LMFDB - Embedded newform 27.14.a.b.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [27,14,Mod(1,27)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("27.1"); S:= CuspForms(chi, 14); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(27, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 14, names="a")

Level:	$ N $	$=$	$ 27 = 3^{3} $
Weight:	$ k $	$=$	$ 14 $
Character orbit:	$[\chi]$	$=$	27.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-117,0,8101] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	yes
Analytic conductor:	$28.9523508170$
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} - 2080x^{2} - 9500x + 13552 $ x^4 - x^3 - 2080x^2 - 9500x + 13552
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{8}\cdot 13 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$48.1619$ of defining polynomial
Character	$\chi$	$=$	27.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+114.486 q^{2} +4914.99 q^{4} -6541.53 q^{5} +3023.86 q^{7} -375171. q^{8} -748912. q^{10} -2.93325e6 q^{11} -1.09205e7 q^{13} +346189. q^{14} -8.32153e7 q^{16} +3.91668e7 q^{17} -2.62272e8 q^{19} -3.21515e7 q^{20} -3.35815e8 q^{22} -5.10689e8 q^{23} -1.17791e9 q^{25} -1.25024e9 q^{26} +1.48623e7 q^{28} -1.28435e9 q^{29} +7.28041e9 q^{31} -6.45357e9 q^{32} +4.48404e9 q^{34} -1.97807e7 q^{35} +1.19549e10 q^{37} -3.00264e10 q^{38} +2.45419e9 q^{40} -4.94440e10 q^{41} +5.84125e8 q^{43} -1.44169e10 q^{44} -5.84666e10 q^{46} -4.28170e10 q^{47} -9.68799e10 q^{49} -1.34854e11 q^{50} -5.36741e10 q^{52} +2.39761e11 q^{53} +1.91879e10 q^{55} -1.13446e9 q^{56} -1.47039e11 q^{58} -3.71617e11 q^{59} +5.07461e11 q^{61} +8.33504e11 q^{62} -5.71420e10 q^{64} +7.14367e10 q^{65} +1.12088e12 q^{67} +1.92505e11 q^{68} -2.26461e9 q^{70} +1.68696e12 q^{71} +4.53212e11 q^{73} +1.36866e12 q^{74} -1.28906e12 q^{76} -8.86974e9 q^{77} +4.27255e11 q^{79} +5.44355e11 q^{80} -5.66063e12 q^{82} -1.21156e12 q^{83} -2.56211e11 q^{85} +6.68740e10 q^{86} +1.10047e12 q^{88} -5.30646e12 q^{89} -3.30221e10 q^{91} -2.51003e12 q^{92} -4.90194e12 q^{94} +1.71566e12 q^{95} +1.48907e13 q^{97} -1.10914e13 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 117 q^{2} + 8101 q^{4} - 11412 q^{5} + 54572 q^{7} - 615375 q^{8} + 969579 q^{10} - 4927212 q^{11} + 12107576 q^{13} - 2015487 q^{14} + 56856625 q^{16} + 34729992 q^{17} + 269848016 q^{19} - 179564463 q^{20}+ \cdots - 3817815378498 q^{98}+O(q^{100}) $ 4 * q - 117 * q^2 + 8101 * q^4 - 11412 * q^5 + 54572 * q^7 - 615375 * q^8 + 969579 * q^10 - 4927212 * q^11 + 12107576 * q^13 - 2015487 * q^14 + 56856625 * q^16 + 34729992 * q^17 + 269848016 * q^19 - 179564463 * q^20 + 971969481 * q^22 - 1296333360 * q^23 - 354407528 * q^25 - 4599933804 * q^26 - 1290887245 * q^28 - 6967785600 * q^29 + 1449752348 * q^31 - 18104036247 * q^32 - 13245631488 * q^34 - 39458246292 * q^35 - 14813291296 * q^37 - 87551260146 * q^38 - 3563985051 * q^40 - 96485867568 * q^41 + 21952024664 * q^43 - 244812106521 * q^44 + 54796717602 * q^46 - 144494756856 * q^47 - 21816898080 * q^49 - 21012587604 * q^50 + 313979782172 * q^52 - 231237557460 * q^53 - 118437139188 * q^55 + 262334662191 * q^56 - 98625574602 * q^58 + 654991446240 * q^59 - 97408780816 * q^61 + 463348969227 * q^62 - 684162915215 * q^64 + 1432017576504 * q^65 - 729275019496 * q^67 + 3678684087600 * q^68 + 1478759994297 * q^70 + 1921547025288 * q^71 - 2662954340932 * q^73 + 4859349629382 * q^74 + 1809674429402 * q^76 + 2884684296924 * q^77 + 3435370655288 * q^79 + 1773900202293 * q^80 - 3142316785446 * q^82 - 3544164633852 * q^83 - 1212132619224 * q^85 - 4370115792870 * q^86 + 20297903695659 * q^88 - 7763565281544 * q^89 - 14738138711480 * q^91 - 10986690530610 * q^92 - 7609182540642 * q^94 - 24364387296720 * q^95 + 24376525074716 * q^97 - 3817815378498 * 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 $

$2$	114.486	1.26490	0.632450	−	0.774601i	$-0.282049\pi$
$2$	114.486	1.26490	0.632450	+	0.774601i	$0.282049\pi$
$3$	0	0
$3$	0	0
$4$	4914.99	0.599974
$5$	−6541.53	−0.187229	−0.0936147	−	0.995609i	$-0.529842\pi$
$5$	−6541.53	−0.187229	−0.0936147	+	0.995609i	$0.529842\pi$
$6$	0	0
$7$	3023.86	0.00971459	0.00485730	−	0.999988i	$-0.498454\pi$
$7$	3023.86	0.00971459	0.00485730	+	0.999988i	$0.498454\pi$
$8$	−375171.	−0.505993
$9$	0	0
$10$	−748912.	−0.236827
$11$	−2.93325e6	−0.499225	−0.249613	−	0.968346i	$-0.580303\pi$
$11$	−2.93325e6	−0.499225	−0.249613	+	0.968346i	$0.580303\pi$
$12$	0	0
$13$	−1.09205e7	−0.627495	−0.313748	−	0.949506i	$-0.601585\pi$
$13$	−1.09205e7	−0.627495	−0.313748	+	0.949506i	$0.601585\pi$
$14$	346189.	0.0122880
$15$	0	0
$16$	−8.32153e7	−1.24001
$17$	3.91668e7	0.393550	0.196775	−	0.980449i	$-0.436953\pi$
$17$	3.91668e7	0.393550	0.196775	+	0.980449i	$0.436953\pi$
$18$	0	0
$19$	−2.62272e8	−1.27895	−0.639475	−	0.768812i	$-0.720848\pi$
$19$	−2.62272e8	−1.27895	−0.639475	+	0.768812i	$0.720848\pi$
$20$	−3.21515e7	−0.112333
$21$	0	0
$22$	−3.35815e8	−0.631471
$23$	−5.10689e8	−0.719325	−0.359663	−	0.933082i	$-0.617108\pi$
$23$	−5.10689e8	−0.719325	−0.359663	+	0.933082i	$0.617108\pi$
$24$	0	0
$25$	−1.17791e9	−0.964945
$26$	−1.25024e9	−0.793719
$27$	0	0
$28$	1.48623e7	0.00582851
$29$	−1.28435e9	−0.400955	−0.200477	−	0.979698i	$-0.564249\pi$
$29$	−1.28435e9	−0.400955	−0.200477	+	0.979698i	$0.564249\pi$
$30$	0	0
$31$	7.28041e9	1.47335	0.736674	−	0.676249i	$-0.236395\pi$
$31$	7.28041e9	1.47335	0.736674	+	0.676249i	$0.236395\pi$
$32$	−6.45357e9	−1.06249
$33$	0	0
$34$	4.48404e9	0.497802
$35$	−1.97807e7	−0.00181886
$36$	0	0
$37$	1.19549e10	0.766008	0.383004	−	0.923747i	$-0.374890\pi$
$37$	1.19549e10	0.766008	0.383004	+	0.923747i	$0.374890\pi$
