LMFDB - Embedded newform 8019.2.a.k.1.17

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			1.17
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)

$f(q)$	$=$	$q-1.25801 q^{2} -0.417404 q^{4} -4.24212 q^{5} -1.43200 q^{7} +3.04112 q^{8} +O(q^{10})$	\(q-1.25801 q^{2} -0.417404 q^{4} -4.24212 q^{5} -1.43200 q^{7} +3.04112 q^{8} +5.33665 q^{10} -1.00000 q^{11} +5.72521 q^{13} +1.80148 q^{14} -2.99097 q^{16} -2.77233 q^{17} +5.16765 q^{19} +1.77068 q^{20} +1.25801 q^{22} +3.12004 q^{23} +12.9956 q^{25} -7.20239 q^{26} +0.597724 q^{28} -1.57035 q^{29} -3.67733 q^{31} -2.31958 q^{32} +3.48763 q^{34} +6.07473 q^{35} +6.11927 q^{37} -6.50098 q^{38} -12.9008 q^{40} +7.00644 q^{41} +7.04076 q^{43} +0.417404 q^{44} -3.92505 q^{46} +0.0564017 q^{47} -4.94937 q^{49} -16.3486 q^{50} -2.38973 q^{52} +7.76096 q^{53} +4.24212 q^{55} -4.35490 q^{56} +1.97552 q^{58} -6.25500 q^{59} -6.68600 q^{61} +4.62613 q^{62} +8.89999 q^{64} -24.2871 q^{65} -13.3673 q^{67} +1.15718 q^{68} -7.64209 q^{70} +10.2446 q^{71} +14.6150 q^{73} -7.69812 q^{74} -2.15700 q^{76} +1.43200 q^{77} -1.30443 q^{79} +12.6880 q^{80} -8.81419 q^{82} +5.40066 q^{83} +11.7606 q^{85} -8.85736 q^{86} -3.04112 q^{88} -12.5796 q^{89} -8.19852 q^{91} -1.30232 q^{92} -0.0709541 q^{94} -21.9218 q^{95} -1.01713 q^{97} +6.22637 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.25801	−0.889549	−0.444775	−	0.895643i	$-0.646716\pi$
$2$	−1.25801	−0.889549	−0.444775	+	0.895643i	$0.646716\pi$
$3$	0	0
$3$	0	0
$4$	−0.417404	−0.208702
$5$	−4.24212	−1.89714	−0.948568	−	0.316574i	$-0.897467\pi$
$5$	−4.24212	−1.89714	−0.948568	+	0.316574i	$0.897467\pi$
$6$	0	0
$7$	−1.43200	−0.541246	−0.270623	−	0.962685i	$-0.587230\pi$
$7$	−1.43200	−0.541246	−0.270623	+	0.962685i	$0.587230\pi$
$8$	3.04112	1.07520
$9$	0	0
$10$	5.33665	1.68760
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	5.72521	1.58789	0.793944	−	0.607991i	$-0.208024\pi$
$13$	5.72521	1.58789	0.793944	+	0.607991i	$0.208024\pi$
$14$	1.80148	0.481465
$15$	0	0
$16$	−2.99097	−0.747741
$17$	−2.77233	−0.672389	−0.336194	−	0.941793i	$-0.609140\pi$
$17$	−2.77233	−0.672389	−0.336194	+	0.941793i	$0.609140\pi$
$18$	0	0
$19$	5.16765	1.18554	0.592771	−	0.805371i	$-0.298034\pi$
$19$	5.16765	1.18554	0.592771	+	0.805371i	$0.298034\pi$
$20$	1.77068	0.395936
$21$	0	0
$22$	1.25801	0.268209
$23$	3.12004	0.650574	0.325287	−	0.945615i	$-0.394539\pi$
$23$	3.12004	0.650574	0.325287	+	0.945615i	$0.394539\pi$
$24$	0	0
$25$	12.9956	2.59912
$26$	−7.20239	−1.41250
$27$	0	0
$28$	0.597724	0.112959
$29$	−1.57035	−0.291606	−0.145803	−	0.989314i	$-0.546577\pi$
$29$	−1.57035	−0.291606	−0.145803	+	0.989314i	$0.546577\pi$
$30$	0	0
$31$	−3.67733	−0.660468	−0.330234	−	0.943899i	$-0.607128\pi$
$31$	−3.67733	−0.660468	−0.330234	+	0.943899i	$0.607128\pi$
$32$	−2.31958	−0.410047
$33$	0	0
$34$	3.48763	0.598123
$35$	6.07473	1.02682
$36$	0	0
$37$	6.11927	1.00600	0.503001	−	0.864286i	$-0.332229\pi$
$37$	6.11927	1.00600	0.503001	+	0.864286i	$0.332229\pi$
$38$	−6.50098	−1.05460
$39$	0	0
$40$	−12.9008	−2.03980
$41$	7.00644	1.09422	0.547111	−	0.837060i	$-0.315728\pi$
$41$	7.00644	1.09422	0.547111	+	0.837060i	$0.315728\pi$
$42$	0	0
$43$	7.04076	1.07371	0.536853	−	0.843676i	$-0.319613\pi$
$43$	7.04076	1.07371	0.536853	+	0.843676i	$0.319613\pi$
$44$	0.417404	0.0629260
$45$	0	0
$46$	−3.92505	−0.578718
$47$	0.0564017	0.00822704	0.00411352	−	0.999992i	$-0.498691\pi$
$47$	0.0564017	0.00822704	0.00411352	+	0.999992i	$0.498691\pi$
$48$	0	0
$49$	−4.94937	−0.707053
$50$	−16.3486	−2.31205
$51$	0	0
$52$	−2.38973	−0.331395
$53$	7.76096	1.06605	0.533025	−	0.846100i	$-0.321055\pi$
$53$	7.76096	1.06605	0.533025	+	0.846100i	$0.321055\pi$
$54$	0	0
$55$	4.24212	0.572008
$56$	−4.35490	−0.581948
$57$	0	0
$58$	1.97552	0.259398
$59$	−6.25500	−0.814332	−0.407166	−	0.913354i	$-0.633483\pi$
$59$	−6.25500	−0.814332	−0.407166	+	0.913354i	$0.633483\pi$
$60$	0	0
$61$	−6.68600	−0.856054	−0.428027	−	0.903766i	$-0.640791\pi$
$61$	−6.68600	−0.856054	−0.428027	+	0.903766i	$0.640791\pi$
$62$	4.62613	0.587519
$63$	0	0
$64$	8.89999	1.11250
$65$	−24.2871	−3.01244
$66$	0	0
$67$	−13.3673	−1.63308	−0.816540	−	0.577289i	$-0.804110\pi$
$67$	−13.3673	−1.63308	−0.816540	+	0.577289i	$0.804110\pi$
$68$	1.15718	0.140329
$69$	0	0
$70$	−7.64209	−0.913405
$71$	10.2446	1.21581	0.607903	−	0.794012i	$-0.292011\pi$
$71$	10.2446	1.21581	0.607903	+	0.794012i	$0.292011\pi$
$72$	0	0
$73$	14.6150	1.71056	0.855278	−	0.518169i	$-0.173386\pi$
$73$	14.6150	1.71056	0.855278	+	0.518169i	$0.173386\pi$
$74$	−7.69812	−0.894888
$75$	0	0
$76$	−2.15700	−0.247425
$77$	1.43200	0.163192
$78$	0	0
$79$	−1.30443	−0.146759	−0.0733797	−	0.997304i	$-0.523378\pi$
$79$	−1.30443	−0.146759	−0.0733797	+	0.997304i	$0.523378\pi$
$80$	12.6880	1.41857
$81$	0	0
$82$	−8.81419	−0.973364
$83$	5.40066	0.592800	0.296400	−	0.955064i	$-0.404214\pi$
