LMFDB - Embedded newform 5200.2.a.ce.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5200,2,Mod(1,5200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5200.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 5200 = 2^{4} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5200.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,0,0,-2,0,5,0,-6] 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:	$41.5222090511$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.940.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 7x - 4 $ x^3 - 7*x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 130)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.29240$ of defining polynomial
Character	$\chi$	$=$	5200.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.29240 q^{3} -1.25511 q^{7} +2.25511 q^{9} -2.00000 q^{11} -1.00000 q^{13} -4.80261 q^{17} +5.09501 q^{19} -2.87720 q^{21} -2.58480 q^{23} -1.70760 q^{27} +5.09501 q^{29} -8.58480 q^{31} -4.58480 q^{33} +7.83991 q^{37} -2.29240 q^{39} -9.67982 q^{41} -10.8772 q^{43} -2.74489 q^{47} -5.42471 q^{49} -11.0095 q^{51} +2.58480 q^{53} +11.6798 q^{57} +5.09501 q^{59} +13.6798 q^{61} -2.83039 q^{63} -8.58480 q^{67} -5.92541 q^{69} -5.38741 q^{71} +6.00000 q^{73} +2.51021 q^{77} -15.0950 q^{79} -10.6798 q^{81} -11.0950 q^{83} +11.6798 q^{87} +5.09501 q^{89} +1.25511 q^{91} -19.6798 q^{93} -6.26462 q^{97} -4.51021 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{7} + 5 q^{9} - 6 q^{11} - 3 q^{13} - 4 q^{17} - 2 q^{19} + 12 q^{21} + 6 q^{23} - 12 q^{27} - 2 q^{29} - 12 q^{31} + 8 q^{37} + 2 q^{41} - 12 q^{43} - 10 q^{47} + 13 q^{49} + 10 q^{51} - 6 q^{53}+ \cdots - 10 q^{99}+O(q^{100}) $ 3 * q - 2 * q^7 + 5 * q^9 - 6 * q^11 - 3 * q^13 - 4 * q^17 - 2 * q^19 + 12 * q^21 + 6 * q^23 - 12 * q^27 - 2 * q^29 - 12 * q^31 + 8 * q^37 + 2 * q^41 - 12 * q^43 - 10 * q^47 + 13 * q^49 + 10 * q^51 - 6 * q^53 + 4 * q^57 - 2 * q^59 + 10 * q^61 - 36 * q^63 - 12 * q^67 - 28 * q^69 + 8 * q^71 + 18 * q^73 + 4 * q^77 - 28 * q^79 - q^81 - 16 * q^83 + 4 * q^87 - 2 * q^89 + 2 * q^91 - 28 * q^93 + 26 * q^97 - 10 * q^99

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
$3$	2.29240	1.32352	0.661759	−	0.749716i	$-0.269810\pi$
$3$	2.29240	1.32352	0.661759	+	0.749716i	$0.269810\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.25511	−0.474385	−0.237193	−	0.971463i	$-0.576227\pi$
$7$	−1.25511	−0.474385	−0.237193	+	0.971463i	$0.576227\pi$
$8$	0	0
$9$	2.25511	0.751702
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.80261	−1.16480	−0.582402	−	0.812901i	$-0.697887\pi$
$17$	−4.80261	−1.16480	−0.582402	+	0.812901i	$0.697887\pi$
$18$	0	0
$19$	5.09501	1.16888	0.584438	−	0.811438i	$-0.301315\pi$
$19$	5.09501	1.16888	0.584438	+	0.811438i	$0.301315\pi$
$20$	0	0
$21$	−2.87720	−0.627858
$22$	0	0
$23$	−2.58480	−0.538969	−0.269484	−	0.963005i	$-0.586853\pi$
$23$	−2.58480	−0.538969	−0.269484	+	0.963005i	$0.586853\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.70760	−0.328627
$28$	0	0
$29$	5.09501	0.946120	0.473060	−	0.881030i	$-0.343149\pi$
$29$	5.09501	0.946120	0.473060	+	0.881030i	$0.343149\pi$
$30$	0	0
$31$	−8.58480	−1.54188	−0.770938	−	0.636910i	$-0.780212\pi$
$31$	−8.58480	−1.54188	−0.770938	+	0.636910i	$0.780212\pi$
$32$	0	0
$33$	−4.58480	−0.798112
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.83991	1.28887	0.644436	−	0.764658i	$-0.277092\pi$
$37$	7.83991	1.28887	0.644436	+	0.764658i	$0.277092\pi$
$38$	0	0
$39$	−2.29240	−0.367078
$40$	0	0
$41$	−9.67982	−1.51173	−0.755867	−	0.654726i	$-0.772784\pi$
$41$	−9.67982	−1.51173	−0.755867	+	0.654726i	$0.772784\pi$
$42$	0	0
$43$	−10.8772	−1.65876	−0.829379	−	0.558686i	$-0.811306\pi$
$43$	−10.8772	−1.65876	−0.829379	+	0.558686i	$0.811306\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.74489	−0.400384	−0.200192	−	0.979757i	$-0.564157\pi$
$47$	−2.74489	−0.400384	−0.200192	+	0.979757i	$0.564157\pi$
$48$	0	0
$49$	−5.42471	−0.774959
$50$	0	0
$51$	−11.0095	−1.54164
$52$	0	0
$53$	2.58480	0.355050	0.177525	−	0.984116i	$-0.443191\pi$
$53$	2.58480	0.355050	0.177525	+	0.984116i	$0.443191\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	11.6798	1.54703
$58$	0	0
$59$	5.09501	0.663314	0.331657	−	0.943400i	$-0.392392\pi$
$59$	5.09501	0.663314	0.331657	+	0.943400i	$0.392392\pi$
$60$	0	0
$61$	13.6798	1.75152	0.875761	−	0.482746i	$-0.160361\pi$
$61$	13.6798	1.75152	0.875761	+	0.482746i	$0.160361\pi$
$62$	0	0
$63$	−2.83039	−0.356596
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−8.58480	−1.04880	−0.524400	−	0.851472i	$-0.675710\pi$
$67$	−8.58480	−1.04880	−0.524400	+	0.851472i	$0.675710\pi$
$68$	0	0
$69$	−5.92541	−0.713335
$70$	0	0
