LMFDB - Embedded newform 688.2.bg.c.81.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [688,2,Mod(17,688)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(688, base_ring=CyclotomicField(42))

chi = DirichletCharacter(H, H._module([0, 0, 38]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("688.17");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 688 = 2^{4} \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	688.bg (of order $21$, degree $12$, 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:	$5.49370765906$
Analytic rank:	$0$
Dimension:	$36$
Relative dimension:	$3$ over $\Q(\zeta_{21})$
Twist minimal:	no (minimal twist has level 43)
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			81.1
Character	$\chi$	$=$	688.81
Dual form			688.2.bg.c.17.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.28860 - 0.397480i) q^{3} +(-1.48781 - 3.79089i) q^{5} +(-1.38920 - 2.40617i) q^{7} +(-0.976224 - 0.665578i) q^{9} +O(q^{10})$	\(q+(-1.28860 - 0.397480i) q^{3} +(-1.48781 - 3.79089i) q^{5} +(-1.38920 - 2.40617i) q^{7} +(-0.976224 - 0.665578i) q^{9} +(0.678323 - 0.326663i) q^{11} +(1.70167 + 0.256486i) q^{13} +(0.410392 + 5.47630i) q^{15} +(-1.18750 + 3.02571i) q^{17} +(0.0395049 - 0.0269340i) q^{19} +(0.833720 + 3.65277i) q^{21} +(-0.152371 + 2.03326i) q^{23} +(-8.49199 + 7.87942i) q^{25} +(3.51575 + 4.40861i) q^{27} +(0.714324 - 0.220340i) q^{29} +(-2.52247 - 2.34051i) q^{31} +(-1.00393 + 0.151318i) q^{33} +(-7.05465 + 8.84626i) q^{35} +(3.91502 - 6.78101i) q^{37} +(-2.09082 - 1.00689i) q^{39} +(-1.86928 + 8.18985i) q^{41} +(4.39142 - 4.86985i) q^{43} +(-1.07069 + 4.69101i) q^{45} +(-7.05780 - 3.39886i) q^{47} +(-0.359777 + 0.623151i) q^{49} +(2.73287 - 3.42691i) q^{51} +(-1.76849 + 0.266557i) q^{53} +(-2.24756 - 2.08543i) q^{55} +(-0.0616116 + 0.0190047i) q^{57} +(-3.60798 - 4.52427i) q^{59} +(2.15836 - 2.00266i) q^{61} +(-0.245321 + 3.27359i) q^{63} +(-1.55946 - 6.83245i) q^{65} +(-4.62708 + 3.15469i) q^{67} +(1.00452 - 2.55948i) q^{69} +(0.543672 + 7.25480i) q^{71} +(-10.0746 - 1.51850i) q^{73} +(14.0747 - 6.77800i) q^{75} +(-1.72834 - 1.17836i) q^{77} +(6.70519 + 11.6137i) q^{79} +(-1.48307 - 3.77880i) q^{81} +(-5.42706 - 1.67402i) q^{83} +13.2369 q^{85} -1.00806 q^{87} +(-11.7786 - 3.63323i) q^{89} +(-1.74682 - 4.45083i) q^{91} +(2.32015 + 4.01861i) q^{93} +(-0.160880 - 0.109686i) q^{95} +(9.41101 - 4.53210i) q^{97} +(-0.879615 - 0.132581i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q + 16 q^{3} - 17 q^{5} - 6 q^{7} - q^{9}+O(q^{10}) $ 36 * q + 16 * q^3 - 17 * q^5 - 6 * q^7 - q^9	$ 36 q + 16 q^{3} - 17 q^{5} - 6 q^{7} - q^{9} + 4 q^{11} + 3 q^{15} - 10 q^{17} - 10 q^{19} - 21 q^{21} - 4 q^{23} - 2 q^{25} + 4 q^{27} + 9 q^{29} - 40 q^{31} - 11 q^{33} - 11 q^{35} - 19 q^{37} + q^{39} - 28 q^{41} + 8 q^{43} - 46 q^{45} + 30 q^{47} + 6 q^{49} - 57 q^{51} - 24 q^{53} - 14 q^{55} + 52 q^{57} + q^{59} - 14 q^{61} - 47 q^{63} + 38 q^{65} - 66 q^{67} - 7 q^{69} + 33 q^{71} + 29 q^{73} + 55 q^{75} - 27 q^{77} + 17 q^{79} + 38 q^{81} + 23 q^{83} - 56 q^{85} + 86 q^{87} - 19 q^{89} + 13 q^{91} - 30 q^{93} - q^{95} - 31 q^{97} + 31 q^{99}+O(q^{100}) $ 36 * q + 16 * q^3 - 17 * q^5 - 6 * q^7 - q^9 + 4 * q^11 + 3 * q^15 - 10 * q^17 - 10 * q^19 - 21 * q^21 - 4 * q^23 - 2 * q^25 + 4 * q^27 + 9 * q^29 - 40 * q^31 - 11 * q^33 - 11 * q^35 - 19 * q^37 + q^39 - 28 * q^41 + 8 * q^43 - 46 * q^45 + 30 * q^47 + 6 * q^49 - 57 * q^51 - 24 * q^53 - 14 * q^55 + 52 * q^57 + q^59 - 14 * q^61 - 47 * q^63 + 38 * q^65 - 66 * q^67 - 7 * q^69 + 33 * q^71 + 29 * q^73 + 55 * q^75 - 27 * q^77 + 17 * q^79 + 38 * q^81 + 23 * q^83 - 56 * q^85 + 86 * q^87 - 19 * q^89 + 13 * q^91 - 30 * q^93 - q^95 - 31 * q^97 + 31 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/688\mathbb{Z}\right)^\times$.

$n$	$431$	$433$	$517$
$\chi(n)$	$1$	$e\left(\frac{2}{21}\right)$	$1$

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
$3$	−1.28860	−	0.397480i	−0.743972	−	0.229485i	−0.100481	−	0.994939i	$-0.532038\pi$
$3$	−1.28860	−	0.397480i	−0.743972	−	0.229485i	−0.643491	+	0.765454i	$0.722514\pi$
$4$		0			0
$5$	−1.48781	−	3.79089i	−0.665371	−	1.69534i	−0.717372	−	0.696690i	$-0.754655\pi$
$5$	−1.48781	−	3.79089i	−0.665371	−	1.69534i	0.0520012	−	0.998647i	$-0.483440\pi$
$6$		0			0
$7$	−1.38920	−	2.40617i	−0.525070	−	0.909448i	−0.999574	−	0.0291942i	$-0.990706\pi$
$7$	−1.38920	−	2.40617i	−0.525070	−	0.909448i	0.474504	−	0.880253i	$-0.342627\pi$
$8$		0			0
$9$	−0.976224	−	0.665578i	−0.325408	−	0.221859i
$10$		0			0
$11$	0.678323	−	0.326663i	0.204522	−	0.0984926i	−0.328819	−	0.944393i	$-0.606651\pi$
$11$	0.678323	−	0.326663i	0.204522	−	0.0984926i	0.533341	+	0.845900i	$0.320936\pi$
$12$		0			0
$13$	1.70167	+	0.256486i	0.471959	+	0.0711364i	0.380715	−	0.924692i	$-0.375678\pi$
$13$	1.70167	+	0.256486i	0.471959	+	0.0711364i	0.0912435	+	0.995829i	$0.470916\pi$
$14$		0			0
$15$	0.410392	+	5.47630i	0.105963	+	1.41398i
$16$		0			0
$17$	−1.18750	+	3.02571i	−0.288012	+	0.733842i	0.711503	+	0.702683i	$0.248015\pi$
$17$	−1.18750	+	3.02571i	−0.288012	+	0.733842i	−0.999514	+	0.0311584i	$0.990080\pi$
$18$		0			0
$19$	0.0395049	−	0.0269340i	0.00906304	−	0.00617908i	−0.558780	−	0.829316i	$-0.688730\pi$
$19$	0.0395049	−	0.0269340i	0.00906304	−	0.00617908i	0.567843	+	0.823137i	$0.307778\pi$
$20$		0			0
$21$	0.833720	+	3.65277i	0.181933	+	0.797099i
$22$		0			0
$23$	−0.152371	+	2.03326i	−0.0317717	+	0.423963i	0.958526	+	0.285004i	$0.0919948\pi$
$23$	−0.152371	+	2.03326i	−0.0317717	+	0.423963i	−0.990298	+	0.138960i	$0.955624\pi$
$24$		0			0
$25$	−8.49199	+	7.87942i	−1.69840	+	1.57588i
$26$		0			0
$27$	3.51575	+	4.40861i	0.676606	+	0.848437i
$28$		0			0
$29$	0.714324	−	0.220340i	0.132647	−	0.0409161i	−0.227721	−	0.973726i	$-0.573127\pi$
$29$	0.714324	−	0.220340i	0.132647	−	0.0409161i	0.360368	+	0.932810i	$0.382651\pi$
$30$		0			0
$31$	−2.52247	−	2.34051i	−0.453050	−	0.420369i	0.420358	−	0.907358i	$-0.361904\pi$
$31$	−2.52247	−	2.34051i	−0.453050	−	0.420369i	−0.873408	+	0.486990i	$0.838095\pi$
$32$		0			0
$33$	−1.00393	+	0.151318i	−0.174761	+	0.0263410i
$34$		0			0
$35$	−7.05465	+	8.84626i	−1.19245	+	1.49529i
$36$		0			0
$37$	3.91502	−	6.78101i	0.643625	−	1.11479i	−0.340992	−	0.940066i	$-0.610763\pi$
$37$	3.91502	−	6.78101i	0.643625	−	1.11479i	0.984617	−	0.174725i	$-0.0559037\pi$
$38$		0			0
$39$	−2.09082	−	1.00689i	−0.334799	−	0.161231i
$40$		0			0
$41$	−1.86928	+	8.18985i	−0.291932	+	1.27904i	0.589900	+	0.807476i	$0.299167\pi$
$41$	−1.86928	+	8.18985i	−0.291932	+	1.27904i	−0.881832	+	0.471563i	$0.843690\pi$
$42$		0			0
$43$	4.39142	−	4.86985i	0.669685	−	0.742645i
$43$	4.39142	−	4.86985i	0.669685	−	0.742645i
$44$		0			0
$45$	−1.07069	+	4.69101i	−0.159610	+	0.699295i
$46$		0			0
$47$	−7.05780	−	3.39886i	−1.02949	−	0.495775i	−0.158644	−	0.987336i	$-0.550712\pi$
$47$	−7.05780	−	3.39886i	−1.02949	−	0.495775i	−0.870843	+	0.491561i	$0.836426\pi$
$48$		0			0
$49$	−0.359777	+	0.623151i	−0.0513966	+	0.0890216i
$50$		0			0
$51$	2.73287	−	3.42691i	0.382678	−	0.479863i
$52$		0			0
$53$	−1.76849	+	0.266557i	−0.242920	+	0.0366144i	−0.269374	−	0.963036i	$-0.586817\pi$
$53$	−1.76849	+	0.266557i	−0.242920	+	0.0366144i	0.0264535	+	0.999650i	$0.491579\pi$
$54$		0			0
