LMFDB - Embedded newform 72.8.d.c.37.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,8,Mod(37,72)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("72.37");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 72 = 2^{3} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	72.d (of order $2$, degree $1$, 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:	$22.4917218349$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 206x^{10} + 24336x^{8} - 1510912x^{6} + 398721024x^{4} - 55297703936x^{2} + 4398046511104 $ x^12 - 206x^10 + 24336x^8 - 1510912x^6 + 398721024x^4 - 55297703936*x^2 + 4398046511104
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{30}\cdot 3^{6}\cdot 5^{2}\cdot 13^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			37.11
Root			$-10.6405 - 3.84452i$ of defining polynomial
Character	$\chi$	$=$	72.37
Dual form			72.8.d.c.37.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(10.6405 - 3.84452i) q^{2} +(98.4393 - 81.8151i) q^{4} +209.469i q^{5} -1301.46 q^{7} +(732.900 - 1249.00i) q^{8} +O(q^{10})$	\(q+(10.6405 - 3.84452i) q^{2} +(98.4393 - 81.8151i) q^{4} +209.469i q^{5} -1301.46 q^{7} +(732.900 - 1249.00i) q^{8} +(805.310 + 2228.85i) q^{10} -7074.87i q^{11} -14331.3i q^{13} +(-13848.1 + 5003.49i) q^{14} +(2996.58 - 16107.6i) q^{16} +19199.4 q^{17} +9617.13i q^{19} +(17137.8 + 20620.0i) q^{20} +(-27199.5 - 75280.0i) q^{22} -86666.4 q^{23} +34247.6 q^{25} +(-55097.0 - 152492. i) q^{26} +(-128115. + 106479. i) q^{28} -101787. i q^{29} +28412.3 q^{31} +(-30041.1 - 182913. i) q^{32} +(204291. - 73812.6i) q^{34} -272616. i q^{35} +76289.0i q^{37} +(36973.3 + 102331. i) q^{38} +(261628. + 153520. i) q^{40} -510168. q^{41} +784099. i q^{43} +(-578831. - 696445. i) q^{44} +(-922172. + 333191. i) q^{46} +180049. q^{47} +870255. q^{49} +(364410. - 131666. i) q^{50} +(-1.17252e6 - 1.41076e6i) q^{52} -880974. i q^{53} +1.48197e6 q^{55} +(-953840. + 1.62553e6i) q^{56} +(-391323. - 1.08306e6i) q^{58} +92535.3i q^{59} +2.40486e6i q^{61} +(302320. - 109232. i) q^{62} +(-1.02287e6 - 1.83079e6i) q^{64} +3.00197e6 q^{65} -1.99319e6i q^{67} +(1.88998e6 - 1.57080e6i) q^{68} +(-1.04808e6 - 2.90076e6i) q^{70} -3.60291e6 q^{71} +1.68585e6 q^{73} +(293295. + 811751. i) q^{74} +(786826. + 946703. i) q^{76} +9.20766e6i q^{77} +4.81609e6 q^{79} +(3.37406e6 + 627693. i) q^{80} +(-5.42843e6 + 1.96135e6i) q^{82} +1.23628e6i q^{83} +4.02169e6i q^{85} +(3.01449e6 + 8.34318e6i) q^{86} +(-8.83654e6 - 5.18518e6i) q^{88} +9.48855e6 q^{89} +1.86516e7i q^{91} +(-8.53138e6 + 7.09062e6i) q^{92} +(1.91580e6 - 692201. i) q^{94} -2.01449e6 q^{95} +1.45734e7 q^{97} +(9.25992e6 - 3.34571e6i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 412 q^{4} + 136 q^{7}+O(q^{10}) $ 12 * q + 412 * q^4 + 136 * q^7	$ 12 q + 412 q^{4} + 136 q^{7} + 4680 q^{10} - 12472 q^{16} + 32624 q^{22} + 229820 q^{25} - 157288 q^{28} - 37224 q^{31} - 74432 q^{34} - 937520 q^{40} - 1264256 q^{46} + 2668188 q^{49} - 1539680 q^{52} + 6928960 q^{55} - 4035448 q^{58} - 3530192 q^{64} - 10228720 q^{70} + 13619048 q^{73} - 2441920 q^{76} + 20470552 q^{79} - 2507200 q^{82} - 26170912 q^{88} - 22132608 q^{94} + 27442456 q^{97}+O(q^{100}) $ 12 * q + 412 * q^4 + 136 * q^7 + 4680 * q^10 - 12472 * q^16 + 32624 * q^22 + 229820 * q^25 - 157288 * q^28 - 37224 * q^31 - 74432 * q^34 - 937520 * q^40 - 1264256 * q^46 + 2668188 * q^49 - 1539680 * q^52 + 6928960 * q^55 - 4035448 * q^58 - 3530192 * q^64 - 10228720 * q^70 + 13619048 * q^73 - 2441920 * q^76 + 20470552 * q^79 - 2507200 * q^82 - 26170912 * q^88 - 22132608 * q^94 + 27442456 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/72\mathbb{Z}\right)^\times$.

$n$	$37$	$55$	$65$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$	10.6405	−	3.84452i	0.940494	−	0.339811i
$2$	10.6405	−	3.84452i	0.940494	−	0.339811i
$3$		0			0
$3$		0			0
$4$	98.4393	−	81.8151i	0.769057	−	0.639180i
$5$			209.469i			0.749420i	0.927142	+	0.374710i	$0.122258\pi$
$5$			209.469i			0.749420i	−0.927142	+	0.374710i	$0.877742\pi$
$6$		0			0
$7$	−1301.46			−1.43413			−0.717064	−	0.697008i	$-0.754514\pi$
$7$	−1301.46			−1.43413			−0.717064	+	0.697008i	$0.754514\pi$
$8$	732.900	−	1249.00i	0.506093	−	0.862479i
$9$		0			0
$10$	805.310	+	2228.85i	0.254661	+	0.704825i
$11$		−	7074.87i		−	1.60267i	−0.598216	−	0.801335i	$-0.704123\pi$
$11$		−	7074.87i		−	1.60267i	0.598216	−	0.801335i	$-0.295877\pi$
$12$		0			0
$13$		−	14331.3i		−	1.80919i	−0.426273	−	0.904594i	$-0.640174\pi$
$13$		−	14331.3i		−	1.80919i	0.426273	−	0.904594i	$-0.359826\pi$
$14$	−13848.1	+	5003.49i	−1.34879	+	0.487332i
$15$		0			0
$16$	2996.58	−	16107.6i	0.182897	−	0.983132i
$17$	19199.4			0.947800			0.473900	−	0.880579i	$-0.342846\pi$
$17$	19199.4			0.947800			0.473900	+	0.880579i	$0.342846\pi$
$18$		0			0
$19$			9617.13i			0.321668i	0.986981	+	0.160834i	$0.0514184\pi$
$19$			9617.13i			0.321668i	−0.986981	+	0.160834i	$0.948582\pi$
$20$	17137.8	+	20620.0i	0.479015	+	0.576347i
$21$		0			0
$22$	−27199.5	−	75280.0i	−0.544605	−	1.50730i
$23$	−86666.4			−1.48526			−0.742632	−	0.669700i	$-0.766423\pi$
$23$	−86666.4			−1.48526			−0.742632	+	0.669700i	$0.766423\pi$
$24$		0			0
$25$	34247.6			0.438369
$26$	−55097.0	−	152492.i	−0.614782	−	1.70153i
$27$		0			0
$28$	−128115.	+	106479.i	−1.10293	+	0.916666i
$29$		−	101787.i		−	0.774997i	−0.921870	−	0.387498i	$-0.873339\pi$
$29$		−	101787.i		−	0.774997i	0.921870	−	0.387498i	$-0.126661\pi$
$30$		0			0
$31$	28412.3			0.171293			0.0856466	−	0.996326i	$-0.472704\pi$
$31$	28412.3			0.171293			0.0856466	+	0.996326i	$0.472704\pi$
$32$	−30041.1	−	182913.i	−0.162066	−	0.986780i
$33$		0			0
$34$	204291.	−	73812.6i	0.891400	−	0.322073i
$35$		−	272616.i		−	1.07476i
$36$		0			0
$37$			76289.0i			0.247603i	0.992307	+	0.123801i	$0.0395086\pi$
$37$			76289.0i			0.247603i	−0.992307	+	0.123801i	$0.960491\pi$
$38$	36973.3	+	102331.i	0.109306	+	0.302527i
$39$		0			0
$40$	261628.	+	153520.i	0.646359	+	0.379276i
$41$	−510168.			−1.15603			−0.578016	−	0.816026i	$-0.696173\pi$
$41$	−510168.			−1.15603			−0.578016	+	0.816026i	$0.696173\pi$
$42$		0			0
$43$			784099.i			1.50394i	0.659196	+	0.751971i	$0.270897\pi$
$43$			784099.i			1.50394i	−0.659196	+	0.751971i	$0.729103\pi$
$44$	−578831.	−	696445.i	−1.02440	−	1.23254i
$45$		0			0
$46$	−922172.	+	333191.i	−1.39688	+	0.504709i
$47$	180049.			0.252957			0.126479	−	0.991969i	$-0.459632\pi$
$47$	180049.			0.252957			0.126479	+	0.991969i	$0.459632\pi$
$48$		0			0
$49$	870255.			1.05672
$50$	364410.	−	131666.i	0.412283	−	0.148963i
$51$		0			0
$52$	−1.17252e6	−	1.41076e6i	−1.15640	−	1.39137i
$53$		−	880974.i		−	0.812826i	−0.913689	−	0.406413i	$-0.866779\pi$
$53$		−	880974.i		−	0.812826i	0.913689	−	0.406413i	$-0.133221\pi$
$54$		0			0
$55$	1.48197e6			1.20107
$56$	−953840.	+	1.62553e6i	−0.725801	+	1.23690i
$57$		0			0
$58$	−391323.	−	1.08306e6i	−0.263352	−	0.728880i
$59$			92535.3i			0.0586577i	0.999570	+	0.0293289i	$0.00933701\pi$
$59$			92535.3i			0.0586577i	−0.999570	+	0.0293289i	$0.990663\pi$
$60$		0			0
