LMFDB - Embedded newform 108.6.i.a.13.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [108,6,Mod(13,108)]

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, 8]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("108.13");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	108.i (of order $9$, 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:	$17.3214525398$
Analytic rank:	$0$
Dimension:	$90$
Relative dimension:	$15$ over $\Q(\zeta_{9})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			13.3
Character	$\chi$	$=$	108.13
Dual form			108.6.i.a.25.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-14.2968 + 6.21296i) q^{3} +(82.0644 + 68.8602i) q^{5} +(4.01162 + 1.46011i) q^{7} +(165.798 - 177.651i) q^{9} +O(q^{10})$	\(q+(-14.2968 + 6.21296i) q^{3} +(82.0644 + 68.8602i) q^{5} +(4.01162 + 1.46011i) q^{7} +(165.798 - 177.651i) q^{9} +(339.748 - 285.083i) q^{11} +(-111.666 + 633.290i) q^{13} +(-1601.09 - 474.620i) q^{15} +(243.562 - 421.862i) q^{17} +(270.567 + 468.636i) q^{19} +(-66.4250 + 4.04908i) q^{21} +(493.552 - 179.638i) q^{23} +(1450.19 + 8224.41i) q^{25} +(-1266.65 + 3569.94i) q^{27} +(1295.10 + 7344.90i) q^{29} +(-2450.39 + 891.870i) q^{31} +(-3086.12 + 6186.62i) q^{33} +(228.667 + 396.064i) q^{35} +(3502.14 - 6065.88i) q^{37} +(-2338.13 - 9747.82i) q^{39} +(-2742.06 + 15551.0i) q^{41} +(1214.60 - 1019.17i) q^{43} +(25839.2 - 3161.92i) q^{45} +(-27318.4 - 9943.09i) q^{47} +(-12860.9 - 10791.6i) q^{49} +(-861.155 + 7544.53i) q^{51} -14618.2 q^{53} +47512.1 q^{55} +(-6779.87 - 5018.98i) q^{57} +(30754.6 + 25806.2i) q^{59} +(13868.6 + 5047.76i) q^{61} +(924.509 - 470.584i) q^{63} +(-52772.3 + 44281.2i) q^{65} +(2941.46 - 16681.8i) q^{67} +(-5940.14 + 5634.67i) q^{69} +(29691.2 - 51426.7i) q^{71} +(-9248.99 - 16019.7i) q^{73} +(-71831.0 - 108573. i) q^{75} +(1779.19 - 647.573i) q^{77} +(16175.9 + 91738.3i) q^{79} +(-4070.82 - 58908.5i) q^{81} +(-7711.19 - 43732.3i) q^{83} +(49037.3 - 17848.1i) q^{85} +(-64149.4 - 96962.3i) q^{87} +(9618.93 + 16660.5i) q^{89} +(-1372.63 + 2377.47i) q^{91} +(29491.7 - 27975.1i) q^{93} +(-10066.4 + 57089.6i) q^{95} +(54592.7 - 45808.7i) q^{97} +(5684.46 - 107623. i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 90 q - 87 q^{5} + 330 q^{9}+O(q^{10}) $ 90 * q - 87 * q^5 + 330 * q^9	$ 90 q - 87 q^{5} + 330 q^{9} - 1257 q^{11} + 531 q^{15} - 3468 q^{17} + 12894 q^{21} + 8106 q^{23} + 4959 q^{25} - 17415 q^{27} + 3468 q^{29} - 6651 q^{31} + 33624 q^{33} - 8229 q^{35} - 10545 q^{39} + 68673 q^{41} + 9459 q^{43} - 53469 q^{45} - 57087 q^{47} - 5490 q^{49} + 42831 q^{51} - 4146 q^{53} + 24624 q^{57} + 5388 q^{59} + 70110 q^{61} - 98115 q^{63} - 172425 q^{65} - 15039 q^{67} + 251037 q^{69} + 67812 q^{71} - 27009 q^{73} - 75273 q^{75} + 23991 q^{77} - 216180 q^{79} + 177822 q^{81} - 76725 q^{83} - 53100 q^{85} - 201483 q^{87} - 98814 q^{89} - 90999 q^{91} + 21765 q^{93} - 143490 q^{95} - 71739 q^{97} + 13635 q^{99}+O(q^{100}) $ 90 * q - 87 * q^5 + 330 * q^9 - 1257 * q^11 + 531 * q^15 - 3468 * q^17 + 12894 * q^21 + 8106 * q^23 + 4959 * q^25 - 17415 * q^27 + 3468 * q^29 - 6651 * q^31 + 33624 * q^33 - 8229 * q^35 - 10545 * q^39 + 68673 * q^41 + 9459 * q^43 - 53469 * q^45 - 57087 * q^47 - 5490 * q^49 + 42831 * q^51 - 4146 * q^53 + 24624 * q^57 + 5388 * q^59 + 70110 * q^61 - 98115 * q^63 - 172425 * q^65 - 15039 * q^67 + 251037 * q^69 + 67812 * q^71 - 27009 * q^73 - 75273 * q^75 + 23991 * q^77 - 216180 * q^79 + 177822 * q^81 - 76725 * q^83 - 53100 * q^85 - 201483 * q^87 - 98814 * q^89 - 90999 * q^91 + 21765 * q^93 - 143490 * q^95 - 71739 * q^97 + 13635 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$29$	$55$
$\chi(n)$	$e\left(\frac{4}{9}\right)$	$1$

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
$3$	−14.2968	+	6.21296i	−0.917142	+	0.398561i
$3$	−14.2968	+	6.21296i	−0.917142	+	0.398561i
$4$		0			0
$5$	82.0644	+	68.8602i	1.46801	+	1.23181i	0.917957	+	0.396681i	$0.129838\pi$
$5$	82.0644	+	68.8602i	1.46801	+	1.23181i	0.550056	+	0.835128i	$0.314607\pi$
$6$		0			0
$7$	4.01162	+	1.46011i	0.0309438	+	0.0112626i	0.357446	−	0.933934i	$-0.383648\pi$
$7$	4.01162	+	1.46011i	0.0309438	+	0.0112626i	−0.326502	+	0.945197i	$0.605870\pi$
$8$		0			0
$9$	165.798	−	177.651i	0.682298	−	0.731074i
$10$		0			0
$11$	339.748	−	285.083i	0.846595	−	0.710377i	−0.112442	−	0.993658i	$-0.535867\pi$
$11$	339.748	−	285.083i	0.846595	−	0.710377i	0.959037	+	0.283281i	$0.0914228\pi$
$12$		0			0
$13$	−111.666	+	633.290i	−0.183258	+	1.03931i	0.744915	+	0.667160i	$0.232490\pi$
$13$	−111.666	+	633.290i	−0.183258	+	1.03931i	−0.928173	+	0.372149i	$0.878621\pi$
$14$		0			0
$15$	−1601.09	−	474.620i	−1.83733	−	0.544650i
$16$		0			0
$17$	243.562	−	421.862i	0.204403	−	0.354037i	−0.745539	−	0.666462i	$-0.767808\pi$
$17$	243.562	−	421.862i	0.204403	−	0.354037i	0.949942	+	0.312425i	$0.101141\pi$
$18$		0			0
$19$	270.567	+	468.636i	0.171946	+	0.297818i	0.939100	−	0.343644i	$-0.111661\pi$
$19$	270.567	+	468.636i	0.171946	+	0.297818i	−0.767154	+	0.641462i	$0.778328\pi$
$20$		0			0
$21$	−66.4250	+	4.04908i	−0.0328687	+	0.00200358i
$22$		0			0
$23$	493.552	−	179.638i	0.194542	−	0.0708074i	−0.242912	−	0.970048i	$-0.578103\pi$
$23$	493.552	−	179.638i	0.194542	−	0.0708074i	0.437454	+	0.899241i	$0.355880\pi$
$24$		0			0
$25$	1450.19	+	8224.41i	0.464060	+	2.63181i
$26$		0			0
$27$	−1266.65	+	3569.94i	−0.334386	+	0.942436i
$28$		0			0
$29$	1295.10	+	7344.90i	0.285963	+	1.62178i	0.701826	+	0.712349i	$0.252368\pi$
$29$	1295.10	+	7344.90i	0.285963	+	1.62178i	−0.415863	+	0.909427i	$0.636520\pi$
$30$		0			0
$31$	−2450.39	+	891.870i	−0.457964	+	0.166685i	−0.560692	−	0.828024i	$-0.689465\pi$
$31$	−2450.39	+	891.870i	−0.457964	+	0.166685i	0.102728	+	0.994709i	$0.467243\pi$
$32$		0			0
$33$	−3086.12	+	6186.62i	−0.493318	+	0.988937i
$34$		0			0
$35$	228.667	+	396.064i	0.0315525	+	0.0546506i
$36$		0			0
$37$	3502.14	−	6065.88i	0.420561	−	0.728432i	−0.575434	−	0.817848i	$-0.695167\pi$
$37$	3502.14	−	6065.88i	0.420561	−	0.728432i	0.995994	+	0.0894160i	$0.0285001\pi$
$38$		0			0
$39$	−2338.13	−	9747.82i	−0.246155	−	1.02623i
$40$		0			0
$41$	−2742.06	+	15551.0i	−0.254752	+	1.44477i	0.541957	+	0.840406i	$0.317684\pi$
$41$	−2742.06	+	15551.0i	−0.254752	+	1.44477i	−0.796709	+	0.604364i	$0.793427\pi$
$42$		0			0
$43$	1214.60	−	1019.17i	0.100175	−	0.0840571i	−0.591324	−	0.806434i	$-0.701395\pi$
$43$	1214.60	−	1019.17i	0.100175	−	0.0840571i	0.691500	+	0.722377i	$0.256950\pi$
$44$		0			0
$45$	25839.2	−	3161.92i	1.90217	−	0.232766i
$46$		0			0
$47$	−27318.4	−	9943.09i	−1.80389	−	0.656564i	−0.997909	−	0.0646276i	$-0.979414\pi$
$47$	−27318.4	−	9943.09i	−1.80389	−	0.656564i	−0.805985	−	0.591936i	$-0.798364\pi$
$48$		0			0
$49$	−12860.9	−	10791.6i	−0.765214	−	0.642091i
$50$		0			0
$51$	−861.155	+	7544.53i	−0.0463614	+	0.406169i
$52$		0			0
$53$	−14618.2			−0.714834			−0.357417	−	0.933945i	$-0.616343\pi$
$53$	−14618.2			−0.714834			−0.357417	+	0.933945i	$0.616343\pi$
$54$		0			0
$55$	47512.1			2.11786
$56$		0			0
$57$	−6779.87	−	5018.98i	−0.276397	−	0.204611i
$58$		0			0
$59$	30754.6	+	25806.2i	1.15022	+	0.965147i	0.999725	−	0.0234639i	$-0.00746948\pi$
$59$	30754.6	+	25806.2i	1.15022	+	0.965147i	0.150493	+	0.988611i	$0.451914\pi$
$60$		0			0
