LMFDB - Embedded newform 168.5.z.b.145.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [168,5,Mod(73,168)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(168, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("168.73");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 168 = 2^{3} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	168.z (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$17.3661537981$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 130 x^{14} + 6137 x^{12} + 133906 x^{10} + 1360384 x^{8} + 5425142 x^{6} + 5784425 x^{4} + \cdots + 117649 $ x^16 + 130x^14 + 6137x^12 + 133906x^10 + 1360384x^8 + 5425142x^6 + 5784425x^4 + 1717862*x^2 + 117649
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{36}\cdot 7^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			145.8
Root			$-0.997354i$ of defining polynomial
Character	$\chi$	$=$	168.145
Dual form			168.5.z.b.73.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.50000 - 2.59808i) q^{3} +(32.1100 + 18.5387i) q^{5} +(1.01846 - 48.9894i) q^{7} +(13.5000 - 23.3827i) q^{9} +O(q^{10})$	\(q+(4.50000 - 2.59808i) q^{3} +(32.1100 + 18.5387i) q^{5} +(1.01846 - 48.9894i) q^{7} +(13.5000 - 23.3827i) q^{9} +(120.631 + 208.939i) q^{11} -291.510i q^{13} +192.660 q^{15} +(-159.313 + 91.9794i) q^{17} +(208.531 + 120.395i) q^{19} +(-122.695 - 223.098i) q^{21} +(-84.1664 + 145.780i) q^{23} +(374.866 + 649.287i) q^{25} -140.296i q^{27} +672.393 q^{29} +(-221.438 + 127.848i) q^{31} +(1085.68 + 626.818i) q^{33} +(940.902 - 1554.17i) q^{35} +(1200.13 - 2078.69i) q^{37} +(-757.366 - 1311.80i) q^{39} -414.009i q^{41} +1501.09 q^{43} +(866.969 - 500.545i) q^{45} +(2331.33 + 1345.99i) q^{47} +(-2398.93 - 99.7877i) q^{49} +(-477.939 + 827.815i) q^{51} +(690.764 + 1196.44i) q^{53} +8945.37i q^{55} +1251.18 q^{57} +(-362.924 + 209.534i) q^{59} +(-1058.36 - 611.042i) q^{61} +(-1131.75 - 685.171i) q^{63} +(5404.22 - 9360.38i) q^{65} +(-931.198 - 1612.88i) q^{67} +874.683i q^{69} -8242.72 q^{71} +(-1897.67 + 1095.62i) q^{73} +(3373.79 + 1947.86i) q^{75} +(10358.7 - 5696.85i) q^{77} +(-3204.95 + 5551.13i) q^{79} +(-364.500 - 631.333i) q^{81} +3622.04i q^{83} -6820.71 q^{85} +(3025.77 - 1746.93i) q^{87} +(-9857.75 - 5691.37i) q^{89} +(-14280.9 - 296.892i) q^{91} +(-664.315 + 1150.63i) q^{93} +(4463.94 + 7731.77i) q^{95} -5492.74i q^{97} +6514.08 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 72 q^{3} - 12 q^{5} + 16 q^{7} + 216 q^{9}+O(q^{10}) $ 16 * q + 72 * q^3 - 12 * q^5 + 16 * q^7 + 216 * q^9	$ 16 q + 72 q^{3} - 12 q^{5} + 16 q^{7} + 216 q^{9} + 252 q^{11} - 72 q^{15} - 696 q^{17} + 156 q^{19} + 108 q^{21} - 672 q^{23} + 84 q^{25} + 1992 q^{29} + 2040 q^{31} + 2268 q^{33} + 2712 q^{35} + 2548 q^{37} - 396 q^{39} + 1304 q^{43} - 324 q^{45} - 744 q^{47} - 5608 q^{49} - 2088 q^{51} - 1164 q^{53} + 936 q^{57} + 8988 q^{59} + 816 q^{61} + 540 q^{63} + 8760 q^{65} + 3044 q^{67} - 4464 q^{71} - 15828 q^{73} + 756 q^{75} + 996 q^{77} - 11144 q^{79} - 5832 q^{81} - 15344 q^{85} + 8964 q^{87} + 22248 q^{89} + 1596 q^{91} + 6120 q^{93} + 3840 q^{95} + 13608 q^{99}+O(q^{100}) $ 16 * q + 72 * q^3 - 12 * q^5 + 16 * q^7 + 216 * q^9 + 252 * q^11 - 72 * q^15 - 696 * q^17 + 156 * q^19 + 108 * q^21 - 672 * q^23 + 84 * q^25 + 1992 * q^29 + 2040 * q^31 + 2268 * q^33 + 2712 * q^35 + 2548 * q^37 - 396 * q^39 + 1304 * q^43 - 324 * q^45 - 744 * q^47 - 5608 * q^49 - 2088 * q^51 - 1164 * q^53 + 936 * q^57 + 8988 * q^59 + 816 * q^61 + 540 * q^63 + 8760 * q^65 + 3044 * q^67 - 4464 * q^71 - 15828 * q^73 + 756 * q^75 + 996 * q^77 - 11144 * q^79 - 5832 * q^81 - 15344 * q^85 + 8964 * q^87 + 22248 * q^89 + 1596 * q^91 + 6120 * q^93 + 3840 * q^95 + 13608 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/168\mathbb{Z}\right)^\times$.

$n$	$73$	$85$	$113$	$127$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$1$	$1$	$1$

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
$3$	4.50000	−	2.59808i	0.500000	−	0.288675i
$3$	4.50000	−	2.59808i	0.500000	−	0.288675i
$4$		0			0
$5$	32.1100	+	18.5387i	1.28440	+	0.741548i	0.977649	−	0.210243i	$-0.0674255\pi$
$5$	32.1100	+	18.5387i	1.28440	+	0.741548i	0.306749	+	0.951790i	$0.400759\pi$
$6$		0			0
$7$	1.01846	−	48.9894i	0.0207849	−	0.999784i
$7$	1.01846	−	48.9894i	0.0207849	−	0.999784i
$8$		0			0
$9$	13.5000	−	23.3827i	0.166667	−	0.288675i
$10$		0			0
$11$	120.631	+	208.939i	0.996952	+	1.72677i	0.566046	+	0.824374i	$0.308473\pi$
$11$	120.631	+	208.939i	0.996952	+	1.72677i	0.430906	+	0.902397i	$0.358194\pi$
$12$		0			0
$13$		−	291.510i		−	1.72491i	−0.506132	−	0.862456i	$-0.668925\pi$
$13$		−	291.510i		−	1.72491i	0.506132	−	0.862456i	$-0.331075\pi$
$14$		0			0
$15$	192.660			0.856265
$16$		0			0
$17$	−159.313	+	91.9794i	−0.551256	+	0.318268i	−0.749628	−	0.661859i	$-0.769768\pi$
$17$	−159.313	+	91.9794i	−0.551256	+	0.318268i	0.198372	+	0.980127i	$0.436434\pi$
$18$		0			0
$19$	208.531	+	120.395i	0.577647	+	0.333505i	0.760198	−	0.649692i	$-0.225102\pi$
$19$	208.531	+	120.395i	0.577647	+	0.333505i	−0.182551	+	0.983196i	$0.558435\pi$
$20$		0			0
$21$	−122.695	−	223.098i	−0.278220	−	0.505892i
$22$		0			0
$23$	−84.1664	+	145.780i	−0.159105	+	0.275577i	−0.934546	−	0.355842i	$-0.884194\pi$
$23$	−84.1664	+	145.780i	−0.159105	+	0.275577i	0.775441	+	0.631420i	$0.217527\pi$
$24$		0			0
$25$	374.866	+	649.287i	0.599786	+	1.03886i
$26$		0			0
$27$		−	140.296i		−	0.192450i
$28$		0			0
$29$	672.393			0.799515			0.399758	−	0.916621i	$-0.369094\pi$
$29$	672.393			0.799515			0.399758	+	0.916621i	$0.369094\pi$
$30$		0			0
$31$	−221.438	+	127.848i	−0.230425	+	0.133036i	−0.610768	−	0.791810i	$-0.709139\pi$
$31$	−221.438	+	127.848i	−0.230425	+	0.133036i	0.380343	+	0.924845i	$0.375806\pi$
$32$		0			0
$33$	1085.68	+	626.818i	0.996952	+	0.575590i
$34$		0			0
$35$	940.902	−	1554.17i	0.768084	−	1.26871i
$36$		0			0
$37$	1200.13	−	2078.69i	0.876649	−	1.51840i	0.0216529	−	0.999766i	$-0.493107\pi$
$37$	1200.13	−	2078.69i	0.876649	−	1.51840i	0.854996	−	0.518635i	$-0.173560\pi$
$38$		0			0
$39$	−757.366	−	1311.80i	−0.497939	−	0.862456i
$40$		0			0
$41$		−	414.009i		−	0.246288i	−0.992389	−	0.123144i	$-0.960702\pi$
$41$		−	414.009i		−	0.246288i	0.992389	−	0.123144i	$-0.0392976\pi$
$42$		0			0
$43$	1501.09			0.811840			0.405920	−	0.913909i	$-0.366951\pi$
$43$	1501.09			0.811840			0.405920	+	0.913909i	$0.366951\pi$
$44$		0			0
$45$	866.969	−	500.545i	0.428133	−	0.247183i
$46$		0			0
$47$	2331.33	+	1345.99i	1.05538	+	0.609322i	0.924150	−	0.382030i	$-0.124775\pi$
$47$	2331.33	+	1345.99i	1.05538	+	0.609322i	0.131227	+	0.991352i	$0.458108\pi$
$48$		0			0
$49$	−2398.93	−	99.7877i	−0.999136	−	0.0415609i
$50$		0			0
$51$	−477.939	+	827.815i	−0.183752	+	0.318268i
$52$		0			0
