LMFDB - Embedded newform 81.10.a.b.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [81,10,Mod(1,81)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(81, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 10, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("81.1"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$ N $	$=$	$ 81 = 3^{4} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	81.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,33] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$41.7179027293$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 1314x^{2} + 10232x + 106624 $ x^4 - x^3 - 1314x^2 + 10232x + 106624
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{3} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$15.7163$ of defining polynomial
Character	$\chi$	$=$	81.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+23.7163 q^{2} +50.4644 q^{4} +2758.02 q^{5} -8237.69 q^{7} -10945.9 q^{8} +65410.1 q^{10} -77786.5 q^{11} +22294.6 q^{13} -195368. q^{14} -285435. q^{16} -279361. q^{17} -282874. q^{19} +139182. q^{20} -1.84481e6 q^{22} +573422. q^{23} +5.65355e6 q^{25} +528746. q^{26} -415710. q^{28} -2.50733e6 q^{29} -4.00346e6 q^{31} -1.16516e6 q^{32} -6.62542e6 q^{34} -2.27197e7 q^{35} +725997. q^{37} -6.70873e6 q^{38} -3.01891e7 q^{40} -1.88672e7 q^{41} -8.38638e6 q^{43} -3.92545e6 q^{44} +1.35995e7 q^{46} +3.96821e6 q^{47} +2.75060e7 q^{49} +1.34082e8 q^{50} +1.12508e6 q^{52} -5.37969e7 q^{53} -2.14537e8 q^{55} +9.01692e7 q^{56} -5.94647e7 q^{58} +9.07115e7 q^{59} -1.40355e8 q^{61} -9.49474e7 q^{62} +1.18510e8 q^{64} +6.14889e7 q^{65} +9.76857e7 q^{67} -1.40978e7 q^{68} -5.38829e8 q^{70} -1.01403e8 q^{71} -5.36025e6 q^{73} +1.72180e7 q^{74} -1.42751e7 q^{76} +6.40782e8 q^{77} +1.85361e8 q^{79} -7.87236e8 q^{80} -4.47460e8 q^{82} -2.36379e8 q^{83} -7.70483e8 q^{85} -1.98894e8 q^{86} +8.51446e8 q^{88} +5.37873e8 q^{89} -1.83656e8 q^{91} +2.89374e7 q^{92} +9.41115e7 q^{94} -7.80172e8 q^{95} +7.11359e8 q^{97} +6.52341e8 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 33 q^{2} + 853 q^{4} + 570 q^{5} - 3238 q^{7} + 4791 q^{8} - 9723 q^{10} - 96690 q^{11} - 141118 q^{13} + 3036 q^{14} - 244463 q^{16} + 285156 q^{17} - 465166 q^{19} - 1041711 q^{20} - 2244480 q^{22}+ \cdots + 735501321 q^{98}+O(q^{100}) $ 4 * q + 33 * q^2 + 853 * q^4 + 570 * q^5 - 3238 * q^7 + 4791 * q^8 - 9723 * q^10 - 96690 * q^11 - 141118 * q^13 + 3036 * q^14 - 244463 * q^16 + 285156 * q^17 - 465166 * q^19 - 1041711 * q^20 - 2244480 * q^22 - 237318 * q^23 + 3417712 * q^25 - 6433737 * q^26 + 769292 * q^28 + 2806926 * q^29 - 8244448 * q^31 + 3896415 * q^32 + 1386189 * q^34 - 31668402 * q^35 - 8204014 * q^37 - 755556 * q^38 - 54180645 * q^40 + 28889292 * q^41 - 39586246 * q^43 - 89567616 * q^44 - 93870624 * q^46 - 17514588 * q^47 - 67913400 * q^49 + 233832372 * q^50 - 207295237 * q^52 - 155826504 * q^53 - 185126142 * q^55 + 133671300 * q^56 - 164356143 * q^58 + 341256108 * q^59 - 238298254 * q^61 - 729569796 * q^62 - 286454975 * q^64 + 379881528 * q^65 - 72601486 * q^67 + 762353217 * q^68 - 780989460 * q^70 - 667543266 * q^71 - 520588288 * q^73 + 183473031 * q^74 - 413336884 * q^76 + 721530228 * q^77 - 137324662 * q^79 - 1181367123 * q^80 - 275042586 * q^82 - 278546844 * q^83 - 1408511358 * q^85 + 420627720 * q^86 - 30700416 * q^88 + 237712932 * q^89 - 762950942 * q^91 - 1238743680 * q^92 - 507333516 * q^94 - 811663194 * q^95 + 700071356 * q^97 + 735501321 * q^98

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 $

$2$	23.7163	1.04812	0.524062	−	0.851680i	$-0.324416\pi$
$2$	23.7163	1.04812	0.524062	+	0.851680i	$0.324416\pi$
$3$	0	0
$3$	0	0
$4$	50.4644	0.0985632
$5$	2758.02	1.97348	0.986739	−	0.162312i	$-0.0518952\pi$
$5$	2758.02	1.97348	0.986739	+	0.162312i	$0.0518952\pi$
$6$	0	0
$7$	−8237.69	−1.29677	−0.648387	−	0.761311i	$-0.724556\pi$
$7$	−8237.69	−1.29677	−0.648387	+	0.761311i	$0.724556\pi$
$8$	−10945.9	−0.944817
$9$	0	0
$10$	65410.1	2.06845
$11$	−77786.5	−1.60191	−0.800954	−	0.598726i	$-0.795674\pi$
$11$	−77786.5	−1.60191	−0.800954	+	0.598726i	$0.795674\pi$
$12$	0	0
$13$	22294.6	0.216498	0.108249	−	0.994124i	$-0.465476\pi$
$13$	22294.6	0.216498	0.108249	+	0.994124i	$0.465476\pi$
$14$	−195368.	−1.35918
$15$	0	0
$16$	−285435.	−1.08885
$17$	−279361.	−0.811233	−0.405616	−	0.914043i	$-0.632943\pi$
$17$	−279361.	−0.811233	−0.405616	+	0.914043i	$0.632943\pi$
$18$	0	0
$19$	−282874.	−0.497968	−0.248984	−	0.968508i	$-0.580097\pi$
$19$	−282874.	−0.497968	−0.248984	+	0.968508i	$0.580097\pi$
$20$	139182.	0.194512
$21$	0	0
$22$	−1.84481e6	−1.67900
$23$	573422.	0.427267	0.213634	−	0.976914i	$-0.431470\pi$
$23$	573422.	0.427267	0.213634	+	0.976914i	$0.431470\pi$
$24$	0	0
$25$	5.65355e6	2.89462
$26$	528746.	0.226917
$27$	0	0
$28$	−415710.	−0.127814
$29$	−2.50733e6	−0.658296	−0.329148	−	0.944278i	$-0.606761\pi$
$29$	−2.50733e6	−0.658296	−0.329148	+	0.944278i	$0.606761\pi$
$30$	0	0
$31$	−4.00346e6	−0.778588	−0.389294	−	0.921114i	$-0.627281\pi$
$31$	−4.00346e6	−0.778588	−0.389294	+	0.921114i	$0.627281\pi$
$32$	−1.16516e6	−0.196431
$33$	0	0
$34$	−6.62542e6	−0.850272
$35$	−2.27197e7	−2.55916
$36$	0	0
$37$	725997.	0.0636836	0.0318418	−	0.999493i	$-0.489863\pi$
$37$	725997.	0.0636836	0.0318418	+	0.999493i	$0.489863\pi$
$38$	−6.70873e6	−0.521932
$39$	0	0
$40$	−3.01891e7	−1.86458
$41$	−1.88672e7	−1.04275	−0.521374	−	0.853328i	$-0.674580\pi$
$41$	−1.88672e7	−1.04275	−0.521374	+	0.853328i	$0.674580\pi$
$42$	0	0
$43$	−8.38638e6	−0.374082	−0.187041	−	0.982352i	$-0.559890\pi$
