LMFDB - Embedded newform 8020.2.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8020,2,Mod(1,8020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8020, 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("8020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8020 = 2^{2} \cdot 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8020.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:	$64.0400224211$
Analytic rank:	$1$
Dimension:	$28$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	8020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.61261 q^{3} -1.00000 q^{5} -1.36796 q^{7} +3.82575 q^{9} +O(q^{10})$	$q-2.61261 q^{3} -1.00000 q^{5} -1.36796 q^{7} +3.82575 q^{9} -3.20956 q^{11} -0.991251 q^{13} +2.61261 q^{15} -6.53588 q^{17} +2.43792 q^{19} +3.57396 q^{21} +2.25201 q^{23} +1.00000 q^{25} -2.15737 q^{27} -4.41202 q^{29} +5.35748 q^{31} +8.38535 q^{33} +1.36796 q^{35} +11.4289 q^{37} +2.58976 q^{39} -4.67265 q^{41} -2.19240 q^{43} -3.82575 q^{45} -0.392606 q^{47} -5.12868 q^{49} +17.0757 q^{51} +9.29821 q^{53} +3.20956 q^{55} -6.36935 q^{57} -1.84032 q^{59} -5.30011 q^{61} -5.23349 q^{63} +0.991251 q^{65} +9.05832 q^{67} -5.88364 q^{69} +1.03723 q^{71} +15.0142 q^{73} -2.61261 q^{75} +4.39056 q^{77} -14.6718 q^{79} -5.84088 q^{81} +1.71945 q^{83} +6.53588 q^{85} +11.5269 q^{87} +13.6317 q^{89} +1.35600 q^{91} -13.9970 q^{93} -2.43792 q^{95} +3.23219 q^{97} -12.2790 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9}+O(q^{10}) $ 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9	$ 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9} + 2 q^{11} + 3 q^{13} - 3 q^{15} - 10 q^{17} - 2 q^{19} - 12 q^{21} - 23 q^{23} + 28 q^{25} + 9 q^{27} - 37 q^{29} - 11 q^{31} + 2 q^{33} + 4 q^{35} - 3 q^{37} - 19 q^{39} - 30 q^{41} + 13 q^{43} - 17 q^{45} - 15 q^{47} + 12 q^{49} - 8 q^{51} - 35 q^{53} - 2 q^{55} - 22 q^{57} - q^{59} - 33 q^{61} - 20 q^{63} - 3 q^{65} + 19 q^{67} - 8 q^{69} - 31 q^{71} + 31 q^{73} + 3 q^{75} - 42 q^{77} - 29 q^{79} - 36 q^{81} + 14 q^{83} + 10 q^{85} - 32 q^{87} - 32 q^{89} - 7 q^{91} - 11 q^{93} + 2 q^{95} + 2 q^{97} - 39 q^{99}+O(q^{100}) $ 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9 + 2 * q^11 + 3 * q^13 - 3 * q^15 - 10 * q^17 - 2 * q^19 - 12 * q^21 - 23 * q^23 + 28 * q^25 + 9 * q^27 - 37 * q^29 - 11 * q^31 + 2 * q^33 + 4 * q^35 - 3 * q^37 - 19 * q^39 - 30 * q^41 + 13 * q^43 - 17 * q^45 - 15 * q^47 + 12 * q^49 - 8 * q^51 - 35 * q^53 - 2 * q^55 - 22 * q^57 - q^59 - 33 * q^61 - 20 * q^63 - 3 * q^65 + 19 * q^67 - 8 * q^69 - 31 * q^71 + 31 * q^73 + 3 * q^75 - 42 * q^77 - 29 * q^79 - 36 * q^81 + 14 * q^83 + 10 * q^85 - 32 * q^87 - 32 * q^89 - 7 * q^91 - 11 * q^93 + 2 * q^95 + 2 * q^97 - 39 * 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
$3$	−2.61261	−1.50839	−0.754197	−	0.656649i	$-0.771974\pi$
$3$	−2.61261	−1.50839	−0.754197	+	0.656649i	$0.771974\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.36796	−0.517041	−0.258521	−	0.966006i	$-0.583235\pi$
$7$	−1.36796	−0.517041	−0.258521	+	0.966006i	$0.583235\pi$
$8$	0	0
$9$	3.82575	1.27525
$10$	0	0
$11$	−3.20956	−0.967720	−0.483860	−	0.875146i	$-0.660766\pi$
$11$	−3.20956	−0.967720	−0.483860	+	0.875146i	$0.660766\pi$
$12$	0	0
$13$	−0.991251	−0.274924	−0.137462	−	0.990507i	$-0.543894\pi$
$13$	−0.991251	−0.274924	−0.137462	+	0.990507i	$0.543894\pi$
$14$	0	0
$15$	2.61261	0.674574
$16$	0	0
$17$	−6.53588	−1.58518	−0.792592	−	0.609753i	$-0.791269\pi$
$17$	−6.53588	−1.58518	−0.792592	+	0.609753i	$0.791269\pi$
$18$	0	0
$19$	2.43792	0.559298	0.279649	−	0.960102i	$-0.409782\pi$
$19$	2.43792	0.559298	0.279649	+	0.960102i	$0.409782\pi$
$20$	0	0
$21$	3.57396	0.779902
$22$	0	0
$23$	2.25201	0.469577	0.234789	−	0.972046i	$-0.424560\pi$
$23$	2.25201	0.469577	0.234789	+	0.972046i	$0.424560\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−2.15737	−0.415186
$28$	0	0
$29$	−4.41202	−0.819291	−0.409646	−	0.912245i	$-0.634348\pi$
$29$	−4.41202	−0.819291	−0.409646	+	0.912245i	$0.634348\pi$
$30$	0	0
$31$	5.35748	0.962232	0.481116	−	0.876657i	$-0.340232\pi$
$31$	5.35748	0.962232	0.481116	+	0.876657i	$0.340232\pi$
$32$	0	0
$33$	8.38535	1.45970
$34$	0	0
$35$	1.36796	0.231228
$36$	0	0
$37$	11.4289	1.87890	0.939449	−	0.342690i	$-0.111338\pi$
$37$	11.4289	1.87890	0.939449	+	0.342690i	$0.111338\pi$
$38$	0	0
$39$	2.58976	0.414693
$40$	0	0
$41$	−4.67265	−0.729745	−0.364873	−	0.931057i	$-0.618887\pi$
$41$	−4.67265	−0.729745	−0.364873	+	0.931057i	$0.618887\pi$
$42$	0	0
$43$	−2.19240	−0.334338	−0.167169	−	0.985928i	$-0.553463\pi$
$43$	−2.19240	−0.334338	−0.167169	+	0.985928i	$0.553463\pi$
$44$	0	0
$45$	−3.82575	−0.570309
$46$	0	0
$47$	−0.392606	−0.0572674	−0.0286337	−	0.999590i	$-0.509116\pi$
$47$	−0.392606	−0.0572674	−0.0286337	+	0.999590i	$0.509116\pi$
$48$	0	0
