LMFDB - Embedded newform 8019.2.a.i.1.5

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:	$1$
Dimension:	$48$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
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)

$f(q)$	$=$	$q-2.49176 q^{2} +4.20888 q^{4} +3.55698 q^{5} -4.04430 q^{7} -5.50400 q^{8} +O(q^{10})$	\(q-2.49176 q^{2} +4.20888 q^{4} +3.55698 q^{5} -4.04430 q^{7} -5.50400 q^{8} -8.86315 q^{10} -1.00000 q^{11} -1.13016 q^{13} +10.0774 q^{14} +5.29691 q^{16} +3.54885 q^{17} +1.44324 q^{19} +14.9709 q^{20} +2.49176 q^{22} +6.96853 q^{23} +7.65210 q^{25} +2.81610 q^{26} -17.0220 q^{28} -8.57363 q^{29} +6.07031 q^{31} -2.19063 q^{32} -8.84289 q^{34} -14.3855 q^{35} -2.27494 q^{37} -3.59621 q^{38} -19.5776 q^{40} -12.3684 q^{41} +0.903928 q^{43} -4.20888 q^{44} -17.3639 q^{46} -7.70589 q^{47} +9.35633 q^{49} -19.0672 q^{50} -4.75673 q^{52} -7.83814 q^{53} -3.55698 q^{55} +22.2598 q^{56} +21.3635 q^{58} +1.49283 q^{59} +12.5087 q^{61} -15.1258 q^{62} -5.13529 q^{64} -4.01997 q^{65} -10.4633 q^{67} +14.9367 q^{68} +35.8452 q^{70} -5.11425 q^{71} +0.0441545 q^{73} +5.66862 q^{74} +6.07442 q^{76} +4.04430 q^{77} -12.7220 q^{79} +18.8410 q^{80} +30.8191 q^{82} +8.30290 q^{83} +12.6232 q^{85} -2.25237 q^{86} +5.50400 q^{88} -1.32126 q^{89} +4.57072 q^{91} +29.3297 q^{92} +19.2012 q^{94} +5.13357 q^{95} +14.3428 q^{97} -23.3138 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 6 q^{2} + 54 q^{4} - 24 q^{5} - 18 q^{8}+O(q^{10}) $ 48 * q - 6 * q^2 + 54 * q^4 - 24 * q^5 - 18 * q^8	$ 48 q - 6 q^{2} + 54 q^{4} - 24 q^{5} - 18 q^{8} - 48 q^{11} - 24 q^{14} + 66 q^{16} - 24 q^{17} - 48 q^{20} + 6 q^{22} - 12 q^{23} + 60 q^{25} - 36 q^{26} - 18 q^{28} - 60 q^{29} + 36 q^{31} - 42 q^{32} + 12 q^{34} - 24 q^{35} + 6 q^{37} - 24 q^{38} - 72 q^{41} - 12 q^{43} - 54 q^{44} - 30 q^{46} - 36 q^{47} + 60 q^{49} - 42 q^{50} - 48 q^{53} + 24 q^{55} - 72 q^{56} + 12 q^{58} - 60 q^{59} - 24 q^{61} - 36 q^{62} + 90 q^{64} - 48 q^{65} - 60 q^{68} - 30 q^{70} - 60 q^{71} - 18 q^{73} - 36 q^{74} - 42 q^{76} - 12 q^{79} - 96 q^{80} + 12 q^{82} - 36 q^{83} + 18 q^{85} - 48 q^{86} + 18 q^{88} - 96 q^{89} + 30 q^{91} - 36 q^{92} - 48 q^{94} - 48 q^{95} + 30 q^{97} - 54 q^{98}+O(q^{100}) $ 48 * q - 6 * q^2 + 54 * q^4 - 24 * q^5 - 18 * q^8 - 48 * q^11 - 24 * q^14 + 66 * q^16 - 24 * q^17 - 48 * q^20 + 6 * q^22 - 12 * q^23 + 60 * q^25 - 36 * q^26 - 18 * q^28 - 60 * q^29 + 36 * q^31 - 42 * q^32 + 12 * q^34 - 24 * q^35 + 6 * q^37 - 24 * q^38 - 72 * q^41 - 12 * q^43 - 54 * q^44 - 30 * q^46 - 36 * q^47 + 60 * q^49 - 42 * q^50 - 48 * q^53 + 24 * q^55 - 72 * q^56 + 12 * q^58 - 60 * q^59 - 24 * q^61 - 36 * q^62 + 90 * q^64 - 48 * q^65 - 60 * q^68 - 30 * q^70 - 60 * q^71 - 18 * q^73 - 36 * q^74 - 42 * q^76 - 12 * q^79 - 96 * q^80 + 12 * q^82 - 36 * q^83 + 18 * q^85 - 48 * q^86 + 18 * q^88 - 96 * q^89 + 30 * q^91 - 36 * q^92 - 48 * q^94 - 48 * q^95 + 30 * q^97 - 54 * 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 $

$2$	−2.49176	−1.76194	−0.880971	−	0.473170i	$-0.843110\pi$
$2$	−2.49176	−1.76194	−0.880971	+	0.473170i	$0.843110\pi$
$3$	0	0
$3$	0	0
$4$	4.20888	2.10444
$5$	3.55698	1.59073	0.795365	−	0.606131i	$-0.207279\pi$
$5$	3.55698	1.59073	0.795365	+	0.606131i	$0.207279\pi$
$6$	0	0
$7$	−4.04430	−1.52860	−0.764300	−	0.644861i	$-0.776915\pi$
$7$	−4.04430	−1.52860	−0.764300	+	0.644861i	$0.776915\pi$
$8$	−5.50400	−1.94596
$9$	0	0
$10$	−8.86315	−2.80277
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.13016	−0.313451	−0.156726	−	0.987642i	$-0.550094\pi$
$13$	−1.13016	−0.313451	−0.156726	+	0.987642i	$0.550094\pi$
$14$	10.0774	2.69330
$15$	0	0
$16$	5.29691	1.32423
$17$	3.54885	0.860723	0.430361	−	0.902657i	$-0.358386\pi$
$17$	3.54885	0.860723	0.430361	+	0.902657i	$0.358386\pi$
$18$	0	0
$19$	1.44324	0.331102	0.165551	−	0.986201i	$-0.447060\pi$
$19$	1.44324	0.331102	0.165551	+	0.986201i	$0.447060\pi$
$20$	14.9709	3.34759
$21$	0	0
$22$	2.49176	0.531245
$23$	6.96853	1.45304	0.726520	−	0.687146i	$-0.241137\pi$
$23$	6.96853	1.45304	0.726520	+	0.687146i	$0.241137\pi$
$24$	0	0
$25$	7.65210	1.53042
$26$	2.81610	0.552283
$27$	0	0
$28$	−17.0220	−3.21685
$29$	−8.57363	−1.59208	−0.796042	−	0.605242i	$-0.793076\pi$
$29$	−8.57363	−1.59208	−0.796042	+	0.605242i	$0.793076\pi$
$30$	0	0
$31$	6.07031	1.09026	0.545130	−	0.838352i	$-0.316480\pi$
$31$	6.07031	1.09026	0.545130	+	0.838352i	$0.316480\pi$
$32$	−2.19063	−0.387251
$33$	0	0
$34$	−8.84289	−1.51654
$35$	−14.3855	−2.43159
$36$	0	0
$37$	−2.27494	−0.373998	−0.186999	−	0.982360i	$-0.559876\pi$
$37$	−2.27494	−0.373998	−0.186999	+	0.982360i	$0.559876\pi$
$38$	−3.59621	−0.583382
$39$	0	0
$40$	−19.5776	−3.09549
$41$	−12.3684	−1.93162	−0.965809	−	0.259256i	$-0.916523\pi$
$41$	−12.3684	−1.93162	−0.965809	+	0.259256i	$0.916523\pi$
$42$	0	0
$43$	0.903928	0.137848	0.0689238	−	0.997622i	$-0.478043\pi$
$43$	0.903928	0.137848	0.0689238	+	0.997622i	$0.478043\pi$
$44$	−4.20888	−0.634512
$45$	0	0
$46$	−17.3639	−2.56017
$47$	−7.70589	−1.12402	−0.562010	−	0.827131i	$-0.689972\pi$
$47$	−7.70589	−1.12402	−0.562010	+	0.827131i	$0.689972\pi$
$48$	0	0
$49$	9.35633	1.33662
$50$	−19.0672	−2.69651
$51$	0	0
$52$	−4.75673	−0.659639
