LMFDB - Embedded newform 84.9.p.b.65.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [84,9,Mod(53,84)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(84, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 4]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("84.53");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 84 = 2^{2} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	84.p (of order $6$, degree $2$, 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:	$34.2198032451$
Analytic rank:	$0$
Dimension:	$40$
Relative dimension:	$20$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			65.11
Character	$\chi$	$=$	84.65
Dual form			84.9.p.b.53.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(21.2854 + 78.1533i) q^{3} +(609.234 + 351.741i) q^{5} +(162.484 + 2395.50i) q^{7} +(-5654.86 + 3327.05i) q^{9} +O(q^{10})$	\(q+(21.2854 + 78.1533i) q^{3} +(609.234 + 351.741i) q^{5} +(162.484 + 2395.50i) q^{7} +(-5654.86 + 3327.05i) q^{9} +(-15990.7 + 9232.23i) q^{11} -5808.82 q^{13} +(-14521.9 + 55100.6i) q^{15} +(98970.4 - 57140.6i) q^{17} +(-34997.2 + 60616.9i) q^{19} +(-183757. + 63687.7i) q^{21} +(255114. + 147290. i) q^{23} +(52131.4 + 90294.2i) q^{25} +(-380385. - 371129. i) q^{27} +635415. i q^{29} +(-410180. - 710453. i) q^{31} +(-1.06190e6 - 1.05321e6i) q^{33} +(-743604. + 1.51657e6i) q^{35} +(332530. - 575958. i) q^{37} +(-123643. - 453978. i) q^{39} -2.64195e6i q^{41} -5.38188e6 q^{43} +(-4.61539e6 + 37899.6i) q^{45} +(3.95171e6 + 2.28152e6i) q^{47} +(-5.71200e6 + 778459. i) q^{49} +(6.57235e6 + 6.51860e6i) q^{51} +(3.92358e6 - 2.26528e6i) q^{53} -1.29894e7 q^{55} +(-5.48234e6 - 1.44489e6i) q^{57} +(-1.48641e7 + 8.58180e6i) q^{59} +(-6.38504e6 + 1.10592e7i) q^{61} +(-8.88875e6 - 1.30056e7i) q^{63} +(-3.53893e6 - 2.04320e6i) q^{65} +(-3.15977e6 - 5.47288e6i) q^{67} +(-6.08101e6 + 2.30731e7i) q^{69} +3.32348e7i q^{71} +(-3.18086e6 - 5.50941e6i) q^{73} +(-5.94715e6 + 5.99619e6i) q^{75} +(-2.47140e7 - 3.68055e7i) q^{77} +(-34639.6 + 59997.6i) q^{79} +(2.09083e7 - 3.76280e7i) q^{81} -8.04512e7i q^{83} +8.03948e7 q^{85} +(-4.96598e7 + 1.35251e7i) q^{87} +(8.74905e7 + 5.05127e7i) q^{89} +(-943840. - 1.39150e7i) q^{91} +(4.67934e7 - 4.71792e7i) q^{93} +(-4.26429e7 + 2.46199e7i) q^{95} +7.56377e7 q^{97} +(5.97091e7 - 1.05409e8i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 81 q^{3} - 34 q^{7} + 4771 q^{9}+O(q^{10}) $ 40 * q + 81 * q^3 - 34 * q^7 + 4771 * q^9	$ 40 q + 81 q^{3} - 34 q^{7} + 4771 q^{9} - 55464 q^{13} + 68482 q^{15} + 311690 q^{19} - 172343 q^{21} + 1766792 q^{25} - 3451932 q^{27} + 31596 q^{31} + 1874885 q^{33} - 1853482 q^{37} + 11217526 q^{39} - 13372600 q^{43} - 527785 q^{45} - 12653462 q^{49} - 1103461 q^{51} + 71577224 q^{55} - 17195214 q^{57} - 21761970 q^{61} + 21945045 q^{63} - 26337350 q^{67} - 5588722 q^{69} + 41115682 q^{73} - 17971730 q^{75} - 120916932 q^{79} - 24550133 q^{81} + 139250060 q^{85} - 16321046 q^{87} + 345074940 q^{91} + 25774675 q^{93} - 707216948 q^{97} - 94510994 q^{99}+O(q^{100}) $ 40 * q + 81 * q^3 - 34 * q^7 + 4771 * q^9 - 55464 * q^13 + 68482 * q^15 + 311690 * q^19 - 172343 * q^21 + 1766792 * q^25 - 3451932 * q^27 + 31596 * q^31 + 1874885 * q^33 - 1853482 * q^37 + 11217526 * q^39 - 13372600 * q^43 - 527785 * q^45 - 12653462 * q^49 - 1103461 * q^51 + 71577224 * q^55 - 17195214 * q^57 - 21761970 * q^61 + 21945045 * q^63 - 26337350 * q^67 - 5588722 * q^69 + 41115682 * q^73 - 17971730 * q^75 - 120916932 * q^79 - 24550133 * q^81 + 139250060 * q^85 - 16321046 * q^87 + 345074940 * q^91 + 25774675 * q^93 - 707216948 * q^97 - 94510994 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/84\mathbb{Z}\right)^\times$.

$n$	$29$	$43$	$73$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{3}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	21.2854	+	78.1533i	0.262783	+	0.964855i
$3$	21.2854	+	78.1533i	0.262783	+	0.964855i
$4$		0			0
$5$	609.234	+	351.741i	0.974774	+	0.562786i	0.900688	−	0.434466i	$-0.143063\pi$
$5$	609.234	+	351.741i	0.974774	+	0.562786i	0.0740858	+	0.997252i	$0.476396\pi$
$6$		0			0
$7$	162.484	+	2395.50i	0.0676734	+	0.997708i
$7$	162.484	+	2395.50i	0.0676734	+	0.997708i
$8$		0			0
$9$	−5654.86	+	3327.05i	−0.861891	+	0.507094i
$10$		0			0
$11$	−15990.7	+	9232.23i	−1.09219	+	0.630573i	−0.934157	−	0.356861i	$-0.883847\pi$
$11$	−15990.7	+	9232.23i	−1.09219	+	0.630573i	−0.158028	+	0.987435i	$0.550514\pi$
$12$		0			0
$13$	−5808.82			−0.203383			−0.101692	−	0.994816i	$-0.532425\pi$
$13$	−5808.82			−0.203383			−0.101692	+	0.994816i	$0.532425\pi$
$14$		0			0
$15$	−14521.9	+	55100.6i	−0.286853	+	1.08841i
$16$		0			0
$17$	98970.4	−	57140.6i	1.18498	−	0.684147i	0.227816	−	0.973704i	$-0.426842\pi$
$17$	98970.4	−	57140.6i	1.18498	−	0.684147i	0.957161	+	0.289558i	$0.0935082\pi$
$18$		0			0
$19$	−34997.2	+	60616.9i	−0.268546	+	0.465135i	−0.968487	−	0.249066i	$-0.919876\pi$
$19$	−34997.2	+	60616.9i	−0.268546	+	0.465135i	0.699941	+	0.714201i	$0.253210\pi$
$20$		0			0
$21$	−183757.	+	63687.7i	−0.944860	+	0.327475i
$22$		0			0
$23$	255114.	+	147290.i	0.911640	+	0.526335i	0.880958	−	0.473194i	$-0.156899\pi$
$23$	255114.	+	147290.i	0.911640	+	0.526335i	0.0306813	+	0.999529i	$0.490232\pi$
$24$		0			0
$25$	52131.4	+	90294.2i	0.133456	+	0.231153i
$26$		0			0
$27$	−380385.	−	371129.i	−0.715762	−	0.698344i
$28$		0			0
$29$			635415.i			0.898391i	0.893433	+	0.449196i	$0.148289\pi$
$29$			635415.i			0.898391i	−0.893433	+	0.449196i	$0.851711\pi$
$30$		0			0
$31$	−410180.	−	710453.i	−0.444148	−	0.769288i	0.553844	−	0.832620i	$-0.313160\pi$
$31$	−410180.	−	710453.i	−0.444148	−	0.769288i	−0.997992	+	0.0633328i	$0.979827\pi$
$32$		0			0
$33$	−1.06190e6	−	1.05321e6i	−0.895419	−	0.888097i
$34$		0			0
$35$	−743604.	+	1.51657e6i	−0.495530	+	1.01063i
$36$		0			0
$37$	332530.	−	575958.i	0.177428	−	0.307315i	−0.763571	−	0.645724i	$-0.776555\pi$
$37$	332530.	−	575958.i	0.177428	−	0.307315i	0.940999	+	0.338409i	$0.109889\pi$
$38$		0			0
$39$	−123643.	−	453978.i	−0.0534455	−	0.196235i
$40$		0			0
$41$		−	2.64195e6i		−	0.934952i	−0.884006	−	0.467476i	$-0.845164\pi$
$41$		−	2.64195e6i		−	0.934952i	0.884006	−	0.467476i	$-0.154836\pi$
$42$		0			0
$43$	−5.38188e6			−1.57420			−0.787101	−	0.616824i	$-0.788419\pi$
$43$	−5.38188e6			−1.57420			−0.787101	+	0.616824i	$0.788419\pi$
$44$		0			0
$45$	−4.61539e6	+	37899.6i	−1.12553	+	0.00924241i
$46$		0			0
$47$	3.95171e6	+	2.28152e6i	0.809830	+	0.467556i	0.846897	−	0.531757i	$-0.178468\pi$
$47$	3.95171e6	+	2.28152e6i	0.809830	+	0.467556i	−0.0370665	+	0.999313i	$0.511801\pi$
$48$		0			0
$49$	−5.71200e6	+	778459.i	−0.990841	+	0.135037i
$50$		0			0
$51$	6.57235e6	+	6.51860e6i	0.971493	+	0.963549i
$52$		0			0
$53$	3.92358e6	−	2.26528e6i	0.497255	−	0.287090i	−0.230324	−	0.973114i	$-0.573979\pi$
$53$	3.92358e6	−	2.26528e6i	0.497255	−	0.287090i	0.727579	+	0.686024i	$0.240645\pi$
$54$		0			0
$55$	−1.29894e7			−1.41951
$56$		0			0
$57$	−5.48234e6	−	1.44489e6i	−0.519357	−	0.136878i
$58$		0			0
$59$	−1.48641e7	+	8.58180e6i	−1.22668	+	0.708224i	−0.966334	−	0.257292i	$-0.917170\pi$
$59$	−1.48641e7	+	8.58180e6i	−1.22668	+	0.708224i	−0.260346	+	0.965515i	$0.583836\pi$
$60$		0			0
