LMFDB - Embedded newform 84.20.a.a.1.4

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			1.4
Root			$149113.$ 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)

$f(q)$	$=$	$q-19683.0 q^{3} +5.19906e6 q^{5} -4.03536e7 q^{7} +3.87420e8 q^{9} +O(q^{10})$	$q-19683.0 q^{3} +5.19906e6 q^{5} -4.03536e7 q^{7} +3.87420e8 q^{9} -3.26005e9 q^{11} +3.52611e10 q^{13} -1.02333e11 q^{15} -3.75431e11 q^{17} +1.86987e12 q^{19} +7.94280e11 q^{21} -1.17148e13 q^{23} +7.95677e12 q^{25} -7.62560e12 q^{27} -5.15283e13 q^{29} +1.60115e14 q^{31} +6.41676e13 q^{33} -2.09801e14 q^{35} +4.14331e14 q^{37} -6.94045e14 q^{39} -4.00796e15 q^{41} -6.55721e14 q^{43} +2.01422e15 q^{45} -1.10623e15 q^{47} +1.62841e15 q^{49} +7.38962e15 q^{51} -1.36187e16 q^{53} -1.69492e16 q^{55} -3.68046e16 q^{57} -8.77533e16 q^{59} -4.09706e16 q^{61} -1.56338e16 q^{63} +1.83325e17 q^{65} +2.97767e17 q^{67} +2.30581e17 q^{69} +3.08941e17 q^{71} +3.48489e16 q^{73} -1.56613e17 q^{75} +1.31555e17 q^{77} +1.97924e18 q^{79} +1.50095e17 q^{81} +2.27598e18 q^{83} -1.95189e18 q^{85} +1.01423e18 q^{87} -3.40640e18 q^{89} -1.42291e18 q^{91} -3.15154e18 q^{93} +9.72155e18 q^{95} -1.32382e18 q^{97} -1.26301e18 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	5.19906e6	1.19045	0.595223	−	0.803560i	$-0.297064\pi$
$5$	5.19906e6	1.19045	0.595223	+	0.803560i	$0.297064\pi$
$6$	0	0
$7$	−4.03536e7	−0.377964
$7$	−4.03536e7	−0.377964
$8$	0	0
$9$	3.87420e8	0.333333
$10$	0	0
$11$	−3.26005e9	−0.416863	−0.208432	−	0.978037i	$-0.566836\pi$
$11$	−3.26005e9	−0.416863	−0.208432	+	0.978037i	$0.566836\pi$
$12$	0	0
$13$	3.52611e10	0.922220	0.461110	−	0.887343i	$-0.347451\pi$
$13$	3.52611e10	0.922220	0.461110	+	0.887343i	$0.347451\pi$
$14$	0	0
$15$	−1.02333e11	−0.687305
$16$	0	0
$17$	−3.75431e11	−0.767831	−0.383916	−	0.923368i	$-0.625425\pi$
$17$	−3.75431e11	−0.767831	−0.383916	+	0.923368i	$0.625425\pi$
$18$	0	0
$19$	1.86987e12	1.32939	0.664693	−	0.747116i	$-0.268562\pi$
$19$	1.86987e12	1.32939	0.664693	+	0.747116i	$0.268562\pi$
$20$	0	0
$21$	7.94280e11	0.218218
$22$	0	0
$23$	−1.17148e13	−1.35618	−0.678092	−	0.734977i	$-0.737193\pi$
$23$	−1.17148e13	−1.35618	−0.678092	+	0.734977i	$0.737193\pi$
$24$	0	0
$25$	7.95677e12	0.417164
$26$	0	0
$27$	−7.62560e12	−0.192450
$28$	0	0
$29$	−5.15283e13	−0.659575	−0.329788	−	0.944055i	$-0.606977\pi$
$29$	−5.15283e13	−0.659575	−0.329788	+	0.944055i	$0.606977\pi$
$30$	0	0
$31$	1.60115e14	1.08767	0.543833	−	0.839193i	$-0.316972\pi$
$31$	1.60115e14	1.08767	0.543833	+	0.839193i	$0.316972\pi$
$32$	0	0
$33$	6.41676e13	0.240676
$34$	0	0
$35$	−2.09801e14	−0.449947
$36$	0	0
$37$	4.14331e14	0.524121	0.262060	−	0.965051i	$-0.415598\pi$
$37$	4.14331e14	0.524121	0.262060	+	0.965051i	$0.415598\pi$
$38$	0	0
$39$	−6.94045e14	−0.532444
$40$	0	0
$41$	−4.00796e15	−1.91195	−0.955976	−	0.293446i	$-0.905198\pi$
$41$	−4.00796e15	−1.91195	−0.955976	+	0.293446i	$0.905198\pi$
$42$	0	0
$43$	−6.55721e14	−0.198961	−0.0994807	−	0.995039i	$-0.531718\pi$
$43$	−6.55721e14	−0.198961	−0.0994807	+	0.995039i	$0.531718\pi$
$44$	0	0
$45$	2.01422e15	0.396816
$46$	0	0
$47$	−1.10623e15	−0.144183	−0.0720916	−	0.997398i	$-0.522967\pi$
$47$	−1.10623e15	−0.144183	−0.0720916	+	0.997398i	$0.522967\pi$
$48$	0	0
$49$	1.62841e15	0.142857
$50$	0	0
$51$	7.38962e15	0.443308
$52$	0	0
$53$	−1.36187e16	−0.566909	−0.283455	−	0.958986i	$-0.591481\pi$
$53$	−1.36187e16	−0.566909	−0.283455	+	0.958986i	$0.591481\pi$
$54$	0	0
$55$	−1.69492e16	−0.496254
$56$	0	0
$57$	−3.68046e16	−0.767522
$58$	0	0
$59$	−8.77533e16	−1.31877	−0.659386	−	0.751805i	$-0.729184\pi$
$59$	−8.77533e16	−1.31877	−0.659386	+	0.751805i	$0.729184\pi$
$60$	0	0
$61$	−4.09706e16	−0.448579	−0.224290	−	0.974523i	$-0.572006\pi$
$61$	−4.09706e16	−0.448579	−0.224290	+	0.974523i	$0.572006\pi$
$62$	0	0
$63$	−1.56338e16	−0.125988
$64$	0	0
$65$	1.83325e17	1.09785
$66$	0	0
$67$	2.97767e17	1.33711	0.668554	−	0.743664i	$-0.266914\pi$
$67$	2.97767e17	1.33711	0.668554	+	0.743664i	$0.266914\pi$
$68$	0	0
$69$	2.30581e17	0.782993
$70$	0	0
$71$	3.08941e17	0.799689	0.399844	−	0.916583i	$-0.369064\pi$
$71$	3.08941e17	0.799689	0.399844	+	0.916583i	$0.369064\pi$
$72$	0	0
$73$	3.48489e16	0.0692822	0.0346411	−	0.999400i	$-0.488971\pi$
$73$	3.48489e16	0.0692822	0.0346411	+	0.999400i	$0.488971\pi$
$74$	0	0
$75$	−1.56613e17	−0.240850
$76$	0	0
$77$	1.31555e17	0.157560
$78$	0	0
$79$	1.97924e18	1.85798	0.928988	−	0.370109i	$-0.120680\pi$
$79$	1.97924e18	1.85798	0.928988	+	0.370109i	$0.120680\pi$
$80$	0	0
$81$	1.50095e17	0.111111
$82$	0	0
$83$	2.27598e18	1.33637	0.668186	−	0.743994i	$-0.267071\pi$
$83$	2.27598e18	1.33637	0.668186	+	0.743994i	$0.267071\pi$
$84$	0	0
$85$	−1.95189e18	−0.914062
$86$	0	0
$87$	1.01423e18	0.380806
