LMFDB - Embedded newform 5.20.b.a.4.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5,20,Mod(4,5)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 20, names="a")

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

chi := DirichletCharacter("5.4");

S:= CuspForms(chi, 20);

N := Newforms(S);

Level:	$ N $	$=$	$ 5 $
Weight:	$ k $	$=$	$ 20 $
Character orbit:	$[\chi]$	$=$	5.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$11.4408348278$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 726881x^{6} + 160513523376x^{4} + 10607307647230976x^{2} + 32429098232548950016 $ x^8 + 726881x^6 + 160513523376x^4 + 10607307647230976*x^2 + 32429098232548950016
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{20}\cdot 3^{8}\cdot 5^{13} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4.5
Root			$56.6657i$ of defining polynomial
Character	$\chi$	$=$	5.4
Dual form			5.20.b.a.4.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+113.331i q^{2} +33157.6i q^{3} +511444. q^{4} +(2.22206e6 - 3.75978e6i) q^{5} -3.75780e6 q^{6} +2.70208e7i q^{7} +1.17381e8i q^{8} +6.28322e7 q^{9} +O(q^{10})$	\(q+113.331i q^{2} +33157.6i q^{3} +511444. q^{4} +(2.22206e6 - 3.75978e6i) q^{5} -3.75780e6 q^{6} +2.70208e7i q^{7} +1.17381e8i q^{8} +6.28322e7 q^{9} +(4.26102e8 + 2.51829e8i) q^{10} +5.56241e9 q^{11} +1.69583e10i q^{12} +4.21072e10i q^{13} -3.06230e9 q^{14} +(1.24665e11 + 7.36781e10i) q^{15} +2.54841e11 q^{16} +5.95087e11i q^{17} +7.12086e9i q^{18} -1.45552e12 q^{19} +(1.13646e12 - 1.92292e12i) q^{20} -8.95945e11 q^{21} +6.30396e11i q^{22} -9.69934e12i q^{23} -3.89208e12 q^{24} +(-9.19842e12 - 1.67089e13i) q^{25} -4.77207e12 q^{26} +4.06212e13i q^{27} +1.38196e13i q^{28} +9.46880e13 q^{29} +(-8.35005e12 + 1.41285e13i) q^{30} +6.27411e12 q^{31} +9.04230e13i q^{32} +1.84437e14i q^{33} -6.74421e13 q^{34} +(1.01592e14 + 6.00417e13i) q^{35} +3.21351e13 q^{36} -1.41692e15i q^{37} -1.64956e14i q^{38} -1.39617e15 q^{39} +(4.41327e14 + 2.60827e14i) q^{40} -9.33298e14 q^{41} -1.01539e14i q^{42} -2.15163e13i q^{43} +2.84486e15 q^{44} +(1.39617e14 - 2.36235e14i) q^{45} +1.09924e15 q^{46} +4.64662e15i q^{47} +8.44993e15i q^{48} +1.06688e16 q^{49} +(1.89364e15 - 1.04247e15i) q^{50} -1.97317e16 q^{51} +2.15355e16i q^{52} -3.42854e16i q^{53} -4.60366e15 q^{54} +(1.23600e16 - 2.09135e16i) q^{55} -3.17173e15 q^{56} -4.82616e16i q^{57} +1.07311e16i q^{58} -3.91436e16 q^{59} +(6.37594e16 + 3.76822e16i) q^{60} -1.44319e17 q^{61} +7.11054e14i q^{62} +1.69777e15i q^{63} +1.23362e17 q^{64} +(1.58314e17 + 9.35644e16i) q^{65} -2.09025e16 q^{66} -3.79832e16i q^{67} +3.04354e17i q^{68} +3.21607e17 q^{69} +(-6.80461e15 + 1.15136e16i) q^{70} -7.32379e17 q^{71} +7.37530e15i q^{72} -5.16836e17i q^{73} +1.60581e17 q^{74} +(5.54027e17 - 3.04998e17i) q^{75} -7.44417e17 q^{76} +1.50301e17i q^{77} -1.58230e17i q^{78} -5.46910e16 q^{79} +(5.66271e17 - 9.58146e17i) q^{80} -1.27388e18 q^{81} -1.05772e17i q^{82} +5.15297e17i q^{83} -4.58226e17 q^{84} +(2.23740e18 + 1.32232e18i) q^{85} +2.43847e15 q^{86} +3.13963e18i q^{87} +6.52922e17i q^{88} +3.08156e18 q^{89} +(2.67729e16 + 1.58230e16i) q^{90} -1.13777e18 q^{91} -4.96067e18i q^{92} +2.08035e17i q^{93} -5.26608e17 q^{94} +(-3.23425e18 + 5.47244e18i) q^{95} -2.99821e18 q^{96} -5.49443e18i q^{97} +1.20911e18i q^{98} +3.49498e17 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 1620744 q^{4} + 147000 q^{5} + 3365736 q^{6} + 345358584 q^{9}+O(q^{10}) $ 8 * q - 1620744 * q^4 + 147000 * q^5 + 3365736 * q^6 + 345358584 * q^9	$ 8 q - 1620744 q^{4} + 147000 q^{5} + 3365736 q^{6} + 345358584 q^{9} - 610691000 q^{10} - 3379575264 q^{11} + 177250591032 q^{14} - 242324628000 q^{15} - 312730276832 q^{16} + 4547188380640 q^{19} - 4180429431000 q^{20} - 2983154334624 q^{21} - 6176642779680 q^{24} + 17715709625000 q^{25} + 15909228128496 q^{26} - 188222300345040 q^{29} + 148501482939000 q^{30} + 72115006686976 q^{31} + 378440221985792 q^{34} - 299115755916000 q^{35} - 964020253238712 q^{36} + 30\!\cdots\!28 q^{39}+ \cdots - 16\!\cdots\!72 q^{99}+O(q^{100}) $ 8 * q - 1620744 * q^4 + 147000 * q^5 + 3365736 * q^6 + 345358584 * q^9 - 610691000 * q^10 - 3379575264 * q^11 + 177250591032 * q^14 - 242324628000 * q^15 - 312730276832 * q^16 + 4547188380640 * q^19 - 4180429431000 * q^20 - 2983154334624 * q^21 - 6176642779680 * q^24 + 17715709625000 * q^25 + 15909228128496 * q^26 - 188222300345040 * q^29 + 148501482939000 * q^30 + 72115006686976 * q^31 + 378440221985792 * q^34 - 299115755916000 * q^35 - 964020253238712 * q^36 + 3011653788049728 * q^39 - 213611427940000 * q^40 - 3314269428857904 * q^41 + 489119304867552 * q^44 + 10198157211261000 * q^45 - 13716870314831944 * q^46 - 11129258011816744 * q^49 + 50375043666750000 * q^50 - 46867071411404544 * q^51 + 86068650834185040 * q^54 + 69821226326124000 * q^55 - 303831778832907360 * q^56 + 259648032778012320 * q^59 + 318110620655844000 * q^60 - 582893562753226064 * q^61 + 525415747276109696 * q^64 + 747545324689032000 * q^65 - 1871067587999976288 * q^66 + 302144302523815008 * q^69 + 2638361065839633000 * q^70 - 2617007600602926144 * q^71 + 1847591013622215312 * q^74 + 2877932428209000000 * q^75 - 7494494299858276320 * q^76 + 3935092091109861760 * q^79 + 5465139505859292000 * q^80 - 8328177293660704152 * q^81 + 7683875738824129632 * q^84 + 3752387009219584000 * q^85 - 14358563079146095224 * q^86 + 2120141101636921680 * q^89 + 7589596726146267000 * q^90 - 3358638219973817664 * q^91 - 10251771654977709928 * q^94 + 6106065789884820000 * q^95 + 15670161523483353216 * q^96 - 16256944625235082272 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

Coefficient data

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

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

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

$2$			113.331i			0.156518i	0.996933	+	0.0782592i	$0.0249362\pi$
$2$			113.331i			0.156518i	−0.996933	+	0.0782592i	$0.975064\pi$
$3$			33157.6i			0.972594i	0.873793	+	0.486297i	$0.161653\pi$
$3$			33157.6i			0.972594i	−0.873793	+	0.486297i	$0.838347\pi$
$4$	511444.			0.975502
$5$	2.22206e6	−	3.75978e6i	0.508792	−	0.860890i
$5$	2.22206e6	−	3.75978e6i	0.508792	−	0.860890i
$6$	−3.75780e6			−0.152229
$7$			2.70208e7i			0.253085i	0.991961	+	0.126542i	$0.0403880\pi$
$7$			2.70208e7i			0.253085i	−0.991961	+	0.126542i	$0.959612\pi$
$8$			1.17381e8i			0.309203i
$9$	6.28322e7			0.0540603
$10$	4.26102e8	+	2.51829e8i	0.134745	+	0.0796353i
$11$	5.56241e9			0.711267			0.355634	−	0.934625i	$-0.384265\pi$
$11$	5.56241e9			0.711267			0.355634	+	0.934625i	$0.384265\pi$
$12$			1.69583e10i			0.948768i
$13$			4.21072e10i			1.10127i	0.834746	+	0.550635i	$0.185615\pi$
$13$			4.21072e10i			1.10127i	−0.834746	+	0.550635i	$0.814385\pi$
$14$	−3.06230e9			−0.0396125
$15$	1.24665e11	+	7.36781e10i	0.837296	+	0.494848i
$16$	2.54841e11			0.927106
$17$			5.95087e11i			1.21707i	0.793527	+	0.608535i	$0.208243\pi$
$17$			5.95087e11i			1.21707i	−0.793527	+	0.608535i	$0.791757\pi$
$18$			7.12086e9i			0.00846143i
$19$	−1.45552e12			−1.03481			−0.517403	−	0.855742i	$-0.673101\pi$
$19$	−1.45552e12			−1.03481			−0.517403	+	0.855742i	$0.673101\pi$
$20$	1.13646e12	−	1.92292e12i	0.496327	−	0.839800i
$21$	−8.95945e11			−0.246149
$22$			6.30396e11i			0.111326i
$23$		−	9.69934e12i		−	1.12287i	−0.827522	−	0.561433i	$-0.810250\pi$
$23$		−	9.69934e12i		−	1.12287i	0.827522	−	0.561433i	$-0.189750\pi$
$24$	−3.89208e12			−0.300729
$25$	−9.19842e12	−	1.67089e13i	−0.482262	−	0.876027i
$26$	−4.77207e12			−0.172369
$27$			4.06212e13i			1.02517i
$28$			1.38196e13i			0.246885i
$29$	9.46880e13			1.21203			0.606016	−	0.795452i	$-0.292767\pi$
$29$	9.46880e13			1.21203			0.606016	+	0.795452i	$0.292767\pi$
$30$	−8.35005e12	+	1.41285e13i	−0.0774528	+	0.131052i
$31$	6.27411e12			0.0426203			0.0213101	−	0.999773i	$-0.493216\pi$
$31$	6.27411e12			0.0426203			0.0213101	+	0.999773i	$0.493216\pi$
$32$			9.04230e13i			0.454312i
$33$			1.84437e14i			0.691774i
$34$	−6.74421e13			−0.190494
$35$	1.01592e14	+	6.00417e13i	0.217878	+	0.128767i
$36$	3.21351e13			0.0527359
