LMFDB - Embedded newform 27.6.a.c.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [27,6,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, 6); 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, 6, names="a")

Level:	$ N $	$=$	$ 27 = 3^{3} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	27.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0] 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:	$4.33036313495$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.44949$ 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)

$f(q)$	$=$	\(q+7.34847 q^{2} +22.0000 q^{4} +58.7878 q^{5} +167.000 q^{7} -73.4847 q^{8} +432.000 q^{10} -764.241 q^{11} -235.000 q^{13} +1227.19 q^{14} -1244.00 q^{16} +176.363 q^{17} +1361.00 q^{19} +1293.33 q^{20} -5616.00 q^{22} -2410.30 q^{23} +331.000 q^{25} -1726.89 q^{26} +3674.00 q^{28} +470.302 q^{29} +3500.00 q^{31} -6789.99 q^{32} +1296.00 q^{34} +9817.55 q^{35} +13115.0 q^{37} +10001.3 q^{38} -4320.00 q^{40} +9406.04 q^{41} +104.000 q^{43} -16813.3 q^{44} -17712.0 q^{46} -20516.9 q^{47} +11082.0 q^{49} +2432.34 q^{50} -5170.00 q^{52} -1058.18 q^{53} -44928.0 q^{55} -12271.9 q^{56} +3456.00 q^{58} +30746.0 q^{59} -7393.00 q^{61} +25719.6 q^{62} -10088.0 q^{64} -13815.1 q^{65} +38861.0 q^{67} +3879.99 q^{68} +72144.0 q^{70} -2469.09 q^{71} +5465.00 q^{73} +96375.2 q^{74} +29942.0 q^{76} -127628. q^{77} -82903.0 q^{79} -73132.0 q^{80} +69120.0 q^{82} -13286.0 q^{83} +10368.0 q^{85} +764.241 q^{86} +56160.0 q^{88} +89768.9 q^{89} -39245.0 q^{91} -53026.6 q^{92} -150768. q^{94} +80010.1 q^{95} -49603.0 q^{97} +81435.7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 44 q^{4} + 334 q^{7} + 864 q^{10} - 470 q^{13} - 2488 q^{16} + 2722 q^{19} - 11232 q^{22} + 662 q^{25} + 7348 q^{28} + 7000 q^{31} + 2592 q^{34} + 26230 q^{37} - 8640 q^{40} + 208 q^{43} - 35424 q^{46}+ \cdots - 99206 q^{97}+O(q^{100}) $ 2 * q + 44 * q^4 + 334 * q^7 + 864 * q^10 - 470 * q^13 - 2488 * q^16 + 2722 * q^19 - 11232 * q^22 + 662 * q^25 + 7348 * q^28 + 7000 * q^31 + 2592 * q^34 + 26230 * q^37 - 8640 * q^40 + 208 * q^43 - 35424 * q^46 + 22164 * q^49 - 10340 * q^52 - 89856 * q^55 + 6912 * q^58 - 14786 * q^61 - 20176 * q^64 + 77722 * q^67 + 144288 * q^70 + 10930 * q^73 + 59884 * q^76 - 165806 * q^79 + 138240 * q^82 + 20736 * q^85 + 112320 * q^88 - 78490 * q^91 - 301536 * q^94 - 99206 * 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 $

$2$	7.34847	1.29904	0.649519	−	0.760345i	$-0.274970\pi$
$2$	7.34847	1.29904	0.649519	+	0.760345i	$0.274970\pi$
$3$	0	0
$3$	0	0
$4$	22.0000	0.687500
$5$	58.7878	1.05163	0.525814	−	0.850600i	$-0.323761\pi$
$5$	58.7878	1.05163	0.525814	+	0.850600i	$0.323761\pi$
$6$	0	0
$7$	167.000	1.28816	0.644082	−	0.764956i	$-0.277239\pi$
$7$	167.000	1.28816	0.644082	+	0.764956i	$0.277239\pi$
$8$	−73.4847	−0.405949
$9$	0	0
$10$	432.000	1.36610
$11$	−764.241	−1.90436	−0.952179	−	0.305541i	$-0.901163\pi$
$11$	−764.241	−1.90436	−0.952179	+	0.305541i	$0.901163\pi$
$12$	0	0
$13$	−235.000	−0.385664	−0.192832	−	0.981232i	$-0.561767\pi$
$13$	−235.000	−0.385664	−0.192832	+	0.981232i	$0.561767\pi$
$14$	1227.19	1.67337
$15$	0	0
$16$	−1244.00	−1.21484
$17$	176.363	0.148008	0.0740041	−	0.997258i	$-0.476422\pi$
$17$	176.363	0.148008	0.0740041	+	0.997258i	$0.476422\pi$
$18$	0	0
$19$	1361.00	0.864916	0.432458	−	0.901654i	$-0.357646\pi$
$19$	1361.00	0.864916	0.432458	+	0.901654i	$0.357646\pi$
$20$	1293.33	0.722994
$21$	0	0
$22$	−5616.00	−2.47383
$23$	−2410.30	−0.950060	−0.475030	−	0.879970i	$-0.657563\pi$
$23$	−2410.30	−0.950060	−0.475030	+	0.879970i	$0.657563\pi$
$24$	0	0
$25$	331.000	0.105920
$26$	−1726.89	−0.500993
$27$	0	0
$28$	3674.00	0.885613
$29$	470.302	0.103844	0.0519221	−	0.998651i	$-0.483465\pi$
$29$	470.302	0.103844	0.0519221	+	0.998651i	$0.483465\pi$
$30$	0	0
$31$	3500.00	0.654130	0.327065	−	0.945002i	$-0.393940\pi$
$31$	3500.00	0.654130	0.327065	+	0.945002i	$0.393940\pi$
$32$	−6789.99	−1.17218
$33$	0	0
$34$	1296.00	0.192268
$35$	9817.55	1.35467
$36$	0	0
$37$	13115.0	1.57494	0.787470	−	0.616353i	$-0.211391\pi$
$37$	13115.0	1.57494	0.787470	+	0.616353i	$0.211391\pi$
$38$	10001.3	1.12356
$39$	0	0
$40$	−4320.00	−0.426907
$41$	9406.04	0.873871	0.436935	−	0.899493i	$-0.356064\pi$
$41$	9406.04	0.873871	0.436935	+	0.899493i	$0.356064\pi$
$42$	0	0
$43$	104.000	0.00857753	0.00428876	−	0.999991i	$-0.498635\pi$
$43$	104.000	0.00857753	0.00428876	+	0.999991i	$0.498635\pi$
$44$	−16813.3	−1.30925
$45$	0	0
$46$	−17712.0	−1.23416
$47$	−20516.9	−1.35478	−0.677388	−	0.735626i	$-0.736888\pi$
$47$	−20516.9	−1.35478	−0.677388	+	0.735626i	$0.736888\pi$
$48$	0	0
$49$	11082.0	0.659368
$50$	2432.34	0.137594
$51$	0	0
$52$	−5170.00	−0.265144
$53$	−1058.18	−0.0517452	−0.0258726	−	0.999665i	$-0.508236\pi$
$53$	−1058.18	−0.0517452	−0.0258726	+	0.999665i	$0.508236\pi$
$54$	0	0
$55$	−44928.0	−2.00267
$56$	−12271.9	−0.522930
$57$	0	0
$58$	3456.00	0.134897
$59$	30746.0	1.14990	0.574948	−	0.818190i	$-0.305022\pi$
$59$	30746.0	1.14990	0.574948	+	0.818190i	$0.305022\pi$
$60$	0	0
$61$	−7393.00	−0.254388	−0.127194	−	0.991878i	$-0.540597\pi$
$61$	−7393.00	−0.254388	−0.127194	+	0.991878i	$0.540597\pi$
$62$	25719.6	0.849739
$63$	0	0
$64$	−10088.0	−0.307861
