LMFDB - Embedded newform 84.9.p.b.65.15

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			65.15
Character	$\chi$	$=$	84.65
Dual form			84.9.p.b.53.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(57.0400 + 57.5103i) q^{3} +(-609.234 - 351.741i) q^{5} +(162.484 + 2395.50i) q^{7} +(-53.8743 + 6560.78i) q^{9} +O(q^{10})$	\(q+(57.0400 + 57.5103i) q^{3} +(-609.234 - 351.741i) q^{5} +(162.484 + 2395.50i) q^{7} +(-53.8743 + 6560.78i) q^{9} +(15990.7 - 9232.23i) q^{11} -5808.82 q^{13} +(-14521.9 - 55100.6i) q^{15} +(-98970.4 + 57140.6i) q^{17} +(-34997.2 + 60616.9i) q^{19} +(-128498. + 145984. i) q^{21} +(-255114. - 147290. i) q^{23} +(52131.4 + 90294.2i) q^{25} +(-380385. + 371129. i) q^{27} -635415. i q^{29} +(-410180. - 710453. i) q^{31} +(1.44306e6 + 393023. i) q^{33} +(743604. - 1.51657e6i) q^{35} +(332530. - 575958. i) q^{37} +(-331335. - 334067. i) q^{39} +2.64195e6i q^{41} -5.38188e6 q^{43} +(2.34052e6 - 3.97810e6i) q^{45} +(-3.95171e6 - 2.28152e6i) q^{47} +(-5.71200e6 + 778459. i) q^{49} +(-8.93145e6 - 2.43252e6i) q^{51} +(-3.92358e6 + 2.26528e6i) q^{53} -1.29894e7 q^{55} +(-5.48234e6 + 1.44489e6i) q^{57} +(1.48641e7 - 8.58180e6i) q^{59} +(-6.38504e6 + 1.10592e7i) q^{61} +(-1.57251e7 + 936965. i) q^{63} +(3.53893e6 + 2.04320e6i) q^{65} +(-3.15977e6 - 5.47288e6i) q^{67} +(-6.08101e6 - 2.30731e7i) q^{69} -3.32348e7i q^{71} +(-3.18086e6 - 5.50941e6i) q^{73} +(-2.21927e6 + 8.14848e6i) q^{75} +(2.47140e7 + 3.68055e7i) q^{77} +(-34639.6 + 59997.6i) q^{79} +(-4.30409e7 - 706915. i) q^{81} +8.04512e7i q^{83} +8.03948e7 q^{85} +(3.65429e7 - 3.62441e7i) q^{87} +(-8.74905e7 - 5.05127e7i) q^{89} +(-943840. - 1.39150e7i) q^{91} +(1.74617e7 - 6.41139e7i) q^{93} +(4.26429e7 - 2.46199e7i) q^{95} +7.56377e7 q^{97} +(5.97091e7 + 1.05409e8i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	57.0400	+	57.5103i	0.704198	+	0.710004i
$3$	57.0400	+	57.5103i	0.704198	+	0.710004i
$4$		0			0
$5$	−609.234	−	351.741i	−0.974774	−	0.562786i	−0.0740858	−	0.997252i	$-0.523604\pi$
$5$	−609.234	−	351.741i	−0.974774	−	0.562786i	−0.900688	+	0.434466i	$0.856937\pi$
$6$		0			0
$7$	162.484	+	2395.50i	0.0676734	+	0.997708i
$7$	162.484	+	2395.50i	0.0676734	+	0.997708i
$8$		0			0
$9$	−53.8743	+	6560.78i	−0.00821130	+	0.999966i
$10$		0			0
$11$	15990.7	−	9232.23i	1.09219	−	0.630573i	0.158028	−	0.987435i	$-0.449486\pi$
$11$	15990.7	−	9232.23i	1.09219	−	0.630573i	0.934157	+	0.356861i	$0.116153\pi$
$12$		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	−0.957161	−	0.289558i	$-0.906492\pi$
$17$	−98970.4	+	57140.6i	−1.18498	+	0.684147i	−0.227816	+	0.973704i	$0.573158\pi$
$18$		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
$21$	−128498.	+	145984.i	−0.660721	+	0.750632i
$22$		0			0
$23$	−255114.	−	147290.i	−0.911640	−	0.526335i	−0.0306813	−	0.999529i	$-0.509768\pi$
$23$	−255114.	−	147290.i	−0.911640	−	0.526335i	−0.880958	+	0.473194i	$0.843101\pi$
$24$		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	$-0.851711\pi$
$29$		−	635415.i		−	0.898391i	0.893433	−	0.449196i	$-0.148289\pi$
$30$		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
$33$	1.44306e6	+	393023.i	1.21682	+	0.331408i
$34$		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
$39$	−331335.	−	334067.i	−0.143222	−	0.144403i
$40$		0			0
$41$			2.64195e6i			0.934952i	0.884006	+	0.467476i	$0.154836\pi$
$41$			2.64195e6i			0.934952i	−0.884006	+	0.467476i	$0.845164\pi$
$42$		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
$45$	2.34052e6	−	3.97810e6i	0.570771	−	0.970120i
$46$		0			0
$47$	−3.95171e6	−	2.28152e6i	−0.809830	−	0.467556i	0.0370665	−	0.999313i	$-0.488199\pi$
$47$	−3.95171e6	−	2.28152e6i	−0.809830	−	0.467556i	−0.846897	+	0.531757i	$0.821532\pi$
$48$		0			0
$49$	−5.71200e6	+	778459.i	−0.990841	+	0.135037i
$50$		0			0
$51$	−8.93145e6	−	2.43252e6i	−1.32020	−	0.359564i
$52$		0			0
$53$	−3.92358e6	+	2.26528e6i	−0.497255	+	0.287090i	−0.727579	−	0.686024i	$-0.759355\pi$
$53$	−3.92358e6	+	2.26528e6i	−0.497255	+	0.287090i	0.230324	+	0.973114i	$0.426021\pi$
$54$		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	0.260346	−	0.965515i	$-0.416164\pi$