$53$	−7.83814	−1.07665	−0.538326	−	0.842737i	$-0.680943\pi$
$53$	−7.83814	−1.07665	−0.538326	+	0.842737i	$0.680943\pi$
$54$	0	0
$55$	−3.55698	−0.479623
$56$	22.2598	2.97459
$57$	0	0
$58$	21.3635	2.80516
$59$	1.49283	0.194350	0.0971751	−	0.995267i	$-0.469019\pi$
$59$	1.49283	0.194350	0.0971751	+	0.995267i	$0.469019\pi$
$60$	0	0
$61$	12.5087	1.60157	0.800785	−	0.598951i	$-0.204416\pi$
$61$	12.5087	1.60157	0.800785	+	0.598951i	$0.204416\pi$
$62$	−15.1258	−1.92097
$63$	0	0
$64$	−5.13529	−0.641912
$65$	−4.01997	−0.498616
$66$	0	0
$67$	−10.4633	−1.27830	−0.639148	−	0.769083i	$-0.720713\pi$
$67$	−10.4633	−1.27830	−0.639148	+	0.769083i	$0.720713\pi$
$68$	14.9367	1.81134
$69$	0	0
$70$	35.8452	4.28432
$71$	−5.11425	−0.606949	−0.303475	−	0.952839i	$-0.598147\pi$
$71$	−5.11425	−0.606949	−0.303475	+	0.952839i	$0.598147\pi$
$72$	0	0
$73$	0.0441545	0.00516789	0.00258394	−	0.999997i	$-0.499178\pi$
$73$	0.0441545	0.00516789	0.00258394	+	0.999997i	$0.499178\pi$
$74$	5.66862	0.658963
$75$	0	0
$76$	6.07442	0.696784
$77$	4.04430	0.460890
$78$	0	0
$79$	−12.7220	−1.43134	−0.715670	−	0.698439i	$-0.753878\pi$
$79$	−12.7220	−1.43134	−0.715670	+	0.698439i	$0.753878\pi$
$80$	18.8410	2.10649
$81$	0	0
$82$	30.8191	3.40340
$83$	8.30290	0.911362	0.455681	−	0.890143i	$-0.349396\pi$
$83$	8.30290	0.911362	0.455681	+	0.890143i	$0.349396\pi$
$84$	0	0
$85$	12.6232	1.36918
$86$	−2.25237	−0.242880
$87$	0	0
$88$	5.50400	0.586729
$89$	−1.32126	−0.140053	−0.0700266	−	0.997545i	$-0.522308\pi$
$89$	−1.32126	−0.140053	−0.0700266	+	0.997545i	$0.522308\pi$
$90$	0	0
$91$	4.57072	0.479142
$92$	29.3297	3.05783
$93$	0	0
$94$	19.2012	1.98046
$95$	5.13357	0.526693
$96$	0	0
$97$	14.3428	1.45629	0.728144	−	0.685424i	$-0.240383\pi$
$97$	14.3428	1.45629	0.728144	+	0.685424i	$0.240383\pi$
$98$	−23.3138	−2.35504
$99$	0	0
$100$	32.2068	3.22068
$101$	−17.6393	−1.75517	−0.877585	−	0.479420i	$-0.840847\pi$
$101$	−17.6393	−1.75517	−0.877585	+	0.479420i	$0.840847\pi$
$102$	0	0
$103$	−0.224768	−0.0221470	−0.0110735	−	0.999939i	$-0.503525\pi$
$103$	−0.224768	−0.0221470	−0.0110735	+	0.999939i	$0.503525\pi$
$104$	6.22043	0.609963
$105$	0	0
$106$	19.5308	1.89700
$107$	10.9847	1.06193	0.530964	−	0.847394i	$-0.321830\pi$
$107$	10.9847	1.06193	0.530964	+	0.847394i	$0.321830\pi$
$108$	0	0
$109$	−1.66827	−0.159791	−0.0798955	−	0.996803i	$-0.525459\pi$
$109$	−1.66827	−0.159791	−0.0798955	+	0.996803i	$0.525459\pi$
$110$	8.86315	0.845068
$111$	0	0
$112$	−21.4223	−2.02421
$113$	−12.6845	−1.19326	−0.596628	−	0.802518i	$-0.703493\pi$
$113$	−12.6845	−1.19326	−0.596628	+	0.802518i	$0.703493\pi$
$114$	0	0
$115$	24.7869	2.31139
$116$	−36.0854	−3.35044
$117$	0	0
$118$	−3.71978	−0.342434
$119$	−14.3526	−1.31570
$120$	0	0
$121$	1.00000	0.0909091
$122$	−31.1686	−2.82187
$123$	0	0
$124$	25.5492	2.29438
$125$	9.43346	0.843754
$126$	0	0
$127$	7.55064	0.670011	0.335005	−	0.942216i	$-0.391262\pi$
$127$	7.55064	0.670011	0.335005	+	0.942216i	$0.391262\pi$
$128$	17.1772	1.51826
$129$	0	0
$130$	10.0168	0.878533
$131$	14.8009	1.29316	0.646582	−	0.762845i	$-0.276198\pi$
$131$	14.8009	1.29316	0.646582	+	0.762845i	$0.276198\pi$
$132$	0	0
$133$	−5.83689	−0.506122
$134$	26.0721	2.25228
$135$	0	0
$136$	−19.5329	−1.67493
$137$	10.5100	0.897930	0.448965	−	0.893549i	$-0.351793\pi$
$137$	10.5100	0.897930	0.448965	+	0.893549i	$0.351793\pi$
$138$	0	0
$139$	7.34442	0.622945	0.311473	−	0.950255i	$-0.399178\pi$
$139$	7.34442	0.622945	0.311473	+	0.950255i	$0.399178\pi$
$140$	−60.5467	−5.11713
$141$	0	0
$142$	12.7435	1.06941
$143$	1.13016	0.0945091
$144$	0	0
$145$	−30.4962	−2.53257
$146$	−0.110022	−0.00910552
$147$	0	0
$148$	−9.57496	−0.787057
$149$	−21.3892	−1.75227	−0.876135	−	0.482066i	$-0.839887\pi$
$149$	−21.3892	−1.75227	−0.876135	+	0.482066i	$0.839887\pi$
$150$	0	0
$151$	−12.8500	−1.04572	−0.522859	−	0.852419i	$-0.675135\pi$
$151$	−12.8500	−1.04572	−0.522859	+	0.852419i	$0.675135\pi$
$152$	−7.94359	−0.644310
$153$	0	0
$154$	−10.0774	−0.812062
$155$	21.5920	1.73431
$156$	0	0
$157$	−11.0264	−0.880002	−0.440001	−	0.897997i	$-0.645022\pi$
$157$	−11.0264	−0.880002	−0.440001	+	0.897997i	$0.645022\pi$
$158$	31.7003	2.52194
$159$	0	0
$160$	−7.79201	−0.616012
$161$	−28.1828	−2.22112
$162$	0	0
$163$	23.4913	1.83998	0.919989	−	0.391943i	$-0.128197\pi$
$163$	23.4913	1.83998	0.919989	+	0.391943i	$0.128197\pi$
$164$	−52.0570	−4.06497
$165$	0	0
$166$	−20.6889	−1.60577
$167$	−1.08711	−0.0841233	−0.0420616	−	0.999115i	$-0.513393\pi$
$167$	−1.08711	−0.0841233	−0.0420616	+	0.999115i	$0.513393\pi$
$168$	0	0
$169$	−11.7227	−0.901748
$170$	−31.4540	−2.41241
$171$	0	0
$172$	3.80452	0.290092
$173$	−16.5096	−1.25520	−0.627600	−	0.778536i	$-0.715963\pi$
$173$	−16.5096	−1.25520	−0.627600	+	0.778536i	$0.715963\pi$
$174$	0	0
$175$	−30.9474	−2.33940
$176$	−5.29691	−0.399269
$177$	0	0
$178$	3.29226	0.246765
$179$	14.5553	1.08791	0.543956	−	0.839114i	$-0.316926\pi$
$179$	14.5553	1.08791	0.543956	+	0.839114i	$0.316926\pi$
$180$	0	0
$181$	−0.337631	−0.0250959	−0.0125480	−	0.999921i	$-0.503994\pi$
$181$	−0.337631	−0.0250959	−0.0125480	+	0.999921i	$0.503994\pi$
$182$	−11.3891	−0.844220
$183$	0	0
$184$	−38.3548	−2.82755
$185$	−8.09192	−0.594930
$186$	0	0
