LMFDB - Embedded newform 27.16.a.d.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [27,16,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, 16); 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, 16, names="a")

Level:	$ N $	$=$	$ 27 = 3^{3} $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	27.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0] 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:	$38.5272463770$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3152x^{4} + 2666232x^{2} - 295612416 $ x^6 - 3152x^4 + 2666232x^2 - 295612416
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{13}\cdot 3^{21}\cdot 5^{3} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$40.1247$ 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)

$f(q)$	$=$	\(q-169.081 q^{2} -4179.66 q^{4} -27639.5 q^{5} -3.84024e6 q^{7} +6.24714e6 q^{8} +4.67332e6 q^{10} -1.22259e8 q^{11} +1.81707e8 q^{13} +6.49310e8 q^{14} -9.19313e8 q^{16} -2.91612e9 q^{17} -1.69610e9 q^{19} +1.15524e8 q^{20} +2.06716e10 q^{22} -1.88617e10 q^{23} -2.97536e10 q^{25} -3.07231e10 q^{26} +1.60509e10 q^{28} -7.90277e10 q^{29} -2.87486e10 q^{31} -4.92681e10 q^{32} +4.93061e11 q^{34} +1.06142e11 q^{35} +2.75482e11 q^{37} +2.86779e11 q^{38} -1.72668e11 q^{40} +8.24004e11 q^{41} +1.79681e12 q^{43} +5.11001e11 q^{44} +3.18916e12 q^{46} +1.50562e12 q^{47} +9.99985e12 q^{49} +5.03077e12 q^{50} -7.59473e11 q^{52} -3.41933e12 q^{53} +3.37918e12 q^{55} -2.39905e13 q^{56} +1.33621e13 q^{58} -1.89591e12 q^{59} +3.35667e13 q^{61} +4.86083e12 q^{62} +3.84543e13 q^{64} -5.02229e12 q^{65} -4.47898e13 q^{67} +1.21884e13 q^{68} -1.79466e13 q^{70} -6.50858e13 q^{71} -9.09481e13 q^{73} -4.65788e13 q^{74} +7.08914e12 q^{76} +4.69503e14 q^{77} -5.11555e13 q^{79} +2.54094e13 q^{80} -1.39323e14 q^{82} -2.70444e14 q^{83} +8.06003e13 q^{85} -3.03806e14 q^{86} -7.63769e14 q^{88} +4.45904e14 q^{89} -6.97797e14 q^{91} +7.88356e13 q^{92} -2.54571e14 q^{94} +4.68795e13 q^{95} +1.12131e15 q^{97} -1.69078e15 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 71124 q^{4} + 2377362 q^{7} + 46852560 q^{10} + 387940494 q^{13} + 1056508872 q^{16} - 1979573730 q^{19} - 16927522128 q^{22} - 792180750 q^{25} + 267175396092 q^{28} + 484649028120 q^{31} + 165585346416 q^{34}+ \cdots - 718669835357730 q^{97}+O(q^{100}) $ 6 * q + 71124 * q^4 + 2377362 * q^7 + 46852560 * q^10 + 387940494 * q^13 + 1056508872 * q^16 - 1979573730 * q^19 - 16927522128 * q^22 - 792180750 * q^25 + 267175396092 * q^28 + 484649028120 * q^31 + 165585346416 * q^34 + 2108295778962 * q^37 + 5048325004320 * q^40 + 4208638642896 * q^43 + 24591340387440 * q^46 + 26450424994284 * q^49 + 56443954337220 * q^52 + 35614339104960 * q^55 + 68411200637952 * q^58 + 81589045842426 * q^61 + 11515176559824 * q^64 + 87445603264854 * q^67 + 83608373597040 * q^70 - 165007694606730 * q^73 - 393810695155932 * q^76 - 740397172026354 * q^79 - 1526116173910272 * q^82 - 913199003104320 * q^85 - 2054825311293216 * q^88 - 1087541878427094 * q^91 + 14553520366896 * q^94 - 718669835357730 * q^97

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 $

$2$	−169.081	−0.934049	−0.467024	−	0.884244i	$-0.654674\pi$
$2$	−169.081	−0.934049	−0.467024	+	0.884244i	$0.654674\pi$
$3$	0	0
$3$	0	0
$4$	−4179.66	−0.127553
$5$	−27639.5	−0.158218	−0.0791090	−	0.996866i	$-0.525208\pi$
$5$	−27639.5	−0.158218	−0.0791090	+	0.996866i	$0.525208\pi$
$6$	0	0
$7$	−3.84024e6	−1.76247	−0.881237	−	0.472675i	$-0.843288\pi$
$7$	−3.84024e6	−1.76247	−0.881237	+	0.472675i	$0.843288\pi$
$8$	6.24714e6	1.05319
$9$	0	0
$10$	4.67332e6	0.147783
$11$	−1.22259e8	−1.89163	−0.945813	−	0.324710i	$-0.894733\pi$
$11$	−1.22259e8	−1.89163	−0.945813	+	0.324710i	$0.894733\pi$
$12$	0	0
$13$	1.81707e8	0.803149	0.401574	−	0.915826i	$-0.368463\pi$
$13$	1.81707e8	0.803149	0.401574	+	0.915826i	$0.368463\pi$
$14$	6.49310e8	1.64624
$15$	0	0
$16$	−9.19313e8	−0.856177
$17$	−2.91612e9	−1.72361	−0.861805	−	0.507240i	$-0.830666\pi$
$17$	−2.91612e9	−1.72361	−0.861805	+	0.507240i	$0.830666\pi$
$18$	0	0
$19$	−1.69610e9	−0.435312	−0.217656	−	0.976026i	$-0.569841\pi$
$19$	−1.69610e9	−0.435312	−0.217656	+	0.976026i	$0.569841\pi$
$20$	1.15524e8	0.0201812
$21$	0	0
$22$	2.06716e10	1.76687
$23$	−1.88617e10	−1.15511	−0.577554	−	0.816352i	$-0.695993\pi$
$23$	−1.88617e10	−1.15511	−0.577554	+	0.816352i	$0.695993\pi$
$24$	0	0
$25$	−2.97536e10	−0.974967
$26$	−3.07231e10	−0.750180
$27$	0	0
$28$	1.60509e10	0.224809
$29$	−7.90277e10	−0.850735	−0.425368	−	0.905021i	$-0.639855\pi$
$29$	−7.90277e10	−0.850735	−0.425368	+	0.905021i	$0.639855\pi$
$30$	0	0
$31$	−2.87486e10	−0.187674	−0.0938369	−	0.995588i	$-0.529913\pi$
$31$	−2.87486e10	−0.187674	−0.0938369	+	0.995588i	$0.529913\pi$
$32$	−4.92681e10	−0.253478
$33$	0	0
$34$	4.93061e11	1.60994
$35$	1.06142e11	0.278855
$36$	0	0
$37$	2.75482e11	0.477069	0.238534	−	0.971134i	$-0.423333\pi$
$37$	2.75482e11	0.477069	0.238534	+	0.971134i	$0.423333\pi$
$38$	2.86779e11	0.406602
$39$	0	0
$40$	−1.72668e11	−0.166633
$41$	8.24004e11	0.660770	0.330385	−	0.943846i	$-0.392821\pi$
$41$	8.24004e11	0.660770	0.330385	+	0.943846i	$0.392821\pi$
$42$	0	0
$43$	1.79681e12	1.00806	0.504032	−	0.863685i	$-0.331849\pi$
$43$	1.79681e12	1.00806	0.504032	+	0.863685i	$0.331849\pi$
$44$	5.11001e11	0.241283
$45$	0	0
$46$	3.18916e12	1.07893
$47$	1.50562e12	0.433492	0.216746	−	0.976228i	$-0.430456\pi$
$47$	1.50562e12	0.433492	0.216746	+	0.976228i	$0.430456\pi$
$48$	0	0
$49$	9.99985e12	2.10631
$50$	5.03077e12	0.910667
