LMFDB - Embedded newform 8019.2.a.l.1.10

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			1.10
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)

$f(q)$	$=$	$q-1.89185 q^{2} +1.57908 q^{4} -1.88472 q^{5} +1.84735 q^{7} +0.796313 q^{8} +O(q^{10})$	\(q-1.89185 q^{2} +1.57908 q^{4} -1.88472 q^{5} +1.84735 q^{7} +0.796313 q^{8} +3.56560 q^{10} +1.00000 q^{11} +1.98640 q^{13} -3.49489 q^{14} -4.66466 q^{16} -2.23532 q^{17} -3.48563 q^{19} -2.97612 q^{20} -1.89185 q^{22} +3.70259 q^{23} -1.44784 q^{25} -3.75797 q^{26} +2.91711 q^{28} -9.14890 q^{29} +9.19902 q^{31} +7.23220 q^{32} +4.22888 q^{34} -3.48172 q^{35} +10.4162 q^{37} +6.59427 q^{38} -1.50083 q^{40} -4.53632 q^{41} +1.35919 q^{43} +1.57908 q^{44} -7.00473 q^{46} -1.15617 q^{47} -3.58732 q^{49} +2.73909 q^{50} +3.13669 q^{52} +0.152162 q^{53} -1.88472 q^{55} +1.47107 q^{56} +17.3083 q^{58} -0.0220593 q^{59} -8.35779 q^{61} -17.4031 q^{62} -4.35288 q^{64} -3.74381 q^{65} -1.58609 q^{67} -3.52975 q^{68} +6.58689 q^{70} -12.7393 q^{71} +12.4829 q^{73} -19.7059 q^{74} -5.50409 q^{76} +1.84735 q^{77} +12.6768 q^{79} +8.79158 q^{80} +8.58201 q^{82} +4.17369 q^{83} +4.21294 q^{85} -2.57138 q^{86} +0.796313 q^{88} +0.143101 q^{89} +3.66957 q^{91} +5.84669 q^{92} +2.18730 q^{94} +6.56942 q^{95} +2.06006 q^{97} +6.78665 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.89185	−1.33774	−0.668869	−	0.743381i	$-0.733221\pi$
$2$	−1.89185	−1.33774	−0.668869	+	0.743381i	$0.733221\pi$
$3$	0	0
$3$	0	0
$4$	1.57908	0.789541
$5$	−1.88472	−0.842871	−0.421436	−	0.906858i	$-0.638474\pi$
$5$	−1.88472	−0.842871	−0.421436	+	0.906858i	$0.638474\pi$
$6$	0	0
$7$	1.84735	0.698231	0.349115	−	0.937080i	$-0.386482\pi$
$7$	1.84735	0.698231	0.349115	+	0.937080i	$0.386482\pi$
$8$	0.796313	0.281539
$9$	0	0
$10$	3.56560	1.12754
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.98640	0.550929	0.275465	−	0.961311i	$-0.411168\pi$
$13$	1.98640	0.550929	0.275465	+	0.961311i	$0.411168\pi$
$14$	−3.49489	−0.934049
$15$	0	0
$16$	−4.66466	−1.16617
$17$	−2.23532	−0.542144	−0.271072	−	0.962559i	$-0.587378\pi$
$17$	−2.23532	−0.542144	−0.271072	+	0.962559i	$0.587378\pi$
$18$	0	0
$19$	−3.48563	−0.799657	−0.399829	−	0.916590i	$-0.630930\pi$
$19$	−3.48563	−0.799657	−0.399829	+	0.916590i	$0.630930\pi$
$20$	−2.97612	−0.665481
$21$	0	0
$22$	−1.89185	−0.403343
$23$	3.70259	0.772044	0.386022	−	0.922490i	$-0.373849\pi$
$23$	3.70259	0.772044	0.386022	+	0.922490i	$0.373849\pi$
$24$	0	0
$25$	−1.44784	−0.289568
$26$	−3.75797	−0.736998
$27$	0	0
$28$	2.91711	0.551282
$29$	−9.14890	−1.69891	−0.849454	−	0.527662i	$-0.823069\pi$
$29$	−9.14890	−1.69891	−0.849454	+	0.527662i	$0.823069\pi$
$30$	0	0
$31$	9.19902	1.65219	0.826097	−	0.563528i	$-0.190556\pi$
$31$	9.19902	1.65219	0.826097	+	0.563528i	$0.190556\pi$
$32$	7.23220	1.27848
$33$	0	0
$34$	4.22888	0.725247
$35$	−3.48172	−0.588519
$36$	0	0
$37$	10.4162	1.71242	0.856208	−	0.516631i	$-0.172814\pi$
$37$	10.4162	1.71242	0.856208	+	0.516631i	$0.172814\pi$
$38$	6.59427	1.06973
$39$	0	0
$40$	−1.50083	−0.237301
$41$	−4.53632	−0.708454	−0.354227	−	0.935160i	$-0.615256\pi$
$41$	−4.53632	−0.708454	−0.354227	+	0.935160i	$0.615256\pi$
$42$	0	0
$43$	1.35919	0.207275	0.103638	−	0.994615i	$-0.466952\pi$
$43$	1.35919	0.207275	0.103638	+	0.994615i	$0.466952\pi$
$44$	1.57908	0.238055
$45$	0	0
$46$	−7.00473	−1.03279
$47$	−1.15617	−0.168645	−0.0843225	−	0.996439i	$-0.526873\pi$
$47$	−1.15617	−0.168645	−0.0843225	+	0.996439i	$0.526873\pi$
$48$	0	0
$49$	−3.58732	−0.512474
$50$	2.73909	0.387366
$51$	0	0
$52$	3.13669	0.434981
$53$	0.152162	0.0209011	0.0104505	−	0.999945i	$-0.496673\pi$
$53$	0.152162	0.0209011	0.0104505	+	0.999945i	$0.496673\pi$
$54$	0	0
$55$	−1.88472	−0.254135
$56$	1.47107	0.196579
$57$	0	0
$58$	17.3083	2.27269
$59$	−0.0220593	−0.00287188	−0.00143594	−	0.999999i	$-0.500457\pi$
$59$	−0.0220593	−0.00287188	−0.00143594	+	0.999999i	$0.500457\pi$
$60$	0	0
$61$	−8.35779	−1.07011	−0.535053	−	0.844819i	$-0.679708\pi$
$61$	−8.35779	−1.07011	−0.535053	+	0.844819i	$0.679708\pi$
$62$	−17.4031	−2.21020
$63$	0	0
$64$	−4.35288	−0.544110
$65$	−3.74381	−0.464362
$66$	0	0
$67$	−1.58609	−0.193772	−0.0968859	−	0.995295i	$-0.530888\pi$
$67$	−1.58609	−0.193772	−0.0968859	+	0.995295i	$0.530888\pi$
$68$	−3.52975	−0.428045
$69$	0	0
$70$	6.58689	0.787283
$71$	−12.7393	−1.51188	−0.755938	−	0.654643i	$-0.772819\pi$
$71$	−12.7393	−1.51188	−0.755938	+	0.654643i	$0.772819\pi$
$72$	0	0
$73$	12.4829	1.46102	0.730508	−	0.682904i	$-0.239283\pi$
$73$	12.4829	1.46102	0.730508	+	0.682904i	$0.239283\pi$
$74$	−19.7059	−2.29076
$75$	0	0
$76$	−5.50409	−0.631362
$77$	1.84735	0.210525
$78$	0	0
$79$	12.6768	1.42625	0.713127	−	0.701035i	$-0.247278\pi$
$79$	12.6768	1.42625	0.713127	+	0.701035i	$0.247278\pi$
$80$	8.79158	0.982928
$81$	0	0
$82$	8.58201	0.947725
$83$	4.17369	0.458122	0.229061	−	0.973412i	$-0.426435\pi$
