LMFDB - Embedded newform 3800.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,2,Mod(1,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3800 = 2^{3} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3800.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:	$30.3431527681$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 760)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	3800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.414214 q^{3} +0.414214 q^{7} -2.82843 q^{9} +O(q^{10})$	$q-0.414214 q^{3} +0.414214 q^{7} -2.82843 q^{9} -1.41421 q^{11} -3.82843 q^{13} +1.00000 q^{17} -1.00000 q^{19} -0.171573 q^{21} -3.24264 q^{23} +2.41421 q^{27} -1.82843 q^{29} +0.585786 q^{31} +0.585786 q^{33} +10.8284 q^{37} +1.58579 q^{39} +7.07107 q^{41} -6.24264 q^{43} +8.00000 q^{47} -6.82843 q^{49} -0.414214 q^{51} +3.82843 q^{53} +0.414214 q^{57} +11.5858 q^{59} +0.585786 q^{61} -1.17157 q^{63} +8.07107 q^{67} +1.34315 q^{69} -11.0711 q^{71} +8.17157 q^{73} -0.585786 q^{77} +4.82843 q^{79} +7.48528 q^{81} +14.4853 q^{83} +0.757359 q^{87} -12.2426 q^{89} -1.58579 q^{91} -0.242641 q^{93} -3.65685 q^{97} +4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{7}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^7	$ 2 q + 2 q^{3} - 2 q^{7} - 2 q^{13} + 2 q^{17} - 2 q^{19} - 6 q^{21} + 2 q^{23} + 2 q^{27} + 2 q^{29} + 4 q^{31} + 4 q^{33} + 16 q^{37} + 6 q^{39} - 4 q^{43} + 16 q^{47} - 8 q^{49} + 2 q^{51} + 2 q^{53} - 2 q^{57} + 26 q^{59} + 4 q^{61} - 8 q^{63} + 2 q^{67} + 14 q^{69} - 8 q^{71} + 22 q^{73} - 4 q^{77} + 4 q^{79} - 2 q^{81} + 12 q^{83} + 10 q^{87} - 16 q^{89} - 6 q^{91} + 8 q^{93} + 4 q^{97} + 8 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^7 - 2 * q^13 + 2 * q^17 - 2 * q^19 - 6 * q^21 + 2 * q^23 + 2 * q^27 + 2 * q^29 + 4 * q^31 + 4 * q^33 + 16 * q^37 + 6 * q^39 - 4 * q^43 + 16 * q^47 - 8 * q^49 + 2 * q^51 + 2 * q^53 - 2 * q^57 + 26 * q^59 + 4 * q^61 - 8 * q^63 + 2 * q^67 + 14 * q^69 - 8 * q^71 + 22 * q^73 - 4 * q^77 + 4 * q^79 - 2 * q^81 + 12 * q^83 + 10 * q^87 - 16 * q^89 - 6 * q^91 + 8 * q^93 + 4 * q^97 + 8 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−0.414214	−0.239146	−0.119573	−	0.992825i	$-0.538153\pi$
$3$	−0.414214	−0.239146	−0.119573	+	0.992825i	$0.538153\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.414214	0.156558	0.0782790	−	0.996931i	$-0.475058\pi$
$7$	0.414214	0.156558	0.0782790	+	0.996931i	$0.475058\pi$
$8$	0	0
$9$	−2.82843	−0.942809
$10$	0	0
$11$	−1.41421	−0.426401	−0.213201	−	0.977008i	$-0.568389\pi$
$11$	−1.41421	−0.426401	−0.213201	+	0.977008i	$0.568389\pi$
$12$	0	0
$13$	−3.82843	−1.06181	−0.530907	−	0.847430i	$-0.678149\pi$
$13$	−3.82843	−1.06181	−0.530907	+	0.847430i	$0.678149\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−0.171573	−0.0374403
$22$	0	0
$23$	−3.24264	−0.676137	−0.338069	−	0.941121i	$-0.609774\pi$
$23$	−3.24264	−0.676137	−0.338069	+	0.941121i	$0.609774\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2.41421	0.464616
$28$	0	0
$29$	−1.82843	−0.339530	−0.169765	−	0.985485i	$-0.554301\pi$
$29$	−1.82843	−0.339530	−0.169765	+	0.985485i	$0.554301\pi$
$30$	0	0
$31$	0.585786	0.105210	0.0526052	−	0.998615i	$-0.483248\pi$
$31$	0.585786	0.105210	0.0526052	+	0.998615i	$0.483248\pi$
$32$	0	0
$33$	0.585786	0.101972
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10.8284	1.78018	0.890091	−	0.455782i	$-0.150640\pi$
$37$	10.8284	1.78018	0.890091	+	0.455782i	$0.150640\pi$
$38$	0	0
$39$	1.58579	0.253929
$40$	0	0
$41$	7.07107	1.10432	0.552158	−	0.833740i	$-0.313805\pi$
$41$	7.07107	1.10432	0.552158	+	0.833740i	$0.313805\pi$
$42$	0	0
$43$	−6.24264	−0.951994	−0.475997	−	0.879447i	$-0.657913\pi$
$43$	−6.24264	−0.951994	−0.475997	+	0.879447i	$0.657913\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	−6.82843	−0.975490
$50$	0	0
$51$	−0.414214	−0.0580015
$52$	0	0
$53$	3.82843	0.525875	0.262937	−	0.964813i	$-0.415309\pi$
$53$	3.82843	0.525875	0.262937	+	0.964813i	$0.415309\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0.414214	0.0548639
$58$	0	0
$59$	11.5858	1.50834	0.754170	−	0.656679i	$-0.228039\pi$
$59$	11.5858	1.50834	0.754170	+	0.656679i	$0.228039\pi$
$60$	0	0
$61$	0.585786	0.0750023	0.0375011	−	0.999297i	$-0.488060\pi$
$61$	0.585786	0.0750023	0.0375011	+	0.999297i	$0.488060\pi$
$62$	0	0
$63$	−1.17157	−0.147604
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.07107	0.986038	0.493019	−	0.870019i	$-0.335893\pi$
$67$	8.07107	0.986038	0.493019	+	0.870019i	$0.335893\pi$
$68$	0	0
$69$	1.34315	0.161696
$70$	0	0
$71$	−11.0711	−1.31389	−0.656947	−	0.753937i	$-0.728152\pi$
$71$	−11.0711	−1.31389	−0.656947	+	0.753937i	$0.728152\pi$