$43$	−8.38638e6	−0.374082	−0.187041	+	0.982352i	$0.559890\pi$
$44$	−3.92545e6	−0.157889
$45$	0	0
$46$	1.35995e7	0.447829
$47$	3.96821e6	0.118619	0.0593096	−	0.998240i	$-0.481110\pi$
$47$	3.96821e6	0.118619	0.0593096	+	0.998240i	$0.481110\pi$
$48$	0	0
$49$	2.75060e7	0.681624
$50$	1.34082e8	3.03392
$51$	0	0
$52$	1.12508e6	0.0213388
$53$	−5.37969e7	−0.936517	−0.468259	−	0.883591i	$-0.655118\pi$
$53$	−5.37969e7	−0.936517	−0.468259	+	0.883591i	$0.655118\pi$
$54$	0	0
$55$	−2.14537e8	−3.16133
$56$	9.01692e7	1.22521
$57$	0	0
$58$	−5.94647e7	−0.689975
$59$	9.07115e7	0.974604	0.487302	−	0.873233i	$-0.337981\pi$
$59$	9.07115e7	0.974604	0.487302	+	0.873233i	$0.337981\pi$
$60$	0	0
$61$	−1.40355e8	−1.29791	−0.648955	−	0.760826i	$-0.724794\pi$
$61$	−1.40355e8	−1.29791	−0.648955	+	0.760826i	$0.724794\pi$
$62$	−9.49474e7	−0.816057
$63$	0	0
$64$	1.18510e8	0.882965
$65$	6.14889e7	0.427255
$66$	0	0
$67$	9.76857e7	0.592235	0.296118	−	0.955151i	$-0.404308\pi$
$67$	9.76857e7	0.592235	0.296118	+	0.955151i	$0.404308\pi$
$68$	−1.40978e7	−0.0799577
$69$	0	0
$70$	−5.38829e8	−2.68231
$71$	−1.01403e8	−0.473576	−0.236788	−	0.971561i	$-0.576095\pi$
$71$	−1.01403e8	−0.473576	−0.236788	+	0.971561i	$0.576095\pi$
$72$	0	0
$73$	−5.36025e6	−0.0220918	−0.0110459	−	0.999939i	$-0.503516\pi$
$73$	−5.36025e6	−0.0220918	−0.0110459	+	0.999939i	$0.503516\pi$
$74$	1.72180e7	0.0667482
$75$	0	0
$76$	−1.42751e7	−0.0490814
$77$	6.40782e8	2.07731
$78$	0	0
$79$	1.85361e8	0.535423	0.267712	−	0.963499i	$-0.413733\pi$
$79$	1.85361e8	0.535423	0.267712	+	0.963499i	$0.413733\pi$
$80$	−7.87236e8	−2.14882
$81$	0	0
$82$	−4.47460e8	−1.09293
$83$	−2.36379e8	−0.546711	−0.273355	−	0.961913i	$-0.588133\pi$
$83$	−2.36379e8	−0.546711	−0.273355	+	0.961913i	$0.588133\pi$
$84$	0	0
$85$	−7.70483e8	−1.60095
$86$	−1.98894e8	−0.392084
$87$	0	0
$88$	8.51446e8	1.51351
$89$	5.37873e8	0.908709	0.454355	−	0.890821i	$-0.349870\pi$
$89$	5.37873e8	0.908709	0.454355	+	0.890821i	$0.349870\pi$
$90$	0	0
$91$	−1.83656e8	−0.280749
$92$	2.89374e7	0.0421128
$93$	0	0
$94$	9.41115e7	0.124327
$95$	−7.80172e8	−0.982730
$96$	0	0
$97$	7.11359e8	0.815861	0.407931	−	0.913013i	$-0.366251\pi$
$97$	7.11359e8	0.815861	0.407931	+	0.913013i	$0.366251\pi$
$98$	6.52341e8	0.714426
$99$	0	0
$100$	2.85303e8	0.285303
$101$	1.32077e9	1.26294	0.631470	−	0.775401i	$-0.282452\pi$
$101$	1.32077e9	1.26294	0.631470	+	0.775401i	$0.282452\pi$
$102$	0	0
$103$	−1.59412e9	−1.39557	−0.697787	−	0.716306i	$-0.745832\pi$
$103$	−1.59412e9	−1.39557	−0.697787	+	0.716306i	$0.745832\pi$
$104$	−2.44035e8	−0.204551
$105$	0	0
$106$	−1.27586e9	−0.981586
$107$	−1.42682e9	−1.05231	−0.526153	−	0.850390i	$-0.676366\pi$
$107$	−1.42682e9	−1.05231	−0.526153	+	0.850390i	$0.676366\pi$
$108$	0	0
$109$	2.12085e9	1.43910	0.719549	−	0.694442i	$-0.244349\pi$
$109$	2.12085e9	1.43910	0.719549	+	0.694442i	$0.244349\pi$
$110$	−5.08803e9	−3.31347
$111$	0	0
$112$	2.35133e9	1.41199
$113$	−1.42632e9	−0.822931	−0.411466	−	0.911425i	$-0.634983\pi$
$113$	−1.42632e9	−0.822931	−0.411466	+	0.911425i	$0.634983\pi$
$114$	0	0
$115$	1.58151e9	0.843202
$116$	−1.26531e8	−0.0648838
$117$	0	0
$118$	2.15134e9	1.02151
$119$	2.30129e9	1.05199
$120$	0	0
$121$	3.69280e9	1.56611
$122$	−3.32871e9	−1.36037
$123$	0	0
$124$	−2.02032e8	−0.0767402
$125$	1.02059e10	3.73899
$126$	0	0
$127$	−3.92014e9	−1.33716	−0.668582	−	0.743639i	$-0.733098\pi$
$127$	−3.92014e9	−1.33716	−0.668582	+	0.743639i	$0.733098\pi$
$128$	3.40717e9	1.12189
$129$	0	0
$130$	1.45829e9	0.447816
$131$	3.87344e9	1.14915	0.574573	−	0.818453i	$-0.305168\pi$
$131$	3.87344e9	1.14915	0.574573	+	0.818453i	$0.305168\pi$
$132$	0	0
$133$	2.33023e9	0.645753
$134$	2.31675e9	0.620736
$135$	0	0
$136$	3.05787e9	0.766467
$137$	2.02888e9	0.492054	0.246027	−	0.969263i	$-0.420875\pi$
$137$	2.02888e9	0.492054	0.246027	+	0.969263i	$0.420875\pi$
$138$	0	0
$139$	5.25556e8	0.119413	0.0597066	−	0.998216i	$-0.480983\pi$
$139$	5.25556e8	0.119413	0.0597066	+	0.998216i	$0.480983\pi$
$140$	−1.14654e9	−0.252239
$141$	0	0
$142$	−2.40491e9	−0.496366
$143$	−1.73422e9	−0.346810
$144$	0	0
$145$	−6.91528e9	−1.29913
$146$	−1.27125e8	−0.0231550
$147$	0	0
$148$	3.66370e7	0.00627686
$149$	−3.23707e9	−0.538040	−0.269020	−	0.963135i	$-0.586700\pi$
$149$	−3.23707e9	−0.538040	−0.269020	+	0.963135i	$0.586700\pi$
$150$	0	0
$151$	1.91182e9	0.299261	0.149630	−	0.988742i	$-0.452192\pi$
$151$	1.91182e9	0.299261	0.149630	+	0.988742i	$0.452192\pi$
$152$	3.09632e9	0.470489
$153$	0	0
$154$	1.51970e10	2.17728
$155$	−1.10416e10	−1.53653
$156$	0	0
$157$	1.20562e10	1.58367	0.791833	−	0.610738i	$-0.209127\pi$
$157$	1.20562e10	1.58367	0.791833	+	0.610738i	$0.209127\pi$
$158$	4.39609e9	0.561190
$159$	0	0
$160$	−3.21353e9	−0.387652
$161$	−4.72368e9	−0.554069
$162$	0	0
$163$	−2.20864e9	−0.245065	−0.122532	−	0.992465i	$-0.539101\pi$
$163$	−2.20864e9	−0.245065	−0.122532	+	0.992465i	$0.539101\pi$
$164$	−9.52120e8	−0.102777
$165$	0	0
$166$	−5.60604e9	−0.573020
$167$	−1.62710e10	−1.61879	−0.809394	−	0.587266i	$-0.800204\pi$
$167$	−1.62710e10	−1.61879	−0.809394	+	0.587266i	$0.800204\pi$
$168$	0	0
$169$	−1.01075e10	−0.953129
$170$	−1.82730e10	−1.67799
$171$	0	0
$172$	−4.23213e8	−0.0368707
$173$	8.94996e9	0.759650	0.379825	−	0.925058i	$-0.375984\pi$
$173$	8.94996e9	0.759650	0.379825	+	0.925058i	$0.375984\pi$
$174$	0	0
$175$	−4.65722e10	−3.75367
$176$	2.22030e10	1.74423
$177$	0	0
$178$	1.27564e10	0.952440
$179$	−1.31320e10	−0.956077	−0.478038	−	0.878339i	$-0.658652\pi$
$179$	−1.31320e10	−0.956077	−0.478038	+	0.878339i	$0.658652\pi$
$180$	0	0
$181$	−2.18286e10	−1.51172	−0.755862	−	0.654731i	$-0.772782\pi$
