LMFDB - Embedded newform 84.20.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [84,20,Mod(1,84)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(84, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 20, names="a")

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

chi := DirichletCharacter("84.1");

S:= CuspForms(chi, 20);

N := Newforms(S);

Level:	$ N $	$=$	$ 84 = 2^{2} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 20 $
Character orbit:	$[\chi]$	$=$	84.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:	$192.206025107$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 134334816755x^{2} + 25646529546394449x - 1331716811709552720606 $ x^4 - x^3 - 134334816755x^2 + 25646529546394449x - 1331716811709552720606
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{13}\cdot 3^{4}\cdot 5\cdot 7 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-445635.$ of defining polynomial
Character	$\chi$	$=$	84.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-19683.0 q^{3} -7.42148e6 q^{5} -4.03536e7 q^{7} +3.87420e8 q^{9} +O(q^{10})$	$q-19683.0 q^{3} -7.42148e6 q^{5} -4.03536e7 q^{7} +3.87420e8 q^{9} +2.41181e9 q^{11} -5.78266e10 q^{13} +1.46077e11 q^{15} -9.21888e10 q^{17} +9.44451e11 q^{19} +7.94280e11 q^{21} -1.02199e13 q^{23} +3.60048e13 q^{25} -7.62560e12 q^{27} -1.23844e14 q^{29} +8.36117e13 q^{31} -4.74717e13 q^{33} +2.99483e14 q^{35} +8.84580e14 q^{37} +1.13820e15 q^{39} +2.76756e15 q^{41} +9.65951e14 q^{43} -2.87523e15 q^{45} +1.28065e16 q^{47} +1.62841e15 q^{49} +1.81455e15 q^{51} +3.08223e16 q^{53} -1.78992e16 q^{55} -1.85896e16 q^{57} -1.08762e16 q^{59} +1.23570e17 q^{61} -1.56338e16 q^{63} +4.29159e17 q^{65} -1.87217e17 q^{67} +2.01159e17 q^{69} -6.87262e17 q^{71} -5.10462e17 q^{73} -7.08683e17 q^{75} -9.73252e16 q^{77} +6.03539e17 q^{79} +1.50095e17 q^{81} +1.64606e18 q^{83} +6.84177e17 q^{85} +2.43763e18 q^{87} -3.23012e18 q^{89} +2.33351e18 q^{91} -1.64573e18 q^{93} -7.00922e18 q^{95} +5.24810e18 q^{97} +9.34385e17 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 78732 q^{3} - 2502150 q^{5} - 161414428 q^{7} + 1549681956 q^{9}+O(q^{10}) $ 4 * q - 78732 * q^3 - 2502150 * q^5 - 161414428 * q^7 + 1549681956 * q^9	$ 4 q - 78732 q^{3} - 2502150 q^{5} - 161414428 q^{7} + 1549681956 q^{9} - 3798155970 q^{11} - 39152617980 q^{13} + 49249818450 q^{15} - 190872643554 q^{17} + 2299240294976 q^{19} + 3177120186324 q^{21} - 10298330967654 q^{23} + 10550140805600 q^{25} - 30502389939948 q^{27} - 77678357293560 q^{29} + 196054920663408 q^{31} + 74759103957510 q^{33} + 100970777755050 q^{35} + 829369260264428 q^{37} + 770640979700340 q^{39} + 22\!\cdots\!02 q^{41}+ \cdots - 14\!\cdots\!30 q^{99}+O(q^{100}) $ 4 * q - 78732 * q^3 - 2502150 * q^5 - 161414428 * q^7 + 1549681956 * q^9 - 3798155970 * q^11 - 39152617980 * q^13 + 49249818450 * q^15 - 190872643554 * q^17 + 2299240294976 * q^19 + 3177120186324 * q^21 - 10298330967654 * q^23 + 10550140805600 * q^25 - 30502389939948 * q^27 - 77678357293560 * q^29 + 196054920663408 * q^31 + 74759103957510 * q^33 + 100970777755050 * q^35 + 829369260264428 * q^37 + 770640979700340 * q^39 + 2210157258858402 * q^41 + 3704496271875056 * q^43 - 969384176551350 * q^45 + 4910647758864108 * q^47 + 6513654391641796 * q^49 + 3756946243073382 * q^51 + 4583139659223684 * q^53 - 49874361740165200 * q^55 - 45255946726012608 * q^57 + 10541900786867604 * q^59 + 67528372286847048 * q^61 - 62535256627415292 * q^63 + 513738934168590300 * q^65 + 232506796904165236 * q^67 + 202702048436333682 * q^69 + 264200775007920198 * q^71 + 668786810310501652 * q^73 - 207658421476624800 * q^75 + 153269293338083790 * q^77 + 1869350307247472268 * q^79 + 600378541187996484 * q^81 + 4190113921662329040 * q^83 + 108669289188945500 * q^85 + 1528943106609141480 * q^87 - 4059161776602418782 * q^89 + 1579949358986053860 * q^91 - 3858949003417859664 * q^93 + 7625582268210966600 * q^95 - 1033481664034434692 * q^97 - 1471483443195669330 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−19683.0	−0.577350
$3$	−19683.0	−0.577350
$4$	0	0
$5$	−7.42148e6	−1.69932	−0.849660	−	0.527331i	$-0.823193\pi$
$5$	−7.42148e6	−1.69932	−0.849660	+	0.527331i	$0.823193\pi$
$6$	0	0
$7$	−4.03536e7	−0.377964
$7$	−4.03536e7	−0.377964
$8$	0	0
$9$	3.87420e8	0.333333
$10$	0	0
$11$	2.41181e9	0.308399	0.154199	−	0.988040i	$-0.450720\pi$
$11$	2.41181e9	0.308399	0.154199	+	0.988040i	$0.450720\pi$
$12$	0	0
$13$	−5.78266e10	−1.51240	−0.756199	−	0.654342i	$-0.772946\pi$
$13$	−5.78266e10	−1.51240	−0.756199	+	0.654342i	$0.772946\pi$
$14$	0	0
$15$	1.46077e11	0.981103
$16$	0	0
$17$	−9.21888e10	−0.188544	−0.0942722	−	0.995546i	$-0.530052\pi$
$17$	−9.21888e10	−0.188544	−0.0942722	+	0.995546i	$0.530052\pi$
$18$	0	0
$19$	9.44451e11	0.671460	0.335730	−	0.941958i	$-0.391017\pi$
$19$	9.44451e11	0.671460	0.335730	+	0.941958i	$0.391017\pi$
$20$	0	0
$21$	7.94280e11	0.218218
$22$	0	0
$23$	−1.02199e13	−1.18313	−0.591566	−	0.806256i	$-0.701490\pi$
$23$	−1.02199e13	−1.18313	−0.591566	+	0.806256i	$0.701490\pi$
$24$	0	0
$25$	3.60048e13	1.88769
$26$	0	0
$27$	−7.62560e12	−0.192450
$28$	0	0
$29$	−1.23844e14	−1.58524	−0.792620	−	0.609716i	$-0.791283\pi$
$29$	−1.23844e14	−1.58524	−0.792620	+	0.609716i	$0.791283\pi$
$30$	0	0
$31$	8.36117e13	0.567977	0.283989	−	0.958828i	$-0.408342\pi$
