LMFDB - Embedded newform 27.14.a.e.1.3

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:	$0$
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			1.3
Root			$-5.80055$ 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)

$f(q)$	$=$	\(q+47.4017 q^{2} -5945.08 q^{4} +47381.4 q^{5} +480775. q^{7} -670121. q^{8} +2.24596e6 q^{10} -6.25107e6 q^{11} -1.57296e7 q^{13} +2.27895e7 q^{14} +1.69373e7 q^{16} +7.30512e7 q^{17} +3.94294e8 q^{19} -2.81687e8 q^{20} -2.96311e8 q^{22} +1.15508e9 q^{23} +1.02430e9 q^{25} -7.45607e8 q^{26} -2.85825e9 q^{28} +1.59461e9 q^{29} -2.33526e8 q^{31} +6.29249e9 q^{32} +3.46275e9 q^{34} +2.27798e10 q^{35} +2.58983e9 q^{37} +1.86902e10 q^{38} -3.17513e10 q^{40} -1.93740e10 q^{41} -2.37958e10 q^{43} +3.71631e10 q^{44} +5.47528e10 q^{46} +1.24721e11 q^{47} +1.34256e11 q^{49} +4.85533e10 q^{50} +9.35135e10 q^{52} +1.30481e11 q^{53} -2.96184e11 q^{55} -3.22178e11 q^{56} +7.55872e10 q^{58} -1.55509e11 q^{59} -2.96704e11 q^{61} -1.10695e10 q^{62} +1.59524e11 q^{64} -7.45289e11 q^{65} -4.11869e11 q^{67} -4.34296e11 q^{68} +1.07980e12 q^{70} -6.51745e11 q^{71} -1.66606e11 q^{73} +1.22762e11 q^{74} -2.34411e12 q^{76} -3.00536e12 q^{77} +1.70061e12 q^{79} +8.02512e11 q^{80} -9.18357e11 q^{82} +3.41254e12 q^{83} +3.46127e12 q^{85} -1.12796e12 q^{86} +4.18897e12 q^{88} -1.30542e12 q^{89} -7.56238e12 q^{91} -6.86706e12 q^{92} +5.91196e12 q^{94} +1.86822e13 q^{95} +2.94200e12 q^{97} +6.36395e12 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	47.4017	0.523719	0.261860	−	0.965106i	$-0.415664\pi$
$2$	47.4017	0.523719	0.261860	+	0.965106i	$0.415664\pi$
$3$	0	0
$3$	0	0
$4$	−5945.08	−0.725718
$5$	47381.4	1.35614	0.678068	−	0.734999i	$-0.262818\pi$
$5$	47381.4	1.35614	0.678068	+	0.734999i	$0.262818\pi$
$6$	0	0
$7$	480775.	1.54456	0.772280	−	0.635282i	$-0.219116\pi$
$7$	480775.	1.54456	0.772280	+	0.635282i	$0.219116\pi$
$8$	−670121.	−0.903792
$9$	0	0
$10$	2.24596e6	0.710234
$11$	−6.25107e6	−1.06390	−0.531951	−	0.846775i	$-0.678541\pi$
$11$	−6.25107e6	−1.06390	−0.531951	+	0.846775i	$0.678541\pi$
$12$	0	0
$13$	−1.57296e7	−0.903826	−0.451913	−	0.892062i	$-0.649258\pi$
$13$	−1.57296e7	−0.903826	−0.451913	+	0.892062i	$0.649258\pi$
$14$	2.27895e7	0.808916
$15$	0	0
$16$	1.69373e7	0.252385
$17$	7.30512e7	0.734023	0.367011	−	0.930216i	$-0.380381\pi$
$17$	7.30512e7	0.734023	0.367011	+	0.930216i	$0.380381\pi$
$18$	0	0
$19$	3.94294e8	1.92274	0.961372	−	0.275253i	$-0.0887616\pi$
$19$	3.94294e8	1.92274	0.961372	+	0.275253i	$0.0887616\pi$
$20$	−2.81687e8	−0.984172
$21$	0	0
$22$	−2.96311e8	−0.557186
$23$	1.15508e9	1.62698	0.813490	−	0.581579i	$-0.197565\pi$
$23$	1.15508e9	1.62698	0.813490	+	0.581579i	$0.197565\pi$
$24$	0	0
$25$	1.02430e9	0.839103
$26$	−7.45607e8	−0.473351
$27$	0	0
$28$	−2.85825e9	−1.12092
$29$	1.59461e9	0.497815	0.248907	−	0.968527i	$-0.419928\pi$
$29$	1.59461e9	0.497815	0.248907	+	0.968527i	$0.419928\pi$
$30$	0	0
$31$	−2.33526e8	−0.0472590	−0.0236295	−	0.999721i	$-0.507522\pi$
$31$	−2.33526e8	−0.0472590	−0.0236295	+	0.999721i	$0.507522\pi$
$32$	6.29249e9	1.03597
$33$	0	0
$34$	3.46275e9	0.384422
$35$	2.27798e10	2.09463
$36$	0	0
$37$	2.58983e9	0.165943	0.0829716	−	0.996552i	$-0.473559\pi$
$37$	2.58983e9	0.165943	0.0829716	+	0.996552i	$0.473559\pi$
$38$	1.86902e10	1.00698
$39$	0	0
$40$	−3.17513e10	−1.22566
$41$	−1.93740e10	−0.636976	−0.318488	−	0.947927i	$-0.603175\pi$
$41$	−1.93740e10	−0.636976	−0.318488	+	0.947927i	$0.603175\pi$
$42$	0	0
$43$	−2.37958e10	−0.574058	−0.287029	−	0.957922i	$-0.592668\pi$
$43$	−2.37958e10	−0.574058	−0.287029	+	0.957922i	$0.592668\pi$
$44$	3.71631e10	0.772093
$45$	0	0
$46$	5.47528e10	0.852081
$47$	1.24721e11	1.68773	0.843863	−	0.536558i	$-0.180276\pi$
$47$	1.24721e11	1.68773	0.843863	+	0.536558i	$0.180276\pi$
$48$	0	0
$49$	1.34256e11	1.38567
$50$	4.85533e10	0.439455
$51$	0	0
$52$	9.35135e10	0.655923
$53$	1.30481e11	0.808641	0.404320	−	0.914617i	$-0.367508\pi$
$53$	1.30481e11	0.808641	0.404320	+	0.914617i	$0.367508\pi$
$54$	0	0
$55$	−2.96184e11	−1.44280
$56$	−3.22178e11	−1.39596
$57$	0	0
$58$	7.55872e10	0.260715
$59$	−1.55509e11	−0.479974	−0.239987	−	0.970776i	$-0.577143\pi$
$59$	−1.55509e11	−0.479974	−0.239987	+	0.970776i	$0.577143\pi$
$60$	0	0
$61$	−2.96704e11	−0.737360	−0.368680	−	0.929556i	$-0.620190\pi$
$61$	−2.96704e11	−0.737360	−0.368680	+	0.929556i	$0.620190\pi$
$62$	−1.10695e10	−0.0247504
$63$	0	0
$64$	1.59524e11	0.290173
$65$	−7.45289e11	−1.22571
$66$	0	0
$67$	−4.11869e11	−0.556254	−0.278127	−	0.960544i	$-0.589714\pi$
$67$	−4.11869e11	−0.556254	−0.278127	+	0.960544i	$0.589714\pi$
$68$	−4.34296e11	−0.532694
$69$	0	0
$70$	1.07980e12	1.09700
$71$	−6.51745e11	−0.603807	−0.301904	−	0.953338i	$-0.597622\pi$
$71$	−6.51745e11	−0.603807	−0.301904	+	0.953338i	$0.597622\pi$
$72$	0	0
$73$	−1.66606e11	−0.128852	−0.0644262	−	0.997922i	$-0.520522\pi$
$73$	−1.66606e11	−0.128852	−0.0644262	+	0.997922i	$0.520522\pi$
$74$	1.22762e11	0.0869077
$75$	0	0
$76$	−2.34411e12	−1.39537
$77$	−3.00536e12	−1.64326
$78$	0	0
$79$	1.70061e12	0.787099	0.393550	−	0.919303i	$-0.371247\pi$
$79$	1.70061e12	0.787099	0.393550	+	0.919303i	$0.371247\pi$
$80$	8.02512e11	0.342268
$81$	0	0
$82$	−9.18357e11	−0.333597
