LMFDB - Embedded newform 16.50.a.b.1.2

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			1.2
Root			$-3.42020e8$ 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)

$f(q)$	$=$	$q+1.33407e11 q^{3} +1.87739e17 q^{5} -8.28497e20 q^{7} -2.21502e23 q^{9} +O(q^{10})$	$q+1.33407e11 q^{3} +1.87739e17 q^{5} -8.28497e20 q^{7} -2.21502e23 q^{9} +3.05508e25 q^{11} -4.92824e25 q^{13} +2.50456e28 q^{15} +2.50818e30 q^{17} -5.26388e30 q^{19} -1.10527e32 q^{21} +7.99827e32 q^{23} +1.74823e34 q^{25} -6.14740e34 q^{27} -2.91459e35 q^{29} -5.98168e36 q^{31} +4.07569e36 q^{33} -1.55541e38 q^{35} +1.31540e38 q^{37} -6.57461e36 q^{39} -2.57335e39 q^{41} -3.01162e39 q^{43} -4.15845e40 q^{45} +1.35707e41 q^{47} +4.29484e41 q^{49} +3.34609e41 q^{51} +3.38284e42 q^{53} +5.73558e42 q^{55} -7.02238e41 q^{57} +1.39260e42 q^{59} -8.43699e43 q^{61} +1.83514e44 q^{63} -9.25222e42 q^{65} -4.28654e44 q^{67} +1.06702e44 q^{69} +1.65460e45 q^{71} -5.25815e45 q^{73} +2.33226e45 q^{75} -2.53113e46 q^{77} -1.45663e46 q^{79} +4.48042e46 q^{81} +9.24216e46 q^{83} +4.70884e47 q^{85} -3.88826e46 q^{87} +8.22598e46 q^{89} +4.08303e46 q^{91} -7.97996e47 q^{93} -9.88236e47 q^{95} -4.76751e48 q^{97} -6.76707e48 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.33407e11	0.272714	0.136357	−	0.990660i	$-0.456461\pi$
$3$	1.33407e11	0.272714	0.136357	+	0.990660i	$0.456461\pi$
$4$	0	0
$5$	1.87739e17	1.40860	0.704302	−	0.709900i	$-0.251260\pi$
$5$	1.87739e17	1.40860	0.704302	+	0.709900i	$0.251260\pi$
$6$	0	0
$7$	−8.28497e20	−1.63452	−0.817258	−	0.576272i	$-0.804507\pi$
$7$	−8.28497e20	−1.63452	−0.817258	+	0.576272i	$0.804507\pi$
$8$	0	0
$9$	−2.21502e23	−0.925627
$10$	0	0
$11$	3.05508e25	0.935195	0.467598	−	0.883941i	$-0.345120\pi$
$11$	3.05508e25	0.935195	0.467598	+	0.883941i	$0.345120\pi$
$12$	0	0
$13$	−4.92824e25	−0.0251814	−0.0125907	−	0.999921i	$-0.504008\pi$
$13$	−4.92824e25	−0.0251814	−0.0125907	+	0.999921i	$0.504008\pi$
$14$	0	0
$15$	2.50456e28	0.384146
$16$	0	0
$17$	2.50818e30	1.79209	0.896047	−	0.443959i	$-0.146427\pi$
$17$	2.50818e30	1.79209	0.896047	+	0.443959i	$0.146427\pi$
$18$	0	0
$19$	−5.26388e30	−0.246515	−0.123257	−	0.992375i	$-0.539334\pi$
$19$	−5.26388e30	−0.246515	−0.123257	+	0.992375i	$0.539334\pi$
$20$	0	0
$21$	−1.10527e32	−0.445755
$22$	0	0
$23$	7.99827e32	0.347268	0.173634	−	0.984810i	$-0.444449\pi$
$23$	7.99827e32	0.347268	0.173634	+	0.984810i	$0.444449\pi$
$24$	0	0
$25$	1.74823e34	0.984168
$26$	0	0
$27$	−6.14740e34	−0.525145
$28$	0	0
$29$	−2.91459e35	−0.432341	−0.216171	−	0.976356i	$-0.569357\pi$
$29$	−2.91459e35	−0.432341	−0.216171	+	0.976356i	$0.569357\pi$
$30$	0	0
$31$	−5.98168e36	−1.73166	−0.865828	−	0.500342i	$-0.833208\pi$
$31$	−5.98168e36	−1.73166	−0.865828	+	0.500342i	$0.833208\pi$
$32$	0	0
$33$	4.07569e36	0.255041
$34$	0	0
$35$	−1.55541e38	−2.30239
$36$	0	0
$37$	1.31540e38	0.499018	0.249509	−	0.968372i	$-0.419731\pi$
$37$	1.31540e38	0.499018	0.249509	+	0.968372i	$0.419731\pi$
$38$	0	0
$39$	−6.57461e36	−0.00686731
$40$	0	0
$41$	−2.57335e39	−0.789394	−0.394697	−	0.918811i	$-0.629150\pi$
$41$	−2.57335e39	−0.789394	−0.394697	+	0.918811i	$0.629150\pi$
$42$	0	0
$43$	−3.01162e39	−0.287622	−0.143811	−	0.989605i	$-0.545936\pi$
$43$	−3.01162e39	−0.287622	−0.143811	+	0.989605i	$0.545936\pi$
$44$	0	0
$45$	−4.15845e40	−1.30384
$46$	0	0
$47$	1.35707e41	1.46624	0.733118	−	0.680102i	$-0.238064\pi$
$47$	1.35707e41	1.46624	0.733118	+	0.680102i	$0.238064\pi$
$48$	0	0
$49$	4.29484e41	1.67164
$50$	0	0
$51$	3.34609e41	0.488729
$52$	0	0
$53$	3.38284e42	1.92541	0.962704	−	0.270557i	$-0.0872079\pi$
$53$	3.38284e42	1.92541	0.962704	+	0.270557i	$0.0872079\pi$
$54$	0	0
$55$	5.73558e42	1.31732
$56$	0	0
$57$	−7.02238e41	−0.0672280
$58$	0	0
$59$	1.39260e42	0.0572732	0.0286366	−	0.999590i	$-0.490883\pi$
$59$	1.39260e42	0.0572732	0.0286366	+	0.999590i	$0.490883\pi$
$60$	0	0
$61$	−8.43699e43	−1.53323	−0.766614	−	0.642108i	$-0.778060\pi$
$61$	−8.43699e43	−1.53323	−0.766614	+	0.642108i	$0.778060\pi$
$62$	0	0
$63$	1.83514e44	1.51295
$64$	0	0
$65$	−9.25222e42	−0.0354706
$66$	0	0
$67$	−4.28654e44	−0.782121	−0.391060	−	0.920365i	$-0.627892\pi$
$67$	−4.28654e44	−0.782121	−0.391060	+	0.920365i	$0.627892\pi$
$68$	0	0
$69$	1.06702e44	0.0947046
$70$	0	0
$71$	1.65460e45	0.729228	0.364614	−	0.931159i	$-0.381201\pi$
$71$	1.65460e45	0.729228	0.364614	+	0.931159i	$0.381201\pi$
$72$	0	0
$73$	−5.25815e45	−1.17334	−0.586669	−	0.809827i	$-0.699561\pi$
$73$	−5.25815e45	−1.17334	−0.586669	+	0.809827i	$0.699561\pi$
$74$	0	0
$75$	2.33226e45	0.268396
$76$	0	0
$77$	−2.53113e46	−1.52859
$78$	0	0
$79$	−1.45663e46	−0.469338	−0.234669	−	0.972075i	$-0.575401\pi$
$79$	−1.45663e46	−0.469338	−0.234669	+	0.972075i	$0.575401\pi$
$80$	0	0
$81$	4.48042e46	0.782413
$82$	0	0
