LMFDB - Embedded newform 16.50.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,50,Mod(1,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 50, names="a")

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 50);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 50 $
Character orbit:	$[\chi]$	$=$	16.a (trivial)

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:	yes
Analytic conductor:	$243.305928158$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 104434803447206332x + 4289992005756109702361620 $ x^3 - x^2 - 104434803447206332*x + 4289992005756109702361620
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{5}\cdot 5^{2}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 2)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.17763e7$ of defining polynomial
Character	$\chi$	$=$	16.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.94355e11 q^{3} -5.65262e16 q^{5} +4.38546e20 q^{7} +3.91700e23 q^{9} +O(q^{10})$	$q-7.94355e11 q^{3} -5.65262e16 q^{5} +4.38546e20 q^{7} +3.91700e23 q^{9} -5.68225e25 q^{11} +1.61873e27 q^{13} +4.49019e28 q^{15} +2.39583e30 q^{17} +9.22680e30 q^{19} -3.48361e32 q^{21} +1.45988e33 q^{23} -1.45684e34 q^{25} -1.21060e35 q^{27} +1.26757e36 q^{29} +2.70852e36 q^{31} +4.51372e37 q^{33} -2.47894e37 q^{35} -7.30159e37 q^{37} -1.28584e39 q^{39} +1.21786e39 q^{41} +1.71091e40 q^{43} -2.21413e40 q^{45} +1.04023e41 q^{47} -6.46007e40 q^{49} -1.90314e42 q^{51} -1.17198e41 q^{53} +3.21196e42 q^{55} -7.32935e42 q^{57} +8.71601e42 q^{59} -2.61290e43 q^{61} +1.71779e44 q^{63} -9.15004e43 q^{65} -3.87604e44 q^{67} -1.15966e45 q^{69} -2.06352e44 q^{71} +4.21892e44 q^{73} +1.15724e46 q^{75} -2.49193e46 q^{77} -8.65650e45 q^{79} +2.43119e45 q^{81} -1.57334e47 q^{83} -1.35427e47 q^{85} -1.00690e48 q^{87} -2.35938e47 q^{89} +7.09886e47 q^{91} -2.15153e48 q^{93} -5.21556e47 q^{95} +1.57542e48 q^{97} -2.22574e49 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 16203614388 q^{3} - 10\!\cdots\!30 q^{5}+ \cdots + 38\!\cdots\!99 q^{9}+O(q^{10}) $ 3 * q + 16203614388 * q^3 - 101813016401840430 * q^5 + 125479179230203797096 * q^7 + 389432931519696922052199 * q^9	$ 3 q + 16203614388 q^{3} - 10\!\cdots\!30 q^{5}+ \cdots - 17\!\cdots\!68 q^{99}+O(q^{100}) $ 3 * q + 16203614388 * q^3 - 101813016401840430 * q^5 + 125479179230203797096 * q^7 + 389432931519696922052199 * q^9 + 25860036791834065092767004 * q^11 + 2945123797958353151076283242 * q^13 - 87846213583565151681178171080 * q^15 + 3910755572889939740390354668854 * q^17 - 14590272981610371514592320211580 * q^19 - 109864176920553355284360667668384 * q^21 + 4953158032634305739950894912759608 * q^23 + 39451396326519132102111422427450525 * q^25 - 196120894647932767255303897131684600 * q^27 + 937305925294643190481854503019954330 * q^29 - 2414809991776326638635666091994140256 * q^31 + 84513992666209345011409783180743651984 * q^33 - 300439012903055674136683294221406953360 * q^35 + 309377978155169809207437071391240173874 * q^37 - 360871814296103457146322338824042742568 * q^39 - 1143109642820632592659114174552039005954 * q^41 + 17579019372148890319048762066144112105148 * q^43 - 114813225075613761568802306330842527416790 * q^45 + 218573936001854642959028674713638582342256 * q^47 + 373628008074429359506251920014708170372651 * q^49 - 2241123510744223933356304325509677976541016 * q^51 + 740200780025011930130058449331923084936802 * q^53 - 3200488473448119911679443277764576117207640 * q^55 - 20594915296543361145254552753753824314692880 * q^57 + 53014363316602767635595514099864508327238540 * q^59 - 124730102606725385116407342714381742110559494 * q^61 + 468292624427612329618287149072325587596605768 * q^63 - 421321721827999315000625667485604880336092420 * q^65 - 1111804215897361218882144911380577470099878924 * q^67 + 770909887444909922528222240570561038770401568 * q^69 + 4013080456871179368491803943114613092268032424 * q^71 - 10015638764428635672946257750737669286684666018 * q^73 + 38646070268253498119014909069282536501398301900 * q^75 - 23360330575813840356220747501745396153372866272 * q^77 - 36077368003505469428776991593224667583970712560 * q^79 - 14427585993163108762153736521422002728666768917 * q^81 - 150749458884727999220762915504103393007529125692 * q^83 + 566911972337878875844467446456481028710158526660 * q^85 - 1072056267743755243448622937355132996639037778920 * q^87 + 542251936064510107856207107418847041962289942670 * q^89 + 1459783720923935217191336434024489699257545191344 * q^91 - 2368291579513879043404530519265491460552020418176 * q^93 + 2813579016011586836853932078889401271258883877400 * q^95 + 4343844254337968328695747517284119056283595359014 * q^97 - 17595360353038055002493296165642581030249046200468 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−7.94355e11	−1.62384	−0.811921	−	0.583767i	$-0.801578\pi$
$3$	−7.94355e11	−1.62384	−0.811921	+	0.583767i	$0.801578\pi$
$4$	0	0
$5$	−5.65262e16	−0.424116	−0.212058	−	0.977257i	$-0.568017\pi$
$5$	−5.65262e16	−0.424116	−0.212058	+	0.977257i	$0.568017\pi$
$6$	0	0
$7$	4.38546e20	0.865194	0.432597	−	0.901587i	$-0.357597\pi$
$7$	4.38546e20	0.865194	0.432597	+	0.901587i	$0.357597\pi$
$8$	0	0
$9$	3.91700e23	1.63686
$10$	0	0
$11$	−5.68225e25	−1.73940	−0.869700	−	0.493580i	$-0.835688\pi$
$11$	−5.68225e25	−1.73940	−0.869700	+	0.493580i	$0.835688\pi$
$12$	0	0
$13$	1.61873e27	0.827106	0.413553	−	0.910480i	$-0.364288\pi$
$13$	1.61873e27	0.827106	0.413553	+	0.910480i	$0.364288\pi$
$14$	0	0
$15$	4.49019e28	0.688698
$16$	0	0