$38$	−3.00264e10	−1.61774
$39$	0	0
$40$	2.45419e9	0.0947367
$41$	−4.94440e10	−1.62562	−0.812809	−	0.582531i	$-0.802063\pi$
$41$	−4.94440e10	−1.62562	−0.812809	+	0.582531i	$0.802063\pi$
$42$	0	0
$43$	5.84125e8	0.0140916	0.00704581	−	0.999975i	$-0.497757\pi$
$43$	5.84125e8	0.0140916	0.00704581	+	0.999975i	$0.497757\pi$
$44$	−1.44169e10	−0.299522
$45$	0	0
$46$	−5.84666e10	−0.909875
$47$	−4.28170e10	−0.579402	−0.289701	−	0.957117i	$-0.593556\pi$
$47$	−4.28170e10	−0.579402	−0.289701	+	0.957117i	$0.593556\pi$
$48$	0	0
$49$	−9.68799e10	−0.999906
$50$	−1.34854e11	−1.22056
$51$	0	0
$52$	−5.36741e10	−0.376481
$53$	2.39761e11	1.48588	0.742942	−	0.669356i	$-0.233430\pi$
$53$	2.39761e11	1.48588	0.742942	+	0.669356i	$0.233430\pi$
$54$	0	0
$55$	1.91879e10	0.0934696
$56$	−1.13446e9	−0.00491551
$57$	0	0
$58$	−1.47039e11	−0.507168
$59$	−3.71617e11	−1.14698	−0.573492	−	0.819211i	$-0.694412\pi$
$59$	−3.71617e11	−1.14698	−0.573492	+	0.819211i	$0.694412\pi$
$60$	0	0
$61$	5.07461e11	1.26113	0.630564	−	0.776138i	$-0.282824\pi$
$61$	5.07461e11	1.26113	0.630564	+	0.776138i	$0.282824\pi$
$62$	8.33504e11	1.86364
$63$	0	0
$64$	−5.71420e10	−0.103941
$65$	7.14367e10	0.117486
$66$	0	0
$67$	1.12088e12	1.51382	0.756908	−	0.653521i	$-0.226709\pi$
$67$	1.12088e12	1.51382	0.756908	+	0.653521i	$0.226709\pi$
$68$	1.92505e11	0.236120
$69$	0	0
$70$	−2.26461e9	−0.00230067
$71$	1.68696e12	1.56288	0.781439	−	0.623982i	$-0.214486\pi$
$71$	1.68696e12	1.56288	0.781439	+	0.623982i	$0.214486\pi$
$72$	0	0
$73$	4.53212e11	0.350512	0.175256	−	0.984523i	$-0.443925\pi$
$73$	4.53212e11	0.350512	0.175256	+	0.984523i	$0.443925\pi$
$74$	1.36866e12	0.968924
$75$	0	0
$76$	−1.28906e12	−0.767337
$77$	−8.86974e9	−0.00484977
$78$	0	0
$79$	4.27255e11	0.197747	0.0988737	−	0.995100i	$-0.468476\pi$
$79$	4.27255e11	0.197747	0.0988737	+	0.995100i	$0.468476\pi$
$80$	5.44355e11	0.232165
$81$	0	0
$82$	−5.66063e12	−2.05625
$83$	−1.21156e12	−0.406759	−0.203380	−	0.979100i	$-0.565193\pi$
$83$	−1.21156e12	−0.406759	−0.203380	+	0.979100i	$0.565193\pi$
$84$	0	0
$85$	−2.56211e11	−0.0736842
$86$	6.68740e10	0.0178245
$87$	0	0
$88$	1.10047e12	0.252604
$89$	−5.30646e12	−1.13180	−0.565900	−	0.824474i	$-0.691471\pi$
$89$	−5.30646e12	−1.13180	−0.565900	+	0.824474i	$0.691471\pi$
$90$	0	0
$91$	−3.30221e10	−0.00609586
$92$	−2.51003e12	−0.431577
$93$	0	0
$94$	−4.90194e12	−0.732886
$95$	1.71566e12	0.239457
$96$	0	0
$97$	1.48907e13	1.81509	0.907544	−	0.419957i	$-0.137955\pi$
$97$	1.48907e13	1.81509	0.907544	+	0.419957i	$0.137955\pi$
$98$	−1.10914e13	−1.26478
$99$	0	0
$100$	−5.78942e12	−0.578942
$101$	−2.31222e12	−0.216741	−0.108370	−	0.994111i	$-0.534563\pi$
$101$	−2.31222e12	−0.216741	−0.108370	+	0.994111i	$0.534563\pi$
$102$	0	0
$103$	3.62363e12	0.299021	0.149511	−	0.988760i	$-0.452230\pi$
$103$	3.62363e12	0.299021	0.149511	+	0.988760i	$0.452230\pi$
$104$	4.09705e12	0.317508
$105$	0	0
$106$	2.74492e13	1.87950
$107$	−2.49571e13	−1.60768	−0.803839	−	0.594847i	$-0.797213\pi$
$107$	−2.49571e13	−1.60768	−0.803839	+	0.594847i	$0.797213\pi$
$108$	0	0
$109$	−6.85838e12	−0.391696	−0.195848	−	0.980634i	$-0.562746\pi$
$109$	−6.85838e12	−0.391696	−0.195848	+	0.980634i	$0.562746\pi$
$110$	2.19674e12	0.118230
$111$	0	0
$112$	−2.51632e11	−0.0120461
$113$	−1.65589e12	−0.0748208	−0.0374104	−	0.999300i	$-0.511911\pi$
$113$	−1.65589e12	−0.0748208	−0.0374104	+	0.999300i	$0.511911\pi$
$114$	0	0
$115$	3.34068e12	0.134679
$116$	−6.31255e12	−0.240563
$117$	0	0
$118$	−4.25449e13	−1.45082
$119$	1.18435e11	0.00382318
$120$	0	0
$121$	−2.59188e13	−0.750774
$122$	5.80971e13	1.59520
$123$	0	0
$124$	3.57832e13	0.883971
$125$	1.56906e13	0.367896
$126$	0	0
$127$	−7.41286e13	−1.56770	−0.783848	−	0.620953i	$-0.786746\pi$
$127$	−7.41286e13	−1.56770	−0.783848	+	0.620953i	$0.786746\pi$
$128$	4.63257e13	0.931016
$129$	0	0
$130$	8.17848e12	0.148608
$131$	−2.34322e13	−0.405088	−0.202544	−	0.979273i	$-0.564921\pi$
$131$	−2.34322e13	−0.405088	−0.202544	+	0.979273i	$0.564921\pi$
$132$	0	0
$133$	−7.93074e11	−0.0124245
$134$	1.28325e14	1.91483
$135$	0	0
$136$	−1.46943e13	−0.199134
$137$	−2.83009e13	−0.365693	−0.182846	−	0.983141i	$-0.558531\pi$
$137$	−2.83009e13	−0.365693	−0.182846	+	0.983141i	$0.558531\pi$
$138$	0	0
$139$	−1.00157e13	−0.117784	−0.0588921	−	0.998264i	$-0.518757\pi$
$139$	−1.00157e13	−0.117784	−0.0588921	+	0.998264i	$0.518757\pi$
$140$	−9.72218e10	−0.00109127
$141$	0	0
$142$	1.93133e14	1.97689
$143$	3.20325e13	0.313261
$144$	0	0
$145$	8.40159e12	0.0750705
$146$	5.18863e13	0.443363
$147$	0	0
$148$	5.87580e13	0.459585
$149$	−2.30649e14	−1.72680	−0.863398	−	0.504523i	$-0.831668\pi$
$149$	−2.30649e14	−1.72680	−0.863398	+	0.504523i	$0.831668\pi$
$150$	0	0
$151$	9.06240e13	0.622147	0.311073	−	0.950386i	$-0.399311\pi$
$151$	9.06240e13	0.622147	0.311073	+	0.950386i	$0.399311\pi$
$152$	9.83968e13	0.647139
$153$	0	0
$154$	−1.01546e12	−0.00613448
$155$	−4.76250e13	−0.275854
$156$	0	0
$157$	−1.09770e14	−0.584971	−0.292485	−	0.956270i	$-0.594482\pi$
$157$	−1.09770e14	−0.584971	−0.292485	+	0.956270i	$0.594482\pi$
$158$	4.89146e13	0.250131
$159$	0	0
$160$	4.22162e13	0.198930
$161$	−1.54425e12	−0.00698795
$162$	0	0
$163$	−1.40380e14	−0.586254	−0.293127	−	0.956073i	$-0.594696\pi$
$163$	−1.40380e14	−0.586254	−0.293127	+	0.956073i	$0.594696\pi$
$164$	−2.43017e14	−0.975329
$165$	0	0
$166$	−1.38706e14	−0.514510
$167$	−4.32365e14	−1.54239	−0.771193	−	0.636601i	$-0.780340\pi$
$167$	−4.32365e14	−1.54239	−0.771193	+	0.636601i	$0.780340\pi$
$168$	0	0
$169$	−1.83618e14	−0.606250
$170$	−2.93325e13	−0.0932032
$171$	0	0
$172$	2.87097e12	0.00845461
$173$	−2.05104e14	−0.581667	−0.290833	−	0.956774i	$-0.593933\pi$