$51$	0	0
$52$	−7.59473e11	−0.102444
$53$	−3.41933e12	−0.399827	−0.199914	−	0.979814i	$-0.564066\pi$
$53$	−3.41933e12	−0.399827	−0.199914	+	0.979814i	$0.564066\pi$
$54$	0	0
$55$	3.37918e12	0.299289
$56$	−2.39905e13	−1.85622
$57$	0	0
$58$	1.33621e13	0.794628
$59$	−1.89591e12	−0.0991806	−0.0495903	−	0.998770i	$-0.515792\pi$
$59$	−1.89591e12	−0.0991806	−0.0495903	+	0.998770i	$0.515792\pi$
$60$	0	0
$61$	3.35667e13	1.36753	0.683763	−	0.729704i	$-0.260342\pi$
$61$	3.35667e13	1.36753	0.683763	+	0.729704i	$0.260342\pi$
$62$	4.86083e12	0.175296
$63$	0	0
$64$	3.84543e13	1.09294
$65$	−5.02229e12	−0.127073
$66$	0	0
$67$	−4.47898e13	−0.902855	−0.451428	−	0.892308i	$-0.649085\pi$
$67$	−4.47898e13	−0.902855	−0.451428	+	0.892308i	$0.649085\pi$
$68$	1.21884e13	0.219852
$69$	0	0
$70$	−1.79466e13	−0.260464
$71$	−6.50858e13	−0.849275	−0.424638	−	0.905363i	$-0.639598\pi$
$71$	−6.50858e13	−0.849275	−0.424638	+	0.905363i	$0.639598\pi$
$72$	0	0
$73$	−9.09481e13	−0.963546	−0.481773	−	0.876296i	$-0.660007\pi$
$73$	−9.09481e13	−0.963546	−0.481773	+	0.876296i	$0.660007\pi$
$74$	−4.65788e13	−0.445605
$75$	0	0
$76$	7.08914e12	0.0555254
$77$	4.69503e14	3.33394
$78$	0	0
$79$	−5.11555e13	−0.299702	−0.149851	−	0.988709i	$-0.547879\pi$
$79$	−5.11555e13	−0.299702	−0.149851	+	0.988709i	$0.547879\pi$
$80$	2.54094e13	0.135463
$81$	0	0
$82$	−1.39323e14	−0.617192
$83$	−2.70444e14	−1.09393	−0.546967	−	0.837154i	$-0.684218\pi$
$83$	−2.70444e14	−1.09393	−0.546967	+	0.837154i	$0.684218\pi$
$84$	0	0
$85$	8.06003e13	0.272706
$86$	−3.03806e14	−0.941581
$87$	0	0
$88$	−7.63769e14	−1.99224
$89$	4.45904e14	1.06860	0.534302	−	0.845294i	$-0.320575\pi$
$89$	4.45904e14	1.06860	0.534302	+	0.845294i	$0.320575\pi$
$90$	0	0
$91$	−6.97797e14	−1.41553
$92$	7.88356e13	0.147338
$93$	0	0
$94$	−2.54571e14	−0.404903
$95$	4.68795e13	0.0688741
$96$	0	0
$97$	1.12131e15	1.40909	0.704546	−	0.709658i	$-0.251151\pi$
$97$	1.12131e15	1.40909	0.704546	+	0.709658i	$0.251151\pi$
$98$	−1.69078e15	−1.96740
$99$	0	0
$100$	1.24360e14	0.124360
$101$	−1.07017e15	−0.993211	−0.496605	−	0.867976i	$-0.665420\pi$
$101$	−1.07017e15	−0.993211	−0.496605	+	0.867976i	$0.665420\pi$
$102$	0	0
$103$	−6.99161e14	−0.560142	−0.280071	−	0.959979i	$-0.590358\pi$
$103$	−6.99161e14	−0.560142	−0.280071	+	0.959979i	$0.590358\pi$
$104$	1.13515e15	0.845868
$105$	0	0
$106$	5.78144e14	0.373458
$107$	1.44920e15	0.872471	0.436235	−	0.899833i	$-0.356312\pi$
$107$	1.44920e15	0.872471	0.436235	+	0.899833i	$0.356312\pi$
$108$	0	0
$109$	−1.00973e15	−0.529063	−0.264532	−	0.964377i	$-0.585217\pi$
$109$	−1.00973e15	−0.529063	−0.264532	+	0.964377i	$0.585217\pi$
$110$	−5.71355e14	−0.279551
$111$	0	0
$112$	3.53038e15	1.50899
$113$	−1.14823e15	−0.459134	−0.229567	−	0.973293i	$-0.573731\pi$
$113$	−1.14823e15	−0.459134	−0.229567	+	0.973293i	$0.573731\pi$
$114$	0	0
$115$	5.21329e14	0.182759
$116$	3.30309e14	0.108514
$117$	0	0
$118$	3.20562e14	0.0926395
$119$	1.11986e16	3.03782
$120$	0	0
$121$	1.07700e16	2.57825
$122$	−5.67549e15	−1.27734
$123$	0	0
$124$	1.20159e14	0.0239384
$125$	1.66587e15	0.312475
$126$	0	0
$127$	−2.73884e15	−0.456077	−0.228038	−	0.973652i	$-0.573231\pi$
$127$	−2.73884e15	−0.456077	−0.228038	+	0.973652i	$0.573231\pi$
$128$	−4.88748e15	−0.767379
$129$	0	0
$130$	8.49173e14	0.118692
$131$	−3.20071e15	−0.422388	−0.211194	−	0.977444i	$-0.567735\pi$
$131$	−3.20071e15	−0.422388	−0.211194	+	0.977444i	$0.567735\pi$
$132$	0	0
$133$	6.51344e15	0.767226
$134$	7.57310e15	0.843311
$135$	0	0
$136$	−1.82174e16	−1.81529
$137$	−5.18798e15	−0.489321	−0.244661	−	0.969609i	$-0.578677\pi$
$137$	−5.18798e15	−0.489321	−0.244661	+	0.969609i	$0.578677\pi$
$138$	0	0
$139$	5.05637e15	0.427787	0.213894	−	0.976857i	$-0.431385\pi$
$139$	5.05637e15	0.427787	0.213894	+	0.976857i	$0.431385\pi$
$140$	−4.43639e14	−0.0355688
$141$	0	0
$142$	1.10048e16	0.793264
$143$	−2.22153e16	−1.51926
$144$	0	0
$145$	2.18429e15	0.134602
$146$	1.53776e16	0.899999
$147$	0	0
$148$	−1.15142e15	−0.0608516
$149$	−1.63384e16	−0.820942	−0.410471	−	0.911874i	$-0.634636\pi$
$149$	−1.63384e16	−0.820942	−0.410471	+	0.911874i	$0.634636\pi$
$150$	0	0
$151$	−2.49188e16	−1.13292	−0.566461	−	0.824089i	$-0.691688\pi$
$151$	−2.49188e16	−1.13292	−0.566461	+	0.824089i	$0.691688\pi$
$152$	−1.05958e16	−0.458466
$153$	0	0
$154$	−7.93840e16	−3.11406
$155$	7.94597e14	0.0296934
$156$	0	0
$157$	1.79217e16	0.608320	0.304160	−	0.952621i	$-0.401624\pi$
$157$	1.79217e16	0.608320	0.304160	+	0.952621i	$0.401624\pi$
$158$	8.64942e15	0.279936
$159$	0	0
$160$	1.36175e15	0.0401048
$161$	7.24335e16	2.03585
$162$	0	0
$163$	−4.54299e16	−1.16395	−0.581976	−	0.813206i	$-0.697720\pi$
$163$	−4.54299e16	−1.16395	−0.581976	+	0.813206i	$0.697720\pi$
$164$	−3.44406e15	−0.0842833
$165$	0	0
$166$	4.57269e16	1.02179
$167$	3.67143e16	0.784263	0.392132	−	0.919909i	$-0.371738\pi$
$167$	3.67143e16	0.784263	0.392132	+	0.919909i	$0.371738\pi$
$168$	0	0
$169$	−1.81685e16	−0.354952
$170$	−1.36280e16	−0.254721
$171$	0	0
$172$	−7.51004e15	−0.128582
$173$	−1.00354e17	−1.64509	−0.822547	−	0.568697i	$-0.807448\pi$
$173$	−1.00354e17	−1.64509	−0.822547	+	0.568697i	$0.807448\pi$
$174$	0	0
$175$	1.14261e17	1.71835
$176$	1.12394e17	1.61957
$177$	0	0
$178$	−7.53939e16	−0.998128
$179$	−1.02308e17	−1.29871	−0.649355	−	0.760485i	$-0.724961\pi$
$179$	−1.02308e17	−1.29871	−0.649355	+	0.760485i	$0.724961\pi$
$180$	0	0
$181$	−6.05521e16	−0.707196	−0.353598	−	0.935397i	$-0.615042\pi$
$181$	−6.05521e16	−0.707196	−0.353598	+	0.935397i	$0.615042\pi$
