LMFDB - Embedded newform 208.4.i.e.113.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,4,Mod(81,208)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(208, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("208.81");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	208.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$12.2723972812$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 5x^{2} + 4x + 16 $ x^4 - x^3 + 5x^2 + 4x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			113.1
Root			$-0.780776 + 1.35234i$ of defining polynomial
Character	$\chi$	$=$	208.113
Dual form			208.4.i.e.81.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.84233 + 3.19101i) q^{3} -17.8078 q^{5} +(2.71922 + 4.70983i) q^{7} +(6.71165 + 11.6249i) q^{9} +O(q^{10})$	\(q+(-1.84233 + 3.19101i) q^{3} -17.8078 q^{5} +(2.71922 + 4.70983i) q^{7} +(6.71165 + 11.6249i) q^{9} +(-11.2116 + 19.4191i) q^{11} +(21.9730 - 41.4027i) q^{13} +(32.8078 - 56.8247i) q^{15} +(-33.9924 - 58.8766i) q^{17} +(-40.4039 - 69.9816i) q^{19} -20.0388 q^{21} +(70.2656 - 121.704i) q^{23} +192.116 q^{25} -148.946 q^{27} +(53.3466 - 92.3990i) q^{29} +276.155 q^{31} +(-41.3111 - 71.5529i) q^{33} +(-48.4233 - 83.8716i) q^{35} +(2.14584 - 3.71670i) q^{37} +(91.6349 + 146.394i) q^{39} +(-113.884 + 197.254i) q^{41} +(13.7647 + 23.8411i) q^{43} +(-119.519 - 207.014i) q^{45} -318.617 q^{47} +(156.712 - 271.433i) q^{49} +250.501 q^{51} -67.6562 q^{53} +(199.654 - 345.811i) q^{55} +297.749 q^{57} +(-145.557 - 252.113i) q^{59} +(-331.655 - 574.444i) q^{61} +(-36.5009 + 63.2215i) q^{63} +(-391.290 + 737.290i) q^{65} +(-212.551 + 368.149i) q^{67} +(258.905 + 448.436i) q^{69} +(-76.4815 - 132.470i) q^{71} +117.268 q^{73} +(-353.942 + 613.045i) q^{75} -121.948 q^{77} -202.462 q^{79} +(93.1932 - 161.415i) q^{81} -336.155 q^{83} +(605.329 + 1048.46i) q^{85} +(196.564 + 340.459i) q^{87} +(-359.097 + 621.974i) q^{89} +(254.750 - 9.09407i) q^{91} +(-508.769 + 881.214i) q^{93} +(719.503 + 1246.22i) q^{95} +(-379.684 - 657.632i) q^{97} -300.994 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 5 q^{3} - 30 q^{5} + 15 q^{7} - 35 q^{9}+O(q^{10}) $ 4 * q + 5 * q^3 - 30 * q^5 + 15 * q^7 - 35 * q^9	$ 4 q + 5 q^{3} - 30 q^{5} + 15 q^{7} - 35 q^{9} + 17 q^{11} + 125 q^{13} + 90 q^{15} - 70 q^{17} - 141 q^{19} + 126 q^{21} + 145 q^{23} + 150 q^{25} - 670 q^{27} - 34 q^{29} + 280 q^{31} - 425 q^{33} - 70 q^{35} + 190 q^{37} + 181 q^{39} - 538 q^{41} + 455 q^{43} - 375 q^{45} - 120 q^{47} + 565 q^{49} + 466 q^{51} + 1090 q^{53} + 510 q^{55} - 450 q^{57} - 809 q^{59} - 502 q^{61} + 390 q^{63} - 555 q^{65} - 475 q^{67} + 479 q^{69} + 127 q^{71} + 1170 q^{73} - 1725 q^{75} + 510 q^{77} - 480 q^{79} - 122 q^{81} - 520 q^{83} + 1205 q^{85} + 1615 q^{87} - 921 q^{89} + 1287 q^{91} - 2200 q^{93} + 1270 q^{95} + 415 q^{97} - 4420 q^{99}+O(q^{100}) $ 4 * q + 5 * q^3 - 30 * q^5 + 15 * q^7 - 35 * q^9 + 17 * q^11 + 125 * q^13 + 90 * q^15 - 70 * q^17 - 141 * q^19 + 126 * q^21 + 145 * q^23 + 150 * q^25 - 670 * q^27 - 34 * q^29 + 280 * q^31 - 425 * q^33 - 70 * q^35 + 190 * q^37 + 181 * q^39 - 538 * q^41 + 455 * q^43 - 375 * q^45 - 120 * q^47 + 565 * q^49 + 466 * q^51 + 1090 * q^53 + 510 * q^55 - 450 * q^57 - 809 * q^59 - 502 * q^61 + 390 * q^63 - 555 * q^65 - 475 * q^67 + 479 * q^69 + 127 * q^71 + 1170 * q^73 - 1725 * q^75 + 510 * q^77 - 480 * q^79 - 122 * q^81 - 520 * q^83 + 1205 * q^85 + 1615 * q^87 - 921 * q^89 + 1287 * q^91 - 2200 * q^93 + 1270 * q^95 + 415 * q^97 - 4420 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/208\mathbb{Z}\right)^\times$.

$n$	$53$	$79$	$145$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−1.84233	+	3.19101i	−0.354556	+	0.614110i	−0.987042	−	0.160462i	$-0.948701\pi$
$3$	−1.84233	+	3.19101i	−0.354556	+	0.614110i	0.632486	+	0.774572i	$0.282035\pi$
$4$		0			0
$5$	−17.8078			−1.59277			−0.796387	−	0.604787i	$-0.793258\pi$
$5$	−17.8078			−1.59277			−0.796387	+	0.604787i	$0.793258\pi$
$6$		0			0
$7$	2.71922	+	4.70983i	0.146824	+	0.254307i	0.930052	−	0.367428i	$-0.119761\pi$
$7$	2.71922	+	4.70983i	0.146824	+	0.254307i	−0.783228	+	0.621735i	$0.786428\pi$
$8$		0			0
$9$	6.71165	+	11.6249i	0.248579	+	0.430552i
$10$		0			0
$11$	−11.2116	+	19.4191i	−0.307313	+	0.532281i	−0.977774	−	0.209664i	$-0.932763\pi$
$11$	−11.2116	+	19.4191i	−0.307313	+	0.532281i	0.670461	+	0.741945i	$0.266096\pi$
$12$		0			0
$13$	21.9730	−	41.4027i	0.468786	−	0.883312i
$13$	21.9730	−	41.4027i	0.468786	−	0.883312i
$14$		0			0
$15$	32.8078	−	56.8247i	0.564729	−	0.978139i
$16$		0			0
$17$	−33.9924	−	58.8766i	−0.484963	−	0.839981i	0.514888	−	0.857258i	$-0.327834\pi$
$17$	−33.9924	−	58.8766i	−0.484963	−	0.839981i	−0.999851	+	0.0172769i	$0.994500\pi$
$18$		0			0
$19$	−40.4039	−	69.9816i	−0.487857	−	0.844993i	0.512045	−	0.858958i	$-0.328888\pi$
$19$	−40.4039	−	69.9816i	−0.487857	−	0.844993i	−0.999902	+	0.0139650i	$0.995555\pi$
$20$		0			0
$21$	−20.0388			−0.208230
$22$		0			0
$23$	70.2656	−	121.704i	0.637017	−	1.10335i	−0.349067	−	0.937098i	$-0.613501\pi$
$23$	70.2656	−	121.704i	0.637017	−	1.10335i	0.986084	−	0.166248i	$-0.0531653\pi$
$24$		0			0
$25$	192.116			1.53693
$26$		0			0
$27$	−148.946			−1.06165
$28$		0			0
$29$	53.3466	−	92.3990i	0.341594	−	0.591657i	−0.643135	−	0.765753i	$-0.722367\pi$
$29$	53.3466	−	92.3990i	0.341594	−	0.591657i	0.984729	+	0.174095i	$0.0557001\pi$
$30$		0			0
$31$	276.155			1.59997			0.799983	−	0.600023i	$-0.204842\pi$
