LMFDB - Embedded newform 5408.2.a.bi.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5408,2,Mod(1,5408)] 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("5408.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5408, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5408 = 2^{5} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5408.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,2,0,0,0,2,0,0,0,0,0,0,0,8,0,0,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)

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

Self dual:	yes
Analytic conductor:	$43.1830974131$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.9248.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 2 $ x^4 - 5*x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 416)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.662153$ of defining polynomial
Character	$\chi$	$=$	5408.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.35829 q^{3} +2.56155 q^{5} +3.68260 q^{7} +2.56155 q^{9} +1.03399 q^{11} +6.04090 q^{15} +6.12311 q^{17} -5.00691 q^{19} +8.68466 q^{21} +7.07488 q^{23} +1.56155 q^{25} -1.03399 q^{27} -5.00000 q^{29} -3.39228 q^{31} +2.43845 q^{33} +9.43318 q^{35} -2.12311 q^{37} +4.12311 q^{41} -8.39919 q^{43} +6.56155 q^{45} +10.7575 q^{47} +6.56155 q^{49} +14.4401 q^{51} -2.56155 q^{53} +2.64861 q^{55} -11.8078 q^{57} +10.4672 q^{59} -11.2462 q^{61} +9.43318 q^{63} -9.72350 q^{67} +16.6847 q^{69} +5.00691 q^{71} +4.31534 q^{73} +3.68260 q^{75} +3.80776 q^{77} -11.5012 q^{79} -10.1231 q^{81} -2.64861 q^{83} +15.6847 q^{85} -11.7915 q^{87} -10.6847 q^{89} -8.00000 q^{93} -12.8255 q^{95} +9.80776 q^{97} +2.64861 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{5} + 2 q^{9} + 8 q^{17} + 10 q^{21} - 2 q^{25} - 20 q^{29} + 18 q^{33} + 8 q^{37} + 18 q^{45} + 18 q^{49} - 2 q^{53} - 6 q^{57} - 12 q^{61} + 42 q^{69} + 42 q^{73} - 26 q^{77} - 24 q^{81} + 38 q^{85}+ \cdots - 2 q^{97}+O(q^{100}) $ 4 * q + 2 * q^5 + 2 * q^9 + 8 * q^17 + 10 * q^21 - 2 * q^25 - 20 * q^29 + 18 * q^33 + 8 * q^37 + 18 * q^45 + 18 * q^49 - 2 * q^53 - 6 * q^57 - 12 * q^61 + 42 * q^69 + 42 * q^73 - 26 * q^77 - 24 * q^81 + 38 * q^85 - 18 * q^89 - 32 * q^93 - 2 * 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
$3$	2.35829	1.36156	0.680781	−	0.732487i	$-0.261641\pi$
$3$	2.35829	1.36156	0.680781	+	0.732487i	$0.261641\pi$
$4$	0	0
$5$	2.56155	1.14556	0.572781	−	0.819709i	$-0.305865\pi$
$5$	2.56155	1.14556	0.572781	+	0.819709i	$0.305865\pi$
$6$	0	0
$7$	3.68260	1.39189	0.695946	−	0.718094i	$-0.254985\pi$
$7$	3.68260	1.39189	0.695946	+	0.718094i	$0.254985\pi$
$8$	0	0
$9$	2.56155	0.853851
$10$	0	0
$11$	1.03399	0.311759	0.155879	−	0.987776i	$-0.450179\pi$
$11$	1.03399	0.311759	0.155879	+	0.987776i	$0.450179\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	6.04090	1.55975
$16$	0	0
$17$	6.12311	1.48507	0.742536	−	0.669807i	$-0.233623\pi$
$17$	6.12311	1.48507	0.742536	+	0.669807i	$0.233623\pi$
$18$	0	0
$19$	−5.00691	−1.14866	−0.574332	−	0.818623i	$-0.694738\pi$
$19$	−5.00691	−1.14866	−0.574332	+	0.818623i	$0.694738\pi$
$20$	0	0
$21$	8.68466	1.89515
$22$	0	0
$23$	7.07488	1.47522	0.737608	−	0.675230i	$-0.235955\pi$
$23$	7.07488	1.47522	0.737608	+	0.675230i	$0.235955\pi$
$24$	0	0
$25$	1.56155	0.312311
$26$	0	0
$27$	−1.03399	−0.198991
$28$	0	0
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	−3.39228	−0.609272	−0.304636	−	0.952469i	$-0.598535\pi$
$31$	−3.39228	−0.609272	−0.304636	+	0.952469i	$0.598535\pi$
$32$	0	0
$33$	2.43845	0.424479
$34$	0	0
$35$	9.43318	1.59450
$36$	0	0
$37$	−2.12311	−0.349036	−0.174518	−	0.984654i	$-0.555837\pi$
$37$	−2.12311	−0.349036	−0.174518	+	0.984654i	$0.555837\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.12311	0.643921	0.321960	−	0.946753i	$-0.395658\pi$
$41$	4.12311	0.643921	0.321960	+	0.946753i	$0.395658\pi$
$42$	0	0
$43$	−8.39919	−1.28086	−0.640432	−	0.768015i	$-0.721245\pi$
$43$	−8.39919	−1.28086	−0.640432	+	0.768015i	$0.721245\pi$
$44$	0	0
$45$	6.56155	0.978139
$46$	0	0
$47$	10.7575	1.56914	0.784570	−	0.620040i	$-0.212884\pi$
$47$	10.7575	1.56914	0.784570	+	0.620040i	$0.212884\pi$
$48$	0	0
$49$	6.56155	0.937365
$50$	0	0
$51$	14.4401	2.02202
$52$	0	0
$53$	−2.56155	−0.351856	−0.175928	−	0.984403i	$-0.556293\pi$
$53$	−2.56155	−0.351856	−0.175928	+	0.984403i	$0.556293\pi$
$54$	0	0
$55$	2.64861	0.357139
$56$	0	0
$57$	−11.8078	−1.56398
$58$	0	0
$59$	10.4672	1.36271	0.681354	−	0.731954i	$-0.261391\pi$
$59$	10.4672	1.36271	0.681354	+	0.731954i	$0.261391\pi$
$60$	0	0
$61$	−11.2462	−1.43993	−0.719965	−	0.694010i	$-0.755842\pi$
$61$	−11.2462	−1.43993	−0.719965	+	0.694010i	$0.755842\pi$
$62$	0	0
$63$	9.43318	1.18847