$182$	1.17984e17	1.32217
$183$	0	0
$184$	−1.17832e17	−1.21655
$185$	−7.61420e15	−0.0754808
$186$	0	0
$187$	3.56522e17	3.26043
$188$	−6.29298e15	−0.0552933
$189$	0	0
$190$	−7.92643e15	−0.0643318
$191$	−1.65278e17	−1.28963	−0.644814	−	0.764339i	$-0.723065\pi$
$191$	−1.65278e17	−1.28963	−0.644814	+	0.764339i	$0.723065\pi$
$192$	0	0
$193$	1.49641e17	1.07987	0.539935	−	0.841706i	$-0.318449\pi$
$193$	1.49641e17	1.07987	0.539935	+	0.841706i	$0.318449\pi$
$194$	−1.89593e17	−1.31616
$195$	0	0
$196$	−4.17960e16	−0.268667
$197$	−4.98992e15	−0.0308743	−0.0154371	−	0.999881i	$-0.504914\pi$
$197$	−4.98992e15	−0.0308743	−0.0154371	+	0.999881i	$0.504914\pi$
$198$	0	0
$199$	1.85140e17	1.06195	0.530973	−	0.847389i	$-0.321827\pi$
$199$	1.85140e17	1.06195	0.530973	+	0.847389i	$0.321827\pi$
$200$	−1.85875e17	−1.02683
$201$	0	0
$202$	1.80945e17	0.927707
$203$	3.03485e17	1.49940
$204$	0	0
$205$	−2.27751e16	−0.104546
$206$	1.18215e17	0.523200
$207$	0	0
$208$	−1.67045e17	−0.687638
$209$	2.07364e17	0.823448
$210$	0	0
$211$	2.98141e16	0.110231	0.0551155	−	0.998480i	$-0.482447\pi$
$211$	2.98141e16	0.110231	0.0551155	+	0.998480i	$0.482447\pi$
$212$	1.42917e16	0.0509992
$213$	0	0
$214$	−2.45032e17	−0.814930
$215$	−4.96629e16	−0.159494
$216$	0	0
$217$	1.10401e17	0.330770
$218$	1.70726e17	0.494171
$219$	0	0
$220$	−1.41238e16	−0.0381753
$221$	−5.29879e17	−1.38431
$222$	0	0
$223$	1.08064e17	0.263872	0.131936	−	0.991258i	$-0.457881\pi$
$223$	1.08064e17	0.263872	0.131936	+	0.991258i	$0.457881\pi$
$224$	1.89201e17	0.446749
$225$	0	0
$226$	1.94143e17	0.428854
$227$	6.50374e17	1.38985	0.694927	−	0.719080i	$-0.255436\pi$
$227$	6.50374e17	1.38985	0.694927	+	0.719080i	$0.255436\pi$
$228$	0	0
$229$	−4.01890e17	−0.804158	−0.402079	−	0.915605i	$-0.631712\pi$
$229$	−4.01890e17	−0.804158	−0.402079	+	0.915605i	$0.631712\pi$
$230$	−8.81468e16	−0.170706
$231$	0	0
$232$	−4.93698e17	−0.895986
$233$	6.27335e17	1.10238	0.551189	−	0.834381i	$-0.314174\pi$
$233$	6.27335e17	1.10238	0.551189	+	0.834381i	$0.314174\pi$
$234$	0	0
$235$	−4.16146e16	−0.0685862
$236$	7.92425e15	0.0126508
$237$	0	0
$238$	−1.89347e18	−2.83747
$239$	−2.41672e17	−0.350947	−0.175474	−	0.984484i	$-0.556146\pi$
$239$	−2.41672e17	−0.350947	−0.175474	+	0.984484i	$0.556146\pi$
$240$	0	0
$241$	2.04932e17	0.279565	0.139782	−	0.990182i	$-0.455360\pi$
$241$	2.04932e17	0.279565	0.139782	+	0.990182i	$0.455360\pi$
$242$	−1.82100e18	−2.40821
$243$	0	0
$244$	−1.40298e17	−0.174432
$245$	−2.76391e17	−0.333256
$246$	0	0
$247$	−3.08194e17	−0.349620
$248$	−1.79596e17	−0.197656
$249$	0	0
$250$	−2.81666e17	−0.291867
$251$	7.70354e17	0.774707	0.387354	−	0.921931i	$-0.373389\pi$
$251$	7.70354e17	0.774707	0.387354	+	0.921931i	$0.373389\pi$
$252$	0	0
$253$	2.30602e18	2.18503
$254$	4.63085e17	0.425998
$255$	0	0
$256$	−4.33693e17	−0.376169
$257$	−6.21658e17	−0.523664	−0.261832	−	0.965113i	$-0.584327\pi$
$257$	−6.21658e17	−0.523664	−0.261832	+	0.965113i	$0.584327\pi$
$258$	0	0
$259$	−1.05792e18	−0.840821
$260$	2.09915e16	0.0162085
$261$	0	0
$262$	5.41179e17	0.394531
$263$	−2.21197e18	−1.56715	−0.783575	−	0.621297i	$-0.786606\pi$
$263$	−2.21197e18	−1.56715	−0.783575	+	0.621297i	$0.786606\pi$
$264$	0	0
$265$	9.45088e16	0.0632599
$266$	−1.10130e18	−0.716626
$267$	0	0
$268$	1.87206e17	0.115162
$269$	2.18962e18	1.30987	0.654933	−	0.755687i	$-0.272697\pi$
$269$	2.18962e18	1.30987	0.654933	+	0.755687i	$0.272697\pi$
$270$	0	0
$271$	7.31740e17	0.414083	0.207041	−	0.978332i	$-0.433616\pi$
$271$	7.31740e17	0.414083	0.207041	+	0.978332i	$0.433616\pi$
$272$	2.68083e18	1.47571
$273$	0	0
$274$	8.77188e17	0.457050
$275$	3.63765e18	1.84427
$276$	0	0
$277$	−3.29858e17	−0.158390	−0.0791952	−	0.996859i	$-0.525235\pi$
$277$	−3.29858e17	−0.158390	−0.0791952	+	0.996859i	$0.525235\pi$
$278$	−8.54935e17	−0.399574
$279$	0	0
$280$	6.63086e17	0.293687
$281$	−1.05923e18	−0.456766	−0.228383	−	0.973571i	$-0.573344\pi$
$281$	−1.05923e18	−0.456766	−0.228383	+	0.973571i	$0.573344\pi$
$282$	0	0
$283$	−2.31294e16	−0.00945727	−0.00472864	−	0.999989i	$-0.501505\pi$
$283$	−2.31294e16	−0.00945727	−0.00472864	+	0.999989i	$0.501505\pi$
$284$	2.72036e17	0.108328
$285$	0	0
$286$	3.75618e18	1.41906
$287$	−3.16437e18	−1.16459
$288$	0	0
$289$	5.64135e18	1.97083
$290$	−3.69322e17	−0.125724
$291$	0	0
$292$	3.80132e17	0.122903
$293$	3.71775e18	1.17158	0.585791	−	0.810462i	$-0.300784\pi$
$293$	3.71775e18	1.17158	0.585791	+	0.810462i	$0.300784\pi$
$294$	0	0
$295$	5.24020e16	0.0156922
$296$	1.72098e18	0.502444
$297$	0	0
$298$	2.76251e18	0.766799
$299$	−3.42730e18	−0.927724
$300$	0	0
$301$	−6.90016e18	−1.77669
$302$	4.21329e18	1.05820
$303$	0	0
$304$	1.55925e18	0.372704
$305$	−9.27769e17	−0.216367
$306$	0	0
$307$	−8.42886e18	−1.87168	−0.935838	−	0.352429i	$-0.885356\pi$
$307$	−8.42886e18	−1.87168	−0.935838	+	0.352429i	$0.885356\pi$
$308$	−1.96236e18	−0.425255
$309$	0	0
$310$	−1.34351e17	−0.0277350
$311$	−7.67091e18	−1.54577	−0.772884	−	0.634548i	$-0.781186\pi$
$311$	−7.67091e18	−1.54577	−0.772884	+	0.634548i	$0.781186\pi$
$312$	0	0
$313$	2.03750e18	0.391306	0.195653	−	0.980673i	$-0.437317\pi$
$313$	2.03750e18	0.391306	0.195653	+	0.980673i	$0.437317\pi$
$314$	−3.03022e18	−0.568201
$315$	0	0
$316$	2.13813e17	0.0382279
$317$	−4.34339e18	−0.758376	−0.379188	−	0.925320i	$-0.623797\pi$
$317$	−4.34339e18	−0.758376	−0.379188	+	0.925320i	$0.623797\pi$
$318$	0	0
$319$	9.66185e18	1.60927
$320$	−1.06286e18	−0.172922
$321$	0	0
$322$	−1.22471e19	−1.90158
$323$	4.94605e18	0.750308
$324$	0	0
