LMFDB - Embedded newform 1680.2.dx.a.31.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1680,2,Mod(31,1680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1680, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1680.31"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1680.dx (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,-2,0,0,0,0,0,-2,0,-6,0,0,0,0,0,18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$13.4148675396$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			31.2
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1680.31
Dual form			1680.2.dx.a.271.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.500000 + 0.866025i) q^{3} +(0.866025 - 0.500000i) q^{5} +(-2.59808 - 0.500000i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(1.09808 + 0.633975i) q^{11} +1.00000i q^{13} +1.00000i q^{15} +(3.63397 + 2.09808i) q^{17} +(-1.23205 - 2.13397i) q^{19} +(1.73205 - 2.00000i) q^{21} +(-2.36603 + 1.36603i) q^{23} +(0.500000 - 0.866025i) q^{25} +1.00000 q^{27} +2.73205 q^{29} +(-3.23205 + 5.59808i) q^{31} +(-1.09808 + 0.633975i) q^{33} +(-2.50000 + 0.866025i) q^{35} +(4.13397 + 7.16025i) q^{37} +(-0.866025 - 0.500000i) q^{39} -4.19615i q^{41} -1.53590i q^{43} +(-0.866025 - 0.500000i) q^{45} +(6.46410 + 11.1962i) q^{47} +(6.50000 + 2.59808i) q^{49} +(-3.63397 + 2.09808i) q^{51} +(-2.73205 + 4.73205i) q^{53} +1.26795 q^{55} +2.46410 q^{57} +(-1.90192 + 3.29423i) q^{59} +(-3.92820 + 2.26795i) q^{61} +(0.866025 + 2.50000i) q^{63} +(0.500000 + 0.866025i) q^{65} +(4.79423 + 2.76795i) q^{67} -2.73205i q^{69} +1.66025i q^{71} +(6.86603 + 3.96410i) q^{73} +(0.500000 + 0.866025i) q^{75} +(-2.53590 - 2.19615i) q^{77} +(3.69615 - 2.13397i) q^{79} +(-0.500000 + 0.866025i) q^{81} -5.26795 q^{83} +4.19615 q^{85} +(-1.36603 + 2.36603i) q^{87} +(-4.09808 + 2.36603i) q^{89} +(0.500000 - 2.59808i) q^{91} +(-3.23205 - 5.59808i) q^{93} +(-2.13397 - 1.23205i) q^{95} +4.92820i q^{97} -1.26795i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} - 2 q^{9} - 6 q^{11} + 18 q^{17} + 2 q^{19} - 6 q^{23} + 2 q^{25} + 4 q^{27} + 4 q^{29} - 6 q^{31} + 6 q^{33} - 10 q^{35} + 20 q^{37} + 12 q^{47} + 26 q^{49} - 18 q^{51} - 4 q^{53} + 12 q^{55}+ \cdots - 12 q^{95}+O(q^{100}) $ 4 * q - 2 * q^3 - 2 * q^9 - 6 * q^11 + 18 * q^17 + 2 * q^19 - 6 * q^23 + 2 * q^25 + 4 * q^27 + 4 * q^29 - 6 * q^31 + 6 * q^33 - 10 * q^35 + 20 * q^37 + 12 * q^47 + 26 * q^49 - 18 * q^51 - 4 * q^53 + 12 * q^55 - 4 * q^57 - 18 * q^59 + 12 * q^61 + 2 * q^65 - 12 * q^67 + 24 * q^73 + 2 * q^75 - 24 * q^77 - 6 * q^79 - 2 * q^81 - 28 * q^83 - 4 * q^85 - 2 * q^87 - 6 * q^89 + 2 * q^91 - 6 * q^93 - 12 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$421$	$1121$	$1471$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$1$	$1$	$1$	$-1$

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
$3$	−0.500000	+	0.866025i	−0.288675	+	0.500000i
$3$	−0.500000	+	0.866025i	−0.288675	+	0.500000i
$4$		0			0
$5$	0.866025	−	0.500000i	0.387298	−	0.223607i
$5$	0.866025	−	0.500000i	0.387298	−	0.223607i
$6$		0			0
$7$	−2.59808	−	0.500000i	−0.981981	−	0.188982i
$7$	−2.59808	−	0.500000i	−0.981981	−	0.188982i
$8$		0			0
$9$	−0.500000	−	0.866025i	−0.166667	−	0.288675i
$10$		0			0
$11$	1.09808	+	0.633975i	0.331082	+	0.191151i	0.656322	−	0.754481i	$-0.272111\pi$
$11$	1.09808	+	0.633975i	0.331082	+	0.191151i	−0.325239	+	0.945632i	$0.605445\pi$
$12$		0			0
$13$			1.00000i			0.277350i	0.990338	+	0.138675i	$0.0442844\pi$
$13$			1.00000i			0.277350i	−0.990338	+	0.138675i	$0.955716\pi$
$14$		0			0
$15$			1.00000i			0.258199i
$16$		0			0
$17$	3.63397	+	2.09808i	0.881368	+	0.508858i	0.871109	−	0.491089i	$-0.163401\pi$
$17$	3.63397	+	2.09808i	0.881368	+	0.508858i	0.0102590	+	0.999947i	$0.496734\pi$
$18$		0			0
$19$	−1.23205	−	2.13397i	−0.282652	−	0.489567i	0.689385	−	0.724395i	$-0.257881\pi$
$19$	−1.23205	−	2.13397i	−0.282652	−	0.489567i	−0.972037	+	0.234828i	$0.924547\pi$
$20$		0			0
$21$	1.73205	−	2.00000i	0.377964	−	0.436436i
$22$		0			0
$23$	−2.36603	+	1.36603i	−0.493350	+	0.284836i	−0.725963	−	0.687733i	$-0.758606\pi$
$23$	−2.36603	+	1.36603i	−0.493350	+	0.284836i	0.232613	+	0.972569i	$0.425272\pi$
$24$		0			0
$25$	0.500000	−	0.866025i	0.100000	−	0.173205i
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	2.73205			0.507329			0.253665	−	0.967292i	$-0.418364\pi$
$29$	2.73205			0.507329			0.253665	+	0.967292i	$0.418364\pi$
$30$		0			0
$31$	−3.23205	+	5.59808i	−0.580493	+	1.00544i	0.414927	+	0.909855i	$0.363807\pi$
$31$	−3.23205	+	5.59808i	−0.580493	+	1.00544i	−0.995421	+	0.0955896i	$0.969526\pi$
$32$		0			0
$33$	−1.09808	+	0.633975i	−0.191151	+	0.110361i
$34$		0			0
$35$	−2.50000	+	0.866025i	−0.422577	+	0.146385i
$36$		0			0
$37$	4.13397	+	7.16025i	0.679621	+	1.17714i	0.975095	+	0.221788i	$0.0711892\pi$
$37$	4.13397	+	7.16025i	0.679621	+	1.17714i	−0.295474	+	0.955351i	$0.595477\pi$
$38$		0			0
$39$	−0.866025	−	0.500000i	−0.138675	−	0.0800641i
$40$		0			0
$41$		−	4.19615i		−	0.655329i	−0.944794	−	0.327664i	$-0.893738\pi$
$41$		−	4.19615i		−	0.655329i	0.944794	−	0.327664i	$-0.106262\pi$
$42$		0			0
$43$		−	1.53590i		−	0.234222i	−0.993119	−	0.117111i	$-0.962637\pi$
$43$		−	1.53590i		−	0.234222i	0.993119	−	0.117111i	$-0.0373634\pi$
$44$		0			0
$45$	−0.866025	−	0.500000i	−0.129099	−	0.0745356i
$46$		0			0
$47$	6.46410	+	11.1962i	0.942886	+	1.63313i	0.759929	+	0.650006i	$0.225234\pi$
$47$	6.46410	+	11.1962i	0.942886	+	1.63313i	0.182957	+	0.983121i	$0.441433\pi$
$48$		0			0
$49$	6.50000	+	2.59808i	0.928571	+	0.371154i
$50$		0			0
$51$	−3.63397	+	2.09808i	−0.508858	+	0.293789i
$52$		0			0
$53$	−2.73205	+	4.73205i	−0.375276	+	0.649997i	−0.990368	−	0.138458i	$-0.955785\pi$
$53$	−2.73205	+	4.73205i	−0.375276	+	0.649997i	0.615092	+	0.788455i	$0.289119\pi$
$54$		0			0
$55$	1.26795			0.170970
$56$		0			0
$57$	2.46410			0.326378
$58$		0			0
$59$	−1.90192	+	3.29423i	−0.247609	+	0.428872i	−0.962862	−	0.269994i	$-0.912978\pi$
$59$	−1.90192	+	3.29423i	−0.247609	+	0.428872i	0.715253	+	0.698866i	$0.246312\pi$
$60$		0			0
$61$	−3.92820	+	2.26795i	−0.502955	+	0.290381i	−0.729933	−	0.683519i	$-0.760449\pi$
$61$	−3.92820	+	2.26795i	−0.502955	+	0.290381i	0.226978	+	0.973900i	$0.427115\pi$
$62$		0			0
$63$	0.866025	+	2.50000i	0.109109	+	0.314970i
$64$		0			0
$65$	0.500000	+	0.866025i	0.0620174	+	0.107417i
$66$		0			0
$67$	4.79423	+	2.76795i	0.585708	+	0.338159i	0.763399	−	0.645928i	$-0.223529\pi$
$67$	4.79423	+	2.76795i	0.585708	+	0.338159i	−0.177690	+	0.984086i	$0.556863\pi$
