LMFDB - Embedded newform 1368.4.a.e.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1368,4,Mod(1,1368)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1368.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1368, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1368 = 2^{3} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1368.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,-2,0,-35,0,0,0,28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$80.7146128879$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.3221.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 9x + 2 $ x^3 - x^2 - 9*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 152)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.44437$ of defining polynomial
Character	$\chi$	$=$	1368.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-14.1468 q^{5} -4.06119 q^{7} +5.53059 q^{11} -45.6697 q^{13} +92.8486 q^{17} +19.0000 q^{19} +57.0689 q^{23} +75.1314 q^{25} +290.147 q^{29} +163.494 q^{31} +57.4527 q^{35} -81.3659 q^{37} -73.1997 q^{41} -346.600 q^{43} -503.383 q^{47} -326.507 q^{49} +164.243 q^{53} -78.2401 q^{55} +763.585 q^{59} +545.832 q^{61} +646.079 q^{65} -510.800 q^{67} +294.139 q^{71} -172.903 q^{73} -22.4608 q^{77} -973.626 q^{79} -510.932 q^{83} -1313.51 q^{85} -845.158 q^{89} +185.473 q^{91} -268.789 q^{95} -863.606 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{5} - 35 q^{7} + 28 q^{11} - 109 q^{13} + 123 q^{17} + 57 q^{19} + 193 q^{23} + 187 q^{25} + 297 q^{29} - 140 q^{31} + 246 q^{35} + 38 q^{37} - 736 q^{41} - 514 q^{43} - 134 q^{47} - 42 q^{49}+ \cdots - 2178 q^{97}+O(q^{100}) $ 3 * q - 2 * q^5 - 35 * q^7 + 28 * q^11 - 109 * q^13 + 123 * q^17 + 57 * q^19 + 193 * q^23 + 187 * q^25 + 297 * q^29 - 140 * q^31 + 246 * q^35 + 38 * q^37 - 736 * q^41 - 514 * q^43 - 134 * q^47 - 42 * q^49 - 311 * q^53 - 130 * q^55 - 199 * q^59 + 56 * q^61 - 1156 * q^65 - 509 * q^67 + 874 * q^71 + 203 * q^73 - 616 * q^77 + 242 * q^79 + 62 * q^83 - 1430 * q^85 - 1764 * q^89 - 687 * q^91 - 38 * q^95 - 2178 * q^97

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−14.1468	−1.26533	−0.632663	−	0.774427i	$-0.718038\pi$
$5$	−14.1468	−1.26533	−0.632663	+	0.774427i	$0.718038\pi$
$6$	0	0
$7$	−4.06119	−0.219283	−0.109642	−	0.993971i	$-0.534970\pi$
$7$	−4.06119	−0.219283	−0.109642	+	0.993971i	$0.534970\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.53059	0.151594	0.0757971	−	0.997123i	$-0.475850\pi$
$11$	5.53059	0.151594	0.0757971	+	0.997123i	$0.475850\pi$
$12$	0	0
$13$	−45.6697	−0.974345	−0.487172	−	0.873306i	$-0.661972\pi$
$13$	−45.6697	−0.974345	−0.487172	+	0.873306i	$0.661972\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	92.8486	1.32465	0.662326	−	0.749216i	$-0.269569\pi$
$17$	92.8486	1.32465	0.662326	+	0.749216i	$0.269569\pi$
$18$	0	0
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	57.0689	0.517378	0.258689	−	0.965961i	$-0.416710\pi$
$23$	57.0689	0.517378	0.258689	+	0.965961i	$0.416710\pi$
$24$	0	0
$25$	75.1314	0.601051
$26$	0	0
$27$	0	0
$28$	0	0
$29$	290.147	1.85790	0.928948	−	0.370209i	$-0.120714\pi$
$29$	290.147	1.85790	0.928948	+	0.370209i	$0.120714\pi$
$30$	0	0
$31$	163.494	0.947240	0.473620	−	0.880729i	$-0.342947\pi$
$31$	163.494	0.947240	0.473620	+	0.880729i	$0.342947\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	57.4527	0.277465
$36$	0	0
$37$	−81.3659	−0.361526	−0.180763	−	0.983527i	$-0.557857\pi$
$37$	−81.3659	−0.361526	−0.180763	+	0.983527i	$0.557857\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−73.1997	−0.278826	−0.139413	−	0.990234i	$-0.544522\pi$
$41$	−73.1997	−0.278826	−0.139413	+	0.990234i	$0.544522\pi$
$42$	0	0
$43$	−346.600	−1.22921	−0.614605	−	0.788835i	$-0.710685\pi$
$43$	−346.600	−1.22921	−0.614605	+	0.788835i	$0.710685\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−503.383	−1.56225	−0.781127	−	0.624372i	$-0.785355\pi$
$47$	−503.383	−1.56225	−0.781127	+	0.624372i	$0.785355\pi$
$48$	0	0
$49$	−326.507	−0.951915
$50$	0	0
$51$	0	0
$52$	0	0
$53$	164.243	0.425670	0.212835	−	0.977088i	$-0.431730\pi$
$53$	164.243	0.425670	0.212835	+	0.977088i	$0.431730\pi$
$54$	0	0
$55$	−78.2401	−0.191816
$56$	0	0
$57$	0	0
$58$	0	0
$59$	763.585	1.68492	0.842460	−	0.538759i	$-0.181107\pi$
$59$	763.585	1.68492	0.842460	+	0.538759i	$0.181107\pi$
$60$	0	0
$61$	545.832	1.14568	0.572841	−	0.819666i	$-0.305841\pi$
$61$	545.832	1.14568	0.572841	+	0.819666i	$0.305841\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	646.079	1.23286
$66$	0	0
$67$	−510.800	−0.931406	−0.465703	−	0.884941i	$-0.654199\pi$
