LMFDB - Embedded newform 8019.2.a.k.1.1

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			1.1
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)

$f(q)$	$=$	$q-2.76871 q^{2} +5.66575 q^{4} -0.801804 q^{5} -0.629042 q^{7} -10.1494 q^{8} +O(q^{10})$	\(q-2.76871 q^{2} +5.66575 q^{4} -0.801804 q^{5} -0.629042 q^{7} -10.1494 q^{8} +2.21996 q^{10} -1.00000 q^{11} +0.705228 q^{13} +1.74163 q^{14} +16.7692 q^{16} -3.00713 q^{17} +4.90258 q^{19} -4.54282 q^{20} +2.76871 q^{22} -2.38332 q^{23} -4.35711 q^{25} -1.95257 q^{26} -3.56399 q^{28} +9.72452 q^{29} +5.97168 q^{31} -26.1302 q^{32} +8.32587 q^{34} +0.504368 q^{35} -5.92391 q^{37} -13.5738 q^{38} +8.13781 q^{40} -2.63380 q^{41} +8.42464 q^{43} -5.66575 q^{44} +6.59871 q^{46} +12.7661 q^{47} -6.60431 q^{49} +12.0636 q^{50} +3.99565 q^{52} -12.1551 q^{53} +0.801804 q^{55} +6.38439 q^{56} -26.9243 q^{58} +10.1710 q^{59} +4.18023 q^{61} -16.5338 q^{62} +38.8086 q^{64} -0.565455 q^{65} -5.37289 q^{67} -17.0376 q^{68} -1.39645 q^{70} -2.56794 q^{71} -4.29869 q^{73} +16.4016 q^{74} +27.7768 q^{76} +0.629042 q^{77} +2.99059 q^{79} -13.4456 q^{80} +7.29223 q^{82} +7.25161 q^{83} +2.41113 q^{85} -23.3254 q^{86} +10.1494 q^{88} -2.20102 q^{89} -0.443618 q^{91} -13.5033 q^{92} -35.3457 q^{94} -3.93091 q^{95} -12.5051 q^{97} +18.2854 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.76871	−1.95777	−0.978886	−	0.204406i	$-0.934474\pi$
$2$	−2.76871	−1.95777	−0.978886	+	0.204406i	$0.934474\pi$
$3$	0	0
$3$	0	0
$4$	5.66575	2.83287
$5$	−0.801804	−0.358578	−0.179289	−	0.983796i	$-0.557380\pi$
$5$	−0.801804	−0.358578	−0.179289	+	0.983796i	$0.557380\pi$
$6$	0	0
$7$	−0.629042	−0.237756	−0.118878	−	0.992909i	$-0.537930\pi$
$7$	−0.629042	−0.237756	−0.118878	+	0.992909i	$0.537930\pi$
$8$	−10.1494	−3.58835
$9$	0	0
$10$	2.21996	0.702013
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	0.705228	0.195595	0.0977976	−	0.995206i	$-0.468820\pi$
$13$	0.705228	0.195595	0.0977976	+	0.995206i	$0.468820\pi$
$14$	1.74163	0.465471
$15$	0	0
$16$	16.7692	4.19230
$17$	−3.00713	−0.729337	−0.364668	−	0.931137i	$-0.618818\pi$
$17$	−3.00713	−0.729337	−0.364668	+	0.931137i	$0.618818\pi$
$18$	0	0
$19$	4.90258	1.12473	0.562365	−	0.826889i	$-0.309892\pi$
$19$	4.90258	1.12473	0.562365	+	0.826889i	$0.309892\pi$
$20$	−4.54282	−1.01580
$21$	0	0
$22$	2.76871	0.590291
$23$	−2.38332	−0.496956	−0.248478	−	0.968638i	$-0.579930\pi$
$23$	−2.38332	−0.496956	−0.248478	+	0.968638i	$0.579930\pi$
$24$	0	0
$25$	−4.35711	−0.871422
$26$	−1.95257	−0.382931
$27$	0	0
$28$	−3.56399	−0.673531
$29$	9.72452	1.80580	0.902899	−	0.429854i	$-0.141435\pi$
$29$	9.72452	1.80580	0.902899	+	0.429854i	$0.141435\pi$
$30$	0	0
$31$	5.97168	1.07255	0.536273	−	0.844045i	$-0.319832\pi$
$31$	5.97168	1.07255	0.536273	+	0.844045i	$0.319832\pi$
$32$	−26.1302	−4.61921
$33$	0	0
$34$	8.32587	1.42788
$35$	0.504368	0.0852538
$36$	0	0
$37$	−5.92391	−0.973885	−0.486943	−	0.873434i	$-0.661888\pi$
$37$	−5.92391	−0.973885	−0.486943	+	0.873434i	$0.661888\pi$
$38$	−13.5738	−2.20196
$39$	0	0
$40$	8.13781	1.28670
$41$	−2.63380	−0.411331	−0.205665	−	0.978622i	$-0.565936\pi$
$41$	−2.63380	−0.411331	−0.205665	+	0.978622i	$0.565936\pi$
$42$	0	0
$43$	8.42464	1.28475	0.642373	−	0.766392i	$-0.277950\pi$
$43$	8.42464	1.28475	0.642373	+	0.766392i	$0.277950\pi$
$44$	−5.66575	−0.854143
$45$	0	0
$46$	6.59871	0.972927
$47$	12.7661	1.86213	0.931067	−	0.364849i	$-0.118880\pi$
$47$	12.7661	1.86213	0.931067	+	0.364849i	$0.118880\pi$
$48$	0	0
$49$	−6.60431	−0.943472
$50$	12.0636	1.70605
$51$	0	0
$52$	3.99565	0.554096
$53$	−12.1551	−1.66963	−0.834814	−	0.550533i	$-0.814425\pi$
$53$	−12.1551	−1.66963	−0.834814	+	0.550533i	$0.814425\pi$
$54$	0	0
$55$	0.801804	0.108115
$56$	6.38439	0.853150
$57$	0	0
$58$	−26.9243	−3.53534
$59$	10.1710	1.32415	0.662075	−	0.749438i	$-0.269676\pi$
$59$	10.1710	1.32415	0.662075	+	0.749438i	$0.269676\pi$
$60$	0	0
$61$	4.18023	0.535224	0.267612	−	0.963527i	$-0.413766\pi$
$61$	4.18023	0.535224	0.267612	+	0.963527i	$0.413766\pi$
$62$	−16.5338	−2.09980
$63$	0	0
$64$	38.8086	4.85107
$65$	−0.565455	−0.0701361
$66$	0	0
$67$	−5.37289	−0.656404	−0.328202	−	0.944608i	$-0.606443\pi$
$67$	−5.37289	−0.656404	−0.328202	+	0.944608i	$0.606443\pi$
$68$	−17.0376	−2.06612
$69$	0	0
$70$	−1.39645	−0.166908
$71$	−2.56794	−0.304758	−0.152379	−	0.988322i	$-0.548693\pi$
$71$	−2.56794	−0.304758	−0.152379	+	0.988322i	$0.548693\pi$
$72$	0	0
$73$	−4.29869	−0.503123	−0.251562	−	0.967841i	$-0.580944\pi$
$73$	−4.29869	−0.503123	−0.251562	+	0.967841i	$0.580944\pi$
$74$	16.4016	1.90665
$75$	0	0
$76$	27.7768	3.18622
$77$	0.629042	0.0716860
$78$	0	0
$79$	2.99059	0.336468	0.168234	−	0.985747i	$-0.446194\pi$
$79$	2.99059	0.336468	0.168234	+	0.985747i	$0.446194\pi$
$80$	−13.4456	−1.50326
$81$	0	0
$82$	7.29223	0.805292
$83$	7.25161	0.795967	0.397984	−	0.917393i	$-0.369710\pi$
$83$	7.25161	0.795967	0.397984	+	0.917393i	$0.369710\pi$
$84$	0	0
$85$	2.41113	0.261524
$86$	−23.3254	−2.51524
$87$	0	0
$88$	10.1494	1.08193
