LMFDB - Embedded newform 27.8.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [27,8,Mod(1,27)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(27, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("27.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 27 = 3^{3} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	27.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$8.43439568807$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ 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)

$f(q)$	$=$	$q+10.3923 q^{2} -20.0000 q^{4} -353.338 q^{5} -559.000 q^{7} -1538.06 q^{8} +O(q^{10})$	\(q+10.3923 q^{2} -20.0000 q^{4} -353.338 q^{5} -559.000 q^{7} -1538.06 q^{8} -3672.00 q^{10} +4718.11 q^{11} -8671.00 q^{13} -5809.30 q^{14} -13424.0 q^{16} +25128.6 q^{17} -32461.0 q^{19} +7066.77 q^{20} +49032.0 q^{22} -82411.0 q^{23} +46723.0 q^{25} -90111.7 q^{26} +11180.0 q^{28} +157797. q^{29} +229892. q^{31} +57365.5 q^{32} +261144. q^{34} +197516. q^{35} -541177. q^{37} -337345. q^{38} +543456. q^{40} +353505. q^{41} -465112. q^{43} -94362.1 q^{44} -856440. q^{46} -830574. q^{47} -511062. q^{49} +485560. q^{50} +173420. q^{52} -1.02622e6 q^{53} -1.66709e6 q^{55} +859776. q^{56} +1.63987e6 q^{58} -785263. q^{59} -137773. q^{61} +2.38911e6 q^{62} +2.31443e6 q^{64} +3.06380e6 q^{65} -314041. q^{67} -502572. q^{68} +2.05265e6 q^{70} -2.80979e6 q^{71} +2.66954e6 q^{73} -5.62408e6 q^{74} +649220. q^{76} -2.63742e6 q^{77} +1.10181e6 q^{79} +4.74321e6 q^{80} +3.67373e6 q^{82} +6.07904e6 q^{83} -8.87890e6 q^{85} -4.83359e6 q^{86} -7.25674e6 q^{88} -3.28636e6 q^{89} +4.84709e6 q^{91} +1.64822e6 q^{92} -8.63158e6 q^{94} +1.14697e7 q^{95} -2.97938e6 q^{97} -5.31111e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 40 q^{4} - 1118 q^{7}+O(q^{10}) $ 2 * q - 40 * q^4 - 1118 * q^7	$ 2 q - 40 q^{4} - 1118 q^{7} - 7344 q^{10} - 17342 q^{13} - 26848 q^{16} - 64922 q^{19} + 98064 q^{22} + 93446 q^{25} + 22360 q^{28} + 459784 q^{31} + 522288 q^{34} - 1082354 q^{37} + 1086912 q^{40} - 930224 q^{43} - 1712880 q^{46} - 1022124 q^{49} + 346840 q^{52} - 3334176 q^{55} + 3279744 q^{58} - 275546 q^{61} + 4628864 q^{64} - 628082 q^{67} + 4105296 q^{70} + 5339074 q^{73} + 1298440 q^{76} + 2203630 q^{79} + 7347456 q^{82} - 17757792 q^{85} - 14513472 q^{88} + 9694178 q^{91} - 17263152 q^{94} - 5958758 q^{97}+O(q^{100}) $ 2 * q - 40 * q^4 - 1118 * q^7 - 7344 * q^10 - 17342 * q^13 - 26848 * q^16 - 64922 * q^19 + 98064 * q^22 + 93446 * q^25 + 22360 * q^28 + 459784 * q^31 + 522288 * q^34 - 1082354 * q^37 + 1086912 * q^40 - 930224 * q^43 - 1712880 * q^46 - 1022124 * q^49 + 346840 * q^52 - 3334176 * q^55 + 3279744 * q^58 - 275546 * q^61 + 4628864 * q^64 - 628082 * q^67 + 4105296 * q^70 + 5339074 * q^73 + 1298440 * q^76 + 2203630 * q^79 + 7347456 * q^82 - 17757792 * q^85 - 14513472 * q^88 + 9694178 * q^91 - 17263152 * q^94 - 5958758 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	10.3923	0.918559	0.459279	−	0.888292i	$-0.348108\pi$
$2$	10.3923	0.918559	0.459279	+	0.888292i	$0.348108\pi$
$3$	0	0
$3$	0	0
$4$	−20.0000	−0.156250
$5$	−353.338	−1.26414	−0.632071	−	0.774911i	$-0.717795\pi$
$5$	−353.338	−1.26414	−0.632071	+	0.774911i	$0.717795\pi$
$6$	0	0
$7$	−559.000	−0.615983	−0.307991	−	0.951389i	$-0.599657\pi$
$7$	−559.000	−0.615983	−0.307991	+	0.951389i	$0.599657\pi$
$8$	−1538.06	−1.06208
$9$	0	0
$10$	−3672.00	−1.16119
$11$	4718.11	1.06879	0.534396	−	0.845234i	$-0.320539\pi$
$11$	4718.11	1.06879	0.534396	+	0.845234i	$0.320539\pi$
$12$	0	0
$13$	−8671.00	−1.09463	−0.547315	−	0.836927i	$-0.684350\pi$
$13$	−8671.00	−1.09463	−0.547315	+	0.836927i	$0.684350\pi$
$14$	−5809.30	−0.565816
$15$	0	0
$16$	−13424.0	−0.819336
$17$	25128.6	1.24050	0.620250	−	0.784404i	$-0.287031\pi$
$17$	25128.6	1.24050	0.620250	+	0.784404i	$0.287031\pi$
$18$	0	0
$19$	−32461.0	−1.08574	−0.542868	−	0.839818i	$-0.682662\pi$
$19$	−32461.0	−1.08574	−0.542868	+	0.839818i	$0.682662\pi$
$20$	7066.77	0.197522
$21$	0	0
$22$	49032.0	0.981748
$23$	−82411.0	−1.41233	−0.706167	−	0.708045i	$-0.749577\pi$
$23$	−82411.0	−1.41233	−0.706167	+	0.708045i	$0.749577\pi$
$24$	0	0
$25$	46723.0	0.598054
$26$	−90111.7	−1.00548
$27$	0	0
$28$	11180.0	0.0962473
$29$	157797.	1.20145	0.600724	−	0.799456i	$-0.294879\pi$
$29$	157797.	1.20145	0.600724	+	0.799456i	$0.294879\pi$
$30$	0	0
$31$	229892.	1.38598	0.692992	−	0.720946i	$-0.256292\pi$
$31$	229892.	1.38598	0.692992	+	0.720946i	$0.256292\pi$
$32$	57365.5	0.309475
$33$	0	0
$34$	261144.	1.13947
$35$	197516.	0.778690
$36$	0	0
$37$	−541177.	−1.75644	−0.878220	−	0.478257i	$-0.841269\pi$
$37$	−541177.	−1.75644	−0.878220	+	0.478257i	$0.841269\pi$
$38$	−337345.	−0.997312
$39$	0	0
$40$	543456.	1.34262
$41$	353505.	0.801035	0.400518	−	0.916289i	$-0.368830\pi$
$41$	353505.	0.801035	0.400518	+	0.916289i	$0.368830\pi$
$42$	0	0
$43$	−465112.	−0.892109	−0.446055	−	0.895006i	$-0.647171\pi$
$43$	−465112.	−0.892109	−0.446055	+	0.895006i	$0.647171\pi$
$44$	−94362.1	−0.166999
$45$	0	0
$46$	−856440.	−1.29731
$47$	−830574.	−1.16691	−0.583453	−	0.812147i	$-0.698299\pi$
$47$	−830574.	−1.16691	−0.583453	+	0.812147i	$0.698299\pi$
$48$	0	0
$49$	−511062.	−0.620565
$50$	485560.	0.549348
$51$	0	0
$52$	173420.	0.171036
$53$	−1.02622e6	−0.946836	−0.473418	−	0.880838i	$-0.656980\pi$
$53$	−1.02622e6	−0.946836	−0.473418	+	0.880838i	$0.656980\pi$
$54$	0	0
$55$	−1.66709e6	−1.35111
$56$	859776.	0.654225
$57$	0	0