$17$	2.39583e30	1.71182	0.855910	−	0.517124i	$-0.172997\pi$
$17$	2.39583e30	1.71182	0.855910	+	0.517124i	$0.172997\pi$
$18$	0	0
$19$	9.22680e30	0.432104	0.216052	−	0.976382i	$-0.430682\pi$
$19$	9.22680e30	0.432104	0.216052	+	0.976382i	$0.430682\pi$
$20$	0	0
$21$	−3.48361e32	−1.40494
$22$	0	0
$23$	1.45988e33	0.633849	0.316925	−	0.948451i	$-0.397350\pi$
$23$	1.45988e33	0.633849	0.316925	+	0.948451i	$0.397350\pi$
$24$	0	0
$25$	−1.45684e34	−0.820125
$26$	0	0
$27$	−1.21060e35	−1.03416
$28$	0	0
$29$	1.26757e36	1.88027	0.940135	−	0.340802i	$-0.110699\pi$
$29$	1.26757e36	1.88027	0.940135	+	0.340802i	$0.110699\pi$
$30$	0	0
$31$	2.70852e36	0.784099	0.392049	−	0.919944i	$-0.371766\pi$
$31$	2.70852e36	0.784099	0.392049	+	0.919944i	$0.371766\pi$
$32$	0	0
$33$	4.51372e37	2.82451
$34$	0	0
$35$	−2.47894e37	−0.366943
$36$	0	0
$37$	−7.30159e37	−0.276997	−0.138499	−	0.990363i	$-0.544228\pi$
$37$	−7.30159e37	−0.276997	−0.138499	+	0.990363i	$0.544228\pi$
$38$	0	0
$39$	−1.28584e39	−1.34309
$40$	0	0
$41$	1.21786e39	0.373587	0.186793	−	0.982399i	$-0.440191\pi$
$41$	1.21786e39	0.373587	0.186793	+	0.982399i	$0.440191\pi$
$42$	0	0
$43$	1.71091e40	1.63400	0.816998	−	0.576640i	$-0.195637\pi$
$43$	1.71091e40	1.63400	0.816998	+	0.576640i	$0.195637\pi$
$44$	0	0
$45$	−2.21413e40	−0.694220
$46$	0	0
$47$	1.04023e41	1.12392	0.561958	−	0.827166i	$-0.310048\pi$
$47$	1.04023e41	1.12392	0.561958	+	0.827166i	$0.310048\pi$
$48$	0	0
$49$	−6.46007e40	−0.251439
$50$	0	0
$51$	−1.90314e42	−2.77973
$52$	0	0
$53$	−1.17198e41	−0.0667057	−0.0333528	−	0.999444i	$-0.510619\pi$
$53$	−1.17198e41	−0.0667057	−0.0333528	+	0.999444i	$0.510619\pi$
$54$	0	0
$55$	3.21196e42	0.737708
$56$	0	0
$57$	−7.32935e42	−0.701668
$58$	0	0
$59$	8.71601e42	0.358462	0.179231	−	0.983807i	$-0.442639\pi$
$59$	8.71601e42	0.358462	0.179231	+	0.983807i	$0.442639\pi$
$60$	0	0
$61$	−2.61290e43	−0.474834	−0.237417	−	0.971408i	$-0.576301\pi$
$61$	−2.61290e43	−0.474834	−0.237417	+	0.971408i	$0.576301\pi$
$62$	0	0
$63$	1.71779e44	1.41620
$64$	0	0
$65$	−9.15004e43	−0.350789
$66$	0	0
$67$	−3.87604e44	−0.707222	−0.353611	−	0.935393i	$-0.615046\pi$
$67$	−3.87604e44	−0.707222	−0.353611	+	0.935393i	$0.615046\pi$
$68$	0	0
$69$	−1.15966e45	−1.02927
$70$	0	0
$71$	−2.06352e44	−0.0909449	−0.0454724	−	0.998966i	$-0.514479\pi$
$71$	−2.06352e44	−0.0909449	−0.0454724	+	0.998966i	$0.514479\pi$
$72$	0	0
$73$	4.21892e44	0.0941438	0.0470719	−	0.998892i	$-0.485011\pi$
$73$	4.21892e44	0.0941438	0.0470719	+	0.998892i	$0.485011\pi$
$74$	0	0
$75$	1.15724e46	1.33175
$76$	0	0
$77$	−2.49193e46	−1.50492
$78$	0	0
$79$	−8.65650e45	−0.278919	−0.139460	−	0.990228i	$-0.544537\pi$
$79$	−8.65650e45	−0.278919	−0.139460	+	0.990228i	$0.544537\pi$
$80$	0	0
$81$	2.43119e45	0.0424557
$82$	0	0
$83$	−1.57334e47	−1.51150	−0.755750	−	0.654860i	$-0.772728\pi$
$83$	−1.57334e47	−1.51150	−0.755750	+	0.654860i	$0.772728\pi$
$84$	0	0
$85$	−1.35427e47	−0.726011
$86$	0	0
$87$	−1.00690e48	−3.05326
$88$	0	0
$89$	−2.35938e47	−0.409962	−0.204981	−	0.978766i	$-0.565713\pi$
$89$	−2.35938e47	−0.409962	−0.204981	+	0.978766i	$0.565713\pi$
$90$	0	0
$91$	7.09886e47	0.715607
$92$	0	0
$93$	−2.15153e48	−1.27325
$94$	0	0
$95$	−5.21556e47	−0.183262
$96$	0	0
$97$	1.57542e48	0.332269	0.166135	−	0.986103i	$-0.446871\pi$
$97$	1.57542e48	0.332269	0.166135	+	0.986103i	$0.446871\pi$
$98$	0	0
$99$	−2.22574e49	−2.84716
$100$	0	0
$101$	−2.10004e49	−1.64571	−0.822857	−	0.568248i	$-0.807621\pi$
$101$	−2.10004e49	−1.64571	−0.822857	+	0.568248i	$0.807621\pi$
$102$	0	0
$103$	3.08538e49	1.49554	0.747769	−	0.663960i	$-0.231125\pi$
$103$	3.08538e49	1.49554	0.747769	+	0.663960i	$0.231125\pi$
$104$	0	0
$105$	1.96915e49	0.595857
$106$	0	0
$107$	1.65194e49	0.314842	0.157421	−	0.987532i	$-0.449682\pi$
$107$	1.65194e49	0.314842	0.157421	+	0.987532i	$0.449682\pi$
$108$	0	0
$109$	−9.19427e49	−1.11319	−0.556593	−	0.830785i	$-0.687892\pi$
$109$	−9.19427e49	−1.11319	−0.556593	+	0.830785i	$0.687892\pi$
$110$	0	0
$111$	5.80005e49	0.449800
$112$	0	0
$113$	6.93766e49	0.347369	0.173685	−	0.984801i	$-0.444433\pi$
$113$	6.93766e49	0.347369	0.173685	+	0.984801i	$0.444433\pi$
$114$	0	0
$115$	−8.25217e49	−0.268826
$116$	0	0
$117$	6.34055e50	1.35386
$118$	0	0
$119$	1.05068e51	1.48106
$120$	0	0
$121$	2.16161e51	2.02552
$122$	0	0
$123$	−9.67409e50	−0.606646
$124$	0	0
$125$	1.82760e51	0.771945
$126$	0	0
$127$	3.58918e51	1.02755	0.513776	−	0.857924i	$-0.328246\pi$
$127$	3.58918e51	1.02755	0.513776	+	0.857924i	$0.328246\pi$
$128$	0	0
$129$	−1.35907e52	−2.65335
$130$	0	0
$131$	−8.30864e51	−1.11271	−0.556357	−	0.830944i	$-0.687801\pi$
$131$	−8.30864e51	−1.11271	−0.556357	+	0.830944i	$0.687801\pi$
$132$	0	0
$133$	4.04638e51	0.373854
$134$	0	0
$135$	6.84308e51	0.438605
$136$	0	0
$137$	1.87006e52	0.835989	0.417995	−	0.908449i	$-0.362733\pi$
$137$	1.87006e52	0.835989	0.417995	+	0.908449i	$0.362733\pi$
$138$	0	0
$139$	−7.10289e51	−0.222624	−0.111312	−	0.993785i	$-0.535505\pi$
$139$	−7.10289e51	−0.222624	−0.111312	+	0.993785i	$0.535505\pi$
$140$	0	0
$141$	−8.26315e52	−1.82506
$142$	0	0
$143$	−9.19800e52	−1.43867
$144$	0	0
$145$	−7.16508e52	−0.797453
$146$	0	0
$147$	5.13159e52	0.408298
$148$	0	0
$149$	−1.54238e53	−0.881308	−0.440654	−	0.897677i	$-0.645253\pi$
$149$	−1.54238e53	−0.881308	−0.440654	+	0.897677i	$0.645253\pi$
$150$	0	0
$151$	2.11963e52	0.0873626	0.0436813	−	0.999046i	$-0.486091\pi$
$151$	2.11963e52	0.0873626	0.0436813	+	0.999046i	$0.486091\pi$
$152$	0	0
$153$	9.38449e53	2.80201
$154$	0	0
$155$	−1.53102e53	−0.332549
$156$	0	0
$157$	2.10086e53	0.333316	0.166658	−	0.986015i	$-0.446702\pi$