$83$	4.17369	0.458122	0.229061	+	0.973412i	$0.426435\pi$
$84$	0	0
$85$	4.21294	0.456958
$86$	−2.57138	−0.277279
$87$	0	0
$88$	0.796313	0.0848873
$89$	0.143101	0.0151686	0.00758432	−	0.999971i	$-0.497586\pi$
$89$	0.143101	0.0151686	0.00758432	+	0.999971i	$0.497586\pi$
$90$	0	0
$91$	3.66957	0.384676
$92$	5.84669	0.609560
$93$	0	0
$94$	2.18730	0.225603
$95$	6.56942	0.674008
$96$	0	0
$97$	2.06006	0.209167	0.104584	−	0.994516i	$-0.466649\pi$
$97$	2.06006	0.209167	0.104584	+	0.994516i	$0.466649\pi$
$98$	6.78665	0.685555
$99$	0	0
$100$	−2.28626	−0.228626
$101$	−2.56986	−0.255710	−0.127855	−	0.991793i	$-0.540809\pi$
$101$	−2.56986	−0.255710	−0.127855	+	0.991793i	$0.540809\pi$
$102$	0	0
$103$	16.4157	1.61749	0.808746	−	0.588159i	$-0.200147\pi$
$103$	16.4157	1.61749	0.808746	+	0.588159i	$0.200147\pi$
$104$	1.58180	0.155108
$105$	0	0
$106$	−0.287867	−0.0279601
$107$	11.5234	1.11401	0.557006	−	0.830508i	$-0.311950\pi$
$107$	11.5234	1.11401	0.557006	+	0.830508i	$0.311950\pi$
$108$	0	0
$109$	−1.39439	−0.133558	−0.0667792	−	0.997768i	$-0.521272\pi$
$109$	−1.39439	−0.133558	−0.0667792	+	0.997768i	$0.521272\pi$
$110$	3.56560	0.339966
$111$	0	0
$112$	−8.61725	−0.814253
$113$	3.36168	0.316241	0.158120	−	0.987420i	$-0.449457\pi$
$113$	3.36168	0.316241	0.158120	+	0.987420i	$0.449457\pi$
$114$	0	0
$115$	−6.97834	−0.650734
$116$	−14.4469	−1.34136
$117$	0	0
$118$	0.0417328	0.00384182
$119$	−4.12940	−0.378542
$120$	0	0
$121$	1.00000	0.0909091
$122$	15.8117	1.43152
$123$	0	0
$124$	14.5260	1.30447
$125$	12.1524	1.08694
$126$	0	0
$127$	−6.26926	−0.556307	−0.278153	−	0.960537i	$-0.589722\pi$
$127$	−6.26926	−0.556307	−0.278153	+	0.960537i	$0.589722\pi$
$128$	−6.22942	−0.550608
$129$	0	0
$130$	7.08271	0.621195
$131$	−1.36777	−0.119502	−0.0597512	−	0.998213i	$-0.519031\pi$
$131$	−1.36777	−0.119502	−0.0597512	+	0.998213i	$0.519031\pi$
$132$	0	0
$133$	−6.43915	−0.558345
$134$	3.00064	0.259216
$135$	0	0
$136$	−1.78001	−0.152635
$137$	−13.1570	−1.12407	−0.562037	−	0.827112i	$-0.689982\pi$
$137$	−13.1570	−1.12407	−0.562037	+	0.827112i	$0.689982\pi$
$138$	0	0
$139$	19.4507	1.64979	0.824893	−	0.565289i	$-0.191236\pi$
$139$	19.4507	1.64979	0.824893	+	0.565289i	$0.191236\pi$
$140$	−5.49793	−0.464660
$141$	0	0
$142$	24.1008	2.02249
$143$	1.98640	0.166111
$144$	0	0
$145$	17.2431	1.43196
$146$	−23.6158	−1.95446
$147$	0	0
$148$	16.4481	1.35202
$149$	−11.5371	−0.945152	−0.472576	−	0.881290i	$-0.656676\pi$
$149$	−11.5371	−0.945152	−0.472576	+	0.881290i	$0.656676\pi$
$150$	0	0
$151$	−6.42843	−0.523138	−0.261569	−	0.965185i	$-0.584240\pi$
$151$	−6.42843	−0.523138	−0.261569	+	0.965185i	$0.584240\pi$
$152$	−2.77565	−0.225135
$153$	0	0
$154$	−3.49489	−0.281626
$155$	−17.3376	−1.39259
$156$	0	0
$157$	1.05064	0.0838502	0.0419251	−	0.999121i	$-0.486651\pi$
$157$	1.05064	0.0838502	0.0419251	+	0.999121i	$0.486651\pi$
$158$	−23.9826	−1.90795
$159$	0	0
$160$	−13.6307	−1.07760
$161$	6.83996	0.539065
$162$	0	0
$163$	4.58907	0.359444	0.179722	−	0.983717i	$-0.442480\pi$
$163$	4.58907	0.359444	0.179722	+	0.983717i	$0.442480\pi$
$164$	−7.16321	−0.559353
$165$	0	0
$166$	−7.89597	−0.612847
$167$	23.3748	1.80880	0.904399	−	0.426688i	$-0.140320\pi$
$167$	23.3748	1.80880	0.904399	+	0.426688i	$0.140320\pi$
$168$	0	0
$169$	−9.05420	−0.696477
$170$	−7.97024	−0.611290
$171$	0	0
$172$	2.14628	0.163652
$173$	−20.3888	−1.55013	−0.775067	−	0.631879i	$-0.782284\pi$
$173$	−20.3888	−1.55013	−0.775067	+	0.631879i	$0.782284\pi$
$174$	0	0
$175$	−2.67466	−0.202185
$176$	−4.66466	−0.351612
$177$	0	0
$178$	−0.270724	−0.0202917
$179$	22.9418	1.71475	0.857375	−	0.514692i	$-0.172094\pi$
$179$	22.9418	1.71475	0.857375	+	0.514692i	$0.172094\pi$
$180$	0	0
$181$	0.315327	0.0234381	0.0117190	−	0.999931i	$-0.496270\pi$
$181$	0.315327	0.0234381	0.0117190	+	0.999931i	$0.496270\pi$
$182$	−6.94227	−0.514595
$183$	0	0
$184$	2.94842	0.217361
$185$	−19.6316	−1.44335
$186$	0	0
$187$	−2.23532	−0.163463
$188$	−1.82569	−0.133152
$189$	0	0
$190$	−12.4283	−0.901646
$191$	−8.05004	−0.582480	−0.291240	−	0.956650i	$-0.594068\pi$
$191$	−8.05004	−0.582480	−0.291240	+	0.956650i	$0.594068\pi$
$192$	0	0
$193$	−13.1424	−0.946008	−0.473004	−	0.881060i	$-0.656830\pi$
$193$	−13.1424	−0.946008	−0.473004	+	0.881060i	$0.656830\pi$
$194$	−3.89732	−0.279811
$195$	0	0
$196$	−5.66466	−0.404619
$197$	12.4887	0.889782	0.444891	−	0.895585i	$-0.353242\pi$
$197$	12.4887	0.889782	0.444891	+	0.895585i	$0.353242\pi$
$198$	0	0
$199$	−0.342694	−0.0242930	−0.0121465	−	0.999926i	$-0.503866\pi$
$199$	−0.342694	−0.0242930	−0.0121465	+	0.999926i	$0.503866\pi$
$200$	−1.15293	−0.0815247
$201$	0	0
$202$	4.86177	0.342073
$203$	−16.9012	−1.18623
$204$	0	0
$205$	8.54968	0.597135
$206$	−31.0561	−2.16378
$207$	0	0
$208$	−9.26591	−0.642475
$209$	−3.48563	−0.241106
$210$	0	0
$211$	−22.2038	−1.52857	−0.764286	−	0.644878i	$-0.776908\pi$
$211$	−22.2038	−1.52857	−0.764286	+	0.644878i	$0.776908\pi$
