LMFDB - Embedded newform 4851.2.a.bl.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4851,2,Mod(1,4851)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4851, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4851.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4851 = 3^{2} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4851.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:	$38.7354300205$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1617)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.254102$ of defining polynomial
Character	$\chi$	$=$	4851.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.254102 q^{2} -1.93543 q^{4} -3.68133 q^{5} -1.00000 q^{8} +O(q^{10})$	$q+0.254102 q^{2} -1.93543 q^{4} -3.68133 q^{5} -1.00000 q^{8} -0.935432 q^{10} -1.00000 q^{11} -2.18953 q^{13} +3.61676 q^{16} +3.36266 q^{17} +3.68133 q^{19} +7.12497 q^{20} -0.254102 q^{22} +8.55220 q^{25} -0.556364 q^{26} -10.0604 q^{29} +8.37907 q^{31} +2.91903 q^{32} +0.854458 q^{34} +0.189534 q^{37} +0.935432 q^{38} +3.68133 q^{40} +8.37907 q^{41} -8.37907 q^{43} +1.93543 q^{44} -5.17313 q^{47} +2.17313 q^{50} +4.23769 q^{52} +1.36266 q^{53} +3.68133 q^{55} -2.55636 q^{58} +4.53579 q^{59} -0.379068 q^{61} +2.12914 q^{62} -6.49180 q^{64} +8.06040 q^{65} +14.4067 q^{67} -6.50820 q^{68} +14.0880 q^{71} +13.9313 q^{73} +0.0481609 q^{74} -7.12497 q^{76} -2.98359 q^{79} -13.3145 q^{80} +2.12914 q^{82} -4.37907 q^{83} -12.3791 q^{85} -2.12914 q^{86} +1.00000 q^{88} -0.637339 q^{89} -1.31450 q^{94} -13.5522 q^{95} +9.39547 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{4} - 4 q^{5} - 3 q^{8}+O(q^{10}) $ 3 * q + 2 * q^4 - 4 * q^5 - 3 * q^8	$ 3 q + 2 q^{4} - 4 q^{5} - 3 q^{8} + 5 q^{10} - 3 q^{11} + 2 q^{13} - 4 q^{16} - 4 q^{17} + 4 q^{19} + 5 q^{20} + 3 q^{25} - 11 q^{26} - 6 q^{29} + 8 q^{31} + 4 q^{32} - 10 q^{34} - 8 q^{37} - 5 q^{38} + 4 q^{40} + 8 q^{41} - 8 q^{43} - 2 q^{44} - 10 q^{47} + q^{50} + 15 q^{52} - 10 q^{53} + 4 q^{55} - 17 q^{58} - 6 q^{59} + 16 q^{61} + 22 q^{62} - 21 q^{64} + 8 q^{67} - 18 q^{68} + 2 q^{73} + 11 q^{74} - 5 q^{76} - 12 q^{79} - 15 q^{80} + 22 q^{82} + 4 q^{83} - 20 q^{85} - 22 q^{86} + 3 q^{88} - 16 q^{89} + 21 q^{94} - 18 q^{95} + 8 q^{97}+O(q^{100}) $ 3 * q + 2 * q^4 - 4 * q^5 - 3 * q^8 + 5 * q^10 - 3 * q^11 + 2 * q^13 - 4 * q^16 - 4 * q^17 + 4 * q^19 + 5 * q^20 + 3 * q^25 - 11 * q^26 - 6 * q^29 + 8 * q^31 + 4 * q^32 - 10 * q^34 - 8 * q^37 - 5 * q^38 + 4 * q^40 + 8 * q^41 - 8 * q^43 - 2 * q^44 - 10 * q^47 + q^50 + 15 * q^52 - 10 * q^53 + 4 * q^55 - 17 * q^58 - 6 * q^59 + 16 * q^61 + 22 * q^62 - 21 * q^64 + 8 * q^67 - 18 * q^68 + 2 * q^73 + 11 * q^74 - 5 * q^76 - 12 * q^79 - 15 * q^80 + 22 * q^82 + 4 * q^83 - 20 * q^85 - 22 * q^86 + 3 * q^88 - 16 * q^89 + 21 * q^94 - 18 * q^95 + 8 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.254102	0.179677	0.0898385	−	0.995956i	$-0.471365\pi$
$2$	0.254102	0.179677	0.0898385	+	0.995956i	$0.471365\pi$
$3$	0	0
$3$	0	0
$4$	−1.93543	−0.967716
$5$	−3.68133	−1.64634	−0.823171	−	0.567794i	$-0.807797\pi$
$5$	−3.68133	−1.64634	−0.823171	+	0.567794i	$0.807797\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−0.935432	−0.295810
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−2.18953	−0.607267	−0.303634	−	0.952789i	$-0.598200\pi$
$13$	−2.18953	−0.607267	−0.303634	+	0.952789i	$0.598200\pi$
$14$	0	0
$15$	0	0
$16$	3.61676	0.904191
$17$	3.36266	0.815565	0.407783	−	0.913079i	$-0.366302\pi$
$17$	3.36266	0.815565	0.407783	+	0.913079i	$0.366302\pi$
$18$	0	0
$19$	3.68133	0.844555	0.422278	−	0.906467i	$-0.361231\pi$
$19$	3.68133	0.844555	0.422278	+	0.906467i	$0.361231\pi$
$20$	7.12497	1.59319
$21$	0	0
$22$	−0.254102	−0.0541747
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	8.55220	1.71044
$26$	−0.556364	−0.109112
$27$	0	0
$28$	0	0
$29$	−10.0604	−1.86817	−0.934085	−	0.357052i	$-0.883782\pi$
$29$	−10.0604	−1.86817	−0.934085	+	0.357052i	$0.883782\pi$
$30$	0	0
$31$	8.37907	1.50493	0.752463	−	0.658635i	$-0.228866\pi$
$31$	8.37907	1.50493	0.752463	+	0.658635i	$0.228866\pi$
$32$	2.91903	0.516016
$33$	0	0
$34$	0.854458	0.146538
$35$	0	0
$36$	0	0
$37$	0.189534	0.0311592	0.0155796	−	0.999879i	$-0.495041\pi$
$37$	0.189534	0.0311592	0.0155796	+	0.999879i	$0.495041\pi$
$38$	0.935432	0.151747
$39$	0	0
$40$	3.68133	0.582069
$41$	8.37907	1.30859	0.654295	−	0.756239i	$-0.272965\pi$
$41$	8.37907	1.30859	0.654295	+	0.756239i	$0.272965\pi$
$42$	0	0
$43$	−8.37907	−1.27780	−0.638898	−	0.769291i	$-0.720609\pi$
$43$	−8.37907	−1.27780	−0.638898	+	0.769291i	$0.720609\pi$
$44$	1.93543	0.291777
$45$	0	0
$46$	0	0
$47$	−5.17313	−0.754578	−0.377289	−	0.926096i	$-0.623144\pi$
$47$	−5.17313	−0.754578	−0.377289	+	0.926096i	$0.623144\pi$
$48$	0	0
$49$	0	0
$50$	2.17313	0.307327
$51$	0	0
$52$	4.23769	0.587663
$53$	1.36266	0.187176	0.0935880	−	0.995611i	$-0.470166\pi$
$53$	1.36266	0.187176	0.0935880	+	0.995611i	$0.470166\pi$
$54$	0	0