$58$	1.63987e6	1.10360
$59$	−785263.	−0.497775	−0.248888	−	0.968532i	$-0.580065\pi$
$59$	−785263.	−0.497775	−0.248888	+	0.968532i	$0.580065\pi$
$60$	0	0
$61$	−137773.	−0.0777159	−0.0388579	−	0.999245i	$-0.512372\pi$
$61$	−137773.	−0.0777159	−0.0388579	+	0.999245i	$0.512372\pi$
$62$	2.38911e6	1.27311
$63$	0	0
$64$	2.31443e6	1.10361
$65$	3.06380e6	1.38377
$66$	0	0
$67$	−314041.	−0.127563	−0.0637815	−	0.997964i	$-0.520316\pi$
$67$	−314041.	−0.127563	−0.0637815	+	0.997964i	$0.520316\pi$
$68$	−502572.	−0.193828
$69$	0	0
$70$	2.05265e6	0.715272
$71$	−2.80979e6	−0.931686	−0.465843	−	0.884868i	$-0.654249\pi$
$71$	−2.80979e6	−0.931686	−0.465843	+	0.884868i	$0.654249\pi$
$72$	0	0
$73$	2.66954e6	0.803167	0.401584	−	0.915822i	$-0.368460\pi$
$73$	2.66954e6	0.803167	0.401584	+	0.915822i	$0.368460\pi$
$74$	−5.62408e6	−1.61339
$75$	0	0
$76$	649220.	0.169646
$77$	−2.63742e6	−0.658358
$78$	0	0
$79$	1.10181e6	0.251428	0.125714	−	0.992067i	$-0.459878\pi$
$79$	1.10181e6	0.251428	0.125714	+	0.992067i	$0.459878\pi$
$80$	4.74321e6	1.03576
$81$	0	0
$82$	3.67373e6	0.735798
$83$	6.07904e6	1.16698	0.583488	−	0.812122i	$-0.301688\pi$
$83$	6.07904e6	1.16698	0.583488	+	0.812122i	$0.301688\pi$
$84$	0	0
$85$	−8.87890e6	−1.56817
$86$	−4.83359e6	−0.819454
$87$	0	0
$88$	−7.25674e6	−1.13515
$89$	−3.28636e6	−0.494140	−0.247070	−	0.968998i	$-0.579468\pi$
$89$	−3.28636e6	−0.494140	−0.247070	+	0.968998i	$0.579468\pi$
$90$	0	0
$91$	4.84709e6	0.674274
$92$	1.64822e6	0.220677
$93$	0	0
$94$	−8.63158e6	−1.07187
$95$	1.14697e7	1.37252
$96$	0	0
$97$	−2.97938e6	−0.331455	−0.165728	−	0.986172i	$-0.552997\pi$
$97$	−2.97938e6	−0.331455	−0.165728	+	0.986172i	$0.552997\pi$
$98$	−5.31111e6	−0.570025
$99$	0	0
$100$	−934460.	−0.0934460
$101$	−2.74739e6	−0.265336	−0.132668	−	0.991161i	$-0.542354\pi$
$101$	−2.74739e6	−0.265336	−0.132668	+	0.991161i	$0.542354\pi$
$102$	0	0
$103$	−5.87109e6	−0.529406	−0.264703	−	0.964330i	$-0.585274\pi$
$103$	−5.87109e6	−0.529406	−0.264703	+	0.964330i	$0.585274\pi$
$104$	1.33365e7	1.16259
$105$	0	0
$106$	−1.06648e7	−0.869724
$107$	5.84848e6	0.461530	0.230765	−	0.973010i	$-0.425877\pi$
$107$	5.84848e6	0.461530	0.230765	+	0.973010i	$0.425877\pi$
$108$	0	0
$109$	8.94694e6	0.661731	0.330866	−	0.943678i	$-0.392659\pi$
$109$	8.94694e6	0.661731	0.330866	+	0.943678i	$0.392659\pi$
$110$	−1.73249e7	−1.24107
$111$	0	0
$112$	7.50402e6	0.504697
$113$	−1.30666e7	−0.851899	−0.425949	−	0.904747i	$-0.640060\pi$
$113$	−1.30666e7	−0.851899	−0.425949	+	0.904747i	$0.640060\pi$
$114$	0	0
$115$	2.91190e7	1.78539
$116$	−3.15594e6	−0.187726
$117$	0	0
$118$	−8.16070e6	−0.457236
$119$	−1.40469e7	−0.764127
$120$	0	0
$121$	2.77336e6	0.142317
$122$	−1.43178e6	−0.0713866
$123$	0	0
$124$	−4.59784e6	−0.216560
$125$	1.10955e7	0.508116
$126$	0	0
$127$	1.96731e7	0.852235	0.426118	−	0.904668i	$-0.359881\pi$
$127$	1.96731e7	0.852235	0.426118	+	0.904668i	$0.359881\pi$
$128$	1.67095e7	0.704253
$129$	0	0
$130$	3.18399e7	1.27107
$131$	8.56750e6	0.332970	0.166485	−	0.986044i	$-0.446758\pi$
$131$	8.56750e6	0.332970	0.166485	+	0.986044i	$0.446758\pi$
$132$	0	0
$133$	1.81457e7	0.668795
$134$	−3.26361e6	−0.117174
$135$	0	0
$136$	−3.86493e7	−1.31752
$137$	5.56139e7	1.84783	0.923913	−	0.382603i	$-0.124972\pi$
$137$	5.56139e7	1.84783	0.923913	+	0.382603i	$0.124972\pi$
$138$	0	0
$139$	2.77989e7	0.877963	0.438981	−	0.898496i	$-0.355339\pi$
$139$	2.77989e7	0.877963	0.438981	+	0.898496i	$0.355339\pi$
$140$	−3.95032e6	−0.121670
$141$	0	0
$142$	−2.92002e7	−0.855808
$143$	−4.09107e7	−1.16993
$144$	0	0
$145$	−5.57556e7	−1.51880
$146$	2.77426e7	0.737756
$147$	0	0
$148$	1.08235e7	0.274444
$149$	−7.78371e7	−1.92768	−0.963839	−	0.266485i	$-0.914138\pi$
$149$	−7.78371e7	−1.92768	−0.963839	+	0.266485i	$0.914138\pi$
$150$	0	0
$151$	1.04632e7	0.247312	0.123656	−	0.992325i	$-0.460538\pi$
$151$	1.04632e7	0.247312	0.123656	+	0.992325i	$0.460538\pi$
$152$	4.99270e7	1.15314
$153$	0	0
$154$	−2.74089e7	−0.604740
$155$	−8.12297e7	−1.75208
$156$	0	0
$157$	−6.57814e7	−1.35661	−0.678304	−	0.734781i	$-0.737285\pi$
$157$	−6.57814e7	−1.35661	−0.678304	+	0.734781i	$0.737285\pi$
$158$	1.14504e7	0.230951
$159$	0	0
$160$	−2.02694e7	−0.391221
$161$	4.60677e7	0.869974
$162$	0	0
$163$	−3.00965e7	−0.544327	−0.272164	−	0.962251i	$-0.587739\pi$
$163$	−3.00965e7	−0.544327	−0.272164	+	0.962251i	$0.587739\pi$
$164$	−7.07009e6	−0.125162
$165$	0	0
$166$	6.31752e7	1.07194
$167$	7.90997e7	1.31422	0.657109	−	0.753796i	$-0.271779\pi$
$167$	7.90997e7	1.31422	0.657109	+	0.753796i	$0.271779\pi$
$168$	0	0
$169$	1.24377e7	0.198215
$170$	−9.22722e7	−1.44045
$171$	0	0
$172$	9.30224e6	0.139392
$173$	−2.81224e7	−0.412944	−0.206472	−	0.978453i	$-0.566198\pi$
$173$	−2.81224e7	−0.412944	−0.206472	+	0.978453i	$0.566198\pi$
$174$	0	0
$175$	−2.61182e7	−0.368391
$176$	−6.33359e7	−0.875700
$177$	0	0
$178$	−3.41528e7	−0.453896
$179$	3.04834e7	0.397263	0.198631	−	0.980074i	$-0.436350\pi$
$179$	3.04834e7	0.397263	0.198631	+	0.980074i	$0.436350\pi$
$180$	0	0
$181$	2.98803e6	0.0374550	0.0187275	−	0.999825i	$-0.494039\pi$
$181$	2.98803e6	0.0374550	0.0187275	+	0.999825i	$0.494039\pi$
$182$	5.03724e7	0.619360
$183$	0	0
$184$	1.26753e8	1.50002
$185$	1.91219e8	2.22039
$186$	0	0
$187$	1.18559e8	1.32584
$188$	1.66115e7	0.182329
$189$	0	0
$190$	1.19197e8	1.26074
$191$	3.39428e7	0.352477	0.176238	−	0.984348i	$-0.443607\pi$
$191$	3.39428e7	0.352477	0.176238	+	0.984348i	$0.443607\pi$
$192$	0	0