$212$	0.240276	0.0165022
$213$	0	0
$214$	−21.8006	−1.49026
$215$	−2.56170	−0.174706
$216$	0	0
$217$	16.9938	1.15361
$218$	2.63797	0.178666
$219$	0	0
$220$	−2.97612	−0.200650
$221$	−4.44024	−0.298683
$222$	0	0
$223$	−24.7559	−1.65777	−0.828887	−	0.559416i	$-0.811026\pi$
$223$	−24.7559	−1.65777	−0.828887	+	0.559416i	$0.811026\pi$
$224$	13.3604	0.892677
$225$	0	0
$226$	−6.35979	−0.423047
$227$	−4.01442	−0.266446	−0.133223	−	0.991086i	$-0.542533\pi$
$227$	−4.01442	−0.266446	−0.133223	+	0.991086i	$0.542533\pi$
$228$	0	0
$229$	−14.6364	−0.967202	−0.483601	−	0.875289i	$-0.660671\pi$
$229$	−14.6364	−0.967202	−0.483601	+	0.875289i	$0.660671\pi$
$230$	13.2019	0.870510
$231$	0	0
$232$	−7.28539	−0.478309
$233$	9.92922	0.650485	0.325242	−	0.945631i	$-0.394554\pi$
$233$	9.92922	0.650485	0.325242	+	0.945631i	$0.394554\pi$
$234$	0	0
$235$	2.17906	0.142146
$236$	−0.0348334	−0.00226746
$237$	0	0
$238$	7.81220	0.506390
$239$	10.5412	0.681856	0.340928	−	0.940089i	$-0.389259\pi$
$239$	10.5412	0.681856	0.340928	+	0.940089i	$0.389259\pi$
$240$	0	0
$241$	−6.65221	−0.428507	−0.214253	−	0.976778i	$-0.568732\pi$
$241$	−6.65221	−0.428507	−0.214253	+	0.976778i	$0.568732\pi$
$242$	−1.89185	−0.121612
$243$	0	0
$244$	−13.1976	−0.844892
$245$	6.76108	0.431949
$246$	0	0
$247$	−6.92386	−0.440555
$248$	7.32530	0.465157
$249$	0	0
$250$	−22.9904	−1.45404
$251$	18.6200	1.17529	0.587643	−	0.809120i	$-0.300056\pi$
$251$	18.6200	1.17529	0.587643	+	0.809120i	$0.300056\pi$
$252$	0	0
$253$	3.70259	0.232780
$254$	11.8605	0.744193
$255$	0	0
$256$	20.4909	1.28068
$257$	24.9231	1.55466	0.777329	−	0.629094i	$-0.216574\pi$
$257$	24.9231	1.55466	0.777329	+	0.629094i	$0.216574\pi$
$258$	0	0
$259$	19.2424	1.19566
$260$	−5.91178	−0.366633
$261$	0	0
$262$	2.58761	0.159863
$263$	16.1417	0.995340	0.497670	−	0.867366i	$-0.334189\pi$
$263$	16.1417	0.995340	0.497670	+	0.867366i	$0.334189\pi$
$264$	0	0
$265$	−0.286782	−0.0176169
$266$	12.1819	0.746919
$267$	0	0
$268$	−2.50456	−0.152991
$269$	−25.1306	−1.53224	−0.766119	−	0.642699i	$-0.777815\pi$
$269$	−25.1306	−1.53224	−0.766119	+	0.642699i	$0.777815\pi$
$270$	0	0
$271$	−26.4133	−1.60449	−0.802247	−	0.596993i	$-0.796362\pi$
$271$	−26.4133	−1.60449	−0.802247	+	0.596993i	$0.796362\pi$
$272$	10.4270	0.632230
$273$	0	0
$274$	24.8909	1.50372
$275$	−1.44784	−0.0873080
$276$	0	0
$277$	17.2946	1.03913	0.519566	−	0.854430i	$-0.326094\pi$
$277$	17.2946	1.03913	0.519566	+	0.854430i	$0.326094\pi$
$278$	−36.7977	−2.20698
$279$	0	0
$280$	−2.77254	−0.165691
$281$	−28.3391	−1.69057	−0.845284	−	0.534317i	$-0.820569\pi$
$281$	−28.3391	−1.69057	−0.845284	+	0.534317i	$0.820569\pi$
$282$	0	0
$283$	15.9283	0.946840	0.473420	−	0.880837i	$-0.343019\pi$
$283$	15.9283	0.946840	0.473420	+	0.880837i	$0.343019\pi$
$284$	−20.1164	−1.19369
$285$	0	0
$286$	−3.75797	−0.222213
$287$	−8.38014	−0.494664
$288$	0	0
$289$	−12.0034	−0.706079
$290$	−32.6213	−1.91559
$291$	0	0
$292$	19.7116	1.15353
$293$	−12.3004	−0.718600	−0.359300	−	0.933222i	$-0.616984\pi$
$293$	−12.3004	−0.718600	−0.359300	+	0.933222i	$0.616984\pi$
$294$	0	0
$295$	0.0415756	0.00242062
$296$	8.29458	0.482113
$297$	0	0
$298$	21.8263	1.26437
$299$	7.35484	0.425341
$300$	0	0
$301$	2.51090	0.144726
$302$	12.1616	0.699821
$303$	0	0
$304$	16.2593	0.932534
$305$	15.7521	0.901962
$306$	0	0
$307$	−4.50360	−0.257034	−0.128517	−	0.991707i	$-0.541022\pi$
$307$	−4.50360	−0.257034	−0.128517	+	0.991707i	$0.541022\pi$
$308$	2.91711	0.166218
$309$	0	0
$310$	32.8000	1.86291
$311$	−13.5573	−0.768764	−0.384382	−	0.923174i	$-0.625586\pi$
$311$	−13.5573	−0.768764	−0.384382	+	0.923174i	$0.625586\pi$
$312$	0	0
$313$	−4.25970	−0.240772	−0.120386	−	0.992727i	$-0.538413\pi$
$313$	−4.25970	−0.240772	−0.120386	+	0.992727i	$0.538413\pi$
$314$	−1.98765	−0.112170
$315$	0	0
$316$	20.0177	1.12609
$317$	34.1606	1.91865	0.959324	−	0.282307i	$-0.0910997\pi$
$317$	34.1606	1.91865	0.959324	+	0.282307i	$0.0910997\pi$
$318$	0	0
$319$	−9.14890	−0.512240
$320$	8.20395	0.458615
$321$	0	0
$322$	−12.9402	−0.721127
$323$	7.79148	0.433530
$324$	0	0
$325$	−2.87599	−0.159531
$326$	−8.68181	−0.480841
$327$	0	0
$328$	−3.61233	−0.199458
$329$	−2.13585	−0.117753
$330$	0	0
$331$	35.3422	1.94258	0.971292	−	0.237890i	$-0.0764557\pi$
$331$	35.3422	1.94258	0.971292	+	0.237890i	$0.0764557\pi$
$332$	6.59059	0.361706
$333$	0	0
$334$	−44.2216	−2.41970
$335$	2.98933	0.163325
$336$	0	0
$337$	6.35161	0.345994	0.172997	−	0.984922i	$-0.444655\pi$
$337$	6.35161	0.345994	0.172997	+	0.984922i	$0.444655\pi$
$338$	17.1292	0.931703
$339$	0	0
$340$	6.65258	0.360787
$341$	9.19902	0.498155
$342$	0	0
$343$	−19.5584	−1.05606
$344$	1.08234	0.0583561
$345$	0	0
$346$	38.5725	2.07367
$347$	−12.0287	−0.645735	−0.322868	−	0.946444i	$-0.604647\pi$
$347$	−12.0287	−0.645735	−0.322868	+	0.946444i	$0.604647\pi$
$348$	0	0
$349$	−0.921513	−0.0493274	−0.0246637	−	0.999696i	$-0.507852\pi$
