LMFDB - Embedded newform 1728.3.q.l.1601.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,3,Mod(449,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 5]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1728.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1728.q (of order $6$, degree $2$, not minimal)

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:	no
Analytic conductor:	$47.0845896815$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 288)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1601.3
Character	$\chi$	$=$	1728.1601
Dual form			1728.3.q.l.449.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.45884 + 2.57431i) q^{5} +(-1.35076 + 2.33958i) q^{7} +O(q^{10})$	$q+(-4.45884 + 2.57431i) q^{5} +(-1.35076 + 2.33958i) q^{7} +(14.7420 + 8.51129i) q^{11} +(-9.54088 - 16.5253i) q^{13} +20.7119i q^{17} -24.5518 q^{19} +(7.30755 - 4.21902i) q^{23} +(0.754187 - 1.30629i) q^{25} +(-12.4116 - 7.16587i) q^{29} +(12.7585 + 22.0983i) q^{31} -13.9091i q^{35} -31.3274 q^{37} +(54.6358 - 31.5440i) q^{41} +(-6.36321 + 11.0214i) q^{43} +(-36.0386 - 20.8069i) q^{47} +(20.8509 + 36.1148i) q^{49} +75.5923i q^{53} -87.6430 q^{55} +(0.818085 - 0.472321i) q^{59} +(43.0393 - 74.5462i) q^{61} +(85.0826 + 49.1225i) q^{65} +(-28.0563 - 48.5949i) q^{67} -0.521073i q^{71} -21.2950 q^{73} +(-39.8258 + 22.9934i) q^{77} +(53.3000 - 92.3183i) q^{79} +(-94.6065 - 54.6211i) q^{83} +(-53.3189 - 92.3510i) q^{85} +1.99321i q^{89} +51.5498 q^{91} +(109.472 - 63.2039i) q^{95} +(28.6557 - 49.6331i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q + 60 q^{25} + 72 q^{29} + 36 q^{41} - 132 q^{49} - 96 q^{61} - 576 q^{65} + 24 q^{73} + 432 q^{77} + 96 q^{85} + 252 q^{97}+O(q^{100}) $ 24 * q + 60 * q^25 + 72 * q^29 + 36 * q^41 - 132 * q^49 - 96 * q^61 - 576 * q^65 + 24 * q^73 + 432 * q^77 + 96 * q^85 + 252 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{6}\right)$

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
$5$	−4.45884	+	2.57431i	−0.891769	+	0.514863i	−0.874521	−	0.484988i	$-0.838824\pi$
$5$	−4.45884	+	2.57431i	−0.891769	+	0.514863i	−0.0172480	+	0.999851i	$0.505490\pi$
$6$		0			0
$7$	−1.35076	+	2.33958i	−0.192966	+	0.334226i	−0.946232	−	0.323490i	$-0.895144\pi$
$7$	−1.35076	+	2.33958i	−0.192966	+	0.334226i	0.753266	+	0.657716i	$0.228477\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	14.7420	+	8.51129i	1.34018	+	0.773754i	0.986834	−	0.161738i	$-0.0517100\pi$
$11$	14.7420	+	8.51129i	1.34018	+	0.773754i	0.353348	+	0.935492i	$0.385043\pi$
$12$		0			0
$13$	−9.54088	−	16.5253i	−0.733914	−	1.27118i	−0.955198	−	0.295967i	$-0.904358\pi$
$13$	−9.54088	−	16.5253i	−0.733914	−	1.27118i	0.221284	−	0.975209i	$-0.428975\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			20.7119i			1.21835i	0.793038	+	0.609173i	$0.208498\pi$
$17$			20.7119i			1.21835i	−0.793038	+	0.609173i	$0.791502\pi$
$18$		0			0
$19$	−24.5518			−1.29220			−0.646099	−	0.763254i	$-0.723601\pi$
$19$	−24.5518			−1.29220			−0.646099	+	0.763254i	$0.723601\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	7.30755	−	4.21902i	0.317720	−	0.183436i	−0.332656	−	0.943048i	$-0.607945\pi$
$23$	7.30755	−	4.21902i	0.317720	−	0.183436i	0.650376	+	0.759613i	$0.274611\pi$
$24$		0			0
$25$	0.754187	−	1.30629i	0.0301675	−	0.0522516i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−12.4116	−	7.16587i	−0.427988	−	0.247099i	0.270501	−	0.962720i	$-0.412811\pi$
$29$	−12.4116	−	7.16587i	−0.427988	−	0.247099i	−0.698489	+	0.715621i	$0.746144\pi$
$30$		0			0
$31$	12.7585	+	22.0983i	0.411563	+	0.712848i	0.995061	−	0.0992664i	$-0.0316496\pi$
$31$	12.7585	+	22.0983i	0.411563	+	0.712848i	−0.583498	+	0.812115i	$0.698316\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	13.9091i		−	0.397403i
$36$		0			0
$37$	−31.3274			−0.846687			−0.423343	−	0.905969i	$-0.639144\pi$
$37$	−31.3274			−0.846687			−0.423343	+	0.905969i	$0.639144\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	54.6358	−	31.5440i	1.33258	−	0.769365i	0.346885	−	0.937908i	$-0.387239\pi$
$41$	54.6358	−	31.5440i	1.33258	−	0.769365i	0.985694	+	0.168542i	$0.0539060\pi$
$42$		0			0
$43$	−6.36321	+	11.0214i	−0.147982	+	0.256312i	−0.930481	−	0.366339i	$-0.880611\pi$
$43$	−6.36321	+	11.0214i	−0.147982	+	0.256312i	0.782500	+	0.622651i	$0.213944\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−36.0386	−	20.8069i	−0.766778	−	0.442700i	0.0649459	−	0.997889i	$-0.479313\pi$
$47$	−36.0386	−	20.8069i	−0.766778	−	0.442700i	−0.831724	+	0.555189i	$0.812646\pi$
$48$		0			0
$49$	20.8509	+	36.1148i	0.425529	+	0.737037i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			75.5923i			1.42627i	0.701027	+	0.713135i	$0.252725\pi$
$53$			75.5923i			1.42627i	−0.701027	+	0.713135i	$0.747275\pi$
$54$		0			0
$55$	−87.6430			−1.59351
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.818085	−	0.472321i	0.0138658	−	0.00800545i	−0.493051	−	0.870000i	$-0.664118\pi$
$59$	0.818085	−	0.472321i	0.0138658	−	0.00800545i	0.506917	+	0.861995i	$0.330785\pi$
$60$		0			0
$61$	43.0393	−	74.5462i	0.705562	−	1.22207i	−0.260927	−	0.965359i	$-0.584028\pi$
$61$	43.0393	−	74.5462i	0.705562	−	1.22207i	0.966488	−	0.256710i	$-0.0826386\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	85.0826	+	49.1225i	1.30896	+	0.755730i
$66$		0			0
$67$	−28.0563	−	48.5949i	−0.418750	−	0.725297i	0.577064	−	0.816699i	$-0.304198\pi$
$67$	−28.0563	−	48.5949i	−0.418750	−	0.725297i	−0.995814	+	0.0914023i	$0.970865\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	0.521073i		−	0.00733905i	−0.999993	−	0.00366953i	$-0.998832\pi$
$71$		−	0.521073i		−	0.00733905i	0.999993	−	0.00366953i	$-0.00116805\pi$
$72$		0			0
$73$	−21.2950			−0.291712			−0.145856	−	0.989306i	$-0.546594\pi$
$73$	−21.2950			−0.291712			−0.145856	+	0.989306i	$0.546594\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−39.8258	+	22.9934i	−0.517218	+	0.298616i
$78$		0			0