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−9.72350	−1.18791	−0.593957	−	0.804497i	$-0.702435\pi$
$67$	−9.72350	−1.18791	−0.593957	+	0.804497i	$0.702435\pi$
$68$	0	0
$69$	16.6847	2.00860
$70$	0	0
$71$	5.00691	0.594211	0.297105	−	0.954845i	$-0.403979\pi$
$71$	5.00691	0.594211	0.297105	+	0.954845i	$0.403979\pi$
$72$	0	0
$73$	4.31534	0.505073	0.252536	−	0.967587i	$-0.418735\pi$
$73$	4.31534	0.505073	0.252536	+	0.967587i	$0.418735\pi$
$74$	0	0
$75$	3.68260	0.425230
$76$	0	0
$77$	3.80776	0.433935
$78$	0	0
$79$	−11.5012	−1.29398	−0.646990	−	0.762498i	$-0.723973\pi$
$79$	−11.5012	−1.29398	−0.646990	+	0.762498i	$0.723973\pi$
$80$	0	0
$81$	−10.1231	−1.12479
$82$	0	0
$83$	−2.64861	−0.290723	−0.145362	−	0.989379i	$-0.546435\pi$
$83$	−2.64861	−0.290723	−0.145362	+	0.989379i	$0.546435\pi$
$84$	0	0
$85$	15.6847	1.70124
$86$	0	0
$87$	−11.7915	−1.26418
$88$	0	0
$89$	−10.6847	−1.13257	−0.566286	−	0.824209i	$-0.691620\pi$
$89$	−10.6847	−1.13257	−0.566286	+	0.824209i	$0.691620\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−8.00000	−0.829561
$94$	0	0
$95$	−12.8255	−1.31586
$96$	0	0
$97$	9.80776	0.995828	0.497914	−	0.867227i	$-0.334100\pi$
$97$	9.80776	0.995828	0.497914	+	0.867227i	$0.334100\pi$
$98$	0	0
$99$	2.64861	0.266196
$100$	0	0
$101$	−0.123106	−0.0122495	−0.00612473	−	0.999981i	$-0.501950\pi$
$101$	−0.123106	−0.0122495	−0.00612473	+	0.999981i	$0.501950\pi$
$102$	0	0
$103$	−8.68951	−0.856203	−0.428101	−	0.903731i	$-0.640817\pi$
$103$	−8.68951	−0.856203	−0.428101	+	0.903731i	$0.640817\pi$
$104$	0	0
$105$	22.2462	2.17101
$106$	0	0
$107$	−9.72350	−0.940006	−0.470003	−	0.882665i	$-0.655747\pi$
$107$	−9.72350	−0.940006	−0.470003	+	0.882665i	$0.655747\pi$
$108$	0	0
$109$	−1.12311	−0.107574	−0.0537870	−	0.998552i	$-0.517129\pi$
$109$	−1.12311	−0.107574	−0.0537870	+	0.998552i	$0.517129\pi$
$110$	0	0
$111$	−5.00691	−0.475235
$112$	0	0
$113$	16.3693	1.53990	0.769948	−	0.638107i	$-0.220282\pi$
$113$	16.3693	1.53990	0.769948	+	0.638107i	$0.220282\pi$
$114$	0	0
$115$	18.1227	1.68995
$116$	0	0
$117$	0	0
$118$	0	0
$119$	22.5490	2.06706
$120$	0	0
$121$	−9.93087	−0.902806
$122$	0	0
$123$	9.72350	0.876738
$124$	0	0
$125$	−8.80776	−0.787790
$126$	0	0
$127$	−0.290319	−0.0257617	−0.0128808	−	0.999917i	$-0.504100\pi$
$127$	−0.290319	−0.0257617	−0.0128808	+	0.999917i	$0.504100\pi$
$128$	0	0
$129$	−19.8078	−1.74398
$130$	0	0
$131$	−11.3381	−0.990616	−0.495308	−	0.868717i	$-0.664945\pi$
$131$	−11.3381	−0.990616	−0.495308	+	0.868717i	$0.664945\pi$
$132$	0	0
$133$	−18.4384	−1.59882
$134$	0	0
$135$	−2.64861	−0.227956
$136$	0	0
$137$	17.0000	1.45241	0.726204	−	0.687479i	$-0.241283\pi$
$137$	17.0000	1.45241	0.726204	+	0.687479i	$0.241283\pi$
$138$	0	0
$139$	20.4810	1.73717	0.868587	−	0.495537i	$-0.165029\pi$
$139$	20.4810	1.73717	0.868587	+	0.495537i	$0.165029\pi$
$140$	0	0
$141$	25.3693	2.13648
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−12.8078	−1.06363
$146$	0	0
$147$	15.4741	1.27628
$148$	0	0
$149$	6.75379	0.553292	0.276646	−	0.960972i	$-0.410777\pi$
$149$	6.75379	0.553292	0.276646	+	0.960972i	$0.410777\pi$
$150$	0	0
$151$	−20.7713	−1.69034	−0.845172	−	0.534494i	$-0.820502\pi$
$151$	−20.7713	−1.69034	−0.845172	+	0.534494i	$0.820502\pi$
$152$	0	0
$153$	15.6847	1.26803
$154$	0	0
$155$	−8.68951	−0.697958
$156$	0	0
$157$	−0.315342	−0.0251670	−0.0125835	−	0.999921i	$-0.504006\pi$
$157$	−0.315342	−0.0251670	−0.0125835	+	0.999921i	$0.504006\pi$
$158$	0	0
$159$	−6.04090	−0.479074
$160$	0	0
$161$	26.0540	2.05334
$162$	0	0
$163$	20.4810	1.60419	0.802097	−	0.597194i	$-0.203718\pi$
$163$	20.4810	1.60419	0.802097	+	0.597194i	$0.203718\pi$
$164$	0	0
$165$	6.24621	0.486267
$166$	0	0
$167$	−10.4672	−0.809974	−0.404987	−	0.914323i	$-0.632724\pi$
$167$	−10.4672	−0.809974	−0.404987	+	0.914323i	$0.632724\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−12.8255	−0.980787
$172$	0	0
$173$	8.43845	0.641563	0.320782	−	0.947153i	$-0.396054\pi$
$173$	8.43845	0.641563	0.320782	+	0.947153i	$0.396054\pi$
$174$	0	0
$175$	5.75058	0.434703
$176$	0	0
$177$	24.6847	1.85541
$178$	0	0
$179$	5.75058	0.429818	0.214909	−	0.976634i	$-0.431054\pi$
$179$	5.75058	0.429818	0.214909	+	0.976634i	$0.431054\pi$
$180$	0	0
$181$	−20.8078	−1.54663	−0.773314	−	0.634023i	$-0.781403\pi$
$181$	−20.8078	−1.54663	−0.773314	+	0.634023i	$0.781403\pi$
$182$	0	0
$183$	−26.5219	−1.96055
$184$	0	0
$185$	−5.43845	−0.399843
$186$	0	0
$187$	6.33122	0.462984
$188$	0	0
$189$	−3.80776	−0.276974
$190$	0	0
$191$	−12.3721	−0.895215	−0.447607	−	0.894230i	$-0.647724\pi$
$191$	−12.3721	−0.895215	−0.447607	+	0.894230i	$0.647724\pi$
$192$	0	0
$193$	1.87689	0.135102	0.0675509	−	0.997716i	$-0.478481\pi$