$83$	9.24216e46	0.887892	0.443946	−	0.896054i	$-0.353578\pi$
$83$	9.24216e46	0.887892	0.443946	+	0.896054i	$0.353578\pi$
$84$	0	0
$85$	4.70884e47	2.52435
$86$	0	0
$87$	−3.88826e46	−0.117905
$88$	0	0
$89$	8.22598e46	0.142933	0.0714665	−	0.997443i	$-0.477232\pi$
$89$	8.22598e46	0.142933	0.0714665	+	0.997443i	$0.477232\pi$
$90$	0	0
$91$	4.08303e46	0.0411594
$92$	0	0
$93$	−7.97996e47	−0.472246
$94$	0	0
$95$	−9.88236e47	−0.347242
$96$	0	0
$97$	−4.76751e48	−1.00551	−0.502753	−	0.864430i	$-0.667679\pi$
$97$	−4.76751e48	−1.00551	−0.502753	+	0.864430i	$0.667679\pi$
$98$	0	0
$99$	−6.76707e48	−0.865642
$100$	0	0
$101$	1.44113e49	1.12935	0.564674	−	0.825314i	$-0.309002\pi$
$101$	1.44113e49	1.12935	0.564674	+	0.825314i	$0.309002\pi$
$102$	0	0
$103$	1.10349e49	0.534879	0.267440	−	0.963575i	$-0.413822\pi$
$103$	1.10349e49	0.534879	0.267440	+	0.963575i	$0.413822\pi$
$104$	0	0
$105$	−2.07502e49	−0.627893
$106$	0	0
$107$	7.92403e49	1.51023	0.755116	−	0.655591i	$-0.227581\pi$
$107$	7.92403e49	1.51023	0.755116	+	0.655591i	$0.227581\pi$
$108$	0	0
$109$	−1.47552e48	−0.0178647	−0.00893236	−	0.999960i	$-0.502843\pi$
$109$	−1.47552e48	−0.0178647	−0.00893236	+	0.999960i	$0.502843\pi$
$110$	0	0
$111$	1.75484e49	0.136089
$112$	0	0
$113$	2.29547e49	0.114935	0.0574673	−	0.998347i	$-0.481698\pi$
$113$	2.29547e49	0.114935	0.0574673	+	0.998347i	$0.481698\pi$
$114$	0	0
$115$	1.50159e50	0.489163
$116$	0	0
$117$	1.09161e49	0.0233086
$118$	0	0
$119$	−2.07802e51	−2.92921
$120$	0	0
$121$	−1.33836e50	−0.125410
$122$	0	0
$123$	−3.43302e50	−0.215279
$124$	0	0
$125$	−5.27986e49	−0.0223011
$126$	0	0
$127$	5.51820e51	1.57981	0.789907	−	0.613227i	$-0.210129\pi$
$127$	5.51820e51	1.57981	0.789907	+	0.613227i	$0.210129\pi$
$128$	0	0
$129$	−4.01770e50	−0.0784385
$130$	0	0
$131$	1.30117e51	0.174256	0.0871282	−	0.996197i	$-0.472231\pi$
$131$	1.30117e51	0.174256	0.0871282	+	0.996197i	$0.472231\pi$
$132$	0	0
$133$	4.36111e51	0.402932
$134$	0	0
$135$	−1.15411e52	−0.739722
$136$	0	0
$137$	−1.00417e52	−0.448902	−0.224451	−	0.974485i	$-0.572059\pi$
$137$	−1.00417e52	−0.448902	−0.224451	+	0.974485i	$0.572059\pi$
$138$	0	0
$139$	3.07477e52	0.963719	0.481860	−	0.876248i	$-0.339962\pi$
$139$	3.07477e52	0.963719	0.481860	+	0.876248i	$0.339962\pi$
$140$	0	0
$141$	1.81042e52	0.399863
$142$	0	0
$143$	−1.50562e51	−0.0235495
$144$	0	0
$145$	−5.47182e52	−0.608998
$146$	0	0
$147$	5.72961e52	0.455880
$148$	0	0
$149$	1.89224e53	1.08122	0.540608	−	0.841275i	$-0.318194\pi$
$149$	1.89224e53	1.08122	0.540608	+	0.841275i	$0.318194\pi$
$150$	0	0
$151$	−2.34497e52	−0.0966500	−0.0483250	−	0.998832i	$-0.515388\pi$
$151$	−2.34497e52	−0.0966500	−0.0483250	+	0.998832i	$0.515388\pi$
$152$	0	0
$153$	−5.55568e53	−1.65881
$154$	0	0
$155$	−1.12299e54	−2.43922
$156$	0	0
$157$	1.13167e54	1.79546	0.897732	−	0.440542i	$-0.145214\pi$
$157$	1.13167e54	1.79546	0.897732	+	0.440542i	$0.145214\pi$
$158$	0	0
$159$	4.51294e53	0.525085
$160$	0	0
$161$	−6.62655e53	−0.567614
$162$	0	0
$163$	2.02699e54	1.28308	0.641542	−	0.767088i	$-0.278295\pi$
$163$	2.02699e54	1.28308	0.641542	+	0.767088i	$0.278295\pi$
$164$	0	0
$165$	7.65165e53	0.359252
$166$	0	0
$167$	2.35877e54	0.824389	0.412194	−	0.911096i	$-0.364762\pi$
$167$	2.35877e54	0.824389	0.412194	+	0.911096i	$0.364762\pi$
$168$	0	0
$169$	−3.82780e54	−0.999366
$170$	0	0
$171$	1.16596e54	0.228181
$172$	0	0
$173$	6.16168e53	0.0906921	0.0453461	−	0.998971i	$-0.485561\pi$
$173$	6.16168e53	0.0906921	0.0453461	+	0.998971i	$0.485561\pi$
$174$	0	0
$175$	−1.44841e55	−1.60864
$176$	0	0
$177$	1.85782e53	0.0156192
$178$	0	0
$179$	−3.10481e54	−0.198215	−0.0991073	−	0.995077i	$-0.531599\pi$
$179$	−3.10481e54	−0.198215	−0.0991073	+	0.995077i	$0.531599\pi$
$180$	0	0
$181$	−8.59693e54	−0.418041	−0.209020	−	0.977911i	$-0.567027\pi$
$181$	−8.59693e54	−0.418041	−0.209020	+	0.977911i	$0.567027\pi$
$182$	0	0
$183$	−1.12555e55	−0.418132
$184$	0	0
$185$	2.46952e55	0.702920
$186$	0	0
$187$	7.66271e55	1.67596
$188$	0	0
$189$	5.09310e55	0.858358
$190$	0	0
$191$	2.27680e55	0.296488	0.148244	−	0.988951i	$-0.452638\pi$
$191$	2.27680e55	0.296488	0.148244	+	0.988951i	$0.452638\pi$
$192$	0	0
$193$	6.99559e55	0.705782	0.352891	−	0.935664i	$-0.385199\pi$
$193$	6.99559e55	0.705782	0.352891	+	0.935664i	$0.385199\pi$
$194$	0	0
$195$	−1.23431e54	−0.00967333
$196$	0	0
$197$	−1.40903e56	−0.860000	−0.430000	−	0.902829i	$-0.641486\pi$
$197$	−1.40903e56	−0.860000	−0.430000	+	0.902829i	$0.641486\pi$
$198$	0	0
$199$	−3.14321e56	−1.49786	−0.748932	−	0.662647i	$-0.769433\pi$
$199$	−3.14321e56	−1.49786	−0.748932	+	0.662647i	$0.769433\pi$
$200$	0	0
$201$	−5.71853e55	−0.213295
$202$	0	0
$203$	2.41473e56	0.706668
$204$	0	0
$205$	−4.83117e56	−1.11194
$206$	0	0
$207$	−1.77163e56	−0.321440
$208$	0	0
$209$	−1.60816e56	−0.230539
