LMFDB - Embedded newform 8019.2.a.k.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8019,2,Mod(1,8019)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8019, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8019.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8019 = 3^{6} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8019.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:	$64.0320373809$
Analytic rank:	$0$
Dimension:	$51$
Twist minimal:	no (minimal twist has level 297)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	8019.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.70504 q^{2} +0.907156 q^{4} -0.871790 q^{5} +5.05699 q^{7} +1.86334 q^{8} +O(q^{10})$	\(q-1.70504 q^{2} +0.907156 q^{4} -0.871790 q^{5} +5.05699 q^{7} +1.86334 q^{8} +1.48644 q^{10} -1.00000 q^{11} -1.15982 q^{13} -8.62237 q^{14} -4.99138 q^{16} +2.29776 q^{17} +5.01721 q^{19} -0.790850 q^{20} +1.70504 q^{22} +2.85479 q^{23} -4.23998 q^{25} +1.97753 q^{26} +4.58748 q^{28} -2.28951 q^{29} -2.50630 q^{31} +4.78381 q^{32} -3.91778 q^{34} -4.40864 q^{35} +2.45279 q^{37} -8.55454 q^{38} -1.62444 q^{40} -9.55457 q^{41} +10.3604 q^{43} -0.907156 q^{44} -4.86753 q^{46} -11.1794 q^{47} +18.5732 q^{49} +7.22933 q^{50} -1.05214 q^{52} +5.20715 q^{53} +0.871790 q^{55} +9.42290 q^{56} +3.90371 q^{58} -6.34757 q^{59} +5.56281 q^{61} +4.27333 q^{62} +1.82617 q^{64} +1.01112 q^{65} +9.14561 q^{67} +2.08443 q^{68} +7.51690 q^{70} +11.2901 q^{71} -5.53019 q^{73} -4.18210 q^{74} +4.55140 q^{76} -5.05699 q^{77} +1.14189 q^{79} +4.35144 q^{80} +16.2909 q^{82} +3.57616 q^{83} -2.00317 q^{85} -17.6648 q^{86} -1.86334 q^{88} +3.08226 q^{89} -5.86519 q^{91} +2.58974 q^{92} +19.0613 q^{94} -4.37396 q^{95} +5.37230 q^{97} -31.6680 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 51 q + 60 q^{4} - 6 q^{5} + 12 q^{7}+O(q^{10}) $ 51 * q + 60 * q^4 - 6 * q^5 + 12 * q^7	$ 51 q + 60 q^{4} - 6 q^{5} + 12 q^{7} + 18 q^{10} - 51 q^{11} + 30 q^{13} - 12 q^{14} + 78 q^{16} + 30 q^{19} - 18 q^{20} - 3 q^{23} + 75 q^{25} - 9 q^{26} + 36 q^{28} + 42 q^{31} + 15 q^{32} + 42 q^{34} + 9 q^{35} + 48 q^{37} + 3 q^{38} + 54 q^{40} + 18 q^{43} - 60 q^{44} + 42 q^{46} - 30 q^{47} + 99 q^{49} + 30 q^{50} + 60 q^{52} - 18 q^{53} + 6 q^{55} - 21 q^{56} + 30 q^{58} - 24 q^{59} + 99 q^{61} + 114 q^{64} + 15 q^{65} + 39 q^{67} + 39 q^{68} + 48 q^{70} - 30 q^{71} + 69 q^{73} + 90 q^{76} - 12 q^{77} + 48 q^{79} - 42 q^{80} + 42 q^{82} + 21 q^{83} + 84 q^{85} - 24 q^{86} - 15 q^{89} + 69 q^{91} + 66 q^{92} + 66 q^{94} + 12 q^{95} + 72 q^{97} + 57 q^{98}+O(q^{100}) $ 51 * q + 60 * q^4 - 6 * q^5 + 12 * q^7 + 18 * q^10 - 51 * q^11 + 30 * q^13 - 12 * q^14 + 78 * q^16 + 30 * q^19 - 18 * q^20 - 3 * q^23 + 75 * q^25 - 9 * q^26 + 36 * q^28 + 42 * q^31 + 15 * q^32 + 42 * q^34 + 9 * q^35 + 48 * q^37 + 3 * q^38 + 54 * q^40 + 18 * q^43 - 60 * q^44 + 42 * q^46 - 30 * q^47 + 99 * q^49 + 30 * q^50 + 60 * q^52 - 18 * q^53 + 6 * q^55 - 21 * q^56 + 30 * q^58 - 24 * q^59 + 99 * q^61 + 114 * q^64 + 15 * q^65 + 39 * q^67 + 39 * q^68 + 48 * q^70 - 30 * q^71 + 69 * q^73 + 90 * q^76 - 12 * q^77 + 48 * q^79 - 42 * q^80 + 42 * q^82 + 21 * q^83 + 84 * q^85 - 24 * q^86 - 15 * q^89 + 69 * q^91 + 66 * q^92 + 66 * q^94 + 12 * q^95 + 72 * q^97 + 57 * q^98

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 $

$2$	−1.70504	−1.20564	−0.602822	−	0.797876i	$-0.705957\pi$
$2$	−1.70504	−1.20564	−0.602822	+	0.797876i	$0.705957\pi$
$3$	0	0
$3$	0	0
$4$	0.907156	0.453578
$5$	−0.871790	−0.389876	−0.194938	−	0.980816i	$-0.562451\pi$
$5$	−0.871790	−0.389876	−0.194938	+	0.980816i	$0.562451\pi$
$6$	0	0
$7$	5.05699	1.91136	0.955682	−	0.294401i	$-0.0951202\pi$
$7$	5.05699	1.91136	0.955682	+	0.294401i	$0.0951202\pi$
$8$	1.86334	0.658790
$9$	0	0
$10$	1.48644	0.470052
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.15982	−0.321675	−0.160838	−	0.986981i	$-0.551420\pi$
$13$	−1.15982	−0.321675	−0.160838	+	0.986981i	$0.551420\pi$
$14$	−8.62237	−2.30442
$15$	0	0
$16$	−4.99138	−1.24785
$17$	2.29776	0.557290	0.278645	−	0.960394i	$-0.410115\pi$
$17$	2.29776	0.557290	0.278645	+	0.960394i	$0.410115\pi$
$18$	0	0
$19$	5.01721	1.15103	0.575514	−	0.817792i	$-0.304802\pi$
$19$	5.01721	1.15103	0.575514	+	0.817792i	$0.304802\pi$
$20$	−0.790850	−0.176839
$21$	0	0
$22$	1.70504	0.363515
$23$	2.85479	0.595265	0.297633	−	0.954680i	$-0.403803\pi$
$23$	2.85479	0.595265	0.297633	+	0.954680i	$0.403803\pi$
$24$	0	0
$25$	−4.23998	−0.847996
$26$	1.97753	0.387826
$27$	0	0
$28$	4.58748	0.866953
$29$	−2.28951	−0.425152	−0.212576	−	0.977145i	$-0.568185\pi$
$29$	−2.28951	−0.425152	−0.212576	+	0.977145i	$0.568185\pi$
$30$	0	0
$31$	−2.50630	−0.450144	−0.225072	−	0.974342i	$-0.572262\pi$
$31$	−2.50630	−0.450144	−0.225072	+	0.974342i	$0.572262\pi$
$32$	4.78381	0.845667
$33$	0	0
$34$	−3.91778	−0.671893
$35$	−4.40864	−0.745196
$36$	0	0
$37$	2.45279	0.403236	0.201618	−	0.979464i	$-0.435380\pi$
$37$	2.45279	0.403236	0.201618	+	0.979464i	$0.435380\pi$
$38$	−8.55454	−1.38773
$39$	0	0
$40$	−1.62444	−0.256847
$41$	−9.55457	−1.49217	−0.746087	−	0.665849i	$-0.768069\pi$
$41$	−9.55457	−1.49217	−0.746087	+	0.665849i	$0.768069\pi$
$42$	0	0
$43$	10.3604	1.57994	0.789971	−	0.613144i	$-0.210096\pi$
$43$	10.3604	1.57994	0.789971	+	0.613144i	$0.210096\pi$
