LMFDB - Embedded newform 10.18.b.a.9.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [10,18,Mod(9,10)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(10, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 18, names="a")

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

chi := DirichletCharacter("10.9");

S:= CuspForms(chi, 18);

N := Newforms(S);

Level:	$ N $	$=$	$ 10 = 2 \cdot 5 $
Weight:	$ k $	$=$	$ 18 $
Character orbit:	$[\chi]$	$=$	10.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:	$18.3222087345$
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} + 556201x^{6} + 76870744104x^{4} + 1868329791349729x^{2} + 78074963590050625 $ x^8 + 556201x^6 + 76870744104x^4 + 1868329791349729*x^2 + 78074963590050625
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{40}\cdot 3^{4}\cdot 5^{11} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			9.3
Root			$414.151i$ of defining polynomial
Character	$\chi$	$=$	10.9
Dual form			10.18.b.a.9.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-256.000i q^{2} +5436.73i q^{3} -65536.0 q^{4} +(-679890. - 548351. i) q^{5} +1.39180e6 q^{6} -7.62753e6i q^{7} +1.67772e7i q^{8} +9.95821e7 q^{9} +O(q^{10})$	\(q-256.000i q^{2} +5436.73i q^{3} -65536.0 q^{4} +(-679890. - 548351. i) q^{5} +1.39180e6 q^{6} -7.62753e6i q^{7} +1.67772e7i q^{8} +9.95821e7 q^{9} +(-1.40378e8 + 1.74052e8i) q^{10} -8.00662e8 q^{11} -3.56302e8i q^{12} +4.04052e9i q^{13} -1.95265e9 q^{14} +(2.98124e9 - 3.69638e9i) q^{15} +4.29497e9 q^{16} +3.05998e10i q^{17} -2.54930e10i q^{18} +1.17425e11 q^{19} +(4.45573e10 + 3.59368e10i) q^{20} +4.14689e10 q^{21} +2.04970e11i q^{22} +2.47340e11i q^{23} -9.12133e10 q^{24} +(1.61561e11 + 7.45637e11i) q^{25} +1.03437e12 q^{26} +1.24350e12i q^{27} +4.99878e11i q^{28} +1.59924e12 q^{29} +(-9.46273e11 - 7.63197e11i) q^{30} -2.34406e12 q^{31} -1.09951e12i q^{32} -4.35299e12i q^{33} +7.83354e12 q^{34} +(-4.18257e12 + 5.18588e12i) q^{35} -6.52621e12 q^{36} -1.55775e13i q^{37} -3.00607e13i q^{38} -2.19672e13 q^{39} +(9.19981e12 - 1.14067e13i) q^{40} -7.85358e13 q^{41} -1.06160e13i q^{42} +4.97304e13i q^{43} +5.24722e13 q^{44} +(-6.77049e13 - 5.46060e13i) q^{45} +6.33191e13 q^{46} +1.34616e14i q^{47} +2.33506e13i q^{48} +1.74451e14 q^{49} +(1.90883e14 - 4.13596e13i) q^{50} -1.66363e14 q^{51} -2.64800e14i q^{52} +4.51163e14i q^{53} +3.18336e14 q^{54} +(5.44362e14 + 4.39044e14i) q^{55} +1.27969e14 q^{56} +6.38406e14i q^{57} -4.09404e14i q^{58} -4.74588e14 q^{59} +(-1.95379e14 + 2.42246e14i) q^{60} -2.35385e15 q^{61} +6.00080e14i q^{62} -7.59565e14i q^{63} -2.81475e14 q^{64} +(2.21563e15 - 2.74711e15i) q^{65} -1.11436e15 q^{66} -2.78074e15i q^{67} -2.00539e15i q^{68} -1.34472e15 q^{69} +(1.32759e15 + 1.07074e15i) q^{70} +2.28139e15 q^{71} +1.67071e15i q^{72} -5.13289e15i q^{73} -3.98785e15 q^{74} +(-4.05383e15 + 8.78364e14i) q^{75} -7.69554e15 q^{76} +6.10708e15i q^{77} +5.62362e15i q^{78} +1.83537e16 q^{79} +(-2.92010e15 - 2.35515e15i) q^{80} +6.09946e15 q^{81} +2.01052e16i q^{82} +2.16368e16i q^{83} -2.71770e15 q^{84} +(1.67794e16 - 2.08045e16i) q^{85} +1.27310e16 q^{86} +8.69462e15i q^{87} -1.34329e16i q^{88} -6.72250e15 q^{89} +(-1.39791e16 + 1.73324e16i) q^{90} +3.08192e16 q^{91} -1.62097e16i q^{92} -1.27440e16i q^{93} +3.44617e16 q^{94} +(-7.98358e16 - 6.43899e16i) q^{95} +5.97775e15 q^{96} +9.88520e16i q^{97} -4.46595e16i q^{98} -7.97316e16 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 524288 q^{4} - 1225560 q^{5} - 974848 q^{6} - 363182504 q^{9}+O(q^{10}) $ 8 * q - 524288 * q^4 - 1225560 * q^5 - 974848 * q^6 - 363182504 * q^9	$ 8 q - 524288 q^{4} - 1225560 q^{5} - 974848 q^{6} - 363182504 q^{9} - 140779520 q^{10} + 146232096 q^{11} + 14260494336 q^{14} - 39815002720 q^{15} + 34359738368 q^{16} - 54264178080 q^{19} + 80318300160 q^{20} + 515442333056 q^{21} + 63887638528 q^{24} + 1013225778600 q^{25} - 2693383569408 q^{26} + 1660243083120 q^{29} + 4536489205760 q^{30} - 6055476993664 q^{31} + 14158246445056 q^{34} - 8725233780960 q^{35} + 23801528582144 q^{36} - 60047234232768 q^{39} + 9226126622720 q^{40} - 219921829971984 q^{41} - 9583466643456 q^{44} + 503517880841080 q^{45} - 136753191067648 q^{46} + 797208944041464 q^{49} + 535409908531200 q^{50} - 26\!\cdots\!24 q^{51}+ \cdots - 23\!\cdots\!48 q^{99}+O(q^{100}) $ 8 * q - 524288 * q^4 - 1225560 * q^5 - 974848 * q^6 - 363182504 * q^9 - 140779520 * q^10 + 146232096 * q^11 + 14260494336 * q^14 - 39815002720 * q^15 + 34359738368 * q^16 - 54264178080 * q^19 + 80318300160 * q^20 + 515442333056 * q^21 + 63887638528 * q^24 + 1013225778600 * q^25 - 2693383569408 * q^26 + 1660243083120 * q^29 + 4536489205760 * q^30 - 6055476993664 * q^31 + 14158246445056 * q^34 - 8725233780960 * q^35 + 23801528582144 * q^36 - 60047234232768 * q^39 + 9226126622720 * q^40 - 219921829971984 * q^41 - 9583466643456 * q^44 + 503517880841080 * q^45 - 136753191067648 * q^46 + 797208944041464 * q^49 + 535409908531200 * q^50 - 2629370351530624 * q^51 + 1870704820633600 * q^54 + 1370380127176480 * q^55 - 934575756804096 * q^56 - 155410557761760 * q^59 + 2609316018257920 * q^60 - 6350444068968944 * q^61 - 2251799813685248 * q^64 + 10508144084782080 * q^65 - 10180424127512576 * q^66 + 20821360231259392 * q^69 - 1051117677240320 * q^70 - 33149026543733184 * q^71 + 6017648345726976 * q^74 + 17427320439323200 * q^75 + 3556257174650880 * q^76 - 19354668791708160 * q^79 - 5263740119285760 * q^80 + 54432374624835848 * q^81 - 33780028739158016 * q^84 - 71530234876895360 * q^85 + 29616028444434432 * q^86 - 43569230299558320 * q^89 - 33153706682736640 * q^90 + 333420307889971776 * q^91 - 11440296922292224 * q^94 - 131478039235197600 * q^95 - 4186940278571008 * q^96 - 236681336419250848 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/10\mathbb{Z}\right)^\times$.

$n$	$7$
$\chi(n)$	$-1$

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 $

$2$		−	256.000i		−	0.707107i
$2$		−	256.000i		−	0.707107i
$3$			5436.73i			0.478418i	0.970968	+	0.239209i	$0.0768881\pi$
$3$			5436.73i			0.478418i	−0.970968	+	0.239209i	$0.923112\pi$
$4$	−65536.0			−0.500000
$5$	−679890.	−	548351.i	−0.778383	−	0.627789i
$5$	−679890.	−	548351.i	−0.778383	−	0.627789i
$6$	1.39180e6			0.338293
$7$		−	7.62753e6i		−	0.500093i	−0.968234	−	0.250046i	$-0.919554\pi$
$7$		−	7.62753e6i		−	0.500093i	0.968234	−	0.250046i	$-0.0804459\pi$
$8$			1.67772e7i			0.353553i
$9$	9.95821e7			0.771116
$10$	−1.40378e8	+	1.74052e8i	−0.443914	+	0.550400i
$11$	−8.00662e8			−1.12619			−0.563095	−	0.826392i	$-0.690389\pi$
$11$	−8.00662e8			−1.12619			−0.563095	+	0.826392i	$0.690389\pi$
$12$		−	3.56302e8i		−	0.239209i
$13$			4.04052e9i			1.37379i	0.726759	+	0.686893i	$0.241026\pi$
$13$			4.04052e9i			1.37379i	−0.726759	+	0.686893i	$0.758974\pi$
$14$	−1.95265e9			−0.353619
$15$	2.98124e9	−	3.69638e9i	0.300346	−	0.372392i
$16$	4.29497e9			0.250000
$17$			3.05998e10i			1.06390i	0.846775	+	0.531952i	$0.178541\pi$
$17$			3.05998e10i			1.06390i	−0.846775	+	0.531952i	$0.821459\pi$
$18$		−	2.54930e10i		−	0.545262i
$19$	1.17425e11			1.58618			0.793092	−	0.609102i	$-0.208470\pi$
$19$	1.17425e11			1.58618			0.793092	+	0.609102i	$0.208470\pi$
$20$	4.45573e10	+	3.59368e10i	0.389192	+	0.313895i
$21$	4.14689e10			0.239253
$22$			2.04970e11i			0.796336i
$23$			2.47340e11i			0.658580i	0.944229	+	0.329290i	$0.106809\pi$
$23$			2.47340e11i			0.658580i	−0.944229	+	0.329290i	$0.893191\pi$
$24$	−9.12133e10			−0.169146
$25$	1.61561e11	+	7.45637e11i	0.211761	+	0.977321i
$26$	1.03437e12			0.971413
$27$			1.24350e12i			0.847334i
$28$			4.99878e11i			0.250046i
$29$	1.59924e12			0.593649			0.296824	−	0.954932i	$-0.404072\pi$
$29$	1.59924e12			0.593649			0.296824	+	0.954932i	$0.404072\pi$
$30$	−9.46273e11	−	7.63197e11i	−0.263321	−	0.212376i
$31$	−2.34406e12			−0.493622			−0.246811	−	0.969064i	$-0.579383\pi$
$31$	−2.34406e12			−0.493622			−0.246811	+	0.969064i	$0.579383\pi$
$32$		−	1.09951e12i		−	0.176777i
$33$		−	4.35299e12i		−	0.538789i
$34$	7.83354e12			0.752294
$35$	−4.18257e12	+	5.18588e12i	−0.313953	+	0.389264i
$36$	−6.52621e12			−0.385558