$210$	0	0
$211$	1.36710e57	1.55196	0.775978	−	0.630760i	$-0.217257\pi$
$211$	1.36710e57	1.55196	0.775978	+	0.630760i	$0.217257\pi$
$212$	0	0
$213$	2.20735e56	0.198870
$214$	0	0
$215$	−5.65398e56	−0.405146
$216$	0	0
$217$	4.95580e57	2.83042
$218$	0	0
$219$	−7.01473e56	−0.319985
$220$	0	0
$221$	−1.23609e56	−0.0451274
$222$	0	0
$223$	−1.75293e56	−0.0513210	−0.0256605	−	0.999671i	$-0.508169\pi$
$223$	−1.75293e56	−0.0513210	−0.0256605	+	0.999671i	$0.508169\pi$
$224$	0	0
$225$	−3.87237e57	−0.910973
$226$	0	0
$227$	4.13704e57	0.783531	0.391765	−	0.920065i	$-0.371865\pi$
$227$	4.13704e57	0.783531	0.391765	+	0.920065i	$0.371865\pi$
$228$	0	0
$229$	6.25222e57	0.955136	0.477568	−	0.878595i	$-0.341518\pi$
$229$	6.25222e57	0.955136	0.477568	+	0.878595i	$0.341518\pi$
$230$	0	0
$231$	−3.37670e57	−0.416868
$232$	0	0
$233$	−5.79416e57	−0.579123	−0.289561	−	0.957159i	$-0.593509\pi$
$233$	−5.79416e57	−0.579123	−0.289561	+	0.957159i	$0.593509\pi$
$234$	0	0
$235$	2.54774e58	2.06535
$236$	0	0
$237$	−1.94324e57	−0.127995
$238$	0	0
$239$	−1.99860e58	−1.07146	−0.535732	−	0.844388i	$-0.679964\pi$
$239$	−1.99860e58	−1.07146	−0.535732	+	0.844388i	$0.679964\pi$
$240$	0	0
$241$	−8.71374e57	−0.380879	−0.190440	−	0.981699i	$-0.560991\pi$
$241$	−8.71374e57	−0.380879	−0.190440	+	0.981699i	$0.560991\pi$
$242$	0	0
$243$	2.06879e58	0.738520
$244$	0	0
$245$	8.06309e58	2.35468
$246$	0	0
$247$	2.59417e56	0.00620759
$248$	0	0
$249$	1.23297e58	0.242140
$250$	0	0
$251$	7.09795e58	1.14585	0.572924	−	0.819609i	$-0.305809\pi$
$251$	7.09795e58	1.14585	0.572924	+	0.819609i	$0.305809\pi$
$252$	0	0
$253$	2.44354e58	0.324763
$254$	0	0
$255$	6.28191e58	0.688426
$256$	0	0
$257$	−3.39451e58	−0.307196	−0.153598	−	0.988133i	$-0.549086\pi$
$257$	−3.39451e58	−0.307196	−0.153598	+	0.988133i	$0.549086\pi$
$258$	0	0
$259$	−1.08981e59	−0.815653
$260$	0	0
$261$	6.45587e58	0.400187
$262$	0	0
$263$	2.34522e59	1.20578	0.602890	−	0.797825i	$-0.294016\pi$
$263$	2.34522e59	1.20578	0.602890	+	0.797825i	$0.294016\pi$
$264$	0	0
$265$	6.35091e59	2.71214
$266$	0	0
$267$	1.09740e58	0.0389798
$268$	0	0
$269$	−2.15592e59	−0.637826	−0.318913	−	0.947784i	$-0.603318\pi$
$269$	−2.15592e59	−0.637826	−0.318913	+	0.947784i	$0.603318\pi$
$270$	0	0
$271$	3.52922e59	0.870825	0.435413	−	0.900231i	$-0.356602\pi$
$271$	3.52922e59	0.870825	0.435413	+	0.900231i	$0.356602\pi$
$272$	0	0
$273$	5.44704e57	0.0112247
$274$	0	0
$275$	5.34100e59	0.920389
$276$	0	0
$277$	2.94410e59	0.424813	0.212406	−	0.977181i	$-0.431870\pi$
$277$	2.94410e59	0.424813	0.212406	+	0.977181i	$0.431870\pi$
$278$	0	0
$279$	1.32495e60	1.60287
$280$	0	0
$281$	4.56150e59	0.463237	0.231618	−	0.972807i	$-0.425598\pi$
$281$	4.56150e59	0.463237	0.231618	+	0.972807i	$0.425598\pi$
$282$	0	0
$283$	−5.39564e59	−0.460548	−0.230274	−	0.973126i	$-0.573962\pi$
$283$	−5.39564e59	−0.460548	−0.230274	+	0.973126i	$0.573962\pi$
$284$	0	0
$285$	−1.31837e59	−0.0946977
$286$	0	0
$287$	2.13201e60	1.29028
$288$	0	0
$289$	4.33215e60	2.21160
$290$	0	0
$291$	−6.36018e59	−0.274215
$292$	0	0
$293$	3.90821e60	1.42469	0.712346	−	0.701828i	$-0.247632\pi$
$293$	3.90821e60	1.42469	0.712346	+	0.701828i	$0.247632\pi$
$294$	0	0
$295$	2.61445e59	0.0806753
$296$	0	0
$297$	−1.87808e60	−0.491113
$298$	0	0
$299$	−3.94174e58	−0.00874468
$300$	0	0
$301$	2.49512e60	0.470123
$302$	0	0
$303$	1.92256e60	0.307989
$304$	0	0
$305$	−1.58395e61	−2.15971
$306$	0	0
$307$	−2.34251e59	−0.0272140	−0.0136070	−	0.999907i	$-0.504331\pi$
$307$	−2.34251e59	−0.0272140	−0.0136070	+	0.999907i	$0.504331\pi$
$308$	0	0
$309$	1.47213e60	0.145869
$310$	0	0
$311$	4.35273e60	0.368241	0.184120	−	0.982904i	$-0.441056\pi$
$311$	4.35273e60	0.368241	0.184120	+	0.982904i	$0.441056\pi$
$312$	0	0
$313$	1.93545e61	1.39941	0.699706	−	0.714431i	$-0.253314\pi$
$313$	1.93545e61	1.39941	0.699706	+	0.714431i	$0.253314\pi$
$314$	0	0
$315$	3.44527e61	2.13115
$316$	0	0
$317$	2.36551e61	1.25306	0.626530	−	0.779397i	$-0.284475\pi$
$317$	2.36551e61	1.25306	0.626530	+	0.779397i	$0.284475\pi$
$318$	0	0
$319$	−8.90431e60	−0.404323
$320$	0	0
$321$	1.05712e61	0.411861
$322$	0	0
$323$	−1.32028e61	−0.441778
$324$	0	0
$325$	−8.61571e59	−0.0247827
$326$	0	0
$327$	−1.96844e59	−0.00487196
$328$	0	0
$329$	−1.12433e62	−2.39658
$330$	0	0
$331$	−6.39438e61	−1.17493	−0.587466	−	0.809249i	$-0.699874\pi$
$331$	−6.39438e61	−1.17493	−0.587466	+	0.809249i	$0.699874\pi$
$332$	0	0
$333$	−2.91364e61	−0.461905
$334$	0	0
$335$	−8.04750e61	−1.10170
$336$	0	0
$337$	9.36305e61	1.10786	0.553930	−	0.832563i	$-0.313128\pi$
$337$	9.36305e61	1.10786	0.553930	+	0.832563i	$0.313128\pi$
$338$	0	0
$339$	3.06232e60	0.0313442
$340$	0	0
$341$	−1.82745e62	−1.61944
$342$	0	0
$343$	−1.42966e62	−1.09781
$344$	0	0
$345$	2.00322e61	0.133401
$346$	0	0