$193$	1.87689	0.135102	0.0675509	+	0.997716i	$0.478481\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5.80776	0.413786	0.206893	−	0.978364i	$-0.433665\pi$
$197$	5.80776	0.413786	0.206893	+	0.978364i	$0.433665\pi$
$198$	0	0
$199$	7.07488	0.501525	0.250763	−	0.968049i	$-0.419319\pi$
$199$	7.07488	0.501525	0.250763	+	0.968049i	$0.419319\pi$
$200$	0	0
$201$	−22.9309	−1.61742
$202$	0	0
$203$	−18.4130	−1.29234
$204$	0	0
$205$	10.5616	0.737651
$206$	0	0
$207$	18.1227	1.25961
$208$	0	0
$209$	−5.17708	−0.358106
$210$	0	0
$211$	2.35829	0.162352	0.0811758	−	0.996700i	$-0.474132\pi$
$211$	2.35829	0.162352	0.0811758	+	0.996700i	$0.474132\pi$
$212$	0	0
$213$	11.8078	0.809055
$214$	0	0
$215$	−21.5150	−1.46731
$216$	0	0
$217$	−12.4924	−0.848041
$218$	0	0
$219$	10.1768	0.687688
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−12.3721	−0.828498	−0.414249	−	0.910164i	$-0.635956\pi$
$223$	−12.3721	−0.828498	−0.414249	+	0.910164i	$0.635956\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	0	0
$227$	−6.49424	−0.431038	−0.215519	−	0.976500i	$-0.569144\pi$
$227$	−6.49424	−0.431038	−0.215519	+	0.976500i	$0.569144\pi$
$228$	0	0
$229$	19.3693	1.27996	0.639980	−	0.768391i	$-0.278943\pi$
$229$	19.3693	1.27996	0.639980	+	0.768391i	$0.278943\pi$
$230$	0	0
$231$	8.97983	0.590829
$232$	0	0
$233$	−15.3693	−1.00688	−0.503439	−	0.864031i	$-0.667932\pi$
$233$	−15.3693	−1.00688	−0.503439	+	0.864031i	$0.667932\pi$
$234$	0	0
$235$	27.5559	1.79755
$236$	0	0
$237$	−27.1231	−1.76184
$238$	0	0
$239$	−6.78456	−0.438857	−0.219428	−	0.975629i	$-0.570419\pi$
$239$	−6.78456	−0.438857	−0.219428	+	0.975629i	$0.570419\pi$
$240$	0	0
$241$	17.8769	1.15155	0.575776	−	0.817607i	$-0.304700\pi$
$241$	17.8769	1.15155	0.575776	+	0.817607i	$0.304700\pi$
$242$	0	0
$243$	−20.7713	−1.33248
$244$	0	0
$245$	16.8078	1.07381
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−6.24621	−0.395838
$250$	0	0
$251$	0.290319	0.0183248	0.00916240	−	0.999958i	$-0.497083\pi$
$251$	0.290319	0.0183248	0.00916240	+	0.999958i	$0.497083\pi$
$252$	0	0
$253$	7.31534	0.459912
$254$	0	0
$255$	36.9890	2.31634
$256$	0	0
$257$	−4.12311	−0.257192	−0.128596	−	0.991697i	$-0.541047\pi$
$257$	−4.12311	−0.257192	−0.128596	+	0.991697i	$0.541047\pi$
$258$	0	0
$259$	−7.81855	−0.485821
$260$	0	0
$261$	−12.8078	−0.792781
$262$	0	0
$263$	−6.91185	−0.426203	−0.213102	−	0.977030i	$-0.568357\pi$
$263$	−6.91185	−0.426203	−0.213102	+	0.977030i	$0.568357\pi$
$264$	0	0
$265$	−6.56155	−0.403073
$266$	0	0
$267$	−25.1976	−1.54207
$268$	0	0
$269$	−18.6847	−1.13922	−0.569612	−	0.821914i	$-0.692906\pi$
$269$	−18.6847	−1.13922	−0.569612	+	0.821914i	$0.692906\pi$
$270$	0	0
$271$	−10.4672	−0.635835	−0.317918	−	0.948118i	$-0.602984\pi$
$271$	−10.4672	−0.635835	−0.317918	+	0.948118i	$0.602984\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.61463	0.0973656
$276$	0	0
$277$	1.63068	0.0979783	0.0489891	−	0.998799i	$-0.484400\pi$
$277$	1.63068	0.0979783	0.0489891	+	0.998799i	$0.484400\pi$
$278$	0	0
$279$	−8.68951	−0.520227
$280$	0	0
$281$	12.3153	0.734672	0.367336	−	0.930088i	$-0.380270\pi$
$281$	12.3153	0.734672	0.367336	+	0.930088i	$0.380270\pi$
$282$	0	0
$283$	−25.1976	−1.49784	−0.748920	−	0.662660i	$-0.769427\pi$
$283$	−25.1976	−1.49784	−0.748920	+	0.662660i	$0.769427\pi$
$284$	0	0
$285$	−30.2462	−1.79163
$286$	0	0
$287$	15.1838	0.896269
$288$	0	0
$289$	20.4924	1.20544
$290$	0	0
$291$	23.1296	1.35588
$292$	0	0
$293$	1.00000	0.0584206	0.0292103	−	0.999573i	$-0.490701\pi$
$293$	1.00000	0.0584206	0.0292103	+	0.999573i	$0.490701\pi$
$294$	0	0
$295$	26.8122	1.56107
$296$	0	0
$297$	−1.06913	−0.0620372
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−30.9309	−1.78283
$302$	0	0
$303$	−0.290319	−0.0166784
$304$	0	0
$305$	−28.8078	−1.64953
$306$	0	0
$307$	−19.6100	−1.11920	−0.559602	−	0.828762i	$-0.689046\pi$
$307$	−19.6100	−1.11920	−0.559602	+	0.828762i	$0.689046\pi$
$308$	0	0
$309$	−20.4924	−1.16577
$310$	0	0
$311$	−10.1768	−0.577076	−0.288538	−	0.957468i	$-0.593169\pi$
$311$	−10.1768	−0.577076	−0.288538	+	0.957468i	$0.593169\pi$
$312$	0	0
$313$	13.1231	0.741762	0.370881	−	0.928680i	$-0.379056\pi$
$313$	13.1231	0.741762	0.370881	+	0.928680i	$0.379056\pi$
$314$	0	0
$315$	24.1636	1.36146
$316$	0	0
$317$	−2.80776	−0.157700	−0.0788499	−	0.996887i	$-0.525125\pi$
$317$	−2.80776	−0.157700	−0.0788499	+	0.996887i	$0.525125\pi$
$318$	0	0
$319$	−5.16994	−0.289461
$320$	0	0
$321$	−22.9309	−1.27988
$322$	0	0
$323$	−30.6578	−1.70585
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−2.64861	−0.146469
$328$	0	0
$329$	39.6155	2.18407
$330$	0	0
$331$	15.1838	0.834575	0.417287	−	0.908775i	$-0.362981\pi$
$331$	15.1838	0.834575	0.417287	+	0.908775i	$0.362981\pi$
$332$	0	0
$333$	−5.43845	−0.298025
$334$	0	0
$335$	−24.9073	−1.36083