$61$	13868.6	+	5047.76i	0.477209	+	0.173690i	0.569415	−	0.822050i	$-0.307170\pi$
$61$	13868.6	+	5047.76i	0.477209	+	0.173690i	−0.0922062	+	0.995740i	$0.529392\pi$
$62$		0			0
$63$	924.509	−	470.584i	0.0293467	−	0.0149378i
$64$		0			0
$65$	−52772.3	+	44281.2i	−1.54925	+	1.29998i
$66$		0			0
$67$	2941.46	−	16681.8i	0.0800526	−	0.454001i	−0.918262	−	0.395972i	$-0.870408\pi$
$67$	2941.46	−	16681.8i	0.0800526	−	0.454001i	0.998315	−	0.0580282i	$-0.0184813\pi$
$68$		0			0
$69$	−5940.14	+	5634.67i	−0.150201	+	0.142477i
$70$		0			0
$71$	29691.2	−	51426.7i	0.699008	−	1.21072i	−0.269803	−	0.962916i	$-0.586959\pi$
$71$	29691.2	−	51426.7i	0.699008	−	1.21072i	0.968811	−	0.247801i	$-0.0797080\pi$
$72$		0			0
$73$	−9248.99	−	16019.7i	−0.203136	−	0.351842i	0.746401	−	0.665496i	$-0.231780\pi$
$73$	−9248.99	−	16019.7i	−0.203136	−	0.351842i	−0.949537	+	0.313654i	$0.898447\pi$
$74$		0			0
$75$	−71831.0	−	108573.i	−1.47455	−	2.22879i
$76$		0			0
$77$	1779.19	−	647.573i	0.0341976	−	0.0124469i
$78$		0			0
$79$	16175.9	+	91738.3i	0.291610	+	1.65380i	0.680672	+	0.732589i	$0.261688\pi$
$79$	16175.9	+	91738.3i	0.291610	+	1.65380i	−0.389062	+	0.921212i	$0.627201\pi$
$80$		0			0
$81$	−4070.82	−	58908.5i	−0.0689396	−	0.997621i
$82$		0			0
$83$	−7711.19	−	43732.3i	−0.122864	−	0.696798i	−0.982553	−	0.185981i	$-0.940454\pi$
$83$	−7711.19	−	43732.3i	−0.122864	−	0.696798i	0.859689	−	0.510818i	$-0.170657\pi$
$84$		0			0
$85$	49037.3	−	17848.1i	0.736172	−	0.267945i
$86$		0			0
$87$	−64149.4	−	96962.3i	−0.908646	−	1.37342i
$88$		0			0
$89$	9618.93	+	16660.5i	0.128722	+	0.222952i	0.923182	−	0.384364i	$-0.125579\pi$
$89$	9618.93	+	16660.5i	0.128722	+	0.222952i	−0.794460	+	0.607317i	$0.792246\pi$
$90$		0			0
$91$	−1372.63	+	2377.47i	−0.0173761	+	0.0300962i
$92$		0			0
$93$	29491.7	−	27975.1i	0.353584	−	0.335401i
$94$		0			0
$95$	−10066.4	+	57089.6i	−0.114437	+	0.649005i
$96$		0			0
$97$	54592.7	−	45808.7i	0.589122	−	0.494332i	−0.298807	−	0.954314i	$-0.596589\pi$
$97$	54592.7	−	45808.7i	0.589122	−	0.494332i	0.887928	+	0.459982i	$0.152144\pi$
$98$		0			0
$99$	5684.46	−	107623.i	0.0582910	−	1.10361i
$100$		0			0
$101$	48770.0	+	17750.8i	0.475718	+	0.173147i	0.568741	−	0.822517i	$-0.307431\pi$
$101$	48770.0	+	17750.8i	0.475718	+	0.173147i	−0.0930231	+	0.995664i	$0.529653\pi$
$102$		0			0
$103$	82454.8	+	69187.8i	0.765813	+	0.642594i	0.939633	−	0.342184i	$-0.111167\pi$
$103$	82454.8	+	69187.8i	0.765813	+	0.642594i	−0.173820	+	0.984778i	$0.555611\pi$
$104$		0			0
$105$	−5729.94	−	4241.75i	−0.0507197	−	0.0375467i
$106$		0			0
$107$	−122016.			−1.03028			−0.515141	−	0.857105i	$-0.672261\pi$
$107$	−122016.			−1.03028			−0.515141	+	0.857105i	$0.672261\pi$
$108$		0			0
$109$	168909.			1.36172			0.680860	−	0.732414i	$-0.261606\pi$
$109$	168909.			1.36172			0.680860	+	0.732414i	$0.261606\pi$
$110$		0			0
$111$	−12382.4	+	108481.i	−0.0953887	+	0.835695i
$112$		0			0
$113$	−55673.6	−	46715.7i	−0.410160	−	0.344165i	0.414245	−	0.910165i	$-0.364046\pi$
$113$	−55673.6	−	46715.7i	−0.410160	−	0.344165i	−0.824405	+	0.566000i	$0.808490\pi$
$114$		0			0
$115$	52872.9	+	19244.2i	0.372811	+	0.135692i
$116$		0			0
$117$	93990.7	+	124836.i	0.634775	+	0.843093i
$118$		0			0
$119$	1593.04	−	1336.72i	0.0103124	−	0.00865314i
$120$		0			0
$121$	6190.57	−	35108.5i	0.0384386	−	0.217996i
$122$		0			0
$123$	−57414.9	−	239366.i	−0.342186	−	1.42659i
$124$		0			0
$125$	−279939.	+	484869.i	−1.60247	+	2.77555i
$126$		0			0
$127$	−158110.	−	273855.i	−0.869863	−	1.50665i	−0.862136	−	0.506677i	$-0.830874\pi$
$127$	−158110.	−	273855.i	−0.869863	−	1.50665i	−0.00772747	−	0.999970i	$-0.502460\pi$
$128$		0			0
$129$	−11032.8	+	22117.1i	−0.0583731	+	0.117018i
$130$		0			0
$131$	−185139.	+	67385.0i	−0.942581	+	0.343072i	−0.767185	−	0.641426i	$-0.778343\pi$
$131$	−185139.	+	67385.0i	−0.942581	+	0.343072i	−0.175397	+	0.984498i	$0.556121\pi$
$132$		0			0
$133$	401.152	+	2275.04i	0.00196644	+	0.0111522i
$134$		0			0
$135$	−349774.	+	205743.i	−1.65178	+	0.971609i
$136$		0			0
$137$	29295.4	+	166142.i	0.133351	+	0.756273i	0.975994	+	0.217799i	$0.0698879\pi$
$137$	29295.4	+	166142.i	0.133351	+	0.756273i	−0.842642	+	0.538474i	$0.819001\pi$
$138$		0			0
$139$	252973.	−	92074.6i	1.11055	−	0.404206i	0.279353	−	0.960189i	$-0.409880\pi$
$139$	252973.	−	92074.6i	1.11055	−	0.404206i	0.831194	+	0.555983i	$0.187658\pi$
$140$		0			0
$141$	452343.	−	27573.5i	1.91611	−	0.116800i
$142$		0			0
$143$	142602.	+	246993.i	0.583156	+	1.01006i
$144$		0			0
$145$	−399489.	+	691936.i	−1.57792	+	2.73304i
$146$		0			0
$147$	250919.	+	74381.3i	0.957722	+	0.283903i
$148$		0			0
$149$	33640.8	−	190786.i	0.124137	−	0.704015i	−0.857680	−	0.514184i	$-0.828095\pi$
$149$	33640.8	−	190786.i	0.124137	−	0.704015i	0.981817	−	0.189831i	$-0.0607940\pi$
$150$		0			0
$151$	121376.	−	101846.i	0.433201	−	0.363498i	−0.399957	−	0.916534i	$-0.630975\pi$
$151$	121376.	−	101846.i	0.433201	−	0.363498i	0.833158	+	0.553035i	$0.186531\pi$
$152$		0			0
$153$	−34562.1	−	113213.i	−0.119363	−	0.390993i
$154$		0			0
$155$	−262504.	−	95543.8i	−0.877621	−	0.319428i
$156$		0			0
$157$	371161.	+	311441.i	1.20175	+	1.00839i	0.999578	+	0.0290361i	$0.00924377\pi$
$157$	371161.	+	311441.i	1.20175	+	1.00839i	0.202170	+	0.979350i	$0.435201\pi$
$158$		0			0
$159$	208994.	−	90822.5i	0.655604	−	0.284905i
$160$		0			0
$161$	2242.23			0.00681735
$162$		0			0
$163$	16219.8			0.0478165			0.0239082	−	0.999714i	$-0.492389\pi$
$163$	16219.8			0.0478165			0.0239082	+	0.999714i	$0.492389\pi$
$164$		0			0
$165$	−679272.	+	295191.i	−1.94238	+	0.844097i
$166$		0			0
$167$	188304.	+	158006.i	0.522478	+	0.438411i	0.865495	−	0.500918i	$-0.167004\pi$
$167$	188304.	+	158006.i	0.522478	+	0.438411i	−0.343017	+	0.939329i	$0.611449\pi$
$168$		0			0
$169$	−39686.2	−	14444.6i	−0.106886	−	0.0389035i
$170$		0			0
$171$	128113.	+	29632.5i	0.335046	+	0.0774959i
$172$		0			0
$173$	548689.	−	460405.i	1.39383	−	1.16957i	0.430075	−	0.902793i	$-0.358487\pi$
$173$	548689.	−	460405.i	1.39383	−	1.16957i	0.963759	−	0.266773i	$-0.0859574\pi$
$174$		0			0
$175$	−6190.95	+	35110.6i	−0.0152814	+	0.0866649i
$176$		0			0
$177$	−600026.	−	177869.i	−1.43958	−	0.426744i
$178$		0			0
$179$	−49592.3	+	85896.3i	−0.115686	+	0.200374i	−0.918054	−	0.396456i	$-0.870240\pi$
$179$	−49592.3	+	85896.3i	−0.115686	+	0.200374i	0.802368	+	0.596830i	$0.203573\pi$
$180$		0			0
$181$	−278101.	−	481685.i	−0.630966	−	1.09286i	−0.987355	−	0.158527i	$-0.949326\pi$
$181$	−278101.	−	481685.i	−0.630966	−	1.09286i	0.356389	−	0.934338i	$-0.384008\pi$
$182$		0			0
$183$	−229639.	+	13998.1i	−0.506894	+	0.0308988i
$184$		0			0
$185$	705098.	−	256635.i	1.51468	−	0.551298i
$186$		0			0
$187$	−37515.8	−	212762.i	−0.0784530	−	0.444929i
$188$		0			0
$189$	−10293.8	+	12471.8i	−0.0209615	+	0.0253965i
$190$		0			0
$191$	−140330.	−	795850.i	−0.278334	−	1.57851i	−0.728167	−	0.685400i	$-0.759628\pi$
$191$	−140330.	−	795850.i	−0.278334	−	1.57851i	0.449833	−	0.893113i	$-0.351484\pi$
$192$		0			0
$193$	380800.	−	138600.i	0.735874	−	0.267836i	0.0532252	−	0.998583i	$-0.483050\pi$
$193$	380800.	−	138600.i	0.735874	−	0.267836i	0.682649	+	0.730746i	$0.260828\pi$
$194$		0			0
$195$	479359.	−	960953.i	0.902765	−	1.80974i
$196$		0			0
$197$	−260584.	−	451345.i	−0.478390	−	0.828596i	0.521303	−	0.853372i	$-0.325446\pi$