$61$	−6.38504e6	+	1.10592e7i	−0.461152	+	0.798739i	−0.999019	−	0.0442911i	$-0.985897\pi$
$61$	−6.38504e6	+	1.10592e7i	−0.461152	+	0.798739i	0.537867	+	0.843030i	$0.319230\pi$
$62$		0			0
$63$	−8.88875e6	−	1.30056e7i	−0.564259	−	0.825598i
$64$		0			0
$65$	−3.53893e6	−	2.04320e6i	−0.198252	−	0.114461i
$66$		0			0
$67$	−3.15977e6	−	5.47288e6i	−0.156804	−	0.271592i	0.776911	−	0.629611i	$-0.216786\pi$
$67$	−3.15977e6	−	5.47288e6i	−0.156804	−	0.271592i	−0.933714	+	0.358019i	$0.883452\pi$
$68$		0			0
$69$	−6.08101e6	+	2.30731e7i	−0.268274	+	1.01791i
$70$		0			0
$71$			3.32348e7i			1.30786i	0.756556	+	0.653929i	$0.226880\pi$
$71$			3.32348e7i			1.30786i	−0.756556	+	0.653929i	$0.773120\pi$
$72$		0			0
$73$	−3.18086e6	−	5.50941e6i	−0.112009	−	0.194005i	0.804571	−	0.593856i	$-0.202395\pi$
$73$	−3.18086e6	−	5.50941e6i	−0.112009	−	0.194005i	−0.916580	+	0.399851i	$0.869062\pi$
$74$		0			0
$75$	−5.94715e6	+	5.99619e6i	−0.187959	+	0.189509i
$76$		0			0
$77$	−2.47140e7	−	3.68055e7i	−0.703040	−	1.04701i
$78$		0			0
$79$	−34639.6	+	59997.6i	−0.000889333	+	0.00154037i	−0.866470	−	0.499230i	$-0.833616\pi$
$79$	−34639.6	+	59997.6i	−0.000889333	+	0.00154037i	0.865580	+	0.500770i	$0.166950\pi$
$80$		0			0
$81$	2.09083e7	−	3.76280e7i	0.485711	−	0.874120i
$82$		0			0
$83$		−	8.04512e7i		−	1.69520i	−0.530637	−	0.847599i	$-0.678047\pi$
$83$		−	8.04512e7i		−	1.69520i	0.530637	−	0.847599i	$-0.321953\pi$
$84$		0			0
$85$	8.03948e7			1.54011
$86$		0			0
$87$	−4.96598e7	+	1.35251e7i	−0.866818	+	0.236082i
$88$		0			0
$89$	8.74905e7	+	5.05127e7i	1.39444	+	0.805082i	0.993803	−	0.111153i	$-0.0354544\pi$
$89$	8.74905e7	+	5.05127e7i	1.39444	+	0.805082i	0.400640	+	0.916235i	$0.368788\pi$
$90$		0			0
$91$	−943840.	−	1.39150e7i	−0.0137636	−	0.202917i
$92$		0			0
$93$	4.67934e7	−	4.71792e7i	0.625537	−	0.630694i
$94$		0			0
$95$	−4.26429e7	+	2.46199e7i	−0.523543	+	0.302268i
$96$		0			0
$97$	7.56377e7			0.854381			0.427190	−	0.904162i	$-0.359503\pi$
$97$	7.56377e7			0.854381			0.427190	+	0.904162i	$0.359503\pi$
$98$		0			0
$99$	5.97091e7	−	1.05409e8i	0.621584	−	1.09733i
$100$		0			0
$101$	−3.87271e7	+	2.23591e7i	−0.372159	+	0.214866i	−0.674401	−	0.738365i	$-0.735598\pi$
$101$	−3.87271e7	+	2.23591e7i	−0.372159	+	0.214866i	0.302242	+	0.953231i	$0.402265\pi$
$102$		0			0
$103$	−8.06813e7	+	1.39744e8i	−0.716843	+	1.24161i	0.245402	+	0.969421i	$0.421080\pi$
$103$	−8.06813e7	+	1.39744e8i	−0.716843	+	1.24161i	−0.962244	+	0.272187i	$0.912253\pi$
$104$		0			0
$105$	−1.34353e8	−	2.58343e7i	−1.10532	−	0.212540i
$106$		0			0
$107$	8.98648e7	+	5.18835e7i	0.685575	+	0.395817i	0.801952	−	0.597388i	$-0.203795\pi$
$107$	8.98648e7	+	5.18835e7i	0.685575	+	0.395817i	−0.116378	+	0.993205i	$0.537128\pi$
$108$		0			0
$109$	1.19274e8	+	2.06589e8i	0.844969	+	1.46353i	0.885648	+	0.464357i	$0.153715\pi$
$109$	1.19274e8	+	2.06589e8i	0.844969	+	1.46353i	−0.0406788	+	0.999172i	$0.512952\pi$
$110$		0			0
$111$	5.20910e7	+	1.37288e7i	0.343140	+	0.0904357i
$112$		0			0
$113$		−	1.31852e8i		−	0.808671i	−0.914611	−	0.404336i	$-0.867503\pi$
$113$		−	1.31852e8i		−	0.808671i	0.914611	−	0.404336i	$-0.132497\pi$
$114$		0			0
$115$	1.03616e8	+	1.79468e8i	0.592429	+	1.02612i
$116$		0			0
$117$	3.28481e7	−	1.93262e7i	0.175294	−	0.103134i
$118$		0			0
$119$	1.52961e8	+	2.27799e8i	0.762770	+	1.13596i
$120$		0			0
$121$	6.32886e7	−	1.09619e8i	0.295246	−	0.511381i
$122$		0			0
$123$	2.06477e8	−	5.62349e7i	0.902093	−	0.245689i
$124$		0			0
$125$		−	2.01451e8i		−	0.825143i
$126$		0			0
$127$	3.54159e8			1.36139			0.680696	−	0.732566i	$-0.261677\pi$
$127$	3.54159e8			1.36139			0.680696	+	0.732566i	$0.261677\pi$
$128$		0			0
$129$	−1.14555e8	−	4.20612e8i	−0.413673	−	1.51888i
$130$		0			0
$131$	−1.49640e8	−	8.63946e7i	−0.508115	−	0.293361i	0.223943	−	0.974602i	$-0.428107\pi$
$131$	−1.49640e8	−	8.63946e7i	−0.508115	−	0.293361i	−0.732059	+	0.681242i	$0.761440\pi$
$132$		0			0
$133$	−1.50894e8	−	7.39863e7i	−0.482242	−	0.236453i
$134$		0			0
$135$	−1.01202e8	−	3.59901e8i	−0.304688	−	1.08355i
$136$		0			0
$137$	−7.38547e7	+	4.26400e7i	−0.209651	+	0.121042i	−0.601149	−	0.799137i	$-0.705290\pi$
$137$	−7.38547e7	+	4.26400e7i	−0.209651	+	0.121042i	0.391498	+	0.920179i	$0.371957\pi$
$138$		0			0
$139$	5.27501e8			1.41307			0.706535	−	0.707678i	$-0.250257\pi$
$139$	5.27501e8			1.41307			0.706535	+	0.707678i	$0.250257\pi$
$140$		0			0
$141$	−9.41947e7	+	3.57402e8i	−0.238314	+	0.904235i
$142$		0			0
$143$	9.28870e7	−	5.36284e7i	0.222132	−	0.128248i
$144$		0			0
$145$	−2.23502e8	+	3.87116e8i	−0.505602	+	0.875729i
$146$		0			0
$147$	−1.82421e8	−	4.29842e8i	−0.390666	−	0.920532i
$148$		0			0
$149$	−5.77780e8	−	3.33581e8i	−1.17224	−	0.676794i	−0.218035	−	0.975941i	$-0.569965\pi$
$149$	−5.77780e8	−	3.33581e8i	−1.17224	−	0.676794i	−0.954207	+	0.299147i	$0.903298\pi$
$150$		0			0
$151$	3.40738e8	+	5.90176e8i	0.655410	+	1.13520i	0.981791	+	0.189965i	$0.0608376\pi$
$151$	3.40738e8	+	5.90176e8i	0.655410	+	1.13520i	−0.326381	+	0.945238i	$0.605829\pi$
$152$		0			0
$153$	−3.69555e8	+	6.52401e8i	−0.674393	+	1.19055i
$154$		0			0
$155$		−	5.77110e8i		−	0.999842i
$156$		0			0
$157$	−1.98763e8	−	3.44267e8i	−0.327142	−	0.566626i	0.654802	−	0.755801i	$-0.272752\pi$
$157$	−1.98763e8	−	3.44267e8i	−0.327142	−	0.566626i	−0.981943	+	0.189175i	$0.939419\pi$
$158$		0			0
$159$	2.60554e8	+	2.58423e8i	0.407671	+	0.404337i
$160$		0			0
$161$	−3.11381e8	+	6.35057e8i	−0.463435	+	0.945169i
$162$		0			0
$163$	−4.50258e8	+	7.79870e8i	−0.637839	+	1.10477i	0.348066	+	0.937470i	$0.386838\pi$
$163$	−4.50258e8	+	7.79870e8i	−0.637839	+	1.10477i	−0.985906	+	0.167301i	$0.946495\pi$
$164$		0			0
$165$	−2.76485e8	−	1.01517e9i	−0.373023	−	1.36962i
$166$		0			0
$167$			5.31526e8i			0.683374i	0.939814	+	0.341687i	$0.110998\pi$
$167$			5.31526e8i			0.683374i	−0.939814	+	0.341687i	$0.889002\pi$
$168$		0			0
$169$	−7.81988e8			−0.958635
$170$		0			0
$171$	−3.77090e6	−	4.59218e8i	−0.00441022	−	0.537074i
$172$		0			0
$173$	8.43588e8	+	4.87046e8i	0.941772	+	0.543732i	0.890515	−	0.454953i	$-0.150344\pi$
$173$	8.43588e8	+	4.87046e8i	0.941772	+	0.543732i	0.0512566	+	0.998686i	$0.483677\pi$
$174$		0			0
$175$	−2.07829e8	+	1.39552e8i	−0.221592	+	0.148793i
$176$		0			0
$177$	−9.87084e8	−	9.79012e8i	−1.00568	−	0.997459i
$178$		0			0
$179$	−2.97615e8	+	1.71828e8i	−0.289897	+	0.167372i	−0.637895	−	0.770123i	$-0.720195\pi$
$179$	−2.97615e8	+	1.71828e8i	−0.289897	+	0.167372i	0.347999	+	0.937495i	$0.386861\pi$
$180$		0			0
$181$	−1.12726e9			−1.05029			−0.525145	−	0.851013i	$-0.675989\pi$
$181$	−1.12726e9			−1.05029			−0.525145	+	0.851013i	$0.675989\pi$
$182$		0			0
$183$	−1.00022e9	−	2.63612e8i	−0.891850	−	0.235050i
$184$		0			0
$185$	4.05176e8	−	2.33929e8i	0.345905	−	0.199709i
$186$		0			0
$187$	−1.05507e9	+	1.82743e9i	−0.862809	+	1.49443i
$188$		0			0
$189$	8.27230e8	−	9.71514e8i	0.648305	−	0.761381i
$190$		0			0
$191$	2.02539e9	+	1.16936e9i	1.52186	+	0.878646i	0.999667	+	0.0258184i	$0.00821916\pi$
$191$	2.02539e9	+	1.16936e9i	1.52186	+	0.878646i	0.522193	+	0.852828i	$0.325114\pi$
$192$		0			0
$193$	−7.87643e7	−	1.36424e8i	−0.0567675	−	0.0983243i	0.836245	−	0.548356i	$-0.184746\pi$
$193$	−7.87643e7	−	1.36424e8i	−0.0567675	−	0.0983243i	−0.893013	+	0.450032i	$0.851413\pi$
$194$		0			0