$181$	−2.18286e10	−1.51172	−0.755862	+	0.654731i	$0.772782\pi$
$182$	−4.35565e9	−0.294260
$183$	0	0
$184$	−6.27664e9	−0.403689
$185$	2.00232e9	0.125678
$186$	0	0
$187$	2.17305e10	1.29952
$188$	2.00253e8	0.0116915
$189$	0	0
$190$	−1.85028e10	−1.03002
$191$	3.25745e10	1.77104	0.885519	−	0.464602i	$-0.153803\pi$
$191$	3.25745e10	1.77104	0.885519	+	0.464602i	$0.153803\pi$
$192$	0	0
$193$	1.20441e10	0.624835	0.312417	−	0.949945i	$-0.398861\pi$
$193$	1.20441e10	0.624835	0.312417	+	0.949945i	$0.398861\pi$
$194$	1.68708e10	0.855123
$195$	0	0
$196$	1.38807e9	0.0671830
$197$	4.12313e9	0.195042	0.0975212	−	0.995233i	$-0.468909\pi$
$197$	4.12313e9	0.195042	0.0975212	+	0.995233i	$0.468909\pi$
$198$	0	0
$199$	1.12675e10	0.509317	0.254658	−	0.967031i	$-0.418037\pi$
$199$	1.12675e10	0.509317	0.254658	+	0.967031i	$0.418037\pi$
$200$	−6.18834e10	−2.73489
$201$	0	0
$202$	3.13239e10	1.32372
$203$	2.06546e10	0.853661
$204$	0	0
$205$	−5.20360e10	−2.05784
$206$	−3.78066e10	−1.46273
$207$	0	0
$208$	−6.36366e9	−0.235734
$209$	2.20038e10	0.797699
$210$	0	0
$211$	−4.04426e10	−1.40465	−0.702325	−	0.711856i	$-0.747855\pi$
$211$	−4.04426e10	−1.40465	−0.702325	+	0.711856i	$0.747855\pi$
$212$	−2.71483e9	−0.0923062
$213$	0	0
$214$	−3.38390e10	−1.10295
$215$	−2.31298e10	−0.738242
$216$	0	0
$217$	3.29793e10	1.00965
$218$	5.02987e10	1.50835
$219$	0	0
$220$	−1.08265e10	−0.311591
$221$	−6.22824e9	−0.175630
$222$	0	0
$223$	−3.52598e10	−0.954790	−0.477395	−	0.878689i	$-0.658419\pi$
$223$	−3.52598e10	−0.954790	−0.477395	+	0.878689i	$0.658419\pi$
$224$	9.59820e9	0.254726
$225$	0	0
$226$	−3.38270e10	−0.862534
$227$	−2.13641e10	−0.534033	−0.267016	−	0.963692i	$-0.586038\pi$
$227$	−2.13641e10	−0.534033	−0.267016	+	0.963692i	$0.586038\pi$
$228$	0	0
$229$	7.06004e10	1.69648	0.848238	−	0.529616i	$-0.177664\pi$
$229$	7.06004e10	1.69648	0.848238	+	0.529616i	$0.177664\pi$
$230$	3.75076e10	0.883780
$231$	0	0
$232$	2.74451e10	0.621969
$233$	6.71540e10	1.49269	0.746346	−	0.665558i	$-0.231806\pi$
$233$	6.71540e10	1.49269	0.746346	+	0.665558i	$0.231806\pi$
$234$	0	0
$235$	1.09444e10	0.234092
$236$	4.57770e9	0.0960601
$237$	0	0
$238$	5.45782e10	1.10261
$239$	−8.31576e10	−1.64858	−0.824292	−	0.566164i	$-0.808427\pi$
$239$	−8.31576e10	−1.64858	−0.824292	+	0.566164i	$0.808427\pi$
$240$	0	0
$241$	−4.69651e10	−0.896806	−0.448403	−	0.893831i	$-0.648007\pi$
$241$	−4.69651e10	−0.896806	−0.448403	+	0.893831i	$0.648007\pi$
$242$	8.75797e10	1.64147
$243$	0	0
$244$	−7.08295e9	−0.127926
$245$	7.58620e10	1.34517
$246$	0	0
$247$	−6.30656e9	−0.107809
$248$	4.38216e10	0.735623
$249$	0	0
$250$	2.42045e11	3.91892
$251$	−2.17422e10	−0.345758	−0.172879	−	0.984943i	$-0.555307\pi$
$251$	−2.17422e10	−0.345758	−0.172879	+	0.984943i	$0.555307\pi$
$252$	0	0
$253$	−4.46045e10	−0.684442
$254$	−9.29713e10	−1.40151
$255$	0	0
$256$	2.01287e10	0.292912
$257$	−2.76626e9	−0.0395544	−0.0197772	−	0.999804i	$-0.506296\pi$
$257$	−2.76626e9	−0.0395544	−0.0197772	+	0.999804i	$0.506296\pi$
$258$	0	0
$259$	−5.98054e9	−0.0825832
$260$	3.10300e9	0.0421116
$261$	0	0
$262$	9.18637e10	1.20445
$263$	−9.82433e10	−1.26620	−0.633100	−	0.774070i	$-0.718218\pi$
$263$	−9.82433e10	−1.26620	−0.633100	+	0.774070i	$0.718218\pi$
$264$	0	0
$265$	−1.48373e11	−1.84820
$266$	5.52645e10	0.676829
$267$	0	0
$268$	4.92965e9	0.0583726
$269$	1.75996e10	0.204935	0.102468	−	0.994736i	$-0.467326\pi$
$269$	1.75996e10	0.204935	0.102468	+	0.994736i	$0.467326\pi$
$270$	0	0
$271$	2.22369e10	0.250445	0.125223	−	0.992129i	$-0.460036\pi$
$271$	2.22369e10	0.250445	0.125223	+	0.992129i	$0.460036\pi$
$272$	7.97394e10	0.883310
$273$	0	0
$274$	4.81175e10	0.515734
$275$	−4.39770e11	−4.63691
$276$	0	0
$277$	−1.00658e11	−1.02728	−0.513640	−	0.858006i	$-0.671703\pi$
$277$	−1.00658e11	−1.02728	−0.513640	+	0.858006i	$0.671703\pi$
$278$	1.24643e10	0.125160
$279$	0	0
$280$	2.48689e11	2.41794
$281$	−2.84220e10	−0.271942	−0.135971	−	0.990713i	$-0.543415\pi$
$281$	−2.84220e10	−0.271942	−0.135971	+	0.990713i	$0.543415\pi$
$282$	0	0
$283$	4.30371e10	0.398845	0.199422	−	0.979914i	$-0.436093\pi$
$283$	4.30371e10	0.398845	0.199422	+	0.979914i	$0.436093\pi$
$284$	−5.11725e9	−0.0466772
$285$	0	0
$286$	−4.11293e10	−0.363500
$287$	1.55422e11	1.35221
$288$	0	0
$289$	−4.05453e10	−0.341901
$290$	−1.64005e11	−1.36165
$291$	0	0
$292$	−2.70502e8	−0.00217744
$293$	−1.73312e10	−0.137380	−0.0686902	−	0.997638i	$-0.521882\pi$
$293$	−1.73312e10	−0.137380	−0.0686902	+	0.997638i	$0.521882\pi$
$294$	0	0
$295$	2.50184e11	1.92336
$296$	−7.94672e9	−0.0601693
$297$	0	0
$298$	−7.67715e10	−0.563932
$299$	1.27842e10	0.0925025
$300$	0	0
$301$	6.90844e10	0.485099
$302$	4.53413e10	0.313662
$303$	0	0
$304$	8.07422e10	0.542212
$305$	−3.87103e11	−2.56140
$306$	0	0
$307$	−2.03269e11	−1.30602	−0.653008	−	0.757351i	$-0.726493\pi$
$307$	−2.03269e11	−1.30602	−0.653008	+	0.757351i	$0.726493\pi$
$308$	3.23366e10	0.204747
$309$	0	0
$310$	−2.61867e11	−1.61047
$311$	2.62568e11	1.59155	0.795774	−	0.605594i	$-0.207064\pi$
$311$	2.62568e11	1.59155	0.795774	+	0.605594i	$0.207064\pi$
$312$	0	0
$313$	1.86044e10	0.109564	0.0547818	−	0.998498i	$-0.482554\pi$
$313$	1.86044e10	0.109564	0.0547818	+	0.998498i	$0.482554\pi$
$314$	2.85930e11	1.65988
$315$	0	0
$316$	9.35414e9	0.0527730
$317$	2.58624e11	1.43847	0.719237	−	0.694764i	$-0.244491\pi$
$317$	2.58624e11	1.43847	0.719237	+	0.694764i	$0.244491\pi$
$318$	0	0
$319$	1.95037e11	1.05453
$320$	3.26852e11	1.74251
$321$	0	0
$322$	−1.12028e11	−0.580733
$323$	7.90239e10	0.403968
$324$	0	0
$325$	1.26044e11	0.626680
$326$	−5.23808e10	−0.256858
$327$	0	0
$328$	2.06519e11	0.985206