$37$		−	1.41692e15i		−	1.79237i	−0.443679	−	0.896186i	$-0.646327\pi$
$37$		−	1.41692e15i		−	1.79237i	0.443679	−	0.896186i	$-0.353673\pi$
$38$		−	1.64956e14i		−	0.161966i
$39$	−1.39617e15			−1.07109
$40$	4.41327e14	+	2.60827e14i	0.266189	+	0.157320i
$41$	−9.33298e14			−0.445219			−0.222610	−	0.974908i	$-0.571458\pi$
$41$	−9.33298e14			−0.445219			−0.222610	+	0.974908i	$0.571458\pi$
$42$		−	1.01539e14i		−	0.0385269i
$43$		−	2.15163e13i		−	0.00652856i	−0.999995	−	0.00326428i	$-0.998961\pi$
$43$		−	2.15163e13i		−	0.00652856i	0.999995	−	0.00326428i	$-0.00103905\pi$
$44$	2.84486e15			0.693842
$45$	1.39617e14	−	2.36235e14i	0.0275054	−	0.0465399i
$46$	1.09924e15			0.175749
$47$			4.64662e15i			0.605630i	0.953049	+	0.302815i	$0.0979265\pi$
$47$			4.64662e15i			0.605630i	−0.953049	+	0.302815i	$0.902074\pi$
$48$			8.44993e15i			0.901698i
$49$	1.06688e16			0.935948
$50$	1.89364e15	−	1.04247e15i	0.137114	−	0.0754830i
$51$	−1.97317e16			−1.18372
$52$			2.15355e16i			1.07429i
$53$		−	3.42854e16i		−	1.42721i	−0.700547	−	0.713606i	$-0.747061\pi$
$53$		−	3.42854e16i		−	1.42721i	0.700547	−	0.713606i	$-0.252939\pi$
$54$	−4.60366e15			−0.160459
$55$	1.23600e16	−	2.09135e16i	0.361887	−	0.612323i
$56$	−3.17173e15			−0.0782545
$57$		−	4.82616e16i		−	1.00645i
$58$			1.07311e16i			0.189705i
$59$	−3.91436e16			−0.588256			−0.294128	−	0.955766i	$-0.595029\pi$
$59$	−3.91436e16			−0.588256			−0.294128	+	0.955766i	$0.595029\pi$
$60$	6.37594e16	+	3.76822e16i	0.816784	+	0.482725i
$61$	−1.44319e17			−1.58011			−0.790057	−	0.613033i	$-0.789949\pi$
$61$	−1.44319e17			−1.58011			−0.790057	+	0.613033i	$0.789949\pi$
$62$			7.11054e14i			0.00667086i
$63$			1.69777e15i			0.0136818i
$64$	1.23362e17			0.855998
$65$	1.58314e17	+	9.35644e16i	0.948073	+	0.560317i
$66$	−2.09025e16			−0.108275
$67$		−	3.79832e16i		−	0.170562i	−0.996357	−	0.0852808i	$-0.972821\pi$
$67$		−	3.79832e16i		−	0.170562i	0.996357	−	0.0852808i	$-0.0271787\pi$
$68$			3.04354e17i			1.18725i
$69$	3.21607e17			1.09209
$70$	−6.80461e15	+	1.15136e16i	−0.0201545	+	0.0341020i
$71$	−7.32379e17			−1.89575			−0.947877	−	0.318637i	$-0.896775\pi$
$71$	−7.32379e17			−1.89575			−0.947877	+	0.318637i	$0.896775\pi$
$72$			7.37530e15i			0.0167156i
$73$		−	5.16836e17i		−	1.02751i	−0.857937	−	0.513754i	$-0.828254\pi$
$73$		−	5.16836e17i		−	1.02751i	0.857937	−	0.513754i	$-0.171746\pi$
$74$	1.60581e17			0.280539
$75$	5.54027e17	−	3.04998e17i	0.852019	−	0.469046i
$76$	−7.44417e17			−1.00946
$77$			1.50301e17i			0.180011i
$78$		−	1.58230e17i		−	0.167645i
$79$	−5.46910e16			−0.0513403			−0.0256702	−	0.999670i	$-0.508172\pi$
$79$	−5.46910e16			−0.0513403			−0.0256702	+	0.999670i	$0.508172\pi$
$80$	5.66271e17	−	9.58146e17i	0.471704	−	0.798136i
$81$	−1.27388e18			−0.943017
$82$		−	1.05772e17i		−	0.0696850i
$83$			5.15297e17i			0.302563i	0.988491	+	0.151281i	$0.0483399\pi$
$83$			5.15297e17i			0.302563i	−0.988491	+	0.151281i	$0.951660\pi$
$84$	−4.58226e17			−0.240119
$85$	2.23740e18	+	1.32232e18i	1.04776	+	0.619235i
$86$	2.43847e15			0.00102184
$87$			3.13963e18i			1.17882i
$88$			6.52922e17i			0.219926i
$89$	3.08156e18			0.932321			0.466160	−	0.884700i	$-0.345637\pi$
$89$	3.08156e18			0.932321			0.466160	+	0.884700i	$0.345637\pi$
$90$	2.67729e16	+	1.58230e16i	0.00728436	+	0.00430510i
$91$	−1.13777e18			−0.278715
$92$		−	4.96067e18i		−	1.09536i
$93$			2.08035e17i			0.0414522i
$94$	−5.26608e17			−0.0947923
$95$	−3.23425e18	+	5.47244e18i	−0.526501	+	0.890854i
$96$	−2.99821e18			−0.441861
$97$		−	5.49443e18i		−	0.733823i	−0.930256	−	0.366912i	$-0.880415\pi$
$97$		−	5.49443e18i		−	0.733823i	0.930256	−	0.366912i	$-0.119585\pi$
$98$			1.20911e18i			0.146493i
$99$	3.49498e17			0.0384513
$100$	−4.70448e18	−	8.54566e18i	−0.470448	−	0.854566i
$101$	8.81595e18			0.802077			0.401038	−	0.916061i	$-0.368649\pi$
$101$	8.81595e18			0.802077			0.401038	+	0.916061i	$0.368649\pi$
$102$		−	2.23622e18i		−	0.185273i
$103$		−	1.69018e19i		−	1.27638i	−0.769878	−	0.638191i	$-0.779683\pi$
$103$		−	1.69018e19i		−	1.27638i	0.769878	−	0.638191i	$-0.220317\pi$
$104$	−4.94258e18			−0.340516
$105$	−1.99084e18	+	3.36856e18i	−0.125239	+	0.211907i
$106$	3.88561e18			0.223385
$107$			2.55780e19i			1.34500i	0.740099	+	0.672498i	$0.234779\pi$
$107$			2.55780e19i			1.34500i	−0.740099	+	0.672498i	$0.765221\pi$
$108$			2.07755e19i			1.00006i
$109$	−2.26374e19			−0.998330			−0.499165	−	0.866507i	$-0.666360\pi$
$109$	−2.26374e19			−0.998330			−0.499165	+	0.866507i	$0.666360\pi$
$110$	2.37015e18	+	1.40078e18i	0.0958398	+	0.0566419i
$111$	4.69816e19			1.74325
$112$			6.88600e18i			0.234637i
$113$			2.99224e18i			0.0937025i	0.998902	+	0.0468512i	$0.0149187\pi$
$113$			2.99224e18i			0.0937025i	−0.998902	+	0.0468512i	$0.985081\pi$
$114$	5.46956e18			0.157527
$115$	−3.64674e19	−	2.15525e19i	−0.966664	−	0.571305i
$116$	4.84276e19			1.18234
$117$			2.64568e18i			0.0595350i
$118$		−	4.43620e18i		−	0.0920729i
$119$	−1.60797e19			−0.308022
$120$	−8.64841e18	+	1.46334e19i	−0.153008	+	0.258894i
$121$	−3.02187e19			−0.494099
$122$		−	1.63558e19i		−	0.247317i
$123$		−	3.09460e19i		−	0.433018i
$124$	3.20886e18			0.0415762
$125$	−8.32612e19	−	2.54401e18i	−0.999534	−	0.0305404i
$126$	−1.92411e17			−0.00214146
$127$		−	1.73870e20i		−	1.79510i	−0.440914	−	0.897549i	$-0.645346\pi$
$127$		−	1.73870e20i		−	1.79510i	0.440914	−	0.897549i	$-0.354654\pi$
$128$			6.13885e19i			0.588291i
$129$	7.13430e17			0.00634964
$130$	−1.06038e19	+	1.79419e19i	−0.0877000	+	0.148391i
$131$	1.18459e20			0.910943			0.455472	−	0.890250i	$-0.349471\pi$
$131$	1.18459e20			0.910943			0.455472	+	0.890250i	$0.349471\pi$
$132$			9.43289e19i			0.674827i
$133$		−	3.93293e19i		−	0.261894i
$134$	4.30470e18			0.0266960
$135$	1.52727e20	+	9.02626e19i	0.882561	+	0.521599i
$136$	−6.98519e19			−0.376321
$137$			1.76822e20i			0.888570i	0.895885	+	0.444285i	$0.146542\pi$
$137$			1.76822e20i			0.888570i	−0.895885	+	0.444285i	$0.853458\pi$
$138$			3.64482e19i			0.170933i
$139$	−3.90655e19			−0.171062			−0.0855309	−	0.996336i	$-0.527259\pi$
$139$	−3.90655e19			−0.171062			−0.0855309	+	0.996336i	$0.527259\pi$
$140$	5.19587e19	+	3.07079e19i	0.212541	+	0.125613i
$141$	−1.54071e20			−0.589033
$142$		−	8.30016e19i		−	0.296720i
$143$			2.34217e20i			0.783298i
$144$	1.60122e19			0.0501196
$145$	2.10402e20	−	3.56006e20i	0.616672	−	1.04343i
$146$	5.85737e19			0.160824
$147$			3.53751e20i			0.910298i
$148$		−	7.24674e20i		−	1.74846i
$149$	2.39602e20			0.542277			0.271139	−	0.962540i	$-0.412600\pi$
$149$	2.39602e20			0.542277			0.271139	+	0.962540i	$0.412600\pi$
$150$	3.45659e19	+	6.27887e19i	0.0734143	+	0.133357i
$151$	−9.81901e20			−1.95788			−0.978941	−	0.204142i	$-0.934560\pi$
$151$	−9.81901e20			−1.95788			−0.978941	+	0.204142i	$0.934560\pi$
$152$		−	1.70850e20i		−	0.319965i
$153$			3.73906e19i			0.0657952i
$154$	−1.70338e19			−0.0281750
$155$	1.39414e19	−	2.35893e19i	0.0216848	−	0.0366914i
$156$	−7.14065e20			−1.04485
$157$		−	5.26858e20i		−	0.725515i	−0.931884	−	0.362758i	$-0.881835\pi$
$157$		−	5.26858e20i		−	0.725515i	0.931884	−	0.362758i	$-0.118165\pi$
$158$		−	6.19821e18i		−	0.00803570i
$159$	1.13682e21			1.38810
$160$	3.39971e20	+	2.00925e20i	0.391112	+	0.231150i
$161$	2.62084e20			0.284180
$162$		−	1.44370e20i		−	0.147600i
$163$			1.47793e21i			1.42519i	0.701576	+	0.712595i	$0.252480\pi$
$163$			1.47793e21i			1.42519i	−0.701576	+	0.712595i	$0.747520\pi$
$164$	−4.77330e20			−0.434312
$165$	6.93441e20	+	4.09828e20i	0.595541	+	0.351969i
$166$	−5.83993e19			−0.0473566
$167$			4.73685e20i			0.362814i	0.983408	+	0.181407i	$0.0580651\pi$
$167$			4.73685e20i			0.362814i	−0.983408	+	0.181407i	$0.941935\pi$
$168$		−	1.05167e20i		−	0.0761099i
$169$	−3.11092e20			−0.212797
$170$	−1.49860e20	+	2.53568e20i	−0.0969217	+	0.163994i
$171$	−9.14535e19			−0.0559419
$172$		−	1.10044e19i		−	0.00636862i
$173$		−	2.66872e20i		−	0.146172i	−0.997326	−	0.0730862i	$-0.976715\pi$
$173$		−	2.66872e20i		−	0.146172i	0.997326	−	0.0730862i	$-0.0232848\pi$
$174$	−3.55819e20			−0.184506
$175$	4.51487e20	−	2.48548e20i	0.221709	−	0.122053i
$176$	1.41753e21			0.659420
$177$		−	1.29791e21i		−	0.572134i
$178$			3.49237e20i			0.145925i