$72$	0	0
$73$	8.17157	0.956410	0.478205	−	0.878248i	$-0.341288\pi$
$73$	8.17157	0.956410	0.478205	+	0.878248i	$0.341288\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.585786	−0.0667566
$78$	0	0
$79$	4.82843	0.543240	0.271620	−	0.962405i	$-0.412441\pi$
$79$	4.82843	0.543240	0.271620	+	0.962405i	$0.412441\pi$
$80$	0	0
$81$	7.48528	0.831698
$82$	0	0
$83$	14.4853	1.58997	0.794983	−	0.606632i	$-0.207480\pi$
$83$	14.4853	1.58997	0.794983	+	0.606632i	$0.207480\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0.757359	0.0811974
$88$	0	0
$89$	−12.2426	−1.29772	−0.648859	−	0.760909i	$-0.724753\pi$
$89$	−12.2426	−1.29772	−0.648859	+	0.760909i	$0.724753\pi$
$90$	0	0
$91$	−1.58579	−0.166236
$92$	0	0
$93$	−0.242641	−0.0251607
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−3.65685	−0.371297	−0.185649	−	0.982616i	$-0.559439\pi$
$97$	−3.65685	−0.371297	−0.185649	+	0.982616i	$0.559439\pi$
$98$	0	0
$99$	4.00000	0.402015
$100$	0	0
$101$	−3.89949	−0.388014	−0.194007	−	0.981000i	$-0.562148\pi$
$101$	−3.89949	−0.388014	−0.194007	+	0.981000i	$0.562148\pi$
$102$	0	0
$103$	9.89949	0.975426	0.487713	−	0.873004i	$-0.337831\pi$
$103$	9.89949	0.975426	0.487713	+	0.873004i	$0.337831\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.414214	0.0400435	0.0200218	−	0.999800i	$-0.493626\pi$
$107$	0.414214	0.0400435	0.0200218	+	0.999800i	$0.493626\pi$
$108$	0	0
$109$	−5.00000	−0.478913	−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000	−0.478913	−0.239457	+	0.970907i	$0.576969\pi$
$110$	0	0
$111$	−4.48528	−0.425724
$112$	0	0
$113$	1.07107	0.100758	0.0503788	−	0.998730i	$-0.483957\pi$
$113$	1.07107	0.100758	0.0503788	+	0.998730i	$0.483957\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	10.8284	1.00109
$118$	0	0
$119$	0.414214	0.0379709
$120$	0	0
$121$	−9.00000	−0.818182
$122$	0	0
$123$	−2.92893	−0.264093
$124$	0	0
$125$	0	0
$126$	0	0
$127$	16.8284	1.49328	0.746641	−	0.665228i	$-0.231665\pi$
$127$	16.8284	1.49328	0.746641	+	0.665228i	$0.231665\pi$
$128$	0	0
$129$	2.58579	0.227666
$130$	0	0
$131$	10.3431	0.903685	0.451842	−	0.892098i	$-0.350767\pi$
$131$	10.3431	0.903685	0.451842	+	0.892098i	$0.350767\pi$
$132$	0	0
$133$	−0.414214	−0.0359169
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.31371	−0.539417	−0.269708	−	0.962942i	$-0.586927\pi$
$137$	−6.31371	−0.539417	−0.269708	+	0.962942i	$0.586927\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	−3.31371	−0.279065
$142$	0	0
$143$	5.41421	0.452759
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2.82843	0.233285
$148$	0	0
$149$	−6.34315	−0.519651	−0.259825	−	0.965656i	$-0.583665\pi$
$149$	−6.34315	−0.519651	−0.259825	+	0.965656i	$0.583665\pi$
$150$	0	0
$151$	16.8284	1.36948	0.684739	−	0.728788i	$-0.259916\pi$
$151$	16.8284	1.36948	0.684739	+	0.728788i	$0.259916\pi$
$152$	0	0
$153$	−2.82843	−0.228665
$154$	0	0
$155$	0	0
$156$	0	0
$157$	11.6569	0.930318	0.465159	−	0.885227i	$-0.345997\pi$
$157$	11.6569	0.930318	0.465159	+	0.885227i	$0.345997\pi$
$158$	0	0
$159$	−1.58579	−0.125761
$160$	0	0
$161$	−1.34315	−0.105855
$162$	0	0
$163$	17.0711	1.33711	0.668555	−	0.743663i	$-0.266913\pi$
$163$	17.0711	1.33711	0.668555	+	0.743663i	$0.266913\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−19.2132	−1.48676	−0.743381	−	0.668868i	$-0.766779\pi$
$167$	−19.2132	−1.48676	−0.743381	+	0.668868i	$0.766779\pi$
$168$	0	0
$169$	1.65685	0.127450
$170$	0	0
$171$	2.82843	0.216295
$172$	0	0
$173$	13.1716	1.00142	0.500708	−	0.865616i	$-0.333073\pi$
$173$	13.1716	1.00142	0.500708	+	0.865616i	$0.333073\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−4.79899	−0.360714
$178$	0	0
$179$	10.9706	0.819978	0.409989	−	0.912090i	$-0.365532\pi$
$179$	10.9706	0.819978	0.409989	+	0.912090i	$0.365532\pi$
$180$	0	0
$181$	−16.4853	−1.22534	−0.612671	−	0.790338i	$-0.709905\pi$
$181$	−16.4853	−1.22534	−0.612671	+	0.790338i	$0.709905\pi$
$182$	0	0
$183$	−0.242641	−0.0179365
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1.41421	−0.103418
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−2.75736	−0.199516	−0.0997578	−	0.995012i	$-0.531807\pi$
$191$	−2.75736	−0.199516	−0.0997578	+	0.995012i	$0.531807\pi$
$192$	0	0
$193$	−3.65685	−0.263226	−0.131613	−	0.991301i	$-0.542016\pi$
$193$	−3.65685	−0.263226	−0.131613	+	0.991301i	$0.542016\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4.72792	−0.336850	−0.168425	−	0.985714i	$-0.553868\pi$
$197$	−4.72792	−0.336850	−0.168425	+	0.985714i	$0.553868\pi$
$198$	0	0
$199$	1.92893	0.136738	0.0683692	−	0.997660i	$-0.478220\pi$
$199$	1.92893	0.136738	0.0683692	+	0.997660i	$0.478220\pi$
$200$	0	0
