LMFDB - Embedded newform 50.18.b.d.49.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [50,18,Mod(49,50)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(50, 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("50.49");

S:= CuspForms(chi, 18);

N := Newforms(S);

Level:	$ N $	$=$	$ 50 = 2 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 18 $
Character orbit:	$[\chi]$	$=$	50.b (of order $2$, degree $1$, not 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:	$91.6110436723$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 41641x^{2} + 433472400 $ x^4 + 41641*x^2 + 433472400
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 10)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.4
Root			$143.792i$ of defining polynomial
Character	$\chi$	$=$	50.49
Dual form			50.18.b.d.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+256.000i q^{2} +8003.53i q^{3} -65536.0 q^{4} -2.04890e6 q^{6} -2.54694e7i q^{7} -1.67772e7i q^{8} +6.50836e7 q^{9} -2.81089e8 q^{11} -5.24520e8i q^{12} -1.52610e9i q^{13} +6.52016e9 q^{14} +4.29497e9 q^{16} -5.46901e10i q^{17} +1.66614e10i q^{18} -6.88947e8 q^{19} +2.03845e11 q^{21} -7.19587e10i q^{22} +3.91035e11i q^{23} +1.34277e11 q^{24} +3.90681e11 q^{26} +1.55448e12i q^{27} +1.66916e12i q^{28} -5.12259e12 q^{29} -7.31204e10 q^{31} +1.09951e12i q^{32} -2.24970e12i q^{33} +1.40007e13 q^{34} -4.26532e12 q^{36} +6.81637e12i q^{37} -1.76370e11i q^{38} +1.22142e13 q^{39} -5.76386e13 q^{41} +5.21843e13i q^{42} +7.57081e13i q^{43} +1.84214e13 q^{44} -1.00105e14 q^{46} +4.60351e13i q^{47} +3.43749e13i q^{48} -4.16059e14 q^{49} +4.37714e14 q^{51} +1.00014e14i q^{52} +6.58261e14i q^{53} -3.97946e14 q^{54} -4.27305e14 q^{56} -5.51401e12i q^{57} -1.31138e15i q^{58} +2.98287e14 q^{59} +8.50623e14 q^{61} -1.87188e13i q^{62} -1.65764e15i q^{63} -2.81475e14 q^{64} +5.75924e14 q^{66} +6.12967e15i q^{67} +3.58417e15i q^{68} -3.12966e15 q^{69} +5.41472e14 q^{71} -1.09192e15i q^{72} -7.16849e15i q^{73} -1.74499e15 q^{74} +4.51508e13 q^{76} +7.15916e15i q^{77} +3.12683e15i q^{78} -5.45373e15 q^{79} -4.03640e15 q^{81} -1.47555e16i q^{82} -3.64723e15i q^{83} -1.33592e16 q^{84} -1.93813e16 q^{86} -4.09988e16i q^{87} +4.71589e15i q^{88} -6.81792e14 q^{89} -3.88687e16 q^{91} -2.56269e16i q^{92} -5.85222e14i q^{93} -1.17850e16 q^{94} -8.79998e15 q^{96} +1.20373e17i q^{97} -1.06511e17i q^{98} -1.82943e16 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 262144 q^{4} + 669696 q^{6} + 215038188 q^{9} - 942962592 q^{11} + 309168128 q^{14} + 17179869184 q^{16} - 257345058800 q^{19} + 870762493248 q^{21} - 43889197056 q^{24} + 789419124736 q^{26} - 5086109499960 q^{29}+ \cdots - 52\!\cdots\!24 q^{99}+O(q^{100}) $ 4 * q - 262144 * q^4 + 669696 * q^6 + 215038188 * q^9 - 942962592 * q^11 + 309168128 * q^14 + 17179869184 * q^16 - 257345058800 * q^19 + 870762493248 * q^21 - 43889197056 * q^24 + 789419124736 * q^26 - 5086109499960 * q^29 + 15678815997488 * q^31 + 16455629088768 * q^34 - 14092742688768 * q^36 + 24135247074576 * q^39 - 75652728528312 * q^41 + 61797996429312 * q^44 - 332984247957504 * q^46 - 1603445169355908 * q^49 + 1295380982531088 * q^51 + 22095889551360 * q^54 - 20261642436608 * q^56 - 3055641434470320 * q^59 - 1263476426244232 * q^61 - 1125899906842624 * q^64 + 244151200161792 * q^66 - 1429906501577664 * q^69 - 6663803321824752 * q^71 - 14236267057948672 * q^74 + 16865365773516800 * q^76 - 16527213684110240 * q^79 - 26865367701318396 * q^81 - 57066290757500928 * q^84 - 15170792410548224 * q^86 - 111756825043809960 * q^89 - 76954793221206032 * q^91 + 113045290507266048 * q^94 + 2876322418262016 * q^96 - 52747342198192224 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/50\mathbb{Z}\right)^\times$.

$n$	$27$
$\chi(n)$	$-1$

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
$3$	8003.53i	0.704289i	0.935946	+	0.352145i	$0.114548\pi$
$3$	8003.53i	0.704289i	−0.935946	+	0.352145i	$0.885452\pi$
$4$	−65536.0	−0.500000
$5$	0	0
$5$	0	0
$6$	−2.04890e6	−0.498008
$7$	− 2.54694e7i	− 1.66988i	−0.550342	−	0.834939i	$-0.685503\pi$
$7$	− 2.54694e7i	− 1.66988i	0.550342	−	0.834939i	$-0.314497\pi$
$8$	− 1.67772e7i	− 0.353553i
$9$	6.50836e7	0.503976
$10$	0	0
$11$	−2.81089e8	−0.395372	−0.197686	−	0.980265i	$-0.563343\pi$
$11$	−2.81089e8	−0.395372	−0.197686	+	0.980265i	$0.563343\pi$
$12$	− 5.24520e8i	− 0.352145i
$13$	− 1.52610e9i	− 0.518876i	−0.965760	−	0.259438i	$-0.916463\pi$
$13$	− 1.52610e9i	− 0.518876i	0.965760	−	0.259438i	$-0.0835373\pi$
$14$	6.52016e9	1.18078
$15$	0	0
$16$	4.29497e9	0.250000
$17$	− 5.46901e10i	− 1.90148i	−0.309984	−	0.950742i	$-0.600324\pi$
$17$	− 5.46901e10i	− 1.90148i	0.309984	−	0.950742i	$-0.399676\pi$
$18$	1.66614e10i	0.356365i
$19$	−6.88947e8	−0.00930637	−0.00465318	−	0.999989i	$-0.501481\pi$
$19$	−6.88947e8	−0.00930637	−0.00465318	+	0.999989i	$0.501481\pi$
$20$	0	0
$21$	2.03845e11	1.17608
$22$	− 7.19587e10i	− 0.279570i
$23$	3.91035e11i	1.04119i	0.853804	+	0.520594i	$0.174290\pi$
$23$	3.91035e11i	1.04119i	−0.853804	+	0.520594i	$0.825710\pi$
$24$	1.34277e11	0.249004
$25$	0	0
$26$	3.90681e11	0.366901
$27$	1.55448e12i	1.05923i
$28$	1.66916e12i	0.834939i
$29$	−5.12259e12	−1.90154	−0.950772	−	0.309892i	$-0.899707\pi$
$29$	−5.12259e12	−1.90154	−0.950772	+	0.309892i	$0.899707\pi$
$30$	0	0
$31$	−7.31204e10	−0.0153980	−0.00769900	−	0.999970i	$-0.502451\pi$
$31$	−7.31204e10	−0.0153980	−0.00769900	+	0.999970i	$0.502451\pi$
$32$	1.09951e12i	0.176777i
$33$	− 2.24970e12i	− 0.278456i
$34$	1.40007e13	1.34455
$35$	0	0
$36$	−4.26532e12	−0.251988
$37$	6.81637e12i	0.319035i	0.987195	+	0.159517i	$0.0509938\pi$
$37$	6.81637e12i	0.319035i	−0.987195	+	0.159517i	$0.949006\pi$
$38$	− 1.76370e11i	− 0.00658059i
$39$	1.22142e13	0.365439
$40$	0	0
$41$	−5.76386e13	−1.12733	−0.563665	−	0.826004i	$-0.690609\pi$
$41$	−5.76386e13	−1.12733	−0.563665	+	0.826004i	$0.690609\pi$
$42$	5.21843e13i	0.831613i
$43$	7.57081e13i	0.987781i	0.869524	+	0.493890i	$0.164426\pi$
$43$	7.57081e13i	0.987781i	−0.869524	+	0.493890i	$0.835574\pi$
$44$	1.84214e13	0.197686
$45$	0	0
$46$	−1.00105e14	−0.736231
$47$	4.60351e13i	0.282006i	0.990009	+	0.141003i	$0.0450327\pi$