$83$	5.40066	0.592800	0.296400	+	0.955064i	$0.404214\pi$
$84$	0	0
$85$	11.7606	1.27561
$86$	−8.85736	−0.955114
$87$	0	0
$88$	−3.04112	−0.324185
$89$	−12.5796	−1.33343	−0.666716	−	0.745312i	$-0.732300\pi$
$89$	−12.5796	−1.33343	−0.666716	+	0.745312i	$0.732300\pi$
$90$	0	0
$91$	−8.19852	−0.859438
$92$	−1.30232	−0.135776
$93$	0	0
$94$	−0.0709541	−0.00731836
$95$	−21.9218	−2.24913
$96$	0	0
$97$	−1.01713	−0.103274	−0.0516372	−	0.998666i	$-0.516444\pi$
$97$	−1.01713	−0.103274	−0.0516372	+	0.998666i	$0.516444\pi$
$98$	6.22637	0.628958
$99$	0	0
$100$	−5.42442	−0.542442
$101$	−9.54195	−0.949459	−0.474730	−	0.880132i	$-0.657454\pi$
$101$	−9.54195	−0.949459	−0.474730	+	0.880132i	$0.657454\pi$
$102$	0	0
$103$	−5.86042	−0.577445	−0.288722	−	0.957413i	$-0.593230\pi$
$103$	−5.86042	−0.577445	−0.288722	+	0.957413i	$0.593230\pi$
$104$	17.4111	1.70730
$105$	0	0
$106$	−9.76338	−0.948303
$107$	−1.03239	−0.0998049	−0.0499025	−	0.998754i	$-0.515891\pi$
$107$	−1.03239	−0.0998049	−0.0499025	+	0.998754i	$0.515891\pi$
$108$	0	0
$109$	1.10304	0.105653	0.0528263	−	0.998604i	$-0.483177\pi$
$109$	1.10304	0.105653	0.0528263	+	0.998604i	$0.483177\pi$
$110$	−5.33665	−0.508829
$111$	0	0
$112$	4.28307	0.404712
$113$	−3.99630	−0.375940	−0.187970	−	0.982175i	$-0.560191\pi$
$113$	−3.99630	−0.375940	−0.187970	+	0.982175i	$0.560191\pi$
$114$	0	0
$115$	−13.2356	−1.23423
$116$	0.655470	0.0608588
$117$	0	0
$118$	7.86887	0.724388
$119$	3.96998	0.363928
$120$	0	0
$121$	1.00000	0.0909091
$122$	8.41107	0.761503
$123$	0	0
$124$	1.53493	0.137841
$125$	−33.9184	−3.03375
$126$	0	0
$127$	−3.09716	−0.274829	−0.137414	−	0.990514i	$-0.543879\pi$
$127$	−3.09716	−0.274829	−0.137414	+	0.990514i	$0.543879\pi$
$128$	−6.55715	−0.579575
$129$	0	0
$130$	30.5534	2.67971
$131$	−6.56536	−0.573619	−0.286809	−	0.957988i	$-0.592595\pi$
$131$	−6.56536	−0.573619	−0.286809	+	0.957988i	$0.592595\pi$
$132$	0	0
$133$	−7.40010	−0.641670
$134$	16.8163	1.45270
$135$	0	0
$136$	−8.43100	−0.722952
$137$	11.8188	1.00975	0.504875	−	0.863192i	$-0.331539\pi$
$137$	11.8188	1.00975	0.504875	+	0.863192i	$0.331539\pi$
$138$	0	0
$139$	11.1246	0.943579	0.471790	−	0.881711i	$-0.343608\pi$
$139$	11.1246	0.943579	0.471790	+	0.881711i	$0.343608\pi$
$140$	−2.53562	−0.214299
$141$	0	0
$142$	−12.8878	−1.08152
$143$	−5.72521	−0.478766
$144$	0	0
$145$	6.66161	0.553217
$146$	−18.3859	−1.52162
$147$	0	0
$148$	−2.55421	−0.209955
$149$	−13.4581	−1.10253	−0.551266	−	0.834330i	$-0.685855\pi$
$149$	−13.4581	−1.10253	−0.551266	+	0.834330i	$0.685855\pi$
$150$	0	0
$151$	−8.94869	−0.728234	−0.364117	−	0.931353i	$-0.618629\pi$
$151$	−8.94869	−0.728234	−0.364117	+	0.931353i	$0.618629\pi$
$152$	15.7155	1.27469
$153$	0	0
$154$	−1.80148	−0.145167
$155$	15.5997	1.25300
$156$	0	0
$157$	16.8556	1.34522	0.672612	−	0.739995i	$-0.265172\pi$
$157$	16.8556	1.34522	0.672612	+	0.739995i	$0.265172\pi$
$158$	1.64098	0.130550
$159$	0	0
$160$	9.83993	0.777915
$161$	−4.46791	−0.352121
$162$	0	0
$163$	6.80232	0.532799	0.266400	−	0.963863i	$-0.414166\pi$
$163$	6.80232	0.532799	0.266400	+	0.963863i	$0.414166\pi$
$164$	−2.92451	−0.228366
$165$	0	0
$166$	−6.79410	−0.527324
$167$	−0.227186	−0.0175802	−0.00879009	−	0.999961i	$-0.502798\pi$
$167$	−0.227186	−0.0175802	−0.00879009	+	0.999961i	$0.502798\pi$
$168$	0	0
$169$	19.7781	1.52139
$170$	−14.7949	−1.13472
$171$	0	0
$172$	−2.93884	−0.224084
$173$	5.32003	0.404475	0.202237	−	0.979337i	$-0.435179\pi$
$173$	5.32003	0.404475	0.202237	+	0.979337i	$0.435179\pi$
$174$	0	0
$175$	−18.6098	−1.40677
$176$	2.99097	0.225453
$177$	0	0
$178$	15.8253	1.18615
$179$	−13.3973	−1.00136	−0.500681	−	0.865632i	$-0.666917\pi$
$179$	−13.3973	−1.00136	−0.500681	+	0.865632i	$0.666917\pi$
$180$	0	0
$181$	10.2946	0.765188	0.382594	−	0.923917i	$-0.375031\pi$
$181$	10.2946	0.765188	0.382594	+	0.923917i	$0.375031\pi$
$182$	10.3138	0.764513
$183$	0	0
$184$	9.48844	0.699497
$185$	−25.9587	−1.90852
$186$	0	0
$187$	2.77233	0.202733
$188$	−0.0235423	−0.00171700
$189$	0	0
$190$	27.5779	2.00071
$191$	−18.3321	−1.32646	−0.663231	−	0.748415i	$-0.730815\pi$
$191$	−18.3321	−1.32646	−0.663231	+	0.748415i	$0.730815\pi$
$192$	0	0
$193$	−15.7674	−1.13496	−0.567480	−	0.823387i	$-0.692082\pi$
$193$	−15.7674	−1.13496	−0.567480	+	0.823387i	$0.692082\pi$
$194$	1.27957	0.0918677
$195$	0	0
$196$	2.06589	0.147563
$197$	−13.9593	−0.994561	−0.497281	−	0.867590i	$-0.665668\pi$
$197$	−13.9593	−0.994561	−0.497281	+	0.867590i	$0.665668\pi$
$198$	0	0
$199$	13.3021	0.942960	0.471480	−	0.881877i	$-0.343720\pi$
$199$	13.3021	0.942960	0.471480	+	0.881877i	$0.343720\pi$
$200$	39.5213	2.79458
$201$	0	0
$202$	12.0039	0.844591
$203$	2.24874	0.157831
$204$	0	0
$205$	−29.7222	−2.07589
$206$	7.37249	0.513666
$207$	0	0
$208$	−17.1239	−1.18733
$209$	−5.16765	−0.357454
$210$	0	0
$211$	−17.3558	−1.19482	−0.597411	−	0.801935i	$-0.703804\pi$
