LMFDB - Embedded newform 108.5.k.a.41.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [108,5,Mod(5,108)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(108, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 5]))

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

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

chi := DirichletCharacter("108.5");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	108.k (of order $18$, degree $6$, 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:	$11.1639560131$
Analytic rank:	$0$
Dimension:	$72$
Relative dimension:	$12$ over $\Q(\zeta_{18})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			41.7
Character	$\chi$	$=$	108.41
Dual form			108.5.k.a.29.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.21143 + 8.91810i) q^{3} +(1.79646 - 2.14093i) q^{5} +(-63.3056 - 23.0413i) q^{7} +(-78.0649 + 21.6073i) q^{9} +O(q^{10})$	\(q+(1.21143 + 8.91810i) q^{3} +(1.79646 - 2.14093i) q^{5} +(-63.3056 - 23.0413i) q^{7} +(-78.0649 + 21.6073i) q^{9} +(-23.1554 - 27.5956i) q^{11} +(8.29634 - 47.0509i) q^{13} +(21.2693 + 13.4274i) q^{15} +(0.309929 + 0.178937i) q^{17} +(-163.845 - 283.789i) q^{19} +(128.795 - 592.478i) q^{21} +(-242.445 - 666.113i) q^{23} +(107.174 + 607.813i) q^{25} +(-287.266 - 670.014i) q^{27} +(-1510.88 + 266.409i) q^{29} +(547.514 - 199.279i) q^{31} +(218.049 - 239.933i) q^{33} +(-163.056 + 94.1403i) q^{35} +(-343.789 + 595.460i) q^{37} +(429.655 + 16.9887i) q^{39} +(-498.566 - 87.9106i) q^{41} +(-1251.43 + 1050.07i) q^{43} +(-93.9804 + 205.948i) q^{45} +(453.321 - 1245.49i) q^{47} +(1637.42 + 1373.96i) q^{49} +(-1.22032 + 2.98074i) q^{51} +1435.60i q^{53} -100.678 q^{55} +(2332.37 - 1804.98i) q^{57} +(-989.454 + 1179.18i) q^{59} +(4824.54 + 1755.99i) q^{61} +(5439.80 + 430.858i) q^{63} +(-85.8289 - 102.287i) q^{65} +(-474.440 + 2690.68i) q^{67} +(5646.75 - 2969.10i) q^{69} +(-4492.65 - 2593.83i) q^{71} +(4367.67 + 7565.02i) q^{73} +(-5290.70 + 1692.11i) q^{75} +(830.030 + 2280.49i) q^{77} +(814.469 + 4619.08i) q^{79} +(5627.25 - 3373.54i) q^{81} +(-9403.81 + 1658.15i) q^{83} +(0.939867 - 0.342084i) q^{85} +(-4206.18 - 13151.4i) q^{87} +(-9672.46 + 5584.40i) q^{89} +(-1609.32 + 2787.43i) q^{91} +(2440.46 + 4641.37i) q^{93} +(-901.914 - 159.032i) q^{95} +(1339.14 - 1123.67i) q^{97} +(2403.89 + 1653.92i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q + 9 q^{5} - 102 q^{9}+O(q^{10}) $ 72 * q + 9 * q^5 - 102 * q^9	$ 72 q + 9 q^{5} - 102 q^{9} + 18 q^{11} - 225 q^{15} - 282 q^{21} - 1278 q^{23} + 441 q^{25} + 54 q^{27} + 1854 q^{29} - 1665 q^{31} - 45 q^{33} - 2673 q^{35} + 6951 q^{39} - 5472 q^{41} + 1260 q^{43} + 5553 q^{45} + 5103 q^{47} - 5904 q^{49} + 1899 q^{51} + 1107 q^{57} - 10944 q^{59} + 8352 q^{61} - 11985 q^{63} + 8757 q^{65} + 378 q^{67} + 5607 q^{69} - 19764 q^{71} + 6111 q^{73} - 3453 q^{75} - 5679 q^{77} - 5652 q^{79} - 20466 q^{81} - 20061 q^{83} + 26100 q^{85} + 40545 q^{87} + 15633 q^{89} - 6039 q^{91} + 40179 q^{93} + 48024 q^{95} - 37530 q^{97} + 12177 q^{99}+O(q^{100}) $ 72 * q + 9 * q^5 - 102 * q^9 + 18 * q^11 - 225 * q^15 - 282 * q^21 - 1278 * q^23 + 441 * q^25 + 54 * q^27 + 1854 * q^29 - 1665 * q^31 - 45 * q^33 - 2673 * q^35 + 6951 * q^39 - 5472 * q^41 + 1260 * q^43 + 5553 * q^45 + 5103 * q^47 - 5904 * q^49 + 1899 * q^51 + 1107 * q^57 - 10944 * q^59 + 8352 * q^61 - 11985 * q^63 + 8757 * q^65 + 378 * q^67 + 5607 * q^69 - 19764 * q^71 + 6111 * q^73 - 3453 * q^75 - 5679 * q^77 - 5652 * q^79 - 20466 * q^81 - 20061 * q^83 + 26100 * q^85 + 40545 * q^87 + 15633 * q^89 - 6039 * q^91 + 40179 * q^93 + 48024 * q^95 - 37530 * q^97 + 12177 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$.

$n$	$29$	$55$
$\chi(n)$	$e\left(\frac{17}{18}\right)$	$1$

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
$3$	1.21143	+	8.91810i	0.134603	+	0.990900i
$3$	1.21143	+	8.91810i	0.134603	+	0.990900i
$4$		0			0
$5$	1.79646	−	2.14093i	0.0718583	−	0.0856374i	−0.728918	−	0.684601i	$-0.759976\pi$
$5$	1.79646	−	2.14093i	0.0718583	−	0.0856374i	0.800776	+	0.598964i	$0.204421\pi$
$6$		0			0
$7$	−63.3056	−	23.0413i	−1.29195	−	0.470232i	−0.397584	−	0.917566i	$-0.630151\pi$
$7$	−63.3056	−	23.0413i	−1.29195	−	0.470232i	−0.894367	+	0.447334i	$0.852373\pi$
$8$		0			0
$9$	−78.0649	+	21.6073i	−0.963764	+	0.266757i
$10$		0			0
$11$	−23.1554	−	27.5956i	−0.191367	−	0.228063i	0.661826	−	0.749657i	$-0.269782\pi$
$11$	−23.1554	−	27.5956i	−0.191367	−	0.228063i	−0.853193	+	0.521595i	$0.825337\pi$
$12$		0			0
$13$	8.29634	−	47.0509i	0.0490908	−	0.278408i	−0.950374	−	0.311109i	$-0.899300\pi$
$13$	8.29634	−	47.0509i	0.0490908	−	0.278408i	0.999465	+	0.0327008i	$0.0104108\pi$
$14$		0			0
$15$	21.2693	+	13.4274i	0.0945304	+	0.0596773i
$16$		0			0
$17$	0.309929	+	0.178937i	0.00107242	+	0.000619161i	0.500536	−	0.865716i	$-0.333136\pi$
$17$	0.309929	+	0.178937i	0.00107242	+	0.000619161i	−0.499464	+	0.866335i	$0.666470\pi$
$18$		0			0
$19$	−163.845	−	283.789i	−0.453865	−	0.786118i	0.544757	−	0.838594i	$-0.316622\pi$
$19$	−163.845	−	283.789i	−0.453865	−	0.786118i	−0.998622	+	0.0524761i	$0.983289\pi$
$20$		0			0
$21$	128.795	−	592.478i	0.292051	−	1.34349i
$22$		0			0
$23$	−242.445	−	666.113i	−0.458309	−	1.25919i	−0.926743	−	0.375695i	$-0.877404\pi$
$23$	−242.445	−	666.113i	−0.458309	−	1.25919i	0.468435	−	0.883498i	$-0.344818\pi$
$24$		0			0
$25$	107.174	+	607.813i	0.171478	+	0.972500i
$26$		0			0
$27$	−287.266	−	670.014i	−0.394055	−	0.919087i
$28$		0			0
$29$	−1510.88	+	266.409i	−1.79653	+	0.316776i	−0.969446	−	0.245306i	$-0.921112\pi$
$29$	−1510.88	+	266.409i	−1.79653	+	0.316776i	−0.827081	+	0.562082i	$0.810000\pi$
$30$		0			0
$31$	547.514	−	199.279i	0.569734	−	0.207366i	−0.0410588	−	0.999157i	$-0.513073\pi$
$31$	547.514	−	199.279i	0.569734	−	0.207366i	0.610793	+	0.791791i	$0.290851\pi$
$32$		0			0
$33$	218.049	−	239.933i	0.200229	−	0.220324i
$34$		0			0
$35$	−163.056	+	94.1403i	−0.133107	+	0.0768492i
$36$		0			0
$37$	−343.789	+	595.460i	−0.251124	+	0.434960i	−0.963836	−	0.266498i	$-0.914134\pi$
$37$	−343.789	+	595.460i	−0.251124	+	0.434960i	0.712711	+	0.701457i	$0.247467\pi$
$38$		0			0
$39$	429.655	+	16.9887i	0.282482	+	0.0111695i
$40$		0			0
$41$	−498.566	−	87.9106i	−0.296589	−	0.0522966i	0.0233738	−	0.999727i	$-0.492559\pi$
$41$	−498.566	−	87.9106i	−0.296589	−	0.0522966i	−0.319963	+	0.947430i	$0.603670\pi$
$42$		0			0
$43$	−1251.43	+	1050.07i	−0.676812	+	0.567913i	−0.915073	−	0.403289i	$-0.867867\pi$
$43$	−1251.43	+	1050.07i	−0.676812	+	0.567913i	0.238261	+	0.971201i	$0.423423\pi$
$44$		0			0
$45$	−93.9804	+	205.948i	−0.0464101	+	0.101703i
$46$		0			0
$47$	453.321	−	1245.49i	0.205215	−	0.563824i	−0.793801	−	0.608178i	$-0.791901\pi$
$47$	453.321	−	1245.49i	0.205215	−	0.563824i	0.999016	+	0.0443538i	$0.0141229\pi$
$48$		0			0
$49$	1637.42	+	1373.96i	0.681974	+	0.572244i
$50$		0			0
$51$	−1.22032	+	2.98074i	−0.000469175	+	0.00114600i
$52$		0			0
$53$			1435.60i			0.511072i	0.966800	+	0.255536i	$0.0822519\pi$
$53$			1435.60i			0.511072i	−0.966800	+	0.255536i	$0.917748\pi$
$54$		0			0
$55$	−100.678			−0.0332820
$56$		0			0
$57$	2332.37	−	1804.98i	0.717872	−	0.555549i
