LMFDB - Embedded newform 5096.2.a.z.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5096,2,Mod(1,5096)] 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("5096.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5096, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5096 = 2^{3} \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5096.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,0,-2,0,2,0,0,0,7,0,8] 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:	$40.6917648700$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 12x^{5} + 19x^{4} + 33x^{3} - 27x^{2} - 24x - 4 $ x^7 - 2x^6 - 12x^5 + 19x^4 + 33x^3 - 27x^2 - 24x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 728)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.33404$ of defining polynomial
Character	$\chi$	$=$	5096.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.33404 q^{3} +1.42174 q^{5} -1.22033 q^{9} -6.47531 q^{11} -1.00000 q^{13} -1.89666 q^{15} -5.48747 q^{17} -2.96375 q^{19} +4.94091 q^{23} -2.97867 q^{25} +5.63010 q^{27} -3.73126 q^{29} -3.98803 q^{31} +8.63834 q^{33} -6.58298 q^{37} +1.33404 q^{39} -10.3060 q^{41} +10.8671 q^{43} -1.73499 q^{45} +10.9919 q^{47} +7.32052 q^{51} +5.45146 q^{53} -9.20618 q^{55} +3.95377 q^{57} +2.14499 q^{59} +2.09880 q^{61} -1.42174 q^{65} +4.94029 q^{67} -6.59139 q^{69} +15.3285 q^{71} +10.6591 q^{73} +3.97367 q^{75} -0.0739765 q^{79} -3.84981 q^{81} +11.4156 q^{83} -7.80174 q^{85} +4.97766 q^{87} -5.97451 q^{89} +5.32021 q^{93} -4.21367 q^{95} -1.49593 q^{97} +7.90202 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 2 q^{3} + 2 q^{5} + 7 q^{9} + 8 q^{11} - 7 q^{13} + 13 q^{15} - 8 q^{17} + 2 q^{19} + 16 q^{23} + 17 q^{25} - 11 q^{27} + 4 q^{29} - q^{31} + 14 q^{33} - 4 q^{37} + 2 q^{39} + 4 q^{41} + 8 q^{43}+ \cdots + 48 q^{99}+O(q^{100}) $ 7 * q - 2 * q^3 + 2 * q^5 + 7 * q^9 + 8 * q^11 - 7 * q^13 + 13 * q^15 - 8 * q^17 + 2 * q^19 + 16 * q^23 + 17 * q^25 - 11 * q^27 + 4 * q^29 - q^31 + 14 * q^33 - 4 * q^37 + 2 * q^39 + 4 * q^41 + 8 * q^43 - 5 * q^45 + 20 * q^47 - 5 * q^51 + 4 * q^53 - 16 * q^55 - 12 * q^57 - 8 * q^59 + 2 * q^61 - 2 * q^65 - 4 * q^67 + 2 * q^69 + 36 * q^71 - 8 * q^73 + 28 * q^75 + 18 * q^79 + 23 * q^81 - 12 * q^83 - q^85 + 26 * q^87 + 34 * q^89 - 12 * q^93 + 58 * q^95 - 2 * q^97 + 48 * 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
$3$	−1.33404	−0.770210	−0.385105	−	0.922873i	$-0.625835\pi$
$3$	−1.33404	−0.770210	−0.385105	+	0.922873i	$0.625835\pi$
$4$	0	0
$5$	1.42174	0.635820	0.317910	−	0.948121i	$-0.397019\pi$
$5$	1.42174	0.635820	0.317910	+	0.948121i	$0.397019\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.22033	−0.406777
$10$	0	0
$11$	−6.47531	−1.95238	−0.976190	−	0.216918i	$-0.930400\pi$
$11$	−6.47531	−1.95238	−0.976190	+	0.216918i	$0.930400\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−1.89666	−0.489715
$16$	0	0
$17$	−5.48747	−1.33091	−0.665454	−	0.746439i	$-0.731762\pi$
$17$	−5.48747	−1.33091	−0.665454	+	0.746439i	$0.731762\pi$
$18$	0	0
$19$	−2.96375	−0.679932	−0.339966	−	0.940438i	$-0.610416\pi$
$19$	−2.96375	−0.679932	−0.339966	+	0.940438i	$0.610416\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.94091	1.03025	0.515126	−	0.857115i	$-0.327745\pi$
$23$	4.94091	1.03025	0.515126	+	0.857115i	$0.327745\pi$
$24$	0	0
$25$	−2.97867	−0.595733
$26$	0	0
$27$	5.63010	1.08351
$28$	0	0
$29$	−3.73126	−0.692878	−0.346439	−	0.938072i	$-0.612609\pi$
$29$	−3.73126	−0.692878	−0.346439	+	0.938072i	$0.612609\pi$
$30$	0	0
$31$	−3.98803	−0.716272	−0.358136	−	0.933669i	$-0.616588\pi$
$31$	−3.98803	−0.716272	−0.358136	+	0.933669i	$0.616588\pi$
$32$	0	0
$33$	8.63834	1.50374
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.58298	−1.08223	−0.541117	−	0.840947i	$-0.681999\pi$
$37$	−6.58298	−1.08223	−0.541117	+	0.840947i	$0.681999\pi$
$38$	0	0
$39$	1.33404	0.213618
$40$	0	0
$41$	−10.3060	−1.60953	−0.804763	−	0.593596i	$-0.797708\pi$
$41$	−10.3060	−1.60953	−0.804763	+	0.593596i	$0.797708\pi$
$42$	0	0
$43$	10.8671	1.65721	0.828607	−	0.559830i	$-0.189134\pi$
$43$	10.8671	1.65721	0.828607	+	0.559830i	$0.189134\pi$
$44$	0	0
$45$	−1.73499	−0.258637
$46$	0	0
$47$	10.9919	1.60334	0.801669	−	0.597769i	$-0.203946\pi$
$47$	10.9919	1.60334	0.801669	+	0.597769i	$0.203946\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	7.32052	1.02508
$52$	0	0
$53$	5.45146	0.748815	0.374408	−	0.927264i	$-0.377846\pi$
$53$	5.45146	0.748815	0.374408	+	0.927264i	$0.377846\pi$
$54$	0	0
$55$	−9.20618	−1.24136
$56$	0	0
$57$	3.95377	0.523690
$58$	0	0
$59$	2.14499	0.279254	0.139627	−	0.990204i	$-0.455410\pi$
$59$	2.14499	0.279254	0.139627	+	0.990204i	$0.455410\pi$
$60$	0	0
