LMFDB - Embedded newform 8019.2.a.f.1.12

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:	$21$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
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)

$f(q)$	$=$	$q+0.878744 q^{2} -1.22781 q^{4} +1.62071 q^{5} +2.93534 q^{7} -2.83642 q^{8} +O(q^{10})$	\(q+0.878744 q^{2} -1.22781 q^{4} +1.62071 q^{5} +2.93534 q^{7} -2.83642 q^{8} +1.42419 q^{10} -1.00000 q^{11} +1.21396 q^{13} +2.57941 q^{14} -0.0368650 q^{16} -1.35267 q^{17} +8.11094 q^{19} -1.98993 q^{20} -0.878744 q^{22} +7.32330 q^{23} -2.37329 q^{25} +1.06676 q^{26} -3.60403 q^{28} +8.00633 q^{29} -6.70455 q^{31} +5.64044 q^{32} -1.18865 q^{34} +4.75734 q^{35} +3.40480 q^{37} +7.12744 q^{38} -4.59702 q^{40} -9.92895 q^{41} +4.20697 q^{43} +1.22781 q^{44} +6.43530 q^{46} +2.63329 q^{47} +1.61620 q^{49} -2.08551 q^{50} -1.49051 q^{52} -7.98667 q^{53} -1.62071 q^{55} -8.32584 q^{56} +7.03551 q^{58} +8.43092 q^{59} -12.6425 q^{61} -5.89158 q^{62} +5.03023 q^{64} +1.96748 q^{65} -9.56383 q^{67} +1.66083 q^{68} +4.18048 q^{70} +7.50475 q^{71} -13.1839 q^{73} +2.99194 q^{74} -9.95869 q^{76} -2.93534 q^{77} +14.6473 q^{79} -0.0597476 q^{80} -8.72501 q^{82} +14.5125 q^{83} -2.19230 q^{85} +3.69685 q^{86} +2.83642 q^{88} -3.74070 q^{89} +3.56337 q^{91} -8.99162 q^{92} +2.31399 q^{94} +13.1455 q^{95} +1.67892 q^{97} +1.42023 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + 6 q^{2} + 18 q^{4} + 12 q^{5} + 18 q^{8}+O(q^{10}) $ 21 * q + 6 * q^2 + 18 * q^4 + 12 * q^5 + 18 * q^8	$ 21 q + 6 q^{2} + 18 q^{4} + 12 q^{5} + 18 q^{8} + 15 q^{10} - 21 q^{11} + 3 q^{13} + 24 q^{14} + 12 q^{16} + 12 q^{17} - 12 q^{19} + 15 q^{20} - 6 q^{22} + 15 q^{25} + 15 q^{26} + 42 q^{28} + 27 q^{29} + 6 q^{31} + 42 q^{32} - 6 q^{34} + 42 q^{35} + 3 q^{37} + 27 q^{38} + 30 q^{40} + 24 q^{41} + 6 q^{43} - 18 q^{44} - 51 q^{46} + 24 q^{47} - 9 q^{49} - 3 q^{50} + 9 q^{52} + 12 q^{53} - 12 q^{55} + 45 q^{56} + 12 q^{58} + 12 q^{59} - 30 q^{61} + 9 q^{62} + 42 q^{64} + 18 q^{67} + 30 q^{68} + 21 q^{70} + 36 q^{71} - 39 q^{73} - 27 q^{76} + 27 q^{79} + 78 q^{80} + 63 q^{82} + 42 q^{83} + 36 q^{85} + 42 q^{86} - 18 q^{88} + 36 q^{89} - 21 q^{91} + 6 q^{92} + 42 q^{94} + 24 q^{95} + 24 q^{97} + 69 q^{98}+O(q^{100}) $ 21 * q + 6 * q^2 + 18 * q^4 + 12 * q^5 + 18 * q^8 + 15 * q^10 - 21 * q^11 + 3 * q^13 + 24 * q^14 + 12 * q^16 + 12 * q^17 - 12 * q^19 + 15 * q^20 - 6 * q^22 + 15 * q^25 + 15 * q^26 + 42 * q^28 + 27 * q^29 + 6 * q^31 + 42 * q^32 - 6 * q^34 + 42 * q^35 + 3 * q^37 + 27 * q^38 + 30 * q^40 + 24 * q^41 + 6 * q^43 - 18 * q^44 - 51 * q^46 + 24 * q^47 - 9 * q^49 - 3 * q^50 + 9 * q^52 + 12 * q^53 - 12 * q^55 + 45 * q^56 + 12 * q^58 + 12 * q^59 - 30 * q^61 + 9 * q^62 + 42 * q^64 + 18 * q^67 + 30 * q^68 + 21 * q^70 + 36 * q^71 - 39 * q^73 - 27 * q^76 + 27 * q^79 + 78 * q^80 + 63 * q^82 + 42 * q^83 + 36 * q^85 + 42 * q^86 - 18 * q^88 + 36 * q^89 - 21 * q^91 + 6 * q^92 + 42 * q^94 + 24 * q^95 + 24 * q^97 + 69 * 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 $

$2$	0.878744	0.621366	0.310683	−	0.950514i	$-0.399442\pi$
$2$	0.878744	0.621366	0.310683	+	0.950514i	$0.399442\pi$
$3$	0	0
$3$	0	0
$4$	−1.22781	−0.613905
$5$	1.62071	0.724805	0.362403	−	0.932022i	$-0.381957\pi$
$5$	1.62071	0.724805	0.362403	+	0.932022i	$0.381957\pi$
$6$	0	0
$7$	2.93534	1.10945	0.554727	−	0.832033i	$-0.312823\pi$
$7$	2.93534	1.10945	0.554727	+	0.832033i	$0.312823\pi$
$8$	−2.83642	−1.00282
$9$	0	0
$10$	1.42419	0.450369
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.21396	0.336691	0.168345	−	0.985728i	$-0.446158\pi$
$13$	1.21396	0.336691	0.168345	+	0.985728i	$0.446158\pi$
$14$	2.57941	0.689376
$15$	0	0
$16$	−0.0368650	−0.00921625
$17$	−1.35267	−0.328072	−0.164036	−	0.986454i	$-0.552451\pi$
$17$	−1.35267	−0.328072	−0.164036	+	0.986454i	$0.552451\pi$
$18$	0	0
$19$	8.11094	1.86078	0.930389	−	0.366574i	$-0.119470\pi$
$19$	8.11094	1.86078	0.930389	+	0.366574i	$0.119470\pi$
$20$	−1.98993	−0.444961
$21$	0	0
$22$	−0.878744	−0.187349
$23$	7.32330	1.52701	0.763507	−	0.645800i	$-0.223476\pi$
$23$	7.32330	1.52701	0.763507	+	0.645800i	$0.223476\pi$
$24$	0	0
$25$	−2.37329	−0.474657
$26$	1.06676	0.209208
$27$	0	0
$28$	−3.60403	−0.681099
$29$	8.00633	1.48674	0.743369	−	0.668881i	$-0.233227\pi$
$29$	8.00633	1.48674	0.743369	+	0.668881i	$0.233227\pi$
$30$	0	0
$31$	−6.70455	−1.20417	−0.602086	−	0.798431i	$-0.705664\pi$
$31$	−6.70455	−1.20417	−0.602086	+	0.798431i	$0.705664\pi$
$32$	5.64044	0.997098
$33$	0	0
$34$	−1.18865	−0.203853
$35$	4.75734	0.804138
$36$	0	0
$37$	3.40480	0.559745	0.279873	−	0.960037i	$-0.409708\pi$
$37$	3.40480	0.559745	0.279873	+	0.960037i	$0.409708\pi$
$38$	7.12744	1.15622
$39$	0	0
$40$	−4.59702	−0.726853
$41$	−9.92895	−1.55064	−0.775321	−	0.631567i	$-0.782412\pi$
$41$	−9.92895	−1.55064	−0.775321	+	0.631567i	$0.782412\pi$
$42$	0	0
$43$	4.20697	0.641557	0.320778	−	0.947154i	$-0.396056\pi$
$43$	4.20697	0.641557	0.320778	+	0.947154i	$0.396056\pi$
$44$	1.22781	0.185099
$45$	0	0
$46$	6.43530	0.948834
$47$	2.63329	0.384105	0.192053	−	0.981385i	$-0.438486\pi$
$47$	2.63329	0.384105	0.192053	+	0.981385i	$0.438486\pi$
$48$	0	0
$49$	1.61620	0.230886
$50$	−2.08551	−0.294936
$51$	0	0
$52$	−1.49051	−0.206696