$61$			2.40486e6i			1.35655i	0.734809	+	0.678274i	$0.237272\pi$
$61$			2.40486e6i			1.35655i	−0.734809	+	0.678274i	$0.762728\pi$
$62$	302320.	−	109232.i	0.161100	−	0.0582073i
$63$		0			0
$64$	−1.02287e6	−	1.83079e6i	−0.487741	−	0.872989i
$65$	3.00197e6			1.35584
$66$		0			0
$67$		−	1.99319e6i		−	0.809630i	−0.914399	−	0.404815i	$-0.867336\pi$
$67$		−	1.99319e6i		−	0.809630i	0.914399	−	0.404815i	$-0.132664\pi$
$68$	1.88998e6	−	1.57080e6i	0.728912	−	0.605815i
$69$		0			0
$70$	−1.04808e6	−	2.90076e6i	−0.365217	−	1.01081i
$71$	−3.60291e6			−1.19467			−0.597337	−	0.801990i	$-0.703775\pi$
$71$	−3.60291e6			−1.19467			−0.597337	+	0.801990i	$0.703775\pi$
$72$		0			0
$73$	1.68585e6			0.507211			0.253605	−	0.967308i	$-0.418384\pi$
$73$	1.68585e6			0.507211			0.253605	+	0.967308i	$0.418384\pi$
$74$	293295.	+	811751.i	0.0841382	+	0.232869i
$75$		0			0
$76$	786826.	+	946703.i	0.205604	+	0.247381i
$77$			9.20766e6i			2.29843i
$78$		0			0
$79$	4.81609e6			1.09901			0.549503	−	0.835492i	$-0.314817\pi$
$79$	4.81609e6			1.09901			0.549503	+	0.835492i	$0.314817\pi$
$80$	3.37406e6	+	627693.i	0.736779	+	0.137067i
$81$		0			0
$82$	−5.42843e6	+	1.96135e6i	−1.08724	+	0.392832i
$83$			1.23628e6i			0.237325i	0.992935	+	0.118662i	$0.0378606\pi$
$83$			1.23628e6i			0.237325i	−0.992935	+	0.118662i	$0.962139\pi$
$84$		0			0
$85$			4.02169e6i			0.710301i
$86$	3.01449e6	+	8.34318e6i	0.511056	+	1.41445i
$87$		0			0
$88$	−8.83654e6	−	5.18518e6i	−1.38227	−	0.811100i
$89$	9.48855e6			1.42671			0.713353	−	0.700805i	$-0.247176\pi$
$89$	9.48855e6			1.42671			0.713353	+	0.700805i	$0.247176\pi$
$90$		0			0
$91$			1.86516e7i			2.59461i
$92$	−8.53138e6	+	7.09062e6i	−1.14225	+	0.949351i
$93$		0			0
$94$	1.91580e6	−	692201.i	0.237905	−	0.0859577i
$95$	−2.01449e6			−0.241065
$96$		0			0
$97$	1.45734e7			1.62129			0.810645	−	0.585538i	$-0.199117\pi$
$97$	1.45734e7			1.62129			0.810645	+	0.585538i	$0.199117\pi$
$98$	9.25992e6	−	3.34571e6i	0.993839	−	0.359085i
$99$		0			0
$100$	3.37131e6	−	2.80197e6i	0.337131	−	0.280197i
$101$		−	1.12238e7i		−	1.08396i	−0.840390	−	0.541982i	$-0.817674\pi$
$101$		−	1.12238e7i		−	1.08396i	0.840390	−	0.541982i	$-0.182326\pi$
$102$		0			0
$103$	1.39417e6			0.125715			0.0628574	−	0.998023i	$-0.479979\pi$
$103$	1.39417e6			0.125715			0.0628574	+	0.998023i	$0.479979\pi$
$104$	−1.78998e7	−	1.05034e7i	−1.56039	−	0.915617i
$105$		0			0
$106$	−3.38693e6	−	9.37398e6i	−0.276207	−	0.764458i
$107$		−	3.22163e6i		−	0.254233i	−0.991888	−	0.127117i	$-0.959428\pi$
$107$		−	3.22163e6i		−	0.254233i	0.991888	−	0.127117i	$-0.0405723\pi$
$108$		0			0
$109$		−	4.22797e6i		−	0.312708i	−0.987701	−	0.156354i	$-0.950026\pi$
$109$		−	4.22797e6i		−	0.312708i	0.987701	−	0.156354i	$-0.0499741\pi$
$110$	1.57688e7	−	5.69746e6i	1.12960	−	0.408138i
$111$		0			0
$112$	−3.89993e6	+	2.09634e7i	−0.262297	+	1.40994i
$113$	2.63919e6			0.172066			0.0860332	−	0.996292i	$-0.472581\pi$
$113$	2.63919e6			0.172066			0.0860332	+	0.996292i	$0.472581\pi$
$114$		0			0
$115$		−	1.81540e7i		−	1.11309i
$116$	−8.32772e6	−	1.00199e7i	−0.495363	−	0.596017i
$117$		0			0
$118$	355754.	+	984619.i	0.0199325	+	0.0551672i
$119$	−2.49873e7			−1.35927
$120$		0			0
$121$	−3.05666e7			−1.56855
$122$	9.24553e6	+	2.55888e7i	0.460970	+	1.27582i
$123$		0			0
$124$	2.79688e6	−	2.32455e6i	0.131734	−	0.109487i
$125$			2.35386e7i			1.07794i
$126$		0			0
$127$	−1.02119e7			−0.442376			−0.221188	−	0.975231i	$-0.570993\pi$
$127$	−1.02119e7			−0.442376			−0.221188	+	0.975231i	$0.570993\pi$
$128$	−1.79223e7	−	1.55480e7i	−0.755368	−	0.655301i
$129$		0			0
$130$	3.19424e7	−	1.15411e7i	1.27516	−	0.460730i
$131$		−	4.73463e7i		−	1.84008i	−0.391825	−	0.920040i	$-0.628156\pi$
$131$		−	4.73463e7i		−	1.84008i	0.391825	−	0.920040i	$-0.371844\pi$
$132$		0			0
$133$		−	1.25163e7i		−	0.461313i
$134$	−7.66286e6	−	2.12085e7i	−0.275121	−	0.761452i
$135$		0			0
$136$	1.40713e7	−	2.39801e7i	0.479675	−	0.817458i
$137$	5.06415e7			1.68261			0.841307	−	0.540557i	$-0.181787\pi$
$137$	5.06415e7			1.68261			0.841307	+	0.540557i	$0.181787\pi$
$138$		0			0
$139$		−	2.58884e7i		−	0.817623i	−0.912619	−	0.408812i	$-0.865943\pi$
$139$		−	2.58884e7i		−	0.817623i	0.912619	−	0.408812i	$-0.134057\pi$
$140$	−2.23041e7	−	2.68361e7i	−0.686968	−	0.826555i
$141$		0			0
$142$	−3.83367e7	+	1.38515e7i	−1.12358	+	0.405964i
$143$	−1.01392e8			−2.89953
$144$		0			0
$145$	2.13213e7			0.580798
$146$	1.79382e7	−	6.48128e6i	0.477029	−	0.172356i
$147$		0			0
$148$	6.24159e6	+	7.50983e6i	0.158263	+	0.190421i
$149$		−	1.48513e7i		−	0.367802i	−0.982945	−	0.183901i	$-0.941127\pi$
$149$		−	1.48513e7i		−	0.367802i	0.982945	−	0.183901i	$-0.0588726\pi$
$150$		0			0
$151$	−2.63661e7			−0.623200			−0.311600	−	0.950213i	$-0.600865\pi$
$151$	−2.63661e7			−0.623200			−0.311600	+	0.950213i	$0.600865\pi$
$152$	1.20118e7	+	7.04840e6i	0.277432	+	0.162794i
$153$		0			0
$154$	3.53991e7	+	9.79739e7i	0.781033	+	2.16166i
$155$			5.95150e6i			0.128371i
$156$		0			0
$157$		−	8.47599e6i		−	0.174800i	−0.996173	−	0.0874001i	$-0.972144\pi$
$157$		−	8.47599e6i		−	0.174800i	0.996173	−	0.0874001i	$-0.0278558\pi$
$158$	5.12455e7	−	1.85156e7i	1.03361	−	0.373454i
$159$		0			0
$160$	3.83147e7	−	6.29269e6i	0.739513	−	0.121455i
$161$	1.12793e8			2.13006
$162$		0			0
$163$			9.54067e6i			0.172553i	0.996271	+	0.0862764i	$0.0274968\pi$
$163$			9.54067e6i			0.172553i	−0.996271	+	0.0862764i	$0.972503\pi$
$164$	−5.02205e7	+	4.17394e7i	−0.889054	+	0.738912i
$165$		0			0
$166$	4.75290e6	+	1.31546e7i	0.0806456	+	0.223202i
$167$	−1.44929e7			−0.240796			−0.120398	−	0.992726i	$-0.538417\pi$
$167$	−1.44929e7			−0.240796			−0.120398	+	0.992726i	$0.538417\pi$
$168$		0			0
$169$	−1.42638e8			−2.27316
$170$	1.54615e7	+	4.27927e7i	0.241368	+	0.668034i
$171$		0			0
$172$	6.41511e7	+	7.71862e7i	0.961291	+	1.15662i
$173$			6.08908e7i			0.894109i	0.894507	+	0.447054i	$0.147527\pi$
$173$			6.08908e7i			0.894109i	−0.894507	+	0.447054i	$0.852473\pi$
$174$		0			0
$175$	−4.45719e7			−0.628677
$176$	−1.13959e8	−	2.12004e7i	−1.57564	−	0.293124i
$177$		0			0
$178$	1.00963e8	−	3.64789e7i	1.34181	−	0.484811i
$179$		−	2.27852e7i		−	0.296939i	−0.988917	−	0.148469i	$-0.952565\pi$
$179$		−	2.27852e7i		−	0.296939i	0.988917	−	0.148469i	$-0.0474347\pi$
$180$		0			0
$181$		−	8.06800e6i		−	0.101132i	−0.998721	−	0.0505662i	$-0.983897\pi$
$181$		−	8.06800e6i		−	0.101132i	0.998721	−	0.0505662i	$-0.0161026\pi$
$182$	7.17066e7	+	1.98462e8i	0.881676	+	2.44021i
$183$		0			0
$184$	−6.35179e7	+	1.08247e8i	−0.751681	+	1.28101i
$185$	−1.59802e7			−0.185559
$186$		0			0
$187$		−	1.35833e8i		−	1.51901i
$188$	1.77239e7	−	1.47307e7i	0.194539	−	0.161685i
$189$		0			0
$190$	−2.14352e7	+	7.74477e6i	−0.226720	+	0.0819164i
$191$	4.57539e7			0.475129			0.237564	−	0.971372i	$-0.423651\pi$
$191$	4.57539e7			0.475129			0.237564	+	0.971372i	$0.423651\pi$
$192$		0			0
$193$	1.21552e8			1.21706			0.608529	−	0.793532i	$-0.291760\pi$
$193$	1.21552e8			1.21706			0.608529	+	0.793532i	$0.291760\pi$
$194$	1.55068e8	−	5.60279e7i	1.52481	−	0.550932i
$195$		0			0
$196$	8.56672e7	−	7.12000e7i	0.812678	−	0.675435i