$49$	−5.12868	−0.732668
$50$	0	0
$51$	17.0757	2.39108
$52$	0	0
$53$	9.29821	1.27721	0.638603	−	0.769536i	$-0.279512\pi$
$53$	9.29821	1.27721	0.638603	+	0.769536i	$0.279512\pi$
$54$	0	0
$55$	3.20956	0.432777
$56$	0	0
$57$	−6.36935	−0.843641
$58$	0	0
$59$	−1.84032	−0.239589	−0.119794	−	0.992799i	$-0.538224\pi$
$59$	−1.84032	−0.239589	−0.119794	+	0.992799i	$0.538224\pi$
$60$	0	0
$61$	−5.30011	−0.678609	−0.339304	−	0.940677i	$-0.610192\pi$
$61$	−5.30011	−0.678609	−0.339304	+	0.940677i	$0.610192\pi$
$62$	0	0
$63$	−5.23349	−0.659357
$64$	0	0
$65$	0.991251	0.122950
$66$	0	0
$67$	9.05832	1.10665	0.553325	−	0.832966i	$-0.313359\pi$
$67$	9.05832	1.10665	0.553325	+	0.832966i	$0.313359\pi$
$68$	0	0
$69$	−5.88364	−0.708307
$70$	0	0
$71$	1.03723	0.123096	0.0615480	−	0.998104i	$-0.480396\pi$
$71$	1.03723	0.123096	0.0615480	+	0.998104i	$0.480396\pi$
$72$	0	0
$73$	15.0142	1.75728	0.878640	−	0.477484i	$-0.158451\pi$
$73$	15.0142	1.75728	0.878640	+	0.477484i	$0.158451\pi$
$74$	0	0
$75$	−2.61261	−0.301679
$76$	0	0
$77$	4.39056	0.500351
$78$	0	0
$79$	−14.6718	−1.65070	−0.825352	−	0.564619i	$-0.809023\pi$
$79$	−14.6718	−1.65070	−0.825352	+	0.564619i	$0.809023\pi$
$80$	0	0
$81$	−5.84088	−0.648986
$82$	0	0
$83$	1.71945	0.188734	0.0943668	−	0.995537i	$-0.469917\pi$
$83$	1.71945	0.188734	0.0943668	+	0.995537i	$0.469917\pi$
$84$	0	0
$85$	6.53588	0.708916
$86$	0	0
$87$	11.5269	1.23581
$88$	0	0
$89$	13.6317	1.44496	0.722481	−	0.691391i	$-0.243002\pi$
$89$	13.6317	1.44496	0.722481	+	0.691391i	$0.243002\pi$
$90$	0	0
$91$	1.35600	0.142147
$92$	0	0
$93$	−13.9970	−1.45142
$94$	0	0
$95$	−2.43792	−0.250126
$96$	0	0
$97$	3.23219	0.328179	0.164090	−	0.986445i	$-0.447531\pi$
$97$	3.23219	0.328179	0.164090	+	0.986445i	$0.447531\pi$
$98$	0	0
$99$	−12.2790	−1.23409
$100$	0	0
$101$	−8.92963	−0.888531	−0.444266	−	0.895895i	$-0.646535\pi$
$101$	−8.92963	−0.888531	−0.444266	+	0.895895i	$0.646535\pi$
$102$	0	0
$103$	16.8692	1.66217	0.831087	−	0.556142i	$-0.187719\pi$
$103$	16.8692	1.66217	0.831087	+	0.556142i	$0.187719\pi$
$104$	0	0
$105$	−3.57396	−0.348783
$106$	0	0
$107$	7.06973	0.683456	0.341728	−	0.939799i	$-0.388988\pi$
$107$	7.06973	0.683456	0.341728	+	0.939799i	$0.388988\pi$
$108$	0	0
$109$	−2.95797	−0.283322	−0.141661	−	0.989915i	$-0.545244\pi$
$109$	−2.95797	−0.283322	−0.141661	+	0.989915i	$0.545244\pi$
$110$	0	0
$111$	−29.8593	−2.83412
$112$	0	0
$113$	4.73283	0.445227	0.222614	−	0.974907i	$-0.428541\pi$
$113$	4.73283	0.445227	0.222614	+	0.974907i	$0.428541\pi$
$114$	0	0
$115$	−2.25201	−0.210001
$116$	0	0
$117$	−3.79228	−0.350597
$118$	0	0
$119$	8.94084	0.819606
$120$	0	0
$121$	−0.698708	−0.0635189
$122$	0	0
$123$	12.2078	1.10074
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−2.56666	−0.227754	−0.113877	−	0.993495i	$-0.536327\pi$
$127$	−2.56666	−0.227754	−0.113877	+	0.993495i	$0.536327\pi$
$128$	0	0
$129$	5.72790	0.504314
$130$	0	0
$131$	−5.32207	−0.464992	−0.232496	−	0.972597i	$-0.574689\pi$
$131$	−5.32207	−0.464992	−0.232496	+	0.972597i	$0.574689\pi$
$132$	0	0
$133$	−3.33499	−0.289180
$134$	0	0
$135$	2.15737	0.185677
$136$	0	0
$137$	−10.5973	−0.905386	−0.452693	−	0.891666i	$-0.649537\pi$
$137$	−10.5973	−0.905386	−0.452693	+	0.891666i	$0.649537\pi$
$138$	0	0
$139$	2.60640	0.221072	0.110536	−	0.993872i	$-0.464743\pi$
$139$	2.60640	0.221072	0.110536	+	0.993872i	$0.464743\pi$
$140$	0	0
$141$	1.02573	0.0863818
$142$	0	0
$143$	3.18148	0.266049
$144$	0	0
$145$	4.41202	0.366398
$146$	0	0
$147$	13.3993	1.10515
$148$	0	0
$149$	−10.1387	−0.830597	−0.415298	−	0.909685i	$-0.636323\pi$
$149$	−10.1387	−0.830597	−0.415298	+	0.909685i	$0.636323\pi$
$150$	0	0
$151$	11.6728	0.949922	0.474961	−	0.880007i	$-0.342462\pi$
$151$	11.6728	0.949922	0.474961	+	0.880007i	$0.342462\pi$
$152$	0	0
$153$	−25.0047	−2.02151
$154$	0	0
$155$	−5.35748	−0.430323
$156$	0	0
$157$	17.1147	1.36590	0.682951	−	0.730465i	$-0.260696\pi$
$157$	17.1147	1.36590	0.682951	+	0.730465i	$0.260696\pi$
$158$	0	0
$159$	−24.2926	−1.92653
$160$	0	0
$161$	−3.08067	−0.242791
$162$	0	0
$163$	12.1992	0.955515	0.477757	−	0.878492i	$-0.341450\pi$
$163$	12.1992	0.955515	0.477757	+	0.878492i	$0.341450\pi$
$164$	0	0
$165$	−8.38535	−0.652798
$166$	0	0
$167$	11.3965	0.881885	0.440942	−	0.897535i	$-0.354644\pi$
$167$	11.3965	0.881885	0.440942	+	0.897535i	$0.354644\pi$
$168$	0	0
$169$	−12.0174	−0.924417
$170$	0	0
$171$	9.32689	0.713245
$172$	0	0
$173$	−16.5170	−1.25577	−0.627883	−	0.778308i	$-0.716078\pi$
$173$	−16.5170	−1.25577	−0.627883	+	0.778308i	$0.716078\pi$
$174$	0	0
$175$	−1.36796	−0.103408
$176$	0	0
$177$	4.80804	0.361394
$178$	0	0
$179$	−21.1787	−1.58297	−0.791486	−	0.611188i	$-0.790692\pi$
$179$	−21.1787	−1.58297	−0.791486	+	0.611188i	$0.790692\pi$
$180$	0	0
$181$	17.7398	1.31859	0.659296	−	0.751883i	$-0.270854\pi$