$83$	3.41254e12	1.14570	0.572850	−	0.819661i	$-0.305838\pi$
$83$	3.41254e12	1.14570	0.572850	+	0.819661i	$0.305838\pi$
$84$	0	0
$85$	3.46127e12	0.995434
$86$	−1.12796e12	−0.300645
$87$	0	0
$88$	4.18897e12	0.961546
$89$	−1.30542e12	−0.278430	−0.139215	−	0.990262i	$-0.544458\pi$
$89$	−1.30542e12	−0.278430	−0.139215	+	0.990262i	$0.544458\pi$
$90$	0	0
$91$	−7.56238e12	−1.39601
$92$	−6.86706e12	−1.18073
$93$	0	0
$94$	5.91196e12	0.883895
$95$	1.86822e13	2.60750
$96$	0	0
$97$	2.94200e12	0.358613	0.179307	−	0.983793i	$-0.442615\pi$
$97$	2.94200e12	0.358613	0.179307	+	0.983793i	$0.442615\pi$
$98$	6.36395e12	0.725700
$99$	0	0
$100$	−6.08953e12	−0.608953
$101$	1.60988e13	1.50905	0.754527	−	0.656269i	$-0.227866\pi$
$101$	1.60988e13	1.50905	0.754527	+	0.656269i	$0.227866\pi$
$102$	0	0
$103$	−2.21067e13	−1.82424	−0.912118	−	0.409927i	$-0.865554\pi$
$103$	−2.21067e13	−1.82424	−0.912118	+	0.409927i	$0.865554\pi$
$104$	1.05407e13	0.816870
$105$	0	0
$106$	6.18504e12	0.423501
$107$	−8.42298e12	−0.542589	−0.271295	−	0.962496i	$-0.587452\pi$
$107$	−8.42298e12	−0.542589	−0.271295	+	0.962496i	$0.587452\pi$
$108$	0	0
$109$	−1.32059e13	−0.754217	−0.377108	−	0.926169i	$-0.623082\pi$
$109$	−1.32059e13	−0.754217	−0.377108	+	0.926169i	$0.623082\pi$
$110$	−1.40396e13	−0.755619
$111$	0	0
$112$	8.14303e12	0.389824
$113$	−1.15824e13	−0.523346	−0.261673	−	0.965157i	$-0.584274\pi$
$113$	−1.15824e13	−0.523346	−0.261673	+	0.965157i	$0.584274\pi$
$114$	0	0
$115$	5.47295e13	2.20641
$116$	−9.48010e12	−0.361273
$117$	0	0
$118$	−7.37139e12	−0.251372
$119$	3.51212e13	1.13374
$120$	0	0
$121$	4.55310e12	0.131887
$122$	−1.40643e13	−0.386169
$123$	0	0
$124$	1.38833e12	0.0342967
$125$	−9.30604e12	−0.218197
$126$	0	0
$127$	−8.93913e13	−1.89048	−0.945238	−	0.326382i	$-0.894171\pi$
$127$	−8.93913e13	−1.89048	−0.945238	+	0.326382i	$0.894171\pi$
$128$	−4.39863e13	−0.884002
$129$	0	0
$130$	−3.53279e13	−0.641928
$131$	−4.38769e13	−0.758530	−0.379265	−	0.925288i	$-0.623823\pi$
$131$	−4.38769e13	−0.758530	−0.379265	+	0.925288i	$0.623823\pi$
$132$	0	0
$133$	1.89567e14	2.96979
$134$	−1.95233e13	−0.291321
$135$	0	0
$136$	−4.89532e13	−0.663404
$137$	−6.39026e13	−0.825724	−0.412862	−	0.910794i	$-0.635471\pi$
$137$	−6.39026e13	−0.825724	−0.412862	+	0.910794i	$0.635471\pi$
$138$	0	0
$139$	−1.89128e13	−0.222413	−0.111206	−	0.993797i	$-0.535471\pi$
$139$	−1.89128e13	−0.222413	−0.111206	+	0.993797i	$0.535471\pi$
$140$	−1.35428e14	−1.52011
$141$	0	0
$142$	−3.08938e13	−0.316225
$143$	9.83265e13	0.961582
$144$	0	0
$145$	7.55550e13	0.675104
$146$	−7.89740e12	−0.0674824
$147$	0	0
$148$	−1.53967e13	−0.120428
$149$	2.29030e14	1.71468	0.857338	−	0.514753i	$-0.172117\pi$
$149$	2.29030e14	1.71468	0.857338	+	0.514753i	$0.172117\pi$
$150$	0	0
$151$	−1.99588e14	−1.37020	−0.685100	−	0.728449i	$-0.740242\pi$
$151$	−1.99588e14	−1.37020	−0.685100	+	0.728449i	$0.740242\pi$
$152$	−2.64224e14	−1.73776
$153$	0	0
$154$	−1.42459e14	−0.860607
$155$	−1.10648e13	−0.0640895
$156$	0	0
$157$	−1.70891e13	−0.0910692	−0.0455346	−	0.998963i	$-0.514499\pi$
$157$	−1.70891e13	−0.0910692	−0.0455346	+	0.998963i	$0.514499\pi$
$158$	8.06119e13	0.412219
$159$	0	0
$160$	2.98147e14	1.40492
$161$	5.55335e14	2.51297
$162$	0	0
$163$	−8.40458e13	−0.350992	−0.175496	−	0.984480i	$-0.556153\pi$
$163$	−8.40458e13	−0.350992	−0.175496	+	0.984480i	$0.556153\pi$
$164$	1.15180e14	0.462265
$165$	0	0
$166$	1.61760e14	0.600025
$167$	−1.36056e14	−0.485356	−0.242678	−	0.970107i	$-0.578026\pi$
$167$	−1.36056e14	−0.485356	−0.242678	+	0.970107i	$0.578026\pi$
$168$	0	0
$169$	−5.54562e13	−0.183099
$170$	1.64070e14	0.521328
$171$	0	0
$172$	1.41468e14	0.416604
$173$	4.60417e14	1.30573	0.652863	−	0.757476i	$-0.273568\pi$
$173$	4.60417e14	1.30573	0.652863	+	0.757476i	$0.273568\pi$
$174$	0	0
$175$	4.92456e14	1.29605
$176$	−1.05876e14	−0.268513
$177$	0	0
$178$	−6.18792e13	−0.145819
$179$	−4.60935e13	−0.104736	−0.0523679	−	0.998628i	$-0.516677\pi$
$179$	−4.60935e13	−0.104736	−0.0523679	+	0.998628i	$0.516677\pi$
$180$	0	0
$181$	2.12381e13	0.0448956	0.0224478	−	0.999748i	$-0.492854\pi$
$181$	2.12381e13	0.0448956	0.0224478	+	0.999748i	$0.492854\pi$
$182$	−3.58469e14	−0.731119
$183$	0	0
$184$	−7.74045e14	−1.47045
$185$	1.22710e14	0.225042
$186$	0	0
$187$	−4.56648e14	−0.780928
$188$	−7.41474e14	−1.22481
$189$	0	0
$190$	8.85567e14	1.36560
$191$	7.26194e14	1.08227	0.541135	−	0.840936i	$-0.317995\pi$
$191$	7.26194e14	1.08227	0.541135	+	0.840936i	$0.317995\pi$
$192$	0	0
$193$	−8.27726e14	−1.15283	−0.576413	−	0.817158i	$-0.695548\pi$
$193$	−8.27726e14	−1.15283	−0.576413	+	0.817158i	$0.695548\pi$
$194$	1.39456e14	0.187813
$195$	0	0
$196$	−7.98163e14	−1.00560
$197$	−2.07518e14	−0.252944	−0.126472	−	0.991970i	$-0.540365\pi$
$197$	−2.07518e14	−0.252944	−0.126472	+	0.991970i	$0.540365\pi$
$198$	0	0
$199$	1.14891e15	1.31142	0.655710	−	0.755013i	$-0.272370\pi$
$199$	1.14891e15	1.31142	0.655710	+	0.755013i	$0.272370\pi$
$200$	−6.86403e14	−0.758375
$201$	0	0
$202$	7.63110e14	0.790320
$203$	7.66650e14	0.768905
$204$	0	0
$205$	−9.17966e14	−0.863826
$206$	−1.04789e15	−0.955388
$207$	0	0
$208$	−2.66416e14	−0.228112
$209$	−2.46476e15	−2.04561
$210$	0	0