$211$	−17.3558	−1.19482	−0.597411	+	0.801935i	$0.703804\pi$
$212$	−3.23945	−0.222487
$213$	0	0
$214$	1.29876	0.0887814
$215$	−29.8678	−2.03696
$216$	0	0
$217$	5.26595	0.357476
$218$	−1.38764	−0.0939831
$219$	0	0
$220$	−1.77068	−0.119379
$221$	−15.8722	−1.06768
$222$	0	0
$223$	7.24010	0.484833	0.242416	−	0.970172i	$-0.422060\pi$
$223$	7.24010	0.484833	0.242416	+	0.970172i	$0.422060\pi$
$224$	3.32164	0.221936
$225$	0	0
$226$	5.02740	0.334418
$227$	−19.2222	−1.27582	−0.637911	−	0.770110i	$-0.720201\pi$
$227$	−19.2222	−1.27582	−0.637911	+	0.770110i	$0.720201\pi$
$228$	0	0
$229$	21.0590	1.39161	0.695807	−	0.718228i	$-0.255047\pi$
$229$	21.0590	1.39161	0.695807	+	0.718228i	$0.255047\pi$
$230$	16.6506	1.09791
$231$	0	0
$232$	−4.77562	−0.313535
$233$	0.602361	0.0394620	0.0197310	−	0.999805i	$-0.493719\pi$
$233$	0.602361	0.0394620	0.0197310	+	0.999805i	$0.493719\pi$
$234$	0	0
$235$	−0.239263	−0.0156078
$236$	2.61086	0.169953
$237$	0	0
$238$	−4.99429	−0.323732
$239$	19.3555	1.25200	0.626001	−	0.779822i	$-0.284691\pi$
$239$	19.3555	1.25200	0.626001	+	0.779822i	$0.284691\pi$
$240$	0	0
$241$	2.69592	0.173659	0.0868296	−	0.996223i	$-0.472326\pi$
$241$	2.69592	0.173659	0.0868296	+	0.996223i	$0.472326\pi$
$242$	−1.25801	−0.0808681
$243$	0	0
$244$	2.79076	0.178660
$245$	20.9958	1.34137
$246$	0	0
$247$	29.5859	1.88251
$248$	−11.1832	−0.710136
$249$	0	0
$250$	42.6698	2.69867
$251$	−14.2474	−0.899288	−0.449644	−	0.893208i	$-0.648449\pi$
$251$	−14.2474	−0.899288	−0.449644	+	0.893208i	$0.648449\pi$
$252$	0	0
$253$	−3.12004	−0.196155
$254$	3.89627	0.244474
$255$	0	0
$256$	−9.55101	−0.596938
$257$	1.51811	0.0946974	0.0473487	−	0.998878i	$-0.484923\pi$
$257$	1.51811	0.0946974	0.0473487	+	0.998878i	$0.484923\pi$
$258$	0	0
$259$	−8.76281	−0.544494
$260$	10.1375	0.628702
$261$	0	0
$262$	8.25931	0.510262
$263$	3.70206	0.228279	0.114139	−	0.993465i	$-0.463589\pi$
$263$	3.70206	0.228279	0.114139	+	0.993465i	$0.463589\pi$
$264$	0	0
$265$	−32.9229	−2.02244
$266$	9.30941	0.570797
$267$	0	0
$268$	5.57958	0.340827
$269$	19.7182	1.20224	0.601120	−	0.799159i	$-0.294722\pi$
$269$	19.7182	1.20224	0.601120	+	0.799159i	$0.294722\pi$
$270$	0	0
$271$	−18.6490	−1.13284	−0.566422	−	0.824116i	$-0.691673\pi$
$271$	−18.6490	−1.13284	−0.566422	+	0.824116i	$0.691673\pi$
$272$	8.29194	0.502773
$273$	0	0
$274$	−14.8682	−0.898223
$275$	−12.9956	−0.783665
$276$	0	0
$277$	−10.7853	−0.648028	−0.324014	−	0.946052i	$-0.605032\pi$
$277$	−10.7853	−0.648028	−0.324014	+	0.946052i	$0.605032\pi$
$278$	−13.9949	−0.839360
$279$	0	0
$280$	18.4740	1.10403
$281$	12.4207	0.740957	0.370479	−	0.928841i	$-0.379194\pi$
$281$	12.4207	0.740957	0.370479	+	0.928841i	$0.379194\pi$
$282$	0	0
$283$	13.3898	0.795942	0.397971	−	0.917398i	$-0.369714\pi$
$283$	13.3898	0.795942	0.397971	+	0.917398i	$0.369714\pi$
$284$	−4.27612	−0.253741
$285$	0	0
$286$	7.20239	0.425886
$287$	−10.0332	−0.592243
$288$	0	0
$289$	−9.31419	−0.547893
$290$	−8.38039	−0.492113
$291$	0	0
$292$	−6.10036	−0.356997
$293$	−9.83174	−0.574377	−0.287188	−	0.957874i	$-0.592721\pi$
$293$	−9.83174	−0.574377	−0.287188	+	0.957874i	$0.592721\pi$
$294$	0	0
$295$	26.5345	1.54490
$296$	18.6095	1.08165
$297$	0	0
$298$	16.9305	0.980757
$299$	17.8629	1.03304
$300$	0	0
$301$	−10.0824	−0.581139
$302$	11.2576	0.647800
$303$	0	0
$304$	−15.4563	−0.886478
$305$	28.3628	1.62405
$306$	0	0
$307$	9.43376	0.538413	0.269207	−	0.963082i	$-0.413239\pi$
$307$	9.43376	0.538413	0.269207	+	0.963082i	$0.413239\pi$
$308$	−0.597724	−0.0340585
$309$	0	0
$310$	−19.6246	−1.11460
$311$	30.6723	1.73927	0.869634	−	0.493697i	$-0.164355\pi$
$311$	30.6723	1.73927	0.869634	+	0.493697i	$0.164355\pi$
$312$	0	0
$313$	−4.99610	−0.282396	−0.141198	−	0.989981i	$-0.545095\pi$
$313$	−4.99610	−0.282396	−0.141198	+	0.989981i	$0.545095\pi$
$314$	−21.2046	−1.19664
$315$	0	0
$316$	0.544472	0.0306290
$317$	25.0014	1.40422	0.702110	−	0.712069i	$-0.252241\pi$
$317$	25.0014	1.40422	0.702110	+	0.712069i	$0.252241\pi$
$318$	0	0
$319$	1.57035	0.0879226
$320$	−37.7549	−2.11056
$321$	0	0
$322$	5.62069	0.313229
$323$	−14.3264	−0.797145
$324$	0	0
$325$	74.4027	4.12712
$326$	−8.55741	−0.473951
$327$	0	0
$328$	21.3074	1.17651
$329$	−0.0807675	−0.00445285
$330$	0	0
$331$	8.00334	0.439903	0.219952	−	0.975511i	$-0.429410\pi$
$331$	8.00334	0.439903	0.219952	+	0.975511i	$0.429410\pi$
$332$	−2.25426	−0.123718
$333$	0	0
$334$	0.285803	0.0156384
$335$	56.7059	3.09817
$336$	0	0
$337$	28.8379	1.57090	0.785450	−	0.618925i	$-0.212432\pi$
$337$	28.8379	1.57090	0.785450	+	0.618925i	$0.212432\pi$
$338$	−24.8810	−1.35335
$339$	0	0
$340$	−4.90891	−0.266223
$341$	3.67733	0.199139
$342$	0	0
$343$	17.1115	0.923936
$344$	21.4118	1.15445
$345$	0	0
$346$	−6.69267	−0.359800
$347$	4.51654	0.242460	0.121230	−	0.992624i	$-0.461316\pi$
$347$	4.51654	0.242460	0.121230	+	0.992624i	$0.461316\pi$
$348$	0	0