$44$	−0.907156	−0.136759
$45$	0	0
$46$	−4.86753	−0.717678
$47$	−11.1794	−1.63068	−0.815342	−	0.578980i	$-0.803451\pi$
$47$	−11.1794	−1.63068	−0.815342	+	0.578980i	$0.803451\pi$
$48$	0	0
$49$	18.5732	2.65331
$50$	7.22933	1.02238
$51$	0	0
$52$	−1.05214	−0.145905
$53$	5.20715	0.715258	0.357629	−	0.933864i	$-0.383585\pi$
$53$	5.20715	0.715258	0.357629	+	0.933864i	$0.383585\pi$
$54$	0	0
$55$	0.871790	0.117552
$56$	9.42290	1.25919
$57$	0	0
$58$	3.90371	0.512582
$59$	−6.34757	−0.826383	−0.413192	−	0.910644i	$-0.635586\pi$
$59$	−6.34757	−0.826383	−0.413192	+	0.910644i	$0.635586\pi$
$60$	0	0
$61$	5.56281	0.712245	0.356122	−	0.934439i	$-0.384099\pi$
$61$	5.56281	0.712245	0.356122	+	0.934439i	$0.384099\pi$
$62$	4.27333	0.542714
$63$	0	0
$64$	1.82617	0.228272
$65$	1.01112	0.125414
$66$	0	0
$67$	9.14561	1.11731	0.558657	−	0.829399i	$-0.311317\pi$
$67$	9.14561	1.11731	0.558657	+	0.829399i	$0.311317\pi$
$68$	2.08443	0.252774
$69$	0	0
$70$	7.51690	0.898441
$71$	11.2901	1.33989	0.669946	−	0.742410i	$-0.266317\pi$
$71$	11.2901	1.33989	0.669946	+	0.742410i	$0.266317\pi$
$72$	0	0
$73$	−5.53019	−0.647260	−0.323630	−	0.946184i	$-0.604903\pi$
$73$	−5.53019	−0.647260	−0.323630	+	0.946184i	$0.604903\pi$
$74$	−4.18210	−0.486159
$75$	0	0
$76$	4.55140	0.522081
$77$	−5.05699	−0.576298
$78$	0	0
$79$	1.14189	0.128473	0.0642365	−	0.997935i	$-0.479539\pi$
$79$	1.14189	0.128473	0.0642365	+	0.997935i	$0.479539\pi$
$80$	4.35144	0.486505
$81$	0	0
$82$	16.2909	1.79903
$83$	3.57616	0.392535	0.196267	−	0.980550i	$-0.437118\pi$
$83$	3.57616	0.392535	0.196267	+	0.980550i	$0.437118\pi$
$84$	0	0
$85$	−2.00317	−0.217274
$86$	−17.6648	−1.90485
$87$	0	0
$88$	−1.86334	−0.198633
$89$	3.08226	0.326719	0.163360	−	0.986567i	$-0.447767\pi$
$89$	3.08226	0.326719	0.163360	+	0.986567i	$0.447767\pi$
$90$	0	0
$91$	−5.86519	−0.614839
$92$	2.58974	0.269999
$93$	0	0
$94$	19.0613	1.96602
$95$	−4.37396	−0.448759
$96$	0	0
$97$	5.37230	0.545474	0.272737	−	0.962089i	$-0.412071\pi$
$97$	5.37230	0.545474	0.272737	+	0.962089i	$0.412071\pi$
$98$	−31.6680	−3.19895
$99$	0	0
$100$	−3.84633	−0.384633
$101$	8.75118	0.870775	0.435387	−	0.900243i	$-0.356611\pi$
$101$	8.75118	0.870775	0.435387	+	0.900243i	$0.356611\pi$
$102$	0	0
$103$	13.3526	1.31568	0.657838	−	0.753160i	$-0.271471\pi$
$103$	13.3526	1.31568	0.657838	+	0.753160i	$0.271471\pi$
$104$	−2.16113	−0.211917
$105$	0	0
$106$	−8.87840	−0.862346
$107$	−12.0568	−1.16558	−0.582788	−	0.812624i	$-0.698038\pi$
$107$	−12.0568	−1.16558	−0.582788	+	0.812624i	$0.698038\pi$
$108$	0	0
$109$	−11.9480	−1.14441	−0.572207	−	0.820109i	$-0.693913\pi$
$109$	−11.9480	−1.14441	−0.572207	+	0.820109i	$0.693913\pi$
$110$	−1.48644	−0.141726
$111$	0	0
$112$	−25.2414	−2.38509
$113$	−3.66140	−0.344435	−0.172218	−	0.985059i	$-0.555093\pi$
$113$	−3.66140	−0.344435	−0.172218	+	0.985059i	$0.555093\pi$
$114$	0	0
$115$	−2.48878	−0.232080
$116$	−2.07694	−0.192839
$117$	0	0
$118$	10.8229	0.996324
$119$	11.6198	1.06518
$120$	0	0
$121$	1.00000	0.0909091
$122$	−9.48481	−0.858714
$123$	0	0
$124$	−2.27360	−0.204176
$125$	8.05533	0.720490
$126$	0	0
$127$	12.5411	1.11284	0.556420	−	0.830901i	$-0.312175\pi$
$127$	12.5411	1.11284	0.556420	+	0.830901i	$0.312175\pi$
$128$	−12.6813	−1.12088
$129$	0	0
$130$	−1.72399	−0.151204
$131$	−5.81462	−0.508026	−0.254013	−	0.967201i	$-0.581751\pi$
$131$	−5.81462	−0.508026	−0.254013	+	0.967201i	$0.581751\pi$
$132$	0	0
$133$	25.3720	2.20003
$134$	−15.5936	−1.34708
$135$	0	0
$136$	4.28152	0.367137
$137$	19.6724	1.68073	0.840365	−	0.542021i	$-0.182341\pi$
$137$	19.6724	1.68073	0.840365	+	0.542021i	$0.182341\pi$
$138$	0	0
$139$	−4.68618	−0.397476	−0.198738	−	0.980053i	$-0.563684\pi$
$139$	−4.68618	−0.397476	−0.198738	+	0.980053i	$0.563684\pi$
$140$	−3.99932	−0.338004
$141$	0	0
$142$	−19.2501	−1.61543
$143$	1.15982	0.0969888
$144$	0	0
$145$	1.99597	0.165757
$146$	9.42919	0.780365
$147$	0	0
$148$	2.22506	0.182899
$149$	−16.9176	−1.38595	−0.692973	−	0.720964i	$-0.743699\pi$
$149$	−16.9176	−1.38595	−0.692973	+	0.720964i	$0.743699\pi$
$150$	0	0
$151$	6.17353	0.502395	0.251197	−	0.967936i	$-0.419176\pi$
$151$	6.17353	0.502395	0.251197	+	0.967936i	$0.419176\pi$
$152$	9.34878	0.758286
$153$	0	0
$154$	8.62237	0.694810
$155$	2.18496	0.175501
$156$	0	0
$157$	−17.8633	−1.42565	−0.712823	−	0.701344i	$-0.752584\pi$
$157$	−17.8633	−1.42565	−0.712823	+	0.701344i	$0.752584\pi$
$158$	−1.94697	−0.154893
$159$	0	0
$160$	−4.17048	−0.329706
$161$	14.4367	1.13777
$162$	0	0
$163$	25.2861	1.98056	0.990281	−	0.139084i	$-0.0444158\pi$
$163$	25.2861	1.98056	0.990281	+	0.139084i	$0.0444158\pi$
$164$	−8.66749	−0.676817
$165$	0	0
$166$	−6.09750	−0.473258
$167$	19.8203	1.53374	0.766872	−	0.641800i	$-0.221812\pi$
$167$	19.8203	1.53374	0.766872	+	0.641800i	$0.221812\pi$
$168$	0	0
$169$	−11.6548	−0.896525
$170$	3.41548	0.261955
$171$	0	0
$172$	9.39848	0.716627
$173$	11.6286	0.884103	0.442051	−	0.896990i	$-0.354251\pi$
$173$	11.6286	0.884103	0.442051	+	0.896990i	$0.354251\pi$
$174$	0	0
$175$	−21.4416	−1.62083
$176$	4.99138	0.376239
$177$	0	0
$178$	−5.25538	−0.393907
$179$	21.7911	1.62875	0.814373	−	0.580342i	$-0.197081\pi$
$179$	21.7911	1.62875	0.814373	+	0.580342i	$0.197081\pi$
$180$	0	0
$181$	−7.01497	−0.521419	−0.260709	−	0.965417i	$-0.583956\pi$
$181$	−7.01497	−0.521419	−0.260709	+	0.965417i	$0.583956\pi$