$349$	−0.921513	−0.0493274	−0.0246637	+	0.999696i	$0.507852\pi$
$350$	5.06004	0.270471
$351$	0	0
$352$	7.23220	0.385478
$353$	−23.1444	−1.23185	−0.615926	−	0.787804i	$-0.711218\pi$
$353$	−23.1444	−1.23185	−0.615926	+	0.787804i	$0.711218\pi$
$354$	0	0
$355$	24.0100	1.27432
$356$	0.225968	0.0119763
$357$	0	0
$358$	−43.4024	−2.29389
$359$	32.4052	1.71028	0.855140	−	0.518397i	$-0.173471\pi$
$359$	32.4052	1.71028	0.855140	+	0.518397i	$0.173471\pi$
$360$	0	0
$361$	−6.85041	−0.360548
$362$	−0.596550	−0.0313540
$363$	0	0
$364$	5.79455	0.303717
$365$	−23.5268	−1.23145
$366$	0	0
$367$	−13.7803	−0.719324	−0.359662	−	0.933083i	$-0.617108\pi$
$367$	−13.7803	−0.719324	−0.359662	+	0.933083i	$0.617108\pi$
$368$	−17.2713	−0.900331
$369$	0	0
$370$	37.1400	1.93082
$371$	0.281096	0.0145938
$372$	0	0
$373$	36.1703	1.87283	0.936414	−	0.350898i	$-0.114124\pi$
$373$	36.1703	1.87283	0.936414	+	0.350898i	$0.114124\pi$
$374$	4.22888	0.218670
$375$	0	0
$376$	−0.920675	−0.0474802
$377$	−18.1734	−0.935978
$378$	0	0
$379$	29.9569	1.53878	0.769392	−	0.638777i	$-0.220559\pi$
$379$	29.9569	1.53878	0.769392	+	0.638777i	$0.220559\pi$
$380$	10.3737	0.532157
$381$	0	0
$382$	15.2294	0.779205
$383$	−12.6943	−0.648648	−0.324324	−	0.945946i	$-0.605137\pi$
$383$	−12.6943	−0.648648	−0.324324	+	0.945946i	$0.605137\pi$
$384$	0	0
$385$	−3.48172	−0.177445
$386$	24.8633	1.26551
$387$	0	0
$388$	3.25300	0.165146
$389$	−25.9380	−1.31511	−0.657555	−	0.753407i	$-0.728409\pi$
$389$	−25.9380	−1.31511	−0.657555	+	0.753407i	$0.728409\pi$
$390$	0	0
$391$	−8.27647	−0.418559
$392$	−2.85663	−0.144281
$393$	0	0
$394$	−23.6267	−1.19029
$395$	−23.8922	−1.20215
$396$	0	0
$397$	3.90320	0.195896	0.0979480	−	0.995192i	$-0.468772\pi$
$397$	3.90320	0.195896	0.0979480	+	0.995192i	$0.468772\pi$
$398$	0.648325	0.0324976
$399$	0	0
$400$	6.75368	0.337684
$401$	−23.4155	−1.16931	−0.584657	−	0.811281i	$-0.698771\pi$
$401$	−23.4155	−1.16931	−0.584657	+	0.811281i	$0.698771\pi$
$402$	0	0
$403$	18.2730	0.910241
$404$	−4.05801	−0.201894
$405$	0	0
$406$	31.9744	1.58686
$407$	10.4162	0.516313
$408$	0	0
$409$	8.66040	0.428229	0.214115	−	0.976809i	$-0.431313\pi$
$409$	8.66040	0.428229	0.214115	+	0.976809i	$0.431313\pi$
$410$	−16.1747	−0.798810
$411$	0	0
$412$	25.9218	1.27707
$413$	−0.0407511	−0.00200523
$414$	0	0
$415$	−7.86622	−0.386138
$416$	14.3661	0.704354
$417$	0	0
$418$	6.59427	0.322536
$419$	25.6163	1.25144	0.625718	−	0.780049i	$-0.284806\pi$
$419$	25.6163	1.25144	0.625718	+	0.780049i	$0.284806\pi$
$420$	0	0
$421$	34.3086	1.67210	0.836049	−	0.548654i	$-0.184860\pi$
$421$	34.3086	1.67210	0.836049	+	0.548654i	$0.184860\pi$
$422$	42.0061	2.04483
$423$	0	0
$424$	0.121169	0.00588447
$425$	3.23638	0.156988
$426$	0	0
$427$	−15.4397	−0.747181
$428$	18.1964	0.879558
$429$	0	0
$430$	4.84633	0.233711
$431$	0.378339	0.0182240	0.00911198	−	0.999958i	$-0.497100\pi$
$431$	0.378339	0.0182240	0.00911198	+	0.999958i	$0.497100\pi$
$432$	0	0
$433$	1.78131	0.0856042	0.0428021	−	0.999084i	$-0.486371\pi$
$433$	1.78131	0.0856042	0.0428021	+	0.999084i	$0.486371\pi$
$434$	−32.1496	−1.54323
$435$	0	0
$436$	−2.20186	−0.105450
$437$	−12.9058	−0.617370
$438$	0	0
$439$	−4.51080	−0.215289	−0.107644	−	0.994189i	$-0.534331\pi$
$439$	−4.51080	−0.215289	−0.107644	+	0.994189i	$0.534331\pi$
$440$	−1.50083	−0.0715491
$441$	0	0
$442$	8.40026	0.399560
$443$	27.6237	1.31244	0.656221	−	0.754569i	$-0.272154\pi$
$443$	27.6237	1.31244	0.656221	+	0.754569i	$0.272154\pi$
$444$	0	0
$445$	−0.269704	−0.0127852
$446$	46.8343	2.21767
$447$	0	0
$448$	−8.04127	−0.379914
$449$	6.93400	0.327236	0.163618	−	0.986524i	$-0.447684\pi$
$449$	6.93400	0.327236	0.163618	+	0.986524i	$0.447684\pi$
$450$	0	0
$451$	−4.53632	−0.213607
$452$	5.30837	0.249685
$453$	0	0
$454$	7.59466	0.356435
$455$	−6.91611	−0.324232
$456$	0	0
$457$	33.5955	1.57153	0.785765	−	0.618525i	$-0.212270\pi$
$457$	33.5955	1.57153	0.785765	+	0.618525i	$0.212270\pi$
$458$	27.6899	1.29386
$459$	0	0
$460$	−11.0194	−0.513781
$461$	19.4587	0.906284	0.453142	−	0.891438i	$-0.350303\pi$
$461$	19.4587	0.906284	0.453142	+	0.891438i	$0.350303\pi$
$462$	0	0
$463$	28.2815	1.31435	0.657176	−	0.753738i	$-0.271751\pi$
$463$	28.2815	1.31435	0.657176	+	0.753738i	$0.271751\pi$
$464$	42.6766	1.98121
$465$	0	0
$466$	−18.7846	−0.870178
$467$	16.1534	0.747491	0.373746	−	0.927531i	$-0.378073\pi$
$467$	16.1534	0.747491	0.373746	+	0.927531i	$0.378073\pi$
$468$	0	0
$469$	−2.93006	−0.135297
$470$	−4.12244	−0.190154
$471$	0	0
$472$	−0.0175661	−0.000808546 0
$473$	1.35919	0.0624958
$474$	0	0
$475$	5.04662	0.231555
$476$	−6.52067	−0.298874
$477$	0	0
$478$	−19.9424	−0.912144
$479$	16.6801	0.762135	0.381067	−	0.924547i	$-0.375557\pi$
$479$	16.6801	0.762135	0.381067	+	0.924547i	$0.375557\pi$
$480$	0	0
$481$	20.6908	0.943420
$482$	12.5850	0.573229
$483$	0	0
$484$	1.57908	0.0717764
$485$	−3.88263	−0.176301
$486$	0	0
$487$	−30.2636	−1.37137	−0.685687	−	0.727897i	$-0.740498\pi$
