−0.00717684
$748$	0	0
$749$	−1.26795	−0.0463299
$750$	0	0
$751$	13.1769	0.480832	0.240416	−	0.970670i	$-0.422716\pi$
$751$	13.1769	0.480832	0.240416	+	0.970670i	$0.422716\pi$
$752$	0	0
$753$	−17.6603	−0.643575
$754$	0	0
$755$	6.92820	0.252143
$756$	0	0
$757$	18.7846	0.682738	0.341369	−	0.939929i	$-0.389109\pi$
$757$	18.7846	0.682738	0.341369	+	0.939929i	$0.389109\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	16.3397	0.592315	0.296158	−	0.955139i	$-0.404295\pi$
$761$	16.3397	0.592315	0.296158	+	0.955139i	$0.404295\pi$
$762$	0	0
$763$	−18.3923	−0.665846
$764$	0	0
$765$	3.46410	0.125245
$766$	0	0
$767$	11.7128	0.422925
$768$	0	0
$769$	27.8564	1.00453	0.502264	−	0.864714i	$-0.332501\pi$
$769$	27.8564	1.00453	0.502264	+	0.864714i	$0.332501\pi$
$770$	0	0
$771$	−4.53590	−0.163356
$772$	0	0
$773$	−14.0000	−0.503545	−0.251773	−	0.967786i	$-0.581013\pi$
$773$	−14.0000	−0.503545	−0.251773	+	0.967786i	$0.581013\pi$
$774$	0	0
$775$	−8.92820	−0.320711
$776$	0	0
$777$	10.0000	0.358748
$778$	0	0
$779$	−4.92820	−0.176571
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−7.66025	−0.273755
$784$	0	0
$785$	10.2487	0.365792
$786$	0	0
$787$	0.143594	0.00511856	0.00255928	−	0.999997i	$-0.499185\pi$
$787$	0.143594	0.00511856	0.00255928	+	0.999997i	$0.499185\pi$
$788$	0	0
$789$	−15.3205	−0.545425
$790$	0	0
$791$	−10.1962	−0.362533
$792$	0	0
$793$	−2.92820	−0.103984
$794$	0	0
$795$	−2.39230	−0.0848463
$796$	0	0
$797$	30.7846	1.09045	0.545223	−	0.838291i	$-0.316445\pi$
$797$	30.7846	1.09045	0.545223	+	0.838291i	$0.316445\pi$
$798$	0	0
$799$	−26.7846	−0.947571
$800$	0	0
$801$	−8.92820	−0.315463
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	31.1769	1.09748
$808$	0	0
$809$	30.7846	1.08233	0.541165	−	0.840917i	$-0.317984\pi$
$809$	30.7846	1.08233	0.541165	+	0.840917i	$0.317984\pi$
$810$	0	0
$811$	4.00000	0.140459	0.0702295	−	0.997531i	$-0.477627\pi$
$811$	4.00000	0.140459	0.0702295	+	0.997531i	$0.477627\pi$
$812$	0	0
$813$	−12.7846	−0.448376
$814$	0	0
$815$	15.3205	0.536654
$816$	0	0
$817$	8.92820	0.312358
$818$	0	0
$819$	1.46410	0.0511599
$820$	0	0
$821$	28.5359	0.995910	0.497955	−	0.867203i	$-0.334085\pi$
$821$	28.5359	0.995910	0.497955	+	0.867203i	$0.334085\pi$
$822$	0	0
$823$	−48.6410	−1.69552	−0.847760	−	0.530381i	$-0.822049\pi$
$823$	−48.6410	−1.69552	−0.847760	+	0.530381i	$0.822049\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	11.9090	0.414115	0.207058	−	0.978329i	$-0.433611\pi$
$827$	11.9090	0.414115	0.207058	+	0.978329i	$0.433611\pi$
$828$	0	0
$829$	17.7128	0.615191	0.307596	−	0.951517i	$-0.400476\pi$
$829$	17.7128	0.615191	0.307596	+	0.951517i	$0.400476\pi$
$830$	0	0
$831$	8.39230	0.291126
$832$	0	0
$833$	−4.73205	−0.163956
$834$	0	0
$835$	17.0718	0.590794
$836$	0	0
$837$	−2.00000	−0.0691301
$838$	0	0
$839$	12.7846	0.441374	0.220687	−	0.975345i	$-0.429170\pi$
$839$	12.7846	0.441374	0.220687	+	0.975345i	$0.429170\pi$
$840$	0	0
$841$	29.6795	1.02343
$842$	0	0
$843$	−6.19615	−0.213407
$844$	0	0
$845$	7.94744	0.273400
$846$	0	0
$847$	−11.0000	−0.377964
$848$	0	0
$849$	−0.392305	−0.0134639
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−43.8564	−1.50161	−0.750807	−	0.660521i	$-0.770335\pi$
$853$	−43.8564	−1.50161	−0.750807	+	0.660521i	$0.770335\pi$