$201$	−3.34315	−0.235807
$202$	0	0
$203$	−0.757359	−0.0531562
$204$	0	0
$205$	0	0
$206$	0	0
$207$	9.17157	0.637468
$208$	0	0
$209$	1.41421	0.0978232
$210$	0	0
$211$	−5.10051	−0.351133	−0.175567	−	0.984468i	$-0.556176\pi$
$211$	−5.10051	−0.351133	−0.175567	+	0.984468i	$0.556176\pi$
$212$	0	0
$213$	4.58579	0.314213
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0.242641	0.0164715
$218$	0	0
$219$	−3.38478	−0.228722
$220$	0	0
$221$	−3.82843	−0.257528
$222$	0	0
$223$	8.82843	0.591195	0.295598	−	0.955313i	$-0.404481\pi$
$223$	8.82843	0.591195	0.295598	+	0.955313i	$0.404481\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−8.75736	−0.581246	−0.290623	−	0.956838i	$-0.593863\pi$
$227$	−8.75736	−0.581246	−0.290623	+	0.956838i	$0.593863\pi$
$228$	0	0
$229$	6.97056	0.460628	0.230314	−	0.973116i	$-0.426025\pi$
$229$	6.97056	0.460628	0.230314	+	0.973116i	$0.426025\pi$
$230$	0	0
$231$	0.242641	0.0159646
$232$	0	0
$233$	−13.6569	−0.894690	−0.447345	−	0.894361i	$-0.647630\pi$
$233$	−13.6569	−0.894690	−0.447345	+	0.894361i	$0.647630\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−2.00000	−0.129914
$238$	0	0
$239$	27.7279	1.79357	0.896785	−	0.442466i	$-0.145896\pi$
$239$	27.7279	1.79357	0.896785	+	0.442466i	$0.145896\pi$
$240$	0	0
$241$	−5.65685	−0.364390	−0.182195	−	0.983262i	$-0.558320\pi$
$241$	−5.65685	−0.364390	−0.182195	+	0.983262i	$0.558320\pi$
$242$	0	0
$243$	−10.3431	−0.663513
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.82843	0.243597
$248$	0	0
$249$	−6.00000	−0.380235
$250$	0	0
$251$	3.55635	0.224475	0.112237	−	0.993681i	$-0.464198\pi$
$251$	3.55635	0.224475	0.112237	+	0.993681i	$0.464198\pi$
$252$	0	0
$253$	4.58579	0.288306
$254$	0	0
$255$	0	0
$256$	0	0
$257$	28.7279	1.79200	0.895999	−	0.444056i	$-0.146461\pi$
$257$	28.7279	1.79200	0.895999	+	0.444056i	$0.146461\pi$
$258$	0	0
$259$	4.48528	0.278702
$260$	0	0
$261$	5.17157	0.320112
$262$	0	0
$263$	−15.6569	−0.965443	−0.482721	−	0.875774i	$-0.660352\pi$
$263$	−15.6569	−0.965443	−0.482721	+	0.875774i	$0.660352\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	5.07107	0.310344
$268$	0	0
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	15.3848	0.934559	0.467279	−	0.884110i	$-0.345234\pi$
$271$	15.3848	0.934559	0.467279	+	0.884110i	$0.345234\pi$
$272$	0	0
$273$	0.656854	0.0397546
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−3.31371	−0.199101	−0.0995507	−	0.995032i	$-0.531741\pi$
$277$	−3.31371	−0.199101	−0.0995507	+	0.995032i	$0.531741\pi$
$278$	0	0
$279$	−1.65685	−0.0991933
$280$	0	0
$281$	−3.75736	−0.224145	−0.112073	−	0.993700i	$-0.535749\pi$
$281$	−3.75736	−0.224145	−0.112073	+	0.993700i	$0.535749\pi$
$282$	0	0
$283$	9.51472	0.565591	0.282796	−	0.959180i	$-0.408738\pi$
$283$	9.51472	0.565591	0.282796	+	0.959180i	$0.408738\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.92893	0.172889
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	1.51472	0.0887944
$292$	0	0
$293$	10.7990	0.630884	0.315442	−	0.948945i	$-0.397847\pi$
$293$	10.7990	0.630884	0.315442	+	0.948945i	$0.397847\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−3.41421	−0.198113
$298$	0	0
$299$	12.4142	0.717933
$300$	0	0
$301$	−2.58579	−0.149042
$302$	0	0
$303$	1.61522	0.0927922
$304$	0	0
$305$	0	0
$306$	0	0
$307$	32.2843	1.84256	0.921280	−	0.388899i	$-0.127145\pi$
$307$	32.2843	1.84256	0.921280	+	0.388899i	$0.127145\pi$
$308$	0	0
$309$	−4.10051	−0.233270
$310$	0	0
$311$	23.2426	1.31797	0.658985	−	0.752156i	$-0.270986\pi$
$311$	23.2426	1.31797	0.658985	+	0.752156i	$0.270986\pi$
$312$	0	0
$313$	6.65685	0.376268	0.188134	−	0.982143i	$-0.439756\pi$
$313$	6.65685	0.376268	0.188134	+	0.982143i	$0.439756\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.79899	0.381869	0.190935	−	0.981603i	$-0.438848\pi$
$317$	6.79899	0.381869	0.190935	+	0.981603i	$0.438848\pi$
$318$	0	0
$319$	2.58579	0.144776
$320$	0	0
$321$	−0.171573	−0.00957626
$322$	0	0
$323$	−1.00000	−0.0556415
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.07107	0.114530
$328$	0	0
$329$	3.31371	0.182691
$330$	0	0
$331$	19.3848	1.06548	0.532742	−	0.846278i	$-0.321162\pi$
$331$	19.3848	1.06548	0.532742	+	0.846278i	$0.321162\pi$
$332$	0	0
$333$	−30.6274	−1.67837
$334$	0	0
$335$	0	0
$336$	0	0
$337$	12.7279	0.693334	0.346667	−	0.937988i	$-0.387313\pi$
$337$	12.7279	0.693334	0.346667	+	0.937988i	$0.387313\pi$
$338$	0	0
$339$	−0.443651	−0.0240958
$340$	0	0
$341$	−0.828427	−0.0448618
$342$	0	0
$343$	−5.72792	−0.309279
$344$	0	0
$345$	0	0
$346$	0	0
$347$	15.1716	0.814453	0.407226	−	0.913327i	$-0.366496\pi$
$347$	15.1716	0.814453	0.407226	+	0.913327i	$0.366496\pi$
$348$	0	0