$31$	276.155			1.59997			0.799983	+	0.600023i	$0.204842\pi$
$32$		0			0
$33$	−41.3111	−	71.5529i	−0.217919	−	0.377447i
$34$		0			0
$35$	−48.4233	−	83.8716i	−0.233858	−	0.405054i
$36$		0			0
$37$	2.14584	−	3.71670i	0.00953442	−	0.0165141i	−0.861219	−	0.508234i	$-0.830298\pi$
$37$	2.14584	−	3.71670i	0.00953442	−	0.0165141i	0.870753	+	0.491720i	$0.163632\pi$
$38$		0			0
$39$	91.6349	+	146.394i	0.376239	+	0.601070i
$40$		0			0
$41$	−113.884	+	197.254i	−0.433799	+	0.751362i	−0.997197	−	0.0748237i	$-0.976161\pi$
$41$	−113.884	+	197.254i	−0.433799	+	0.751362i	0.563398	+	0.826186i	$0.309494\pi$
$42$		0			0
$43$	13.7647	+	23.8411i	0.0488162	+	0.0845521i	0.889401	−	0.457128i	$-0.151122\pi$
$43$	13.7647	+	23.8411i	0.0488162	+	0.0845521i	−0.840585	+	0.541680i	$0.817788\pi$
$44$		0			0
$45$	−119.519	−	207.014i	−0.395931	−	0.685773i
$46$		0			0
$47$	−318.617			−0.988832			−0.494416	−	0.869225i	$-0.664618\pi$
$47$	−318.617			−0.988832			−0.494416	+	0.869225i	$0.664618\pi$
$48$		0			0
$49$	156.712	−	271.433i	0.456885	−	0.791348i
$50$		0			0
$51$	250.501			0.687787
$52$		0			0
$53$	−67.6562			−0.175345			−0.0876726	−	0.996149i	$-0.527943\pi$
$53$	−67.6562			−0.175345			−0.0876726	+	0.996149i	$0.527943\pi$
$54$		0			0
$55$	199.654	−	345.811i	0.489480	−	0.847804i
$56$		0			0
$57$	297.749			0.691892
$58$		0			0
$59$	−145.557	−	252.113i	−0.321186	−	0.556310i	0.659547	−	0.751663i	$-0.270748\pi$
$59$	−145.557	−	252.113i	−0.321186	−	0.556310i	−0.980733	+	0.195353i	$0.937415\pi$
$60$		0			0
$61$	−331.655	−	574.444i	−0.696133	−	1.20574i	−0.969797	−	0.243912i	$-0.921569\pi$
$61$	−331.655	−	574.444i	−0.696133	−	1.20574i	0.273664	−	0.961825i	$-0.411764\pi$
$62$		0			0
$63$	−36.5009	+	63.2215i	−0.0729950	+	0.126431i
$64$		0			0
$65$	−391.290	+	737.290i	−0.746670	+	1.40692i
$66$		0			0
$67$	−212.551	+	368.149i	−0.387570	+	0.671291i	−0.992122	−	0.125275i	$-0.960019\pi$
$67$	−212.551	+	368.149i	−0.387570	+	0.671291i	0.604552	+	0.796566i	$0.293352\pi$
$68$		0			0
$69$	258.905	+	448.436i	0.451717	+	0.782397i
$70$		0			0
$71$	−76.4815	−	132.470i	−0.127841	−	0.221427i	0.794999	−	0.606611i	$-0.207471\pi$
$71$	−76.4815	−	132.470i	−0.127841	−	0.221427i	−0.922840	+	0.385184i	$0.874138\pi$
$72$		0			0
$73$	117.268			0.188016			0.0940081	−	0.995571i	$-0.470032\pi$
$73$	117.268			0.188016			0.0940081	+	0.995571i	$0.470032\pi$
$74$		0			0
$75$	−353.942	+	613.045i	−0.544929	+	0.943845i
$76$		0			0
$77$	−121.948			−0.180484
$78$		0			0
$79$	−202.462			−0.288339			−0.144169	−	0.989553i	$-0.546051\pi$
$79$	−202.462			−0.288339			−0.144169	+	0.989553i	$0.546051\pi$
$80$		0			0
$81$	93.1932	−	161.415i	0.127837	−	0.221420i
$82$		0			0
$83$	−336.155			−0.444552			−0.222276	−	0.974984i	$-0.571349\pi$
$83$	−336.155			−0.444552			−0.222276	+	0.974984i	$0.571349\pi$
$84$		0			0
$85$	605.329	+	1048.46i	0.772437	+	1.33790i
$86$		0			0
$87$	196.564	+	340.459i	0.242228	+	0.419552i
$88$		0			0
$89$	−359.097	+	621.974i	−0.427688	+	0.740777i	−0.996667	−	0.0815748i	$-0.974005\pi$
$89$	−359.097	+	621.974i	−0.427688	+	0.740777i	0.568979	+	0.822352i	$0.307338\pi$
$90$		0			0
$91$	254.750	−	9.09407i	0.293462	−	0.0104760i
$92$		0			0
$93$	−508.769	+	881.214i	−0.567278	+	0.982555i
$94$		0			0
$95$	719.503	+	1246.22i	0.777047	+	1.34588i
$96$		0			0
$97$	−379.684	−	657.632i	−0.397434	−	0.688376i	0.595975	−	0.803003i	$-0.296766\pi$
$97$	−379.684	−	657.632i	−0.397434	−	0.688376i	−0.993409	+	0.114628i	$0.963433\pi$
$98$		0			0
$99$	−300.994			−0.305566
$100$		0			0
$101$	174.348	−	301.980i	0.171766	−	0.297507i	−0.767272	−	0.641322i	$-0.778386\pi$
$101$	174.348	−	301.980i	0.171766	−	0.297507i	0.939037	+	0.343816i	$0.111720\pi$
$102$		0			0
$103$	580.303			0.555136			0.277568	−	0.960706i	$-0.410472\pi$
$103$	580.303			0.555136			0.277568	+	0.960706i	$0.410472\pi$
$104$		0			0
$105$	356.847			0.331663
$106$		0			0
$107$	285.747	−	494.928i	0.258170	−	0.447163i	−0.707582	−	0.706631i	$-0.750214\pi$
$107$	285.747	−	494.928i	0.258170	−	0.447163i	0.965752	+	0.259468i	$0.0835473\pi$
$108$		0			0
$109$	176.004			0.154661			0.0773307	−	0.997005i	$-0.475360\pi$
$109$	176.004			0.154661			0.0773307	+	0.997005i	$0.475360\pi$
$110$		0			0
$111$	7.90668	+	13.6948i	0.00676098	+	0.0117104i
$112$		0			0
$113$	−632.441	−	1095.42i	−0.526505	−	0.911933i	−0.999523	−	0.0308807i	$-0.990169\pi$
$113$	−632.441	−	1095.42i	−0.526505	−	0.911933i	0.473018	−	0.881053i	$-0.343165\pi$
$114$		0			0
$115$	−1251.27	+	2167.27i	−1.01462	+	1.75738i
$116$		0			0
$117$	628.778	−	22.4462i	0.496842	−	0.0177363i
$118$		0			0
$119$	184.866	−	320.197i	0.142409	−	0.246659i
$120$		0			0
$121$	414.098	+	717.239i	0.311118	+	0.538872i
$122$		0			0
$123$	−419.625	−	726.812i	−0.307613	−	0.532801i
$124$		0			0
$125$	−1195.19			−0.855211
$126$		0			0
$127$	1302.05	−	2255.22i	0.909752	−	1.57574i	0.0953448	−	0.995444i	$-0.469605\pi$
$127$	1302.05	−	2255.22i	0.909752	−	1.57574i	0.814408	−	0.580293i	$-0.197062\pi$
$128$		0			0
$129$	−101.436			−0.0692323
$130$		0			0
$131$	−2131.70			−1.42174			−0.710870	−	0.703324i	$-0.751698\pi$
$131$	−2131.70			−1.42174			−0.710870	+	0.703324i	$0.751698\pi$
$132$		0			0
$133$	219.734	−	380.591i	0.143259	−	0.248131i
$134$		0			0
$135$	2652.40			1.69098
$136$		0			0
$137$	−343.992	−	595.812i	−0.214520	−	0.371560i	0.738604	−	0.674140i	$-0.235485\pi$
$137$	−343.992	−	595.812i	−0.214520	−	0.371560i	−0.953124	+	0.302580i	$0.902152\pi$
$138$		0			0
