LMFDB - Embedded newform 9072.2.a.by.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9072,2,Mod(1,9072)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9072, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9072.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9072 = 2^{4} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9072.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$72.4402847137$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.321.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 1 $ x^3 - x^2 - 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 252)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.239123$ of defining polynomial
Character	$\chi$	$=$	9072.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.239123 q^{5} +1.00000 q^{7} +O(q^{10})$	$q+0.239123 q^{5} +1.00000 q^{7} +5.12476 q^{11} -4.88564 q^{13} +3.70370 q^{17} -3.66019 q^{19} -7.42107 q^{23} -4.94282 q^{25} +3.46457 q^{29} +0.717370 q^{31} +0.239123 q^{35} +4.60301 q^{37} +5.60301 q^{41} -12.4887 q^{43} -4.33981 q^{47} +1.00000 q^{49} -0.942820 q^{53} +1.22545 q^{55} -7.57893 q^{59} +5.50808 q^{61} -1.16827 q^{65} +0.660190 q^{67} -13.7414 q^{71} -3.66019 q^{73} +5.12476 q^{77} +6.22545 q^{79} -9.70370 q^{83} +0.885640 q^{85} -7.48865 q^{89} -4.88564 q^{91} -0.875237 q^{95} -17.1488 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{5} + 3 q^{7}+O(q^{10}) $ 3 * q + q^5 + 3 * q^7	$ 3 q + q^{5} + 3 q^{7} - 2 q^{11} + 3 q^{13} + 2 q^{17} - 3 q^{19} - 14 q^{23} - 6 q^{25} + q^{29} + 3 q^{31} + q^{35} - 3 q^{37} - 3 q^{43} - 21 q^{47} + 3 q^{49} + 6 q^{53} - 6 q^{55} - 31 q^{59} + 6 q^{61} + 15 q^{65} - 6 q^{67} - 17 q^{71} - 3 q^{73} - 2 q^{77} + 9 q^{79} - 20 q^{83} - 15 q^{85} + 12 q^{89} + 3 q^{91} - 20 q^{95} - 9 q^{97}+O(q^{100}) $ 3 * q + q^5 + 3 * q^7 - 2 * q^11 + 3 * q^13 + 2 * q^17 - 3 * q^19 - 14 * q^23 - 6 * q^25 + q^29 + 3 * q^31 + q^35 - 3 * q^37 - 3 * q^43 - 21 * q^47 + 3 * q^49 + 6 * q^53 - 6 * q^55 - 31 * q^59 + 6 * q^61 + 15 * q^65 - 6 * q^67 - 17 * q^71 - 3 * q^73 - 2 * q^77 + 9 * q^79 - 20 * q^83 - 15 * q^85 + 12 * q^89 + 3 * q^91 - 20 * q^95 - 9 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0.239123	0.106939	0.0534696	−	0.998569i	$-0.482972\pi$
$5$	0.239123	0.106939	0.0534696	+	0.998569i	$0.482972\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.12476	1.54517	0.772587	−	0.634909i	$-0.218962\pi$
$11$	5.12476	1.54517	0.772587	+	0.634909i	$0.218962\pi$
$12$	0	0
$13$	−4.88564	−1.35503	−0.677516	−	0.735508i	$-0.736944\pi$
$13$	−4.88564	−1.35503	−0.677516	+	0.735508i	$0.736944\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.70370	0.898278	0.449139	−	0.893462i	$-0.351731\pi$
$17$	3.70370	0.898278	0.449139	+	0.893462i	$0.351731\pi$
$18$	0	0
$19$	−3.66019	−0.839705	−0.419853	−	0.907592i	$-0.637918\pi$
$19$	−3.66019	−0.839705	−0.419853	+	0.907592i	$0.637918\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.42107	−1.54740	−0.773700	−	0.633553i	$-0.781596\pi$
$23$	−7.42107	−1.54740	−0.773700	+	0.633553i	$0.781596\pi$
$24$	0	0
$25$	−4.94282	−0.988564
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.46457	0.643355	0.321678	−	0.946849i	$-0.395753\pi$
$29$	3.46457	0.643355	0.321678	+	0.946849i	$0.395753\pi$
$30$	0	0
$31$	0.717370	0.128843	0.0644217	−	0.997923i	$-0.479480\pi$
$31$	0.717370	0.128843	0.0644217	+	0.997923i	$0.479480\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.239123	0.0404192
$36$	0	0
$37$	4.60301	0.756730	0.378365	−	0.925656i	$-0.376486\pi$
$37$	4.60301	0.756730	0.378365	+	0.925656i	$0.376486\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.60301	0.875043	0.437522	−	0.899208i	$-0.355856\pi$
$41$	5.60301	0.875043	0.437522	+	0.899208i	$0.355856\pi$
$42$	0	0
$43$	−12.4887	−1.90450	−0.952251	−	0.305317i	$-0.901237\pi$
$43$	−12.4887	−1.90450	−0.952251	+	0.305317i	$0.901237\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.33981	−0.633026	−0.316513	−	0.948588i	$-0.602512\pi$
$47$	−4.33981	−0.633026	−0.316513	+	0.948588i	$0.602512\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−0.942820	−0.129506	−0.0647531	−	0.997901i	$-0.520626\pi$
$53$	−0.942820	−0.129506	−0.0647531	+	0.997901i	$0.520626\pi$
$54$	0	0
$55$	1.22545	0.165240
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−7.57893	−0.986693	−0.493347	−	0.869833i	$-0.664227\pi$
$59$	−7.57893	−0.986693	−0.493347	+	0.869833i	$0.664227\pi$
$60$	0	0
$61$	5.50808	0.705237	0.352619	−	0.935767i	$-0.385291\pi$
$61$	5.50808	0.705237	0.352619	+	0.935767i	$0.385291\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.16827	−0.144906
$66$	0	0
$67$	0.660190	0.0806550	0.0403275	−	0.999187i	$-0.487160\pi$
$67$	0.660190	0.0806550	0.0403275	+	0.999187i	$0.487160\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−13.7414	−1.63081	−0.815405	−	0.578891i	$-0.803486\pi$