$31$	8.36117e13	0.567977	0.283989	+	0.958828i	$0.408342\pi$
$32$	0	0
$33$	−4.74717e13	−0.178054
$34$	0	0
$35$	2.99483e14	0.642283
$36$	0	0
$37$	8.84580e14	1.11898	0.559488	−	0.828839i	$-0.310998\pi$
$37$	8.84580e14	1.11898	0.559488	+	0.828839i	$0.310998\pi$
$38$	0	0
$39$	1.13820e15	0.873183
$40$	0	0
$41$	2.76756e15	1.32023	0.660117	−	0.751163i	$-0.270507\pi$
$41$	2.76756e15	1.32023	0.660117	+	0.751163i	$0.270507\pi$
$42$	0	0
$43$	9.65951e14	0.293093	0.146546	−	0.989204i	$-0.453184\pi$
$43$	9.65951e14	0.293093	0.146546	+	0.989204i	$0.453184\pi$
$44$	0	0
$45$	−2.87523e15	−0.566440
$46$	0	0
$47$	1.28065e16	1.66917	0.834584	−	0.550881i	$-0.185708\pi$
$47$	1.28065e16	1.66917	0.834584	+	0.550881i	$0.185708\pi$
$48$	0	0
$49$	1.62841e15	0.142857
$50$	0	0
$51$	1.81455e15	0.108856
$52$	0	0
$53$	3.08223e16	1.28305	0.641526	−	0.767101i	$-0.278301\pi$
$53$	3.08223e16	1.28305	0.641526	+	0.767101i	$0.278301\pi$
$54$	0	0
$55$	−1.78992e16	−0.524068
$56$	0	0
$57$	−1.85896e16	−0.387668
$58$	0	0
$59$	−1.08762e16	−0.163450	−0.0817249	−	0.996655i	$-0.526043\pi$
$59$	−1.08762e16	−0.163450	−0.0817249	+	0.996655i	$0.526043\pi$
$60$	0	0
$61$	1.23570e17	1.35294	0.676469	−	0.736471i	$-0.263509\pi$
$61$	1.23570e17	1.35294	0.676469	+	0.736471i	$0.263509\pi$
$62$	0	0
$63$	−1.56338e16	−0.125988
$64$	0	0
$65$	4.29159e17	2.57005
$66$	0	0
$67$	−1.87217e17	−0.840688	−0.420344	−	0.907365i	$-0.638091\pi$
$67$	−1.87217e17	−0.840688	−0.420344	+	0.907365i	$0.638091\pi$
$68$	0	0
$69$	2.01159e17	0.683082
$70$	0	0
$71$	−6.87262e17	−1.77897	−0.889483	−	0.456968i	$-0.848935\pi$
$71$	−6.87262e17	−1.77897	−0.889483	+	0.456968i	$0.848935\pi$
$72$	0	0
$73$	−5.10462e17	−1.01484	−0.507419	−	0.861700i	$-0.669400\pi$
$73$	−5.10462e17	−1.01484	−0.507419	+	0.861700i	$0.669400\pi$
$74$	0	0
$75$	−7.08683e17	−1.08986
$76$	0	0
$77$	−9.73252e16	−0.116564
$78$	0	0
$79$	6.03539e17	0.566562	0.283281	−	0.959037i	$-0.408577\pi$
$79$	6.03539e17	0.566562	0.283281	+	0.959037i	$0.408577\pi$
$80$	0	0
$81$	1.50095e17	0.111111
$82$	0	0
$83$	1.64606e18	0.966504	0.483252	−	0.875481i	$-0.339455\pi$
$83$	1.64606e18	0.966504	0.483252	+	0.875481i	$0.339455\pi$
$84$	0	0
$85$	6.84177e17	0.320397
$86$	0	0
$87$	2.43763e18	0.915239
$88$	0	0
$89$	−3.23012e18	−0.977269	−0.488634	−	0.872489i	$-0.662505\pi$
$89$	−3.23012e18	−0.977269	−0.488634	+	0.872489i	$0.662505\pi$
$90$	0	0
$91$	2.33351e18	0.571633
$92$	0	0
$93$	−1.64573e18	−0.327922
$94$	0	0
$95$	−7.00922e18	−1.14103
$96$	0	0
$97$	5.24810e18	0.700924	0.350462	−	0.936577i	$-0.386025\pi$
$97$	5.24810e18	0.700924	0.350462	+	0.936577i	$0.386025\pi$
$98$	0	0
$99$	9.34385e17	0.102800
$100$	0	0
$101$	1.67523e19	1.52412	0.762062	−	0.647504i	$-0.224187\pi$
$101$	1.67523e19	1.52412	0.762062	+	0.647504i	$0.224187\pi$
$102$	0	0
$103$	8.47103e18	0.639709	0.319855	−	0.947467i	$-0.396366\pi$
$103$	8.47103e18	0.639709	0.319855	+	0.947467i	$0.396366\pi$
$104$	0	0
$105$	−5.89473e18	−0.370822
$106$	0	0
$107$	9.14609e18	0.480939	0.240469	−	0.970657i	$-0.422699\pi$
$107$	9.14609e18	0.480939	0.240469	+	0.970657i	$0.422699\pi$
$108$	0	0
$109$	2.10514e18	0.0928389	0.0464194	−	0.998922i	$-0.485219\pi$
$109$	2.10514e18	0.0928389	0.0464194	+	0.998922i	$0.485219\pi$
$110$	0	0
$111$	−1.74112e19	−0.646041
$112$	0	0
$113$	2.24463e19	0.702909	0.351455	−	0.936205i	$-0.385687\pi$
$113$	2.24463e19	0.702909	0.351455	+	0.936205i	$0.385687\pi$
$114$	0	0
$115$	7.58469e19	2.01052
$116$	0	0
$117$	−2.24032e19	−0.504133
$118$	0	0
$119$	3.72015e18	0.0712630
$120$	0	0
$121$	−5.53423e19	−0.904890
$122$	0	0
$123$	−5.44739e19	−0.762237
$124$	0	0
$125$	−1.25656e20	−1.50847
$126$	0	0
$127$	−1.74797e20	−1.80467	−0.902334	−	0.431037i	$-0.858148\pi$
$127$	−1.74797e20	−1.80467	−0.902334	+	0.431037i	$0.858148\pi$
$128$	0	0
$129$	−1.90128e19	−0.169217
$130$	0	0
$131$	−8.32280e19	−0.640018	−0.320009	−	0.947415i	$-0.603686\pi$
$131$	−8.32280e19	−0.640018	−0.320009	+	0.947415i	$0.603686\pi$
$132$	0	0
$133$	−3.81120e19	−0.253788
$134$	0	0
$135$	5.65932e19	0.327034
$136$	0	0
$137$	−3.20908e20	−1.61263	−0.806316	−	0.591485i	$-0.798542\pi$
$137$	−3.20908e20	−1.61263	−0.806316	+	0.591485i	$0.798542\pi$
$138$	0	0
$139$	2.89495e20	1.26765	0.633827	−	0.773475i	$-0.281483\pi$
$139$	2.89495e20	1.26765	0.633827	+	0.773475i	$0.281483\pi$
$140$	0	0
$141$	−2.52070e20	−0.963695
$142$	0	0
$143$	−1.39467e20	−0.466422
$144$	0	0
$145$	9.19107e20	2.69383
$146$	0	0
$147$	−3.20521e19	−0.0824786
$148$	0	0
$149$	1.26314e20	0.285880	0.142940	−	0.989731i	$-0.454344\pi$
$149$	1.26314e20	0.285880	0.142940	+	0.989731i	$0.454344\pi$
$150$	0	0
$151$	−3.74041e20	−0.745828	−0.372914	−	0.927866i	$-0.621641\pi$
$151$	−3.74041e20	−0.745828	−0.372914	+	0.927866i	$0.621641\pi$
$152$	0	0
$153$	−3.57158e19	−0.0628481
$154$	0	0
$155$	−6.20522e20	−0.965175
$156$	0	0
$157$	1.03116e21	1.41997	0.709986	−	0.704215i	$-0.248701\pi$
$157$	1.03116e21	1.41997	0.709986	+	0.704215i	$0.248701\pi$
$158$	0	0
$159$	−6.06676e20	−0.740771
$160$	0	0
$161$	4.12411e20	0.447182
$162$	0	0
$163$	4.77294e20	0.460261	0.230130	−	0.973160i	$-0.426085\pi$
$163$	4.77294e20	0.460261	0.230130	+	0.973160i	$0.426085\pi$
$164$	0	0
$165$	3.52310e20	0.302571
$166$	0	0
$167$	−2.48669e20	−0.190465	−0.0952326	−	0.995455i	$-0.530359\pi$
$167$	−2.48669e20	−0.190465	−0.0952326	+	0.995455i	$0.530359\pi$