$197$			1.71007e8i			1.59361i	0.604238	+	0.796804i	$0.293478\pi$
$197$			1.71007e8i			1.59361i	−0.604238	+	0.796804i	$0.706522\pi$
$198$		0			0
$199$	−1.51211e7			−0.136018			−0.0680091	−	0.997685i	$-0.521665\pi$
$199$	−1.51211e7			−0.136018			−0.0680091	+	0.997685i	$0.521665\pi$
$200$	2.51001e7	−	4.27753e7i	0.221855	−	0.378084i
$201$		0			0
$202$	−4.31501e7	−	1.19426e8i	−0.368343	−	1.01946i
$203$			1.32472e8i			1.11144i
$204$		0			0
$205$		−	1.06865e8i		−	0.866353i
$206$	1.48347e7	−	5.35993e6i	0.118234	−	0.0427193i
$207$		0			0
$208$	−2.30843e8	−	4.29449e7i	−1.77867	−	0.330895i
$209$	6.80399e7			0.515528
$210$		0			0
$211$			2.28562e8i			1.67500i	0.546437	+	0.837500i	$0.315984\pi$
$211$			2.28562e8i			1.67500i	−0.546437	+	0.837500i	$0.684016\pi$
$212$	−7.20770e7	−	8.67225e7i	−0.519543	−	0.625110i
$213$		0			0
$214$	−1.23856e7	−	3.42797e7i	−0.0863913	−	0.239105i
$215$	−1.64245e8			−1.12709
$216$		0			0
$217$	−3.69774e7			−0.245656
$218$	−1.62545e7	−	4.49876e7i	−0.106262	−	0.294100i
$219$		0			0
$220$	1.45884e8	−	1.21247e8i	0.923694	−	0.767703i
$221$		−	2.75153e8i		−	1.71475i
$222$		0			0
$223$	5.65483e7			0.341470			0.170735	−	0.985317i	$-0.445386\pi$
$223$	5.65483e7			0.341470			0.170735	+	0.985317i	$0.445386\pi$
$224$	3.90973e7	+	2.38054e8i	0.232423	+	1.41517i
$225$		0			0
$226$	2.80822e7	−	1.01464e7i	0.161827	−	0.0584700i
$227$		−	1.78749e8i		−	1.01427i	−0.861867	−	0.507135i	$-0.830704\pi$
$227$		−	1.78749e8i		−	1.01427i	0.861867	−	0.507135i	$-0.169296\pi$
$228$		0			0
$229$		−	5.90997e7i		−	0.325208i	−0.986691	−	0.162604i	$-0.948011\pi$
$229$		−	5.90997e7i		−	0.325208i	0.986691	−	0.162604i	$-0.0519893\pi$
$230$	−6.97934e7	−	1.93167e8i	−0.378239	−	1.04685i
$231$		0			0
$232$	−1.27132e8	−	7.45998e7i	−0.668419	−	0.392220i
$233$	1.82022e8			0.942709			0.471355	−	0.881944i	$-0.343765\pi$
$233$	1.82022e8			0.942709			0.471355	+	0.881944i	$0.343765\pi$
$234$		0			0
$235$			3.77147e7i			0.189571i
$236$	7.57078e6	+	9.10910e6i	0.0374929	+	0.0451111i
$237$		0			0
$238$	−2.65876e8	+	9.60642e7i	−1.27838	+	0.461894i
$239$	−6.73710e7			−0.319213			−0.159606	−	0.987181i	$-0.551023\pi$
$239$	−6.73710e7			−0.319213			−0.159606	+	0.987181i	$0.551023\pi$
$240$		0			0
$241$	1.70435e8			0.784333			0.392166	−	0.919894i	$-0.371726\pi$
$241$	1.70435e8			0.784333			0.392166	+	0.919894i	$0.371726\pi$
$242$	−3.25243e8	+	1.17514e8i	−1.47521	+	0.533011i
$243$		0			0
$244$	1.96754e8	+	2.36732e8i	0.867078	+	1.04326i
$245$			1.82292e8i			0.791928i
$246$		0			0
$247$	1.37826e8			0.581958
$248$	2.08234e7	−	3.54870e7i	0.0866902	−	0.147737i
$249$		0			0
$250$	9.04948e7	+	2.50462e8i	0.366297	+	1.01380i
$251$		−	9.22637e7i		−	0.368275i	−0.982900	−	0.184138i	$-0.941051\pi$
$251$		−	9.22637e7i		−	0.368275i	0.982900	−	0.184138i	$-0.0589492\pi$
$252$		0			0
$253$			6.13154e8i			2.38039i
$254$	−1.08659e8	+	3.92597e7i	−0.416052	+	0.150324i
$255$		0			0
$256$	−2.50476e8	−	9.65358e7i	−0.933097	−	0.359624i
$257$	−4.30168e8			−1.58078			−0.790391	−	0.612603i	$-0.790123\pi$
$257$	−4.30168e8			−1.58078			−0.790391	+	0.612603i	$0.790123\pi$
$258$		0			0
$259$		−	9.92870e7i		−	0.355094i
$260$	2.95512e8	−	2.45606e8i	1.04272	−	0.866628i
$261$		0			0
$262$	−1.82024e8	−	5.03787e8i	−0.625279	−	1.73058i
$263$	1.87195e8			0.634526			0.317263	−	0.948338i	$-0.397236\pi$
$263$	1.87195e8			0.634526			0.317263	+	0.948338i	$0.397236\pi$
$264$		0			0
$265$	1.84537e8			0.609149
$266$	−4.81192e7	−	1.33179e8i	−0.156759	−	0.433862i
$267$		0			0
$268$	−1.63073e8	−	1.96208e8i	−0.517500	−	0.622652i
$269$		−	3.68317e8i		−	1.15369i	−0.816854	−	0.576845i	$-0.804284\pi$
$269$		−	3.68317e8i		−	1.15369i	0.816854	−	0.576845i	$-0.195716\pi$
$270$		0			0
$271$	4.53681e8			1.38471			0.692354	−	0.721558i	$-0.256574\pi$
$271$	4.53681e8			1.38471			0.692354	+	0.721558i	$0.256574\pi$
$272$	5.75327e7	−	3.09257e8i	0.173350	−	0.931813i
$273$		0			0
$274$	5.38850e8	−	1.94692e8i	1.58249	−	0.571771i
$275$		−	2.42297e8i		−	0.702561i
$276$		0			0
$277$			4.63451e8i			1.31016i	0.755559	+	0.655081i	$0.227365\pi$
$277$			4.63451e8i			1.31016i	−0.755559	+	0.655081i	$0.772635\pi$
$278$	−9.95285e7	−	2.75465e8i	−0.277837	−	0.768969i
$279$		0			0
$280$	−3.40498e8	−	1.99800e8i	−0.926962	−	0.543930i
$281$	−5.49202e8			−1.47659			−0.738295	−	0.674478i	$-0.764369\pi$
$281$	−5.49202e8			−1.47659			−0.738295	+	0.674478i	$0.764369\pi$
$282$		0			0
$283$			4.01427e8i			1.05282i	0.850231	+	0.526410i	$0.176462\pi$
$283$			4.01427e8i			1.05282i	−0.850231	+	0.526410i	$0.823538\pi$
$284$	−3.54668e8	+	2.94773e8i	−0.918772	+	0.763612i
$285$		0			0
$286$	−1.07886e9	+	3.89804e8i	−2.72699	+	0.985293i
$287$	6.63963e8			1.65790
$288$		0			0
$289$	−4.17209e7			−0.101674
$290$	2.26869e8	−	8.19702e7i	0.546237	−	0.197362i
$291$		0			0
$292$	1.65954e8	−	1.37928e8i	0.390074	−	0.324199i
$293$		−	4.20278e8i		−	0.976112i	−0.872812	−	0.488056i	$-0.837706\pi$
$293$		−	4.20278e8i		−	0.976112i	0.872812	−	0.488056i	$-0.162294\pi$
$294$		0			0
$295$	−1.93833e7			−0.0439593
$296$	9.52852e7	+	5.59122e7i	0.213552	+	0.125310i
$297$		0			0
$298$	−5.70964e7	−	1.58025e8i	−0.124983	−	0.345915i
$299$			1.24204e9i			2.68712i
$300$		0			0
$301$		−	1.02047e9i		−	2.15685i
$302$	−2.80548e8	+	1.01365e8i	−0.586115	+	0.211770i
$303$		0			0
$304$	1.54909e8	+	2.88185e7i	0.316242	+	0.0588321i
$305$	−5.03744e8			−1.01662
$306$		0			0
$307$		−	4.12131e8i		−	0.812925i	−0.913667	−	0.406463i	$-0.866762\pi$
$307$		−	4.12131e8i		−	0.812925i	0.913667	−	0.406463i	$-0.133238\pi$
$308$	7.53326e8	+	9.06396e8i	1.46911	+	1.76763i
$309$		0			0
$310$	2.28807e7	+	6.33268e7i	0.0436218	+	0.120732i
$311$	6.54901e8			1.23457			0.617283	−	0.786741i	$-0.288233\pi$
$311$	6.54901e8			1.23457			0.617283	+	0.786741i	$0.288233\pi$
$312$		0			0
$313$	3.92213e8			0.722963			0.361482	−	0.932379i	$-0.382271\pi$
$313$	3.92213e8			0.722963			0.361482	+	0.932379i	$0.382271\pi$
$314$	−3.25861e7	−	9.01886e7i	−0.0593990	−	0.164398i
$315$		0			0
$316$	4.74092e8	−	3.94029e8i	0.845197	−	0.702462i
$317$			6.67005e8i			1.17604i	0.808847	+	0.588019i	$0.200092\pi$
$317$			6.67005e8i			1.17604i	−0.808847	+	0.588019i	$0.799908\pi$
$318$		0			0
$319$	−7.20131e8			−1.24206
$320$	3.83494e8	−	2.14259e8i	0.654236	−	0.365523i
$321$		0			0
$322$	1.20017e9	−	4.33635e8i	2.00331	−	0.723817i
$323$			1.84643e8i			0.304877i
$324$		0			0
$325$		−	4.90812e8i		−	0.793092i
$326$	3.66793e7	+	1.01517e8i	0.0586354	+	0.162285i
$327$		0			0
$328$	−3.73902e8	+	6.37201e8i	−0.585059	+	0.997053i
$329$	−2.34326e8			−0.362773
$330$		0			0
$331$			5.23035e8i			0.792744i	0.918090	+	0.396372i	$0.129731\pi$
$331$			5.23035e8i			0.792744i	−0.918090	+	0.396372i	$0.870269\pi$
$332$	1.01146e8	+	1.21698e8i	0.151693	+	0.182516i
$333$		0			0
$334$	−1.54212e8	+	5.57184e7i	−0.226467	+	0.0818250i
$335$	4.17512e8			0.606754
$336$		0			0
$337$	−1.13571e9			−1.61645			−0.808224	−	0.588875i	$-0.799571\pi$
$337$	−1.13571e9			−1.61645			−0.808224	+	0.588875i	$0.799571\pi$
$338$	−1.51773e9	+	5.48374e8i	−2.13790	+	0.772446i
$339$		0			0
