LMFDB - Embedded newform 8019.2.a.l.1.16

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			1.16
Character	$\chi$	$=$	8019.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.44748 q^{2} +0.0951963 q^{4} -3.47143 q^{5} +5.22882 q^{7} +2.75716 q^{8} +O(q^{10})$	\(q-1.44748 q^{2} +0.0951963 q^{4} -3.47143 q^{5} +5.22882 q^{7} +2.75716 q^{8} +5.02482 q^{10} +1.00000 q^{11} -1.87034 q^{13} -7.56861 q^{14} -4.18133 q^{16} -4.92244 q^{17} +0.177331 q^{19} -0.330467 q^{20} -1.44748 q^{22} -4.23324 q^{23} +7.05082 q^{25} +2.70727 q^{26} +0.497764 q^{28} -5.22733 q^{29} -7.91980 q^{31} +0.538061 q^{32} +7.12513 q^{34} -18.1515 q^{35} +0.472654 q^{37} -0.256683 q^{38} -9.57130 q^{40} -2.32100 q^{41} -0.0255766 q^{43} +0.0951963 q^{44} +6.12753 q^{46} +12.0748 q^{47} +20.3406 q^{49} -10.2059 q^{50} -0.178049 q^{52} +3.18605 q^{53} -3.47143 q^{55} +14.4167 q^{56} +7.56645 q^{58} -5.14747 q^{59} +1.47673 q^{61} +11.4637 q^{62} +7.58383 q^{64} +6.49274 q^{65} +7.34442 q^{67} -0.468598 q^{68} +26.2739 q^{70} +3.52855 q^{71} +7.21290 q^{73} -0.684157 q^{74} +0.0168813 q^{76} +5.22882 q^{77} -13.0021 q^{79} +14.5152 q^{80} +3.35959 q^{82} -1.86629 q^{83} +17.0879 q^{85} +0.0370216 q^{86} +2.75716 q^{88} -16.4859 q^{89} -9.77964 q^{91} -0.402989 q^{92} -17.4780 q^{94} -0.615593 q^{95} -2.08477 q^{97} -29.4425 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 51 q + 60 q^{4} + 6 q^{5} + 12 q^{7}+O(q^{10}) $ 51 * q + 60 * q^4 + 6 * q^5 + 12 * q^7	$ 51 q + 60 q^{4} + 6 q^{5} + 12 q^{7} + 18 q^{10} + 51 q^{11} + 30 q^{13} + 12 q^{14} + 78 q^{16} + 30 q^{19} + 18 q^{20} + 3 q^{23} + 75 q^{25} + 9 q^{26} + 36 q^{28} + 42 q^{31} - 15 q^{32} + 42 q^{34} - 9 q^{35} + 48 q^{37} - 3 q^{38} + 54 q^{40} + 18 q^{43} + 60 q^{44} + 42 q^{46} + 30 q^{47} + 99 q^{49} - 30 q^{50} + 60 q^{52} + 18 q^{53} + 6 q^{55} + 21 q^{56} + 30 q^{58} + 24 q^{59} + 99 q^{61} + 114 q^{64} - 15 q^{65} + 39 q^{67} - 39 q^{68} + 48 q^{70} + 30 q^{71} + 69 q^{73} + 90 q^{76} + 12 q^{77} + 48 q^{79} + 42 q^{80} + 42 q^{82} - 21 q^{83} + 84 q^{85} + 24 q^{86} + 15 q^{89} + 69 q^{91} - 66 q^{92} + 66 q^{94} - 12 q^{95} + 72 q^{97} - 57 q^{98}+O(q^{100}) $ 51 * q + 60 * q^4 + 6 * q^5 + 12 * q^7 + 18 * q^10 + 51 * q^11 + 30 * q^13 + 12 * q^14 + 78 * q^16 + 30 * q^19 + 18 * q^20 + 3 * q^23 + 75 * q^25 + 9 * q^26 + 36 * q^28 + 42 * q^31 - 15 * q^32 + 42 * q^34 - 9 * q^35 + 48 * q^37 - 3 * q^38 + 54 * q^40 + 18 * q^43 + 60 * q^44 + 42 * q^46 + 30 * q^47 + 99 * q^49 - 30 * q^50 + 60 * q^52 + 18 * q^53 + 6 * q^55 + 21 * q^56 + 30 * q^58 + 24 * q^59 + 99 * q^61 + 114 * q^64 - 15 * q^65 + 39 * q^67 - 39 * q^68 + 48 * q^70 + 30 * q^71 + 69 * q^73 + 90 * q^76 + 12 * q^77 + 48 * q^79 + 42 * q^80 + 42 * q^82 - 21 * q^83 + 84 * q^85 + 24 * q^86 + 15 * q^89 + 69 * q^91 - 66 * q^92 + 66 * q^94 - 12 * q^95 + 72 * q^97 - 57 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.44748	−1.02352	−0.511761	−	0.859128i	$-0.671007\pi$
$2$	−1.44748	−1.02352	−0.511761	+	0.859128i	$0.671007\pi$
$3$	0	0
$3$	0	0
$4$	0.0951963	0.0475982
$5$	−3.47143	−1.55247	−0.776235	−	0.630444i	$-0.782873\pi$
$5$	−3.47143	−1.55247	−0.776235	+	0.630444i	$0.782873\pi$
$6$	0	0
$7$	5.22882	1.97631	0.988154	−	0.153466i	$-0.0490434\pi$
$7$	5.22882	1.97631	0.988154	+	0.153466i	$0.0490434\pi$
$8$	2.75716	0.974805
$9$	0	0
$10$	5.02482	1.58899
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.87034	−0.518738	−0.259369	−	0.965778i	$-0.583515\pi$
$13$	−1.87034	−0.518738	−0.259369	+	0.965778i	$0.583515\pi$
$14$	−7.56861	−2.02280
$15$	0	0
$16$	−4.18133	−1.04533
$17$	−4.92244	−1.19387	−0.596934	−	0.802290i	$-0.703615\pi$
$17$	−4.92244	−1.19387	−0.596934	+	0.802290i	$0.703615\pi$
$18$	0	0
$19$	0.177331	0.0406826	0.0203413	−	0.999793i	$-0.493525\pi$
$19$	0.177331	0.0406826	0.0203413	+	0.999793i	$0.493525\pi$
$20$	−0.330467	−0.0738947
$21$	0	0
$22$	−1.44748	−0.308604
$23$	−4.23324	−0.882692	−0.441346	−	0.897337i	$-0.645499\pi$
$23$	−4.23324	−0.882692	−0.441346	+	0.897337i	$0.645499\pi$
$24$	0	0
$25$	7.05082	1.41016
$26$	2.70727	0.530940
$27$	0	0
$28$	0.497764	0.0940686
$29$	−5.22733	−0.970690	−0.485345	−	0.874323i	$-0.661306\pi$
$29$	−5.22733	−0.970690	−0.485345	+	0.874323i	$0.661306\pi$
$30$	0	0
$31$	−7.91980	−1.42244	−0.711219	−	0.702971i	$-0.751857\pi$
$31$	−7.91980	−1.42244	−0.711219	+	0.702971i	$0.751857\pi$
$32$	0.538061	0.0951167
$33$	0	0
$34$	7.12513	1.22195
$35$	−18.1515	−3.06816
$36$	0	0
$37$	0.472654	0.0777038	0.0388519	−	0.999245i	$-0.487630\pi$
$37$	0.472654	0.0777038	0.0388519	+	0.999245i	$0.487630\pi$
$38$	−0.256683	−0.0416395
$39$	0	0
$40$	−9.57130	−1.51336
$41$	−2.32100	−0.362479	−0.181239	−	0.983439i	$-0.558011\pi$
$41$	−2.32100	−0.362479	−0.181239	+	0.983439i	$0.558011\pi$
$42$	0	0
$43$	−0.0255766	−0.00390040	−0.00195020	−	0.999998i	$-0.500621\pi$
$43$	−0.0255766	−0.00390040	−0.00195020	+	0.999998i	$0.500621\pi$
$44$	0.0951963	0.0143514
$45$	0	0
$46$	6.12753	0.903455
$47$	12.0748	1.76129	0.880643	−	0.473781i	$-0.157111\pi$