$187$	−3.54885	−0.259518
$188$	−32.4332	−2.36543
$189$	0	0
$190$	−12.7916	−0.928003
$191$	−15.4383	−1.11708	−0.558539	−	0.829478i	$-0.688638\pi$
$191$	−15.4383	−1.11708	−0.558539	+	0.829478i	$0.688638\pi$
$192$	0	0
$193$	−8.06174	−0.580297	−0.290149	−	0.956982i	$-0.593705\pi$
$193$	−8.06174	−0.580297	−0.290149	+	0.956982i	$0.593705\pi$
$194$	−35.7388	−2.56590
$195$	0	0
$196$	39.3797	2.81283
$197$	15.1189	1.07718	0.538588	−	0.842569i	$-0.318958\pi$
$197$	15.1189	1.07718	0.538588	+	0.842569i	$0.318958\pi$
$198$	0	0
$199$	3.10776	0.220303	0.110152	−	0.993915i	$-0.464866\pi$
$199$	3.10776	0.220303	0.110152	+	0.993915i	$0.464866\pi$
$200$	−42.1172	−2.97813
$201$	0	0
$202$	43.9528	3.09251
$203$	34.6743	2.43366
$204$	0	0
$205$	−43.9941	−3.07268
$206$	0.560068	0.0390218
$207$	0	0
$208$	−5.98637	−0.415080
$209$	−1.44324	−0.0998310
$210$	0	0
$211$	3.85962	0.265707	0.132854	−	0.991136i	$-0.457586\pi$
$211$	3.85962	0.265707	0.132854	+	0.991136i	$0.457586\pi$
$212$	−32.9898	−2.26575
$213$	0	0
$214$	−27.3712	−1.87106
$215$	3.21525	0.219278
$216$	0	0
$217$	−24.5501	−1.66657
$218$	4.15693	0.281543
$219$	0	0
$220$	−14.9709	−1.00934
$221$	−4.01079	−0.269795
$222$	0	0
$223$	−2.44463	−0.163704	−0.0818522	−	0.996644i	$-0.526084\pi$
$223$	−2.44463	−0.163704	−0.0818522	+	0.996644i	$0.526084\pi$
$224$	8.85954	0.591953
$225$	0	0
$226$	31.6067	2.10245
$227$	−23.4510	−1.55650	−0.778248	−	0.627957i	$-0.783891\pi$
$227$	−23.4510	−1.55650	−0.778248	+	0.627957i	$0.783891\pi$
$228$	0	0
$229$	−12.8582	−0.849694	−0.424847	−	0.905265i	$-0.639672\pi$
$229$	−12.8582	−0.849694	−0.424847	+	0.905265i	$0.639672\pi$
$230$	−61.7631	−4.07254
$231$	0	0
$232$	47.1893	3.09813
$233$	25.8700	1.69480	0.847401	−	0.530953i	$-0.178166\pi$
$233$	25.8700	1.69480	0.847401	+	0.530953i	$0.178166\pi$
$234$	0	0
$235$	−27.4097	−1.78801
$236$	6.28315	0.408998
$237$	0	0
$238$	35.7633	2.31819
$239$	10.1639	0.657446	0.328723	−	0.944426i	$-0.393382\pi$
$239$	10.1639	0.657446	0.328723	+	0.944426i	$0.393382\pi$
$240$	0	0
$241$	19.4894	1.25542	0.627711	−	0.778447i	$-0.283992\pi$
$241$	19.4894	1.25542	0.627711	+	0.778447i	$0.283992\pi$
$242$	−2.49176	−0.160177
$243$	0	0
$244$	52.6475	3.37041
$245$	33.2803	2.12620
$246$	0	0
$247$	−1.63110	−0.103784
$248$	−33.4110	−2.12160
$249$	0	0
$250$	−23.5059	−1.48665
$251$	21.4674	1.35501	0.677504	−	0.735519i	$-0.263062\pi$
$251$	21.4674	1.35501	0.677504	+	0.735519i	$0.263062\pi$
$252$	0	0
$253$	−6.96853	−0.438108
$254$	−18.8144	−1.18052
$255$	0	0
$256$	−32.5309	−2.03318
$257$	4.59154	0.286412	0.143206	−	0.989693i	$-0.454259\pi$
$257$	4.59154	0.286412	0.143206	+	0.989693i	$0.454259\pi$
$258$	0	0
$259$	9.20054	0.571694
$260$	−16.9196	−1.04931
$261$	0	0
$262$	−36.8804	−2.27848
$263$	−27.9475	−1.72332	−0.861659	−	0.507488i	$-0.830574\pi$
$263$	−27.9475	−1.72332	−0.861659	+	0.507488i	$0.830574\pi$
$264$	0	0
$265$	−27.8801	−1.71266
$266$	14.5441	0.891758
$267$	0	0
$268$	−44.0388	−2.69010
$269$	−21.1416	−1.28903	−0.644514	−	0.764593i	$-0.722940\pi$
$269$	−21.1416	−1.28903	−0.644514	+	0.764593i	$0.722940\pi$
$270$	0	0
$271$	−9.41312	−0.571806	−0.285903	−	0.958259i	$-0.592294\pi$
$271$	−9.41312	−0.571806	−0.285903	+	0.958259i	$0.592294\pi$
$272$	18.7979	1.13979
$273$	0	0
$274$	−26.1884	−1.58210
$275$	−7.65210	−0.461439
$276$	0	0
$277$	8.26221	0.496428	0.248214	−	0.968705i	$-0.420156\pi$
$277$	8.26221	0.496428	0.248214	+	0.968705i	$0.420156\pi$
$278$	−18.3005	−1.09759
$279$	0	0
$280$	79.1777	4.73177
$281$	−6.06213	−0.361637	−0.180818	−	0.983517i	$-0.557875\pi$
$281$	−6.06213	−0.361637	−0.180818	+	0.983517i	$0.557875\pi$
$282$	0	0
$283$	5.10053	0.303195	0.151598	−	0.988442i	$-0.451558\pi$
$283$	5.10053	0.303195	0.151598	+	0.988442i	$0.451558\pi$
$284$	−21.5252	−1.27729
$285$	0	0
$286$	−2.81610	−0.166520
$287$	50.0214	2.95267
$288$	0	0
$289$	−4.40565	−0.259156
$290$	75.9893	4.46225
$291$	0	0
$292$	0.185841	0.0108755
$293$	−12.5744	−0.734604	−0.367302	−	0.930102i	$-0.619718\pi$
$293$	−12.5744	−0.734604	−0.367302	+	0.930102i	$0.619718\pi$
$294$	0	0
$295$	5.30997	0.309158
$296$	12.5213	0.727785
$297$	0	0
$298$	53.2968	3.08740
$299$	−7.87559	−0.455457
$300$	0	0
$301$	−3.65575	−0.210714
$302$	32.0191	1.84249
$303$	0	0
$304$	7.64470	0.438454
$305$	44.4931	2.54767
$306$	0	0
$307$	−4.55428	−0.259926	−0.129963	−	0.991519i	$-0.541486\pi$
$307$	−4.55428	−0.259926	−0.129963	+	0.991519i	$0.541486\pi$
$308$	17.0220	0.969916
$309$	0	0
$310$	−53.8020	−3.05575
$311$	−34.4374	−1.95277	−0.976384	−	0.216043i	$-0.930685\pi$
$311$	−34.4374	−1.95277	−0.976384	+	0.216043i	$0.930685\pi$
$312$	0	0
$313$	21.2113	1.19893	0.599466	−	0.800400i	$-0.295379\pi$
$313$	21.2113	1.19893	0.599466	+	0.800400i	$0.295379\pi$
$314$	27.4752	1.55051
$315$	0	0
$316$	−53.5455	−3.01217
$317$	−13.1654	−0.739445	−0.369722	−	0.929142i	$-0.620547\pi$
$317$	−13.1654	−0.739445	−0.369722	+	0.929142i	$0.620547\pi$
$318$	0	0
$319$	8.57363	0.480031
$320$	−18.2661	−1.02111
$321$	0	0
$322$	70.2249	3.91348
$323$	5.12184	0.284987
$324$	0	0
$325$	−8.64813	−0.479712
$326$	−58.5347	−3.24194
$327$	0	0
$328$	68.0756	3.75885
$329$	31.1649	1.71818
$330$	0	0
$331$	9.96584	0.547772	0.273886	−	0.961762i	$-0.411691\pi$