$340$	3.29035e8	+	3.95892e8i	0.454010	+	0.546262i
$341$		−	2.01013e8i		−	0.274527i
$342$		0			0
$343$	−6.07934e7			−0.0813443
$344$	9.79342e8	+	5.74667e8i	1.29712	+	0.761134i
$345$		0			0
$346$	2.34096e8	+	6.47907e8i	0.303828	+	0.840904i
$347$		−	9.94124e8i		−	1.27728i	−0.769504	−	0.638642i	$-0.779497\pi$
$347$		−	9.94124e8i		−	1.27728i	0.769504	−	0.638642i	$-0.220503\pi$
$348$		0			0
$349$			8.70786e8i			1.09653i	0.836303	+	0.548267i	$0.184712\pi$
$349$			8.70786e8i			1.09653i	−0.836303	+	0.548267i	$0.815288\pi$
$350$	−4.74266e8	+	1.71358e8i	−0.591267	+	0.213631i
$351$		0			0
$352$	−1.29409e9	+	2.12537e8i	−1.58148	+	0.259738i
$353$	7.47327e8			0.904273			0.452136	−	0.891949i	$-0.350662\pi$
$353$	7.47327e8			0.904273			0.452136	+	0.891949i	$0.350662\pi$
$354$		0			0
$355$		−	7.54700e8i		−	0.895313i
$356$	9.34046e8	−	7.76306e8i	1.09722	−	0.911923i
$357$		0			0
$358$	−8.75982e7	−	2.42445e8i	−0.100903	−	0.279269i
$359$	8.39765e8			0.957915			0.478958	−	0.877838i	$-0.341015\pi$
$359$	8.39765e8			0.957915			0.478958	+	0.877838i	$0.341015\pi$
$360$		0			0
$361$	8.01383e8			0.896530
$362$	−3.10176e7	−	8.58473e7i	−0.0343659	−	0.0951145i
$363$		0			0
$364$	1.52598e9	+	1.83605e9i	1.65842	+	1.99540i
$365$			3.53134e8i			0.380114i
$366$		0			0
$367$	2.85579e8			0.301575			0.150788	−	0.988566i	$-0.451819\pi$
$367$	2.85579e8			0.301575			0.150788	+	0.988566i	$0.451819\pi$
$368$	−2.59703e8	+	1.39599e9i	−0.271650	+	1.46021i
$369$		0			0
$370$	−1.70037e8	+	6.14363e7i	−0.174517	+	0.0630549i
$371$			1.14655e9i			1.16570i
$372$		0			0
$373$		−	2.13291e8i		−	0.212810i	−0.994323	−	0.106405i	$-0.966066\pi$
$373$		−	2.13291e8i		−	0.212810i	0.994323	−	0.106405i	$-0.0339340\pi$
$374$	−5.22215e8	−	1.44533e9i	−0.516177	−	1.42862i
$375$		0			0
$376$	1.31958e8	−	2.24881e8i	0.128020	−	0.218170i
$377$	−1.45874e9			−1.40212
$378$		0			0
$379$		−	1.77655e8i		−	0.167625i	−0.996482	−	0.0838127i	$-0.973290\pi$
$379$		−	1.77655e8i		−	0.167625i	0.996482	−	0.0838127i	$-0.0267098\pi$
$380$	−1.98305e8	+	1.64816e8i	−0.185392	+	0.154084i
$381$		0			0
$382$	4.86843e8	−	1.75902e8i	0.446856	−	0.161454i
$383$	−1.03999e9			−0.945876			−0.472938	−	0.881096i	$-0.656807\pi$
$383$	−1.03999e9			−0.945876			−0.472938	+	0.881096i	$0.656807\pi$
$384$		0			0
$385$	−1.92872e9			−1.72249
$386$	1.29337e9	−	4.67309e8i	1.14464	−	0.413570i
$387$		0			0
$388$	1.43460e9	−	1.19233e9i	1.24686	−	1.03630i
$389$		−	1.52613e9i		−	1.31452i	−0.753663	−	0.657261i	$-0.771715\pi$
$389$		−	1.52613e9i		−	1.31452i	0.753663	−	0.657261i	$-0.228285\pi$
$390$		0			0
$391$	−1.66395e9			−1.40773
$392$	6.37810e8	−	1.08695e9i	0.534798	−	0.911399i
$393$		0			0
$394$	6.57439e8	+	1.81959e9i	0.541525	+	1.49878i
$395$			1.00882e9i			0.823617i
$396$		0			0
$397$		−	1.40941e9i		−	1.13050i	−0.824919	−	0.565250i	$-0.808780\pi$
$397$		−	1.40941e9i		−	1.13050i	0.824919	−	0.565250i	$-0.191220\pi$
$398$	−1.60895e8	+	5.81334e7i	−0.127924	+	0.0462205i
$399$		0			0
$400$	1.02626e8	−	5.51648e8i	0.0801764	−	0.430975i
$401$	−1.43905e9			−1.11448			−0.557238	−	0.830353i	$-0.688139\pi$
$401$	−1.43905e9			−1.11448			−0.557238	+	0.830353i	$0.688139\pi$
$402$		0			0
$403$		−	4.07185e8i		−	0.309902i
$404$	−9.18276e8	−	1.10486e9i	−0.692849	−	0.833630i
$405$		0			0
$406$	5.09291e8	+	1.40956e9i	0.377681	+	1.04531i
$407$	5.39735e8			0.396826
$408$		0			0
$409$	−2.10100e8			−0.151843			−0.0759214	−	0.997114i	$-0.524190\pi$
$409$	−2.10100e8			−0.151843			−0.0759214	+	0.997114i	$0.524190\pi$
$410$	−4.10843e8	−	1.13709e9i	−0.294396	−	0.814800i
$411$		0			0
$412$	1.37241e8	−	1.14064e8i	0.0966818	−	0.0803544i
$413$		−	1.20431e8i		−	0.0841226i
$414$		0			0
$415$	−2.58962e8			−0.177856
$416$	−2.62139e9	+	4.30528e8i	−1.78527	+	0.293207i
$417$		0			0
$418$	7.23977e8	−	2.61581e8i	0.484850	−	0.175182i
$419$		−	2.02032e9i		−	1.34175i	−0.741572	−	0.670873i	$-0.765920\pi$
$419$		−	2.02032e9i		−	1.34175i	0.741572	−	0.670873i	$-0.234080\pi$
$420$		0			0
$421$		−	8.78781e8i		−	0.573975i	−0.957934	−	0.286988i	$-0.907346\pi$
$421$		−	8.78781e8i		−	0.573975i	0.957934	−	0.286988i	$-0.0926539\pi$
$422$	8.78711e8	+	2.43200e9i	0.569184	+	1.57533i
$423$		0			0
$424$	−1.10034e9	−	6.45666e8i	−0.701046	−	0.411365i
$425$	6.57534e8			0.415486
$426$		0			0
$427$		−	3.12983e9i		−	1.94546i
$428$	−2.63578e8	−	3.17135e8i	−0.162501	−	0.195520i
$429$		0			0
$430$	−1.74764e9	+	6.31443e8i	−1.06002	+	0.382996i
$431$	1.53441e9			0.923150			0.461575	−	0.887101i	$-0.347285\pi$
$431$	1.53441e9			0.923150			0.461575	+	0.887101i	$0.347285\pi$
$432$		0			0
$433$	4.25446e8			0.251847			0.125924	−	0.992040i	$-0.459811\pi$
$433$	4.25446e8			0.251847			0.125924	+	0.992040i	$0.459811\pi$
$434$	−3.93457e8	+	1.42161e8i	−0.231038	+	0.0834767i
$435$		0			0
$436$	−3.45912e8	−	4.16198e8i	−0.199877	−	0.240490i
$437$		−	8.33482e8i		−	0.477762i
$438$		0			0
$439$	3.51493e8			0.198285			0.0991427	−	0.995073i	$-0.468390\pi$
$439$	3.51493e8			0.198285			0.0991427	+	0.995073i	$0.468390\pi$
$440$	1.08614e9	−	1.85098e9i	0.607855	−	1.03590i
$441$		0			0
$442$	−1.05783e9	−	2.92775e9i	−0.582691	−	1.61271i
$443$		−	3.38530e9i		−	1.85005i	−0.379904	−	0.925026i	$-0.624043\pi$
$443$		−	3.38530e9i		−	1.85005i	0.379904	−	0.925026i	$-0.375957\pi$
$444$		0			0
$445$			1.98756e9i			1.06920i
$446$	6.01700e8	−	2.17401e8i	0.321150	−	0.116035i
$447$		0			0
$448$	1.33122e9	+	2.38270e9i	0.699482	+	1.25198i
$449$	−3.28992e7			−0.0171523			−0.00857617	−	0.999963i	$-0.502730\pi$
$449$	−3.28992e7			−0.0171523			−0.00857617	+	0.999963i	$0.502730\pi$
$450$		0			0
$451$			3.60937e9i			1.85274i
$452$	2.59800e8	−	2.15925e8i	0.132329	−	0.109981i
$453$		0			0
$454$	−6.87206e8	−	1.90198e9i	−0.344660	−	0.953915i
$455$	−3.90694e9			−1.94445
$456$		0			0
$457$	7.54142e8			0.369612			0.184806	−	0.982775i	$-0.440834\pi$
$457$	7.54142e8			0.369612			0.184806	+	0.982775i	$0.440834\pi$
$458$	−2.27210e8	−	6.28849e8i	−0.110509	−	0.305856i
$459$		0			0
$460$	−1.48527e9	−	1.78706e9i	−0.711463	−	0.856027i
$461$			1.59847e9i			0.759893i	0.925008	+	0.379946i	$0.124058\pi$
$461$			1.59847e9i			0.759893i	−0.925008	+	0.379946i	$0.875942\pi$
$462$		0			0
$463$	2.90911e9			1.36216			0.681079	−	0.732210i	$-0.261511\pi$
$463$	2.90911e9			1.36216			0.681079	+	0.732210i	$0.261511\pi$
$464$	−1.63955e9	−	3.05014e8i	−0.761924	−	0.141745i
$465$		0			0
$466$	1.93680e9	−	6.99787e8i	0.886612	−	0.320343i
$467$			2.61345e9i			1.18742i	0.804679	+	0.593711i	$0.202338\pi$
$467$			2.61345e9i			1.18742i	−0.804679	+	0.593711i	$0.797662\pi$
$468$		0			0
$469$			2.59406e9i			1.16111i
$470$	1.44995e8	+	4.01302e8i	0.0644185	+	0.178291i
$471$		0			0
$472$	1.15577e8	+	6.78191e7i	0.0505911	+	0.0296862i
$473$	5.54740e9			2.41032
$474$		0			0
$475$			3.29363e8i			0.141009i
$476$	−2.45973e9	+	2.04434e9i	−1.04535	+	0.868816i
$477$		0			0
$478$	−7.16859e8	+	2.59009e8i	−0.300218	+	0.108472i
$479$	−3.07163e9			−1.27701			−0.638506	−	0.769617i	$-0.720447\pi$
$479$	−3.07163e9			−1.27701			−0.638506	+	0.769617i	$0.720447\pi$
$480$		0			0
$481$	1.09332e9			0.447960