$71$	−5.38741	−0.639369	−0.319684	−	0.947524i	$-0.603577\pi$
$71$	−5.38741	−0.639369	−0.319684	+	0.947524i	$0.603577\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.51021	0.286065
$78$	0	0
$79$	−15.0950	−1.69832	−0.849161	−	0.528134i	$-0.822892\pi$
$79$	−15.0950	−1.69832	−0.849161	+	0.528134i	$0.822892\pi$
$80$	0	0
$81$	−10.6798	−1.18665
$82$	0	0
$83$	−11.0950	−1.21784	−0.608918	−	0.793233i	$-0.708396\pi$
$83$	−11.0950	−1.21784	−0.608918	+	0.793233i	$0.708396\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	11.6798	1.25221
$88$	0	0
$89$	5.09501	0.540070	0.270035	−	0.962850i	$-0.412965\pi$
$89$	5.09501	0.540070	0.270035	+	0.962850i	$0.412965\pi$
$90$	0	0
$91$	1.25511	0.131571
$92$	0	0
$93$	−19.6798	−2.04070
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−6.26462	−0.636076	−0.318038	−	0.948078i	$-0.603024\pi$
$97$	−6.26462	−0.636076	−0.318038	+	0.948078i	$0.603024\pi$
$98$	0	0
$99$	−4.51021	−0.453293
$100$	0	0
$101$	−12.6594	−1.25966	−0.629829	−	0.776734i	$-0.716875\pi$
$101$	−12.6594	−1.25966	−0.629829	+	0.776734i	$0.716875\pi$
$102$	0	0
$103$	7.60522	0.749365	0.374682	−	0.927153i	$-0.377752\pi$
$103$	7.60522	0.749365	0.374682	+	0.927153i	$0.377752\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.58480	0.443230	0.221615	−	0.975134i	$-0.428867\pi$
$107$	4.58480	0.443230	0.221615	+	0.975134i	$0.428867\pi$
$108$	0	0
$109$	14.8772	1.42498	0.712489	−	0.701683i	$-0.247568\pi$
$109$	14.8772	1.42498	0.712489	+	0.701683i	$0.247568\pi$
$110$	0	0
$111$	17.9722	1.70585
$112$	0	0
$113$	−4.00000	−0.376288	−0.188144	−	0.982141i	$-0.560247\pi$
$113$	−4.00000	−0.376288	−0.188144	+	0.982141i	$0.560247\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−2.25511	−0.208485
$118$	0	0
$119$	6.02778	0.552566
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−22.1900	−2.00081
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1.41520	0.125578	0.0627892	−	0.998027i	$-0.480000\pi$
$127$	1.41520	0.125578	0.0627892	+	0.998027i	$0.480000\pi$
$128$	0	0
$129$	−24.9349	−2.19540
$130$	0	0
$131$	−11.0095	−0.961906	−0.480953	−	0.876746i	$-0.659709\pi$
$131$	−11.0095	−0.961906	−0.480953	+	0.876746i	$0.659709\pi$
$132$	0	0
$133$	−6.39478	−0.554497
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−14.5848	−1.24606	−0.623032	−	0.782196i	$-0.714099\pi$
$137$	−14.5848	−1.24606	−0.623032	+	0.782196i	$0.714099\pi$
$138$	0	0
$139$	−7.32970	−0.621697	−0.310848	−	0.950459i	$-0.600613\pi$
$139$	−7.32970	−0.621697	−0.310848	+	0.950459i	$0.600613\pi$
$140$	0	0
$141$	−6.29240	−0.529916
$142$	0	0
$143$	2.00000	0.167248
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−12.4356	−1.02567
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	14.5570	1.18463	0.592317	−	0.805705i	$-0.298213\pi$
$151$	14.5570	1.18463	0.592317	+	0.805705i	$0.298213\pi$
$152$	0	0
$153$	−10.8304	−0.875585
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−13.0950	−1.04510	−0.522548	−	0.852610i	$-0.675018\pi$
$157$	−13.0950	−1.04510	−0.522548	+	0.852610i	$0.675018\pi$
$158$	0	0
$159$	5.92541	0.469915
$160$	0	0
$161$	3.24420	0.255679
$162$	0	0
$163$	12.2646	0.960639	0.480320	−	0.877094i	$-0.340521\pi$
$163$	12.2646	0.960639	0.480320	+	0.877094i	$0.340521\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−18.3392	−1.41913	−0.709565	−	0.704640i	$-0.751109\pi$
$167$	−18.3392	−1.41913	−0.709565	+	0.704640i	$0.751109\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	11.4898	0.878646
$172$	0	0
$173$	9.41520	0.715824	0.357912	−	0.933755i	$-0.383489\pi$
$173$	9.41520	0.715824	0.357912	+	0.933755i	$0.383489\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	11.6798	0.877909
$178$	0	0
$179$	7.32970	0.547847	0.273924	−	0.961751i	$-0.411678\pi$
$179$	7.32970	0.547847	0.273924	+	0.961751i	$0.411678\pi$
$180$	0	0
$181$	11.7544	0.873698	0.436849	−	0.899535i	$-0.356094\pi$
$181$	11.7544	0.873698	0.436849	+	0.899535i	$0.356094\pi$
$182$	0	0
$183$	31.3596	2.31817
$184$	0	0
$185$	0	0
$186$	0	0
$187$	9.60522	0.702403
$188$	0	0
$189$	2.14322	0.155896
$190$	0	0
$191$	1.60522	0.116150	0.0580749	−	0.998312i	$-0.481504\pi$
$191$	1.60522	0.116150	0.0580749	+	0.998312i	$0.481504\pi$
$192$	0	0
$193$	15.7544	1.13403	0.567014	−	0.823708i	$-0.308099\pi$
$193$	15.7544	1.13403	0.567014	+	0.823708i	$0.308099\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.00951	0.0719249	0.0359625	−	0.999353i	$-0.488550\pi$
$197$	1.00951	0.0719249	0.0359625	+	0.999353i	$0.488550\pi$
$198$	0	0
$199$	−0.435617	−0.0308801	−0.0154400	−	0.999881i	$-0.504915\pi$
$199$	−0.435617	−0.0308801	−0.0154400	+	0.999881i	$0.504915\pi$
$200$	0	0
$201$	−19.6798	−1.38811
$202$	0	0
$203$	−6.39478	−0.448825
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−5.82900	−0.405144