$58$		0			0
$59$	−989.454	+	1179.18i	−0.284244	+	0.338749i	−0.889207	−	0.457505i	$-0.848743\pi$
$59$	−989.454	+	1179.18i	−0.284244	+	0.338749i	0.604963	+	0.796253i	$0.293188\pi$
$60$		0			0
$61$	4824.54	+	1755.99i	1.29657	+	0.471913i	0.895878	−	0.444300i	$-0.146547\pi$
$61$	4824.54	+	1755.99i	1.29657	+	0.471913i	0.400692	+	0.916213i	$0.368770\pi$
$62$		0			0
$63$	5439.80	+	430.858i	1.37057	+	0.108556i
$64$		0			0
$65$	−85.8289	−	102.287i	−0.0203145	−	0.0242099i
$66$		0			0
$67$	−474.440	+	2690.68i	−0.105689	+	0.599394i	0.885253	+	0.465109i	$0.153985\pi$
$67$	−474.440	+	2690.68i	−0.105689	+	0.599394i	−0.990943	+	0.134285i	$0.957126\pi$
$68$		0			0
$69$	5646.75	−	2969.10i	1.18604	−	0.623629i
$70$		0			0
$71$	−4492.65	−	2593.83i	−0.891222	−	0.514548i	−0.0168803	−	0.999858i	$-0.505373\pi$
$71$	−4492.65	−	2593.83i	−0.891222	−	0.514548i	−0.874342	+	0.485310i	$0.838707\pi$
$72$		0			0
$73$	4367.67	+	7565.02i	0.819604	+	1.41960i	0.905975	+	0.423332i	$0.139140\pi$
$73$	4367.67	+	7565.02i	0.819604	+	1.41960i	−0.0863709	+	0.996263i	$0.527527\pi$
$74$		0			0
$75$	−5290.70	+	1692.11i	−0.940569	+	0.300819i
$76$		0			0
$77$	830.030	+	2280.49i	0.139995	+	0.384633i
$78$		0			0
$79$	814.469	+	4619.08i	0.130503	+	0.740119i	0.977886	+	0.209138i	$0.0670656\pi$
$79$	814.469	+	4619.08i	0.130503	+	0.740119i	−0.847383	+	0.530982i	$0.821823\pi$
$80$		0			0
$81$	5627.25	−	3373.54i	0.857682	−	0.514181i
$82$		0			0
$83$	−9403.81	+	1658.15i	−1.36505	+	0.240695i	−0.807704	−	0.589588i	$-0.799290\pi$
$83$	−9403.81	+	1658.15i	−1.36505	+	0.240695i	−0.557343	+	0.830282i	$0.688179\pi$
$84$		0			0
$85$	0.939867	−	0.342084i	0.000130085	−	4.73472e-5i
$86$		0			0
$87$	−4206.18	−	13151.4i	−0.555712	−	1.73754i
$88$		0			0
$89$	−9672.46	+	5584.40i	−1.22112	+	0.705012i	−0.965156	−	0.261676i	$-0.915725\pi$
$89$	−9672.46	+	5584.40i	−1.22112	+	0.705012i	−0.255960	+	0.966687i	$0.582392\pi$
$90$		0			0
$91$	−1609.32	+	2787.43i	−0.194339	+	0.336605i
$92$		0			0
$93$	2440.46	+	4641.37i	0.282167	+	0.536637i
$94$		0			0
$95$	−901.914	−	159.032i	−0.0999351	−	0.0176213i
$96$		0			0
$97$	1339.14	−	1123.67i	0.142325	−	0.119425i	−0.568845	−	0.822444i	$-0.692610\pi$
$97$	1339.14	−	1123.67i	0.142325	−	0.119425i	0.711171	+	0.703019i	$0.248165\pi$
$98$		0			0
$99$	2403.89	+	1653.92i	0.245270	+	0.168750i
$100$		0			0
$101$	5798.66	−	15931.7i	0.568440	−	1.56178i	−0.238500	−	0.971142i	$-0.576656\pi$
$101$	5798.66	−	15931.7i	0.568440	−	1.56178i	0.806940	−	0.590633i	$-0.201122\pi$
$102$		0			0
$103$	−614.976	−	516.026i	−0.0579674	−	0.0486404i	0.613344	−	0.789816i	$-0.289824\pi$
$103$	−614.976	−	516.026i	−0.0579674	−	0.0486404i	−0.671311	+	0.741176i	$0.734269\pi$
$104$		0			0
$105$	−1037.08	−	1340.10i	−0.0940665	−	0.121551i
$106$		0			0
$107$			110.997i			0.00969492i	0.999988	+	0.00484746i	$0.00154300\pi$
$107$			110.997i			0.00969492i	−0.999988	+	0.00484746i	$0.998457\pi$
$108$		0			0
$109$	−8515.84			−0.716761			−0.358381	−	0.933576i	$-0.616671\pi$
$109$	−8515.84			−0.716761			−0.358381	+	0.933576i	$0.616671\pi$
$110$		0			0
$111$	−5726.84	−	2344.58i	−0.464803	−	0.190292i
$112$		0			0
$113$	5198.29	−	6195.08i	0.407102	−	0.485165i	−0.523070	−	0.852290i	$-0.675213\pi$
$113$	5198.29	−	6195.08i	0.407102	−	0.485165i	0.930172	+	0.367124i	$0.119658\pi$
$114$		0			0
$115$	−1861.65	−	677.584i	−0.140767	−	0.0512351i
$116$		0			0
$117$	368.989	+	3852.28i	0.0269552	+	0.281415i
$118$		0			0
$119$	−15.4973	−	18.4689i	−0.00109436	−	0.00130421i
$120$		0			0
$121$	2317.04	−	13140.6i	0.158257	−	0.897520i
$122$		0			0
$123$	180.018	−	4552.75i	0.0118989	−	0.300929i
$124$		0			0
$125$	3006.55	+	1735.83i	0.192419	+	0.111093i
$126$		0			0
$127$	−15006.6	−	25992.2i	−0.930410	−	1.61152i	−0.782621	−	0.622498i	$-0.786118\pi$
$127$	−15006.6	−	25992.2i	−0.930410	−	1.61152i	−0.147789	−	0.989019i	$-0.547216\pi$
$128$		0			0
$129$	−10880.6	−	9888.24i	−0.653845	−	0.594210i
$130$		0			0
$131$	−3367.32	−	9251.64i	−0.196219	−	0.539108i	0.802092	−	0.597201i	$-0.203720\pi$
$131$	−3367.32	−	9251.64i	−0.196219	−	0.539108i	−0.998311	+	0.0580924i	$0.981498\pi$
$132$		0			0
$133$	3833.46	+	21740.6i	0.216714	+	1.22905i
$134$		0			0
$135$	−1950.52	−	588.635i	−0.107024	−	0.0322982i
$136$		0			0
$137$	−27173.2	+	4791.37i	−1.44777	+	0.255281i	−0.841620	−	0.540070i	$-0.818398\pi$
$137$	−27173.2	+	4791.37i	−1.44777	+	0.255281i	−0.606150	+	0.795351i	$0.707287\pi$
$138$		0			0
$139$	23150.6	−	8426.12i	1.19821	−	0.436112i	0.335609	−	0.942001i	$-0.391058\pi$
$139$	23150.6	−	8426.12i	1.19821	−	0.436112i	0.862598	+	0.505889i	$0.168836\pi$
$140$		0			0
$141$	11656.5	+	2533.94i	0.586316	+	0.127455i
$142$		0			0
$143$	−1490.50	+	860.542i	−0.0728888	+	0.0420824i
$144$		0			0
$145$	−2143.87	+	3713.29i	−0.101967	+	0.176613i
$146$		0			0
$147$	−10269.5	+	16267.1i	−0.475241	+	0.752794i
$148$		0			0
$149$	37247.5	+	6567.74i	1.67774	+	0.295831i	0.929835	−	0.367976i	$-0.119949\pi$
$149$	37247.5	+	6567.74i	1.67774	+	0.295831i	0.747904	+	0.663807i	$0.231060\pi$
$150$		0			0
$151$	−12653.1	+	10617.2i	−0.554936	+	0.465646i	−0.876608	−	0.481205i	$-0.840199\pi$
$151$	−12653.1	+	10617.2i	−0.554936	+	0.465646i	0.321673	+	0.946851i	$0.395755\pi$
$152$		0			0
$153$	−28.0609	−	7.27201i	−0.00119872	−	0.000310650i
$154$		0			0
$155$	556.943	−	1530.19i	0.0231818	−	0.0636915i
$156$		0			0
$157$	3919.19	+	3288.59i	0.159000	+	0.133417i	0.718817	−	0.695200i	$-0.244684\pi$
$157$	3919.19	+	3288.59i	0.159000	+	0.133417i	−0.559817	+	0.828616i	$0.689129\pi$
$158$		0			0
$159$	−12802.8	+	1739.13i	−0.506421	+	0.0687920i
$160$		0			0
$161$			47754.9i			1.84233i
$162$		0			0
$163$	−36016.6			−1.35559			−0.677794	−	0.735252i	$-0.737064\pi$
$163$	−36016.6			−1.35559			−0.677794	+	0.735252i	$0.737064\pi$
$164$		0			0
$165$	−121.964	−	897.857i	−0.00447987	−	0.0329791i
$166$		0			0
$167$	12971.2	−	15458.5i	0.465102	−	0.554287i	−0.481602	−	0.876390i	$-0.659945\pi$
$167$	12971.2	−	15458.5i	0.465102	−	0.554287i	0.946705	+	0.322102i	$0.104390\pi$
$168$		0			0
$169$	24693.6	+	8987.74i	0.864592	+	0.314686i
$170$		0			0
$171$	18922.5	+	18613.7i	0.647121	+	0.636561i
$172$		0			0
$173$	−20392.5	−	24302.9i	−0.681363	−	0.812017i	0.308919	−	0.951088i	$-0.400033\pi$
$173$	−20392.5	−	24302.9i	−0.681363	−	0.812017i	−0.990282	+	0.139071i	$0.955588\pi$
$174$		0			0
$175$	7220.12	−	40947.4i	0.235759	−	1.33706i
$176$		0			0
$177$	−11714.7	−	7395.54i	−0.373926	−	0.236061i
$178$		0			0
$179$	47563.9	+	27461.1i	1.48447	+	0.857060i	0.999844	−	0.0176610i	$-0.00562197\pi$
$179$	47563.9	+	27461.1i	1.48447	+	0.857060i	0.484627	+	0.874721i	$0.338955\pi$
$180$		0			0
$181$	−10313.7	−	17863.9i	−0.314817	−	0.545279i	0.664582	−	0.747216i	$-0.268610\pi$
$181$	−10313.7	−	17863.9i	−0.314817	−	0.545279i	−0.979399	+	0.201937i	$0.935276\pi$
$182$		0			0
$183$	−9815.48	+	45152.9i	−0.293096	+	1.34829i
$184$		0			0
$185$	657.238	+	1805.75i	0.0192035	+	0.0527611i
$186$		0			0
$187$	−2.23865	−	12.6960i	−6.40183e−5	−	0.000363066i
$188$		0			0
$189$	2747.51	+	49034.6i	0.0769157	+	1.37271i
$190$		0			0
$191$	63011.6	−	11110.6i	1.72724	−	0.304560i	0.780169	−	0.625569i	$-0.215133\pi$
