LMFDB - Embedded newform 5096.2.a.ba.1.5

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)

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")

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);

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			1.5
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	$-0.374165\pi$
$3$	1.33404	0.770210	0.385105	+	0.922873i	$0.374165\pi$
$4$	0	0
$5$	−1.42174	−0.635820	−0.317910	−	0.948121i	$-0.602981\pi$
$5$	−1.42174	−0.635820	−0.317910	+	0.948121i	$0.602981\pi$
$6$	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	$-0.268238\pi$
$17$	5.48747	1.33091	0.665454	+	0.746439i	$0.268238\pi$
$18$	0	0
$19$	2.96375	0.679932	0.339966	−	0.940438i	$-0.389584\pi$
$19$	2.96375	0.679932	0.339966	+	0.940438i	$0.389584\pi$
$20$	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	$-0.383412\pi$
$31$	3.98803	0.716272	0.358136	+	0.933669i	$0.383412\pi$
$32$	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	$-0.202292\pi$
$41$	10.3060	1.60953	0.804763	+	0.593596i	$0.202292\pi$
$42$	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	$-0.796054\pi$
$47$	−10.9919	−1.60334	−0.801669	+	0.597769i	$0.796054\pi$
$48$	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	$-0.544590\pi$
$59$	−2.14499	−0.279254	−0.139627	+	0.990204i	$0.544590\pi$
$60$	0	0
$61$	−2.09880	−0.268724	−0.134362	−	0.990932i	$-0.542899\pi$
$61$	−2.09880	−0.268724	−0.134362	+	0.990932i	$0.542899\pi$
$62$	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	$-0.714402\pi$
$73$	−10.6591	−1.24755	−0.623776	+	0.781603i	$0.714402\pi$
$74$	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	$-0.715518\pi$
$83$	−11.4156	−1.25303	−0.626513	+	0.779411i	$0.715518\pi$
$84$	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	$-0.397442\pi$
$89$	5.97451	0.633297	0.316649	+	0.948543i	$0.397442\pi$
$90$	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	$-0.475803\pi$
$97$	1.49593	0.151889	0.0759445	+	0.997112i	$0.475803\pi$
$98$	0	0
$99$	7.90202	0.794182
$100$	0	0
$101$	4.41174	0.438985	0.219492	−	0.975614i	$-0.429560\pi$
$101$	4.41174	0.438985	0.219492	+	0.975614i	$0.429560\pi$
$102$	0	0
$103$	10.5788	1.04236	0.521181	−	0.853446i	$-0.325492\pi$
$103$	10.5788	1.04236	0.521181	+	0.853446i	$0.325492\pi$
$104$	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	$-0.452203\pi$
$131$	3.42439	0.299190	0.149595	+	0.988747i	$0.452203\pi$
$132$	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	$-0.218684\pi$
$139$	18.2304	1.54628	0.773142	+	0.634232i	$0.218684\pi$
$140$	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	$-0.395354\pi$
$157$	8.09097	0.645729	0.322865	+	0.946445i	$0.395354\pi$
$158$	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	$-0.0704312\pi$
$167$	25.2156	1.95124	0.975620	+	0.219465i	$0.0704312\pi$
$168$	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	$-0.341150\pi$
$173$	12.5896	0.957167	0.478583	+	0.878042i	$0.341150\pi$
$174$	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	$-0.410707\pi$
$181$	7.44951	0.553718	0.276859	+	0.960911i	$0.410707\pi$
$182$	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	$-0.525258\pi$
$199$	−2.23636	−0.158531	−0.0792656	+	0.996854i	$0.525258\pi$
$200$	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	$-0.336629\pi$
$223$	14.6646	0.982012	0.491006	+	0.871156i	$0.336629\pi$
$224$	0	0
$225$	3.63496	0.242330
$226$	0	0
$227$	−12.9287	−0.858109	−0.429055	−	0.903279i	$-0.641153\pi$
$227$	−12.9287	−0.858109	−0.429055	+	0.903279i	$0.641153\pi$
$228$	0	0
$229$	−0.212322	−0.0140306	−0.00701531	−	0.999975i	$-0.502233\pi$
$229$	−0.212322	−0.0140306	−0.00701531	+	0.999975i	$0.502233\pi$
$230$	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	$-0.498547\pi$
$241$	0.141723	0.00912918	0.00456459	+	0.999990i	$0.498547\pi$
$242$	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	$-0.406226\pi$
$251$	9.20022	0.580712	0.290356	+	0.956919i	$0.406226\pi$
$252$	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	$-0.836800\pi$
$257$	−27.9399	−1.74284	−0.871420	+	0.490537i	$0.836800\pi$
$258$	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	$-0.619872\pi$
$269$	−12.0631	−0.735501	−0.367751	+	0.929924i	$0.619872\pi$
$270$	0	0