$59$	1.48641e7	−	8.58180e6i	1.22668	−	0.708224i	0.966334	+	0.257292i	$0.0828302\pi$
$60$		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
$63$	−1.57251e7	+	936965.i	−0.998230	+	0.0594787i
$64$		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	$-0.773120\pi$
$71$		−	3.32348e7i		−	1.30786i	0.756556	−	0.653929i	$-0.226880\pi$
$72$		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
$75$	−2.21927e6	+	8.14848e6i	−0.0701400	+	0.257532i
$76$		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
$81$	−4.30409e7	−	706915.i	−0.999865	−	0.0164220i
$82$		0			0
$83$			8.04512e7i			1.69520i	0.530637	+	0.847599i	$0.321953\pi$
$83$			8.04512e7i			1.69520i	−0.530637	+	0.847599i	$0.678047\pi$
$84$		0			0
$85$	8.03948e7			1.54011
$86$		0			0
$87$	3.65429e7	−	3.62441e7i	0.637862	−	0.632645i
$88$		0			0
$89$	−8.74905e7	−	5.05127e7i	−1.39444	−	0.805082i	−0.400640	−	0.916235i	$-0.631212\pi$
$89$	−8.74905e7	−	5.05127e7i	−1.39444	−	0.805082i	−0.993803	+	0.111153i	$0.964546\pi$
$90$		0			0
$91$	−943840.	−	1.39150e7i	−0.0137636	−	0.202917i
$92$		0			0
$93$	1.74617e7	−	6.41139e7i	0.233429	−	0.857078i
$94$		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	−0.302242	−	0.953231i	$-0.597735\pi$
$101$	3.87271e7	−	2.23591e7i	0.372159	−	0.214866i	0.674401	+	0.738365i	$0.264402\pi$
$102$		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
$105$	1.29634e8	−	4.37402e7i	1.06650	−	0.359852i
$106$		0			0
$107$	−8.98648e7	−	5.18835e7i	−0.685575	−	0.395817i	0.116378	−	0.993205i	$-0.462872\pi$
$107$	−8.98648e7	−	5.18835e7i	−0.685575	−	0.395817i	−0.801952	+	0.597388i	$0.796205\pi$
$108$		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	$0.132497\pi$
$113$			1.31852e8i			0.808671i	−0.914611	+	0.404336i	$0.867503\pi$
$114$		0			0
$115$	1.03616e8	+	1.79468e8i	0.592429	+	1.02612i
$116$		0			0
$117$	312946.	−	3.81104e7i	0.00167004	−	0.203376i
$118$		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
$123$	−1.51939e8	+	1.50697e8i	−0.663819	+	0.658391i
$124$		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
$129$	−3.06983e8	−	3.09514e8i	−1.10855	−	1.11769i
$130$		0			0
$131$	1.49640e8	+	8.63946e7i	0.508115	+	0.293361i	0.732059	−	0.681242i	$-0.238560\pi$
$131$	1.49640e8	+	8.63946e7i	0.508115	+	0.293361i	−0.223943	+	0.974602i	$0.571893\pi$
$132$		0			0
$133$	−1.50894e8	−	7.39863e7i	−0.482242	−	0.236453i
$134$		0			0
$135$	3.62285e8	−	9.23068e7i	1.09072	−	0.277906i
$136$		0			0
$137$	7.38547e7	−	4.26400e7i	0.209651	−	0.121042i	−0.391498	−	0.920179i	$-0.628043\pi$
$137$	7.38547e7	−	4.26400e7i	0.209651	−	0.121042i	0.601149	+	0.799137i	$0.294710\pi$
$138$		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
$147$	−3.70582e8	−	2.84096e8i	−0.793624	−	0.608408i
$148$		0			0
$149$	5.77780e8	+	3.33581e8i	1.17224	+	0.676794i	0.954207	−	0.299147i	$-0.0967020\pi$
$149$	5.77780e8	+	3.33581e8i	1.17224	+	0.676794i	0.218035	+	0.975941i	$0.430035\pi$
$150$		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
$159$	−3.54078e8	−	9.64348e7i	−0.554001	−	0.150885i
$160$		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
$165$	−7.40917e8	−	7.47026e8i	−0.999617	−	1.00786i
$166$		0			0
$167$		−	5.31526e8i		−	0.683374i	−0.939814	−	0.341687i	$-0.889002\pi$
$167$		−	5.31526e8i		−	0.683374i	0.939814	−	0.341687i	$-0.110998\pi$
$168$		0			0
$169$	−7.81988e8			−0.958635
$170$		0			0
$171$	−3.95809e8	−	2.32874e8i	−0.462914	−	0.272356i
$172$		0			0
$173$	−8.43588e8	−	4.87046e8i	−0.941772	−	0.543732i	−0.0512566	−	0.998686i	$-0.516323\pi$
$173$	−8.43588e8	−	4.87046e8i	−0.941772	−	0.543732i	−0.890515	+	0.454953i	$0.849656\pi$
$174$		0			0
$175$	−2.07829e8	+	1.39552e8i	−0.221592	+	0.148793i
$176$		0			0
$177$	1.34139e9	+	3.65334e8i	1.36667	+	0.372218i
$178$		0			0
$179$	2.97615e8	−	1.71828e8i	0.289897	−	0.167372i	−0.347999	−	0.937495i	$-0.613139\pi$
$179$	2.97615e8	−	1.71828e8i	0.289897	−	0.167372i	0.637895	+	0.770123i	$0.279805\pi$
$180$		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
$189$	−9.50843e8	−	8.50909e8i	−0.745181	−	0.666862i
$190$		0			0
$191$	−2.02539e9	−	1.16936e9i	−1.52186	−	0.878646i	−0.999667	−	0.0258184i	$-0.991781\pi$