$191$	63011.6	−	11110.6i	1.72724	−	0.304560i	0.947076	+	0.321009i	$0.104022\pi$
$192$		0			0
$193$	31715.8	−	11543.6i	0.851454	−	0.309904i	0.120821	−	0.992674i	$-0.461447\pi$
$193$	31715.8	−	11543.6i	0.851454	−	0.309904i	0.730633	+	0.682770i	$0.239225\pi$
$194$		0			0
$195$	808.229	−	889.344i	0.0212552	−	0.0233884i
$196$		0			0
$197$	−7886.51	+	4553.28i	−0.203213	+	0.117325i	−0.598153	−	0.801382i	$-0.704099\pi$
$197$	−7886.51	+	4553.28i	−0.203213	+	0.117325i	0.394940	+	0.918707i	$0.370765\pi$
$198$		0			0
$199$	9023.29	−	15628.8i	0.227855	−	0.394656i	−0.729317	−	0.684176i	$-0.760162\pi$
$199$	9023.29	−	15628.8i	0.227855	−	0.394656i	0.957172	+	0.289519i	$0.0934954\pi$
$200$		0			0
$201$	−24570.5	−	971.528i	−0.608166	−	0.0240471i
$202$		0			0
$203$	101786.	+	17947.5i	2.46998	+	0.435525i
$204$		0			0
$205$	−1083.86	+	909.469i	−0.0257909	+	0.0216411i
$206$		0			0
$207$	33319.4	+	46761.4i	0.777599	+	1.09131i
$208$		0			0
$209$	−4037.40	+	11092.7i	−0.0924292	+	0.253947i
$210$		0			0
$211$	−32956.5	−	27653.8i	−0.740247	−	0.621141i	0.192657	−	0.981266i	$-0.438290\pi$
$211$	−32956.5	−	27653.8i	−0.740247	−	0.621141i	−0.932904	+	0.360125i	$0.882734\pi$
$212$		0			0
$213$	17689.5	−	43208.2i	0.389903	−	0.952372i
$214$		0			0
$215$			4565.63i			0.0987696i
$216$		0			0
$217$	−39252.4			−0.833578
$218$		0			0
$219$	−62174.5	+	48115.8i	−1.29635	+	1.00323i
$220$		0			0
$221$	10.9904	−	13.0979i	0.000225025	−	0.000268174i
$222$		0			0
$223$	−13939.3	−	5073.49i	−0.280305	−	0.102023i	0.198042	−	0.980194i	$-0.436542\pi$
$223$	−13939.3	−	5073.49i	−0.280305	−	0.102023i	−0.478347	+	0.878171i	$0.658764\pi$
$224$		0			0
$225$	−21499.7	−	45133.1i	−0.424685	−	0.891518i
$226$		0			0
$227$	−37883.9	−	45148.2i	−0.735195	−	0.876171i	0.260817	−	0.965388i	$-0.416008\pi$
$227$	−37883.9	−	45148.2i	−0.735195	−	0.876171i	−0.996012	+	0.0892169i	$0.971564\pi$
$228$		0			0
$229$	6889.02	−	39069.6i	0.131367	−	0.745019i	−0.845954	−	0.533256i	$-0.820968\pi$
$229$	6889.02	−	39069.6i	0.131367	−	0.745019i	0.977321	−	0.211763i	$-0.0679206\pi$
$230$		0			0
$231$	−19332.1	+	10164.9i	−0.362289	+	0.190494i
$232$		0			0
$233$	−56676.4	−	32722.1i	−1.04398	−	0.602740i	−0.123019	−	0.992404i	$-0.539258\pi$
$233$	−56676.4	−	32722.1i	−1.04398	−	0.602740i	−0.920957	+	0.389665i	$0.872591\pi$
$234$		0			0
$235$	−1852.14	−	3208.00i	−0.0335380	−	0.0580896i
$236$		0			0
$237$	−40206.8	+	12859.2i	−0.715818	+	0.228938i
$238$		0			0
$239$	14949.7	+	41074.0i	0.261720	+	0.719069i	0.999052	+	0.0435386i	$0.0138632\pi$
$239$	14949.7	+	41074.0i	0.261720	+	0.719069i	−0.737332	+	0.675531i	$0.763915\pi$
$240$		0			0
$241$	8549.32	+	48485.6i	0.147197	+	0.834793i	0.965577	+	0.260117i	$0.0837610\pi$
$241$	8549.32	+	48485.6i	0.147197	+	0.834793i	−0.818381	+	0.574677i	$0.805128\pi$
$242$		0			0
$243$	36902.6	+	46097.6i	0.624948	+	0.780666i
$244$		0			0
$245$	5883.11	−	1037.35i	0.0980110	−	0.0172820i
$246$		0			0
$247$	−14711.8	+	5354.67i	−0.241142	+	0.0877685i
$248$		0			0
$249$	−26179.6	−	81855.4i	−0.422244	−	1.32023i
$250$		0			0
$251$	−55699.9	+	32158.3i	−0.884111	+	0.510442i	−0.872012	−	0.489485i	$-0.837185\pi$
$251$	−55699.9	+	32158.3i	−0.884111	+	0.510442i	−0.0120994	+	0.999927i	$0.503851\pi$
$252$		0			0
$253$	−12767.8	+	22114.6i	−0.199470	+	0.345491i
$254$		0			0
$255$	4.18932	+	7.96742i	6.44263e−5	+	0.000122528i
$256$		0			0
$257$	−20894.2	−	3684.22i	−0.316344	−	0.0557800i	0.0132216	−	0.999913i	$-0.495791\pi$
$257$	−20894.2	−	3684.22i	−0.316344	−	0.0557800i	−0.329566	+	0.944133i	$0.606902\pi$
$258$		0			0
$259$	35483.9	−	29774.6i	0.528972	−	0.443860i
$260$		0			0
$261$	112190.	−	53443.2i	1.64693	−	0.784533i
$262$		0			0
$263$	18444.1	−	50674.9i	0.266653	−	0.732624i	−0.732027	−	0.681275i	$-0.761426\pi$
$263$	18444.1	−	50674.9i	0.266653	−	0.732624i	0.998681	−	0.0513488i	$-0.0163520\pi$
$264$		0			0
$265$	3073.53	+	2579.00i	0.0437669	+	0.0367248i
$266$		0			0
$267$	−61519.7	−	79494.8i	−0.862962	−	1.11511i
$268$		0			0
$269$			68360.6i			0.944717i	0.881407	+	0.472358i	$0.156597\pi$
$269$			68360.6i			0.944717i	−0.881407	+	0.472358i	$0.843403\pi$
$270$		0			0
$271$	−46387.1			−0.631624			−0.315812	−	0.948822i	$-0.602277\pi$
$271$	−46387.1			−0.631624			−0.315812	+	0.948822i	$0.602277\pi$
$272$		0			0
$273$	−26808.1	−	10975.3i	−0.359700	−	0.147262i
$274$		0			0
$275$	14291.3	−	17031.7i	0.188976	−	0.225213i
$276$		0			0
$277$	29407.5	+	10703.4i	0.383264	+	0.139497i	0.526465	−	0.850197i	$-0.323517\pi$
$277$	29407.5	+	10703.4i	0.383264	+	0.139497i	−0.143201	+	0.989694i	$0.545739\pi$
$278$		0			0
$279$	−38435.8	+	27387.0i	−0.493773	+	0.351832i
$280$		0			0
$281$	−82866.8	−	98756.8i	−1.04947	−	1.25070i	−0.967179	−	0.254095i	$-0.918222\pi$
$281$	−82866.8	−	98756.8i	−1.04947	−	1.25070i	−0.0822858	−	0.996609i	$-0.526222\pi$
$282$		0			0
$283$	8633.45	−	48962.7i	0.107798	−	0.611354i	−0.882268	−	0.470748i	$-0.843984\pi$
$283$	8633.45	−	48962.7i	0.107798	−	0.611354i	0.990066	−	0.140605i	$-0.0449049\pi$
$284$		0			0
$285$	325.656	−	8236.01i	0.00400930	−	0.101398i
$286$		0			0
$287$	29536.4	+	17052.9i	0.358586	+	0.207030i
$288$		0			0
$289$	−41760.4	−	72331.2i	−0.499999	−	0.866024i
$290$		0			0
$291$	11643.3	+	10581.3i	0.137496	+	0.124955i
$292$		0			0
$293$	−41261.9	−	113366.i	−0.480633	−	1.32053i	−0.908952	−	0.416901i	$-0.863116\pi$
$293$	−41261.9	−	113366.i	−0.480633	−	1.32053i	0.428319	−	0.903628i	$-0.359106\pi$
$294$		0			0
$295$	747.046	+	4236.71i	0.00858427	+	0.0486838i
$296$		0			0
$297$	−11837.7	+	23441.8i	−0.134200	+	0.265752i
$298$		0			0
$299$	−33352.6	+	5880.97i	−0.373068	+	0.0657819i
$300$		0			0
$301$	103417.	−	37640.8i	1.14146	−	0.415457i
$302$		0			0
$303$	149105.	+	32412.9i	1.62408	+	0.353047i
$304$		0			0
$305$	12426.5	−	7174.46i	0.133583	−	0.0771240i
$306$		0			0
$307$	−42628.5	+	73834.8i	−0.452297	+	0.783401i	−0.998528	−	0.0542330i	$-0.982729\pi$
$307$	−42628.5	+	73834.8i	−0.452297	+	0.783401i	0.546231	+	0.837634i	$0.316062\pi$
$308$		0			0
$309$	3856.97	−	6109.54i	0.0403952	−	0.0639870i
$310$		0			0
$311$	128622.	+	22679.6i	1.32983	+	0.234484i	0.793007	−	0.609212i	$-0.208514\pi$
$311$	128622.	+	22679.6i	1.32983	+	0.234484i	0.536820	+	0.843697i	$0.319625\pi$
$312$		0			0
$313$	−131429.	+	110282.i	−1.34153	+	1.12568i	−0.360303	+	0.932835i	$0.617327\pi$
$313$	−131429.	+	110282.i	−1.34153	+	1.12568i	−0.981229	+	0.192844i	$0.938229\pi$
$314$		0			0
$315$	10694.8	−	10872.2i	0.107783	−	0.109572i
$316$		0			0
$317$	−44695.5	+	122800.i	−0.444780	+	1.22202i	0.491534	+	0.870859i	$0.336437\pi$
$317$	−44695.5	+	122800.i	−0.444780	+	1.22202i	−0.936314	+	0.351165i	$0.885786\pi$
$318$		0			0
$319$	42336.8	+	35524.8i	0.416041	+	0.349100i
$320$		0			0
$321$	−989.883	+	134.465i	−0.00960669	+	0.00130497i
$322$		0			0
$323$		−	117.272i		−	0.00112406i
$324$		0			0
$325$	29487.3			0.279170
$326$		0			0
$327$	−10316.3	−	75945.1i	−0.0964784	−	0.710239i
$328$		0			0
$329$	−57395.4	+	68401.2i	−0.530256	+	0.631934i
$330$		0			0
$331$	146219.	+	53219.2i	1.33459	+	0.485750i	0.908104	−	0.418746i	$-0.137530\pi$
$331$	146219.	+	53219.2i	1.33459	+	0.485750i	0.426483	+	0.904496i	$0.359752\pi$
$332$		0			0