$79$	53.3000	−	92.3183i	0.674683	−	1.16859i	−0.301878	−	0.953347i	$-0.597614\pi$
$79$	53.3000	−	92.3183i	0.674683	−	1.16859i	0.976561	−	0.215239i	$-0.0690531\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−94.6065	−	54.6211i	−1.13984	−	0.658085i	−0.193447	−	0.981111i	$-0.561967\pi$
$83$	−94.6065	−	54.6211i	−1.13984	−	0.658085i	−0.946390	+	0.323026i	$0.895300\pi$
$84$		0			0
$85$	−53.3189	−	92.3510i	−0.627281	−	1.08648i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			1.99321i			0.0223956i	0.999937	+	0.0111978i	$0.00356444\pi$
$89$			1.99321i			0.0223956i	−0.999937	+	0.0111978i	$0.996436\pi$
$90$		0			0
$91$	51.5498			0.566481
$92$		0			0
$93$		0			0
$94$		0			0
$95$	109.472	−	63.2039i	1.15234	−	0.665305i
$96$		0			0
$97$	28.6557	−	49.6331i	0.295420	−	0.511682i	−0.679663	−	0.733525i	$-0.737874\pi$
$97$	28.6557	−	49.6331i	0.295420	−	0.511682i	0.975082	+	0.221843i	$0.0712072\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−109.992	−	63.5036i	−1.08902	−	0.628749i	−0.155708	−	0.987803i	$-0.549766\pi$
$101$	−109.992	−	63.5036i	−1.08902	−	0.628749i	−0.933317	+	0.359054i	$0.883099\pi$
$102$		0			0
$103$	−61.4675	−	106.465i	−0.596772	−	1.03364i	−0.993294	−	0.115614i	$-0.963116\pi$
$103$	−61.4675	−	106.465i	−0.596772	−	1.03364i	0.396522	−	0.918025i	$-0.370217\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	173.812i		−	1.62441i	−0.583370	−	0.812207i	$-0.698266\pi$
$107$		−	173.812i		−	1.62441i	0.583370	−	0.812207i	$-0.301734\pi$
$108$		0			0
$109$	−186.685			−1.71270			−0.856352	−	0.516392i	$-0.827275\pi$
$109$	−186.685			−1.71270			−0.856352	+	0.516392i	$0.827275\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	17.0778	−	9.85990i	0.151131	−	0.0872557i	−0.422527	−	0.906350i	$-0.638857\pi$
$113$	17.0778	−	9.85990i	0.151131	−	0.0872557i	0.573659	+	0.819094i	$0.305524\pi$
$114$		0			0
$115$	−21.7222	+	37.6239i	−0.188888	+	0.327164i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−48.4572	−	27.9768i	−0.407203	−	0.235099i
$120$		0			0
$121$	84.3842	+	146.158i	0.697390	+	1.20792i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	120.950i		−	0.967597i
$126$		0			0
$127$	−164.219			−1.29307			−0.646533	−	0.762886i	$-0.723782\pi$
$127$	−164.219			−1.29307			−0.646533	+	0.762886i	$0.723782\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	111.466	−	64.3551i	0.850888	−	0.491261i	−0.0100621	−	0.999949i	$-0.503203\pi$
$131$	111.466	−	64.3551i	0.850888	−	0.491261i	0.860950	+	0.508689i	$0.169870\pi$
$132$		0			0
$133$	33.1635	−	57.4409i	0.249350	−	0.431887i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−57.2994	−	33.0818i	−0.418244	−	0.241473i	0.276082	−	0.961134i	$-0.410964\pi$
$137$	−57.2994	−	33.0818i	−0.418244	−	0.241473i	−0.694326	+	0.719661i	$0.744297\pi$
$138$		0			0
$139$	−138.404	−	239.723i	−0.995711	−	1.72462i	−0.577975	−	0.816055i	$-0.696157\pi$
$139$	−138.404	−	239.723i	−0.995711	−	1.72462i	−0.417737	−	0.908568i	$-0.637177\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	324.821i		−	2.27148i
$144$		0			0
$145$	73.7888			0.508888
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−43.1200	+	24.8954i	−0.289396	+	0.167083i	−0.637669	−	0.770310i	$-0.720101\pi$
$149$	−43.1200	+	24.8954i	−0.289396	+	0.167083i	0.348273	+	0.937393i	$0.386768\pi$
$150$		0			0
$151$	−54.8348	+	94.9767i	−0.363144	+	0.628984i	−0.988477	−	0.151374i	$-0.951630\pi$
$151$	−54.8348	+	94.9767i	−0.363144	+	0.628984i	0.625332	+	0.780359i	$0.284963\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−113.776	−	65.6886i	−0.734038	−	0.423797i
$156$		0			0
$157$	135.581	+	234.833i	0.863571	+	1.49575i	0.868459	+	0.495761i	$0.165111\pi$
$157$	135.581	+	234.833i	0.863571	+	1.49575i	−0.00488805	+	0.999988i	$0.501556\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			22.7955i			0.141587i
$162$		0			0
$163$	−98.5286			−0.604470			−0.302235	−	0.953233i	$-0.597733\pi$
$163$	−98.5286			−0.604470			−0.302235	+	0.953233i	$0.597733\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−177.135	+	102.269i	−1.06069	+	0.612390i	−0.925623	−	0.378446i	$-0.876459\pi$
$167$	−177.135	+	102.269i	−1.06069	+	0.612390i	−0.135068	+	0.990836i	$0.543125\pi$
$168$		0			0
$169$	−97.5569	+	168.973i	−0.577260	+	0.999843i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	62.2657	+	35.9491i	0.359917	+	0.207798i	0.669044	−	0.743222i	$-0.266704\pi$
$173$	62.2657	+	35.9491i	0.359917	+	0.207798i	−0.309127	+	0.951021i	$0.600037\pi$
$174$		0			0
$175$	2.03745	+	3.52897i	0.0116426	+	0.0201655i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			23.3010i			0.130173i	0.997880	+	0.0650866i	$0.0207324\pi$
$179$			23.3010i			0.130173i	−0.997880	+	0.0650866i	$0.979268\pi$
$180$		0			0
$181$	315.260			1.74177			0.870883	−	0.491490i	$-0.163547\pi$
$181$	315.260			1.74177			0.870883	+	0.491490i	$0.163547\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	139.684	−	80.6466i	0.755049	−	0.435928i
$186$		0			0
$187$	−176.285	+	305.334i	−0.942700	+	1.63280i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	29.2391	+	16.8812i	0.153084	+	0.0883833i	0.574585	−	0.818445i	$-0.305163\pi$
$191$	29.2391	+	16.8812i	0.153084	+	0.0883833i	−0.421501	+	0.906828i	$0.638497\pi$
$192$		0			0
$193$	−45.3269	−	78.5084i	−0.234854	−	0.406779i	0.724376	−	0.689405i	$-0.242128\pi$
$193$	−45.3269	−	78.5084i	−0.234854	−	0.406779i	−0.959230	+	0.282626i	$0.908795\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			133.523i			0.677783i	0.940826	+	0.338891i	$0.110052\pi$
$197$			133.523i			0.677783i	−0.940826	+	0.338891i	$0.889948\pi$
$198$		0			0
$199$	−68.1279			−0.342351			−0.171176	−	0.985241i	$-0.554757\pi$
$199$	−68.1279			−0.342351			−0.171176	+	0.985241i	$0.554757\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	33.5303	−	19.3587i	0.165174	−	0.0953632i
$204$		0			0
$205$	−162.408	+	281.299i	−0.792235	+	1.37219i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−361.942	−	208.967i	−1.73178	−	0.999843i
