LMFDB - Embedded newform 80.16.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [80,16,Mod(1,80)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(80, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 16, names="a")

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

chi := DirichletCharacter("80.1");

S:= CuspForms(chi, 16);

N := Newforms(S);

Level:	$ N $	$=$	$ 80 = 2^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	80.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:	$114.154804080$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 10)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	80.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1302.00 q^{3} -78125.0 q^{5} +90706.0 q^{7} -1.26537e7 q^{9} +6.94116e7 q^{11} -3.70575e7 q^{13} -1.01719e8 q^{15} -1.37203e9 q^{17} +5.06876e9 q^{19} +1.18099e8 q^{21} +1.23426e10 q^{23} +6.10352e9 q^{25} -3.51574e10 q^{27} +5.78257e10 q^{29} -2.33728e11 q^{31} +9.03740e10 q^{33} -7.08641e9 q^{35} +7.42248e11 q^{37} -4.82488e10 q^{39} -7.72700e11 q^{41} +4.05994e11 q^{43} +9.88571e11 q^{45} -1.62301e12 q^{47} -4.73933e12 q^{49} -1.78638e12 q^{51} -5.36560e12 q^{53} -5.42278e12 q^{55} +6.59953e12 q^{57} -9.24016e12 q^{59} +9.44308e11 q^{61} -1.14777e12 q^{63} +2.89511e12 q^{65} +7.05676e13 q^{67} +1.60701e13 q^{69} +8.25347e13 q^{71} -1.78432e14 q^{73} +7.94678e12 q^{75} +6.29605e12 q^{77} -3.07261e14 q^{79} +1.35792e14 q^{81} -4.31597e13 q^{83} +1.07190e14 q^{85} +7.52891e13 q^{87} +6.81678e14 q^{89} -3.36133e12 q^{91} -3.04314e14 q^{93} -3.95997e14 q^{95} -2.36520e14 q^{97} -8.78314e14 q^{99} +O(q^{100})$

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
$3$	1302.00	0.343717	0.171859	−	0.985122i	$-0.445023\pi$
$3$	1302.00	0.343717	0.171859	+	0.985122i	$0.445023\pi$
$4$	0	0
$5$	−78125.0	−0.447214
$5$	−78125.0	−0.447214
$6$	0	0
$7$	90706.0	0.0416295	0.0208147	−	0.999783i	$-0.493374\pi$
$7$	90706.0	0.0416295	0.0208147	+	0.999783i	$0.493374\pi$
$8$	0	0
$9$	−1.26537e7	−0.881858
$10$	0	0
$11$	6.94116e7	1.07396	0.536979	−	0.843596i	$-0.319565\pi$
$11$	6.94116e7	1.07396	0.536979	+	0.843596i	$0.319565\pi$
$12$	0	0
$13$	−3.70575e7	−0.163795	−0.0818975	−	0.996641i	$-0.526098\pi$
$13$	−3.70575e7	−0.163795	−0.0818975	+	0.996641i	$0.526098\pi$
$14$	0	0
$15$	−1.01719e8	−0.153715
$16$	0	0
$17$	−1.37203e9	−0.810954	−0.405477	−	0.914105i	$-0.632894\pi$
$17$	−1.37203e9	−0.810954	−0.405477	+	0.914105i	$0.632894\pi$
$18$	0	0
$19$	5.06876e9	1.30092	0.650459	−	0.759542i	$-0.274577\pi$
$19$	5.06876e9	1.30092	0.650459	+	0.759542i	$0.274577\pi$
$20$	0	0
$21$	1.18099e8	0.0143088
$22$	0	0
$23$	1.23426e10	0.755873	0.377937	−	0.925831i	$-0.376634\pi$
$23$	1.23426e10	0.755873	0.377937	+	0.925831i	$0.376634\pi$
$24$	0	0
$25$	6.10352e9	0.200000
$26$	0	0
$27$	−3.51574e10	−0.646828
$28$	0	0
$29$	5.78257e10	0.622495	0.311248	−	0.950329i	$-0.399253\pi$
$29$	5.78257e10	0.622495	0.311248	+	0.950329i	$0.399253\pi$
$30$	0	0
$31$	−2.33728e11	−1.52580	−0.762901	−	0.646516i	$-0.776225\pi$
$31$	−2.33728e11	−1.52580	−0.762901	+	0.646516i	$0.776225\pi$
$32$	0	0
$33$	9.03740e10	0.369138
$34$	0	0
$35$	−7.08641e9	−0.0186173
$36$	0	0
$37$	7.42248e11	1.28539	0.642696	−	0.766121i	$-0.277816\pi$
$37$	7.42248e11	1.28539	0.642696	+	0.766121i	$0.277816\pi$
$38$	0	0
$39$	−4.82488e10	−0.0562992
$40$	0	0
$41$	−7.72700e11	−0.619629	−0.309815	−	0.950797i	$-0.600267\pi$
$41$	−7.72700e11	−0.619629	−0.309815	+	0.950797i	$0.600267\pi$
$42$	0	0
$43$	4.05994e11	0.227775	0.113888	−	0.993494i	$-0.463670\pi$
$43$	4.05994e11	0.227775	0.113888	+	0.993494i	$0.463670\pi$
$44$	0	0
$45$	9.88571e11	0.394379
$46$	0	0
$47$	−1.62301e12	−0.467291	−0.233646	−	0.972322i	$-0.575066\pi$
$47$	−1.62301e12	−0.467291	−0.233646	+	0.972322i	$0.575066\pi$
$48$	0	0
$49$	−4.73933e12	−0.998267
$50$	0	0
$51$	−1.78638e12	−0.278739
$52$	0	0
$53$	−5.36560e12	−0.627407	−0.313703	−	0.949521i	$-0.601570\pi$
$53$	−5.36560e12	−0.627407	−0.313703	+	0.949521i	$0.601570\pi$
$54$	0	0
$55$	−5.42278e12	−0.480289
$56$	0	0
$57$	6.59953e12	0.447148
$58$	0	0
$59$	−9.24016e12	−0.483381	−0.241690	−	0.970353i	$-0.577702\pi$
$59$	−9.24016e12	−0.483381	−0.241690	+	0.970353i	$0.577702\pi$
$60$	0	0
$61$	9.44308e11	0.0384716	0.0192358	−	0.999815i	$-0.493877\pi$
$61$	9.44308e11	0.0384716	0.0192358	+	0.999815i	$0.493877\pi$
$62$	0	0
$63$	−1.14777e12	−0.0367113
$64$	0	0
$65$	2.89511e12	0.0732513
$66$	0	0
$67$	7.05676e13	1.42247	0.711237	−	0.702953i	$-0.248135\pi$
$67$	7.05676e13	1.42247	0.711237	+	0.702953i	$0.248135\pi$
$68$	0	0
$69$	1.60701e13	0.259807
$70$	0	0
$71$	8.25347e13	1.07696	0.538479	−	0.842639i	$-0.318999\pi$
$71$	8.25347e13	1.07696	0.538479	+	0.842639i	$0.318999\pi$
$72$	0	0