$89$	−2.20102	−0.233307	−0.116654	−	0.993173i	$-0.537217\pi$
$89$	−2.20102	−0.233307	−0.116654	+	0.993173i	$0.537217\pi$
$90$	0	0
$91$	−0.443618	−0.0465038
$92$	−13.5033	−1.40781
$93$	0	0
$94$	−35.3457	−3.64563
$95$	−3.93091	−0.403303
$96$	0	0
$97$	−12.5051	−1.26970	−0.634852	−	0.772634i	$-0.718939\pi$
$97$	−12.5051	−1.26970	−0.634852	+	0.772634i	$0.718939\pi$
$98$	18.2854	1.84710
$99$	0	0
$100$	−24.6863	−2.46863
$101$	−10.2902	−1.02391	−0.511955	−	0.859012i	$-0.671078\pi$
$101$	−10.2902	−1.02391	−0.511955	+	0.859012i	$0.671078\pi$
$102$	0	0
$103$	4.25835	0.419588	0.209794	−	0.977746i	$-0.432721\pi$
$103$	4.25835	0.419588	0.209794	+	0.977746i	$0.432721\pi$
$104$	−7.15763	−0.701864
$105$	0	0
$106$	33.6538	3.26875
$107$	−7.84745	−0.758641	−0.379321	−	0.925265i	$-0.623842\pi$
$107$	−7.84745	−0.758641	−0.379321	+	0.925265i	$0.623842\pi$
$108$	0	0
$109$	−12.9523	−1.24061	−0.620303	−	0.784362i	$-0.712991\pi$
$109$	−12.9523	−1.24061	−0.620303	+	0.784362i	$0.712991\pi$
$110$	−2.21996	−0.211665
$111$	0	0
$112$	−10.5485	−0.996742
$113$	−4.88935	−0.459952	−0.229976	−	0.973196i	$-0.573865\pi$
$113$	−4.88935	−0.459952	−0.229976	+	0.973196i	$0.573865\pi$
$114$	0	0
$115$	1.91095	0.178197
$116$	55.0966	5.11559
$117$	0	0
$118$	−28.1605	−2.59238
$119$	1.89161	0.173404
$120$	0	0
$121$	1.00000	0.0909091
$122$	−11.5738	−1.04785
$123$	0	0
$124$	33.8340	3.03839
$125$	7.50257	0.671050
$126$	0	0
$127$	7.57724	0.672371	0.336186	−	0.941796i	$-0.390863\pi$
$127$	7.57724	0.672371	0.336186	+	0.941796i	$0.390863\pi$
$128$	−55.1892	−4.87808
$129$	0	0
$130$	1.56558	0.137310
$131$	−1.83357	−0.160200	−0.0801001	−	0.996787i	$-0.525524\pi$
$131$	−1.83357	−0.160200	−0.0801001	+	0.996787i	$0.525524\pi$
$132$	0	0
$133$	−3.08393	−0.267411
$134$	14.8760	1.28509
$135$	0	0
$136$	30.5205	2.61711
$137$	−9.84199	−0.840858	−0.420429	−	0.907325i	$-0.638120\pi$
$137$	−9.84199	−0.840858	−0.420429	+	0.907325i	$0.638120\pi$
$138$	0	0
$139$	10.8804	0.922860	0.461430	−	0.887177i	$-0.347337\pi$
$139$	10.8804	0.922860	0.461430	+	0.887177i	$0.347337\pi$
$140$	2.85762	0.241513
$141$	0	0
$142$	7.10987	0.596647
$143$	−0.705228	−0.0589742
$144$	0	0
$145$	−7.79716	−0.647518
$146$	11.9018	0.985001
$147$	0	0
$148$	−33.5634	−2.75889
$149$	−13.6238	−1.11611	−0.558054	−	0.829804i	$-0.688452\pi$
$149$	−13.6238	−1.11611	−0.558054	+	0.829804i	$0.688452\pi$
$150$	0	0
$151$	8.82042	0.717795	0.358898	−	0.933377i	$-0.383153\pi$
$151$	8.82042	0.717795	0.358898	+	0.933377i	$0.383153\pi$
$152$	−49.7582	−4.03592
$153$	0	0
$154$	−1.74163	−0.140345
$155$	−4.78812	−0.384591
$156$	0	0
$157$	−12.5075	−0.998204	−0.499102	−	0.866543i	$-0.666337\pi$
$157$	−12.5075	−0.998204	−0.499102	+	0.866543i	$0.666337\pi$
$158$	−8.28008	−0.658728
$159$	0	0
$160$	20.9513	1.65635
$161$	1.49921	0.118154
$162$	0	0
$163$	−2.15927	−0.169127	−0.0845637	−	0.996418i	$-0.526950\pi$
$163$	−2.15927	−0.169127	−0.0845637	+	0.996418i	$0.526950\pi$
$164$	−14.9225	−1.16525
$165$	0	0
$166$	−20.0776	−1.55832
$167$	−4.57698	−0.354177	−0.177089	−	0.984195i	$-0.556668\pi$
$167$	−4.57698	−0.354177	−0.177089	+	0.984195i	$0.556668\pi$
$168$	0	0
$169$	−12.5027	−0.961743
$170$	−6.67572	−0.512004
$171$	0	0
$172$	47.7319	3.63952
$173$	−5.68521	−0.432239	−0.216119	−	0.976367i	$-0.569340\pi$
$173$	−5.68521	−0.432239	−0.216119	+	0.976367i	$0.569340\pi$
$174$	0	0
$175$	2.74081	0.207185
$176$	−16.7692	−1.26402
$177$	0	0
$178$	6.09397	0.456762
$179$	16.9115	1.26402	0.632012	−	0.774959i	$-0.282229\pi$
$179$	16.9115	1.26402	0.632012	+	0.774959i	$0.282229\pi$
$180$	0	0
$181$	17.4723	1.29871	0.649354	−	0.760487i	$-0.275040\pi$
$181$	17.4723	1.29871	0.649354	+	0.760487i	$0.275040\pi$
$182$	1.22825	0.0910439
$183$	0	0
$184$	24.1892	1.78325
$185$	4.74982	0.349214
$186$	0	0
$187$	3.00713	0.219903
$188$	72.3297	5.27519
$189$	0	0
$190$	10.8835	0.789575
$191$	8.57576	0.620520	0.310260	−	0.950652i	$-0.399584\pi$
$191$	8.57576	0.620520	0.310260	+	0.950652i	$0.399584\pi$
$192$	0	0
$193$	19.6980	1.41789	0.708945	−	0.705264i	$-0.249171\pi$
$193$	19.6980	1.41789	0.708945	+	0.705264i	$0.249171\pi$
$194$	34.6230	2.48579
$195$	0	0
$196$	−37.4183	−2.67274
$197$	24.4745	1.74374	0.871869	−	0.489739i	$-0.162908\pi$
$197$	24.4745	1.74374	0.871869	+	0.489739i	$0.162908\pi$
$198$	0	0
$199$	−9.00778	−0.638545	−0.319272	−	0.947663i	$-0.603438\pi$
$199$	−9.00778	−0.638545	−0.319272	+	0.947663i	$0.603438\pi$
$200$	44.2220	3.12697
$201$	0	0
$202$	28.4905	2.00458
$203$	−6.11713	−0.429338
$204$	0	0
$205$	2.11179	0.147494
$206$	−11.7901	−0.821457
$207$	0	0
$208$	11.8261	0.819993
$209$	−4.90258	−0.339119
$210$	0	0
$211$	−5.09600	−0.350823	−0.175411	−	0.984495i	$-0.556126\pi$
$211$	−5.09600	−0.350823	−0.175411	+	0.984495i	$0.556126\pi$
$212$	−68.8675	−4.72984
$213$	0	0
$214$	21.7273	1.48525
$215$	−6.75491	−0.460681
$216$	0	0
$217$	−3.75644	−0.255004
$218$	35.8612	2.42883
$219$	0	0
$220$	4.54282	0.306277
$221$	−2.12072	−0.142655
$222$	0	0
$223$	−5.48259	−0.367141	−0.183571	−	0.983007i	$-0.558766\pi$
$223$	−5.48259	−0.367141	−0.183571	+	0.983007i	$0.558766\pi$