$347$	9.31722e61	0.538535	0.269268	−	0.963065i	$-0.413218\pi$
$347$	9.31722e61	0.538535	0.269268	+	0.963065i	$0.413218\pi$
$348$	0	0
$349$	2.03454e62	1.02151	0.510755	−	0.859726i	$-0.329366\pi$
$349$	2.03454e62	1.02151	0.510755	+	0.859726i	$0.329366\pi$
$350$	0	0
$351$	3.02959e60	0.0132239
$352$	0	0
$353$	−2.78134e62	−1.05627	−0.528133	−	0.849161i	$-0.677108\pi$
$353$	−2.78134e62	−1.05627	−0.528133	+	0.849161i	$0.677108\pi$
$354$	0	0
$355$	3.10633e62	1.02719
$356$	0	0
$357$	−2.77222e62	−0.798835
$358$	0	0
$359$	−7.25567e62	−1.82332	−0.911662	−	0.410942i	$-0.865200\pi$
$359$	−7.25567e62	−1.82332	−0.911662	+	0.410942i	$0.865200\pi$
$360$	0	0
$361$	−4.28251e62	−0.939230
$362$	0	0
$363$	−1.78546e61	−0.0342010
$364$	0	0
$365$	−9.87160e62	−1.65277
$366$	0	0
$367$	3.66239e62	0.536347	0.268174	−	0.963371i	$-0.413580\pi$
$367$	3.66239e62	0.536347	0.268174	+	0.963371i	$0.413580\pi$
$368$	0	0
$369$	5.70001e62	0.730685
$370$	0	0
$371$	−2.80267e63	−3.14711
$372$	0	0
$373$	1.01303e63	0.997140	0.498570	−	0.866850i	$-0.333859\pi$
$373$	1.01303e63	0.997140	0.498570	+	0.866850i	$0.333859\pi$
$374$	0	0
$375$	−7.04369e60	−0.00608183
$376$	0	0
$377$	1.43638e61	0.0108869
$378$	0	0
$379$	4.10768e62	0.273486	0.136743	−	0.990607i	$-0.456336\pi$
$379$	4.10768e62	0.273486	0.136743	+	0.990607i	$0.456336\pi$
$380$	0	0
$381$	7.36165e62	0.430837
$382$	0	0
$383$	1.90644e63	0.981415	0.490708	−	0.871324i	$-0.336738\pi$
$383$	1.90644e63	0.981415	0.490708	+	0.871324i	$0.336738\pi$
$384$	0	0
$385$	−4.75191e63	−2.15318
$386$	0	0
$387$	6.67079e62	0.266231
$388$	0	0
$389$	3.21862e63	1.13215	0.566076	−	0.824353i	$-0.308461\pi$
$389$	3.21862e63	1.13215	0.566076	+	0.824353i	$0.308461\pi$
$390$	0	0
$391$	2.00611e63	0.622336
$392$	0	0
$393$	1.73586e62	0.0475221
$394$	0	0
$395$	−2.73466e63	−0.661111
$396$	0	0
$397$	−1.66988e63	−0.356713	−0.178356	−	0.983966i	$-0.557078\pi$
$397$	−1.66988e63	−0.356713	−0.178356	+	0.983966i	$0.557078\pi$
$398$	0	0
$399$	5.81802e62	0.109885
$400$	0	0
$401$	−3.21505e63	−0.537218	−0.268609	−	0.963249i	$-0.586564\pi$
$401$	−3.21505e63	−0.537218	−0.268609	+	0.963249i	$0.586564\pi$
$402$	0	0
$403$	2.94791e62	0.0436055
$404$	0	0
$405$	8.41150e63	1.10211
$406$	0	0
$407$	4.01867e63	0.466680
$408$	0	0
$409$	1.21477e64	1.25104	0.625522	−	0.780207i	$-0.284886\pi$
$409$	1.21477e64	1.25104	0.625522	+	0.780207i	$0.284886\pi$
$410$	0	0
$411$	−1.33962e63	−0.122422
$412$	0	0
$413$	−1.15376e63	−0.0936139
$414$	0	0
$415$	1.73511e64	1.25069
$416$	0	0
$417$	4.10196e63	0.262820
$418$	0	0
$419$	9.43742e63	0.537786	0.268893	−	0.963170i	$-0.413342\pi$
$419$	9.43742e63	0.537786	0.268893	+	0.963170i	$0.413342\pi$
$420$	0	0
$421$	−7.07535e63	−0.358787	−0.179394	−	0.983777i	$-0.557414\pi$
$421$	−7.07535e63	−0.358787	−0.179394	+	0.983777i	$0.557414\pi$
$422$	0	0
$423$	−3.00593e64	−1.35719
$424$	0	0
$425$	4.38489e64	1.76372
$426$	0	0
$427$	6.99002e64	2.50608
$428$	0	0
$429$	−2.00860e62	−0.00642228
$430$	0	0
$431$	−1.47866e64	−0.421867	−0.210933	−	0.977500i	$-0.567650\pi$
$431$	−1.47866e64	−0.421867	−0.210933	+	0.977500i	$0.567650\pi$
$432$	0	0
$433$	5.40840e64	1.37757	0.688787	−	0.724963i	$-0.258143\pi$
$433$	5.40840e64	1.37757	0.688787	+	0.724963i	$0.258143\pi$
$434$	0	0
$435$	−7.29978e63	−0.166082
$436$	0	0
$437$	−4.21020e63	−0.0856066
$438$	0	0
$439$	3.69191e63	0.0671229	0.0335614	−	0.999437i	$-0.489315\pi$
$439$	3.69191e63	0.0671229	0.0335614	+	0.999437i	$0.489315\pi$
$440$	0	0
$441$	−9.51316e64	−1.54732
$442$	0	0
$443$	7.49563e64	1.09123	0.545616	−	0.838036i	$-0.316296\pi$
$443$	7.49563e64	1.09123	0.545616	+	0.838036i	$0.316296\pi$
$444$	0	0
$445$	1.54434e64	0.201336
$446$	0	0
$447$	2.52437e64	0.294862
$448$	0	0
$449$	−5.65052e64	−0.591634	−0.295817	−	0.955245i	$-0.595592\pi$
$449$	−5.65052e64	−0.591634	−0.295817	+	0.955245i	$0.595592\pi$
$450$	0	0
$451$	−7.86179e64	−0.738238
$452$	0	0
$453$	−3.12835e63	−0.0263578
$454$	0	0
$455$	7.66544e63	0.0579773
$456$	0	0
$457$	5.93488e64	0.403148	0.201574	−	0.979473i	$-0.435394\pi$
$457$	5.93488e64	0.403148	0.201574	+	0.979473i	$0.435394\pi$
$458$	0	0
$459$	−1.54188e65	−0.941109
$460$	0	0
$461$	1.06042e65	0.581840	0.290920	−	0.956747i	$-0.406039\pi$
$461$	1.06042e65	0.581840	0.290920	+	0.956747i	$0.406039\pi$
$462$	0	0
$463$	−1.86764e65	−0.921637	−0.460818	−	0.887495i	$-0.652444\pi$
$463$	−1.86764e65	−0.921637	−0.460818	+	0.887495i	$0.652444\pi$
$464$	0	0
$465$	−1.49815e65	−0.665209
$466$	0	0
$467$	2.28278e64	0.0912432	0.0456216	−	0.998959i	$-0.485473\pi$
$467$	2.28278e64	0.0912432	0.0456216	+	0.998959i	$0.485473\pi$
$468$	0	0
$469$	3.55138e65	1.27839
$470$	0	0
$471$	1.50972e65	0.489648
$472$	0	0
$473$	−9.20074e64	−0.268983
$474$	0	0
$475$	−9.20250e64	−0.242612
$476$	0	0
$477$	−7.49306e65	−1.78221