$67$	−510.800	−0.931406	−0.465703	+	0.884941i	$0.654199\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	294.139	0.491659	0.245830	−	0.969313i	$-0.420940\pi$
$71$	294.139	0.491659	0.245830	+	0.969313i	$0.420940\pi$
$72$	0	0
$73$	−172.903	−0.277217	−0.138608	−	0.990347i	$-0.544263\pi$
$73$	−172.903	−0.277217	−0.138608	+	0.990347i	$0.544263\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−22.4608	−0.0332421
$78$	0	0
$79$	−973.626	−1.38660	−0.693300	−	0.720649i	$-0.743844\pi$
$79$	−973.626	−1.38660	−0.693300	+	0.720649i	$0.743844\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−510.932	−0.675688	−0.337844	−	0.941202i	$-0.609698\pi$
$83$	−510.932	−0.675688	−0.337844	+	0.941202i	$0.609698\pi$
$84$	0	0
$85$	−1313.51	−1.67612
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−845.158	−1.00659	−0.503295	−	0.864114i	$-0.667879\pi$
$89$	−845.158	−1.00659	−0.503295	+	0.864114i	$0.667879\pi$
$90$	0	0
$91$	185.473	0.213658
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−268.789	−0.290286
$96$	0	0
$97$	−863.606	−0.903979	−0.451989	−	0.892023i	$-0.649286\pi$
$97$	−863.606	−0.903979	−0.451989	+	0.892023i	$0.649286\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1016.53	1.00147	0.500735	−	0.865600i	$-0.333063\pi$
$101$	1016.53	1.00147	0.500735	+	0.865600i	$0.333063\pi$
$102$	0	0
$103$	−185.045	−0.177020	−0.0885100	−	0.996075i	$-0.528211\pi$
$103$	−185.045	−0.177020	−0.0885100	+	0.996075i	$0.528211\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	396.796	0.358502	0.179251	−	0.983803i	$-0.442633\pi$
$107$	396.796	0.358502	0.179251	+	0.983803i	$0.442633\pi$
$108$	0	0
$109$	1355.57	1.19119	0.595595	−	0.803285i	$-0.296916\pi$
$109$	1355.57	1.19119	0.595595	+	0.803285i	$0.296916\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1915.38	−1.59455	−0.797275	−	0.603616i	$-0.793726\pi$
$113$	−1915.38	−1.59455	−0.797275	+	0.603616i	$0.793726\pi$
$114$	0	0
$115$	−807.341	−0.654652
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−377.075	−0.290474
$120$	0	0
$121$	−1300.41	−0.977019
$122$	0	0
$123$	0	0
$124$	0	0
$125$	705.481	0.504801
$126$	0	0
$127$	985.870	0.688833	0.344417	−	0.938817i	$-0.388077\pi$
$127$	985.870	0.688833	0.344417	+	0.938817i	$0.388077\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2296.18	−1.53144	−0.765718	−	0.643176i	$-0.777616\pi$
$131$	−2296.18	−1.53144	−0.765718	+	0.643176i	$0.777616\pi$
$132$	0	0
$133$	−77.1626	−0.0503071
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1344.91	−0.838712	−0.419356	−	0.907822i	$-0.637744\pi$
$137$	−1344.91	−0.838712	−0.419356	+	0.907822i	$0.637744\pi$
$138$	0	0
$139$	402.094	0.245361	0.122681	−	0.992446i	$-0.460851\pi$
$139$	402.094	0.245361	0.122681	+	0.992446i	$0.460851\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−252.580	−0.147705
$144$	0	0
$145$	−4104.65	−2.35085
$146$	0	0
$147$	0	0
$148$	0	0
$149$	738.796	0.406205	0.203102	−	0.979158i	$-0.434898\pi$
$149$	738.796	0.406205	0.203102	+	0.979158i	$0.434898\pi$
$150$	0	0
$151$	819.258	0.441525	0.220762	−	0.975328i	$-0.429145\pi$
$151$	819.258	0.441525	0.220762	+	0.975328i	$0.429145\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2312.92	−1.19857
$156$	0	0
$157$	−1469.33	−0.746913	−0.373456	−	0.927648i	$-0.621827\pi$
$157$	−1469.33	−0.746913	−0.373456	+	0.927648i	$0.621827\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−231.767	−0.113452
$162$	0	0
$163$	−582.247	−0.279786	−0.139893	−	0.990167i	$-0.544676\pi$
$163$	−582.247	−0.279786	−0.139893	+	0.990167i	$0.544676\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2401.31	−1.11269	−0.556345	−	0.830952i	$-0.687797\pi$
$167$	−2401.31	−1.11269	−0.556345	+	0.830952i	$0.687797\pi$
$168$	0	0
$169$	−111.282	−0.0506519
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1104.21	−0.485269	−0.242635	−	0.970118i	$-0.578012\pi$
$173$	−1104.21	−0.485269	−0.242635	+	0.970118i	$0.578012\pi$
$174$	0	0
$175$	−305.122	−0.131801
$176$	0	0
$177$	0	0
$178$	0	0
$179$	822.030	0.343248	0.171624	−	0.985163i	$-0.445099\pi$
$179$	822.030	0.343248	0.171624	+	0.985163i	$0.445099\pi$
$180$	0	0
$181$	151.476	0.0622052	0.0311026	−	0.999516i	$-0.490098\pi$
$181$	151.476	0.0622052	0.0311026	+	0.999516i	$0.490098\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1151.07	0.457449
$186$	0	0
$187$	513.508	0.200810
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−477.884	−0.181039	−0.0905195	−	0.995895i	$-0.528853\pi$
$191$	−477.884	−0.181039	−0.0905195	+	0.995895i	$0.528853\pi$
$192$	0	0
$193$	−2169.48	−0.809134	−0.404567	−	0.914508i	$-0.632578\pi$