$197$	−260584.	−	451345.i	−0.478390	−	0.828596i	−0.999693	+	0.0247755i	$0.992113\pi$
$198$		0			0
$199$	−161176.	+	279165.i	−0.288515	+	0.499722i	−0.973455	−	0.228877i	$-0.926495\pi$
$199$	−161176.	+	279165.i	−0.288515	+	0.499722i	0.684941	+	0.728599i	$0.259828\pi$
$200$		0			0
$201$	61590.0	+	256772.i	0.107528	+	0.448289i
$202$		0			0
$203$	−5528.89	+	31355.9i	−0.00941668	+	0.0534047i
$204$		0			0
$205$	−1.29587e6	+	1.08736e6i	−2.15366	+	1.80713i
$206$		0			0
$207$	49917.1	−	117464.i	0.0809700	−	0.190536i
$208$		0			0
$209$	225525.	+	82084.3i	0.357132	+	0.129985i
$210$		0			0
$211$	−652607.	−	547602.i	−1.00913	−	0.846757i	−0.0209033	−	0.999782i	$-0.506654\pi$
$211$	−652607.	−	547602.i	−1.00913	−	0.846757i	−0.988222	+	0.153024i	$0.951099\pi$
$212$		0			0
$213$	−104978.	+	919708.i	−0.158544	+	1.38900i
$214$		0			0
$215$	169855.			0.250601
$216$		0			0
$217$	−11132.3			−0.0160485
$218$		0			0
$219$	231761.	+	171568.i	0.326535	+	0.241727i
$220$		0			0
$221$	239964.	+	201353.i	0.330495	+	0.277318i
$222$		0			0
$223$	1.07685e6	+	391941.i	1.45008	+	0.527787i	0.942615	−	0.333883i	$-0.108359\pi$
$223$	1.07685e6	+	391941.i	1.45008	+	0.527787i	0.507469	+	0.861670i	$0.330581\pi$
$224$		0			0
$225$	1.70151e6	+	1.10597e6i	2.24068	+	1.45642i
$226$		0			0
$227$	−763955.	+	641035.i	−0.984019	+	0.825690i	−0.984691	−	0.174310i	$-0.944230\pi$
$227$	−763955.	+	641035.i	−0.984019	+	0.825690i	0.000672269		1.00000i	$0.499786\pi$
$228$		0			0
$229$	251017.	−	1.42359e6i	0.316312	−	1.79389i	−0.248457	−	0.968643i	$-0.579923\pi$
$229$	251017.	−	1.42359e6i	0.316312	−	1.79389i	0.564769	−	0.825249i	$-0.308965\pi$
$230$		0			0
$231$	−21413.4	+	20312.3i	−0.0264032	+	0.0250454i
$232$		0			0
$233$	140339.	−	243074.i	0.169351	−	0.293325i	−0.768841	−	0.639440i	$-0.779166\pi$
$233$	140339.	−	243074.i	0.169351	−	0.293325i	0.938192	+	0.346116i	$0.112500\pi$
$234$		0			0
$235$	−1.55719e6	−	2.69713e6i	−1.83938	−	3.18590i
$236$		0			0
$237$	−801231.	−	1.21107e6i	−0.926588	−	1.40054i
$238$		0			0
$239$	712225.	−	259229.i	0.806534	−	0.293554i	0.0943425	−	0.995540i	$-0.469925\pi$
$239$	712225.	−	259229.i	0.806534	−	0.293554i	0.712191	+	0.701986i	$0.247703\pi$
$240$		0			0
$241$	173449.	+	983676.i	0.192366	+	1.09096i	0.916120	+	0.400904i	$0.131304\pi$
$241$	173449.	+	983676.i	0.192366	+	1.09096i	−0.723754	+	0.690058i	$0.757585\pi$
$242$		0			0
$243$	424196.	+	816913.i	0.460841	+	0.887483i
$244$		0			0
$245$	−312313.	−	1.77121e6i	−0.332411	−	1.88519i
$246$		0			0
$247$	−326996.	+	119017.i	−0.341036	+	0.124127i
$248$		0			0
$249$	381952.	+	577324.i	0.390401	+	0.590094i
$250$		0			0
$251$	93241.9	+	161500.i	0.0934172	+	0.161803i	0.908947	−	0.416912i	$-0.136888\pi$
$251$	93241.9	+	161500.i	0.0934172	+	0.161803i	−0.815530	+	0.578715i	$0.803554\pi$
$252$		0			0
$253$	116472.	−	201735.i	0.114398	−	0.198143i
$254$		0			0
$255$	−590188.	+	559838.i	−0.568382	+	0.539153i
$256$		0			0
$257$	175984.	−	998056.i	0.166204	−	0.942588i	−0.781611	−	0.623766i	$-0.785602\pi$
$257$	175984.	−	998056.i	0.166204	−	0.942588i	0.947815	−	0.318822i	$-0.103287\pi$
$258$		0			0
$259$	22906.1	−	19220.5i	0.0212178	−	0.0178039i
$260$		0			0
$261$	1.51956e6	+	987696.i	1.38075	+	0.897474i
$262$		0			0
$263$	−1.48088e6	−	538995.i	−1.32017	−	0.480502i	−0.416656	−	0.909064i	$-0.636798\pi$
$263$	−1.48088e6	−	538995.i	−1.32017	−	0.480502i	−0.903512	+	0.428562i	$0.859020\pi$
$264$		0			0
$265$	−1.19964e6	−	1.00661e6i	−1.04939	−	0.880539i
$266$		0			0
$267$	−241031.	−	178430.i	−0.206916	−	0.153176i
$268$		0			0
$269$	−1.26419e6			−1.06520			−0.532602	−	0.846366i	$-0.678786\pi$
$269$	−1.26419e6			−1.06520			−0.532602	+	0.846366i	$0.678786\pi$
$270$		0			0
$271$	684705.			0.566344			0.283172	−	0.959069i	$-0.408613\pi$
$271$	684705.			0.566344			0.283172	+	0.959069i	$0.408613\pi$
$272$		0			0
$273$	4853.18	−	42518.4i	0.00394112	−	0.0345279i
$274$		0			0
$275$	2.83734e6	+	2.38081e6i	2.26245	+	1.89842i
$276$		0			0
$277$	−1.67343e6	−	609079.i	−1.31041	−	0.476952i	−0.410040	−	0.912068i	$-0.634485\pi$
$277$	−1.67343e6	−	609079.i	−1.31041	−	0.476952i	−0.900374	+	0.435116i	$0.856707\pi$
$278$		0			0
$279$	−247829.	+	583186.i	−0.190609	+	0.448535i
$280$		0			0
$281$	840722.	−	705450.i	0.635165	−	0.532967i	−0.267364	−	0.963596i	$-0.586153\pi$
$281$	840722.	−	705450.i	0.635165	−	0.532967i	0.902529	+	0.430629i	$0.141708\pi$
$282$		0			0
$283$	−167744.	+	951321.i	−0.124503	+	0.706092i	0.857099	+	0.515152i	$0.172265\pi$
$283$	−167744.	+	951321.i	−0.124503	+	0.706092i	−0.981602	+	0.190940i	$0.938847\pi$
$284$		0			0
$285$	−210777.	−	878743.i	−0.153713	−	0.640840i
$286$		0			0
$287$	−33706.2	+	58380.9i	−0.0241549	+	0.0418376i
$288$		0			0
$289$	591283.	+	1.02413e6i	0.416439	+	0.721293i
$290$		0			0
$291$	−495894.	+	994101.i	−0.343287	+	0.688173i
$292$		0			0
$293$	1.92886e6	−	702047.i	1.31260	−	0.477746i	0.411518	−	0.911402i	$-0.364999\pi$
$293$	1.92886e6	−	702047.i	1.31260	−	0.477746i	0.901079	+	0.433656i	$0.142777\pi$
$294$		0			0
$295$	746839.	+	4.23553e6i	0.499657	+	2.83370i
$296$		0			0
$297$	587387.	+	1.57398e6i	0.386396	+	1.03540i
$298$		0			0
$299$	58650.1	+	332621.i	0.0379394	+	0.215165i
$300$		0			0
$301$	6360.59	−	2315.07i	0.00404652	−	0.00147281i
$302$		0			0
$303$	−807541.	+	49225.4i	−0.505310	+	0.0308023i
$304$		0			0
$305$	790530.	+	1.36924e6i	0.486596	+	0.842809i
$306$		0			0
$307$	−559210.	+	968580.i	−0.338633	+	0.586529i	−0.984176	−	0.177195i	$-0.943298\pi$
$307$	−559210.	+	968580.i	−0.338633	+	0.586529i	0.645543	+	0.763724i	$0.276631\pi$
$308$		0			0
$309$	−1.60870e6	−	476878.i	−0.958472	−	0.284126i
$310$		0			0
$311$	−87131.1	+	494145.i	−0.0510825	+	0.289703i	−0.999638	−	0.0269079i	$-0.991434\pi$
$311$	−87131.1	+	494145.i	−0.0510825	+	0.289703i	0.948555	+	0.316611i	$0.102545\pi$
$312$		0			0
$313$	−991770.	+	832194.i	−0.572203	+	0.480135i	−0.882376	−	0.470545i	$-0.844057\pi$
$313$	−991770.	+	832194.i	−0.572203	+	0.480135i	0.310173	+	0.950680i	$0.399613\pi$
$314$		0			0
$315$	108274.	+	25043.7i	0.0614819	+	0.0142207i
$316$		0			0
$317$	−1.58038e6	−	575213.i	−0.883313	−	0.321500i	−0.139767	−	0.990184i	$-0.544635\pi$
$317$	−1.58038e6	−	575213.i	−0.883313	−	0.321500i	−0.743546	+	0.668685i	$0.766858\pi$
$318$		0			0
$319$	2.53391e6	+	2.12621e6i	1.39417	+	1.16985i
$320$		0			0
$321$	1.74444e6	−	758078.i	0.944915	−	0.410631i
$322$		0			0
$323$	263600.			0.140585
$324$		0			0
$325$	−5.37038e6			−2.82031
$326$		0			0
$327$	−2.41487e6	+	1.04943e6i	−1.24889	+	0.542729i
$328$		0			0
$329$	−95073.0	−	79775.8i	−0.0484248	−	0.0406332i
$330$		0			0
$331$	1.23180e6	+	448337.i	0.617973	+	0.224924i	0.631988	−	0.774978i	$-0.282239\pi$
$331$	1.23180e6	+	448337.i	0.617973	+	0.224924i	−0.0140154	+	0.999902i	$0.504461\pi$
$332$		0			0
$333$	−496961.	−	1.62787e6i	−0.245591	−	0.804469i
$334$		0			0
$335$	1.39010e6	−	1.16643e6i	0.676760	−	0.567869i
$336$		0			0
$337$	547553.	−	3.10533e6i	0.262634	−	1.48947i	−0.513054	−	0.858356i	$-0.671486\pi$
$337$	547553.	−	3.10533e6i	0.262634	−	1.48947i	0.775688	−	0.631116i	$-0.217403\pi$
$338$		0			0
$339$	1.08620e6	+	321989.i	0.513346	+	0.152174i
$340$		0			0
$341$	−578260.	+	1.00158e6i	−0.269301	+	0.466442i
$342$		0			0
$343$	−71711.4	−	124208.i	−0.0329119	−	0.0570051i