$210$		0			0
$211$	−194.598	−	337.053i	−0.922264	−	1.59741i	−0.795904	−	0.605423i	$-0.793004\pi$
$211$	−194.598	−	337.053i	−0.922264	−	1.59741i	−0.126360	−	0.991984i	$-0.540329\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	65.5236i		−	0.304761i
$216$		0			0
$217$	−68.9344			−0.317670
$218$		0			0
$219$		0			0
$220$		0			0
$221$	342.270	−	197.610i	1.54873	−	0.894161i
$222$		0			0
$223$	182.010	−	315.250i	0.816188	−	1.41368i	−0.0922839	−	0.995733i	$-0.529417\pi$
$223$	182.010	−	315.250i	0.816188	−	1.41368i	0.908472	−	0.417946i	$-0.137250\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	234.090	+	135.152i	1.03123	+	0.595383i	0.917338	−	0.398110i	$-0.130334\pi$
$227$	234.090	+	135.152i	1.03123	+	0.595383i	0.113896	+	0.993493i	$0.463667\pi$
$228$		0			0
$229$	16.2206	+	28.0949i	0.0708323	+	0.122685i	0.899266	−	0.437402i	$-0.144101\pi$
$229$	16.2206	+	28.0949i	0.0708323	+	0.122685i	−0.828434	+	0.560087i	$0.810768\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	5.02699i		−	0.0215751i	−0.999942	−	0.0107875i	$-0.996566\pi$
$233$		−	5.02699i		−	0.0215751i	0.999942	−	0.0107875i	$-0.00343384\pi$
$234$		0			0
$235$	214.254			0.911718
$236$		0			0
$237$		0			0
$238$		0			0
$239$	97.4122	−	56.2410i	0.407583	−	0.235318i	−0.282168	−	0.959365i	$-0.591054\pi$
$239$	97.4122	−	56.2410i	0.407583	−	0.235318i	0.689750	+	0.724047i	$0.257720\pi$
$240$		0			0
$241$	129.981	−	225.133i	0.539339	−	0.934162i	−0.459601	−	0.888125i	$-0.652008\pi$
$241$	129.981	−	225.133i	0.539339	−	0.934162i	0.998940	−	0.0460364i	$-0.0146590\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−185.942	−	107.354i	−0.758946	−	0.438178i
$246$		0			0
$247$	234.245	+	405.725i	0.948362	+	1.64261i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			41.6594i			0.165974i	0.996551	+	0.0829869i	$0.0264460\pi$
$251$			41.6594i			0.165974i	−0.996551	+	0.0829869i	$0.973554\pi$
$252$		0			0
$253$	143.637			0.567736
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−241.532	+	139.448i	−0.939812	+	0.542601i	−0.889901	−	0.456153i	$-0.849227\pi$
$257$	−241.532	+	139.448i	−0.939812	+	0.542601i	−0.0499107	+	0.998754i	$0.515894\pi$
$258$		0			0
$259$	42.3158	−	73.2931i	0.163382	−	0.282985i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−96.6987	−	55.8290i	−0.367676	−	0.212278i	0.304767	−	0.952427i	$-0.401421\pi$
$263$	−96.6987	−	55.8290i	−0.367676	−	0.212278i	−0.672442	+	0.740149i	$0.734755\pi$
$264$		0			0
$265$	−194.598	−	337.054i	−0.734333	−	1.27190i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			134.254i			0.499087i	0.968364	+	0.249544i	$0.0802806\pi$
$269$			134.254i			0.499087i	−0.968364	+	0.249544i	$0.919719\pi$
$270$		0			0
$271$	−484.716			−1.78862			−0.894311	−	0.447446i	$-0.852334\pi$
$271$	−484.716			−1.78862			−0.894311	+	0.447446i	$0.852334\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	22.2364	−	12.8382i	0.0808598	−	0.0466844i
$276$		0			0
$277$	−34.7966	+	60.2694i	−0.125619	+	0.217579i	−0.921975	−	0.387250i	$-0.873425\pi$
$277$	−34.7966	+	60.2694i	−0.125619	+	0.217579i	0.796355	+	0.604829i	$0.206758\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−190.262	−	109.848i	−0.677087	−	0.390917i	0.121669	−	0.992571i	$-0.461175\pi$
$281$	−190.262	−	109.848i	−0.677087	−	0.390917i	−0.798757	+	0.601654i	$0.794509\pi$
$282$		0			0
$283$	138.132	+	239.251i	0.488099	+	0.845411i	0.999906	−	0.0136886i	$-0.00435734\pi$
$283$	138.132	+	239.251i	0.488099	+	0.845411i	−0.511808	+	0.859100i	$0.671024\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			170.433i			0.593844i
$288$		0			0
$289$	−139.982			−0.484365
$290$		0			0
$291$		0			0
$292$		0			0
$293$	475.539	−	274.553i	1.62300	−	0.937040i	0.636889	−	0.770956i	$-0.280221\pi$
$293$	475.539	−	274.553i	1.62300	−	0.937040i	0.986112	−	0.166084i	$-0.0531123\pi$
$294$		0			0
$295$	−2.43181	+	4.21201i	−0.00824341	+	0.0142780i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−139.441	−	80.5063i	−0.466358	−	0.269252i
$300$		0			0
$301$	−17.1903	−	29.7745i	−0.0571107	−	0.0989187i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			443.186i			1.45307i
$306$		0			0
$307$	−129.673			−0.422387			−0.211193	−	0.977444i	$-0.567735\pi$
$307$	−129.673			−0.422387			−0.211193	+	0.977444i	$0.567735\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	157.804	−	91.1081i	0.507408	−	0.292952i	−0.224360	−	0.974506i	$-0.572029\pi$
$311$	157.804	−	91.1081i	0.507408	−	0.292952i	0.731768	+	0.681554i	$0.238696\pi$
$312$		0			0
$313$	−70.7617	+	122.563i	−0.226076	+	0.391575i	−0.956642	−	0.291267i	$-0.905923\pi$
$313$	−70.7617	+	122.563i	−0.226076	+	0.391575i	0.730566	+	0.682842i	$0.239256\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−115.170	−	66.4936i	−0.363313	−	0.209759i	0.307220	−	0.951639i	$-0.400601\pi$
$317$	−115.170	−	66.4936i	−0.363313	−	0.209759i	−0.670533	+	0.741879i	$0.733935\pi$
$318$		0			0
$319$	−121.982	−	211.278i	−0.382388	−	0.662315i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	508.513i		−	1.57434i
$324$		0			0
$325$	−28.7824			−0.0885614
$326$		0			0
$327$		0			0
$328$		0			0
$329$	97.3589	−	56.2102i	0.295924	−	0.170852i
$330$		0			0
$331$	−135.804	+	235.219i	−0.410283	+	0.710630i	−0.994920	−	0.100664i	$-0.967903\pi$
$331$	−135.804	+	235.219i	−0.410283	+	0.710630i	0.584638	+	0.811294i	$0.301237\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	250.197	+	144.451i	0.746857	+	0.431198i
$336$		0			0
$337$	−72.8100	−	126.111i	−0.216053	−	0.374215i	0.737545	−	0.675298i	$-0.235985\pi$
$337$	−72.8100	−	126.111i	−0.216053	−	0.374215i	−0.953598	+	0.301083i	$0.902652\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			434.364i			1.27379i
$342$		0			0
$343$	−245.033			−0.714381
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−451.582	+	260.721i	−1.30139	+	0.751358i	−0.980642	−	0.195807i	$-0.937267\pi$
$347$	−451.582	+	260.721i	−1.30139	+	0.751358i	−0.320748	+	0.947165i	$0.603934\pi$