$193$	−2169.48	−0.809134	−0.404567	+	0.914508i	$0.632578\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1444.93	−0.522574	−0.261287	−	0.965261i	$-0.584147\pi$
$197$	−1444.93	−0.522574	−0.261287	+	0.965261i	$0.584147\pi$
$198$	0	0
$199$	1490.17	0.530830	0.265415	−	0.964134i	$-0.414491\pi$
$199$	1490.17	0.530830	0.265415	+	0.964134i	$0.414491\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1178.34	−0.407406
$204$	0	0
$205$	1035.54	0.352806
$206$	0	0
$207$	0	0
$208$	0	0
$209$	105.081	0.0347781
$210$	0	0
$211$	−2270.00	−0.740630	−0.370315	−	0.928906i	$-0.620750\pi$
$211$	−2270.00	−0.740630	−0.370315	+	0.928906i	$0.620750\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4903.28	1.55535
$216$	0	0
$217$	−663.981	−0.207714
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−4240.36	−1.29067
$222$	0	0
$223$	2380.30	0.714783	0.357392	−	0.933955i	$-0.383666\pi$
$223$	2380.30	0.714783	0.357392	+	0.933955i	$0.383666\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1972.94	0.576865	0.288432	−	0.957500i	$-0.406866\pi$
$227$	1972.94	0.576865	0.288432	+	0.957500i	$0.406866\pi$
$228$	0	0
$229$	21.4526	0.00619053	0.00309526	−	0.999995i	$-0.499015\pi$
$229$	21.4526	0.00619053	0.00309526	+	0.999995i	$0.499015\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1711.23	−0.481143	−0.240571	−	0.970631i	$-0.577335\pi$
$233$	−1711.23	−0.481143	−0.240571	+	0.970631i	$0.577335\pi$
$234$	0	0
$235$	7121.24	1.97676
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2193.99	0.593796	0.296898	−	0.954909i	$-0.404048\pi$
$239$	2193.99	0.593796	0.296898	+	0.954909i	$0.404048\pi$
$240$	0	0
$241$	726.890	0.194287	0.0971434	−	0.995270i	$-0.469029\pi$
$241$	726.890	0.194287	0.0971434	+	0.995270i	$0.469029\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	4619.02	1.20448
$246$	0	0
$247$	−867.724	−0.223530
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−3168.49	−0.796785	−0.398393	−	0.917215i	$-0.630432\pi$
$251$	−3168.49	−0.796785	−0.398393	+	0.917215i	$0.630432\pi$
$252$	0	0
$253$	315.625	0.0784315
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4341.53	−1.05376	−0.526881	−	0.849939i	$-0.676639\pi$
$257$	−4341.53	−1.05376	−0.526881	+	0.849939i	$0.676639\pi$
$258$	0	0
$259$	330.442	0.0792767
$260$	0	0
$261$	0	0
$262$	0	0
$263$	6418.23	1.50481	0.752405	−	0.658701i	$-0.228894\pi$
$263$	6418.23	1.50481	0.752405	+	0.658701i	$0.228894\pi$
$264$	0	0
$265$	−2323.51	−0.538612
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−6476.36	−1.46792	−0.733961	−	0.679192i	$-0.762330\pi$
$269$	−6476.36	−1.46792	−0.733961	+	0.679192i	$0.762330\pi$
$270$	0	0
$271$	−6832.73	−1.53158	−0.765791	−	0.643090i	$-0.777652\pi$
$271$	−6832.73	−1.53158	−0.765791	+	0.643090i	$0.777652\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	415.521	0.0911158
$276$	0	0
$277$	−3243.37	−0.703521	−0.351761	−	0.936090i	$-0.614417\pi$
$277$	−3243.37	−0.703521	−0.351761	+	0.936090i	$0.614417\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−4524.91	−0.960618	−0.480309	−	0.877099i	$-0.659476\pi$
$281$	−4524.91	−0.960618	−0.480309	+	0.877099i	$0.659476\pi$
$282$	0	0
$283$	−363.865	−0.0764293	−0.0382147	−	0.999270i	$-0.512167\pi$
$283$	−363.865	−0.0764293	−0.0382147	+	0.999270i	$0.512167\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	297.278	0.0611420
$288$	0	0
$289$	3707.85	0.754703
$290$	0	0
$291$	0	0
$292$	0	0
$293$	4565.73	0.910350	0.455175	−	0.890402i	$-0.349577\pi$
$293$	4565.73	0.910350	0.455175	+	0.890402i	$0.349577\pi$
$294$	0	0
$295$	−10802.3	−2.13197
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2606.32	−0.504104
$300$	0	0
$301$	1407.61	0.269546
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−7721.77	−1.44966
$306$	0	0
$307$	8554.23	1.59028	0.795139	−	0.606427i	$-0.207398\pi$
$307$	8554.23	1.59028	0.795139	+	0.606427i	$0.207398\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−2620.19	−0.477740	−0.238870	−	0.971052i	$-0.576777\pi$
$311$	−2620.19	−0.477740	−0.238870	+	0.971052i	$0.576777\pi$
$312$	0	0
$313$	−5233.04	−0.945013	−0.472507	−	0.881327i	$-0.656651\pi$
$313$	−5233.04	−0.945013	−0.472507	+	0.881327i	$0.656651\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8199.12	−1.45271	−0.726354	−	0.687320i	$-0.758787\pi$
$317$	−8199.12	−1.45271	−0.726354	+	0.687320i	$0.758787\pi$
$318$	0	0
$319$	1604.69	0.281646
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1764.12	0.303896
$324$	0	0
$325$	−3431.22	−0.585631
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2044.33	0.342576
$330$	0	0