$211$	9.94611e14	0.775921	0.387960	−	0.921676i	$-0.373180\pi$
$211$	9.94611e14	0.775921	0.387960	+	0.921676i	$0.373180\pi$
$212$	−7.75723e14	−0.586845
$213$	0	0
$214$	−3.99263e14	−0.284164
$215$	−1.12748e15	−0.778500
$216$	0	0
$217$	−1.12274e14	−0.0729943
$218$	−6.25982e14	−0.394998
$219$	0	0
$220$	1.76084e15	1.04706
$221$	−1.14906e15	−0.663428
$222$	0	0
$223$	−2.41276e15	−1.31381	−0.656905	−	0.753973i	$-0.728135\pi$
$223$	−2.41276e15	−1.31381	−0.656905	+	0.753973i	$0.728135\pi$
$224$	3.02527e15	1.60012
$225$	0	0
$226$	−5.49026e14	−0.274087
$227$	3.06024e15	1.48452	0.742261	−	0.670111i	$-0.233753\pi$
$227$	3.06024e15	1.48452	0.742261	+	0.670111i	$0.233753\pi$
$228$	0	0
$229$	−3.23848e14	−0.148392	−0.0741960	−	0.997244i	$-0.523639\pi$
$229$	−3.23848e14	−0.148392	−0.0741960	+	0.997244i	$0.523639\pi$
$230$	2.59427e15	1.15554
$231$	0	0
$232$	−1.06858e15	−0.449921
$233$	−1.42436e15	−0.583183	−0.291592	−	0.956543i	$-0.594185\pi$
$233$	−1.42436e15	−0.583183	−0.291592	+	0.956543i	$0.594185\pi$
$234$	0	0
$235$	5.90944e15	2.28879
$236$	9.24514e14	0.348326
$237$	0	0
$238$	1.66480e15	0.593763
$239$	−4.22145e15	−1.46513	−0.732564	−	0.680699i	$-0.761676\pi$
$239$	−4.22145e15	−1.46513	−0.732564	+	0.680699i	$0.761676\pi$
$240$	0	0
$241$	1.09833e15	0.361095	0.180548	−	0.983566i	$-0.442213\pi$
$241$	1.09833e15	0.361095	0.180548	+	0.983566i	$0.442213\pi$
$242$	2.15825e14	0.0690718
$243$	0	0
$244$	1.76393e15	0.535115
$245$	6.36124e15	1.87915
$246$	0	0
$247$	−6.20206e15	−1.73782
$248$	1.56491e14	0.0427123
$249$	0	0
$250$	−4.41122e14	−0.114274
$251$	7.33521e15	1.85154	0.925771	−	0.378084i	$-0.123417\pi$
$251$	7.33521e15	1.85154	0.925771	+	0.378084i	$0.123417\pi$
$252$	0	0
$253$	−7.22050e15	−1.73095
$254$	−4.23730e15	−0.990079
$255$	0	0
$256$	−3.39185e15	−0.753141
$257$	1.31250e15	0.284141	0.142070	−	0.989857i	$-0.454624\pi$
$257$	1.31250e15	0.284141	0.142070	+	0.989857i	$0.454624\pi$
$258$	0	0
$259$	1.24513e15	0.256309
$260$	4.43080e15	0.889520
$261$	0	0
$262$	−2.07984e15	−0.397257
$263$	−6.36587e15	−1.18617	−0.593083	−	0.805141i	$-0.702090\pi$
$263$	−6.36587e15	−1.18617	−0.593083	+	0.805141i	$0.702090\pi$
$264$	0	0
$265$	6.18240e15	1.09663
$266$	8.98577e15	1.55534
$267$	0	0
$268$	2.44860e15	0.403684
$269$	−6.30700e15	−1.01492	−0.507461	−	0.861675i	$-0.669416\pi$
$269$	−6.30700e15	−1.01492	−0.507461	+	0.861675i	$0.669416\pi$
$270$	0	0
$271$	4.75698e15	0.729510	0.364755	−	0.931104i	$-0.381153\pi$
$271$	4.75698e15	0.729510	0.364755	+	0.931104i	$0.381153\pi$
$272$	1.23729e15	0.185256
$273$	0	0
$274$	−3.02909e15	−0.432447
$275$	−6.40294e15	−0.892724
$276$	0	0
$277$	1.25033e16	1.66306	0.831529	−	0.555482i	$-0.187466\pi$
$277$	1.25033e16	1.66306	0.831529	+	0.555482i	$0.187466\pi$
$278$	−8.96499e14	−0.116482
$279$	0	0
$280$	−1.52652e16	−1.89311
$281$	−1.03066e15	−0.124889	−0.0624446	−	0.998048i	$-0.519890\pi$
$281$	−1.03066e15	−0.124889	−0.0624446	+	0.998048i	$0.519890\pi$
$282$	0	0
$283$	7.60681e15	0.880219	0.440109	−	0.897944i	$-0.354940\pi$
$283$	7.60681e15	0.880219	0.440109	+	0.897944i	$0.354940\pi$
$284$	3.87468e15	0.438194
$285$	0	0
$286$	4.66084e15	0.503599
$287$	−9.31452e15	−0.983848
$288$	0	0
$289$	−4.56810e15	−0.461211
$290$	3.58143e15	0.353565
$291$	0	0
$292$	9.90487e14	0.0935105
$293$	−1.08336e16	−1.00031	−0.500154	−	0.865937i	$-0.666723\pi$
$293$	−1.08336e16	−1.00031	−0.500154	+	0.865937i	$0.666723\pi$
$294$	0	0
$295$	−7.36824e15	−0.650910
$296$	−1.73550e15	−0.149978
$297$	0	0
$298$	1.08564e16	0.898009
$299$	−1.81689e16	−1.47051
$300$	0	0
$301$	−1.14404e16	−0.886667
$302$	−9.46080e15	−0.717600
$303$	0	0
$304$	6.67826e15	0.485272
$305$	−1.40583e16	−0.999959
$306$	0	0
$307$	−1.39379e16	−0.950166	−0.475083	−	0.879941i	$-0.657582\pi$
$307$	−1.39379e16	−0.950166	−0.475083	+	0.879941i	$0.657582\pi$
$308$	1.78671e16	1.19254
$309$	0	0
$310$	−5.24489e14	−0.0335649
$311$	−1.10565e16	−0.692908	−0.346454	−	0.938067i	$-0.612614\pi$
$311$	−1.10565e16	−0.692908	−0.346454	+	0.938067i	$0.612614\pi$
$312$	0	0
$313$	1.81823e16	1.09298	0.546488	−	0.837467i	$-0.315964\pi$
$313$	1.81823e16	1.09298	0.546488	+	0.837467i	$0.315964\pi$
$314$	−8.10052e14	−0.0476947
$315$	0	0
$316$	−1.01103e16	−0.571212
$317$	1.09738e16	0.607394	0.303697	−	0.952769i	$-0.401779\pi$
$317$	1.09738e16	0.607394	0.303697	+	0.952769i	$0.401779\pi$
$318$	0	0
$319$	−9.96802e15	−0.529626
$320$	7.55848e15	0.393513
$321$	0	0
$322$	2.63238e16	1.31609
$323$	2.88036e16	1.41134
$324$	0	0
$325$	−1.61117e16	−0.758403
$326$	−3.98391e15	−0.183821
$327$	0	0
$328$	1.29829e16	0.575694
$329$	5.99626e16	2.60680
$330$	0	0
$331$	−2.14295e16	−0.895632	−0.447816	−	0.894126i	$-0.647798\pi$
$331$	−2.14295e16	−0.895632	−0.447816	+	0.894126i	$0.647798\pi$
$332$	−2.02879e16	−0.831455
$333$	0	0
$334$	−6.44927e15	−0.254190
$335$	−1.95150e16	−0.754356
$336$	0	0
$337$	−1.47789e16	−0.549602	−0.274801	−	0.961501i	$-0.588612\pi$
$337$	−1.47789e16	−0.549602	−0.274801	+	0.961501i	$0.588612\pi$
$338$	−2.62872e15	−0.0958926
$339$	0	0
$340$	−2.05775e16	−0.722405
$341$	1.45979e15	0.0502789
$342$	0	0
$343$	1.79651e16	0.595686
$344$	1.59461e16	0.518829
$345$	0	0
$346$	2.18245e16	0.683833
$347$	−5.99115e15	−0.184234	−0.0921168	−	0.995748i	$-0.529363\pi$