$47$	12.0748	1.76129	0.880643	+	0.473781i	$0.157111\pi$
$48$	0	0
$49$	20.3406	2.90579
$50$	−10.2059	−1.44333
$51$	0	0
$52$	−0.178049	−0.0246910
$53$	3.18605	0.437638	0.218819	−	0.975766i	$-0.429780\pi$
$53$	3.18605	0.437638	0.218819	+	0.975766i	$0.429780\pi$
$54$	0	0
$55$	−3.47143	−0.468087
$56$	14.4167	1.92651
$57$	0	0
$58$	7.56645	0.993523
$59$	−5.14747	−0.670144	−0.335072	−	0.942193i	$-0.608761\pi$
$59$	−5.14747	−0.670144	−0.335072	+	0.942193i	$0.608761\pi$
$60$	0	0
$61$	1.47673	0.189076	0.0945378	−	0.995521i	$-0.469863\pi$
$61$	1.47673	0.189076	0.0945378	+	0.995521i	$0.469863\pi$
$62$	11.4637	1.45590
$63$	0	0
$64$	7.58383	0.947979
$65$	6.49274	0.805325
$66$	0	0
$67$	7.34442	0.897263	0.448632	−	0.893717i	$-0.351912\pi$
$67$	7.34442	0.897263	0.448632	+	0.893717i	$0.351912\pi$
$68$	−0.468598	−0.0568259
$69$	0	0
$70$	26.2739	3.14033
$71$	3.52855	0.418762	0.209381	−	0.977834i	$-0.432855\pi$
$71$	3.52855	0.418762	0.209381	+	0.977834i	$0.432855\pi$
$72$	0	0
$73$	7.21290	0.844206	0.422103	−	0.906548i	$-0.361292\pi$
$73$	7.21290	0.844206	0.422103	+	0.906548i	$0.361292\pi$
$74$	−0.684157	−0.0795316
$75$	0	0
$76$	0.0168813	0.00193642
$77$	5.22882	0.595879
$78$	0	0
$79$	−13.0021	−1.46285	−0.731425	−	0.681922i	$-0.761144\pi$
$79$	−13.0021	−1.46285	−0.731425	+	0.681922i	$0.761144\pi$
$80$	14.5152	1.62285
$81$	0	0
$82$	3.35959	0.371005
$83$	−1.86629	−0.204852	−0.102426	−	0.994741i	$-0.532661\pi$
$83$	−1.86629	−0.204852	−0.102426	+	0.994741i	$0.532661\pi$
$84$	0	0
$85$	17.0879	1.85344
$86$	0.0370216	0.00399215
$87$	0	0
$88$	2.75716	0.293915
$89$	−16.4859	−1.74750	−0.873749	−	0.486378i	$-0.838318\pi$
$89$	−16.4859	−1.74750	−0.873749	+	0.486378i	$0.838318\pi$
$90$	0	0
$91$	−9.77964	−1.02519
$92$	−0.402989	−0.0420145
$93$	0	0
$94$	−17.4780	−1.80272
$95$	−0.615593	−0.0631585
$96$	0	0
$97$	−2.08477	−0.211676	−0.105838	−	0.994383i	$-0.533753\pi$
$97$	−2.08477	−0.211676	−0.105838	+	0.994383i	$0.533753\pi$
$98$	−29.4425	−2.97414
$99$	0	0
$100$	0.671212	0.0671212
$101$	−3.41903	−0.340206	−0.170103	−	0.985426i	$-0.554410\pi$
$101$	−3.41903	−0.340206	−0.170103	+	0.985426i	$0.554410\pi$
$102$	0	0
$103$	−0.630409	−0.0621160	−0.0310580	−	0.999518i	$-0.509888\pi$
$103$	−0.630409	−0.0621160	−0.0310580	+	0.999518i	$0.509888\pi$
$104$	−5.15682	−0.505668
$105$	0	0
$106$	−4.61174	−0.447932
$107$	0.553773	0.0535352	0.0267676	−	0.999642i	$-0.491479\pi$
$107$	0.553773	0.0535352	0.0267676	+	0.999642i	$0.491479\pi$
$108$	0	0
$109$	18.0126	1.72529	0.862645	−	0.505809i	$-0.168806\pi$
$109$	18.0126	1.72529	0.862645	+	0.505809i	$0.168806\pi$
$110$	5.02482	0.479098
$111$	0	0
$112$	−21.8634	−2.06590
$113$	0.751734	0.0707172	0.0353586	−	0.999375i	$-0.488743\pi$
$113$	0.751734	0.0707172	0.0353586	+	0.999375i	$0.488743\pi$
$114$	0	0
$115$	14.6954	1.37035
$116$	−0.497622	−0.0462031
$117$	0	0
$118$	7.45086	0.685907
$119$	−25.7386	−2.35945
$120$	0	0
$121$	1.00000	0.0909091
$122$	−2.13753	−0.193523
$123$	0	0
$124$	−0.753936	−0.0677054
$125$	−7.11927	−0.636767
$126$	0	0
$127$	−6.77638	−0.601307	−0.300653	−	0.953734i	$-0.597205\pi$
$127$	−6.77638	−0.601307	−0.300653	+	0.953734i	$0.597205\pi$
$128$	−12.0536	−1.06539
$129$	0	0
$130$	−9.39810	−0.824268
$131$	16.0918	1.40594	0.702972	−	0.711218i	$-0.251856\pi$
$131$	16.0918	1.40594	0.702972	+	0.711218i	$0.251856\pi$
$132$	0	0
$133$	0.927233	0.0804013
$134$	−10.6309	−0.918369
$135$	0	0
$136$	−13.5720	−1.16379
$137$	−9.00281	−0.769162	−0.384581	−	0.923091i	$-0.625654\pi$
$137$	−9.00281	−0.769162	−0.384581	+	0.923091i	$0.625654\pi$
$138$	0	0
$139$	−0.731921	−0.0620807	−0.0310403	−	0.999518i	$-0.509882\pi$
$139$	−0.731921	−0.0620807	−0.0310403	+	0.999518i	$0.509882\pi$
$140$	−1.72795	−0.146039
$141$	0	0
$142$	−5.10750	−0.428612
$143$	−1.87034	−0.156405
$144$	0	0
$145$	18.1463	1.50697
$146$	−10.4405	−0.864064
$147$	0	0
$148$	0.0449949	0.00369856
$149$	8.92563	0.731216	0.365608	−	0.930769i	$-0.380861\pi$
$149$	8.92563	0.731216	0.365608	+	0.930769i	$0.380861\pi$
$150$	0	0
$151$	−3.72104	−0.302814	−0.151407	−	0.988472i	$-0.548380\pi$
$151$	−3.72104	−0.302814	−0.151407	+	0.988472i	$0.548380\pi$
$152$	0.488931	0.0396576
$153$	0	0
$154$	−7.56861	−0.609896
$155$	27.4930	2.20829
$156$	0	0
$157$	2.03258	0.162217	0.0811086	−	0.996705i	$-0.474154\pi$
$157$	2.03258	0.162217	0.0811086	+	0.996705i	$0.474154\pi$
$158$	18.8203	1.49726
$159$	0	0
$160$	−1.86784	−0.147666
$161$	−22.1349	−1.74447
$162$	0	0
$163$	−0.831494	−0.0651277	−0.0325638	−	0.999470i	$-0.510367\pi$
$163$	−0.831494	−0.0651277	−0.0325638	+	0.999470i	$0.510367\pi$
$164$	−0.220950	−0.0172533
$165$	0	0
$166$	2.70142	0.209671
$167$	−0.665771	−0.0515189	−0.0257595	−	0.999668i	$-0.508200\pi$
$167$	−0.665771	−0.0515189	−0.0257595	+	0.999668i	$0.508200\pi$
$168$	0	0
$169$	−9.50185	−0.730911
$170$	−24.7344	−1.89704
$171$	0	0
$172$	−0.00243480	−0.000185652 0
$173$	5.56970	0.423457	0.211728	−	0.977329i	$-0.432091\pi$
$173$	5.56970	0.423457	0.211728	+	0.977329i	$0.432091\pi$
$174$	0	0
$175$	36.8675	2.78692
$176$	−4.18133	−0.315180
