LMFDB - Embedded newform 45.18.b.d.19.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,18,Mod(19,45)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("45.19");

S:= CuspForms(chi, 18);

N := Newforms(S);

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 18 $
Character orbit:	$[\chi]$	$=$	45.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:	$82.4499393051$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 1479925 x^{14} + 856740725236 x^{12} + \cdots + 35\!\cdots\!36 $ x^16 + 1479925x^14 + 856740725236x^12 + 246346732853742400x^10 + 36846316485794632155136x^8 + 2705282734342187411139788800x^6 + 81161927553510012423196446294016x^4 + 937303132294728639909584019914752000*x^2 + 3553135874433297828333352298414117748736
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{35}\cdot 3^{62}\cdot 5^{23} $
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			19.4
Root			$-419.421i$ of defining polynomial
Character	$\chi$	$=$	45.19
Dual form			45.18.b.d.19.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-419.421i q^{2} -44841.8 q^{4} +(808072. - 331600. i) q^{5} +1.07280e7i q^{7} -3.61668e7i q^{8} +O(q^{10})$	\(q-419.421i q^{2} -44841.8 q^{4} +(808072. - 331600. i) q^{5} +1.07280e7i q^{7} -3.61668e7i q^{8} +(-1.39080e8 - 3.38922e8i) q^{10} +3.34405e8 q^{11} -1.71011e8i q^{13} +4.49954e9 q^{14} -2.10466e10 q^{16} +3.73698e9i q^{17} -3.46862e10 q^{19} +(-3.62354e10 + 1.48695e10i) q^{20} -1.40256e11i q^{22} -5.95580e10i q^{23} +(5.43022e11 - 5.35914e11i) q^{25} -7.17257e10 q^{26} -4.81062e11i q^{28} -2.08236e12 q^{29} +5.85106e12 q^{31} +4.08693e12i q^{32} +1.56737e12 q^{34} +(3.55740e12 + 8.66899e12i) q^{35} -2.29506e13i q^{37} +1.45481e13i q^{38} +(-1.19929e13 - 2.92253e13i) q^{40} +3.39080e13 q^{41} -7.65153e13i q^{43} -1.49953e13 q^{44} -2.49798e13 q^{46} -2.72333e14i q^{47} +1.17541e14 q^{49} +(-2.24773e14 - 2.27755e14i) q^{50} +7.66845e12i q^{52} -4.91865e14i q^{53} +(2.70223e14 - 1.10889e14i) q^{55} +3.87996e14 q^{56} +8.73384e14i q^{58} +2.15693e15 q^{59} -1.43236e15 q^{61} -2.45405e15i q^{62} -1.04448e15 q^{64} +(-5.67074e13 - 1.38190e14i) q^{65} -3.26290e15i q^{67} -1.67573e14i q^{68} +(3.63595e15 - 1.49205e15i) q^{70} -1.54056e15 q^{71} +1.33066e15i q^{73} -9.62597e15 q^{74} +1.55539e15 q^{76} +3.58749e15i q^{77} -2.64034e16 q^{79} +(-1.70072e16 + 6.97905e15i) q^{80} -1.42217e16i q^{82} +3.48504e16i q^{83} +(1.23918e15 + 3.01975e15i) q^{85} -3.20921e16 q^{86} -1.20943e16i q^{88} +6.75463e16 q^{89} +1.83461e15 q^{91} +2.67068e15i q^{92} -1.14222e17 q^{94} +(-2.80289e16 + 1.15019e16i) q^{95} +7.30588e16i q^{97} -4.92991e16i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 862698 q^{4} - 292740 q^{5}+O(q^{10}) $ 16 * q - 862698 * q^4 - 292740 * q^5	$ 16 q - 862698 q^{4} - 292740 q^{5} - 158256910 q^{10} - 907386144 q^{11} - 38248328748 q^{14} + 64410639650 q^{16} + 44375877728 q^{19} - 124652464020 q^{20} + 1631404456600 q^{25} - 2940223801452 q^{26} + 5348468604504 q^{29} - 16935964052224 q^{31} + 60848757680284 q^{34} - 25456583995440 q^{35} + 246023763543250 q^{40} + 4873344779184 q^{41} - 224598182313996 q^{44} - 10\!\cdots\!24 q^{46}+ \cdots + 43\!\cdots\!00 q^{95}+O(q^{100}) $ 16 * q - 862698 * q^4 - 292740 * q^5 - 158256910 * q^10 - 907386144 * q^11 - 38248328748 * q^14 + 64410639650 * q^16 + 44375877728 * q^19 - 124652464020 * q^20 + 1631404456600 * q^25 - 2940223801452 * q^26 + 5348468604504 * q^29 - 16935964052224 * q^31 + 60848757680284 * q^34 - 25456583995440 * q^35 + 246023763543250 * q^40 + 4873344779184 * q^41 - 224598182313996 * q^44 - 1042392874396424 * q^46 + 85575569613760 * q^49 - 669894278364600 * q^50 + 2661435553553280 * q^55 + 8893968141042660 * q^56 - 2882362957102272 * q^59 - 6340947108203680 * q^61 + 2538830661242878 * q^64 - 13690132732535880 * q^65 + 10824491372201640 * q^70 + 11739285749899008 * q^71 + 6749027807219172 * q^74 - 32687788759104192 * q^76 + 28655274740363840 * q^79 + 7449390713147460 * q^80 + 27636069676463240 * q^85 - 261659254088221464 * q^86 + 119080233211377312 * q^89 + 201912942604136256 * q^91 - 186654825194113808 * q^94 + 434666709576297600 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$.

$n$	$11$	$37$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$		−	419.421i		−	1.15850i	−0.815151	−	0.579249i	$-0.803346\pi$
$2$		−	419.421i		−	1.15850i	0.815151	−	0.579249i	$-0.196654\pi$
$3$		0			0
$3$		0			0
$4$	−44841.8			−0.342115
$5$	808072.	−	331600.i	0.925135	−	0.379638i
$5$	808072.	−	331600.i	0.925135	−	0.379638i
$6$		0			0
$7$			1.07280e7i			0.703372i	0.936118	+	0.351686i	$0.114391\pi$
$7$			1.07280e7i			0.703372i	−0.936118	+	0.351686i	$0.885609\pi$
$8$		−	3.61668e7i		−	0.762157i
$9$		0			0
$10$	−1.39080e8	−	3.38922e8i	−0.439809	−	1.07177i
$11$	3.34405e8			0.470365			0.235182	−	0.971951i	$-0.424431\pi$
$11$	3.34405e8			0.470365			0.235182	+	0.971951i	$0.424431\pi$
$12$		0			0
$13$		−	1.71011e8i		−	0.0581442i	−0.999577	−	0.0290721i	$-0.990745\pi$
$13$		−	1.71011e8i		−	0.0581442i	0.999577	−	0.0290721i	$-0.00925524\pi$
$14$	4.49954e9			0.814854
$15$		0			0
$16$	−2.10466e10			−1.22507
$17$			3.73698e9i			0.129929i	0.997888	+	0.0649643i	$0.0206934\pi$
$17$			3.73698e9i			0.129929i	−0.997888	+	0.0649643i	$0.979307\pi$
$18$		0			0
$19$	−3.46862e10			−0.468545			−0.234272	−	0.972171i	$-0.575271\pi$
$19$	−3.46862e10			−0.468545			−0.234272	+	0.972171i	$0.575271\pi$
$20$	−3.62354e10	+	1.48695e10i	−0.316503	+	0.129880i
$21$		0			0
$22$		−	1.40256e11i		−	0.544916i
$23$		−	5.95580e10i		−	0.158582i	−0.996852	−	0.0792909i	$-0.974734\pi$
$23$		−	5.95580e10i		−	0.158582i	0.996852	−	0.0792909i	$-0.0252656\pi$
$24$		0			0
$25$	5.43022e11	−	5.35914e11i	0.711750	−	0.702433i
$26$	−7.17257e10			−0.0673599
$27$		0			0
$28$		−	4.81062e11i		−	0.240634i
$29$	−2.08236e12			−0.772987			−0.386494	−	0.922292i	$-0.626314\pi$
$29$	−2.08236e12			−0.772987			−0.386494	+	0.922292i	$0.626314\pi$
$30$		0			0
$31$	5.85106e12			1.23214			0.616070	−	0.787692i	$-0.288724\pi$
$31$	5.85106e12			1.23214			0.616070	+	0.787692i	$0.288724\pi$
$32$			4.08693e12i			0.657086i
$33$		0			0
$34$	1.56737e12			0.150522
$35$	3.55740e12	+	8.66899e12i	0.267027	+	0.650714i
$36$		0			0
$37$		−	2.29506e13i		−	1.07419i	−0.843523	−	0.537093i	$-0.819522\pi$
$37$		−	2.29506e13i		−	1.07419i	0.843523	−	0.537093i	$-0.180478\pi$
$38$			1.45481e13i			0.542808i
$39$		0			0
$40$	−1.19929e13	−	2.92253e13i	−0.289344	−	0.705098i
$41$	3.39080e13			0.663193			0.331596	−	0.943421i	$-0.392413\pi$
$41$	3.39080e13			0.663193			0.331596	+	0.943421i	$0.392413\pi$
$42$		0			0
$43$		−	7.65153e13i		−	0.998312i	−0.866512	−	0.499156i	$-0.833643\pi$
$43$		−	7.65153e13i		−	0.998312i	0.866512	−	0.499156i	$-0.166357\pi$
$44$	−1.49953e13			−0.160919