$71$	−13.7414	−1.63081	−0.815405	+	0.578891i	$0.803486\pi$
$72$	0	0
$73$	−3.66019	−0.428393	−0.214196	−	0.976791i	$-0.568713\pi$
$73$	−3.66019	−0.428393	−0.214196	+	0.976791i	$0.568713\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.12476	0.584021
$78$	0	0
$79$	6.22545	0.700418	0.350209	−	0.936672i	$-0.386111\pi$
$79$	6.22545	0.700418	0.350209	+	0.936672i	$0.386111\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−9.70370	−1.06512	−0.532560	−	0.846393i	$-0.678770\pi$
$83$	−9.70370	−1.06512	−0.532560	+	0.846393i	$0.678770\pi$
$84$	0	0
$85$	0.885640	0.0960612
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−7.48865	−0.793795	−0.396898	−	0.917863i	$-0.629913\pi$
$89$	−7.48865	−0.793795	−0.396898	+	0.917863i	$0.629913\pi$
$90$	0	0
$91$	−4.88564	−0.512154
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−0.875237	−0.0897974
$96$	0	0
$97$	−17.1488	−1.74120	−0.870600	−	0.491991i	$-0.836269\pi$
$97$	−17.1488	−1.74120	−0.870600	+	0.491991i	$0.836269\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	7.18194	0.714630	0.357315	−	0.933984i	$-0.383692\pi$
$101$	7.18194	0.714630	0.357315	+	0.933984i	$0.383692\pi$
$102$	0	0
$103$	12.8285	1.26403	0.632013	−	0.774958i	$-0.282229\pi$
$103$	12.8285	1.26403	0.632013	+	0.774958i	$0.282229\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.56526	−0.731361	−0.365681	−	0.930740i	$-0.619164\pi$
$107$	−7.56526	−0.731361	−0.365681	+	0.930740i	$0.619164\pi$
$108$	0	0
$109$	−6.98057	−0.668617	−0.334309	−	0.942464i	$-0.608503\pi$
$109$	−6.98057	−0.668617	−0.334309	+	0.942464i	$0.608503\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−19.5699	−1.84098	−0.920491	−	0.390764i	$-0.872211\pi$
$113$	−19.5699	−1.84098	−0.920491	+	0.390764i	$0.872211\pi$
$114$	0	0
$115$	−1.77455	−0.165478
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.70370	0.339517
$120$	0	0
$121$	15.2632	1.38756
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−2.37756	−0.212655
$126$	0	0
$127$	16.8090	1.49156	0.745780	−	0.666192i	$-0.232077\pi$
$127$	16.8090	1.49156	0.745780	+	0.666192i	$0.232077\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−4.89931	−0.428055	−0.214027	−	0.976828i	$-0.568658\pi$
$131$	−4.89931	−0.428055	−0.214027	+	0.976828i	$0.568658\pi$
$132$	0	0
$133$	−3.66019	−0.317379
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.44514	0.465210	0.232605	−	0.972571i	$-0.425275\pi$
$137$	5.44514	0.465210	0.232605	+	0.972571i	$0.425275\pi$
$138$	0	0
$139$	−5.66019	−0.480091	−0.240046	−	0.970762i	$-0.577162\pi$
$139$	−5.66019	−0.480091	−0.240046	+	0.970762i	$0.577162\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−25.0377	−2.09376
$144$	0	0
$145$	0.828460	0.0687999
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.28263	0.187000	0.0935002	−	0.995619i	$-0.470194\pi$
$149$	2.28263	0.187000	0.0935002	+	0.995619i	$0.470194\pi$
$150$	0	0
$151$	−11.2632	−0.916586	−0.458293	−	0.888801i	$-0.651539\pi$
$151$	−11.2632	−0.916586	−0.458293	+	0.888801i	$0.651539\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.171540	0.0137784
$156$	0	0
$157$	5.54583	0.442605	0.221303	−	0.975205i	$-0.428969\pi$
$157$	5.54583	0.442605	0.221303	+	0.975205i	$0.428969\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−7.42107	−0.584862
$162$	0	0
$163$	−6.66019	−0.521666	−0.260833	−	0.965384i	$-0.583997\pi$
$163$	−6.66019	−0.521666	−0.260833	+	0.965384i	$0.583997\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−4.40739	−0.341054	−0.170527	−	0.985353i	$-0.554547\pi$
$167$	−4.40739	−0.341054	−0.170527	+	0.985353i	$0.554547\pi$
$168$	0	0
$169$	10.8695	0.836114
$170$	0	0
$171$	0	0
$172$	0	0
$173$	25.3308	1.92586	0.962932	−	0.269745i	$-0.0869393\pi$
$173$	25.3308	1.92586	0.962932	+	0.269745i	$0.0869393\pi$
$174$	0	0
$175$	−4.94282	−0.373642
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−9.54583	−0.713489	−0.356744	−	0.934202i	$-0.616113\pi$
$179$	−9.54583	−0.713489	−0.356744	+	0.934202i	$0.616113\pi$
$180$	0	0
$181$	12.3743	0.919774	0.459887	−	0.887978i	$-0.347890\pi$
$181$	12.3743	0.919774	0.459887	+	0.887978i	$0.347890\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.10069	0.0809241
$186$	0	0
$187$	18.9806	1.38800
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−13.1683	−0.952823	−0.476411	−	0.879223i	$-0.658063\pi$
$191$	−13.1683	−0.952823	−0.476411	+	0.879223i	$0.658063\pi$
$192$	0	0
$193$	11.1488	0.802511	0.401256	−	0.915966i	$-0.368574\pi$
$193$	11.1488	0.802511	0.401256	+	0.915966i	$0.368574\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−0.144194	−0.0102734	−0.00513669	−	0.999987i	$-0.501635\pi$
$197$	−0.144194	−0.0102734	−0.00513669	+	0.999987i	$0.501635\pi$
$198$	0	0
$199$	19.4692	1.38014	0.690068	−	0.723744i	$-0.257581\pi$
$199$	19.4692	1.38014	0.690068	+	0.723744i	$0.257581\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3.46457	0.243165