$331$	4942.10	0.820672	0.410336	−	0.911934i	$-0.365411\pi$
$331$	4942.10	0.820672	0.410336	+	0.911934i	$0.365411\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	7226.18	1.17853
$336$	0	0
$337$	−11622.4	−1.87868	−0.939339	−	0.342991i	$-0.888560\pi$
$337$	−11622.4	−1.87868	−0.939339	+	0.342991i	$0.888560\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	904.221	0.143596
$342$	0	0
$343$	2718.99	0.428023
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6401.72	0.990382	0.495191	−	0.868784i	$-0.335098\pi$
$347$	6401.72	0.990382	0.495191	+	0.868784i	$0.335098\pi$
$348$	0	0
$349$	−9713.15	−1.48978	−0.744890	−	0.667188i	$-0.767498\pi$
$349$	−9713.15	−1.48978	−0.744890	+	0.667188i	$0.767498\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	10674.0	1.60941	0.804705	−	0.593675i	$-0.202323\pi$
$353$	10674.0	1.60941	0.804705	+	0.593675i	$0.202323\pi$
$354$	0	0
$355$	−4161.11	−0.622110
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4991.30	0.733791	0.366895	−	0.930262i	$-0.380421\pi$
$359$	4991.30	0.733791	0.366895	+	0.930262i	$0.380421\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2446.03	0.350770
$366$	0	0
$367$	−7742.21	−1.10120	−0.550599	−	0.834770i	$-0.685601\pi$
$367$	−7742.21	−1.10120	−0.550599	+	0.834770i	$0.685601\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−667.022	−0.0933425
$372$	0	0
$373$	−1644.48	−0.228279	−0.114139	−	0.993465i	$-0.536411\pi$
$373$	−1644.48	−0.228279	−0.114139	+	0.993465i	$0.536411\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−13250.9	−1.81023
$378$	0	0
$379$	−1385.48	−0.187776	−0.0938881	−	0.995583i	$-0.529930\pi$
$379$	−1385.48	−0.187776	−0.0938881	+	0.995583i	$0.529930\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	6651.19	0.887363	0.443682	−	0.896185i	$-0.353672\pi$
$383$	6651.19	0.887363	0.443682	+	0.896185i	$0.353672\pi$
$384$	0	0
$385$	317.748	0.0420621
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−9322.00	−1.21502	−0.607512	−	0.794311i	$-0.707832\pi$
$389$	−9322.00	−1.21502	−0.607512	+	0.794311i	$0.707832\pi$
$390$	0	0
$391$	5298.76	0.685345
$392$	0	0
$393$	0	0
$394$	0	0
$395$	13773.7	1.75450
$396$	0	0
$397$	11403.0	1.44157	0.720783	−	0.693160i	$-0.243782\pi$
$397$	11403.0	1.44157	0.720783	+	0.693160i	$0.243782\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−3794.17	−0.472498	−0.236249	−	0.971693i	$-0.575918\pi$
$401$	−3794.17	−0.472498	−0.236249	+	0.971693i	$0.575918\pi$
$402$	0	0
$403$	−7466.73	−0.922939
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−450.002	−0.0548053
$408$	0	0
$409$	3210.32	0.388118	0.194059	−	0.980990i	$-0.437835\pi$
$409$	3210.32	0.388118	0.194059	+	0.980990i	$0.437835\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−3101.06	−0.369475
$414$	0	0
$415$	7228.05	0.854966
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−5646.89	−0.658398	−0.329199	−	0.944261i	$-0.606779\pi$
$419$	−5646.89	−0.658398	−0.329199	+	0.944261i	$0.606779\pi$
$420$	0	0
$421$	−1912.56	−0.221407	−0.110704	−	0.993853i	$-0.535310\pi$
$421$	−1912.56	−0.221407	−0.110704	+	0.993853i	$0.535310\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	6975.84	0.796183
$426$	0	0
$427$	−2216.73	−0.251229
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4568.25	0.510545	0.255273	−	0.966869i	$-0.417835\pi$
$431$	4568.25	0.510545	0.255273	+	0.966869i	$0.417835\pi$
$432$	0	0
$433$	7432.60	0.824915	0.412458	−	0.910977i	$-0.364671\pi$
$433$	7432.60	0.824915	0.412458	+	0.910977i	$0.364671\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1084.31	0.118695
$438$	0	0
$439$	−5937.86	−0.645555	−0.322777	−	0.946475i	$-0.604617\pi$
$439$	−5937.86	−0.645555	−0.322777	+	0.946475i	$0.604617\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−11714.5	−1.25638	−0.628188	−	0.778062i	$-0.716203\pi$
$443$	−11714.5	−1.25638	−0.628188	+	0.778062i	$0.716203\pi$
$444$	0	0
$445$	11956.3	1.27367
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−2007.87	−0.211040	−0.105520	−	0.994417i	$-0.533651\pi$
$449$	−2007.87	−0.211040	−0.105520	+	0.994417i	$0.533651\pi$
$450$	0	0
$451$	−404.838	−0.0422684
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2623.85	−0.270347
$456$	0	0
$457$	1447.39	0.148153	0.0740764	−	0.997253i	$-0.476399\pi$
$457$	1447.39	0.148153	0.0740764	+	0.997253i	$0.476399\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	2893.39	0.292318	0.146159	−	0.989261i	$-0.453309\pi$
$461$	2893.39	0.292318	0.146159	+	0.989261i	$0.453309\pi$
$462$	0	0
$463$	−13259.1	−1.33089	−0.665445	−	0.746447i	$-0.731758\pi$