$347$	−5.99115e15	−0.184234	−0.0921168	+	0.995748i	$0.529363\pi$
$348$	0	0
$349$	5.41396e16	1.60380	0.801900	−	0.597459i	$-0.203823\pi$
$349$	5.41396e16	1.60380	0.801900	+	0.597459i	$0.203823\pi$
$350$	2.33432e16	0.678764
$351$	0	0
$352$	−3.93347e16	−1.10217
$353$	1.22511e16	0.337008	0.168504	−	0.985701i	$-0.446106\pi$
$353$	1.22511e16	0.337008	0.168504	+	0.985701i	$0.446106\pi$
$354$	0	0
$355$	−3.08806e16	−0.818844
$356$	7.76084e15	0.202062
$357$	0	0
$358$	−2.18491e15	−0.0548522
$359$	2.85295e16	0.703365	0.351683	−	0.936119i	$-0.385610\pi$
$359$	2.85295e16	0.703365	0.351683	+	0.936119i	$0.385610\pi$
$360$	0	0
$361$	1.13415e17	2.69694
$362$	1.00672e15	0.0235127
$363$	0	0
$364$	4.49590e16	1.01311
$365$	−7.89403e15	−0.174741
$366$	0	0
$367$	−3.02329e16	−0.645878	−0.322939	−	0.946420i	$-0.604671\pi$
$367$	−3.02329e16	−0.645878	−0.322939	+	0.946420i	$0.604671\pi$
$368$	1.95640e16	0.410626
$369$	0	0
$370$	5.81664e15	0.117859
$371$	6.27323e16	1.24899
$372$	0	0
$373$	−6.45406e16	−1.24087	−0.620434	−	0.784259i	$-0.713044\pi$
$373$	−6.45406e16	−1.24087	−0.620434	+	0.784259i	$0.713044\pi$
$374$	−2.16459e16	−0.408987
$375$	0	0
$376$	−8.35779e16	−1.52535
$377$	−2.50825e16	−0.449938
$378$	0	0
$379$	3.08975e16	0.535511	0.267755	−	0.963487i	$-0.413718\pi$
$379$	3.08975e16	0.535511	0.267755	+	0.963487i	$0.413718\pi$
$380$	−1.11067e17	−1.89231
$381$	0	0
$382$	3.44228e16	0.566806
$383$	−7.84138e16	−1.26941	−0.634703	−	0.772756i	$-0.718877\pi$
$383$	−7.84138e16	−1.26941	−0.634703	+	0.772756i	$0.718877\pi$
$384$	0	0
$385$	−1.42398e17	−2.22848
$386$	−3.92356e16	−0.603758
$387$	0	0
$388$	−1.74904e16	−0.260252
$389$	−2.94320e16	−0.430672	−0.215336	−	0.976540i	$-0.569085\pi$
$389$	−2.94320e16	−0.430672	−0.215336	+	0.976540i	$0.569085\pi$
$390$	0	0
$391$	8.43802e16	1.19424
$392$	−8.99677e16	−1.25235
$393$	0	0
$394$	−9.83667e15	−0.132472
$395$	8.05775e16	1.06741
$396$	0	0
$397$	−3.94887e16	−0.506214	−0.253107	−	0.967438i	$-0.581452\pi$
$397$	−3.94887e16	−0.506214	−0.253107	+	0.967438i	$0.581452\pi$
$398$	5.44603e16	0.686816
$399$	0	0
$400$	1.73488e16	0.211777
$401$	2.71623e16	0.326233	0.163117	−	0.986607i	$-0.447845\pi$
$401$	2.71623e16	0.326233	0.163117	+	0.986607i	$0.447845\pi$
$402$	0	0
$403$	3.67326e15	0.0427139
$404$	−9.57087e16	−1.09515
$405$	0	0
$406$	3.63405e16	0.402690
$407$	−1.61892e16	−0.176547
$408$	0	0
$409$	6.86344e16	0.725004	0.362502	−	0.931983i	$-0.381923\pi$
$409$	6.86344e16	0.725004	0.362502	+	0.931983i	$0.381923\pi$
$410$	−4.35131e16	−0.452402
$411$	0	0
$412$	1.31426e17	1.32388
$413$	−7.47649e16	−0.741349
$414$	0	0
$415$	1.61691e17	1.55372
$416$	−9.89780e16	−0.936337
$417$	0	0
$418$	−1.16833e17	−1.07133
$419$	1.20340e17	1.08647	0.543237	−	0.839579i	$-0.317198\pi$
$419$	1.20340e17	1.08647	0.543237	+	0.839579i	$0.317198\pi$
$420$	0	0
$421$	−1.94529e17	−1.70275	−0.851373	−	0.524561i	$-0.824229\pi$
$421$	−1.94529e17	−1.70275	−0.851373	+	0.524561i	$0.824229\pi$
$422$	4.71462e16	0.406365
$423$	0	0
$424$	−8.74384e16	−0.730843
$425$	7.48261e16	0.615921
$426$	0	0
$427$	−1.42648e17	−1.13890
$428$	5.00753e16	0.393767
$429$	0	0
$430$	−5.34444e16	−0.407715
$431$	−1.46134e17	−1.09812	−0.549059	−	0.835784i	$-0.685014\pi$
$431$	−1.46134e17	−1.09812	−0.549059	+	0.835784i	$0.685014\pi$
$432$	0	0
$433$	2.42578e17	1.76880	0.884401	−	0.466727i	$-0.154567\pi$
$433$	2.42578e17	1.76880	0.884401	+	0.466727i	$0.154567\pi$
$434$	−5.32195e15	−0.0382285
$435$	0	0
$436$	7.85102e16	0.547349
$437$	4.55442e17	3.12827
$438$	0	0
$439$	1.23684e17	0.824697	0.412349	−	0.911026i	$-0.364709\pi$
$439$	1.23684e17	0.824697	0.412349	+	0.911026i	$0.364709\pi$
$440$	1.98479e17	1.30399
$441$	0	0
$442$	−5.44675e16	−0.347450
$443$	−1.35638e16	−0.0852622	−0.0426311	−	0.999091i	$-0.513574\pi$
$443$	−1.35638e16	−0.0852622	−0.0426311	+	0.999091i	$0.513574\pi$
$444$	0	0
$445$	−6.18528e16	−0.377589
$446$	−1.14369e17	−0.688068
$447$	0	0
$448$	7.66952e16	0.448189
$449$	8.80752e15	0.0507286	0.0253643	−	0.999678i	$-0.491925\pi$
$449$	8.80752e15	0.0507286	0.0253643	+	0.999678i	$0.491925\pi$
$450$	0	0
$451$	1.21108e17	0.677680
$452$	6.88584e16	0.379802
$453$	0	0
$454$	1.45060e17	0.777473
$455$	−3.58316e17	−1.89318
$456$	0	0
$457$	−1.03296e17	−0.530430	−0.265215	−	0.964189i	$-0.585443\pi$
$457$	−1.03296e17	−0.530430	−0.265215	+	0.964189i	$0.585443\pi$
$458$	−1.53509e16	−0.0777158
$459$	0	0
$460$	−3.25371e17	−1.60123
$461$	1.99772e17	0.969344	0.484672	−	0.874696i	$-0.338939\pi$
$461$	1.99772e17	0.969344	0.484672	+	0.874696i	$0.338939\pi$
$462$	0	0
$463$	−9.36783e16	−0.441939	−0.220970	−	0.975281i	$-0.570922\pi$
$463$	−9.36783e16	−0.441939	−0.220970	+	0.975281i	$0.570922\pi$
$464$	2.70084e16	0.125641
$465$	0	0
$466$	−6.75168e16	−0.305424
$467$	2.49900e17	1.11482	0.557412	−	0.830236i	$-0.311795\pi$
$467$	2.49900e17	1.11482	0.557412	+	0.830236i	$0.311795\pi$
$468$	0	0
$469$	−1.98017e17	−0.859168
$470$	2.80117e17	1.19868
$471$	0	0
$472$	1.04210e17	0.433796
$473$	1.48749e17	0.610741
$474$	0	0
$475$	4.03874e17	1.61338
$476$	−2.08799e17	−0.822778
$477$	0	0
$478$	−2.00104e17	−0.767315
$479$	−2.24020e17	−0.847435	−0.423718	−	0.905794i	$-0.639275\pi$