$336$	0	0
$337$	34.4233	1.87516	0.937578	−	0.347775i	$-0.113063\pi$
$337$	34.4233	1.87516	0.937578	+	0.347775i	$0.113063\pi$
$338$	0	0
$339$	38.6037	2.09666
$340$	0	0
$341$	−3.50758	−0.189946
$342$	0	0
$343$	−1.61463	−0.0871816
$344$	0	0
$345$	42.7386	2.30097
$346$	0	0
$347$	−5.00691	−0.268785	−0.134392	−	0.990928i	$-0.542908\pi$
$347$	−5.00691	−0.268785	−0.134392	+	0.990928i	$0.542908\pi$
$348$	0	0
$349$	28.4384	1.52228	0.761138	−	0.648590i	$-0.224641\pi$
$349$	28.4384	1.52228	0.761138	+	0.648590i	$0.224641\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	13.4924	0.718129	0.359065	−	0.933313i	$-0.383096\pi$
$353$	13.4924	0.718129	0.359065	+	0.933313i	$0.383096\pi$
$354$	0	0
$355$	12.8255	0.680705
$356$	0	0
$357$	53.1771	2.81443
$358$	0	0
$359$	−1.48734	−0.0784986	−0.0392493	−	0.999229i	$-0.512497\pi$
$359$	−1.48734	−0.0784986	−0.0392493	+	0.999229i	$0.512497\pi$
$360$	0	0
$361$	6.06913	0.319428
$362$	0	0
$363$	−23.4199	−1.22923
$364$	0	0
$365$	11.0540	0.578592
$366$	0	0
$367$	−6.49424	−0.338997	−0.169498	−	0.985530i	$-0.554215\pi$
$367$	−6.49424	−0.338997	−0.169498	+	0.985530i	$0.554215\pi$
$368$	0	0
$369$	10.5616	0.549812
$370$	0	0
$371$	−9.43318	−0.489746
$372$	0	0
$373$	24.3693	1.26180	0.630898	−	0.775866i	$-0.282687\pi$
$373$	24.3693	1.26180	0.630898	+	0.775866i	$0.282687\pi$
$374$	0	0
$375$	−20.7713	−1.07263
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−2.35829	−0.121137	−0.0605687	−	0.998164i	$-0.519291\pi$
$379$	−2.35829	−0.121137	−0.0605687	+	0.998164i	$0.519291\pi$
$380$	0	0
$381$	−0.684658	−0.0350761
$382$	0	0
$383$	−13.6964	−0.699854	−0.349927	−	0.936777i	$-0.613794\pi$
$383$	−13.6964	−0.699854	−0.349927	+	0.936777i	$0.613794\pi$
$384$	0	0
$385$	9.75379	0.497099
$386$	0	0
$387$	−21.5150	−1.09367
$388$	0	0
$389$	6.80776	0.345167	0.172584	−	0.984995i	$-0.444788\pi$
$389$	6.80776	0.345167	0.172584	+	0.984995i	$0.444788\pi$
$390$	0	0
$391$	43.3203	2.19080
$392$	0	0
$393$	−26.7386	−1.34879
$394$	0	0
$395$	−29.4608	−1.48233
$396$	0	0
$397$	18.6847	0.937756	0.468878	−	0.883263i	$-0.344658\pi$
$397$	18.6847	0.937756	0.468878	+	0.883263i	$0.344658\pi$
$398$	0	0
$399$	−43.4833	−2.17689
$400$	0	0
$401$	14.3693	0.717569	0.358785	−	0.933420i	$-0.383191\pi$
$401$	14.3693	0.717569	0.358785	+	0.933420i	$0.383191\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−25.9309	−1.28852
$406$	0	0
$407$	−2.19526	−0.108815
$408$	0	0
$409$	4.12311	0.203874	0.101937	−	0.994791i	$-0.467496\pi$
$409$	4.12311	0.203874	0.101937	+	0.994791i	$0.467496\pi$
$410$	0	0
$411$	40.0910	1.97754
$412$	0	0
$413$	38.5464	1.89674
$414$	0	0
$415$	−6.78456	−0.333041
$416$	0	0
$417$	48.3002	2.36527
$418$	0	0
$419$	27.1025	1.32404	0.662022	−	0.749484i	$-0.269698\pi$
$419$	27.1025	1.32404	0.662022	+	0.749484i	$0.269698\pi$
$420$	0	0
$421$	−15.6847	−0.764423	−0.382212	−	0.924075i	$-0.624838\pi$
$421$	−15.6847	−0.764423	−0.382212	+	0.924075i	$0.624838\pi$
$422$	0	0
$423$	27.5559	1.33981
$424$	0	0
$425$	9.56155	0.463803
$426$	0	0
$427$	−41.4153	−2.00423
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−21.9683	−1.05818	−0.529088	−	0.848567i	$-0.677466\pi$
$431$	−21.9683	−1.05818	−0.529088	+	0.848567i	$0.677466\pi$
$432$	0	0
$433$	−38.3693	−1.84391	−0.921956	−	0.387295i	$-0.873409\pi$
$433$	−38.3693	−1.84391	−0.921956	+	0.387295i	$0.873409\pi$
$434$	0	0
$435$	−30.2045	−1.44819
$436$	0	0
$437$	−35.4233	−1.69453
$438$	0	0
$439$	15.1838	0.724681	0.362341	−	0.932046i	$-0.381978\pi$
$439$	15.1838	0.724681	0.362341	+	0.932046i	$0.381978\pi$
$440$	0	0
$441$	16.8078	0.800370
$442$	0	0
$443$	−2.64861	−0.125839	−0.0629197	−	0.998019i	$-0.520041\pi$
$443$	−2.64861	−0.125839	−0.0629197	+	0.998019i	$0.520041\pi$
$444$	0	0
$445$	−27.3693	−1.29743
$446$	0	0
$447$	15.9274	0.753341
$448$	0	0
$449$	8.43845	0.398235	0.199117	−	0.979976i	$-0.436192\pi$
$449$	8.43845	0.398235	0.199117	+	0.979976i	$0.436192\pi$
$450$	0	0
$451$	4.26324	0.200748
$452$	0	0
$453$	−48.9848	−2.30151
$454$	0	0
$455$	0	0
$456$	0	0
$457$	37.4924	1.75382	0.876911	−	0.480653i	$-0.159600\pi$
$457$	37.4924	1.75382	0.876911	+	0.480653i	$0.159600\pi$
$458$	0	0
$459$	−6.33122	−0.295516
$460$	0	0
$461$	−6.12311	−0.285181	−0.142591	−	0.989782i	$-0.545543\pi$
$461$	−6.12311	−0.285181	−0.142591	+	0.989782i	$0.545543\pi$
$462$	0	0
$463$	−6.78456	−0.315305	−0.157653	−	0.987495i	$-0.550393\pi$
$463$	−6.78456	−0.315305	−0.157653	+	0.987495i	$0.550393\pi$
$464$	0	0
$465$	−20.4924	−0.950313
$466$	0	0
$467$	11.5012	0.532210	0.266105	−	0.963944i	$-0.414263\pi$
$467$	11.5012	0.532210	0.266105	+	0.963944i	$0.414263\pi$
$468$	0	0
$469$	−35.8078	−1.65345
$470$	0	0
$471$	−0.743668	−0.0342664
$472$	0	0
$473$	−8.68466	−0.399321
$474$	0	0