$157$	2.10086e53	0.333316	0.166658	+	0.986015i	$0.446702\pi$
$158$	0	0
$159$	9.30971e52	0.108319
$160$	0	0
$161$	6.40226e53	0.548403
$162$	0	0
$163$	2.57326e54	1.62887	0.814436	−	0.580253i	$-0.197046\pi$
$163$	2.57326e54	1.62887	0.814436	+	0.580253i	$0.197046\pi$
$164$	0	0
$165$	−2.55144e54	−1.19792
$166$	0	0
$167$	−3.95831e53	−0.138343	−0.0691713	−	0.997605i	$-0.522036\pi$
$167$	−3.95831e53	−0.138343	−0.0691713	+	0.997605i	$0.522036\pi$
$168$	0	0
$169$	−1.20995e54	−0.315896
$170$	0	0
$171$	3.61414e54	0.707294
$172$	0	0
$173$	8.07344e54	1.18831	0.594154	−	0.804351i	$-0.297487\pi$
$173$	8.07344e54	1.18831	0.594154	+	0.804351i	$0.297487\pi$
$174$	0	0
$175$	−6.38890e54	−0.709568
$176$	0	0
$177$	−6.92361e54	−0.582086
$178$	0	0
$179$	−1.08682e55	−0.693841	−0.346921	−	0.937894i	$-0.612773\pi$
$179$	−1.08682e55	−0.693841	−0.346921	+	0.937894i	$0.612773\pi$
$180$	0	0
$181$	−9.05176e54	−0.440158	−0.220079	−	0.975482i	$-0.570631\pi$
$181$	−9.05176e54	−0.440158	−0.220079	+	0.975482i	$0.570631\pi$
$182$	0	0
$183$	2.07557e55	0.771055
$184$	0	0
$185$	4.12731e54	0.117479
$186$	0	0
$187$	−1.36137e56	−2.97754
$188$	0	0
$189$	−5.30905e55	−0.894752
$190$	0	0
$191$	9.68165e55	1.26076	0.630379	−	0.776287i	$-0.282899\pi$
$191$	9.68165e55	1.26076	0.630379	+	0.776287i	$0.282899\pi$
$192$	0	0
$193$	−5.34429e55	−0.539183	−0.269591	−	0.962975i	$-0.586889\pi$
$193$	−5.34429e55	−0.539183	−0.269591	+	0.962975i	$0.586889\pi$
$194$	0	0
$195$	7.26838e55	0.569626
$196$	0	0
$197$	2.48206e56	1.51492	0.757461	−	0.652880i	$-0.226439\pi$
$197$	2.48206e56	1.51492	0.757461	+	0.652880i	$0.226439\pi$
$198$	0	0
$199$	−5.78065e55	−0.275471	−0.137736	−	0.990469i	$-0.543982\pi$
$199$	−5.78065e55	−0.275471	−0.137736	+	0.990469i	$0.543982\pi$
$200$	0	0
$201$	3.07895e56	1.14842
$202$	0	0
$203$	5.55887e56	1.62680
$204$	0	0
$205$	−6.88408e55	−0.158444
$206$	0	0
$207$	5.71836e56	1.03752
$208$	0	0
$209$	−5.24290e56	−0.751602
$210$	0	0
$211$	−5.32965e54	−0.00605032	−0.00302516	−	0.999995i	$-0.500963\pi$
$211$	−5.32965e54	−0.00605032	−0.00302516	+	0.999995i	$0.500963\pi$
$212$	0	0
$213$	1.63916e56	0.147680
$214$	0	0
$215$	−9.67115e56	−0.693004
$216$	0	0
$217$	1.18781e57	0.678397
$218$	0	0
$219$	−3.35132e56	−0.152875
$220$	0	0
$221$	3.87820e57	1.41586
$222$	0	0
$223$	−4.37480e57	−1.28082	−0.640412	−	0.768032i	$-0.721236\pi$
$223$	−4.37480e57	−1.28082	−0.640412	+	0.768032i	$0.721236\pi$
$224$	0	0
$225$	−5.70642e57	−1.34243
$226$	0	0
$227$	−2.82668e57	−0.535357	−0.267679	−	0.963508i	$-0.586257\pi$
$227$	−2.82668e57	−0.535357	−0.267679	+	0.963508i	$0.586257\pi$
$228$	0	0
$229$	−7.37233e57	−1.12625	−0.563126	−	0.826371i	$-0.690401\pi$
$229$	−7.37233e57	−1.12625	−0.563126	+	0.826371i	$0.690401\pi$
$230$	0	0
$231$	1.97948e58	2.44375
$232$	0	0
$233$	5.28015e57	0.527747	0.263874	−	0.964557i	$-0.415000\pi$
$233$	5.28015e57	0.527747	0.263874	+	0.964557i	$0.415000\pi$
$234$	0	0
$235$	−5.88005e57	−0.476671
$236$	0	0
$237$	6.87633e57	0.452921
$238$	0	0
$239$	1.17485e58	0.629845	0.314923	−	0.949117i	$-0.398021\pi$
$239$	1.17485e58	0.629845	0.314923	+	0.949117i	$0.398021\pi$
$240$	0	0
$241$	2.71621e58	1.18726	0.593631	−	0.804738i	$-0.297694\pi$
$241$	2.71621e58	1.18726	0.593631	+	0.804738i	$0.297694\pi$
$242$	0	0
$243$	2.70384e58	0.965222
$244$	0	0
$245$	3.65163e57	0.106639
$246$	0	0
$247$	1.49357e58	0.357395
$248$	0	0
$249$	1.24979e59	2.45444
$250$	0	0
$251$	3.65004e58	0.589239	0.294620	−	0.955615i	$-0.404807\pi$
$251$	3.65004e58	0.589239	0.294620	+	0.955615i	$0.404807\pi$
$252$	0	0
$253$	−8.29542e58	−1.10252
$254$	0	0
$255$	1.07577e59	1.17893
$256$	0	0
$257$	−9.35018e57	−0.0846172	−0.0423086	−	0.999105i	$-0.513471\pi$
$257$	−9.35018e57	−0.0846172	−0.0423086	+	0.999105i	$0.513471\pi$
$258$	0	0
$259$	−3.20209e58	−0.239656
$260$	0	0
$261$	4.96506e59	3.07774
$262$	0	0
$263$	8.31210e58	0.427361	0.213681	−	0.976904i	$-0.431455\pi$
$263$	8.31210e58	0.427361	0.213681	+	0.976904i	$0.431455\pi$
$264$	0	0
$265$	6.62478e57	0.0282910
$266$	0	0
$267$	1.87419e59	0.665713
$268$	0	0
$269$	−4.44312e59	−1.31449	−0.657244	−	0.753678i	$-0.728278\pi$
$269$	−4.44312e59	−1.31449	−0.657244	+	0.753678i	$0.728278\pi$
$270$	0	0
$271$	−6.43014e59	−1.58662	−0.793309	−	0.608819i	$-0.791644\pi$
$271$	−6.43014e59	−1.58662	−0.793309	+	0.608819i	$0.791644\pi$
$272$	0	0
$273$	−5.63901e59	−1.16203
$274$	0	0
$275$	8.27811e59	1.42653
$276$	0	0
$277$	1.24422e60	1.79533	0.897663	−	0.440682i	$-0.145263\pi$
$277$	1.24422e60	1.79533	0.897663	+	0.440682i	$0.145263\pi$
$278$	0	0
$279$	1.06093e60	1.28346
$280$	0	0
$281$	8.54992e59	0.868275	0.434137	−	0.900847i	$-0.357053\pi$
$281$	8.54992e59	0.868275	0.434137	+	0.900847i	$0.357053\pi$
$282$	0	0
$283$	1.94873e60	1.66335	0.831677	−	0.555260i	$-0.187381\pi$
$283$	1.94873e60	1.66335	0.831677	+	0.555260i	$0.187381\pi$
$284$	0	0
$285$	4.14301e59	0.297589
$286$	0	0
$287$	5.34086e59	0.323225
$288$	0	0
$289$	3.78119e60	1.93033
$290$	0	0
$291$	−1.25145e60	−0.539553
$292$	0	0
$293$	−1.71967e60	−0.626887	−0.313443	−	0.949607i	$-0.601483\pi$
$293$	−1.71967e60	−0.626887	−0.313443	+	0.949607i	$0.601483\pi$
$294$	0	0
$295$	−4.92683e59	−0.152030
$296$	0	0
$297$	6.87895e60	1.79882
$298$	0	0
$299$	2.36315e60	0.524260
$300$	0	0
$301$	7.50315e60	1.41372
$302$	0	0
$303$	1.66818e61	2.67238
$304$	0	0
$305$	1.47697e60	0.201385
$306$	0	0
$307$	4.30346e59	0.0499951	0.0249976	−	0.999688i	$-0.492042\pi$
$307$	4.30346e59	0.0499951	0.0249976	+	0.999688i	$0.492042\pi$
$308$	0	0
$309$	−2.45089e61	−2.42852
$310$	0	0