$191$	−2.02539e9	−	1.16936e9i	−1.52186	−	0.878646i	−0.522193	−	0.852828i	$-0.674886\pi$
$192$		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	$-0.959302\pi$
$197$		−	3.84095e8i		−	0.255019i	0.991837	−	0.127510i	$-0.0406984\pi$
$198$		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
$201$	1.34514e8	−	4.93893e8i	0.0824106	−	0.302586i
$202$		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
$207$	9.80083e8	−	1.66581e9i	0.533803	−	0.907287i
$208$		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
$213$	1.91135e9	−	1.89572e9i	0.928584	−	0.920990i
$214$		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
$219$	1.35412e8	−	4.97189e8i	0.0588681	−	0.216145i
$220$		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	−0.998896	−	0.0469848i	$-0.985039\pi$
$227$	−3.87042e9	+	2.23459e9i	−1.45765	+	0.841577i	−0.458758	+	0.888561i	$0.651705\pi$
$228$		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
$231$	−7.07012e8	+	3.52070e9i	−0.248301	+	1.23646i
$232$		0			0
$233$	−2.59276e9	−	1.49693e9i	−0.879708	−	0.507900i	−0.00914618	−	0.999958i	$-0.502911\pi$
$233$	−2.59276e9	−	1.49693e9i	−0.879708	−	0.507900i	−0.870562	+	0.492058i	$0.836245\pi$
$234$		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	$-0.728118\pi$
$239$		−	4.92036e9i		−	1.50801i	0.656866	−	0.754007i	$-0.271882\pi$
$240$		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
$243$	−2.41440e9	−	2.51562e9i	−0.692443	−	0.721473i
$244$		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
$249$	−4.62678e9	+	4.58894e9i	−1.20360	+	1.19375i
$250$		0			0
$251$			4.80868e9i			1.21152i	0.795647	+	0.605761i	$0.207131\pi$
$251$			4.80868e9i			1.21152i	−0.795647	+	0.605761i	$0.792869\pi$
$252$		0			0
$253$	−5.43927e9			−1.32757
$254$		0			0
$255$	4.58572e9	+	4.62353e9i	1.08454	+	1.09349i
$256$		0			0
$257$	1.01266e8	+	5.84662e7i	0.0232131	+	0.0134021i	0.511562	−	0.859247i	$-0.329067\pi$
$257$	1.01266e8	+	5.84662e7i	0.0232131	+	0.0134021i	−0.488349	+	0.872649i	$0.662401\pi$
$258$		0			0
$259$	1.43374e9	+	7.02989e8i	0.318618	+	0.156225i
$260$		0			0
$261$	4.16882e9	+	3.42326e7i	0.898361	+	0.00737696i
$262$		0			0
$263$	−2.23432e9	+	1.28998e9i	−0.467005	+	0.269626i	−0.714985	−	0.699140i	$-0.753567\pi$
$263$	−2.23432e9	+	1.28998e9i	−0.467005	+	0.269626i	0.247980	+	0.968765i	$0.420233\pi$
$264$		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	0.157367	−	0.987540i	$-0.449700\pi$
$269$	5.71409e9	−	3.29903e9i	1.09128	−	0.630054i	0.933918	+	0.357487i	$0.116366\pi$
$270$		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
$273$	7.46420e8	−	8.47993e8i	0.134379	−	0.152666i
$274$		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	$0.124029\pi$
$281$			4.73676e9i			0.759724i	−0.925043	+	0.379862i	$0.875971\pi$
$282$		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
$285$	3.84825e9	+	1.04809e9i	0.583289	+	0.158862i
$286$		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
$291$	4.31438e9	+	4.34995e9i	0.601653	+	0.606614i
$292$		0			0
$293$		−	1.21546e9i		−	0.164919i	−0.996594	−	0.0824597i	$-0.973722\pi$
$293$		−	1.21546e9i		−	0.164919i	0.996594	−	0.0824597i	$-0.0262776\pi$
$294$		0			0
$295$	−1.20743e10			−1.59431
$296$		0			0
$297$	−2.65628e9	+	9.44641e9i	−0.341388	+	1.21406i
$298$		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
$303$	3.49487e9	+	9.51844e8i	0.414630	+	0.112926i
$304$		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	−0.867309	−	0.497770i	$-0.834152\pi$
$311$	−8.13770e9	+	4.69830e9i	−0.869881	+	0.502226i	−0.00257258	+	0.999997i	$0.500819\pi$
$312$		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
$315$	9.90981e9	+	4.96033e9i	1.00652	+	0.503811i
$316$		0			0
$317$	−1.01457e10	−	5.85761e9i	−1.00472	−	0.580074i	−0.0950772	−	0.995470i	$-0.530310\pi$
$317$	−1.01457e10	−	5.85761e9i	−1.00472	−	0.580074i	−0.909641	+	0.415396i	$0.863643\pi$
$318$		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
$327$	−5.07760e9	+	1.86433e10i	−0.444086	+	1.63055i
$328$		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
$333$	3.76082e9	+	2.21268e9i	0.305848	+	0.179946i
$334$		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
$339$	−7.58283e9	+	7.52082e9i	−0.574160	+	0.569464i