$329$	−3.26889e10	−0.153822
$330$	0	0
$331$	−3.93033e11	−1.79971	−0.899857	−	0.436185i	$-0.856329\pi$
$331$	−3.93033e11	−1.79971	−0.899857	+	0.436185i	$0.856329\pi$
$332$	−1.19287e10	−0.0538856
$333$	0	0
$334$	−3.85888e11	−1.69669
$335$	2.69419e11	1.16876
$336$	0	0
$337$	−1.67090e11	−0.705695	−0.352847	−	0.935681i	$-0.614787\pi$
$337$	−1.67090e11	−0.705695	−0.352847	+	0.935681i	$0.614787\pi$
$338$	−2.39712e11	−0.998997
$339$	0	0
$340$	−3.88820e10	−0.157795
$341$	3.11415e11	1.24723
$342$	0	0
$343$	1.05835e11	0.412862
$344$	9.17967e10	0.353439
$345$	0	0
$346$	2.12260e11	0.796207
$347$	3.23689e11	1.19852	0.599261	−	0.800554i	$-0.295461\pi$
$347$	3.23689e11	1.19852	0.599261	+	0.800554i	$0.295461\pi$
$348$	0	0
$349$	−3.11506e11	−1.12396	−0.561981	−	0.827150i	$-0.689961\pi$
$349$	−3.11506e11	−1.12396	−0.561981	+	0.827150i	$0.689961\pi$
$350$	−1.10452e12	−3.93431
$351$	0	0
$352$	9.06335e10	0.314664
$353$	−2.07753e11	−0.712134	−0.356067	−	0.934461i	$-0.615882\pi$
$353$	−2.07753e11	−0.712134	−0.356067	+	0.934461i	$0.615882\pi$
$354$	0	0
$355$	−2.79672e11	−0.934592
$356$	2.71434e10	0.0895653
$357$	0	0
$358$	−3.11443e11	−1.00209
$359$	−1.67386e11	−0.531855	−0.265928	−	0.963993i	$-0.585678\pi$
$359$	−1.67386e11	−0.531855	−0.265928	+	0.963993i	$0.585678\pi$
$360$	0	0
$361$	−2.42670e11	−0.752027
$362$	−5.17694e11	−1.58447
$363$	0	0
$364$	−9.26808e9	−0.0276716
$365$	−1.47837e10	−0.0435978
$366$	0	0
$367$	−3.29735e11	−0.948785	−0.474393	−	0.880313i	$-0.657332\pi$
$367$	−3.29735e11	−0.948785	−0.474393	+	0.880313i	$0.657332\pi$
$368$	−1.63675e11	−0.465229
$369$	0	0
$370$	4.74876e10	0.131726
$371$	4.43162e11	1.21445
$372$	0	0
$373$	−6.21116e11	−1.66143	−0.830717	−	0.556695i	$-0.812069\pi$
$373$	−6.21116e11	−1.66143	−0.830717	+	0.556695i	$0.812069\pi$
$374$	5.15368e11	1.36206
$375$	0	0
$376$	−4.34358e10	−0.112073
$377$	−5.59000e10	−0.142520
$378$	0	0
$379$	−3.01833e10	−0.0751433	−0.0375717	−	0.999294i	$-0.511962\pi$
$379$	−3.01833e10	−0.0751433	−0.0375717	+	0.999294i	$0.511962\pi$
$380$	−3.93709e10	−0.0968611
$381$	0	0
$382$	7.72548e11	1.85627
$383$	4.46863e11	1.06116	0.530579	−	0.847636i	$-0.321975\pi$
$383$	4.46863e11	1.06116	0.530579	+	0.847636i	$0.321975\pi$
$384$	0	0
$385$	1.76729e12	4.09953
$386$	2.85641e11	0.654904
$387$	0	0
$388$	3.58983e10	0.0804139
$389$	2.16823e10	0.0480100	0.0240050	−	0.999712i	$-0.492358\pi$
$389$	2.16823e10	0.0480100	0.0240050	+	0.999712i	$0.492358\pi$
$390$	0	0
$391$	−1.60192e11	−0.346613
$392$	−3.01079e11	−0.644010
$393$	0	0
$394$	9.77855e10	0.204428
$395$	5.11230e11	1.05665
$396$	0	0
$397$	−4.74648e11	−0.958990	−0.479495	−	0.877545i	$-0.659180\pi$
$397$	−4.74648e11	−0.958990	−0.479495	+	0.877545i	$0.659180\pi$
$398$	2.67223e11	0.533827
$399$	0	0
$400$	−1.61372e12	−3.15180
$401$	−6.33960e11	−1.22437	−0.612185	−	0.790715i	$-0.709709\pi$
$401$	−6.33960e11	−1.22437	−0.612185	+	0.790715i	$0.709709\pi$
$402$	0	0
$403$	−8.92555e10	−0.168563
$404$	6.66521e10	0.124479
$405$	0	0
$406$	4.89852e11	0.894742
$407$	−5.64728e10	−0.102015
$408$	0	0
$409$	−1.41003e11	−0.249158	−0.124579	−	0.992210i	$-0.539758\pi$
$409$	−1.41003e11	−0.249158	−0.124579	+	0.992210i	$0.539758\pi$
$410$	−1.23410e12	−2.15687
$411$	0	0
$412$	−8.04461e10	−0.137552
$413$	−7.47253e11	−1.26384
$414$	0	0
$415$	−6.51938e11	−1.07892
$416$	−2.59767e10	−0.0425269
$417$	0	0
$418$	5.21849e11	0.836088
$419$	6.27484e11	0.994580	0.497290	−	0.867584i	$-0.334329\pi$
$419$	6.27484e11	0.994580	0.497290	+	0.867584i	$0.334329\pi$
$420$	0	0
$421$	−1.06714e10	−0.0165559	−0.00827795	−	0.999966i	$-0.502635\pi$
$421$	−1.06714e10	−0.0165559	−0.00827795	+	0.999966i	$0.502635\pi$
$422$	−9.59151e11	−1.47225
$423$	0	0
$424$	5.88857e11	0.884838
$425$	−1.57938e12	−2.34821
$426$	0	0
$427$	1.15620e12	1.68310
$428$	−7.20036e10	−0.103719
$429$	0	0
$430$	−5.48554e11	−0.773769
$431$	8.33459e11	1.16342	0.581710	−	0.813396i	$-0.302384\pi$
$431$	8.33459e11	1.16342	0.581710	+	0.813396i	$0.302384\pi$
$432$	0	0
$433$	1.25996e12	1.72251	0.861253	−	0.508177i	$-0.169680\pi$
$433$	1.25996e12	1.72251	0.861253	+	0.508177i	$0.169680\pi$
$434$	7.82147e11	1.05824
$435$	0	0
$436$	1.07027e11	0.141842
$437$	−1.62206e11	−0.212765
$438$	0	0
$439$	3.79345e11	0.487466	0.243733	−	0.969842i	$-0.421628\pi$
$439$	3.79345e11	0.487466	0.243733	+	0.969842i	$0.421628\pi$
$440$	2.34831e12	2.98688
$441$	0	0
$442$	−1.47711e11	−0.184082
$443$	−3.72934e11	−0.460061	−0.230030	−	0.973183i	$-0.573883\pi$
$443$	−3.72934e11	−0.460061	−0.230030	+	0.973183i	$0.573883\pi$
$444$	0	0
$445$	1.48347e12	1.79332
$446$	−8.36233e11	−1.00074
$447$	0	0
$448$	−9.76245e11	−1.14501
$449$	−9.47204e10	−0.109985	−0.0549927	−	0.998487i	$-0.517514\pi$
$449$	−9.47204e10	−0.109985	−0.0549927	+	0.998487i	$0.517514\pi$
$450$	0	0
$451$	1.46761e12	1.67039
$452$	−7.19783e10	−0.0811108
$453$	0	0
$454$	−5.06678e11	−0.559732
$455$	−5.06527e11	−0.554053
$456$	0	0
$457$	9.25235e11	0.992269	0.496134	−	0.868246i	$-0.334752\pi$
$457$	9.25235e11	0.992269	0.496134	+	0.868246i	$0.334752\pi$
$458$	1.67438e12	1.77812
$459$	0	0
$460$	7.98100e10	0.0831088
$461$	8.29299e11	0.855179	0.427589	−	0.903973i	$-0.359363\pi$
$461$	8.29299e11	0.855179	0.427589	+	0.903973i	$0.359363\pi$
$462$	0	0
$463$	1.28894e12	1.30352	0.651761	−	0.758424i	$-0.274030\pi$
$463$	1.28894e12	1.30352	0.651761	+	0.758424i	$0.274030\pi$
$464$	7.15681e11	0.716784
$465$	0	0
$466$	1.59265e12	1.56453
$467$	−8.24491e11	−0.802158	−0.401079	−	0.916044i	$-0.631365\pi$
$467$	−8.24491e11	−0.802158	−0.401079	+	0.916044i	$0.631365\pi$
$468$	0	0
$469$	−8.04704e11	−0.767995
$470$	2.59561e11	0.245358