$487$	−30.2636	−1.37137	−0.685687	+	0.727897i	$0.740498\pi$
$488$	−6.65542	−0.301277
$489$	0	0
$490$	−12.7909	−0.577835
$491$	−26.5806	−1.19957	−0.599783	−	0.800162i	$-0.704747\pi$
$491$	−26.5806	−1.19957	−0.599783	+	0.800162i	$0.704747\pi$
$492$	0	0
$493$	20.4507	0.921054
$494$	13.0989	0.589346
$495$	0	0
$496$	−42.9104	−1.92673
$497$	−23.5339	−1.05564
$498$	0	0
$499$	38.8183	1.73775	0.868873	−	0.495035i	$-0.164845\pi$
$499$	38.8183	1.73775	0.868873	+	0.495035i	$0.164845\pi$
$500$	19.1896	0.858183
$501$	0	0
$502$	−35.2262	−1.57222
$503$	29.8325	1.33017	0.665083	−	0.746769i	$-0.268396\pi$
$503$	29.8325	1.33017	0.665083	+	0.746769i	$0.268396\pi$
$504$	0	0
$505$	4.84345	0.215531
$506$	−7.00473	−0.311398
$507$	0	0
$508$	−9.89967	−0.439227
$509$	27.5611	1.22162	0.610812	−	0.791776i	$-0.290843\pi$
$509$	27.5611	1.22162	0.610812	+	0.791776i	$0.290843\pi$
$510$	0	0
$511$	23.0603	1.02013
$512$	−26.3067	−1.16260
$513$	0	0
$514$	−47.1506	−2.07972
$515$	−30.9390	−1.36334
$516$	0	0
$517$	−1.15617	−0.0508484
$518$	−36.4036	−1.59948
$519$	0	0
$520$	−2.98125	−0.130736
$521$	4.16421	0.182437	0.0912187	−	0.995831i	$-0.470924\pi$
$521$	4.16421	0.182437	0.0912187	+	0.995831i	$0.470924\pi$
$522$	0	0
$523$	−7.59511	−0.332111	−0.166056	−	0.986116i	$-0.553103\pi$
$523$	−7.59511	−0.332111	−0.166056	+	0.986116i	$0.553103\pi$
$524$	−2.15982	−0.0943520
$525$	0	0
$526$	−30.5376	−1.33150
$527$	−20.5627	−0.895727
$528$	0	0
$529$	−9.29082	−0.403949
$530$	0.542548	0.0235668
$531$	0	0
$532$	−10.1679	−0.440836
$533$	−9.01095	−0.390308
$534$	0	0
$535$	−21.7184	−0.938969
$536$	−1.26302	−0.0545543
$537$	0	0
$538$	47.5432	2.04973
$539$	−3.58732	−0.154517
$540$	0	0
$541$	−32.8221	−1.41113	−0.705566	−	0.708644i	$-0.749307\pi$
$541$	−32.8221	−1.41113	−0.705566	+	0.708644i	$0.749307\pi$
$542$	49.9699	2.14639
$543$	0	0
$544$	−16.1663	−0.693123
$545$	2.62803	0.112573
$546$	0	0
$547$	−5.30970	−0.227026	−0.113513	−	0.993536i	$-0.536210\pi$
$547$	−5.30970	−0.227026	−0.113513	+	0.993536i	$0.536210\pi$
$548$	−20.7759	−0.887503
$549$	0	0
$550$	2.73909	0.116795
$551$	31.8896	1.35854
$552$	0	0
$553$	23.4185	0.995854
$554$	−32.7187	−1.39009
$555$	0	0
$556$	30.7142	1.30257
$557$	0.774739	0.0328268	0.0164134	−	0.999865i	$-0.494775\pi$
$557$	0.774739	0.0328268	0.0164134	+	0.999865i	$0.494775\pi$
$558$	0	0
$559$	2.69991	0.114194
$560$	16.2411	0.686311
$561$	0	0
$562$	53.6132	2.26154
$563$	30.2169	1.27349	0.636746	−	0.771074i	$-0.280280\pi$
$563$	30.2169	1.27349	0.636746	+	0.771074i	$0.280280\pi$
$564$	0	0
$565$	−6.33582	−0.266550
$566$	−30.1339	−1.26662
$567$	0	0
$568$	−10.1445	−0.425653
$569$	−6.89883	−0.289214	−0.144607	−	0.989489i	$-0.546192\pi$
$569$	−6.89883	−0.289214	−0.144607	+	0.989489i	$0.546192\pi$
$570$	0	0
$571$	−7.82574	−0.327497	−0.163749	−	0.986502i	$-0.552359\pi$
$571$	−7.82574	−0.327497	−0.163749	+	0.986502i	$0.552359\pi$
$572$	3.13669	0.131152
$573$	0	0
$574$	15.8539	0.661731
$575$	−5.36076	−0.223559
$576$	0	0
$577$	−15.9767	−0.665118	−0.332559	−	0.943082i	$-0.607912\pi$
$577$	−15.9767	−0.665118	−0.332559	+	0.943082i	$0.607912\pi$
$578$	22.7085	0.944549
$579$	0	0
$580$	27.2283	1.13059
$581$	7.71024	0.319875
$582$	0	0
$583$	0.152162	0.00630191
$584$	9.94033	0.411334
$585$	0	0
$586$	23.2705	0.961297
$587$	−2.06603	−0.0852742	−0.0426371	−	0.999091i	$-0.513576\pi$
$587$	−2.06603	−0.0852742	−0.0426371	+	0.999091i	$0.513576\pi$
$588$	0	0
$589$	−32.0644	−1.32119
$590$	−0.0786546	−0.00323816
$591$	0	0
$592$	−48.5882	−1.99696
$593$	6.00920	0.246768	0.123384	−	0.992359i	$-0.460625\pi$
$593$	6.00920	0.246768	0.123384	+	0.992359i	$0.460625\pi$
$594$	0	0
$595$	7.78276	0.319062
$596$	−18.2179	−0.746236
$597$	0	0
$598$	−13.9142	−0.568995
$599$	7.94592	0.324662	0.162331	−	0.986736i	$-0.448099\pi$
$599$	7.94592	0.324662	0.162331	+	0.986736i	$0.448099\pi$
$600$	0	0
$601$	5.23675	0.213611	0.106806	−	0.994280i	$-0.465938\pi$
$601$	5.23675	0.213611	0.106806	+	0.994280i	$0.465938\pi$
$602$	−4.75023	−0.193605
$603$	0	0
$604$	−10.1510	−0.413039
$605$	−1.88472	−0.0766247
$606$	0	0
$607$	23.3824	0.949061	0.474531	−	0.880239i	$-0.342618\pi$
$607$	23.3824	0.949061	0.474531	+	0.880239i	$0.342618\pi$
$608$	−25.2087	−1.02235
$609$	0	0
$610$	−29.8005	−1.20659
$611$	−2.29662	−0.0929115
$612$	0	0
$613$	−20.1710	−0.814699	−0.407350	−	0.913272i	$-0.633547\pi$
$613$	−20.1710	−0.814699	−0.407350	+	0.913272i	$0.633547\pi$
$614$	8.52011	0.343844
$615$	0	0
$616$	1.47107	0.0592709
$617$	27.1175	1.09171	0.545854	−	0.837880i	$-0.316205\pi$
$617$	27.1175	1.09171	0.545854	+	0.837880i	$0.316205\pi$
$618$	0	0
$619$	44.6837	1.79599	0.897995	−	0.440006i	$-0.145024\pi$
$619$	44.6837	1.79599	0.897995	+	0.440006i	$0.145024\pi$
$620$	−27.3774	−1.09950
$621$	0	0
$622$	25.6483	1.02840
$623$	0.264356	0.0105912
$624$	0	0
$625$	−15.6646	−0.626583
$626$	8.05869	0.322090
$627$	0	0
$628$	1.65905	0.0662031