$168$	0	0
$169$	1.88200e21	1.28735
$170$	0	0
$171$	3.65900e20	0.223820
$172$	0	0
$173$	−6.33575e20	−0.347025	−0.173512	−	0.984832i	$-0.555512\pi$
$173$	−6.33575e20	−0.347025	−0.173512	+	0.984832i	$0.555512\pi$
$174$	0	0
$175$	−1.45292e21	−0.713480
$176$	0	0
$177$	2.14077e20	0.0943678
$178$	0	0
$179$	1.52239e21	0.603145	0.301573	−	0.953443i	$-0.402488\pi$
$179$	1.52239e21	0.603145	0.301573	+	0.953443i	$0.402488\pi$
$180$	0	0
$181$	−1.88011e21	−0.670249	−0.335124	−	0.942174i	$-0.608778\pi$
$181$	−1.88011e21	−0.670249	−0.335124	+	0.942174i	$0.608778\pi$
$182$	0	0
$183$	−2.43222e21	−0.781120
$184$	0	0
$185$	−6.56489e21	−1.90150
$186$	0	0
$187$	−2.22342e20	−0.0581468
$188$	0	0
$189$	3.07720e20	0.0727393
$190$	0	0
$191$	−6.09496e21	−1.30363	−0.651814	−	0.758379i	$-0.725992\pi$
$191$	−6.09496e21	−1.30363	−0.651814	+	0.758379i	$0.725992\pi$
$192$	0	0
$193$	4.44395e21	0.860943	0.430472	−	0.902604i	$-0.358347\pi$
$193$	4.44395e21	0.860943	0.430472	+	0.902604i	$0.358347\pi$
$194$	0	0
$195$	−8.44714e21	−1.48382
$196$	0	0
$197$	2.91733e21	0.465111	0.232556	−	0.972583i	$-0.425291\pi$
$197$	2.91733e21	0.465111	0.232556	+	0.972583i	$0.425291\pi$
$198$	0	0
$199$	−8.79289e21	−1.27358	−0.636791	−	0.771036i	$-0.719739\pi$
$199$	−8.79289e21	−1.27358	−0.636791	+	0.771036i	$0.719739\pi$
$200$	0	0
$201$	3.68500e21	0.485372
$202$	0	0
$203$	4.99756e21	0.599164
$204$	0	0
$205$	−2.05394e22	−2.24350
$206$	0	0
$207$	−3.95941e21	−0.394377
$208$	0	0
$209$	2.27784e21	0.207078
$210$	0	0
$211$	−2.01127e22	−1.67027	−0.835137	−	0.550042i	$-0.814612\pi$
$211$	−2.01127e22	−1.67027	−0.835137	+	0.550042i	$0.814612\pi$
$212$	0	0
$213$	1.35274e22	1.02709
$214$	0	0
$215$	−7.16878e21	−0.498058
$216$	0	0
$217$	−3.37403e21	−0.214675
$218$	0	0
$219$	1.00474e22	0.585916
$220$	0	0
$221$	5.33097e21	0.285154
$222$	0	0
$223$	2.35075e22	1.15428	0.577138	−	0.816646i	$-0.304169\pi$
$223$	2.35075e22	1.15428	0.577138	+	0.816646i	$0.304169\pi$
$224$	0	0
$225$	1.39490e22	0.629230
$226$	0	0
$227$	−4.01075e22	−1.66334	−0.831668	−	0.555273i	$-0.812614\pi$
$227$	−4.01075e22	−1.66334	−0.831668	+	0.555273i	$0.812614\pi$
$228$	0	0
$229$	−7.43737e21	−0.283780	−0.141890	−	0.989882i	$-0.545318\pi$
$229$	−7.43737e21	−0.283780	−0.141890	+	0.989882i	$0.545318\pi$
$230$	0	0
$231$	1.91565e21	0.0672981
$232$	0	0
$233$	−2.88178e22	−0.932781	−0.466391	−	0.884579i	$-0.654446\pi$
$233$	−2.88178e22	−0.932781	−0.466391	+	0.884579i	$0.654446\pi$
$234$	0	0
$235$	−9.50430e22	−2.83645
$236$	0	0
$237$	−1.18794e22	−0.327105
$238$	0	0
$239$	3.31483e22	0.842716	0.421358	−	0.906894i	$-0.361554\pi$
$239$	3.31483e22	0.842716	0.421358	+	0.906894i	$0.361554\pi$
$240$	0	0
$241$	−2.17044e22	−0.509784	−0.254892	−	0.966969i	$-0.582040\pi$
$241$	−2.17044e22	−0.509784	−0.254892	+	0.966969i	$0.582040\pi$
$242$	0	0
$243$	−2.95431e21	−0.0641500
$244$	0	0
$245$	−1.20852e22	−0.242760
$246$	0	0
$247$	−5.46145e22	−1.01552
$248$	0	0
$249$	−3.23994e22	−0.558011
$250$	0	0
$251$	2.76000e22	0.440564	0.220282	−	0.975436i	$-0.429302\pi$
$251$	2.76000e22	0.440564	0.220282	+	0.975436i	$0.429302\pi$
$252$	0	0
$253$	−2.46485e22	−0.364876
$254$	0	0
$255$	−1.34667e22	−0.184981
$256$	0	0
$257$	4.51935e22	0.576383	0.288192	−	0.957573i	$-0.406946\pi$
$257$	4.51935e22	0.576383	0.288192	+	0.957573i	$0.406946\pi$
$258$	0	0
$259$	−3.56960e22	−0.422933
$260$	0	0
$261$	−4.79798e22	−0.528413
$262$	0	0
$263$	−1.68101e23	−1.72183	−0.860914	−	0.508750i	$-0.830108\pi$
$263$	−1.68101e23	−1.72183	−0.860914	+	0.508750i	$0.830108\pi$
$264$	0	0
$265$	−2.28747e23	−2.18032
$266$	0	0
$267$	6.35785e22	0.564226
$268$	0	0
$269$	−9.80208e22	−0.810348	−0.405174	−	0.914240i	$-0.632789\pi$
$269$	−9.80208e22	−0.810348	−0.405174	+	0.914240i	$0.632789\pi$
$270$	0	0
$271$	−4.24236e22	−0.326888	−0.163444	−	0.986553i	$-0.552260\pi$
$271$	−4.24236e22	−0.326888	−0.163444	+	0.986553i	$0.552260\pi$
$272$	0	0
$273$	−4.59305e22	−0.330032
$274$	0	0
$275$	8.68368e22	0.582161
$276$	0	0
$277$	2.25454e23	1.41091	0.705457	−	0.708753i	$-0.250742\pi$
$277$	2.25454e23	1.41091	0.705457	+	0.708753i	$0.250742\pi$
$278$	0	0
$279$	3.23929e22	0.189326
$280$	0	0
$281$	−2.05444e23	−1.12197	−0.560987	−	0.827825i	$-0.689578\pi$
$281$	−2.05444e23	−1.12197	−0.560987	+	0.827825i	$0.689578\pi$
$282$	0	0
$283$	3.44948e23	1.76109	0.880547	−	0.473958i	$-0.157175\pi$
$283$	3.44948e23	1.76109	0.880547	+	0.473958i	$0.157175\pi$
$284$	0	0
$285$	1.37963e23	0.658772
$286$	0	0
$287$	−1.11681e23	−0.499001
$288$	0	0
$289$	−2.30574e23	−0.964451
$290$	0	0
$291$	−1.03298e23	−0.404679
$292$	0	0
$293$	−2.07798e23	−0.762780	−0.381390	−	0.924414i	$-0.624555\pi$
$293$	−2.07798e23	−0.762780	−0.381390	+	0.924414i	$0.624555\pi$
$294$	0	0
$295$	8.07177e22	0.277753
$296$	0	0
$297$	−1.83915e22	−0.0593514
$298$	0	0
$299$	5.90984e23	1.78937
$300$	0	0
$301$	−3.89796e22	−0.110779
$302$	0	0
$303$	−3.29735e23	−0.879953
$304$	0	0
$305$	−9.17069e23	−2.29908
$306$	0	0
$307$	7.84092e23	1.84736	0.923682	−	0.383160i	$-0.125164\pi$
$307$	7.84092e23	1.84736	0.923682	+	0.383160i	$0.125164\pi$
$308$	0	0
$309$	−1.66735e23	−0.369336
$310$	0	0
$311$	−5.03827e23	−1.04968	−0.524841	−	0.851201i	$-0.675875\pi$
$311$	−5.03827e23	−1.04968	−0.524841	+	0.851201i	$0.675875\pi$
$312$	0	0
$313$	5.78910e23	1.13485	0.567427	−	0.823424i	$-0.307939\pi$