$73$	−1.78432e14	−1.89039	−0.945197	−	0.326502i	$-0.894130\pi$
$73$	−1.78432e14	−1.89039	−0.945197	+	0.326502i	$0.894130\pi$
$74$	0	0
$75$	7.94678e12	0.0687435
$76$	0	0
$77$	6.29605e12	0.0447083
$78$	0	0
$79$	−3.07261e14	−1.80013	−0.900067	−	0.435752i	$-0.856482\pi$
$79$	−3.07261e14	−1.80013	−0.900067	+	0.435752i	$0.856482\pi$
$80$	0	0
$81$	1.35792e14	0.659532
$82$	0	0
$83$	−4.31597e13	−0.174579	−0.0872897	−	0.996183i	$-0.527821\pi$
$83$	−4.31597e13	−0.174579	−0.0872897	+	0.996183i	$0.527821\pi$
$84$	0	0
$85$	1.07190e14	0.362669
$86$	0	0
$87$	7.52891e13	0.213962
$88$	0	0
$89$	6.81678e14	1.63363	0.816816	−	0.576899i	$-0.195737\pi$
$89$	6.81678e14	1.63363	0.816816	+	0.576899i	$0.195737\pi$
$90$	0	0
$91$	−3.36133e12	−0.00681869
$92$	0	0
$93$	−3.04314e14	−0.524445
$94$	0	0
$95$	−3.95997e14	−0.581788
$96$	0	0
$97$	−2.36520e14	−0.297222	−0.148611	−	0.988896i	$-0.547480\pi$
$97$	−2.36520e14	−0.297222	−0.148611	+	0.988896i	$0.547480\pi$
$98$	0	0
$99$	−8.78314e14	−0.947079
$100$	0	0
$101$	−1.39643e15	−1.29601	−0.648005	−	0.761636i	$-0.724397\pi$
$101$	−1.39643e15	−1.29601	−0.648005	+	0.761636i	$0.724397\pi$
$102$	0	0
$103$	1.46886e15	1.17680	0.588398	−	0.808572i	$-0.299759\pi$
$103$	1.46886e15	1.17680	0.588398	+	0.808572i	$0.299759\pi$
$104$	0	0
$105$	−9.22650e12	−0.00639908
$106$	0	0
$107$	−1.19960e15	−0.722203	−0.361102	−	0.932526i	$-0.617599\pi$
$107$	−1.19960e15	−0.722203	−0.361102	+	0.932526i	$0.617599\pi$
$108$	0	0
$109$	2.63914e14	0.138281	0.0691407	−	0.997607i	$-0.477974\pi$
$109$	2.63914e14	0.138281	0.0691407	+	0.997607i	$0.477974\pi$
$110$	0	0
$111$	9.66406e14	0.441812
$112$	0	0
$113$	−4.67332e15	−1.86869	−0.934346	−	0.356368i	$-0.884015\pi$
$113$	−4.67332e15	−1.86869	−0.934346	+	0.356368i	$0.884015\pi$
$114$	0	0
$115$	−9.64268e14	−0.338037
$116$	0	0
$117$	4.68914e14	0.144444
$118$	0	0
$119$	−1.24451e14	−0.0337596
$120$	0	0
$121$	6.40729e14	0.153385
$122$	0	0
$123$	−1.00606e15	−0.212977
$124$	0	0
$125$	−4.76837e14	−0.0894427
$126$	0	0
$127$	−5.53712e15	−0.922053	−0.461027	−	0.887386i	$-0.652519\pi$
$127$	−5.53712e15	−0.922053	−0.461027	+	0.887386i	$0.652519\pi$
$128$	0	0
$129$	5.28605e14	0.0782903
$130$	0	0
$131$	−7.83095e15	−1.03343	−0.516714	−	0.856158i	$-0.672845\pi$
$131$	−7.83095e15	−1.03343	−0.516714	+	0.856158i	$0.672845\pi$
$132$	0	0
$133$	4.59767e14	0.0541565
$134$	0	0
$135$	2.74667e15	0.289270
$136$	0	0
$137$	−1.88872e16	−1.78140	−0.890702	−	0.454587i	$-0.849787\pi$
$137$	−1.88872e16	−1.78140	−0.890702	+	0.454587i	$0.849787\pi$
$138$	0	0
$139$	−6.08888e15	−0.515142	−0.257571	−	0.966259i	$-0.582922\pi$
$139$	−6.08888e15	−0.515142	−0.257571	+	0.966259i	$0.582922\pi$
$140$	0	0
$141$	−2.11316e15	−0.160616
$142$	0	0
$143$	−2.57222e15	−0.175909
$144$	0	0
$145$	−4.51763e15	−0.278388
$146$	0	0
$147$	−6.17061e15	−0.343122
$148$	0	0
$149$	−2.97847e16	−1.49657	−0.748285	−	0.663378i	$-0.769122\pi$
$149$	−2.97847e16	−1.49657	−0.748285	+	0.663378i	$0.769122\pi$
$150$	0	0
$151$	−3.43750e16	−1.56284	−0.781421	−	0.624004i	$-0.785505\pi$
$151$	−3.43750e16	−1.56284	−0.781421	+	0.624004i	$0.785505\pi$
$152$	0	0
$153$	1.73612e16	0.715146
$154$	0	0
$155$	1.82600e16	0.682359
$156$	0	0
$157$	3.24674e16	1.10205	0.551024	−	0.834490i	$-0.314237\pi$
$157$	3.24674e16	1.10205	0.551024	+	0.834490i	$0.314237\pi$
$158$	0	0
$159$	−6.98601e15	−0.215651
$160$	0	0
$161$	1.11955e15	0.0314666
$162$	0	0
$163$	−2.42386e16	−0.621013	−0.310506	−	0.950571i	$-0.600499\pi$
$163$	−2.42386e16	−0.621013	−0.310506	+	0.950571i	$0.600499\pi$
$164$	0	0
$165$	−7.06047e15	−0.165084
$166$	0	0
$167$	−3.26384e15	−0.0697196	−0.0348598	−	0.999392i	$-0.511098\pi$
$167$	−3.26384e15	−0.0697196	−0.0348598	+	0.999392i	$0.511098\pi$
$168$	0	0
$169$	−4.98126e16	−0.973171
$170$	0	0
$171$	−6.41386e16	−1.14722
$172$	0	0
$173$	−9.54205e16	−1.56421	−0.782106	−	0.623145i	$-0.785855\pi$
$173$	−9.54205e16	−1.56421	−0.782106	+	0.623145i	$0.785855\pi$
$174$	0	0
$175$	5.53625e14	0.00832589
$176$	0	0
$177$	−1.20307e16	−0.166146
$178$	0	0
$179$	1.08211e17	1.37365	0.686823	−	0.726824i	$-0.259005\pi$
$179$	1.08211e17	1.37365	0.686823	+	0.726824i	$0.259005\pi$
$180$	0	0
$181$	2.21984e16	0.259258	0.129629	−	0.991563i	$-0.458621\pi$
$181$	2.21984e16	0.259258	0.129629	+	0.991563i	$0.458621\pi$
$182$	0	0
$183$	1.22949e15	0.0132234
$184$	0	0
$185$	−5.79881e16	−0.574845
$186$	0	0
$187$	−9.52347e16	−0.870930
$188$	0	0
$189$	−3.18899e15	−0.0269271
$190$	0	0
$191$	3.42389e16	0.267159	0.133580	−	0.991038i	$-0.457353\pi$
$191$	3.42389e16	0.267159	0.133580	+	0.991038i	$0.457353\pi$
$192$	0	0
$193$	−2.27131e17	−1.63907	−0.819533	−	0.573031i	$-0.805767\pi$
$193$	−2.27131e17	−1.63907	−0.819533	+	0.573031i	$0.805767\pi$
$194$	0	0