$47$	4.60351e13i	0.282006i	−0.990009	+	0.141003i	$0.954967\pi$
$48$	3.43749e13i	0.176072i
$49$	−4.16059e14	−1.78849
$50$	0	0
$51$	4.37714e14	1.33919
$52$	1.00014e14i	0.259438i
$53$	6.58261e14i	1.45229i	0.687541	+	0.726145i	$0.258690\pi$
$53$	6.58261e14i	1.45229i	−0.687541	+	0.726145i	$0.741310\pi$
$54$	−3.97946e14	−0.748992
$55$	0	0
$56$	−4.27305e14	−0.590391
$57$	− 5.51401e12i	− 0.00655437i
$58$	− 1.31138e15i	− 1.34459i
$59$	2.98287e14	0.264479	0.132240	−	0.991218i	$-0.457783\pi$
$59$	2.98287e14	0.264479	0.132240	+	0.991218i	$0.457783\pi$
$60$	0	0
$61$	8.50623e14	0.568111	0.284056	−	0.958808i	$-0.408320\pi$
$61$	8.50623e14	0.568111	0.284056	+	0.958808i	$0.408320\pi$
$62$	− 1.87188e13i	− 0.0108880i
$63$	− 1.65764e15i	− 0.841580i
$64$	−2.81475e14	−0.125000
$65$	0	0
$66$	5.75924e14	0.196898
$67$	6.12967e15i	1.84417i	0.386987	+	0.922085i	$0.373516\pi$
$67$	6.12967e15i	1.84417i	−0.386987	+	0.922085i	$0.626484\pi$
$68$	3.58417e15i	0.950742i
$69$	−3.12966e15	−0.733298
$70$	0	0
$71$	5.41472e14	0.0995131	0.0497565	−	0.998761i	$-0.484155\pi$
$71$	5.41472e14	0.0995131	0.0497565	+	0.998761i	$0.484155\pi$
$72$	− 1.09192e15i	− 0.178183i
$73$	− 7.16849e15i	− 1.04036i	−0.854057	−	0.520180i	$-0.825865\pi$
$73$	− 7.16849e15i	− 1.04036i	0.854057	−	0.520180i	$-0.174135\pi$
$74$	−1.74499e15	−0.225592
$75$	0	0
$76$	4.51508e13	0.00465318
$77$	7.15916e15i	0.660223i
$78$	3.12683e15i	0.258404i
$79$	−5.45373e15	−0.404448	−0.202224	−	0.979339i	$-0.564817\pi$
$79$	−5.45373e15	−0.404448	−0.202224	+	0.979339i	$0.564817\pi$
$80$	0	0
$81$	−4.03640e15	−0.242031
$82$	− 1.47555e16i	− 0.797142i
$83$	− 3.64723e15i	− 0.177746i	−0.996043	−	0.0888728i	$-0.971674\pi$
$83$	− 3.64723e15i	− 0.177746i	0.996043	−	0.0888728i	$-0.0283265\pi$
$84$	−1.33592e16	−0.588039
$85$	0	0
$86$	−1.93813e16	−0.698467
$87$	− 4.09988e16i	− 1.33924i
$88$	4.71589e15i	0.139785i
$89$	−6.81792e14	−0.0183585	−0.00917925	−	0.999958i	$-0.502922\pi$
$89$	−6.81792e14	−0.0183585	−0.00917925	+	0.999958i	$0.502922\pi$
$90$	0	0
$91$	−3.88687e16	−0.866460
$92$	− 2.56269e16i	− 0.520594i
$93$	− 5.85222e14i	− 0.0108446i
$94$	−1.17850e16	−0.199408
$95$	0	0
$96$	−8.79998e15	−0.124502
$97$	1.20373e17i	1.55944i	0.626129	+	0.779720i	$0.284638\pi$
$97$	1.20373e17i	1.55944i	−0.626129	+	0.779720i	$0.715362\pi$
$98$	− 1.06511e17i	− 1.26466i
$99$	−1.82943e16	−0.199258
$100$	0	0
$101$	7.74092e16	0.711314	0.355657	−	0.934617i	$-0.384257\pi$
$101$	7.74092e16	0.711314	0.355657	+	0.934617i	$0.384257\pi$
$102$	1.12055e17i	0.946954i
$103$	− 1.39309e17i	− 1.08359i	−0.840512	−	0.541793i	$-0.817746\pi$
$103$	− 1.39309e17i	− 1.08359i	0.840512	−	0.541793i	$-0.182254\pi$
$104$	−2.56037e16	−0.183450
$105$	0	0
$106$	−1.68515e17	−1.02692
$107$	− 2.75662e17i	− 1.55101i	−0.631341	−	0.775505i	$-0.717495\pi$
$107$	− 2.75662e17i	− 1.55101i	0.631341	−	0.775505i	$-0.282505\pi$
$108$	− 1.01874e17i	− 0.529617i
$109$	−1.37550e17	−0.661202	−0.330601	−	0.943771i	$-0.607251\pi$
$109$	−1.37550e17	−0.661202	−0.330601	+	0.943771i	$0.607251\pi$
$110$	0	0
$111$	−5.45550e16	−0.224693
$112$	− 1.09390e17i	− 0.417470i
$113$	− 3.90258e17i	− 1.38097i	−0.723345	−	0.690486i	$-0.757396\pi$
$113$	− 3.90258e17i	− 1.38097i	0.723345	−	0.690486i	$-0.242604\pi$
$114$	1.41159e15	0.00463464
$115$	0	0
$116$	3.35714e17	0.950772
$117$	− 9.93239e16i	− 0.261501i
$118$	7.63614e16i	0.187015i
$119$	−1.39292e18	−3.17525
$120$	0	0
$121$	−4.26436e17	−0.843681
$122$	2.17759e17i	0.401715i
$123$	− 4.61313e17i	− 0.793966i
$124$	4.79202e15	0.00769900
$125$	0	0
$126$	4.24355e17	0.595087
$127$	7.37824e17i	0.967433i	0.875225	+	0.483717i	$0.160713\pi$
$127$	7.37824e17i	0.967433i	−0.875225	+	0.483717i	$0.839287\pi$
$128$	− 7.20576e16i	− 0.0883883i
$129$	−6.05933e17	−0.695684
$130$	0	0
$131$	−1.45408e18	−1.46482	−0.732409	−	0.680865i	$-0.761604\pi$
$131$	−1.45408e18	−1.46482	−0.732409	+	0.680865i	$0.761604\pi$
$132$	1.47437e17i	0.139228i
$133$	1.75470e16i	0.0155405i
$134$	−1.56919e18	−1.30403
$135$	0	0
$136$	−9.17547e17	−0.672276
$137$	2.27534e18i	1.56647i	0.621725	+	0.783236i	$0.286432\pi$
$137$	2.27534e18i	1.56647i	−0.621725	+	0.783236i	$0.713568\pi$
$138$	− 8.01194e17i	− 0.518520i
$139$	−1.41598e17	−0.0861849	−0.0430924	−	0.999071i	$-0.513721\pi$
$139$	−1.41598e17	−0.0861849	−0.0430924	+	0.999071i	$0.513721\pi$
$140$	0	0
$141$	−3.68444e17	−0.198614
$142$	1.38617e17i	0.0703664i
$143$	4.28969e17i	0.205149i
$144$	2.79532e17	0.125994
$145$	0	0
$146$	1.83513e18	0.735646
$147$	− 3.32994e18i	− 1.25962i
$148$	− 4.46717e17i	− 0.159517i
$149$	1.31089e18	0.442063	0.221032	−	0.975267i	$-0.429058\pi$
$149$	1.31089e18	0.442063	0.221032	+	0.975267i	$0.429058\pi$
$150$	0	0
$151$	−2.76548e18	−0.832658	−0.416329	−	0.909214i	$-0.636684\pi$
$151$	−2.76548e18	−0.832658	−0.416329	+	0.909214i	$0.636684\pi$
$152$	1.15586e16i	0.00329030i
$153$	− 3.55943e18i	− 0.958303i
$154$	−1.83274e18	−0.466848
$155$	0	0
$156$	−8.00468e17	−0.182719
$157$	3.81235e18i	0.824225i	0.911133	+	0.412112i	$0.135209\pi$
$157$	3.81235e18i	0.824225i	−0.911133	+	0.412112i	$0.864791\pi$
$158$	− 1.39615e18i	− 0.285988i
$159$	−5.26842e18	−1.02283
$160$	0	0
$161$	9.95942e18	1.73866
$162$	− 1.03332e18i	− 0.171142i
$163$	1.03773e19i	1.63114i	0.578658	+	0.815570i	$0.303577\pi$
$163$	1.03773e19i	1.63114i	−0.578658	+	0.815570i	$0.696423\pi$
$164$	3.77740e18	0.563665
$165$	0	0
$166$	9.33690e17	0.125685
$167$	− 1.21920e19i	− 1.55949i	−0.626095	−	0.779747i	$-0.715348\pi$
$167$	− 1.21920e19i	− 1.55949i	0.626095	−	0.779747i	$-0.284652\pi$
$168$	− 3.41995e18i	− 0.415806i
$169$	6.32145e18	0.730768
$170$	0	0
$171$	−4.48391e16	−0.00469019
$172$	− 4.96161e18i	− 0.493890i
$173$	− 3.88210e18i	− 0.367853i	−0.982940	−	0.183927i	$-0.941119\pi$
$173$	− 3.88210e18i	− 0.367853i	0.982940	−	0.183927i	$-0.0588809\pi$
$174$	1.04957e19	0.946983
$175$	0	0
$176$	−1.20727e18	−0.0988430
$177$	2.38735e18i	0.186270i
$178$	− 1.74539e17i	− 0.0129814i
$179$	1.62354e19	1.15136	0.575680	−	0.817675i	$-0.304737\pi$
$179$	1.62354e19	1.15136	0.575680	+	0.817675i	$0.304737\pi$
$180$	0	0
$181$	−2.32195e19	−1.49825	−0.749127	−	0.662427i	$-0.769527\pi$
$181$	−2.32195e19	−1.49825	−0.749127	+	0.662427i	$0.769527\pi$
$182$	− 9.95039e18i	− 0.612680i