$55$	3.68133	0.496391
$56$	0	0
$57$	0	0
$58$	−2.55636	−0.335667
$59$	4.53579	0.590509	0.295255	−	0.955419i	$-0.404595\pi$
$59$	4.53579	0.590509	0.295255	+	0.955419i	$0.404595\pi$
$60$	0	0
$61$	−0.379068	−0.0485347	−0.0242673	−	0.999706i	$-0.507725\pi$
$61$	−0.379068	−0.0485347	−0.0242673	+	0.999706i	$0.507725\pi$
$62$	2.12914	0.270400
$63$	0	0
$64$	−6.49180	−0.811475
$65$	8.06040	0.999769
$66$	0	0
$67$	14.4067	1.76005	0.880026	−	0.474925i	$-0.157525\pi$
$67$	14.4067	1.76005	0.880026	+	0.474925i	$0.157525\pi$
$68$	−6.50820	−0.789236
$69$	0	0
$70$	0	0
$71$	14.0880	1.67194	0.835968	−	0.548778i	$-0.184907\pi$
$71$	14.0880	1.67194	0.835968	+	0.548778i	$0.184907\pi$
$72$	0	0
$73$	13.9313	1.63053	0.815266	−	0.579087i	$-0.196591\pi$
$73$	13.9313	1.63053	0.815266	+	0.579087i	$0.196591\pi$
$74$	0.0481609	0.00559859
$75$	0	0
$76$	−7.12497	−0.817290
$77$	0	0
$78$	0	0
$79$	−2.98359	−0.335680	−0.167840	−	0.985814i	$-0.553679\pi$
$79$	−2.98359	−0.335680	−0.167840	+	0.985814i	$0.553679\pi$
$80$	−13.3145	−1.48861
$81$	0	0
$82$	2.12914	0.235124
$83$	−4.37907	−0.480665	−0.240333	−	0.970691i	$-0.577257\pi$
$83$	−4.37907	−0.480665	−0.240333	+	0.970691i	$0.577257\pi$
$84$	0	0
$85$	−12.3791	−1.34270
$86$	−2.12914	−0.229591
$87$	0	0
$88$	1.00000	0.106600
$89$	−0.637339	−0.0675578	−0.0337789	−	0.999429i	$-0.510754\pi$
$89$	−0.637339	−0.0675578	−0.0337789	+	0.999429i	$0.510754\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	−1.31450	−0.135580
$95$	−13.5522	−1.39043
$96$	0	0
$97$	9.39547	0.953966	0.476983	−	0.878913i	$-0.341730\pi$
$97$	9.39547	0.953966	0.476983	+	0.878913i	$0.341730\pi$
$98$	0	0
$99$	0	0
$100$	−16.5522	−1.65522
$101$	−7.10439	−0.706913	−0.353457	−	0.935451i	$-0.614994\pi$
$101$	−7.10439	−0.706913	−0.353457	+	0.935451i	$0.614994\pi$
$102$	0	0
$103$	0.258271	0.0254482	0.0127241	−	0.999919i	$-0.495950\pi$
$103$	0.258271	0.0254482	0.0127241	+	0.999919i	$0.495950\pi$
$104$	2.18953	0.214701
$105$	0	0
$106$	0.346255	0.0336312
$107$	−12.5358	−1.21188	−0.605940	−	0.795510i	$-0.707203\pi$
$107$	−12.5358	−1.21188	−0.605940	+	0.795510i	$0.707203\pi$
$108$	0	0
$109$	−17.7417	−1.69935	−0.849675	−	0.527307i	$-0.823202\pi$
$109$	−17.7417	−1.69935	−0.849675	+	0.527307i	$0.823202\pi$
$110$	0.935432	0.0891900
$111$	0	0
$112$	0	0
$113$	−18.1208	−1.70466	−0.852331	−	0.523003i	$-0.824812\pi$
$113$	−18.1208	−1.70466	−0.852331	+	0.523003i	$0.824812\pi$
$114$	0	0
$115$	0	0
$116$	19.4712	1.80786
$117$	0	0
$118$	1.15255	0.106101
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−0.0963218	−0.00872057
$123$	0	0
$124$	−16.2171	−1.45634
$125$	−13.0768	−1.16963
$126$	0	0
$127$	−10.9836	−0.974636	−0.487318	−	0.873224i	$-0.662025\pi$
$127$	−10.9836	−0.974636	−0.487318	+	0.873224i	$0.662025\pi$
$128$	−7.48763	−0.661819
$129$	0	0
$130$	2.04816	0.179636
$131$	−16.1208	−1.40848	−0.704240	−	0.709962i	$-0.748712\pi$
$131$	−16.1208	−1.40848	−0.704240	+	0.709962i	$0.748712\pi$
$132$	0	0
$133$	0	0
$134$	3.66075	0.316241
$135$	0	0
$136$	−3.36266	−0.288346
$137$	4.98359	0.425777	0.212889	−	0.977076i	$-0.431713\pi$
$137$	4.98359	0.425777	0.212889	+	0.977076i	$0.431713\pi$
$138$	0	0
$139$	−21.4506	−1.81942	−0.909710	−	0.415244i	$-0.863696\pi$
$139$	−21.4506	−1.81942	−0.909710	+	0.415244i	$0.863696\pi$
$140$	0	0
$141$	0	0
$142$	3.57978	0.300409
$143$	2.18953	0.183098
$144$	0	0
$145$	37.0357	3.07564
$146$	3.53996	0.292969
$147$	0	0
$148$	−0.366830	−0.0301533
$149$	−5.30226	−0.434378	−0.217189	−	0.976130i	$-0.569689\pi$
$149$	−5.30226	−0.434378	−0.217189	+	0.976130i	$0.569689\pi$
$150$	0	0
$151$	0.637339	0.0518659	0.0259329	−	0.999664i	$-0.491744\pi$
$151$	0.637339	0.0518659	0.0259329	+	0.999664i	$0.491744\pi$
$152$	−3.68133	−0.298595
$153$	0	0
$154$	0	0
$155$	−30.8461	−2.47762
$156$	0	0
$157$	−9.70892	−0.774856	−0.387428	−	0.921900i	$-0.626636\pi$
$157$	−9.70892	−0.774856	−0.387428	+	0.921900i	$0.626636\pi$
$158$	−0.758136	−0.0603141
$159$	0	0
$160$	−10.7459	−0.849538
$161$	0	0
$162$	0	0
$163$	7.68133	0.601648	0.300824	−	0.953680i	$-0.402738\pi$
$163$	7.68133	0.601648	0.300824	+	0.953680i	$0.402738\pi$
$164$	−16.2171	−1.26634
$165$	0	0
$166$	−1.11273	−0.0863645
$167$	−16.0000	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$168$	0	0
$169$	−8.20594	−0.631226
$170$	−3.14554	−0.241252
$171$	0	0
$172$	16.2171	1.23654
$173$	13.7089	1.04227	0.521135	−	0.853474i	$-0.325509\pi$
$173$	13.7089	1.04227	0.521135	+	0.853474i	$0.325509\pi$
$174$	0	0
$175$	0	0
$176$	−3.61676	−0.272624
$177$	0	0
$178$	−0.161949	−0.0121386
$179$	7.74173	0.578644	0.289322	−	0.957232i	$-0.406570\pi$
$179$	7.74173	0.578644	0.289322	+	0.957232i	$0.406570\pi$
$180$	0	0
$181$	20.3791	1.51476	0.757382	−	0.652972i	$-0.226478\pi$
$181$	20.3791	1.51476	0.757382	+	0.652972i	$0.226478\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−0.697737	−0.0512987
$186$	0	0
$187$	−3.36266	−0.245902
$188$	10.0122	0.730217
$189$	0	0
$190$	−3.44364	−0.249828
