LMFDB - Embedded newform 1680.2.dx.a.31.1

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			31.1
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1680.31
Dual form			1680.2.dx.a.271.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.500000 + 0.866025i) q^{3} +(-0.866025 + 0.500000i) q^{5} +(2.59808 + 0.500000i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(-4.09808 - 2.36603i) q^{11} -1.00000i q^{13} -1.00000i q^{15} +(5.36603 + 3.09808i) q^{17} +(2.23205 + 3.86603i) q^{19} +(-1.73205 + 2.00000i) q^{21} +(-0.633975 + 0.366025i) q^{23} +(0.500000 - 0.866025i) q^{25} +1.00000 q^{27} -0.732051 q^{29} +(0.232051 - 0.401924i) q^{31} +(4.09808 - 2.36603i) q^{33} +(-2.50000 + 0.866025i) q^{35} +(5.86603 + 10.1603i) q^{37} +(0.866025 + 0.500000i) q^{39} -6.19615i q^{41} +8.46410i q^{43} +(0.866025 + 0.500000i) q^{45} +(-0.464102 - 0.803848i) q^{47} +(6.50000 + 2.59808i) q^{49} +(-5.36603 + 3.09808i) q^{51} +(0.732051 - 1.26795i) q^{53} +4.73205 q^{55} -4.46410 q^{57} +(-7.09808 + 12.2942i) q^{59} +(9.92820 - 5.73205i) q^{61} +(-0.866025 - 2.50000i) q^{63} +(0.500000 + 0.866025i) q^{65} +(-10.7942 - 6.23205i) q^{67} -0.732051i q^{69} +15.6603i q^{71} +(5.13397 + 2.96410i) q^{73} +(0.500000 + 0.866025i) q^{75} +(-9.46410 - 8.19615i) q^{77} +(-6.69615 + 3.86603i) q^{79} +(-0.500000 + 0.866025i) q^{81} -8.73205 q^{83} -6.19615 q^{85} +(0.366025 - 0.633975i) q^{87} +(1.09808 - 0.633975i) q^{89} +(0.500000 - 2.59808i) q^{91} +(0.232051 + 0.401924i) q^{93} +(-3.86603 - 2.23205i) q^{95} +8.92820i q^{97} +4.73205i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$11$	−4.09808	−	2.36603i	−1.23562	−	0.713384i	−0.267421	−	0.963580i	$-0.586172\pi$
$11$	−4.09808	−	2.36603i	−1.23562	−	0.713384i	−0.968195	+	0.250196i	$0.919505\pi$
$12$		0			0
$13$		−	1.00000i		−	0.277350i	−0.990338	−	0.138675i	$-0.955716\pi$
$13$		−	1.00000i		−	0.277350i	0.990338	−	0.138675i	$-0.0442844\pi$
$14$		0			0
$15$		−	1.00000i		−	0.258199i
$16$		0			0
$17$	5.36603	+	3.09808i	1.30145	+	0.751394i	0.980653	−	0.195753i	$-0.0627152\pi$
$17$	5.36603	+	3.09808i	1.30145	+	0.751394i	0.320799	+	0.947147i	$0.396049\pi$
$18$		0			0
$19$	2.23205	+	3.86603i	0.512068	+	0.886927i	0.999902	+	0.0139909i	$0.00445360\pi$
$19$	2.23205	+	3.86603i	0.512068	+	0.886927i	−0.487835	+	0.872936i	$0.662213\pi$
$20$		0			0
$21$	−1.73205	+	2.00000i	−0.377964	+	0.436436i
$22$		0			0
$23$	−0.633975	+	0.366025i	−0.132193	+	0.0763216i	−0.564638	−	0.825339i	$-0.690984\pi$
$23$	−0.633975	+	0.366025i	−0.132193	+	0.0763216i	0.432445	+	0.901660i	$0.357651\pi$
$24$		0			0
$25$	0.500000	−	0.866025i	0.100000	−	0.173205i
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	−0.732051			−0.135938			−0.0679692	−	0.997687i	$-0.521652\pi$
$29$	−0.732051			−0.135938			−0.0679692	+	0.997687i	$0.521652\pi$
$30$		0			0
$31$	0.232051	−	0.401924i	0.0416776	−	0.0721876i	−0.844434	−	0.535659i	$-0.820063\pi$
$31$	0.232051	−	0.401924i	0.0416776	−	0.0721876i	0.886112	+	0.463472i	$0.153396\pi$
$32$		0			0
$33$	4.09808	−	2.36603i	0.713384	−	0.411872i
$34$		0			0
$35$	−2.50000	+	0.866025i	−0.422577	+	0.146385i
$36$		0			0
$37$	5.86603	+	10.1603i	0.964369	+	1.67034i	0.711302	+	0.702887i	$0.248106\pi$
$37$	5.86603	+	10.1603i	0.964369	+	1.67034i	0.253067	+	0.967449i	$0.418561\pi$
$38$		0			0
$39$	0.866025	+	0.500000i	0.138675	+	0.0800641i
$40$		0			0
$41$		−	6.19615i		−	0.967676i	−0.875157	−	0.483838i	$-0.839242\pi$
$41$		−	6.19615i		−	0.967676i	0.875157	−	0.483838i	$-0.160758\pi$
$42$		0			0
$43$			8.46410i			1.29076i	0.763860	+	0.645382i	$0.223302\pi$
$43$			8.46410i			1.29076i	−0.763860	+	0.645382i	$0.776698\pi$
$44$		0			0
$45$	0.866025	+	0.500000i	0.129099	+	0.0745356i
$46$		0			0
$47$	−0.464102	−	0.803848i	−0.0676962	−	0.117253i	0.830191	−	0.557480i	$-0.188232\pi$
$47$	−0.464102	−	0.803848i	−0.0676962	−	0.117253i	−0.897887	+	0.440226i	$0.854898\pi$
$48$		0			0
$49$	6.50000	+	2.59808i	0.928571	+	0.371154i
$50$		0			0
$51$	−5.36603	+	3.09808i	−0.751394	+	0.433817i
$52$		0			0
$53$	0.732051	−	1.26795i	0.100555	−	0.174166i	−0.811359	−	0.584549i	$-0.801272\pi$
$53$	0.732051	−	1.26795i	0.100555	−	0.174166i	0.911913	+	0.410383i	$0.134605\pi$
$54$		0			0
$55$	4.73205			0.638070
$56$		0			0
$57$	−4.46410			−0.591285
$58$		0			0
$59$	−7.09808	+	12.2942i	−0.924091	+	1.60057i	−0.131074	+	0.991373i	$0.541843\pi$
$59$	−7.09808	+	12.2942i	−0.924091	+	1.60057i	−0.793017	+	0.609200i	$0.791491\pi$
$60$		0			0
$61$	9.92820	−	5.73205i	1.27118	−	0.733914i	0.295967	−	0.955198i	$-0.404358\pi$
$61$	9.92820	−	5.73205i	1.27118	−	0.733914i	0.975209	+	0.221284i	$0.0710249\pi$
$62$		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
$67$	−10.7942	−	6.23205i	−1.31872	−	0.761366i	−0.335201	−	0.942146i	$-0.608804\pi$
$67$	−10.7942	−	6.23205i	−1.31872	−	0.761366i	−0.983524	+	0.180780i	$0.942138\pi$
$68$		0			0
$69$		−	0.732051i		−	0.0881286i
$70$		0			0
$71$			15.6603i			1.85853i	0.369414	+	0.929265i	$0.379559\pi$
$71$			15.6603i			1.85853i	−0.369414	+	0.929265i	$0.620441\pi$
$72$		0			0
$73$	5.13397	+	2.96410i	0.600886	+	0.346922i	0.769390	−	0.638779i	$-0.220560\pi$
$73$	5.13397	+	2.96410i	0.600886	+	0.346922i	−0.168504	+	0.985701i	$0.553894\pi$
$74$		0			0
$75$	0.500000	+	0.866025i	0.0577350	+	0.100000i
$76$		0			0
$77$	−9.46410	−	8.19615i	−1.07853	−	0.934038i
$78$		0			0
$79$	−6.69615	+	3.86603i	−0.753376	+	0.434962i	−0.826912	−	0.562331i	$-0.809905\pi$
$79$	−6.69615	+	3.86603i	−0.753376	+	0.434962i	0.0735364	+	0.997293i	$0.476571\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	−8.73205			−0.958467			−0.479234	−	0.877687i	$-0.659085\pi$
$83$	−8.73205			−0.958467			−0.479234	+	0.877687i	$0.659085\pi$
$84$		0			0
$85$	−6.19615			−0.672067
$86$		0			0
$87$	0.366025	−	0.633975i	0.0392420	−	0.0679692i
$88$		0			0
$89$	1.09808	−	0.633975i	0.116396	−	0.0672012i	−0.440672	−	0.897668i	$-0.645260\pi$
$89$	1.09808	−	0.633975i	0.116396	−	0.0672012i	0.557068	+	0.830467i	$0.311926\pi$
$90$		0			0
$91$	0.500000	−	2.59808i	0.0524142	−	0.272352i
$92$		0			0
$93$	0.232051	+	0.401924i	0.0240625	+	0.0416776i