$179$	−1.71409e21			−0.679095			−0.339547	−	0.940589i	$-0.610274\pi$
$179$	−1.71409e21			−0.679095			−0.339547	+	0.940589i	$0.610274\pi$
$180$	7.14061e19	−	1.20821e20i	0.0268316	−	0.0453998i
$181$	−3.11223e20			−0.110949			−0.0554747	−	0.998460i	$-0.517667\pi$
$181$	−3.11223e20			−0.110949			−0.0554747	+	0.998460i	$0.517667\pi$
$182$		−	1.28945e20i		−	0.0436240i
$183$		−	4.78526e21i		−	1.53681i
$184$	1.13852e21			0.347193
$185$	−5.32730e21	−	3.14847e21i	−1.54303	−	0.911943i
$186$	−2.35769e19			−0.00648804
$187$			3.31012e21i			0.865662i
$188$			2.37649e21i			0.590793i
$189$	−1.09762e21			−0.259456
$190$	−6.20199e20	−	3.66542e20i	−0.139435	−	0.0824071i
$191$	6.54783e21			1.40049			0.700246	−	0.713902i	$-0.253074\pi$
$191$	6.54783e21			1.40049			0.700246	+	0.713902i	$0.253074\pi$
$192$			4.09040e21i			0.832539i
$193$			7.76020e20i			0.150341i	0.997171	+	0.0751706i	$0.0239501\pi$
$193$			7.76020e20i			0.150341i	−0.997171	+	0.0751706i	$0.976050\pi$
$194$	6.22692e20			0.114857
$195$	−3.10238e21	+	5.24931e21i	−0.544961	+	0.922090i
$196$	5.45648e21			0.913019
$197$		−	5.13756e21i		−	0.819082i	−0.912292	−	0.409541i	$-0.865689\pi$
$197$		−	5.13756e21i		−	0.819082i	0.912292	−	0.409541i	$-0.134311\pi$
$198$			3.96092e19i			0.00601834i
$199$	4.15884e21			0.602376			0.301188	−	0.953565i	$-0.402617\pi$
$199$	4.15884e21			0.602376			0.301188	+	0.953565i	$0.402617\pi$
$200$	1.96131e21	−	1.07972e21i	0.270870	−	0.149117i
$201$	1.25943e21			0.165887
$202$			9.99124e20i			0.125540i
$203$			2.55854e21i			0.306747i
$204$	−1.00917e22			−1.15472
$205$	−2.07384e21	+	3.50900e21i	−0.226524	+	0.383285i
$206$	1.91551e21			0.199777
$207$		−	6.09431e20i		−	0.0607024i
$208$			1.07306e22i			1.02099i
$209$	−8.09620e21			−0.736024
$210$	−3.81764e20	−	2.25625e20i	−0.0331674	−	0.0196021i
$211$	5.95290e21			0.494362			0.247181	−	0.968969i	$-0.420496\pi$
$211$	5.95290e21			0.494362			0.247181	+	0.968969i	$0.420496\pi$
$212$		−	1.75351e22i		−	1.39225i
$213$		−	2.42840e22i		−	1.84380i
$214$	−2.89880e21			−0.210517
$215$	−8.08966e19	−	4.78104e19i	−0.00562037	−	0.00332167i
$216$	−4.76816e21			−0.316986
$217$			1.69531e20i			0.0107865i
$218$		−	2.56553e21i		−	0.156257i
$219$	1.71371e22			0.999349
$220$	6.32144e21	−	1.06961e22i	0.353021	−	0.597322i
$221$	−2.50574e22			−1.34032
$222$			5.32450e21i			0.272851i
$223$			3.76323e22i			1.84784i	0.382584	+	0.923921i	$0.375034\pi$
$223$			3.76323e22i			1.84784i	−0.382584	+	0.923921i	$0.624966\pi$
$224$	−2.44330e21			−0.114979
$225$	−5.77957e20	−	1.04986e21i	−0.0260712	−	0.0473582i
$226$	−3.39115e20			−0.0146662
$227$			3.15704e22i			1.30929i	0.755938	+	0.654644i	$0.227181\pi$
$227$			3.15704e22i			1.30929i	−0.755938	+	0.654644i	$0.772819\pi$
$228$		−	2.46831e22i		−	0.981791i
$229$	2.99352e22			1.14221			0.571104	−	0.820878i	$-0.306515\pi$
$229$	2.99352e22			1.14221			0.571104	+	0.820878i	$0.306515\pi$
$230$	2.44257e21	−	4.13290e21i	0.0894197	−	0.151301i
$231$	−4.98362e21			−0.175078
$232$			1.11146e22i			0.374763i
$233$			9.59429e20i			0.0310550i	0.999879	+	0.0155275i	$0.00494275\pi$
$233$			9.59429e20i			0.0310550i	−0.999879	+	0.0155275i	$0.995057\pi$
$234$	−2.99839e20			−0.00931832
$235$	1.74703e22	+	1.03250e22i	0.521381	+	0.308140i
$236$	−2.00197e22			−0.573845
$237$		−	1.81342e21i		−	0.0499333i
$238$		−	1.82234e21i		−	0.0482112i
$239$	4.17690e22			1.06188			0.530938	−	0.847411i	$-0.321840\pi$
$239$	4.17690e22			1.06188			0.530938	+	0.847411i	$0.321840\pi$
$240$	3.17699e22	+	1.87762e22i	0.776263	+	0.458776i
$241$	−4.86489e22			−1.14264			−0.571322	−	0.820726i	$-0.693569\pi$
$241$	−4.86489e22			−1.14264			−0.571322	+	0.820726i	$0.693569\pi$
$242$		−	3.42472e21i		−	0.0773356i
$243$			4.97374e21i			0.108000i
$244$	−7.38108e22			−1.54140
$245$	2.37066e22	−	4.01123e22i	0.476202	−	0.805748i
$246$	3.50715e21			0.0677752
$247$		−	6.12878e22i		−	1.13960i
$248$			7.36462e20i			0.0131783i
$249$	−1.70860e22			−0.294271
$250$	2.88317e20	−	9.43611e21i	0.00478013	−	0.156445i
$251$	−4.95879e22			−0.791545			−0.395772	−	0.918349i	$-0.629523\pi$
$251$	−4.95879e22			−0.791545			−0.395772	+	0.918349i	$0.629523\pi$
$252$			8.68316e20i			0.0133467i
$253$		−	5.39517e22i		−	0.798657i
$254$	1.97049e22			0.280966
$255$	−4.38449e22	+	7.41868e22i	−0.602265	+	1.01905i
$256$	5.77201e22			0.763919
$257$		−	6.03408e22i		−	0.769568i	−0.923007	−	0.384784i	$-0.874276\pi$
$257$		−	6.03408e22i		−	0.769568i	0.923007	−	0.384784i	$-0.125724\pi$
$258$			8.08540e19i			0.000993836i
$259$	3.82862e22			0.453622
$260$	8.09686e22	+	4.78530e22i	0.924847	+	0.546591i
$261$	5.94945e21			0.0655228
$262$			1.34252e22i			0.142579i
$263$			1.28674e23i			1.31798i	0.752151	+	0.658991i	$0.229017\pi$
$263$			1.28674e23i			1.31798i	−0.752151	+	0.658991i	$0.770983\pi$
$264$	−2.16493e22			−0.213898
$265$	−1.28906e23	−	7.61841e22i	−1.22867	−	0.726153i
$266$	4.45724e21			0.0409912
$267$			1.02177e23i			0.906770i
$268$		−	1.94263e22i		−	0.166383i
$269$	9.53759e22			0.788482			0.394241	−	0.919007i	$-0.371007\pi$
$269$	9.53759e22			0.788482			0.394241	+	0.919007i	$0.371007\pi$
$270$	−1.02296e22	+	1.73088e22i	−0.0816399	+	0.138137i
$271$	5.86695e22			0.452068			0.226034	−	0.974119i	$-0.427424\pi$
$271$	5.86695e22			0.452068			0.226034	+	0.974119i	$0.427424\pi$
$272$			1.51653e23i			1.12835i
$273$		−	3.77257e22i		−	0.271077i
$274$	−2.00395e22			−0.139078
$275$	−5.11654e22	−	9.29417e22i	−0.343017	−	0.623089i
$276$	1.64484e23			1.06534
$277$		−	8.11673e22i		−	0.507953i	−0.967210	−	0.253976i	$-0.918261\pi$
$277$		−	8.11673e22i		−	0.507953i	0.967210	−	0.253976i	$-0.0817386\pi$
$278$		−	4.42735e21i		−	0.0267743i
$279$	3.94216e20			0.00230406
$280$	−7.04775e21	+	1.19250e22i	−0.0398152	+	0.0673685i
$281$	9.85616e22			0.538267			0.269134	−	0.963103i	$-0.413263\pi$
$281$	9.85616e22			0.538267			0.269134	+	0.963103i	$0.413263\pi$
$282$		−	1.74611e22i		−	0.0921945i
$283$		−	8.61893e22i		−	0.440030i	−0.975497	−	0.220015i	$-0.929389\pi$
$283$		−	8.61893e22i		−	0.440030i	0.975497	−	0.220015i	$-0.0706106\pi$
$284$	−3.74571e23			−1.84931
$285$	−1.81453e23	−	1.07240e23i	−0.866440	−	0.512072i
$286$	−2.65442e22			−0.122601
$287$		−	2.52184e22i		−	0.112678i
$288$			5.68147e21i			0.0245602i
$289$	−1.15056e23			−0.481261
$290$	4.03467e22	+	2.38452e22i	0.163315	+	0.0965205i
$291$	1.82182e23			0.713712
$292$		−	2.64332e23i		−	1.00234i
$293$		−	5.65789e22i		−	0.207688i	−0.994594	−	0.103844i	$-0.966886\pi$
$293$		−	5.65789e22i		−	0.207688i	0.994594	−	0.103844i	$-0.0331143\pi$
$294$	−4.00912e22			−0.142478
$295$	−8.69792e22	+	1.47171e23i	−0.299300	+	0.506424i
$296$	1.66319e23			0.554206
$297$			2.25952e23i			0.729172i
$298$			2.71545e22i			0.0848764i
$299$	4.08412e23			1.23658
$300$	2.83354e23	−	1.55989e23i	0.831146	−	0.457555i
$301$	5.81387e20			0.00165228
$302$		−	1.11280e23i		−	0.306445i
$303$			2.92316e23i			0.780095i
$304$	−3.70926e23			−0.959375
$305$	−3.20684e23	+	5.42606e23i	−0.803949	+	1.36030i
$306$	−4.23753e21			−0.0102982
$307$			8.75128e21i			0.0206185i	0.999947	+	0.0103092i	$0.00328159\pi$
$307$			8.75128e21i			0.0206185i	−0.999947	+	0.0103092i	$0.996718\pi$
$308$			7.68704e22i			0.175601i
$309$	5.60425e23			1.24140
$310$	2.67341e21	+	1.58000e21i	0.00574287	+	0.00339408i
$311$	−3.00525e23			−0.626119			−0.313060	−	0.949733i	$-0.601354\pi$
$311$	−3.00525e23			−0.626119			−0.313060	+	0.949733i	$0.601354\pi$
$312$		−	1.63884e23i		−	0.331184i
$313$		−	3.65350e22i		−	0.0716206i	−0.999359	−	0.0358103i	$-0.988599\pi$
$313$		−	3.65350e22i		−	0.0716206i	0.999359	−	0.0358103i	$-0.0114012\pi$
$314$	5.97096e22			0.113557
$315$	6.38326e21	+	3.77255e21i	0.0117786	+	0.00696120i
$316$	−2.79714e22			−0.0500826
$317$			4.43734e23i			0.771009i	0.922706	+	0.385504i	$0.125973\pi$
$317$			4.43734e23i			0.771009i	−0.922706	+	0.385504i	$0.874027\pi$
$318$			1.28838e23i			0.217263i
$319$	5.26694e23			0.862079
$320$	2.74118e23	−	4.63815e23i	0.435524	−	0.736920i
$321$	−8.48107e23			−1.30814
$322$			2.97023e22i			0.0444795i
$323$		−	8.66161e23i		−	1.25943i
$324$	−6.51516e23			−0.919915
$325$	7.03564e23	−	3.87319e23i	0.964743	−	0.531101i
$326$	−1.67496e23			−0.223069