$340$		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
$345$	−4.41102e9	+	1.61959e10i	−0.311360	+	1.14322i
$346$		0			0
$347$	1.51320e10	−	8.73645e9i	1.04371	−	0.602584i	0.122825	−	0.992428i	$-0.460805\pi$
$347$	1.51320e10	−	8.73645e9i	1.04371	−	0.602584i	0.920881	+	0.389845i	$0.127471\pi$
$348$		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	−0.816750	−	0.576992i	$-0.804226\pi$
$353$	−1.12641e10	+	6.50335e9i	−0.725436	+	0.418830i	0.0913145	+	0.995822i	$0.470893\pi$
$354$		0			0
$355$	−1.16901e10	+	2.02478e10i	−0.736044	+	1.27487i
$356$		0			0
$357$	4.37588e9	−	2.17905e10i	0.269397	−	1.34151i
$358$		0			0
$359$	2.59268e10	+	1.49689e10i	1.56089	+	0.901179i	0.997167	+	0.0752131i	$0.0239637\pi$
$359$	2.59268e10	+	1.49689e10i	1.56089	+	0.901179i	0.563720	+	0.825966i	$0.309370\pi$
$360$		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
$369$	−1.73332e10	−	1.42333e8i	−0.934920	−	0.00767716i
$370$		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
$375$	−1.15855e10	+	1.14908e10i	−0.585855	+	0.581064i
$376$		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
$381$	2.02012e10	+	2.03678e10i	0.958690	+	0.966594i
$382$		0			0
$383$	9.56810e8	+	5.52414e8i	0.0444662	+	0.0256726i	0.522068	−	0.852904i	$-0.325161\pi$
$383$	9.56810e8	+	5.52414e8i	0.0444662	+	0.0256726i	−0.477602	+	0.878576i	$0.658494\pi$
$384$		0			0
$385$	−2.11057e9	−	3.11161e10i	−0.0960632	−	1.41626i
$386$		0			0
$387$	2.89945e8	−	3.53093e10i	0.0129262	−	1.57415i
$388$		0			0
$389$	−3.07289e10	+	1.77414e10i	−1.34199	+	0.774798i	−0.987099	−	0.160110i	$-0.948815\pi$
$389$	−3.07289e10	+	1.77414e10i	−1.34199	+	0.774798i	−0.354890	+	0.934908i	$0.615482\pi$
$390$		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
$399$	−4.35202e9	−	1.28981e10i	−0.171711	−	0.508904i
$400$		0			0
$401$	4.10788e10	+	2.37168e10i	1.58869	+	0.917232i	0.993523	+	0.113632i	$0.0362486\pi$
$401$	4.10788e10	+	2.37168e10i	1.58869	+	0.917232i	0.595170	+	0.803600i	$0.297085\pi$
$402$		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
$411$	6.66492e9	+	1.81522e9i	0.233576	+	0.0636154i
$412$		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
$417$	3.00887e10	+	3.03367e10i	0.995081	+	1.00329i
$418$		0			0
$419$		−	2.91611e10i		−	0.946124i	−0.881029	−	0.473062i	$-0.843149\pi$
$419$		−	2.91611e10i		−	0.946124i	0.881029	−	0.473062i	$-0.156851\pi$
$420$		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
$423$	1.51815e10	−	2.58034e10i	0.474190	−	0.805964i
$424$		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
$429$	−8.38246e9	−	2.28300e9i	−0.247481	−	0.0674027i
$430$		0			0
$431$	−1.56064e10	+	9.01037e9i	−0.452266	+	0.261116i	−0.708787	−	0.705423i	$-0.750757\pi$
$431$	−1.56064e10	+	9.01037e9i	−0.452266	+	0.261116i	0.256521	+	0.966539i	$0.417424\pi$
$432$		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
$441$	−4.79957e9	−	3.75171e10i	−0.126896	−	0.991916i
$442$		0			0
$443$	−2.29358e10	−	1.32420e10i	−0.595525	−	0.343826i	0.171754	−	0.985140i	$-0.445056\pi$
$443$	−2.29358e10	−	1.32420e10i	−0.595525	−	0.343826i	−0.767279	+	0.641313i	$0.778390\pi$
$444$		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	$-0.791597\pi$
$449$		−	4.94979e10i		−	1.21787i	0.793220	−	0.608935i	$-0.208403\pi$
$450$		0			0
$451$	2.43911e10	+	4.22466e10i	0.589556	+	1.02114i
$452$		0			0
$453$	−1.45055e10	+	5.32596e10i	−0.344461	+	1.26475i
$454$		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
$459$	1.64404e10	−	5.84662e10i	0.370392	−	1.31721i
$460$		0			0
$461$			3.37260e10i			0.746726i	0.927685	+	0.373363i	$0.121795\pi$
$461$			3.37260e10i			0.746726i	−0.927685	+	0.373363i	$0.878205\pi$
$462$		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
$465$	−3.31898e10	+	3.29183e10i	−0.709892	+	0.704087i
$466$		0			0
$467$	−3.72153e10	−	2.14863e10i	−0.782445	−	0.451745i	0.0548509	−	0.998495i	$-0.482532\pi$
$467$	−3.72153e10	−	2.14863e10i	−0.782445	−	0.451745i	−0.837296	+	0.546750i	$0.815865\pi$
$468$		0			0
$469$	1.25969e10	−	8.45847e9i	0.260358	−	0.174824i
$470$		0			0
$471$	8.46148e9	−	3.10679e10i	0.171934	−	0.631289i