$88$	0	0
$89$	−3.40640e18	−1.03060	−0.515301	−	0.857009i	$-0.672320\pi$
$89$	−3.40640e18	−1.03060	−0.515301	+	0.857009i	$0.672320\pi$
$90$	0	0
$91$	−1.42291e18	−0.348566
$92$	0	0
$93$	−3.15154e18	−0.627965
$94$	0	0
$95$	9.72155e18	1.58256
$96$	0	0
$97$	−1.32382e18	−0.176806	−0.0884032	−	0.996085i	$-0.528176\pi$
$97$	−1.32382e18	−0.176806	−0.0884032	+	0.996085i	$0.528176\pi$
$98$	0	0
$99$	−1.26301e18	−0.138954
$100$	0	0
$101$	4.65030e18	0.423085	0.211543	−	0.977369i	$-0.432151\pi$
$101$	4.65030e18	0.423085	0.211543	+	0.977369i	$0.432151\pi$
$102$	0	0
$103$	−1.44636e19	−1.09225	−0.546127	−	0.837702i	$-0.683899\pi$
$103$	−1.44636e19	−1.09225	−0.546127	+	0.837702i	$0.683899\pi$
$104$	0	0
$105$	4.12951e18	0.259777
$106$	0	0
$107$	−1.28069e19	−0.673439	−0.336719	−	0.941605i	$-0.609317\pi$
$107$	−1.28069e19	−0.673439	−0.336719	+	0.941605i	$0.609317\pi$
$108$	0	0
$109$	1.47325e19	0.649719	0.324859	−	0.945762i	$-0.394683\pi$
$109$	1.47325e19	0.649719	0.324859	+	0.945762i	$0.394683\pi$
$110$	0	0
$111$	−8.15529e18	−0.302601
$112$	0	0
$113$	6.28019e18	0.196665	0.0983326	−	0.995154i	$-0.468649\pi$
$113$	6.28019e18	0.196665	0.0983326	+	0.995154i	$0.468649\pi$
$114$	0	0
$115$	−6.09057e19	−1.61447
$116$	0	0
$117$	1.36609e19	0.307407
$118$	0	0
$119$	1.51500e19	0.290213
$120$	0	0
$121$	−5.05312e19	−0.826225
$122$	0	0
$123$	7.88887e19	1.10387
$124$	0	0
$125$	−5.77965e19	−0.693836
$126$	0	0
$127$	2.05715e19	0.212388	0.106194	−	0.994345i	$-0.466134\pi$
$127$	2.05715e19	0.212388	0.106194	+	0.994345i	$0.466134\pi$
$128$	0	0
$129$	1.29066e19	0.114870
$130$	0	0
$131$	1.44217e20	1.10902	0.554508	−	0.832178i	$-0.312907\pi$
$131$	1.44217e20	1.10902	0.554508	+	0.832178i	$0.312907\pi$
$132$	0	0
$133$	−7.54558e19	−0.502461
$134$	0	0
$135$	−3.96460e19	−0.229102
$136$	0	0
$137$	−3.07958e19	−0.154756	−0.0773778	−	0.997002i	$-0.524655\pi$
$137$	−3.07958e19	−0.154756	−0.0773778	+	0.997002i	$0.524655\pi$
$138$	0	0
$139$	9.24955e19	0.405023	0.202512	−	0.979280i	$-0.435090\pi$
$139$	9.24955e19	0.405023	0.202512	+	0.979280i	$0.435090\pi$
$140$	0	0
$141$	2.17739e19	0.0832442
$142$	0	0
$143$	−1.14953e20	−0.384440
$144$	0	0
$145$	−2.67899e20	−0.785189
$146$	0	0
$147$	−3.20521e19	−0.0824786
$148$	0	0
$149$	−2.49848e20	−0.565466	−0.282733	−	0.959199i	$-0.591241\pi$
$149$	−2.49848e20	−0.565466	−0.282733	+	0.959199i	$0.591241\pi$
$150$	0	0
$151$	−3.91441e20	−0.780523	−0.390262	−	0.920704i	$-0.627615\pi$
$151$	−3.91441e20	−0.780523	−0.390262	+	0.920704i	$0.627615\pi$
$152$	0	0
$153$	−1.45450e20	−0.255944
$154$	0	0
$155$	8.32448e20	1.29481
$156$	0	0
$157$	−9.39003e20	−1.29306	−0.646532	−	0.762887i	$-0.723781\pi$
$157$	−9.39003e20	−1.29306	−0.646532	+	0.762887i	$0.723781\pi$
$158$	0	0
$159$	2.68056e20	0.327305
$160$	0	0
$161$	4.72733e20	0.512589
$162$	0	0
$163$	−5.98389e20	−0.577034	−0.288517	−	0.957475i	$-0.593162\pi$
$163$	−5.98389e20	−0.577034	−0.288517	+	0.957475i	$0.593162\pi$
$164$	0	0
$165$	3.33611e20	0.286512
$166$	0	0
$167$	2.65939e20	0.203693	0.101846	−	0.994800i	$-0.467525\pi$
$167$	2.65939e20	0.203693	0.101846	+	0.994800i	$0.467525\pi$
$168$	0	0
$169$	−2.18572e20	−0.149510
$170$	0	0
$171$	7.24424e20	0.443129
$172$	0	0
$173$	−2.51325e21	−1.37657	−0.688283	−	0.725442i	$-0.741635\pi$
$173$	−2.51325e21	−1.37657	−0.688283	+	0.725442i	$0.741635\pi$
$174$	0	0
$175$	−3.21084e20	−0.157673
$176$	0	0
$177$	1.72725e21	0.761393
$178$	0	0
$179$	−2.32007e21	−0.919175	−0.459587	−	0.888133i	$-0.652003\pi$
$179$	−2.32007e21	−0.919175	−0.459587	+	0.888133i	$0.652003\pi$
$180$	0	0
$181$	−3.09952e21	−1.10496	−0.552481	−	0.833525i	$-0.686319\pi$
$181$	−3.09952e21	−1.10496	−0.552481	+	0.833525i	$0.686319\pi$
$182$	0	0
$183$	8.06425e20	0.258987
$184$	0	0
$185$	2.15413e21	0.623938
$186$	0	0
$187$	1.22393e21	0.320081
$188$	0	0
$189$	3.07720e20	0.0727393
$190$	0	0
$191$	1.34867e21	0.288461	0.144231	−	0.989544i	$-0.453929\pi$
$191$	1.34867e21	0.288461	0.144231	+	0.989544i	$0.453929\pi$
$192$	0	0
$193$	5.24158e21	1.01547	0.507736	−	0.861513i	$-0.330483\pi$
$193$	5.24158e21	1.01547	0.507736	+	0.861513i	$0.330483\pi$
$194$	0	0
$195$	−3.60838e21	−0.633846
$196$	0	0
$197$	−6.82393e21	−1.08794	−0.543971	−	0.839104i	$-0.683080\pi$
$197$	−6.82393e21	−1.08794	−0.543971	+	0.839104i	$0.683080\pi$
$198$	0	0
$199$	1.22657e22	1.77659	0.888297	−	0.459269i	$-0.151889\pi$
$199$	1.22657e22	1.77659	0.888297	+	0.459269i	$0.151889\pi$
$200$	0	0
$201$	−5.86096e21	−0.771980
$202$	0	0
$203$	2.07935e21	0.249296
$204$	0	0
$205$	−2.08376e22	−2.27608
$206$	0	0
$207$	−4.53854e21	−0.452061
$208$	0	0
$209$	−6.09586e21	−0.554173
$210$	0	0
$211$	−2.19976e22	−1.82681	−0.913404	−	0.407055i	$-0.866556\pi$
$211$	−2.19976e22	−1.82681	−0.913404	+	0.407055i	$0.866556\pi$