$191$	−9.70892	−0.702512	−0.351256	−	0.936279i	$-0.614245\pi$
$191$	−9.70892	−0.702512	−0.351256	+	0.936279i	$0.614245\pi$
$192$	0	0
$193$	−20.0880	−1.44596	−0.722982	−	0.690866i	$-0.757229\pi$
$193$	−20.0880	−1.44596	−0.722982	+	0.690866i	$0.757229\pi$
$194$	2.38741	0.171406
$195$	0	0
$196$	0	0
$197$	−2.25827	−0.160895	−0.0804476	−	0.996759i	$-0.525635\pi$
$197$	−2.25827	−0.160895	−0.0804476	+	0.996759i	$0.525635\pi$
$198$	0	0
$199$	−17.1372	−1.21482	−0.607412	−	0.794387i	$-0.707792\pi$
$199$	−17.1372	−1.21482	−0.607412	+	0.794387i	$0.707792\pi$
$200$	−8.55220	−0.604732
$201$	0	0
$202$	−1.80524	−0.127016
$203$	0	0
$204$	0	0
$205$	−30.8461	−2.15439
$206$	0.0656270	0.00457245
$207$	0	0
$208$	−7.91903	−0.549086
$209$	−3.68133	−0.254643
$210$	0	0
$211$	−9.65375	−0.664591	−0.332296	−	0.943175i	$-0.607823\pi$
$211$	−9.65375	−0.664591	−0.332296	+	0.943175i	$0.607823\pi$
$212$	−2.63734	−0.181133
$213$	0	0
$214$	−3.18537	−0.217747
$215$	30.8461	2.10369
$216$	0	0
$217$	0	0
$218$	−4.50820	−0.305334
$219$	0	0
$220$	−7.12497	−0.480365
$221$	−7.36266	−0.495266
$222$	0	0
$223$	1.65375	0.110743	0.0553715	−	0.998466i	$-0.482366\pi$
$223$	1.65375	0.110743	0.0553715	+	0.998466i	$0.482366\pi$
$224$	0	0
$225$	0	0
$226$	−4.60453	−0.306288
$227$	4.69251	0.311453	0.155726	−	0.987800i	$-0.450228\pi$
$227$	4.69251	0.311453	0.155726	+	0.987800i	$0.450228\pi$
$228$	0	0
$229$	−8.00000	−0.528655	−0.264327	−	0.964433i	$-0.585150\pi$
$229$	−8.00000	−0.528655	−0.264327	+	0.964433i	$0.585150\pi$
$230$	0	0
$231$	0	0
$232$	10.0604	0.660498
$233$	15.0164	0.983758	0.491879	−	0.870664i	$-0.336310\pi$
$233$	15.0164	0.983758	0.491879	+	0.870664i	$0.336310\pi$
$234$	0	0
$235$	19.0440	1.24229
$236$	−8.77871	−0.571445
$237$	0	0
$238$	0	0
$239$	20.2775	1.31164	0.655822	−	0.754916i	$-0.272322\pi$
$239$	20.2775	1.31164	0.655822	+	0.754916i	$0.272322\pi$
$240$	0	0
$241$	−21.1731	−1.36388	−0.681941	−	0.731408i	$-0.738864\pi$
$241$	−21.1731	−1.36388	−0.681941	+	0.731408i	$0.738864\pi$
$242$	0.254102	0.0163343
$243$	0	0
$244$	0.733661	0.0469678
$245$	0	0
$246$	0	0
$247$	−8.06040	−0.512871
$248$	−8.37907	−0.532071
$249$	0	0
$250$	−3.32284	−0.210155
$251$	−7.08514	−0.447210	−0.223605	−	0.974680i	$-0.571783\pi$
$251$	−7.08514	−0.447210	−0.223605	+	0.974680i	$0.571783\pi$
$252$	0	0
$253$	0	0
$254$	−2.79095	−0.175120
$255$	0	0
$256$	11.0810	0.692561
$257$	23.5439	1.46863	0.734313	−	0.678811i	$-0.237505\pi$
$257$	23.5439	1.46863	0.734313	+	0.678811i	$0.237505\pi$
$258$	0	0
$259$	0	0
$260$	−15.6004	−0.967493
$261$	0	0
$262$	−4.09632	−0.253072
$263$	9.17313	0.565639	0.282820	−	0.959173i	$-0.408730\pi$
$263$	9.17313	0.565639	0.282820	+	0.959173i	$0.408730\pi$
$264$	0	0
$265$	−5.01641	−0.308155
$266$	0	0
$267$	0	0
$268$	−27.8831	−1.70323
$269$	17.3955	1.06062	0.530310	−	0.847804i	$-0.322075\pi$
$269$	17.3955	1.06062	0.530310	+	0.847804i	$0.322075\pi$
$270$	0	0
$271$	22.1484	1.34542	0.672709	−	0.739907i	$-0.265131\pi$
$271$	22.1484	1.34542	0.672709	+	0.739907i	$0.265131\pi$
$272$	12.1619	0.737426
$273$	0	0
$274$	1.26634	0.0765024
$275$	−8.55220	−0.515717
$276$	0	0
$277$	−14.6925	−0.882787	−0.441394	−	0.897314i	$-0.645516\pi$
$277$	−14.6925	−0.882787	−0.441394	+	0.897314i	$0.645516\pi$
$278$	−5.45065	−0.326908
$279$	0	0
$280$	0	0
$281$	27.8901	1.66378	0.831892	−	0.554937i	$-0.187258\pi$
$281$	27.8901	1.66378	0.831892	+	0.554937i	$0.187258\pi$
$282$	0	0
$283$	16.3738	0.973324	0.486662	−	0.873590i	$-0.338214\pi$
$283$	16.3738	0.973324	0.486662	+	0.873590i	$0.338214\pi$
$284$	−27.2663	−1.61796
$285$	0	0
$286$	0.556364	0.0328985
$287$	0	0
$288$	0	0
$289$	−5.69251	−0.334853
$290$	9.41082	0.552623
$291$	0	0
$292$	−26.9630	−1.57789
$293$	−12.7581	−0.745338	−0.372669	−	0.927964i	$-0.621557\pi$
$293$	−12.7581	−0.745338	−0.372669	+	0.927964i	$0.621557\pi$
$294$	0	0
$295$	−16.6977	−0.972180
$296$	−0.189534	−0.0110164
$297$	0	0
$298$	−1.34731	−0.0780478
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0.161949	0.00931911
$303$	0	0
$304$	13.3145	0.763639
$305$	1.39547	0.0799047
$306$	0	0
$307$	29.4506	1.68084	0.840419	−	0.541938i	$-0.182309\pi$
$307$	29.4506	1.68084	0.840419	+	0.541938i	$0.182309\pi$
$308$	0	0
$309$	0	0
$310$	−7.83805	−0.445171
$311$	−12.7581	−0.723448	−0.361724	−	0.932285i	$-0.617812\pi$
$311$	−12.7581	−0.723448	−0.361724	+	0.932285i	$0.617812\pi$
$312$	0	0
$313$	18.3463	1.03699	0.518496	−	0.855080i	$-0.326492\pi$
$313$	18.3463	1.03699	0.518496	+	0.855080i	$0.326492\pi$
$314$	−2.46705	−0.139224
$315$	0	0
$316$	5.77454	0.324843
$317$	−3.01641	−0.169418	−0.0847091	−	0.996406i	$-0.526996\pi$
$317$	−3.01641	−0.169418	−0.0847091	+	0.996406i	$0.526996\pi$
$318$	0	0
$319$	10.0604	0.563274
$320$	23.8984	1.33596
$321$	0	0
$322$	0	0
$323$	12.3791	0.688790
$324$	0	0
$325$	−18.7253	−1.03869
$326$	1.95184	0.108102
$327$	0	0
$328$	−8.37907	−0.462657
$329$	0	0
$330$	0	0
$331$	−14.7909	−0.812984	−0.406492	−	0.913654i	$-0.633248\pi$
