LMFDB - Embedded newform 80.10.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [80,10,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, 10, names="a")

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

chi := DirichletCharacter("80.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 80 = 2^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 10 $
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:	$41.2028668931$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{46}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 46 $ x^2 - 46
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{3}\cdot 3 $
Twist minimal:	no (minimal twist has level 40)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$6.78233$ of defining polynomial
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)

$f(q)$	$=$	$q+108.776 q^{3} -625.000 q^{5} +2570.09 q^{7} -7850.80 q^{9} +O(q^{10})$	$q+108.776 q^{3} -625.000 q^{5} +2570.09 q^{7} -7850.80 q^{9} +9251.97 q^{11} -15727.2 q^{13} -67984.9 q^{15} +77471.9 q^{17} -510747. q^{19} +279564. q^{21} -1.54785e6 q^{23} +390625. q^{25} -2.99501e6 q^{27} +66275.5 q^{29} -2.44676e6 q^{31} +1.00639e6 q^{33} -1.60630e6 q^{35} +9.93458e6 q^{37} -1.71074e6 q^{39} +8.93602e6 q^{41} -7.40844e6 q^{43} +4.90675e6 q^{45} -4.23348e7 q^{47} -3.37483e7 q^{49} +8.42707e6 q^{51} -9.26398e7 q^{53} -5.78248e6 q^{55} -5.55570e7 q^{57} +5.25366e7 q^{59} -1.93659e8 q^{61} -2.01772e7 q^{63} +9.82950e6 q^{65} -1.34688e8 q^{67} -1.68369e8 q^{69} -8.98365e7 q^{71} -3.66295e7 q^{73} +4.24906e7 q^{75} +2.37784e7 q^{77} -1.00757e8 q^{79} -1.71258e8 q^{81} -3.39285e8 q^{83} -4.84199e7 q^{85} +7.20918e6 q^{87} +1.03368e9 q^{89} -4.04203e7 q^{91} -2.66148e8 q^{93} +3.19217e8 q^{95} +7.87506e8 q^{97} -7.26354e7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 108 q^{3} - 1250 q^{5} + 908 q^{7} + 19458 q^{9}+O(q^{10}) $ 2 * q - 108 * q^3 - 1250 * q^5 + 908 * q^7 + 19458 * q^9	$ 2 q - 108 q^{3} - 1250 q^{5} + 908 q^{7} + 19458 q^{9} - 25120 q^{11} + 146948 q^{13} + 67500 q^{15} + 169268 q^{17} + 25480 q^{19} + 639864 q^{21} - 1782748 q^{23} + 781250 q^{25} - 4648104 q^{27} + 7323340 q^{29} - 10677272 q^{31} + 8457408 q^{33} - 567500 q^{35} - 5750460 q^{37} - 36974808 q^{39} - 7795764 q^{41} - 16770524 q^{43} - 12161250 q^{45} - 15393892 q^{47} - 71339334 q^{49} - 11472120 q^{51} - 58529292 q^{53} + 15700000 q^{55} - 171798192 q^{57} - 59618264 q^{59} - 188163772 q^{61} - 65566836 q^{63} - 91842500 q^{65} + 105998252 q^{67} - 117448344 q^{69} + 168665592 q^{71} - 390212412 q^{73} - 42187500 q^{75} + 80907584 q^{77} - 466946256 q^{79} - 350427222 q^{81} - 329535164 q^{83} - 105792500 q^{85} - 1565947656 q^{87} - 108334860 q^{89} - 310800616 q^{91} + 1518028560 q^{93} - 15925000 q^{95} + 2043058628 q^{97} - 1011292704 q^{99}+O(q^{100}) $ 2 * q - 108 * q^3 - 1250 * q^5 + 908 * q^7 + 19458 * q^9 - 25120 * q^11 + 146948 * q^13 + 67500 * q^15 + 169268 * q^17 + 25480 * q^19 + 639864 * q^21 - 1782748 * q^23 + 781250 * q^25 - 4648104 * q^27 + 7323340 * q^29 - 10677272 * q^31 + 8457408 * q^33 - 567500 * q^35 - 5750460 * q^37 - 36974808 * q^39 - 7795764 * q^41 - 16770524 * q^43 - 12161250 * q^45 - 15393892 * q^47 - 71339334 * q^49 - 11472120 * q^51 - 58529292 * q^53 + 15700000 * q^55 - 171798192 * q^57 - 59618264 * q^59 - 188163772 * q^61 - 65566836 * q^63 - 91842500 * q^65 + 105998252 * q^67 - 117448344 * q^69 + 168665592 * q^71 - 390212412 * q^73 - 42187500 * q^75 + 80907584 * q^77 - 466946256 * q^79 - 350427222 * q^81 - 329535164 * q^83 - 105792500 * q^85 - 1565947656 * q^87 - 108334860 * q^89 - 310800616 * q^91 + 1518028560 * q^93 - 15925000 * q^95 + 2043058628 * q^97 - 1011292704 * 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
$3$	108.776	0.775331	0.387665	−	0.921800i	$-0.373282\pi$
$3$	108.776	0.775331	0.387665	+	0.921800i	$0.373282\pi$
$4$	0	0
$5$	−625.000	−0.447214
$5$	−625.000	−0.447214
$6$	0	0
$7$	2570.09	0.404582	0.202291	−	0.979325i	$-0.435161\pi$
$7$	2570.09	0.404582	0.202291	+	0.979325i	$0.435161\pi$
$8$	0	0
$9$	−7850.80	−0.398862
$10$	0	0
$11$	9251.97	0.190532	0.0952659	−	0.995452i	$-0.469630\pi$
$11$	9251.97	0.190532	0.0952659	+	0.995452i	$0.469630\pi$
$12$	0	0
$13$	−15727.2	−0.152724	−0.0763618	−	0.997080i	$-0.524330\pi$
$13$	−15727.2	−0.152724	−0.0763618	+	0.997080i	$0.524330\pi$
$14$	0	0
$15$	−67984.9	−0.346739
$16$	0	0
$17$	77471.9	0.224970	0.112485	−	0.993653i	$-0.464119\pi$
$17$	77471.9	0.224970	0.112485	+	0.993653i	$0.464119\pi$
$18$	0	0
$19$	−510747.	−0.899114	−0.449557	−	0.893252i	$-0.648418\pi$
$19$	−510747.	−0.899114	−0.449557	+	0.893252i	$0.648418\pi$
$20$	0	0
$21$	279564.	0.313685
$22$	0	0
$23$	−1.54785e6	−1.15333	−0.576665	−	0.816981i	$-0.695646\pi$
$23$	−1.54785e6	−1.15333	−0.576665	+	0.816981i	$0.695646\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	0	0
$27$	−2.99501e6	−1.08458
$28$	0	0
$29$	66275.5	0.0174005	0.00870025	−	0.999962i	$-0.497231\pi$
$29$	66275.5	0.0174005	0.00870025	+	0.999962i	$0.497231\pi$
$30$	0	0
$31$	−2.44676e6	−0.475843	−0.237921	−	0.971284i	$-0.576466\pi$
$31$	−2.44676e6	−0.475843	−0.237921	+	0.971284i	$0.576466\pi$
$32$	0	0
$33$	1.00639e6	0.147725
$34$	0	0
$35$	−1.60630e6	−0.180935
$36$	0	0
$37$	9.93458e6	0.871449	0.435724	−	0.900080i	$-0.356492\pi$
$37$	9.93458e6	0.871449	0.435724	+	0.900080i	$0.356492\pi$