$181$	17.7398	1.31859	0.659296	+	0.751883i	$0.270854\pi$
$182$	0	0
$183$	13.8471	1.02361
$184$	0	0
$185$	−11.4289	−0.840268
$186$	0	0
$187$	20.9773	1.53401
$188$	0	0
$189$	2.95121	0.214669
$190$	0	0
$191$	8.75806	0.633711	0.316855	−	0.948474i	$-0.397373\pi$
$191$	8.75806	0.633711	0.316855	+	0.948474i	$0.397373\pi$
$192$	0	0
$193$	−19.2440	−1.38521	−0.692606	−	0.721316i	$-0.743538\pi$
$193$	−19.2440	−1.38521	−0.692606	+	0.721316i	$0.743538\pi$
$194$	0	0
$195$	−2.58976	−0.185456
$196$	0	0
$197$	−10.4914	−0.747480	−0.373740	−	0.927533i	$-0.621925\pi$
$197$	−10.4914	−0.747480	−0.373740	+	0.927533i	$0.621925\pi$
$198$	0	0
$199$	21.6944	1.53787	0.768936	−	0.639325i	$-0.220786\pi$
$199$	21.6944	1.53787	0.768936	+	0.639325i	$0.220786\pi$
$200$	0	0
$201$	−23.6659	−1.66926
$202$	0	0
$203$	6.03548	0.423607
$204$	0	0
$205$	4.67265	0.326352
$206$	0	0
$207$	8.61565	0.598829
$208$	0	0
$209$	−7.82467	−0.541244
$210$	0	0
$211$	28.2075	1.94188	0.970942	−	0.239317i	$-0.0769234\pi$
$211$	28.2075	1.94188	0.970942	+	0.239317i	$0.0769234\pi$
$212$	0	0
$213$	−2.70987	−0.185677
$214$	0	0
$215$	2.19240	0.149521
$216$	0	0
$217$	−7.32884	−0.497514
$218$	0	0
$219$	−39.2263	−2.65067
$220$	0	0
$221$	6.47870	0.435804
$222$	0	0
$223$	−28.7403	−1.92459	−0.962297	−	0.272000i	$-0.912315\pi$
$223$	−28.7403	−1.92459	−0.962297	+	0.272000i	$0.912315\pi$
$224$	0	0
$225$	3.82575	0.255050
$226$	0	0
$227$	26.8221	1.78025	0.890123	−	0.455721i	$-0.150618\pi$
$227$	26.8221	1.78025	0.890123	+	0.455721i	$0.150618\pi$
$228$	0	0
$229$	25.8803	1.71022	0.855109	−	0.518449i	$-0.173490\pi$
$229$	25.8803	1.71022	0.855109	+	0.518449i	$0.173490\pi$
$230$	0	0
$231$	−11.4708	−0.754726
$232$	0	0
$233$	−17.8218	−1.16755	−0.583774	−	0.811916i	$-0.698425\pi$
$233$	−17.8218	−1.16755	−0.583774	+	0.811916i	$0.698425\pi$
$234$	0	0
$235$	0.392606	0.0256108
$236$	0	0
$237$	38.3317	2.48991
$238$	0	0
$239$	20.9133	1.35277	0.676385	−	0.736548i	$-0.263545\pi$
$239$	20.9133	1.35277	0.676385	+	0.736548i	$0.263545\pi$
$240$	0	0
$241$	2.51678	0.162120	0.0810600	−	0.996709i	$-0.474169\pi$
$241$	2.51678	0.162120	0.0810600	+	0.996709i	$0.474169\pi$
$242$	0	0
$243$	21.7321	1.39411
$244$	0	0
$245$	5.12868	0.327659
$246$	0	0
$247$	−2.41659	−0.153764
$248$	0	0
$249$	−4.49225	−0.284685
$250$	0	0
$251$	−15.9940	−1.00953	−0.504767	−	0.863255i	$-0.668422\pi$
$251$	−15.9940	−1.00953	−0.504767	+	0.863255i	$0.668422\pi$
$252$	0	0
$253$	−7.22798	−0.454419
$254$	0	0
$255$	−17.0757	−1.06932
$256$	0	0
$257$	4.01518	0.250460	0.125230	−	0.992128i	$-0.460033\pi$
$257$	4.01518	0.250460	0.125230	+	0.992128i	$0.460033\pi$
$258$	0	0
$259$	−15.6343	−0.971468
$260$	0	0
$261$	−16.8793	−1.04480
$262$	0	0
$263$	7.13676	0.440071	0.220036	−	0.975492i	$-0.429383\pi$
$263$	7.13676	0.440071	0.220036	+	0.975492i	$0.429383\pi$
$264$	0	0
$265$	−9.29821	−0.571184
$266$	0	0
$267$	−35.6145	−2.17957
$268$	0	0
$269$	3.46899	0.211508	0.105754	−	0.994392i	$-0.466274\pi$
$269$	3.46899	0.211508	0.105754	+	0.994392i	$0.466274\pi$
$270$	0	0
$271$	−4.79503	−0.291277	−0.145639	−	0.989338i	$-0.546524\pi$
$271$	−4.79503	−0.291277	−0.145639	+	0.989338i	$0.546524\pi$
$272$	0	0
$273$	−3.54269	−0.214413
$274$	0	0
$275$	−3.20956	−0.193544
$276$	0	0
$277$	−12.9177	−0.776147	−0.388074	−	0.921628i	$-0.626859\pi$
$277$	−12.9177	−0.776147	−0.388074	+	0.921628i	$0.626859\pi$
$278$	0	0
$279$	20.4964	1.22709
$280$	0	0
$281$	−28.6750	−1.71061	−0.855304	−	0.518127i	$-0.826630\pi$
$281$	−28.6750	−1.71061	−0.855304	+	0.518127i	$0.826630\pi$
$282$	0	0
$283$	5.74356	0.341419	0.170710	−	0.985321i	$-0.445394\pi$
$283$	5.74356	0.341419	0.170710	+	0.985321i	$0.445394\pi$
$284$	0	0
$285$	6.36935	0.377288
$286$	0	0
$287$	6.39201	0.377308
$288$	0	0
$289$	25.7177	1.51281
$290$	0	0
$291$	−8.44447	−0.495024
$292$	0	0
$293$	−11.6349	−0.679721	−0.339860	−	0.940476i	$-0.610380\pi$
$293$	−11.6349	−0.679721	−0.339860	+	0.940476i	$0.610380\pi$
$294$	0	0
$295$	1.84032	0.107147
$296$	0	0
$297$	6.92422	0.401784
$298$	0	0
$299$	−2.23231	−0.129098
$300$	0	0
$301$	2.99913	0.172867
$302$	0	0
$303$	23.3297	1.34026
$304$	0	0
$305$	5.30011	0.303483
$306$	0	0
$307$	11.7444	0.670290	0.335145	−	0.942167i	$-0.391215\pi$
$307$	11.7444	0.670290	0.335145	+	0.942167i	$0.391215\pi$
$308$	0	0
$309$	−44.0728	−2.50721
$310$	0	0
$311$	−26.1365	−1.48206	−0.741031	−	0.671470i	$-0.765663\pi$
$311$	−26.1365	−1.48206	−0.741031	+	0.671470i	$0.765663\pi$
$312$	0	0
$313$	−11.4762	−0.648675	−0.324337	−	0.945941i	$-0.605141\pi$
$313$	−11.4762	−0.648675	−0.324337	+	0.945941i	$0.605141\pi$
$314$	0	0
$315$	5.23349	0.294874
$316$	0	0
$317$	−13.6955	−0.769213	−0.384607	−	0.923081i	$-0.625663\pi$
$317$	−13.6955	−0.769213	−0.384607	+	0.923081i	$0.625663\pi$
$318$	0	0
$319$	14.1606	0.792844
$320$	0	0
$321$	−18.4705	−1.03092
$322$	0	0
$323$	−15.9340	−0.886590
$324$	0	0