$37$		−	1.55775e13i		−	0.729094i	−0.931185	−	0.364547i	$-0.881224\pi$
$37$		−	1.55775e13i		−	0.729094i	0.931185	−	0.364547i	$-0.118776\pi$
$38$		−	3.00607e13i		−	1.12160i
$39$	−2.19672e13			−0.657244
$40$	9.19981e12	−	1.14067e13i	0.221957	−	0.275200i
$41$	−7.85358e13			−1.53605			−0.768024	−	0.640421i	$-0.778760\pi$
$41$	−7.85358e13			−1.53605			−0.768024	+	0.640421i	$0.778760\pi$
$42$		−	1.06160e13i		−	0.169178i
$43$			4.97304e13i			0.648843i	0.945913	+	0.324422i	$0.105170\pi$
$43$			4.97304e13i			0.648843i	−0.945913	+	0.324422i	$0.894830\pi$
$44$	5.24722e13			0.563095
$45$	−6.77049e13	−	5.46060e13i	−0.600224	−	0.484099i
$46$	6.33191e13			0.465686
$47$			1.34616e14i			0.824640i	0.911039	+	0.412320i	$0.135281\pi$
$47$			1.34616e14i			0.824640i	−0.911039	+	0.412320i	$0.864719\pi$
$48$			2.33506e13i			0.119604i
$49$	1.74451e14			0.749907
$50$	1.90883e14	−	4.13596e13i	0.691071	−	0.149738i
$51$	−1.66363e14			−0.508991
$52$		−	2.64800e14i		−	0.686893i
$53$			4.51163e14i			0.995378i	0.867355	+	0.497689i	$0.165818\pi$
$53$			4.51163e14i			0.995378i	−0.867355	+	0.497689i	$0.834182\pi$
$54$	3.18336e14			0.599155
$55$	5.44362e14	+	4.39044e14i	0.876607	+	0.707010i
$56$	1.27969e14			0.176810
$57$			6.38406e14i			0.758858i
$58$		−	4.09404e14i		−	0.419773i
$59$	−4.74588e14			−0.420799			−0.210399	−	0.977615i	$-0.567476\pi$
$59$	−4.74588e14			−0.420799			−0.210399	+	0.977615i	$0.567476\pi$
$60$	−1.95379e14	+	2.42246e14i	−0.150173	+	0.186196i
$61$	−2.35385e15			−1.57208			−0.786042	−	0.618174i	$-0.787873\pi$
$61$	−2.35385e15			−1.57208			−0.786042	+	0.618174i	$0.787873\pi$
$62$			6.00080e14i			0.349044i
$63$		−	7.59565e14i		−	0.385630i
$64$	−2.81475e14			−0.125000
$65$	2.21563e15	−	2.74711e15i	0.862448	−	1.06933i
$66$	−1.11436e15			−0.380982
$67$		−	2.78074e15i		−	0.836613i	−0.908306	−	0.418306i	$-0.862624\pi$
$67$		−	2.78074e15i		−	0.836613i	0.908306	−	0.418306i	$-0.137376\pi$
$68$		−	2.00539e15i		−	0.531952i
$69$	−1.34472e15			−0.315076
$70$	1.32759e15	+	1.07074e15i	0.275251	+	0.221998i
$71$	2.28139e15			0.419279			0.209640	−	0.977779i	$-0.432771\pi$
$71$	2.28139e15			0.419279			0.209640	+	0.977779i	$0.432771\pi$
$72$			1.67071e15i			0.272631i
$73$		−	5.13289e15i		−	0.744934i	−0.928045	−	0.372467i	$-0.878512\pi$
$73$		−	5.13289e15i		−	0.744934i	0.928045	−	0.372467i	$-0.121488\pi$
$74$	−3.98785e15			−0.515547
$75$	−4.05383e15	+	8.78364e14i	−0.467568	+	0.101310i
$76$	−7.69554e15			−0.793092
$77$			6.10708e15i			0.563199i
$78$			5.62362e15i			0.464742i
$79$	1.83537e16			1.36111			0.680556	−	0.732696i	$-0.261738\pi$
$79$	1.83537e16			1.36111			0.680556	+	0.732696i	$0.261738\pi$
$80$	−2.92010e15	−	2.35515e15i	−0.194596	−	0.156947i
$81$	6.09946e15			0.365737
$82$			2.01052e16i			1.08615i
$83$			2.16368e16i			1.05446i	0.849723	+	0.527230i	$0.176769\pi$
$83$			2.16368e16i			1.05446i	−0.849723	+	0.527230i	$0.823231\pi$
$84$	−2.71770e15			−0.119627
$85$	1.67794e16	−	2.08045e16i	0.667907	−	0.828125i
$86$	1.27310e16			0.458802
$87$			8.69462e15i			0.284012i
$88$		−	1.34329e16i		−	0.398168i
$89$	−6.72250e15			−0.181015			−0.0905077	−	0.995896i	$-0.528849\pi$
$89$	−6.72250e15			−0.181015			−0.0905077	+	0.995896i	$0.528849\pi$
$90$	−1.39791e16	+	1.73324e16i	−0.342309	+	0.424423i
$91$	3.08192e16			0.687021
$92$		−	1.62097e16i		−	0.329290i
$93$		−	1.27440e16i		−	0.236158i
$94$	3.44617e16			0.583109
$95$	−7.98358e16	−	6.43899e16i	−1.23466	−	0.995789i
$96$	5.97775e15			0.0845731
$97$			9.88520e16i			1.28064i	0.768110	+	0.640318i	$0.221197\pi$
$97$			9.88520e16i			1.28064i	−0.768110	+	0.640318i	$0.778803\pi$
$98$		−	4.46595e16i		−	0.530264i
$99$	−7.97316e16			−0.868423
$100$	−1.05881e16	−	4.88661e16i	−0.105881	−	0.488661i
$101$	−4.97720e16			−0.457355			−0.228678	−	0.973502i	$-0.573440\pi$
$101$	−4.97720e16			−0.457355			−0.228678	+	0.973502i	$0.573440\pi$
$102$			4.25889e16i			0.359911i
$103$		−	1.98696e17i		−	1.54551i	−0.634705	−	0.772755i	$-0.718878\pi$
$103$		−	1.98696e17i		−	1.54551i	0.634705	−	0.772755i	$-0.281122\pi$
$104$	−6.77887e16			−0.485707
$105$	−2.81943e16	−	2.27395e16i	−0.186231	−	0.150201i
$106$	1.15498e17			0.703839
$107$			2.70729e17i			1.52325i	0.648016	+	0.761627i	$0.275599\pi$
$107$			2.70729e17i			1.52325i	−0.648016	+	0.761627i	$0.724401\pi$
$108$		−	8.14941e16i		−	0.423667i
$109$	−1.26486e17			−0.608019			−0.304010	−	0.952669i	$-0.598325\pi$
$109$	−1.26486e17			−0.608019			−0.304010	+	0.952669i	$0.598325\pi$
$110$	1.12395e17	−	1.39357e17i	0.499931	−	0.619855i
$111$	8.46908e16			0.348812
$112$		−	3.27600e16i		−	0.125023i
$113$			2.28445e17i			0.808379i	0.914675	+	0.404190i	$0.132446\pi$
$113$			2.28445e17i			0.808379i	−0.914675	+	0.404190i	$0.867554\pi$
$114$	1.63432e17			0.536594
$115$	1.35629e17	−	1.68164e17i	0.413449	−	0.512627i
$116$	−1.04808e17			−0.296824
$117$			4.02364e17i			1.05935i
$118$			1.21495e17i			0.297550i
$119$	2.33401e17			0.532051
$120$	6.20150e16	+	5.00169e16i	0.131661	+	0.106188i
$121$	1.35613e17			0.268304
$122$			6.02586e17i			1.11163i
$123$		−	4.26978e17i		−	0.734873i
$124$	1.53620e17			0.246811
$125$	2.99027e17	−	5.95543e17i	0.448720	−	0.893672i
$126$	−1.94449e17			−0.272681
$127$			1.11625e18i			1.46363i	0.681506	+	0.731813i	$0.261325\pi$
$127$			1.11625e18i			1.46363i	−0.681506	+	0.731813i	$0.738675\pi$
$128$			7.20576e16i			0.0883883i
$129$	−2.70371e17			−0.310418
$130$	−7.03260e17	−	5.67200e17i	−0.756132	−	0.609843i
$131$	−9.11182e17			−0.917908			−0.458954	−	0.888460i	$-0.651776\pi$
$131$	−9.11182e17			−0.917908			−0.458954	+	0.888460i	$0.651776\pi$
$132$			2.85277e17i			0.269395i
$133$		−	8.95660e17i		−	0.793239i
$134$	−7.11869e17			−0.591575
$135$	6.81876e17	−	8.45444e17i	0.531947	−	0.659550i
$136$	−5.13379e17			−0.376147
$137$		−	2.01389e16i		−	0.0138647i	−0.999976	−	0.00693235i	$-0.997793\pi$
$137$		−	2.01389e16i		−	0.0138647i	0.999976	−	0.00693235i	$-0.00220665\pi$
$138$			3.44249e17i			0.222793i
$139$	2.23214e18			1.35861			0.679305	−	0.733856i	$-0.262281\pi$
$139$	2.23214e18			1.35861			0.679305	+	0.733856i	$0.262281\pi$
$140$	2.74109e17	−	3.39862e17i	0.156976	−	0.194632i
$141$	−7.31871e17			−0.394523
$142$		−	5.84036e17i		−	0.296475i
$143$		−	3.23509e18i		−	1.54714i
$144$	4.27702e17			0.192779
$145$	−1.08730e18	−	8.76943e17i	−0.462086	−	0.372686i
$146$	−1.31402e18			−0.526748
$147$			9.48445e17i			0.358769i
$148$			1.02089e18i			0.364547i
$149$	−2.44433e18			−0.824284			−0.412142	−	0.911120i	$-0.635219\pi$
$149$	−2.44433e18			−0.824284			−0.412142	+	0.911120i	$0.635219\pi$
$150$	2.24861e17	+	1.03778e18i	0.0716373	+	0.330621i
$151$	2.30058e18			0.692681			0.346341	−	0.938109i	$-0.387424\pi$
$151$	2.30058e18			0.692681			0.346341	+	0.938109i	$0.387424\pi$
$152$			1.97006e18i			0.560801i
$153$			3.04719e18i			0.820394i
$154$	1.56341e18			0.398242
$155$	1.59370e18	+	1.28537e18i	0.384227	+	0.309891i
$156$	1.43965e18			0.328622
$157$			6.87105e18i			1.48551i	0.669563	+	0.742755i	$0.266481\pi$
$157$			6.87105e18i			1.48551i	−0.669563	+	0.742755i	$0.733519\pi$
$158$		−	4.69855e18i		−	0.962451i
$159$	−2.45285e18			−0.476207
$160$	−6.02919e17	+	7.47547e17i	−0.110979	+	0.137600i
$161$	1.88660e18			0.329351
$162$		−	1.56146e18i		−	0.258615i
$163$			6.49765e18i			1.02132i	0.859783	+	0.510660i	$0.170599\pi$
$163$			6.49765e18i			1.02132i	−0.859783	+	0.510660i	$0.829401\pi$
$164$	5.14692e18			0.768024
$165$	−2.38697e18	+	2.95955e18i	−0.338246	+	0.419385i
$166$	5.53903e18			0.745616
$167$		−	1.47639e19i		−	1.88847i	−0.329266	−	0.944237i	$-0.606801\pi$
$167$		−	1.47639e19i		−	1.88847i	0.329266	−	0.944237i	$-0.393199\pi$
$168$			6.95732e17i			0.0845888i
$169$	−7.67541e18			−0.887288
$170$	−5.32595e18	−	4.29553e18i	−0.585573	−	0.472282i
$171$	1.16934e19			1.22313
$172$		−	3.25913e18i		−	0.324422i
$173$			6.26986e18i			0.594109i	0.954861	+	0.297054i	$0.0960042\pi$
$173$			6.26986e18i			0.594109i	−0.954861	+	0.297054i	$0.903996\pi$
$174$	2.22582e18			0.200827
$175$	5.68737e18	−	1.23231e18i	0.488751	−	0.105900i
$176$	−3.43882e18			−0.281547
$177$		−	2.58021e18i		−	0.201318i
$178$			1.72096e18i			0.127997i