$94$		0			0
$95$	−3.86603	−	2.23205i	−0.396646	−	0.229004i
$96$		0			0
$97$			8.92820i			0.906522i	0.891378	+	0.453261i	$0.149739\pi$
$97$			8.92820i			0.906522i	−0.891378	+	0.453261i	$0.850261\pi$
$98$		0			0
$99$			4.73205i			0.475589i
$100$		0			0
$101$	−7.56218	−	4.36603i	−0.752465	−	0.434436i	0.0741190	−	0.997249i	$-0.476386\pi$
$101$	−7.56218	−	4.36603i	−0.752465	−	0.434436i	−0.826584	+	0.562814i	$0.809719\pi$
$102$		0			0
$103$	1.13397	+	1.96410i	0.111734	+	0.193529i	0.916469	−	0.400105i	$-0.131026\pi$
$103$	1.13397	+	1.96410i	0.111734	+	0.193529i	−0.804736	+	0.593633i	$0.797693\pi$
$104$		0			0
$105$	0.500000	−	2.59808i	0.0487950	−	0.253546i
$106$		0			0
$107$	14.4904	−	8.36603i	1.40084	−	0.808774i	0.406359	−	0.913713i	$-0.366798\pi$
$107$	14.4904	−	8.36603i	1.40084	−	0.808774i	0.994479	+	0.104939i	$0.0334648\pi$
$108$		0			0
$109$	−8.96410	+	15.5263i	−0.858605	+	1.48715i	0.0146544	+	0.999893i	$0.495335\pi$
$109$	−8.96410	+	15.5263i	−0.858605	+	1.48715i	−0.873260	+	0.487255i	$0.837998\pi$
$110$		0			0
$111$	−11.7321			−1.11356
$112$		0			0
$113$	14.3923			1.35391			0.676957	−	0.736022i	$-0.263298\pi$
$113$	14.3923			1.35391			0.676957	+	0.736022i	$0.263298\pi$
$114$		0			0
$115$	0.366025	−	0.633975i	0.0341320	−	0.0591184i
$116$		0			0
$117$	−0.866025	+	0.500000i	−0.0800641	+	0.0462250i
$118$		0			0
$119$	12.3923	+	10.7321i	1.13600	+	0.983805i
$120$		0			0
$121$	5.69615	+	9.86603i	0.517832	+	0.896911i
$122$		0			0
$123$	5.36603	+	3.09808i	0.483838	+	0.279344i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$			3.39230i			0.301018i	0.988609	+	0.150509i	$0.0480913\pi$
$127$			3.39230i			0.301018i	−0.988609	+	0.150509i	$0.951909\pi$
$128$		0			0
$129$	−7.33013	−	4.23205i	−0.645382	−	0.372611i
$130$		0			0
$131$	3.73205	+	6.46410i	0.326071	+	0.564771i	0.981728	−	0.190287i	$-0.0609419\pi$
$131$	3.73205	+	6.46410i	0.326071	+	0.564771i	−0.655658	+	0.755058i	$0.727609\pi$
$132$		0			0
$133$	3.86603	+	11.1603i	0.335227	+	0.967717i
$134$		0			0
$135$	−0.866025	+	0.500000i	−0.0745356	+	0.0430331i
$136$		0			0
$137$	1.16987	−	2.02628i	0.0999490	−	0.173117i	−0.811714	−	0.584055i	$-0.801465\pi$
$137$	1.16987	−	2.02628i	0.0999490	−	0.173117i	0.911663	+	0.410938i	$0.134799\pi$
$138$		0			0
$139$	5.92820			0.502824			0.251412	−	0.967880i	$-0.419105\pi$
$139$	5.92820			0.502824			0.251412	+	0.967880i	$0.419105\pi$
$140$		0			0
$141$	0.928203			0.0781688
$142$		0			0
$143$	−2.36603	+	4.09808i	−0.197857	+	0.342698i
$144$		0			0
$145$	0.633975	−	0.366025i	0.0526487	−	0.0303968i
$146$		0			0
$147$	−5.50000	+	4.33013i	−0.453632	+	0.357143i
$148$		0			0
$149$	11.6603	+	20.1962i	0.955245	+	1.65453i	0.733806	+	0.679359i	$0.237742\pi$
$149$	11.6603	+	20.1962i	0.955245	+	1.65453i	0.221439	+	0.975174i	$0.428925\pi$
$150$		0			0
$151$	15.4641	+	8.92820i	1.25845	+	0.726567i	0.972773	−	0.231759i	$-0.0744481\pi$
$151$	15.4641	+	8.92820i	1.25845	+	0.726567i	0.285677	+	0.958326i	$0.407781\pi$
$152$		0			0
$153$		−	6.19615i		−	0.500929i
$154$		0			0
$155$			0.464102i			0.0372775i
$156$		0			0
$157$	−17.3205	−	10.0000i	−1.38233	−	0.798087i	−0.389892	−	0.920860i	$-0.627488\pi$
$157$	−17.3205	−	10.0000i	−1.38233	−	0.798087i	−0.992435	+	0.122774i	$0.960821\pi$
$158$		0			0
$159$	0.732051	+	1.26795i	0.0580554	+	0.100555i
$160$		0			0
$161$	−1.83013	+	0.633975i	−0.144234	+	0.0499642i
$162$		0			0
$163$	7.73205	−	4.46410i	0.605621	−	0.349655i	−0.165629	−	0.986188i	$-0.552965\pi$
$163$	7.73205	−	4.46410i	0.605621	−	0.349655i	0.771250	+	0.636533i	$0.219632\pi$
$164$		0			0
$165$	−2.36603	+	4.09808i	−0.184195	+	0.319035i
$166$		0			0
$167$	−17.1244			−1.32512			−0.662561	−	0.749008i	$-0.730531\pi$
$167$	−17.1244			−1.32512			−0.662561	+	0.749008i	$0.730531\pi$
$168$		0			0
$169$	12.0000			0.923077
$170$		0			0
$171$	2.23205	−	3.86603i	0.170689	−	0.295642i
$172$		0			0
$173$	−3.80385	+	2.19615i	−0.289201	+	0.166970i	−0.637582	−	0.770383i	$-0.720065\pi$
$173$	−3.80385	+	2.19615i	−0.289201	+	0.166970i	0.348380	+	0.937353i	$0.386732\pi$
$174$		0			0
$175$	1.73205	−	2.00000i	0.130931	−	0.151186i
$176$		0			0
$177$	−7.09808	−	12.2942i	−0.533524	−	0.924091i
$178$		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
$181$			5.58846i			0.415387i	0.978194	+	0.207693i	$0.0665956\pi$
$181$			5.58846i			0.415387i	−0.978194	+	0.207693i	$0.933404\pi$
$182$		0			0
$183$			11.4641i			0.847451i
$184$		0			0
$185$	−10.1603	−	5.86603i	−0.746997	−	0.431279i
$186$		0			0
$187$	−14.6603	−	25.3923i	−1.07206	−	1.85687i
$188$		0			0
$189$	2.59808	+	0.500000i	0.188982	+	0.0363696i
$190$		0			0
$191$	1.73205	−	1.00000i	0.125327	−	0.0723575i	−0.436026	−	0.899934i	$-0.643614\pi$
$191$	1.73205	−	1.00000i	0.125327	−	0.0723575i	0.561353	+	0.827577i	$0.310281\pi$
$192$		0			0
$193$	4.59808	−	7.96410i	0.330977	−	0.573269i	−0.651727	−	0.758454i	$-0.725955\pi$
$193$	4.59808	−	7.96410i	0.330977	−	0.573269i	0.982704	+	0.185185i	$0.0592885\pi$
$194$		0			0
$195$	−1.00000			−0.0716115
$196$		0			0
$197$	−13.2679			−0.945302			−0.472651	−	0.881250i	$-0.656703\pi$
$197$	−13.2679			−0.945302			−0.472651	+	0.881250i	$0.656703\pi$
$198$		0			0
$199$	−1.92820	+	3.33975i	−0.136687	+	0.236748i	−0.926241	−	0.376933i	$-0.876979\pi$
$199$	−1.92820	+	3.33975i	−0.136687	+	0.236748i	0.789554	+	0.613681i	$0.210312\pi$
$200$		0			0
$201$	10.7942	−	6.23205i	0.761366	−	0.439575i
$202$		0			0
$203$	−1.90192	−	0.366025i	−0.133489	−	0.0256899i
$204$		0			0
$205$	3.09808	+	5.36603i	0.216379	+	0.374779i
$206$		0			0
$207$	0.633975	+	0.366025i	0.0440643	+	0.0254405i
$208$		0			0
$209$		−	21.1244i		−	1.46120i
$210$		0			0
$211$		−	24.7846i		−	1.70624i	−0.521712	−	0.853121i	$-0.674707\pi$
$211$		−	24.7846i		−	1.70624i	0.521712	−	0.853121i	$-0.325293\pi$
$212$		0			0
$213$	−13.5622	−	7.83013i	−0.929265	−	0.536511i
$214$		0			0
$215$	−4.23205	−	7.33013i	−0.288623	−	0.499911i
$216$		0			0
$217$	0.803848	−	0.928203i	0.0545687	−	0.0630105i
$218$		0			0
$219$	−5.13397	+	2.96410i	−0.346922	+	0.200295i
$220$		0			0
$221$	3.09808	−	5.36603i	0.208399	−	0.360958i
$222$		0			0