$331$	−14.7909	−0.812984	−0.406492	+	0.913654i	$0.633248\pi$
$332$	8.47539	0.465147
$333$	0	0
$334$	−4.06563	−0.222461
$335$	−53.0357	−2.89765
$336$	0	0
$337$	−19.0716	−1.03890	−0.519448	−	0.854502i	$-0.673862\pi$
$337$	−19.0716	−1.03890	−0.519448	+	0.854502i	$0.673862\pi$
$338$	−2.08514	−0.113417
$339$	0	0
$340$	23.9588	1.29935
$341$	−8.37907	−0.453752
$342$	0	0
$343$	0	0
$344$	8.37907	0.451769
$345$	0	0
$346$	3.48346	0.187272
$347$	−9.45065	−0.507337	−0.253669	−	0.967291i	$-0.581637\pi$
$347$	−9.45065	−0.507337	−0.253669	+	0.967291i	$0.581637\pi$
$348$	0	0
$349$	−28.9805	−1.55129	−0.775645	−	0.631170i	$-0.782575\pi$
$349$	−28.9805	−1.55129	−0.775645	+	0.631170i	$0.782575\pi$
$350$	0	0
$351$	0	0
$352$	−2.91903	−0.155585
$353$	18.7306	0.996927	0.498463	−	0.866911i	$-0.333898\pi$
$353$	18.7306	0.996927	0.498463	+	0.866911i	$0.333898\pi$
$354$	0	0
$355$	−51.8625	−2.75258
$356$	1.23353	0.0653767
$357$	0	0
$358$	1.96719	0.103969
$359$	−7.48346	−0.394962	−0.197481	−	0.980307i	$-0.563276\pi$
$359$	−7.48346	−0.394962	−0.197481	+	0.980307i	$0.563276\pi$
$360$	0	0
$361$	−5.44780	−0.286727
$362$	5.17836	0.272168
$363$	0	0
$364$	0	0
$365$	−51.2856	−2.68441
$366$	0	0
$367$	15.4283	0.805350	0.402675	−	0.915343i	$-0.368080\pi$
$367$	15.4283	0.805350	0.402675	+	0.915343i	$0.368080\pi$
$368$	0	0
$369$	0	0
$370$	−0.177296	−0.00921719
$371$	0	0
$372$	0	0
$373$	−26.1208	−1.35248	−0.676242	−	0.736680i	$-0.736393\pi$
$373$	−26.1208	−1.35248	−0.676242	+	0.736680i	$0.736393\pi$
$374$	−0.854458	−0.0441830
$375$	0	0
$376$	5.17313	0.266784
$377$	22.0276	1.13448
$378$	0	0
$379$	−13.6485	−0.701077	−0.350539	−	0.936548i	$-0.614001\pi$
$379$	−13.6485	−0.701077	−0.350539	+	0.936548i	$0.614001\pi$
$380$	26.2294	1.34554
$381$	0	0
$382$	−2.46705	−0.126225
$383$	−25.4506	−1.30047	−0.650234	−	0.759734i	$-0.725329\pi$
$383$	−25.4506	−1.30047	−0.650234	+	0.759734i	$0.725329\pi$
$384$	0	0
$385$	0	0
$386$	−5.10439	−0.259807
$387$	0	0
$388$	−18.1843	−0.923168
$389$	24.7909	1.25695	0.628476	−	0.777829i	$-0.283679\pi$
$389$	24.7909	1.25695	0.628476	+	0.777829i	$0.283679\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	−0.573830	−0.0289092
$395$	10.9836	0.552645
$396$	0	0
$397$	−19.8625	−0.996872	−0.498436	−	0.866927i	$-0.666092\pi$
$397$	−19.8625	−0.996872	−0.498436	+	0.866927i	$0.666092\pi$
$398$	−4.35459	−0.218276
$399$	0	0
$400$	30.9313	1.54656
$401$	−5.24186	−0.261766	−0.130883	−	0.991398i	$-0.541781\pi$
$401$	−5.24186	−0.261766	−0.130883	+	0.991398i	$0.541781\pi$
$402$	0	0
$403$	−18.3463	−0.913892
$404$	13.7501	0.684091
$405$	0	0
$406$	0	0
$407$	−0.189534	−0.00939485
$408$	0	0
$409$	5.07158	0.250773	0.125387	−	0.992108i	$-0.459983\pi$
$409$	5.07158	0.250773	0.125387	+	0.992108i	$0.459983\pi$
$410$	−7.83805	−0.387094
$411$	0	0
$412$	−0.499865	−0.0246266
$413$	0	0
$414$	0	0
$415$	16.1208	0.791339
$416$	−6.39131	−0.313360
$417$	0	0
$418$	−0.935432	−0.0457535
$419$	6.25516	0.305585	0.152792	−	0.988258i	$-0.451173\pi$
$419$	6.25516	0.305585	0.152792	+	0.988258i	$0.451173\pi$
$420$	0	0
$421$	−10.2223	−0.498207	−0.249103	−	0.968477i	$-0.580136\pi$
$421$	−10.2223	−0.498207	−0.249103	+	0.968477i	$0.580136\pi$
$422$	−2.45303	−0.119412
$423$	0	0
$424$	−1.36266	−0.0661767
$425$	28.7581	1.39497
$426$	0	0
$427$	0	0
$428$	24.2622	1.17276
$429$	0	0
$430$	7.83805	0.377984
$431$	−2.76125	−0.133005	−0.0665023	−	0.997786i	$-0.521184\pi$
$431$	−2.76125	−0.133005	−0.0665023	+	0.997786i	$0.521184\pi$
$432$	0	0
$433$	27.7417	1.33318	0.666591	−	0.745423i	$-0.267753\pi$
$433$	27.7417	1.33318	0.666591	+	0.745423i	$0.267753\pi$
$434$	0	0
$435$	0	0
$436$	34.3379	1.64449
$437$	0	0
$438$	0	0
$439$	−35.4782	−1.69328	−0.846642	−	0.532163i	$-0.821379\pi$
$439$	−35.4782	−1.69328	−0.846642	+	0.532163i	$0.821379\pi$
$440$	−3.68133	−0.175501
$441$	0	0
$442$	−1.87086	−0.0889880
$443$	3.68656	0.175154	0.0875769	−	0.996158i	$-0.472088\pi$
$443$	3.68656	0.175154	0.0875769	+	0.996158i	$0.472088\pi$
$444$	0	0
$445$	2.34625	0.111223
$446$	0.420220	0.0198980
$447$	0	0
$448$	0	0
$449$	−4.22546	−0.199412	−0.0997058	−	0.995017i	$-0.531790\pi$
$449$	−4.22546	−0.199412	−0.0997058	+	0.995017i	$0.531790\pi$
$450$	0	0
$451$	−8.37907	−0.394555
$452$	35.0716	1.64963
$453$	0	0
$454$	1.19237	0.0559609
$455$	0	0
$456$	0	0
$457$	28.5878	1.33728	0.668642	−	0.743585i	$-0.266876\pi$
$457$	28.5878	1.33728	0.668642	+	0.743585i	$0.266876\pi$
$458$	−2.03281	−0.0949871
$459$	0	0
$460$	0	0
$461$	36.8789	1.71762	0.858812	−	0.512292i	$-0.171203\pi$
$461$	36.8789	1.71762	0.858812	+	0.512292i	$0.171203\pi$
$462$	0	0
$463$	−38.2692	−1.77852	−0.889260	−	0.457402i	$-0.848780\pi$
$463$	−38.2692	−1.77852	−0.889260	+	0.457402i	$0.848780\pi$
$464$	−36.3861	−1.68918
$465$	0	0
$466$	3.81569	0.176759
$467$	−22.8820	−1.05885	−0.529427	−	0.848355i	$-0.677593\pi$
$467$	−22.8820	−1.05885	−0.529427	+	0.848355i	$0.677593\pi$
$468$	0	0
$469$	0	0
$470$	4.83911	0.223212
$471$	0	0
$472$	−4.53579	−0.208777