$53$	690.764	+	1196.44i	0.245911	+	0.425930i	0.962387	−	0.271681i	$-0.0875795\pi$
$53$	690.764	+	1196.44i	0.245911	+	0.425930i	−0.716476	+	0.697611i	$0.754246\pi$
$54$		0			0
$55$			8945.37i			2.95715i
$56$		0			0
$57$	1251.18			0.385098
$58$		0			0
$59$	−362.924	+	209.534i	−0.104258	+	0.0601937i	−0.551223	−	0.834358i	$-0.685838\pi$
$59$	−362.924	+	209.534i	−0.104258	+	0.0601937i	0.446964	+	0.894552i	$0.352505\pi$
$60$		0			0
$61$	−1058.36	−	611.042i	−0.284428	−	0.164214i	0.350999	−	0.936376i	$-0.385842\pi$
$61$	−1058.36	−	611.042i	−0.284428	−	0.164214i	−0.635426	+	0.772162i	$0.719176\pi$
$62$		0			0
$63$	−1131.75	−	685.171i	−0.285149	−	0.172631i
$64$		0			0
$65$	5404.22	−	9360.38i	1.27910	−	2.21547i
$66$		0			0
$67$	−931.198	−	1612.88i	−0.207440	−	0.359297i	0.743467	−	0.668772i	$-0.233180\pi$
$67$	−931.198	−	1612.88i	−0.207440	−	0.359297i	−0.950907	+	0.309476i	$0.899847\pi$
$68$		0			0
$69$			874.683i			0.183718i
$70$		0			0
$71$	−8242.72			−1.63514			−0.817568	−	0.575831i	$-0.804678\pi$
$71$	−8242.72			−1.63514			−0.817568	+	0.575831i	$0.804678\pi$
$72$		0			0
$73$	−1897.67	+	1095.62i	−0.356103	+	0.205596i	−0.667370	−	0.744726i	$-0.732580\pi$
$73$	−1897.67	+	1095.62i	−0.356103	+	0.205596i	0.311267	+	0.950323i	$0.399247\pi$
$74$		0			0
$75$	3373.79	+	1947.86i	0.599786	+	0.346286i
$76$		0			0
$77$	10358.7	−	5696.85i	1.74712	−	0.960845i
$78$		0			0
$79$	−3204.95	+	5551.13i	−0.513531	+	0.889462i	0.486346	+	0.873767i	$0.338330\pi$
$79$	−3204.95	+	5551.13i	−0.513531	+	0.889462i	−0.999877	+	0.0156956i	$0.995004\pi$
$80$		0			0
$81$	−364.500	−	631.333i	−0.0555556	−	0.0962250i
$82$		0			0
$83$			3622.04i			0.525771i	0.964827	+	0.262886i	$0.0846742\pi$
$83$			3622.04i			0.525771i	−0.964827	+	0.262886i	$0.915326\pi$
$84$		0			0
$85$	−6820.71			−0.944043
$86$		0			0
$87$	3025.77	−	1746.93i	0.399758	−	0.230800i
$88$		0			0
$89$	−9857.75	−	5691.37i	−1.24451	−	0.718517i	−0.274499	−	0.961587i	$-0.588512\pi$
$89$	−9857.75	−	5691.37i	−1.24451	−	0.718517i	−0.970009	+	0.243070i	$0.921845\pi$
$90$		0			0
$91$	−14280.9	−	296.892i	−1.72454	−	0.0358522i
$92$		0			0
$93$	−664.315	+	1150.63i	−0.0768083	+	0.133036i
$94$		0			0
$95$	4463.94	+	7731.77i	0.494619	+	0.856706i
$96$		0			0
$97$		−	5492.74i		−	0.583775i	−0.956453	−	0.291887i	$-0.905717\pi$
$97$		−	5492.74i		−	0.583775i	0.956453	−	0.291887i	$-0.0942833\pi$
$98$		0			0
$99$	6514.08			0.664634
$100$		0			0
$101$	−11736.9	+	6776.31i	−1.15056	+	0.664279i	−0.949025	−	0.315201i	$-0.897928\pi$
$101$	−11736.9	+	6776.31i	−1.15056	+	0.664279i	−0.201540	+	0.979480i	$0.564595\pi$
$102$		0			0
$103$	−17827.3	−	10292.6i	−1.68040	−	0.970178i	−0.961396	−	0.275170i	$-0.911266\pi$
$103$	−17827.3	−	10292.6i	−1.68040	−	0.970178i	−0.719002	−	0.695008i	$-0.755401\pi$
$104$		0			0
$105$	196.217	−	9438.29i	0.0177974	−	0.856080i
$106$		0			0
$107$	−7269.37	+	12590.9i	−0.634935	+	1.09974i	0.351594	+	0.936153i	$0.385640\pi$
$107$	−7269.37	+	12590.9i	−0.634935	+	1.09974i	−0.986529	+	0.163587i	$0.947694\pi$
$108$		0			0
$109$	7264.18	+	12581.9i	0.611412	+	1.05900i	0.991003	+	0.133842i	$0.0427314\pi$
$109$	7264.18	+	12581.9i	0.611412	+	1.05900i	−0.379591	+	0.925154i	$0.623935\pi$
$110$		0			0
$111$		−	12472.1i		−	1.01227i
$112$		0			0
$113$	−639.919			−0.0501151			−0.0250575	−	0.999686i	$-0.507977\pi$
$113$	−639.919			−0.0501151			−0.0250575	+	0.999686i	$0.507977\pi$
$114$		0			0
$115$	−5405.16	+	3120.67i	−0.408708	+	0.235967i
$116$		0			0
$117$	−6816.29	−	3935.39i	−0.497939	−	0.287485i
$118$		0			0
$119$	4343.76	+	7898.33i	0.306741	+	0.557752i
$120$		0			0
$121$	−21783.2	+	37729.7i	−1.48782	+	2.57699i
$122$		0			0
$123$	−1075.63	−	1863.04i	−0.0710971	−	0.123144i
$124$		0			0
$125$			4624.74i			0.295983i
$126$		0			0
$127$	12094.7			0.749870			0.374935	−	0.927051i	$-0.377665\pi$
$127$	12094.7			0.749870			0.374935	+	0.927051i	$0.377665\pi$
$128$		0			0
$129$	6754.91	−	3899.95i	0.405920	−	0.234358i
$130$		0			0
$131$	−8240.60	−	4757.71i	−0.480194	−	0.277240i	0.240304	−	0.970698i	$-0.422753\pi$
$131$	−8240.60	−	4757.71i	−0.480194	−	0.277240i	−0.720497	+	0.693458i	$0.756086\pi$
$132$		0			0
$133$	6110.47	−	10093.2i	0.345439	−	0.570591i
$134$		0			0
$135$	2600.91	−	4504.90i	0.142711	−	0.247183i
$136$		0			0
$137$	−12492.1	−	21636.9i	−0.665570	−	1.15280i	−0.979130	−	0.203233i	$-0.934855\pi$
$137$	−12492.1	−	21636.9i	−0.665570	−	1.15280i	0.313560	−	0.949568i	$-0.398478\pi$
$138$		0			0
$139$		−	6194.01i		−	0.320584i	−0.987070	−	0.160292i	$-0.948756\pi$
$139$		−	6194.01i		−	0.320584i	0.987070	−	0.160292i	$-0.0512436\pi$
$140$		0			0
$141$	13988.0			0.703584
$142$		0			0
$143$	60907.9	−	35165.2i	2.97853	−	1.71965i
$144$		0			0
$145$	21590.5	+	12465.3i	1.02690	+	0.592879i
$146$		0			0
$147$	−11054.4	+	5783.55i	−0.511566	+	0.267645i
$148$		0			0
$149$	15758.3	−	27294.3i	0.709804	−	1.22942i	−0.255126	−	0.966908i	$-0.582117\pi$
$149$	15758.3	−	27294.3i	0.709804	−	1.22942i	0.964930	−	0.262508i	$-0.0845497\pi$
$150$		0			0
$151$	−1708.72	−	2959.59i	−0.0749406	−	0.129801i	0.826120	−	0.563494i	$-0.190543\pi$
$151$	−1708.72	−	2959.59i	−0.0749406	−	0.129801i	−0.901060	+	0.433694i	$0.857210\pi$
$152$		0			0
$153$			4966.89i			0.212179i
$154$		0			0
$155$	−9480.50			−0.394610
$156$		0			0
$157$	2773.89	−	1601.51i	0.112536	−	0.0649725i	−0.442676	−	0.896682i	$-0.645971\pi$
$157$	2773.89	−	1601.51i	0.112536	−	0.0649725i	0.555211	+	0.831709i	$0.312637\pi$
$158$		0			0
$159$	6216.88	+	3589.32i	0.245911	+	0.141977i
$160$		0			0
$161$	7055.98	+	4271.73i	0.272211	+	0.164798i
$162$		0			0
$163$	−17096.6	+	29612.3i	−0.643481	+	1.11454i	0.341169	+	0.940002i	$0.389177\pi$
$163$	−17096.6	+	29612.3i	−0.643481	+	1.11454i	−0.984650	+	0.174540i	$0.944156\pi$
$164$		0			0
$165$	23240.8	+	40254.2i	0.853655	+	1.47857i
$166$		0			0
$167$			48829.4i			1.75085i	0.483357	+	0.875423i	$0.339417\pi$
$167$			48829.4i			1.75085i	−0.483357	+	0.875423i	$0.660583\pi$
$168$		0			0
$169$	−56417.2			−1.97532
$170$		0			0
$171$	5630.33	−	3250.67i	0.192549	−	0.111168i
$172$		0			0
$173$	−5809.64	−	3354.20i	−0.194114	−	0.112072i	0.399793	−	0.916605i	$-0.369082\pi$
$173$	−5809.64	−	3354.20i	−0.194114	−	0.112072i	−0.593907	+	0.804534i	$0.702415\pi$
$174$		0			0
$175$	32190.0	−	17703.2i	1.05110	−	0.578064i
$176$		0			0
$177$	−1088.77	+	1885.81i	−0.0347528	+	0.0601937i
$178$		0			0
$179$	929.252	+	1609.51i	0.0290020	+	0.0502329i	0.880162	−	0.474673i	$-0.157434\pi$
$179$	929.252	+	1609.51i	0.0290020	+	0.0502329i	−0.851160	+	0.524906i	$0.824100\pi$
$180$		0			0
$181$		−	8647.41i		−	0.263955i	−0.991253	−	0.131977i	$-0.957867\pi$
$181$		−	8647.41i		−	0.263955i	0.991253	−	0.131977i	$-0.0421326\pi$
$182$		0			0
$183$	−6350.13			−0.189618
$184$		0			0
$185$	77072.4	−	44497.8i	2.25193	−	1.30015i
$186$		0			0
$187$	−38436.2	−	22191.2i	−1.09915	−	0.634595i