$472$		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	−0.0901438	−	0.995929i	$-0.528733\pi$
$479$	3.82865e10	−	2.21047e10i	0.727284	−	0.419898i	0.817428	+	0.576031i	$0.195399\pi$
$480$		0			0
$481$	−1.93160e9	+	3.34564e9i	−0.0360859	+	0.0625027i
$482$		0			0
$483$	5.42835e10	−	1.83160e10i	0.997423	−	0.336545i
$484$		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	$0.189559\pi$
$491$			6.52034e10i			1.12187i	−0.827858	+	0.560937i	$0.810441\pi$
$492$		0			0
$493$	3.63080e10	+	6.28873e10i	0.614631	+	1.06457i
$494$		0			0
$495$	6.99796e8	−	8.52207e10i	0.0116560	−	1.41946i
$496$		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
$501$	3.05682e10	−	3.03182e10i	0.485198	−	0.481230i
$502$		0			0
$503$		−	3.15775e10i		−	0.493294i	−0.969105	−	0.246647i	$-0.920671\pi$
$503$		−	3.15775e10i		−	0.493294i	0.969105	−	0.246647i	$-0.0793288\pi$
$504$		0			0
$505$	−3.14584e10			−0.483695
$506$		0			0
$507$	−4.46046e10	−	4.49724e10i	−0.675069	−	0.680635i
$508$		0			0
$509$	9.84004e9	+	5.68115e9i	0.146597	+	0.0846379i	0.571504	−	0.820599i	$-0.306360\pi$
$509$	9.84004e9	+	5.68115e9i	0.146597	+	0.0846379i	−0.424907	+	0.905237i	$0.639693\pi$
$510$		0			0
$511$	1.26809e10	−	8.51493e9i	0.185981	−	0.124881i
$512$		0			0
$513$	−9.18424e9	−	3.60462e10i	−0.132609	−	0.520464i
$514$		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	−0.503184	−	0.864179i	$-0.667838\pi$
$521$	−4.69731e8	+	2.71199e8i	−0.00637527	+	0.00368076i	0.496809	+	0.867860i	$0.334505\pi$
$522$		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
$525$	−1.98802e10	−	3.99227e9i	−0.261688	−	0.0525512i
$526$		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
$537$	2.68579e10	+	7.31487e9i	0.322979	+	0.0879649i
$538$		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
$543$	−6.42989e10	−	6.48290e10i	−0.739612	−	0.745710i
$544$		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
$549$	−7.22130e10	−	4.24866e10i	−0.794925	−	0.467695i
$550$		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
$555$	−3.65646e10	−	9.95853e9i	−0.385380	−	0.104960i
$556$		0			0
$557$	−8.45787e8	+	4.88316e8i	−0.00878700	+	0.00507317i	−0.504387	−	0.863478i	$-0.668282\pi$
$557$	−8.45787e8	+	4.88316e8i	−0.00878700	+	0.00507317i	0.495600	+	0.868551i	$0.334948\pi$
$558$		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	−0.753918	−	0.656968i	$-0.771839\pi$
$563$	−5.64564e10	+	3.25951e10i	−0.561926	+	0.324428i	0.191992	+	0.981396i	$0.438505\pi$
$564$		0			0
$565$	4.63777e10	−	8.03285e10i	0.455109	−	0.788272i
$566$		0			0
$567$	−5.30004e9	−	1.03219e11i	−0.0512799	−	0.998684i
$568$		0			0
$569$	3.37409e10	+	1.94803e10i	0.321890	+	0.185844i	0.652235	−	0.758017i	$-0.273832\pi$
$569$	3.37409e10	+	1.94803e10i	0.321890	+	0.185844i	−0.330344	+	0.943860i	$0.607165\pi$
$570$		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
$579$	3.35306e9	−	1.23114e10i	0.0298351	−	0.109545i
$580$		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
$585$	−1.35957e10	+	2.31081e10i	−0.116085	+	0.197306i
$586$		0			0
$587$			9.21090e10i			0.775800i	0.921702	+	0.387900i	$0.126799\pi$
$587$			9.21090e10i			0.775800i	−0.921702	+	0.387900i	$0.873201\pi$
$588$		0			0
$589$	5.74206e10			0.477097
$590$		0			0
$591$	2.20894e10	−	2.19088e10i	0.181065	−	0.179584i
$592$		0			0
$593$	1.75779e11	+	1.01486e11i	1.42150	+	0.820704i	0.996427	−	0.0844539i	$-0.0269146\pi$
$593$	1.75779e11	+	1.01486e11i	1.42150	+	0.820704i	0.425074	+	0.905158i	$0.360248\pi$
$594$		0			0
$595$	1.30629e10	+	1.92585e11i	0.104225	+	1.53658i
$596$		0			0
$597$	−5.69142e10	+	2.08971e11i	−0.448047	+	1.64509i
$598$		0			0
$599$	−4.83697e10	+	2.79263e10i	−0.375722	+	0.216923i	−0.675955	−	0.736943i	$-0.736269\pi$
$599$	−4.83697e10	+	2.79263e10i	−0.375722	+	0.216923i	0.300233	+	0.953866i	$0.402935\pi$
$600$		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
$609$	9.27602e10	+	8.16494e10i	0.674361	+	0.593586i
$610$		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	$-0.812482\pi$
$617$		−	1.61044e11i		−	1.11123i	0.831439	−	0.555616i	$-0.187518\pi$
$618$		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
$621$	1.51705e11	−	3.86531e10i	1.02008	−	0.259907i