$177$	0	0
$178$	23.8629	1.78860
$179$	−11.0531	−0.826146	−0.413073	−	0.910698i	$-0.635545\pi$
$179$	−11.0531	−0.826146	−0.413073	+	0.910698i	$0.635545\pi$
$180$	0	0
$181$	2.87100	0.213400	0.106700	−	0.994291i	$-0.465972\pi$
$181$	2.87100	0.213400	0.106700	+	0.994291i	$0.465972\pi$
$182$	14.1558	1.04930
$183$	0	0
$184$	−11.6717	−0.860452
$185$	−1.64078	−0.120633
$186$	0	0
$187$	−4.92244	−0.359965
$188$	1.14947	0.0838339
$189$	0	0
$190$	0.891058	0.0646441
$191$	2.95682	0.213948	0.106974	−	0.994262i	$-0.465884\pi$
$191$	2.95682	0.213948	0.106974	+	0.994262i	$0.465884\pi$
$192$	0	0
$193$	−25.4932	−1.83504	−0.917520	−	0.397690i	$-0.869812\pi$
$193$	−25.4932	−1.83504	−0.917520	+	0.397690i	$0.869812\pi$
$194$	3.01766	0.216656
$195$	0	0
$196$	1.93635	0.138310
$197$	18.3614	1.30819	0.654097	−	0.756410i	$-0.273049\pi$
$197$	18.3614	1.30819	0.654097	+	0.756410i	$0.273049\pi$
$198$	0	0
$199$	−10.1260	−0.717816	−0.358908	−	0.933373i	$-0.616851\pi$
$199$	−10.1260	−0.717816	−0.358908	+	0.933373i	$0.616851\pi$
$200$	19.4403	1.37463
$201$	0	0
$202$	4.94898	0.348209
$203$	−27.3327	−1.91838
$204$	0	0
$205$	8.05718	0.562738
$206$	0.912503	0.0635771
$207$	0	0
$208$	7.82049	0.542253
$209$	0.177331	0.0122663
$210$	0	0
$211$	8.78192	0.604572	0.302286	−	0.953217i	$-0.402250\pi$
$211$	8.78192	0.604572	0.302286	+	0.953217i	$0.402250\pi$
$212$	0.303300	0.0208307
$213$	0	0
$214$	−0.801575	−0.0547945
$215$	0.0887875	0.00605525
$216$	0	0
$217$	−41.4112	−2.81118
$218$	−26.0728	−1.76587
$219$	0	0
$220$	−0.330467	−0.0222801
$221$	9.20662	0.619304
$222$	0	0
$223$	13.9213	0.932241	0.466120	−	0.884721i	$-0.345651\pi$
$223$	13.9213	0.932241	0.466120	+	0.884721i	$0.345651\pi$
$224$	2.81342	0.187980
$225$	0	0
$226$	−1.08812	−0.0723806
$227$	4.73352	0.314175	0.157087	−	0.987585i	$-0.449790\pi$
$227$	4.73352	0.314175	0.157087	+	0.987585i	$0.449790\pi$
$228$	0	0
$229$	6.41867	0.424157	0.212079	−	0.977253i	$-0.431977\pi$
$229$	6.41867	0.424157	0.212079	+	0.977253i	$0.431977\pi$
$230$	−21.2713	−1.40259
$231$	0	0
$232$	−14.4126	−0.946233
$233$	20.2212	1.32474	0.662369	−	0.749178i	$-0.269551\pi$
$233$	20.2212	1.32474	0.662369	+	0.749178i	$0.269551\pi$
$234$	0	0
$235$	−41.9167	−2.73434
$236$	−0.490020	−0.0318976
$237$	0	0
$238$	37.2560	2.41495
$239$	10.0688	0.651299	0.325650	−	0.945491i	$-0.394417\pi$
$239$	10.0688	0.651299	0.325650	+	0.945491i	$0.394417\pi$
$240$	0	0
$241$	17.7253	1.14179	0.570894	−	0.821024i	$-0.306597\pi$
$241$	17.7253	1.14179	0.570894	+	0.821024i	$0.306597\pi$
$242$	−1.44748	−0.0930475
$243$	0	0
$244$	0.140579	0.00899965
$245$	−70.6108	−4.51116
$246$	0	0
$247$	−0.331669	−0.0211036
$248$	−21.8362	−1.38660
$249$	0	0
$250$	10.3050	0.651746
$251$	4.79564	0.302698	0.151349	−	0.988480i	$-0.451638\pi$
$251$	4.79564	0.302698	0.151349	+	0.988480i	$0.451638\pi$
$252$	0	0
$253$	−4.23324	−0.266142
$254$	9.80867	0.615451
$255$	0	0
$256$	2.27962	0.142476
$257$	24.1910	1.50899	0.754495	−	0.656305i	$-0.227882\pi$
$257$	24.1910	1.50899	0.754495	+	0.656305i	$0.227882\pi$
$258$	0	0
$259$	2.47142	0.153567
$260$	0.618084	0.0383320
$261$	0	0
$262$	−23.2925	−1.43901
$263$	−14.8673	−0.916756	−0.458378	−	0.888757i	$-0.651569\pi$
$263$	−14.8673	−0.916756	−0.458378	+	0.888757i	$0.651569\pi$
$264$	0	0
$265$	−11.0601	−0.679419
$266$	−1.34215	−0.0822926
$267$	0	0
$268$	0.699161	0.0427081
$269$	−17.4439	−1.06358	−0.531788	−	0.846878i	$-0.678480\pi$
$269$	−17.4439	−1.06358	−0.531788	+	0.846878i	$0.678480\pi$
$270$	0	0
$271$	−12.4981	−0.759208	−0.379604	−	0.925149i	$-0.623940\pi$
$271$	−12.4981	−0.759208	−0.379604	+	0.925149i	$0.623940\pi$
$272$	20.5824	1.24799
$273$	0	0
$274$	13.0314	0.787255
$275$	7.05082	0.425180
$276$	0	0
$277$	−14.7081	−0.883724	−0.441862	−	0.897083i	$-0.645682\pi$
$277$	−14.7081	−0.883724	−0.441862	+	0.897083i	$0.645682\pi$
$278$	1.05944	0.0635410
$279$	0	0
$280$	−50.0466	−2.99086
$281$	8.46547	0.505008	0.252504	−	0.967596i	$-0.418746\pi$
$281$	8.46547	0.505008	0.252504	+	0.967596i	$0.418746\pi$
$282$	0	0
$283$	−25.0537	−1.48929	−0.744645	−	0.667460i	$-0.767381\pi$
$283$	−25.0537	−1.48929	−0.744645	+	0.667460i	$0.767381\pi$
$284$	0.335905	0.0199323
$285$	0	0
$286$	2.70727	0.160084
$287$	−12.1361	−0.716370
$288$	0	0
$289$	7.23045	0.425321
$290$	−26.2664	−1.54242
$291$	0	0
$292$	0.686641	0.0401826
$293$	6.86403	0.401001	0.200500	−	0.979694i	$-0.435743\pi$
$293$	6.86403	0.401001	0.200500	+	0.979694i	$0.435743\pi$
$294$	0	0
$295$	17.8691	1.04038
$296$	1.30318	0.0757461
$297$	0	0
$298$	−12.9197	−0.748416
$299$	7.91758	0.457886
$300$	0	0
$301$	−0.133736	−0.00770839
$302$	5.38613	0.309937
$303$	0	0
$304$	−0.741481	−0.0425268
$305$	−5.12636	−0.293534
$306$	0	0
$307$	25.2812	1.44287	0.721437	−	0.692480i	$-0.243482\pi$
$307$	25.2812	1.44287	0.721437	+	0.692480i	$0.243482\pi$
$308$	0.497764	0.0283628
$309$	0	0
$310$	−39.7956	−2.26024
$311$	0.468279	0.0265537	0.0132768	−	0.999912i	$-0.495774\pi$
$311$	0.468279	0.0265537	0.0132768	+	0.999912i	$0.495774\pi$
$312$	0	0
$313$	25.6502	1.44984	0.724918	−	0.688835i	$-0.241878\pi$