$331$	9.96584	0.547772	0.273886	+	0.961762i	$0.411691\pi$
$332$	34.9459	1.91791
$333$	0	0
$334$	2.70883	0.148220
$335$	−37.2178	−2.03342
$336$	0	0
$337$	11.3916	0.620542	0.310271	−	0.950648i	$-0.399580\pi$
$337$	11.3916	0.620542	0.310271	+	0.950648i	$0.399580\pi$
$338$	29.2103	1.58883
$339$	0	0
$340$	53.1295	2.88135
$341$	−6.07031	−0.328726
$342$	0	0
$343$	−9.52970	−0.514555
$344$	−4.97522	−0.268246
$345$	0	0
$346$	41.1380	2.21159
$347$	0.0685434	0.00367960	0.00183980	−	0.999998i	$-0.499414\pi$
$347$	0.0685434	0.00367960	0.00183980	+	0.999998i	$0.499414\pi$
$348$	0	0
$349$	23.4171	1.25349	0.626746	−	0.779224i	$-0.284387\pi$
$349$	23.4171	1.25349	0.626746	+	0.779224i	$0.284387\pi$
$350$	77.1134	4.12189
$351$	0	0
$352$	2.19063	0.116761
$353$	−16.3768	−0.871647	−0.435824	−	0.900032i	$-0.643543\pi$
$353$	−16.3768	−0.871647	−0.435824	+	0.900032i	$0.643543\pi$
$354$	0	0
$355$	−18.1913	−0.965492
$356$	−5.56102	−0.294733
$357$	0	0
$358$	−36.2683	−1.91684
$359$	−20.1715	−1.06461	−0.532307	−	0.846552i	$-0.678675\pi$
$359$	−20.1715	−1.06461	−0.532307	+	0.846552i	$0.678675\pi$
$360$	0	0
$361$	−16.9171	−0.890372
$362$	0.841297	0.0442176
$363$	0	0
$364$	19.2376	1.00832
$365$	0.157057	0.00822071
$366$	0	0
$367$	9.37461	0.489351	0.244675	−	0.969605i	$-0.421319\pi$
$367$	9.37461	0.489351	0.244675	+	0.969605i	$0.421319\pi$
$368$	36.9117	1.92415
$369$	0	0
$370$	20.1632	1.04823
$371$	31.6998	1.64577
$372$	0	0
$373$	−5.89142	−0.305046	−0.152523	−	0.988300i	$-0.548740\pi$
$373$	−5.89142	−0.305046	−0.152523	+	0.988300i	$0.548740\pi$
$374$	8.84289	0.457255
$375$	0	0
$376$	42.4132	2.18729
$377$	9.68961	0.499040
$378$	0	0
$379$	−29.8024	−1.53084	−0.765422	−	0.643528i	$-0.777470\pi$
$379$	−29.8024	−1.53084	−0.765422	+	0.643528i	$0.777470\pi$
$380$	21.6066	1.10839
$381$	0	0
$382$	38.4687	1.96823
$383$	13.4645	0.688004	0.344002	−	0.938969i	$-0.388217\pi$
$383$	13.4645	0.688004	0.344002	+	0.938969i	$0.388217\pi$
$384$	0	0
$385$	14.3855	0.733152
$386$	20.0879	1.02245
$387$	0	0
$388$	60.3670	3.06467
$389$	9.02119	0.457392	0.228696	−	0.973498i	$-0.426554\pi$
$389$	9.02119	0.457392	0.228696	+	0.973498i	$0.426554\pi$
$390$	0	0
$391$	24.7303	1.25066
$392$	−51.4973	−2.60100
$393$	0	0
$394$	−37.6727	−1.89792
$395$	−45.2520	−2.27687
$396$	0	0
$397$	13.7477	0.689978	0.344989	−	0.938607i	$-0.387883\pi$
$397$	13.7477	0.689978	0.344989	+	0.938607i	$0.387883\pi$
$398$	−7.74380	−0.388162
$399$	0	0
$400$	40.5324	2.02662
$401$	−12.2438	−0.611426	−0.305713	−	0.952124i	$-0.598895\pi$
$401$	−12.2438	−0.611426	−0.305713	+	0.952124i	$0.598895\pi$
$402$	0	0
$403$	−6.86044	−0.341743
$404$	−74.2415	−3.69365
$405$	0	0
$406$	−86.4001	−4.28797
$407$	2.27494	0.112765
$408$	0	0
$409$	−12.1889	−0.602700	−0.301350	−	0.953514i	$-0.597437\pi$
$409$	−12.1889	−0.602700	−0.301350	+	0.953514i	$0.597437\pi$
$410$	109.623	5.41388
$411$	0	0
$412$	−0.946021	−0.0466071
$413$	−6.03745	−0.297084
$414$	0	0
$415$	29.5333	1.44973
$416$	2.47577	0.121384
$417$	0	0
$418$	3.59621	0.175896
$419$	14.1451	0.691034	0.345517	−	0.938413i	$-0.387704\pi$
$419$	14.1451	0.691034	0.345517	+	0.938413i	$0.387704\pi$
$420$	0	0
$421$	−0.498284	−0.0242849	−0.0121424	−	0.999926i	$-0.503865\pi$
$421$	−0.498284	−0.0242849	−0.0121424	+	0.999926i	$0.503865\pi$
$422$	−9.61726	−0.468161
$423$	0	0
$424$	43.1411	2.09512
$425$	27.1562	1.31727
$426$	0	0
$427$	−50.5888	−2.44816
$428$	46.2331	2.23476
$429$	0	0
$430$	−8.01164	−0.386356
$431$	−26.8748	−1.29451	−0.647257	−	0.762272i	$-0.724084\pi$
$431$	−26.8748	−1.29451	−0.647257	+	0.762272i	$0.724084\pi$
$432$	0	0
$433$	−33.3027	−1.60042	−0.800212	−	0.599717i	$-0.795280\pi$
$433$	−33.3027	−1.60042	−0.800212	+	0.599717i	$0.795280\pi$
$434$	61.1731	2.93640
$435$	0	0
$436$	−7.02154	−0.336271
$437$	10.0573	0.481104
$438$	0	0
$439$	8.87523	0.423592	0.211796	−	0.977314i	$-0.432069\pi$
$439$	8.87523	0.423592	0.211796	+	0.977314i	$0.432069\pi$
$440$	19.5776	0.933326
$441$	0	0
$442$	9.99392	0.475362
$443$	−17.1808	−0.816287	−0.408143	−	0.912918i	$-0.633824\pi$
$443$	−17.1808	−0.816287	−0.408143	+	0.912918i	$0.633824\pi$
$444$	0	0
$445$	−4.69969	−0.222787
$446$	6.09143	0.288438
$447$	0	0
$448$	20.7686	0.981226
$449$	−9.12237	−0.430511	−0.215256	−	0.976558i	$-0.569058\pi$
$449$	−9.12237	−0.430511	−0.215256	+	0.976558i	$0.569058\pi$
$450$	0	0
$451$	12.3684	0.582405
$452$	−53.3874	−2.51113
$453$	0	0
$454$	58.4343	2.74246
$455$	16.2580	0.762185
$456$	0	0
$457$	−16.9728	−0.793953	−0.396977	−	0.917829i	$-0.629941\pi$
$457$	−16.9728	−0.793953	−0.396977	+	0.917829i	$0.629941\pi$
$458$	32.0396	1.49711
$459$	0	0
$460$	104.325	4.86419
$461$	2.64451	0.123167	0.0615836	−	0.998102i	$-0.480385\pi$
$461$	2.64451	0.123167	0.0615836	+	0.998102i	$0.480385\pi$
$462$	0	0
$463$	−11.5538	−0.536949	−0.268475	−	0.963287i	$-0.586520\pi$
$463$	−11.5538	−0.536949	−0.268475	+	0.963287i	$0.586520\pi$
$464$	−45.4137	−2.10828
$465$	0	0
$466$	−64.4620	−2.98614
$467$	−25.6864	−1.18862	−0.594312	−	0.804235i	$-0.702576\pi$
$467$	−25.6864	−1.18862	−0.594312	+	0.804235i	$0.702576\pi$
$468$	0	0
$469$	42.3167	1.95400
$470$	68.2984	3.15037
$471$	0	0
$472$	−8.21655	−0.378197
$473$	−0.903928	−0.0415626
$474$	0	0
$475$	11.0438	0.506725