$349$	−24.7246	−1.32348	−0.661740	−	0.749733i	$-0.730182\pi$
$349$	−24.7246	−1.32348	−0.661740	+	0.749733i	$0.730182\pi$
$350$	23.4113	1.25139
$351$	0	0
$352$	2.31958	0.123634
$353$	33.6355	1.79023	0.895117	−	0.445830i	$-0.147092\pi$
$353$	33.6355	1.79023	0.895117	+	0.445830i	$0.147092\pi$
$354$	0	0
$355$	−43.4587	−2.30655
$356$	5.25076	0.278290
$357$	0	0
$358$	16.8540	0.890760
$359$	20.7304	1.09411	0.547055	−	0.837097i	$-0.315749\pi$
$359$	20.7304	1.09411	0.547055	+	0.837097i	$0.315749\pi$
$360$	0	0
$361$	7.70465	0.405508
$362$	−12.9507	−0.680672
$363$	0	0
$364$	3.42210	0.179367
$365$	−61.9987	−3.24516
$366$	0	0
$367$	−9.60287	−0.501266	−0.250633	−	0.968082i	$-0.580639\pi$
$367$	−9.60287	−0.501266	−0.250633	+	0.968082i	$0.580639\pi$
$368$	−9.33194	−0.486461
$369$	0	0
$370$	32.6564	1.69772
$371$	−11.1137	−0.576995
$372$	0	0
$373$	21.5115	1.11382	0.556911	−	0.830572i	$-0.311987\pi$
$373$	21.5115	1.11382	0.556911	+	0.830572i	$0.311987\pi$
$374$	−3.48763	−0.180341
$375$	0	0
$376$	0.171525	0.00884571
$377$	−8.99058	−0.463038
$378$	0	0
$379$	−3.33979	−0.171554	−0.0857768	−	0.996314i	$-0.527337\pi$
$379$	−3.33979	−0.171554	−0.0857768	+	0.996314i	$0.527337\pi$
$380$	9.15026	0.469398
$381$	0	0
$382$	23.0620	1.17995
$383$	−0.602249	−0.0307735	−0.0153867	−	0.999882i	$-0.504898\pi$
$383$	−0.602249	−0.0307735	−0.0153867	+	0.999882i	$0.504898\pi$
$384$	0	0
$385$	−6.07473	−0.309597
$386$	19.8356	1.00960
$387$	0	0
$388$	0.424556	0.0215536
$389$	10.9470	0.555034	0.277517	−	0.960721i	$-0.410489\pi$
$389$	10.9470	0.555034	0.277517	+	0.960721i	$0.410489\pi$
$390$	0	0
$391$	−8.64979	−0.437439
$392$	−15.0516	−0.760223
$393$	0	0
$394$	17.5610	0.884711
$395$	5.53353	0.278422
$396$	0	0
$397$	13.0791	0.656419	0.328209	−	0.944605i	$-0.393555\pi$
$397$	13.0791	0.656419	0.328209	+	0.944605i	$0.393555\pi$
$398$	−16.7342	−0.838809
$399$	0	0
$400$	−38.8694	−1.94347
$401$	−16.6487	−0.831398	−0.415699	−	0.909502i	$-0.636463\pi$
$401$	−16.6487	−0.831398	−0.415699	+	0.909502i	$0.636463\pi$
$402$	0	0
$403$	−21.0535	−1.04875
$404$	3.98285	0.198154
$405$	0	0
$406$	−2.82895	−0.140398
$407$	−6.11927	−0.303321
$408$	0	0
$409$	−26.7850	−1.32443	−0.662216	−	0.749313i	$-0.730384\pi$
$409$	−26.7850	−1.32443	−0.662216	+	0.749313i	$0.730384\pi$
$410$	37.3909	1.84660
$411$	0	0
$412$	2.44616	0.120514
$413$	8.95718	0.440754
$414$	0	0
$415$	−22.9103	−1.12462
$416$	−13.2801	−0.651109
$417$	0	0
$418$	6.50098	0.317973
$419$	20.6626	1.00943	0.504717	−	0.863285i	$-0.331597\pi$
$419$	20.6626	1.00943	0.504717	+	0.863285i	$0.331597\pi$
$420$	0	0
$421$	37.2746	1.81665	0.908327	−	0.418261i	$-0.137360\pi$
$421$	37.2746	1.81665	0.908327	+	0.418261i	$0.137360\pi$
$422$	21.8338	1.06285
$423$	0	0
$424$	23.6020	1.14622
$425$	−36.0281	−1.74762
$426$	0	0
$427$	9.57437	0.463336
$428$	0.430924	0.0208295
$429$	0	0
$430$	37.5740	1.81198
$431$	−34.7950	−1.67602	−0.838009	−	0.545656i	$-0.816281\pi$
$431$	−34.7950	−1.67602	−0.838009	+	0.545656i	$0.816281\pi$
$432$	0	0
$433$	−5.84558	−0.280921	−0.140460	−	0.990086i	$-0.544858\pi$
$433$	−5.84558	−0.280921	−0.140460	+	0.990086i	$0.544858\pi$
$434$	−6.62463	−0.317992
$435$	0	0
$436$	−0.460415	−0.0220499
$437$	16.1233	0.771282
$438$	0	0
$439$	0.658797	0.0314426	0.0157213	−	0.999876i	$-0.494996\pi$
$439$	0.658797	0.0314426	0.0157213	+	0.999876i	$0.494996\pi$
$440$	12.9008	0.615023
$441$	0	0
$442$	19.9674	0.949752
$443$	12.9031	0.613043	0.306522	−	0.951864i	$-0.400835\pi$
$443$	12.9031	0.613043	0.306522	+	0.951864i	$0.400835\pi$
$444$	0	0
$445$	53.3641	2.52970
$446$	−9.10814	−0.431283
$447$	0	0
$448$	−12.7448	−0.602136
$449$	3.70636	0.174914	0.0874569	−	0.996168i	$-0.472126\pi$
$449$	3.70636	0.174914	0.0874569	+	0.996168i	$0.472126\pi$
$450$	0	0
$451$	−7.00644	−0.329920
$452$	1.66807	0.0784595
$453$	0	0
$454$	24.1817	1.13491
$455$	34.7791	1.63047
$456$	0	0
$457$	−21.9123	−1.02501	−0.512506	−	0.858683i	$-0.671283\pi$
$457$	−21.9123	−1.02501	−0.512506	+	0.858683i	$0.671283\pi$
$458$	−26.4924	−1.23791
$459$	0	0
$460$	5.52460	0.257586
$461$	−20.3496	−0.947775	−0.473887	−	0.880586i	$-0.657150\pi$
$461$	−20.3496	−0.947775	−0.473887	+	0.880586i	$0.657150\pi$
$462$	0	0
$463$	33.8857	1.57480	0.787401	−	0.616441i	$-0.211426\pi$
$463$	33.8857	1.57480	0.787401	+	0.616441i	$0.211426\pi$
$464$	4.69686	0.218046
$465$	0	0
$466$	−0.757777	−0.0351034
$467$	−14.0709	−0.651125	−0.325563	−	0.945520i	$-0.605554\pi$
$467$	−14.0709	−0.651125	−0.325563	+	0.945520i	$0.605554\pi$
$468$	0	0
$469$	19.1421	0.883898
$470$	0.300996	0.0138839
$471$	0	0
$472$	−19.0222	−0.875570
$473$	−7.04076	−0.323734
$474$	0	0
$475$	67.1568	3.08137
$476$	−1.65709	−0.0759525
$477$	0	0
$478$	−24.3494	−1.11372
$479$	36.8192	1.68231	0.841156	−	0.540792i	$-0.181875\pi$
$479$	36.8192	1.68231	0.841156	+	0.540792i	$0.181875\pi$
$480$	0	0
$481$	35.0341	1.59742
$482$	−3.39150	−0.154478
$483$	0	0