$53$	−7.98667	−1.09705	−0.548527	−	0.836133i	$-0.684811\pi$
$53$	−7.98667	−1.09705	−0.548527	+	0.836133i	$0.684811\pi$
$54$	0	0
$55$	−1.62071	−0.218537
$56$	−8.32584	−1.11259
$57$	0	0
$58$	7.03551	0.923808
$59$	8.43092	1.09761	0.548806	−	0.835950i	$-0.315082\pi$
$59$	8.43092	1.09761	0.548806	+	0.835950i	$0.315082\pi$
$60$	0	0
$61$	−12.6425	−1.61870	−0.809351	−	0.587325i	$-0.800181\pi$
$61$	−12.6425	−1.61870	−0.809351	+	0.587325i	$0.800181\pi$
$62$	−5.89158	−0.748231
$63$	0	0
$64$	5.03023	0.628779
$65$	1.96748	0.244035
$66$	0	0
$67$	−9.56383	−1.16841	−0.584204	−	0.811607i	$-0.698593\pi$
$67$	−9.56383	−1.16841	−0.584204	+	0.811607i	$0.698593\pi$
$68$	1.66083	0.201405
$69$	0	0
$70$	4.18048	0.499664
$71$	7.50475	0.890649	0.445325	−	0.895369i	$-0.353088\pi$
$71$	7.50475	0.890649	0.445325	+	0.895369i	$0.353088\pi$
$72$	0	0
$73$	−13.1839	−1.54306	−0.771529	−	0.636194i	$-0.780508\pi$
$73$	−13.1839	−1.54306	−0.771529	+	0.636194i	$0.780508\pi$
$74$	2.99194	0.347806
$75$	0	0
$76$	−9.95869	−1.14234
$77$	−2.93534	−0.334513
$78$	0	0
$79$	14.6473	1.64795	0.823975	−	0.566627i	$-0.191752\pi$
$79$	14.6473	1.64795	0.823975	+	0.566627i	$0.191752\pi$
$80$	−0.0597476	−0.00667999
$81$	0	0
$82$	−8.72501	−0.963516
$83$	14.5125	1.59295	0.796477	−	0.604669i	$-0.206695\pi$
$83$	14.5125	1.59295	0.796477	+	0.604669i	$0.206695\pi$
$84$	0	0
$85$	−2.19230	−0.237788
$86$	3.69685	0.398641
$87$	0	0
$88$	2.83642	0.302363
$89$	−3.74070	−0.396514	−0.198257	−	0.980150i	$-0.563528\pi$
$89$	−3.74070	−0.396514	−0.198257	+	0.980150i	$0.563528\pi$
$90$	0	0
$91$	3.56337	0.373543
$92$	−8.99162	−0.937441
$93$	0	0
$94$	2.31399	0.238670
$95$	13.1455	1.34870
$96$	0	0
$97$	1.67892	0.170469	0.0852344	−	0.996361i	$-0.472836\pi$
$97$	1.67892	0.170469	0.0852344	+	0.996361i	$0.472836\pi$
$98$	1.42023	0.143465
$99$	0	0
$100$	2.91394	0.291394
$101$	4.04701	0.402693	0.201346	−	0.979520i	$-0.435468\pi$
$101$	4.04701	0.402693	0.201346	+	0.979520i	$0.435468\pi$
$102$	0	0
$103$	13.7069	1.35058	0.675288	−	0.737554i	$-0.264019\pi$
$103$	13.7069	1.35058	0.675288	+	0.737554i	$0.264019\pi$
$104$	−3.44329	−0.337642
$105$	0	0
$106$	−7.01823	−0.681671
$107$	3.00545	0.290548	0.145274	−	0.989391i	$-0.453594\pi$
$107$	3.00545	0.290548	0.145274	+	0.989391i	$0.453594\pi$
$108$	0	0
$109$	−19.8277	−1.89915	−0.949575	−	0.313539i	$-0.898485\pi$
$109$	−19.8277	−1.89915	−0.949575	+	0.313539i	$0.898485\pi$
$110$	−1.42419	−0.135791
$111$	0	0
$112$	−0.108211	−0.0102250
$113$	7.21976	0.679178	0.339589	−	0.940574i	$-0.389712\pi$
$113$	7.21976	0.679178	0.339589	+	0.940574i	$0.389712\pi$
$114$	0	0
$115$	11.8690	1.10679
$116$	−9.83025	−0.912716
$117$	0	0
$118$	7.40862	0.682019
$119$	−3.97056	−0.363980
$120$	0	0
$121$	1.00000	0.0909091
$122$	−11.1095	−1.00581
$123$	0	0
$124$	8.23191	0.739247
$125$	−11.9500	−1.06884
$126$	0	0
$127$	11.1310	0.987717	0.493858	−	0.869542i	$-0.335586\pi$
$127$	11.1310	0.987717	0.493858	+	0.869542i	$0.335586\pi$
$128$	−6.86060	−0.606397
$129$	0	0
$130$	1.72891	0.151635
$131$	−1.22185	−0.106753	−0.0533766	−	0.998574i	$-0.516998\pi$
$131$	−1.22185	−0.106753	−0.0533766	+	0.998574i	$0.516998\pi$
$132$	0	0
$133$	23.8083	2.06445
$134$	−8.40416	−0.726009
$135$	0	0
$136$	3.83675	0.328999
$137$	17.3594	1.48312	0.741558	−	0.670889i	$-0.234087\pi$
$137$	17.3594	1.48312	0.741558	+	0.670889i	$0.234087\pi$
$138$	0	0
$139$	20.4680	1.73608	0.868039	−	0.496497i	$-0.165381\pi$
$139$	20.4680	1.73608	0.868039	+	0.496497i	$0.165381\pi$
$140$	−5.84111	−0.493664
$141$	0	0
$142$	6.59475	0.553419
$143$	−1.21396	−0.101516
$144$	0	0
$145$	12.9760	1.07760
$146$	−11.5853	−0.958803
$147$	0	0
$148$	−4.18044	−0.343630
$149$	0.230662	0.0188966	0.00944830	−	0.999955i	$-0.496992\pi$
$149$	0.230662	0.0188966	0.00944830	+	0.999955i	$0.496992\pi$
$150$	0	0
$151$	3.69451	0.300655	0.150327	−	0.988636i	$-0.451967\pi$
$151$	3.69451	0.300655	0.150327	+	0.988636i	$0.451967\pi$
$152$	−23.0060	−1.86603
$153$	0	0
$154$	−2.57941	−0.207855
$155$	−10.8662	−0.872791
$156$	0	0
$157$	9.13250	0.728853	0.364427	−	0.931232i	$-0.381265\pi$
$157$	9.13250	0.728853	0.364427	+	0.931232i	$0.381265\pi$
$158$	12.8712	1.02398
$159$	0	0
$160$	9.14154	0.722702
$161$	21.4964	1.69415
$162$	0	0
$163$	3.05921	0.239615	0.119808	−	0.992797i	$-0.461772\pi$
$163$	3.05921	0.239615	0.119808	+	0.992797i	$0.461772\pi$
$164$	12.1909	0.951947
$165$	0	0
$166$	12.7528	0.989807
$167$	10.3486	0.800801	0.400401	−	0.916340i	$-0.368871\pi$
$167$	10.3486	0.800801	0.400401	+	0.916340i	$0.368871\pi$
$168$	0	0
$169$	−11.5263	−0.886639
$170$	−1.92647	−0.147753
$171$	0	0
$172$	−5.16536	−0.393855
$173$	−5.72452	−0.435227	−0.217613	−	0.976035i	$-0.569827\pi$
$173$	−5.72452	−0.435227	−0.217613	+	0.976035i	$0.569827\pi$
$174$	0	0
$175$	−6.96639	−0.526610
$176$	0.0368650	0.00277881
$177$	0	0
$178$	−3.28712	−0.246380
$179$	−5.48032	−0.409618	−0.204809	−	0.978802i	$-0.565657\pi$
$179$	−5.48032	−0.409618	−0.204809	+	0.978802i	$0.565657\pi$
$180$	0	0
$181$	8.90967	0.662251	0.331125	−	0.943587i	$-0.392572\pi$
$181$	8.90967	0.662251	0.331125	+	0.943587i	$0.392572\pi$
$182$	3.13129	0.232107
$183$	0	0
$184$	−20.7719	−1.53133
$185$	5.51820	0.405706
$186$	0	0
$187$	1.35267	0.0989174
$188$	−3.23318	−0.235804
$189$	0	0