$476$	−60.4084	−2.76881
$477$	0	0
$478$	−25.3259	−1.15838
$479$	18.6554	0.852389	0.426195	−	0.904632i	$-0.359854\pi$
$479$	18.6554	0.852389	0.426195	+	0.904632i	$0.359854\pi$
$480$	0	0
$481$	2.57106	0.117230
$482$	−48.5629	−2.21198
$483$	0	0
$484$	4.20888	0.191313
$485$	51.0170	2.31656
$486$	0	0
$487$	19.1526	0.867887	0.433944	−	0.900940i	$-0.357122\pi$
$487$	19.1526	0.867887	0.433944	+	0.900940i	$0.357122\pi$
$488$	−68.8477	−3.11659
$489$	0	0
$490$	−82.9265	−3.74624
$491$	−26.9879	−1.21795	−0.608973	−	0.793191i	$-0.708418\pi$
$491$	−26.9879	−1.21795	−0.608973	+	0.793191i	$0.708418\pi$
$492$	0	0
$493$	−30.4265	−1.37034
$494$	4.06431	0.182862
$495$	0	0
$496$	32.1538	1.44375
$497$	20.6835	0.927783
$498$	0	0
$499$	−28.8838	−1.29302	−0.646509	−	0.762906i	$-0.723772\pi$
$499$	−28.8838	−1.29302	−0.646509	+	0.762906i	$0.723772\pi$
$500$	39.7043	1.77563
$501$	0	0
$502$	−53.4916	−2.38745
$503$	44.4141	1.98033	0.990163	−	0.139921i	$-0.0446847\pi$
$503$	44.4141	1.98033	0.990163	+	0.139921i	$0.0446847\pi$
$504$	0	0
$505$	−62.7424	−2.79200
$506$	17.3639	0.771921
$507$	0	0
$508$	31.7797	1.41000
$509$	−8.06604	−0.357521	−0.178761	−	0.983893i	$-0.557209\pi$
$509$	−8.06604	−0.357521	−0.178761	+	0.983893i	$0.557209\pi$
$510$	0	0
$511$	−0.178574	−0.00789964
$512$	46.7048	2.06408
$513$	0	0
$514$	−11.4410	−0.504642
$515$	−0.799495	−0.0352300
$516$	0	0
$517$	7.70589	0.338905
$518$	−22.9256	−1.00729
$519$	0	0
$520$	22.1259	0.970286
$521$	−8.31554	−0.364311	−0.182155	−	0.983270i	$-0.558307\pi$
$521$	−8.31554	−0.364311	−0.182155	+	0.983270i	$0.558307\pi$
$522$	0	0
$523$	−6.78458	−0.296669	−0.148334	−	0.988937i	$-0.547391\pi$
$523$	−6.78458	−0.296669	−0.148334	+	0.988937i	$0.547391\pi$
$524$	62.2953	2.72138
$525$	0	0
$526$	69.6386	3.03639
$527$	21.5426	0.938411
$528$	0	0
$529$	25.5605	1.11132
$530$	69.4706	3.01761
$531$	0	0
$532$	−24.5668	−1.06510
$533$	13.9783	0.605468
$534$	0	0
$535$	39.0722	1.68924
$536$	57.5901	2.48751
$537$	0	0
$538$	52.6799	2.27119
$539$	−9.35633	−0.403006
$540$	0	0
$541$	13.5668	0.583283	0.291642	−	0.956528i	$-0.405799\pi$
$541$	13.5668	0.583283	0.291642	+	0.956528i	$0.405799\pi$
$542$	23.4553	1.00749
$543$	0	0
$544$	−7.77420	−0.333316
$545$	−5.93399	−0.254184
$546$	0	0
$547$	3.57049	0.152663	0.0763315	−	0.997082i	$-0.475679\pi$
$547$	3.57049	0.152663	0.0763315	+	0.997082i	$0.475679\pi$
$548$	44.2353	1.88964
$549$	0	0
$550$	19.0672	0.813029
$551$	−12.3738	−0.527142
$552$	0	0
$553$	51.4516	2.18795
$554$	−20.5875	−0.874678
$555$	0	0
$556$	30.9118	1.31095
$557$	−17.1342	−0.725998	−0.362999	−	0.931789i	$-0.618247\pi$
$557$	−17.1342	−0.725998	−0.362999	+	0.931789i	$0.618247\pi$
$558$	0	0
$559$	−1.02159	−0.0432085
$560$	−76.1985	−3.21997
$561$	0	0
$562$	15.1054	0.637183
$563$	−14.6007	−0.615346	−0.307673	−	0.951492i	$-0.599550\pi$
$563$	−14.6007	−0.615346	−0.307673	+	0.951492i	$0.599550\pi$
$564$	0	0
$565$	−45.1184	−1.89815
$566$	−12.7093	−0.534212
$567$	0	0
$568$	28.1488	1.18110
$569$	−23.1163	−0.969086	−0.484543	−	0.874767i	$-0.661014\pi$
$569$	−23.1163	−0.969086	−0.484543	+	0.874767i	$0.661014\pi$
$570$	0	0
$571$	6.84505	0.286457	0.143228	−	0.989690i	$-0.454252\pi$
$571$	6.84505	0.286457	0.143228	+	0.989690i	$0.454252\pi$
$572$	4.75673	0.198889
$573$	0	0
$574$	−124.641	−5.20243
$575$	53.3239	2.22376
$576$	0	0
$577$	23.3045	0.970178	0.485089	−	0.874465i	$-0.338787\pi$
$577$	23.3045	0.970178	0.485089	+	0.874465i	$0.338787\pi$
$578$	10.9778	0.456618
$579$	0	0
$580$	−128.355	−5.32965
$581$	−33.5794	−1.39311
$582$	0	0
$583$	7.83814	0.324623
$584$	−0.243026	−0.0100565
$585$	0	0
$586$	31.3324	1.29433
$587$	11.5686	0.477488	0.238744	−	0.971083i	$-0.423264\pi$
$587$	11.5686	0.477488	0.238744	+	0.971083i	$0.423264\pi$
$588$	0	0
$589$	8.76091	0.360987
$590$	−13.2312	−0.544719
$591$	0	0
$592$	−12.0502	−0.495258
$593$	−8.26278	−0.339312	−0.169656	−	0.985503i	$-0.554266\pi$
$593$	−8.26278	−0.339312	−0.169656	+	0.985503i	$0.554266\pi$
$594$	0	0
$595$	−51.0519	−2.09292
$596$	−90.0245	−3.68755
$597$	0	0
$598$	19.6241	0.802489
$599$	−25.9828	−1.06163	−0.530814	−	0.847488i	$-0.678114\pi$
$599$	−25.9828	−1.06163	−0.530814	+	0.847488i	$0.678114\pi$
$600$	0	0
$601$	−17.3993	−0.709731	−0.354866	−	0.934917i	$-0.615473\pi$
$601$	−17.3993	−0.709731	−0.354866	+	0.934917i	$0.615473\pi$
$602$	9.10926	0.371266
$603$	0	0
$604$	−54.0841	−2.20065
$605$	3.55698	0.144612
$606$	0	0
$607$	30.0036	1.21781	0.608904	−	0.793244i	$-0.291610\pi$
$607$	30.0036	1.21781	0.608904	+	0.793244i	$0.291610\pi$
$608$	−3.16160	−0.128220
$609$	0	0
$610$	−110.866	−4.48884
$611$	8.70892	0.352325
$612$	0	0
$613$	−30.2708	−1.22263	−0.611314	−	0.791388i	$-0.709359\pi$
$613$	−30.2708	−1.22263	−0.611314	+	0.791388i	$0.709359\pi$
$614$	11.3482	0.457975
$615$	0	0
$616$	−22.2598	−0.896873
$617$	−29.8601	−1.20212	−0.601061	−	0.799203i	$-0.705255\pi$
$617$	−29.8601	−1.20212	−0.601061	+	0.799203i	$0.705255\pi$
$618$	0	0
$619$	7.77160	0.312367	0.156184	−	0.987728i	$-0.450081\pi$
$619$	7.77160	0.312367	0.156184	+	0.987728i	$0.450081\pi$
$620$	90.8779	3.64975
$621$	0	0
$622$	85.8099	3.44066
$623$	5.34356	0.214085
$624$	0	0
$625$	−4.70588	−0.188235
$626$	−52.8535	−2.11245
$627$	0	0
$628$	−46.4088	−1.85191