$182$	10.0004	0.741277
$183$	0	0
$184$	5.31945	0.392155
$185$	−2.13832	−0.157212
$186$	0	0
$187$	−2.29776	−0.168029
$188$	−10.1415	−0.739643
$189$	0	0
$190$	7.45777	0.541043
$191$	−6.82425	−0.493785	−0.246893	−	0.969043i	$-0.579409\pi$
$191$	−6.82425	−0.493785	−0.246893	+	0.969043i	$0.579409\pi$
$192$	0	0
$193$	22.9510	1.65205	0.826025	−	0.563634i	$-0.190597\pi$
$193$	22.9510	1.65205	0.826025	+	0.563634i	$0.190597\pi$
$194$	−9.15998	−0.657648
$195$	0	0
$196$	16.8488	1.20348
$197$	−1.19763	−0.0853279	−0.0426639	−	0.999089i	$-0.513584\pi$
$197$	−1.19763	−0.0853279	−0.0426639	+	0.999089i	$0.513584\pi$
$198$	0	0
$199$	5.91835	0.419540	0.209770	−	0.977751i	$-0.432728\pi$
$199$	5.91835	0.419540	0.209770	+	0.977751i	$0.432728\pi$
$200$	−7.90053	−0.558652
$201$	0	0
$202$	−14.9211	−1.04984
$203$	−11.5780	−0.812619
$204$	0	0
$205$	8.32958	0.581763
$206$	−22.7668	−1.58624
$207$	0	0
$208$	5.78909	0.401401
$209$	−5.01721	−0.347048
$210$	0	0
$211$	−25.5087	−1.75609	−0.878044	−	0.478580i	$-0.841152\pi$
$211$	−25.5087	−1.75609	−0.878044	+	0.478580i	$0.841152\pi$
$212$	4.72370	0.324425
$213$	0	0
$214$	20.5573	1.40527
$215$	−9.03207	−0.615982
$216$	0	0
$217$	−12.6743	−0.860389
$218$	20.3719	1.37976
$219$	0	0
$220$	0.790850	0.0533191
$221$	−2.66499	−0.179266
$222$	0	0
$223$	4.62587	0.309771	0.154886	−	0.987932i	$-0.450499\pi$
$223$	4.62587	0.309771	0.154886	+	0.987932i	$0.450499\pi$
$224$	24.1917	1.61638
$225$	0	0
$226$	6.24282	0.415266
$227$	12.0940	0.802707	0.401353	−	0.915923i	$-0.368540\pi$
$227$	12.0940	0.802707	0.401353	+	0.915923i	$0.368540\pi$
$228$	0	0
$229$	−25.3953	−1.67817	−0.839083	−	0.544003i	$-0.816908\pi$
$229$	−25.3953	−1.67817	−0.839083	+	0.544003i	$0.816908\pi$
$230$	4.24346	0.279806
$231$	0	0
$232$	−4.26614	−0.280086
$233$	−19.8397	−1.29974	−0.649870	−	0.760045i	$-0.725177\pi$
$233$	−19.8397	−1.29974	−0.649870	+	0.760045i	$0.725177\pi$
$234$	0	0
$235$	9.74609	0.635765
$236$	−5.75824	−0.374829
$237$	0	0
$238$	−19.8122	−1.28423
$239$	−8.14757	−0.527022	−0.263511	−	0.964656i	$-0.584881\pi$
$239$	−8.14757	−0.527022	−0.263511	+	0.964656i	$0.584881\pi$
$240$	0	0
$241$	−12.9251	−0.832577	−0.416289	−	0.909233i	$-0.636669\pi$
$241$	−12.9251	−0.832577	−0.416289	+	0.909233i	$0.636669\pi$
$242$	−1.70504	−0.109604
$243$	0	0
$244$	5.04634	0.323059
$245$	−16.1919	−1.03446
$246$	0	0
$247$	−5.81905	−0.370257
$248$	−4.67008	−0.296551
$249$	0	0
$250$	−13.7346	−0.868655
$251$	18.3060	1.15547	0.577733	−	0.816226i	$-0.303938\pi$
$251$	18.3060	1.15547	0.577733	+	0.816226i	$0.303938\pi$
$252$	0	0
$253$	−2.85479	−0.179479
$254$	−21.3830	−1.34169
$255$	0	0
$256$	17.9698	1.12311
$257$	6.40022	0.399235	0.199617	−	0.979874i	$-0.436030\pi$
$257$	6.40022	0.399235	0.199617	+	0.979874i	$0.436030\pi$
$258$	0	0
$259$	12.4037	0.770730
$260$	0.917241	0.0568849
$261$	0	0
$262$	9.91415	0.612498
$263$	−18.6751	−1.15155	−0.575777	−	0.817607i	$-0.695300\pi$
$263$	−18.6751	−1.15155	−0.575777	+	0.817607i	$0.695300\pi$
$264$	0	0
$265$	−4.53955	−0.278862
$266$	−43.2603	−2.65246
$267$	0	0
$268$	8.29650	0.506790
$269$	−13.2429	−0.807437	−0.403718	−	0.914883i	$-0.632282\pi$
$269$	−13.2429	−0.807437	−0.403718	+	0.914883i	$0.632282\pi$
$270$	0	0
$271$	17.5582	1.06658	0.533292	−	0.845931i	$-0.320955\pi$
$271$	17.5582	1.06658	0.533292	+	0.845931i	$0.320955\pi$
$272$	−11.4690	−0.695411
$273$	0	0
$274$	−33.5423	−2.02636
$275$	4.23998	0.255681
$276$	0	0
$277$	16.8743	1.01388	0.506940	−	0.861981i	$-0.330776\pi$
$277$	16.8743	1.01388	0.506940	+	0.861981i	$0.330776\pi$
$278$	7.99011	0.479215
$279$	0	0
$280$	−8.21479	−0.490928
$281$	−15.0662	−0.898776	−0.449388	−	0.893337i	$-0.648358\pi$
$281$	−15.0662	−0.898776	−0.449388	+	0.893337i	$0.648358\pi$
$282$	0	0
$283$	8.06894	0.479649	0.239824	−	0.970816i	$-0.422910\pi$
$283$	8.06894	0.479649	0.239824	+	0.970816i	$0.422910\pi$
$284$	10.2419	0.607746
$285$	0	0
$286$	−1.97753	−0.116934
$287$	−48.3174	−2.85209
$288$	0	0
$289$	−11.7203	−0.689428
$290$	−3.40321	−0.199843
$291$	0	0
$292$	−5.01675	−0.293583
$293$	0.183657	0.0107294	0.00536468	−	0.999986i	$-0.498292\pi$
$293$	0.183657	0.0107294	0.00536468	+	0.999986i	$0.498292\pi$
$294$	0	0
$295$	5.53375	0.322187
$296$	4.57038	0.265648
$297$	0	0
$298$	28.8452	1.67096
$299$	−3.31104	−0.191482
$300$	0	0
$301$	52.3923	3.01984
$302$	−10.5261	−0.605709
$303$	0	0
$304$	−25.0428	−1.43630
$305$	−4.84960	−0.277687
$306$	0	0
$307$	22.3142	1.27354	0.636770	−	0.771054i	$-0.280270\pi$
$307$	22.3142	1.27354	0.636770	+	0.771054i	$0.280270\pi$
$308$	−4.58748	−0.261396
$309$	0	0
$310$	−3.72545	−0.211591
$311$	−18.5792	−1.05353	−0.526766	−	0.850011i	$-0.676595\pi$
$311$	−18.5792	−1.05353	−0.526766	+	0.850011i	$0.676595\pi$
$312$	0	0
$313$	−26.9055	−1.52079	−0.760393	−	0.649463i	$-0.774994\pi$
$313$	−26.9055	−1.52079	−0.760393	+	0.649463i	$0.774994\pi$
$314$	30.4576	1.71882
$315$	0	0
$316$	1.03587	0.0582725
$317$	3.67466	0.206390	0.103195	−	0.994661i	$-0.467093\pi$
$317$	3.67466	0.206390	0.103195	+	0.994661i	$0.467093\pi$
$318$	0	0
$319$	2.28951	0.128188
$320$	−1.59204	−0.0889977
$321$	0	0
$322$	−24.6151	−1.37174
$323$	11.5284	0.641456
$324$	0	0
$325$	4.91760	0.272780
$326$	−43.1138	−2.38785
$327$	0	0
$328$	−17.8034	−0.983029
$329$	−56.5342	−3.11683
$330$	0	0
$331$	−6.80903	−0.374258	−0.187129	−	0.982335i	$-0.559918\pi$