$183$	6.80799e18i	0.400115i
$184$	6.56048e18	0.368116
$185$	0	0
$186$	1.49817e17	0.00766832
$187$	1.53728e19i	0.751793i
$188$	− 3.01696e18i	− 0.141003i
$189$	3.95915e19	1.76879
$190$	0	0
$191$	4.47928e19	1.82989	0.914944	−	0.403580i	$-0.132234\pi$
$191$	4.47928e19	1.82989	0.914944	+	0.403580i	$0.132234\pi$
$192$	− 2.25279e18i	− 0.0880362i
$193$	3.61154e19i	1.35038i	0.737645	+	0.675189i	$0.235938\pi$
$193$	3.61154e19i	1.35038i	−0.737645	+	0.675189i	$0.764062\pi$
$194$	−3.08154e19	−1.10269
$195$	0	0
$196$	2.72668e19	0.894247
$197$	1.80503e19i	0.566920i	0.958984	+	0.283460i	$0.0914823\pi$
$197$	1.80503e19i	0.566920i	−0.958984	+	0.283460i	$0.908518\pi$
$198$	− 4.68333e18i	− 0.140897i
$199$	6.39636e18	0.184366	0.0921832	−	0.995742i	$-0.470615\pi$
$199$	6.39636e18	0.184366	0.0921832	+	0.995742i	$0.470615\pi$
$200$	0	0
$201$	−4.90590e19	−1.29883
$202$	1.98168e19i	0.502975i
$203$	1.30469e20i	3.17535i
$204$	−2.86860e19	−0.669597
$205$	0	0
$206$	3.56631e19	0.766211
$207$	2.54500e19i	0.524734i
$208$	− 6.55453e18i	− 0.129719i
$209$	1.93655e17	0.00367948
$210$	0	0
$211$	7.72101e18	0.135292	0.0676462	−	0.997709i	$-0.478451\pi$
$211$	7.72101e18	0.135292	0.0676462	+	0.997709i	$0.478451\pi$
$212$	− 4.31398e19i	− 0.726145i
$213$	4.33369e18i	0.0700860i
$214$	7.05695e19	1.09673
$215$	0	0
$216$	2.60798e19	0.374496
$217$	1.86233e18i	0.0257128i
$218$	− 3.52127e19i	− 0.467541i
$219$	5.73733e19	0.732715
$220$	0	0
$221$	−8.34623e19	−0.986634
$222$	− 1.39661e19i	− 0.158882i
$223$	− 1.08588e20i	− 1.18902i	−0.804088	−	0.594510i	$-0.797346\pi$
$223$	− 1.08588e20i	− 1.18902i	0.804088	−	0.594510i	$-0.202654\pi$
$224$	2.80039e19	0.295196
$225$	0	0
$226$	9.99061e19	0.976495
$227$	− 1.95539e19i	− 0.184083i	−0.995755	−	0.0920415i	$-0.970661\pi$
$227$	− 1.95539e19i	− 0.184083i	0.995755	−	0.0920415i	$-0.0293392\pi$
$228$	3.61366e17i	0.00327719i
$229$	−1.20601e20	−1.05378	−0.526890	−	0.849933i	$-0.676642\pi$
$229$	−1.20601e20	−1.05378	−0.526890	+	0.849933i	$0.676642\pi$
$230$	0	0
$231$	−5.72986e19	−0.464988
$232$	8.59427e19i	0.672297i
$233$	7.20388e19i	0.543302i	0.962396	+	0.271651i	$0.0875696\pi$
$233$	7.20388e19i	0.543302i	−0.962396	+	0.271651i	$0.912430\pi$
$234$	2.54269e19	0.184909
$235$	0	0
$236$	−1.95485e19	−0.132240
$237$	− 4.36491e19i	− 0.284849i
$238$	− 3.56588e20i	− 2.24524i
$239$	−1.70425e20	−1.03550	−0.517750	−	0.855532i	$-0.673230\pi$
$239$	−1.70425e20	−1.03550	−0.517750	+	0.855532i	$0.673230\pi$
$240$	0	0
$241$	−1.71129e20	−0.968679	−0.484340	−	0.874880i	$-0.660940\pi$
$241$	−1.71129e20	−0.968679	−0.484340	+	0.874880i	$0.660940\pi$
$242$	− 1.09168e20i	− 0.596573i
$243$	1.68440e20i	0.888775i
$244$	−5.57464e19	−0.284056
$245$	0	0
$246$	1.18096e20	0.561419
$247$	1.05140e18i	0.00482885i
$248$	1.22676e18i	0.00544401i
$249$	2.91907e19	0.125184
$250$	0	0
$251$	1.85831e20	0.744546	0.372273	−	0.928123i	$-0.378578\pi$
$251$	1.85831e20	0.744546	0.372273	+	0.928123i	$0.378578\pi$
$252$	1.08635e20i	0.420790i
$253$	− 1.09916e20i	− 0.411657i
$254$	−1.88883e20	−0.684078
$255$	0	0
$256$	1.84467e19	0.0625000
$257$	− 1.68273e19i	− 0.0551547i	−0.999620	−	0.0275773i	$-0.991221\pi$
$257$	− 1.68273e19i	− 0.0551547i	0.999620	−	0.0275773i	$-0.00877925\pi$
$258$	− 1.55119e20i	− 0.491923i
$259$	1.73609e20	0.532750
$260$	0	0
$261$	−3.33396e20	−0.958333
$262$	− 3.72245e20i	− 1.03578i
$263$	2.77268e20i	0.746924i	0.927645	+	0.373462i	$0.121829\pi$
$263$	2.77268e20i	0.746924i	−0.927645	+	0.373462i	$0.878171\pi$
$264$	−3.77438e19	−0.0984491
$265$	0	0
$266$	−4.49204e18	−0.0109888
$267$	− 5.45675e18i	− 0.0129297i
$268$	− 4.01714e20i	− 0.922085i
$269$	−9.40799e19	−0.209220	−0.104610	−	0.994513i	$-0.533359\pi$
$269$	−9.40799e19	−0.209220	−0.104610	+	0.994513i	$0.533359\pi$
$270$	0	0
$271$	−6.17628e20	−1.28970	−0.644849	−	0.764310i	$-0.723080\pi$
$271$	−6.17628e20	−1.28970	−0.644849	+	0.764310i	$0.723080\pi$
$272$	− 2.34892e20i	− 0.475371i
$273$	− 3.11087e20i	− 0.610238i
$274$	−5.82488e20	−1.10766
$275$	0	0
$276$	2.05106e20	0.366649
$277$	6.67865e20i	1.15774i	0.815420	+	0.578869i	$0.196506\pi$
$277$	6.67865e20i	1.15774i	−0.815420	+	0.578869i	$0.803494\pi$
$278$	− 3.62491e19i	− 0.0609419i
$279$	−4.75894e18	−0.00776023
$280$	0	0
$281$	−1.13640e20	−0.174393	−0.0871963	−	0.996191i	$-0.527791\pi$
$281$	−1.13640e20	−0.174393	−0.0871963	+	0.996191i	$0.527791\pi$
$282$	− 9.43216e19i	− 0.140441i
$283$	− 3.76957e20i	− 0.544637i	−0.962207	−	0.272318i	$-0.912210\pi$
$283$	− 3.76957e20i	− 0.544637i	0.962207	−	0.272318i	$-0.0877904\pi$
$284$	−3.54859e19	−0.0497565
$285$	0	0
$286$	−1.09816e20	−0.145062
$287$	1.46802e21i	1.88250i
$288$	7.15602e19i	0.0890913i
$289$	−2.16376e21	−2.61564
$290$	0	0
$291$	−9.63408e20	−1.09830
$292$	4.69794e20i	0.520180i
$293$	− 3.57350e20i	− 0.384344i	−0.981361	−	0.192172i	$-0.938447\pi$
$293$	− 3.57350e20i	− 0.384344i	0.981361	−	0.192172i	$-0.0615531\pi$
$294$	8.52464e20	0.890684
$295$	0	0
$296$	1.14360e20	0.112796
$297$	− 4.36946e20i	− 0.418792i
$298$	3.35589e20i	0.312586i
$299$	5.96757e20	0.540247
$300$	0	0
$301$	1.92824e21	1.64947
$302$	− 7.07963e20i	− 0.588778i
$303$	6.19547e20i	0.500971i
$304$	−2.95900e18	−0.00232659
$305$	0	0
$306$	9.11213e20	0.677623
$307$	− 2.25658e21i	− 1.63220i	−0.577908	−	0.816102i	$-0.696131\pi$
$307$	− 2.25658e21i	− 1.63220i	0.577908	−	0.816102i	$-0.303869\pi$
$308$	− 4.69182e20i	− 0.330112i
$309$	1.11497e21	0.763158
$310$	0	0
$311$	5.76844e20	0.373762	0.186881	−	0.982383i	$-0.440162\pi$
$311$	5.76844e20	0.373762	0.186881	+	0.982383i	$0.440162\pi$
$312$	− 2.04920e20i	− 0.129202i
$313$	2.63218e21i	1.61506i	0.589827	+	0.807530i	$0.299196\pi$
$313$	2.63218e21i	1.61506i	−0.589827	+	0.807530i	$0.700804\pi$
$314$	−9.75961e20	−0.582815
$315$	0	0
$316$	3.57415e20	0.202224
$317$	9.95998e20i	0.548599i	0.961644	+	0.274299i	$0.0884459\pi$
$317$	9.95998e20i	0.548599i	−0.961644	+	0.274299i	$0.911554\pi$
$318$	− 1.34871e21i	− 0.723252i
$319$	1.43990e21	0.751817
$320$	0	0
$321$	2.20627e21	1.09236
$322$	2.54961e21i	1.22942i
$323$	3.76786e19i	0.0176959i
$324$	2.64529e20	0.121016
$325$	0	0
$326$	−2.65660e21	−1.15339
$327$	− 1.10088e21i	− 0.465678i
$328$	9.67016e20i	0.398571i
$329$	1.17249e21	0.470915