$629$	−8.07343	−0.321909
$630$	0	0
$631$	−34.7673	−1.38406	−0.692032	−	0.721867i	$-0.743284\pi$
$631$	−34.7673	−1.38406	−0.692032	+	0.721867i	$0.743284\pi$
$632$	70.0220	2.78533
$633$	0	0
$634$	32.8052	1.30286
$635$	26.8575	1.06581
$636$	0	0
$637$	−10.5742	−0.418965
$638$	−21.3635	−0.845787
$639$	0	0
$640$	61.0989	2.41514
$641$	12.5705	0.496504	0.248252	−	0.968695i	$-0.420144\pi$
$641$	12.5705	0.496504	0.248252	+	0.968695i	$0.420144\pi$
$642$	0	0
$643$	3.39819	0.134012	0.0670058	−	0.997753i	$-0.478655\pi$
$643$	3.39819	0.134012	0.0670058	+	0.997753i	$0.478655\pi$
$644$	−118.618	−4.67421
$645$	0	0
$646$	−12.7624	−0.502130
$647$	9.29088	0.365262	0.182631	−	0.983182i	$-0.441539\pi$
$647$	9.29088	0.365262	0.182631	+	0.983182i	$0.441539\pi$
$648$	0	0
$649$	−1.49283	−0.0585988
$650$	21.5491	0.845225
$651$	0	0
$652$	98.8719	3.87212
$653$	9.77783	0.382636	0.191318	−	0.981528i	$-0.438724\pi$
$653$	9.77783	0.382636	0.191318	+	0.981528i	$0.438724\pi$
$654$	0	0
$655$	52.6466	2.05707
$656$	−65.5142	−2.55790
$657$	0	0
$658$	−77.6555	−3.02733
$659$	−1.60770	−0.0626270	−0.0313135	−	0.999510i	$-0.509969\pi$
$659$	−1.60770	−0.0626270	−0.0313135	+	0.999510i	$0.509969\pi$
$660$	0	0
$661$	−27.7651	−1.07994	−0.539968	−	0.841686i	$-0.681564\pi$
$661$	−27.7651	−1.07994	−0.539968	+	0.841686i	$0.681564\pi$
$662$	−24.8325	−0.965143
$663$	0	0
$664$	−45.6992	−1.77347
$665$	−20.7617	−0.805104
$666$	0	0
$667$	−59.7456	−2.31336
$668$	−4.57552	−0.177032
$669$	0	0
$670$	92.7379	3.58278
$671$	−12.5087	−0.482892
$672$	0	0
$673$	−42.4797	−1.63747	−0.818735	−	0.574171i	$-0.805324\pi$
$673$	−42.4797	−1.63747	−0.818735	+	0.574171i	$0.805324\pi$
$674$	−28.3852	−1.09336
$675$	0	0
$676$	−49.3395	−1.89767
$677$	40.8506	1.57001	0.785007	−	0.619486i	$-0.212659\pi$
$677$	40.8506	1.57001	0.785007	+	0.619486i	$0.212659\pi$
$678$	0	0
$679$	−58.0064	−2.22608
$680$	−69.4781	−2.66436
$681$	0	0
$682$	15.1258	0.579195
$683$	−22.3549	−0.855387	−0.427694	−	0.903924i	$-0.640674\pi$
$683$	−22.3549	−0.855387	−0.427694	+	0.903924i	$0.640674\pi$
$684$	0	0
$685$	37.3839	1.42836
$686$	23.7457	0.906617
$687$	0	0
$688$	4.78802	0.182541
$689$	8.85839	0.337478
$690$	0	0
$691$	13.5531	0.515584	0.257792	−	0.966200i	$-0.417005\pi$
$691$	13.5531	0.515584	0.257792	+	0.966200i	$0.417005\pi$
$692$	−69.4868	−2.64149
$693$	0	0
$694$	−0.170794	−0.00648325
$695$	26.1239	0.990938
$696$	0	0
$697$	−43.8936	−1.66259
$698$	−58.3500	−2.20858
$699$	0	0
$700$	−130.254	−4.92313
$701$	−21.3138	−0.805010	−0.402505	−	0.915418i	$-0.631860\pi$
$701$	−21.3138	−0.805010	−0.402505	+	0.915418i	$0.631860\pi$
$702$	0	0
$703$	−3.28329	−0.123832
$704$	5.13529	0.193544
$705$	0	0
$706$	40.8070	1.53579
$707$	71.3383	2.68295
$708$	0	0
$709$	−21.6579	−0.813380	−0.406690	−	0.913566i	$-0.633317\pi$
$709$	−21.6579	−0.813380	−0.406690	+	0.913566i	$0.633317\pi$
$710$	45.3283	1.70114
$711$	0	0
$712$	7.27221	0.272538
$713$	42.3011	1.58419
$714$	0	0
$715$	4.01997	0.150338
$716$	61.2614	2.28944
$717$	0	0
$718$	50.2627	1.87579
$719$	23.9696	0.893914	0.446957	−	0.894555i	$-0.352508\pi$
$719$	23.9696	0.893914	0.446957	+	0.894555i	$0.352508\pi$
$720$	0	0
$721$	0.909028	0.0338540
$722$	42.1533	1.56878
$723$	0	0
$724$	−1.42105	−0.0528129
$725$	−65.6063	−2.43656
$726$	0	0
$727$	24.7920	0.919483	0.459741	−	0.888053i	$-0.347942\pi$
$727$	24.7920	0.919483	0.459741	+	0.888053i	$0.347942\pi$
$728$	−25.1572	−0.932390
$729$	0	0
$730$	−0.391348	−0.0144844
$731$	3.20790	0.118649
$732$	0	0
$733$	−44.3602	−1.63848	−0.819241	−	0.573450i	$-0.805605\pi$
$733$	−44.3602	−1.63848	−0.819241	+	0.573450i	$0.805605\pi$
$734$	−23.3593	−0.862208
$735$	0	0
$736$	−15.2654	−0.562692
$737$	10.4633	0.385421
$738$	0	0
$739$	−2.94545	−0.108350	−0.0541750	−	0.998531i	$-0.517253\pi$
$739$	−2.94545	−0.108350	−0.0541750	+	0.998531i	$0.517253\pi$
$740$	−34.0579	−1.25199
$741$	0	0
$742$	−78.9883	−2.89975
$743$	−16.0660	−0.589406	−0.294703	−	0.955589i	$-0.595221\pi$
$743$	−16.0660	−0.589406	−0.294703	+	0.955589i	$0.595221\pi$
$744$	0	0
$745$	−76.0809	−2.78739
$746$	14.6800	0.537474
$747$	0	0
$748$	−14.9367	−0.546139
$749$	−44.4252	−1.62326
$750$	0	0
$751$	23.0165	0.839883	0.419942	−	0.907551i	$-0.362051\pi$
$751$	23.0165	0.839883	0.419942	+	0.907551i	$0.362051\pi$
$752$	−40.8174	−1.48846
$753$	0	0
$754$	−24.1442	−0.879280
$755$	−45.7072	−1.66345
$756$	0	0
$757$	−1.52510	−0.0554307	−0.0277154	−	0.999616i	$-0.508823\pi$
$757$	−1.52510	−0.0554307	−0.0277154	+	0.999616i	$0.508823\pi$
$758$	74.2604	2.69726
$759$	0	0
$760$	−28.2552	−1.02492
$761$	−0.747409	−0.0270936	−0.0135468	−	0.999908i	$-0.504312\pi$
$761$	−0.747409	−0.0270936	−0.0135468	+	0.999908i	$0.504312\pi$
$762$	0	0
$763$	6.74697	0.244257
$764$	−64.9781	−2.35082
$765$	0	0
$766$	−33.5503	−1.21222
$767$	−1.68715	−0.0609193
$768$	0	0
$769$	38.8736	1.40182	0.700909	−	0.713251i	$-0.252778\pi$
$769$	38.8736	1.40182	0.700909	+	0.713251i	$0.252778\pi$
$770$	−35.8452	−1.29177
$771$	0	0
$772$	−33.9309	−1.22120
$773$	−35.7887	−1.28723	−0.643616	−	0.765349i	$-0.722567\pi$
$773$	−35.7887	−1.28723	−0.643616	+	0.765349i	$0.722567\pi$
$774$	0	0
$775$	46.4506	1.66855
$776$	−78.9427	−2.83388
$777$	0	0
$778$	−22.4787	−0.805899