$478$	0	0
$479$	1.07480e65	0.230732	0.115366	−	0.993323i	$-0.463196\pi$
$479$	1.07480e65	0.230732	0.115366	+	0.993323i	$0.463196\pi$
$480$	0	0
$481$	−6.48262e63	−0.0125660
$482$	0	0
$483$	−8.84026e64	−0.154796
$484$	0	0
$485$	−8.95047e65	−1.41636
$486$	0	0
$487$	−9.52950e64	−0.136336	−0.0681680	−	0.997674i	$-0.521715\pi$
$487$	−9.52950e64	−0.136336	−0.0681680	+	0.997674i	$0.521715\pi$
$488$	0	0
$489$	2.70414e65	0.349915
$490$	0	0
$491$	−7.75386e65	−0.907865	−0.453933	−	0.891036i	$-0.649979\pi$
$491$	−7.75386e65	−0.907865	−0.453933	+	0.891036i	$0.649979\pi$
$492$	0	0
$493$	−7.31033e65	−0.774796
$494$	0	0
$495$	−1.27044e66	−1.21935
$496$	0	0
$497$	−1.37083e66	−1.19193
$498$	0	0
$499$	−1.56740e66	−1.23514	−0.617569	−	0.786517i	$-0.711882\pi$
$499$	−1.56740e66	−1.23514	−0.617569	+	0.786517i	$0.711882\pi$
$500$	0	0
$501$	3.14676e65	0.224822
$502$	0	0
$503$	5.11802e65	0.331655	0.165827	−	0.986155i	$-0.446971\pi$
$503$	5.11802e65	0.331655	0.165827	+	0.986155i	$0.446971\pi$
$504$	0	0
$505$	2.70555e66	1.59081
$506$	0	0
$507$	−5.10654e65	−0.272541
$508$	0	0
$509$	−1.63079e66	−0.790335	−0.395167	−	0.918609i	$-0.629313\pi$
$509$	−1.63079e66	−0.790335	−0.395167	+	0.918609i	$0.629313\pi$
$510$	0	0
$511$	4.35636e66	1.91784
$512$	0	0
$513$	3.23592e65	0.129456
$514$	0	0
$515$	2.07168e66	0.753434
$516$	0	0
$517$	4.14595e66	1.37122
$518$	0	0
$519$	8.22010e64	0.0247330
$520$	0	0
$521$	3.56843e65	0.0977133	0.0488566	−	0.998806i	$-0.484442\pi$
$521$	3.56843e65	0.0977133	0.0488566	+	0.998806i	$0.484442\pi$
$522$	0	0
$523$	7.86554e65	0.196082	0.0980408	−	0.995182i	$-0.468742\pi$
$523$	7.86554e65	0.196082	0.0980408	+	0.995182i	$0.468742\pi$
$524$	0	0
$525$	−1.93227e66	−0.438698
$526$	0	0
$527$	−1.50031e67	−3.10329
$528$	0	0
$529$	−4.66501e66	−0.879405
$530$	0	0
$531$	−3.08463e65	−0.0530136
$532$	0	0
$533$	1.26821e65	0.0198780
$534$	0	0
$535$	1.48765e67	2.12732
$536$	0	0
$537$	−4.14202e65	−0.0540558
$538$	0	0
$539$	1.31211e67	1.56331
$540$	0	0
$541$	9.49189e66	1.03281	0.516404	−	0.856345i	$-0.327270\pi$
$541$	9.49189e66	1.03281	0.516404	+	0.856345i	$0.327270\pi$
$542$	0	0
$543$	−1.14689e66	−0.114005
$544$	0	0
$545$	−2.77013e65	−0.0251643
$546$	0	0
$547$	1.44217e67	1.19764	0.598819	−	0.800884i	$-0.295637\pi$
$547$	1.44217e67	1.19764	0.598819	+	0.800884i	$0.295637\pi$
$548$	0	0
$549$	1.86881e67	1.41920
$550$	0	0
$551$	1.53421e66	0.106578
$552$	0	0
$553$	1.20681e67	0.767140
$554$	0	0
$555$	3.29451e66	0.191696
$556$	0	0
$557$	7.89683e65	0.0420727	0.0210364	−	0.999779i	$-0.493303\pi$
$557$	7.89683e65	0.0420727	0.0210364	+	0.999779i	$0.493303\pi$
$558$	0	0
$559$	1.48420e65	0.00724273
$560$	0	0
$561$	1.02226e67	0.457057
$562$	0	0
$563$	−2.05636e67	−0.842643	−0.421321	−	0.906911i	$-0.638434\pi$
$563$	−2.05636e67	−0.842643	−0.421321	+	0.906911i	$0.638434\pi$
$564$	0	0
$565$	4.30950e66	0.161897
$566$	0	0
$567$	−3.71202e67	−1.27887
$568$	0	0
$569$	−2.99189e67	−0.945573	−0.472787	−	0.881177i	$-0.656752\pi$
$569$	−2.99189e67	−0.945573	−0.472787	+	0.881177i	$0.656752\pi$
$570$	0	0
$571$	1.50438e67	0.436289	0.218144	−	0.975917i	$-0.430000\pi$
$571$	1.50438e67	0.436289	0.218144	+	0.975917i	$0.430000\pi$
$572$	0	0
$573$	3.03741e66	0.0808564
$574$	0	0
$575$	1.39828e67	0.341770
$576$	0	0
$577$	1.42499e67	0.319893	0.159947	−	0.987126i	$-0.448868\pi$
$577$	1.42499e67	0.319893	0.159947	+	0.987126i	$0.448868\pi$
$578$	0	0
$579$	9.33259e66	0.192476
$580$	0	0
$581$	−7.65710e67	−1.45127
$582$	0	0
$583$	1.03349e68	1.80063
$584$	0	0
$585$	2.04939e66	0.0328326
$586$	0	0
$587$	4.53396e67	0.668104	0.334052	−	0.942555i	$-0.391584\pi$
$587$	4.53396e67	0.668104	0.334052	+	0.942555i	$0.391584\pi$
$588$	0	0
$589$	3.14869e67	0.426879
$590$	0	0
$591$	−1.87974e67	−0.234534
$592$	0	0
$593$	5.39408e67	0.619552	0.309776	−	0.950810i	$-0.399746\pi$
$593$	5.39408e67	0.619552	0.309776	+	0.950810i	$0.399746\pi$
$594$	0	0
$595$	−3.90126e68	−4.12609
$596$	0	0
$597$	−4.19325e67	−0.408488
$598$	0	0
$599$	1.84268e68	1.65383	0.826917	−	0.562323i	$-0.190092\pi$
$599$	1.84268e68	1.65383	0.826917	+	0.562323i	$0.190092\pi$
$600$	0	0
$601$	−6.15406e67	−0.509023	−0.254511	−	0.967070i	$-0.581915\pi$
$601$	−6.15406e67	−0.509023	−0.254511	+	0.967070i	$0.581915\pi$
$602$	0	0
$603$	9.49476e67	0.723952
$604$	0	0
$605$	−2.51262e67	−0.176653
$606$	0	0
$607$	−2.91666e68	−1.89131	−0.945654	−	0.325175i	$-0.894577\pi$
$607$	−2.91666e68	−1.89131	−0.945654	+	0.325175i	$0.894577\pi$
$608$	0	0
$609$	3.22141e67	0.192718
$610$	0	0
$611$	−6.68795e66	−0.0369218
$612$	0	0
$613$	−1.34057e68	−0.683139	−0.341569	−	0.939857i	$-0.610958\pi$
$613$	−1.34057e68	−0.683139	−0.341569	+	0.939857i	$0.610958\pi$
$614$	0	0
$615$	−6.44511e67	−0.303243
$616$	0	0
$617$	−2.63476e68	−1.14487	−0.572433	−	0.819951i	$-0.694000\pi$