$333$	13971.6	−	53912.8i	0.125996	−	0.486187i
$334$		0			0
$335$	4908.26	+	5849.44i	0.0437359	+	0.0521224i
$336$		0			0
$337$	−19651.5	+	111449.i	−0.173036	+	0.981335i	0.767351	+	0.641228i	$0.221575\pi$
$337$	−19651.5	+	111449.i	−0.173036	+	0.981335i	−0.940387	+	0.340107i	$0.889537\pi$
$338$		0			0
$339$	61545.7	+	38853.9i	0.535547	+	0.338092i
$340$		0			0
$341$	−18177.2	−	10494.6i	−0.156321	−	0.0902519i
$342$		0			0
$343$	8875.78	+	15373.3i	0.0754429	+	0.130671i
$344$		0			0
$345$	3787.51	−	17423.2i	0.0318211	−	0.146383i
$346$		0			0
$347$	−43939.9	−	120724.i	−0.364922	−	1.00262i	−0.977265	−	0.212023i	$-0.931995\pi$
$347$	−43939.9	−	120724.i	−0.364922	−	1.00262i	0.612342	−	0.790593i	$-0.290227\pi$
$348$		0			0
$349$	30410.3	+	172466.i	0.249672	+	1.41596i	0.809386	+	0.587276i	$0.199800\pi$
$349$	30410.3	+	172466.i	0.249672	+	1.41596i	−0.559714	+	0.828686i	$0.689089\pi$
$350$		0			0
$351$	−33908.0	+	7957.45i	−0.275225	+	0.0645892i
$352$		0			0
$353$	−162310.	+	28619.6i	−1.30255	+	0.229675i	−0.781531	−	0.623867i	$-0.785561\pi$
$353$	−162310.	+	28619.6i	−1.30255	+	0.229675i	−0.521023	+	0.853542i	$0.674450\pi$
$354$		0			0
$355$	−13624.1	+	4958.76i	−0.108106	+	0.0393475i
$356$		0			0
$357$	145.934	−	160.580i	0.00114504	−	0.00125995i
$358$		0			0
$359$	−114576.	+	66150.4i	−0.889005	+	0.513267i	−0.873617	−	0.486614i	$-0.838232\pi$
$359$	−114576.	+	66150.4i	−0.889005	+	0.513267i	−0.0153883	+	0.999882i	$0.504898\pi$
$360$		0			0
$361$	11469.8	−	19866.4i	0.0880123	−	0.152442i
$362$		0			0
$363$	119996.	+	4744.69i	0.910654	+	0.0360077i
$364$		0			0
$365$	24042.5	+	4239.35i	0.180466	+	0.0318210i
$366$		0			0
$367$	−9070.65	+	7611.18i	−0.0673452	+	0.0565093i	−0.675839	−	0.737049i	$-0.736219\pi$
$367$	−9070.65	+	7611.18i	−0.0673452	+	0.0565093i	0.608494	+	0.793559i	$0.291774\pi$
$368$		0			0
$369$	40820.0	−	3909.92i	0.299792	−	0.0287154i
$370$		0			0
$371$	33078.2	−	90881.6i	0.240322	−	0.660280i
$372$		0			0
$373$	174210.	+	146180.i	1.25215	+	1.05068i	0.996473	+	0.0839156i	$0.0267426\pi$
$373$	174210.	+	146180.i	1.25215	+	1.05068i	0.255676	+	0.966762i	$0.417702\pi$
$374$		0			0
$375$	−11838.1	+	28915.5i	−0.0841819	+	0.205621i
$376$		0			0
$377$			73298.5i			0.515718i
$378$		0			0
$379$	55108.6			0.383655			0.191828	−	0.981429i	$-0.438559\pi$
$379$	55108.6			0.383655			0.191828	+	0.981429i	$0.438559\pi$
$380$		0			0
$381$	213621.	−	165318.i	1.47162	−	1.13886i
$382$		0			0
$383$	−115966.	+	138203.i	−0.790559	+	0.942152i	−0.999359	−	0.0358069i	$-0.988600\pi$
$383$	−115966.	+	138203.i	−0.790559	+	0.942152i	0.208800	+	0.977958i	$0.433044\pi$
$384$		0			0
$385$	6373.49	+	2319.76i	0.0429987	+	0.0156503i
$386$		0			0
$387$	75003.2	−	109014.i	0.500792	−	0.727878i
$388$		0			0
$389$	42524.9	+	50679.2i	0.281024	+	0.334912i	0.888030	−	0.459786i	$-0.152074\pi$
$389$	42524.9	+	50679.2i	0.281024	+	0.334912i	−0.607005	+	0.794698i	$0.707629\pi$
$390$		0			0
$391$	44.0518	−	249.830i	0.000288144	−	0.00163415i
$392$		0			0
$393$	78427.7	−	41237.8i	0.507790	−	0.266999i
$394$		0			0
$395$	11352.3	+	6554.26i	0.0727596	+	0.0420078i
$396$		0			0
$397$	101672.	+	176102.i	0.645092	+	1.11733i	0.984280	+	0.176614i	$0.0565145\pi$
$397$	101672.	+	176102.i	0.645092	+	1.11733i	−0.339188	+	0.940719i	$0.610152\pi$
$398$		0			0
$399$	−189241.	+	60524.4i	−1.18869	+	0.380176i
$400$		0			0
$401$	−66530.2	−	182790.i	−0.413743	−	1.13675i	−0.955185	−	0.296010i	$-0.904344\pi$
$401$	−66530.2	−	182790.i	−0.413743	−	1.13675i	0.541442	−	0.840738i	$-0.317879\pi$
$402$		0			0
$403$	−4833.89	−	27414.3i	−0.0297637	−	0.168798i
$404$		0			0
$405$	2886.59	−	18108.0i	0.0175984	−	0.110398i
$406$		0			0
$407$	24392.6	−	4301.08i	0.147255	−	0.0259650i
$408$		0			0
$409$	22329.3	−	8127.22i	0.133484	−	0.0485842i	−0.274414	−	0.961612i	$-0.588484\pi$
$409$	22329.3	−	8127.22i	0.133484	−	0.0485842i	0.407898	+	0.913027i	$0.366262\pi$
$410$		0			0
$411$	−75648.3	−	236529.i	−0.447832	−	1.40023i
$412$		0			0
$413$	89807.9	−	51850.6i	0.526520	−	0.303986i
$414$		0			0
$415$	−13343.6	+	23111.7i	−0.0774775	+	0.134195i
$416$		0			0
$417$	103190.	+	196251.i	0.593426	+	1.12860i
$418$		0			0
$419$	85498.0	+	15075.6i	0.486999	+	0.0858710i	0.411756	−	0.911294i	$-0.364916\pi$
$419$	85498.0	+	15075.6i	0.486999	+	0.0858710i	0.0752430	+	0.997165i	$0.476027\pi$
$420$		0			0
$421$	151352.	−	126999.i	0.853934	−	0.716536i	−0.106718	−	0.994289i	$-0.534034\pi$
$421$	151352.	−	126999.i	0.853934	−	0.716536i	0.960652	+	0.277754i	$0.0895899\pi$
$422$		0			0
$423$	−8476.79	+	107024.i	−0.0473752	+	0.598136i
$424$		0			0
$425$	−75.5442	+	207.556i	−0.000418238	+	0.00114910i
$426$		0			0
$427$	−264960.	−	222328.i	−1.45320	−	1.21938i
$428$		0			0
$429$	−9480.04	−	12250.0i	−0.0515105	−	0.0665610i
$430$		0			0
$431$		−	328870.i		−	1.77039i	−0.465219	−	0.885196i	$-0.654024\pi$
$431$		−	328870.i		−	1.77039i	0.465219	−	0.885196i	$-0.345976\pi$
$432$		0			0
$433$	−72397.6			−0.386143			−0.193072	−	0.981185i	$-0.561845\pi$
$433$	−72397.6			−0.386143			−0.193072	+	0.981185i	$0.561845\pi$
$434$		0			0
$435$	−35712.6	−	14620.8i	−0.188731	−	0.0772669i
$436$		0			0
$437$	−149312.	+	177943.i	−0.781864	+	0.931789i
$438$		0			0
$439$	−337287.	−	122762.i	−1.75013	−	0.636995i	−0.750417	−	0.660965i	$-0.770147\pi$
$439$	−337287.	−	122762.i	−1.75013	−	0.636995i	−0.999713	+	0.0239702i	$0.992369\pi$
$440$		0			0
$441$	−157513.	−	71877.7i	−0.809912	−	0.369587i
$442$		0			0
$443$	110730.	+	131963.i	0.564232	+	0.672425i	0.970436	−	0.241357i	$-0.0775925\pi$
$443$	110730.	+	131963.i	0.564232	+	0.672425i	−0.406205	+	0.913782i	$0.633148\pi$
$444$		0			0
$445$	−5420.33	+	30740.2i	−0.0273720	+	0.155234i
$446$		0			0
$447$	−13449.0	+	340133.i	−0.0673093	+	1.70229i
$448$		0			0
$449$	−118104.	−	68187.2i	−0.585829	−	0.338228i	0.177618	−	0.984100i	$-0.443161\pi$
$449$	−118104.	−	68187.2i	−0.585829	−	0.338228i	−0.763446	+	0.645871i	$0.776494\pi$
$450$		0			0
$451$	9118.57	+	15793.8i	0.0448305	+	0.0776487i
$452$		0			0
$453$	−110014.	−	99979.5i	−0.536105	−	0.487208i
$454$		0			0
$455$	3076.62	+	8452.94i	0.0148611	+	0.0408305i
$456$		0			0
$457$	−27641.6	−	156763.i	−0.132352	−	0.750605i	−0.976667	−	0.214758i	$-0.931104\pi$
$457$	−27641.6	−	156763.i	−0.132352	−	0.750605i	0.844315	−	0.535847i	$-0.180007\pi$
$458$		0			0
$459$	30.8587	−	259.059i	0.000146471	−	0.00122963i
$460$		0			0
$461$	26862.1	−	4736.51i	0.126397	−	0.0222872i	−0.110092	−	0.993921i	$-0.535114\pi$
$461$	26862.1	−	4736.51i	0.126397	−	0.0222872i	0.236489	+	0.971634i	$0.424003\pi$
$462$		0			0
$463$	−376701.	+	137108.i	−1.75726	+	0.639589i	−0.999910	−	0.0134236i	$-0.995727\pi$
$463$	−376701.	+	137108.i	−1.75726	+	0.639589i	−0.757347	+	0.653013i	$0.773505\pi$
$464$		0			0
$465$	14321.1	+	3113.16i	0.0662322	+	0.0143978i
$466$		0			0
$467$	−10596.7	+	6117.99i	−0.0485887	+	0.0280527i	−0.524098	−	0.851658i	$-0.675597\pi$
$467$	−10596.7	+	6117.99i	−0.0485887	+	0.0280527i	0.475509	+	0.879711i	$0.342264\pi$
$468$		0			0
$469$	92031.6	−	159403.i	0.418400	−	0.724689i
$470$		0			0
$471$	−24580.1	+	38935.6i	−0.110801	+	0.175511i
$472$		0			0
$473$	57954.6	+	10219.0i	0.259039	+	0.0456756i