$195$	3.76944e15	0.0251778
$196$	0	0
$197$	2.63870e16	0.163265	0.0816325	−	0.996662i	$-0.473987\pi$
$197$	2.63870e16	0.163265	0.0816325	+	0.996662i	$0.473987\pi$
$198$	0	0
$199$	−1.99493e17	−1.14427	−0.572137	−	0.820158i	$-0.693885\pi$
$199$	−1.99493e17	−1.14427	−0.572137	+	0.820158i	$0.693885\pi$
$200$	0	0
$201$	9.18790e16	0.488929
$202$	0	0
$203$	5.24514e15	0.0259141
$204$	0	0
$205$	6.03672e16	0.277107
$206$	0	0
$207$	−1.56180e17	−0.666573
$208$	0	0
$209$	3.51831e17	1.39713
$210$	0	0
$211$	−6.45604e16	−0.238697	−0.119349	−	0.992852i	$-0.538081\pi$
$211$	−6.45604e16	−0.238697	−0.119349	+	0.992852i	$0.538081\pi$
$212$	0	0
$213$	1.07460e17	0.370170
$214$	0	0
$215$	−3.17183e16	−0.101864
$216$	0	0
$217$	−2.12005e16	−0.0635183
$218$	0	0
$219$	−2.32319e17	−0.649761
$220$	0	0
$221$	5.08439e16	0.132830
$222$	0	0
$223$	6.71970e17	1.64083	0.820415	−	0.571769i	$-0.193743\pi$
$223$	6.71970e17	1.64083	0.820415	+	0.571769i	$0.193743\pi$
$224$	0	0
$225$	−7.72321e16	−0.176372
$226$	0	0
$227$	5.40955e17	1.15602	0.578012	−	0.816028i	$-0.303829\pi$
$227$	5.40955e17	1.15602	0.578012	+	0.816028i	$0.303829\pi$
$228$	0	0
$229$	7.52317e17	1.50534	0.752671	−	0.658397i	$-0.228765\pi$
$229$	7.52317e17	1.50534	0.752671	+	0.658397i	$0.228765\pi$
$230$	0	0
$231$	8.19746e15	0.0153670
$232$	0	0
$233$	−7.64780e17	−1.34390	−0.671951	−	0.740596i	$-0.734543\pi$
$233$	−7.64780e17	−1.34390	−0.671951	+	0.740596i	$0.734543\pi$
$234$	0	0
$235$	1.26798e17	0.208979
$236$	0	0
$237$	−4.00054e17	−0.618737
$238$	0	0
$239$	6.74103e17	0.978908	0.489454	−	0.872029i	$-0.337196\pi$
$239$	6.74103e17	0.978908	0.489454	+	0.872029i	$0.337196\pi$
$240$	0	0
$241$	3.16938e17	0.432361	0.216180	−	0.976353i	$-0.430640\pi$
$241$	3.16938e17	0.432361	0.216180	+	0.976353i	$0.430640\pi$
$242$	0	0
$243$	6.81271e17	0.873520
$244$	0	0
$245$	3.70260e17	0.446439
$246$	0	0
$247$	−1.87835e17	−0.213084
$248$	0	0
$249$	−5.61940e16	−0.0600060
$250$	0	0
$251$	1.54964e17	0.155840	0.0779200	−	0.996960i	$-0.475172\pi$
$251$	1.54964e17	0.155840	0.0779200	+	0.996960i	$0.475172\pi$
$252$	0	0
$253$	8.56722e17	0.811776
$254$	0	0
$255$	1.39561e17	0.124656
$256$	0	0
$257$	2.25253e18	1.89746	0.948731	−	0.316086i	$-0.102369\pi$
$257$	2.25253e18	1.89746	0.948731	+	0.316086i	$0.102369\pi$
$258$	0	0
$259$	6.73263e16	0.0535102
$260$	0	0
$261$	−7.31710e17	−0.548953
$262$	0	0
$263$	−1.00925e18	−0.715044	−0.357522	−	0.933905i	$-0.616378\pi$
$263$	−1.00925e18	−0.715044	−0.357522	+	0.933905i	$0.616378\pi$
$264$	0	0
$265$	4.19187e17	0.280585
$266$	0	0
$267$	8.87544e17	0.561508
$268$	0	0
$269$	−9.29818e17	−0.556232	−0.278116	−	0.960548i	$-0.589710\pi$
$269$	−9.29818e17	−0.556232	−0.278116	+	0.960548i	$0.589710\pi$
$270$	0	0
$271$	6.51376e16	0.0368606	0.0184303	−	0.999830i	$-0.494133\pi$
$271$	6.51376e16	0.0368606	0.0184303	+	0.999830i	$0.494133\pi$
$272$	0	0
$273$	−4.37646e15	−0.00234370
$274$	0	0
$275$	4.23655e17	0.214792
$276$	0	0
$277$	−8.74228e16	−0.0419784	−0.0209892	−	0.999780i	$-0.506682\pi$
$277$	−8.74228e16	−0.0419784	−0.0209892	+	0.999780i	$0.506682\pi$
$278$	0	0
$279$	2.95752e18	1.34554
$280$	0	0
$281$	1.10585e18	0.476867	0.238433	−	0.971159i	$-0.423366\pi$
$281$	1.10585e18	0.476867	0.238433	+	0.971159i	$0.423366\pi$
$282$	0	0
$283$	−2.15159e18	−0.879756	−0.439878	−	0.898058i	$-0.644978\pi$
$283$	−2.15159e18	−0.879756	−0.439878	+	0.898058i	$0.644978\pi$
$284$	0	0
$285$	−5.15588e17	−0.199971
$286$	0	0
$287$	−7.00885e16	−0.0257948
$288$	0	0
$289$	−9.79963e17	−0.342354
$290$	0	0
$291$	−3.07949e17	−0.102160
$292$	0	0
$293$	5.52332e18	1.74058	0.870288	−	0.492543i	$-0.163933\pi$
$293$	5.52332e18	1.74058	0.870288	+	0.492543i	$0.163933\pi$
$294$	0	0
$295$	7.21887e17	0.216174
$296$	0	0
$297$	−2.44033e18	−0.694665
$298$	0	0
$299$	−4.57387e17	−0.123808
$300$	0	0
$301$	3.68261e16	0.00948216
$302$	0	0
$303$	−1.81815e18	−0.445461
$304$	0	0
$305$	−7.37741e16	−0.0172050
$306$	0	0
$307$	−9.87586e17	−0.219299	−0.109650	−	0.993970i	$-0.534973\pi$
$307$	−9.87586e17	−0.219299	−0.109650	+	0.993970i	$0.534973\pi$
$308$	0	0
$309$	1.91245e18	0.404485
$310$	0	0
$311$	3.09644e18	0.623965	0.311982	−	0.950088i	$-0.399007\pi$
$311$	3.09644e18	0.623965	0.311982	+	0.950088i	$0.399007\pi$
$312$	0	0
$313$	1.19355e18	0.229224	0.114612	−	0.993410i	$-0.463438\pi$
$313$	1.19355e18	0.229224	0.114612	+	0.993410i	$0.463438\pi$
$314$	0	0
$315$	8.96693e16	0.0164178
$316$	0	0
$317$	−2.70108e18	−0.471621	−0.235811	−	0.971799i	$-0.575774\pi$
$317$	−2.70108e18	−0.471621	−0.235811	+	0.971799i	$0.575774\pi$
$318$	0	0
$319$	4.01378e18	0.668534
$320$	0	0
$321$	−1.56188e18	−0.248234
$322$	0	0
$323$	−6.95448e18	−1.05498
$324$	0	0
$325$	−2.26181e17	−0.0327590
$326$	0	0
$327$	3.43616e17	0.0475297