$208$	0	0
$209$	−10.1900	−0.704859
$210$	0	0
$211$	−16.3501	−1.12559	−0.562794	−	0.826597i	$-0.690274\pi$
$211$	−16.3501	−1.12559	−0.562794	+	0.826597i	$0.690274\pi$
$212$	0	0
$213$	−12.3501	−0.846216
$214$	0	0
$215$	0	0
$216$	0	0
$217$	10.7748	0.731443
$218$	0	0
$219$	13.7544	0.929437
$220$	0	0
$221$	4.80261	0.323059
$222$	0	0
$223$	−21.7843	−1.45879	−0.729394	−	0.684094i	$-0.760198\pi$
$223$	−21.7843	−1.45879	−0.729394	+	0.684094i	$0.760198\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−9.02042	−0.598706	−0.299353	−	0.954142i	$-0.596771\pi$
$227$	−9.02042	−0.598706	−0.299353	+	0.954142i	$0.596771\pi$
$228$	0	0
$229$	4.68718	0.309737	0.154869	−	0.987935i	$-0.450505\pi$
$229$	4.68718	0.309737	0.154869	+	0.987935i	$0.450505\pi$
$230$	0	0
$231$	5.75441	0.378612
$232$	0	0
$233$	−13.5366	−0.886812	−0.443406	−	0.896321i	$-0.646230\pi$
$233$	−13.5366	−0.886812	−0.443406	+	0.896321i	$0.646230\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−34.6038	−2.24776
$238$	0	0
$239$	−3.19739	−0.206822	−0.103411	−	0.994639i	$-0.532976\pi$
$239$	−3.19739	−0.206822	−0.103411	+	0.994639i	$0.532976\pi$
$240$	0	0
$241$	14.1154	0.909255	0.454627	−	0.890682i	$-0.349772\pi$
$241$	14.1154	0.909255	0.454627	+	0.890682i	$0.349772\pi$
$242$	0	0
$243$	−19.3596	−1.24192
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−5.09501	−0.324188
$248$	0	0
$249$	−25.4342	−1.61183
$250$	0	0
$251$	21.1696	1.33621	0.668107	−	0.744065i	$-0.267105\pi$
$251$	21.1696	1.33621	0.668107	+	0.744065i	$0.267105\pi$
$252$	0	0
$253$	5.16961	0.325010
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−8.21781	−0.512613	−0.256306	−	0.966596i	$-0.582506\pi$
$257$	−8.21781	−0.512613	−0.256306	+	0.966596i	$0.582506\pi$
$258$	0	0
$259$	−9.83991	−0.611422
$260$	0	0
$261$	11.4898	0.711200
$262$	0	0
$263$	−4.51021	−0.278111	−0.139056	−	0.990285i	$-0.544407\pi$
$263$	−4.51021	−0.278111	−0.139056	+	0.990285i	$0.544407\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	11.6798	0.714793
$268$	0	0
$269$	−8.51021	−0.518877	−0.259438	−	0.965760i	$-0.583537\pi$
$269$	−8.51021	−0.518877	−0.259438	+	0.965760i	$0.583537\pi$
$270$	0	0
$271$	−4.95180	−0.300800	−0.150400	−	0.988625i	$-0.548056\pi$
$271$	−4.95180	−0.300800	−0.150400	+	0.988625i	$0.548056\pi$
$272$	0	0
$273$	2.87720	0.174136
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−19.8698	−1.19386	−0.596932	−	0.802292i	$-0.703614\pi$
$277$	−19.8698	−1.19386	−0.596932	+	0.802292i	$0.703614\pi$
$278$	0	0
$279$	−19.3596	−1.15903
$280$	0	0
$281$	30.2646	1.80544	0.902718	−	0.430233i	$-0.141569\pi$
$281$	30.2646	1.80544	0.902718	+	0.430233i	$0.141569\pi$
$282$	0	0
$283$	26.9240	1.60047	0.800233	−	0.599689i	$-0.204709\pi$
$283$	26.9240	1.60047	0.800233	+	0.599689i	$0.204709\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	12.1492	0.717144
$288$	0	0
$289$	6.06508	0.356769
$290$	0	0
$291$	−14.3610	−0.841858
$292$	0	0
$293$	−5.18051	−0.302649	−0.151324	−	0.988484i	$-0.548354\pi$
$293$	−5.18051	−0.302649	−0.151324	+	0.988484i	$0.548354\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3.41520	0.198170
$298$	0	0
$299$	2.58480	0.149483
$300$	0	0
$301$	13.6520	0.786890
$302$	0	0
$303$	−29.0204	−1.66718
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−0.320184	−0.0182738	−0.00913692	−	0.999958i	$-0.502908\pi$
$307$	−0.320184	−0.0182738	−0.00913692	+	0.999958i	$0.502908\pi$
$308$	0	0
$309$	17.4342	0.991798
$310$	0	0
$311$	9.86984	0.559667	0.279834	−	0.960048i	$-0.409721\pi$
$311$	9.86984	0.559667	0.279834	+	0.960048i	$0.409721\pi$
$312$	0	0
$313$	−10.6126	−0.599859	−0.299929	−	0.953961i	$-0.596963\pi$
$313$	−10.6126	−0.599859	−0.299929	+	0.953961i	$0.596963\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	12.1900	0.684660	0.342330	−	0.939580i	$-0.388784\pi$
$317$	12.1900	0.684660	0.342330	+	0.939580i	$0.388784\pi$
$318$	0	0
$319$	−10.1900	−0.570532
$320$	0	0
$321$	10.5102	0.586623
$322$	0	0
$323$	−24.4694	−1.36151
$324$	0	0
$325$	0	0
$326$	0	0
$327$	34.1045	1.88598
$328$	0	0
$329$	3.44513	0.189936
$330$	0	0
$331$	17.9444	0.986315	0.493158	−	0.869940i	$-0.335843\pi$
$331$	17.9444	0.986315	0.493158	+	0.869940i	$0.335843\pi$
$332$	0	0
$333$	17.6798	0.968848
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−20.9518	−1.14132	−0.570659	−	0.821187i	$-0.693312\pi$
$337$	−20.9518	−1.14132	−0.570659	+	0.821187i	$0.693312\pi$
$338$	0	0
$339$	−9.16961	−0.498025
$340$	0	0
$341$	17.1696	0.929786
$342$	0	0
$343$	15.5943	0.842014
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3.51757	−0.188833	−0.0944166	−	0.995533i	$-0.530099\pi$
$347$	−3.51757	−0.188833	−0.0944166	+	0.995533i	$0.530099\pi$
$348$	0	0