$45$		0			0
$46$	−2.49798e13			−0.183716
$47$		−	2.72333e14i		−	1.66828i	−0.551556	−	0.834138i	$-0.685966\pi$
$47$		−	2.72333e14i		−	1.66828i	0.551556	−	0.834138i	$-0.314034\pi$
$48$		0			0
$49$	1.17541e14			0.505268
$50$	−2.24773e14	−	2.27755e14i	−0.813766	−	0.824560i
$51$		0			0
$52$			7.66845e12i			0.0198920i
$53$		−	4.91865e14i		−	1.08518i	−0.839998	−	0.542589i	$-0.817444\pi$
$53$		−	4.91865e14i		−	1.08518i	0.839998	−	0.542589i	$-0.182556\pi$
$54$		0			0
$55$	2.70223e14	−	1.10889e14i	0.435151	−	0.178568i
$56$	3.87996e14			0.536080
$57$		0			0
$58$			8.73384e14i			0.895503i
$59$	2.15693e15			1.91247			0.956234	−	0.292604i	$-0.0945219\pi$
$59$	2.15693e15			1.91247			0.956234	+	0.292604i	$0.0945219\pi$
$60$		0			0
$61$	−1.43236e15			−0.956637			−0.478319	−	0.878186i	$-0.658754\pi$
$61$	−1.43236e15			−0.956637			−0.478319	+	0.878186i	$0.658754\pi$
$62$		−	2.45405e15i		−	1.42743i
$63$		0			0
$64$	−1.04448e15			−0.463841
$65$	−5.67074e13	−	1.38190e14i	−0.0220738	−	0.0537913i
$66$		0			0
$67$		−	3.26290e15i		−	0.981676i	−0.871251	−	0.490838i	$-0.836691\pi$
$67$		−	3.26290e15i		−	0.981676i	0.871251	−	0.490838i	$-0.163309\pi$
$68$		−	1.67573e14i		−	0.0444506i
$69$		0			0
$70$	3.63595e15	−	1.49205e15i	0.753850	−	0.309350i
$71$	−1.54056e15			−0.283128			−0.141564	−	0.989929i	$-0.545213\pi$
$71$	−1.54056e15			−0.283128			−0.141564	+	0.989929i	$0.545213\pi$
$72$		0			0
$73$			1.33066e15i			0.193118i	0.995327	+	0.0965591i	$0.0307837\pi$
$73$			1.33066e15i			0.193118i	−0.995327	+	0.0965591i	$0.969216\pi$
$74$	−9.62597e15			−1.24444
$75$		0			0
$76$	1.55539e15			0.160296
$77$			3.58749e15i			0.330841i
$78$		0			0
$79$	−2.64034e16			−1.95808			−0.979039	−	0.203672i	$-0.934712\pi$
$79$	−2.64034e16			−1.95808			−0.979039	+	0.203672i	$0.934712\pi$
$80$	−1.70072e16	+	6.97905e15i	−1.13336	+	0.465084i
$81$		0			0
$82$		−	1.42217e16i		−	0.768307i
$83$			3.48504e16i			1.69842i	0.528057	+	0.849209i	$0.322921\pi$
$83$			3.48504e16i			1.69842i	−0.528057	+	0.849209i	$0.677079\pi$
$84$		0			0
$85$	1.23918e15	+	3.01975e15i	0.0493259	+	0.120202i
$86$	−3.20921e16			−1.15654
$87$		0			0
$88$		−	1.20943e16i		−	0.358492i
$89$	6.75463e16			1.81881			0.909403	−	0.415917i	$-0.136539\pi$
$89$	6.75463e16			1.81881			0.909403	+	0.415917i	$0.136539\pi$
$90$		0			0
$91$	1.83461e15			0.0408970
$92$			2.67068e15i			0.0542533i
$93$		0			0
$94$	−1.14222e17			−1.93269
$95$	−2.80289e16	+	1.15019e16i	−0.433467	+	0.177877i
$96$		0			0
$97$			7.30588e16i			0.946482i	0.880933	+	0.473241i	$0.156916\pi$
$97$			7.30588e16i			0.946482i	−0.880933	+	0.473241i	$0.843084\pi$
$98$		−	4.92991e16i		−	0.585352i
$99$		0			0
$100$	−2.43501e16	+	2.40313e16i	−0.243501	+	0.240313i
$101$	−1.78572e17			−1.64090			−0.820448	−	0.571722i	$-0.806276\pi$
$101$	−1.78572e17			−1.64090			−0.820448	+	0.571722i	$0.806276\pi$
$102$		0			0
$103$		−	1.71813e16i		−	0.133641i	−0.997765	−	0.0668205i	$-0.978715\pi$
$103$		−	1.71813e16i		−	0.133641i	0.997765	−	0.0668205i	$-0.0212855\pi$
$104$	−6.18493e15			−0.0443150
$105$		0			0
$106$	−2.06298e17			−1.25718
$107$		−	2.34539e17i		−	1.31963i	−0.751427	−	0.659816i	$-0.770634\pi$
$107$		−	2.34539e17i		−	1.31963i	0.751427	−	0.659816i	$-0.229366\pi$
$108$		0			0
$109$	−1.61438e17			−0.776032			−0.388016	−	0.921653i	$-0.626839\pi$
$109$	−1.61438e17			−0.776032			−0.388016	+	0.921653i	$0.626839\pi$
$110$	−4.65090e16	−	1.13337e17i	−0.206871	−	0.504121i
$111$		0			0
$112$		−	2.25787e17i		−	0.861681i
$113$		−	2.96020e17i		−	1.04750i	−0.851872	−	0.523751i	$-0.824532\pi$
$113$		−	2.96020e17i		−	1.04750i	0.851872	−	0.523751i	$-0.175468\pi$
$114$		0			0
$115$	−1.97494e16	−	4.81271e16i	−0.0602037	−	0.146710i
$116$	9.33766e16			0.264451
$117$		0			0
$118$		−	9.04661e17i		−	2.21559i
$119$	−4.00903e16			−0.0913882
$120$		0			0
$121$	−3.93620e17			−0.778757
$122$			6.00760e17i			1.10826i
$123$		0			0
$124$	−2.62372e17			−0.421534
$125$	2.61092e17	−	6.13123e17i	0.391795	−	0.920053i
$126$		0			0
$127$			1.16061e17i			0.152180i	0.997101	+	0.0760898i	$0.0242436\pi$
$127$			1.16061e17i			0.152180i	−0.997101	+	0.0760898i	$0.975756\pi$
$128$			9.73757e17i			1.19444i
$129$		0			0
$130$	−5.79596e16	+	2.37843e16i	−0.0623170	+	0.0255724i
$131$	−3.06668e17			−0.308932			−0.154466	−	0.987998i	$-0.549366\pi$
$131$	−3.06668e17			−0.308932			−0.154466	+	0.987998i	$0.549366\pi$
$132$		0			0
$133$		−	3.72113e17i		−	0.329561i
$134$	−1.36853e18			−1.13727
$135$		0			0
$136$	1.35154e17			0.0990261
$137$		−	1.12091e18i		−	0.771695i	−0.922563	−	0.385848i	$-0.873909\pi$
$137$		−	1.12091e18i		−	0.771695i	0.922563	−	0.385848i	$-0.126091\pi$
$138$		0			0
$139$	1.86832e18			1.13717			0.568585	−	0.822624i	$-0.307491\pi$
$139$	1.86832e18			1.13717			0.568585	+	0.822624i	$0.307491\pi$
$140$	−1.59520e17	−	3.88733e17i	−0.0913539	−	0.222619i
$141$		0			0
$142$			6.46142e17i			0.328002i
$143$		−	5.71870e16i		−	0.0273490i
$144$		0			0
$145$	−1.68270e18	+	6.90510e17i	−0.715118	+	0.293455i
$146$	5.58107e17			0.223727
$147$		0			0
$148$			1.02915e18i			0.367496i
$149$	−3.24828e17			−0.109539			−0.0547696	−	0.998499i	$-0.517442\pi$
$149$	−3.24828e17			−0.109539			−0.0547696	+	0.998499i	$0.517442\pi$
$150$		0			0
$151$	−5.29857e18			−1.59535			−0.797673	−	0.603090i	$-0.793936\pi$
$151$	−5.29857e18			−1.59535			−0.797673	+	0.603090i	$0.793936\pi$
$152$			1.25449e18i			0.357105i
$153$		0			0
$154$	1.50467e18			0.383279
$155$	4.72808e18	−	1.94021e18i	1.13990	−	0.467767i
$156$		0			0
$157$			6.65127e18i			1.43800i	0.695012	+	0.718998i	$0.255399\pi$
$157$			6.65127e18i			1.43800i	−0.695012	+	0.718998i	$0.744601\pi$
$158$			1.10741e19i			2.26843i
$159$		0			0
$160$	1.35523e18	+	3.30253e18i	0.249455	+	0.607893i
$161$	6.38937e17			0.111542
$162$		0			0
$163$		−	2.57771e18i		−	0.405172i	−0.979265	−	0.202586i	$-0.935065\pi$
$163$		−	2.57771e18i		−	0.405172i	0.979265	−	0.202586i	$-0.0649345\pi$
$164$	−1.52050e18			−0.226889
$165$		0			0
$166$	1.46170e19			1.96761
$167$		−	5.28879e18i		−	0.676498i	−0.941057	−	0.338249i	$-0.890165\pi$
$167$		−	5.28879e18i		−	0.676498i	0.941057	−	0.338249i	$-0.109835\pi$
$168$		0			0
$169$	8.62117e18			0.996619
$170$	1.26655e18	−	5.19739e17i	0.139253	−	0.0571439i
$171$		0			0
$172$			3.43108e18i			0.341538i
$173$			5.73330e18i			0.543266i	0.962401	+	0.271633i	$0.0875637\pi$
$173$			5.73330e18i			0.543266i	−0.962401	+	0.271633i	$0.912436\pi$
$174$		0			0
$175$	5.74928e18	+	5.82553e18i	0.494071	+	0.500625i
$176$	−7.03808e18			−0.576231
$177$		0			0
$178$		−	2.83303e19i		−	2.10708i
$179$	1.50499e19			1.06729			0.533645	−	0.845709i	$-0.320822\pi$
$179$	1.50499e19			1.06729			0.533645	+	0.845709i	$0.320822\pi$
$180$		0			0
$181$	1.29917e19			0.838296			0.419148	−	0.907918i	$-0.362329\pi$