$212$	0	0
$213$	−6.08088e21	−0.461701
$214$	0	0
$215$	−3.40914e21	−0.236853
$216$	0	0
$217$	−6.46122e21	−0.411099
$218$	0	0
$219$	−6.85930e20	−0.0400001
$220$	0	0
$221$	−1.32381e22	−0.708109
$222$	0	0
$223$	1.59540e22	0.783384	0.391692	−	0.920096i	$-0.371890\pi$
$223$	1.59540e22	0.783384	0.391692	+	0.920096i	$0.371890\pi$
$224$	0	0
$225$	3.08261e21	0.139055
$226$	0	0
$227$	−3.49203e22	−1.44821	−0.724106	−	0.689688i	$-0.757747\pi$
$227$	−3.49203e22	−1.44821	−0.724106	+	0.689688i	$0.757747\pi$
$228$	0	0
$229$	−2.29723e22	−0.876532	−0.438266	−	0.898845i	$-0.644407\pi$
$229$	−2.29723e22	−0.876532	−0.438266	+	0.898845i	$0.644407\pi$
$230$	0	0
$231$	−2.58939e21	−0.0909671
$232$	0	0
$233$	−1.26198e22	−0.408479	−0.204240	−	0.978921i	$-0.565472\pi$
$233$	−1.26198e22	−0.408479	−0.204240	+	0.978921i	$0.565472\pi$
$234$	0	0
$235$	−5.75134e21	−0.171642
$236$	0	0
$237$	−3.89573e22	−1.07270
$238$	0	0
$239$	−7.03904e22	−1.78951	−0.894754	−	0.446560i	$-0.852649\pi$
$239$	−7.03904e22	−1.78951	−0.894754	+	0.446560i	$0.852649\pi$
$240$	0	0
$241$	−3.15996e22	−0.742198	−0.371099	−	0.928593i	$-0.621019\pi$
$241$	−3.15996e22	−0.742198	−0.371099	+	0.928593i	$0.621019\pi$
$242$	0	0
$243$	−2.95431e21	−0.0641500
$244$	0	0
$245$	8.46622e21	0.170064
$246$	0	0
$247$	6.59336e22	1.22599
$248$	0	0
$249$	−4.47982e22	−0.771555
$250$	0	0
$251$	−3.26282e22	−0.520826	−0.260413	−	0.965497i	$-0.583859\pi$
$251$	−3.26282e22	−0.520826	−0.260413	+	0.965497i	$0.583859\pi$
$252$	0	0
$253$	3.81907e22	0.565344
$254$	0	0
$255$	3.84191e22	0.527734
$256$	0	0
$257$	4.55565e22	0.581012	0.290506	−	0.956873i	$-0.406176\pi$
$257$	4.55565e22	0.581012	0.290506	+	0.956873i	$0.406176\pi$
$258$	0	0
$259$	−1.67198e22	−0.198099
$260$	0	0
$261$	−1.99631e22	−0.219858
$262$	0	0
$263$	2.80215e22	0.287019	0.143510	−	0.989649i	$-0.454161\pi$
$263$	2.80215e22	0.287019	0.143510	+	0.989649i	$0.454161\pi$
$264$	0	0
$265$	−7.08043e22	−0.674875
$266$	0	0
$267$	6.70483e22	0.595019
$268$	0	0
$269$	−1.18963e23	−0.983478	−0.491739	−	0.870743i	$-0.663639\pi$
$269$	−1.18963e23	−0.983478	−0.491739	+	0.870743i	$0.663639\pi$
$270$	0	0
$271$	−2.36090e23	−1.81915	−0.909576	−	0.415538i	$-0.863593\pi$
$271$	−2.36090e23	−1.81915	−0.909576	+	0.415538i	$0.863593\pi$
$272$	0	0
$273$	2.80072e22	0.201245
$274$	0	0
$275$	−2.59395e22	−0.173900
$276$	0	0
$277$	2.00105e23	1.25228	0.626138	−	0.779712i	$-0.284635\pi$
$277$	2.00105e23	1.25228	0.626138	+	0.779712i	$0.284635\pi$
$278$	0	0
$279$	6.20318e22	0.362556
$280$	0	0
$281$	−1.64346e23	−0.897532	−0.448766	−	0.893649i	$-0.648136\pi$
$281$	−1.64346e23	−0.897532	−0.448766	+	0.893649i	$0.648136\pi$
$282$	0	0
$283$	−1.03986e23	−0.530889	−0.265445	−	0.964126i	$-0.585519\pi$
$283$	−1.03986e23	−0.530889	−0.265445	+	0.964126i	$0.585519\pi$
$284$	0	0
$285$	−1.91349e23	−0.913694
$286$	0	0
$287$	1.61736e23	0.722650
$288$	0	0
$289$	−9.81237e22	−0.410435
$290$	0	0
$291$	2.60568e22	0.102079
$292$	0	0
$293$	−1.51892e23	−0.557562	−0.278781	−	0.960355i	$-0.589930\pi$
$293$	−1.51892e23	−0.557562	−0.278781	+	0.960355i	$0.589930\pi$
$294$	0	0
$295$	−4.56235e23	−1.56993
$296$	0	0
$297$	2.48598e22	0.0802254
$298$	0	0
$299$	−4.13076e23	−1.25070
$300$	0	0
$301$	2.64607e22	0.0752004
$302$	0	0
$303$	−9.15319e22	−0.244268
$304$	0	0
$305$	−2.13009e23	−0.534010
$306$	0	0
$307$	−2.77612e23	−0.654068	−0.327034	−	0.945013i	$-0.606049\pi$
$307$	−2.77612e23	−0.654068	−0.327034	+	0.945013i	$0.606049\pi$
$308$	0	0
$309$	2.84688e23	0.630614
$310$	0	0
$311$	5.65690e23	1.17857	0.589284	−	0.807926i	$-0.299410\pi$
$311$	5.65690e23	1.17857	0.589284	+	0.807926i	$0.299410\pi$
$312$	0	0
$313$	−8.81988e23	−1.72899	−0.864493	−	0.502645i	$-0.832360\pi$
$313$	−8.81988e23	−1.72899	−0.864493	+	0.502645i	$0.832360\pi$
$314$	0	0
$315$	−8.12812e22	−0.149982
$316$	0	0
$317$	5.18653e23	0.901184	0.450592	−	0.892730i	$-0.351213\pi$
$317$	5.18653e23	0.901184	0.450592	+	0.892730i	$0.351213\pi$
$318$	0	0
$319$	1.67985e23	0.274953
$320$	0	0
$321$	2.52078e23	0.388810
$322$	0	0
$323$	−7.02006e23	−1.02074
$324$	0	0
$325$	2.80565e23	0.384717
$326$	0	0
$327$	−2.89980e23	−0.375115
$328$	0	0
$329$	4.46402e22	0.0544961
$330$	0	0
$331$	8.39440e23	0.967439	0.483720	−	0.875223i	$-0.339285\pi$
$331$	8.39440e23	0.967439	0.483720	+	0.875223i	$0.339285\pi$
$332$	0	0
$333$	1.60520e23	0.174707
$334$	0	0
$335$	1.54811e24	1.59176
$336$	0	0
$337$	8.13042e23	0.790003	0.395001	−	0.918680i	$-0.370744\pi$
$337$	8.13042e23	0.790003	0.395001	+	0.918680i	$0.370744\pi$
$338$	0	0
$339$	−1.23613e23	−0.113545
$340$	0	0
$341$	−5.21983e23	−0.453409
$342$	0	0
$343$	−6.57124e22	−0.0539949
$344$	0	0
$345$	1.19881e24	0.932112
$346$	0	0
$347$	1.44971e24	1.06697	0.533483	−	0.845811i	$-0.320883\pi$