$325$	−0.991251	−0.0549847
$326$	0	0
$327$	7.72802	0.427361
$328$	0	0
$329$	0.537070	0.0296096
$330$	0	0
$331$	−29.3218	−1.61167	−0.805837	−	0.592138i	$-0.798284\pi$
$331$	−29.3218	−1.61167	−0.805837	+	0.592138i	$0.798284\pi$
$332$	0	0
$333$	43.7241	2.39606
$334$	0	0
$335$	−9.05832	−0.494909
$336$	0	0
$337$	−13.0685	−0.711887	−0.355943	−	0.934507i	$-0.615840\pi$
$337$	−13.0685	−0.711887	−0.355943	+	0.934507i	$0.615840\pi$
$338$	0	0
$339$	−12.3651	−0.671578
$340$	0	0
$341$	−17.1952	−0.931171
$342$	0	0
$343$	16.5916	0.895861
$344$	0	0
$345$	5.88364	0.316765
$346$	0	0
$347$	−0.973751	−0.0522737	−0.0261368	−	0.999658i	$-0.508321\pi$
$347$	−0.973751	−0.0522737	−0.0261368	+	0.999658i	$0.508321\pi$
$348$	0	0
$349$	5.23236	0.280082	0.140041	−	0.990146i	$-0.455277\pi$
$349$	5.23236	0.280082	0.140041	+	0.990146i	$0.455277\pi$
$350$	0	0
$351$	2.13850	0.114145
$352$	0	0
$353$	9.74109	0.518466	0.259233	−	0.965815i	$-0.416530\pi$
$353$	9.74109	0.518466	0.259233	+	0.965815i	$0.416530\pi$
$354$	0	0
$355$	−1.03723	−0.0550502
$356$	0	0
$357$	−23.3590	−1.23629
$358$	0	0
$359$	25.4789	1.34473	0.672363	−	0.740222i	$-0.265279\pi$
$359$	25.4789	1.34473	0.672363	+	0.740222i	$0.265279\pi$
$360$	0	0
$361$	−13.0565	−0.687186
$362$	0	0
$363$	1.82545	0.0958115
$364$	0	0
$365$	−15.0142	−0.785880
$366$	0	0
$367$	9.77869	0.510443	0.255222	−	0.966883i	$-0.417852\pi$
$367$	9.77869	0.510443	0.255222	+	0.966883i	$0.417852\pi$
$368$	0	0
$369$	−17.8764	−0.930608
$370$	0	0
$371$	−12.7196	−0.660369
$372$	0	0
$373$	−25.0304	−1.29602	−0.648011	−	0.761631i	$-0.724399\pi$
$373$	−25.0304	−1.29602	−0.648011	+	0.761631i	$0.724399\pi$
$374$	0	0
$375$	2.61261	0.134915
$376$	0	0
$377$	4.37342	0.225242
$378$	0	0
$379$	−16.3935	−0.842076	−0.421038	−	0.907043i	$-0.638334\pi$
$379$	−16.3935	−0.842076	−0.421038	+	0.907043i	$0.638334\pi$
$380$	0	0
$381$	6.70569	0.343543
$382$	0	0
$383$	−15.1666	−0.774976	−0.387488	−	0.921875i	$-0.626657\pi$
$383$	−15.1666	−0.774976	−0.387488	+	0.921875i	$0.626657\pi$
$384$	0	0
$385$	−4.39056	−0.223764
$386$	0	0
$387$	−8.38759	−0.426365
$388$	0	0
$389$	34.9845	1.77378	0.886891	−	0.461978i	$-0.152860\pi$
$389$	34.9845	1.77378	0.886891	+	0.461978i	$0.152860\pi$
$390$	0	0
$391$	−14.7189	−0.744366
$392$	0	0
$393$	13.9045	0.701391
$394$	0	0
$395$	14.6718	0.738217
$396$	0	0
$397$	−11.8801	−0.596244	−0.298122	−	0.954528i	$-0.596360\pi$
$397$	−11.8801	−0.596244	−0.298122	+	0.954528i	$0.596360\pi$
$398$	0	0
$399$	8.71304	0.436197
$400$	0	0
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	0	0
$403$	−5.31061	−0.264540
$404$	0	0
$405$	5.84088	0.290235
$406$	0	0
$407$	−36.6817	−1.81825
$408$	0	0
$409$	−34.1894	−1.69056	−0.845278	−	0.534326i	$-0.820565\pi$
$409$	−34.1894	−1.69056	−0.845278	+	0.534326i	$0.820565\pi$
$410$	0	0
$411$	27.6866	1.36568
$412$	0	0
$413$	2.51749	0.123877
$414$	0	0
$415$	−1.71945	−0.0844043
$416$	0	0
$417$	−6.80952	−0.333464
$418$	0	0
$419$	39.2267	1.91635	0.958174	−	0.286185i	$-0.0923872\pi$
$419$	39.2267	1.91635	0.958174	+	0.286185i	$0.0923872\pi$
$420$	0	0
$421$	−4.98906	−0.243152	−0.121576	−	0.992582i	$-0.538795\pi$
$421$	−4.98906	−0.243152	−0.121576	+	0.992582i	$0.538795\pi$
$422$	0	0
$423$	−1.50201	−0.0730303
$424$	0	0
$425$	−6.53588	−0.317037
$426$	0	0
$427$	7.25035	0.350869
$428$	0	0
$429$	−8.31199	−0.401307
$430$	0	0
$431$	−3.28749	−0.158353	−0.0791764	−	0.996861i	$-0.525229\pi$
$431$	−3.28749	−0.158353	−0.0791764	+	0.996861i	$0.525229\pi$
$432$	0	0
$433$	−38.6793	−1.85881	−0.929404	−	0.369065i	$-0.879678\pi$
$433$	−38.6793	−1.85881	−0.929404	+	0.369065i	$0.879678\pi$
$434$	0	0
$435$	−11.5269	−0.552672
$436$	0	0
$437$	5.49024	0.262634
$438$	0	0
$439$	−1.64495	−0.0785091	−0.0392545	−	0.999229i	$-0.512498\pi$
$439$	−1.64495	−0.0785091	−0.0392545	+	0.999229i	$0.512498\pi$
$440$	0	0
$441$	−19.6210	−0.934336
$442$	0	0
$443$	−2.62465	−0.124701	−0.0623504	−	0.998054i	$-0.519860\pi$
$443$	−2.62465	−0.124701	−0.0623504	+	0.998054i	$0.519860\pi$
$444$	0	0
$445$	−13.6317	−0.646206
$446$	0	0
$447$	26.4886	1.25287
$448$	0	0
$449$	−34.3438	−1.62079	−0.810393	−	0.585887i	$-0.800746\pi$
$449$	−34.3438	−1.62079	−0.810393	+	0.585887i	$0.800746\pi$
$450$	0	0
$451$	14.9972	0.706189
$452$	0	0
$453$	−30.4966	−1.43286
$454$	0	0
$455$	−1.35600	−0.0635700
$456$	0	0
$457$	−28.4786	−1.33217	−0.666087	−	0.745874i	$-0.732032\pi$
$457$	−28.4786	−1.33217	−0.666087	+	0.745874i	$0.732032\pi$
$458$	0	0
$459$	14.1003	0.658147
$460$	0	0
$461$	−35.0769	−1.63369	−0.816847	−	0.576855i	$-0.804280\pi$
$461$	−35.0769	−1.63369	−0.816847	+	0.576855i	$0.804280\pi$
$462$	0	0
$463$	4.28748	0.199256	0.0996281	−	0.995025i	$-0.468235\pi$
$463$	4.28748	0.199256	0.0996281	+	0.995025i	$0.468235\pi$
$464$	0	0
$465$	13.9970	0.649097
$466$	0	0
$467$	2.88913	0.133693	0.0668465	−	0.997763i	$-0.478706\pi$