$349$	17.8568	0.955852	0.477926	−	0.878400i	$-0.341389\pi$
$349$	17.8568	0.955852	0.477926	+	0.878400i	$0.341389\pi$
$350$	0	0
$351$	1.70760	0.0911449
$352$	0	0
$353$	−36.9240	−1.96527	−0.982634	−	0.185557i	$-0.940591\pi$
$353$	−36.9240	−1.96527	−0.982634	+	0.185557i	$0.940591\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	13.8181	0.731331
$358$	0	0
$359$	6.77483	0.357562	0.178781	−	0.983889i	$-0.442785\pi$
$359$	6.77483	0.357562	0.178781	+	0.983889i	$0.442785\pi$
$360$	0	0
$361$	6.95916	0.366272
$362$	0	0
$363$	−16.0468	−0.842239
$364$	0	0
$365$	0	0
$366$	0	0
$367$	7.80997	0.407677	0.203839	−	0.979004i	$-0.434658\pi$
$367$	7.80997	0.407677	0.203839	+	0.979004i	$0.434658\pi$
$368$	0	0
$369$	−21.8290	−1.13637
$370$	0	0
$371$	−3.24420	−0.168430
$372$	0	0
$373$	−15.8698	−0.821709	−0.410855	−	0.911701i	$-0.634770\pi$
$373$	−15.8698	−0.821709	−0.410855	+	0.911701i	$0.634770\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5.09501	−0.262407
$378$	0	0
$379$	16.7748	0.861665	0.430833	−	0.902432i	$-0.358220\pi$
$379$	16.7748	0.861665	0.430833	+	0.902432i	$0.358220\pi$
$380$	0	0
$381$	3.24420	0.166205
$382$	0	0
$383$	16.6147	0.848973	0.424487	−	0.905434i	$-0.360455\pi$
$383$	16.6147	0.848973	0.424487	+	0.905434i	$0.360455\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−24.5292	−1.24689
$388$	0	0
$389$	36.7193	1.86174	0.930870	−	0.365350i	$-0.119051\pi$
$389$	36.7193	1.86174	0.930870	+	0.365350i	$0.119051\pi$
$390$	0	0
$391$	12.4138	0.627793
$392$	0	0
$393$	−25.2382	−1.27310
$394$	0	0
$395$	0	0
$396$	0	0
$397$	29.2104	1.46603	0.733015	−	0.680212i	$-0.238112\pi$
$397$	29.2104	1.46603	0.733015	+	0.680212i	$0.238112\pi$
$398$	0	0
$399$	−14.6594	−0.733888
$400$	0	0
$401$	15.6052	0.779288	0.389644	−	0.920966i	$-0.372598\pi$
$401$	15.6052	0.779288	0.389644	+	0.920966i	$0.372598\pi$
$402$	0	0
$403$	8.58480	0.427640
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−15.6798	−0.777220
$408$	0	0
$409$	13.7952	0.682131	0.341066	−	0.940039i	$-0.389212\pi$
$409$	13.7952	0.682131	0.341066	+	0.940039i	$0.389212\pi$
$410$	0	0
$411$	−33.4342	−1.64919
$412$	0	0
$413$	−6.39478	−0.314666
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−16.8026	−0.822827
$418$	0	0
$419$	30.6893	1.49927	0.749636	−	0.661850i	$-0.230229\pi$
$419$	30.6893	1.49927	0.749636	+	0.661850i	$0.230229\pi$
$420$	0	0
$421$	−16.9180	−0.824535	−0.412268	−	0.911063i	$-0.635263\pi$
$421$	−16.9180	−0.824535	−0.412268	+	0.911063i	$0.635263\pi$
$422$	0	0
$423$	−6.19003	−0.300969
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−17.1696	−0.830895
$428$	0	0
$429$	4.58480	0.221356
$430$	0	0
$431$	17.9722	0.865691	0.432846	−	0.901468i	$-0.357510\pi$
$431$	17.9722	0.865691	0.432846	+	0.901468i	$0.357510\pi$
$432$	0	0
$433$	2.61259	0.125553	0.0627764	−	0.998028i	$-0.480004\pi$
$433$	2.61259	0.125553	0.0627764	+	0.998028i	$0.480004\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−13.1696	−0.629988
$438$	0	0
$439$	−14.6594	−0.699655	−0.349827	−	0.936814i	$-0.613760\pi$
$439$	−14.6594	−0.699655	−0.349827	+	0.936814i	$0.613760\pi$
$440$	0	0
$441$	−12.2333	−0.582538
$442$	0	0
$443$	−8.33324	−0.395924	−0.197962	−	0.980210i	$-0.563432\pi$
$443$	−8.33324	−0.395924	−0.197962	+	0.980210i	$0.563432\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−4.04084	−0.190699	−0.0953495	−	0.995444i	$-0.530397\pi$
$449$	−4.04084	−0.190699	−0.0953495	+	0.995444i	$0.530397\pi$
$450$	0	0
$451$	19.3596	0.911610
$452$	0	0
$453$	33.3705	1.56788
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−0.979580	−0.0458228	−0.0229114	−	0.999737i	$-0.507294\pi$
$457$	−0.979580	−0.0458228	−0.0229114	+	0.999737i	$0.507294\pi$
$458$	0	0
$459$	8.20093	0.382787
$460$	0	0
$461$	10.7280	0.499654	0.249827	−	0.968291i	$-0.419626\pi$
$461$	10.7280	0.499654	0.249827	+	0.968291i	$0.419626\pi$
$462$	0	0
$463$	−23.2104	−1.07868	−0.539340	−	0.842088i	$-0.681326\pi$
$463$	−23.2104	−1.07868	−0.539340	+	0.842088i	$0.681326\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−28.5848	−1.32275	−0.661373	−	0.750057i	$-0.730026\pi$
$467$	−28.5848	−1.32275	−0.661373	+	0.750057i	$0.730026\pi$
$468$	0	0
$469$	10.7748	0.497535
$470$	0	0
$471$	−30.0190	−1.38320
$472$	0	0
$473$	21.7544	1.00027
$474$	0	0
$475$	0	0
$476$	0	0
$477$	5.82900	0.266892
$478$	0	0
$479$	30.4078	1.38937	0.694685	−	0.719314i	$-0.255544\pi$
$479$	30.4078	1.38937	0.694685	+	0.719314i	$0.255544\pi$
$480$	0	0
$481$	−7.83991	−0.357469
$482$	0	0
$483$	7.43701	0.338396
$484$	0	0
$485$	0	0
$486$	0	0
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	0	0
$489$	28.1154	1.27142
$490$	0	0
$491$	−22.6893	−1.02396	−0.511978	−	0.858999i	$-0.671087\pi$