$204$	0	0
$205$	1.33981	0.0935764
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−18.7576	−1.29749
$210$	0	0
$211$	3.22872	0.222274	0.111137	−	0.993805i	$-0.464551\pi$
$211$	3.22872	0.222274	0.111137	+	0.993805i	$0.464551\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2.98633	−0.203666
$216$	0	0
$217$	0.717370	0.0486982
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−18.0949	−1.21720
$222$	0	0
$223$	−20.7713	−1.39095	−0.695474	−	0.718551i	$-0.744806\pi$
$223$	−20.7713	−1.39095	−0.695474	+	0.718551i	$0.744806\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−21.9428	−1.45640	−0.728198	−	0.685367i	$-0.759642\pi$
$227$	−21.9428	−1.45640	−0.728198	+	0.685367i	$0.759642\pi$
$228$	0	0
$229$	22.7713	1.50477	0.752384	−	0.658724i	$-0.228904\pi$
$229$	22.7713	1.50477	0.752384	+	0.658724i	$0.228904\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−25.7817	−1.68901	−0.844507	−	0.535544i	$-0.820106\pi$
$233$	−25.7817	−1.68901	−0.844507	+	0.535544i	$0.820106\pi$
$234$	0	0
$235$	−1.03775	−0.0676953
$236$	0	0
$237$	0	0
$238$	0	0
$239$	27.2977	1.76574	0.882870	−	0.469617i	$-0.155608\pi$
$239$	27.2977	1.76574	0.882870	+	0.469617i	$0.155608\pi$
$240$	0	0
$241$	10.0345	0.646378	0.323189	−	0.946334i	$-0.395245\pi$
$241$	10.0345	0.646378	0.323189	+	0.946334i	$0.395245\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0.239123	0.0152770
$246$	0	0
$247$	17.8824	1.13783
$248$	0	0
$249$	0	0
$250$	0	0
$251$	28.3171	1.78736	0.893680	−	0.448705i	$-0.148115\pi$
$251$	28.3171	1.78736	0.893680	+	0.448705i	$0.148115\pi$
$252$	0	0
$253$	−38.0312	−2.39100
$254$	0	0
$255$	0	0
$256$	0	0
$257$	28.8629	1.80042	0.900210	−	0.435455i	$-0.143413\pi$
$257$	28.8629	1.80042	0.900210	+	0.435455i	$0.143413\pi$
$258$	0	0
$259$	4.60301	0.286017
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1.20929	0.0745680	0.0372840	−	0.999305i	$-0.488129\pi$
$263$	1.20929	0.0745680	0.0372840	+	0.999305i	$0.488129\pi$
$264$	0	0
$265$	−0.225450	−0.0138493
$266$	0	0
$267$	0	0
$268$	0	0
$269$	9.01367	0.549573	0.274787	−	0.961505i	$-0.411393\pi$
$269$	9.01367	0.549573	0.274787	+	0.961505i	$0.411393\pi$
$270$	0	0
$271$	−17.6030	−1.06931	−0.534653	−	0.845071i	$-0.679558\pi$
$271$	−17.6030	−1.06931	−0.534653	+	0.845071i	$0.679558\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−25.3308	−1.52750
$276$	0	0
$277$	1.45417	0.0873726	0.0436863	−	0.999045i	$-0.486090\pi$
$277$	1.45417	0.0873726	0.0436863	+	0.999045i	$0.486090\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−20.2963	−1.21078	−0.605388	−	0.795931i	$-0.706982\pi$
$281$	−20.2963	−1.21078	−0.605388	+	0.795931i	$0.706982\pi$
$282$	0	0
$283$	4.60301	0.273621	0.136810	−	0.990597i	$-0.456315\pi$
$283$	4.60301	0.273621	0.136810	+	0.990597i	$0.456315\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.60301	0.330735
$288$	0	0
$289$	−3.28263	−0.193096
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−7.07334	−0.413229	−0.206614	−	0.978422i	$-0.566245\pi$
$293$	−7.07334	−0.413229	−0.206614	+	0.978422i	$0.566245\pi$
$294$	0	0
$295$	−1.81230	−0.105516
$296$	0	0
$297$	0	0
$298$	0	0
$299$	36.2567	2.09678
$300$	0	0
$301$	−12.4887	−0.719834
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.31711	0.0754175
$306$	0	0
$307$	−15.7518	−0.899006	−0.449503	−	0.893279i	$-0.648399\pi$
$307$	−15.7518	−0.899006	−0.449503	+	0.893279i	$0.648399\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	19.6238	1.11276	0.556382	−	0.830926i	$-0.312189\pi$
$311$	19.6238	1.11276	0.556382	+	0.830926i	$0.312189\pi$
$312$	0	0
$313$	−25.4854	−1.44052	−0.720259	−	0.693705i	$-0.755977\pi$
$313$	−25.4854	−1.44052	−0.720259	+	0.693705i	$0.755977\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.28263	0.465199	0.232599	−	0.972573i	$-0.425277\pi$
$317$	8.28263	0.465199	0.232599	+	0.972573i	$0.425277\pi$
$318$	0	0
$319$	17.7551	0.994096
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−13.5562	−0.754289
$324$	0	0
$325$	24.1488	1.33954
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−4.33981	−0.239261
$330$	0	0
$331$	11.9806	0.658512	0.329256	−	0.944241i	$-0.393202\pi$
$331$	11.9806	0.658512	0.329256	+	0.944241i	$0.393202\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0.157867	0.00862518
$336$	0	0
$337$	−12.9201	−0.703804	−0.351902	−	0.936037i	$-0.614465\pi$
$337$	−12.9201	−0.703804	−0.351902	+	0.936037i	$0.614465\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	3.67635	0.199086
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−16.1866	−0.868942	−0.434471	−	0.900686i	$-0.643065\pi$
$347$	−16.1866	−0.868942	−0.434471	+	0.900686i	$0.643065\pi$
$348$	0	0
$349$	−18.1144	−0.969639	−0.484820	−	0.874614i	$-0.661115\pi$
$349$	−18.1144	−0.969639	−0.484820	+	0.874614i	$0.661115\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−11.6979	−0.622618	−0.311309	−	0.950309i	$-0.600767\pi$