$179$	−1.83152e19			−1.29885			−0.649427	−	0.760424i	$-0.724991\pi$
$179$	−1.83152e19			−1.29885			−0.649427	+	0.760424i	$0.724991\pi$
$180$	4.43711e18	+	3.57866e18i	0.300112	+	0.242049i
$181$	−2.43403e19			−1.57057			−0.785286	−	0.619133i	$-0.787484\pi$
$181$	−2.43403e19			−1.57057			−0.785286	+	0.619133i	$0.787484\pi$
$182$		−	7.88972e18i		−	0.485797i
$183$		−	1.27973e19i		−	0.752113i
$184$	−4.14968e18			−0.232843
$185$	−8.54195e18	+	1.05910e19i	−0.457718	+	0.567515i
$186$	−3.26247e18			−0.166989
$187$		−	2.45001e19i		−	1.19816i
$188$		−	8.82219e18i		−	0.412320i
$189$	9.48485e18			0.423745
$190$	−1.64838e19	+	2.04380e19i	−0.704129	+	0.873036i
$191$	2.16535e19			0.884593			0.442297	−	0.896869i	$-0.354164\pi$
$191$	2.16535e19			0.884593			0.442297	+	0.896869i	$0.354164\pi$
$192$		−	1.53030e18i		−	0.0598022i
$193$			3.13407e19i			1.17185i	0.810366	+	0.585923i	$0.199268\pi$
$193$			3.13407e19i			1.17185i	−0.810366	+	0.585923i	$0.800732\pi$
$194$	2.53061e19			0.905547
$195$	1.49353e19	+	1.20458e19i	0.511588	+	0.412611i
$196$	−1.14328e19			−0.374954
$197$		−	5.00104e19i		−	1.57072i	−0.619042	−	0.785358i	$-0.712479\pi$
$197$		−	5.00104e19i		−	1.57072i	0.619042	−	0.785358i	$-0.287521\pi$
$198$			2.04113e19i			0.614068i
$199$	4.15817e19			1.19853			0.599267	−	0.800549i	$-0.295459\pi$
$199$	4.15817e19			1.19853			0.599267	+	0.800549i	$0.295459\pi$
$200$	−1.25097e19	+	2.71054e18i	−0.345535	+	0.0748689i
$201$	1.51181e19			0.400250
$202$			1.27416e19i			0.323399i
$203$		−	1.21982e19i		−	0.296880i
$204$	1.09028e19			0.254495
$205$	5.33957e19	+	4.30652e19i	1.19563	+	0.964315i
$206$	−5.08661e19			−1.09284
$207$			2.46307e19i			0.507841i
$208$			1.73539e19i			0.343447i
$209$	−9.40174e19			−1.78634
$210$	−5.82131e18	+	7.21773e18i	−0.106208	+	0.131685i
$211$	3.68930e19			0.646462			0.323231	−	0.946320i	$-0.395231\pi$
$211$	3.68930e19			0.646462			0.323231	+	0.946320i	$0.395231\pi$
$212$		−	2.95674e19i		−	0.497689i
$213$			1.24033e19i			0.200591i
$214$	6.93066e19			1.07710
$215$	2.72697e19	−	3.38112e19i	0.407337	−	0.505049i
$216$	−2.08625e19			−0.299578
$217$			1.78794e19i			0.246857i
$218$			3.23804e19i			0.429934i
$219$	2.79062e19			0.356390
$220$	−3.56753e19	−	2.87732e19i	−0.438304	−	0.353505i
$221$	−1.23639e20			−1.46158
$222$		−	2.16809e19i		−	0.246647i
$223$		−	3.64270e19i		−	0.398871i	−0.979911	−	0.199435i	$-0.936089\pi$
$223$		−	3.64270e19i		−	0.398871i	0.979911	−	0.199435i	$-0.0639108\pi$
$224$	−8.38656e18			−0.0884048
$225$	1.60886e19	+	7.42521e19i	0.163293	+	0.753629i
$226$	5.84820e19			0.571611
$227$		−	1.70407e20i		−	1.60424i	−0.597164	−	0.802119i	$-0.703706\pi$
$227$		−	1.70407e20i		−	1.60424i	0.597164	−	0.802119i	$-0.296294\pi$
$228$		−	4.18386e19i		−	0.379429i
$229$	−1.77339e20			−1.54954			−0.774768	−	0.632246i	$-0.782133\pi$
$229$	−1.77339e20			−1.54954			−0.774768	+	0.632246i	$0.782133\pi$
$230$	−4.30500e19	−	3.47211e19i	−0.362482	−	0.292353i
$231$	−3.32025e19			−0.269445
$232$			2.68307e19i			0.209887i
$233$		−	3.76345e19i		−	0.283832i	−0.989879	−	0.141916i	$-0.954674\pi$
$233$		−	3.76345e19i		−	0.283832i	0.989879	−	0.141916i	$-0.0453262\pi$
$234$	1.03005e20			0.749073
$235$	7.38168e19	−	9.15240e19i	0.517700	−	0.641886i
$236$	3.11026e19			0.210399
$237$			9.97843e19i			0.651180i
$238$		−	5.97506e19i		−	0.376217i
$239$	−1.88943e20			−1.14802			−0.574009	−	0.818849i	$-0.694613\pi$
$239$	−1.88943e20			−1.14802			−0.574009	+	0.818849i	$0.694613\pi$
$240$	1.28043e19	−	1.58758e19i	0.0750864	−	0.0930981i
$241$	3.08575e20			1.74669			0.873345	−	0.487101i	$-0.161946\pi$
$241$	3.08575e20			1.74669			0.873345	+	0.487101i	$0.161946\pi$
$242$		−	3.47170e19i		−	0.189719i
$243$			1.93747e20i			1.02231i
$244$	1.54262e20			0.786042
$245$	−1.18608e20	−	9.56606e19i	−0.583715	−	0.470784i
$246$	−1.09306e20			−0.519634
$247$			4.74457e20i			2.17908i
$248$		−	3.93268e19i		−	0.174522i
$249$	−1.17634e20			−0.504472
$250$	−1.52459e20	−	7.65510e19i	−0.631922	−	0.317293i
$251$	1.35005e20			0.540909			0.270454	−	0.962733i	$-0.412826\pi$
$251$	1.35005e20			0.540909			0.270454	+	0.962733i	$0.412826\pi$
$252$			4.97789e19i			0.192815i
$253$		−	1.98036e20i		−	0.741686i
$254$	2.85760e20			1.03494
$255$	1.13108e20	+	9.12253e19i	0.396190	+	0.319539i
$256$	1.84467e19			0.0625000
$257$			3.93269e20i			1.28902i	0.764597	+	0.644508i	$0.222938\pi$
$257$			3.93269e20i			1.28902i	−0.764597	+	0.644508i	$0.777062\pi$
$258$			6.92149e19i			0.219499i
$259$	−1.18818e20			−0.364615
$260$	−1.45203e20	+	1.80035e20i	−0.431224	+	0.534666i
$261$	1.59255e20			0.457772
$262$			2.33263e20i			0.649059i
$263$		−	4.73601e20i		−	1.27582i	−0.770112	−	0.637909i	$-0.779800\pi$
$263$		−	4.73601e20i		−	1.27582i	0.770112	−	0.637909i	$-0.220200\pi$
$264$	7.30310e19			0.190491
$265$	2.47396e20	−	3.06741e20i	0.624888	−	0.774786i
$266$	−2.29289e20			−0.560905
$267$		−	3.65484e19i		−	0.0866010i
$268$			1.82239e20i			0.418306i
$269$	1.78967e20			0.397996			0.198998	−	0.980000i	$-0.436231\pi$
$269$	1.78967e20			0.397996			0.198998	+	0.980000i	$0.436231\pi$
$270$	−2.16434e20	−	1.74560e20i	−0.466373	−	0.376143i
$271$	−3.56661e19			−0.0744760			−0.0372380	−	0.999306i	$-0.511856\pi$
$271$	−3.56661e19			−0.0744760			−0.0372380	+	0.999306i	$0.511856\pi$
$272$			1.31425e20i			0.265976i
$273$			1.67556e20i			0.328683i
$274$	−5.15555e18			−0.00980383
$275$	−1.29356e20	−	5.97004e20i	−0.238483	−	1.10065i
$276$	8.81278e19			0.157538
$277$		−	4.48775e20i		−	0.777948i	−0.921249	−	0.388974i	$-0.872830\pi$
$277$		−	4.48775e20i		−	0.777948i	0.921249	−	0.388974i	$-0.127170\pi$
$278$		−	5.71427e20i		−	0.960683i
$279$	−2.33427e20			−0.380640
$280$	−8.70046e19	−	7.01718e19i	−0.137626	−	0.110999i
$281$	−7.28586e19			−0.111809			−0.0559046	−	0.998436i	$-0.517804\pi$
$281$	−7.28586e19			−0.111809			−0.0559046	+	0.998436i	$0.517804\pi$
$282$			1.87359e20i			0.278970i
$283$		−	4.64318e18i		−	0.00670859i	−0.999994	−	0.00335429i	$-0.998932\pi$
$283$		−	4.64318e18i		−	0.00670859i	0.999994	−	0.00335429i	$-0.00106771\pi$
$284$	−1.49513e20			−0.209640
$285$	3.50071e20	−	4.34046e20i	0.476403	−	0.590683i
$286$	−8.28184e20			−1.09400
$287$			5.99034e20i			0.768167i
$288$		−	1.09492e20i		−	0.136315i
$289$	−1.09106e20			−0.131891
$290$	−2.24497e20	+	2.78350e20i	−0.263529	+	0.326744i
$291$	−5.37432e20			−0.612679
$292$			3.36389e20i			0.372467i
$293$			5.01390e20i			0.539264i	0.962963	+	0.269632i	$0.0869020\pi$
$293$			5.01390e20i			0.539264i	−0.962963	+	0.269632i	$0.913098\pi$
$294$	2.42802e20			0.253688
$295$	3.22668e20	+	2.60241e20i	0.327543	+	0.264173i
$296$	2.61347e20			0.257774
$297$		−	9.95625e20i		−	0.954259i
$298$			6.25749e20i			0.582856i
$299$	−9.99384e20			−0.904747
$300$	2.65672e20	−	5.75645e19i	0.233784	−	0.0506552i
$301$	3.79320e20			0.324482
$302$		−	5.88948e20i		−	0.489799i
$303$		−	2.70597e20i		−	0.218807i
$304$	5.04335e20			0.396546
$305$	1.60036e21	+	1.29074e21i	1.22368	+	0.986937i
$306$	7.80081e20			0.580106
$307$			2.05269e21i			1.48473i	0.669995	+	0.742366i	$0.266296\pi$
$307$			2.05269e21i			1.48473i	−0.669995	+	0.742366i	$0.733704\pi$
$308$		−	4.00233e20i		−	0.281600i
$309$	1.08025e21			0.739399
$310$	3.29055e20	−	4.07988e20i	0.219126	−	0.271690i
$311$	1.57005e20			0.101730			0.0508651	−	0.998706i	$-0.483802\pi$
$311$	1.57005e20			0.101730			0.0508651	+	0.998706i	$0.483802\pi$
$312$		−	3.68549e20i		−	0.232371i
$313$		−	7.02650e20i		−	0.431134i	−0.976489	−	0.215567i	$-0.930840\pi$
$313$		−	7.02650e20i		−	0.431134i	0.976489	−	0.215567i	$-0.0691599\pi$
$314$	1.75899e21			1.05041
$315$	−4.16509e20	+	5.16421e20i	−0.242094	+	0.300168i
$316$	−1.20283e21			−0.680556
$317$		−	1.48179e21i		−	0.816177i	−0.912942	−	0.408088i	$-0.866196\pi$
$317$		−	1.48179e21i		−	0.816177i	0.912942	−	0.408088i	$-0.133804\pi$
$318$			6.27930e20i			0.336729i
$319$	−1.28045e21			−0.668561
$320$	1.91372e20	+	1.54347e20i	0.0972979	+	0.0784737i
$321$	−1.47188e21			−0.728752
$322$		−	4.82968e20i		−	0.232886i
$323$			3.59317e21i			1.68755i
$324$	−3.99734e20			−0.182868
$325$	−3.01276e21	+	6.52791e20i	−1.34263	+	0.290915i
$326$	1.66340e21			0.722182