$347$	1.44971e24	1.06697	0.533483	+	0.845811i	$0.320883\pi$
$348$	0	0
$349$	−2.22891e24	−1.55329	−0.776643	−	0.629941i	$-0.783079\pi$
$349$	−2.22891e24	−1.55329	−0.776643	+	0.629941i	$0.783079\pi$
$350$	0	0
$351$	−2.68887e23	−0.177481
$352$	0	0
$353$	1.26147e24	0.788891	0.394445	−	0.918919i	$-0.370937\pi$
$353$	1.26147e24	0.788891	0.394445	+	0.918919i	$0.370937\pi$
$354$	0	0
$355$	1.60620e24	0.951987
$356$	0	0
$357$	−2.98198e23	−0.167555
$358$	0	0
$359$	−1.57576e24	−0.839640	−0.419820	−	0.907607i	$-0.637907\pi$
$359$	−1.57576e24	−0.839640	−0.419820	+	0.907607i	$0.637907\pi$
$360$	0	0
$361$	1.51798e24	0.767268
$362$	0	0
$363$	9.94605e23	0.477021
$364$	0	0
$365$	1.81181e23	0.0824767
$366$	0	0
$367$	1.48268e24	0.640796	0.320398	−	0.947283i	$-0.396183\pi$
$367$	1.48268e24	0.640796	0.320398	+	0.947283i	$0.396183\pi$
$368$	0	0
$369$	−1.55277e24	−0.637317
$370$	0	0
$371$	5.49562e23	0.214272
$372$	0	0
$373$	4.26725e24	1.58094	0.790468	−	0.612504i	$-0.209838\pi$
$373$	4.26725e24	1.58094	0.790468	+	0.612504i	$0.209838\pi$
$374$	0	0
$375$	1.13761e24	0.400586
$376$	0	0
$377$	−1.81694e24	−0.608274
$378$	0	0
$379$	−1.61263e24	−0.513406	−0.256703	−	0.966490i	$-0.582636\pi$
$379$	−1.61263e24	−0.513406	−0.256703	+	0.966490i	$0.582636\pi$
$380$	0	0
$381$	−4.04909e23	−0.122622
$382$	0	0
$383$	−1.93019e24	−0.556175	−0.278087	−	0.960556i	$-0.589701\pi$
$383$	−1.93019e24	−0.556175	−0.278087	+	0.960556i	$0.589701\pi$
$384$	0	0
$385$	6.83962e23	0.187566
$386$	0	0
$387$	−2.54040e23	−0.0663205
$388$	0	0
$389$	4.29378e24	1.06738	0.533689	−	0.845681i	$-0.320805\pi$
$389$	4.29378e24	1.06738	0.533689	+	0.845681i	$0.320805\pi$
$390$	0	0
$391$	4.39809e24	1.04132
$392$	0	0
$393$	−2.83861e24	−0.640291
$394$	0	0
$395$	1.02902e25	2.21182
$396$	0	0
$397$	1.14962e24	0.235529	0.117765	−	0.993042i	$-0.462427\pi$
$397$	1.14962e24	0.235529	0.117765	+	0.993042i	$0.462427\pi$
$398$	0	0
$399$	1.48520e24	0.290096
$400$	0	0
$401$	−2.20900e24	−0.411457	−0.205728	−	0.978609i	$-0.565956\pi$
$401$	−2.20900e24	−0.411457	−0.205728	+	0.978609i	$0.565956\pi$
$402$	0	0
$403$	5.64584e24	1.00307
$404$	0	0
$405$	7.80351e23	0.132272
$406$	0	0
$407$	−1.35074e24	−0.218487
$408$	0	0
$409$	−1.70797e23	−0.0263699	−0.0131849	−	0.999913i	$-0.504197\pi$
$409$	−1.70797e23	−0.0263699	−0.0131849	+	0.999913i	$0.504197\pi$
$410$	0	0
$411$	6.06154e23	0.0893482
$412$	0	0
$413$	3.54116e24	0.498449
$414$	0	0
$415$	1.18330e25	1.59088
$416$	0	0
$417$	−1.82059e24	−0.233840
$418$	0	0
$419$	−8.43434e24	−1.03518	−0.517592	−	0.855628i	$-0.673172\pi$
$419$	−8.43434e24	−1.03518	−0.517592	+	0.855628i	$0.673172\pi$
$420$	0	0
$421$	2.18689e24	0.256535	0.128268	−	0.991740i	$-0.459058\pi$
$421$	2.18689e24	0.256535	0.128268	+	0.991740i	$0.459058\pi$
$422$	0	0
$423$	−4.28575e23	−0.0480610
$424$	0	0
$425$	−2.98722e24	−0.320311
$426$	0	0
$427$	1.65331e24	0.169547
$428$	0	0
$429$	2.26262e24	0.221956
$430$	0	0
$431$	1.32191e25	1.24070	0.620351	−	0.784325i	$-0.286990\pi$
$431$	1.32191e25	1.24070	0.620351	+	0.784325i	$0.286990\pi$
$432$	0	0
$433$	1.80991e25	1.62563	0.812813	−	0.582524i	$-0.197935\pi$
$433$	1.80991e25	1.62563	0.812813	+	0.582524i	$0.197935\pi$
$434$	0	0
$435$	5.27305e24	0.453329
$436$	0	0
$437$	−2.19050e25	−1.80289
$438$	0	0
$439$	−1.99275e25	−1.57050	−0.785252	−	0.619176i	$-0.787467\pi$
$439$	−1.99275e25	−1.57050	−0.785252	+	0.619176i	$0.787467\pi$
$440$	0	0
$441$	6.30881e23	0.0476190
$442$	0	0
$443$	−1.83932e25	−1.32991	−0.664954	−	0.746885i	$-0.731549\pi$
$443$	−1.83932e25	−1.32991	−0.664954	+	0.746885i	$0.731549\pi$
$444$	0	0
$445$	−1.77101e25	−1.22688
$446$	0	0
$447$	4.91776e24	0.326472
$448$	0	0
$449$	1.17920e25	0.750323	0.375161	−	0.926959i	$-0.377587\pi$
$449$	1.17920e25	0.750323	0.375161	+	0.926959i	$0.377587\pi$
$450$	0	0
$451$	1.30662e25	0.797023
$452$	0	0
$453$	7.70474e24	0.450635
$454$	0	0
$455$	−7.39782e24	−0.414950
$456$	0	0
$457$	−2.48745e25	−1.33829	−0.669144	−	0.743133i	$-0.733339\pi$
$457$	−2.48745e25	−1.33829	−0.669144	+	0.743133i	$0.733339\pi$
$458$	0	0
$459$	2.86289e24	0.147769
$460$	0	0
$461$	1.44985e25	0.718066	0.359033	−	0.933325i	$-0.383107\pi$
$461$	1.44985e25	0.718066	0.359033	+	0.933325i	$0.383107\pi$
$462$	0	0
$463$	−3.83979e25	−1.82511	−0.912554	−	0.408956i	$-0.865893\pi$
$463$	−3.83979e25	−1.82511	−0.912554	+	0.408956i	$0.865893\pi$
$464$	0	0
$465$	−1.63851e25	−0.747559
$466$	0	0
$467$	7.17206e24	0.314148	0.157074	−	0.987587i	$-0.449794\pi$
$467$	7.17206e24	0.314148	0.157074	+	0.987587i	$0.449794\pi$
$468$	0	0
$469$	−1.20160e25	−0.505379
$470$	0	0
$471$	1.84824e25	0.746551
$472$	0	0
$473$	2.13768e24	0.0829398
$474$	0	0
$475$	1.48781e25	0.554572
$476$	0	0
$477$	−5.27615e24	−0.188970
$478$	0	0
$479$	1.26967e25	0.437024	0.218512	−	0.975834i	$-0.429880\pi$