$473$	8.37907	0.385270
$474$	0	0
$475$	31.4835	1.44456
$476$	0	0
$477$	0	0
$478$	5.15255	0.235672
$479$	5.77454	0.263846	0.131923	−	0.991260i	$-0.457885\pi$
$479$	5.77454	0.263846	0.131923	+	0.991260i	$0.457885\pi$
$480$	0	0
$481$	−0.414991	−0.0189220
$482$	−5.38013	−0.245058
$483$	0	0
$484$	−1.93543	−0.0879742
$485$	−34.5878	−1.57055
$486$	0	0
$487$	−12.6925	−0.575152	−0.287576	−	0.957758i	$-0.592849\pi$
$487$	−12.6925	−0.575152	−0.287576	+	0.957758i	$0.592849\pi$
$488$	0.379068	0.0171596
$489$	0	0
$490$	0	0
$491$	−20.0192	−0.903456	−0.451728	−	0.892156i	$-0.649192\pi$
$491$	−20.0192	−0.903456	−0.451728	+	0.892156i	$0.649192\pi$
$492$	0	0
$493$	−33.8297	−1.52361
$494$	−2.04816	−0.0921511
$495$	0	0
$496$	30.3051	1.36074
$497$	0	0
$498$	0	0
$499$	17.0768	0.764463	0.382231	−	0.924067i	$-0.375156\pi$
$499$	17.0768	0.764463	0.382231	+	0.924067i	$0.375156\pi$
$500$	25.3093	1.13187
$501$	0	0
$502$	−1.80035	−0.0803534
$503$	−0.313441	−0.0139756	−0.00698782	−	0.999976i	$-0.502224\pi$
$503$	−0.313441	−0.0139756	−0.00698782	+	0.999976i	$0.502224\pi$
$504$	0	0
$505$	26.1536	1.16382
$506$	0	0
$507$	0	0
$508$	21.2580	0.943171
$509$	16.1208	0.714542	0.357271	−	0.934001i	$-0.383707\pi$
$509$	16.1208	0.714542	0.357271	+	0.934001i	$0.383707\pi$
$510$	0	0
$511$	0	0
$512$	17.7909	0.786256
$513$	0	0
$514$	5.98253	0.263878
$515$	−0.950780	−0.0418964
$516$	0	0
$517$	5.17313	0.227514
$518$	0	0
$519$	0	0
$520$	−8.06040	−0.353472
$521$	19.9948	0.875987	0.437993	−	0.898978i	$-0.355689\pi$
$521$	19.9948	0.875987	0.437993	+	0.898978i	$0.355689\pi$
$522$	0	0
$523$	−6.98882	−0.305600	−0.152800	−	0.988257i	$-0.548829\pi$
$523$	−6.98882	−0.305600	−0.152800	+	0.988257i	$0.548829\pi$
$524$	31.2007	1.36301
$525$	0	0
$526$	2.33091	0.101632
$527$	28.1760	1.22736
$528$	0	0
$529$	−23.0000	−1.00000
$530$	−1.27468	−0.0553684
$531$	0	0
$532$	0	0
$533$	−18.3463	−0.794664
$534$	0	0
$535$	46.1484	1.99517
$536$	−14.4067	−0.622273
$537$	0	0
$538$	4.42022	0.190569
$539$	0	0
$540$	0	0
$541$	7.08203	0.304480	0.152240	−	0.988344i	$-0.451351\pi$
$541$	7.08203	0.304480	0.152240	+	0.988344i	$0.451351\pi$
$542$	5.62794	0.241741
$543$	0	0
$544$	9.81569	0.420844
$545$	65.3132	2.79771
$546$	0	0
$547$	31.5386	1.34849	0.674247	−	0.738506i	$-0.264468\pi$
$547$	31.5386	1.34849	0.674247	+	0.738506i	$0.264468\pi$
$548$	−9.64541	−0.412031
$549$	0	0
$550$	−2.17313	−0.0926625
$551$	−37.0357	−1.57777
$552$	0	0
$553$	0	0
$554$	−3.73339	−0.158617
$555$	0	0
$556$	41.5163	1.76068
$557$	14.8737	0.630219	0.315110	−	0.949055i	$-0.397959\pi$
$557$	14.8737	0.630219	0.315110	+	0.949055i	$0.397959\pi$
$558$	0	0
$559$	18.3463	0.775964
$560$	0	0
$561$	0	0
$562$	7.08692	0.298944
$563$	37.8953	1.59710	0.798549	−	0.601930i	$-0.205601\pi$
$563$	37.8953	1.59710	0.798549	+	0.601930i	$0.205601\pi$
$564$	0	0
$565$	66.7086	2.80645
$566$	4.16062	0.174884
$567$	0	0
$568$	−14.0880	−0.591119
$569$	41.9177	1.75728	0.878641	−	0.477484i	$-0.158451\pi$
$569$	41.9177	1.75728	0.878641	+	0.477484i	$0.158451\pi$
$570$	0	0
$571$	−28.1208	−1.17682	−0.588409	−	0.808563i	$-0.700246\pi$
$571$	−28.1208	−1.17682	−0.588409	+	0.808563i	$0.700246\pi$
$572$	−4.23769	−0.177187
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−47.6043	−1.98179	−0.990896	−	0.134633i	$-0.957015\pi$
$577$	−47.6043	−1.98179	−0.990896	+	0.134633i	$0.957015\pi$
$578$	−1.44648	−0.0601655
$579$	0	0
$580$	−71.6800	−2.97635
$581$	0	0
$582$	0	0
$583$	−1.36266	−0.0564357
$584$	−13.9313	−0.576480
$585$	0	0
$586$	−3.24186	−0.133920
$587$	−39.3267	−1.62319	−0.811594	−	0.584222i	$-0.801400\pi$
$587$	−39.3267	−1.62319	−0.811594	+	0.584222i	$0.801400\pi$
$588$	0	0
$589$	30.8461	1.27099
$590$	−4.24292	−0.174678
$591$	0	0
$592$	0.685500	0.0281739
$593$	−29.3955	−1.20713	−0.603564	−	0.797315i	$-0.706253\pi$
$593$	−29.3955	−1.20713	−0.603564	+	0.797315i	$0.706253\pi$
$594$	0	0
$595$	0	0
$596$	10.2622	0.420355
$597$	0	0
$598$	0	0
$599$	−26.4671	−1.08141	−0.540707	−	0.841211i	$-0.681843\pi$
$599$	−26.4671	−1.08141	−0.540707	+	0.841211i	$0.681843\pi$
$600$	0	0
$601$	−33.5522	−1.36862	−0.684311	−	0.729190i	$-0.739897\pi$
$601$	−33.5522	−1.36862	−0.684311	+	0.729190i	$0.739897\pi$
$602$	0	0
$603$	0	0
$604$	−1.23353	−0.0501915
$605$	−3.68133	−0.149667
$606$	0	0
$607$	17.9724	0.729478	0.364739	−	0.931110i	$-0.381158\pi$
$607$	17.9724	0.729478	0.364739	+	0.931110i	$0.381158\pi$
$608$	10.7459	0.435804
$609$	0	0
$610$	0.354593	0.0143570
$611$	11.3267	0.458231
$612$	0	0
$613$	28.5878	1.15465	0.577326	−	0.816514i	$-0.304096\pi$
$613$	28.5878	1.15465	0.577326	+	0.816514i	$0.304096\pi$
$614$	7.48346	0.302008
$615$	0	0
$616$	0	0
$617$	31.0164	1.24867	0.624337	−	0.781155i	$-0.285369\pi$
$617$	31.0164	1.24867	0.624337	+	0.781155i	$0.285369\pi$
$618$	0	0
$619$	−11.4835	−0.461559	−0.230780	−	0.973006i	$-0.574128\pi$
$619$	−11.4835	−0.461559	−0.230780	+	0.973006i	$0.574128\pi$
$620$	59.7006	2.39763
$621$	0	0
$622$	−3.24186	−0.129987