$224$	16.4370	1.09824
$225$	0	0
$226$	13.5372	0.900481
$227$	3.16040	0.209763	0.104882	−	0.994485i	$-0.466554\pi$
$227$	3.16040	0.209763	0.104882	+	0.994485i	$0.466554\pi$
$228$	0	0
$229$	−11.1911	−0.739531	−0.369765	−	0.929125i	$-0.620562\pi$
$229$	−11.1911	−0.739531	−0.369765	+	0.929125i	$0.620562\pi$
$230$	−5.29087	−0.348870
$231$	0	0
$232$	−98.6978	−6.47983
$233$	19.5374	1.27994	0.639970	−	0.768400i	$-0.278947\pi$
$233$	19.5374	1.27994	0.639970	+	0.768400i	$0.278947\pi$
$234$	0	0
$235$	−10.2359	−0.667719
$236$	57.6262	3.75115
$237$	0	0
$238$	−5.23732	−0.339485
$239$	14.4460	0.934434	0.467217	−	0.884143i	$-0.345257\pi$
$239$	14.4460	0.934434	0.467217	+	0.884143i	$0.345257\pi$
$240$	0	0
$241$	25.7685	1.65990	0.829948	−	0.557841i	$-0.188370\pi$
$241$	25.7685	1.65990	0.829948	+	0.557841i	$0.188370\pi$
$242$	−2.76871	−0.177979
$243$	0	0
$244$	23.6841	1.51622
$245$	5.29536	0.338308
$246$	0	0
$247$	3.45744	0.219992
$248$	−60.6089	−3.84867
$249$	0	0
$250$	−20.7724	−1.31376
$251$	−22.2904	−1.40696	−0.703480	−	0.710715i	$-0.748372\pi$
$251$	−22.2904	−1.40696	−0.703480	+	0.710715i	$0.748372\pi$
$252$	0	0
$253$	2.38332	0.149838
$254$	−20.9792	−1.31635
$255$	0	0
$256$	75.1857	4.69910
$257$	23.8429	1.48728	0.743641	−	0.668580i	$-0.233097\pi$
$257$	23.8429	1.48728	0.743641	+	0.668580i	$0.233097\pi$
$258$	0	0
$259$	3.72639	0.231547
$260$	−3.20372	−0.198687
$261$	0	0
$262$	5.07663	0.313636
$263$	11.8405	0.730116	0.365058	−	0.930985i	$-0.381049\pi$
$263$	11.8405	0.730116	0.365058	+	0.930985i	$0.381049\pi$
$264$	0	0
$265$	9.74598	0.598691
$266$	8.53851	0.523529
$267$	0	0
$268$	−30.4415	−1.85951
$269$	0.474565	0.0289347	0.0144674	−	0.999895i	$-0.495395\pi$
$269$	0.474565	0.0289347	0.0144674	+	0.999895i	$0.495395\pi$
$270$	0	0
$271$	−1.57509	−0.0956797	−0.0478398	−	0.998855i	$-0.515234\pi$
$271$	−1.57509	−0.0956797	−0.0478398	+	0.998855i	$0.515234\pi$
$272$	−50.4272	−3.05760
$273$	0	0
$274$	27.2496	1.64621
$275$	4.35711	0.262744
$276$	0	0
$277$	−2.61613	−0.157188	−0.0785942	−	0.996907i	$-0.525043\pi$
$277$	−2.61613	−0.157188	−0.0785942	+	0.996907i	$0.525043\pi$
$278$	−30.1245	−1.80675
$279$	0	0
$280$	−5.11903	−0.305920
$281$	15.0170	0.895838	0.447919	−	0.894074i	$-0.352165\pi$
$281$	15.0170	0.895838	0.447919	+	0.894074i	$0.352165\pi$
$282$	0	0
$283$	7.74954	0.460662	0.230331	−	0.973112i	$-0.426019\pi$
$283$	7.74954	0.460662	0.230331	+	0.973112i	$0.426019\pi$
$284$	−14.5493	−0.863340
$285$	0	0
$286$	1.95257	0.115458
$287$	1.65677	0.0977962
$288$	0	0
$289$	−7.95715	−0.468068
$290$	21.5880	1.26769
$291$	0	0
$292$	−24.3553	−1.42528
$293$	−2.59269	−0.151467	−0.0757333	−	0.997128i	$-0.524130\pi$
$293$	−2.59269	−0.151467	−0.0757333	+	0.997128i	$0.524130\pi$
$294$	0	0
$295$	−8.15514	−0.474811
$296$	60.1241	3.49464
$297$	0	0
$298$	37.7204	2.18509
$299$	−1.68078	−0.0972022
$300$	0	0
$301$	−5.29945	−0.305455
$302$	−24.4212	−1.40528
$303$	0	0
$304$	82.2123	4.71520
$305$	−3.35172	−0.191919
$306$	0	0
$307$	−4.86910	−0.277894	−0.138947	−	0.990300i	$-0.544372\pi$
$307$	−4.86910	−0.277894	−0.138947	+	0.990300i	$0.544372\pi$
$308$	3.56399	0.203077
$309$	0	0
$310$	13.2569	0.752941
$311$	−29.1770	−1.65447	−0.827237	−	0.561853i	$-0.810089\pi$
$311$	−29.1770	−1.65447	−0.827237	+	0.561853i	$0.810089\pi$
$312$	0	0
$313$	26.9090	1.52099	0.760495	−	0.649344i	$-0.224957\pi$
$313$	26.9090	1.52099	0.760495	+	0.649344i	$0.224957\pi$
$314$	34.6295	1.95426
$315$	0	0
$316$	16.9439	0.953171
$317$	−1.32561	−0.0744539	−0.0372269	−	0.999307i	$-0.511852\pi$
$317$	−1.32561	−0.0744539	−0.0372269	+	0.999307i	$0.511852\pi$
$318$	0	0
$319$	−9.72452	−0.544468
$320$	−31.1169	−1.73949
$321$	0	0
$322$	−4.15087	−0.231319
$323$	−14.7427	−0.820307
$324$	0	0
$325$	−3.07276	−0.170446
$326$	5.97840	0.331113
$327$	0	0
$328$	26.7315	1.47600
$329$	−8.03044	−0.442732
$330$	0	0
$331$	14.8443	0.815916	0.407958	−	0.913001i	$-0.366241\pi$
$331$	14.8443	0.815916	0.407958	+	0.913001i	$0.366241\pi$
$332$	41.0858	2.25487
$333$	0	0
$334$	12.6723	0.693399
$335$	4.30801	0.235372
$336$	0	0
$337$	22.2747	1.21338	0.606691	−	0.794938i	$-0.292496\pi$
$337$	22.2747	1.21338	0.606691	+	0.794938i	$0.292496\pi$
$338$	34.6162	1.88287
$339$	0	0
$340$	13.6609	0.740864
$341$	−5.97168	−0.323385
$342$	0	0
$343$	8.55768	0.462071
$344$	−85.5049	−4.61011
$345$	0	0
$346$	15.7407	0.846225
$347$	19.0460	1.02244	0.511221	−	0.859449i	$-0.329193\pi$
$347$	19.0460	1.02244	0.511221	+	0.859449i	$0.329193\pi$
$348$	0	0
$349$	−28.3997	−1.52020	−0.760099	−	0.649807i	$-0.774850\pi$
$349$	−28.3997	−1.52020	−0.760099	+	0.649807i	$0.774850\pi$
$350$	−7.58849	−0.405622
$351$	0	0
$352$	26.1302	1.39275
$353$	20.2730	1.07902	0.539510	−	0.841979i	$-0.318609\pi$
$353$	20.2730	1.07902	0.539510	+	0.841979i	$0.318609\pi$
$354$	0	0
$355$	2.05898	0.109279
$356$	−12.4704	−0.660930
$357$	0	0
$358$	−46.8230	−2.47467
$359$	9.14130	0.482459	0.241230	−	0.970468i	$-0.422449\pi$
$359$	9.14130	0.482459	0.241230	+	0.970468i	$0.422449\pi$
$360$	0	0
$361$	5.03532	0.265017