$313$	5.78910e23	1.13485	0.567427	+	0.823424i	$0.307939\pi$
$314$	0	0
$315$	1.16026e23	0.214094
$316$	0	0
$317$	2.39798e23	0.416661	0.208331	−	0.978058i	$-0.433197\pi$
$317$	2.39798e23	0.416661	0.208331	+	0.978058i	$0.433197\pi$
$318$	0	0
$319$	−2.98689e23	−0.488886
$320$	0	0
$321$	−1.80023e23	−0.277670
$322$	0	0
$323$	−8.70678e22	−0.126600
$324$	0	0
$325$	−2.08204e24	−2.85494
$326$	0	0
$327$	−4.14355e22	−0.0536006
$328$	0	0
$329$	−5.16788e23	−0.630886
$330$	0	0
$331$	−1.61656e23	−0.186306	−0.0931530	−	0.995652i	$-0.529695\pi$
$331$	−1.61656e23	−0.186306	−0.0931530	+	0.995652i	$0.529695\pi$
$332$	0	0
$333$	3.42704e23	0.372992
$334$	0	0
$335$	1.38943e24	1.42860
$336$	0	0
$337$	1.81253e24	1.76117	0.880583	−	0.473892i	$-0.157151\pi$
$337$	1.81253e24	1.76117	0.880583	+	0.473892i	$0.157151\pi$
$338$	0	0
$339$	−4.41810e23	−0.405825
$340$	0	0
$341$	2.01655e23	0.175163
$342$	0	0
$343$	−6.57124e22	−0.0539949
$344$	0	0
$345$	−1.49290e24	−1.16077
$346$	0	0
$347$	−4.40964e23	−0.324544	−0.162272	−	0.986746i	$-0.551882\pi$
$347$	−4.40964e23	−0.324544	−0.162272	+	0.986746i	$0.551882\pi$
$348$	0	0
$349$	−8.09131e23	−0.563868	−0.281934	−	0.959434i	$-0.590976\pi$
$349$	−8.09131e23	−0.563868	−0.281934	+	0.959434i	$0.590976\pi$
$350$	0	0
$351$	4.40963e23	0.291061
$352$	0	0
$353$	1.19548e24	0.747621	0.373810	−	0.927505i	$-0.378051\pi$
$353$	1.19548e24	0.747621	0.373810	+	0.927505i	$0.378051\pi$
$354$	0	0
$355$	5.10050e24	3.02303
$356$	0	0
$357$	−7.32237e22	−0.0411437
$358$	0	0
$359$	−5.19682e23	−0.276911	−0.138456	−	0.990369i	$-0.544214\pi$
$359$	−5.19682e23	−0.276911	−0.138456	+	0.990369i	$0.544214\pi$
$360$	0	0
$361$	−1.08643e24	−0.549141
$362$	0	0
$363$	1.08930e24	0.522439
$364$	0	0
$365$	3.78838e24	1.72453
$366$	0	0
$367$	−2.96383e23	−0.128093	−0.0640465	−	0.997947i	$-0.520401\pi$
$367$	−2.96383e23	−0.128093	−0.0640465	+	0.997947i	$0.520401\pi$
$368$	0	0
$369$	1.07221e24	0.440078
$370$	0	0
$371$	−1.24379e24	−0.484948
$372$	0	0
$373$	−3.40901e24	−1.26297	−0.631487	−	0.775387i	$-0.717555\pi$
$373$	−3.40901e24	−1.26297	−0.631487	+	0.775387i	$0.717555\pi$
$374$	0	0
$375$	2.47328e24	0.870915
$376$	0	0
$377$	7.16150e24	2.39751
$378$	0	0
$379$	3.96907e24	1.26362	0.631810	−	0.775123i	$-0.282312\pi$
$379$	3.96907e24	1.26362	0.631810	+	0.775123i	$0.282312\pi$
$380$	0	0
$381$	3.44052e24	1.04193
$382$	0	0
$383$	1.83069e24	0.527506	0.263753	−	0.964590i	$-0.415040\pi$
$383$	1.83069e24	0.527506	0.263753	+	0.964590i	$0.415040\pi$
$384$	0	0
$385$	7.22297e23	0.198079
$386$	0	0
$387$	3.74229e23	0.0976975
$388$	0	0
$389$	−9.01500e23	−0.224101	−0.112051	−	0.993702i	$-0.535742\pi$
$389$	−9.01500e23	−0.224101	−0.112051	+	0.993702i	$0.535742\pi$
$390$	0	0
$391$	9.42163e23	0.223073
$392$	0	0
$393$	1.63818e24	0.369514
$394$	0	0
$395$	−4.47915e24	−0.962771
$396$	0	0
$397$	5.42683e24	1.11182	0.555912	−	0.831241i	$-0.312369\pi$
$397$	5.42683e24	1.11182	0.555912	+	0.831241i	$0.312369\pi$
$398$	0	0
$399$	7.50159e23	0.146525
$400$	0	0
$401$	4.62223e24	0.860954	0.430477	−	0.902602i	$-0.358345\pi$
$401$	4.62223e24	0.860954	0.430477	+	0.902602i	$0.358345\pi$
$402$	0	0
$403$	−4.83498e24	−0.859007
$404$	0	0
$405$	−1.11392e24	−0.188813
$406$	0	0
$407$	2.13344e24	0.345091
$408$	0	0
$409$	9.17087e24	1.41592	0.707961	−	0.706252i	$-0.249615\pi$
$409$	9.17087e24	1.41592	0.707961	+	0.706252i	$0.249615\pi$
$410$	0	0
$411$	6.31643e24	0.931053
$412$	0	0
$413$	4.38895e23	0.0617782
$414$	0	0
$415$	−1.22162e25	−1.64240
$416$	0	0
$417$	−5.69813e24	−0.731880
$418$	0	0
$419$	−2.20754e24	−0.270942	−0.135471	−	0.990781i	$-0.543255\pi$
$419$	−2.20754e24	−0.270942	−0.135471	+	0.990781i	$0.543255\pi$
$420$	0	0
$421$	−1.56078e25	−1.83089	−0.915443	−	0.402449i	$-0.868159\pi$
$421$	−1.56078e25	−1.83089	−0.915443	+	0.402449i	$0.868159\pi$
$422$	0	0
$423$	4.96149e24	0.556389
$424$	0	0
$425$	−3.31924e24	−0.355913
$426$	0	0
$427$	−4.98648e24	−0.511363
$428$	0	0
$429$	2.74513e24	0.269289
$430$	0	0
$431$	1.65214e23	0.0155064	0.00775322	−	0.999970i	$-0.497532\pi$
$431$	1.65214e23	0.0155064	0.00775322	+	0.999970i	$0.497532\pi$
$432$	0	0
$433$	1.66619e24	0.149655	0.0748274	−	0.997197i	$-0.476159\pi$
$433$	1.66619e24	0.149655	0.0748274	+	0.997197i	$0.476159\pi$
$434$	0	0
$435$	−1.80908e25	−1.55528
$436$	0	0
$437$	−9.65222e24	−0.794427
$438$	0	0
$439$	2.11912e24	0.167010	0.0835051	−	0.996507i	$-0.473389\pi$
$439$	2.11912e24	0.167010	0.0835051	+	0.996507i	$0.473389\pi$
$440$	0	0
$441$	6.30881e23	0.0476190
$442$	0	0
$443$	1.58223e25	1.14403	0.572013	−	0.820245i	$-0.306163\pi$
$443$	1.58223e25	1.14403	0.572013	+	0.820245i	$0.306163\pi$
$444$	0	0
$445$	2.39723e25	1.66069
$446$	0	0
$447$	−2.48625e24	−0.165053
$448$	0	0
$449$	1.22780e25	0.781246	0.390623	−	0.920551i	$-0.372260\pi$
$449$	1.22780e25	0.781246	0.390623	+	0.920551i	$0.372260\pi$
$450$	0	0
$451$	6.67483e24	0.407158
$452$	0	0
$453$	7.36226e24	0.430604
$454$	0	0
$455$	−1.73181e25	−0.971387
$456$	0	0
$457$	6.45971e24	0.347543	0.173772	−	0.984786i	$-0.444405\pi$
$457$	6.45971e24	0.347543	0.173772	+	0.984786i	$0.444405\pi$
$458$	0	0
$459$	7.02995e23	0.0362854
$460$	0	0
$461$	−9.45492e24	−0.468273	−0.234136	−	0.972204i	$-0.575226\pi$
$461$	−9.45492e24	−0.468273	−0.234136	+	0.972204i	$0.575226\pi$
$462$	0	0
$463$	2.70061e25	1.28364	0.641819	−	0.766856i	$-0.278180\pi$