$139$	−339.790	−	588.534i	−0.207343	−	0.359128i	0.743534	−	0.668698i	$-0.233148\pi$
$139$	−339.790	−	588.534i	−0.207343	−	0.359128i	−0.950877	+	0.309570i	$0.899815\pi$
$140$		0			0
$141$	586.998	−	1016.71i	0.350597	−	0.607252i
$142$		0			0
$143$	557.652	+	890.890i	0.326106	+	0.520979i
$144$		0			0
$145$	−949.983	+	1645.42i	−0.544082	+	0.942377i
$146$		0			0
$147$	577.429	+	1000.14i	0.323983	+	0.561155i
$148$		0			0
$149$	987.731	+	1710.80i	0.543074	+	0.940632i	0.998725	+	0.0504739i	$0.0160732\pi$
$149$	987.731	+	1710.80i	0.543074	+	0.940632i	−0.455651	+	0.890159i	$0.650593\pi$
$150$		0			0
$151$	−1803.24			−0.971824			−0.485912	−	0.874008i	$-0.661513\pi$
$151$	−1803.24			−0.971824			−0.485912	+	0.874008i	$0.661513\pi$
$152$		0			0
$153$	456.290	−	790.318i	0.241104	−	0.417604i
$154$		0			0
$155$	−4917.71			−2.54839
$156$		0			0
$157$	−397.168			−0.201894			−0.100947	−	0.994892i	$-0.532187\pi$
$157$	−397.168			−0.201894			−0.100947	+	0.994892i	$0.532187\pi$
$158$		0			0
$159$	124.645	−	215.892i	0.0621698	−	0.107681i
$160$		0			0
$161$	764.272			0.374118
$162$		0			0
$163$	470.696	+	815.270i	0.226183	+	0.391760i	0.956674	−	0.291162i	$-0.0940420\pi$
$163$	470.696	+	815.270i	0.226183	+	0.391760i	−0.730491	+	0.682922i	$0.760709\pi$
$164$		0			0
$165$	735.658	+	1274.20i	0.347096	+	0.601189i
$166$		0			0
$167$	1840.22	−	3187.35i	0.852696	−	1.47691i	−0.0260704	−	0.999660i	$-0.508299\pi$
$167$	1840.22	−	3187.35i	0.852696	−	1.47691i	0.878766	−	0.477252i	$-0.158367\pi$
$168$		0			0
$169$	−1231.37	−	1819.49i	−0.560479	−	0.828168i
$170$		0			0
$171$	542.353	−	939.383i	0.242543	−	0.420096i
$172$		0			0
$173$	−711.387	−	1232.16i	−0.312634	−	0.541499i	0.666297	−	0.745686i	$-0.267878\pi$
$173$	−711.387	−	1232.16i	−0.312634	−	0.541499i	−0.978932	+	0.204187i	$0.934545\pi$
$174$		0			0
$175$	522.408	+	904.837i	0.225659	+	0.390853i
$176$		0			0
$177$	1072.66			0.455514
$178$		0			0
$179$	583.946	−	1011.42i	0.243833	−	0.422331i	−0.717970	−	0.696074i	$-0.754928\pi$
$179$	583.946	−	1011.42i	0.243833	−	0.422331i	0.961803	+	0.273743i	$0.0882617\pi$
$180$		0			0
$181$	−1133.96			−0.465673			−0.232836	−	0.972516i	$-0.574801\pi$
$181$	−1133.96			−0.465673			−0.232836	+	0.972516i	$0.574801\pi$
$182$		0			0
$183$	2444.07			0.987274
$184$		0			0
$185$	−38.2126	+	66.1861i	−0.0151862	+	0.0263032i
$186$		0			0
$187$	1524.44			0.596141
$188$		0			0
$189$	−405.018	−	701.511i	−0.155877	−	0.269986i
$190$		0			0
$191$	1341.06	+	2322.78i	0.508040	+	0.879952i	0.999957	+	0.00930919i	$0.00296325\pi$
$191$	1341.06	+	2322.78i	0.508040	+	0.879952i	−0.491916	+	0.870642i	$0.663703\pi$
$192$		0			0
$193$	985.333	−	1706.65i	0.367491	−	0.636514i	−0.621681	−	0.783270i	$-0.713550\pi$
$193$	985.333	−	1706.65i	0.367491	−	0.636514i	0.989173	+	0.146757i	$0.0468834\pi$
$194$		0			0
$195$	−1631.81	−	2606.94i	−0.599265	−	0.957369i
$196$		0			0
$197$	2008.02	−	3478.00i	0.726222	−	1.25785i	−0.232247	−	0.972657i	$-0.574608\pi$
$197$	2008.02	−	3478.00i	0.726222	−	1.25785i	0.958469	−	0.285197i	$-0.0920590\pi$
$198$		0			0
$199$	−2113.03	−	3659.87i	−0.752707	−	1.30373i	−0.946506	−	0.322685i	$-0.895415\pi$
$199$	−2113.03	−	3659.87i	−0.752707	−	1.30373i	0.193800	−	0.981041i	$-0.437919\pi$
$200$		0			0
$201$	−783.177	−	1356.50i	−0.274831	−	0.476021i
$202$		0			0
$203$	580.245			0.200617
$204$		0			0
$205$	2028.03	−	3512.65i	0.690944	−	1.19675i
$206$		0			0
$207$	1886.39			0.633398
$208$		0			0
$209$	1811.98			0.599699
$210$		0			0
$211$	682.334	−	1181.84i	0.222625	−	0.385597i	−0.732980	−	0.680251i	$-0.761871\pi$
$211$	682.334	−	1181.84i	0.222625	−	0.385597i	0.955604	+	0.294654i	$0.0952043\pi$
$212$		0			0
$213$	563.617			0.181307
$214$		0			0
$215$	−245.118	−	424.557i	−0.0777532	−	0.134672i
$216$		0			0
$217$	750.928	+	1300.65i	0.234914	+	0.406883i
$218$		0			0
$219$	−216.046	+	374.203i	−0.0666624	+	0.115463i
$220$		0			0
$221$	−3184.57	+	113.683i	−0.969309	+	0.0346025i
$222$		0			0
$223$	−529.734	+	917.527i	−0.159075	+	0.275525i	−0.934535	−	0.355871i	$-0.884184\pi$
$223$	−529.734	+	917.527i	−0.159075	+	0.275525i	0.775461	+	0.631396i	$0.217518\pi$
$224$		0			0
$225$	1289.42	+	2233.34i	0.382050	+	0.661729i
$226$		0			0
$227$	1732.10	+	3000.08i	0.506446	+	0.877190i	0.999972	+	0.00745930i	$0.00237439\pi$
$227$	1732.10	+	3000.08i	0.506446	+	0.877190i	−0.493526	+	0.869731i	$0.664292\pi$
$228$		0			0
$229$	−2324.64			−0.670815			−0.335407	−	0.942073i	$-0.608874\pi$
$229$	−2324.64			−0.670815			−0.335407	+	0.942073i	$0.608874\pi$
$230$		0			0
$231$	224.668	−	389.137i	0.0639917	−	0.110837i
$232$		0			0
$233$	−3731.01			−1.04904			−0.524521	−	0.851398i	$-0.675755\pi$
$233$	−3731.01			−1.04904			−0.524521	+	0.851398i	$0.675755\pi$
$234$		0			0
$235$	5673.86			1.57499
$236$		0			0
$237$	373.002	−	646.058i	0.102232	−	0.177072i
$238$		0			0
$239$	−6044.47			−1.63592			−0.817958	−	0.575278i	$-0.804894\pi$
$239$	−6044.47			−1.63592			−0.817958	+	0.575278i	$0.804894\pi$
$240$		0			0
$241$	2586.98	+	4480.78i	0.691461	+	1.19765i	0.971359	+	0.237616i	$0.0763659\pi$
$241$	2586.98	+	4480.78i	0.691461	+	1.19765i	−0.279898	+	0.960030i	$0.590301\pi$
$242$		0			0
$243$	−1667.39	−	2888.00i	−0.440176	−	0.762408i
$244$		0			0
$245$	−2790.68	+	4833.61i	−0.727715	+	1.26044i
$246$		0			0
$247$	−3785.22	+	135.125i	−0.975093	+	0.0348090i
$248$		0			0
$249$	619.309	−	1072.67i	0.157619	−	0.273004i
$250$		0			0
$251$	2810.37	+	4867.70i	0.706728	+	1.22409i	0.966064	+	0.258301i	$0.0831628\pi$