$65$	−13815.1	−0.405575
$66$	0	0
$67$	38861.0	1.05761	0.528807	−	0.848742i	$-0.322640\pi$
$67$	38861.0	1.05761	0.528807	+	0.848742i	$0.322640\pi$
$68$	3879.99	0.101756
$69$	0	0
$70$	72144.0	1.75977
$71$	−2469.09	−0.0581287	−0.0290643	−	0.999578i	$-0.509253\pi$
$71$	−2469.09	−0.0581287	−0.0290643	+	0.999578i	$0.509253\pi$
$72$	0	0
$73$	5465.00	0.120028	0.0600141	−	0.998198i	$-0.480885\pi$
$73$	5465.00	0.120028	0.0600141	+	0.998198i	$0.480885\pi$
$74$	96375.2	2.04591
$75$	0	0
$76$	29942.0	0.594630
$77$	−127628.	−2.45313
$78$	0	0
$79$	−82903.0	−1.49452	−0.747261	−	0.664530i	$-0.768632\pi$
$79$	−82903.0	−1.49452	−0.747261	+	0.664530i	$0.768632\pi$
$80$	−73132.0	−1.27756
$81$	0	0
$82$	69120.0	1.13519
$83$	−13286.0	−0.211690	−0.105845	−	0.994383i	$-0.533755\pi$
$83$	−13286.0	−0.211690	−0.105845	+	0.994383i	$0.533755\pi$
$84$	0	0
$85$	10368.0	0.155649
$86$	764.241	0.0111425
$87$	0	0
$88$	56160.0	0.773073
$89$	89768.9	1.20130	0.600649	−	0.799513i	$-0.294909\pi$
$89$	89768.9	1.20130	0.600649	+	0.799513i	$0.294909\pi$
$90$	0	0
$91$	−39245.0	−0.496799
$92$	−53026.6	−0.653166
$93$	0	0
$94$	−150768.	−1.75991
$95$	80010.1	0.909570
$96$	0	0
$97$	−49603.0	−0.535277	−0.267639	−	0.963519i	$-0.586243\pi$
$97$	−49603.0	−0.535277	−0.267639	+	0.963519i	$0.586243\pi$
$98$	81435.7	0.856544
$99$	0	0
$100$	7282.00	0.0728200
$101$	138739.	1.35330	0.676652	−	0.736303i	$-0.263430\pi$
$101$	138739.	1.35330	0.676652	+	0.736303i	$0.263430\pi$
$102$	0	0
$103$	−94933.0	−0.881707	−0.440853	−	0.897579i	$-0.645324\pi$
$103$	−94933.0	−0.881707	−0.440853	+	0.897579i	$0.645324\pi$
$104$	17268.9	0.156560
$105$	0	0
$106$	−7776.00	−0.0672189
$107$	−168780.	−1.42515	−0.712575	−	0.701596i	$-0.752471\pi$
$107$	−168780.	−1.42515	−0.712575	+	0.701596i	$0.752471\pi$
$108$	0	0
$109$	124850.	1.00652	0.503260	−	0.864135i	$-0.332134\pi$
$109$	124850.	1.00652	0.503260	+	0.864135i	$0.332134\pi$
$110$	−330152.	−2.60155
$111$	0	0
$112$	−207748.	−1.56492
$113$	172542.	1.27116	0.635578	−	0.772037i	$-0.280762\pi$
$113$	172542.	1.27116	0.635578	+	0.772037i	$0.280762\pi$
$114$	0	0
$115$	−141696.	−0.999109
$116$	10346.6	0.0713928
$117$	0	0
$118$	225936.	1.49376
$119$	29452.7	0.190659
$120$	0	0
$121$	423013.	2.62658
$122$	−54327.2	−0.330459
$123$	0	0
$124$	77000.0	0.449714
$125$	−164253.	−0.940239
$126$	0	0
$127$	−70108.0	−0.385708	−0.192854	−	0.981227i	$-0.561774\pi$
$127$	−70108.0	−0.385708	−0.192854	+	0.981227i	$0.561774\pi$
$128$	143148.	0.772255
$129$	0	0
$130$	−101520.	−0.526858
$131$	−139445.	−0.709943	−0.354971	−	0.934877i	$-0.615509\pi$
$131$	−139445.	−0.709943	−0.354971	+	0.934877i	$0.615509\pi$
$132$	0	0
$133$	227287.	1.11415
$134$	285569.	1.37388
$135$	0	0
$136$	−12960.0	−0.0600838
$137$	9464.83	0.0430835	0.0215418	−	0.999768i	$-0.493143\pi$
$137$	9464.83	0.0430835	0.0215418	+	0.999768i	$0.493143\pi$
$138$	0	0
$139$	−150913.	−0.662506	−0.331253	−	0.943542i	$-0.607471\pi$
$139$	−150913.	−0.662506	−0.331253	+	0.943542i	$0.607471\pi$
$140$	215986.	0.931335
$141$	0	0
$142$	−18144.0	−0.0755113
$143$	179597.	0.734443
$144$	0	0
$145$	27648.0	0.109205
$146$	40159.4	0.155921
$147$	0	0
$148$	288530.	1.08277
$149$	−230801.	−0.851670	−0.425835	−	0.904801i	$-0.640020\pi$
$149$	−230801.	−0.851670	−0.425835	+	0.904801i	$0.640020\pi$
$150$	0	0
$151$	129461.	0.462058	0.231029	−	0.972947i	$-0.425791\pi$
$151$	129461.	0.462058	0.231029	+	0.972947i	$0.425791\pi$
$152$	−100013.	−0.351112
$153$	0	0
$154$	−937872.	−3.18670
$155$	205757.	0.687901
$156$	0	0
$157$	257822.	0.834778	0.417389	−	0.908728i	$-0.362945\pi$
$157$	257822.	0.834778	0.417389	+	0.908728i	$0.362945\pi$
$158$	−609210.	−1.94144
$159$	0	0
$160$	−399168.	−1.23270
$161$	−402520.	−1.22383
$162$	0	0
$163$	−131569.	−0.387869	−0.193934	−	0.981015i	$-0.562125\pi$
$163$	−131569.	−0.387869	−0.193934	+	0.981015i	$0.562125\pi$
$164$	206933.	0.600786
$165$	0	0
$166$	−97632.0	−0.274993
$167$	−257079.	−0.713305	−0.356652	−	0.934237i	$-0.616082\pi$
$167$	−257079.	−0.713305	−0.356652	+	0.934237i	$0.616082\pi$
$168$	0	0
$169$	−316068.	−0.851263
$170$	76188.9	0.202195
$171$	0	0
$172$	2288.00	0.00589705
$173$	−167428.	−0.425316	−0.212658	−	0.977127i	$-0.568212\pi$
$173$	−167428.	−0.425316	−0.212658	+	0.977127i	$0.568212\pi$
$174$	0	0
$175$	55277.0	0.136442
$176$	950716.	2.31350
$177$	0	0
$178$	659664.	1.56053
$179$	389057.	0.907572	0.453786	−	0.891111i	$-0.350073\pi$
$179$	389057.	0.907572	0.453786	+	0.891111i	$0.350073\pi$
$180$	0	0
$181$	165305.	0.375050	0.187525	−	0.982260i	$-0.439953\pi$
$181$	165305.	0.375050	0.187525	+	0.982260i	$0.439953\pi$
$182$	−288391.	−0.645361
$183$	0	0
$184$	177120.	0.385676
$185$	771001.	1.65625
$186$	0	0
$187$	−134784.	−0.281861
$188$	−451372.	−0.931409
$189$	0	0
$190$	587952.	1.18157
$191$	−343967.	−0.682234	−0.341117	−	0.940021i	$-0.610805\pi$
$191$	−343967.	−0.682234	−0.341117	+	0.940021i	$0.610805\pi$
$192$	0	0
$193$	−251785.	−0.486560	−0.243280	−	0.969956i	$-0.578223\pi$
$193$	−251785.	−0.486560	−0.243280	+	0.969956i	$0.578223\pi$
$194$	−364506.	−0.695345
$195$	0	0
$196$	243804.	0.453316
$197$	820971.	1.50717	0.753585	−	0.657350i	$-0.228323\pi$
$197$	820971.	1.50717	0.753585	+	0.657350i	$0.228323\pi$
$198$	0	0