$463$	2.70061e25	1.28364	0.641819	+	0.766856i	$0.278180\pi$
$464$	0	0
$465$	1.22137e25	0.557244
$466$	0	0
$467$	−1.26862e25	−0.555677	−0.277839	−	0.960628i	$-0.589618\pi$
$467$	−1.26862e25	−0.555677	−0.277839	+	0.960628i	$0.589618\pi$
$468$	0	0
$469$	7.55489e24	0.317750
$470$	0	0
$471$	−2.02964e25	−0.819822
$472$	0	0
$473$	2.32969e24	0.0903894
$474$	0	0
$475$	3.40048e25	1.26751
$476$	0	0
$477$	1.19412e25	0.427684
$478$	0	0
$479$	−2.23471e25	−0.769191	−0.384596	−	0.923085i	$-0.625659\pi$
$479$	−2.23471e25	−0.769191	−0.384596	+	0.923085i	$0.625659\pi$
$480$	0	0
$481$	−5.11523e25	−1.69234
$482$	0	0
$483$	−8.11748e24	−0.258181
$484$	0	0
$485$	−3.89487e25	−1.19109
$486$	0	0
$487$	2.10184e25	0.618123	0.309061	−	0.951042i	$-0.399985\pi$
$487$	2.10184e25	0.618123	0.309061	+	0.951042i	$0.399985\pi$
$488$	0	0
$489$	−9.39458e24	−0.265732
$490$	0	0
$491$	−3.24164e25	−0.882044	−0.441022	−	0.897496i	$-0.645384\pi$
$491$	−3.24164e25	−0.882044	−0.441022	+	0.897496i	$0.645384\pi$
$492$	0	0
$493$	1.14171e25	0.298888
$494$	0	0
$495$	−6.93451e24	−0.174689
$496$	0	0
$497$	2.77335e25	0.672386
$498$	0	0
$499$	3.50988e25	0.819101	0.409551	−	0.912287i	$-0.365686\pi$
$499$	3.50988e25	0.819101	0.409551	+	0.912287i	$0.365686\pi$
$500$	0	0
$501$	4.89456e24	0.109965
$502$	0	0
$503$	1.55427e25	0.336226	0.168113	−	0.985768i	$-0.446233\pi$
$503$	1.55427e25	0.336226	0.168113	+	0.985768i	$0.446233\pi$
$504$	0	0
$505$	−1.24326e26	−2.58997
$506$	0	0
$507$	−3.70434e25	−0.743251
$508$	0	0
$509$	−8.78767e25	−1.69846	−0.849229	−	0.528025i	$-0.822933\pi$
$509$	−8.78767e25	−1.69846	−0.849229	+	0.528025i	$0.822933\pi$
$510$	0	0
$511$	2.05990e25	0.383572
$512$	0	0
$513$	−7.20201e24	−0.129223
$514$	0	0
$515$	−6.28676e25	−1.08707
$516$	0	0
$517$	3.08868e25	0.514769
$518$	0	0
$519$	1.24707e25	0.200355
$520$	0	0
$521$	1.38719e25	0.214870	0.107435	−	0.994212i	$-0.465736\pi$
$521$	1.38719e25	0.214870	0.107435	+	0.994212i	$0.465736\pi$
$522$	0	0
$523$	−9.05520e25	−1.35248	−0.676242	−	0.736680i	$-0.736393\pi$
$523$	−9.05520e25	−1.35248	−0.676242	+	0.736680i	$0.736393\pi$
$524$	0	0
$525$	2.85979e25	0.411928
$526$	0	0
$527$	−7.70806e24	−0.107089
$528$	0	0
$529$	2.98314e25	0.399802
$530$	0	0
$531$	−4.21367e24	−0.0544832
$532$	0	0
$533$	−1.60039e26	−1.99672
$534$	0	0
$535$	−6.78775e25	−0.817269
$536$	0	0
$537$	−2.99652e25	−0.348226
$538$	0	0
$539$	3.92742e24	0.0440570
$540$	0	0
$541$	−1.30366e26	−1.41186	−0.705929	−	0.708283i	$-0.749470\pi$
$541$	−1.30366e26	−1.41186	−0.705929	+	0.708283i	$0.749470\pi$
$542$	0	0
$543$	3.70061e25	0.386968
$544$	0	0
$545$	−1.56233e25	−0.157763
$546$	0	0
$547$	1.40637e26	1.37157	0.685786	−	0.727803i	$-0.259459\pi$
$547$	1.40637e26	1.37157	0.685786	+	0.727803i	$0.259459\pi$
$548$	0	0
$549$	4.78734e25	0.450980
$550$	0	0
$551$	−1.16965e26	−1.06443
$552$	0	0
$553$	−2.43550e25	−0.214140
$554$	0	0
$555$	1.29217e26	1.09783
$556$	0	0
$557$	2.01464e26	1.65414	0.827071	−	0.562097i	$-0.190005\pi$
$557$	2.01464e26	1.65414	0.827071	+	0.562097i	$0.190005\pi$
$558$	0	0
$559$	−5.58577e25	−0.443273
$560$	0	0
$561$	4.37635e24	0.0335711
$562$	0	0
$563$	1.55598e26	1.15392	0.576958	−	0.816774i	$-0.304239\pi$
$563$	1.55598e26	1.15392	0.576958	+	0.816774i	$0.304239\pi$
$564$	0	0
$565$	−1.66585e26	−1.19447
$566$	0	0
$567$	−6.05686e24	−0.0419961
$568$	0	0
$569$	1.62590e26	1.09026	0.545128	−	0.838353i	$-0.316481\pi$
$569$	1.62590e26	1.09026	0.545128	+	0.838353i	$0.316481\pi$
$570$	0	0
$571$	1.25420e26	0.813437	0.406719	−	0.913553i	$-0.366673\pi$
$571$	1.25420e26	0.813437	0.406719	+	0.913553i	$0.366673\pi$
$572$	0	0
$573$	1.19967e26	0.752650
$574$	0	0
$575$	−3.67967e26	−2.23339
$576$	0	0
$577$	−4.43622e25	−0.260521	−0.130261	−	0.991480i	$-0.541581\pi$
$577$	−4.43622e25	−0.260521	−0.130261	+	0.991480i	$0.541581\pi$
$578$	0	0
$579$	−8.74703e25	−0.497066
$580$	0	0
$581$	−6.64244e25	−0.365304
$582$	0	0
$583$	7.43376e25	0.395692
$584$	0	0
$585$	1.66265e26	0.856683
$586$	0	0
$587$	−3.21627e25	−0.160432	−0.0802159	−	0.996778i	$-0.525561\pi$
$587$	−3.21627e25	−0.160432	−0.0802159	+	0.996778i	$0.525561\pi$
$588$	0	0
$589$	7.89672e25	0.381374
$590$	0	0
$591$	−5.74218e25	−0.268532
$592$	0	0
$593$	−3.03595e26	−1.37491	−0.687456	−	0.726226i	$-0.741272\pi$
$593$	−3.03595e26	−1.37491	−0.687456	+	0.726226i	$0.741272\pi$
$594$	0	0
$595$	−2.76090e25	−0.121099
$596$	0	0
$597$	1.73070e26	0.735303
$598$	0	0
$599$	−2.20580e26	−0.907845	−0.453922	−	0.891041i	$-0.649976\pi$
$599$	−2.20580e26	−0.907845	−0.453922	+	0.891041i	$0.649976\pi$
$600$	0	0
$601$	−1.96572e26	−0.783817	−0.391909	−	0.920004i	$-0.628185\pi$
$601$	−1.96572e26	−0.783817	−0.391909	+	0.920004i	$0.628185\pi$
$602$	0	0
$603$	−7.25318e25	−0.280229
$604$	0	0
$605$	4.10721e26	1.53770
$606$	0	0
$607$	2.04278e26	0.741188	0.370594	−	0.928795i	$-0.379154\pi$
$607$	2.04278e26	0.741188	0.370594	+	0.928795i	$0.379154\pi$
$608$	0	0
$609$	−9.83670e25	−0.345928
$610$	0	0
$611$	−7.40556e26	−2.52445
$612$	0	0
$613$	−8.50047e25	−0.280911	−0.140455	−	0.990087i	$-0.544857\pi$
$613$	−8.50047e25	−0.280911	−0.140455	+	0.990087i	$0.544857\pi$
$614$	0	0
$615$	4.04277e26	1.29528
$616$	0	0
$617$	−6.89030e25	−0.214057	−0.107028	−	0.994256i	$-0.534134\pi$
$617$	−6.89030e25	−0.214057	−0.107028	+	0.994256i	$0.534134\pi$
$618$	0	0
$619$	5.37779e26	1.62010	0.810051	−	0.586359i	$-0.199439\pi$