$68$		0			0
$69$		−	2.73205i		−	0.328900i
$70$		0			0
$71$			1.66025i			0.197036i	0.995135	+	0.0985180i	$0.0314102\pi$
$71$			1.66025i			0.197036i	−0.995135	+	0.0985180i	$0.968590\pi$
$72$		0			0
$73$	6.86603	+	3.96410i	0.803607	+	0.463963i	0.844731	−	0.535191i	$-0.179760\pi$
$73$	6.86603	+	3.96410i	0.803607	+	0.463963i	−0.0411235	+	0.999154i	$0.513094\pi$
$74$		0			0
$75$	0.500000	+	0.866025i	0.0577350	+	0.100000i
$76$		0			0
$77$	−2.53590	−	2.19615i	−0.288992	−	0.250275i
$78$		0			0
$79$	3.69615	−	2.13397i	0.415850	−	0.240091i	−0.277450	−	0.960740i	$-0.589489\pi$
$79$	3.69615	−	2.13397i	0.415850	−	0.240091i	0.693300	+	0.720649i	$0.256156\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	−5.26795			−0.578233			−0.289116	−	0.957294i	$-0.593361\pi$
$83$	−5.26795			−0.578233			−0.289116	+	0.957294i	$0.593361\pi$
$84$		0			0
$85$	4.19615			0.455137
$86$		0			0
$87$	−1.36603	+	2.36603i	−0.146453	+	0.253665i
$88$		0			0
$89$	−4.09808	+	2.36603i	−0.434395	+	0.250798i	−0.701217	−	0.712948i	$-0.747360\pi$
$89$	−4.09808	+	2.36603i	−0.434395	+	0.250798i	0.266822	+	0.963746i	$0.414026\pi$
$90$		0			0
$91$	0.500000	−	2.59808i	0.0524142	−	0.272352i
$92$		0			0
$93$	−3.23205	−	5.59808i	−0.335148	−	0.580493i
$94$		0			0
$95$	−2.13397	−	1.23205i	−0.218941	−	0.126406i
$96$		0			0
$97$			4.92820i			0.500383i	0.968196	+	0.250192i	$0.0804936\pi$
$97$			4.92820i			0.500383i	−0.968196	+	0.250192i	$0.919506\pi$
$98$		0			0
$99$		−	1.26795i		−	0.127434i
$100$		0			0
$101$	4.56218	+	2.63397i	0.453954	+	0.262090i	0.709499	−	0.704707i	$-0.248922\pi$
$101$	4.56218	+	2.63397i	0.453954	+	0.262090i	−0.255545	+	0.966797i	$0.582255\pi$
$102$		0			0
$103$	2.86603	+	4.96410i	0.282398	+	0.489127i	0.971975	−	0.235085i	$-0.0755368\pi$
$103$	2.86603	+	4.96410i	0.282398	+	0.489127i	−0.689577	+	0.724212i	$0.742204\pi$
$104$		0			0
$105$	0.500000	−	2.59808i	0.0487950	−	0.253546i
$106$		0			0
$107$	−11.4904	+	6.63397i	−1.11082	+	0.641331i	−0.939041	−	0.343805i	$-0.888284\pi$
$107$	−11.4904	+	6.63397i	−1.11082	+	0.641331i	−0.171776	+	0.985136i	$0.554951\pi$
$108$		0			0
$109$	−2.03590	+	3.52628i	−0.195004	+	0.337756i	−0.946902	−	0.321523i	$-0.895805\pi$
$109$	−2.03590	+	3.52628i	−0.195004	+	0.337756i	0.751898	+	0.659279i	$0.229139\pi$
$110$		0			0
$111$	−8.26795			−0.784759
$112$		0			0
$113$	−6.39230			−0.601337			−0.300669	−	0.953729i	$-0.597210\pi$
$113$	−6.39230			−0.601337			−0.300669	+	0.953729i	$0.597210\pi$
$114$		0			0
$115$	−1.36603	+	2.36603i	−0.127383	+	0.220633i
$116$		0			0
$117$	0.866025	−	0.500000i	0.0800641	−	0.0462250i
$118$		0			0
$119$	−8.39230	−	7.26795i	−0.769321	−	0.666252i
$120$		0			0
$121$	−4.69615	−	8.13397i	−0.426923	−	0.739452i
$122$		0			0
$123$	3.63397	+	2.09808i	0.327664	+	0.189177i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$			17.3923i			1.54332i	0.636037	+	0.771659i	$0.280573\pi$
$127$			17.3923i			1.54332i	−0.636037	+	0.771659i	$0.719427\pi$
$128$		0			0
$129$	1.33013	+	0.767949i	0.117111	+	0.0676142i
$130$		0			0
$131$	0.267949	+	0.464102i	0.0234108	+	0.0405487i	0.877493	−	0.479589i	$-0.159214\pi$
$131$	0.267949	+	0.464102i	0.0234108	+	0.0405487i	−0.854083	+	0.520137i	$0.825881\pi$
$132$		0			0
$133$	2.13397	+	6.16025i	0.185039	+	0.534162i
$134$		0			0
$135$	0.866025	−	0.500000i	0.0745356	−	0.0430331i
$136$		0			0
$137$	9.83013	−	17.0263i	0.839844	−	1.45465i	−0.0501799	−	0.998740i	$-0.515979\pi$
$137$	9.83013	−	17.0263i	0.839844	−	1.45465i	0.890024	−	0.455913i	$-0.150687\pi$
$138$		0			0
$139$	−7.92820			−0.672461			−0.336231	−	0.941780i	$-0.609152\pi$
$139$	−7.92820			−0.672461			−0.336231	+	0.941780i	$0.609152\pi$
$140$		0			0
$141$	−12.9282			−1.08875
$142$		0			0
$143$	−0.633975	+	1.09808i	−0.0530156	+	0.0918257i
$144$		0			0
$145$	2.36603	−	1.36603i	0.196488	−	0.113442i
$146$		0			0
$147$	−5.50000	+	4.33013i	−0.453632	+	0.357143i
$148$		0			0
$149$	−5.66025	−	9.80385i	−0.463706	−	0.803162i	0.535436	−	0.844576i	$-0.320147\pi$
$149$	−5.66025	−	9.80385i	−0.463706	−	0.803162i	−0.999142	+	0.0414133i	$0.986814\pi$
$150$		0			0
$151$	8.53590	+	4.92820i	0.694642	+	0.401051i	0.805349	−	0.592802i	$-0.201978\pi$
$151$	8.53590	+	4.92820i	0.694642	+	0.401051i	−0.110707	+	0.993853i	$0.535312\pi$
$152$		0			0
$153$		−	4.19615i		−	0.339239i
$154$		0			0
$155$			6.46410i			0.519209i
$156$		0			0
$157$	17.3205	+	10.0000i	1.38233	+	0.798087i	0.992435	−	0.122774i	$-0.0391789\pi$
$157$	17.3205	+	10.0000i	1.38233	+	0.798087i	0.389892	+	0.920860i	$0.372512\pi$
$158$		0			0
$159$	−2.73205	−	4.73205i	−0.216666	−	0.375276i
$160$		0			0
$161$	6.83013	−	2.36603i	0.538289	−	0.186469i
$162$		0			0
$163$	4.26795	−	2.46410i	0.334292	−	0.193003i	−0.323453	−	0.946244i	$-0.604844\pi$
$163$	4.26795	−	2.46410i	0.334292	−	0.193003i	0.657745	+	0.753241i	$0.271511\pi$
$164$		0			0
$165$	−0.633975	+	1.09808i	−0.0493549	+	0.0854851i
$166$		0			0
$167$	7.12436			0.551299			0.275650	−	0.961258i	$-0.411107\pi$
$167$	7.12436			0.551299			0.275650	+	0.961258i	$0.411107\pi$
$168$		0			0
$169$	12.0000			0.923077
$170$		0			0
$171$	−1.23205	+	2.13397i	−0.0942173	+	0.163189i
$172$		0			0
$173$	−14.1962	+	8.19615i	−1.07931	+	0.623142i	−0.930711	−	0.365755i	$-0.880811\pi$
$173$	−14.1962	+	8.19615i	−1.07931	+	0.623142i	−0.148602	+	0.988897i	$0.547477\pi$
$174$		0			0
$175$	−1.73205	+	2.00000i	−0.130931	+	0.151186i
$176$		0			0
$177$	−1.90192	−	3.29423i	−0.142957	−	0.247609i
$178$		0			0
$179$	3.00000	+	1.73205i	0.224231	+	0.129460i	0.607908	−	0.794008i	$-0.292009\pi$
$179$	3.00000	+	1.73205i	0.224231	+	0.129460i	−0.383677	+	0.923467i	$0.625342\pi$
$180$		0			0
$181$			25.5885i			1.90198i	0.309229	+	0.950988i	$0.399929\pi$
$181$			25.5885i			1.90198i	−0.309229	+	0.950988i	$0.600071\pi$
$182$		0			0
$183$		−	4.53590i		−	0.335303i
$184$		0			0
$185$	7.16025	+	4.13397i	0.526432	+	0.303936i
$186$		0			0
$187$	2.66025	+	4.60770i	0.194537	+	0.336948i
$188$		0			0
$189$	−2.59808	−	0.500000i	−0.188982	−	0.0363696i
$190$		0			0
$191$	−1.73205	+	1.00000i	−0.125327	+	0.0723575i	−0.561353	−	0.827577i	$-0.689719\pi$
$191$	−1.73205	+	1.00000i	−0.125327	+	0.0723575i	0.436026	+	0.899934i	$0.356386\pi$
$192$		0			0
$193$	−0.598076	+	1.03590i	−0.0430505	+	0.0745656i	−0.886748	−	0.462254i	$-0.847041\pi$
$193$	−0.598076	+	1.03590i	−0.0430505	+	0.0745656i	0.843697	+	0.536819i	$0.180374\pi$
$194$		0			0
$195$	−1.00000			−0.0716115
$196$		0			0
$197$	−16.7321			−1.19211			−0.596055	−	0.802944i	$-0.703266\pi$
$197$	−16.7321			−1.19211			−0.596055	+	0.802944i	$0.703266\pi$