$38$	0	0
$39$	−1.71074e6	−0.118411
$40$	0	0
$41$	8.93602e6	0.493875	0.246937	−	0.969031i	$-0.420576\pi$
$41$	8.93602e6	0.493875	0.246937	+	0.969031i	$0.420576\pi$
$42$	0	0
$43$	−7.40844e6	−0.330460	−0.165230	−	0.986255i	$-0.552837\pi$
$43$	−7.40844e6	−0.330460	−0.165230	+	0.986255i	$0.552837\pi$
$44$	0	0
$45$	4.90675e6	0.178376
$46$	0	0
$47$	−4.23348e7	−1.26549	−0.632743	−	0.774362i	$-0.718071\pi$
$47$	−4.23348e7	−1.26549	−0.632743	+	0.774362i	$0.718071\pi$
$48$	0	0
$49$	−3.37483e7	−0.836313
$50$	0	0
$51$	8.42707e6	0.174426
$52$	0	0
$53$	−9.26398e7	−1.61271	−0.806355	−	0.591432i	$-0.798563\pi$
$53$	−9.26398e7	−1.61271	−0.806355	+	0.591432i	$0.798563\pi$
$54$	0	0
$55$	−5.78248e6	−0.0852084
$56$	0	0
$57$	−5.55570e7	−0.697111
$58$	0	0
$59$	5.25366e7	0.564453	0.282227	−	0.959348i	$-0.408927\pi$
$59$	5.25366e7	0.564453	0.282227	+	0.959348i	$0.408927\pi$
$60$	0	0
$61$	−1.93659e8	−1.79083	−0.895413	−	0.445236i	$-0.853120\pi$
$61$	−1.93659e8	−1.79083	−0.895413	+	0.445236i	$0.853120\pi$
$62$	0	0
$63$	−2.01772e7	−0.161372
$64$	0	0
$65$	9.82950e6	0.0683001
$66$	0	0
$67$	−1.34688e8	−0.816565	−0.408283	−	0.912856i	$-0.633872\pi$
$67$	−1.34688e8	−0.816565	−0.408283	+	0.912856i	$0.633872\pi$
$68$	0	0
$69$	−1.68369e8	−0.894212
$70$	0	0
$71$	−8.98365e7	−0.419556	−0.209778	−	0.977749i	$-0.567274\pi$
$71$	−8.98365e7	−0.419556	−0.209778	+	0.977749i	$0.567274\pi$
$72$	0	0
$73$	−3.66295e7	−0.150966	−0.0754829	−	0.997147i	$-0.524050\pi$
$73$	−3.66295e7	−0.150966	−0.0754829	+	0.997147i	$0.524050\pi$
$74$	0	0
$75$	4.24906e7	0.155066
$76$	0	0
$77$	2.37784e7	0.0770857
$78$	0	0
$79$	−1.00757e8	−0.291040	−0.145520	−	0.989355i	$-0.546485\pi$
$79$	−1.00757e8	−0.291040	−0.145520	+	0.989355i	$0.546485\pi$
$80$	0	0
$81$	−1.71258e8	−0.442047
$82$	0	0
$83$	−3.39285e8	−0.784717	−0.392358	−	0.919812i	$-0.628341\pi$
$83$	−3.39285e8	−0.784717	−0.392358	+	0.919812i	$0.628341\pi$
$84$	0	0
$85$	−4.84199e7	−0.100609
$86$	0	0
$87$	7.20918e6	0.0134912
$88$	0	0
$89$	1.03368e9	1.74635	0.873174	−	0.487408i	$-0.162058\pi$
$89$	1.03368e9	1.74635	0.873174	+	0.487408i	$0.162058\pi$
$90$	0	0
$91$	−4.04203e7	−0.0617893
$92$	0	0
$93$	−2.66148e8	−0.368936
$94$	0	0
$95$	3.19217e8	0.402096
$96$	0	0
$97$	7.87506e8	0.903194	0.451597	−	0.892222i	$-0.350854\pi$
$97$	7.87506e8	0.903194	0.451597	+	0.892222i	$0.350854\pi$
$98$	0	0
$99$	−7.26354e7	−0.0759958
$100$	0	0
$101$	−1.74591e8	−0.166946	−0.0834729	−	0.996510i	$-0.526601\pi$
$101$	−1.74591e8	−0.166946	−0.0834729	+	0.996510i	$0.526601\pi$
$102$	0	0
$103$	−1.05496e9	−0.923564	−0.461782	−	0.886993i	$-0.652790\pi$
$103$	−1.05496e9	−0.923564	−0.461782	+	0.886993i	$0.652790\pi$
$104$	0	0
$105$	−1.74727e8	−0.140284
$106$	0	0
$107$	−2.20574e8	−0.162677	−0.0813386	−	0.996687i	$-0.525920\pi$
$107$	−2.20574e8	−0.162677	−0.0813386	+	0.996687i	$0.525920\pi$
$108$	0	0
$109$	2.01691e9	1.36857	0.684285	−	0.729215i	$-0.260115\pi$
$109$	2.01691e9	1.36857	0.684285	+	0.729215i	$0.260115\pi$
$110$	0	0
$111$	1.08064e9	0.675661
$112$	0	0
$113$	7.73490e8	0.446274	0.223137	−	0.974787i	$-0.428370\pi$
$113$	7.73490e8	0.446274	0.223137	+	0.974787i	$0.428370\pi$
$114$	0	0
$115$	9.67406e8	0.515785
$116$	0	0
$117$	1.23471e8	0.0609157
$118$	0	0
$119$	1.99109e8	0.0910187
$120$	0	0
$121$	−2.27235e9	−0.963698
$122$	0	0
$123$	9.72024e8	0.382916
$124$	0	0
$125$	−2.44141e8	−0.0894427
$126$	0	0
$127$	1.54910e8	0.0528398	0.0264199	−	0.999651i	$-0.491589\pi$
$127$	1.54910e8	0.0528398	0.0264199	+	0.999651i	$0.491589\pi$
$128$	0	0
$129$	−8.05860e8	−0.256216
$130$	0	0
$131$	6.30550e9	1.87068	0.935338	−	0.353754i	$-0.115095\pi$
$131$	6.30550e9	1.87068	0.935338	+	0.353754i	$0.115095\pi$
$132$	0	0
$133$	−1.31267e9	−0.363766
$134$	0	0
$135$	1.87188e9	0.485039
$136$	0	0
$137$	4.41067e9	1.06970	0.534850	−	0.844947i	$-0.320368\pi$
$137$	4.41067e9	1.06970	0.534850	+	0.844947i	$0.320368\pi$
$138$	0	0
$139$	3.97342e9	0.902814	0.451407	−	0.892318i	$-0.350922\pi$
$139$	3.97342e9	0.902814	0.451407	+	0.892318i	$0.350922\pi$
$140$	0	0
$141$	−4.60501e9	−0.981171
$142$	0	0
$143$	−1.45508e8	−0.0290987
$144$	0	0
$145$	−4.14222e7	−0.00778174
$146$	0	0
$147$	−3.67100e9	−0.648420
$148$	0	0
$149$	4.64678e9	0.772350	0.386175	−	0.922426i	$-0.373796\pi$
$149$	4.64678e9	0.772350	0.386175	+	0.922426i	$0.373796\pi$
$150$	0	0
$151$	4.13148e9	0.646709	0.323355	−	0.946278i	$-0.395189\pi$
$151$	4.13148e9	0.646709	0.323355	+	0.946278i	$0.395189\pi$
$152$	0	0
$153$	−6.08216e8	−0.0897318
$154$	0	0
$155$	1.52922e9	0.212803
$156$	0	0
$157$	1.43532e10	1.88538	0.942690	−	0.333670i	$-0.108287\pi$
$157$	1.43532e10	1.88538	0.942690	+	0.333670i	$0.108287\pi$
$158$	0	0
$159$	−1.00770e10	−1.25038
$160$	0	0
$161$	−3.97811e9	−0.466616
$162$	0	0
$163$	3.03607e9	0.336874	0.168437	−	0.985712i	$-0.446128\pi$
$163$	3.03607e9	0.336874	0.168437	+	0.985712i	$0.446128\pi$
$164$	0	0
$165$	−6.28995e8	−0.0660647
$166$	0	0
$167$	−1.66473e10	−1.65623	−0.828113	−	0.560561i	$-0.810586\pi$
$167$	−1.66473e10	−1.65623	−0.828113	+	0.560561i	$0.810586\pi$
$168$	0	0
$169$	−1.03572e10	−0.976675
$170$	0	0
$171$	4.00978e9	0.358622
$172$	0	0
$173$	3.55392e9	0.301648	0.150824	−	0.988561i	$-0.451807\pi$
$173$	3.55392e9	0.301648	0.150824	+	0.988561i	$0.451807\pi$
$174$	0	0
$175$	1.00394e9	0.0809164
$176$	0	0
$177$	5.71472e9	0.437638
$178$	0	0
$179$	−1.35153e10	−0.983979	−0.491990	−	0.870601i	$-0.663730\pi$