$61$	2.09880	0.268724	0.134362	−	0.990932i	$-0.457101\pi$
$61$	2.09880	0.268724	0.134362	+	0.990932i	$0.457101\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.42174	−0.176345
$66$	0	0
$67$	4.94029	0.603553	0.301776	−	0.953379i	$-0.402420\pi$
$67$	4.94029	0.603553	0.301776	+	0.953379i	$0.402420\pi$
$68$	0	0
$69$	−6.59139	−0.793510
$70$	0	0
$71$	15.3285	1.81916	0.909581	−	0.415527i	$-0.136403\pi$
$71$	15.3285	1.81916	0.909581	+	0.415527i	$0.136403\pi$
$72$	0	0
$73$	10.6591	1.24755	0.623776	−	0.781603i	$-0.285598\pi$
$73$	10.6591	1.24755	0.623776	+	0.781603i	$0.285598\pi$
$74$	0	0
$75$	3.97367	0.458840
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−0.0739765	−0.00832301	−0.00416151	−	0.999991i	$-0.501325\pi$
$79$	−0.0739765	−0.00832301	−0.00416151	+	0.999991i	$0.501325\pi$
$80$	0	0
$81$	−3.84981	−0.427756
$82$	0	0
$83$	11.4156	1.25303	0.626513	−	0.779411i	$-0.284482\pi$
$83$	11.4156	1.25303	0.626513	+	0.779411i	$0.284482\pi$
$84$	0	0
$85$	−7.80174	−0.846217
$86$	0	0
$87$	4.97766	0.533662
$88$	0	0
$89$	−5.97451	−0.633297	−0.316649	−	0.948543i	$-0.602558\pi$
$89$	−5.97451	−0.633297	−0.316649	+	0.948543i	$0.602558\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	5.32021	0.551680
$94$	0	0
$95$	−4.21367	−0.432314
$96$	0	0
$97$	−1.49593	−0.151889	−0.0759445	−	0.997112i	$-0.524197\pi$
$97$	−1.49593	−0.151889	−0.0759445	+	0.997112i	$0.524197\pi$
$98$	0	0
$99$	7.90202	0.794182
$100$	0	0
$101$	−4.41174	−0.438985	−0.219492	−	0.975614i	$-0.570440\pi$
$101$	−4.41174	−0.438985	−0.219492	+	0.975614i	$0.570440\pi$
$102$	0	0
$103$	−10.5788	−1.04236	−0.521181	−	0.853446i	$-0.674508\pi$
$103$	−10.5788	−1.04236	−0.521181	+	0.853446i	$0.674508\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.55588	−0.440433	−0.220217	−	0.975451i	$-0.570676\pi$
$107$	−4.55588	−0.440433	−0.220217	+	0.975451i	$0.570676\pi$
$108$	0	0
$109$	7.54809	0.722976	0.361488	−	0.932377i	$-0.382269\pi$
$109$	7.54809	0.722976	0.361488	+	0.932377i	$0.382269\pi$
$110$	0	0
$111$	8.78197	0.833548
$112$	0	0
$113$	7.02619	0.660969	0.330484	−	0.943811i	$-0.392788\pi$
$113$	7.02619	0.660969	0.330484	+	0.943811i	$0.392788\pi$
$114$	0	0
$115$	7.02467	0.655054
$116$	0	0
$117$	1.22033	0.112820
$118$	0	0
$119$	0	0
$120$	0	0
$121$	30.9297	2.81179
$122$	0	0
$123$	13.7486	1.23967
$124$	0	0
$125$	−11.3436	−1.01460
$126$	0	0
$127$	16.5956	1.47262	0.736310	−	0.676644i	$-0.236566\pi$
$127$	16.5956	1.47262	0.736310	+	0.676644i	$0.236566\pi$
$128$	0	0
$129$	−14.4972	−1.27640
$130$	0	0
$131$	−3.42439	−0.299190	−0.149595	−	0.988747i	$-0.547797\pi$
$131$	−3.42439	−0.299190	−0.149595	+	0.988747i	$0.547797\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	8.00452	0.688919
$136$	0	0
$137$	14.1334	1.20750	0.603750	−	0.797174i	$-0.293673\pi$
$137$	14.1334	1.20750	0.603750	+	0.797174i	$0.293673\pi$
$138$	0	0
$139$	−18.2304	−1.54628	−0.773142	−	0.634232i	$-0.781316\pi$
$139$	−18.2304	−1.54628	−0.773142	+	0.634232i	$0.781316\pi$
$140$	0	0
$141$	−14.6637	−1.23491
$142$	0	0
$143$	6.47531	0.541493
$144$	0	0
$145$	−5.30487	−0.440546
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−13.1853	−1.08018	−0.540090	−	0.841607i	$-0.681610\pi$
$149$	−13.1853	−1.08018	−0.540090	+	0.841607i	$0.681610\pi$
$150$	0	0
$151$	10.1575	0.826609	0.413305	−	0.910593i	$-0.364374\pi$
$151$	10.1575	0.826609	0.413305	+	0.910593i	$0.364374\pi$
$152$	0	0
$153$	6.69652	0.541382
$154$	0	0
$155$	−5.66993	−0.455420
$156$	0	0
$157$	−8.09097	−0.645729	−0.322865	−	0.946445i	$-0.604646\pi$
$157$	−8.09097	−0.645729	−0.322865	+	0.946445i	$0.604646\pi$
$158$	0	0
$159$	−7.27248	−0.576745
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−4.36497	−0.341891	−0.170945	−	0.985280i	$-0.554682\pi$
$163$	−4.36497	−0.341891	−0.170945	+	0.985280i	$0.554682\pi$
$164$	0	0
$165$	12.2814	0.956109
$166$	0	0
$167$	−25.2156	−1.95124	−0.975620	−	0.219465i	$-0.929569\pi$
$167$	−25.2156	−1.95124	−0.975620	+	0.219465i	$0.929569\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	3.61676	0.276580
$172$	0	0
$173$	−12.5896	−0.957167	−0.478583	−	0.878042i	$-0.658850\pi$
$173$	−12.5896	−0.957167	−0.478583	+	0.878042i	$0.658850\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2.86151	−0.215084
$178$	0	0
$179$	−13.1806	−0.985167	−0.492583	−	0.870265i	$-0.663947\pi$
$179$	−13.1806	−0.985167	−0.492583	+	0.870265i	$0.663947\pi$
$180$	0	0
$181$	−7.44951	−0.553718	−0.276859	−	0.960911i	$-0.589293\pi$
$181$	−7.44951	−0.553718	−0.276859	+	0.960911i	$0.589293\pi$
$182$	0	0
$183$	−2.79989	−0.206974
$184$	0	0
$185$	−9.35926	−0.688106
$186$	0	0
$187$	35.5331	2.59844
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−7.52867	−0.544755	−0.272378	−	0.962190i	$-0.587810\pi$