$198$		0			0
$199$	11.9282	−	20.6603i	0.845568	−	1.46457i	−0.0395592	−	0.999217i	$-0.512595\pi$
$199$	11.9282	−	20.6603i	0.845568	−	1.46457i	0.885127	−	0.465349i	$-0.154071\pi$
$200$		0			0
$201$	−4.79423	+	2.76795i	−0.338159	+	0.195236i
$202$		0			0
$203$	−7.09808	−	1.36603i	−0.498187	−	0.0958762i
$204$		0			0
$205$	−2.09808	−	3.63397i	−0.146536	−	0.253808i
$206$		0			0
$207$	2.36603	+	1.36603i	0.164450	+	0.0949453i
$208$		0			0
$209$		−	3.12436i		−	0.216116i
$210$		0			0
$211$		−	16.7846i		−	1.15550i	−0.816214	−	0.577750i	$-0.803931\pi$
$211$		−	16.7846i		−	1.15550i	0.816214	−	0.577750i	$-0.196069\pi$
$212$		0			0
$213$	−1.43782	−	0.830127i	−0.0985180	−	0.0568794i
$214$		0			0
$215$	−0.767949	−	1.33013i	−0.0523737	−	0.0907139i
$216$		0			0
$217$	11.1962	−	12.9282i	0.760044	−	0.877624i
$218$		0			0
$219$	−6.86603	+	3.96410i	−0.463963	+	0.267869i
$220$		0			0
$221$	−2.09808	+	3.63397i	−0.141132	+	0.244448i
$222$		0			0
$223$	16.3923			1.09771			0.548855	−	0.835918i	$-0.315064\pi$
$223$	16.3923			1.09771			0.548855	+	0.835918i	$0.315064\pi$
$224$		0			0
$225$	−1.00000			−0.0666667
$226$		0			0
$227$	−11.0263	+	19.0981i	−0.731840	+	1.26758i	0.224256	+	0.974530i	$0.428005\pi$
$227$	−11.0263	+	19.0981i	−0.731840	+	1.26758i	−0.956096	+	0.293054i	$0.905329\pi$
$228$		0			0
$229$	−14.8923	+	8.59808i	−0.984111	+	0.568177i	−0.903509	−	0.428569i	$-0.859018\pi$
$229$	−14.8923	+	8.59808i	−0.984111	+	0.568177i	−0.0806024	+	0.996746i	$0.525684\pi$
$230$		0			0
$231$	3.16987	−	1.09808i	0.208562	−	0.0722481i
$232$		0			0
$233$	−3.00000	−	5.19615i	−0.196537	−	0.340411i	0.750867	−	0.660454i	$-0.229636\pi$
$233$	−3.00000	−	5.19615i	−0.196537	−	0.340411i	−0.947403	+	0.320043i	$0.896303\pi$
$234$		0			0
$235$	11.1962	+	6.46410i	0.730356	+	0.421671i
$236$		0			0
$237$			4.26795i			0.277233i
$238$		0			0
$239$		−	29.3205i		−	1.89659i	−0.317396	−	0.948293i	$-0.602809\pi$
$239$		−	29.3205i		−	1.89659i	0.317396	−	0.948293i	$-0.397191\pi$
$240$		0			0
$241$	13.7321	+	7.92820i	0.884559	+	0.510700i	0.872159	−	0.489223i	$-0.162719\pi$
$241$	13.7321	+	7.92820i	0.884559	+	0.510700i	0.0124002	+	0.999923i	$0.496053\pi$
$242$		0			0
$243$	−0.500000	−	0.866025i	−0.0320750	−	0.0555556i
$244$		0			0
$245$	6.92820	−	1.00000i	0.442627	−	0.0638877i
$246$		0			0
$247$	2.13397	−	1.23205i	0.135782	−	0.0783935i
$248$		0			0
$249$	2.63397	−	4.56218i	0.166921	−	0.289116i
$250$		0			0
$251$	19.1244			1.20712			0.603559	−	0.797318i	$-0.293749\pi$
$251$	19.1244			1.20712			0.603559	+	0.797318i	$0.293749\pi$
$252$		0			0
$253$	−3.46410			−0.217786
$254$		0			0
$255$	−2.09808	+	3.63397i	−0.131387	+	0.227568i
$256$		0			0
$257$	−0.294229	+	0.169873i	−0.0183535	+	0.0105964i	−0.509149	−	0.860679i	$-0.670040\pi$
$257$	−0.294229	+	0.169873i	−0.0183535	+	0.0105964i	0.490795	+	0.871275i	$0.336706\pi$
$258$		0			0
$259$	−7.16025	−	20.6699i	−0.444917	−	1.28436i
$260$		0			0
$261$	−1.36603	−	2.36603i	−0.0845548	−	0.146453i
$262$		0			0
$263$	−9.80385	−	5.66025i	−0.604531	−	0.349026i	0.166291	−	0.986077i	$-0.446821\pi$
$263$	−9.80385	−	5.66025i	−0.604531	−	0.349026i	−0.770822	+	0.637051i	$0.780154\pi$
$264$		0			0
$265$			5.46410i			0.335657i
$266$		0			0
$267$		−	4.73205i		−	0.289597i
$268$		0			0
$269$	−13.7321	−	7.92820i	−0.837258	−	0.483391i	0.0190733	−	0.999818i	$-0.493928\pi$
$269$	−13.7321	−	7.92820i	−0.837258	−	0.483391i	−0.856331	+	0.516427i	$0.827262\pi$
$270$		0			0
$271$	−1.00000	−	1.73205i	−0.0607457	−	0.105215i	0.834053	−	0.551684i	$-0.186015\pi$
$271$	−1.00000	−	1.73205i	−0.0607457	−	0.105215i	−0.894799	+	0.446469i	$0.852681\pi$
$272$		0			0
$273$	2.00000	+	1.73205i	0.121046	+	0.104828i
$274$		0			0
$275$	1.09808	−	0.633975i	0.0662165	−	0.0382301i
$276$		0			0
$277$	−5.59808	+	9.69615i	−0.336356	+	0.582585i	−0.983744	−	0.179575i	$-0.942528\pi$
$277$	−5.59808	+	9.69615i	−0.336356	+	0.582585i	0.647389	+	0.762160i	$0.275861\pi$
$278$		0			0
$279$	6.46410			0.386996
$280$		0			0
$281$	13.8564			0.826604			0.413302	−	0.910594i	$-0.364375\pi$
$281$	13.8564			0.826604			0.413302	+	0.910594i	$0.364375\pi$
$282$		0			0
$283$	16.5981	−	28.7487i	0.986653	−	1.70893i	0.352307	−	0.935885i	$-0.385397\pi$
$283$	16.5981	−	28.7487i	0.986653	−	1.70893i	0.634346	−	0.773049i	$-0.281269\pi$
$284$		0			0
$285$	2.13397	−	1.23205i	0.126406	−	0.0729804i
$286$		0			0
$287$	−2.09808	+	10.9019i	−0.123846	+	0.643520i
$288$		0			0
$289$	0.303848	+	0.526279i	0.0178734	+	0.0309576i
$290$		0			0
$291$	−4.26795	−	2.46410i	−0.250192	−	0.144448i
$292$		0			0
$293$		−	17.4641i		−	1.02026i	−0.860096	−	0.510132i	$-0.829597\pi$
$293$		−	17.4641i		−	1.02026i	0.860096	−	0.510132i	$-0.170403\pi$
$294$		0			0
$295$			3.80385i			0.221469i
$296$		0			0
$297$	1.09808	+	0.633975i	0.0637168	+	0.0367869i
$298$		0			0
$299$	−1.36603	−	2.36603i	−0.0789993	−	0.136831i
$300$		0			0
$301$	−0.767949	+	3.99038i	−0.0442639	+	0.230002i
$302$		0			0
$303$	−4.56218	+	2.63397i	−0.262090	+	0.151318i
$304$		0			0
$305$	−2.26795	+	3.92820i	−0.129862	+	0.224928i
$306$		0			0
$307$	−4.12436			−0.235389			−0.117695	−	0.993050i	$-0.537550\pi$
$307$	−4.12436			−0.235389			−0.117695	+	0.993050i	$0.537550\pi$
$308$		0			0
$309$	−5.73205			−0.326085
$310$		0			0
$311$	9.09808	−	15.7583i	0.515905	−	0.893573i	−0.483925	−	0.875109i	$-0.660789\pi$
$311$	9.09808	−	15.7583i	0.515905	−	0.893573i	0.999830	−	0.0184634i	$-0.00587743\pi$
$312$		0			0
$313$	−16.4545	+	9.50000i	−0.930062	+	0.536972i	−0.886831	−	0.462093i	$-0.847098\pi$
$313$	−16.4545	+	9.50000i	−0.930062	+	0.536972i	−0.0432311	+	0.999065i	$0.513765\pi$
$314$		0			0
$315$	2.00000	+	1.73205i	0.112687	+	0.0975900i
$316$		0			0
$317$	−14.0263	−	24.2942i	−0.787794	−	1.36450i	−0.927315	−	0.374281i	$-0.877890\pi$
$317$	−14.0263	−	24.2942i	−0.787794	−	1.36450i	0.139521	−	0.990219i	$-0.455444\pi$
$318$		0			0
$319$	3.00000	+	1.73205i	0.167968	+	0.0969762i
$320$		0			0
$321$		−	13.2679i		−	0.740545i
$322$		0			0
$323$		−	10.3397i		−	0.575319i
$324$		0			0
$325$	0.866025	+	0.500000i	0.0480384	+	0.0277350i
$326$		0			0
$327$	−2.03590	−	3.52628i	−0.112585	−	0.195004i
$328$		0			0
$329$	−11.1962	−	32.3205i	−0.617264	−	1.78189i
$330$		0			0
$331$	7.50000	−	4.33013i	0.412237	−	0.238005i	−0.279513	−	0.960142i	$-0.590173\pi$
$331$	7.50000	−	4.33013i	0.412237	−	0.238005i	0.691751	+	0.722137i	$0.256840\pi$
$332$		0			0
$333$	4.13397	−	7.16025i	0.226540	−	0.392380i
$334$		0			0
$335$	5.53590			0.302458
$336$		0			0
$337$	33.4449			1.82186			0.910929	−	0.412563i	$-0.135366\pi$