$199$	−336157.	−0.601741	−0.300870	−	0.953665i	$-0.597277\pi$
$199$	−336157.	−0.601741	−0.300870	+	0.953665i	$0.597277\pi$
$200$	−24323.4	−0.0429982
$201$	0	0
$202$	1.01952e6	1.75799
$203$	78540.4	0.133768
$204$	0	0
$205$	552960.	0.918986
$206$	−697612.	−1.14537
$207$	0	0
$208$	292340.	0.468522
$209$	−1.04013e6	−1.64711
$210$	0	0
$211$	−821557.	−1.27037	−0.635187	−	0.772358i	$-0.719077\pi$
$211$	−821557.	−1.27037	−0.635187	+	0.772358i	$0.719077\pi$
$212$	−23280.0	−0.0355748
$213$	0	0
$214$	−1.24027e6	−1.85132
$215$	6113.93	0.00902036
$216$	0	0
$217$	584500.	0.842627
$218$	917456.	1.30751
$219$	0	0
$220$	−988416.	−1.37684
$221$	−41445.4	−0.0570815
$222$	0	0
$223$	670388.	0.902743	0.451371	−	0.892336i	$-0.350935\pi$
$223$	670388.	0.902743	0.451371	+	0.892336i	$0.350935\pi$
$224$	−1.13393e6	−1.50996
$225$	0	0
$226$	1.26792e6	1.65128
$227$	860065.	1.10781	0.553907	−	0.832579i	$-0.313136\pi$
$227$	860065.	1.10781	0.553907	+	0.832579i	$0.313136\pi$
$228$	0	0
$229$	−1.43277e6	−1.80546	−0.902732	−	0.430203i	$-0.858442\pi$
$229$	−1.43277e6	−1.80546	−0.902732	+	0.430203i	$0.858442\pi$
$230$	−1.04125e6	−1.29788
$231$	0	0
$232$	−34560.0	−0.0421555
$233$	407046.	0.491195	0.245598	−	0.969372i	$-0.421016\pi$
$233$	407046.	0.491195	0.245598	+	0.969372i	$0.421016\pi$
$234$	0	0
$235$	−1.20614e6	−1.42472
$236$	676412.	0.790553
$237$	0	0
$238$	216432.	0.247673
$239$	115577.	0.130881	0.0654404	−	0.997856i	$-0.479155\pi$
$239$	115577.	0.130881	0.0654404	+	0.997856i	$0.479155\pi$
$240$	0	0
$241$	−1.55192e6	−1.72118	−0.860590	−	0.509298i	$-0.829905\pi$
$241$	−1.55192e6	−1.72118	−0.860590	+	0.509298i	$0.829905\pi$
$242$	3.10850e6	3.41202
$243$	0	0
$244$	−162646.	−0.174892
$245$	651486.	0.693410
$246$	0	0
$247$	−319835.	−0.333567
$248$	−257196.	−0.265544
$249$	0	0
$250$	−1.20701e6	−1.22141
$251$	−1.70226e6	−1.70546	−0.852729	−	0.522353i	$-0.825054\pi$
$251$	−1.70226e6	−1.70546	−0.852729	+	0.522353i	$0.825054\pi$
$252$	0	0
$253$	1.84205e6	1.80925
$254$	−515186.	−0.501049
$255$	0	0
$256$	1.37474e6	1.31105
$257$	−1.38210e6	−1.30529	−0.652645	−	0.757664i	$-0.726340\pi$
$257$	−1.38210e6	−1.30529	−0.652645	+	0.757664i	$0.726340\pi$
$258$	0	0
$259$	2.19020e6	2.02878
$260$	−303933.	−0.278833
$261$	0	0
$262$	−1.02470e6	−0.922243
$263$	−1.61678e6	−1.44132	−0.720662	−	0.693286i	$-0.756162\pi$
$263$	−1.61678e6	−1.44132	−0.720662	+	0.693286i	$0.756162\pi$
$264$	0	0
$265$	−62208.0	−0.0544166
$266$	1.67021e6	1.44733
$267$	0	0
$268$	854942.	0.727109
$269$	−406870.	−0.342827	−0.171413	−	0.985199i	$-0.554833\pi$
$269$	−406870.	−0.342827	−0.171413	+	0.985199i	$0.554833\pi$
$270$	0	0
$271$	246053.	0.203519	0.101760	−	0.994809i	$-0.467553\pi$
$271$	246053.	0.203519	0.101760	+	0.994809i	$0.467553\pi$
$272$	−219396.	−0.179807
$273$	0	0
$274$	69552.0	0.0559672
$275$	−252964.	−0.201710
$276$	0	0
$277$	−347350.	−0.271999	−0.136000	−	0.990709i	$-0.543425\pi$
$277$	−347350.	−0.271999	−0.136000	+	0.990709i	$0.543425\pi$
$278$	−1.10898e6	−0.860620
$279$	0	0
$280$	−721440.	−0.549927
$281$	611393.	0.461907	0.230953	−	0.972965i	$-0.425815\pi$
$281$	611393.	0.461907	0.230953	+	0.972965i	$0.425815\pi$
$282$	0	0
$283$	2.05827e6	1.52770	0.763848	−	0.645397i	$-0.223308\pi$
$283$	2.05827e6	1.52770	0.763848	+	0.645397i	$0.223308\pi$
$284$	−54319.9	−0.0399635
$285$	0	0
$286$	1.31976e6	0.954069
$287$	1.57081e6	1.12569
$288$	0	0
$289$	−1.38875e6	−0.978094
$290$	203170.	0.141862
$291$	0	0
$292$	120230.	0.0825193
$293$	2.26844e6	1.54369	0.771843	−	0.635813i	$-0.219335\pi$
$293$	2.26844e6	1.54369	0.771843	+	0.635813i	$0.219335\pi$
$294$	0	0
$295$	1.80749e6	1.20926
$296$	−963752.	−0.639346
$297$	0	0
$298$	−1.69603e6	−1.10635
$299$	566420.	0.366404
$300$	0	0
$301$	17368.0	0.0110493
$302$	951340.	0.600231
$303$	0	0
$304$	−1.69308e6	−1.05074
$305$	−434618.	−0.267521
$306$	0	0
$307$	902576.	0.546560	0.273280	−	0.961935i	$-0.411891\pi$
$307$	902576.	0.546560	0.273280	+	0.961935i	$0.411891\pi$
$308$	−2.80782e6	−1.68652
$309$	0	0
$310$	1.51200e6	0.893609
$311$	1.45964e6	0.855746	0.427873	−	0.903839i	$-0.359263\pi$
$311$	1.45964e6	0.855746	0.427873	+	0.903839i	$0.359263\pi$
$312$	0	0
$313$	−2.75992e6	−1.59234	−0.796169	−	0.605075i	$-0.793143\pi$
$313$	−2.75992e6	−1.59234	−0.796169	+	0.605075i	$0.793143\pi$
$314$	1.89460e6	1.08441
$315$	0	0
$316$	−1.82387e6	−1.02748
$317$	17048.4	0.00952877	0.00476438	−	0.999989i	$-0.498483\pi$
$317$	17048.4	0.00952877	0.00476438	+	0.999989i	$0.498483\pi$
$318$	0	0
$319$	−359424.	−0.197756
$320$	−593051.	−0.323755
$321$	0	0
$322$	−2.95790e6	−1.58981
$323$	240030.	0.128015
$324$	0	0
$325$	−77785.0	−0.0408496
$326$	−966831.	−0.503856
$327$	0	0
$328$	−691200.	−0.354747
$329$	−3.42633e6	−1.74518
$330$	0	0
$331$	3.37236e6	1.69186	0.845929	−	0.533296i	$-0.179047\pi$
$331$	3.37236e6	1.69186	0.845929	+	0.533296i	$0.179047\pi$
$332$	−292293.	−0.145537
$333$	0	0
$334$	−1.88914e6	−0.926610
$335$	2.28455e6	1.11222
$336$	0	0
$337$	−360523.	−0.172925	−0.0864626	−	0.996255i	$-0.527556\pi$
$337$	−360523.	−0.172925	−0.0864626	+	0.996255i	$0.527556\pi$
$338$	−2.32262e6	−1.10582
$339$	0	0
$340$	228096.	0.107009
$341$	−2.67484e6	−1.24570
$342$	0	0
$343$	−956075.	−0.438790
$344$	−7642.41	−0.00348204
$345$	0	0
$346$	−1.23034e6	−0.552502