$325$	−5.40644e18	−0.783044
$326$	7.68133e18	1.08719
$327$	0	0
$328$	5.14767e18	0.695916
$329$	−5.78194e18	−0.764019
$330$	0	0
$331$	−3.39424e18	−0.428581	−0.214291	−	0.976770i	$-0.568744\pi$
$331$	−3.39424e18	−0.428581	−0.214291	+	0.976770i	$0.568744\pi$
$332$	1.13036e18	0.139535
$333$	0	0
$334$	−6.20769e18	−0.732540
$335$	1.23797e18	0.142848
$336$	0	0
$337$	−1.37227e19	−1.51431	−0.757157	−	0.653233i	$-0.773412\pi$
$337$	−1.37227e19	−1.51431	−0.757157	+	0.653233i	$0.773412\pi$
$338$	3.07195e18	0.331542
$339$	0	0
$340$	−3.36882e17	−0.0347845
$341$	3.51477e18	0.355009
$342$	0	0
$343$	−2.01700e19	−1.94985
$344$	1.12249e19	1.06168
$345$	0	0
$346$	1.69680e19	1.53660
$347$	8.41324e18	0.745577	0.372788	−	0.927916i	$-0.378402\pi$
$347$	8.41324e18	0.745577	0.372788	+	0.927916i	$0.378402\pi$
$348$	0	0
$349$	−3.99165e18	−0.338814	−0.169407	−	0.985546i	$-0.554185\pi$
$349$	−3.99165e18	−0.338814	−0.169407	+	0.985546i	$0.554185\pi$
$350$	−1.93193e19	−1.60503
$351$	0	0
$352$	6.02347e18	0.479487
$353$	−7.73764e18	−0.602974	−0.301487	−	0.953470i	$-0.597483\pi$
$353$	−7.73764e18	−0.602974	−0.301487	+	0.953470i	$0.597483\pi$
$354$	0	0
$355$	1.79894e18	0.134371
$356$	−1.86373e18	−0.136304
$357$	0	0
$358$	1.72983e19	1.21306
$359$	2.11510e19	1.45252	0.726260	−	0.687420i	$-0.241257\pi$
$359$	2.11510e19	1.45252	0.726260	+	0.687420i	$0.241257\pi$
$360$	0	0
$361$	−1.23044e19	−0.810504
$362$	1.02382e19	0.660556
$363$	0	0
$364$	2.91655e18	0.180555
$365$	2.51376e18	0.152450
$366$	0	0
$367$	2.04039e19	1.18773	0.593866	−	0.804564i	$-0.297601\pi$
$367$	2.04039e19	1.18773	0.593866	+	0.804564i	$0.297601\pi$
$368$	1.73398e19	0.988978
$369$	0	0
$370$	1.28742e18	0.0705027
$371$	1.31311e19	0.704685
$372$	0	0
$373$	−1.09276e18	−0.0563260	−0.0281630	−	0.999603i	$-0.508966\pi$
$373$	−1.09276e18	−0.0563260	−0.0281630	+	0.999603i	$0.508966\pi$
$374$	−6.02811e19	−3.04540
$375$	0	0
$376$	9.40582e18	0.456550
$377$	−1.43599e19	−0.683267
$378$	0	0
$379$	−1.69674e19	−0.775927	−0.387964	−	0.921675i	$-0.626821\pi$
$379$	−1.69674e19	−0.775927	−0.387964	+	0.921675i	$0.626821\pi$
$380$	−1.95940e17	−0.00878511
$381$	0	0
$382$	2.79453e19	1.20458
$383$	−2.92156e19	−1.23488	−0.617439	−	0.786619i	$-0.711830\pi$
$383$	−2.92156e19	−1.23488	−0.617439	+	0.786619i	$0.711830\pi$
$384$	0	0
$385$	−1.29768e19	−0.527489
$386$	−2.53015e19	−1.00865
$387$	0	0
$388$	−4.68671e18	−0.179734
$389$	−3.11343e18	−0.117116	−0.0585581	−	0.998284i	$-0.518650\pi$
$389$	−3.11343e18	−0.117116	−0.0585581	+	0.998284i	$0.518650\pi$
$390$	0	0
$391$	5.50031e19	1.99096
$392$	6.24705e19	2.21835
$393$	0	0
$394$	8.43700e17	0.0288381
$395$	1.41392e18	0.0474182
$396$	0	0
$397$	5.25085e19	1.69551	0.847756	−	0.530387i	$-0.177953\pi$
$397$	5.25085e19	1.69551	0.847756	+	0.530387i	$0.177953\pi$
$398$	−3.13037e19	−0.991909
$399$	0	0
$400$	2.73529e19	0.834744
$401$	5.18347e19	1.55252	0.776261	−	0.630412i	$-0.217114\pi$
$401$	5.18347e19	1.55252	0.776261	+	0.630412i	$0.217114\pi$
$402$	0	0
$403$	−5.22381e18	−0.150730
$404$	4.47294e18	0.126687
$405$	0	0
$406$	−5.13135e19	−1.40051
$407$	−3.36802e19	−0.902436
$408$	0	0
$409$	−2.77271e19	−0.716109	−0.358054	−	0.933701i	$-0.616560\pi$
$409$	−2.77271e19	−0.716109	−0.358054	+	0.933701i	$0.616560\pi$
$410$	3.85083e18	0.0976508
$411$	0	0
$412$	2.92225e18	0.0714478
$413$	7.28073e18	0.174803
$414$	0	0
$415$	7.47494e18	0.173080
$416$	−8.95235e18	−0.203581
$417$	0	0
$418$	−3.50613e19	−0.769140
$419$	1.43078e19	0.308297	0.154148	−	0.988048i	$-0.450737\pi$
$419$	1.43078e19	0.308297	0.154148	+	0.988048i	$0.450737\pi$
$420$	0	0
$421$	−4.00981e18	−0.0833696	−0.0416848	−	0.999131i	$-0.513273\pi$
$421$	−4.00981e18	−0.0833696	−0.0416848	+	0.999131i	$0.513273\pi$
$422$	−5.04100e18	−0.102961
$423$	0	0
$424$	−2.13611e19	−0.421094
$425$	8.67652e19	1.68046
$426$	0	0
$427$	−1.28904e20	−2.41023
$428$	−6.05718e18	−0.111286
$429$	0	0
$430$	8.39705e18	0.148975
$431$	2.81435e19	0.490680	0.245340	−	0.969437i	$-0.421100\pi$
$431$	2.81435e19	0.490680	0.245340	+	0.969437i	$0.421100\pi$
$432$	0	0
$433$	5.48889e19	0.924326	0.462163	−	0.886795i	$-0.347073\pi$
$433$	5.48889e19	0.924326	0.462163	+	0.886795i	$0.347073\pi$
$434$	−1.86668e19	−0.308955
$435$	0	0
$436$	4.22034e18	0.0674837
$437$	3.19915e19	0.502832
$438$	0	0
$439$	−5.42441e18	−0.0823889	−0.0411945	−	0.999151i	$-0.513116\pi$
$439$	−5.42441e18	−0.0823889	−0.0411945	+	0.999151i	$0.513116\pi$
$440$	2.11102e19	0.315208
$441$	0	0
$442$	8.95924e19	1.29302
$443$	−8.37943e19	−1.18901	−0.594506	−	0.804091i	$-0.702652\pi$
$443$	−8.37943e19	−1.18901	−0.594506	+	0.804091i	$0.702652\pi$
$444$	0	0
$445$	−1.23246e19	−0.169072
$446$	−1.82715e19	−0.246470
$447$	0	0
$448$	−1.47674e20	−1.92627
$449$	−9.00272e19	−1.15485	−0.577426	−	0.816443i	$-0.695943\pi$
$449$	−9.00272e19	−1.15485	−0.577426	+	0.816443i	$0.695943\pi$
$450$	0	0
$451$	−1.00742e20	−1.24993
$452$	4.79920e18	0.0585640
$453$	0	0
$454$	−1.09966e20	−1.29819
$455$	1.92868e19	0.223962
$456$	0	0
$457$	8.14199e19	0.914870	0.457435	−	0.889243i	$-0.348768\pi$
$457$	8.14199e19	0.914870	0.457435	+	0.889243i	$0.348768\pi$
$458$	6.79520e19	0.751123
$459$	0	0
$460$	−2.17898e18	−0.0233115
$461$	−1.24454e20	−1.30995	−0.654973	−	0.755652i	$-0.727320\pi$
$461$	−1.24454e20	−1.30995	−0.654973	+	0.755652i	$0.727320\pi$
$462$	0	0
$463$	1.05312e19	0.107305	0.0536524	−	0.998560i	$-0.482914\pi$
$463$	1.05312e19	0.107305	0.0536524	+	0.998560i	$0.482914\pi$
$464$	7.26512e19	0.728380
$465$	0	0
$466$	−1.06070e20	−1.02967