$482$	1.81351e9	−	6.55243e8i	0.737660	−	0.266525i
$483$		0			0
$484$	−3.00896e9	+	2.50081e9i	−1.20631	+	1.00259i
$485$			3.05269e9i			1.21503i
$486$		0			0
$487$	5.06292e8			0.198632			0.0993161	−	0.995056i	$-0.468334\pi$
$487$	5.06292e8			0.198632			0.0993161	+	0.995056i	$0.468334\pi$
$488$	3.00367e9	+	1.76252e9i	1.16999	+	0.686538i
$489$		0			0
$490$	7.00825e8	+	1.93967e9i	0.269106	+	0.744803i
$491$			1.83863e9i			0.700985i	0.936566	+	0.350492i	$0.113986\pi$
$491$			1.83863e9i			0.700985i	−0.936566	+	0.350492i	$0.886014\pi$
$492$		0			0
$493$		−	1.95425e9i		−	0.734542i
$494$	1.46653e9	−	5.29875e8i	0.547328	−	0.197756i
$495$		0			0
$496$	8.51398e7	−	4.57655e8i	0.0313290	−	0.168404i
$497$	4.68905e9			1.71331
$498$		0			0
$499$			3.44032e9i			1.23950i	0.784799	+	0.619750i	$0.212766\pi$
$499$			3.44032e9i			1.23950i	−0.784799	+	0.619750i	$0.787234\pi$
$500$	1.92581e9	+	2.31712e9i	0.689000	+	0.829000i
$501$		0			0
$502$	−3.54710e8	−	9.81729e8i	−0.125144	−	0.346361i
$503$	−2.78650e9			−0.976271			−0.488136	−	0.872768i	$-0.662323\pi$
$503$	−2.78650e9			−0.976271			−0.488136	+	0.872768i	$0.662323\pi$
$504$		0			0
$505$	2.35104e9			0.812345
$506$	2.35728e9	+	6.52425e9i	0.808882	+	2.23874i
$507$		0			0
$508$	−1.00525e9	+	8.35484e8i	−0.340212	+	0.282758i
$509$		−	7.16492e8i		−	0.240824i	−0.992724	−	0.120412i	$-0.961579\pi$
$509$		−	7.16492e8i		−	0.240824i	0.992724	−	0.120412i	$-0.0384215\pi$
$510$		0			0
$511$	−2.19406e9			−0.727405
$512$	−3.03632e9	−	6.42236e7i	−0.999776	−	0.0211471i
$513$		0			0
$514$	−4.57719e9	+	1.65379e9i	−1.48672	+	0.537167i
$515$			2.92036e8i			0.0942132i
$516$		0			0
$517$		−	1.27382e9i		−	0.405407i
$518$	−3.81711e8	−	1.05646e9i	−0.120665	−	0.333964i
$519$		0			0
$520$	2.20014e9	−	3.74947e9i	0.686182	−	1.16939i
$521$	−1.71400e8			−0.0530980			−0.0265490	−	0.999648i	$-0.508452\pi$
$521$	−1.71400e8			−0.0530980			−0.0265490	+	0.999648i	$0.508452\pi$
$522$		0			0
$523$			4.14809e9i			1.26792i	0.773365	+	0.633961i	$0.218572\pi$
$523$			4.14809e9i			1.26792i	−0.773365	+	0.633961i	$0.781428\pi$
$524$	−3.87364e9	−	4.66074e9i	−1.17614	−	1.41513i
$525$		0			0
$526$	1.99184e9	−	7.19676e8i	0.596767	−	0.215619i
$527$	5.45499e8			0.162352
$528$		0			0
$529$	4.10625e9			1.20601
$530$	1.96356e9	−	7.09457e8i	0.572900	−	0.206995i
$531$		0			0
$532$	−1.02402e9	−	1.23210e9i	−0.294862	−	0.354776i
$533$			7.31137e9i			2.09148i
$534$		0			0
$535$	6.74833e8			0.190528
$536$	−2.48950e9	−	1.46081e9i	−0.698289	−	0.409748i
$537$		0			0
$538$	−1.41600e9	−	3.91907e9i	−0.392036	−	1.08504i
$539$		−	6.15694e9i		−	1.69357i
$540$		0			0
$541$			4.07583e9i			1.10669i	0.832953	+	0.553344i	$0.186649\pi$
$541$			4.07583e9i			1.10669i	−0.832953	+	0.553344i	$0.813351\pi$
$542$	4.82738e9	−	1.74419e9i	1.30231	−	0.470539i
$543$		0			0
$544$	−5.76772e8	−	3.51183e9i	−0.153606	−	0.935271i
$545$	8.85630e8			0.234350
$546$		0			0
$547$		−	3.64381e9i		−	0.951919i	−0.879467	−	0.475960i	$-0.842101\pi$
$547$		−	3.64381e9i		−	0.951919i	0.879467	−	0.475960i	$-0.157899\pi$
$548$	4.98511e9	−	4.14324e9i	1.29403	−	1.07549i
$549$		0			0
$550$	−9.31517e8	−	2.57816e9i	−0.238738	−	0.660754i
$551$	9.78900e8			0.249292
$552$		0			0
$553$	−6.26795e9			−1.57611
$554$	1.78175e9	+	4.93134e9i	0.445207	+	1.23220i
$555$		0			0
$556$	−2.11806e9	−	2.54843e9i	−0.522609	−	0.628799i
$557$		−	4.47720e9i		−	1.09777i	−0.835896	−	0.548887i	$-0.815052\pi$
$557$		−	4.47720e9i		−	1.09777i	0.835896	−	0.548887i	$-0.184948\pi$
$558$		0			0
$559$	1.12372e10			2.72092
$560$	−4.39120e9	−	8.16917e8i	−1.05664	−	0.196571i
$561$		0			0
$562$	−5.84376e9	+	2.11142e9i	−1.38872	+	0.501761i
$563$			3.55388e9i			0.839311i	0.907683	+	0.419656i	$0.137849\pi$
$563$			3.55388e9i			0.839311i	−0.907683	+	0.419656i	$0.862151\pi$
$564$		0			0
$565$			5.52829e8i			0.128950i
$566$	1.54329e9	+	4.27137e9i	0.357760	+	0.990170i
$567$		0			0
$568$	−2.64058e9	+	4.50005e9i	−0.604616	+	1.03038i
$569$	−7.15092e8			−0.162731			−0.0813653	−	0.996684i	$-0.525928\pi$
$569$	−7.15092e8			−0.162731			−0.0813653	+	0.996684i	$0.525928\pi$
$570$		0			0
$571$		−	5.46695e9i		−	1.22891i	−0.788954	−	0.614453i	$-0.789377\pi$
$571$		−	5.46695e9i		−	1.22891i	0.788954	−	0.614453i	$-0.210623\pi$
$572$	−9.98097e9	+	8.29540e9i	−2.22991	+	1.85332i
$573$		0			0
$574$	7.06488e9	−	2.55262e9i	1.55924	−	0.563371i
$575$	−2.96812e9			−0.651094
$576$		0			0
$577$	5.63907e9			1.22206			0.611030	−	0.791608i	$-0.290756\pi$
$577$	5.63907e9			1.22206			0.611030	+	0.791608i	$0.290756\pi$
$578$	−4.43930e8	+	1.60397e8i	−0.0956240	+	0.0345500i
$579$		0			0
$580$	2.09885e9	−	1.74440e9i	0.446667	−	0.371235i
$581$		−	1.60897e9i		−	0.340354i
$582$		0			0
$583$	−6.23278e9			−1.30269
$584$	1.23556e9	−	2.10563e9i	0.256696	−	0.437459i
$585$		0			0
$586$	−1.61577e9	−	4.47195e9i	−0.331694	−	0.918027i
$587$		−	1.70729e9i		−	0.348396i	−0.984711	−	0.174198i	$-0.944267\pi$
$587$		−	1.70729e9i		−	0.348396i	0.984711	−	0.174198i	$-0.0557332\pi$
$588$		0			0
$589$			2.73245e8i			0.0550995i
$590$	−2.06247e8	+	7.45196e7i	−0.0413434	+	0.0149379i
$591$		0			0
$592$	1.22884e9	+	2.28606e8i	0.243426	+	0.0452858i
$593$	4.95214e9			0.975218			0.487609	−	0.873062i	$-0.337869\pi$
$593$	4.95214e9			0.975218			0.487609	+	0.873062i	$0.337869\pi$
$594$		0			0
$595$		−	5.23407e9i		−	1.01866i
$596$	−1.21506e9	−	1.46196e9i	−0.235092	−	0.282861i
$597$		0			0
$598$	4.77506e9	+	1.32159e10i	0.913114	+	2.52722i
$599$	−3.14394e8			−0.0597697			−0.0298848	−	0.999553i	$-0.509514\pi$
$599$	−3.14394e8			−0.0597697			−0.0298848	+	0.999553i	$0.509514\pi$
$600$		0			0
$601$	−7.52006e9			−1.41306			−0.706530	−	0.707683i	$-0.749741\pi$
$601$	−7.52006e9			−1.41306			−0.706530	+	0.707683i	$0.749741\pi$
$602$	−3.92323e9	−	1.08583e10i	−0.732920	−	2.02850i
$603$		0			0
$604$	−2.59546e9	+	2.15715e9i	−0.479276	+	0.398337i
$605$		−	6.40277e9i		−	1.17550i
$606$		0			0
$607$	−9.76620e9			−1.77241			−0.886207	−	0.463290i	$-0.846669\pi$
$607$	−9.76620e9			−1.77241			−0.886207	+	0.463290i	$0.846669\pi$
$608$	1.75910e9	−	2.88909e8i	0.317415	−	0.0521313i
$609$		0			0
$610$	−5.36007e9	+	1.93666e9i	−0.956129	+	0.345460i
$611$		−	2.58033e9i		−	0.457648i
$612$		0			0
$613$			4.97523e9i			0.872371i	0.899857	+	0.436185i	$0.143671\pi$
$613$			4.97523e9i			0.872371i	−0.899857	+	0.436185i	$0.856329\pi$
$614$	−1.58445e9	−	4.38526e9i	−0.276241	−	0.764551i
$615$		0			0
$616$	1.15004e10	+	6.74830e9i	1.98235	+	1.16322i
$617$	−2.86237e9			−0.490600			−0.245300	−	0.969447i	$-0.578887\pi$
$617$	−2.86237e9			−0.490600			−0.245300	+	0.969447i	$0.578887\pi$
$618$		0			0
$619$			8.71027e9i			1.47609i	0.674749	+	0.738047i	$0.264252\pi$
$619$			8.71027e9i			1.47609i	−0.674749	+	0.738047i	$0.735748\pi$
$620$	4.86923e8	+	5.85862e8i	0.0820520	+	0.0987243i
$621$		0			0
$622$	6.96846e9	−	2.51778e9i	1.16110	−	0.419519i
$623$	−1.23490e10			−2.04608
$624$		0			0
$625$	−2.25503e9			−0.369464
$626$	4.17333e9	−	1.50787e9i	0.679942	−	0.245671i
$627$		0			0
$628$	−6.93464e8	−	8.34371e8i	−0.111729	−	0.134431i
$629$			1.46470e9i			0.234678i
$630$		0			0