$179$	−1.35153e10	−0.983979	−0.491990	+	0.870601i	$0.663730\pi$
$180$	0	0
$181$	−1.74861e10	−1.21098	−0.605492	−	0.795851i	$-0.707024\pi$
$181$	−1.74861e10	−1.21098	−0.605492	+	0.795851i	$0.707024\pi$
$182$	0	0
$183$	−2.10654e10	−1.38848
$184$	0	0
$185$	−6.20911e9	−0.389724
$186$	0	0
$187$	7.16768e8	0.0428638
$188$	0	0
$189$	−7.69745e9	−0.438802
$190$	0	0
$191$	−7.82637e9	−0.425511	−0.212755	−	0.977106i	$-0.568244\pi$
$191$	−7.82637e9	−0.425511	−0.212755	+	0.977106i	$0.568244\pi$
$192$	0	0
$193$	−3.50211e9	−0.181686	−0.0908431	−	0.995865i	$-0.528956\pi$
$193$	−3.50211e9	−0.181686	−0.0908431	+	0.995865i	$0.528956\pi$
$194$	0	0
$195$	1.06921e9	0.0529552
$196$	0	0
$197$	1.83251e10	0.866858	0.433429	−	0.901188i	$-0.357303\pi$
$197$	1.83251e10	0.866858	0.433429	+	0.901188i	$0.357303\pi$
$198$	0	0
$199$	−3.19364e10	−1.44360	−0.721800	−	0.692102i	$-0.756685\pi$
$199$	−3.19364e10	−1.44360	−0.721800	+	0.692102i	$0.756685\pi$
$200$	0	0
$201$	−1.46508e10	−0.633108
$202$	0	0
$203$	1.70334e8	0.00703993
$204$	0	0
$205$	−5.58501e9	−0.220868
$206$	0	0
$207$	1.21519e10	0.460019
$208$	0	0
$209$	−4.72542e9	−0.171310
$210$	0	0
$211$	3.95237e10	1.37273	0.686367	−	0.727255i	$-0.259204\pi$
$211$	3.95237e10	1.37273	0.686367	+	0.727255i	$0.259204\pi$
$212$	0	0
$213$	−9.77204e9	−0.325295
$214$	0	0
$215$	4.63028e9	0.147786
$216$	0	0
$217$	−6.28838e9	−0.192517
$218$	0	0
$219$	−3.98441e9	−0.117048
$220$	0	0
$221$	−1.21842e9	−0.0343582
$222$	0	0
$223$	1.95302e10	0.528853	0.264427	−	0.964406i	$-0.414817\pi$
$223$	1.95302e10	0.528853	0.264427	+	0.964406i	$0.414817\pi$
$224$	0	0
$225$	−3.06672e9	−0.0797724
$226$	0	0
$227$	6.76270e10	1.69045	0.845227	−	0.534407i	$-0.179465\pi$
$227$	6.76270e10	1.69045	0.845227	+	0.534407i	$0.179465\pi$
$228$	0	0
$229$	−3.75826e10	−0.903082	−0.451541	−	0.892250i	$-0.649125\pi$
$229$	−3.75826e10	−0.903082	−0.451541	+	0.892250i	$0.649125\pi$
$230$	0	0
$231$	2.58651e9	0.0597669
$232$	0	0
$233$	2.30519e10	0.512395	0.256197	−	0.966624i	$-0.417530\pi$
$233$	2.30519e10	0.512395	0.256197	+	0.966624i	$0.417530\pi$
$234$	0	0
$235$	2.64593e10	0.565943
$236$	0	0
$237$	−1.09599e10	−0.225652
$238$	0	0
$239$	−5.93741e10	−1.17708	−0.588540	−	0.808468i	$-0.700297\pi$
$239$	−5.93741e10	−1.17708	−0.588540	+	0.808468i	$0.700297\pi$
$240$	0	0
$241$	3.82138e10	0.729699	0.364850	−	0.931066i	$-0.381120\pi$
$241$	3.82138e10	0.729699	0.364850	+	0.931066i	$0.381120\pi$
$242$	0	0
$243$	4.03221e10	0.741848
$244$	0	0
$245$	2.10927e10	0.374011
$246$	0	0
$247$	8.03263e9	0.137316
$248$	0	0
$249$	−3.69060e10	−0.608415
$250$	0	0
$251$	−4.14452e10	−0.659087	−0.329543	−	0.944140i	$-0.606895\pi$
$251$	−4.14452e10	−0.659087	−0.329543	+	0.944140i	$0.606895\pi$
$252$	0	0
$253$	−1.43207e10	−0.219746
$254$	0	0
$255$	−5.26692e9	−0.0780056
$256$	0	0
$257$	1.01455e11	1.45069	0.725344	−	0.688386i	$-0.241681\pi$
$257$	1.01455e11	1.45069	0.725344	+	0.688386i	$0.241681\pi$
$258$	0	0
$259$	2.55327e10	0.352573
$260$	0	0
$261$	−5.20316e8	−0.00694040
$262$	0	0
$263$	−1.15290e11	−1.48590	−0.742952	−	0.669345i	$-0.766575\pi$
$263$	−1.15290e11	−1.48590	−0.742952	+	0.669345i	$0.766575\pi$
$264$	0	0
$265$	5.78999e10	0.721226
$266$	0	0
$267$	1.12439e11	1.35400
$268$	0	0
$269$	3.28063e10	0.382008	0.191004	−	0.981589i	$-0.438826\pi$
$269$	3.28063e10	0.382008	0.191004	+	0.981589i	$0.438826\pi$
$270$	0	0
$271$	−9.59659e10	−1.08082	−0.540412	−	0.841400i	$-0.681732\pi$
$271$	−9.59659e10	−1.08082	−0.540412	+	0.841400i	$0.681732\pi$
$272$	0	0
$273$	−4.39675e9	−0.0479071
$274$	0	0
$275$	3.61405e9	0.0381063
$276$	0	0
$277$	7.63563e10	0.779267	0.389633	−	0.920970i	$-0.372602\pi$
$277$	7.63563e10	0.779267	0.389633	+	0.920970i	$0.372602\pi$
$278$	0	0
$279$	1.92090e10	0.189796
$280$	0	0
$281$	1.86577e11	1.78517	0.892586	−	0.450878i	$-0.148889\pi$
$281$	1.86577e11	1.78517	0.892586	+	0.450878i	$0.148889\pi$
$282$	0	0
$283$	2.66169e10	0.246672	0.123336	−	0.992365i	$-0.460641\pi$
$283$	2.66169e10	0.246672	0.123336	+	0.992365i	$0.460641\pi$
$284$	0	0
$285$	3.47231e10	0.311758
$286$	0	0
$287$	2.29664e10	0.199813
$288$	0	0
$289$	−1.12586e11	−0.949389
$290$	0	0
$291$	8.56617e10	0.700275
$292$	0	0
$293$	−3.08895e10	−0.244853	−0.122427	−	0.992478i	$-0.539068\pi$
$293$	−3.08895e10	−0.244853	−0.122427	+	0.992478i	$0.539068\pi$
$294$	0	0
$295$	−3.28354e10	−0.252431
$296$	0	0
$297$	−2.77098e10	−0.206647
$298$	0	0
$299$	2.43433e10	0.176141
$300$	0	0
$301$	−1.90403e10	−0.133698
$302$	0	0
$303$	−1.89913e10	−0.129438
$304$	0	0
$305$	1.21037e11	0.800882
$306$	0	0
$307$	2.55079e11	1.63890	0.819449	−	0.573152i	$-0.194280\pi$
$307$	2.55079e11	1.63890	0.819449	+	0.573152i	$0.194280\pi$
$308$	0	0
$309$	−1.14754e11	−0.716068
$310$	0	0
$311$	−2.86687e11	−1.73774	−0.868872	−	0.495036i	$-0.835155\pi$
$311$	−2.86687e11	−1.73774	−0.868872	+	0.495036i	$0.835155\pi$
$312$	0	0
$313$	−2.53253e11	−1.49144	−0.745718	−	0.666262i	$-0.767893\pi$
$313$	−2.53253e11	−1.49144	−0.745718	+	0.666262i	$0.767893\pi$
$314$	0	0
$315$	1.26108e10	0.0721679
$316$	0	0
$317$	1.91250e11	1.06374	0.531869	−	0.846827i	$-0.321490\pi$
$317$	1.91250e11	1.06374	0.531869	+	0.846827i	$0.321490\pi$
$318$	0	0
$319$	6.13179e8	0.00331535
$320$	0	0
$321$	−2.39931e10	−0.126129
$322$	0	0
$323$	−3.95685e10	−0.202273
$324$	0	0
$325$	−6.14344e9	−0.0305447
$326$	0	0
$327$	2.19391e11	1.06109