$330$	0	0
$331$	−3.56520e21	−1.36002	−0.680011	−	0.733202i	$-0.738025\pi$
$331$	−3.56520e21	−1.36002	−0.680011	+	0.733202i	$0.738025\pi$
$332$	2.39025e20i	0.0888728i
$333$	4.43634e20i	0.160786i
$334$	3.12114e21	1.10273
$335$	0	0
$336$	8.75508e20	0.294019
$337$	− 3.65283e20i	− 0.119612i	−0.998210	−	0.0598060i	$-0.980952\pi$
$337$	− 3.65283e20i	− 0.119612i	0.998210	−	0.0598060i	$-0.0190482\pi$
$338$	1.61829e21i	0.516731i
$339$	3.12344e21	0.972605
$340$	0	0
$341$	2.05533e19	0.00608793
$342$	− 1.14788e19i	− 0.00331646i
$343$	4.67180e21i	1.31669i
$344$	1.27017e21	0.349233
$345$	0	0
$346$	9.93818e20	0.260112
$347$	2.06934e21i	0.528483i	0.964457	+	0.264242i	$0.0851217\pi$
$347$	2.06934e21i	0.528483i	−0.964457	+	0.264242i	$0.914878\pi$
$348$	2.68690e21i	0.669618i
$349$	−3.13803e20	−0.0763204	−0.0381602	−	0.999272i	$-0.512150\pi$
$349$	−3.13803e20	−0.0763204	−0.0381602	+	0.999272i	$0.512150\pi$
$350$	0	0
$351$	2.37228e21	0.549611
$352$	− 3.09060e20i	− 0.0698925i
$353$	− 6.87901e21i	− 1.51859i	−0.650746	−	0.759296i	$-0.725544\pi$
$353$	− 6.87901e21i	− 1.51859i	0.650746	−	0.759296i	$-0.274456\pi$
$354$	−6.11161e20	−0.131713
$355$	0	0
$356$	4.46820e19	0.00917925
$357$	− 1.11483e22i	− 2.23629i
$358$	4.15625e21i	0.814134i
$359$	9.50907e21	1.81901	0.909505	−	0.415693i	$-0.136461\pi$
$359$	9.50907e21	1.81901	0.909505	+	0.415693i	$0.136461\pi$
$360$	0	0
$361$	−5.47991e21	−0.999913
$362$	− 5.94420e21i	− 1.05943i
$363$	− 3.41300e21i	− 0.594196i
$364$	2.54730e21	0.433230
$365$	0	0
$366$	−1.74285e21	−0.282924
$367$	− 1.88008e20i	− 0.0298205i	−0.999889	−	0.0149103i	$-0.995254\pi$
$367$	− 1.88008e20i	− 0.0298205i	0.999889	−	0.0149103i	$-0.00474626\pi$
$368$	1.67948e21i	0.260297i
$369$	−3.75133e21	−0.568147
$370$	0	0
$371$	1.67655e22	2.42515
$372$	3.83531e19i	0.00542232i
$373$	9.66745e20i	0.133594i	0.997767	+	0.0667970i	$0.0212780\pi$
$373$	9.66745e20i	0.133594i	−0.997767	+	0.0667970i	$0.978722\pi$
$374$	−3.93543e21	−0.531598
$375$	0	0
$376$	7.72342e20	0.0997040
$377$	7.81756e21i	0.986665i
$378$	1.01354e22i	1.25073i
$379$	−1.09478e22	−1.32097	−0.660484	−	0.750840i	$-0.729649\pi$
$379$	−1.09478e22	−1.32097	−0.660484	+	0.750840i	$0.729649\pi$
$380$	0	0
$381$	−5.90520e21	−0.681353
$382$	1.14670e22i	1.29393i
$383$	− 9.48135e21i	− 1.04636i	−0.852223	−	0.523179i	$-0.824746\pi$
$383$	− 9.48135e21i	− 1.04636i	0.852223	−	0.523179i	$-0.175254\pi$
$384$	5.76715e20	0.0622510
$385$	0	0
$386$	−9.24554e21	−0.954861
$387$	4.92736e21i	0.497818i
$388$	− 7.88875e21i	− 0.779720i
$389$	−8.68048e21	−0.839406	−0.419703	−	0.907661i	$-0.637866\pi$
$389$	−8.68048e21	−0.839406	−0.419703	+	0.907661i	$0.637866\pi$
$390$	0	0
$391$	2.13857e22	1.97980
$392$	6.98030e21i	0.632328i
$393$	− 1.16378e22i	− 1.03166i
$394$	−4.62088e21	−0.400873
$395$	0	0
$396$	1.19893e21	0.0996291
$397$	5.61938e21i	0.457055i	0.973537	+	0.228528i	$0.0733911\pi$
$397$	5.61938e21i	0.457055i	−0.973537	+	0.228528i	$0.926609\pi$
$398$	1.63747e21i	0.130367i
$399$	−1.40438e20	−0.0109450
$400$	0	0
$401$	−4.97857e21	−0.371858	−0.185929	−	0.982563i	$-0.559530\pi$
$401$	−4.97857e21	−0.371858	−0.185929	+	0.982563i	$0.559530\pi$
$402$	− 1.25591e22i	− 0.918411i
$403$	1.11589e20i	0.00798965i
$404$	−5.07309e21	−0.355657
$405$	0	0
$406$	−3.34001e22	−2.24531
$407$	− 1.91600e21i	− 0.126137i
$408$	− 7.34362e21i	− 0.473477i
$409$	−2.65693e20	−0.0167777	−0.00838883	−	0.999965i	$-0.502670\pi$
$409$	−2.65693e20	−0.0167777	−0.00838883	+	0.999965i	$0.502670\pi$
$410$	0	0
$411$	−1.82108e22	−1.10325
$412$	9.12977e21i	0.541793i
$413$	− 7.59717e21i	− 0.441648i
$414$	−6.51519e21	−0.371043
$415$	0	0
$416$	1.67796e21	0.0917252
$417$	− 1.13328e21i	− 0.0606991i
$418$	4.95757e19i	0.00260178i
$419$	−1.71242e22	−0.880626	−0.440313	−	0.897844i	$-0.645133\pi$
$419$	−1.71242e22	−0.880626	−0.440313	+	0.897844i	$0.645133\pi$
$420$	0	0
$421$	2.71839e22	1.34250	0.671249	−	0.741231i	$-0.265758\pi$
$421$	2.71839e22	1.34250	0.671249	+	0.741231i	$0.265758\pi$
$422$	1.97658e21i	0.0956662i
$423$	2.99613e21i	0.142124i
$424$	1.10438e22	0.513462
$425$	0	0
$426$	−1.10942e21	−0.0495583
$427$	− 2.16648e22i	− 0.948677i
$428$	1.80658e22i	0.775505i
$429$	−3.43327e21	−0.144484
$430$	0	0
$431$	−2.34759e22	−0.949656	−0.474828	−	0.880079i	$-0.657490\pi$
$431$	−2.34759e22	−0.949656	−0.474828	+	0.880079i	$0.657490\pi$
$432$	6.67643e21i	0.264809i
$433$	− 1.88732e22i	− 0.734003i	−0.930220	−	0.367002i	$-0.880384\pi$
$433$	− 1.88732e22i	− 0.734003i	0.930220	−	0.367002i	$-0.119616\pi$
$434$	−4.76757e20	−0.0181817
$435$	0	0
$436$	9.01446e21	0.330601
$437$	− 2.69402e20i	− 0.00968968i
$438$	1.46876e22i	0.518107i
$439$	−5.51895e22	−1.90945	−0.954724	−	0.297494i	$-0.903849\pi$
$439$	−5.51895e22	−1.90945	−0.954724	+	0.297494i	$0.903849\pi$
$440$	0	0
$441$	−2.70786e22	−0.901359
$442$	− 2.13664e22i	− 0.697656i
$443$	− 1.20735e22i	− 0.386723i	−0.981128	−	0.193362i	$-0.938061\pi$
$443$	− 1.20735e22i	− 0.386723i	0.981128	−	0.193362i	$-0.0619391\pi$
$444$	3.57532e21	0.112346
$445$	0	0
$446$	2.77984e22	0.840764
$447$	1.04918e22i	0.311341i
$448$	7.16899e21i	0.208735i
$449$	−2.57466e22	−0.735573	−0.367787	−	0.929910i	$-0.619884\pi$
$449$	−2.57466e22	−0.735573	−0.367787	+	0.929910i	$0.619884\pi$
$450$	0	0
$451$	1.62016e22	0.445714
$452$	2.55760e22i	0.690486i
$453$	− 2.21336e22i	− 0.586432i
$454$	5.00580e21	0.130166
$455$	0	0
$456$	−9.25097e19	−0.00231732
$457$	1.01069e22i	0.248503i	0.992251	+	0.124252i	$0.0396530\pi$
$457$	1.01069e22i	0.248503i	−0.992251	+	0.124252i	$0.960347\pi$
$458$	− 3.08739e22i	− 0.745136i
$459$	8.50144e22	2.01412
$460$	0	0
$461$	1.68599e21	0.0384942	0.0192471	−	0.999815i	$-0.493873\pi$
$461$	1.68599e21	0.0384942	0.0192471	+	0.999815i	$0.493873\pi$
$462$	− 1.46684e22i	− 0.328796i
$463$	− 2.77945e22i	− 0.611674i	−0.952084	−	0.305837i	$-0.901064\pi$
$463$	− 2.77945e22i	− 0.611674i	0.952084	−	0.305837i	$-0.0989363\pi$
$464$	−2.20013e22	−0.475386
$465$	0	0
$466$	−1.84419e22	−0.384172
$467$	− 8.26643e22i	− 1.69093i	−0.534034	−	0.845463i	$-0.679324\pi$
$467$	− 8.26643e22i	− 1.69093i	0.534034	−	0.845463i	$-0.320676\pi$
$468$	6.50929e21i	0.130751i
$469$	1.56119e23	3.07954
$470$	0	0
$471$	−3.05123e22	−0.580493