$471$	0	0
$472$	−9.92922e11	−0.920823
$473$	6.52347e11	0.599244
$474$	0	0
$475$	−1.59924e12	−1.44143
$476$	1.16133e11	0.103687
$477$	0	0
$478$	−1.97219e12	−1.72792
$479$	3.04858e11	0.264599	0.132300	−	0.991210i	$-0.457764\pi$
$479$	3.04858e11	0.264599	0.132300	+	0.991210i	$0.457764\pi$
$480$	0	0
$481$	1.61858e10	0.0137874
$482$	−1.11384e12	−0.939964
$483$	0	0
$484$	1.86355e11	0.154361
$485$	1.96194e12	1.61008
$486$	0	0
$487$	−2.42855e12	−1.95644	−0.978222	−	0.207563i	$-0.933447\pi$
$487$	−2.42855e12	−1.95644	−0.978222	+	0.207563i	$0.933447\pi$
$488$	1.53632e12	1.22629
$489$	0	0
$490$	1.79917e12	1.40990
$491$	1.40710e12	1.09260	0.546298	−	0.837591i	$-0.316037\pi$
$491$	1.40710e12	1.09260	0.546298	+	0.837591i	$0.316037\pi$
$492$	0	0
$493$	7.00451e11	0.534031
$494$	−1.49568e11	−0.112997
$495$	0	0
$496$	1.14273e12	0.847764
$497$	8.35329e11	0.614121
$498$	0	0
$499$	−1.72797e11	−0.124762	−0.0623810	−	0.998052i	$-0.519869\pi$
$499$	−1.72797e11	−0.124762	−0.0623810	+	0.998052i	$0.519869\pi$
$500$	5.15032e11	0.368527
$501$	0	0
$502$	−5.15646e11	−0.362397
$503$	−9.49276e11	−0.661206	−0.330603	−	0.943770i	$-0.607252\pi$
$503$	−9.49276e11	−0.661206	−0.330603	+	0.943770i	$0.607252\pi$
$504$	0	0
$505$	3.64272e12	2.49238
$506$	−1.05786e12	−0.717380
$507$	0	0
$508$	−1.97827e11	−0.131795
$509$	−2.16570e12	−1.43011	−0.715054	−	0.699069i	$-0.753598\pi$
$509$	−2.16570e12	−1.43011	−0.715054	+	0.699069i	$0.753598\pi$
$510$	0	0
$511$	4.41561e10	0.0286481
$512$	−1.26709e12	−0.814880
$513$	0	0
$514$	−6.56056e10	−0.0414579
$515$	−4.39661e12	−2.75413
$516$	0	0
$517$	−3.08674e11	−0.190017
$518$	−1.41837e11	−0.0865574
$519$	0	0
$520$	−6.73054e11	−0.403678
$521$	−5.46646e11	−0.325040	−0.162520	−	0.986705i	$-0.551962\pi$
$521$	−5.46646e11	−0.325040	−0.162520	+	0.986705i	$0.551962\pi$
$522$	0	0
$523$	8.52704e11	0.498357	0.249178	−	0.968458i	$-0.419839\pi$
$523$	8.52704e11	0.498357	0.249178	+	0.968458i	$0.419839\pi$
$524$	1.95471e11	0.113264
$525$	0	0
$526$	−2.32997e12	−1.32713
$527$	1.11841e12	0.631616
$528$	0	0
$529$	−1.47234e12	−0.817443
$530$	−3.51886e12	−1.93714
$531$	0	0
$532$	1.17594e11	0.0636475
$533$	−4.20636e11	−0.225753
$534$	0	0
$535$	−3.93520e12	−2.07671
$536$	−1.06926e12	−0.559554
$537$	0	0
$538$	4.17398e11	0.214798
$539$	−2.13959e12	−1.09190
$540$	0	0
$541$	2.62781e12	1.31888	0.659442	−	0.751755i	$-0.270793\pi$
$541$	2.62781e12	1.31888	0.659442	+	0.751755i	$0.270793\pi$
$542$	5.27378e11	0.262497
$543$	0	0
$544$	3.25499e11	0.159351
$545$	5.84934e12	2.84003
$546$	0	0
$547$	1.44897e12	0.692017	0.346008	−	0.938231i	$-0.387537\pi$
$547$	1.44897e12	0.692017	0.346008	+	0.938231i	$0.387537\pi$
$548$	1.02386e11	0.0484985
$549$	0	0
$550$	−1.04297e13	−4.86006
$551$	7.09259e11	0.327811
$552$	0	0
$553$	−1.52695e12	−0.694323
$554$	−2.38723e12	−1.07672
$555$	0	0
$556$	2.65218e10	0.0117697
$557$	−2.91860e12	−1.28477	−0.642387	−	0.766381i	$-0.722056\pi$
$557$	−2.91860e12	−1.28477	−0.642387	+	0.766381i	$0.722056\pi$
$558$	0	0
$559$	−1.86971e11	−0.0809880
$560$	6.48501e12	2.78653
$561$	0	0
$562$	−6.74066e11	−0.285029
$563$	−1.02936e12	−0.431797	−0.215899	−	0.976416i	$-0.569268\pi$
$563$	−1.02936e12	−0.431797	−0.215899	+	0.976416i	$0.569268\pi$
$564$	0	0
$565$	−3.93382e12	−1.62404
$566$	1.02068e12	0.418039
$567$	0	0
$568$	1.10995e12	0.447443
$569$	−3.66470e12	−1.46566	−0.732830	−	0.680411i	$-0.761801\pi$
$569$	−3.66470e12	−1.46566	−0.732830	+	0.680411i	$0.761801\pi$
$570$	0	0
$571$	8.50009e11	0.334627	0.167314	−	0.985904i	$-0.446491\pi$
$571$	8.50009e11	0.334627	0.167314	+	0.985904i	$0.446491\pi$
$572$	−8.75163e10	−0.0341827
$573$	0	0
$574$	3.68604e12	1.41728
$575$	3.24187e12	1.23678
$576$	0	0
$577$	−1.82898e12	−0.686938	−0.343469	−	0.939164i	$-0.611602\pi$
$577$	−1.82898e12	−0.686938	−0.343469	+	0.939164i	$0.611602\pi$
$578$	−9.61586e11	−0.358355
$579$	0	0
$580$	−3.48975e11	−0.128047
$581$	1.94722e12	0.708960
$582$	0	0
$583$	4.18467e12	1.50021
$584$	5.86729e10	0.0208728
$585$	0	0
$586$	−4.11033e11	−0.143992
$587$	−2.04313e12	−0.710271	−0.355135	−	0.934815i	$-0.615565\pi$
$587$	−2.04313e12	−0.710271	−0.355135	+	0.934815i	$0.615565\pi$
$588$	0	0
$589$	1.13247e12	0.387712
$590$	5.93345e12	2.01592
$591$	0	0
$592$	−2.07225e11	−0.0693417
$593$	−4.13707e12	−1.37387	−0.686937	−	0.726717i	$-0.741045\pi$
$593$	−4.13707e12	−1.37387	−0.686937	+	0.726717i	$0.741045\pi$
$594$	0	0
$595$	6.34700e12	2.07607
$596$	−1.63357e11	−0.0530309
$597$	0	0
$598$	3.03195e11	0.0969541
$599$	−1.61690e12	−0.513170	−0.256585	−	0.966522i	$-0.582597\pi$
$599$	−1.61690e12	−0.513170	−0.256585	+	0.966522i	$0.582597\pi$
$600$	0	0
$601$	−2.26777e12	−0.709030	−0.354515	−	0.935050i	$-0.615354\pi$
$601$	−2.26777e12	−0.709030	−0.354515	+	0.935050i	$0.615354\pi$
$602$	1.63843e12	0.508444
$603$	0	0
$604$	9.64787e10	0.0294961
$605$	1.01848e13	3.09068
$606$	0	0
$607$	1.88066e12	0.562290	0.281145	−	0.959665i	$-0.409286\pi$
$607$	1.88066e12	0.562290	0.281145	+	0.959665i	$0.409286\pi$
$608$	3.29592e11	0.0978163
$609$	0	0
$610$	−9.18066e12	−2.68466
$611$	8.84697e10	0.0256808
$612$	0	0
$613$	−5.78740e12	−1.65543	−0.827717	−	0.561146i	$-0.810360\pi$
$613$	−5.78740e12	−1.65543	−0.827717	+	0.561146i	$0.810360\pi$
$614$	−4.82080e12	−1.36887
$615$	0	0
$616$	−7.01395e12	−1.96268
$617$	1.70164e12	0.472699	0.236349	−	0.971668i	$-0.424049\pi$
$617$	1.70164e12	0.472699	0.236349	+	0.971668i	$0.424049\pi$
$618$	0	0
$619$	−5.62977e12	−1.54128	−0.770642	−	0.637268i	$-0.780065\pi$
$619$	−5.62977e12	−1.54128	−0.770642	+	0.637268i	$0.780065\pi$
$620$	−5.57209e11	−0.151445
$621$	0	0