$328$	0	0
$329$	−1.47217e17	−0.0194531
$330$	0	0
$331$	9.04958e17	0.114266	0.0571332	−	0.998367i	$-0.481804\pi$
$331$	9.04958e17	0.114266	0.0571332	+	0.998367i	$0.481804\pi$
$332$	0	0
$333$	−9.39218e18	−1.13353
$334$	0	0
$335$	−5.51309e18	−0.636150
$336$	0	0
$337$	−6.84140e17	−0.0754955	−0.0377477	−	0.999287i	$-0.512018\pi$
$337$	−6.84140e17	−0.0754955	−0.0377477	+	0.999287i	$0.512018\pi$
$338$	0	0
$339$	−6.08467e18	−0.642302
$340$	0	0
$341$	−1.62234e19	−1.63865
$342$	0	0
$343$	−8.60518e17	−0.0831868
$344$	0	0
$345$	−1.25548e18	−0.116189
$346$	0	0
$347$	1.18426e19	1.04948	0.524742	−	0.851261i	$-0.324162\pi$
$347$	1.18426e19	1.04948	0.524742	+	0.851261i	$0.324162\pi$
$348$	0	0
$349$	−4.67865e18	−0.397127	−0.198564	−	0.980088i	$-0.563628\pi$
$349$	−4.67865e18	−0.397127	−0.198564	+	0.980088i	$0.563628\pi$
$350$	0	0
$351$	1.30284e18	0.105947
$352$	0	0
$353$	−6.21571e18	−0.484374	−0.242187	−	0.970230i	$-0.577865\pi$
$353$	−6.21571e18	−0.484374	−0.242187	+	0.970230i	$0.577865\pi$
$354$	0	0
$355$	−6.44803e18	−0.481631
$356$	0	0
$357$	−1.62035e17	−0.0116037
$358$	0	0
$359$	−8.67422e18	−0.595693	−0.297846	−	0.954614i	$-0.596268\pi$
$359$	−8.67422e18	−0.595693	−0.297846	+	0.954614i	$0.596268\pi$
$360$	0	0
$361$	1.05112e19	0.692386
$362$	0	0
$363$	8.34229e17	0.0527212
$364$	0	0
$365$	1.39400e19	0.845410
$366$	0	0
$367$	−2.60076e19	−1.51393	−0.756964	−	0.653457i	$-0.773318\pi$
$367$	−2.60076e19	−1.51393	−0.756964	+	0.653457i	$0.773318\pi$
$368$	0	0
$369$	9.77751e18	0.546425
$370$	0	0
$371$	−4.86692e17	−0.0261186
$372$	0	0
$373$	3.24522e18	0.167274	0.0836369	−	0.996496i	$-0.473346\pi$
$373$	3.24522e18	0.167274	0.0836369	+	0.996496i	$0.473346\pi$
$374$	0	0
$375$	−6.20842e17	−0.0307430
$376$	0	0
$377$	−2.14287e18	−0.101962
$378$	0	0
$379$	−3.97275e19	−1.81676	−0.908379	−	0.418148i	$-0.862679\pi$
$379$	−3.97275e19	−1.81676	−0.908379	+	0.418148i	$0.862679\pi$
$380$	0	0
$381$	−7.20933e18	−0.316926
$382$	0	0
$383$	1.39053e19	0.587747	0.293874	−	0.955844i	$-0.405056\pi$
$383$	1.39053e19	0.587747	0.293874	+	0.955844i	$0.405056\pi$
$384$	0	0
$385$	−4.91879e17	−0.0199941
$386$	0	0
$387$	−5.13733e18	−0.200866
$388$	0	0
$389$	−2.40519e19	−0.904749	−0.452374	−	0.891828i	$-0.649423\pi$
$389$	−2.40519e19	−0.904749	−0.452374	+	0.891828i	$0.649423\pi$
$390$	0	0
$391$	−1.69344e19	−0.612978
$392$	0	0
$393$	−1.01959e19	−0.355207
$394$	0	0
$395$	2.40048e19	0.805044
$396$	0	0
$397$	4.68747e17	0.0151359	0.00756797	−	0.999971i	$-0.497591\pi$
$397$	4.68747e17	0.0151359	0.00756797	+	0.999971i	$0.497591\pi$
$398$	0	0
$399$	5.98617e17	0.0186145
$400$	0	0
$401$	−4.59281e19	−1.37561	−0.687806	−	0.725894i	$-0.741426\pi$
$401$	−4.59281e19	−1.37561	−0.687806	+	0.725894i	$0.741426\pi$
$402$	0	0
$403$	8.66137e18	0.249919
$404$	0	0
$405$	−1.06087e19	−0.294952
$406$	0	0
$407$	5.15206e19	1.38046
$408$	0	0
$409$	−2.87071e18	−0.0741421	−0.0370710	−	0.999313i	$-0.511803\pi$
$409$	−2.87071e18	−0.0741421	−0.0370710	+	0.999313i	$0.511803\pi$
$410$	0	0
$411$	−2.45911e19	−0.612300
$412$	0	0
$413$	−8.38138e17	−0.0201229
$414$	0	0
$415$	3.37185e18	0.0780743
$416$	0	0
$417$	−7.92773e18	−0.177063
$418$	0	0
$419$	1.95181e19	0.420563	0.210282	−	0.977641i	$-0.432562\pi$
$419$	1.95181e19	0.420563	0.210282	+	0.977641i	$0.432562\pi$
$420$	0	0
$421$	7.84258e19	1.63058	0.815292	−	0.579049i	$-0.196576\pi$
$421$	7.84258e19	1.63058	0.815292	+	0.579049i	$0.196576\pi$
$422$	0	0
$423$	2.05371e19	0.412084
$424$	0	0
$425$	−8.37419e18	−0.162191
$426$	0	0
$427$	8.56544e16	0.00160155
$428$	0	0
$429$	−3.34903e18	−0.0604629
$430$	0	0
$431$	5.24106e19	0.913776	0.456888	−	0.889524i	$-0.348964\pi$
$431$	5.24106e19	0.913776	0.456888	+	0.889524i	$0.348964\pi$
$432$	0	0
$433$	6.72089e19	1.13179	0.565897	−	0.824476i	$-0.308530\pi$
$433$	6.72089e19	1.13179	0.565897	+	0.824476i	$0.308530\pi$
$434$	0	0
$435$	−5.88196e18	−0.0956869
$436$	0	0
$437$	6.25618e19	0.983329
$438$	0	0
$439$	6.41938e19	0.975010	0.487505	−	0.873120i	$-0.337907\pi$
$439$	6.41938e19	0.975010	0.487505	+	0.873120i	$0.337907\pi$
$440$	0	0
$441$	5.99701e19	0.880330
$442$	0	0
$443$	−8.09420e19	−1.14854	−0.574270	−	0.818666i	$-0.694714\pi$
$443$	−8.09420e19	−1.14854	−0.574270	+	0.818666i	$0.694714\pi$
$444$	0	0
$445$	−5.32561e19	−0.730582
$446$	0	0
$447$	−3.87797e19	−0.514397
$448$	0	0
$449$	−5.63441e19	−0.722771	−0.361386	−	0.932416i	$-0.617696\pi$
$449$	−5.63441e19	−0.722771	−0.361386	+	0.932416i	$0.617696\pi$
$450$	0	0
$451$	−5.36344e19	−0.665456
$452$	0	0
$453$	−4.47562e19	−0.537176
$454$	0	0
$455$	2.62604e17	0.00304941
$456$	0	0
$457$	8.12666e19	0.913147	0.456573	−	0.889686i	$-0.349077\pi$
$457$	8.12666e19	0.913147	0.456573	+	0.889686i	$0.349077\pi$
$458$	0	0
$459$	4.82369e19	0.524547