$467$	−5.50879e19	−0.526234	−0.263117	−	0.964764i	$-0.584751\pi$
$467$	−5.50879e19	−0.526234	−0.263117	+	0.964764i	$0.584751\pi$
$468$	0	0
$469$	1.72003e20	1.59126
$470$	7.03624e18	0.0640629
$471$	0	0
$472$	−1.18440e19	−0.104456
$473$	−2.19676e20	−1.90688
$474$	0	0
$475$	5.04653e19	0.424415
$476$	−4.68063e19	−0.387483
$477$	0	0
$478$	4.08621e19	0.327802
$479$	1.73904e20	1.37339	0.686693	−	0.726947i	$-0.259061\pi$
$479$	1.73904e20	1.37339	0.686693	+	0.726947i	$0.259061\pi$
$480$	0	0
$481$	5.00570e19	0.383157
$482$	−3.46501e19	−0.261127
$483$	0	0
$484$	−4.50149e19	−0.328864
$485$	−3.09926e19	−0.222944
$486$	0	0
$487$	2.52282e20	1.75962	0.879809	−	0.475327i	$-0.157670\pi$
$487$	2.52282e20	1.75962	0.879809	+	0.475327i	$0.157670\pi$
$488$	2.09696e20	1.44026
$489$	0	0
$490$	4.67325e19	0.311278
$491$	3.93011e19	0.257806	0.128903	−	0.991657i	$-0.458854\pi$
$491$	3.93011e19	0.257806	0.128903	+	0.991657i	$0.458854\pi$
$492$	0	0
$493$	2.30455e20	1.46634
$494$	5.21096e19	0.326562
$495$	0	0
$496$	2.64289e19	0.160682
$497$	2.49945e20	1.49683
$498$	0	0
$499$	−2.13250e20	−1.23918	−0.619591	−	0.784925i	$-0.712701\pi$
$499$	−2.13250e20	−1.23918	−0.619591	+	0.784925i	$0.712701\pi$
$500$	−6.96276e18	−0.0398572
$501$	0	0
$502$	−1.30252e20	−0.723614
$503$	8.53406e19	0.467085	0.233543	−	0.972347i	$-0.424968\pi$
$503$	8.53406e19	0.467085	0.233543	+	0.972347i	$0.424968\pi$
$504$	0	0
$505$	2.95789e19	0.157144
$506$	−3.89903e20	−2.04093
$507$	0	0
$508$	1.14474e19	0.0581740
$509$	−2.52433e20	−1.26405	−0.632023	−	0.774950i	$-0.717775\pi$
$509$	−2.52433e20	−1.26405	−0.632023	+	0.774950i	$0.717775\pi$
$510$	0	0
$511$	3.49262e20	1.69822
$512$	2.33482e20	1.11874
$513$	0	0
$514$	1.05110e20	0.489128
$515$	1.93245e19	0.0886245
$516$	0	0
$517$	−1.84075e20	−0.820006
$518$	1.78874e20	0.785367
$519$	0	0
$520$	−3.13750e19	−0.133831
$521$	2.93399e20	1.23360	0.616802	−	0.787118i	$-0.288428\pi$
$521$	2.93399e20	1.23360	0.616802	+	0.787118i	$0.288428\pi$
$522$	0	0
$523$	1.12634e20	0.460156	0.230078	−	0.973172i	$-0.426102\pi$
$523$	1.12634e20	0.460156	0.230078	+	0.973172i	$0.426102\pi$
$524$	1.33779e19	0.0538769
$525$	0	0
$526$	3.74001e20	1.46379
$527$	8.38344e19	0.323476
$528$	0	0
$529$	8.91297e19	0.334276
$530$	−1.59796e19	−0.0590878
$531$	0	0
$532$	−2.72240e19	−0.0978620
$533$	1.49727e20	0.530697
$534$	0	0
$535$	−4.00553e19	−0.138040
$536$	−2.79808e20	−0.950878
$537$	0	0
$538$	−3.70223e20	−1.22348
$539$	−1.22257e21	−3.98436
$540$	0	0
$541$	−5.49763e20	−1.74259	−0.871297	−	0.490756i	$-0.836720\pi$
$541$	−5.49763e20	−1.74259	−0.871297	+	0.490756i	$0.836720\pi$
$542$	−1.23723e20	−0.386774
$543$	0	0
$544$	1.43672e20	0.436898
$545$	2.79085e19	0.0837073
$546$	0	0
$547$	−5.79095e20	−1.68984	−0.844919	−	0.534895i	$-0.820351\pi$
$547$	−5.79095e20	−1.68984	−0.844919	+	0.534895i	$0.820351\pi$
$548$	2.16840e19	0.0624144
$549$	0	0
$550$	−6.15057e20	−1.72264
$551$	1.34039e20	0.370335
$552$	0	0
$553$	1.96449e20	0.528217
$554$	5.57727e19	0.147944
$555$	0	0
$556$	−2.11339e19	−0.0545656
$557$	5.79591e20	1.47641	0.738207	−	0.674575i	$-0.235673\pi$
$557$	5.79591e20	1.47641	0.738207	+	0.674575i	$0.235673\pi$
$558$	0	0
$559$	3.26492e20	0.809625
$560$	−9.75780e19	−0.238749
$561$	0	0
$562$	1.79096e20	0.426641
$563$	2.20743e20	0.518888	0.259444	−	0.965758i	$-0.416461\pi$
$563$	2.20743e20	0.518888	0.259444	+	0.965758i	$0.416461\pi$
$564$	0	0
$565$	3.17365e19	0.0726433
$566$	3.91074e18	0.00883355
$567$	0	0
$568$	−4.06600e20	−0.894448
$569$	−5.10121e20	−1.10747	−0.553734	−	0.832694i	$-0.686798\pi$
$569$	−5.10121e20	−1.10747	−0.553734	+	0.832694i	$0.686798\pi$
$570$	0	0
$571$	−3.83960e20	−0.811924	−0.405962	−	0.913890i	$-0.633063\pi$
$571$	−3.83960e20	−0.811924	−0.405962	+	0.913890i	$0.633063\pi$
$572$	9.28523e19	0.193786
$573$	0	0
$574$	5.35034e20	1.08778
$575$	5.61205e20	1.12619
$576$	0	0
$577$	−5.62677e20	−1.10012	−0.550061	−	0.835125i	$-0.685395\pi$
$577$	−5.62677e20	−1.10012	−0.550061	+	0.835125i	$0.685395\pi$
$578$	−9.53844e20	−1.84085
$579$	0	0
$580$	−9.12959e18	−0.0171689
$581$	1.03857e21	1.92803
$582$	0	0
$583$	4.18044e20	0.756324
$584$	−5.68166e20	−1.01480
$585$	0	0
$586$	−6.28600e20	−1.09431
$587$	9.29291e20	1.59722	0.798612	−	0.601846i	$-0.205568\pi$
$587$	9.29291e20	1.59722	0.798612	+	0.601846i	$0.205568\pi$
$588$	0	0
$589$	4.87606e19	0.0816966
$590$	−8.86017e18	−0.0146572
$591$	0	0
$592$	−2.53254e20	−0.408455
$593$	−6.48210e20	−1.03230	−0.516149	−	0.856499i	$-0.672635\pi$
$593$	−6.48210e20	−1.03230	−0.516149	+	0.856499i	$0.672635\pi$
$594$	0	0
$595$	−3.09524e20	−0.480637
$596$	6.82889e19	0.104714
$597$	0	0
$598$	5.79492e20	0.866539
$599$	1.05168e20	0.155304	0.0776521	−	0.996981i	$-0.475258\pi$
$599$	1.05168e20	0.155304	0.0776521	+	0.996981i	$0.475258\pi$
$600$	0	0
$601$	4.20944e20	0.606270	0.303135	−	0.952948i	$-0.401967\pi$
$601$	4.20944e20	0.606270	0.303135	+	0.952948i	$0.401967\pi$
$602$	1.16669e21	1.65951
$603$	0	0
$604$	1.04152e20	0.144508
$605$	−2.97678e20	−0.407926
$606$	0	0
$607$	−1.00492e21	−1.34344	−0.671720	−	0.740805i	$-0.734444\pi$
$607$	−1.00492e21	−1.34344	−0.671720	+	0.740805i	$0.734444\pi$
$608$	8.35638e19	0.110342
$609$	0	0
$610$	1.56868e20	0.202097
$611$	2.73581e20	0.348159
$612$	0	0
$613$	5.01545e20	0.622811	0.311406	−	0.950277i	$-0.399200\pi$
$613$	5.01545e20	0.622811	0.311406	+	0.950277i	$0.399200\pi$
$614$	1.42516e21	1.74824
$615$	0	0
$616$	2.93305e21	3.51127
$617$	−1.10413e21	−1.30581	−0.652905	−	0.757440i	$-0.726450\pi$