$337$	33.4449			1.82186			0.910929	+	0.412563i	$0.135366\pi$
$338$		0			0
$339$	3.19615	−	5.53590i	0.173591	−	0.300669i
$340$		0			0
$341$	−7.09808	+	4.09808i	−0.384382	+	0.221923i
$342$		0			0
$343$	−15.5885	−	10.0000i	−0.841698	−	0.539949i
$344$		0			0
$345$	−1.36603	−	2.36603i	−0.0735443	−	0.127383i
$346$		0			0
$347$	8.19615	+	4.73205i	0.439993	+	0.254030i	0.703594	−	0.710602i	$-0.251577\pi$
$347$	8.19615	+	4.73205i	0.439993	+	0.254030i	−0.263602	+	0.964632i	$0.584911\pi$
$348$		0			0
$349$			13.0718i			0.699717i	0.936803	+	0.349859i	$0.113770\pi$
$349$			13.0718i			0.699717i	−0.936803	+	0.349859i	$0.886230\pi$
$350$		0			0
$351$			1.00000i			0.0533761i
$352$		0			0
$353$	9.75833	+	5.63397i	0.519384	+	0.299866i	0.736682	−	0.676239i	$-0.236391\pi$
$353$	9.75833	+	5.63397i	0.519384	+	0.299866i	−0.217299	+	0.976105i	$0.569725\pi$
$354$		0			0
$355$	0.830127	+	1.43782i	0.0440586	+	0.0763117i
$356$		0			0
$357$	10.4904	−	3.63397i	0.555210	−	0.192330i
$358$		0			0
$359$	10.0981	−	5.83013i	0.532956	−	0.307702i	−0.209263	−	0.977859i	$-0.567107\pi$
$359$	10.0981	−	5.83013i	0.532956	−	0.307702i	0.742219	+	0.670157i	$0.233773\pi$
$360$		0			0
$361$	6.46410	−	11.1962i	0.340216	−	0.589271i
$362$		0			0
$363$	9.39230			0.492968
$364$		0			0
$365$	7.92820			0.414981
$366$		0			0
$367$	6.40192	−	11.0885i	0.334178	−	0.578813i	−0.649149	−	0.760661i	$-0.724875\pi$
$367$	6.40192	−	11.0885i	0.334178	−	0.578813i	0.983327	+	0.181849i	$0.0582081\pi$
$368$		0			0
$369$	−3.63397	+	2.09808i	−0.189177	+	0.109221i
$370$		0			0
$371$	9.46410	−	10.9282i	0.491352	−	0.567364i
$372$		0			0
$373$	−10.9904	−	19.0359i	−0.569060	−	0.985641i	−0.996659	−	0.0816729i	$-0.973974\pi$
$373$	−10.9904	−	19.0359i	−0.569060	−	0.985641i	0.427599	−	0.903969i	$-0.359360\pi$
$374$		0			0
$375$	0.866025	+	0.500000i	0.0447214	+	0.0258199i
$376$		0			0
$377$			2.73205i			0.140708i
$378$		0			0
$379$		−	32.5167i		−	1.67027i	−0.550046	−	0.835134i	$-0.685390\pi$
$379$		−	32.5167i		−	1.67027i	0.550046	−	0.835134i	$-0.314610\pi$
$380$		0			0
$381$	−15.0622	−	8.69615i	−0.771659	−	0.445517i
$382$		0			0
$383$	−10.9282	−	18.9282i	−0.558405	−	0.967186i	−0.997630	−	0.0688088i	$-0.978080\pi$
$383$	−10.9282	−	18.9282i	−0.558405	−	0.967186i	0.439225	−	0.898377i	$-0.355253\pi$
$384$		0			0
$385$	−3.29423	−	0.633975i	−0.167889	−	0.0323103i
$386$		0			0
$387$	−1.33013	+	0.767949i	−0.0676142	+	0.0390371i
$388$		0			0
$389$	−14.7583	+	25.5622i	−0.748277	+	1.29605i	0.200371	+	0.979720i	$0.435785\pi$
$389$	−14.7583	+	25.5622i	−0.748277	+	1.29605i	−0.948648	+	0.316334i	$0.897548\pi$
$390$		0			0
$391$	−11.4641			−0.579765
$392$		0			0
$393$	−0.535898			−0.0270325
$394$		0			0
$395$	2.13397	−	3.69615i	0.107372	−	0.185974i
$396$		0			0
$397$	4.79423	−	2.76795i	0.240615	−	0.138919i	−0.374844	−	0.927088i	$-0.622304\pi$
$397$	4.79423	−	2.76795i	0.240615	−	0.138919i	0.615460	+	0.788168i	$0.288970\pi$
$398$		0			0
$399$	−6.40192	−	1.23205i	−0.320497	−	0.0616797i
$400$		0			0
$401$	−6.00000	−	10.3923i	−0.299626	−	0.518967i	0.676425	−	0.736512i	$-0.263528\pi$
$401$	−6.00000	−	10.3923i	−0.299626	−	0.518967i	−0.976050	+	0.217545i	$0.930195\pi$
$402$		0			0
$403$	−5.59808	−	3.23205i	−0.278860	−	0.161000i
$404$		0			0
$405$			1.00000i			0.0496904i
$406$		0			0
$407$			10.4833i			0.519640i
$408$		0			0
$409$	−8.42820	−	4.86603i	−0.416748	−	0.240609i	0.276937	−	0.960888i	$-0.410681\pi$
$409$	−8.42820	−	4.86603i	−0.416748	−	0.240609i	−0.693685	+	0.720279i	$0.744014\pi$
$410$		0			0
$411$	9.83013	+	17.0263i	0.484884	+	0.839844i
$412$		0			0
$413$	6.58846	−	7.60770i	0.324197	−	0.374350i
$414$		0			0
$415$	−4.56218	+	2.63397i	−0.223949	+	0.129297i
$416$		0			0
$417$	3.96410	−	6.86603i	0.194123	−	0.336231i
$418$		0			0
$419$	−4.92820			−0.240758			−0.120379	−	0.992728i	$-0.538411\pi$
$419$	−4.92820			−0.240758			−0.120379	+	0.992728i	$0.538411\pi$
$420$		0			0
$421$	−25.9282			−1.26366			−0.631832	−	0.775106i	$-0.717697\pi$
$421$	−25.9282			−1.26366			−0.631832	+	0.775106i	$0.717697\pi$
$422$		0			0
$423$	6.46410	−	11.1962i	0.314295	−	0.544376i
$424$		0			0
$425$	3.63397	−	2.09808i	0.176274	−	0.101772i
$426$		0			0
$427$	11.3397	−	3.92820i	0.548769	−	0.190099i
$428$		0			0
$429$	−0.633975	−	1.09808i	−0.0306086	−	0.0530156i
$430$		0			0
$431$	−14.6603	−	8.46410i	−0.706160	−	0.407701i	0.103478	−	0.994632i	$-0.467003\pi$
$431$	−14.6603	−	8.46410i	−0.706160	−	0.407701i	−0.809637	+	0.586930i	$0.800336\pi$
$432$		0			0
$433$			19.5359i			0.938835i	0.882976	+	0.469418i	$0.155536\pi$
$433$			19.5359i			0.938835i	−0.882976	+	0.469418i	$0.844464\pi$
$434$		0			0
$435$			2.73205i			0.130992i
$436$		0			0
$437$	5.83013	+	3.36603i	0.278893	+	0.161019i
$438$		0			0
$439$	5.73205	+	9.92820i	0.273576	+	0.473847i	0.969775	−	0.244002i	$-0.0784602\pi$
$439$	5.73205	+	9.92820i	0.273576	+	0.473847i	−0.696199	+	0.717849i	$0.745127\pi$
$440$		0			0
$441$	−1.00000	−	6.92820i	−0.0476190	−	0.329914i
$442$		0			0
$443$	−4.39230	+	2.53590i	−0.208685	+	0.120484i	−0.600700	−	0.799475i	$-0.705111\pi$
$443$	−4.39230	+	2.53590i	−0.208685	+	0.120484i	0.392015	+	0.919959i	$0.371778\pi$
$444$		0			0
$445$	−2.36603	+	4.09808i	−0.112160	+	0.194267i
$446$		0			0
$447$	11.3205			0.535442
$448$		0			0
$449$	−13.3205			−0.628634			−0.314317	−	0.949318i	$-0.601775\pi$
$449$	−13.3205			−0.628634			−0.314317	+	0.949318i	$0.601775\pi$
$450$		0			0
$451$	2.66025	−	4.60770i	0.125266	−	0.216968i
$452$		0			0
$453$	−8.53590	+	4.92820i	−0.401051	+	0.231547i
$454$		0			0
$455$	−0.866025	−	2.50000i	−0.0405999	−	0.117202i
$456$		0			0
$457$	−2.86603	−	4.96410i	−0.134067	−	0.232211i	0.791174	−	0.611592i	$-0.209470\pi$
$457$	−2.86603	−	4.96410i	−0.134067	−	0.232211i	−0.925241	+	0.379381i	$0.876137\pi$
$458$		0			0
$459$	3.63397	+	2.09808i	0.169619	+	0.0979298i
$460$		0			0
$461$			21.8038i			1.01551i	0.861503	+	0.507753i	$0.169524\pi$
$461$			21.8038i			1.01551i	−0.861503	+	0.507753i	$0.830476\pi$
$462$		0			0
$463$			17.7846i			0.826521i	0.910613	+	0.413260i	$0.135610\pi$
$463$			17.7846i			0.826521i	−0.910613	+	0.413260i	$0.864390\pi$
$464$		0			0
$465$	−5.59808	−	3.23205i	−0.259605	−	0.149883i
$466$		0			0
$467$	20.8564	+	36.1244i	0.965119	+	1.67164i	0.709294	+	0.704913i	$0.249014\pi$
$467$	20.8564	+	36.1244i	0.965119	+	1.67164i	0.255825	+	0.966723i	$0.417653\pi$
$468$		0			0
$469$	−11.0718	−	9.58846i	−0.511248	−	0.442754i
$470$		0			0
$471$	−17.3205	+	10.0000i	−0.798087	+	0.460776i
$472$		0			0
$473$	0.973721	−	1.68653i	0.0447717	−	0.0775469i