$460$	0	0
$461$	1.12727e20	1.18651	0.593255	−	0.805015i	$-0.297843\pi$
$461$	1.12727e20	1.18651	0.593255	+	0.805015i	$0.297843\pi$
$462$	0	0
$463$	1.79852e20	1.83256	0.916278	−	0.400542i	$-0.131178\pi$
$463$	1.79852e20	1.83256	0.916278	+	0.400542i	$0.131178\pi$
$464$	0	0
$465$	2.37745e19	0.234539
$466$	0	0
$467$	−3.50796e19	−0.335103	−0.167552	−	0.985863i	$-0.553586\pi$
$467$	−3.50796e19	−0.335103	−0.167552	+	0.985863i	$0.553586\pi$
$468$	0	0
$469$	6.40090e18	0.0592168
$470$	0	0
$471$	4.22726e19	0.378793
$472$	0	0
$473$	2.81807e19	0.244621
$474$	0	0
$475$	3.09373e19	0.260184
$476$	0	0
$477$	6.78947e19	0.553284
$478$	0	0
$479$	−1.32716e19	−0.104811	−0.0524053	−	0.998626i	$-0.516689\pi$
$479$	−1.32716e19	−0.104811	−0.0524053	+	0.998626i	$0.516689\pi$
$480$	0	0
$481$	−2.75058e19	−0.210541
$482$	0	0
$483$	1.45766e18	0.0108156
$484$	0	0
$485$	1.84781e19	0.132922
$486$	0	0
$487$	2.27726e20	1.58835	0.794173	−	0.607692i	$-0.207904\pi$
$487$	2.27726e20	1.58835	0.794173	+	0.607692i	$0.207904\pi$
$488$	0	0
$489$	−3.15587e19	−0.213453
$490$	0	0
$491$	−2.06518e19	−0.135471	−0.0677356	−	0.997703i	$-0.521577\pi$
$491$	−2.06518e19	−0.135471	−0.0677356	+	0.997703i	$0.521577\pi$
$492$	0	0
$493$	−7.93385e19	−0.504815
$494$	0	0
$495$	6.86183e19	0.423546
$496$	0	0
$497$	7.48639e18	0.0448332
$498$	0	0
$499$	−2.08538e20	−1.21180	−0.605901	−	0.795540i	$-0.707187\pi$
$499$	−2.08538e20	−1.21180	−0.605901	+	0.795540i	$0.707187\pi$
$500$	0	0
$501$	−4.24952e18	−0.0239638
$502$	0	0
$503$	−1.61975e20	−0.886516	−0.443258	−	0.896394i	$-0.646178\pi$
$503$	−1.61975e20	−0.886516	−0.443258	+	0.896394i	$0.646178\pi$
$504$	0	0
$505$	1.09096e20	0.579594
$506$	0	0
$507$	−6.48561e19	−0.334496
$508$	0	0
$509$	1.84945e20	0.926102	0.463051	−	0.886332i	$-0.346755\pi$
$509$	1.84945e20	0.926102	0.463051	+	0.886332i	$0.346755\pi$
$510$	0	0
$511$	−1.61849e19	−0.0786960
$512$	0	0
$513$	−1.78204e20	−0.841469
$514$	0	0
$515$	−1.14755e20	−0.526279
$516$	0	0
$517$	−1.12656e20	−0.501851
$518$	0	0
$519$	−1.24237e20	−0.537647
$520$	0	0
$521$	−1.19106e20	−0.500785	−0.250392	−	0.968144i	$-0.580560\pi$
$521$	−1.19106e20	−0.500785	−0.250392	+	0.968144i	$0.580560\pi$
$522$	0	0
$523$	2.07240e20	0.846663	0.423331	−	0.905975i	$-0.360861\pi$
$523$	2.07240e20	0.846663	0.423331	+	0.905975i	$0.360861\pi$
$524$	0	0
$525$	7.20820e17	0.00286175
$526$	0	0
$527$	3.20681e20	1.23735
$528$	0	0
$529$	−1.14295e20	−0.428656
$530$	0	0
$531$	1.16922e20	0.426273
$532$	0	0
$533$	2.86343e19	0.101492
$534$	0	0
$535$	9.37190e19	0.322979
$536$	0	0
$537$	1.40891e20	0.472146
$538$	0	0
$539$	−3.28965e20	−1.07210
$540$	0	0
$541$	−4.26438e20	−1.35169	−0.675844	−	0.737045i	$-0.736221\pi$
$541$	−4.26438e20	−1.35169	−0.675844	+	0.737045i	$0.736221\pi$
$542$	0	0
$543$	2.89023e19	0.0891114
$544$	0	0
$545$	−2.06183e19	−0.0618413
$546$	0	0
$547$	1.46134e20	0.426427	0.213214	−	0.977006i	$-0.431607\pi$
$547$	1.46134e20	0.426427	0.213214	+	0.977006i	$0.431607\pi$
$548$	0	0
$549$	−1.19490e19	−0.0339265
$550$	0	0
$551$	2.93105e20	0.809815
$552$	0	0
$553$	−2.78704e19	−0.0749386
$554$	0	0
$555$	−7.55005e19	−0.197584
$556$	0	0
$557$	3.01056e20	0.766891	0.383445	−	0.923564i	$-0.374737\pi$
$557$	3.01056e20	0.766891	0.383445	+	0.923564i	$0.374737\pi$
$558$	0	0
$559$	−1.50451e19	−0.0373084
$560$	0	0
$561$	−1.23996e20	−0.299354
$562$	0	0
$563$	−5.43653e20	−1.27793	−0.638967	−	0.769234i	$-0.720638\pi$
$563$	−5.43653e20	−1.27793	−0.638967	+	0.769234i	$0.720638\pi$
$564$	0	0
$565$	3.65103e20	0.835704
$566$	0	0
$567$	1.23171e19	0.0274560
$568$	0	0
$569$	2.77606e19	0.0602681	0.0301340	−	0.999546i	$-0.490407\pi$
$569$	2.77606e19	0.0602681	0.0301340	+	0.999546i	$0.490407\pi$
$570$	0	0
$571$	−7.86073e20	−1.66223	−0.831116	−	0.556098i	$-0.812298\pi$
$571$	−7.86073e20	−1.66223	−0.831116	+	0.556098i	$0.812298\pi$
$572$	0	0
$573$	4.45791e19	0.0918272
$574$	0	0
$575$	7.53334e19	0.151175
$576$	0	0
$577$	−9.30552e19	−0.181937	−0.0909687	−	0.995854i	$-0.528996\pi$
$577$	−9.30552e19	−0.181937	−0.0909687	+	0.995854i	$0.528996\pi$
$578$	0	0
$579$	−2.95725e20	−0.563376
$580$	0	0
$581$	−3.91485e18	−0.00726765
$582$	0	0
$583$	−3.72435e20	−0.673808
$584$	0	0
$585$	−3.66339e19	−0.0645973
$586$	0	0
$587$	−9.75604e20	−1.67682	−0.838412	−	0.545036i	$-0.816516\pi$
$587$	−9.75604e20	−1.67682	−0.838412	+	0.545036i	$0.816516\pi$
$588$	0	0
$589$	−1.18471e21	−1.98494
$590$	0	0
$591$	3.43559e19	0.0561171
$592$	0	0
$593$	1.11954e20	0.178292	0.0891459	−	0.996019i	$-0.471586\pi$
$593$	1.11954e20	0.178292	0.0891459	+	0.996019i	$0.471586\pi$
$594$	0	0
$595$	9.72275e18	0.0150977
$596$	0	0
$597$	−2.59740e20	−0.393307
$598$	0	0
$599$	7.49099e20	1.10621	0.553106	−	0.833111i	$-0.313442\pi$