$251$	2810.37	+	4867.70i	0.706728	+	1.22409i	−0.259337	+	0.965787i	$0.583504\pi$
$252$		0			0
$253$	1575.59	+	2729.00i	0.391527	+	0.678144i
$254$		0			0
$255$	−4460.86			−1.09549
$256$		0			0
$257$	837.070	−	1449.85i	0.203171	−	0.351903i	−0.746377	−	0.665523i	$-0.768209\pi$
$257$	837.070	−	1449.85i	0.203171	−	0.351903i	0.949549	+	0.313620i	$0.101542\pi$
$258$		0			0
$259$	23.3401			0.00559954
$260$		0			0
$261$	1432.17			0.339653
$262$		0			0
$263$	−3154.59	+	5463.91i	−0.739622	+	1.28106i	0.213044	+	0.977043i	$0.431662\pi$
$263$	−3154.59	+	5463.91i	−0.739622	+	1.28106i	−0.952666	+	0.304020i	$0.901671\pi$
$264$		0			0
$265$	1204.81			0.279285
$266$		0			0
$267$	−1323.15	−	2291.76i	−0.303279	−	0.525294i
$268$		0			0
$269$	1241.37	+	2150.11i	0.281366	+	0.487340i	0.971721	−	0.236131i	$-0.0758793\pi$
$269$	1241.37	+	2150.11i	0.281366	+	0.487340i	−0.690356	+	0.723470i	$0.742546\pi$
$270$		0			0
$271$	1417.86	−	2455.81i	0.317819	−	0.550478i	−0.662214	−	0.749315i	$-0.730383\pi$
$271$	1417.86	−	2455.81i	0.317819	−	0.550478i	0.980033	+	0.198837i	$0.0637163\pi$
$272$		0			0
$273$	−440.313	+	829.662i	−0.0976153	+	0.183932i
$274$		0			0
$275$	−2153.94	+	3730.74i	−0.472318	+	0.818080i
$276$		0			0
$277$	1918.76	+	3323.38i	0.416198	+	0.720876i	0.995553	−	0.0941989i	$-0.0300290\pi$
$277$	1918.76	+	3323.38i	0.416198	+	0.720876i	−0.579355	+	0.815075i	$0.696696\pi$
$278$		0			0
$279$	1853.46	+	3210.28i	0.397719	+	0.688869i
$280$		0			0
$281$	−9122.13			−1.93659			−0.968293	−	0.249819i	$-0.919629\pi$
$281$	−9122.13			−1.93659			−0.968293	+	0.249819i	$0.919629\pi$
$282$		0			0
$283$	1063.92	−	1842.77i	0.223476	−	0.387072i	−0.732385	−	0.680891i	$-0.761593\pi$
$283$	1063.92	−	1842.77i	0.223476	−	0.387072i	0.955861	+	0.293819i	$0.0949262\pi$
$284$		0			0
$285$	−5302.24			−1.10203
$286$		0			0
$287$	−1238.71			−0.254769
$288$		0			0
$289$	145.530	−	252.066i	0.0296215	−	0.0513059i
$290$		0			0
$291$	2798.01			0.563651
$292$		0			0
$293$	−4137.38	−	7166.16i	−0.824944	−	1.42884i	−0.901962	−	0.431815i	$-0.857873\pi$
$293$	−4137.38	−	7166.16i	−0.824944	−	1.42884i	0.0770183	−	0.997030i	$-0.475460\pi$
$294$		0			0
$295$	2592.05	+	4489.56i	0.511576	+	0.886076i
$296$		0			0
$297$	1669.93	−	2892.40i	0.326260	−	0.565099i
$298$		0			0
$299$	−3494.92	−	5583.38i	−0.675974	−	1.07992i
$300$		0			0
$301$	−74.8585	+	129.659i	−0.0143348	+	0.0248286i
$302$		0			0
$303$	642.414	+	1112.69i	0.121801	+	0.210966i
$304$		0			0
$305$	5906.04	+	10229.6i	1.10878	+	1.92047i
$306$		0			0
$307$	3610.49			0.671211			0.335605	−	0.942003i	$-0.391059\pi$
$307$	3610.49			0.671211			0.335605	+	0.942003i	$0.391059\pi$
$308$		0			0
$309$	−1069.11	+	1851.75i	−0.196827	+	0.340914i
$310$		0			0
$311$	−3331.06			−0.607354			−0.303677	−	0.952775i	$-0.598214\pi$
$311$	−3331.06			−0.607354			−0.303677	+	0.952775i	$0.598214\pi$
$312$		0			0
$313$	−358.125			−0.0646724			−0.0323362	−	0.999477i	$-0.510295\pi$
$313$	−358.125			−0.0646724			−0.0323362	+	0.999477i	$0.510295\pi$
$314$		0			0
$315$	650.000	−	1125.83i	0.116265	−	0.201376i
$316$		0			0
$317$	3047.46			0.539944			0.269972	−	0.962868i	$-0.412986\pi$
$317$	3047.46			0.539944			0.269972	+	0.962868i	$0.412986\pi$
$318$		0			0
$319$	1196.21	+	2071.89i	0.209952	+	0.363647i
$320$		0			0
$321$	1052.88	+	1823.64i	0.183072	+	0.317089i
$322$		0			0
$323$	−2746.85	+	4757.69i	−0.473185	+	0.819581i
$324$		0			0
$325$	4221.38	−	7954.15i	0.720492	−	1.35759i
$326$		0			0
$327$	−324.257	+	561.629i	−0.0548362	+	0.0949791i
$328$		0			0
$329$	−866.392	−	1500.63i	−0.145185	−	0.251467i
$330$		0			0
$331$	3847.39	+	6663.87i	0.638887	+	1.10658i	0.985677	+	0.168641i	$0.0539380\pi$
$331$	3847.39	+	6663.87i	0.638887	+	1.10658i	−0.346791	+	0.937942i	$0.612729\pi$
$332$		0			0
$333$	57.6084			0.00948025
$334$		0			0
$335$	3785.05	−	6555.90i	0.617312	−	1.06922i
$336$		0			0
$337$	4712.21			0.761693			0.380846	−	0.924638i	$-0.375633\pi$
$337$	4712.21			0.761693			0.380846	+	0.924638i	$0.375633\pi$
$338$		0			0
$339$	4660.66			0.746703
$340$		0			0
$341$	−3096.16	+	5362.70i	−0.491690	+	0.851632i
$342$		0			0
$343$	3569.92			0.561976
$344$		0			0
$345$	−4610.52	−	7985.65i	−0.719484	−	1.24618i
$346$		0			0
$347$	−2630.99	−	4557.01i	−0.407029	−	0.704995i	0.587526	−	0.809205i	$-0.300102\pi$
$347$	−2630.99	−	4557.01i	−0.407029	−	0.704995i	−0.994555	+	0.104210i	$0.966768\pi$
$348$		0			0
$349$	−25.1672	+	43.5909i	−0.00386009	+	0.00668587i	−0.867949	−	0.496653i	$-0.834562\pi$
$349$	−25.1672	+	43.5909i	−0.00386009	+	0.00668587i	0.864089	+	0.503339i	$0.167895\pi$
$350$		0			0
$351$	−3272.79	+	6166.77i	−0.497689	+	0.937772i
$352$		0			0
$353$	−4528.82	+	7844.14i	−0.682846	+	1.18272i	0.291263	+	0.956643i	$0.405925\pi$
$353$	−4528.82	+	7844.14i	−0.682846	+	1.18272i	−0.974109	+	0.226081i	$0.927409\pi$
$354$		0			0
$355$	1361.96	+	2358.99i	0.203621	+	0.352683i
$356$		0			0
$357$	681.168	+	1179.82i	0.100984	+	0.174909i
$358$		0			0
$359$	−7177.86			−1.05525			−0.527623	−	0.849479i	$-0.676917\pi$
$359$	−7177.86			−1.05525			−0.527623	+	0.849479i	$0.676917\pi$
$360$		0			0
$361$	164.553	−	285.014i	0.0239908	−	0.0415532i
$362$		0			0
$363$	−3051.62			−0.441235
$364$		0			0
$365$	−2088.28			−0.299467
$366$		0			0
$367$	2002.07	−	3467.69i	0.284761	−	0.493221i	−0.687790	−	0.725910i	$-0.741419\pi$
$367$	2002.07	−	3467.69i	0.284761	−	0.493221i	0.972551	+	0.232689i	$0.0747524\pi$
$368$		0			0
$369$	−3057.41			−0.431334
$370$		0			0
$371$	−183.972	−	318.649i	−0.0257449	−	0.0445915i