$629$	−23.2836	−0.928377
$630$	0	0
$631$	38.5624	1.53515	0.767573	−	0.640962i	$-0.221464\pi$
$631$	38.5624	1.53515	0.767573	+	0.640962i	$0.221464\pi$
$632$	10.0947	0.401546
$633$	0	0
$634$	−64.6265	−2.56665
$635$	11.8158	0.468895
$636$	0	0
$637$	−7.12586	−0.282337
$638$	17.3083	0.685243
$639$	0	0
$640$	11.7407	0.464092
$641$	28.2299	1.11501	0.557507	−	0.830172i	$-0.311758\pi$
$641$	28.2299	1.11501	0.557507	+	0.830172i	$0.311758\pi$
$642$	0	0
$643$	27.7704	1.09516	0.547579	−	0.836754i	$-0.315550\pi$
$643$	27.7704	1.09516	0.547579	+	0.836754i	$0.315550\pi$
$644$	10.8009	0.425613
$645$	0	0
$646$	−14.7403	−0.579949
$647$	13.0106	0.511499	0.255749	−	0.966743i	$-0.417678\pi$
$647$	13.0106	0.511499	0.255749	+	0.966743i	$0.417678\pi$
$648$	0	0
$649$	−0.0220593	−0.000865903 0
$650$	5.44093	0.213411
$651$	0	0
$652$	7.24651	0.283795
$653$	−42.4788	−1.66232	−0.831161	−	0.556031i	$-0.812323\pi$
$653$	−42.4788	−1.66232	−0.831161	+	0.556031i	$0.812323\pi$
$654$	0	0
$655$	2.57786	0.100725
$656$	21.1604	0.826175
$657$	0	0
$658$	4.04070	0.157523
$659$	28.2287	1.09963	0.549817	−	0.835285i	$-0.314697\pi$
$659$	28.2287	1.09963	0.549817	+	0.835285i	$0.314697\pi$
$660$	0	0
$661$	−11.1942	−0.435403	−0.217701	−	0.976015i	$-0.569856\pi$
$661$	−11.1942	−0.435403	−0.217701	+	0.976015i	$0.569856\pi$
$662$	−66.8620	−2.59867
$663$	0	0
$664$	3.32356	0.128979
$665$	12.1360	0.470613
$666$	0	0
$667$	−33.8746	−1.31163
$668$	36.9107	1.42812
$669$	0	0
$670$	−5.65536	−0.218485
$671$	−8.35779	−0.322649
$672$	0	0
$673$	23.2849	0.897568	0.448784	−	0.893640i	$-0.351857\pi$
$673$	23.2849	0.897568	0.448784	+	0.893640i	$0.351857\pi$
$674$	−12.0163	−0.462849
$675$	0	0
$676$	−14.2973	−0.549897
$677$	46.8955	1.80234	0.901170	−	0.433466i	$-0.142710\pi$
$677$	46.8955	1.80234	0.901170	+	0.433466i	$0.142710\pi$
$678$	0	0
$679$	3.80564	0.146047
$680$	3.35482	0.128652
$681$	0	0
$682$	−17.4031	−0.666401
$683$	17.9056	0.685138	0.342569	−	0.939493i	$-0.388703\pi$
$683$	17.9056	0.685138	0.342569	+	0.939493i	$0.388703\pi$
$684$	0	0
$685$	24.7971	0.947450
$686$	37.0015	1.41273
$687$	0	0
$688$	−6.34018	−0.241717
$689$	0.302255	0.0115150
$690$	0	0
$691$	−44.8077	−1.70457	−0.852283	−	0.523081i	$-0.824783\pi$
$691$	−44.8077	−1.70457	−0.852283	+	0.523081i	$0.824783\pi$
$692$	−32.1956	−1.22389
$693$	0	0
$694$	22.7565	0.863824
$695$	−36.6590	−1.39056
$696$	0	0
$697$	10.1401	0.384084
$698$	1.74336	0.0659871
$699$	0	0
$700$	−4.22350	−0.159633
$701$	45.1900	1.70680	0.853402	−	0.521253i	$-0.174535\pi$
$701$	45.1900	1.70680	0.853402	+	0.521253i	$0.174535\pi$
$702$	0	0
$703$	−36.3071	−1.36935
$704$	−4.35288	−0.164055
$705$	0	0
$706$	43.7856	1.64789
$707$	−4.74741	−0.178545
$708$	0	0
$709$	35.1939	1.32173	0.660867	−	0.750503i	$-0.270189\pi$
$709$	35.1939	1.32173	0.660867	+	0.750503i	$0.270189\pi$
$710$	−45.4232	−1.70470
$711$	0	0
$712$	0.113953	0.00427057
$713$	34.0602	1.27557
$714$	0	0
$715$	−3.74381	−0.140011
$716$	36.2270	1.35387
$717$	0	0
$718$	−61.3056	−2.28791
$719$	4.14463	0.154569	0.0772843	−	0.997009i	$-0.475375\pi$
$719$	4.14463	0.154569	0.0772843	+	0.997009i	$0.475375\pi$
$720$	0	0
$721$	30.3255	1.12938
$722$	12.9599	0.482318
$723$	0	0
$724$	0.497927	0.0185053
$725$	13.2461	0.491949
$726$	0	0
$727$	5.13877	0.190587	0.0952933	−	0.995449i	$-0.469621\pi$
$727$	5.13877	0.190587	0.0952933	+	0.995449i	$0.469621\pi$
$728$	2.92213	0.108301
$729$	0	0
$730$	44.5091	1.64736
$731$	−3.03823	−0.112373
$732$	0	0
$733$	−33.7645	−1.24712	−0.623560	−	0.781775i	$-0.714314\pi$
$733$	−33.7645	−1.24712	−0.623560	+	0.781775i	$0.714314\pi$
$734$	26.0702	0.962267
$735$	0	0
$736$	26.7779	0.987046
$737$	−1.58609	−0.0584244
$738$	0	0
$739$	43.3680	1.59532	0.797659	−	0.603109i	$-0.206071\pi$
$739$	43.3680	1.59532	0.797659	+	0.603109i	$0.206071\pi$
$740$	−31.0000	−1.13958
$741$	0	0
$742$	−0.531790	−0.0195226
$743$	−22.8304	−0.837566	−0.418783	−	0.908086i	$-0.637543\pi$
$743$	−22.8304	−0.837566	−0.418783	+	0.908086i	$0.637543\pi$
$744$	0	0
$745$	21.7441	0.796642
$746$	−68.4286	−2.50535
$747$	0	0
$748$	−3.52975	−0.129060
$749$	21.2878	0.777838
$750$	0	0
$751$	−10.2470	−0.373920	−0.186960	−	0.982368i	$-0.559863\pi$
$751$	−10.2470	−0.373920	−0.186960	+	0.982368i	$0.559863\pi$
$752$	5.39316	0.196668
$753$	0	0
$754$	34.3813	1.25209
$755$	12.1158	0.440938
$756$	0	0
$757$	−19.7162	−0.716599	−0.358299	−	0.933607i	$-0.616643\pi$
$757$	−19.7162	−0.716599	−0.358299	+	0.933607i	$0.616643\pi$
$758$	−56.6739	−2.05849
$759$	0	0
$760$	5.23132	0.189760
$761$	−17.4458	−0.632411	−0.316205	−	0.948691i	$-0.602409\pi$
$761$	−17.4458	−0.632411	−0.316205	+	0.948691i	$0.602409\pi$
$762$	0	0
$763$	−2.57592	−0.0932546
$764$	−12.7117	−0.459892
$765$	0	0
$766$	24.0156	0.867720
$767$	−0.0438187	−0.00158220
$768$	0	0
$769$	−29.0537	−1.04770	−0.523852	−	0.851809i	$-0.675506\pi$
$769$	−29.0537	−1.04770	−0.523852	+	0.851809i	$0.675506\pi$
$770$	6.58689	0.237375