$599$	7.49099e20	1.10621	0.553106	+	0.833111i	$0.313442\pi$
$600$	0	0
$601$	−1.59205e20	−0.229297	−0.114649	−	0.993406i	$-0.536574\pi$
$601$	−1.59205e20	−0.229297	−0.114649	+	0.993406i	$0.536574\pi$
$602$	0	0
$603$	−8.92941e20	−1.25442
$604$	0	0
$605$	−5.00569e19	−0.0685960
$606$	0	0
$607$	8.11583e20	1.08497	0.542485	−	0.840065i	$-0.317483\pi$
$607$	8.11583e20	1.08497	0.542485	+	0.840065i	$0.317483\pi$
$608$	0	0
$609$	6.82917e18	0.00890714
$610$	0	0
$611$	6.01447e19	0.0765399
$612$	0	0
$613$	5.55939e19	0.0690357	0.0345179	−	0.999404i	$-0.489010\pi$
$613$	5.55939e19	0.0690357	0.0345179	+	0.999404i	$0.489010\pi$
$614$	0	0
$615$	7.85981e19	0.0952464
$616$	0	0
$617$	−6.97947e19	−0.0825437	−0.0412718	−	0.999148i	$-0.513141\pi$
$617$	−6.97947e19	−0.0825437	−0.0412718	+	0.999148i	$0.513141\pi$
$618$	0	0
$619$	6.03520e20	0.696646	0.348323	−	0.937375i	$-0.386751\pi$
$619$	6.03520e20	0.696646	0.348323	+	0.937375i	$0.386751\pi$
$620$	0	0
$621$	−4.33935e20	−0.488920
$622$	0	0
$623$	6.18323e19	0.0680072
$624$	0	0
$625$	3.72529e19	0.0400000
$626$	0	0
$627$	4.58084e20	0.480218
$628$	0	0
$629$	−1.01838e21	−1.04239
$630$	0	0
$631$	3.64878e20	0.364693	0.182346	−	0.983234i	$-0.441631\pi$
$631$	3.64878e20	0.364693	0.182346	+	0.983234i	$0.441631\pi$
$632$	0	0
$633$	−8.40576e19	−0.0820445
$634$	0	0
$635$	4.32587e20	0.412355
$636$	0	0
$637$	1.75628e20	0.163511
$638$	0	0
$639$	−1.04437e21	−0.949725
$640$	0	0
$641$	2.06269e20	0.183231	0.0916156	−	0.995794i	$-0.470797\pi$
$641$	2.06269e20	0.183231	0.0916156	+	0.995794i	$0.470797\pi$
$642$	0	0
$643$	−2.11836e21	−1.83830	−0.919152	−	0.393903i	$-0.871125\pi$
$643$	−2.11836e21	−1.83830	−0.919152	+	0.393903i	$0.871125\pi$
$644$	0	0
$645$	−4.12972e19	−0.0350125
$646$	0	0
$647$	2.29485e21	1.90096	0.950480	−	0.310787i	$-0.100593\pi$
$647$	2.29485e21	1.90096	0.950480	+	0.310787i	$0.100593\pi$
$648$	0	0
$649$	−6.41375e20	−0.519130
$650$	0	0
$651$	−2.76031e19	−0.0218323
$652$	0	0
$653$	−5.84612e20	−0.451876	−0.225938	−	0.974142i	$-0.572545\pi$
$653$	−5.84612e20	−0.451876	−0.225938	+	0.974142i	$0.572545\pi$
$654$	0	0
$655$	6.11793e20	0.462163
$656$	0	0
$657$	2.25783e21	1.66706
$658$	0	0
$659$	−1.79922e21	−1.29850	−0.649252	−	0.760573i	$-0.724918\pi$
$659$	−1.79922e21	−1.29850	−0.649252	+	0.760573i	$0.724918\pi$
$660$	0	0
$661$	1.48620e21	1.04849	0.524247	−	0.851566i	$-0.324347\pi$
$661$	1.48620e21	1.04849	0.524247	+	0.851566i	$0.324347\pi$
$662$	0	0
$663$	6.61987e19	0.0456560
$664$	0	0
$665$	−3.59193e19	−0.0242195
$666$	0	0
$667$	7.13722e20	0.470528
$668$	0	0
$669$	8.74905e20	0.563982
$670$	0	0
$671$	6.55460e19	0.0413168
$672$	0	0
$673$	−1.01586e21	−0.626214	−0.313107	−	0.949718i	$-0.601370\pi$
$673$	−1.01586e21	−0.626214	−0.313107	+	0.949718i	$0.601370\pi$
$674$	0	0
$675$	−2.14584e20	−0.129366
$676$	0	0
$677$	2.64661e21	1.56054	0.780269	−	0.625445i	$-0.215082\pi$
$677$	2.64661e21	1.56054	0.780269	+	0.625445i	$0.215082\pi$
$678$	0	0
$679$	−2.14538e19	−0.0123732
$680$	0	0
$681$	7.04323e20	0.397346
$682$	0	0
$683$	1.97807e21	1.09166	0.545830	−	0.837896i	$-0.316214\pi$
$683$	1.97807e21	1.09166	0.545830	+	0.837896i	$0.316214\pi$
$684$	0	0
$685$	1.47556e21	0.796668
$686$	0	0
$687$	9.79517e20	0.517412
$688$	0	0
$689$	1.98835e20	0.102766
$690$	0	0
$691$	−2.52634e21	−1.27763	−0.638817	−	0.769359i	$-0.720576\pi$
$691$	−2.52634e21	−1.27763	−0.638817	+	0.769359i	$0.720576\pi$
$692$	0	0
$693$	−7.96684e19	−0.0394264
$694$	0	0
$695$	4.75694e20	0.230378
$696$	0	0
$697$	1.06017e21	0.502491
$698$	0	0
$699$	−9.95744e20	−0.461922
$700$	0	0
$701$	−2.18514e21	−0.992190	−0.496095	−	0.868268i	$-0.665233\pi$
$701$	−2.18514e21	−0.992190	−0.496095	+	0.868268i	$0.665233\pi$
$702$	0	0
$703$	3.76228e21	1.67219
$704$	0	0
$705$	1.65091e20	0.0718297
$706$	0	0
$707$	−1.26664e20	−0.0539522
$708$	0	0
$709$	−2.66228e19	−0.0111021	−0.00555107	−	0.999985i	$-0.501767\pi$
$709$	−2.66228e19	−0.0111021	−0.00555107	+	0.999985i	$0.501767\pi$
$710$	0	0
$711$	3.88799e21	1.58746
$712$	0	0
$713$	−2.88482e21	−1.15331
$714$	0	0
$715$	2.00955e20	0.0786688
$716$	0	0
$717$	8.77682e20	0.336468
$718$	0	0
$719$	−9.86146e20	−0.370233	−0.185116	−	0.982717i	$-0.559266\pi$
$719$	−9.86146e20	−0.370233	−0.185116	+	0.982717i	$0.559266\pi$
$720$	0	0
$721$	1.33234e20	0.0489894
$722$	0	0
$723$	4.12653e20	0.148610
$724$	0	0
$725$	3.52940e20	0.124499
$726$	0	0
$727$	2.40230e21	0.830078	0.415039	−	0.909804i	$-0.363768\pi$
$727$	2.40230e21	0.830078	0.415039	+	0.909804i	$0.363768\pi$
$728$	0	0
$729$	−1.06145e21	−0.359288
$730$	0	0
$731$	−5.57036e20	−0.184715
$732$	0	0
$733$	1.87874e21	0.610361	0.305181	−	0.952295i	$-0.401283\pi$
$733$	1.87874e21	0.610361	0.305181	+	0.952295i	$0.401283\pi$
$734$	0	0