$362$	−48.3757	−2.54257
$363$	0	0
$364$	−2.51343	−0.131739
$365$	3.44670	0.180409
$366$	0	0
$367$	−16.7919	−0.876528	−0.438264	−	0.898846i	$-0.644406\pi$
$367$	−16.7919	−0.876528	−0.438264	+	0.898846i	$0.644406\pi$
$368$	−39.9663	−2.08339
$369$	0	0
$370$	−13.1509	−0.683681
$371$	7.64605	0.396963
$372$	0	0
$373$	−31.9684	−1.65526	−0.827631	−	0.561272i	$-0.810312\pi$
$373$	−31.9684	−1.65526	−0.827631	+	0.561272i	$0.810312\pi$
$374$	−8.32587	−0.430521
$375$	0	0
$376$	−129.568	−6.68198
$377$	6.85801	0.353205
$378$	0	0
$379$	3.57139	0.183450	0.0917250	−	0.995784i	$-0.470762\pi$
$379$	3.57139	0.183450	0.0917250	+	0.995784i	$0.470762\pi$
$380$	−22.2715	−1.14251
$381$	0	0
$382$	−23.7438	−1.21484
$383$	−0.354912	−0.0181351	−0.00906757	−	0.999959i	$-0.502886\pi$
$383$	−0.354912	−0.0181351	−0.00906757	+	0.999959i	$0.502886\pi$
$384$	0	0
$385$	−0.504368	−0.0257050
$386$	−54.5379	−2.77591
$387$	0	0
$388$	−70.8509	−3.59691
$389$	8.52918	0.432447	0.216223	−	0.976344i	$-0.430626\pi$
$389$	8.52918	0.432447	0.216223	+	0.976344i	$0.430626\pi$
$390$	0	0
$391$	7.16695	0.362448
$392$	67.0296	3.38551
$393$	0	0
$394$	−67.7628	−3.41384
$395$	−2.39787	−0.120650
$396$	0	0
$397$	7.32166	0.367464	0.183732	−	0.982976i	$-0.441182\pi$
$397$	7.32166	0.367464	0.183732	+	0.982976i	$0.441182\pi$
$398$	24.9399	1.25013
$399$	0	0
$400$	−73.0652	−3.65326
$401$	11.9699	0.597747	0.298873	−	0.954293i	$-0.403389\pi$
$401$	11.9699	0.597747	0.298873	+	0.954293i	$0.403389\pi$
$402$	0	0
$403$	4.21140	0.209785
$404$	−58.3014	−2.90061
$405$	0	0
$406$	16.9365	0.840547
$407$	5.92391	0.293637
$408$	0	0
$409$	9.68697	0.478990	0.239495	−	0.970898i	$-0.423018\pi$
$409$	9.68697	0.478990	0.239495	+	0.970898i	$0.423018\pi$
$410$	−5.84694	−0.288760
$411$	0	0
$412$	24.1267	1.18864
$413$	−6.39798	−0.314824
$414$	0	0
$415$	−5.81437	−0.285416
$416$	−18.4278	−0.903496
$417$	0	0
$418$	13.5738	0.663917
$419$	−7.13143	−0.348393	−0.174197	−	0.984711i	$-0.555733\pi$
$419$	−7.13143	−0.348393	−0.174197	+	0.984711i	$0.555733\pi$
$420$	0	0
$421$	−15.8569	−0.772816	−0.386408	−	0.922328i	$-0.626284\pi$
$421$	−15.8569	−0.772816	−0.386408	+	0.922328i	$0.626284\pi$
$422$	14.1093	0.686831
$423$	0	0
$424$	123.366	5.99120
$425$	13.1024	0.635560
$426$	0	0
$427$	−2.62954	−0.127252
$428$	−44.4616	−2.14913
$429$	0	0
$430$	18.7024	0.901909
$431$	−19.2561	−0.927532	−0.463766	−	0.885958i	$-0.653502\pi$
$431$	−19.2561	−0.927532	−0.463766	+	0.885958i	$0.653502\pi$
$432$	0	0
$433$	−35.5315	−1.70754	−0.853768	−	0.520654i	$-0.825688\pi$
$433$	−35.5315	−1.70754	−0.853768	+	0.520654i	$0.825688\pi$
$434$	10.4005	0.499239
$435$	0	0
$436$	−73.3845	−3.51448
$437$	−11.6844	−0.558941
$438$	0	0
$439$	−28.3758	−1.35430	−0.677152	−	0.735843i	$-0.736786\pi$
$439$	−28.3758	−1.35430	−0.677152	+	0.735843i	$0.736786\pi$
$440$	−8.13781	−0.387955
$441$	0	0
$442$	5.87164	0.279286
$443$	24.3809	1.15837	0.579185	−	0.815196i	$-0.303371\pi$
$443$	24.3809	1.15837	0.579185	+	0.815196i	$0.303371\pi$
$444$	0	0
$445$	1.76478	0.0836587
$446$	15.1797	0.718779
$447$	0	0
$448$	−24.4122	−1.15337
$449$	−9.93191	−0.468716	−0.234358	−	0.972150i	$-0.575299\pi$
$449$	−9.93191	−0.468716	−0.234358	+	0.972150i	$0.575299\pi$
$450$	0	0
$451$	2.63380	0.124021
$452$	−27.7018	−1.30298
$453$	0	0
$454$	−8.75023	−0.410669
$455$	0.355695	0.0166752
$456$	0	0
$457$	12.0453	0.563455	0.281727	−	0.959494i	$-0.409093\pi$
$457$	12.0453	0.563455	0.281727	+	0.959494i	$0.409093\pi$
$458$	30.9850	1.44783
$459$	0	0
$460$	10.8270	0.504810
$461$	35.1708	1.63807	0.819034	−	0.573745i	$-0.194510\pi$
$461$	35.1708	1.63807	0.819034	+	0.573745i	$0.194510\pi$
$462$	0	0
$463$	22.6992	1.05492	0.527462	−	0.849579i	$-0.323144\pi$
$463$	22.6992	1.05492	0.527462	+	0.849579i	$0.323144\pi$
$464$	163.072	7.57044
$465$	0	0
$466$	−54.0935	−2.50583
$467$	−15.2847	−0.707291	−0.353645	−	0.935380i	$-0.615058\pi$
$467$	−15.2847	−0.707291	−0.353645	+	0.935380i	$0.615058\pi$
$468$	0	0
$469$	3.37978	0.156064
$470$	28.3403	1.30724
$471$	0	0
$472$	−103.229	−4.75151
$473$	−8.42464	−0.387365
$474$	0	0
$475$	−21.3611	−0.980114
$476$	10.7174	0.491231
$477$	0	0
$478$	−39.9968	−1.82941
$479$	29.9932	1.37042	0.685212	−	0.728344i	$-0.259710\pi$
$479$	29.9932	1.37042	0.685212	+	0.728344i	$0.259710\pi$
$480$	0	0
$481$	−4.17771	−0.190487
$482$	−71.3455	−3.24970
$483$	0	0
$484$	5.66575	0.257534
$485$	10.0267	0.455287
$486$	0	0
$487$	2.26984	0.102856	0.0514281	−	0.998677i	$-0.483623\pi$
$487$	2.26984	0.102856	0.0514281	+	0.998677i	$0.483623\pi$
$488$	−42.4267	−1.92057
$489$	0	0
$490$	−14.6613	−0.662330
$491$	7.17293	0.323710	0.161855	−	0.986815i	$-0.448252\pi$
$491$	7.17293	0.323710	0.161855	+	0.986815i	$0.448252\pi$
$492$	0	0
$493$	−29.2429	−1.31703
$494$	−9.57265	−0.430694
$495$	0	0
$496$	100.140	4.49643
$497$	1.61534	0.0724579
$498$	0	0
$499$	−19.7524	−0.884239	−0.442120	−	0.896956i	$-0.645773\pi$
$499$	−19.7524	−0.884239	−0.442120	+	0.896956i	$0.645773\pi$
$500$	42.5076	1.90100
$501$	0	0
$502$	61.7157	2.75451
$503$	−25.6401	−1.14323	−0.571617	−	0.820521i	$-0.693684\pi$