$349$	28.6274	1.53239	0.766195	−	0.642608i	$-0.222148\pi$
$349$	28.6274	1.53239	0.766195	+	0.642608i	$0.222148\pi$
$350$	0	0
$351$	−9.24264	−0.493336
$352$	0	0
$353$	36.1127	1.92208	0.961042	−	0.276401i	$-0.0891417\pi$
$353$	36.1127	1.92208	0.961042	+	0.276401i	$0.0891417\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−0.171573	−0.00908060
$358$	0	0
$359$	−6.07107	−0.320419	−0.160209	−	0.987083i	$-0.551217\pi$
$359$	−6.07107	−0.320419	−0.160209	+	0.987083i	$0.551217\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	3.72792	0.195665
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−33.4558	−1.74638	−0.873190	−	0.487379i	$-0.837953\pi$
$367$	−33.4558	−1.74638	−0.873190	+	0.487379i	$0.837953\pi$
$368$	0	0
$369$	−20.0000	−1.04116
$370$	0	0
$371$	1.58579	0.0823299
$372$	0	0
$373$	−27.9706	−1.44826	−0.724130	−	0.689663i	$-0.757759\pi$
$373$	−27.9706	−1.44826	−0.724130	+	0.689663i	$0.757759\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7.00000	0.360518
$378$	0	0
$379$	−3.38478	−0.173864	−0.0869321	−	0.996214i	$-0.527706\pi$
$379$	−3.38478	−0.173864	−0.0869321	+	0.996214i	$0.527706\pi$
$380$	0	0
$381$	−6.97056	−0.357113
$382$	0	0
$383$	−20.7279	−1.05915	−0.529574	−	0.848264i	$-0.677648\pi$
$383$	−20.7279	−1.05915	−0.529574	+	0.848264i	$0.677648\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	17.6569	0.897548
$388$	0	0
$389$	7.89949	0.400520	0.200260	−	0.979743i	$-0.435821\pi$
$389$	7.89949	0.400520	0.200260	+	0.979743i	$0.435821\pi$
$390$	0	0
$391$	−3.24264	−0.163987
$392$	0	0
$393$	−4.28427	−0.216113
$394$	0	0
$395$	0	0
$396$	0	0
$397$	22.6274	1.13564	0.567819	−	0.823154i	$-0.307787\pi$
$397$	22.6274	1.13564	0.567819	+	0.823154i	$0.307787\pi$
$398$	0	0
$399$	0.171573	0.00858939
$400$	0	0
$401$	−6.10051	−0.304645	−0.152322	−	0.988331i	$-0.548675\pi$
$401$	−6.10051	−0.304645	−0.152322	+	0.988331i	$0.548675\pi$
$402$	0	0
$403$	−2.24264	−0.111714
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−15.3137	−0.759072
$408$	0	0
$409$	2.58579	0.127859	0.0639295	−	0.997954i	$-0.479637\pi$
$409$	2.58579	0.127859	0.0639295	+	0.997954i	$0.479637\pi$
$410$	0	0
$411$	2.61522	0.128999
$412$	0	0
$413$	4.79899	0.236143
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1.65685	0.0811365
$418$	0	0
$419$	−27.2132	−1.32945	−0.664726	−	0.747087i	$-0.731452\pi$
$419$	−27.2132	−1.32945	−0.664726	+	0.747087i	$0.731452\pi$
$420$	0	0
$421$	13.1421	0.640508	0.320254	−	0.947332i	$-0.396232\pi$
$421$	13.1421	0.640508	0.320254	+	0.947332i	$0.396232\pi$
$422$	0	0
$423$	−22.6274	−1.10018
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0.242641	0.0117422
$428$	0	0
$429$	−2.24264	−0.108276
$430$	0	0
$431$	−5.41421	−0.260793	−0.130397	−	0.991462i	$-0.541625\pi$
$431$	−5.41421	−0.260793	−0.130397	+	0.991462i	$0.541625\pi$
$432$	0	0
$433$	5.07107	0.243700	0.121850	−	0.992549i	$-0.461117\pi$
$433$	5.07107	0.243700	0.121850	+	0.992549i	$0.461117\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.24264	0.155117
$438$	0	0
$439$	−30.7279	−1.46656	−0.733282	−	0.679925i	$-0.762012\pi$
$439$	−30.7279	−1.46656	−0.733282	+	0.679925i	$0.762012\pi$
$440$	0	0
$441$	19.3137	0.919700
$442$	0	0
$443$	15.5563	0.739104	0.369552	−	0.929210i	$-0.379511\pi$
$443$	15.5563	0.739104	0.369552	+	0.929210i	$0.379511\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	2.62742	0.124273
$448$	0	0
$449$	−35.9411	−1.69617	−0.848083	−	0.529863i	$-0.822243\pi$
$449$	−35.9411	−1.69617	−0.848083	+	0.529863i	$0.822243\pi$
$450$	0	0
$451$	−10.0000	−0.470882
$452$	0	0
$453$	−6.97056	−0.327506
$454$	0	0
$455$	0	0
$456$	0	0
$457$	16.3137	0.763123	0.381562	−	0.924343i	$-0.375386\pi$
$457$	16.3137	0.763123	0.381562	+	0.924343i	$0.375386\pi$
$458$	0	0
$459$	2.41421	0.112686
$460$	0	0
$461$	−7.27208	−0.338694	−0.169347	−	0.985556i	$-0.554166\pi$
$461$	−7.27208	−0.338694	−0.169347	+	0.985556i	$0.554166\pi$
$462$	0	0
$463$	22.1421	1.02903	0.514516	−	0.857481i	$-0.327972\pi$
$463$	22.1421	1.02903	0.514516	+	0.857481i	$0.327972\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−21.8995	−1.01339	−0.506694	−	0.862126i	$-0.669133\pi$
$467$	−21.8995	−1.01339	−0.506694	+	0.862126i	$0.669133\pi$
$468$	0	0
$469$	3.34315	0.154372
$470$	0	0
$471$	−4.82843	−0.222482
$472$	0	0
$473$	8.82843	0.405932
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−10.8284	−0.495800
$478$	0	0
$479$	33.4558	1.52864	0.764318	−	0.644839i	$-0.223076\pi$
$479$	33.4558	1.52864	0.764318	+	0.644839i	$0.223076\pi$
$480$	0	0
$481$	−41.4558	−1.89022
$482$	0	0
$483$	0.556349	0.0253148
$484$	0	0
$485$	0	0
$486$	0	0
$487$	7.17157	0.324975	0.162487	−	0.986711i	$-0.448048\pi$