$190$	11.5515	0.838037
$191$	−24.1213	−1.74536	−0.872678	−	0.488296i	$-0.837619\pi$
$191$	−24.1213	−1.74536	−0.872678	+	0.488296i	$0.837619\pi$
$192$	0	0
$193$	3.70865	0.266954	0.133477	−	0.991052i	$-0.457386\pi$
$193$	3.70865	0.266954	0.133477	+	0.991052i	$0.457386\pi$
$194$	1.47534	0.105923
$195$	0	0
$196$	−1.98439	−0.141742
$197$	0.264521	0.0188464	0.00942318	−	0.999956i	$-0.497000\pi$
$197$	0.264521	0.0188464	0.00942318	+	0.999956i	$0.497000\pi$
$198$	0	0
$199$	0.0295833	0.00209711	0.00104855	−	0.999999i	$-0.499666\pi$
$199$	0.0295833	0.00209711	0.00104855	+	0.999999i	$0.499666\pi$
$200$	6.73163	0.475998
$201$	0	0
$202$	3.55628	0.250219
$203$	23.5013	1.64947
$204$	0	0
$205$	−16.0920	−1.12391
$206$	12.0448	0.839202
$207$	0	0
$208$	−0.0447525	−0.00310303
$209$	−8.11094	−0.561046
$210$	0	0
$211$	−18.9772	−1.30645	−0.653224	−	0.757165i	$-0.726584\pi$
$211$	−18.9772	−1.30645	−0.653224	+	0.757165i	$0.726584\pi$
$212$	9.80611	0.673486
$213$	0	0
$214$	2.64102	0.180536
$215$	6.81829	0.465004
$216$	0	0
$217$	−19.6801	−1.33597
$218$	−17.4235	−1.18007
$219$	0	0
$220$	1.98993	0.134161
$221$	−1.64209	−0.110459
$222$	0	0
$223$	5.15986	0.345530	0.172765	−	0.984963i	$-0.444730\pi$
$223$	5.15986	0.345530	0.172765	+	0.984963i	$0.444730\pi$
$224$	16.5566	1.10623
$225$	0	0
$226$	6.34432	0.422018
$227$	−20.1891	−1.33999	−0.669997	−	0.742363i	$-0.733705\pi$
$227$	−20.1891	−1.33999	−0.669997	+	0.742363i	$0.733705\pi$
$228$	0	0
$229$	24.6580	1.62945	0.814724	−	0.579849i	$-0.196888\pi$
$229$	24.6580	1.62945	0.814724	+	0.579849i	$0.196888\pi$
$230$	10.4298	0.687720
$231$	0	0
$232$	−22.7093	−1.49094
$233$	−1.78094	−0.116673	−0.0583367	−	0.998297i	$-0.518580\pi$
$233$	−1.78094	−0.116673	−0.0583367	+	0.998297i	$0.518580\pi$
$234$	0	0
$235$	4.26781	0.278401
$236$	−10.3516	−0.673830
$237$	0	0
$238$	−3.48910	−0.226165
$239$	14.3496	0.928198	0.464099	−	0.885783i	$-0.346378\pi$
$239$	14.3496	0.928198	0.464099	+	0.885783i	$0.346378\pi$
$240$	0	0
$241$	7.15703	0.461025	0.230512	−	0.973069i	$-0.425960\pi$
$241$	7.15703	0.461025	0.230512	+	0.973069i	$0.425960\pi$
$242$	0.878744	0.0564878
$243$	0	0
$244$	15.5225	0.993729
$245$	2.61941	0.167348
$246$	0	0
$247$	9.84633	0.626507
$248$	19.0169	1.20757
$249$	0	0
$250$	−10.5010	−0.664140
$251$	20.8126	1.31368	0.656840	−	0.754030i	$-0.271893\pi$
$251$	20.8126	1.31368	0.656840	+	0.754030i	$0.271893\pi$
$252$	0	0
$253$	−7.32330	−0.460412
$254$	9.78130	0.613733
$255$	0	0
$256$	−16.0892	−1.00557
$257$	15.1921	0.947655	0.473827	−	0.880618i	$-0.342872\pi$
$257$	15.1921	0.947655	0.473827	+	0.880618i	$0.342872\pi$
$258$	0	0
$259$	9.99423	0.621011
$260$	−2.41569	−0.149814
$261$	0	0
$262$	−1.07369	−0.0663328
$263$	12.8538	0.792598	0.396299	−	0.918121i	$-0.370294\pi$
$263$	12.8538	0.792598	0.396299	+	0.918121i	$0.370294\pi$
$264$	0	0
$265$	−12.9441	−0.795150
$266$	20.9214	1.28278
$267$	0	0
$268$	11.7426	0.717291
$269$	22.8920	1.39575	0.697875	−	0.716220i	$-0.254129\pi$
$269$	22.8920	1.39575	0.697875	+	0.716220i	$0.254129\pi$
$270$	0	0
$271$	−31.6229	−1.92095	−0.960476	−	0.278364i	$-0.910208\pi$
$271$	−31.6229	−1.92095	−0.960476	+	0.278364i	$0.910208\pi$
$272$	0.0498664	0.00302359
$273$	0	0
$274$	15.2545	0.921557
$275$	2.37329	0.143115
$276$	0	0
$277$	−8.42294	−0.506086	−0.253043	−	0.967455i	$-0.581431\pi$
$277$	−8.42294	−0.506086	−0.253043	+	0.967455i	$0.581431\pi$
$278$	17.9862	1.07874
$279$	0	0
$280$	−13.4938	−0.806409
$281$	3.44786	0.205682	0.102841	−	0.994698i	$-0.467207\pi$
$281$	3.44786	0.205682	0.102841	+	0.994698i	$0.467207\pi$
$282$	0	0
$283$	−4.13787	−0.245971	−0.122985	−	0.992408i	$-0.539247\pi$
$283$	−4.13787	−0.245971	−0.122985	+	0.992408i	$0.539247\pi$
$284$	−9.21440	−0.546774
$285$	0	0
$286$	−1.06676	−0.0630786
$287$	−29.1448	−1.72036
$288$	0	0
$289$	−15.1703	−0.892369
$290$	11.4026	0.669581
$291$	0	0
$292$	16.1873	0.947291
$293$	1.08359	0.0633040	0.0316520	−	0.999499i	$-0.489923\pi$
$293$	1.08359	0.0633040	0.0316520	+	0.999499i	$0.489923\pi$
$294$	0	0
$295$	13.6641	0.795556
$296$	−9.65743	−0.561326
$297$	0	0
$298$	0.202693	0.0117417
$299$	8.89017	0.514132
$300$	0	0
$301$	12.3489	0.711777
$302$	3.24653	0.186817
$303$	0	0
$304$	−0.299010	−0.0171494
$305$	−20.4898	−1.17324
$306$	0	0
$307$	−14.0702	−0.803026	−0.401513	−	0.915853i	$-0.631516\pi$
$307$	−14.0702	−0.803026	−0.401513	+	0.915853i	$0.631516\pi$
$308$	3.60403	0.205359
$309$	0	0
$310$	−9.54856	−0.542322
$311$	21.9681	1.24570	0.622848	−	0.782343i	$-0.285975\pi$
$311$	21.9681	1.24570	0.622848	+	0.782343i	$0.285975\pi$
$312$	0	0
$313$	26.9911	1.52563	0.762815	−	0.646617i	$-0.223817\pi$
$313$	26.9911	1.52563	0.762815	+	0.646617i	$0.223817\pi$
$314$	8.02513	0.452884
$315$	0	0
$316$	−17.9841	−1.01168
$317$	16.1355	0.906258	0.453129	−	0.891445i	$-0.350308\pi$
$317$	16.1355	0.906258	0.453129	+	0.891445i	$0.350308\pi$
$318$	0	0
$319$	−8.00633	−0.448269
$320$	8.15257	0.455742
$321$	0	0
$322$	18.8898	1.05269
$323$	−10.9715	−0.610469
$324$	0	0
$325$	−2.88106	−0.159813
$326$	2.68826	0.148889
$327$	0	0
$328$	28.1627	1.55502
$329$	7.72960	0.426147
$330$	0	0
$331$	−10.7626	−0.591567	−0.295784	−	0.955255i	$-0.595581\pi$
$331$	−10.7626	−0.591567	−0.295784	+	0.955255i	$0.595581\pi$
$332$	−17.8186	−0.977922
$333$	0	0