$617$	−1.10413e21	−1.30581	−0.652905	+	0.757440i	$0.726450\pi$
$618$	0	0
$619$	−6.37648e20	−0.736039	−0.368020	−	0.929818i	$-0.619964\pi$
$619$	−6.37648e20	−0.736039	−0.368020	+	0.929818i	$0.619964\pi$
$620$	−3.32115e18	−0.00378748
$621$	0	0
$622$	1.29700e21	1.44382
$623$	−1.71238e21	−1.88339
$624$	0	0
$625$	8.61965e20	0.925528
$626$	−3.44503e20	−0.365498
$627$	0	0
$628$	−7.49067e19	−0.0775931
$629$	−8.03340e20	−0.822280
$630$	0	0
$631$	−1.89099e21	−1.89003	−0.945015	−	0.327028i	$-0.893953\pi$
$631$	−1.89099e21	−1.89003	−0.945015	+	0.327028i	$0.893953\pi$
$632$	−3.19576e20	−0.315643
$633$	0	0
$634$	7.34385e20	0.708360
$635$	7.57002e19	0.0721595
$636$	0	0
$637$	1.81704e21	1.69168
$638$	−1.63363e21	−1.50314
$639$	0	0
$640$	1.35088e20	0.121413
$641$	8.20388e20	0.728759	0.364380	−	0.931251i	$-0.381281\pi$
$641$	8.20388e20	0.728759	0.364380	+	0.931251i	$0.381281\pi$
$642$	0	0
$643$	−1.68828e21	−1.46508	−0.732541	−	0.680723i	$-0.761666\pi$
$643$	−1.68828e21	−1.46508	−0.732541	+	0.680723i	$0.761666\pi$
$644$	−3.02747e20	−0.259679
$645$	0	0
$646$	−8.36282e20	−0.700824
$647$	1.34343e21	1.11284	0.556419	−	0.830902i	$-0.312175\pi$
$647$	1.34343e21	1.11284	0.556419	+	0.830902i	$0.312175\pi$
$648$	0	0
$649$	2.31792e20	0.187613
$650$	9.14125e20	0.731401
$651$	0	0
$652$	1.89882e20	0.148466
$653$	2.18575e20	0.168948	0.0844738	−	0.996426i	$-0.473079\pi$
$653$	2.18575e20	0.168948	0.0844738	+	0.996426i	$0.473079\pi$
$654$	0	0
$655$	8.84662e19	0.0668294
$656$	−7.57518e20	−0.565736
$657$	0	0
$658$	9.77615e20	0.713631
$659$	6.75320e19	0.0487382	0.0243691	−	0.999703i	$-0.492242\pi$
$659$	6.75320e19	0.0487382	0.0243691	+	0.999703i	$0.492242\pi$
$660$	0	0
$661$	−2.43099e20	−0.171503	−0.0857517	−	0.996317i	$-0.527329\pi$
$661$	−2.43099e20	−0.171503	−0.0857517	+	0.996317i	$0.527329\pi$
$662$	5.73902e20	0.400316
$663$	0	0
$664$	−1.68950e21	−1.15212
$665$	−1.80028e20	−0.121389
$666$	0	0
$667$	1.49060e21	0.982692
$668$	−1.53453e20	−0.100035
$669$	0	0
$670$	−2.09317e20	−0.133427
$671$	−4.10383e21	−2.58685
$672$	0	0
$673$	2.42357e21	1.49397	0.746986	−	0.664840i	$-0.231500\pi$
$673$	2.42357e21	1.49397	0.746986	+	0.664840i	$0.231500\pi$
$674$	2.32025e21	1.41444
$675$	0	0
$676$	7.59383e19	0.0452752
$677$	9.73991e20	0.574302	0.287151	−	0.957885i	$-0.407292\pi$
$677$	9.73991e20	0.574302	0.287151	+	0.957885i	$0.407292\pi$
$678$	0	0
$679$	−4.30611e21	−2.48349
$680$	5.03521e20	0.287211
$681$	0	0
$682$	−5.94280e20	−0.331596
$683$	2.57604e21	1.42167	0.710833	−	0.703360i	$-0.248318\pi$
$683$	2.57604e21	1.42167	0.710833	+	0.703360i	$0.248318\pi$
$684$	0	0
$685$	1.43393e20	0.0774194
$686$	3.41037e21	1.82125
$687$	0	0
$688$	−1.65183e21	−0.863081
$689$	−6.21316e20	−0.321121
$690$	0	0
$691$	−1.30645e21	−0.660704	−0.330352	−	0.943858i	$-0.607167\pi$
$691$	−1.30645e21	−0.660704	−0.330352	+	0.943858i	$0.607167\pi$
$692$	4.19447e20	0.209837
$693$	0	0
$694$	−1.42252e21	−0.696405
$695$	−1.39756e20	−0.0676836
$696$	0	0
$697$	−2.40290e21	−1.13891
$698$	6.74911e20	0.316469
$699$	0	0
$700$	−4.77572e20	−0.219181
$701$	−3.27047e21	−1.48499	−0.742497	−	0.669849i	$-0.766359\pi$
$701$	−3.27047e21	−1.48499	−0.742497	+	0.669849i	$0.766359\pi$
$702$	0	0
$703$	−4.67247e20	−0.207674
$704$	−4.70139e21	−2.06743
$705$	0	0
$706$	1.30829e21	0.563207
$707$	4.10970e21	1.75051
$708$	0	0
$709$	−1.81669e21	−0.757592	−0.378796	−	0.925480i	$-0.623662\pi$
$709$	−1.81669e21	−0.757592	−0.378796	+	0.925480i	$0.623662\pi$
$710$	−3.04166e20	−0.125509
$711$	0	0
$712$	2.78563e21	1.12544
$713$	5.42248e20	0.216784
$714$	0	0
$715$	6.14020e20	0.240374
$716$	4.27613e20	0.165655
$717$	0	0
$718$	−3.57623e21	−1.35672
$719$	−2.05441e21	−0.771296	−0.385648	−	0.922646i	$-0.626022\pi$
$719$	−2.05441e21	−0.771296	−0.385648	+	0.922646i	$0.626022\pi$
$720$	0	0
$721$	2.68494e21	0.987235
$722$	2.08043e21	0.757050
$723$	0	0
$724$	2.53087e20	0.0902051
$725$	2.35136e21	0.829439
$726$	0	0
$727$	6.34657e20	0.219296	0.109648	−	0.993970i	$-0.465028\pi$
$727$	6.34657e20	0.219296	0.109648	+	0.993970i	$0.465028\pi$
$728$	−4.35924e21	−1.49082
$729$	0	0
$730$	−4.25029e20	−0.142396
$731$	−5.23971e21	−1.73751
$732$	0	0
$733$	1.26674e21	0.411534	0.205767	−	0.978601i	$-0.434031\pi$
$733$	1.26674e21	0.411534	0.205767	+	0.978601i	$0.434031\pi$
$734$	−3.44991e21	−1.10940
$735$	0	0
$736$	9.29282e20	0.292795
$737$	5.47596e21	1.70787
$738$	0	0
$739$	1.83101e21	0.559574	0.279787	−	0.960062i	$-0.409736\pi$
$739$	1.83101e21	0.559574	0.279787	+	0.960062i	$0.409736\pi$
$740$	3.18248e19	0.00962781
$741$	0	0
$742$	−2.22021e21	−0.658210
$743$	4.76308e19	0.0139789	0.00698943	−	0.999976i	$-0.497775\pi$
$743$	4.76308e19	0.0139789	0.00698943	+	0.999976i	$0.497775\pi$
$744$	0	0
$745$	4.51586e20	0.129888
$746$	1.84765e20	0.0526112
$747$	0	0
$748$	−1.49014e21	−0.415877
$749$	−5.56528e21	−1.53771
$750$	0	0
$751$	−5.58937e21	−1.51378	−0.756891	−	0.653541i	$-0.773283\pi$
$751$	−5.58937e21	−1.51378	−0.756891	+	0.653541i	$0.773283\pi$
$752$	−1.38414e21	−0.371146
$753$	0	0
$754$	2.42798e21	0.638205
$755$	6.88744e20	0.179248
$756$	0	0
$757$	7.22893e21	1.84440	0.922200	−	0.386714i	$-0.126390\pi$
$757$	7.22893e21	1.84440	0.922200	+	0.386714i	$0.126390\pi$
$758$	2.86886e21	0.724754
$759$	0	0
$760$	2.92863e20	0.0725375
$761$	3.85444e21	0.945313	0.472657	−	0.881247i	$-0.343295\pi$
$761$	3.85444e21	0.945313	0.472657	+	0.881247i	$0.343295\pi$
$762$	0	0
$763$	3.87761e21	0.932460
$764$	6.90805e20	0.164496
$765$	0	0
$766$	4.93980e21	1.15344
$767$	−3.44499e20	−0.0796568
$768$	0	0