$188$		0			0
$189$	−6873.02	−	142.886i	−0.192409	−	0.00400006i
$190$		0			0
$191$	9959.38	−	17250.1i	0.273002	−	0.472853i	−0.696627	−	0.717433i	$-0.745317\pi$
$191$	9959.38	−	17250.1i	0.273002	−	0.472853i	0.969629	+	0.244580i	$0.0786502\pi$
$192$		0			0
$193$	22909.1	+	39679.7i	0.615026	+	1.06526i	0.990380	+	0.138375i	$0.0441879\pi$
$193$	22909.1	+	39679.7i	0.615026	+	1.06526i	−0.375354	+	0.926882i	$0.622479\pi$
$194$		0			0
$195$		−	56162.3i		−	1.47698i
$196$		0			0
$197$	−62474.8			−1.60980			−0.804901	−	0.593409i	$-0.797782\pi$
$197$	−62474.8			−1.60980			−0.804901	+	0.593409i	$0.797782\pi$
$198$		0			0
$199$	47414.9	−	27375.0i	1.19732	−	0.691271i	0.237360	−	0.971422i	$-0.423718\pi$
$199$	47414.9	−	27375.0i	1.19732	−	0.691271i	0.959956	+	0.280151i	$0.0903845\pi$
$200$		0			0
$201$	−8380.78	−	4838.65i	−0.207440	−	0.119766i
$202$		0			0
$203$	684.806	−	32940.1i	0.0166179	−	0.799343i
$204$		0			0
$205$	7675.19	−	13293.8i	0.182634	−	0.316331i
$206$		0			0
$207$	2272.49	+	3936.07i	0.0530349	+	0.0918591i
$208$		0			0
$209$			58093.7i			1.32995i
$210$		0			0
$211$	36869.0			0.828126			0.414063	−	0.910248i	$-0.364109\pi$
$211$	36869.0			0.828126			0.414063	+	0.910248i	$0.364109\pi$
$212$		0			0
$213$	−37092.3	+	21415.2i	−0.817568	+	0.472023i
$214$		0			0
$215$	48200.0	+	27828.3i	1.04273	+	0.602018i
$216$		0			0
$217$	6037.65	+	10978.3i	0.128218	+	0.233140i
$218$		0			0
$219$	−5693.02	+	9860.60i	−0.118701	+	0.205596i
$220$		0			0
$221$	26812.9	+	46441.4i	0.548984	+	0.950868i
$222$		0			0
$223$		−	68084.4i		−	1.36911i	−0.728962	−	0.684554i	$-0.759997\pi$
$223$		−	68084.4i		−	1.36911i	0.728962	−	0.684554i	$-0.240003\pi$
$224$		0			0
$225$	20242.8			0.399857
$226$		0			0
$227$	−46349.4	+	26759.8i	−0.899482	+	0.519316i	−0.877032	−	0.480432i	$-0.840480\pi$
$227$	−46349.4	+	26759.8i	−0.899482	+	0.519316i	−0.0224499	+	0.999748i	$0.507147\pi$
$228$		0			0
$229$	46190.0	+	26667.8i	0.880800	+	0.508530i	0.870922	−	0.491421i	$-0.163522\pi$
$229$	46190.0	+	26667.8i	0.880800	+	0.508530i	0.00987779	+	0.999951i	$0.496856\pi$
$230$		0			0
$231$	31813.2	−	52548.5i	0.596187	−	0.984773i
$232$		0			0
$233$	9375.06	−	16238.1i	0.172688	−	0.299104i	−0.766671	−	0.642041i	$-0.778088\pi$
$233$	9375.06	−	16238.1i	0.172688	−	0.299104i	0.939359	+	0.342936i	$0.111421\pi$
$234$		0			0
$235$	49905.9	+	86439.5i	0.903683	+	1.56522i
$236$		0			0
$237$			33306.8i			0.592975i
$238$		0			0
$239$	−63451.0			−1.11082			−0.555409	−	0.831577i	$-0.687438\pi$
$239$	−63451.0			−1.11082			−0.555409	+	0.831577i	$0.687438\pi$
$240$		0			0
$241$	34087.7	−	19680.6i	0.586900	−	0.338847i	−0.176971	−	0.984216i	$-0.556630\pi$
$241$	34087.7	−	19680.6i	0.586900	−	0.338847i	0.763871	+	0.645369i	$0.223296\pi$
$242$		0			0
$243$	−3280.50	−	1894.00i	−0.0555556	−	0.0320750i
$244$		0			0
$245$	−75179.5	−	47677.1i	−1.25247	−	0.794288i
$246$		0			0
$247$	35096.4	−	60788.8i	0.575267	−	0.996391i
$248$		0			0
$249$	9410.33	+	16299.2i	0.151777	+	0.262886i
$250$		0			0
$251$		−	47201.9i		−	0.749225i	−0.927181	−	0.374612i	$-0.877776\pi$
$251$		−	47201.9i		−	0.749225i	0.927181	−	0.374612i	$-0.122224\pi$
$252$		0			0
$253$	−40612.3			−0.634479
$254$		0			0
$255$	−30693.2	+	17720.7i	−0.472022	+	0.272522i
$256$		0			0
$257$	−25252.2	−	14579.4i	−0.382325	−	0.220735i	0.296504	−	0.955031i	$-0.404179\pi$
$257$	−25252.2	−	14579.4i	−0.382325	−	0.220735i	−0.678829	+	0.734296i	$0.737512\pi$
$258$		0			0
$259$	−100612.	−	60910.8i	−1.49985	−	0.908019i
$260$		0			0
$261$	9077.30	−	15722.3i	0.133253	−	0.230800i
$262$		0			0
$263$	45384.3	+	78608.0i	0.656137	+	1.13646i	0.981607	+	0.190911i	$0.0611440\pi$
$263$	45384.3	+	78608.0i	0.656137	+	1.13646i	−0.325470	+	0.945552i	$0.605523\pi$
$264$		0			0
$265$			51223.5i			0.729419i
$266$		0			0
$267$	−59146.5			−0.829672
$268$		0			0
$269$	−20000.0	+	11547.0i	−0.276392	+	0.159575i	−0.631789	−	0.775141i	$-0.717679\pi$
$269$	−20000.0	+	11547.0i	−0.276392	+	0.159575i	0.355397	+	0.934715i	$0.384346\pi$
$270$		0			0
$271$	−8039.72	−	4641.73i	−0.109472	−	0.0632036i	0.444264	−	0.895896i	$-0.353465\pi$
$271$	−8039.72	−	4641.73i	−0.109472	−	0.0632036i	−0.553736	+	0.832692i	$0.686798\pi$
$272$		0			0
$273$	−65035.5	+	35766.9i	−0.872619	+	0.479906i
$274$		0			0
$275$	−90441.0	+	156648.i	−1.19591	+	2.07138i
$276$		0			0
$277$	−48037.8	−	83204.0i	−0.626071	−	1.08439i	−0.988333	−	0.152311i	$-0.951329\pi$
$277$	−48037.8	−	83204.0i	−0.626071	−	1.08439i	0.362261	−	0.932077i	$-0.382005\pi$
$278$		0			0
$279$			6903.77i			0.0886906i
$280$		0			0
$281$	−99427.9			−1.25920			−0.629601	−	0.776919i	$-0.716782\pi$
$281$	−99427.9			−1.25920			−0.629601	+	0.776919i	$0.716782\pi$
$282$		0			0
$283$	52989.5	−	30593.5i	0.661633	−	0.381994i	−0.131266	−	0.991347i	$-0.541904\pi$
$283$	52989.5	−	30593.5i	0.661633	−	0.381994i	0.792899	+	0.609353i	$0.208571\pi$
$284$		0			0
$285$	40175.5	+	23195.3i	0.494619	+	0.285569i
$286$		0			0
$287$	−20282.1	−	421.653i	−0.246234	−	0.00511907i
$288$		0			0
$289$	−24840.1	+	43024.3i	−0.297411	+	0.515131i
$290$		0			0
$291$	−14270.6	−	24717.3i	−0.168521	−	0.291887i
$292$		0			0
$293$			39342.6i			0.458277i	0.973394	+	0.229139i	$0.0735909\pi$
$293$			39342.6i			0.458277i	−0.973394	+	0.229139i	$0.926409\pi$
$294$		0			0
$295$	−15538.0			−0.178546
$296$		0			0
$297$	29313.4	−	16924.1i	0.332317	−	0.191863i
$298$		0			0
$299$	42496.5	+	24535.4i	0.475347	+	0.274442i
$300$		0			0
$301$	1528.80	−	73537.6i	0.0168740	−	0.811664i
$302$		0			0
$303$	−35210.7	+	60986.8i	−0.383522	+	0.664279i
$304$		0			0
$305$	−22655.8	−	39241.0i	−0.243546	−	0.421833i
$306$		0			0
$307$		−	90747.5i		−	0.962848i	−0.876488	−	0.481424i	$-0.840120\pi$
$307$		−	90747.5i		−	0.962848i	0.876488	−	0.481424i	$-0.159880\pi$
$308$		0			0
$309$	−106964.			−1.12026
$310$		0			0
$311$	−59621.1	+	34422.3i	−0.616424	+	0.355892i	−0.775475	−	0.631378i	$-0.782490\pi$
$311$	−59621.1	+	34422.3i	−0.616424	+	0.355892i	0.159052	+	0.987270i	$0.449156\pi$
$312$		0			0
$313$	−89751.3	−	51817.9i	−0.916119	−	0.528922i	−0.0337243	−	0.999431i	$-0.510737\pi$
$313$	−89751.3	−	51817.9i	−0.916119	−	0.528922i	−0.882395	+	0.470509i	$0.844070\pi$
$314$		0			0
$315$	−23638.4	−	42982.1i	−0.238230	−	0.433178i
$316$		0			0
$317$	41592.3	−	72040.0i	0.413899	−	0.716894i	−0.581413	−	0.813608i	$-0.697500\pi$
$317$	41592.3	−	72040.0i	0.413899	−	0.716894i	0.995312	+	0.0967144i	$0.0308333\pi$
$318$		0			0
$319$	81111.5	+	140489.i	0.797078	+	1.38058i
$320$		0			0
$321$			75545.5i			0.733160i
$322$		0			0
$323$	−44295.5			−0.424576
$324$		0			0
$325$	189274.	−	109277.i	1.79194	−	1.03458i
$326$		0			0
$327$	65377.6	+	37745.8i	0.611412	+	0.352999i
$328$		0			0
$329$	68313.7	−	112840.i	0.631126	−	1.04248i
$330$		0			0
$331$	13612.9	−	23578.3i	0.124250	−	0.215207i	−0.797190	−	0.603729i	$-0.793681\pi$