$55$	−2.24756	−	2.08543i	−0.303061	−	0.281200i
$56$		0			0
$57$	−0.0616116	+	0.0190047i	−0.00816065	+	0.00251723i
$58$		0			0
$59$	−3.60798	−	4.52427i	−0.469719	−	0.589009i	0.489383	−	0.872069i	$-0.337222\pi$
$59$	−3.60798	−	4.52427i	−0.469719	−	0.589009i	−0.959102	+	0.283060i	$0.908651\pi$
$60$		0			0
$61$	2.15836	−	2.00266i	0.276349	−	0.256415i	−0.529806	−	0.848119i	$-0.677735\pi$
$61$	2.15836	−	2.00266i	0.276349	−	0.256415i	0.806155	+	0.591704i	$0.201545\pi$
$62$		0			0
$63$	−0.245321	+	3.27359i	−0.0309076	+	0.412433i
$64$		0			0
$65$	−1.55946	−	6.83245i	−0.193428	−	0.847462i
$66$		0			0
$67$	−4.62708	+	3.15469i	−0.565288	+	0.385407i	−0.811978	−	0.583688i	$-0.801609\pi$
$67$	−4.62708	+	3.15469i	−0.565288	+	0.385407i	0.246691	+	0.969094i	$0.420657\pi$
$68$		0			0
$69$	1.00452	−	2.55948i	0.120930	−	0.308126i
$70$		0			0
$71$	0.543672	+	7.25480i	0.0645220	+	0.860987i	0.931756	+	0.363086i	$0.118277\pi$
$71$	0.543672	+	7.25480i	0.0645220	+	0.860987i	−0.867234	+	0.497901i	$0.834104\pi$
$72$		0			0
$73$	−10.0746	−	1.51850i	−1.17914	−	0.177727i	−0.469918	−	0.882710i	$-0.655716\pi$
$73$	−10.0746	−	1.51850i	−1.17914	−	0.177727i	−0.709224	+	0.704983i	$0.750955\pi$
$74$		0			0
$75$	14.0747	−	6.77800i	1.62520	−	0.782656i
$76$		0			0
$77$	−1.72834	−	1.17836i	−0.196962	−	0.134287i
$78$		0			0
$79$	6.70519	+	11.6137i	0.754393	+	1.30665i	0.945676	+	0.325112i	$0.105402\pi$
$79$	6.70519	+	11.6137i	0.754393	+	1.30665i	−0.191283	+	0.981535i	$0.561265\pi$
$80$		0			0
$81$	−1.48307	−	3.77880i	−0.164785	−	0.419866i
$82$		0			0
$83$	−5.42706	−	1.67402i	−0.595697	−	0.183748i	−0.0177773	−	0.999842i	$-0.505659\pi$
$83$	−5.42706	−	1.67402i	−0.595697	−	0.183748i	−0.577919	+	0.816094i	$0.696135\pi$
$84$		0			0
$85$	13.2369			1.43574
$86$		0			0
$87$	−1.00806			−0.108075
$88$		0			0
$89$	−11.7786	−	3.63323i	−1.24853	−	0.385121i	−0.401105	−	0.916032i	$-0.631374\pi$
$89$	−11.7786	−	3.63323i	−1.24853	−	0.385121i	−0.847428	+	0.530911i	$0.821850\pi$
$90$		0			0
$91$	−1.74682	−	4.45083i	−0.183117	−	0.466573i
$92$		0			0
$93$	2.32015	+	4.01861i	0.240588	+	0.416710i
$94$		0			0
$95$	−0.160880	−	0.109686i	−0.0165059	−	0.0112535i
$96$		0			0
$97$	9.41101	−	4.53210i	0.955543	−	0.460165i	0.109917	−	0.993941i	$-0.464941\pi$
$97$	9.41101	−	4.53210i	0.955543	−	0.460165i	0.845626	+	0.533775i	$0.179227\pi$
$98$		0			0
$99$	−0.879615	−	0.132581i	−0.0884046	−	0.0133249i
$100$		0			0
$101$	0.171298	+	2.28582i	0.0170448	+	0.227447i	0.999220	+	0.0394811i	$0.0125705\pi$
$101$	0.171298	+	2.28582i	0.0170448	+	0.227447i	−0.982175	+	0.187966i	$0.939810\pi$
$102$		0			0
$103$	1.94001	−	4.94307i	0.191155	−	0.487056i	−0.802877	−	0.596145i	$-0.796698\pi$
$103$	1.94001	−	4.94307i	0.191155	−	0.487056i	0.994032	+	0.109090i	$0.0347935\pi$
$104$		0			0
$105$	12.6068	−	8.59518i	1.23030	−	0.838804i
$106$		0			0
$107$	−0.800089	−	3.50542i	−0.0773475	−	0.338882i	0.921417	−	0.388575i	$-0.127033\pi$
$107$	−0.800089	−	3.50542i	−0.0773475	−	0.338882i	−0.998765	+	0.0496936i	$0.984176\pi$
$108$		0			0
$109$	0.415847	−	5.54910i	0.0398310	−	0.531507i	−0.941225	−	0.337781i	$-0.890324\pi$
$109$	0.415847	−	5.54910i	0.0398310	−	0.531507i	0.981056	−	0.193726i	$-0.0620573\pi$
$110$		0			0
$111$	−7.74019	+	7.18185i	−0.734667	+	0.681671i
$112$		0			0
$113$	0.0874119	+	0.109611i	0.00822302	+	0.0103113i	0.785926	−	0.618321i	$-0.212187\pi$
$113$	0.0874119	+	0.109611i	0.00822302	+	0.0103113i	−0.777703	+	0.628632i	$0.783615\pi$
$114$		0			0
$115$	7.93455	−	2.44748i	0.739901	−	0.228229i
$116$		0			0
$117$	−1.49050	−	1.38298i	−0.137797	−	0.127857i
$118$		0			0
$119$	8.93005	−	1.34599i	0.818617	−	0.123387i
$120$		0			0
$121$	−6.50497	+	8.15698i	−0.591361	+	0.741544i
$122$		0			0
$123$	5.66405	−	9.81041i	0.510710	−	0.884575i
$124$		0			0
$125$	24.1590	+	11.6344i	2.16084	+	1.04061i
$126$		0			0
$127$	0.384858	−	1.68617i	0.0341507	−	0.149624i	−0.954978	−	0.296677i	$-0.904122\pi$
$127$	0.384858	−	1.68617i	0.0341507	−	0.149624i	0.989129	+	0.147053i	$0.0469788\pi$
$128$		0			0
$129$	−7.59444	+	4.52977i	−0.668653	+	0.398824i
$130$		0			0
$131$	−4.87229	+	21.3469i	−0.425694	+	1.86509i	0.0714858	+	0.997442i	$0.477226\pi$
$131$	−4.87229	+	21.3469i	−0.425694	+	1.86509i	−0.497180	+	0.867647i	$0.665631\pi$
$132$		0			0
$133$	−0.119688	−	0.0576388i	−0.0103783	−	0.00499791i
$134$		0			0
$135$	11.4818	−	19.8870i	0.988193	−	1.71160i
$136$		0			0
$137$	13.3197	−	16.7024i	1.13798	−	1.42698i	0.249314	−	0.968423i	$-0.419795\pi$
$137$	13.3197	−	16.7024i	1.13798	−	1.42698i	0.888665	−	0.458557i	$-0.151634\pi$
$138$		0			0
$139$	12.0494	−	1.81616i	1.02202	−	0.154044i	0.383411	−	0.923578i	$-0.374749\pi$
$139$	12.0494	−	1.81616i	1.02202	−	0.154044i	0.638606	+	0.769534i	$0.279511\pi$
$140$		0			0
$141$	7.74369	+	7.18509i	0.652136	+	0.605094i
$142$		0			0
$143$	1.23807	−	0.381893i	0.103532	−	0.0319355i
$144$		0			0
$145$	−1.89807	−	2.38010i	−0.157626	−	0.197656i
$146$		0			0
$147$	0.711297	−	0.659987i	0.0586668	−	0.0544348i
$148$		0			0
$149$	−0.617820	+	8.24424i	−0.0506138	+	0.675395i	0.912977	+	0.408012i	$0.133778\pi$
$149$	−0.617820	+	8.24424i	−0.0506138	+	0.675395i	−0.963590	+	0.267383i	$0.913841\pi$
$150$		0			0
$151$	−1.01707	−	4.45608i	−0.0827681	−	0.362631i	0.916535	−	0.399954i	$-0.130974\pi$
$151$	−1.01707	−	4.45608i	−0.0827681	−	0.362631i	−0.999303	+	0.0373233i	$0.988117\pi$
$152$		0			0
$153$	3.17311	−	2.16339i	0.256531	−	0.174900i
$154$		0			0
$155$	−5.11966	+	13.0447i	−0.411221	+	1.04777i
$156$		0			0
$157$	−0.748623	−	9.98968i	−0.0597466	−	0.797263i	−0.943935	−	0.330132i	$-0.892907\pi$
$157$	−0.748623	−	9.98968i	−0.0597466	−	0.797263i	0.884188	−	0.467131i	$-0.154712\pi$
$158$		0			0
$159$	2.38482	+	0.359453i	0.189128	+	0.0285065i
$160$		0			0
$161$	5.10404	−	2.45798i	0.402255	−	0.193716i
$162$		0			0
$163$	−17.6610	−	12.0410i	−1.38331	−	0.943128i	−0.999808	−	0.0195896i	$-0.993764\pi$
$163$	−17.6610	−	12.0410i	−1.38331	−	0.943128i	−0.383506	−	0.923538i	$-0.625284\pi$
$164$		0			0
$165$	2.06728	+	3.58064i	0.160938	+	0.278753i
$166$		0			0
$167$	−8.02372	−	20.4441i	−0.620894	−	1.58201i	−0.801197	−	0.598401i	$-0.795803\pi$
$167$	−8.02372	−	20.4441i	−0.620894	−	1.58201i	0.180302	−	0.983611i	$-0.442292\pi$
$168$		0			0
$169$	−9.59254	−	2.95891i	−0.737888	−	0.227608i
$170$		0			0
$171$	−0.0564923			−0.00432007
$172$		0			0
$173$	−15.5770			−1.18430			−0.592151	−	0.805827i	$-0.701721\pi$
$173$	−15.5770			−1.18430			−0.592151	+	0.805827i	$0.701721\pi$
$174$		0			0
$175$	30.7563	+	9.48708i	2.32496	+	0.717155i
$176$		0			0
$177$	2.85093	+	7.26405i	0.214289	+	0.546000i
$178$		0			0
$179$	−3.49011	−	6.04505i	−0.260863	−	0.451828i	0.705608	−	0.708602i	$-0.250674\pi$
$179$	−3.49011	−	6.04505i	−0.260863	−	0.451828i	−0.966472	+	0.256774i	$0.917340\pi$
$180$		0			0
$181$	−4.54803	−	3.10080i	−0.338053	−	0.230480i	0.382380	−	0.924005i	$-0.375104\pi$
$181$	−4.54803	−	3.10080i	−0.338053	−	0.230480i	−0.720433	+	0.693525i	$0.756057\pi$
$182$		0			0
$183$	−3.57727	+	1.72272i	−0.264439	+	0.127347i
$184$		0			0
$185$	−31.5309	−	4.75252i	−2.31820	−	0.349412i
$186$		0			0
$187$	0.182877	+	2.44032i	0.0133733	+	0.178454i
$188$		0			0
$189$	5.72378	−	14.5840i	0.416344	−	1.06083i
$190$		0			0
$191$	15.7034	−	10.7064i	1.13626	−	0.774689i	0.159115	−	0.987260i	$-0.449136\pi$