$223$	−4.39230			−0.294130			−0.147065	−	0.989127i	$-0.546983\pi$
$223$	−4.39230			−0.294130			−0.147065	+	0.989127i	$0.546983\pi$
$224$		0			0
$225$	−1.00000			−0.0666667
$226$		0			0
$227$	8.02628	−	13.9019i	0.532723	−	0.922703i	−0.466547	−	0.884496i	$-0.654502\pi$
$227$	8.02628	−	13.9019i	0.532723	−	0.922703i	0.999270	−	0.0382067i	$-0.0121645\pi$
$228$		0			0
$229$	5.89230	−	3.40192i	0.389374	−	0.224805i	−0.292515	−	0.956261i	$-0.594492\pi$
$229$	5.89230	−	3.40192i	0.389374	−	0.224805i	0.681889	+	0.731456i	$0.261159\pi$
$230$		0			0
$231$	11.8301	−	4.09808i	0.778365	−	0.269634i
$232$		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
$235$	0.803848	+	0.464102i	0.0524372	+	0.0302747i
$236$		0			0
$237$		−	7.73205i		−	0.502251i
$238$		0			0
$239$		−	5.32051i		−	0.344155i	−0.985083	−	0.172078i	$-0.944952\pi$
$239$		−	5.32051i		−	0.344155i	0.985083	−	0.172078i	$-0.0550480\pi$
$240$		0			0
$241$	10.2679	+	5.92820i	0.661417	+	0.381869i	0.792817	−	0.609460i	$-0.208614\pi$
$241$	10.2679	+	5.92820i	0.661417	+	0.381869i	−0.131400	+	0.991329i	$0.541947\pi$
$242$		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
$247$	3.86603	−	2.23205i	0.245989	−	0.142022i
$248$		0			0
$249$	4.36603	−	7.56218i	0.276686	−	0.479234i
$250$		0			0
$251$	−5.12436			−0.323446			−0.161723	−	0.986836i	$-0.551705\pi$
$251$	−5.12436			−0.323446			−0.161723	+	0.986836i	$0.551705\pi$
$252$		0			0
$253$	3.46410			0.217786
$254$		0			0
$255$	3.09808	−	5.36603i	0.194009	−	0.336034i
$256$		0			0
$257$	15.2942	−	8.83013i	0.954028	−	0.550808i	0.0596980	−	0.998216i	$-0.480986\pi$
$257$	15.2942	−	8.83013i	0.954028	−	0.550808i	0.894330	+	0.447408i	$0.147653\pi$
$258$		0			0
$259$	10.1603	+	29.3301i	0.631327	+	1.82249i
$260$		0			0
$261$	0.366025	+	0.633975i	0.0226564	+	0.0392420i
$262$		0			0
$263$	−20.1962	−	11.6603i	−1.24535	−	0.719002i	−0.275170	−	0.961396i	$-0.588734\pi$
$263$	−20.1962	−	11.6603i	−1.24535	−	0.719002i	−0.970178	+	0.242393i	$0.922068\pi$
$264$		0			0
$265$			1.46410i			0.0899390i
$266$		0			0
$267$			1.26795i			0.0775972i
$268$		0			0
$269$	−10.2679	−	5.92820i	−0.626048	−	0.361449i	0.153172	−	0.988200i	$-0.451051\pi$
$269$	−10.2679	−	5.92820i	−0.626048	−	0.361449i	−0.779220	+	0.626751i	$0.784384\pi$
$270$		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
$275$	−4.09808	+	2.36603i	−0.247123	+	0.142677i
$276$		0			0
$277$	−0.401924	+	0.696152i	−0.0241493	+	0.0418277i	−0.877847	−	0.478940i	$-0.841021\pi$
$277$	−0.401924	+	0.696152i	−0.0241493	+	0.0418277i	0.853698	+	0.520768i	$0.174354\pi$
$278$		0			0
$279$	−0.464102			−0.0277850
$280$		0			0
$281$	−13.8564			−0.826604			−0.413302	−	0.910594i	$-0.635625\pi$
$281$	−13.8564			−0.826604			−0.413302	+	0.910594i	$0.635625\pi$
$282$		0			0
$283$	11.4019	−	19.7487i	0.677774	−	1.17394i	−0.297876	−	0.954605i	$-0.596278\pi$
$283$	11.4019	−	19.7487i	0.677774	−	1.17394i	0.975650	−	0.219334i	$-0.0703886\pi$
$284$		0			0
$285$	3.86603	−	2.23205i	0.229004	−	0.132215i
$286$		0			0
$287$	3.09808	−	16.0981i	0.182874	−	0.950239i
$288$		0			0
$289$	10.6962	+	18.5263i	0.629185	+	1.08978i
$290$		0			0
$291$	−7.73205	−	4.46410i	−0.453261	−	0.261690i
$292$		0			0
$293$			10.5359i			0.615514i	0.951465	+	0.307757i	$0.0995783\pi$
$293$			10.5359i			0.615514i	−0.951465	+	0.307757i	$0.900422\pi$
$294$		0			0
$295$		−	14.1962i		−	0.826532i
$296$		0			0
$297$	−4.09808	−	2.36603i	−0.237795	−	0.137291i
$298$		0			0
$299$	0.366025	+	0.633975i	0.0211678	+	0.0366637i
$300$		0			0
$301$	−4.23205	+	21.9904i	−0.243931	+	1.26750i
$302$		0			0
$303$	7.56218	−	4.36603i	0.434436	−	0.250822i
$304$		0			0
$305$	−5.73205	+	9.92820i	−0.328216	+	0.568487i
$306$		0			0
$307$	20.1244			1.14856			0.574279	−	0.818660i	$-0.305283\pi$
$307$	20.1244			1.14856			0.574279	+	0.818660i	$0.305283\pi$
$308$		0			0
$309$	−2.26795			−0.129019
$310$		0			0
$311$	3.90192	−	6.75833i	0.221258	−	0.383230i	−0.733932	−	0.679223i	$-0.762317\pi$
$311$	3.90192	−	6.75833i	0.221258	−	0.383230i	0.955190	+	0.295993i	$0.0956504\pi$
$312$		0			0
$313$	16.4545	−	9.50000i	0.930062	−	0.536972i	0.0432311	−	0.999065i	$-0.486235\pi$
$313$	16.4545	−	9.50000i	0.930062	−	0.536972i	0.886831	+	0.462093i	$0.152902\pi$
$314$		0			0
$315$	2.00000	+	1.73205i	0.112687	+	0.0975900i
$316$		0			0
$317$	5.02628	+	8.70577i	0.282304	+	0.488965i	0.971952	−	0.235180i	$-0.0755680\pi$
$317$	5.02628	+	8.70577i	0.282304	+	0.488965i	−0.689648	+	0.724145i	$0.742235\pi$
$318$		0			0
$319$	3.00000	+	1.73205i	0.167968	+	0.0969762i
$320$		0			0
$321$			16.7321i			0.933892i
$322$		0			0
$323$			27.6603i			1.53906i
$324$		0			0
$325$	−0.866025	−	0.500000i	−0.0480384	−	0.0277350i
$326$		0			0
$327$	−8.96410	−	15.5263i	−0.495716	−	0.858605i
$328$		0			0
$329$	−0.803848	−	2.32051i	−0.0443176	−	0.127934i
$330$		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
$333$	5.86603	−	10.1603i	0.321456	−	0.556778i
$334$		0			0
$335$	12.4641			0.680987
$336$		0			0
$337$	−25.4449			−1.38607			−0.693035	−	0.720904i	$-0.743727\pi$
$337$	−25.4449			−1.38607			−0.693035	+	0.720904i	$0.743727\pi$
$338$		0			0
$339$	−7.19615	+	12.4641i	−0.390841	+	0.676957i
$340$		0			0
$341$	−1.90192	+	1.09808i	−0.102995	+	0.0594642i
$342$		0			0
$343$	15.5885	+	10.0000i	0.841698	+	0.539949i
$344$		0			0
$345$	0.366025	+	0.633975i	0.0197061	+	0.0341320i
$346$		0			0
$347$	−2.19615	−	1.26795i	−0.117896	−	0.0680671i	0.439893	−	0.898050i	$-0.355017\pi$
$347$	−2.19615	−	1.26795i	−0.117896	−	0.0680671i	−0.557788	+	0.829983i	$0.688350\pi$
$348$		0			0
$349$		−	26.9282i		−	1.44143i	−0.693230	−	0.720717i	$-0.743813\pi$
$349$		−	26.9282i		−	1.44143i	0.693230	−	0.720717i	$-0.256187\pi$
$350$		0			0
$351$		−	1.00000i		−	0.0533761i
$352$		0			0
$353$	−12.7583	−	7.36603i	−0.679057	−	0.392054i	0.120442	−	0.992720i	$-0.461569\pi$
$353$	−12.7583	−	7.36603i	−0.679057	−	0.392054i	−0.799500	+	0.600666i	$0.794902\pi$
$354$		0			0
$355$	−7.83013	−	13.5622i	−0.415580	−	0.719806i
$356$		0			0
$357$	−15.4904	+	5.36603i	−0.819838	+	0.284000i
$358$		0			0
$359$	4.90192	−	2.83013i	0.258714	−	0.149368i	−0.365034	−	0.930994i	$-0.618943\pi$