$474$		0			0
$475$	154930.	−	130002.i	0.686672	−	0.576186i
$476$		0			0
$477$	−31019.5	−	112070.i	−0.136332	−	0.492553i
$478$		0			0
$479$	113858.	−	312822.i	0.496241	−	1.36341i	−0.398641	−	0.917107i	$-0.630518\pi$
$479$	113858.	−	312822.i	0.496241	−	1.36341i	0.894882	−	0.446304i	$-0.147260\pi$
$480$		0			0
$481$	25164.7	+	21115.7i	0.108768	+	0.0912674i
$482$		0			0
$483$	−425883.	+	57851.7i	−1.82556	+	0.247983i
$484$		0			0
$485$		−	4885.63i		−	0.0207700i
$486$		0			0
$487$	308646.			1.30137			0.650687	−	0.759346i	$-0.274481\pi$
$487$	308646.			1.30137			0.650687	+	0.759346i	$0.274481\pi$
$488$		0			0
$489$	−43631.6	−	321199.i	−0.182466	−	1.34325i
$490$		0			0
$491$	−232265.	+	276803.i	−0.963433	+	1.14817i	0.0254796	+	0.999675i	$0.491889\pi$
$491$	−232265.	+	276803.i	−0.963433	+	1.14817i	−0.988913	+	0.148499i	$0.952556\pi$
$492$		0			0
$493$	−515.935	−	187.785i	−0.00212276	−	0.000772623i
$494$		0			0
$495$	7859.43	−	2175.38i	0.0320760	−	0.00887820i
$496$		0			0
$497$	224645.	+	267721.i	0.909459	+	1.08385i
$498$		0			0
$499$	−43062.5	+	244220.i	−0.172941	+	0.980797i	0.767553	+	0.640986i	$0.221474\pi$
$499$	−43062.5	+	244220.i	−0.172941	+	0.980797i	−0.940494	+	0.339811i	$0.889637\pi$
$500$		0			0
$501$	153574.	+	96951.9i	0.611847	+	0.386261i
$502$		0			0
$503$	−92432.5	−	53365.9i	−0.365333	−	0.210925i	0.306085	−	0.952004i	$-0.400981\pi$
$503$	−92432.5	−	53365.9i	−0.365333	−	0.210925i	−0.671417	+	0.741079i	$0.734314\pi$
$504$		0			0
$505$	−23691.6	−	41035.1i	−0.0928993	−	0.160906i
$506$		0			0
$507$	−50238.9	+	231108.i	−0.195445	+	0.899081i
$508$		0			0
$509$	−62475.8	−	171651.i	−0.241144	−	0.662537i	−0.999937	−	0.0112058i	$-0.996433\pi$
$509$	−62475.8	−	171651.i	−0.241144	−	0.662537i	0.758793	−	0.651331i	$-0.225789\pi$
$510$		0			0
$511$	−102189.	−	579545.i	−0.391349	−	2.21945i
$512$		0			0
$513$	−143075.	+	191302.i	−0.543663	+	0.726915i
$514$		0			0
$515$	−2209.56	+	389.604i	−0.00833087	+	0.00146896i
$516$		0			0
$517$	−44866.8	+	16330.2i	−0.167859	+	0.0610956i
$518$		0			0
$519$	192031.	−	211304.i	0.712913	−	0.784462i
$520$		0			0
$521$	346945.	−	200309.i	1.27816	−	0.737945i	0.301649	−	0.953419i	$-0.402463\pi$
$521$	346945.	−	200309.i	1.27816	−	0.737945i	0.976510	+	0.215474i	$0.0691297\pi$
$522$		0			0
$523$	−1640.33	+	2841.14i	−0.00599692	+	0.0103870i	−0.869008	−	0.494798i	$-0.835242\pi$
$523$	−1640.33	+	2841.14i	−0.00599692	+	0.0103870i	0.863011	+	0.505184i	$0.168576\pi$
$524$		0			0
$525$	373919.	+	14784.9i	1.35662	+	0.0536414i
$526$		0			0
$527$	205.349	+	36.2085i	0.000739386	+	0.000130374i
$528$		0			0
$529$	−170556.	+	143114.i	−0.609475	+	0.511410i
$530$		0			0
$531$	51762.6	−	113432.i	0.183581	−	0.402298i
$532$		0			0
$533$	−8272.54	+	22728.6i	−0.0291195	+	0.0800053i
$534$		0			0
$535$	237.638	+	199.402i	0.000830247	+	0.000696660i
$536$		0			0
$537$	−187280.	+	457447.i	−0.649446	+	1.58632i
$538$		0			0
$539$		−	77000.2i		−	0.265042i
$540$		0			0
$541$	197783.			0.675764			0.337882	−	0.941189i	$-0.390290\pi$
$541$	197783.			0.675764			0.337882	+	0.941189i	$0.390290\pi$
$542$		0			0
$543$	146817.	−	113619.i	0.497941	−	0.385348i
$544$		0			0
$545$	−15298.3	+	18231.9i	−0.0515053	+	0.0613816i
$546$		0			0
$547$	−17508.9	−	6372.72i	−0.0585173	−	0.0212986i	0.312596	−	0.949886i	$-0.398802\pi$
$547$	−17508.9	−	6372.72i	−0.0585173	−	0.0212986i	−0.371113	+	0.928588i	$0.621024\pi$
$548$		0			0
$549$	−414569.	−	32835.8i	−1.37547	−	0.108944i
$550$		0			0
$551$	323154.	+	385120.i	1.06441	+	1.26851i
$552$		0			0
$553$	54869.5	−	311180.i	0.179424	−	1.01756i
$554$		0			0
$555$	−15307.6	+	8048.85i	−0.0496961	+	0.0261305i
$556$		0			0
$557$	464794.	+	268349.i	1.49813	+	0.864948i	0.999998	−	0.00215163i	$-0.000684886\pi$
$557$	464794.	+	268349.i	1.49813	+	0.864948i	0.498135	+	0.867099i	$0.334018\pi$
$558$		0			0
$559$	39024.5	+	67592.4i	0.124886	+	0.216309i
$560$		0			0
$561$	110.513	−	35.3449i	0.000351144	−	0.000112305i
$562$		0			0
$563$	−105660.	−	290297.i	−0.333343	−	0.915853i	−0.987236	−	0.159266i	$-0.949087\pi$
$563$	−105660.	−	290297.i	−0.333343	−	0.915853i	0.653892	−	0.756588i	$-0.273135\pi$
$564$		0			0
$565$	−3924.75	−	22258.4i	−0.0122946	−	0.0697263i
$566$		0			0
$567$	−433967.	+	83904.5i	−1.34987	+	0.260987i
$568$		0			0
$569$	111616.	−	19680.9i	0.344748	−	0.0607884i	0.00140673	−	0.999999i	$-0.499552\pi$
$569$	111616.	−	19680.9i	0.344748	−	0.0607884i	0.343342	+	0.939211i	$0.388441\pi$
$570$		0			0
$571$	167605.	−	61003.4i	0.514063	−	0.187103i	−0.0719457	−	0.997409i	$-0.522921\pi$
$571$	167605.	−	61003.4i	0.514063	−	0.187103i	0.586008	+	0.810305i	$0.300699\pi$
$572$		0			0
$573$	175420.	+	548484.i	0.534281	+	1.67053i
$574$		0			0
$575$	378888.	−	218751.i	1.14598	−	0.661629i
$576$		0			0
$577$	−33755.5	+	58466.2i	−0.101389	+	0.175612i	−0.912257	−	0.409618i	$-0.865662\pi$
$577$	−33755.5	+	58466.2i	−0.101389	+	0.175612i	0.810868	+	0.585229i	$0.198995\pi$
$578$		0			0
$579$	141369.	+	268860.i	0.421692	+	0.801992i
$580$		0			0
$581$	633520.	+	111707.i	1.87676	+	0.330923i
$582$		0			0
$583$	39616.3	−	33242.0i	0.116556	−	0.0978025i
$584$		0			0
$585$	8910.36	+	6130.48i	0.0260366	+	0.0179136i
$586$		0			0
$587$	−225369.	+	619195.i	−0.654059	+	1.79701i	−0.0518795	+	0.998653i	$0.516521\pi$
$587$	−225369.	+	619195.i	−0.654059	+	1.79701i	−0.602180	+	0.798360i	$0.705701\pi$
$588$		0			0
$589$	−146261.	−	122727.i	−0.421597	−	0.353762i
$590$		0			0
$591$	−50160.5	−	64816.7i	−0.143611	−	0.185572i
$592$		0			0
$593$			27873.9i			0.0792664i	0.999214	+	0.0396332i	$0.0126189\pi$
$593$			27873.9i			0.0792664i	−0.999214	+	0.0396332i	$0.987381\pi$
$594$		0			0
$595$	−67.3809			−0.000190328
$596$		0			0
$597$	150310.	+	61537.4i	0.421735	+	0.172659i
$598$		0			0
$599$	−354548.	+	422534.i	−0.988147	+	1.17763i	−0.00405057	+	0.999992i	$0.501289\pi$
$599$	−354548.	+	422534.i	−0.988147	+	1.17763i	−0.984096	+	0.177636i	$0.943155\pi$
$600$		0			0
$601$	−251293.	−	91463.1i	−0.695715	−	0.253219i	−0.0301346	−	0.999546i	$-0.509594\pi$
$601$	−251293.	−	91463.1i	−0.695715	−	0.253219i	−0.665580	+	0.746326i	$0.731816\pi$
$602$		0			0
$603$	−21101.3	−	220299.i	−0.0580328	−	0.605868i
$604$		0			0
$605$	−23970.7	−	28567.2i	−0.0654892	−	0.0780470i
$606$		0			0
$607$	−74129.4	+	420409.i	−0.201193	+	1.14102i	0.702125	+	0.712053i	$0.252235\pi$
$607$	−74129.4	+	420409.i	−0.201193	+	1.14102i	−0.903319	+	0.428970i	$0.858877\pi$
$608$		0			0
$609$	−36751.8	+	929475.i	−0.0990934	+	2.50613i
$610$		0			0
$611$	−54840.4	−	31662.1i	−0.146899	−	0.0848121i
$612$		0			0
$613$	−269887.	−	467458.i	−0.718226	−	1.24400i	−0.961702	−	0.274097i	$-0.911621\pi$
$613$	−269887.	−	467458.i	−0.718226	−	1.24400i	0.243476	−	0.969907i	$-0.421712\pi$
$614$		0			0
$615$	−9423.75	−	8564.23i	−0.0249157	−	0.0226432i
$616$		0			0
$617$	−184549.	−	507044.i	−0.484776	−	1.33191i	−0.905355	−	0.424655i	$-0.860395\pi$
$617$	−184549.	−	507044.i	−0.484776	−	1.33191i	0.420579	−	0.907256i	$-0.361827\pi$
$618$		0			0
$619$	12734.3	+	72219.7i	0.0332348	+	0.188484i	0.996906	−	0.0786093i	$-0.0250480\pi$
$619$	12734.3	+	72219.7i	0.0332348	+	0.188484i	−0.963671	+	0.267093i	$0.913937\pi$
$620$		0			0