$503$	−25.6401	−1.14323	−0.571617	+	0.820521i	$0.693684\pi$
$504$	0	0
$505$	8.25069	0.367151
$506$	−6.59871	−0.293348
$507$	0	0
$508$	42.9307	1.90474
$509$	−2.81421	−0.124737	−0.0623687	−	0.998053i	$-0.519865\pi$
$509$	−2.81421	−0.124737	−0.0623687	+	0.998053i	$0.519865\pi$
$510$	0	0
$511$	2.70406	0.119620
$512$	−97.7887	−4.32169
$513$	0	0
$514$	−66.0141	−2.91176
$515$	−3.41436	−0.150455
$516$	0	0
$517$	−12.7661	−0.561454
$518$	−10.3173	−0.453316
$519$	0	0
$520$	5.73902	0.251673
$521$	4.82951	0.211584	0.105792	−	0.994388i	$-0.466262\pi$
$521$	4.82951	0.211584	0.105792	+	0.994388i	$0.466262\pi$
$522$	0	0
$523$	−16.8796	−0.738092	−0.369046	−	0.929411i	$-0.620316\pi$
$523$	−16.8796	−0.738092	−0.369046	+	0.929411i	$0.620316\pi$
$524$	−10.3886	−0.453827
$525$	0	0
$526$	−32.7829	−1.42940
$527$	−17.9576	−0.782247
$528$	0	0
$529$	−17.3198	−0.753035
$530$	−26.9838	−1.17210
$531$	0	0
$532$	−17.4728	−0.757541
$533$	−1.85743	−0.0804543
$534$	0	0
$535$	6.29212	0.272032
$536$	54.5316	2.35540
$537$	0	0
$538$	−1.31393	−0.0566476
$539$	6.60431	0.284468
$540$	0	0
$541$	5.76471	0.247844	0.123922	−	0.992292i	$-0.460453\pi$
$541$	5.76471	0.247844	0.123922	+	0.992292i	$0.460453\pi$
$542$	4.36095	0.187319
$543$	0	0
$544$	78.5770	3.36896
$545$	10.3852	0.444854
$546$	0	0
$547$	12.8846	0.550906	0.275453	−	0.961314i	$-0.411172\pi$
$547$	12.8846	0.550906	0.275453	+	0.961314i	$0.411172\pi$
$548$	−55.7622	−2.38204
$549$	0	0
$550$	−12.0636	−0.514392
$551$	47.6752	2.03103
$552$	0	0
$553$	−1.88121	−0.0799972
$554$	7.24331	0.307739
$555$	0	0
$556$	61.6453	2.61434
$557$	5.56989	0.236004	0.118002	−	0.993013i	$-0.462351\pi$
$557$	5.56989	0.236004	0.118002	+	0.993013i	$0.462351\pi$
$558$	0	0
$559$	5.94130	0.251290
$560$	8.45785	0.357409
$561$	0	0
$562$	−41.5776	−1.75385
$563$	24.3171	1.02485	0.512423	−	0.858733i	$-0.328748\pi$
$563$	24.3171	1.02485	0.512423	+	0.858733i	$0.328748\pi$
$564$	0	0
$565$	3.92030	0.164928
$566$	−21.4562	−0.901872
$567$	0	0
$568$	26.0630	1.09358
$569$	37.6523	1.57847	0.789233	−	0.614094i	$-0.210478\pi$
$569$	37.6523	1.57847	0.789233	+	0.614094i	$0.210478\pi$
$570$	0	0
$571$	−23.1929	−0.970592	−0.485296	−	0.874350i	$-0.661288\pi$
$571$	−23.1929	−0.970592	−0.485296	+	0.874350i	$0.661288\pi$
$572$	−3.99565	−0.167066
$573$	0	0
$574$	−4.58712	−0.191463
$575$	10.3844	0.433058
$576$	0	0
$577$	−5.14483	−0.214182	−0.107091	−	0.994249i	$-0.534154\pi$
$577$	−5.14483	−0.214182	−0.107091	+	0.994249i	$0.534154\pi$
$578$	22.0310	0.916370
$579$	0	0
$580$	−44.1767	−1.83434
$581$	−4.56156	−0.189246
$582$	0	0
$583$	12.1551	0.503412
$584$	43.6290	1.80538
$585$	0	0
$586$	7.17840	0.296537
$587$	−19.1430	−0.790116	−0.395058	−	0.918656i	$-0.629275\pi$
$587$	−19.1430	−0.790116	−0.395058	+	0.918656i	$0.629275\pi$
$588$	0	0
$589$	29.2767	1.20632
$590$	22.5792	0.929571
$591$	0	0
$592$	−99.3392	−4.08282
$593$	−23.4013	−0.960978	−0.480489	−	0.877001i	$-0.659541\pi$
$593$	−23.4013	−0.960978	−0.480489	+	0.877001i	$0.659541\pi$
$594$	0	0
$595$	−1.51670	−0.0621787
$596$	−77.1892	−3.16179
$597$	0	0
$598$	4.65360	0.190300
$599$	0.800913	0.0327244	0.0163622	−	0.999866i	$-0.494792\pi$
$599$	0.800913	0.0327244	0.0163622	+	0.999866i	$0.494792\pi$
$600$	0	0
$601$	18.8876	0.770441	0.385221	−	0.922825i	$-0.374125\pi$
$601$	18.8876	0.770441	0.385221	+	0.922825i	$0.374125\pi$
$602$	14.6726	0.598012
$603$	0	0
$604$	49.9742	2.03342
$605$	−0.801804	−0.0325980
$606$	0	0
$607$	−2.78565	−0.113066	−0.0565330	−	0.998401i	$-0.518005\pi$
$607$	−2.78565	−0.113066	−0.0565330	+	0.998401i	$0.518005\pi$
$608$	−128.106	−5.19537
$609$	0	0
$610$	9.27995	0.375734
$611$	9.00305	0.364224
$612$	0	0
$613$	4.06116	0.164029	0.0820145	−	0.996631i	$-0.473865\pi$
$613$	4.06116	0.164029	0.0820145	+	0.996631i	$0.473865\pi$
$614$	13.4811	0.544053
$615$	0	0
$616$	−6.38439	−0.257234
$617$	42.5183	1.71172	0.855860	−	0.517207i	$-0.173028\pi$
$617$	42.5183	1.71172	0.855860	+	0.517207i	$0.173028\pi$
$618$	0	0
$619$	−6.08562	−0.244602	−0.122301	−	0.992493i	$-0.539027\pi$
$619$	−6.08562	−0.244602	−0.122301	+	0.992493i	$0.539027\pi$
$620$	−27.1283	−1.08950
$621$	0	0
$622$	80.7825	3.23908
$623$	1.38453	0.0554701
$624$	0	0
$625$	15.7700	0.630799
$626$	−74.5033	−2.97775
$627$	0	0
$628$	−70.8641	−2.82779
$629$	17.8140	0.710290
$630$	0	0
$631$	33.8586	1.34789	0.673945	−	0.738781i	$-0.264598\pi$
$631$	33.8586	1.34789	0.673945	+	0.738781i	$0.264598\pi$
$632$	−30.3527	−1.20736
$633$	0	0
$634$	3.67024	0.145764
$635$	−6.07546	−0.241097
$636$	0	0
$637$	−4.65754	−0.184539
$638$	26.9243	1.06595
$639$	0	0
$640$	44.2509	1.74917
$641$	21.6330	0.854452	0.427226	−	0.904145i	$-0.359491\pi$
$641$	21.6330	0.854452	0.427226	+	0.904145i	$0.359491\pi$
$642$	0	0
$643$	42.6281	1.68109	0.840544	−	0.541744i	$-0.182236\pi$
$643$	42.6281	1.68109	0.840544	+	0.541744i	$0.182236\pi$
$644$	8.49413	0.334715
$645$	0	0
$646$	40.8183	1.60597
$647$	11.5796	0.455239	0.227620	−	0.973750i	$-0.426906\pi$
$647$	11.5796	0.455239	0.227620	+	0.973750i	$0.426906\pi$
$648$	0	0