$347$	−1.44206e6	−0.642926	−0.321463	−	0.946922i	$-0.604175\pi$
$347$	−1.44206e6	−0.642926	−0.321463	+	0.946922i	$0.604175\pi$
$348$	0	0
$349$	677579.	0.297781	0.148890	−	0.988854i	$-0.452430\pi$
$349$	677579.	0.297781	0.148890	+	0.988854i	$0.452430\pi$
$350$	406201.	0.177244
$351$	0	0
$352$	5.18918e6	2.23225
$353$	3.84484e6	1.64226	0.821129	−	0.570743i	$-0.193345\pi$
$353$	3.84484e6	1.64226	0.821129	+	0.570743i	$0.193345\pi$
$354$	0	0
$355$	−145152.	−0.0611297
$356$	1.97492e6	0.825893
$357$	0	0
$358$	2.85898e6	1.17897
$359$	1.36699e6	0.559796	0.279898	−	0.960030i	$-0.409699\pi$
$359$	1.36699e6	0.559796	0.279898	+	0.960030i	$0.409699\pi$
$360$	0	0
$361$	−623778.	−0.251920
$362$	1.21474e6	0.487205
$363$	0	0
$364$	−863390.	−0.341549
$365$	321275.	0.126225
$366$	0	0
$367$	−1.16951e6	−0.453251	−0.226625	−	0.973982i	$-0.572769\pi$
$367$	−1.16951e6	−0.453251	−0.226625	+	0.973982i	$0.572769\pi$
$368$	2.99841e6	1.15417
$369$	0	0
$370$	5.66568e6	2.15153
$371$	−176716.	−0.0666563
$372$	0	0
$373$	2.52666e6	0.940318	0.470159	−	0.882582i	$-0.344197\pi$
$373$	2.52666e6	0.940318	0.470159	+	0.882582i	$0.344197\pi$
$374$	−990456.	−0.366148
$375$	0	0
$376$	1.50768e6	0.549971
$377$	−110521.	−0.0400490
$378$	0	0
$379$	219269.	0.0784114	0.0392057	−	0.999231i	$-0.487517\pi$
$379$	219269.	0.0784114	0.0392057	+	0.999231i	$0.487517\pi$
$380$	1.76022e6	0.625329
$381$	0	0
$382$	−2.52763e6	−0.886248
$383$	−59728.4	−0.0208058	−0.0104029	−	0.999946i	$-0.503311\pi$
$383$	−59728.4	−0.0208058	−0.0104029	+	0.999946i	$0.503311\pi$
$384$	0	0
$385$	−7.50298e6	−2.57977
$386$	−1.85023e6	−0.632060
$387$	0	0
$388$	−1.09127e6	−0.368003
$389$	−2.70653e6	−0.906857	−0.453428	−	0.891293i	$-0.649799\pi$
$389$	−2.70653e6	−0.906857	−0.453428	+	0.891293i	$0.649799\pi$
$390$	0	0
$391$	−425088.	−0.140617
$392$	−814357.	−0.267670
$393$	0	0
$394$	6.03288e6	1.95787
$395$	−4.87368e6	−1.57168
$396$	0	0
$397$	4.43128e6	1.41108	0.705542	−	0.708668i	$-0.250704\pi$
$397$	4.43128e6	1.41108	0.705542	+	0.708668i	$0.250704\pi$
$398$	−2.47024e6	−0.781684
$399$	0	0
$400$	−411764.	−0.128676
$401$	2.37585e6	0.737832	0.368916	−	0.929463i	$-0.379729\pi$
$401$	2.37585e6	0.737832	0.368916	+	0.929463i	$0.379729\pi$
$402$	0	0
$403$	−822500.	−0.252274
$404$	3.05226e6	0.930397
$405$	0	0
$406$	577152.	0.173770
$407$	−1.00230e7	−2.99925
$408$	0	0
$409$	−1.71805e6	−0.507840	−0.253920	−	0.967225i	$-0.581720\pi$
$409$	−1.71805e6	−0.507840	−0.253920	+	0.967225i	$0.581720\pi$
$410$	4.06341e6	1.19380
$411$	0	0
$412$	−2.08853e6	−0.606173
$413$	5.13458e6	1.48126
$414$	0	0
$415$	−781056.	−0.222619
$416$	1.59565e6	0.452068
$417$	0	0
$418$	−7.64338e6	−2.13966
$419$	5.34633e6	1.48772	0.743860	−	0.668336i	$-0.232993\pi$
$419$	5.34633e6	1.48772	0.743860	+	0.668336i	$0.232993\pi$
$420$	0	0
$421$	−5.68202e6	−1.56242	−0.781209	−	0.624269i	$-0.785397\pi$
$421$	−5.68202e6	−1.56242	−0.781209	+	0.624269i	$0.785397\pi$
$422$	−6.03719e6	−1.65026
$423$	0	0
$424$	77760.0	0.0210059
$425$	58376.2	0.0156770
$426$	0	0
$427$	−1.23463e6	−0.327693
$428$	−3.71315e6	−0.979791
$429$	0	0
$430$	44928.0	0.0117178
$431$	−2.29890e6	−0.596109	−0.298055	−	0.954549i	$-0.596338\pi$
$431$	−2.29890e6	−0.596109	−0.298055	+	0.954549i	$0.596338\pi$
$432$	0	0
$433$	−4.72608e6	−1.21138	−0.605691	−	0.795700i	$-0.707103\pi$
$433$	−4.72608e6	−1.21138	−0.605691	+	0.795700i	$0.707103\pi$
$434$	4.29518e6	1.09460
$435$	0	0
$436$	2.74670e6	0.691982
$437$	−3.28042e6	−0.821723
$438$	0	0
$439$	6.12223e6	1.51617	0.758086	−	0.652155i	$-0.226135\pi$
$439$	6.12223e6	1.51617	0.758086	+	0.652155i	$0.226135\pi$
$440$	3.30152e6	0.812985
$441$	0	0
$442$	−304560.	−0.0741510
$443$	6.68429e6	1.61825	0.809125	−	0.587636i	$-0.199941\pi$
$443$	6.68429e6	1.61825	0.809125	+	0.587636i	$0.199941\pi$
$444$	0	0
$445$	5.27731e6	1.26332
$446$	4.92633e6	1.17270
$447$	0	0
$448$	−1.68470e6	−0.396576
$449$	5.01701e6	1.17443	0.587217	−	0.809429i	$-0.300223\pi$
$449$	5.01701e6	1.17443	0.587217	+	0.809429i	$0.300223\pi$
$450$	0	0
$451$	−7.18848e6	−1.66416
$452$	3.79593e6	0.873920
$453$	0	0
$454$	6.32016e6	1.43909
$455$	−2.30713e6	−0.522448
$456$	0	0
$457$	683222.	0.153028	0.0765141	−	0.997069i	$-0.475621\pi$
$457$	683222.	0.153028	0.0765141	+	0.997069i	$0.475621\pi$
$458$	−1.05287e7	−2.34537
$459$	0	0
$460$	−3.11731e6	−0.686888
$461$	−5.04123e6	−1.10480	−0.552400	−	0.833579i	$-0.686288\pi$
$461$	−5.04123e6	−1.10480	−0.552400	+	0.833579i	$0.686288\pi$
$462$	0	0
$463$	5.04086e6	1.09283	0.546415	−	0.837515i	$-0.315992\pi$
$463$	5.04086e6	1.09283	0.546415	+	0.837515i	$0.315992\pi$
$464$	−585056.	−0.126154
$465$	0	0
$466$	2.99117e6	0.638081
$467$	8.12347e6	1.72365	0.861825	−	0.507205i	$-0.169321\pi$
$467$	8.12347e6	1.72365	0.861825	+	0.507205i	$0.169321\pi$
$468$	0	0
$469$	6.48979e6	1.36238
$470$	−8.86331e6	−1.85077
$471$	0	0
$472$	−2.25936e6	−0.466800
$473$	−79481.0	−0.0163347
$474$	0	0
$475$	450491.	0.0916119
$476$	647959.	0.131078
$477$	0	0
$478$	849312.	0.170019
$479$	6.60739e6	1.31580	0.657902	−	0.753104i	$-0.271444\pi$
$479$	6.60739e6	1.31580	0.657902	+	0.753104i	$0.271444\pi$
$480$	0	0
$481$	−3.08202e6	−0.607398
$482$	−1.14042e7	−2.23588
$483$	0	0
$484$	9.30629e6	1.80577
$485$	−2.91605e6	−0.562912
$486$	0	0