$334$	9.09380	0.497591
$335$	−15.5002	−0.846868
$336$	0	0
$337$	−12.0680	−0.657386	−0.328693	−	0.944437i	$-0.606608\pi$
$337$	−12.0680	−0.657386	−0.328693	+	0.944437i	$0.606608\pi$
$338$	−10.1287	−0.550927
$339$	0	0
$340$	2.69173	0.145979
$341$	6.70455	0.363072
$342$	0	0
$343$	−15.8033	−0.853296
$344$	−11.9327	−0.643369
$345$	0	0
$346$	−5.03038	−0.270435
$347$	−6.96392	−0.373843	−0.186921	−	0.982375i	$-0.559851\pi$
$347$	−6.96392	−0.373843	−0.186921	+	0.982375i	$0.559851\pi$
$348$	0	0
$349$	−0.378819	−0.0202777	−0.0101389	−	0.999949i	$-0.503227\pi$
$349$	−0.378819	−0.0202777	−0.0101389	+	0.999949i	$0.503227\pi$
$350$	−6.12168	−0.327217
$351$	0	0
$352$	−5.64044	−0.300636
$353$	−6.88115	−0.366247	−0.183123	−	0.983090i	$-0.558621\pi$
$353$	−6.88115	−0.366247	−0.183123	+	0.983090i	$0.558621\pi$
$354$	0	0
$355$	12.1630	0.645548
$356$	4.59287	0.243422
$357$	0	0
$358$	−4.81579	−0.254523
$359$	26.7485	1.41173	0.705867	−	0.708345i	$-0.250558\pi$
$359$	26.7485	1.41173	0.705867	+	0.708345i	$0.250558\pi$
$360$	0	0
$361$	46.7874	2.46249
$362$	7.82932	0.411500
$363$	0	0
$364$	−4.37514	−0.229320
$365$	−21.3673	−1.11842
$366$	0	0
$367$	7.27318	0.379657	0.189828	−	0.981817i	$-0.439207\pi$
$367$	7.27318	0.379657	0.189828	+	0.981817i	$0.439207\pi$
$368$	−0.269974	−0.0140733
$369$	0	0
$370$	4.84909	0.252092
$371$	−23.4436	−1.21713
$372$	0	0
$373$	29.1370	1.50866	0.754328	−	0.656497i	$-0.227963\pi$
$373$	29.1370	1.50866	0.754328	+	0.656497i	$0.227963\pi$
$374$	1.18865	0.0614639
$375$	0	0
$376$	−7.46911	−0.385190
$377$	9.71934	0.500571
$378$	0	0
$379$	−12.8220	−0.658623	−0.329311	−	0.944221i	$-0.606817\pi$
$379$	−12.8220	−0.658623	−0.329311	+	0.944221i	$0.606817\pi$
$380$	−16.1402	−0.827974
$381$	0	0
$382$	−21.1964	−1.08450
$383$	−0.363385	−0.0185681	−0.00928404	−	0.999957i	$-0.502955\pi$
$383$	−0.363385	−0.0185681	−0.00928404	+	0.999957i	$0.502955\pi$
$384$	0	0
$385$	−4.75734	−0.242457
$386$	3.25895	0.165876
$387$	0	0
$388$	−2.06140	−0.104652
$389$	−19.3261	−0.979873	−0.489936	−	0.871758i	$-0.662980\pi$
$389$	−19.3261	−0.979873	−0.489936	+	0.871758i	$0.662980\pi$
$390$	0	0
$391$	−9.90604	−0.500970
$392$	−4.58423	−0.231539
$393$	0	0
$394$	0.232447	0.0117105
$395$	23.7391	1.19444
$396$	0	0
$397$	−17.3256	−0.869549	−0.434774	−	0.900539i	$-0.643172\pi$
$397$	−17.3256	−0.869549	−0.434774	+	0.900539i	$0.643172\pi$
$398$	0.0259962	0.00130307
$399$	0	0
$400$	0.0874912	0.00437456
$401$	16.1636	0.807170	0.403585	−	0.914942i	$-0.367764\pi$
$401$	16.1636	0.807170	0.403585	+	0.914942i	$0.367764\pi$
$402$	0	0
$403$	−8.13903	−0.405434
$404$	−4.96896	−0.247215
$405$	0	0
$406$	20.6516	1.02492
$407$	−3.40480	−0.168770
$408$	0	0
$409$	2.90298	0.143543	0.0717716	−	0.997421i	$-0.477135\pi$
$409$	2.90298	0.143543	0.0717716	+	0.997421i	$0.477135\pi$
$410$	−14.1407	−0.698361
$411$	0	0
$412$	−16.8294	−0.829126
$413$	24.7476	1.21775
$414$	0	0
$415$	23.5206	1.15458
$416$	6.84725	0.335714
$417$	0	0
$418$	−7.12744	−0.348614
$419$	−0.942966	−0.0460669	−0.0230334	−	0.999735i	$-0.507332\pi$
$419$	−0.942966	−0.0460669	−0.0230334	+	0.999735i	$0.507332\pi$
$420$	0	0
$421$	13.4878	0.657357	0.328679	−	0.944442i	$-0.393397\pi$
$421$	13.4878	0.657357	0.328679	+	0.944442i	$0.393397\pi$
$422$	−16.6761	−0.811781
$423$	0	0
$424$	22.6535	1.10015
$425$	3.21028	0.155722
$426$	0	0
$427$	−37.1099	−1.79587
$428$	−3.69012	−0.178369
$429$	0	0
$430$	5.99153	0.288937
$431$	13.7381	0.661739	0.330870	−	0.943677i	$-0.392658\pi$
$431$	13.7381	0.661739	0.330870	+	0.943677i	$0.392658\pi$
$432$	0	0
$433$	−3.64741	−0.175283	−0.0876417	−	0.996152i	$-0.527933\pi$
$433$	−3.64741	−0.175283	−0.0876417	+	0.996152i	$0.527933\pi$
$434$	−17.2938	−0.830128
$435$	0	0
$436$	24.3447	1.16590
$437$	59.3989	2.84143
$438$	0	0
$439$	−18.4732	−0.881676	−0.440838	−	0.897587i	$-0.645319\pi$
$439$	−18.4732	−0.881676	−0.440838	+	0.897587i	$0.645319\pi$
$440$	4.59702	0.219154
$441$	0	0
$442$	−1.44297	−0.0686353
$443$	35.9859	1.70974	0.854870	−	0.518842i	$-0.173637\pi$
$443$	35.9859	1.70974	0.854870	+	0.518842i	$0.173637\pi$
$444$	0	0
$445$	−6.06261	−0.287395
$446$	4.53419	0.214700
$447$	0	0
$448$	14.7654	0.697601
$449$	−11.5787	−0.546435	−0.273217	−	0.961952i	$-0.588088\pi$
$449$	−11.5787	−0.546435	−0.273217	+	0.961952i	$0.588088\pi$
$450$	0	0
$451$	9.92895	0.467536
$452$	−8.86449	−0.416951
$453$	0	0
$454$	−17.7410	−0.832627
$455$	5.77520	0.270746
$456$	0	0
$457$	13.6070	0.636506	0.318253	−	0.948006i	$-0.396904\pi$
$457$	13.6070	0.636506	0.318253	+	0.948006i	$0.396904\pi$
$458$	21.6681	1.01248
$459$	0	0
$460$	−14.5728	−0.679462
$461$	17.2620	0.803971	0.401986	−	0.915646i	$-0.368320\pi$
$461$	17.2620	0.803971	0.401986	+	0.915646i	$0.368320\pi$
$462$	0	0
$463$	−32.9869	−1.53303	−0.766516	−	0.642225i	$-0.778011\pi$
$463$	−32.9869	−1.53303	−0.766516	+	0.642225i	$0.778011\pi$
$464$	−0.295154	−0.0137022
$465$	0	0
$466$	−1.56499	−0.0724968
$467$	−22.3097	−1.03237	−0.516185	−	0.856477i	$-0.672648\pi$
$467$	−22.3097	−1.03237	−0.516185	+	0.856477i	$0.672648\pi$
$468$	0	0
$469$	−28.0731	−1.29629
$470$	3.75031	0.172989
$471$	0	0
$472$	−23.9136	−1.10071
$473$	−4.20697	−0.193437
$474$	0	0
$475$	−19.2496	−0.883231
$476$	4.87509	0.223449
$477$	0	0
$478$	12.6096	0.576750