$191$	−7.52867	−0.544755	−0.272378	+	0.962190i	$0.587810\pi$
$192$	0	0
$193$	25.5924	1.84218	0.921092	−	0.389345i	$-0.127299\pi$
$193$	25.5924	1.84218	0.921092	+	0.389345i	$0.127299\pi$
$194$	0	0
$195$	1.89666	0.135822
$196$	0	0
$197$	12.4346	0.885929	0.442964	−	0.896539i	$-0.353927\pi$
$197$	12.4346	0.885929	0.442964	+	0.896539i	$0.353927\pi$
$198$	0	0
$199$	2.23636	0.158531	0.0792656	−	0.996854i	$-0.474742\pi$
$199$	2.23636	0.158531	0.0792656	+	0.996854i	$0.474742\pi$
$200$	0	0
$201$	−6.59056	−0.464862
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−14.6524	−1.02337
$206$	0	0
$207$	−6.02954	−0.419082
$208$	0	0
$209$	19.1912	1.32748
$210$	0	0
$211$	9.82507	0.676386	0.338193	−	0.941077i	$-0.390184\pi$
$211$	9.82507	0.676386	0.338193	+	0.941077i	$0.390184\pi$
$212$	0	0
$213$	−20.4489	−1.40114
$214$	0	0
$215$	15.4501	1.05369
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−14.2197	−0.960876
$220$	0	0
$221$	5.48747	0.369127
$222$	0	0
$223$	−14.6646	−0.982012	−0.491006	−	0.871156i	$-0.663371\pi$
$223$	−14.6646	−0.982012	−0.491006	+	0.871156i	$0.663371\pi$
$224$	0	0
$225$	3.63496	0.242330
$226$	0	0
$227$	12.9287	0.858109	0.429055	−	0.903279i	$-0.358847\pi$
$227$	12.9287	0.858109	0.429055	+	0.903279i	$0.358847\pi$
$228$	0	0
$229$	0.212322	0.0140306	0.00701531	−	0.999975i	$-0.497767\pi$
$229$	0.212322	0.0140306	0.00701531	+	0.999975i	$0.497767\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−17.6453	−1.15598	−0.577990	−	0.816044i	$-0.696163\pi$
$233$	−17.6453	−1.15598	−0.577990	+	0.816044i	$0.696163\pi$
$234$	0	0
$235$	15.6276	1.01943
$236$	0	0
$237$	0.0986879	0.00641047
$238$	0	0
$239$	−27.0954	−1.75266	−0.876329	−	0.481713i	$-0.840015\pi$
$239$	−27.0954	−1.75266	−0.876329	+	0.481713i	$0.840015\pi$
$240$	0	0
$241$	−0.141723	−0.00912918	−0.00456459	−	0.999990i	$-0.501453\pi$
$241$	−0.141723	−0.00912918	−0.00456459	+	0.999990i	$0.501453\pi$
$242$	0	0
$243$	−11.7545	−0.754051
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.96375	0.188579
$248$	0	0
$249$	−15.2289	−0.965092
$250$	0	0
$251$	−9.20022	−0.580712	−0.290356	−	0.956919i	$-0.593774\pi$
$251$	−9.20022	−0.580712	−0.290356	+	0.956919i	$0.593774\pi$
$252$	0	0
$253$	−31.9939	−2.01144
$254$	0	0
$255$	10.4079	0.651765
$256$	0	0
$257$	27.9399	1.74284	0.871420	−	0.490537i	$-0.163200\pi$
$257$	27.9399	1.74284	0.871420	+	0.490537i	$0.163200\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	4.55337	0.281847
$262$	0	0
$263$	23.0438	1.42094	0.710471	−	0.703727i	$-0.248482\pi$
$263$	23.0438	1.42094	0.710471	+	0.703727i	$0.248482\pi$
$264$	0	0
$265$	7.75054	0.476112
$266$	0	0
$267$	7.97026	0.487772
$268$	0	0
$269$	12.0631	0.735501	0.367751	−	0.929924i	$-0.380128\pi$
$269$	12.0631	0.735501	0.367751	+	0.929924i	$0.380128\pi$
$270$	0	0
$271$	17.5000	1.06305	0.531526	−	0.847042i	$-0.321619\pi$
$271$	17.5000	1.06305	0.531526	+	0.847042i	$0.321619\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	19.2878	1.16310
$276$	0	0
$277$	−12.7586	−0.766589	−0.383295	−	0.923626i	$-0.625211\pi$
$277$	−12.7586	−0.766589	−0.383295	+	0.923626i	$0.625211\pi$
$278$	0	0
$279$	4.86672	0.291363
$280$	0	0
$281$	−21.2289	−1.26641	−0.633205	−	0.773984i	$-0.718261\pi$
$281$	−21.2289	−1.26641	−0.633205	+	0.773984i	$0.718261\pi$
$282$	0	0
$283$	−15.2121	−0.904264	−0.452132	−	0.891951i	$-0.649336\pi$
$283$	−15.2121	−0.904264	−0.452132	+	0.891951i	$0.649336\pi$
$284$	0	0
$285$	5.62122	0.332972
$286$	0	0
$287$	0	0
$288$	0	0
$289$	13.1123	0.771314
$290$	0	0
$291$	1.99564	0.116986
$292$	0	0
$293$	24.3078	1.42008	0.710040	−	0.704162i	$-0.248677\pi$
$293$	24.3078	1.42008	0.710040	+	0.704162i	$0.248677\pi$
$294$	0	0
$295$	3.04961	0.177555
$296$	0	0
$297$	−36.4567	−2.11543
$298$	0	0
$299$	−4.94091	−0.285740
$300$	0	0
$301$	0	0
$302$	0	0
$303$	5.88545	0.338111
$304$	0	0
$305$	2.98395	0.170860
$306$	0	0
$307$	28.1957	1.60922	0.804608	−	0.593806i	$-0.202375\pi$
$307$	28.1957	1.60922	0.804608	+	0.593806i	$0.202375\pi$
$308$	0	0
$309$	14.1126	0.802838
$310$	0	0
$311$	−1.94825	−0.110475	−0.0552374	−	0.998473i	$-0.517592\pi$
$311$	−1.94825	−0.110475	−0.0552374	+	0.998473i	$0.517592\pi$
$312$	0	0
$313$	23.9236	1.35224	0.676120	−	0.736791i	$-0.263660\pi$
$313$	23.9236	1.35224	0.676120	+	0.736791i	$0.263660\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6.71089	−0.376921	−0.188461	−	0.982081i	$-0.560350\pi$
$317$	−6.71089	−0.376921	−0.188461	+	0.982081i	$0.560350\pi$
$318$	0	0
$319$	24.1611	1.35276
$320$	0	0
$321$	6.07773	0.339226
$322$	0	0
$323$	16.2635	0.904926
$324$	0	0
$325$	2.97867	0.165227
$326$	0	0
$327$	−10.0695	−0.556843
$328$	0	0
$329$	0	0
$330$	0	0
$331$	16.5268	0.908393	0.454196	−	0.890902i	$-0.349926\pi$