$467$	2.88913	0.133693	0.0668465	+	0.997763i	$0.478706\pi$
$468$	0	0
$469$	−12.3914	−0.572184
$470$	0	0
$471$	−44.7141	−2.06032
$472$	0	0
$473$	7.03665	0.323546
$474$	0	0
$475$	2.43792	0.111860
$476$	0	0
$477$	35.5726	1.62876
$478$	0	0
$479$	−33.5805	−1.53433	−0.767165	−	0.641450i	$-0.778333\pi$
$479$	−33.5805	−1.53433	−0.767165	+	0.641450i	$0.778333\pi$
$480$	0	0
$481$	−11.3289	−0.516553
$482$	0	0
$483$	8.04861	0.366224
$484$	0	0
$485$	−3.23219	−0.146766
$486$	0	0
$487$	−10.1239	−0.458758	−0.229379	−	0.973337i	$-0.573670\pi$
$487$	−10.1239	−0.458758	−0.229379	+	0.973337i	$0.573670\pi$
$488$	0	0
$489$	−31.8718	−1.44129
$490$	0	0
$491$	−6.81466	−0.307541	−0.153771	−	0.988107i	$-0.549142\pi$
$491$	−6.81466	−0.307541	−0.153771	+	0.988107i	$0.549142\pi$
$492$	0	0
$493$	28.8364	1.29873
$494$	0	0
$495$	12.2790	0.551900
$496$	0	0
$497$	−1.41889	−0.0636457
$498$	0	0
$499$	−28.4108	−1.27184	−0.635921	−	0.771754i	$-0.719380\pi$
$499$	−28.4108	−1.27184	−0.635921	+	0.771754i	$0.719380\pi$
$500$	0	0
$501$	−29.7746	−1.33023
$502$	0	0
$503$	21.4288	0.955464	0.477732	−	0.878506i	$-0.341459\pi$
$503$	21.4288	0.955464	0.477732	+	0.878506i	$0.341459\pi$
$504$	0	0
$505$	8.92963	0.397363
$506$	0	0
$507$	31.3969	1.39438
$508$	0	0
$509$	13.9952	0.620325	0.310163	−	0.950684i	$-0.399617\pi$
$509$	13.9952	0.620325	0.310163	+	0.950684i	$0.399617\pi$
$510$	0	0
$511$	−20.5389	−0.908587
$512$	0	0
$513$	−5.25951	−0.232213
$514$	0	0
$515$	−16.8692	−0.743347
$516$	0	0
$517$	1.26009	0.0554188
$518$	0	0
$519$	43.1526	1.89419
$520$	0	0
$521$	15.4782	0.678113	0.339056	−	0.940766i	$-0.389892\pi$
$521$	15.4782	0.678113	0.339056	+	0.940766i	$0.389892\pi$
$522$	0	0
$523$	−13.4262	−0.587085	−0.293542	−	0.955946i	$-0.594834\pi$
$523$	−13.4262	−0.587085	−0.293542	+	0.955946i	$0.594834\pi$
$524$	0	0
$525$	3.57396	0.155980
$526$	0	0
$527$	−35.0158	−1.52531
$528$	0	0
$529$	−17.9284	−0.779497
$530$	0	0
$531$	−7.04060	−0.305536
$532$	0	0
$533$	4.63177	0.200624
$534$	0	0
$535$	−7.06973	−0.305651
$536$	0	0
$537$	55.3318	2.38774
$538$	0	0
$539$	16.4608	0.709017
$540$	0	0
$541$	33.2993	1.43165	0.715824	−	0.698280i	$-0.246051\pi$
$541$	33.2993	1.43165	0.715824	+	0.698280i	$0.246051\pi$
$542$	0	0
$543$	−46.3474	−1.98896
$544$	0	0
$545$	2.95797	0.126705
$546$	0	0
$547$	−43.1252	−1.84390	−0.921950	−	0.387308i	$-0.873405\pi$
$547$	−43.1252	−1.84390	−0.921950	+	0.387308i	$0.873405\pi$
$548$	0	0
$549$	−20.2769	−0.865397
$550$	0	0
$551$	−10.7562	−0.458228
$552$	0	0
$553$	20.0704	0.853482
$554$	0	0
$555$	29.8593	1.26746
$556$	0	0
$557$	12.7461	0.540070	0.270035	−	0.962851i	$-0.412965\pi$
$557$	12.7461	0.540070	0.270035	+	0.962851i	$0.412965\pi$
$558$	0	0
$559$	2.17322	0.0919175
$560$	0	0
$561$	−54.8056	−2.31390
$562$	0	0
$563$	7.69223	0.324189	0.162094	−	0.986775i	$-0.448175\pi$
$563$	7.69223	0.324189	0.162094	+	0.986775i	$0.448175\pi$
$564$	0	0
$565$	−4.73283	−0.199112
$566$	0	0
$567$	7.99010	0.335553
$568$	0	0
$569$	14.7351	0.617726	0.308863	−	0.951107i	$-0.400052\pi$
$569$	14.7351	0.617726	0.308863	+	0.951107i	$0.400052\pi$
$570$	0	0
$571$	34.0058	1.42310	0.711549	−	0.702637i	$-0.247994\pi$
$571$	34.0058	1.42310	0.711549	+	0.702637i	$0.247994\pi$
$572$	0	0
$573$	−22.8814	−0.955885
$574$	0	0
$575$	2.25201	0.0939155
$576$	0	0
$577$	−0.842175	−0.0350602	−0.0175301	−	0.999846i	$-0.505580\pi$
$577$	−0.842175	−0.0350602	−0.0175301	+	0.999846i	$0.505580\pi$
$578$	0	0
$579$	50.2771	2.08945
$580$	0	0
$581$	−2.35214	−0.0975831
$582$	0	0
$583$	−29.8432	−1.23598
$584$	0	0
$585$	3.79228	0.156792
$586$	0	0
$587$	−17.1065	−0.706063	−0.353031	−	0.935611i	$-0.614849\pi$
$587$	−17.1065	−0.706063	−0.353031	+	0.935611i	$0.614849\pi$
$588$	0	0
$589$	13.0611	0.538174
$590$	0	0
$591$	27.4099	1.12749
$592$	0	0
$593$	−33.8043	−1.38818	−0.694089	−	0.719890i	$-0.744192\pi$
$593$	−33.8043	−1.38818	−0.694089	+	0.719890i	$0.744192\pi$
$594$	0	0
$595$	−8.94084	−0.366539
$596$	0	0
$597$	−56.6790	−2.31972
$598$	0	0
$599$	40.3297	1.64783	0.823914	−	0.566715i	$-0.191786\pi$
$599$	40.3297	1.64783	0.823914	+	0.566715i	$0.191786\pi$
$600$	0	0
$601$	−25.4728	−1.03906	−0.519529	−	0.854453i	$-0.673893\pi$
$601$	−25.4728	−1.03906	−0.519529	+	0.854453i	$0.673893\pi$
$602$	0	0
$603$	34.6549	1.41126
$604$	0	0
$605$	0.698708	0.0284065
$606$	0	0
$607$	−34.7204	−1.40926	−0.704629	−	0.709576i	$-0.748887\pi$
$607$	−34.7204	−1.40926	−0.704629	+	0.709576i	$0.748887\pi$
$608$	0	0
$609$	−15.7684	−0.638967
$610$	0	0
$611$	0.389171	0.0157442
$612$	0	0
$613$	19.2832	0.778840	0.389420	−	0.921060i	$-0.372675\pi$
$613$	19.2832	0.778840	0.389420	+	0.921060i	$0.372675\pi$
$614$	0	0
$615$	−12.2078	−0.492267
$616$	0	0
$617$	−33.5601	−1.35108	−0.675539	−	0.737325i	$-0.736089\pi$
$617$	−33.5601	−1.35108	−0.675539	+	0.737325i	$0.736089\pi$
$618$	0	0
$619$	−6.70266	−0.269403	−0.134701	−	0.990886i	$-0.543007\pi$