$271$	−17.5000	−1.06305	−0.531526	−	0.847042i	$-0.678381\pi$
$271$	−17.5000	−1.06305	−0.531526	+	0.847042i	$0.678381\pi$
$272$	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	$-0.350664\pi$
$283$	15.2121	0.904264	0.452132	+	0.891951i	$0.350664\pi$
$284$	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	$-0.751323\pi$
$293$	−24.3078	−1.42008	−0.710040	+	0.704162i	$0.751323\pi$
$294$	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	$-0.797625\pi$
$307$	−28.1957	−1.60922	−0.804608	+	0.593806i	$0.797625\pi$
$308$	0	0
$309$	14.1126	0.802838
$310$	0	0
$311$	1.94825	0.110475	0.0552374	−	0.998473i	$-0.482408\pi$
$311$	1.94825	0.110475	0.0552374	+	0.998473i	$0.482408\pi$
$312$	0	0
$313$	−23.9236	−1.35224	−0.676120	−	0.736791i	$-0.736340\pi$
$313$	−23.9236	−1.35224	−0.676120	+	0.736791i	$0.736340\pi$
$314$	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	$-0.611160\pi$
$349$	−12.7843	−0.684327	−0.342163	+	0.939640i	$0.611160\pi$
$350$	0	0
$351$	−5.63010	−0.300513
$352$	0	0
$353$	−8.44261	−0.449355	−0.224677	−	0.974433i	$-0.572133\pi$
$353$	−8.44261	−0.449355	−0.224677	+	0.974433i	$0.572133\pi$
$354$	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	$-0.344886\pi$
$367$	17.9406	0.936490	0.468245	+	0.883599i	$0.344886\pi$
$368$	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	$-0.485891\pi$
$383$	1.73439	0.0886232	0.0443116	+	0.999018i	$0.485891\pi$
$384$	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	$-0.275299\pi$
$397$	25.8518	1.29747	0.648734	+	0.761016i	$0.275299\pi$
$398$	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	$-0.613480\pi$
$409$	−14.1163	−0.698007	−0.349003	+	0.937121i	$0.613480\pi$
$410$	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	$-0.529823\pi$
$419$	−3.82997	−0.187106	−0.0935532	+	0.995614i	$0.529823\pi$
$420$	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	$-0.513280\pi$
$433$	−1.73578	−0.0834164	−0.0417082	+	0.999130i	$0.513280\pi$
$434$	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	$-0.439592\pi$
$439$	7.90495	0.377283	0.188641	+	0.982046i	$0.439592\pi$
$440$	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	$-0.505197\pi$
$461$	−0.701091	−0.0326530	−0.0163265	+	0.999867i	$0.505197\pi$
$462$	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	$-0.455380\pi$
$467$	6.03876	0.279440	0.139720	+	0.990191i	$0.455380\pi$
$468$	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	$-0.612319\pi$
$479$	−15.1269	−0.691165	−0.345583	+	0.938388i	$0.612319\pi$
$480$	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	$-0.739752\pi$
$503$	−30.6801	−1.36796	−0.683979	+	0.729502i	$0.739752\pi$
$504$	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	$-0.628381\pi$
$509$	−17.7092	−0.784948	−0.392474	+	0.919763i	$0.628381\pi$
$510$	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	$-0.496509\pi$
$521$	0.500698	0.0219360	0.0109680	+	0.999940i	$0.496509\pi$
$522$	0	0
$523$	6.90377	0.301881	0.150940	−	0.988543i	$-0.451770\pi$
$523$	6.90377	0.301881	0.150940	+	0.988543i	$0.451770\pi$
$524$	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	$-0.561545\pi$
$563$	−9.11831	−0.384291	−0.192145	+	0.981366i	$0.561545\pi$
$564$	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	$-0.462729\pi$
$577$	5.61243	0.233648	0.116824	+	0.993153i	$0.462729\pi$
$578$	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	$-0.451739\pi$
$587$	7.31862	0.302072	0.151036	+	0.988528i	$0.451739\pi$
$588$	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	$-0.513579\pi$
$593$	−2.07707	−0.0852949	−0.0426475	+	0.999090i	$0.513579\pi$
$594$	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	$-0.646183\pi$
$601$	−21.7340	−0.886549	−0.443275	+	0.896386i	$0.646183\pi$
$602$	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	$-0.414533\pi$
$607$	13.0720	0.530576	0.265288	+	0.964169i	$0.414533\pi$
$608$	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	$-0.546749\pi$
$619$	−7.28177	−0.292679	−0.146339	+	0.989234i	$0.546749\pi$
$620$	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	$-0.486093\pi$
$643$	2.21508	0.0873542	0.0436771	+	0.999046i	$0.486093\pi$
$644$	0	0
$645$	−20.6111	−0.811562
$646$	0	0
$647$	41.5732	1.63441	0.817205	−	0.576347i	$-0.195522\pi$