$854$	0	0
$855$	0.732051	0.0250356
$856$	0	0
$857$	3.07180	0.104931	0.0524653	−	0.998623i	$-0.483292\pi$
$857$	3.07180	0.104931	0.0524653	+	0.998623i	$0.483292\pi$
$858$	0	0
$859$	18.1436	0.619051	0.309526	−	0.950891i	$-0.399830\pi$
$859$	18.1436	0.619051	0.309526	+	0.950891i	$0.399830\pi$
$860$	0	0
$861$	−4.92820	−0.167953
$862$	0	0
$863$	8.98076	0.305709	0.152854	−	0.988249i	$-0.451153\pi$
$863$	8.98076	0.305709	0.152854	+	0.988249i	$0.451153\pi$
$864$	0	0
$865$	2.24871	0.0764585
$866$	0	0
$867$	−5.39230	−0.183132
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−9.56922	−0.324241
$872$	0	0
$873$	2.00000	0.0676897
$874$	0	0
$875$	6.92820	0.234216
$876$	0	0
$877$	1.71281	0.0578376	0.0289188	−	0.999582i	$-0.490794\pi$
$877$	1.71281	0.0578376	0.0289188	+	0.999582i	$0.490794\pi$
$878$	0	0
$879$	6.00000	0.202375
$880$	0	0
$881$	0.732051	0.0246634	0.0123317	−	0.999924i	$-0.496075\pi$
$881$	0.732051	0.0246634	0.0123317	+	0.999924i	$0.496075\pi$
$882$	0	0
$883$	22.9282	0.771595	0.385798	−	0.922583i	$-0.373926\pi$
$883$	22.9282	0.771595	0.385798	+	0.922583i	$0.373926\pi$
$884$	0	0
$885$	5.85641	0.196861
$886$	0	0
$887$	−6.24871	−0.209811	−0.104906	−	0.994482i	$-0.533454\pi$
$887$	−6.24871	−0.209811	−0.104906	+	0.994482i	$0.533454\pi$
$888$	0	0
$889$	−0.392305	−0.0131575
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−5.66025	−0.189413
$894$	0	0
$895$	2.78461	0.0930792
$896$	0	0
$897$	0	0
$898$	0	0
$899$	15.3205	0.510968
$900$	0	0
$901$	15.4641	0.515184
$902$	0	0
$903$	8.92820	0.297112
$904$	0	0
$905$	6.53590	0.217261
$906$	0	0
$907$	−54.6410	−1.81433	−0.907163	−	0.420780i	$-0.861756\pi$
$907$	−54.6410	−1.81433	−0.907163	+	0.420780i	$0.861756\pi$
$908$	0	0
$909$	4.73205	0.156952
$910$	0	0
$911$	−9.26795	−0.307061	−0.153530	−	0.988144i	$-0.549064\pi$
$911$	−9.26795	−0.307061	−0.153530	+	0.988144i	$0.549064\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−1.46410	−0.0484017
$916$	0	0
$917$	−16.5885	−0.547799
$918$	0	0
$919$	−2.14359	−0.0707106	−0.0353553	−	0.999375i	$-0.511256\pi$
$919$	−2.14359	−0.0707106	−0.0353553	+	0.999375i	$0.511256\pi$
$920$	0	0
$921$	22.7846	0.750778
$922$	0	0
$923$	1.85641	0.0611044
$924$	0	0
$925$	44.6410	1.46779
$926$	0	0
$927$	10.9282	0.358929
$928$	0	0
$929$	46.9808	1.54139	0.770694	−	0.637205i	$-0.219910\pi$
$929$	46.9808	1.54139	0.770694	+	0.637205i	$0.219910\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	12.5885	0.412128
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−20.6410	−0.674313	−0.337156	−	0.941449i	$-0.609465\pi$
$937$	−20.6410	−0.674313	−0.337156	+	0.941449i	$0.609465\pi$
$938$	0	0
$939$	2.00000	0.0652675
$940$	0	0
$941$	−44.5359	−1.45183	−0.725914	−	0.687785i	$-0.758583\pi$
$941$	−44.5359	−1.45183	−0.725914	+	0.687785i	$0.758583\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0.732051	0.0238136
$946$	0	0
$947$	22.9282	0.745066	0.372533	−	0.928019i	$-0.378489\pi$
$947$	22.9282	0.745066	0.372533	+	0.928019i	$0.378489\pi$
$948$	0	0
$949$	−15.2154	−0.493912
$950$	0	0
$951$	−1.80385	−0.0584938
$952$	0	0
$953$	29.8038	0.965441	0.482721	−	0.875774i	$-0.339649\pi$
$953$	29.8038	0.965441	0.482721	+	0.875774i	$0.339649\pi$
$954$	0	0
$955$	0.784610	0.0253894
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−0.928203	−0.0299732