$619$	−6.70266	−0.269403	−0.134701	+	0.990886i	$0.543007\pi$
$620$	0	0
$621$	−4.85843	−0.194962
$622$	0	0
$623$	−18.6477	−0.747105
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	20.4428	0.816408
$628$	0	0
$629$	−74.6978	−2.97840
$630$	0	0
$631$	−29.8511	−1.18835	−0.594177	−	0.804334i	$-0.702522\pi$
$631$	−29.8511	−1.18835	−0.594177	+	0.804334i	$0.702522\pi$
$632$	0	0
$633$	−73.6953	−2.92912
$634$	0	0
$635$	2.56666	0.101855
$636$	0	0
$637$	5.08381	0.201428
$638$	0	0
$639$	3.96817	0.156978
$640$	0	0
$641$	−27.7421	−1.09575	−0.547874	−	0.836561i	$-0.684563\pi$
$641$	−27.7421	−1.09575	−0.547874	+	0.836561i	$0.684563\pi$
$642$	0	0
$643$	48.0533	1.89504	0.947519	−	0.319699i	$-0.103582\pi$
$643$	48.0533	1.89504	0.947519	+	0.319699i	$0.103582\pi$
$644$	0	0
$645$	−5.72790	−0.225536
$646$	0	0
$647$	−18.5956	−0.731070	−0.365535	−	0.930798i	$-0.619114\pi$
$647$	−18.5956	−0.731070	−0.365535	+	0.930798i	$0.619114\pi$
$648$	0	0
$649$	5.90661	0.231855
$650$	0	0
$651$	19.1474	0.750447
$652$	0	0
$653$	3.53683	0.138407	0.0692034	−	0.997603i	$-0.477954\pi$
$653$	3.53683	0.138407	0.0692034	+	0.997603i	$0.477954\pi$
$654$	0	0
$655$	5.32207	0.207951
$656$	0	0
$657$	57.4407	2.24097
$658$	0	0
$659$	29.4430	1.14694	0.573469	−	0.819228i	$-0.305597\pi$
$659$	29.4430	1.14694	0.573469	+	0.819228i	$0.305597\pi$
$660$	0	0
$661$	3.94650	0.153501	0.0767506	−	0.997050i	$-0.475545\pi$
$661$	3.94650	0.153501	0.0767506	+	0.997050i	$0.475545\pi$
$662$	0	0
$663$	−16.9263	−0.657365
$664$	0	0
$665$	3.33499	0.129325
$666$	0	0
$667$	−9.93593	−0.384721
$668$	0	0
$669$	75.0874	2.90305
$670$	0	0
$671$	17.0110	0.656703
$672$	0	0
$673$	13.3389	0.514177	0.257088	−	0.966388i	$-0.417237\pi$
$673$	13.3389	0.514177	0.257088	+	0.966388i	$0.417237\pi$
$674$	0	0
$675$	−2.15737	−0.0830373
$676$	0	0
$677$	−3.07411	−0.118148	−0.0590739	−	0.998254i	$-0.518815\pi$
$677$	−3.07411	−0.118148	−0.0590739	+	0.998254i	$0.518815\pi$
$678$	0	0
$679$	−4.42152	−0.169682
$680$	0	0
$681$	−70.0758	−2.68531
$682$	0	0
$683$	19.5880	0.749516	0.374758	−	0.927123i	$-0.377726\pi$
$683$	19.5880	0.749516	0.374758	+	0.927123i	$0.377726\pi$
$684$	0	0
$685$	10.5973	0.404901
$686$	0	0
$687$	−67.6152	−2.57968
$688$	0	0
$689$	−9.21686	−0.351134
$690$	0	0
$691$	−3.34403	−0.127213	−0.0636064	−	0.997975i	$-0.520260\pi$
$691$	−3.34403	−0.127213	−0.0636064	+	0.997975i	$0.520260\pi$
$692$	0	0
$693$	16.7972	0.638073
$694$	0	0
$695$	−2.60640	−0.0988664
$696$	0	0
$697$	30.5399	1.15678
$698$	0	0
$699$	46.5616	1.76112
$700$	0	0
$701$	−24.1053	−0.910447	−0.455223	−	0.890377i	$-0.650441\pi$
$701$	−24.1053	−0.910447	−0.455223	+	0.890377i	$0.650441\pi$
$702$	0	0
$703$	27.8627	1.05086
$704$	0	0
$705$	−1.02573	−0.0386311
$706$	0	0
$707$	12.2154	0.459408
$708$	0	0
$709$	11.4991	0.431860	0.215930	−	0.976409i	$-0.430722\pi$
$709$	11.4991	0.431860	0.215930	+	0.976409i	$0.430722\pi$
$710$	0	0
$711$	−56.1306	−2.10506
$712$	0	0
$713$	12.0651	0.451842
$714$	0	0
$715$	−3.18148	−0.118981
$716$	0	0
$717$	−54.6384	−2.04051
$718$	0	0
$719$	29.8653	1.11379	0.556894	−	0.830584i	$-0.311993\pi$
$719$	29.8653	1.11379	0.556894	+	0.830584i	$0.311993\pi$
$720$	0	0
$721$	−23.0765	−0.859413
$722$	0	0
$723$	−6.57538	−0.244541
$724$	0	0
$725$	−4.41202	−0.163858
$726$	0	0
$727$	−32.4968	−1.20524	−0.602620	−	0.798028i	$-0.705877\pi$
$727$	−32.4968	−1.20524	−0.602620	+	0.798028i	$0.705877\pi$
$728$	0	0
$729$	−39.2549	−1.45388
$730$	0	0
$731$	14.3293	0.529988
$732$	0	0
$733$	47.4215	1.75155	0.875777	−	0.482716i	$-0.160350\pi$
$733$	47.4215	1.75155	0.875777	+	0.482716i	$0.160350\pi$
$734$	0	0
$735$	−13.3993	−0.494239
$736$	0	0
$737$	−29.0732	−1.07093
$738$	0	0
$739$	−50.7770	−1.86786	−0.933932	−	0.357451i	$-0.883646\pi$
$739$	−50.7770	−1.86786	−0.933932	+	0.357451i	$0.883646\pi$
$740$	0	0
$741$	6.31363	0.231937
$742$	0	0
$743$	−37.4366	−1.37342	−0.686709	−	0.726933i	$-0.740945\pi$
$743$	−37.4366	−1.37342	−0.686709	+	0.726933i	$0.740945\pi$
$744$	0	0
$745$	10.1387	0.371454
$746$	0	0
$747$	6.57817	0.240683
$748$	0	0
$749$	−9.67113	−0.353375
$750$	0	0
$751$	−35.4765	−1.29456	−0.647279	−	0.762254i	$-0.724093\pi$
$751$	−35.4765	−1.29456	−0.647279	+	0.762254i	$0.724093\pi$
$752$	0	0
$753$	41.7863	1.52278
$754$	0	0
$755$	−11.6728	−0.424818
$756$	0	0
$757$	21.6604	0.787260	0.393630	−	0.919269i	$-0.371219\pi$
$757$	21.6604	0.787260	0.393630	+	0.919269i	$0.371219\pi$
$758$	0	0
$759$	18.8839	0.685443
$760$	0	0
$761$	−33.0889	−1.19947	−0.599736	−	0.800198i	$-0.704728\pi$
$761$	−33.0889	−1.19947	−0.599736	+	0.800198i	$0.704728\pi$
$762$	0	0
$763$	4.04639	0.146489
$764$	0	0
$765$	25.0047	0.904045
$766$	0	0
$767$	1.82422	0.0658687
$768$	0	0
$769$	18.9234	0.682395	0.341197	−	0.939992i	$-0.389168\pi$
$769$	18.9234	0.682395	0.341197	+	0.939992i	$0.389168\pi$
$770$	0	0