$359$	4.90192	−	2.83013i	0.258714	−	0.149368i	0.623748	+	0.781626i	$0.285609\pi$
$360$		0			0
$361$	−0.464102	+	0.803848i	−0.0244264	+	0.0423078i
$362$		0			0
$363$	−11.3923			−0.597941
$364$		0			0
$365$	−5.92820			−0.310296
$366$		0			0
$367$	11.5981	−	20.0885i	0.605415	−	1.04861i	−0.386571	−	0.922260i	$-0.626341\pi$
$367$	11.5981	−	20.0885i	0.605415	−	1.04861i	0.991986	−	0.126349i	$-0.0403260\pi$
$368$		0			0
$369$	−5.36603	+	3.09808i	−0.279344	+	0.161279i
$370$		0			0
$371$	2.53590	−	2.92820i	0.131657	−	0.152025i
$372$		0			0
$373$	14.9904	+	25.9641i	0.776173	+	1.34437i	0.934133	+	0.356925i	$0.116175\pi$
$373$	14.9904	+	25.9641i	0.776173	+	1.34437i	−0.157961	+	0.987445i	$0.550492\pi$
$374$		0			0
$375$	−0.866025	−	0.500000i	−0.0447214	−	0.0258199i
$376$		0			0
$377$			0.732051i			0.0377025i
$378$		0			0
$379$		−	12.5167i		−	0.642938i	−0.946920	−	0.321469i	$-0.895823\pi$
$379$		−	12.5167i		−	0.642938i	0.946920	−	0.321469i	$-0.104177\pi$
$380$		0			0
$381$	−2.93782	−	1.69615i	−0.150509	−	0.0868965i
$382$		0			0
$383$	2.92820	+	5.07180i	0.149624	+	0.259157i	0.931089	−	0.364793i	$-0.118860\pi$
$383$	2.92820	+	5.07180i	0.149624	+	0.259157i	−0.781464	+	0.623950i	$0.785527\pi$
$384$		0			0
$385$	12.2942	+	2.36603i	0.626572	+	0.120584i
$386$		0			0
$387$	7.33013	−	4.23205i	0.372611	−	0.215127i
$388$		0			0
$389$	7.75833	−	13.4378i	0.393363	−	0.681325i	−0.599528	−	0.800354i	$-0.704645\pi$
$389$	7.75833	−	13.4378i	0.393363	−	0.681325i	0.992891	+	0.119029i	$0.0379782\pi$
$390$		0			0
$391$	−4.53590			−0.229390
$392$		0			0
$393$	−7.46410			−0.376514
$394$		0			0
$395$	3.86603	−	6.69615i	0.194521	−	0.336920i
$396$		0			0
$397$	−10.7942	+	6.23205i	−0.541747	+	0.312778i	−0.745787	−	0.666185i	$-0.767926\pi$
$397$	−10.7942	+	6.23205i	−0.541747	+	0.312778i	0.204040	+	0.978963i	$0.434593\pi$
$398$		0			0
$399$	−11.5981	−	2.23205i	−0.580630	−	0.111742i
$400$		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
$403$	−0.401924	−	0.232051i	−0.0200212	−	0.0115593i
$404$		0			0
$405$		−	1.00000i		−	0.0496904i
$406$		0			0
$407$		−	55.5167i		−	2.75186i
$408$		0			0
$409$	5.42820	+	3.13397i	0.268407	+	0.154965i	0.628164	−	0.778081i	$-0.283807\pi$
$409$	5.42820	+	3.13397i	0.268407	+	0.154965i	−0.359756	+	0.933046i	$0.617140\pi$
$410$		0			0
$411$	1.16987	+	2.02628i	0.0577056	+	0.0999490i
$412$		0			0
$413$	−24.5885	+	28.3923i	−1.20992	+	1.39709i
$414$		0			0
$415$	7.56218	−	4.36603i	0.371213	−	0.214320i
$416$		0			0
$417$	−2.96410	+	5.13397i	−0.145153	+	0.251412i
$418$		0			0
$419$	8.92820			0.436171			0.218086	−	0.975930i	$-0.430019\pi$
$419$	8.92820			0.436171			0.218086	+	0.975930i	$0.430019\pi$
$420$		0			0
$421$	−12.0718			−0.588343			−0.294172	−	0.955753i	$-0.595044\pi$
$421$	−12.0718			−0.588343			−0.294172	+	0.955753i	$0.595044\pi$
$422$		0			0
$423$	−0.464102	+	0.803848i	−0.0225654	+	0.0390844i
$424$		0			0
$425$	5.36603	−	3.09808i	0.260290	−	0.150279i
$426$		0			0
$427$	28.6603	−	9.92820i	1.38697	−	0.480459i
$428$		0			0
$429$	−2.36603	−	4.09808i	−0.114233	−	0.197857i
$430$		0			0
$431$	2.66025	+	1.53590i	0.128140	+	0.0739816i	0.562700	−	0.826661i	$-0.309763\pi$
$431$	2.66025	+	1.53590i	0.128140	+	0.0739816i	−0.434560	+	0.900643i	$0.643096\pi$
$432$		0			0
$433$		−	26.4641i		−	1.27178i	−0.771778	−	0.635892i	$-0.780632\pi$
$433$		−	26.4641i		−	1.27178i	0.771778	−	0.635892i	$-0.219368\pi$
$434$		0			0
$435$			0.732051i			0.0350991i
$436$		0			0
$437$	−2.83013	−	1.63397i	−0.135383	−	0.0781636i
$438$		0			0
$439$	2.26795	+	3.92820i	0.108243	+	0.187483i	0.915059	−	0.403321i	$-0.132144\pi$
$439$	2.26795	+	3.92820i	0.108243	+	0.187483i	−0.806815	+	0.590804i	$0.798811\pi$
$440$		0			0
$441$	−1.00000	−	6.92820i	−0.0476190	−	0.329914i
$442$		0			0
$443$	16.3923	−	9.46410i	0.778822	−	0.449653i	−0.0571907	−	0.998363i	$-0.518214\pi$
$443$	16.3923	−	9.46410i	0.778822	−	0.449653i	0.836013	+	0.548710i	$0.184881\pi$
$444$		0			0
$445$	−0.633975	+	1.09808i	−0.0300533	+	0.0520538i
$446$		0			0
$447$	−23.3205			−1.10302
$448$		0			0
$449$	21.3205			1.00618			0.503088	−	0.864235i	$-0.332197\pi$
$449$	21.3205			1.00618			0.503088	+	0.864235i	$0.332197\pi$
$450$		0			0
$451$	−14.6603	+	25.3923i	−0.690324	+	1.19568i
$452$		0			0
$453$	−15.4641	+	8.92820i	−0.726567	+	0.419484i
$454$		0			0
$455$	0.866025	+	2.50000i	0.0405999	+	0.117202i
$456$		0			0
$457$	−1.13397	−	1.96410i	−0.0530451	−	0.0918768i	0.838284	−	0.545234i	$-0.183559\pi$
$457$	−1.13397	−	1.96410i	−0.0530451	−	0.0918768i	−0.891329	+	0.453358i	$0.850226\pi$
$458$		0			0
$459$	5.36603	+	3.09808i	0.250465	+	0.144606i
$460$		0			0
$461$		−	32.1962i		−	1.49952i	−0.661707	−	0.749762i	$-0.730168\pi$
$461$		−	32.1962i		−	1.49952i	0.661707	−	0.749762i	$-0.269832\pi$
$462$		0			0
$463$			23.7846i			1.10536i	0.833392	+	0.552682i	$0.186396\pi$
$463$			23.7846i			1.10536i	−0.833392	+	0.552682i	$0.813604\pi$
$464$		0			0
$465$	−0.401924	−	0.232051i	−0.0186388	−	0.0107611i
$466$		0			0
$467$	−6.85641	−	11.8756i	−0.317277	−	0.549539i	0.662642	−	0.748936i	$-0.269435\pi$
$467$	−6.85641	−	11.8756i	−0.317277	−	0.549539i	−0.979919	+	0.199397i	$0.936102\pi$
$468$		0			0
$469$	−24.9282	−	21.5885i	−1.15108	−	0.996862i
$470$		0			0
$471$	17.3205	−	10.0000i	0.798087	−	0.460776i
$472$		0			0
$473$	20.0263	−	34.6865i	0.920809	−	1.59489i
$474$		0			0
$475$	4.46410			0.204827
$476$		0			0
$477$	−1.46410			−0.0670366
$478$		0			0
$479$	−16.9282	+	29.3205i	−0.773469	+	1.33969i	0.162181	+	0.986761i	$0.448147\pi$
$479$	−16.9282	+	29.3205i	−0.773469	+	1.33969i	−0.935651	+	0.352927i	$0.885186\pi$
$480$		0			0
$481$	10.1603	−	5.86603i	0.463268	−	0.267468i
$482$		0			0
$483$	0.366025	−	1.90192i	0.0166547	−	0.0865405i
$484$		0			0
$485$	−4.46410	−	7.73205i	−0.202704	−	0.351094i
$486$		0			0
$487$	−27.6506	−	15.9641i	−1.25297	−	0.723402i	−0.281272	−	0.959628i	$-0.590756\pi$
$487$	−27.6506	−	15.9641i	−1.25297	−	0.723402i	−0.971698	+	0.236226i	$0.924090\pi$
$488$		0			0
$489$			8.92820i			0.403747i
$490$		0			0
$491$			37.4641i			1.69073i	0.534188	+	0.845366i	$0.320617\pi$