$479$	1.26967e25	0.437024	0.218512	+	0.975834i	$0.429880\pi$
$480$	0	0
$481$	1.46098e25	0.483355
$482$	0	0
$483$	−9.30479e24	−0.295944
$484$	0	0
$485$	−6.88263e24	−0.210479
$486$	0	0
$487$	−2.96867e25	−0.873046	−0.436523	−	0.899693i	$-0.643790\pi$
$487$	−2.96867e25	−0.873046	−0.436523	+	0.899693i	$0.643790\pi$
$488$	0	0
$489$	1.17781e25	0.333151
$490$	0	0
$491$	1.10326e25	0.300195	0.150098	−	0.988671i	$-0.452041\pi$
$491$	1.10326e25	0.300195	0.150098	+	0.988671i	$0.452041\pi$
$492$	0	0
$493$	1.93453e25	0.506443
$494$	0	0
$495$	−6.56647e24	−0.165418
$496$	0	0
$497$	−1.24669e25	−0.302254
$498$	0	0
$499$	−6.00880e25	−1.40227	−0.701137	−	0.713027i	$-0.747324\pi$
$499$	−6.00880e25	−1.40227	−0.701137	+	0.713027i	$0.747324\pi$
$500$	0	0
$501$	−5.23448e24	−0.117602
$502$	0	0
$503$	−4.92141e25	−1.06462	−0.532309	−	0.846550i	$-0.678676\pi$
$503$	−4.92141e25	−1.06462	−0.532309	+	0.846550i	$0.678676\pi$
$504$	0	0
$505$	2.41772e25	0.503661
$506$	0	0
$507$	4.30216e24	0.0863198
$508$	0	0
$509$	2.32168e25	0.448728	0.224364	−	0.974505i	$-0.427970\pi$
$509$	2.32168e25	0.448728	0.224364	+	0.974505i	$0.427970\pi$
$510$	0	0
$511$	−1.40628e24	−0.0261862
$512$	0	0
$513$	−1.42588e25	−0.255841
$514$	0	0
$515$	−7.51974e25	−1.30027
$516$	0	0
$517$	3.60635e24	0.0601047
$518$	0	0
$519$	4.94682e25	0.794761
$520$	0	0
$521$	−3.41264e25	−0.528607	−0.264303	−	0.964440i	$-0.585142\pi$
$521$	−3.41264e25	−0.528607	−0.264303	+	0.964440i	$0.585142\pi$
$522$	0	0
$523$	1.69364e25	0.252961	0.126481	−	0.991969i	$-0.459632\pi$
$523$	1.69364e25	0.252961	0.126481	+	0.991969i	$0.459632\pi$
$524$	0	0
$525$	6.31990e24	0.0910326
$526$	0	0
$527$	−6.01122e25	−0.835145
$528$	0	0
$529$	6.26200e25	0.839236
$530$	0	0
$531$	−3.39974e25	−0.439590
$532$	0	0
$533$	−1.41325e26	−1.76324
$534$	0	0
$535$	−6.65839e25	−0.801693
$536$	0	0
$537$	4.56660e25	0.530686
$538$	0	0
$539$	−5.30871e24	−0.0595519
$540$	0	0
$541$	4.89045e25	0.529632	0.264816	−	0.964299i	$-0.414689\pi$
$541$	4.89045e25	0.529632	0.264816	+	0.964299i	$0.414689\pi$
$542$	0	0
$543$	6.10078e25	0.637950
$544$	0	0
$545$	7.65953e25	0.773456
$546$	0	0
$547$	−4.50177e25	−0.439040	−0.219520	−	0.975608i	$-0.570449\pi$
$547$	−4.50177e25	−0.439040	−0.219520	+	0.975608i	$0.570449\pi$
$548$	0	0
$549$	−1.58729e25	−0.149526
$550$	0	0
$551$	−9.63509e25	−0.876830
$552$	0	0
$553$	−7.98693e25	−0.702249
$554$	0	0
$555$	−4.23998e25	−0.360231
$556$	0	0
$557$	−1.45911e26	−1.19802	−0.599011	−	0.800741i	$-0.704439\pi$
$557$	−1.45911e26	−1.19802	−0.599011	+	0.800741i	$0.704439\pi$
$558$	0	0
$559$	−2.31215e25	−0.183486
$560$	0	0
$561$	−2.40905e25	−0.184799
$562$	0	0
$563$	−1.39246e26	−1.03265	−0.516323	−	0.856394i	$-0.672700\pi$
$563$	−1.39246e26	−1.03265	−0.516323	+	0.856394i	$0.672700\pi$
$564$	0	0
$565$	3.26511e25	0.234120
$566$	0	0
$567$	−6.05686e24	−0.0419961
$568$	0	0
$569$	−1.55899e26	−1.04538	−0.522692	−	0.852522i	$-0.675072\pi$
$569$	−1.55899e26	−1.04538	−0.522692	+	0.852522i	$0.675072\pi$
$570$	0	0
$571$	−1.64256e26	−1.06531	−0.532657	−	0.846331i	$-0.678806\pi$
$571$	−1.64256e26	−1.06531	−0.532657	+	0.846331i	$0.678806\pi$
$572$	0	0
$573$	−2.65458e25	−0.166543
$574$	0	0
$575$	−9.32115e25	−0.565751
$576$	0	0
$577$	1.14813e26	0.674248	0.337124	−	0.941460i	$-0.390546\pi$
$577$	1.14813e26	0.674248	0.337124	+	0.941460i	$0.390546\pi$
$578$	0	0
$579$	−1.03170e26	−0.586283
$580$	0	0
$581$	−9.18442e25	−0.505101
$582$	0	0
$583$	4.43975e25	0.236324
$584$	0	0
$585$	7.10238e25	0.365951
$586$	0	0
$587$	3.78010e26	1.88556	0.942782	−	0.333410i	$-0.108199\pi$
$587$	3.78010e26	1.88556	0.942782	+	0.333410i	$0.108199\pi$
$588$	0	0
$589$	2.99394e26	1.44593
$590$	0	0
$591$	1.34315e26	0.628123
$592$	0	0
$593$	−1.06740e25	−0.0483400	−0.0241700	−	0.999708i	$-0.507694\pi$
$593$	−1.06740e25	−0.0483400	−0.0241700	+	0.999708i	$0.507694\pi$
$594$	0	0
$595$	7.87659e25	0.345483
$596$	0	0
$597$	−2.41426e26	−1.02572
$598$	0	0
$599$	1.16470e26	0.479355	0.239678	−	0.970853i	$-0.422958\pi$
$599$	1.16470e26	0.479355	0.239678	+	0.970853i	$0.422958\pi$
$600$	0	0
$601$	3.94573e26	1.57333	0.786665	−	0.617380i	$-0.211806\pi$
$601$	3.94573e26	1.57333	0.786665	+	0.617380i	$0.211806\pi$
$602$	0	0
$603$	1.15361e26	0.445703
$604$	0	0
$605$	−2.62715e26	−0.983577
$606$	0	0
$607$	4.66795e26	1.69369	0.846845	−	0.531840i	$-0.178499\pi$
$607$	4.66795e26	1.69369	0.846845	+	0.531840i	$0.178499\pi$
$608$	0	0
$609$	−4.09279e25	−0.143931
$610$	0	0
$611$	−3.90068e25	−0.132969
$612$	0	0
$613$	2.50502e26	0.827821	0.413911	−	0.910318i	$-0.364163\pi$
$613$	2.50502e26	0.827821	0.413911	+	0.910318i	$0.364163\pi$
$614$	0	0
$615$	4.10147e26	1.31409
$616$	0	0
$617$	−2.80020e25	−0.0869922	−0.0434961	−	0.999054i	$-0.513850\pi$