$771$	−10.4901	−0.377792
$772$	0	0
$773$	−6.25909	−0.225124	−0.112562	−	0.993645i	$-0.535906\pi$
$773$	−6.25909	−0.225124	−0.112562	+	0.993645i	$0.535906\pi$
$774$	0	0
$775$	5.35748	0.192446
$776$	0	0
$777$	40.8464	1.46536
$778$	0	0
$779$	−11.3916	−0.408145
$780$	0	0
$781$	−3.32904	−0.119122
$782$	0	0
$783$	9.51836	0.340159
$784$	0	0
$785$	−17.1147	−0.610850
$786$	0	0
$787$	−27.4763	−0.979423	−0.489711	−	0.871885i	$-0.662898\pi$
$787$	−27.4763	−0.979423	−0.489711	+	0.871885i	$0.662898\pi$
$788$	0	0
$789$	−18.6456	−0.663800
$790$	0	0
$791$	−6.47434	−0.230201
$792$	0	0
$793$	5.25374	0.186566
$794$	0	0
$795$	24.2926	0.861571
$796$	0	0
$797$	33.4892	1.18625	0.593123	−	0.805112i	$-0.297895\pi$
$797$	33.4892	1.18625	0.593123	+	0.805112i	$0.297895\pi$
$798$	0	0
$799$	2.56602	0.0907794
$800$	0	0
$801$	52.1516	1.84269
$802$	0	0
$803$	−48.1891	−1.70055
$804$	0	0
$805$	3.08067	0.108579
$806$	0	0
$807$	−9.06314	−0.319038
$808$	0	0
$809$	45.0476	1.58379	0.791895	−	0.610657i	$-0.209095\pi$
$809$	45.0476	1.58379	0.791895	+	0.610657i	$0.209095\pi$
$810$	0	0
$811$	−25.4986	−0.895378	−0.447689	−	0.894189i	$-0.647753\pi$
$811$	−25.4986	−0.895378	−0.447689	+	0.894189i	$0.647753\pi$
$812$	0	0
$813$	12.5276	0.439361
$814$	0	0
$815$	−12.1992	−0.427319
$816$	0	0
$817$	−5.34491	−0.186995
$818$	0	0
$819$	5.18770	0.181273
$820$	0	0
$821$	26.2944	0.917682	0.458841	−	0.888518i	$-0.348265\pi$
$821$	26.2944	0.917682	0.458841	+	0.888518i	$0.348265\pi$
$822$	0	0
$823$	23.8950	0.832927	0.416464	−	0.909152i	$-0.363269\pi$
$823$	23.8950	0.832927	0.416464	+	0.909152i	$0.363269\pi$
$824$	0	0
$825$	8.38535	0.291940
$826$	0	0
$827$	19.9050	0.692165	0.346083	−	0.938204i	$-0.387512\pi$
$827$	19.9050	0.692165	0.346083	+	0.938204i	$0.387512\pi$
$828$	0	0
$829$	−3.63035	−0.126087	−0.0630436	−	0.998011i	$-0.520081\pi$
$829$	−3.63035	−0.126087	−0.0630436	+	0.998011i	$0.520081\pi$
$830$	0	0
$831$	33.7489	1.17074
$832$	0	0
$833$	33.5204	1.16141
$834$	0	0
$835$	−11.3965	−0.394391
$836$	0	0
$837$	−11.5581	−0.399506
$838$	0	0
$839$	36.3796	1.25596	0.627981	−	0.778229i	$-0.283882\pi$
$839$	36.3796	1.25596	0.627981	+	0.778229i	$0.283882\pi$
$840$	0	0
$841$	−9.53410	−0.328762
$842$	0	0
$843$	74.9167	2.58027
$844$	0	0
$845$	12.0174	0.413412
$846$	0	0
$847$	0.955806	0.0328419
$848$	0	0
$849$	−15.0057	−0.514995
$850$	0	0
$851$	25.7380	0.882288
$852$	0	0
$853$	−1.90614	−0.0652649	−0.0326325	−	0.999467i	$-0.510389\pi$
$853$	−1.90614	−0.0652649	−0.0326325	+	0.999467i	$0.510389\pi$
$854$	0	0
$855$	−9.32689	−0.318973
$856$	0	0
$857$	15.5649	0.531687	0.265843	−	0.964016i	$-0.414350\pi$
$857$	15.5649	0.531687	0.265843	+	0.964016i	$0.414350\pi$
$858$	0	0
$859$	22.8591	0.779942	0.389971	−	0.920827i	$-0.372485\pi$
$859$	22.8591	0.779942	0.389971	+	0.920827i	$0.372485\pi$
$860$	0	0
$861$	−16.6999	−0.569130
$862$	0	0
$863$	14.3912	0.489884	0.244942	−	0.969538i	$-0.421231\pi$
$863$	14.3912	0.489884	0.244942	+	0.969538i	$0.421231\pi$
$864$	0	0
$865$	16.5170	0.561596
$866$	0	0
$867$	−67.1905	−2.28191
$868$	0	0
$869$	47.0900	1.59742
$870$	0	0
$871$	−8.97907	−0.304244
$872$	0	0
$873$	12.3656	0.418511
$874$	0	0
$875$	1.36796	0.0462456
$876$	0	0
$877$	−13.4726	−0.454938	−0.227469	−	0.973785i	$-0.573045\pi$
$877$	−13.4726	−0.454938	−0.227469	+	0.973785i	$0.573045\pi$
$878$	0	0
$879$	30.3976	1.02529
$880$	0	0
$881$	46.5168	1.56719	0.783595	−	0.621272i	$-0.213384\pi$
$881$	46.5168	1.56719	0.783595	+	0.621272i	$0.213384\pi$
$882$	0	0
$883$	19.8116	0.666714	0.333357	−	0.942801i	$-0.391818\pi$
$883$	19.8116	0.666714	0.333357	+	0.942801i	$0.391818\pi$
$884$	0	0
$885$	−4.80804	−0.161620
$886$	0	0
$887$	−14.2450	−0.478302	−0.239151	−	0.970982i	$-0.576869\pi$
$887$	−14.2450	−0.478302	−0.239151	+	0.970982i	$0.576869\pi$
$888$	0	0
$889$	3.51110	0.117758
$890$	0	0
$891$	18.7467	0.628037
$892$	0	0
$893$	−0.957142	−0.0320295
$894$	0	0
$895$	21.1787	0.707926
$896$	0	0
$897$	5.83217	0.194730
$898$	0	0
$899$	−23.6373	−0.788348
$900$	0	0
$901$	−60.7720	−2.02461
$902$	0	0
$903$	−7.83556	−0.260751
$904$	0	0
$905$	−17.7398	−0.589692
$906$	0	0
$907$	13.6509	0.453270	0.226635	−	0.973980i	$-0.427227\pi$
$907$	13.6509	0.453270	0.226635	+	0.973980i	$0.427227\pi$
$908$	0	0
$909$	−34.1626	−1.13310
$910$	0	0
$911$	−36.0471	−1.19429	−0.597146	−	0.802132i	$-0.703699\pi$
$911$	−36.0471	−1.19429	−0.597146	+	0.802132i	$0.703699\pi$
$912$	0	0
$913$	−5.51867	−0.182641
$914$	0	0
$915$	−13.8471	−0.457772
$916$	0	0
$917$	7.28040	0.240420
$918$	0	0
$919$	−39.4256	−1.30053	−0.650265	−	0.759708i	$-0.725342\pi$
$919$	−39.4256	−1.30053	−0.650265	+	0.759708i	$0.725342\pi$
$920$	0	0
$921$	−30.6836	−1.01106
$922$	0	0
$923$	−1.02815	−0.0338420
$924$	0	0
$925$	11.4289	0.375779
$926$	0	0
$927$	64.5375	2.11969
$928$	0	0
$929$	11.9514	0.392114	0.196057	−	0.980593i	$-0.437186\pi$