$353$	−11.6979	−0.622618	−0.311309	+	0.950309i	$0.600767\pi$
$354$	0	0
$355$	−3.28590	−0.174397
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−35.7245	−1.88547	−0.942734	−	0.333547i	$-0.891755\pi$
$359$	−35.7245	−1.88547	−0.942734	+	0.333547i	$0.891755\pi$
$360$	0	0
$361$	−5.60301	−0.294895
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−0.875237	−0.0458120
$366$	0	0
$367$	−17.0539	−0.890207	−0.445103	−	0.895479i	$-0.646833\pi$
$367$	−17.0539	−0.890207	−0.445103	+	0.895479i	$0.646833\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−0.942820	−0.0489488
$372$	0	0
$373$	−25.9234	−1.34226	−0.671131	−	0.741339i	$-0.734191\pi$
$373$	−25.9234	−1.34226	−0.671131	+	0.741339i	$0.734191\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−16.9267	−0.871767
$378$	0	0
$379$	−26.8446	−1.37892	−0.689458	−	0.724326i	$-0.742151\pi$
$379$	−26.8446	−1.37892	−0.689458	+	0.724326i	$0.742151\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−24.8525	−1.26991	−0.634953	−	0.772551i	$-0.718980\pi$
$383$	−24.8525	−1.26991	−0.634953	+	0.772551i	$0.718980\pi$
$384$	0	0
$385$	1.22545	0.0624547
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−18.2528	−0.925454	−0.462727	−	0.886501i	$-0.653129\pi$
$389$	−18.2528	−0.925454	−0.462727	+	0.886501i	$0.653129\pi$
$390$	0	0
$391$	−27.4854	−1.39000
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.48865	0.0749021
$396$	0	0
$397$	12.3743	0.621048	0.310524	−	0.950566i	$-0.399496\pi$
$397$	12.3743	0.621048	0.310524	+	0.950566i	$0.399496\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−10.9623	−0.547429	−0.273714	−	0.961811i	$-0.588252\pi$
$401$	−10.9623	−0.547429	−0.273714	+	0.961811i	$0.588252\pi$
$402$	0	0
$403$	−3.50481	−0.174587
$404$	0	0
$405$	0	0
$406$	0	0
$407$	23.5893	1.16928
$408$	0	0
$409$	−13.3204	−0.658650	−0.329325	−	0.944217i	$-0.606821\pi$
$409$	−13.3204	−0.658650	−0.329325	+	0.944217i	$0.606821\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−7.57893	−0.372935
$414$	0	0
$415$	−2.32038	−0.113903
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−4.71410	−0.230299	−0.115149	−	0.993348i	$-0.536735\pi$
$419$	−4.71410	−0.230299	−0.115149	+	0.993348i	$0.536735\pi$
$420$	0	0
$421$	19.3171	0.941458	0.470729	−	0.882278i	$-0.343991\pi$
$421$	19.3171	0.941458	0.470729	+	0.882278i	$0.343991\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−18.3067	−0.888006
$426$	0	0
$427$	5.50808	0.266555
$428$	0	0
$429$	0	0
$430$	0	0
$431$	30.2794	1.45851	0.729253	−	0.684244i	$-0.239868\pi$
$431$	30.2794	1.45851	0.729253	+	0.684244i	$0.239868\pi$
$432$	0	0
$433$	34.2060	1.64384	0.821918	−	0.569606i	$-0.192904\pi$
$433$	34.2060	1.64384	0.821918	+	0.569606i	$0.192904\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	27.1625	1.29936
$438$	0	0
$439$	0.622440	0.0297075	0.0148537	−	0.999890i	$-0.495272\pi$
$439$	0.622440	0.0297075	0.0148537	+	0.999890i	$0.495272\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	5.17867	0.246046	0.123023	−	0.992404i	$-0.460741\pi$
$443$	5.17867	0.246046	0.123023	+	0.992404i	$0.460741\pi$
$444$	0	0
$445$	−1.79071	−0.0848878
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−10.4977	−0.495416	−0.247708	−	0.968835i	$-0.579677\pi$
$449$	−10.4977	−0.495416	−0.247708	+	0.968835i	$0.579677\pi$
$450$	0	0
$451$	28.7141	1.35209
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.16827	−0.0547694
$456$	0	0
$457$	−3.20929	−0.150124	−0.0750621	−	0.997179i	$-0.523916\pi$
$457$	−3.20929	−0.150124	−0.0750621	+	0.997179i	$0.523916\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−36.2301	−1.68740	−0.843702	−	0.536812i	$-0.819628\pi$
$461$	−36.2301	−1.68740	−0.843702	+	0.536812i	$0.819628\pi$
$462$	0	0
$463$	−29.0506	−1.35010	−0.675049	−	0.737773i	$-0.735877\pi$
$463$	−29.0506	−1.35010	−0.675049	+	0.737773i	$0.735877\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	37.4933	1.73498	0.867491	−	0.497452i	$-0.165731\pi$
$467$	37.4933	1.73498	0.867491	+	0.497452i	$0.165731\pi$
$468$	0	0
$469$	0.660190	0.0304847
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−64.0014	−2.94279
$474$	0	0
$475$	18.0917	0.830102
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−29.9097	−1.36661	−0.683305	−	0.730133i	$-0.739458\pi$
$479$	−29.9097	−1.36661	−0.683305	+	0.730133i	$0.739458\pi$
$480$	0	0
$481$	−22.4887	−1.02539
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.10069	−0.186203
$486$	0	0
$487$	−21.2632	−0.963528	−0.481764	−	0.876301i	$-0.660004\pi$
$487$	−21.2632	−0.963528	−0.481764	+	0.876301i	$0.660004\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	21.3970	0.965633	0.482816	−	0.875722i	$-0.339614\pi$
$491$	21.3970	0.965633	0.482816	+	0.875722i	$0.339614\pi$
$492$	0	0
$493$	12.8317	0.577912
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−13.7414	−0.616388
$498$	0	0