$474$		0			0
$475$	−2.46410			−0.113061
$476$		0			0
$477$	5.46410			0.250184
$478$		0			0
$479$	−3.07180	+	5.32051i	−0.140354	+	0.243100i	−0.927630	−	0.373501i	$-0.878157\pi$
$479$	−3.07180	+	5.32051i	−0.140354	+	0.243100i	0.787276	+	0.616601i	$0.211491\pi$
$480$		0			0
$481$	−7.16025	+	4.13397i	−0.326479	+	0.188493i
$482$		0			0
$483$	−1.36603	+	7.09808i	−0.0621563	+	0.322974i
$484$		0			0
$485$	2.46410	+	4.26795i	0.111889	+	0.193798i
$486$		0			0
$487$	15.6506	+	9.03590i	0.709198	+	0.409456i	0.810764	−	0.585373i	$-0.199052\pi$
$487$	15.6506	+	9.03590i	0.709198	+	0.409456i	−0.101566	+	0.994829i	$0.532385\pi$
$488$		0			0
$489$			4.92820i			0.222861i
$490$		0			0
$491$		−	30.5359i		−	1.37807i	−0.724730	−	0.689033i	$-0.758036\pi$
$491$		−	30.5359i		−	1.37807i	0.724730	−	0.689033i	$-0.241964\pi$
$492$		0			0
$493$	9.92820	+	5.73205i	0.447144	+	0.258159i
$494$		0			0
$495$	−0.633975	−	1.09808i	−0.0284950	−	0.0493549i
$496$		0			0
$497$	0.830127	−	4.31347i	0.0372363	−	0.193485i
$498$		0			0
$499$	19.1603	−	11.0622i	0.857731	−	0.495211i	−0.00552096	−	0.999985i	$-0.501757\pi$
$499$	19.1603	−	11.0622i	0.857731	−	0.495211i	0.863252	+	0.504774i	$0.168424\pi$
$500$		0			0
$501$	−3.56218	+	6.16987i	−0.159146	+	0.275650i
$502$		0			0
$503$	36.9282			1.64655			0.823274	−	0.567645i	$-0.192145\pi$
$503$	36.9282			1.64655			0.823274	+	0.567645i	$0.192145\pi$
$504$		0			0
$505$	5.26795			0.234421
$506$		0			0
$507$	−6.00000	+	10.3923i	−0.266469	+	0.461538i
$508$		0			0
$509$	−19.9808	+	11.5359i	−0.885632	+	0.511320i	−0.872511	−	0.488594i	$-0.837510\pi$
$509$	−19.9808	+	11.5359i	−0.885632	+	0.511320i	−0.0131206	+	0.999914i	$0.504177\pi$
$510$		0			0
$511$	−15.8564	−	13.7321i	−0.701446	−	0.607470i
$512$		0			0
$513$	−1.23205	−	2.13397i	−0.0543964	−	0.0942173i
$514$		0			0
$515$	4.96410	+	2.86603i	0.218744	+	0.126292i
$516$		0			0
$517$			16.3923i			0.720933i
$518$		0			0
$519$		−	16.3923i		−	0.719542i
$520$		0			0
$521$	29.9090	+	17.2679i	1.31034	+	0.756523i	0.982152	−	0.188091i	$-0.0602301\pi$
$521$	29.9090	+	17.2679i	1.31034	+	0.756523i	0.328184	+	0.944614i	$0.393563\pi$
$522$		0			0
$523$	−8.40192	−	14.5526i	−0.367390	−	0.636339i	0.621766	−	0.783203i	$-0.286415\pi$
$523$	−8.40192	−	14.5526i	−0.367390	−	0.636339i	−0.989157	+	0.146864i	$0.953082\pi$
$524$		0			0
$525$	−0.866025	−	2.50000i	−0.0377964	−	0.109109i
$526$		0			0
$527$	−23.4904	+	13.5622i	−1.02326	+	0.590778i
$528$		0			0
$529$	−7.76795	+	13.4545i	−0.337737	+	0.584978i
$530$		0			0
$531$	3.80385			0.165073
$532$		0			0
$533$	4.19615			0.181756
$534$		0			0
$535$	−6.63397	+	11.4904i	−0.286812	+	0.496772i
$536$		0			0
$537$	−3.00000	+	1.73205i	−0.129460	+	0.0747435i
$538$		0			0
$539$	5.49038	+	6.97372i	0.236487	+	0.300379i
$540$		0			0
$541$	−17.8923	−	30.9904i	−0.769250	−	1.33238i	−0.937970	−	0.346716i	$-0.887297\pi$
$541$	−17.8923	−	30.9904i	−0.769250	−	1.33238i	0.168720	−	0.985664i	$-0.446037\pi$
$542$		0			0
$543$	−22.1603	−	12.7942i	−0.950988	−	0.549053i
$544$		0			0
$545$			4.07180i			0.174417i
$546$		0			0
$547$			5.60770i			0.239768i	0.992788	+	0.119884i	$0.0382522\pi$
$547$			5.60770i			0.239768i	−0.992788	+	0.119884i	$0.961748\pi$
$548$		0			0
$549$	3.92820	+	2.26795i	0.167652	+	0.0967937i
$550$		0			0
$551$	−3.36603	−	5.83013i	−0.143398	−	0.248372i
$552$		0			0
$553$	−10.6699	+	3.69615i	−0.453729	+	0.157176i
$554$		0			0
$555$	−7.16025	+	4.13397i	−0.303936	+	0.175477i
$556$		0			0
$557$	−6.26795	+	10.8564i	−0.265582	+	0.460001i	−0.967716	−	0.252044i	$-0.918897\pi$
$557$	−6.26795	+	10.8564i	−0.265582	+	0.460001i	0.702134	+	0.712045i	$0.252231\pi$
$558$		0			0
$559$	1.53590			0.0649616
$560$		0			0
$561$	−5.32051			−0.224632
$562$		0			0
$563$	7.19615	−	12.4641i	0.303282	−	0.525299i	−0.673596	−	0.739100i	$-0.735251\pi$
$563$	7.19615	−	12.4641i	0.303282	−	0.525299i	0.976877	+	0.213801i	$0.0685844\pi$
$564$		0			0
$565$	−5.53590	+	3.19615i	−0.232897	+	0.134463i
$566$		0			0
$567$	1.73205	−	2.00000i	0.0727393	−	0.0839921i
$568$		0			0
$569$	−11.2942	−	19.5622i	−0.473479	−	0.820089i	0.526060	−	0.850447i	$-0.323669\pi$
$569$	−11.2942	−	19.5622i	−0.473479	−	0.820089i	−0.999539	+	0.0303581i	$0.990335\pi$
$570$		0			0
$571$	7.62436	+	4.40192i	0.319069	+	0.184215i	0.650978	−	0.759097i	$-0.274359\pi$
$571$	7.62436	+	4.40192i	0.319069	+	0.184215i	−0.331908	+	0.943312i	$0.607692\pi$
$572$		0			0
$573$		−	2.00000i		−	0.0835512i
$574$		0			0
$575$			2.73205i			0.113934i
$576$		0			0
$577$	9.65064	+	5.57180i	0.401761	+	0.231957i	0.687244	−	0.726427i	$-0.258820\pi$
$577$	9.65064	+	5.57180i	0.401761	+	0.231957i	−0.285482	+	0.958384i	$0.592154\pi$
$578$		0			0
$579$	−0.598076	−	1.03590i	−0.0248552	−	0.0430505i
$580$		0			0
$581$	13.6865	+	2.63397i	0.567813	+	0.109276i
$582$		0			0
$583$	−6.00000	+	3.46410i	−0.248495	+	0.143468i
$584$		0			0
$585$	0.500000	−	0.866025i	0.0206725	−	0.0358057i
$586$		0			0
$587$	−7.66025			−0.316173			−0.158086	−	0.987425i	$-0.550532\pi$
$587$	−7.66025			−0.316173			−0.158086	+	0.987425i	$0.550532\pi$
$588$		0			0
$589$	15.9282			0.656310
$590$		0			0
$591$	8.36603	−	14.4904i	0.344132	−	0.596055i
$592$		0			0
$593$	−35.2750	+	20.3660i	−1.44857	+	0.836332i	−0.998396	−	0.0566085i	$-0.981971\pi$
$593$	−35.2750	+	20.3660i	−1.44857	+	0.836332i	−0.450174	+	0.892941i	$0.648638\pi$
$594$		0			0
$595$	−10.9019	−	2.09808i	−0.446935	−	0.0860127i
$596$		0			0
$597$	11.9282	+	20.6603i	0.488189	+	0.845568i
$598$		0			0
$599$	−16.3923	−	9.46410i	−0.669771	−	0.386693i	0.126219	−	0.992002i	$-0.459716\pi$
$599$	−16.3923	−	9.46410i	−0.669771	−	0.386693i	−0.795990	+	0.605310i	$0.793049\pi$
$600$		0			0
$601$		−	28.6603i		−	1.16908i	−0.811366	−	0.584538i	$-0.801276\pi$
$601$		−	28.6603i		−	1.16908i	0.811366	−	0.584538i	$-0.198724\pi$
$602$		0			0
$603$		−	5.53590i		−	0.225439i
$604$		0			0
$605$	−8.13397	−	4.69615i	−0.330693	−	0.190926i
$606$		0			0
$607$	16.4019	+	28.4090i	0.665734	+	1.15308i	0.979086	+	0.203447i	$0.0652146\pi$
$607$	16.4019	+	28.4090i	0.665734	+	1.15308i	−0.313352	+	0.949637i	$0.601452\pi$
$608$		0			0
$609$	4.73205	−	5.46410i	0.191752	−	0.221417i
$610$		0			0
$611$	−11.1962	+	6.46410i	−0.452948	+	0.261510i
$612$		0			0
$613$	15.3205	−	26.5359i	0.618789	−	1.07177i	−0.370917	−	0.928666i	$-0.620957\pi$
$613$	15.3205	−	26.5359i	0.618789	−	1.07177i	0.989707	−	0.143109i	$-0.0457099\pi$
$614$		0			0
$615$	4.19615			0.169205
$616$		0			0
$617$	6.00000			0.241551			0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000			0.241551			0.120775	+	0.992680i	$0.461462\pi$