$311$	4.46188e59	0.0377475	0.0188737	−	0.999822i	$-0.493992\pi$
$311$	4.46188e59	0.0377475	0.0188737	+	0.999822i	$0.493992\pi$
$312$	0	0
$313$	3.50202e60	0.253211	0.126605	−	0.991953i	$-0.459592\pi$
$313$	3.50202e60	0.253211	0.126605	+	0.991953i	$0.459592\pi$
$314$	0	0
$315$	−9.71000e60	−0.600635
$316$	0	0
$317$	8.94498e59	0.0473836	0.0236918	−	0.999719i	$-0.492458\pi$
$317$	8.94498e59	0.0473836	0.0236918	+	0.999719i	$0.492458\pi$
$318$	0	0
$319$	−7.20264e61	−3.27054
$320$	0	0
$321$	−1.31223e61	−0.511253
$322$	0	0
$323$	2.21059e61	0.739684
$324$	0	0
$325$	−2.35822e61	−0.678330
$326$	0	0
$327$	7.30351e61	1.80764
$328$	0	0
$329$	4.56191e61	0.972405
$330$	0	0
$331$	−2.61998e61	−0.481407	−0.240704	−	0.970599i	$-0.577378\pi$
$331$	−2.61998e61	−0.481407	−0.240704	+	0.970599i	$0.577378\pi$
$332$	0	0
$333$	−2.86003e61	−0.453406
$334$	0	0
$335$	2.19098e61	0.299944
$336$	0	0
$337$	1.36805e62	1.61872	0.809358	−	0.587316i	$-0.199815\pi$
$337$	1.36805e62	1.61872	0.809358	+	0.587316i	$0.199815\pi$
$338$	0	0
$339$	−5.51096e61	−0.564073
$340$	0	0
$341$	−1.53905e62	−1.36386
$342$	0	0
$343$	−1.41003e62	−1.08274
$344$	0	0
$345$	6.55515e61	0.436531
$346$	0	0
$347$	7.99219e61	0.461949	0.230974	−	0.972960i	$-0.425809\pi$
$347$	7.99219e61	0.461949	0.230974	+	0.972960i	$0.425809\pi$
$348$	0	0
$349$	−1.83544e62	−0.921548	−0.460774	−	0.887517i	$-0.652428\pi$
$349$	−1.83544e62	−0.921548	−0.460774	+	0.887517i	$0.652428\pi$
$350$	0	0
$351$	−1.95963e62	−0.855362
$352$	0	0
$353$	1.40284e61	0.0532756	0.0266378	−	0.999645i	$-0.491520\pi$
$353$	1.40284e61	0.0532756	0.0266378	+	0.999645i	$0.491520\pi$
$354$	0	0
$355$	1.16643e61	0.0385712
$356$	0	0
$357$	−8.34616e62	−2.40500
$358$	0	0
$359$	5.18325e62	1.30253	0.651265	−	0.758850i	$-0.274239\pi$
$359$	5.18325e62	1.30253	0.651265	+	0.758850i	$0.274239\pi$
$360$	0	0
$361$	−3.70826e62	−0.813286
$362$	0	0
$363$	−1.71708e63	−3.28912
$364$	0	0
$365$	−2.38480e61	−0.0399279
$366$	0	0
$367$	−1.17851e62	−0.172590	−0.0862948	−	0.996270i	$-0.527503\pi$
$367$	−1.17851e62	−0.172590	−0.0862948	+	0.996270i	$0.527503\pi$
$368$	0	0
$369$	4.77034e62	0.611510
$370$	0	0
$371$	−5.13969e61	−0.0577134
$372$	0	0
$373$	−1.26964e63	−1.24972	−0.624860	−	0.780737i	$-0.714844\pi$
$373$	−1.26964e63	−1.24972	−0.624860	+	0.780737i	$0.714844\pi$
$374$	0	0
$375$	−1.45176e63	−1.25352
$376$	0	0
$377$	2.05184e63	1.55518
$378$	0	0
$379$	8.52756e61	0.0567759	0.0283880	−	0.999597i	$-0.490963\pi$
$379$	8.52756e61	0.0567759	0.0283880	+	0.999597i	$0.490963\pi$
$380$	0	0
$381$	−2.85108e63	−1.66858
$382$	0	0
$383$	−2.26804e63	−1.16756	−0.583781	−	0.811911i	$-0.698427\pi$
$383$	−2.26804e63	−1.16756	−0.583781	+	0.811911i	$0.698427\pi$
$384$	0	0
$385$	1.40859e63	0.638261
$386$	0	0
$387$	6.70165e63	2.67463
$388$	0	0
$389$	−3.69467e62	−0.129960	−0.0649802	−	0.997887i	$-0.520698\pi$
$389$	−3.69467e62	−0.129960	−0.0649802	+	0.997887i	$0.520698\pi$
$390$	0	0
$391$	3.49764e63	1.08504
$392$	0	0
$393$	6.60001e63	1.80687
$394$	0	0
$395$	4.89319e62	0.118294
$396$	0	0
$397$	−6.48767e63	−1.38587	−0.692933	−	0.721002i	$-0.743682\pi$
$397$	−6.48767e63	−1.38587	−0.692933	+	0.721002i	$0.743682\pi$
$398$	0	0
$399$	−3.21426e63	−0.607079
$400$	0	0
$401$	6.26348e63	1.04659	0.523297	−	0.852150i	$-0.324702\pi$
$401$	6.26348e63	1.04659	0.523297	+	0.852150i	$0.324702\pi$
$402$	0	0
$403$	4.38435e63	0.648532
$404$	0	0
$405$	−1.37426e62	−0.0180061
$406$	0	0
$407$	4.14895e63	0.481809
$408$	0	0
$409$	4.09059e63	0.421274	0.210637	−	0.977564i	$-0.432446\pi$
$409$	4.09059e63	0.421274	0.210637	+	0.977564i	$0.432446\pi$
$410$	0	0
$411$	−1.48549e64	−1.35751
$412$	0	0
$413$	3.82238e63	0.310139
$414$	0	0
$415$	8.89347e63	0.641052
$416$	0	0
$417$	5.64221e63	0.361507
$418$	0	0
$419$	−4.65451e63	−0.265235	−0.132617	−	0.991167i	$-0.542338\pi$
$419$	−4.65451e63	−0.265235	−0.132617	+	0.991167i	$0.542338\pi$
$420$	0	0
$421$	−6.78354e63	−0.343990	−0.171995	−	0.985098i	$-0.555021\pi$
$421$	−6.78354e63	−0.343990	−0.171995	+	0.985098i	$0.555021\pi$
$422$	0	0
$423$	4.07460e64	1.83970
$424$	0	0
$425$	−3.49034e64	−1.40391
$426$	0	0
$427$	−1.14588e64	−0.410824
$428$	0	0
$429$	7.30648e64	2.33617
$430$	0	0
$431$	−5.56035e64	−1.58639	−0.793193	−	0.608971i	$-0.791583\pi$
$431$	−5.56035e64	−1.58639	−0.793193	+	0.608971i	$0.791583\pi$
$432$	0	0
$433$	3.47009e64	0.883869	0.441934	−	0.897047i	$-0.354292\pi$
$433$	3.47009e64	0.883869	0.441934	+	0.897047i	$0.354292\pi$
$434$	0	0
$435$	5.69162e64	1.29494
$436$	0	0
$437$	1.34701e64	0.273889
$438$	0	0
$439$	4.09271e64	0.744099	0.372049	−	0.928213i	$-0.378655\pi$
$439$	4.09271e64	0.744099	0.372049	+	0.928213i	$0.378655\pi$
$440$	0	0
$441$	−2.53041e64	−0.411572
$442$	0	0
$443$	1.21785e65	1.77298	0.886488	−	0.462751i	$-0.153138\pi$
$443$	1.21785e65	1.77298	0.886488	+	0.462751i	$0.153138\pi$
$444$	0	0
$445$	1.33367e64	0.173871
$446$	0	0
$447$	1.22520e65	1.43110
$448$	0	0
$449$	5.54542e64	0.580629	0.290315	−	0.956931i	$-0.406240\pi$
$449$	5.54542e64	0.580629	0.290315	+	0.956931i	$0.406240\pi$
$450$	0	0
$451$	−6.92016e64	−0.649817
$452$	0	0
$453$	−1.68374e64	−0.141863
$454$	0	0
$455$	−4.01272e64	−0.303500
$456$	0	0
$457$	−1.52223e65	−1.03403	−0.517014	−	0.855977i	$-0.672956\pi$
$457$	−1.52223e65	−1.03403	−0.517014	+	0.855977i	$0.672956\pi$
$458$	0	0
$459$	−2.90040e65	−1.77030
$460$	0	0
$461$	1.88013e65	1.03161	0.515806	−	0.856706i	$-0.327493\pi$
$461$	1.88013e65	1.03161	0.515806	+	0.856706i	$0.327493\pi$
$462$	0	0
$463$	−3.45913e65	−1.70699	−0.853497	−	0.521098i	$-0.825523\pi$
$463$	−3.45913e65	−1.70699	−0.853497	+	0.521098i	$0.825523\pi$