$191$	15.7034	−	10.7064i	1.13626	−	0.774689i	0.977146	+	0.212571i	$0.0681837\pi$
$192$		0			0
$193$	0.685255	+	3.00230i	0.0493257	+	0.216110i	0.993584	−	0.113095i	$-0.0360764\pi$
$193$	0.685255	+	3.00230i	0.0493257	+	0.216110i	−0.944258	+	0.329205i	$0.893219\pi$
$194$		0			0
$195$	−0.706241	+	9.42413i	−0.0505750	+	0.674876i
$196$		0			0
$197$	−8.60862	+	7.98763i	−0.613339	+	0.569095i	−0.924462	−	0.381274i	$-0.875485\pi$
$197$	−8.60862	+	7.98763i	−0.613339	+	0.569095i	0.311123	+	0.950370i	$0.399295\pi$
$198$		0			0
$199$	−0.166901	−	0.209287i	−0.0118313	−	0.0148360i	0.775881	−	0.630880i	$-0.217306\pi$
$199$	−0.166901	−	0.209287i	−0.0118313	−	0.0148360i	−0.787712	+	0.616044i	$0.788734\pi$
$200$		0			0
$201$	7.21637	−	2.22595i	0.509003	−	0.157007i
$202$		0			0
$203$	−1.52252	−	1.41269i	−0.106860	−	0.0991514i
$204$		0			0
$205$	33.8280	−	5.09874i	2.36265	−	0.356112i
$206$		0			0
$207$	1.50204	−	1.88350i	0.104399	−	0.130912i
$208$		0			0
$209$	0.0179987	−	0.0311747i	0.00124500	−	0.00215640i
$210$		0			0
$211$	14.7666	+	7.11124i	1.01658	+	0.489558i	0.866532	−	0.499122i	$-0.166344\pi$
$211$	14.7666	+	7.11124i	1.01658	+	0.489558i	0.150045	+	0.988679i	$0.452058\pi$
$212$		0			0
$213$	2.18306	−	9.56462i	0.149581	−	0.655357i
$214$		0			0
$215$	−24.9947	−	9.40196i	−1.70462	−	0.641208i
$216$		0			0
$217$	−2.12745	+	9.32096i	−0.144421	+	0.632748i
$218$		0			0
$219$	12.3785	+	5.96118i	0.836463	+	0.402819i
$220$		0			0
$221$	−2.79679	+	4.84418i	−0.188132	+	0.325855i
$222$		0			0
$223$	−3.03608	+	3.80712i	−0.203311	+	0.254944i	−0.873025	−	0.487675i	$-0.837845\pi$
$223$	−3.03608	+	3.80712i	−0.203311	+	0.254944i	0.669714	+	0.742619i	$0.266417\pi$
$224$		0			0
$225$	13.5345	−	2.03999i	0.902297	−	0.135999i
$226$		0			0
$227$	−14.1316	−	13.1122i	−0.937947	−	0.870288i	0.0537930	−	0.998552i	$-0.482869\pi$
$227$	−14.1316	−	13.1122i	−0.937947	−	0.870288i	−0.991740	+	0.128265i	$0.959059\pi$
$228$		0			0
$229$	−6.90123	+	2.12875i	−0.456046	+	0.140672i	−0.514263	−	0.857632i	$-0.671935\pi$
$229$	−6.90123	+	2.12875i	−0.456046	+	0.140672i	0.0582171	+	0.998304i	$0.481458\pi$
$230$		0			0
$231$	1.75875	+	2.20541i	0.115718	+	0.145105i
$232$		0			0
$233$	−6.45459	+	5.98898i	−0.422854	+	0.392351i	−0.862695	−	0.505724i	$-0.831225\pi$
$233$	−6.45459	+	5.98898i	−0.422854	+	0.392351i	0.439841	+	0.898076i	$0.355035\pi$
$234$		0			0
$235$	−2.38400	+	31.8122i	−0.155515	+	2.07520i
$236$		0			0
$237$	−4.02407	−	17.6306i	−0.261391	−	1.14523i
$238$		0			0
$239$	−4.97430	+	3.39142i	−0.321761	+	0.219373i	−0.713418	−	0.700738i	$-0.752854\pi$
$239$	−4.97430	+	3.39142i	−0.321761	+	0.219373i	0.391658	+	0.920111i	$0.371902\pi$
$240$		0			0
$241$	−6.42983	+	16.3830i	−0.414182	+	1.05532i	0.559787	+	0.828637i	$0.310883\pi$
$241$	−6.42983	+	16.3830i	−0.414182	+	1.05532i	−0.973969	+	0.226682i	$0.927212\pi$
$242$		0			0
$243$	−0.855086	−	11.4103i	−0.0548538	−	0.731973i
$244$		0			0
$245$	2.89758	+	0.436740i	0.185119	+	0.0279023i
$246$		0			0
$247$	0.0741325	−	0.0357003i	0.00471694	−	0.00227156i
$248$		0			0
$249$	6.32790	+	4.31429i	0.401014	+	0.273407i
$250$		0			0
$251$	−8.69184	−	15.0547i	−0.548624	−	0.950245i	−0.998369	−	0.0570882i	$-0.981818\pi$
$251$	−8.69184	−	15.0547i	−0.548624	−	0.950245i	0.449745	−	0.893157i	$-0.351515\pi$
$252$		0			0
$253$	0.560833	+	1.42898i	0.0352592	+	0.0898391i
$254$		0			0
$255$	−17.0570	−	5.26140i	−1.06815	−	0.329481i
$256$		0			0
$257$	−24.3517			−1.51902			−0.759508	−	0.650498i	$-0.774560\pi$
$257$	−24.3517			−1.51902			−0.759508	+	0.650498i	$0.774560\pi$
$258$		0			0
$259$	−21.7550			−1.35179
$260$		0			0
$261$	−0.843994	−	0.260338i	−0.0522419	−	0.0161145i
$262$		0			0
$263$	7.49599	+	19.0995i	0.462223	+	1.17772i	0.952060	+	0.305910i	$0.0989608\pi$
$263$	7.49599	+	19.0995i	0.462223	+	1.17772i	−0.489838	+	0.871814i	$0.662944\pi$
$264$		0			0
$265$	3.64167	+	6.30755i	0.223706	+	0.387470i
$266$		0			0
$267$	13.7338	+	9.36353i	0.840493	+	0.573039i
$268$		0			0
$269$	15.3360	−	7.38545i	0.935055	−	0.450299i	0.0966331	−	0.995320i	$-0.469193\pi$
$269$	15.3360	−	7.38545i	0.935055	−	0.450299i	0.838422	+	0.545021i	$0.183478\pi$
$270$		0			0
$271$	3.41724	+	0.515067i	0.207583	+	0.0312881i	0.252010	−	0.967725i	$-0.418908\pi$
$271$	3.41724	+	0.515067i	0.207583	+	0.0312881i	−0.0444275	+	0.999013i	$0.514146\pi$
$272$		0			0
$273$	0.481835	+	6.42965i	0.0291620	+	0.389140i
$274$		0			0
$275$	−3.18640	+	8.11881i	−0.192147	+	0.489582i
$276$		0			0
$277$	19.9087	−	13.5735i	1.19619	−	0.815552i	0.209448	−	0.977820i	$-0.432833\pi$
$277$	19.9087	−	13.5735i	1.19619	−	0.815552i	0.986747	+	0.162268i	$0.0518809\pi$
$278$		0			0
$279$	0.904705	+	3.96377i	0.0541632	+	0.237305i
$280$		0			0
$281$	0.429939	−	5.73714i	0.0256480	−	0.342249i	−0.969587	−	0.244746i	$-0.921295\pi$
$281$	0.429939	−	5.73714i	0.0256480	−	0.342249i	0.995235	−	0.0975031i	$-0.0310856\pi$
$282$		0			0
$283$	−11.6970	+	10.8533i	−0.695317	+	0.645160i	−0.946472	−	0.322787i	$-0.895380\pi$
$283$	−11.6970	+	10.8533i	−0.695317	+	0.645160i	0.251154	+	0.967947i	$0.419190\pi$
$284$		0			0
$285$	0.163711	+	0.205287i	0.00969741	+	0.0121602i
$286$		0			0
$287$	22.3030	−	6.87956i	1.31650	−	0.406088i
$288$		0			0
$289$	4.71715	+	4.37687i	0.277479	+	0.257463i
$290$		0			0
$291$	−13.9284	+	2.09937i	−0.816498	+	0.123067i
$292$		0			0
$293$	−3.77347	+	4.73178i	−0.220449	+	0.276434i	−0.879741	−	0.475453i	$-0.842284\pi$
$293$	−3.77347	+	4.73178i	−0.220449	+	0.276434i	0.659293	+	0.751886i	$0.270856\pi$
$294$		0			0
$295$	−11.7830	+	20.4087i	−0.686032	+	1.18824i
$296$		0			0
$297$	3.82494	+	1.84199i	0.221946	+	0.106883i
$298$		0			0
$299$	−0.780788	+	3.42085i	−0.0451541	+	0.197833i
$300$		0			0
$301$	−17.8183	−	3.80130i	−1.02703	−	0.219103i
$302$		0			0
$303$	0.687831	−	3.01359i	0.0395149	−	0.173126i
$304$		0			0
$305$	−10.8031	−	5.20250i	−0.618584	−	0.297894i
$306$		0			0
$307$	11.7476	−	20.3475i	0.670472	−	1.16129i	−0.307298	−	0.951613i	$-0.599425\pi$
$307$	11.7476	−	20.3475i	0.670472	−	1.16129i	0.977770	−	0.209679i	$-0.0672418\pi$
$308$		0			0
$309$	−4.46467	+	5.59852i	−0.253986	+	0.318488i
$310$		0			0
$311$	−1.16489	+	0.175579i	−0.0660550	+	0.00995619i	−0.181987	−	0.983301i	$-0.558253\pi$
$311$	−1.16489	+	0.175579i	−0.0660550	+	0.00995619i	0.115932	+	0.993257i	$0.463015\pi$
$312$		0			0
$313$	21.8138	+	20.2403i	1.23299	+	1.14405i	0.984495	+	0.175415i	$0.0561268\pi$
$313$	21.8138	+	20.2403i	1.23299	+	1.14405i	0.248496	+	0.968633i	$0.420064\pi$
$314$		0			0
$315$	12.7748	−	3.94050i	0.719778	−	0.222022i
$316$		0			0
$317$	−16.7284	−	20.9767i	−0.939560	−	1.17817i	−0.983822	−	0.179151i	$-0.942665\pi$
$317$	−16.7284	−	20.9767i	−0.939560	−	1.17817i	0.0442619	−	0.999020i	$-0.485906\pi$
$318$		0			0
$319$	0.412565	−	0.382805i	0.0230992	−	0.0214330i
$320$		0			0
$321$	−0.362340	+	4.83509i	−0.0202239	+	0.269869i
$322$		0			0
$323$	0.0345821	+	0.151514i	0.00192420	+	0.00843048i
$324$		0			0
$325$	−16.4715	+	11.2301i	−0.913677	+	0.622934i
$326$		0			0
$327$	−2.74151	+	6.98526i	−0.151606	+	0.386286i
$328$		0			0
$329$	1.62649	+	21.7040i	0.0896713	+	1.19658i
$330$		0			0
$331$	−2.75598	−	0.415397i	−0.151482	−	0.0228323i	0.0728632	−	0.997342i	$-0.476786\pi$
$331$	−2.75598	−	0.415397i	−0.151482	−	0.0228323i	−0.224346	+	0.974510i	$0.572024\pi$