$348$		0			0
$349$	54.7294	−	94.7940i	0.156818	−	0.271616i	−0.776902	−	0.629622i	$-0.783210\pi$
$349$	54.7294	−	94.7940i	0.156818	−	0.271616i	0.933719	+	0.358006i	$0.116543\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−125.200	−	72.2844i	−0.354675	−	0.204772i	0.312068	−	0.950060i	$-0.398978\pi$
$353$	−125.200	−	72.2844i	−0.354675	−	0.204772i	−0.666742	+	0.745288i	$0.732312\pi$
$354$		0			0
$355$	1.34140	+	2.32338i	0.00377861	+	0.00654474i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			281.683i			0.784632i	0.919831	+	0.392316i	$0.128326\pi$
$359$			281.683i			0.784632i	−0.919831	+	0.392316i	$0.871674\pi$
$360$		0			0
$361$	241.789			0.669776
$362$		0			0
$363$		0			0
$364$		0			0
$365$	94.9510	−	54.8200i	0.260140	−	0.150192i
$366$		0			0
$367$	60.0387	−	103.990i	0.163593	−	0.283352i	−0.772562	−	0.634940i	$-0.781025\pi$
$367$	60.0387	−	103.990i	0.163593	−	0.283352i	0.936155	+	0.351588i	$0.114358\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−176.855	−	102.107i	−0.476697	−	0.275221i
$372$		0			0
$373$	−7.26251	−	12.5790i	−0.0194705	−	0.0337240i	0.856126	−	0.516767i	$-0.172865\pi$
$373$	−7.26251	−	12.5790i	−0.0194705	−	0.0337240i	−0.875597	+	0.483043i	$0.839531\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			273.475i			0.725397i
$378$		0			0
$379$	126.300			0.333245			0.166622	−	0.986021i	$-0.446714\pi$
$379$	126.300			0.333245			0.166622	+	0.986021i	$0.446714\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	100.614	−	58.0893i	0.262699	−	0.151669i	−0.362866	−	0.931841i	$-0.618202\pi$
$383$	100.614	−	58.0893i	0.262699	−	0.151669i	0.625565	+	0.780172i	$0.284868\pi$
$384$		0			0
$385$	118.385	−	205.048i	0.307492	−	0.532593i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	490.406	+	283.136i	1.26068	+	0.727856i	0.973207	−	0.229932i	$-0.0738503\pi$
$389$	490.406	+	283.136i	1.26068	+	0.727856i	0.287477	+	0.957788i	$0.407184\pi$
$390$		0			0
$391$	87.3838	+	151.353i	0.223488	+	0.387092i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			548.844i			1.38948i
$396$		0			0
$397$	−13.9822			−0.0352198			−0.0176099	−	0.999845i	$-0.505606\pi$
$397$	−13.9822			−0.0352198			−0.0176099	+	0.999845i	$0.505606\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	469.310	−	270.956i	1.17035	−	0.675701i	0.216586	−	0.976263i	$-0.430508\pi$
$401$	469.310	−	270.956i	1.17035	−	0.675701i	0.953762	+	0.300562i	$0.0971744\pi$
$402$		0			0
$403$	243.454	−	421.675i	0.604104	−	1.04634i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−461.829	−	266.637i	−1.13471	−	0.655127i
$408$		0			0
$409$	−251.236	−	435.154i	−0.614270	−	1.06395i	−0.990512	−	0.137425i	$-0.956117\pi$
$409$	−251.236	−	435.154i	−0.614270	−	1.06395i	0.376243	−	0.926521i	$-0.377216\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			2.55197i			0.00617911i
$414$		0			0
$415$	562.447			1.35529
$416$		0			0
$417$		0			0
$418$		0			0
$419$	580.810	−	335.331i	1.38618	−	0.800312i	0.393299	−	0.919411i	$-0.371334\pi$
$419$	580.810	−	335.331i	1.38618	−	0.800312i	0.992882	+	0.119099i	$0.0380005\pi$
$420$		0			0
$421$	−61.1786	+	105.965i	−0.145317	+	0.251697i	−0.929491	−	0.368844i	$-0.879754\pi$
$421$	−61.1786	+	105.965i	−0.145317	+	0.251697i	0.784174	+	0.620541i	$0.213087\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	27.0557	+	15.6206i	0.0636605	+	0.0367544i
$426$		0			0
$427$	116.271	+	201.388i	0.272298	+	0.471635i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	83.3635i		−	0.193419i	−0.995313	−	0.0967094i	$-0.969168\pi$
$431$		−	83.3635i		−	0.193419i	0.995313	−	0.0967094i	$-0.0308318\pi$
$432$		0			0
$433$	−485.126			−1.12038			−0.560191	−	0.828363i	$-0.689272\pi$
$433$	−485.126			−1.12038			−0.560191	+	0.828363i	$0.689272\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−179.413	+	103.584i	−0.410557	+	0.237035i
$438$		0			0
$439$	186.444	−	322.931i	0.424702	−	0.735605i	−0.571691	−	0.820469i	$-0.693712\pi$
$439$	186.444	−	322.931i	0.424702	−	0.735605i	0.996393	+	0.0848640i	$0.0270456\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	325.598	+	187.984i	0.734985	+	0.424344i	0.820243	−	0.572015i	$-0.193838\pi$
$443$	325.598	+	187.984i	0.734985	+	0.424344i	−0.0852580	+	0.996359i	$0.527171\pi$
$444$		0			0
$445$	−5.13114	−	8.88740i	−0.0115307	−	0.0199717i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			389.751i			0.868042i	0.900903	+	0.434021i	$0.142906\pi$
$449$			389.751i			0.868042i	−0.900903	+	0.434021i	$0.857094\pi$
$450$		0			0
$451$	1073.92			2.38120
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−229.852	+	132.705i	−0.505170	+	0.291660i
$456$		0			0
$457$	−115.772	+	200.522i	−0.253329	+	0.438780i	−0.964440	−	0.264300i	$-0.914859\pi$
$457$	−115.772	+	200.522i	−0.253329	+	0.438780i	0.711111	+	0.703080i	$0.248192\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−116.580	−	67.3074i	−0.252885	−	0.146003i	0.368200	−	0.929747i	$-0.379974\pi$
$461$	−116.580	−	67.3074i	−0.252885	−	0.146003i	−0.621084	+	0.783744i	$0.713308\pi$
$462$		0			0
$463$	116.874	+	202.431i	0.252427	+	0.437216i	0.964193	−	0.265200i	$-0.0854379\pi$
$463$	116.874	+	202.431i	0.252427	+	0.437216i	−0.711767	+	0.702416i	$0.752105\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	12.8139i		−	0.0274387i	−0.999906	−	0.0137194i	$-0.995633\pi$
$467$		−	12.8139i		−	0.0274387i	0.999906	−	0.0137194i	$-0.00436714\pi$
$468$		0			0
$469$	151.589			0.323218
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−187.613	+	108.318i	−0.396644	+	0.229003i
$474$		0			0
$475$	−18.5166	+	32.0717i	−0.0389824	+	0.0675194i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	296.584	+	171.233i	0.619174	+	0.357480i	0.776547	−	0.630059i	$-0.216969\pi$
$479$	296.584	+	171.233i	0.619174	+	0.357480i	−0.157373	+	0.987539i	$0.550303\pi$
$480$		0			0
$481$	298.891	+	517.695i	0.621395	+	1.07629i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			295.075i			0.608402i