$487$	7.17157	0.324975	0.162487	+	0.986711i	$0.448048\pi$
$488$	0	0
$489$	−7.07107	−0.319765
$490$	0	0
$491$	17.7574	0.801378	0.400689	−	0.916214i	$-0.368771\pi$
$491$	17.7574	0.801378	0.400689	+	0.916214i	$0.368771\pi$
$492$	0	0
$493$	−1.82843	−0.0823482
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4.58579	−0.205701
$498$	0	0
$499$	18.9289	0.847375	0.423688	−	0.905808i	$-0.360735\pi$
$499$	18.9289	0.847375	0.423688	+	0.905808i	$0.360735\pi$
$500$	0	0
$501$	7.95837	0.355554
$502$	0	0
$503$	32.8995	1.46692	0.733458	−	0.679735i	$-0.237905\pi$
$503$	32.8995	1.46692	0.733458	+	0.679735i	$0.237905\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−0.686292	−0.0304793
$508$	0	0
$509$	9.65685	0.428033	0.214016	−	0.976830i	$-0.431345\pi$
$509$	9.65685	0.428033	0.214016	+	0.976830i	$0.431345\pi$
$510$	0	0
$511$	3.38478	0.149734
$512$	0	0
$513$	−2.41421	−0.106590
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−11.3137	−0.497576
$518$	0	0
$519$	−5.45584	−0.239485
$520$	0	0
$521$	−28.0000	−1.22670	−0.613351	−	0.789810i	$-0.710179\pi$
$521$	−28.0000	−1.22670	−0.613351	+	0.789810i	$0.710179\pi$
$522$	0	0
$523$	−20.2132	−0.883862	−0.441931	−	0.897049i	$-0.645706\pi$
$523$	−20.2132	−0.883862	−0.441931	+	0.897049i	$0.645706\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.585786	0.0255173
$528$	0	0
$529$	−12.4853	−0.542838
$530$	0	0
$531$	−32.7696	−1.42208
$532$	0	0
$533$	−27.0711	−1.17258
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−4.54416	−0.196095
$538$	0	0
$539$	9.65685	0.415950
$540$	0	0
$541$	30.0416	1.29159	0.645795	−	0.763511i	$-0.276526\pi$
$541$	30.0416	1.29159	0.645795	+	0.763511i	$0.276526\pi$
$542$	0	0
$543$	6.82843	0.293036
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−23.9411	−1.02365	−0.511824	−	0.859090i	$-0.671030\pi$
$547$	−23.9411	−1.02365	−0.511824	+	0.859090i	$0.671030\pi$
$548$	0	0
$549$	−1.65685	−0.0707128
$550$	0	0
$551$	1.82843	0.0778936
$552$	0	0
$553$	2.00000	0.0850487
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−43.3137	−1.83526	−0.917630	−	0.397435i	$-0.869900\pi$
$557$	−43.3137	−1.83526	−0.917630	+	0.397435i	$0.869900\pi$
$558$	0	0
$559$	23.8995	1.01084
$560$	0	0
$561$	0.585786	0.0247319
$562$	0	0
$563$	−44.9706	−1.89528	−0.947642	−	0.319336i	$-0.896540\pi$
$563$	−44.9706	−1.89528	−0.947642	+	0.319336i	$0.896540\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3.10051	0.130209
$568$	0	0
$569$	−8.97056	−0.376066	−0.188033	−	0.982163i	$-0.560211\pi$
$569$	−8.97056	−0.376066	−0.188033	+	0.982163i	$0.560211\pi$
$570$	0	0
$571$	47.6985	1.99612	0.998060	−	0.0622637i	$-0.0198320\pi$
$571$	47.6985	1.99612	0.998060	+	0.0622637i	$0.0198320\pi$
$572$	0	0
$573$	1.14214	0.0477134
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−25.0000	−1.04076	−0.520382	−	0.853934i	$-0.674210\pi$
$577$	−25.0000	−1.04076	−0.520382	+	0.853934i	$0.674210\pi$
$578$	0	0
$579$	1.51472	0.0629496
$580$	0	0
$581$	6.00000	0.248922
$582$	0	0
$583$	−5.41421	−0.224234
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−32.3848	−1.33666	−0.668331	−	0.743864i	$-0.732991\pi$
$587$	−32.3848	−1.33666	−0.668331	+	0.743864i	$0.732991\pi$
$588$	0	0
$589$	−0.585786	−0.0241369
$590$	0	0
$591$	1.95837	0.0805566
$592$	0	0
$593$	−42.0000	−1.72473	−0.862367	−	0.506284i	$-0.831019\pi$
$593$	−42.0000	−1.72473	−0.862367	+	0.506284i	$0.831019\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−0.798990	−0.0327005
$598$	0	0
$599$	−33.5563	−1.37108	−0.685538	−	0.728037i	$-0.740433\pi$
$599$	−33.5563	−1.37108	−0.685538	+	0.728037i	$0.740433\pi$
$600$	0	0
$601$	0.443651	0.0180969	0.00904845	−	0.999959i	$-0.497120\pi$
$601$	0.443651	0.0180969	0.00904845	+	0.999959i	$0.497120\pi$
$602$	0	0
$603$	−22.8284	−0.929645
$604$	0	0
$605$	0	0
$606$	0	0
$607$	30.4853	1.23736	0.618680	−	0.785643i	$-0.287668\pi$
$607$	30.4853	1.23736	0.618680	+	0.785643i	$0.287668\pi$
$608$	0	0
$609$	0.313708	0.0127121
$610$	0	0
$611$	−30.6274	−1.23905
$612$	0	0
$613$	−2.72792	−0.110180	−0.0550899	−	0.998481i	$-0.517545\pi$
$613$	−2.72792	−0.110180	−0.0550899	+	0.998481i	$0.517545\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−31.7990	−1.28018	−0.640090	−	0.768300i	$-0.721103\pi$
$617$	−31.7990	−1.28018	−0.640090	+	0.768300i	$0.721103\pi$
$618$	0	0
$619$	6.38478	0.256626	0.128313	−	0.991734i	$-0.459044\pi$
$619$	6.38478	0.256626	0.128313	+	0.991734i	$0.459044\pi$
$620$	0	0
$621$	−7.82843	−0.314144
$622$	0	0
$623$	−5.07107	−0.203168
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−0.585786	−0.0233941
$628$	0	0
$629$	10.8284	0.431758
$630$	0	0
$631$	−5.02944	−0.200219	−0.100109	−	0.994976i	$-0.531919\pi$