$181$	1.29917e19			0.838296			0.419148	+	0.907918i	$0.362329\pi$
$182$		−	7.69473e17i		−	0.0473791i
$183$		0			0
$184$	−2.15402e18			−0.120864
$185$	−7.61043e18	−	1.85458e19i	−0.407802	−	0.993768i
$186$		0			0
$187$			1.24966e18i			0.0611139i
$188$			1.22119e19i			0.570743i
$189$		0			0
$190$	4.82415e18	+	1.17559e19i	0.206070	+	0.502170i
$191$	2.59643e19			1.06070			0.530350	−	0.847779i	$-0.322061\pi$
$191$	2.59643e19			1.06070			0.530350	+	0.847779i	$0.322061\pi$
$192$		0			0
$193$		−	2.61210e19i		−	0.976681i	−0.872653	−	0.488341i	$-0.837602\pi$
$193$		−	2.61210e19i		−	0.976681i	0.872653	−	0.488341i	$-0.162398\pi$
$194$	3.06424e19			1.09650
$195$		0			0
$196$	−5.27074e18			−0.172860
$197$		−	7.78588e18i		−	0.244537i	−0.992497	−	0.122268i	$-0.960983\pi$
$197$		−	7.78588e18i		−	0.244537i	0.992497	−	0.122268i	$-0.0390169\pi$
$198$		0			0
$199$	−6.77156e18			−0.195181			−0.0975905	−	0.995227i	$-0.531114\pi$
$199$	−6.77156e18			−0.195181			−0.0975905	+	0.995227i	$0.531114\pi$
$200$	−1.93823e19	−	1.96393e19i	−0.535364	−	0.542465i
$201$		0			0
$202$			7.48967e19i			1.90097i
$203$		−	2.23395e19i		−	0.543697i
$204$		0			0
$205$	2.74001e19	−	1.12439e19i	0.613543	−	0.251773i
$206$	−7.20620e18			−0.154823
$207$		0			0
$208$			3.59921e18i			0.0712309i
$209$	−1.15992e19			−0.220387
$210$		0			0
$211$	2.18258e19			0.382445			0.191222	−	0.981547i	$-0.438755\pi$
$211$	2.18258e19			0.382445			0.191222	+	0.981547i	$0.438755\pi$
$212$			2.20561e19i			0.371256i
$213$		0			0
$214$	−9.83705e19			−1.52879
$215$	−2.53725e19	−	6.18299e19i	−0.378997	−	0.923573i
$216$		0			0
$217$			6.27701e19i			0.866652i
$218$			6.77103e19i			0.899031i
$219$		0			0
$220$	−1.21173e19	+	4.97244e18i	−0.148872	+	0.0610910i
$221$	6.39066e17			0.00755460
$222$		0			0
$223$		−	1.29260e20i		−	1.41538i	−0.706522	−	0.707691i	$-0.749737\pi$
$223$		−	1.29260e20i		−	1.41538i	0.706522	−	0.707691i	$-0.250263\pi$
$224$	−4.38445e19			−0.462175
$225$		0			0
$226$	−1.24157e20			−1.21353
$227$			8.45139e19i			0.795625i	0.917467	+	0.397813i	$0.130231\pi$
$227$			8.45139e19i			0.795625i	−0.917467	+	0.397813i	$0.869769\pi$
$228$		0			0
$229$	1.99781e19			0.174564			0.0872818	−	0.996184i	$-0.472182\pi$
$229$	1.99781e19			0.174564			0.0872818	+	0.996184i	$0.472182\pi$
$230$	−2.01855e19	+	8.28332e18i	−0.169963	+	0.0697458i
$231$		0			0
$232$			7.53121e19i			0.589138i
$233$			1.10729e20i			0.835096i	0.908655	+	0.417548i	$0.137110\pi$
$233$			1.10729e20i			0.835096i	−0.908655	+	0.417548i	$0.862890\pi$
$234$		0			0
$235$	−9.03055e19	−	2.20064e20i	−0.633341	−	1.54338i
$236$	−9.67205e19			−0.654285
$237$		0			0
$238$			1.68147e19i			0.105873i
$239$	−2.16598e20			−1.31605			−0.658024	−	0.752997i	$-0.728608\pi$
$239$	−2.16598e20			−1.31605			−0.658024	+	0.752997i	$0.728608\pi$
$240$		0			0
$241$	−1.37231e20			−0.776799			−0.388400	−	0.921491i	$-0.626972\pi$
$241$	−1.37231e20			−0.776799			−0.388400	+	0.921491i	$0.626972\pi$
$242$			1.65093e20i			0.902188i
$243$		0			0
$244$	6.42294e19			0.327280
$245$	9.49815e19	−	3.89765e19i	0.467441	−	0.191819i
$246$		0			0
$247$			5.93173e18i			0.0272432i
$248$		−	2.11614e20i		−	0.939084i
$249$		0			0
$250$	−2.57157e20	−	1.09507e20i	−1.06588	−	0.453893i
$251$	6.71142e19			0.268898			0.134449	−	0.990921i	$-0.457074\pi$
$251$	6.71142e19			0.268898			0.134449	+	0.990921i	$0.457074\pi$
$252$		0			0
$253$		−	1.99165e19i		−	0.0745913i
$254$	4.86786e19			0.176300
$255$		0			0
$256$	2.71512e20			0.919919
$257$		−	8.17440e19i		−	0.267932i	−0.990986	−	0.133966i	$-0.957229\pi$
$257$		−	8.17440e19i		−	0.267932i	0.990986	−	0.133966i	$-0.0427713\pi$
$258$		0			0
$259$	2.46214e20			0.755553
$260$	2.54286e18	+	6.19666e18i	0.00755177	+	0.0184028i
$261$		0			0
$262$			1.28623e20i			0.357896i
$263$		−	5.79577e20i		−	1.56130i	−0.624966	−	0.780652i	$-0.714887\pi$
$263$		−	5.79577e20i		−	1.56130i	0.624966	−	0.780652i	$-0.285113\pi$
$264$		0			0
$265$	−1.63102e20	−	3.97462e20i	−0.411975	−	1.00394i
$266$	−1.56072e20			−0.381795
$267$		0			0
$268$			1.46314e20i			0.335847i
$269$	1.64818e20			0.366530			0.183265	−	0.983064i	$-0.441333\pi$
$269$	1.64818e20			0.366530			0.183265	+	0.983064i	$0.441333\pi$
$270$		0			0
$271$	−3.20470e20			−0.669187			−0.334594	−	0.942362i	$-0.608599\pi$
$271$	−3.20470e20			−0.669187			−0.334594	+	0.942362i	$0.608599\pi$
$272$		−	7.86507e19i		−	0.159172i
$273$		0			0
$274$	−4.70133e20			−0.894007
$275$	1.81589e20	−	1.79212e20i	0.334782	−	0.330400i
$276$		0			0
$277$			8.56736e20i			1.48514i	0.669766	+	0.742572i	$0.266394\pi$
$277$			8.56736e20i			1.48514i	−0.669766	+	0.742572i	$0.733606\pi$
$278$		−	7.83613e20i		−	1.31741i
$279$		0			0
$280$	3.13529e20	−	1.28660e20i	0.495946	−	0.203516i
$281$	7.81806e20			1.19976			0.599881	−	0.800089i	$-0.295214\pi$
$281$	7.81806e20			1.19976			0.599881	+	0.800089i	$0.295214\pi$
$282$		0			0
$283$		−	7.37354e18i		−	0.0106535i	−0.999986	−	0.00532674i	$-0.998304\pi$
$283$		−	7.37354e18i		−	0.0106535i	0.999986	−	0.00532674i	$-0.00169556\pi$
$284$	6.90813e19			0.0968623
$285$		0			0
$286$	−2.39854e19			−0.0316837
$287$			3.63765e20i			0.466471i
$288$		0			0
$289$	8.13275e20			0.983119
$290$	2.89614e20	+	7.05757e20i	0.339967	+	0.828462i
$291$		0			0
$292$		−	5.96692e19i		−	0.0660687i
$293$			2.80055e20i			0.301209i	0.988594	+	0.150605i	$0.0481220\pi$
$293$			2.80055e20i			0.301209i	−0.988594	+	0.150605i	$0.951878\pi$
$294$		0			0
$295$	1.74296e21	−	7.15238e20i	1.76929	−	0.726045i
$296$	−8.30049e20			−0.818699
$297$		0			0
$298$			1.36239e20i			0.126901i
$299$	−1.01851e19			−0.00922061
$300$		0			0
$301$	8.20855e20			0.702184
$302$			2.22233e21i			1.84820i
$303$		0			0
$304$	7.30026e20			0.574001
$305$	−1.15745e21	+	4.74969e20i	−0.885019	+	0.363176i
$306$		0			0
$307$			2.37978e21i			1.72132i	0.509180	+	0.860660i	$0.329949\pi$
$307$			2.37978e21i			1.72132i	−0.509180	+	0.860660i	$0.670051\pi$
$308$		−	1.60869e20i		−	0.113186i
$309$		0			0
$310$	−8.13765e20	−	1.98305e21i	−0.541907	−	1.32057i
$311$	−5.06135e20			−0.327947			−0.163973	−	0.986465i	$-0.552431\pi$
$311$	−5.06135e20			−0.327947			−0.163973	+	0.986465i	$0.552431\pi$
$312$		0			0
$313$			2.02180e21i			1.24054i	0.784388	+	0.620271i	$0.212977\pi$
$313$			2.02180e21i			1.24054i	−0.784388	+	0.620271i	$0.787023\pi$
$314$	2.78968e21			1.66591
$315$		0			0
$316$	1.18398e21			0.669889
$317$			2.79743e21i			1.54083i	0.637542	+	0.770415i	$0.279951\pi$
$317$			2.79743e21i			1.54083i	−0.637542	+	0.770415i	$0.720049\pi$
$318$		0			0
$319$	−6.96350e20			−0.363586
$320$	−8.44013e20	+	3.46349e20i	−0.429115	+	0.176092i
$321$		0			0
$322$		−	2.67983e20i		−	0.129221i
$323$		−	1.29622e20i		−	0.0608774i
$324$		0			0
$325$	−9.16473e19	−	9.28630e19i	−0.0408424	−	0.0413841i
$326$	−1.08114e21			−0.469390
$327$		0			0