$618$		0			0
$619$	−3.50000	+	6.06218i	−0.140677	+	0.243659i	−0.927752	−	0.373198i	$-0.878261\pi$
$619$	−3.50000	+	6.06218i	−0.140677	+	0.243659i	0.787075	+	0.616858i	$0.211595\pi$
$620$		0			0
$621$	−2.36603	+	1.36603i	−0.0949453	+	0.0548167i
$622$		0			0
$623$	11.8301	−	4.09808i	0.473964	−	0.164186i
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.0200000	−	0.0346410i
$626$		0			0
$627$	2.70577	+	1.56218i	0.108058	+	0.0623874i
$628$		0			0
$629$			34.6936i			1.38332i
$630$		0			0
$631$			31.4641i			1.25257i	0.779596	+	0.626283i	$0.215425\pi$
$631$			31.4641i			1.25257i	−0.779596	+	0.626283i	$0.784575\pi$
$632$		0			0
$633$	14.5359	+	8.39230i	0.577750	+	0.333564i
$634$		0			0
$635$	8.69615	+	15.0622i	0.345096	+	0.597724i
$636$		0			0
$637$	−2.59808	+	6.50000i	−0.102940	+	0.257539i
$638$		0			0
$639$	1.43782	−	0.830127i	0.0568794	−	0.0328393i
$640$		0			0
$641$	11.2224	−	19.4378i	0.443259	−	0.767748i	−0.554670	−	0.832071i	$-0.687155\pi$
$641$	11.2224	−	19.4378i	0.443259	−	0.767748i	0.997929	+	0.0643228i	$0.0204887\pi$
$642$		0			0
$643$	−19.9808			−0.787964			−0.393982	−	0.919118i	$-0.628903\pi$
$643$	−19.9808			−0.787964			−0.393982	+	0.919118i	$0.628903\pi$
$644$		0			0
$645$	1.53590			0.0604759
$646$		0			0
$647$	−2.09808	+	3.63397i	−0.0824839	+	0.142866i	−0.904316	−	0.426863i	$-0.859619\pi$
$647$	−2.09808	+	3.63397i	−0.0824839	+	0.142866i	0.821832	+	0.569729i	$0.192952\pi$
$648$		0			0
$649$	−4.17691	+	2.41154i	−0.163958	+	0.0946613i
$650$		0			0
$651$	5.59808	+	16.1603i	0.219406	+	0.633370i
$652$		0			0
$653$	−20.2942	−	35.1506i	−0.794175	−	1.37555i	−0.923362	−	0.383930i	$-0.874570\pi$
$653$	−20.2942	−	35.1506i	−0.794175	−	1.37555i	0.129187	−	0.991620i	$-0.458763\pi$
$654$		0			0
$655$	0.464102	+	0.267949i	0.0181340	+	0.0104696i
$656$		0			0
$657$		−	7.92820i		−	0.309309i
$658$		0			0
$659$			8.78461i			0.342200i	0.985254	+	0.171100i	$0.0547321\pi$
$659$			8.78461i			0.342200i	−0.985254	+	0.171100i	$0.945268\pi$
$660$		0			0
$661$	3.44744	+	1.99038i	0.134090	+	0.0774169i	0.565544	−	0.824718i	$-0.308666\pi$
$661$	3.44744	+	1.99038i	0.134090	+	0.0774169i	−0.431454	+	0.902135i	$0.641999\pi$
$662$		0			0
$663$	−2.09808	−	3.63397i	−0.0814825	−	0.141132i
$664$		0			0
$665$	4.92820	+	4.26795i	0.191108	+	0.165504i
$666$		0			0
$667$	−6.46410	+	3.73205i	−0.250291	+	0.144506i
$668$		0			0
$669$	−8.19615	+	14.1962i	−0.316882	+	0.548855i
$670$		0			0
$671$	−5.75129			−0.222026
$672$		0			0
$673$	29.4449			1.13502			0.567508	−	0.823368i	$-0.307908\pi$
$673$	29.4449			1.13502			0.567508	+	0.823368i	$0.307908\pi$
$674$		0			0
$675$	0.500000	−	0.866025i	0.0192450	−	0.0333333i
$676$		0			0
$677$	−27.8827	+	16.0981i	−1.07162	+	0.618699i	−0.928623	−	0.371024i	$-0.879007\pi$
$677$	−27.8827	+	16.0981i	−1.07162	+	0.618699i	−0.142995	+	0.989723i	$0.545673\pi$
$678$		0			0
$679$	2.46410	−	12.8038i	0.0945635	−	0.491367i
$680$		0			0
$681$	−11.0263	−	19.0981i	−0.422528	−	0.731840i
$682$		0			0
$683$	−2.95448	−	1.70577i	−0.113050	−	0.0652695i	0.442409	−	0.896814i	$-0.354124\pi$
$683$	−2.95448	−	1.70577i	−0.113050	−	0.0652695i	−0.555459	+	0.831544i	$0.687457\pi$
$684$		0			0
$685$		−	19.6603i		−	0.751180i
$686$		0			0
$687$		−	17.1962i		−	0.656074i
$688$		0			0
$689$	−4.73205	−	2.73205i	−0.180277	−	0.104083i
$690$		0			0
$691$	23.2846	+	40.3301i	0.885788	+	1.53423i	0.844807	+	0.535070i	$0.179715\pi$
$691$	23.2846	+	40.3301i	0.885788	+	1.53423i	0.0409808	+	0.999160i	$0.486952\pi$
$692$		0			0
$693$	−0.633975	+	3.29423i	−0.0240827	+	0.125137i
$694$		0			0
$695$	−6.86603	+	3.96410i	−0.260443	+	0.150367i
$696$		0			0
$697$	8.80385	−	15.2487i	0.333470	−	0.577586i
$698$		0			0
$699$	6.00000			0.226941
$700$		0			0
$701$	19.9090			0.751951			0.375976	−	0.926630i	$-0.377308\pi$
$701$	19.9090			0.751951			0.375976	+	0.926630i	$0.377308\pi$
$702$		0			0
$703$	10.1865	−	17.6436i	0.384192	−	0.665441i
$704$		0			0
$705$	−11.1962	+	6.46410i	−0.421671	+	0.243452i
$706$		0			0
$707$	−10.5359	−	9.12436i	−0.396243	−	0.343157i
$708$		0			0
$709$	−5.85641	−	10.1436i	−0.219942	−	0.380951i	0.734848	−	0.678232i	$-0.237253\pi$
$709$	−5.85641	−	10.1436i	−0.219942	−	0.380951i	−0.954790	+	0.297281i	$0.903920\pi$
$710$		0			0
$711$	−3.69615	−	2.13397i	−0.138617	−	0.0800303i
$712$		0			0
$713$		−	17.6603i		−	0.661382i
$714$		0			0
$715$			1.26795i			0.0474186i
$716$		0			0
$717$	25.3923	+	14.6603i	0.948293	+	0.547497i
$718$		0			0
$719$	−12.4641	−	21.5885i	−0.464833	−	0.805114i	0.534361	−	0.845256i	$-0.320552\pi$
$719$	−12.4641	−	21.5885i	−0.464833	−	0.805114i	−0.999194	+	0.0401425i	$0.987219\pi$
$720$		0			0
$721$	−4.96410	−	14.3301i	−0.184873	−	0.533682i
$722$		0			0
$723$	−13.7321	+	7.92820i	−0.510700	+	0.294853i
$724$		0			0
$725$	1.36603	−	2.36603i	0.0507329	−	0.0878720i
$726$		0			0
$727$	−27.7321			−1.02852			−0.514262	−	0.857633i	$-0.671934\pi$
$727$	−27.7321			−1.02852			−0.514262	+	0.857633i	$0.671934\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	3.22243	−	5.58142i	0.119186	−	0.206436i
$732$		0			0
$733$	13.4545	−	7.76795i	0.496953	−	0.286916i	−0.230502	−	0.973072i	$-0.574037\pi$
$733$	13.4545	−	7.76795i	0.496953	−	0.286916i	0.727454	+	0.686156i	$0.240703\pi$
$734$		0			0
$735$	−2.59808	+	6.50000i	−0.0958315	+	0.239756i
$736$		0			0
$737$	3.50962	+	6.07884i	0.129278	+	0.223917i
$738$		0			0
$739$	7.03590	+	4.06218i	0.258820	+	0.149430i	0.623796	−	0.781587i	$-0.285590\pi$
$739$	7.03590	+	4.06218i	0.258820	+	0.149430i	−0.364976	+	0.931017i	$0.618923\pi$
$740$		0			0
$741$			2.46410i			0.0905210i
$742$		0			0
$743$			48.4449i			1.77727i	0.458615	+	0.888635i	$0.348346\pi$
$743$			48.4449i			1.77727i	−0.458615	+	0.888635i	$0.651654\pi$
$744$		0			0
$745$	−9.80385	−	5.66025i	−0.359185	−	0.207376i
$746$		0			0
$747$	2.63397	+	4.56218i	0.0963721	+	0.166921i
$748$		0			0
$749$	33.1699	−	11.4904i	1.21200	−	0.419849i
$750$		0			0
$751$	−19.9641	+	11.5263i	−0.728500	+	0.420600i	−0.817873	−	0.575398i	$-0.804847\pi$
$751$	−19.9641	+	11.5263i	−0.728500	+	0.420600i	0.0893729	+	0.995998i	$0.471514\pi$
$752$		0			0
$753$	−9.56218	+	16.5622i	−0.348465	+	0.603559i
$754$		0			0
$755$	9.85641			0.358711
$756$		0			0
$757$	35.8564			1.30322			0.651612	−	0.758553i	$-0.274093\pi$
$757$	35.8564			1.30322			0.651612	+	0.758553i	$0.274093\pi$
$758$		0			0
$759$	1.73205	−	3.00000i	0.0628695	−	0.108893i
$760$		0			0
$761$	36.2942	−	20.9545i	1.31566	−	0.759599i	0.332637	−	0.943055i	$-0.392062\pi$