$464$	0	0
$465$	1.21618e65	0.540007
$466$	0	0
$467$	1.96639e65	0.785970	0.392985	−	0.919545i	$-0.371442\pi$
$467$	1.96639e65	0.785970	0.392985	+	0.919545i	$0.371442\pi$
$468$	0	0
$469$	−1.69982e65	−0.611884
$470$	0	0
$471$	−1.66883e65	−0.541252
$472$	0	0
$473$	−9.72185e65	−2.84217
$474$	0	0
$475$	−1.34419e65	−0.354379
$476$	0	0
$477$	−4.59066e64	−0.109188
$478$	0	0
$479$	−5.20394e64	−0.111715	−0.0558576	−	0.998439i	$-0.517789\pi$
$479$	−5.20394e64	−0.111715	−0.0558576	+	0.998439i	$0.517789\pi$
$480$	0	0
$481$	−1.18193e65	−0.229106
$482$	0	0
$483$	−5.08567e65	−0.890519
$484$	0	0
$485$	−8.90528e64	−0.140921
$486$	0	0
$487$	−1.99942e65	−0.286051	−0.143026	−	0.989719i	$-0.545683\pi$
$487$	−1.99942e65	−0.286051	−0.143026	+	0.989719i	$0.545683\pi$
$488$	0	0
$489$	−2.04408e66	−2.64503
$490$	0	0
$491$	3.18295e65	0.372677	0.186339	−	0.982486i	$-0.440338\pi$
$491$	3.18295e65	0.372677	0.186339	+	0.982486i	$0.440338\pi$
$492$	0	0
$493$	3.03688e66	3.21869
$494$	0	0
$495$	1.25813e66	1.20753
$496$	0	0
$497$	−9.04947e64	−0.0786849
$498$	0	0
$499$	−6.39909e65	−0.504260	−0.252130	−	0.967693i	$-0.581131\pi$
$499$	−6.39909e65	−0.504260	−0.252130	+	0.967693i	$0.581131\pi$
$500$	0	0
$501$	3.14430e65	0.224647
$502$	0	0
$503$	−1.03311e66	−0.669471	−0.334735	−	0.942312i	$-0.608647\pi$
$503$	−1.03311e66	−0.669471	−0.334735	+	0.942312i	$0.608647\pi$
$504$	0	0
$505$	1.18708e66	0.697974
$506$	0	0
$507$	9.61133e65	0.512966
$508$	0	0
$509$	−1.63284e66	−0.791333	−0.395666	−	0.918394i	$-0.629486\pi$
$509$	−1.63284e66	−0.791333	−0.395666	+	0.918394i	$0.629486\pi$
$510$	0	0
$511$	1.85019e65	0.0814526
$512$	0	0
$513$	−1.11700e66	−0.446866
$514$	0	0
$515$	−1.74405e66	−0.634281
$516$	0	0
$517$	−5.91088e66	−1.95494
$518$	0	0
$519$	−6.41318e66	−1.92963
$520$	0	0
$521$	3.93193e66	1.07667	0.538334	−	0.842732i	$-0.319054\pi$
$521$	3.93193e66	1.07667	0.538334	+	0.842732i	$0.319054\pi$
$522$	0	0
$523$	2.10835e66	0.525596	0.262798	−	0.964851i	$-0.415355\pi$
$523$	2.10835e66	0.525596	0.262798	+	0.964851i	$0.415355\pi$
$524$	0	0
$525$	5.07505e66	1.15223
$526$	0	0
$527$	6.48917e66	1.34224
$528$	0	0
$529$	−3.17348e66	−0.598235
$530$	0	0
$531$	3.41406e66	0.586753
$532$	0	0
$533$	1.97137e66	0.308996
$534$	0	0
$535$	−9.33781e65	−0.133530
$536$	0	0
$537$	8.63324e66	1.12669
$538$	0	0
$539$	3.67077e66	0.437354
$540$	0	0
$541$	−8.41780e66	−0.915937	−0.457968	−	0.888968i	$-0.651423\pi$
$541$	−8.41780e66	−0.915937	−0.457968	+	0.888968i	$0.651423\pi$
$542$	0	0
$543$	7.19031e66	0.714746
$544$	0	0
$545$	5.19717e66	0.472120
$546$	0	0
$547$	−5.67745e66	−0.471481	−0.235740	−	0.971816i	$-0.575752\pi$
$547$	−5.67745e66	−0.471481	−0.235740	+	0.971816i	$0.575752\pi$
$548$	0	0
$549$	−1.02347e67	−0.777238
$550$	0	0
$551$	1.16956e67	0.812472
$552$	0	0
$553$	−3.79628e66	−0.241319
$554$	0	0
$555$	−3.27855e66	−0.190767
$556$	0	0
$557$	2.68265e67	1.42926	0.714631	−	0.699502i	$-0.246595\pi$
$557$	2.68265e67	1.42926	0.714631	+	0.699502i	$0.246595\pi$
$558$	0	0
$559$	2.76950e67	1.35149
$560$	0	0
$561$	1.08141e68	4.83506
$562$	0	0
$563$	2.46352e67	1.00949	0.504744	−	0.863269i	$-0.331587\pi$
$563$	2.46352e67	1.00949	0.504744	+	0.863269i	$0.331587\pi$
$564$	0	0
$565$	−3.92160e66	−0.147325
$566$	0	0
$567$	1.06619e66	0.0367324
$568$	0	0
$569$	−2.62664e67	−0.830137	−0.415069	−	0.909790i	$-0.636242\pi$
$569$	−2.62664e67	−0.830137	−0.415069	+	0.909790i	$0.636242\pi$
$570$	0	0
$571$	−5.51879e67	−1.60051	−0.800257	−	0.599658i	$-0.795303\pi$
$571$	−5.51879e67	−1.60051	−0.800257	+	0.599658i	$0.795303\pi$
$572$	0	0
$573$	−7.69066e67	−2.04727
$574$	0	0
$575$	−2.12681e67	−0.519836
$576$	0	0
$577$	−4.65003e66	−0.104387	−0.0521937	−	0.998637i	$-0.516621\pi$
$577$	−4.65003e66	−0.104387	−0.0521937	+	0.998637i	$0.516621\pi$
$578$	0	0
$579$	4.24526e67	0.875548
$580$	0	0
$581$	−6.89981e67	−1.30774
$582$	0	0
$583$	6.65951e66	0.116028
$584$	0	0
$585$	−3.58407e67	−0.574193
$586$	0	0
$587$	9.45616e67	1.39342	0.696709	−	0.717354i	$-0.254647\pi$
$587$	9.45616e67	1.39342	0.696709	+	0.717354i	$0.254647\pi$
$588$	0	0
$589$	2.49910e67	0.338812
$590$	0	0
$591$	−1.97164e68	−2.45999
$592$	0	0
$593$	−9.11419e67	−1.04684	−0.523418	−	0.852076i	$-0.675343\pi$
$593$	−9.11419e67	−1.04684	−0.523418	+	0.852076i	$0.675343\pi$
$594$	0	0
$595$	−5.93912e67	−0.628140
$596$	0	0
$597$	4.59188e67	0.447321
$598$	0	0
$599$	−1.33989e68	−1.20258	−0.601289	−	0.799032i	$-0.705346\pi$
$599$	−1.33989e68	−1.20258	−0.601289	+	0.799032i	$0.705346\pi$
$600$	0	0
$601$	1.90045e68	1.57193	0.785963	−	0.618274i	$-0.212168\pi$
$601$	1.90045e68	1.57193	0.785963	+	0.618274i	$0.212168\pi$
$602$	0	0
$603$	−1.51825e68	−1.15762
$604$	0	0
$605$	−1.22188e68	−0.859054
$606$	0	0
$607$	2.65855e68	1.72394	0.861968	−	0.506962i	$-0.169232\pi$
$607$	2.65855e68	1.72394	0.861968	+	0.506962i	$0.169232\pi$
$608$	0	0
$609$	−4.41572e68	−2.64166
$610$	0	0
$611$	1.68385e68	0.929597
$612$	0	0
$613$	3.75059e68	1.91125	0.955627	−	0.294580i	$-0.0951798\pi$
$613$	3.75059e68	1.91125	0.955627	+	0.294580i	$0.0951798\pi$
$614$	0	0
$615$	5.46840e67	0.257288
$616$	0	0
$617$	1.14520e68	0.497617	0.248809	−	0.968553i	$-0.419961\pi$
$617$	1.14520e68	0.497617	0.248809	+	0.968553i	$0.419961\pi$
$618$	0	0
$619$	−4.65278e68	−1.86763	−0.933817	−	0.357750i	$-0.883544\pi$
$619$	−4.65278e68	−1.86763	−0.933817	+	0.357750i	$0.883544\pi$
$620$	0	0
$621$	−1.76734e68	−0.655504
$622$	0	0
$623$	−1.03470e68	−0.354697
$624$	0	0
$625$	1.55479e68	0.492731
$626$	0	0
$627$	4.16472e68	1.22048
$628$	0	0
$629$	−1.74934e68	−0.474170
$630$	0	0