$621$	−376659.	+	353793.i	−0.976709	+	0.917417i
$622$		0			0
$623$	740993.	−	130657.i	1.90914	−	0.336633i
$624$		0			0
$625$	−353363.	+	128613.i	−0.904608	+	0.329250i
$626$		0			0
$627$	−103816.	−	22567.9i	−0.264077	−	0.0574059i
$628$		0			0
$629$	−213.100	+	123.033i	−0.000538620	+	0.000310972i
$630$		0			0
$631$	331505.	−	574184.i	0.832591	−	1.44209i	−0.0633861	−	0.997989i	$-0.520190\pi$
$631$	331505.	−	574184.i	0.832591	−	1.44209i	0.895977	−	0.444101i	$-0.146477\pi$
$632$		0			0
$633$	206695.	−	327410.i	0.515849	−	0.817118i
$634$		0			0
$635$	−82606.2	−	14565.7i	−0.204864	−	0.0361230i
$636$		0			0
$637$	78230.6	−	65643.3i	0.192796	−	0.161775i
$638$		0			0
$639$	406764.	+	105413.i	0.996187	+	0.258163i
$640$		0			0
$641$	−148039.	+	406733.i	−0.360296	+	0.989904i	0.618629	+	0.785683i	$0.287688\pi$
$641$	−148039.	+	406733.i	−0.360296	+	0.989904i	−0.978925	+	0.204221i	$0.934534\pi$
$642$		0			0
$643$	240171.	+	201528.i	0.580897	+	0.487431i	0.885242	−	0.465131i	$-0.153993\pi$
$643$	240171.	+	201528.i	0.580897	+	0.487431i	−0.304344	+	0.952562i	$0.598437\pi$
$644$		0			0
$645$	−40716.7	+	5530.93i	−0.0978708	+	0.0132947i
$646$		0			0
$647$		−	417006.i		−	0.996170i	−0.867128	−	0.498085i	$-0.834037\pi$
$647$		−	417006.i		−	0.996170i	0.867128	−	0.498085i	$-0.165963\pi$
$648$		0			0
$649$	55451.5			0.131651
$650$		0			0
$651$	−47551.5	−	350056.i	−0.112202	−	0.825992i
$652$		0			0
$653$	−165991.	+	197820.i	−0.389276	+	0.463921i	−0.924719	−	0.380650i	$-0.875700\pi$
$653$	−165991.	+	197820.i	−0.389276	+	0.463921i	0.535443	+	0.844571i	$0.320145\pi$
$654$		0			0
$655$	−25856.4	−	9410.96i	−0.0602678	−	0.0219357i
$656$		0			0
$657$	−504421.	−	496189.i	−1.16859	−	1.14952i
$658$		0			0
$659$	−187100.	−	222978.i	−0.430828	−	0.513441i	0.506333	−	0.862338i	$-0.331001\pi$
$659$	−187100.	−	222978.i	−0.430828	−	0.513441i	−0.937161	+	0.348897i	$0.886556\pi$
$660$		0			0
$661$	−31474.8	+	178503.i	−0.0720378	+	0.408547i	0.927370	+	0.374145i	$0.122064\pi$
$661$	−31474.8	+	178503.i	−0.0720378	+	0.408547i	−0.999408	+	0.0344018i	$0.989047\pi$
$662$		0			0
$663$	130.122	+	82.1466i	0.000296023	+	0.000186880i
$664$		0			0
$665$	53431.9	+	30848.9i	0.120825	+	0.0697584i
$666$		0			0
$667$	543764.	+	941827.i	1.22225	+	2.11699i
$668$		0			0
$669$	28359.4	−	130458.i	0.0633644	−	0.291487i
$670$		0			0
$671$	−63256.8	−	173797.i	−0.140495	−	0.386008i
$672$		0			0
$673$	−77824.8	−	441366.i	−0.171826	−	0.974472i	−0.941744	−	0.336330i	$-0.890814\pi$
$673$	−77824.8	−	441366.i	−0.171826	−	0.974472i	0.769919	−	0.638142i	$-0.220297\pi$
$674$		0			0
$675$	376456.	−	246412.i	0.826241	−	0.540822i
$676$		0			0
$677$	−27774.3	+	4897.36i	−0.0605991	+	0.0106853i	−0.203866	−	0.978999i	$-0.565351\pi$
$677$	−27774.3	+	4897.36i	−0.0605991	+	0.0106853i	0.143266	+	0.989684i	$0.454239\pi$
$678$		0			0
$679$	−110666.	+	40279.1i	−0.240035	+	0.0873654i
$680$		0			0
$681$	356743.	−	392546.i	0.769238	−	0.846440i
$682$		0			0
$683$	−202638.	+	116993.i	−0.434390	+	0.250795i	−0.701215	−	0.712950i	$-0.747359\pi$
$683$	−202638.	+	116993.i	−0.434390	+	0.250795i	0.266825	+	0.963745i	$0.414025\pi$
$684$		0			0
$685$	−38557.5	+	66783.5i	−0.0821727	+	0.142327i
$686$		0			0
$687$	356772.	+	14106.9i	0.755922	+	0.0298895i
$688$		0			0
$689$	67546.3	+	11910.2i	0.142286	+	0.0250889i
$690$		0			0
$691$	156806.	−	131576.i	0.328403	−	0.275563i	−0.463646	−	0.886021i	$-0.653459\pi$
$691$	156806.	−	131576.i	0.328403	−	0.275563i	0.792049	+	0.610458i	$0.209014\pi$
$692$		0			0
$693$	−114071.	−	160091.i	−0.237525	−	0.333351i
$694$		0			0
$695$	23549.2	−	64701.0i	0.0487537	−	0.133950i
$696$		0			0
$697$	−138.789	−	116.458i	−0.000285687	−	0.000239720i
$698$		0			0
$699$	223160.	−	545086.i	0.456732	−	1.11561i
$700$		0			0
$701$		−	299004.i		−	0.608473i	−0.952597	−	0.304237i	$-0.901599\pi$
$701$		−	299004.i		−	0.608473i	0.952597	−	0.304237i	$-0.0984013\pi$
$702$		0			0
$703$	225313.			0.455906
$704$		0			0
$705$	26365.5	−	20403.8i	0.0530466	−	0.0410519i
$706$		0			0
$707$	−734174.	+	874955.i	−1.46879	+	1.75044i
$708$		0			0
$709$	−374736.	−	136393.i	−0.745474	−	0.271330i	−0.0587740	−	0.998271i	$-0.518719\pi$
$709$	−374736.	−	136393.i	−0.745474	−	0.271330i	−0.686700	+	0.726941i	$0.740941\pi$
$710$		0			0
$711$	−163387.	−	342990.i	−0.323206	−	0.678488i
$712$		0			0
$713$	−265485.	−	316392.i	−0.522228	−	0.622367i
$714$		0			0
$715$	−835.260	+	4737.00i	−0.00163384	+	0.00926597i
$716$		0			0
$717$	−348191.	+	183081.i	−0.677297	+	0.356127i
$718$		0			0
$719$	−664619.	−	383718.i	−1.28563	−	0.742257i	−0.307756	−	0.951465i	$-0.599578\pi$
$719$	−664619.	−	383718.i	−1.28563	−	0.742257i	−0.977871	+	0.209208i	$0.932911\pi$
$720$		0			0
$721$	27041.5	+	46837.2i	0.0520187	+	0.0900991i
$722$		0			0
$723$	−422043.	+	134981.i	−0.807383	+	0.258223i
$724$		0			0
$725$	−323853.	−	889780.i	−0.616130	−	1.69280i
$726$		0			0
$727$	−2474.26	−	14032.3i	−0.00468142	−	0.0265496i	0.982378	−	0.186907i	$-0.0598464\pi$
$727$	−2474.26	−	14032.3i	−0.00468142	−	0.0265496i	−0.987059	+	0.160358i	$0.948735\pi$
$728$		0			0
$729$	−366398.	+	384945.i	−0.689442	+	0.724341i
$730$		0			0
$731$	−575.750	+	101.520i	−0.00107745	+	0.000189984i
$732$		0			0
$733$	−100373.	+	36532.6i	−0.186813	+	0.0679944i	−0.433733	−	0.901042i	$-0.642804\pi$
$733$	−100373.	+	36532.6i	−0.186813	+	0.0679944i	0.246920	+	0.969036i	$0.420582\pi$
$734$		0			0
$735$	16378.2	+	51209.5i	0.0303173	+	0.0947929i
$736$		0			0
$737$	85236.8	−	49211.5i	0.156925	−	0.0906007i
$738$		0			0
$739$	−276629.	+	479136.i	−0.506535	+	0.877344i	0.493436	+	0.869782i	$0.335741\pi$
$739$	−276629.	+	479136.i	−0.506535	+	0.877344i	−0.999971	+	0.00756237i	$0.997593\pi$
$740$		0			0
$741$	−65575.8	−	124715.i	−0.119428	−	0.227134i
$742$		0			0
$743$	192790.	+	33994.1i	0.349227	+	0.0615781i	0.345510	−	0.938415i	$-0.387706\pi$
$743$	192790.	+	33994.1i	0.349227	+	0.0615781i	0.00371646	+	0.999993i	$0.498817\pi$
$744$		0			0
$745$	80974.6	−	67945.8i	0.145894	−	0.122419i
$746$		0			0
$747$	698279.	−	332634.i	1.25138	−	0.596108i
$748$		0			0
$749$	2557.52	−	7026.74i	0.00455886	−	0.0125254i
$750$		0			0
$751$	−284195.	−	238468.i	−0.503891	−	0.422815i	0.355082	−	0.934835i	$-0.384453\pi$
$751$	−284195.	−	238468.i	−0.503891	−	0.422815i	−0.858974	+	0.512020i	$0.828897\pi$
$752$		0			0
$753$	−354268.	−	457779.i	−0.624801	−	0.807358i
$754$		0			0
$755$			46162.8i			0.0809838i
$756$		0			0
$757$	−467363.			−0.815572			−0.407786	−	0.913078i	$-0.633699\pi$
$757$	−467363.			−0.815572			−0.407786	+	0.913078i	$0.633699\pi$
$758$		0			0
$759$	−212687.	−	87074.7i	−0.369197	−	0.151150i
$760$		0			0
$761$	−193920.	+	231105.i	−0.334852	+	0.399061i	−0.907028	−	0.421069i	$-0.861655\pi$
$761$	−193920.	+	231105.i	−0.334852	+	0.399061i	0.572176	+	0.820131i	$0.306099\pi$
$762$		0			0
$763$	539100.	+	196216.i	0.926020	+	0.337044i
$764$		0			0
$765$	−65.9791	+	47.0127i	−0.000112741	+	8.03327e-5i
$766$		0			0
$767$	47272.9	+	56337.6i	0.0803565	+	0.0957652i
$768$		0			0
$769$	−43909.6	+	249024.i	−0.0742518	+	0.421103i	0.924911	+	0.380185i	$0.124140\pi$
$769$	−43909.6	+	249024.i	−0.0742518	+	0.421103i	−0.999162	+	0.0409186i	$0.986972\pi$