$499$	−14.5653	−0.652031	−0.326015	−	0.945364i	$-0.605706\pi$
$499$	−14.5653	−0.652031	−0.326015	+	0.945364i	$0.605706\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2.92339	−0.130348	−0.0651738	−	0.997874i	$-0.520760\pi$
$503$	−2.92339	−0.130348	−0.0651738	+	0.997874i	$0.520760\pi$
$504$	0	0
$505$	1.71737	0.0764220
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−19.2405	−0.852820	−0.426410	−	0.904530i	$-0.640222\pi$
$509$	−19.2405	−0.852820	−0.426410	+	0.904530i	$0.640222\pi$
$510$	0	0
$511$	−3.66019	−0.161917
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3.06758	0.135174
$516$	0	0
$517$	−22.2405	−0.978136
$518$	0	0
$519$	0	0
$520$	0	0
$521$	27.7486	1.21569	0.607844	−	0.794057i	$-0.292035\pi$
$521$	27.7486	1.21569	0.607844	+	0.794057i	$0.292035\pi$
$522$	0	0
$523$	2.73680	0.119672	0.0598360	−	0.998208i	$-0.480942\pi$
$523$	2.73680	0.119672	0.0598360	+	0.998208i	$0.480942\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.65692	0.115737
$528$	0	0
$529$	32.0722	1.39444
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−27.3743	−1.18571
$534$	0	0
$535$	−1.80903	−0.0782112
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5.12476	0.220739
$540$	0	0
$541$	−11.9773	−0.514944	−0.257472	−	0.966286i	$-0.582890\pi$
$541$	−11.9773	−0.514944	−0.257472	+	0.966286i	$0.582890\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1.66922	−0.0715014
$546$	0	0
$547$	21.4692	0.917958	0.458979	−	0.888447i	$-0.348215\pi$
$547$	21.4692	0.917958	0.458979	+	0.888447i	$0.348215\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−12.6810	−0.540229
$552$	0	0
$553$	6.22545	0.264733
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−31.8493	−1.34950	−0.674748	−	0.738048i	$-0.735748\pi$
$557$	−31.8493	−1.34950	−0.674748	+	0.738048i	$0.735748\pi$
$558$	0	0
$559$	61.0150	2.58066
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−35.5483	−1.49818	−0.749091	−	0.662467i	$-0.769510\pi$
$563$	−35.5483	−1.49818	−0.749091	+	0.662467i	$0.769510\pi$
$564$	0	0
$565$	−4.67962	−0.196873
$566$	0	0
$567$	0	0
$568$	0	0
$569$	21.7486	0.911748	0.455874	−	0.890044i	$-0.349327\pi$
$569$	21.7486	0.911748	0.455874	+	0.890044i	$0.349327\pi$
$570$	0	0
$571$	9.59974	0.401737	0.200868	−	0.979618i	$-0.435624\pi$
$571$	9.59974	0.401737	0.200868	+	0.979618i	$0.435624\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	36.6810	1.52970
$576$	0	0
$577$	13.0183	0.541960	0.270980	−	0.962585i	$-0.412652\pi$
$577$	13.0183	0.541960	0.270980	+	0.962585i	$0.412652\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−9.70370	−0.402577
$582$	0	0
$583$	−4.83173	−0.200110
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−17.2873	−0.713522	−0.356761	−	0.934196i	$-0.616119\pi$
$587$	−17.2873	−0.713522	−0.356761	+	0.934196i	$0.616119\pi$
$588$	0	0
$589$	−2.62571	−0.108190
$590$	0	0
$591$	0	0
$592$	0	0
$593$	12.4153	0.509836	0.254918	−	0.966963i	$-0.417952\pi$
$593$	12.4153	0.509836	0.254918	+	0.966963i	$0.417952\pi$
$594$	0	0
$595$	0.885640	0.0363077
$596$	0	0
$597$	0	0
$598$	0	0
$599$	7.88564	0.322199	0.161099	−	0.986938i	$-0.448496\pi$
$599$	7.88564	0.322199	0.161099	+	0.986938i	$0.448496\pi$
$600$	0	0
$601$	22.2826	0.908927	0.454464	−	0.890765i	$-0.349831\pi$
$601$	22.2826	0.908927	0.454464	+	0.890765i	$0.349831\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	3.64979	0.148385
$606$	0	0
$607$	−22.0917	−0.896673	−0.448336	−	0.893865i	$-0.647983\pi$
$607$	−22.0917	−0.896673	−0.448336	+	0.893865i	$0.647983\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	21.2028	0.857771
$612$	0	0
$613$	−29.5264	−1.19256	−0.596280	−	0.802777i	$-0.703355\pi$
$613$	−29.5264	−1.19256	−0.596280	+	0.802777i	$0.703355\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−10.0331	−0.403918	−0.201959	−	0.979394i	$-0.564731\pi$
$617$	−10.0331	−0.403918	−0.201959	+	0.979394i	$0.564731\pi$
$618$	0	0
$619$	38.2567	1.53767	0.768833	−	0.639450i	$-0.220838\pi$
$619$	38.2567	1.53767	0.768833	+	0.639450i	$0.220838\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−7.48865	−0.300026
$624$	0	0
$625$	24.1456	0.965823
$626$	0	0
$627$	0	0
$628$	0	0
$629$	17.0482	0.679754
$630$	0	0
$631$	23.0377	0.917118	0.458559	−	0.888664i	$-0.348366\pi$
$631$	23.0377	0.917118	0.458559	+	0.888664i	$0.348366\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4.01943	0.159506
$636$	0	0
$637$	−4.88564	−0.193576
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−17.3729	−0.686189	−0.343094	−	0.939301i	$-0.611475\pi$
$641$	−17.3729	−0.686189	−0.343094	+	0.939301i	$0.611475\pi$
$642$	0	0
$643$	18.9590	0.747669	0.373835	−	0.927495i	$-0.378043\pi$
$643$	18.9590	0.747669	0.373835	+	0.927495i	$0.378043\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−19.0194	−0.747731	−0.373865	−	0.927483i	$-0.621968\pi$
$647$	−19.0194	−0.747731	−0.373865	+	0.927483i	$0.621968\pi$