$631$	−4.71150e67	−0.118152	−0.0590758	−	0.998253i	$-0.518815\pi$
$631$	−4.71150e67	−0.118152	−0.0590758	+	0.998253i	$0.518815\pi$
$632$	0	0
$633$	4.23363e66	0.00982476
$634$	0	0
$635$	−2.02883e68	−0.435802
$636$	0	0
$637$	−1.04571e68	−0.207967
$638$	0	0
$639$	−8.08279e67	−0.148864
$640$	0	0
$641$	−1.14710e69	−1.95694	−0.978471	−	0.206383i	$-0.933831\pi$
$641$	−1.14710e69	−1.95694	−0.978471	+	0.206383i	$0.933831\pi$
$642$	0	0
$643$	−5.88590e68	−0.930345	−0.465173	−	0.885220i	$-0.654008\pi$
$643$	−5.88590e68	−0.930345	−0.465173	+	0.885220i	$0.654008\pi$
$644$	0	0
$645$	7.68233e68	1.12533
$646$	0	0
$647$	8.23336e68	1.11795	0.558975	−	0.829185i	$-0.311195\pi$
$647$	8.23336e68	1.11795	0.558975	+	0.829185i	$0.311195\pi$
$648$	0	0
$649$	−4.95266e68	−0.623510
$650$	0	0
$651$	−9.43544e68	−1.10161
$652$	0	0
$653$	−4.04310e68	−0.437866	−0.218933	−	0.975740i	$-0.570258\pi$
$653$	−4.04310e68	−0.437866	−0.218933	+	0.975740i	$0.570258\pi$
$654$	0	0
$655$	4.69656e68	0.471920
$656$	0	0
$657$	1.65255e68	0.154100
$658$	0	0
$659$	1.59048e69	1.37669	0.688343	−	0.725385i	$-0.258338\pi$
$659$	1.59048e69	1.37669	0.688343	+	0.725385i	$0.258338\pi$
$660$	0	0
$661$	1.64318e69	1.32053	0.660266	−	0.751032i	$-0.270444\pi$
$661$	1.64318e69	1.32053	0.660266	+	0.751032i	$0.270444\pi$
$662$	0	0
$663$	−3.08066e69	−2.29913
$664$	0	0
$665$	−2.28727e68	−0.158557
$666$	0	0
$667$	1.85050e69	1.19181
$668$	0	0
$669$	3.47515e69	2.07986
$670$	0	0
$671$	1.48472e69	0.825927
$672$	0	0
$673$	−1.44435e69	−0.746968	−0.373484	−	0.927637i	$-0.621837\pi$
$673$	−1.44435e69	−0.746968	−0.373484	+	0.927637i	$0.621837\pi$
$674$	0	0
$675$	1.76365e69	0.848144
$676$	0	0
$677$	4.42761e69	1.98037	0.990186	−	0.139755i	$-0.0446314\pi$
$677$	4.42761e69	1.98037	0.990186	+	0.139755i	$0.0446314\pi$
$678$	0	0
$679$	6.90897e68	0.287477
$680$	0	0
$681$	2.24539e69	0.869336
$682$	0	0
$683$	6.24013e68	0.224847	0.112423	−	0.993660i	$-0.464139\pi$
$683$	6.24013e68	0.224847	0.112423	+	0.993660i	$0.464139\pi$
$684$	0	0
$685$	−1.05707e69	−0.354557
$686$	0	0
$687$	5.85624e69	1.82885
$688$	0	0
$689$	−1.89712e68	−0.0551726
$690$	0	0
$691$	1.65576e69	0.448524	0.224262	−	0.974529i	$-0.428003\pi$
$691$	1.65576e69	0.448524	0.224262	+	0.974529i	$0.428003\pi$
$692$	0	0
$693$	−9.76089e69	−2.46335
$694$	0	0
$695$	4.01499e68	0.0944185
$696$	0	0
$697$	2.91778e69	0.639513
$698$	0	0
$699$	−4.19431e69	−0.856978
$700$	0	0
$701$	3.50945e69	0.668572	0.334286	−	0.942472i	$-0.391505\pi$
$701$	3.50945e69	0.668572	0.334286	+	0.942472i	$0.391505\pi$
$702$	0	0
$703$	−6.73704e68	−0.119692
$704$	0	0
$705$	4.67085e69	0.774038
$706$	0	0
$707$	−9.20966e69	−1.42386
$708$	0	0
$709$	−7.32363e69	−1.05656	−0.528280	−	0.849070i	$-0.677163\pi$
$709$	−7.32363e69	−1.05656	−0.528280	+	0.849070i	$0.677163\pi$
$710$	0	0
$711$	−3.39075e69	−0.456552
$712$	0	0
$713$	3.95412e69	0.497000
$714$	0	0
$715$	5.19928e69	0.610162
$716$	0	0
$717$	−9.33248e69	−1.02277
$718$	0	0
$719$	−5.39270e69	−0.552012	−0.276006	−	0.961156i	$-0.589011\pi$
$719$	−5.39270e69	−0.552012	−0.276006	+	0.961156i	$0.589011\pi$
$720$	0	0
$721$	1.35308e70	1.29393
$722$	0	0
$723$	−2.15764e70	−1.92792
$724$	0	0
$725$	−1.84664e70	−1.54206
$726$	0	0
$727$	−1.12907e70	−0.881309	−0.440655	−	0.897677i	$-0.645254\pi$
$727$	−1.12907e70	−0.881309	−0.440655	+	0.897677i	$0.645254\pi$
$728$	0	0
$729$	−2.20599e70	−1.60982
$730$	0	0
$731$	4.09907e70	2.79711
$732$	0	0
$733$	4.27091e69	0.272567	0.136283	−	0.990670i	$-0.456484\pi$
$733$	4.27091e69	0.272567	0.136283	+	0.990670i	$0.456484\pi$
$734$	0	0
$735$	−2.90069e69	−0.173166
$736$	0	0
$737$	2.20246e70	1.23014
$738$	0	0
$739$	2.03599e70	1.06411	0.532055	−	0.846710i	$-0.321420\pi$
$739$	2.03599e70	1.06411	0.532055	+	0.846710i	$0.321420\pi$
$740$	0	0
$741$	−1.18642e70	−0.580354
$742$	0	0
$743$	−1.12304e70	−0.514243	−0.257122	−	0.966379i	$-0.582774\pi$
$743$	−1.12304e70	−0.514243	−0.257122	+	0.966379i	$0.582774\pi$
$744$	0	0
$745$	8.71849e69	0.373777
$746$	0	0
$747$	−6.16276e70	−2.47412
$748$	0	0
$749$	7.24453e69	0.272399
$750$	0	0
$751$	1.72919e69	0.0609066	0.0304533	−	0.999536i	$-0.490305\pi$
$751$	1.72919e69	0.0609066	0.0304533	+	0.999536i	$0.490305\pi$
$752$	0	0
$753$	−2.89943e70	−0.956831
$754$	0	0
$755$	−1.19815e69	−0.0370519
$756$	0	0
$757$	−1.67080e69	−0.0484256	−0.0242128	−	0.999707i	$-0.507708\pi$
$757$	−1.67080e69	−0.0484256	−0.0242128	+	0.999707i	$0.507708\pi$
$758$	0	0
$759$	6.58951e70	1.79031
$760$	0	0
$761$	−1.82952e70	−0.466030	−0.233015	−	0.972473i	$-0.574859\pi$
$761$	−1.82952e70	−0.466030	−0.233015	+	0.972473i	$0.574859\pi$
$762$	0	0
$763$	−4.03211e70	−0.963122
$764$	0	0
$765$	−5.30470e70	−1.18838
$766$	0	0
$767$	1.41088e70	0.296486
$768$	0	0
$769$	1.74967e70	0.344954	0.172477	−	0.985014i	$-0.444823\pi$
$769$	1.74967e70	0.344954	0.172477	+	0.985014i	$0.444823\pi$
$770$	0	0
$771$	7.42736e69	0.137405
$772$	0	0
$773$	5.68036e70	0.986229	0.493115	−	0.869964i	$-0.335858\pi$
$773$	5.68036e70	0.986229	0.493115	+	0.869964i	$0.335858\pi$
$774$	0	0
$775$	−3.94587e70	−0.643059
$776$	0	0
$777$	2.54359e70	0.389164
$778$	0	0
$779$	1.12369e70	0.161428
$780$	0	0
$781$	1.17254e70	0.158190
$782$	0	0
$783$	−1.53452e71	−1.94451
$784$	0	0
$785$	−1.18754e70	−0.141365
$786$	0	0
$787$	−2.11456e70	−0.236505	−0.118252	−	0.992984i	$-0.537729\pi$
$787$	−2.11456e70	−0.236505	−0.118252	+	0.992984i	$0.537729\pi$
$788$	0	0
$789$	−6.60276e70	−0.693967
$790$	0	0
$791$	3.04249e70	0.300542
$792$	0	0
$793$	−4.22957e70	−0.392738
$794$	0	0
$795$	−5.26243e69	−0.0459400
$796$	0	0