$619$	5.37779e26	1.62010	0.810051	+	0.586359i	$0.199439\pi$
$620$	0	0
$621$	7.79330e25	0.227694
$622$	0	0
$623$	1.30347e26	0.369373
$624$	0	0
$625$	2.45812e26	0.675683
$626$	0	0
$627$	−4.48347e25	−0.119556
$628$	0	0
$629$	−8.15483e25	−0.210976
$630$	0	0
$631$	2.60646e25	0.0654293	0.0327146	−	0.999465i	$-0.489585\pi$
$631$	2.60646e25	0.0654293	0.0327146	+	0.999465i	$0.489585\pi$
$632$	0	0
$633$	3.95879e26	0.964333
$634$	0	0
$635$	1.29725e27	3.06671
$636$	0	0
$637$	−9.41657e25	−0.216057
$638$	0	0
$639$	−2.66259e26	−0.592989
$640$	0	0
$641$	6.08937e26	1.31650	0.658250	−	0.752799i	$-0.271297\pi$
$641$	6.08937e26	1.31650	0.658250	+	0.752799i	$0.271297\pi$
$642$	0	0
$643$	6.30417e26	1.32319	0.661597	−	0.749860i	$-0.269879\pi$
$643$	6.30417e26	1.32319	0.661597	+	0.749860i	$0.269879\pi$
$644$	0	0
$645$	1.41103e26	0.287554
$646$	0	0
$647$	−7.10508e26	−1.40598	−0.702989	−	0.711201i	$-0.748152\pi$
$647$	−7.10508e26	−1.40598	−0.702989	+	0.711201i	$0.748152\pi$
$648$	0	0
$649$	−2.62314e25	−0.0504077
$650$	0	0
$651$	6.64111e25	0.123943
$652$	0	0
$653$	−4.33211e26	−0.785279	−0.392640	−	0.919692i	$-0.628438\pi$
$653$	−4.33211e26	−0.785279	−0.392640	+	0.919692i	$0.628438\pi$
$654$	0	0
$655$	6.17674e26	1.08760
$656$	0	0
$657$	−1.97763e26	−0.338279
$658$	0	0
$659$	5.56808e25	0.0925325	0.0462662	−	0.998929i	$-0.485268\pi$
$659$	5.56808e25	0.0925325	0.0462662	+	0.998929i	$0.485268\pi$
$660$	0	0
$661$	2.24486e26	0.362473	0.181236	−	0.983440i	$-0.441990\pi$
$661$	2.24486e26	0.362473	0.181236	+	0.983440i	$0.441990\pi$
$662$	0	0
$663$	−1.04929e26	−0.164634
$664$	0	0
$665$	2.82847e26	0.431267
$666$	0	0
$667$	1.26568e27	1.87555
$668$	0	0
$669$	−4.62698e26	−0.666422
$670$	0	0
$671$	2.98026e26	0.417245
$672$	0	0
$673$	5.77140e26	0.785485	0.392743	−	0.919648i	$-0.371526\pi$
$673$	5.77140e26	0.785485	0.392743	+	0.919648i	$0.371526\pi$
$674$	0	0
$675$	−2.74558e26	−0.363286
$676$	0	0
$677$	−1.02234e27	−1.31524	−0.657619	−	0.753351i	$-0.728436\pi$
$677$	−1.02234e27	−1.31524	−0.657619	+	0.753351i	$0.728436\pi$
$678$	0	0
$679$	−2.11780e26	−0.264924
$680$	0	0
$681$	7.89436e26	0.960328
$682$	0	0
$683$	1.29370e24	0.00153051	0.000765254	−	1.00000i	$-0.499756\pi$
$683$	1.29370e24	0.00153051	0.000765254		1.00000i	$0.499756\pi$
$684$	0	0
$685$	2.38161e27	2.74038
$686$	0	0
$687$	1.46390e26	0.163841
$688$	0	0
$689$	−1.78235e27	−1.94049
$690$	0	0
$691$	9.73654e26	1.03125	0.515624	−	0.856815i	$-0.327560\pi$
$691$	9.73654e26	1.03125	0.515624	+	0.856815i	$0.327560\pi$
$692$	0	0
$693$	−3.77058e25	−0.0388546
$694$	0	0
$695$	−2.14848e27	−2.15415
$696$	0	0
$697$	−2.55138e26	−0.248922
$698$	0	0
$699$	5.67221e26	0.538541
$700$	0	0
$701$	6.09036e26	0.562758	0.281379	−	0.959597i	$-0.409208\pi$
$701$	6.09036e26	0.562758	0.281379	+	0.959597i	$0.409208\pi$
$702$	0	0
$703$	8.35442e26	0.751348
$704$	0	0
$705$	1.87073e27	1.63763
$706$	0	0
$707$	−6.76014e26	−0.576065
$708$	0	0
$709$	−2.13406e27	−1.77039	−0.885193	−	0.465224i	$-0.845974\pi$
$709$	−2.13406e27	−1.77039	−0.885193	+	0.465224i	$0.845974\pi$
$710$	0	0
$711$	2.33823e26	0.188854
$712$	0	0
$713$	−8.54505e26	−0.671992
$714$	0	0
$715$	1.03505e27	0.792600
$716$	0	0
$717$	−6.52458e26	−0.486542
$718$	0	0
$719$	−4.89933e26	−0.355805	−0.177903	−	0.984048i	$-0.556931\pi$
$719$	−4.89933e26	−0.355805	−0.177903	+	0.984048i	$0.556931\pi$
$720$	0	0
$721$	−3.41837e26	−0.241787
$722$	0	0
$723$	4.27208e26	0.294324
$724$	0	0
$725$	−4.45899e27	−2.99244
$726$	0	0
$727$	−2.31317e27	−1.51227	−0.756137	−	0.654413i	$-0.772916\pi$
$727$	−2.31317e27	−1.51227	−0.756137	+	0.654413i	$0.772916\pi$
$728$	0	0
$729$	5.81497e25	0.0370370
$730$	0	0
$731$	−8.90499e25	−0.0552609
$732$	0	0
$733$	−1.05392e27	−0.637263	−0.318632	−	0.947879i	$-0.603223\pi$
$733$	−1.05392e27	−0.637263	−0.318632	+	0.947879i	$0.603223\pi$
$734$	0	0
$735$	2.37874e26	0.140158
$736$	0	0
$737$	−4.51532e26	−0.259267
$738$	0	0
$739$	4.28830e26	0.239973	0.119987	−	0.992776i	$-0.461715\pi$
$739$	4.28830e26	0.239973	0.119987	+	0.992776i	$0.461715\pi$
$740$	0	0
$741$	1.07498e27	0.586308
$742$	0	0
$743$	1.83083e26	0.0973319	0.0486659	−	0.998815i	$-0.484503\pi$
$743$	1.83083e26	0.0973319	0.0486659	+	0.998815i	$0.484503\pi$
$744$	0	0
$745$	−9.37440e26	−0.485802
$746$	0	0
$747$	6.37717e26	0.322168
$748$	0	0
$749$	−3.69078e26	−0.181778
$750$	0	0
$751$	−8.91365e23	−0.000428032 0	−0.000214016	−	1.00000i	$-0.500068\pi$
$751$	−8.91365e23	−0.000428032 0	−0.000214016		1.00000i	$0.500068\pi$
$752$	0	0
$753$	−5.43251e26	−0.254360
$754$	0	0
$755$	2.77594e27	1.26740
$756$	0	0
$757$	2.67239e27	1.18984	0.594920	−	0.803785i	$-0.297184\pi$
$757$	2.67239e27	1.18984	0.594920	+	0.803785i	$0.297184\pi$
$758$	0	0
$759$	4.85157e26	0.210662
$760$	0	0
$761$	3.13662e27	1.32833	0.664167	−	0.747584i	$-0.268786\pi$
$761$	3.13662e27	1.32833	0.664167	+	0.747584i	$0.268786\pi$
$762$	0	0
$763$	−8.49501e25	−0.0350898
$764$	0	0
$765$	2.65064e26	0.106799
$766$	0	0
$767$	6.28936e26	0.247201
$768$	0	0
$769$	−1.67473e27	−0.642162	−0.321081	−	0.947052i	$-0.604046\pi$
$769$	−1.67473e27	−0.642162	−0.321081	+	0.947052i	$0.604046\pi$
$770$	0	0
$771$	−8.89543e26	−0.332775
$772$	0	0
$773$	−1.49727e27	−0.546507	−0.273253	−	0.961942i	$-0.588100\pi$
$773$	−1.49727e27	−0.546507	−0.273253	+	0.961942i	$0.588100\pi$
$774$	0	0
$775$	3.01042e27	1.07216
$776$	0	0