$332$		0			0
$333$	−8.33523	+	4.01403i	−0.456768	+	0.219968i
$334$		0			0
$335$	18.8433	+	12.8472i	1.02952	+	0.701915i
$336$		0			0
$337$	−4.52447	−	7.83661i	−0.246463	−	0.426887i	0.716079	−	0.698020i	$-0.245935\pi$
$337$	−4.52447	−	7.83661i	−0.246463	−	0.426887i	−0.962542	+	0.271132i	$0.912602\pi$
$338$		0			0
$339$	−0.0690706	−	0.175989i	−0.00375140	−	0.00955841i
$340$		0			0
$341$	−2.47561	−	0.763625i	−0.134062	−	0.0413526i
$342$		0			0
$343$	−17.4496			−0.942192
$344$		0			0
$345$	−11.1973			−0.602840
$346$		0			0
$347$	−10.0705	−	3.10634i	−0.540613	−	0.166757i	0.0124100	−	0.999923i	$-0.496050\pi$
$347$	−10.0705	−	3.10634i	−0.540613	−	0.166757i	−0.553023	+	0.833166i	$0.686526\pi$
$348$		0			0
$349$	−4.48874	−	11.4371i	−0.240277	−	0.612215i	0.758966	−	0.651130i	$-0.225705\pi$
$349$	−4.48874	−	11.4371i	−0.240277	−	0.612215i	−0.999242	+	0.0389157i	$0.987610\pi$
$350$		0			0
$351$	4.85190	+	8.40374i	0.258975	+	0.448559i
$352$		0			0
$353$	3.53890	+	2.41278i	0.188356	+	0.128419i	0.653827	−	0.756644i	$-0.273162\pi$
$353$	3.53890	+	2.41278i	0.188356	+	0.128419i	−0.465470	+	0.885063i	$0.654115\pi$
$354$		0			0
$355$	26.6933	−	12.8548i	1.41673	−	0.682262i
$356$		0			0
$357$	−12.0422	−	1.81508i	−0.637343	−	0.0960640i
$358$		0			0
$359$	2.55838	+	34.1392i	0.135026	+	1.80180i	0.495062	+	0.868858i	$0.335145\pi$
$359$	2.55838	+	34.1392i	0.135026	+	1.80180i	−0.360036	+	0.932938i	$0.617236\pi$
$360$		0			0
$361$	−6.94064	+	17.6845i	−0.365297	+	0.930762i
$362$		0			0
$363$	11.6245	−	7.92547i	0.610129	−	0.415979i
$364$		0			0
$365$	9.23266	+	40.4509i	0.483259	+	2.11730i
$366$		0			0
$367$	2.31138	−	30.8433i	0.120653	−	1.61000i	−0.528219	−	0.849108i	$-0.677140\pi$
$367$	2.31138	−	30.8433i	0.120653	−	1.61000i	0.648873	−	0.760897i	$-0.275241\pi$
$368$		0			0
$369$	7.27582	−	6.75098i	0.378764	−	0.351442i
$370$		0			0
$371$	3.09817	+	3.88498i	0.160849	+	0.201698i
$372$		0			0
$373$	7.58815	−	2.34063i	0.392900	−	0.121193i	−0.0920093	−	0.995758i	$-0.529329\pi$
$373$	7.58815	−	2.34063i	0.392900	−	0.121193i	0.484909	+	0.874565i	$0.338853\pi$
$374$		0			0
$375$	−26.5068	−	24.5947i	−1.36880	−	1.27006i
$376$		0			0
$377$	1.27206	−	0.191732i	0.0655144	−	0.00987471i
$378$		0			0
$379$	−13.9219	+	17.4576i	−0.715122	+	0.896735i	−0.998051	−	0.0624060i	$-0.980123\pi$
$379$	−13.9219	+	17.4576i	−0.715122	+	0.896735i	0.282929	+	0.959141i	$0.408694\pi$
$380$		0			0
$381$	−1.16615	+	2.01983i	−0.0597435	+	0.103479i
$382$		0			0
$383$	−22.6340	−	10.8999i	−1.15654	−	0.556961i	−0.245549	−	0.969384i	$-0.578968\pi$
$383$	−22.6340	−	10.8999i	−1.15654	−	0.556961i	−0.910992	+	0.412423i	$0.864682\pi$
$384$		0			0
$385$	−1.89559	+	8.30511i	−0.0966081	+	0.423268i
$386$		0			0
$387$	−7.52828	+	1.83123i	−0.382684	+	0.0930867i
$388$		0			0
$389$	3.91826	−	17.1670i	0.198664	−	0.870403i	−0.773070	−	0.634321i	$-0.781280\pi$
$389$	3.91826	−	17.1670i	0.198664	−	0.870403i	0.971733	−	0.236081i	$-0.0758632\pi$
$390$		0			0
$391$	−5.97109	−	2.87553i	−0.301971	−	0.145422i
$392$		0			0
$393$	14.7634	−	25.5709i	0.744714	−	1.28988i
$394$		0			0
$395$	34.0503	−	42.6977i	1.71326	−	2.14836i
$396$		0			0
$397$	−1.43147	+	0.215759i	−0.0718434	+	0.0108287i	−0.184866	−	0.982764i	$-0.559185\pi$
$397$	−1.43147	+	0.215759i	−0.0718434	+	0.0108287i	0.113022	+	0.993592i	$0.463947\pi$
$398$		0			0
$399$	0.131320	+	0.121847i	0.00657420	+	0.00609996i
$400$		0			0
$401$	7.37812	−	2.27585i	0.368446	−	0.113650i	−0.105003	−	0.994472i	$-0.533485\pi$
$401$	7.37812	−	2.27585i	0.368446	−	0.113650i	0.473448	+	0.880822i	$0.343009\pi$
$402$		0			0
$403$	−3.69211	−	4.62976i	−0.183917	−	0.230625i
$404$		0			0
$405$	−12.1185	+	11.2443i	−0.602172	+	0.558734i
$406$		0			0
$407$	0.440541	−	5.87861i	0.0218368	−	0.291392i
$408$		0			0
$409$	−7.12781	−	31.2290i	−0.352448	−	1.54417i	−0.771511	−	0.636216i	$-0.780499\pi$
$409$	−7.12781	−	31.2290i	−0.352448	−	1.54417i	0.419063	−	0.907957i	$-0.362359\pi$
$410$		0			0
$411$	−23.8026	+	16.2283i	−1.17409	+	0.800484i
$412$		0			0
$413$	−5.87394	+	14.9666i	−0.289038	+	0.736456i
$414$		0			0
$415$	1.72841	+	23.0640i	0.0848442	+	1.13217i
$416$		0			0
$417$	−16.2487	−	2.44910i	−0.795703	−	0.119933i
$418$		0			0
$419$	−19.9028	+	9.58467i	−0.972314	+	0.468242i	−0.851455	−	0.524428i	$-0.824279\pi$
$419$	−19.9028	+	9.58467i	−0.972314	+	0.468242i	−0.120859	+	0.992670i	$0.538565\pi$
$420$		0			0
$421$	−12.5748	−	8.57335i	−0.612858	−	0.417840i	0.216717	−	0.976234i	$-0.430465\pi$
$421$	−12.5748	−	8.57335i	−0.612858	−	0.417840i	−0.829575	+	0.558395i	$0.811417\pi$
$422$		0			0
$423$	4.62779	+	8.01557i	0.225011	+	0.389730i
$424$		0			0
$425$	−13.7565	−	35.0511i	−0.667290	−	1.70023i
$426$		0			0
$427$	−7.81715	−	2.41127i	−0.378298	−	0.116690i
$428$		0			0
$429$	−1.74716			−0.0843539
$430$		0			0
$431$	−16.5109			−0.795301			−0.397651	−	0.917537i	$-0.630174\pi$
$431$	−16.5109			−0.795301			−0.397651	+	0.917537i	$0.630174\pi$
$432$		0			0
$433$	24.1153	+	7.43859i	1.15891	+	0.357476i	0.813833	−	0.581099i	$-0.197377\pi$
$433$	24.1153	+	7.43859i	1.15891	+	0.357476i	0.345075	+	0.938575i	$0.387853\pi$
$434$		0			0
$435$	1.49980	+	3.82143i	0.0719100	+	0.183224i
$436$		0			0
$437$	0.0487442	+	0.0844275i	0.00233175	+	0.00403872i
$438$		0			0
$439$	−15.6707	−	10.6841i	−0.747922	−	0.509924i	0.128358	−	0.991728i	$-0.459029\pi$
$439$	−15.6707	−	10.6841i	−0.747922	−	0.509924i	−0.876279	+	0.481804i	$0.839982\pi$
$440$		0			0
$441$	0.765978	−	0.368876i	0.0364752	−	0.0175655i
$442$		0			0
$443$	37.0756	+	5.58825i	1.76152	+	0.265506i	0.948510	−	0.316748i	$-0.102591\pi$
$443$	37.0756	+	5.58825i	1.76152	+	0.265506i	0.813007	+	0.582254i	$0.197829\pi$
$444$		0			0
$445$	3.75126	+	50.0571i	0.177827	+	2.37293i
$446$		0			0
$447$	4.07304	−	10.3779i	0.192648	−	0.490859i
$448$		0			0
$449$	−14.9462	+	10.1902i	−0.705357	+	0.480904i	−0.862102	−	0.506735i	$-0.830852\pi$
$449$	−14.9462	+	10.1902i	−0.705357	+	0.480904i	0.156745	+	0.987639i	$0.449900\pi$
$450$		0			0
$451$	1.40735	+	6.16599i	0.0662693	+	0.290345i
$452$		0			0
$453$	−0.460606	+	6.14636i	−0.0216412	+	0.288781i
$454$		0			0
$455$	−14.2736	+	13.2440i	−0.669159	+	0.620889i
$456$		0			0
$457$	4.42658	+	5.55075i	0.207067	+	0.259653i	0.874510	−	0.485007i	$-0.161183\pi$
$457$	4.42658	+	5.55075i	0.207067	+	0.259653i	−0.667444	+	0.744660i	$0.732612\pi$
$458$		0			0
$459$	−17.5141	+	5.40239i	−0.817489	+	0.252162i
$460$		0			0
$461$	1.48177	+	1.37488i	0.0690130	+	0.0640347i	0.713925	−	0.700222i	$-0.246916\pi$
$461$	1.48177	+	1.37488i	0.0690130	+	0.0640347i	−0.644912	+	0.764257i	$0.723106\pi$
$462$		0			0
$463$	23.5610	−	3.55125i	1.09497	−	0.165041i	0.423374	−	0.905955i	$-0.360846\pi$
$463$	23.5610	−	3.55125i	1.09497	−	0.165041i	0.671600	+	0.740914i	$0.265608\pi$
$464$		0			0
$465$	11.7822	−	14.7744i	0.546385	−	0.685145i
$466$		0			0
$467$	8.68583	−	15.0443i	0.401932	−	0.696167i	−0.592027	−	0.805918i	$-0.701672\pi$
$467$	8.68583	−	15.0443i	0.401932	−	0.696167i	0.993959	+	0.109751i	$0.0350054\pi$
$468$		0			0
$469$	14.0187	+	6.75104i	0.647323	+	0.311734i
$470$		0			0
$471$	−3.00602	+	13.1702i	−0.138510	+	0.606852i
$472$		0			0
$473$	1.38800	−	4.73784i	0.0638203	−	0.217846i
$474$		0			0
$475$	−0.123251	+	0.539998i	−0.00565515	+	0.0247768i
$476$		0			0