$631$	−1.83120e8			−0.0290157			−0.0145078	−	0.999895i	$-0.504618\pi$
$631$	−1.83120e8			−0.0290157			−0.0145078	+	0.999895i	$0.504618\pi$
$632$	3.52971e9	−	6.01531e9i	0.556198	−	0.947869i
$633$		0			0
$634$	2.56432e9	+	7.09724e9i	0.399631	+	1.10606i
$635$		−	2.13907e9i		−	0.331526i
$636$		0			0
$637$		−	1.24719e10i		−	1.91181i
$638$	−7.66253e9	+	2.76856e9i	−1.16815	+	0.422067i
$639$		0			0
$640$	3.25684e9	−	3.75417e9i	0.491096	−	0.566088i
$641$	−2.45117e9			−0.367596			−0.183798	−	0.982964i	$-0.558839\pi$
$641$	−2.45117e9			−0.367596			−0.183798	+	0.982964i	$0.558839\pi$
$642$		0			0
$643$		−	3.25861e9i		−	0.483387i	−0.970353	−	0.241693i	$-0.922297\pi$
$643$		−	3.25861e9i		−	0.483387i	0.970353	−	0.241693i	$-0.0777028\pi$
$644$	1.11033e10	−	9.22816e9i	1.63813	−	1.36149i
$645$		0			0
$646$	7.09865e8	+	1.96469e9i	0.103601	+	0.286735i
$647$	6.42294e8			0.0932328			0.0466164	−	0.998913i	$-0.485156\pi$
$647$	6.42294e8			0.0932328			0.0466164	+	0.998913i	$0.485156\pi$
$648$		0			0
$649$	6.54675e8			0.0940090
$650$	−1.88694e9	−	5.22248e9i	−0.269502	−	0.745898i
$651$		0			0
$652$	7.80571e8	+	9.39177e8i	0.110292	+	0.132703i
$653$			1.32066e10i			1.85607i	0.372491	+	0.928036i	$0.378504\pi$
$653$			1.32066e10i			1.85607i	−0.372491	+	0.928036i	$0.621496\pi$
$654$		0			0
$655$	9.91760e9			1.37899
$656$	−1.52876e9	+	8.21760e9i	−0.211435	+	1.13653i
$657$		0			0
$658$	−2.49334e9	+	9.00872e8i	−0.341186	+	0.123274i
$659$		−	6.38485e9i		−	0.869064i	−0.900656	−	0.434532i	$-0.856914\pi$
$659$		−	6.38485e9i		−	0.869064i	0.900656	−	0.434532i	$-0.143086\pi$
$660$		0			0
$661$		−	1.12927e10i		−	1.52087i	−0.649416	−	0.760433i	$-0.724987\pi$
$661$		−	1.12927e10i		−	1.52087i	0.649416	−	0.760433i	$-0.275013\pi$
$662$	2.01082e9	+	5.56534e9i	0.269383	+	0.745571i
$663$		0			0
$664$	1.54412e9	+	9.06069e8i	0.204688	+	0.120108i
$665$	2.62178e9			0.345717
$666$		0			0
$667$			8.82153e9i			1.15107i
$668$	−1.42667e9	+	1.18574e9i	−0.185186	+	0.153912i
$669$		0			0
$670$	4.44253e9	−	1.60514e9i	0.570648	−	0.206182i
$671$	1.70141e10			2.17410
$672$		0			0
$673$	1.21934e9			0.154195			0.0770977	−	0.997024i	$-0.475435\pi$
$673$	1.21934e9			0.154195			0.0770977	+	0.997024i	$0.475435\pi$
$674$	−1.20845e10	+	4.36625e9i	−1.52026	+	0.549287i
$675$		0			0
$676$	−1.40412e10	+	1.16699e10i	−1.74819	+	1.45296i
$677$		−	1.32664e10i		−	1.64321i	−0.570054	−	0.821607i	$-0.693078\pi$
$677$		−	1.32664e10i		−	1.64321i	0.570054	−	0.821607i	$-0.306922\pi$
$678$		0			0
$679$	−1.89667e10			−2.32514
$680$	5.02310e9	+	2.94750e9i	0.612620	+	0.359478i
$681$		0			0
$682$	−7.72800e8	−	2.13888e9i	−0.0932872	−	0.258190i
$683$			8.29925e9i			0.996704i	0.866975	+	0.498352i	$0.166061\pi$
$683$			8.29925e9i			0.996704i	−0.866975	+	0.498352i	$0.833939\pi$
$684$		0			0
$685$			1.06078e10i			1.26099i
$686$	−6.46871e8	+	2.33722e8i	−0.0765038	+	0.0276417i
$687$		0			0
$688$	1.26300e10	+	2.34962e9i	1.47857	+	0.275067i
$689$	−1.26255e10			−1.47056
$690$		0			0
$691$		−	1.05496e10i		−	1.21636i	−0.793798	−	0.608182i	$-0.791899\pi$
$691$		−	1.05496e10i		−	1.21636i	0.793798	−	0.608182i	$-0.208101\pi$
$692$	4.98179e9	+	5.99405e9i	0.571497	+	0.687620i
$693$		0			0
$694$	−3.82193e9	−	1.05779e10i	−0.434035	−	1.20128i
$695$	5.42282e9			0.612743
$696$		0			0
$697$	−9.79493e9			−1.09569
$698$	3.34776e9	+	9.26557e9i	0.372615	+	1.03128i
$699$		0			0
$700$	−4.38762e9	+	3.64665e9i	−0.483488	+	0.401838i
$701$		−	3.64147e8i		−	0.0399267i	−0.999801	−	0.0199633i	$-0.993645\pi$
$701$		−	3.64147e8i		−	0.0399267i	0.999801	−	0.0199633i	$-0.00635495\pi$
$702$		0			0
$703$	−7.33681e8			−0.0796459
$704$	−1.29526e10	+	7.23665e9i	−1.39911	+	0.781687i
$705$		0			0
$706$	7.95192e9	−	2.87312e9i	0.850463	−	0.307282i
$707$			1.46073e10i			1.55454i
$708$		0			0
$709$		−	3.73743e9i		−	0.393832i	−0.980420	−	0.196916i	$-0.936907\pi$
$709$		−	3.73743e9i		−	0.393832i	0.980420	−	0.196916i	$-0.0630927\pi$
$710$	−2.90146e9	−	8.03036e9i	−0.304237	−	0.842037i
$711$		0			0
$712$	6.95416e9	−	1.18512e10i	0.722046	−	1.23050i
$713$	−2.46239e9			−0.254416
$714$		0			0
$715$		−	2.12385e10i		−	2.17297i
$716$	−1.86417e9	−	2.24296e9i	−0.189798	−	0.228363i
$717$		0			0
$718$	8.93549e9	−	3.22849e9i	0.900913	−	0.325510i
$719$	1.38910e10			1.39374			0.696870	−	0.717197i	$-0.254575\pi$
$719$	1.38910e10			1.39374			0.696870	+	0.717197i	$0.254575\pi$
$720$		0			0
$721$	−1.81446e9			−0.180291
$722$	8.52709e9	−	3.08093e9i	0.843181	−	0.304651i
$723$		0			0
$724$	−6.60084e8	−	7.94208e8i	−0.0646419	−	0.0777766i
$725$		−	3.48596e9i		−	0.339735i
$726$		0			0
$727$	−4.81417e9			−0.464677			−0.232338	−	0.972635i	$-0.574638\pi$
$727$	−4.81417e9			−0.464677			−0.232338	+	0.972635i	$0.574638\pi$
$728$	2.32959e10	+	1.36698e10i	2.23779	+	1.31311i
$729$		0			0
$730$	1.35763e9	+	3.75751e9i	0.129167	+	0.357495i
$731$			1.50542e10i			1.42544i
$732$		0			0
$733$		−	1.05175e10i		−	0.986390i	−0.869919	−	0.493195i	$-0.835829\pi$
$733$		−	1.05175e10i		−	0.986390i	0.869919	−	0.493195i	$-0.164171\pi$
$734$	3.03870e9	−	1.09792e9i	0.283630	−	0.102479i
$735$		0			0
$736$	2.60356e9	+	1.58524e10i	0.240710	+	1.46563i
$737$	−1.41016e10			−1.29757
$738$		0			0
$739$			1.01835e10i			0.928203i	0.885782	+	0.464102i	$0.153623\pi$
$739$			1.01835e10i			0.928203i	−0.885782	+	0.464102i	$0.846377\pi$
$740$	−1.57308e9	+	1.30742e9i	−0.142705	+	0.118605i
$741$		0			0
$742$	4.40795e9	+	1.21999e10i	0.396116	+	1.09633i
$743$	1.88665e10			1.68744			0.843722	−	0.536780i	$-0.180360\pi$
$743$	1.88665e10			1.68744			0.843722	+	0.536780i	$0.180360\pi$
$744$		0			0
$745$	3.11090e9			0.275638
$746$	−8.20003e8	−	2.26952e9i	−0.0723152	−	0.200146i
$747$		0			0
$748$	−1.11132e10	−	1.33713e10i	−0.970922	−	1.16821i
$749$			4.19282e9i			0.364603i
$750$		0			0
$751$	1.93566e10			1.66759			0.833795	−	0.552074i	$-0.186164\pi$
$751$	1.93566e10			1.66759			0.833795	+	0.552074i	$0.186164\pi$
$752$	5.39531e8	−	2.90016e9i	0.0462651	−	0.248691i
$753$		0			0
$754$	−1.55217e10	+	5.60817e9i	−1.31868	+	0.476454i
$755$		−	5.52290e9i		−	0.467039i
$756$		0			0
$757$			7.46403e8i			0.0625371i	0.999511	+	0.0312686i	$0.00995472\pi$
$757$			7.46403e8i			0.0625371i	−0.999511	+	0.0312686i	$0.990045\pi$
$758$	−6.82998e8	−	1.89033e9i	−0.0569610	−	0.157651i
$759$		0			0
$760$	−1.47642e9	+	2.51611e9i	−0.122001	+	0.207913i
$761$	−7.94454e9			−0.653465			−0.326733	−	0.945117i	$-0.605948\pi$
$761$	−7.94454e9			−0.653465			−0.326733	+	0.945117i	$0.605948\pi$
$762$		0			0
$763$			5.50253e9i			0.448463i
$764$	4.50398e9	−	3.74336e9i	0.365401	−	0.303693i
$765$		0			0
$766$	−1.10660e10	+	3.99827e9i	−0.889591	+	0.321419i
$767$	1.32615e9			0.106123
$768$		0			0
$769$	−5.43124e9			−0.430682			−0.215341	−	0.976539i	$-0.569086\pi$
$769$	−5.43124e9			−0.430682			−0.215341	+	0.976539i	$0.569086\pi$
$770$	−2.05225e10	+	7.41502e9i	−1.61999	+	0.585322i
$771$		0			0
$772$	1.19655e10	−	9.94478e9i	0.935987	−	0.777920i
$773$		−	6.10644e9i		−	0.475510i	−0.971325	−	0.237755i	$-0.923588\pi$
$773$		−	6.10644e9i		−	0.475510i	0.971325	−	0.237755i	$-0.0764116\pi$
$774$		0			0
$775$	9.73052e8			0.0750896