$331$	16.5268	0.908393	0.454196	+	0.890902i	$0.349926\pi$
$332$	0	0
$333$	8.03340	0.440228
$334$	0	0
$335$	7.02379	0.383751
$336$	0	0
$337$	20.3139	1.10657	0.553283	−	0.832993i	$-0.313375\pi$
$337$	20.3139	1.10657	0.553283	+	0.832993i	$0.313375\pi$
$338$	0	0
$339$	−9.37324	−0.509085
$340$	0	0
$341$	25.8238	1.39844
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−9.37121	−0.504529
$346$	0	0
$347$	9.90756	0.531866	0.265933	−	0.963992i	$-0.414320\pi$
$347$	9.90756	0.531866	0.265933	+	0.963992i	$0.414320\pi$
$348$	0	0
$349$	12.7843	0.684327	0.342163	−	0.939640i	$-0.388840\pi$
$349$	12.7843	0.684327	0.342163	+	0.939640i	$0.388840\pi$
$350$	0	0
$351$	−5.63010	−0.300513
$352$	0	0
$353$	8.44261	0.449355	0.224677	−	0.974433i	$-0.427867\pi$
$353$	8.44261	0.449355	0.224677	+	0.974433i	$0.427867\pi$
$354$	0	0
$355$	21.7931	1.15666
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.69653	0.247874	0.123937	−	0.992290i	$-0.460448\pi$
$359$	4.69653	0.247874	0.123937	+	0.992290i	$0.460448\pi$
$360$	0	0
$361$	−10.2162	−0.537693
$362$	0	0
$363$	−41.2615	−2.16567
$364$	0	0
$365$	15.1544	0.793218
$366$	0	0
$367$	−17.9406	−0.936490	−0.468245	−	0.883599i	$-0.655114\pi$
$367$	−17.9406	−0.936490	−0.468245	+	0.883599i	$0.655114\pi$
$368$	0	0
$369$	12.5767	0.654718
$370$	0	0
$371$	0	0
$372$	0	0
$373$	13.9156	0.720521	0.360261	−	0.932852i	$-0.382688\pi$
$373$	13.9156	0.720521	0.360261	+	0.932852i	$0.382688\pi$
$374$	0	0
$375$	15.1328	0.781454
$376$	0	0
$377$	3.73126	0.192170
$378$	0	0
$379$	2.04903	0.105252	0.0526258	−	0.998614i	$-0.483241\pi$
$379$	2.04903	0.105252	0.0526258	+	0.998614i	$0.483241\pi$
$380$	0	0
$381$	−22.1392	−1.13423
$382$	0	0
$383$	−1.73439	−0.0886232	−0.0443116	−	0.999018i	$-0.514109\pi$
$383$	−1.73439	−0.0886232	−0.0443116	+	0.999018i	$0.514109\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−13.2614	−0.674116
$388$	0	0
$389$	−15.6261	−0.792277	−0.396138	−	0.918191i	$-0.629650\pi$
$389$	−15.6261	−0.792277	−0.396138	+	0.918191i	$0.629650\pi$
$390$	0	0
$391$	−27.1131	−1.37117
$392$	0	0
$393$	4.56828	0.230439
$394$	0	0
$395$	−0.105175	−0.00529193
$396$	0	0
$397$	−25.8518	−1.29747	−0.648734	−	0.761016i	$-0.724701\pi$
$397$	−25.8518	−1.29747	−0.648734	+	0.761016i	$0.724701\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−17.5100	−0.874408	−0.437204	−	0.899362i	$-0.644031\pi$
$401$	−17.5100	−0.874408	−0.437204	+	0.899362i	$0.644031\pi$
$402$	0	0
$403$	3.98803	0.198658
$404$	0	0
$405$	−5.47341	−0.271976
$406$	0	0
$407$	42.6268	2.11293
$408$	0	0
$409$	14.1163	0.698007	0.349003	−	0.937121i	$-0.386520\pi$
$409$	14.1163	0.698007	0.349003	+	0.937121i	$0.386520\pi$
$410$	0	0
$411$	−18.8546	−0.930028
$412$	0	0
$413$	0	0
$414$	0	0
$415$	16.2300	0.796698
$416$	0	0
$417$	24.3202	1.19096
$418$	0	0
$419$	3.82997	0.187106	0.0935532	−	0.995614i	$-0.470177\pi$
$419$	3.82997	0.187106	0.0935532	+	0.995614i	$0.470177\pi$
$420$	0	0
$421$	−6.50545	−0.317056	−0.158528	−	0.987354i	$-0.550675\pi$
$421$	−6.50545	−0.317056	−0.158528	+	0.987354i	$0.550675\pi$
$422$	0	0
$423$	−13.4138	−0.652200
$424$	0	0
$425$	16.3453	0.792866
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−8.63834	−0.417063
$430$	0	0
$431$	−11.9134	−0.573846	−0.286923	−	0.957954i	$-0.592632\pi$
$431$	−11.9134	−0.573846	−0.286923	+	0.957954i	$0.592632\pi$
$432$	0	0
$433$	1.73578	0.0834164	0.0417082	−	0.999130i	$-0.486720\pi$
$433$	1.73578	0.0834164	0.0417082	+	0.999130i	$0.486720\pi$
$434$	0	0
$435$	7.07693	0.339313
$436$	0	0
$437$	−14.6436	−0.700500
$438$	0	0
$439$	−7.90495	−0.377283	−0.188641	−	0.982046i	$-0.560408\pi$
$439$	−7.90495	−0.377283	−0.188641	+	0.982046i	$0.560408\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	30.0346	1.42698	0.713492	−	0.700663i	$-0.247112\pi$
$443$	30.0346	1.42698	0.713492	+	0.700663i	$0.247112\pi$
$444$	0	0
$445$	−8.49418	−0.402663
$446$	0	0
$447$	17.5897	0.831966
$448$	0	0
$449$	−7.38395	−0.348470	−0.174235	−	0.984704i	$-0.555745\pi$
$449$	−7.38395	−0.348470	−0.174235	+	0.984704i	$0.555745\pi$
$450$	0	0
$451$	66.7345	3.14241
$452$	0	0
$453$	−13.5506	−0.636663
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−0.264593	−0.0123771	−0.00618856	−	0.999981i	$-0.501970\pi$
$457$	−0.264593	−0.0123771	−0.00618856	+	0.999981i	$0.501970\pi$
$458$	0	0
$459$	−30.8950	−1.44206
$460$	0	0
$461$	0.701091	0.0326530	0.0163265	−	0.999867i	$-0.494803\pi$
$461$	0.701091	0.0326530	0.0163265	+	0.999867i	$0.494803\pi$
$462$	0	0
$463$	38.5664	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$463$	38.5664	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$464$	0	0
$465$	7.56393	0.350769
$466$	0	0
$467$	−6.03876	−0.279440	−0.139720	−	0.990191i	$-0.544620\pi$