$617$	−2.63476e68	−1.14487	−0.572433	+	0.819951i	$0.694000\pi$
$618$	0	0
$619$	−1.18534e68	−0.475799	−0.237899	−	0.971290i	$-0.576459\pi$
$619$	−1.18534e68	−0.475799	−0.237899	+	0.971290i	$0.576459\pi$
$620$	0	0
$621$	−4.91686e67	−0.182366
$622$	0	0
$623$	−6.81520e67	−0.233626
$624$	0	0
$625$	−3.20461e68	−1.01558
$626$	0	0
$627$	−2.14539e67	−0.0628713
$628$	0	0
$629$	3.29927e68	0.894288
$630$	0	0
$631$	−4.68843e68	−1.17573	−0.587866	−	0.808959i	$-0.700032\pi$
$631$	−4.68843e68	−1.17573	−0.587866	+	0.808959i	$0.700032\pi$
$632$	0	0
$633$	1.82380e68	0.423240
$634$	0	0
$635$	1.03598e69	2.22533
$636$	0	0
$637$	−2.11660e67	−0.0420943
$638$	0	0
$639$	−3.66497e68	−0.674993
$640$	0	0
$641$	−2.45988e68	−0.419654	−0.209827	−	0.977738i	$-0.567290\pi$
$641$	−2.45988e68	−0.419654	−0.209827	+	0.977738i	$0.567290\pi$
$642$	0	0
$643$	−6.14694e68	−0.971605	−0.485803	−	0.874069i	$-0.661473\pi$
$643$	−6.14694e68	−0.971605	−0.485803	+	0.874069i	$0.661473\pi$
$644$	0	0
$645$	−7.54279e67	−0.110489
$646$	0	0
$647$	9.77317e68	1.32703	0.663515	−	0.748163i	$-0.269064\pi$
$647$	9.77317e68	1.32703	0.663515	+	0.748163i	$0.269064\pi$
$648$	0	0
$649$	4.25450e67	0.0535616
$650$	0	0
$651$	6.61138e68	0.771894
$652$	0	0
$653$	8.35256e67	0.0904580	0.0452290	−	0.998977i	$-0.485598\pi$
$653$	8.35256e67	0.0904580	0.0452290	+	0.998977i	$0.485598\pi$
$654$	0	0
$655$	2.44281e68	0.245459
$656$	0	0
$657$	1.16469e69	1.08607
$658$	0	0
$659$	3.35474e68	0.290380	0.145190	−	0.989404i	$-0.453621\pi$
$659$	3.35474e68	0.290380	0.145190	+	0.989404i	$0.453621\pi$
$660$	0	0
$661$	2.63520e68	0.211776	0.105888	−	0.994378i	$-0.466231\pi$
$661$	2.63520e68	0.211776	0.105888	+	0.994378i	$0.466231\pi$
$662$	0	0
$663$	−1.64903e67	−0.0123069
$664$	0	0
$665$	8.18751e68	0.567572
$666$	0	0
$667$	−2.33117e68	−0.150138
$668$	0	0
$669$	−2.33852e67	−0.0139959
$670$	0	0
$671$	−2.57757e69	−1.43387
$672$	0	0
$673$	3.25131e69	1.68147	0.840735	−	0.541447i	$-0.182123\pi$
$673$	3.25131e69	1.68147	0.840735	+	0.541447i	$0.182123\pi$
$674$	0	0
$675$	−1.07471e69	−0.516831
$676$	0	0
$677$	−2.82926e69	−1.26547	−0.632733	−	0.774370i	$-0.718067\pi$
$677$	−2.82926e69	−1.26547	−0.632733	+	0.774370i	$0.718067\pi$
$678$	0	0
$679$	3.94987e69	1.64351
$680$	0	0
$681$	5.51909e68	0.213680
$682$	0	0
$683$	2.71804e69	0.979374	0.489687	−	0.871898i	$-0.337111\pi$
$683$	2.71804e69	0.979374	0.489687	+	0.871898i	$0.337111\pi$
$684$	0	0
$685$	−1.88521e69	−0.632325
$686$	0	0
$687$	8.34089e68	0.260479
$688$	0	0
$689$	−1.66715e68	−0.0484845
$690$	0	0
$691$	2.43872e69	0.660618	0.330309	−	0.943873i	$-0.392847\pi$
$691$	2.43872e69	0.660618	0.330309	+	0.943873i	$0.392847\pi$
$692$	0	0
$693$	5.60650e69	1.41491
$694$	0	0
$695$	5.77255e69	1.35750
$696$	0	0
$697$	−6.45443e69	−1.41467
$698$	0	0
$699$	−7.72980e68	−0.157935
$700$	0	0
$701$	7.63989e69	1.45544	0.727722	−	0.685872i	$-0.240579\pi$
$701$	7.63989e69	1.45544	0.727722	+	0.685872i	$0.240579\pi$
$702$	0	0
$703$	−6.92413e68	−0.123015
$704$	0	0
$705$	3.39886e69	0.563248
$706$	0	0
$707$	−1.19397e70	−1.84594
$708$	0	0
$709$	2.36736e69	0.341533	0.170766	−	0.985312i	$-0.445376\pi$
$709$	2.36736e69	0.341533	0.170766	+	0.985312i	$0.445376\pi$
$710$	0	0
$711$	3.22646e69	0.434432
$712$	0	0
$713$	−4.78431e69	−0.601348
$714$	0	0
$715$	−2.82663e68	−0.0331720
$716$	0	0
$717$	−2.66627e69	−0.292203
$718$	0	0
$719$	−9.01007e69	−0.922296	−0.461148	−	0.887323i	$-0.652562\pi$
$719$	−9.01007e69	−0.922296	−0.461148	+	0.887323i	$0.652562\pi$
$720$	0	0
$721$	−9.14237e69	−0.874269
$722$	0	0
$723$	−1.16247e69	−0.103871
$724$	0	0
$725$	−5.09538e69	−0.425496
$726$	0	0
$727$	−1.12644e70	−0.879257	−0.439628	−	0.898180i	$-0.644890\pi$
$727$	−1.12644e70	−0.879257	−0.439628	+	0.898180i	$0.644890\pi$
$728$	0	0
$729$	−7.96172e69	−0.581008
$730$	0	0
$731$	−7.55369e69	−0.515446
$732$	0	0
$733$	4.32996e69	0.276335	0.138168	−	0.990409i	$-0.455879\pi$
$733$	4.32996e69	0.276335	0.138168	+	0.990409i	$0.455879\pi$
$734$	0	0
$735$	1.07567e70	0.642154
$736$	0	0
$737$	−1.30957e70	−0.731436
$738$	0	0
$739$	2.10376e70	1.09953	0.549766	−	0.835319i	$-0.314717\pi$
$739$	2.10376e70	1.09953	0.549766	+	0.835319i	$0.314717\pi$
$740$	0	0
$741$	3.46080e67	0.00169289
$742$	0	0
$743$	−3.09494e69	−0.141718	−0.0708592	−	0.997486i	$-0.522574\pi$
$743$	−3.09494e69	−0.141718	−0.0708592	+	0.997486i	$0.522574\pi$
$744$	0	0
$745$	3.55247e70	1.52301
$746$	0	0
$747$	−2.04716e70	−0.821857
$748$	0	0
$749$	−6.56504e70	−2.46850
$750$	0	0
$751$	−2.36292e70	−0.832281	−0.416140	−	0.909300i	$-0.636617\pi$
$751$	−2.36292e70	−0.832281	−0.416140	+	0.909300i	$0.636617\pi$
$752$	0	0
$753$	9.46915e69	0.312488
$754$	0	0
$755$	−4.40242e69	−0.136142
$756$	0	0
$757$	4.84445e70	1.40409	0.702045	−	0.712132i	$-0.252270\pi$