$769$	3.65675e21	0.829179	0.414589	−	0.910009i	$-0.363925\pi$
$769$	3.65675e21	0.829179	0.414589	+	0.910009i	$0.363925\pi$
$770$	2.19414e21	0.492701
$771$	0	0
$772$	−6.25450e20	−0.137741
$773$	2.41516e21	0.526745	0.263372	−	0.964694i	$-0.415165\pi$
$773$	2.41516e21	0.526745	0.263372	+	0.964694i	$0.415165\pi$
$774$	0	0
$775$	8.55375e20	0.182976
$776$	7.00501e21	1.48404
$777$	0	0
$778$	5.26421e20	0.109392
$779$	−1.39760e21	−0.287641
$780$	0	0
$781$	7.95732e21	1.60651
$782$	−9.29998e21	−1.85965
$783$	0	0
$784$	−9.19300e21	−1.80338
$785$	−4.95348e20	−0.0962472
$786$	0	0
$787$	6.48388e21	1.23602	0.618009	−	0.786171i	$-0.287940\pi$
$787$	6.48388e21	1.23602	0.618009	+	0.786171i	$0.287940\pi$
$788$	2.08562e19	0.00393811
$789$	0	0
$790$	−2.39066e20	−0.0442909
$791$	4.40947e21	0.809212
$792$	0	0
$793$	6.09931e21	1.09833
$794$	−8.87818e21	−1.58369
$795$	0	0
$796$	−7.73823e20	−0.135455
$797$	7.56562e21	1.31192	0.655959	−	0.754796i	$-0.272264\pi$
$797$	7.56562e21	1.31192	0.655959	+	0.754796i	$0.272264\pi$
$798$	0	0
$799$	−4.39057e21	−0.747171
$800$	1.46590e21	0.247133
$801$	0	0
$802$	−8.76425e21	−1.45013
$803$	1.11192e22	1.82267
$804$	0	0
$805$	−2.00203e21	−0.322108
$806$	8.83247e20	0.140789
$807$	0	0
$808$	−6.68549e21	−1.04604
$809$	1.66091e21	0.257473	0.128736	−	0.991679i	$-0.458908\pi$
$809$	1.66091e21	0.257473	0.128736	+	0.991679i	$0.458908\pi$
$810$	0	0
$811$	−7.92902e21	−1.20660	−0.603300	−	0.797514i	$-0.706148\pi$
$811$	−7.92902e21	−1.20660	−0.603300	+	0.797514i	$0.706148\pi$
$812$	−1.26846e21	−0.191253
$813$	0	0
$814$	5.69467e21	0.842919
$815$	1.25566e21	0.184158
$816$	0	0
$817$	−3.04757e21	−0.438822
$818$	4.68811e21	0.668880
$819$	0	0
$820$	9.51921e19	0.0133351
$821$	−6.66282e21	−0.924878	−0.462439	−	0.886651i	$-0.653026\pi$
$821$	−6.66282e21	−0.924878	−0.462439	+	0.886651i	$0.653026\pi$
$822$	0	0
$823$	6.96585e21	0.949457	0.474729	−	0.880132i	$-0.342546\pi$
$823$	6.96585e21	0.949457	0.474729	+	0.880132i	$0.342546\pi$
$824$	−4.36776e21	−0.589935
$825$	0	0
$826$	−1.23103e21	−0.163275
$827$	−1.15750e22	−1.52136	−0.760678	−	0.649130i	$-0.775133\pi$
$827$	−1.15750e22	−1.52136	−0.760678	+	0.649130i	$0.775133\pi$
$828$	0	0
$829$	2.92212e21	0.377172	0.188586	−	0.982057i	$-0.439610\pi$
$829$	2.92212e21	0.377172	0.188586	+	0.982057i	$0.439610\pi$
$830$	−1.26387e21	−0.161665
$831$	0	0
$832$	6.98742e21	0.877792
$833$	−2.91608e22	−3.63046
$834$	0	0
$835$	−1.01477e21	−0.124084
$836$	−8.66711e20	−0.105033
$837$	0	0
$838$	−2.41918e21	−0.287964
$839$	1.11314e22	1.31322	0.656608	−	0.754232i	$-0.271991\pi$
$839$	1.11314e22	1.31322	0.656608	+	0.754232i	$0.271991\pi$
$840$	0	0
$841$	−2.38381e21	−0.276249
$842$	6.77982e20	0.0778713
$843$	0	0
$844$	−1.24613e20	−0.0140603
$845$	5.02170e20	0.0561598
$846$	0	0
$847$	−4.13593e22	−4.54410
$848$	3.14344e21	0.342323
$849$	0	0
$850$	−1.46703e22	−1.56963
$851$	−5.19607e21	−0.551066
$852$	0	0
$853$	1.58068e22	1.64712	0.823561	−	0.567228i	$-0.191984\pi$
$853$	1.58068e22	1.64712	0.823561	+	0.567228i	$0.191984\pi$
$854$	2.17952e22	2.25127
$855$	0	0
$856$	9.05338e21	0.918877
$857$	1.08467e22	1.09130	0.545648	−	0.838014i	$-0.316284\pi$
$857$	1.08467e22	1.09130	0.545648	+	0.838014i	$0.316284\pi$
$858$	0	0
$859$	−2.89651e21	−0.286370	−0.143185	−	0.989696i	$-0.545734\pi$
$859$	−2.89651e21	−0.286370	−0.143185	+	0.989696i	$0.545734\pi$
$860$	2.07574e20	0.0203439
$861$	0	0
$862$	−4.75853e21	−0.458319
$863$	−9.65661e21	−0.922027	−0.461014	−	0.887393i	$-0.652514\pi$
$863$	−9.65661e21	−0.922027	−0.461014	+	0.887393i	$0.652514\pi$
$864$	0	0
$865$	2.77375e21	0.260283
$866$	−9.28066e21	−0.863365
$867$	0	0
$868$	−4.61440e20	−0.0421908
$869$	6.25422e21	0.566924
$870$	0	0
$871$	−8.13861e21	−0.725127
$872$	−6.30794e21	−0.557204
$873$	0	0
$874$	−5.40914e21	−0.469670
$875$	−6.39733e21	−0.550729
$876$	0	0
$877$	7.14527e21	0.604674	0.302337	−	0.953201i	$-0.402233\pi$
$877$	7.14527e21	0.604674	0.302337	+	0.953201i	$0.402233\pi$
$878$	9.17164e20	0.0769553
$879$	0	0
$880$	−3.10652e21	−0.256245
$881$	−3.76635e21	−0.308036	−0.154018	−	0.988068i	$-0.549221\pi$
$881$	−3.76635e21	−0.308036	−0.154018	+	0.988068i	$0.549221\pi$
$882$	0	0
$883$	3.49878e21	0.281327	0.140663	−	0.990057i	$-0.455076\pi$
$883$	3.49878e21	0.281327	0.140663	+	0.990057i	$0.455076\pi$
$884$	2.21472e21	0.176574
$885$	0	0
$886$	1.41680e22	1.11060
$887$	1.55181e22	1.20618	0.603089	−	0.797674i	$-0.293936\pi$
$887$	1.55181e22	1.20618	0.603089	+	0.797674i	$0.293936\pi$
$888$	0	0
$889$	1.05178e22	0.803824
$890$	2.08385e21	0.157922
$891$	0	0
$892$	−4.51670e20	−0.0336577
$893$	−2.55369e21	−0.188704
$894$	0	0
$895$	2.82775e21	0.205479
$896$	1.87691e22	1.35249
$897$	0	0
$898$	1.52219e22	1.07869
$899$	2.27193e21	0.159661
$900$	0	0
$901$	9.97120e21	0.689146
$902$	1.70335e22	1.16750
$903$	0	0
$904$	−7.17314e21	−0.483555
$905$	1.67363e21	0.111891
$906$	0	0
$907$	−1.55853e22	−1.02485	−0.512425	−	0.858732i	$-0.671253\pi$
$907$	−1.55853e22	−1.02485	−0.512425	+	0.858732i	$0.671253\pi$
$908$	−2.71834e21	−0.177280
$909$	0	0
$910$	−3.26103e21	−0.209191
$911$	−1.12565e22	−0.716167	−0.358083	−	0.933690i	$-0.616570\pi$
$911$	−1.12565e22	−0.716167	−0.358083	+	0.933690i	$0.616570\pi$
$912$	0	0
$913$	3.30642e22	2.06932
$914$	−1.37666e22	−0.854533
$915$	0	0
$916$	1.67977e21	0.102573
$917$	1.22915e22	0.744448
$918$	0	0
$919$	−1.18424e22	−0.705624	−0.352812	−	0.935694i	$-0.614774\pi$
$919$	−1.18424e22	−0.705624	−0.352812	+	0.935694i	$0.614774\pi$
$920$	3.25682e21	0.192480
$921$	0	0
$922$	2.10429e22	1.22355