$475$	−7.81855	−0.358740
$476$	0	0
$477$	−6.56155	−0.300433
$478$	0	0
$479$	−11.7915	−0.538766	−0.269383	−	0.963033i	$-0.586820\pi$
$479$	−11.7915	−0.538766	−0.269383	+	0.963033i	$0.586820\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	61.4429	2.79575
$484$	0	0
$485$	25.1231	1.14078
$486$	0	0
$487$	−36.1181	−1.63667	−0.818333	−	0.574744i	$-0.805102\pi$
$487$	−36.1181	−1.63667	−0.818333	+	0.574744i	$0.805102\pi$
$488$	0	0
$489$	48.3002	2.18421
$490$	0	0
$491$	4.84388	0.218601	0.109301	−	0.994009i	$-0.465139\pi$
$491$	4.84388	0.218601	0.109301	+	0.994009i	$0.465139\pi$
$492$	0	0
$493$	−30.6155	−1.37885
$494$	0	0
$495$	6.78456	0.304943
$496$	0	0
$497$	18.4384	0.827077
$498$	0	0
$499$	31.6918	1.41872	0.709360	−	0.704846i	$-0.248984\pi$
$499$	31.6918	1.41872	0.709360	+	0.704846i	$0.248984\pi$
$500$	0	0
$501$	−24.6847	−1.10283
$502$	0	0
$503$	32.5628	1.45190	0.725951	−	0.687746i	$-0.241400\pi$
$503$	32.5628	1.45190	0.725951	+	0.687746i	$0.241400\pi$
$504$	0	0
$505$	−0.315342	−0.0140325
$506$	0	0
$507$	0	0
$508$	0	0
$509$	10.3693	0.459612	0.229806	−	0.973236i	$-0.426191\pi$
$509$	10.3693	0.459612	0.229806	+	0.973236i	$0.426191\pi$
$510$	0	0
$511$	15.8917	0.703007
$512$	0	0
$513$	5.17708	0.228574
$514$	0	0
$515$	−22.2586	−0.980833
$516$	0	0
$517$	11.1231	0.489194
$518$	0	0
$519$	19.9003	0.873528
$520$	0	0
$521$	33.4384	1.46496	0.732482	−	0.680786i	$-0.238362\pi$
$521$	33.4384	1.46496	0.732482	+	0.680786i	$0.238362\pi$
$522$	0	0
$523$	−29.9142	−1.30805	−0.654027	−	0.756471i	$-0.726922\pi$
$523$	−29.9142	−1.30805	−0.654027	+	0.756471i	$0.726922\pi$
$524$	0	0
$525$	13.5616	0.591875
$526$	0	0
$527$	−20.7713	−0.904812
$528$	0	0
$529$	27.0540	1.17626
$530$	0	0
$531$	26.8122	1.16355
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−24.9073	−1.07683
$536$	0	0
$537$	13.5616	0.585224
$538$	0	0
$539$	6.78456	0.292232
$540$	0	0
$541$	10.1771	0.437547	0.218773	−	0.975776i	$-0.429794\pi$
$541$	10.1771	0.437547	0.218773	+	0.975776i	$0.429794\pi$
$542$	0	0
$543$	−49.0708	−2.10583
$544$	0	0
$545$	−2.87689	−0.123233
$546$	0	0
$547$	−19.6100	−0.838464	−0.419232	−	0.907879i	$-0.637701\pi$
$547$	−19.6100	−0.838464	−0.419232	+	0.907879i	$0.637701\pi$
$548$	0	0
$549$	−28.8078	−1.22949
$550$	0	0
$551$	25.0345	1.06651
$552$	0	0
$553$	−42.3542	−1.80108
$554$	0	0
$555$	−12.8255	−0.544410
$556$	0	0
$557$	−35.9848	−1.52473	−0.762363	−	0.647149i	$-0.775961\pi$
$557$	−35.9848	−1.52473	−0.762363	+	0.647149i	$0.775961\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	14.9309	0.630382
$562$	0	0
$563$	20.4810	0.863170	0.431585	−	0.902072i	$-0.357954\pi$
$563$	20.4810	0.863170	0.431585	+	0.902072i	$0.357954\pi$
$564$	0	0
$565$	41.9309	1.76404
$566$	0	0
$567$	−37.2794	−1.56559
$568$	0	0
$569$	35.6695	1.49534	0.747672	−	0.664069i	$-0.231172\pi$
$569$	35.6695	1.49534	0.747672	+	0.664069i	$0.231172\pi$
$570$	0	0
$571$	13.2431	0.554205	0.277103	−	0.960840i	$-0.410626\pi$
$571$	13.2431	0.554205	0.277103	+	0.960840i	$0.410626\pi$
$572$	0	0
$573$	−29.1771	−1.21889
$574$	0	0
$575$	11.0478	0.460725
$576$	0	0
$577$	−13.9309	−0.579950	−0.289975	−	0.957034i	$-0.593647\pi$
$577$	−13.9309	−0.579950	−0.289975	+	0.957034i	$0.593647\pi$
$578$	0	0
$579$	4.42627	0.183949
$580$	0	0
$581$	−9.75379	−0.404655
$582$	0	0
$583$	−2.64861	−0.109694
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−42.5766	−1.75732	−0.878662	−	0.477444i	$-0.841563\pi$
$587$	−42.5766	−1.75732	−0.878662	+	0.477444i	$0.841563\pi$
$588$	0	0
$589$	16.9848	0.699848
$590$	0	0
$591$	13.6964	0.563395
$592$	0	0
$593$	11.0540	0.453932	0.226966	−	0.973903i	$-0.427119\pi$
$593$	11.0540	0.453932	0.226966	+	0.973903i	$0.427119\pi$
$594$	0	0
$595$	57.7603	2.36794
$596$	0	0
$597$	16.6847	0.682858
$598$	0	0
$599$	1.32431	0.0541097	0.0270549	−	0.999634i	$-0.491387\pi$
$599$	1.32431	0.0541097	0.0270549	+	0.999634i	$0.491387\pi$
$600$	0	0
$601$	6.12311	0.249767	0.124883	−	0.992171i	$-0.460144\pi$
$601$	6.12311	0.249767	0.124883	+	0.992171i	$0.460144\pi$
$602$	0	0
$603$	−24.9073	−1.01430
$604$	0	0
$605$	−25.4384	−1.03422
$606$	0	0
$607$	27.4286	1.11329	0.556646	−	0.830750i	$-0.312088\pi$
$607$	27.4286	1.11329	0.556646	+	0.830750i	$0.312088\pi$
$608$	0	0
$609$	−43.4233	−1.75960
$610$	0	0
$611$	0	0
$612$	0	0
$613$	26.3693	1.06505	0.532523	−	0.846415i	$-0.321244\pi$
$613$	26.3693	1.06505	0.532523	+	0.846415i	$0.321244\pi$
$614$	0	0
$615$	24.9073	1.00436
$616$	0	0
$617$	−9.24621	−0.372238	−0.186119	−	0.982527i	$-0.559591\pi$
$617$	−9.24621	−0.372238	−0.186119	+	0.982527i	$0.559591\pi$
$618$	0	0
$619$	−2.64861	−0.106457	−0.0532284	−	0.998582i	$-0.516951\pi$
$619$	−2.64861	−0.106457	−0.0532284	+	0.998582i	$0.516951\pi$
$620$	0	0
$621$	−7.31534	−0.293555