$757$	4.84445e70	1.40409	0.702045	+	0.712132i	$0.252270\pi$
$758$	0	0
$759$	3.25985e69	0.0885673
$760$	0	0
$761$	−1.48623e70	−0.378584	−0.189292	−	0.981921i	$-0.560619\pi$
$761$	−1.48623e70	−0.378584	−0.189292	+	0.981921i	$0.560619\pi$
$762$	0	0
$763$	1.22247e69	0.0292002
$764$	0	0
$765$	−1.04302e71	−2.33661
$766$	0	0
$767$	−6.86305e67	−0.00144222
$768$	0	0
$769$	−6.67570e70	−1.31614	−0.658069	−	0.752957i	$-0.728627\pi$
$769$	−6.67570e70	−1.31614	−0.658069	+	0.752957i	$0.728627\pi$
$770$	0	0
$771$	−4.52850e69	−0.0837765
$772$	0	0
$773$	8.33587e70	1.44728	0.723642	−	0.690176i	$-0.242467\pi$
$773$	8.33587e70	1.44728	0.723642	+	0.690176i	$0.242467\pi$
$774$	0	0
$775$	−1.04574e71	−1.70424
$776$	0	0
$777$	−1.45388e70	−0.222440
$778$	0	0
$779$	1.35458e70	0.194597
$780$	0	0
$781$	5.05494e70	0.681970
$782$	0	0
$783$	1.79172e70	0.227042
$784$	0	0
$785$	2.12458e71	2.52910
$786$	0	0
$787$	1.52479e71	1.70541	0.852706	−	0.522390i	$-0.174960\pi$
$787$	1.52479e71	1.70541	0.852706	+	0.522390i	$0.174960\pi$
$788$	0	0
$789$	3.12868e70	0.328833
$790$	0	0
$791$	−1.90179e70	−0.187862
$792$	0	0
$793$	4.15795e69	0.0386088
$794$	0	0
$795$	8.47254e70	0.739638
$796$	0	0
$797$	8.15352e70	0.669293	0.334647	−	0.942344i	$-0.391383\pi$
$797$	8.15352e70	0.669293	0.334647	+	0.942344i	$0.391383\pi$
$798$	0	0
$799$	3.40377e71	2.62763
$800$	0	0
$801$	−1.82207e70	−0.132303
$802$	0	0
$803$	−1.60641e71	−1.09730
$804$	0	0
$805$	−1.24406e71	−0.799544
$806$	0	0
$807$	−2.87615e70	−0.173944
$808$	0	0
$809$	−2.92840e71	−1.66683	−0.833416	−	0.552646i	$-0.813618\pi$
$809$	−2.92840e71	−1.66683	−0.833416	+	0.552646i	$0.813618\pi$
$810$	0	0
$811$	−2.95951e71	−1.58565	−0.792827	−	0.609446i	$-0.791392\pi$
$811$	−2.95951e71	−1.58565	−0.792827	+	0.609446i	$0.791392\pi$
$812$	0	0
$813$	4.70822e70	0.237486
$814$	0	0
$815$	3.80545e71	1.80736
$816$	0	0
$817$	1.58528e70	0.0709031
$818$	0	0
$819$	−9.04400e69	−0.0380982
$820$	0	0
$821$	2.31367e71	0.918107	0.459053	−	0.888409i	$-0.348189\pi$
$821$	2.31367e71	0.918107	0.459053	+	0.888409i	$0.348189\pi$
$822$	0	0
$823$	−1.40957e70	−0.0526976	−0.0263488	−	0.999653i	$-0.508388\pi$
$823$	−1.40957e70	−0.0526976	−0.0263488	+	0.999653i	$0.508388\pi$
$824$	0	0
$825$	7.12525e70	0.251003
$826$	0	0
$827$	−3.13935e71	−1.04221	−0.521105	−	0.853493i	$-0.674480\pi$
$827$	−3.13935e71	−1.04221	−0.521105	+	0.853493i	$0.674480\pi$
$828$	0	0
$829$	−1.00068e71	−0.313118	−0.156559	−	0.987669i	$-0.550040\pi$
$829$	−1.00068e71	−0.313118	−0.156559	+	0.987669i	$0.550040\pi$
$830$	0	0
$831$	3.92762e70	0.115852
$832$	0	0
$833$	1.07722e72	2.99574
$834$	0	0
$835$	4.42833e71	1.16124
$836$	0	0
$837$	3.67718e71	0.909370
$838$	0	0
$839$	−7.28357e70	−0.169893	−0.0849466	−	0.996386i	$-0.527072\pi$
$839$	−7.28357e70	−0.169893	−0.0849466	+	0.996386i	$0.527072\pi$
$840$	0	0
$841$	−3.69518e71	−0.813081
$842$	0	0
$843$	6.08535e70	0.126331
$844$	0	0
$845$	−7.18626e71	−1.40771
$846$	0	0
$847$	1.10883e71	0.204984
$848$	0	0
$849$	−7.19814e70	−0.125598
$850$	0	0
$851$	1.05209e71	0.173293
$852$	0	0
$853$	−3.12687e71	−0.486249	−0.243125	−	0.969995i	$-0.578172\pi$
$853$	−3.12687e71	−0.486249	−0.243125	+	0.969995i	$0.578172\pi$
$854$	0	0
$855$	2.18896e71	0.321417
$856$	0	0
$857$	−1.07117e72	−1.48535	−0.742675	−	0.669652i	$-0.766443\pi$
$857$	−1.07117e72	−1.48535	−0.742675	+	0.669652i	$0.766443\pi$
$858$	0	0
$859$	−6.94017e71	−0.908945	−0.454473	−	0.890761i	$-0.650172\pi$
$859$	−6.94017e71	−0.908945	−0.454473	+	0.890761i	$0.650172\pi$
$860$	0	0
$861$	2.84425e71	0.351876
$862$	0	0
$863$	−1.51616e72	−1.77207	−0.886037	−	0.463614i	$-0.846552\pi$
$863$	−1.51616e72	−1.77207	−0.886037	+	0.463614i	$0.846552\pi$
$864$	0	0
$865$	1.15679e71	0.127749
$866$	0	0
$867$	5.77938e71	0.603134
$868$	0	0
$869$	−4.45012e71	−0.438922
$870$	0	0
$871$	2.11251e70	0.0196949
$872$	0	0
$873$	1.05601e72	0.930723
$874$	0	0
$875$	4.37435e70	0.0364516
$876$	0	0
$877$	−1.82968e72	−1.44173	−0.720867	−	0.693073i	$-0.756256\pi$
$877$	−1.82968e72	−1.44173	−0.720867	+	0.693073i	$0.756256\pi$
$878$	0	0
$879$	5.21382e71	0.388533
$880$	0	0
$881$	8.23192e71	0.580218	0.290109	−	0.956994i	$-0.406308\pi$
$881$	8.23192e71	0.580218	0.290109	+	0.956994i	$0.406308\pi$
$882$	0	0
$883$	−2.32646e72	−1.55116	−0.775582	−	0.631247i	$-0.782544\pi$
$883$	−2.32646e72	−1.55116	−0.775582	+	0.631247i	$0.782544\pi$
$884$	0	0
$885$	3.48785e70	0.0220013
$886$	0	0
$887$	−2.02329e72	−1.20762	−0.603809	−	0.797129i	$-0.706351\pi$
$887$	−2.02329e72	−1.20762	−0.603809	+	0.797129i	$0.706351\pi$
$888$	0	0
$889$	−4.57181e72	−2.58223
$890$	0	0
$891$	1.36881e72	0.731709
$892$	0	0
$893$	−7.14344e71	−0.361449
$894$	0	0
$895$	−5.82893e71	−0.279206
$896$	0	0
$897$	−5.25855e69	−0.00238479
$898$	0	0
$899$	1.74341e72	0.748666
$900$	0	0
$901$	8.48479e72	3.45051
$902$	0	0