$328$	0	0
$329$	−1.08804e11	−0.511993
$330$	0	0
$331$	−2.58825e11	−1.18517	−0.592583	−	0.805509i	$-0.701892\pi$
$331$	−2.58825e11	−1.18517	−0.592583	+	0.805509i	$0.701892\pi$
$332$	0	0
$333$	−7.79944e10	−0.347588
$334$	0	0
$335$	8.41797e10	0.365179
$336$	0	0
$337$	−3.49169e11	−1.47469	−0.737347	−	0.675515i	$-0.763922\pi$
$337$	−3.49169e11	−1.47469	−0.737347	+	0.675515i	$0.763922\pi$
$338$	0	0
$339$	8.41371e10	0.346010
$340$	0	0
$341$	−2.26373e10	−0.0906632
$342$	0	0
$343$	−1.90448e11	−0.742939
$344$	0	0
$345$	1.05230e11	0.399904
$346$	0	0
$347$	1.67410e11	0.619867	0.309933	−	0.950758i	$-0.399693\pi$
$347$	1.67410e11	0.619867	0.309933	+	0.950758i	$0.399693\pi$
$348$	0	0
$349$	−3.00153e11	−1.08300	−0.541500	−	0.840701i	$-0.682143\pi$
$349$	−3.00153e11	−1.08300	−0.541500	+	0.840701i	$0.682143\pi$
$350$	0	0
$351$	4.71032e10	0.165641
$352$	0	0
$353$	−3.04604e11	−1.04412	−0.522059	−	0.852910i	$-0.674836\pi$
$353$	−3.04604e11	−1.04412	−0.522059	+	0.852910i	$0.674836\pi$
$354$	0	0
$355$	5.61478e10	0.187631
$356$	0	0
$357$	2.16583e10	0.0705696
$358$	0	0
$359$	−3.61955e11	−1.15008	−0.575042	−	0.818124i	$-0.695014\pi$
$359$	−3.61955e11	−1.15008	−0.575042	+	0.818124i	$0.695014\pi$
$360$	0	0
$361$	−6.18248e10	−0.191593
$362$	0	0
$363$	−2.47177e11	−0.747185
$364$	0	0
$365$	2.28935e10	0.0675139
$366$	0	0
$367$	2.27133e11	0.653557	0.326778	−	0.945101i	$-0.394037\pi$
$367$	2.27133e11	0.653557	0.326778	+	0.945101i	$0.394037\pi$
$368$	0	0
$369$	−7.01549e10	−0.196988
$370$	0	0
$371$	−2.38092e11	−0.652474
$372$	0	0
$373$	−1.89711e10	−0.0507460	−0.0253730	−	0.999678i	$-0.508077\pi$
$373$	−1.89711e10	−0.0507460	−0.0253730	+	0.999678i	$0.508077\pi$
$374$	0	0
$375$	−2.65566e10	−0.0693477
$376$	0	0
$377$	−1.04233e9	−0.00265747
$378$	0	0
$379$	−6.86720e11	−1.70964	−0.854818	−	0.518929i	$-0.826331\pi$
$379$	−6.86720e11	−1.70964	−0.854818	+	0.518929i	$0.826331\pi$
$380$	0	0
$381$	1.68504e10	0.0409684
$382$	0	0
$383$	−1.01644e11	−0.241371	−0.120686	−	0.992691i	$-0.538509\pi$
$383$	−1.01644e11	−0.241371	−0.120686	+	0.992691i	$0.538509\pi$
$384$	0	0
$385$	−1.48615e10	−0.0344738
$386$	0	0
$387$	5.81622e10	0.131808
$388$	0	0
$389$	2.42170e11	0.536226	0.268113	−	0.963388i	$-0.413600\pi$
$389$	2.42170e11	0.536226	0.268113	+	0.963388i	$0.413600\pi$
$390$	0	0
$391$	−1.19915e11	−0.259464
$392$	0	0
$393$	6.85887e11	1.45039
$394$	0	0
$395$	6.29729e10	0.130157
$396$	0	0
$397$	5.48896e11	1.10900	0.554502	−	0.832182i	$-0.312909\pi$
$397$	5.48896e11	1.10900	0.554502	+	0.832182i	$0.312909\pi$
$398$	0	0
$399$	−1.42786e11	−0.282039
$400$	0	0
$401$	4.84257e11	0.935247	0.467623	−	0.883928i	$-0.345110\pi$
$401$	4.84257e11	0.935247	0.467623	+	0.883928i	$0.345110\pi$
$402$	0	0
$403$	3.84807e10	0.0726725
$404$	0	0
$405$	1.07036e11	0.197690
$406$	0	0
$407$	9.19145e10	0.166039
$408$	0	0
$409$	−2.42077e11	−0.427759	−0.213880	−	0.976860i	$-0.568610\pi$
$409$	−2.42077e11	−0.427759	−0.213880	+	0.976860i	$0.568610\pi$
$410$	0	0
$411$	4.79775e11	0.829372
$412$	0	0
$413$	1.35024e11	0.228368
$414$	0	0
$415$	2.12053e11	0.350936
$416$	0	0
$417$	4.32213e11	0.699979
$418$	0	0
$419$	2.42716e11	0.384711	0.192355	−	0.981325i	$-0.438387\pi$
$419$	2.42716e11	0.384711	0.192355	+	0.981325i	$0.438387\pi$
$420$	0	0
$421$	−5.24744e11	−0.814100	−0.407050	−	0.913406i	$-0.633443\pi$
$421$	−5.24744e11	−0.814100	−0.407050	+	0.913406i	$0.633443\pi$
$422$	0	0
$423$	3.32362e11	0.504755
$424$	0	0
$425$	3.02624e10	0.0449939
$426$	0	0
$427$	−4.97721e11	−0.724536
$428$	0	0
$429$	−1.58277e10	−0.0225611
$430$	0	0
$431$	1.12716e12	1.57340	0.786698	−	0.617338i	$-0.211789\pi$
$431$	1.12716e12	1.57340	0.786698	+	0.617338i	$0.211789\pi$
$432$	0	0
$433$	−5.16558e11	−0.706194	−0.353097	−	0.935587i	$-0.614871\pi$
$433$	−5.16558e11	−0.706194	−0.353097	+	0.935587i	$0.614871\pi$
$434$	0	0
$435$	−4.50574e9	−0.00603343
$436$	0	0
$437$	7.90560e11	1.03698
$438$	0	0
$439$	−8.41124e10	−0.108086	−0.0540430	−	0.998539i	$-0.517211\pi$
$439$	−8.41124e10	−0.108086	−0.0540430	+	0.998539i	$0.517211\pi$
$440$	0	0
$441$	2.64951e11	0.333574
$442$	0	0
$443$	6.08740e11	0.750957	0.375478	−	0.926831i	$-0.377478\pi$
$443$	6.08740e11	0.750957	0.375478	+	0.926831i	$0.377478\pi$
$444$	0	0
$445$	−6.46050e11	−0.780991
$446$	0	0
$447$	5.05458e11	0.598827
$448$	0	0
$449$	−3.98270e11	−0.462454	−0.231227	−	0.972900i	$-0.574274\pi$
$449$	−3.98270e11	−0.462454	−0.231227	+	0.972900i	$0.574274\pi$
$450$	0	0
$451$	8.26758e10	0.0940988
$452$	0	0
$453$	4.49405e11	0.501414
$454$	0	0
$455$	2.52627e10	0.0276330
$456$	0	0
$457$	6.05041e11	0.648876	0.324438	−	0.945907i	$-0.394825\pi$
$457$	6.05041e11	0.648876	0.324438	+	0.945907i	$0.394825\pi$
$458$	0	0
$459$	−2.32029e11	−0.243998
$460$	0	0
$461$	−1.38199e12	−1.42512	−0.712560	−	0.701611i	$-0.752464\pi$
$461$	−1.38199e12	−1.42512	−0.712560	+	0.701611i	$0.752464\pi$
$462$	0	0
$463$	2.97456e11	0.300821	0.150410	−	0.988624i	$-0.451941\pi$
$463$	2.97456e11	0.300821	0.150410	+	0.988624i	$0.451941\pi$
$464$	0	0
$465$	1.66343e11	0.164993
$466$	0	0
$467$	−7.91234e11	−0.769802	−0.384901	−	0.922958i	$-0.625764\pi$
$467$	−7.91234e11	−0.769802	−0.384901	+	0.922958i	$0.625764\pi$
$468$	0	0
$469$	−3.46159e11	−0.330368
$470$	0	0
$471$	1.56128e12	1.46179
$472$	0	0
$473$	−6.85427e10	−0.0629631
$474$	0	0
$475$	−1.99511e11	−0.179823
$476$	0	0
$477$	7.27297e11	0.643249