$631$	−5.02944	−0.200219	−0.100109	+	0.994976i	$0.531919\pi$
$632$	0	0
$633$	2.11270	0.0839722
$634$	0	0
$635$	0	0
$636$	0	0
$637$	26.1421	1.03579
$638$	0	0
$639$	31.3137	1.23875
$640$	0	0
$641$	3.21320	0.126914	0.0634570	−	0.997985i	$-0.479787\pi$
$641$	3.21320	0.126914	0.0634570	+	0.997985i	$0.479787\pi$
$642$	0	0
$643$	36.8284	1.45237	0.726186	−	0.687499i	$-0.241291\pi$
$643$	36.8284	1.45237	0.726186	+	0.687499i	$0.241291\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−2.07107	−0.0814221	−0.0407110	−	0.999171i	$-0.512962\pi$
$647$	−2.07107	−0.0814221	−0.0407110	+	0.999171i	$0.512962\pi$
$648$	0	0
$649$	−16.3848	−0.643159
$650$	0	0
$651$	−0.100505	−0.00393910
$652$	0	0
$653$	24.2843	0.950317	0.475158	−	0.879900i	$-0.342391\pi$
$653$	24.2843	0.950317	0.475158	+	0.879900i	$0.342391\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−23.1127	−0.901712
$658$	0	0
$659$	10.8995	0.424584	0.212292	−	0.977206i	$-0.431907\pi$
$659$	10.8995	0.424584	0.212292	+	0.977206i	$0.431907\pi$
$660$	0	0
$661$	26.5147	1.03130	0.515652	−	0.856798i	$-0.327550\pi$
$661$	26.5147	1.03130	0.515652	+	0.856798i	$0.327550\pi$
$662$	0	0
$663$	1.58579	0.0615868
$664$	0	0
$665$	0	0
$666$	0	0
$667$	5.92893	0.229569
$668$	0	0
$669$	−3.65685	−0.141382
$670$	0	0
$671$	−0.828427	−0.0319811
$672$	0	0
$673$	28.0000	1.07932	0.539660	−	0.841883i	$-0.318553\pi$
$673$	28.0000	1.07932	0.539660	+	0.841883i	$0.318553\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−32.6569	−1.25510	−0.627552	−	0.778574i	$-0.715943\pi$
$677$	−32.6569	−1.25510	−0.627552	+	0.778574i	$0.715943\pi$
$678$	0	0
$679$	−1.51472	−0.0581296
$680$	0	0
$681$	3.62742	0.139003
$682$	0	0
$683$	0.686292	0.0262602	0.0131301	−	0.999914i	$-0.495820\pi$
$683$	0.686292	0.0262602	0.0131301	+	0.999914i	$0.495820\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−2.88730	−0.110157
$688$	0	0
$689$	−14.6569	−0.558382
$690$	0	0
$691$	−43.4558	−1.65314	−0.826569	−	0.562835i	$-0.809711\pi$
$691$	−43.4558	−1.65314	−0.826569	+	0.562835i	$0.809711\pi$
$692$	0	0
$693$	1.65685	0.0629387
$694$	0	0
$695$	0	0
$696$	0	0
$697$	7.07107	0.267836
$698$	0	0
$699$	5.65685	0.213962
$700$	0	0
$701$	−0.970563	−0.0366576	−0.0183288	−	0.999832i	$-0.505835\pi$
$701$	−0.970563	−0.0366576	−0.0183288	+	0.999832i	$0.505835\pi$
$702$	0	0
$703$	−10.8284	−0.408402
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.61522	−0.0607467
$708$	0	0
$709$	33.3137	1.25112	0.625561	−	0.780175i	$-0.284870\pi$
$709$	33.3137	1.25112	0.625561	+	0.780175i	$0.284870\pi$
$710$	0	0
$711$	−13.6569	−0.512172
$712$	0	0
$713$	−1.89949	−0.0711366
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−11.4853	−0.428926
$718$	0	0
$719$	27.5858	1.02878	0.514388	−	0.857557i	$-0.328019\pi$
$719$	27.5858	1.02878	0.514388	+	0.857557i	$0.328019\pi$
$720$	0	0
$721$	4.10051	0.152711
$722$	0	0
$723$	2.34315	0.0871425
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−44.2132	−1.63978	−0.819888	−	0.572523i	$-0.805965\pi$
$727$	−44.2132	−1.63978	−0.819888	+	0.572523i	$0.805965\pi$
$728$	0	0
$729$	−18.1716	−0.673021
$730$	0	0
$731$	−6.24264	−0.230892
$732$	0	0
$733$	34.6274	1.27899	0.639496	−	0.768794i	$-0.279143\pi$
$733$	34.6274	1.27899	0.639496	+	0.768794i	$0.279143\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−11.4142	−0.420448
$738$	0	0
$739$	4.58579	0.168691	0.0843454	−	0.996437i	$-0.473120\pi$
$739$	4.58579	0.168691	0.0843454	+	0.996437i	$0.473120\pi$
$740$	0	0
$741$	−1.58579	−0.0582553
$742$	0	0
$743$	5.75736	0.211217	0.105609	−	0.994408i	$-0.466321\pi$
$743$	5.75736	0.211217	0.105609	+	0.994408i	$0.466321\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−40.9706	−1.49903
$748$	0	0
$749$	0.171573	0.00626914
$750$	0	0
$751$	38.2426	1.39549	0.697747	−	0.716344i	$-0.254186\pi$
$751$	38.2426	1.39549	0.697747	+	0.716344i	$0.254186\pi$
$752$	0	0
$753$	−1.47309	−0.0536823
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−27.6569	−1.00521	−0.502603	−	0.864517i	$-0.667624\pi$
$757$	−27.6569	−1.00521	−0.502603	+	0.864517i	$0.667624\pi$
$758$	0	0
$759$	−1.89949	−0.0689473
$760$	0	0
$761$	−37.9706	−1.37643	−0.688216	−	0.725506i	$-0.741606\pi$
$761$	−37.9706	−1.37643	−0.688216	+	0.725506i	$0.741606\pi$
$762$	0	0
$763$	−2.07107	−0.0749777
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−44.3553	−1.60158
$768$	0	0
$769$	8.79899	0.317300	0.158650	−	0.987335i	$-0.449286\pi$
$769$	8.79899	0.317300	0.158650	+	0.987335i	$0.449286\pi$
$770$	0	0
$771$	−11.8995	−0.428550
$772$	0	0
$773$	9.00000	0.323708	0.161854	−	0.986815i	$-0.448253\pi$
$773$	9.00000	0.323708	0.161854	+	0.986815i	$0.448253\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−1.85786	−0.0666505