$327$		−	7.50602e23i		−	0.970970i
$328$		−	1.09552e23i		−	0.137663i
$329$	−1.25555e23			−0.153276
$330$	−4.64464e22	+	7.85887e22i	−0.0550896	+	0.0932132i
$331$	−5.86541e23			−0.675978			−0.337989	−	0.941150i	$-0.609747\pi$
$331$	−5.86541e23			−0.675978			−0.337989	+	0.941150i	$0.609747\pi$
$332$			2.63545e23i			0.295150i
$333$		−	8.90279e22i		−	0.0968961i
$334$	−5.36835e22			−0.0567870
$335$	−1.42809e23	−	8.44009e22i	−0.146835	−	0.0867803i
$336$	−2.28324e23			−0.228206
$337$			1.61676e24i			1.57094i	0.618897	+	0.785472i	$0.287580\pi$
$337$			1.61676e24i			1.57094i	−0.618897	+	0.785472i	$0.712420\pi$
$338$		−	3.52566e22i		−	0.0333067i
$339$	−9.92156e22			−0.0911345
$340$	1.14430e24	+	6.76291e23i	1.02210	+	0.604065i
$341$	3.48992e22			0.0303144
$342$		−	1.03646e22i		−	0.00875594i
$343$			5.96285e23i			0.489959i
$344$	2.52561e21			0.00201865
$345$	7.14629e23	−	1.20917e24i	0.555648	−	0.940171i
$346$	3.02450e22			0.0228787
$347$		−	1.75971e24i		−	1.29512i	−0.762014	−	0.647560i	$-0.775789\pi$
$347$		−	1.75971e24i		−	1.29512i	0.762014	−	0.647560i	$-0.224211\pi$
$348$			1.60575e24i			1.14994i
$349$	2.36703e24			1.64954			0.824768	−	0.565472i	$-0.191306\pi$
$349$	2.36703e24			1.64954			0.824768	+	0.565472i	$0.191306\pi$
$350$	2.81684e22	+	5.11677e22i	0.0191036	+	0.0347016i
$351$	−1.71044e24			−1.12899
$352$			5.02970e23i			0.323137i
$353$			1.43847e23i			0.0899585i	0.998988	+	0.0449793i	$0.0143222\pi$
$353$			1.43847e23i			0.0899585i	−0.998988	+	0.0449793i	$0.985678\pi$
$354$	1.47094e23			0.0895496
$355$	−1.62739e24	+	2.75359e24i	−0.964543	+	1.63203i
$356$	1.57604e24			0.909481
$357$		−	5.33165e23i		−	0.299581i
$358$		−	1.94261e23i		−	0.106291i
$359$	−1.16835e24			−0.622550			−0.311275	−	0.950320i	$-0.600756\pi$
$359$	−1.16835e24			−0.622550			−0.311275	+	0.950320i	$0.600756\pi$
$360$	2.77295e22	+	1.63883e22i	0.0143903	+	0.00850474i
$361$	1.40119e23			0.0708238
$362$		−	3.52714e22i		−	0.0173656i
$363$		−	1.00198e24i		−	0.480558i
$364$	−5.81904e23			−0.271887
$365$	−1.94319e24	−	1.14844e24i	−0.884571	−	0.522788i
$366$	5.42321e23			0.240539
$367$		−	2.80054e24i		−	1.21036i	−0.796089	−	0.605180i	$-0.793101\pi$
$367$		−	2.80054e24i		−	1.21036i	0.796089	−	0.605180i	$-0.206899\pi$
$368$		−	2.47179e24i		−	1.04102i
$369$	−5.86412e22			−0.0240687
$370$	3.56820e23	−	6.03750e23i	0.142736	−	0.241513i
$371$	9.26418e23			0.361206
$372$			1.06398e23i			0.0404367i
$373$			1.42432e24i			0.527682i	0.964566	+	0.263841i	$0.0849894\pi$
$373$			1.42432e24i			0.527682i	−0.964566	+	0.263841i	$0.915011\pi$
$374$	−3.75141e23			−0.135492
$375$	8.43535e22	−	2.76074e24i	0.0297034	−	0.972141i
$376$	−5.45425e23			−0.187262
$377$			3.98704e24i			1.33478i
$378$		−	1.24394e23i		−	0.0406096i
$379$	−2.87049e24			−0.913868			−0.456934	−	0.889501i	$-0.651052\pi$
$379$	−2.87049e24			−0.913868			−0.456934	+	0.889501i	$0.651052\pi$
$380$	−1.65414e24	+	2.79885e24i	−0.513602	+	0.869030i
$381$	5.76511e24			1.74590
$382$			7.42076e23i			0.219203i
$383$			5.69586e23i			0.164124i	0.996627	+	0.0820618i	$0.0261505\pi$
$383$			5.69586e23i			0.164124i	−0.996627	+	0.0820618i	$0.973850\pi$
$384$	−2.03550e24			−0.572169
$385$	5.65098e23	+	3.33976e23i	0.154970	+	0.0915881i
$386$	−8.79474e22			−0.0235312
$387$		−	1.35192e21i		−	0.000352936i
$388$		−	2.81009e24i		−	0.715846i
$389$	4.66196e23			0.115890			0.0579452	−	0.998320i	$-0.481545\pi$
$389$	4.66196e23			0.115890			0.0579452	+	0.998320i	$0.481545\pi$
$390$	−5.94912e23	−	3.51597e23i	−0.144324	−	0.0852965i
$391$	5.77195e24			1.36661
$392$			1.25231e24i			0.289397i
$393$			3.92783e24i			0.885978i
$394$	5.82247e23			0.128201
$395$	−1.21526e23	+	2.05626e23i	−0.0261215	+	0.0441983i
$396$	1.78749e23			0.0375093
$397$			4.81447e24i			0.986367i	0.869925	+	0.493184i	$0.164167\pi$
$397$			4.81447e24i			0.986367i	−0.869925	+	0.493184i	$0.835833\pi$
$398$			4.71327e23i			0.0942830i
$399$	1.30407e24			0.254717
$400$	−2.34414e24	−	4.25811e24i	−0.447108	−	0.812170i
$401$	−1.07502e24			−0.200237			−0.100118	−	0.994976i	$-0.531922\pi$
$401$	−1.07502e24			−0.200237			−0.100118	+	0.994976i	$0.531922\pi$
$402$			1.42734e23i			0.0259644i
$403$			2.64185e23i			0.0469365i
$404$	4.50886e24			0.782427
$405$	−2.83062e24	+	4.78950e24i	−0.479799	+	0.811834i
$406$	−2.89964e23			−0.0480116
$407$		−	7.88148e24i		−	1.27485i
$408$		−	2.31613e24i		−	0.366008i
$409$	−6.08999e24			−0.940254			−0.470127	−	0.882599i	$-0.655792\pi$
$409$	−6.08999e24			−0.940254			−0.470127	+	0.882599i	$0.655792\pi$
$410$	−3.97680e23	−	2.35031e23i	−0.0599911	−	0.0354551i
$411$	−5.86301e24			−0.864218
$412$		−	8.64435e24i		−	1.24511i
$413$		−	1.05769e24i		−	0.148879i
$414$	6.90677e22			0.00950105
$415$	1.93740e24	+	1.14502e24i	0.260473	+	0.153941i
$416$	−3.80745e24			−0.500320
$417$		−	1.29532e24i		−	0.166374i
$418$		−	9.17555e23i		−	0.115201i
$419$	6.17193e24			0.757509			0.378754	−	0.925497i	$-0.376352\pi$
$419$	6.17193e24			0.757509			0.378754	+	0.925497i	$0.376352\pi$
$420$	−1.01820e24	+	1.72283e24i	−0.122170	+	0.206716i
$421$	−2.02089e24			−0.237062			−0.118531	−	0.992950i	$-0.537818\pi$
$421$	−2.02089e24			−0.237062			−0.118531	+	0.992950i	$0.537818\pi$
$422$			6.74651e23i			0.0773768i
$423$			2.91957e23i			0.0327405i
$424$	4.02445e24			0.441297
$425$	9.94324e24	−	5.47386e24i	1.06619	−	0.586947i
$426$	2.75214e24			0.288589
$427$		−	3.89960e24i		−	0.399903i
$428$			1.30817e25i			1.31205i
$429$	−7.76610e24			−0.761831
$430$	5.41842e21	−	9.16813e21i	0.000519903	−	0.000879691i
$431$	−5.28591e24			−0.496119			−0.248059	−	0.968745i	$-0.579793\pi$
$431$	−5.28591e24			−0.496119			−0.248059	+	0.968745i	$0.579793\pi$
$432$			1.03520e25i			0.950444i
$433$		−	8.30646e24i		−	0.746073i	−0.927817	−	0.373036i	$-0.878317\pi$
$433$		−	8.30646e24i		−	0.746073i	0.927817	−	0.373036i	$-0.121683\pi$
$434$	−1.92132e22			−0.00168829
$435$	1.18043e25	+	6.97644e24i	1.01483	+	0.599771i
$436$	−1.15777e25			−0.973873
$437$			1.41176e25i			1.16195i
$438$			1.94217e24i			0.156417i
$439$	−8.06653e24			−0.635732			−0.317866	−	0.948136i	$-0.602966\pi$
$439$	−8.06653e24			−0.635732			−0.317866	+	0.948136i	$0.602966\pi$
$440$	2.45484e24	+	1.45083e24i	0.189332	+	0.111896i
$441$	6.70342e23			0.0505976
$442$		−	2.83980e24i		−	0.209785i
$443$			2.08283e24i			0.150597i	0.997161	+	0.0752987i	$0.0239910\pi$
$443$			2.08283e24i			0.150597i	−0.997161	+	0.0752987i	$0.976009\pi$
$444$	2.40285e25			1.70054
$445$	6.84739e24	−	1.15860e25i	0.474357	−	0.802625i
$446$	−4.26493e24			−0.289221
$447$			7.94464e24i			0.527416i
$448$			3.33334e24i			0.216640i
$449$	6.94790e24			0.442093			0.221046	−	0.975263i	$-0.429053\pi$
$449$	6.94790e24			0.442093			0.221046	+	0.975263i	$0.429053\pi$
$450$	1.18982e23	−	6.55007e22i	0.00741244	−	0.00408063i
$451$	−5.19139e24			−0.316670
$452$			1.53036e24i			0.0914069i
$453$		−	3.25575e25i		−	1.90423i
$454$	−3.57792e24			−0.204928
$455$	−2.52818e24	+	4.27776e24i	−0.141808	+	0.239943i
$456$	5.66500e24			0.311196
$457$		−	8.71540e24i		−	0.468904i	−0.972128	−	0.234452i	$-0.924671\pi$
$457$		−	8.71540e24i		−	0.468904i	0.972128	−	0.234452i	$-0.0753295\pi$
$458$			3.39260e24i			0.178777i
$459$	−2.41732e25			−1.24771
$460$	−1.86510e25	−	1.10229e25i	−0.942982	−	0.557309i
$461$	−2.94818e25			−1.46014			−0.730072	−	0.683371i	$-0.760513\pi$
$461$	−2.94818e25			−1.46014			−0.730072	+	0.683371i	$0.760513\pi$
$462$		−	5.64801e23i		−	0.0274029i
$463$			1.35704e25i			0.645019i	0.946566	+	0.322509i	$0.104526\pi$
$463$			1.35704e25i			0.645019i	−0.946566	+	0.322509i	$0.895474\pi$
$464$	2.41304e25			1.12368
$465$	7.82165e23	+	4.62265e23i	0.0356858	+	0.0210905i
$466$	−1.08733e23			−0.00486068
$467$		−	2.43556e25i		−	1.06681i	−0.845859	−	0.533407i	$-0.820911\pi$
$467$		−	2.43556e25i		−	1.06681i	0.845859	−	0.533407i	$-0.179089\pi$
$468$			1.35312e24i			0.0580765i
$469$	1.02634e24			0.0431666
$470$	−1.17015e24	+	1.97993e24i	−0.0482295	+	0.0816057i
$471$	1.74694e25			0.705632
$472$		−	4.59471e24i		−	0.181890i
$473$		−	1.19683e23i		−	0.00464355i
$474$	2.05518e23			0.00781548
$475$	1.33885e25	+	2.43201e25i	0.499048	+	0.906518i
$476$	−8.22387e24			−0.300476