$344$		0			0
$345$	−875478.	+	53366.7i	−0.396002	+	0.0241392i
$346$		0			0
$347$	−2.15442e6	+	784146.i	−0.960522	+	0.349601i	−0.774238	−	0.632895i	$-0.781867\pi$
$347$	−2.15442e6	+	784146.i	−0.960522	+	0.349601i	−0.186284	+	0.982496i	$0.559644\pi$
$348$		0			0
$349$	−321695.	−	1.82443e6i	−0.141378	−	0.801794i	−0.970204	−	0.242288i	$-0.922102\pi$
$349$	−321695.	−	1.82443e6i	−0.141378	−	0.801794i	0.828827	−	0.559506i	$-0.189009\pi$
$350$		0			0
$351$	−2.11937e6	−	1.20080e6i	−0.918203	−	0.520239i
$352$		0			0
$353$	−89595.8	−	508123.i	−0.0382693	−	0.217036i	0.959676	−	0.281109i	$-0.0907022\pi$
$353$	−89595.8	−	508123.i	−0.0382693	−	0.217036i	−0.997945	+	0.0640726i	$0.979591\pi$
$354$		0			0
$355$	5.97784e6	−	2.17576e6i	2.51752	−	0.916304i
$356$		0			0
$357$	−14470.5	+	29008.4i	−0.00600913	+	0.0120463i
$358$		0			0
$359$	650691.	+	1.12703e6i	0.266464	+	0.461529i	0.967946	−	0.251158i	$-0.0808113\pi$
$359$	650691.	+	1.12703e6i	0.266464	+	0.461529i	−0.701482	+	0.712687i	$0.747478\pi$
$360$		0			0
$361$	1.09164e6	−	1.89077e6i	0.440869	−	0.763608i
$362$		0			0
$363$	129622.	+	540402.i	0.0516312	+	0.215253i
$364$		0			0
$365$	344108.	−	1.95154e6i	0.135196	−	0.766733i
$366$		0			0
$367$	1.00157e6	−	840413.i	0.388163	−	0.325708i	−0.427734	−	0.903905i	$-0.640688\pi$
$367$	1.00157e6	−	840413.i	0.388163	−	0.325708i	0.815897	+	0.578197i	$0.196244\pi$
$368$		0			0
$369$	2.30802e6	+	3.06546e6i	0.882418	+	1.17201i
$370$		0			0
$371$	−58642.8	−	21344.2i	−0.0221197	−	0.00805092i
$372$		0			0
$373$	1.72088e6	+	1.44399e6i	0.640439	+	0.537393i	0.904153	−	0.427208i	$-0.140503\pi$
$373$	1.72088e6	+	1.44399e6i	0.640439	+	0.537393i	−0.263714	+	0.964601i	$0.584947\pi$
$374$		0			0
$375$	989773.	−	8.67134e6i	0.363461	−	3.18426i
$376$		0			0
$377$	−4.79607e6			−1.73793
$378$		0			0
$379$	−2.30140e6			−0.822988			−0.411494	−	0.911412i	$-0.634993\pi$
$379$	−2.30140e6			−0.822988			−0.411494	+	0.911412i	$0.634993\pi$
$380$		0			0
$381$	3.96193e6	+	2.93293e6i	1.39828	+	1.03512i
$382$		0			0
$383$	62754.5	+	52657.3i	0.0218599	+	0.0183426i	0.653652	−	0.756795i	$-0.273236\pi$
$383$	62754.5	+	52657.3i	0.0218599	+	0.0183426i	−0.631792	+	0.775138i	$0.717680\pi$
$384$		0			0
$385$	190600.	+	69372.8i	0.0655347	+	0.0238527i
$386$		0			0
$387$	20321.9	−	384751.i	0.00689742	−	0.130588i
$388$		0			0
$389$	645375.	−	541534.i	0.216241	−	0.181448i	−0.528232	−	0.849100i	$-0.677145\pi$
$389$	645375.	−	541534.i	0.216241	−	0.181448i	0.744473	+	0.667652i	$0.232701\pi$
$390$		0			0
$391$	44428.0	−	251964.i	0.0146965	−	0.0833483i
$392$		0			0
$393$	2.22823e6	−	2.11365e6i	0.727746	−	0.690322i
$394$		0			0
$395$	−4.98965e6	+	8.64233e6i	−1.60908	+	2.78701i
$396$		0			0
$397$	456649.	+	790939.i	0.145414	+	0.251864i	0.929527	−	0.368753i	$-0.120215\pi$
$397$	456649.	+	790939.i	0.145414	+	0.251864i	−0.784113	+	0.620618i	$0.786882\pi$
$398$		0			0
$399$	−19870.0	−	30033.6i	−0.00624834	−	0.00944441i
$400$		0			0
$401$	−1.68898e6	+	614738.i	−0.524522	+	0.190910i	−0.590691	−	0.806898i	$-0.701145\pi$
$401$	−1.68898e6	+	614738.i	−0.524522	+	0.190910i	0.0661689	+	0.997808i	$0.478922\pi$
$402$		0			0
$403$	−291187.	−	1.65140e6i	−0.0893118	−	0.506513i
$404$		0			0
$405$	3.72238e6	−	5.11461e6i	1.12767	−	1.54944i
$406$		0			0
$407$	−539432.	−	3.05927e6i	−0.161417	−	0.915444i
$408$		0			0
$409$	−115478.	+	42030.6i	−0.0341343	+	0.0124239i	−0.359031	−	0.933326i	$-0.616893\pi$
$409$	−115478.	+	42030.6i	−0.0341343	+	0.0124239i	0.324897	+	0.945750i	$0.394671\pi$
$410$		0			0
$411$	−1.45107e6	−	2.19330e6i	−0.423723	−	0.640461i
$412$		0			0
$413$	85695.8	+	148429.i	0.0247220	+	0.0428198i
$414$		0			0
$415$	2.37860e6	−	4.11986e6i	0.677956	−	1.17425i
$416$		0			0
$417$	−3.04465e6	+	2.88808e6i	−0.857428	+	0.813335i
$418$		0			0
$419$	379555.	−	2.15256e6i	0.105618	−	0.598991i	−0.885353	−	0.464919i	$-0.846083\pi$
$419$	379555.	−	2.15256e6i	0.105618	−	0.598991i	0.990972	−	0.134072i	$-0.0428054\pi$
$420$		0			0
$421$	3.64218e6	−	3.05616e6i	1.00151	−	0.840369i	0.0143200	−	0.999897i	$-0.495442\pi$
$421$	3.64218e6	−	3.05616e6i	1.00151	−	0.840369i	0.987193	+	0.159528i	$0.0509972\pi$
$422$		0			0
$423$	−6.29575e6	+	3.20460e6i	−1.71079	+	0.870809i
$424$		0			0
$425$	3.82278e6	+	1.39138e6i	1.02661	+	0.373657i
$426$		0			0
$427$	48265.3	+	40499.4i	0.0128105	+	0.0107493i
$428$		0			0
$429$	−3.57331e6	−	2.64524e6i	−0.937406	−	0.693941i
$430$		0			0
$431$	184976.			0.0479648			0.0239824	−	0.999712i	$-0.492365\pi$
$431$	184976.			0.0479648			0.0239824	+	0.999712i	$0.492365\pi$
$432$		0			0
$433$	−4.37007e6			−1.12013			−0.560065	−	0.828449i	$-0.689224\pi$
$433$	−4.37007e6			−1.12013			−0.560065	+	0.828449i	$0.689224\pi$
$434$		0			0
$435$	1.41246e6	−	1.23745e7i	0.357893	−	3.13548i
$436$		0			0
$437$	217724.	+	182692.i	0.0545384	+	0.0457631i
$438$		0			0
$439$	2.04856e6	+	745614.i	0.507325	+	0.184651i	0.582986	−	0.812482i	$-0.301884\pi$
$439$	2.04856e6	+	745614.i	0.507325	+	0.184651i	−0.0756604	+	0.997134i	$0.524107\pi$
$440$		0			0
$441$	−4.04947e6	+	495529.i	−0.991520	+	0.121331i
$442$		0			0
$443$	−4.21582e6	+	3.53749e6i	−1.02064	+	0.856418i	−0.989708	−	0.143103i	$-0.954292\pi$
$443$	−4.21582e6	+	3.53749e6i	−1.02064	+	0.856418i	−0.0309318	+	0.999521i	$0.509847\pi$
$444$		0			0
$445$	−357872.	+	2.02959e6i	−0.0856698	+	0.485857i
$446$		0			0
$447$	704391.	+	2.93665e6i	0.166742	+	0.695157i
$448$		0			0
$449$	166745.	−	288812.i	0.0390336	−	0.0676081i	−0.845849	−	0.533423i	$-0.820905\pi$
$449$	166745.	−	288812.i	0.0390336	−	0.0676081i	0.884882	+	0.465815i	$0.154239\pi$
$450$		0			0
$451$	3.50171e6	+	6.06514e6i	0.810660	+	1.40410i
$452$		0			0
$453$	−1.10252e6	+	2.21018e6i	−0.252430	+	0.506037i
$454$		0			0
$455$	−276358.	+	100586.i	−0.0625811	+	0.0227777i
$456$		0			0
$457$	1.21689e6	+	6.90130e6i	0.272558	+	1.54575i	0.746612	+	0.665259i	$0.231679\pi$
$457$	1.21689e6	+	6.90130e6i	0.272558	+	1.54575i	−0.474054	+	0.880496i	$0.657210\pi$
$458$		0			0
$459$	1.19752e6	+	1.40386e6i	0.265308	+	0.311022i
$460$		0			0
$461$	−602960.	−	3.41956e6i	−0.132141	−	0.749406i	−0.976808	−	0.214116i	$-0.931313\pi$
$461$	−602960.	−	3.41956e6i	−0.132141	−	0.749406i	0.844668	−	0.535291i	$-0.179798\pi$
$462$		0			0
$463$	5.06303e6	−	1.84279e6i	1.09763	−	0.399506i	0.271191	−	0.962526i	$-0.412583\pi$
$463$	5.06303e6	−	1.84279e6i	1.09763	−	0.399506i	0.826444	+	0.563019i	$0.190360\pi$
$464$		0			0
$465$	4.34659e6	−	264956.i	0.932215	−	0.0568252i
$466$		0			0
$467$	782405.	+	1.35517e6i	0.166012	+	0.287541i	0.937014	−	0.349291i	$-0.113578\pi$
$467$	782405.	+	1.35517e6i	0.166012	+	0.287541i	−0.771002	+	0.636832i	$0.780244\pi$
$468$		0			0
$469$	36157.3	−	62626.2i	0.00759038	−	0.0131469i
$470$		0			0
$471$	−7.24140e6	−	2.14661e6i	−1.50408	−	0.445863i
$472$		0			0
$473$	122110.	−	692521.i	0.0250957	−	0.142325i
$474$		0			0
$475$	−3.46188e6	+	2.90487e6i	−0.704009	+	0.590734i
$476$		0			0
$477$	−2.42368e6	+	2.59695e6i	−0.487730	+	0.522597i
$478$		0			0
$479$	−2.04735e6	−	745173.i	−0.407711	−	0.148395i	0.130020	−	0.991511i	$-0.458496\pi$
$479$	−2.04735e6	−	745173.i	−0.407711	−	0.148395i	−0.537731	+	0.843117i	$0.680718\pi$
$480$		0			0
$481$	3.45039e6	+	2.89522e6i	0.679995	+	0.570584i
$482$		0			0
$483$	−32056.8	+	13930.9i	−0.00625248	+	0.00271713i
$484$		0			0