$647$	41.5732	1.63441	0.817205	+	0.576347i	$0.195522\pi$
$648$	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	$-0.499907\pi$
$661$	0.0150773	0.000586439 0	0.000293219		1.00000i	$0.499907\pi$
$662$	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	$-0.148600\pi$
$677$	46.4701	1.78599	0.892995	+	0.450067i	$0.148600\pi$
$678$	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	$-0.241021\pi$
$691$	38.2090	1.45354	0.726770	+	0.686881i	$0.241021\pi$
$692$	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	$-0.530437\pi$
$719$	−5.12016	−0.190950	−0.0954749	+	0.995432i	$0.530437\pi$
$720$	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	$-0.425039\pi$
$727$	12.5823	0.466652	0.233326	+	0.972399i	$0.425039\pi$
$728$	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	$-0.659132\pi$
$733$	−25.9564	−0.958723	−0.479362	+	0.877618i	$0.659132\pi$
$734$	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	$-0.817405\pi$
$761$	−46.3411	−1.67986	−0.839932	+	0.542691i	$0.817405\pi$
$762$	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	$-0.751634\pi$
$769$	−39.4181	−1.42145	−0.710727	+	0.703468i	$0.751634\pi$
$770$	0	0
$771$	−37.2730	−1.34235
$772$	0	0
$773$	13.1642	0.473483	0.236741	−	0.971573i	$-0.423921\pi$
$773$	13.1642	0.473483	0.236741	+	0.971573i	$0.423921\pi$
$774$	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	$-0.855407\pi$
$787$	−50.4172	−1.79718	−0.898590	+	0.438790i	$0.855407\pi$
$788$	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	$-0.279525\pi$
$797$	36.0554	1.27715	0.638574	+	0.769561i	$0.279525\pi$
$798$	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	$-0.443003\pi$
$811$	10.1443	0.356214	0.178107	+	0.984011i	$0.443003\pi$
$812$	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	$-0.553189\pi$
$829$	−9.57754	−0.332642	−0.166321	+	0.986072i	$0.553189\pi$
$830$	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	$-0.562911\pi$
$839$	−11.3751	−0.392711	−0.196355	+	0.980533i	$0.562911\pi$
$840$	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	$-0.206185\pi$
$853$	46.5806	1.59489	0.797445	+	0.603392i	$0.206185\pi$
$854$	0	0
$855$	5.14207	0.175855
$856$	0	0
$857$	−23.2008	−0.792525	−0.396263	−	0.918137i	$-0.629693\pi$
$857$	−23.2008	−0.792525	−0.396263	+	0.918137i	$0.629693\pi$
$858$	0	0
$859$	26.1724	0.892992	0.446496	−	0.894786i	$-0.352672\pi$
$859$	26.1724	0.892992	0.446496	+	0.894786i	$0.352672\pi$
$860$	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	$-0.461267\pi$
$881$	7.20574	0.242767	0.121384	+	0.992606i	$0.461267\pi$
$882$	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	$-0.474642\pi$
$887$	4.74029	0.159164	0.0795818	+	0.996828i	$0.474642\pi$
$888$	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	$-0.477621\pi$
$929$	4.28232	0.140499	0.0702493	+	0.997529i	$0.477621\pi$
$930$	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	$-0.529017\pi$
$937$	−5.57308	−0.182065	−0.0910324	+	0.995848i	$0.529017\pi$
$938$	0	0
$939$	−31.9151	−1.04151
$940$	0	0
$941$	43.4173	1.41536	0.707681	−	0.706532i	$-0.249741\pi$
$941$	43.4173	1.41536	0.707681	+	0.706532i	$0.249741\pi$
$942$	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	$-0.596863\pi$
$971$	−18.6733	−0.599256	−0.299628	+	0.954056i	$0.596863\pi$
$972$	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	$-0.574767\pi$
$983$	−14.5936	−0.465464	−0.232732	+	0.972541i	$0.574767\pi$
$984$	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	$-0.943477\pi$
$997$	−62.1576	−1.96855	−0.984275	+	0.176640i	$0.943477\pi$
$998$	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.ba.1.5		7
7.3	odd	6		728.2.r.f.625.5	yes	14
7.5	odd	6		728.2.r.f.417.5	✓	14
7.6	odd	2		5096.2.a.z.1.3		7
28.3	even	6		1456.2.r.q.625.3		14
28.19	even	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	7.5	odd	6
728.2.r.f.625.5	yes	14	7.3	odd	6
1456.2.r.q.417.3		14	28.19	even	6
1456.2.r.q.625.3		14	28.3	even	6
5096.2.a.z.1.3		7	7.6	odd	2
5096.2.a.ba.1.5		7	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:	\( N \)	\(=\)	\( 5096 = 2^{3} \cdot 7^{2} \cdot 13 \)
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

Introduction

L-functions

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