$467$	−6.03876	−0.279440	−0.139720	+	0.990191i	$0.544620\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	10.7937	0.497347
$472$	0	0
$473$	−70.3677	−3.23551
$474$	0	0
$475$	8.82803	0.405058
$476$	0	0
$477$	−6.65258	−0.304601
$478$	0	0
$479$	15.1269	0.691165	0.345583	−	0.938388i	$-0.387681\pi$
$479$	15.1269	0.691165	0.345583	+	0.938388i	$0.387681\pi$
$480$	0	0
$481$	6.58298	0.300158
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.12682	−0.0965740
$486$	0	0
$487$	29.6489	1.34352	0.671761	−	0.740768i	$-0.265538\pi$
$487$	29.6489	1.34352	0.671761	+	0.740768i	$0.265538\pi$
$488$	0	0
$489$	5.82306	0.263328
$490$	0	0
$491$	−18.2627	−0.824186	−0.412093	−	0.911142i	$-0.635202\pi$
$491$	−18.2627	−0.824186	−0.412093	+	0.911142i	$0.635202\pi$
$492$	0	0
$493$	20.4752	0.922157
$494$	0	0
$495$	11.2346	0.504957
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−10.7488	−0.481184	−0.240592	−	0.970626i	$-0.577342\pi$
$499$	−10.7488	−0.481184	−0.240592	+	0.970626i	$0.577342\pi$
$500$	0	0
$501$	33.6387	1.50287
$502$	0	0
$503$	30.6801	1.36796	0.683979	−	0.729502i	$-0.260248\pi$
$503$	30.6801	1.36796	0.683979	+	0.729502i	$0.260248\pi$
$504$	0	0
$505$	−6.27233	−0.279115
$506$	0	0
$507$	−1.33404	−0.0592469
$508$	0	0
$509$	17.7092	0.784948	0.392474	−	0.919763i	$-0.371619\pi$
$509$	17.7092	0.784948	0.392474	+	0.919763i	$0.371619\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−16.6862	−0.736715
$514$	0	0
$515$	−15.0403	−0.662755
$516$	0	0
$517$	−71.1762	−3.13032
$518$	0	0
$519$	16.7950	0.737219
$520$	0	0
$521$	−0.500698	−0.0219360	−0.0109680	−	0.999940i	$-0.503491\pi$
$521$	−0.500698	−0.0219360	−0.0109680	+	0.999940i	$0.503491\pi$
$522$	0	0
$523$	−6.90377	−0.301881	−0.150940	−	0.988543i	$-0.548230\pi$
$523$	−6.90377	−0.301881	−0.150940	+	0.988543i	$0.548230\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	21.8842	0.953292
$528$	0	0
$529$	1.41260	0.0614175
$530$	0	0
$531$	−2.61760	−0.113594
$532$	0	0
$533$	10.3060	0.446402
$534$	0	0
$535$	−6.47725	−0.280036
$536$	0	0
$537$	17.5835	0.758785
$538$	0	0
$539$	0	0
$540$	0	0
$541$	14.8371	0.637898	0.318949	−	0.947772i	$-0.396670\pi$
$541$	14.8371	0.637898	0.318949	+	0.947772i	$0.396670\pi$
$542$	0	0
$543$	9.93796	0.426479
$544$	0	0
$545$	10.7314	0.459682
$546$	0	0
$547$	4.73378	0.202402	0.101201	−	0.994866i	$-0.467732\pi$
$547$	4.73378	0.202402	0.101201	+	0.994866i	$0.467732\pi$
$548$	0	0
$549$	−2.56123	−0.109311
$550$	0	0
$551$	11.0585	0.471110
$552$	0	0
$553$	0	0
$554$	0	0
$555$	12.4856	0.529986
$556$	0	0
$557$	12.6064	0.534149	0.267075	−	0.963676i	$-0.413943\pi$
$557$	12.6064	0.534149	0.267075	+	0.963676i	$0.413943\pi$
$558$	0	0
$559$	−10.8671	−0.459629
$560$	0	0
$561$	−47.4027	−2.00134
$562$	0	0
$563$	9.11831	0.384291	0.192145	−	0.981366i	$-0.438455\pi$
$563$	9.11831	0.384291	0.192145	+	0.981366i	$0.438455\pi$
$564$	0	0
$565$	9.98939	0.420257
$566$	0	0
$567$	0	0
$568$	0	0
$569$	8.22590	0.344848	0.172424	−	0.985023i	$-0.444840\pi$
$569$	8.22590	0.344848	0.172424	+	0.985023i	$0.444840\pi$
$570$	0	0
$571$	29.6725	1.24175	0.620877	−	0.783908i	$-0.286777\pi$
$571$	29.6725	1.24175	0.620877	+	0.783908i	$0.286777\pi$
$572$	0	0
$573$	10.0436	0.419576
$574$	0	0
$575$	−14.7173	−0.613755
$576$	0	0
$577$	−5.61243	−0.233648	−0.116824	−	0.993153i	$-0.537271\pi$
$577$	−5.61243	−0.233648	−0.116824	+	0.993153i	$0.537271\pi$
$578$	0	0
$579$	−34.1414	−1.41887
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−35.2999	−1.46197
$584$	0	0
$585$	1.73499	0.0717329
$586$	0	0
$587$	−7.31862	−0.302072	−0.151036	−	0.988528i	$-0.548261\pi$
$587$	−7.31862	−0.302072	−0.151036	+	0.988528i	$0.548261\pi$
$588$	0	0
$589$	11.8196	0.487016
$590$	0	0
$591$	−16.5883	−0.682351
$592$	0	0
$593$	2.07707	0.0852949	0.0426475	−	0.999090i	$-0.486421\pi$
$593$	2.07707	0.0852949	0.0426475	+	0.999090i	$0.486421\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−2.98340	−0.122102
$598$	0	0
$599$	20.3274	0.830554	0.415277	−	0.909695i	$-0.363685\pi$
$599$	20.3274	0.830554	0.415277	+	0.909695i	$0.363685\pi$
$600$	0	0
$601$	21.7340	0.886549	0.443275	−	0.896386i	$-0.353817\pi$
$601$	21.7340	0.886549	0.443275	+	0.896386i	$0.353817\pi$
$602$	0	0
$603$	−6.02878	−0.245511
$604$	0	0
$605$	43.9738	1.78779
$606$	0	0
$607$	−13.0720	−0.530576	−0.265288	−	0.964169i	$-0.585467\pi$
$607$	−13.0720	−0.530576	−0.265288	+	0.964169i	$0.585467\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−10.9919	−0.444686
$612$	0	0
$613$	−26.6950	−1.07820	−0.539100	−	0.842242i	$-0.681236\pi$
$613$	−26.6950	−1.07820	−0.539100	+	0.842242i	$0.681236\pi$
$614$	0	0
$615$	19.5469	0.788209
$616$	0	0
$617$	−30.0516	−1.20983	−0.604916	−	0.796290i	$-0.706793\pi$