$313$	25.6502	1.44984	0.724918	+	0.688835i	$0.241878\pi$
$314$	−2.94211	−0.166033
$315$	0	0
$316$	−1.23775	−0.0696290
$317$	−10.8662	−0.610306	−0.305153	−	0.952303i	$-0.598708\pi$
$317$	−10.8662	−0.610306	−0.305153	+	0.952303i	$0.598708\pi$
$318$	0	0
$319$	−5.22733	−0.292674
$320$	−26.3267	−1.47171
$321$	0	0
$322$	32.0398	1.78551
$323$	−0.872903	−0.0485696
$324$	0	0
$325$	−13.1874	−0.731505
$326$	1.20357	0.0666596
$327$	0	0
$328$	−6.39937	−0.353346
$329$	63.1368	3.48084
$330$	0	0
$331$	13.9210	0.765168	0.382584	−	0.923921i	$-0.375034\pi$
$331$	13.9210	0.765168	0.382584	+	0.923921i	$0.375034\pi$
$332$	−0.177664	−0.00975059
$333$	0	0
$334$	0.963690	0.0527308
$335$	−25.4956	−1.39297
$336$	0	0
$337$	19.4235	1.05807	0.529033	−	0.848601i	$-0.322555\pi$
$337$	19.4235	1.05807	0.529033	+	0.848601i	$0.322555\pi$
$338$	13.7537	0.748104
$339$	0	0
$340$	1.62671	0.0882205
$341$	−7.91980	−0.428881
$342$	0	0
$343$	69.7553	3.76643
$344$	−0.0705190	−0.00380213
$345$	0	0
$346$	−8.06203	−0.433417
$347$	−2.11548	−0.113565	−0.0567824	−	0.998387i	$-0.518084\pi$
$347$	−2.11548	−0.113565	−0.0567824	+	0.998387i	$0.518084\pi$
$348$	0	0
$349$	14.7761	0.790946	0.395473	−	0.918478i	$-0.370581\pi$
$349$	14.7761	0.790946	0.395473	+	0.918478i	$0.370581\pi$
$350$	−53.3649	−2.85247
$351$	0	0
$352$	0.538061	0.0286788
$353$	25.2483	1.34383	0.671916	−	0.740627i	$-0.265471\pi$
$353$	25.2483	1.34383	0.671916	+	0.740627i	$0.265471\pi$
$354$	0	0
$355$	−12.2491	−0.650115
$356$	−1.56939	−0.0831776
$357$	0	0
$358$	15.9991	0.845579
$359$	−11.0046	−0.580802	−0.290401	−	0.956905i	$-0.593789\pi$
$359$	−11.0046	−0.580802	−0.290401	+	0.956905i	$0.593789\pi$
$360$	0	0
$361$	−18.9686	−0.998345
$362$	−4.15572	−0.218420
$363$	0	0
$364$	−0.930986	−0.0487969
$365$	−25.0391	−1.31060
$366$	0	0
$367$	25.6398	1.33839	0.669193	−	0.743089i	$-0.266640\pi$
$367$	25.6398	1.33839	0.669193	+	0.743089i	$0.266640\pi$
$368$	17.7006	0.922707
$369$	0	0
$370$	2.37500	0.123470
$371$	16.6593	0.864907
$372$	0	0
$373$	−25.8933	−1.34071	−0.670353	−	0.742042i	$-0.733857\pi$
$373$	−25.8933	−1.34071	−0.670353	+	0.742042i	$0.733857\pi$
$374$	7.12513	0.368432
$375$	0	0
$376$	33.2921	1.71691
$377$	9.77685	0.503534
$378$	0	0
$379$	11.1342	0.571923	0.285962	−	0.958241i	$-0.407687\pi$
$379$	11.1342	0.571923	0.285962	+	0.958241i	$0.407687\pi$
$380$	−0.0586022	−0.00300623
$381$	0	0
$382$	−4.27994	−0.218981
$383$	13.0865	0.668687	0.334343	−	0.942451i	$-0.391485\pi$
$383$	13.0865	0.668687	0.334343	+	0.942451i	$0.391485\pi$
$384$	0	0
$385$	−18.1515	−0.925085
$386$	36.9009	1.87820
$387$	0	0
$388$	−0.198462	−0.0100754
$389$	−18.7778	−0.952071	−0.476036	−	0.879426i	$-0.657927\pi$
$389$	−18.7778	−0.952071	−0.476036	+	0.879426i	$0.657927\pi$
$390$	0	0
$391$	20.8379	1.05382
$392$	56.0822	2.83258
$393$	0	0
$394$	−26.5777	−1.33897
$395$	45.1359	2.27103
$396$	0	0
$397$	−27.6829	−1.38936	−0.694682	−	0.719317i	$-0.744455\pi$
$397$	−27.6829	−1.38936	−0.694682	+	0.719317i	$0.744455\pi$
$398$	14.6572	0.734701
$399$	0	0
$400$	−29.4818	−1.47409
$401$	−36.5005	−1.82275	−0.911373	−	0.411582i	$-0.864976\pi$
$401$	−36.5005	−1.82275	−0.911373	+	0.411582i	$0.864976\pi$
$402$	0	0
$403$	14.8127	0.737872
$404$	−0.325479	−0.0161932
$405$	0	0
$406$	39.5636	1.96351
$407$	0.472654	0.0234286
$408$	0	0
$409$	−0.685776	−0.0339094	−0.0169547	−	0.999856i	$-0.505397\pi$
$409$	−0.685776	−0.0339094	−0.0169547	+	0.999856i	$0.505397\pi$
$410$	−11.6626	−0.575974
$411$	0	0
$412$	−0.0600126	−0.00295661
$413$	−26.9152	−1.32441
$414$	0	0
$415$	6.47871	0.318027
$416$	−1.00635	−0.0493406
$417$	0	0
$418$	−0.256683	−0.0125548
$419$	26.5951	1.29926	0.649629	−	0.760252i	$-0.274924\pi$
$419$	26.5951	1.29926	0.649629	+	0.760252i	$0.274924\pi$
$420$	0	0
$421$	5.20701	0.253774	0.126887	−	0.991917i	$-0.459501\pi$
$421$	5.20701	0.253774	0.126887	+	0.991917i	$0.459501\pi$
$422$	−12.7117	−0.618794
$423$	0	0
$424$	8.78446	0.426611
$425$	−34.7073	−1.68355
$426$	0	0
$427$	7.72154	0.373672
$428$	0.0527171	0.00254818
$429$	0	0
$430$	−0.128518	−0.00619769
$431$	20.6042	0.992468	0.496234	−	0.868189i	$-0.334716\pi$
$431$	20.6042	0.992468	0.496234	+	0.868189i	$0.334716\pi$
$432$	0	0
$433$	6.87020	0.330161	0.165080	−	0.986280i	$-0.447212\pi$
$433$	6.87020	0.330161	0.165080	+	0.986280i	$0.447212\pi$
$434$	59.9418	2.87730
$435$	0	0
$436$	1.71473	0.0821207
$437$	−0.750686	−0.0359102
$438$	0	0
$439$	38.8679	1.85506	0.927532	−	0.373743i	$-0.121926\pi$
$439$	38.8679	1.85506	0.927532	+	0.373743i	$0.121926\pi$
$440$	−9.57130	−0.456294
$441$	0	0
$442$	−13.3264	−0.633872
$443$	20.2273	0.961029	0.480514	−	0.876987i	$-0.340450\pi$
$443$	20.2273	0.961029	0.480514	+	0.876987i	$0.340450\pi$
$444$	0	0
$445$	57.2295	2.71294
$446$	−20.1508	−0.954170
$447$	0	0
$448$	39.6545	1.87350
$449$	−28.9185	−1.36475	−0.682373	−	0.731004i	$-0.739052\pi$
$449$	−28.9185	−1.36475	−0.682373	+	0.731004i	$0.739052\pi$
$450$	0	0
$451$	−2.32100	−0.109291
$452$	0.0715623	0.00336601
$453$	0	0
$454$	−6.85168	−0.321565