$331$	13612.9	−	23578.3i	0.124250	−	0.215207i	0.921439	+	0.388522i	$0.127014\pi$
$332$		0			0
$333$	−32403.6	−	56124.6i	−0.292216	−	0.506133i
$334$		0			0
$335$		−	69052.8i		−	0.615307i
$336$		0			0
$337$	84734.0			0.746102			0.373051	−	0.927811i	$-0.378312\pi$
$337$	84734.0			0.746102			0.373051	+	0.927811i	$0.378312\pi$
$338$		0			0
$339$	−2879.64	+	1662.56i	−0.0250575	+	0.0144670i
$340$		0			0
$341$	−53424.7	−	30844.8i	−0.459445	−	0.265261i
$342$		0			0
$343$	−7331.76	+	117420.i	−0.0623189	+	0.998056i
$344$		0			0
$345$	−16215.5	+	28086.0i	−0.136236	+	0.235967i
$346$		0			0
$347$	58629.8	+	101550.i	0.486922	+	0.843374i	0.999887	−	0.0150355i	$-0.00478612\pi$
$347$	58629.8	+	101550.i	0.486922	+	0.843374i	−0.512965	+	0.858410i	$0.671453\pi$
$348$		0			0
$349$			216124.i			1.77440i	0.461386	+	0.887200i	$0.347352\pi$
$349$			216124.i			1.77440i	−0.461386	+	0.887200i	$0.652648\pi$
$350$		0			0
$351$	−40897.7			−0.331960
$352$		0			0
$353$	95068.1	−	54887.6i	0.762931	−	0.440479i	−0.0674160	−	0.997725i	$-0.521475\pi$
$353$	95068.1	−	54887.6i	0.762931	−	0.440479i	0.830347	+	0.557246i	$0.188142\pi$
$354$		0			0
$355$	−264674.	−	152809.i	−2.10017	−	1.21253i
$356$		0			0
$357$	40067.4	+	24257.1i	0.314380	+	0.190328i
$358$		0			0
$359$	−8169.28	+	14149.6i	−0.0633863	+	0.109788i	−0.895977	−	0.444100i	$-0.853523\pi$
$359$	−8169.28	+	14149.6i	−0.0633863	+	0.109788i	0.832591	+	0.553889i	$0.186857\pi$
$360$		0			0
$361$	−36170.5	−	62649.1i	−0.277549	−	0.480729i
$362$		0			0
$363$			226378.i			1.71799i
$364$		0			0
$365$	−81245.6			−0.609837
$366$		0			0
$367$	167531.	−	96724.1i	1.24384	−	0.718129i	0.273964	−	0.961740i	$-0.411665\pi$
$367$	167531.	−	96724.1i	1.24384	−	0.718129i	0.969873	+	0.243611i	$0.0783319\pi$
$368$		0			0
$369$	−9680.65	−	5589.13i	−0.0710971	−	0.0410479i
$370$		0			0
$371$	59316.3	−	32621.6i	0.430950	−	0.237005i
$372$		0			0
$373$	−30304.0	+	52488.1i	−0.217812	+	0.377262i	−0.954139	−	0.299364i	$-0.903225\pi$
$373$	−30304.0	+	52488.1i	−0.217812	+	0.377262i	0.736327	+	0.676626i	$0.236559\pi$
$374$		0			0
$375$	12015.4	+	20811.3i	0.0854431	+	0.147992i
$376$		0			0
$377$		−	196009.i		−	1.37909i
$378$		0			0
$379$	−83368.9			−0.580398			−0.290199	−	0.956966i	$-0.593721\pi$
$379$	−83368.9			−0.580398			−0.290199	+	0.956966i	$0.593721\pi$
$380$		0			0
$381$	54426.0	−	31422.8i	0.374935	−	0.216469i
$382$		0			0
$383$	−48495.2	−	27998.7i	−0.330599	−	0.190871i	0.325508	−	0.945539i	$-0.394465\pi$
$383$	−48495.2	−	27998.7i	−0.330599	−	0.190871i	−0.656107	+	0.754668i	$0.727798\pi$
$384$		0			0
$385$	438229.	+	9110.52i	2.95651	+	0.0614641i
$386$		0			0
$387$	20264.7	−	35099.6i	0.135307	−	0.234358i
$388$		0			0
$389$	113829.	+	197157.i	0.752232	+	1.30290i	0.946739	+	0.322003i	$0.104356\pi$
$389$	113829.	+	197157.i	0.752232	+	1.30290i	−0.194506	+	0.980901i	$0.562310\pi$
$390$		0			0
$391$		−	30966.3i		−	0.202552i
$392$		0			0
$393$	−49443.6			−0.320129
$394$		0			0
$395$	−205821.	+	118831.i	−1.31916	+	0.761616i
$396$		0			0
$397$	137056.	+	79129.6i	0.869598	+	0.502062i	0.867214	−	0.497935i	$-0.165908\pi$
$397$	137056.	+	79129.6i	0.869598	+	0.502062i	0.00238308	+	0.999997i	$0.499241\pi$
$398$		0			0
$399$	1274.28	−	61294.8i	0.00800424	−	0.385015i
$400$		0			0
$401$	−40914.4	+	70865.8i	−0.254441	+	0.440705i	−0.964744	−	0.263192i	$-0.915225\pi$
$401$	−40914.4	+	70865.8i	−0.254441	+	0.440705i	0.710303	+	0.703897i	$0.248558\pi$
$402$		0			0
$403$	37268.8	+	64551.5i	0.229475	+	0.397463i
$404$		0			0
$405$		−	27029.4i		−	0.164788i
$406$		0			0
$407$	579093.			3.49591
$408$		0			0
$409$	106157.	−	61290.1i	0.634606	−	0.366390i	−0.147928	−	0.988998i	$-0.547260\pi$
$409$	106157.	−	61290.1i	0.634606	−	0.366390i	0.782534	+	0.622608i	$0.213927\pi$
$410$		0			0
$411$	−112429.	−	64910.8i	−0.665570	−	0.384267i
$412$		0			0
$413$	9895.33	+	17992.8i	0.0580136	+	0.105487i
$414$		0			0
$415$	−67147.9	+	116304.i	−0.389885	+	0.675300i
$416$		0			0
$417$	−16092.5	−	27873.0i	−0.0925447	−	0.160292i
$418$		0			0
$419$			69442.3i			0.395545i	0.980248	+	0.197772i	$0.0633707\pi$
$419$			69442.3i			0.395545i	−0.980248	+	0.197772i	$0.936629\pi$
$420$		0			0
$421$	9457.17			0.0533577			0.0266788	−	0.999644i	$-0.491507\pi$
$421$	9457.17			0.0533577			0.0266788	+	0.999644i	$0.491507\pi$
$422$		0			0
$423$	62945.8	−	36341.8i	0.351792	−	0.203107i
$424$		0			0
$425$	−119442.	−	68959.9i	−0.661271	−	0.381785i
$426$		0			0
$427$	−31012.5	+	51225.9i	−0.170091	+	0.280953i
$428$		0			0
$429$	182724.	−	316487.i	0.992843	−	1.71965i
$430$		0			0
$431$	10313.2	+	17863.0i	0.0555187	+	0.0961612i	0.892449	−	0.451148i	$-0.148985\pi$
$431$	10313.2	+	17863.0i	0.0555187	+	0.0961612i	−0.836930	+	0.547309i	$0.815652\pi$
$432$		0			0
$433$		−	329996.i		−	1.76008i	−0.474898	−	0.880041i	$-0.657515\pi$
$433$		−	329996.i		−	1.76008i	0.474898	−	0.880041i	$-0.342485\pi$
$434$		0			0
$435$	129543.			0.684597
$436$		0			0
$437$	−35102.5	+	20266.5i	−0.183813	+	0.106124i
$438$		0			0
$439$	188361.	+	108750.i	0.977376	+	0.564289i	0.901477	−	0.432827i	$-0.142484\pi$
$439$	188361.	+	108750.i	0.977376	+	0.564289i	0.0758994	+	0.997115i	$0.475817\pi$
$440$		0			0
$441$	−34718.8	+	54746.2i	−0.178520	+	0.281499i
$442$		0			0
$443$	−93158.6	+	161355.i	−0.474696	+	0.822197i	−0.999580	−	0.0289763i	$-0.990775\pi$
$443$	−93158.6	+	161355.i	−0.474696	+	0.822197i	0.524884	+	0.851174i	$0.324109\pi$
$444$		0			0
$445$	−211021.	−	365499.i	−1.06563	−	1.84572i
$446$		0			0
$447$		−	163766.i		−	0.819611i
$448$		0			0
$449$	171209.			0.849248			0.424624	−	0.905370i	$-0.360406\pi$
$449$	171209.			0.849248			0.424624	+	0.905370i	$0.360406\pi$
$450$		0			0
$451$	86502.8	−	49942.4i	0.425282	−	0.245537i
$452$		0			0
$453$	−15378.5	−	8878.77i	−0.0749406	−	0.0432669i
$454$		0			0
$455$	−453055.	−	274283.i	−2.18841	−	1.32488i
$456$		0			0
$457$	−158334.	+	274243.i	−0.758129	+	1.31312i	0.185675	+	0.982611i	$0.440553\pi$
$457$	−158334.	+	274243.i	−0.758129	+	1.31312i	−0.943804	+	0.330507i	$0.892780\pi$
$458$		0			0
$459$	12904.4	+	22351.0i	0.0612507	+	0.106089i
$460$		0			0
$461$		−	12814.3i		−	0.0602968i	−0.999545	−	0.0301484i	$-0.990402\pi$
$461$		−	12814.3i		−	0.0602968i	0.999545	−	0.0301484i	$-0.00959799\pi$
$462$		0			0
$463$	242588.			1.13164			0.565820	−	0.824529i	$-0.308560\pi$
$463$	242588.			1.13164			0.565820	+	0.824529i	$0.308560\pi$
$464$		0			0
$465$	−42662.3	+	24631.1i	−0.197305	+	0.113914i
$466$		0			0
$467$	216599.	+	125053.i	0.993167	+	0.573405i	0.906220	−	0.422807i	$-0.138955\pi$
$467$	216599.	+	125053.i	0.993167	+	0.573405i	0.0869478	+	0.996213i	$0.472289\pi$
$468$		0			0
$469$	−79962.6	+	43976.2i	−0.363531	+	0.199927i
$470$		0			0
$471$	8321.67	−	14413.6i	0.0375119	−	0.0649725i
$472$		0			0