$193$	−1.57244e8	−1.57443	−0.787214	−	0.616680i	$-0.788477\pi$
$193$	−1.57244e8	−1.57443	−0.787214	+	0.616680i	$0.788477\pi$
$194$	−3.09626e7	−0.304461
$195$	0	0
$196$	1.02212e7	0.0969633
$197$	−6.30072e7	−0.587163	−0.293581	−	0.955934i	$-0.594847\pi$
$197$	−6.30072e7	−0.587163	−0.293581	+	0.955934i	$0.594847\pi$
$198$	0	0
$199$	1.92815e8	1.73443	0.867213	−	0.497937i	$-0.165909\pi$
$199$	1.92815e8	1.73443	0.867213	+	0.497937i	$0.165909\pi$
$200$	−7.18628e7	−0.635184
$201$	0	0
$202$	−2.85517e7	−0.243727
$203$	−8.82084e7	−0.740072
$204$	0	0
$205$	−1.24907e8	−1.01262
$206$	−6.10142e7	−0.486290
$207$	0	0
$208$	1.16400e8	0.896870
$209$	−1.53154e8	−1.16043
$210$	0	0
$211$	−1.27805e8	−0.936610	−0.468305	−	0.883567i	$-0.655135\pi$
$211$	−1.27805e8	−0.936610	−0.468305	+	0.883567i	$0.655135\pi$
$212$	2.05244e7	0.147943
$213$	0	0
$214$	6.07792e7	0.423942
$215$	1.64342e8	1.12775
$216$	0	0
$217$	−1.28510e8	−0.853742
$218$	9.29793e7	0.607839
$219$	0	0
$220$	3.33418e7	0.211110
$221$	−2.17890e8	−1.35789
$222$	0	0
$223$	−6.65653e7	−0.401958	−0.200979	−	0.979596i	$-0.564412\pi$
$223$	−6.65653e7	−0.401958	−0.200979	+	0.979596i	$0.564412\pi$
$224$	−3.20673e7	−0.190632
$225$	0	0
$226$	−1.35792e8	−0.782519
$227$	1.46415e8	0.830795	0.415397	−	0.909640i	$-0.363643\pi$
$227$	1.46415e8	0.830795	0.415397	+	0.909640i	$0.363643\pi$
$228$	0	0
$229$	−2.55728e8	−1.40719	−0.703597	−	0.710599i	$-0.748424\pi$
$229$	−2.55728e8	−1.40719	−0.703597	+	0.710599i	$0.748424\pi$
$230$	3.02613e8	1.63999
$231$	0	0
$232$	−2.42701e8	−1.27604
$233$	5.51954e7	0.285862	0.142931	−	0.989733i	$-0.454347\pi$
$233$	5.51954e7	0.285862	0.142931	+	0.989733i	$0.454347\pi$
$234$	0	0
$235$	2.93474e8	1.47513
$236$	1.57053e7	0.0777774
$237$	0	0
$238$	−1.45979e8	−0.701896
$239$	−1.82590e8	−0.865134	−0.432567	−	0.901602i	$-0.642392\pi$
$239$	−1.82590e8	−0.865134	−0.432567	+	0.901602i	$0.642392\pi$
$240$	0	0
$241$	1.72443e7	0.0793571	0.0396786	−	0.999212i	$-0.487367\pi$
$241$	1.72443e7	0.0793571	0.0396786	+	0.999212i	$0.487367\pi$
$242$	2.88216e7	0.130727
$243$	0	0
$244$	2.75546e6	0.0121431
$245$	1.80578e8	0.784482
$246$	0	0
$247$	2.81469e8	1.18848
$248$	−3.53588e8	−1.47203
$249$	0	0
$250$	1.15308e8	0.466735
$251$	−4.88338e7	−0.194923	−0.0974613	−	0.995239i	$-0.531072\pi$
$251$	−4.88338e7	−0.194923	−0.0974613	+	0.995239i	$0.531072\pi$
$252$	0	0
$253$	−3.88824e8	−1.50949
$254$	2.04449e8	0.782828
$255$	0	0
$256$	−1.22597e8	−0.456710
$257$	8.21424e7	0.301857	0.150929	−	0.988545i	$-0.451774\pi$
$257$	8.21424e7	0.301857	0.150929	+	0.988545i	$0.451774\pi$
$258$	0	0
$259$	3.02518e8	1.08194
$260$	−6.12759e7	−0.216214
$261$	0	0
$262$	8.90361e7	0.305852
$263$	−2.07617e8	−0.703748	−0.351874	−	0.936047i	$-0.614455\pi$
$263$	−2.07617e8	−0.703748	−0.351874	+	0.936047i	$0.614455\pi$
$264$	0	0
$265$	3.62603e8	1.19693
$266$	1.88576e8	0.614327
$267$	0	0
$268$	6.28082e6	0.0199317
$269$	−3.42054e8	−1.07143	−0.535713	−	0.844400i	$-0.679957\pi$
$269$	−3.42054e8	−1.07143	−0.535713	+	0.844400i	$0.679957\pi$
$270$	0	0
$271$	6.74035e7	0.205726	0.102863	−	0.994696i	$-0.467200\pi$
$271$	6.74035e7	0.205726	0.102863	+	0.994696i	$0.467200\pi$
$272$	−3.37326e8	−1.01639
$273$	0	0
$274$	5.77956e8	1.69734
$275$	2.20444e8	0.639196
$276$	0	0
$277$	2.08092e8	0.588270	0.294135	−	0.955764i	$-0.404968\pi$
$277$	2.08092e8	0.588270	0.294135	+	0.955764i	$0.404968\pi$
$278$	2.88895e8	0.806460
$279$	0	0
$280$	−3.03792e8	−0.827033
$281$	−2.54709e8	−0.684814	−0.342407	−	0.939552i	$-0.611242\pi$
$281$	−2.54709e8	−0.684814	−0.342407	+	0.939552i	$0.611242\pi$
$282$	0	0
$283$	1.82076e8	0.477529	0.238764	−	0.971078i	$-0.423258\pi$
$283$	1.82076e8	0.477529	0.238764	+	0.971078i	$0.423258\pi$
$284$	5.61958e7	0.145576
$285$	0	0
$286$	−4.25156e8	−1.07465
$287$	−1.97609e8	−0.493424
$288$	0	0
$289$	2.21108e8	0.538842
$290$	−5.79430e8	−1.39511
$291$	0	0
$292$	−5.33907e7	−0.125495
$293$	5.44321e8	1.26421	0.632104	−	0.774884i	$-0.282192\pi$
$293$	5.44321e8	1.26421	0.632104	+	0.774884i	$0.282192\pi$
$294$	0	0
$295$	2.77464e8	0.629258
$296$	8.32363e8	1.86549
$297$	0	0
$298$	−8.08907e8	−1.77069
$299$	7.14586e8	1.54598
$300$	0	0
$301$	2.59998e8	0.549524
$302$	1.08737e8	0.227171
$303$	0	0
$304$	4.35756e8	0.889583
$305$	4.86805e7	0.0982439
$306$	0	0
$307$	−5.34564e8	−1.05442	−0.527212	−	0.849734i	$-0.676763\pi$
$307$	−5.34564e8	−1.05442	−0.527212	+	0.849734i	$0.676763\pi$
$308$	5.27484e7	0.102868
$309$	0	0
$310$	−8.44163e8	−1.60939
$311$	−3.99687e8	−0.753457	−0.376729	−	0.926324i	$-0.622951\pi$
$311$	−3.99687e8	−0.753457	−0.376729	+	0.926324i	$0.622951\pi$
$312$	0	0
$313$	−9.13258e8	−1.68340	−0.841702	−	0.539943i	$-0.818446\pi$
$313$	−9.13258e8	−1.68340	−0.841702	+	0.539943i	$0.818446\pi$
$314$	−6.83621e8	−1.24612
$315$	0	0
$316$	−2.20363e7	−0.0392856
$317$	4.10657e8	0.724055	0.362027	−	0.932167i	$-0.382085\pi$
$317$	4.10657e8	0.724055	0.362027	+	0.932167i	$0.382085\pi$
$318$	0	0
$319$	7.44502e8	1.28410
$320$	−8.17778e8	−1.39512
$321$	0	0
$322$	4.78750e8	0.799122
$323$	−8.15699e8	−1.34686
$324$	0	0
$325$	−4.05135e8	−0.654648
$326$	−3.12772e8	−0.499996
$327$	0	0
$328$	−5.43712e8	−0.850766
$329$	4.64291e8	0.718794
$330$	0	0
$331$	−9.29745e8	−1.40918	−0.704589	−	0.709616i	$-0.748869\pi$
$331$	−9.29745e8	−1.40918	−0.704589	+	0.709616i	$0.748869\pi$
$332$	−1.21581e8	−0.182340
$333$	0	0
$334$	8.22028e8	1.20719
$335$	1.10963e8	0.161258
$336$	0	0
$337$	−5.73925e8	−0.816866	−0.408433	−	0.912788i	$-0.633925\pi$