$617$	−2.80020e25	−0.0869922	−0.0434961	+	0.999054i	$0.513850\pi$
$618$	0	0
$619$	−1.75767e25	−0.0529511	−0.0264756	−	0.999649i	$-0.508428\pi$
$619$	−1.75767e25	−0.0529511	−0.0264756	+	0.999649i	$0.508428\pi$
$620$	0	0
$621$	8.93320e25	0.260998
$622$	0	0
$623$	1.37461e26	0.389531
$624$	0	0
$625$	−4.52251e26	−1.24314
$626$	0	0
$627$	1.19985e26	0.319952
$628$	0	0
$629$	−1.55553e26	−0.402437
$630$	0	0
$631$	−2.98471e25	−0.0749245	−0.0374622	−	0.999298i	$-0.511927\pi$
$631$	−2.98471e25	−0.0749245	−0.0374622	+	0.999298i	$0.511927\pi$
$632$	0	0
$633$	4.32980e26	1.05471
$634$	0	0
$635$	1.06952e26	0.252837
$636$	0	0
$637$	5.74197e25	0.131746
$638$	0	0
$639$	1.19690e26	0.266563
$640$	0	0
$641$	−1.34989e26	−0.291841	−0.145920	−	0.989296i	$-0.546614\pi$
$641$	−1.34989e26	−0.291841	−0.145920	+	0.989296i	$0.546614\pi$
$642$	0	0
$643$	−7.90960e25	−0.166016	−0.0830080	−	0.996549i	$-0.526453\pi$
$643$	−7.90960e25	−0.166016	−0.0830080	+	0.996549i	$0.526453\pi$
$644$	0	0
$645$	6.71020e25	0.136747
$646$	0	0
$647$	4.68915e26	0.927904	0.463952	−	0.885860i	$-0.346431\pi$
$647$	4.68915e26	0.927904	0.463952	+	0.885860i	$0.346431\pi$
$648$	0	0
$649$	2.86080e26	0.549748
$650$	0	0
$651$	1.27176e26	0.237348
$652$	0	0
$653$	1.91185e26	0.346561	0.173280	−	0.984873i	$-0.444563\pi$
$653$	1.91185e26	0.346561	0.173280	+	0.984873i	$0.444563\pi$
$654$	0	0
$655$	7.49791e26	1.32022
$656$	0	0
$657$	1.35012e25	0.0230941
$658$	0	0
$659$	−2.03494e26	−0.338175	−0.169087	−	0.985601i	$-0.554082\pi$
$659$	−2.03494e26	−0.338175	−0.169087	+	0.985601i	$0.554082\pi$
$660$	0	0
$661$	3.97041e26	0.641093	0.320546	−	0.947233i	$-0.396133\pi$
$661$	3.97041e26	0.641093	0.320546	+	0.947233i	$0.396133\pi$
$662$	0	0
$663$	2.60566e26	0.408827
$664$	0	0
$665$	−3.92300e26	−0.598153
$666$	0	0
$667$	6.03641e26	0.894506
$668$	0	0
$669$	−3.14023e26	−0.452287
$670$	0	0
$671$	1.33566e26	0.186996
$672$	0	0
$673$	−1.23624e27	−1.68252	−0.841258	−	0.540634i	$-0.818185\pi$
$673$	−1.23624e27	−1.68252	−0.841258	+	0.540634i	$0.818185\pi$
$674$	0	0
$675$	−6.06751e25	−0.0802832
$676$	0	0
$677$	−1.54693e27	−1.99012	−0.995058	−	0.0992913i	$-0.968342\pi$
$677$	−1.54693e27	−1.99012	−0.995058	+	0.0992913i	$0.968342\pi$
$678$	0	0
$679$	5.34210e25	0.0668265
$680$	0	0
$681$	6.87336e26	0.836126
$682$	0	0
$683$	3.34528e26	0.395764	0.197882	−	0.980226i	$-0.436594\pi$
$683$	3.34528e26	0.395764	0.197882	+	0.980226i	$0.436594\pi$
$684$	0	0
$685$	−1.60109e26	−0.184228
$686$	0	0
$687$	4.52164e26	0.506066
$688$	0	0
$689$	−4.80209e26	−0.522815
$690$	0	0
$691$	−1.29617e27	−1.37285	−0.686423	−	0.727203i	$-0.740820\pi$
$691$	−1.29617e27	−1.37285	−0.686423	+	0.727203i	$0.740820\pi$
$692$	0	0
$693$	5.09670e25	0.0525199
$694$	0	0
$695$	4.80890e26	0.482159
$696$	0	0
$697$	1.50471e27	1.46806
$698$	0	0
$699$	2.48395e26	0.235836
$700$	0	0
$701$	4.42534e26	0.408908	0.204454	−	0.978876i	$-0.434458\pi$
$701$	4.42534e26	0.408908	0.204454	+	0.978876i	$0.434458\pi$
$702$	0	0
$703$	7.74744e26	0.696759
$704$	0	0
$705$	1.13204e26	0.0990978
$706$	0	0
$707$	−1.87656e26	−0.159911
$708$	0	0
$709$	4.53170e26	0.375943	0.187972	−	0.982174i	$-0.439809\pi$
$709$	4.53170e26	0.375943	0.187972	+	0.982174i	$0.439809\pi$
$710$	0	0
$711$	7.66797e26	0.619326
$712$	0	0
$713$	−1.87571e27	−1.47508
$714$	0	0
$715$	−5.97648e26	−0.457655
$716$	0	0
$717$	1.38549e27	1.03317
$718$	0	0
$719$	−1.68830e27	−1.22610	−0.613048	−	0.790046i	$-0.710057\pi$
$719$	−1.68830e27	−1.22610	−0.613048	+	0.790046i	$0.710057\pi$
$720$	0	0
$721$	5.83660e26	0.412834
$722$	0	0
$723$	6.21975e26	0.428508
$724$	0	0
$725$	−4.09998e26	−0.275151
$726$	0	0
$727$	2.39919e27	1.56851	0.784255	−	0.620438i	$-0.213045\pi$
$727$	2.39919e27	1.56851	0.784255	+	0.620438i	$0.213045\pi$
$728$	0	0
$729$	5.81497e25	0.0370370
$730$	0	0
$731$	2.46178e26	0.152769
$732$	0	0
$733$	9.91474e25	0.0599506	0.0299753	−	0.999551i	$-0.490457\pi$
$733$	9.91474e25	0.0599506	0.0299753	+	0.999551i	$0.490457\pi$
$734$	0	0
$735$	−1.66641e26	−0.0981864
$736$	0	0
$737$	−9.70737e26	−0.557392
$738$	0	0
$739$	−1.00367e27	−0.561656	−0.280828	−	0.959758i	$-0.590609\pi$
$739$	−1.00367e27	−0.561656	−0.280828	+	0.959758i	$0.590609\pi$
$740$	0	0
$741$	−1.29777e27	−0.707824
$742$	0	0
$743$	2.49791e26	0.132796	0.0663978	−	0.997793i	$-0.478849\pi$
$743$	2.49791e26	0.132796	0.0663978	+	0.997793i	$0.478849\pi$
$744$	0	0
$745$	−1.29898e27	−0.673158
$746$	0	0
$747$	8.81763e26	0.445457
$748$	0	0
$749$	5.16805e26	0.254536
$750$	0	0
$751$	2.80773e27	1.34827	0.674133	−	0.738610i	$-0.264518\pi$
$751$	2.80773e27	1.34827	0.674133	+	0.738610i	$0.264518\pi$
$752$	0	0
$753$	6.42221e26	0.300699
$754$	0	0
$755$	−2.03513e27	−0.929171
$756$	0	0
$757$	2.49101e26	0.110909	0.0554543	−	0.998461i	$-0.482339\pi$