$473$	181078.	+	313637.i	0.809365	+	1.40186i
$474$		0			0
$475$			180528.i			0.800126i
$476$		0			0
$477$	37301.3			0.163941
$478$		0			0
$479$	128573.	−	74231.6i	0.560375	−	0.323532i	−0.192921	−	0.981214i	$-0.561796\pi$
$479$	128573.	−	74231.6i	0.560375	−	0.323532i	0.753296	+	0.657682i	$0.228463\pi$
$480$		0			0
$481$	−605959.	−	349851.i	−2.61911	−	1.51214i
$482$		0			0
$483$	42850.2	+	890.831i	0.183679	+	0.00381857i
$484$		0			0
$485$	101828.	−	176372.i	0.432897	−	0.749799i
$486$		0			0
$487$	−96989.3	−	167990.i	−0.408946	−	0.708315i	0.585826	−	0.810437i	$-0.300770\pi$
$487$	−96989.3	−	167990.i	−0.408946	−	0.708315i	−0.994772	+	0.102122i	$0.967437\pi$
$488$		0			0
$489$			177674.i			0.743028i
$490$		0			0
$491$	−49269.8			−0.204370			−0.102185	−	0.994765i	$-0.532583\pi$
$491$	−49269.8			−0.204370			−0.102185	+	0.994765i	$0.532583\pi$
$492$		0			0
$493$	−107121.	+	61846.3i	−0.440738	+	0.254460i
$494$		0			0
$495$	209167.	+	120763.i	0.853655	+	0.492858i
$496$		0			0
$497$	−8394.90	+	403806.i	−0.0339862	+	1.63478i
$498$		0			0
$499$	−163620.	+	283398.i	−0.657106	+	1.13814i	0.324255	+	0.945970i	$0.394886\pi$
$499$	−163620.	+	283398.i	−0.657106	+	1.13814i	−0.981361	+	0.192172i	$0.938447\pi$
$500$		0			0
$501$	126862.	+	219732.i	0.505426	+	0.875423i
$502$		0			0
$503$		−	339695.i		−	1.34262i	−0.741177	−	0.671310i	$-0.765732\pi$
$503$		−	339695.i		−	1.34262i	0.741177	−	0.671310i	$-0.234268\pi$
$504$		0			0
$505$	−502496.			−1.97038
$506$		0			0
$507$	−253877.	+	146576.i	−0.987661	+	0.570226i
$508$		0			0
$509$	207478.	+	119787.i	0.800822	+	0.462355i	0.843759	−	0.536723i	$-0.180338\pi$
$509$	207478.	+	119787.i	0.800822	+	0.462355i	−0.0429364	+	0.999078i	$0.513671\pi$
$510$		0			0
$511$	51741.2	+	94081.7i	0.198150	+	0.360299i
$512$		0			0
$513$	16891.0	−	29256.0i	0.0641830	−	0.111168i
$514$		0			0
$515$	−381623.	−	660991.i	−1.43887	−	2.49219i
$516$		0			0
$517$			649474.i			2.42986i
$518$		0			0
$519$	−34857.8			−0.129409
$520$		0			0
$521$	104509.	−	60338.3i	0.385015	−	0.222289i	−0.294983	−	0.955503i	$-0.595314\pi$
$521$	104509.	−	60338.3i	0.385015	−	0.222289i	0.679998	+	0.733214i	$0.261981\pi$
$522$		0			0
$523$	35753.3	+	20642.1i	0.130711	+	0.0754660i	0.563930	−	0.825823i	$-0.309289\pi$
$523$	35753.3	+	20642.1i	0.130711	+	0.0754660i	−0.433219	+	0.901289i	$0.642622\pi$
$524$		0			0
$525$	98860.7	−	163296.i	0.358678	−	0.592459i
$526$		0			0
$527$	23518.7	−	40735.5i	0.0846821	−	0.146674i
$528$		0			0
$529$	125753.	+	217810.i	0.449371	+	0.778334i
$530$		0			0
$531$			11314.8i			0.0401291i
$532$		0			0
$533$	−120688.			−0.424824
$534$		0			0
$535$	−466838.	+	269529.i	−1.63102	+	0.941669i
$536$		0			0
$537$	8363.27	+	4828.54i	0.0290020	+	0.0167443i
$538$		0			0
$539$	−268536.	−	513267.i	−0.924324	−	1.76671i
$540$		0			0
$541$	43278.9	−	74961.3i	0.147871	−	0.256120i	−0.782570	−	0.622563i	$-0.786091\pi$
$541$	43278.9	−	74961.3i	0.147871	−	0.256120i	0.930440	+	0.366444i	$0.119425\pi$
$542$		0			0
$543$	−22466.6	−	38913.4i	−0.0761971	−	0.131977i
$544$		0			0
$545$			538674.i			1.81356i
$546$		0			0
$547$	155182.			0.518639			0.259320	−	0.965792i	$-0.416502\pi$
$547$	155182.			0.518639			0.259320	+	0.965792i	$0.416502\pi$
$548$		0			0
$549$	−28575.6	+	16498.1i	−0.0948092	+	0.0547381i
$550$		0			0
$551$	140214.	+	80952.9i	0.461838	+	0.266642i
$552$		0			0
$553$	268683.	+	162662.i	0.878596	+	0.531908i
$554$		0			0
$555$	231217.	−	400480.i	0.750644	−	1.30015i
$556$		0			0
$557$	−227259.	−	393624.i	−0.732505	−	1.26874i	−0.955809	−	0.293988i	$-0.905018\pi$
$557$	−227259.	−	393624.i	−0.732505	−	1.26874i	0.223304	−	0.974749i	$-0.428316\pi$
$558$		0			0
$559$		−	437583.i		−	1.40035i
$560$		0			0
$561$	−230617.			−0.732768
$562$		0			0
$563$	426700.	−	246356.i	1.34619	−	0.777223i	0.358482	−	0.933537i	$-0.383294\pi$
$563$	426700.	−	246356.i	1.34619	−	0.777223i	0.987707	+	0.156314i	$0.0499610\pi$
$564$		0			0
$565$	−20547.8	−	11863.3i	−0.0643677	−	0.0371627i
$566$		0			0
$567$	−31299.8	+	17213.7i	−0.0973590	+	0.0535435i
$568$		0			0
$569$	228239.	−	395322.i	0.704962	−	1.22103i	−0.261744	−	0.965137i	$-0.584298\pi$
$569$	228239.	−	395322.i	0.704962	−	1.22103i	0.966705	−	0.255892i	$-0.0823692\pi$
$570$		0			0
$571$	459.462	+	795.811i	0.00140921	+	0.00244083i	0.866729	−	0.498779i	$-0.166218\pi$
$571$	459.462	+	795.811i	0.00140921	+	0.00244083i	−0.865320	+	0.501220i	$0.832885\pi$
$572$		0			0
$573$		−	103501.i		−	0.315235i
$574$		0			0
$575$	−126204.			−0.381715
$576$		0			0
$577$	−492629.	+	284420.i	−1.47968	+	0.854295i	−0.999735	−	0.0230027i	$-0.992677\pi$
$577$	−492629.	+	284420.i	−1.47968	+	0.854295i	−0.479947	+	0.877298i	$0.659344\pi$
$578$		0			0
$579$	206182.	+	119039.i	0.615026	+	0.355085i
$580$		0			0
$581$	177442.	+	3688.91i	0.525658	+	0.0109281i
$582$		0			0
$583$	−166655.	+	288656.i	−0.490323	+	0.849264i
$584$		0			0
$585$	−145914.	−	252730.i	−0.426368	−	0.738491i
$586$		0			0
$587$			29836.4i			0.0865904i	0.999062	+	0.0432952i	$0.0137856\pi$
$587$			29836.4i			0.0865904i	−0.999062	+	0.0432952i	$0.986214\pi$
$588$		0			0
$589$	−61568.9			−0.177472
$590$		0			0
$591$	−281137.	+	162314.i	−0.804901	+	0.464710i
$592$		0			0
$593$	146689.	+	84691.1i	0.417147	+	0.240840i	0.693856	−	0.720114i	$-0.255910\pi$
$593$	146689.	+	84691.1i	0.417147	+	0.240840i	−0.276709	+	0.960954i	$0.589244\pi$
$594$		0			0
$595$	−6946.64	+	334143.i	−0.0196219	+	0.943839i
$596$		0			0
$597$	142245.	−	246375.i	0.399105	−	0.691271i
$598$		0			0
$599$	184887.	+	320234.i	0.515292	+	0.892512i	0.999842	+	0.0177484i	$0.00564979\pi$
$599$	184887.	+	320234.i	0.515292	+	0.892512i	−0.484551	+	0.874763i	$0.661017\pi$
$600$		0			0
$601$		−	290795.i		−	0.805077i	−0.915403	−	0.402539i	$-0.868128\pi$
$601$		−	290795.i		−	0.805077i	0.915403	−	0.402539i	$-0.131872\pi$
$602$		0			0
$603$	−50284.7			−0.138293
$604$		0			0
$605$	−1.39892e6	+	807666.i	−3.82192	+	2.20659i
$606$		0			0
$607$	−259389.	−	149758.i	−0.704002	−	0.406456i	0.104834	−	0.994490i	$-0.466569\pi$
$607$	−259389.	−	149758.i	−0.704002	−	0.406456i	−0.808836	+	0.588034i	$0.799902\pi$
$608$		0			0
$609$	−82499.3	−	150010.i	−0.222441	−	0.404469i
$610$		0			0
$611$	392370.	−	679606.i	1.05103	−	1.82043i
$612$		0			0
$613$	66128.8	+	114539.i	0.175983	+	0.304811i	0.940501	−	0.339791i	$-0.110356\pi$
$613$	66128.8	+	114539.i	0.175983	+	0.304811i	−0.764518	+	0.644602i	$0.777023\pi$
$614$		0			0
$615$		−	79762.9i		−	0.210888i
$616$		0			0
$617$	−15578.9			−0.0409228			−0.0204614	−	0.999791i	$-0.506514\pi$
$617$	−15578.9			−0.0409228			−0.0204614	+	0.999791i	$0.506514\pi$
$618$		0			0
$619$	−349359.	+	201703.i	−0.911782	+	0.526418i	−0.881004	−	0.473109i	$-0.843132\pi$
$619$	−349359.	+	201703.i	−0.911782	+	0.526418i	−0.0307781	+	0.999526i	$0.509799\pi$
$620$		0			0