$485$	7.63451e6			1.47376
$486$		0			0
$487$	1.16659e6			0.222893			0.111446	−	0.993770i	$-0.464452\pi$
$487$	1.16659e6			0.222893			0.111446	+	0.993770i	$0.464452\pi$
$488$		0			0
$489$	−231892.	+	100773.i	−0.0438545	+	0.0190578i
$490$		0			0
$491$	5.55952e6	+	4.66499e6i	1.04072	+	0.873268i	0.992087	−	0.125549i	$-0.0400693\pi$
$491$	5.55952e6	+	4.66499e6i	1.04072	+	0.873268i	0.0486324	+	0.998817i	$0.484514\pi$
$492$		0			0
$493$	3.41397e6	+	1.24259e6i	0.632620	+	0.230255i
$494$		0			0
$495$	7.87743e6	−	8.44057e6i	1.44501	−	1.54831i
$496$		0			0
$497$	194198.	−	162952.i	0.0352658	−	0.0295916i
$498$		0			0
$499$	180832.	−	1.02555e6i	0.0325106	−	0.184377i	−0.964228	−	0.265074i	$-0.914604\pi$
$499$	180832.	−	1.02555e6i	0.0325106	−	0.184377i	0.996739	+	0.0806973i	$0.0257147\pi$
$500$		0			0
$501$	−3.67383e6	−	1.08906e6i	−0.653920	−	0.193846i
$502$		0			0
$503$	623490.	−	1.07992e6i	0.109878	−	0.190314i	−0.805843	−	0.592129i	$-0.798287\pi$
$503$	623490.	−	1.07992e6i	0.109878	−	0.190314i	0.915721	+	0.401816i	$0.131621\pi$
$504$		0			0
$505$	2.77995e6	+	4.81502e6i	0.485075	+	0.840175i
$506$		0			0
$507$	657130.	−	40056.8i	0.113535	−	0.00692080i
$508$		0			0
$509$	−4.59935e6	+	1.67403e6i	−0.786868	+	0.286397i	−0.704034	−	0.710167i	$-0.748620\pi$
$509$	−4.59935e6	+	1.67403e6i	−0.786868	+	0.286397i	−0.0828346	+	0.996563i	$0.526397\pi$
$510$		0			0
$511$	−13712.9	−	77769.5i	−0.00232314	−	0.0131752i
$512$		0			0
$513$	−2.01572e6	+	372311.i	−0.338171	+	0.0624615i
$514$		0			0
$515$	2.00232e6	+	1.13557e7i	0.332671	+	1.88667i
$516$		0			0
$517$	−1.21160e7	+	4.40986e6i	−1.99358	+	0.725602i
$518$		0			0
$519$	−4.98404e6	+	9.99131e6i	−0.812200	+	1.62819i
$520$		0			0
$521$	−4.84233e6	−	8.38717e6i	−0.781556	−	1.35370i	−0.931035	−	0.364930i	$-0.881093\pi$
$521$	−4.84233e6	−	8.38717e6i	−0.781556	−	1.35370i	0.149479	−	0.988765i	$-0.452240\pi$
$522$		0			0
$523$	5.00106e6	−	8.66209e6i	0.799481	−	1.38474i	−0.120474	−	0.992716i	$-0.538441\pi$
$523$	5.00106e6	−	8.66209e6i	0.799481	−	1.38474i	0.919955	−	0.392025i	$-0.128225\pi$
$524$		0			0
$525$	−129630.	−	540434.i	−0.0205261	−	0.0855746i
$526$		0			0
$527$	−220577.	+	1.25095e6i	−0.0345966	+	0.196207i
$528$		0			0
$529$	−4.71920e6	+	3.95988e6i	−0.733212	+	0.615238i
$530$		0			0
$531$	9.68356e6	−	1.18497e6i	1.49039	−	0.182377i
$532$		0			0
$533$	−9.54211e6	−	3.47304e6i	−1.45488	−	0.529532i
$534$		0			0
$535$	−1.00131e7	−	8.40203e6i	−1.51247	−	1.26911i
$536$		0			0
$537$	175342.	−	1.53616e6i	0.0262391	−	0.229880i
$538$		0			0
$539$	−7.44599e6			−1.10395
$540$		0			0
$541$	1.07747e6			0.158275			0.0791376	−	0.996864i	$-0.474783\pi$
$541$	1.07747e6			0.158275			0.0791376	+	0.996864i	$0.474783\pi$
$542$		0			0
$543$	6.96864e6	+	5.15873e6i	1.01426	+	0.750833i
$544$		0			0
$545$	1.38614e7	+	1.16311e7i	1.99902	+	1.67738i
$546$		0			0
$547$	−2.44692e6	−	890605.i	−0.349664	−	0.127267i	0.161217	−	0.986919i	$-0.448458\pi$
$547$	−2.44692e6	−	890605.i	−0.349664	−	0.127267i	−0.510880	+	0.859652i	$0.670680\pi$
$548$		0			0
$549$	3.19614e6	−	1.62686e6i	0.452579	−	0.230367i
$550$		0			0
$551$	−3.09167e6	+	2.59422e6i	−0.433825	+	0.364022i
$552$		0			0
$553$	−69056.3	+	391637.i	−0.00960263	+	0.0544592i
$554$		0			0
$555$	−8.48620e6	+	8.04980e6i	−1.16945	+	1.10931i
$556$		0			0
$557$	6.11584e6	−	1.05929e7i	0.835253	−	1.44670i	−0.0585707	−	0.998283i	$-0.518654\pi$
$557$	6.11584e6	−	1.05929e7i	0.835253	−	1.44670i	0.893824	−	0.448418i	$-0.148012\pi$
$558$		0			0
$559$	509800.	+	882999.i	0.0690034	+	0.119517i
$560$		0			0
$561$	1.85824e6	+	2.80874e6i	0.249284	+	0.376795i
$562$		0			0
$563$	−1.01307e7	+	3.68727e6i	−1.34700	+	0.490268i	−0.912011	−	0.410166i	$-0.865471\pi$
$563$	−1.01307e7	+	3.68727e6i	−1.34700	+	0.490268i	−0.434991	+	0.900435i	$0.643248\pi$
$564$		0			0
$565$	−1.35197e6	−	7.66739e6i	−0.178174	−	1.01048i
$566$		0			0
$567$	69682.3	−	242262.i	0.00910258	−	0.0316467i
$568$		0			0
$569$	442234.	+	2.50804e6i	0.0572627	+	0.324753i	0.999960	−	0.00889505i	$-0.00283142\pi$
$569$	442234.	+	2.50804e6i	0.0572627	+	0.324753i	−0.942698	+	0.333648i	$0.891720\pi$
$570$		0			0
$571$	1.00007e7	−	3.63995e6i	1.28363	−	0.467203i	0.391998	−	0.919966i	$-0.371784\pi$
$571$	1.00007e7	−	3.63995e6i	1.28363	−	0.467203i	0.891631	+	0.452763i	$0.149562\pi$
$572$		0			0
$573$	6.95086e6	+	1.05063e7i	0.884406	+	1.33679i
$574$		0			0
$575$	2.19316e6	+	3.79866e6i	0.276631	+	0.479139i
$576$		0			0
$577$	1.13203e6	−	1.96073e6i	0.141552	−	0.245176i	−0.786529	−	0.617553i	$-0.788124\pi$
$577$	1.13203e6	−	1.96073e6i	0.141552	−	0.245176i	0.928081	+	0.372378i	$0.121457\pi$
$578$		0			0
$579$	−4.58312e6	+	4.34743e6i	−0.568152	+	0.538935i
$580$		0			0
$581$	32919.6	−	186696.i	0.00404589	−	0.0229454i
$582$		0			0
$583$	−4.96652e6	+	4.16741e6i	−0.605175	+	0.507802i
$584$		0			0
$585$	−882954.	+	1.67168e7i	−0.106672	+	2.01959i
$586$		0			0
$587$	−387154.	−	140913.i	−0.0463755	−	0.0168793i	0.318728	−	0.947846i	$-0.396744\pi$
$587$	−387154.	−	140913.i	−0.0463755	−	0.0168793i	−0.365104	+	0.930967i	$0.618966\pi$
$588$		0			0
$589$	−1.08096e6	−	907032.i	−0.128387	−	0.107729i
$590$		0			0
$591$	6.52971e6	+	4.83380e6i	0.768998	+	0.569272i
$592$		0			0
$593$	8.45756e6			0.987662			0.493831	−	0.869558i	$-0.335596\pi$
$593$	8.45756e6			0.987662			0.493831	+	0.869558i	$0.335596\pi$
$594$		0			0
$595$	222779.			0.0257978
$596$		0			0
$597$	569865.	−	4.99256e6i	0.0654389	−	0.573307i
$598$		0			0
$599$	−1.16637e7	−	9.78699e6i	−1.32822	−	1.11451i	−0.984491	−	0.175435i	$-0.943867\pi$
$599$	−1.16637e7	−	9.78699e6i	−1.32822	−	1.11451i	−0.343725	−	0.939071i	$-0.611689\pi$
$600$		0			0
$601$	7.86028e6	+	2.86091e6i	0.887670	+	0.323086i	0.745301	−	0.666728i	$-0.232306\pi$
$601$	7.86028e6	+	2.86091e6i	0.887670	+	0.323086i	0.142369	+	0.989814i	$0.454528\pi$
$602$		0			0
$603$	−2.47586e6	−	3.28837e6i	−0.277289	−	0.368288i
$604$		0			0
$605$	2.92560e6	−	2.45487e6i	0.324958	−	0.272672i
$606$		0			0
$607$	−212833.	+	1.20703e6i	−0.0234459	+	0.132968i	−0.994284	−	0.106768i	$-0.965950\pi$
$607$	−212833.	+	1.20703e6i	−0.0234459	+	0.132968i	0.970838	+	0.239736i	$0.0770609\pi$
$608$		0			0
$609$	−115767.	−	482641.i	−0.0126486	−	0.0527328i
$610$		0			0
$611$	9.34741e6	−	1.61902e7i	1.01295	−	1.75448i
$612$		0			0
$613$	−6.16287e6	−	1.06744e7i	−0.662418	−	1.14734i	−0.979978	−	0.199104i	$-0.936197\pi$
$613$	−6.16287e6	−	1.06744e7i	−0.662418	−	1.14734i	0.317560	−	0.948238i	$-0.397136\pi$
$614$		0			0
$615$	1.17711e7	−	2.35970e7i	1.25496	−	2.51576i
$616$		0			0
$617$	−2.53339e6	+	922079.i	−0.267910	+	0.0975114i	−0.472482	−	0.881340i	$-0.656642\pi$
$617$	−2.53339e6	+	922079.i	−0.267910	+	0.0975114i	0.204572	+	0.978852i	$0.434420\pi$
$618$		0			0
$619$	−1.65276e6	−	9.37327e6i	−0.173374	−	0.983251i	−0.940004	−	0.341163i	$-0.889179\pi$
$619$	−1.65276e6	−	9.37327e6i	−0.173374	−	0.983251i	0.766631	−	0.642088i	$-0.221932\pi$
$620$		0			0
$621$	16140.2	+	1.98949e6i	0.00167950	+	0.207020i
$622$		0			0
$623$	14261.3	+	80880.1i	0.00147211	+	0.00834875i
$624$		0			0
$625$	−3.18374e7	+	1.15879e7i	−3.26015	+	1.18660i
$626$		0			0
$627$	−3.73427e6	+	227631.i	−0.379348	+	0.0231239i
$628$		0			0
$629$	−1.70598e6	−	2.95484e6i	−0.171928	−	0.297788i
$630$		0			0
$631$	−4.87406e6	+	8.44212e6i	−0.487324	+	0.844070i	−0.999894	−	0.0145758i	$-0.995360\pi$