$478$	0	0
$479$	9.22160e11	0.800381	0.400190	−	0.916432i	$-0.368944\pi$
$479$	9.22160e11	0.800381	0.400190	+	0.916432i	$0.368944\pi$
$480$	0	0
$481$	−1.56243e11	−0.133091
$482$	0	0
$483$	−4.32722e11	−0.361782
$484$	0	0
$485$	−4.92191e11	−0.403921
$486$	0	0
$487$	−6.94283e11	−0.559315	−0.279657	−	0.960100i	$-0.590221\pi$
$487$	−6.94283e11	−0.559315	−0.279657	+	0.960100i	$0.590221\pi$
$488$	0	0
$489$	3.30251e11	0.261189
$490$	0	0
$491$	6.59800e10	0.0512325	0.0256163	−	0.999672i	$-0.491845\pi$
$491$	6.59800e10	0.0512325	0.0256163	+	0.999672i	$0.491845\pi$
$492$	0	0
$493$	5.13449e9	0.00391459
$494$	0	0
$495$	4.53971e10	0.0339864
$496$	0	0
$497$	−2.30888e11	−0.169745
$498$	0	0
$499$	−1.30514e12	−0.942331	−0.471166	−	0.882045i	$-0.656167\pi$
$499$	−1.30514e12	−0.942331	−0.471166	+	0.882045i	$0.656167\pi$
$500$	0	0
$501$	−1.81083e12	−1.28412
$502$	0	0
$503$	1.44209e12	1.00447	0.502233	−	0.864732i	$-0.332512\pi$
$503$	1.44209e12	1.00447	0.502233	+	0.864732i	$0.332512\pi$
$504$	0	0
$505$	1.09119e11	0.0746604
$506$	0	0
$507$	−1.12661e12	−0.757247
$508$	0	0
$509$	1.79176e12	1.18318	0.591590	−	0.806239i	$-0.298501\pi$
$509$	1.79176e12	1.18318	0.591590	+	0.806239i	$0.298501\pi$
$510$	0	0
$511$	−9.41411e10	−0.0610780
$512$	0	0
$513$	1.52970e12	0.975162
$514$	0	0
$515$	6.59348e11	0.413030
$516$	0	0
$517$	−3.91681e11	−0.241115
$518$	0	0
$519$	3.86581e11	0.233877
$520$	0	0
$521$	4.38474e11	0.260720	0.130360	−	0.991467i	$-0.458387\pi$
$521$	4.38474e11	0.260720	0.130360	+	0.991467i	$0.458387\pi$
$522$	0	0
$523$	−1.83422e12	−1.07200	−0.535999	−	0.844219i	$-0.680065\pi$
$523$	−1.83422e12	−1.07200	−0.535999	+	0.844219i	$0.680065\pi$
$524$	0	0
$525$	1.09205e11	0.0627370
$526$	0	0
$527$	−1.89555e11	−0.107050
$528$	0	0
$529$	5.94685e11	0.330169
$530$	0	0
$531$	−4.12454e11	−0.225139
$532$	0	0
$533$	−1.40539e11	−0.0754264
$534$	0	0
$535$	1.37859e11	0.0727515
$536$	0	0
$537$	−1.47014e12	−0.762910
$538$	0	0
$539$	−3.12238e11	−0.159344
$540$	0	0
$541$	1.02360e12	0.513741	0.256870	−	0.966446i	$-0.417309\pi$
$541$	1.02360e12	0.513741	0.256870	+	0.966446i	$0.417309\pi$
$542$	0	0
$543$	−1.90206e12	−0.938914
$544$	0	0
$545$	−1.26057e12	−0.612043
$546$	0	0
$547$	−3.78808e12	−1.80916	−0.904579	−	0.426307i	$-0.859814\pi$
$547$	−3.78808e12	−1.80916	−0.904579	+	0.426307i	$0.859814\pi$
$548$	0	0
$549$	1.52038e12	0.714293
$550$	0	0
$551$	−3.38500e10	−0.0156450
$552$	0	0
$553$	−2.58954e11	−0.117749
$554$	0	0
$555$	−6.75402e11	−0.302165
$556$	0	0
$557$	−9.90336e11	−0.435948	−0.217974	−	0.975955i	$-0.569945\pi$
$557$	−9.90336e11	−0.435948	−0.217974	+	0.975955i	$0.569945\pi$
$558$	0	0
$559$	1.16514e11	0.0504691
$560$	0	0
$561$	7.79671e10	0.0332337
$562$	0	0
$563$	6.55168e11	0.274830	0.137415	−	0.990514i	$-0.456121\pi$
$563$	6.55168e11	0.274830	0.137415	+	0.990514i	$0.456121\pi$
$564$	0	0
$565$	−4.83431e11	−0.199580
$566$	0	0
$567$	−4.40148e11	−0.178844
$568$	0	0
$569$	8.66683e11	0.346621	0.173311	−	0.984867i	$-0.444554\pi$
$569$	8.66683e11	0.346621	0.173311	+	0.984867i	$0.444554\pi$
$570$	0	0
$571$	−3.20349e12	−1.26113	−0.630566	−	0.776135i	$-0.717177\pi$
$571$	−3.20349e12	−1.26113	−0.630566	+	0.776135i	$0.717177\pi$
$572$	0	0
$573$	−8.51321e11	−0.329912
$574$	0	0
$575$	−6.04629e11	−0.230666
$576$	0	0
$577$	−4.45477e12	−1.67315	−0.836573	−	0.547855i	$-0.815445\pi$
$577$	−4.45477e12	−1.67315	−0.836573	+	0.547855i	$0.815445\pi$
$578$	0	0
$579$	−3.80945e11	−0.140867
$580$	0	0
$581$	−8.71991e11	−0.317482
$582$	0	0
$583$	−8.57101e11	−0.307272
$584$	0	0
$585$	−7.71695e10	−0.0272423
$586$	0	0
$587$	2.76564e12	0.961443	0.480721	−	0.876873i	$-0.340375\pi$
$587$	2.76564e12	0.961443	0.480721	+	0.876873i	$0.340375\pi$
$588$	0	0
$589$	1.24968e12	0.427837
$590$	0	0
$591$	1.99333e12	0.672102
$592$	0	0
$593$	−3.04035e12	−1.00966	−0.504832	−	0.863218i	$-0.668446\pi$
$593$	−3.04035e12	−1.00966	−0.504832	+	0.863218i	$0.668446\pi$
$594$	0	0
$595$	−1.24443e11	−0.0407048
$596$	0	0
$597$	−3.47391e12	−1.11927
$598$	0	0
$599$	6.11443e12	1.94060	0.970299	−	0.241910i	$-0.0777739\pi$
$599$	6.11443e12	1.94060	0.970299	+	0.241910i	$0.0777739\pi$
$600$	0	0
$601$	−1.11504e12	−0.348623	−0.174312	−	0.984691i	$-0.555770\pi$
$601$	−1.11504e12	−0.348623	−0.174312	+	0.984691i	$0.555770\pi$
$602$	0	0
$603$	1.05740e12	0.325697
$604$	0	0
$605$	1.42022e12	0.430979
$606$	0	0
$607$	−1.91395e12	−0.572244	−0.286122	−	0.958193i	$-0.592366\pi$
$607$	−1.91395e12	−0.572244	−0.286122	+	0.958193i	$0.592366\pi$
$608$	0	0
$609$	1.85282e10	0.00545828
$610$	0	0
$611$	6.65809e11	0.193270
$612$	0	0
$613$	−8.43474e11	−0.241268	−0.120634	−	0.992697i	$-0.538493\pi$
$613$	−8.43474e11	−0.241268	−0.120634	+	0.992697i	$0.538493\pi$
$614$	0	0
$615$	−6.07515e11	−0.171245
$616$	0	0
$617$	−1.92088e12	−0.533602	−0.266801	−	0.963752i	$-0.585967\pi$
$617$	−1.92088e12	−0.533602	−0.266801	+	0.963752i	$0.585967\pi$
$618$	0	0
$619$	5.88086e12	1.61003	0.805013	−	0.593257i	$-0.202158\pi$
$619$	5.88086e12	1.61003	0.805013	+	0.593257i	$0.202158\pi$
$620$	0	0
$621$	4.63583e12	1.25088
$622$	0	0
$623$	2.65665e12	0.706541
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	0	0
$627$	−5.14012e11	−0.132822
$628$	0	0
$629$	7.69651e11	0.196050
$630$	0	0
$631$	−2.74699e12	−0.689804	−0.344902	−	0.938639i	$-0.612088\pi$
$631$	−2.74699e12	−0.689804	−0.344902	+	0.938639i	$0.612088\pi$
$632$	0	0