$337$	−5.73925e8	−0.816866	−0.408433	+	0.912788i	$0.633925\pi$
$338$	1.29257e8	0.182072
$339$	0	0
$340$	1.77578e8	0.245026
$341$	1.08465e9	1.48133
$342$	0	0
$343$	7.46044e8	0.998240
$344$	7.15371e8	0.947494
$345$	0	0
$346$	−2.92257e8	−0.379313
$347$	−1.09295e9	−1.40427	−0.702133	−	0.712046i	$-0.747769\pi$
$347$	−1.09295e9	−1.40427	−0.702133	+	0.712046i	$0.747769\pi$
$348$	0	0
$349$	7.28528e8	0.917397	0.458699	−	0.888592i	$-0.348316\pi$
$349$	7.28528e8	0.917397	0.458699	+	0.888592i	$0.348316\pi$
$350$	−2.71428e8	−0.338389
$351$	0	0
$352$	2.70657e8	0.330765
$353$	−5.86912e8	−0.710169	−0.355084	−	0.934834i	$-0.615548\pi$
$353$	−5.86912e8	−0.710169	−0.355084	+	0.934834i	$0.615548\pi$
$354$	0	0
$355$	9.92806e8	1.17778
$356$	6.57272e7	0.0772094
$357$	0	0
$358$	3.16793e8	0.364909
$359$	1.19985e9	1.36866	0.684330	−	0.729173i	$-0.260095\pi$
$359$	1.19985e9	1.36866	0.684330	+	0.729173i	$0.260095\pi$
$360$	0	0
$361$	1.59845e8	0.178823
$362$	3.10525e7	0.0344046
$363$	0	0
$364$	−9.69418e7	−0.105355
$365$	−9.43250e8	−1.01532
$366$	0	0
$367$	−1.31551e9	−1.38919	−0.694595	−	0.719401i	$-0.744416\pi$
$367$	−1.31551e9	−1.38919	−0.694595	+	0.719401i	$0.744416\pi$
$368$	1.10628e9	1.15718
$369$	0	0
$370$	1.98720e9	2.03956
$371$	5.73657e8	0.583235
$372$	0	0
$373$	−1.07725e9	−1.07482	−0.537411	−	0.843321i	$-0.680598\pi$
$373$	−1.07725e9	−1.07482	−0.537411	+	0.843321i	$0.680598\pi$
$374$	1.23211e9	1.21786
$375$	0	0
$376$	1.27747e9	1.23935
$377$	−1.36826e9	−1.31514
$378$	0	0
$379$	7.73954e8	0.730261	0.365130	−	0.930956i	$-0.381024\pi$
$379$	7.73954e8	0.730261	0.365130	+	0.930956i	$0.381024\pi$
$380$	−2.29394e8	−0.214457
$381$	0	0
$382$	3.52744e8	0.323771
$383$	1.27039e9	1.15542	0.577710	−	0.816242i	$-0.303946\pi$
$383$	1.27039e9	1.15542	0.577710	+	0.816242i	$0.303946\pi$
$384$	0	0
$385$	9.31902e8	0.832258
$386$	−1.63413e9	−1.44620
$387$	0	0
$388$	5.95876e7	0.0517899
$389$	−1.08638e9	−0.935743	−0.467871	−	0.883797i	$-0.654979\pi$
$389$	−1.08638e9	−0.935743	−0.467871	+	0.883797i	$0.654979\pi$
$390$	0	0
$391$	−2.07087e9	−1.75200
$392$	7.86045e8	0.659092
$393$	0	0
$394$	−6.54790e8	−0.539344
$395$	−3.89314e8	−0.317841
$396$	0	0
$397$	9.52195e7	0.0763764	0.0381882	−	0.999271i	$-0.487841\pi$
$397$	9.52195e7	0.0763764	0.0381882	+	0.999271i	$0.487841\pi$
$398$	2.00380e9	1.59317
$399$	0	0
$400$	−6.27210e8	−0.490007
$401$	−1.44316e9	−1.11766	−0.558828	−	0.829284i	$-0.688749\pi$
$401$	−1.44316e9	−1.11766	−0.558828	+	0.829284i	$0.688749\pi$
$402$	0	0
$403$	−1.99339e9	−1.51714
$404$	5.49479e7	0.0414587
$405$	0	0
$406$	−9.16688e8	−0.679799
$407$	−2.55333e9	−1.87727
$408$	0	0
$409$	2.38724e9	1.72530	0.862650	−	0.505801i	$-0.168803\pi$
$409$	2.38724e9	1.72530	0.862650	+	0.505801i	$0.168803\pi$
$410$	−1.29807e9	−0.930153
$411$	0	0
$412$	1.17422e8	0.0827196
$413$	4.38962e8	0.306621
$414$	0	0
$415$	−2.14796e9	−1.47522
$416$	−4.97416e8	−0.338761
$417$	0	0
$418$	−1.59163e9	−1.06592
$419$	2.30536e9	1.53105	0.765527	−	0.643404i	$-0.222478\pi$
$419$	2.30536e9	1.53105	0.765527	+	0.643404i	$0.222478\pi$
$420$	0	0
$421$	−1.20938e9	−0.789904	−0.394952	−	0.918702i	$-0.629239\pi$
$421$	−1.20938e9	−0.789904	−0.394952	+	0.918702i	$0.629239\pi$
$422$	−1.32819e9	−0.860332
$423$	0	0
$424$	1.57839e9	1.00562
$425$	1.17408e9	0.741887
$426$	0	0
$427$	7.70151e7	0.0478716
$428$	−1.16970e8	−0.0721140
$429$	0	0
$430$	1.70789e9	1.03591
$431$	2.87283e9	1.72838	0.864189	−	0.503167i	$-0.167832\pi$
$431$	2.87283e9	1.72838	0.864189	+	0.503167i	$0.167832\pi$
$432$	0	0
$433$	−1.79302e9	−1.06140	−0.530699	−	0.847560i	$-0.678070\pi$
$433$	−1.79302e9	−1.06140	−0.530699	+	0.847560i	$0.678070\pi$
$434$	−1.33551e9	−0.784212
$435$	0	0
$436$	−1.78939e8	−0.103396
$437$	2.67514e9	1.53342
$438$	0	0
$439$	−3.10244e8	−0.175016	−0.0875081	−	0.996164i	$-0.527890\pi$
$439$	−3.10244e8	−0.175016	−0.0875081	+	0.996164i	$0.527890\pi$
$440$	2.56408e9	1.43499
$441$	0	0
$442$	−2.26438e9	−1.24730
$443$	−6.12654e8	−0.334813	−0.167406	−	0.985888i	$-0.553539\pi$
$443$	−6.12654e8	−0.334813	−0.167406	+	0.985888i	$0.553539\pi$
$444$	0	0
$445$	1.16120e9	0.624663
$446$	−6.91766e8	−0.369222
$447$	0	0
$448$	−1.29377e9	−0.679803
$449$	−8.72232e8	−0.454747	−0.227374	−	0.973808i	$-0.573014\pi$
$449$	−8.72232e8	−0.454747	−0.227374	+	0.973808i	$0.573014\pi$
$450$	0	0
$451$	1.66787e9	0.856140
$452$	2.61332e8	0.133109
$453$	0	0
$454$	1.52158e9	0.763134
$455$	−1.71266e9	−0.852377
$456$	0	0
$457$	3.90056e9	1.91170	0.955851	−	0.293851i	$-0.0949369\pi$
$457$	3.90056e9	1.91170	0.955851	+	0.293851i	$0.0949369\pi$
$458$	−2.65760e9	−1.29259
$459$	0	0
$460$	−5.82379e8	−0.278967
$461$	2.10841e9	1.00231	0.501155	−	0.865358i	$-0.332909\pi$
$461$	2.10841e9	1.00231	0.501155	+	0.865358i	$0.332909\pi$
$462$	0	0
$463$	1.44349e9	0.675895	0.337948	−	0.941165i	$-0.390267\pi$
$463$	1.44349e9	0.675895	0.337948	+	0.941165i	$0.390267\pi$
$464$	−2.11826e9	−0.984390
$465$	0	0
$466$	5.73607e8	0.262581
$467$	−2.03696e9	−0.925493	−0.462746	−	0.886491i	$-0.653136\pi$
$467$	−2.03696e9	−0.925493	−0.462746	+	0.886491i	$0.653136\pi$
$468$	0	0
$469$	1.75549e8	0.0785766
$470$	3.04987e9	1.35500
$471$	0	0
$472$	1.20778e9	0.528679
$473$	−2.19445e9	−0.953479
$474$	0	0
$475$	−1.51668e9	−0.649329
$476$	2.80938e8	0.119395
$477$	0	0
$478$	−1.89753e9	−0.794676
$479$	−7.00832e8	−0.291367	−0.145683	−	0.989331i	$-0.546538\pi$
$479$	−7.00832e8	−0.291367	−0.145683	+	0.989331i	$0.546538\pi$
$480$	0	0
$481$	4.69255e9	1.92265