$648$	0	0
$649$	−38.8402	−1.52461
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−7.18659	−0.281233	−0.140616	−	0.990064i	$-0.544908\pi$
$653$	−7.18659	−0.281233	−0.140616	+	0.990064i	$0.544908\pi$
$654$	0	0
$655$	−1.17154	−0.0457758
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−25.4523	−0.991480	−0.495740	−	0.868471i	$-0.665103\pi$
$659$	−25.4523	−0.991480	−0.495740	+	0.868471i	$0.665103\pi$
$660$	0	0
$661$	8.28590	0.322284	0.161142	−	0.986931i	$-0.448482\pi$
$661$	8.28590	0.322284	0.161142	+	0.986931i	$0.448482\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−0.875237	−0.0339402
$666$	0	0
$667$	−25.7108	−0.995527
$668$	0	0
$669$	0	0
$670$	0	0
$671$	28.2276	1.08971
$672$	0	0
$673$	−11.8317	−0.456080	−0.228040	−	0.973652i	$-0.573232\pi$
$673$	−11.8317	−0.456080	−0.228040	+	0.973652i	$0.573232\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	13.6063	0.522932	0.261466	−	0.965213i	$-0.415794\pi$
$677$	13.6063	0.522932	0.261466	+	0.965213i	$0.415794\pi$
$678$	0	0
$679$	−17.1488	−0.658112
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−3.58142	−0.137039	−0.0685196	−	0.997650i	$-0.521828\pi$
$683$	−3.58142	−0.137039	−0.0685196	+	0.997650i	$0.521828\pi$
$684$	0	0
$685$	1.30206	0.0497492
$686$	0	0
$687$	0	0
$688$	0	0
$689$	4.60628	0.175485
$690$	0	0
$691$	−11.7174	−0.445750	−0.222875	−	0.974847i	$-0.571544\pi$
$691$	−11.7174	−0.445750	−0.222875	+	0.974847i	$0.571544\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.35348	−0.0513405
$696$	0	0
$697$	20.7518	0.786032
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−10.5926	−0.400077	−0.200039	−	0.979788i	$-0.564107\pi$
$701$	−10.5926	−0.400077	−0.200039	+	0.979788i	$0.564107\pi$
$702$	0	0
$703$	−16.8479	−0.635430
$704$	0	0
$705$	0	0
$706$	0	0
$707$	7.18194	0.270105
$708$	0	0
$709$	38.2977	1.43830	0.719150	−	0.694855i	$-0.244532\pi$
$709$	38.2977	1.43830	0.719150	+	0.694855i	$0.244532\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−5.32365	−0.199372
$714$	0	0
$715$	−5.98711	−0.223905
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−41.6752	−1.55422	−0.777112	−	0.629362i	$-0.783316\pi$
$719$	−41.6752	−1.55422	−0.777112	+	0.629362i	$0.783316\pi$
$720$	0	0
$721$	12.8285	0.477757
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−17.1248	−0.635998
$726$	0	0
$727$	32.8252	1.21742	0.608709	−	0.793393i	$-0.291688\pi$
$727$	32.8252	1.21742	0.608709	+	0.793393i	$0.291688\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−46.2542	−1.71077
$732$	0	0
$733$	9.29768	0.343418	0.171709	−	0.985148i	$-0.445071\pi$
$733$	9.29768	0.343418	0.171709	+	0.985148i	$0.445071\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.38332	0.124626
$738$	0	0
$739$	−11.3776	−0.418530	−0.209265	−	0.977859i	$-0.567107\pi$
$739$	−11.3776	−0.418530	−0.209265	+	0.977859i	$0.567107\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	2.32365	0.0852464	0.0426232	−	0.999091i	$-0.486428\pi$
$743$	2.32365	0.0852464	0.0426232	+	0.999091i	$0.486428\pi$
$744$	0	0
$745$	0.545830	0.0199977
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−7.56526	−0.276429
$750$	0	0
$751$	11.1338	0.406278	0.203139	−	0.979150i	$-0.434886\pi$
$751$	11.1338	0.406278	0.203139	+	0.979150i	$0.434886\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−2.69329	−0.0980190
$756$	0	0
$757$	52.1639	1.89593	0.947964	−	0.318376i	$-0.103138\pi$
$757$	52.1639	1.89593	0.947964	+	0.318376i	$0.103138\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−47.0255	−1.70467	−0.852336	−	0.522995i	$-0.824815\pi$
$761$	−47.0255	−1.70467	−0.852336	+	0.522995i	$0.824815\pi$
$762$	0	0
$763$	−6.98057	−0.252714
$764$	0	0
$765$	0	0
$766$	0	0
$767$	37.0279	1.33700
$768$	0	0
$769$	−6.60628	−0.238229	−0.119114	−	0.992881i	$-0.538005\pi$
$769$	−6.60628	−0.238229	−0.119114	+	0.992881i	$0.538005\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	19.0870	0.686512	0.343256	−	0.939242i	$-0.388470\pi$
$773$	19.0870	0.686512	0.343256	+	0.939242i	$0.388470\pi$
$774$	0	0
$775$	−3.54583	−0.127370
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−20.5081	−0.734778
$780$	0	0
$781$	−70.4217	−2.51989
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.32614	0.0473319
$786$	0	0
$787$	−50.9007	−1.81441	−0.907207	−	0.420685i	$-0.861790\pi$
$787$	−50.9007	−1.81441	−0.907207	+	0.420685i	$0.861790\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−19.5699	−0.695826
$792$	0	0
$793$	−26.9105	−0.955620
$794$	0	0
$795$	0	0
$796$	0	0
$797$	8.77455	0.310810	0.155405	−	0.987851i	$-0.450332\pi$
$797$	8.77455	0.310810	0.155405	+	0.987851i	$0.450332\pi$
$798$	0	0
$799$	−16.0733	−0.568634
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−18.7576	−0.661942
$804$	0	0
$805$	−1.77455	−0.0625447
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−9.51384	−0.334489	−0.167244	−	0.985915i	$-0.553487\pi$