$372$		0			0
$373$	5007.09	+	8672.53i	0.695060	+	1.20388i	0.970161	+	0.242464i	$0.0779555\pi$
$373$	5007.09	+	8672.53i	0.695060	+	1.20388i	−0.275101	+	0.961415i	$0.588711\pi$
$374$		0			0
$375$	2201.94	−	3813.87i	0.303221	−	0.525194i
$376$		0			0
$377$	−2653.39	−	4238.98i	−0.362484	−	0.579094i
$378$		0			0
$379$	−4084.56	+	7074.66i	−0.553587	+	0.958842i	0.444425	+	0.895816i	$0.353408\pi$
$379$	−4084.56	+	7074.66i	−0.553587	+	0.958842i	−0.998012	+	0.0630252i	$0.979925\pi$
$380$		0			0
$381$	4797.62	+	8309.73i	0.645117	+	1.11738i
$382$		0			0
$383$	−3655.12	−	6330.86i	−0.487645	−	0.844626i	0.512254	−	0.858834i	$-0.328811\pi$
$383$	−3655.12	−	6330.86i	−0.487645	−	0.844626i	−0.999899	+	0.0142079i	$0.995477\pi$
$384$		0			0
$385$	2171.62			0.287470
$386$		0			0
$387$	−184.767	+	320.027i	−0.0242694	+	0.0420358i
$388$		0			0
$389$	8785.47			1.14509			0.572546	−	0.819872i	$-0.305956\pi$
$389$	8785.47			1.14509			0.572546	+	0.819872i	$0.305956\pi$
$390$		0			0
$391$	−9553.99			−1.23572
$392$		0			0
$393$	3927.30	−	6802.29i	0.504087	−	0.873104i
$394$		0			0
$395$	3605.40			0.459259
$396$		0			0
$397$	−5633.40	−	9757.33i	−0.712171	−	1.23352i	−0.964040	−	0.265756i	$-0.914379\pi$
$397$	−5633.40	−	9757.33i	−0.712171	−	1.23352i	0.251869	−	0.967761i	$-0.418955\pi$
$398$		0			0
$399$	809.646	+	1402.35i	0.101586	+	0.175953i
$400$		0			0
$401$	−788.117	+	1365.06i	−0.0981464	+	0.169995i	−0.910917	−	0.412589i	$-0.864625\pi$
$401$	−788.117	+	1365.06i	−0.0981464	+	0.169995i	0.812771	+	0.582583i	$0.197958\pi$
$402$		0			0
$403$	6067.96	−	11433.6i	0.750042	−	1.41327i
$404$		0			0
$405$	−1659.56	+	2874.45i	−0.203616	+	0.352672i
$406$		0			0
$407$	48.1168	+	83.3407i	0.00586010	+	0.0101500i
$408$		0			0
$409$	−3377.89	−	5850.68i	−0.408377	−	0.707329i	0.586331	−	0.810071i	$-0.300572\pi$
$409$	−3377.89	−	5850.68i	−0.408377	−	0.707329i	−0.994708	+	0.102742i	$0.967238\pi$
$410$		0			0
$411$	2534.99			0.304238
$412$		0			0
$413$	791.606	−	1371.10i	0.0943157	−	0.163360i
$414$		0			0
$415$	5986.17			0.708072
$416$		0			0
$417$	2504.02			0.294059
$418$		0			0
$419$	5378.09	−	9315.13i	0.627057	−	1.08610i	−0.361082	−	0.932534i	$-0.617592\pi$
$419$	5378.09	−	9315.13i	0.627057	−	1.08610i	0.988139	−	0.153561i	$-0.0490742\pi$
$420$		0			0
$421$	7886.03			0.912925			0.456463	−	0.889743i	$-0.349116\pi$
$421$	7886.03			0.912925			0.456463	+	0.889743i	$0.349116\pi$
$422$		0			0
$423$	−2138.45	−	3703.90i	−0.245803	−	0.425744i
$424$		0			0
$425$	−6530.50	−	11311.2i	−0.745355	−	1.29099i
$426$		0			0
$427$	1803.69	−	3124.08i	0.204418	−	0.354063i
$428$		0			0
$429$	−3870.22	+	138.159i	−0.435561	+	0.0155487i
$430$		0			0
$431$	−7042.31	+	12197.6i	−0.787044	+	1.36320i	0.140726	+	0.990049i	$0.455056\pi$
$431$	−7042.31	+	12197.6i	−0.787044	+	1.36320i	−0.927770	+	0.373152i	$0.878277\pi$
$432$		0			0
$433$	−932.072	−	1614.40i	−0.103447	−	0.179175i	0.809656	−	0.586905i	$-0.199654\pi$
$433$	−932.072	−	1614.40i	−0.103447	−	0.179175i	−0.913103	+	0.407730i	$0.866321\pi$
$434$		0			0
$435$	−3500.36	−	6062.81i	−0.385815	−	0.668252i
$436$		0			0
$437$	−11356.0			−1.24309
$438$		0			0
$439$	3077.24	−	5329.94i	0.334553	−	0.579463i	−0.648846	−	0.760920i	$-0.724748\pi$
$439$	3077.24	−	5329.94i	0.334553	−	0.579463i	0.983399	+	0.181457i	$0.0580813\pi$
$440$		0			0
$441$	4207.17			0.454289
$442$		0			0
$443$	14539.3			1.55933			0.779663	−	0.626200i	$-0.215391\pi$
$443$	14539.3			1.55933			0.779663	+	0.626200i	$0.215391\pi$
$444$		0			0
$445$	6394.72	−	11076.0i	0.681210	−	1.17989i
$446$		0			0
$447$	−7278.90			−0.770202
$448$		0			0
$449$	3521.93	+	6100.17i	0.370179	+	0.641169i	0.989593	−	0.143896i	$-0.0459630\pi$
$449$	3521.93	+	6100.17i	0.370179	+	0.641169i	−0.619414	+	0.785065i	$0.712630\pi$
$450$		0			0
$451$	−2553.66	−	4423.08i	−0.266624	−	0.461806i
$452$		0			0
$453$	3322.16	−	5754.15i	0.344567	−	0.596807i
$454$		0			0
$455$	−4536.52	+	161.945i	−0.467418	+	0.0166859i
$456$		0			0
$457$	−7049.43	+	12210.0i	−0.721572	+	1.24980i	0.238798	+	0.971069i	$0.423247\pi$
$457$	−7049.43	+	12210.0i	−0.721572	+	1.24980i	−0.960370	+	0.278730i	$0.910087\pi$
$458$		0			0
$459$	5063.04	+	8769.44i	0.514863	+	0.891770i
$460$		0			0
$461$	−7224.85	−	12513.8i	−0.729924	−	1.26426i	−0.956915	−	0.290368i	$-0.906222\pi$
$461$	−7224.85	−	12513.8i	−0.729924	−	1.26426i	0.226991	−	0.973897i	$-0.427111\pi$
$462$		0			0
$463$	−15806.5			−1.58659			−0.793293	−	0.608840i	$-0.791635\pi$
$463$	−15806.5			−1.58659			−0.793293	+	0.608840i	$0.791635\pi$
$464$		0			0
$465$	9060.04	−	15692.4i	0.903547	−	1.56499i
$466$		0			0
$467$	15071.3			1.49340			0.746699	−	0.665162i	$-0.231638\pi$
$467$	15071.3			1.49340			0.746699	+	0.665162i	$0.231638\pi$
$468$		0			0
$469$	−2311.89			−0.227619
$470$		0			0
$471$	731.713	−	1267.36i	0.0715830	−	0.123985i
$472$		0			0
$473$	−617.299			−0.0600073
$474$		0			0
$475$	−7762.25	−	13444.6i	−0.749803	−	1.29870i
$476$		0			0
$477$	−454.085	−	786.498i	−0.0435872	−	0.0754953i
$478$		0			0
$479$	−196.272	+	339.954i	−0.0187222	+	0.0324277i	−0.875235	−	0.483698i	$-0.839293\pi$
$479$	−196.272	+	339.954i	−0.0187222	+	0.0324277i	0.856513	+	0.516126i	$0.172626\pi$
$480$		0			0
$481$	−106.731	−	170.511i	−0.0101175	−	0.0161634i
$482$		0			0
$483$	−1408.04	+	2438.80i	−0.132646	+	0.229750i
$484$		0			0
$485$	6761.33	+	11711.0i	0.633023	+	1.09643i
$486$		0			0
$487$	4748.94	+	8225.41i	0.441879	+	0.765357i	0.997829	−	0.0658588i	$-0.0209787\pi$