$778$	0	0
$779$	−7.07107	−0.253347
$780$	0	0
$781$	15.6569	0.560246
$782$	0	0
$783$	−4.41421	−0.157751
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−31.2426	−1.11368	−0.556840	−	0.830620i	$-0.687986\pi$
$787$	−31.2426	−1.11368	−0.556840	+	0.830620i	$0.687986\pi$
$788$	0	0
$789$	6.48528	0.230882
$790$	0	0
$791$	0.443651	0.0157744
$792$	0	0
$793$	−2.24264	−0.0796385
$794$	0	0
$795$	0	0
$796$	0	0
$797$	6.85786	0.242918	0.121459	−	0.992596i	$-0.461243\pi$
$797$	6.85786	0.242918	0.121459	+	0.992596i	$0.461243\pi$
$798$	0	0
$799$	8.00000	0.283020
$800$	0	0
$801$	34.6274	1.22350
$802$	0	0
$803$	−11.5563	−0.407815
$804$	0	0
$805$	0	0
$806$	0	0
$807$	4.14214	0.145810
$808$	0	0
$809$	−25.4853	−0.896015	−0.448007	−	0.894030i	$-0.647866\pi$
$809$	−25.4853	−0.896015	−0.448007	+	0.894030i	$0.647866\pi$
$810$	0	0
$811$	40.0122	1.40502	0.702509	−	0.711675i	$-0.252063\pi$
$811$	40.0122	1.40502	0.702509	+	0.711675i	$0.252063\pi$
$812$	0	0
$813$	−6.37258	−0.223496
$814$	0	0
$815$	0	0
$816$	0	0
$817$	6.24264	0.218402
$818$	0	0
$819$	4.48528	0.156728
$820$	0	0
$821$	27.5980	0.963176	0.481588	−	0.876398i	$-0.340060\pi$
$821$	27.5980	0.963176	0.481588	+	0.876398i	$0.340060\pi$
$822$	0	0
$823$	17.5269	0.610950	0.305475	−	0.952200i	$-0.401185\pi$
$823$	17.5269	0.610950	0.305475	+	0.952200i	$0.401185\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	19.8701	0.690950	0.345475	−	0.938428i	$-0.387718\pi$
$827$	19.8701	0.690950	0.345475	+	0.938428i	$0.387718\pi$
$828$	0	0
$829$	−47.7696	−1.65911	−0.829553	−	0.558429i	$-0.811404\pi$
$829$	−47.7696	−1.65911	−0.829553	+	0.558429i	$0.811404\pi$
$830$	0	0
$831$	1.37258	0.0476144
$832$	0	0
$833$	−6.82843	−0.236591
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.41421	0.0488824
$838$	0	0
$839$	17.9411	0.619396	0.309698	−	0.950835i	$-0.399772\pi$
$839$	17.9411	0.619396	0.309698	+	0.950835i	$0.399772\pi$
$840$	0	0
$841$	−25.6569	−0.884719
$842$	0	0
$843$	1.55635	0.0536035
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−3.72792	−0.128093
$848$	0	0
$849$	−3.94113	−0.135259
$850$	0	0
$851$	−35.1127	−1.20365
$852$	0	0
$853$	47.9411	1.64147	0.820736	−	0.571307i	$-0.193563\pi$
$853$	47.9411	1.64147	0.820736	+	0.571307i	$0.193563\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	16.0000	0.546550	0.273275	−	0.961936i	$-0.411893\pi$
$857$	16.0000	0.546550	0.273275	+	0.961936i	$0.411893\pi$
$858$	0	0
$859$	−37.2132	−1.26970	−0.634849	−	0.772636i	$-0.718938\pi$
$859$	−37.2132	−1.26970	−0.634849	+	0.772636i	$0.718938\pi$
$860$	0	0
$861$	−1.21320	−0.0413459
$862$	0	0
$863$	−20.9289	−0.712429	−0.356215	−	0.934404i	$-0.615933\pi$
$863$	−20.9289	−0.712429	−0.356215	+	0.934404i	$0.615933\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	6.62742	0.225079
$868$	0	0
$869$	−6.82843	−0.231639
$870$	0	0
$871$	−30.8995	−1.04699
$872$	0	0
$873$	10.3431	0.350062
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−12.5147	−0.422592	−0.211296	−	0.977422i	$-0.567768\pi$
$877$	−12.5147	−0.422592	−0.211296	+	0.977422i	$0.567768\pi$
$878$	0	0
$879$	−4.47309	−0.150874
$880$	0	0
$881$	−44.2843	−1.49198	−0.745988	−	0.665960i	$-0.768022\pi$
$881$	−44.2843	−1.49198	−0.745988	+	0.665960i	$0.768022\pi$
$882$	0	0
$883$	−35.4558	−1.19318	−0.596592	−	0.802545i	$-0.703479\pi$
$883$	−35.4558	−1.19318	−0.596592	+	0.802545i	$0.703479\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	15.0711	0.506037	0.253018	−	0.967461i	$-0.418577\pi$
$887$	15.0711	0.506037	0.253018	+	0.967461i	$0.418577\pi$
$888$	0	0
$889$	6.97056	0.233785
$890$	0	0
$891$	−10.5858	−0.354637
$892$	0	0
$893$	−8.00000	−0.267710
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−5.14214	−0.171691
$898$	0	0
$899$	−1.07107	−0.0357221
$900$	0	0
$901$	3.82843	0.127543
$902$	0	0
$903$	1.07107	0.0356429
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−22.5563	−0.748971	−0.374486	−	0.927233i	$-0.622181\pi$
$907$	−22.5563	−0.748971	−0.374486	+	0.927233i	$0.622181\pi$
$908$	0	0
$909$	11.0294	0.365823
$910$	0	0
$911$	1.65685	0.0548940	0.0274470	−	0.999623i	$-0.491262\pi$
$911$	1.65685	0.0548940	0.0274470	+	0.999623i	$0.491262\pi$
$912$	0	0
$913$	−20.4853	−0.677964
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.28427	0.141479
$918$	0	0
$919$	−40.8406	−1.34721	−0.673604	−	0.739093i	$-0.735255\pi$
$919$	−40.8406	−1.34721	−0.673604	+	0.739093i	$0.735255\pi$
$920$	0	0
$921$	−13.3726	−0.440642
$922$	0	0
$923$	42.3848	1.39511
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−28.0000	−0.919641
$928$	0	0
$929$	8.51472	0.279359	0.139679	−	0.990197i	$-0.455393\pi$
$929$	8.51472	0.279359	0.139679	+	0.990197i	$0.455393\pi$