$491$			37.4641i			1.69073i	−0.534188	+	0.845366i	$0.679383\pi$
$492$		0			0
$493$	−3.92820	−	2.26795i	−0.176917	−	0.102143i
$494$		0			0
$495$	−2.36603	−	4.09808i	−0.106345	−	0.184195i
$496$		0			0
$497$	−7.83013	+	40.6865i	−0.351229	+	1.82504i
$498$		0			0
$499$	1.83975	−	1.06218i	0.0823583	−	0.0475496i	−0.458255	−	0.888821i	$-0.651525\pi$
$499$	1.83975	−	1.06218i	0.0823583	−	0.0475496i	0.540614	+	0.841271i	$0.318192\pi$
$500$		0			0
$501$	8.56218	−	14.8301i	0.382530	−	0.662561i
$502$		0			0
$503$	23.0718			1.02872			0.514360	−	0.857574i	$-0.328029\pi$
$503$	23.0718			1.02872			0.514360	+	0.857574i	$0.328029\pi$
$504$		0			0
$505$	8.73205			0.388571
$506$		0			0
$507$	−6.00000	+	10.3923i	−0.266469	+	0.461538i
$508$		0			0
$509$	31.9808	−	18.4641i	1.41752	−	0.818407i	0.421442	−	0.906856i	$-0.361524\pi$
$509$	31.9808	−	18.4641i	1.41752	−	0.818407i	0.996081	+	0.0884486i	$0.0281909\pi$
$510$		0			0
$511$	11.8564	+	10.2679i	0.524497	+	0.454227i
$512$		0			0
$513$	2.23205	+	3.86603i	0.0985475	+	0.170689i
$514$		0			0
$515$	−1.96410	−	1.13397i	−0.0865487	−	0.0499689i
$516$		0			0
$517$			4.39230i			0.193173i
$518$		0			0
$519$		−	4.39230i		−	0.192801i
$520$		0			0
$521$	−35.9090	−	20.7321i	−1.57320	−	0.908288i	−0.995773	−	0.0918454i	$-0.970723\pi$
$521$	−35.9090	−	20.7321i	−1.57320	−	0.908288i	−0.577427	−	0.816442i	$-0.695943\pi$
$522$		0			0
$523$	−13.5981	−	23.5526i	−0.594602	−	1.02988i	−0.993603	−	0.112931i	$-0.963976\pi$
$523$	−13.5981	−	23.5526i	−0.594602	−	1.02988i	0.399001	−	0.916951i	$-0.369357\pi$
$524$		0			0
$525$	0.866025	+	2.50000i	0.0377964	+	0.109109i
$526$		0			0
$527$	2.49038	−	1.43782i	0.108483	−	0.0626325i
$528$		0			0
$529$	−11.2321	+	19.4545i	−0.488350	+	0.845847i
$530$		0			0
$531$	14.1962			0.616061
$532$		0			0
$533$	−6.19615			−0.268385
$534$		0			0
$535$	−8.36603	+	14.4904i	−0.361695	+	0.626474i
$536$		0			0
$537$	−3.00000	+	1.73205i	−0.129460	+	0.0747435i
$538$		0			0
$539$	−20.4904	−	26.0263i	−0.882583	−	1.12103i
$540$		0			0
$541$	2.89230	+	5.00962i	0.124350	+	0.215380i	0.921479	−	0.388429i	$-0.126982\pi$
$541$	2.89230	+	5.00962i	0.124350	+	0.215380i	−0.797129	+	0.603809i	$0.793649\pi$
$542$		0			0
$543$	−4.83975	−	2.79423i	−0.207693	−	0.119912i
$544$		0			0
$545$		−	17.9282i		−	0.767960i
$546$		0			0
$547$		−	26.3923i		−	1.12845i	−0.825620	−	0.564227i	$-0.809174\pi$
$547$		−	26.3923i		−	1.12845i	0.825620	−	0.564227i	$-0.190826\pi$
$548$		0			0
$549$	−9.92820	−	5.73205i	−0.423725	−	0.244638i
$550$		0			0
$551$	−1.63397	−	2.83013i	−0.0696097	−	0.120567i
$552$		0			0
$553$	−19.3301	+	6.69615i	−0.822001	+	0.284749i
$554$		0			0
$555$	10.1603	−	5.86603i	0.431279	−	0.248999i
$556$		0			0
$557$	−9.73205	+	16.8564i	−0.412360	+	0.714229i	−0.995147	−	0.0983960i	$-0.968629\pi$
$557$	−9.73205	+	16.8564i	−0.412360	+	0.714229i	0.582787	+	0.812625i	$0.301962\pi$
$558$		0			0
$559$	8.46410			0.357993
$560$		0			0
$561$	29.3205			1.23791
$562$		0			0
$563$	−3.19615	+	5.53590i	−0.134702	+	0.233310i	−0.925483	−	0.378788i	$-0.876341\pi$
$563$	−3.19615	+	5.53590i	−0.134702	+	0.233310i	0.790782	+	0.612098i	$0.209674\pi$
$564$		0			0
$565$	−12.4641	+	7.19615i	−0.524369	+	0.302744i
$566$		0			0
$567$	−1.73205	+	2.00000i	−0.0727393	+	0.0839921i
$568$		0			0
$569$	4.29423	+	7.43782i	0.180023	+	0.311810i	0.941888	−	0.335926i	$-0.109049\pi$
$569$	4.29423	+	7.43782i	0.180023	+	0.311810i	−0.761865	+	0.647736i	$0.775716\pi$
$570$		0			0
$571$	−16.6244	−	9.59808i	−0.695708	−	0.401667i	0.110039	−	0.993927i	$-0.464902\pi$
$571$	−16.6244	−	9.59808i	−0.695708	−	0.401667i	−0.805747	+	0.592260i	$0.798236\pi$
$572$		0			0
$573$			2.00000i			0.0835512i
$574$		0			0
$575$			0.732051i			0.0305286i
$576$		0			0
$577$	−33.6506	−	19.4282i	−1.40089	−	0.808807i	−0.406410	−	0.913691i	$-0.633219\pi$
$577$	−33.6506	−	19.4282i	−1.40089	−	0.808807i	−0.994484	+	0.104884i	$0.966553\pi$
$578$		0			0
$579$	4.59808	+	7.96410i	0.191090	+	0.330977i
$580$		0			0
$581$	−22.6865	−	4.36603i	−0.941196	−	0.181133i
$582$		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
$587$	9.66025			0.398721			0.199361	−	0.979926i	$-0.436113\pi$
$587$	9.66025			0.398721			0.199361	+	0.979926i	$0.436113\pi$
$588$		0			0
$589$	2.07180			0.0853669
$590$		0			0
$591$	6.63397	−	11.4904i	0.272885	−	0.472651i
$592$		0			0
$593$	32.2750	−	18.6340i	1.32538	−	0.765206i	0.340795	−	0.940138i	$-0.389304\pi$
$593$	32.2750	−	18.6340i	1.32538	−	0.765206i	0.984580	+	0.174932i	$0.0559706\pi$
$594$		0			0
$595$	−16.0981	−	3.09808i	−0.659957	−	0.127009i
$596$		0			0
$597$	−1.92820	−	3.33975i	−0.0789161	−	0.136687i
$598$		0			0
$599$	4.39230	+	2.53590i	0.179465	+	0.103614i	0.587041	−	0.809557i	$-0.300293\pi$
$599$	4.39230	+	2.53590i	0.179465	+	0.103614i	−0.407576	+	0.913171i	$0.633626\pi$
$600$		0			0
$601$			11.3397i			0.462558i	0.972887	+	0.231279i	$0.0742910\pi$
$601$			11.3397i			0.462558i	−0.972887	+	0.231279i	$0.925709\pi$
$602$		0			0
$603$			12.4641i			0.507577i
$604$		0			0
$605$	−9.86603	−	5.69615i	−0.401111	−	0.231582i
$606$		0			0
$607$	21.5981	+	37.4090i	0.876639	+	1.51838i	0.855007	+	0.518617i	$0.173553\pi$
$607$	21.5981	+	37.4090i	0.876639	+	1.51838i	0.0216322	+	0.999766i	$0.493114\pi$
$608$		0			0
$609$	1.26795	−	1.46410i	0.0513799	−	0.0593284i
$610$		0			0
$611$	−0.803848	+	0.464102i	−0.0325202	+	0.0187755i
$612$		0			0
$613$	−19.3205	+	33.4641i	−0.780348	+	1.35160i	0.151391	+	0.988474i	$0.451625\pi$
$613$	−19.3205	+	33.4641i	−0.780348	+	1.35160i	−0.931739	+	0.363128i	$0.881709\pi$
$614$		0			0
$615$	−6.19615			−0.249853
$616$		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
$621$	−0.633975	+	0.366025i	−0.0254405	+	0.0146881i
$622$		0			0
$623$	3.16987	−	1.09808i	0.126998	−	0.0439935i
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.0200000	−	0.0346410i
$626$		0			0
$627$	18.2942	+	10.5622i	0.730601	+	0.421813i
$628$		0			0
$629$			72.6936i			2.89848i
$630$		0			0
$631$		−	24.5359i		−	0.976759i	−0.872631	−	0.488379i	$-0.837588\pi$
$631$		−	24.5359i		−	0.976759i	0.872631	−	0.488379i	$-0.162412\pi$