$327$		−	6.87671e20i		−	0.290887i
$328$		−	1.31761e21i		−	0.543075i
$329$	1.02679e21			0.412397
$330$	7.57645e20	+	6.11064e20i	0.296550	+	0.239176i
$331$	5.79418e20			0.221031			0.110516	−	0.993874i	$-0.464750\pi$
$331$	5.79418e20			0.221031			0.110516	+	0.993874i	$0.464750\pi$
$332$		−	1.41799e21i		−	0.527230i
$333$		−	1.55124e21i		−	0.562216i
$334$	−3.77956e21			−1.33535
$335$	−1.52482e21	+	1.89060e21i	−0.525216	+	0.651205i
$336$	1.78107e20			0.0598133
$337$		−	5.90081e21i		−	1.93222i	−0.258123	−	0.966112i	$-0.583104\pi$
$337$		−	5.90081e21i		−	1.93222i	0.258123	−	0.966112i	$-0.416896\pi$
$338$			1.96491e21i			0.627408i
$339$	−1.24200e21			−0.386743
$340$	−1.09966e21	+	1.36344e21i	−0.333954	+	0.414063i
$341$	1.87680e21			0.555912
$342$		−	2.99351e21i		−	0.864885i
$343$		−	3.10503e21i		−	0.875116i
$344$	−8.34337e20			−0.229401
$345$	9.14264e20	+	7.37381e20i	0.245250	+	0.197801i
$346$	1.60508e21			0.420098
$347$		−	1.87410e21i		−	0.478621i	−0.970943	−	0.239310i	$-0.923079\pi$
$347$		−	1.87410e21i		−	0.478621i	0.970943	−	0.239310i	$-0.0769214\pi$
$348$		−	5.69811e20i		−	0.142006i
$349$	4.16245e21			1.01236			0.506178	−	0.862429i	$-0.331058\pi$
$349$	4.16245e21			1.01236			0.506178	+	0.862429i	$0.331058\pi$
$350$	−3.15472e20	−	1.45597e21i	−0.0748828	−	0.345599i
$351$	−5.02440e21			−1.16406
$352$			8.80338e20i			0.199084i
$353$			4.19140e21i			0.925283i	0.886545	+	0.462641i	$0.153098\pi$
$353$			4.19140e21i			0.925283i	−0.886545	+	0.462641i	$0.846902\pi$
$354$	−6.60533e20			−0.142353
$355$	−1.55109e21	−	1.25100e21i	−0.326360	−	0.263219i
$356$	4.40566e20			0.0905077
$357$			1.26894e21i			0.254543i
$358$			4.68868e21i			0.918428i
$359$	8.08860e21			1.54729			0.773643	−	0.633622i	$-0.218432\pi$
$359$	8.08860e21			1.54729			0.773643	+	0.633622i	$0.218432\pi$
$360$	9.16136e20	−	1.13590e21i	0.171155	−	0.212211i
$361$	8.30814e21			1.51598
$362$			6.23111e21i			1.11056i
$363$			7.37293e20i			0.128361i
$364$	−2.01977e21			−0.343510
$365$	−2.81463e21	+	3.48980e21i	−0.467662	+	0.579844i
$366$	−3.27610e21			−0.531824
$367$		−	5.82699e21i		−	0.924236i	−0.886818	−	0.462118i	$-0.847090\pi$
$367$		−	5.82699e21i		−	0.924236i	0.886818	−	0.462118i	$-0.152910\pi$
$368$			1.06232e21i			0.164645i
$369$	−7.82076e21			−1.18447
$370$	2.71130e21	+	2.18674e21i	0.401294	+	0.323655i
$371$	3.44126e21			0.497782
$372$			8.35193e20i			0.118079i
$373$			5.94224e20i			0.0821154i	0.999157	+	0.0410577i	$0.0130728\pi$
$373$			5.94224e20i			0.0821154i	−0.999157	+	0.0410577i	$0.986927\pi$
$374$	−6.27202e21			−0.847225
$375$	3.23781e21	+	1.62573e21i	0.427549	+	0.214676i
$376$	−2.25848e21			−0.291554
$377$			6.46175e21i			0.815547i
$378$		−	2.42812e21i		−	0.299633i
$379$	7.98387e21			0.963341			0.481671	−	0.876352i	$-0.340030\pi$
$379$	7.98387e21			0.963341			0.481671	+	0.876352i	$0.340030\pi$
$380$	5.23212e21	+	4.21986e21i	0.617329	+	0.497894i
$381$	−6.06875e21			−0.700224
$382$		−	5.54329e21i		−	0.625502i
$383$			5.20909e20i			0.0574874i	0.999587	+	0.0287437i	$0.00915066\pi$
$383$			5.20909e20i			0.0574874i	−0.999587	+	0.0287437i	$0.990849\pi$
$384$	−3.91758e20			−0.0422866
$385$	3.34882e21	−	4.15214e21i	0.353571	−	0.438385i
$386$	8.02321e21			0.828621
$387$			4.95226e21i			0.500334i
$388$		−	6.47837e21i		−	0.640318i
$389$	5.53394e21			0.535134			0.267567	−	0.963539i	$-0.413780\pi$
$389$	5.53394e21			0.535134			0.267567	+	0.963539i	$0.413780\pi$
$390$	3.08372e21	−	3.82344e21i	0.291760	−	0.361747i
$391$	−7.56856e21			−0.700665
$392$			2.92681e21i			0.265132i
$393$		−	4.95385e21i		−	0.439144i
$394$	−1.28027e22			−1.11066
$395$	−1.24785e22	−	1.00643e22i	−1.05947	−	0.854491i
$396$	5.22529e21			0.434212
$397$			2.12491e21i			0.172831i	0.996259	+	0.0864154i	$0.0275412\pi$
$397$			2.12491e21i			0.172831i	−0.996259	+	0.0864154i	$0.972459\pi$
$398$		−	1.06449e22i		−	0.847492i
$399$	4.86946e21			0.379500
$400$	6.93899e20	+	3.20249e21i	0.0529403	+	0.244330i
$401$	−2.46231e22			−1.83915			−0.919573	−	0.392920i	$-0.871465\pi$
$401$	−2.46231e22			−1.83915			−0.919573	+	0.392920i	$0.871465\pi$
$402$		−	3.87024e21i		−	0.283020i
$403$		−	9.47124e21i		−	0.678131i
$404$	3.26186e21			0.228678
$405$	−4.14696e21	−	3.34465e21i	−0.284683	−	0.229606i
$406$	−3.12275e21			−0.209926
$407$			1.24723e22i			0.821099i
$408$		−	2.79110e21i		−	0.179955i
$409$	−2.57342e21			−0.162504			−0.0812518	−	0.996694i	$-0.525892\pi$
$409$	−2.57342e21			−0.162504			−0.0812518	+	0.996694i	$0.525892\pi$
$410$	1.10247e22	−	1.36693e22i	0.681874	−	0.845441i
$411$	1.09490e20			0.00663312
$412$			1.30217e22i			0.772755i
$413$			3.61993e21i			0.210439i
$414$	6.30545e21			0.359098
$415$	1.18646e22	−	1.47107e22i	0.661978	−	0.820774i
$416$	4.44260e21			0.242853
$417$			1.21355e22i			0.649984i
$418$			2.40685e22i			1.26314i
$419$	3.27601e21			0.168471			0.0842356	−	0.996446i	$-0.473155\pi$
$419$	3.27601e21			0.168471			0.0842356	+	0.996446i	$0.473155\pi$
$420$	1.84774e21	+	1.49026e21i	0.0931154	+	0.0751003i
$421$	−1.30423e22			−0.644103			−0.322051	−	0.946722i	$-0.604372\pi$
$421$	−1.30423e22			−0.644103			−0.322051	+	0.946722i	$0.604372\pi$
$422$		−	9.44461e21i		−	0.457118i
$423$			1.34053e22i			0.635894i
$424$	−7.56925e21			−0.351919
$425$	−2.28163e22	+	4.94373e21i	−1.03978	+	0.225294i
$426$	3.17525e21			0.141839
$427$			1.79541e22i			0.786188i
$428$		−	1.77425e22i		−	0.761627i
$429$	1.75883e22			0.740181
$430$	−8.65566e21	−	6.98105e21i	−0.357124	−	0.288031i
$431$	2.81338e22			1.13808			0.569038	−	0.822311i	$-0.307316\pi$
$431$	2.81338e22			1.13808			0.569038	+	0.822311i	$0.307316\pi$
$432$			5.34080e21i			0.211833i
$433$			4.89549e21i			0.190392i	0.995459	+	0.0951961i	$0.0303478\pi$
$433$			4.89549e21i			0.190392i	−0.995459	+	0.0951961i	$0.969652\pi$
$434$	4.57713e21			0.174554
$435$	4.76771e21	−	5.91138e21i	0.178300	−	0.221070i
$436$	8.28939e21			0.304010
$437$			2.90438e22i			1.04463i
$438$		−	7.14398e21i		−	0.252006i
$439$	1.56115e22			0.540129			0.270065	−	0.962842i	$-0.412955\pi$
$439$	1.56115e22			0.540129			0.270065	+	0.962842i	$0.412955\pi$
$440$	−7.36594e21	+	9.13288e21i	−0.249966	+	0.309928i
$441$	1.73722e22			0.578266
$442$			3.16516e22i			1.03349i
$443$		−	3.75412e22i		−	1.20248i	−0.799069	−	0.601239i	$-0.794674\pi$
$443$		−	3.75412e22i		−	1.20248i	0.799069	−	0.601239i	$-0.205326\pi$
$444$	−5.55030e21			−0.174406
$445$	4.57056e21	+	3.68629e21i	0.140899	+	0.113640i
$446$	−9.32531e21			−0.282044
$447$		−	1.32892e22i		−	0.394352i
$448$			2.14696e21i			0.0625116i
$449$	−4.81955e21			−0.137693			−0.0688466	−	0.997627i	$-0.521932\pi$
$449$	−4.81955e21			−0.137693			−0.0688466	+	0.997627i	$0.521932\pi$
$450$	1.90085e22	−	4.11868e21i	0.532896	−	0.115465i
$451$	6.28807e22			1.72988
$452$		−	1.49714e22i		−	0.404190i
$453$			1.25076e22i			0.331391i
$454$	−4.36243e22			−1.13437
$455$	−2.09537e22	−	1.68998e22i	−0.534765	−	0.431304i
$456$	−1.07107e22			−0.268297
$457$		−	4.08818e22i		−	1.00518i	−0.864526	−	0.502588i	$-0.832381\pi$
$457$		−	4.08818e22i		−	1.00518i	0.864526	−	0.502588i	$-0.167619\pi$
$458$			4.53987e22i			1.09569i
$459$	−3.80509e22			−0.901482
$460$	−8.88861e21	+	1.10208e22i	−0.206725	+	0.256314i
$461$	−4.33277e22			−0.989253			−0.494626	−	0.869106i	$-0.664695\pi$
$461$	−4.33277e22			−0.989253			−0.494626	+	0.869106i	$0.664695\pi$
$462$			8.49985e21i			0.190526i
$463$		−	2.76914e22i		−	0.609405i	−0.952448	−	0.304702i	$-0.901443\pi$
$463$		−	2.76914e22i		−	0.609405i	0.952448	−	0.304702i	$-0.0985570\pi$
$464$	6.86867e21			0.148412
$465$	−6.98821e21	+	8.66454e21i	−0.148257	+	0.183821i
$466$	−9.63443e21			−0.200699
$467$		−	5.60313e22i		−	1.14614i	−0.819507	−	0.573070i	$-0.805752\pi$
$467$		−	5.60313e22i		−	1.14614i	0.819507	−	0.573070i	$-0.194248\pi$
$468$		−	2.63693e22i		−	0.529674i
$469$	−2.12102e22			−0.418384
$470$	−2.34301e22	−	1.88971e22i	−0.453882	−	0.366069i
$471$	−3.73560e22			−0.710695
$472$		−	7.96227e21i		−	0.148775i
$473$		−	3.98172e22i		−	0.730721i
$474$	2.55448e22			0.460454
$475$	1.89712e22	+	8.75561e22i	0.335892	+	1.55021i
$476$	−1.52961e22			−0.266025
$477$			4.49277e22i			0.767552i
$478$			4.83694e22i			0.811772i