$479$	−2.24020e17	−0.847435	−0.423718	+	0.905794i	$0.639275\pi$
$480$	0	0
$481$	−4.07368e16	−0.149984
$482$	5.20626e16	0.189112
$483$	0	0
$484$	−2.70686e16	−0.0957129
$485$	1.39396e17	0.486328
$486$	0	0
$487$	−4.62933e17	−1.57246	−0.786230	−	0.617934i	$-0.787970\pi$
$487$	−4.62933e17	−1.57246	−0.786230	+	0.617934i	$0.787970\pi$
$488$	1.98828e17	0.666419
$489$	0	0
$490$	3.01533e17	0.984148
$491$	−2.84047e17	−0.914873	−0.457436	−	0.889242i	$-0.651232\pi$
$491$	−2.84047e17	−0.914873	−0.457436	+	0.889242i	$0.651232\pi$
$492$	0	0
$493$	1.16488e17	0.365407
$494$	−2.93988e17	−0.910132
$495$	0	0
$496$	−3.95529e15	−0.0119275
$497$	−3.13343e17	−0.932617
$498$	0	0
$499$	−4.52566e17	−1.31229	−0.656143	−	0.754637i	$-0.727813\pi$
$499$	−4.52566e17	−1.31229	−0.656143	+	0.754637i	$0.727813\pi$
$500$	5.53252e16	0.158350
$501$	0	0
$502$	3.47701e17	0.969688
$503$	−4.43954e17	−1.22221	−0.611105	−	0.791549i	$-0.709275\pi$
$503$	−4.43954e17	−1.22221	−0.611105	+	0.791549i	$0.709275\pi$
$504$	0	0
$505$	7.62784e17	2.04648
$506$	−3.42264e17	−0.906530
$507$	0	0
$508$	5.31439e17	1.37195
$509$	−3.83569e17	−0.977638	−0.488819	−	0.872385i	$-0.662572\pi$
$509$	−3.83569e17	−0.977638	−0.488819	+	0.872385i	$0.662572\pi$
$510$	0	0
$511$	−8.01001e16	−0.199020
$512$	1.99557e17	0.489567
$513$	0	0
$514$	6.22146e16	0.148810
$515$	−1.04744e18	−2.47391
$516$	0	0
$517$	−7.79636e17	−1.79558
$518$	5.90210e16	0.134234
$519$	0	0
$520$	4.99434e17	1.10779
$521$	7.08020e17	1.55096	0.775479	−	0.631373i	$-0.217508\pi$
$521$	7.08020e17	1.55096	0.775479	+	0.631373i	$0.217508\pi$
$522$	0	0
$523$	−1.91474e17	−0.409119	−0.204560	−	0.978854i	$-0.565576\pi$
$523$	−1.91474e17	−0.409119	−0.204560	+	0.978854i	$0.565576\pi$
$524$	2.60852e17	0.550479
$525$	0	0
$526$	−3.01753e17	−0.621218
$527$	−1.70594e16	−0.0346891
$528$	0	0
$529$	8.30180e17	1.64706
$530$	2.93056e17	0.574324
$531$	0	0
$532$	−1.12699e18	−2.15523
$533$	3.04744e17	0.575715
$534$	0	0
$535$	−3.99093e17	−0.735824
$536$	2.76002e17	0.502738
$537$	0	0
$538$	−2.98962e17	−0.531534
$539$	−8.39242e17	−1.47421
$540$	0	0
$541$	−1.65933e17	−0.284544	−0.142272	−	0.989828i	$-0.545441\pi$
$541$	−1.65933e17	−0.284544	−0.142272	+	0.989828i	$0.545441\pi$
$542$	2.25489e17	0.382058
$543$	0	0
$544$	4.59674e17	0.760426
$545$	−6.25715e17	−1.02282
$546$	0	0
$547$	−1.62847e17	−0.259934	−0.129967	−	0.991518i	$-0.541487\pi$
$547$	−1.62847e17	−0.259934	−0.129967	+	0.991518i	$0.541487\pi$
$548$	3.79906e17	0.599243
$549$	0	0
$550$	−3.03510e17	−0.467537
$551$	6.28745e17	0.957170
$552$	0	0
$553$	8.17613e17	1.21572
$554$	5.92679e17	0.870975
$555$	0	0
$556$	1.12438e17	0.161409
$557$	−6.57534e17	−0.932953	−0.466476	−	0.884534i	$-0.654477\pi$
$557$	−6.57534e17	−0.932953	−0.466476	+	0.884534i	$0.654477\pi$
$558$	0	0
$559$	3.74298e17	0.518848
$560$	3.85828e17	0.528654
$561$	0	0
$562$	−4.88551e16	−0.0654069
$563$	−1.83396e17	−0.242708	−0.121354	−	0.992609i	$-0.538724\pi$
$563$	−1.83396e17	−0.242708	−0.121354	+	0.992609i	$0.538724\pi$
$564$	0	0
$565$	−5.48791e17	−0.709729
$566$	3.60575e17	0.460988
$567$	0	0
$568$	4.36748e17	0.545716
$569$	9.52474e17	1.17658	0.588292	−	0.808648i	$-0.299800\pi$
$569$	9.52474e17	1.17658	0.588292	+	0.808648i	$0.299800\pi$
$570$	0	0
$571$	2.25209e17	0.271926	0.135963	−	0.990714i	$-0.456587\pi$
$571$	2.25209e17	0.271926	0.135963	+	0.990714i	$0.456587\pi$
$572$	−5.84559e17	−0.697837
$573$	0	0
$574$	−4.41524e17	−0.515260
$575$	1.18315e18	1.36520
$576$	0	0
$577$	7.13907e17	0.805377	0.402688	−	0.915337i	$-0.368076\pi$
$577$	7.13907e17	0.805377	0.402688	+	0.915337i	$0.368076\pi$
$578$	−2.16535e17	−0.241545
$579$	0	0
$580$	−4.49181e17	−0.489936
$581$	1.64067e18	1.76960
$582$	0	0
$583$	−8.15648e17	−0.860315
$584$	1.11646e17	0.116456
$585$	0	0
$586$	−5.13531e17	−0.523880
$587$	1.12264e17	0.113264	0.0566321	−	0.998395i	$-0.481964\pi$
$587$	1.12264e17	0.113264	0.0566321	+	0.998395i	$0.481964\pi$
$588$	0	0
$589$	−9.20778e16	−0.0908669
$590$	−3.49267e17	−0.340894
$591$	0	0
$592$	4.38646e16	0.0418816
$593$	−9.96270e16	−0.0940852	−0.0470426	−	0.998893i	$-0.514980\pi$
$593$	−9.96270e16	−0.0940852	−0.0470426	+	0.998893i	$0.514980\pi$
$594$	0	0
$595$	1.66409e18	1.53751
$596$	−1.36160e18	−1.24437
$597$	0	0
$598$	−8.61238e17	−0.770132
$599$	4.32020e17	0.382146	0.191073	−	0.981576i	$-0.438803\pi$
$599$	4.32020e17	0.382146	0.191073	+	0.981576i	$0.438803\pi$
$600$	0	0
$601$	−1.10657e18	−0.957841	−0.478921	−	0.877858i	$-0.658972\pi$
$601$	−1.10657e18	−0.957841	−0.478921	+	0.877858i	$0.658972\pi$
$602$	−5.42296e17	−0.464365
$603$	0	0
$604$	1.18657e18	0.994379
$605$	2.15732e17	0.178857
$606$	0	0
$607$	1.71192e18	1.38918	0.694588	−	0.719408i	$-0.255587\pi$
$607$	1.71192e18	1.38918	0.694588	+	0.719408i	$0.255587\pi$
$608$	2.48109e18	1.99191
$609$	0	0
$610$	−6.66384e17	−0.523698
$611$	−1.96180e18	−1.52541
$612$	0	0
$613$	6.48443e17	0.493604	0.246802	−	0.969066i	$-0.420620\pi$
$613$	6.48443e17	0.493604	0.246802	+	0.969066i	$0.420620\pi$
$614$	−6.60682e17	−0.497620
$615$	0	0
$616$	2.01395e18	1.48517
$617$	−6.76815e17	−0.493874	−0.246937	−	0.969032i	$-0.579424\pi$
$617$	−6.76815e17	−0.493874	−0.246937	+	0.969032i	$0.579424\pi$
$618$	0	0
$619$	1.89819e16	0.0135628	0.00678142	−	0.999977i	$-0.497841\pi$