$173$	−2.05104e14	−0.581667	−0.290833	+	0.956774i	$0.593933\pi$
$174$	0	0
$175$	−3.56184e12	−0.00937405
$176$	2.44091e14	0.619042
$177$	0	0
$178$	−6.07514e14	−1.43161
$179$	−5.14530e14	−1.16914	−0.584569	−	0.811344i	$-0.698736\pi$
$179$	−5.14530e14	−1.16914	−0.584569	+	0.811344i	$0.698736\pi$
$180$	0	0
$181$	2.47164e14	0.522487	0.261243	−	0.965273i	$-0.415867\pi$
$181$	2.47164e14	0.522487	0.261243	+	0.965273i	$0.415867\pi$
$182$	−3.78055e12	−0.00771066
$183$	0	0
$184$	1.91596e14	0.363973
$185$	−7.82030e13	−0.143419
$186$	0	0
$187$	−1.14886e14	−0.196470
$188$	−2.10445e14	−0.347626
$189$	0	0
$190$	1.96418e14	0.302889
$191$	6.83084e14	1.01802	0.509011	−	0.860760i	$-0.330011\pi$
$191$	6.83084e14	1.01802	0.509011	+	0.860760i	$0.330011\pi$
$192$	0	0
$193$	1.01525e15	1.41400	0.707001	−	0.707212i	$-0.250048\pi$
$193$	1.01525e15	1.41400	0.707001	+	0.707212i	$0.250048\pi$
$194$	1.70477e15	2.29591
$195$	0	0
$196$	−4.76164e14	−0.599918
$197$	−5.56249e14	−0.678014	−0.339007	−	0.940784i	$-0.610091\pi$
$197$	−5.56249e14	−0.678014	−0.339007	+	0.940784i	$0.610091\pi$
$198$	0	0
$199$	1.42117e15	1.62219	0.811094	−	0.584916i	$-0.198873\pi$
$199$	1.42117e15	1.62219	0.811094	+	0.584916i	$0.198873\pi$
$200$	4.41918e14	0.488255
$201$	0	0
$202$	−2.64717e14	−0.274156
$203$	−3.88369e12	−0.00389511
$204$	0	0
$205$	3.23439e14	0.304363
$206$	4.14854e14	0.378232
$207$	0	0
$208$	9.08752e14	0.778097
$209$	7.69309e14	0.638484
$210$	0	0
$211$	−2.60015e14	−0.202844	−0.101422	−	0.994843i	$-0.532339\pi$
$211$	−2.60015e14	−0.202844	−0.101422	+	0.994843i	$0.532339\pi$
$212$	1.17842e15	0.891492
$213$	0	0
$214$	−2.85723e15	−2.03355
$215$	−3.82107e12	−0.00263837
$216$	0	0
$217$	2.20150e13	0.0143130
$218$	−7.85187e14	−0.495457
$219$	0	0
$220$	9.43085e13	0.0560794
$221$	−4.27721e14	−0.246951
$222$	0	0
$223$	−1.04630e15	−0.569738	−0.284869	−	0.958567i	$-0.591950\pi$
$223$	−1.04630e15	−0.569738	−0.284869	+	0.958567i	$0.591950\pi$
$224$	−1.95147e13	−0.0103217
$225$	0	0
$226$	−1.89576e14	−0.0946409
$227$	7.53498e14	0.365523	0.182761	−	0.983157i	$-0.441496\pi$
$227$	7.53498e14	0.365523	0.182761	+	0.983157i	$0.441496\pi$
$228$	0	0
$229$	−2.15536e15	−0.987618	−0.493809	−	0.869570i	$-0.664396\pi$
$229$	−2.15536e15	−0.987618	−0.493809	+	0.869570i	$0.664396\pi$
$230$	3.82461e14	0.170355
$231$	0	0
$232$	4.81850e14	0.202880
$233$	1.82751e15	0.748248	0.374124	−	0.927379i	$-0.377943\pi$
$233$	1.82751e15	0.748248	0.374124	+	0.927379i	$0.377943\pi$
$234$	0	0
$235$	2.80088e14	0.108481
$236$	−1.82649e15	−0.688162
$237$	0	0
$238$	1.35591e13	0.00483595
$239$	−4.09058e15	−1.41971	−0.709853	−	0.704350i	$-0.751239\pi$
$239$	−4.09058e15	−1.41971	−0.709853	+	0.704350i	$0.751239\pi$
$240$	0	0
$241$	−2.10327e15	−0.691489	−0.345744	−	0.938329i	$-0.612374\pi$
$241$	−2.10327e15	−0.691489	−0.345744	+	0.938329i	$0.612374\pi$
$242$	−2.96733e15	−0.949655
$243$	0	0
$244$	2.49417e15	0.756644
$245$	6.33742e14	0.187212
$246$	0	0
$247$	2.86414e15	0.802534
$248$	−2.73140e15	−0.745503
$249$	0	0
$250$	1.79635e15	0.465351
$251$	4.82134e15	1.21699	0.608497	−	0.793556i	$-0.291773\pi$
$251$	4.82134e15	1.21699	0.608497	+	0.793556i	$0.291773\pi$
$252$	0	0
$253$	1.49798e15	0.359105
$254$	−8.48667e15	−1.98298
$255$	0	0
$256$	5.77174e15	1.28158
$257$	−6.87431e15	−1.48821	−0.744104	−	0.668064i	$-0.767123\pi$
$257$	−6.87431e15	−1.48821	−0.744104	+	0.668064i	$0.767123\pi$
$258$	0	0
$259$	3.61498e13	0.00744145
$260$	3.51111e14	0.0704883
$261$	0	0
$262$	−2.68265e15	−0.512396
$263$	4.29952e15	0.801139	0.400570	−	0.916266i	$-0.368812\pi$
$263$	4.29952e15	0.801139	0.400570	+	0.916266i	$0.368812\pi$
$264$	0	0
$265$	−1.56840e15	−0.278201
$266$	−9.07957e13	−0.0157157
$267$	0	0
$268$	5.50912e15	0.908251
$269$	4.51075e15	0.725871	0.362935	−	0.931814i	$-0.381775\pi$
$269$	4.51075e15	0.725871	0.362935	+	0.931814i	$0.381775\pi$
$270$	0	0
$271$	−6.31461e15	−0.968381	−0.484190	−	0.874963i	$-0.660886\pi$
$271$	−6.31461e15	−0.968381	−0.484190	+	0.874963i	$0.660886\pi$
$272$	−3.25928e15	−0.488004
$273$	0	0
$274$	−3.24005e15	−0.462565
$275$	3.45511e15	0.481725
$276$	0	0
$277$	2.04896e15	0.272530	0.136265	−	0.990672i	$-0.456490\pi$
$277$	2.04896e15	0.272530	0.136265	+	0.990672i	$0.456490\pi$
$278$	−1.14666e15	−0.148985
$279$	0	0
$280$	7.42113e12	0.000920329 0
$281$	1.13826e16	1.37928	0.689640	−	0.724153i	$-0.257769\pi$
$281$	1.13826e16	1.37928	0.689640	+	0.724153i	$0.257769\pi$
$282$	0	0
$283$	−1.26106e16	−1.45923	−0.729614	−	0.683859i	$-0.760300\pi$
$283$	−1.26106e16	−1.45923	−0.729614	+	0.683859i	$0.760300\pi$
$284$	8.29138e15	0.937687
$285$	0	0
$286$	3.66727e15	0.396245
$287$	−1.49512e14	−0.0157922
$288$	0	0
$289$	−8.37054e15	−0.845118
$290$	9.61863e14	0.0949568
$291$	0	0
$292$	2.22753e15	0.210298
$293$	3.42022e15	0.315801	0.157901	−	0.987455i	$-0.449527\pi$
$293$	3.42022e15	0.315801	0.157901	+	0.987455i	$0.449527\pi$
$294$	0	0
$295$	2.43094e15	0.214749
$296$	−4.48512e15	−0.387594
$297$	0	0
$298$	−2.64060e16	−2.18423
$299$	5.57697e15	0.451373
$300$	0	0
$301$	1.76631e12	0.000136894 0
$302$	1.03752e16	0.786954
$303$	0	0
$304$	2.18250e16	1.58590
$305$	−3.31957e15	−0.236120
$306$	0	0
$307$	1.73522e16	1.18292	0.591461	−	0.806334i	$-0.298551\pi$
$307$	1.73522e16	1.18292	0.591461	+	0.806334i	$0.298551\pi$
$308$	−4.35947e13	−0.00290974
$309$	0	0
$310$	−5.45239e15	−0.348928
$311$	1.96489e16	1.23139	0.615695	−	0.787985i	$-0.288875\pi$
$311$	1.96489e16	1.23139	0.615695	+	0.787985i	$0.288875\pi$
$312$	0	0
$313$	1.85002e16	1.11209	0.556043	−	0.831154i	$-0.312319\pi$
$313$	1.85002e16	1.11209	0.556043	+	0.831154i	$0.312319\pi$
$314$	−1.25670e16	−0.739930
$315$	0	0
$316$	2.09995e15	0.118643
$317$	7.67121e15	0.424599	0.212299	−	0.977205i	$-0.431905\pi$