$633$	4.29922e12	1.06432
$634$	0	0
$635$	−9.68185e10	−0.0236307
$636$	0	0
$637$	5.30766e11	0.127725
$638$	0	0
$639$	7.05288e11	0.167345
$640$	0	0
$641$	2.23697e12	0.523358	0.261679	−	0.965155i	$-0.415724\pi$
$641$	2.23697e12	0.523358	0.261679	+	0.965155i	$0.415724\pi$
$642$	0	0
$643$	2.69028e12	0.620652	0.310326	−	0.950630i	$-0.399562\pi$
$643$	2.69028e12	0.620652	0.310326	+	0.950630i	$0.399562\pi$
$644$	0	0
$645$	5.03663e11	0.114583
$646$	0	0
$647$	2.70149e12	0.606085	0.303043	−	0.952977i	$-0.401998\pi$
$647$	2.70149e12	0.606085	0.303043	+	0.952977i	$0.401998\pi$
$648$	0	0
$649$	4.86067e11	0.107546
$650$	0	0
$651$	−6.84025e11	−0.149265
$652$	0	0
$653$	4.06920e12	0.875790	0.437895	−	0.899026i	$-0.355724\pi$
$653$	4.06920e12	0.875790	0.437895	+	0.899026i	$0.355724\pi$
$654$	0	0
$655$	−3.94094e12	−0.836592
$656$	0	0
$657$	2.87571e11	0.0602145
$658$	0	0
$659$	4.45718e12	0.920611	0.460305	−	0.887761i	$-0.347740\pi$
$659$	4.45718e12	0.920611	0.460305	+	0.887761i	$0.347740\pi$
$660$	0	0
$661$	1.34009e12	0.273041	0.136521	−	0.990637i	$-0.456408\pi$
$661$	1.34009e12	0.273041	0.136521	+	0.990637i	$0.456408\pi$
$662$	0	0
$663$	−1.32534e11	−0.0266390
$664$	0	0
$665$	8.20416e11	0.162681
$666$	0	0
$667$	−1.02584e11	−0.0200685
$668$	0	0
$669$	2.12442e12	0.410036
$670$	0	0
$671$	−1.79173e12	−0.341209
$672$	0	0
$673$	−3.39056e11	−0.0637094	−0.0318547	−	0.999493i	$-0.510141\pi$
$673$	−3.39056e11	−0.0637094	−0.0318547	+	0.999493i	$0.510141\pi$
$674$	0	0
$675$	−1.16993e12	−0.216916
$676$	0	0
$677$	−2.59404e12	−0.474600	−0.237300	−	0.971436i	$-0.576262\pi$
$677$	−2.59404e12	−0.474600	−0.237300	+	0.971436i	$0.576262\pi$
$678$	0	0
$679$	2.02396e12	0.365416
$680$	0	0
$681$	7.35619e12	1.31066
$682$	0	0
$683$	8.24479e12	1.44973	0.724864	−	0.688892i	$-0.241903\pi$
$683$	8.24479e12	1.44973	0.724864	+	0.688892i	$0.241903\pi$
$684$	0	0
$685$	−2.75667e12	−0.478385
$686$	0	0
$687$	−4.08808e12	−0.700187
$688$	0	0
$689$	1.45697e12	0.246299
$690$	0	0
$691$	−6.27189e12	−1.04652	−0.523259	−	0.852173i	$-0.675284\pi$
$691$	−6.27189e12	−1.04652	−0.523259	+	0.852173i	$0.675284\pi$
$692$	0	0
$693$	−1.86679e11	−0.0307466
$694$	0	0
$695$	−2.48339e12	−0.403751
$696$	0	0
$697$	6.92290e11	0.111107
$698$	0	0
$699$	2.50749e12	0.397276
$700$	0	0
$701$	6.28225e12	0.982616	0.491308	−	0.870986i	$-0.336519\pi$
$701$	6.28225e12	0.982616	0.491308	+	0.870986i	$0.336519\pi$
$702$	0	0
$703$	−5.07406e12	−0.783532
$704$	0	0
$705$	2.87813e12	0.438793
$706$	0	0
$707$	−4.48714e11	−0.0675433
$708$	0	0
$709$	−6.61259e12	−0.982796	−0.491398	−	0.870935i	$-0.663514\pi$
$709$	−6.61259e12	−0.982796	−0.491398	+	0.870935i	$0.663514\pi$
$710$	0	0
$711$	7.91021e11	0.116085
$712$	0	0
$713$	3.78721e12	0.548804
$714$	0	0
$715$	9.09423e10	0.0130133
$716$	0	0
$717$	−6.45847e12	−0.912627
$718$	0	0
$719$	−8.37720e12	−1.16901	−0.584506	−	0.811390i	$-0.698712\pi$
$719$	−8.37720e12	−1.16901	−0.584506	+	0.811390i	$0.698712\pi$
$720$	0	0
$721$	−2.71133e12	−0.373657
$722$	0	0
$723$	4.15674e12	0.565758
$724$	0	0
$725$	2.58889e10	0.00348010
$726$	0	0
$727$	1.08045e12	0.143450	0.0717250	−	0.997424i	$-0.477150\pi$
$727$	1.08045e12	0.143450	0.0717250	+	0.997424i	$0.477150\pi$
$728$	0	0
$729$	7.75695e12	1.01722
$730$	0	0
$731$	−5.73946e11	−0.0743435
$732$	0	0
$733$	3.43980e12	0.440114	0.220057	−	0.975487i	$-0.429376\pi$
$733$	3.43980e12	0.440114	0.220057	+	0.975487i	$0.429376\pi$
$734$	0	0
$735$	2.29437e12	0.289982
$736$	0	0
$737$	−1.24613e12	−0.155582
$738$	0	0
$739$	−2.69367e12	−0.332234	−0.166117	−	0.986106i	$-0.553123\pi$
$739$	−2.69367e12	−0.332234	−0.166117	+	0.986106i	$0.553123\pi$
$740$	0	0
$741$	8.73756e11	0.106465
$742$	0	0
$743$	−6.62213e12	−0.797165	−0.398582	−	0.917132i	$-0.630498\pi$
$743$	−6.62213e12	−0.797165	−0.398582	+	0.917132i	$0.630498\pi$
$744$	0	0
$745$	−2.90424e12	−0.345405
$746$	0	0
$747$	2.66366e12	0.312994
$748$	0	0
$749$	−5.66894e11	−0.0658163
$750$	0	0
$751$	−1.12044e12	−0.128532	−0.0642658	−	0.997933i	$-0.520471\pi$
$751$	−1.12044e12	−0.128532	−0.0642658	+	0.997933i	$0.520471\pi$
$752$	0	0
$753$	−4.50824e12	−0.511010
$754$	0	0
$755$	−2.58217e12	−0.289217
$756$	0	0
$757$	−1.33327e13	−1.47566	−0.737831	−	0.674985i	$-0.764150\pi$
$757$	−1.33327e13	−1.47566	−0.737831	+	0.674985i	$0.764150\pi$
$758$	0	0
$759$	−1.55774e12	−0.170376
$760$	0	0
$761$	−6.19570e12	−0.669668	−0.334834	−	0.942277i	$-0.608680\pi$
$761$	−6.19570e12	−0.669668	−0.334834	+	0.942277i	$0.608680\pi$
$762$	0	0
$763$	5.18363e12	0.553699
$764$	0	0
$765$	3.80135e11	0.0401293
$766$	0	0
$767$	−8.26254e11	−0.0862054
$768$	0	0
$769$	−1.15631e13	−1.19236	−0.596180	−	0.802851i	$-0.703315\pi$
$769$	−1.15631e13	−1.19236	−0.596180	+	0.802851i	$0.703315\pi$
$770$	0	0
$771$	1.10359e13	1.12476
$772$	0	0
$773$	−1.13061e12	−0.113895	−0.0569477	−	0.998377i	$-0.518137\pi$
$773$	−1.13061e12	−0.113895	−0.0569477	+	0.998377i	$0.518137\pi$
$774$	0	0
$775$	−9.55765e11	−0.0951686
$776$	0	0
$777$	2.77735e12	0.273360
$778$	0	0
$779$	−4.56405e12	−0.444050
$780$	0	0
$781$	−8.31165e11	−0.0799387
$782$	0	0
$783$	−1.98496e11	−0.0188723
$784$	0	0
$785$	−8.97072e12	−0.843168
$786$	0	0
$787$	5.47647e12	0.508879	0.254440	−	0.967089i	$-0.418109\pi$
$787$	5.47647e12	0.508879	0.254440	+	0.967089i	$0.418109\pi$
$788$	0	0
$789$	−1.25408e13	−1.15207
$790$	0	0
$791$	1.98794e12	0.180555
$792$	0	0
$793$	3.04572e12	0.273502