$619$	1.89819e16	0.0135628	0.00678142	+	0.999977i	$0.497841\pi$
$620$	6.57811e16	0.0465110
$621$	0	0
$622$	−5.24097e17	−0.362889
$623$	−6.27615e17	−0.430052
$624$	0	0
$625$	−1.69129e18	−1.13501
$626$	8.61872e17	0.572413
$627$	0	0
$628$	1.01596e17	0.0660906
$629$	1.89190e17	0.121806
$630$	0	0
$631$	−2.82684e15	−0.00178283	−0.000891415	−	1.00000i	$-0.500284\pi$
$631$	−2.82684e15	−0.00178283	−0.000891415		1.00000i	$0.500284\pi$
$632$	−1.13962e18	−0.711374
$633$	0	0
$634$	5.20174e17	0.318104
$635$	−4.23549e18	−2.56374
$636$	0	0
$637$	−2.11179e18	−1.25240
$638$	−4.72501e17	−0.277375
$639$	0	0
$640$	−2.08414e18	−1.19883
$641$	2.48831e18	1.41686	0.708431	−	0.705780i	$-0.249403\pi$
$641$	2.48831e18	1.41686	0.708431	+	0.705780i	$0.249403\pi$
$642$	0	0
$643$	−3.13945e18	−1.75179	−0.875895	−	0.482501i	$-0.839728\pi$
$643$	−3.13945e18	−1.75179	−0.875895	+	0.482501i	$0.839728\pi$
$644$	−3.30151e18	−1.82371
$645$	0	0
$646$	1.36534e18	0.739145
$647$	−3.01794e18	−1.61745	−0.808727	−	0.588184i	$-0.799843\pi$
$647$	−3.01794e18	−1.61745	−0.808727	+	0.588184i	$0.799843\pi$
$648$	0	0
$649$	9.72097e17	0.510645
$650$	−7.63722e17	−0.397190
$651$	0	0
$652$	4.99659e17	0.254721
$653$	1.86528e18	0.941475	0.470738	−	0.882273i	$-0.343988\pi$
$653$	1.86528e18	0.941475	0.470738	+	0.882273i	$0.343988\pi$
$654$	0	0
$655$	−2.07895e18	−1.02867
$656$	−3.28142e17	−0.160763
$657$	0	0
$658$	2.84233e18	1.36523
$659$	−1.62853e18	−0.774533	−0.387267	−	0.921968i	$-0.626581\pi$
$659$	−1.62853e18	−0.774533	−0.387267	+	0.921968i	$0.626581\pi$
$660$	0	0
$661$	1.66052e18	0.774347	0.387173	−	0.922007i	$-0.373451\pi$
$661$	1.66052e18	0.774347	0.387173	+	0.922007i	$0.373451\pi$
$662$	−1.01579e18	−0.469060
$663$	0	0
$664$	−2.28682e18	−1.03547
$665$	8.98194e18	4.02744
$666$	0	0
$667$	1.84191e18	0.809935
$668$	8.08864e17	0.352232
$669$	0	0
$670$	−9.25041e17	−0.395071
$671$	1.85472e18	0.784478
$672$	0	0
$673$	3.52806e18	1.46365	0.731826	−	0.681491i	$-0.238668\pi$
$673$	3.52806e18	1.46365	0.731826	+	0.681491i	$0.238668\pi$
$674$	−7.00545e17	−0.287837
$675$	0	0
$676$	3.29692e17	0.132879
$677$	1.75400e18	0.700169	0.350084	−	0.936718i	$-0.386153\pi$
$677$	1.75400e18	0.700169	0.350084	+	0.936718i	$0.386153\pi$
$678$	0	0
$679$	1.41444e18	0.553899
$680$	−2.31947e18	−0.899665
$681$	0	0
$682$	6.91963e16	0.0263320
$683$	−7.47953e17	−0.281929	−0.140964	−	0.990015i	$-0.545020\pi$
$683$	−7.47953e17	−0.281929	−0.140964	+	0.990015i	$0.545020\pi$
$684$	0	0
$685$	−3.02780e18	−1.11979
$686$	8.51574e17	0.311972
$687$	0	0
$688$	−4.03036e17	−0.144884
$689$	−2.05242e18	−0.730870
$690$	0	0
$691$	3.09486e18	1.08152	0.540760	−	0.841177i	$-0.318137\pi$
$691$	3.09486e18	1.08152	0.540760	+	0.841177i	$0.318137\pi$
$692$	−2.73722e18	−0.947588
$693$	0	0
$694$	−2.83990e17	−0.0964867
$695$	−8.96116e17	−0.301622
$696$	0	0
$697$	−1.41529e18	−0.467555
$698$	2.56631e18	0.839940
$699$	0	0
$700$	−2.92769e18	−0.940564
$701$	3.92104e18	1.24805	0.624027	−	0.781402i	$-0.285495\pi$
$701$	3.92104e18	1.24805	0.624027	+	0.781402i	$0.285495\pi$
$702$	0	0
$703$	1.02115e18	0.319066
$704$	−9.97195e17	−0.308715
$705$	0	0
$706$	5.80723e17	0.176498
$707$	7.73991e18	2.33082
$708$	0	0
$709$	9.54184e17	0.282119	0.141059	−	0.990001i	$-0.454949\pi$
$709$	9.54184e17	0.282119	0.141059	+	0.990001i	$0.454949\pi$
$710$	−1.46379e18	−0.428844
$711$	0	0
$712$	8.74791e17	0.251643
$713$	−2.69742e17	−0.0768894
$714$	0	0
$715$	4.65885e18	1.30404
$716$	2.74030e17	0.0760087
$717$	0	0
$718$	1.35235e18	0.368366
$719$	−1.29564e18	−0.349740	−0.174870	−	0.984592i	$-0.555950\pi$
$719$	−1.29564e18	−0.349740	−0.174870	+	0.984592i	$0.555950\pi$
$720$	0	0
$721$	−1.06283e19	−2.81764
$722$	5.37603e18	1.41244
$723$	0	0
$724$	−1.26262e17	−0.0325816
$725$	1.63335e18	0.417718
$726$	0	0
$727$	−2.40973e18	−0.605334	−0.302667	−	0.953096i	$-0.597877\pi$
$727$	−2.40973e18	−0.605334	−0.302667	+	0.953096i	$0.597877\pi$
$728$	5.06771e18	1.26171
$729$	0	0
$730$	−3.74190e17	−0.0915153
$731$	−1.73831e18	−0.421371
$732$	0	0
$733$	−4.02839e18	−0.959303	−0.479651	−	0.877459i	$-0.659237\pi$
$733$	−4.02839e18	−0.959303	−0.479651	+	0.877459i	$0.659237\pi$
$734$	−1.43309e18	−0.338259
$735$	0	0
$736$	7.26834e18	1.68550
$737$	2.57462e18	0.591800
$738$	0	0
$739$	−4.93863e18	−1.11537	−0.557683	−	0.830054i	$-0.688310\pi$
$739$	−4.93863e18	−1.11537	−0.557683	+	0.830054i	$0.688310\pi$
$740$	−7.29520e17	−0.163317
$741$	0	0
$742$	2.97361e18	0.654123
$743$	−2.94560e18	−0.642312	−0.321156	−	0.947026i	$-0.604071\pi$
$743$	−2.94560e18	−0.642312	−0.321156	+	0.947026i	$0.604071\pi$
$744$	0	0
$745$	1.08518e19	2.32533
$746$	−3.05933e18	−0.649866
$747$	0	0
$748$	2.71481e18	0.566734
$749$	−4.04956e18	−0.838062
$750$	0	0
$751$	−5.55685e17	−0.113024	−0.0565118	−	0.998402i	$-0.517998\pi$
$751$	−5.55685e17	−0.113024	−0.0565118	+	0.998402i	$0.517998\pi$
$752$	2.11243e18	0.425957
$753$	0	0
$754$	−1.18895e18	−0.235641
$755$	−9.45676e18	−1.85818
$756$	0	0
$757$	1.46981e18	0.283882	0.141941	−	0.989875i	$-0.454666\pi$
$757$	1.46981e18	0.283882	0.141941	+	0.989875i	$0.454666\pi$
$758$	1.46459e18	0.280457
$759$	0	0
$760$	−1.25193e19	−2.35664
$761$	4.18599e17	0.0781265	0.0390632	−	0.999237i	$-0.487563\pi$