$903$	3.32865e71	0.128209
$904$	0	0
$905$	−1.61398e72	−0.588854
$906$	0	0
$907$	−5.38576e72	−1.86152	−0.930762	−	0.365627i	$-0.880855\pi$
$907$	−5.38576e72	−1.86152	−0.930762	+	0.365627i	$0.880855\pi$
$908$	0	0
$909$	−3.19212e72	−1.04536
$910$	0	0
$911$	4.00433e72	1.24260	0.621298	−	0.783574i	$-0.286606\pi$
$911$	4.00433e72	1.24260	0.621298	+	0.783574i	$0.286606\pi$
$912$	0	0
$913$	2.82356e72	0.830352
$914$	0	0
$915$	−2.11310e72	−0.588983
$916$	0	0
$917$	−1.07802e72	−0.284825
$918$	0	0
$919$	6.26307e72	1.56876	0.784381	−	0.620279i	$-0.212981\pi$
$919$	6.26307e72	1.56876	0.784381	+	0.620279i	$0.212981\pi$
$920$	0	0
$921$	−3.12507e70	−0.00742162
$922$	0	0
$923$	−8.15426e70	−0.0183630
$924$	0	0
$925$	2.29963e72	0.491118
$926$	0	0
$927$	−2.44425e72	−0.495099
$928$	0	0
$929$	−3.00047e72	−0.576508	−0.288254	−	0.957554i	$-0.593075\pi$
$929$	−3.00047e72	−0.576508	−0.288254	+	0.957554i	$0.593075\pi$
$930$	0	0
$931$	−2.26075e72	−0.412084
$932$	0	0
$933$	5.80684e71	0.100424
$934$	0	0
$935$	1.43859e73	2.36076
$936$	0	0
$937$	−3.72864e72	−0.580671	−0.290336	−	0.956925i	$-0.593767\pi$
$937$	−3.72864e72	−0.580671	−0.290336	+	0.956925i	$0.593767\pi$
$938$	0	0
$939$	2.58202e72	0.381639
$940$	0	0
$941$	−2.09160e72	−0.293449	−0.146724	−	0.989177i	$-0.546873\pi$
$941$	−2.09160e72	−0.293449	−0.146724	+	0.989177i	$0.546873\pi$
$942$	0	0
$943$	−2.05823e72	−0.274131
$944$	0	0
$945$	9.56174e72	1.20909
$946$	0	0
$947$	−8.20140e72	−0.984721	−0.492361	−	0.870391i	$-0.663866\pi$
$947$	−8.20140e72	−0.984721	−0.492361	+	0.870391i	$0.663866\pi$
$948$	0	0
$949$	2.59134e71	0.0295463
$950$	0	0
$951$	3.15574e72	0.341727
$952$	0	0
$953$	1.17749e73	1.21110	0.605549	−	0.795808i	$-0.292953\pi$
$953$	1.17749e73	1.21110	0.605549	+	0.795808i	$0.292953\pi$
$954$	0	0
$955$	4.27444e72	0.417635
$956$	0	0
$957$	−1.18790e72	−0.110265
$958$	0	0
$959$	8.31948e72	0.733737
$960$	0	0
$961$	2.38482e73	1.99863
$962$	0	0
$963$	−1.75519e73	−1.39791
$964$	0	0
$965$	1.31334e73	0.994168
$966$	0	0
$967$	3.48982e72	0.251105	0.125552	−	0.992087i	$-0.459930\pi$
$967$	3.48982e72	0.251105	0.125552	+	0.992087i	$0.459930\pi$
$968$	0	0
$969$	−1.76134e72	−0.120479
$970$	0	0
$971$	2.02593e73	1.31751	0.658753	−	0.752359i	$-0.271084\pi$
$971$	2.02593e73	1.31751	0.658753	+	0.752359i	$0.271084\pi$
$972$	0	0
$973$	−2.54744e73	−1.57521
$974$	0	0
$975$	−1.14939e71	−0.00675859
$976$	0	0
$977$	−1.10677e73	−0.618926	−0.309463	−	0.950911i	$-0.600149\pi$
$977$	−1.10677e73	−0.618926	−0.309463	+	0.950911i	$0.600149\pi$
$978$	0	0
$979$	2.51311e72	0.133670
$980$	0	0
$981$	3.26831e71	0.0165361
$982$	0	0
$983$	6.73348e72	0.324100	0.162050	−	0.986783i	$-0.448189\pi$
$983$	6.73348e72	0.324100	0.162050	+	0.986783i	$0.448189\pi$
$984$	0	0
$985$	−2.64530e73	−1.21140
$986$	0	0
$987$	−1.49993e73	−0.653582
$988$	0	0
$989$	−2.40877e72	−0.0998819
$990$	0	0
$991$	2.62295e73	1.03511	0.517554	−	0.855650i	$-0.326843\pi$
$991$	2.62295e73	1.03511	0.517554	+	0.855650i	$0.326843\pi$
$992$	0	0
$993$	−8.53053e72	−0.320420
$994$	0	0
$995$	−5.90102e73	−2.10990
$996$	0	0
$997$	2.54966e73	0.867864	0.433932	−	0.900946i	$-0.357126\pi$
$997$	2.54966e73	0.867864	0.433932	+	0.900946i	$0.357126\pi$
$998$	0	0
$999$	−8.08631e72	−0.262057

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.50.a.b.1.2		3
4.3	odd	2		2.50.a.b.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2.50.a.b.1.2	✓	3	4.3	odd	2
16.50.a.b.1.2		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)

\(f(q)\)	\(=\)	\(q+1.33407e11 q^{3} +1.87739e17 q^{5} -8.28497e20 q^{7} -2.21502e23 q^{9} +O(q^{10})\)	\(q+1.33407e11 q^{3} +1.87739e17 q^{5} -8.28497e20 q^{7} -2.21502e23 q^{9} +3.05508e25 q^{11} -4.92824e25 q^{13} +2.50456e28 q^{15} +2.50818e30 q^{17} -5.26388e30 q^{19} -1.10527e32 q^{21} +7.99827e32 q^{23} +1.74823e34 q^{25} -6.14740e34 q^{27} -2.91459e35 q^{29} -5.98168e36 q^{31} +4.07569e36 q^{33} -1.55541e38 q^{35} +1.31540e38 q^{37} -6.57461e36 q^{39} -2.57335e39 q^{41} -3.01162e39 q^{43} -4.15845e40 q^{45} +1.35707e41 q^{47} +4.29484e41 q^{49} +3.34609e41 q^{51} +3.38284e42 q^{53} +5.73558e42 q^{55} -7.02238e41 q^{57} +1.39260e42 q^{59} -8.43699e43 q^{61} +1.83514e44 q^{63} -9.25222e42 q^{65} -4.28654e44 q^{67} +1.06702e44 q^{69} +1.65460e45 q^{71} -5.25815e45 q^{73} +2.33226e45 q^{75} -2.53113e46 q^{77} -1.45663e46 q^{79} +4.48042e46 q^{81} +9.24216e46 q^{83} +4.70884e47 q^{85} -3.88826e46 q^{87} +8.22598e46 q^{89} +4.08303e46 q^{91} -7.97996e47 q^{93} -9.88236e47 q^{95} -4.76751e48 q^{97} -6.76707e48 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 16203614388 q^{3} - 10\!\cdots\!30 q^{5}+ \cdots + 38\!\cdots\!99 q^{9}+O(q^{10}) \) 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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