$331$	−6.80903	−0.374258	−0.187129	+	0.982335i	$0.559918\pi$
$332$	3.24414	0.178045
$333$	0	0
$334$	−33.7945	−1.84915
$335$	−7.97306	−0.435615
$336$	0	0
$337$	−18.6186	−1.01422	−0.507110	−	0.861881i	$-0.669286\pi$
$337$	−18.6186	−1.01422	−0.507110	+	0.861881i	$0.669286\pi$
$338$	19.8719	1.08089
$339$	0	0
$340$	−1.81719	−0.0985508
$341$	2.50630	0.135724
$342$	0	0
$343$	58.5255	3.16008
$344$	19.3049	1.04085
$345$	0	0
$346$	−19.8271	−1.06591
$347$	9.05768	0.486242	0.243121	−	0.969996i	$-0.421829\pi$
$347$	9.05768	0.486242	0.243121	+	0.969996i	$0.421829\pi$
$348$	0	0
$349$	15.2265	0.815057	0.407529	−	0.913192i	$-0.366391\pi$
$349$	15.2265	0.815057	0.407529	+	0.913192i	$0.366391\pi$
$350$	36.5587	1.95414
$351$	0	0
$352$	−4.78381	−0.254978
$353$	18.7774	0.999422	0.499711	−	0.866192i	$-0.333440\pi$
$353$	18.7774	0.999422	0.499711	+	0.866192i	$0.333440\pi$
$354$	0	0
$355$	−9.84262	−0.522392
$356$	2.79609	0.148193
$357$	0	0
$358$	−37.1547	−1.96369
$359$	−25.3569	−1.33828	−0.669142	−	0.743135i	$-0.733338\pi$
$359$	−25.3569	−1.33828	−0.669142	+	0.743135i	$0.733338\pi$
$360$	0	0
$361$	6.17244	0.324865
$362$	11.9608	0.628645
$363$	0	0
$364$	−5.32064	−0.278877
$365$	4.82117	0.252351
$366$	0	0
$367$	−2.95483	−0.154241	−0.0771205	−	0.997022i	$-0.524573\pi$
$367$	−2.95483	−0.154241	−0.0771205	+	0.997022i	$0.524573\pi$
$368$	−14.2494	−0.742799
$369$	0	0
$370$	3.64591	0.189542
$371$	26.3325	1.36712
$372$	0	0
$373$	−1.71745	−0.0889262	−0.0444631	−	0.999011i	$-0.514158\pi$
$373$	−1.71745	−0.0889262	−0.0444631	+	0.999011i	$0.514158\pi$
$374$	3.91778	0.202583
$375$	0	0
$376$	−20.8310	−1.07428
$377$	2.65541	0.136761
$378$	0	0
$379$	−18.7278	−0.961984	−0.480992	−	0.876725i	$-0.659723\pi$
$379$	−18.7278	−0.961984	−0.480992	+	0.876725i	$0.659723\pi$
$380$	−3.96786	−0.203547
$381$	0	0
$382$	11.6356	0.595329
$383$	14.4964	0.740729	0.370365	−	0.928886i	$-0.379233\pi$
$383$	14.4964	0.740729	0.370365	+	0.928886i	$0.379233\pi$
$384$	0	0
$385$	4.40864	0.224685
$386$	−39.1323	−1.99178
$387$	0	0
$388$	4.87352	0.247415
$389$	16.0454	0.813534	0.406767	−	0.913532i	$-0.366656\pi$
$389$	16.0454	0.813534	0.406767	+	0.913532i	$0.366656\pi$
$390$	0	0
$391$	6.55964	0.331735
$392$	34.6082	1.74798
$393$	0	0
$394$	2.04201	0.102875
$395$	−0.995490	−0.0500886
$396$	0	0
$397$	1.27272	0.0638758	0.0319379	−	0.999490i	$-0.489832\pi$
$397$	1.27272	0.0638758	0.0319379	+	0.999490i	$0.489832\pi$
$398$	−10.0910	−0.505816
$399$	0	0
$400$	21.1634	1.05817
$401$	15.6180	0.779928	0.389964	−	0.920830i	$-0.372487\pi$
$401$	15.6180	0.779928	0.389964	+	0.920830i	$0.372487\pi$
$402$	0	0
$403$	2.90684	0.144800
$404$	7.93869	0.394964
$405$	0	0
$406$	19.7410	0.979730
$407$	−2.45279	−0.121580
$408$	0	0
$409$	1.91494	0.0946876	0.0473438	−	0.998879i	$-0.484924\pi$
$409$	1.91494	0.0946876	0.0473438	+	0.998879i	$0.484924\pi$
$410$	−14.2023	−0.701399
$411$	0	0
$412$	12.1129	0.596762
$413$	−32.0996	−1.57952
$414$	0	0
$415$	−3.11766	−0.153040
$416$	−5.54835	−0.272030
$417$	0	0
$418$	8.55454	0.418416
$419$	−5.80374	−0.283531	−0.141766	−	0.989900i	$-0.545278\pi$
$419$	−5.80374	−0.283531	−0.141766	+	0.989900i	$0.545278\pi$
$420$	0	0
$421$	−1.71356	−0.0835138	−0.0417569	−	0.999128i	$-0.513295\pi$
$421$	−1.71356	−0.0835138	−0.0417569	+	0.999128i	$0.513295\pi$
$422$	43.4932	2.11722
$423$	0	0
$424$	9.70270	0.471205
$425$	−9.74248	−0.472580
$426$	0	0
$427$	28.1311	1.36136
$428$	−10.9374	−0.528680
$429$	0	0
$430$	15.4000	0.742655
$431$	−28.8643	−1.39035	−0.695173	−	0.718842i	$-0.744672\pi$
$431$	−28.8643	−1.39035	−0.695173	+	0.718842i	$0.744672\pi$
$432$	0	0
$433$	−2.35069	−0.112967	−0.0564835	−	0.998404i	$-0.517989\pi$
$433$	−2.35069	−0.112967	−0.0564835	+	0.998404i	$0.517989\pi$
$434$	21.6102	1.03732
$435$	0	0
$436$	−10.8387	−0.519081
$437$	14.3231	0.685167
$438$	0	0
$439$	27.3917	1.30734	0.653668	−	0.756781i	$-0.273229\pi$
$439$	27.3917	1.30734	0.653668	+	0.756781i	$0.273229\pi$
$440$	1.62444	0.0774422
$441$	0	0
$442$	4.54390	0.216131
$443$	10.6658	0.506748	0.253374	−	0.967368i	$-0.418460\pi$
$443$	10.6658	0.506748	0.253374	+	0.967368i	$0.418460\pi$
$444$	0	0
$445$	−2.68709	−0.127380
$446$	−7.88729	−0.373474
$447$	0	0
$448$	9.23494	0.436310
$449$	−8.05790	−0.380276	−0.190138	−	0.981757i	$-0.560894\pi$
$449$	−8.05790	−0.380276	−0.190138	+	0.981757i	$0.560894\pi$
$450$	0	0
$451$	9.55457	0.449907
$452$	−3.32146	−0.156228
$453$	0	0
$454$	−20.6207	−0.967779
$455$	5.11321	0.239711
$456$	0	0
$457$	−5.21726	−0.244053	−0.122027	−	0.992527i	$-0.538939\pi$
$457$	−5.21726	−0.244053	−0.122027	+	0.992527i	$0.538939\pi$
$458$	43.2999	2.02327
$459$	0	0
$460$	−2.25771	−0.105266
$461$	17.0015	0.791837	0.395919	−	0.918286i	$-0.370426\pi$
$461$	17.0015	0.791837	0.395919	+	0.918286i	$0.370426\pi$
$462$	0	0
$463$	22.1736	1.03049	0.515247	−	0.857042i	$-0.327700\pi$
$463$	22.1736	1.03049	0.515247	+	0.857042i	$0.327700\pi$
$464$	11.4278	0.530523
$465$	0	0
$466$	33.8274	1.56702
$467$	30.4271	1.40800	0.703999	−	0.710201i	$-0.251396\pi$
$467$	30.4271	1.40800	0.703999	+	0.710201i	$0.251396\pi$
$468$	0	0
$469$	46.2493	2.13559
$470$	−16.6175	−0.766507
$471$	0	0
$472$	−11.8277	−0.544413
$473$	−10.3604	−0.476370
$474$	0	0
$475$	−21.2729	−0.976067
$476$	10.5410	0.483144
$477$	0	0
$478$	13.8919	0.635402