$482$	1.79208e8	0.0728942
$483$	0	0
$484$	−5.54671e7	−0.0222370
$485$	1.05273e9	0.419006
$486$	0	0
$487$	−2.15479e8	−0.0845384	−0.0422692	−	0.999106i	$-0.513459\pi$
$487$	−2.15479e8	−0.0845384	−0.0422692	+	0.999106i	$0.513459\pi$
$488$	2.11903e8	0.0825407
$489$	0	0
$490$	1.87662e9	0.720593
$491$	−1.15709e9	−0.441146	−0.220573	−	0.975370i	$-0.570793\pi$
$491$	−1.15709e9	−0.441146	−0.220573	+	0.975370i	$0.570793\pi$
$492$	0	0
$493$	3.96521e9	1.49040
$494$	2.92512e9	1.09169
$495$	0	0
$496$	−3.08607e9	−1.13559
$497$	1.57067e9	0.573902
$498$	0	0
$499$	−3.24480e9	−1.16906	−0.584529	−	0.811373i	$-0.698721\pi$
$499$	−3.24480e9	−1.16906	−0.584529	+	0.811373i	$0.698721\pi$
$500$	−2.21911e8	−0.0793932
$501$	0	0
$502$	−5.07495e8	−0.179048
$503$	−1.71289e9	−0.600125	−0.300062	−	0.953920i	$-0.597008\pi$
$503$	−1.71289e9	−0.600125	−0.300062	+	0.953920i	$0.597008\pi$
$504$	0	0
$505$	9.70759e8	0.335422
$506$	−4.04078e9	−1.38656
$507$	0	0
$508$	−3.93462e8	−0.133162
$509$	−2.27447e8	−0.0764485	−0.0382242	−	0.999269i	$-0.512170\pi$
$509$	−2.27447e8	−0.0764485	−0.0382242	+	0.999269i	$0.512170\pi$
$510$	0	0
$511$	−1.49227e9	−0.494737
$512$	−3.41288e9	−1.12377
$513$	0	0
$514$	8.53649e8	0.277273
$515$	2.07448e9	0.669244
$516$	0	0
$517$	−3.91874e9	−1.24718
$518$	3.14386e9	0.993823
$519$	0	0
$520$	−4.71231e9	−1.46968
$521$	4.19551e9	1.29973	0.649864	−	0.760050i	$-0.274826\pi$
$521$	4.19551e9	1.29973	0.649864	+	0.760050i	$0.274826\pi$
$522$	0	0
$523$	1.77394e9	0.542228	0.271114	−	0.962547i	$-0.412608\pi$
$523$	1.77394e9	0.542228	0.271114	+	0.962547i	$0.412608\pi$
$524$	−1.71350e8	−0.0520265
$525$	0	0
$526$	−2.15762e9	−0.646434
$527$	5.77686e9	1.71931
$528$	0	0
$529$	3.38674e9	0.994689
$530$	3.76828e9	1.09945
$531$	0	0
$532$	−3.62914e8	−0.104499
$533$	−3.06524e9	−0.876837
$534$	0	0
$535$	−2.06649e9	−0.583439
$536$	4.83014e8	0.135482
$537$	0	0
$538$	−3.55473e9	−0.984168
$539$	−2.41124e9	−0.663255
$540$	0	0
$541$	−3.74560e9	−1.01702	−0.508511	−	0.861055i	$-0.669804\pi$
$541$	−3.74560e9	−1.01702	−0.508511	+	0.861055i	$0.669804\pi$
$542$	7.00478e8	0.188972
$543$	0	0
$544$	1.44151e9	0.383904
$545$	−3.16130e9	−0.836522
$546$	0	0
$547$	−2.14352e9	−0.559980	−0.279990	−	0.960003i	$-0.590331\pi$
$547$	−2.14352e9	−0.559980	−0.279990	+	0.960003i	$0.590331\pi$
$548$	−1.11228e9	−0.288723
$549$	0	0
$550$	2.29092e9	0.587139
$551$	−5.12224e9	−1.30446
$552$	0	0
$553$	−6.15915e8	−0.154875
$554$	2.16256e9	0.540360
$555$	0	0
$556$	−5.55978e8	−0.137182
$557$	−3.54423e9	−0.869019	−0.434509	−	0.900667i	$-0.643078\pi$
$557$	−3.54423e9	−0.869019	−0.434509	+	0.900667i	$0.643078\pi$
$558$	0	0
$559$	4.03299e9	0.976530
$560$	−2.65146e9	−0.638008
$561$	0	0
$562$	−2.64702e9	−0.629042
$563$	−1.35364e9	−0.319686	−0.159843	−	0.987142i	$-0.551099\pi$
$563$	−1.35364e9	−0.319686	−0.159843	+	0.987142i	$0.551099\pi$
$564$	0	0
$565$	4.61693e9	1.07692
$566$	1.89219e9	0.438638
$567$	0	0
$568$	4.32163e9	0.989528
$569$	1.11107e9	0.252843	0.126421	−	0.991977i	$-0.459651\pi$
$569$	1.11107e9	0.252843	0.126421	+	0.991977i	$0.459651\pi$
$570$	0	0
$571$	−1.85765e9	−0.417578	−0.208789	−	0.977961i	$-0.566952\pi$
$571$	−1.85765e9	−0.417578	−0.208789	+	0.977961i	$0.566952\pi$
$572$	8.18214e8	0.182802
$573$	0	0
$574$	−2.05361e9	−0.453239
$575$	−3.85049e9	−0.844653
$576$	0	0
$577$	3.26056e9	0.706606	0.353303	−	0.935509i	$-0.385059\pi$
$577$	3.26056e9	0.706606	0.353303	+	0.935509i	$0.385059\pi$
$578$	2.29782e9	0.494958
$579$	0	0
$580$	1.11511e9	0.237313
$581$	−3.39818e9	−0.718837
$582$	0	0
$583$	−4.84181e9	−1.01197
$584$	−4.10591e9	−0.853031
$585$	0	0
$586$	5.65675e9	1.16125
$587$	8.49773e9	1.73408	0.867041	−	0.498237i	$-0.166019\pi$
$587$	8.49773e9	1.73408	0.867041	+	0.498237i	$0.166019\pi$
$588$	0	0
$589$	−7.46252e9	−1.50481
$590$	2.88349e9	0.578011
$591$	0	0
$592$	7.26476e9	1.43911
$593$	4.65379e9	0.916464	0.458232	−	0.888832i	$-0.348483\pi$
$593$	4.65379e9	0.916464	0.458232	+	0.888832i	$0.348483\pi$
$594$	0	0
$595$	4.96330e9	0.965965
$596$	1.55674e9	0.301200
$597$	0	0
$598$	7.42619e9	1.42008
$599$	−1.19483e9	−0.227149	−0.113575	−	0.993529i	$-0.536230\pi$
$599$	−1.19483e9	−0.227149	−0.113575	+	0.993529i	$0.536230\pi$
$600$	0	0
$601$	7.18932e9	1.35091	0.675457	−	0.737400i	$-0.263947\pi$
$601$	7.18932e9	1.35091	0.675457	+	0.737400i	$0.263947\pi$
$602$	2.70197e9	0.504770
$603$	0	0
$604$	−2.09264e8	−0.0386426
$605$	−9.79933e8	−0.179909
$606$	0	0
$607$	−3.73407e8	−0.0677676	−0.0338838	−	0.999426i	$-0.510788\pi$
$607$	−3.73407e8	−0.0677676	−0.0338838	+	0.999426i	$0.510788\pi$
$608$	−1.86214e9	−0.336009
$609$	0	0
$610$	5.05902e8	0.0902428
$611$	7.20191e9	1.27733
$612$	0	0
$613$	−2.57761e9	−0.451965	−0.225982	−	0.974131i	$-0.572559\pi$
$613$	−2.57761e9	−0.451965	−0.225982	+	0.974131i	$0.572559\pi$
$614$	−5.55535e9	−0.968551
$615$	0	0
$616$	4.05652e9	0.699231
$617$	8.58580e9	1.47158	0.735788	−	0.677212i	$-0.236812\pi$
$617$	8.58580e9	1.47158	0.735788	+	0.677212i	$0.236812\pi$
$618$	0	0
$619$	−7.91264e9	−1.34092	−0.670462	−	0.741944i	$-0.733904\pi$
$619$	−7.91264e9	−1.34092	−0.670462	+	0.741944i	$0.733904\pi$
$620$	1.62459e9	0.273762
$621$	0	0
$622$	−4.15367e9	−0.692095
$623$	1.83707e9	0.304382
$624$	0	0
$625$	−7.57071e9	−1.24039
$626$	−9.49086e9	−1.54631
$627$	0	0
$628$	1.31563e9	0.211970
$629$	−1.35990e10	−2.17886
$630$	0	0
$631$	1.12326e9	0.177983	0.0889916	−	0.996032i	$-0.471636\pi$
$631$	1.12326e9	0.177983	0.0889916	+	0.996032i	$0.471636\pi$