$622$	6.22714e12	1.66814
$623$	−4.43084e12	−1.17839
$624$	0	0
$625$	1.71059e13	4.48420
$626$	4.41229e11	0.114836
$627$	0	0
$628$	6.08411e11	0.156091
$629$	−2.02815e11	−0.0516622
$630$	0	0
$631$	−2.93043e12	−0.735868	−0.367934	−	0.929852i	$-0.619935\pi$
$631$	−2.93043e12	−0.735868	−0.367934	+	0.929852i	$0.619935\pi$
$632$	−2.02895e12	−0.505877
$633$	0	0
$634$	6.13361e12	1.50770
$635$	−1.08118e13	−2.63886
$636$	0	0
$637$	6.13234e11	0.147570
$638$	4.62556e12	1.10528
$639$	0	0
$640$	9.39705e12	2.21402
$641$	5.53422e12	1.29478	0.647389	−	0.762160i	$-0.275861\pi$
$641$	5.53422e12	1.29478	0.647389	+	0.762160i	$0.275861\pi$
$642$	0	0
$643$	−7.28546e12	−1.68077	−0.840383	−	0.541993i	$-0.817670\pi$
$643$	−7.28546e12	−1.68077	−0.840383	+	0.541993i	$0.817670\pi$
$644$	−2.38377e11	−0.0546108
$645$	0	0
$646$	1.87416e12	0.423409
$647$	5.73701e12	1.28711	0.643556	−	0.765399i	$-0.277458\pi$
$647$	5.73701e12	1.28711	0.643556	+	0.765399i	$0.277458\pi$
$648$	0	0
$649$	−7.05613e12	−1.56123
$650$	2.98929e12	0.656838
$651$	0	0
$652$	−1.11458e11	−0.0241544
$653$	−3.54942e12	−0.763920	−0.381960	−	0.924179i	$-0.624751\pi$
$653$	−3.54942e12	−0.763920	−0.381960	+	0.924179i	$0.624751\pi$
$654$	0	0
$655$	1.06830e13	2.26782
$656$	5.38535e12	1.13539
$657$	0	0
$658$	−7.75261e11	−0.161225
$659$	−3.26894e12	−0.675185	−0.337593	−	0.941292i	$-0.609613\pi$
$659$	−3.26894e12	−0.675185	−0.337593	+	0.941292i	$0.609613\pi$
$660$	0	0
$661$	−7.06519e12	−1.43952	−0.719760	−	0.694223i	$-0.755748\pi$
$661$	−7.06519e12	−1.43952	−0.719760	+	0.694223i	$0.755748\pi$
$662$	−9.32131e12	−1.88632
$663$	0	0
$664$	2.58739e12	0.516542
$665$	6.42682e12	1.27438
$666$	0	0
$667$	−1.43776e12	−0.281268
$668$	−8.21106e11	−0.159553
$669$	0	0
$670$	6.38963e12	1.22501
$671$	1.09178e13	2.07913
$672$	0	0
$673$	7.72311e12	1.45119	0.725595	−	0.688122i	$-0.241565\pi$
$673$	7.72311e12	1.45119	0.725595	+	0.688122i	$0.241565\pi$
$674$	−3.96277e12	−0.739656
$675$	0	0
$676$	−5.10066e11	−0.0939434
$677$	−3.13880e12	−0.574268	−0.287134	−	0.957890i	$-0.592703\pi$
$677$	−3.13880e12	−0.574268	−0.287134	+	0.957890i	$0.592703\pi$
$678$	0	0
$679$	−5.85996e12	−1.05799
$680$	8.43366e12	1.51261
$681$	0	0
$682$	7.38563e12	1.30725
$683$	−8.82159e11	−0.155115	−0.0775575	−	0.996988i	$-0.524712\pi$
$683$	−8.82159e11	−0.155115	−0.0775575	+	0.996988i	$0.524712\pi$
$684$	0	0
$685$	5.59568e12	0.971059
$686$	2.51001e12	0.432731
$687$	0	0
$688$	2.39377e12	0.407318
$689$	−1.19938e12	−0.202754
$690$	0	0
$691$	−6.02737e12	−1.00572	−0.502860	−	0.864368i	$-0.667719\pi$
$691$	−6.02737e12	−1.00572	−0.502860	+	0.864368i	$0.667719\pi$
$692$	4.51654e11	0.0748735
$693$	0	0
$694$	7.67672e12	1.25620
$695$	1.44949e12	0.235659
$696$	0	0
$697$	5.27075e12	0.845912
$698$	−7.38778e12	−1.17805
$699$	0	0
$700$	−2.35024e12	−0.369974
$701$	−1.41432e11	−0.0221216	−0.0110608	−	0.999939i	$-0.503521\pi$
$701$	−1.41432e11	−0.0221216	−0.0110608	+	0.999939i	$0.503521\pi$
$702$	0	0
$703$	−2.05366e11	−0.0317124
$704$	−9.21845e12	−1.41443
$705$	0	0
$706$	−4.92714e12	−0.746404
$707$	−1.08801e13	−1.63775
$708$	0	0
$709$	3.93409e12	0.584704	0.292352	−	0.956311i	$-0.405562\pi$
$709$	3.93409e12	0.584704	0.292352	+	0.956311i	$0.405562\pi$
$710$	−6.63280e12	−0.979568
$711$	0	0
$712$	−5.88753e12	−0.858564
$713$	−2.29567e12	−0.332665
$714$	0	0
$715$	−4.78301e12	−0.684422
$716$	−6.62699e11	−0.0942340
$717$	0	0
$718$	−3.96978e12	−0.557450
$719$	6.55887e12	0.915270	0.457635	−	0.889140i	$-0.348697\pi$
$719$	6.55887e12	0.915270	0.457635	+	0.889140i	$0.348697\pi$
$720$	0	0
$721$	1.31318e13	1.80974
$722$	−5.75524e12	−0.788218
$723$	0	0
$724$	−1.10157e12	−0.149000
$725$	−1.41753e13	−1.90552
$726$	0	0
$727$	4.37762e12	0.581210	0.290605	−	0.956843i	$-0.406144\pi$
$727$	4.37762e12	0.581210	0.290605	+	0.956843i	$0.406144\pi$
$728$	2.01029e12	0.265257
$729$	0	0
$730$	−3.50614e11	−0.0456959
$731$	2.34283e12	0.303467
$732$	0	0
$733$	−2.26970e12	−0.290402	−0.145201	−	0.989402i	$-0.546383\pi$
$733$	−2.26970e12	−0.290402	−0.145201	+	0.989402i	$0.546383\pi$
$734$	−7.82011e12	−0.994444
$735$	0	0
$736$	−6.68127e11	−0.0839284
$737$	−7.59863e12	−0.948706
$738$	0	0
$739$	−9.68674e12	−1.19475	−0.597376	−	0.801961i	$-0.703790\pi$
$739$	−9.68674e12	−1.19475	−0.597376	+	0.801961i	$0.703790\pi$
$740$	1.01046e11	0.0123872
$741$	0	0
$742$	1.05102e13	1.27290
$743$	1.22572e13	1.47550	0.737752	−	0.675071i	$-0.235887\pi$
$743$	1.22572e13	1.47550	0.737752	+	0.675071i	$0.235887\pi$
$744$	0	0
$745$	−8.92792e12	−1.06181
$746$	−1.47306e13	−1.74139
$747$	0	0
$748$	1.09662e12	0.128085
$749$	1.17537e13	1.36460
$750$	0	0
$751$	−1.28832e13	−1.47790	−0.738948	−	0.673763i	$-0.764677\pi$
$751$	−1.28832e13	−1.47790	−0.738948	+	0.673763i	$0.764677\pi$
$752$	−1.13267e12	−0.129158
$753$	0	0
$754$	−1.32574e12	−0.149378
$755$	5.27283e12	0.590585
$756$	0	0
$757$	−4.80929e12	−0.532291	−0.266145	−	0.963933i	$-0.585750\pi$
$757$	−4.80929e12	−0.532291	−0.266145	+	0.963933i	$0.585750\pi$
$758$	−7.15837e11	−0.0787595
$759$	0	0
$760$	8.53971e12	0.928500
$761$	1.65944e13	1.79362	0.896812	−	0.442412i	$-0.145877\pi$
$761$	1.65944e13	1.79362	0.896812	+	0.442412i	$0.145877\pi$
$762$	0	0
$763$	−1.74709e13	−1.86619
$764$	1.64385e12	0.174559
$765$	0	0
$766$	1.05979e13	1.11222
$767$	2.02238e12	0.211000
$768$	0	0
$769$	−7.94565e12	−0.819334	−0.409667	−	0.912235i	$-0.634355\pi$
$769$	−7.94565e12	−0.819334	−0.409667	+	0.912235i	$0.634355\pi$
$770$	4.19136e13	4.29682
$771$	0	0
$772$	6.07796e11	0.0615857
$773$	1.28741e12	0.129690	0.0648452	−	0.997895i	$-0.479345\pi$
$773$	1.28741e12	0.129690	0.0648452	+	0.997895i	$0.479345\pi$