$479$	42.3396	1.93455	0.967273	−	0.253739i	$-0.0816603\pi$
$479$	42.3396	1.93455	0.967273	+	0.253739i	$0.0816603\pi$
$480$	0	0
$481$	−2.84478	−0.129711
$482$	22.0377	1.00379
$483$	0	0
$484$	0.907156	0.0412344
$485$	−4.68352	−0.212668
$486$	0	0
$487$	−0.743237	−0.0336793	−0.0168396	−	0.999858i	$-0.505360\pi$
$487$	−0.743237	−0.0336793	−0.0168396	+	0.999858i	$0.505360\pi$
$488$	10.3654	0.469220
$489$	0	0
$490$	27.6078	1.24719
$491$	25.2271	1.13848	0.569241	−	0.822171i	$-0.307237\pi$
$491$	25.2271	1.13848	0.569241	+	0.822171i	$0.307237\pi$
$492$	0	0
$493$	−5.26076	−0.236933
$494$	9.92170	0.446399
$495$	0	0
$496$	12.5099	0.561710
$497$	57.0941	2.56102
$498$	0	0
$499$	26.4898	1.18585	0.592924	−	0.805258i	$-0.297973\pi$
$499$	26.4898	1.18585	0.592924	+	0.805258i	$0.297973\pi$
$500$	7.30744	0.326799
$501$	0	0
$502$	−31.2125	−1.39308
$503$	16.2654	0.725240	0.362620	−	0.931937i	$-0.381882\pi$
$503$	16.2654	0.725240	0.362620	+	0.931937i	$0.381882\pi$
$504$	0	0
$505$	−7.62919	−0.339494
$506$	4.86753	0.216388
$507$	0	0
$508$	11.3767	0.504760
$509$	−35.3486	−1.56680	−0.783399	−	0.621519i	$-0.786516\pi$
$509$	−35.3486	−1.56680	−0.783399	+	0.621519i	$0.786516\pi$
$510$	0	0
$511$	−27.9661	−1.23715
$512$	−5.27656	−0.233193
$513$	0	0
$514$	−10.9126	−0.481335
$515$	−11.6407	−0.512951
$516$	0	0
$517$	11.1794	0.491670
$518$	−21.1488	−0.929227
$519$	0	0
$520$	1.88406	0.0826213
$521$	3.49524	0.153129	0.0765646	−	0.997065i	$-0.475605\pi$
$521$	3.49524	0.153129	0.0765646	+	0.997065i	$0.475605\pi$
$522$	0	0
$523$	18.8456	0.824062	0.412031	−	0.911170i	$-0.364820\pi$
$523$	18.8456	0.824062	0.412031	+	0.911170i	$0.364820\pi$
$524$	−5.27477	−0.230429
$525$	0	0
$526$	31.8417	1.38836
$527$	−5.75888	−0.250861
$528$	0	0
$529$	−14.8502	−0.645659
$530$	7.74010	0.336208
$531$	0	0
$532$	23.0164	0.997887
$533$	11.0816	0.479995
$534$	0	0
$535$	10.5110	0.454431
$536$	17.0414	0.736076
$537$	0	0
$538$	22.5797	0.973481
$539$	−18.5732	−0.800003
$540$	0	0
$541$	−29.4096	−1.26442	−0.632209	−	0.774798i	$-0.717852\pi$
$541$	−29.4096	−1.26442	−0.632209	+	0.774798i	$0.717852\pi$
$542$	−29.9374	−1.28592
$543$	0	0
$544$	10.9921	0.471282
$545$	10.4162	0.446180
$546$	0	0
$547$	−3.71218	−0.158722	−0.0793608	−	0.996846i	$-0.525288\pi$
$547$	−3.71218	−0.158722	−0.0793608	+	0.996846i	$0.525288\pi$
$548$	17.8460	0.762343
$549$	0	0
$550$	−7.22933	−0.308260
$551$	−11.4870	−0.489361
$552$	0	0
$553$	5.77454	0.245558
$554$	−28.7714	−1.22238
$555$	0	0
$556$	−4.25109	−0.180287
$557$	28.4515	1.20553	0.602764	−	0.797919i	$-0.294066\pi$
$557$	28.4515	1.20553	0.602764	+	0.797919i	$0.294066\pi$
$558$	0	0
$559$	−12.0161	−0.508228
$560$	22.0052	0.929889
$561$	0	0
$562$	25.6885	1.08360
$563$	−19.8389	−0.836110	−0.418055	−	0.908422i	$-0.637288\pi$
$563$	−19.8389	−0.836110	−0.418055	+	0.908422i	$0.637288\pi$
$564$	0	0
$565$	3.19197	0.134287
$566$	−13.7579	−0.578286
$567$	0	0
$568$	21.0373	0.882707
$569$	−4.29607	−0.180101	−0.0900504	−	0.995937i	$-0.528703\pi$
$569$	−4.29607	−0.180101	−0.0900504	+	0.995937i	$0.528703\pi$
$570$	0	0
$571$	4.46351	0.186792	0.0933960	−	0.995629i	$-0.470228\pi$
$571$	4.46351	0.186792	0.0933960	+	0.995629i	$0.470228\pi$
$572$	1.05214	0.0439920
$573$	0	0
$574$	82.3830	3.43860
$575$	−12.1043	−0.504783
$576$	0	0
$577$	−6.86448	−0.285772	−0.142886	−	0.989739i	$-0.545638\pi$
$577$	−6.86448	−0.285772	−0.142886	+	0.989739i	$0.545638\pi$
$578$	19.9835	0.831205
$579$	0	0
$580$	1.81066	0.0751836
$581$	18.0846	0.750277
$582$	0	0
$583$	−5.20715	−0.215658
$584$	−10.3046	−0.426409
$585$	0	0
$586$	−0.313142	−0.0129358
$587$	9.66286	0.398829	0.199414	−	0.979915i	$-0.436096\pi$
$587$	9.66286	0.398829	0.199414	+	0.979915i	$0.436096\pi$
$588$	0	0
$589$	−12.5746	−0.518128
$590$	−9.43526	−0.388443
$591$	0	0
$592$	−12.2428	−0.503176
$593$	5.70073	0.234101	0.117050	−	0.993126i	$-0.462656\pi$
$593$	5.70073	0.234101	0.117050	+	0.993126i	$0.462656\pi$
$594$	0	0
$595$	−10.1300	−0.415290
$596$	−15.3469	−0.628635
$597$	0	0
$598$	5.64544	0.230859
$599$	31.3831	1.28228	0.641140	−	0.767424i	$-0.278462\pi$
$599$	31.3831	1.28228	0.641140	+	0.767424i	$0.278462\pi$
$600$	0	0
$601$	15.1204	0.616774	0.308387	−	0.951261i	$-0.400211\pi$
$601$	15.1204	0.616774	0.308387	+	0.951261i	$0.400211\pi$
$602$	−89.3309	−3.64086
$603$	0	0
$604$	5.60036	0.227875
$605$	−0.871790	−0.0354433
$606$	0	0
$607$	27.6446	1.12206	0.561030	−	0.827795i	$-0.310405\pi$
$607$	27.6446	1.12206	0.561030	+	0.827795i	$0.310405\pi$
$608$	24.0014	0.973386
$609$	0	0
$610$	8.26876	0.334792
$611$	12.9661	0.524551
$612$	0	0
$613$	−43.8922	−1.77279	−0.886394	−	0.462931i	$-0.846798\pi$
$613$	−43.8922	−1.77279	−0.886394	+	0.462931i	$0.846798\pi$
$614$	−38.0466	−1.53544
$615$	0	0
$616$	−9.42290	−0.379659
$617$	29.5088	1.18798	0.593989	−	0.804473i	$-0.297552\pi$
$617$	29.5088	1.18798	0.593989	+	0.804473i	$0.297552\pi$
$618$	0	0
$619$	3.66744	0.147407	0.0737034	−	0.997280i	$-0.476518\pi$
$619$	3.66744	0.147407	0.0737034	+	0.997280i	$0.476518\pi$
$620$	1.98210	0.0796032
$621$	0	0
$622$	31.6783	1.27018
$623$	15.5870	0.624479
$624$	0	0
$625$	14.1774	0.567094
$626$	45.8749	1.83353
$627$	0	0
$628$	−16.2048	−0.646642
$629$	5.63593	0.224719
$630$	0	0
$631$	−6.02716	−0.239937	−0.119969	−	0.992778i	$-0.538279\pi$
$631$	−6.02716	−0.239937	−0.119969	+	0.992778i	$0.538279\pi$