$317$	7.67121e15	0.424599	0.212299	+	0.977205i	$0.431905\pi$
$318$	0	0
$319$	3.76731e15	0.200167
$320$	3.73796e14	0.0194608
$321$	0	0
$322$	−1.76795e14	−0.00883907
$323$	−1.02724e16	−0.503331
$324$	0	0
$325$	1.28634e16	0.605498
$326$	−1.60715e16	−0.741554
$327$	0	0
$328$	1.85499e16	0.822551
$329$	−1.29473e14	−0.00562866
$330$	0	0
$331$	4.44416e15	0.185741	0.0928705	−	0.995678i	$-0.470396\pi$
$331$	4.44416e15	0.185741	0.0928705	+	0.995678i	$0.470396\pi$
$332$	−5.95481e15	−0.244045
$333$	0	0
$334$	−4.94996e16	−1.95097
$335$	−7.33227e15	−0.283431
$336$	0	0
$337$	1.01752e16	0.378398	0.189199	−	0.981939i	$-0.439411\pi$
$337$	1.01752e16	0.378398	0.189199	+	0.981939i	$0.439411\pi$
$338$	−2.10216e16	−0.766846
$339$	0	0
$340$	−1.25927e15	−0.0442086
$341$	−2.13553e16	−0.735532
$342$	0	0
$343$	−5.85930e14	−0.0194283
$344$	−2.19147e14	−0.00713026
$345$	0	0
$346$	−2.34815e16	−0.735751
$347$	−3.71340e16	−1.14190	−0.570952	−	0.820983i	$-0.693426\pi$
$347$	−3.71340e16	−1.14190	−0.570952	+	0.820983i	$0.693426\pi$
$348$	0	0
$349$	−3.31685e16	−0.982563	−0.491282	−	0.871001i	$-0.663471\pi$
$349$	−3.31685e16	−0.982563	−0.491282	+	0.871001i	$0.663471\pi$
$350$	−4.07780e14	−0.0118572
$351$	0	0
$352$	1.89299e16	0.530422
$353$	−3.87859e16	−1.06694	−0.533468	−	0.845820i	$-0.679112\pi$
$353$	−3.87859e16	−1.06694	−0.533468	+	0.845820i	$0.679112\pi$
$354$	0	0
$355$	−1.10353e16	−0.292617
$356$	−2.60812e16	−0.679051
$357$	0	0
$358$	−5.89063e16	−1.47884
$359$	3.95144e16	0.974185	0.487093	−	0.873350i	$-0.338057\pi$
$359$	3.95144e16	0.974185	0.487093	+	0.873350i	$0.338057\pi$
$360$	0	0
$361$	2.67335e16	0.635711
$362$	2.82968e16	0.660894
$363$	0	0
$364$	−1.62303e14	−0.00365736
$365$	−2.96470e15	−0.0656262
$366$	0	0
$367$	1.58923e16	0.339514	0.169757	−	0.985486i	$-0.445702\pi$
$367$	1.58923e16	0.339514	0.169757	+	0.985486i	$0.445702\pi$
$368$	4.24971e16	0.891967
$369$	0	0
$370$	−8.95313e15	−0.181411
$371$	7.25003e14	0.0144348
$372$	0	0
$373$	−2.97259e16	−0.571514	−0.285757	−	0.958302i	$-0.592245\pi$
$373$	−2.97259e16	−0.571514	−0.285757	+	0.958302i	$0.592245\pi$
$374$	−1.31528e16	−0.248515
$375$	0	0
$376$	1.60637e16	0.293173
$377$	1.40257e16	0.251597
$378$	0	0
$379$	3.32748e16	0.576715	0.288357	−	0.957523i	$-0.406891\pi$
$379$	3.32748e16	0.576715	0.288357	+	0.957523i	$0.406891\pi$
$380$	8.43244e15	0.143668
$381$	0	0
$382$	7.82034e16	1.28770
$383$	2.19430e16	0.355225	0.177612	−	0.984101i	$-0.443163\pi$
$383$	2.19430e16	0.355225	0.177612	+	0.984101i	$0.443163\pi$
$384$	0	0
$385$	5.80216e13	0.000908020 0
$386$	1.16232e17	1.78857
$387$	0	0
$388$	7.31875e16	1.08901
$389$	−7.78907e16	−1.13976	−0.569880	−	0.821728i	$-0.693010\pi$
$389$	−7.78907e16	−1.13976	−0.569880	+	0.821728i	$0.693010\pi$
$390$	0	0
$391$	−2.00020e16	−0.283091
$392$	3.63465e16	0.505945
$393$	0	0
$394$	−6.36825e16	−0.857620
$395$	−2.79490e15	−0.0370241
$396$	0	0
$397$	1.14310e17	1.46537	0.732684	−	0.680569i	$-0.238267\pi$
$397$	1.14310e17	1.46537	0.732684	+	0.680569i	$0.238267\pi$
$398$	1.62704e17	2.05191
$399$	0	0
$400$	9.80203e16	1.19654
$401$	−7.06722e16	−0.848809	−0.424405	−	0.905473i	$-0.639517\pi$
$401$	−7.06722e16	−0.848809	−0.424405	+	0.905473i	$0.639517\pi$
$402$	0	0
$403$	−7.95057e16	−0.924518
$404$	−1.13646e16	−0.130039
$405$	0	0
$406$	−4.44627e14	−0.00492693
$407$	−3.50666e16	−0.382410
$408$	0	0
$409$	−1.48247e17	−1.56597	−0.782987	−	0.622038i	$-0.786305\pi$
$409$	−1.48247e17	−1.56597	−0.782987	+	0.622038i	$0.786305\pi$
$410$	3.70292e16	0.384990
$411$	0	0
$412$	1.78101e16	0.179405
$413$	−1.12372e15	−0.0111425
$414$	0	0
$415$	7.92545e15	0.0761573
$416$	7.04762e16	0.666708
$417$	0	0
$418$	8.80749e16	0.807619
$419$	1.93096e17	1.74334	0.871671	−	0.490091i	$-0.163037\pi$
$419$	1.93096e17	1.74334	0.871671	+	0.490091i	$0.163037\pi$
$420$	0	0
$421$	−1.84851e17	−1.61804	−0.809018	−	0.587783i	$-0.800001\pi$
$421$	−1.84851e17	−1.61804	−0.809018	+	0.587783i	$0.800001\pi$
$422$	−2.97680e16	−0.256578
$423$	0	0
$424$	−8.99512e16	−0.751846
$425$	−4.61350e16	−0.379754
$426$	0	0
$427$	1.53449e15	0.0122513
$428$	−1.22664e17	−0.964566
$429$	0	0
$430$	−4.37458e14	−0.00333727
$431$	−2.61733e17	−1.96678	−0.983392	−	0.181494i	$-0.941907\pi$
$431$	−2.61733e17	−1.96678	−0.983392	+	0.181494i	$0.941907\pi$
$432$	0	0
$433$	1.95120e15	0.0142275	0.00711376	−	0.999975i	$-0.497736\pi$
$433$	1.95120e15	0.0142275	0.00711376	+	0.999975i	$0.497736\pi$
$434$	2.52040e15	0.0181045
$435$	0	0
$436$	−3.37089e16	−0.235008
$437$	1.33939e17	0.919980
$438$	0	0
$439$	−1.62018e17	−1.08030	−0.540149	−	0.841569i	$-0.681632\pi$
$439$	−1.62018e17	−1.08030	−0.540149	+	0.841569i	$0.681632\pi$
$440$	−7.19875e15	−0.0472950
$441$	0	0
$442$	−4.89679e16	−0.312368
$443$	−1.00283e17	−0.630382	−0.315191	−	0.949028i	$-0.602069\pi$
$443$	−1.00283e17	−0.630382	−0.315191	+	0.949028i	$0.602069\pi$
$444$	0	0
$445$	3.47123e16	0.211906
$446$	−1.19787e17	−0.720662
$447$	0	0
$448$	−1.72789e14	−0.00100974
$449$	−2.07644e17	−1.19596	−0.597982	−	0.801510i	$-0.704031\pi$
$449$	−2.07644e17	−1.19596	−0.597982	+	0.801510i	$0.704031\pi$
$450$	0	0
$451$	1.45032e17	0.811549
$452$	−8.13870e15	−0.0448906
$453$	0	0
$454$	8.62648e16	0.462350
$455$	2.16015e14	0.00114132
$456$	0	0
$457$	3.67342e17	1.88632	0.943160	−	0.332340i	$-0.107838\pi$
$457$	3.67342e17	1.88632	0.943160	+	0.332340i	$0.107838\pi$
$458$	−2.46758e17	−1.24924
$459$	0	0
$460$	1.64194e16	0.0808038
$461$	−2.67782e17	−1.29935	−0.649674	−	0.760213i	$-0.725095\pi$
$461$	−2.67782e17	−1.29935	−0.649674	+	0.760213i	$0.725095\pi$
$462$	0	0
$463$	−1.53329e17	−0.723349	−0.361675	−	0.932304i	$-0.617795\pi$
$463$	−1.53329e17	−0.723349	−0.361675	+	0.932304i	$0.617795\pi$