$479$	9.17774e21			0.151316			0.0756578	−	0.997134i	$-0.475894\pi$
$479$	9.17774e21			0.151316			0.0756578	+	0.997134i	$0.475894\pi$
$480$	−4.06421e21	−	3.27791e21i	−0.0658303	−	0.0530941i
$481$	6.29413e22			1.00162
$482$		−	7.89952e22i		−	1.23510i
$483$			1.02569e22i			0.157567i
$484$	−8.88755e21			−0.134152
$485$	5.42056e22	−	6.72085e22i	0.803970	−	0.996826i
$486$	4.95993e22			0.722881
$487$		−	3.59313e22i		−	0.514608i	−0.966331	−	0.257304i	$-0.917166\pi$
$487$		−	3.59313e22i		−	0.514608i	0.966331	−	0.257304i	$-0.0828342\pi$
$488$		−	3.94911e22i		−	0.555815i
$489$	−3.53260e22			−0.488617
$490$	−2.44891e22	+	3.03636e22i	−0.332894	+	0.412749i
$491$	−7.54007e22			−1.00736			−0.503678	−	0.863892i	$-0.668020\pi$
$491$	−7.54007e22			−1.00736			−0.503678	+	0.863892i	$0.668020\pi$
$492$			2.79825e22i			0.367437i
$493$			4.89363e22i			0.631585i
$494$	1.21461e23			1.54084
$495$	5.42087e22	+	4.37210e22i	0.675966	+	0.545187i
$496$	−1.00677e22			−0.123406
$497$		−	1.74014e22i		−	0.209679i
$498$			3.01142e22i			0.356716i
$499$	−2.71251e22			−0.315876			−0.157938	−	0.987449i	$-0.550485\pi$
$499$	−2.71251e22			−0.315876			−0.157938	+	0.987449i	$0.550485\pi$
$500$	−1.95971e22	+	3.90295e22i	−0.224360	+	0.446836i
$501$	8.02674e22			0.903480
$502$		−	3.45613e22i		−	0.382480i
$503$			1.48369e23i			1.61442i	0.590267	+	0.807208i	$0.299023\pi$
$503$			1.48369e23i			1.61442i	−0.590267	+	0.807208i	$0.700977\pi$
$504$	1.27434e22			0.136341
$505$	3.38395e22	+	2.72926e22i	0.355998	+	0.287123i
$506$	−5.06972e22			−0.524451
$507$		−	4.17292e22i		−	0.424495i
$508$		−	7.31546e22i		−	0.731813i
$509$	−1.48619e22			−0.146209			−0.0731045	−	0.997324i	$-0.523291\pi$
$509$	−1.48619e22			−0.146209			−0.0731045	+	0.997324i	$0.523291\pi$
$510$	2.33537e22	−	2.89557e22i	0.225948	−	0.280149i
$511$	−3.91513e22			−0.372536
$512$		−	4.72237e21i		−	0.0441942i
$513$			1.46018e23i			1.34403i
$514$	1.00677e23			0.911472
$515$	−1.08955e23	+	1.35091e23i	−0.970254	+	1.20300i
$516$	1.77190e22			0.155209
$517$		−	1.07782e23i		−	0.928701i
$518$			3.04174e22i			0.257822i
$519$	−3.40875e22			−0.284232
$520$	4.60889e22	+	3.71720e22i	0.378066	+	0.304921i
$521$	−1.22940e23			−0.992141			−0.496070	−	0.868282i	$-0.665224\pi$
$521$	−1.22940e23			−0.992141			−0.496070	+	0.868282i	$0.665224\pi$
$522$		−	4.07694e22i		−	0.323694i
$523$			6.01560e22i			0.469910i	0.972006	+	0.234955i	$0.0754942\pi$
$523$			6.01560e22i			0.469910i	−0.972006	+	0.234955i	$0.924506\pi$
$524$	5.97152e22			0.458954
$525$	6.69975e21	+	3.09207e22i	0.0506646	+	0.233827i
$526$	−1.21242e23			−0.902139
$527$		−	7.17278e22i		−	0.525167i
$528$		−	1.86959e22i		−	0.134697i
$529$	7.98728e22			0.566273
$530$	−7.85257e22	−	6.33333e22i	−0.547856	−	0.441862i
$531$	−4.72605e22			−0.324485
$532$			5.86979e22i			0.396619i
$533$		−	3.17326e23i		−	2.11020i
$534$	−9.35640e21			−0.0612362
$535$	1.48454e23	−	1.84066e23i	0.956282	−	1.18568i
$536$	4.66531e22			0.295787
$537$		−	9.95747e22i		−	0.621395i
$538$		−	4.58156e22i		−	0.281426i
$539$	−1.39677e23			−0.844538
$540$	−4.46874e22	+	5.54070e22i	−0.265973	+	0.329775i
$541$	1.09898e23			0.643892			0.321946	−	0.946758i	$-0.395663\pi$
$541$	1.09898e23			0.643892			0.321946	+	0.946758i	$0.395663\pi$
$542$			9.13051e21i			0.0526625i
$543$		−	1.32332e23i		−	0.751390i
$544$	3.36448e22			0.188073
$545$	8.59965e22	+	6.93588e22i	0.473272	+	0.381708i
$546$	4.28943e22			0.232414
$547$			2.00742e23i			1.07089i	0.844570	+	0.535445i	$0.179856\pi$
$547$			2.00742e23i			1.07089i	−0.844570	+	0.535445i	$0.820144\pi$
$548$			1.31982e21i			0.00693235i
$549$	−2.34402e23			−1.21226
$550$	−1.52833e23	+	3.31151e22i	−0.778277	+	0.168633i
$551$	1.87790e23			0.941636
$552$		−	2.25607e22i		−	0.111396i
$553$		−	1.39994e23i		−	0.680682i
$554$	−1.14886e23			−0.550092
$555$	−5.75804e22	−	4.64403e22i	−0.271509	−	0.218980i
$556$	−1.46285e23			−0.679305
$557$			7.56562e22i			0.346000i	0.984922	+	0.173000i	$0.0553460\pi$
$557$			7.56562e22i			0.346000i	−0.984922	+	0.173000i	$0.944654\pi$
$558$			5.97572e22i			0.269153i
$559$	−2.00937e23			−0.891372
$560$	−1.79640e22	+	2.22732e22i	−0.0784882	+	0.0973160i
$561$	1.33200e23			0.573220
$562$			1.86518e22i			0.0790610i
$563$			4.12084e23i			1.72054i	0.509840	+	0.860269i	$0.329704\pi$
$563$			4.12084e23i			1.72054i	−0.509840	+	0.860269i	$0.670296\pi$
$564$	4.79639e22			0.197261
$565$	1.25268e23	−	1.55318e23i	0.507492	−	0.629229i
$566$	−1.18866e21			−0.00474369
$567$		−	4.65238e22i		−	0.182902i
$568$			3.82754e22i			0.148238i
$569$	4.07810e23			1.55598			0.777990	−	0.628277i	$-0.216240\pi$
$569$	4.07810e23			1.55598			0.777990	+	0.628277i	$0.216240\pi$
$570$	−1.11116e23	−	8.96181e22i	−0.417676	−	0.336868i
$571$	1.55923e23			0.577436			0.288718	−	0.957414i	$-0.406771\pi$
$571$	1.55923e23			0.577436			0.288718	+	0.957414i	$0.406771\pi$
$572$			2.12015e23i			0.773572i
$573$			1.17724e23i			0.423205i
$574$	1.53353e23			0.543176
$575$	−1.84426e23	+	3.99606e22i	−0.643644	+	0.139462i
$576$	−2.80299e22			−0.0963895
$577$			1.83765e23i			0.622686i	0.950298	+	0.311343i	$0.100779\pi$
$577$			1.83765e23i			0.622686i	−0.950298	+	0.311343i	$0.899221\pi$
$578$			2.79311e22i			0.0932614i
$579$	−1.70391e23			−0.560632
$580$	7.12576e22	+	5.74714e22i	0.231043	+	0.186343i
$581$	1.65036e23			0.527328
$582$			1.37583e23i			0.433230i
$583$		−	3.61229e23i		−	1.12098i
$584$	8.61157e22			0.263374
$585$	2.20637e23	−	2.73563e23i	0.665048	−	0.824580i
$586$	1.28356e23			0.381317
$587$			4.55166e23i			1.33274i	0.745620	+	0.666371i	$0.232153\pi$
$587$			4.55166e23i			1.33274i	−0.745620	+	0.666371i	$0.767847\pi$
$588$		−	6.21573e22i		−	0.179384i
$589$	−2.75250e23			−0.782975
$590$	6.66217e22	−	8.26029e22i	0.186799	−	0.231608i
$591$	2.71893e23			0.751459
$592$		−	6.69049e22i		−	0.182274i
$593$		−	4.29433e22i		−	0.115327i	−0.998336	−	0.0576635i	$-0.981635\pi$
$593$		−	4.29433e22i		−	0.115327i	0.998336	−	0.0576635i	$-0.0183650\pi$
$594$	−2.54880e23			−0.674763
$595$	−1.58687e23	−	1.27986e23i	−0.414139	−	0.334016i
$596$	1.60192e23			0.412142
$597$			2.26068e23i			0.573400i
$598$			2.55842e23i			0.639753i
$599$	5.70533e23			1.40654			0.703271	−	0.710922i	$-0.251722\pi$
$599$	5.70533e23			1.40654			0.703271	+	0.710922i	$0.251722\pi$
$600$	−1.47365e22	−	6.80120e22i	−0.0358186	−	0.165310i
$601$	−6.23141e23			−1.49332			−0.746660	−	0.665205i	$-0.768344\pi$
$601$	−6.23141e23			−1.49332			−0.746660	+	0.665205i	$0.768344\pi$
$602$		−	9.71059e22i		−	0.229443i
$603$		−	2.76912e23i		−	0.645126i
$604$	−1.50771e23			−0.346341
$605$	−9.22021e22	−	7.43637e22i	−0.208843	−	0.168438i
$606$	−6.92729e22			−0.154720
$607$			1.75622e23i			0.386790i	0.981121	+	0.193395i	$0.0619498\pi$
$607$			1.75622e23i			0.386790i	−0.981121	+	0.193395i	$0.938050\pi$
$608$		−	1.29110e23i		−	0.280400i
$609$	6.63185e22			0.142032
$610$	3.30429e23	−	4.09692e23i	0.697870	−	0.865275i
$611$	−5.43919e23			−1.13288
$612$		−	1.99701e23i		−	0.410197i
$613$		−	2.95997e23i		−	0.599615i	−0.954000	−	0.299808i	$-0.903078\pi$
$613$		−	2.95997e23i		−	0.599615i	0.954000	−	0.299808i	$-0.0969225\pi$
$614$	5.25490e23			1.04986
$615$	−2.34134e23	+	2.90298e23i	−0.461345	+	0.572013i
$616$	−1.02460e23			−0.199121
$617$			2.38209e23i			0.456598i	0.973591	+	0.228299i	$0.0733164\pi$
$617$			2.38209e23i			0.456598i	−0.973591	+	0.228299i	$0.926684\pi$
$618$		−	2.76545e23i		−	0.522834i
$619$	−6.87488e23			−1.28202			−0.641010	−	0.767533i	$-0.721484\pi$
$619$	−6.87488e23			−1.28202			−0.641010	+	0.767533i	$0.721484\pi$
$620$	−1.04445e23	−	8.42380e22i	−0.192114	−	0.154945i
$621$	−3.07568e23			−0.558037
$622$		−	4.01932e22i		−	0.0719340i
$623$			5.12760e22i			0.0905245i
$624$	−9.43486e22			−0.164311
$625$	−5.29873e23	+	2.40932e23i	−0.910314	+	0.413918i
$626$	−1.79878e23			−0.304858
$627$		−	5.11148e23i		−	0.854619i
$628$		−	4.50301e23i		−	0.742755i
$629$	4.76669e23			0.775686
$630$	1.32204e23	+	1.06626e23i	0.212251	+	0.171186i
$631$	2.51988e23			0.399145			0.199573	−	0.979883i	$-0.436045\pi$
$631$	2.51988e23			0.399145			0.199573	+	0.979883i	$0.436045\pi$
$632$			3.07924e23i			0.481226i