$477$	1.90385	+	0.916848i	0.0871715	+	0.0419796i
$478$		0			0
$479$	3.86370	−	6.69213i	0.176537	−	0.305771i	−0.764155	−	0.645033i	$-0.776844\pi$
$479$	3.86370	−	6.69213i	0.176537	−	0.305771i	0.940692	+	0.339261i	$0.110177\pi$
$480$		0			0
$481$	8.40131	−	10.5349i	0.383067	−	0.480350i
$482$		0			0
$483$	−7.55405	+	1.13859i	−0.343721	+	0.0518076i
$484$		0			0
$485$	−31.1825	−	28.9332i	−1.41593	−	1.31379i
$486$		0			0
$487$	−16.7051	+	5.15285i	−0.756982	+	0.233498i	−0.649139	−	0.760670i	$-0.724871\pi$
$487$	−16.7051	+	5.15285i	−0.756982	+	0.233498i	−0.107843	+	0.994168i	$0.534394\pi$
$488$		0			0
$489$	17.9718	+	22.5359i	0.812713	+	1.01911i
$490$		0			0
$491$	−15.9175	+	14.7693i	−0.718346	+	0.666528i	−0.952107	−	0.305766i	$-0.901087\pi$
$491$	−15.9175	+	14.7693i	−0.718346	+	0.666528i	0.233760	+	0.972294i	$0.424897\pi$
$492$		0			0
$493$	−0.181578	+	2.42299i	−0.00817786	+	0.109126i
$494$		0			0
$495$	0.806105	+	3.53178i	0.0362317	+	0.158742i
$496$		0			0
$497$	16.7010	−	11.3866i	0.749144	−	0.510758i
$498$		0			0
$499$	13.1125	−	33.4100i	0.586995	−	1.49564i	−0.260842	−	0.965382i	$-0.584000\pi$
$499$	13.1125	−	33.4100i	0.586995	−	1.49564i	0.847837	−	0.530257i	$-0.177905\pi$
$500$		0			0
$501$	2.21323	+	29.5335i	0.0988797	+	1.31946i
$502$		0			0
$503$	−0.0899578	−	0.0135590i	−0.00401102	−	0.000604564i	0.147036	−	0.989131i	$-0.453027\pi$
$503$	−0.0899578	−	0.0135590i	−0.00401102	−	0.000604564i	−0.151047	+	0.988527i	$0.548265\pi$
$504$		0			0
$505$	8.41042	−	4.05025i	0.374259	−	0.180234i
$506$		0			0
$507$	11.1848	+	7.62568i	0.496735	+	0.338668i
$508$		0			0
$509$	−11.3425	−	19.6459i	−0.502749	−	0.870787i	−0.999995	−	0.00317746i	$-0.998989\pi$
$509$	−11.3425	−	19.6459i	−0.502749	−	0.870787i	0.497246	−	0.867610i	$-0.334345\pi$
$510$		0			0
$511$	10.3419	+	26.3507i	0.457498	+	1.16569i
$512$		0			0
$513$	0.257630	+	0.0794685i	0.0113747	+	0.00350862i
$514$		0			0
$515$	−21.6250			−0.952913
$516$		0			0
$517$	−5.89775			−0.259383
$518$		0			0
$519$	20.0725	+	6.19156i	0.881087	+	0.271779i
$520$		0			0
$521$	1.72346	+	4.39129i	0.0755060	+	0.192386i	0.963643	−	0.267192i	$-0.0860958\pi$
$521$	1.72346	+	4.39129i	0.0755060	+	0.192386i	−0.888137	+	0.459578i	$0.848001\pi$
$522$		0			0
$523$	0.853098	+	1.47761i	0.0373034	+	0.0646113i	0.884074	−	0.467346i	$-0.154790\pi$
$523$	0.853098	+	1.47761i	0.0373034	+	0.0646113i	−0.846771	+	0.531958i	$0.821457\pi$
$524$		0			0
$525$	−35.8616	−	24.4500i	−1.56513	−	1.06709i
$526$		0			0
$527$	10.0771	−	4.85290i	0.438968	−	0.211396i
$528$		0			0
$529$	18.6322	+	2.80835i	0.810095	+	0.122102i
$530$		0			0
$531$	0.510945	+	6.81809i	0.0221731	+	0.295880i
$532$		0			0
$533$	−5.28148	+	13.4570i	−0.228766	+	0.582887i
$534$		0			0
$535$	−12.0983	+	8.24846i	−0.523054	+	0.356612i
$536$		0			0
$537$	2.09456	+	9.17688i	0.0903870	+	0.396011i
$538$		0			0
$539$	−0.0404841	+	0.540223i	−0.00174378	+	0.0232691i
$540$		0			0
$541$	−8.33253	+	7.73145i	−0.358243	+	0.332401i	−0.838676	−	0.544630i	$-0.816670\pi$
$541$	−8.33253	+	7.73145i	−0.358243	+	0.332401i	0.480433	+	0.877031i	$0.340480\pi$
$542$		0			0
$543$	4.62808	+	5.80343i	0.198610	+	0.249049i
$544$		0			0
$545$	−21.6547	+	6.67960i	−0.927586	+	0.286123i
$546$		0			0
$547$	17.1076	+	15.8735i	0.731467	+	0.678703i	0.955204	−	0.295947i	$-0.0956352\pi$
$547$	17.1076	+	15.8735i	0.731467	+	0.678703i	−0.223737	+	0.974650i	$0.571826\pi$
$548$		0			0
$549$	−3.43997	+	0.518492i	−0.146814	+	0.0221287i
$550$		0			0
$551$	0.0222847	−	0.0279441i	0.000949358	−	0.00119046i
$552$		0			0
$553$	18.6298	−	32.2677i	0.792218	−	1.37216i
$554$		0			0
$555$	38.7416	+	18.6570i	1.64449	+	0.791944i
$556$		0			0
$557$	−1.28534	+	5.63146i	−0.0544618	+	0.238613i	−0.994831	−	0.101542i	$-0.967622\pi$
$557$	−1.28534	+	5.63146i	−0.0544618	+	0.238613i	0.940369	+	0.340155i	$0.110479\pi$
$558$		0			0
$559$	8.72180	−	7.16055i	0.368893	−	0.302859i
$560$		0			0
$561$	0.734322	−	3.21728i	0.0310031	−	0.135834i
$562$		0			0
$563$	3.42474	+	1.64927i	0.144336	+	0.0695084i	0.504657	−	0.863320i	$-0.331619\pi$
$563$	3.42474	+	1.64927i	0.144336	+	0.0695084i	−0.360322	+	0.932828i	$0.617333\pi$
$564$		0			0
$565$	0.285471	−	0.494450i	0.0120098	−	0.0208017i
$566$		0			0
$567$	−7.03215	+	8.81804i	−0.295323	+	0.370323i
$568$		0			0
$569$	35.8718	−	5.40680i	1.50382	−	0.226665i	0.655116	−	0.755528i	$-0.272620\pi$
$569$	35.8718	−	5.40680i	1.50382	−	0.226665i	0.848707	+	0.528863i	$0.177381\pi$
$570$		0			0
$571$	19.4633	+	18.0593i	0.814514	+	0.755758i	0.972803	−	0.231635i	$-0.0744073\pi$
$571$	19.4633	+	18.0593i	0.814514	+	0.755758i	−0.158289	+	0.987393i	$0.550598\pi$
$572$		0			0
$573$	−24.4910	+	7.55446i	−1.02312	+	0.315592i
$574$		0			0
$575$	−14.7269	−	18.4670i	−0.614156	−	0.770127i
$576$		0			0
$577$	9.54816	−	8.85940i	0.397495	−	0.368821i	−0.455949	−	0.890006i	$-0.650700\pi$
$577$	9.54816	−	8.85940i	0.397495	−	0.368821i	0.853444	+	0.521184i	$0.174510\pi$
$578$		0			0
$579$	0.310335	−	4.14113i	0.0128971	−	0.172099i
$580$		0			0
$581$	3.51130	+	15.3840i	0.145673	+	0.638236i
$582$		0			0
$583$	−1.11253	+	0.758511i	−0.0460763	+	0.0314143i
$584$		0			0
$585$	−3.02515	+	7.70795i	−0.125074	+	0.318685i
$586$		0			0
$587$	0.430770	+	5.74822i	0.0177798	+	0.237255i	0.999013	+	0.0444278i	$0.0141465\pi$
$587$	0.430770	+	5.74822i	0.0177798	+	0.237255i	−0.981233	+	0.192827i	$0.938234\pi$
$588$		0			0
$589$	−0.162689	−	0.0245215i	−0.00670350	−	0.00101039i
$590$		0			0
$591$	14.2680	−	6.87109i	0.586905	−	0.282639i
$592$		0			0
$593$	6.69370	+	4.56369i	0.274877	+	0.187408i	0.692911	−	0.721024i	$-0.256328\pi$
$593$	6.69370	+	4.56369i	0.274877	+	0.187408i	−0.418033	+	0.908432i	$0.637280\pi$
$594$		0			0
$595$	−18.3888	−	31.8503i	−0.753866	−	1.30573i
$596$		0			0
$597$	0.131881	+	0.336026i	0.00539752	+	0.0137526i
$598$		0			0
$599$	8.12129	+	2.50509i	0.331827	+	0.102355i	0.456192	−	0.889882i	$-0.349213\pi$
$599$	8.12129	+	2.50509i	0.331827	+	0.102355i	−0.124365	+	0.992237i	$0.539689\pi$
$600$		0			0
$601$	35.6791			1.45538			0.727691	−	0.685905i	$-0.240594\pi$
$601$	35.6791			1.45538			0.727691	+	0.685905i	$0.240594\pi$
$602$		0			0
$603$	6.61676			0.269455
$604$		0			0
$605$	40.6004	+	12.5236i	1.65064	+	0.509155i
$606$		0			0
$607$	−12.0041	−	30.5861i	−0.487233	−	1.24145i	−0.937321	−	0.348466i	$-0.886703\pi$
$607$	−12.0041	−	30.5861i	−0.487233	−	1.24145i	0.450088	−	0.892984i	$-0.351393\pi$
$608$		0			0
$609$	1.40040	+	2.42556i	0.0567469	+	0.0982886i
$610$		0			0
$611$	−11.1383	−	7.59397i	−0.450608	−	0.307219i
$612$		0			0
$613$	2.01111	−	0.968499i	0.0812280	−	0.0391173i	−0.392829	−	0.919612i	$-0.628503\pi$
$613$	2.01111	−	0.968499i	0.0812280	−	0.0391173i	0.474057	+	0.880494i	$0.342789\pi$
$614$		0			0
$615$	−45.6172	−	6.87569i	−1.83946	−	0.277255i
$616$		0			0
$617$	−0.983300	−	13.1212i	−0.0395862	−	0.528241i	−0.981383	−	0.192059i	$-0.938483\pi$
$617$	−0.983300	−	13.1212i	−0.0395862	−	0.528241i	0.941797	−	0.336182i	$-0.109136\pi$
$618$		0			0
$619$	5.06713	−	12.9108i	0.203665	−	0.518930i	−0.792149	−	0.610327i	$-0.791038\pi$
$619$	5.06713	−	12.9108i	0.203665	−	0.518930i	0.995815	+	0.0913968i	$0.0291332\pi$
$620$		0			0
$621$	−9.49953	+	6.47667i	−0.381203	+	0.259900i
$622$		0			0
$623$	7.62076	+	33.3887i	0.305319	+	1.33769i
$624$		0			0
$625$	3.83193	−	51.1336i	0.153277	−	2.04534i