$770$		0			0
$771$	7544.31	−	190800.i	0.0126914	−	0.320974i
$772$		0			0
$773$	−84150.9	−	48584.5i	−0.140831	−	0.0813091i	0.427929	−	0.903812i	$-0.359243\pi$
$773$	−84150.9	−	48584.5i	−0.140831	−	0.0813091i	−0.568760	+	0.822503i	$0.692577\pi$
$774$		0			0
$775$	179803.	+	311429.i	0.299361	+	0.518508i
$776$		0			0
$777$	308519.	+	280379.i	0.511022	+	0.464413i
$778$		0			0
$779$	56739.7	+	155891.i	0.0935001	+	0.256889i
$780$		0			0
$781$	32451.0	+	184039.i	0.0532018	+	0.301722i
$782$		0			0
$783$	612522.	+	935781.i	0.999075	+	1.52634i
$784$		0			0
$785$	14081.3	−	2482.91i	0.0228509	−	0.00402923i
$786$		0			0
$787$	−813299.	+	296017.i	−1.31311	+	0.477933i	−0.901244	−	0.433312i	$-0.857345\pi$
$787$	−813299.	+	296017.i	−1.31311	+	0.477933i	−0.411865	+	0.911245i	$0.635123\pi$
$788$		0			0
$789$	474267.	+	103098.i	0.761849	+	0.165613i
$790$		0			0
$791$	−471823.	+	272407.i	−0.754096	+	0.435377i
$792$		0			0
$793$	122647.	−	212431.i	0.195034	−	0.337809i
$794$		0			0
$795$	−19276.4	+	30534.3i	−0.0304994	+	0.0483119i
$796$		0			0
$797$	92570.1	+	16322.6i	0.145732	+	0.0256964i	0.246038	−	0.969260i	$-0.420871\pi$
$797$	92570.1	+	16322.6i	0.145732	+	0.0256964i	−0.100306	+	0.994957i	$0.531982\pi$
$798$		0			0
$799$	363.361	−	304.896i	0.000569174	−	0.000477594i
$800$		0			0
$801$	634416.	−	644941.i	0.988801	−	1.00521i
$802$		0			0
$803$	107626.	−	295700.i	0.166911	−	0.458585i
$804$		0			0
$805$	102240.	+	85789.7i	0.157772	+	0.132386i
$806$		0			0
$807$	−609647.	+	82814.1i	−0.936119	+	0.127162i
$808$		0			0
$809$			68486.9i			0.104643i	0.998630	+	0.0523215i	$0.0166621\pi$
$809$			68486.9i			0.104643i	−0.998630	+	0.0523215i	$0.983338\pi$
$810$		0			0
$811$	1.17479e6			1.78615			0.893076	−	0.449907i	$-0.148543\pi$
$811$	1.17479e6			1.78615			0.893076	+	0.449907i	$0.148543\pi$
$812$		0			0
$813$	−56194.7	−	413684.i	−0.0850186	−	0.625875i
$814$		0			0
$815$	−64702.3	+	77109.2i	−0.0974102	+	0.116089i
$816$		0			0
$817$	503038.	+	183091.i	0.753628	+	0.274298i
$818$		0			0
$819$	65402.7	−	252373.i	0.0975053	−	0.376249i
$820$		0			0
$821$	−247025.	−	294393.i	−0.366484	−	0.436759i	0.551016	−	0.834495i	$-0.314240\pi$
$821$	−247025.	−	294393.i	−0.366484	−	0.436759i	−0.917500	+	0.397736i	$0.869796\pi$
$822$		0			0
$823$	−104830.	+	594519.i	−0.154769	+	0.877741i	0.804227	+	0.594322i	$0.202580\pi$
$823$	−104830.	+	594519.i	−0.154769	+	0.877741i	−0.958996	+	0.283419i	$0.908531\pi$
$824$		0			0
$825$	169203.	+	106818.i	0.248600	+	0.156942i
$826$		0			0
$827$	−485502.	−	280305.i	−0.709873	−	0.409845i	0.101141	−	0.994872i	$-0.467751\pi$
$827$	−485502.	−	280305.i	−0.709873	−	0.409845i	−0.811014	+	0.585027i	$0.801084\pi$
$828$		0			0
$829$	−272233.	−	471522.i	−0.396125	−	0.686109i	0.597119	−	0.802153i	$-0.296312\pi$
$829$	−272233.	−	471522.i	−0.396125	−	0.686109i	−0.993244	+	0.116044i	$0.962979\pi$
$830$		0			0
$831$	−59829.3	+	275225.i	−0.0866387	+	0.398553i
$832$		0			0
$833$	261.631	+	718.825i	0.000377050	+	0.00103594i
$834$		0			0
$835$	−9793.40	−	55541.2i	−0.0140463	−	0.0796603i
$836$		0			0
$837$	−290802.	−	309596.i	−0.415094	−	0.441921i
$838$		0			0
$839$	−223372.	+	39386.5i	−0.317326	+	0.0559531i	−0.330043	−	0.943966i	$-0.607063\pi$
$839$	−223372.	+	39386.5i	−0.317326	+	0.0559531i	0.0127169	+	0.999919i	$0.495952\pi$
$840$		0			0
$841$	1.54716e6	−	563119.i	2.18747	−	0.796174i
$842$		0			0
$843$	780335.	−	858651.i	1.09806	−	1.20826i
$844$		0			0
$845$	63603.2	−	36721.3i	0.0890769	−	0.0514286i
$846$		0			0
$847$	−449459.	+	778485.i	−0.626503	+	1.08513i
$848$		0			0
$849$	447113.	+	17679.0i	0.620300	+	0.0245269i
$850$		0			0
$851$	479993.	+	84635.8i	0.662790	+	0.116868i
$852$		0			0
$853$	−1.09396e6	+	917945.i	−1.50351	+	1.26159i	−0.628171	+	0.778075i	$0.716196\pi$
$853$	−1.09396e6	+	917945.i	−1.50351	+	1.26159i	−0.875335	+	0.483517i	$0.839359\pi$
$854$		0			0
$855$	73844.1	−	7073.12i	0.101014	−	0.00967562i
$856$		0			0
$857$	−318552.	+	875214.i	−0.433729	+	1.19166i	0.509778	+	0.860306i	$0.329728\pi$
$857$	−318552.	+	875214.i	−0.433729	+	1.19166i	−0.943506	+	0.331354i	$0.892495\pi$
$858$		0			0
$859$	782685.	+	656751.i	1.06072	+	0.890050i	0.994180	−	0.107734i	$-0.0343595\pi$
$859$	782685.	+	656751.i	1.06072	+	0.890050i	0.0665401	+	0.997784i	$0.478804\pi$
$860$		0			0
$861$	−116298.	+	284067.i	−0.156879	+	0.383190i
$862$		0			0
$863$			181110.i			0.243175i	0.992581	+	0.121588i	$0.0387986\pi$
$863$			181110.i			0.243175i	−0.992581	+	0.121588i	$0.961201\pi$
$864$		0			0
$865$	−88665.1			−0.118501
$866$		0			0
$867$	594467.	−	460048.i	0.790841	−	0.612019i
$868$		0			0
$869$	108607.	−	129433.i	0.143820	−	0.171398i
$870$		0			0
$871$	122663.	+	44645.6i	0.161688	+	0.0588495i
$872$		0			0
$873$	−80260.2	+	116654.i	−0.105310	+	0.153064i
$874$		0			0
$875$	−150335.	−	179163.i	−0.196356	−	0.234008i
$876$		0			0
$877$	164581.	−	933383.i	0.213983	−	1.21356i	−0.668679	−	0.743551i	$-0.733140\pi$
$877$	164581.	−	933383.i	0.213983	−	1.21356i	0.882662	−	0.470008i	$-0.155749\pi$
$878$		0			0
$879$	961023.	−	505312.i	1.24382	−	0.654006i
$880$		0			0
$881$	1.14130e6	+	658931.i	1.47044	+	0.848962i	0.999450	−	0.0331740i	$-0.0105616\pi$
$881$	1.14130e6	+	658931.i	1.47044	+	0.848962i	0.470995	+	0.882136i	$0.343895\pi$
$882$		0			0
$883$	270281.	+	468140.i	0.346652	+	0.600419i	0.985653	−	0.168787i	$-0.0539851\pi$
$883$	270281.	+	468140.i	0.346652	+	0.600419i	−0.639000	+	0.769207i	$0.720652\pi$
$884$		0			0
$885$	−36878.4	+	11794.7i	−0.0470853	+	0.0150592i
$886$		0			0
$887$	−11366.4	−	31228.9i	−0.0144469	−	0.0396926i	0.932260	−	0.361790i	$-0.117834\pi$
$887$	−11366.4	−	31228.9i	−0.0144469	−	0.0396926i	−0.946706	+	0.322098i	$0.895612\pi$
$888$		0			0
$889$	351106.	+	1.99122e6i	0.444257	+	2.51951i
$890$		0			0
$891$	−223396.	−	77171.4i	−0.281398	−	0.0972078i
$892$		0			0
$893$	−427730.	+	75420.3i	−0.536373	+	0.0945770i
$894$		0			0
$895$	144239.	−	52498.7i	0.180068	−	0.0655394i
$896$		0			0
$897$	−92851.4	−	290318.i	−0.115399	−	0.360818i
$898$		0			0
$899$	−774138.	+	446949.i	−0.957854	+	0.553017i
$900$		0			0
$901$	−256.883	+	444.934i	−0.000316436	+	0.000548083i
$902$		0			0
$903$	460967.	+	876686.i	0.565320	+	1.07515i
$904$		0			0
$905$	−56773.5	−	10010.7i	−0.0693184	−	0.0122227i
$906$		0			0
$907$	920499.	−	772390.i	1.11894	−	0.938906i	0.120394	−	0.992726i	$-0.461584\pi$
$907$	920499.	−	772390.i	1.11894	−	0.938906i	0.998551	+	0.0538202i	$0.0171398\pi$
$908$		0			0
$909$	−108431.	+	1.36900e6i	−0.131228	+	1.65682i
$910$		0			0
$911$	350174.	−	962095.i	0.421937	−	1.15926i	−0.528660	−	0.848833i	$-0.677306\pi$
$911$	350174.	−	962095.i	0.421937	−	1.15926i	0.950597	−	0.310428i	$-0.100472\pi$
$912$		0			0
$913$	263507.	+	221109.i	0.316119	+	0.265255i
$914$		0			0
$915$	79036.4	+	102130.i	0.0944028	+	0.121986i
$916$		0			0
$917$			663268.i			0.788770i
$918$		0			0
$919$	−1.16310e6			−1.37716			−0.688581	−	0.725160i	$-0.741766\pi$
$919$	−1.16310e6			−1.37716			−0.688581	+	0.725160i	$0.741766\pi$
$920$		0			0
$921$	−710107.	−	290720.i	−0.837153	−	0.342732i
$922$		0			0
$923$	−159315.	+	189864.i	−0.187005	+	0.222864i
$924$		0			0