$797$	1.36404e71	1.11969	0.559844	−	0.828598i	$-0.310861\pi$
$797$	1.36404e71	1.11969	0.559844	+	0.828598i	$0.310861\pi$
$798$	0	0
$799$	2.49223e71	1.92394
$800$	0	0
$801$	−9.24171e70	−0.671051
$802$	0	0
$803$	−2.39730e70	−0.163754
$804$	0	0
$805$	−3.61896e70	−0.232586
$806$	0	0
$807$	3.52941e71	2.13452
$808$	0	0
$809$	1.21224e71	0.690002	0.345001	−	0.938602i	$-0.387879\pi$
$809$	1.21224e71	0.690002	0.345001	+	0.938602i	$0.387879\pi$
$810$	0	0
$811$	2.96765e71	1.59001	0.795007	−	0.606600i	$-0.207467\pi$
$811$	2.96765e71	1.59001	0.795007	+	0.606600i	$0.207467\pi$
$812$	0	0
$813$	5.10781e71	2.57642
$814$	0	0
$815$	−1.45457e71	−0.690831
$816$	0	0
$817$	1.57863e71	0.706056
$818$	0	0
$819$	2.78062e71	1.17135
$820$	0	0
$821$	4.81615e71	1.91114	0.955570	−	0.294766i	$-0.0952416\pi$
$821$	4.81615e71	1.91114	0.955570	+	0.294766i	$0.0952416\pi$
$822$	0	0
$823$	1.34527e71	0.502935	0.251468	−	0.967866i	$-0.419087\pi$
$823$	1.34527e71	0.502935	0.251468	+	0.967866i	$0.419087\pi$
$824$	0	0
$825$	−6.57575e71	−2.31645
$826$	0	0
$827$	−7.76898e70	−0.257916	−0.128958	−	0.991650i	$-0.541163\pi$
$827$	−7.76898e70	−0.257916	−0.128958	+	0.991650i	$0.541163\pi$
$828$	0	0
$829$	−5.87735e70	−0.183906	−0.0919529	−	0.995763i	$-0.529311\pi$
$829$	−5.87735e70	−0.183906	−0.0919529	+	0.995763i	$0.529311\pi$
$830$	0	0
$831$	−9.88353e71	−2.91533
$832$	0	0
$833$	−1.54773e71	−0.430419
$834$	0	0
$835$	2.23748e70	0.0586733
$836$	0	0
$837$	−3.27894e71	−0.810886
$838$	0	0
$839$	2.93145e70	0.0683777	0.0341888	−	0.999415i	$-0.489115\pi$
$839$	2.93145e70	0.0683777	0.0341888	+	0.999415i	$0.489115\pi$
$840$	0	0
$841$	1.15226e72	2.53542
$842$	0	0
$843$	−6.79167e71	−1.40994
$844$	0	0
$845$	6.83941e70	0.133977
$846$	0	0
$847$	9.47966e71	1.75246
$848$	0	0
$849$	−1.54799e72	−2.70102
$850$	0	0
$851$	−1.06595e71	−0.175574
$852$	0	0
$853$	−5.36884e71	−0.834889	−0.417445	−	0.908702i	$-0.637074\pi$
$853$	−5.36884e71	−0.834889	−0.417445	+	0.908702i	$0.637074\pi$
$854$	0	0
$855$	−2.04294e71	−0.299975
$856$	0	0
$857$	2.07429e71	0.287634	0.143817	−	0.989604i	$-0.454062\pi$
$857$	2.07429e71	0.287634	0.143817	+	0.989604i	$0.454062\pi$
$858$	0	0
$859$	7.03695e71	0.921620	0.460810	−	0.887499i	$-0.347559\pi$
$859$	7.03695e71	0.921620	0.460810	+	0.887499i	$0.347559\pi$
$860$	0	0
$861$	−4.24254e71	−0.524866
$862$	0	0
$863$	3.68596e71	0.430810	0.215405	−	0.976525i	$-0.430893\pi$
$863$	3.68596e71	0.430810	0.215405	+	0.976525i	$0.430893\pi$
$864$	0	0
$865$	−4.56361e71	−0.503981
$866$	0	0
$867$	−3.00361e72	−3.13455
$868$	0	0
$869$	4.91884e71	0.485152
$870$	0	0
$871$	−6.27425e71	−0.584947
$872$	0	0
$873$	6.17094e71	0.543879
$874$	0	0
$875$	8.01488e71	0.667882
$876$	0	0
$877$	−1.93572e72	−1.52529	−0.762645	−	0.646817i	$-0.776100\pi$
$877$	−1.93572e72	−1.52529	−0.762645	+	0.646817i	$0.776100\pi$
$878$	0	0
$879$	1.36603e72	1.01796
$880$	0	0
$881$	5.95854e71	0.419982	0.209991	−	0.977703i	$-0.432657\pi$
$881$	5.95854e71	0.419982	0.209991	+	0.977703i	$0.432657\pi$
$882$	0	0
$883$	−1.87124e72	−1.24765	−0.623826	−	0.781564i	$-0.714422\pi$
$883$	−1.87124e72	−1.24765	−0.623826	+	0.781564i	$0.714422\pi$
$884$	0	0
$885$	3.91365e71	0.246872
$886$	0	0
$887$	2.12365e72	1.26752	0.633758	−	0.773531i	$-0.281511\pi$
$887$	2.12365e72	1.26752	0.633758	+	0.773531i	$0.281511\pi$
$888$	0	0
$889$	1.57402e72	0.889033
$890$	0	0
$891$	−1.38146e71	−0.0738474
$892$	0	0
$893$	9.59804e71	0.485648
$894$	0	0
$895$	6.14341e71	0.294269
$896$	0	0
$897$	−1.87718e72	−0.851316
$898$	0	0
$899$	3.43323e72	1.47432
$900$	0	0
$901$	−2.80788e71	−0.114188
$902$	0	0
$903$	−5.96016e72	−2.29566
$904$	0	0
$905$	5.11662e71	0.186678
$906$	0	0
$907$	−2.10319e72	−0.726943	−0.363471	−	0.931605i	$-0.618409\pi$
$907$	−2.10319e72	−0.726943	−0.363471	+	0.931605i	$0.618409\pi$
$908$	0	0
$909$	−8.22587e72	−2.69381
$910$	0	0
$911$	−4.91507e72	−1.52521	−0.762604	−	0.646866i	$-0.776079\pi$
$911$	−4.91507e72	−1.52521	−0.762604	+	0.646866i	$0.776079\pi$
$912$	0	0
$913$	8.94009e72	2.62910
$914$	0	0
$915$	−1.17324e72	−0.327017
$916$	0	0
$917$	−3.64372e72	−0.962713
$918$	0	0
$919$	1.76130e72	0.441166	0.220583	−	0.975368i	$-0.429204\pi$
$919$	1.76130e72	0.441166	0.220583	+	0.975368i	$0.429204\pi$
$920$	0	0
$921$	−3.41847e71	−0.0811842
$922$	0	0
$923$	−3.34026e71	−0.0752210
$924$	0	0
$925$	1.06372e72	0.227172
$926$	0	0
$927$	1.20854e73	2.44799
$928$	0	0
$929$	−4.47943e72	−0.860671	−0.430336	−	0.902669i	$-0.641605\pi$
$929$	−4.47943e72	−0.860671	−0.430336	+	0.902669i	$0.641605\pi$
$930$	0	0
$931$	−5.96058e71	−0.108648
$932$	0	0
$933$	−3.54431e71	−0.0612959
$934$	0	0
$935$	7.69533e72	1.26282
$936$	0	0
$937$	6.98453e72	1.08772	0.543860	−	0.839176i	$-0.316962\pi$
$937$	6.98453e72	1.08772	0.543860	+	0.839176i	$0.316962\pi$
$938$	0	0
$939$	−2.78185e72	−0.411174
$940$	0	0
$941$	−1.60971e72	−0.225840	−0.112920	−	0.993604i	$-0.536020\pi$
$941$	−1.60971e72	−0.225840	−0.112920	+	0.993604i	$0.536020\pi$
$942$	0	0
$943$	1.77793e72	0.236798
$944$	0	0
$945$	3.00101e72	0.379479
$946$	0	0
$947$	1.31016e72	0.157307	0.0786537	−	0.996902i	$-0.474938\pi$
$947$	1.31016e72	0.157307	0.0786537	+	0.996902i	$0.474938\pi$
$948$	0	0
$949$	6.82928e71	0.0778668
$950$	0	0
$951$	−7.10549e71	−0.0769434
$952$	0	0
$953$	−4.69763e72	−0.483173	−0.241587	−	0.970379i	$-0.577668\pi$
$953$	−4.69763e72	−0.483173	−0.241587	+	0.970379i	$0.577668\pi$
$954$	0	0
$955$	−5.47267e72	−0.534708
$956$	0	0
$957$	5.72145e73	5.31084
$958$	0	0
$959$	8.20106e72	0.723293
$960$	0	0
$961$	−4.59618e72	−0.385189
$962$	0	0
$963$	6.47066e72	0.515353
$964$	0	0
$965$	3.02093e72	0.228676