$472$	− 5.00442e21i	− 0.0935075i
$473$	− 2.12807e22i	− 0.390541i
$474$	1.11742e22	0.201418
$475$	0	0
$476$	9.12865e22	1.58762
$477$	4.28420e22i	0.731920i
$478$	− 4.36287e22i	− 0.732209i
$479$	−2.59191e22	−0.427335	−0.213667	−	0.976906i	$-0.568541\pi$
$479$	−2.59191e22	−0.427335	−0.213667	+	0.976906i	$0.568541\pi$
$480$	0	0
$481$	1.04024e22	0.165540
$482$	− 4.38091e22i	− 0.684960i
$483$	7.97106e22i	1.22452i
$484$	2.79469e22	0.421841
$485$	0	0
$486$	−4.31206e22	−0.628459
$487$	4.14828e22i	0.594116i	0.954859	+	0.297058i	$0.0960055\pi$
$487$	4.14828e22i	0.594116i	−0.954859	+	0.297058i	$0.903994\pi$
$488$	− 1.42711e22i	− 0.200858i
$489$	−8.30554e22	−1.14879
$490$	0	0
$491$	3.08644e22	0.412349	0.206174	−	0.978515i	$-0.433899\pi$
$491$	3.08644e22	0.412349	0.206174	+	0.978515i	$0.433899\pi$
$492$	3.02326e22i	0.396983i
$493$	2.80155e23i	3.61575i
$494$	−2.69158e20	−0.00341451
$495$	0	0
$496$	−3.14050e20	−0.00384950
$497$	− 1.37910e22i	− 0.166175i
$498$	7.47282e21i	0.0885187i
$499$	−3.24182e22	−0.377516	−0.188758	−	0.982024i	$-0.560446\pi$
$499$	−3.24182e22	−0.377516	−0.188758	+	0.982024i	$0.560446\pi$
$500$	0	0
$501$	9.75788e22	1.09833
$502$	4.75728e22i	0.526474i
$503$	6.39324e22i	0.695654i	0.937559	+	0.347827i	$0.113080\pi$
$503$	6.39324e22i	0.695654i	−0.937559	+	0.347827i	$0.886920\pi$
$504$	−2.78106e22	−0.297543
$505$	0	0
$506$	2.81384e22	0.291085
$507$	5.05939e22i	0.514672i
$508$	− 4.83540e22i	− 0.483717i
$509$	9.33355e22	0.918218	0.459109	−	0.888380i	$-0.348169\pi$
$509$	9.33355e22	0.918218	0.459109	+	0.888380i	$0.348169\pi$
$510$	0	0
$511$	−1.82577e23	−1.73728
$512$	4.72237e21i	0.0441942i
$513$	− 1.07095e21i	− 0.00985763i
$514$	4.30778e21	0.0390003
$515$	0	0
$516$	3.97104e22	0.347842
$517$	− 1.29400e22i	− 0.111497i
$518$	4.44438e22i	0.376711i
$519$	3.10705e22	0.259075
$520$	0	0
$521$	3.30201e22	0.266476	0.133238	−	0.991084i	$-0.457463\pi$
$521$	3.30201e22	0.266476	0.133238	+	0.991084i	$0.457463\pi$
$522$	− 8.53495e22i	− 0.677644i
$523$	4.06002e22i	0.317149i	0.987347	+	0.158575i	$0.0506898\pi$
$523$	4.06002e22i	0.317149i	−0.987347	+	0.158575i	$0.949310\pi$
$524$	9.52948e22	0.732409
$525$	0	0
$526$	−7.09807e22	−0.528155
$527$	3.99896e21i	0.0292790i
$528$	− 9.66240e21i	− 0.0696140i
$529$	−1.18585e22	−0.0840729
$530$	0	0
$531$	1.94136e22	0.133291
$532$	− 1.14996e21i	− 0.00777025i
$533$	8.79621e22i	0.584944i
$534$	1.39693e21	0.00914268
$535$	0	0
$536$	1.02839e23	0.652013
$537$	1.29940e23i	0.810890i
$538$	− 2.40844e22i	− 0.147941i
$539$	1.16949e23	0.707121
$540$	0	0
$541$	−4.20460e22	−0.246347	−0.123174	−	0.992385i	$-0.539307\pi$
$541$	−4.20460e22	−0.246347	−0.123174	+	0.992385i	$0.539307\pi$
$542$	− 1.58113e23i	− 0.911954i
$543$	− 1.85838e23i	− 1.05520i
$544$	6.01324e22	0.336138
$545$	0	0
$546$	7.96383e22	0.431504
$547$	2.01110e23i	1.07285i	0.843947	+	0.536427i	$0.180226\pi$
$547$	2.01110e23i	1.07285i	−0.843947	+	0.536427i	$0.819774\pi$
$548$	− 1.49117e23i	− 0.783236i
$549$	5.53616e22	0.286315
$550$	0	0
$551$	3.52919e21	0.0176965
$552$	5.25070e22i	0.259260i
$553$	1.38903e23i	0.675380i
$554$	−1.70973e23	−0.818645
$555$	0	0
$556$	9.27976e21	0.0430924
$557$	− 1.46727e23i	− 0.671030i	−0.942035	−	0.335515i	$-0.891090\pi$
$557$	− 1.46727e23i	− 0.671030i	0.942035	−	0.335515i	$-0.108910\pi$
$558$	− 1.21829e21i	− 0.00548731i
$559$	1.15538e23	0.512536
$560$	0	0
$561$	−1.23036e23	−0.529480
$562$	− 2.90919e22i	− 0.123314i
$563$	− 3.39706e23i	− 1.41834i	−0.705036	−	0.709172i	$-0.749069\pi$
$563$	− 3.39706e23i	− 1.41834i	0.705036	−	0.709172i	$-0.250931\pi$
$564$	2.41463e22	0.0993068
$565$	0	0
$566$	9.65010e22	0.385116
$567$	1.02805e23i	0.404163i
$568$	− 9.08439e21i	− 0.0351832i
$569$	−2.72602e23	−1.04010	−0.520050	−	0.854136i	$-0.674087\pi$
$569$	−2.72602e23	−1.04010	−0.520050	+	0.854136i	$0.674087\pi$
$570$	0	0
$571$	−2.79882e22	−0.103650	−0.0518249	−	0.998656i	$-0.516504\pi$
$571$	−2.79882e22	−0.103650	−0.0518249	+	0.998656i	$0.516504\pi$
$572$	− 2.81129e22i	− 0.102574i
$573$	3.58501e23i	1.28877i
$574$	−3.75813e23	−1.33113
$575$	0	0
$576$	−1.83194e22	−0.0629971
$577$	− 3.86678e23i	− 1.31025i	−0.755519	−	0.655127i	$-0.772615\pi$
$577$	− 3.86678e23i	− 1.31025i	0.755519	−	0.655127i	$-0.227385\pi$
$578$	− 5.53923e23i	− 1.84954i
$579$	−2.89051e23	−0.951056
$580$	0	0
$581$	−9.28926e22	−0.296814
$582$	− 2.46632e23i	− 0.776613i
$583$	− 1.85030e23i	− 0.574195i
$584$	−1.20267e23	−0.367823
$585$	0	0
$586$	9.14817e22	0.271772
$587$	− 3.64152e23i	− 1.06625i	−0.846036	−	0.533125i	$-0.821017\pi$
$587$	− 3.64152e23i	− 1.06625i	0.846036	−	0.533125i	$-0.178983\pi$
$588$	2.18231e23i	0.629809i
$589$	5.03761e19	0.000143299 0
$590$	0	0
$591$	−1.44466e23	−0.399276
$592$	2.92761e22i	0.0797587i
$593$	− 6.86689e23i	− 1.84415i	−0.387017	−	0.922073i	$-0.626494\pi$
$593$	− 6.86689e23i	− 1.84415i	0.387017	−	0.922073i	$-0.373506\pi$
$594$	1.11858e23	0.296130
$595$	0	0
$596$	−8.59108e22	−0.221032
$597$	5.11935e22i	0.129847i
$598$	1.52770e23i	0.382013i
$599$	3.82850e23	0.943846	0.471923	−	0.881640i	$-0.343560\pi$
$599$	3.82850e23	0.943846	0.471923	+	0.881640i	$0.343560\pi$
$600$	0	0
$601$	−2.66402e23	−0.638417	−0.319209	−	0.947684i	$-0.603417\pi$
$601$	−2.66402e23	−0.638417	−0.319209	+	0.947684i	$0.603417\pi$
$602$	4.93629e23i	1.16635i
$603$	3.98941e23i	0.929418i
$604$	1.81239e23	0.416329
$605$	0	0
$606$	−1.58604e23	−0.354240
$607$	1.11277e23i	0.245076i	0.992464	+	0.122538i	$0.0391034\pi$
$607$	1.11277e23i	0.245076i	−0.992464	+	0.122538i	$0.960897\pi$
$608$	− 7.57505e20i	− 0.00164515i
$609$	−1.04421e24	−2.23636
$610$	0	0
$611$	7.02541e22	0.146326
$612$	2.33271e23i	0.479152i
$613$	− 3.91229e23i	− 0.792532i	−0.918136	−	0.396266i	$-0.870306\pi$
$613$	− 3.91229e23i	− 0.792532i	0.918136	−	0.396266i	$-0.129694\pi$
$614$	5.77684e23	1.15414
$615$	0	0
$616$	1.20111e23	0.233424
$617$	5.72626e23i	1.09761i	0.835951	+	0.548804i	$0.184917\pi$
$617$	5.72626e23i	1.09761i	−0.835951	+	0.548804i	$0.815083\pi$
$618$	2.85431e23i	0.539634i
$619$	−7.86155e23	−1.46601	−0.733007	−	0.680222i	$-0.761884\pi$
$619$	−7.86155e23	−1.46601	−0.733007	+	0.680222i	$0.761884\pi$
$620$	0	0
$621$	−6.07855e23	−1.10286
$622$	1.47672e23i	0.264290i
$623$	1.73648e22i	0.0306565i