$632$	2.12773	0.0846367
$633$	0	0
$634$	−6.26544	−0.248833
$635$	−10.9332	−0.433870
$636$	0	0
$637$	−21.5415	−0.853505
$638$	−3.90371	−0.154549
$639$	0	0
$640$	11.0555	0.437005
$641$	44.3883	1.75323	0.876615	−	0.481192i	$-0.159796\pi$
$641$	44.3883	1.75323	0.876615	+	0.481192i	$0.159796\pi$
$642$	0	0
$643$	49.1136	1.93685	0.968426	−	0.249300i	$-0.0802005\pi$
$643$	49.1136	1.93685	0.968426	+	0.249300i	$0.0802005\pi$
$644$	13.0963	0.516067
$645$	0	0
$646$	−19.6563	−0.773368
$647$	10.1571	0.399317	0.199658	−	0.979866i	$-0.436017\pi$
$647$	10.1571	0.399317	0.199658	+	0.979866i	$0.436017\pi$
$648$	0	0
$649$	6.34757	0.249164
$650$	−8.38470	−0.328875
$651$	0	0
$652$	22.9385	0.898339
$653$	6.91460	0.270589	0.135295	−	0.990805i	$-0.456802\pi$
$653$	6.91460	0.270589	0.135295	+	0.990805i	$0.456802\pi$
$654$	0	0
$655$	5.06913	0.198067
$656$	47.6905	1.86200
$657$	0	0
$658$	96.3929	3.75779
$659$	−25.7049	−1.00132	−0.500661	−	0.865643i	$-0.666910\pi$
$659$	−25.7049	−1.00132	−0.500661	+	0.865643i	$0.666910\pi$
$660$	0	0
$661$	−19.9564	−0.776212	−0.388106	−	0.921615i	$-0.626871\pi$
$661$	−19.9564	−0.776212	−0.388106	+	0.921615i	$0.626871\pi$
$662$	11.6097	0.451222
$663$	0	0
$664$	6.66361	0.258598
$665$	−22.1191	−0.857741
$666$	0	0
$667$	−6.53608	−0.253078
$668$	17.9802	0.695673
$669$	0	0
$670$	13.5944	0.525196
$671$	−5.56281	−0.214750
$672$	0	0
$673$	−48.0361	−1.85166	−0.925829	−	0.377943i	$-0.876631\pi$
$673$	−48.0361	−1.85166	−0.925829	+	0.377943i	$0.876631\pi$
$674$	31.7454	1.22279
$675$	0	0
$676$	−10.5727	−0.406644
$677$	12.4971	0.480304	0.240152	−	0.970735i	$-0.422803\pi$
$677$	12.4971	0.480304	0.240152	+	0.970735i	$0.422803\pi$
$678$	0	0
$679$	27.1677	1.04260
$680$	−3.73258	−0.143138
$681$	0	0
$682$	−4.27333	−0.163634
$683$	−18.6768	−0.714646	−0.357323	−	0.933981i	$-0.616311\pi$
$683$	−18.6768	−0.714646	−0.357323	+	0.933981i	$0.616311\pi$
$684$	0	0
$685$	−17.1502	−0.655277
$686$	−99.7882	−3.80993
$687$	0	0
$688$	−51.7126	−1.97152
$689$	−6.03935	−0.230081
$690$	0	0
$691$	−7.44631	−0.283271	−0.141636	−	0.989919i	$-0.545236\pi$
$691$	−7.44631	−0.283271	−0.141636	+	0.989919i	$0.545236\pi$
$692$	10.5489	0.401010
$693$	0	0
$694$	−15.4437	−0.586235
$695$	4.08536	0.154967
$696$	0	0
$697$	−21.9542	−0.831573
$698$	−25.9618	−0.982669
$699$	0	0
$700$	−19.4508	−0.735173
$701$	−22.6298	−0.854718	−0.427359	−	0.904082i	$-0.640556\pi$
$701$	−22.6298	−0.854718	−0.427359	+	0.904082i	$0.640556\pi$
$702$	0	0
$703$	12.3062	0.464136
$704$	−1.82617	−0.0688264
$705$	0	0
$706$	−32.0162	−1.20495
$707$	44.2546	1.66437
$708$	0	0
$709$	−18.4078	−0.691319	−0.345660	−	0.938360i	$-0.612345\pi$
$709$	−18.4078	−0.691319	−0.345660	+	0.938360i	$0.612345\pi$
$710$	16.7820	0.629819
$711$	0	0
$712$	5.74331	0.215239
$713$	−7.15495	−0.267955
$714$	0	0
$715$	−1.01112	−0.0378136
$716$	19.7680	0.738764
$717$	0	0
$718$	43.2344	1.61349
$719$	37.0045	1.38004	0.690018	−	0.723793i	$-0.257603\pi$
$719$	37.0045	1.38004	0.690018	+	0.723793i	$0.257603\pi$
$720$	0	0
$721$	67.5242	2.51473
$722$	−10.5242	−0.391672
$723$	0	0
$724$	−6.36367	−0.236504
$725$	9.70749	0.360527
$726$	0	0
$727$	19.1332	0.709611	0.354805	−	0.934940i	$-0.384547\pi$
$727$	19.1332	0.709611	0.354805	+	0.934940i	$0.384547\pi$
$728$	−10.9288	−0.405050
$729$	0	0
$730$	−8.22027	−0.304246
$731$	23.8057	0.880485
$732$	0	0
$733$	−11.6318	−0.429629	−0.214814	−	0.976655i	$-0.568915\pi$
$733$	−11.6318	−0.429629	−0.214814	+	0.976655i	$0.568915\pi$
$734$	5.03811	0.185960
$735$	0	0
$736$	13.6568	0.503396
$737$	−9.14561	−0.336883
$738$	0	0
$739$	4.19462	0.154302	0.0771508	−	0.997019i	$-0.475418\pi$
$739$	4.19462	0.154302	0.0771508	+	0.997019i	$0.475418\pi$
$740$	−1.93979	−0.0713080
$741$	0	0
$742$	−44.8980	−1.64826
$743$	14.4032	0.528404	0.264202	−	0.964467i	$-0.414891\pi$
$743$	14.4032	0.528404	0.264202	+	0.964467i	$0.414891\pi$
$744$	0	0
$745$	14.7486	0.540347
$746$	2.92832	0.107213
$747$	0	0
$748$	−2.08443	−0.0762144
$749$	−60.9713	−2.22784
$750$	0	0
$751$	2.64251	0.0964265	0.0482132	−	0.998837i	$-0.484647\pi$
$751$	2.64251	0.0964265	0.0482132	+	0.998837i	$0.484647\pi$
$752$	55.8007	2.03484
$753$	0	0
$754$	−4.52758	−0.164885
$755$	−5.38202	−0.195872
$756$	0	0
$757$	−0.414828	−0.0150772	−0.00753859	−	0.999972i	$-0.502400\pi$
$757$	−0.414828	−0.0150772	−0.00753859	+	0.999972i	$0.502400\pi$
$758$	31.9317	1.15981
$759$	0	0
$760$	−8.15017	−0.295638
$761$	34.0779	1.23532	0.617662	−	0.786443i	$-0.288080\pi$
$761$	34.0779	1.23532	0.617662	+	0.786443i	$0.288080\pi$
$762$	0	0
$763$	−60.4211	−2.18739
$764$	−6.19066	−0.223970
$765$	0	0
$766$	−24.7168	−0.893056
$767$	7.36202	0.265827
$768$	0	0
$769$	37.5853	1.35536	0.677680	−	0.735356i	$-0.262985\pi$
$769$	37.5853	1.35536	0.677680	+	0.735356i	$0.262985\pi$
$770$	−7.51690	−0.270890
$771$	0	0
$772$	20.8201	0.749334
$773$	30.6963	1.10407	0.552035	−	0.833821i	$-0.313852\pi$
$773$	30.6963	1.10407	0.552035	+	0.833821i	$0.313852\pi$
$774$	0	0
$775$	10.6267	0.381721
$776$	10.0104	0.359353
$777$	0	0
$778$	−27.3580	−0.980833
$779$	−47.9373	−1.71753
$780$	0	0
$781$	−11.2901	−0.403992
$782$	−11.1844	−0.399955
$783$	0	0
$784$	−92.7058	−3.31092
$785$	15.5731	0.555826
$786$	0	0
$787$	−23.8454	−0.849997	−0.424998	−	0.905194i	$-0.639725\pi$
$787$	−23.8454	−0.849997	−0.424998	+	0.905194i	$0.639725\pi$