$649$	−10.1710	−0.399246
$650$	8.50757	0.333694
$651$	0	0
$652$	−12.2339	−0.479116
$653$	13.2799	0.519681	0.259841	−	0.965652i	$-0.416330\pi$
$653$	13.2799	0.519681	0.259841	+	0.965652i	$0.416330\pi$
$654$	0	0
$655$	1.47017	0.0574442
$656$	−44.1667	−1.72442
$657$	0	0
$658$	22.2339	0.866769
$659$	−3.31843	−0.129267	−0.0646337	−	0.997909i	$-0.520588\pi$
$659$	−3.31843	−0.129267	−0.0646337	+	0.997909i	$0.520588\pi$
$660$	0	0
$661$	47.0928	1.83170	0.915850	−	0.401521i	$-0.131518\pi$
$661$	47.0928	1.83170	0.915850	+	0.401521i	$0.131518\pi$
$662$	−41.0995	−1.59738
$663$	0	0
$664$	−73.5993	−2.85621
$665$	2.47271	0.0958875
$666$	0	0
$667$	−23.1766	−0.897402
$668$	−25.9320	−1.00334
$669$	0	0
$670$	−11.9276	−0.460804
$671$	−4.18023	−0.161376
$672$	0	0
$673$	−17.1510	−0.661124	−0.330562	−	0.943784i	$-0.607238\pi$
$673$	−17.1510	−0.661124	−0.330562	+	0.943784i	$0.607238\pi$
$674$	−61.6723	−2.37553
$675$	0	0
$676$	−70.8369	−2.72449
$677$	−15.4248	−0.592822	−0.296411	−	0.955061i	$-0.595790\pi$
$677$	−15.4248	−0.592822	−0.296411	+	0.955061i	$0.595790\pi$
$678$	0	0
$679$	7.86625	0.301879
$680$	−24.4715	−0.938439
$681$	0	0
$682$	16.5338	0.633114
$683$	14.3473	0.548983	0.274491	−	0.961590i	$-0.411491\pi$
$683$	14.3473	0.548983	0.274491	+	0.961590i	$0.411491\pi$
$684$	0	0
$685$	7.89135	0.301513
$686$	−23.6937	−0.904631
$687$	0	0
$688$	141.274	5.38603
$689$	−8.57210	−0.326571
$690$	0	0
$691$	21.0729	0.801652	0.400826	−	0.916154i	$-0.368723\pi$
$691$	21.0729	0.801652	0.400826	+	0.916154i	$0.368723\pi$
$692$	−32.2110	−1.22448
$693$	0	0
$694$	−52.7328	−2.00171
$695$	−8.72391	−0.330917
$696$	0	0
$697$	7.92019	0.299999
$698$	78.6304	2.97620
$699$	0	0
$700$	15.5287	0.586930
$701$	15.3167	0.578503	0.289251	−	0.957253i	$-0.406594\pi$
$701$	15.3167	0.578503	0.289251	+	0.957253i	$0.406594\pi$
$702$	0	0
$703$	−29.0425	−1.09536
$704$	−38.8086	−1.46265
$705$	0	0
$706$	−56.1299	−2.11248
$707$	6.47295	0.243440
$708$	0	0
$709$	30.9106	1.16087	0.580437	−	0.814305i	$-0.302882\pi$
$709$	30.9106	1.16087	0.580437	+	0.814305i	$0.302882\pi$
$710$	−5.70072	−0.213944
$711$	0	0
$712$	22.3389	0.837187
$713$	−14.2324	−0.533008
$714$	0	0
$715$	0.565455	0.0211468
$716$	95.8162	3.58082
$717$	0	0
$718$	−25.3096	−0.944545
$719$	0.183121	0.00682925	0.00341462	−	0.999994i	$-0.498913\pi$
$719$	0.183121	0.00682925	0.00341462	+	0.999994i	$0.498913\pi$
$720$	0	0
$721$	−2.67868	−0.0997593
$722$	−13.9413	−0.518843
$723$	0	0
$724$	98.9937	3.67907
$725$	−42.3708	−1.57361
$726$	0	0
$727$	−37.6267	−1.39550	−0.697749	−	0.716342i	$-0.745815\pi$
$727$	−37.6267	−1.39550	−0.697749	+	0.716342i	$0.745815\pi$
$728$	4.50245	0.166872
$729$	0	0
$730$	−9.54292	−0.353199
$731$	−25.3340	−0.937012
$732$	0	0
$733$	37.2639	1.37637	0.688187	−	0.725534i	$-0.258407\pi$
$733$	37.2639	1.37637	0.688187	+	0.725534i	$0.258407\pi$
$734$	46.4918	1.71604
$735$	0	0
$736$	62.2766	2.29555
$737$	5.37289	0.197913
$738$	0	0
$739$	−47.5216	−1.74811	−0.874055	−	0.485827i	$-0.838519\pi$
$739$	−47.5216	−1.74811	−0.874055	+	0.485827i	$0.838519\pi$
$740$	26.9113	0.989277
$741$	0	0
$742$	−21.1697	−0.777164
$743$	23.7430	0.871047	0.435524	−	0.900177i	$-0.356563\pi$
$743$	23.7430	0.871047	0.435524	+	0.900177i	$0.356563\pi$
$744$	0	0
$745$	10.9237	0.400212
$746$	88.5112	3.24063
$747$	0	0
$748$	17.0376	0.622958
$749$	4.93638	0.180371
$750$	0	0
$751$	23.5058	0.857739	0.428869	−	0.903367i	$-0.358912\pi$
$751$	23.5058	0.857739	0.428869	+	0.903367i	$0.358912\pi$
$752$	214.078	7.80661
$753$	0	0
$754$	−18.9878	−0.691495
$755$	−7.07224	−0.257385
$756$	0	0
$757$	19.2291	0.698892	0.349446	−	0.936956i	$-0.386370\pi$
$757$	19.2291	0.698892	0.349446	+	0.936956i	$0.386370\pi$
$758$	−9.88814	−0.359153
$759$	0	0
$760$	39.8963	1.44719
$761$	4.07835	0.147840	0.0739201	−	0.997264i	$-0.476449\pi$
$761$	4.07835	0.147840	0.0739201	+	0.997264i	$0.476449\pi$
$762$	0	0
$763$	8.14755	0.294961
$764$	48.5881	1.75785
$765$	0	0
$766$	0.982647	0.0355045
$767$	7.17287	0.258997
$768$	0	0
$769$	5.97768	0.215561	0.107780	−	0.994175i	$-0.465626\pi$
$769$	5.97768	0.215561	0.107780	+	0.994175i	$0.465626\pi$
$770$	1.39645	0.0503245
$771$	0	0
$772$	111.604	4.01670
$773$	−33.0103	−1.18730	−0.593650	−	0.804723i	$-0.702314\pi$
$773$	−33.0103	−1.18730	−0.593650	+	0.804723i	$0.702314\pi$
$774$	0	0
$775$	−26.0193	−0.934640
$776$	126.919	4.55614
$777$	0	0
$778$	−23.6148	−0.846632
$779$	−12.9124	−0.462636
$780$	0	0
$781$	2.56794	0.0918880
$782$	−19.8432	−0.709591
$783$	0	0
$784$	−110.749	−3.95531
$785$	10.0285	0.357934
$786$	0	0
$787$	−22.0786	−0.787017	−0.393509	−	0.919321i	$-0.628739\pi$
$787$	−22.0786	−0.787017	−0.393509	+	0.919321i	$0.628739\pi$
$788$	138.666	4.93979
$789$	0	0
$790$	6.63900	0.236205
$791$	3.07561	0.109356
$792$	0	0
$793$	2.94802	0.104687
$794$	−20.2715	−0.719410
$795$	0	0
$796$	−51.0358	−1.80892
$797$	16.3929	0.580666	0.290333	−	0.956926i	$-0.406234\pi$
$797$	16.3929	0.580666	0.290333	+	0.956926i	$0.406234\pi$
$798$	0	0
$799$	−38.3895	−1.35812
$800$	113.852	4.02528
$801$	0	0
$802$	−33.1411	−1.17025