$622$		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
$627$	−7.43268e10	+	7.37189e10i	−0.480923	+	0.476990i
$628$		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
$633$	−2.95500e10	−	2.97936e10i	−0.184053	−	0.185570i
$634$		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
$639$	2.18046e11	+	1.79050e9i	1.30781	+	0.0107392i
$640$		0			0
$641$	1.89291e11	−	1.09287e11i	1.12124	−	0.647346i	0.179520	−	0.983754i	$-0.442546\pi$
$641$	1.89291e11	−	1.09287e11i	1.12124	−	0.647346i	0.941716	+	0.336408i	$0.109212\pi$
$642$		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	−0.759030	−	0.651055i	$-0.774327\pi$
$647$	−1.00709e11	+	5.81445e10i	−0.574715	+	0.331812i	0.184316	+	0.982867i	$0.440993\pi$
$648$		0			0
$649$	1.58458e11	−	2.74458e11i	0.893174	−	1.54702i
$650$		0			0
$651$	1.56422e11	+	3.14120e10i	0.870910	+	0.174892i
$652$		0			0
$653$	−5.69948e9	−	3.29059e9i	−0.0313460	−	0.0180976i	0.484245	−	0.874932i	$-0.339094\pi$
$653$	−5.69948e9	−	3.29059e9i	−0.0313460	−	0.0180976i	−0.515591	+	0.856835i	$0.672428\pi$
$654$		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	$0.187182\pi$
$659$			2.09248e11i			1.10948i	−0.832024	+	0.554739i	$0.812818\pi$
$660$		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
$663$	5.18812e10	+	1.41301e10i	0.268507	+	0.0731291i
$664$		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
$669$	1.49340e11	+	1.50571e11i	0.745542	+	0.751689i
$670$		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
$675$	−5.33408e10	−	1.49992e10i	−0.256947	−	0.0722523i
$676$		0			0
$677$	−4.96379e10	−	2.86584e10i	−0.236297	−	0.136426i	0.377177	−	0.926141i	$-0.376895\pi$
$677$	−4.96379e10	−	2.86584e10i	−0.236297	−	0.136426i	−0.613474	+	0.789715i	$0.710228\pi$
$678$		0			0
$679$	1.22899e10	+	1.81190e11i	0.0578189	+	0.852422i
$680$		0			0
$681$	−3.49280e11	−	9.51281e10i	−1.62400	−	0.442303i
$682$		0			0
$683$	1.92706e10	−	1.11259e10i	0.0885549	−	0.0511272i	−0.455069	−	0.890456i	$-0.650385\pi$
$683$	1.92706e10	−	1.11259e10i	0.0885549	−	0.0511272i	0.543624	+	0.839329i	$0.317052\pi$
$684$		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
$693$	−2.42804e11	+	1.60160e11i	−1.05275	+	0.694419i
$694$		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	$-0.959024\pi$
$701$		−	6.19979e10i		−	0.256747i	0.991726	−	0.128373i	$-0.0409756\pi$
$702$		0			0
$703$	2.32752e10	+	4.03138e10i	0.0952954	+	0.165057i
$704$		0			0
$705$	−6.83266e10	+	2.50874e11i	−0.276588	+	1.01554i
$706$		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
$711$	−3.91765e8	−	2.30495e8i	−0.00153302	−	0.000901952i
$712$		0			0
$713$			2.41662e11i			0.935084i
$714$		0			0
$715$	7.54532e10			0.288705
$716$		0			0
$717$	2.82972e11	−	2.80657e11i	1.07070	−	1.06194i
$718$		0			0
$719$	1.64981e11	+	9.52517e10i	0.617330	+	0.356416i	0.775829	−	0.630943i	$-0.217332\pi$
$719$	1.64981e11	+	9.52517e10i	0.617330	+	0.356416i	−0.158499	+	0.987359i	$0.550665\pi$
$720$		0			0
$721$	−3.47866e11	−	1.70566e11i	−1.28727	−	0.631176i
$722$		0			0
$723$	−1.14597e11	+	4.20766e11i	−0.419393	+	1.53988i
$724$		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
$735$	1.25843e11	+	3.03430e11i	0.431201	+	1.03970i
$736$		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	$0.206408\pi$
$743$			3.68118e11i			1.20790i	−0.797021	+	0.603952i	$0.793592\pi$
$744$		0			0
$745$	−2.34669e11	−	4.06458e11i	−0.761781	−	1.31944i
$746$		0			0
$747$	−5.27823e11	−	4.33426e9i	−1.69514	−	0.0139198i
$748$		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
$753$	−2.76549e11	+	2.74287e11i	−0.860185	+	0.853151i
$754$		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
$759$	−3.10256e11	−	3.12814e11i	−0.934874	−	0.942582i
$760$		0			0
$761$	−4.03325e10	−	2.32860e10i	−0.120259	−	0.0694314i	0.438664	−	0.898651i	$-0.355452\pi$
$761$	−4.03325e10	−	2.32860e10i	−0.120259	−	0.0694314i	−0.558923	+	0.829220i	$0.688785\pi$
$762$		0			0
$763$	−4.75503e11	+	3.19288e11i	−1.40299	+	0.942074i
$764$		0			0
$765$	−4.33122e9	+	5.27453e11i	−0.0126463	+	1.54006i
$766$		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	−0.681060	−	0.732228i	$-0.738481\pi$