$491$	−22.6893	−1.02396	−0.511978	+	0.858999i	$0.671087\pi$
$492$	0	0
$493$	−24.4694	−1.10204
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.76177	0.303307
$498$	0	0
$499$	−31.8698	−1.42669	−0.713345	−	0.700813i	$-0.752821\pi$
$499$	−31.8698	−1.42669	−0.713345	+	0.700813i	$0.752821\pi$
$500$	0	0
$501$	−42.0408	−1.87825
$502$	0	0
$503$	−14.2646	−0.636028	−0.318014	−	0.948086i	$-0.603016\pi$
$503$	−14.2646	−0.636028	−0.318014	+	0.948086i	$0.603016\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	2.29240	0.101809
$508$	0	0
$509$	23.3596	1.03540	0.517699	−	0.855563i	$-0.326789\pi$
$509$	23.3596	1.03540	0.517699	+	0.855563i	$0.326789\pi$
$510$	0	0
$511$	−7.53063	−0.333135
$512$	0	0
$513$	−8.70024	−0.384125
$514$	0	0
$515$	0	0
$516$	0	0
$517$	5.48979	0.241441
$518$	0	0
$519$	21.5834	0.947407
$520$	0	0
$521$	−40.4247	−1.77104	−0.885519	−	0.464602i	$-0.846197\pi$
$521$	−40.4247	−1.77104	−0.885519	+	0.464602i	$0.846197\pi$
$522$	0	0
$523$	37.9853	1.66098	0.830490	−	0.557033i	$-0.188060\pi$
$523$	37.9853	1.66098	0.830490	+	0.557033i	$0.188060\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	41.2295	1.79598
$528$	0	0
$529$	−16.3188	−0.709513
$530$	0	0
$531$	11.4898	0.498614
$532$	0	0
$533$	9.67982	0.419279
$534$	0	0
$535$	0	0
$536$	0	0
$537$	16.8026	0.725086
$538$	0	0
$539$	10.8494	0.467318
$540$	0	0
$541$	26.8772	1.15554	0.577771	−	0.816199i	$-0.303923\pi$
$541$	26.8772	1.15554	0.577771	+	0.816199i	$0.303923\pi$
$542$	0	0
$543$	26.9458	1.15636
$544$	0	0
$545$	0	0
$546$	0	0
$547$	19.8976	0.850761	0.425381	−	0.905015i	$-0.360140\pi$
$547$	19.8976	0.850761	0.425381	+	0.905015i	$0.360140\pi$
$548$	0	0
$549$	30.8494	1.31662
$550$	0	0
$551$	25.9592	1.10590
$552$	0	0
$553$	18.9458	0.805659
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.50070	0.0635865	0.0317933	−	0.999494i	$-0.489878\pi$
$557$	1.50070	0.0635865	0.0317933	+	0.999494i	$0.489878\pi$
$558$	0	0
$559$	10.8772	0.460057
$560$	0	0
$561$	22.0190	0.929644
$562$	0	0
$563$	−20.6872	−0.871861	−0.435930	−	0.899980i	$-0.643581\pi$
$563$	−20.6872	−0.871861	−0.435930	+	0.899980i	$0.643581\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	13.4043	0.562927
$568$	0	0
$569$	−6.08550	−0.255117	−0.127559	−	0.991831i	$-0.540714\pi$
$569$	−6.08550	−0.255117	−0.127559	+	0.991831i	$0.540714\pi$
$570$	0	0
$571$	9.98909	0.418031	0.209015	−	0.977912i	$-0.432974\pi$
$571$	9.98909	0.418031	0.209015	+	0.977912i	$0.432974\pi$
$572$	0	0
$573$	3.67982	0.153727
$574$	0	0
$575$	0	0
$576$	0	0
$577$	8.33921	0.347166	0.173583	−	0.984819i	$-0.444466\pi$
$577$	8.33921	0.347166	0.173583	+	0.984819i	$0.444466\pi$
$578$	0	0
$579$	36.1154	1.50091
$580$	0	0
$581$	13.9254	0.577723
$582$	0	0
$583$	−5.16961	−0.214103
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−23.2442	−0.959391	−0.479695	−	0.877435i	$-0.659253\pi$
$587$	−23.2442	−0.959391	−0.479695	+	0.877435i	$0.659253\pi$
$588$	0	0
$589$	−43.7397	−1.80226
$590$	0	0
$591$	2.31421	0.0951940
$592$	0	0
$593$	34.2646	1.40708	0.703540	−	0.710656i	$-0.251602\pi$
$593$	34.2646	1.40708	0.703540	+	0.710656i	$0.251602\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−0.998609	−0.0408703
$598$	0	0
$599$	31.1886	1.27433	0.637167	−	0.770726i	$-0.280106\pi$
$599$	31.1886	1.27433	0.637167	+	0.770726i	$0.280106\pi$
$600$	0	0
$601$	−3.40429	−0.138864	−0.0694320	−	0.997587i	$-0.522119\pi$
$601$	−3.40429	−0.138864	−0.0694320	+	0.997587i	$0.522119\pi$
$602$	0	0
$603$	−19.3596	−0.788385
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−0.945827	−0.0383899	−0.0191950	−	0.999816i	$-0.506110\pi$
$607$	−0.945827	−0.0383899	−0.0191950	+	0.999816i	$0.506110\pi$
$608$	0	0
$609$	−14.6594	−0.594029
$610$	0	0
$611$	2.74489	0.111047
$612$	0	0
$613$	−40.7193	−1.64464	−0.822318	−	0.569028i	$-0.807319\pi$
$613$	−40.7193	−1.64464	−0.822318	+	0.569028i	$0.807319\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−12.6594	−0.509648	−0.254824	−	0.966987i	$-0.582018\pi$
$617$	−12.6594	−0.509648	−0.254824	+	0.966987i	$0.582018\pi$
$618$	0	0
$619$	−5.03945	−0.202553	−0.101276	−	0.994858i	$-0.532293\pi$
$619$	−5.03945	−0.202553	−0.101276	+	0.994858i	$0.532293\pi$
$620$	0	0
$621$	4.41381	0.177120
$622$	0	0
$623$	−6.39478	−0.256201
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−23.3596	−0.932894
$628$	0	0
$629$	−37.6520	−1.50128
$630$	0	0
$631$	1.00736	0.0401024	0.0200512	−	0.999799i	$-0.493617\pi$
$631$	1.00736	0.0401024	0.0200512	+	0.999799i	$0.493617\pi$
$632$	0	0
$633$	−37.4810	−1.48974
$634$	0	0
$635$	0	0
$636$	0	0
$637$	5.42471	0.214935
$638$	0	0
$639$	−12.1492	−0.480614
$640$	0	0
$641$	−41.3596	−1.63361	−0.816804	−	0.576916i	$-0.804256\pi$
$641$	−41.3596	−1.63361	−0.816804	+	0.576916i	$0.804256\pi$