$624$	5.24594e22	0.0913597
$625$	0	0
$626$	−6.73838e23	−1.14202
$627$	1.54993e21i	0.00259142i
$628$	− 2.49846e23i	− 0.412112i
$629$	3.72788e23	0.606640
$630$	0	0
$631$	6.52712e23	1.03389	0.516943	−	0.856020i	$-0.327070\pi$
$631$	6.52712e23	1.03389	0.516943	+	0.856020i	$0.327070\pi$
$632$	9.14983e22i	0.142994i
$633$	6.17954e22i	0.0952850i
$634$	−2.54975e23	−0.387918
$635$	0	0
$636$	3.45271e23	0.511416
$637$	6.34945e23i	0.928007i
$638$	3.68615e23i	0.531615i
$639$	3.52410e22	0.0501522
$640$	0	0
$641$	7.10734e23	0.984950	0.492475	−	0.870327i	$-0.336092\pi$
$641$	7.10734e23	0.984950	0.492475	+	0.870327i	$0.336092\pi$
$642$	5.64805e23i	0.772415i
$643$	2.34278e23i	0.316183i	0.987424	+	0.158092i	$0.0505341\pi$
$643$	2.34278e23i	0.316183i	−0.987424	+	0.158092i	$0.949466\pi$
$644$	−6.52701e23	−0.869329
$645$	0	0
$646$	−9.64571e21	−0.0125129
$647$	− 1.50806e23i	− 0.193078i	−0.995329	−	0.0965391i	$-0.969223\pi$
$647$	− 1.50806e23i	− 0.193078i	0.995329	−	0.0965391i	$-0.0307773\pi$
$648$	6.77195e22i	0.0855710i
$649$	−8.38450e22	−0.104568
$650$	0	0
$651$	−1.49052e22	−0.0181092
$652$	− 6.80089e23i	− 0.815570i
$653$	6.74375e22i	0.0798251i	0.999203	+	0.0399126i	$0.0127079\pi$
$653$	6.74375e22i	0.0798251i	−0.999203	+	0.0399126i	$0.987292\pi$
$654$	2.81826e23	0.329284
$655$	0	0
$656$	−2.47556e23	−0.281832
$657$	− 4.66551e23i	− 0.524317i
$658$	3.00157e23i	0.332987i
$659$	−7.11301e22	−0.0778981	−0.0389491	−	0.999241i	$-0.512401\pi$
$659$	−7.11301e22	−0.0778981	−0.0389491	+	0.999241i	$0.512401\pi$
$660$	0	0
$661$	1.69805e24	1.81233	0.906164	−	0.422926i	$-0.138997\pi$
$661$	1.69805e24	1.81233	0.906164	+	0.422926i	$0.138997\pi$
$662$	− 9.12691e23i	− 0.961680i
$663$	− 6.67994e23i	− 0.694876i
$664$	−6.11903e22	−0.0628426
$665$	0	0
$666$	−1.13570e23	−0.113693
$667$	− 2.00311e24i	− 1.97986i
$668$	7.99012e23i	0.779747i
$669$	8.69085e23	0.837414
$670$	0	0
$671$	−2.39101e23	−0.224615
$672$	2.24130e23i	0.207903i
$673$	− 1.39848e24i	− 1.28094i	−0.767984	−	0.640469i	$-0.778740\pi$
$673$	− 1.39848e24i	− 1.28094i	0.767984	−	0.640469i	$-0.221260\pi$
$674$	9.35124e22	0.0845785
$675$	0	0
$676$	−4.14282e23	−0.365384
$677$	9.38639e23i	0.817514i	0.912643	+	0.408757i	$0.134038\pi$
$677$	9.38639e23i	0.817514i	−0.912643	+	0.408757i	$0.865962\pi$
$678$	7.99602e23i	0.687735i
$679$	3.06582e24	2.60408
$680$	0	0
$681$	1.56500e23	0.129648
$682$	5.26165e21i	0.00430482i
$683$	1.23254e24i	0.995922i	0.867199	+	0.497961i	$0.165918\pi$
$683$	1.23254e24i	0.995922i	−0.867199	+	0.497961i	$0.834082\pi$
$684$	2.93858e21	0.00234509
$685$	0	0
$686$	−1.19598e24	−0.931041
$687$	− 9.65236e23i	− 0.742167i
$688$	3.25164e23i	0.246945i
$689$	1.00457e24	0.753559
$690$	0	0
$691$	3.83055e23	0.280348	0.140174	−	0.990127i	$-0.455234\pi$
$691$	3.83055e23	0.280348	0.140174	+	0.990127i	$0.455234\pi$
$692$	2.54417e23i	0.183927i
$693$	4.65944e23i	0.332737i
$694$	−5.29751e23	−0.373694
$695$	0	0
$696$	−6.87845e23	−0.473492
$697$	3.15226e24i	2.14360i
$698$	− 8.03335e22i	− 0.0539667i
$699$	−5.76565e23	−0.382642
$700$	0	0
$701$	1.63235e24	1.05733	0.528665	−	0.848831i	$-0.322693\pi$
$701$	1.63235e24	1.05733	0.528665	+	0.848831i	$0.322693\pi$
$702$	6.07304e23i	0.388634i
$703$	− 4.69611e21i	− 0.00296906i
$704$	7.91195e22	0.0494215
$705$	0	0
$706$	1.76103e24	1.07381
$707$	− 1.97156e24i	− 1.18781i
$708$	− 1.56457e23i	− 0.0931350i
$709$	6.09431e23	0.358453	0.179226	−	0.983808i	$-0.442641\pi$
$709$	6.09431e23	0.358453	0.179226	+	0.983808i	$0.442641\pi$
$710$	0	0
$711$	−3.54948e23	−0.203832
$712$	1.14386e22i	0.00649071i
$713$	− 2.85926e22i	− 0.0160322i
$714$	2.85396e24	1.58130
$715$	0	0
$716$	−1.06400e24	−0.575680
$717$	− 1.36400e24i	− 0.729292i
$718$	2.43432e24i	1.28623i
$719$	4.49148e23	0.234527	0.117264	−	0.993101i	$-0.462588\pi$
$719$	4.49148e23	0.234527	0.117264	+	0.993101i	$0.462588\pi$
$720$	0	0
$721$	−3.54812e24	−1.80946
$722$	− 1.40286e24i	− 0.707046i
$723$	− 1.36964e24i	− 0.682230i
$724$	1.52171e24	0.749127
$725$	0	0
$726$	8.73727e23	0.420160
$727$	5.00713e23i	0.237983i	0.992895	+	0.118992i	$0.0379662\pi$
$727$	5.00713e23i	0.237983i	−0.992895	+	0.118992i	$0.962034\pi$
$728$	6.52109e23i	0.306340i
$729$	−1.86938e24	−0.867986
$730$	0	0
$731$	4.14048e24	1.87825
$732$	− 4.46168e23i	− 0.200057i
$733$	1.20329e24i	0.533320i	0.963791	+	0.266660i	$0.0859201\pi$
$733$	1.20329e24i	0.533320i	−0.963791	+	0.266660i	$0.914080\pi$
$734$	4.81301e22	0.0210863
$735$	0	0
$736$	−4.29948e23	−0.184058
$737$	− 1.72298e24i	− 0.729133i
$738$	− 9.60340e23i	− 0.401741i
$739$	2.19426e24	0.907426	0.453713	−	0.891148i	$-0.350099\pi$
$739$	2.19426e24	0.907426	0.453713	+	0.891148i	$0.350099\pi$
$740$	0	0
$741$	−8.41491e21	−0.00340091
$742$	4.29197e24i	1.71484i
$743$	− 9.89633e22i	− 0.0390903i	−0.999809	−	0.0195452i	$-0.993778\pi$
$743$	− 9.89633e22i	− 0.0390903i	0.999809	−	0.0195452i	$-0.00622181\pi$
$744$	−9.81839e21	−0.00383416
$745$	0	0
$746$	−2.47487e23	−0.0944652
$747$	− 2.37375e23i	− 0.0895796i
$748$	− 1.00747e24i	− 0.375897i
$749$	−7.02094e24	−2.59000
$750$	0	0
$751$	−1.01856e24	−0.367322	−0.183661	−	0.982990i	$-0.558795\pi$
$751$	−1.01856e24	−0.367322	−0.183661	+	0.982990i	$0.558795\pi$
$752$	1.97719e23i	0.0705014i
$753$	1.48731e24i	0.524376i
$754$	−2.00130e24	−0.697677
$755$	0	0
$756$	−2.59467e24	−0.884397
$757$	1.43989e24i	0.485305i	0.970113	+	0.242653i	$0.0780175\pi$
$757$	1.43989e24i	0.485305i	−0.970113	+	0.242653i	$0.921982\pi$
$758$	− 2.80263e24i	− 0.934066i
$759$	8.79713e23	0.289925
$760$	0	0
$761$	−7.15424e23	−0.230565	−0.115283	−	0.993333i	$-0.536777\pi$
$761$	−7.15424e23	−0.230565	−0.115283	+	0.993333i	$0.536777\pi$
$762$	− 1.51173e24i	− 0.481789i
$763$	3.50330e24i	1.10413i
$764$	−2.93554e24	−0.914944
$765$	0	0
$766$	2.42722e24	0.739887
$767$	− 4.55214e23i	− 0.137232i
$768$	1.47639e23i	0.0440181i
$769$	−1.87177e24	−0.551922	−0.275961	−	0.961169i	$-0.588996\pi$
$769$	−1.87177e24	−0.551922	−0.275961	+	0.961169i	$0.588996\pi$
$770$	0	0
$771$	1.34678e23	0.0388449
$772$	− 2.36686e24i	− 0.675189i
$773$	− 1.86999e24i	− 0.527611i	−0.964576	−	0.263805i	$-0.915022\pi$
$773$	− 1.86999e24i	− 0.527611i	0.964576	−	0.263805i	$-0.0849777\pi$
$774$	−1.26140e24	−0.352011