$328$		−	1.22634e21i		−	0.505457i
$329$	2.92158e21			1.17342
$330$		0			0
$331$	−1.11907e21			−0.426894			−0.213447	−	0.976955i	$-0.568469\pi$
$331$	−1.11907e21			−0.426894			−0.213447	+	0.976955i	$0.568469\pi$
$332$		−	1.56276e21i		−	0.581055i
$333$		0			0
$334$	−2.21823e21			−0.783721
$335$	−1.08198e21	−	2.63666e21i	−0.372682	−	0.908183i
$336$		0			0
$337$			3.91312e20i			0.128135i	0.997946	+	0.0640677i	$0.0204074\pi$
$337$			3.91312e20i			0.128135i	−0.997946	+	0.0640677i	$0.979593\pi$
$338$		−	3.61590e21i		−	1.15458i
$339$		0			0
$340$	−5.55672e19	−	1.35411e20i	−0.0168751	−	0.0411228i
$341$	1.95662e21			0.579555
$342$		0			0
$343$			3.75663e21i			1.05876i
$344$	−2.76731e21			−0.760871
$345$		0			0
$346$	2.40466e21			0.629372
$347$		−	2.57931e21i		−	0.658724i	−0.944204	−	0.329362i	$-0.893166\pi$
$347$		−	2.57931e21i		−	0.658724i	0.944204	−	0.329362i	$-0.106834\pi$
$348$		0			0
$349$	3.74101e21			0.909856			0.454928	−	0.890528i	$-0.349665\pi$
$349$	3.74101e21			0.909856			0.454928	+	0.890528i	$0.349665\pi$
$350$	2.44335e21	−	2.41137e21i	0.579972	−	0.572380i
$351$		0			0
$352$			1.36669e21i			0.309070i
$353$			2.90176e21i			0.640585i	0.947319	+	0.320292i	$0.103781\pi$
$353$			2.90176e21i			0.640585i	−0.947319	+	0.320292i	$0.896219\pi$
$354$		0			0
$355$	−1.24488e21	+	5.10849e20i	−0.261931	+	0.107486i
$356$	−3.02889e21			−0.622241
$357$		0			0
$358$		−	6.31223e21i		−	1.23645i
$359$	−1.30144e21			−0.248955			−0.124477	−	0.992222i	$-0.539725\pi$
$359$	−1.30144e21			−0.248955			−0.124477	+	0.992222i	$0.539725\pi$
$360$		0			0
$361$	−4.27726e21			−0.780466
$362$		−	5.44898e21i		−	0.971164i
$363$		0			0
$364$	−8.22670e19			−0.0139915
$365$	4.41247e20	+	1.07527e21i	0.0733150	+	0.178660i
$366$		0			0
$367$		−	3.71583e21i		−	0.589379i	−0.955593	−	0.294689i	$-0.904784\pi$
$367$		−	3.71583e21i		−	0.589379i	0.955593	−	0.294689i	$-0.0952162\pi$
$368$			1.25349e21i			0.194274i
$369$		0			0
$370$	−7.77848e21	+	3.19197e21i	−1.15128	+	0.472438i
$371$	5.27672e21			0.763284
$372$		0			0
$373$		−	1.11016e22i		−	1.53412i	−0.641576	−	0.767060i	$-0.721719\pi$
$373$		−	1.11016e22i		−	1.53412i	0.641576	−	0.767060i	$-0.278281\pi$
$374$	5.24135e20			0.0708002
$375$		0			0
$376$	−9.84938e21			−1.27149
$377$			3.56107e20i			0.0449447i
$378$		0			0
$379$	9.29176e20			0.112115			0.0560576	−	0.998428i	$-0.482147\pi$
$379$	9.29176e20			0.112115			0.0560576	+	0.998428i	$0.482147\pi$
$380$	1.25687e21	−	5.15767e20i	0.148296	−	0.0608546i
$381$		0			0
$382$		−	1.08900e22i		−	1.22882i
$383$			9.75771e21i			1.07686i	0.842671	+	0.538429i	$0.180982\pi$
$383$			9.75771e21i			1.07686i	−0.842671	+	0.538429i	$0.819018\pi$
$384$		0			0
$385$	1.18961e21	+	2.89895e21i	0.125600	+	0.306073i
$386$	−1.09557e22			−1.13148
$387$		0			0
$388$		−	3.27608e21i		−	0.323806i
$389$	−9.36571e21			−0.905668			−0.452834	−	0.891595i	$-0.649587\pi$
$389$	−9.36571e21			−0.905668			−0.452834	+	0.891595i	$0.649587\pi$
$390$		0			0
$391$	2.22567e20			0.0206043
$392$		−	4.25107e21i		−	0.385094i
$393$		0			0
$394$	−3.26556e21			−0.283295
$395$	−2.13359e22	+	8.75538e21i	−1.81149	+	0.743361i
$396$		0			0
$397$			4.74176e21i			0.385674i	0.981231	+	0.192837i	$0.0617689\pi$
$397$			4.74176e21i			0.385674i	−0.981231	+	0.192837i	$0.938231\pi$
$398$			2.84013e21i			0.226117i
$399$		0			0
$400$	−1.14288e22	+	1.12792e22i	−0.871945	+	0.860531i
$401$	5.99579e21			0.447836			0.223918	−	0.974608i	$-0.428115\pi$
$401$	5.99579e21			0.447836			0.223918	+	0.974608i	$0.428115\pi$
$402$		0			0
$403$		−	1.00060e21i		−	0.0716418i
$404$	8.00747e21			0.561376
$405$		0			0
$406$	−9.36965e21			−0.629872
$407$		−	7.67480e21i		−	0.505260i
$408$		0			0
$409$	2.55749e22			1.61498			0.807489	−	0.589883i	$-0.200826\pi$
$409$	2.55749e22			1.61498			0.807489	+	0.589883i	$0.200826\pi$
$410$	−4.71593e21	−	1.14922e22i	−0.291678	−	0.710788i
$411$		0			0
$412$			7.70440e20i			0.0457207i
$413$			2.31395e22i			1.34518i
$414$		0			0
$415$	1.15564e22	+	2.81617e22i	0.644784	+	1.57127i
$416$	6.98911e20			0.0382057
$417$		0			0
$418$			4.86496e21i			0.255318i
$419$	2.84831e22			1.46477			0.732383	−	0.680893i	$-0.238408\pi$
$419$	2.84831e22			1.46477			0.732383	+	0.680893i	$0.238408\pi$
$420$		0			0
$421$	1.41976e22			0.701158			0.350579	−	0.936533i	$-0.385985\pi$
$421$	1.41976e22			0.701158			0.350579	+	0.936533i	$0.385985\pi$
$422$		−	9.15418e21i		−	0.443061i
$423$		0			0
$424$	−1.77892e22			−0.827076
$425$	2.00270e21	+	2.02926e21i	0.0912662	+	0.0924767i
$426$		0			0
$427$		−	1.53663e22i		−	0.672872i
$428$			1.05171e22i			0.451467i
$429$		0			0
$430$	−2.59327e22	+	1.06417e22i	−1.06996	+	0.439067i
$431$	4.51333e21			0.182575			0.0912873	−	0.995825i	$-0.470902\pi$
$431$	4.51333e21			0.182575			0.0912873	+	0.995825i	$0.470902\pi$
$432$		0			0
$433$			2.65630e22i			1.03307i	0.856265	+	0.516536i	$0.172779\pi$
$433$			2.65630e22i			1.03307i	−0.856265	+	0.516536i	$0.827221\pi$
$434$	2.63271e22			1.00401
$435$		0			0
$436$	7.23915e21			0.265492
$437$			2.06584e21i			0.0743026i
$438$		0			0
$439$	−1.24991e21			−0.0432446			−0.0216223	−	0.999766i	$-0.506883\pi$
$439$	−1.24991e21			−0.0432446			−0.0216223	+	0.999766i	$0.506883\pi$
$440$	−4.01048e21	−	9.77310e21i	−0.136097	−	0.331653i
$441$		0			0
$442$		−	2.68038e20i		−	0.00875198i
$443$		−	2.13227e22i		−	0.682984i	−0.939885	−	0.341492i	$-0.889068\pi$
$443$		−	2.13227e22i		−	0.682984i	0.939885	−	0.341492i	$-0.110932\pi$
$444$		0			0
$445$	5.45823e22	−	2.23983e22i	1.68264	−	0.690488i
$446$	−5.42145e22			−1.63972
$447$		0			0
$448$		−	1.12051e22i		−	0.326252i
$449$	4.98797e22			1.42505			0.712525	−	0.701647i	$-0.247552\pi$
$449$	4.98797e22			1.42505			0.712525	+	0.701647i	$0.247552\pi$
$450$		0			0
$451$	1.13390e22			0.311942
$452$			1.32741e22i			0.358366i
$453$		0			0
$454$	3.54469e22			0.921729
$455$	1.48250e21	−	6.08356e20i	0.0378352	−	0.0155261i
$456$		0			0
$457$		−	5.82003e22i		−	1.43099i	−0.698616	−	0.715497i	$-0.746200\pi$
$457$		−	5.82003e22i		−	1.43099i	0.698616	−	0.715497i	$-0.253800\pi$
$458$		−	8.37925e21i		−	0.202231i
$459$		0			0
$460$	8.85599e20	+	2.15811e21i	0.0205966	+	0.0501916i
$461$	−1.16830e22			−0.266744			−0.133372	−	0.991066i	$-0.542581\pi$
$461$	−1.16830e22			−0.266744			−0.133372	+	0.991066i	$0.542581\pi$
$462$		0			0
$463$			7.28447e21i			0.160310i	0.996782	+	0.0801548i	$0.0255415\pi$
$463$			7.28447e21i			0.160310i	−0.996782	+	0.0801548i	$0.974459\pi$
$464$	4.38265e22			0.946965
$465$		0			0
$466$	4.64421e22			0.967457
$467$		−	6.85242e22i		−	1.40169i	−0.713316	−	0.700843i	$-0.752807\pi$
$467$		−	6.85242e22i		−	1.40169i	0.713316	−	0.700843i	$-0.247193\pi$
$468$		0			0
$469$	3.50044e22			0.690483
$470$	−9.22996e22	+	3.78760e22i	−1.78800	+	0.733723i
$471$		0			0
$472$		−	7.80092e22i		−	1.45760i