$633$			2.00577e23i			0.309279i
$634$	−3.79339e23			−0.577124
$635$	6.12097e23	−	7.58927e23i	0.918848	−	1.13926i
$636$	1.60750e23			0.238103
$637$			7.04874e23i			1.03021i
$638$			3.27795e23i			0.472744i
$639$	2.27186e23			0.323313
$640$	3.95129e22	−	4.89912e22i	0.0554893	−	0.0688000i
$641$	−3.86173e23			−0.535167			−0.267583	−	0.963535i	$-0.586225\pi$
$641$	−3.86173e23			−0.535167			−0.267583	+	0.963535i	$0.586225\pi$
$642$			3.76801e23i			0.515305i
$643$			1.74107e23i			0.234976i	0.993074	+	0.117488i	$0.0374842\pi$
$643$			1.74107e23i			0.234976i	−0.993074	+	0.117488i	$0.962516\pi$
$644$	−1.23640e23			−0.164675
$645$	1.83822e23	+	1.48258e23i	0.241624	+	0.194877i
$646$	9.19850e23			1.19328
$647$		−	6.96954e23i		−	0.892314i	−0.894955	−	0.446157i	$-0.852792\pi$
$647$		−	6.96954e23i		−	0.892314i	0.894955	−	0.446157i	$-0.147208\pi$
$648$			1.02332e23i			0.129307i
$649$	3.79985e23			0.473899
$650$	1.67115e23	+	7.71268e23i	0.205708	+	0.949383i
$651$	−9.72055e22			−0.118101
$652$		−	4.25830e23i		−	0.510660i
$653$			6.36904e23i			0.753896i	0.926234	+	0.376948i	$0.123027\pi$
$653$			6.36904e23i			0.753896i	−0.926234	+	0.376948i	$0.876973\pi$
$654$	−1.76044e23			−0.205688
$655$	6.19504e23	+	4.99648e23i	0.714484	+	0.576253i
$656$	−3.37309e23			−0.384012
$657$		−	5.11144e23i		−	0.574431i
$658$		−	2.62857e23i		−	0.291608i
$659$	1.27336e24			1.39452			0.697261	−	0.716817i	$-0.254402\pi$
$659$	1.27336e24			1.39452			0.697261	+	0.716817i	$0.254402\pi$
$660$	1.56432e23	−	1.93957e23i	0.169123	−	0.209692i
$661$	−1.00536e24			−1.07302			−0.536511	−	0.843893i	$-0.680258\pi$
$661$	−1.00536e24			−1.07302			−0.536511	+	0.843893i	$0.680258\pi$
$662$		−	1.48331e23i		−	0.156293i
$663$		−	6.72193e23i		−	0.699244i
$664$	−3.63006e23			−0.372808
$665$	−4.91136e23	+	6.08950e23i	−0.497987	+	0.617444i
$666$	−3.97118e23			−0.397547
$667$			3.95556e23i			0.390965i
$668$			9.67567e23i			0.944237i
$669$	1.98044e23			0.190827
$670$	4.83993e23	+	3.90354e23i	0.460472	+	0.371384i
$671$	1.88464e24			1.77046
$672$		−	4.55955e22i		−	0.0422944i
$673$			7.09328e23i			0.649709i	0.945764	+	0.324855i	$0.105315\pi$
$673$			7.09328e23i			0.649709i	−0.945764	+	0.324855i	$0.894685\pi$
$674$	−1.51061e24			−1.36629
$675$	−9.27201e23	+	2.00901e23i	−0.828117	+	0.179432i
$676$	5.03016e23			0.443644
$677$		−	1.30786e24i		−	1.13909i	−0.821962	−	0.569543i	$-0.807120\pi$
$677$		−	1.30786e24i		−	1.13909i	0.821962	−	0.569543i	$-0.192880\pi$
$678$			3.17951e23i			0.273469i
$679$	7.53997e23			0.640437
$680$	3.49041e23	+	2.81512e23i	0.292786	+	0.236141i
$681$	9.26460e23			0.767496
$682$		−	4.80461e23i		−	0.393089i
$683$			8.57872e23i			0.693181i	0.938017	+	0.346590i	$0.112661\pi$
$683$			8.57872e23i			0.693181i	−0.938017	+	0.346590i	$0.887339\pi$
$684$	−7.66338e23			−0.611566
$685$	−1.10432e22	+	1.36922e22i	−0.00870411	+	0.0107921i
$686$	−7.94887e23			−0.618800
$687$		−	9.64142e23i		−	0.741325i
$688$			2.13590e23i			0.162211i
$689$	−1.82293e24			−1.36744
$690$	1.88769e23	−	2.34052e23i	0.139867	−	0.173418i
$691$	−5.12718e23			−0.375245			−0.187623	−	0.982241i	$-0.560078\pi$
$691$	−5.12718e23			−0.375245			−0.187623	+	0.982241i	$0.560078\pi$
$692$		−	4.10901e23i		−	0.297054i
$693$			6.08155e23i			0.434292i
$694$	−4.79769e23			−0.338436
$695$	−1.51761e24	−	1.22400e24i	−1.05752	−	0.852921i
$696$	−1.45872e23			−0.100413
$697$		−	2.40318e24i		−	1.63421i
$698$		−	1.06559e24i		−	0.715844i
$699$	2.04609e23			0.135790
$700$	−3.72727e23	+	8.07608e22i	−0.244376	+	0.0529501i
$701$	−8.92190e23			−0.577902			−0.288951	−	0.957344i	$-0.593306\pi$
$701$	−8.92190e23			−0.577902			−0.288951	+	0.957344i	$0.593306\pi$
$702$			1.28625e24i			0.823111i
$703$		−	1.82918e24i		−	1.15648i
$704$	2.25366e23			0.140774
$705$	4.97591e23	+	4.01322e23i	0.307090	+	0.247677i
$706$	1.07300e24			0.654274
$707$			3.79638e23i			0.228720i
$708$			1.69097e23i			0.100659i
$709$	1.23118e24			0.724151			0.362076	−	0.932149i	$-0.382068\pi$
$709$	1.23118e24			0.724151			0.362076	+	0.932149i	$0.382068\pi$
$710$	−3.20257e23	+	3.97080e23i	−0.186124	+	0.230771i
$711$	1.82770e24			1.04958
$712$		−	1.12785e23i		−	0.0639986i
$713$		−	5.79781e23i		−	0.325089i
$714$	3.24848e23			0.179989
$715$	−1.77397e24	+	2.19951e24i	−0.971280	+	1.20427i
$716$	1.20030e24			0.649427
$717$		−	1.02723e24i		−	0.549233i
$718$		−	2.07068e24i		−	1.09410i
$719$	1.37291e24			0.716881			0.358440	−	0.933553i	$-0.383309\pi$
$719$	1.37291e24			0.716881			0.358440	+	0.933553i	$0.383309\pi$
$720$	−2.90790e23	−	2.34531e23i	−0.150056	−	0.121025i
$721$	−1.51556e24			−0.772898
$722$		−	2.12688e24i		−	1.07196i
$723$			1.67764e24i			0.835648i
$724$	1.59517e24			0.785286
$725$	2.58374e23	+	1.19245e24i	0.125712	+	0.580186i
$726$	1.88747e23			0.0907651
$727$			2.59538e23i			0.123355i	0.998096	+	0.0616776i	$0.0196451\pi$
$727$			2.59538e23i			0.123355i	−0.998096	+	0.0616776i	$0.980355\pi$
$728$			5.17061e23i			0.242898i
$729$	−2.65667e23			−0.123354
$730$	8.93390e23	+	7.20545e23i	0.410012	+	0.330687i
$731$	−1.52174e24			−0.690307
$732$			8.38682e23i			0.376056i
$733$			2.58771e24i			1.14692i	0.819234	+	0.573459i	$0.194399\pi$
$733$			2.58771e24i			1.14692i	−0.819234	+	0.573459i	$0.805601\pi$
$734$	−1.49171e24			−0.653534
$735$	5.20081e23	−	6.44838e23i	0.225231	−	0.279260i
$736$	2.71954e23			0.116422
$737$			2.22643e24i			0.942185i
$738$			2.00211e24i			0.837548i
$739$	−2.73860e24			−1.13253			−0.566267	−	0.824222i	$-0.691613\pi$
$739$	−2.73860e24			−1.13253			−0.566267	+	0.824222i	$0.691613\pi$
$740$	5.59806e23	−	6.94092e23i	0.228859	−	0.283757i
$741$	−2.57949e24			−1.04251
$742$		−	8.80962e23i		−	0.351985i
$743$		−	2.67510e24i		−	1.05666i	−0.849039	−	0.528330i	$-0.822818\pi$
$743$		−	2.67510e24i		−	1.05666i	0.849039	−	0.528330i	$-0.177182\pi$
$744$	2.13809e23			0.0834943
$745$	1.66188e24	+	1.34035e24i	0.641609	+	0.517476i
$746$	1.52121e23			0.0580644
$747$			2.15464e24i			0.813111i
$748$			1.60564e24i			0.599079i
$749$	2.06499e24			0.761768
$750$	4.16187e23	−	8.28879e23i	0.151799	−	0.302323i
$751$	4.30433e24			1.55227			0.776133	−	0.630570i	$-0.217179\pi$
$751$	4.30433e24			1.55227			0.776133	+	0.630570i	$0.217179\pi$
$752$			5.78171e23i			0.206160i
$753$			7.33988e23i			0.258780i
$754$	1.65421e24			0.576678
$755$	−1.56414e24	−	1.26153e24i	−0.539171	−	0.434858i
$756$	−6.21599e23			−0.211873
$757$			3.71187e24i			1.25106i	0.780201	+	0.625529i	$0.215117\pi$
$757$			3.71187e24i			1.25106i	−0.780201	+	0.625529i	$0.784883\pi$
$758$		−	2.04387e24i		−	0.681185i
$759$	1.07667e24			0.354836
$760$	1.08028e24	−	1.33942e24i	0.352065	−	0.436518i
$761$	−3.67110e23			−0.118312			−0.0591558	−	0.998249i	$-0.518841\pi$
$761$	−3.67110e23			−0.118312			−0.0591558	+	0.998249i	$0.518841\pi$
$762$			1.55360e24i			0.495133i
$763$			9.64776e23i			0.304066i
$764$	−1.41908e24			−0.442297
$765$	1.67093e24	−	2.07175e24i	0.515034	−	0.638581i
$766$	1.33353e23			0.0406497
$767$		−	1.91758e24i		−	0.578088i
$768$			1.00290e23i			0.0299011i
$769$	2.99462e24			0.883016			0.441508	−	0.897257i	$-0.354444\pi$
$769$	2.99462e24			0.883016			0.441508	+	0.897257i	$0.354444\pi$
$770$	−1.06295e24	−	8.57299e23i	−0.309985	−	0.250012i
$771$	−2.13810e24			−0.616688
$772$		−	2.05394e24i		−	0.585923i
$773$		−	4.98422e24i		−	1.40628i	−0.711053	−	0.703139i	$-0.751781\pi$
$773$		−	4.98422e24i		−	1.40628i	0.711053	−	0.703139i	$-0.248219\pi$
$774$	1.26778e24			0.353789
$775$	−3.78709e23	−	1.74782e24i	−0.104530	−	0.482427i
$776$	−1.65846e24			−0.452773
$777$		−	6.45982e23i		−	0.174438i
$778$		−	1.41669e24i		−	0.378397i
$779$	−9.22204e24			−2.43646
$780$	−9.78800e23	−	7.89432e23i	−0.255794	−	0.206305i
$781$	−1.82662e24			−0.472188
$782$			1.93755e24i			0.495445i
$783$			1.98865e24i			0.503019i
$784$	7.49263e23			0.187477
$785$	3.76775e24	−	4.67155e24i	0.932588	−	1.15630i
$786$	−1.26819e24			−0.310521
$787$			4.79172e24i			1.16066i	0.814380	+	0.580332i	$0.197077\pi$
$787$			4.79172e24i			1.16066i	−0.814380	+	0.580332i	$0.802923\pi$
$788$			3.27748e24i			0.785358i
$789$	2.57484e24			0.610374
$790$	−2.57646e24	+	3.19450e24i	−0.604217	+	0.749156i