$617$	−30.0516	−1.20983	−0.604916	+	0.796290i	$0.706793\pi$
$618$	0	0
$619$	7.28177	0.292679	0.146339	−	0.989234i	$-0.453251\pi$
$619$	7.28177	0.292679	0.146339	+	0.989234i	$0.453251\pi$
$620$	0	0
$621$	27.8178	1.11629
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−1.23422	−0.0493686
$626$	0	0
$627$	−25.6019	−1.02244
$628$	0	0
$629$	36.1239	1.44035
$630$	0	0
$631$	7.66163	0.305005	0.152502	−	0.988303i	$-0.451267\pi$
$631$	7.66163	0.305005	0.152502	+	0.988303i	$0.451267\pi$
$632$	0	0
$633$	−13.1071	−0.520959
$634$	0	0
$635$	23.5946	0.936321
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−18.7059	−0.739992
$640$	0	0
$641$	12.7398	0.503193	0.251597	−	0.967832i	$-0.419044\pi$
$641$	12.7398	0.503193	0.251597	+	0.967832i	$0.419044\pi$
$642$	0	0
$643$	−2.21508	−0.0873542	−0.0436771	−	0.999046i	$-0.513907\pi$
$643$	−2.21508	−0.0873542	−0.0436771	+	0.999046i	$0.513907\pi$
$644$	0	0
$645$	−20.6111	−0.811562
$646$	0	0
$647$	−41.5732	−1.63441	−0.817205	−	0.576347i	$-0.804478\pi$
$647$	−41.5732	−1.63441	−0.817205	+	0.576347i	$0.804478\pi$
$648$	0	0
$649$	−13.8895	−0.545210
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−36.5089	−1.42871	−0.714353	−	0.699786i	$-0.753279\pi$
$653$	−36.5089	−1.42871	−0.714353	+	0.699786i	$0.753279\pi$
$654$	0	0
$655$	−4.86857	−0.190231
$656$	0	0
$657$	−13.0076	−0.507475
$658$	0	0
$659$	−24.4320	−0.951736	−0.475868	−	0.879517i	$-0.657866\pi$
$659$	−24.4320	−0.951736	−0.475868	+	0.879517i	$0.657866\pi$
$660$	0	0
$661$	−0.0150773	−0.000586439 0	−0.000293219	−	1.00000i	$-0.500093\pi$
$661$	−0.0150773	−0.000586439 0	−0.000293219		1.00000i	$0.500093\pi$
$662$	0	0
$663$	−7.32052	−0.284305
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−18.4358	−0.713839
$668$	0	0
$669$	19.5632	0.756355
$670$	0	0
$671$	−13.5904	−0.524652
$672$	0	0
$673$	−22.5358	−0.868691	−0.434346	−	0.900746i	$-0.643020\pi$
$673$	−22.5358	−0.868691	−0.434346	+	0.900746i	$0.643020\pi$
$674$	0	0
$675$	−16.7702	−0.645485
$676$	0	0
$677$	−46.4701	−1.78599	−0.892995	−	0.450067i	$-0.851400\pi$
$677$	−46.4701	−1.78599	−0.892995	+	0.450067i	$0.851400\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−17.2475	−0.660924
$682$	0	0
$683$	−38.2450	−1.46340	−0.731701	−	0.681625i	$-0.761273\pi$
$683$	−38.2450	−1.46340	−0.731701	+	0.681625i	$0.761273\pi$
$684$	0	0
$685$	20.0940	0.767752
$686$	0	0
$687$	−0.283247	−0.0108065
$688$	0	0
$689$	−5.45146	−0.207684
$690$	0	0
$691$	−38.2090	−1.45354	−0.726770	−	0.686881i	$-0.758979\pi$
$691$	−38.2090	−1.45354	−0.726770	+	0.686881i	$0.758979\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−25.9189	−0.983158
$696$	0	0
$697$	56.5539	2.14213
$698$	0	0
$699$	23.5395	0.890348
$700$	0	0
$701$	33.3498	1.25960	0.629802	−	0.776756i	$-0.283136\pi$
$701$	33.3498	1.25960	0.629802	+	0.776756i	$0.283136\pi$
$702$	0	0
$703$	19.5103	0.735846
$704$	0	0
$705$	−20.8479	−0.785178
$706$	0	0
$707$	0	0
$708$	0	0
$709$	16.9687	0.637273	0.318637	−	0.947877i	$-0.396775\pi$
$709$	16.9687	0.637273	0.318637	+	0.947877i	$0.396775\pi$
$710$	0	0
$711$	0.0902758	0.00338561
$712$	0	0
$713$	−19.7045	−0.737940
$714$	0	0
$715$	9.20618	0.344292
$716$	0	0
$717$	36.1465	1.34991
$718$	0	0
$719$	5.12016	0.190950	0.0954749	−	0.995432i	$-0.469563\pi$
$719$	5.12016	0.190950	0.0954749	+	0.995432i	$0.469563\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0.189065	0.00703139
$724$	0	0
$725$	11.1142	0.412771
$726$	0	0
$727$	−12.5823	−0.466652	−0.233326	−	0.972399i	$-0.574961\pi$
$727$	−12.5823	−0.466652	−0.233326	+	0.972399i	$0.574961\pi$
$728$	0	0
$729$	27.2304	1.00853
$730$	0	0
$731$	−59.6328	−2.20560
$732$	0	0
$733$	25.9564	0.958723	0.479362	−	0.877618i	$-0.340868\pi$
$733$	25.9564	0.958723	0.479362	+	0.877618i	$0.340868\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−31.9899	−1.17836
$738$	0	0
$739$	−26.2344	−0.965049	−0.482525	−	0.875882i	$-0.660280\pi$
$739$	−26.2344	−0.965049	−0.482525	+	0.875882i	$0.660280\pi$
$740$	0	0
$741$	−3.95377	−0.145245
$742$	0	0
$743$	21.7174	0.796735	0.398368	−	0.917226i	$-0.369577\pi$
$743$	21.7174	0.796735	0.398368	+	0.917226i	$0.369577\pi$
$744$	0	0
$745$	−18.7460	−0.686800
$746$	0	0
$747$	−13.9308	−0.509701
$748$	0	0
$749$	0	0
$750$	0	0
$751$	17.2631	0.629938	0.314969	−	0.949102i	$-0.398006\pi$
$751$	17.2631	0.629938	0.314969	+	0.949102i	$0.398006\pi$
$752$	0	0
$753$	12.2735	0.447271
$754$	0	0
$755$	14.4413	0.525574
$756$	0	0
$757$	2.66371	0.0968143	0.0484071	−	0.998828i	$-0.484586\pi$
$757$	2.66371	0.0968143	0.0484071	+	0.998828i	$0.484586\pi$
$758$	0	0
$759$	42.6813	1.54923
$760$	0	0
$761$	46.3411	1.67986	0.839932	−	0.542691i	$-0.182595\pi$
$761$	46.3411	1.67986	0.839932	+	0.542691i	$0.182595\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	9.52069	0.344221