$455$	33.9493	1.59157
$456$	0	0
$457$	−31.0078	−1.45048	−0.725241	−	0.688495i	$-0.758272\pi$
$457$	−31.0078	−1.45048	−0.725241	+	0.688495i	$0.758272\pi$
$458$	−9.29089	−0.434135
$459$	0	0
$460$	1.39895	0.0652263
$461$	28.1796	1.31245	0.656227	−	0.754563i	$-0.272151\pi$
$461$	28.1796	1.31245	0.656227	+	0.754563i	$0.272151\pi$
$462$	0	0
$463$	9.11474	0.423598	0.211799	−	0.977313i	$-0.432068\pi$
$463$	9.11474	0.423598	0.211799	+	0.977313i	$0.432068\pi$
$464$	21.8572	1.01469
$465$	0	0
$466$	−29.2698	−1.35590
$467$	33.9341	1.57028	0.785141	−	0.619317i	$-0.212590\pi$
$467$	33.9341	1.57028	0.785141	+	0.619317i	$0.212590\pi$
$468$	0	0
$469$	38.4026	1.77327
$470$	60.6735	2.79866
$471$	0	0
$472$	−14.1924	−0.653259
$473$	−0.0255766	−0.00117601
$474$	0	0
$475$	1.25033	0.0573691
$476$	−2.45022	−0.112306
$477$	0	0
$478$	−14.5744	−0.666619
$479$	−17.3178	−0.791270	−0.395635	−	0.918408i	$-0.629475\pi$
$479$	−17.3178	−0.791270	−0.395635	+	0.918408i	$0.629475\pi$
$480$	0	0
$481$	−0.884021	−0.0403079
$482$	−25.6570	−1.16864
$483$	0	0
$484$	0.0951963	0.00432711
$485$	7.23713	0.328621
$486$	0	0
$487$	20.8186	0.943380	0.471690	−	0.881765i	$-0.343644\pi$
$487$	20.8186	0.943380	0.471690	+	0.881765i	$0.343644\pi$
$488$	4.07158	0.184312
$489$	0	0
$490$	102.208	4.61727
$491$	−5.03021	−0.227010	−0.113505	−	0.993537i	$-0.536208\pi$
$491$	−5.03021	−0.227010	−0.113505	+	0.993537i	$0.536208\pi$
$492$	0	0
$493$	25.7312	1.15888
$494$	0.480084	0.0216000
$495$	0	0
$496$	33.1153	1.48692
$497$	18.4501	0.827602
$498$	0	0
$499$	−32.4697	−1.45354	−0.726772	−	0.686879i	$-0.758980\pi$
$499$	−32.4697	−1.45354	−0.726772	+	0.686879i	$0.758980\pi$
$500$	−0.677729	−0.0303089
$501$	0	0
$502$	−6.94158	−0.309818
$503$	−5.78982	−0.258155	−0.129078	−	0.991634i	$-0.541202\pi$
$503$	−5.78982	−0.258155	−0.129078	+	0.991634i	$0.541202\pi$
$504$	0	0
$505$	11.8689	0.528160
$506$	6.12753	0.272402
$507$	0	0
$508$	−0.645087	−0.0286211
$509$	25.5354	1.13184	0.565918	−	0.824462i	$-0.308522\pi$
$509$	25.5354	1.13184	0.565918	+	0.824462i	$0.308522\pi$
$510$	0	0
$511$	37.7149	1.66841
$512$	20.8074	0.919566
$513$	0	0
$514$	−35.0159	−1.54449
$515$	2.18842	0.0964333
$516$	0	0
$517$	12.0748	0.531048
$518$	−3.57733	−0.157179
$519$	0	0
$520$	17.9015	0.785034
$521$	−10.6639	−0.467196	−0.233598	−	0.972333i	$-0.575050\pi$
$521$	−10.6639	−0.467196	−0.233598	+	0.972333i	$0.575050\pi$
$522$	0	0
$523$	34.7913	1.52132	0.760660	−	0.649151i	$-0.224876\pi$
$523$	34.7913	1.52132	0.760660	+	0.649151i	$0.224876\pi$
$524$	1.53188	0.0669203
$525$	0	0
$526$	21.5201	0.938321
$527$	38.9848	1.69820
$528$	0	0
$529$	−5.07965	−0.220855
$530$	16.0093	0.695401
$531$	0	0
$532$	0.0882692	0.00382695
$533$	4.34104	0.188031
$534$	0	0
$535$	−1.92238	−0.0831119
$536$	20.2498	0.874656
$537$	0	0
$538$	25.2497	1.08859
$539$	20.3406	0.876130
$540$	0	0
$541$	32.6115	1.40208	0.701039	−	0.713123i	$-0.252720\pi$
$541$	32.6115	1.40208	0.701039	+	0.713123i	$0.252720\pi$
$542$	18.0908	0.777067
$543$	0	0
$544$	−2.64858	−0.113557
$545$	−62.5293	−2.67846
$546$	0	0
$547$	37.2827	1.59409	0.797046	−	0.603919i	$-0.206395\pi$
$547$	37.2827	1.59409	0.797046	+	0.603919i	$0.206395\pi$
$548$	−0.857035	−0.0366107
$549$	0	0
$550$	−10.2059	−0.435182
$551$	−0.926969	−0.0394902
$552$	0	0
$553$	−67.9856	−2.89104
$554$	21.2897	0.904511
$555$	0	0
$556$	−0.0696761	−0.00295493
$557$	7.25677	0.307479	0.153740	−	0.988111i	$-0.450868\pi$
$557$	7.25677	0.307479	0.153740	+	0.988111i	$0.450868\pi$
$558$	0	0
$559$	0.0478369	0.00202328
$560$	75.8973	3.20725
$561$	0	0
$562$	−12.2536	−0.516887
$563$	31.2063	1.31519	0.657594	−	0.753372i	$-0.271574\pi$
$563$	31.2063	1.31519	0.657594	+	0.753372i	$0.271574\pi$
$564$	0	0
$565$	−2.60959	−0.109786
$566$	36.2648	1.52432
$567$	0	0
$568$	9.72879	0.408211
$569$	16.5422	0.693485	0.346742	−	0.937960i	$-0.387288\pi$
$569$	16.5422	0.693485	0.346742	+	0.937960i	$0.387288\pi$
$570$	0	0
$571$	−19.1322	−0.800659	−0.400329	−	0.916371i	$-0.631104\pi$
$571$	−19.1322	−0.800659	−0.400329	+	0.916371i	$0.631104\pi$
$572$	−0.178049	−0.00744460
$573$	0	0
$574$	17.5667	0.733220
$575$	−29.8478	−1.24474
$576$	0	0
$577$	26.2272	1.09185	0.545926	−	0.837833i	$-0.316178\pi$
$577$	26.2272	1.09185	0.545926	+	0.837833i	$0.316178\pi$
$578$	−10.4659	−0.435325
$579$	0	0
$580$	1.72746	0.0717289
$581$	−9.75851	−0.404851
$582$	0	0
$583$	3.18605	0.131953
$584$	19.8871	0.822936
$585$	0	0
$586$	−9.93553	−0.410433
$587$	−45.2777	−1.86881	−0.934406	−	0.356209i	$-0.884069\pi$
$587$	−45.2777	−1.86881	−0.934406	+	0.356209i	$0.884069\pi$
$588$	0	0
$589$	−1.40443	−0.0578684
$590$	−25.8651	−1.06485
$591$	0	0
$592$	−1.97632	−0.0812264
$593$	−0.859721	−0.0353045	−0.0176523	−	0.999844i	$-0.505619\pi$
$593$	−0.859721	−0.0353045	−0.0176523	+	0.999844i	$0.505619\pi$
$594$	0	0
$595$	89.3496	3.66298
$596$	0.849687	0.0348045
$597$	0	0
$598$	−11.4605	−0.468656
$599$	−2.28191	−0.0932364	−0.0466182	−	0.998913i	$-0.514844\pi$
$599$	−2.28191	−0.0932364	−0.0466182	+	0.998913i	$0.514844\pi$
$600$	0	0