$487$	−2.49806e6	−0.477289	−0.238644	−	0.971107i	$-0.576703\pi$
$487$	−2.49806e6	−0.477289	−0.238644	+	0.971107i	$0.576703\pi$
$488$	543272.	0.103269
$489$	0	0
$490$	4.78742e6	0.900765
$491$	−5.70141e6	−1.06728	−0.533640	−	0.845711i	$-0.679176\pi$
$491$	−5.70141e6	−1.06728	−0.533640	+	0.845711i	$0.679176\pi$
$492$	0	0
$493$	82944.0	0.0153698
$494$	−2.35030e6	−0.433317
$495$	0	0
$496$	−4.35400e6	−0.794665
$497$	−412337.	−0.0748793
$498$	0	0
$499$	4.19754e6	0.754646	0.377323	−	0.926082i	$-0.376845\pi$
$499$	4.19754e6	0.754646	0.377323	+	0.926082i	$0.376845\pi$
$500$	−3.61357e6	−0.646414
$501$	0	0
$502$	−1.25090e7	−2.21546
$503$	5.37784e6	0.947738	0.473869	−	0.880595i	$-0.342857\pi$
$503$	5.37784e6	0.947738	0.473869	+	0.880595i	$0.342857\pi$
$504$	0	0
$505$	8.15616e6	1.42317
$506$	1.35362e7	2.35029
$507$	0	0
$508$	−1.54238e6	−0.265174
$509$	4.35623e6	0.745275	0.372637	−	0.927977i	$-0.378454\pi$
$509$	4.35623e6	0.745275	0.372637	+	0.927977i	$0.378454\pi$
$510$	0	0
$511$	912655.	0.154616
$512$	5.52146e6	0.930849
$513$	0	0
$514$	−1.01563e7	−1.69562
$515$	−5.58090e6	−0.927227
$516$	0	0
$517$	1.56799e7	2.57998
$518$	1.60947e7	2.63546
$519$	0	0
$520$	1.01520e6	0.164643
$521$	−4.29639e6	−0.693440	−0.346720	−	0.937969i	$-0.612705\pi$
$521$	−4.29639e6	−0.693440	−0.346720	+	0.937969i	$0.612705\pi$
$522$	0	0
$523$	−1.07625e7	−1.72052	−0.860261	−	0.509854i	$-0.829699\pi$
$523$	−1.07625e7	−1.72052	−0.860261	+	0.509854i	$0.829699\pi$
$524$	−3.06778e6	−0.488086
$525$	0	0
$526$	−1.18809e7	−1.87234
$527$	617271.	0.0968166
$528$	0	0
$529$	−626807.	−0.0973856
$530$	−457134.	−0.0706893
$531$	0	0
$532$	5.00031e6	0.765981
$533$	−2.21042e6	−0.337021
$534$	0	0
$535$	−9.92218e6	−1.49873
$536$	−2.85569e6	−0.429338
$537$	0	0
$538$	−2.98987e6	−0.445345
$539$	−8.46932e6	−1.25567
$540$	0	0
$541$	7.49825e6	1.10146	0.550728	−	0.834685i	$-0.314350\pi$
$541$	7.49825e6	1.10146	0.550728	+	0.834685i	$0.314350\pi$
$542$	1.80811e6	0.264379
$543$	0	0
$544$	−1.19750e6	−0.173492
$545$	7.33965e6	1.05848
$546$	0	0
$547$	−3.63295e6	−0.519148	−0.259574	−	0.965723i	$-0.583582\pi$
$547$	−3.63295e6	−0.519148	−0.259574	+	0.965723i	$0.583582\pi$
$548$	208226.	0.0296199
$549$	0	0
$550$	−1.85890e6	−0.262028
$551$	640081.	0.0898165
$552$	0	0
$553$	−1.38448e7	−1.92519
$554$	−2.55249e6	−0.353338
$555$	0	0
$556$	−3.32009e6	−0.455473
$557$	−7.12137e6	−0.972581	−0.486290	−	0.873797i	$-0.661650\pi$
$557$	−7.12137e6	−0.972581	−0.486290	+	0.873797i	$0.661650\pi$
$558$	0	0
$559$	−24440.0	−0.00330805
$560$	−1.22130e7	−1.64571
$561$	0	0
$562$	4.49280e6	0.600035
$563$	−3.41768e6	−0.454424	−0.227212	−	0.973845i	$-0.572961\pi$
$563$	−3.41768e6	−0.454424	−0.227212	+	0.973845i	$0.572961\pi$
$564$	0	0
$565$	1.01434e7	1.33678
$566$	1.51251e7	1.98453
$567$	0	0
$568$	181440.	0.0235973
$569$	−5.43699e6	−0.704008	−0.352004	−	0.935999i	$-0.614500\pi$
$569$	−5.43699e6	−0.704008	−0.352004	+	0.935999i	$0.614500\pi$
$570$	0	0
$571$	−3.92190e6	−0.503391	−0.251696	−	0.967806i	$-0.580988\pi$
$571$	−3.92190e6	−0.503391	−0.251696	+	0.967806i	$0.580988\pi$
$572$	3.95112e6	0.504929
$573$	0	0
$574$	1.15430e7	1.46231
$575$	−797809.	−0.100630
$576$	0	0
$577$	1.07034e7	1.33839	0.669193	−	0.743088i	$-0.266640\pi$
$577$	1.07034e7	1.33839	0.669193	+	0.743088i	$0.266640\pi$
$578$	−1.02052e7	−1.27058
$579$	0	0
$580$	608256.	0.0750786
$581$	−2.21877e6	−0.272691
$582$	0	0
$583$	808704.	0.0985413
$584$	−401594.	−0.0487253
$585$	0	0
$586$	1.66696e7	2.00531
$587$	2.99424e6	0.358667	0.179333	−	0.983788i	$-0.442606\pi$
$587$	2.99424e6	0.358667	0.179333	+	0.983788i	$0.442606\pi$
$588$	0	0
$589$	4.76350e6	0.565767
$590$	1.32823e7	1.57088
$591$	0	0
$592$	−1.63151e7	−1.91331
$593$	−5.85279e6	−0.683481	−0.341740	−	0.939794i	$-0.611016\pi$
$593$	−5.85279e6	−0.683481	−0.341740	+	0.939794i	$0.611016\pi$
$594$	0	0
$595$	1.73146e6	0.200502
$596$	−5.07762e6	−0.585523
$597$	0	0
$598$	4.16232e6	0.475973
$599$	−1.93059e6	−0.219848	−0.109924	−	0.993940i	$-0.535061\pi$
$599$	−1.93059e6	−0.219848	−0.109924	+	0.993940i	$0.535061\pi$
$600$	0	0
$601$	−6.64461e6	−0.750384	−0.375192	−	0.926947i	$-0.622423\pi$
$601$	−6.64461e6	−0.750384	−0.375192	+	0.926947i	$0.622423\pi$
$602$	127628.	0.0143534
$603$	0	0
$604$	2.84814e6	0.317665
$605$	2.48680e7	2.76218
$606$	0	0
$607$	−6.72545e6	−0.740883	−0.370441	−	0.928856i	$-0.620794\pi$
$607$	−6.72545e6	−0.740883	−0.370441	+	0.928856i	$0.620794\pi$
$608$	−9.24117e6	−1.01384
$609$	0	0
$610$	−3.19378e6	−0.347520
$611$	4.82148e6	0.522489
$612$	0	0
$613$	3.03643e6	0.326372	0.163186	−	0.986595i	$-0.447823\pi$
$613$	3.03643e6	0.326372	0.163186	+	0.986595i	$0.447823\pi$
$614$	6.63255e6	0.710002
$615$	0	0
$616$	9.37872e6	0.995845
$617$	5.93504e6	0.627640	0.313820	−	0.949483i	$-0.398391\pi$
$617$	5.93504e6	0.627640	0.313820	+	0.949483i	$0.398391\pi$
$618$	0	0
$619$	1.21004e7	1.26932	0.634661	−	0.772791i	$-0.281140\pi$
$619$	1.21004e7	1.26932	0.634661	+	0.772791i	$0.281140\pi$
$620$	4.52666e6	0.472932
$621$	0	0
$622$	1.07261e7	1.11165
$623$	1.49914e7	1.54747
$624$	0	0
$625$	−1.06904e7	−1.09470
$626$	−2.02812e7	−2.06851
$627$	0	0
$628$	5.67208e6	0.573910
$629$	2.31300e6	0.233104
$630$	0	0
$631$	−6.17037e6	−0.616933	−0.308466	−	0.951235i	$-0.599816\pi$