$788$	−1.08644	−0.0387029
$789$	0	0
$790$	1.69735	0.0603890
$791$	−18.5157	−0.658341
$792$	0	0
$793$	−6.45184	−0.229112
$794$	−2.17003	−0.0770115
$795$	0	0
$796$	5.36886	0.190294
$797$	42.8545	1.51798	0.758991	−	0.651101i	$-0.225692\pi$
$797$	42.8545	1.51798	0.758991	+	0.651101i	$0.225692\pi$
$798$	0	0
$799$	−25.6876	−0.908763
$800$	−20.2833	−0.717122
$801$	0	0
$802$	−26.6294	−0.940315
$803$	5.53019	0.195156
$804$	0	0
$805$	−12.5857	−0.443589
$806$	−4.95628	−0.174578
$807$	0	0
$808$	16.3064	0.573658
$809$	−24.1252	−0.848196	−0.424098	−	0.905616i	$-0.639409\pi$
$809$	−24.1252	−0.848196	−0.424098	+	0.905616i	$0.639409\pi$
$810$	0	0
$811$	−6.53449	−0.229457	−0.114728	−	0.993397i	$-0.536600\pi$
$811$	−6.53449	−0.229457	−0.114728	+	0.993397i	$0.536600\pi$
$812$	−10.5031	−0.368586
$813$	0	0
$814$	4.18210	0.146582
$815$	−22.0442	−0.772174
$816$	0	0
$817$	51.9802	1.81856
$818$	−3.26504	−0.114160
$819$	0	0
$820$	7.55623	0.263875
$821$	−20.1321	−0.702615	−0.351307	−	0.936260i	$-0.614263\pi$
$821$	−20.1321	−0.702615	−0.351307	+	0.936260i	$0.614263\pi$
$822$	0	0
$823$	17.6775	0.616200	0.308100	−	0.951354i	$-0.400307\pi$
$823$	17.6775	0.616200	0.308100	+	0.951354i	$0.400307\pi$
$824$	24.8805	0.866754
$825$	0	0
$826$	54.7311	1.90434
$827$	24.7650	0.861165	0.430582	−	0.902551i	$-0.358308\pi$
$827$	24.7650	0.861165	0.430582	+	0.902551i	$0.358308\pi$
$828$	0	0
$829$	−18.3129	−0.636035	−0.318017	−	0.948085i	$-0.603017\pi$
$829$	−18.3129	−0.636035	−0.318017	+	0.948085i	$0.603017\pi$
$830$	5.31574	0.184512
$831$	0	0
$832$	−2.11803	−0.0734293
$833$	42.6768	1.47866
$834$	0	0
$835$	−17.2792	−0.597971
$836$	−4.55140	−0.157413
$837$	0	0
$838$	9.89560	0.341838
$839$	9.73722	0.336166	0.168083	−	0.985773i	$-0.446242\pi$
$839$	9.73722	0.336166	0.168083	+	0.985773i	$0.446242\pi$
$840$	0	0
$841$	−23.7581	−0.819246
$842$	2.92168	0.100688
$843$	0	0
$844$	−23.1403	−0.796523
$845$	10.1606	0.349534
$846$	0	0
$847$	5.05699	0.173760
$848$	−25.9909	−0.892531
$849$	0	0
$850$	16.6113	0.569763
$851$	7.00220	0.240032
$852$	0	0
$853$	17.0185	0.582701	0.291350	−	0.956616i	$-0.405895\pi$
$853$	17.0185	0.582701	0.291350	+	0.956616i	$0.405895\pi$
$854$	−47.9646	−1.64131
$855$	0	0
$856$	−22.4660	−0.767871
$857$	−31.1976	−1.06569	−0.532845	−	0.846213i	$-0.678877\pi$
$857$	−31.1976	−1.06569	−0.532845	+	0.846213i	$0.678877\pi$
$858$	0	0
$859$	16.2099	0.553075	0.276537	−	0.961003i	$-0.410813\pi$
$859$	16.2099	0.553075	0.276537	+	0.961003i	$0.410813\pi$
$860$	−8.19350	−0.279396
$861$	0	0
$862$	49.2148	1.67626
$863$	41.3758	1.40845	0.704224	−	0.709978i	$-0.251295\pi$
$863$	41.3758	1.40845	0.704224	+	0.709978i	$0.251295\pi$
$864$	0	0
$865$	−10.1377	−0.344691
$866$	4.00802	0.136198
$867$	0	0
$868$	−11.4976	−0.390254
$869$	−1.14189	−0.0387360
$870$	0	0
$871$	−10.6072	−0.359413
$872$	−22.2633	−0.753929
$873$	0	0
$874$	−24.4214	−0.826067
$875$	40.7357	1.37712
$876$	0	0
$877$	41.7591	1.41010	0.705052	−	0.709155i	$-0.250924\pi$
$877$	41.7591	1.41010	0.705052	+	0.709155i	$0.250924\pi$
$878$	−46.7040	−1.57618
$879$	0	0
$880$	−4.35144	−0.146687
$881$	−3.91649	−0.131950	−0.0659749	−	0.997821i	$-0.521016\pi$
$881$	−3.91649	−0.131950	−0.0659749	+	0.997821i	$0.521016\pi$
$882$	0	0
$883$	41.5401	1.39794	0.698968	−	0.715153i	$-0.253643\pi$
$883$	41.5401	1.39794	0.698968	+	0.715153i	$0.253643\pi$
$884$	−2.41756	−0.0813113
$885$	0	0
$886$	−18.1856	−0.610958
$887$	25.6663	0.861789	0.430895	−	0.902402i	$-0.358198\pi$
$887$	25.6663	0.861789	0.430895	+	0.902402i	$0.358198\pi$
$888$	0	0
$889$	63.4201	2.12704
$890$	4.58159	0.153575
$891$	0	0
$892$	4.19639	0.140506
$893$	−56.0895	−1.87696
$894$	0	0
$895$	−18.9973	−0.635010
$896$	−64.1294	−2.14241
$897$	0	0
$898$	13.7390	0.458478
$899$	5.73819	0.191379
$900$	0	0
$901$	11.9648	0.398606
$902$	−16.2909	−0.542428
$903$	0	0
$904$	−6.82243	−0.226911
$905$	6.11558	0.203289
$906$	0	0
$907$	−11.6827	−0.387918	−0.193959	−	0.981010i	$-0.562133\pi$
$907$	−11.6827	−0.387918	−0.193959	+	0.981010i	$0.562133\pi$
$908$	10.9711	0.364090
$909$	0	0
$910$	−8.71822	−0.289006
$911$	14.9652	0.495820	0.247910	−	0.968783i	$-0.420256\pi$
$911$	14.9652	0.495820	0.247910	+	0.968783i	$0.420256\pi$
$912$	0	0
$913$	−3.57616	−0.118354
$914$	8.89563	0.294241
$915$	0	0
$916$	−23.0375	−0.761180
$917$	−29.4045	−0.971022
$918$	0	0
$919$	13.3553	0.440550	0.220275	−	0.975438i	$-0.429305\pi$
$919$	13.3553	0.440550	0.220275	+	0.975438i	$0.429305\pi$
$920$	−4.63744	−0.152892
$921$	0	0
$922$	−28.9882	−0.954674
$923$	−13.0945	−0.431010
$924$	0	0
$925$	−10.3998	−0.341943
$926$	−37.8068	−1.24241
$927$	0	0
$928$	−10.9526	−0.359537
$929$	−9.16741	−0.300773	−0.150386	−	0.988627i	$-0.548052\pi$
$929$	−9.16741	−0.300773	−0.150386	+	0.988627i	$0.548052\pi$
$930$	0	0
$931$	93.1856	3.05404
$932$	−17.9977	−0.589534
$933$	0	0
$934$	−51.8794	−1.69754
$935$	2.00317	0.0655106
$936$	0	0
$937$	−45.5522	−1.48812	−0.744062	−	0.668110i	$-0.767103\pi$
$937$	−45.5522	−1.48812	−0.744062	+	0.668110i	$0.767103\pi$
$938$	−78.8569	−2.57477
$939$	0	0
$940$	8.84123	0.288369
$941$	26.6945	0.870217	0.435108	−	0.900378i	$-0.356710\pi$
$941$	26.6945	0.870217	0.435108	+	0.900378i	$0.356710\pi$
$942$	0	0
$943$	−27.2763	−0.888239
$944$	31.6831	1.03120
$945$	0	0
$946$	17.6648	0.574333
$947$	−49.1892	−1.59844	−0.799218	−	0.601042i	$-0.794752\pi$