$774$	0	0
$775$	−2.26338e13	−2.25372
$776$	−7.78649e12	−0.770840
$777$	0	0
$778$	5.14224e11	0.0503204
$779$	5.33703e12	0.519256
$780$	0	0
$781$	7.88781e12	0.758625
$782$	−3.79916e12	−0.363293
$783$	0	0
$784$	−7.85117e12	−0.742185
$785$	3.32514e13	3.12533
$786$	0	0
$787$	1.64637e13	1.52982	0.764911	−	0.644136i	$-0.222783\pi$
$787$	1.64637e13	1.52982	0.764911	+	0.644136i	$0.222783\pi$
$788$	2.08071e11	0.0192240
$789$	0	0
$790$	1.21245e13	1.10750
$791$	1.17496e13	1.06716
$792$	0	0
$793$	−3.12917e12	−0.280995
$794$	−1.12569e13	−1.00514
$795$	0	0
$796$	5.68606e11	0.0501999
$797$	−1.52321e13	−1.33720	−0.668602	−	0.743621i	$-0.733107\pi$
$797$	−1.52321e13	−1.33720	−0.668602	+	0.743621i	$0.733107\pi$
$798$	0	0
$799$	−1.10856e12	−0.0962277
$800$	−6.58727e12	−0.568592
$801$	0	0
$802$	−1.50352e13	−1.28329
$803$	4.16955e11	0.0353891
$804$	0	0
$805$	−1.30280e13	−1.09344
$806$	−2.11681e12	−0.176675
$807$	0	0
$808$	−1.44571e13	−1.19325
$809$	−6.05897e12	−0.497313	−0.248657	−	0.968592i	$-0.579989\pi$
$809$	−6.05897e12	−0.497313	−0.248657	+	0.968592i	$0.579989\pi$
$810$	0	0
$811$	1.47666e13	1.19863	0.599317	−	0.800512i	$-0.295439\pi$
$811$	1.47666e13	1.19863	0.599317	+	0.800512i	$0.295439\pi$
$812$	1.04232e12	0.0841396
$813$	0	0
$814$	−1.33933e12	−0.106925
$815$	−6.09147e12	−0.483630
$816$	0	0
$817$	2.37229e12	0.186281
$818$	−3.34408e12	−0.261148
$819$	0	0
$820$	−2.62597e12	−0.202828
$821$	−7.00160e12	−0.537840	−0.268920	−	0.963163i	$-0.586667\pi$
$821$	−7.00160e12	−0.537840	−0.268920	+	0.963163i	$0.586667\pi$
$822$	0	0
$823$	1.18700e13	0.901889	0.450944	−	0.892552i	$-0.351087\pi$
$823$	1.18700e13	0.901889	0.450944	+	0.892552i	$0.351087\pi$
$824$	1.74491e13	1.31856
$825$	0	0
$826$	−1.77221e13	−1.32466
$827$	−2.53412e13	−1.88388	−0.941938	−	0.335788i	$-0.890998\pi$
$827$	−2.53412e13	−1.88388	−0.941938	+	0.335788i	$0.890998\pi$
$828$	0	0
$829$	7.44726e12	0.547647	0.273824	−	0.961780i	$-0.411712\pi$
$829$	7.44726e12	0.547647	0.273824	+	0.961780i	$0.411712\pi$
$830$	−1.54616e13	−1.13084
$831$	0	0
$832$	2.64212e12	0.191160
$833$	−7.68410e12	−0.552956
$834$	0	0
$835$	−4.48758e13	−3.19465
$836$	1.11041e12	0.0786238
$837$	0	0
$838$	1.48816e13	1.04244
$839$	1.75645e13	1.22379	0.611895	−	0.790939i	$-0.290408\pi$
$839$	1.75645e13	1.22379	0.611895	+	0.790939i	$0.290408\pi$
$840$	0	0
$841$	−8.22043e12	−0.566647
$842$	−2.53087e11	−0.0173526
$843$	0	0
$844$	−2.04091e12	−0.138447
$845$	−2.78766e13	−1.88098
$846$	0	0
$847$	−3.04201e13	−2.03089
$848$	1.53555e13	1.01973
$849$	0	0
$850$	−3.74571e13	−2.46121
$851$	4.16303e11	0.0272099
$852$	0	0
$853$	1.77887e13	1.15046	0.575232	−	0.817990i	$-0.304912\pi$
$853$	1.77887e13	1.15046	0.575232	+	0.817990i	$0.304912\pi$
$854$	2.74209e13	1.76409
$855$	0	0
$856$	1.56179e13	0.994238
$857$	−1.35605e13	−0.858738	−0.429369	−	0.903129i	$-0.641264\pi$
$857$	−1.35605e13	−0.858738	−0.429369	+	0.903129i	$0.641264\pi$
$858$	0	0
$859$	1.41652e13	0.887674	0.443837	−	0.896107i	$-0.353617\pi$
$859$	1.41652e13	0.887674	0.443837	+	0.896107i	$0.353617\pi$
$860$	−1.16723e12	−0.0727635
$861$	0	0
$862$	1.97666e13	1.21941
$863$	6.29108e12	0.386079	0.193040	−	0.981191i	$-0.438165\pi$
$863$	6.29108e12	0.386079	0.193040	+	0.981191i	$0.438165\pi$
$864$	0	0
$865$	2.46842e13	1.49915
$866$	2.98816e13	1.80540
$867$	0	0
$868$	1.66428e12	0.0995147
$869$	−1.44186e13	−0.857698
$870$	0	0
$871$	2.17786e12	0.128218
$872$	−2.32147e13	−1.35968
$873$	0	0
$874$	−3.84694e12	−0.223005
$875$	−8.40727e13	−4.84863
$876$	0	0
$877$	1.28376e13	0.732800	0.366400	−	0.930457i	$-0.380590\pi$
$877$	1.28376e13	0.732800	0.366400	+	0.930457i	$0.380590\pi$
$878$	8.99667e12	0.510924
$879$	0	0
$880$	6.12364e13	3.44221
$881$	1.70945e13	0.956013	0.478006	−	0.878356i	$-0.341360\pi$
$881$	1.70945e13	0.956013	0.478006	+	0.878356i	$0.341360\pi$
$882$	0	0
$883$	−3.34104e12	−0.184952	−0.0924758	−	0.995715i	$-0.529478\pi$
$883$	−3.34104e12	−0.184952	−0.0924758	+	0.995715i	$0.529478\pi$
$884$	−3.14304e11	−0.0173107
$885$	0	0
$886$	−8.84462e12	−0.482200
$887$	1.76903e13	0.959574	0.479787	−	0.877385i	$-0.340714\pi$
$887$	1.76903e13	0.959574	0.479787	+	0.877385i	$0.340714\pi$
$888$	0	0
$889$	3.22929e13	1.73400
$890$	3.51824e13	1.87962
$891$	0	0
$892$	−1.77936e12	−0.0941072
$893$	−1.12250e12	−0.0590686
$894$	0	0
$895$	−3.62184e13	−1.88680
$896$	−2.80672e13	−1.45483
$897$	0	0
$898$	−2.24642e12	−0.115278
$899$	1.00380e13	0.512541
$900$	0	0
$901$	1.50287e13	0.759734
$902$	3.48064e13	1.75077
$903$	0	0
$904$	1.56124e13	0.777520
$905$	−6.02037e13	−2.98335
$906$	0	0
$907$	−1.47850e13	−0.725418	−0.362709	−	0.931902i	$-0.618148\pi$
$907$	−1.47850e13	−0.725418	−0.362709	+	0.931902i	$0.618148\pi$
$908$	−1.07813e12	−0.0526360
$909$	0	0
$910$	−1.20130e13	−0.580716
$911$	2.68687e13	1.29245	0.646225	−	0.763147i	$-0.276347\pi$
$911$	2.68687e13	1.29245	0.646225	+	0.763147i	$0.276347\pi$
$912$	0	0
$913$	1.83871e13	0.875780
$914$	2.19432e13	1.04002
$915$	0	0
$916$	3.56281e12	0.167210
$917$	−3.19082e13	−1.49018
$918$	0	0
$919$	−3.94629e13	−1.82503	−0.912514	−	0.409046i	$-0.865861\pi$
$919$	−3.94629e13	−1.82503	−0.912514	+	0.409046i	$0.865861\pi$
$920$	−1.73111e13	−0.796672
$921$	0	0
$922$	1.96679e13	0.896333
$923$	−2.26074e12	−0.102528
$924$	0	0
$925$	4.10446e12	0.184340
$926$	3.05689e13	1.36625
$927$	0	0
$928$	2.92144e12	0.129310
$929$	1.25262e12	0.0551760	0.0275880	−	0.999619i	$-0.491217\pi$
$929$	1.25262e12	0.0551760	0.0275880	+	0.999619i	$0.491217\pi$
$930$	0	0
$931$	−7.78072e12	−0.339427
$932$	3.38889e12	0.147125
$933$	0	0
$934$	−1.95539e13	−0.840760
$935$	5.99332e13	2.56458