$484$	−0.417404	−0.0189729
$485$	4.31481	0.195925
$486$	0	0
$487$	−20.1025	−0.910929	−0.455465	−	0.890254i	$-0.650527\pi$
$487$	−20.1025	−0.910929	−0.455465	+	0.890254i	$0.650527\pi$
$488$	−20.3330	−0.920430
$489$	0	0
$490$	−26.4130	−1.19322
$491$	27.5385	1.24280	0.621398	−	0.783495i	$-0.286565\pi$
$491$	27.5385	1.24280	0.621398	+	0.783495i	$0.286565\pi$
$492$	0	0
$493$	4.35352	0.196073
$494$	−37.2195	−1.67458
$495$	0	0
$496$	10.9988	0.493859
$497$	−14.6702	−0.658050
$498$	0	0
$499$	15.9211	0.712725	0.356363	−	0.934348i	$-0.384017\pi$
$499$	15.9211	0.712725	0.356363	+	0.934348i	$0.384017\pi$
$500$	14.1577	0.633150
$501$	0	0
$502$	17.9234	0.799961
$503$	6.62774	0.295516	0.147758	−	0.989024i	$-0.452794\pi$
$503$	6.62774	0.295516	0.147758	+	0.989024i	$0.452794\pi$
$504$	0	0
$505$	40.4781	1.80125
$506$	3.92505	0.174490
$507$	0	0
$508$	1.29277	0.0573573
$509$	15.8767	0.703721	0.351861	−	0.936052i	$-0.385549\pi$
$509$	15.8767	0.703721	0.351861	+	0.936052i	$0.385549\pi$
$510$	0	0
$511$	−20.9287	−0.925832
$512$	25.1296	1.11058
$513$	0	0
$514$	−1.90981	−0.0842380
$515$	24.8606	1.09549
$516$	0	0
$517$	−0.0564017	−0.00248055
$518$	11.0237	0.484355
$519$	0	0
$520$	−73.8600	−3.23897
$521$	21.6570	0.948812	0.474406	−	0.880306i	$-0.342663\pi$
$521$	21.6570	0.948812	0.474406	+	0.880306i	$0.342663\pi$
$522$	0	0
$523$	−5.51245	−0.241043	−0.120521	−	0.992711i	$-0.538457\pi$
$523$	−5.51245	−0.241043	−0.120521	+	0.992711i	$0.538457\pi$
$524$	2.74041	0.119715
$525$	0	0
$526$	−4.65723	−0.203065
$527$	10.1948	0.444091
$528$	0	0
$529$	−13.2653	−0.576753
$530$	41.4175	1.79906
$531$	0	0
$532$	3.08883	0.133918
$533$	40.1133	1.73750
$534$	0	0
$535$	4.37953	0.189343
$536$	−40.6517	−1.75589
$537$	0	0
$538$	−24.8057	−1.06945
$539$	4.94937	0.213184
$540$	0	0
$541$	−12.6432	−0.543574	−0.271787	−	0.962357i	$-0.587615\pi$
$541$	−12.6432	−0.543574	−0.271787	+	0.962357i	$0.587615\pi$
$542$	23.4606	1.00772
$543$	0	0
$544$	6.43063	0.275711
$545$	−4.67925	−0.200437
$546$	0	0
$547$	−36.7897	−1.57301	−0.786506	−	0.617582i	$-0.788112\pi$
$547$	−36.7897	−1.57301	−0.786506	+	0.617582i	$0.788112\pi$
$548$	−4.93322	−0.210737
$549$	0	0
$550$	16.3486	0.697109
$551$	−8.11502	−0.345711
$552$	0	0
$553$	1.86794	0.0794329
$554$	13.5681	0.576453
$555$	0	0
$556$	−4.64347	−0.196927
$557$	−40.2514	−1.70551	−0.852754	−	0.522313i	$-0.825069\pi$
$557$	−40.2514	−1.70551	−0.852754	+	0.522313i	$0.825069\pi$
$558$	0	0
$559$	40.3098	1.70492
$560$	−18.1693	−0.767794
$561$	0	0
$562$	−15.6254	−0.659118
$563$	41.4334	1.74621	0.873105	−	0.487531i	$-0.162103\pi$
$563$	41.4334	1.74621	0.873105	+	0.487531i	$0.162103\pi$
$564$	0	0
$565$	16.9528	0.713210
$566$	−16.8446	−0.708030
$567$	0	0
$568$	31.1550	1.30723
$569$	−4.29902	−0.180224	−0.0901122	−	0.995932i	$-0.528723\pi$
$569$	−4.29902	−0.180224	−0.0901122	+	0.995932i	$0.528723\pi$
$570$	0	0
$571$	−28.7576	−1.20347	−0.601734	−	0.798697i	$-0.705523\pi$
$571$	−28.7576	−1.20347	−0.601734	+	0.798697i	$0.705523\pi$
$572$	2.38973	0.0999195
$573$	0	0
$574$	12.6219	0.526829
$575$	40.5469	1.69092
$576$	0	0
$577$	14.8871	0.619758	0.309879	−	0.950776i	$-0.399712\pi$
$577$	14.8871	0.619758	0.309879	+	0.950776i	$0.399712\pi$
$578$	11.7174	0.487378
$579$	0	0
$580$	−2.78058	−0.115457
$581$	−7.73376	−0.320850
$582$	0	0
$583$	−7.76096	−0.321426
$584$	44.4461	1.83919
$585$	0	0
$586$	12.3685	0.510936
$587$	1.34722	0.0556057	0.0278029	−	0.999613i	$-0.491149\pi$
$587$	1.34722	0.0556057	0.0278029	+	0.999613i	$0.491149\pi$
$588$	0	0
$589$	−19.0032	−0.783012
$590$	−33.3807	−1.37426
$591$	0	0
$592$	−18.3025	−0.752229
$593$	22.6058	0.928309	0.464154	−	0.885754i	$-0.346358\pi$
$593$	22.6058	0.928309	0.464154	+	0.885754i	$0.346358\pi$
$594$	0	0
$595$	−16.8412	−0.690420
$596$	5.61747	0.230101
$597$	0	0
$598$	−22.4718	−0.918939
$599$	−34.7528	−1.41996	−0.709980	−	0.704222i	$-0.751296\pi$
$599$	−34.7528	−1.41996	−0.709980	+	0.704222i	$0.751296\pi$
$600$	0	0
$601$	39.2242	1.59999	0.799994	−	0.600008i	$-0.204836\pi$
$601$	39.2242	1.59999	0.799994	+	0.600008i	$0.204836\pi$
$602$	12.6838	0.516952
$603$	0	0
$604$	3.73522	0.151984
$605$	−4.24212	−0.172467
$606$	0	0
$607$	−12.2928	−0.498951	−0.249475	−	0.968381i	$-0.580258\pi$
$607$	−12.2928	−0.498951	−0.249475	+	0.968381i	$0.580258\pi$
$608$	−11.9868	−0.486128
$609$	0	0
$610$	−35.6808	−1.44467
$611$	0.322912	0.0130636
$612$	0	0
$613$	13.9570	0.563718	0.281859	−	0.959456i	$-0.409049\pi$
$613$	13.9570	0.563718	0.281859	+	0.959456i	$0.409049\pi$
$614$	−11.8678	−0.478945
$615$	0	0
$616$	4.35490	0.175464
$617$	15.7318	0.633340	0.316670	−	0.948536i	$-0.397435\pi$
$617$	15.7318	0.633340	0.316670	+	0.948536i	$0.397435\pi$
$618$	0	0
$619$	−42.1761	−1.69520	−0.847601	−	0.530635i	$-0.821954\pi$
$619$	−42.1761	−1.69520	−0.847601	+	0.530635i	$0.821954\pi$
$620$	−6.51138	−0.261503
$621$	0	0
$622$	−38.5862	−1.54716
$623$	18.0140	0.721715