$791$	1.74247e24			0.404265
$792$		−	1.33767e24i		−	0.307034i
$793$		−	9.51079e24i		−	2.15971i
$794$	5.43977e23			0.122210
$795$	1.66767e24	+	1.34502e24i	0.370671	+	0.298957i
$796$	−2.72510e24			−0.599267
$797$		−	2.41579e24i		−	0.525610i	−0.964849	−	0.262805i	$-0.915352\pi$
$797$		−	2.41579e24i		−	0.525610i	0.964849	−	0.262805i	$-0.0846476\pi$
$798$		−	1.24658e24i		−	0.268347i
$799$	−4.11922e24			−0.877338
$800$	8.19837e23	−	1.77638e23i	0.172768	−	0.0374345i
$801$	−6.69440e23			−0.139584
$802$			6.30352e24i			1.30047i
$803$			4.10972e24i			0.838937i
$804$	−9.90782e23			−0.200125
$805$	−1.28268e24	−	1.03452e24i	−0.256361	−	0.206763i
$806$	−2.42464e24			−0.479511
$807$			9.72996e23i			0.190408i
$808$		−	8.35036e23i		−	0.161699i
$809$	−4.17740e24			−0.800467			−0.400234	−	0.916413i	$-0.631071\pi$
$809$	−4.17740e24			−0.800467			−0.400234	+	0.916413i	$0.631071\pi$
$810$	−8.56230e23	+	1.06162e24i	−0.162356	+	0.201302i
$811$	3.24336e24			0.608580			0.304290	−	0.952579i	$-0.401581\pi$
$811$	3.24336e24			0.608580			0.304290	+	0.952579i	$0.401581\pi$
$812$			7.99423e23i			0.148440i
$813$		−	1.93907e23i		−	0.0356306i
$814$	3.19292e24			0.580604
$815$	3.56299e24	−	4.41768e24i	0.641173	−	0.794978i
$816$	−7.14523e23			−0.127248
$817$			5.83957e24i			1.02918i
$818$			6.58797e23i			0.114907i
$819$	3.06904e24			0.529773
$820$	−3.49934e24	−	2.82232e24i	−0.597817	−	0.482157i
$821$	7.10530e24			1.20134			0.600670	−	0.799497i	$-0.294901\pi$
$821$	7.10530e24			1.20134			0.600670	+	0.799497i	$0.294901\pi$
$822$		−	2.80294e22i		−	0.00469033i
$823$		−	1.29946e24i		−	0.215212i	−0.994194	−	0.107606i	$-0.965682\pi$
$823$		−	1.29946e24i		−	0.215212i	0.994194	−	0.107606i	$-0.0343184\pi$
$824$	3.33356e24			0.546420
$825$	3.24575e24	−	7.03273e23i	0.526570	−	0.114095i
$826$	9.26703e23			0.148803
$827$			3.17524e24i			0.504638i	0.967644	+	0.252319i	$0.0811933\pi$
$827$			3.17524e24i			0.504638i	−0.967644	+	0.252319i	$0.918807\pi$
$828$		−	1.61420e24i		−	0.253921i
$829$	4.53431e24			0.705988			0.352994	−	0.935626i	$-0.385164\pi$
$829$	4.53431e24			0.705988			0.352994	+	0.935626i	$0.385164\pi$
$830$	−3.76593e24	−	3.03733e24i	−0.580375	−	0.468089i
$831$	2.43987e24			0.372184
$832$		−	1.13731e24i		−	0.171723i
$833$			5.33817e24i			0.797829i
$834$	3.10670e24			0.459608
$835$	−8.09580e24	+	1.00378e25i	−1.18556	+	1.46996i
$836$	6.16153e24			0.893172
$837$		−	2.91485e24i		−	0.418263i
$838$		−	8.38658e23i		−	0.119127i
$839$	2.29905e24			0.323275			0.161637	−	0.986850i	$-0.448323\pi$
$839$	2.29905e24			0.323275			0.161637	+	0.986850i	$0.448323\pi$
$840$	3.81506e23	−	4.73021e23i	0.0531040	−	0.0658425i
$841$	−4.69959e24			−0.647581
$842$			3.33882e24i			0.455449i
$843$		−	3.96113e23i		−	0.0534915i
$844$	−2.41782e24			−0.323231
$845$	5.21843e24	+	4.20882e24i	0.690650	+	0.557030i
$846$	3.43176e24			0.449645
$847$		−	1.03439e24i		−	0.134177i
$848$			1.93773e24i			0.248845i
$849$	2.52438e22			0.00320951
$850$	1.26560e24	+	5.84098e24i	0.159307	+	0.735233i
$851$	3.85295e24			0.480167
$852$		−	8.12863e23i		−	0.100295i
$853$		−	1.51968e24i		−	0.185646i	−0.995683	−	0.0928228i	$-0.970411\pi$
$853$		−	1.51968e24i		−	0.185646i	0.995683	−	0.0928228i	$-0.0295890\pi$
$854$	4.59624e24			0.555919
$855$	−7.95021e24	−	6.41208e24i	−0.952066	−	0.767869i
$856$	−4.54207e24			−0.538552
$857$			2.14832e23i			0.0252210i	0.999920	+	0.0126105i	$0.00401415\pi$
$857$			2.14832e23i			0.0252210i	−0.999920	+	0.0126105i	$0.995986\pi$
$858$		−	4.50262e24i		−	0.523387i
$859$	−1.66184e25			−1.91270			−0.956352	−	0.292218i	$-0.905607\pi$
$859$	−1.66184e25			−1.91270			−0.956352	+	0.292218i	$0.905607\pi$
$860$	−1.78715e24	+	2.21585e24i	−0.203668	+	0.252524i
$861$	−3.25679e24			−0.367505
$862$		−	7.20224e24i		−	0.804741i
$863$			7.51987e24i			0.831991i	0.909367	+	0.415995i	$0.136567\pi$
$863$			7.51987e24i			0.831991i	−0.909367	+	0.415995i	$0.863433\pi$
$864$	1.36724e24			0.149789
$865$	3.43808e24	−	4.26281e24i	0.372975	−	0.462444i
$866$	1.25325e24			0.134628
$867$		−	5.93180e23i		−	0.0630992i
$868$		−	1.17174e24i		−	0.123428i
$869$	−1.46951e25			−1.53287
$870$	−1.51331e24	−	1.22053e24i	−0.156320	−	0.126077i
$871$	1.12356e25			1.14933
$872$		−	2.12208e24i		−	0.214967i
$873$			9.84389e24i			0.987519i
$874$	7.43522e24			0.738664
$875$	−4.54252e24	−	2.28084e24i	−0.446919	−	0.224402i
$876$	−1.82886e24			−0.178195
$877$		−	1.17579e25i		−	1.13457i	−0.823521	−	0.567285i	$-0.807994\pi$
$877$		−	1.17579e25i		−	1.13457i	0.823521	−	0.567285i	$-0.192006\pi$
$878$		−	3.99656e24i		−	0.381929i
$879$	−2.72593e24			−0.257993
$880$	2.33802e24	+	1.88568e24i	0.219152	+	0.176752i
$881$	−8.93818e24			−0.829763			−0.414881	−	0.909875i	$-0.636177\pi$
$881$	−8.93818e24			−0.829763			−0.414881	+	0.909875i	$0.636177\pi$
$882$		−	4.44729e24i		−	0.408896i
$883$		−	1.46768e25i		−	1.33649i	−0.743943	−	0.668243i	$-0.767047\pi$
$883$		−	1.46768e25i		−	1.33649i	0.743943	−	0.668243i	$-0.232953\pi$
$884$	8.10281e24			0.730788
$885$	−1.41486e24	+	1.75426e24i	−0.126385	+	0.156702i
$886$	−9.61056e24			−0.850280
$887$		−	8.69989e24i		−	0.762365i	−0.924500	−	0.381183i	$-0.875517\pi$
$887$		−	8.69989e24i		−	0.762365i	0.924500	−	0.381183i	$-0.124483\pi$
$888$			1.42088e24i			0.123324i
$889$	8.51423e24			0.731949
$890$	9.43690e23	−	1.17006e24i	0.0803553	−	0.0996309i
$891$	−4.88361e24			−0.411889
$892$			2.38728e24i			0.199435i
$893$			1.58072e25i			1.30803i
$894$	−3.40203e24			−0.278849
$895$	1.24523e25	+	1.00432e25i	1.01101	+	0.815407i
$896$	5.49622e23			0.0442024
$897$		−	5.43339e24i		−	0.432847i
$898$			1.23381e24i			0.0973639i
$899$	−3.74871e24			−0.293038
$900$	−1.05438e24	−	4.86619e24i	−0.0816463	−	0.376814i
$901$	−1.38055e25			−1.05899
$902$		−	1.60975e25i		−	1.22321i
$903$			2.06226e24i			0.155238i
$904$	−3.83267e24			−0.285805
$905$	1.65487e25	+	1.33470e25i	1.22251	+	0.985988i
$906$	3.20196e24			0.234329
$907$		−	1.22949e25i		−	0.891377i	−0.895188	−	0.445689i	$-0.852959\pi$
$907$		−	1.22949e25i		−	0.891377i	0.895188	−	0.445689i	$-0.147041\pi$
$908$			1.11678e25i			0.802119i
$909$	−4.95640e24			−0.352674
$910$	−4.32634e24	+	5.36414e24i	−0.304978	+	0.378136i
$911$	2.07466e25			1.44891			0.724454	−	0.689324i	$-0.242092\pi$
$911$	2.07466e25			1.44891			0.724454	+	0.689324i	$0.242092\pi$
$912$			2.74193e24i			0.189715i
$913$		−	1.73238e25i		−	1.18752i
$914$	−1.04657e25			−0.710767
$915$	−7.01740e24	+	8.70073e24i	−0.472168	+	0.585432i
$916$	1.16221e25			0.774768
$917$			6.95007e24i			0.459039i
$918$			9.74103e24i			0.637444i
$919$	1.71429e25			1.11148			0.555740	−	0.831356i	$-0.312435\pi$
$919$	1.71429e25			1.11148			0.555740	+	0.831356i	$0.312435\pi$
$920$	2.82133e24	+	2.27548e24i	0.181241	+	0.146176i
$921$	−1.11599e25			−0.710322
$922$			1.10919e25i			0.699507i
$923$			9.21801e24i			0.576000i
$924$	2.17596e24			0.134722
$925$	1.16152e25	−	2.51672e24i	0.712559	−	0.154394i
$926$	−7.08899e24			−0.430914
$927$		−	1.97865e25i		−	1.19177i
$928$		−	1.75838e24i		−	0.104943i
$929$	2.04085e25			1.20692			0.603459	−	0.797394i	$-0.293789\pi$
$929$	2.04085e25			1.20692			0.603459	+	0.797394i	$0.293789\pi$
$930$	2.21812e24	+	1.78898e24i	0.129981	+	0.104834i
$931$	2.04849e25			1.18949
$932$			2.46641e24i			0.141916i
$933$			8.53593e23i			0.0486695i
$934$	−1.43440e25			−0.810443
$935$	−1.34347e25	+	1.66574e25i	−0.752191	+	0.932626i
$936$	−6.75054e24			−0.374536
$937$			2.43882e25i			1.34089i	0.741960	+	0.670444i	$0.233897\pi$
$937$			2.43882e25i			1.34089i	−0.741960	+	0.670444i	$0.766103\pi$
$938$			5.42980e24i			0.295842i
$939$	3.82012e24			0.206262
$940$	−4.83766e24	+	5.99811e24i	−0.258850	+	0.320943i
$941$	−3.32898e25			−1.76522			−0.882612	−	0.470102i	$-0.844217\pi$
$941$	−3.32898e25			−1.76522			−0.882612	+	0.470102i	$0.844217\pi$
$942$			9.56315e24i			0.502537i
$943$		−	1.94251e25i		−	1.01161i
$944$	−2.03834e24			−0.105200
$945$	−6.44865e24	−	5.20103e24i	−0.329836	−	0.266023i
$946$	−1.01932e25			−0.516698
$947$			6.45878e24i			0.324471i	0.986752	+	0.162236i	$0.0518704\pi$
$947$			6.45878e24i			0.324471i	−0.986752	+	0.162236i	$0.948130\pi$
$948$		−	6.53946e24i		−	0.325590i