$735$	4.82079e20	0.153449
$736$	0	0
$737$	4.89821e21	1.52768
$738$	0	0
$739$	−5.37988e21	−1.64414	−0.822072	−	0.569384i	$-0.807182\pi$
$739$	−5.37988e21	−1.64414	−0.822072	+	0.569384i	$0.807182\pi$
$740$	0	0
$741$	−2.44562e20	−0.0732406
$742$	0	0
$743$	2.67434e21	0.784876	0.392438	−	0.919778i	$-0.371632\pi$
$743$	2.67434e21	0.784876	0.392438	+	0.919778i	$0.371632\pi$
$744$	0	0
$745$	2.32693e21	0.669286
$746$	0	0
$747$	5.46130e20	0.153954
$748$	0	0
$749$	−1.08811e20	−0.0300649
$750$	0	0
$751$	−4.11797e21	−1.11528	−0.557640	−	0.830083i	$-0.688293\pi$
$751$	−4.11797e21	−1.11528	−0.557640	+	0.830083i	$0.688293\pi$
$752$	0	0
$753$	2.01764e20	0.0535649
$754$	0	0
$755$	2.68554e21	0.698924
$756$	0	0
$757$	−3.83231e21	−0.977781	−0.488890	−	0.872345i	$-0.662598\pi$
$757$	−3.83231e21	−0.977781	−0.488890	+	0.872345i	$0.662598\pi$
$758$	0	0
$759$	1.11545e21	0.279022
$760$	0	0
$761$	3.19772e21	0.784253	0.392126	−	0.919911i	$-0.371740\pi$
$761$	3.19772e21	0.784253	0.392126	+	0.919911i	$0.371740\pi$
$762$	0	0
$763$	2.39386e19	0.00575658
$764$	0	0
$765$	−1.35635e21	−0.319823
$766$	0	0
$767$	3.42417e20	0.0791753
$768$	0	0
$769$	−5.42196e21	−1.22944	−0.614722	−	0.788744i	$-0.710732\pi$
$769$	−5.42196e21	−1.22944	−0.614722	+	0.788744i	$0.710732\pi$
$770$	0	0
$771$	2.93280e21	0.652191
$772$	0	0
$773$	−4.69088e21	−1.02308	−0.511538	−	0.859261i	$-0.670924\pi$
$773$	−4.69088e21	−1.02308	−0.511538	+	0.859261i	$0.670924\pi$
$774$	0	0
$775$	−1.42656e21	−0.305160
$776$	0	0
$777$	8.76589e19	0.0183924
$778$	0	0
$779$	−3.91663e21	−0.806087
$780$	0	0
$781$	5.72887e21	1.15661
$782$	0	0
$783$	−2.03300e21	−0.402647
$784$	0	0
$785$	−2.53652e21	−0.492851
$786$	0	0
$787$	9.27131e21	1.76738	0.883691	−	0.468070i	$-0.155051\pi$
$787$	9.27131e21	1.76738	0.883691	+	0.468070i	$0.155051\pi$
$788$	0	0
$789$	−1.31405e21	−0.245773
$790$	0	0
$791$	−4.23899e20	−0.0777926
$792$	0	0
$793$	−3.49937e19	−0.00630145
$794$	0	0
$795$	5.45782e20	0.0964419
$796$	0	0
$797$	8.35314e21	1.44848	0.724240	−	0.689548i	$-0.242191\pi$
$797$	8.35314e21	1.44848	0.724240	+	0.689548i	$0.242191\pi$
$798$	0	0
$799$	2.22682e21	0.378951
$800$	0	0
$801$	−8.62575e21	−1.44063
$802$	0	0
$803$	−1.23853e22	−2.03020
$804$	0	0
$805$	−8.74649e19	−0.0140723
$806$	0	0
$807$	−1.21062e21	−0.191187
$808$	0	0
$809$	−5.86882e21	−0.909781	−0.454890	−	0.890547i	$-0.650322\pi$
$809$	−5.86882e21	−0.909781	−0.454890	+	0.890547i	$0.650322\pi$
$810$	0	0
$811$	−9.20718e21	−1.40110	−0.700552	−	0.713602i	$-0.747063\pi$
$811$	−9.20718e21	−1.40110	−0.700552	+	0.713602i	$0.747063\pi$
$812$	0	0
$813$	8.48091e19	0.0126696
$814$	0	0
$815$	1.89364e21	0.277725
$816$	0	0
$817$	2.05789e21	0.296317
$818$	0	0
$819$	4.25333e19	0.00601312
$820$	0	0
$821$	2.78078e21	0.386005	0.193003	−	0.981198i	$-0.438177\pi$
$821$	2.78078e21	0.386005	0.193003	+	0.981198i	$0.438177\pi$
$822$	0	0
$823$	−1.08620e22	−1.48051	−0.740257	−	0.672324i	$-0.765296\pi$
$823$	−1.08620e22	−1.48051	−0.740257	+	0.672324i	$0.765296\pi$
$824$	0	0
$825$	5.51599e20	0.0738276
$826$	0	0
$827$	4.48033e21	0.588869	0.294435	−	0.955672i	$-0.404869\pi$
$827$	4.48033e21	0.588869	0.294435	+	0.955672i	$0.404869\pi$
$828$	0	0
$829$	−1.15562e22	−1.49161	−0.745807	−	0.666163i	$-0.767936\pi$
$829$	−1.15562e22	−1.49161	−0.745807	+	0.666163i	$0.767936\pi$
$830$	0	0
$831$	−1.13824e20	−0.0144287
$832$	0	0
$833$	6.50250e21	0.809548
$834$	0	0
$835$	2.54987e20	0.0311796
$836$	0	0
$837$	8.21727e21	0.986930
$838$	0	0
$839$	−1.16410e22	−1.37333	−0.686666	−	0.726973i	$-0.740926\pi$
$839$	−1.16410e22	−1.37333	−0.686666	+	0.726973i	$0.740926\pi$
$840$	0	0
$841$	−5.28537e21	−0.612500
$842$	0	0
$843$	1.43981e21	0.163907
$844$	0	0
$845$	3.89161e21	0.435215
$846$	0	0
$847$	5.81179e19	0.00638535
$848$	0	0
$849$	−2.80138e21	−0.302387
$850$	0	0
$851$	9.16129e21	0.971594
$852$	0	0
$853$	−8.05337e21	−0.839189	−0.419595	−	0.907712i	$-0.637828\pi$
$853$	−8.05337e21	−0.839189	−0.419595	+	0.907712i	$0.637828\pi$
$854$	0	0
$855$	5.01083e21	0.513055
$856$	0	0
$857$	3.51278e21	0.353423	0.176712	−	0.984263i	$-0.443454\pi$
$857$	3.51278e21	0.353423	0.176712	+	0.984263i	$0.443454\pi$
$858$	0	0
$859$	−4.79620e21	−0.474186	−0.237093	−	0.971487i	$-0.576195\pi$
$859$	−4.79620e21	−0.474186	−0.237093	+	0.971487i	$0.576195\pi$
$860$	0	0
$861$	−9.12552e19	−0.00886613
$862$	0	0
$863$	−6.27772e21	−0.599406	−0.299703	−	0.954033i	$-0.596888\pi$
$863$	−6.27772e21	−0.599406	−0.299703	+	0.954033i	$0.596888\pi$
$864$	0	0
$865$	7.45472e21	0.699537
$866$	0	0
$867$	−1.27591e21	−0.117673
$868$	0	0
$869$	−2.13275e22	−1.93327
$870$	0	0
$871$	−2.61506e21	−0.232994
$872$	0	0
$873$	2.99286e21	0.262107
$874$	0	0
$875$	−4.32520e19	−0.00372345
$876$	0	0