$477$		−	2.15423e24i		−	0.0771554i
$478$			4.73374e24i			0.166203i
$479$	−1.71994e23			−0.00592005			−0.00296003	−	0.999996i	$-0.500942\pi$
$479$	−1.71994e23			−0.00592005			−0.00296003	+	0.999996i	$0.500942\pi$
$480$	−6.66219e24	+	1.12726e25i	−0.224815	+	0.380394i
$481$	5.96623e25			1.97389
$482$		−	5.51345e24i		−	0.178845i
$483$			8.69008e24i			0.276392i
$484$	−1.54551e25			−0.481995
$485$	−2.06579e25	−	1.22089e25i	−0.631741	−	0.373363i
$486$	−5.63681e23			−0.0169040
$487$			5.44742e25i			1.60201i	0.598657	+	0.801006i	$0.295701\pi$
$487$			5.44742e25i			1.60201i	−0.598657	+	0.801006i	$0.704299\pi$
$488$		−	1.69403e25i		−	0.488575i
$489$	−4.90048e25			−1.38613
$490$	4.54598e24	+	2.68670e24i	0.126114	+	0.0745345i
$491$	6.23616e25			1.69685			0.848425	−	0.529316i	$-0.177551\pi$
$491$	6.23616e25			1.69685			0.848425	+	0.529316i	$0.177551\pi$
$492$		−	1.58271e25i		−	0.422410i
$493$			5.63476e25i			1.47513i
$494$	6.94584e24			0.178369
$495$	7.76605e23	−	1.31404e24i	0.0195637	−	0.0331023i
$496$	1.59890e24			0.0395135
$497$		−	1.97895e25i		−	0.479787i
$498$		−	1.93638e24i		−	0.0460588i
$499$	9.50774e24			0.221882			0.110941	−	0.993827i	$-0.464614\pi$
$499$	9.50774e24			0.221882			0.110941	+	0.993827i	$0.464614\pi$
$500$	−4.25834e25	−	1.30112e24i	−0.975047	−	0.0297922i
$501$	−1.57063e25			−0.352870
$502$		−	5.61987e24i		−	0.123891i
$503$			1.28370e25i			0.277694i	0.990314	+	0.138847i	$0.0443397\pi$
$503$			1.28370e25i			0.277694i	−0.990314	+	0.138847i	$0.955660\pi$
$504$	−1.99286e23			−0.00423046
$505$	1.95895e25	−	3.31460e25i	0.408090	−	0.690500i
$506$	6.11443e24			0.125005
$507$		−	1.03151e25i		−	0.206965i
$508$		−	8.89246e25i		−	1.75112i
$509$	−2.61312e25			−0.505058			−0.252529	−	0.967589i	$-0.581262\pi$
$509$	−2.61312e25			−0.505058			−0.252529	+	0.967589i	$0.581262\pi$
$510$	−8.40770e24	−	4.96901e24i	−0.159500	−	0.0942655i
$511$	1.39653e25			0.260047
$512$			3.87268e25i			0.707859i
$513$		−	5.91250e25i		−	1.06086i
$514$	6.83852e24			0.120452
$515$	−6.35473e25	−	3.75568e25i	−1.09882	−	0.649412i
$516$	3.64879e23			0.00619408
$517$			2.58464e25i			0.430765i
$518$			4.33903e24i			0.0710002i
$519$	8.84885e24			0.142166
$520$	−1.09827e25	+	1.85830e25i	−0.173251	+	0.293146i
$521$	−1.39805e25			−0.216554			−0.108277	−	0.994121i	$-0.534533\pi$
$521$	−1.39805e25			−0.216554			−0.108277	+	0.994121i	$0.534533\pi$
$522$			6.74260e23i			0.0102555i
$523$			7.37992e25i			1.10226i	0.834418	+	0.551132i	$0.185804\pi$
$523$			7.37992e25i			1.10226i	−0.834418	+	0.551132i	$0.814196\pi$
$524$	6.05852e25			0.888627
$525$	8.24128e24	+	1.49702e25i	0.118708	+	0.215633i
$526$	−1.45828e25			−0.206289
$527$			3.73364e24i			0.0518719i
$528$			4.70020e25i			0.641348i
$529$	−1.94618e25			−0.260827
$530$	8.63405e24	−	1.46091e25i	0.113656	−	0.192310i
$531$	−2.45948e24			−0.0318013
$532$		−	2.01147e25i		−	0.255478i
$533$		−	3.92985e25i		−	0.490307i
$534$	−1.15799e25			−0.141926
$535$	9.61678e25	+	5.68358e25i	1.15789	+	0.684323i
$536$	4.45851e24			0.0527381
$537$		−	5.68353e25i		−	0.660484i
$538$			1.08091e25i			0.123412i
$539$	5.93441e25			0.665709
$540$	7.81112e25	+	4.61643e25i	0.860940	+	0.508821i
$541$	−2.49490e25			−0.270197			−0.135098	−	0.990832i	$-0.543135\pi$
$541$	−2.49490e25			−0.270197			−0.135098	+	0.990832i	$0.543135\pi$
$542$			6.64910e24i			0.0707570i
$543$		−	1.03194e25i		−	0.107909i
$544$	−5.38095e25			−0.552929
$545$	−5.03015e25	+	8.51115e25i	−0.507942	+	0.859452i
$546$	4.27551e24			0.0424285
$547$			5.47088e25i			0.533553i	0.963758	+	0.266776i	$0.0859585\pi$
$547$			5.47088e25i			0.533553i	−0.963758	+	0.266776i	$0.914041\pi$
$548$			9.04347e25i			0.866802i
$549$	−9.06784e24			−0.0854214
$550$	1.05332e25	−	5.79865e24i	0.0975249	−	0.0536885i
$551$	−1.37820e26			−1.25422
$552$			3.77506e25i			0.337678i
$553$		−	1.47779e24i		−	0.0129935i
$554$	9.19881e24			0.0795040
$555$	1.04396e26	−	1.76641e26i	0.886951	−	1.50075i
$556$	−1.99798e25			−0.166871
$557$		−	2.16234e26i		−	1.77541i	−0.460412	−	0.887705i	$-0.652298\pi$
$557$		−	2.16234e26i		−	1.77541i	0.460412	−	0.887705i	$-0.347702\pi$
$558$			4.46771e22i			0.000360628i
$559$	9.05990e23			0.00718971
$560$	2.58899e25	+	1.53011e25i	0.201996	+	0.119381i
$561$	−1.09756e26			−0.841938
$562$			1.11701e25i			0.0842487i
$563$		−	1.99170e26i		−	1.47705i	−0.674227	−	0.738524i	$-0.735523\pi$
$563$		−	1.99170e26i		−	1.47705i	0.674227	−	0.738524i	$-0.264477\pi$
$564$	−7.87987e25			−0.574602
$565$	1.12502e25	+	6.64892e24i	0.0806675	+	0.0476750i
$566$	9.76796e24			0.0688728
$567$		−	3.44211e25i		−	0.238663i
$568$		−	8.59675e25i		−	0.586172i
$569$	2.58945e26			1.73636			0.868182	−	0.496246i	$-0.165289\pi$
$569$	2.58945e26			1.73636			0.868182	+	0.496246i	$0.165289\pi$
$570$	1.21537e25	−	2.05643e25i	0.0801486	−	0.135614i
$571$	−1.85085e26			−1.20040			−0.600202	−	0.799849i	$-0.704913\pi$
$571$	−1.85085e26			−1.20040			−0.600202	+	0.799849i	$0.704913\pi$
$572$			1.19789e26i			0.764108i
$573$			2.17111e26i			1.36211i
$574$	2.85804e24			0.0176362
$575$	−1.62065e26	+	8.92187e25i	−0.983660	+	0.541516i
$576$	7.75112e24			0.0462755
$577$			1.24419e26i			0.730663i	0.930878	+	0.365331i	$0.119044\pi$
$577$			1.24419e26i			0.730663i	−0.930878	+	0.365331i	$0.880956\pi$
$578$		−	1.30395e25i		−	0.0753262i
$579$	−2.57310e25			−0.146221
$580$	1.07609e26	−	1.82077e26i	0.601564	−	1.01786i
$581$	−1.39237e25			−0.0765741
$582$			2.06470e25i			0.111709i
$583$		−	1.90710e26i		−	1.01513i
$584$	6.06667e25			0.317708
$585$	9.94719e24	+	5.87886e24i	0.0512531	+	0.0302909i
$586$	6.41217e24			0.0325071
$587$			3.19668e26i			1.59454i	0.603620	+	0.797272i	$0.293725\pi$
$587$			3.19668e26i			1.59454i	−0.603620	+	0.797272i	$0.706275\pi$
$588$			1.80924e26i			0.887997i
$589$	−9.13210e24			−0.0441037
$590$	−1.66791e25	−	9.85748e24i	−0.0792646	−	0.0468459i
$591$	1.70349e26			0.796635
$592$		−	3.61088e26i		−	1.66172i
$593$			1.78032e26i			0.806265i	0.915142	+	0.403132i	$0.132079\pi$
$593$			1.78032e26i			0.806265i	−0.915142	+	0.403132i	$0.867921\pi$
$594$	−2.56075e25			−0.114129
$595$	−3.57300e25	+	6.04562e25i	−0.156719	+	0.265173i
$596$	1.22543e26			0.528992
$597$			1.37897e26i			0.585868i
$598$			4.62859e25i			0.193547i
$599$	1.92332e26			0.791584			0.395792	−	0.918340i	$-0.370470\pi$
$599$	1.92332e26			0.791584			0.395792	+	0.918340i	$0.370470\pi$
$600$	3.58010e25	+	6.50323e25i	0.145030	+	0.263446i
$601$	3.81486e26			1.52114			0.760572	−	0.649253i	$-0.224918\pi$
$601$	3.81486e26			1.52114			0.760572	+	0.649253i	$0.224918\pi$
$602$			6.58894e22i			0.000258612i
$603$		−	2.38657e24i		−	0.00922061i
$604$	−5.02187e26			−1.90992
$605$	−6.71475e25	+	1.13616e26i	−0.251393	+	0.425365i
$606$	−3.31286e25			−0.122099
$607$		−	2.42669e26i		−	0.880483i	−0.897879	−	0.440242i	$-0.854893\pi$
$607$		−	2.42669e26i		−	0.880483i	0.897879	−	0.440242i	$-0.145107\pi$
$608$		−	1.31612e26i		−	0.470125i
$609$	−8.48353e25			−0.298340
$610$	−6.14943e25	−	3.63436e25i	−0.212913	−	0.125833i
$611$	−1.95656e26			−0.666963
$612$			1.91232e25i			0.0641833i
$613$		−	2.73712e26i		−	0.904524i	−0.891885	−	0.452262i	$-0.850617\pi$
$613$		−	2.73712e26i		−	0.904524i	0.891885	−	0.452262i	$-0.149383\pi$
$614$	−9.91795e23			−0.00322717
$615$	−1.16350e26	−	6.87637e25i	−0.372780	−	0.220316i
$616$	−1.76424e25			−0.0556599
$617$		−	2.32043e26i		−	0.720873i	−0.932784	−	0.360436i	$-0.882628\pi$
$617$		−	2.32043e26i		−	0.720873i	0.932784	−	0.360436i	$-0.117372\pi$
$618$			6.35138e25i			0.194302i
$619$	−3.05361e26			−0.919924			−0.459962	−	0.887939i	$-0.652137\pi$
$619$	−3.05361e26			−0.919924			−0.459962	+	0.887939i	$0.652137\pi$
$620$	7.13026e24	−	1.20646e25i	0.0211536	−	0.0357925i
$621$	3.93999e26			1.15113
$622$		−	3.40590e25i		−	0.0979992i
$623$			8.32661e25i			0.235956i
$624$	−3.55802e26			−0.993014
$625$	−1.94576e26	+	3.07391e26i	−0.534846	+	0.844949i
$626$	4.14057e24			0.0112099
$627$		−	2.68451e26i		−	0.715852i
$628$		−	2.69458e26i		−	0.707742i
$629$	8.43189e26			2.18144
$630$	−4.27548e23	+	7.23424e23i	−0.00108956	+	0.00184356i
$631$	3.28337e26			0.824216			0.412108	−	0.911135i	$-0.364793\pi$
$631$	3.28337e26			0.824216			0.412108	+	0.911135i	$0.364793\pi$