$632$	−1.69466e9	−0.267038
$633$	0	0
$634$	4.26767e9	0.665087
$635$	−6.95126e9	−1.07735
$636$	0	0
$637$	4.43142e9	0.679289
$638$	7.73709e9	1.17952
$639$	0	0
$640$	−5.90411e9	−0.890275
$641$	4.61389e9	0.691933	0.345967	−	0.938247i	$-0.387551\pi$
$641$	4.61389e9	0.691933	0.345967	+	0.938247i	$0.387551\pi$
$642$	0	0
$643$	−2.82326e8	−0.0418805	−0.0209402	−	0.999781i	$-0.506666\pi$
$643$	−2.82326e8	−0.0418805	−0.0209402	+	0.999781i	$0.506666\pi$
$644$	−9.21355e8	−0.135933
$645$	0	0
$646$	−8.47700e9	−1.23717
$647$	1.09978e10	1.59639	0.798197	−	0.602396i	$-0.205787\pi$
$647$	1.09978e10	1.59639	0.798197	+	0.602396i	$0.205787\pi$
$648$	0	0
$649$	−3.70496e9	−0.532018
$650$	−4.21029e9	−0.601333
$651$	0	0
$652$	6.01931e8	0.0850511
$653$	1.55218e9	0.218145	0.109073	−	0.994034i	$-0.465212\pi$
$653$	1.55218e9	0.218145	0.109073	+	0.994034i	$0.465212\pi$
$654$	0	0
$655$	−3.02723e9	−0.420921
$656$	−4.74545e9	−0.656317
$657$	0	0
$658$	4.82505e9	0.660254
$659$	8.20063e9	1.11622	0.558108	−	0.829768i	$-0.311527\pi$
$659$	8.20063e9	1.11622	0.558108	+	0.829768i	$0.311527\pi$
$660$	0	0
$661$	−8.23211e8	−0.110868	−0.0554340	−	0.998462i	$-0.517654\pi$
$661$	−8.23211e8	−0.110868	−0.0554340	+	0.998462i	$0.517654\pi$
$662$	−9.66219e9	−1.29441
$663$	0	0
$664$	−9.34994e9	−1.23943
$665$	−6.41157e9	−0.845452
$666$	0	0
$667$	−1.30042e10	−1.69685
$668$	−1.58199e9	−0.205347
$669$	0	0
$670$	1.15316e9	0.148125
$671$	−6.50028e8	−0.0830621
$672$	0	0
$673$	1.46721e9	0.185541	0.0927707	−	0.995688i	$-0.470428\pi$
$673$	1.46721e9	0.185541	0.0927707	+	0.995688i	$0.470428\pi$
$674$	−5.96440e9	−0.750339
$675$	0	0
$676$	−2.48754e8	−0.0309712
$677$	−3.49168e9	−0.432487	−0.216244	−	0.976339i	$-0.569381\pi$
$677$	−3.49168e9	−0.432487	−0.216244	+	0.976339i	$0.569381\pi$
$678$	0	0
$679$	1.66547e9	0.204171
$680$	1.36563e10	1.66553
$681$	0	0
$682$	1.12721e10	1.36069
$683$	−1.51044e10	−1.81398	−0.906988	−	0.421156i	$-0.861624\pi$
$683$	−1.51044e10	−1.81398	−0.906988	+	0.421156i	$0.861624\pi$
$684$	0	0
$685$	−1.96505e10	−2.33591
$686$	7.75312e9	0.916942
$687$	0	0
$688$	6.24366e9	0.730937
$689$	8.89835e9	1.03644
$690$	0	0
$691$	1.26280e10	1.45601	0.728003	−	0.685574i	$-0.240449\pi$
$691$	1.26280e10	1.45601	0.728003	+	0.685574i	$0.240449\pi$
$692$	5.62448e8	0.0645225
$693$	0	0
$694$	−1.13583e10	−1.28990
$695$	−9.82242e9	−1.10987
$696$	0	0
$697$	8.88307e9	0.993685
$698$	7.57109e9	0.842683
$699$	0	0
$700$	5.22363e8	0.0575611
$701$	1.52554e10	1.67267	0.836336	−	0.548217i	$-0.184693\pi$
$701$	1.52554e10	1.67267	0.836336	+	0.548217i	$0.184693\pi$
$702$	0	0
$703$	1.75671e10	1.90703
$704$	1.09197e10	1.17953
$705$	0	0
$706$	−6.09937e9	−0.652331
$707$	1.53579e9	0.163442
$708$	0	0
$709$	−1.75998e10	−1.85458	−0.927292	−	0.374339i	$-0.877870\pi$
$709$	−1.75998e10	−1.85458	−0.927292	+	0.374339i	$0.877870\pi$
$710$	1.03175e10	1.08186
$711$	0	0
$712$	5.05462e9	0.524818
$713$	−1.89456e10	−1.95747
$714$	0	0
$715$	1.44553e10	1.47896
$716$	−6.09668e8	−0.0620723
$717$	0	0
$718$	1.24692e10	1.25719
$719$	1.01872e10	1.02212	0.511061	−	0.859544i	$-0.329253\pi$
$719$	1.01872e10	1.02212	0.511061	+	0.859544i	$0.329253\pi$
$720$	0	0
$721$	3.28194e9	0.326105
$722$	1.66116e9	0.164259
$723$	0	0
$724$	−5.97606e7	−0.00585234
$725$	7.37274e9	0.718531
$726$	0	0
$727$	7.36507e8	0.0710896	0.0355448	−	0.999368i	$-0.488683\pi$
$727$	7.36507e8	0.0710896	0.0355448	+	0.999368i	$0.488683\pi$
$728$	−7.45512e9	−0.716135
$729$	0	0
$730$	−9.80254e9	−0.932628
$731$	−1.16876e10	−1.10666
$732$	0	0
$733$	1.65404e9	0.155125	0.0775625	−	0.996987i	$-0.475286\pi$
$733$	1.65404e9	0.155125	0.0775625	+	0.996987i	$0.475286\pi$
$734$	−1.36711e10	−1.27605
$735$	0	0
$736$	−4.72755e9	−0.437083
$737$	−1.48168e9	−0.136338
$738$	0	0
$739$	1.56333e10	1.42493	0.712467	−	0.701706i	$-0.247578\pi$
$739$	1.56333e10	1.42493	0.712467	+	0.701706i	$0.247578\pi$
$740$	−3.82437e9	−0.346936
$741$	0	0
$742$	5.96161e9	0.535735
$743$	−1.15063e10	−1.02914	−0.514571	−	0.857448i	$-0.672049\pi$
$743$	−1.15063e10	−1.02914	−0.514571	+	0.857448i	$0.672049\pi$
$744$	0	0
$745$	2.75028e10	2.43686
$746$	−1.11951e10	−0.987287
$747$	0	0
$748$	−2.37119e9	−0.207162
$749$	−3.26930e9	−0.284294
$750$	0	0
$751$	1.12163e10	0.966294	0.483147	−	0.875539i	$-0.339494\pi$
$751$	1.12163e10	0.966294	0.483147	+	0.875539i	$0.339494\pi$
$752$	1.11496e10	0.956088
$753$	0	0
$754$	−1.42193e10	−1.20803
$755$	−3.69706e9	−0.312638
$756$	0	0
$757$	−1.70722e10	−1.43039	−0.715194	−	0.698926i	$-0.753661\pi$
$757$	−1.70722e10	−1.43039	−0.715194	+	0.698926i	$0.753661\pi$
$758$	8.04316e9	0.670787
$759$	0	0
$760$	−1.76411e10	−1.45774
$761$	−2.70766e9	−0.222714	−0.111357	−	0.993780i	$-0.535520\pi$
$761$	−2.70766e9	−0.222714	−0.111357	+	0.993780i	$0.535520\pi$
$762$	0	0
$763$	−5.00134e9	−0.407615
$764$	−6.78856e8	−0.0550745
$765$	0	0
$766$	1.32022e10	1.06132
$767$	6.80902e9	0.544880
$768$	0	0
$769$	−8.21338e9	−0.651298	−0.325649	−	0.945491i	$-0.605583\pi$
$769$	−8.21338e9	−0.651298	−0.325649	+	0.945491i	$0.605583\pi$
$770$	9.68461e9	0.764477
$771$	0	0
$772$	3.14488e9	0.246004
$773$	1.01337e10	0.789115	0.394557	−	0.918871i	$-0.370898\pi$
$773$	1.01337e10	0.789115	0.394557	+	0.918871i	$0.370898\pi$
$774$	0	0
$775$	1.07412e10	0.828893
$776$	4.58247e9	0.352033
$777$	0	0
$778$	−1.12899e10	−0.859535
$779$	−1.14751e10	−0.869713
$780$	0	0
$781$	−1.32569e10	−0.995778
$782$	−2.15211e10	−1.60932
$783$	0	0
$784$	6.86050e9	0.508451
$785$	2.32431e10	1.71495