$794$	0	0
$795$	6.29811e12	0.559189
$796$	0	0
$797$	4.60260e12	0.404055	0.202028	−	0.979380i	$-0.435247\pi$
$797$	4.60260e12	0.404055	0.202028	+	0.979380i	$0.435247\pi$
$798$	0	0
$799$	−3.27976e12	−0.284696
$800$	0	0
$801$	−8.11521e12	−0.696552
$802$	0	0
$803$	−3.38895e11	−0.0287638
$804$	0	0
$805$	2.48632e12	0.208677
$806$	0	0
$807$	3.56854e12	0.296183
$808$	0	0
$809$	9.55665e12	0.784399	0.392200	−	0.919880i	$-0.371714\pi$
$809$	9.55665e12	0.784399	0.392200	+	0.919880i	$0.371714\pi$
$810$	0	0
$811$	3.00510e12	0.243930	0.121965	−	0.992534i	$-0.461081\pi$
$811$	3.00510e12	0.243930	0.121965	+	0.992534i	$0.461081\pi$
$812$	0	0
$813$	−1.04388e13	−0.837997
$814$	0	0
$815$	−1.89754e12	−0.150655
$816$	0	0
$817$	3.78384e12	0.297121
$818$	0	0
$819$	3.17332e11	0.0246454
$820$	0	0
$821$	9.91597e10	0.00761712	0.00380856	−	0.999993i	$-0.498788\pi$
$821$	9.91597e10	0.00761712	0.00380856	+	0.999993i	$0.498788\pi$
$822$	0	0
$823$	−2.24519e13	−1.70590	−0.852950	−	0.521993i	$-0.825189\pi$
$823$	−2.24519e13	−1.70590	−0.852950	+	0.521993i	$0.825189\pi$
$824$	0	0
$825$	3.93122e11	0.0295450
$826$	0	0
$827$	−1.24385e13	−0.924686	−0.462343	−	0.886701i	$-0.652991\pi$
$827$	−1.24385e13	−0.924686	−0.462343	+	0.886701i	$0.652991\pi$
$828$	0	0
$829$	4.88385e12	0.359143	0.179571	−	0.983745i	$-0.442529\pi$
$829$	4.88385e12	0.359143	0.179571	+	0.983745i	$0.442529\pi$
$830$	0	0
$831$	8.30573e12	0.604189
$832$	0	0
$833$	−2.61454e12	−0.188145
$834$	0	0
$835$	1.04046e13	0.740687
$836$	0	0
$837$	7.32808e12	0.516090
$838$	0	0
$839$	−1.97525e13	−1.37624	−0.688118	−	0.725599i	$-0.741563\pi$
$839$	−1.97525e13	−1.37624	−0.688118	+	0.725599i	$0.741563\pi$
$840$	0	0
$841$	−1.45028e13	−0.999697
$842$	0	0
$843$	2.02951e13	1.38410
$844$	0	0
$845$	6.47322e12	0.436783
$846$	0	0
$847$	−5.84013e12	−0.389895
$848$	0	0
$849$	2.89528e12	0.191252
$850$	0	0
$851$	−1.53772e13	−1.00507
$852$	0	0
$853$	1.86458e12	0.120590	0.0602949	−	0.998181i	$-0.480796\pi$
$853$	1.86458e12	0.120590	0.0602949	+	0.998181i	$0.480796\pi$
$854$	0	0
$855$	−2.50611e12	−0.160381
$856$	0	0
$857$	2.99648e13	1.89757	0.948786	−	0.315919i	$-0.102313\pi$
$857$	2.99648e13	1.89757	0.948786	+	0.315919i	$0.102313\pi$
$858$	0	0
$859$	3.49272e12	0.218874	0.109437	−	0.993994i	$-0.465095\pi$
$859$	3.49272e12	0.218874	0.109437	+	0.993994i	$0.465095\pi$
$860$	0	0
$861$	2.49819e12	0.154921
$862$	0	0
$863$	−1.51658e13	−0.930716	−0.465358	−	0.885122i	$-0.654074\pi$
$863$	−1.51658e13	−0.930716	−0.465358	+	0.885122i	$0.654074\pi$
$864$	0	0
$865$	−2.22120e12	−0.134901
$866$	0	0
$867$	−1.22466e13	−0.736090
$868$	0	0
$869$	−9.32198e11	−0.0554523
$870$	0	0
$871$	2.11826e12	0.124709
$872$	0	0
$873$	−6.18255e12	−0.360250
$874$	0	0
$875$	−6.27463e11	−0.0361869
$876$	0	0
$877$	2.89333e13	1.65158	0.825791	−	0.563977i	$-0.190729\pi$
$877$	2.89333e13	1.65158	0.825791	+	0.563977i	$0.190729\pi$
$878$	0	0
$879$	−3.36003e12	−0.189842
$880$	0	0
$881$	−3.24456e13	−1.81453	−0.907266	−	0.420557i	$-0.861835\pi$
$881$	−3.24456e13	−1.81453	−0.907266	+	0.420557i	$0.861835\pi$
$882$	0	0
$883$	−1.91045e12	−0.105758	−0.0528788	−	0.998601i	$-0.516840\pi$
$883$	−1.91045e12	−0.105758	−0.0528788	+	0.998601i	$0.516840\pi$
$884$	0	0
$885$	−3.57170e12	−0.195718
$886$	0	0
$887$	−1.40998e12	−0.0764815	−0.0382408	−	0.999269i	$-0.512175\pi$
$887$	−1.40998e12	−0.0764815	−0.0382408	+	0.999269i	$0.512175\pi$
$888$	0	0
$889$	3.98131e11	0.0213781
$890$	0	0
$891$	−1.58448e12	−0.0842240
$892$	0	0
$893$	2.16224e13	1.13782
$894$	0	0
$895$	8.44704e12	0.440049
$896$	0	0
$897$	2.64797e12	0.136567
$898$	0	0
$899$	−1.62160e11	−0.00827991
$900$	0	0
$901$	−7.17698e12	−0.362811
$902$	0	0
$903$	−2.07113e12	−0.103660
$904$	0	0
$905$	1.09288e13	0.541569
$906$	0	0
$907$	1.19566e13	0.586644	0.293322	−	0.956014i	$-0.405239\pi$
$907$	1.19566e13	0.586644	0.293322	+	0.956014i	$0.405239\pi$
$908$	0	0
$909$	1.37068e12	0.0665883
$910$	0	0
$911$	3.17877e13	1.52907	0.764534	−	0.644583i	$-0.222969\pi$
$911$	3.17877e13	1.52907	0.764534	+	0.644583i	$0.222969\pi$
$912$	0	0
$913$	−3.13905e12	−0.149513
$914$	0	0
$915$	1.31659e13	0.620949
$916$	0	0
$917$	1.62057e13	0.756842
$918$	0	0
$919$	2.97224e13	1.37456	0.687281	−	0.726392i	$-0.258804\pi$
$919$	2.97224e13	1.37456	0.687281	+	0.726392i	$0.258804\pi$
$920$	0	0
$921$	2.77464e13	1.27069
$922$	0	0
$923$	1.41288e12	0.0640762
$924$	0	0
$925$	3.88070e12	0.174290
$926$	0	0
$927$	8.28225e12	0.368375
$928$	0	0
$929$	−3.18271e11	−0.0140193	−0.00700965	−	0.999975i	$-0.502231\pi$
$929$	−3.18271e11	−0.0140193	−0.00700965	+	0.999975i	$0.502231\pi$
$930$	0	0
$931$	1.72368e13	0.751941
$932$	0	0
$933$	−3.11846e13	−1.34733
$934$	0	0
$935$	−4.47980e11	−0.0191693
$936$	0	0
$937$	3.26024e13	1.38172	0.690862	−	0.722986i	$-0.257231\pi$
$937$	3.26024e13	1.38172	0.690862	+	0.722986i	$0.257231\pi$
$938$	0	0
$939$	−2.75478e13	−1.15636
$940$	0	0
$941$	3.81764e12	0.158724	0.0793619	−	0.996846i	$-0.474712\pi$
$941$	3.81764e12	0.158724	0.0793619	+	0.996846i	$0.474712\pi$
$942$	0	0
$943$	−1.38316e13	−0.569600
$944$	0	0
$945$	4.81090e12	0.196238
$946$	0	0
$947$	1.17529e13	0.474863	0.237432	−	0.971404i	$-0.423694\pi$
$947$	1.17529e13	0.474863	0.237432	+	0.971404i	$0.423694\pi$
$948$	0	0
$949$	5.76080e11	0.0230560
$950$	0	0
$951$	2.08034e13	0.824748
$952$	0	0
$953$	−3.67264e11	−0.0144232	−0.00721158	−	0.999974i	$-0.502296\pi$