$479$	−6.80029	−0.310713	−0.155357	−	0.987858i	$-0.549653\pi$
$479$	−6.80029	−0.310713	−0.155357	+	0.987858i	$0.549653\pi$
$480$	0	0
$481$	4.13327	0.188461
$482$	6.28919	0.286465
$483$	0	0
$484$	−1.22781	−0.0558095
$485$	2.72105	0.123557
$486$	0	0
$487$	−32.3365	−1.46530	−0.732652	−	0.680603i	$-0.761718\pi$
$487$	−32.3365	−1.46530	−0.732652	+	0.680603i	$0.761718\pi$
$488$	35.8593	1.62327
$489$	0	0
$490$	2.30179	0.103984
$491$	−26.6248	−1.20156	−0.600781	−	0.799413i	$-0.705144\pi$
$491$	−26.6248	−1.20156	−0.600781	+	0.799413i	$0.705144\pi$
$492$	0	0
$493$	−10.8300	−0.487757
$494$	8.65240	0.389290
$495$	0	0
$496$	0.247163	0.0110980
$497$	22.0290	0.988134
$498$	0	0
$499$	28.6102	1.28077	0.640383	−	0.768056i	$-0.278776\pi$
$499$	28.6102	1.28077	0.640383	+	0.768056i	$0.278776\pi$
$500$	14.6723	0.656166
$501$	0	0
$502$	18.2890	0.816276
$503$	27.8305	1.24090	0.620450	−	0.784246i	$-0.286950\pi$
$503$	27.8305	1.24090	0.620450	+	0.784246i	$0.286950\pi$
$504$	0	0
$505$	6.55905	0.291874
$506$	−6.43530	−0.286084
$507$	0	0
$508$	−13.6667	−0.606364
$509$	−13.0513	−0.578488	−0.289244	−	0.957255i	$-0.593404\pi$
$509$	−13.0513	−0.578488	−0.289244	+	0.957255i	$0.593404\pi$
$510$	0	0
$511$	−38.6992	−1.71195
$512$	−0.417064	−0.0184318
$513$	0	0
$514$	13.3499	0.588840
$515$	22.2149	0.978905
$516$	0	0
$517$	−2.63329	−0.115812
$518$	8.78236	0.385875
$519$	0	0
$520$	−5.58058	−0.244725
$521$	−28.2015	−1.23553	−0.617766	−	0.786362i	$-0.711962\pi$
$521$	−28.2015	−1.23553	−0.617766	+	0.786362i	$0.711962\pi$
$522$	0	0
$523$	−9.42284	−0.412032	−0.206016	−	0.978549i	$-0.566050\pi$
$523$	−9.42284	−0.412032	−0.206016	+	0.978549i	$0.566050\pi$
$524$	1.50020	0.0655363
$525$	0	0
$526$	11.2952	0.492493
$527$	9.06907	0.395055
$528$	0	0
$529$	30.6307	1.33177
$530$	−11.3746	−0.494079
$531$	0	0
$532$	−29.2321	−1.26737
$533$	−12.0533	−0.522087
$534$	0	0
$535$	4.87098	0.210591
$536$	27.1270	1.17171
$537$	0	0
$538$	20.1162	0.867271
$539$	−1.61620	−0.0696149
$540$	0	0
$541$	−9.20285	−0.395661	−0.197831	−	0.980236i	$-0.563390\pi$
$541$	−9.20285	−0.395661	−0.197831	+	0.980236i	$0.563390\pi$
$542$	−27.7884	−1.19361
$543$	0	0
$544$	−7.62968	−0.327120
$545$	−32.1351	−1.37651
$546$	0	0
$547$	−9.90402	−0.423465	−0.211733	−	0.977328i	$-0.567911\pi$
$547$	−9.90402	−0.423465	−0.211733	+	0.977328i	$0.567911\pi$
$548$	−21.3141	−0.910492
$549$	0	0
$550$	2.08551	0.0889264
$551$	64.9389	2.76649
$552$	0	0
$553$	42.9947	1.82832
$554$	−7.40161	−0.314464
$555$	0	0
$556$	−25.1309	−1.06579
$557$	9.70056	0.411026	0.205513	−	0.978654i	$-0.434114\pi$
$557$	9.70056	0.411026	0.205513	+	0.978654i	$0.434114\pi$
$558$	0	0
$559$	5.10708	0.216006
$560$	−0.175379	−0.00741114
$561$	0	0
$562$	3.02979	0.127804
$563$	10.8123	0.455684	0.227842	−	0.973698i	$-0.426833\pi$
$563$	10.8123	0.455684	0.227842	+	0.973698i	$0.426833\pi$
$564$	0	0
$565$	11.7012	0.492272
$566$	−3.63613	−0.152838
$567$	0	0
$568$	−21.2866	−0.893166
$569$	−7.98882	−0.334909	−0.167454	−	0.985880i	$-0.553555\pi$
$569$	−7.98882	−0.334909	−0.167454	+	0.985880i	$0.553555\pi$
$570$	0	0
$571$	−7.01760	−0.293678	−0.146839	−	0.989160i	$-0.546910\pi$
$571$	−7.01760	−0.293678	−0.146839	+	0.989160i	$0.546910\pi$
$572$	1.49051	0.0623212
$573$	0	0
$574$	−25.6108	−1.06898
$575$	−17.3803	−0.724808
$576$	0	0
$577$	40.0014	1.66528	0.832641	−	0.553814i	$-0.186828\pi$
$577$	40.0014	1.66528	0.832641	+	0.553814i	$0.186828\pi$
$578$	−13.3308	−0.554487
$579$	0	0
$580$	−15.9320	−0.661541
$581$	42.5991	1.76731
$582$	0	0
$583$	7.98667	0.330774
$584$	37.3950	1.54742
$585$	0	0
$586$	0.952198	0.0393349
$587$	39.7301	1.63984	0.819918	−	0.572481i	$-0.194019\pi$
$587$	39.7301	1.63984	0.819918	+	0.572481i	$0.194019\pi$
$588$	0	0
$589$	−54.3802	−2.24070
$590$	12.0073	0.494331
$591$	0	0
$592$	−0.125518	−0.00515875
$593$	−7.75484	−0.318453	−0.159227	−	0.987242i	$-0.550900\pi$
$593$	−7.75484	−0.318453	−0.159227	+	0.987242i	$0.550900\pi$
$594$	0	0
$595$	−6.43514	−0.263815
$596$	−0.283209	−0.0116007
$597$	0	0
$598$	7.81218	0.319464
$599$	31.0222	1.26753	0.633766	−	0.773525i	$-0.281508\pi$
$599$	31.0222	1.26753	0.633766	+	0.773525i	$0.281508\pi$
$600$	0	0
$601$	−14.5917	−0.595209	−0.297605	−	0.954689i	$-0.596188\pi$
$601$	−14.5917	−0.595209	−0.297605	+	0.954689i	$0.596188\pi$
$602$	10.8515	0.442274
$603$	0	0
$604$	−4.53615	−0.184574
$605$	1.62071	0.0658914
$606$	0	0
$607$	11.1857	0.454013	0.227006	−	0.973893i	$-0.427106\pi$
$607$	11.1857	0.454013	0.227006	+	0.973893i	$0.427106\pi$
$608$	45.7493	1.85538
$609$	0	0
$610$	−18.0053	−0.729013
$611$	3.19670	0.129325
$612$	0	0
$613$	−5.58013	−0.225379	−0.112690	−	0.993630i	$-0.535947\pi$
$613$	−5.58013	−0.225379	−0.112690	+	0.993630i	$0.535947\pi$
$614$	−12.3641	−0.498973
$615$	0	0
$616$	8.32584	0.335458
$617$	38.8872	1.56554	0.782771	−	0.622310i	$-0.213806\pi$
$617$	38.8872	1.56554	0.782771	+	0.622310i	$0.213806\pi$
$618$	0	0
$619$	13.2129	0.531070	0.265535	−	0.964101i	$-0.414451\pi$
$619$	13.2129	0.531070	0.265535	+	0.964101i	$0.414451\pi$
$620$	13.3416	0.535810
$621$	0	0
$622$	19.3043	0.774033
$623$	−10.9802	−0.439913
$624$	0	0
$625$	−7.50109	−0.300043
$626$	23.7183	0.947974
$627$	0	0
$628$	−11.2130	−0.447446
$629$	−4.60558	−0.183637
$630$	0	0