$486$		0			0
$487$	759.492			1.55953			0.779766	−	0.626071i	$-0.215338\pi$
$487$	759.492			1.55953			0.779766	+	0.626071i	$0.215338\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−806.683	+	465.739i	−1.64294	+	0.948552i	−0.663159	+	0.748479i	$0.730785\pi$
$491$	−806.683	+	465.739i	−1.64294	+	0.948552i	−0.979781	+	0.200073i	$0.935882\pi$
$492$		0			0
$493$	148.419	−	257.068i	0.301052	−	0.521437i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1.21909	+	0.703844i	0.00245290	+	0.00141619i
$498$		0			0
$499$	−213.090	−	369.083i	−0.427034	−	0.739645i	0.569574	−	0.821940i	$-0.307108\pi$
$499$	−213.090	−	369.083i	−0.427034	−	0.739645i	−0.996608	+	0.0822952i	$0.973775\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			489.397i			0.972957i	0.873693	+	0.486478i	$0.161719\pi$
$503$			489.397i			0.972957i	−0.873693	+	0.486478i	$0.838281\pi$
$504$		0			0
$505$	653.913			1.29488
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−420.860	+	242.984i	−0.826837	+	0.477375i	−0.852768	−	0.522289i	$-0.825078\pi$
$509$	−420.860	+	242.984i	−0.826837	+	0.477375i	0.0259314	+	0.999664i	$0.491745\pi$
$510$		0			0
$511$	28.7644	−	49.8214i	0.0562904	−	0.0974978i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	548.148	+	316.473i	1.06436	+	0.614511i
$516$		0			0
$517$	−354.187	−	613.470i	−0.685081	−	1.18660i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	552.405i		−	1.06028i	−0.847911	−	0.530139i	$-0.822140\pi$
$521$		−	552.405i		−	1.06028i	0.847911	−	0.530139i	$-0.177860\pi$
$522$		0			0
$523$	−610.963			−1.16819			−0.584095	−	0.811685i	$-0.698550\pi$
$523$	−610.963			−1.16819			−0.584095	+	0.811685i	$0.698550\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−457.697	+	264.252i	−0.868495	+	0.501426i
$528$		0			0
$529$	−228.900	+	396.466i	−0.432703	+	0.749463i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1042.55	−	601.915i	−1.95600	−	1.12930i
$534$		0			0
$535$	447.447	+	775.002i	0.836350	+	1.44860i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			709.872i			1.31702i
$540$		0			0
$541$	−514.884			−0.951727			−0.475863	−	0.879519i	$-0.657864\pi$
$541$	−514.884			−0.951727			−0.475863	+	0.879519i	$0.657864\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	832.398	−	480.585i	1.52734	−	0.881808i
$546$		0			0
$547$	−218.276	+	378.065i	−0.399042	+	0.691162i	−0.993608	−	0.112885i	$-0.963991\pi$
$547$	−218.276	+	378.065i	−0.399042	+	0.691162i	0.594566	+	0.804047i	$0.297324\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	304.728	+	175.935i	0.553045	+	0.319301i
$552$		0			0
$553$	143.991	+	249.400i	0.260381	+	0.450994i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			294.918i			0.529475i	0.964321	+	0.264738i	$0.0852853\pi$
$557$			294.918i			0.529475i	−0.964321	+	0.264738i	$0.914715\pi$
$558$		0			0
$559$	242.843			0.434423
$560$		0			0
$561$		0			0
$562$		0			0
$563$	82.1450	−	47.4264i	0.145906	−	0.0842387i	−0.425270	−	0.905067i	$-0.639821\pi$
$563$	82.1450	−	47.4264i	0.145906	−	0.0842387i	0.571176	+	0.820828i	$0.306487\pi$
$564$		0			0
$565$	−50.7650	+	87.9275i	−0.0898495	+	0.155624i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	416.600	+	240.524i	0.732161	+	0.422713i	0.819212	−	0.573491i	$-0.194411\pi$
$569$	416.600	+	240.524i	0.732161	+	0.422713i	−0.0870513	+	0.996204i	$0.527744\pi$
$570$		0			0
$571$	267.895	+	464.007i	0.469168	+	0.812622i	0.999379	−	0.0352436i	$-0.0112207\pi$
$571$	267.895	+	464.007i	0.469168	+	0.812622i	−0.530211	+	0.847866i	$0.677887\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	12.7277i		−	0.0221352i
$576$		0			0
$577$	244.105			0.423058			0.211529	−	0.977372i	$-0.432156\pi$
$577$	244.105			0.423058			0.211529	+	0.977372i	$0.432156\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	255.581	−	147.560i	0.439899	−	0.253976i
$582$		0			0
$583$	−643.388	+	1114.38i	−1.10358	+	1.91146i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−278.748	−	160.935i	−0.474868	−	0.274165i	0.243407	−	0.969924i	$-0.421735\pi$
$587$	−278.748	−	160.935i	−0.474868	−	0.274165i	−0.718276	+	0.695759i	$0.755068\pi$
$588$		0			0
$589$	−313.243	−	542.552i	−0.531821	−	0.921141i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			467.495i			0.788355i	0.919034	+	0.394178i	$0.128971\pi$
$593$			467.495i			0.788355i	−0.919034	+	0.394178i	$0.871029\pi$
$594$		0			0
$595$	288.084			0.484175
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−605.536	+	349.607i	−1.01091	+	0.583650i	−0.911459	−	0.411392i	$-0.865043\pi$
$599$	−605.536	+	349.607i	−1.01091	+	0.583650i	−0.0994536	+	0.995042i	$0.531709\pi$
$600$		0			0
$601$	185.246	−	320.855i	0.308229	−	0.533868i	−0.669746	−	0.742590i	$-0.733597\pi$
$601$	185.246	−	320.855i	0.308229	−	0.533868i	0.977975	+	0.208722i	$0.0669304\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−752.512	−	434.463i	−1.24382	−	0.718121i
$606$		0			0
$607$	−484.541	−	839.250i	−0.798256	−	1.38262i	−0.920751	−	0.390151i	$-0.872423\pi$
$607$	−484.541	−	839.250i	−0.798256	−	1.38262i	0.122495	−	0.992469i	$-0.460910\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			794.064i			1.29961i
$612$		0			0
$613$	−575.128			−0.938219			−0.469109	−	0.883140i	$-0.655425\pi$
$613$	−575.128			−0.938219			−0.469109	+	0.883140i	$0.655425\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−243.619	+	140.653i	−0.394844	+	0.227963i	−0.684257	−	0.729241i	$-0.739873\pi$
$617$	−243.619	+	140.653i	−0.394844	+	0.227963i	0.289413	+	0.957204i	$0.406540\pi$
$618$		0			0
$619$	−99.7624	+	172.793i	−0.161167	+	0.279149i	−0.935287	−	0.353889i	$-0.884859\pi$
$619$	−99.7624	+	172.793i	−0.161167	+	0.279149i	0.774121	+	0.633038i	$0.218192\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−4.66328	−	2.69234i	−0.00748520	−	0.00432158i
$624$		0			0
$625$	330.217	+	571.953i	0.528347	+	0.915124i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	648.849i		−	1.03156i