$624$	0	0
$625$	78.9079	3.15632
$626$	6.28516	0.251205
$627$	0	0
$628$	−7.03560	−0.280751
$629$	−16.9646	−0.676424
$630$	0	0
$631$	−9.90348	−0.394251	−0.197126	−	0.980378i	$-0.563161\pi$
$631$	−9.90348	−0.394251	−0.197126	+	0.980378i	$0.563161\pi$
$632$	−3.96692	−0.157796
$633$	0	0
$634$	−31.4521	−1.24912
$635$	13.1386	0.521387
$636$	0	0
$637$	−28.3362	−1.12272
$638$	−1.97552	−0.0782115
$639$	0	0
$640$	27.8162	1.09953
$641$	−12.8109	−0.505999	−0.253000	−	0.967466i	$-0.581417\pi$
$641$	−12.8109	−0.505999	−0.253000	+	0.967466i	$0.581417\pi$
$642$	0	0
$643$	6.43724	0.253860	0.126930	−	0.991912i	$-0.459488\pi$
$643$	6.43724	0.253860	0.126930	+	0.991912i	$0.459488\pi$
$644$	1.86492	0.0734883
$645$	0	0
$646$	18.0228	0.709099
$647$	−42.4517	−1.66895	−0.834474	−	0.551048i	$-0.814228\pi$
$647$	−42.4517	−1.66895	−0.834474	+	0.551048i	$0.814228\pi$
$648$	0	0
$649$	6.25500	0.245530
$650$	−93.5995	−3.67127
$651$	0	0
$652$	−2.83932	−0.111196
$653$	−0.424913	−0.0166281	−0.00831407	−	0.999965i	$-0.502646\pi$
$653$	−0.424913	−0.0166281	−0.00831407	+	0.999965i	$0.502646\pi$
$654$	0	0
$655$	27.8511	1.08823
$656$	−20.9560	−0.818195
$657$	0	0
$658$	0.101606	0.00396103
$659$	28.3262	1.10343	0.551716	−	0.834032i	$-0.313973\pi$
$659$	28.3262	1.10343	0.551716	+	0.834032i	$0.313973\pi$
$660$	0	0
$661$	−13.2216	−0.514260	−0.257130	−	0.966377i	$-0.582777\pi$
$661$	−13.2216	−0.514260	−0.257130	+	0.966377i	$0.582777\pi$
$662$	−10.0683	−0.391316
$663$	0	0
$664$	16.4241	0.637378
$665$	31.3921	1.21733
$666$	0	0
$667$	−4.89955	−0.189711
$668$	0.0948284	0.00366902
$669$	0	0
$670$	−71.3367	−2.75598
$671$	6.68600	0.258110
$672$	0	0
$673$	4.91131	0.189317	0.0946586	−	0.995510i	$-0.469824\pi$
$673$	4.91131	0.189317	0.0946586	+	0.995510i	$0.469824\pi$
$674$	−36.2784	−1.39739
$675$	0	0
$676$	−8.25544	−0.317517
$677$	0.603587	0.0231977	0.0115989	−	0.999933i	$-0.496308\pi$
$677$	0.603587	0.0231977	0.0115989	+	0.999933i	$0.496308\pi$
$678$	0	0
$679$	1.45654	0.0558969
$680$	35.7654	1.37154
$681$	0	0
$682$	−4.62613	−0.177144
$683$	3.49517	0.133739	0.0668695	−	0.997762i	$-0.478699\pi$
$683$	3.49517	0.133739	0.0668695	+	0.997762i	$0.478699\pi$
$684$	0	0
$685$	−50.1369	−1.91563
$686$	−21.5265	−0.821886
$687$	0	0
$688$	−21.0587	−0.802854
$689$	44.4331	1.69277
$690$	0	0
$691$	34.9538	1.32970	0.664852	−	0.746975i	$-0.268495\pi$
$691$	34.9538	1.32970	0.664852	+	0.746975i	$0.268495\pi$
$692$	−2.22060	−0.0844147
$693$	0	0
$694$	−5.68186	−0.215681
$695$	−47.1921	−1.79010
$696$	0	0
$697$	−19.4242	−0.735742
$698$	31.1039	1.17730
$699$	0	0
$700$	7.76779	0.293595
$701$	−7.32738	−0.276751	−0.138376	−	0.990380i	$-0.544188\pi$
$701$	−7.32738	−0.276751	−0.138376	+	0.990380i	$0.544188\pi$
$702$	0	0
$703$	31.6223	1.19266
$704$	−8.89999	−0.335431
$705$	0	0
$706$	−42.3138	−1.59250
$707$	13.6641	0.513891
$708$	0	0
$709$	−5.67495	−0.213127	−0.106564	−	0.994306i	$-0.533985\pi$
$709$	−5.67495	−0.213127	−0.106564	+	0.994306i	$0.533985\pi$
$710$	54.6716	2.05179
$711$	0	0
$712$	−38.2560	−1.43371
$713$	−11.4734	−0.429684
$714$	0	0
$715$	24.2871	0.908285
$716$	5.59209	0.208986
$717$	0	0
$718$	−26.0791	−0.973265
$719$	25.1154	0.936646	0.468323	−	0.883557i	$-0.344858\pi$
$719$	25.1154	0.936646	0.468323	+	0.883557i	$0.344858\pi$
$720$	0	0
$721$	8.39214	0.312540
$722$	−9.69255	−0.360719
$723$	0	0
$724$	−4.29699	−0.159696
$725$	−20.4076	−0.757921
$726$	0	0
$727$	−3.41977	−0.126832	−0.0634161	−	0.997987i	$-0.520200\pi$
$727$	−3.41977	−0.126832	−0.0634161	+	0.997987i	$0.520200\pi$
$728$	−24.9327	−0.924068
$729$	0	0
$730$	77.9951	2.88673
$731$	−19.5193	−0.721947
$732$	0	0
$733$	14.1205	0.521551	0.260775	−	0.965400i	$-0.416022\pi$
$733$	14.1205	0.521551	0.260775	+	0.965400i	$0.416022\pi$
$734$	12.0805	0.445900
$735$	0	0
$736$	−7.23718	−0.266766
$737$	13.3673	0.492392
$738$	0	0
$739$	−33.1140	−1.21812	−0.609059	−	0.793125i	$-0.708453\pi$
$739$	−33.1140	−1.21812	−0.609059	+	0.793125i	$0.708453\pi$
$740$	10.8353	0.398312
$741$	0	0
$742$	13.9812	0.513266
$743$	18.4528	0.676969	0.338484	−	0.940972i	$-0.390086\pi$
$743$	18.4528	0.676969	0.338484	+	0.940972i	$0.390086\pi$
$744$	0	0
$745$	57.0910	2.09165
$746$	−27.0617	−0.990799
$747$	0	0
$748$	−1.15718	−0.0423108
$749$	1.47839	0.0540190
$750$	0	0
$751$	29.4600	1.07501	0.537505	−	0.843260i	$-0.319367\pi$
$751$	29.4600	1.07501	0.537505	+	0.843260i	$0.319367\pi$
$752$	−0.168696	−0.00615170
$753$	0	0
$754$	11.3103	0.411895
$755$	37.9615	1.38156
$756$	0	0
$757$	−33.3390	−1.21173	−0.605863	−	0.795569i	$-0.707172\pi$
$757$	−33.3390	−1.21173	−0.605863	+	0.795569i	$0.707172\pi$
$758$	4.20150	0.152605
$759$	0	0
$760$	−66.6670	−2.41827
$761$	2.69799	0.0978021	0.0489011	−	0.998804i	$-0.484428\pi$
$761$	2.69799	0.0978021	0.0489011	+	0.998804i	$0.484428\pi$
$762$	0	0
$763$	−1.57956	−0.0571840
$764$	7.65188	0.276835
$765$	0	0
$766$	0.757637	0.0273745