$601$	−22.2403	−0.907199	−0.453600	−	0.891206i	$-0.649860\pi$
$601$	−22.2403	−0.907199	−0.453600	+	0.891206i	$0.649860\pi$
$602$	0.193579	0.00788971
$603$	0	0
$604$	−0.354229	−0.0144134
$605$	−3.47143	−0.141134
$606$	0	0
$607$	11.9010	0.483046	0.241523	−	0.970395i	$-0.422353\pi$
$607$	11.9010	0.483046	0.241523	+	0.970395i	$0.422353\pi$
$608$	0.0954151	0.00386959
$609$	0	0
$610$	7.42030	0.300439
$611$	−22.5839	−0.913645
$612$	0	0
$613$	−19.1619	−0.773940	−0.386970	−	0.922092i	$-0.626478\pi$
$613$	−19.1619	−0.773940	−0.386970	+	0.922092i	$0.626478\pi$
$614$	−36.5940	−1.47681
$615$	0	0
$616$	14.4167	0.580866
$617$	15.4433	0.621722	0.310861	−	0.950455i	$-0.399383\pi$
$617$	15.4433	0.621722	0.310861	+	0.950455i	$0.399383\pi$
$618$	0	0
$619$	−17.0188	−0.684044	−0.342022	−	0.939692i	$-0.611112\pi$
$619$	−17.0188	−0.684044	−0.342022	+	0.939692i	$0.611112\pi$
$620$	2.61723	0.105111
$621$	0	0
$622$	−0.677824	−0.0271783
$623$	−86.2016	−3.45359
$624$	0	0
$625$	−10.5400	−0.421602
$626$	−37.1282	−1.48394
$627$	0	0
$628$	0.193494	0.00772124
$629$	−2.32661	−0.0927681
$630$	0	0
$631$	−22.7984	−0.907591	−0.453795	−	0.891106i	$-0.649930\pi$
$631$	−22.7984	−0.907591	−0.453795	+	0.891106i	$0.649930\pi$
$632$	−35.8489	−1.42599
$633$	0	0
$634$	15.7286	0.624662
$635$	23.5237	0.933511
$636$	0	0
$637$	−38.0436	−1.50734
$638$	7.56645	0.299559
$639$	0	0
$640$	41.8431	1.65399
$641$	−25.8431	−1.02074	−0.510371	−	0.859954i	$-0.670492\pi$
$641$	−25.8431	−1.02074	−0.510371	+	0.859954i	$0.670492\pi$
$642$	0	0
$643$	−33.5085	−1.32145	−0.660724	−	0.750629i	$-0.729751\pi$
$643$	−33.5085	−1.32145	−0.660724	+	0.750629i	$0.729751\pi$
$644$	−2.10716	−0.0830336
$645$	0	0
$646$	1.26351	0.0497121
$647$	31.9807	1.25729	0.628646	−	0.777691i	$-0.283609\pi$
$647$	31.9807	1.25729	0.628646	+	0.777691i	$0.283609\pi$
$648$	0	0
$649$	−5.14747	−0.202056
$650$	19.0885	0.748712
$651$	0	0
$652$	−0.0791552	−0.00309996
$653$	−0.816043	−0.0319342	−0.0159671	−	0.999873i	$-0.505083\pi$
$653$	−0.816043	−0.0319342	−0.0159671	+	0.999873i	$0.505083\pi$
$654$	0	0
$655$	−55.8614	−2.18268
$656$	9.70485	0.378911
$657$	0	0
$658$	−91.3892	−3.56272
$659$	5.71681	0.222695	0.111348	−	0.993782i	$-0.464483\pi$
$659$	5.71681	0.222695	0.111348	+	0.993782i	$0.464483\pi$
$660$	0	0
$661$	0.0828712	0.00322332	0.00161166	−	0.999999i	$-0.499487\pi$
$661$	0.0828712	0.00322332	0.00161166	+	0.999999i	$0.499487\pi$
$662$	−20.1504	−0.783166
$663$	0	0
$664$	−5.14568	−0.199691
$665$	−3.21882	−0.124821
$666$	0	0
$667$	22.1285	0.856821
$668$	−0.0633790	−0.00245221
$669$	0	0
$670$	36.9044	1.42574
$671$	1.47673	0.0570085
$672$	0	0
$673$	−12.6872	−0.489055	−0.244528	−	0.969642i	$-0.578633\pi$
$673$	−12.6872	−0.489055	−0.244528	+	0.969642i	$0.578633\pi$
$674$	−28.1151	−1.08295
$675$	0	0
$676$	−0.904541	−0.0347900
$677$	11.3222	0.435146	0.217573	−	0.976044i	$-0.430186\pi$
$677$	11.3222	0.435146	0.217573	+	0.976044i	$0.430186\pi$
$678$	0	0
$679$	−10.9009	−0.418338
$680$	47.1142	1.80675
$681$	0	0
$682$	11.4637	0.438969
$683$	24.3669	0.932373	0.466187	−	0.884686i	$-0.345627\pi$
$683$	24.3669	0.932373	0.466187	+	0.884686i	$0.345627\pi$
$684$	0	0
$685$	31.2526	1.19410
$686$	−100.969	−3.85503
$687$	0	0
$688$	0.106944	0.00407721
$689$	−5.95898	−0.227019
$690$	0	0
$691$	−31.3274	−1.19175	−0.595875	−	0.803077i	$-0.703194\pi$
$691$	−31.3274	−1.19175	−0.595875	+	0.803077i	$0.703194\pi$
$692$	0.530215	0.0201557
$693$	0	0
$694$	3.06211	0.116236
$695$	2.54081	0.0963784
$696$	0	0
$697$	11.4250	0.432752
$698$	−21.3881	−0.809551
$699$	0	0
$700$	3.50965	0.132652
$701$	−19.0621	−0.719966	−0.359983	−	0.932959i	$-0.617217\pi$
$701$	−19.0621	−0.719966	−0.359983	+	0.932959i	$0.617217\pi$
$702$	0	0
$703$	0.0838163	0.00316119
$704$	7.58383	0.285826
$705$	0	0
$706$	−36.5464	−1.37544
$707$	−17.8775	−0.672352
$708$	0	0
$709$	−27.5237	−1.03367	−0.516837	−	0.856084i	$-0.672891\pi$
$709$	−27.5237	−1.03367	−0.516837	+	0.856084i	$0.672891\pi$
$710$	17.7303	0.665408
$711$	0	0
$712$	−45.4542	−1.70347
$713$	33.5264	1.25557
$714$	0	0
$715$	6.49274	0.242815
$716$	−1.05221	−0.0393230
$717$	0	0
$718$	15.9290	0.594464
$719$	34.9237	1.30243	0.651217	−	0.758892i	$-0.274259\pi$
$719$	34.9237	1.30243	0.651217	+	0.758892i	$0.274259\pi$
$720$	0	0
$721$	−3.29629	−0.122760
$722$	27.4566	1.02183
$723$	0	0
$724$	0.273309	0.0101574
$725$	−36.8569	−1.36883
$726$	0	0
$727$	43.6152	1.61760	0.808798	−	0.588086i	$-0.200118\pi$
$727$	43.6152	1.61760	0.808798	+	0.588086i	$0.200118\pi$
$728$	−26.9641	−0.999355
$729$	0	0
$730$	36.2435	1.34143
$731$	0.125900	0.00465656
$732$	0	0
$733$	44.9091	1.65876	0.829378	−	0.558688i	$-0.188695\pi$
$733$	44.9091	1.65876	0.829378	+	0.558688i	$0.188695\pi$
$734$	−37.1131	−1.36987
$735$	0	0
$736$	−2.27774	−0.0839588
$737$	7.34442	0.270535
$738$	0	0
$739$	−16.8006	−0.618022	−0.309011	−	0.951059i	$-0.599998\pi$
$739$	−16.8006	−0.618022	−0.309011	+	0.951059i	$0.599998\pi$
$740$	−0.156197	−0.00574190
$741$	0	0
$742$	−24.1140	−0.885251
$743$	−8.19911	−0.300796	−0.150398	−	0.988626i	$-0.548056\pi$