$473$		−	2.55871e22i		−	0.469571i
$474$		0			0
$475$	−1.88354e22	+	1.85888e22i	−0.333487	+	0.329121i
$476$	1.79772e21			0.0312653
$477$		0			0
$478$			9.08456e22i			1.52464i
$479$	1.05339e23			1.73676			0.868379	−	0.495901i	$-0.165162\pi$
$479$	1.05339e23			1.73676			0.868379	+	0.495901i	$0.165162\pi$
$480$		0			0
$481$	−3.92482e21			−0.0624577
$482$			5.75577e22i			0.899920i
$483$		0			0
$484$	1.76506e22			0.266425
$485$	2.42263e22	+	5.90368e22i	0.359321	+	0.875624i
$486$		0			0
$487$			1.06266e23i			1.52195i	0.648783	+	0.760973i	$0.275278\pi$
$487$			1.06266e23i			1.52195i	−0.648783	+	0.760973i	$0.724722\pi$
$488$			5.18037e22i			0.729108i
$489$		0			0
$490$	−1.63476e22	−	3.98372e22i	−0.222222	−	0.541529i
$491$	1.66492e22			0.222434			0.111217	−	0.993796i	$-0.464525\pi$
$491$	1.66492e22			0.222434			0.111217	+	0.993796i	$0.464525\pi$
$492$		0			0
$493$		−	7.78173e21i		−	0.100433i
$494$	2.48789e21			0.0315611
$495$		0			0
$496$	−1.23145e23			−1.50946
$497$		−	1.65271e22i		−	0.199144i
$498$		0			0
$499$	8.56565e22			0.997484			0.498742	−	0.866751i	$-0.333796\pi$
$499$	8.56565e22			0.997484			0.498742	+	0.866751i	$0.333796\pi$
$500$	−1.17078e22	+	2.74935e22i	−0.134039	+	0.314764i
$501$		0			0
$502$		−	2.81491e22i		−	0.311518i
$503$		−	1.14040e23i		−	1.24087i	−0.784256	−	0.620437i	$-0.786955\pi$
$503$		−	1.14040e23i		−	1.24087i	0.784256	−	0.620437i	$-0.213045\pi$
$504$		0			0
$505$	−1.44299e23	+	5.92144e22i	−1.51805	+	0.622946i
$506$	−8.35338e21			−0.0864138
$507$		0			0
$508$		−	5.20440e21i		−	0.0520630i
$509$	−8.98075e22			−0.883510			−0.441755	−	0.897136i	$-0.645644\pi$
$509$	−8.98075e22			−0.883510			−0.441755	+	0.897136i	$0.645644\pi$
$510$		0			0
$511$	−1.42753e22			−0.135834
$512$			1.37544e22i			0.128720i
$513$		0			0
$514$	−3.42851e22			−0.310398
$515$	−5.69732e21	−	1.38837e22i	−0.0507352	−	0.123636i
$516$		0			0
$517$		−	9.10693e22i		−	0.784698i
$518$		−	1.03267e23i		−	0.875306i
$519$		0			0
$520$	−4.99787e21	+	2.05092e21i	−0.0409974	+	0.0168237i
$521$	−1.07887e23			−0.870663			−0.435331	−	0.900270i	$-0.643369\pi$
$521$	−1.07887e23			−0.870663			−0.435331	+	0.900270i	$0.643369\pi$
$522$		0			0
$523$			1.89182e23i			1.47780i	0.673816	+	0.738899i	$0.264654\pi$
$523$			1.89182e23i			1.47780i	−0.673816	+	0.738899i	$0.735346\pi$
$524$	1.37515e22			0.105690
$525$		0			0
$526$	−2.43087e23			−1.80877
$527$			2.18653e22i			0.160090i
$528$		0			0
$529$	1.37503e23			0.974852
$530$	−1.66704e23	+	6.84086e22i	−1.16306	+	0.477272i
$531$		0			0
$532$			1.66862e22i			0.112748i
$533$		−	5.79866e21i		−	0.0385608i
$534$		0			0
$535$	−7.77732e22	−	1.89525e23i	−0.500983	−	1.22084i
$536$	−1.18009e23			−0.748192
$537$		0			0
$538$		−	6.91280e22i		−	0.424624i
$539$	3.93062e22			0.237660
$540$		0			0
$541$	2.66057e22			0.155883			0.0779413	−	0.996958i	$-0.475165\pi$
$541$	2.66057e22			0.155883			0.0779413	+	0.996958i	$0.475165\pi$
$542$			1.34412e23i			0.775252i
$543$		0			0
$544$	−1.52728e22			−0.0853743
$545$	−1.30453e23	+	5.35327e22i	−0.717934	+	0.294611i
$546$		0			0
$547$			1.81383e23i			0.967617i	0.875174	+	0.483809i	$0.160747\pi$
$547$			1.81383e23i			0.967617i	−0.875174	+	0.483809i	$0.839253\pi$
$548$			5.02636e22i			0.264009i
$549$		0			0
$550$	−7.51653e22	−	7.61623e22i	−0.382767	−	0.387844i
$551$	7.22290e22			0.362179
$552$		0			0
$553$		−	2.83256e23i		−	1.37726i
$554$	3.59333e23			1.72054
$555$		0			0
$556$	−8.37788e22			−0.389044
$557$			3.44839e22i			0.157705i	0.996886	+	0.0788527i	$0.0251257\pi$
$557$			3.44839e22i			0.157705i	−0.996886	+	0.0788527i	$0.974874\pi$
$558$		0			0
$559$	−1.30850e22			−0.0580461
$560$	−7.48712e22	−	1.82453e23i	−0.327127	−	0.797172i
$561$		0			0
$562$		−	3.27906e23i		−	1.38992i
$563$		−	1.06909e22i		−	0.0446367i	−0.999751	−	0.0223184i	$-0.992895\pi$
$563$		−	1.06909e22i		−	0.0446367i	0.999751	−	0.0223184i	$-0.00710475\pi$
$564$		0			0
$565$	−9.81603e22	−	2.39206e23i	−0.397671	−	0.969080i
$566$	−3.09261e21			−0.0123420
$567$		0			0
$568$			5.57170e22i			0.215788i
$569$	4.26503e23			1.62730			0.813651	−	0.581354i	$-0.197477\pi$
$569$	4.26503e23			1.62730			0.813651	+	0.581354i	$0.197477\pi$
$570$		0			0
$571$	3.00545e23			1.11302			0.556509	−	0.830842i	$-0.312141\pi$
$571$	3.00545e23			1.11302			0.556509	+	0.830842i	$0.312141\pi$
$572$			2.56437e21i			0.00935651i
$573$		0			0
$574$	1.52571e23			0.540405
$575$	−3.19179e22	−	3.23413e22i	−0.111393	−	0.112871i
$576$		0			0
$577$			4.83070e23i			1.63687i	0.574596	+	0.818437i	$0.305159\pi$
$577$			4.83070e23i			1.63687i	−0.574596	+	0.818437i	$0.694841\pi$
$578$		−	3.41105e23i		−	1.13894i
$579$		0			0
$580$	7.54550e22	−	3.09637e22i	0.244653	−	0.100396i
$581$	−3.73875e23			−1.19462
$582$		0			0
$583$		−	1.64482e23i		−	0.510429i
$584$	4.81257e22			0.147186
$585$		0			0
$586$	1.17461e23			0.348950
$587$		−	5.06179e22i		−	0.148211i	−0.997250	−	0.0741054i	$-0.976390\pi$
$587$		−	5.06179e22i		−	0.148211i	0.997250	−	0.0741054i	$-0.0236101\pi$
$588$		0			0
$589$	−2.02951e23			−0.577312
$590$	−2.99986e23	−	7.31032e23i	−0.841121	−	2.04972i
$591$		0			0
$592$			4.83032e23i			1.31596i
$593$			4.53419e23i			1.21769i	0.793291	+	0.608843i	$0.208366\pi$
$593$			4.53419e23i			1.21769i	−0.793291	+	0.608843i	$0.791634\pi$
$594$		0			0
$595$	−3.23958e22	+	1.32939e22i	−0.0845464	+	0.0346944i
$596$	1.45658e22			0.0374751
$597$		0			0
$598$			4.27184e21i			0.0106821i
$599$	3.08340e23			0.760154			0.380077	−	0.924955i	$-0.375898\pi$
$599$	3.08340e23			0.760154			0.380077	+	0.924955i	$0.375898\pi$
$600$		0			0
$601$	9.69990e22			0.232453			0.116226	−	0.993223i	$-0.462920\pi$
$601$	9.69990e22			0.232453			0.116226	+	0.993223i	$0.462920\pi$
$602$		−	3.44284e23i		−	0.813479i
$603$		0			0
$604$	2.37597e23			0.545792
$605$	−3.18074e23	+	1.30525e23i	−0.720456	+	0.295646i
$606$		0			0
$607$			6.98262e23i			1.53785i	0.639338	+	0.768926i	$0.279208\pi$
$607$			6.98262e23i			1.53785i	−0.639338	+	0.768926i	$0.720792\pi$
$608$		−	1.41760e23i		−	0.307874i
$609$		0			0
$610$	1.99212e23	+	4.85457e23i	0.420738	+	1.02529i
$611$	−4.65720e22			−0.0970006
$612$		0			0
$613$		−	4.47675e23i		−	0.906879i	−0.891287	−	0.453439i	$-0.850197\pi$
$613$		−	4.47675e23i		−	0.906879i	0.891287	−	0.453439i	$-0.149803\pi$
$614$	9.98131e23			1.99414
$615$		0			0
$616$	1.29748e23			0.252153
$617$			7.12716e23i			1.36613i	0.730357	+	0.683066i	$0.239354\pi$
$617$			7.12716e23i			1.36613i	−0.730357	+	0.683066i	$0.760646\pi$
$618$		0			0
$619$	−2.79945e23			−0.522038			−0.261019	−	0.965334i	$-0.584059\pi$
$619$	−2.79945e23			−0.522038			−0.261019	+	0.965334i	$0.584059\pi$
$620$	−2.12015e23	+	8.70025e22i	−0.389976	+	0.160030i
$621$		0			0
$622$			2.12284e23i			0.379925i
$623$			7.24635e23i			1.27930i
$624$		0			0
$625$	7.66954e21	−	5.82026e23i	0.0131762	−	0.999913i