$464$	1.06877e17	0.497186
$465$	0	0
$466$	2.09224e17	0.946460
$467$	2.51521e17	1.12206	0.561028	−	0.827797i	$-0.310406\pi$
$467$	2.51521e17	1.12206	0.561028	+	0.827797i	$0.310406\pi$
$468$	0	0
$469$	3.38939e15	0.0147061
$470$	3.20661e16	0.137218
$471$	0	0
$472$	1.39420e17	0.580366
$473$	−1.71338e15	−0.00703489
$474$	0	0
$475$	3.08933e17	1.23412
$476$	5.82107e14	0.00229381
$477$	0	0
$478$	−4.68313e17	−1.79579
$479$	4.67447e17	1.76828	0.884141	−	0.467220i	$-0.154744\pi$
$479$	4.67447e17	1.76828	0.884141	+	0.467220i	$0.154744\pi$
$480$	0	0
$481$	−1.30553e17	−0.480666
$482$	−2.40795e17	−0.874665
$483$	0	0
$484$	−1.27390e17	−0.450445
$485$	−9.74076e16	−0.339838
$486$	0	0
$487$	4.25409e17	1.44500	0.722501	−	0.691370i	$-0.242992\pi$
$487$	4.25409e17	1.44500	0.722501	+	0.691370i	$0.242992\pi$
$488$	−1.90385e17	−0.638121
$489$	0	0
$490$	7.25545e16	0.236804
$491$	2.71463e17	0.874343	0.437171	−	0.899378i	$-0.355980\pi$
$491$	2.71463e17	0.874343	0.437171	+	0.899378i	$0.355980\pi$
$492$	0	0
$493$	−5.03038e16	−0.157796
$494$	3.27903e17	1.01513
$495$	0	0
$496$	−6.05842e17	−1.82696
$497$	5.10113e15	0.0151827
$498$	0	0
$499$	−3.49875e17	−1.01452	−0.507259	−	0.861794i	$-0.669341\pi$
$499$	−3.49875e17	−1.01452	−0.507259	+	0.861794i	$0.669341\pi$
$500$	7.71192e16	0.220728
$501$	0	0
$502$	5.51975e17	1.53938
$503$	4.96554e17	1.36702	0.683510	−	0.729941i	$-0.260453\pi$
$503$	4.96554e17	1.36702	0.683510	+	0.729941i	$0.260453\pi$
$504$	0	0
$505$	1.51255e16	0.0405803
$506$	1.71497e17	0.454233
$507$	0	0
$508$	−3.64342e17	−0.940577
$509$	−6.10574e17	−1.55623	−0.778113	−	0.628124i	$-0.783823\pi$
$509$	−6.10574e17	−1.55623	−0.778113	+	0.628124i	$0.783823\pi$
$510$	0	0
$511$	1.37045e15	0.00340508
$512$	2.81282e17	0.690061
$513$	0	0
$514$	−7.87010e17	−1.88244
$515$	−2.37041e16	−0.0559856
$516$	0	0
$517$	1.25593e17	0.289252
$518$	4.13864e15	0.00941270
$519$	0	0
$520$	−2.68010e16	−0.0594468
$521$	7.37351e17	1.61521	0.807605	−	0.589724i	$-0.200763\pi$
$521$	7.37351e17	1.61521	0.807605	+	0.589724i	$0.200763\pi$
$522$	0	0
$523$	−1.38691e17	−0.296338	−0.148169	−	0.988962i	$-0.547338\pi$
$523$	−1.38691e17	−0.296338	−0.148169	+	0.988962i	$0.547338\pi$
$524$	−1.15169e17	−0.243042
$525$	0	0
$526$	4.92234e17	1.01336
$527$	2.85151e17	0.579836
$528$	0	0
$529$	−2.43234e17	−0.482571
$530$	−1.79560e17	−0.351897
$531$	0	0
$532$	−3.89795e15	−0.00745437
$533$	5.39953e17	1.02007
$534$	0	0
$535$	1.63257e17	0.301005
$536$	−4.20522e17	−0.765980
$537$	0	0
$538$	5.16417e17	0.918155
$539$	2.84173e17	0.499178
$540$	0	0
$541$	5.71783e17	0.980504	0.490252	−	0.871581i	$-0.336905\pi$
$541$	5.71783e17	0.980504	0.490252	+	0.871581i	$0.336905\pi$
$542$	−7.22933e17	−1.22491
$543$	0	0
$544$	−2.52766e17	−0.418144
$545$	4.48643e16	0.0733370
$546$	0	0
$547$	3.67486e17	0.586575	0.293287	−	0.956024i	$-0.405251\pi$
$547$	3.67486e17	0.586575	0.293287	+	0.956024i	$0.405251\pi$
$548$	−1.39099e17	−0.219406
$549$	0	0
$550$	3.95561e17	0.609334
$551$	3.36848e17	0.512801
$552$	0	0
$553$	1.29196e15	0.00192104
$554$	2.34576e17	0.344724
$555$	0	0
$556$	−4.92273e16	−0.0706675
$557$	−1.10324e18	−1.56535	−0.782677	−	0.622428i	$-0.786146\pi$
$557$	−1.10324e18	−1.56535	−0.782677	+	0.622428i	$0.786146\pi$
$558$	0	0
$559$	−6.37893e15	−0.00884242
$560$	1.64606e15	0.00225539
$561$	0	0
$562$	1.30315e18	1.74465
$563$	−4.85768e17	−0.642872	−0.321436	−	0.946931i	$-0.604165\pi$
$563$	−4.85768e17	−0.642872	−0.321436	+	0.946931i	$0.604165\pi$
$564$	0	0
$565$	1.08321e16	0.0140087
$566$	−1.44373e18	−1.84578
$567$	0	0
$568$	−6.32898e17	−0.790805
$569$	−8.39125e17	−1.03657	−0.518283	−	0.855209i	$-0.673429\pi$
$569$	−8.39125e17	−1.03657	−0.518283	+	0.855209i	$0.673429\pi$
$570$	0	0
$571$	−1.01671e18	−1.22762	−0.613808	−	0.789456i	$-0.710363\pi$
$571$	−1.01671e18	−1.22762	−0.613808	+	0.789456i	$0.710363\pi$
$572$	1.57440e17	0.187949
$573$	0	0
$574$	−1.71170e16	−0.0199756
$575$	6.01546e17	0.694109
$576$	0	0
$577$	6.57622e17	0.741880	0.370940	−	0.928657i	$-0.379036\pi$
$577$	6.57622e17	0.741880	0.370940	+	0.928657i	$0.379036\pi$
$578$	−9.58308e17	−1.06899
$579$	0	0
$580$	4.12937e16	0.0450404
$581$	−3.66359e15	−0.00395150
$582$	0	0
$583$	−7.03278e17	−0.741790
$584$	−1.70032e17	−0.177357
$585$	0	0
$586$	3.91566e17	0.399457
$587$	1.64594e18	1.66060	0.830302	−	0.557314i	$-0.188168\pi$
$587$	1.64594e18	1.66060	0.830302	+	0.557314i	$0.188168\pi$
$588$	0	0
$589$	−1.90945e18	−1.88434
$590$	2.78308e17	0.271637
$591$	0	0
$592$	−9.94828e17	−0.949854
$593$	−1.74184e18	−1.64495	−0.822473	−	0.568804i	$-0.807406\pi$
$593$	−1.74184e18	−1.64495	−0.822473	+	0.568804i	$0.807406\pi$
$594$	0	0
$595$	−7.74746e14	−0.000715812 0
$596$	−1.13364e18	−1.03603
$597$	0	0
$598$	6.38484e17	0.570942
$599$	9.58228e17	0.847607	0.423803	−	0.905754i	$-0.360695\pi$
$599$	9.58228e17	0.847607	0.423803	+	0.905754i	$0.360695\pi$
$600$	0	0
$601$	−8.44262e17	−0.730791	−0.365395	−	0.930852i	$-0.619066\pi$
$601$	−8.44262e17	−0.730791	−0.365395	+	0.930852i	$0.619066\pi$
$602$	2.02218e14	0.000173158 0
$603$	0	0
$604$	4.45416e17	0.373272
$605$	1.69548e17	0.140567
$606$	0	0
$607$	2.78894e17	0.226315	0.113157	−	0.993577i	$-0.463904\pi$
$607$	2.78894e17	0.226315	0.113157	+	0.993577i	$0.463904\pi$
$608$	1.69259e18	1.35887
$609$	0	0
$610$	−3.80044e17	−0.298669
$611$	4.67583e17	0.363572
$612$	0	0
$613$	−5.82115e17	−0.443114	−0.221557	−	0.975147i	$-0.571114\pi$
$613$	−5.82115e17	−0.443114	−0.221557	+	0.975147i	$0.571114\pi$
$614$	1.98658e18	1.49628
$615$	0	0
$616$	3.32767e15	0.00245395
$617$	−2.23431e18	−1.63038	−0.815192	−	0.579190i	$-0.803369\pi$
$617$	−2.23431e18	−1.63038	−0.815192	+	0.579190i	$0.803369\pi$
$618$	0	0