$761$	36.2942	−	20.9545i	1.31566	−	0.759599i	0.983028	+	0.183456i	$0.0587284\pi$
$762$		0			0
$763$	7.05256	−	8.14359i	0.255320	−	0.294818i
$764$		0			0
$765$	−2.09808	−	3.63397i	−0.0758561	−	0.131387i
$766$		0			0
$767$	−3.29423	−	1.90192i	−0.118948	−	0.0686745i
$768$		0			0
$769$			46.1244i			1.66329i	0.555310	+	0.831643i	$0.312600\pi$
$769$			46.1244i			1.66329i	−0.555310	+	0.831643i	$0.687400\pi$
$770$		0			0
$771$		−	0.339746i		−	0.0122357i
$772$		0			0
$773$	36.6340	+	21.1506i	1.31763	+	0.760735i	0.983347	−	0.181736i	$-0.0581718\pi$
$773$	36.6340	+	21.1506i	1.31763	+	0.760735i	0.334285	+	0.942472i	$0.391505\pi$
$774$		0			0
$775$	3.23205	+	5.59808i	0.116099	+	0.201089i
$776$		0			0
$777$	21.4808	+	4.13397i	0.770618	+	0.148306i
$778$		0			0
$779$	−8.95448	+	5.16987i	−0.320828	+	0.185230i
$780$		0			0
$781$	−1.05256	+	1.82309i	−0.0376635	+	0.0652351i
$782$		0			0
$783$	2.73205			0.0976355
$784$		0			0
$785$	20.0000			0.713831
$786$		0			0
$787$	5.12436	−	8.87564i	0.182664	−	0.316383i	−0.760123	−	0.649779i	$-0.774861\pi$
$787$	5.12436	−	8.87564i	0.182664	−	0.316383i	0.942787	+	0.333396i	$0.108195\pi$
$788$		0			0
$789$	9.80385	−	5.66025i	0.349026	−	0.201510i
$790$		0			0
$791$	16.6077	+	3.19615i	0.590502	+	0.113642i
$792$		0			0
$793$	−2.26795	−	3.92820i	−0.0805372	−	0.139495i
$794$		0			0
$795$	−4.73205	−	2.73205i	−0.167829	−	0.0968959i
$796$		0			0
$797$		−	33.3731i		−	1.18213i	−0.806622	−	0.591067i	$-0.798707\pi$
$797$		−	33.3731i		−	1.18213i	0.806622	−	0.591067i	$-0.201293\pi$
$798$		0			0
$799$			54.2487i			1.91918i
$800$		0			0
$801$	4.09808	+	2.36603i	0.144798	+	0.0835994i
$802$		0			0
$803$	5.02628	+	8.70577i	0.177374	+	0.307220i
$804$		0			0
$805$	4.73205	−	5.46410i	0.166783	−	0.192584i
$806$		0			0
$807$	13.7321	−	7.92820i	0.483391	−	0.279086i
$808$		0			0
$809$	−23.1962	+	40.1769i	−0.815533	+	1.41255i	0.0934111	+	0.995628i	$0.470223\pi$
$809$	−23.1962	+	40.1769i	−0.815533	+	1.41255i	−0.908944	+	0.416917i	$0.863110\pi$
$810$		0			0
$811$	−50.1051			−1.75943			−0.879714	−	0.475503i	$-0.842266\pi$
$811$	−50.1051			−1.75943			−0.879714	+	0.475503i	$0.842266\pi$
$812$		0			0
$813$	2.00000			0.0701431
$814$		0			0
$815$	2.46410	−	4.26795i	0.0863137	−	0.149500i
$816$		0			0
$817$	−3.27757	+	1.89230i	−0.114668	+	0.0662034i
$818$		0			0
$819$	−2.50000	+	0.866025i	−0.0873571	+	0.0302614i
$820$		0			0
$821$	20.9545	+	36.2942i	0.731316	+	1.26668i	0.956321	+	0.292320i	$0.0944271\pi$
$821$	20.9545	+	36.2942i	0.731316	+	1.26668i	−0.225004	+	0.974358i	$0.572240\pi$
$822$		0			0
$823$	−18.9282	−	10.9282i	−0.659796	−	0.380933i	0.132403	−	0.991196i	$-0.457731\pi$
$823$	−18.9282	−	10.9282i	−0.659796	−	0.380933i	−0.792199	+	0.610263i	$0.791064\pi$
$824$		0			0
$825$			1.26795i			0.0441443i
$826$		0			0
$827$		−	0.143594i		−	0.00499324i	−0.999997	−	0.00249662i	$-0.999205\pi$
$827$		−	0.143594i		−	0.00499324i	0.999997	−	0.00249662i	$-0.000794699\pi$
$828$		0			0
$829$	−32.3038	−	18.6506i	−1.12196	−	0.647763i	−0.180059	−	0.983656i	$-0.557629\pi$
$829$	−32.3038	−	18.6506i	−1.12196	−	0.647763i	−0.941900	+	0.335893i	$0.890962\pi$
$830$		0			0
$831$	−5.59808	−	9.69615i	−0.194195	−	0.336356i
$832$		0			0
$833$	18.1699	+	23.0788i	0.629549	+	0.799634i
$834$		0			0
$835$	6.16987	−	3.56218i	0.213517	−	0.123274i
$836$		0			0
$837$	−3.23205	+	5.59808i	−0.111716	+	0.193498i
$838$		0			0
$839$	32.9808			1.13862			0.569311	−	0.822122i	$-0.307210\pi$
$839$	32.9808			1.13862			0.569311	+	0.822122i	$0.307210\pi$
$840$		0			0
$841$	−21.5359			−0.742617
$842$		0			0
$843$	−6.92820	+	12.0000i	−0.238620	+	0.413302i
$844$		0			0
$845$	10.3923	−	6.00000i	0.357506	−	0.206406i
$846$		0			0
$847$	8.13397	+	23.4808i	0.279487	+	0.806809i
$848$		0			0
$849$	16.5981	+	28.7487i	0.569645	+	0.986653i
$850$		0			0
$851$	−19.5622	−	11.2942i	−0.670583	−	0.387161i
$852$		0			0
$853$			8.60770i			0.294722i	0.989083	+	0.147361i	$0.0470779\pi$
$853$			8.60770i			0.294722i	−0.989083	+	0.147361i	$0.952922\pi$
$854$		0			0
$855$			2.46410i			0.0842705i
$856$		0			0
$857$	−21.8827	−	12.6340i	−0.747498	−	0.431568i	0.0772910	−	0.997009i	$-0.475373\pi$
$857$	−21.8827	−	12.6340i	−0.747498	−	0.431568i	−0.824789	+	0.565440i	$0.808706\pi$
$858$		0			0
$859$	−8.26795	−	14.3205i	−0.282099	−	0.488609i	0.689803	−	0.723997i	$-0.257697\pi$
$859$	−8.26795	−	14.3205i	−0.282099	−	0.488609i	−0.971901	+	0.235388i	$0.924364\pi$
$860$		0			0
$861$	−8.39230	−	7.26795i	−0.286009	−	0.247691i
$862$		0			0
$863$	−3.92820	+	2.26795i	−0.133718	+	0.0772019i	−0.565367	−	0.824840i	$-0.691265\pi$
$863$	−3.92820	+	2.26795i	−0.133718	+	0.0772019i	0.431649	+	0.902042i	$0.357932\pi$
$864$		0			0
$865$	−8.19615	+	14.1962i	−0.278678	+	0.482684i
$866$		0			0
$867$	−0.607695			−0.0206384
$868$		0			0
$869$	5.41154			0.183574
$870$		0			0
$871$	−2.76795	+	4.79423i	−0.0937884	+	0.162446i
$872$		0			0
$873$	4.26795	−	2.46410i	0.144448	−	0.0833972i
$874$		0			0
$875$	−0.500000	+	2.59808i	−0.0169031	+	0.0878310i
$876$		0			0
$877$	−19.3923	−	33.5885i	−0.654832	−	1.13420i	−0.981936	−	0.189213i	$-0.939406\pi$
$877$	−19.3923	−	33.5885i	−0.654832	−	1.13420i	0.327104	−	0.944988i	$-0.393927\pi$
$878$		0			0
$879$	15.1244	+	8.73205i	0.510132	+	0.294525i
$880$		0			0
$881$			48.4974i			1.63392i	0.576695	+	0.816960i	$0.304342\pi$
$881$			48.4974i			1.63392i	−0.576695	+	0.816960i	$0.695658\pi$
$882$		0			0
$883$			34.4641i			1.15981i	0.814684	+	0.579905i	$0.196910\pi$
$883$			34.4641i			1.15981i	−0.814684	+	0.579905i	$0.803090\pi$
$884$		0			0
$885$	−3.29423	−	1.90192i	−0.110734	−	0.0639325i
$886$		0			0
$887$	3.56218	+	6.16987i	0.119606	+	0.207164i	0.919612	−	0.392829i	$-0.128503\pi$
$887$	3.56218	+	6.16987i	0.119606	+	0.207164i	−0.800005	+	0.599993i	$0.795170\pi$
$888$		0			0
$889$	8.69615	−	45.1865i	0.291660	−	1.51551i
$890$		0			0
$891$	−1.09808	+	0.633975i	−0.0367869	+	0.0212389i
$892$		0			0
$893$	15.9282	−	27.5885i	0.533017	−	0.923213i
$894$		0			0
$895$	3.46410			0.115792
$896$		0			0
$897$	2.73205			0.0912205
$898$		0			0
$899$	−8.83013	+	15.2942i	−0.294501	+	0.510091i
$900$		0			0
$901$	−19.8564	+	11.4641i	−0.661513	+	0.381925i
$902$		0			0
$903$	−3.07180	−	2.66025i	−0.102223	−	0.0885277i
$904$		0			0
$905$	12.7942	+	22.1603i	0.425295	+	0.736632i
$906$		0			0
$907$	−29.5981	−	17.0885i	−0.982788	−	0.567413i	−0.0796773	−	0.996821i	$-0.525389\pi$
$907$	−29.5981	−	17.0885i	−0.982788	−	0.567413i	−0.903111	+	0.429408i	$0.858722\pi$
$908$		0			0
$909$		−	5.26795i		−	0.174727i
$910$		0			0
$911$		−	43.3731i		−	1.43701i	−0.695520	−	0.718507i	$-0.744826\pi$