$786$	0	0
$787$	−7.90806e9	−0.578307	−0.289153	−	0.957283i	$-0.593374\pi$
$787$	−7.90806e9	−0.578307	−0.289153	+	0.957283i	$0.593374\pi$
$788$	1.26014e9	0.0917442
$789$	0	0
$790$	−4.04586e9	−0.291955
$791$	7.30423e9	0.524755
$792$	0	0
$793$	1.19463e9	0.0850701
$794$	9.89550e8	0.0701562
$795$	0	0
$796$	−3.85631e9	−0.271004
$797$	−1.36876e10	−0.957686	−0.478843	−	0.877900i	$-0.658944\pi$
$797$	−1.36876e10	−0.957686	−0.478843	+	0.877900i	$0.658944\pi$
$798$	0	0
$799$	−2.08712e10	−1.44755
$800$	2.68029e9	0.185083
$801$	0	0
$802$	−1.49977e10	−1.02663
$803$	1.25952e10	0.858419
$804$	0	0
$805$	−1.62775e10	−1.09977
$806$	−2.07160e10	−1.39358
$807$	0	0
$808$	4.22566e9	0.281809
$809$	9.65209e9	0.640917	0.320458	−	0.947263i	$-0.396163\pi$
$809$	9.65209e9	0.640917	0.320458	+	0.947263i	$0.396163\pi$
$810$	0	0
$811$	2.93565e10	1.93256	0.966278	−	0.257502i	$-0.0828994\pi$
$811$	2.93565e10	1.93256	0.966278	+	0.257502i	$0.0828994\pi$
$812$	1.76417e9	0.115636
$813$	0	0
$814$	−2.65350e10	−1.72438
$815$	1.06343e10	0.688107
$816$	0	0
$817$	1.50980e10	0.968595
$818$	2.48089e10	1.58479
$819$	0	0
$820$	2.49814e9	0.158222
$821$	−1.46088e9	−0.0921323	−0.0460662	−	0.998938i	$-0.514669\pi$
$821$	−1.46088e9	−0.0921323	−0.0460662	+	0.998938i	$0.514669\pi$
$822$	0	0
$823$	2.29819e10	1.43710	0.718548	−	0.695478i	$-0.244807\pi$
$823$	2.29819e10	1.43710	0.718548	+	0.695478i	$0.244807\pi$
$824$	9.03010e9	0.562273
$825$	0	0
$826$	4.56183e9	0.281649
$827$	−1.26270e10	−0.776300	−0.388150	−	0.921596i	$-0.626886\pi$
$827$	−1.26270e10	−0.776300	−0.388150	+	0.921596i	$0.626886\pi$
$828$	0	0
$829$	−1.63249e10	−0.995198	−0.497599	−	0.867407i	$-0.665785\pi$
$829$	−1.63249e10	−0.995198	−0.497599	+	0.867407i	$0.665785\pi$
$830$	−2.23222e10	−1.35508
$831$	0	0
$832$	−2.00684e10	−1.20804
$833$	−1.28423e10	−0.769811
$834$	0	0
$835$	−2.79490e10	−1.66136
$836$	3.06309e9	0.181317
$837$	0	0
$838$	2.39580e10	1.40636
$839$	−1.16654e10	−0.681918	−0.340959	−	0.940078i	$-0.610752\pi$
$839$	−1.16654e10	−0.681918	−0.340959	+	0.940078i	$0.610752\pi$
$840$	0	0
$841$	7.64994e9	0.443478
$842$	−1.25682e10	−0.725573
$843$	0	0
$844$	2.55610e9	0.146345
$845$	−4.39473e9	−0.250572
$846$	0	0
$847$	−1.55031e9	−0.0876649
$848$	1.37760e10	0.775777
$849$	0	0
$850$	1.22014e10	0.681467
$851$	4.45989e10	2.48068
$852$	0	0
$853$	−1.22702e9	−0.0676906	−0.0338453	−	0.999427i	$-0.510775\pi$
$853$	−1.22702e9	−0.0676906	−0.0338453	+	0.999427i	$0.510775\pi$
$854$	8.00364e8	0.0439729
$855$	0	0
$856$	−8.99532e9	−0.490183
$857$	−2.22932e10	−1.20987	−0.604935	−	0.796275i	$-0.706801\pi$
$857$	−2.22932e10	−1.20987	−0.604935	+	0.796275i	$0.706801\pi$
$858$	0	0
$859$	1.11325e10	0.599263	0.299632	−	0.954055i	$-0.403136\pi$
$859$	1.11325e10	0.599263	0.299632	+	0.954055i	$0.403136\pi$
$860$	−3.28684e9	−0.176211
$861$	0	0
$862$	2.98553e10	1.58762
$863$	−1.07748e10	−0.570653	−0.285327	−	0.958430i	$-0.592102\pi$
$863$	−1.07748e10	−0.570653	−0.285327	+	0.958430i	$0.592102\pi$
$864$	0	0
$865$	9.93673e9	0.522020
$866$	−1.86336e10	−0.974956
$867$	0	0
$868$	2.57019e9	0.133397
$869$	5.19848e9	0.268724
$870$	0	0
$871$	2.72305e9	0.139634
$872$	−1.37609e10	−0.702814
$873$	0	0
$874$	2.78009e10	1.40854
$875$	−6.20240e9	−0.312991
$876$	0	0
$877$	−4.31886e9	−0.216207	−0.108104	−	0.994140i	$-0.534478\pi$
$877$	−4.31886e9	−0.216207	−0.108104	+	0.994140i	$0.534478\pi$
$878$	−3.22415e9	−0.160763
$879$	0	0
$880$	2.23790e10	1.10701
$881$	−2.49223e10	−1.22793	−0.613963	−	0.789335i	$-0.710426\pi$
$881$	−2.49223e10	−1.22793	−0.613963	+	0.789335i	$0.710426\pi$
$882$	0	0
$883$	8.75421e9	0.427912	0.213956	−	0.976843i	$-0.431365\pi$
$883$	8.75421e9	0.427912	0.213956	+	0.976843i	$0.431365\pi$
$884$	4.35780e9	0.212170
$885$	0	0
$886$	−6.36689e9	−0.307545
$887$	−7.18370e8	−0.0345633	−0.0172817	−	0.999851i	$-0.505501\pi$
$887$	−7.18370e8	−0.0345633	−0.0172817	+	0.999851i	$0.505501\pi$
$888$	0	0
$889$	−1.09973e10	−0.524962
$890$	1.20675e10	0.573789
$891$	0	0
$892$	1.33131e9	0.0628059
$893$	2.69613e10	1.26695
$894$	0	0
$895$	−1.07710e10	−0.502197
$896$	−9.34061e9	−0.433808
$897$	0	0
$898$	−9.06451e9	−0.417712
$899$	3.62762e10	1.66519
$900$	0	0
$901$	−2.57874e10	−1.17455
$902$	1.73330e10	0.786415
$903$	0	0
$904$	2.00972e10	0.904788
$905$	−1.05579e9	−0.0473484
$906$	0	0
$907$	2.02975e10	0.903271	0.451635	−	0.892203i	$-0.350841\pi$
$907$	2.02975e10	0.903271	0.451635	+	0.892203i	$0.350841\pi$
$908$	−2.92829e9	−0.129812
$909$	0	0
$910$	−1.77985e10	−0.782959
$911$	2.91617e10	1.27790	0.638951	−	0.769247i	$-0.279368\pi$
$911$	2.91617e10	1.27790	0.638951	+	0.769247i	$0.279368\pi$
$912$	0	0
$913$	2.86816e10	1.24725
$914$	4.05358e10	1.75601
$915$	0	0
$916$	5.11456e9	0.219874
$917$	−4.78923e9	−0.205104
$918$	0	0
$919$	−2.07384e10	−0.881394	−0.440697	−	0.897656i	$-0.645269\pi$
$919$	−2.07384e10	−0.881394	−0.440697	+	0.897656i	$0.645269\pi$
$920$	−4.47867e10	−1.89623
$921$	0	0
$922$	2.19112e10	0.920680
$923$	2.43637e10	1.01985
$924$	0	0
$925$	−2.52854e10	−1.05045
$926$	1.50012e10	0.620849
$927$	0	0
$928$	9.05209e9	0.371819
$929$	−8.26175e9	−0.338078	−0.169039	−	0.985609i	$-0.554066\pi$
$929$	−8.26175e9	−0.338078	−0.169039	+	0.985609i	$0.554066\pi$
$930$	0	0
$931$	1.65896e10	0.673770
$932$	−1.10391e9	−0.0446660
$933$	0	0
$934$	−2.11687e10	−0.850120
$935$	−4.18916e10	−1.67605
$936$	0	0
$937$	−4.78884e10	−1.90170	−0.950849	−	0.309655i	$-0.899786\pi$
$937$	−4.78884e10	−1.90170	−0.950849	+	0.309655i	$0.899786\pi$