$463$	−13259.1	−1.33089	−0.665445	+	0.746447i	$0.731758\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4031.59	−0.399486	−0.199743	−	0.979848i	$-0.564011\pi$
$467$	−4031.59	−0.399486	−0.199743	+	0.979848i	$0.564011\pi$
$468$	0	0
$469$	2074.46	0.204242
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1916.91	−0.186341
$474$	0	0
$475$	1427.50	0.137891
$476$	0	0
$477$	0	0
$478$	0	0
$479$	8616.81	0.821946	0.410973	−	0.911647i	$-0.365189\pi$
$479$	8616.81	0.821946	0.410973	+	0.911647i	$0.365189\pi$
$480$	0	0
$481$	3715.95	0.352251
$482$	0	0
$483$	0	0
$484$	0	0
$485$	12217.2	1.14383
$486$	0	0
$487$	−10149.9	−0.944429	−0.472215	−	0.881484i	$-0.656545\pi$
$487$	−10149.9	−0.944429	−0.472215	+	0.881484i	$0.656545\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	5982.00	0.549825	0.274912	−	0.961469i	$-0.411351\pi$
$491$	5982.00	0.549825	0.274912	+	0.961469i	$0.411351\pi$
$492$	0	0
$493$	26939.8	2.46107
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1194.55	−0.107813
$498$	0	0
$499$	2022.13	0.181409	0.0907044	−	0.995878i	$-0.471088\pi$
$499$	2022.13	0.181409	0.0907044	+	0.995878i	$0.471088\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1621.54	0.143739	0.0718697	−	0.997414i	$-0.477103\pi$
$503$	1621.54	0.143739	0.0718697	+	0.997414i	$0.477103\pi$
$504$	0	0
$505$	−14380.6	−1.26719
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1108.58	0.0965358	0.0482679	−	0.998834i	$-0.484630\pi$
$509$	1108.58	0.0965358	0.0482679	+	0.998834i	$0.484630\pi$
$510$	0	0
$511$	702.193	0.0607890
$512$	0	0
$513$	0	0
$514$	0	0
$515$	2617.80	0.223988
$516$	0	0
$517$	−2784.01	−0.236829
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−5249.04	−0.441391	−0.220696	−	0.975343i	$-0.570833\pi$
$521$	−5249.04	−0.441391	−0.220696	+	0.975343i	$0.570833\pi$
$522$	0	0
$523$	6191.33	0.517645	0.258822	−	0.965925i	$-0.416666\pi$
$523$	6191.33	0.517645	0.258822	+	0.965925i	$0.416666\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	15180.2	1.25476
$528$	0	0
$529$	−8910.14	−0.732320
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3343.01	0.271673
$534$	0	0
$535$	−5613.38	−0.453622
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1805.78	−0.144305
$540$	0	0
$541$	−16975.2	−1.34903	−0.674513	−	0.738263i	$-0.735646\pi$
$541$	−16975.2	−1.34903	−0.674513	+	0.738263i	$0.735646\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−19176.9	−1.50724
$546$	0	0
$547$	16676.3	1.30352	0.651762	−	0.758424i	$-0.274030\pi$
$547$	16676.3	1.30352	0.651762	+	0.758424i	$0.274030\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	5512.80	0.426231
$552$	0	0
$553$	3954.08	0.304059
$554$	0	0
$555$	0	0
$556$	0	0
$557$	12448.6	0.946973	0.473486	−	0.880801i	$-0.342995\pi$
$557$	12448.6	0.946973	0.473486	+	0.880801i	$0.342995\pi$
$558$	0	0
$559$	15829.1	1.19768
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−12829.5	−0.960386	−0.480193	−	0.877163i	$-0.659433\pi$
$563$	−12829.5	−0.960386	−0.480193	+	0.877163i	$0.659433\pi$
$564$	0	0
$565$	27096.5	2.01763
$566$	0	0
$567$	0	0
$568$	0	0
$569$	916.141	0.0674985	0.0337492	−	0.999430i	$-0.489255\pi$
$569$	916.141	0.0674985	0.0337492	+	0.999430i	$0.489255\pi$
$570$	0	0
$571$	1247.99	0.0914654	0.0457327	−	0.998954i	$-0.485438\pi$
$571$	1247.99	0.0914654	0.0457327	+	0.998954i	$0.485438\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4287.66	0.310970
$576$	0	0
$577$	12669.4	0.914095	0.457047	−	0.889442i	$-0.348907\pi$
$577$	12669.4	0.914095	0.457047	+	0.889442i	$0.348907\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	2074.99	0.148167
$582$	0	0
$583$	908.362	0.0645292
$584$	0	0
$585$	0	0
$586$	0	0
$587$	20350.0	1.43089	0.715446	−	0.698668i	$-0.246223\pi$
$587$	20350.0	1.43089	0.715446	+	0.698668i	$0.246223\pi$
$588$	0	0
$589$	3106.39	0.217312
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−2306.21	−0.159705	−0.0798523	−	0.996807i	$-0.525445\pi$
$593$	−2306.21	−0.159705	−0.0798523	+	0.996807i	$0.525445\pi$
$594$	0	0
$595$	5334.40	0.367545
$596$	0	0
$597$	0	0
$598$	0	0
$599$	21458.5	1.46372	0.731861	−	0.681454i	$-0.238652\pi$
$599$	21458.5	1.46372	0.731861	+	0.681454i	$0.238652\pi$
$600$	0	0
$601$	8415.81	0.571195	0.285598	−	0.958350i	$-0.407808\pi$
$601$	8415.81	0.571195	0.285598	+	0.958350i	$0.407808\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	18396.6	1.23625
$606$	0	0
$607$	−3515.81	−0.235095	−0.117547	−	0.993067i	$-0.537503\pi$
$607$	−3515.81	−0.235095	−0.117547	+	0.993067i	$0.537503\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	22989.3	1.52217
$612$	0	0
$613$	16618.5	1.09497	0.547484	−	0.836816i	$-0.315586\pi$