$743$	−8.19911	−0.300796	−0.150398	+	0.988626i	$0.548056\pi$
$744$	0	0
$745$	−30.9847	−1.13519
$746$	37.4801	1.37224
$747$	0	0
$748$	−0.468598	−0.0171337
$749$	2.89558	0.105802
$750$	0	0
$751$	−16.8701	−0.615600	−0.307800	−	0.951451i	$-0.599593\pi$
$751$	−16.8701	−0.615600	−0.307800	+	0.951451i	$0.599593\pi$
$752$	−50.4886	−1.84113
$753$	0	0
$754$	−14.1518	−0.515378
$755$	12.9173	0.470109
$756$	0	0
$757$	5.28903	0.192233	0.0961166	−	0.995370i	$-0.469358\pi$
$757$	5.28903	0.192233	0.0961166	+	0.995370i	$0.469358\pi$
$758$	−16.1165	−0.585376
$759$	0	0
$760$	−1.69729	−0.0615672
$761$	−9.83518	−0.356525	−0.178262	−	0.983983i	$-0.557048\pi$
$761$	−9.83518	−0.356525	−0.178262	+	0.983983i	$0.557048\pi$
$762$	0	0
$763$	94.1844	3.40971
$764$	0.281478	0.0101835
$765$	0	0
$766$	−18.9424	−0.684416
$767$	9.62750	0.347629
$768$	0	0
$769$	5.52605	0.199275	0.0996373	−	0.995024i	$-0.468232\pi$
$769$	5.52605	0.199275	0.0996373	+	0.995024i	$0.468232\pi$
$770$	26.2739	0.946845
$771$	0	0
$772$	−2.42686	−0.0873445
$773$	19.1244	0.687856	0.343928	−	0.938996i	$-0.388242\pi$
$773$	19.1244	0.687856	0.343928	+	0.938996i	$0.388242\pi$
$774$	0	0
$775$	−55.8411	−2.00587
$776$	−5.74805	−0.206343
$777$	0	0
$778$	27.1804	0.974466
$779$	−0.411585	−0.0147466
$780$	0	0
$781$	3.52855	0.126261
$782$	−30.1624	−1.07861
$783$	0	0
$784$	−85.0506	−3.03752
$785$	−7.05594	−0.251837
$786$	0	0
$787$	−7.23739	−0.257985	−0.128993	−	0.991646i	$-0.541174\pi$
$787$	−7.23739	−0.257985	−0.128993	+	0.991646i	$0.541174\pi$
$788$	1.74794	0.0622677
$789$	0	0
$790$	−65.3332	−2.32445
$791$	3.93068	0.139759
$792$	0	0
$793$	−2.76198	−0.0980807
$794$	40.0704	1.42204
$795$	0	0
$796$	−0.963962	−0.0341667
$797$	8.00195	0.283443	0.141722	−	0.989907i	$-0.454736\pi$
$797$	8.00195	0.283443	0.141722	+	0.989907i	$0.454736\pi$
$798$	0	0
$799$	−59.4374	−2.10274
$800$	3.79377	0.134130
$801$	0	0
$802$	52.8337	1.86562
$803$	7.21290	0.254538
$804$	0	0
$805$	76.8396	2.70824
$806$	−21.4410	−0.755228
$807$	0	0
$808$	−9.42683	−0.331635
$809$	−4.15831	−0.146198	−0.0730992	−	0.997325i	$-0.523289\pi$
$809$	−4.15831	−0.146198	−0.0730992	+	0.997325i	$0.523289\pi$
$810$	0	0
$811$	52.4755	1.84266	0.921332	−	0.388776i	$-0.127102\pi$
$811$	52.4755	1.84266	0.921332	+	0.388776i	$0.127102\pi$
$812$	−2.60198	−0.0913115
$813$	0	0
$814$	−0.684157	−0.0239797
$815$	2.88647	0.101109
$816$	0	0
$817$	−0.00453554	−0.000158678 0
$818$	0.992646	0.0347071
$819$	0	0
$820$	0.767013	0.0267853
$821$	−42.9013	−1.49727	−0.748633	−	0.662985i	$-0.769289\pi$
$821$	−42.9013	−1.49727	−0.748633	+	0.662985i	$0.769289\pi$
$822$	0	0
$823$	−31.5883	−1.10110	−0.550549	−	0.834803i	$-0.685582\pi$
$823$	−31.5883	−1.10110	−0.550549	+	0.834803i	$0.685582\pi$
$824$	−1.73814	−0.0605510
$825$	0	0
$826$	38.9592	1.35556
$827$	46.7686	1.62631	0.813153	−	0.582050i	$-0.197749\pi$
$827$	46.7686	1.62631	0.813153	+	0.582050i	$0.197749\pi$
$828$	0	0
$829$	48.9146	1.69888	0.849438	−	0.527689i	$-0.176941\pi$
$829$	48.9146	1.69888	0.849438	+	0.527689i	$0.176941\pi$
$830$	−9.37779	−0.325508
$831$	0	0
$832$	−14.1843	−0.491752
$833$	−100.125	−3.46913
$834$	0	0
$835$	2.31118	0.0799816
$836$	0.0168813	0.000583851 0
$837$	0	0
$838$	−38.4959	−1.32982
$839$	13.8116	0.476830	0.238415	−	0.971163i	$-0.423372\pi$
$839$	13.8116	0.476830	0.238415	+	0.971163i	$0.423372\pi$
$840$	0	0
$841$	−1.67506	−0.0577606
$842$	−7.53703	−0.259743
$843$	0	0
$844$	0.836007	0.0287765
$845$	32.9850	1.13472
$846$	0	0
$847$	5.22882	0.179664
$848$	−13.3219	−0.457477
$849$	0	0
$850$	50.2380	1.72315
$851$	−2.00086	−0.0685886
$852$	0	0
$853$	−10.5021	−0.359585	−0.179792	−	0.983705i	$-0.557543\pi$
$853$	−10.5021	−0.359585	−0.179792	+	0.983705i	$0.557543\pi$
$854$	−11.1768	−0.382461
$855$	0	0
$856$	1.52684	0.0521864
$857$	−20.2454	−0.691571	−0.345786	−	0.938314i	$-0.612388\pi$
$857$	−20.2454	−0.691571	−0.345786	+	0.938314i	$0.612388\pi$
$858$	0	0
$859$	−55.0986	−1.87994	−0.939970	−	0.341258i	$-0.889147\pi$
$859$	−55.0986	−1.87994	−0.939970	+	0.341258i	$0.889147\pi$
$860$	0.00845224	0.000288219 0
$861$	0	0
$862$	−29.8241	−1.01581
$863$	−25.0704	−0.853406	−0.426703	−	0.904392i	$-0.640325\pi$
$863$	−25.0704	−0.853406	−0.426703	+	0.904392i	$0.640325\pi$
$864$	0	0
$865$	−19.3348	−0.657404
$866$	−9.94448	−0.337927
$867$	0	0
$868$	−3.94219	−0.133807
$869$	−13.0021	−0.441066
$870$	0	0
$871$	−13.7365	−0.465444
$872$	49.6636	1.68182
$873$	0	0
$874$	1.08660	0.0367549
$875$	−37.2254	−1.25845
$876$	0	0
$877$	−5.73882	−0.193786	−0.0968932	−	0.995295i	$-0.530891\pi$
$877$	−5.73882	−0.193786	−0.0968932	+	0.995295i	$0.530891\pi$
$878$	−56.2605	−1.89870
$879$	0	0
$880$	14.5152	0.489307
$881$	−47.6576	−1.60563	−0.802813	−	0.596230i	$-0.796664\pi$
$881$	−47.6576	−1.60563	−0.802813	+	0.596230i	$0.796664\pi$
$882$	0	0
$883$	29.4769	0.991978	0.495989	−	0.868329i	$-0.334806\pi$
$883$	29.4769	0.991978	0.495989	+	0.868329i	$0.334806\pi$
$884$	0.876436	0.0294777
$885$	0	0
$886$	−29.2786	−0.983635
$887$	31.0340	1.04202	0.521011	−	0.853550i	$-0.325555\pi$