$773$	−1.38340e11	+	7.98706e10i	−0.387462	+	0.223702i	0.293598	+	0.955929i	$0.405147\pi$
$774$		0			0
$775$	4.27666e10	−	7.40738e10i	0.118549	−	0.205333i
$776$		0			0
$777$	4.13512e10	+	1.22553e11i	0.113450	+	0.336233i
$778$		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
$789$	−2.01633e11	−	5.49156e10i	−0.520299	−	0.141706i
$790$		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
$795$	1.81796e11	+	1.83295e11i	0.455110	+	0.458863i
$796$		0			0
$797$			1.43490e11i			0.355623i	0.984065	+	0.177811i	$0.0569017\pi$
$797$			1.43490e11i			0.355623i	−0.984065	+	0.177811i	$0.943098\pi$
$798$		0			0
$799$	5.21470e11			1.27951
$800$		0			0
$801$	3.36116e11	−	5.71285e11i	0.816505	−	1.38779i
$802$		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
$807$	5.15660e11	+	1.40442e11i	1.21582	+	0.331134i
$808$		0			0
$809$	−2.06797e11	+	1.19394e11i	−0.482781	+	0.278734i	−0.721575	−	0.692337i	$-0.756581\pi$
$809$	−2.06797e11	+	1.19394e11i	−0.482781	+	0.278734i	0.238794	+	0.971070i	$0.423248\pi$
$810$		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
$819$	9.13441e10	−	5.44266e9i	0.203023	−	0.0120970i
$820$		0			0
$821$	−1.29258e11	−	7.46272e10i	−0.284502	−	0.164257i	0.350958	−	0.936391i	$-0.385856\pi$
$821$	−1.29258e11	−	7.46272e10i	−0.284502	−	0.164257i	−0.635460	+	0.772134i	$0.719189\pi$
$822$		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	$-0.922433\pi$
$827$		−	2.25722e11i		−	0.482561i	0.970455	−	0.241280i	$-0.0775674\pi$
$828$		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
$831$	1.30484e11	−	4.79095e11i	0.273623	−	1.00466i
$832$		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
$837$	4.19696e11	+	1.18016e11i	0.855132	+	0.240459i
$838$		0			0
$839$			4.05393e10i			0.0818141i	0.999163	+	0.0409071i	$0.0130248\pi$
$839$			4.05393e10i			0.0818141i	−0.999163	+	0.0409071i	$0.986975\pi$
$840$		0			0
$841$	9.64939e10			0.192893
$842$		0			0
$843$	−2.72412e11	+	2.70185e11i	−0.539407	+	0.534996i
$844$		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
$849$	−1.12119e11	+	4.11667e11i	−0.215799	+	0.792346i
$850$		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	$0.679666\pi$
$857$	−8.27521e11	+	4.77770e11i	−1.53411	+	0.885717i	−0.999166	+	0.0408282i	$0.987000\pi$
$858$		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
$861$	−3.85681e11	−	3.39484e11i	−0.701804	−	0.617742i
$862$		0			0
$863$	−5.78218e11	−	3.33834e11i	−1.04243	−	0.601849i	−0.121912	−	0.992541i	$-0.538902\pi$
$863$	−5.78218e11	−	3.33834e11i	−1.04243	−	0.601849i	−0.920521	+	0.390692i	$0.872236\pi$
$864$		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
$873$	−4.07493e9	+	4.96242e11i	−0.00701557	+	0.854352i
$874$		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
$879$	6.99018e10	−	6.93301e10i	0.117093	−	0.116136i
$880$		0			0
$881$		−	5.96835e11i		−	0.990720i	−0.868688	−	0.495360i	$-0.835036\pi$
$881$		−	5.96835e11i		−	0.990720i	0.868688	−	0.495360i	$-0.164964\pi$
$882$		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
$885$	−6.88718e11	−	6.94397e11i	−1.12271	−	1.13197i
$886$		0			0
$887$	6.44471e11	+	3.72086e11i	1.04114	+	0.601103i	0.920156	−	0.391552i	$-0.128062\pi$
$887$	6.44471e11	+	3.72086e11i	1.04114	+	0.601103i	0.120984	+	0.992654i	$0.461395\pi$
$888$		0			0
$889$	5.75451e10	+	8.48386e11i	0.0921301	+	1.35827i
$890$		0			0
$891$	−6.94780e11	+	3.86059e11i	−1.10239	+	0.612553i
$892$		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
$903$	6.91559e11	−	7.85667e11i	1.04011	−	1.18165i
$904$		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	$0.259014\pi$
$911$			1.00126e12i			1.45369i	−0.686802	+	0.726845i	$0.740986\pi$
$912$		0			0
$913$	7.42744e11	+	1.28647e12i	1.06895	+	1.85147i
$914$		0			0
$915$	7.02092e11	+	1.91218e11i	1.00164	+	0.272800i
$916$		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
$921$	−2.48434e11	−	2.50483e11i	−0.345281	−	0.348128i
$922$		0			0
$923$			1.93055e11i			0.265996i
$924$		0			0
$925$	6.93409e10			0.0947158
$926$		0			0
$927$	−9.12483e11	−	5.36861e11i	−1.23568	−	0.727014i
$928$		0			0
$929$	−1.23462e12	−	7.12809e11i	−1.65757	−	0.956997i	−0.973833	−	0.227266i	$-0.927021\pi$