$642$	0	0
$643$	−3.26601	−0.128799	−0.0643994	−	0.997924i	$-0.520513\pi$
$643$	−3.26601	−0.128799	−0.0643994	+	0.997924i	$0.520513\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	19.4342	0.764038	0.382019	−	0.924154i	$-0.375229\pi$
$647$	19.4342	0.764038	0.382019	+	0.924154i	$0.375229\pi$
$648$	0	0
$649$	−10.1900	−0.399994
$650$	0	0
$651$	24.7002	0.968079
$652$	0	0
$653$	4.51021	0.176498	0.0882491	−	0.996098i	$-0.471873\pi$
$653$	4.51021	0.176498	0.0882491	+	0.996098i	$0.471873\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	13.5306	0.527880
$658$	0	0
$659$	−27.2104	−1.05997	−0.529984	−	0.848007i	$-0.677802\pi$
$659$	−27.2104	−1.05997	−0.529984	+	0.848007i	$0.677802\pi$
$660$	0	0
$661$	−24.5292	−0.954077	−0.477038	−	0.878882i	$-0.658290\pi$
$661$	−24.5292	−0.954077	−0.477038	+	0.878882i	$0.658290\pi$
$662$	0	0
$663$	11.0095	0.427574
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−13.1696	−0.509929
$668$	0	0
$669$	−49.9385	−1.93073
$670$	0	0
$671$	−27.3596	−1.05621
$672$	0	0
$673$	8.95180	0.345066	0.172533	−	0.985004i	$-0.444805\pi$
$673$	8.95180	0.345066	0.172533	+	0.985004i	$0.444805\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	32.6256	1.25391	0.626953	−	0.779057i	$-0.284302\pi$
$677$	32.6256	1.25391	0.626953	+	0.779057i	$0.284302\pi$
$678$	0	0
$679$	7.86276	0.301745
$680$	0	0
$681$	−20.6784	−0.792399
$682$	0	0
$683$	−11.0950	−0.424539	−0.212269	−	0.977211i	$-0.568085\pi$
$683$	−11.0950	−0.424539	−0.212269	+	0.977211i	$0.568085\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	10.7449	0.409943
$688$	0	0
$689$	−2.58480	−0.0984732
$690$	0	0
$691$	13.5306	0.514729	0.257365	−	0.966314i	$-0.417146\pi$
$691$	13.5306	0.514729	0.257365	+	0.966314i	$0.417146\pi$
$692$	0	0
$693$	5.66079	0.215036
$694$	0	0
$695$	0	0
$696$	0	0
$697$	46.4884	1.76087
$698$	0	0
$699$	−31.0313	−1.17371
$700$	0	0
$701$	−48.1345	−1.81801	−0.909007	−	0.416781i	$-0.863158\pi$
$701$	−48.1345	−1.81801	−0.909007	+	0.416781i	$0.863158\pi$
$702$	0	0
$703$	39.9444	1.50653
$704$	0	0
$705$	0	0
$706$	0	0
$707$	15.8889	0.597563
$708$	0	0
$709$	16.5292	0.620769	0.310384	−	0.950611i	$-0.399542\pi$
$709$	16.5292	0.620769	0.310384	+	0.950611i	$0.399542\pi$
$710$	0	0
$711$	−34.0408	−1.27663
$712$	0	0
$713$	22.1900	0.831023
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−7.32970	−0.273733
$718$	0	0
$719$	6.39478	0.238485	0.119242	−	0.992865i	$-0.461953\pi$
$719$	6.39478	0.238485	0.119242	+	0.992865i	$0.461953\pi$
$720$	0	0
$721$	−9.54535	−0.355488
$722$	0	0
$723$	32.3582	1.20342
$724$	0	0
$725$	0	0
$726$	0	0
$727$	35.0204	1.29884	0.649418	−	0.760432i	$-0.275013\pi$
$727$	35.0204	1.29884	0.649418	+	0.760432i	$0.275013\pi$
$728$	0	0
$729$	−12.3406	−0.457059
$730$	0	0
$731$	52.2390	1.93213
$732$	0	0
$733$	22.6485	0.836541	0.418271	−	0.908322i	$-0.362636\pi$
$733$	22.6485	0.836541	0.418271	+	0.908322i	$0.362636\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	17.1696	0.632451
$738$	0	0
$739$	−25.0394	−0.921091	−0.460546	−	0.887636i	$-0.652346\pi$
$739$	−25.0394	−0.921091	−0.460546	+	0.887636i	$0.652346\pi$
$740$	0	0
$741$	−11.6798	−0.429069
$742$	0	0
$743$	−35.8252	−1.31430	−0.657149	−	0.753760i	$-0.728238\pi$
$743$	−35.8252	−1.31430	−0.657149	+	0.753760i	$0.728238\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−25.0204	−0.915449
$748$	0	0
$749$	−5.75441	−0.210262
$750$	0	0
$751$	−41.6988	−1.52161	−0.760806	−	0.648979i	$-0.775196\pi$
$751$	−41.6988	−1.52161	−0.760806	+	0.648979i	$0.775196\pi$
$752$	0	0
$753$	48.5292	1.76850
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−11.6052	−0.421799	−0.210900	−	0.977508i	$-0.567639\pi$
$757$	−11.6052	−0.421799	−0.210900	+	0.977508i	$0.567639\pi$
$758$	0	0
$759$	11.8508	0.430157
$760$	0	0
$761$	30.2646	1.09709	0.548546	−	0.836121i	$-0.315182\pi$
$761$	30.2646	1.09709	0.548546	+	0.836121i	$0.315182\pi$
$762$	0	0
$763$	−18.6725	−0.675988
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5.09501	−0.183970
$768$	0	0
$769$	24.6594	0.889241	0.444620	−	0.895719i	$-0.353339\pi$
$769$	24.6594	0.889241	0.444620	+	0.895719i	$0.353339\pi$
$770$	0	0
$771$	−18.8385	−0.678453
$772$	0	0
$773$	30.3501	1.09162	0.545809	−	0.837910i	$-0.316222\pi$
$773$	30.3501	1.09162	0.545809	+	0.837910i	$0.316222\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−22.5570	−0.809229
$778$	0	0
$779$	−49.3188	−1.76703
$780$	0	0
$781$	10.7748	0.385554
$782$	0	0
$783$	−8.70024	−0.310921
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−11.1288	−0.396698	−0.198349	−	0.980131i	$-0.563558\pi$
$787$	−11.1288	−0.396698	−0.198349	+	0.980131i	$0.563558\pi$
$788$	0	0
$789$	−10.3392	−0.368086
$790$	0	0
$791$	5.02042	0.178506