$803$	4.29869	0.151697
$804$	0	0
$805$	−1.20207	−0.0423674
$806$	−11.6601	−0.410711
$807$	0	0
$808$	104.439	3.67414
$809$	−0.100443	−0.00353138	−0.00176569	−	0.999998i	$-0.500562\pi$
$809$	−0.100443	−0.00353138	−0.00176569	+	0.999998i	$0.500562\pi$
$810$	0	0
$811$	29.3186	1.02951	0.514757	−	0.857336i	$-0.327882\pi$
$811$	29.3186	1.02951	0.514757	+	0.857336i	$0.327882\pi$
$812$	−34.6581	−1.21626
$813$	0	0
$814$	−16.4016	−0.574875
$815$	1.73131	0.0606453
$816$	0	0
$817$	41.3025	1.44499
$818$	−26.8204	−0.937753
$819$	0	0
$820$	11.9649	0.417832
$821$	17.2759	0.602934	0.301467	−	0.953477i	$-0.402524\pi$
$821$	17.2759	0.602934	0.301467	+	0.953477i	$0.402524\pi$
$822$	0	0
$823$	26.5823	0.926602	0.463301	−	0.886201i	$-0.346665\pi$
$823$	26.5823	0.926602	0.463301	+	0.886201i	$0.346665\pi$
$824$	−43.2196	−1.50563
$825$	0	0
$826$	17.7141	0.616354
$827$	−20.6225	−0.717116	−0.358558	−	0.933507i	$-0.616731\pi$
$827$	−20.6225	−0.717116	−0.358558	+	0.933507i	$0.616731\pi$
$828$	0	0
$829$	8.87546	0.308257	0.154129	−	0.988051i	$-0.450743\pi$
$829$	8.87546	0.308257	0.154129	+	0.988051i	$0.450743\pi$
$830$	16.0983	0.558780
$831$	0	0
$832$	27.3689	0.948846
$833$	19.8600	0.688109
$834$	0	0
$835$	3.66984	0.127000
$836$	−27.7768	−0.960680
$837$	0	0
$838$	19.7449	0.682075
$839$	−37.1184	−1.28147	−0.640735	−	0.767762i	$-0.721370\pi$
$839$	−37.1184	−1.28147	−0.640735	+	0.767762i	$0.721370\pi$
$840$	0	0
$841$	65.5662	2.26090
$842$	43.9030	1.51300
$843$	0	0
$844$	−28.8726	−0.993837
$845$	10.0247	0.344859
$846$	0	0
$847$	−0.629042	−0.0216141
$848$	−203.831	−6.99957
$849$	0	0
$850$	−36.2767	−1.24428
$851$	14.1186	0.483978
$852$	0	0
$853$	4.37401	0.149763	0.0748816	−	0.997192i	$-0.476142\pi$
$853$	4.37401	0.149763	0.0748816	+	0.997192i	$0.476142\pi$
$854$	7.28043	0.249131
$855$	0	0
$856$	79.6467	2.72227
$857$	0.135420	0.00462585	0.00231292	−	0.999997i	$-0.499264\pi$
$857$	0.135420	0.00462585	0.00231292	+	0.999997i	$0.499264\pi$
$858$	0	0
$859$	32.2770	1.10128	0.550638	−	0.834744i	$-0.314384\pi$
$859$	32.2770	1.10128	0.550638	+	0.834744i	$0.314384\pi$
$860$	−38.2716	−1.30505
$861$	0	0
$862$	53.3144	1.81590
$863$	39.1121	1.33139	0.665696	−	0.746223i	$-0.268135\pi$
$863$	39.1121	1.33139	0.665696	+	0.746223i	$0.268135\pi$
$864$	0	0
$865$	4.55842	0.154991
$866$	98.3764	3.34297
$867$	0	0
$868$	−21.2830	−0.722393
$869$	−2.99059	−0.101449
$870$	0	0
$871$	−3.78912	−0.128389
$872$	131.458	4.45173
$873$	0	0
$874$	32.3507	1.09428
$875$	−4.71943	−0.159546
$876$	0	0
$877$	47.3435	1.59868	0.799338	−	0.600882i	$-0.205184\pi$
$877$	47.3435	1.59868	0.799338	+	0.600882i	$0.205184\pi$
$878$	78.5644	2.65142
$879$	0	0
$880$	13.4456	0.453251
$881$	−39.7909	−1.34059	−0.670294	−	0.742095i	$-0.733832\pi$
$881$	−39.7909	−1.34059	−0.670294	+	0.742095i	$0.733832\pi$
$882$	0	0
$883$	−36.4775	−1.22757	−0.613783	−	0.789475i	$-0.710353\pi$
$883$	−36.4775	−1.22757	−0.613783	+	0.789475i	$0.710353\pi$
$884$	−12.0154	−0.404123
$885$	0	0
$886$	−67.5035	−2.26783
$887$	−16.5688	−0.556326	−0.278163	−	0.960534i	$-0.589726\pi$
$887$	−16.5688	−0.556326	−0.278163	+	0.960534i	$0.589726\pi$
$888$	0	0
$889$	−4.76640	−0.159860
$890$	−4.88617	−0.163785
$891$	0	0
$892$	−31.0630	−1.04006
$893$	62.5871	2.09440
$894$	0	0
$895$	−13.5597	−0.453251
$896$	34.7163	1.15979
$897$	0	0
$898$	27.4986	0.917639
$899$	58.0717	1.93680
$900$	0	0
$901$	36.5519	1.21772
$902$	−7.29223	−0.242805
$903$	0	0
$904$	49.6239	1.65047
$905$	−14.0094	−0.465687
$906$	0	0
$907$	−20.3298	−0.675039	−0.337519	−	0.941319i	$-0.609588\pi$
$907$	−20.3298	−0.675039	−0.337519	+	0.941319i	$0.609588\pi$
$908$	17.9060	0.594232
$909$	0	0
$910$	−0.984816	−0.0326463
$911$	57.7715	1.91405	0.957027	−	0.289998i	$-0.0936545\pi$
$911$	57.7715	1.91405	0.957027	+	0.289998i	$0.0936545\pi$
$912$	0	0
$913$	−7.25161	−0.239993
$914$	−33.3499	−1.10312
$915$	0	0
$916$	−63.4061	−2.09500
$917$	1.15340	0.0380885
$918$	0	0
$919$	29.3071	0.966751	0.483376	−	0.875413i	$-0.339411\pi$
$919$	29.3071	0.966751	0.483376	+	0.875413i	$0.339411\pi$
$920$	−19.3950	−0.639434
$921$	0	0
$922$	−97.3777	−3.20696
$923$	−1.81098	−0.0596092
$924$	0	0
$925$	25.8111	0.848665
$926$	−62.8476	−2.06530
$927$	0	0
$928$	−254.104	−8.34136
$929$	33.1031	1.08608	0.543038	−	0.839708i	$-0.317274\pi$
$929$	33.1031	1.08608	0.543038	+	0.839708i	$0.317274\pi$
$930$	0	0
$931$	−32.3782	−1.06115
$932$	110.694	3.62591
$933$	0	0
$934$	42.3188	1.38471
$935$	−2.41113	−0.0788524
$936$	0	0
$937$	37.9274	1.23903	0.619517	−	0.784983i	$-0.287328\pi$
$937$	37.9274	1.23903	0.619517	+	0.784983i	$0.287328\pi$
$938$	−9.35762	−0.305537
$939$	0	0
$940$	−57.9942	−1.89156
$941$	−50.1923	−1.63622	−0.818112	−	0.575059i	$-0.804979\pi$
$941$	−50.1923	−1.63622	−0.818112	+	0.575059i	$0.804979\pi$
$942$	0	0
$943$	6.27719	0.204413
$944$	170.559	5.55123
$945$	0	0
$946$	23.3254	0.758373
$947$	43.1041	1.40070	0.700348	−	0.713802i	$-0.253028\pi$
$947$	43.1041	1.40070	0.700348	+	0.713802i	$0.253028\pi$
$948$	0	0
$949$	−3.03156	−0.0984085
$950$	59.1426	1.91884