$630$		0			0
$631$	262.288			0.415670			0.207835	−	0.978164i	$-0.433358\pi$
$631$	262.288			0.415670			0.207835	+	0.978164i	$0.433358\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	732.228	−	422.752i	1.15311	−	0.665751i
$636$		0			0
$637$	397.872	−	689.134i	0.624603	−	1.08184i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−467.214	−	269.746i	−0.728883	−	0.420821i	0.0891306	−	0.996020i	$-0.471591\pi$
$641$	−467.214	−	269.746i	−0.728883	−	0.420821i	−0.818013	+	0.575199i	$0.804924\pi$
$642$		0			0
$643$	362.961	+	628.667i	0.564480	+	0.977709i	0.997098	+	0.0761311i	$0.0242568\pi$
$643$	362.961	+	628.667i	0.564480	+	0.977709i	−0.432617	+	0.901578i	$0.642410\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			564.865i			0.873053i	0.899691	+	0.436526i	$0.143791\pi$
$647$			564.865i			0.873053i	−0.899691	+	0.436526i	$0.856209\pi$
$648$		0			0
$649$	16.0803			0.0247770
$650$		0			0
$651$		0			0
$652$		0			0
$653$	642.332	−	370.850i	0.983663	−	0.567918i	0.0802887	−	0.996772i	$-0.474416\pi$
$653$	642.332	−	370.850i	0.983663	−	0.567918i	0.903374	+	0.428854i	$0.141082\pi$
$654$		0			0
$655$	−331.341	+	573.899i	−0.505864	+	0.876182i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	72.0489	+	41.5974i	0.109331	+	0.0631221i	0.553668	−	0.832737i	$-0.313228\pi$
$659$	72.0489	+	41.5974i	0.109331	+	0.0631221i	−0.444338	+	0.895859i	$0.646561\pi$
$660$		0			0
$661$	381.186	+	660.233i	0.576680	+	0.998839i	0.995857	+	0.0909346i	$0.0289854\pi$
$661$	381.186	+	660.233i	0.576680	+	0.998839i	−0.419177	+	0.907905i	$0.637681\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			341.493i			0.513524i
$666$		0			0
$667$	−120.932			−0.181307
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1268.97	−	732.640i	1.89116	−	1.09186i
$672$		0			0
$673$	−167.260	+	289.702i	−0.248529	+	0.430464i	−0.963118	−	0.269080i	$-0.913280\pi$
$673$	−167.260	+	289.702i	−0.248529	+	0.430464i	0.714589	+	0.699544i	$0.246614\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−473.678	−	273.478i	−0.699672	−	0.403956i	0.107553	−	0.994199i	$-0.465698\pi$
$677$	−473.678	−	273.478i	−0.699672	−	0.403956i	−0.807225	+	0.590243i	$0.799032\pi$
$678$		0			0
$679$	77.4139	+	134.085i	0.114012	+	0.197474i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			597.583i			0.874938i	0.899233	+	0.437469i	$0.144125\pi$
$683$			597.583i			0.874938i	−0.899233	+	0.437469i	$0.855875\pi$
$684$		0			0
$685$	340.652			0.497302
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1249.18	−	721.217i	1.81304	−	1.04676i
$690$		0			0
$691$	460.492	−	797.595i	0.666414	−	1.15426i	−0.312486	−	0.949922i	$-0.601162\pi$
$691$	460.492	−	797.595i	0.666414	−	1.15426i	0.978900	−	0.204340i	$-0.0655048\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1234.24	+	712.590i	1.77589	+	1.02531i
$696$		0			0
$697$	653.335	+	1131.61i	0.937353	+	1.62354i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			1008.60i			1.43880i	0.694595	+	0.719401i	$0.255584\pi$
$701$			1008.60i			1.43880i	−0.694595	+	0.719401i	$0.744416\pi$
$702$		0			0
$703$	769.143			1.09409
$704$		0			0
$705$		0			0
$706$		0			0
$707$	297.144	−	171.556i	0.420289	−	0.242654i
$708$		0			0
$709$	−352.213	+	610.052i	−0.496775	+	0.860439i	−0.999993	−	0.00371995i	$-0.998816\pi$
$709$	−352.213	+	610.052i	−0.496775	+	0.860439i	0.503218	+	0.864159i	$0.332149\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	186.466	+	107.656i	0.261524	+	0.150991i
$714$		0			0
$715$	836.191	+	1448.33i	1.16950	+	2.02563i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			185.398i			0.257855i	0.991654	+	0.128928i	$0.0411535\pi$
$719$			185.398i			0.257855i	−0.991654	+	0.128928i	$0.958846\pi$
$720$		0			0
$721$	332.111			0.460626
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−18.7214	+	10.8088i	−0.0258226	+	0.0149087i
$726$		0			0
$727$	182.447	−	316.008i	0.250959	−	0.434673i	−0.712831	−	0.701336i	$-0.752587\pi$
$727$	182.447	−	316.008i	0.250959	−	0.434673i	0.963790	+	0.266662i	$0.0859208\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−228.274	−	131.794i	−0.312276	−	0.180293i
$732$		0			0
$733$	460.539	+	797.676i	0.628293	+	1.08823i	0.987894	+	0.155129i	$0.0495793\pi$
$733$	460.539	+	797.676i	0.628293	+	1.08823i	−0.359602	+	0.933106i	$0.617087\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	955.181i		−	1.29604i
$738$		0			0
$739$	−379.356			−0.513338			−0.256669	−	0.966499i	$-0.582625\pi$
$739$	−379.356			−0.513338			−0.256669	+	0.966499i	$0.582625\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−844.830	+	487.763i	−1.13705	+	0.656478i	−0.945699	−	0.325044i	$-0.894621\pi$
$743$	−844.830	+	487.763i	−1.13705	+	0.656478i	−0.191353	+	0.981521i	$0.561288\pi$
$744$		0			0
$745$	128.177	−	222.009i	0.172050	−	0.297999i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	406.648	+	234.779i	0.542922	+	0.313456i
$750$		0			0
$751$	−478.108	−	828.107i	−0.636628	−	1.10267i	−0.986168	−	0.165751i	$-0.946995\pi$
$751$	−478.108	−	828.107i	−0.636628	−	1.10267i	0.349539	−	0.936922i	$-0.386338\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	564.648i		−	0.747878i
$756$		0			0
$757$	453.529			0.599113			0.299557	−	0.954078i	$-0.403161\pi$
$757$	453.529			0.599113			0.299557	+	0.954078i	$0.403161\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	532.418	−	307.392i	0.699630	−	0.403931i	−0.107580	−	0.994196i	$-0.534310\pi$
$761$	532.418	−	307.392i	0.699630	−	0.403931i	0.807210	+	0.590265i	$0.200977\pi$
$762$		0			0
$763$	252.166	−	436.765i	0.330493	−	0.572431i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−15.6105	−	9.01272i	−0.0203527	−	0.0117506i
$768$		0			0
$769$	112.434	+	194.741i	0.146208	+	0.253239i	0.929823	−	0.368007i	$-0.119960\pi$
$769$	112.434	+	194.741i	0.146208	+	0.253239i	−0.783615	+	0.621246i	$0.786627\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			102.096i			0.132078i	0.997817	+	0.0660388i	$0.0210361\pi$