$809$	−9.51384	−0.334489	−0.167244	+	0.985915i	$0.553487\pi$
$810$	0	0
$811$	25.0118	0.878282	0.439141	−	0.898418i	$-0.355283\pi$
$811$	25.0118	0.878282	0.439141	+	0.898418i	$0.355283\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−1.59261	−0.0557866
$816$	0	0
$817$	45.7108	1.59922
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−35.5940	−1.24224	−0.621119	−	0.783716i	$-0.713322\pi$
$821$	−35.5940	−1.24224	−0.621119	+	0.783716i	$0.713322\pi$
$822$	0	0
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	25.4531	0.885090	0.442545	−	0.896746i	$-0.354076\pi$
$827$	25.4531	0.885090	0.442545	+	0.896746i	$0.354076\pi$
$828$	0	0
$829$	−17.5458	−0.609392	−0.304696	−	0.952450i	$-0.598555\pi$
$829$	−17.5458	−0.609392	−0.304696	+	0.952450i	$0.598555\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3.70370	0.128325
$834$	0	0
$835$	−1.05391	−0.0364721
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−24.1125	−0.832455	−0.416227	−	0.909261i	$-0.636648\pi$
$839$	−24.1125	−0.832455	−0.416227	+	0.909261i	$0.636648\pi$
$840$	0	0
$841$	−16.9967	−0.586094
$842$	0	0
$843$	0	0
$844$	0	0
$845$	2.59915	0.0894133
$846$	0	0
$847$	15.2632	0.524450
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−34.1592	−1.17096
$852$	0	0
$853$	32.5231	1.11357	0.556785	−	0.830656i	$-0.312035\pi$
$853$	32.5231	1.11357	0.556785	+	0.830656i	$0.312035\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−0.599740	−0.0204867	−0.0102434	−	0.999948i	$-0.503261\pi$
$857$	−0.599740	−0.0204867	−0.0102434	+	0.999948i	$0.503261\pi$
$858$	0	0
$859$	26.4347	0.901942	0.450971	−	0.892539i	$-0.351078\pi$
$859$	26.4347	0.901942	0.450971	+	0.892539i	$0.351078\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−19.8454	−0.675545	−0.337773	−	0.941228i	$-0.609674\pi$
$863$	−19.8454	−0.675545	−0.337773	+	0.941228i	$0.609674\pi$
$864$	0	0
$865$	6.05718	0.205950
$866$	0	0
$867$	0	0
$868$	0	0
$869$	31.9040	1.08227
$870$	0	0
$871$	−3.22545	−0.109290
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−2.37756	−0.0803762
$876$	0	0
$877$	20.4703	0.691234	0.345617	−	0.938376i	$-0.387670\pi$
$877$	20.4703	0.691234	0.345617	+	0.938376i	$0.387670\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−31.1683	−1.05009	−0.525043	−	0.851076i	$-0.675951\pi$
$881$	−31.1683	−1.05009	−0.525043	+	0.851076i	$0.675951\pi$
$882$	0	0
$883$	2.64187	0.0889060	0.0444530	−	0.999011i	$-0.485846\pi$
$883$	2.64187	0.0889060	0.0444530	+	0.999011i	$0.485846\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	23.1650	0.777805	0.388902	−	0.921279i	$-0.372854\pi$
$887$	23.1650	0.777805	0.388902	+	0.921279i	$0.372854\pi$
$888$	0	0
$889$	16.8090	0.563757
$890$	0	0
$891$	0	0
$892$	0	0
$893$	15.8845	0.531555
$894$	0	0
$895$	−2.28263	−0.0762999
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.48538	0.0828921
$900$	0	0
$901$	−3.49192	−0.116333
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.95898	0.0983599
$906$	0	0
$907$	50.0528	1.66198	0.830988	−	0.556290i	$-0.187776\pi$
$907$	50.0528	1.66198	0.830988	+	0.556290i	$0.187776\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	10.8446	0.359298	0.179649	−	0.983731i	$-0.442504\pi$
$911$	10.8446	0.359298	0.179649	+	0.983731i	$0.442504\pi$
$912$	0	0
$913$	−49.7292	−1.64579
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−4.89931	−0.161790
$918$	0	0
$919$	−11.1910	−0.369156	−0.184578	−	0.982818i	$-0.559092\pi$
$919$	−11.1910	−0.369156	−0.184578	+	0.982818i	$0.559092\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	67.1358	2.20980
$924$	0	0
$925$	−22.7518	−0.748076
$926$	0	0
$927$	0	0
$928$	0	0
$929$	41.2955	1.35486	0.677431	−	0.735586i	$-0.263093\pi$
$929$	41.2955	1.35486	0.677431	+	0.735586i	$0.263093\pi$
$930$	0	0
$931$	−3.66019	−0.119958
$932$	0	0
$933$	0	0
$934$	0	0
$935$	4.53870	0.148431
$936$	0	0
$937$	−33.5620	−1.09642	−0.548211	−	0.836340i	$-0.684691\pi$
$937$	−33.5620	−1.09642	−0.548211	+	0.836340i	$0.684691\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	50.4224	1.64372	0.821862	−	0.569686i	$-0.192935\pi$
$941$	50.4224	1.64372	0.821862	+	0.569686i	$0.192935\pi$
$942$	0	0
$943$	−41.5803	−1.35404
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−20.2424	−0.657789	−0.328895	−	0.944367i	$-0.606676\pi$
$947$	−20.2424	−0.657789	−0.328895	+	0.944367i	$0.606676\pi$
$948$	0	0
$949$	17.8824	0.580486
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−29.3685	−0.951340	−0.475670	−	0.879624i	$-0.657794\pi$
$953$	−29.3685	−0.951340	−0.475670	+	0.879624i	$0.657794\pi$
$954$	0	0
$955$	−3.14884	−0.101894
$956$	0	0
$957$	0	0
$958$	0	0
$959$	5.44514	0.175833
$960$	0	0
$961$	−30.4854	−0.983399
$962$	0	0
$963$	0	0
$964$	0	0
$965$	2.66595	0.0858199
$966$	0	0
$967$	30.4315	0.978610	0.489305	−	0.872113i	$-0.337250\pi$