$877$	−4.69963e21	−0.397711	−0.198855	−	0.980029i	$-0.563722\pi$
$877$	−4.69963e21	−0.397711	−0.198855	+	0.980029i	$0.563722\pi$
$878$	0	0
$879$	7.19136e21	0.598266
$880$	0	0
$881$	−1.89876e22	−1.55293	−0.776465	−	0.630161i	$-0.782989\pi$
$881$	−1.89876e22	−1.55293	−0.776465	+	0.630161i	$0.782989\pi$
$882$	0	0
$883$	5.43744e21	0.437209	0.218605	−	0.975814i	$-0.429849\pi$
$883$	5.43744e21	0.437209	0.218605	+	0.975814i	$0.429849\pi$
$884$	0	0
$885$	9.39897e20	0.0743029
$886$	0	0
$887$	1.43529e22	1.11561	0.557807	−	0.829971i	$-0.311643\pi$
$887$	1.43529e22	1.11561	0.557807	+	0.829971i	$0.311643\pi$
$888$	0	0
$889$	−5.02250e20	−0.0383846
$890$	0	0
$891$	9.42554e21	0.708310
$892$	0	0
$893$	−8.22665e21	−0.607907
$894$	0	0
$895$	−8.45401e21	−0.614314
$896$	0	0
$897$	−5.95517e20	−0.0425551
$898$	0	0
$899$	−1.35155e22	−0.949804
$900$	0	0
$901$	7.36175e21	0.508798
$902$	0	0
$903$	4.79476e19	0.00325918
$904$	0	0
$905$	−1.73425e21	−0.115944
$906$	0	0
$907$	1.44404e22	0.949564	0.474782	−	0.880103i	$-0.342527\pi$
$907$	1.44404e22	0.949564	0.474782	+	0.880103i	$0.342527\pi$
$908$	0	0
$909$	1.76700e22	1.14290
$910$	0	0
$911$	6.00661e21	0.382157	0.191078	−	0.981575i	$-0.438802\pi$
$911$	6.00661e21	0.382157	0.191078	+	0.981575i	$0.438802\pi$
$912$	0	0
$913$	−2.99579e21	−0.187491
$914$	0	0
$915$	−9.60538e19	−0.00591366
$916$	0	0
$917$	−7.10315e20	−0.0430210
$918$	0	0
$919$	−2.25875e22	−1.34586	−0.672932	−	0.739704i	$-0.734966\pi$
$919$	−2.25875e22	−1.34586	−0.672932	+	0.739704i	$0.734966\pi$
$920$	0	0
$921$	−1.28584e21	−0.0753770
$922$	0	0
$923$	−3.05853e21	−0.176400
$924$	0	0
$925$	4.53032e21	0.257079
$926$	0	0
$927$	−1.85865e22	−1.03777
$928$	0	0
$929$	3.26962e22	1.79630	0.898151	−	0.439686i	$-0.144910\pi$
$929$	3.26962e22	1.79630	0.898151	+	0.439686i	$0.144910\pi$
$930$	0	0
$931$	−2.40225e22	−1.29866
$932$	0	0
$933$	4.03157e21	0.214468
$934$	0	0
$935$	7.44021e21	0.389492
$936$	0	0
$937$	2.67083e21	0.137594	0.0687970	−	0.997631i	$-0.478084\pi$
$937$	2.67083e21	0.137594	0.0687970	+	0.997631i	$0.478084\pi$
$938$	0	0
$939$	1.55401e21	0.0787881
$940$	0	0
$941$	−2.61162e22	−1.30313	−0.651565	−	0.758593i	$-0.725887\pi$
$941$	−2.61162e22	−1.30313	−0.651565	+	0.758593i	$0.725887\pi$
$942$	0	0
$943$	−9.53715e21	−0.468361
$944$	0	0
$945$	2.49140e20	0.0120422
$946$	0	0
$947$	1.03859e22	0.494105	0.247053	−	0.969002i	$-0.420538\pi$
$947$	1.03859e22	0.494105	0.247053	+	0.969002i	$0.420538\pi$
$948$	0	0
$949$	6.61225e21	0.309637
$950$	0	0
$951$	−3.51681e21	−0.162104
$952$	0	0
$953$	1.16131e20	0.00526929	0.00263464	−	0.999997i	$-0.499161\pi$
$953$	1.16131e20	0.00526929	0.00263464	+	0.999997i	$0.499161\pi$
$954$	0	0
$955$	−2.67492e21	−0.119477
$956$	0	0
$957$	5.22594e21	0.229787
$958$	0	0
$959$	−1.71318e21	−0.0741589
$960$	0	0
$961$	3.11635e22	1.32807
$962$	0	0
$963$	1.51794e22	0.636881
$964$	0	0
$965$	1.77446e22	0.733013
$966$	0	0
$967$	3.55619e22	1.44639	0.723197	−	0.690642i	$-0.242672\pi$
$967$	3.55619e22	1.44639	0.723197	+	0.690642i	$0.242672\pi$
$968$	0	0
$969$	−9.05473e21	−0.362616
$970$	0	0
$971$	4.08883e22	1.61233	0.806166	−	0.591690i	$-0.201539\pi$
$971$	4.08883e22	1.61233	0.806166	+	0.591690i	$0.201539\pi$
$972$	0	0
$973$	−5.52298e20	−0.0214451
$974$	0	0
$975$	−2.94487e20	−0.0112598
$976$	0	0
$977$	1.47056e22	0.553700	0.276850	−	0.960913i	$-0.410709\pi$
$977$	1.47056e22	0.553700	0.276850	+	0.960913i	$0.410709\pi$
$978$	0	0
$979$	4.73164e22	1.75445
$980$	0	0
$981$	−3.33949e21	−0.121945
$982$	0	0
$983$	2.02443e22	0.728034	0.364017	−	0.931392i	$-0.381405\pi$
$983$	2.02443e22	0.728034	0.364017	+	0.931392i	$0.381405\pi$
$984$	0	0
$985$	−2.06148e21	−0.0730144
$986$	0	0
$987$	−1.91676e20	−0.00668636
$988$	0	0
$989$	5.01104e21	0.172169
$990$	0	0
$991$	−1.99356e22	−0.674649	−0.337324	−	0.941389i	$-0.609522\pi$
$991$	−1.99356e22	−0.674649	−0.337324	+	0.941389i	$0.609522\pi$
$992$	0	0
$993$	1.17825e21	0.0392753
$994$	0	0
$995$	1.55854e22	0.511735
$996$	0	0
$997$	−4.00115e21	−0.129411	−0.0647056	−	0.997904i	$-0.520611\pi$
$997$	−4.00115e21	−0.129411	−0.0647056	+	0.997904i	$0.520611\pi$
$998$	0	0
$999$	−2.60955e22	−0.831427

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.16.a.b.1.1		1
4.3	odd	2		10.16.a.c.1.1	✓	1
12.11	even	2		90.16.a.d.1.1		1
20.3	even	4		50.16.b.b.49.1		2
20.7	even	4		50.16.b.b.49.2		2
20.19	odd	2		50.16.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.16.a.c.1.1	✓	1	4.3	odd	2
50.16.a.a.1.1		1	20.19	odd	2
50.16.b.b.49.1		2	20.3	even	4
50.16.b.b.49.2		2	20.7	even	4
80.16.a.b.1.1		1	1.1	even	1	trivial
90.16.a.d.1.1		1	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	80.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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