$626$		0			0
$627$	−0.0355844	+	0.0330175i	−0.00142110	+	0.00131859i
$628$		0			0
$629$	15.8683	+	19.8982i	0.632709	+	0.793392i
$630$		0			0
$631$	5.11794	−	1.57867i	0.203742	−	0.0628461i	−0.191204	−	0.981550i	$-0.561239\pi$
$631$	5.11794	−	1.57867i	0.203742	−	0.0628461i	0.394946	+	0.918704i	$0.370763\pi$
$632$		0			0
$633$	−16.2017	−	15.0330i	−0.643959	−	0.597506i
$634$		0			0
$635$	−6.96470	+	1.04976i	−0.276386	+	0.0416584i
$636$		0			0
$637$	−0.772051	+	0.968121i	−0.0305898	+	0.0383584i
$638$		0			0
$639$	4.29789	−	7.44417i	0.170022	−	0.294487i
$640$		0			0
$641$	−4.56273	−	2.19729i	−0.180217	−	0.0867879i	0.341600	−	0.939845i	$-0.389031\pi$
$641$	−4.56273	−	2.19729i	−0.180217	−	0.0867879i	−0.521817	+	0.853058i	$0.674746\pi$
$642$		0			0
$643$	4.59652	−	20.1387i	0.181269	−	0.794191i	−0.799758	−	0.600322i	$-0.795039\pi$
$643$	4.59652	−	20.1387i	0.181269	−	0.794191i	0.981027	−	0.193869i	$-0.0621037\pi$
$644$		0			0
$645$	28.4710	+	22.0502i	1.12104	+	0.868226i
$646$		0			0
$647$	6.78518	−	29.7278i	0.266753	−	1.16872i	−0.647013	−	0.762479i	$-0.723982\pi$
$647$	6.78518	−	29.7278i	0.266753	−	1.16872i	0.913766	−	0.406242i	$-0.133161\pi$
$648$		0			0
$649$	−3.92529	−	1.89032i	−0.154081	−	0.0742015i
$650$		0			0
$651$	6.44631	−	11.1653i	0.252651	−	0.437604i
$652$		0			0
$653$	25.9574	−	32.5496i	1.01579	−	1.27376i	0.0544173	−	0.998518i	$-0.482670\pi$
$653$	25.9574	−	32.5496i	1.01579	−	1.27376i	0.961374	−	0.275244i	$-0.0887587\pi$
$654$		0			0
$655$	88.1729	−	13.2899i	3.44520	−	0.519280i
$656$		0			0
$657$	8.82438	+	8.18783i	0.344272	+	0.319438i
$658$		0			0
$659$	20.9158	−	6.45166i	0.814762	−	0.251321i	0.140754	−	0.990045i	$-0.455047\pi$
$659$	20.9158	−	6.45166i	0.814762	−	0.251321i	0.674008	+	0.738724i	$0.264571\pi$
$660$		0			0
$661$	−3.40742	−	4.27277i	−0.132533	−	0.166191i	0.711136	−	0.703054i	$-0.248181\pi$
$661$	−3.40742	−	4.27277i	−0.132533	−	0.166191i	−0.843670	+	0.536863i	$0.819609\pi$
$662$		0			0
$663$	5.52940	−	5.13053i	0.214744	−	0.199253i
$664$		0			0
$665$	−0.0404284	+	0.539480i	−0.00156775	+	0.0209201i
$666$		0			0
$667$	0.339165	+	1.48598i	0.0131325	+	0.0575373i
$668$		0			0
$669$	5.42554	−	3.69907i	0.209763	−	0.143014i
$670$		0			0
$671$	0.809867	−	2.06351i	0.0312646	−	0.0796608i
$672$		0			0
$673$	0.265593	+	3.54410i	0.0102379	+	0.136615i	0.999980	−	0.00624712i	$-0.00198853\pi$
$673$	0.265593	+	3.54410i	0.0102379	+	0.136615i	−0.989743	+	0.142862i	$0.954369\pi$
$674$		0			0
$675$	−64.5930	−	9.73582i	−2.48618	−	0.374732i
$676$		0			0
$677$	−8.78571	+	4.23097i	−0.337662	+	0.162610i	−0.595029	−	0.803704i	$-0.702859\pi$
$677$	−8.78571	+	4.23097i	−0.337662	+	0.162610i	0.257367	+	0.966314i	$0.417145\pi$
$678$		0			0
$679$	−23.9788	−	16.3485i	−0.920223	−	0.627398i
$680$		0			0
$681$	12.9981	+	22.5134i	0.498088	+	0.862714i
$682$		0			0
$683$	7.66440	+	19.5286i	0.293270	+	0.747240i	0.999239	+	0.0389974i	$0.0124164\pi$
$683$	7.66440	+	19.5286i	0.293270	+	0.747240i	−0.705969	+	0.708242i	$0.749488\pi$
$684$		0			0
$685$	−83.1341	−	25.6435i	−3.17639	−	0.979787i
$686$		0			0
$687$	9.73904			0.371568
$688$		0			0
$689$	−3.07775			−0.117253
$690$		0			0
$691$	−8.48595	−	2.61757i	−0.322821	−	0.0995770i	0.129109	−	0.991630i	$-0.458788\pi$
$691$	−8.48595	−	2.61757i	−0.322821	−	0.0995770i	−0.451930	+	0.892053i	$0.649264\pi$
$692$		0			0
$693$	0.902953	+	2.30069i	0.0343003	+	0.0873958i
$694$		0			0
$695$	−24.8121	−	42.9758i	−0.941177	−	1.63017i
$696$		0			0
$697$	−22.5603	−	15.3814i	−0.854532	−	0.582610i
$698$		0			0
$699$	10.6979	−	5.15182i	0.404630	−	0.194860i
$700$		0			0
$701$	24.8182	+	3.74074i	0.937370	+	0.141286i	0.599922	−	0.800058i	$-0.295198\pi$
$701$	24.8182	+	3.74074i	0.937370	+	0.141286i	0.337448	+	0.941344i	$0.390436\pi$
$702$		0			0
$703$	−0.0279772	−	0.373330i	−0.00105518	−	0.0140804i
$704$		0			0
$705$	15.7167	−	40.0456i	0.591926	−	1.50820i
$706$		0			0
$707$	5.26210	−	3.58764i	0.197902	−	0.134927i
$708$		0			0
$709$	−11.1254	−	48.7436i	−0.417824	−	1.83060i	−0.544629	−	0.838677i	$-0.683330\pi$
$709$	−11.1254	−	48.7436i	−0.417824	−	1.83060i	0.126806	−	0.991928i	$-0.459528\pi$
$710$		0			0
$711$	1.18408	−	15.8004i	0.0444064	−	0.592563i
$712$		0			0
$713$	5.14322	−	4.77221i	0.192615	−	0.178721i
$714$		0			0
$715$	−3.28973	−	4.12519i	−0.123029	−	0.154273i
$716$		0			0
$717$	7.75789	−	2.39299i	0.289724	−	0.0893679i
$718$		0			0
$719$	8.04392	+	7.46367i	0.299988	+	0.278348i	0.815776	−	0.578368i	$-0.196310\pi$
$719$	8.04392	+	7.46367i	0.299988	+	0.278348i	−0.515788	+	0.856716i	$0.672501\pi$
$720$		0			0
$721$	−14.5890	+	2.19893i	−0.543321	+	0.0818925i
$722$		0			0
$723$	14.7974	−	18.5553i	0.550319	−	0.690079i
$724$		0			0
$725$	−4.32989	+	7.49958i	−0.160808	+	0.278527i
$726$		0			0
$727$	21.2553	+	10.2360i	0.788316	+	0.379633i	0.784318	−	0.620359i	$-0.213013\pi$
$727$	21.2553	+	10.2360i	0.788316	+	0.379633i	0.00399815	+	0.999992i	$0.498727\pi$
$728$		0			0
$729$	−6.14342	+	26.9161i	−0.227534	+	0.996892i
$730$		0			0
$731$	9.51991	+	19.0701i	0.352107	+	0.705333i
$732$		0			0
$733$	7.85287	−	34.4057i	0.290052	−	1.27080i	−0.594399	−	0.804170i	$-0.702610\pi$
$733$	7.85287	−	34.4057i	0.290052	−	1.27080i	0.884451	−	0.466633i	$-0.154533\pi$
$734$		0			0
$735$	−3.56022	−	1.71451i	−0.131321	−	0.0632406i
$736$		0			0
$737$	−2.10813	+	3.65139i	−0.0776541	+	0.134501i
$738$		0			0
$739$	13.7042	−	17.1846i	0.504119	−	0.632145i	−0.463034	−	0.886340i	$-0.653239\pi$
$739$	13.7042	−	17.1846i	0.504119	−	0.632145i	0.967153	+	0.254196i	$0.0818107\pi$
$740$		0			0
$741$	−0.109717	+	0.0165372i	−0.00403056	+	0.000607509i
$742$		0			0
$743$	1.34948	+	1.25213i	0.0495075	+	0.0459362i	0.704539	−	0.709666i	$-0.251154\pi$
$743$	1.34948	+	1.25213i	0.0495075	+	0.0459362i	−0.655031	+	0.755602i	$0.727344\pi$
$744$		0			0
$745$	32.1722	−	9.92381i	1.17870	−	0.363580i
$746$		0			0
$747$	4.18383	+	5.24635i	0.153078	+	0.191954i
$748$		0			0
$749$	−7.32316	+	6.79490i	−0.267582	+	0.248280i
$750$		0			0
$751$	2.39276	−	31.9291i	0.0873129	−	1.16511i	−0.765692	−	0.643207i	$-0.777603\pi$
$751$	2.39276	−	31.9291i	0.0873129	−	1.16511i	0.853005	−	0.521903i	$-0.174778\pi$
$752$		0			0
$753$	5.21634	+	22.8543i	0.190094	+	0.832857i
$754$		0			0
$755$	−15.3793	+	10.4854i	−0.559710	+	0.381604i
$756$		0			0
$757$	6.22743	−	15.8672i	0.226340	−	0.576704i	−0.771917	−	0.635723i	$-0.780702\pi$
$757$	6.22743	−	15.8672i	0.226340	−	0.576704i	0.998257	+	0.0590188i	$0.0187972\pi$
$758$		0			0
$759$	−0.154698	−	2.06430i	−0.00561517	−	0.0749292i
$760$		0			0
$761$	−2.81970	−	0.425001i	−0.102214	−	0.0154063i	0.0977361	−	0.995212i	$-0.468840\pi$
$761$	−2.81970	−	0.425001i	−0.102214	−	0.0154063i	−0.199950	+	0.979806i	$0.564078\pi$
$762$		0			0
$763$	−13.9298	+	6.70823i	−0.504292	+	0.242854i
$764$		0			0
$765$	−12.9222	−	8.81019i	−0.467202	−	0.318533i
$766$		0			0
$767$	−4.97919	−	8.62421i	−0.179788	−	0.311402i
$768$		0			0
$769$	17.5058	+	44.6040i	0.631275	+	1.60846i	0.784400	+	0.620255i	$0.212971\pi$
$769$	17.5058	+	44.6040i	0.631275	+	1.60846i	−0.153126	+	0.988207i	$0.548934\pi$
$770$		0			0
$771$	31.3795	+	9.67930i	1.13011	+	0.348591i
$772$		0			0
$773$	12.7995			0.460366			0.230183	−	0.973147i	$-0.426067\pi$
$773$	12.7995			0.460366			0.230183	+	0.973147i	$0.426067\pi$