$195$	8.43554e7	−	3.20069e8i	0.0583411	−	0.221363i
$196$		0			0
$197$			3.84095e8i			0.255019i	0.991837	+	0.127510i	$0.0406984\pi$
$197$			3.84095e8i			0.255019i	−0.991837	+	0.127510i	$0.959302\pi$
$198$		0			0
$199$	1.33693e9	+	2.31563e9i	0.852505	+	1.47658i	0.878941	+	0.476931i	$0.158251\pi$
$199$	1.33693e9	+	2.31563e9i	0.852505	+	1.47658i	−0.0264357	+	0.999651i	$0.508416\pi$
$200$		0			0
$201$	3.60467e8	−	3.63439e8i	0.220842	−	0.222662i
$202$		0			0
$203$	−1.52213e9	+	1.03245e8i	−0.896332	+	0.0607972i
$204$		0			0
$205$	9.29283e8	−	1.60957e9i	0.526178	−	0.911367i
$206$		0			0
$207$	−1.93268e9	+	1.58703e7i	−1.05264	+	0.00864379i
$208$		0			0
$209$		−	1.29241e9i		−	0.677352i
$210$		0			0
$211$	−5.18057e8			−0.261365			−0.130683	−	0.991424i	$-0.541717\pi$
$211$	−5.18057e8			−0.261365			−0.130683	+	0.991424i	$0.541717\pi$
$212$		0			0
$213$	−2.59741e9	+	7.07417e8i	−1.26189	+	0.343682i
$214$		0			0
$215$	−3.27882e9	−	1.89303e9i	−1.53449	−	0.885939i
$216$		0			0
$217$	1.63524e9	−	1.09802e9i	0.737467	−	0.495191i
$218$		0			0
$219$	3.62873e8	−	3.65865e8i	0.157753	−	0.159054i
$220$		0			0
$221$	−5.74902e8	+	3.31920e8i	−0.241004	+	0.139144i
$222$		0			0
$223$	2.61816e9			1.05871			0.529355	−	0.848400i	$-0.322434\pi$
$223$	2.61816e9			1.05871			0.529355	+	0.848400i	$0.322434\pi$
$224$		0			0
$225$	−5.95209e8	−	3.37158e8i	−0.232241	−	0.131554i
$226$		0			0
$227$	3.87042e9	−	2.23459e9i	1.45765	−	0.841577i	0.458758	−	0.888561i	$-0.348295\pi$
$227$	3.87042e9	−	2.23459e9i	1.45765	−	0.841577i	0.998896	+	0.0469848i	$0.0149612\pi$
$228$		0			0
$229$	1.87168e9	−	3.24185e9i	0.680598	−	1.17883i	−0.294200	−	0.955744i	$-0.595053\pi$
$229$	1.87168e9	−	3.24185e9i	0.680598	−	1.17883i	0.974799	−	0.223087i	$-0.0716133\pi$
$230$		0			0
$231$	2.35043e9	−	2.71490e9i	0.825465	−	0.953467i
$232$		0			0
$233$	2.59276e9	+	1.49693e9i	0.879708	+	0.507900i	0.870562	−	0.492058i	$-0.163755\pi$
$233$	2.59276e9	+	1.49693e9i	0.879708	+	0.507900i	0.00914618	+	0.999958i	$0.497089\pi$
$234$		0			0
$235$	1.60501e9	+	2.77996e9i	0.526268	+	0.911523i
$236$		0			0
$237$	−5.42632e6	−	1.43013e6i	−0.00171994	−	0.000453295i
$238$		0			0
$239$			4.92036e9i			1.50801i	0.656866	+	0.754007i	$0.271882\pi$
$239$			4.92036e9i			1.50801i	−0.656866	+	0.754007i	$0.728118\pi$
$240$		0			0
$241$	2.69193e9	+	4.66255e9i	0.797985	+	1.38215i	0.920926	+	0.389737i	$0.127434\pi$
$241$	2.69193e9	+	4.66255e9i	0.797985	+	1.38215i	−0.122941	+	0.992414i	$0.539232\pi$
$242$		0			0
$243$	3.38579e9	+	8.33122e8i	0.971035	+	0.238937i
$244$		0			0
$245$	−3.75376e9	−	1.53488e9i	−1.04184	−	0.426001i
$246$		0			0
$247$	2.03292e8	−	3.52113e8i	0.0546177	−	0.0946006i
$248$		0			0
$249$	6.28753e9	−	1.71244e9i	1.63562	−	0.445469i
$250$		0			0
$251$		−	4.80868e9i		−	1.21152i	−0.795647	−	0.605761i	$-0.792869\pi$
$251$		−	4.80868e9i		−	1.21152i	0.795647	−	0.605761i	$-0.207131\pi$
$252$		0			0
$253$	−5.43927e9			−1.32757
$254$		0			0
$255$	1.71124e9	+	6.28312e9i	0.404715	+	1.48599i
$256$		0			0
$257$	−1.01266e8	−	5.84662e7i	−0.0232131	−	0.0134021i	0.488349	−	0.872649i	$-0.337599\pi$
$257$	−1.01266e8	−	5.84662e7i	−0.0232131	−	0.0134021i	−0.511562	+	0.859247i	$0.670933\pi$
$258$		0			0
$259$	1.43374e9	+	7.02989e8i	0.318618	+	0.156225i
$260$		0			0
$261$	−2.11406e9	−	3.59319e9i	−0.455569	−	0.774315i
$262$		0			0
$263$	2.23432e9	−	1.28998e9i	0.467005	−	0.269626i	−0.247980	−	0.968765i	$-0.579767\pi$
$263$	2.23432e9	−	1.28998e9i	0.467005	−	0.269626i	0.714985	+	0.699140i	$0.246433\pi$
$264$		0			0
$265$	3.18717e9			0.646282
$266$		0			0
$267$	−2.08546e9	+	7.91285e9i	−0.410352	+	1.55700i
$268$		0			0
$269$	−5.71409e9	+	3.29903e9i	−1.09128	+	0.630054i	−0.933918	−	0.357487i	$-0.883634\pi$
$269$	−5.71409e9	+	3.29903e9i	−1.09128	+	0.630054i	−0.157367	+	0.987540i	$0.550300\pi$
$270$		0			0
$271$	7.84735e8	−	1.35920e9i	0.145494	−	0.252003i	−0.784063	−	0.620681i	$-0.786856\pi$
$271$	7.84735e8	−	1.35920e9i	0.145494	−	0.252003i	0.929557	+	0.368678i	$0.120189\pi$
$272$		0			0
$273$	1.06741e9	−	3.69951e8i	0.192168	−	0.0666029i
$274$		0			0
$275$	−1.66723e9	−	9.62578e8i	−0.291518	−	0.168308i
$276$		0			0
$277$	−3.06510e9	−	5.30891e9i	−0.520626	−	0.901750i	−0.999712	−	0.0239826i	$-0.992365\pi$
$277$	−3.06510e9	−	5.30891e9i	−0.520626	−	0.901750i	0.479087	−	0.877768i	$-0.340968\pi$
$278$		0			0
$279$	4.68323e9	+	2.65283e9i	0.772909	+	0.437817i
$280$		0			0
$281$		−	4.73676e9i		−	0.759724i	−0.925043	−	0.379862i	$-0.875971\pi$
$281$		−	4.73676e9i		−	0.759724i	0.925043	−	0.379862i	$-0.124029\pi$
$282$		0			0
$283$	2.63371e9	+	4.56173e9i	0.410604	+	0.711186i	0.994956	−	0.100314i	$-0.0319847\pi$
$283$	2.63371e9	+	4.56173e9i	0.410604	+	0.711186i	−0.584352	+	0.811500i	$0.698651\pi$
$284$		0			0
$285$	−2.83180e9	−	2.80864e9i	−0.429223	−	0.425713i
$286$		0			0
$287$	6.32878e9	−	4.29274e8i	0.932808	−	0.0632714i
$288$		0			0
$289$	3.04222e9	−	5.26928e9i	0.436113	−	0.755370i
$290$		0			0
$291$	1.60998e9	+	5.91133e9i	0.224516	+	0.824354i
$292$		0			0
$293$			1.21546e9i			0.164919i	0.996594	+	0.0824597i	$0.0262776\pi$
$293$			1.21546e9i			0.164919i	−0.996594	+	0.0824597i	$0.973722\pi$
$294$		0			0
$295$	−1.20743e10			−1.59431
$296$		0			0
$297$	9.50897e9	+	2.42280e9i	1.22210	+	0.311380i
$298$		0			0
$299$	−1.48191e9	−	8.55583e8i	−0.185412	−	0.107048i
$300$		0			0
$301$	−8.74469e8	−	1.28923e10i	−0.106532	−	1.57059i
$302$		0			0
$303$	−2.57176e9	−	2.55072e9i	−0.305112	−	0.302617i
$304$		0			0
$305$	−7.77996e9	+	4.49176e9i	−0.899038	+	0.519060i
$306$		0			0
$307$	−4.35544e9			−0.490318			−0.245159	−	0.969483i	$-0.578840\pi$
$307$	−4.35544e9			−0.490318			−0.245159	+	0.969483i	$0.578840\pi$
$308$		0			0
$309$	−1.26388e10	−	3.33100e9i	−1.38635	−	0.365376i
$310$		0			0
$311$	8.13770e9	−	4.69830e9i	0.869881	−	0.502226i	0.00257258	−	0.999997i	$-0.499181\pi$
$311$	8.13770e9	−	4.69830e9i	0.869881	−	0.502226i	0.867309	+	0.497770i	$0.165848\pi$
$312$		0			0
$313$	−1.08369e9	+	1.87700e9i	−0.112908	+	0.195563i	−0.916942	−	0.399021i	$-0.869350\pi$
$313$	−1.08369e9	+	1.87700e9i	−0.112908	+	0.195563i	0.804033	+	0.594584i	$0.202683\pi$
$314$		0			0
$315$	−8.40716e8	−	1.10500e10i	−0.0853900	−	1.12233i
$316$		0			0
$317$	1.01457e10	+	5.85761e9i	1.00472	+	0.580074i	0.909641	−	0.415396i	$-0.136357\pi$
$317$	1.01457e10	+	5.85761e9i	1.00472	+	0.580074i	0.0950772	+	0.995470i	$0.469690\pi$
$318$		0			0
$319$	−5.86630e9	−	1.01607e10i	−0.566502	−	0.981210i
$320$		0			0
$321$	−2.14206e9	+	8.12759e9i	−0.201749	+	0.765494i
$322$		0			0
$323$			7.99904e9i			0.734899i
$324$		0			0
$325$	−3.02822e8	−	5.24503e8i	−0.0271428	−	0.0470126i
$326$		0			0
$327$	−1.36068e10	+	1.37190e10i	−1.19005	+	1.19986i
$328$		0			0
$329$	−4.82329e9	+	9.83703e9i	−0.411680	+	0.839615i
$330$		0			0
$331$	7.27030e9	−	1.25925e10i	0.605676	−	1.04906i	−0.386268	−	0.922386i	$-0.626236\pi$
$331$	7.27030e9	−	1.25925e10i	0.605676	−	1.04906i	0.991944	−	0.126675i	$-0.0404305\pi$
$332$		0			0
$333$	3.58296e7	+	4.36331e9i	0.00291384	+	0.354845i
$334$		0			0
$335$		−	4.44569e9i		−	0.352988i
$336$		0			0
$337$	−2.40080e10			−1.86138			−0.930692	−	0.365805i	$-0.880794\pi$
$337$	−2.40080e10			−1.86138			−0.930692	+	0.365805i	$0.880794\pi$
$338$		0			0
$339$	1.03046e10	−	2.80652e9i	0.780250	−	0.212505i