$767$	−35.8112	−1.29307
$768$	0	0
$769$	−40.2509	−1.45148	−0.725742	−	0.687966i	$-0.758504\pi$
$769$	−40.2509	−1.45148	−0.725742	+	0.687966i	$0.758504\pi$
$770$	7.64209	0.275402
$771$	0	0
$772$	6.58137	0.236869
$773$	4.61283	0.165912	0.0829560	−	0.996553i	$-0.473564\pi$
$773$	4.61283	0.165912	0.0829560	+	0.996553i	$0.473564\pi$
$774$	0	0
$775$	−47.7892	−1.71664
$776$	−3.09323	−0.111041
$777$	0	0
$778$	−13.7714	−0.493730
$779$	36.2068	1.29724
$780$	0	0
$781$	−10.2446	−0.366579
$782$	10.8815	0.389123
$783$	0	0
$784$	14.8034	0.528693
$785$	−71.5036	−2.55207
$786$	0	0
$787$	−24.9182	−0.888237	−0.444119	−	0.895968i	$-0.646483\pi$
$787$	−24.9182	−0.888237	−0.444119	+	0.895968i	$0.646483\pi$
$788$	5.82668	0.207567
$789$	0	0
$790$	−6.96126	−0.247670
$791$	5.72271	0.203476
$792$	0	0
$793$	−38.2788	−1.35932
$794$	−16.4536	−0.583917
$795$	0	0
$796$	−5.55234	−0.196798
$797$	−7.22600	−0.255958	−0.127979	−	0.991777i	$-0.540849\pi$
$797$	−7.22600	−0.255958	−0.127979	+	0.991777i	$0.540849\pi$
$798$	0	0
$799$	−0.156364	−0.00553177
$800$	−30.1443	−1.06576
$801$	0	0
$802$	20.9443	0.739570
$803$	−14.6150	−0.515752
$804$	0	0
$805$	18.9534	0.668021
$806$	26.4856	0.932915
$807$	0	0
$808$	−29.0183	−1.02086
$809$	−5.60624	−0.197105	−0.0985524	−	0.995132i	$-0.531421\pi$
$809$	−5.60624	−0.197105	−0.0985524	+	0.995132i	$0.531421\pi$
$810$	0	0
$811$	2.81266	0.0987658	0.0493829	−	0.998780i	$-0.484275\pi$
$811$	2.81266	0.0987658	0.0493829	+	0.998780i	$0.484275\pi$
$812$	−0.938634	−0.0329396
$813$	0	0
$814$	7.69812	0.269819
$815$	−28.8563	−1.01079
$816$	0	0
$817$	36.3842	1.27292
$818$	33.6959	1.17815
$819$	0	0
$820$	12.4062	0.433242
$821$	10.5765	0.369124	0.184562	−	0.982821i	$-0.440913\pi$
$821$	10.5765	0.369124	0.184562	+	0.982821i	$0.440913\pi$
$822$	0	0
$823$	29.2347	1.01906	0.509528	−	0.860454i	$-0.329820\pi$
$823$	29.2347	1.01906	0.509528	+	0.860454i	$0.329820\pi$
$824$	−17.8223	−0.620869
$825$	0	0
$826$	−11.2682	−0.392072
$827$	−44.2606	−1.53909	−0.769546	−	0.638592i	$-0.779517\pi$
$827$	−44.2606	−1.53909	−0.769546	+	0.638592i	$0.779517\pi$
$828$	0	0
$829$	−47.9626	−1.66581	−0.832905	−	0.553416i	$-0.813324\pi$
$829$	−47.9626	−1.66581	−0.832905	+	0.553416i	$0.813324\pi$
$830$	28.8214	1.00041
$831$	0	0
$832$	50.9543	1.76652
$833$	13.7213	0.475414
$834$	0	0
$835$	0.963752	0.0333520
$836$	2.15700	0.0746014
$837$	0	0
$838$	−25.9938	−0.897941
$839$	−48.4932	−1.67417	−0.837086	−	0.547071i	$-0.815743\pi$
$839$	−48.4932	−1.67417	−0.837086	+	0.547071i	$0.815743\pi$
$840$	0	0
$841$	−26.5340	−0.914966
$842$	−46.8919	−1.61600
$843$	0	0
$844$	7.24438	0.249362
$845$	−83.9010	−2.88628
$846$	0	0
$847$	−1.43200	−0.0492042
$848$	−23.2128	−0.797129
$849$	0	0
$850$	45.3238	1.55460
$851$	19.0924	0.654479
$852$	0	0
$853$	−18.9697	−0.649509	−0.324754	−	0.945798i	$-0.605282\pi$
$853$	−18.9697	−0.649509	−0.324754	+	0.945798i	$0.605282\pi$
$854$	−12.0447	−0.412160
$855$	0	0
$856$	−3.13963	−0.107310
$857$	4.62735	0.158067	0.0790337	−	0.996872i	$-0.474817\pi$
$857$	4.62735	0.158067	0.0790337	+	0.996872i	$0.474817\pi$
$858$	0	0
$859$	−13.7775	−0.470083	−0.235041	−	0.971985i	$-0.575523\pi$
$859$	−13.7775	−0.470083	−0.235041	+	0.971985i	$0.575523\pi$
$860$	12.4669	0.425119
$861$	0	0
$862$	43.7726	1.49090
$863$	−0.480729	−0.0163642	−0.00818211	−	0.999967i	$-0.502604\pi$
$863$	−0.480729	−0.0163642	−0.00818211	+	0.999967i	$0.502604\pi$
$864$	0	0
$865$	−22.5682	−0.767343
$866$	7.35381	0.249893
$867$	0	0
$868$	−2.19803	−0.0746059
$869$	1.30443	0.0442496
$870$	0	0
$871$	−76.5308	−2.59315
$872$	3.35450	0.113598
$873$	0	0
$874$	−20.2833	−0.686094
$875$	48.5712	1.64201
$876$	0	0
$877$	53.3468	1.80139	0.900697	−	0.434448i	$-0.143056\pi$
$877$	53.3468	1.80139	0.900697	+	0.434448i	$0.143056\pi$
$878$	−0.828774	−0.0279698
$879$	0	0
$880$	−12.6880	−0.427714
$881$	−8.91798	−0.300454	−0.150227	−	0.988651i	$-0.548001\pi$
$881$	−8.91798	−0.300454	−0.150227	+	0.988651i	$0.548001\pi$
$882$	0	0
$883$	−45.5949	−1.53439	−0.767195	−	0.641413i	$-0.778348\pi$
$883$	−45.5949	−1.53439	−0.767195	+	0.641413i	$0.778348\pi$
$884$	6.62511	0.222827
$885$	0	0
$886$	−16.2322	−0.545332
$887$	42.4249	1.42449	0.712244	−	0.701932i	$-0.247679\pi$
$887$	42.4249	1.42449	0.712244	+	0.701932i	$0.247679\pi$
$888$	0	0
$889$	4.43515	0.148750
$890$	−67.1327	−2.25029
$891$	0	0
$892$	−3.02205	−0.101186
$893$	0.291465	0.00975350
$894$	0	0
$895$	56.8330	1.89972
$896$	9.38985	0.313693
$897$	0	0
$898$	−4.66264	−0.155594
$899$	5.77469	0.192597
$900$	0	0
$901$	−21.5159	−0.716800
$902$	8.81419	0.293480
$903$	0	0
$904$	−12.1533	−0.404211
$905$	−43.6708	−1.45167
$906$	0	0
$907$	37.9421	1.25985	0.629924	−	0.776657i	$-0.283086\pi$
$907$	37.9421	1.25985	0.629924	+	0.776657i	$0.283086\pi$
$908$	8.02342	0.266266
$909$	0	0
$910$	−43.7526	−1.45038
$911$	−45.3984	−1.50412	−0.752059	−	0.659096i	$-0.770939\pi$