$631$	−4.87406e6	+	8.44212e6i	−0.487324	+	0.844070i	0.512570	+	0.858645i	$0.328694\pi$
$632$		0			0
$633$	1.27324e7	+	3.77435e6i	1.26300	+	0.374398i
$634$		0			0
$635$	5.88249e6	−	3.33613e7i	0.578931	−	3.28328i
$636$		0			0
$637$	8.27036e6	−	6.93966e6i	0.807562	−	0.677625i
$638$		0			0
$639$	−4.21325e6	−	1.38011e7i	−0.408193	−	1.33710i
$640$		0			0
$641$	1.62414e7	+	5.91138e6i	1.56127	+	0.568256i	0.971027	−	0.238969i	$-0.0768095\pi$
$641$	1.62414e7	+	5.91138e6i	1.56127	+	0.568256i	0.590243	+	0.807225i	$0.299032\pi$
$642$		0			0
$643$	−5.13832e6	−	4.31156e6i	−0.490110	−	0.411251i	0.363956	−	0.931416i	$-0.381426\pi$
$643$	−5.13832e6	−	4.31156e6i	−0.490110	−	0.411251i	−0.854066	+	0.520165i	$0.825870\pi$
$644$		0			0
$645$	−2.42839e6	+	1.05530e6i	−0.229837	+	0.0998799i
$646$		0			0
$647$	1.08528e6			0.101925			0.0509624	−	0.998701i	$-0.483771\pi$
$647$	1.08528e6			0.101925			0.0509624	+	0.998701i	$0.483771\pi$
$648$		0			0
$649$	1.78057e7			1.65939
$650$		0			0
$651$	159156.	−	69164.3i	0.0147187	−	0.00639631i
$652$		0			0
$653$	3.49003e6	+	2.92848e6i	0.320292	+	0.268757i	0.788730	−	0.614739i	$-0.210739\pi$
$653$	3.49003e6	+	2.92848e6i	0.320292	+	0.268757i	−0.468438	+	0.883496i	$0.655183\pi$
$654$		0			0
$655$	−1.98334e7	−	7.21878e6i	−1.80632	−	0.657447i
$656$		0			0
$657$	−4.37939e6	−	1.01295e6i	−0.395822	−	0.0915534i
$658$		0			0
$659$	1.05618e7	−	8.86244e6i	0.947384	−	0.794950i	−0.0314710	−	0.999505i	$-0.510019\pi$
$659$	1.05618e7	−	8.86244e6i	0.947384	−	0.794950i	0.978855	+	0.204555i	$0.0655747\pi$
$660$		0			0
$661$	−8598.59	+	48765.0i	−0.000765462	+	0.00434115i	−0.985188	−	0.171477i	$-0.945146\pi$
$661$	−8598.59	+	48765.0i	−0.000765462	+	0.00434115i	0.984423	+	0.175818i	$0.0562571\pi$
$662$		0			0
$663$	−4.68172e6	−	1.38783e6i	−0.413639	−	0.122618i
$664$		0			0
$665$	−123740.	+	214324.i	−0.0108506	+	0.0187939i
$666$		0			0
$667$	1.95862e6	+	3.39244e6i	0.170466	+	0.295255i
$668$		0			0
$669$	−1.78306e7	+	1.08691e6i	−1.54029	+	0.0938916i
$670$		0			0
$671$	6.15087e6	−	2.23873e6i	0.527388	−	0.191954i
$672$		0			0
$673$	−785552.	−	4.45509e6i	−0.0668555	−	0.379156i	−0.999816	−	0.0191779i	$-0.993895\pi$
$673$	−785552.	−	4.45509e6i	−0.0668555	−	0.379156i	0.932961	−	0.359979i	$-0.117216\pi$
$674$		0			0
$675$	−3.11976e7	−	5.24038e6i	−2.63549	−	0.442694i
$676$		0			0
$677$	−1.83799e6	−	1.04237e7i	−0.154124	−	0.874082i	−0.959582	−	0.281429i	$-0.909192\pi$
$677$	−1.83799e6	−	1.04237e7i	−0.154124	−	0.874082i	0.805458	−	0.592653i	$-0.201919\pi$
$678$		0			0
$679$	285890.	−	104056.i	0.0237972	−	0.00866146i
$680$		0			0
$681$	6.93941e6	−	1.39112e7i	0.573396	−	1.14947i
$682$		0			0
$683$	3.05141e6	+	5.28521e6i	0.250293	+	0.433521i	0.963607	−	0.267325i	$-0.0861396\pi$
$683$	3.05141e6	+	5.28521e6i	0.250293	+	0.433521i	−0.713313	+	0.700845i	$0.752806\pi$
$684$		0			0
$685$	−9.03648e6	+	1.56517e7i	−0.735823	+	1.27448i
$686$		0			0
$687$	5.25595e6	+	2.19124e7i	0.424874	+	1.77132i
$688$		0			0
$689$	1.63236e6	−	9.25759e6i	0.130999	−	0.742934i
$690$		0			0
$691$	1.14705e7	−	9.62490e6i	0.913876	−	0.766833i	−0.0589760	−	0.998259i	$-0.518784\pi$
$691$	1.14705e7	−	9.62490e6i	0.913876	−	0.766833i	0.972853	+	0.231426i	$0.0743391\pi$
$692$		0			0
$693$	179945.	−	423442.i	0.0142333	−	0.0334935i
$694$		0			0
$695$	2.71003e7	+	9.86371e6i	2.12820	+	0.774602i
$696$		0			0
$697$	5.89252e6	+	4.94441e6i	0.459430	+	0.385507i
$698$		0			0
$699$	−496191.	+	4.34710e6i	−0.0384110	+	0.336517i
$700$		0			0
$701$	2.75313e6			0.211608			0.105804	−	0.994387i	$-0.466258\pi$
$701$	2.75313e6			0.211608			0.105804	+	0.994387i	$0.466258\pi$
$702$		0			0
$703$	3.79025e6			0.289254
$704$		0			0
$705$	3.90199e7	+	2.88856e7i	2.95675	+	2.18881i
$706$		0			0
$707$	169728.	+	142419.i	0.0127704	+	0.0107157i
$708$		0			0
$709$	−2.20559e7	−	8.02770e6i	−1.64782	−	0.599757i	−0.659439	−	0.751758i	$-0.729206\pi$
$709$	−2.20559e7	−	8.02770e6i	−1.64782	−	0.599757i	−0.988380	+	0.152001i	$0.951428\pi$
$710$		0			0
$711$	1.89794e7	+	1.23364e7i	1.40802	+	0.915196i
$712$		0			0
$713$	−1.04918e6	+	880368.i	−0.0772906	+	0.0648545i
$714$		0			0
$715$	−5.30549e6	+	3.00890e7i	−0.388115	+	2.20111i
$716$		0			0
$717$	−8.57198e6	+	8.13117e6i	−0.622706	+	0.590684i
$718$		0			0
$719$	−3.08542e6	+	5.34410e6i	−0.222583	+	0.385525i	−0.955592	−	0.294694i	$-0.904782\pi$
$719$	−3.08542e6	+	5.34410e6i	−0.222583	+	0.385525i	0.733009	+	0.680219i	$0.238115\pi$
$720$		0			0
$721$	229755.	+	397948.i	0.0164599	+	0.0285094i
$722$		0			0
$723$	−8.59130e6	−	1.29858e7i	−0.611242	−	0.923897i
$724$		0			0
$725$	−5.85294e7	+	2.13029e7i	−4.13551	+	1.50520i
$726$		0			0
$727$	−1.28434e6	−	7.28387e6i	−0.0901249	−	0.511124i	−0.996133	−	0.0878625i	$-0.971996\pi$
$727$	−1.28434e6	−	7.28387e6i	−0.0901249	−	0.511124i	0.906008	−	0.423261i	$-0.139115\pi$
$728$		0			0
$729$	−1.11401e7	−	9.04375e6i	−0.776372	−	0.630274i
$730$		0			0
$731$	−134118.	−	760624.i	−0.00928315	−	0.0526473i
$732$		0			0
$733$	6.48069e6	−	2.35878e6i	0.445514	−	0.162154i	−0.109514	−	0.993985i	$-0.534930\pi$
$733$	6.48069e6	−	2.35878e6i	0.445514	−	0.162154i	0.555028	+	0.831831i	$0.312707\pi$
$734$		0			0
$735$	1.54696e7	+	2.33824e7i	1.05623	+	1.59650i
$736$		0			0
$737$	−3.75634e6	−	6.50618e6i	−0.254740	−	0.441222i
$738$		0			0
$739$	−1.18607e7	+	2.05433e7i	−0.798910	+	1.38375i	0.121416	+	0.992602i	$0.461257\pi$
$739$	−1.18607e7	+	2.05433e7i	−0.798910	+	1.38375i	−0.920326	+	0.391152i	$0.872077\pi$
$740$		0			0
$741$	3.93556e6	−	3.73317e6i	0.263306	−	0.249766i
$742$		0			0
$743$	2.45138e6	−	1.39025e7i	0.162907	−	0.923889i	−0.788290	−	0.615304i	$-0.789033\pi$
$743$	2.45138e6	−	1.39025e7i	0.162907	−	0.923889i	0.951197	−	0.308585i	$-0.0998555\pi$
$744$		0			0
$745$	1.58983e7	−	1.33403e7i	1.04945	−	0.880589i
$746$		0			0
$747$	−9.04759e6	−	5.88084e6i	−0.593242	−	0.385601i
$748$		0			0
$749$	−489480.	−	178156.i	−0.0318809	−	0.0116037i
$750$		0			0
$751$	5.93985e6	+	4.98412e6i	0.384304	+	0.322470i	0.814389	−	0.580319i	$-0.197072\pi$
$751$	5.93985e6	+	4.98412e6i	0.384304	+	0.322470i	−0.430085	+	0.902788i	$0.641516\pi$
$752$		0			0
$753$	−2.33645e6	−	1.72963e6i	−0.150165	−	0.111164i
$754$		0			0
$755$	1.69738e7			1.08370
$756$		0			0
$757$	−2.39155e6			−0.151684			−0.0758420	−	0.997120i	$-0.524164\pi$
$757$	−2.39155e6			−0.151684			−0.0758420	+	0.997120i	$0.524164\pi$
$758$		0			0
$759$	−411805.	+	3.60780e6i	−0.0259470	+	0.227320i
$760$		0			0
$761$	−4.03192e6	−	3.38319e6i	−0.252377	−	0.211770i	0.507818	−	0.861464i	$-0.330452\pi$
$761$	−4.03192e6	−	3.38319e6i	−0.252377	−	0.211770i	−0.760195	+	0.649695i	$0.774897\pi$
$762$		0			0
$763$	677599.	+	246626.i	0.0421368	+	0.0153365i
$764$		0			0
$765$	4.95957e6	−	1.16707e7i	0.306401	−	0.721015i
$766$		0			0
$767$	−1.97771e7	+	1.65949e7i	−1.21387	+	1.01856i
$768$		0			0
$769$	4.21546e6	−	2.39071e7i	0.257057	−	1.45784i	−0.533680	−	0.845687i	$-0.679191\pi$
$769$	4.21546e6	−	2.39071e7i	0.257057	−	1.45784i	0.790737	−	0.612156i	$-0.209698\pi$
$770$		0			0
$771$	3.68486e6	+	1.53624e7i	0.223247	+	0.930729i
$772$		0			0
$773$	6.07327e6	−	1.05192e7i	0.365573	−	0.633191i	−0.623295	−	0.781987i	$-0.714206\pi$
$773$	6.07327e6	−	1.05192e7i	0.365573	−	0.633191i	0.988868	+	0.148796i	$0.0475397\pi$