$340$		0			0
$341$	1.31181e10	+	7.57376e9i	0.970185	+	0.560136i
$342$		0			0
$343$	−2.79290e9	−	1.35566e10i	−0.201781	−	0.979431i
$344$		0			0
$345$	−1.18205e10	+	1.19180e10i	−0.834374	+	0.841253i
$346$		0			0
$347$	−1.51320e10	+	8.73645e9i	−1.04371	+	0.602584i	−0.920881	−	0.389845i	$-0.872529\pi$
$347$	−1.51320e10	+	8.73645e9i	−1.04371	+	0.602584i	−0.122825	+	0.992428i	$0.539195\pi$
$348$		0			0
$349$	−1.04318e10			−0.703162			−0.351581	−	0.936157i	$-0.614356\pi$
$349$	−1.04318e10			−0.703162			−0.351581	+	0.936157i	$0.614356\pi$
$350$		0			0
$351$	2.20959e9	+	2.15582e9i	0.145574	+	0.142031i
$352$		0			0
$353$	1.12641e10	−	6.50335e9i	0.725436	−	0.418830i	−0.0913145	−	0.995822i	$-0.529107\pi$
$353$	1.12641e10	−	6.50335e9i	0.725436	−	0.418830i	0.816750	+	0.576992i	$0.195774\pi$
$354$		0			0
$355$	−1.16901e10	+	2.02478e10i	−0.736044	+	1.27487i
$356$		0			0
$357$	−1.45474e10	+	1.68032e10i	−0.895596	+	1.03447i
$358$		0			0
$359$	−2.59268e10	−	1.49689e10i	−1.56089	−	0.901179i	−0.997167	−	0.0752131i	$-0.976036\pi$
$359$	−2.59268e10	−	1.49689e10i	−1.56089	−	0.901179i	−0.563720	−	0.825966i	$-0.690630\pi$
$360$		0			0
$361$	6.04218e9	+	1.04654e10i	0.355766	+	0.616205i
$362$		0			0
$363$	9.91421e9	+	2.61292e9i	0.570994	+	0.150487i
$364$		0			0
$365$		−	4.47536e9i		−	0.252149i
$366$		0			0
$367$	−2.87771e9	−	4.98434e9i	−0.158629	−	0.274753i	0.775746	−	0.631046i	$-0.217374\pi$
$367$	−2.87771e9	−	4.98434e9i	−0.158629	−	0.274753i	−0.934375	+	0.356292i	$0.884041\pi$
$368$		0			0
$369$	8.78989e9	+	1.49399e10i	0.474109	+	0.805826i
$370$		0			0
$371$	6.06399e9	+	9.03085e9i	0.320083	+	0.476687i
$372$		0			0
$373$	7.81228e9	−	1.35313e10i	0.403592	−	0.699041i	−0.590565	−	0.806990i	$-0.701095\pi$
$373$	7.81228e9	−	1.35313e10i	0.403592	−	0.699041i	0.994156	+	0.107949i	$0.0344283\pi$
$374$		0			0
$375$	1.57440e10	−	4.28796e9i	0.796143	−	0.216833i
$376$		0			0
$377$		−	3.69101e9i		−	0.182718i
$378$		0			0
$379$	−2.80127e10			−1.35768			−0.678841	−	0.734286i	$-0.737517\pi$
$379$	−2.80127e10			−1.35768			−0.678841	+	0.734286i	$0.737517\pi$
$380$		0			0
$381$	7.53841e9	+	2.76787e10i	0.357750	+	1.31355i
$382$		0			0
$383$	−9.56810e8	−	5.52414e8i	−0.0444662	−	0.0256726i	0.477602	−	0.878576i	$-0.341506\pi$
$383$	−9.56810e8	−	5.52414e8i	−0.0444662	−	0.0256726i	−0.522068	+	0.852904i	$0.674839\pi$
$384$		0			0
$385$	−2.11057e9	−	3.11161e10i	−0.0960632	−	1.41626i
$386$		0			0
$387$	3.04338e10	−	1.79058e10i	1.35679	−	0.798269i
$388$		0			0
$389$	3.07289e10	−	1.77414e10i	1.34199	−	0.774798i	0.354890	−	0.934908i	$-0.384518\pi$
$389$	3.07289e10	−	1.77414e10i	1.34199	−	0.774798i	0.987099	+	0.160110i	$0.0511850\pi$
$390$		0			0
$391$	3.36650e10			1.44036
$392$		0			0
$393$	3.56688e9	−	1.35338e10i	0.149526	−	0.567348i
$394$		0			0
$395$	−4.22072e7	+	2.43684e7i	−0.00173380	+	0.00100101i
$396$		0			0
$397$	1.14535e9	−	1.98380e9i	0.0461078	−	0.0798611i	−0.842050	−	0.539399i	$-0.818652\pi$
$397$	1.14535e9	−	1.98380e9i	0.0461078	−	0.0798611i	0.888158	+	0.459538i	$0.151985\pi$
$398$		0			0
$399$	2.57043e9	−	1.33677e10i	0.101418	−	0.527430i
$400$		0			0
$401$	−4.10788e10	−	2.37168e10i	−1.58869	−	0.917232i	−0.993523	−	0.113632i	$-0.963751\pi$
$401$	−4.10788e10	−	2.37168e10i	−1.58869	−	0.917232i	−0.595170	−	0.803600i	$-0.702915\pi$
$402$		0			0
$403$	2.38266e9	+	4.12690e9i	0.0903322	+	0.156460i
$404$		0			0
$405$	2.59733e10	−	1.55699e10i	0.965401	−	0.578718i
$406$		0			0
$407$			1.22800e10i			0.447527i
$408$		0			0
$409$	1.93658e10	+	3.35426e10i	0.692059	+	1.19868i	0.971162	+	0.238420i	$0.0766294\pi$
$409$	1.93658e10	+	3.35426e10i	0.692059	+	1.19868i	−0.279103	+	0.960261i	$0.590037\pi$
$410$		0			0
$411$	−4.90449e9	−	4.86438e9i	−0.171880	−	0.170475i
$412$		0			0
$413$	−2.29728e10	−	3.42125e10i	−0.789614	−	1.17594i
$414$		0			0
$415$	2.82980e10	−	4.90136e10i	0.954034	−	1.65243i
$416$		0			0
$417$	1.12281e10	+	4.12259e10i	0.371330	+	1.36341i
$418$		0			0
$419$			2.91611e10i			0.946124i	0.881029	+	0.473062i	$0.156851\pi$
$419$			2.91611e10i			0.946124i	−0.881029	+	0.473062i	$0.843149\pi$
$420$		0			0
$421$	3.55004e10			1.13007			0.565034	−	0.825067i	$-0.308863\pi$
$421$	3.55004e10			1.13007			0.565034	+	0.825067i	$0.308863\pi$
$422$		0			0
$423$	−2.99371e10	+	2.45831e8i	−0.935080	+	0.00767848i
$424$		0			0
$425$	1.03189e10	+	5.95764e9i	0.316285	+	0.182607i
$426$		0			0
$427$	−2.75298e10	−	1.34984e10i	−0.828116	−	0.406042i
$428$		0			0
$429$	6.16837e9	+	6.11792e9i	0.182113	+	0.180624i
$430$		0			0
$431$	1.56064e10	−	9.01037e9i	0.452266	−	0.261116i	−0.256521	−	0.966539i	$-0.582576\pi$
$431$	1.56064e10	−	9.01037e9i	0.452266	−	0.261116i	0.708787	+	0.705423i	$0.249243\pi$
$432$		0			0
$433$	4.23529e10			1.20485			0.602424	−	0.798176i	$-0.294202\pi$
$433$	4.23529e10			1.20485			0.602424	+	0.798176i	$0.294202\pi$
$434$		0			0
$435$	−3.50117e10	−	9.22747e9i	−0.977815	−	0.257707i
$436$		0			0
$437$	−1.78566e10	+	1.03095e10i	−0.489634	+	0.282691i
$438$		0			0
$439$	−2.36711e10	+	4.09995e10i	−0.637324	+	1.10388i	0.348694	+	0.937237i	$0.386625\pi$
$439$	−2.36711e10	+	4.09995e10i	−0.637324	+	1.10388i	−0.986018	+	0.166641i	$0.946708\pi$
$440$		0			0
$441$	2.97106e10	−	2.34062e10i	0.785520	−	0.618836i
$442$		0			0
$443$	2.29358e10	+	1.32420e10i	0.595525	+	0.343826i	0.767279	−	0.641313i	$-0.221610\pi$
$443$	2.29358e10	+	1.32420e10i	0.595525	+	0.343826i	−0.171754	+	0.985140i	$0.554944\pi$
$444$		0			0
$445$	3.55348e10	+	6.15481e10i	0.906178	+	1.56955i
$446$		0			0
$447$	1.37722e10	−	5.22558e10i	0.344964	−	1.30889i
$448$		0			0
$449$			4.94979e10i			1.21787i	0.793220	+	0.608935i	$0.208403\pi$
$449$			4.94979e10i			1.21787i	−0.793220	+	0.608935i	$0.791597\pi$
$450$		0			0
$451$	2.43911e10	+	4.22466e10i	0.589556	+	1.02114i
$452$		0			0
$453$	−3.88714e10	+	3.91919e10i	−0.923077	+	0.930688i
$454$		0			0
$455$	4.31946e9	−	8.80948e9i	0.100782	−	0.205544i
$456$		0			0
$457$	8.41985e9	−	1.45836e10i	0.193037	−	0.334349i	−0.753219	−	0.657770i	$-0.771500\pi$
$457$	8.41985e9	−	1.45836e10i	0.193037	−	0.334349i	0.946255	+	0.323421i	$0.104833\pi$
$458$		0			0
$459$	−5.88534e10	−	1.49953e10i	−1.32593	−	0.337835i
$460$		0			0
$461$		−	3.37260e10i		−	0.746726i	−0.927685	−	0.373363i	$-0.878205\pi$
$461$		−	3.37260e10i		−	0.746726i	0.927685	−	0.373363i	$-0.121795\pi$
$462$		0			0
$463$	−3.43863e10			−0.748276			−0.374138	−	0.927373i	$-0.622061\pi$
$463$	−3.43863e10			−0.748276			−0.374138	+	0.927373i	$0.622061\pi$
$464$		0			0
$465$	4.51030e10	−	1.22840e10i	0.964703	−	0.262741i
$466$		0			0
$467$	3.72153e10	+	2.14863e10i	0.782445	+	0.451745i	0.837296	−	0.546750i	$-0.184135\pi$
$467$	3.72153e10	+	2.14863e10i	0.782445	+	0.451745i	−0.0548509	+	0.998495i	$0.517468\pi$
$468$		0			0
$469$	1.25969e10	−	8.45847e9i	0.260358	−	0.174824i
$470$		0			0
$471$	2.26748e10	−	2.28618e10i	0.460745	−	0.464544i
$472$		0			0
$473$	8.60600e10	−	4.96868e10i	1.71932	−	0.992650i
$474$		0			0
$475$	−7.29781e9			−0.143357
$476$		0			0
$477$	−1.46506e10	+	2.58638e10i	−0.282998	+	0.499596i
$478$		0			0
$479$	−3.82865e10	+	2.21047e10i	−0.727284	+	0.419898i	−0.817428	−	0.576031i	$-0.804601\pi$
$479$	−3.82865e10	+	2.21047e10i	−0.727284	+	0.419898i	0.0901438	+	0.995929i	$0.471267\pi$