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	−1.26795	−0.0408591
$964$	0	0
$965$	−2.82309	−0.0908783
$966$	0	0
$967$	32.6410	1.04966	0.524832	−	0.851206i	$-0.324128\pi$
$967$	32.6410	1.04966	0.524832	+	0.851206i	$0.324128\pi$
$968$	0	0
$969$	−4.73205	−0.152015
$970$	0	0
$971$	59.7128	1.91628	0.958138	−	0.286308i	$-0.0924280\pi$
$971$	59.7128	1.91628	0.958138	+	0.286308i	$0.0924280\pi$
$972$	0	0
$973$	19.3205	0.619387
$974$	0	0
$975$	6.53590	0.209316
$976$	0	0
$977$	58.3013	1.86522	0.932611	−	0.360882i	$-0.117524\pi$
$977$	58.3013	1.86522	0.932611	+	0.360882i	$0.117524\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−18.3923	−0.587221
$982$	0	0
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	0	0
$985$	13.1769	0.419851
$986$	0	0
$987$	−5.66025	−0.180168
$988$	0	0
$989$	0	0
$990$	0	0
$991$	28.7846	0.914373	0.457187	−	0.889371i	$-0.348857\pi$
$991$	28.7846	0.914373	0.457187	+	0.889371i	$0.348857\pi$
$992$	0	0
$993$	26.2487	0.832978
$994$	0	0
$995$	−6.14359	−0.194765
$996$	0	0
$997$	7.85641	0.248815	0.124407	−	0.992231i	$-0.460297\pi$
$997$	7.85641	0.248815	0.124407	+	0.992231i	$0.460297\pi$
$998$	0	0
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3192.2.a.q.1.1	✓	2
3.2	odd	2		9576.2.a.bj.1.2		2
4.3	odd	2		6384.2.a.bs.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3192.2.a.q.1.1	✓	2	1.1	even	1	trivial
6384.2.a.bs.1.1		2	4.3	odd	2
9576.2.a.bj.1.2		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3192.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.732051 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -0.732051 q^{5} +1.00000 q^{7} +1.00000 q^{9} +1.46410 q^{13} +0.732051 q^{15} -4.73205 q^{17} -1.00000 q^{19} -1.00000 q^{21} -4.46410 q^{25} -1.00000 q^{27} +7.66025 q^{29} +2.00000 q^{31} -0.732051 q^{35} -10.0000 q^{37} -1.46410 q^{39} +4.92820 q^{41} -8.92820 q^{43} -0.732051 q^{45} +5.66025 q^{47} +1.00000 q^{49} +4.73205 q^{51} -3.26795 q^{53} +1.00000 q^{57} +8.00000 q^{59} -2.00000 q^{61} +1.00000 q^{63} -1.07180 q^{65} -6.53590 q^{67} +1.26795 q^{71} -10.3923 q^{73} +4.46410 q^{75} +4.00000 q^{79} +1.00000 q^{81} -0.196152 q^{83} +3.46410 q^{85} -7.66025 q^{87} -8.92820 q^{89} +1.46410 q^{91} -2.00000 q^{93} +0.732051 q^{95} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} - 4 q^{13} - 2 q^{15} - 6 q^{17} - 2 q^{19} - 2 q^{21} - 2 q^{25} - 2 q^{27} - 2 q^{29} + 4 q^{31} + 2 q^{35} - 20 q^{37} + 4 q^{39} - 4 q^{41} - 4 q^{43} + 2 q^{45} - 6 q^{47} + 2 q^{49} + 6 q^{51} - 10 q^{53} + 2 q^{57} + 16 q^{59} - 4 q^{61} + 2 q^{63} - 16 q^{65} - 20 q^{67} + 6 q^{71} + 2 q^{75} + 8 q^{79} + 2 q^{81} + 10 q^{83} + 2 q^{87} - 4 q^{89} - 4 q^{91} - 4 q^{93} - 2 q^{95} + 4 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 - 4 * q^13 - 2 * q^15 - 6 * q^17 - 2 * q^19 - 2 * q^21 - 2 * q^25 - 2 * q^27 - 2 * q^29 + 4 * q^31 + 2 * q^35 - 20 * q^37 + 4 * q^39 - 4 * q^41 - 4 * q^43 + 2 * q^45 - 6 * q^47 + 2 * q^49 + 6 * q^51 - 10 * q^53 + 2 * q^57 + 16 * q^59 - 4 * q^61 + 2 * q^63 - 16 * q^65 - 20 * q^67 + 6 * q^71 + 2 * q^75 + 8 * q^79 + 2 * q^81 + 10 * q^83 + 2 * q^87 - 4 * q^89 - 4 * q^91 - 4 * q^93 - 2 * q^95 + 4 * q^97

Introduction

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