$771$	0	0
$772$	−20.7529	−0.746912
$773$	−10.1054	−0.363466	−0.181733	−	0.983348i	$-0.558171\pi$
$773$	−10.1054	−0.363466	−0.181733	+	0.983348i	$0.558171\pi$
$774$	0	0
$775$	−13.3187	−0.478422
$776$	1.64045	0.0588889
$777$	0	0
$778$	49.0707	1.75927
$779$	15.8119	0.566520
$780$	0	0
$781$	−12.7393	−0.455848
$782$	15.6578	0.559922
$783$	0	0
$784$	16.7336	0.597630
$785$	−1.98016	−0.0706749
$786$	0	0
$787$	−42.0328	−1.49831	−0.749154	−	0.662396i	$-0.769540\pi$
$787$	−42.0328	−1.49831	−0.749154	+	0.662396i	$0.769540\pi$
$788$	19.7206	0.702519
$789$	0	0
$790$	45.2004	1.60816
$791$	6.21019	0.220809
$792$	0	0
$793$	−16.6019	−0.589552
$794$	−7.38425	−0.262057
$795$	0	0
$796$	−0.541142	−0.0191803
$797$	23.4811	0.831742	0.415871	−	0.909424i	$-0.363477\pi$
$797$	23.4811	0.831742	0.415871	+	0.909424i	$0.363477\pi$
$798$	0	0
$799$	2.58441	0.0914300
$800$	−10.4711	−0.370208
$801$	0	0
$802$	44.2985	1.56423
$803$	12.4829	0.440513
$804$	0	0
$805$	−12.8914	−0.454362
$806$	−34.5696	−1.21766
$807$	0	0
$808$	−2.04641	−0.0719924
$809$	−25.3285	−0.890504	−0.445252	−	0.895405i	$-0.646886\pi$
$809$	−25.3285	−0.890504	−0.445252	+	0.895405i	$0.646886\pi$
$810$	0	0
$811$	−5.26646	−0.184930	−0.0924652	−	0.995716i	$-0.529475\pi$
$811$	−5.26646	−0.184930	−0.0924652	+	0.995716i	$0.529475\pi$
$812$	−26.6883	−0.936577
$813$	0	0
$814$	−19.7059	−0.690691
$815$	−8.64910	−0.302965
$816$	0	0
$817$	−4.73764	−0.165749
$818$	−16.3841	−0.572858
$819$	0	0
$820$	13.5006	0.471463
$821$	−25.1273	−0.876949	−0.438475	−	0.898744i	$-0.644481\pi$
$821$	−25.1273	−0.876949	−0.438475	+	0.898744i	$0.644481\pi$
$822$	0	0
$823$	−7.08358	−0.246918	−0.123459	−	0.992350i	$-0.539399\pi$
$823$	−7.08358	−0.246918	−0.123459	+	0.992350i	$0.539399\pi$
$824$	13.0721	0.455387
$825$	0	0
$826$	0.0770949	0.00268247
$827$	13.8258	0.480769	0.240384	−	0.970678i	$-0.422726\pi$
$827$	13.8258	0.480769	0.240384	+	0.970678i	$0.422726\pi$
$828$	0	0
$829$	−8.31407	−0.288760	−0.144380	−	0.989522i	$-0.546119\pi$
$829$	−8.31407	−0.288760	−0.144380	+	0.989522i	$0.546119\pi$
$830$	14.8817	0.516551
$831$	0	0
$832$	−8.64658	−0.299766
$833$	8.01879	0.277835
$834$	0	0
$835$	−44.0549	−1.52458
$836$	−5.50409	−0.190363
$837$	0	0
$838$	−48.4620	−1.67409
$839$	27.6979	0.956237	0.478118	−	0.878295i	$-0.341319\pi$
$839$	27.6979	0.956237	0.478118	+	0.878295i	$0.341319\pi$
$840$	0	0
$841$	54.7024	1.88629
$842$	−64.9066	−2.23683
$843$	0	0
$844$	−35.0616	−1.20687
$845$	17.0646	0.587041
$846$	0	0
$847$	1.84735	0.0634755
$848$	−0.709785	−0.0243741
$849$	0	0
$850$	−6.12273	−0.210008
$851$	38.5670	1.32206
$852$	0	0
$853$	30.6617	1.04984	0.524918	−	0.851153i	$-0.324096\pi$
$853$	30.6617	1.04984	0.524918	+	0.851153i	$0.324096\pi$
$854$	29.2096	0.999531
$855$	0	0
$856$	9.17626	0.313638
$857$	39.9494	1.36465	0.682323	−	0.731051i	$-0.260970\pi$
$857$	39.9494	1.36465	0.682323	+	0.731051i	$0.260970\pi$
$858$	0	0
$859$	4.78837	0.163377	0.0816886	−	0.996658i	$-0.473969\pi$
$859$	4.78837	0.163377	0.0816886	+	0.996658i	$0.473969\pi$
$860$	−4.04513	−0.137938
$861$	0	0
$862$	−0.715760	−0.0243789
$863$	−15.3411	−0.522218	−0.261109	−	0.965309i	$-0.584088\pi$
$863$	−15.3411	−0.522218	−0.261109	+	0.965309i	$0.584088\pi$
$864$	0	0
$865$	38.4272	1.30656
$866$	−3.36996	−0.114516
$867$	0	0
$868$	26.8345	0.910824
$869$	12.6768	0.430032
$870$	0	0
$871$	−3.15061	−0.106754
$872$	−1.11037	−0.0376019
$873$	0	0
$874$	24.4159	0.825879
$875$	22.4496	0.758935
$876$	0	0
$877$	−38.7111	−1.30718	−0.653590	−	0.756848i	$-0.726738\pi$
$877$	−38.7111	−1.30718	−0.653590	+	0.756848i	$0.726738\pi$
$878$	8.53375	0.288000
$879$	0	0
$880$	8.79158	0.296364
$881$	−0.626915	−0.0211213	−0.0105606	−	0.999944i	$-0.503362\pi$
$881$	−0.626915	−0.0211213	−0.0105606	+	0.999944i	$0.503362\pi$
$882$	0	0
$883$	6.49737	0.218654	0.109327	−	0.994006i	$-0.465130\pi$
$883$	6.49737	0.218654	0.109327	+	0.994006i	$0.465130\pi$
$884$	−7.01151	−0.235822
$885$	0	0
$886$	−52.2598	−1.75570
$887$	−3.76578	−0.126443	−0.0632213	−	0.998000i	$-0.520137\pi$
$887$	−3.76578	−0.126443	−0.0632213	+	0.998000i	$0.520137\pi$
$888$	0	0
$889$	−11.5815	−0.388431
$890$	0.510239	0.0171033
$891$	0	0
$892$	−39.0915	−1.30888
$893$	4.02998	0.134858
$894$	0	0
$895$	−43.2388	−1.44531
$896$	−11.5079	−0.384452
$897$	0	0
$898$	−13.1181	−0.437755
$899$	−84.1610	−2.80693
$900$	0	0
$901$	−0.340131	−0.0113314
$902$	8.58201	0.285750
$903$	0	0
$904$	2.67695	0.0890341
$905$	−0.594302	−0.0197553
$906$	0	0
$907$	−5.40480	−0.179463	−0.0897317	−	0.995966i	$-0.528601\pi$
$907$	−5.40480	−0.179463	−0.0897317	+	0.995966i	$0.528601\pi$
$908$	−6.33910	−0.210370
$909$	0	0
$910$	13.0842	0.433737
$911$	30.0931	0.997030	0.498515	−	0.866881i	$-0.333879\pi$
$911$	30.0931	0.997030	0.498515	+	0.866881i	$0.333879\pi$
$912$	0	0
$913$	4.17369	0.138129
$914$	−63.5574	−2.10229
$915$	0	0
$916$	−23.1121	−0.763645
$917$	−2.52674	−0.0834403
$918$	0	0