$923$	−1.18265e22	−0.682094
$924$	0	0
$925$	−8.19660e21	−0.465126
$926$	−1.78062e21	−0.100228
$927$	0	0
$928$	3.89355e21	0.215643
$929$	1.03209e22	0.567020	0.283510	−	0.958969i	$-0.408501\pi$
$929$	1.03209e22	0.567020	0.283510	+	0.958969i	$0.408501\pi$
$930$	0	0
$931$	−1.69608e22	−0.916903
$932$	−2.62205e21	−0.140612
$933$	0	0
$934$	9.31430e21	0.491529
$935$	−9.85410e21	−0.515858
$936$	0	0
$937$	2.45364e22	1.26405	0.632023	−	0.774949i	$-0.282225\pi$
$937$	2.45364e22	1.26405	0.632023	+	0.774949i	$0.282225\pi$
$938$	−2.90825e22	−1.48631
$939$	0	0
$940$	1.73935e20	0.00874839
$941$	1.56936e22	0.783069	0.391534	−	0.920163i	$-0.371944\pi$
$941$	1.56936e22	0.783069	0.391534	+	0.920163i	$0.371944\pi$
$942$	0	0
$943$	−1.55421e22	−0.763261
$944$	1.74293e21	0.0849162
$945$	0	0
$946$	3.71430e22	1.78112
$947$	−3.40234e22	−1.61865	−0.809326	−	0.587360i	$-0.800167\pi$
$947$	−3.40234e22	−1.61865	−0.809326	+	0.587360i	$0.800167\pi$
$948$	0	0
$949$	−1.65259e22	−0.773871
$950$	−8.53271e21	−0.396424
$951$	0	0
$952$	6.99592e22	3.19940
$953$	−1.40772e22	−0.638735	−0.319367	−	0.947631i	$-0.603470\pi$
$953$	−1.40772e22	−0.638735	−0.319367	+	0.947631i	$0.603470\pi$
$954$	0	0
$955$	4.56820e21	0.204042
$956$	1.01011e21	0.0447644
$957$	0	0
$958$	−2.94038e22	−1.28281
$959$	1.99231e22	0.862416
$960$	0	0
$961$	−2.26388e22	−0.964779
$962$	−8.46368e21	−0.357887
$963$	0	0
$964$	−8.56546e20	−0.0356593
$965$	−4.13602e21	−0.170855
$966$	0	0
$967$	−1.37191e22	−0.557989	−0.278995	−	0.960293i	$-0.590001\pi$
$967$	−1.37191e22	−0.557989	−0.278995	+	0.960293i	$0.590001\pi$
$968$	6.72817e22	2.71539
$969$	0	0
$970$	5.24026e21	0.208240
$971$	2.19551e22	0.865749	0.432874	−	0.901454i	$-0.357499\pi$
$971$	2.19551e22	0.865749	0.432874	+	0.901454i	$0.357499\pi$
$972$	0	0
$973$	−1.94176e22	−0.753964
$974$	−4.26560e22	−1.64357
$975$	0	0
$976$	−3.08584e22	−1.17084
$977$	−3.50788e22	−1.32079	−0.660397	−	0.750917i	$-0.729612\pi$
$977$	−3.50788e22	−1.32079	−0.660397	+	0.750917i	$0.729612\pi$
$978$	0	0
$979$	−5.45158e22	−2.02140
$980$	1.15522e21	0.0425079
$981$	0	0
$982$	−6.64506e21	−0.240803
$983$	3.62351e22	1.30310	0.651551	−	0.758605i	$-0.274119\pi$
$983$	3.62351e22	1.30310	0.651551	+	0.758605i	$0.274119\pi$
$984$	0	0
$985$	1.37919e20	0.00488486
$986$	−3.89655e22	−1.36963
$987$	0	0
$988$	1.28814e21	0.0445951
$989$	−3.38909e22	−1.16442
$990$	0	0
$991$	4.96008e22	1.67856	0.839279	−	0.543701i	$-0.182977\pi$
$991$	4.96008e22	1.67856	0.839279	+	0.543701i	$0.182977\pi$
$992$	1.41639e21	0.0475713
$993$	0	0
$994$	−4.22609e22	−1.39811
$995$	−5.11719e21	−0.168019
$996$	0	0
$997$	2.15470e21	0.0696904	0.0348452	−	0.999393i	$-0.488906\pi$
$997$	2.15470e21	0.0696904	0.0348452	+	0.999393i	$0.488906\pi$
$998$	3.60565e22	1.15746
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.16.a.d.1.2	✓	6
3.2	odd	2	inner	27.16.a.d.1.5	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.16.a.d.1.2	✓	6	1.1	even	1	trivial
27.16.a.d.1.5	yes	6	3.2	odd	2	inner

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:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	27.a (trivial)

\(f(q)\)	\(=\)	\(q-169.081 q^{2} -4179.66 q^{4} -27639.5 q^{5} -3.84024e6 q^{7} +6.24714e6 q^{8} +4.67332e6 q^{10} -1.22259e8 q^{11} +1.81707e8 q^{13} +6.49310e8 q^{14} -9.19313e8 q^{16} -2.91612e9 q^{17} -1.69610e9 q^{19} +1.15524e8 q^{20} +2.06716e10 q^{22} -1.88617e10 q^{23} -2.97536e10 q^{25} -3.07231e10 q^{26} +1.60509e10 q^{28} -7.90277e10 q^{29} -2.87486e10 q^{31} -4.92681e10 q^{32} +4.93061e11 q^{34} +1.06142e11 q^{35} +2.75482e11 q^{37} +2.86779e11 q^{38} -1.72668e11 q^{40} +8.24004e11 q^{41} +1.79681e12 q^{43} +5.11001e11 q^{44} +3.18916e12 q^{46} +1.50562e12 q^{47} +9.99985e12 q^{49} +5.03077e12 q^{50} -7.59473e11 q^{52} -3.41933e12 q^{53} +3.37918e12 q^{55} -2.39905e13 q^{56} +1.33621e13 q^{58} -1.89591e12 q^{59} +3.35667e13 q^{61} +4.86083e12 q^{62} +3.84543e13 q^{64} -5.02229e12 q^{65} -4.47898e13 q^{67} +1.21884e13 q^{68} -1.79466e13 q^{70} -6.50858e13 q^{71} -9.09481e13 q^{73} -4.65788e13 q^{74} +7.08914e12 q^{76} +4.69503e14 q^{77} -5.11555e13 q^{79} +2.54094e13 q^{80} -1.39323e14 q^{82} -2.70444e14 q^{83} +8.06003e13 q^{85} -3.03806e14 q^{86} -7.63769e14 q^{88} +4.45904e14 q^{89} -6.97797e14 q^{91} +7.88356e13 q^{92} -2.54571e14 q^{94} +4.68795e13 q^{95} +1.12131e15 q^{97} -1.69078e15 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 71124 q^{4} + 2377362 q^{7} + 46852560 q^{10} + 387940494 q^{13} + 1056508872 q^{16} - 1979573730 q^{19} - 16927522128 q^{22} - 792180750 q^{25} + 267175396092 q^{28} + 484649028120 q^{31} + 165585346416 q^{34}+ \cdots - 718669835357730 q^{97}+O(q^{100}) \) 6 * q + 71124 * q^4 + 2377362 * q^7 + 46852560 * q^10 + 387940494 * q^13 + 1056508872 * q^16 - 1979573730 * q^19 - 16927522128 * q^22 - 792180750 * q^25 + 267175396092 * q^28 + 484649028120 * q^31 + 165585346416 * q^34 + 2108295778962 * q^37 + 5048325004320 * q^40 + 4208638642896 * q^43 + 24591340387440 * q^46 + 26450424994284 * q^49 + 56443954337220 * q^52 + 35614339104960 * q^55 + 68411200637952 * q^58 + 81589045842426 * q^61 + 11515176559824 * q^64 + 87445603264854 * q^67 + 83608373597040 * q^70 - 165007694606730 * q^73 - 393810695155932 * q^76 - 740397172026354 * q^79 - 1526116173910272 * q^82 - 913199003104320 * q^85 - 2054825311293216 * q^88 - 1087541878427094 * q^91 + 14553520366896 * q^94 - 718669835357730 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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