$623$	0	0
$624$	0	0
$625$	5.37907	0.215163
$626$	4.66181	0.186324
$627$	0	0
$628$	18.7909	0.749841
$629$	0.637339	0.0254124
$630$	0	0
$631$	−10.6597	−0.424356	−0.212178	−	0.977231i	$-0.568056\pi$
$631$	−10.6597	−0.424356	−0.212178	+	0.977231i	$0.568056\pi$
$632$	2.98359	0.118681
$633$	0	0
$634$	−0.766474	−0.0304406
$635$	40.4342	1.60458
$636$	0	0
$637$	0	0
$638$	2.55636	0.101207
$639$	0	0
$640$	27.5644	1.08958
$641$	−2.69251	−0.106348	−0.0531739	−	0.998585i	$-0.516934\pi$
$641$	−2.69251	−0.106348	−0.0531739	+	0.998585i	$0.516934\pi$
$642$	0	0
$643$	40.3791	1.59239	0.796197	−	0.605037i	$-0.206842\pi$
$643$	40.3791	1.59239	0.796197	+	0.605037i	$0.206842\pi$
$644$	0	0
$645$	0	0
$646$	3.14554	0.123760
$647$	−18.0687	−0.710355	−0.355178	−	0.934799i	$-0.615580\pi$
$647$	−18.0687	−0.710355	−0.355178	+	0.934799i	$0.615580\pi$
$648$	0	0
$649$	−4.53579	−0.178045
$650$	−4.75814	−0.186629
$651$	0	0
$652$	−14.8667	−0.582225
$653$	−29.1044	−1.13894	−0.569471	−	0.822011i	$-0.692852\pi$
$653$	−29.1044	−1.13894	−0.569471	+	0.822011i	$0.692852\pi$
$654$	0	0
$655$	59.3460	2.31884
$656$	30.3051	1.18322
$657$	0	0
$658$	0	0
$659$	−38.5686	−1.50242	−0.751210	−	0.660064i	$-0.770529\pi$
$659$	−38.5686	−1.50242	−0.751210	+	0.660064i	$0.770529\pi$
$660$	0	0
$661$	23.1596	0.900803	0.450401	−	0.892826i	$-0.351281\pi$
$661$	23.1596	0.900803	0.450401	+	0.892826i	$0.351281\pi$
$662$	−3.75841	−0.146075
$663$	0	0
$664$	4.37907	0.169941
$665$	0	0
$666$	0	0
$667$	0	0
$668$	30.9669	1.19815
$669$	0	0
$670$	−13.4764	−0.520641
$671$	0.379068	0.0146338
$672$	0	0
$673$	−29.5491	−1.13903	−0.569517	−	0.821980i	$-0.692870\pi$
$673$	−29.5491	−1.13903	−0.569517	+	0.821980i	$0.692870\pi$
$674$	−4.84612	−0.186666
$675$	0	0
$676$	15.8820	0.610848
$677$	−37.9505	−1.45856	−0.729278	−	0.684217i	$-0.760144\pi$
$677$	−37.9505	−1.45856	−0.729278	+	0.684217i	$0.760144\pi$
$678$	0	0
$679$	0	0
$680$	12.3791	0.474716
$681$	0	0
$682$	−2.12914	−0.0815288
$683$	42.9669	1.64408	0.822042	−	0.569427i	$-0.192835\pi$
$683$	42.9669	1.64408	0.822042	+	0.569427i	$0.192835\pi$
$684$	0	0
$685$	−18.3463	−0.700974
$686$	0	0
$687$	0	0
$688$	−30.3051	−1.15537
$689$	−2.98359	−0.113666
$690$	0	0
$691$	−18.0880	−0.688099	−0.344050	−	0.938951i	$-0.611799\pi$
$691$	−18.0880	−0.688099	−0.344050	+	0.938951i	$0.611799\pi$
$692$	−26.5327	−1.00862
$693$	0	0
$694$	−2.40142	−0.0911568
$695$	78.9669	2.99539
$696$	0	0
$697$	28.1760	1.06724
$698$	−7.36399	−0.278731
$699$	0	0
$700$	0	0
$701$	16.4671	0.621952	0.310976	−	0.950418i	$-0.399344\pi$
$701$	16.4671	0.621952	0.310976	+	0.950418i	$0.399344\pi$
$702$	0	0
$703$	0.697737	0.0263157
$704$	6.49180	0.244669
$705$	0	0
$706$	4.75946	0.179125
$707$	0	0
$708$	0	0
$709$	13.1955	0.495567	0.247783	−	0.968815i	$-0.420298\pi$
$709$	13.1955	0.495567	0.247783	+	0.968815i	$0.420298\pi$
$710$	−13.1784	−0.494575
$711$	0	0
$712$	0.637339	0.0238853
$713$	0	0
$714$	0	0
$715$	−8.06040	−0.301442
$716$	−14.9836	−0.559963
$717$	0	0
$718$	−1.90156	−0.0709656
$719$	−41.7938	−1.55865	−0.779323	−	0.626623i	$-0.784437\pi$
$719$	−41.7938	−1.55865	−0.779323	+	0.626623i	$0.784437\pi$
$720$	0	0
$721$	0	0
$722$	−1.38430	−0.0515182
$723$	0	0
$724$	−39.4423	−1.46586
$725$	−86.0385	−3.19539
$726$	0	0
$727$	−46.9013	−1.73947	−0.869736	−	0.493517i	$-0.835711\pi$
$727$	−46.9013	−1.73947	−0.869736	+	0.493517i	$0.835711\pi$
$728$	0	0
$729$	0	0
$730$	−13.0318	−0.482327
$731$	−28.1760	−1.04213
$732$	0	0
$733$	−5.07158	−0.187323	−0.0936615	−	0.995604i	$-0.529857\pi$
$733$	−5.07158	−0.187323	−0.0936615	+	0.995604i	$0.529857\pi$
$734$	3.92035	0.144703
$735$	0	0
$736$	0	0
$737$	−14.4067	−0.530676
$738$	0	0
$739$	−23.4178	−0.861439	−0.430719	−	0.902486i	$-0.641740\pi$
$739$	−23.4178	−0.861439	−0.430719	+	0.902486i	$0.641740\pi$
$740$	1.35042	0.0496426
$741$	0	0
$742$	0	0
$743$	11.2059	0.411106	0.205553	−	0.978646i	$-0.434101\pi$
$743$	11.2059	0.411106	0.205553	+	0.978646i	$0.434101\pi$
$744$	0	0
$745$	19.5194	0.715135
$746$	−6.63734	−0.243010
$747$	0	0
$748$	6.50820	0.237963
$749$	0	0
$750$	0	0
$751$	44.2468	1.61459	0.807295	−	0.590148i	$-0.200931\pi$
$751$	44.2468	1.61459	0.807295	+	0.590148i	$0.200931\pi$
$752$	−18.7100	−0.682283
$753$	0	0
$754$	5.59725	0.203840
$755$	−2.34625	−0.0853889
$756$	0	0
$757$	20.2552	0.736186	0.368093	−	0.929789i	$-0.380011\pi$
$757$	20.2552	0.736186	0.368093	+	0.929789i	$0.380011\pi$
$758$	−3.46811	−0.125967
$759$	0	0
$760$	13.5522	0.491590
$761$	−41.7745	−1.51433	−0.757163	−	0.653226i	$-0.773415\pi$
$761$	−41.7745	−1.51433	−0.757163	+	0.653226i	$0.773415\pi$
$762$	0	0
$763$	0	0
$764$	18.7909	0.679833
$765$	0	0
$766$	−6.46705	−0.233664
$767$	−9.93126	−0.358597
$768$	0	0
$769$	−0.997160	−0.0359585	−0.0179792	−	0.999838i	$-0.505723\pi$
$769$	−0.997160	−0.0359585	−0.0179792	+	0.999838i	$0.505723\pi$
$770$	0	0
$771$	0	0
$772$	38.8789	1.39928
$773$	−45.8245	−1.64819	−0.824096	−	0.566450i	$-0.808316\pi$