$776$	1.06809e10	−	1.82023e10i	0.820523	−	1.39833i
$777$		0			0
$778$	−5.86724e9	−	1.62387e10i	−0.446689	−	1.23630i
$779$		−	4.90635e9i		−	0.371858i
$780$		0			0
$781$			2.54901e10i			1.91467i
$782$	−1.77052e10	+	6.39708e9i	−1.32396	+	0.478363i
$783$		0			0
$784$	2.60779e9	−	1.40177e10i	0.193271	−	1.03890i
$785$	1.77546e9			0.130999
$786$		0			0
$787$			2.55127e10i			1.86571i	0.360249	+	0.932856i	$0.382692\pi$
$787$			2.55127e10i			1.86571i	−0.360249	+	0.932856i	$0.617308\pi$
$788$	1.39909e10	+	1.68338e10i	1.01860	+	1.22557i
$789$		0			0
$790$	3.87844e9	+	1.07344e10i	0.279874	+	0.774606i
$791$	−3.43480e9			−0.246765
$792$		0			0
$793$	3.44647e10			2.45425
$794$	−5.41851e9	−	1.49968e10i	−0.384157	−	1.06323i
$795$		0			0
$796$	−1.48851e9	+	1.23713e9i	−0.104606	+	0.0869402i
$797$			1.11066e10i			0.777102i	0.921427	+	0.388551i	$0.127024\pi$
$797$			1.11066e10i			0.777102i	−0.921427	+	0.388551i	$0.872976\pi$
$798$		0			0
$799$	3.45683e9			0.239753
$800$	−1.02884e9	−	6.26434e9i	−0.0710446	−	0.432574i
$801$		0			0
$802$	−1.53122e10	+	5.53246e9i	−1.04816	+	0.378711i
$803$		−	1.19272e10i		−	0.812892i
$804$		0			0
$805$			2.36267e10i			1.59631i
$806$	−1.56543e9	−	4.33264e9i	−0.105308	−	0.291461i
$807$		0			0
$808$	−1.40186e10	−	8.22592e9i	−0.934897	−	0.548586i
$809$	−1.80950e10			−1.20154			−0.600769	−	0.799422i	$-0.705139\pi$
$809$	−1.80950e10			−1.20154			−0.600769	+	0.799422i	$0.705139\pi$
$810$		0			0
$811$		−	3.12449e9i		−	0.205686i	−0.994698	−	0.102843i	$-0.967206\pi$
$811$		−	3.12449e9i		−	0.205686i	0.994698	−	0.102843i	$-0.0327940\pi$
$812$	1.08382e10	+	1.30404e10i	0.710413	+	0.854764i
$813$		0			0
$814$	5.74303e9	−	2.07502e9i	0.373212	−	0.134846i
$815$	−1.99848e9			−0.129315
$816$		0			0
$817$	−7.54078e9			−0.483770
$818$	−2.23556e9	+	8.07734e8i	−0.142807	+	0.0515979i
$819$		0			0
$820$	−8.74313e9	−	1.05197e10i	−0.553756	−	0.666275i
$821$			1.65303e9i			0.104251i	0.998641	+	0.0521256i	$0.0165996\pi$
$821$			1.65303e9i			0.104251i	−0.998641	+	0.0521256i	$0.983400\pi$
$822$		0			0
$823$	−1.07490e9			−0.0672153			−0.0336077	−	0.999435i	$-0.510700\pi$
$823$	−1.07490e9			−0.0672153			−0.0336077	+	0.999435i	$0.510700\pi$
$824$	1.02179e9	−	1.74133e9i	0.0636233	−	0.108426i
$825$		0			0
$826$	−4.63000e8	−	1.28144e9i	−0.0285858	−	0.0791168i
$827$			1.76287e10i			1.08380i	0.840442	+	0.541902i	$0.182296\pi$
$827$			1.76287e10i			1.08380i	−0.840442	+	0.541902i	$0.817704\pi$
$828$		0			0
$829$		−	1.43986e10i		−	0.877767i	−0.898544	−	0.438884i	$-0.855374\pi$
$829$		−	1.43986e10i		−	0.877767i	0.898544	−	0.438884i	$-0.144626\pi$
$830$	−2.75548e9	+	9.95587e8i	−0.167272	+	0.0604374i
$831$		0			0
$832$	−2.62376e10	+	1.46590e10i	−1.57940	+	0.882415i
$833$	1.67084e10			1.00156
$834$		0			0
$835$		−	3.03583e9i		−	0.180457i
$836$	6.69780e9	−	5.56669e9i	0.396470	−	0.329515i
$837$		0			0
$838$	−7.76715e9	−	2.14971e10i	−0.455940	−	1.26190i
$839$	−2.19576e10			−1.28357			−0.641783	−	0.766886i	$-0.721805\pi$
$839$	−2.19576e10			−1.28357			−0.641783	+	0.766886i	$0.721805\pi$
$840$		0			0
$841$	6.88926e9			0.399380
$842$	−3.37849e9	−	9.35064e9i	−0.195043	−	0.539820i
$843$		0			0
$844$	1.86998e10	+	2.24995e10i	1.07063	+	1.28817i
$845$		−	2.98782e10i		−	1.70356i
$846$		0			0
$847$	3.97812e10			2.24950
$848$	−1.41904e10	−	2.63991e9i	−0.799116	−	0.148663i
$849$		0			0
$850$	6.99647e9	−	2.52790e9i	0.390762	−	0.141187i
$851$		−	6.61169e9i		−	0.367756i
$852$		0			0
$853$			5.54683e9i			0.306001i	0.988226	+	0.153001i	$0.0488936\pi$
$853$			5.54683e9i			0.306001i	−0.988226	+	0.153001i	$0.951106\pi$
$854$	−1.20327e10	−	3.33028e10i	−0.661089	−	1.82969i
$855$		0			0
$856$	−4.02383e9	−	2.36114e9i	−0.219271	−	0.128666i
$857$	−4.05471e8			−0.0220053			−0.0110027	−	0.999939i	$-0.503502\pi$
$857$	−4.05471e8			−0.0220053			−0.0110027	+	0.999939i	$0.503502\pi$
$858$		0			0
$859$		−	2.10697e10i		−	1.13418i	−0.823655	−	0.567091i	$-0.808069\pi$
$859$		−	2.10697e10i		−	1.13418i	0.823655	−	0.567091i	$-0.191931\pi$
$860$	−1.61681e10	+	1.34377e10i	−0.866793	+	0.720411i
$861$		0			0
$862$	1.63269e10	−	5.89909e9i	0.868217	−	0.313697i
$863$	2.45154e10			1.29838			0.649190	−	0.760626i	$-0.275108\pi$
$863$	2.45154e10			1.29838			0.649190	+	0.760626i	$0.275108\pi$
$864$		0			0
$865$	−1.27548e10			−0.670063
$866$	4.52695e9	−	1.63564e9i	0.236861	−	0.0855804i
$867$		0			0
$868$	−3.64003e9	+	3.02531e9i	−0.188924	+	0.157019i
$869$		−	3.40732e10i		−	1.76134i
$870$		0			0
$871$	−2.85650e10			−1.46477
$872$	−5.28075e9	−	3.09868e9i	−0.269704	−	0.158259i
$873$		0			0
$874$	−3.20434e9	−	8.86864e9i	−0.162349	−	0.449332i
$875$		−	3.06346e10i		−	1.54591i
$876$		0			0
$877$		−	8.74791e9i		−	0.437931i	−0.975733	−	0.218965i	$-0.929732\pi$
$877$		−	8.74791e9i		−	0.437931i	0.975733	−	0.218965i	$-0.0702682\pi$
$878$	3.74005e9	−	1.35132e9i	0.186486	−	0.0673795i
$879$		0			0
$880$	4.44084e9	−	2.38710e10i	0.219673	−	1.18081i
$881$	−2.32920e10			−1.14760			−0.573801	−	0.818995i	$-0.694532\pi$
$881$	−2.32920e10			−1.14760			−0.573801	+	0.818995i	$0.694532\pi$
$882$		0			0
$883$			1.88930e10i			0.923502i	0.887010	+	0.461751i	$0.152779\pi$
$883$			1.88930e10i			0.923502i	−0.887010	+	0.461751i	$0.847221\pi$
$884$	−2.25116e10	−	2.70858e10i	−1.09603	−	1.31874i
$885$		0			0
$886$	−1.30149e10	−	3.60212e10i	−0.628668	−	1.73996i
$887$	6.20541e9			0.298564			0.149282	−	0.988795i	$-0.452304\pi$
$887$	6.20541e9			0.298564			0.149282	+	0.988795i	$0.452304\pi$
$888$		0			0
$889$	1.32903e10			0.634424
$890$	7.64122e9	+	2.11486e10i	0.363327	+	1.00558i
$891$		0			0
$892$	5.56657e9	−	4.62650e9i	0.262610	−	0.218261i
$893$			1.73155e9i			0.0813683i
$894$		0			0
$895$	4.77280e9			0.222532
$896$	2.33251e10	+	2.02351e10i	1.08329	+	0.939784i
$897$		0			0
$898$	−3.50063e8	+	1.26482e8i	−0.0161317	+	0.00582855i
$899$		−	2.89200e9i		−	0.132752i
$900$		0			0
$901$		−	1.69142e10i		−	0.770397i
$902$	1.38763e10	+	3.84054e10i	0.629580	+	1.74249i
$903$		0			0
$904$	1.93426e9	−	3.29636e9i	0.0870815	−	0.148404i
$905$	1.69000e9			0.0757908
$906$		0			0
$907$			2.35965e10i			1.05008i	0.851078	+	0.525040i	$0.175950\pi$
$907$			2.35965e10i			1.05008i	−0.851078	+	0.525040i	$0.824050\pi$
$908$	−1.46244e10	−	1.75959e10i	−0.648302	−	0.780032i
$909$		0			0
$910$	−4.15717e10	+	1.50203e10i	−1.82874	+	0.660746i
$911$	−1.75129e10			−0.767437			−0.383718	−	0.923450i	$-0.625357\pi$
$911$	−1.75129e10			−0.767437			−0.383718	+	0.923450i	$0.625357\pi$
$912$		0			0
$913$	8.74651e9			0.380353
$914$	8.02442e9	−	2.89932e9i	0.347618	−	0.125598i
$915$		0			0
$916$	−4.83525e9	−	5.81773e9i	−0.207867	−	0.250103i
$917$			6.16193e10i			2.63891i
$918$		0			0
$919$	−2.35772e10			−1.00205			−0.501024	−	0.865434i	$-0.667043\pi$
$919$	−2.35772e10			−1.00205			−0.501024	+	0.865434i	$0.667043\pi$
$920$	−2.26744e10	−	1.33050e10i	−0.960014	−	0.563325i
$921$		0			0
$922$	6.14537e9	+	1.70085e10i	0.258220	+	0.714675i
$923$			5.16344e10i			2.16139i
$924$		0			0
$925$			2.61271e9i			0.108541i
$926$	3.09544e10	−	1.11842e10i	1.28110	−	0.462876i
$927$		0			0