$930$	0	0
$931$	6.82843	0.223793
$932$	0	0
$933$	−9.62742	−0.315187
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−0.313708	−0.0102484	−0.00512420	−	0.999987i	$-0.501631\pi$
$937$	−0.313708	−0.0102484	−0.00512420	+	0.999987i	$0.501631\pi$
$938$	0	0
$939$	−2.75736	−0.0899830
$940$	0	0
$941$	28.8579	0.940739	0.470370	−	0.882469i	$-0.344121\pi$
$941$	28.8579	0.940739	0.470370	+	0.882469i	$0.344121\pi$
$942$	0	0
$943$	−22.9289	−0.746669
$944$	0	0
$945$	0	0
$946$	0	0
$947$	49.2548	1.60057	0.800284	−	0.599622i	$-0.204682\pi$
$947$	49.2548	1.60057	0.800284	+	0.599622i	$0.204682\pi$
$948$	0	0
$949$	−31.2843	−1.01553
$950$	0	0
$951$	−2.81623	−0.0913226
$952$	0	0
$953$	−23.2132	−0.751949	−0.375975	−	0.926630i	$-0.622692\pi$
$953$	−23.2132	−0.751949	−0.375975	+	0.926630i	$0.622692\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−1.07107	−0.0346227
$958$	0	0
$959$	−2.61522	−0.0844500
$960$	0	0
$961$	−30.6569	−0.988931
$962$	0	0
$963$	−1.17157	−0.0377534
$964$	0	0
$965$	0	0
$966$	0	0
$967$	21.8579	0.702902	0.351451	−	0.936206i	$-0.385688\pi$
$967$	21.8579	0.702902	0.351451	+	0.936206i	$0.385688\pi$
$968$	0	0
$969$	0.414214	0.0133065
$970$	0	0
$971$	43.5980	1.39913	0.699563	−	0.714571i	$-0.253378\pi$
$971$	43.5980	1.39913	0.699563	+	0.714571i	$0.253378\pi$
$972$	0	0
$973$	−1.65685	−0.0531163
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−38.3848	−1.22804	−0.614019	−	0.789291i	$-0.710448\pi$
$977$	−38.3848	−1.22804	−0.614019	+	0.789291i	$0.710448\pi$
$978$	0	0
$979$	17.3137	0.553349
$980$	0	0
$981$	14.1421	0.451524
$982$	0	0
$983$	−14.2010	−0.452942	−0.226471	−	0.974018i	$-0.572719\pi$
$983$	−14.2010	−0.452942	−0.226471	+	0.974018i	$0.572719\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−1.37258	−0.0436898
$988$	0	0
$989$	20.2426	0.643679
$990$	0	0
$991$	−54.5269	−1.73210	−0.866052	−	0.499954i	$-0.833350\pi$
$991$	−54.5269	−1.73210	−0.866052	+	0.499954i	$0.833350\pi$
$992$	0	0
$993$	−8.02944	−0.254806
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−3.55635	−0.112631	−0.0563154	−	0.998413i	$-0.517935\pi$
$997$	−3.55635	−0.112631	−0.0563154	+	0.998413i	$0.517935\pi$
$998$	0	0
$999$	26.1421	0.827101

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.a.p.1.1		2
4.3	odd	2		7600.2.a.x.1.2		2
5.2	odd	4		760.2.d.c.609.3	yes	4
5.3	odd	4		760.2.d.c.609.2	✓	4
5.4	even	2		3800.2.a.l.1.2		2
20.3	even	4		1520.2.d.d.609.3		4
20.7	even	4		1520.2.d.d.609.2		4
20.19	odd	2		7600.2.a.bc.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.d.c.609.2	✓	4	5.3	odd	4
760.2.d.c.609.3	yes	4	5.2	odd	4
1520.2.d.d.609.2		4	20.7	even	4
1520.2.d.d.609.3		4	20.3	even	4
3800.2.a.l.1.2		2	5.4	even	2
3800.2.a.p.1.1		2	1.1	even	1	trivial
7600.2.a.x.1.2		2	4.3	odd	2
7600.2.a.bc.1.1		2	20.19	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:	\( N \)	\(=\)	\( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3800.a (trivial)

\(f(q)\)	\(=\)	\(q-0.414214 q^{3} +0.414214 q^{7} -2.82843 q^{9} +O(q^{10})\)	\(q-0.414214 q^{3} +0.414214 q^{7} -2.82843 q^{9} -1.41421 q^{11} -3.82843 q^{13} +1.00000 q^{17} -1.00000 q^{19} -0.171573 q^{21} -3.24264 q^{23} +2.41421 q^{27} -1.82843 q^{29} +0.585786 q^{31} +0.585786 q^{33} +10.8284 q^{37} +1.58579 q^{39} +7.07107 q^{41} -6.24264 q^{43} +8.00000 q^{47} -6.82843 q^{49} -0.414214 q^{51} +3.82843 q^{53} +0.414214 q^{57} +11.5858 q^{59} +0.585786 q^{61} -1.17157 q^{63} +8.07107 q^{67} +1.34315 q^{69} -11.0711 q^{71} +8.17157 q^{73} -0.585786 q^{77} +4.82843 q^{79} +7.48528 q^{81} +14.4853 q^{83} +0.757359 q^{87} -12.2426 q^{89} -1.58579 q^{91} -0.242641 q^{93} -3.65685 q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{7}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^7	\( 2 q + 2 q^{3} - 2 q^{7} - 2 q^{13} + 2 q^{17} - 2 q^{19} - 6 q^{21} + 2 q^{23} + 2 q^{27} + 2 q^{29} + 4 q^{31} + 4 q^{33} + 16 q^{37} + 6 q^{39} - 4 q^{43} + 16 q^{47} - 8 q^{49} + 2 q^{51} + 2 q^{53} - 2 q^{57} + 26 q^{59} + 4 q^{61} - 8 q^{63} + 2 q^{67} + 14 q^{69} - 8 q^{71} + 22 q^{73} - 4 q^{77} + 4 q^{79} - 2 q^{81} + 12 q^{83} + 10 q^{87} - 16 q^{89} - 6 q^{91} + 8 q^{93} + 4 q^{97} + 8 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^7 - 2 * q^13 + 2 * q^17 - 2 * q^19 - 6 * q^21 + 2 * q^23 + 2 * q^27 + 2 * q^29 + 4 * q^31 + 4 * q^33 + 16 * q^37 + 6 * q^39 - 4 * q^43 + 16 * q^47 - 8 * q^49 + 2 * q^51 + 2 * q^53 - 2 * q^57 + 26 * q^59 + 4 * q^61 - 8 * q^63 + 2 * q^67 + 14 * q^69 - 8 * q^71 + 22 * q^73 - 4 * q^77 + 4 * q^79 - 2 * q^81 + 12 * q^83 + 10 * q^87 - 16 * q^89 - 6 * q^91 + 8 * q^93 + 4 * q^97 + 8 * q^99

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