$951$	0	0
$952$	−19.1987	−0.622233
$953$	2.77548	0.0899066	0.0449533	−	0.998989i	$-0.485686\pi$
$953$	2.77548	0.0899066	0.0449533	+	0.998989i	$0.485686\pi$
$954$	0	0
$955$	−6.87608	−0.222505
$956$	81.8474	2.64713
$957$	0	0
$958$	−83.0424	−2.68298
$959$	6.19103	0.199919
$960$	0	0
$961$	4.66099	0.150354
$962$	11.5669	0.372931
$963$	0	0
$964$	145.998	4.70227
$965$	−15.7939	−0.508424
$966$	0	0
$967$	46.4191	1.49274	0.746368	−	0.665533i	$-0.231796\pi$
$967$	46.4191	1.49274	0.746368	+	0.665533i	$0.231796\pi$
$968$	−10.1494	−0.326213
$969$	0	0
$970$	−27.7609	−0.891349
$971$	20.8120	0.667890	0.333945	−	0.942592i	$-0.391620\pi$
$971$	20.8120	0.667890	0.333945	+	0.942592i	$0.391620\pi$
$972$	0	0
$973$	−6.84420	−0.219415
$974$	−6.28451	−0.201369
$975$	0	0
$976$	70.0990	2.24382
$977$	−3.52448	−0.112758	−0.0563790	−	0.998409i	$-0.517956\pi$
$977$	−3.52448	−0.112758	−0.0563790	+	0.998409i	$0.517956\pi$
$978$	0	0
$979$	2.20102	0.0703448
$980$	30.0022	0.958384
$981$	0	0
$982$	−19.8598	−0.633750
$983$	−4.17562	−0.133182	−0.0665908	−	0.997780i	$-0.521212\pi$
$983$	−4.17562	−0.133182	−0.0665908	+	0.997780i	$0.521212\pi$
$984$	0	0
$985$	−19.6238	−0.625266
$986$	80.9651	2.57845
$987$	0	0
$988$	19.5890	0.623209
$989$	−20.0786	−0.638462
$990$	0	0
$991$	33.4780	1.06346	0.531732	−	0.846912i	$-0.321541\pi$
$991$	33.4780	1.06346	0.531732	+	0.846912i	$0.321541\pi$
$992$	−156.041	−4.95432
$993$	0	0
$994$	−4.47240	−0.141856
$995$	7.22248	0.228968
$996$	0	0
$997$	58.7466	1.86052	0.930262	−	0.366897i	$-0.119580\pi$
$997$	58.7466	1.86052	0.930262	+	0.366897i	$0.119580\pi$
$998$	54.6886	1.73114
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8019.2.a.k.1.1		51
3.2	odd	2		8019.2.a.l.1.51		51
27.2	odd	18		297.2.j.c.166.1	yes	102
27.13	even	9		891.2.j.c.100.17		102
27.14	odd	18		297.2.j.c.34.1	✓	102
27.25	even	9		891.2.j.c.793.17		102

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
297.2.j.c.34.1	✓	102	27.14	odd	18
297.2.j.c.166.1	yes	102	27.2	odd	18
891.2.j.c.100.17		102	27.13	even	9
891.2.j.c.793.17		102	27.25	even	9
8019.2.a.k.1.1		51	1.1	even	1	trivial
8019.2.a.l.1.51		51	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8019 = 3^{6} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8019.a (trivial)

\(f(q)\)	\(=\)	\(q-2.76871 q^{2} +5.66575 q^{4} -0.801804 q^{5} -0.629042 q^{7} -10.1494 q^{8} +O(q^{10})\)	\(q-2.76871 q^{2} +5.66575 q^{4} -0.801804 q^{5} -0.629042 q^{7} -10.1494 q^{8} +2.21996 q^{10} -1.00000 q^{11} +0.705228 q^{13} +1.74163 q^{14} +16.7692 q^{16} -3.00713 q^{17} +4.90258 q^{19} -4.54282 q^{20} +2.76871 q^{22} -2.38332 q^{23} -4.35711 q^{25} -1.95257 q^{26} -3.56399 q^{28} +9.72452 q^{29} +5.97168 q^{31} -26.1302 q^{32} +8.32587 q^{34} +0.504368 q^{35} -5.92391 q^{37} -13.5738 q^{38} +8.13781 q^{40} -2.63380 q^{41} +8.42464 q^{43} -5.66575 q^{44} +6.59871 q^{46} +12.7661 q^{47} -6.60431 q^{49} +12.0636 q^{50} +3.99565 q^{52} -12.1551 q^{53} +0.801804 q^{55} +6.38439 q^{56} -26.9243 q^{58} +10.1710 q^{59} +4.18023 q^{61} -16.5338 q^{62} +38.8086 q^{64} -0.565455 q^{65} -5.37289 q^{67} -17.0376 q^{68} -1.39645 q^{70} -2.56794 q^{71} -4.29869 q^{73} +16.4016 q^{74} +27.7768 q^{76} +0.629042 q^{77} +2.99059 q^{79} -13.4456 q^{80} +7.29223 q^{82} +7.25161 q^{83} +2.41113 q^{85} -23.3254 q^{86} +10.1494 q^{88} -2.20102 q^{89} -0.443618 q^{91} -13.5033 q^{92} -35.3457 q^{94} -3.93091 q^{95} -12.5051 q^{97} +18.2854 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 51 q + 60 q^{4} - 6 q^{5} + 12 q^{7}+O(q^{10}) \) 51 * q + 60 * q^4 - 6 * q^5 + 12 * q^7	\( 51 q + 60 q^{4} - 6 q^{5} + 12 q^{7} + 18 q^{10} - 51 q^{11} + 30 q^{13} - 12 q^{14} + 78 q^{16} + 30 q^{19} - 18 q^{20} - 3 q^{23} + 75 q^{25} - 9 q^{26} + 36 q^{28} + 42 q^{31} + 15 q^{32} + 42 q^{34} + 9 q^{35} + 48 q^{37} + 3 q^{38} + 54 q^{40} + 18 q^{43} - 60 q^{44} + 42 q^{46} - 30 q^{47} + 99 q^{49} + 30 q^{50} + 60 q^{52} - 18 q^{53} + 6 q^{55} - 21 q^{56} + 30 q^{58} - 24 q^{59} + 99 q^{61} + 114 q^{64} + 15 q^{65} + 39 q^{67} + 39 q^{68} + 48 q^{70} - 30 q^{71} + 69 q^{73} + 90 q^{76} - 12 q^{77} + 48 q^{79} - 42 q^{80} + 42 q^{82} + 21 q^{83} + 84 q^{85} - 24 q^{86} - 15 q^{89} + 69 q^{91} + 66 q^{92} + 66 q^{94} + 12 q^{95} + 72 q^{97} + 57 q^{98}+O(q^{100}) \) 51 * q + 60 * q^4 - 6 * q^5 + 12 * q^7 + 18 * q^10 - 51 * q^11 + 30 * q^13 - 12 * q^14 + 78 * q^16 + 30 * q^19 - 18 * q^20 - 3 * q^23 + 75 * q^25 - 9 * q^26 + 36 * q^28 + 42 * q^31 + 15 * q^32 + 42 * q^34 + 9 * q^35 + 48 * q^37 + 3 * q^38 + 54 * q^40 + 18 * q^43 - 60 * q^44 + 42 * q^46 - 30 * q^47 + 99 * q^49 + 30 * q^50 + 60 * q^52 - 18 * q^53 + 6 * q^55 - 21 * q^56 + 30 * q^58 - 24 * q^59 + 99 * q^61 + 114 * q^64 + 15 * q^65 + 39 * q^67 + 39 * q^68 + 48 * q^70 - 30 * q^71 + 69 * q^73 + 90 * q^76 - 12 * q^77 + 48 * q^79 - 42 * q^80 + 42 * q^82 + 21 * q^83 + 84 * q^85 - 24 * q^86 - 15 * q^89 + 69 * q^91 + 66 * q^92 + 66 * q^94 + 12 * q^95 + 72 * q^97 + 57 * q^98

Introduction

L-functions

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