$792$	0	0
$793$	−13.6798	−0.485785
$794$	0	0
$795$	0	0
$796$	0	0
$797$	46.7939	1.65752	0.828762	−	0.559602i	$-0.189046\pi$
$797$	46.7939	1.65752	0.828762	+	0.559602i	$0.189046\pi$
$798$	0	0
$799$	13.1827	0.466369
$800$	0	0
$801$	11.4898	0.405972
$802$	0	0
$803$	−12.0000	−0.423471
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−19.5088	−0.686743
$808$	0	0
$809$	50.4437	1.77351	0.886754	−	0.462242i	$-0.152955\pi$
$809$	50.4437	1.77351	0.886754	+	0.462242i	$0.152955\pi$
$810$	0	0
$811$	36.3991	1.27814	0.639072	−	0.769147i	$-0.279318\pi$
$811$	36.3991	1.27814	0.639072	+	0.769147i	$0.279318\pi$
$812$	0	0
$813$	−11.3515	−0.398115
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−55.4195	−1.93888
$818$	0	0
$819$	2.83039	0.0989020
$820$	0	0
$821$	−6.23684	−0.217667	−0.108834	−	0.994060i	$-0.534712\pi$
$821$	−6.23684	−0.217667	−0.108834	+	0.994060i	$0.534712\pi$
$822$	0	0
$823$	−33.7734	−1.17727	−0.588634	−	0.808400i	$-0.700334\pi$
$823$	−33.7734	−1.17727	−0.588634	+	0.808400i	$0.700334\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−37.2295	−1.29460	−0.647298	−	0.762237i	$-0.724101\pi$
$827$	−37.2295	−1.29460	−0.647298	+	0.762237i	$0.724101\pi$
$828$	0	0
$829$	−38.2646	−1.32899	−0.664493	−	0.747295i	$-0.731352\pi$
$829$	−38.2646	−1.32899	−0.664493	+	0.747295i	$0.731352\pi$
$830$	0	0
$831$	−45.5497	−1.58010
$832$	0	0
$833$	26.0528	0.902675
$834$	0	0
$835$	0	0
$836$	0	0
$837$	14.6594	0.506703
$838$	0	0
$839$	−50.1345	−1.73083	−0.865417	−	0.501052i	$-0.832946\pi$
$839$	−50.1345	−1.73083	−0.865417	+	0.501052i	$0.832946\pi$
$840$	0	0
$841$	−3.04084	−0.104857
$842$	0	0
$843$	69.3787	2.38953
$844$	0	0
$845$	0	0
$846$	0	0
$847$	8.78574	0.301881
$848$	0	0
$849$	61.7207	2.11825
$850$	0	0
$851$	−20.2646	−0.694662
$852$	0	0
$853$	−12.3910	−0.424258	−0.212129	−	0.977242i	$-0.568040\pi$
$853$	−12.3910	−0.424258	−0.212129	+	0.977242i	$0.568040\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−25.4005	−0.867664	−0.433832	−	0.900994i	$-0.642839\pi$
$857$	−25.4005	−0.867664	−0.433832	+	0.900994i	$0.642839\pi$
$858$	0	0
$859$	50.1900	1.71246	0.856231	−	0.516593i	$-0.172800\pi$
$859$	50.1900	1.71246	0.856231	+	0.516593i	$0.172800\pi$
$860$	0	0
$861$	27.8508	0.949153
$862$	0	0
$863$	43.4451	1.47889	0.739445	−	0.673217i	$-0.235088\pi$
$863$	43.4451	1.47889	0.739445	+	0.673217i	$0.235088\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	13.9036	0.472191
$868$	0	0
$869$	30.1900	1.02413
$870$	0	0
$871$	8.58480	0.290885
$872$	0	0
$873$	−14.1274	−0.478139
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−11.6689	−0.394031	−0.197016	−	0.980400i	$-0.563125\pi$
$877$	−11.6689	−0.394031	−0.197016	+	0.980400i	$0.563125\pi$
$878$	0	0
$879$	−11.8758	−0.400561
$880$	0	0
$881$	−0.0446585	−0.00150458	−0.000752291	−	1.00000i	$-0.500239\pi$
$881$	−0.0446585	−0.00150458	−0.000752291		1.00000i	$0.500239\pi$
$882$	0	0
$883$	−24.4269	−0.822029	−0.411015	−	0.911629i	$-0.634826\pi$
$883$	−24.4269	−0.822029	−0.411015	+	0.911629i	$0.634826\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−47.2295	−1.58581	−0.792905	−	0.609345i	$-0.791433\pi$
$887$	−47.2295	−1.58581	−0.792905	+	0.609345i	$0.791433\pi$
$888$	0	0
$889$	−1.77622	−0.0595725
$890$	0	0
$891$	21.3596	0.715575
$892$	0	0
$893$	−13.9853	−0.467999
$894$	0	0
$895$	0	0
$896$	0	0
$897$	5.92541	0.197844
$898$	0	0
$899$	−43.7397	−1.45880
$900$	0	0
$901$	−12.4138	−0.413564
$902$	0	0
$903$	31.2959	1.04146
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−44.7133	−1.48468	−0.742340	−	0.670023i	$-0.766284\pi$
$907$	−44.7133	−1.48468	−0.742340	+	0.670023i	$0.766284\pi$
$908$	0	0
$909$	−28.5483	−0.946886
$910$	0	0
$911$	−51.9444	−1.72100	−0.860498	−	0.509454i	$-0.829847\pi$
$911$	−51.9444	−1.72100	−0.860498	+	0.509454i	$0.829847\pi$
$912$	0	0
$913$	22.1900	0.734383
$914$	0	0
$915$	0	0
$916$	0	0
$917$	13.8181	0.456314
$918$	0	0
$919$	13.5497	0.446962	0.223481	−	0.974708i	$-0.428258\pi$
$919$	13.5497	0.446962	0.223481	+	0.974708i	$0.428258\pi$
$920$	0	0
$921$	−0.733989	−0.0241858
$922$	0	0
$923$	5.38741	0.177329
$924$	0	0
$925$	0	0
$926$	0	0
$927$	17.1506	0.563299
$928$	0	0
$929$	−44.1682	−1.44911	−0.724556	−	0.689216i	$-0.757955\pi$
$929$	−44.1682	−1.44911	−0.724556	+	0.689216i	$0.757955\pi$
$930$	0	0
$931$	−27.6390	−0.905831
$932$	0	0
$933$	22.6256	0.740730
$934$	0	0
$935$	0	0
$936$	0	0
$937$	41.6988	1.36224	0.681121	−	0.732171i	$-0.261493\pi$
$937$	41.6988	1.36224	0.681121	+	0.732171i	$0.261493\pi$
$938$	0	0
$939$	−24.3283	−0.793924
$940$	0	0
$941$	37.4473	1.22075	0.610373	−	0.792114i	$-0.291019\pi$
$941$	37.4473	1.22075	0.610373	+	0.792114i	$0.291019\pi$
$942$	0	0
$943$	25.0204	0.814777