$613$	16618.5	1.09497	0.547484	+	0.836816i	$0.315586\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−29469.3	−1.92284	−0.961419	−	0.275088i	$-0.911293\pi$
$617$	−29469.3	−1.92284	−0.961419	+	0.275088i	$0.911293\pi$
$618$	0	0
$619$	29408.3	1.90957	0.954783	−	0.297304i	$-0.0960876\pi$
$619$	29408.3	1.90957	0.954783	+	0.297304i	$0.0960876\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	3432.34	0.220729
$624$	0	0
$625$	−19371.7	−1.23979
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−7554.70	−0.478896
$630$	0	0
$631$	1315.79	0.0830124	0.0415062	−	0.999138i	$-0.486784\pi$
$631$	1315.79	0.0830124	0.0415062	+	0.999138i	$0.486784\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−13946.9	−0.871599
$636$	0	0
$637$	14911.5	0.927493
$638$	0	0
$639$	0	0
$640$	0	0
$641$	28499.5	1.75610	0.878052	−	0.478565i	$-0.158843\pi$
$641$	28499.5	1.75610	0.878052	+	0.478565i	$0.158843\pi$
$642$	0	0
$643$	19337.7	1.18601	0.593006	−	0.805198i	$-0.297941\pi$
$643$	19337.7	1.18601	0.593006	+	0.805198i	$0.297941\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−23409.8	−1.42246	−0.711232	−	0.702957i	$-0.751863\pi$
$647$	−23409.8	−1.42246	−0.711232	+	0.702957i	$0.751863\pi$
$648$	0	0
$649$	4223.08	0.255424
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−11324.7	−0.678665	−0.339332	−	0.940667i	$-0.610201\pi$
$653$	−11324.7	−0.678665	−0.339332	+	0.940667i	$0.610201\pi$
$654$	0	0
$655$	32483.5	1.93777
$656$	0	0
$657$	0	0
$658$	0	0
$659$	6171.58	0.364811	0.182406	−	0.983223i	$-0.441612\pi$
$659$	6171.58	0.364811	0.182406	+	0.983223i	$0.441612\pi$
$660$	0	0
$661$	−10178.3	−0.598926	−0.299463	−	0.954108i	$-0.596807\pi$
$661$	−10178.3	−0.598926	−0.299463	+	0.954108i	$0.596807\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1091.60	0.0636549
$666$	0	0
$667$	16558.4	0.961234
$668$	0	0
$669$	0	0
$670$	0	0
$671$	3018.78	0.173679
$672$	0	0
$673$	−1615.15	−0.0925106	−0.0462553	−	0.998930i	$-0.514729\pi$
$673$	−1615.15	−0.0925106	−0.0462553	+	0.998930i	$0.514729\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−33333.2	−1.89232	−0.946158	−	0.323706i	$-0.895071\pi$
$677$	−33333.2	−1.89232	−0.946158	+	0.323706i	$0.895071\pi$
$678$	0	0
$679$	3507.27	0.198228
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−11254.4	−0.630507	−0.315253	−	0.949008i	$-0.602089\pi$
$683$	−11254.4	−0.630507	−0.315253	+	0.949008i	$0.602089\pi$
$684$	0	0
$685$	19026.2	1.06124
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−7500.93	−0.414750
$690$	0	0
$691$	−15873.6	−0.873891	−0.436945	−	0.899488i	$-0.643940\pi$
$691$	−15873.6	−0.873891	−0.436945	+	0.899488i	$0.643940\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−5688.34	−0.310462
$696$	0	0
$697$	−6796.49	−0.369348
$698$	0	0
$699$	0	0
$700$	0	0
$701$	14151.8	0.762494	0.381247	−	0.924473i	$-0.375495\pi$
$701$	14151.8	0.762494	0.381247	+	0.924473i	$0.375495\pi$
$702$	0	0
$703$	−1545.95	−0.0829398
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4128.32	−0.219606
$708$	0	0
$709$	−26805.8	−1.41991	−0.709953	−	0.704249i	$-0.751284\pi$
$709$	−26805.8	−1.41991	−0.709953	+	0.704249i	$0.751284\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	9330.44	0.490081
$714$	0	0
$715$	3573.20	0.186895
$716$	0	0
$717$	0	0
$718$	0	0
$719$	254.907	0.0132217	0.00661087	−	0.999978i	$-0.497896\pi$
$719$	254.907	0.0132217	0.00661087	+	0.999978i	$0.497896\pi$
$720$	0	0
$721$	751.504	0.0388176
$722$	0	0
$723$	0	0
$724$	0	0
$725$	21799.2	1.11669
$726$	0	0
$727$	10874.1	0.554745	0.277373	−	0.960762i	$-0.410536\pi$
$727$	10874.1	0.554745	0.277373	+	0.960762i	$0.410536\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−32181.3	−1.62828
$732$	0	0
$733$	−14048.2	−0.707888	−0.353944	−	0.935267i	$-0.615160\pi$
$733$	−14048.2	−0.707888	−0.353944	+	0.935267i	$0.615160\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2825.03	−0.141196
$738$	0	0
$739$	−334.036	−0.0166275	−0.00831374	−	0.999965i	$-0.502646\pi$
$739$	−334.036	−0.0166275	−0.00831374	+	0.999965i	$0.502646\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−10340.4	−0.510571	−0.255286	−	0.966866i	$-0.582169\pi$
$743$	−10340.4	−0.510571	−0.255286	+	0.966866i	$0.582169\pi$
$744$	0	0
$745$	−10451.6	−0.513981
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1611.46	−0.0786135
$750$	0	0
$751$	31144.7	1.51330	0.756648	−	0.653822i	$-0.226836\pi$
$751$	31144.7	1.51330	0.756648	+	0.653822i	$0.226836\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−11589.9	−0.558673
$756$	0	0
$757$	−1688.02	−0.0810465	−0.0405233	−	0.999179i	$-0.512902\pi$