$774$		0			0
$775$	39.8627			1.43191
$776$		0			0
$777$	28.0335	+	8.64718i	1.00570	+	0.310216i
$778$		0			0
$779$	0.146739	+	0.373886i	0.00525749	+	0.0133959i
$780$		0			0
$781$	2.73866	+	4.74350i	0.0979970	+	0.169736i
$782$		0			0
$783$	3.48278	+	2.37452i	0.124464	+	0.0848583i
$784$		0			0
$785$	−36.7559	+	17.7007i	−1.31188	+	0.631766i
$786$		0			0
$787$	13.0915	+	1.97322i	0.466660	+	0.0703377i	0.378164	−	0.925739i	$-0.376556\pi$
$787$	13.0915	+	1.97322i	0.466660	+	0.0703377i	0.0884966	+	0.996076i	$0.471794\pi$
$788$		0			0
$789$	−2.06766	−	27.5910i	−0.0736107	−	0.982267i
$790$		0			0
$791$	0.142310	−	0.362600i	0.00505997	−	0.0128926i
$792$		0			0
$793$	4.18647	−	2.85429i	0.148666	−	0.101359i
$794$		0			0
$795$	−2.18552	−	9.57538i	−0.0775123	−	0.339604i
$796$		0			0
$797$	0.866491	−	11.5625i	0.0306927	−	0.409566i	−0.960604	−	0.277920i	$-0.910355\pi$
$797$	0.866491	−	11.5625i	0.0306927	−	0.409566i	0.991297	−	0.131645i	$-0.0420260\pi$
$798$		0			0
$799$	18.6651	−	17.3187i	0.660324	−	0.612691i
$800$		0			0
$801$	9.08039	+	11.3864i	0.320840	+	0.402320i
$802$		0			0
$803$	−7.32986	+	2.26096i	−0.258665	+	0.0797876i
$804$		0			0
$805$	−16.9118	−	15.6918i	−0.596062	−	0.553065i
$806$		0			0
$807$	−22.6976	+	3.42111i	−0.798992	+	0.120429i
$808$		0			0
$809$	−15.0173	+	18.8311i	−0.527980	+	0.662066i	−0.972282	−	0.233811i	$-0.924880\pi$
$809$	−15.0173	+	18.8311i	−0.527980	+	0.662066i	0.444302	+	0.895877i	$0.353452\pi$
$810$		0			0
$811$	−9.67307	+	16.7542i	−0.339667	+	0.588321i	−0.984370	−	0.176112i	$-0.943648\pi$
$811$	−9.67307	+	16.7542i	−0.339667	+	0.588321i	0.644703	+	0.764433i	$0.276981\pi$
$812$		0			0
$813$	−4.19872	−	2.02200i	−0.147256	−	0.0709146i
$814$		0			0
$815$	−19.3700	+	84.8657i	−0.678503	+	2.97271i
$816$		0			0
$817$	0.0423181	−	0.310661i	0.00148052	−	0.0108687i
$818$		0			0
$819$	−1.25709	+	5.50765i	−0.0439261	+	0.192453i
$820$		0			0
$821$	4.68010	+	2.25382i	0.163336	+	0.0786587i	0.513766	−	0.857930i	$-0.328250\pi$
$821$	4.68010	+	2.25382i	0.163336	+	0.0786587i	−0.350429	+	0.936589i	$0.613964\pi$
$822$		0			0
$823$	−27.1185	+	46.9706i	−0.945290	+	1.63729i	−0.190122	+	0.981761i	$0.560888\pi$
$823$	−27.1185	+	46.9706i	−0.945290	+	1.63729i	−0.755169	+	0.655530i	$0.772445\pi$
$824$		0			0
$825$	7.33304	−	9.19534i	0.255304	−	0.320141i
$826$		0			0
$827$	3.32662	−	0.501407i	0.115678	−	0.0174356i	−0.0909482	−	0.995856i	$-0.528990\pi$
$827$	3.32662	−	0.501407i	0.115678	−	0.0174356i	0.206626	+	0.978420i	$0.433752\pi$
$828$		0			0
$829$	−29.6523	−	27.5133i	−1.02987	−	0.955578i	−0.0308341	−	0.999525i	$-0.509816\pi$
$829$	−29.6523	−	27.5133i	−1.02987	−	0.955578i	−0.999034	+	0.0439467i	$0.986007\pi$
$830$		0			0
$831$	−31.0494	+	9.57747i	−1.07709	+	0.332239i
$832$		0			0
$833$	−1.45824	−	1.82857i	−0.0505249	−	0.0633562i
$834$		0			0
$835$	−65.5635	+	60.8341i	−2.26892	+	2.10525i
$836$		0			0
$837$	1.45003	−	19.3493i	0.0501202	−	0.668808i
$838$		0			0
$839$	−0.0182394	−	0.0799120i	−0.000629694	−	0.00275887i	0.974612	−	0.223900i	$-0.0718790\pi$
$839$	−0.0182394	−	0.0799120i	−0.000629694	−	0.00275887i	−0.975242	+	0.221141i	$0.929022\pi$
$840$		0			0
$841$	−23.4992	+	16.0215i	−0.810318	+	0.552465i
$842$		0			0
$843$	−2.83442	+	7.22197i	−0.0976225	+	0.248738i
$844$		0			0
$845$	3.05503	+	40.7666i	0.105096	+	1.40241i
$846$		0			0
$847$	28.6638	+	4.32038i	0.984901	+	0.148450i
$848$		0			0
$849$	19.3867	−	9.33616i	0.665351	−	0.320416i
$850$		0			0
$851$	13.1910	+	8.99347i	0.452181	+	0.308292i
$852$		0			0
$853$	12.0750	+	20.9145i	0.413441	+	0.716101i	0.995263	−	0.0972152i	$-0.0309935\pi$
$853$	12.0750	+	20.9145i	0.413441	+	0.716101i	−0.581823	+	0.813316i	$0.697660\pi$
$854$		0			0
$855$	0.0840500	+	0.214156i	0.00287445	+	0.00732398i
$856$		0			0
$857$	40.2162	+	12.4051i	1.37376	+	0.423749i	0.891794	−	0.452441i	$-0.149447\pi$
$857$	40.2162	+	12.4051i	1.37376	+	0.423749i	0.481966	+	0.876190i	$0.339923\pi$
$858$		0			0
$859$	2.35350			0.0803004			0.0401502	−	0.999194i	$-0.487216\pi$
$859$	2.35350			0.0803004			0.0401502	+	0.999194i	$0.487216\pi$
$860$		0			0
$861$	−31.4741			−1.07263
$862$		0			0
$863$	29.2992	+	9.03761i	0.997356	+	0.307644i	0.750113	−	0.661310i	$-0.229999\pi$
$863$	29.2992	+	9.03761i	0.997356	+	0.307644i	0.247243	+	0.968953i	$0.420475\pi$
$864$		0			0
$865$	23.1758	+	59.0509i	0.787999	+	2.00779i
$866$		0			0
$867$	−4.33878	−	7.51499i	−0.147353	−	0.255222i
$868$		0			0
$869$	8.34206	+	5.68752i	0.282985	+	0.192936i
$870$		0			0
$871$	−8.68290	+	4.18147i	−0.294209	+	0.141684i
$872$		0			0
$873$	−12.2037	−	1.83942i	−0.413033	−	0.0622548i
$874$		0			0
$875$	−5.56750	−	74.2932i	−0.188216	−	2.51157i
$876$		0			0
$877$	−16.8677	+	42.9782i	−0.569582	+	1.45127i	0.298287	+	0.954476i	$0.403585\pi$
$877$	−16.8677	+	42.9782i	−0.569582	+	1.45127i	−0.867868	+	0.496794i	$0.834510\pi$
$878$		0			0
$879$	6.74327	−	4.59748i	0.227445	−	0.155069i
$880$		0			0
$881$	−2.29329	−	10.0475i	−0.0772627	−	0.338510i	0.921492	−	0.388397i	$-0.126971\pi$
$881$	−2.29329	−	10.0475i	−0.0772627	−	0.338510i	−0.998755	+	0.0498868i	$0.984114\pi$
$882$		0			0
$883$	−0.882701	+	11.7788i	−0.0297053	+	0.396389i	0.962457	+	0.271434i	$0.0874978\pi$
$883$	−0.882701	+	11.7788i	−0.0297053	+	0.396389i	−0.992162	+	0.124955i	$0.960121\pi$
$884$		0			0
$885$	23.2956	−	21.6151i	0.783072	−	0.726585i
$886$		0			0
$887$	13.8680	+	17.3899i	0.465640	+	0.583895i	0.958098	−	0.286442i	$-0.0924726\pi$
$887$	13.8680	+	17.3899i	0.465640	+	0.583895i	−0.492457	+	0.870337i	$0.663901\pi$
$888$		0			0
$889$	−4.59187	+	1.41641i	−0.154006	+	0.0475047i
$890$		0			0
$891$	−2.24039	−	2.07878i	−0.0750560	−	0.0696418i
$892$		0			0
$893$	−0.370362	+	0.0558232i	−0.0123937	+	0.00186805i
$894$		0			0
$895$	−17.7235	+	22.2245i	−0.592430	+	0.742884i
$896$		0			0
$897$	2.36584	−	4.09776i	0.0789931	−	0.136820i
$898$		0			0
$899$	−2.31757	−	1.11608i	−0.0772954	−	0.0372235i
$900$		0			0
$901$	1.29356	−	5.66746i	0.0430947	−	0.188810i
$902$		0			0
$903$	21.4496	+	11.9807i	0.713799	+	0.398694i
$904$		0			0
$905$	−4.98815	+	21.8545i	−0.165812	+	0.726468i
$906$		0			0
$907$	31.5843	+	15.2102i	1.04874	+	0.505046i	0.877197	−	0.480131i	$-0.159411\pi$
$907$	31.5843	+	15.2102i	1.04874	+	0.505046i	0.171541	+	0.985177i	$0.445125\pi$
$908$		0			0
$909$	1.35416	−	2.34548i	0.0449148	−	0.0777947i
$910$		0			0
$911$	−34.7823	+	43.6156i	−1.15239	+	1.44505i	−0.277498	+	0.960726i	$0.589505\pi$
$911$	−34.7823	+	43.6156i	−1.15239	+	1.44505i	−0.874890	+	0.484322i	$0.839066\pi$
$912$		0			0
$913$	−4.22814	+	0.637289i	−0.139931	+	0.0210912i
$914$		0			0
$915$	11.8530	+	10.9979i	0.391847	+	0.363581i
$916$		0			0
$917$	58.1330	−	17.9316i	1.91972	−	0.592155i
$918$		0			0
$919$	−27.4108	−	34.3720i	−0.904198	−	1.13383i	−0.990494	−	0.137559i	$-0.956074\pi$
$919$	−27.4108	−	34.3720i	−0.904198	−	1.13383i	0.0862951	−	0.996270i	$-0.472497\pi$
$920$		0			0
$921$	−23.2257	+	21.5503i	−0.765312	+	0.710105i
$922$		0			0
$923$	−0.935602	+	12.4847i	−0.0307957	+	0.410940i
$924$		0			0
$925$	20.1841	+	88.4323i	0.663649	+	2.90764i
$926$		0			0
$927$	−5.18389	+	3.53432i	−0.170261	+	0.116082i
$928$		0			0
$929$	−2.10522	+	5.36400i	−0.0690699	+	0.175987i	−0.961240	−	0.275711i	$-0.911087\pi$
$929$	−2.10522	+	5.36400i	−0.0690699	+	0.175987i	0.892171	+	0.451699i	$0.149182\pi$