$774$		0			0
$775$	−1.08886e7	−	1.88597e7i	−0.651207	−	1.12792i
$776$		0			0
$777$	−208068.	+	417106.i	−0.0123638	+	0.0247853i
$778$		0			0
$779$	−8.02967e6	+	2.92256e6i	−0.474083	+	0.172552i
$780$		0			0
$781$	−4.57332e6	−	2.59366e7i	−0.268290	−	1.52155i
$782$		0			0
$783$	−2.78613e7	−	4.67998e6i	−1.62404	−	0.272797i
$784$		0			0
$785$	9.01321e6	+	5.11165e7i	0.522042	+	2.96065i
$786$		0			0
$787$	1.36692e7	−	4.97520e6i	0.786697	−	0.286334i	0.0827347	−	0.996572i	$-0.473635\pi$
$787$	1.36692e7	−	4.97520e6i	0.786697	−	0.286334i	0.703962	+	0.710237i	$0.251412\pi$
$788$		0			0
$789$	2.45206e7	−	1.49470e6i	1.40229	−	0.0854797i
$790$		0			0
$791$	−155131.	−	268695.i	−0.00881572	−	0.0152693i
$792$		0			0
$793$	−4.74536e6	+	8.21920e6i	−0.267970	+	0.464137i
$794$		0			0
$795$	2.34051e7	+	6.93810e6i	1.31338	+	0.389335i
$796$		0			0
$797$	−2.83721e6	+	1.60906e7i	−0.158214	+	0.897277i	0.797574	+	0.603221i	$0.206116\pi$
$797$	−2.83721e6	+	1.60906e7i	−0.158214	+	0.897277i	−0.955788	+	0.294056i	$0.904995\pi$
$798$		0			0
$799$	−1.08484e7	+	9.10285e6i	−0.601170	+	0.504441i
$800$		0			0
$801$	4.55455e6	+	1.05347e6i	0.250821	+	0.0580148i
$802$		0			0
$803$	−7.70927e6	−	2.80595e6i	−0.421915	−	0.153564i
$804$		0			0
$805$	184007.	+	154400.i	0.0100080	+	0.00839767i
$806$		0			0
$807$	1.80739e7	−	7.85438e6i	0.976943	−	0.424549i
$808$		0			0
$809$	−1.09868e7			−0.590203			−0.295101	−	0.955466i	$-0.595353\pi$
$809$	−1.09868e7			−0.590203			−0.295101	+	0.955466i	$0.595353\pi$
$810$		0			0
$811$	3.23848e7			1.72897			0.864487	−	0.502655i	$-0.167643\pi$
$811$	3.23848e7			1.72897			0.864487	+	0.502655i	$0.167643\pi$
$812$		0			0
$813$	−9.78911e6	+	4.25405e6i	−0.519418	+	0.225723i
$814$		0			0
$815$	1.33107e6	+	1.11690e6i	0.0701952	+	0.0589008i
$816$		0			0
$817$	806249.	+	293451.i	0.0422585	+	0.0153808i
$818$		0			0
$819$	194780.	+	638031.i	0.0101469	+	0.0332378i
$820$		0			0
$821$	1.76175e7	−	1.47829e7i	0.912194	−	0.765421i	−0.0603413	−	0.998178i	$-0.519219\pi$
$821$	1.76175e7	−	1.47829e7i	0.912194	−	0.765421i	0.972535	+	0.232756i	$0.0747745\pi$
$822$		0			0
$823$	−4.65842e6	+	2.64192e7i	−0.239739	+	1.35963i	0.592660	+	0.805453i	$0.298078\pi$
$823$	−4.65842e6	+	2.64192e7i	−0.239739	+	1.35963i	−0.832399	+	0.554177i	$0.813033\pi$
$824$		0			0
$825$	−5.53568e7	−	1.64097e7i	−2.83163	−	0.839396i
$826$		0			0
$827$	−4.58500e6	+	7.94146e6i	−0.233118	+	0.403772i	−0.958724	−	0.284338i	$-0.908226\pi$
$827$	−4.58500e6	+	7.94146e6i	−0.233118	+	0.403772i	0.725606	+	0.688110i	$0.241559\pi$
$828$		0			0
$829$	1.01005e7	+	1.74946e7i	0.510455	+	0.884134i	0.999927	+	0.0121149i	$0.00385639\pi$
$829$	1.01005e7	+	1.74946e7i	0.510455	+	0.884134i	−0.489471	+	0.872019i	$0.662810\pi$
$830$		0			0
$831$	2.77089e7	−	1.68906e6i	1.39193	−	0.0848481i
$832$		0			0
$833$	−7.68502e6	+	2.79712e6i	−0.383736	+	0.139668i
$834$		0			0
$835$	4.57274e6	+	2.59333e7i	0.226966	+	1.28719i
$836$		0			0
$837$	−80133.1	−	9.87745e6i	−0.00395365	−	0.487339i
$838$		0			0
$839$	1.75344e6	+	9.94424e6i	0.0859974	+	0.487716i	0.997137	+	0.0756175i	$0.0240928\pi$
$839$	1.75344e6	+	9.94424e6i	0.0859974	+	0.487716i	−0.911139	+	0.412098i	$0.864796\pi$
$840$		0			0
$841$	−3.29961e7	+	1.20096e7i	−1.60869	+	0.585515i
$842$		0			0
$843$	−7.63673e6	+	1.53091e7i	−0.370117	+	0.741959i
$844$		0			0
$845$	−2.26217e6	−	3.91819e6i	−0.108989	−	0.188774i
$846$		0			0
$847$	76096.4	−	131803.i	0.00364465	−	0.00631272i
$848$		0			0
$849$	−3.51232e6	−	1.46431e7i	−0.167234	−	0.697208i
$850$		0			0
$851$	638822.	−	3.62294e6i	0.0302382	−	0.171489i
$852$		0			0
$853$	2.07128e7	−	1.73801e7i	0.974691	−	0.817863i	−0.00858901	−	0.999963i	$-0.502734\pi$
$853$	2.07128e7	−	1.73801e7i	0.974691	−	0.817863i	0.983280	+	0.182100i	$0.0582896\pi$
$854$		0			0
$855$	8.47304e6	+	1.12537e7i	0.396391	+	0.526477i
$856$		0			0
$857$	−6.18729e6	−	2.25199e6i	−0.287772	−	0.104740i	0.194101	−	0.980982i	$-0.437821\pi$
$857$	−6.18729e6	−	2.25199e6i	−0.287772	−	0.104740i	−0.481873	+	0.876241i	$0.660043\pi$
$858$		0			0
$859$	−2.86976e6	−	2.40801e6i	−0.132697	−	0.111346i	0.574023	−	0.818839i	$-0.305382\pi$
$859$	−2.86976e6	−	2.40801e6i	−0.132697	−	0.111346i	−0.706721	+	0.707493i	$0.749826\pi$
$860$		0			0
$861$	119174.	−	1.04408e6i	0.00547866	−	0.0479982i
$862$		0			0
$863$	1.18247e7			0.540458			0.270229	−	0.962796i	$-0.412901\pi$
$863$	1.18247e7			0.540458			0.270229	+	0.962796i	$0.412901\pi$
$864$		0			0
$865$	7.67314e7			3.48685
$866$		0			0
$867$	−1.48164e7	−	1.09682e7i	−0.669413	−	0.495551i
$868$		0			0
$869$	3.16488e7	+	2.65565e7i	1.42170	+	1.19295i
$870$		0			0
$871$	1.02360e7	+	3.72559e6i	0.457177	+	0.166399i
$872$		0			0
$873$	913411.	−	1.72935e7i	0.0405630	−	0.767973i
$874$		0			0
$875$	−1.83097e6	+	1.53637e6i	−0.0808466	+	0.0678383i
$876$		0			0
$877$	−1.96926e6	+	1.11682e7i	−0.0864578	+	0.490327i	0.910575	+	0.413345i	$0.135640\pi$
$877$	−1.96926e6	+	1.11682e7i	−0.0864578	+	0.490327i	−0.997032	+	0.0769820i	$0.975472\pi$
$878$		0			0
$879$	−2.32148e7	+	2.20209e7i	−1.01343	+	0.961311i
$880$		0			0
$881$	−1.96923e6	+	3.41080e6i	−0.0854784	+	0.148053i	−0.905595	−	0.424144i	$-0.860575\pi$
$881$	−1.96923e6	+	3.41080e6i	−0.0854784	+	0.148053i	0.820117	+	0.572196i	$0.193909\pi$
$882$		0			0
$883$	−5.88849e6	−	1.01992e7i	−0.254157	−	0.440213i	0.710509	−	0.703688i	$-0.248465\pi$
$883$	−5.88849e6	−	1.01992e7i	−0.254157	−	0.440213i	−0.964666	+	0.263475i	$0.915131\pi$
$884$		0			0
$885$	−3.69926e7	−	5.59146e7i	−1.58766	−	2.39976i
$886$		0			0
$887$	−3.43239e7	+	1.24929e7i	−1.46483	+	0.533155i	−0.946692	−	0.322141i	$-0.895597\pi$
$887$	−3.43239e7	+	1.24929e7i	−1.46483	+	0.533155i	−0.518139	+	0.855296i	$0.673375\pi$
$888$		0			0
$889$	−234420.	−	1.32946e6i	−0.00994809	−	0.0564184i
$890$		0			0
$891$	−1.81769e7	−	1.88536e7i	−0.767051	−	0.795608i
$892$		0			0
$893$	−2.73178e6	−	1.54927e7i	−0.114635	−	0.650126i
$894$		0			0
$895$	−9.98459e6	+	3.63410e6i	−0.416651	+	0.151649i
$896$		0			0
$897$	−2.90507e6	−	4.39103e6i	−0.120552	−	0.182216i
$898$		0			0
$899$	−9.72421e6	−	1.68428e7i	−0.401287	−	0.695049i
$900$		0			0
$901$	−3.56045e6	+	6.16689e6i	−0.146114	+	0.253078i
$902$		0			0
$903$	−76552.9	+	72616.2i	−0.00312422	+	0.00296356i
$904$		0			0
$905$	1.03467e7	−	5.86792e7i	0.419935	−	2.38157i
$906$		0			0
$907$	−1.88953e7	+	1.58551e7i	−0.762670	+	0.639956i	−0.938820	−	0.344407i	$-0.888080\pi$
$907$	−1.88953e7	+	1.58551e7i	−0.762670	+	0.639956i	0.176151	+	0.984363i	$0.443635\pi$
$908$		0			0
$909$	1.12394e7	−	5.72098e6i	0.451164	−	0.229647i
$910$		0			0
$911$	−3.67345e7	−	1.33703e7i	−1.46649	−	0.533758i	−0.519343	−	0.854566i	$-0.673823\pi$
$911$	−3.67345e7	−	1.33703e7i	−1.46649	−	0.533758i	−0.947144	+	0.320808i	$0.896046\pi$
$912$		0			0
$913$	−1.50872e7	−	1.26597e7i	−0.599006	−	0.502626i
$914$		0			0
$915$	−1.98091e7	−	1.46642e7i	−0.782189	−	0.579037i
$916$		0			0
$917$	−841094.			−0.0330310
$918$		0			0
$919$	−1.95110e7			−0.762063			−0.381031	−	0.924562i	$-0.624431\pi$
$919$	−1.95110e7			−0.762063			−0.381031	+	0.924562i	$0.624431\pi$
$920$		0			0
$921$	1.97718e6	−	1.73220e7i	0.0768064	−	0.672896i
$922$		0			0
$923$	2.92525e7	+	2.45458e7i	1.13021	+	0.948359i
$924$		0			0
$925$	5.49670e7	+	2.00064e7i	2.11226	+	0.768801i
$926$		0			0
$927$	2.59622e7	−	3.17696e6i	0.992297	−	0.121426i