$480$		0			0
$481$	−1.93160e9	+	3.34564e9i	−0.0360859	+	0.0625027i
$482$		0			0
$483$	−5.62597e10	−	1.08180e10i	−1.03373	−	0.198774i
$484$		0			0
$485$	4.60811e10	+	2.66049e10i	0.832828	+	0.480834i
$486$		0			0
$487$	−2.97796e10	−	5.15799e10i	−0.529424	−	0.916989i	−0.999411	−	0.0343160i	$-0.989075\pi$
$487$	−2.97796e10	−	5.15799e10i	−0.529424	−	0.916989i	0.469987	−	0.882673i	$-0.344259\pi$
$488$		0			0
$489$	−7.05333e10	−	1.85893e10i	−1.23356	−	0.325108i
$490$		0			0
$491$		−	6.52034e10i		−	1.12187i	−0.827858	−	0.560937i	$-0.810441\pi$
$491$		−	6.52034e10i		−	1.12187i	0.827858	−	0.560937i	$-0.189559\pi$
$492$		0			0
$493$	3.63080e10	+	6.28873e10i	0.614631	+	1.06457i
$494$		0			0
$495$	7.34534e10	−	4.32164e10i	1.22346	−	0.719826i
$496$		0			0
$497$	−7.96139e10	+	5.40013e9i	−1.30486	+	0.0885072i
$498$		0			0
$499$	−1.42092e10	+	2.46110e10i	−0.229175	+	0.396942i	−0.957564	−	0.288221i	$-0.906936\pi$
$499$	−1.42092e10	+	2.46110e10i	−0.229175	+	0.396942i	0.728389	+	0.685164i	$0.240269\pi$
$500$		0			0
$501$	−4.15405e10	+	1.13137e10i	−0.659357	+	0.179579i
$502$		0			0
$503$			3.15775e10i			0.493294i	0.969105	+	0.246647i	$0.0793288\pi$
$503$			3.15775e10i			0.493294i	−0.969105	+	0.246647i	$0.920671\pi$
$504$		0			0
$505$	−3.14584e10			−0.483695
$506$		0			0
$507$	−1.66449e10	−	6.11149e10i	−0.251913	−	0.924944i
$508$		0			0
$509$	−9.84004e9	−	5.68115e9i	−0.146597	−	0.0846379i	0.424907	−	0.905237i	$-0.360307\pi$
$509$	−9.84004e9	−	5.68115e9i	−0.146597	−	0.0846379i	−0.571504	+	0.820599i	$0.693640\pi$
$510$		0			0
$511$	1.26809e10	−	8.51493e9i	0.185981	−	0.124881i
$512$		0			0
$513$	3.58091e10	−	1.00693e10i	0.517039	−	0.145389i
$514$		0			0
$515$	−9.83075e10	+	5.67579e10i	−1.39752	+	0.806858i
$516$		0			0
$517$	−8.42542e10			−1.17931
$518$		0			0
$519$	−2.01081e10	+	7.62961e10i	−0.277142	+	1.05156i
$520$		0			0
$521$	4.69731e8	−	2.71199e8i	0.00637527	−	0.00368076i	−0.496809	−	0.867860i	$-0.665495\pi$
$521$	4.69731e8	−	2.71199e8i	0.00637527	−	0.00368076i	0.503184	+	0.864179i	$0.332162\pi$
$522$		0			0
$523$	−4.11546e10	+	7.12818e10i	−0.550061	+	0.952734i	0.448208	+	0.893929i	$0.352062\pi$
$523$	−4.11546e10	+	7.12818e10i	−0.550061	+	0.952734i	−0.998269	+	0.0588050i	$0.981271\pi$
$524$		0			0
$525$	−1.53302e10	−	1.32721e10i	−0.201795	−	0.174704i
$526$		0			0
$527$	−8.11915e10	−	4.68759e10i	−1.05261	−	0.607725i
$528$		0			0
$529$	4.23333e9	+	7.33235e9i	0.0540580	+	0.0936312i
$530$		0			0
$531$	5.55025e10	−	9.79825e10i	0.698127	−	1.23245i
$532$		0			0
$533$			1.53466e10i			0.190153i
$534$		0			0
$535$	3.64991e10	+	6.32184e10i	0.445520	+	0.771664i
$536$		0			0
$537$	−1.97638e10	−	1.96022e10i	−0.237669	−	0.235726i
$538$		0			0
$539$	8.41519e10	−	6.51826e10i	0.997031	−	0.772283i
$540$		0			0
$541$	−2.57238e9	+	4.45549e9i	−0.0300294	+	0.0520124i	−0.880650	−	0.473768i	$-0.842893\pi$
$541$	−2.57238e9	+	4.45549e9i	−0.0300294	+	0.0520124i	0.850620	+	0.525781i	$0.176227\pi$
$542$		0			0
$543$	−2.39942e10	−	8.80990e10i	−0.275998	−	1.01338i
$544$		0			0
$545$			1.67815e11i			1.90215i
$546$		0			0
$547$	−1.34225e11			−1.49929			−0.749643	−	0.661842i	$-0.769775\pi$
$547$	−1.34225e11			−1.49929			−0.749643	+	0.661842i	$0.769775\pi$
$548$		0			0
$549$	−6.87979e8	−	8.37817e10i	−0.00757331	−	0.922273i
$550$		0			0
$551$	−3.85169e10	−	2.22377e10i	−0.417874	−	0.241259i
$552$		0			0
$553$	−1.49352e8	−	7.32304e7i	−0.00159702	−	0.000783052i
$554$		0			0
$555$	2.69066e10	+	2.66866e10i	0.283588	+	0.281269i
$556$		0			0
$557$	8.45787e8	−	4.88316e8i	0.00878700	−	0.00507317i	−0.495600	−	0.868551i	$-0.665052\pi$
$557$	8.45787e8	−	4.88316e8i	0.00878700	−	0.00507317i	0.504387	+	0.863478i	$0.331718\pi$
$558$		0			0
$559$	3.12624e10			0.320166
$560$		0			0
$561$	−1.65278e11	−	4.35595e10i	−1.66864	−	0.439776i
$562$		0			0
$563$	5.64564e10	−	3.25951e10i	0.561926	−	0.324428i	−0.191992	−	0.981396i	$-0.561495\pi$
$563$	5.64564e10	−	3.25951e10i	0.561926	−	0.324428i	0.753918	+	0.656968i	$0.228161\pi$
$564$		0			0
$565$	4.63777e10	−	8.03285e10i	0.455109	−	0.788272i
$566$		0			0
$567$	9.35349e10	+	4.39717e10i	0.904985	+	0.425443i
$568$		0			0
$569$	−3.37409e10	−	1.94803e10i	−0.321890	−	0.185844i	0.330344	−	0.943860i	$-0.392835\pi$
$569$	−3.37409e10	−	1.94803e10i	−0.321890	−	0.185844i	−0.652235	+	0.758017i	$0.726168\pi$
$570$		0			0
$571$	7.02714e9	+	1.21714e10i	0.0661050	+	0.114497i	0.897184	−	0.441658i	$-0.145609\pi$
$571$	7.02714e9	+	1.21714e10i	0.0661050	+	0.114497i	−0.831079	+	0.556155i	$0.812276\pi$
$572$		0			0
$573$	−4.82780e10	+	1.83181e11i	−0.447848	+	1.69927i
$574$		0			0
$575$			3.07138e10i			0.280971i
$576$		0			0
$577$	2.30836e10	+	3.99819e10i	0.208257	+	0.360712i	0.951166	−	0.308681i	$-0.0998876\pi$
$577$	2.30836e10	+	3.99819e10i	0.208257	+	0.360712i	−0.742908	+	0.669393i	$0.766554\pi$
$578$		0			0
$579$	8.98543e9	−	9.05952e9i	0.0799511	−	0.0806104i
$580$		0			0
$581$	1.92721e11	−	1.30720e10i	1.69131	−	0.114720i
$582$		0			0
$583$	−4.18272e10	+	7.24468e10i	−0.362063	+	0.627112i
$584$		0			0
$585$	2.68100e10	−	2.20152e8i	0.228915	−	0.00187975i
$586$		0			0
$587$		−	9.21090e10i		−	0.775800i	−0.921702	−	0.387900i	$-0.873201\pi$
$587$		−	9.21090e10i		−	0.775800i	0.921702	−	0.387900i	$-0.126799\pi$
$588$		0			0
$589$	5.74206e10			0.477097
$590$		0			0
$591$	−3.00182e10	+	8.17560e9i	−0.246057	+	0.0670147i
$592$		0			0
$593$	−1.75779e11	−	1.01486e11i	−1.42150	−	0.820704i	−0.425074	−	0.905158i	$-0.639752\pi$
$593$	−1.75779e11	−	1.01486e11i	−1.42150	−	0.820704i	−0.996427	+	0.0844539i	$0.973085\pi$
$594$		0			0
$595$	1.30629e10	+	1.92585e11i	0.104225	+	1.53658i
$596$		0			0
$597$	−1.52517e11	+	1.53775e11i	−1.20066	+	1.21056i
$598$		0			0
$599$	4.83697e10	−	2.79263e10i	0.375722	−	0.216923i	−0.300233	−	0.953866i	$-0.597065\pi$
$599$	4.83697e10	−	2.79263e10i	0.375722	−	0.216923i	0.675955	+	0.736943i	$0.263731\pi$
$600$		0			0
$601$	−4.50162e9			−0.0345041			−0.0172521	−	0.999851i	$-0.505492\pi$
$601$	−4.50162e9			−0.0345041			−0.0172521	+	0.999851i	$0.505492\pi$
$602$		0			0
$603$	3.60766e10	+	2.04357e10i	0.272870	+	0.154568i
$604$		0			0
$605$	7.71151e10	−	4.45224e10i	0.575596	−	0.332321i
$606$		0			0
$607$	4.96900e10	−	8.60655e10i	0.366028	−	0.633978i	−0.622913	−	0.782291i	$-0.714051\pi$
$607$	4.96900e10	−	8.60655e10i	0.366028	−	0.633978i	0.988941	+	0.148313i	$0.0473842\pi$
$608$		0			0
$609$	−4.04681e10	−	1.16762e11i	−0.294201	−	0.848854i
$610$		0			0
$611$	−2.29548e10	−	1.32530e10i	−0.164706	−	0.0950929i
$612$		0			0
$613$	3.94467e10	+	6.83237e10i	0.279363	+	0.483871i	0.971227	−	0.238157i	$-0.0765434\pi$
$613$	3.94467e10	+	6.83237e10i	0.279363	+	0.483871i	−0.691864	+	0.722028i	$0.743210\pi$
$614$		0			0
$615$	1.45573e11	+	3.83663e10i	1.01761	+	0.268194i
$616$		0			0
$617$			1.61044e11i			1.11123i	0.831439	+	0.555616i	$0.187518\pi$
$617$			1.61044e11i			1.11123i	−0.831439	+	0.555616i	$0.812482\pi$
$618$		0			0
$619$	4.81929e10	+	8.34725e10i	0.328262	+	0.568566i	0.982167	−	0.188010i	$-0.0602038\pi$
$619$	4.81929e10	+	8.34725e10i	0.328262	+	0.568566i	−0.653905	+	0.756576i	$0.726870\pi$
$620$		0			0
$621$	−4.23781e10	−	1.50707e11i	−0.284954	−	1.01337i
$622$		0			0
$623$	−1.06787e11	+	2.17791e11i	−0.708870	+	1.44573i
$624$		0			0