$757$	−1688.02	−0.0810465	−0.0405233	+	0.999179i	$0.512902\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−11804.3	−0.562296	−0.281148	−	0.959664i	$-0.590715\pi$
$761$	−11804.3	−0.562296	−0.281148	+	0.959664i	$0.590715\pi$
$762$	0	0
$763$	−5505.21	−0.261208
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−34872.7	−1.64169
$768$	0	0
$769$	20685.2	0.969995	0.484998	−	0.874516i	$-0.338821\pi$
$769$	20685.2	0.969995	0.484998	+	0.874516i	$0.338821\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	24226.8	1.12727	0.563634	−	0.826024i	$-0.309403\pi$
$773$	24226.8	1.12727	0.563634	+	0.826024i	$0.309403\pi$
$774$	0	0
$775$	12283.6	0.569340
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1390.79	−0.0639671
$780$	0	0
$781$	1626.76	0.0745327
$782$	0	0
$783$	0	0
$784$	0	0
$785$	20786.3	0.945089
$786$	0	0
$787$	−2974.74	−0.134737	−0.0673685	−	0.997728i	$-0.521460\pi$
$787$	−2974.74	−0.134737	−0.0673685	+	0.997728i	$0.521460\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	7778.73	0.349659
$792$	0	0
$793$	−24928.0	−1.11629
$794$	0	0
$795$	0	0
$796$	0	0
$797$	3979.23	0.176853	0.0884264	−	0.996083i	$-0.471816\pi$
$797$	3979.23	0.176853	0.0884264	+	0.996083i	$0.471816\pi$
$798$	0	0
$799$	−46738.4	−2.06944
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−956.259	−0.0420245
$804$	0	0
$805$	3278.76	0.143554
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−27125.7	−1.17885	−0.589425	−	0.807823i	$-0.700646\pi$
$809$	−27125.7	−1.17885	−0.589425	+	0.807823i	$0.700646\pi$
$810$	0	0
$811$	8095.42	0.350516	0.175258	−	0.984523i	$-0.443924\pi$
$811$	8095.42	0.350516	0.175258	+	0.984523i	$0.443924\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	8236.91	0.354020
$816$	0	0
$817$	−6585.41	−0.282000
$818$	0	0
$819$	0	0
$820$	0	0
$821$	19575.6	0.832148	0.416074	−	0.909331i	$-0.363406\pi$
$821$	19575.6	0.832148	0.416074	+	0.909331i	$0.363406\pi$
$822$	0	0
$823$	−857.808	−0.0363321	−0.0181660	−	0.999835i	$-0.505783\pi$
$823$	−857.808	−0.0363321	−0.0181660	+	0.999835i	$0.505783\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	33329.2	1.40142	0.700708	−	0.713448i	$-0.252867\pi$
$827$	33329.2	1.40142	0.700708	+	0.713448i	$0.252867\pi$
$828$	0	0
$829$	−37545.6	−1.57299	−0.786497	−	0.617594i	$-0.788107\pi$
$829$	−37545.6	−1.57299	−0.786497	+	0.617594i	$0.788107\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−30315.7	−1.26096
$834$	0	0
$835$	33970.8	1.40791
$836$	0	0
$837$	0	0
$838$	0	0
$839$	8699.05	0.357955	0.178978	−	0.983853i	$-0.442721\pi$
$839$	8699.05	0.357955	0.178978	+	0.983853i	$0.442721\pi$
$840$	0	0
$841$	59796.5	2.45178
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1574.28	0.0640911
$846$	0	0
$847$	5281.22	0.214244
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−4643.46	−0.187046
$852$	0	0
$853$	−39492.9	−1.58524	−0.792621	−	0.609715i	$-0.791284\pi$
$853$	−39492.9	−1.58524	−0.792621	+	0.609715i	$0.791284\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	49878.9	1.98813	0.994067	−	0.108774i	$-0.0346925\pi$
$857$	49878.9	1.98813	0.994067	+	0.108774i	$0.0346925\pi$
$858$	0	0
$859$	−21676.4	−0.860988	−0.430494	−	0.902593i	$-0.641661\pi$
$859$	−21676.4	−0.860988	−0.430494	+	0.902593i	$0.641661\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	19826.9	0.782058	0.391029	−	0.920378i	$-0.372119\pi$
$863$	19826.9	0.782058	0.391029	+	0.920378i	$0.372119\pi$
$864$	0	0
$865$	15621.0	0.614024
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−5384.73	−0.210201
$870$	0	0
$871$	23328.1	0.907511
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−2865.09	−0.110694
$876$	0	0
$877$	−28960.6	−1.11509	−0.557543	−	0.830148i	$-0.688256\pi$
$877$	−28960.6	−1.11509	−0.557543	+	0.830148i	$0.688256\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−9612.36	−0.367592	−0.183796	−	0.982964i	$-0.558839\pi$
$881$	−9612.36	−0.367592	−0.183796	+	0.982964i	$0.558839\pi$
$882$	0	0
$883$	−37634.8	−1.43433	−0.717164	−	0.696905i	$-0.754560\pi$
$883$	−37634.8	−1.43433	−0.717164	+	0.696905i	$0.754560\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−11501.8	−0.435390	−0.217695	−	0.976017i	$-0.569854\pi$
$887$	−11501.8	−0.435390	−0.217695	+	0.976017i	$0.569854\pi$
$888$	0	0
$889$	−4003.80	−0.151050
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−9564.27	−0.358406
$894$	0	0
$895$	−11629.1	−0.434321
$896$	0	0
$897$	0	0
$898$	0	0
$899$	47437.4	1.75987
$900$	0	0
$901$	15249.7	0.563865
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2142.90	−0.0787099
$906$	0	0
$907$	−18651.0	−0.682796	−0.341398	−	0.939919i	$-0.610900\pi$