$773$	−45.8245	−1.64819	−0.824096	+	0.566450i	$0.808316\pi$
$774$	0	0
$775$	71.6594	2.57408
$776$	−9.39547	−0.337278
$777$	0	0
$778$	6.29942	0.225845
$779$	30.8461	1.10518
$780$	0	0
$781$	−14.0880	−0.504108
$782$	0	0
$783$	0	0
$784$	0	0
$785$	35.7417	1.27568
$786$	0	0
$787$	19.8021	0.705870	0.352935	−	0.935648i	$-0.385184\pi$
$787$	19.8021	0.705870	0.352935	+	0.935648i	$0.385184\pi$
$788$	4.37073	0.155701
$789$	0	0
$790$	2.79095	0.0992975
$791$	0	0
$792$	0	0
$793$	0.829982	0.0294735
$794$	−5.04710	−0.179115
$795$	0	0
$796$	33.1679	1.17561
$797$	−29.8349	−1.05681	−0.528404	−	0.848993i	$-0.677209\pi$
$797$	−29.8349	−1.05681	−0.528404	+	0.848993i	$0.677209\pi$
$798$	0	0
$799$	−17.3955	−0.615408
$800$	24.9641	0.882613
$801$	0	0
$802$	−1.33197	−0.0470334
$803$	−13.9313	−0.491624
$804$	0	0
$805$	0	0
$806$	−4.66181	−0.164205
$807$	0	0
$808$	7.10439	0.249932
$809$	−19.0112	−0.668397	−0.334199	−	0.942503i	$-0.608466\pi$
$809$	−19.0112	−0.668397	−0.334199	+	0.942503i	$0.608466\pi$
$810$	0	0
$811$	−20.4395	−0.717727	−0.358863	−	0.933390i	$-0.616836\pi$
$811$	−20.4395	−0.717727	−0.358863	+	0.933390i	$0.616836\pi$
$812$	0	0
$813$	0	0
$814$	−0.0481609	−0.00168804
$815$	−28.2775	−0.990518
$816$	0	0
$817$	−30.8461	−1.07917
$818$	1.28870	0.0450582
$819$	0	0
$820$	59.7006	2.08483
$821$	28.0932	0.980460	0.490230	−	0.871593i	$-0.336913\pi$
$821$	28.0932	0.980460	0.490230	+	0.871593i	$0.336913\pi$
$822$	0	0
$823$	−28.1260	−0.980412	−0.490206	−	0.871607i	$-0.663078\pi$
$823$	−28.1260	−0.980412	−0.490206	+	0.871607i	$0.663078\pi$
$824$	−0.258271	−0.00899729
$825$	0	0
$826$	0	0
$827$	0.914857	0.0318127	0.0159063	−	0.999873i	$-0.494937\pi$
$827$	0.914857	0.0318127	0.0159063	+	0.999873i	$0.494937\pi$
$828$	0	0
$829$	29.6922	1.03125	0.515627	−	0.856813i	$-0.327559\pi$
$829$	29.6922	1.03125	0.515627	+	0.856813i	$0.327559\pi$
$830$	4.09632	0.142185
$831$	0	0
$832$	14.2140	0.492782
$833$	0	0
$834$	0	0
$835$	58.9013	2.03836
$836$	7.12497	0.246422
$837$	0	0
$838$	1.58945	0.0549065
$839$	32.2775	1.11434	0.557172	−	0.830397i	$-0.311886\pi$
$839$	32.2775	1.11434	0.557172	+	0.830397i	$0.311886\pi$
$840$	0	0
$841$	72.2116	2.49006
$842$	−2.59752	−0.0895163
$843$	0	0
$844$	18.6842	0.643136
$845$	30.2088	1.03921
$846$	0	0
$847$	0	0
$848$	4.92842	0.169243
$849$	0	0
$850$	7.30749	0.250645
$851$	0	0
$852$	0	0
$853$	−9.65375	−0.330538	−0.165269	−	0.986249i	$-0.552849\pi$
$853$	−9.65375	−0.330538	−0.165269	+	0.986249i	$0.552849\pi$
$854$	0	0
$855$	0	0
$856$	12.5358	0.428464
$857$	−17.1372	−0.585396	−0.292698	−	0.956205i	$-0.594553\pi$
$857$	−17.1372	−0.585396	−0.292698	+	0.956205i	$0.594553\pi$
$858$	0	0
$859$	−29.7089	−1.01366	−0.506828	−	0.862047i	$-0.669182\pi$
$859$	−29.7089	−1.01366	−0.506828	+	0.862047i	$0.669182\pi$
$860$	−59.7006	−2.03577
$861$	0	0
$862$	−0.701637	−0.0238979
$863$	49.0716	1.67042	0.835208	−	0.549934i	$-0.185347\pi$
$863$	49.0716	1.67042	0.835208	+	0.549934i	$0.185347\pi$
$864$	0	0
$865$	−50.4671	−1.71593
$866$	7.04922	0.239542
$867$	0	0
$868$	0	0
$869$	2.98359	0.101211
$870$	0	0
$871$	−31.5439	−1.06882
$872$	17.7417	0.600811
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−26.3791	−0.890758	−0.445379	−	0.895342i	$-0.646931\pi$
$877$	−26.3791	−0.890758	−0.445379	+	0.895342i	$0.646931\pi$
$878$	−9.01508	−0.304244
$879$	0	0
$880$	13.3145	0.448832
$881$	−16.9393	−0.570701	−0.285350	−	0.958423i	$-0.592110\pi$
$881$	−16.9393	−0.570701	−0.285350	+	0.958423i	$0.592110\pi$
$882$	0	0
$883$	33.0768	1.11312	0.556562	−	0.830806i	$-0.312120\pi$
$883$	33.0768	1.11312	0.556562	+	0.830806i	$0.312120\pi$
$884$	14.2499	0.479277
$885$	0	0
$886$	0.936761	0.0314711
$887$	−0.516541	−0.0173438	−0.00867188	−	0.999962i	$-0.502760\pi$
$887$	−0.516541	−0.0173438	−0.00867188	+	0.999962i	$0.502760\pi$
$888$	0	0
$889$	0	0
$890$	0.596187	0.0199842
$891$	0	0
$892$	−3.20071	−0.107168
$893$	−19.0440	−0.637283
$894$	0	0
$895$	−28.4999	−0.952645
$896$	0	0
$897$	0	0
$898$	−1.07370	−0.0358297
$899$	−84.2968	−2.81145
$900$	0	0
$901$	4.58217	0.152654
$902$	−2.12914	−0.0708925
$903$	0	0
$904$	18.1208	0.602689
$905$	−75.0221	−2.49382
$906$	0	0
$907$	−2.72532	−0.0904929	−0.0452464	−	0.998976i	$-0.514407\pi$
$907$	−2.72532	−0.0904929	−0.0452464	+	0.998976i	$0.514407\pi$
$908$	−9.08203	−0.301398
$909$	0	0
$910$	0	0
$911$	−42.4671	−1.40700	−0.703498	−	0.710697i	$-0.748380\pi$
$911$	−42.4671	−1.40700	−0.703498	+	0.710697i	$0.748380\pi$
$912$	0	0
$913$	4.37907	0.144926
$914$	7.26422	0.240279
$915$	0	0
$916$	15.4835	0.511588
$917$	0	0
$918$	0	0
$919$	7.56576	0.249571	0.124786	−	0.992184i	$-0.460176\pi$
$919$	7.56576	0.249571	0.124786	+	0.992184i	$0.460176\pi$
$920$	0	0
$921$	0	0
$922$	9.37100	0.308617
$923$	−30.8461	−1.01531
$924$	0	0
$925$	1.62093	0.0532959
$926$	−9.72426	−0.319559
$927$	0	0
$928$	−29.3666	−0.964005
$929$	−9.33508	−0.306274	−0.153137	−	0.988205i	$-0.548938\pi$
$929$	−9.33508	−0.306274	−0.153137	+	0.988205i	$0.548938\pi$
$930$	0	0
$931$	0	0