$773$			102.096i			0.132078i	−0.997817	+	0.0660388i	$0.978964\pi$
$774$		0			0
$775$	38.4891			0.0496633
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1341.40	+	774.460i	−1.72196	+	0.994172i
$780$		0			0
$781$	4.43500	−	7.68165i	0.00567862	−	0.00983566i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1209.07	−	698.054i	−1.54021	−	0.889241i
$786$		0			0
$787$	−296.213	−	513.056i	−0.376383	−	0.651914i	0.614150	−	0.789189i	$-0.289499\pi$
$787$	−296.213	−	513.056i	−0.376383	−	0.651914i	−0.990533	+	0.137275i	$0.956165\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			53.2734i			0.0673494i
$792$		0			0
$793$	−1642.53			−2.07129
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1264.63	+	730.134i	−1.58674	+	0.916102i	−0.592896	+	0.805279i	$0.702015\pi$
$797$	−1264.63	+	730.134i	−1.58674	+	0.916102i	−0.993840	+	0.110823i	$0.964651\pi$
$798$		0			0
$799$	430.949	−	746.426i	0.539361	−	0.934201i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−313.930	−	181.248i	−0.390947	−	0.225713i
$804$		0			0
$805$	−58.6828	−	101.642i	−0.0728979	−	0.126263i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			680.689i			0.841396i	0.907201	+	0.420698i	$0.138215\pi$
$809$			680.689i			0.841396i	−0.907201	+	0.420698i	$0.861785\pi$
$810$		0			0
$811$	441.852			0.544823			0.272412	−	0.962181i	$-0.412179\pi$
$811$	441.852			0.544823			0.272412	+	0.962181i	$0.412179\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	439.324	−	253.644i	0.539048	−	0.311219i
$816$		0			0
$817$	156.228	−	270.595i	0.191222	−	0.331205i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−348.996	−	201.493i	−0.425086	−	0.245424i	0.272165	−	0.962251i	$-0.412260\pi$
$821$	−348.996	−	201.493i	−0.425086	−	0.245424i	−0.697251	+	0.716827i	$0.745594\pi$
$822$		0			0
$823$	165.140	+	286.031i	0.200656	+	0.347547i	0.948740	−	0.316057i	$-0.102359\pi$
$823$	165.140	+	286.031i	0.200656	+	0.347547i	−0.748084	+	0.663604i	$0.769026\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	949.014i		−	1.14754i	−0.819017	−	0.573769i	$-0.805481\pi$
$827$		−	949.014i		−	1.14754i	0.819017	−	0.573769i	$-0.194519\pi$
$828$		0			0
$829$	−239.894			−0.289377			−0.144689	−	0.989477i	$-0.546218\pi$
$829$	−239.894			−0.289377			−0.144689	+	0.989477i	$0.546218\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−748.005	+	431.861i	−0.897966	+	0.518441i
$834$		0			0
$835$	526.546	−	912.004i	0.630594	−	1.09222i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−1382.65	−	798.271i	−1.64797	−	0.951455i	−0.977878	−	0.209175i	$-0.932922\pi$
$839$	−1382.65	−	798.271i	−1.64797	−	0.951455i	−0.670090	−	0.742280i	$-0.733744\pi$
$840$		0			0
$841$	−317.801	−	550.447i	−0.377884	−	0.654515i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	1004.57i		−	1.18884i
$846$		0			0
$847$	−455.931			−0.538290
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−228.927	+	132.171i	−0.269009	+	0.155313i
$852$		0			0
$853$	540.685	−	936.494i	0.633863	−	1.09788i	−0.352892	−	0.935664i	$-0.614802\pi$
$853$	540.685	−	936.494i	0.633863	−	1.09788i	0.986755	−	0.162218i	$-0.0518649\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−659.652	−	380.850i	−0.769722	−	0.444399i	0.0630533	−	0.998010i	$-0.479916\pi$
$857$	−659.652	−	380.850i	−0.769722	−	0.444399i	−0.832775	+	0.553611i	$0.813250\pi$
$858$		0			0
$859$	307.213	+	532.109i	0.357640	+	0.619451i	0.987566	−	0.157204i	$-0.0502480\pi$
$859$	307.213	+	532.109i	0.357640	+	0.619451i	−0.629926	+	0.776655i	$0.716915\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			394.879i			0.457566i	0.973477	+	0.228783i	$0.0734746\pi$
$863$			394.879i			0.457566i	−0.973477	+	0.228783i	$0.926525\pi$
$864$		0			0
$865$	−370.177			−0.427951
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1571.50	−	907.304i	1.80840	−	1.04408i
$870$		0			0
$871$	−535.363	+	927.276i	−0.614653	+	1.06461i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	282.972	+	163.374i	0.323396	+	0.186713i
$876$		0			0
$877$	199.542	+	345.618i	0.227528	+	0.394091i	0.957075	−	0.289840i	$-0.0936022\pi$
$877$	199.542	+	345.618i	0.227528	+	0.394091i	−0.729547	+	0.683931i	$0.760269\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			874.437i			0.992550i	0.868165	+	0.496275i	$0.165299\pi$
$881$			874.437i			0.992550i	−0.868165	+	0.496275i	$0.834701\pi$
$882$		0			0
$883$	−1565.39			−1.77280			−0.886402	−	0.462917i	$-0.846803\pi$
$883$	−1565.39			−1.77280			−0.886402	+	0.462917i	$0.846803\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−459.193	+	265.115i	−0.517692	+	0.298889i	−0.735990	−	0.676993i	$-0.763283\pi$
$887$	−459.193	+	265.115i	−0.517692	+	0.298889i	0.218298	+	0.975882i	$0.429950\pi$
$888$		0			0
$889$	221.821	−	384.205i	0.249517	−	0.432176i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	884.810	+	510.846i	0.990829	+	0.572055i
$894$		0			0
$895$	−59.9841	−	103.896i	−0.0670214	−	0.116084i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	365.702i		−	0.406787i
$900$		0			0
$901$	−1565.66			−1.73769
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1405.69	+	811.578i	−1.55325	+	0.896771i
$906$		0			0
$907$	145.559	−	252.116i	0.160484	−	0.277967i	−0.774558	−	0.632503i	$-0.782028\pi$
$907$	145.559	−	252.116i	0.160484	−	0.277967i	0.935042	+	0.354536i	$0.115361\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−425.207	−	245.493i	−0.466747	−	0.269477i	0.248130	−	0.968727i	$-0.420184\pi$
$911$	−425.207	−	245.493i	−0.466747	−	0.269477i	−0.714877	+	0.699250i	$0.753517\pi$
$912$		0			0
$913$	−929.792	−	1610.45i	−1.01839	−	1.76391i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			347.713i			0.379186i
$918$		0			0
$919$	−186.789			−0.203253			−0.101626	−	0.994823i	$-0.532405\pi$
$919$	−186.789			−0.203253			−0.101626	+	0.994823i	$0.532405\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−8.61088	+	4.97149i	−0.00932923	+	0.00538623i
$924$		0			0