$631$	−6.17037e6	−0.616933	−0.308466	+	0.951235i	$0.599816\pi$
$632$	6.09210e6	0.606701
$633$	0	0
$634$	125280.	0.0123782
$635$	−4.12149e6	−0.405621
$636$	0	0
$637$	−2.60427e6	−0.254295
$638$	−2.64122e6	−0.256893
$639$	0	0
$640$	8.41536e6	0.812125
$641$	−1.13478e7	−1.09085	−0.545427	−	0.838158i	$-0.683632\pi$
$641$	−1.13478e7	−1.09085	−0.545427	+	0.838158i	$0.683632\pi$
$642$	0	0
$643$	−1.00778e7	−0.961257	−0.480629	−	0.876924i	$-0.659592\pi$
$643$	−1.00778e7	−0.961257	−0.480629	+	0.876924i	$0.659592\pi$
$644$	−8.85543e6	−0.841386
$645$	0	0
$646$	1.76386e6	0.166296
$647$	−7.07287e6	−0.664255	−0.332128	−	0.943234i	$-0.607767\pi$
$647$	−7.07287e6	−0.664255	−0.332128	+	0.943234i	$0.607767\pi$
$648$	0	0
$649$	−2.34973e7	−2.18981
$650$	−571601.	−0.0530651
$651$	0	0
$652$	−2.89452e6	−0.266660
$653$	−1.84773e7	−1.69573	−0.847865	−	0.530213i	$-0.822112\pi$
$653$	−1.84773e7	−1.69573	−0.847865	+	0.530213i	$0.822112\pi$
$654$	0	0
$655$	−8.19763e6	−0.746595
$656$	−1.17011e7	−1.06162
$657$	0	0
$658$	−2.51783e7	−2.26705
$659$	−1.53893e7	−1.38040	−0.690202	−	0.723616i	$-0.742479\pi$
$659$	−1.53893e7	−1.38040	−0.690202	+	0.723616i	$0.742479\pi$
$660$	0	0
$661$	5.21093e6	0.463886	0.231943	−	0.972729i	$-0.425492\pi$
$661$	5.21093e6	0.463886	0.231943	+	0.972729i	$0.425492\pi$
$662$	2.47817e7	2.19779
$663$	0	0
$664$	976320.	0.0859354
$665$	1.33617e7	1.17168
$666$	0	0
$667$	−1.13357e6	−0.0986582
$668$	−5.65573e6	−0.490397
$669$	0	0
$670$	1.67880e7	1.44481
$671$	5.65003e6	0.484445
$672$	0	0
$673$	1.60404e7	1.36514	0.682569	−	0.730821i	$-0.260863\pi$
$673$	1.60404e7	1.36514	0.682569	+	0.730821i	$0.260863\pi$
$674$	−2.64929e6	−0.224636
$675$	0	0
$676$	−6.95350e6	−0.585243
$677$	1.48811e7	1.24785	0.623927	−	0.781482i	$-0.285536\pi$
$677$	1.48811e7	1.24785	0.623927	+	0.781482i	$0.285536\pi$
$678$	0	0
$679$	−8.28370e6	−0.689525
$680$	−761889.	−0.0631858
$681$	0	0
$682$	−1.96560e7	−1.61821
$683$	5.62881e6	0.461705	0.230853	−	0.972989i	$-0.425848\pi$
$683$	5.62881e6	0.461705	0.230853	+	0.972989i	$0.425848\pi$
$684$	0	0
$685$	556416.	0.0453078
$686$	−7.02569e6	−0.570005
$687$	0	0
$688$	−129376.	−0.0104204
$689$	248672.	0.0199563
$690$	0	0
$691$	−1.01916e7	−0.811987	−0.405993	−	0.913876i	$-0.633074\pi$
$691$	−1.01916e7	−0.811987	−0.405993	+	0.913876i	$0.633074\pi$
$692$	−3.68341e6	−0.292405
$693$	0	0
$694$	−1.05970e7	−0.835185
$695$	−8.87184e6	−0.696709
$696$	0	0
$697$	1.65888e6	0.129340
$698$	4.97917e6	0.386828
$699$	0	0
$700$	1.21609e6	0.0938041
$701$	2.39524e7	1.84100	0.920501	−	0.390739i	$-0.127781\pi$
$701$	2.39524e7	1.84100	0.920501	+	0.390739i	$0.127781\pi$
$702$	0	0
$703$	1.78495e7	1.36219
$704$	7.70966e6	0.586278
$705$	0	0
$706$	2.82537e7	2.13335
$707$	2.31694e7	1.74328
$708$	0	0
$709$	1.49472e7	1.11672	0.558360	−	0.829599i	$-0.311431\pi$
$709$	1.49472e7	1.11672	0.558360	+	0.829599i	$0.311431\pi$
$710$	−1.06665e6	−0.0794098
$711$	0	0
$712$	−6.59664e6	−0.487666
$713$	−8.43604e6	−0.621463
$714$	0	0
$715$	1.05581e7	0.772360
$716$	8.55926e6	0.623956
$717$	0	0
$718$	1.00453e7	0.727196
$719$	−2.61670e7	−1.88770	−0.943848	−	0.330380i	$-0.892823\pi$
$719$	−2.61670e7	−1.88770	−0.943848	+	0.330380i	$0.892823\pi$
$720$	0	0
$721$	−1.58538e7	−1.13578
$722$	−4.58381e6	−0.327253
$723$	0	0
$724$	3.63671e6	0.257847
$725$	155670.	0.0109992
$726$	0	0
$727$	−2.91140e6	−0.204299	−0.102149	−	0.994769i	$-0.532572\pi$
$727$	−2.91140e6	−0.204299	−0.102149	+	0.994769i	$0.532572\pi$
$728$	2.88391e6	0.201675
$729$	0	0
$730$	2.36088e6	0.163971
$731$	18341.8	0.00126954
$732$	0	0
$733$	6.78250e6	0.466262	0.233131	−	0.972445i	$-0.425103\pi$
$733$	6.78250e6	0.466262	0.233131	+	0.972445i	$0.425103\pi$
$734$	−8.59410e6	−0.588790
$735$	0	0
$736$	1.63659e7	1.11364
$737$	−2.96992e7	−2.01407
$738$	0	0
$739$	1.52588e7	1.02780	0.513901	−	0.857850i	$-0.328200\pi$
$739$	1.52588e7	1.02780	0.513901	+	0.857850i	$0.328200\pi$
$740$	1.69620e7	1.13867
$741$	0	0
$742$	−1.29859e6	−0.0865890
$743$	1.96863e6	0.130825	0.0654125	−	0.997858i	$-0.479164\pi$
$743$	1.96863e6	0.130825	0.0654125	+	0.997858i	$0.479164\pi$
$744$	0	0
$745$	−1.35683e7	−0.895640
$746$	1.85671e7	1.22151
$747$	0	0
$748$	−2.96525e6	−0.193779
$749$	−2.81862e7	−1.83583
$750$	0	0
$751$	−1.62379e7	−1.05058	−0.525290	−	0.850924i	$-0.676043\pi$
$751$	−1.62379e7	−1.05058	−0.525290	+	0.850924i	$0.676043\pi$
$752$	2.55231e7	1.64584
$753$	0	0
$754$	−812160.	−0.0520251
$755$	7.61072e6	0.485913
$756$	0	0
$757$	9.17096e6	0.581668	0.290834	−	0.956774i	$-0.406067\pi$
$757$	9.17096e6	0.581668	0.290834	+	0.956774i	$0.406067\pi$
$758$	1.61129e6	0.101859
$759$	0	0
$760$	−5.87952e6	−0.369239
$761$	−2.31222e7	−1.44733	−0.723666	−	0.690151i	$-0.757544\pi$
$761$	−2.31222e7	−1.44733	−0.723666	+	0.690151i	$0.757544\pi$
$762$	0	0
$763$	2.08500e7	1.29656
$764$	−7.56728e6	−0.469036
$765$	0	0
$766$	−438912.	−0.0270275
$767$	−7.22531e6	−0.443474
$768$	0	0
$769$	9.38318e6	0.572182	0.286091	−	0.958202i	$-0.407644\pi$
$769$	9.38318e6	0.572182	0.286091	+	0.958202i	$0.407644\pi$
$770$	−5.51354e7	−3.35123
$771$	0	0
$772$	−5.53927e6	−0.334510
$773$	−89239.8	−0.00537168	−0.00268584	−	0.999996i	$-0.500855\pi$
$773$	−89239.8	−0.00537168	−0.00268584	+	0.999996i	$0.500855\pi$
$774$	0	0