$932$	−29.0632	−0.951998
$933$	0	0
$934$	−5.81437	−0.190252
$935$	12.3791	0.404839
$936$	0	0
$937$	0.379068	0.0123836	0.00619181	−	0.999981i	$-0.498029\pi$
$937$	0.379068	0.0123836	0.00619181	+	0.999981i	$0.498029\pi$
$938$	0	0
$939$	0	0
$940$	−36.8584	−1.20219
$941$	−50.6430	−1.65092	−0.825458	−	0.564464i	$-0.809083\pi$
$941$	−50.6430	−1.65092	−0.825458	+	0.564464i	$0.809083\pi$
$942$	0	0
$943$	0	0
$944$	16.4049	0.533933
$945$	0	0
$946$	2.12914	0.0692242
$947$	−42.6535	−1.38605	−0.693026	−	0.720913i	$-0.743723\pi$
$947$	−42.6535	−1.38605	−0.693026	+	0.720913i	$0.743723\pi$
$948$	0	0
$949$	−30.5030	−0.990168
$950$	8.00000	0.259554
$951$	0	0
$952$	0	0
$953$	−14.5603	−0.471653	−0.235827	−	0.971795i	$-0.575780\pi$
$953$	−14.5603	−0.471653	−0.235827	+	0.971795i	$0.575780\pi$
$954$	0	0
$955$	35.7417	1.15658
$956$	−39.2458	−1.26930
$957$	0	0
$958$	1.46732	0.0474070
$959$	0	0
$960$	0	0
$961$	39.2088	1.26480
$962$	−0.105450	−0.00339984
$963$	0	0
$964$	40.9792	1.31985
$965$	73.9505	2.38055
$966$	0	0
$967$	10.4671	0.336598	0.168299	−	0.985736i	$-0.446173\pi$
$967$	10.4671	0.336598	0.168299	+	0.985736i	$0.446173\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	−8.78883	−0.282192
$971$	5.49702	0.176408	0.0882040	−	0.996102i	$-0.471887\pi$
$971$	5.49702	0.176408	0.0882040	+	0.996102i	$0.471887\pi$
$972$	0	0
$973$	0	0
$974$	−3.22519	−0.103342
$975$	0	0
$976$	−1.37100	−0.0438846
$977$	−39.2028	−1.25421	−0.627105	−	0.778935i	$-0.715760\pi$
$977$	−39.2028	−1.25421	−0.627105	+	0.778935i	$0.715760\pi$
$978$	0	0
$979$	0.637339	0.0203694
$980$	0	0
$981$	0	0
$982$	−5.08692	−0.162330
$983$	−18.7253	−0.597245	−0.298623	−	0.954371i	$-0.596527\pi$
$983$	−18.7253	−0.597245	−0.298623	+	0.954371i	$0.596527\pi$
$984$	0	0
$985$	8.31344	0.264888
$986$	−8.59619	−0.273758
$987$	0	0
$988$	15.6004	0.496313
$989$	0	0
$990$	0	0
$991$	−18.8513	−0.598833	−0.299416	−	0.954123i	$-0.596792\pi$
$991$	−18.8513	−0.598833	−0.299416	+	0.954123i	$0.596792\pi$
$992$	24.4587	0.776565
$993$	0	0
$994$	0	0
$995$	63.0877	2.00002
$996$	0	0
$997$	−13.0716	−0.413981	−0.206991	−	0.978343i	$-0.566367\pi$
$997$	−13.0716	−0.413981	−0.206991	+	0.978343i	$0.566367\pi$
$998$	4.33925	0.137356
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4851.2.a.bl.1.2		3
3.2	odd	2		1617.2.a.r.1.2	yes	3
7.6	odd	2		4851.2.a.bm.1.2		3
21.20	even	2		1617.2.a.q.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1617.2.a.q.1.2	✓	3	21.20	even	2
1617.2.a.r.1.2	yes	3	3.2	odd	2
4851.2.a.bl.1.2		3	1.1	even	1	trivial
4851.2.a.bm.1.2		3	7.6	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4851 = 3^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4851.a (trivial)

\(f(q)\)	\(=\)	\(q+0.254102 q^{2} -1.93543 q^{4} -3.68133 q^{5} -1.00000 q^{8} +O(q^{10})\)	\(q+0.254102 q^{2} -1.93543 q^{4} -3.68133 q^{5} -1.00000 q^{8} -0.935432 q^{10} -1.00000 q^{11} -2.18953 q^{13} +3.61676 q^{16} +3.36266 q^{17} +3.68133 q^{19} +7.12497 q^{20} -0.254102 q^{22} +8.55220 q^{25} -0.556364 q^{26} -10.0604 q^{29} +8.37907 q^{31} +2.91903 q^{32} +0.854458 q^{34} +0.189534 q^{37} +0.935432 q^{38} +3.68133 q^{40} +8.37907 q^{41} -8.37907 q^{43} +1.93543 q^{44} -5.17313 q^{47} +2.17313 q^{50} +4.23769 q^{52} +1.36266 q^{53} +3.68133 q^{55} -2.55636 q^{58} +4.53579 q^{59} -0.379068 q^{61} +2.12914 q^{62} -6.49180 q^{64} +8.06040 q^{65} +14.4067 q^{67} -6.50820 q^{68} +14.0880 q^{71} +13.9313 q^{73} +0.0481609 q^{74} -7.12497 q^{76} -2.98359 q^{79} -13.3145 q^{80} +2.12914 q^{82} -4.37907 q^{83} -12.3791 q^{85} -2.12914 q^{86} +1.00000 q^{88} -0.637339 q^{89} -1.31450 q^{94} -13.5522 q^{95} +9.39547 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{4} - 4 q^{5} - 3 q^{8}+O(q^{10}) \) 3 * q + 2 * q^4 - 4 * q^5 - 3 * q^8	\( 3 q + 2 q^{4} - 4 q^{5} - 3 q^{8} + 5 q^{10} - 3 q^{11} + 2 q^{13} - 4 q^{16} - 4 q^{17} + 4 q^{19} + 5 q^{20} + 3 q^{25} - 11 q^{26} - 6 q^{29} + 8 q^{31} + 4 q^{32} - 10 q^{34} - 8 q^{37} - 5 q^{38} + 4 q^{40} + 8 q^{41} - 8 q^{43} - 2 q^{44} - 10 q^{47} + q^{50} + 15 q^{52} - 10 q^{53} + 4 q^{55} - 17 q^{58} - 6 q^{59} + 16 q^{61} + 22 q^{62} - 21 q^{64} + 8 q^{67} - 18 q^{68} + 2 q^{73} + 11 q^{74} - 5 q^{76} - 12 q^{79} - 15 q^{80} + 22 q^{82} + 4 q^{83} - 20 q^{85} - 22 q^{86} + 3 q^{88} - 16 q^{89} + 21 q^{94} - 18 q^{95} + 8 q^{97}+O(q^{100}) \) 3 * q + 2 * q^4 - 4 * q^5 - 3 * q^8 + 5 * q^10 - 3 * q^11 + 2 * q^13 - 4 * q^16 - 4 * q^17 + 4 * q^19 + 5 * q^20 + 3 * q^25 - 11 * q^26 - 6 * q^29 + 8 * q^31 + 4 * q^32 - 10 * q^34 - 8 * q^37 - 5 * q^38 + 4 * q^40 + 8 * q^41 - 8 * q^43 - 2 * q^44 - 10 * q^47 + q^50 + 15 * q^52 - 10 * q^53 + 4 * q^55 - 17 * q^58 - 6 * q^59 + 16 * q^61 + 22 * q^62 - 21 * q^64 + 8 * q^67 - 18 * q^68 + 2 * q^73 + 11 * q^74 - 5 * q^76 - 12 * q^79 - 15 * q^80 + 22 * q^82 + 4 * q^83 - 20 * q^85 - 22 * q^86 + 3 * q^88 - 16 * q^89 + 21 * q^94 - 18 * q^95 + 8 * q^97

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