$930$		0			0
$931$	0.00257101	+	0.0343077i	8.42614e−5	+	0.00112439i
$932$		0			0
$933$	1.57087	+	0.236770i	0.0514278	+	0.00775150i
$934$		0			0
$935$	8.97889	−	4.32401i	0.293641	−	0.141410i
$936$		0			0
$937$	−18.1263	−	12.3583i	−0.592161	−	0.403729i	0.229828	−	0.973231i	$-0.426184\pi$
$937$	−18.1263	−	12.3583i	−0.592161	−	0.403729i	−0.821990	+	0.569503i	$0.807136\pi$
$938$		0			0
$939$	−20.0641	−	34.7521i	−0.654769	−	1.13409i
$940$		0			0
$941$	−6.91668	−	17.6234i	−0.225477	−	0.574507i	0.772706	−	0.634765i	$-0.218903\pi$
$941$	−6.91668	−	17.6234i	−0.225477	−	0.574507i	−0.998183	+	0.0602578i	$0.980808\pi$
$942$		0			0
$943$	−16.3672	−	5.04862i	−0.532991	−	0.164406i
$944$		0			0
$945$	−63.8021			−2.07548
$946$		0			0
$947$	−40.2434			−1.30774			−0.653868	−	0.756609i	$-0.726855\pi$
$947$	−40.2434			−1.30774			−0.653868	+	0.756609i	$0.726855\pi$
$948$		0			0
$949$	−16.7542	−	5.16798i	−0.543864	−	0.167760i
$950$		0			0
$951$	13.2183	+	33.6797i	0.428634	+	1.09214i
$952$		0			0
$953$	−18.3255	−	31.7407i	−0.593621	−	1.02818i	−0.993740	−	0.111718i	$-0.964365\pi$
$953$	−18.3255	−	31.7407i	−0.593621	−	1.02818i	0.400119	−	0.916463i	$-0.368969\pi$
$954$		0			0
$955$	−63.9506	−	43.6008i	−2.06939	−	1.41089i
$956$		0			0
$957$	−0.683788	+	0.329295i	−0.0221037	+	0.0106446i
$958$		0			0
$959$	−58.6926	−	8.84648i	−1.89528	−	0.285668i
$960$		0			0
$961$	−1.43176	−	19.1056i	−0.0461859	−	0.616308i
$962$		0			0
$963$	−1.55207	+	3.95460i	−0.0500146	+	0.127435i
$964$		0			0
$965$	10.3619	−	7.06459i	0.333560	−	0.227417i
$966$		0			0
$967$	−1.90500	−	8.34635i	−0.0612607	−	0.268400i	0.935017	−	0.354603i	$-0.115384\pi$
$967$	−1.90500	−	8.34635i	−0.0612607	−	0.268400i	−0.996278	+	0.0862025i	$0.972527\pi$
$968$		0			0
$969$	0.0156614	−	0.208987i	0.000503116	−	0.00671362i
$970$		0			0
$971$	−36.2742	+	33.6575i	−1.16409	+	1.08012i	−0.168562	+	0.985691i	$0.553912\pi$
$971$	−36.2742	+	33.6575i	−1.16409	+	1.08012i	−0.995531	+	0.0944301i	$0.969897\pi$
$972$		0			0
$973$	−21.1091	−	26.4699i	−0.676725	−	0.848587i
$974$		0			0
$975$	25.6889	−	7.92398i	0.822704	−	0.253770i
$976$		0			0
$977$	−13.4546	−	12.4840i	−0.430450	−	0.399399i	0.434971	−	0.900444i	$-0.356759\pi$
$977$	−13.4546	−	12.4840i	−0.430450	−	0.399399i	−0.865421	+	0.501045i	$0.832949\pi$
$978$		0			0
$979$	−9.17655	+	1.38314i	−0.293284	+	0.0442054i
$980$		0			0
$981$	−4.09932	+	5.14038i	−0.130881	+	0.164120i
$982$		0			0
$983$	−2.35817	+	4.08447i	−0.0752140	+	0.130274i	−0.901179	−	0.433446i	$-0.857297\pi$
$983$	−2.35817	+	4.08447i	−0.0752140	+	0.130274i	0.825965	+	0.563721i	$0.190631\pi$
$984$		0			0
$985$	43.0882	+	20.7502i	1.37291	+	0.661157i
$986$		0			0
$987$	6.53101	−	28.6142i	0.207884	−	0.910800i
$988$		0			0
$989$	9.23252	+	9.67091i	0.293577	+	0.307517i
$990$		0			0
$991$	10.0778	−	44.1536i	0.320131	−	1.40259i	−0.517188	−	0.855872i	$-0.673021\pi$
$991$	10.0778	−	44.1536i	0.320131	−	1.40259i	0.837319	−	0.546714i	$-0.184122\pi$
$992$		0			0
$993$	3.38623	+	1.63072i	0.107459	+	0.0517495i
$994$		0			0
$995$	−0.545067	+	0.944083i	−0.0172798	+	0.0299295i
$996$		0			0
$997$	−18.2570	+	22.8935i	−0.578204	+	0.725045i	−0.981805	−	0.189890i	$-0.939187\pi$
$997$	−18.2570	+	22.8935i	−0.578204	+	0.725045i	0.403601	+	0.914935i	$0.367758\pi$
$998$		0			0
$999$	43.6590	−	6.58054i	1.38131	−	0.208199i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.2.bg.c.81.1		36
4.3	odd	2		43.2.g.a.38.1	yes	36
12.11	even	2		387.2.y.c.253.3		36
43.17	even	21	inner	688.2.bg.c.17.1		36
172.19	even	42		1849.2.a.o.1.1		18
172.67	odd	42		1849.2.a.n.1.18		18
172.103	odd	42		43.2.g.a.17.1	✓	36
516.275	even	42		387.2.y.c.361.3		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.2.g.a.17.1	✓	36	172.103	odd	42
43.2.g.a.38.1	yes	36	4.3	odd	2
387.2.y.c.253.3		36	12.11	even	2
387.2.y.c.361.3		36	516.275	even	42
688.2.bg.c.17.1		36	43.17	even	21	inner
688.2.bg.c.81.1		36	1.1	even	1	trivial
1849.2.a.n.1.18		18	172.67	odd	42
1849.2.a.o.1.1		18	172.19	even	42

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:	\( N \)	\(=\)	\( 688 = 2^{4} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	688.bg (of order \(21\), degree \(12\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.28860 - 0.397480i) q^{3} +(-1.48781 - 3.79089i) q^{5} +(-1.38920 - 2.40617i) q^{7} +(-0.976224 - 0.665578i) q^{9} +O(q^{10})\)	\(q+(-1.28860 - 0.397480i) q^{3} +(-1.48781 - 3.79089i) q^{5} +(-1.38920 - 2.40617i) q^{7} +(-0.976224 - 0.665578i) q^{9} +(0.678323 - 0.326663i) q^{11} +(1.70167 + 0.256486i) q^{13} +(0.410392 + 5.47630i) q^{15} +(-1.18750 + 3.02571i) q^{17} +(0.0395049 - 0.0269340i) q^{19} +(0.833720 + 3.65277i) q^{21} +(-0.152371 + 2.03326i) q^{23} +(-8.49199 + 7.87942i) q^{25} +(3.51575 + 4.40861i) q^{27} +(0.714324 - 0.220340i) q^{29} +(-2.52247 - 2.34051i) q^{31} +(-1.00393 + 0.151318i) q^{33} +(-7.05465 + 8.84626i) q^{35} +(3.91502 - 6.78101i) q^{37} +(-2.09082 - 1.00689i) q^{39} +(-1.86928 + 8.18985i) q^{41} +(4.39142 - 4.86985i) q^{43} +(-1.07069 + 4.69101i) q^{45} +(-7.05780 - 3.39886i) q^{47} +(-0.359777 + 0.623151i) q^{49} +(2.73287 - 3.42691i) q^{51} +(-1.76849 + 0.266557i) q^{53} +(-2.24756 - 2.08543i) q^{55} +(-0.0616116 + 0.0190047i) q^{57} +(-3.60798 - 4.52427i) q^{59} +(2.15836 - 2.00266i) q^{61} +(-0.245321 + 3.27359i) q^{63} +(-1.55946 - 6.83245i) q^{65} +(-4.62708 + 3.15469i) q^{67} +(1.00452 - 2.55948i) q^{69} +(0.543672 + 7.25480i) q^{71} +(-10.0746 - 1.51850i) q^{73} +(14.0747 - 6.77800i) q^{75} +(-1.72834 - 1.17836i) q^{77} +(6.70519 + 11.6137i) q^{79} +(-1.48307 - 3.77880i) q^{81} +(-5.42706 - 1.67402i) q^{83} +13.2369 q^{85} -1.00806 q^{87} +(-11.7786 - 3.63323i) q^{89} +(-1.74682 - 4.45083i) q^{91} +(2.32015 + 4.01861i) q^{93} +(-0.160880 - 0.109686i) q^{95} +(9.41101 - 4.53210i) q^{97} +(-0.879615 - 0.132581i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 16 q^{3} - 17 q^{5} - 6 q^{7} - q^{9}+O(q^{10}) \) 36 * q + 16 * q^3 - 17 * q^5 - 6 * q^7 - q^9	\( 36 q + 16 q^{3} - 17 q^{5} - 6 q^{7} - q^{9} + 4 q^{11} + 3 q^{15} - 10 q^{17} - 10 q^{19} - 21 q^{21} - 4 q^{23} - 2 q^{25} + 4 q^{27} + 9 q^{29} - 40 q^{31} - 11 q^{33} - 11 q^{35} - 19 q^{37} + q^{39} - 28 q^{41} + 8 q^{43} - 46 q^{45} + 30 q^{47} + 6 q^{49} - 57 q^{51} - 24 q^{53} - 14 q^{55} + 52 q^{57} + q^{59} - 14 q^{61} - 47 q^{63} + 38 q^{65} - 66 q^{67} - 7 q^{69} + 33 q^{71} + 29 q^{73} + 55 q^{75} - 27 q^{77} + 17 q^{79} + 38 q^{81} + 23 q^{83} - 56 q^{85} + 86 q^{87} - 19 q^{89} + 13 q^{91} - 30 q^{93} - q^{95} - 31 q^{97} + 31 q^{99}+O(q^{100}) \) 36 * q + 16 * q^3 - 17 * q^5 - 6 * q^7 - q^9 + 4 * q^11 + 3 * q^15 - 10 * q^17 - 10 * q^19 - 21 * q^21 - 4 * q^23 - 2 * q^25 + 4 * q^27 + 9 * q^29 - 40 * q^31 - 11 * q^33 - 11 * q^35 - 19 * q^37 + q^39 - 28 * q^41 + 8 * q^43 - 46 * q^45 + 30 * q^47 + 6 * q^49 - 57 * q^51 - 24 * q^53 - 14 * q^55 + 52 * q^57 + q^59 - 14 * q^61 - 47 * q^63 + 38 * q^65 - 66 * q^67 - 7 * q^69 + 33 * q^71 + 29 * q^73 + 55 * q^75 - 27 * q^77 + 17 * q^79 + 38 * q^81 + 23 * q^83 - 56 * q^85 + 86 * q^87 - 19 * q^89 + 13 * q^91 - 30 * q^93 - q^95 - 31 * q^97 + 31 * q^99

Introduction

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Varieties

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