$919$	3.00956	0.0992763	0.0496381	−	0.998767i	$-0.484193\pi$
$919$	3.00956	0.0992763	0.0496381	+	0.998767i	$0.484193\pi$
$920$	−5.55694	−0.183207
$921$	0	0
$922$	−36.8129	−1.21237
$923$	−25.3054	−0.832937
$924$	0	0
$925$	−15.0810	−0.495861
$926$	−53.5042	−1.75826
$927$	0	0
$928$	−66.1667	−2.17203
$929$	−17.8050	−0.584162	−0.292081	−	0.956394i	$-0.594348\pi$
$929$	−17.8050	−0.584162	−0.292081	+	0.956394i	$0.594348\pi$
$930$	0	0
$931$	12.5040	0.409803
$932$	15.6790	0.513584
$933$	0	0
$934$	−30.5598	−0.999947
$935$	4.21294	0.137778
$936$	0	0
$937$	30.5106	0.996737	0.498369	−	0.866965i	$-0.333933\pi$
$937$	30.5106	0.996737	0.498369	+	0.866965i	$0.333933\pi$
$938$	5.54321	0.180992
$939$	0	0
$940$	3.44091	0.112230
$941$	27.9482	0.911085	0.455542	−	0.890214i	$-0.349445\pi$
$941$	27.9482	0.911085	0.455542	+	0.890214i	$0.349445\pi$
$942$	0	0
$943$	−16.7961	−0.546957
$944$	0.102899	0.00334909
$945$	0	0
$946$	−2.57138	−0.0836029
$947$	17.9009	0.581701	0.290851	−	0.956768i	$-0.406062\pi$
$947$	17.9009	0.581701	0.290851	+	0.956768i	$0.406062\pi$
$948$	0	0
$949$	24.7961	0.804917
$950$	−9.54744	−0.309760
$951$	0	0
$952$	−3.28830	−0.106574
$953$	−4.30751	−0.139534	−0.0697669	−	0.997563i	$-0.522226\pi$
$953$	−4.30751	−0.139534	−0.0697669	+	0.997563i	$0.522226\pi$
$954$	0	0
$955$	15.1720	0.490956
$956$	16.6455	0.538353
$957$	0	0
$958$	−31.5563	−1.01954
$959$	−24.3054	−0.784864
$960$	0	0
$961$	53.6220	1.72974
$962$	−39.1438	−1.26205
$963$	0	0
$964$	−10.5044	−0.338323
$965$	24.7696	0.797363
$966$	0	0
$967$	30.8464	0.991952	0.495976	−	0.868336i	$-0.334810\pi$
$967$	30.8464	0.991952	0.495976	+	0.868336i	$0.334810\pi$
$968$	0.796313	0.0255945
$969$	0	0
$970$	7.34534	0.235845
$971$	54.6961	1.75528	0.877641	−	0.479318i	$-0.159116\pi$
$971$	54.6961	1.75528	0.877641	+	0.479318i	$0.159116\pi$
$972$	0	0
$973$	35.9321	1.15193
$974$	57.2540	1.83454
$975$	0	0
$976$	38.9863	1.24792
$977$	6.98080	0.223336	0.111668	−	0.993746i	$-0.464381\pi$
$977$	6.98080	0.223336	0.111668	+	0.993746i	$0.464381\pi$
$978$	0	0
$979$	0.143101	0.00457352
$980$	10.6763	0.341042
$981$	0	0
$982$	50.2865	1.60471
$983$	31.9533	1.01915	0.509576	−	0.860426i	$-0.329802\pi$
$983$	31.9533	1.01915	0.509576	+	0.860426i	$0.329802\pi$
$984$	0	0
$985$	−23.5376	−0.749972
$986$	−38.6896	−1.23213
$987$	0	0
$988$	−10.9333	−0.347836
$989$	5.03254	0.160025
$990$	0	0
$991$	9.33178	0.296434	0.148217	−	0.988955i	$-0.452647\pi$
$991$	9.33178	0.296434	0.148217	+	0.988955i	$0.452647\pi$
$992$	66.5292	2.11230
$993$	0	0
$994$	44.5225	1.41217
$995$	0.645882	0.0204758
$996$	0	0
$997$	−34.9856	−1.10800	−0.554002	−	0.832515i	$-0.686900\pi$
$997$	−34.9856	−1.10800	−0.554002	+	0.832515i	$0.686900\pi$
$998$	−73.4383	−2.32465
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8019.2.a.l.1.10		51
3.2	odd	2		8019.2.a.k.1.42		51
27.4	even	9		297.2.j.c.232.4	✓	102
27.7	even	9		297.2.j.c.265.4	yes	102
27.20	odd	18		891.2.j.c.199.14		102
27.23	odd	18		891.2.j.c.694.14		102

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
297.2.j.c.232.4	✓	102	27.4	even	9
297.2.j.c.265.4	yes	102	27.7	even	9
891.2.j.c.199.14		102	27.20	odd	18
891.2.j.c.694.14		102	27.23	odd	18
8019.2.a.k.1.42		51	3.2	odd	2
8019.2.a.l.1.10		51	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:	\( N \)	\(=\)	\( 8019 = 3^{6} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8019.a (trivial)

\(f(q)\)	\(=\)	\(q-1.89185 q^{2} +1.57908 q^{4} -1.88472 q^{5} +1.84735 q^{7} +0.796313 q^{8} +O(q^{10})\)	\(q-1.89185 q^{2} +1.57908 q^{4} -1.88472 q^{5} +1.84735 q^{7} +0.796313 q^{8} +3.56560 q^{10} +1.00000 q^{11} +1.98640 q^{13} -3.49489 q^{14} -4.66466 q^{16} -2.23532 q^{17} -3.48563 q^{19} -2.97612 q^{20} -1.89185 q^{22} +3.70259 q^{23} -1.44784 q^{25} -3.75797 q^{26} +2.91711 q^{28} -9.14890 q^{29} +9.19902 q^{31} +7.23220 q^{32} +4.22888 q^{34} -3.48172 q^{35} +10.4162 q^{37} +6.59427 q^{38} -1.50083 q^{40} -4.53632 q^{41} +1.35919 q^{43} +1.57908 q^{44} -7.00473 q^{46} -1.15617 q^{47} -3.58732 q^{49} +2.73909 q^{50} +3.13669 q^{52} +0.152162 q^{53} -1.88472 q^{55} +1.47107 q^{56} +17.3083 q^{58} -0.0220593 q^{59} -8.35779 q^{61} -17.4031 q^{62} -4.35288 q^{64} -3.74381 q^{65} -1.58609 q^{67} -3.52975 q^{68} +6.58689 q^{70} -12.7393 q^{71} +12.4829 q^{73} -19.7059 q^{74} -5.50409 q^{76} +1.84735 q^{77} +12.6768 q^{79} +8.79158 q^{80} +8.58201 q^{82} +4.17369 q^{83} +4.21294 q^{85} -2.57138 q^{86} +0.796313 q^{88} +0.143101 q^{89} +3.66957 q^{91} +5.84669 q^{92} +2.18730 q^{94} +6.56942 q^{95} +2.06006 q^{97} +6.78665 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 51 q + 60 q^{4} + 6 q^{5} + 12 q^{7}+O(q^{10}) \) 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

Varieties

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Representations

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Database

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