$631$	−45.2856	−1.80279	−0.901395	−	0.432998i	$-0.857456\pi$
$631$	−45.2856	−1.80279	−0.901395	+	0.432998i	$0.857456\pi$
$632$	−41.5458	−1.65260
$633$	0	0
$634$	14.1789	0.563118
$635$	18.0402	0.715902
$636$	0	0
$637$	1.96200	0.0777373
$638$	−7.03551	−0.278539
$639$	0	0
$640$	−11.1191	−0.439520
$641$	13.9630	0.551504	0.275752	−	0.961229i	$-0.411073\pi$
$641$	13.9630	0.551504	0.275752	+	0.961229i	$0.411073\pi$
$642$	0	0
$643$	3.07322	0.121196	0.0605979	−	0.998162i	$-0.480699\pi$
$643$	3.07322	0.121196	0.0605979	+	0.998162i	$0.480699\pi$
$644$	−26.3934	−1.04005
$645$	0	0
$646$	−9.64110	−0.379324
$647$	−14.4302	−0.567308	−0.283654	−	0.958927i	$-0.591547\pi$
$647$	−14.4302	−0.567308	−0.283654	+	0.958927i	$0.591547\pi$
$648$	0	0
$649$	−8.43092	−0.330943
$650$	−2.53172	−0.0993021
$651$	0	0
$652$	−3.75612	−0.147101
$653$	−26.9069	−1.05295	−0.526475	−	0.850191i	$-0.676486\pi$
$653$	−26.9069	−1.05295	−0.526475	+	0.850191i	$0.676486\pi$
$654$	0	0
$655$	−1.98026	−0.0773753
$656$	0.366031	0.0142911
$657$	0	0
$658$	6.79234	0.264793
$659$	25.0845	0.977152	0.488576	−	0.872521i	$-0.337516\pi$
$659$	25.0845	0.977152	0.488576	+	0.872521i	$0.337516\pi$
$660$	0	0
$661$	−17.9296	−0.697380	−0.348690	−	0.937238i	$-0.613373\pi$
$661$	−17.9296	−0.697380	−0.348690	+	0.937238i	$0.613373\pi$
$662$	−9.45759	−0.367580
$663$	0	0
$664$	−41.1635	−1.59745
$665$	38.5865	1.49632
$666$	0	0
$667$	58.6328	2.27027
$668$	−12.7062	−0.491616
$669$	0	0
$670$	−13.6207	−0.526215
$671$	12.6425	0.488057
$672$	0	0
$673$	5.04171	0.194344	0.0971718	−	0.995268i	$-0.469020\pi$
$673$	5.04171	0.194344	0.0971718	+	0.995268i	$0.469020\pi$
$674$	−10.6047	−0.408477
$675$	0	0
$676$	14.1521	0.544312
$677$	34.8128	1.33797	0.668983	−	0.743278i	$-0.266730\pi$
$677$	34.8128	1.33797	0.668983	+	0.743278i	$0.266730\pi$
$678$	0	0
$679$	4.92821	0.189127
$680$	6.21827	0.238460
$681$	0	0
$682$	5.89158	0.225600
$683$	−26.6749	−1.02069	−0.510343	−	0.859971i	$-0.670481\pi$
$683$	−26.6749	−1.02069	−0.510343	+	0.859971i	$0.670481\pi$
$684$	0	0
$685$	28.1347	1.07497
$686$	−13.8870	−0.530209
$687$	0	0
$688$	−0.155090	−0.00591275
$689$	−9.69546	−0.369368
$690$	0	0
$691$	−8.66639	−0.329685	−0.164843	−	0.986320i	$-0.552712\pi$
$691$	−8.66639	−0.329685	−0.164843	+	0.986320i	$0.552712\pi$
$692$	7.02862	0.267188
$693$	0	0
$694$	−6.11950	−0.232293
$695$	33.1729	1.25832
$696$	0	0
$697$	13.4306	0.508722
$698$	−0.332885	−0.0125999
$699$	0	0
$700$	8.55340	0.323288
$701$	−38.4201	−1.45111	−0.725554	−	0.688165i	$-0.758416\pi$
$701$	−38.4201	−1.45111	−0.725554	+	0.688165i	$0.758416\pi$
$702$	0	0
$703$	27.6161	1.04156
$704$	−5.03023	−0.189584
$705$	0	0
$706$	−6.04677	−0.227573
$707$	11.8793	0.446769
$708$	0	0
$709$	−6.17387	−0.231865	−0.115932	−	0.993257i	$-0.536986\pi$
$709$	−6.17387	−0.231865	−0.115932	+	0.993257i	$0.536986\pi$
$710$	10.6882	0.401121
$711$	0	0
$712$	10.6102	0.397634
$713$	−49.0994	−1.83879
$714$	0	0
$715$	−1.96748	−0.0735794
$716$	6.72879	0.251467
$717$	0	0
$718$	23.5051	0.877203
$719$	1.24110	0.0462851	0.0231425	−	0.999732i	$-0.492633\pi$
$719$	1.24110	0.0462851	0.0231425	+	0.999732i	$0.492633\pi$
$720$	0	0
$721$	40.2342	1.49840
$722$	41.1141	1.53011
$723$	0	0
$724$	−10.9394	−0.406559
$725$	−19.0013	−0.705691
$726$	0	0
$727$	−36.1135	−1.33938	−0.669688	−	0.742642i	$-0.733572\pi$
$727$	−36.1135	−1.33938	−0.669688	+	0.742642i	$0.733572\pi$
$728$	−10.1072	−0.374598
$729$	0	0
$730$	−18.7764	−0.694946
$731$	−5.69066	−0.210477
$732$	0	0
$733$	−4.07545	−0.150530	−0.0752651	−	0.997164i	$-0.523980\pi$
$733$	−4.07545	−0.150530	−0.0752651	+	0.997164i	$0.523980\pi$
$734$	6.39126	0.235906
$735$	0	0
$736$	41.3066	1.52258
$737$	9.56383	0.352288
$738$	0	0
$739$	−15.1772	−0.558303	−0.279151	−	0.960247i	$-0.590053\pi$
$739$	−15.1772	−0.558303	−0.279151	+	0.960247i	$0.590053\pi$
$740$	−6.77530	−0.249065
$741$	0	0
$742$	−20.6009	−0.756282
$743$	2.54334	0.0933060	0.0466530	−	0.998911i	$-0.485144\pi$
$743$	2.54334	0.0933060	0.0466530	+	0.998911i	$0.485144\pi$
$744$	0	0
$745$	0.373838	0.0136964
$746$	25.6040	0.937427
$747$	0	0
$748$	−1.66083	−0.0607258
$749$	8.82201	0.322349
$750$	0	0
$751$	44.4989	1.62379	0.811894	−	0.583804i	$-0.198436\pi$
$751$	44.4989	1.62379	0.811894	+	0.583804i	$0.198436\pi$
$752$	−0.0970763	−0.00354001
$753$	0	0
$754$	8.54081	0.311038
$755$	5.98774	0.217916
$756$	0	0
$757$	−50.0588	−1.81942	−0.909708	−	0.415248i	$-0.863695\pi$
$757$	−50.0588	−1.81942	−0.909708	+	0.415248i	$0.863695\pi$
$758$	−11.2673	−0.409246
$759$	0	0
$760$	−37.2862	−1.35251
$761$	−15.0651	−0.546111	−0.273055	−	0.961998i	$-0.588034\pi$
$761$	−15.0651	−0.546111	−0.273055	+	0.961998i	$0.588034\pi$
$762$	0	0
$763$	−58.2010	−2.10702
$764$	29.6164	1.07148
$765$	0	0
$766$	−0.319322	−0.0115376
$767$	10.2348	0.369556
$768$	0	0
$769$	−20.5699	−0.741770	−0.370885	−	0.928679i	$-0.620946\pi$
$769$	−20.5699	−0.741770	−0.370885	+	0.928679i	$0.620946\pi$
$770$	−4.18048	−0.150654
$771$	0	0
$772$	−4.55351	−0.163884
$773$	−54.0398	−1.94368	−0.971839	−	0.235646i	$-0.924279\pi$
$773$	−54.0398	−1.94368	−0.971839	+	0.235646i	$0.924279\pi$
$774$	0	0
$775$	15.9118	0.571569
$776$	−4.76213	−0.170950
$777$	0	0
$778$	−16.9827	−0.608859
$779$	−80.5332	−2.88540
$780$	0	0
$781$	−7.50475	−0.268541
$782$	−8.70487	−0.311286