$929$	11.9514	0.392114	0.196057	+	0.980593i	$0.437186\pi$
$930$	0	0
$931$	−12.5033	−0.409780
$932$	0	0
$933$	68.2845	2.23553
$934$	0	0
$935$	−20.9773	−0.686031
$936$	0	0
$937$	28.1498	0.919614	0.459807	−	0.888019i	$-0.347919\pi$
$937$	28.1498	0.919614	0.459807	+	0.888019i	$0.347919\pi$
$938$	0	0
$939$	29.9830	0.978457
$940$	0	0
$941$	41.4984	1.35281	0.676405	−	0.736530i	$-0.263537\pi$
$941$	41.4984	1.35281	0.676405	+	0.736530i	$0.263537\pi$
$942$	0	0
$943$	−10.5229	−0.342672
$944$	0	0
$945$	−2.95121	−0.0960027
$946$	0	0
$947$	−54.5087	−1.77129	−0.885647	−	0.464358i	$-0.846285\pi$
$947$	−54.5087	−1.77129	−0.885647	+	0.464358i	$0.846285\pi$
$948$	0	0
$949$	−14.8829	−0.483118
$950$	0	0
$951$	35.7809	1.16028
$952$	0	0
$953$	52.2598	1.69286	0.846431	−	0.532499i	$-0.178747\pi$
$953$	52.2598	1.69286	0.846431	+	0.532499i	$0.178747\pi$
$954$	0	0
$955$	−8.75806	−0.283404
$956$	0	0
$957$	−36.9963	−1.19592
$958$	0	0
$959$	14.4967	0.468122
$960$	0	0
$961$	−2.29740	−0.0741095
$962$	0	0
$963$	27.0470	0.871578
$964$	0	0
$965$	19.2440	0.619486
$966$	0	0
$967$	40.4831	1.30185	0.650924	−	0.759143i	$-0.274382\pi$
$967$	40.4831	1.30185	0.650924	+	0.759143i	$0.274382\pi$
$968$	0	0
$969$	41.6293	1.33733
$970$	0	0
$971$	26.6327	0.854683	0.427342	−	0.904090i	$-0.359450\pi$
$971$	26.6327	0.854683	0.427342	+	0.904090i	$0.359450\pi$
$972$	0	0
$973$	−3.56546	−0.114303
$974$	0	0
$975$	2.58976	0.0829386
$976$	0	0
$977$	8.29124	0.265260	0.132630	−	0.991166i	$-0.457658\pi$
$977$	8.29124	0.265260	0.132630	+	0.991166i	$0.457658\pi$
$978$	0	0
$979$	−43.7519	−1.39832
$980$	0	0
$981$	−11.3164	−0.361306
$982$	0	0
$983$	−15.5643	−0.496423	−0.248211	−	0.968706i	$-0.579843\pi$
$983$	−15.5643	−0.496423	−0.248211	+	0.968706i	$0.579843\pi$
$984$	0	0
$985$	10.4914	0.334283
$986$	0	0
$987$	−1.40316	−0.0446630
$988$	0	0
$989$	−4.93732	−0.156998
$990$	0	0
$991$	8.40509	0.266996	0.133498	−	0.991049i	$-0.457379\pi$
$991$	8.40509	0.266996	0.133498	+	0.991049i	$0.457379\pi$
$992$	0	0
$993$	76.6066	2.43104
$994$	0	0
$995$	−21.6944	−0.687758
$996$	0	0
$997$	58.8897	1.86506	0.932528	−	0.361096i	$-0.117598\pi$
$997$	58.8897	1.86506	0.932528	+	0.361096i	$0.117598\pi$
$998$	0	0
$999$	−24.6564	−0.780093

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.c.1.2	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.c.1.2	✓	28	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:	\( N \)	\(=\)	\( 8020 = 2^{2} \cdot 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8020.a (trivial)

\(f(q)\)	\(=\)	\(q-2.61261 q^{3} -1.00000 q^{5} -1.36796 q^{7} +3.82575 q^{9} +O(q^{10})\)	\(q-2.61261 q^{3} -1.00000 q^{5} -1.36796 q^{7} +3.82575 q^{9} -3.20956 q^{11} -0.991251 q^{13} +2.61261 q^{15} -6.53588 q^{17} +2.43792 q^{19} +3.57396 q^{21} +2.25201 q^{23} +1.00000 q^{25} -2.15737 q^{27} -4.41202 q^{29} +5.35748 q^{31} +8.38535 q^{33} +1.36796 q^{35} +11.4289 q^{37} +2.58976 q^{39} -4.67265 q^{41} -2.19240 q^{43} -3.82575 q^{45} -0.392606 q^{47} -5.12868 q^{49} +17.0757 q^{51} +9.29821 q^{53} +3.20956 q^{55} -6.36935 q^{57} -1.84032 q^{59} -5.30011 q^{61} -5.23349 q^{63} +0.991251 q^{65} +9.05832 q^{67} -5.88364 q^{69} +1.03723 q^{71} +15.0142 q^{73} -2.61261 q^{75} +4.39056 q^{77} -14.6718 q^{79} -5.84088 q^{81} +1.71945 q^{83} +6.53588 q^{85} +11.5269 q^{87} +13.6317 q^{89} +1.35600 q^{91} -13.9970 q^{93} -2.43792 q^{95} +3.23219 q^{97} -12.2790 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9}+O(q^{10}) \) 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9	\( 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9} + 2 q^{11} + 3 q^{13} - 3 q^{15} - 10 q^{17} - 2 q^{19} - 12 q^{21} - 23 q^{23} + 28 q^{25} + 9 q^{27} - 37 q^{29} - 11 q^{31} + 2 q^{33} + 4 q^{35} - 3 q^{37} - 19 q^{39} - 30 q^{41} + 13 q^{43} - 17 q^{45} - 15 q^{47} + 12 q^{49} - 8 q^{51} - 35 q^{53} - 2 q^{55} - 22 q^{57} - q^{59} - 33 q^{61} - 20 q^{63} - 3 q^{65} + 19 q^{67} - 8 q^{69} - 31 q^{71} + 31 q^{73} + 3 q^{75} - 42 q^{77} - 29 q^{79} - 36 q^{81} + 14 q^{83} + 10 q^{85} - 32 q^{87} - 32 q^{89} - 7 q^{91} - 11 q^{93} + 2 q^{95} + 2 q^{97} - 39 q^{99}+O(q^{100}) \) 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9 + 2 * q^11 + 3 * q^13 - 3 * q^15 - 10 * q^17 - 2 * q^19 - 12 * q^21 - 23 * q^23 + 28 * q^25 + 9 * q^27 - 37 * q^29 - 11 * q^31 + 2 * q^33 + 4 * q^35 - 3 * q^37 - 19 * q^39 - 30 * q^41 + 13 * q^43 - 17 * q^45 - 15 * q^47 + 12 * q^49 - 8 * q^51 - 35 * q^53 - 2 * q^55 - 22 * q^57 - q^59 - 33 * q^61 - 20 * q^63 - 3 * q^65 + 19 * q^67 - 8 * q^69 - 31 * q^71 + 31 * q^73 + 3 * q^75 - 42 * q^77 - 29 * q^79 - 36 * q^81 + 14 * q^83 + 10 * q^85 - 32 * q^87 - 32 * q^89 - 7 * q^91 - 11 * q^93 + 2 * q^95 + 2 * q^97 - 39 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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