$766$	0	0
$767$	−2.14499	−0.0774512
$768$	0	0
$769$	39.4181	1.42145	0.710727	−	0.703468i	$-0.248366\pi$
$769$	39.4181	1.42145	0.710727	+	0.703468i	$0.248366\pi$
$770$	0	0
$771$	−37.2730	−1.34235
$772$	0	0
$773$	−13.1642	−0.473483	−0.236741	−	0.971573i	$-0.576079\pi$
$773$	−13.1642	−0.473483	−0.236741	+	0.971573i	$0.576079\pi$
$774$	0	0
$775$	11.8790	0.426707
$776$	0	0
$777$	0	0
$778$	0	0
$779$	30.5444	1.09437
$780$	0	0
$781$	−99.2570	−3.55169
$782$	0	0
$783$	−21.0074	−0.750743
$784$	0	0
$785$	−11.5032	−0.410568
$786$	0	0
$787$	50.4172	1.79718	0.898590	−	0.438790i	$-0.144593\pi$
$787$	50.4172	1.79718	0.898590	+	0.438790i	$0.144593\pi$
$788$	0	0
$789$	−30.7414	−1.09442
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−2.09880	−0.0745307
$794$	0	0
$795$	−10.3395	−0.366706
$796$	0	0
$797$	−36.0554	−1.27715	−0.638574	−	0.769561i	$-0.720475\pi$
$797$	−36.0554	−1.27715	−0.638574	+	0.769561i	$0.720475\pi$
$798$	0	0
$799$	−60.3179	−2.13389
$800$	0	0
$801$	7.29088	0.257610
$802$	0	0
$803$	−69.0209	−2.43569
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−16.0927	−0.566490
$808$	0	0
$809$	28.4774	1.00121	0.500606	−	0.865675i	$-0.333111\pi$
$809$	28.4774	1.00121	0.500606	+	0.865675i	$0.333111\pi$
$810$	0	0
$811$	−10.1443	−0.356214	−0.178107	−	0.984011i	$-0.556997\pi$
$811$	−10.1443	−0.356214	−0.178107	+	0.984011i	$0.556997\pi$
$812$	0	0
$813$	−23.3458	−0.818773
$814$	0	0
$815$	−6.20584	−0.217381
$816$	0	0
$817$	−32.2073	−1.12679
$818$	0	0
$819$	0	0
$820$	0	0
$821$	23.7524	0.828964	0.414482	−	0.910058i	$-0.363963\pi$
$821$	23.7524	0.828964	0.414482	+	0.910058i	$0.363963\pi$
$822$	0	0
$823$	7.50884	0.261742	0.130871	−	0.991399i	$-0.458223\pi$
$823$	7.50884	0.261742	0.130871	+	0.991399i	$0.458223\pi$
$824$	0	0
$825$	−25.7307	−0.895829
$826$	0	0
$827$	9.15858	0.318475	0.159238	−	0.987240i	$-0.449096\pi$
$827$	9.15858	0.318475	0.159238	+	0.987240i	$0.449096\pi$
$828$	0	0
$829$	9.57754	0.332642	0.166321	−	0.986072i	$-0.446811\pi$
$829$	9.57754	0.332642	0.166321	+	0.986072i	$0.446811\pi$
$830$	0	0
$831$	17.0205	0.590435
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−35.8499	−1.24064
$836$	0	0
$837$	−22.4530	−0.776090
$838$	0	0
$839$	11.3751	0.392711	0.196355	−	0.980533i	$-0.437089\pi$
$839$	11.3751	0.392711	0.196355	+	0.980533i	$0.437089\pi$
$840$	0	0
$841$	−15.0777	−0.519920
$842$	0	0
$843$	28.3203	0.975402
$844$	0	0
$845$	1.42174	0.0489092
$846$	0	0
$847$	0	0
$848$	0	0
$849$	20.2936	0.696473
$850$	0	0
$851$	−32.5259	−1.11497
$852$	0	0
$853$	−46.5806	−1.59489	−0.797445	−	0.603392i	$-0.793815\pi$
$853$	−46.5806	−1.59489	−0.797445	+	0.603392i	$0.793815\pi$
$854$	0	0
$855$	5.14207	0.175855
$856$	0	0
$857$	23.2008	0.792525	0.396263	−	0.918137i	$-0.370307\pi$
$857$	23.2008	0.792525	0.396263	+	0.918137i	$0.370307\pi$
$858$	0	0
$859$	−26.1724	−0.892992	−0.446496	−	0.894786i	$-0.647328\pi$
$859$	−26.1724	−0.892992	−0.446496	+	0.894786i	$0.647328\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	54.1703	1.84398	0.921989	−	0.387215i	$-0.126563\pi$
$863$	54.1703	1.84398	0.921989	+	0.387215i	$0.126563\pi$
$864$	0	0
$865$	−17.8990	−0.608585
$866$	0	0
$867$	−17.4924	−0.594074
$868$	0	0
$869$	0.479021	0.0162497
$870$	0	0
$871$	−4.94029	−0.167395
$872$	0	0
$873$	1.82553	0.0617849
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−26.7496	−0.903269	−0.451635	−	0.892203i	$-0.649159\pi$
$877$	−26.7496	−0.903269	−0.451635	+	0.892203i	$0.649159\pi$
$878$	0	0
$879$	−32.4277	−1.09376
$880$	0	0
$881$	−7.20574	−0.242767	−0.121384	−	0.992606i	$-0.538733\pi$
$881$	−7.20574	−0.242767	−0.121384	+	0.992606i	$0.538733\pi$
$882$	0	0
$883$	22.6737	0.763031	0.381515	−	0.924362i	$-0.375402\pi$
$883$	22.6737	0.763031	0.381515	+	0.924362i	$0.375402\pi$
$884$	0	0
$885$	−4.06831	−0.136755
$886$	0	0
$887$	−4.74029	−0.159164	−0.0795818	−	0.996828i	$-0.525358\pi$
$887$	−4.74029	−0.159164	−0.0795818	+	0.996828i	$0.525358\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	24.9287	0.835143
$892$	0	0
$893$	−32.5774	−1.09016
$894$	0	0
$895$	−18.7394	−0.626388
$896$	0	0
$897$	6.59139	0.220080
$898$	0	0
$899$	14.8804	0.496289
$900$	0	0
$901$	−29.9147	−0.996604
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−10.5912	−0.352065
$906$	0	0
$907$	−30.8995	−1.02600	−0.513000	−	0.858388i	$-0.671466\pi$
$907$	−30.8995	−1.02600	−0.513000	+	0.858388i	$0.671466\pi$
$908$	0	0
$909$	5.38378	0.178569
$910$	0	0
$911$	6.38121	0.211419	0.105710	−	0.994397i	$-0.466289\pi$
$911$	6.38121	0.211419	0.105710	+	0.994397i	$0.466289\pi$
$912$	0	0
$913$	−73.9196	−2.44638
$914$	0	0
$915$	−3.98071	−0.131598
$916$	0	0
$917$	0	0
$918$	0	0