$936$	0	0
$937$	1.15393e13	0.489046	0.244523	−	0.969644i	$-0.421369\pi$
$937$	1.15393e13	0.489046	0.244523	+	0.969644i	$0.421369\pi$
$938$	−1.90846e13	−0.804954
$939$	0	0
$940$	5.52303e11	0.0230729
$941$	2.36134e13	0.981760	0.490880	−	0.871227i	$-0.336675\pi$
$941$	2.36134e13	0.981760	0.490880	+	0.871227i	$0.336675\pi$
$942$	0	0
$943$	−1.08189e13	−0.445532
$944$	−2.58922e13	−1.06120
$945$	0	0
$946$	1.54713e13	0.628082
$947$	7.08605e12	0.286305	0.143153	−	0.989701i	$-0.454276\pi$
$947$	7.08605e12	0.286305	0.143153	+	0.989701i	$0.454276\pi$
$948$	0	0
$949$	−1.19504e11	−0.00478284
$950$	−3.79282e13	−1.51080
$951$	0	0
$952$	−2.51898e13	−0.993935
$953$	−9.78924e12	−0.384442	−0.192221	−	0.981352i	$-0.561569\pi$
$953$	−9.78924e12	−0.384442	−0.192221	+	0.981352i	$0.561569\pi$
$954$	0	0
$955$	8.98412e13	3.49511
$956$	−4.19650e12	−0.162490
$957$	0	0
$958$	7.23012e12	0.277333
$959$	−1.67133e13	−0.638084
$960$	0	0
$961$	−1.04119e13	−0.393801
$962$	3.83868e11	0.0144509
$963$	0	0
$964$	−2.37006e12	−0.0883921
$965$	3.32178e13	1.23310
$966$	0	0
$967$	3.95029e13	1.45281	0.726406	−	0.687265i	$-0.241189\pi$
$967$	3.95029e13	1.45281	0.726406	+	0.687265i	$0.241189\pi$
$968$	−4.04211e13	−1.47969
$969$	0	0
$970$	4.65301e13	1.68757
$971$	4.12585e13	1.48945	0.744727	−	0.667369i	$-0.232580\pi$
$971$	4.12585e13	1.48945	0.744727	+	0.667369i	$0.232580\pi$
$972$	0	0
$973$	−4.32937e12	−0.154852
$974$	−5.75964e13	−2.05059
$975$	0	0
$976$	4.00624e13	1.41323
$977$	2.31785e13	0.813881	0.406940	−	0.913455i	$-0.366596\pi$
$977$	2.31785e13	0.813881	0.406940	+	0.913455i	$0.366596\pi$
$978$	0	0
$979$	−4.18393e13	−1.45567
$980$	3.82833e12	0.132584
$981$	0	0
$982$	3.33714e13	1.14518
$983$	−1.46986e13	−0.502093	−0.251047	−	0.967975i	$-0.580775\pi$
$983$	−1.46986e13	−0.502093	−0.251047	+	0.967975i	$0.580775\pi$
$984$	0	0
$985$	1.13717e13	0.384912
$986$	1.66121e13	0.559731
$987$	0	0
$988$	−3.18257e11	−0.0106260
$989$	−4.80893e12	−0.159833
$990$	0	0
$991$	1.69616e13	0.558646	0.279323	−	0.960197i	$-0.409890\pi$
$991$	1.69616e13	0.558646	0.279323	+	0.960197i	$0.409890\pi$
$992$	4.66466e12	0.152939
$993$	0	0
$994$	1.98109e13	0.643675
$995$	3.10759e13	1.00513
$996$	0	0
$997$	6.45914e12	0.207036	0.103518	−	0.994628i	$-0.466990\pi$
$997$	6.45914e12	0.207036	0.103518	+	0.994628i	$0.466990\pi$
$998$	−4.09810e12	−0.130766
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.10.a.b.1.3	yes	4
3.2	odd	2		81.10.a.a.1.2	✓	4
9.2	odd	6		81.10.c.k.28.3		8
9.4	even	3		81.10.c.i.55.2		8
9.5	odd	6		81.10.c.k.55.3		8
9.7	even	3		81.10.c.i.28.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.10.a.a.1.2	✓	4	3.2	odd	2
81.10.a.b.1.3	yes	4	1.1	even	1	trivial
81.10.c.i.28.2		8	9.7	even	3
81.10.c.i.55.2		8	9.4	even	3
81.10.c.k.28.3		8	9.2	odd	6
81.10.c.k.55.3		8	9.5	odd	6

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	81.a (trivial)

\(f(q)\)	\(=\)	\(q+23.7163 q^{2} +50.4644 q^{4} +2758.02 q^{5} -8237.69 q^{7} -10945.9 q^{8} +65410.1 q^{10} -77786.5 q^{11} +22294.6 q^{13} -195368. q^{14} -285435. q^{16} -279361. q^{17} -282874. q^{19} +139182. q^{20} -1.84481e6 q^{22} +573422. q^{23} +5.65355e6 q^{25} +528746. q^{26} -415710. q^{28} -2.50733e6 q^{29} -4.00346e6 q^{31} -1.16516e6 q^{32} -6.62542e6 q^{34} -2.27197e7 q^{35} +725997. q^{37} -6.70873e6 q^{38} -3.01891e7 q^{40} -1.88672e7 q^{41} -8.38638e6 q^{43} -3.92545e6 q^{44} +1.35995e7 q^{46} +3.96821e6 q^{47} +2.75060e7 q^{49} +1.34082e8 q^{50} +1.12508e6 q^{52} -5.37969e7 q^{53} -2.14537e8 q^{55} +9.01692e7 q^{56} -5.94647e7 q^{58} +9.07115e7 q^{59} -1.40355e8 q^{61} -9.49474e7 q^{62} +1.18510e8 q^{64} +6.14889e7 q^{65} +9.76857e7 q^{67} -1.40978e7 q^{68} -5.38829e8 q^{70} -1.01403e8 q^{71} -5.36025e6 q^{73} +1.72180e7 q^{74} -1.42751e7 q^{76} +6.40782e8 q^{77} +1.85361e8 q^{79} -7.87236e8 q^{80} -4.47460e8 q^{82} -2.36379e8 q^{83} -7.70483e8 q^{85} -1.98894e8 q^{86} +8.51446e8 q^{88} +5.37873e8 q^{89} -1.83656e8 q^{91} +2.89374e7 q^{92} +9.41115e7 q^{94} -7.80172e8 q^{95} +7.11359e8 q^{97} +6.52341e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 33 q^{2} + 853 q^{4} + 570 q^{5} - 3238 q^{7} + 4791 q^{8} - 9723 q^{10} - 96690 q^{11} - 141118 q^{13} + 3036 q^{14} - 244463 q^{16} + 285156 q^{17} - 465166 q^{19} - 1041711 q^{20} - 2244480 q^{22}+ \cdots + 735501321 q^{98}+O(q^{100}) \) 4 * q + 33 * q^2 + 853 * q^4 + 570 * q^5 - 3238 * q^7 + 4791 * q^8 - 9723 * q^10 - 96690 * q^11 - 141118 * q^13 + 3036 * q^14 - 244463 * q^16 + 285156 * q^17 - 465166 * q^19 - 1041711 * q^20 - 2244480 * q^22 - 237318 * q^23 + 3417712 * q^25 - 6433737 * q^26 + 769292 * q^28 + 2806926 * q^29 - 8244448 * q^31 + 3896415 * q^32 + 1386189 * q^34 - 31668402 * q^35 - 8204014 * q^37 - 755556 * q^38 - 54180645 * q^40 + 28889292 * q^41 - 39586246 * q^43 - 89567616 * q^44 - 93870624 * q^46 - 17514588 * q^47 - 67913400 * q^49 + 233832372 * q^50 - 207295237 * q^52 - 155826504 * q^53 - 185126142 * q^55 + 133671300 * q^56 - 164356143 * q^58 + 341256108 * q^59 - 238298254 * q^61 - 729569796 * q^62 - 286454975 * q^64 + 379881528 * q^65 - 72601486 * q^67 + 762353217 * q^68 - 780989460 * q^70 - 667543266 * q^71 - 520588288 * q^73 + 183473031 * q^74 - 413336884 * q^76 + 721530228 * q^77 - 137324662 * q^79 - 1181367123 * q^80 - 275042586 * q^82 - 278546844 * q^83 - 1408511358 * q^85 + 420627720 * q^86 - 30700416 * q^88 + 237712932 * q^89 - 762950942 * q^91 - 1238743680 * q^92 - 507333516 * q^94 - 811663194 * q^95 + 700071356 * q^97 + 735501321 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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