$632$		−	6.41968e24i		−	0.0158746i
$633$			1.97384e26i			0.480814i
$634$	−5.02890e25			−0.120677
$635$	−6.53712e26	−	3.86348e26i	−1.54538	−	0.913331i
$636$	5.81421e26			1.35409
$637$			4.49232e26i			1.03073i
$638$			5.96910e25i			0.134931i
$639$	−4.60170e25			−0.102485
$640$	2.30807e26	+	1.36409e26i	0.506454	+	0.299318i
$641$	−6.76321e26			−1.46218			−0.731091	−	0.682280i	$-0.760988\pi$
$641$	−6.76321e26			−1.46218			−0.731091	+	0.682280i	$0.760988\pi$
$642$		−	9.61172e25i		−	0.204747i
$643$			3.72363e26i			0.781560i	0.920484	+	0.390780i	$0.127795\pi$
$643$			3.72363e26i			0.781560i	−0.920484	+	0.390780i	$0.872205\pi$
$644$	1.34041e26			0.277219
$645$	1.58528e24	−	2.68234e24i	0.00323064	−	0.00546634i
$646$	9.81633e25			0.197124
$647$			5.88512e26i			1.16457i	0.812986	+	0.582284i	$0.197841\pi$
$647$			5.88512e26i			1.16457i	−0.812986	+	0.582284i	$0.802159\pi$
$648$		−	1.49529e26i		−	0.291583i
$649$	−2.17733e26			−0.418407
$650$	4.38955e25	+	7.97359e25i	0.0831272	+	0.151000i
$651$	−5.62126e24			−0.0104909
$652$			7.55881e26i			1.39028i
$653$			1.34507e25i			0.0243821i	0.999926	+	0.0121911i	$0.00388063\pi$
$653$			1.34507e25i			0.0243821i	−0.999926	+	0.0121911i	$0.996119\pi$
$654$	8.50668e25			0.151975
$655$	2.63223e26	−	4.45381e26i	0.463480	−	0.784222i
$656$	−2.37843e26			−0.412765
$657$		−	3.24739e25i		−	0.0555474i
$658$		−	1.42294e25i		−	0.0239905i
$659$	7.10095e26			1.18006			0.590031	−	0.807380i	$-0.299115\pi$
$659$	7.10095e26			1.18006			0.590031	+	0.807380i	$0.299115\pi$
$660$	3.54656e26	+	2.09604e26i	0.580952	+	0.343346i
$661$	3.17692e26			0.512971			0.256486	−	0.966548i	$-0.417435\pi$
$661$	3.17692e26			0.512971			0.256486	+	0.966548i	$0.417435\pi$
$662$		−	6.64736e25i		−	0.105803i
$663$		−	8.30845e26i		−	1.30359i
$664$	−6.04860e25			−0.0935531
$665$	−1.47869e26	−	8.73918e25i	−0.225462	−	0.133249i
$666$	1.00897e25			0.0151660
$667$		−	9.18412e26i		−	1.36095i
$668$			2.42264e26i			0.353925i
$669$	−1.24780e27			−1.79720
$670$	9.56528e24	−	1.61847e25i	0.0135827	−	0.0229824i
$671$	−8.02759e26			−1.12388
$672$		−	8.10140e25i		−	0.111828i
$673$			8.95316e26i			1.21852i	0.792970	+	0.609261i	$0.208534\pi$
$673$			8.95316e26i			1.21852i	−0.792970	+	0.609261i	$0.791466\pi$
$674$	−1.83229e26			−0.245882
$675$	6.78735e26	−	3.73651e26i	0.898079	−	0.494402i
$676$	−1.59106e26			−0.207584
$677$		−	1.10734e27i		−	1.42459i	−0.701882	−	0.712293i	$-0.747657\pi$
$677$		−	1.10734e27i		−	1.42459i	0.701882	−	0.712293i	$-0.252343\pi$
$678$		−	1.12442e25i		−	0.0142642i
$679$	1.48464e26			0.185720
$680$	−1.55215e26	+	2.62628e26i	−0.191469	+	0.323971i
$681$	−1.04680e27			−1.27341
$682$			3.95518e24i			0.00474476i
$683$		−	9.58956e26i		−	1.13449i	−0.823548	−	0.567247i	$-0.808009\pi$
$683$		−	9.58956e26i		−	1.13449i	0.823548	−	0.567247i	$-0.191991\pi$
$684$	−4.67733e25			−0.0545714
$685$	6.64813e26	+	3.92909e26i	0.764961	+	0.452097i
$686$	−6.75779e25			−0.0766877
$687$			9.92581e26i			1.11090i
$688$		−	5.48323e24i		−	0.00605267i
$689$	1.44366e27			1.57175
$690$	1.37037e26	+	8.09900e25i	0.147154	+	0.0869691i
$691$	−5.16103e26			−0.546632			−0.273316	−	0.961924i	$-0.588120\pi$
$691$	−5.16103e26			−0.546632			−0.273316	+	0.961924i	$0.588120\pi$
$692$		−	1.36490e26i		−	0.142591i
$693$			9.44372e24i			0.00973144i
$694$	1.99430e26			0.202710
$695$	−8.68058e25	+	1.46878e26i	−0.0870348	+	0.147265i
$696$	−3.68533e26			−0.364493
$697$		−	5.55394e26i		−	0.541863i
$698$			2.68259e26i			0.258183i
$699$	−3.18124e25			−0.0302039
$700$	2.30910e26	−	1.27119e26i	0.216278	−	0.119063i
$701$	3.31094e26			0.305936			0.152968	−	0.988231i	$-0.451117\pi$
$701$	3.31094e26			0.305936			0.152968	+	0.988231i	$0.451117\pi$
$702$		−	1.93847e26i		−	0.176708i
$703$			2.06235e27i			1.85476i
$704$	6.86192e26			0.608843
$705$	−3.42354e26	+	5.79273e26i	−0.299695	+	0.507092i
$706$	−1.63024e25			−0.0140802
$707$			2.38214e26i			0.202994i
$708$		−	6.63808e26i		−	0.558118i
$709$	2.83893e26			0.235513			0.117757	−	0.993042i	$-0.462430\pi$
$709$	2.83893e26			0.235513			0.117757	+	0.993042i	$0.462430\pi$
$710$	−3.12068e26	−	1.84434e26i	−0.255444	−	0.150969i
$711$	−3.43635e24			−0.00277547
$712$			3.61716e26i			0.288276i
$713$		−	6.08548e25i		−	0.0478568i
$714$	6.04244e25			0.0468899
$715$	8.80606e26	+	5.20444e26i	0.674333	+	0.398535i
$716$	−8.76662e26			−0.662458
$717$			1.38496e27i			1.03277i
$718$		−	1.32410e26i		−	0.0974406i
$719$	3.71131e26			0.269527			0.134764	−	0.990878i	$-0.456973\pi$
$719$	3.71131e26			0.269527			0.134764	+	0.990878i	$0.456973\pi$
$720$	3.55800e25	−	6.02024e25i	0.0255004	−	0.0431475i
$721$	4.56701e26			0.323033
$722$			1.58799e25i			0.0110852i
$723$		−	1.61308e27i		−	1.11133i
$724$	−1.59173e26			−0.108231
$725$	−8.70981e26	−	1.58213e27i	−0.584517	−	1.06177i
$726$	1.13556e26			0.0752162
$727$			1.58762e27i			1.03793i	0.854795	+	0.518966i	$0.173683\pi$
$727$			1.58762e27i			1.03793i	−0.854795	+	0.518966i	$0.826317\pi$
$728$		−	1.33552e26i		−	0.0861794i
$729$	−1.64549e27			−1.04806
$730$	1.30154e26	−	2.20224e26i	0.0818259	−	0.138452i
$731$	1.28041e25			0.00794572
$732$		−	2.44739e27i		−	1.49916i
$733$		−	2.28793e27i		−	1.38342i	−0.722174	−	0.691711i	$-0.756857\pi$
$733$		−	2.28793e27i		−	1.38342i	0.722174	−	0.691711i	$-0.243143\pi$
$734$	3.17390e26			0.189444
$735$	1.33003e27	+	7.86055e26i	0.783666	+	0.463152i
$736$	8.77043e26			0.510131
$737$		−	2.11278e26i		−	0.121315i
$738$		−	6.64589e24i		−	0.00376719i
$739$	−3.00171e27			−1.67976			−0.839878	−	0.542775i	$-0.817374\pi$
$739$	−3.00171e27			−1.67976			−0.839878	+	0.542775i	$0.817374\pi$
$740$	−2.72461e27	−	1.61026e27i	−1.50523	−	0.889602i
$741$	2.03216e27			1.10837
$742$			1.04992e26i			0.0565354i
$743$			2.74458e27i			1.45909i	0.683933	+	0.729545i	$0.260268\pi$
$743$			2.74458e27i			1.45909i	−0.683933	+	0.729545i	$0.739732\pi$
$744$	−2.44193e25			−0.0128171
$745$	5.32409e26	−	9.00852e26i	0.275906	−	0.466841i
$746$	−1.61420e26			−0.0825920
$747$			3.23772e25i			0.0163566i
$748$			1.69294e27i			0.844455i
$749$	−6.91138e26			−0.340398
$750$	3.12879e26	+	9.55990e24i	0.152158	+	0.00464913i
$751$	1.30272e27			0.625562			0.312781	−	0.949825i	$-0.398739\pi$
$751$	1.30272e27			0.625562			0.312781	+	0.949825i	$0.398739\pi$
$752$			1.18415e27i			0.561483i
$753$		−	1.64422e27i		−	0.769852i
$754$	−4.51858e26			−0.208917
$755$	−2.18184e27	+	3.69173e27i	−0.996154	+	1.68552i
$756$	−5.61369e26			−0.253100
$757$		−	1.66513e27i		−	0.741372i	−0.928758	−	0.370686i	$-0.879123\pi$
$757$		−	1.66513e27i		−	0.741372i	0.928758	−	0.370686i	$-0.120877\pi$
$758$		−	3.25317e26i		−	0.143037i
$759$	1.78891e27			0.776770
$760$	−6.42360e26	−	3.79639e26i	−0.275454	−	0.162795i
$761$	3.74821e27			1.58734			0.793669	−	0.608349i	$-0.208168\pi$
$761$	3.74821e27			1.58734			0.793669	+	0.608349i	$0.208168\pi$
$762$			6.53368e26i			0.273266i
$763$		−	6.11679e26i		−	0.252662i
$764$	3.34885e27			1.36618
$765$	1.40581e26	+	8.30840e25i	0.0566424	+	0.0334760i
$766$	−6.45520e25			−0.0256884
$767$		−	1.64822e27i		−	0.647829i
$768$			1.91386e27i			0.742984i
$769$	−2.72740e27			−1.04580			−0.522900	−	0.852394i	$-0.675150\pi$
$769$	−2.72740e27			−1.04580			−0.522900	+	0.852394i	$0.675150\pi$
$770$	−3.78500e25	+	6.40434e25i	−0.0143352	+	0.0242556i
$771$	2.00076e27			0.748477
$772$			3.96891e26i			0.146658i
$773$			1.98940e27i			0.726135i	0.931763	+	0.363067i	$0.118271\pi$
$773$			1.98940e27i			0.726135i	−0.931763	+	0.363067i	$0.881729\pi$
$774$	1.53215e23			5.52409e−5
$775$	−5.77119e25	−	1.04833e26i	−0.0205542	−	0.0373365i
$776$	6.44942e26			0.226900
$777$			1.26948e27i			0.441190i
$778$			5.28347e25i			0.0181390i
$779$	1.35843e27			0.460715
$780$	−1.58669e27	+	2.68473e27i	−0.531611	+	0.899501i
$781$	−4.07380e27			−1.34839
$782$			6.54144e26i			0.213899i
$783$			3.84634e27i			1.24254i
$784$	2.71884e27			0.867723
$785$	−1.98087e27	−	1.17071e27i	−0.624589	−	0.369136i
$786$	−4.45146e26			−0.138672
$787$		−	5.25578e27i		−	1.61762i	−0.588068	−	0.808812i	$-0.700111\pi$
$787$		−	5.25578e27i		−	1.61762i	0.588068	−	0.808812i	$-0.299889\pi$
$788$		−	2.62757e27i		−	0.799016i
$789$	−4.26651e27			−1.28186