$919$	41.2856	1.36189	0.680943	−	0.732336i	$-0.261570\pi$
$919$	41.2856	1.36189	0.680943	+	0.732336i	$0.261570\pi$
$920$	0	0
$921$	−37.6143	−1.23943
$922$	0	0
$923$	−15.3285	−0.504545
$924$	0	0
$925$	19.6085	0.644723
$926$	0	0
$927$	12.9097	0.424009
$928$	0	0
$929$	−4.28232	−0.140499	−0.0702493	−	0.997529i	$-0.522379\pi$
$929$	−4.28232	−0.140499	−0.0702493	+	0.997529i	$0.522379\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	2.59904	0.0850888
$934$	0	0
$935$	50.5187	1.65214
$936$	0	0
$937$	5.57308	0.182065	0.0910324	−	0.995848i	$-0.470983\pi$
$937$	5.57308	0.182065	0.0910324	+	0.995848i	$0.470983\pi$
$938$	0	0
$939$	−31.9151	−1.04151
$940$	0	0
$941$	−43.4173	−1.41536	−0.707681	−	0.706532i	$-0.750259\pi$
$941$	−43.4173	−1.41536	−0.707681	+	0.706532i	$0.750259\pi$
$942$	0	0
$943$	−50.9210	−1.65822
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−27.7856	−0.902912	−0.451456	−	0.892293i	$-0.649095\pi$
$947$	−27.7856	−0.902912	−0.451456	+	0.892293i	$0.649095\pi$
$948$	0	0
$949$	−10.6591	−0.346008
$950$	0	0
$951$	8.95262	0.290308
$952$	0	0
$953$	−43.3574	−1.40448	−0.702241	−	0.711939i	$-0.747817\pi$
$953$	−43.3574	−1.40448	−0.702241	+	0.711939i	$0.747817\pi$
$954$	0	0
$955$	−10.7038	−0.346366
$956$	0	0
$957$	−32.2319	−1.04191
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−15.0956	−0.486954
$962$	0	0
$963$	5.55967	0.179158
$964$	0	0
$965$	36.3857	1.17130
$966$	0	0
$967$	40.6261	1.30645	0.653223	−	0.757165i	$-0.273416\pi$
$967$	40.6261	1.30645	0.653223	+	0.757165i	$0.273416\pi$
$968$	0	0
$969$	−21.6962	−0.696983
$970$	0	0
$971$	18.6733	0.599256	0.299628	−	0.954056i	$-0.403137\pi$
$971$	18.6733	0.599256	0.299628	+	0.954056i	$0.403137\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−3.97367	−0.127259
$976$	0	0
$977$	32.5506	1.04139	0.520693	−	0.853744i	$-0.325674\pi$
$977$	32.5506	1.04139	0.520693	+	0.853744i	$0.325674\pi$
$978$	0	0
$979$	38.6868	1.23644
$980$	0	0
$981$	−9.21116	−0.294090
$982$	0	0
$983$	14.5936	0.465464	0.232732	−	0.972541i	$-0.425233\pi$
$983$	14.5936	0.465464	0.232732	+	0.972541i	$0.425233\pi$
$984$	0	0
$985$	17.6787	0.563291
$986$	0	0
$987$	0	0
$988$	0	0
$989$	53.6933	1.70735
$990$	0	0
$991$	−16.0304	−0.509222	−0.254611	−	0.967044i	$-0.581947\pi$
$991$	−16.0304	−0.509222	−0.254611	+	0.967044i	$0.581947\pi$
$992$	0	0
$993$	−22.0474	−0.699653
$994$	0	0
$995$	3.17951	0.100797
$996$	0	0
$997$	62.1576	1.96855	0.984275	−	0.176640i	$-0.0565230\pi$
$997$	62.1576	1.96855	0.984275	+	0.176640i	$0.0565230\pi$
$998$	0	0
$999$	−37.0628	−1.17262

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5096.2.a.z.1.3		7
7.2	even	3		728.2.r.f.417.5	✓	14
7.4	even	3		728.2.r.f.625.5	yes	14
7.6	odd	2		5096.2.a.ba.1.5		7
28.11	odd	6		1456.2.r.q.625.3		14
28.23	odd	6		1456.2.r.q.417.3		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.r.f.417.5	✓	14	7.2	even	3
728.2.r.f.625.5	yes	14	7.4	even	3
1456.2.r.q.417.3		14	28.23	odd	6
1456.2.r.q.625.3		14	28.11	odd	6
5096.2.a.z.1.3		7	1.1	even	1	trivial
5096.2.a.ba.1.5		7	7.6	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:	\( N \)	\(=\)	\( 5096 = 2^{3} \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5096.a (trivial)

\(f(q)\)	\(=\)	\(q-1.33404 q^{3} +1.42174 q^{5} -1.22033 q^{9} -6.47531 q^{11} -1.00000 q^{13} -1.89666 q^{15} -5.48747 q^{17} -2.96375 q^{19} +4.94091 q^{23} -2.97867 q^{25} +5.63010 q^{27} -3.73126 q^{29} -3.98803 q^{31} +8.63834 q^{33} -6.58298 q^{37} +1.33404 q^{39} -10.3060 q^{41} +10.8671 q^{43} -1.73499 q^{45} +10.9919 q^{47} +7.32052 q^{51} +5.45146 q^{53} -9.20618 q^{55} +3.95377 q^{57} +2.14499 q^{59} +2.09880 q^{61} -1.42174 q^{65} +4.94029 q^{67} -6.59139 q^{69} +15.3285 q^{71} +10.6591 q^{73} +3.97367 q^{75} -0.0739765 q^{79} -3.84981 q^{81} +11.4156 q^{83} -7.80174 q^{85} +4.97766 q^{87} -5.97451 q^{89} +5.32021 q^{93} -4.21367 q^{95} -1.49593 q^{97} +7.90202 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 2 q^{3} + 2 q^{5} + 7 q^{9} + 8 q^{11} - 7 q^{13} + 13 q^{15} - 8 q^{17} + 2 q^{19} + 16 q^{23} + 17 q^{25} - 11 q^{27} + 4 q^{29} - q^{31} + 14 q^{33} - 4 q^{37} + 2 q^{39} + 4 q^{41} + 8 q^{43}+ \cdots + 48 q^{99}+O(q^{100}) \) 7 * q - 2 * q^3 + 2 * q^5 + 7 * q^9 + 8 * q^11 - 7 * q^13 + 13 * q^15 - 8 * q^17 + 2 * q^19 + 16 * q^23 + 17 * q^25 - 11 * q^27 + 4 * q^29 - q^31 + 14 * q^33 - 4 * q^37 + 2 * q^39 + 4 * q^41 + 8 * q^43 - 5 * q^45 + 20 * q^47 - 5 * q^51 + 4 * q^53 - 16 * q^55 - 12 * q^57 - 8 * q^59 + 2 * q^61 - 2 * q^65 - 4 * q^67 + 2 * q^69 + 36 * q^71 - 8 * q^73 + 28 * q^75 + 18 * q^79 + 23 * q^81 - 12 * q^83 - q^85 + 26 * q^87 + 34 * q^89 - 12 * q^93 + 58 * q^95 - 2 * q^97 + 48 * q^99

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