$911$	−45.3984	−1.50412	−0.752059	+	0.659096i	$0.770939\pi$
$912$	0	0
$913$	−5.40066	−0.178736
$914$	27.5659	0.911799
$915$	0	0
$916$	−8.79009	−0.290433
$917$	9.40162	0.310469
$918$	0	0
$919$	40.9833	1.35191	0.675956	−	0.736942i	$-0.263731\pi$
$919$	40.9833	1.35191	0.675956	+	0.736942i	$0.263731\pi$
$920$	−40.2511	−1.32704
$921$	0	0
$922$	25.6000	0.843092
$923$	58.6523	1.93056
$924$	0	0
$925$	79.5237	2.61472
$926$	−42.6286	−1.40086
$927$	0	0
$928$	3.64254	0.119572
$929$	25.3805	0.832706	0.416353	−	0.909203i	$-0.363308\pi$
$929$	25.3805	0.832706	0.416353	+	0.909203i	$0.363308\pi$
$930$	0	0
$931$	−25.5766	−0.838240
$932$	−0.251428	−0.00823579
$933$	0	0
$934$	17.7014	0.579208
$935$	−11.7606	−0.384612
$936$	0	0
$937$	23.4875	0.767303	0.383652	−	0.923478i	$-0.374666\pi$
$937$	23.4875	0.767303	0.383652	+	0.923478i	$0.374666\pi$
$938$	−24.0809	−0.786271
$939$	0	0
$940$	0.0998694	0.00325738
$941$	20.5134	0.668719	0.334360	−	0.942446i	$-0.391480\pi$
$941$	20.5134	0.668719	0.334360	+	0.942446i	$0.391480\pi$
$942$	0	0
$943$	21.8604	0.711872
$944$	18.7085	0.608910
$945$	0	0
$946$	8.85736	0.287978
$947$	−7.64514	−0.248434	−0.124217	−	0.992255i	$-0.539642\pi$
$947$	−7.64514	−0.248434	−0.124217	+	0.992255i	$0.539642\pi$
$948$	0	0
$949$	83.6740	2.71617
$950$	−84.4842	−2.74103
$951$	0	0
$952$	12.0732	0.391295
$953$	−48.8816	−1.58343	−0.791715	−	0.610891i	$-0.790811\pi$
$953$	−48.8816	−1.58343	−0.791715	+	0.610891i	$0.790811\pi$
$954$	0	0
$955$	77.7669	2.51648
$956$	−8.07905	−0.261295
$957$	0	0
$958$	−46.3190	−1.49650
$959$	−16.9246	−0.546523
$960$	0	0
$961$	−17.4772	−0.563782
$962$	−44.0734	−1.42098
$963$	0	0
$964$	−1.12529	−0.0362430
$965$	66.8872	2.15317
$966$	0	0
$967$	−35.5792	−1.14415	−0.572075	−	0.820202i	$-0.693861\pi$
$967$	−35.5792	−1.14415	−0.572075	+	0.820202i	$0.693861\pi$
$968$	3.04112	0.0977455
$969$	0	0
$970$	−5.42809	−0.174285
$971$	−13.1235	−0.421154	−0.210577	−	0.977577i	$-0.567534\pi$
$971$	−13.1235	−0.421154	−0.210577	+	0.977577i	$0.567534\pi$
$972$	0	0
$973$	−15.9305	−0.510709
$974$	25.2892	0.810317
$975$	0	0
$976$	19.9976	0.640107
$977$	−7.98148	−0.255350	−0.127675	−	0.991816i	$-0.540751\pi$
$977$	−7.98148	−0.255350	−0.127675	+	0.991816i	$0.540751\pi$
$978$	0	0
$979$	12.5796	0.402045
$980$	−8.76375	−0.279948
$981$	0	0
$982$	−34.6438	−1.10553
$983$	−13.6685	−0.435957	−0.217979	−	0.975954i	$-0.569946\pi$
$983$	−13.6685	−0.435957	−0.217979	+	0.975954i	$0.569946\pi$
$984$	0	0
$985$	59.2172	1.88682
$986$	−5.47679	−0.174416
$987$	0	0
$988$	−12.3493	−0.392883
$989$	21.9675	0.698525
$990$	0	0
$991$	33.9392	1.07812	0.539058	−	0.842269i	$-0.318780\pi$
$991$	33.9392	1.07812	0.539058	+	0.842269i	$0.318780\pi$
$992$	8.52985	0.270823
$993$	0	0
$994$	18.4553	0.585368
$995$	−56.4291	−1.78892
$996$	0	0
$997$	−53.1315	−1.68269	−0.841346	−	0.540497i	$-0.818236\pi$
$997$	−53.1315	−1.68269	−0.841346	+	0.540497i	$0.818236\pi$
$998$	−20.0289	−0.634004
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8019.2.a.k.1.17		51
3.2	odd	2		8019.2.a.l.1.35		51
27.2	odd	18		297.2.j.c.166.6	yes	102
27.13	even	9		891.2.j.c.100.12		102
27.14	odd	18		297.2.j.c.34.6	✓	102
27.25	even	9		891.2.j.c.793.12		102

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
297.2.j.c.34.6	✓	102	27.14	odd	18
297.2.j.c.166.6	yes	102	27.2	odd	18
891.2.j.c.100.12		102	27.13	even	9
891.2.j.c.793.12		102	27.25	even	9
8019.2.a.k.1.17		51	1.1	even	1	trivial
8019.2.a.l.1.35		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)

\(f(q)\)	\(=\)	\(q-1.25801 q^{2} -0.417404 q^{4} -4.24212 q^{5} -1.43200 q^{7} +3.04112 q^{8} +O(q^{10})\)	\(q-1.25801 q^{2} -0.417404 q^{4} -4.24212 q^{5} -1.43200 q^{7} +3.04112 q^{8} +5.33665 q^{10} -1.00000 q^{11} +5.72521 q^{13} +1.80148 q^{14} -2.99097 q^{16} -2.77233 q^{17} +5.16765 q^{19} +1.77068 q^{20} +1.25801 q^{22} +3.12004 q^{23} +12.9956 q^{25} -7.20239 q^{26} +0.597724 q^{28} -1.57035 q^{29} -3.67733 q^{31} -2.31958 q^{32} +3.48763 q^{34} +6.07473 q^{35} +6.11927 q^{37} -6.50098 q^{38} -12.9008 q^{40} +7.00644 q^{41} +7.04076 q^{43} +0.417404 q^{44} -3.92505 q^{46} +0.0564017 q^{47} -4.94937 q^{49} -16.3486 q^{50} -2.38973 q^{52} +7.76096 q^{53} +4.24212 q^{55} -4.35490 q^{56} +1.97552 q^{58} -6.25500 q^{59} -6.68600 q^{61} +4.62613 q^{62} +8.89999 q^{64} -24.2871 q^{65} -13.3673 q^{67} +1.15718 q^{68} -7.64209 q^{70} +10.2446 q^{71} +14.6150 q^{73} -7.69812 q^{74} -2.15700 q^{76} +1.43200 q^{77} -1.30443 q^{79} +12.6880 q^{80} -8.81419 q^{82} +5.40066 q^{83} +11.7606 q^{85} -8.85736 q^{86} -3.04112 q^{88} -12.5796 q^{89} -8.19852 q^{91} -1.30232 q^{92} -0.0709541 q^{94} -21.9218 q^{95} -1.01713 q^{97} +6.22637 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 51 q + 60 q^{4} - 6 q^{5} + 12 q^{7}+O(q^{10}) \) 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

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