$928$		0			0
$929$	−2.99867e7	+	2.51618e7i	−1.13996	+	0.956540i	−0.999437	−	0.0335374i	$-0.989323\pi$
$929$	−2.99867e7	+	2.51618e7i	−1.13996	+	0.956540i	−0.140523	+	0.990077i	$0.544878\pi$
$930$		0			0
$931$	1.57759e6	−	8.94696e6i	0.0596513	−	0.338299i
$932$		0			0
$933$	−1.82440e6	−	7.60604e6i	−0.0686146	−	0.286058i
$934$		0			0
$935$	1.15722e7	−	2.00436e7i	0.432898	−	0.749801i
$936$		0			0
$937$	1.27358e7	+	2.20591e7i	0.473890	+	0.820802i	0.999553	−	0.0298911i	$-0.00951606\pi$
$937$	1.27358e7	+	2.20591e7i	0.473890	+	0.820802i	−0.525663	+	0.850693i	$0.676183\pi$
$938$		0			0
$939$	9.00878e6	−	1.80596e7i	0.333428	−	0.668410i
$940$		0			0
$941$	−1.50394e7	+	5.47388e6i	−0.553675	+	0.201521i	−0.603679	−	0.797228i	$-0.706299\pi$
$941$	−1.50394e7	+	5.47388e6i	−0.553675	+	0.201521i	0.0500034	+	0.998749i	$0.484077\pi$
$942$		0			0
$943$	1.44020e6	+	8.16780e6i	0.0527406	+	0.299107i
$944$		0			0
$945$	−1.70357e6	+	314655.i	−0.0620554	+	0.0114619i
$946$		0			0
$947$	2.54893e6	+	1.44557e7i	0.0923598	+	0.523799i	0.995524	+	0.0945043i	$0.0301266\pi$
$947$	2.54893e6	+	1.44557e7i	0.0923598	+	0.523799i	−0.903165	+	0.429294i	$0.858762\pi$
$948$		0			0
$949$	1.11779e7	−	4.06844e6i	0.402899	−	0.146643i
$950$		0			0
$951$	2.61682e7	−	1.59514e6i	0.938260	−	0.0571937i
$952$		0			0
$953$	−975361.	−	1.68937e6i	−0.0347883	−	0.0602551i	0.848107	−	0.529825i	$-0.177742\pi$
$953$	−975361.	−	1.68937e6i	−0.0347883	−	0.0602551i	−0.882895	+	0.469570i	$0.844409\pi$
$954$		0			0
$955$	4.32863e7	−	7.49741e7i	1.53583	−	2.66013i
$956$		0			0
$957$	−4.94369e7	−	1.46549e7i	−1.74490	−	0.517253i
$958$		0			0
$959$	−125064.	+	709274.i	−0.00439123	+	0.0249039i
$960$		0			0
$961$	−1.67222e7	+	1.40316e7i	−0.584097	+	0.490116i
$962$		0			0
$963$	−2.02300e7	+	2.16762e7i	−0.702960	+	0.753213i
$964$		0			0
$965$	4.07941e7	+	1.48479e7i	1.41020	+	0.513269i
$966$		0			0
$967$	−2.03651e7	−	1.70883e7i	−0.700358	−	0.587670i	0.221517	−	0.975156i	$-0.428899\pi$
$967$	−2.03651e7	−	1.70883e7i	−0.700358	−	0.587670i	−0.921876	+	0.387486i	$0.873344\pi$
$968$		0			0
$969$	−3.76864e6	+	1.63773e6i	−0.128936	+	0.0560317i
$970$		0			0
$971$	5.11477e7			1.74092			0.870458	−	0.492242i	$-0.163823\pi$
$971$	5.11477e7			1.74092			0.870458	+	0.492242i	$0.163823\pi$
$972$		0			0
$973$	1.14927e6			0.0389170
$974$		0			0
$975$	7.67794e7	−	3.33659e7i	2.58662	−	1.12407i
$976$		0			0
$977$	3.06081e6	+	2.56832e6i	0.102589	+	0.0860821i	0.692639	−	0.721284i	$-0.256448\pi$
$977$	3.06081e6	+	2.56832e6i	0.102589	+	0.0860821i	−0.590051	+	0.807366i	$0.700892\pi$
$978$		0			0
$979$	8.01763e6	+	2.91818e6i	0.267356	+	0.0973095i
$980$		0			0
$981$	2.80049e7	−	3.00069e7i	0.929098	−	0.995518i
$982$		0			0
$983$	−2.59484e7	+	2.17733e7i	−0.856497	+	0.718687i	−0.961211	−	0.275816i	$-0.911052\pi$
$983$	−2.59484e7	+	2.17733e7i	−0.856497	+	0.718687i	0.104713	+	0.994502i	$0.466608\pi$
$984$		0			0
$985$	9.69502e6	−	5.49832e7i	0.318389	−	1.80567i
$986$		0			0
$987$	1.85489e6	+	549855.i	0.0606072	+	0.0179662i
$988$		0			0
$989$	416385.	−	721200.i	0.0135364	−	0.0234458i
$990$		0			0
$991$	−4.99176e6	−	8.64598e6i	−0.161462	−	0.279660i	0.773931	−	0.633269i	$-0.218287\pi$
$991$	−4.99176e6	−	8.64598e6i	−0.161462	−	0.279660i	−0.935393	+	0.353610i	$0.884954\pi$
$992$		0			0
$993$	−2.03963e7	+	1.24330e6i	−0.656414	+	0.0400131i
$994$		0			0
$995$	−3.24502e7	+	1.18109e7i	−1.03911	+	0.378203i
$996$		0			0
$997$	−4.79955e6	−	2.72196e7i	−0.152919	−	0.867249i	−0.960663	−	0.277716i	$-0.910423\pi$
$997$	−4.79955e6	−	2.72196e7i	−0.152919	−	0.867249i	0.807744	−	0.589534i	$-0.200688\pi$
$998$		0			0
$999$	1.72189e7	+	2.01858e7i	0.545872	+	0.639929i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.6.i.a.13.3	✓	90
3.2	odd	2		324.6.i.a.253.1		90
27.2	odd	18		324.6.i.a.73.1		90
27.25	even	9	inner	108.6.i.a.25.3	yes	90

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.6.i.a.13.3	✓	90	1.1	even	1	trivial
108.6.i.a.25.3	yes	90	27.25	even	9	inner
324.6.i.a.73.1		90	27.2	odd	18
324.6.i.a.253.1		90	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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	108.i (of order \(9\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(-14.2968 + 6.21296i) q^{3} +(82.0644 + 68.8602i) q^{5} +(4.01162 + 1.46011i) q^{7} +(165.798 - 177.651i) q^{9} +O(q^{10})\)	\(q+(-14.2968 + 6.21296i) q^{3} +(82.0644 + 68.8602i) q^{5} +(4.01162 + 1.46011i) q^{7} +(165.798 - 177.651i) q^{9} +(339.748 - 285.083i) q^{11} +(-111.666 + 633.290i) q^{13} +(-1601.09 - 474.620i) q^{15} +(243.562 - 421.862i) q^{17} +(270.567 + 468.636i) q^{19} +(-66.4250 + 4.04908i) q^{21} +(493.552 - 179.638i) q^{23} +(1450.19 + 8224.41i) q^{25} +(-1266.65 + 3569.94i) q^{27} +(1295.10 + 7344.90i) q^{29} +(-2450.39 + 891.870i) q^{31} +(-3086.12 + 6186.62i) q^{33} +(228.667 + 396.064i) q^{35} +(3502.14 - 6065.88i) q^{37} +(-2338.13 - 9747.82i) q^{39} +(-2742.06 + 15551.0i) q^{41} +(1214.60 - 1019.17i) q^{43} +(25839.2 - 3161.92i) q^{45} +(-27318.4 - 9943.09i) q^{47} +(-12860.9 - 10791.6i) q^{49} +(-861.155 + 7544.53i) q^{51} -14618.2 q^{53} +47512.1 q^{55} +(-6779.87 - 5018.98i) q^{57} +(30754.6 + 25806.2i) q^{59} +(13868.6 + 5047.76i) q^{61} +(924.509 - 470.584i) q^{63} +(-52772.3 + 44281.2i) q^{65} +(2941.46 - 16681.8i) q^{67} +(-5940.14 + 5634.67i) q^{69} +(29691.2 - 51426.7i) q^{71} +(-9248.99 - 16019.7i) q^{73} +(-71831.0 - 108573. i) q^{75} +(1779.19 - 647.573i) q^{77} +(16175.9 + 91738.3i) q^{79} +(-4070.82 - 58908.5i) q^{81} +(-7711.19 - 43732.3i) q^{83} +(49037.3 - 17848.1i) q^{85} +(-64149.4 - 96962.3i) q^{87} +(9618.93 + 16660.5i) q^{89} +(-1372.63 + 2377.47i) q^{91} +(29491.7 - 27975.1i) q^{93} +(-10066.4 + 57089.6i) q^{95} +(54592.7 - 45808.7i) q^{97} +(5684.46 - 107623. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 90 q - 87 q^{5} + 330 q^{9}+O(q^{10}) \) 90 * q - 87 * q^5 + 330 * q^9	\( 90 q - 87 q^{5} + 330 q^{9} - 1257 q^{11} + 531 q^{15} - 3468 q^{17} + 12894 q^{21} + 8106 q^{23} + 4959 q^{25} - 17415 q^{27} + 3468 q^{29} - 6651 q^{31} + 33624 q^{33} - 8229 q^{35} - 10545 q^{39} + 68673 q^{41} + 9459 q^{43} - 53469 q^{45} - 57087 q^{47} - 5490 q^{49} + 42831 q^{51} - 4146 q^{53} + 24624 q^{57} + 5388 q^{59} + 70110 q^{61} - 98115 q^{63} - 172425 q^{65} - 15039 q^{67} + 251037 q^{69} + 67812 q^{71} - 27009 q^{73} - 75273 q^{75} + 23991 q^{77} - 216180 q^{79} + 177822 q^{81} - 76725 q^{83} - 53100 q^{85} - 201483 q^{87} - 98814 q^{89} - 90999 q^{91} + 21765 q^{93} - 143490 q^{95} - 71739 q^{97} + 13635 q^{99}+O(q^{100}) \) 90 * q - 87 * q^5 + 330 * q^9 - 1257 * q^11 + 531 * q^15 - 3468 * q^17 + 12894 * q^21 + 8106 * q^23 + 4959 * q^25 - 17415 * q^27 + 3468 * q^29 - 6651 * q^31 + 33624 * q^33 - 8229 * q^35 - 10545 * q^39 + 68673 * q^41 + 9459 * q^43 - 53469 * q^45 - 57087 * q^47 - 5490 * q^49 + 42831 * q^51 - 4146 * q^53 + 24624 * q^57 + 5388 * q^59 + 70110 * q^61 - 98115 * q^63 - 172425 * q^65 - 15039 * q^67 + 251037 * q^69 + 67812 * q^71 - 27009 * q^73 - 75273 * q^75 + 23991 * q^77 - 216180 * q^79 + 177822 * q^81 - 76725 * q^83 - 53100 * q^85 - 201483 * q^87 - 98814 * q^89 - 90999 * q^91 + 21765 * q^93 - 143490 * q^95 - 71739 * q^97 + 13635 * q^99

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