$632$		0			0
$633$	21.4641	+	12.3923i	0.853121	+	0.492550i
$634$		0			0
$635$	−1.69615	−	2.93782i	−0.0673098	−	0.116584i
$636$		0			0
$637$	2.59808	−	6.50000i	0.102940	−	0.257539i
$638$		0			0
$639$	13.5622	−	7.83013i	0.536511	−	0.309755i
$640$		0			0
$641$	−18.2224	+	31.5622i	−0.719743	+	1.24663i	0.241359	+	0.970436i	$0.422407\pi$
$641$	−18.2224	+	31.5622i	−0.719743	+	1.24663i	−0.961102	+	0.276195i	$0.910926\pi$
$642$		0			0
$643$	31.9808			1.26120			0.630599	−	0.776109i	$-0.282809\pi$
$643$	31.9808			1.26120			0.630599	+	0.776109i	$0.282809\pi$
$644$		0			0
$645$	8.46410			0.333274
$646$		0			0
$647$	3.09808	−	5.36603i	0.121798	−	0.210960i	−0.798679	−	0.601758i	$-0.794467\pi$
$647$	3.09808	−	5.36603i	0.121798	−	0.210960i	0.920477	+	0.390797i	$0.127801\pi$
$648$		0			0
$649$	58.1769	−	33.5885i	2.28364	−	1.31846i
$650$		0			0
$651$	0.401924	+	1.16025i	0.0157526	+	0.0454739i
$652$		0			0
$653$	−4.70577	−	8.15064i	−0.184151	−	0.318959i	0.759139	−	0.650928i	$-0.225620\pi$
$653$	−4.70577	−	8.15064i	−0.184151	−	0.318959i	−0.943290	+	0.331969i	$0.892287\pi$
$654$		0			0
$655$	−6.46410	−	3.73205i	−0.252573	−	0.145823i
$656$		0			0
$657$		−	5.92820i		−	0.231281i
$658$		0			0
$659$			32.7846i			1.27711i	0.769577	+	0.638554i	$0.220467\pi$
$659$			32.7846i			1.27711i	−0.769577	+	0.638554i	$0.779533\pi$
$660$		0			0
$661$	41.5526	+	23.9904i	1.61621	+	0.933118i	0.987889	+	0.155161i	$0.0495897\pi$
$661$	41.5526	+	23.9904i	1.61621	+	0.933118i	0.628318	+	0.777957i	$0.283744\pi$
$662$		0			0
$663$	3.09808	+	5.36603i	0.120319	+	0.208399i
$664$		0			0
$665$	−8.92820	−	7.73205i	−0.346221	−	0.299836i
$666$		0			0
$667$	0.464102	−	0.267949i	0.0179701	−	0.0103750i
$668$		0			0
$669$	2.19615	−	3.80385i	0.0849082	−	0.147065i
$670$		0			0
$671$	−54.2487			−2.09425
$672$		0			0
$673$	−29.4449			−1.13502			−0.567508	−	0.823368i	$-0.692092\pi$
$673$	−29.4449			−1.13502			−0.567508	+	0.823368i	$0.692092\pi$
$674$		0			0
$675$	0.500000	−	0.866025i	0.0192450	−	0.0333333i
$676$		0			0
$677$	18.8827	−	10.9019i	0.725721	−	0.418995i	−0.0911340	−	0.995839i	$-0.529049\pi$
$677$	18.8827	−	10.9019i	0.725721	−	0.418995i	0.816855	+	0.576844i	$0.195716\pi$
$678$		0			0
$679$	−4.46410	+	23.1962i	−0.171316	+	0.890187i
$680$		0			0
$681$	8.02628	+	13.9019i	0.307568	+	0.532723i
$682$		0			0
$683$	29.9545	+	17.2942i	1.14618	+	0.661745i	0.947953	−	0.318412i	$-0.103149\pi$
$683$	29.9545	+	17.2942i	1.14618	+	0.661745i	0.198224	+	0.980157i	$0.436483\pi$
$684$		0			0
$685$			2.33975i			0.0893971i
$686$		0			0
$687$			6.80385i			0.259583i
$688$		0			0
$689$	−1.26795	−	0.732051i	−0.0483050	−	0.0278889i
$690$		0			0
$691$	−18.2846	−	31.6699i	−0.695579	−	1.20478i	−0.969985	−	0.243165i	$-0.921814\pi$
$691$	−18.2846	−	31.6699i	−0.695579	−	1.20478i	0.274406	−	0.961614i	$-0.411519\pi$
$692$		0			0
$693$	−2.36603	+	12.2942i	−0.0898779	+	0.467019i
$694$		0			0
$695$	−5.13397	+	2.96410i	−0.194743	+	0.112435i
$696$		0			0
$697$	19.1962	−	33.2487i	0.727106	−	1.25938i
$698$		0			0
$699$	6.00000			0.226941
$700$		0			0
$701$	−45.9090			−1.73396			−0.866979	−	0.498345i	$-0.833941\pi$
$701$	−45.9090			−1.73396			−0.866979	+	0.498345i	$0.833941\pi$
$702$		0			0
$703$	−26.1865	+	45.3564i	−0.987644	+	1.71065i
$704$		0			0
$705$	−0.803848	+	0.464102i	−0.0302747	+	0.0174791i
$706$		0			0
$707$	−17.4641	−	15.1244i	−0.656805	−	0.568810i
$708$		0			0
$709$	21.8564	+	37.8564i	0.820835	+	1.42173i	0.905062	+	0.425281i	$0.139825\pi$
$709$	21.8564	+	37.8564i	0.820835	+	1.42173i	−0.0842270	+	0.996447i	$0.526842\pi$
$710$		0			0
$711$	6.69615	+	3.86603i	0.251125	+	0.144987i
$712$		0			0
$713$			0.339746i			0.0127236i
$714$		0			0
$715$		−	4.73205i		−	0.176969i
$716$		0			0
$717$	4.60770	+	2.66025i	0.172078	+	0.0993490i
$718$		0			0
$719$	−5.53590	−	9.58846i	−0.206454	−	0.357589i	0.744141	−	0.668023i	$-0.232859\pi$
$719$	−5.53590	−	9.58846i	−0.206454	−	0.357589i	−0.950595	+	0.310434i	$0.899526\pi$
$720$		0			0
$721$	1.96410	+	5.66987i	0.0731470	+	0.211157i
$722$		0			0
$723$	−10.2679	+	5.92820i	−0.381869	+	0.220472i
$724$		0			0
$725$	−0.366025	+	0.633975i	−0.0135938	+	0.0235452i
$726$		0			0
$727$	−24.2679			−0.900048			−0.450024	−	0.893016i	$-0.648585\pi$
$727$	−24.2679			−0.900048			−0.450024	+	0.893016i	$0.648585\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−26.2224	+	45.4186i	−0.969872	+	1.67987i
$732$		0			0
$733$	−19.4545	+	11.2321i	−0.718568	+	0.414865i	−0.814225	−	0.580549i	$-0.802838\pi$
$733$	−19.4545	+	11.2321i	−0.718568	+	0.414865i	0.0956576	+	0.995414i	$0.469505\pi$
$734$		0			0
$735$	2.59808	−	6.50000i	0.0958315	−	0.239756i
$736$		0			0
$737$	29.4904	+	51.0788i	1.08629	+	1.88151i
$738$		0			0
$739$	13.9641	+	8.06218i	0.513678	+	0.296572i	0.734344	−	0.678777i	$-0.237490\pi$
$739$	13.9641	+	8.06218i	0.513678	+	0.296572i	−0.220666	+	0.975349i	$0.570823\pi$
$740$		0			0
$741$			4.46410i			0.163993i
$742$		0			0
$743$			10.4449i			0.383185i	0.981475	+	0.191592i	$0.0613652\pi$
$743$			10.4449i			0.383185i	−0.981475	+	0.191592i	$0.938635\pi$
$744$		0			0
$745$	−20.1962	−	11.6603i	−0.739930	−	0.427199i
$746$		0			0
$747$	4.36603	+	7.56218i	0.159745	+	0.276686i
$748$		0			0
$749$	41.8301	−	14.4904i	1.52844	−	0.529467i
$750$		0			0
$751$	−13.0359	+	7.52628i	−0.475687	+	0.274638i	−0.718617	−	0.695406i	$-0.755225\pi$
$751$	−13.0359	+	7.52628i	−0.475687	+	0.274638i	0.242931	+	0.970044i	$0.421891\pi$
$752$		0			0
$753$	2.56218	−	4.43782i	0.0933710	−	0.161723i
$754$		0			0
$755$	−17.8564			−0.649861
$756$		0			0
$757$	8.14359			0.295984			0.147992	−	0.988989i	$-0.452719\pi$
$757$	8.14359			0.295984			0.147992	+	0.988989i	$0.452719\pi$
$758$		0			0
$759$	−1.73205	+	3.00000i	−0.0628695	+	0.108893i
$760$		0			0
$761$	20.7058	−	11.9545i	0.750584	−	0.433350i	−0.0753211	−	0.997159i	$-0.523998\pi$
$761$	20.7058	−	11.9545i	0.750584	−	0.433350i	0.825905	+	0.563810i	$0.190665\pi$
$762$		0			0
$763$	−31.0526	+	35.8564i	−1.12418	+	1.29809i
$764$		0			0
$765$	3.09808	+	5.36603i	0.112011	+	0.194009i
$766$		0			0
$767$	12.2942	+	7.09808i	0.443919	+	0.256297i
$768$		0			0