$947$	−49.1892	−1.59844	−0.799218	+	0.601042i	$0.794752\pi$
$948$	0	0
$949$	6.41401	0.208208
$950$	36.2711	1.17679
$951$	0	0
$952$	21.6516	0.701733
$953$	45.1010	1.46097	0.730483	−	0.682931i	$-0.239295\pi$
$953$	45.1010	1.46097	0.730483	+	0.682931i	$0.239295\pi$
$954$	0	0
$955$	5.94931	0.192515
$956$	−7.39112	−0.239046
$957$	0	0
$958$	−72.1906	−2.33237
$959$	99.4834	3.21249
$960$	0	0
$961$	−24.7185	−0.797370
$962$	4.85047	0.156385
$963$	0	0
$964$	−11.7251	−0.377639
$965$	−20.0085	−0.644095
$966$	0	0
$967$	1.76540	0.0567715	0.0283857	−	0.999597i	$-0.490963\pi$
$967$	1.76540	0.0567715	0.0283857	+	0.999597i	$0.490963\pi$
$968$	1.86334	0.0598900
$969$	0	0
$970$	7.98558	0.256401
$971$	−22.7488	−0.730044	−0.365022	−	0.930999i	$-0.618939\pi$
$971$	−22.7488	−0.730044	−0.365022	+	0.930999i	$0.618939\pi$
$972$	0	0
$973$	−23.6980	−0.759722
$974$	1.26725	0.0406052
$975$	0	0
$976$	−27.7661	−0.888771
$977$	−20.2912	−0.649174	−0.324587	−	0.945856i	$-0.605225\pi$
$977$	−20.2912	−0.649174	−0.324587	+	0.945856i	$0.605225\pi$
$978$	0	0
$979$	−3.08226	−0.0985096
$980$	−14.6886	−0.469210
$981$	0	0
$982$	−43.0132	−1.37261
$983$	−55.3443	−1.76521	−0.882605	−	0.470116i	$-0.844212\pi$
$983$	−55.3443	−1.76521	−0.882605	+	0.470116i	$0.844212\pi$
$984$	0	0
$985$	1.04409	0.0332673
$986$	8.96979	0.285656
$987$	0	0
$988$	−5.27879	−0.167941
$989$	29.5767	0.940484
$990$	0	0
$991$	−9.88818	−0.314108	−0.157054	−	0.987590i	$-0.550200\pi$
$991$	−9.88818	−0.314108	−0.157054	+	0.987590i	$0.550200\pi$
$992$	−11.9897	−0.380672
$993$	0	0
$994$	−97.3476	−3.08768
$995$	−5.15955	−0.163569
$996$	0	0
$997$	−3.62436	−0.114785	−0.0573923	−	0.998352i	$-0.518279\pi$
$997$	−3.62436	−0.114785	−0.0573923	+	0.998352i	$0.518279\pi$
$998$	−45.1662	−1.42971
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8019.2.a.k.1.14		51
3.2	odd	2		8019.2.a.l.1.38		51
27.2	odd	18		297.2.j.c.166.4	yes	102
27.13	even	9		891.2.j.c.100.14		102
27.14	odd	18		297.2.j.c.34.4	✓	102
27.25	even	9		891.2.j.c.793.14		102

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
297.2.j.c.34.4	✓	102	27.14	odd	18
297.2.j.c.166.4	yes	102	27.2	odd	18
891.2.j.c.100.14		102	27.13	even	9
891.2.j.c.793.14		102	27.25	even	9
8019.2.a.k.1.14		51	1.1	even	1	trivial
8019.2.a.l.1.38		51	3.2	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:	\( N \)	\(=\)	\( 8019 = 3^{6} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8019.a (trivial)

\(f(q)\)	\(=\)	\(q-1.70504 q^{2} +0.907156 q^{4} -0.871790 q^{5} +5.05699 q^{7} +1.86334 q^{8} +O(q^{10})\)	\(q-1.70504 q^{2} +0.907156 q^{4} -0.871790 q^{5} +5.05699 q^{7} +1.86334 q^{8} +1.48644 q^{10} -1.00000 q^{11} -1.15982 q^{13} -8.62237 q^{14} -4.99138 q^{16} +2.29776 q^{17} +5.01721 q^{19} -0.790850 q^{20} +1.70504 q^{22} +2.85479 q^{23} -4.23998 q^{25} +1.97753 q^{26} +4.58748 q^{28} -2.28951 q^{29} -2.50630 q^{31} +4.78381 q^{32} -3.91778 q^{34} -4.40864 q^{35} +2.45279 q^{37} -8.55454 q^{38} -1.62444 q^{40} -9.55457 q^{41} +10.3604 q^{43} -0.907156 q^{44} -4.86753 q^{46} -11.1794 q^{47} +18.5732 q^{49} +7.22933 q^{50} -1.05214 q^{52} +5.20715 q^{53} +0.871790 q^{55} +9.42290 q^{56} +3.90371 q^{58} -6.34757 q^{59} +5.56281 q^{61} +4.27333 q^{62} +1.82617 q^{64} +1.01112 q^{65} +9.14561 q^{67} +2.08443 q^{68} +7.51690 q^{70} +11.2901 q^{71} -5.53019 q^{73} -4.18210 q^{74} +4.55140 q^{76} -5.05699 q^{77} +1.14189 q^{79} +4.35144 q^{80} +16.2909 q^{82} +3.57616 q^{83} -2.00317 q^{85} -17.6648 q^{86} -1.86334 q^{88} +3.08226 q^{89} -5.86519 q^{91} +2.58974 q^{92} +19.0613 q^{94} -4.37396 q^{95} +5.37230 q^{97} -31.6680 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 51 q + 60 q^{4} - 6 q^{5} + 12 q^{7}+O(q^{10}) \) 51 * q + 60 * q^4 - 6 * q^5 + 12 * q^7	\( 51 q + 60 q^{4} - 6 q^{5} + 12 q^{7} + 18 q^{10} - 51 q^{11} + 30 q^{13} - 12 q^{14} + 78 q^{16} + 30 q^{19} - 18 q^{20} - 3 q^{23} + 75 q^{25} - 9 q^{26} + 36 q^{28} + 42 q^{31} + 15 q^{32} + 42 q^{34} + 9 q^{35} + 48 q^{37} + 3 q^{38} + 54 q^{40} + 18 q^{43} - 60 q^{44} + 42 q^{46} - 30 q^{47} + 99 q^{49} + 30 q^{50} + 60 q^{52} - 18 q^{53} + 6 q^{55} - 21 q^{56} + 30 q^{58} - 24 q^{59} + 99 q^{61} + 114 q^{64} + 15 q^{65} + 39 q^{67} + 39 q^{68} + 48 q^{70} - 30 q^{71} + 69 q^{73} + 90 q^{76} - 12 q^{77} + 48 q^{79} - 42 q^{80} + 42 q^{82} + 21 q^{83} + 84 q^{85} - 24 q^{86} - 15 q^{89} + 69 q^{91} + 66 q^{92} + 66 q^{94} + 12 q^{95} + 72 q^{97} + 57 q^{98}+O(q^{100}) \) 51 * q + 60 * q^4 - 6 * q^5 + 12 * q^7 + 18 * q^10 - 51 * q^11 + 30 * q^13 - 12 * q^14 + 78 * q^16 + 30 * q^19 - 18 * q^20 - 3 * q^23 + 75 * q^25 - 9 * q^26 + 36 * q^28 + 42 * q^31 + 15 * q^32 + 42 * q^34 + 9 * q^35 + 48 * q^37 + 3 * q^38 + 54 * q^40 + 18 * q^43 - 60 * q^44 + 42 * q^46 - 30 * q^47 + 99 * q^49 + 30 * q^50 + 60 * q^52 - 18 * q^53 + 6 * q^55 - 21 * q^56 + 30 * q^58 - 24 * q^59 + 99 * q^61 + 114 * q^64 + 15 * q^65 + 39 * q^67 + 39 * q^68 + 48 * q^70 - 30 * q^71 + 69 * q^73 + 90 * q^76 - 12 * q^77 + 48 * q^79 - 42 * q^80 + 42 * q^82 + 21 * q^83 + 84 * q^85 - 24 * q^86 - 15 * q^89 + 69 * q^91 + 66 * q^92 + 66 * q^94 + 12 * q^95 + 72 * q^97 + 57 * q^98

Introduction

L-functions

Modular forms

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Database

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