$775$	0	0
$776$	2.01952e24	0.551345
$777$	1.38948e24i	0.375210i
$778$	− 2.22220e24i	− 0.593550i
$779$	3.97099e22	0.0104913
$780$	0	0
$781$	−1.52202e23	−0.0393447
$782$	5.47475e24i	1.39993i
$783$	− 7.96294e24i	− 2.01418i
$784$	−1.78696e24	−0.447124
$785$	0	0
$786$	2.97928e24	0.729490
$787$	3.29304e24i	0.797648i	0.917027	+	0.398824i	$0.130582\pi$
$787$	3.29304e24i	0.797648i	−0.917027	+	0.398824i	$0.869418\pi$
$788$	− 1.18295e24i	− 0.283460i
$789$	−2.21913e24	−0.526051
$790$	0	0
$791$	−9.93963e24	−2.30606
$792$	3.06927e23i	0.0704484i
$793$	− 1.29813e24i	− 0.294779i
$794$	−1.43856e24	−0.323187
$795$	0	0
$796$	−4.19192e23	−0.0921832
$797$	3.80089e24i	0.826969i	0.910511	+	0.413485i	$0.135688\pi$
$797$	3.80089e24i	0.826969i	−0.910511	+	0.413485i	$0.864312\pi$
$798$	− 3.59522e22i	− 0.00773929i
$799$	2.51767e24	0.536229
$800$	0	0
$801$	−4.43735e22	−0.00925225
$802$	− 1.27451e24i	− 0.262944i
$803$	2.01498e24i	0.411329i
$804$	3.21513e24	0.649415
$805$	0	0
$806$	−2.85667e22	−0.00564953
$807$	− 7.52972e23i	− 0.147351i
$808$	− 1.29871e24i	− 0.251487i
$809$	−1.38713e24	−0.265799	−0.132900	−	0.991129i	$-0.542429\pi$
$809$	−1.38713e24	−0.265799	−0.132900	+	0.991129i	$0.542429\pi$
$810$	0	0
$811$	2.43418e24	0.456747	0.228374	−	0.973574i	$-0.426659\pi$
$811$	2.43418e24	0.456747	0.228374	+	0.973574i	$0.426659\pi$
$812$	− 8.55042e24i	− 1.58767i
$813$	− 4.94321e24i	− 0.908321i
$814$	4.90497e23	0.0891926
$815$	0	0
$816$	1.87997e24	0.334799
$817$	− 5.21589e22i	− 0.00919265i
$818$	− 6.80174e22i	− 0.0118636i
$819$	−2.52972e24	−0.436675
$820$	0	0
$821$	−7.07675e24	−1.19651	−0.598256	−	0.801305i	$-0.704140\pi$
$821$	−7.07675e24	−1.19651	−0.598256	+	0.801305i	$0.704140\pi$
$822$	− 4.66196e24i	− 0.780115i
$823$	− 5.31425e24i	− 0.880124i	−0.897968	−	0.440062i	$-0.854957\pi$
$823$	− 5.31425e24i	− 0.880124i	0.897968	−	0.440062i	$-0.145043\pi$
$824$	−2.33722e24	−0.383105
$825$	0	0
$826$	1.94488e24	0.312292
$827$	− 5.13487e24i	− 0.816080i	−0.912964	−	0.408040i	$-0.866212\pi$
$827$	− 5.13487e24i	− 0.816080i	0.912964	−	0.408040i	$-0.133788\pi$
$828$	− 1.66789e24i	− 0.262367i
$829$	−1.06738e25	−1.66191	−0.830954	−	0.556341i	$-0.812205\pi$
$829$	−1.06738e25	−1.66191	−0.830954	+	0.556341i	$0.812205\pi$
$830$	0	0
$831$	−5.34528e24	−0.815383
$832$	4.29558e23i	0.0648595i
$833$	2.27543e25i	3.40079i
$834$	2.90121e23	0.0429208
$835$	0	0
$836$	−1.26914e22	−0.00183974
$837$	− 1.13664e23i	− 0.0163101i
$838$	− 4.38380e24i	− 0.622697i
$839$	6.47302e24	0.910186	0.455093	−	0.890444i	$-0.349606\pi$
$839$	6.47302e24	0.910186	0.455093	+	0.890444i	$0.349606\pi$
$840$	0	0
$841$	1.89837e25	2.61587
$842$	6.95908e24i	0.949290i
$843$	− 9.09523e23i	− 0.122823i
$844$	−5.06004e23	−0.0676462
$845$	0	0
$846$	−7.67010e23	−0.100497
$847$	1.08611e25i	1.40885i
$848$	2.82721e24i	0.363073i
$849$	3.01699e24	0.383582
$850$	0	0
$851$	−2.66544e24	−0.332175
$852$	− 2.84013e23i	− 0.0350430i
$853$	2.69306e24i	0.328988i	0.986378	+	0.164494i	$0.0525991\pi$
$853$	2.69306e24i	0.328988i	−0.986378	+	0.164494i	$0.947401\pi$
$854$	5.54620e24	0.670816
$855$	0	0
$856$	−4.62484e24	−0.548365
$857$	1.86785e24i	0.219283i	0.993971	+	0.109641i	$0.0349702\pi$
$857$	1.86785e24i	0.219283i	−0.993971	+	0.109641i	$0.965030\pi$
$858$	− 8.78916e23i	− 0.102166i
$859$	1.13124e25	1.30200	0.651000	−	0.759077i	$-0.274350\pi$
$859$	1.13124e25	1.30200	0.651000	+	0.759077i	$0.274350\pi$
$860$	0	0
$861$	−1.17493e25	−1.32583
$862$	− 6.00984e24i	− 0.671508i
$863$	− 2.37061e24i	− 0.262282i	−0.991364	−	0.131141i	$-0.958136\pi$
$863$	− 2.37061e24i	− 0.262282i	0.991364	−	0.131141i	$-0.0418640\pi$
$864$	−1.70917e24	−0.187248
$865$	0	0
$866$	4.83153e24	0.519019
$867$	− 1.73178e25i	− 1.84217i
$868$	− 1.22050e23i	− 0.0128564i
$869$	1.53298e24	0.159907
$870$	0	0
$871$	9.35446e24	0.956895
$872$	2.30770e24i	0.233770i
$873$	7.83429e24i	0.785921i
$874$	6.89670e22	0.00685164
$875$	0	0
$876$	−3.76002e24	−0.366357
$877$	6.79534e24i	0.655714i	0.944727	+	0.327857i	$0.106326\pi$
$877$	6.79534e24i	0.655714i	−0.944727	+	0.327857i	$0.893674\pi$
$878$	− 1.41285e25i	− 1.35018i
$879$	2.86007e24	0.270689
$880$	0	0
$881$	−1.88402e25	−1.74900	−0.874501	−	0.485024i	$-0.838811\pi$
$881$	−1.88402e25	−1.74900	−0.874501	+	0.485024i	$0.838811\pi$
$882$	− 6.93212e24i	− 0.637357i
$883$	3.85491e24i	0.351033i	0.984476	+	0.175517i	$0.0561596\pi$
$883$	3.85491e24i	0.351033i	−0.984476	+	0.175517i	$0.943840\pi$
$884$	5.46979e24	0.493317
$885$	0	0
$886$	3.09081e24	0.273455
$887$	− 8.32727e24i	− 0.729712i	−0.931064	−	0.364856i	$-0.881118\pi$
$887$	− 8.32727e24i	− 0.729712i	0.931064	−	0.364856i	$-0.118882\pi$
$888$	9.15281e23i	0.0794410i
$889$	1.87919e25	1.61550
$890$	0	0
$891$	1.13459e24	0.0956924
$892$	7.11640e24i	0.594510i
$893$	− 3.17158e22i	− 0.00262445i
$894$	−2.68590e24	−0.220151
$895$	0	0
$896$	−1.83526e24	−0.147598
$897$	4.77617e24i	0.380491i
$898$	− 6.59113e24i	− 0.520129i
$899$	3.74565e23	0.0292799
$900$	0	0
$901$	3.60004e25	2.76151
$902$	4.14760e24i	0.315168i
$903$	1.54327e25i	1.16171i
$904$	−6.54744e24	−0.488248
$905$	0	0
$906$	5.66621e24	0.414670
$907$	− 1.92435e25i	− 1.39515i	−0.716511	−	0.697576i	$-0.754262\pi$
$907$	− 1.92435e25i	− 1.39515i	0.716511	−	0.697576i	$-0.245738\pi$
$908$	1.28148e24i	0.0920415i
$909$	5.03807e24	0.358485
$910$	0	0
$911$	−1.57682e25	−1.10123	−0.550613	−	0.834761i	$-0.685606\pi$
$911$	−1.57682e25	−1.10123	−0.550613	+	0.834761i	$0.685606\pi$
$912$	− 2.36825e22i	− 0.00163859i
$913$	1.02519e24i	0.0702756i
$914$	−2.58738e24	−0.175718
$915$	0	0
$916$	7.90373e24	0.526890
$917$	3.70346e25i	2.44607i
$918$	2.17637e25i	1.42420i
$919$	−1.03058e25	−0.668189	−0.334094	−	0.942540i	$-0.608430\pi$
$919$	−1.03058e25	−0.668189	−0.334094	+	0.942540i	$0.608430\pi$
$920$	0	0
$921$	1.80606e25	1.14954
$922$	4.31612e23i	0.0272195i
$923$	− 8.26339e23i	− 0.0516349i
$924$	3.75512e24	0.232494
$925$	0	0
$926$	7.11539e24	0.432519
$927$	− 9.06674e24i	− 0.546102i
$928$	− 5.63234e24i	− 0.336149i
$929$	−9.73688e24	−0.575819	−0.287910	−	0.957658i	$-0.592960\pi$
$929$	−9.73688e24	−0.575819	−0.287910	+	0.957658i	$0.592960\pi$
$930$	0	0
$931$	2.86642e23	0.0166444
$932$	− 4.72113e24i	− 0.271651i
$933$	4.61679e24i	0.263236i