$777$	7.02604e26	0.244180
$778$	0	0
$779$	2.61383e27	0.886484
$780$	0	0
$781$	−1.65754e27	−0.548631
$782$	0	0
$783$	9.44387e26	0.305080
$784$	0	0
$785$	−7.65274e27	−2.41299
$786$	0	0
$787$	1.28307e27	0.394903	0.197451	−	0.980313i	$-0.436734\pi$
$787$	1.28307e27	0.394903	0.197451	+	0.980313i	$0.436734\pi$
$788$	0	0
$789$	3.30873e27	0.994098
$790$	0	0
$791$	−9.05789e26	−0.265675
$792$	0	0
$793$	−7.14561e27	−2.04618
$794$	0	0
$795$	4.50243e27	1.25881
$796$	0	0
$797$	−5.98236e27	−1.63312	−0.816561	−	0.577259i	$-0.804122\pi$
$797$	−5.98236e27	−1.63312	−0.816561	+	0.577259i	$0.804122\pi$
$798$	0	0
$799$	−1.18061e27	−0.314712
$800$	0	0
$801$	−1.25142e27	−0.325756
$802$	0	0
$803$	−1.23114e27	−0.312974
$804$	0	0
$805$	−3.06070e27	−0.759905
$806$	0	0
$807$	1.92934e27	0.467855
$808$	0	0
$809$	−6.24360e27	−1.47885	−0.739425	−	0.673239i	$-0.764902\pi$
$809$	−6.24360e27	−1.47885	−0.739425	+	0.673239i	$0.764902\pi$
$810$	0	0
$811$	−1.32089e27	−0.305611	−0.152806	−	0.988256i	$-0.548831\pi$
$811$	−1.32089e27	−0.305611	−0.152806	+	0.988256i	$0.548831\pi$
$812$	0	0
$813$	8.35024e26	0.188729
$814$	0	0
$815$	−3.54223e27	−0.782131
$816$	0	0
$817$	9.12294e26	0.196800
$818$	0	0
$819$	9.04051e26	0.190544
$820$	0	0
$821$	5.57605e27	1.14833	0.574165	−	0.818740i	$-0.305327\pi$
$821$	5.57605e27	1.14833	0.574165	+	0.818740i	$0.305327\pi$
$822$	0	0
$823$	−1.85671e27	−0.373633	−0.186816	−	0.982395i	$-0.559817\pi$
$823$	−1.85671e27	−0.373633	−0.186816	+	0.982395i	$0.559817\pi$
$824$	0	0
$825$	−1.70921e27	−0.336111
$826$	0	0
$827$	−3.23221e27	−0.621151	−0.310575	−	0.950549i	$-0.600522\pi$
$827$	−3.23221e27	−0.621151	−0.310575	+	0.950549i	$0.600522\pi$
$828$	0	0
$829$	3.25532e27	0.611400	0.305700	−	0.952128i	$-0.401110\pi$
$829$	3.25532e27	0.611400	0.305700	+	0.952128i	$0.401110\pi$
$830$	0	0
$831$	−4.43761e27	−0.814591
$832$	0	0
$833$	−1.50121e26	−0.0269349
$834$	0	0
$835$	1.84549e27	0.323661
$836$	0	0
$837$	−6.37589e26	−0.109307
$838$	0	0
$839$	−5.84958e27	−0.980361	−0.490181	−	0.871621i	$-0.663069\pi$
$839$	−5.84958e27	−0.980361	−0.490181	+	0.871621i	$0.663069\pi$
$840$	0	0
$841$	9.23414e27	1.51299
$842$	0	0
$843$	4.04375e27	0.647772
$844$	0	0
$845$	−1.39672e28	−2.18762
$846$	0	0
$847$	2.23326e27	0.342016
$848$	0	0
$849$	−6.78962e27	−1.01677
$850$	0	0
$851$	−9.04034e27	−1.32390
$852$	0	0
$853$	−5.86863e27	−0.840467	−0.420234	−	0.907416i	$-0.638052\pi$
$853$	−5.86863e27	−0.840467	−0.420234	+	0.907416i	$0.638052\pi$
$854$	0	0
$855$	−2.71552e27	−0.380342
$856$	0	0
$857$	−3.60144e27	−0.493354	−0.246677	−	0.969098i	$-0.579339\pi$
$857$	−3.60144e27	−0.493354	−0.246677	+	0.969098i	$0.579339\pi$
$858$	0	0
$859$	−1.42178e28	−1.90501	−0.952504	−	0.304525i	$-0.901502\pi$
$859$	−1.42178e28	−1.90501	−0.952504	+	0.304525i	$0.901502\pi$
$860$	0	0
$861$	2.19822e27	0.288098
$862$	0	0
$863$	−3.42429e27	−0.439003	−0.219502	−	0.975612i	$-0.570443\pi$
$863$	−3.42429e27	−0.439003	−0.219502	+	0.975612i	$0.570443\pi$
$864$	0	0
$865$	4.70206e27	0.589706
$866$	0	0
$867$	4.53838e27	0.556826
$868$	0	0
$869$	1.45562e27	0.174727
$870$	0	0
$871$	1.08261e28	1.27146
$872$	0	0
$873$	2.03322e27	0.233641
$874$	0	0
$875$	5.07065e27	0.570148
$876$	0	0
$877$	1.03299e28	1.13658	0.568289	−	0.822829i	$-0.307605\pi$
$877$	1.03299e28	1.13658	0.568289	+	0.822829i	$0.307605\pi$
$878$	0	0
$879$	4.09009e27	0.440391
$880$	0	0
$881$	−3.37402e26	−0.0355531	−0.0177765	−	0.999842i	$-0.505659\pi$
$881$	−3.37402e26	−0.0355531	−0.0177765	+	0.999842i	$0.505659\pi$
$882$	0	0
$883$	1.84028e28	1.89783	0.948916	−	0.315529i	$-0.102182\pi$
$883$	1.84028e28	1.89783	0.948916	+	0.315529i	$0.102182\pi$
$884$	0	0
$885$	−1.58877e27	−0.160361
$886$	0	0
$887$	−4.84349e27	−0.478502	−0.239251	−	0.970958i	$-0.576902\pi$
$887$	−4.84349e27	−0.478502	−0.239251	+	0.970958i	$0.576902\pi$
$888$	0	0
$889$	7.05367e27	0.682101
$890$	0	0
$891$	3.62000e26	0.0342665
$892$	0	0
$893$	1.20951e28	1.12078
$894$	0	0
$895$	−1.12984e28	−1.02494
$896$	0	0
$897$	−1.16323e28	−1.03309
$898$	0	0
$899$	−1.03548e28	−0.900380
$900$	0	0
$901$	−2.84147e27	−0.241912
$902$	0	0
$903$	7.67236e26	0.0639580
$904$	0	0
$905$	1.39532e28	1.13897
$906$	0	0
$907$	1.67206e28	1.33655	0.668273	−	0.743916i	$-0.267034\pi$
$907$	1.67206e28	1.33655	0.668273	+	0.743916i	$0.267034\pi$
$908$	0	0
$909$	6.49017e27	0.508041
$910$	0	0
$911$	−1.27638e28	−0.978486	−0.489243	−	0.872147i	$-0.662727\pi$
$911$	−1.27638e28	−0.978486	−0.489243	+	0.872147i	$0.662727\pi$
$912$	0	0
$913$	3.96998e27	0.298069
$914$	0	0
$915$	1.80507e28	1.32737
$916$	0	0
$917$	3.35855e27	0.241904
$918$	0	0
$919$	7.21209e27	0.508820	0.254410	−	0.967097i	$-0.418119\pi$
$919$	7.21209e27	0.508820	0.254410	+	0.967097i	$0.418119\pi$
$920$	0	0
$921$	−1.54333e28	−1.06658
$922$	0	0
$923$	3.97420e28	2.69051
$924$	0	0
$925$	3.18491e28	2.11228
$926$	0	0
$927$	3.28185e27	0.213236
$928$	0	0
$929$	1.44975e28	0.922877	0.461439	−	0.887172i	$-0.347333\pi$
$929$	1.44975e28	0.922877	0.461439	+	0.887172i	$0.347333\pi$
$930$	0	0
$931$	1.53796e27	0.0959229
$932$	0	0
$933$	9.91682e27	0.606034
$934$	0	0
$935$	1.65010e27	0.0988101
$936$	0	0
$937$	−2.33430e28	−1.36971	−0.684857	−	0.728677i	$-0.740135\pi$
$937$	−2.33430e28	−1.36971	−0.684857	+	0.728677i	$0.740135\pi$
$938$	0	0
$939$	−1.13947e28	−0.655208
$940$	0	0
$941$	−1.86717e28	−1.05216	−0.526080	−	0.850435i	$-0.676339\pi$