$487$	4748.94	+	8225.41i	0.441879	+	0.765357i	−0.555950	+	0.831216i	$0.687645\pi$
$488$		0			0
$489$	−3468.71			−0.320778
$490$		0			0
$491$	−946.912	+	1640.10i	−0.0870337	+	0.150747i	−0.906256	−	0.422729i	$-0.861072\pi$
$491$	−946.912	+	1640.10i	−0.0870337	+	0.150747i	0.819222	+	0.573476i	$0.194405\pi$
$492$		0			0
$493$	−7253.52			−0.662641
$494$		0			0
$495$	5360.04			0.486699
$496$		0			0
$497$	415.941	−	720.430i	0.0375402	−	0.0650216i
$498$		0			0
$499$	13370.1			1.19945			0.599727	−	0.800205i	$-0.295276\pi$
$499$	13370.1			1.19945			0.599727	+	0.800205i	$0.295276\pi$
$500$		0			0
$501$	6780.57	+	11744.3i	0.604658	+	1.04730i
$502$		0			0
$503$	2777.36	+	4810.52i	0.246195	+	0.426423i	0.962467	−	0.271399i	$-0.0874862\pi$
$503$	2777.36	+	4810.52i	0.246195	+	0.426423i	−0.716272	+	0.697822i	$0.754153\pi$
$504$		0			0
$505$	−3104.76	+	5377.60i	−0.273584	+	0.473861i
$506$		0			0
$507$	8074.59	−	577.231i	0.707308	−	0.0505635i
$508$		0			0
$509$	−1098.78	+	1903.13i	−0.0956824	+	0.165727i	−0.909893	−	0.414843i	$-0.863837\pi$
$509$	−1098.78	+	1903.13i	−0.0956824	+	0.165727i	0.814211	+	0.580569i	$0.197170\pi$
$510$		0			0
$511$	318.878	+	552.313i	0.0276053	+	0.0478139i
$512$		0			0
$513$	6018.00	+	10423.5i	0.517936	+	0.897091i
$514$		0			0
$515$	−10333.9			−0.884206
$516$		0			0
$517$	3572.23	−	6187.28i	0.303881	−	0.526337i
$518$		0			0
$519$	5242.44			0.443386
$520$		0			0
$521$	17005.2			1.42997			0.714983	−	0.699142i	$-0.246435\pi$
$521$	17005.2			1.42997			0.714983	+	0.699142i	$0.246435\pi$
$522$		0			0
$523$	−7243.11	+	12545.4i	−0.605581	+	1.04890i	0.386378	+	0.922341i	$0.373726\pi$
$523$	−7243.11	+	12545.4i	−0.605581	+	1.04890i	−0.991959	+	0.126557i	$0.959607\pi$
$524$		0			0
$525$	−3849.79			−0.320035
$526$		0			0
$527$	−9387.19	−	16259.1i	−0.775925	−	1.34394i
$528$		0			0
$529$	−3791.02	−	6566.23i	−0.311582	−	0.539675i
$530$		0			0
$531$	1953.86	−	3384.18i	0.159680	−	0.276574i
$532$		0			0
$533$	5664.46	+	9049.39i	0.460328	+	0.735408i
$534$		0			0
$535$	−5088.51	+	8813.56i	−0.411206	+	0.712230i
$536$		0			0
$537$	2151.64	+	3726.75i	0.172905	+	0.299481i
$538$		0			0
$539$	3513.99	+	6086.41i	0.280813	+	0.486383i
$540$		0			0
$541$	−15266.7			−1.21325			−0.606623	−	0.794990i	$-0.707476\pi$
$541$	−15266.7			−1.21325			−0.606623	+	0.794990i	$0.707476\pi$
$542$		0			0
$543$	2089.13	−	3618.48i	0.165107	−	0.285974i
$544$		0			0
$545$	−3134.23			−0.246341
$546$		0			0
$547$	−15260.5			−1.19286			−0.596430	−	0.802665i	$-0.703414\pi$
$547$	−15260.5			−1.19286			−0.596430	+	0.802665i	$0.703414\pi$
$548$		0			0
$549$	4451.91	−	7710.93i	0.346089	−	0.599443i
$550$		0			0
$551$	−8621.64			−0.666595
$552$		0			0
$553$	−550.540	−	953.563i	−0.0423351	−	0.0733266i
$554$		0			0
$555$	−140.800	−	243.873i	−0.0107687	−	0.0186520i
$556$		0			0
$557$	5221.05	−	9043.12i	0.397169	−	0.687916i	−0.596207	−	0.802831i	$-0.703326\pi$
$557$	5221.05	−	9043.12i	0.397169	−	0.687916i	0.993375	+	0.114915i	$0.0366595\pi$
$558$		0			0
$559$	1289.54	−	46.0341i	0.0975702	−	0.00348307i
$560$		0			0
$561$	−2808.53	+	4864.51i	−0.211366	+	0.366096i
$562$		0			0
$563$	3572.63	+	6187.98i	0.267440	+	0.463219i	0.968200	−	0.250178i	$-0.0804891\pi$
$563$	3572.63	+	6187.98i	0.267440	+	0.463219i	−0.700760	+	0.713397i	$0.747156\pi$
$564$		0			0
$565$	11262.4	+	19507.0i	0.838604	+	1.45250i
$566$		0			0
$567$	1013.65			0.0750783
$568$		0			0
$569$	−2219.43	+	3844.17i	−0.163521	+	0.283226i	−0.936129	−	0.351657i	$-0.885618\pi$
$569$	−2219.43	+	3844.17i	−0.163521	+	0.283226i	0.772608	+	0.634883i	$0.218952\pi$
$570$		0			0
$571$	−10117.3			−0.741497			−0.370748	−	0.928733i	$-0.620899\pi$
$571$	−10117.3			−0.741497			−0.370748	+	0.928733i	$0.620899\pi$
$572$		0			0
$573$	−9882.70			−0.720516
$574$		0			0
$575$	13499.2	−	23381.3i	0.979052	−	1.69577i
$576$		0			0
$577$	3105.60			0.224069			0.112035	−	0.993704i	$-0.464263\pi$
$577$	3105.60			0.224069			0.112035	+	0.993704i	$0.464263\pi$
$578$		0			0
$579$	3630.62	+	6288.41i	0.260593	+	0.451360i
$580$		0			0
$581$	−914.081	−	1583.24i	−0.0652711	−	0.113053i
$582$		0			0
$583$	758.538	−	1313.83i	0.0538858	−	0.0933329i
$584$		0			0
$585$	−11197.1	+	399.716i	−0.791358	+	0.0282500i
$586$		0			0
$587$	9831.16	−	17028.1i	0.691270	−	1.19731i	−0.280152	−	0.959956i	$-0.590385\pi$
$587$	9831.16	−	17028.1i	0.691270	−	1.19731i	0.971422	−	0.237359i	$-0.0762818\pi$
$588$		0			0
$589$	−11157.7	−	19325.8i	−0.780555	−	1.35196i
$590$		0			0
$591$	7398.88	+	12815.2i	0.514974	+	0.891960i
$592$		0			0
$593$	6395.51			0.442888			0.221444	−	0.975173i	$-0.428923\pi$
$593$	6395.51			0.442888			0.221444	+	0.975173i	$0.428923\pi$
$594$		0			0
$595$	−3292.05	+	5702.00i	−0.226825	+	0.392872i
$596$		0			0
$597$	15571.6			1.06751
$598$		0			0
$599$	−8878.48			−0.605618			−0.302809	−	0.953051i	$-0.597924\pi$
$599$	−8878.48			−0.605618			−0.302809	+	0.953051i	$0.597924\pi$
$600$		0			0
$601$	−9550.29	+	16541.6i	−0.648194	+	1.12270i	0.335360	+	0.942090i	$0.391142\pi$
$601$	−9550.29	+	16541.6i	−0.648194	+	1.12270i	−0.983554	+	0.180615i	$0.942191\pi$
$602$		0			0
$603$	−5706.26			−0.385368
$604$		0			0
$605$	−7374.16	−	12772.4i	−0.495541	−	0.858302i
$606$		0			0
$607$	8297.88	+	14372.4i	0.554861	+	0.961047i	0.997914	+	0.0645522i	$0.0205619\pi$
$607$	8297.88	+	14372.4i	0.554861	+	0.961047i	−0.443053	+	0.896495i	$0.646105\pi$
$608$		0			0
$609$	−1069.00	+	1851.57i	−0.0711300	+	0.123201i
$610$		0			0
$611$	−7000.98	+	13191.6i	−0.463551	+	0.873447i