$938$	1.82436e9	0.0721772
$939$	0	0
$940$	−5.86947e9	−0.230490
$941$	−2.54037e10	−0.993878	−0.496939	−	0.867786i	$-0.665543\pi$
$941$	−2.54037e10	−0.993878	−0.496939	+	0.867786i	$0.665543\pi$
$942$	0	0
$943$	−2.91327e10	−1.13133
$944$	1.05414e10	0.407845
$945$	0	0
$946$	−2.28054e10	−0.875827
$947$	−4.11954e10	−1.57625	−0.788123	−	0.615518i	$-0.788947\pi$
$947$	−4.11954e10	−1.57625	−0.788123	+	0.615518i	$0.788947\pi$
$948$	0	0
$949$	−2.31476e10	−0.879171
$950$	−1.57618e10	−0.596447
$951$	0	0
$952$	2.16050e10	0.811567
$953$	−1.74074e10	−0.651491	−0.325745	−	0.945458i	$-0.605615\pi$
$953$	−1.74074e10	−0.651491	−0.325745	+	0.945458i	$0.605615\pi$
$954$	0	0
$955$	−1.19933e10	−0.445581
$956$	3.65179e9	0.135177
$957$	0	0
$958$	−7.28326e9	−0.267637
$959$	−3.10881e10	−1.13823
$960$	0	0
$961$	2.53377e10	0.920949
$962$	4.87664e10	1.76607
$963$	0	0
$964$	−3.44886e8	−0.0123996
$965$	5.55603e10	1.99030
$966$	0	0
$967$	−9.30580e9	−0.330949	−0.165474	−	0.986214i	$-0.552916\pi$
$967$	−9.30580e9	−0.330949	−0.165474	+	0.986214i	$0.552916\pi$
$968$	−4.26559e9	−0.151153
$969$	0	0
$970$	1.09403e10	0.384882
$971$	−2.04072e10	−0.715347	−0.357673	−	0.933847i	$-0.616430\pi$
$971$	−2.04072e10	−0.715347	−0.357673	+	0.933847i	$0.616430\pi$
$972$	0	0
$973$	−1.55396e10	−0.540810
$974$	−2.23933e9	−0.0776535
$975$	0	0
$976$	1.84946e9	0.0636754
$977$	−8.37841e9	−0.287429	−0.143715	−	0.989619i	$-0.545905\pi$
$977$	−8.37841e9	−0.287429	−0.143715	+	0.989619i	$0.545905\pi$
$978$	0	0
$979$	−1.55054e10	−0.528133
$980$	−3.61156e9	−0.122575
$981$	0	0
$982$	−1.20249e10	−0.405219
$983$	−3.20951e10	−1.07771	−0.538855	−	0.842398i	$-0.681143\pi$
$983$	−3.20951e10	−1.07771	−0.538855	+	0.842398i	$0.681143\pi$
$984$	0	0
$985$	2.22629e10	0.742257
$986$	4.12077e10	1.36902
$987$	0	0
$988$	−5.62939e9	−0.185700
$989$	3.83303e10	1.25996
$990$	0	0
$991$	−2.26449e9	−0.0739115	−0.0369558	−	0.999317i	$-0.511766\pi$
$991$	−2.26449e9	−0.0739115	−0.0369558	+	0.999317i	$0.511766\pi$
$992$	1.31879e10	0.428928
$993$	0	0
$994$	1.63229e10	0.527163
$995$	−6.81291e10	−2.19256
$996$	0	0
$997$	9.29325e9	0.296985	0.148493	−	0.988914i	$-0.452558\pi$
$997$	9.29325e9	0.296985	0.148493	+	0.988914i	$0.452558\pi$
$998$	−3.37209e10	−1.07385
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.8.a.c.1.2	yes	2
3.2	odd	2	inner	27.8.a.c.1.1	✓	2
4.3	odd	2		432.8.a.n.1.1		2
9.2	odd	6		81.8.c.g.28.2		4
9.4	even	3		81.8.c.g.55.1		4
9.5	odd	6		81.8.c.g.55.2		4
9.7	even	3		81.8.c.g.28.1		4
12.11	even	2		432.8.a.n.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.8.a.c.1.1	✓	2	3.2	odd	2	inner
27.8.a.c.1.2	yes	2	1.1	even	1	trivial
81.8.c.g.28.1		4	9.7	even	3
81.8.c.g.28.2		4	9.2	odd	6
81.8.c.g.55.1		4	9.4	even	3
81.8.c.g.55.2		4	9.5	odd	6
432.8.a.n.1.1		2	4.3	odd	2
432.8.a.n.1.2		2	12.11	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	27.a (trivial)

\(f(q)\)	\(=\)	\(q+10.3923 q^{2} -20.0000 q^{4} -353.338 q^{5} -559.000 q^{7} -1538.06 q^{8} +O(q^{10})\)	\(q+10.3923 q^{2} -20.0000 q^{4} -353.338 q^{5} -559.000 q^{7} -1538.06 q^{8} -3672.00 q^{10} +4718.11 q^{11} -8671.00 q^{13} -5809.30 q^{14} -13424.0 q^{16} +25128.6 q^{17} -32461.0 q^{19} +7066.77 q^{20} +49032.0 q^{22} -82411.0 q^{23} +46723.0 q^{25} -90111.7 q^{26} +11180.0 q^{28} +157797. q^{29} +229892. q^{31} +57365.5 q^{32} +261144. q^{34} +197516. q^{35} -541177. q^{37} -337345. q^{38} +543456. q^{40} +353505. q^{41} -465112. q^{43} -94362.1 q^{44} -856440. q^{46} -830574. q^{47} -511062. q^{49} +485560. q^{50} +173420. q^{52} -1.02622e6 q^{53} -1.66709e6 q^{55} +859776. q^{56} +1.63987e6 q^{58} -785263. q^{59} -137773. q^{61} +2.38911e6 q^{62} +2.31443e6 q^{64} +3.06380e6 q^{65} -314041. q^{67} -502572. q^{68} +2.05265e6 q^{70} -2.80979e6 q^{71} +2.66954e6 q^{73} -5.62408e6 q^{74} +649220. q^{76} -2.63742e6 q^{77} +1.10181e6 q^{79} +4.74321e6 q^{80} +3.67373e6 q^{82} +6.07904e6 q^{83} -8.87890e6 q^{85} -4.83359e6 q^{86} -7.25674e6 q^{88} -3.28636e6 q^{89} +4.84709e6 q^{91} +1.64822e6 q^{92} -8.63158e6 q^{94} +1.14697e7 q^{95} -2.97938e6 q^{97} -5.31111e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 40 q^{4} - 1118 q^{7}+O(q^{10}) \) 2 * q - 40 * q^4 - 1118 * q^7	\( 2 q - 40 q^{4} - 1118 q^{7} - 7344 q^{10} - 17342 q^{13} - 26848 q^{16} - 64922 q^{19} + 98064 q^{22} + 93446 q^{25} + 22360 q^{28} + 459784 q^{31} + 522288 q^{34} - 1082354 q^{37} + 1086912 q^{40} - 930224 q^{43} - 1712880 q^{46} - 1022124 q^{49} + 346840 q^{52} - 3334176 q^{55} + 3279744 q^{58} - 275546 q^{61} + 4628864 q^{64} - 628082 q^{67} + 4105296 q^{70} + 5339074 q^{73} + 1298440 q^{76} + 2203630 q^{79} + 7347456 q^{82} - 17757792 q^{85} - 14513472 q^{88} + 9694178 q^{91} - 17263152 q^{94} - 5958758 q^{97}+O(q^{100}) \) 2 * q - 40 * q^4 - 1118 * q^7 - 7344 * q^10 - 17342 * q^13 - 26848 * q^16 - 64922 * q^19 + 98064 * q^22 + 93446 * q^25 + 22360 * q^28 + 459784 * q^31 + 522288 * q^34 - 1082354 * q^37 + 1086912 * q^40 - 930224 * q^43 - 1712880 * q^46 - 1022124 * q^49 + 346840 * q^52 - 3334176 * q^55 + 3279744 * q^58 - 275546 * q^61 + 4628864 * q^64 - 628082 * q^67 + 4105296 * q^70 + 5339074 * q^73 + 1298440 * q^76 + 2203630 * q^79 + 7347456 * q^82 - 17757792 * q^85 - 14513472 * q^88 + 9694178 * q^91 - 17263152 * q^94 - 5958758 * q^97

Introduction

L-functions

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Database

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