$925$	−23.6267	+	40.9227i	−0.0255424	+	0.0442408i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	916.609	+	529.204i	0.986662	+	0.569650i	0.904275	−	0.426951i	$-0.140412\pi$
$929$	916.609	+	529.204i	0.986662	+	0.569650i	0.0823871	+	0.996600i	$0.473746\pi$
$930$		0			0
$931$	−511.926	−	886.682i	−0.549867	−	0.952398i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	1815.25i		−	1.94144i
$936$		0			0
$937$	1154.84			1.23248			0.616241	−	0.787558i	$-0.288655\pi$
$937$	1154.84			1.23248			0.616241	+	0.787558i	$0.288655\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−521.534	+	301.108i	−0.554234	+	0.319987i	−0.750828	−	0.660498i	$-0.770345\pi$
$941$	−521.534	+	301.108i	−0.554234	+	0.319987i	0.196594	+	0.980485i	$0.437012\pi$
$942$		0			0
$943$	266.169	−	461.019i	0.282258	−	0.488885i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	964.044	+	556.591i	1.01800	+	0.587742i	0.913524	−	0.406786i	$-0.133351\pi$
$947$	964.044	+	556.591i	1.01800	+	0.587742i	0.104475	+	0.994528i	$0.466684\pi$
$948$		0			0
$949$	203.173	+	351.906i	0.214092	+	0.370817i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	188.454i		−	0.197749i	−0.995100	−	0.0988743i	$-0.968476\pi$
$953$		−	188.454i		−	0.197749i	0.995100	−	0.0988743i	$-0.0315242\pi$
$954$		0			0
$955$	−173.830			−0.182021
$956$		0			0
$957$		0			0
$958$		0			0
$959$	154.795	−	89.3712i	0.161413	−	0.0931921i
$960$		0			0
$961$	154.943	−	268.370i	0.161231	−	0.279261i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	404.211	+	233.371i	0.418871	+	0.241835i
$966$		0			0
$967$	−284.943	−	493.536i	−0.294667	−	0.510379i	0.680240	−	0.732989i	$-0.261875\pi$
$967$	−284.943	−	493.536i	−0.294667	−	0.510379i	−0.974907	+	0.222611i	$0.928542\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1376.53i		−	1.41764i	−0.705391	−	0.708818i	$-0.749229\pi$
$971$		−	1376.53i		−	1.41764i	0.705391	−	0.708818i	$-0.250771\pi$
$972$		0			0
$973$	747.802			0.768552
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−273.790	+	158.073i	−0.280235	+	0.161794i	−0.633530	−	0.773718i	$-0.718395\pi$
$977$	−273.790	+	158.073i	−0.280235	+	0.161794i	0.353295	+	0.935512i	$0.385061\pi$
$978$		0			0
$979$	−16.9648	+	29.3839i	−0.0173287	+	0.0300142i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1113.02	−	642.604i	−1.13227	−	0.653717i	−0.187766	−	0.982214i	$-0.560125\pi$
$983$	−1113.02	−	642.604i	−1.13227	−	0.653717i	−0.944505	+	0.328496i	$0.893458\pi$
$984$		0			0
$985$	−343.731	−	595.359i	−0.348965	−	0.604425i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			107.386i			0.108580i
$990$		0			0
$991$	1743.97			1.75981			0.879903	−	0.475153i	$-0.157607\pi$
$991$	1743.97			1.75981			0.879903	+	0.475153i	$0.157607\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	303.771	−	175.382i	0.305298	−	0.176264i
$996$		0			0
$997$	−781.754	+	1354.04i	−0.784107	+	1.35811i	0.145425	+	0.989369i	$0.453545\pi$
$997$	−781.754	+	1354.04i	−0.784107	+	1.35811i	−0.929531	+	0.368743i	$0.879788\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.3.q.l.1601.3		24
3.2	odd	2		576.3.q.k.65.7		24
4.3	odd	2	inner	1728.3.q.l.1601.4		24
8.3	odd	2		864.3.q.b.737.9		24
8.5	even	2		864.3.q.b.737.10		24
9.4	even	3		576.3.q.k.257.7		24
9.5	odd	6	inner	1728.3.q.l.449.3		24
12.11	even	2		576.3.q.k.65.6		24
24.5	odd	2		288.3.q.a.65.6	✓	24
24.11	even	2		288.3.q.a.65.7	yes	24
36.23	even	6	inner	1728.3.q.l.449.4		24
36.31	odd	6		576.3.q.k.257.6		24
72.5	odd	6		864.3.q.b.449.10		24
72.11	even	6		2592.3.e.j.161.7		24
72.13	even	6		288.3.q.a.257.6	yes	24
72.29	odd	6		2592.3.e.j.161.17		24
72.43	odd	6		2592.3.e.j.161.8		24
72.59	even	6		864.3.q.b.449.9		24
72.61	even	6		2592.3.e.j.161.18		24
72.67	odd	6		288.3.q.a.257.7	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
288.3.q.a.65.6	✓	24	24.5	odd	2
288.3.q.a.65.7	yes	24	24.11	even	2
288.3.q.a.257.6	yes	24	72.13	even	6
288.3.q.a.257.7	yes	24	72.67	odd	6
576.3.q.k.65.6		24	12.11	even	2
576.3.q.k.65.7		24	3.2	odd	2
576.3.q.k.257.6		24	36.31	odd	6
576.3.q.k.257.7		24	9.4	even	3
864.3.q.b.449.9		24	72.59	even	6
864.3.q.b.449.10		24	72.5	odd	6
864.3.q.b.737.9		24	8.3	odd	2
864.3.q.b.737.10		24	8.5	even	2
1728.3.q.l.449.3		24	9.5	odd	6	inner
1728.3.q.l.449.4		24	36.23	even	6	inner
1728.3.q.l.1601.3		24	1.1	even	1	trivial
1728.3.q.l.1601.4		24	4.3	odd	2	inner
2592.3.e.j.161.7		24	72.11	even	6
2592.3.e.j.161.8		24	72.43	odd	6
2592.3.e.j.161.17		24	72.29	odd	6
2592.3.e.j.161.18		24	72.61	even	6

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:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1728.q (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-4.45884 + 2.57431i) q^{5} +(-1.35076 + 2.33958i) q^{7} +O(q^{10})\)	\(q+(-4.45884 + 2.57431i) q^{5} +(-1.35076 + 2.33958i) q^{7} +(14.7420 + 8.51129i) q^{11} +(-9.54088 - 16.5253i) q^{13} +20.7119i q^{17} -24.5518 q^{19} +(7.30755 - 4.21902i) q^{23} +(0.754187 - 1.30629i) q^{25} +(-12.4116 - 7.16587i) q^{29} +(12.7585 + 22.0983i) q^{31} -13.9091i q^{35} -31.3274 q^{37} +(54.6358 - 31.5440i) q^{41} +(-6.36321 + 11.0214i) q^{43} +(-36.0386 - 20.8069i) q^{47} +(20.8509 + 36.1148i) q^{49} +75.5923i q^{53} -87.6430 q^{55} +(0.818085 - 0.472321i) q^{59} +(43.0393 - 74.5462i) q^{61} +(85.0826 + 49.1225i) q^{65} +(-28.0563 - 48.5949i) q^{67} -0.521073i q^{71} -21.2950 q^{73} +(-39.8258 + 22.9934i) q^{77} +(53.3000 - 92.3183i) q^{79} +(-94.6065 - 54.6211i) q^{83} +(-53.3189 - 92.3510i) q^{85} +1.99321i q^{89} +51.5498 q^{91} +(109.472 - 63.2039i) q^{95} +(28.6557 - 49.6331i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q + 60 q^{25} + 72 q^{29} + 36 q^{41} - 132 q^{49} - 96 q^{61} - 576 q^{65} + 24 q^{73} + 432 q^{77} + 96 q^{85} + 252 q^{97}+O(q^{100}) \) 24 * q + 60 * q^25 + 72 * q^29 + 36 * q^41 - 132 * q^49 - 96 * q^61 - 576 * q^65 + 24 * q^73 + 432 * q^77 + 96 * q^85 + 252 * q^97

Introduction

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