$769$		−	21.8756i		−	0.788856i	−0.918927	−	0.394428i	$-0.870943\pi$
$769$		−	21.8756i		−	0.788856i	0.918927	−	0.394428i	$-0.129057\pi$
$770$		0			0
$771$			17.6603i			0.636019i
$772$		0			0
$773$	38.3660	+	22.1506i	1.37993	+	0.796703i	0.992151	−	0.125049i	$-0.0399089\pi$
$773$	38.3660	+	22.1506i	1.37993	+	0.796703i	0.387779	+	0.921752i	$0.373242\pi$
$774$		0			0
$775$	−0.232051	−	0.401924i	−0.00833551	−	0.0144375i
$776$		0			0
$777$	−30.4808	−	5.86603i	−1.09349	−	0.210442i
$778$		0			0
$779$	23.9545	−	13.8301i	0.858258	−	0.495516i
$780$		0			0
$781$	37.0526	−	64.1769i	1.32584	−	2.29643i
$782$		0			0
$783$	−0.732051			−0.0261614
$784$		0			0
$785$	20.0000			0.713831
$786$		0			0
$787$	−19.1244	+	33.1244i	−0.681710	+	1.18076i	0.292749	+	0.956189i	$0.405430\pi$
$787$	−19.1244	+	33.1244i	−0.681710	+	1.18076i	−0.974459	+	0.224566i	$0.927903\pi$
$788$		0			0
$789$	20.1962	−	11.6603i	0.719002	−	0.415116i
$790$		0			0
$791$	37.3923	+	7.19615i	1.32952	+	0.255866i
$792$		0			0
$793$	−5.73205	−	9.92820i	−0.203551	−	0.352561i
$794$		0			0
$795$	−1.26795	−	0.732051i	−0.0449695	−	0.0259632i
$796$		0			0
$797$		−	39.3731i		−	1.39467i	−0.716747	−	0.697333i	$-0.754370\pi$
$797$		−	39.3731i		−	1.39467i	0.716747	−	0.697333i	$-0.245630\pi$
$798$		0			0
$799$		−	5.75129i		−	0.203466i
$800$		0			0
$801$	−1.09808	−	0.633975i	−0.0387986	−	0.0224004i
$802$		0			0
$803$	−14.0263	−	24.2942i	−0.494977	−	0.857325i
$804$		0			0
$805$	1.26795	−	1.46410i	0.0446893	−	0.0516028i
$806$		0			0
$807$	10.2679	−	5.92820i	0.361449	−	0.208683i
$808$		0			0
$809$	−12.8038	+	22.1769i	−0.450159	+	0.779699i	−0.998396	−	0.0566247i	$-0.981966\pi$
$809$	−12.8038	+	22.1769i	−0.450159	+	0.779699i	0.548236	+	0.836324i	$0.315299\pi$
$810$		0			0
$811$	26.1051			0.916675			0.458337	−	0.888778i	$-0.348445\pi$
$811$	26.1051			0.916675			0.458337	+	0.888778i	$0.348445\pi$
$812$		0			0
$813$	2.00000			0.0701431
$814$		0			0
$815$	−4.46410	+	7.73205i	−0.156371	+	0.270842i
$816$		0			0
$817$	−32.7224	+	18.8923i	−1.14481	+	0.660958i
$818$		0			0
$819$	−2.50000	+	0.866025i	−0.0873571	+	0.0302614i
$820$		0			0
$821$	−11.9545	−	20.7058i	−0.417214	−	0.722636i	0.578444	−	0.815722i	$-0.303660\pi$
$821$	−11.9545	−	20.7058i	−0.417214	−	0.722636i	−0.995658	+	0.0930859i	$0.970327\pi$
$822$		0			0
$823$	−5.07180	−	2.92820i	−0.176792	−	0.102071i	0.408993	−	0.912538i	$-0.365880\pi$
$823$	−5.07180	−	2.92820i	−0.176792	−	0.102071i	−0.585784	+	0.810467i	$0.699213\pi$
$824$		0			0
$825$		−	4.73205i		−	0.164749i
$826$		0			0
$827$			27.8564i			0.968662i	0.874885	+	0.484331i	$0.160937\pi$
$827$			27.8564i			0.968662i	−0.874885	+	0.484331i	$0.839063\pi$
$828$		0			0
$829$	−42.6962	−	24.6506i	−1.48290	−	0.856152i	−0.483087	−	0.875572i	$-0.660485\pi$
$829$	−42.6962	−	24.6506i	−1.48290	−	0.856152i	−0.999811	+	0.0194203i	$0.993818\pi$
$830$		0			0
$831$	−0.401924	−	0.696152i	−0.0139426	−	0.0241493i
$832$		0			0
$833$	26.8301	+	34.0788i	0.929609	+	1.18076i
$834$		0			0
$835$	14.8301	−	8.56218i	0.513218	−	0.296306i
$836$		0			0
$837$	0.232051	−	0.401924i	0.00802085	−	0.0138925i
$838$		0			0
$839$	−18.9808			−0.655289			−0.327644	−	0.944801i	$-0.606255\pi$
$839$	−18.9808			−0.655289			−0.327644	+	0.944801i	$0.606255\pi$
$840$		0			0
$841$	−28.4641			−0.981521
$842$		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
$847$	9.86603	+	28.4808i	0.339001	+	0.978611i
$848$		0			0
$849$	11.4019	+	19.7487i	0.391313	+	0.677774i
$850$		0			0
$851$	−7.43782	−	4.29423i	−0.254965	−	0.147204i
$852$		0			0
$853$		−	29.3923i		−	1.00637i	−0.864178	−	0.503187i	$-0.832161\pi$
$853$		−	29.3923i		−	1.00637i	0.864178	−	0.503187i	$-0.167839\pi$
$854$		0			0
$855$			4.46410i			0.152669i
$856$		0			0
$857$	24.8827	+	14.3660i	0.849976	+	0.490734i	0.860643	−	0.509209i	$-0.170062\pi$
$857$	24.8827	+	14.3660i	0.849976	+	0.490734i	−0.0106665	+	0.999943i	$0.503395\pi$
$858$		0			0
$859$	−11.7321	−	20.3205i	−0.400292	−	0.693327i	0.593469	−	0.804857i	$-0.297758\pi$
$859$	−11.7321	−	20.3205i	−0.400292	−	0.693327i	−0.993761	+	0.111530i	$0.964425\pi$
$860$		0			0
$861$	12.3923	+	10.7321i	0.422329	+	0.365747i
$862$		0			0
$863$	9.92820	−	5.73205i	0.337960	−	0.195121i	−0.321410	−	0.946940i	$-0.604157\pi$
$863$	9.92820	−	5.73205i	0.337960	−	0.195121i	0.659370	+	0.751819i	$0.270823\pi$
$864$		0			0
$865$	2.19615	−	3.80385i	0.0746714	−	0.129335i
$866$		0			0
$867$	−21.3923			−0.726521
$868$		0			0
$869$	36.5885			1.24118
$870$		0			0
$871$	−6.23205	+	10.7942i	−0.211165	+	0.365748i
$872$		0			0
$873$	7.73205	−	4.46410i	0.261690	−	0.151087i
$874$		0			0
$875$	−0.500000	+	2.59808i	−0.0169031	+	0.0878310i
$876$		0			0
$877$	1.39230	+	2.41154i	0.0470148	+	0.0814320i	0.888575	−	0.458731i	$-0.151696\pi$
$877$	1.39230	+	2.41154i	0.0470148	+	0.0814320i	−0.841560	+	0.540163i	$0.818363\pi$
$878$		0			0
$879$	−9.12436	−	5.26795i	−0.307757	−	0.177684i
$880$		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
$883$		−	27.5359i		−	0.926657i	−0.886187	−	0.463328i	$-0.846655\pi$
$883$		−	27.5359i		−	0.926657i	0.886187	−	0.463328i	$-0.153345\pi$
$884$		0			0
$885$	12.2942	+	7.09808i	0.413266	+	0.238599i
$886$		0			0
$887$	−8.56218	−	14.8301i	−0.287490	−	0.497947i	0.685720	−	0.727865i	$-0.259487\pi$
$887$	−8.56218	−	14.8301i	−0.287490	−	0.497947i	−0.973210	+	0.229918i	$0.926154\pi$
$888$		0			0
$889$	−1.69615	+	8.81347i	−0.0568871	+	0.295594i
$890$		0			0
$891$	4.09808	−	2.36603i	0.137291	−	0.0792648i
$892$		0			0
$893$	2.07180	−	3.58846i	0.0693300	−	0.120083i
$894$		0			0
$895$	−3.46410			−0.115792
$896$		0			0
$897$	−0.732051			−0.0244425
$898$		0			0
$899$	−0.169873	+	0.294229i	−0.00566558	+	0.00981307i
$900$		0			0
$901$	7.85641	−	4.53590i	0.261735	−	0.151113i
$902$		0			0
$903$	−16.9282	−	14.6603i	−0.563335	−	0.487863i
$904$		0			0
$905$	−2.79423	−	4.83975i	−0.0928833	−	0.160879i
$906$		0			0
$907$	−24.4019	−	14.0885i	−0.810253	−	0.467800i	0.0367910	−	0.999323i	$-0.488286\pi$
$907$	−24.4019	−	14.0885i	−0.810253	−	0.467800i	−0.847044	+	0.531523i	$0.821620\pi$
$908$		0			0
$909$			8.73205i			0.289624i
$910$		0			0