$925$	−398773.	−	145142.i	−0.466061	−	0.169632i
$926$		0			0
$927$	59157.9	+	26995.5i	0.0688420	+	0.0314147i
$928$		0			0
$929$	39446.9	+	47011.0i	0.0457069	+	0.0544713i	0.788414	−	0.615146i	$-0.210903\pi$
$929$	39446.9	+	47011.0i	0.0457069	+	0.0544713i	−0.742707	+	0.669617i	$0.766458\pi$
$930$		0			0
$931$	121630.	−	689798.i	0.140327	−	0.795834i
$932$		0			0
$933$	−46441.8	+	1.17454e6i	−0.0533514	+	1.34929i
$934$		0			0
$935$	−31.2030	−	18.0151i	−3.56922e−5	−	2.06069e-5i
$936$		0			0
$937$	−727314.	−	1.25974e6i	−0.828405	−	1.43484i	−0.899289	−	0.437355i	$-0.855915\pi$
$937$	−727314.	−	1.25974e6i	−0.828405	−	1.43484i	0.0708843	−	0.997485i	$-0.477418\pi$
$938$		0			0
$939$	−1.14272e6	−	1.03849e6i	−1.29601	−	1.17780i
$940$		0			0
$941$	−515545.	−	1.41645e6i	−0.582220	−	1.59964i	−0.784377	−	0.620284i	$-0.787017\pi$
$941$	−515545.	−	1.41645e6i	−0.582220	−	1.59964i	0.202157	−	0.979353i	$-0.435205\pi$
$942$		0			0
$943$	62316.5	+	353415.i	0.0700777	+	0.397430i
$944$		0			0
$945$	109916.	+	82206.4i	0.123082	+	0.0920539i
$946$		0			0
$947$	−1.31161e6	+	231272.i	−1.46253	+	0.257883i	−0.847572	−	0.530681i	$-0.821936\pi$
$947$	−1.31161e6	+	231272.i	−1.46253	+	0.257883i	−0.614954	+	0.788563i	$0.710825\pi$
$948$		0			0
$949$	392177.	−	142741.i	0.435461	−	0.158495i
$950$		0			0
$951$	−1.14929e6	−	249835.i	−1.27077	−	0.276244i
$952$		0			0
$953$	608131.	−	351105.i	0.669594	−	0.386590i	−0.126329	−	0.991988i	$-0.540319\pi$
$953$	608131.	−	351105.i	0.669594	−	0.386590i	0.795923	+	0.605398i	$0.206986\pi$
$954$		0			0
$955$	89410.5	−	154864.i	0.0980352	−	0.169802i
$956$		0			0
$957$	−265525.	+	420599.i	−0.289923	+	0.459245i
$958$		0			0
$959$	1.83061e6	+	322787.i	1.99049	+	0.350977i
$960$		0			0
$961$	−447398.	+	375412.i	−0.484448	+	0.406501i
$962$		0			0
$963$	−2398.35	−	8664.98i	−0.00258618	−	0.00934361i
$964$		0			0
$965$	32262.0	−	88639.1i	0.0346447	−	0.0951855i
$966$		0			0
$967$	−938183.	−	787229.i	−1.00331	−	0.841876i	−0.0158695	−	0.999874i	$-0.505052\pi$
$967$	−938183.	−	787229.i	−1.00331	−	0.841876i	−0.987439	+	0.157998i	$0.949496\pi$
$968$		0			0
$969$	1045.85	−	142.067i	0.00111383	−	0.000151303i
$970$		0			0
$971$			1.35101e6i			1.43292i	0.697630	+	0.716458i	$0.254238\pi$
$971$			1.35101e6i			1.43292i	−0.697630	+	0.716458i	$0.745762\pi$
$972$		0			0
$973$	−1.65971e6			−1.75310
$974$		0			0
$975$	35721.8	+	262970.i	0.0375771	+	0.276629i
$976$		0			0
$977$	862941.	−	1.02841e6i	0.904049	−	1.07740i	−0.0926082	−	0.995703i	$-0.529520\pi$
$977$	862941.	−	1.02841e6i	0.904049	−	1.07740i	0.996657	−	0.0817007i	$-0.0260352\pi$
$978$		0			0
$979$	378075.	+	137608.i	0.394469	+	0.143575i
$980$		0			0
$981$	664788.	−	184004.i	0.690789	−	0.191201i
$982$		0			0
$983$	−186399.	−	222142.i	−0.192902	−	0.229892i	0.660920	−	0.750456i	$-0.270166\pi$
$983$	−186399.	−	222142.i	−0.192902	−	0.229892i	−0.853823	+	0.520564i	$0.825722\pi$
$984$		0			0
$985$	−4419.51	+	25064.3i	−0.00455514	+	0.0258335i
$986$		0			0
$987$	−679539.	−	428995.i	−0.697558	−	0.440370i
$988$		0			0
$989$	1.00287e6	+	579006.i	1.02530	+	0.591957i
$990$		0			0
$991$	−477578.	−	827190.i	−0.486292	−	0.842283i	0.513584	−	0.858040i	$-0.328318\pi$
$991$	−477578.	−	827190.i	−0.486292	−	0.842283i	−0.999876	+	0.0157566i	$0.994984\pi$
$992$		0			0
$993$	−297481.	+	1.36846e6i	−0.301690	+	1.38782i
$994$		0			0
$995$	−17250.3	−	47394.7i	−0.0174241	−	0.0478722i
$996$		0			0
$997$	−111523.	−	632480.i	−0.112195	−	0.636292i	−0.988101	−	0.153809i	$-0.950846\pi$
$997$	−111523.	−	632480.i	−0.112195	−	0.636292i	0.875905	−	0.482483i	$-0.160265\pi$
$998$		0			0
$999$	497725.	+	59288.1i	0.498722	+	0.0594069i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.5.k.a.41.7	yes	72
3.2	odd	2		324.5.k.a.233.6		72
27.2	odd	18	inner	108.5.k.a.29.7	✓	72
27.25	even	9		324.5.k.a.89.6		72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.5.k.a.29.7	✓	72	27.2	odd	18	inner
108.5.k.a.41.7	yes	72	1.1	even	1	trivial
324.5.k.a.89.6		72	27.25	even	9
324.5.k.a.233.6		72	3.2	odd	2

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:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	108.k (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(1.21143 + 8.91810i) q^{3} +(1.79646 - 2.14093i) q^{5} +(-63.3056 - 23.0413i) q^{7} +(-78.0649 + 21.6073i) q^{9} +O(q^{10})\)	\(q+(1.21143 + 8.91810i) q^{3} +(1.79646 - 2.14093i) q^{5} +(-63.3056 - 23.0413i) q^{7} +(-78.0649 + 21.6073i) q^{9} +(-23.1554 - 27.5956i) q^{11} +(8.29634 - 47.0509i) q^{13} +(21.2693 + 13.4274i) q^{15} +(0.309929 + 0.178937i) q^{17} +(-163.845 - 283.789i) q^{19} +(128.795 - 592.478i) q^{21} +(-242.445 - 666.113i) q^{23} +(107.174 + 607.813i) q^{25} +(-287.266 - 670.014i) q^{27} +(-1510.88 + 266.409i) q^{29} +(547.514 - 199.279i) q^{31} +(218.049 - 239.933i) q^{33} +(-163.056 + 94.1403i) q^{35} +(-343.789 + 595.460i) q^{37} +(429.655 + 16.9887i) q^{39} +(-498.566 - 87.9106i) q^{41} +(-1251.43 + 1050.07i) q^{43} +(-93.9804 + 205.948i) q^{45} +(453.321 - 1245.49i) q^{47} +(1637.42 + 1373.96i) q^{49} +(-1.22032 + 2.98074i) q^{51} +1435.60i q^{53} -100.678 q^{55} +(2332.37 - 1804.98i) q^{57} +(-989.454 + 1179.18i) q^{59} +(4824.54 + 1755.99i) q^{61} +(5439.80 + 430.858i) q^{63} +(-85.8289 - 102.287i) q^{65} +(-474.440 + 2690.68i) q^{67} +(5646.75 - 2969.10i) q^{69} +(-4492.65 - 2593.83i) q^{71} +(4367.67 + 7565.02i) q^{73} +(-5290.70 + 1692.11i) q^{75} +(830.030 + 2280.49i) q^{77} +(814.469 + 4619.08i) q^{79} +(5627.25 - 3373.54i) q^{81} +(-9403.81 + 1658.15i) q^{83} +(0.939867 - 0.342084i) q^{85} +(-4206.18 - 13151.4i) q^{87} +(-9672.46 + 5584.40i) q^{89} +(-1609.32 + 2787.43i) q^{91} +(2440.46 + 4641.37i) q^{93} +(-901.914 - 159.032i) q^{95} +(1339.14 - 1123.67i) q^{97} +(2403.89 + 1653.92i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 9 q^{5} - 102 q^{9}+O(q^{10}) \) 72 * q + 9 * q^5 - 102 * q^9	\( 72 q + 9 q^{5} - 102 q^{9} + 18 q^{11} - 225 q^{15} - 282 q^{21} - 1278 q^{23} + 441 q^{25} + 54 q^{27} + 1854 q^{29} - 1665 q^{31} - 45 q^{33} - 2673 q^{35} + 6951 q^{39} - 5472 q^{41} + 1260 q^{43} + 5553 q^{45} + 5103 q^{47} - 5904 q^{49} + 1899 q^{51} + 1107 q^{57} - 10944 q^{59} + 8352 q^{61} - 11985 q^{63} + 8757 q^{65} + 378 q^{67} + 5607 q^{69} - 19764 q^{71} + 6111 q^{73} - 3453 q^{75} - 5679 q^{77} - 5652 q^{79} - 20466 q^{81} - 20061 q^{83} + 26100 q^{85} + 40545 q^{87} + 15633 q^{89} - 6039 q^{91} + 40179 q^{93} + 48024 q^{95} - 37530 q^{97} + 12177 q^{99}+O(q^{100}) \) 72 * q + 9 * q^5 - 102 * q^9 + 18 * q^11 - 225 * q^15 - 282 * q^21 - 1278 * q^23 + 441 * q^25 + 54 * q^27 + 1854 * q^29 - 1665 * q^31 - 45 * q^33 - 2673 * q^35 + 6951 * q^39 - 5472 * q^41 + 1260 * q^43 + 5553 * q^45 + 5103 * q^47 - 5904 * q^49 + 1899 * q^51 + 1107 * q^57 - 10944 * q^59 + 8352 * q^61 - 11985 * q^63 + 8757 * q^65 + 378 * q^67 + 5607 * q^69 - 19764 * q^71 + 6111 * q^73 - 3453 * q^75 - 5679 * q^77 - 5652 * q^79 - 20466 * q^81 - 20061 * q^83 + 26100 * q^85 + 40545 * q^87 + 15633 * q^89 - 6039 * q^91 + 40179 * q^93 + 48024 * q^95 - 37530 * q^97 + 12177 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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