$622$	0	0
$623$	−39.3473	−1.57642
$624$	0	0
$625$	−30.3693	−1.21477
$626$	0	0
$627$	−12.2091	−0.487584
$628$	0	0
$629$	−13.0000	−0.518344
$630$	0	0
$631$	−21.0616	−0.838450	−0.419225	−	0.907882i	$-0.637698\pi$
$631$	−21.0616	−0.838450	−0.419225	+	0.907882i	$0.637698\pi$
$632$	0	0
$633$	5.56155	0.221052
$634$	0	0
$635$	−0.743668	−0.0295116
$636$	0	0
$637$	0	0
$638$	0	0
$639$	12.8255	0.507367
$640$	0	0
$641$	10.6155	0.419288	0.209644	−	0.977778i	$-0.432769\pi$
$641$	10.6155	0.419288	0.209644	+	0.977778i	$0.432769\pi$
$642$	0	0
$643$	32.5628	1.28415	0.642075	−	0.766642i	$-0.278074\pi$
$643$	32.5628	1.28415	0.642075	+	0.766642i	$0.278074\pi$
$644$	0	0
$645$	−50.7386	−1.99783
$646$	0	0
$647$	49.5242	1.94700	0.973498	−	0.228694i	$-0.0734456\pi$
$647$	49.5242	1.94700	0.973498	+	0.228694i	$0.0734456\pi$
$648$	0	0
$649$	10.8229	0.424837
$650$	0	0
$651$	−29.4608	−1.15466
$652$	0	0
$653$	3.56155	0.139374	0.0696872	−	0.997569i	$-0.477800\pi$
$653$	3.56155	0.139374	0.0696872	+	0.997569i	$0.477800\pi$
$654$	0	0
$655$	−29.0432	−1.13481
$656$	0	0
$657$	11.0540	0.431257
$658$	0	0
$659$	−37.8600	−1.47482	−0.737408	−	0.675447i	$-0.763951\pi$
$659$	−37.8600	−1.47482	−0.737408	+	0.675447i	$0.763951\pi$
$660$	0	0
$661$	24.1231	0.938280	0.469140	−	0.883124i	$-0.344564\pi$
$661$	24.1231	0.938280	0.469140	+	0.883124i	$0.344564\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−47.2311	−1.83154
$666$	0	0
$667$	−35.3744	−1.36970
$668$	0	0
$669$	−29.1771	−1.12805
$670$	0	0
$671$	−11.6284	−0.448911
$672$	0	0
$673$	−23.7386	−0.915057	−0.457529	−	0.889195i	$-0.651265\pi$
$673$	−23.7386	−0.915057	−0.457529	+	0.889195i	$0.651265\pi$
$674$	0	0
$675$	−1.61463	−0.0621470
$676$	0	0
$677$	−14.8769	−0.571765	−0.285883	−	0.958265i	$-0.592287\pi$
$677$	−14.8769	−0.571765	−0.285883	+	0.958265i	$0.592287\pi$
$678$	0	0
$679$	36.1181	1.38608
$680$	0	0
$681$	−15.3153	−0.586885
$682$	0	0
$683$	−6.49424	−0.248495	−0.124248	−	0.992251i	$-0.539652\pi$
$683$	−6.49424	−0.248495	−0.124248	+	0.992251i	$0.539652\pi$
$684$	0	0
$685$	43.5464	1.66382
$686$	0	0
$687$	45.6786	1.74275
$688$	0	0
$689$	0	0
$690$	0	0
$691$	9.30589	0.354013	0.177006	−	0.984210i	$-0.443359\pi$
$691$	9.30589	0.354013	0.177006	+	0.984210i	$0.443359\pi$
$692$	0	0
$693$	9.75379	0.370516
$694$	0	0
$695$	52.4631	1.99004
$696$	0	0
$697$	25.2462	0.956268
$698$	0	0
$699$	−36.2454	−1.37093
$700$	0	0
$701$	−14.8769	−0.561893	−0.280946	−	0.959723i	$-0.590648\pi$
$701$	−14.8769	−0.561893	−0.280946	+	0.959723i	$0.590648\pi$
$702$	0	0
$703$	10.6302	0.400925
$704$	0	0
$705$	64.9848	2.44747
$706$	0	0
$707$	−0.453349	−0.0170499
$708$	0	0
$709$	20.1231	0.755739	0.377870	−	0.925859i	$-0.376657\pi$
$709$	20.1231	0.755739	0.377870	+	0.925859i	$0.376657\pi$
$710$	0	0
$711$	−29.4608	−1.10487
$712$	0	0
$713$	−24.0000	−0.898807
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−16.0000	−0.597531
$718$	0	0
$719$	−14.2771	−0.532444	−0.266222	−	0.963912i	$-0.585775\pi$
$719$	−14.2771	−0.532444	−0.266222	+	0.963912i	$0.585775\pi$
$720$	0	0
$721$	−32.0000	−1.19174
$722$	0	0
$723$	42.1590	1.56791
$724$	0	0
$725$	−7.80776	−0.289973
$726$	0	0
$727$	25.6509	0.951340	0.475670	−	0.879624i	$-0.342206\pi$
$727$	25.6509	0.951340	0.475670	+	0.879624i	$0.342206\pi$
$728$	0	0
$729$	−18.6155	−0.689464
$730$	0	0
$731$	−51.4291	−1.90218
$732$	0	0
$733$	11.9309	0.440677	0.220338	−	0.975424i	$-0.429284\pi$
$733$	11.9309	0.440677	0.220338	+	0.975424i	$0.429284\pi$
$734$	0	0
$735$	39.6377	1.46206
$736$	0	0
$737$	−10.0540	−0.370343
$738$	0	0
$739$	7.23791	0.266251	0.133125	−	0.991099i	$-0.457499\pi$
$739$	7.23791	0.266251	0.133125	+	0.991099i	$0.457499\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	27.8462	1.02158	0.510789	−	0.859706i	$-0.329353\pi$
$743$	27.8462	1.02158	0.510789	+	0.859706i	$0.329353\pi$
$744$	0	0
$745$	17.3002	0.633830
$746$	0	0
$747$	−6.78456	−0.248234
$748$	0	0
$749$	−35.8078	−1.30839
$750$	0	0
$751$	−21.9683	−0.801635	−0.400818	−	0.916158i	$-0.631274\pi$
$751$	−21.9683	−0.801635	−0.400818	+	0.916158i	$0.631274\pi$
$752$	0	0
$753$	0.684658	0.0249503
$754$	0	0
$755$	−53.2068	−1.93639
$756$	0	0
$757$	−13.4233	−0.487878	−0.243939	−	0.969791i	$-0.578440\pi$
$757$	−13.4233	−0.487878	−0.243939	+	0.969791i	$0.578440\pi$
$758$	0	0
$759$	17.2517	0.626198
$760$	0	0
$761$	−3.06913	−0.111256	−0.0556279	−	0.998452i	$-0.517716\pi$
$761$	−3.06913	−0.111256	−0.0556279	+	0.998452i	$0.517716\pi$
$762$	0	0
$763$	−4.13595	−0.149731
$764$	0	0
$765$	40.1771	1.45261
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−4.05398	−0.146190	−0.0730950	−	0.997325i	$-0.523288\pi$
$769$	−4.05398	−0.146190	−0.0730950	+	0.997325i	$0.523288\pi$
$770$	0	0
$771$	−9.72350	−0.350183