$911$		−	43.3731i		−	1.43701i	0.695520	−	0.718507i	$-0.255174\pi$
$912$		0			0
$913$	−5.78461	−	3.33975i	−0.191443	−	0.110529i
$914$		0			0
$915$	−2.26795	−	3.92820i	−0.0749761	−	0.129862i
$916$		0			0
$917$	−0.464102	−	1.33975i	−0.0153260	−	0.0442423i
$918$		0			0
$919$	−27.5718	+	15.9186i	−0.909510	+	0.525106i	−0.880273	−	0.474467i	$-0.842641\pi$
$919$	−27.5718	+	15.9186i	−0.909510	+	0.525106i	−0.0292363	+	0.999573i	$0.509308\pi$
$920$		0			0
$921$	2.06218	−	3.57180i	0.0679511	−	0.117695i
$922$		0			0
$923$	−1.66025			−0.0546479
$924$		0			0
$925$	8.26795			0.271848
$926$		0			0
$927$	2.86603	−	4.96410i	0.0941326	−	0.163042i
$928$		0			0
$929$	4.90192	−	2.83013i	0.160827	−	0.0928535i	−0.417426	−	0.908711i	$-0.637068\pi$
$929$	4.90192	−	2.83013i	0.160827	−	0.0928535i	0.578253	+	0.815857i	$0.303735\pi$
$930$		0			0
$931$	−2.46410	−	17.0718i	−0.0807577	−	0.559506i
$932$		0			0
$933$	9.09808	+	15.7583i	0.297858	+	0.515905i
$934$		0			0
$935$	4.60770	+	2.66025i	0.150688	+	0.0869996i
$936$		0			0
$937$			23.5359i			0.768884i	0.923149	+	0.384442i	$0.125606\pi$
$937$			23.5359i			0.768884i	−0.923149	+	0.384442i	$0.874394\pi$
$938$		0			0
$939$		−	19.0000i		−	0.620042i
$940$		0			0
$941$	52.6865	+	30.4186i	1.71753	+	0.991618i	0.923370	+	0.383911i	$0.125423\pi$
$941$	52.6865	+	30.4186i	1.71753	+	0.991618i	0.794162	+	0.607706i	$0.207910\pi$
$942$		0			0
$943$	5.73205	+	9.92820i	0.186661	+	0.323307i
$944$		0			0
$945$	−2.50000	+	0.866025i	−0.0813250	+	0.0281718i
$946$		0			0
$947$	−50.1506	+	28.9545i	−1.62968	+	0.940894i	−0.645488	+	0.763770i	$0.723346\pi$
$947$	−50.1506	+	28.9545i	−1.62968	+	0.940894i	−0.984189	+	0.177124i	$0.943321\pi$
$948$		0			0
$949$	−3.96410	+	6.86603i	−0.128680	+	0.222881i
$950$		0			0
$951$	28.0526			0.909667
$952$		0			0
$953$	−40.3923			−1.30844			−0.654218	−	0.756306i	$-0.727002\pi$
$953$	−40.3923			−1.30844			−0.654218	+	0.756306i	$0.727002\pi$
$954$		0			0
$955$	−1.00000	+	1.73205i	−0.0323592	+	0.0560478i
$956$		0			0
$957$	−3.00000	+	1.73205i	−0.0969762	+	0.0559893i
$958$		0			0
$959$	−34.0526	+	39.3205i	−1.09961	+	1.26973i
$960$		0			0
$961$	−5.39230	−	9.33975i	−0.173945	−	0.301282i
$962$		0			0
$963$	11.4904	+	6.63397i	0.370272	+	0.213777i
$964$		0			0
$965$			1.19615i			0.0385055i
$966$		0			0
$967$			16.4641i			0.529450i	0.964324	+	0.264725i	$0.0852812\pi$
$967$			16.4641i			0.529450i	−0.964324	+	0.264725i	$0.914719\pi$
$968$		0			0
$969$	8.95448	+	5.16987i	0.287659	+	0.166080i
$970$		0			0
$971$	14.5359	+	25.1769i	0.466479	+	0.807966i	0.999267	−	0.0382833i	$-0.0121889\pi$
$971$	14.5359	+	25.1769i	0.466479	+	0.807966i	−0.532788	+	0.846249i	$0.678856\pi$
$972$		0			0
$973$	20.5981	+	3.96410i	0.660344	+	0.127083i
$974$		0			0
$975$	−0.866025	+	0.500000i	−0.0277350	+	0.0160128i
$976$		0			0
$977$	−2.83013	+	4.90192i	−0.0905438	+	0.156826i	−0.907740	−	0.419533i	$-0.862194\pi$
$977$	−2.83013	+	4.90192i	−0.0905438	+	0.156826i	0.817196	+	0.576359i	$0.195527\pi$
$978$		0			0
$979$	−6.00000			−0.191761
$980$		0			0
$981$	4.07180			0.130002
$982$		0			0
$983$	14.9019	−	25.8109i	0.475298	−	0.823240i	−0.524302	−	0.851532i	$-0.675674\pi$
$983$	14.9019	−	25.8109i	0.475298	−	0.823240i	0.999600	+	0.0282928i	$0.00900708\pi$
$984$		0			0
$985$	−14.4904	+	8.36603i	−0.461702	+	0.266564i
$986$		0			0
$987$	33.5885	+	6.46410i	1.06913	+	0.205755i
$988$		0			0
$989$	2.09808	+	3.63397i	0.0667149	+	0.115554i
$990$		0			0
$991$	24.2321	+	13.9904i	0.769756	+	0.444419i	0.832788	−	0.553592i	$-0.186743\pi$
$991$	24.2321	+	13.9904i	0.769756	+	0.444419i	−0.0630313	+	0.998012i	$0.520077\pi$
$992$		0			0
$993$			8.66025i			0.274825i
$994$		0			0
$995$		−	23.8564i		−	0.756299i
$996$		0			0
$997$	21.3109	+	12.3038i	0.674923	+	0.389667i	0.797939	−	0.602738i	$-0.205924\pi$
$997$	21.3109	+	12.3038i	0.674923	+	0.389667i	−0.123017	+	0.992405i	$0.539257\pi$
$998$		0			0
$999$	4.13397	+	7.16025i	0.130793	+	0.226540i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.2.dx.a.31.2	✓	4
4.3	odd	2		1680.2.dx.d.31.2	yes	4
7.5	odd	6		1680.2.dx.d.271.2	yes	4
28.19	even	6	inner	1680.2.dx.a.271.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1680.2.dx.a.31.2	✓	4	1.1	even	1	trivial
1680.2.dx.a.271.2	yes	4	28.19	even	6	inner
1680.2.dx.d.31.2	yes	4	4.3	odd	2
1680.2.dx.d.271.2	yes	4	7.5	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1680.dx (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{3} +(0.866025 - 0.500000i) q^{5} +(-2.59808 - 0.500000i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(1.09808 + 0.633975i) q^{11} +1.00000i q^{13} +1.00000i q^{15} +(3.63397 + 2.09808i) q^{17} +(-1.23205 - 2.13397i) q^{19} +(1.73205 - 2.00000i) q^{21} +(-2.36603 + 1.36603i) q^{23} +(0.500000 - 0.866025i) q^{25} +1.00000 q^{27} +2.73205 q^{29} +(-3.23205 + 5.59808i) q^{31} +(-1.09808 + 0.633975i) q^{33} +(-2.50000 + 0.866025i) q^{35} +(4.13397 + 7.16025i) q^{37} +(-0.866025 - 0.500000i) q^{39} -4.19615i q^{41} -1.53590i q^{43} +(-0.866025 - 0.500000i) q^{45} +(6.46410 + 11.1962i) q^{47} +(6.50000 + 2.59808i) q^{49} +(-3.63397 + 2.09808i) q^{51} +(-2.73205 + 4.73205i) q^{53} +1.26795 q^{55} +2.46410 q^{57} +(-1.90192 + 3.29423i) q^{59} +(-3.92820 + 2.26795i) q^{61} +(0.866025 + 2.50000i) q^{63} +(0.500000 + 0.866025i) q^{65} +(4.79423 + 2.76795i) q^{67} -2.73205i q^{69} +1.66025i q^{71} +(6.86603 + 3.96410i) q^{73} +(0.500000 + 0.866025i) q^{75} +(-2.53590 - 2.19615i) q^{77} +(3.69615 - 2.13397i) q^{79} +(-0.500000 + 0.866025i) q^{81} -5.26795 q^{83} +4.19615 q^{85} +(-1.36603 + 2.36603i) q^{87} +(-4.09808 + 2.36603i) q^{89} +(0.500000 - 2.59808i) q^{91} +(-3.23205 - 5.59808i) q^{93} +(-2.13397 - 1.23205i) q^{95} +4.92820i q^{97} -1.26795i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - 2 q^{9} - 6 q^{11} + 18 q^{17} + 2 q^{19} - 6 q^{23} + 2 q^{25} + 4 q^{27} + 4 q^{29} - 6 q^{31} + 6 q^{33} - 10 q^{35} + 20 q^{37} + 12 q^{47} + 26 q^{49} - 18 q^{51} - 4 q^{53} + 12 q^{55}+ \cdots - 12 q^{95}+O(q^{100}) \) 4 * q - 2 * q^3 - 2 * q^9 - 6 * q^11 + 18 * q^17 + 2 * q^19 - 6 * q^23 + 2 * q^25 + 4 * q^27 + 4 * q^29 - 6 * q^31 + 6 * q^33 - 10 * q^35 + 20 * q^37 + 12 * q^47 + 26 * q^49 - 18 * q^51 - 4 * q^53 + 12 * q^55 - 4 * q^57 - 18 * q^59 + 12 * q^61 + 2 * q^65 - 12 * q^67 + 24 * q^73 + 2 * q^75 - 24 * q^77 - 6 * q^79 - 2 * q^81 - 28 * q^83 - 4 * q^85 - 2 * q^87 - 6 * q^89 + 2 * q^91 - 6 * q^93 - 12 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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