$967$	30.4315	0.978610	0.489305	+	0.872113i	$0.337250\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−7.18659	−0.230629	−0.115314	−	0.993329i	$-0.536788\pi$
$971$	−7.18659	−0.230629	−0.115314	+	0.993329i	$0.536788\pi$
$972$	0	0
$973$	−5.66019	−0.181457
$974$	0	0
$975$	0	0
$976$	0	0
$977$	28.5426	0.913157	0.456579	−	0.889683i	$-0.349075\pi$
$977$	28.5426	0.913157	0.456579	+	0.889683i	$0.349075\pi$
$978$	0	0
$979$	−38.3776	−1.22655
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−4.41642	−0.140862	−0.0704310	−	0.997517i	$-0.522437\pi$
$983$	−4.41642	−0.140862	−0.0704310	+	0.997517i	$0.522437\pi$
$984$	0	0
$985$	−0.0344801	−0.00109863
$986$	0	0
$987$	0	0
$988$	0	0
$989$	92.6791	2.94702
$990$	0	0
$991$	5.81341	0.184669	0.0923345	−	0.995728i	$-0.470567\pi$
$991$	5.81341	0.184669	0.0923345	+	0.995728i	$0.470567\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	4.65554	0.147591
$996$	0	0
$997$	−52.6408	−1.66715	−0.833575	−	0.552407i	$-0.813710\pi$
$997$	−52.6408	−1.66715	−0.833575	+	0.552407i	$0.813710\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9072.2.a.by.1.2		3
3.2	odd	2		9072.2.a.bv.1.2		3
4.3	odd	2		2268.2.a.i.1.2		3
9.2	odd	6		3024.2.r.j.1009.2		6
9.4	even	3		1008.2.r.j.673.3		6
9.5	odd	6		3024.2.r.j.2017.2		6
9.7	even	3		1008.2.r.j.337.3		6
12.11	even	2		2268.2.a.h.1.2		3
36.7	odd	6		252.2.j.a.85.1	✓	6
36.11	even	6		756.2.j.b.253.2		6
36.23	even	6		756.2.j.b.505.2		6
36.31	odd	6		252.2.j.a.169.1	yes	6
252.11	even	6		5292.2.i.f.1549.2		6
252.23	even	6		5292.2.i.f.2125.2		6
252.31	even	6		1764.2.l.f.961.1		6
252.47	odd	6		5292.2.l.f.361.2		6
252.59	odd	6		5292.2.l.f.3313.2		6
252.67	odd	6		1764.2.l.e.961.3		6
252.79	odd	6		1764.2.l.e.949.3		6
252.83	odd	6		5292.2.j.d.1765.2		6
252.95	even	6		5292.2.l.e.3313.2		6
252.103	even	6		1764.2.i.d.1537.2		6
252.115	even	6		1764.2.i.d.373.2		6
252.131	odd	6		5292.2.i.e.2125.2		6
252.139	even	6		1764.2.j.e.1177.3		6
252.151	odd	6		1764.2.i.g.373.2		6
252.167	odd	6		5292.2.j.d.3529.2		6
252.187	even	6		1764.2.l.f.949.1		6
252.191	even	6		5292.2.l.e.361.2		6
252.223	even	6		1764.2.j.e.589.3		6
252.227	odd	6		5292.2.i.e.1549.2		6
252.247	odd	6		1764.2.i.g.1537.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.j.a.85.1	✓	6	36.7	odd	6
252.2.j.a.169.1	yes	6	36.31	odd	6
756.2.j.b.253.2		6	36.11	even	6
756.2.j.b.505.2		6	36.23	even	6
1008.2.r.j.337.3		6	9.7	even	3
1008.2.r.j.673.3		6	9.4	even	3
1764.2.i.d.373.2		6	252.115	even	6
1764.2.i.d.1537.2		6	252.103	even	6
1764.2.i.g.373.2		6	252.151	odd	6
1764.2.i.g.1537.2		6	252.247	odd	6
1764.2.j.e.589.3		6	252.223	even	6
1764.2.j.e.1177.3		6	252.139	even	6
1764.2.l.e.949.3		6	252.79	odd	6
1764.2.l.e.961.3		6	252.67	odd	6
1764.2.l.f.949.1		6	252.187	even	6
1764.2.l.f.961.1		6	252.31	even	6
2268.2.a.h.1.2		3	12.11	even	2
2268.2.a.i.1.2		3	4.3	odd	2
3024.2.r.j.1009.2		6	9.2	odd	6
3024.2.r.j.2017.2		6	9.5	odd	6
5292.2.i.e.1549.2		6	252.227	odd	6
5292.2.i.e.2125.2		6	252.131	odd	6
5292.2.i.f.1549.2		6	252.11	even	6
5292.2.i.f.2125.2		6	252.23	even	6
5292.2.j.d.1765.2		6	252.83	odd	6
5292.2.j.d.3529.2		6	252.167	odd	6
5292.2.l.e.361.2		6	252.191	even	6
5292.2.l.e.3313.2		6	252.95	even	6
5292.2.l.f.361.2		6	252.47	odd	6
5292.2.l.f.3313.2		6	252.59	odd	6
9072.2.a.bv.1.2		3	3.2	odd	2
9072.2.a.by.1.2		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9072 = 2^{4} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9072.a (trivial)

\(f(q)\)	\(=\)	\(q+0.239123 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q+0.239123 q^{5} +1.00000 q^{7} +5.12476 q^{11} -4.88564 q^{13} +3.70370 q^{17} -3.66019 q^{19} -7.42107 q^{23} -4.94282 q^{25} +3.46457 q^{29} +0.717370 q^{31} +0.239123 q^{35} +4.60301 q^{37} +5.60301 q^{41} -12.4887 q^{43} -4.33981 q^{47} +1.00000 q^{49} -0.942820 q^{53} +1.22545 q^{55} -7.57893 q^{59} +5.50808 q^{61} -1.16827 q^{65} +0.660190 q^{67} -13.7414 q^{71} -3.66019 q^{73} +5.12476 q^{77} +6.22545 q^{79} -9.70370 q^{83} +0.885640 q^{85} -7.48865 q^{89} -4.88564 q^{91} -0.875237 q^{95} -17.1488 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{5} + 3 q^{7}+O(q^{10}) \) 3 * q + q^5 + 3 * q^7	\( 3 q + q^{5} + 3 q^{7} - 2 q^{11} + 3 q^{13} + 2 q^{17} - 3 q^{19} - 14 q^{23} - 6 q^{25} + q^{29} + 3 q^{31} + q^{35} - 3 q^{37} - 3 q^{43} - 21 q^{47} + 3 q^{49} + 6 q^{53} - 6 q^{55} - 31 q^{59} + 6 q^{61} + 15 q^{65} - 6 q^{67} - 17 q^{71} - 3 q^{73} - 2 q^{77} + 9 q^{79} - 20 q^{83} - 15 q^{85} + 12 q^{89} + 3 q^{91} - 20 q^{95} - 9 q^{97}+O(q^{100}) \) 3 * q + q^5 + 3 * q^7 - 2 * q^11 + 3 * q^13 + 2 * q^17 - 3 * q^19 - 14 * q^23 - 6 * q^25 + q^29 + 3 * q^31 + q^35 - 3 * q^37 - 3 * q^43 - 21 * q^47 + 3 * q^49 + 6 * q^53 - 6 * q^55 - 31 * q^59 + 6 * q^61 + 15 * q^65 - 6 * q^67 - 17 * q^71 - 3 * q^73 - 2 * q^77 + 9 * q^79 - 20 * q^83 - 15 * q^85 + 12 * q^89 + 3 * q^91 - 20 * q^95 - 9 * q^97

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