$779$	−17.8505	−0.639562
$780$	0	0
$781$	5.11425	0.183002
$782$	−61.6220	−2.20360
$783$	0	0
$784$	49.5596	1.76999
$785$	−39.2207	−1.39985
$786$	0	0
$787$	−35.8481	−1.27785	−0.638923	−	0.769271i	$-0.720620\pi$
$787$	−35.8481	−1.27785	−0.638923	+	0.769271i	$0.720620\pi$
$788$	63.6336	2.26685
$789$	0	0
$790$	112.757	4.01172
$791$	51.2998	1.82401
$792$	0	0
$793$	−14.1369	−0.502014
$794$	−34.2560	−1.21570
$795$	0	0
$796$	13.0802	0.463615
$797$	12.9598	0.459061	0.229530	−	0.973301i	$-0.426281\pi$
$797$	12.9598	0.459061	0.229530	+	0.973301i	$0.426281\pi$
$798$	0	0
$799$	−27.3471	−0.967469
$800$	−16.7629	−0.592657
$801$	0	0
$802$	30.5086	1.07730
$803$	−0.0441545	−0.00155818
$804$	0	0
$805$	−100.246	−3.53320
$806$	17.0946	0.602131
$807$	0	0
$808$	97.0865	3.41549
$809$	1.36202	0.0478862	0.0239431	−	0.999713i	$-0.492378\pi$
$809$	1.36202	0.0478862	0.0239431	+	0.999713i	$0.492378\pi$
$810$	0	0
$811$	22.9479	0.805810	0.402905	−	0.915242i	$-0.368000\pi$
$811$	22.9479	0.805810	0.402905	+	0.915242i	$0.368000\pi$
$812$	145.940	5.12149
$813$	0	0
$814$	−5.66862	−0.198685
$815$	83.5580	2.92691
$816$	0	0
$817$	1.30458	0.0456416
$818$	30.3717	1.06192
$819$	0	0
$820$	−185.166	−6.46627
$821$	−44.9029	−1.56712	−0.783560	−	0.621316i	$-0.786598\pi$
$821$	−44.9029	−1.56712	−0.783560	+	0.621316i	$0.786598\pi$
$822$	0	0
$823$	7.93606	0.276634	0.138317	−	0.990388i	$-0.455831\pi$
$823$	7.93606	0.276634	0.138317	+	0.990388i	$0.455831\pi$
$824$	1.23712	0.0430972
$825$	0	0
$826$	15.0439	0.523444
$827$	4.69900	0.163400	0.0817001	−	0.996657i	$-0.473965\pi$
$827$	4.69900	0.163400	0.0817001	+	0.996657i	$0.473965\pi$
$828$	0	0
$829$	23.0285	0.799814	0.399907	−	0.916556i	$-0.369042\pi$
$829$	23.0285	0.799814	0.399907	+	0.916556i	$0.369042\pi$
$830$	−73.5899	−2.55434
$831$	0	0
$832$	5.80373	0.201208
$833$	33.2042	1.15046
$834$	0	0
$835$	−3.86684	−0.133817
$836$	−6.07442	−0.210088
$837$	0	0
$838$	−35.2462	−1.21756
$839$	7.26968	0.250977	0.125489	−	0.992095i	$-0.459950\pi$
$839$	7.26968	0.250977	0.125489	+	0.992095i	$0.459950\pi$
$840$	0	0
$841$	44.5072	1.53473
$842$	1.24160	0.0427885
$843$	0	0
$844$	16.2447	0.559165
$845$	−41.6975	−1.43444
$846$	0	0
$847$	−4.04430	−0.138964
$848$	−41.5179	−1.42573
$849$	0	0
$850$	−67.6667	−2.32095
$851$	−15.8530	−0.543434
$852$	0	0
$853$	−17.7251	−0.606896	−0.303448	−	0.952848i	$-0.598138\pi$
$853$	−17.7251	−0.606896	−0.303448	+	0.952848i	$0.598138\pi$
$854$	126.055	4.31352
$855$	0	0
$856$	−60.4596	−2.06647
$857$	−35.8056	−1.22309	−0.611547	−	0.791208i	$-0.709453\pi$
$857$	−35.8056	−1.22309	−0.611547	+	0.791208i	$0.709453\pi$
$858$	0	0
$859$	−20.4905	−0.699128	−0.349564	−	0.936913i	$-0.613670\pi$
$859$	−20.4905	−0.699128	−0.349564	+	0.936913i	$0.613670\pi$
$860$	13.5326	0.461458
$861$	0	0
$862$	66.9656	2.28086
$863$	−54.1687	−1.84392	−0.921962	−	0.387280i	$-0.873415\pi$
$863$	−54.1687	−1.84392	−0.921962	+	0.387280i	$0.873415\pi$
$864$	0	0
$865$	−58.7242	−1.99668
$866$	82.9824	2.81986
$867$	0	0
$868$	−103.328	−3.50720
$869$	12.7220	0.431565
$870$	0	0
$871$	11.8253	0.400684
$872$	9.18215	0.310947
$873$	0	0
$874$	−25.0603	−0.847677
$875$	−38.1517	−1.28976
$876$	0	0
$877$	1.86949	0.0631281	0.0315641	−	0.999502i	$-0.489951\pi$
$877$	1.86949	0.0631281	0.0315641	+	0.999502i	$0.489951\pi$
$878$	−22.1150	−0.746344
$879$	0	0
$880$	−18.8410	−0.635129
$881$	−38.4939	−1.29689	−0.648447	−	0.761260i	$-0.724581\pi$
$881$	−38.4939	−1.29689	−0.648447	+	0.761260i	$0.724581\pi$
$882$	0	0
$883$	−49.9524	−1.68103	−0.840516	−	0.541787i	$-0.817748\pi$
$883$	−49.9524	−1.68103	−0.840516	+	0.541787i	$0.817748\pi$
$884$	−16.8809	−0.567766
$885$	0	0
$886$	42.8106	1.43825
$887$	22.6818	0.761580	0.380790	−	0.924662i	$-0.375652\pi$
$887$	22.6818	0.761580	0.380790	+	0.924662i	$0.375652\pi$
$888$	0	0
$889$	−30.5370	−1.02418
$890$	11.7105	0.392537
$891$	0	0
$892$	−10.2891	−0.344506
$893$	−11.1214	−0.372165
$894$	0	0
$895$	51.7728	1.73057
$896$	−69.4696	−2.32082
$897$	0	0
$898$	22.7308	0.758535
$899$	−52.0446	−1.73578
$900$	0	0
$901$	−27.8164	−0.926699
$902$	−30.8191	−1.02616
$903$	0	0
$904$	69.8154	2.32203
$905$	−1.20095	−0.0399209
$906$	0	0
$907$	30.3826	1.00884	0.504419	−	0.863459i	$-0.331707\pi$
$907$	30.3826	1.00884	0.504419	+	0.863459i	$0.331707\pi$
$908$	−98.7023	−3.27555
$909$	0	0
$910$	−40.5110	−1.34292
$911$	−31.4540	−1.04212	−0.521059	−	0.853520i	$-0.674463\pi$
$911$	−31.4540	−1.04212	−0.521059	+	0.853520i	$0.674463\pi$
$912$	0	0
$913$	−8.30290	−0.274786
$914$	42.2921	1.39890
$915$	0	0
$916$	−54.1186	−1.78813
$917$	−59.8593	−1.97673
$918$	0	0
$919$	−2.38371	−0.0786315	−0.0393157	−	0.999227i	$-0.512518\pi$
$919$	−2.38371	−0.0786315	−0.0393157	+	0.999227i	$0.512518\pi$
$920$	−136.427	−4.49787
$921$	0	0
$922$	−6.58949	−0.217013
$923$	5.77994	0.190249
$924$	0	0
$925$	−17.4081	−0.572374
$926$	28.7892	0.946073
$927$	0	0
$928$	18.7816	0.616537
$929$	−19.4516	−0.638185	−0.319092	−	0.947724i	$-0.603378\pi$
$929$	−19.4516	−0.638185	−0.319092	+	0.947724i	$0.603378\pi$
$930$	0	0
$931$	13.5034	0.442557
$932$	108.884	3.56661
$933$	0	0
$934$	64.0043	2.09429
$935$	−12.6232	−0.412822
$936$	0	0
$937$	50.3754	1.64569	0.822847	−	0.568263i	$-0.192384\pi$