$621$	20452.4	+	11808.2i	0.0530349	+	0.0306197i
$622$		0			0
$623$	−288857.	+	477129.i	−0.744229	+	1.22930i
$624$		0			0
$625$	148555.	−	257304.i	0.380300	−	0.658699i
$626$		0			0
$627$	150932.	+	261421.i	0.383924	+	0.664976i
$628$		0			0
$629$			441550.i			1.11604i
$630$		0			0
$631$	−351516.			−0.882850			−0.441425	−	0.897298i	$-0.645527\pi$
$631$	−351516.			−0.882850			−0.441425	+	0.897298i	$0.645527\pi$
$632$		0			0
$633$	165910.	−	95788.4i	0.414063	−	0.239059i
$634$		0			0
$635$	388359.	+	224219.i	0.963132	+	0.556064i
$636$		0			0
$637$	−29089.1	+	699311.i	−0.0716889	+	1.72342i
$638$		0			0
$639$	−111277.	+	192737.i	−0.272523	+	0.472023i
$640$		0			0
$641$	−308091.	−	533629.i	−0.749830	−	1.29874i	−0.947904	−	0.318557i	$-0.896802\pi$
$641$	−308091.	−	533629.i	−0.749830	−	1.29874i	0.198073	−	0.980187i	$-0.436532\pi$
$642$		0			0
$643$		−	371943.i		−	0.899610i	−0.893127	−	0.449805i	$-0.851494\pi$
$643$		−	371943.i		−	0.899610i	0.893127	−	0.449805i	$-0.148506\pi$
$644$		0			0
$645$	289200.			0.695150
$646$		0			0
$647$	−45856.5	+	26475.3i	−0.109545	+	0.0632458i	−0.553771	−	0.832669i	$-0.686812\pi$
$647$	−45856.5	+	26475.3i	−0.109545	+	0.0632458i	0.444227	+	0.895914i	$0.353479\pi$
$648$		0			0
$649$	−87559.8	−	50552.7i	−0.207881	−	0.120020i
$650$		0			0
$651$	55692.0	+	33716.3i	0.131411	+	0.0795569i
$652$		0			0
$653$	−147682.	+	255794.i	−0.346340	+	0.599878i	−0.985596	−	0.169115i	$-0.945909\pi$
$653$	−147682.	+	255794.i	−0.346340	+	0.599878i	0.639256	+	0.768994i	$0.279242\pi$
$654$		0			0
$655$	−176404.	−	305540.i	−0.411173	−	0.712173i
$656$		0			0
$657$			59163.6i			0.137064i
$658$		0			0
$659$	198597.			0.457300			0.228650	−	0.973509i	$-0.426569\pi$
$659$	198597.			0.457300			0.228650	+	0.973509i	$0.426569\pi$
$660$		0			0
$661$	726530.	−	419462.i	1.66284	−	0.960041i	0.691491	−	0.722385i	$-0.256954\pi$
$661$	726530.	−	419462.i	1.66284	−	0.960041i	0.971349	−	0.237657i	$-0.0763793\pi$
$662$		0			0
$663$	241316.	+	139324.i	0.548984	+	0.316956i
$664$		0			0
$665$	383321.	−	210811.i	0.866802	−	0.476706i
$666$		0			0
$667$	−56592.8	+	98021.7i	−0.127207	+	0.220328i
$668$		0			0
$669$	−176888.	−	306380.i	−0.395227	−	0.684554i
$670$		0			0
$671$		−	294843.i		−	0.654855i
$672$		0			0
$673$	306271.			0.676201			0.338100	−	0.941110i	$-0.390216\pi$
$673$	306271.			0.676201			0.338100	+	0.941110i	$0.390216\pi$
$674$		0			0
$675$	91092.5	−	52592.3i	0.199929	−	0.115429i
$676$		0			0
$677$	649203.	+	374817.i	1.41646	+	0.817791i	0.995985	−	0.0895154i	$-0.0285318\pi$
$677$	649203.	+	374817.i	1.41646	+	0.817791i	0.420470	+	0.907306i	$0.361865\pi$
$678$		0			0
$679$	−269086.	−	5594.14i	−0.583649	−	0.0121337i
$680$		0			0
$681$	−139048.	+	240839.i	−0.299827	+	0.519316i
$682$		0			0
$683$	−201254.	−	348582.i	−0.431422	−	0.747245i	0.565574	−	0.824698i	$-0.308655\pi$
$683$	−201254.	−	348582.i	−0.431422	−	0.747245i	−0.996996	+	0.0774523i	$0.975321\pi$
$684$		0			0
$685$		−	926348.i		−	1.97421i
$686$		0			0
$687$	277140.			0.587200
$688$		0			0
$689$	348774.	−	201365.i	0.734693	−	0.424175i
$690$		0			0
$691$	−333675.	−	192647.i	−0.698824	−	0.403466i	0.108085	−	0.994142i	$-0.465528\pi$
$691$	−333675.	−	192647.i	−0.698824	−	0.403466i	−0.806909	+	0.590675i	$0.798861\pi$
$692$		0			0
$693$	6634.34	−	319121.i	0.0138144	−	0.664491i
$694$		0			0
$695$	114829.	−	198889.i	0.237729	−	0.411758i
$696$		0			0
$697$	38080.3	+	65957.1i	0.0783854	+	0.135768i
$698$		0			0
$699$		−	97428.5i		−	0.199403i
$700$		0			0
$701$	−113793.			−0.231569			−0.115784	−	0.993274i	$-0.536938\pi$
$701$	−113793.			−0.231569			−0.115784	+	0.993274i	$0.536938\pi$
$702$		0			0
$703$	500529.	−	288980.i	1.01279	−	0.584733i
$704$		0			0
$705$	449153.	+	259318.i	0.903683	+	0.521741i
$706$		0			0
$707$	320014.	+	581886.i	0.640221	+	1.16412i
$708$		0			0
$709$	−69701.1	+	120726.i	−0.138659	+	0.240164i	−0.926989	−	0.375088i	$-0.877612\pi$
$709$	−69701.1	+	120726.i	−0.138659	+	0.240164i	0.788330	+	0.615252i	$0.210946\pi$
$710$		0			0
$711$	86533.6	+	149881.i	0.171177	+	0.296487i
$712$		0			0
$713$		−	43041.8i		−	0.0846665i
$714$		0			0
$715$	2.60767e6			5.10082
$716$		0			0
$717$	−285530.	+	164851.i	−0.555409	+	0.320665i
$718$		0			0
$719$	−305577.	−	176425.i	−0.591102	−	0.341273i	0.174431	−	0.984669i	$-0.444191\pi$
$719$	−305577.	−	176425.i	−0.591102	−	0.341273i	−0.765533	+	0.643396i	$0.777525\pi$
$720$		0			0
$721$	−522386.	+	862868.i	−1.00490	+	1.65987i
$722$		0			0
$723$	102263.	−	177125.i	0.195633	−	0.338847i
$724$		0			0
$725$	252057.	+	436576.i	0.479538	+	0.830584i
$726$		0			0
$727$			120300.i			0.227612i	0.993503	+	0.113806i	$0.0363042\pi$
$727$			120300.i			0.227612i	−0.993503	+	0.113806i	$0.963696\pi$
$728$		0			0
$729$	−19683.0			−0.0370370
$730$		0			0
$731$	−239143.	+	138070.i	−0.447532	+	0.258382i
$732$		0			0
$733$	−298466.	−	172319.i	−0.555504	−	0.320720i	0.195835	−	0.980637i	$-0.437258\pi$
$733$	−298466.	−	172319.i	−0.555504	−	0.320720i	−0.751339	+	0.659917i	$0.770592\pi$
$734$		0			0
$735$	−462176.	−	19225.1i	−0.855526	−	0.0355872i
$736$		0			0
$737$	224663.	−	389128.i	0.413615	−	0.716403i
$738$		0			0
$739$	169904.	+	294282.i	0.311110	+	0.538859i	0.978603	−	0.205757i	$-0.0659658\pi$
$739$	169904.	+	294282.i	0.311110	+	0.538859i	−0.667493	+	0.744616i	$0.732632\pi$
$740$		0			0
$741$		−	364733.i		−	0.664261i
$742$		0			0
$743$	−337620.			−0.611577			−0.305788	−	0.952100i	$-0.598920\pi$
$743$	−337620.			−0.611577			−0.305788	+	0.952100i	$0.598920\pi$
$744$		0			0
$745$	1.01200e6	−	584278.i	1.82334	−	1.05271i
$746$		0			0
$747$	84693.0	+	48897.5i	0.151777	+	0.0876286i
$748$		0			0
$749$	609418.	+	368946.i	1.08630	+	0.657656i
$750$		0			0
$751$	−324860.	+	562675.i	−0.575993	+	0.997648i	0.419940	+	0.907552i	$0.362051\pi$
$751$	−324860.	+	562675.i	−0.575993	+	0.997648i	−0.995933	+	0.0900967i	$0.971282\pi$
$752$		0			0
$753$	−122634.	−	212409.i	−0.216283	−	0.374612i
$754$		0			0
$755$		−	126710.i		−	0.222288i
$756$		0			0
$757$	468455.			0.817477			0.408739	−	0.912651i	$-0.365969\pi$
$757$	468455.			0.817477			0.408739	+	0.912651i	$0.365969\pi$
$758$		0			0
$759$	−182756.	+	105514.i	−0.317239	+	0.183158i
$760$		0			0
$761$	362089.	+	209052.i	0.625238	+	0.360981i	0.778906	−	0.627141i	$-0.215775\pi$
$761$	362089.	+	209052.i	0.625238	+	0.360981i	−0.153667	+	0.988123i	$0.549108\pi$
$762$		0			0
$763$	623780.	−	343054.i	1.07148	−	0.589268i
$764$		0			0
$765$	−92079.6	+	159487.i	−0.157341	+	0.272522i
$766$		0			0
$767$	61081.3	+	105796.i	0.103829	+	0.179837i
$768$		0			0
$769$			793335.i			1.34154i	0.741665	+	0.670770i	$0.234036\pi$
$769$			793335.i			1.34154i	−0.741665	+	0.670770i	$0.765964\pi$
$770$		0			0
$771$	−151513.			−0.254883
$772$		0			0
$773$	−361267.	+	208578.i	−0.604602	+	0.349067i	−0.770850	−	0.637017i	$-0.780168\pi$