$790$	−2.33039e25	−	1.37728e25i	−0.00691786	−	0.00408850i
$791$	−8.08526e25			−0.0237147
$792$			4.10245e25i			0.0118892i
$793$		−	6.07684e27i		−	1.74013i
$794$	−5.45631e26			−0.154385
$795$	2.52608e27	−	4.27421e27i	0.706253	−	1.19500i
$796$	2.12701e27			0.587619
$797$			2.82824e27i			0.772081i	0.922482	+	0.386040i	$0.126157\pi$
$797$			2.82824e27i			0.772081i	−0.922482	+	0.386040i	$0.873843\pi$
$798$			1.47792e26i			0.0398678i
$799$	−2.76514e27			−0.737095
$800$	1.51087e27	−	8.31749e26i	0.397989	−	0.219097i
$801$	1.93621e26			0.0504015
$802$		−	1.21833e26i		−	0.0313408i
$803$		−	2.87485e27i		−	0.730833i
$804$	6.44130e26			0.161823
$805$	5.82365e26	−	9.85377e26i	0.144589	−	0.244648i
$806$	−2.99405e25			−0.00734642
$807$			3.16244e27i			0.766873i
$808$			1.03482e27i			0.248004i
$809$	2.81690e26			0.0667207			0.0333603	−	0.999443i	$-0.489379\pi$
$809$	2.81690e26			0.0667207			0.0333603	+	0.999443i	$0.489379\pi$
$810$	−5.42801e26	−	3.20799e26i	−0.127067	−	0.0750974i
$811$	−2.86731e27			−0.663401			−0.331700	−	0.943385i	$-0.607622\pi$
$811$	−2.86731e27			−0.663401			−0.331700	+	0.943385i	$0.607622\pi$
$812$			1.30855e27i			0.299232i
$813$			1.94534e27i			0.439679i
$814$	8.93219e26			0.199538
$815$	5.55671e27	+	3.28405e27i	1.22693	+	0.725125i
$816$	−5.02844e27			−1.09743
$817$			3.13174e25i			0.00675579i
$818$		−	6.90188e26i		−	0.147167i
$819$	−7.14884e25			−0.0150674
$820$	−1.06065e27	+	1.79466e27i	−0.220974	+	0.373895i
$821$	−4.13541e27			−0.851646			−0.425823	−	0.904807i	$-0.640015\pi$
$821$	−4.13541e27			−0.851646			−0.425823	+	0.904807i	$0.640015\pi$
$822$		−	6.64464e26i		−	0.135266i
$823$		−	4.19567e26i		−	0.0844313i	−0.999109	−	0.0422156i	$-0.986558\pi$
$823$		−	4.19567e26i		−	0.0844313i	0.999109	−	0.0422156i	$-0.0134416\pi$
$824$	1.98396e27			0.394660
$825$	3.08173e27	−	1.69653e27i	0.606013	−	0.333617i
$826$	1.19870e26			0.0233023
$827$		−	8.90302e26i		−	0.171094i	−0.996334	−	0.0855470i	$-0.972736\pi$
$827$		−	8.90302e26i		−	0.171094i	0.996334	−	0.0855470i	$-0.0272638\pi$
$828$		−	3.11690e26i		−	0.0592153i
$829$	2.26694e27			0.425768			0.212884	−	0.977078i	$-0.431714\pi$
$829$	2.26694e27			0.425768			0.212884	+	0.977078i	$0.431714\pi$
$830$	−1.29767e26	+	2.19569e26i	−0.0240947	+	0.0407688i
$831$	2.69132e27			0.494032
$832$			5.19444e27i			0.942685i
$833$			6.34885e27i			1.13911i
$834$	1.46801e26			0.0260406
$835$	1.78095e27	+	1.05256e27i	0.312343	+	0.184597i
$836$	−4.14075e27			−0.717992
$837$			2.54862e26i			0.0436932i
$838$			6.99474e26i			0.118564i
$839$	6.09159e27			1.02092			0.510461	−	0.859901i	$-0.329475\pi$
$839$	6.09159e27			1.02092			0.510461	+	0.859901i	$0.329475\pi$
$840$	−3.95405e26	−	2.33687e26i	−0.0655222	−	0.0387241i
$841$	2.86256e27			0.469022
$842$		−	2.29030e26i		−	0.0371046i
$843$			3.26807e27i			0.523516i
$844$	3.04457e27			0.482251
$845$	−6.91265e26	+	1.16964e27i	−0.108269	+	0.183195i
$846$	−3.30879e25			−0.00512450
$847$		−	8.16531e26i		−	0.125049i
$848$		−	8.73732e27i		−	1.32318i
$849$	2.85783e27			0.427970
$850$	6.20361e26	+	1.12688e27i	0.0918681	+	0.166878i
$851$	−1.37432e28			−2.01259
$852$		−	1.24199e28i		−	1.79863i
$853$			1.32363e27i			0.189562i	0.995498	+	0.0947810i	$0.0302151\pi$
$853$			1.32363e27i			0.189562i	−0.995498	+	0.0947810i	$0.969785\pi$
$854$	4.41947e26			0.0625922
$855$	−2.03215e26	+	3.43845e26i	−0.0284628	+	0.0481598i
$856$	−3.00238e27			−0.415876
$857$		−	2.10638e27i		−	0.288548i	−0.989538	−	0.144274i	$-0.953915\pi$
$857$		−	2.10638e27i		−	0.288548i	0.989538	−	0.144274i	$-0.0460847\pi$
$858$		−	8.80143e26i		−	0.119241i
$859$	−7.62793e27			−1.02205			−0.511025	−	0.859566i	$-0.670734\pi$
$859$	−7.62793e27			−1.02205			−0.511025	+	0.859566i	$0.670734\pi$
$860$	−4.13741e25	−	2.44523e25i	−0.00548268	−	0.00324030i
$861$	8.36184e26			0.109590
$862$		−	5.99060e26i		−	0.0776518i
$863$			2.70442e27i			0.346714i	0.984859	+	0.173357i	$0.0554615\pi$
$863$			2.70442e27i			0.346714i	−0.984859	+	0.173357i	$0.944539\pi$
$864$	−3.67309e27			−0.465748
$865$	−1.00338e27	−	5.93005e26i	−0.125838	−	0.0743713i
$866$	9.41384e26			0.116774
$867$		−	3.81499e27i		−	0.468072i
$868$			8.67058e25i			0.0105223i
$869$	−3.04214e26			−0.0365167
$870$	−7.90650e26	+	1.33780e27i	−0.0938753	+	0.158840i
$871$	1.59937e27			0.187835
$872$		−	2.65720e27i		−	0.308686i
$873$		−	3.45227e26i		−	0.0396707i
$874$	−1.59997e27			−0.181866
$875$	6.87412e25	−	2.24978e27i	0.00772931	−	0.252967i
$876$	8.76464e27			0.974867
$877$			9.43005e27i			1.03757i	0.854905	+	0.518785i	$0.173616\pi$
$877$			9.43005e27i			1.03757i	−0.854905	+	0.518785i	$0.826384\pi$
$878$		−	9.14192e26i		−	0.0995037i
$879$	1.87602e27			0.201997
$880$	3.14983e27	−	5.32961e27i	0.335507	−	0.567688i
$881$	1.59431e28			1.67998			0.839988	−	0.542605i	$-0.182562\pi$
$881$	1.59431e28			1.67998			0.839988	+	0.542605i	$0.182562\pi$
$882$			7.59709e25i			0.00791946i
$883$		−	3.02275e27i		−	0.311728i	−0.987779	−	0.155864i	$-0.950184\pi$
$883$		−	3.02275e27i		−	0.311728i	0.987779	−	0.155864i	$-0.0498162\pi$
$884$	−1.28155e28			−1.30749
$885$	−4.87985e27	−	2.88403e27i	−0.492545	−	0.291097i
$886$	−2.36050e26			−0.0235713
$887$		−	9.50319e26i		−	0.0938847i	−0.998898	−	0.0469424i	$-0.985052\pi$
$887$		−	9.50319e26i		−	0.0938847i	0.998898	−	0.0469424i	$-0.0149477\pi$
$888$			5.51475e27i			0.539017i
$889$	4.69809e27			0.454312
$890$	1.31306e27	+	7.76025e26i	0.125626	+	0.0742456i
$891$	−7.08583e27			−0.670737
$892$			1.92468e28i			1.80257i
$893$		−	6.76325e27i		−	0.626710i
$894$	−9.00378e26			−0.0825503
$895$	−3.80881e27	+	6.44461e27i	−0.345518	+	0.584626i
$896$	−1.65876e27			−0.148888
$897$			1.35420e28i			1.20269i
$898$			7.87416e26i			0.0691957i
$899$	5.94083e26			0.0516571
$900$	−2.95593e26	−	5.36942e26i	−0.0254325	−	0.0461981i
$901$	2.04028e28			1.73702
$902$		−	5.88348e26i		−	0.0495647i
$903$			1.92774e25i			0.00160700i
$904$	−3.51232e26			−0.0289730
$905$	−6.91555e26	+	1.17013e27i	−0.0564502	+	0.0955153i
$906$	3.68979e27			0.298046
$907$		−	9.98796e27i		−	0.798376i	−0.916869	−	0.399188i	$-0.869292\pi$
$907$		−	9.98796e27i		−	0.798376i	0.916869	−	0.399188i	$-0.130708\pi$
$908$			1.61465e28i			1.27721i
$909$	5.53925e26			0.0433605
$910$	−4.84805e26	−	2.86523e26i	−0.0375555	−	0.0221955i
$911$	−2.89381e27			−0.221843			−0.110921	−	0.993829i	$-0.535380\pi$
$911$	−2.89381e27			−0.221843			−0.110921	+	0.993829i	$0.535380\pi$
$912$		−	1.22990e28i		−	0.933083i
$913$			2.86629e27i			0.215203i
$914$	9.87729e26			0.0733921
$915$	−1.79915e28	−	1.06331e28i	−1.32302	−	0.781916i
$916$	1.53102e28			1.11423
$917$			3.20086e27i			0.230546i
$918$		−	2.73958e27i		−	0.195289i
$919$	1.36611e28			0.963800			0.481900	−	0.876226i	$-0.339947\pi$
$919$	1.36611e28			0.963800			0.481900	+	0.876226i	$0.339947\pi$
$920$	2.52985e27	−	4.28058e27i	0.176649	−	0.298895i
$921$	−2.90172e26			−0.0200534
$922$		−	3.34122e27i		−	0.228539i
$923$		−	3.08384e28i		−	2.08774i
$924$	−2.54884e27			−0.170789
$925$	−2.36751e28	+	1.30334e28i	−1.57017	+	0.864393i
$926$	−1.53795e27			−0.100957
$927$		−	1.06198e27i		−	0.0690015i
$928$			8.56197e27i			0.550640i
$929$	−2.48929e28			−1.58462			−0.792311	−	0.610118i	$-0.791122\pi$
$929$	−2.48929e28			−1.58462			−0.792311	+	0.610118i	$0.791122\pi$
$930$	−5.23892e25	+	8.86439e25i	−0.00330106	+	0.00558549i
$931$	−1.55286e28			−0.968525
$932$			4.90694e26i			0.0302942i
$933$		−	9.96471e27i		−	0.608960i
$934$	2.76026e27			0.166976
$935$	1.24453e28	+	7.35527e27i	0.745240	+	0.440442i
$936$	−3.10553e26			−0.0184084
$937$		−	1.77582e28i		−	1.04201i	−0.853553	−	0.521006i	$-0.825557\pi$
$937$		−	1.77582e28i		−	1.04201i	0.853553	−	0.521006i	$-0.174443\pi$
$938$			1.16316e26i			0.00675637i
$939$	1.21141e27			0.0696578
$940$	8.93507e27	+	5.28068e27i	0.508608	+	0.300591i
$941$	1.32109e28			0.744439			0.372220	−	0.928145i	$-0.378597\pi$
$941$	1.32109e28			0.744439			0.372220	+	0.928145i	$0.378597\pi$
$942$			1.97983e27i			0.110444i
$943$			9.05238e27i			0.499921i
$944$	−9.97539e27			−0.545376
$945$	−2.43897e27	+	4.12680e27i	−0.132009	+	0.223363i
$946$	1.35638e25			0.000726801
$947$		−	1.92300e28i		−	1.02013i	−0.860137	−	0.510063i	$-0.829622\pi$