$772$	0	0
$773$	18.6847	0.672040	0.336020	−	0.941855i	$-0.390919\pi$
$773$	18.6847	0.672040	0.336020	+	0.941855i	$0.390919\pi$
$774$	0	0
$775$	−5.29723	−0.190282
$776$	0	0
$777$	−18.4384	−0.661476
$778$	0	0
$779$	−20.6440	−0.739648
$780$	0	0
$781$	5.17708	0.185251
$782$	0	0
$783$	5.16994	0.184759
$784$	0	0
$785$	−0.807764	−0.0288303
$786$	0	0
$787$	−17.6693	−0.629844	−0.314922	−	0.949118i	$-0.601978\pi$
$787$	−17.6693	−0.629844	−0.314922	+	0.949118i	$0.601978\pi$
$788$	0	0
$789$	−16.3002	−0.580302
$790$	0	0
$791$	60.2817	2.14337
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−15.4741	−0.548809
$796$	0	0
$797$	−52.0540	−1.84385	−0.921923	−	0.387373i	$-0.873383\pi$
$797$	−52.0540	−1.84385	−0.921923	+	0.387373i	$0.873383\pi$
$798$	0	0
$799$	65.8692	2.33029
$800$	0	0
$801$	−27.3693	−0.967047
$802$	0	0
$803$	4.46201	0.157461
$804$	0	0
$805$	66.7386	2.35223
$806$	0	0
$807$	−44.0639	−1.55112
$808$	0	0
$809$	32.3693	1.13804	0.569022	−	0.822322i	$-0.307322\pi$
$809$	32.3693	1.13804	0.569022	+	0.822322i	$0.307322\pi$
$810$	0	0
$811$	−14.8934	−0.522979	−0.261490	−	0.965206i	$-0.584214\pi$
$811$	−14.8934	−0.522979	−0.261490	+	0.965206i	$0.584214\pi$
$812$	0	0
$813$	−24.6847	−0.865729
$814$	0	0
$815$	52.4631	1.83770
$816$	0	0
$817$	42.0540	1.47128
$818$	0	0
$819$	0	0
$820$	0	0
$821$	21.8078	0.761096	0.380548	−	0.924761i	$-0.375735\pi$
$821$	21.8078	0.761096	0.380548	+	0.924761i	$0.375735\pi$
$822$	0	0
$823$	−3.10196	−0.108128	−0.0540638	−	0.998537i	$-0.517217\pi$
$823$	−3.10196	−0.108128	−0.0540638	+	0.998537i	$0.517217\pi$
$824$	0	0
$825$	3.80776	0.132569
$826$	0	0
$827$	38.8940	1.35248	0.676238	−	0.736683i	$-0.263609\pi$
$827$	38.8940	1.35248	0.676238	+	0.736683i	$0.263609\pi$
$828$	0	0
$829$	−45.9848	−1.59712	−0.798560	−	0.601915i	$-0.794404\pi$
$829$	−45.9848	−1.59712	−0.798560	+	0.601915i	$0.794404\pi$
$830$	0	0
$831$	3.84563	0.133403
$832$	0	0
$833$	40.1771	1.39205
$834$	0	0
$835$	−26.8122	−0.927874
$836$	0	0
$837$	3.50758	0.121240
$838$	0	0
$839$	42.7396	1.47554	0.737768	−	0.675055i	$-0.235880\pi$
$839$	42.7396	1.47554	0.737768	+	0.675055i	$0.235880\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	0	0
$843$	29.0432	1.00030
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−36.5714	−1.25661
$848$	0	0
$849$	−59.4233	−2.03940
$850$	0	0
$851$	−15.0207	−0.514904
$852$	0	0
$853$	23.0540	0.789353	0.394677	−	0.918820i	$-0.370857\pi$
$853$	23.0540	0.789353	0.394677	+	0.918820i	$0.370857\pi$
$854$	0	0
$855$	−32.8531	−1.12355
$856$	0	0
$857$	17.4384	0.595686	0.297843	−	0.954615i	$-0.403733\pi$
$857$	17.4384	0.595686	0.297843	+	0.954615i	$0.403733\pi$
$858$	0	0
$859$	−23.4199	−0.799077	−0.399539	−	0.916716i	$-0.630830\pi$
$859$	−23.4199	−0.799077	−0.399539	+	0.916716i	$0.630830\pi$
$860$	0	0
$861$	35.8078	1.22033
$862$	0	0
$863$	−10.5945	−0.360639	−0.180320	−	0.983608i	$-0.557713\pi$
$863$	−10.5945	−0.360639	−0.180320	+	0.983608i	$0.557713\pi$
$864$	0	0
$865$	21.6155	0.734950
$866$	0	0
$867$	48.3272	1.64128
$868$	0	0
$869$	−11.8920	−0.403410
$870$	0	0
$871$	0	0
$872$	0	0
$873$	25.1231	0.850288
$874$	0	0
$875$	−32.4355	−1.09652
$876$	0	0
$877$	−9.24621	−0.312222	−0.156111	−	0.987739i	$-0.549896\pi$
$877$	−9.24621	−0.312222	−0.156111	+	0.987739i	$0.549896\pi$
$878$	0	0
$879$	2.35829	0.0795433
$880$	0	0
$881$	28.8617	0.972377	0.486188	−	0.873854i	$-0.338387\pi$
$881$	28.8617	0.972377	0.486188	+	0.873854i	$0.338387\pi$
$882$	0	0
$883$	−16.6354	−0.559824	−0.279912	−	0.960026i	$-0.590305\pi$
$883$	−16.6354	−0.559824	−0.279912	+	0.960026i	$0.590305\pi$
$884$	0	0
$885$	63.2311	2.12549
$886$	0	0
$887$	−36.1181	−1.21273	−0.606363	−	0.795188i	$-0.707372\pi$
$887$	−36.1181	−1.21273	−0.606363	+	0.795188i	$0.707372\pi$
$888$	0	0
$889$	−1.06913	−0.0358575
$890$	0	0
$891$	−10.4672	−0.350663
$892$	0	0
$893$	−53.8617	−1.80241
$894$	0	0
$895$	14.7304	0.492383
$896$	0	0
$897$	0	0
$898$	0	0
$899$	16.9614	0.565695
$900$	0	0
$901$	−15.6847	−0.522532
$902$	0	0
$903$	−72.9441	−2.42743
$904$	0	0
$905$	−53.3002	−1.77176
$906$	0	0
$907$	24.6169	0.817392	0.408696	−	0.912671i	$-0.365984\pi$
$907$	24.6169	0.817392	0.408696	+	0.912671i	$0.365984\pi$
$908$	0	0
$909$	−0.315342	−0.0104592
$910$	0	0
$911$	15.8917	0.526515	0.263257	−	0.964726i	$-0.415203\pi$
$911$	15.8917	0.526515	0.263257	+	0.964726i	$0.415203\pi$
$912$	0	0
$913$	−2.73863	−0.0906355
$914$	0	0
$915$	−67.9372	−2.24593
$916$	0	0
$917$	−41.7538	−1.37883
$918$	0	0
$919$	7.98158	0.263288	0.131644	−	0.991297i	$-0.457974\pi$
$919$	7.98158	0.263288	0.131644	+	0.991297i	$0.457974\pi$
$920$	0	0
$921$	−46.2462	−1.52386
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−3.31534	−0.109008
$926$	0	0
$927$	−22.2586	−0.731070