$911$		−	29.3731i		−	0.973173i	−0.873632	−	0.486587i	$-0.838242\pi$
$911$		−	29.3731i		−	0.973173i	0.873632	−	0.486587i	$-0.161758\pi$
$912$		0			0
$913$	35.7846	+	20.6603i	1.18430	+	0.683755i
$914$		0			0
$915$	−5.73205	−	9.92820i	−0.189496	−	0.328216i
$916$		0			0
$917$	6.46410	+	18.6603i	0.213463	+	0.616216i
$918$		0			0
$919$	−41.4282	+	23.9186i	−1.36659	+	0.789001i	−0.990491	−	0.137578i	$-0.956068\pi$
$919$	−41.4282	+	23.9186i	−1.36659	+	0.789001i	−0.376099	+	0.926579i	$0.622735\pi$
$920$		0			0
$921$	−10.0622	+	17.4282i	−0.331560	+	0.574279i
$922$		0			0
$923$	15.6603			0.515464
$924$		0			0
$925$	11.7321			0.385747
$926$		0			0
$927$	1.13397	−	1.96410i	0.0372446	−	0.0645096i
$928$		0			0
$929$	10.0981	−	5.83013i	0.331307	−	0.191280i	−0.325114	−	0.945675i	$-0.605403\pi$
$929$	10.0981	−	5.83013i	0.331307	−	0.191280i	0.656421	+	0.754395i	$0.272069\pi$
$930$		0			0
$931$	4.46410	+	30.9282i	0.146305	+	1.01363i
$932$		0			0
$933$	3.90192	+	6.75833i	0.127743	+	0.221258i
$934$		0			0
$935$	25.3923	+	14.6603i	0.830417	+	0.479442i
$936$		0			0
$937$		−	30.4641i		−	0.995219i	−0.867401	−	0.497609i	$-0.834211\pi$
$937$		−	30.4641i		−	0.995219i	0.867401	−	0.497609i	$-0.165789\pi$
$938$		0			0
$939$			19.0000i			0.620042i
$940$		0			0
$941$	16.3135	+	9.41858i	0.531804	+	0.307037i	0.741751	−	0.670676i	$-0.233996\pi$
$941$	16.3135	+	9.41858i	0.531804	+	0.307037i	−0.209947	+	0.977713i	$0.567329\pi$
$942$		0			0
$943$	2.26795	+	3.92820i	0.0738546	+	0.127920i
$944$		0			0
$945$	−2.50000	+	0.866025i	−0.0813250	+	0.0281718i
$946$		0			0
$947$	−6.84936	+	3.95448i	−0.222574	+	0.128503i	−0.607142	−	0.794594i	$-0.707684\pi$
$947$	−6.84936	+	3.95448i	−0.222574	+	0.128503i	0.384567	+	0.923097i	$0.374351\pi$
$948$		0			0
$949$	2.96410	−	5.13397i	0.0962188	−	0.166656i
$950$		0			0
$951$	−10.0526			−0.325977
$952$		0			0
$953$	−19.6077			−0.635156			−0.317578	−	0.948232i	$-0.602869\pi$
$953$	−19.6077			−0.635156			−0.317578	+	0.948232i	$0.602869\pi$
$954$		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
$959$	4.05256	−	4.67949i	0.130864	−	0.151109i
$960$		0			0
$961$	15.3923	+	26.6603i	0.496526	+	0.860008i
$962$		0			0
$963$	−14.4904	−	8.36603i	−0.466946	−	0.269591i
$964$		0			0
$965$			9.19615i			0.296035i
$966$		0			0
$967$		−	9.53590i		−	0.306654i	−0.988176	−	0.153327i	$-0.951001\pi$
$967$		−	9.53590i		−	0.306654i	0.988176	−	0.153327i	$-0.0489988\pi$
$968$		0			0
$969$	−23.9545	−	13.8301i	−0.769529	−	0.444288i
$970$		0			0
$971$	21.4641	+	37.1769i	0.688816	+	1.19306i	0.972221	+	0.234064i	$0.0752025\pi$
$971$	21.4641	+	37.1769i	0.688816	+	1.19306i	−0.283405	+	0.959000i	$0.591464\pi$
$972$		0			0
$973$	15.4019	+	2.96410i	0.493763	+	0.0950247i
$974$		0			0
$975$	0.866025	−	0.500000i	0.0277350	−	0.0160128i
$976$		0			0
$977$	5.83013	−	10.0981i	0.186522	−	0.323066i	−0.757566	−	0.652758i	$-0.773612\pi$
$977$	5.83013	−	10.0981i	0.186522	−	0.323066i	0.944088	+	0.329692i	$0.106945\pi$
$978$		0			0
$979$	−6.00000			−0.191761
$980$		0			0
$981$	17.9282			0.572403
$982$		0			0
$983$	20.0981	−	34.8109i	0.641029	−	1.11029i	−0.344174	−	0.938906i	$-0.611841\pi$
$983$	20.0981	−	34.8109i	0.641029	−	1.11029i	0.985203	−	0.171389i	$-0.0548255\pi$
$984$		0			0
$985$	11.4904	−	6.63397i	0.366114	−	0.211376i
$986$		0			0
$987$	2.41154	+	0.464102i	0.0767603	+	0.0147725i
$988$		0			0
$989$	−3.09808	−	5.36603i	−0.0985131	−	0.170630i
$990$		0			0
$991$	20.7679	+	11.9904i	0.659716	+	0.380887i	0.792169	−	0.610302i	$-0.208952\pi$
$991$	20.7679	+	11.9904i	0.659716	+	0.380887i	−0.132453	+	0.991189i	$0.542285\pi$
$992$		0			0
$993$			8.66025i			0.274825i
$994$		0			0
$995$		−	3.85641i		−	0.122256i
$996$		0			0
$997$	−39.3109	−	22.6962i	−1.24499	−	0.718794i	−0.274883	−	0.961478i	$-0.588639\pi$
$997$	−39.3109	−	22.6962i	−1.24499	−	0.718794i	−0.970106	+	0.242683i	$0.921972\pi$
$998$		0			0
$999$	5.86603	+	10.1603i	0.185593	+	0.321456i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.2.dx.a.31.1	✓	4
4.3	odd	2		1680.2.dx.d.31.1	yes	4
7.5	odd	6		1680.2.dx.d.271.1	yes	4
28.19	even	6	inner	1680.2.dx.a.271.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1680.2.dx.a.31.1	✓	4	1.1	even	1	trivial
1680.2.dx.a.271.1	yes	4	28.19	even	6	inner
1680.2.dx.d.31.1	yes	4	4.3	odd	2
1680.2.dx.d.271.1	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)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{3} +(-0.866025 + 0.500000i) q^{5} +(2.59808 + 0.500000i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(-4.09808 - 2.36603i) q^{11} -1.00000i q^{13} -1.00000i q^{15} +(5.36603 + 3.09808i) q^{17} +(2.23205 + 3.86603i) q^{19} +(-1.73205 + 2.00000i) q^{21} +(-0.633975 + 0.366025i) q^{23} +(0.500000 - 0.866025i) q^{25} +1.00000 q^{27} -0.732051 q^{29} +(0.232051 - 0.401924i) q^{31} +(4.09808 - 2.36603i) q^{33} +(-2.50000 + 0.866025i) q^{35} +(5.86603 + 10.1603i) q^{37} +(0.866025 + 0.500000i) q^{39} -6.19615i q^{41} +8.46410i q^{43} +(0.866025 + 0.500000i) q^{45} +(-0.464102 - 0.803848i) q^{47} +(6.50000 + 2.59808i) q^{49} +(-5.36603 + 3.09808i) q^{51} +(0.732051 - 1.26795i) q^{53} +4.73205 q^{55} -4.46410 q^{57} +(-7.09808 + 12.2942i) q^{59} +(9.92820 - 5.73205i) q^{61} +(-0.866025 - 2.50000i) q^{63} +(0.500000 + 0.866025i) q^{65} +(-10.7942 - 6.23205i) q^{67} -0.732051i q^{69} +15.6603i q^{71} +(5.13397 + 2.96410i) q^{73} +(0.500000 + 0.866025i) q^{75} +(-9.46410 - 8.19615i) q^{77} +(-6.69615 + 3.86603i) q^{79} +(-0.500000 + 0.866025i) q^{81} -8.73205 q^{83} -6.19615 q^{85} +(0.366025 - 0.633975i) q^{87} +(1.09808 - 0.633975i) q^{89} +(0.500000 - 2.59808i) q^{91} +(0.232051 + 0.401924i) q^{93} +(-3.86603 - 2.23205i) q^{95} +8.92820i q^{97} +4.73205i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - 2 q^{9} - 6 q^{11} + 18 q^{17} + 2 q^{19} - 6 q^{23} + 2 q^{25} + 4 q^{27} + 4 q^{29} - 6 q^{31} + 6 q^{33} - 10 q^{35} + 20 q^{37} + 12 q^{47} + 26 q^{49} - 18 q^{51} - 4 q^{53} + 12 q^{55}+ \cdots - 12 q^{95}+O(q^{100}) \) 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

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