$928$	−1.86182e10	+	3.05780e9i	−0.764751	+	0.125600i
$929$	2.00974e10			0.822404			0.411202	−	0.911544i	$-0.365109\pi$
$929$	2.00974e10			0.822404			0.411202	+	0.911544i	$0.365109\pi$
$930$		0			0
$931$			8.36935e9i			0.339913i
$932$	1.79181e10	−	1.48921e10i	0.724997	−	0.602561i
$933$		0			0
$934$	1.00475e10	+	2.78083e10i	0.403499	+	1.11676i
$935$	2.84529e10			1.13838
$936$		0			0
$937$	1.82209e10			0.723572			0.361786	−	0.932261i	$-0.382167\pi$
$937$	1.82209e10			0.723572			0.361786	+	0.932261i	$0.382167\pi$
$938$	9.97291e9	+	2.76020e10i	0.394559	+	1.09202i
$939$		0			0
$940$	3.08563e9	+	3.71261e9i	0.121170	+	0.145791i
$941$			2.27533e10i			0.890186i	0.895484	+	0.445093i	$0.146829\pi$
$941$			2.27533e10i			0.890186i	−0.895484	+	0.445093i	$0.853171\pi$
$942$		0			0
$943$	4.42144e10			1.71701
$944$	1.49052e9	+	2.77290e8i	0.0576683	+	0.0107283i
$945$		0			0
$946$	5.90270e10	−	2.13271e10i	2.26690	−	0.819055i
$947$		−	2.13355e10i		−	0.816355i	−0.912903	−	0.408177i	$-0.866164\pi$
$947$		−	2.13355e10i		−	0.816355i	0.912903	−	0.408177i	$-0.133836\pi$
$948$		0			0
$949$		−	2.41604e10i		−	0.917640i
$950$	1.26625e9	+	3.50458e9i	0.0479165	+	0.132618i
$951$		0			0
$952$	−1.83132e10	+	3.12092e10i	−0.687915	+	1.17234i
$953$	−6.44765e9			−0.241310			−0.120655	−	0.992694i	$-0.538500\pi$
$953$	−6.44765e9			−0.241310			−0.120655	+	0.992694i	$0.538500\pi$
$954$		0			0
$955$			9.58404e9i			0.356071i
$956$	−6.63195e9	+	5.51196e9i	−0.245493	+	0.204035i
$957$		0			0
$958$	−3.26836e10	+	1.18090e10i	−1.20102	+	0.433943i
$959$	−6.59079e10			−2.41308
$960$		0			0
$961$	−2.67054e10			−0.970659
$962$	1.16334e10	−	4.20330e9i	0.421304	−	0.152222i
$963$		0			0
$964$	1.67775e10	−	1.39442e10i	0.603196	−	0.501330i
$965$			2.54614e10i			0.912088i
$966$		0			0
$967$	−5.49318e10			−1.95358			−0.976789	−	0.214202i	$-0.931285\pi$
$967$	−5.49318e10			−1.95358			−0.976789	+	0.214202i	$0.931285\pi$
$968$	−2.24023e10	+	3.81778e10i	−0.793832	+	1.35284i
$969$		0			0
$970$	1.17361e10	+	3.24820e10i	0.412880	+	1.14273i
$971$		−	5.77679e8i		−	0.0202497i	−0.999949	−	0.0101249i	$-0.996777\pi$
$971$		−	5.77679e8i		−	0.0202497i	0.999949	−	0.0101249i	$-0.00322290\pi$
$972$		0			0
$973$			3.36927e10i			1.17258i
$974$	5.38719e9	−	1.94645e9i	0.186812	−	0.0674975i
$975$		0			0
$976$	3.87366e10	+	7.20636e9i	1.33367	+	0.248108i
$977$	4.47160e10			1.53402			0.767011	−	0.641634i	$-0.221743\pi$
$977$	4.47160e10			1.53402			0.767011	+	0.641634i	$0.221743\pi$
$978$		0			0
$979$		−	6.71302e10i		−	2.28654i
$980$	1.49142e10	+	1.79447e10i	0.506185	+	0.609038i
$981$		0			0
$982$	7.06865e9	+	1.95639e10i	0.238202	+	0.659272i
$983$	−2.34992e10			−0.789071			−0.394536	−	0.918881i	$-0.629094\pi$
$983$	−2.34992e10			−0.789071			−0.394536	+	0.918881i	$0.629094\pi$
$984$		0			0
$985$	−3.58207e10			−1.19428
$986$	−7.51318e9	−	2.07942e10i	−0.249606	−	0.690832i
$987$		0			0
$988$	1.35675e10	−	1.12762e10i	0.447559	−	0.371976i
$989$		−	6.79551e10i		−	2.23375i
$990$		0			0
$991$	8.91884e9			0.291106			0.145553	−	0.989350i	$-0.453504\pi$
$991$	8.91884e9			0.291106			0.145553	+	0.989350i	$0.453504\pi$
$992$	−8.53537e8	−	5.19698e9i	−0.0277608	−	0.169029i
$993$		0			0
$994$	4.98937e10	−	1.80271e10i	1.61136	−	0.582203i
$995$		−	3.16740e9i		−	0.101935i
$996$		0			0
$997$			1.98963e10i			0.635828i	0.948120	+	0.317914i	$0.102982\pi$
$997$			1.98963e10i			0.635828i	−0.948120	+	0.317914i	$0.897018\pi$
$998$	1.32264e10	+	3.66066e10i	0.421196	+	1.16574i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.8.d.c.37.11	yes	12
3.2	odd	2	inner	72.8.d.c.37.2	yes	12
4.3	odd	2		288.8.d.c.145.9		12
8.3	odd	2		288.8.d.c.145.4		12
8.5	even	2	inner	72.8.d.c.37.12	yes	12
12.11	even	2		288.8.d.c.145.3		12
24.5	odd	2	inner	72.8.d.c.37.1	✓	12
24.11	even	2		288.8.d.c.145.10		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.8.d.c.37.1	✓	12	24.5	odd	2	inner
72.8.d.c.37.2	yes	12	3.2	odd	2	inner
72.8.d.c.37.11	yes	12	1.1	even	1	trivial
72.8.d.c.37.12	yes	12	8.5	even	2	inner
288.8.d.c.145.3		12	12.11	even	2
288.8.d.c.145.4		12	8.3	odd	2
288.8.d.c.145.9		12	4.3	odd	2
288.8.d.c.145.10		12	24.11	even	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:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	72.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(10.6405 - 3.84452i) q^{2} +(98.4393 - 81.8151i) q^{4} +209.469i q^{5} -1301.46 q^{7} +(732.900 - 1249.00i) q^{8} +O(q^{10})\)	\(q+(10.6405 - 3.84452i) q^{2} +(98.4393 - 81.8151i) q^{4} +209.469i q^{5} -1301.46 q^{7} +(732.900 - 1249.00i) q^{8} +(805.310 + 2228.85i) q^{10} -7074.87i q^{11} -14331.3i q^{13} +(-13848.1 + 5003.49i) q^{14} +(2996.58 - 16107.6i) q^{16} +19199.4 q^{17} +9617.13i q^{19} +(17137.8 + 20620.0i) q^{20} +(-27199.5 - 75280.0i) q^{22} -86666.4 q^{23} +34247.6 q^{25} +(-55097.0 - 152492. i) q^{26} +(-128115. + 106479. i) q^{28} -101787. i q^{29} +28412.3 q^{31} +(-30041.1 - 182913. i) q^{32} +(204291. - 73812.6i) q^{34} -272616. i q^{35} +76289.0i q^{37} +(36973.3 + 102331. i) q^{38} +(261628. + 153520. i) q^{40} -510168. q^{41} +784099. i q^{43} +(-578831. - 696445. i) q^{44} +(-922172. + 333191. i) q^{46} +180049. q^{47} +870255. q^{49} +(364410. - 131666. i) q^{50} +(-1.17252e6 - 1.41076e6i) q^{52} -880974. i q^{53} +1.48197e6 q^{55} +(-953840. + 1.62553e6i) q^{56} +(-391323. - 1.08306e6i) q^{58} +92535.3i q^{59} +2.40486e6i q^{61} +(302320. - 109232. i) q^{62} +(-1.02287e6 - 1.83079e6i) q^{64} +3.00197e6 q^{65} -1.99319e6i q^{67} +(1.88998e6 - 1.57080e6i) q^{68} +(-1.04808e6 - 2.90076e6i) q^{70} -3.60291e6 q^{71} +1.68585e6 q^{73} +(293295. + 811751. i) q^{74} +(786826. + 946703. i) q^{76} +9.20766e6i q^{77} +4.81609e6 q^{79} +(3.37406e6 + 627693. i) q^{80} +(-5.42843e6 + 1.96135e6i) q^{82} +1.23628e6i q^{83} +4.02169e6i q^{85} +(3.01449e6 + 8.34318e6i) q^{86} +(-8.83654e6 - 5.18518e6i) q^{88} +9.48855e6 q^{89} +1.86516e7i q^{91} +(-8.53138e6 + 7.09062e6i) q^{92} +(1.91580e6 - 692201. i) q^{94} -2.01449e6 q^{95} +1.45734e7 q^{97} +(9.25992e6 - 3.34571e6i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 412 q^{4} + 136 q^{7}+O(q^{10}) \) 12 * q + 412 * q^4 + 136 * q^7	\( 12 q + 412 q^{4} + 136 q^{7} + 4680 q^{10} - 12472 q^{16} + 32624 q^{22} + 229820 q^{25} - 157288 q^{28} - 37224 q^{31} - 74432 q^{34} - 937520 q^{40} - 1264256 q^{46} + 2668188 q^{49} - 1539680 q^{52} + 6928960 q^{55} - 4035448 q^{58} - 3530192 q^{64} - 10228720 q^{70} + 13619048 q^{73} - 2441920 q^{76} + 20470552 q^{79} - 2507200 q^{82} - 26170912 q^{88} - 22132608 q^{94} + 27442456 q^{97}+O(q^{100}) \) 12 * q + 412 * q^4 + 136 * q^7 + 4680 * q^10 - 12472 * q^16 + 32624 * q^22 + 229820 * q^25 - 157288 * q^28 - 37224 * q^31 - 74432 * q^34 - 937520 * q^40 - 1264256 * q^46 + 2668188 * q^49 - 1539680 * q^52 + 6928960 * q^55 - 4035448 * q^58 - 3530192 * q^64 - 10228720 * q^70 + 13619048 * q^73 - 2441920 * q^76 + 20470552 * q^79 - 2507200 * q^82 - 26170912 * q^88 - 22132608 * q^94 + 27442456 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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