$757$	2.49101e26	0.110909	0.0554543	+	0.998461i	$0.482339\pi$
$758$	0	0
$759$	−7.51707e26	−0.326401
$760$	0	0
$761$	−3.21096e27	−1.35982	−0.679909	−	0.733297i	$-0.737981\pi$
$761$	−3.21096e27	−1.35982	−0.679909	+	0.733297i	$0.737981\pi$
$762$	0	0
$763$	−5.94511e26	−0.245571
$764$	0	0
$765$	−7.56203e26	−0.304687
$766$	0	0
$767$	−3.09428e27	−1.21620
$768$	0	0
$769$	3.67676e27	1.40982	0.704911	−	0.709295i	$-0.250987\pi$
$769$	3.67676e27	1.40982	0.704911	+	0.709295i	$0.250987\pi$
$770$	0	0
$771$	−8.96688e26	−0.335448
$772$	0	0
$773$	8.93598e26	0.326165	0.163082	−	0.986612i	$-0.447856\pi$
$773$	8.93598e26	0.326165	0.163082	+	0.986612i	$0.447856\pi$
$774$	0	0
$775$	1.27400e27	0.453735
$776$	0	0
$777$	3.29095e26	0.114373
$778$	0	0
$779$	−7.49435e27	−2.54172
$780$	0	0
$781$	−1.00716e27	−0.333361
$782$	0	0
$783$	3.92934e26	0.126935
$784$	0	0
$785$	−4.88193e27	−1.53932
$786$	0	0
$787$	−6.90396e26	−0.212490	−0.106245	−	0.994340i	$-0.533883\pi$
$787$	−6.90396e26	−0.212490	−0.106245	+	0.994340i	$0.533883\pi$
$788$	0	0
$789$	−5.51547e26	−0.165711
$790$	0	0
$791$	−2.53428e26	−0.0743325
$792$	0	0
$793$	−1.44467e27	−0.413689
$794$	0	0
$795$	1.39364e27	0.389639
$796$	0	0
$797$	−4.88555e27	−1.33371	−0.666853	−	0.745190i	$-0.732359\pi$
$797$	−4.88555e27	−1.33371	−0.666853	+	0.745190i	$0.732359\pi$
$798$	0	0
$799$	4.15312e26	0.110708
$800$	0	0
$801$	−1.31971e27	−0.343534
$802$	0	0
$803$	−1.13609e26	−0.0288812
$804$	0	0
$805$	2.45777e27	0.610211
$806$	0	0
$807$	2.34154e27	0.567811
$808$	0	0
$809$	1.15202e27	0.272865	0.136432	−	0.990649i	$-0.456436\pi$
$809$	1.15202e27	0.272865	0.136432	+	0.990649i	$0.456436\pi$
$810$	0	0
$811$	1.54408e26	0.0357250	0.0178625	−	0.999840i	$-0.494314\pi$
$811$	1.54408e26	0.0357250	0.0178625	+	0.999840i	$0.494314\pi$
$812$	0	0
$813$	4.64695e27	1.05029
$814$	0	0
$815$	−3.11106e27	−0.686928
$816$	0	0
$817$	−1.22611e27	−0.264497
$818$	0	0
$819$	−5.51266e26	−0.116189
$820$	0	0
$821$	1.54014e27	0.317176	0.158588	−	0.987345i	$-0.449306\pi$
$821$	1.54014e27	0.317176	0.158588	+	0.987345i	$0.449306\pi$
$822$	0	0
$823$	1.18328e27	0.238117	0.119059	−	0.992887i	$-0.462012\pi$
$823$	1.18328e27	0.238117	0.119059	+	0.992887i	$0.462012\pi$
$824$	0	0
$825$	5.10566e26	0.100401
$826$	0	0
$827$	1.41777e27	0.272460	0.136230	−	0.990677i	$-0.456501\pi$
$827$	1.41777e27	0.272460	0.136230	+	0.990677i	$0.456501\pi$
$828$	0	0
$829$	−8.49524e27	−1.59554	−0.797769	−	0.602963i	$-0.793987\pi$
$829$	−8.49524e27	−1.59554	−0.797769	+	0.602963i	$0.793987\pi$
$830$	0	0
$831$	−3.93866e27	−0.723002
$832$	0	0
$833$	−6.11358e26	−0.109690
$834$	0	0
$835$	1.38263e27	0.242485
$836$	0	0
$837$	−1.22097e27	−0.209322
$838$	0	0
$839$	−8.01734e26	−0.134367	−0.0671833	−	0.997741i	$-0.521401\pi$
$839$	−8.01734e26	−0.134367	−0.0671833	+	0.997741i	$0.521401\pi$
$840$	0	0
$841$	−3.44810e27	−0.564960
$842$	0	0
$843$	3.23483e27	0.518190
$844$	0	0
$845$	−1.13637e27	−0.177984
$846$	0	0
$847$	2.03911e27	0.312284
$848$	0	0
$849$	2.04676e27	0.306509
$850$	0	0
$851$	−4.85379e27	−0.710805
$852$	0	0
$853$	−5.21416e27	−0.746738	−0.373369	−	0.927683i	$-0.621797\pi$
$853$	−5.21416e27	−0.746738	−0.373369	+	0.927683i	$0.621797\pi$
$854$	0	0
$855$	3.76633e27	0.527521
$856$	0	0
$857$	−2.91978e27	−0.399974	−0.199987	−	0.979799i	$-0.564090\pi$
$857$	−2.91978e27	−0.399974	−0.199987	+	0.979799i	$0.564090\pi$
$858$	0	0
$859$	1.09946e28	1.47314	0.736568	−	0.676363i	$-0.236445\pi$
$859$	1.09946e28	1.47314	0.736568	+	0.676363i	$0.236445\pi$
$860$	0	0
$861$	−3.18344e27	−0.417222
$862$	0	0
$863$	4.35327e27	0.558101	0.279051	−	0.960276i	$-0.409980\pi$
$863$	4.35327e27	0.558101	0.279051	+	0.960276i	$0.409980\pi$
$864$	0	0
$865$	−1.30665e28	−1.63873
$866$	0	0
$867$	1.93137e27	0.236965
$868$	0	0
$869$	−6.45241e27	−0.774523
$870$	0	0
$871$	1.04996e28	1.23311
$872$	0	0
$873$	−5.12875e26	−0.0589354
$874$	0	0
$875$	2.33230e27	0.262245
$876$	0	0
$877$	1.44659e27	0.159165	0.0795826	−	0.996828i	$-0.474641\pi$
$877$	1.44659e27	0.159165	0.0795826	+	0.996828i	$0.474641\pi$
$878$	0	0
$879$	2.98969e27	0.321909
$880$	0	0
$881$	−1.47605e28	−1.55536	−0.777679	−	0.628662i	$-0.783603\pi$
$881$	−1.47605e28	−1.55536	−0.777679	+	0.628662i	$0.783603\pi$
$882$	0	0
$883$	3.35356e27	0.345844	0.172922	−	0.984936i	$-0.444679\pi$
$883$	3.35356e27	0.345844	0.172922	+	0.984936i	$0.444679\pi$
$884$	0	0
$885$	8.98007e27	0.906398
$886$	0	0
$887$	1.22830e28	1.21347	0.606736	−	0.794904i	$-0.292479\pi$
$887$	1.22830e28	1.21347	0.606736	+	0.794904i	$0.292479\pi$
$888$	0	0
$889$	−8.30134e26	−0.0802752
$890$	0	0
$891$	−4.89316e26	−0.0463182
$892$	0	0
$893$	−2.06850e27	−0.191675
$894$	0	0
$895$	−1.20622e28	−1.09423
$896$	0	0
$897$	8.13057e27	0.722092
$898$	0	0
$899$	−8.25045e27	−0.717398