$625$	9.12224e10	−	1.58002e11i	0.597835	−	1.03548i
$626$		0			0
$627$	1.01006e11	−	2.75094e10i	0.653546	−	0.177996i
$628$		0			0
$629$		−	7.60038e10i		−	0.485548i
$630$		0			0
$631$	2.33586e11			1.47343			0.736713	−	0.676205i	$-0.236377\pi$
$631$	2.33586e11			1.47343			0.736713	+	0.676205i	$0.236377\pi$
$632$		0			0
$633$	−1.10271e10	−	4.04879e10i	−0.0686823	−	0.252180i
$634$		0			0
$635$	2.15766e11	+	1.24572e11i	1.32705	+	0.766173i
$636$		0			0
$637$	3.31800e10	−	4.52193e9i	0.201520	−	0.0274641i
$638$		0			0
$639$	−1.10574e11	−	1.87939e11i	−0.663207	−	1.12723i
$640$		0			0
$641$	−1.89291e11	+	1.09287e11i	−1.12124	+	0.647346i	−0.941716	−	0.336408i	$-0.890788\pi$
$641$	−1.89291e11	+	1.09287e11i	−1.12124	+	0.647346i	−0.179520	+	0.983754i	$0.557454\pi$
$642$		0			0
$643$	−1.74202e10			−0.101908			−0.0509542	−	0.998701i	$-0.516226\pi$
$643$	−1.74202e10			−0.101908			−0.0509542	+	0.998701i	$0.516226\pi$
$644$		0			0
$645$	7.81554e10	−	2.96545e11i	0.451565	−	1.71337i
$646$		0			0
$647$	1.00709e11	−	5.81445e10i	0.574715	−	0.331812i	−0.184316	−	0.982867i	$-0.559007\pi$
$647$	1.00709e11	−	5.81445e10i	0.574715	−	0.331812i	0.759030	+	0.651055i	$0.225673\pi$
$648$		0			0
$649$	1.58458e11	−	2.74458e11i	0.893174	−	1.54702i
$650$		0			0
$651$	1.20621e11	+	1.04428e11i	0.671581	+	0.581421i
$652$		0			0
$653$	5.69948e9	+	3.29059e9i	0.0313460	+	0.0180976i	0.515591	−	0.856835i	$-0.327572\pi$
$653$	5.69948e9	+	3.29059e9i	0.0313460	+	0.0180976i	−0.484245	+	0.874932i	$0.660906\pi$
$654$		0			0
$655$	−6.07771e10	−	1.05269e11i	−0.330198	−	0.571920i
$656$		0			0
$657$	3.63174e10	+	2.05721e10i	0.194919	+	0.110412i
$658$		0			0
$659$		−	2.09248e11i		−	1.10948i	−0.832024	−	0.554739i	$-0.812818\pi$
$659$		−	2.09248e11i		−	1.10948i	0.832024	−	0.554739i	$-0.187182\pi$
$660$		0			0
$661$	−1.12959e11	−	1.95650e11i	−0.591716	−	1.02488i	−0.994001	−	0.109368i	$-0.965117\pi$
$661$	−1.12959e11	−	1.95650e11i	−0.591716	−	1.02488i	0.402285	−	0.915514i	$-0.368216\pi$
$662$		0			0
$663$	−3.81776e10	−	3.78654e10i	−0.197585	−	0.195969i
$664$		0			0
$665$	−6.59057e10	−	9.81506e10i	−0.337005	−	0.501888i
$666$		0			0
$667$	−9.35905e10	+	1.62103e11i	−0.472855	+	0.819009i
$668$		0			0
$669$	5.57286e10	+	2.04618e11i	0.278211	+	1.02150i
$670$		0			0
$671$		−	2.35792e11i		−	1.16316i
$672$		0			0
$673$	2.24850e11			1.09606			0.548028	−	0.836460i	$-0.315379\pi$
$673$	2.24850e11			1.09606			0.548028	+	0.836460i	$0.315379\pi$
$674$		0			0
$675$	1.36807e10	−	5.36941e10i	0.0659014	−	0.258649i
$676$		0			0
$677$	4.96379e10	+	2.86584e10i	0.236297	+	0.136426i	0.613474	−	0.789715i	$-0.289772\pi$
$677$	4.96379e10	+	2.86584e10i	0.236297	+	0.136426i	−0.377177	+	0.926141i	$0.623105\pi$
$678$		0			0
$679$	1.22899e10	+	1.81190e11i	0.0578189	+	0.852422i
$680$		0			0
$681$	2.57024e11	+	2.54922e11i	1.19505	+	1.18527i
$682$		0			0
$683$	−1.92706e10	+	1.11259e10i	−0.0885549	+	0.0511272i	−0.543624	−	0.839329i	$-0.682948\pi$
$683$	−1.92706e10	+	1.11259e10i	−0.0885549	+	0.0511272i	0.455069	+	0.890456i	$0.349615\pi$
$684$		0			0
$685$	−5.99931e10			−0.272483
$686$		0			0
$687$	2.93201e11	+	7.72741e10i	1.31625	+	0.346902i
$688$		0			0
$689$	−2.27914e10	+	1.31586e10i	−0.101133	+	0.0583893i
$690$		0			0
$691$	−1.42600e11	+	2.46990e11i	−0.625471	+	1.08335i	0.362979	+	0.931797i	$0.381760\pi$
$691$	−1.42600e11	+	2.46990e11i	−0.625471	+	1.08335i	−0.988450	+	0.151550i	$0.951574\pi$
$692$		0			0
$693$	2.62208e11	+	1.25906e11i	1.13688	+	0.545899i
$694$		0			0
$695$	3.21371e11	+	1.85544e11i	1.37742	+	0.795257i
$696$		0			0
$697$	−1.50963e11	−	2.61475e11i	−0.639644	−	1.10790i
$698$		0			0
$699$	−6.18021e10	+	2.34495e11i	−0.258878	+	0.982258i
$700$		0			0
$701$			6.19979e10i			0.256747i	0.991726	+	0.128373i	$0.0409756\pi$
$701$			6.19979e10i			0.256747i	−0.991726	+	0.128373i	$0.959024\pi$
$702$		0			0
$703$	2.32752e10	+	4.03138e10i	0.0952954	+	0.165057i
$704$		0			0
$705$	−1.83100e11	+	1.84610e11i	−0.741193	+	0.747305i
$706$		0			0
$707$	−5.98536e10	−	8.91375e10i	−0.239559	−	0.356766i
$708$		0			0
$709$	6.25605e10	−	1.08358e11i	0.247580	−	0.428821i	−0.715274	−	0.698844i	$-0.753698\pi$
$709$	6.25605e10	−	1.08358e11i	0.247580	−	0.428821i	0.962854	+	0.270023i	$0.0870315\pi$
$710$		0			0
$711$	−3.73237e6	−	4.54526e8i	−1.46052e−5	−	0.00177861i
$712$		0			0
$713$		−	2.41662e11i		−	0.935084i
$714$		0			0
$715$	7.54532e10			0.288705
$716$		0			0
$717$	−3.84542e11	+	1.04732e11i	−1.45501	+	0.396280i
$718$		0			0
$719$	−1.64981e11	−	9.52517e10i	−0.617330	−	0.356416i	0.158499	−	0.987359i	$-0.449335\pi$
$719$	−1.64981e11	−	9.52517e10i	−0.617330	−	0.356416i	−0.775829	+	0.630943i	$0.782668\pi$
$720$		0			0
$721$	−3.47866e11	−	1.70566e11i	−1.28727	−	0.631176i
$722$		0			0
$723$	−3.07095e11	+	3.09627e11i	−1.12388	+	1.13315i
$724$		0			0
$725$	−5.73743e10	+	3.31251e10i	−0.207666	+	0.119896i
$726$		0			0
$727$	−4.27788e11			−1.53141			−0.765704	−	0.643193i	$-0.777609\pi$
$727$	−4.27788e11			−1.53141			−0.765704	+	0.643193i	$0.777609\pi$
$728$		0			0
$729$	6.95671e9	+	2.82344e11i	0.0246317	+	0.999697i
$730$		0			0
$731$	−5.32647e11	+	3.07524e11i	−1.86539	+	1.07698i
$732$		0			0
$733$	1.01219e11	−	1.75317e11i	0.350628	−	0.607305i	−0.635732	−	0.771910i	$-0.719302\pi$
$733$	1.01219e11	−	1.75317e11i	0.350628	−	0.607305i	0.986360	+	0.164605i	$0.0526349\pi$
$734$		0			0
$735$	4.00558e10	−	3.26039e11i	0.137251	−	1.11717i
$736$		0			0
$737$	1.01054e11	+	5.83434e10i	0.342517	+	0.197752i
$738$		0			0
$739$	−9.09328e9	−	1.57500e10i	−0.0304890	−	0.0528085i	0.850378	−	0.526172i	$-0.176373\pi$
$739$	−9.09328e9	−	1.57500e10i	−0.0304890	−	0.0528085i	−0.880867	+	0.473363i	$0.843040\pi$
$740$		0			0
$741$	3.18459e10	+	8.39310e9i	0.105628	+	0.0278388i
$742$		0			0
$743$		−	3.68118e11i		−	1.20790i	−0.797021	−	0.603952i	$-0.793592\pi$
$743$		−	3.68118e11i		−	1.20790i	0.797021	−	0.603952i	$-0.206408\pi$
$744$		0			0
$745$	−2.34669e11	−	4.06458e11i	−0.761781	−	1.31944i
$746$		0			0
$747$	2.67665e11	+	4.54941e11i	0.859625	+	1.46107i
$748$		0			0
$749$	−1.09685e11	+	2.23701e11i	−0.348514	+	0.710789i
$750$		0			0
$751$	9.42942e10	−	1.63322e11i	0.296432	−	0.513435i	−0.678885	−	0.734245i	$-0.737536\pi$
$751$	9.42942e10	−	1.63322e11i	0.296432	−	0.513435i	0.975317	+	0.220809i	$0.0708698\pi$
$752$		0			0
$753$	3.75814e11	−	1.02355e11i	1.16894	−	0.318367i
$754$		0			0
$755$			4.79407e11i			1.47542i
$756$		0			0
$757$	5.07735e11			1.54616			0.773079	−	0.634310i	$-0.218716\pi$
$757$	5.07735e11			1.54616			0.773079	+	0.634310i	$0.218716\pi$
$758$		0			0
$759$	−1.15777e11	−	4.25096e11i	−0.348863	−	1.28092i
$760$		0			0
$761$	4.03325e10	+	2.32860e10i	0.120259	+	0.0694314i	0.558923	−	0.829220i	$-0.311215\pi$
$761$	4.03325e10	+	2.32860e10i	0.120259	+	0.0694314i	−0.438664	+	0.898651i	$0.644548\pi$
$762$		0			0
$763$	−4.75503e11	+	3.19288e11i	−1.40299	+	0.942074i
$764$		0			0
$765$	−4.54622e11	+	2.67477e11i	−1.32741	+	0.780982i
$766$		0			0
$767$	8.63430e10	−	4.98502e10i	0.249486	−	0.144041i
$768$		0			0
$769$	−3.17047e11			−0.906605			−0.453303	−	0.891357i	$-0.649754\pi$
$769$	−3.17047e11			−0.906605			−0.453303	+	0.891357i	$0.649754\pi$
$770$		0			0
$771$	2.41383e9	−	9.15877e9i	0.00683107	−	0.0259191i
$772$		0			0