$761$	4.18599e17	0.0781265	0.0390632	+	0.999237i	$0.487563\pi$
$762$	0	0
$763$	−6.34908e18	−1.16493
$764$	−4.31728e18	−0.785423
$765$	0	0
$766$	−3.71694e18	−0.664812
$767$	2.44609e18	0.433813
$768$	0	0
$769$	−3.05614e17	−0.0532907	−0.0266454	−	0.999645i	$-0.508482\pi$
$769$	−3.05614e17	−0.0532907	−0.0266454	+	0.999645i	$0.508482\pi$
$770$	−6.74991e18	−1.16710
$771$	0	0
$772$	4.92090e18	0.836627
$773$	−1.24126e17	−0.0209264	−0.0104632	−	0.999945i	$-0.503331\pi$
$773$	−1.24126e17	−0.0209264	−0.0104632	+	0.999945i	$0.503331\pi$
$774$	0	0
$775$	−2.39200e17	−0.0396552
$776$	−1.97149e18	−0.324111
$777$	0	0
$778$	−1.39512e18	−0.225551
$779$	−7.63903e18	−1.22474
$780$	0	0
$781$	4.07410e18	0.642392
$782$	3.99976e18	0.625447
$783$	0	0
$784$	2.27393e18	0.349722
$785$	−8.09706e17	−0.123502
$786$	0	0
$787$	1.31170e18	0.196788	0.0983941	−	0.995148i	$-0.468629\pi$
$787$	1.31170e18	0.196788	0.0983941	+	0.995148i	$0.468629\pi$
$788$	1.23371e18	0.183566
$789$	0	0
$790$	3.81951e18	0.559025
$791$	−5.56854e18	−0.808340
$792$	0	0
$793$	4.66702e18	0.666444
$794$	−1.87183e18	−0.265114
$795$	0	0
$796$	−6.83038e18	−0.951721
$797$	2.90092e18	0.400920	0.200460	−	0.979702i	$-0.435756\pi$
$797$	2.90092e18	0.400920	0.200460	+	0.979702i	$0.435756\pi$
$798$	0	0
$799$	9.11099e18	1.23883
$800$	6.44537e18	0.869287
$801$	0	0
$802$	1.28754e18	0.170855
$803$	1.04147e18	0.137086
$804$	0	0
$805$	2.63126e19	3.40793
$806$	1.74119e17	0.0223701
$807$	0	0
$808$	−1.07881e19	−1.36387
$809$	−1.28697e19	−1.61400	−0.807001	−	0.590550i	$-0.798911\pi$
$809$	−1.28697e19	−1.61400	−0.807001	+	0.590550i	$0.798911\pi$
$810$	0	0
$811$	−1.15282e19	−1.42275	−0.711373	−	0.702814i	$-0.751926\pi$
$811$	−1.15282e19	−1.42275	−0.711373	+	0.702814i	$0.751926\pi$
$812$	−4.55780e18	−0.558009
$813$	0	0
$814$	−7.67394e17	−0.0924613
$815$	−3.98221e18	−0.475992
$816$	0	0
$817$	−9.38254e18	−1.10377
$818$	3.25338e18	0.379698
$819$	0	0
$820$	5.45738e18	0.626894
$821$	1.62326e18	0.184994	0.0924969	−	0.995713i	$-0.470515\pi$
$821$	1.62326e18	0.184994	0.0924969	+	0.995713i	$0.470515\pi$
$822$	0	0
$823$	2.78739e18	0.312679	0.156339	−	0.987703i	$-0.450031\pi$
$823$	2.78739e18	0.312679	0.156339	+	0.987703i	$0.450031\pi$
$824$	1.48141e19	1.64873
$825$	0	0
$826$	−3.54398e18	−0.388258
$827$	5.09011e18	0.553276	0.276638	−	0.960974i	$-0.410780\pi$
$827$	5.09011e18	0.553276	0.276638	+	0.960974i	$0.410780\pi$
$828$	0	0
$829$	−1.26916e19	−1.35804	−0.679021	−	0.734119i	$-0.737595\pi$
$829$	−1.26916e19	−1.35804	−0.679021	+	0.734119i	$0.737595\pi$
$830$	7.66443e18	0.813715
$831$	0	0
$832$	−2.50924e18	−0.262265
$833$	9.80756e18	1.01711
$834$	0	0
$835$	−6.44652e18	−0.658208
$836$	1.46532e19	1.48454
$837$	0	0
$838$	5.70432e18	0.569008
$839$	−6.85426e18	−0.678434	−0.339217	−	0.940708i	$-0.610162\pi$
$839$	−6.85426e18	−0.678434	−0.339217	+	0.940708i	$0.610162\pi$
$840$	0	0
$841$	−7.71784e18	−0.752180
$842$	−9.22098e18	−0.891761
$843$	0	0
$844$	−5.91305e18	−0.563100
$845$	−2.62760e18	−0.248308
$846$	0	0
$847$	2.18902e18	0.203708
$848$	2.21000e18	0.204089
$849$	0	0
$850$	3.54688e18	0.322570
$851$	2.99147e18	0.269986
$852$	0	0
$853$	2.01942e18	0.179497	0.0897485	−	0.995964i	$-0.471394\pi$
$853$	2.01942e18	0.179497	0.0897485	+	0.995964i	$0.471394\pi$
$854$	−6.76175e18	−0.596462
$855$	0	0
$856$	5.64441e18	0.490388
$857$	−1.04505e19	−0.901078	−0.450539	−	0.892757i	$-0.648768\pi$
$857$	−1.04505e19	−0.901078	−0.450539	+	0.892757i	$0.648768\pi$
$858$	0	0
$859$	−1.15457e19	−0.980538	−0.490269	−	0.871571i	$-0.663102\pi$
$859$	−1.15457e19	−0.980538	−0.490269	+	0.871571i	$0.663102\pi$
$860$	6.70296e18	0.564972
$861$	0	0
$862$	−6.92699e18	−0.575105
$863$	2.27301e19	1.87297	0.936485	−	0.350707i	$-0.114059\pi$
$863$	2.27301e19	1.87297	0.936485	+	0.350707i	$0.114059\pi$
$864$	0	0
$865$	2.18152e19	1.77074
$866$	1.14986e19	0.926356
$867$	0	0
$868$	6.67475e17	0.0529733
$869$	−1.06306e19	−0.837397
$870$	0	0
$871$	6.47852e18	0.502757
$872$	8.84956e18	0.681655
$873$	0	0
$874$	2.15887e19	1.63833
$875$	−4.47411e18	−0.337019
$876$	0	0
$877$	8.07848e18	0.599560	0.299780	−	0.954008i	$-0.403087\pi$
$877$	8.07848e18	0.599560	0.299780	+	0.954008i	$0.403087\pi$
$878$	5.86283e18	0.431910
$879$	0	0
$880$	−5.01656e18	−0.364140
$881$	3.87936e18	0.279523	0.139761	−	0.990185i	$-0.455366\pi$
$881$	3.87936e18	0.279523	0.139761	+	0.990185i	$0.455366\pi$
$882$	0	0
$883$	−7.80475e18	−0.554133	−0.277067	−	0.960851i	$-0.589362\pi$
$883$	−7.80475e18	−0.554133	−0.277067	+	0.960851i	$0.589362\pi$
$884$	6.83128e18	0.481462
$885$	0	0
$886$	−6.42945e17	−0.0446534
$887$	−1.49223e19	−1.02880	−0.514401	−	0.857550i	$-0.671986\pi$
$887$	−1.49223e19	−1.02880	−0.514401	+	0.857550i	$0.671986\pi$
$888$	0	0
$889$	−4.29771e19	−2.91995
$890$	−2.93192e18	−0.197750
$891$	0	0
$892$	1.43441e19	0.953456
$893$	4.91765e19	3.24507
$894$	0	0
$895$	−2.18398e18	−0.142036
$896$	−2.11476e19	−1.36539
$897$	0	0
$898$	4.17491e17	0.0265675
$899$	−3.72383e17	−0.0235262
$900$	0	0
$901$	9.53183e18	0.593561
$902$	5.74071e18	0.354914
$903$	0	0
$904$	7.76162e18	0.472996
$905$	1.00629e18	0.0608846
$906$	0	0
$907$	2.75513e19	1.64322	0.821608	−	0.570053i	$-0.193077\pi$