$907$	−18651.0	−0.682796	−0.341398	+	0.939919i	$0.610900\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−42680.1	−1.55220	−0.776100	−	0.630610i	$-0.782805\pi$
$911$	−42680.1	−1.55220	−0.776100	+	0.630610i	$0.782805\pi$
$912$	0	0
$913$	−2825.76	−0.102430
$914$	0	0
$915$	0	0
$916$	0	0
$917$	9325.21	0.335819
$918$	0	0
$919$	−5918.21	−0.212431	−0.106215	−	0.994343i	$-0.533873\pi$
$919$	−5918.21	−0.212431	−0.106215	+	0.994343i	$0.533873\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−13433.2	−0.479046
$924$	0	0
$925$	−6113.13	−0.217296
$926$	0	0
$927$	0	0
$928$	0	0
$929$	21425.2	0.756661	0.378331	−	0.925671i	$-0.376498\pi$
$929$	21425.2	0.756661	0.378331	+	0.925671i	$0.376498\pi$
$930$	0	0
$931$	−6203.63	−0.218384
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−7264.48	−0.254090
$936$	0	0
$937$	−26280.8	−0.916283	−0.458142	−	0.888879i	$-0.651485\pi$
$937$	−26280.8	−0.916283	−0.458142	+	0.888879i	$0.651485\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−17105.3	−0.592581	−0.296290	−	0.955098i	$-0.595750\pi$
$941$	−17105.3	−0.592581	−0.296290	+	0.955098i	$0.595750\pi$
$942$	0	0
$943$	−4177.43	−0.144258
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−30353.4	−1.04155	−0.520777	−	0.853692i	$-0.674358\pi$
$947$	−30353.4	−1.04155	−0.520777	+	0.853692i	$0.674358\pi$
$948$	0	0
$949$	7896.44	0.270105
$950$	0	0
$951$	0	0
$952$	0	0
$953$	16580.1	0.563569	0.281784	−	0.959478i	$-0.409074\pi$
$953$	16580.1	0.563569	0.281784	+	0.959478i	$0.409074\pi$
$954$	0	0
$955$	6760.52	0.229074
$956$	0	0
$957$	0	0
$958$	0	0
$959$	5461.94	0.183916
$960$	0	0
$961$	−3060.60	−0.102736
$962$	0	0
$963$	0	0
$964$	0	0
$965$	30691.2	1.02382
$966$	0	0
$967$	1243.10	0.0413396	0.0206698	−	0.999786i	$-0.493420\pi$
$967$	1243.10	0.0413396	0.0206698	+	0.999786i	$0.493420\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−19859.0	−0.656341	−0.328170	−	0.944619i	$-0.606432\pi$
$971$	−19859.0	−0.656341	−0.328170	+	0.944619i	$0.606432\pi$
$972$	0	0
$973$	−1632.98	−0.0538036
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−21378.0	−0.700043	−0.350021	−	0.936742i	$-0.613826\pi$
$977$	−21378.0	−0.700043	−0.350021	+	0.936742i	$0.613826\pi$
$978$	0	0
$979$	−4674.23	−0.152593
$980$	0	0
$981$	0	0
$982$	0	0
$983$	37387.5	1.21310	0.606549	−	0.795046i	$-0.292553\pi$
$983$	37387.5	1.21310	0.606549	+	0.795046i	$0.292553\pi$
$984$	0	0
$985$	20441.1	0.661227
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−19780.1	−0.635966
$990$	0	0
$991$	−56543.4	−1.81247	−0.906236	−	0.422771i	$-0.861057\pi$
$991$	−56543.4	−1.81247	−0.906236	+	0.422771i	$0.861057\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−21081.1	−0.671674
$996$	0	0
$997$	17828.4	0.566330	0.283165	−	0.959071i	$-0.408616\pi$
$997$	17828.4	0.566330	0.283165	+	0.959071i	$0.408616\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1368.4.a.e.1.1		3
3.2	odd	2		152.4.a.b.1.1	✓	3
12.11	even	2		304.4.a.j.1.3		3
24.5	odd	2		1216.4.a.x.1.3		3
24.11	even	2		1216.4.a.q.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.4.a.b.1.1	✓	3	3.2	odd	2
304.4.a.j.1.3		3	12.11	even	2
1216.4.a.q.1.1		3	24.11	even	2
1216.4.a.x.1.3		3	24.5	odd	2
1368.4.a.e.1.1		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1368.a (trivial)

\(f(q)\)	\(=\)	\(q-14.1468 q^{5} -4.06119 q^{7} +5.53059 q^{11} -45.6697 q^{13} +92.8486 q^{17} +19.0000 q^{19} +57.0689 q^{23} +75.1314 q^{25} +290.147 q^{29} +163.494 q^{31} +57.4527 q^{35} -81.3659 q^{37} -73.1997 q^{41} -346.600 q^{43} -503.383 q^{47} -326.507 q^{49} +164.243 q^{53} -78.2401 q^{55} +763.585 q^{59} +545.832 q^{61} +646.079 q^{65} -510.800 q^{67} +294.139 q^{71} -172.903 q^{73} -22.4608 q^{77} -973.626 q^{79} -510.932 q^{83} -1313.51 q^{85} -845.158 q^{89} +185.473 q^{91} -268.789 q^{95} -863.606 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{5} - 35 q^{7} + 28 q^{11} - 109 q^{13} + 123 q^{17} + 57 q^{19} + 193 q^{23} + 187 q^{25} + 297 q^{29} - 140 q^{31} + 246 q^{35} + 38 q^{37} - 736 q^{41} - 514 q^{43} - 134 q^{47} - 42 q^{49}+ \cdots - 2178 q^{97}+O(q^{100}) \) 3 * q - 2 * q^5 - 35 * q^7 + 28 * q^11 - 109 * q^13 + 123 * q^17 + 57 * q^19 + 193 * q^23 + 187 * q^25 + 297 * q^29 - 140 * q^31 + 246 * q^35 + 38 * q^37 - 736 * q^41 - 514 * q^43 - 134 * q^47 - 42 * q^49 - 311 * q^53 - 130 * q^55 - 199 * q^59 + 56 * q^61 - 1156 * q^65 - 509 * q^67 + 874 * q^71 + 203 * q^73 - 616 * q^77 + 242 * q^79 + 62 * q^83 - 1430 * q^85 - 1764 * q^89 - 687 * q^91 - 38 * q^95 - 2178 * q^97

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