$929$	−1.23462e12	−	7.12809e11i	−1.65757	−	0.956997i	−0.683734	−	0.729731i	$-0.739645\pi$
$930$		0			0
$931$	1.52716e11	−	3.73488e11i	0.203276	−	0.497138i
$932$		0			0
$933$	−7.34375e11	−	2.00010e11i	−0.969151	−	0.263953i
$934$		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	−0.129086	−	0.991633i	$-0.541204\pi$
$941$	5.21530e11	−	3.01105e11i	0.665151	−	0.384025i	0.794237	+	0.607608i	$0.207871\pi$
$942$		0			0
$943$	3.89133e11	−	6.73999e11i	0.492098	−	0.852339i
$944$		0			0
$945$	2.79986e11	+	8.52854e11i	0.351082	+	1.06942i
$946$		0			0
$947$	8.89542e11	+	5.13577e11i	1.10603	+	0.638566i	0.937798	−	0.347181i	$-0.112861\pi$
$947$	8.89542e11	+	5.13577e11i	1.10603	+	0.638566i	0.168231	+	0.985748i	$0.446195\pi$
$948$		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	$0.00708012\pi$
$953$			3.66907e10i			0.0444820i	−0.999753	+	0.0222410i	$0.992920\pi$
$954$		0			0
$955$	8.22623e11	+	1.42482e12i	0.988979	+	1.71296i
$956$		0			0
$957$	2.49733e11	−	9.16941e11i	0.297734	−	1.09318i
$958$		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
$963$	3.45238e11	−	5.86788e11i	0.401433	−	0.682301i
$964$		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
$969$	4.60027e11	−	4.56265e11i	0.521781	−	0.517514i
$970$		0			0
$971$	5.62250e11	+	3.24615e11i	0.632489	+	0.365168i	0.781715	−	0.623635i	$-0.214345\pi$
$971$	5.62250e11	+	3.24615e11i	0.632489	+	0.365168i	−0.149227	+	0.988803i	$0.547678\pi$
$972$		0			0
$973$	8.57104e10	+	1.26363e12i	0.0956273	+	1.40983i
$974$		0			0
$975$	1.28914e10	−	4.73331e10i	0.0142653	−	0.0523777i
$976$		0			0
$977$	−1.07179e12	+	6.18797e11i	−1.17633	+	0.679156i	−0.955163	−	0.296079i	$-0.904321\pi$
$977$	−1.07179e12	+	6.18797e11i	−1.17633	+	0.679156i	−0.221170	+	0.975235i	$0.570988\pi$
$978$		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	−0.629792	−	0.776764i	$-0.716860\pi$
$983$	−2.53961e11	+	1.46625e11i	−0.271990	+	0.157034i	0.357802	+	0.933798i	$0.383526\pi$
$984$		0			0
$985$	−1.35102e11	+	2.34003e11i	−0.143521	+	0.248586i
$986$		0			0
$987$	8.40851e11	−	2.83715e11i	0.886034	−	0.298961i
$988$		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
$999$	8.72651e10	+	3.42497e11i	0.0876150	+	0.343871i

Twists

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

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

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)

\(f(q)\)	\(=\)	\(q+(57.0400 + 57.5103i) q^{3} +(-609.234 - 351.741i) q^{5} +(162.484 + 2395.50i) q^{7} +(-53.8743 + 6560.78i) q^{9} +O(q^{10})\)	\(q+(57.0400 + 57.5103i) q^{3} +(-609.234 - 351.741i) q^{5} +(162.484 + 2395.50i) q^{7} +(-53.8743 + 6560.78i) q^{9} +(15990.7 - 9232.23i) q^{11} -5808.82 q^{13} +(-14521.9 - 55100.6i) q^{15} +(-98970.4 + 57140.6i) q^{17} +(-34997.2 + 60616.9i) q^{19} +(-128498. + 145984. i) q^{21} +(-255114. - 147290. i) q^{23} +(52131.4 + 90294.2i) q^{25} +(-380385. + 371129. i) q^{27} -635415. i q^{29} +(-410180. - 710453. i) q^{31} +(1.44306e6 + 393023. i) q^{33} +(743604. - 1.51657e6i) q^{35} +(332530. - 575958. i) q^{37} +(-331335. - 334067. i) q^{39} +2.64195e6i q^{41} -5.38188e6 q^{43} +(2.34052e6 - 3.97810e6i) q^{45} +(-3.95171e6 - 2.28152e6i) q^{47} +(-5.71200e6 + 778459. i) q^{49} +(-8.93145e6 - 2.43252e6i) q^{51} +(-3.92358e6 + 2.26528e6i) q^{53} -1.29894e7 q^{55} +(-5.48234e6 + 1.44489e6i) q^{57} +(1.48641e7 - 8.58180e6i) q^{59} +(-6.38504e6 + 1.10592e7i) q^{61} +(-1.57251e7 + 936965. i) q^{63} +(3.53893e6 + 2.04320e6i) q^{65} +(-3.15977e6 - 5.47288e6i) q^{67} +(-6.08101e6 - 2.30731e7i) q^{69} -3.32348e7i q^{71} +(-3.18086e6 - 5.50941e6i) q^{73} +(-2.21927e6 + 8.14848e6i) q^{75} +(2.47140e7 + 3.68055e7i) q^{77} +(-34639.6 + 59997.6i) q^{79} +(-4.30409e7 - 706915. i) q^{81} +8.04512e7i q^{83} +8.03948e7 q^{85} +(3.65429e7 - 3.62441e7i) q^{87} +(-8.74905e7 - 5.05127e7i) q^{89} +(-943840. - 1.39150e7i) q^{91} +(1.74617e7 - 6.41139e7i) q^{93} +(4.26429e7 - 2.46199e7i) q^{95} +7.56377e7 q^{97} +(5.97091e7 + 1.05409e8i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 81 q^{3} - 34 q^{7} + 4771 q^{9}+O(q^{10}) \) 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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