$907$	2.75513e19	1.64322	0.821608	+	0.570053i	$0.193077\pi$
$908$	−1.81934e19	−1.07735
$909$	0	0
$910$	−1.69848e19	−0.991496
$911$	−3.30596e19	−1.91614	−0.958071	−	0.286530i	$-0.907498\pi$
$911$	−3.30596e19	−1.91614	−0.958071	+	0.286530i	$0.907498\pi$
$912$	0	0
$913$	−2.13320e19	−1.21891
$914$	−4.89640e18	−0.277796
$915$	0	0
$916$	1.92530e18	0.107691
$917$	−2.10949e19	−1.17160
$918$	0	0
$919$	−1.99555e19	−1.09273	−0.546365	−	0.837547i	$-0.683989\pi$
$919$	−1.99555e19	−1.09273	−0.546365	+	0.837547i	$0.683989\pi$
$920$	−3.66754e19	−1.99413
$921$	0	0
$922$	9.46950e18	0.507664
$923$	1.02517e19	0.545736
$924$	0	0
$925$	2.65275e18	0.139244
$926$	−4.44051e18	−0.231452
$927$	0	0
$928$	1.00341e19	0.515722
$929$	−2.23597e19	−1.14120	−0.570602	−	0.821227i	$-0.693290\pi$
$929$	−2.23597e19	−1.14120	−0.570602	+	0.821227i	$0.693290\pi$
$930$	0	0
$931$	5.29363e19	2.66428
$932$	8.46792e18	0.423227
$933$	0	0
$934$	1.18457e19	0.583854
$935$	−2.16366e19	−1.05904
$936$	0	0
$937$	−1.36964e19	−0.661149	−0.330574	−	0.943780i	$-0.607242\pi$
$937$	−1.36964e19	−0.661149	−0.330574	+	0.943780i	$0.607242\pi$
$938$	−9.38632e18	−0.449963
$939$	0	0
$940$	−3.51321e19	−1.66101
$941$	2.10985e19	0.990649	0.495325	−	0.868708i	$-0.335049\pi$
$941$	2.10985e19	0.990649	0.495325	+	0.868708i	$0.335049\pi$
$942$	0	0
$943$	−2.23785e19	−1.03635
$944$	−2.63390e18	−0.121138
$945$	0	0
$946$	7.05096e18	0.319857
$947$	1.78205e19	0.802868	0.401434	−	0.915888i	$-0.368512\pi$
$947$	1.78205e19	0.802868	0.401434	+	0.915888i	$0.368512\pi$
$948$	0	0
$949$	2.62064e18	0.116460
$950$	1.91443e19	0.844959
$951$	0	0
$952$	−2.35355e19	−1.02467
$953$	1.66045e19	0.717996	0.358998	−	0.933338i	$-0.383119\pi$
$953$	1.66045e19	0.717996	0.358998	+	0.933338i	$0.383119\pi$
$954$	0	0
$955$	3.44081e19	1.46771
$956$	2.50969e19	1.06327
$957$	0	0
$958$	−1.06189e19	−0.443818
$959$	−3.07228e19	−1.27538
$960$	0	0
$961$	−2.43630e19	−0.997767
$962$	−1.93099e18	−0.0785494
$963$	0	0
$964$	−6.52966e18	−0.262053
$965$	−3.92188e19	−1.56339
$966$	0	0
$967$	7.37484e18	0.290055	0.145028	−	0.989428i	$-0.453673\pi$
$967$	7.37484e18	0.290055	0.145028	+	0.989428i	$0.453673\pi$
$968$	−3.05113e18	−0.119199
$969$	0	0
$970$	6.60760e18	0.254699
$971$	3.63645e19	1.39236	0.696182	−	0.717865i	$-0.254881\pi$
$971$	3.63645e19	1.39236	0.696182	+	0.717865i	$0.254881\pi$
$972$	0	0
$973$	−9.09282e18	−0.343530
$974$	−2.19438e19	−0.823528
$975$	0	0
$976$	−5.02536e18	−0.186099
$977$	4.29777e18	0.158099	0.0790493	−	0.996871i	$-0.474812\pi$
$977$	4.29777e18	0.158099	0.0790493	+	0.996871i	$0.474812\pi$
$978$	0	0
$979$	8.16028e18	0.296222
$980$	−3.78181e19	−1.36373
$981$	0	0
$982$	−1.34643e19	−0.479136
$983$	4.43027e19	1.56615	0.783074	−	0.621929i	$-0.213651\pi$
$983$	4.43027e19	1.56615	0.783074	+	0.621929i	$0.213651\pi$
$984$	0	0
$985$	−9.83248e18	−0.343026
$986$	5.52174e18	0.191371
$987$	0	0
$988$	3.68718e19	1.26117
$989$	−2.74861e19	−0.933981
$990$	0	0
$991$	−4.66719e19	−1.56523	−0.782613	−	0.622509i	$-0.786113\pi$
$991$	−4.66719e19	−1.56523	−0.782613	+	0.622509i	$0.786113\pi$
$992$	−1.46946e18	−0.0489589
$993$	0	0
$994$	−1.48530e19	−0.488429
$995$	5.44371e19	1.77846
$996$	0	0
$997$	5.52570e19	1.78184	0.890920	−	0.454159i	$-0.150060\pi$
$997$	5.52570e19	1.78184	0.890920	+	0.454159i	$0.150060\pi$
$998$	−2.14524e19	−0.687269
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.14.a.e.1.3	yes	4
3.2	odd	2		27.14.a.b.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.14.a.b.1.2	✓	4	3.2	odd	2
27.14.a.e.1.3	yes	4	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+47.4017 q^{2} -5945.08 q^{4} +47381.4 q^{5} +480775. q^{7} -670121. q^{8} +2.24596e6 q^{10} -6.25107e6 q^{11} -1.57296e7 q^{13} +2.27895e7 q^{14} +1.69373e7 q^{16} +7.30512e7 q^{17} +3.94294e8 q^{19} -2.81687e8 q^{20} -2.96311e8 q^{22} +1.15508e9 q^{23} +1.02430e9 q^{25} -7.45607e8 q^{26} -2.85825e9 q^{28} +1.59461e9 q^{29} -2.33526e8 q^{31} +6.29249e9 q^{32} +3.46275e9 q^{34} +2.27798e10 q^{35} +2.58983e9 q^{37} +1.86902e10 q^{38} -3.17513e10 q^{40} -1.93740e10 q^{41} -2.37958e10 q^{43} +3.71631e10 q^{44} +5.47528e10 q^{46} +1.24721e11 q^{47} +1.34256e11 q^{49} +4.85533e10 q^{50} +9.35135e10 q^{52} +1.30481e11 q^{53} -2.96184e11 q^{55} -3.22178e11 q^{56} +7.55872e10 q^{58} -1.55509e11 q^{59} -2.96704e11 q^{61} -1.10695e10 q^{62} +1.59524e11 q^{64} -7.45289e11 q^{65} -4.11869e11 q^{67} -4.34296e11 q^{68} +1.07980e12 q^{70} -6.51745e11 q^{71} -1.66606e11 q^{73} +1.22762e11 q^{74} -2.34411e12 q^{76} -3.00536e12 q^{77} +1.70061e12 q^{79} +8.02512e11 q^{80} -9.18357e11 q^{82} +3.41254e12 q^{83} +3.46127e12 q^{85} -1.12796e12 q^{86} +4.18897e12 q^{88} -1.30542e12 q^{89} -7.56238e12 q^{91} -6.86706e12 q^{92} +5.91196e12 q^{94} +1.86822e13 q^{95} +2.94200e12 q^{97} +6.36395e12 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 117 q^{2} + 8101 q^{4} + 11412 q^{5} + 54572 q^{7} + 615375 q^{8} + 969579 q^{10} + 4927212 q^{11} + 12107576 q^{13} + 2015487 q^{14} + 56856625 q^{16} - 34729992 q^{17} + 269848016 q^{19} + 179564463 q^{20}+ \cdots + 3817815378498 q^{98}+O(q^{100}) \) 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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