$773$	1.38340e11	−	7.98706e10i	0.387462	−	0.223702i	−0.293598	−	0.955929i	$-0.594853\pi$
$773$	1.38340e11	−	7.98706e10i	0.387462	−	0.223702i	0.681060	+	0.732228i	$0.261519\pi$
$774$		0			0
$775$	4.27666e10	−	7.40738e10i	0.118549	−	0.205333i
$776$		0			0
$777$	−2.44233e10	+	1.27015e11i	−0.0670069	+	0.348473i
$778$		0			0
$779$	1.60147e11	+	9.24608e10i	0.434879	+	0.251077i
$780$		0			0
$781$	−3.06832e11	−	5.31448e11i	−0.824700	−	1.42842i
$782$		0			0
$783$	2.35821e11	−	2.41703e11i	0.627386	−	0.643035i
$784$		0			0
$785$		−	2.79652e11i		−	0.736443i
$786$		0			0
$787$	−1.84155e11	−	3.18967e11i	−0.480049	−	0.831470i	0.519689	−	0.854356i	$-0.326048\pi$
$787$	−1.84155e11	−	3.18967e11i	−0.480049	−	0.831470i	−0.999738	+	0.0228860i	$0.992715\pi$
$788$		0			0
$789$	1.48375e11	+	1.47161e11i	0.382870	+	0.379739i
$790$		0			0
$791$	3.15850e11	−	2.14238e10i	0.806817	−	0.0547255i
$792$		0			0
$793$	3.70896e10	−	6.42410e10i	0.0937905	−	0.162450i
$794$		0			0
$795$	6.78402e10	+	2.49088e11i	0.169832	+	0.623568i
$796$		0			0
$797$		−	1.43490e11i		−	0.355623i	−0.984065	−	0.177811i	$-0.943098\pi$
$797$		−	1.43490e11i		−	0.355623i	0.984065	−	0.177811i	$-0.0569017\pi$
$798$		0			0
$799$	5.21470e11			1.27951
$800$		0			0
$801$	−6.62805e11	+	5.44267e9i	−1.61011	+	0.0132215i
$802$		0			0
$803$	1.01728e11	+	5.87328e10i	0.244669	+	0.141260i
$804$		0			0
$805$	−4.13080e11	+	2.77373e11i	−0.983672	+	0.660511i
$806$		0			0
$807$	−3.79457e11	−	3.76354e11i	−0.894681	−	0.887365i
$808$		0			0
$809$	2.06797e11	−	1.19394e11i	0.482781	−	0.278734i	−0.238794	−	0.971070i	$-0.576752\pi$
$809$	2.06797e11	−	1.19394e11i	0.482781	−	0.278734i	0.721575	+	0.692337i	$0.243419\pi$
$810$		0			0
$811$	−5.41244e11			−1.25115			−0.625575	−	0.780164i	$-0.715136\pi$
$811$	−5.41244e11			−1.25115			−0.625575	+	0.780164i	$0.715136\pi$
$812$		0			0
$813$	1.22929e11	+	3.23985e10i	0.281380	+	0.0741587i
$814$		0			0
$815$	−5.48625e11	+	3.16749e11i	−1.24350	+	0.717934i
$816$		0			0
$817$	1.88351e11	−	3.26233e11i	0.422746	−	0.732217i
$818$		0			0
$819$	5.16332e10	+	7.55473e10i	0.114761	+	0.167913i
$820$		0			0
$821$	1.29258e11	+	7.46272e10i	0.284502	+	0.164257i	0.635460	−	0.772134i	$-0.280811\pi$
$821$	1.29258e11	+	7.46272e10i	0.284502	+	0.164257i	−0.350958	+	0.936391i	$0.614144\pi$
$822$		0			0
$823$	−1.20563e10	−	2.08821e10i	−0.0262793	−	0.0455171i	0.852587	−	0.522586i	$-0.175033\pi$
$823$	−1.20563e10	−	2.08821e10i	−0.0262793	−	0.0455171i	−0.878866	+	0.477069i	$0.841699\pi$
$824$		0			0
$825$	3.97409e10	−	1.50789e11i	0.0857870	−	0.325501i
$826$		0			0
$827$			2.25722e11i			0.482561i	0.970455	+	0.241280i	$0.0775674\pi$
$827$			2.25722e11i			0.482561i	−0.970455	+	0.241280i	$0.922433\pi$
$828$		0			0
$829$	−1.57764e11	−	2.73256e11i	−0.334034	−	0.578564i	0.649265	−	0.760562i	$-0.275077\pi$
$829$	−1.57764e11	−	2.73256e11i	−0.334034	−	0.578564i	−0.983299	+	0.181999i	$0.941743\pi$
$830$		0			0
$831$	3.49667e11	−	3.52550e11i	0.733247	−	0.739293i
$832$		0			0
$833$	−5.20837e11	+	4.03431e11i	−1.08174	+	0.837895i
$834$		0			0
$835$	−1.86960e11	+	3.23823e11i	−0.384593	+	0.666135i
$836$		0			0
$837$	−1.07643e11	+	4.22476e11i	−0.219323	+	0.860796i
$838$		0			0
$839$		−	4.05393e10i		−	0.0818141i	−0.999163	−	0.0409071i	$-0.986975\pi$
$839$		−	4.05393e10i		−	0.0818141i	0.999163	−	0.0409071i	$-0.0130248\pi$
$840$		0			0
$841$	9.64939e10			0.192893
$842$		0			0
$843$	3.70193e11	−	1.00824e11i	0.733024	−	0.199642i
$844$		0			0
$845$	−4.76414e11	−	2.75058e11i	−0.934453	−	0.539507i
$846$		0			0
$847$	2.72875e11	+	1.33796e11i	0.530189	+	0.259962i
$848$		0			0
$849$	−3.00454e11	+	3.02931e11i	−0.578292	+	0.583060i
$850$		0			0
$851$	1.69666e11	−	9.79567e10i	0.323502	−	0.186774i
$852$		0			0
$853$	6.56671e10			0.124037			0.0620185	−	0.998075i	$-0.480246\pi$
$853$	6.56671e10			0.124037			0.0620185	+	0.998075i	$0.480246\pi$
$854$		0			0
$855$	1.59228e11	−	2.81097e11i	0.297959	−	0.526008i
$856$		0			0
$857$	8.27521e11	−	4.77770e11i	1.53411	−	0.885717i	0.534941	−	0.844889i	$-0.320334\pi$
$857$	8.27521e11	−	4.77770e11i	1.53411	−	0.885717i	0.999166	−	0.0408282i	$-0.0129996\pi$
$858$		0			0
$859$	−2.13682e11	+	3.70109e11i	−0.392461	+	0.679762i	−0.992773	−	0.120003i	$-0.961709\pi$
$859$	−2.13682e11	+	3.70109e11i	−0.392461	+	0.679762i	0.600313	+	0.799765i	$0.295043\pi$
$860$		0			0
$861$	1.68260e11	+	4.85477e11i	0.306174	+	0.883398i
$862$		0			0
$863$	5.78218e11	+	3.33834e11i	1.04243	+	0.601849i	0.920521	−	0.390692i	$-0.127764\pi$
$863$	5.78218e11	+	3.33834e11i	1.04243	+	0.601849i	0.121912	+	0.992541i	$0.461098\pi$
$864$		0			0
$865$	3.42628e11	+	5.93449e11i	0.612010	+	1.06003i
$866$		0			0
$867$	4.76566e11	+	1.25601e11i	0.843425	+	0.222288i
$868$		0			0
$869$		−	1.27920e9i		−	0.00224316i
$870$		0			0
$871$	1.83545e10	+	3.17910e10i	0.0318912	+	0.0552372i
$872$		0			0
$873$	−4.27721e11	+	2.51650e11i	−0.736383	+	0.433252i
$874$		0			0
$875$	4.82575e11	−	3.27325e10i	0.823251	−	0.0558402i
$876$		0			0
$877$	7.18711e10	−	1.24484e11i	0.121494	−	0.210434i	−0.798863	−	0.601513i	$-0.794565\pi$
$877$	7.18711e10	−	1.24484e11i	0.121494	−	0.210434i	0.920357	+	0.391079i	$0.127898\pi$
$878$		0			0
$879$	−9.49925e10	+	2.58717e10i	−0.159123	+	0.0433380i
$880$		0			0
$881$			5.96835e11i			0.990720i	0.868688	+	0.495360i	$0.164964\pi$
$881$			5.96835e11i			0.990720i	−0.868688	+	0.495360i	$0.835036\pi$
$882$		0			0
$883$	1.17561e12			1.93384			0.966922	−	0.255071i	$-0.0820989\pi$
$883$	1.17561e12			1.93384			0.966922	+	0.255071i	$0.0820989\pi$
$884$		0			0
$885$	−2.57006e11	−	9.43646e11i	−0.418958	−	1.53828i
$886$		0			0
$887$	−6.44471e11	−	3.72086e11i	−1.04114	−	0.601103i	−0.120984	−	0.992654i	$-0.538605\pi$
$887$	−6.44471e11	−	3.72086e11i	−1.04114	−	0.601103i	−0.920156	+	0.391552i	$0.871938\pi$
$888$		0			0
$889$	5.75451e10	+	8.48386e11i	0.0921301	+	1.35827i
$890$		0			0
$891$	1.30528e10	+	7.94727e11i	0.0207106	+	1.26098i
$892$		0			0
$893$	−2.76598e11	+	1.59694e11i	−0.434953	+	0.251120i
$894$		0			0
$895$	−2.41756e11			−0.376778
$896$		0			0
$897$	3.53235e10	−	1.34028e11i	0.0545624	−	0.207026i
$898$		0			0
$899$	4.51433e11	−	2.60635e11i	0.691121	−	0.399019i
$900$		0			0
$901$	2.58879e11	−	4.48392e11i	0.392824	−	0.680391i
$902$		0			0
$903$	9.88960e11	−	3.42760e11i	1.48740	−	0.515512i
$904$		0			0
$905$	−6.86764e11	−	3.96504e11i	−1.02380	−	0.591089i
$906$		0			0
$907$	−1.91309e11	−	3.31357e11i	−0.282687	−	0.489629i	0.689358	−	0.724420i	$-0.257893\pi$
$907$	−1.91309e11	−	3.31357e11i	−0.282687	−	0.489629i	−0.972046	+	0.234792i	$0.924559\pi$
$908$		0			0
$909$	1.44607e11	−	2.55284e11i	0.211803	−	0.373911i
$910$		0			0
$911$		−	1.00126e12i		−	1.45369i	−0.686802	−	0.726845i	$-0.740986\pi$
$911$		−	1.00126e12i		−	1.45369i	0.686802	−	0.726845i	$-0.259014\pi$
$912$		0			0
$913$	7.42744e11	+	1.28647e12i	1.06895	+	1.85147i
$914$		0			0
$915$	−5.16646e11	−	5.12420e11i	−0.737069	−	0.731042i
$916$		0			0
$917$	1.82644e11	−	3.72500e11i	0.258302	−	0.526803i
$918$		0			0
$919$	4.46267e11	−	7.72957e11i	0.625652	−	1.08366i	−0.362763	−	0.931882i	$-0.618166\pi$
$919$	4.46267e11	−	7.72957e11i	0.625652	−	1.08366i	0.988414	−	0.151779i	$-0.0485003\pi$
$920$		0			0
$921$	−9.27072e10	−	3.40392e11i	−0.128847	−	0.473086i
$922$		0			0