$612$		0			0
$613$	−8234.58	+	14262.7i	−0.542564	+	0.939748i	0.456192	+	0.889881i	$0.349213\pi$
$613$	−8234.58	+	14262.7i	−0.542564	+	0.939748i	−0.998756	+	0.0498668i	$0.984120\pi$
$614$		0			0
$615$	7472.59	+	12942.9i	0.489958	+	0.848631i
$616$		0			0
$617$	−5057.99	−	8760.69i	−0.330027	−	0.571624i	0.652489	−	0.757798i	$-0.273725\pi$
$617$	−5057.99	−	8760.69i	−0.330027	−	0.571624i	−0.982517	+	0.186174i	$0.940391\pi$
$618$		0			0
$619$	−18854.8			−1.22430			−0.612148	−	0.790743i	$-0.709694\pi$
$619$	−18854.8			−1.22430			−0.612148	+	0.790743i	$0.709694\pi$
$620$		0			0
$621$	−10465.8	+	18127.3i	−0.676292	+	1.17137i
$622$		0			0
$623$	−3905.86			−0.251180
$624$		0			0
$625$	−2730.82			−0.174773
$626$		0			0
$627$	−3338.26	+	5782.03i	−0.212627	+	0.368281i
$628$		0			0
$629$	−291.769			−0.0184954
$630$		0			0
$631$	9473.12	+	16407.9i	0.597653	+	1.03517i	0.993167	+	0.116705i	$0.0372333\pi$
$631$	9473.12	+	16407.9i	0.597653	+	1.03517i	−0.395514	+	0.918460i	$0.629433\pi$
$632$		0			0
$633$	2514.17	+	4354.66i	0.157866	+	0.273432i
$634$		0			0
$635$	−23186.7	+	40160.5i	−1.44903	+	2.50979i
$636$		0			0
$637$	−7794.62	−	12452.5i	−0.484826	−	0.774545i
$638$		0			0
$639$	1026.63	−	1778.18i	0.0635571	−	0.110084i
$640$		0			0
$641$	11793.5	+	20426.9i	0.726698	+	1.25868i	0.958271	+	0.285861i	$0.0922795\pi$
$641$	11793.5	+	20426.9i	0.726698	+	1.25868i	−0.231573	+	0.972818i	$0.574387\pi$
$642$		0			0
$643$	−13576.5	−	23515.2i	−0.832669	−	1.44222i	−0.895915	−	0.444226i	$-0.853479\pi$
$643$	−13576.5	−	23515.2i	−0.832669	−	1.44222i	0.0632461	−	0.997998i	$-0.479855\pi$
$644$		0			0
$645$	1806.35			0.110272
$646$		0			0
$647$	3428.36	−	5938.09i	0.208319	−	0.360820i	−0.742866	−	0.669440i	$-0.766534\pi$
$647$	3428.36	−	5938.09i	0.208319	−	0.360820i	0.951185	+	0.308620i	$0.0998673\pi$
$648$		0			0
$649$	6527.75			0.394817
$650$		0			0
$651$	−5533.83			−0.333161
$652$		0			0
$653$	4036.95	−	6992.20i	0.241926	−	0.419029i	−0.719337	−	0.694662i	$-0.755554\pi$
$653$	4036.95	−	6992.20i	0.241926	−	0.419029i	0.961263	+	0.275633i	$0.0888874\pi$
$654$		0			0
$655$	37960.9			2.26451
$656$		0			0
$657$	787.061	+	1363.23i	0.0467370	+	0.0809508i
$658$		0			0
$659$	2652.86	+	4594.89i	0.156815	+	0.271611i	0.933718	−	0.358008i	$-0.116544\pi$
$659$	2652.86	+	4594.89i	0.156815	+	0.271611i	−0.776904	+	0.629620i	$0.783211\pi$
$660$		0			0
$661$	−12924.2	+	22385.3i	−0.760502	+	1.31723i	0.182091	+	0.983282i	$0.441714\pi$
$661$	−12924.2	+	22385.3i	−0.760502	+	1.31723i	−0.942592	+	0.333946i	$0.891620\pi$
$662$		0			0
$663$	5504.26	−	10371.4i	0.322425	−	0.607531i
$664$		0			0
$665$	−3912.98	+	6777.48i	−0.228179	+	0.395217i
$666$		0			0
$667$	−7496.86	−	12984.9i	−0.435202	−	0.753792i
$668$		0			0
$669$	−1951.89	−	3380.77i	−0.112802	−	0.195379i
$670$		0			0
$671$	14873.6			0.855722
$672$		0			0
$673$	7264.55	−	12582.6i	0.416089	−	0.720687i	−0.579453	−	0.815005i	$-0.696734\pi$
$673$	7264.55	−	12582.6i	0.416089	−	0.720687i	0.995542	+	0.0943186i	$0.0300673\pi$
$674$		0			0
$675$	−28615.0			−1.63169
$676$		0			0
$677$	12058.1			0.684535			0.342267	−	0.939603i	$-0.388805\pi$
$677$	12058.1			0.684535			0.342267	+	0.939603i	$0.388805\pi$
$678$		0			0
$679$	2064.89	−	3576.50i	0.116706	−	0.202140i
$680$		0			0
$681$	−12764.4			−0.718255
$682$		0			0
$683$	15014.4	+	26005.7i	0.841156	+	1.45693i	0.888918	+	0.458067i	$0.151458\pi$
$683$	15014.4	+	26005.7i	0.841156	+	1.45693i	−0.0477615	+	0.998859i	$0.515209\pi$
$684$		0			0
$685$	6125.74	+	10610.1i	0.341682	+	0.591811i
$686$		0			0
$687$	4282.75	−	7417.94i	0.237842	−	0.411954i
$688$		0			0
$689$	−1486.61	+	2801.15i	−0.0821994	+	0.154884i
$690$		0			0
$691$	224.848	−	389.448i	0.0123786	−	0.0214404i	−0.859770	−	0.510682i	$-0.829393\pi$
$691$	224.848	−	389.448i	0.0123786	−	0.0214404i	0.872148	+	0.489241i	$0.162726\pi$
$692$		0			0
$693$	−818.471	−	1417.63i	−0.0448646	−	0.0777077i
$694$		0			0
$695$	6050.90	+	10480.5i	0.330250	+	0.572010i
$696$		0			0
$697$	15484.8			0.841506
$698$		0			0
$699$	6873.75	−	11905.7i	0.371944	−	0.644227i
$700$		0			0
$701$	−26986.0			−1.45399			−0.726994	−	0.686644i	$-0.759083\pi$
$701$	−26986.0			−1.45399			−0.726994	+	0.686644i	$0.759083\pi$
$702$		0			0
$703$	−346.801			−0.0186057
$704$		0			0
$705$	−10453.1	+	18105.3i	−0.558422	+	0.967215i
$706$		0			0
$707$	1896.37			0.100877
$708$		0			0
$709$	4549.44	+	7879.85i	0.240984	+	0.417396i	0.960995	−	0.276566i	$-0.0891965\pi$
$709$	4549.44	+	7879.85i	0.240984	+	0.417396i	−0.720011	+	0.693963i	$0.755863\pi$
$710$		0			0
$711$	−1358.85	−	2353.60i	−0.0716751	−	0.124145i
$712$		0			0
$713$	19404.2	−	33609.1i	1.01921	−	1.76532i
$714$		0			0
$715$	−9930.53	−	15864.8i	−0.519414	−	0.829802i
$716$		0			0
$717$	11135.9	−	19287.9i	0.580025	−	1.00463i
$718$		0			0
$719$	−3146.78	−	5450.38i	−0.163220	−	0.282705i	0.772802	−	0.634647i	$-0.218855\pi$
$719$	−3146.78	−	5450.38i	−0.163220	−	0.282705i	−0.936022	+	0.351942i	$0.885521\pi$
$720$		0			0
$721$	1577.97	+	2733.13i	0.0815074	+	0.141175i
$722$		0			0
$723$	−19064.3			−0.980648
$724$		0			0
$725$	10248.8	−	17751.4i	0.525006	−	0.909337i
$726$		0			0
$727$	−18070.7			−0.921878			−0.460939	−	0.887432i	$-0.652487\pi$
$727$	−18070.7			−0.921878			−0.460939	+	0.887432i	$0.652487\pi$
$728$		0			0
$729$	17319.9			0.879944
$730$		0			0
$731$	935.790	−	1620.84i	0.0473481	−	0.0820093i
$732$		0			0
$733$	34771.5			1.75214			0.876068	−	0.482188i	$-0.160158\pi$