$941$	−1.86717e28	−1.05216	−0.526080	+	0.850435i	$0.676339\pi$
$942$	0	0
$943$	−2.82843e28	−1.56201
$944$	0	0
$945$	−2.28374e27	−0.123607
$946$	0	0
$947$	9.36298e27	0.496694	0.248347	−	0.968671i	$-0.420113\pi$
$947$	9.36298e27	0.496694	0.248347	+	0.968671i	$0.420113\pi$
$948$	0	0
$949$	2.95183e28	1.53484
$950$	0	0
$951$	−4.71995e27	−0.240560
$952$	0	0
$953$	−3.83907e28	−1.91798	−0.958989	−	0.283443i	$-0.908523\pi$
$953$	−3.83907e28	−1.91798	−0.958989	+	0.283443i	$0.908523\pi$
$954$	0	0
$955$	4.52336e28	2.21528
$956$	0	0
$957$	5.87909e27	0.282258
$958$	0	0
$959$	1.29498e28	0.609517
$960$	0	0
$961$	−1.46798e28	−0.677402
$962$	0	0
$963$	3.54338e27	0.160313
$964$	0	0
$965$	−3.29807e28	−1.46302
$966$	0	0
$967$	−1.31854e28	−0.573513	−0.286757	−	0.958003i	$-0.592577\pi$
$967$	−1.31854e28	−0.573513	−0.286757	+	0.958003i	$0.592577\pi$
$968$	0	0
$969$	1.71376e27	0.0730926
$970$	0	0
$971$	2.40387e28	1.00537	0.502687	−	0.864468i	$-0.332345\pi$
$971$	2.40387e28	1.00537	0.502687	+	0.864468i	$0.332345\pi$
$972$	0	0
$973$	−1.16822e28	−0.479128
$974$	0	0
$975$	4.09808e28	1.64830
$976$	0	0
$977$	1.18191e28	0.466214	0.233107	−	0.972451i	$-0.425111\pi$
$977$	1.18191e28	0.466214	0.233107	+	0.972451i	$0.425111\pi$
$978$	0	0
$979$	−7.79044e27	−0.301388
$980$	0	0
$981$	8.15576e26	0.0309463
$982$	0	0
$983$	1.06950e28	0.398034	0.199017	−	0.979996i	$-0.436225\pi$
$983$	1.06950e28	0.398034	0.199017	+	0.979996i	$0.436225\pi$
$984$	0	0
$985$	−2.16509e28	−0.790373
$986$	0	0
$987$	1.01719e28	0.364242
$988$	0	0
$989$	−9.87195e27	−0.346767
$990$	0	0
$991$	−1.40617e28	−0.484549	−0.242274	−	0.970208i	$-0.577893\pi$
$991$	−1.40617e28	−0.484549	−0.242274	+	0.970208i	$0.577893\pi$
$992$	0	0
$993$	3.18188e27	0.107564
$994$	0	0
$995$	6.52562e28	2.16423
$996$	0	0
$997$	−1.74406e28	−0.567489	−0.283744	−	0.958900i	$-0.591577\pi$
$997$	−1.74406e28	−0.567489	−0.283744	+	0.958900i	$0.591577\pi$
$998$	0	0
$999$	−6.74545e27	−0.215347

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.20.a.a.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.20.a.a.1.1	✓	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 20 \)
Character orbit:	\([\chi]\)	\(=\)	84.a (trivial)

\(f(q)\)	\(=\)	\(q-19683.0 q^{3} -7.42148e6 q^{5} -4.03536e7 q^{7} +3.87420e8 q^{9} +O(q^{10})\)	\(q-19683.0 q^{3} -7.42148e6 q^{5} -4.03536e7 q^{7} +3.87420e8 q^{9} +2.41181e9 q^{11} -5.78266e10 q^{13} +1.46077e11 q^{15} -9.21888e10 q^{17} +9.44451e11 q^{19} +7.94280e11 q^{21} -1.02199e13 q^{23} +3.60048e13 q^{25} -7.62560e12 q^{27} -1.23844e14 q^{29} +8.36117e13 q^{31} -4.74717e13 q^{33} +2.99483e14 q^{35} +8.84580e14 q^{37} +1.13820e15 q^{39} +2.76756e15 q^{41} +9.65951e14 q^{43} -2.87523e15 q^{45} +1.28065e16 q^{47} +1.62841e15 q^{49} +1.81455e15 q^{51} +3.08223e16 q^{53} -1.78992e16 q^{55} -1.85896e16 q^{57} -1.08762e16 q^{59} +1.23570e17 q^{61} -1.56338e16 q^{63} +4.29159e17 q^{65} -1.87217e17 q^{67} +2.01159e17 q^{69} -6.87262e17 q^{71} -5.10462e17 q^{73} -7.08683e17 q^{75} -9.73252e16 q^{77} +6.03539e17 q^{79} +1.50095e17 q^{81} +1.64606e18 q^{83} +6.84177e17 q^{85} +2.43763e18 q^{87} -3.23012e18 q^{89} +2.33351e18 q^{91} -1.64573e18 q^{93} -7.00922e18 q^{95} +5.24810e18 q^{97} +9.34385e17 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 78732 q^{3} - 2502150 q^{5} - 161414428 q^{7} + 1549681956 q^{9}+O(q^{10}) \) 4 * q - 78732 * q^3 - 2502150 * q^5 - 161414428 * q^7 + 1549681956 * q^9	\( 4 q - 78732 q^{3} - 2502150 q^{5} - 161414428 q^{7} + 1549681956 q^{9} - 3798155970 q^{11} - 39152617980 q^{13} + 49249818450 q^{15} - 190872643554 q^{17} + 2299240294976 q^{19} + 3177120186324 q^{21} - 10298330967654 q^{23} + 10550140805600 q^{25} - 30502389939948 q^{27} - 77678357293560 q^{29} + 196054920663408 q^{31} + 74759103957510 q^{33} + 100970777755050 q^{35} + 829369260264428 q^{37} + 770640979700340 q^{39} + 22\!\cdots\!02 q^{41}+ \cdots - 14\!\cdots\!30 q^{99}+O(q^{100}) \) 4 * q - 78732 * q^3 - 2502150 * q^5 - 161414428 * q^7 + 1549681956 * q^9 - 3798155970 * q^11 - 39152617980 * q^13 + 49249818450 * q^15 - 190872643554 * q^17 + 2299240294976 * q^19 + 3177120186324 * q^21 - 10298330967654 * q^23 + 10550140805600 * q^25 - 30502389939948 * q^27 - 77678357293560 * q^29 + 196054920663408 * q^31 + 74759103957510 * q^33 + 100970777755050 * q^35 + 829369260264428 * q^37 + 770640979700340 * q^39 + 2210157258858402 * q^41 + 3704496271875056 * q^43 - 969384176551350 * q^45 + 4910647758864108 * q^47 + 6513654391641796 * q^49 + 3756946243073382 * q^51 + 4583139659223684 * q^53 - 49874361740165200 * q^55 - 45255946726012608 * q^57 + 10541900786867604 * q^59 + 67528372286847048 * q^61 - 62535256627415292 * q^63 + 513738934168590300 * q^65 + 232506796904165236 * q^67 + 202702048436333682 * q^69 + 264200775007920198 * q^71 + 668786810310501652 * q^73 - 207658421476624800 * q^75 + 153269293338083790 * q^77 + 1869350307247472268 * q^79 + 600378541187996484 * q^81 + 4190113921662329040 * q^83 + 108669289188945500 * q^85 + 1528943106609141480 * q^87 - 4059161776602418782 * q^89 + 1579949358986053860 * q^91 - 3858949003417859664 * q^93 + 7625582268210966600 * q^95 - 1033481664034434692 * q^97 - 1471483443195669330 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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