$626$	8.47985e23			1.43716
$627$		0			0
$628$		−	2.98255e23i		−	0.491961i
$629$	8.57660e22			0.139568
$630$		0			0
$631$	−5.21339e23			−0.825792			−0.412896	−	0.910778i	$-0.635483\pi$
$631$	−5.21339e23			−0.825792			−0.412896	+	0.910778i	$0.635483\pi$
$632$			9.54926e23i			1.49236i
$633$		0			0
$634$	1.17330e24			1.78505
$635$	3.84860e22	+	9.37860e22i	0.0577731	+	0.140787i
$636$		0			0
$637$		−	2.01008e22i		−	0.0293784i
$638$			2.92064e23i			0.421213i
$639$		0			0
$640$	3.22898e23	+	7.86866e23i	0.453456	+	1.10502i
$641$	−8.46920e23			−1.17368			−0.586839	−	0.809703i	$-0.699628\pi$
$641$	−8.46920e23			−1.17368			−0.586839	+	0.809703i	$0.699628\pi$
$642$		0			0
$643$			4.26351e23i			0.575406i	0.957720	+	0.287703i	$0.0928915\pi$
$643$			4.26351e23i			0.575406i	−0.957720	+	0.287703i	$0.907108\pi$
$644$	−2.86511e22			−0.0381602
$645$		0			0
$646$	−5.43660e22			−0.0705263
$647$		−	1.01754e24i		−	1.30276i	−0.758751	−	0.651381i	$-0.774190\pi$
$647$		−	1.01754e24i		−	1.30276i	0.758751	−	0.651381i	$-0.225810\pi$
$648$		0			0
$649$	7.21288e23			0.899557
$650$	−3.89487e22	+	3.84388e22i	−0.0479434	+	0.0473158i
$651$		0			0
$652$			1.15589e23i			0.138615i
$653$			8.08343e23i			0.956827i	0.878135	+	0.478414i	$0.158788\pi$
$653$			8.08343e23i			0.956827i	−0.878135	+	0.478414i	$0.841212\pi$
$654$		0			0
$655$	−2.47810e23	+	1.01691e23i	−0.285804	+	0.117282i
$656$	−7.13648e23			−0.812459
$657$		0			0
$658$		−	1.22537e24i		−	1.35940i
$659$	−3.59315e23			−0.393504			−0.196752	−	0.980453i	$-0.563039\pi$
$659$	−3.59315e23			−0.393504			−0.196752	+	0.980453i	$0.563039\pi$
$660$		0			0
$661$	4.13902e23			0.441759			0.220879	−	0.975301i	$-0.429107\pi$
$661$	4.13902e23			0.441759			0.220879	+	0.975301i	$0.429107\pi$
$662$			4.69362e23i			0.494555i
$663$		0			0
$664$	1.26043e24			1.29446
$665$	−1.23393e23	−	3.00694e23i	−0.125114	−	0.304888i
$666$		0			0
$667$			1.24021e23i			0.122582i
$668$			2.37159e23i			0.231440i
$669$		0			0
$670$	−1.10587e24	+	4.53804e23i	−1.05213	+	0.431750i
$671$	−4.78987e23			−0.449968
$672$		0			0
$673$		−	1.16691e24i		−	1.06884i	−0.845220	−	0.534418i	$-0.820531\pi$
$673$		−	1.16691e24i		−	1.06884i	0.845220	−	0.534418i	$-0.179469\pi$
$674$	1.64124e23			0.148444
$675$		0			0
$676$	−3.86588e23			−0.340959
$677$		−	1.51625e24i		−	1.32059i	−0.751007	−	0.660294i	$-0.770432\pi$
$677$		−	1.51625e24i		−	1.32059i	0.751007	−	0.660294i	$-0.229568\pi$
$678$		0			0
$679$	−7.83773e23			−0.665729
$680$	1.09215e23	−	4.48172e22i	0.0916125	−	0.0375941i
$681$		0			0
$682$		−	8.20648e23i		−	0.671413i
$683$		−	1.21958e24i		−	0.985447i	−0.870186	−	0.492723i	$-0.836001\pi$
$683$		−	1.21958e24i		−	0.985447i	0.870186	−	0.492723i	$-0.163999\pi$
$684$		0			0
$685$	−3.71694e23	−	9.05776e23i	−0.292965	−	0.713922i
$686$	1.57561e24			1.22657
$687$		0			0
$688$			1.61039e24i			1.22300i
$689$	−8.41145e22			−0.0630968
$690$		0			0
$691$	−1.99985e24			−1.46364			−0.731820	−	0.681498i	$-0.761329\pi$
$691$	−1.99985e24			−1.46364			−0.731820	+	0.681498i	$0.761329\pi$
$692$		−	2.57091e23i		−	0.185860i
$693$		0			0
$694$	−1.08182e24			−0.763130
$695$	1.50974e24	−	6.19535e23i	1.05204	−	0.431713i
$696$		0			0
$697$			1.26714e23i			0.0861678i
$698$		−	1.56906e24i		−	1.05407i
$699$		0			0
$700$	−2.57808e23	−	2.61227e23i	−0.169029	−	0.171272i
$701$	−1.54812e23			−0.100277			−0.0501387	−	0.998742i	$-0.515966\pi$
$701$	−1.54812e23			−0.100277			−0.0501387	+	0.998742i	$0.515966\pi$
$702$		0			0
$703$			7.96070e23i			0.503304i
$704$	−3.49278e23			−0.218174
$705$		0			0
$706$	1.21706e24			0.742116
$707$		−	1.91571e24i		−	1.15416i
$708$		0			0
$709$	−1.48852e24			−0.875508			−0.437754	−	0.899095i	$-0.644226\pi$
$709$	−1.48852e24			−0.875508			−0.437754	+	0.899095i	$0.644226\pi$
$710$	2.14261e23	+	5.22130e23i	0.124522	+	0.303447i
$711$		0			0
$712$		−	2.44293e24i		−	1.38622i
$713$		−	3.48477e23i		−	0.195395i
$714$		0			0
$715$	−1.89632e22	−	4.62113e22i	−0.0103827	−	0.0253015i
$716$	−6.74863e23			−0.365136
$717$		0			0
$718$			5.45850e23i			0.288414i
$719$	−7.17786e23			−0.374800			−0.187400	−	0.982284i	$-0.560006\pi$
$719$	−7.17786e23			−0.374800			−0.187400	+	0.982284i	$0.560006\pi$
$720$		0			0
$721$	1.84321e23			0.0939993
$722$			1.79397e24i			0.904168i
$723$		0			0
$724$	−5.82570e23			−0.286794
$725$	−1.13077e24	+	1.11596e24i	−0.550174	+	0.542972i
$726$		0			0
$727$		−	4.29185e23i		−	0.203987i	−0.994785	−	0.101993i	$-0.967478\pi$
$727$		−	4.29185e23i		−	0.203987i	0.994785	−	0.101993i	$-0.0325221\pi$
$728$		−	6.63518e22i		−	0.0311699i
$729$		0			0
$730$	4.50991e23	−	1.85068e23i	0.206978	−	0.0849352i
$731$	2.85936e23			0.129709
$732$		0			0
$733$			3.92543e24i			1.73982i	0.493213	+	0.869908i	$0.335822\pi$
$733$			3.92543e24i			1.73982i	−0.493213	+	0.869908i	$0.664178\pi$
$734$	−1.55850e24			−0.682794
$735$		0			0
$736$	2.43409e23			0.104202
$737$		−	1.09113e24i		−	0.461746i
$738$		0			0
$739$	−2.27046e22			−0.00938935			−0.00469468	−	0.999989i	$-0.501494\pi$
$739$	−2.27046e22			−0.00938935			−0.00469468	+	0.999989i	$0.501494\pi$
$740$	3.41265e23	+	8.31625e23i	0.139515	+	0.339983i
$741$		0			0
$742$		−	2.21317e24i		−	0.884262i
$743$		−	4.93045e23i		−	0.194752i	−0.995248	−	0.0973759i	$-0.968955\pi$
$743$		−	4.93045e23i		−	0.194752i	0.995248	−	0.0973759i	$-0.0310449\pi$
$744$		0			0
$745$	−2.62484e23	+	1.07713e23i	−0.101339	+	0.0415853i
$746$	−4.65623e24			−1.77727
$747$		0			0
$748$		−	5.60372e22i		−	0.0209080i
$749$	2.51613e24			0.928192
$750$		0			0
$751$	−3.69489e24			−1.33248			−0.666242	−	0.745736i	$-0.732098\pi$
$751$	−3.69489e24			−1.33248			−0.666242	+	0.745736i	$0.732098\pi$
$752$			5.73167e24i			2.04376i
$753$		0			0
$754$	1.49359e23			0.0520683
$755$	−4.28163e24	+	1.75701e24i	−1.47591	+	0.605654i
$756$		0			0
$757$		−	4.49458e24i		−	1.51487i	−0.652913	−	0.757433i	$-0.726453\pi$
$757$		−	4.49458e24i		−	1.51487i	0.652913	−	0.757433i	$-0.273547\pi$
$758$		−	3.89716e23i		−	0.129885i
$759$		0			0
$760$	4.15988e23	+	1.01372e24i	0.135570	+	0.330370i
$761$	3.27781e24			1.05637			0.528183	−	0.849130i	$-0.322873\pi$
$761$	3.27781e24			1.05637			0.528183	+	0.849130i	$0.322873\pi$
$762$		0			0
$763$		−	1.73190e24i		−	0.545839i
$764$	−1.16428e24			−0.362882
$765$		0			0
$766$	4.09259e24			1.24754
$767$		−	3.68860e23i		−	0.111199i
$768$		0			0
$769$	−2.02659e24			−0.597574			−0.298787	−	0.954320i	$-0.596582\pi$
$769$	−2.02659e24			−0.597574			−0.298787	+	0.954320i	$0.596582\pi$
$770$	1.21588e24	−	4.98948e23i	0.354584	−	0.145507i
$771$		0			0
$772$			1.17131e24i			0.334138i
$773$			1.38279e24i			0.390149i	0.980788	+	0.195075i	$0.0624949\pi$
$773$			1.38279e24i			0.390149i	−0.980788	+	0.195075i	$0.937505\pi$
$774$		0			0
$775$	3.17725e24	−	3.13566e24i	0.876975	−	0.865495i