$733$	34771.5			1.75214			0.876068	+	0.482188i	$0.160158\pi$
$734$		0			0
$735$	−10282.7	−	17810.2i	−0.516032	−	0.893794i
$736$		0			0
$737$	−4766.09	−	8255.10i	−0.238210	−	0.412592i
$738$		0			0
$739$	11815.7	−	20465.4i	0.588158	−	1.01872i	−0.406316	−	0.913733i	$-0.633187\pi$
$739$	11815.7	−	20465.4i	0.588158	−	1.01872i	0.994474	−	0.104986i	$-0.0334799\pi$
$740$		0			0
$741$	6542.44	−	12327.6i	0.324349	−	0.611156i
$742$		0			0
$743$	16251.4	−	28148.3i	0.802431	−	1.38985i	−0.115581	−	0.993298i	$-0.536873\pi$
$743$	16251.4	−	28148.3i	0.802431	−	1.38985i	0.918012	−	0.396553i	$-0.129794\pi$
$744$		0			0
$745$	−17589.3	−	30465.5i	−0.864995	−	1.49822i
$746$		0			0
$747$	−2256.16	−	3907.78i	−0.110507	−	0.191403i
$748$		0			0
$749$	3108.04			0.151622
$750$		0			0
$751$	1010.43	−	1750.12i	0.0490960	−	0.0850368i	−0.840433	−	0.541915i	$-0.817699\pi$
$751$	1010.43	−	1750.12i	0.0490960	−	0.0850368i	0.889529	+	0.456879i	$0.151033\pi$
$752$		0			0
$753$	−20710.5			−1.00230
$754$		0			0
$755$	32111.6			1.54790
$756$		0			0
$757$	−6284.11	+	10884.4i	−0.301717	+	0.522589i	−0.976525	−	0.215404i	$-0.930893\pi$
$757$	−6284.11	+	10884.4i	−0.301717	+	0.522589i	0.674808	+	0.737993i	$0.264226\pi$
$758$		0			0
$759$	−11611.0			−0.555273
$760$		0			0
$761$	4352.40	+	7538.59i	0.207325	+	0.359098i	0.950871	−	0.309587i	$-0.100191\pi$
$761$	4352.40	+	7538.59i	0.207325	+	0.359098i	−0.743546	+	0.668685i	$0.766857\pi$
$762$		0			0
$763$	478.593	+	828.948i	0.0227081	+	0.0393315i
$764$		0			0
$765$	−8125.51	+	14073.8i	−0.384024	+	0.665149i
$766$		0			0
$767$	−13636.5	+	486.797i	−0.641962	+	0.0229168i
$768$		0			0
$769$	10957.9	−	18979.7i	0.513853	−	0.890020i	−0.486018	−	0.873949i	$-0.661551\pi$
$769$	10957.9	−	18979.7i	0.513853	−	0.890020i	0.999871	−	0.0160706i	$-0.00511566\pi$
$770$		0			0
$771$	3084.32	+	5342.19i	0.144071	+	0.249539i
$772$		0			0
$773$	11538.8	+	19985.7i	0.536896	+	0.929930i	0.999069	+	0.0431408i	$0.0137364\pi$
$773$	11538.8	+	19985.7i	0.536896	+	0.929930i	−0.462173	+	0.886790i	$0.652930\pi$
$774$		0			0
$775$	53054.0			2.45904
$776$		0			0
$777$	−43.0001	+	74.4783i	−0.00198535	+	0.00343873i
$778$		0			0
$779$	18405.5			0.846528
$780$		0			0
$781$	3429.94			0.157148
$782$		0			0
$783$	−7945.76	+	13762.5i	−0.362654	+	0.628136i
$784$		0			0
$785$	7072.67			0.321572
$786$		0			0
$787$	−8261.21	−	14308.8i	−0.374181

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:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	208.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.84233 + 3.19101i) q^{3} -17.8078 q^{5} +(2.71922 + 4.70983i) q^{7} +(6.71165 + 11.6249i) q^{9} +O(q^{10})\)	\(q+(-1.84233 + 3.19101i) q^{3} -17.8078 q^{5} +(2.71922 + 4.70983i) q^{7} +(6.71165 + 11.6249i) q^{9} +(-11.2116 + 19.4191i) q^{11} +(21.9730 - 41.4027i) q^{13} +(32.8078 - 56.8247i) q^{15} +(-33.9924 - 58.8766i) q^{17} +(-40.4039 - 69.9816i) q^{19} -20.0388 q^{21} +(70.2656 - 121.704i) q^{23} +192.116 q^{25} -148.946 q^{27} +(53.3466 - 92.3990i) q^{29} +276.155 q^{31} +(-41.3111 - 71.5529i) q^{33} +(-48.4233 - 83.8716i) q^{35} +(2.14584 - 3.71670i) q^{37} +(91.6349 + 146.394i) q^{39} +(-113.884 + 197.254i) q^{41} +(13.7647 + 23.8411i) q^{43} +(-119.519 - 207.014i) q^{45} -318.617 q^{47} +(156.712 - 271.433i) q^{49} +250.501 q^{51} -67.6562 q^{53} +(199.654 - 345.811i) q^{55} +297.749 q^{57} +(-145.557 - 252.113i) q^{59} +(-331.655 - 574.444i) q^{61} +(-36.5009 + 63.2215i) q^{63} +(-391.290 + 737.290i) q^{65} +(-212.551 + 368.149i) q^{67} +(258.905 + 448.436i) q^{69} +(-76.4815 - 132.470i) q^{71} +117.268 q^{73} +(-353.942 + 613.045i) q^{75} -121.948 q^{77} -202.462 q^{79} +(93.1932 - 161.415i) q^{81} -336.155 q^{83} +(605.329 + 1048.46i) q^{85} +(196.564 + 340.459i) q^{87} +(-359.097 + 621.974i) q^{89} +(254.750 - 9.09407i) q^{91} +(-508.769 + 881.214i) q^{93} +(719.503 + 1246.22i) q^{95} +(-379.684 - 657.632i) q^{97} -300.994 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 5 q^{3} - 30 q^{5} + 15 q^{7} - 35 q^{9}+O(q^{10}) \) 4 * q + 5 * q^3 - 30 * q^5 + 15 * q^7 - 35 * q^9	\( 4 q + 5 q^{3} - 30 q^{5} + 15 q^{7} - 35 q^{9} + 17 q^{11} + 125 q^{13} + 90 q^{15} - 70 q^{17} - 141 q^{19} + 126 q^{21} + 145 q^{23} + 150 q^{25} - 670 q^{27} - 34 q^{29} + 280 q^{31} - 425 q^{33} - 70 q^{35} + 190 q^{37} + 181 q^{39} - 538 q^{41} + 455 q^{43} - 375 q^{45} - 120 q^{47} + 565 q^{49} + 466 q^{51} + 1090 q^{53} + 510 q^{55} - 450 q^{57} - 809 q^{59} - 502 q^{61} + 390 q^{63} - 555 q^{65} - 475 q^{67} + 479 q^{69} + 127 q^{71} + 1170 q^{73} - 1725 q^{75} + 510 q^{77} - 480 q^{79} - 122 q^{81} - 520 q^{83} + 1205 q^{85} + 1615 q^{87} - 921 q^{89} + 1287 q^{91} - 2200 q^{93} + 1270 q^{95} + 415 q^{97} - 4420 q^{99}+O(q^{100}) \) 4 * q + 5 * q^3 - 30 * q^5 + 15 * q^7 - 35 * q^9 + 17 * q^11 + 125 * q^13 + 90 * q^15 - 70 * q^17 - 141 * q^19 + 126 * q^21 + 145 * q^23 + 150 * q^25 - 670 * q^27 - 34 * q^29 + 280 * q^31 - 425 * q^33 - 70 * q^35 + 190 * q^37 + 181 * q^39 - 538 * q^41 + 455 * q^43 - 375 * q^45 - 120 * q^47 + 565 * q^49 + 466 * q^51 + 1090 * q^53 + 510 * q^55 - 450 * q^57 - 809 * q^59 - 502 * q^61 + 390 * q^63 - 555 * q^65 - 475 * q^67 + 479 * q^69 + 127 * q^71 + 1170 * q^73 - 1725 * q^75 + 510 * q^77 - 480 * q^79 - 122 * q^81 - 520 * q^83 + 1205 * q^85 + 1615 * q^87 - 921 * q^89 + 1287 * q^91 - 2200 * q^93 + 1270 * q^95 + 415 * q^97 - 4420 * q^99

Introduction

L-functions

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