$928$	0	0
$929$	−30.1231	−0.988307	−0.494154	−	0.869375i	$-0.664522\pi$
$929$	−30.1231	−0.988307	−0.494154	+	0.869375i	$0.664522\pi$
$930$	0	0
$931$	−32.8531	−1.07672
$932$	0	0
$933$	−24.0000	−0.785725
$934$	0	0
$935$	16.2177	0.530377
$936$	0	0
$937$	−48.8078	−1.59448	−0.797240	−	0.603662i	$-0.793708\pi$
$937$	−48.8078	−1.59448	−0.797240	+	0.603662i	$0.793708\pi$
$938$	0	0
$939$	30.9481	1.00995
$940$	0	0
$941$	2.00000	0.0651981	0.0325991	−	0.999469i	$-0.489622\pi$
$941$	2.00000	0.0651981	0.0325991	+	0.999469i	$0.489622\pi$
$942$	0	0
$943$	29.1705	0.949922
$944$	0	0
$945$	−9.75379	−0.317291
$946$	0	0
$947$	−31.8191	−1.03398	−0.516991	−	0.855991i	$-0.672948\pi$
$947$	−31.8191	−1.03398	−0.516991	+	0.855991i	$0.672948\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−6.62153	−0.214718
$952$	0	0
$953$	−34.7926	−1.12704	−0.563522	−	0.826101i	$-0.690554\pi$
$953$	−34.7926	−1.12704	−0.563522	+	0.826101i	$0.690554\pi$
$954$	0	0
$955$	−31.6918	−1.02552
$956$	0	0
$957$	−12.1922	−0.394119
$958$	0	0
$959$	62.6042	2.02160
$960$	0	0
$961$	−19.4924	−0.628788
$962$	0	0
$963$	−24.9073	−0.802625
$964$	0	0
$965$	4.80776	0.154767
$966$	0	0
$967$	−8.68951	−0.279436	−0.139718	−	0.990191i	$-0.544620\pi$
$967$	−8.68951	−0.279436	−0.139718	+	0.990191i	$0.544620\pi$
$968$	0	0
$969$	−72.3002	−2.32262
$970$	0	0
$971$	−18.5760	−0.596133	−0.298067	−	0.954545i	$-0.596342\pi$
$971$	−18.5760	−0.596133	−0.298067	+	0.954545i	$0.596342\pi$
$972$	0	0
$973$	75.4233	2.41796
$974$	0	0
$975$	0	0
$976$	0	0
$977$	15.7386	0.503524	0.251762	−	0.967789i	$-0.418990\pi$
$977$	15.7386	0.503524	0.251762	+	0.967789i	$0.418990\pi$
$978$	0	0
$979$	−11.0478	−0.353089
$980$	0	0
$981$	−2.87689	−0.0918522
$982$	0	0
$983$	−30.5305	−0.973773	−0.486886	−	0.873465i	$-0.661867\pi$
$983$	−30.5305	−0.973773	−0.486886	+	0.873465i	$0.661867\pi$
$984$	0	0
$985$	14.8769	0.474017
$986$	0	0
$987$	93.4251	2.97375
$988$	0	0
$989$	−59.4233	−1.88955
$990$	0	0
$991$	17.0887	0.542840	0.271420	−	0.962461i	$-0.412507\pi$
$991$	17.0887	0.542840	0.271420	+	0.962461i	$0.412507\pi$
$992$	0	0
$993$	35.8078	1.13633
$994$	0	0
$995$	18.1227	0.574528
$996$	0	0
$997$	−28.2311	−0.894087	−0.447043	−	0.894512i	$-0.647523\pi$
$997$	−28.2311	−0.894087	−0.447043	+	0.894512i	$0.647523\pi$
$998$	0	0
$999$	2.19526	0.0694551

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5408.2.a.bi.1.4		4
4.3	odd	2	inner	5408.2.a.bi.1.1		4
13.4	even	6		416.2.i.g.289.1	✓	8
13.10	even	6		416.2.i.g.321.1	yes	8
13.12	even	2		5408.2.a.bh.1.4		4
52.23	odd	6		416.2.i.g.321.4	yes	8
52.43	odd	6		416.2.i.g.289.4	yes	8
52.51	odd	2		5408.2.a.bh.1.1		4
104.43	odd	6		832.2.i.q.705.1		8
104.69	even	6		832.2.i.q.705.4		8
104.75	odd	6		832.2.i.q.321.1		8
104.101	even	6		832.2.i.q.321.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
416.2.i.g.289.1	✓	8	13.4	even	6
416.2.i.g.289.4	yes	8	52.43	odd	6
416.2.i.g.321.1	yes	8	13.10	even	6
416.2.i.g.321.4	yes	8	52.23	odd	6
832.2.i.q.321.1		8	104.75	odd	6
832.2.i.q.321.4		8	104.101	even	6
832.2.i.q.705.1		8	104.43	odd	6
832.2.i.q.705.4		8	104.69	even	6
5408.2.a.bh.1.1		4	52.51	odd	2
5408.2.a.bh.1.4		4	13.12	even	2
5408.2.a.bi.1.1		4	4.3	odd	2	inner
5408.2.a.bi.1.4		4	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:	\( N \)	\(=\)	\( 5408 = 2^{5} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5408.a (trivial)

\(f(q)\)	\(=\)	\(q+2.35829 q^{3} +2.56155 q^{5} +3.68260 q^{7} +2.56155 q^{9} +1.03399 q^{11} +6.04090 q^{15} +6.12311 q^{17} -5.00691 q^{19} +8.68466 q^{21} +7.07488 q^{23} +1.56155 q^{25} -1.03399 q^{27} -5.00000 q^{29} -3.39228 q^{31} +2.43845 q^{33} +9.43318 q^{35} -2.12311 q^{37} +4.12311 q^{41} -8.39919 q^{43} +6.56155 q^{45} +10.7575 q^{47} +6.56155 q^{49} +14.4401 q^{51} -2.56155 q^{53} +2.64861 q^{55} -11.8078 q^{57} +10.4672 q^{59} -11.2462 q^{61} +9.43318 q^{63} -9.72350 q^{67} +16.6847 q^{69} +5.00691 q^{71} +4.31534 q^{73} +3.68260 q^{75} +3.80776 q^{77} -11.5012 q^{79} -10.1231 q^{81} -2.64861 q^{83} +15.6847 q^{85} -11.7915 q^{87} -10.6847 q^{89} -8.00000 q^{93} -12.8255 q^{95} +9.80776 q^{97} +2.64861 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{5} + 2 q^{9} + 8 q^{17} + 10 q^{21} - 2 q^{25} - 20 q^{29} + 18 q^{33} + 8 q^{37} + 18 q^{45} + 18 q^{49} - 2 q^{53} - 6 q^{57} - 12 q^{61} + 42 q^{69} + 42 q^{73} - 26 q^{77} - 24 q^{81} + 38 q^{85}+ \cdots - 2 q^{97}+O(q^{100}) \) 4 * q + 2 * q^5 + 2 * q^9 + 8 * q^17 + 10 * q^21 - 2 * q^25 - 20 * q^29 + 18 * q^33 + 8 * q^37 + 18 * q^45 + 18 * q^49 - 2 * q^53 - 6 * q^57 - 12 * q^61 + 42 * q^69 + 42 * q^73 - 26 * q^77 - 24 * q^81 + 38 * q^85 - 18 * q^89 - 32 * q^93 - 2 * q^97

Introduction

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