$776$	2.64230e24			0.721368
$777$		0			0
$778$			3.92817e24i			1.04921i
$779$	−1.17614e24			−0.310735
$780$		0			0
$781$	−5.15170e23			−0.133173
$782$		−	9.33492e22i		−	0.0238700i
$783$		0			0
$784$	−2.47383e24			−0.618990
$785$	2.20556e24	+	5.37471e24i	0.545918	+	1.33034i
$786$		0			0
$787$			3.07765e23i			0.0745476i	0.999305	+	0.0372738i	$0.0118674\pi$
$787$			3.07765e23i			0.0745476i	−0.999305	+	0.0372738i	$0.988133\pi$
$788$			3.49132e23i			0.0836599i
$789$		0			0
$790$	3.67219e24	+	8.94871e24i	0.861181	+	2.09860i
$791$	3.17570e24			0.736783
$792$		0			0
$793$			2.44949e23i			0.0556229i
$794$	1.98879e24			0.446802
$795$		0			0
$796$	3.03649e23			0.0667745
$797$		−	8.31053e24i		−	1.80814i	−0.427379	−	0.904072i	$-0.640563\pi$
$797$		−	8.31053e24i		−	1.80814i	0.427379	−	0.904072i	$-0.359437\pi$
$798$		0			0
$799$	1.01770e24			0.216757
$800$	2.19024e24	+	2.21929e24i	0.461558	+	0.467681i
$801$		0			0
$802$		−	2.51476e24i		−	0.518817i
$803$			4.44979e23i			0.0908360i
$804$		0			0
$805$	5.16307e23	−	2.11872e23i	0.103191	−	0.0423455i
$806$	−4.19671e23			−0.0829968
$807$		0			0
$808$			6.45836e24i			1.25062i
$809$	−7.72799e24			−1.48083			−0.740413	−	0.672152i	$-0.765370\pi$
$809$	−7.72799e24			−1.48083			−0.740413	+	0.672152i	$0.765370\pi$
$810$		0			0
$811$	7.39633e24			1.38784			0.693920	−	0.720052i	$-0.255882\pi$
$811$	7.39633e24			1.38784			0.693920	+	0.720052i	$0.255882\pi$
$812$			1.00174e24i			0.186007i
$813$		0			0
$814$	−3.21897e24			−0.585342
$815$	−8.54768e23	−	2.08297e24i	−0.153819	−	0.374838i
$816$		0			0
$817$			2.65402e24i			0.467754i
$818$		−	1.07267e25i		−	1.87095i
$819$		0			0
$820$	−1.22867e24	+	5.04197e23i	−0.209903	+	0.0861355i
$821$	7.64184e24			1.29205			0.646027	−	0.763314i	$-0.276429\pi$
$821$	7.64184e24			1.29205			0.646027	+	0.763314i	$0.276429\pi$
$822$		0			0
$823$			1.25140e24i			0.207252i	0.994616	+	0.103626i	$0.0330445\pi$
$823$			1.25140e24i			0.207252i	−0.994616	+	0.103626i	$0.966956\pi$
$824$	−6.21392e23			−0.101856
$825$		0			0
$826$	9.70519e24			1.55838
$827$		−	4.37351e24i		−	0.695077i	−0.937666	−	0.347539i	$-0.887018\pi$
$827$		−	4.37351e24i		−	0.695077i	0.937666	−	0.347539i	$-0.112982\pi$
$828$		0			0
$829$	7.31119e24			1.13835			0.569174	−	0.822217i	$-0.307263\pi$
$829$	7.31119e24			1.13835			0.569174	+	0.822217i	$0.307263\pi$
$830$	1.18116e25	−	4.84700e24i	1.82031	−	0.746980i
$831$		0			0
$832$			1.78617e23i			0.0269697i
$833$			4.39248e23i			0.0656488i
$834$		0			0
$835$	−1.75376e24	−	4.27372e24i	−0.256824	−	0.625852i
$836$	5.20130e23			0.0753977
$837$		0			0
$838$		−	1.19464e25i		−	1.69693i
$839$	−7.56155e24			−1.06325			−0.531624	−	0.846981i	$-0.678418\pi$
$839$	−7.56155e24			−1.06325			−0.531624	+	0.846981i	$0.678418\pi$
$840$		0			0
$841$	−2.92094e24			−0.402491
$842$		−	5.95475e24i		−	0.812289i
$843$		0			0
$844$	−9.78705e23			−0.130840
$845$	6.96653e24	−	2.85878e24i	0.922007	−	0.378355i
$846$		0			0
$847$		−	4.22276e24i		−	0.547756i
$848$			1.03521e25i			1.32942i
$849$		0			0
$850$	8.51115e23	−	8.39974e23i	0.107134	−	0.105732i
$851$	−1.36689e24			−0.170346
$852$		0			0
$853$			3.98595e23i			0.0486929i	0.999704	+	0.0243464i	$0.00775048\pi$
$853$			3.98595e23i			0.0486929i	−0.999704	+	0.0243464i	$0.992250\pi$
$854$	−6.44494e24			−0.779520
$855$		0			0
$856$	−8.48252e24			−1.00577
$857$			6.59250e24i			0.773950i	0.922090	+	0.386975i	$0.126480\pi$
$857$			6.59250e24i			0.773950i	−0.922090	+	0.386975i	$0.873520\pi$
$858$		0			0
$859$	2.27761e24			0.262143			0.131071	−	0.991373i	$-0.458158\pi$
$859$	2.27761e24			0.262143			0.131071	+	0.991373i	$0.458158\pi$
$860$	1.13775e24	+	2.77256e24i	0.129661	+	0.315969i
$861$		0			0
$862$		−	1.89298e24i		−	0.211512i
$863$			1.31683e25i			1.45693i	0.685082	+	0.728466i	$0.259766\pi$
$863$			1.31683e25i			1.45693i	−0.685082	+	0.728466i	$0.740234\pi$
$864$		0			0
$865$	1.90116e24	+	4.63292e24i	0.206244	+	0.502594i
$866$	1.11411e25			1.19681
$867$		0			0
$868$		−	2.81472e24i		−	0.296495i
$869$	−8.82943e24			−0.921011
$870$		0			0
$871$	−5.57993e23			−0.0570788
$872$			5.83867e24i			0.591458i
$873$		0			0
$874$	8.66455e23			0.0860794
$875$	6.57758e24	+	2.80099e24i	0.647139	+	0.275577i
$876$		0			0
$877$			1.30212e25i			1.25648i	0.778020	+	0.628239i	$0.216224\pi$
$877$			1.30212e25i			1.25648i	−0.778020	+	0.628239i	$0.783776\pi$
$878$			5.24240e23i			0.0500987i
$879$		0			0
$880$	−5.68728e24	+	2.33383e24i	−0.533091	+	0.218759i
$881$	−1.89740e25			−1.76142			−0.880712	−	0.473651i	$-0.842936\pi$
$881$	−1.89740e25			−1.76142			−0.880712	+	0.473651i	$0.842936\pi$
$882$		0			0
$883$		−	4.83659e24i		−	0.440426i	−0.975452	−	0.220213i	$-0.929325\pi$
$883$		−	4.83659e24i		−	0.440426i	0.975452	−	0.220213i	$-0.0706752\pi$
$884$	−2.86569e22			−0.00258455
$885$		0			0
$886$	−8.94319e24			−0.791235
$887$			1.40113e25i			1.22780i	0.789385	+	0.613898i	$0.210399\pi$
$887$			1.40113e25i			1.22780i	−0.789385	+	0.613898i	$0.789601\pi$
$888$		0			0
$889$	−1.24511e24			−0.107039
$890$	−9.39433e24	−	2.28929e25i	−0.799928	−	1.94933i
$891$		0			0
$892$			5.79626e24i			0.484224i
$893$			9.44618e24i			0.781661i
$894$		0			0
$895$	1.21614e25	−	4.99054e24i	0.987388	−	0.405184i
$896$	−1.04464e25			−0.840138
$897$		0			0
$898$		−	2.09206e25i		−	1.65092i
$899$	−1.21840e25			−0.952428
$900$		0			0
$901$	1.83809e24			0.140996
$902$		−	4.75582e24i		−	0.361384i
$903$		0			0
$904$	−1.07061e25			−0.798361
$905$	1.04982e25	−	4.30804e24i	0.775537	−	0.318249i
$906$		0			0
$907$			1.00093e25i			0.725674i	0.931853	+	0.362837i	$0.118192\pi$
$907$			1.00093e25i			0.725674i	−0.931853	+	0.362837i	$0.881808\pi$
$908$		−	3.78975e24i		−	0.272196i
$909$		0			0
$910$	−2.55157e23	−	6.21789e23i	−0.0179869	−	0.0438320i
$911$	2.06992e25			1.44560			0.722800	−	0.691057i	$-0.242855\pi$
$911$	2.06992e25			1.44560			0.722800	+	0.691057i	$0.242855\pi$
$912$		0			0
$913$			1.16542e25i			0.798876i
$914$	−2.44104e25			−1.65780
$915$		0			0
$916$	−8.95855e23			−0.0597209
$917$		−	3.28993e24i		−	0.217294i
$918$		0			0
$919$	−2.83913e25			−1.84078			−0.920392	−	0.390996i	$-0.872130\pi$
$919$	−2.83913e25			−1.84078			−0.920392	+	0.390996i	$0.872130\pi$
$920$	−1.74060e24	+	7.14273e23i	−0.111816	+	0.0458847i
$921$		0			0
$922$			4.90007e24i			0.309022i
$923$			2.63453e23i			0.0164622i
$924$		0			0
$925$	−1.22996e25	−	1.24627e25i	−0.754544	−	0.764552i
$926$	3.05526e24			0.185718
$927$		0			0
$928$		−	8.51044e24i		−	0.507919i
$929$	2.96280e25			1.75214			0.876069	−	0.482186i	$-0.160157\pi$
$929$	2.96280e25			1.75214			0.876069	+	0.482186i	$0.160157\pi$
$930$		0			0
$931$	−4.07704e24			−0.236741
$932$		−	4.96529e24i		−	0.285699i
$933$		0			0
$934$	−2.87405e25			−1.62385
$935$	4.14389e23	+	1.00982e24i	0.0232011	+	0.0565386i
$936$		0			0