$773$	−361267.	+	208578.i	−0.604602	+	0.349067i	0.166248	+	0.986084i	$0.446835\pi$
$774$		0			0
$775$	−166019.	−	95851.4i	−0.276411	−	0.159586i
$776$		0			0
$777$	−611003.	−	12702.4i	−1.01205	−	0.0210399i
$778$		0			0
$779$	49844.8	−	86333.7i	0.0821381	−	0.142267i
$780$		0			0
$781$	−994329.	−	1.72223e6i	−1.63015	−	2.82351i
$782$		0			0
$783$		−	94334.1i		−	0.153867i
$784$		0			0
$785$	118759.			0.192721
$786$		0			0
$787$	−227617.	+	131414.i	−0.367497	+	0.212175i	−0.672365	−	0.740220i	$-0.734721\pi$
$787$	−227617.	+	131414.i	−0.367497	+	0.212175i	0.304867	+	0.952395i	$0.401388\pi$
$788$		0			0
$789$	408459.	+	235824.i	0.656137	+	0.378821i
$790$		0			0
$791$	−651.734	+	31349.3i	−0.00104164	+	0.0501043i
$792$		0			0
$793$	−178125.	+	308521.i	−0.283255	+	0.490613i
$794$		0			0
$795$	133082.	+	230506.i	0.210565	+	0.364710i
$796$		0			0
$797$			4683.81i			0.00737366i	0.999993	+	0.00368683i	$0.00117356\pi$
$797$			4683.81i			0.00737366i	−0.999993	+	0.00368683i	$0.998826\pi$
$798$		0			0
$799$	−495214.			−0.775710
$800$		0			0
$801$	−266159.	+	153667.i	−0.414836	+	0.239506i
$802$		0			0
$803$	−457837.	−	264332.i	−0.710035	−	0.409939i
$804$		0			0
$805$	147375.	+	267974.i	0.227421	+	0.413524i
$806$		0			0
$807$	−60000.0	+	103923.i	−0.0921306	+	0.159575i
$808$		0			0
$809$	−276216.	−	478420.i	−0.422038	−	0.730992i	0.574100	−	0.818785i	$-0.305352\pi$
$809$	−276216.	−	478420.i	−0.422038	−	0.730992i	−0.996139	+	0.0877930i	$0.972019\pi$
$810$		0			0
$811$		−	1.11332e6i		−	1.69270i	−0.532630	−	0.846348i	$-0.678796\pi$
$811$		−	1.11332e6i		−	1.69270i	0.532630	−	0.846348i	$-0.321204\pi$
$812$		0			0
$813$	−48238.3			−0.0729812
$814$		0			0
$815$	−1.09794e6	+	633899.i	−1.65297	+	0.954343i
$816$		0			0
$817$	313024.	+	180724.i	0.468957	+	0.270752i
$818$		0			0
$819$	−199734.	+	329918.i	−0.297773	+	0.491856i
$820$		0			0
$821$	437652.	−	758036.i	0.649296	−	1.12461i	−0.333995	−	0.942575i	$-0.608397\pi$
$821$	437652.	−	758036.i	0.649296	−	1.12461i	0.983291	−	0.182040i	$-0.0582699\pi$
$822$		0			0
$823$	−110188.	−	190852.i	−0.162680	−	0.281771i	0.773149	−	0.634225i	$-0.218681\pi$
$823$	−110188.	−	190852.i	−0.162680	−	0.281771i	−0.935829	+	0.352454i	$0.885347\pi$
$824$		0			0
$825$			939891.i			1.38092i
$826$		0			0
$827$	361288.			0.528254			0.264127	−	0.964488i	$-0.414916\pi$
$827$	361288.			0.528254			0.264127	+	0.964488i	$0.414916\pi$
$828$		0			0
$829$	872945.	−	503995.i	1.27022	−	0.733360i	0.295188	−	0.955439i	$-0.404618\pi$
$829$	872945.	−	503995.i	1.27022	−	0.733360i	0.975029	+	0.222080i	$0.0712845\pi$
$830$		0			0
$831$	−432340.	−	249612.i	−0.626071	−	0.361462i
$832$		0			0
$833$	391358.	−	204754.i	0.564007	−	0.295082i
$834$		0			0
$835$	−905232.	+	1.56791e6i	−1.29834	+	2.24878i
$836$		0			0
$837$	17936.5	+	31066.9i	0.0256028	+	0.0443453i
$838$		0			0
$839$			243197.i			0.345489i	0.984967	+	0.172744i	$0.0552635\pi$
$839$			243197.i			0.345489i	−0.984967	+	0.172744i	$0.944737\pi$
$840$		0			0
$841$	−255169.			−0.360775
$842$		0			0
$843$	−447425.	+	258321.i	−0.629601	+	0.363500i
$844$		0			0
$845$	−1.81155e6	−	1.04590e6i	−2.53710	−	1.46480i
$846$		0			0
$847$	1.82617e6	+	1.10557e6i	2.54551	+	1.54107i
$848$		0			0
$849$	158969.	−	275342.i	0.220544	−	0.381994i
$850$		0			0
$851$	202022.	+	349912.i	0.278958	+	0.483169i
$852$		0			0
$853$		−	652144.i		−	0.896283i	−0.893963	−	0.448142i	$-0.852086\pi$
$853$		−	652144.i		−	0.896283i	0.893963	−	0.448142i	$-0.147914\pi$
$854$		0			0
$855$	241053.			0.329746
$856$		0			0
$857$	158295.	−	91391.6i	0.215529	−	0.124436i	−0.388349	−	0.921512i	$-0.626955\pi$
$857$	158295.	−	91391.6i	0.215529	−	0.124436i	0.603878	+	0.797077i	$0.293621\pi$
$858$		0			0
$859$	−84693.4	−	48897.8i	−0.114779	−	0.0662678i	0.441511	−	0.897256i	$-0.354443\pi$
$859$	−84693.4	−	48897.8i	−0.114779	−	0.0662678i	−0.556291	+	0.830988i	$0.687776\pi$
$860$		0			0
$861$	−92364.8	+	50796.9i	−0.124595	+	0.0685222i
$862$		0			0
$863$	264832.	−	458702.i	0.355589	−	0.615899i	−0.631629	−	0.775271i	$-0.717614\pi$
$863$	264832.	−	458702.i	0.355589	−	0.615899i	0.987219	+	0.159372i	$0.0509468\pi$
$864$		0			0
$865$	−124365.	−	215406.i	−0.166213	−	0.287890i
$866$		0			0
$867$			258146.i			0.343421i
$868$		0			0
$869$	−1.54647e6			−2.04786
$870$		0			0
$871$	−470172.	+	271454.i	−0.619755	+	0.357816i
$872$		0			0
$873$	−128435.	−	74152.0i	−0.168521	−	0.0972958i
$874$		0			0
$875$	226563.	+	4710.12i	0.295919	+	0.00615200i
$876$		0			0
$877$	766714.	−	1.32799e6i	0.996860	−	1.72661i	0.429859	−	0.902896i	$-0.358563\pi$
$877$	766714.	−	1.32799e6i	0.996860	−	1.72661i	0.567002	−	0.823717i	$-0.308103\pi$
$878$		0			0
$879$	102215.	+	177042.i	0.132293	+	0.229139i
$880$		0			0
$881$		−	598324.i		−	0.770876i	−0.922734	−	0.385438i	$-0.874050\pi$
$881$		−	598324.i		−	0.770876i	0.922734	−	0.385438i	$-0.125950\pi$
$882$		0			0
$883$	−264388.			−0.339094			−0.169547	−	0.985522i	$-0.554230\pi$
$883$	−264388.			−0.339094			−0.169547	+	0.985522i	$0.554230\pi$
$884$		0			0
$885$	−69920.8	+	40368.8i	−0.0892729	+	0.0515417i
$886$		0			0
$887$	467367.	+	269835.i	0.594034	+	0.342966i	0.766691	−	0.642016i	$-0.221902\pi$
$887$	467367.	+	269835.i	0.594034	+	0.342966i	−0.172657	+	0.984982i	$0.555235\pi$
$888$		0			0
$889$	12317.9	−	592510.i	0.0155860	−	0.749708i
$890$		0			0
$891$	87940.1	−	152317.i	0.110772	−	0.191863i
$892$		0			0
$893$	324102.	+	561361.i	0.406424	+	0.703946i
$894$		0			0
$895$			68908.5i			0.0860254i
$896$		0			0
$897$	254979.			0.316898
$898$		0			0
$899$	−148894.	+	85963.7i	−0.184228	+	0.106364i
$900$		0			0
$901$	−220095.	−	127072.i	−0.271120	−	0.156531i
$902$		0			0
$903$	−184177.	−	334891.i	−0.225870	−	0.410703i
$904$		0			0
$905$	160312.	−	277668.i	0.195735	−	0.339023i
$906$		0			0
$907$	−339627.	−	588251.i	−0.412846	−	0.715070i	0.582354	−	0.812935i	$-0.302132\pi$
$907$	−339627.	−	588251.i	−0.412846	−	0.715070i	−0.995200	+	0.0978656i	$0.968798\pi$
$908$		0			0
$909$			365921.i			0.442853i
$910$		0			0
$911$	−1.04261e6			−1.25628			−0.628139	−	0.778101i	$-0.716183\pi$
$911$	−1.04261e6			−1.25628			−0.628139	+	0.778101i	$0.716183\pi$
$912$		0			0
$913$	−756786.	+	436931.i	−0.907887	+	0.524169i
$914$		0			0
$915$	−203902.	−	117723.i	−0.243546	−	0.140611i
$916$		0			0
$917$	−241470.	+	398857.i	−0.287161	+	0.474327i
$918$		0			0
$919$	1496.27	−	2591.62i	0.00177166	−	0.00306860i	−0.865138	−	0.501534i	$-0.832769\pi$
$919$	1496.27	−	2591.62i	0.00177166	−	0.00306860i	0.866910	+	0.498465i	$0.166103\pi$
$920$		0			0
$921$	−235769.	−	408364.i	−0.277950	−	0.481424i
$922$		0			0
$923$			2.40284e6i			2.82047i
$924$		0			0
$925$	1.79956e6			2.10321
$926$		0			0
$927$	−481338.	+	277901.i	−0.560132	+	0.323393i
$928$		0			0
$929$	1.22590e6	+	707774.i	1.42044	+	0.820093i	0.996337	−	0.0855192i	$-0.0272549\pi$