$783$	0	0
$784$	−0.0595814	−0.00212791
$785$	14.8012	0.528277
$786$	0	0
$787$	40.4738	1.44274	0.721368	−	0.692552i	$-0.243514\pi$
$787$	40.4738	1.44274	0.721368	+	0.692552i	$0.243514\pi$
$788$	−0.324782	−0.0115699
$789$	0	0
$790$	20.8606	0.742186
$791$	21.1924	0.753516
$792$	0	0
$793$	−15.3474	−0.545002
$794$	−15.2248	−0.540308
$795$	0	0
$796$	−0.0363227	−0.00128742
$797$	−41.4088	−1.46677	−0.733387	−	0.679811i	$-0.762062\pi$
$797$	−41.4088	−1.46677	−0.733387	+	0.679811i	$0.762062\pi$
$798$	0	0
$799$	−3.56199	−0.126014
$800$	−13.3864	−0.473280
$801$	0	0
$802$	14.2036	0.501548
$803$	13.1839	0.465250
$804$	0	0
$805$	34.8394	1.22793
$806$	−7.15212	−0.251923
$807$	0	0
$808$	−11.4790	−0.403830
$809$	−38.8634	−1.36637	−0.683183	−	0.730247i	$-0.739405\pi$
$809$	−38.8634	−1.36637	−0.683183	+	0.730247i	$0.739405\pi$
$810$	0	0
$811$	25.9832	0.912393	0.456197	−	0.889879i	$-0.349211\pi$
$811$	25.9832	0.912393	0.456197	+	0.889879i	$0.349211\pi$
$812$	−28.8551	−1.01262
$813$	0	0
$814$	−2.99194	−0.104868
$815$	4.95810	0.173675
$816$	0	0
$817$	34.1225	1.19379
$818$	2.55098	0.0891929
$819$	0	0
$820$	19.7579	0.689976
$821$	39.7876	1.38860	0.694298	−	0.719688i	$-0.255715\pi$
$821$	39.7876	1.38860	0.694298	+	0.719688i	$0.255715\pi$
$822$	0	0
$823$	46.3202	1.61462	0.807311	−	0.590126i	$-0.200922\pi$
$823$	46.3202	1.61462	0.807311	+	0.590126i	$0.200922\pi$
$824$	−38.8784	−1.35439
$825$	0	0
$826$	21.7468	0.756668
$827$	20.3503	0.707649	0.353824	−	0.935312i	$-0.384881\pi$
$827$	20.3503	0.707649	0.353824	+	0.935312i	$0.384881\pi$
$828$	0	0
$829$	−9.86089	−0.342483	−0.171241	−	0.985229i	$-0.554778\pi$
$829$	−9.86089	−0.342483	−0.171241	+	0.985229i	$0.554778\pi$
$830$	20.6686	0.717417
$831$	0	0
$832$	6.10648	0.211704
$833$	−2.18620	−0.0757473
$834$	0	0
$835$	16.7722	0.580425
$836$	9.95869	0.344428
$837$	0	0
$838$	−0.828625	−0.0286244
$839$	−11.0824	−0.382608	−0.191304	−	0.981531i	$-0.561272\pi$
$839$	−11.0824	−0.382608	−0.191304	+	0.981531i	$0.561272\pi$
$840$	0	0
$841$	35.1014	1.21039
$842$	11.8524	0.408459
$843$	0	0
$844$	23.3004	0.802034
$845$	−18.6809	−0.642641
$846$	0	0
$847$	2.93534	0.100859
$848$	0.294429	0.0101107
$849$	0	0
$850$	2.82102	0.0967601
$851$	24.9344	0.854739
$852$	0	0
$853$	−39.1646	−1.34097	−0.670486	−	0.741922i	$-0.733914\pi$
$853$	−39.1646	−1.34097	−0.670486	+	0.741922i	$0.733914\pi$
$854$	−32.6101	−1.11589
$855$	0	0
$856$	−8.52471	−0.291369
$857$	−2.50632	−0.0856143	−0.0428071	−	0.999083i	$-0.513630\pi$
$857$	−2.50632	−0.0856143	−0.0428071	+	0.999083i	$0.513630\pi$
$858$	0	0
$859$	−11.8961	−0.405889	−0.202944	−	0.979190i	$-0.565051\pi$
$859$	−11.8961	−0.405889	−0.202944	+	0.979190i	$0.565051\pi$
$860$	−8.37157	−0.285468
$861$	0	0
$862$	12.0722	0.411182
$863$	1.14237	0.0388868	0.0194434	−	0.999811i	$-0.493811\pi$
$863$	1.14237	0.0388868	0.0194434	+	0.999811i	$0.493811\pi$
$864$	0	0
$865$	−9.27781	−0.315455
$866$	−3.20514	−0.108915
$867$	0	0
$868$	24.1634	0.820160
$869$	−14.6473	−0.496875
$870$	0	0
$871$	−11.6101	−0.393392
$872$	56.2397	1.90452
$873$	0	0
$874$	52.1964	1.76557
$875$	−35.0772	−1.18583
$876$	0	0
$877$	−28.8615	−0.974583	−0.487291	−	0.873239i	$-0.662015\pi$
$877$	−28.8615	−0.974583	−0.487291	+	0.873239i	$0.662015\pi$
$878$	−16.2332	−0.547843
$879$	0	0
$880$	0.0597476	0.00201409
$881$	−5.46260	−0.184040	−0.0920198	−	0.995757i	$-0.529332\pi$
$881$	−5.46260	−0.184040	−0.0920198	+	0.995757i	$0.529332\pi$
$882$	0	0
$883$	−4.99742	−0.168177	−0.0840884	−	0.996458i	$-0.526798\pi$
$883$	−4.99742	−0.168177	−0.0840884	+	0.996458i	$0.526798\pi$
$884$	2.01617	0.0678112
$885$	0	0
$886$	31.6224	1.06237
$887$	33.1958	1.11461	0.557303	−	0.830309i	$-0.311836\pi$
$887$	33.1958	1.11461	0.557303	+	0.830309i	$0.311836\pi$
$888$	0	0
$889$	32.6732	1.09583
$890$	−5.32748	−0.178577
$891$	0	0
$892$	−6.33532	−0.212122
$893$	21.3585	0.714734
$894$	0	0
$895$	−8.88203	−0.296893
$896$	−20.1382	−0.672769
$897$	0	0
$898$	−10.1747	−0.339536
$899$	−53.6788	−1.79029
$900$	0	0
$901$	10.8034	0.359912
$902$	8.72501	0.290511
$903$	0	0
$904$	−20.4783	−0.681097
$905$	14.4400	0.480003
$906$	0	0
$907$	−11.0423	−0.366653	−0.183327	−	0.983052i	$-0.558687\pi$
$907$	−11.0423	−0.366653	−0.183327	+	0.983052i	$0.558687\pi$
$908$	24.7883	0.822629
$909$	0	0
$910$	5.07493	0.168232
$911$	−56.9068	−1.88541	−0.942703	−	0.333634i	$-0.891725\pi$
$911$	−56.9068	−1.88541	−0.942703	+	0.333634i	$0.891725\pi$
$912$	0	0
$913$	−14.5125	−0.480294
$914$	11.9570	0.395503
$915$	0	0
$916$	−30.2754	−1.00033
$917$	−3.58653	−0.118438
$918$	0	0
$919$	−2.02666	−0.0668534	−0.0334267	−	0.999441i	$-0.510642\pi$
$919$	−2.02666	−0.0668534	−0.0334267	+	0.999441i	$0.510642\pi$
$920$	−33.6654	−1.10991
$921$	0	0
$922$	15.1689	0.499560
$923$	9.11043	0.299874
$924$	0	0
$925$	−8.08056	−0.265687
$926$	−28.9870	−0.952573
$927$	0	0
$928$	45.1592	1.48242
$929$	5.78276	0.189726	0.0948630	−	0.995490i	$-0.469759\pi$
$929$	5.78276	0.189726	0.0948630	+	0.995490i	$0.469759\pi$
$930$	0	0
$931$	13.1089	0.429628
$932$	2.18666	0.0716263
$933$	0	0
$934$	−19.6045	−0.641479
$935$	2.19230	0.0716958
$936$	0	0
$937$	−19.1763	−0.626463	−0.313232	−	0.949677i	$-0.601412\pi$
$937$	−19.1763	−0.626463	−0.313232	+	0.949677i	$0.601412\pi$
$938$	−24.6690	−0.805473