$953$	−3.67264e11	−0.0144232	−0.00721158	+	0.999974i	$0.502296\pi$
$954$	0	0
$955$	4.89148e12	0.190294
$956$	0	0
$957$	6.66991e10	0.00257049
$958$	0	0
$959$	1.13358e13	0.432782
$960$	0	0
$961$	−2.04530e13	−0.773574
$962$	0	0
$963$	1.73168e12	0.0648857
$964$	0	0
$965$	2.18882e12	0.0812525
$966$	0	0
$967$	4.49704e13	1.65389	0.826947	−	0.562280i	$-0.190076\pi$
$967$	4.49704e13	1.65389	0.826947	+	0.562280i	$0.190076\pi$
$968$	0	0
$969$	−4.30411e12	−0.156829
$970$	0	0
$971$	−3.49682e13	−1.26237	−0.631185	−	0.775632i	$-0.717431\pi$
$971$	−3.49682e13	−1.26237	−0.631185	+	0.775632i	$0.717431\pi$
$972$	0	0
$973$	1.02120e13	0.365262
$974$	0	0
$975$	−6.68258e11	−0.0236823
$976$	0	0
$977$	−3.89696e12	−0.136836	−0.0684180	−	0.997657i	$-0.521795\pi$
$977$	−3.89696e12	−0.136836	−0.0684180	+	0.997657i	$0.521795\pi$
$978$	0	0
$979$	9.56358e12	0.332735
$980$	0	0
$981$	−1.58343e13	−0.545870
$982$	0	0
$983$	−6.27073e12	−0.214204	−0.107102	−	0.994248i	$-0.534157\pi$
$983$	−6.27073e12	−0.214204	−0.107102	+	0.994248i	$0.534157\pi$
$984$	0	0
$985$	−1.14532e13	−0.387671
$986$	0	0
$987$	−1.18353e13	−0.396964
$988$	0	0
$989$	1.14672e13	0.381129
$990$	0	0
$991$	1.85239e13	0.610101	0.305050	−	0.952336i	$-0.401327\pi$
$991$	1.85239e13	0.610101	0.305050	+	0.952336i	$0.401327\pi$
$992$	0	0
$993$	−2.81539e13	−0.918897
$994$	0	0
$995$	1.99602e13	0.645598
$996$	0	0
$997$	−2.60256e13	−0.834206	−0.417103	−	0.908859i	$-0.636955\pi$
$997$	−2.60256e13	−0.834206	−0.417103	+	0.908859i	$0.636955\pi$
$998$	0	0
$999$	−2.97542e13	−0.945157

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.10.a.g.1.2		2
4.3	odd	2		40.10.a.c.1.1	✓	2
5.2	odd	4		400.10.c.m.49.2		4
5.3	odd	4		400.10.c.m.49.3		4
5.4	even	2		400.10.a.r.1.1		2
8.3	odd	2		320.10.a.n.1.2		2
8.5	even	2		320.10.a.q.1.1		2
12.11	even	2		360.10.a.g.1.1		2
20.3	even	4		200.10.c.d.49.2		4
20.7	even	4		200.10.c.d.49.3		4
20.19	odd	2		200.10.a.c.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.10.a.c.1.1	✓	2	4.3	odd	2
80.10.a.g.1.2		2	1.1	even	1	trivial
200.10.a.c.1.2		2	20.19	odd	2
200.10.c.d.49.2		4	20.3	even	4
200.10.c.d.49.3		4	20.7	even	4
320.10.a.n.1.2		2	8.3	odd	2
320.10.a.q.1.1		2	8.5	even	2
360.10.a.g.1.1		2	12.11	even	2
400.10.a.r.1.1		2	5.4	even	2
400.10.c.m.49.2		4	5.2	odd	4
400.10.c.m.49.3		4	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	80.a (trivial)

\(f(q)\)	\(=\)	\(q+108.776 q^{3} -625.000 q^{5} +2570.09 q^{7} -7850.80 q^{9} +O(q^{10})\)	\(q+108.776 q^{3} -625.000 q^{5} +2570.09 q^{7} -7850.80 q^{9} +9251.97 q^{11} -15727.2 q^{13} -67984.9 q^{15} +77471.9 q^{17} -510747. q^{19} +279564. q^{21} -1.54785e6 q^{23} +390625. q^{25} -2.99501e6 q^{27} +66275.5 q^{29} -2.44676e6 q^{31} +1.00639e6 q^{33} -1.60630e6 q^{35} +9.93458e6 q^{37} -1.71074e6 q^{39} +8.93602e6 q^{41} -7.40844e6 q^{43} +4.90675e6 q^{45} -4.23348e7 q^{47} -3.37483e7 q^{49} +8.42707e6 q^{51} -9.26398e7 q^{53} -5.78248e6 q^{55} -5.55570e7 q^{57} +5.25366e7 q^{59} -1.93659e8 q^{61} -2.01772e7 q^{63} +9.82950e6 q^{65} -1.34688e8 q^{67} -1.68369e8 q^{69} -8.98365e7 q^{71} -3.66295e7 q^{73} +4.24906e7 q^{75} +2.37784e7 q^{77} -1.00757e8 q^{79} -1.71258e8 q^{81} -3.39285e8 q^{83} -4.84199e7 q^{85} +7.20918e6 q^{87} +1.03368e9 q^{89} -4.04203e7 q^{91} -2.66148e8 q^{93} +3.19217e8 q^{95} +7.87506e8 q^{97} -7.26354e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 108 q^{3} - 1250 q^{5} + 908 q^{7} + 19458 q^{9}+O(q^{10}) \) 2 * q - 108 * q^3 - 1250 * q^5 + 908 * q^7 + 19458 * q^9	\( 2 q - 108 q^{3} - 1250 q^{5} + 908 q^{7} + 19458 q^{9} - 25120 q^{11} + 146948 q^{13} + 67500 q^{15} + 169268 q^{17} + 25480 q^{19} + 639864 q^{21} - 1782748 q^{23} + 781250 q^{25} - 4648104 q^{27} + 7323340 q^{29} - 10677272 q^{31} + 8457408 q^{33} - 567500 q^{35} - 5750460 q^{37} - 36974808 q^{39} - 7795764 q^{41} - 16770524 q^{43} - 12161250 q^{45} - 15393892 q^{47} - 71339334 q^{49} - 11472120 q^{51} - 58529292 q^{53} + 15700000 q^{55} - 171798192 q^{57} - 59618264 q^{59} - 188163772 q^{61} - 65566836 q^{63} - 91842500 q^{65} + 105998252 q^{67} - 117448344 q^{69} + 168665592 q^{71} - 390212412 q^{73} - 42187500 q^{75} + 80907584 q^{77} - 466946256 q^{79} - 350427222 q^{81} - 329535164 q^{83} - 105792500 q^{85} - 1565947656 q^{87} - 108334860 q^{89} - 310800616 q^{91} + 1518028560 q^{93} - 15925000 q^{95} + 2043058628 q^{97} - 1011292704 q^{99}+O(q^{100}) \) 2 * q - 108 * q^3 - 1250 * q^5 + 908 * q^7 + 19458 * q^9 - 25120 * q^11 + 146948 * q^13 + 67500 * q^15 + 169268 * q^17 + 25480 * q^19 + 639864 * q^21 - 1782748 * q^23 + 781250 * q^25 - 4648104 * q^27 + 7323340 * q^29 - 10677272 * q^31 + 8457408 * q^33 - 567500 * q^35 - 5750460 * q^37 - 36974808 * q^39 - 7795764 * q^41 - 16770524 * q^43 - 12161250 * q^45 - 15393892 * q^47 - 71339334 * q^49 - 11472120 * q^51 - 58529292 * q^53 + 15700000 * q^55 - 171798192 * q^57 - 59618264 * q^59 - 188163772 * q^61 - 65566836 * q^63 - 91842500 * q^65 + 105998252 * q^67 - 117448344 * q^69 + 168665592 * q^71 - 390212412 * q^73 - 42187500 * q^75 + 80907584 * q^77 - 466946256 * q^79 - 350427222 * q^81 - 329535164 * q^83 - 105792500 * q^85 - 1565947656 * q^87 - 108334860 * q^89 - 310800616 * q^91 + 1518028560 * q^93 - 15925000 * q^95 + 2043058628 * q^97 - 1011292704 * q^99

Introduction

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