$947$		−	1.92300e28i		−	1.02013i	0.860137	−	0.510063i	$-0.170378\pi$
$948$		−	9.27465e26i		−	0.0487100i
$949$	2.17625e28			1.13156
$950$	−2.75623e27	+	1.51734e27i	−0.141887	+	0.0781102i
$951$	−1.47132e28			−0.749879
$952$		−	1.88745e27i		−	0.0952413i
$953$			2.59851e28i			1.29820i	0.760704	+	0.649099i	$0.224854\pi$
$953$			2.59851e28i			1.29820i	−0.760704	+	0.649099i	$0.775146\pi$
$954$	2.44142e26			0.0120763
$955$	1.45496e28	−	2.46184e28i	0.712558	−	1.20567i
$956$	2.13625e28			1.03586
$957$			1.74639e28i			0.838453i
$958$		−	1.94923e25i		−	0.000926597i
$959$	−4.77788e27			−0.224884
$960$	1.53790e28	+	9.08910e27i	0.716724	+	0.423589i
$961$	−2.16313e28			−0.998184
$962$			6.76162e27i			0.308950i
$963$			1.60712e27i			0.0727109i
$964$	−2.48812e28			−1.11465
$965$	2.91766e27	+	1.72436e27i	0.129427	+	0.0764924i
$966$	−9.84859e26			−0.0432605
$967$			2.59918e28i			1.13054i	0.824907	+	0.565269i	$0.191228\pi$
$967$			2.59918e28i			1.13054i	−0.824907	+	0.565269i	$0.808772\pi$
$968$		−	3.54710e27i		−	0.152777i
$969$	2.87199e28			1.22492
$970$	1.38366e27	−	2.34119e27i	0.0584382	−	0.0988791i
$971$	1.02365e28			0.428122			0.214061	−	0.976820i	$-0.431331\pi$
$971$	1.02365e28			0.428122			0.214061	+	0.976820i	$0.431331\pi$
$972$			2.54379e27i			0.105354i
$973$		−	1.05558e27i		−	0.0432932i
$974$	−6.17364e27			−0.250744
$975$	1.28426e28	+	2.33285e28i	0.516546	+	0.938303i
$976$	−3.67783e28			−1.46493
$977$			5.73585e27i			0.226256i	0.993580	+	0.113128i	$0.0360870\pi$
$977$			5.73585e27i			0.226256i	−0.993580	+	0.113128i	$0.963913\pi$
$978$		−	5.55379e27i		−	0.216955i
$979$	1.71409e28			0.663129
$980$	1.21246e28	−	2.05152e28i	0.464536	−	0.786009i
$981$	−1.42235e27			−0.0539700
$982$			7.06754e27i			0.265588i
$983$			2.73216e28i			1.01683i	0.861113	+	0.508414i	$0.169768\pi$
$983$			2.73216e28i			1.01683i	−0.861113	+	0.508414i	$0.830232\pi$
$984$	3.63247e27			0.133890
$985$	−1.93161e28	−	1.14159e28i	−0.705140	−	0.416742i
$986$	−6.38596e27			−0.230885
$987$		−	4.16312e27i		−	0.149075i
$988$		−	3.13453e28i		−	1.11168i
$989$	−2.08694e26			−0.00733069
$990$	1.48922e26	+	8.80138e25i	0.00518112	+	0.00306208i
$991$	−3.38695e28			−1.16710			−0.583551	−	0.812077i	$-0.698337\pi$
$991$	−3.38695e28			−1.16710			−0.583551	+	0.812077i	$0.698337\pi$
$992$			5.67324e26i			0.0193629i
$993$		−	1.94483e28i		−	0.657453i
$994$	2.24277e27			0.0750955
$995$	9.24117e27	−	1.56363e28i	0.306484	−	0.518580i
$996$	−8.73854e27			−0.287062
$997$		−	4.64239e28i		−	1.51056i	−0.655403	−	0.755279i	$-0.727501\pi$
$997$		−	4.64239e28i		−	1.51056i	0.655403	−	0.755279i	$-0.272499\pi$
$998$			1.07753e27i			0.0347286i
$999$	5.75569e28			1.83749

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5.20.b.a.4.5	yes	8
3.2	odd	2		45.20.b.b.19.4		8
4.3	odd	2		80.20.c.a.49.3		8
5.2	odd	4		25.20.a.f.1.4		8
5.3	odd	4		25.20.a.f.1.5		8
5.4	even	2	inner	5.20.b.a.4.4	✓	8
15.14	odd	2		45.20.b.b.19.5		8
20.19	odd	2		80.20.c.a.49.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.20.b.a.4.4	✓	8	5.4	even	2	inner
5.20.b.a.4.5	yes	8	1.1	even	1	trivial
25.20.a.f.1.4		8	5.2	odd	4
25.20.a.f.1.5		8	5.3	odd	4
45.20.b.b.19.4		8	3.2	odd	2
45.20.b.b.19.5		8	15.14	odd	2
80.20.c.a.49.3		8	4.3	odd	2
80.20.c.a.49.6		8	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5 \)
Weight:	\( k \)	\(=\)	\( 20 \)
Character orbit:	\([\chi]\)	\(=\)	5.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+113.331i q^{2} +33157.6i q^{3} +511444. q^{4} +(2.22206e6 - 3.75978e6i) q^{5} -3.75780e6 q^{6} +2.70208e7i q^{7} +1.17381e8i q^{8} +6.28322e7 q^{9} +O(q^{10})\)	\(q+113.331i q^{2} +33157.6i q^{3} +511444. q^{4} +(2.22206e6 - 3.75978e6i) q^{5} -3.75780e6 q^{6} +2.70208e7i q^{7} +1.17381e8i q^{8} +6.28322e7 q^{9} +(4.26102e8 + 2.51829e8i) q^{10} +5.56241e9 q^{11} +1.69583e10i q^{12} +4.21072e10i q^{13} -3.06230e9 q^{14} +(1.24665e11 + 7.36781e10i) q^{15} +2.54841e11 q^{16} +5.95087e11i q^{17} +7.12086e9i q^{18} -1.45552e12 q^{19} +(1.13646e12 - 1.92292e12i) q^{20} -8.95945e11 q^{21} +6.30396e11i q^{22} -9.69934e12i q^{23} -3.89208e12 q^{24} +(-9.19842e12 - 1.67089e13i) q^{25} -4.77207e12 q^{26} +4.06212e13i q^{27} +1.38196e13i q^{28} +9.46880e13 q^{29} +(-8.35005e12 + 1.41285e13i) q^{30} +6.27411e12 q^{31} +9.04230e13i q^{32} +1.84437e14i q^{33} -6.74421e13 q^{34} +(1.01592e14 + 6.00417e13i) q^{35} +3.21351e13 q^{36} -1.41692e15i q^{37} -1.64956e14i q^{38} -1.39617e15 q^{39} +(4.41327e14 + 2.60827e14i) q^{40} -9.33298e14 q^{41} -1.01539e14i q^{42} -2.15163e13i q^{43} +2.84486e15 q^{44} +(1.39617e14 - 2.36235e14i) q^{45} +1.09924e15 q^{46} +4.64662e15i q^{47} +8.44993e15i q^{48} +1.06688e16 q^{49} +(1.89364e15 - 1.04247e15i) q^{50} -1.97317e16 q^{51} +2.15355e16i q^{52} -3.42854e16i q^{53} -4.60366e15 q^{54} +(1.23600e16 - 2.09135e16i) q^{55} -3.17173e15 q^{56} -4.82616e16i q^{57} +1.07311e16i q^{58} -3.91436e16 q^{59} +(6.37594e16 + 3.76822e16i) q^{60} -1.44319e17 q^{61} +7.11054e14i q^{62} +1.69777e15i q^{63} +1.23362e17 q^{64} +(1.58314e17 + 9.35644e16i) q^{65} -2.09025e16 q^{66} -3.79832e16i q^{67} +3.04354e17i q^{68} +3.21607e17 q^{69} +(-6.80461e15 + 1.15136e16i) q^{70} -7.32379e17 q^{71} +7.37530e15i q^{72} -5.16836e17i q^{73} +1.60581e17 q^{74} +(5.54027e17 - 3.04998e17i) q^{75} -7.44417e17 q^{76} +1.50301e17i q^{77} -1.58230e17i q^{78} -5.46910e16 q^{79} +(5.66271e17 - 9.58146e17i) q^{80} -1.27388e18 q^{81} -1.05772e17i q^{82} +5.15297e17i q^{83} -4.58226e17 q^{84} +(2.23740e18 + 1.32232e18i) q^{85} +2.43847e15 q^{86} +3.13963e18i q^{87} +6.52922e17i q^{88} +3.08156e18 q^{89} +(2.67729e16 + 1.58230e16i) q^{90} -1.13777e18 q^{91} -4.96067e18i q^{92} +2.08035e17i q^{93} -5.26608e17 q^{94} +(-3.23425e18 + 5.47244e18i) q^{95} -2.99821e18 q^{96} -5.49443e18i q^{97} +1.20911e18i q^{98} +3.49498e17 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 1620744 q^{4} + 147000 q^{5} + 3365736 q^{6} + 345358584 q^{9}+O(q^{10}) \) 8 * q - 1620744 * q^4 + 147000 * q^5 + 3365736 * q^6 + 345358584 * q^9	\( 8 q - 1620744 q^{4} + 147000 q^{5} + 3365736 q^{6} + 345358584 q^{9} - 610691000 q^{10} - 3379575264 q^{11} + 177250591032 q^{14} - 242324628000 q^{15} - 312730276832 q^{16} + 4547188380640 q^{19} - 4180429431000 q^{20} - 2983154334624 q^{21} - 6176642779680 q^{24} + 17715709625000 q^{25} + 15909228128496 q^{26} - 188222300345040 q^{29} + 148501482939000 q^{30} + 72115006686976 q^{31} + 378440221985792 q^{34} - 299115755916000 q^{35} - 964020253238712 q^{36} + 30\!\cdots\!28 q^{39}+ \cdots - 16\!\cdots\!72 q^{99}+O(q^{100}) \) 8 * q - 1620744 * q^4 + 147000 * q^5 + 3365736 * q^6 + 345358584 * q^9 - 610691000 * q^10 - 3379575264 * q^11 + 177250591032 * q^14 - 242324628000 * q^15 - 312730276832 * q^16 + 4547188380640 * q^19 - 4180429431000 * q^20 - 2983154334624 * q^21 - 6176642779680 * q^24 + 17715709625000 * q^25 + 15909228128496 * q^26 - 188222300345040 * q^29 + 148501482939000 * q^30 + 72115006686976 * q^31 + 378440221985792 * q^34 - 299115755916000 * q^35 - 964020253238712 * q^36 + 3011653788049728 * q^39 - 213611427940000 * q^40 - 3314269428857904 * q^41 + 489119304867552 * q^44 + 10198157211261000 * q^45 - 13716870314831944 * q^46 - 11129258011816744 * q^49 + 50375043666750000 * q^50 - 46867071411404544 * q^51 + 86068650834185040 * q^54 + 69821226326124000 * q^55 - 303831778832907360 * q^56 + 259648032778012320 * q^59 + 318110620655844000 * q^60 - 582893562753226064 * q^61 + 525415747276109696 * q^64 + 747545324689032000 * q^65 - 1871067587999976288 * q^66 + 302144302523815008 * q^69 + 2638361065839633000 * q^70 - 2617007600602926144 * q^71 + 1847591013622215312 * q^74 + 2877932428209000000 * q^75 - 7494494299858276320 * q^76 + 3935092091109861760 * q^79 + 5465139505859292000 * q^80 - 8328177293660704152 * q^81 + 7683875738824129632 * q^84 + 3752387009219584000 * q^85 - 14358563079146095224 * q^86 + 2120141101636921680 * q^89 + 7589596726146267000 * q^90 - 3358638219973817664 * q^91 - 10251771654977709928 * q^94 + 6106065789884820000 * q^95 + 15670161523483353216 * q^96 - 16256944625235082272 * q^99

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