$949$	2.07396e25			1.02338
$950$	2.24144e25	−	4.85664e24i	1.09616	−	0.237512i
$951$	8.05612e24			0.390474
$952$			3.91581e24i			0.188108i
$953$			5.03035e24i			0.239502i	0.992804	+	0.119751i	$0.0382095\pi$
$953$			5.03035e24i			0.239502i	−0.992804	+	0.119751i	$0.961790\pi$
$954$	1.15015e25			0.542742
$955$	−1.47220e25	−	1.18737e25i	−0.688553	−	0.555338i
$956$	1.23826e25			0.574009
$957$		−	6.96146e24i		−	0.319852i
$958$		−	2.34950e24i		−	0.106996i
$959$	−1.53610e23			−0.00693364
$960$	−8.39145e23	+	1.04044e24i	−0.0375432	+	0.0465491i
$961$	−1.70555e25			−0.756337
$962$		−	1.61130e25i		−	0.708252i
$963$			2.69597e25i			1.17461i
$964$	−2.02228e25			−0.873345
$965$	1.71857e25	−	2.13082e25i	0.735673	−	0.912146i
$966$	2.62577e24			0.111417
$967$		−	3.12808e25i		−	1.31569i	−0.753155	−	0.657844i	$-0.771469\pi$
$967$		−	3.12808e25i		−	1.31569i	0.753155	−	0.657844i	$-0.228531\pi$
$968$			2.27521e24i			0.0948596i
$969$	−1.95351e25			−0.807352
$970$	−1.72054e25	−	1.38766e25i	−0.704862	−	0.568492i
$971$	3.53883e25			1.43713			0.718566	−	0.695459i	$-0.244799\pi$
$971$	3.53883e25			1.43713			0.718566	+	0.695459i	$0.244799\pi$
$972$		−	1.26974e25i		−	0.511154i
$973$		−	1.70257e25i		−	0.679431i
$974$	−9.19841e24			−0.363883
$975$	−3.54905e24	−	1.63796e25i	−0.139179	−	0.642338i
$976$	−1.01097e25			−0.393021
$977$			1.94607e24i			0.0749989i	0.999297	+	0.0374995i	$0.0119392\pi$
$977$			1.94607e24i			0.0749989i	−0.999297	+	0.0374995i	$0.988061\pi$
$978$			9.04345e24i			0.345505i
$979$	5.38245e24			0.203858
$980$	7.77307e24	+	6.26921e24i	0.291858	+	0.235392i
$981$	−1.25957e25			−0.468853
$982$			1.93026e25i			0.712308i
$983$		−	2.46859e25i		−	0.903117i	−0.892242	−	0.451558i	$-0.850868\pi$
$983$		−	2.46859e25i		−	0.903117i	0.892242	−	0.451558i	$-0.149132\pi$
$984$	7.16351e24			0.259817
$985$	−2.74233e25	+	3.40016e25i	−0.986079	+	1.22262i
$986$	1.25277e25			0.446598
$987$			5.58237e24i			0.197298i
$988$		−	3.10940e25i		−	1.08954i
$989$	−1.23003e25			−0.427315
$990$	1.11926e25	−	1.38774e25i	0.385505	−	0.477980i
$991$	−1.11187e25			−0.379687			−0.189844	−	0.981814i	$-0.560798\pi$
$991$	−1.11187e25			−0.379687			−0.189844	+	0.981814i	$0.560798\pi$
$992$			2.57732e24i			0.0872609i
$993$			3.15014e24i			0.105745i
$994$	−4.45475e24			−0.148265
$995$	−2.82709e25	−	2.28014e25i	−0.932919	−	0.752427i
$996$	7.70924e24			0.252236
$997$			4.78013e25i			1.55071i	0.631524	+	0.775356i	$0.282430\pi$
$997$			4.78013e25i			1.55071i	−0.631524	+	0.775356i	$0.717570\pi$
$998$			6.94403e24i			0.223358i
$999$	1.93707e25			0.617786

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10.18.b.a.9.3	✓	8
3.2	odd	2		90.18.c.b.19.7		8
4.3	odd	2		80.18.c.a.49.3		8
5.2	odd	4		50.18.a.k.1.3		4
5.3	odd	4		50.18.a.j.1.2		4
5.4	even	2	inner	10.18.b.a.9.6	yes	8
15.14	odd	2		90.18.c.b.19.3		8
20.19	odd	2		80.18.c.a.49.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.18.b.a.9.3	✓	8	1.1	even	1	trivial
10.18.b.a.9.6	yes	8	5.4	even	2	inner
50.18.a.j.1.2		4	5.3	odd	4
50.18.a.k.1.3		4	5.2	odd	4
80.18.c.a.49.3		8	4.3	odd	2
80.18.c.a.49.6		8	20.19	odd	2
90.18.c.b.19.3		8	15.14	odd	2
90.18.c.b.19.7		8	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 10 = 2 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	10.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-256.000i q^{2} +5436.73i q^{3} -65536.0 q^{4} +(-679890. - 548351. i) q^{5} +1.39180e6 q^{6} -7.62753e6i q^{7} +1.67772e7i q^{8} +9.95821e7 q^{9} +O(q^{10})\)	\(q-256.000i q^{2} +5436.73i q^{3} -65536.0 q^{4} +(-679890. - 548351. i) q^{5} +1.39180e6 q^{6} -7.62753e6i q^{7} +1.67772e7i q^{8} +9.95821e7 q^{9} +(-1.40378e8 + 1.74052e8i) q^{10} -8.00662e8 q^{11} -3.56302e8i q^{12} +4.04052e9i q^{13} -1.95265e9 q^{14} +(2.98124e9 - 3.69638e9i) q^{15} +4.29497e9 q^{16} +3.05998e10i q^{17} -2.54930e10i q^{18} +1.17425e11 q^{19} +(4.45573e10 + 3.59368e10i) q^{20} +4.14689e10 q^{21} +2.04970e11i q^{22} +2.47340e11i q^{23} -9.12133e10 q^{24} +(1.61561e11 + 7.45637e11i) q^{25} +1.03437e12 q^{26} +1.24350e12i q^{27} +4.99878e11i q^{28} +1.59924e12 q^{29} +(-9.46273e11 - 7.63197e11i) q^{30} -2.34406e12 q^{31} -1.09951e12i q^{32} -4.35299e12i q^{33} +7.83354e12 q^{34} +(-4.18257e12 + 5.18588e12i) q^{35} -6.52621e12 q^{36} -1.55775e13i q^{37} -3.00607e13i q^{38} -2.19672e13 q^{39} +(9.19981e12 - 1.14067e13i) q^{40} -7.85358e13 q^{41} -1.06160e13i q^{42} +4.97304e13i q^{43} +5.24722e13 q^{44} +(-6.77049e13 - 5.46060e13i) q^{45} +6.33191e13 q^{46} +1.34616e14i q^{47} +2.33506e13i q^{48} +1.74451e14 q^{49} +(1.90883e14 - 4.13596e13i) q^{50} -1.66363e14 q^{51} -2.64800e14i q^{52} +4.51163e14i q^{53} +3.18336e14 q^{54} +(5.44362e14 + 4.39044e14i) q^{55} +1.27969e14 q^{56} +6.38406e14i q^{57} -4.09404e14i q^{58} -4.74588e14 q^{59} +(-1.95379e14 + 2.42246e14i) q^{60} -2.35385e15 q^{61} +6.00080e14i q^{62} -7.59565e14i q^{63} -2.81475e14 q^{64} +(2.21563e15 - 2.74711e15i) q^{65} -1.11436e15 q^{66} -2.78074e15i q^{67} -2.00539e15i q^{68} -1.34472e15 q^{69} +(1.32759e15 + 1.07074e15i) q^{70} +2.28139e15 q^{71} +1.67071e15i q^{72} -5.13289e15i q^{73} -3.98785e15 q^{74} +(-4.05383e15 + 8.78364e14i) q^{75} -7.69554e15 q^{76} +6.10708e15i q^{77} +5.62362e15i q^{78} +1.83537e16 q^{79} +(-2.92010e15 - 2.35515e15i) q^{80} +6.09946e15 q^{81} +2.01052e16i q^{82} +2.16368e16i q^{83} -2.71770e15 q^{84} +(1.67794e16 - 2.08045e16i) q^{85} +1.27310e16 q^{86} +8.69462e15i q^{87} -1.34329e16i q^{88} -6.72250e15 q^{89} +(-1.39791e16 + 1.73324e16i) q^{90} +3.08192e16 q^{91} -1.62097e16i q^{92} -1.27440e16i q^{93} +3.44617e16 q^{94} +(-7.98358e16 - 6.43899e16i) q^{95} +5.97775e15 q^{96} +9.88520e16i q^{97} -4.46595e16i q^{98} -7.97316e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 524288 q^{4} - 1225560 q^{5} - 974848 q^{6} - 363182504 q^{9}+O(q^{10}) \) 8 * q - 524288 * q^4 - 1225560 * q^5 - 974848 * q^6 - 363182504 * q^9	\( 8 q - 524288 q^{4} - 1225560 q^{5} - 974848 q^{6} - 363182504 q^{9} - 140779520 q^{10} + 146232096 q^{11} + 14260494336 q^{14} - 39815002720 q^{15} + 34359738368 q^{16} - 54264178080 q^{19} + 80318300160 q^{20} + 515442333056 q^{21} + 63887638528 q^{24} + 1013225778600 q^{25} - 2693383569408 q^{26} + 1660243083120 q^{29} + 4536489205760 q^{30} - 6055476993664 q^{31} + 14158246445056 q^{34} - 8725233780960 q^{35} + 23801528582144 q^{36} - 60047234232768 q^{39} + 9226126622720 q^{40} - 219921829971984 q^{41} - 9583466643456 q^{44} + 503517880841080 q^{45} - 136753191067648 q^{46} + 797208944041464 q^{49} + 535409908531200 q^{50} - 26\!\cdots\!24 q^{51}+ \cdots - 23\!\cdots\!48 q^{99}+O(q^{100}) \) 8 * q - 524288 * q^4 - 1225560 * q^5 - 974848 * q^6 - 363182504 * q^9 - 140779520 * q^10 + 146232096 * q^11 + 14260494336 * q^14 - 39815002720 * q^15 + 34359738368 * q^16 - 54264178080 * q^19 + 80318300160 * q^20 + 515442333056 * q^21 + 63887638528 * q^24 + 1013225778600 * q^25 - 2693383569408 * q^26 + 1660243083120 * q^29 + 4536489205760 * q^30 - 6055476993664 * q^31 + 14158246445056 * q^34 - 8725233780960 * q^35 + 23801528582144 * q^36 - 60047234232768 * q^39 + 9226126622720 * q^40 - 219921829971984 * q^41 - 9583466643456 * q^44 + 503517880841080 * q^45 - 136753191067648 * q^46 + 797208944041464 * q^49 + 535409908531200 * q^50 - 2629370351530624 * q^51 + 1870704820633600 * q^54 + 1370380127176480 * q^55 - 934575756804096 * q^56 - 155410557761760 * q^59 + 2609316018257920 * q^60 - 6350444068968944 * q^61 - 2251799813685248 * q^64 + 10508144084782080 * q^65 - 10180424127512576 * q^66 + 20821360231259392 * q^69 - 1051117677240320 * q^70 - 33149026543733184 * q^71 + 6017648345726976 * q^74 + 17427320439323200 * q^75 + 3556257174650880 * q^76 - 19354668791708160 * q^79 - 5263740119285760 * q^80 + 54432374624835848 * q^81 - 33780028739158016 * q^84 - 71530234876895360 * q^85 + 29616028444434432 * q^86 - 43569230299558320 * q^89 - 33153706682736640 * q^90 + 333420307889971776 * q^91 - 11440296922292224 * q^94 - 131478039235197600 * q^95 - 4186940278571008 * q^96 - 236681336419250848 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more