$775$	1.15850e6	0.0692854
$776$	3.64506e6	0.217295
$777$	0	0
$778$	−1.98888e7	−1.17804
$779$	1.28016e7	0.755825
$780$	0	0
$781$	1.88698e6	0.110698
$782$	−3.12375e6	−0.182666
$783$	0	0
$784$	−1.37860e7	−0.801029
$785$	1.51568e7	0.877875
$786$	0	0
$787$	3.04100e7	1.75017	0.875084	−	0.483971i	$-0.160806\pi$
$787$	3.04100e7	1.75017	0.875084	+	0.483971i	$0.160806\pi$
$788$	1.80614e7	1.03618
$789$	0	0
$790$	−3.58141e7	−2.04167
$791$	2.88145e7	1.63746
$792$	0	0
$793$	1.73736e6	0.0981083
$794$	3.25631e7	1.83305
$795$	0	0
$796$	−7.39545e6	−0.413697
$797$	2.44534e6	0.136362	0.0681809	−	0.997673i	$-0.478281\pi$
$797$	2.44534e6	0.136362	0.0681809	+	0.997673i	$0.478281\pi$
$798$	0	0
$799$	−3.61843e6	−0.200518
$800$	−2.24749e6	−0.124157
$801$	0	0
$802$	1.74588e7	0.958473
$803$	−4.17658e6	−0.228576
$804$	0	0
$805$	−2.36632e7	−1.28702
$806$	−6.04412e6	−0.327714
$807$	0	0
$808$	−1.01952e7	−0.549373
$809$	2.11075e7	1.13388	0.566938	−	0.823760i	$-0.308128\pi$
$809$	2.11075e7	1.13388	0.566938	+	0.823760i	$0.308128\pi$
$810$	0	0
$811$	−1.56012e7	−0.832925	−0.416462	−	0.909153i	$-0.636730\pi$
$811$	−1.56012e7	−0.832925	−0.416462	+	0.909153i	$0.636730\pi$
$812$	1.72789e6	0.0919657
$813$	0	0
$814$	−7.36538e7	−3.89614
$815$	−7.73465e6	−0.407893
$816$	0	0
$817$	141544.	0.00741885
$818$	−1.26250e7	−0.659703
$819$	0	0
$820$	1.21651e7	0.631803
$821$	4.04677e6	0.209532	0.104766	−	0.994497i	$-0.466591\pi$
$821$	4.04677e6	0.209532	0.104766	+	0.994497i	$0.466591\pi$
$822$	0	0
$823$	2.21529e7	1.14007	0.570035	−	0.821620i	$-0.306929\pi$
$823$	2.21529e7	1.14007	0.570035	+	0.821620i	$0.306929\pi$
$824$	6.97612e6	0.357928
$825$	0	0
$826$	3.77313e7	1.92421
$827$	1.72252e7	0.875792	0.437896	−	0.899026i	$-0.355724\pi$
$827$	1.72252e7	0.875792	0.437896	+	0.899026i	$0.355724\pi$
$828$	0	0
$829$	−2.31834e7	−1.17163	−0.585816	−	0.810444i	$-0.699226\pi$
$829$	−2.31834e7	−1.17163	−0.585816	+	0.810444i	$0.699226\pi$
$830$	−5.73957e6	−0.289190
$831$	0	0
$832$	2.37068e6	0.118731
$833$	1.95446e6	0.0975919
$834$	0	0
$835$	−1.51131e7	−0.750131
$836$	−2.28829e7	−1.13239
$837$	0	0
$838$	3.92874e7	1.93260
$839$	1.07272e7	0.526118	0.263059	−	0.964780i	$-0.415269\pi$
$839$	1.07272e7	0.526118	0.263059	+	0.964780i	$0.415269\pi$
$840$	0	0
$841$	−2.02900e7	−0.989216
$842$	−4.17541e7	−2.02964
$843$	0	0
$844$	−1.80743e7	−0.873382
$845$	−1.85809e7	−0.895211
$846$	0	0
$847$	7.06432e7	3.38346
$848$	1.31638e6	0.0628623
$849$	0	0
$850$	428976.	0.0203651
$851$	−3.16111e7	−1.49629
$852$	0	0
$853$	−421069.	−0.0198144	−0.00990719	−	0.999951i	$-0.503154\pi$
$853$	−421069.	−0.0198144	−0.00990719	+	0.999951i	$0.503154\pi$
$854$	−9.07265e6	−0.425686
$855$	0	0
$856$	1.24027e7	0.578539
$857$	2.71932e7	1.26476	0.632381	−	0.774658i	$-0.282078\pi$
$857$	2.71932e7	1.26476	0.632381	+	0.774658i	$0.282078\pi$
$858$	0	0
$859$	2.01529e7	0.931869	0.465934	−	0.884819i	$-0.345718\pi$
$859$	2.01529e7	0.931869	0.465934	+	0.884819i	$0.345718\pi$
$860$	134506.	0.00620150
$861$	0	0
$862$	−1.68934e7	−0.774369
$863$	−1.88086e7	−0.859666	−0.429833	−	0.902908i	$-0.641428\pi$
$863$	−1.88086e7	−0.859666	−0.429833	+	0.902908i	$0.641428\pi$
$864$	0	0
$865$	−9.84269e6	−0.447274
$866$	−3.47294e7	−1.57363
$867$	0	0
$868$	1.28590e7	0.579306
$869$	6.33579e7	2.84611
$870$	0	0
$871$	−9.13234e6	−0.407884
$872$	−9.17456e6	−0.408596
$873$	0	0
$874$	−2.41060e7	−1.06745
$875$	−2.74302e7	−1.21118
$876$	0	0
$877$	−2.21143e7	−0.970898	−0.485449	−	0.874265i	$-0.661344\pi$
$877$	−2.21143e7	−0.970898	−0.485449	+	0.874265i	$0.661344\pi$
$878$	4.49890e7	1.96956
$879$	0	0
$880$	5.58904e7	2.43294
$881$	1.00370e7	0.435677	0.217838	−	0.975985i	$-0.430099\pi$
$881$	1.00370e7	0.435677	0.217838	+	0.975985i	$0.430099\pi$
$882$	0	0
$883$	−2.12042e7	−0.915207	−0.457604	−	0.889156i	$-0.651292\pi$
$883$	−2.12042e7	−0.915207	−0.457604	+	0.889156i	$0.651292\pi$
$884$	−911798.	−0.0392435
$885$	0	0
$886$	4.91193e7	2.10217
$887$	−1.73507e7	−0.740472	−0.370236	−	0.928938i	$-0.620723\pi$
$887$	−1.73507e7	−0.740472	−0.370236	+	0.928938i	$0.620723\pi$
$888$	0	0
$889$	−1.17080e7	−0.496855
$890$	3.87802e7	1.64110
$891$	0	0
$892$	1.47485e7	0.620636
$893$	−2.79235e7	−1.17177
$894$	0	0
$895$	2.28718e7	0.954427
$896$	2.39057e7	0.994792
$897$	0	0
$898$	3.68673e7	1.52563
$899$	1.64606e6	0.0679275
$900$	0	0
$901$	−186624.	−0.00765871
$902$	−5.28243e7	−2.16181
$903$	0	0
$904$	−1.26792e7	−0.516025
$905$	9.71791e6	0.394413
$906$	0	0
$907$	−1.02757e7	−0.414756	−0.207378	−	0.978261i	$-0.566493\pi$
$907$	−1.02757e7	−0.414756	−0.207378	+	0.978261i	$0.566493\pi$
$908$	1.89214e7	0.761622
$909$	0	0
$910$	−1.69538e7	−0.678679
$911$	3.94713e7	1.57574	0.787871	−	0.615840i	$-0.211183\pi$
$911$	3.94713e7	1.57574	0.787871	+	0.615840i	$0.211183\pi$
$912$	0	0
$913$	1.01537e7	0.403133
$914$	5.02064e6	0.198789
$915$	0	0
$916$	−3.15210e7	−1.24126
$917$	−2.32872e7	−0.914523
$918$	0	0
$919$	−1.65850e6	−0.0647779	−0.0323889	−	0.999475i	$-0.510312\pi$
$919$	−1.65850e6	−0.0647779	−0.0323889	+	0.999475i	$0.510312\pi$
$920$	1.04125e7	0.405588
$921$	0	0
$922$	−3.70453e7	−1.43518
$923$	580235.	0.0224181
$924$	0	0
$925$	4.34106e6	0.166818
$926$	3.70426e7	1.41963
$927$	0	0
$928$	−3.19334e6	−0.121724
$929$	−4.65517e6	−0.176969	−0.0884843	−	0.996078i	$-0.528202\pi$