$619$	−1.72028e18	−1.22916	−0.614582	−	0.788853i	$-0.710675\pi$
$619$	−1.72028e18	−1.22916	−0.614582	+	0.788853i	$0.710675\pi$
$620$	−2.34077e17	−0.165505
$621$	0	0
$622$	2.24952e18	1.55759
$623$	−1.60460e16	−0.0109950
$624$	0	0
$625$	1.33524e18	0.896064
$626$	2.11801e18	1.40668
$627$	0	0
$628$	−5.39516e17	−0.350967
$629$	4.68234e17	0.301463
$630$	0	0
$631$	1.43832e18	0.907121	0.453560	−	0.891225i	$-0.350154\pi$
$631$	1.43832e18	0.907121	0.453560	+	0.891225i	$0.350154\pi$
$632$	−1.60294e17	−0.100059
$633$	0	0
$634$	8.78245e17	0.537076
$635$	4.84914e17	0.293519
$636$	0	0
$637$	1.05798e18	0.627436
$638$	4.31303e17	0.253191
$639$	0	0
$640$	−3.03041e17	−0.174314
$641$	−8.11058e16	−0.0461822	−0.0230911	−	0.999733i	$-0.507351\pi$
$641$	−8.11058e16	−0.0461822	−0.0230911	+	0.999733i	$0.507351\pi$
$642$	0	0
$643$	1.89202e18	1.05573	0.527866	−	0.849328i	$-0.322992\pi$
$643$	1.89202e18	1.05573	0.527866	+	0.849328i	$0.322992\pi$
$644$	−7.58998e15	−0.00419259
$645$	0	0
$646$	−1.17604e18	−0.636664
$647$	7.37029e17	0.395009	0.197504	−	0.980302i	$-0.436716\pi$
$647$	7.37029e17	0.395009	0.197504	+	0.980302i	$0.436716\pi$
$648$	0	0
$649$	1.09005e18	0.572604
$650$	1.47267e18	0.765895
$651$	0	0
$652$	−6.89967e17	−0.351738
$653$	−2.07702e18	−1.04835	−0.524173	−	0.851612i	$-0.675625\pi$
$653$	−2.07702e18	−1.04835	−0.524173	+	0.851612i	$0.675625\pi$
$654$	0	0
$655$	1.53282e17	0.0758443
$656$	4.11450e18	2.01577
$657$	0	0
$658$	−1.48228e16	−0.00711969
$659$	1.26324e18	0.600800	0.300400	−	0.953813i	$-0.402880\pi$
$659$	1.26324e18	0.600800	0.300400	+	0.953813i	$0.402880\pi$
$660$	0	0
$661$	−5.47092e17	−0.255124	−0.127562	−	0.991831i	$-0.540715\pi$
$661$	−5.47092e17	−0.255124	−0.127562	+	0.991831i	$0.540715\pi$
$662$	5.08793e17	0.234944
$663$	0	0
$664$	4.54542e17	0.205817
$665$	5.18791e15	0.00232623
$666$	0	0
$667$	6.55901e17	0.288417
$668$	−2.12507e18	−0.925393
$669$	0	0
$670$	−8.39440e17	−0.358512
$671$	−1.48851e18	−0.629586
$672$	0	0
$673$	−1.49579e18	−0.620542	−0.310271	−	0.950648i	$-0.600420\pi$
$673$	−1.49579e18	−0.620542	−0.310271	+	0.950648i	$0.600420\pi$
$674$	1.16492e18	0.478636
$675$	0	0
$676$	−9.02481e17	−0.363734
$677$	−3.21910e18	−1.28501	−0.642506	−	0.766280i	$-0.722105\pi$
$677$	−3.21910e18	−1.28501	−0.642506	+	0.766280i	$0.722105\pi$
$678$	0	0
$679$	4.50273e16	0.0176328
$680$	9.61228e16	0.0372837
$681$	0	0
$682$	−2.44487e18	−0.930375
$683$	1.37368e18	0.517786	0.258893	−	0.965906i	$-0.416642\pi$
$683$	1.37368e18	0.517786	0.258893	+	0.965906i	$0.416642\pi$
$684$	0	0
$685$	1.85131e17	0.0684685
$686$	−6.70807e16	−0.0245748
$687$	0	0
$688$	−4.86082e16	−0.0174737
$689$	−2.61830e18	−0.932385
$690$	0	0
$691$	2.87602e18	1.00504	0.502521	−	0.864565i	$-0.332406\pi$
$691$	2.87602e18	1.00504	0.502521	+	0.864565i	$0.332406\pi$
$692$	−1.00808e18	−0.348985
$693$	0	0
$694$	−4.25131e18	−1.44440
$695$	6.55183e16	0.0220527
$696$	0	0
$697$	−1.93656e18	−0.639762
$698$	−3.79732e18	−1.24285
$699$	0	0
$700$	−1.75064e16	−0.00562419
$701$	−3.51627e18	−1.11922	−0.559610	−	0.828756i	$-0.689049\pi$
$701$	−3.51627e18	−1.11922	−0.559610	+	0.828756i	$0.689049\pi$
$702$	0	0
$703$	−3.13542e18	−0.979685
$704$	1.67612e17	0.0518898
$705$	0	0
$706$	−4.44043e18	−1.34957
$707$	−6.99184e15	−0.00210555
$708$	0	0
$709$	3.33008e18	0.984586	0.492293	−	0.870430i	$-0.336159\pi$
$709$	3.33008e18	0.984586	0.492293	+	0.870430i	$0.336159\pi$
$710$	−1.26338e18	−0.370131
$711$	0	0
$712$	1.99083e18	0.572682
$713$	−3.71802e18	−1.05982
$714$	0	0
$715$	−2.09542e17	−0.0586517
$716$	−2.52891e18	−0.701453
$717$	0	0
$718$	4.52384e18	1.23225
$719$	1.59173e18	0.429666	0.214833	−	0.976651i	$-0.431079\pi$
$719$	1.59173e18	0.429666	0.214833	+	0.976651i	$0.431079\pi$
$720$	0	0
$721$	1.09574e16	0.00290487
$722$	3.06061e18	0.804111
$723$	0	0
$724$	1.21481e18	0.313479
$725$	1.51285e18	0.386899
$726$	0	0
$727$	1.40771e18	0.353622	0.176811	−	0.984245i	$-0.443422\pi$
$727$	1.40771e18	0.353622	0.176811	+	0.984245i	$0.443422\pi$
$728$	1.23889e16	0.00308446
$729$	0	0
$730$	−3.39416e17	−0.0830106
$731$	2.28783e16	0.00554576
$732$	0	0
$733$	5.31732e18	1.26624	0.633121	−	0.774052i	$-0.281773\pi$
$733$	5.31732e18	1.26624	0.633121	+	0.774052i	$0.281773\pi$
$734$	1.81944e18	0.429451
$735$	0	0
$736$	3.29576e18	0.764276
$737$	−3.28782e18	−0.755735
$738$	0	0
$739$	1.55754e17	0.0351764	0.0175882	−	0.999845i	$-0.494401\pi$
$739$	1.55754e17	0.0351764	0.0175882	+	0.999845i	$0.494401\pi$
$740$	−3.84367e17	−0.0860478
$741$	0	0
$742$	8.30025e16	0.0182585
$743$	4.30761e18	0.939310	0.469655	−	0.882850i	$-0.344378\pi$
$743$	4.30761e18	0.939310	0.469655	+	0.882850i	$0.344378\pi$
$744$	0	0
$745$	1.50880e18	0.323307
$746$	−3.40319e18	−0.722909
$747$	0	0
$748$	−5.64664e17	−0.117877
$749$	−7.54667e16	−0.0156179
$750$	0	0
$751$	1.97069e18	0.400828	0.200414	−	0.979711i	$-0.435771\pi$
$751$	1.97069e18	0.400828	0.200414	+	0.979711i	$0.435771\pi$
$752$	3.56303e18	0.718462
$753$	0	0
$754$	1.60574e18	0.318246
$755$	−5.92819e17	−0.116484
$756$	0	0
$757$	−8.51821e18	−1.64522	−0.822612	−	0.568603i	$-0.807484\pi$
$757$	−8.51821e18	−1.64522	−0.822612	+	0.568603i	$0.807484\pi$
$758$	3.80950e18	0.729487
$759$	0	0
$760$	−6.43665e17	−0.121163
$761$	3.33558e18	0.622546	0.311273	−	0.950321i	$-0.399245\pi$
$761$	3.33558e18	0.622546	0.311273	+	0.950321i	$0.399245\pi$
$762$	0	0
$763$	−2.07388e16	−0.00380517
$764$	3.35735e18	0.610787
$765$	0	0
$766$	2.51216e18	0.449324
$767$	4.05824e18	0.719727
$768$	0	0
$769$	4.94092e18	0.861563	0.430781	−	0.902456i	$-0.358238\pi$
$769$	4.94092e18	0.861563	0.430781	+	0.902456i	$0.358238\pi$
$770$	6.64265e15	0.00114855
$771$	0	0
$772$	4.98994e18	0.848365
$773$	−5.55567e18	−0.936633	−0.468317	−	0.883561i	$-0.655139\pi$