$966$	0	0
$967$	1.35157e73	0.972503	0.486252	−	0.873819i	$-0.338364\pi$
$967$	1.35157e73	0.972503	0.486252	+	0.873819i	$0.338364\pi$
$968$	0	0
$969$	−1.75599e73	−1.20113
$970$	0	0
$971$	−1.73168e73	−1.12615	−0.563074	−	0.826406i	$-0.690381\pi$
$971$	−1.73168e73	−1.12615	−0.563074	+	0.826406i	$0.690381\pi$
$972$	0	0
$973$	−3.11495e72	−0.192613
$974$	0	0
$975$	1.87326e73	1.10150
$976$	0	0
$977$	−1.61768e72	−0.0904639	−0.0452320	−	0.998977i	$-0.514403\pi$
$977$	−1.61768e72	−0.0904639	−0.0452320	+	0.998977i	$0.514403\pi$
$978$	0	0
$979$	1.34066e73	0.713088
$980$	0	0
$981$	−3.60139e73	−1.82213
$982$	0	0
$983$	1.86672e72	0.0898499	0.0449249	−	0.998990i	$-0.485695\pi$
$983$	1.86672e72	0.0898499	0.0449249	+	0.998990i	$0.485695\pi$
$984$	0	0
$985$	−1.40302e73	−0.642503
$986$	0	0
$987$	−3.62378e73	−1.57903
$988$	0	0
$989$	2.49773e73	1.03571
$990$	0	0
$991$	−2.80003e73	−1.10499	−0.552494	−	0.833517i	$-0.686324\pi$
$991$	−2.80003e73	−1.10499	−0.552494	+	0.833517i	$0.686324\pi$
$992$	0	0
$993$	2.08120e73	0.781729
$994$	0	0
$995$	3.26758e72	0.116832
$996$	0	0
$997$	−1.83032e73	−0.623011	−0.311505	−	0.950244i	$-0.600833\pi$
$997$	−1.83032e73	−0.623011	−0.311505	+	0.950244i	$0.600833\pi$
$998$	0	0
$999$	8.83932e72	0.286460

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.50.a.b.1.1		3
4.3	odd	2		2.50.a.b.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2.50.a.b.1.3	✓	3	4.3	odd	2
16.50.a.b.1.1		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 50 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

\(f(q)\)	\(=\)	\(q-7.94355e11 q^{3} -5.65262e16 q^{5} +4.38546e20 q^{7} +3.91700e23 q^{9} +O(q^{10})\)	\(q-7.94355e11 q^{3} -5.65262e16 q^{5} +4.38546e20 q^{7} +3.91700e23 q^{9} -5.68225e25 q^{11} +1.61873e27 q^{13} +4.49019e28 q^{15} +2.39583e30 q^{17} +9.22680e30 q^{19} -3.48361e32 q^{21} +1.45988e33 q^{23} -1.45684e34 q^{25} -1.21060e35 q^{27} +1.26757e36 q^{29} +2.70852e36 q^{31} +4.51372e37 q^{33} -2.47894e37 q^{35} -7.30159e37 q^{37} -1.28584e39 q^{39} +1.21786e39 q^{41} +1.71091e40 q^{43} -2.21413e40 q^{45} +1.04023e41 q^{47} -6.46007e40 q^{49} -1.90314e42 q^{51} -1.17198e41 q^{53} +3.21196e42 q^{55} -7.32935e42 q^{57} +8.71601e42 q^{59} -2.61290e43 q^{61} +1.71779e44 q^{63} -9.15004e43 q^{65} -3.87604e44 q^{67} -1.15966e45 q^{69} -2.06352e44 q^{71} +4.21892e44 q^{73} +1.15724e46 q^{75} -2.49193e46 q^{77} -8.65650e45 q^{79} +2.43119e45 q^{81} -1.57334e47 q^{83} -1.35427e47 q^{85} -1.00690e48 q^{87} -2.35938e47 q^{89} +7.09886e47 q^{91} -2.15153e48 q^{93} -5.21556e47 q^{95} +1.57542e48 q^{97} -2.22574e49 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 16203614388 q^{3} - 10\!\cdots\!30 q^{5}+ \cdots + 38\!\cdots\!99 q^{9}+O(q^{10}) \) 3 * q + 16203614388 * q^3 - 101813016401840430 * q^5 + 125479179230203797096 * q^7 + 389432931519696922052199 * q^9	\( 3 q + 16203614388 q^{3} - 10\!\cdots\!30 q^{5}+ \cdots - 17\!\cdots\!68 q^{99}+O(q^{100}) \) 3 * q + 16203614388 * q^3 - 101813016401840430 * q^5 + 125479179230203797096 * q^7 + 389432931519696922052199 * q^9 + 25860036791834065092767004 * q^11 + 2945123797958353151076283242 * q^13 - 87846213583565151681178171080 * q^15 + 3910755572889939740390354668854 * q^17 - 14590272981610371514592320211580 * q^19 - 109864176920553355284360667668384 * q^21 + 4953158032634305739950894912759608 * q^23 + 39451396326519132102111422427450525 * q^25 - 196120894647932767255303897131684600 * q^27 + 937305925294643190481854503019954330 * q^29 - 2414809991776326638635666091994140256 * q^31 + 84513992666209345011409783180743651984 * q^33 - 300439012903055674136683294221406953360 * q^35 + 309377978155169809207437071391240173874 * q^37 - 360871814296103457146322338824042742568 * q^39 - 1143109642820632592659114174552039005954 * q^41 + 17579019372148890319048762066144112105148 * q^43 - 114813225075613761568802306330842527416790 * q^45 + 218573936001854642959028674713638582342256 * q^47 + 373628008074429359506251920014708170372651 * q^49 - 2241123510744223933356304325509677976541016 * q^51 + 740200780025011930130058449331923084936802 * q^53 - 3200488473448119911679443277764576117207640 * q^55 - 20594915296543361145254552753753824314692880 * q^57 + 53014363316602767635595514099864508327238540 * q^59 - 124730102606725385116407342714381742110559494 * q^61 + 468292624427612329618287149072325587596605768 * q^63 - 421321721827999315000625667485604880336092420 * q^65 - 1111804215897361218882144911380577470099878924 * q^67 + 770909887444909922528222240570561038770401568 * q^69 + 4013080456871179368491803943114613092268032424 * q^71 - 10015638764428635672946257750737669286684666018 * q^73 + 38646070268253498119014909069282536501398301900 * q^75 - 23360330575813840356220747501745396153372866272 * q^77 - 36077368003505469428776991593224667583970712560 * q^79 - 14427585993163108762153736521422002728666768917 * q^81 - 150749458884727999220762915504103393007529125692 * q^83 + 566911972337878875844467446456481028710158526660 * q^85 - 1072056267743755243448622937355132996639037778920 * q^87 + 542251936064510107856207107418847041962289942670 * q^89 + 1459783720923935217191336434024489699257545191344 * q^91 - 2368291579513879043404530519265491460552020418176 * q^93 + 2813579016011586836853932078889401271258883877400 * q^95 + 4343844254337968328695747517284119056283595359014 * q^97 - 17595360353038055002493296165642581030249046200468 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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