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−11.5644	−0.375792	−0.187896	−	0.982189i	$-0.560167\pi$
$947$	−11.5644	−0.375792	−0.187896	+	0.982189i	$0.560167\pi$
$948$	0	0
$949$	−6.00000	−0.194768
$950$	0	0
$951$	27.9444	0.906160
$952$	0	0
$953$	−53.4810	−1.73242	−0.866210	−	0.499680i	$-0.833451\pi$
$953$	−53.4810	−1.73242	−0.866210	+	0.499680i	$0.833451\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−23.3596	−0.755110
$958$	0	0
$959$	18.3055	0.591114
$960$	0	0
$961$	42.6988	1.37738
$962$	0	0
$963$	10.3392	0.333176
$964$	0	0
$965$	0	0
$966$	0	0
$967$	34.8941	1.12212	0.561059	−	0.827776i	$-0.310394\pi$
$967$	34.8941	1.12212	0.561059	+	0.827776i	$0.310394\pi$
$968$	0	0
$969$	−56.0936	−1.80199
$970$	0	0
$971$	−31.8807	−1.02310	−0.511551	−	0.859253i	$-0.670929\pi$
$971$	−31.8807	−1.02310	−0.511551	+	0.859253i	$0.670929\pi$
$972$	0	0
$973$	9.19954	0.294924
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−2.03375	−0.0650655	−0.0325328	−	0.999471i	$-0.510357\pi$
$977$	−2.03375	−0.0650655	−0.0325328	+	0.999471i	$0.510357\pi$
$978$	0	0
$979$	−10.1900	−0.325675
$980$	0	0
$981$	33.5497	1.07116
$982$	0	0
$983$	29.4043	0.937851	0.468926	−	0.883238i	$-0.344641\pi$
$983$	29.4043	0.937851	0.468926	+	0.883238i	$0.344641\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	7.89762	0.251384
$988$	0	0
$989$	28.1154	0.894019
$990$	0	0
$991$	−32.9986	−1.04824	−0.524118	−	0.851646i	$-0.675605\pi$
$991$	−32.9986	−1.04824	−0.524118	+	0.851646i	$0.675605\pi$
$992$	0	0
$993$	41.1359	1.30541
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−31.1696	−0.987151	−0.493576	−	0.869703i	$-0.664310\pi$
$997$	−31.1696	−0.987151	−0.493576	+	0.869703i	$0.664310\pi$
$998$	0	0
$999$	−13.3874	−0.423559

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5200.2.a.ce.1.3		3
4.3	odd	2		650.2.a.n.1.1		3
5.2	odd	4		1040.2.d.b.209.2		6
5.3	odd	4		1040.2.d.b.209.5		6
5.4	even	2		5200.2.a.cf.1.1		3
12.11	even	2		5850.2.a.cs.1.2		3
20.3	even	4		130.2.b.a.79.4	yes	6
20.7	even	4		130.2.b.a.79.3	✓	6
20.19	odd	2		650.2.a.o.1.3		3
52.51	odd	2		8450.2.a.cc.1.1		3
60.23	odd	4		1170.2.e.f.469.3		6
60.47	odd	4		1170.2.e.f.469.6		6
60.59	even	2		5850.2.a.cp.1.2		3
260.47	odd	4		1690.2.c.d.1689.5		6
260.83	odd	4		1690.2.c.d.1689.2		6
260.103	even	4		1690.2.b.a.339.1		6
260.187	odd	4		1690.2.c.a.1689.5		6
260.203	odd	4		1690.2.c.a.1689.2		6
260.207	even	4		1690.2.b.a.339.6		6
260.259	odd	2		8450.2.a.bs.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.2.b.a.79.3	✓	6	20.7	even	4
130.2.b.a.79.4	yes	6	20.3	even	4
650.2.a.n.1.1		3	4.3	odd	2
650.2.a.o.1.3		3	20.19	odd	2
1040.2.d.b.209.2		6	5.2	odd	4
1040.2.d.b.209.5		6	5.3	odd	4
1170.2.e.f.469.3		6	60.23	odd	4
1170.2.e.f.469.6		6	60.47	odd	4
1690.2.b.a.339.1		6	260.103	even	4
1690.2.b.a.339.6		6	260.207	even	4
1690.2.c.a.1689.2		6	260.203	odd	4
1690.2.c.a.1689.5		6	260.187	odd	4
1690.2.c.d.1689.2		6	260.83	odd	4
1690.2.c.d.1689.5		6	260.47	odd	4
5200.2.a.ce.1.3		3	1.1	even	1	trivial
5200.2.a.cf.1.1		3	5.4	even	2
5850.2.a.cp.1.2		3	60.59	even	2
5850.2.a.cs.1.2		3	12.11	even	2
8450.2.a.bs.1.3		3	260.259	odd	2
8450.2.a.cc.1.1		3	52.51	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5200 = 2^{4} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5200.a (trivial)

\(f(q)\)	\(=\)	\(q+2.29240 q^{3} -1.25511 q^{7} +2.25511 q^{9} -2.00000 q^{11} -1.00000 q^{13} -4.80261 q^{17} +5.09501 q^{19} -2.87720 q^{21} -2.58480 q^{23} -1.70760 q^{27} +5.09501 q^{29} -8.58480 q^{31} -4.58480 q^{33} +7.83991 q^{37} -2.29240 q^{39} -9.67982 q^{41} -10.8772 q^{43} -2.74489 q^{47} -5.42471 q^{49} -11.0095 q^{51} +2.58480 q^{53} +11.6798 q^{57} +5.09501 q^{59} +13.6798 q^{61} -2.83039 q^{63} -8.58480 q^{67} -5.92541 q^{69} -5.38741 q^{71} +6.00000 q^{73} +2.51021 q^{77} -15.0950 q^{79} -10.6798 q^{81} -11.0950 q^{83} +11.6798 q^{87} +5.09501 q^{89} +1.25511 q^{91} -19.6798 q^{93} -6.26462 q^{97} -4.51021 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{7} + 5 q^{9} - 6 q^{11} - 3 q^{13} - 4 q^{17} - 2 q^{19} + 12 q^{21} + 6 q^{23} - 12 q^{27} - 2 q^{29} - 12 q^{31} + 8 q^{37} + 2 q^{41} - 12 q^{43} - 10 q^{47} + 13 q^{49} + 10 q^{51} - 6 q^{53}+ \cdots - 10 q^{99}+O(q^{100}) \) 3 * q - 2 * q^7 + 5 * q^9 - 6 * q^11 - 3 * q^13 - 4 * q^17 - 2 * q^19 + 12 * q^21 + 6 * q^23 - 12 * q^27 - 2 * q^29 - 12 * q^31 + 8 * q^37 + 2 * q^41 - 12 * q^43 - 10 * q^47 + 13 * q^49 + 10 * q^51 - 6 * q^53 + 4 * q^57 - 2 * q^59 + 10 * q^61 - 36 * q^63 - 12 * q^67 - 28 * q^69 + 8 * q^71 + 18 * q^73 + 4 * q^77 - 28 * q^79 - q^81 - 16 * q^83 + 4 * q^87 - 2 * q^89 + 2 * q^91 - 28 * q^93 + 26 * q^97 - 10 * q^99

Introduction

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