$937$	50.3754	1.64569	0.822847	+	0.568263i	$0.192384\pi$
$938$	−105.443	−3.44284
$939$	0	0
$940$	−115.364	−3.76276
$941$	−8.43399	−0.274940	−0.137470	−	0.990506i	$-0.543897\pi$
$941$	−8.43399	−0.274940	−0.137470	+	0.990506i	$0.543897\pi$
$942$	0	0
$943$	−86.1895	−2.80672
$944$	7.90739	0.257364
$945$	0	0
$946$	2.25237	0.0732310
$947$	57.5031	1.86860	0.934300	−	0.356488i	$-0.116026\pi$
$947$	57.5031	1.86860	0.934300	+	0.356488i	$0.116026\pi$
$948$	0	0
$949$	−0.0499018	−0.00161988
$950$	−27.5185	−0.892820
$951$	0	0
$952$	78.9968	2.56030
$953$	41.9638	1.35934	0.679670	−	0.733518i	$-0.262123\pi$
$953$	41.9638	1.35934	0.679670	+	0.733518i	$0.262123\pi$
$954$	0	0
$955$	−54.9138	−1.77697
$956$	42.7785	1.38355
$957$	0	0
$958$	−46.4849	−1.50186
$959$	−42.5056	−1.37258
$960$	0	0
$961$	5.84862	0.188665
$962$	−6.40647	−0.206553
$963$	0	0
$964$	82.0285	2.64196
$965$	−28.6755	−0.923096
$966$	0	0
$967$	11.7040	0.376374	0.188187	−	0.982133i	$-0.439739\pi$
$967$	11.7040	0.376374	0.188187	+	0.982133i	$0.439739\pi$
$968$	−5.50400	−0.176905
$969$	0	0
$970$	−127.122	−4.08165
$971$	55.7282	1.78840	0.894201	−	0.447666i	$-0.147745\pi$
$971$	55.7282	1.78840	0.894201	+	0.447666i	$0.147745\pi$
$972$	0	0
$973$	−29.7030	−0.952235
$974$	−47.7237	−1.52917
$975$	0	0
$976$	66.2572	2.12084
$977$	49.1119	1.57123	0.785614	−	0.618717i	$-0.212347\pi$
$977$	49.1119	1.57123	0.785614	+	0.618717i	$0.212347\pi$
$978$	0	0
$979$	1.32126	0.0422276
$980$	140.073	4.47446
$981$	0	0
$982$	67.2474	2.14595
$983$	7.09582	0.226322	0.113161	−	0.993577i	$-0.463902\pi$
$983$	7.09582	0.226322	0.113161	+	0.993577i	$0.463902\pi$
$984$	0	0
$985$	53.7776	1.71350
$986$	75.8157	2.41446
$987$	0	0
$988$	−6.86509	−0.218408
$989$	6.29905	0.200298
$990$	0	0
$991$	−28.5678	−0.907485	−0.453743	−	0.891133i	$-0.649911\pi$
$991$	−28.5678	−0.907485	−0.453743	+	0.891133i	$0.649911\pi$
$992$	−13.2978	−0.422204
$993$	0	0
$994$	−51.5384	−1.63470
$995$	11.0542	0.350443
$996$	0	0
$997$	−18.7096	−0.592539	−0.296270	−	0.955104i	$-0.595743\pi$
$997$	−18.7096	−0.592539	−0.296270	+	0.955104i	$0.595743\pi$
$998$	71.9717	2.27822
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8019.2.a.i.1.5	✓	48
3.2	odd	2		8019.2.a.j.1.44	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8019.2.a.i.1.5	✓	48	1.1	even	1	trivial
8019.2.a.j.1.44	yes	48	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)

\(f(q)\)	\(=\)	\(q-2.49176 q^{2} +4.20888 q^{4} +3.55698 q^{5} -4.04430 q^{7} -5.50400 q^{8} +O(q^{10})\)	\(q-2.49176 q^{2} +4.20888 q^{4} +3.55698 q^{5} -4.04430 q^{7} -5.50400 q^{8} -8.86315 q^{10} -1.00000 q^{11} -1.13016 q^{13} +10.0774 q^{14} +5.29691 q^{16} +3.54885 q^{17} +1.44324 q^{19} +14.9709 q^{20} +2.49176 q^{22} +6.96853 q^{23} +7.65210 q^{25} +2.81610 q^{26} -17.0220 q^{28} -8.57363 q^{29} +6.07031 q^{31} -2.19063 q^{32} -8.84289 q^{34} -14.3855 q^{35} -2.27494 q^{37} -3.59621 q^{38} -19.5776 q^{40} -12.3684 q^{41} +0.903928 q^{43} -4.20888 q^{44} -17.3639 q^{46} -7.70589 q^{47} +9.35633 q^{49} -19.0672 q^{50} -4.75673 q^{52} -7.83814 q^{53} -3.55698 q^{55} +22.2598 q^{56} +21.3635 q^{58} +1.49283 q^{59} +12.5087 q^{61} -15.1258 q^{62} -5.13529 q^{64} -4.01997 q^{65} -10.4633 q^{67} +14.9367 q^{68} +35.8452 q^{70} -5.11425 q^{71} +0.0441545 q^{73} +5.66862 q^{74} +6.07442 q^{76} +4.04430 q^{77} -12.7220 q^{79} +18.8410 q^{80} +30.8191 q^{82} +8.30290 q^{83} +12.6232 q^{85} -2.25237 q^{86} +5.50400 q^{88} -1.32126 q^{89} +4.57072 q^{91} +29.3297 q^{92} +19.2012 q^{94} +5.13357 q^{95} +14.3428 q^{97} -23.3138 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 6 q^{2} + 54 q^{4} - 24 q^{5} - 18 q^{8}+O(q^{10}) \) 48 * q - 6 * q^2 + 54 * q^4 - 24 * q^5 - 18 * q^8	\( 48 q - 6 q^{2} + 54 q^{4} - 24 q^{5} - 18 q^{8} - 48 q^{11} - 24 q^{14} + 66 q^{16} - 24 q^{17} - 48 q^{20} + 6 q^{22} - 12 q^{23} + 60 q^{25} - 36 q^{26} - 18 q^{28} - 60 q^{29} + 36 q^{31} - 42 q^{32} + 12 q^{34} - 24 q^{35} + 6 q^{37} - 24 q^{38} - 72 q^{41} - 12 q^{43} - 54 q^{44} - 30 q^{46} - 36 q^{47} + 60 q^{49} - 42 q^{50} - 48 q^{53} + 24 q^{55} - 72 q^{56} + 12 q^{58} - 60 q^{59} - 24 q^{61} - 36 q^{62} + 90 q^{64} - 48 q^{65} - 60 q^{68} - 30 q^{70} - 60 q^{71} - 18 q^{73} - 36 q^{74} - 42 q^{76} - 12 q^{79} - 96 q^{80} + 12 q^{82} - 36 q^{83} + 18 q^{85} - 48 q^{86} + 18 q^{88} - 96 q^{89} + 30 q^{91} - 36 q^{92} - 48 q^{94} - 48 q^{95} + 30 q^{97} - 54 q^{98}+O(q^{100}) \) 48 * q - 6 * q^2 + 54 * q^4 - 24 * q^5 - 18 * q^8 - 48 * q^11 - 24 * q^14 + 66 * q^16 - 24 * q^17 - 48 * q^20 + 6 * q^22 - 12 * q^23 + 60 * q^25 - 36 * q^26 - 18 * q^28 - 60 * q^29 + 36 * q^31 - 42 * q^32 + 12 * q^34 - 24 * q^35 + 6 * q^37 - 24 * q^38 - 72 * q^41 - 12 * q^43 - 54 * q^44 - 30 * q^46 - 36 * q^47 + 60 * q^49 - 42 * q^50 - 48 * q^53 + 24 * q^55 - 72 * q^56 + 12 * q^58 - 60 * q^59 - 24 * q^61 - 36 * q^62 + 90 * q^64 - 48 * q^65 - 60 * q^68 - 30 * q^70 - 60 * q^71 - 18 * q^73 - 36 * q^74 - 42 * q^76 - 12 * q^79 - 96 * q^80 + 12 * q^82 - 36 * q^83 + 18 * q^85 - 48 * q^86 + 18 * q^88 - 96 * q^89 + 30 * q^91 - 36 * q^92 - 48 * q^94 - 48 * q^95 + 30 * q^97 - 54 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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