$929$	−4.65517e6	−0.176969	−0.0884843	+	0.996078i	$0.528202\pi$
$930$	0	0
$931$	1.50826e7	0.570298
$932$	8.95502e6	0.337697
$933$	0	0
$934$	5.96951e7	2.23909
$935$	−7.92365e6	−0.296412
$936$	0	0
$937$	−3.86106e7	−1.43667	−0.718337	−	0.695696i	$-0.755096\pi$
$937$	−3.86106e7	−1.43667	−0.718337	+	0.695696i	$0.755096\pi$
$938$	4.76900e7	1.76978
$939$	0	0
$940$	−2.65352e7	−0.979495
$941$	−3.14452e7	−1.15766	−0.578828	−	0.815449i	$-0.696490\pi$
$941$	−3.14452e7	−1.15766	−0.578828	+	0.815449i	$0.696490\pi$
$942$	0	0
$943$	−2.26714e7	−0.830230
$944$	−3.82480e7	−1.39694
$945$	0	0
$946$	−584064.	−0.0212194
$947$	−9.34273e6	−0.338531	−0.169266	−	0.985570i	$-0.554140\pi$
$947$	−9.34273e6	−0.338531	−0.169266	+	0.985570i	$0.554140\pi$
$948$	0	0
$949$	−1.28428e6	−0.0462906
$950$	3.31042e6	0.119007
$951$	0	0
$952$	−2.16432e6	−0.0773979
$953$	−4.89849e6	−0.174715	−0.0873574	−	0.996177i	$-0.527842\pi$
$953$	−4.89849e6	−0.174715	−0.0873574	+	0.996177i	$0.527842\pi$
$954$	0	0
$955$	−2.02211e7	−0.717456
$956$	2.54269e6	0.0899805
$957$	0	0
$958$	4.85542e7	1.70928
$959$	1.58063e6	0.0554987
$960$	0	0
$961$	−1.63792e7	−0.572114
$962$	−2.26482e7	−0.789033
$963$	0	0
$964$	−3.41422e7	−1.18331
$965$	−1.48019e7	−0.511680
$966$	0	0
$967$	−1.13925e7	−0.391791	−0.195896	−	0.980625i	$-0.562761\pi$
$967$	−1.13925e7	−0.391791	−0.195896	+	0.980625i	$0.562761\pi$
$968$	−3.10850e7	−1.06626
$969$	0	0
$970$	−2.14285e7	−0.731244
$971$	−1.48394e7	−0.505089	−0.252544	−	0.967585i	$-0.581267\pi$
$971$	−1.48394e7	−0.505089	−0.252544	+	0.967585i	$0.581267\pi$
$972$	0	0
$973$	−2.52025e7	−0.853416
$974$	−1.83570e7	−0.620016
$975$	0	0
$976$	9.19689e6	0.309041
$977$	−2.53280e7	−0.848915	−0.424458	−	0.905448i	$-0.639535\pi$
$977$	−2.53280e7	−0.848915	−0.424458	+	0.905448i	$0.639535\pi$
$978$	0	0
$979$	−6.86051e7	−2.28770
$980$	1.43327e7	0.476719
$981$	0	0
$982$	−4.18967e7	−1.38644
$983$	3.48906e7	1.15166	0.575830	−	0.817569i	$-0.304679\pi$
$983$	3.48906e7	1.15166	0.575830	+	0.817569i	$0.304679\pi$
$984$	0	0
$985$	4.82630e7	1.58498
$986$	609511.	0.0199659
$987$	0	0
$988$	−7.03637e6	−0.229328
$989$	−250671.	−0.00814917
$990$	0	0
$991$	2.38192e7	0.770447	0.385224	−	0.922823i	$-0.374124\pi$
$991$	2.38192e7	0.770447	0.385224	+	0.922823i	$0.374124\pi$
$992$	−2.37649e7	−0.766757
$993$	0	0
$994$	−3.03005e6	−0.0972710
$995$	−1.97619e7	−0.632807
$996$	0	0
$997$	3.78624e7	1.20634	0.603171	−	0.797612i	$-0.293904\pi$
$997$	3.78624e7	1.20634	0.603171	+	0.797612i	$0.293904\pi$
$998$	3.08455e7	0.980313
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.6.a.c.1.2	yes	2
3.2	odd	2	inner	27.6.a.c.1.1	✓	2
4.3	odd	2		432.6.a.o.1.2		2
5.4	even	2		675.6.a.j.1.1		2
9.2	odd	6		81.6.c.f.28.2		4
9.4	even	3		81.6.c.f.55.1		4
9.5	odd	6		81.6.c.f.55.2		4
9.7	even	3		81.6.c.f.28.1		4
12.11	even	2		432.6.a.o.1.1		2
15.14	odd	2		675.6.a.j.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.6.a.c.1.1	✓	2	3.2	odd	2	inner
27.6.a.c.1.2	yes	2	1.1	even	1	trivial
81.6.c.f.28.1		4	9.7	even	3
81.6.c.f.28.2		4	9.2	odd	6
81.6.c.f.55.1		4	9.4	even	3
81.6.c.f.55.2		4	9.5	odd	6
432.6.a.o.1.1		2	12.11	even	2
432.6.a.o.1.2		2	4.3	odd	2
675.6.a.j.1.1		2	5.4	even	2
675.6.a.j.1.2		2	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	27.a (trivial)

\(f(q)\)	\(=\)	\(q+7.34847 q^{2} +22.0000 q^{4} +58.7878 q^{5} +167.000 q^{7} -73.4847 q^{8} +432.000 q^{10} -764.241 q^{11} -235.000 q^{13} +1227.19 q^{14} -1244.00 q^{16} +176.363 q^{17} +1361.00 q^{19} +1293.33 q^{20} -5616.00 q^{22} -2410.30 q^{23} +331.000 q^{25} -1726.89 q^{26} +3674.00 q^{28} +470.302 q^{29} +3500.00 q^{31} -6789.99 q^{32} +1296.00 q^{34} +9817.55 q^{35} +13115.0 q^{37} +10001.3 q^{38} -4320.00 q^{40} +9406.04 q^{41} +104.000 q^{43} -16813.3 q^{44} -17712.0 q^{46} -20516.9 q^{47} +11082.0 q^{49} +2432.34 q^{50} -5170.00 q^{52} -1058.18 q^{53} -44928.0 q^{55} -12271.9 q^{56} +3456.00 q^{58} +30746.0 q^{59} -7393.00 q^{61} +25719.6 q^{62} -10088.0 q^{64} -13815.1 q^{65} +38861.0 q^{67} +3879.99 q^{68} +72144.0 q^{70} -2469.09 q^{71} +5465.00 q^{73} +96375.2 q^{74} +29942.0 q^{76} -127628. q^{77} -82903.0 q^{79} -73132.0 q^{80} +69120.0 q^{82} -13286.0 q^{83} +10368.0 q^{85} +764.241 q^{86} +56160.0 q^{88} +89768.9 q^{89} -39245.0 q^{91} -53026.6 q^{92} -150768. q^{94} +80010.1 q^{95} -49603.0 q^{97} +81435.7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 44 q^{4} + 334 q^{7} + 864 q^{10} - 470 q^{13} - 2488 q^{16} + 2722 q^{19} - 11232 q^{22} + 662 q^{25} + 7348 q^{28} + 7000 q^{31} + 2592 q^{34} + 26230 q^{37} - 8640 q^{40} + 208 q^{43} - 35424 q^{46}+ \cdots - 99206 q^{97}+O(q^{100}) \) 2 * q + 44 * q^4 + 334 * q^7 + 864 * q^10 - 470 * q^13 - 2488 * q^16 + 2722 * q^19 - 11232 * q^22 + 662 * q^25 + 7348 * q^28 + 7000 * q^31 + 2592 * q^34 + 26230 * q^37 - 8640 * q^40 + 208 * q^43 - 35424 * q^46 + 22164 * q^49 - 10340 * q^52 - 89856 * q^55 + 6912 * q^58 - 14786 * q^61 - 20176 * q^64 + 77722 * q^67 + 144288 * q^70 + 10930 * q^73 + 59884 * q^76 - 165806 * q^79 + 138240 * q^82 + 20736 * q^85 + 112320 * q^88 - 78490 * q^91 - 301536 * q^94 - 99206 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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