$934$	2.11621e25	1.19567
$935$	0	0
$936$	−1.66638e24	−0.0924547
$937$	3.22145e25i	1.77119i	0.464459	+	0.885594i	$0.346249\pi$
$937$	3.22145e25i	1.77119i	−0.464459	+	0.885594i	$0.653751\pi$
$938$	3.99664e25i	2.17756i
$939$	−2.10667e25	−1.13747
$940$	0	0
$941$	−7.90374e24	−0.419103	−0.209552	−	0.977798i	$-0.567200\pi$
$941$	−7.90374e24	−0.419103	−0.209552	+	0.977798i	$0.567200\pi$
$942$	− 7.81114e24i	− 0.410470i
$943$	− 2.25387e25i	− 1.17376i
$944$	1.28113e24	0.0661198
$945$	0	0
$946$	5.44786e24	0.276154
$947$	− 2.36682e25i	− 1.18902i	−0.804086	−	0.594512i	$-0.797345\pi$
$947$	− 2.36682e25i	− 1.18902i	0.804086	−	0.594512i	$-0.202655\pi$
$948$	2.86059e24i	0.142424i
$949$	−1.09398e25	−0.539818
$950$	0	0
$951$	−7.97150e24	−0.386372
$952$	2.33693e25i	1.12262i
$953$	9.09477e24i	0.433014i	0.976281	+	0.216507i	$0.0694664\pi$
$953$	9.09477e24i	0.433014i	−0.976281	+	0.216507i	$0.930534\pi$
$954$	−1.09676e25	−0.517546
$955$	0	0
$956$	1.11690e25	0.517750
$957$	1.15243e25i	0.529496i
$958$	− 6.63529e24i	− 0.302171i
$959$	5.79516e25	2.61582
$960$	0	0
$961$	−2.25448e25	−0.999763
$962$	2.66302e24i	0.117054i
$963$	− 1.79411e25i	− 0.781673i
$964$	1.12151e25	0.484340
$965$	0	0
$966$	−2.04059e25	−0.865865
$967$	− 3.62970e25i	− 1.52667i	−0.646001	−	0.763337i	$-0.723560\pi$
$967$	− 3.62970e25i	− 1.52667i	0.646001	−	0.763337i	$-0.276440\pi$
$968$	7.15441e24i	0.298286i
$969$	−3.01562e23	−0.0124630
$970$	0	0
$971$	−9.70396e24	−0.394081	−0.197040	−	0.980395i	$-0.563133\pi$
$971$	−9.70396e24	−0.394081	−0.197040	+	0.980395i	$0.563133\pi$
$972$	− 1.10389e25i	− 0.444387i
$973$	3.60641e24i	0.143918i
$974$	−1.06196e25	−0.420104
$975$	0	0
$976$	3.65340e24	0.142028
$977$	− 3.70141e25i	− 1.42647i	−0.700924	−	0.713236i	$-0.747229\pi$
$977$	− 3.70141e25i	− 1.42647i	0.700924	−	0.713236i	$-0.252771\pi$
$978$	− 2.12622e25i	− 0.812321i
$979$	1.91644e23	0.00725843
$980$	0	0
$981$	−8.95223e24	−0.333230
$982$	7.90127e24i	0.291574i
$983$	− 6.43184e24i	− 0.235305i	−0.993055	−	0.117652i	$-0.962463\pi$
$983$	− 6.43184e24i	− 0.235305i	0.993055	−	0.117652i	$-0.0375368\pi$
$984$	−7.73954e24	−0.280709
$985$	0	0
$986$	−7.17196e25	−2.55672
$987$	9.38403e24i	0.331661i
$988$	− 6.89045e22i	− 0.00241442i
$989$	−2.96045e25	−1.02847
$990$	0	0
$991$	5.17070e25	1.76573	0.882863	−	0.469631i	$-0.155613\pi$
$991$	5.17070e25	1.76573	0.882863	+	0.469631i	$0.155613\pi$
$992$	− 8.03967e22i	− 0.00272201i
$993$	− 2.85342e25i	− 0.957848i
$994$	3.53048e24	0.117503
$995$	0	0
$996$	−1.91304e24	−0.0625922
$997$	2.77457e25i	0.900093i	0.893005	+	0.450047i	$0.148593\pi$
$997$	2.77457e25i	0.900093i	−0.893005	+	0.450047i	$0.851407\pi$
$998$	− 8.29907e24i	− 0.266944i
$999$	−1.05959e25	−0.337933

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.18.b.d.49.4		4
5.2	odd	4		10.18.a.c.1.2	✓	2
5.3	odd	4		50.18.a.f.1.1		2
5.4	even	2	inner	50.18.b.d.49.1		4
15.2	even	4		90.18.a.k.1.2		2
20.7	even	4		80.18.a.d.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.18.a.c.1.2	✓	2	5.2	odd	4
50.18.a.f.1.1		2	5.3	odd	4
50.18.b.d.49.1		4	5.4	even	2	inner
50.18.b.d.49.4		4	1.1	even	1	trivial
80.18.a.d.1.1		2	20.7	even	4
90.18.a.k.1.2		2	15.2	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:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	50.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+256.000i q^{2} +8003.53i q^{3} -65536.0 q^{4} -2.04890e6 q^{6} -2.54694e7i q^{7} -1.67772e7i q^{8} +6.50836e7 q^{9} -2.81089e8 q^{11} -5.24520e8i q^{12} -1.52610e9i q^{13} +6.52016e9 q^{14} +4.29497e9 q^{16} -5.46901e10i q^{17} +1.66614e10i q^{18} -6.88947e8 q^{19} +2.03845e11 q^{21} -7.19587e10i q^{22} +3.91035e11i q^{23} +1.34277e11 q^{24} +3.90681e11 q^{26} +1.55448e12i q^{27} +1.66916e12i q^{28} -5.12259e12 q^{29} -7.31204e10 q^{31} +1.09951e12i q^{32} -2.24970e12i q^{33} +1.40007e13 q^{34} -4.26532e12 q^{36} +6.81637e12i q^{37} -1.76370e11i q^{38} +1.22142e13 q^{39} -5.76386e13 q^{41} +5.21843e13i q^{42} +7.57081e13i q^{43} +1.84214e13 q^{44} -1.00105e14 q^{46} +4.60351e13i q^{47} +3.43749e13i q^{48} -4.16059e14 q^{49} +4.37714e14 q^{51} +1.00014e14i q^{52} +6.58261e14i q^{53} -3.97946e14 q^{54} -4.27305e14 q^{56} -5.51401e12i q^{57} -1.31138e15i q^{58} +2.98287e14 q^{59} +8.50623e14 q^{61} -1.87188e13i q^{62} -1.65764e15i q^{63} -2.81475e14 q^{64} +5.75924e14 q^{66} +6.12967e15i q^{67} +3.58417e15i q^{68} -3.12966e15 q^{69} +5.41472e14 q^{71} -1.09192e15i q^{72} -7.16849e15i q^{73} -1.74499e15 q^{74} +4.51508e13 q^{76} +7.15916e15i q^{77} +3.12683e15i q^{78} -5.45373e15 q^{79} -4.03640e15 q^{81} -1.47555e16i q^{82} -3.64723e15i q^{83} -1.33592e16 q^{84} -1.93813e16 q^{86} -4.09988e16i q^{87} +4.71589e15i q^{88} -6.81792e14 q^{89} -3.88687e16 q^{91} -2.56269e16i q^{92} -5.85222e14i q^{93} -1.17850e16 q^{94} -8.79998e15 q^{96} +1.20373e17i q^{97} -1.06511e17i q^{98} -1.82943e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 262144 q^{4} + 669696 q^{6} + 215038188 q^{9} - 942962592 q^{11} + 309168128 q^{14} + 17179869184 q^{16} - 257345058800 q^{19} + 870762493248 q^{21} - 43889197056 q^{24} + 789419124736 q^{26} - 5086109499960 q^{29}+ \cdots - 52\!\cdots\!24 q^{99}+O(q^{100}) \) 4 * q - 262144 * q^4 + 669696 * q^6 + 215038188 * q^9 - 942962592 * q^11 + 309168128 * q^14 + 17179869184 * q^16 - 257345058800 * q^19 + 870762493248 * q^21 - 43889197056 * q^24 + 789419124736 * q^26 - 5086109499960 * q^29 + 15678815997488 * q^31 + 16455629088768 * q^34 - 14092742688768 * q^36 + 24135247074576 * q^39 - 75652728528312 * q^41 + 61797996429312 * q^44 - 332984247957504 * q^46 - 1603445169355908 * q^49 + 1295380982531088 * q^51 + 22095889551360 * q^54 - 20261642436608 * q^56 - 3055641434470320 * q^59 - 1263476426244232 * q^61 - 1125899906842624 * q^64 + 244151200161792 * q^66 - 1429906501577664 * q^69 - 6663803321824752 * q^71 - 14236267057948672 * q^74 + 16865365773516800 * q^76 - 16527213684110240 * q^79 - 26865367701318396 * q^81 - 57066290757500928 * q^84 - 15170792410548224 * q^86 - 111756825043809960 * q^89 - 76954793221206032 * q^91 + 113045290507266048 * q^94 + 2876322418262016 * q^96 - 52747342198192224 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more