$929$	1.22590e6	+	707774.i	1.42044	+	0.820093i	0.424106	+	0.905612i	$0.360588\pi$
$930$		0			0
$931$	−488236.	−	309628.i	−0.563287	−	0.357224i
$932$		0			0
$933$	−178863.	+	309800.i	−0.205475	+	0.355892i
$934$		0			0
$935$	−822790.	−	1.42511e6i	−0.941165	−	1.63015i
$936$		0			0
$937$			255935.i			0.291508i	0.989321	+	0.145754i	$0.0465608\pi$
$937$			255935.i			0.291508i	−0.989321	+	0.145754i	$0.953439\pi$
$938$		0			0
$939$	−538508.			−0.610746
$940$		0			0
$941$	−1.13971e6	+	658009.i	−1.28710	+	0.743110i	−0.978137	−	0.207963i	$-0.933317\pi$
$941$	−1.13971e6	+	658009.i	−1.28710	+	0.743110i	−0.308967	+	0.951073i	$0.599983\pi$
$942$		0			0
$943$	60354.5	+	34845.7i	0.0678713	+	0.0391855i
$944$		0			0
$945$	−218044.	−	132005.i	−0.244163	−	0.147818i
$946$		0			0
$947$	−723052.	+	1.25236e6i	−0.806250	+	1.39647i	0.109194	+	0.994020i	$0.465173\pi$
$947$	−723052.	+	1.25236e6i	−0.806250	+	1.39647i	−0.915444	+	0.402445i	$0.868160\pi$
$948$		0			0
$949$	319385.	+	553191.i	0.354635	+	0.614246i
$950$		0			0
$951$		−	432240.i		−	0.477929i
$952$		0			0
$953$	−97524.3			−0.107381			−0.0536904	−	0.998558i	$-0.517098\pi$
$953$	−97524.3			−0.107381			−0.0536904	+	0.998558i	$0.517098\pi$
$954$		0			0
$955$	639590.	−	369268.i	0.701286	−	0.404888i
$956$		0			0
$957$	730003.	+	421468.i	0.797078	+	0.460193i
$958$		0			0
$959$	−1.07270e6	+	589944.i	−1.16639	+	0.641466i
$960$		0			0
$961$	−429071.	+	743172.i	−0.464603	+	0.804716i
$962$		0			0
$963$	196273.	+	339955.i	0.211645	+	0.366580i
$964$		0			0
$965$			1.69882e6i			1.82428i
$966$		0			0
$967$	−1.06989e6			−1.14416			−0.572078	−	0.820199i	$-0.693863\pi$
$967$	−1.06989e6			−1.14416			−0.572078	+	0.820199i	$0.693863\pi$
$968$		0			0
$969$	−199330.	+	115083.i	−0.212288	+	0.122564i
$970$		0			0
$971$	272189.	+	157149.i	0.288691	+	0.166676i	0.637351	−	0.770573i	$-0.280030\pi$
$971$	272189.	+	157149.i	0.288691	+	0.166676i	−0.348661	+	0.937249i	$0.613363\pi$
$972$		0			0
$973$	−303441.	−	6308.36i	−0.320515	−	0.00666333i
$974$		0			0
$975$	567821.	−	983495.i	0.597314	−	1.03458i
$976$		0			0
$977$	687675.	+	1.19109e6i	0.720434	+	1.24783i	0.960826	+	0.277152i	$0.0893905\pi$
$977$	687675.	+	1.19109e6i	0.720434	+	1.24783i	−0.240393	+	0.970676i	$0.577276\pi$
$978$		0			0
$979$		−	2.74623e6i		−	2.86531i
$980$		0			0
$981$	392266.			0.407608
$982$		0			0
$983$	268162.	−	154823.i	0.277517	−	0.160225i	−0.354782	−	0.934949i	$-0.615445\pi$
$983$	268162.	−	154823.i	0.277517	−	0.160225i	0.632299	+	0.774725i	$0.282111\pi$
$984$		0			0
$985$	−2.00606e6	−	1.15820e6i	−2.06763	−	1.19374i
$986$		0			0
$987$	14246.2	−	685262.i	0.0146240	−	0.703432i
$988$		0			0
$989$	−126341.	+	218830.i	−0.129167	+	0.223725i
$990$		0			0
$991$	−151852.	−	263016.i	−0.154623	−	0.267815i	0.778299	−	0.627894i	$-0.216083\pi$
$991$	−151852.	−	263016.i	−0.154623	−	0.267815i	−0.932922	+	0.360079i	$0.882750\pi$
$992$		0			0
$993$		−	141470.i		−	0.143471i
$994$		0			0
$995$	2.02999e6			2.05044
$996$		0			0
$997$	−446842.	+	257984.i	−0.449535	+	0.259539i	−0.707634	−	0.706579i	$-0.750237\pi$
$997$	−446842.	+	257984.i	−0.449535	+	0.259539i	0.258099	+	0.966119i	$0.416904\pi$
$998$		0			0
$999$	−291632.	−	168374.i	−0.292216	−	0.168711i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	168.5.z.b.145.8	yes	16
3.2	odd	2		504.5.by.c.145.1		16
4.3	odd	2		336.5.bh.h.145.8		16
7.2	even	3		1176.5.f.a.97.9		16
7.3	odd	6	inner	168.5.z.b.73.8	✓	16
7.5	odd	6		1176.5.f.a.97.8		16
21.17	even	6		504.5.by.c.73.1		16
28.3	even	6		336.5.bh.h.241.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.5.z.b.73.8	✓	16	7.3	odd	6	inner
168.5.z.b.145.8	yes	16	1.1	even	1	trivial
336.5.bh.h.145.8		16	4.3	odd	2
336.5.bh.h.241.8		16	28.3	even	6
504.5.by.c.73.1		16	21.17	even	6
504.5.by.c.145.1		16	3.2	odd	2
1176.5.f.a.97.8		16	7.5	odd	6
1176.5.f.a.97.9		16	7.2	even	3

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	168.z (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(4.50000 - 2.59808i) q^{3} +(32.1100 + 18.5387i) q^{5} +(1.01846 - 48.9894i) q^{7} +(13.5000 - 23.3827i) q^{9} +O(q^{10})\)	\(q+(4.50000 - 2.59808i) q^{3} +(32.1100 + 18.5387i) q^{5} +(1.01846 - 48.9894i) q^{7} +(13.5000 - 23.3827i) q^{9} +(120.631 + 208.939i) q^{11} -291.510i q^{13} +192.660 q^{15} +(-159.313 + 91.9794i) q^{17} +(208.531 + 120.395i) q^{19} +(-122.695 - 223.098i) q^{21} +(-84.1664 + 145.780i) q^{23} +(374.866 + 649.287i) q^{25} -140.296i q^{27} +672.393 q^{29} +(-221.438 + 127.848i) q^{31} +(1085.68 + 626.818i) q^{33} +(940.902 - 1554.17i) q^{35} +(1200.13 - 2078.69i) q^{37} +(-757.366 - 1311.80i) q^{39} -414.009i q^{41} +1501.09 q^{43} +(866.969 - 500.545i) q^{45} +(2331.33 + 1345.99i) q^{47} +(-2398.93 - 99.7877i) q^{49} +(-477.939 + 827.815i) q^{51} +(690.764 + 1196.44i) q^{53} +8945.37i q^{55} +1251.18 q^{57} +(-362.924 + 209.534i) q^{59} +(-1058.36 - 611.042i) q^{61} +(-1131.75 - 685.171i) q^{63} +(5404.22 - 9360.38i) q^{65} +(-931.198 - 1612.88i) q^{67} +874.683i q^{69} -8242.72 q^{71} +(-1897.67 + 1095.62i) q^{73} +(3373.79 + 1947.86i) q^{75} +(10358.7 - 5696.85i) q^{77} +(-3204.95 + 5551.13i) q^{79} +(-364.500 - 631.333i) q^{81} +3622.04i q^{83} -6820.71 q^{85} +(3025.77 - 1746.93i) q^{87} +(-9857.75 - 5691.37i) q^{89} +(-14280.9 - 296.892i) q^{91} +(-664.315 + 1150.63i) q^{93} +(4463.94 + 7731.77i) q^{95} -5492.74i q^{97} +6514.08 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 72 q^{3} - 12 q^{5} + 16 q^{7} + 216 q^{9}+O(q^{10}) \) 16 * q + 72 * q^3 - 12 * q^5 + 16 * q^7 + 216 * q^9	\( 16 q + 72 q^{3} - 12 q^{5} + 16 q^{7} + 216 q^{9} + 252 q^{11} - 72 q^{15} - 696 q^{17} + 156 q^{19} + 108 q^{21} - 672 q^{23} + 84 q^{25} + 1992 q^{29} + 2040 q^{31} + 2268 q^{33} + 2712 q^{35} + 2548 q^{37} - 396 q^{39} + 1304 q^{43} - 324 q^{45} - 744 q^{47} - 5608 q^{49} - 2088 q^{51} - 1164 q^{53} + 936 q^{57} + 8988 q^{59} + 816 q^{61} + 540 q^{63} + 8760 q^{65} + 3044 q^{67} - 4464 q^{71} - 15828 q^{73} + 756 q^{75} + 996 q^{77} - 11144 q^{79} - 5832 q^{81} - 15344 q^{85} + 8964 q^{87} + 22248 q^{89} + 1596 q^{91} + 6120 q^{93} + 3840 q^{95} + 13608 q^{99}+O(q^{100}) \) 16 * q + 72 * q^3 - 12 * q^5 + 16 * q^7 + 216 * q^9 + 252 * q^11 - 72 * q^15 - 696 * q^17 + 156 * q^19 + 108 * q^21 - 672 * q^23 + 84 * q^25 + 1992 * q^29 + 2040 * q^31 + 2268 * q^33 + 2712 * q^35 + 2548 * q^37 - 396 * q^39 + 1304 * q^43 - 324 * q^45 - 744 * q^47 - 5608 * q^49 - 2088 * q^51 - 1164 * q^53 + 936 * q^57 + 8988 * q^59 + 816 * q^61 + 540 * q^63 + 8760 * q^65 + 3044 * q^67 - 4464 * q^71 - 15828 * q^73 + 756 * q^75 + 996 * q^77 - 11144 * q^79 - 5832 * q^81 - 15344 * q^85 + 8964 * q^87 + 22248 * q^89 + 1596 * q^91 + 6120 * q^93 + 3840 * q^95 + 13608 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more