$923$		−	1.93055e11i		−	0.265996i
$924$		0			0
$925$	6.93409e10			0.0947158
$926$		0			0
$927$	−8.69330e9	−	1.05866e12i	−0.0117724	−	1.43364i
$928$		0			0
$929$	1.23462e12	+	7.12809e11i	1.65757	+	0.956997i	0.973833	+	0.227266i	$0.0729787\pi$
$929$	1.23462e12	+	7.12809e11i	1.65757	+	0.956997i	0.683734	+	0.729731i	$0.260355\pi$
$930$		0			0
$931$	1.52716e11	−	3.73488e11i	0.203276	−	0.497138i
$932$		0			0
$933$	5.40402e11	+	5.35982e11i	0.713165	+	0.707333i
$934$		0			0
$935$	−1.28557e12	+	7.42223e11i	−1.68209	+	0.971154i
$936$		0			0
$937$	1.25608e12			1.62952			0.814761	−	0.579797i	$-0.196868\pi$
$937$	1.25608e12			1.62952			0.814761	+	0.579797i	$0.196868\pi$
$938$		0			0
$939$	−1.69760e11	−	4.47409e10i	−0.218360	−	0.0575496i
$940$		0			0
$941$	−5.21530e11	+	3.01105e11i	−0.665151	+	0.384025i	−0.794237	−	0.607608i	$-0.792129\pi$
$941$	−5.21530e11	+	3.01105e11i	−0.665151	+	0.384025i	0.129086	+	0.991633i	$0.458796\pi$
$942$		0			0
$943$	3.89133e11	−	6.73999e11i	0.492098	−	0.852339i
$944$		0			0
$945$	8.45698e11	−	3.00908e11i	1.06045	−	0.377317i
$946$		0			0
$947$	−8.89542e11	−	5.13577e11i	−1.10603	−	0.638566i	−0.168231	−	0.985748i	$-0.553805\pi$
$947$	−8.89542e11	−	5.13577e11i	−1.10603	−	0.638566i	−0.937798	+	0.347181i	$0.887139\pi$
$948$		0			0
$949$	1.84770e10	+	3.20032e10i	0.0227807	+	0.0394574i
$950$		0			0
$951$	−2.41837e11	+	9.17600e11i	−0.295665	+	1.12184i
$952$		0			0
$953$		−	3.66907e10i		−	0.0444820i	−0.999753	−	0.0222410i	$-0.992920\pi$
$953$		−	3.66907e10i		−	0.0444820i	0.999753	−	0.0222410i	$-0.00708012\pi$
$954$		0			0
$955$	8.22623e11	+	1.42482e12i	0.988979	+	1.71296i
$956$		0			0
$957$	6.69227e11	−	6.74745e11i	0.797859	−	0.804437i
$958$		0			0
$959$	−1.14144e11	−	1.69990e11i	−0.134952	−	0.200979i
$960$		0			0
$961$	8.99496e10	−	1.55797e11i	0.105464	−	0.182670i
$962$		0			0
$963$	−6.80792e11	+	5.59037e9i	−0.791607	+	0.00650034i
$964$		0			0
$965$		−	1.10819e11i		−	0.127792i
$966$		0			0
$967$	4.01353e11			0.459008			0.229504	−	0.973308i	$-0.426290\pi$
$967$	4.01353e11			0.459008			0.229504	+	0.973308i	$0.426290\pi$
$968$		0			0
$969$	−6.25151e11	+	1.70263e11i	−0.709071	+	0.193119i
$970$		0			0
$971$	−5.62250e11	−	3.24615e11i	−0.632489	−	0.365168i	0.149227	−	0.988803i	$-0.452322\pi$
$971$	−5.62250e11	−	3.24615e11i	−0.632489	−	0.365168i	−0.781715	+	0.623635i	$0.785655\pi$
$972$		0			0
$973$	8.57104e10	+	1.26363e12i	0.0956273	+	1.40983i
$974$		0			0
$975$	3.45459e10	−	3.48308e10i	0.0382277	−	0.0385429i
$976$		0			0
$977$	1.07179e12	−	6.18797e11i	1.17633	−	0.679156i	0.221170	−	0.975235i	$-0.429012\pi$
$977$	1.07179e12	−	6.18797e11i	1.17633	−	0.679156i	0.955163	+	0.296079i	$0.0956792\pi$
$978$		0			0
$979$	−1.86538e12			−2.03065
$980$		0			0
$981$	−1.36181e12	−	7.71402e11i	−1.47042	−	0.832923i
$982$		0			0
$983$	2.53961e11	−	1.46625e11i	0.271990	−	0.157034i	−0.357802	−	0.933798i	$-0.616474\pi$
$983$	2.53961e11	−	1.46625e11i	0.271990	−	0.157034i	0.629792	+	0.776764i	$0.283140\pi$
$984$		0			0
$985$	−1.35102e11	+	2.34003e11i	−0.143521	+	0.248586i
$986$		0			0
$987$	−8.71461e11	−	1.67571e11i	−0.918289	−	0.176575i
$988$		0			0
$989$	−1.37299e12	−	7.92699e11i	−1.43510	−	0.828558i
$990$		0			0
$991$	−6.59894e11	−	1.14297e12i	−0.684194	−	1.18506i	−0.973689	−	0.227880i	$-0.926821\pi$
$991$	−6.59894e11	−	1.14297e12i	−0.684194	−	1.18506i	0.289495	−	0.957180i	$-0.406513\pi$
$992$		0			0
$993$	1.13890e12	+	3.00161e11i	1.17135	+	0.308714i
$994$		0			0
$995$			1.88102e12i			1.91911i
$996$		0			0
$997$	4.15131e11	+	7.19027e11i	0.420150	+	0.727721i	0.995954	−	0.0898671i	$-0.0286442\pi$
$997$	4.15131e11	+	7.19027e11i	0.420150	+	0.727721i	−0.575804	+	0.817588i	$0.695311\pi$
$998$		0			0
$999$	−3.40244e11	+	9.56749e10i	−0.341608	+	0.0960585i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.9.p.b.65.11	yes	40
3.2	odd	2	inner	84.9.p.b.65.15	yes	40
7.4	even	3	inner	84.9.p.b.53.15	yes	40
21.11	odd	6	inner	84.9.p.b.53.11	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.9.p.b.53.11	✓	40	21.11	odd	6	inner
84.9.p.b.53.15	yes	40	7.4	even	3	inner
84.9.p.b.65.11	yes	40	1.1	even	1	trivial
84.9.p.b.65.15	yes	40	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	84.p (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(21.2854 + 78.1533i) q^{3} +(609.234 + 351.741i) q^{5} +(162.484 + 2395.50i) q^{7} +(-5654.86 + 3327.05i) q^{9} +O(q^{10})\)	\(q+(21.2854 + 78.1533i) q^{3} +(609.234 + 351.741i) q^{5} +(162.484 + 2395.50i) q^{7} +(-5654.86 + 3327.05i) q^{9} +(-15990.7 + 9232.23i) q^{11} -5808.82 q^{13} +(-14521.9 + 55100.6i) q^{15} +(98970.4 - 57140.6i) q^{17} +(-34997.2 + 60616.9i) q^{19} +(-183757. + 63687.7i) q^{21} +(255114. + 147290. i) q^{23} +(52131.4 + 90294.2i) q^{25} +(-380385. - 371129. i) q^{27} +635415. i q^{29} +(-410180. - 710453. i) q^{31} +(-1.06190e6 - 1.05321e6i) q^{33} +(-743604. + 1.51657e6i) q^{35} +(332530. - 575958. i) q^{37} +(-123643. - 453978. i) q^{39} -2.64195e6i q^{41} -5.38188e6 q^{43} +(-4.61539e6 + 37899.6i) q^{45} +(3.95171e6 + 2.28152e6i) q^{47} +(-5.71200e6 + 778459. i) q^{49} +(6.57235e6 + 6.51860e6i) q^{51} +(3.92358e6 - 2.26528e6i) q^{53} -1.29894e7 q^{55} +(-5.48234e6 - 1.44489e6i) q^{57} +(-1.48641e7 + 8.58180e6i) q^{59} +(-6.38504e6 + 1.10592e7i) q^{61} +(-8.88875e6 - 1.30056e7i) q^{63} +(-3.53893e6 - 2.04320e6i) q^{65} +(-3.15977e6 - 5.47288e6i) q^{67} +(-6.08101e6 + 2.30731e7i) q^{69} +3.32348e7i q^{71} +(-3.18086e6 - 5.50941e6i) q^{73} +(-5.94715e6 + 5.99619e6i) q^{75} +(-2.47140e7 - 3.68055e7i) q^{77} +(-34639.6 + 59997.6i) q^{79} +(2.09083e7 - 3.76280e7i) q^{81} -8.04512e7i q^{83} +8.03948e7 q^{85} +(-4.96598e7 + 1.35251e7i) q^{87} +(8.74905e7 + 5.05127e7i) q^{89} +(-943840. - 1.39150e7i) q^{91} +(4.67934e7 - 4.71792e7i) q^{93} +(-4.26429e7 + 2.46199e7i) q^{95} +7.56377e7 q^{97} +(5.97091e7 - 1.05409e8i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 81 q^{3} - 34 q^{7} + 4771 q^{9}+O(q^{10}) \) 40 * q + 81 * q^3 - 34 * q^7 + 4771 * q^9	\( 40 q + 81 q^{3} - 34 q^{7} + 4771 q^{9} - 55464 q^{13} + 68482 q^{15} + 311690 q^{19} - 172343 q^{21} + 1766792 q^{25} - 3451932 q^{27} + 31596 q^{31} + 1874885 q^{33} - 1853482 q^{37} + 11217526 q^{39} - 13372600 q^{43} - 527785 q^{45} - 12653462 q^{49} - 1103461 q^{51} + 71577224 q^{55} - 17195214 q^{57} - 21761970 q^{61} + 21945045 q^{63} - 26337350 q^{67} - 5588722 q^{69} + 41115682 q^{73} - 17971730 q^{75} - 120916932 q^{79} - 24550133 q^{81} + 139250060 q^{85} - 16321046 q^{87} + 345074940 q^{91} + 25774675 q^{93} - 707216948 q^{97} - 94510994 q^{99}+O(q^{100}) \) 40 * q + 81 * q^3 - 34 * q^7 + 4771 * q^9 - 55464 * q^13 + 68482 * q^15 + 311690 * q^19 - 172343 * q^21 + 1766792 * q^25 - 3451932 * q^27 + 31596 * q^31 + 1874885 * q^33 - 1853482 * q^37 + 11217526 * q^39 - 13372600 * q^43 - 527785 * q^45 - 12653462 * q^49 - 1103461 * q^51 + 71577224 * q^55 - 17195214 * q^57 - 21761970 * q^61 + 21945045 * q^63 - 26337350 * q^67 - 5588722 * q^69 + 41115682 * q^73 - 17971730 * q^75 - 120916932 * q^79 - 24550133 * q^81 + 139250060 * q^85 - 16321046 * q^87 + 345074940 * q^91 + 25774675 * q^93 - 707216948 * q^97 - 94510994 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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