$937$			3.01741e25i			1.65901i	0.558503	+	0.829503i	$0.311376\pi$
$937$			3.01741e25i			1.65901i	−0.558503	+	0.829503i	$0.688624\pi$
$938$		−	1.46816e25i		−	0.799923i
$939$		0			0
$940$	4.04946e24	+	9.86807e24i	0.216676	+	0.528014i
$941$	2.71509e24			0.143970			0.0719851	−	0.997406i	$-0.477067\pi$
$941$	2.71509e24			0.143970			0.0719851	+	0.997406i	$0.477067\pi$
$942$		0			0
$943$		−	2.01949e24i		−	0.105170i
$944$	−4.53960e25			−2.34291
$945$		0			0
$946$	−1.07318e25			−0.543996
$947$			1.64526e25i			0.826533i	0.910610	+	0.413267i	$0.135612\pi$
$947$			1.64526e25i			0.826533i	−0.910610	+	0.413267i	$0.864388\pi$
$948$		0			0
$949$	2.27558e23			0.0112287
$950$	7.79653e24	+	7.89994e24i	0.381286	+	0.386343i
$951$		0			0
$952$			1.44994e24i			0.0696522i
$953$		−	2.20027e25i		−	1.04758i	−0.851848	−	0.523789i	$-0.824518\pi$
$953$		−	2.20027e25i		−	1.04758i	0.851848	−	0.523789i	$-0.175482\pi$
$954$		0			0
$955$	2.09810e25	−	8.60975e24i	0.981291	−	0.402682i
$956$	9.71263e24			0.450241
$957$		0			0
$958$		−	4.41816e25i		−	2.01203i
$959$	1.20251e25			0.542789
$960$		0			0
$961$	1.16848e25			0.518168
$962$			1.64615e24i			0.0723571i
$963$		0			0
$964$	6.15370e24			0.265755
$965$	−8.66173e24	−	2.11077e25i	−0.370785	−	0.903562i
$966$		0			0
$967$		−	1.65769e24i		−	0.0697235i	−0.999392	−	0.0348618i	$-0.988901\pi$
$967$		−	1.65769e24i		−	0.0697235i	0.999392	−	0.0348618i	$-0.0110991\pi$
$968$			1.42360e25i			0.593535i
$969$		0			0
$970$	2.47612e25	−	1.01610e25i	1.01441	−	0.416272i
$971$	−1.82700e25			−0.741952			−0.370976	−	0.928642i	$-0.620977\pi$
$971$	−1.82700e25			−0.741952			−0.370976	+	0.928642i	$0.620977\pi$
$972$		0			0
$973$			2.00433e25i			0.799854i
$974$	4.45703e25			1.76317
$975$		0			0
$976$	3.01462e25			1.17195
$977$		−	3.22266e25i		−	1.24197i	−0.783823	−	0.620984i	$-0.786733\pi$
$977$		−	3.22266e25i		−	1.24197i	0.783823	−	0.620984i	$-0.213267\pi$
$978$		0			0
$979$	2.25878e25			0.855502
$980$	−4.25914e24	+	1.74778e24i	−0.159919	+	0.0656242i
$981$		0			0
$982$		−	6.98303e24i		−	0.257689i
$983$			1.74885e25i			0.639804i	0.947451	+	0.319902i	$0.103650\pi$
$983$			1.74885e25i			0.639804i	−0.947451	+	0.319902i	$0.896350\pi$
$984$		0			0
$985$	−2.58180e24	−	6.29155e24i	−0.0928355	−	0.226230i
$986$	−3.26382e24			−0.116352
$987$		0			0
$988$		−	2.65989e23i		−	0.00932031i
$989$	−4.55709e24			−0.158314
$990$		0			0
$991$	2.45273e25			0.837575			0.418787	−	0.908084i	$-0.362455\pi$
$991$	2.45273e25			0.837575			0.418787	+	0.908084i	$0.362455\pi$
$992$			2.39128e25i			0.809621i
$993$		0			0
$994$	−6.93180e24			−0.230708
$995$	−5.47191e24	+	2.24545e24i	−0.180569	+	0.0740981i
$996$		0			0
$997$		−	2.80715e25i		−	0.910660i	−0.890323	−	0.455330i	$-0.849521\pi$
$997$		−	2.80715e25i		−	0.910660i	0.890323	−	0.455330i	$-0.150479\pi$
$998$		−	3.59261e25i		−	1.15558i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.18.b.d.19.4		16
3.2	odd	2		15.18.b.a.4.13	yes	16
5.4	even	2	inner	45.18.b.d.19.13		16
15.2	even	4		75.18.a.k.1.2		8
15.8	even	4		75.18.a.l.1.7		8
15.14	odd	2		15.18.b.a.4.4	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.18.b.a.4.4	✓	16	15.14	odd	2
15.18.b.a.4.13	yes	16	3.2	odd	2
45.18.b.d.19.4		16	1.1	even	1	trivial
45.18.b.d.19.13		16	5.4	even	2	inner
75.18.a.k.1.2		8	15.2	even	4
75.18.a.l.1.7		8	15.8	even	4

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:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	45.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-419.421i q^{2} -44841.8 q^{4} +(808072. - 331600. i) q^{5} +1.07280e7i q^{7} -3.61668e7i q^{8} +O(q^{10})\)	\(q-419.421i q^{2} -44841.8 q^{4} +(808072. - 331600. i) q^{5} +1.07280e7i q^{7} -3.61668e7i q^{8} +(-1.39080e8 - 3.38922e8i) q^{10} +3.34405e8 q^{11} -1.71011e8i q^{13} +4.49954e9 q^{14} -2.10466e10 q^{16} +3.73698e9i q^{17} -3.46862e10 q^{19} +(-3.62354e10 + 1.48695e10i) q^{20} -1.40256e11i q^{22} -5.95580e10i q^{23} +(5.43022e11 - 5.35914e11i) q^{25} -7.17257e10 q^{26} -4.81062e11i q^{28} -2.08236e12 q^{29} +5.85106e12 q^{31} +4.08693e12i q^{32} +1.56737e12 q^{34} +(3.55740e12 + 8.66899e12i) q^{35} -2.29506e13i q^{37} +1.45481e13i q^{38} +(-1.19929e13 - 2.92253e13i) q^{40} +3.39080e13 q^{41} -7.65153e13i q^{43} -1.49953e13 q^{44} -2.49798e13 q^{46} -2.72333e14i q^{47} +1.17541e14 q^{49} +(-2.24773e14 - 2.27755e14i) q^{50} +7.66845e12i q^{52} -4.91865e14i q^{53} +(2.70223e14 - 1.10889e14i) q^{55} +3.87996e14 q^{56} +8.73384e14i q^{58} +2.15693e15 q^{59} -1.43236e15 q^{61} -2.45405e15i q^{62} -1.04448e15 q^{64} +(-5.67074e13 - 1.38190e14i) q^{65} -3.26290e15i q^{67} -1.67573e14i q^{68} +(3.63595e15 - 1.49205e15i) q^{70} -1.54056e15 q^{71} +1.33066e15i q^{73} -9.62597e15 q^{74} +1.55539e15 q^{76} +3.58749e15i q^{77} -2.64034e16 q^{79} +(-1.70072e16 + 6.97905e15i) q^{80} -1.42217e16i q^{82} +3.48504e16i q^{83} +(1.23918e15 + 3.01975e15i) q^{85} -3.20921e16 q^{86} -1.20943e16i q^{88} +6.75463e16 q^{89} +1.83461e15 q^{91} +2.67068e15i q^{92} -1.14222e17 q^{94} +(-2.80289e16 + 1.15019e16i) q^{95} +7.30588e16i q^{97} -4.92991e16i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 862698 q^{4} - 292740 q^{5}+O(q^{10}) \) 16 * q - 862698 * q^4 - 292740 * q^5	\( 16 q - 862698 q^{4} - 292740 q^{5} - 158256910 q^{10} - 907386144 q^{11} - 38248328748 q^{14} + 64410639650 q^{16} + 44375877728 q^{19} - 124652464020 q^{20} + 1631404456600 q^{25} - 2940223801452 q^{26} + 5348468604504 q^{29} - 16935964052224 q^{31} + 60848757680284 q^{34} - 25456583995440 q^{35} + 246023763543250 q^{40} + 4873344779184 q^{41} - 224598182313996 q^{44} - 10\!\cdots\!24 q^{46}+ \cdots + 43\!\cdots\!00 q^{95}+O(q^{100}) \) 16 * q - 862698 * q^4 - 292740 * q^5 - 158256910 * q^10 - 907386144 * q^11 - 38248328748 * q^14 + 64410639650 * q^16 + 44375877728 * q^19 - 124652464020 * q^20 + 1631404456600 * q^25 - 2940223801452 * q^26 + 5348468604504 * q^29 - 16935964052224 * q^31 + 60848757680284 * q^34 - 25456583995440 * q^35 + 246023763543250 * q^40 + 4873344779184 * q^41 - 224598182313996 * q^44 - 1042392874396424 * q^46 + 85575569613760 * q^49 - 669894278364600 * q^50 + 2661435553553280 * q^55 + 8893968141042660 * q^56 - 2882362957102272 * q^59 - 6340947108203680 * q^61 + 2538830661242878 * q^64 - 13690132732535880 * q^65 + 10824491372201640 * q^70 + 11739285749899008 * q^71 + 6749027807219172 * q^74 - 32687788759104192 * q^76 + 28655274740363840 * q^79 + 7449390713147460 * q^80 + 27636069676463240 * q^85 - 261659254088221464 * q^86 + 119080233211377312 * q^89 + 201912942604136256 * q^91 - 186654825194113808 * q^94 + 434666709576297600 * q^95

Introduction

L-functions

Modular forms

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Database

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