$900$	0	0
$901$	5.11287e27	0.435291
$902$	0	0
$903$	−5.20826e26	−0.0434170
$904$	0	0
$905$	−1.61146e28	−1.31540
$906$	0	0
$907$	−1.26864e28	−1.01407	−0.507037	−	0.861924i	$-0.669259\pi$
$907$	−1.26864e28	−1.01407	−0.507037	+	0.861924i	$0.669259\pi$
$908$	0	0
$909$	1.80162e27	0.141028
$910$	0	0
$911$	1.03607e28	0.794261	0.397131	−	0.917762i	$-0.370006\pi$
$911$	1.03607e28	0.794261	0.397131	+	0.917762i	$0.370006\pi$
$912$	0	0
$913$	−7.41982e27	−0.557085
$914$	0	0
$915$	4.19265e27	0.308311
$916$	0	0
$917$	−5.81966e27	−0.419169
$918$	0	0
$919$	1.91586e28	1.35166	0.675828	−	0.737060i	$-0.263786\pi$
$919$	1.91586e28	1.35166	0.675828	+	0.737060i	$0.263786\pi$
$920$	0	0
$921$	5.46423e27	0.377626
$922$	0	0
$923$	1.08936e28	0.737489
$924$	0	0
$925$	3.29674e27	0.218644
$926$	0	0
$927$	−5.60351e27	−0.364085
$928$	0	0
$929$	1.26967e28	0.808242	0.404121	−	0.914706i	$-0.367578\pi$
$929$	1.26967e28	0.808242	0.404121	+	0.914706i	$0.367578\pi$
$930$	0	0
$931$	3.04491e27	0.189912
$932$	0	0
$933$	−1.11345e28	−0.680446
$934$	0	0
$935$	6.36326e27	0.381039
$936$	0	0
$937$	3.09817e28	1.81794	0.908968	−	0.416866i	$-0.136872\pi$
$937$	3.09817e28	1.81794	0.908968	+	0.416866i	$0.136872\pi$
$938$	0	0
$939$	1.73602e28	0.998230
$940$	0	0
$941$	1.64498e28	0.926957	0.463478	−	0.886108i	$-0.346601\pi$
$941$	1.64498e28	0.926957	0.463478	+	0.886108i	$0.346601\pi$
$942$	0	0
$943$	4.69523e28	2.59296
$944$	0	0
$945$	1.59986e27	0.0865923
$946$	0	0
$947$	−1.21568e28	−0.644902	−0.322451	−	0.946586i	$-0.604507\pi$
$947$	−1.21568e28	−0.644902	−0.322451	+	0.946586i	$0.604507\pi$
$948$	0	0
$949$	1.22881e27	0.0638934
$950$	0	0
$951$	−1.02086e28	−0.520299
$952$	0	0
$953$	2.58383e28	1.29086	0.645432	−	0.763818i	$-0.276677\pi$
$953$	2.58383e28	1.29086	0.645432	+	0.763818i	$0.276677\pi$
$954$	0	0
$955$	7.01180e27	0.343398
$956$	0	0
$957$	−3.30644e27	−0.158744
$958$	0	0
$959$	1.24272e27	0.0584921
$960$	0	0
$961$	3.96615e27	0.183019
$962$	0	0
$963$	−4.96165e27	−0.224480
$964$	0	0
$965$	2.72513e28	1.20886
$966$	0	0
$967$	−2.16830e28	−0.943124	−0.471562	−	0.881833i	$-0.656310\pi$
$967$	−2.16830e28	−0.943124	−0.471562	+	0.881833i	$0.656310\pi$
$968$	0	0
$969$	1.38176e28	0.589327
$970$	0	0
$971$	5.43068e27	0.227128	0.113564	−	0.993531i	$-0.463773\pi$
$971$	5.43068e27	0.227128	0.113564	+	0.993531i	$0.463773\pi$
$972$	0	0
$973$	−3.73253e27	−0.153084
$974$	0	0
$975$	−5.52235e27	−0.222116
$976$	0	0
$977$	−1.65468e28	−0.652703	−0.326352	−	0.945248i	$-0.605819\pi$
$977$	−1.65468e28	−0.652703	−0.326352	+	0.945248i	$0.605819\pi$
$978$	0	0
$979$	1.11051e28	0.429620
$980$	0	0
$981$	5.70768e27	0.216573
$982$	0	0
$983$	2.42674e28	0.903159	0.451579	−	0.892231i	$-0.350861\pi$
$983$	2.42674e28	0.903159	0.451579	+	0.892231i	$0.350861\pi$
$984$	0	0
$985$	−3.54781e28	−1.29514
$986$	0	0
$987$	−8.78654e26	−0.0314633
$988$	0	0
$989$	7.68161e27	0.269828
$990$	0	0
$991$	3.07002e28	1.05789	0.528947	−	0.848655i	$-0.322587\pi$
$991$	3.07002e28	1.05789	0.528947	+	0.848655i	$0.322587\pi$
$992$	0	0
$993$	−1.65227e28	−0.558551
$994$	0	0
$995$	6.37702e28	2.11494
$996$	0	0
$997$	−4.45806e28	−1.45058	−0.725291	−	0.688442i	$-0.758295\pi$
$997$	−4.45806e28	−1.45058	−0.725291	+	0.688442i	$0.758295\pi$
$998$	0	0
$999$	−3.15952e27	−0.100867

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.20.a.a.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.20.a.a.1.4	✓	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)

\(f(q)\)	\(=\)	\(q-19683.0 q^{3} +5.19906e6 q^{5} -4.03536e7 q^{7} +3.87420e8 q^{9} +O(q^{10})\)	\(q-19683.0 q^{3} +5.19906e6 q^{5} -4.03536e7 q^{7} +3.87420e8 q^{9} -3.26005e9 q^{11} +3.52611e10 q^{13} -1.02333e11 q^{15} -3.75431e11 q^{17} +1.86987e12 q^{19} +7.94280e11 q^{21} -1.17148e13 q^{23} +7.95677e12 q^{25} -7.62560e12 q^{27} -5.15283e13 q^{29} +1.60115e14 q^{31} +6.41676e13 q^{33} -2.09801e14 q^{35} +4.14331e14 q^{37} -6.94045e14 q^{39} -4.00796e15 q^{41} -6.55721e14 q^{43} +2.01422e15 q^{45} -1.10623e15 q^{47} +1.62841e15 q^{49} +7.38962e15 q^{51} -1.36187e16 q^{53} -1.69492e16 q^{55} -3.68046e16 q^{57} -8.77533e16 q^{59} -4.09706e16 q^{61} -1.56338e16 q^{63} +1.83325e17 q^{65} +2.97767e17 q^{67} +2.30581e17 q^{69} +3.08941e17 q^{71} +3.48489e16 q^{73} -1.56613e17 q^{75} +1.31555e17 q^{77} +1.97924e18 q^{79} +1.50095e17 q^{81} +2.27598e18 q^{83} -1.95189e18 q^{85} +1.01423e18 q^{87} -3.40640e18 q^{89} -1.42291e18 q^{91} -3.15154e18 q^{93} +9.72155e18 q^{95} -1.32382e18 q^{97} -1.26301e18 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 78732 q^{3} - 2502150 q^{5} - 161414428 q^{7} + 1549681956 q^{9}+O(q^{10}) \) 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

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