$773$	−5.55567e18	−0.936633	−0.468317	+	0.883561i	$0.655139\pi$
$774$	0	0
$775$	−8.57568e18	−1.42170
$776$	−5.58654e18	−0.918421
$777$	0	0
$778$	−8.91738e18	−1.44168
$779$	1.29678e19	2.07908
$780$	0	0
$781$	−4.94827e18	−0.780228
$782$	−2.28995e18	−0.358082
$783$	0	0
$784$	8.06189e18	1.23989
$785$	7.18060e17	0.109524
$786$	0	0
$787$	−1.08869e19	−1.63331	−0.816657	−	0.577124i	$-0.804175\pi$
$787$	−1.08869e19	−1.63331	−0.816657	+	0.577124i	$0.804175\pi$
$788$	−2.73396e18	−0.406791
$789$	0	0
$790$	−3.19976e17	−0.0468319
$791$	−5.00719e15	−0.000726853 0
$792$	0	0
$793$	−5.54172e18	−0.791351
$794$	1.30869e19	1.85355
$795$	0	0
$796$	6.98504e18	0.973271
$797$	4.46024e18	0.616424	0.308212	−	0.951318i	$-0.400269\pi$
$797$	4.46024e18	0.616424	0.308212	+	0.951318i	$0.400269\pi$
$798$	0	0
$799$	−1.67701e18	−0.228024
$800$	7.60174e18	1.02525
$801$	0	0
$802$	−8.09096e18	−1.07366
$803$	−1.32938e18	−0.174984
$804$	0	0
$805$	1.01018e16	0.00130835
$806$	−9.10227e18	−1.16942
$807$	0	0
$808$	8.67479e17	0.109669
$809$	1.01200e19	1.26916	0.634579	−	0.772858i	$-0.281174\pi$
$809$	1.01200e19	1.26916	0.634579	+	0.772858i	$0.281174\pi$
$810$	0	0
$811$	4.80568e18	0.593088	0.296544	−	0.955019i	$-0.404166\pi$
$811$	4.80568e18	0.593088	0.296544	+	0.955019i	$0.404166\pi$
$812$	−1.90883e16	−0.00233697
$813$	0	0
$814$	−4.01462e18	−0.483711
$815$	9.18300e17	0.109764
$816$	0	0
$817$	−1.53200e17	−0.0180225
$818$	−1.69722e19	−1.98080
$819$	0	0
$820$	1.58970e18	0.182610
$821$	−8.52222e18	−0.971230	−0.485615	−	0.874173i	$-0.661404\pi$
$821$	−8.52222e18	−0.971230	−0.485615	+	0.874173i	$0.661404\pi$
$822$	0	0
$823$	1.05317e19	1.18140	0.590702	−	0.806890i	$-0.298851\pi$
$823$	1.05317e19	1.18140	0.590702	+	0.806890i	$0.298851\pi$
$824$	−1.35948e18	−0.151303
$825$	0	0
$826$	−1.28650e17	−0.0140941
$827$	5.41391e18	0.588471	0.294235	−	0.955733i	$-0.404935\pi$
$827$	5.41391e18	0.588471	0.294235	+	0.955733i	$0.404935\pi$
$828$	0	0
$829$	4.47079e17	0.0478388	0.0239194	−	0.999714i	$-0.492385\pi$
$829$	4.47079e17	0.0478388	0.0239194	+	0.999714i	$0.492385\pi$
$830$	9.07351e17	0.0963314
$831$	0	0
$832$	6.24019e17	0.0652223
$833$	−3.79448e18	−0.393513
$834$	0	0
$835$	2.82833e18	0.288780
$836$	3.78115e18	0.383074
$837$	0	0
$838$	2.21068e19	2.20516
$839$	3.35229e18	0.331810	0.165905	−	0.986142i	$-0.446946\pi$
$839$	3.35229e18	0.331810	0.165905	+	0.986142i	$0.446946\pi$
$840$	0	0
$841$	−8.61108e18	−0.839235
$842$	−2.11628e19	−2.04666
$843$	0	0
$844$	−1.27797e18	−0.121701
$845$	1.20114e18	0.113508
$846$	0	0
$847$	−7.83747e16	−0.00729347
$848$	−1.99518e19	−1.84250
$849$	0	0
$850$	−5.28181e18	−0.480352
$851$	−6.10521e18	−0.551009
$852$	0	0
$853$	1.17906e19	1.04801	0.524007	−	0.851714i	$-0.324437\pi$
$853$	1.17906e19	1.04801	0.524007	+	0.851714i	$0.324437\pi$
$854$	1.75678e17	0.0154967
$855$	0	0
$856$	9.36317e18	0.813473
$857$	−6.74326e18	−0.581426	−0.290713	−	0.956810i	$-0.593892\pi$
$857$	−6.74326e18	−0.581426	−0.290713	+	0.956810i	$0.593892\pi$
$858$	0	0
$859$	1.95628e19	1.66140	0.830700	−	0.556720i	$-0.187940\pi$
$859$	1.95628e19	1.66140	0.830700	+	0.556720i	$0.187940\pi$
$860$	−1.87805e16	−0.00158295
$861$	0	0
$862$	−2.99647e19	−2.48779
$863$	8.84466e18	0.728805	0.364402	−	0.931242i	$-0.381273\pi$
$863$	8.84466e18	0.728805	0.364402	+	0.931242i	$0.381273\pi$
$864$	0	0
$865$	1.34169e18	0.108905
$866$	2.23384e17	0.0179964
$867$	0	0
$868$	1.08203e17	0.00858742
$869$	−1.25325e18	−0.0987205
$870$	0	0
$871$	−1.22406e19	−0.949912
$872$	2.57306e18	0.198195
$873$	0	0
$874$	1.53341e19	1.16368
$875$	4.74462e16	0.00357396
$876$	0	0
$877$	8.52671e18	0.632826	0.316413	−	0.948622i	$-0.397522\pi$
$877$	8.52671e18	0.632826	0.316413	+	0.948622i	$0.397522\pi$
$878$	−1.85487e19	−1.36647
$879$	0	0
$880$	−1.59673e18	−0.115903
$881$	4.36783e18	0.314719	0.157359	−	0.987541i	$-0.449702\pi$
$881$	4.36783e18	0.314719	0.157359	+	0.987541i	$0.449702\pi$
$882$	0	0
$883$	−2.24110e19	−1.59117	−0.795585	−	0.605842i	$-0.792836\pi$
$883$	−2.24110e19	−1.59117	−0.795585	+	0.605842i	$0.792836\pi$
$884$	−2.10224e18	−0.148164
$885$	0	0
$886$	−1.14810e19	−0.797371
$887$	1.04948e19	0.723554	0.361777	−	0.932265i	$-0.382170\pi$
$887$	1.04948e19	0.723554	0.361777	+	0.932265i	$0.382170\pi$
$888$	0	0
$889$	−2.24155e17	−0.0152295
$890$	3.97407e18	0.268040
$891$	0	0
$892$	−5.14256e18	−0.341828
$893$	1.12297e19	0.741026
$894$	0	0
$895$	3.36581e18	0.218897
$896$	1.40083e17	0.00904444
$897$	0	0
$898$	−2.37723e19	−1.51278
$899$	−9.35058e18	−0.590746
$900$	0	0
$901$	9.39066e18	0.584770
$902$	1.66040e19	1.02653
$903$	0	0
$904$	6.21243e17	0.0378588
$905$	−1.61683e18	−0.0978249
$906$	0	0
$907$	−1.42478e19	−0.849767	−0.424884	−	0.905248i	$-0.639685\pi$
$907$	−1.42478e19	−0.849767	−0.424884	+	0.905248i	$0.639685\pi$
$908$	3.70344e18	0.219304
$909$	0	0
$910$	2.47306e16	0.00144366
$911$	−1.87944e19	−1.08933	−0.544664	−	0.838654i	$-0.683343\pi$
$911$	−1.87944e19	−1.08933	−0.544664	+	0.838654i	$0.683343\pi$
$912$	0	0
$913$	3.55381e18	0.203064
$914$	4.20554e19	2.38601
$915$	0	0
$916$	−1.05936e19	−0.592546
$917$	−7.08556e16	−0.00393526
$918$	0	0
$919$	2.11122e19	1.15606	0.578032	−	0.816014i	$-0.303821\pi$
$919$	2.11122e19	1.15606	0.578032	+	0.816014i	$0.303821\pi$
$920$	−1.25333e18	−0.0681465
$921$	0	0
$922$	−3.06572e19	−1.64355
$923$	−1.84224e19	−0.980698
$924$	0	0
$925$	−1.40818e19	−0.739156
$926$	−1.75540e19	−0.914965
$927$	0	0
$928$	8.28863e18	0.426011
$929$	2.93826e19	1.49964	0.749822	−	0.661639i	$-0.230139\pi$
$929$	2.93826e19	1.49964	0.749822	+	0.661639i	$0.230139\pi$
$930$	0	0
$931$	2.54089e19	1.27883
$932$	8.98218e18	0.448930
$933$	0	0
$934$	2.87956e19	1.41929
$935$	7.51530e17	0.0367850