$939$	0	0
$940$	−5.24006	−0.170912
$941$	−48.6632	−1.58638	−0.793188	−	0.608977i	$-0.791580\pi$
$941$	−48.6632	−1.58638	−0.793188	+	0.608977i	$0.791580\pi$
$942$	0	0
$943$	−72.7127	−2.36785
$944$	−0.310806	−0.0101159
$945$	0	0
$946$	−3.69685	−0.120195
$947$	45.5873	1.48139	0.740694	−	0.671843i	$-0.234497\pi$
$947$	45.5873	1.48139	0.740694	+	0.671843i	$0.234497\pi$
$948$	0	0
$949$	−16.0047	−0.519534
$950$	−16.9154	−0.548810
$951$	0	0
$952$	11.2622	0.365008
$953$	39.9871	1.29531	0.647654	−	0.761934i	$-0.275750\pi$
$953$	39.9871	1.29531	0.647654	+	0.761934i	$0.275750\pi$
$954$	0	0
$955$	−39.0937	−1.26504
$956$	−17.6186	−0.569825
$957$	0	0
$958$	−5.97571	−0.193066
$959$	50.9558	1.64545
$960$	0	0
$961$	13.9510	0.450031
$962$	3.63209	0.117103
$963$	0	0
$964$	−8.78747	−0.283025
$965$	6.01065	0.193490
$966$	0	0
$967$	13.3998	0.430907	0.215454	−	0.976514i	$-0.430877\pi$
$967$	13.3998	0.430907	0.215454	+	0.976514i	$0.430877\pi$
$968$	−2.83642	−0.0911659
$969$	0	0
$970$	2.39111	0.0767739
$971$	−28.9427	−0.928816	−0.464408	−	0.885621i	$-0.653733\pi$
$971$	−28.9427	−0.928816	−0.464408	+	0.885621i	$0.653733\pi$
$972$	0	0
$973$	60.0806	1.92610
$974$	−28.4155	−0.910490
$975$	0	0
$976$	0.466065	0.0149184
$977$	−49.7732	−1.59239	−0.796193	−	0.605043i	$-0.793156\pi$
$977$	−49.7732	−1.59239	−0.796193	+	0.605043i	$0.793156\pi$
$978$	0	0
$979$	3.74070	0.119553
$980$	−3.21613	−0.102736
$981$	0	0
$982$	−23.3964	−0.746610
$983$	−1.71769	−0.0547859	−0.0273929	−	0.999625i	$-0.508721\pi$
$983$	−1.71769	−0.0547859	−0.0273929	+	0.999625i	$0.508721\pi$
$984$	0	0
$985$	0.428714	0.0136599
$986$	−9.51676	−0.303075
$987$	0	0
$988$	−12.0894	−0.384615
$989$	30.8089	0.979666
$990$	0	0
$991$	−19.8277	−0.629848	−0.314924	−	0.949117i	$-0.601979\pi$
$991$	−19.8277	−0.629848	−0.314924	+	0.949117i	$0.601979\pi$
$992$	−37.8166	−1.20068
$993$	0	0
$994$	19.3578	0.613992
$995$	0.0479461	0.00151999
$996$	0	0
$997$	−46.4201	−1.47014	−0.735070	−	0.677991i	$-0.762851\pi$
$997$	−46.4201	−1.47014	−0.735070	+	0.677991i	$0.762851\pi$
$998$	25.1410	0.795824
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8019.2.a.f.1.12	yes	21
3.2	odd	2		8019.2.a.c.1.10	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8019.2.a.c.1.10	✓	21	3.2	odd	2
8019.2.a.f.1.12	yes	21	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)

\(f(q)\)	\(=\)	\(q+0.878744 q^{2} -1.22781 q^{4} +1.62071 q^{5} +2.93534 q^{7} -2.83642 q^{8} +O(q^{10})\)	\(q+0.878744 q^{2} -1.22781 q^{4} +1.62071 q^{5} +2.93534 q^{7} -2.83642 q^{8} +1.42419 q^{10} -1.00000 q^{11} +1.21396 q^{13} +2.57941 q^{14} -0.0368650 q^{16} -1.35267 q^{17} +8.11094 q^{19} -1.98993 q^{20} -0.878744 q^{22} +7.32330 q^{23} -2.37329 q^{25} +1.06676 q^{26} -3.60403 q^{28} +8.00633 q^{29} -6.70455 q^{31} +5.64044 q^{32} -1.18865 q^{34} +4.75734 q^{35} +3.40480 q^{37} +7.12744 q^{38} -4.59702 q^{40} -9.92895 q^{41} +4.20697 q^{43} +1.22781 q^{44} +6.43530 q^{46} +2.63329 q^{47} +1.61620 q^{49} -2.08551 q^{50} -1.49051 q^{52} -7.98667 q^{53} -1.62071 q^{55} -8.32584 q^{56} +7.03551 q^{58} +8.43092 q^{59} -12.6425 q^{61} -5.89158 q^{62} +5.03023 q^{64} +1.96748 q^{65} -9.56383 q^{67} +1.66083 q^{68} +4.18048 q^{70} +7.50475 q^{71} -13.1839 q^{73} +2.99194 q^{74} -9.95869 q^{76} -2.93534 q^{77} +14.6473 q^{79} -0.0597476 q^{80} -8.72501 q^{82} +14.5125 q^{83} -2.19230 q^{85} +3.69685 q^{86} +2.83642 q^{88} -3.74070 q^{89} +3.56337 q^{91} -8.99162 q^{92} +2.31399 q^{94} +13.1455 q^{95} +1.67892 q^{97} +1.42023 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 6 q^{2} + 18 q^{4} + 12 q^{5} + 18 q^{8}+O(q^{10}) \) 21 * q + 6 * q^2 + 18 * q^4 + 12 * q^5 + 18 * q^8	\( 21 q + 6 q^{2} + 18 q^{4} + 12 q^{5} + 18 q^{8} + 15 q^{10} - 21 q^{11} + 3 q^{13} + 24 q^{14} + 12 q^{16} + 12 q^{17} - 12 q^{19} + 15 q^{20} - 6 q^{22} + 15 q^{25} + 15 q^{26} + 42 q^{28} + 27 q^{29} + 6 q^{31} + 42 q^{32} - 6 q^{34} + 42 q^{35} + 3 q^{37} + 27 q^{38} + 30 q^{40} + 24 q^{41} + 6 q^{43} - 18 q^{44} - 51 q^{46} + 24 q^{47} - 9 q^{49} - 3 q^{50} + 9 q^{52} + 12 q^{53} - 12 q^{55} + 45 q^{56} + 12 q^{58} + 12 q^{59} - 30 q^{61} + 9 q^{62} + 42 q^{64} + 18 q^{67} + 30 q^{68} + 21 q^{70} + 36 q^{71} - 39 q^{73} - 27 q^{76} + 27 q^{79} + 78 q^{80} + 63 q^{82} + 42 q^{83} + 36 q^{85} + 42 q^{86} - 18 q^{88} + 36 q^{89} - 21 q^{91} + 6 q^{92} + 42 q^{94} + 24 q^{95} + 24 q^{97} + 69 q^{98}+O(q^{100}) \) 21 * q + 6 * q^2 + 18 * q^4 + 12 * q^5 + 18 * q^8 + 15 * q^10 - 21 * q^11 + 3 * q^13 + 24 * q^14 + 12 * q^16 + 12 * q^17 - 12 * q^19 + 15 * q^20 - 6 * q^22 + 15 * q^25 + 15 * q^26 + 42 * q^28 + 27 * q^29 + 6 * q^31 + 42 * q^32 - 6 * q^34 + 42 * q^35 + 3 * q^37 + 27 * q^38 + 30 * q^40 + 24 * q^41 + 6 * q^43 - 18 * q^44 - 51 * q^46 + 24 * q^47 - 9 * q^49 - 3 * q^50 + 9 * q^52 + 12 * q^53 - 12 * q^55 + 45 * q^56 + 12 * q^58 + 12 * q^59 - 30 * q^61 + 9 * q^62 + 42 * q^64 + 18 * q^67 + 30 * q^68 + 21 * q^70 + 36 * q^71 - 39 * q^73 - 27 * q^76 + 27 * q^79 + 78 * q^80 + 63 * q^82 + 42 * q^83 + 36 * q^85 + 42 * q^86 - 18 * q^88 + 36 * q^89 - 21 * q^91 + 6 * q^92 + 42 * q^94 + 24 * q^95 + 24 * q^97 + 69 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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