$936$	0	0
$937$	−2.32101e18	−0.112039	−0.0560195	−	0.998430i	$-0.517841\pi$
$937$	−2.32101e18	−0.112039	−0.0560195	+	0.998430i	$0.517841\pi$
$938$	3.88037e17	0.0186018
$939$	0	0
$940$	1.37663e18	0.0650859
$941$	1.70723e19	0.801601	0.400800	−	0.916165i	$-0.368732\pi$
$941$	1.70723e19	0.801601	0.400800	+	0.916165i	$0.368732\pi$
$942$	0	0
$943$	2.52505e19	1.16935
$944$	3.09242e19	1.42227
$945$	0	0
$946$	−1.96158e17	−0.00889844
$947$	1.83288e19	0.825768	0.412884	−	0.910784i	$-0.364521\pi$
$947$	1.83288e19	0.825768	0.412884	+	0.910784i	$0.364521\pi$
$948$	0	0
$949$	−4.94930e18	−0.219945
$950$	3.53684e19	1.56103
$951$	0	0
$952$	−4.44334e16	−0.00193450
$953$	−3.07542e19	−1.32984	−0.664922	−	0.746913i	$-0.731535\pi$
$953$	−3.07542e19	−1.32984	−0.664922	+	0.746913i	$0.731535\pi$
$954$	0	0
$955$	−4.46841e18	−0.190604
$956$	−2.01051e19	−0.851787
$957$	0	0
$958$	5.35160e19	2.23670
$959$	−8.55780e16	−0.00355256
$960$	0	0
$961$	2.85869e19	1.17075
$962$	−1.49465e19	−0.607995
$963$	0	0
$964$	−1.03376e19	−0.414876
$965$	−6.64128e18	−0.264743
$966$	0	0
$967$	−3.02790e19	−1.19088	−0.595442	−	0.803398i	$-0.703023\pi$
$967$	−3.02790e19	−1.19088	−0.595442	+	0.803398i	$0.703023\pi$
$968$	9.72397e18	0.379886
$969$	0	0
$970$	−1.11518e19	−0.429861
$971$	−5.19323e18	−0.198844	−0.0994221	−	0.995045i	$-0.531699\pi$
$971$	−5.19323e18	−0.198844	−0.0994221	+	0.995045i	$0.531699\pi$
$972$	0	0
$973$	−3.02862e16	−0.00114423
$974$	4.87033e19	1.82778
$975$	0	0
$976$	−4.22285e19	−1.56380
$977$	−1.76022e19	−0.647517	−0.323758	−	0.946140i	$-0.604947\pi$
$977$	−1.76022e19	−0.647517	−0.323758	+	0.946140i	$0.604947\pi$
$978$	0	0
$979$	1.55652e19	0.565023
$980$	3.11484e18	0.112322
$981$	0	0
$982$	3.10787e19	1.10596
$983$	−1.35519e19	−0.479074	−0.239537	−	0.970887i	$-0.576996\pi$
$983$	−1.35519e19	−0.479074	−0.239537	+	0.970887i	$0.576996\pi$
$984$	0	0
$985$	3.63871e18	0.126944
$986$	−5.75907e18	−0.199596
$987$	0	0
$988$	1.40772e19	0.481500
$989$	−2.98306e17	−0.0101365
$990$	0	0
$991$	−3.58254e19	−1.20147	−0.600734	−	0.799449i	$-0.705125\pi$
$991$	−3.58254e19	−1.20147	−0.600734	+	0.799449i	$0.705125\pi$
$992$	−4.69847e19	−1.56542
$993$	0	0
$994$	5.84007e17	0.0192046
$995$	−9.29662e18	−0.303721
$996$	0	0
$997$	−3.39036e19	−1.09327	−0.546635	−	0.837371i	$-0.684091\pi$
$997$	−3.39036e19	−1.09327	−0.546635	+	0.837371i	$0.684091\pi$
$998$	−4.00557e19	−1.28326
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.14.a.b.1.4	✓	4
3.2	odd	2		27.14.a.e.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.14.a.b.1.4	✓	4	1.1	even	1	trivial
27.14.a.e.1.1	yes	4	3.2	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	27.a (trivial)

\(f(q)\)	\(=\)	\(q+114.486 q^{2} +4914.99 q^{4} -6541.53 q^{5} +3023.86 q^{7} -375171. q^{8} -748912. q^{10} -2.93325e6 q^{11} -1.09205e7 q^{13} +346189. q^{14} -8.32153e7 q^{16} +3.91668e7 q^{17} -2.62272e8 q^{19} -3.21515e7 q^{20} -3.35815e8 q^{22} -5.10689e8 q^{23} -1.17791e9 q^{25} -1.25024e9 q^{26} +1.48623e7 q^{28} -1.28435e9 q^{29} +7.28041e9 q^{31} -6.45357e9 q^{32} +4.48404e9 q^{34} -1.97807e7 q^{35} +1.19549e10 q^{37} -3.00264e10 q^{38} +2.45419e9 q^{40} -4.94440e10 q^{41} +5.84125e8 q^{43} -1.44169e10 q^{44} -5.84666e10 q^{46} -4.28170e10 q^{47} -9.68799e10 q^{49} -1.34854e11 q^{50} -5.36741e10 q^{52} +2.39761e11 q^{53} +1.91879e10 q^{55} -1.13446e9 q^{56} -1.47039e11 q^{58} -3.71617e11 q^{59} +5.07461e11 q^{61} +8.33504e11 q^{62} -5.71420e10 q^{64} +7.14367e10 q^{65} +1.12088e12 q^{67} +1.92505e11 q^{68} -2.26461e9 q^{70} +1.68696e12 q^{71} +4.53212e11 q^{73} +1.36866e12 q^{74} -1.28906e12 q^{76} -8.86974e9 q^{77} +4.27255e11 q^{79} +5.44355e11 q^{80} -5.66063e12 q^{82} -1.21156e12 q^{83} -2.56211e11 q^{85} +6.68740e10 q^{86} +1.10047e12 q^{88} -5.30646e12 q^{89} -3.30221e10 q^{91} -2.51003e12 q^{92} -4.90194e12 q^{94} +1.71566e12 q^{95} +1.48907e13 q^{97} -1.10914e13 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 117 q^{2} + 8101 q^{4} - 11412 q^{5} + 54572 q^{7} - 615375 q^{8} + 969579 q^{10} - 4927212 q^{11} + 12107576 q^{13} - 2015487 q^{14} + 56856625 q^{16} + 34729992 q^{17} + 269848016 q^{19} - 179564463 q^{20}+ \cdots - 3817815378498 q^{98}+O(q^{100}) \) 4 * q - 117 * q^2 + 8101 * q^4 - 11412 * q^5 + 54572 * q^7 - 615375 * q^8 + 969579 * q^10 - 4927212 * q^11 + 12107576 * q^13 - 2015487 * q^14 + 56856625 * q^16 + 34729992 * q^17 + 269848016 * q^19 - 179564463 * q^20 + 971969481 * q^22 - 1296333360 * q^23 - 354407528 * q^25 - 4599933804 * q^26 - 1290887245 * q^28 - 6967785600 * q^29 + 1449752348 * q^31 - 18104036247 * q^32 - 13245631488 * q^34 - 39458246292 * q^35 - 14813291296 * q^37 - 87551260146 * q^38 - 3563985051 * q^40 - 96485867568 * q^41 + 21952024664 * q^43 - 244812106521 * q^44 + 54796717602 * q^46 - 144494756856 * q^47 - 21816898080 * q^49 - 21012587604 * q^50 + 313979782172 * q^52 - 231237557460 * q^53 - 118437139188 * q^55 + 262334662191 * q^56 - 98625574602 * q^58 + 654991446240 * q^59 - 97408780816 * q^61 + 463348969227 * q^62 - 684162915215 * q^64 + 1432017576504 * q^65 - 729275019496 * q^67 + 3678684087600 * q^68 + 1478759994297 * q^70 + 1921547025288 * q^71 - 2662954340932 * q^73 + 4859349629382 * q^74 + 1809674429402 * q^76 + 2884684296924 * q^77 + 3435370655288 * q^79 + 1773900202293 * q^80 - 3142316785446 * q^82 - 3544164633852 * q^83 - 1212132619224 * q^85 - 4370115792870 * q^86 + 20297903695659 * q^88 - 7763565281544 * q^89 - 14738138711480 * q^91 - 10986690530610 * q^92 - 7609182540642 * q^94 - 24364387296720 * q^95 + 24376525074716 * q^97 - 3817815378498 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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