$887$	31.0340	1.04202	0.521011	+	0.853550i	$0.325555\pi$
$888$	0	0
$889$	−35.4325	−1.18837
$890$	−82.8385	−2.77675
$891$	0	0
$892$	1.32526	0.0443730
$893$	2.14123	0.0716537
$894$	0	0
$895$	38.3700	1.28257
$896$	−63.0259	−2.10555
$897$	0	0
$898$	41.8589	1.39685
$899$	41.3994	1.38075
$900$	0	0
$901$	−15.6831	−0.522481
$902$	3.35959	0.111862
$903$	0	0
$904$	2.07265	0.0689355
$905$	−9.96648	−0.331297
$906$	0	0
$907$	41.7888	1.38757	0.693787	−	0.720180i	$-0.255941\pi$
$907$	41.7888	1.38757	0.693787	+	0.720180i	$0.255941\pi$
$908$	0.450614	0.0149541
$909$	0	0
$910$	−49.1410	−1.62901
$911$	38.0078	1.25925	0.629627	−	0.776898i	$-0.283208\pi$
$911$	38.0078	1.25925	0.629627	+	0.776898i	$0.283208\pi$
$912$	0	0
$913$	−1.86629	−0.0617653
$914$	44.8831	1.48460
$915$	0	0
$916$	0.611033	0.0201891
$917$	84.1409	2.77858
$918$	0	0
$919$	34.3707	1.13378	0.566892	−	0.823792i	$-0.308146\pi$
$919$	34.3707	1.13378	0.566892	+	0.823792i	$0.308146\pi$
$920$	40.5176	1.33583
$921$	0	0
$922$	−40.7894	−1.34333
$923$	−6.59957	−0.217228
$924$	0	0
$925$	3.33260	0.109575
$926$	−13.1934	−0.433562
$927$	0	0
$928$	−2.81262	−0.0923288
$929$	55.4656	1.81977	0.909883	−	0.414864i	$-0.136171\pi$
$929$	55.4656	1.81977	0.909883	+	0.414864i	$0.136171\pi$
$930$	0	0
$931$	3.60702	0.118215
$932$	1.92499	0.0630551
$933$	0	0
$934$	−49.1189	−1.60722
$935$	17.0879	0.558835
$936$	0	0
$937$	−30.6977	−1.00285	−0.501425	−	0.865201i	$-0.667191\pi$
$937$	−30.6977	−1.00285	−0.501425	+	0.865201i	$0.667191\pi$
$938$	−55.5870	−1.81498
$939$	0	0
$940$	−3.99031	−0.130150
$941$	27.3870	0.892791	0.446396	−	0.894836i	$-0.352707\pi$
$941$	27.3870	0.892791	0.446396	+	0.894836i	$0.352707\pi$
$942$	0	0
$943$	9.82534	0.319957
$944$	21.5233	0.700523
$945$	0	0
$946$	0.0370216	0.00120368
$947$	54.3089	1.76480	0.882402	−	0.470497i	$-0.155925\pi$
$947$	54.3089	1.76480	0.882402	+	0.470497i	$0.155925\pi$
$948$	0	0
$949$	−13.4905	−0.437921
$950$	−1.80983	−0.0587186
$951$	0	0
$952$	−70.9655	−2.30000
$953$	−13.9550	−0.452046	−0.226023	−	0.974122i	$-0.572573\pi$
$953$	−13.9550	−0.452046	−0.226023	+	0.974122i	$0.572573\pi$
$954$	0	0
$955$	−10.2644	−0.332148
$956$	0.958516	0.0310006
$957$	0	0
$958$	25.0671	0.809882
$959$	−47.0741	−1.52010
$960$	0	0
$961$	31.7232	1.02333
$962$	1.27960	0.0412560
$963$	0	0
$964$	1.68738	0.0543470
$965$	88.4978	2.84884
$966$	0	0
$967$	3.28921	0.105774	0.0528870	−	0.998601i	$-0.483158\pi$
$967$	3.28921	0.105774	0.0528870	+	0.998601i	$0.483158\pi$
$968$	2.75716	0.0886186
$969$	0	0
$970$	−10.4756	−0.336351
$971$	37.5442	1.20485	0.602425	−	0.798175i	$-0.294201\pi$
$971$	37.5442	1.20485	0.602425	+	0.798175i	$0.294201\pi$
$972$	0	0
$973$	−3.82708	−0.122691
$974$	−30.1345	−0.965570
$975$	0	0
$976$	−6.17469	−0.197647
$977$	−39.3509	−1.25895	−0.629473	−	0.777022i	$-0.716729\pi$
$977$	−39.3509	−1.25895	−0.629473	+	0.777022i	$0.716729\pi$
$978$	0	0
$979$	−16.4859	−0.526890
$980$	−6.72189	−0.214723
$981$	0	0
$982$	7.28112	0.232350
$983$	0.570790	0.0182054	0.00910269	−	0.999959i	$-0.497102\pi$
$983$	0.570790	0.0182054	0.00910269	+	0.999959i	$0.497102\pi$
$984$	0	0
$985$	−63.7403	−2.03093
$986$	−37.2454	−1.18614
$987$	0	0
$988$	−0.0315737	−0.00100449
$989$	0.108272	0.00344285
$990$	0	0
$991$	45.0572	1.43129	0.715644	−	0.698465i	$-0.246133\pi$
$991$	45.0572	1.43129	0.715644	+	0.698465i	$0.246133\pi$
$992$	−4.26134	−0.135298
$993$	0	0
$994$	−26.7062	−0.847070
$995$	35.1518	1.11439
$996$	0	0
$997$	21.8619	0.692373	0.346186	−	0.938166i	$-0.387476\pi$
$997$	21.8619	0.692373	0.346186	+	0.938166i	$0.387476\pi$
$998$	46.9992	1.48773
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8019.2.a.l.1.16		51
3.2	odd	2		8019.2.a.k.1.36		51
27.5	odd	18		891.2.j.c.397.12		102
27.11	odd	18		891.2.j.c.496.12		102
27.16	even	9		297.2.j.c.67.6	✓	102
27.22	even	9		297.2.j.c.133.6	yes	102

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
297.2.j.c.67.6	✓	102	27.16	even	9
297.2.j.c.133.6	yes	102	27.22	even	9
891.2.j.c.397.12		102	27.5	odd	18
891.2.j.c.496.12		102	27.11	odd	18
8019.2.a.k.1.36		51	3.2	odd	2
8019.2.a.l.1.16		51	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.44748 q^{2} +0.0951963 q^{4} -3.47143 q^{5} +5.22882 q^{7} +2.75716 q^{8} +O(q^{10})\)	\(q-1.44748 q^{2} +0.0951963 q^{4} -3.47143 q^{5} +5.22882 q^{7} +2.75716 q^{8} +5.02482 q^{10} +1.00000 q^{11} -1.87034 q^{13} -7.56861 q^{14} -4.18133 q^{16} -4.92244 q^{17} +0.177331 q^{19} -0.330467 q^{20} -1.44748 q^{22} -4.23324 q^{23} +7.05082 q^{25} +2.70727 q^{26} +0.497764 q^{28} -5.22733 q^{29} -7.91980 q^{31} +0.538061 q^{32} +7.12513 q^{34} -18.1515 q^{35} +0.472654 q^{37} -0.256683 q^{38} -9.57130 q^{40} -2.32100 q^{41} -0.0255766 q^{43} +0.0951963 q^{44} +6.12753 q^{46} +12.0748 q^{47} +20.3406 q^{49} -10.2059 q^{50} -0.178049 q^{52} +3.18605 q^{53} -3.47143 q^{55} +14.4167 q^{56} +7.56645 q^{58} -5.14747 q^{59} +1.47673 q^{61} +11.4637 q^{62} +7.58383 q^{64} +6.49274 q^{65} +7.34442 q^{67} -0.468598 q^{68} +26.2739 q^{70} +3.52855 q^{71} +7.21290 q^{73} -0.684157 q^{74} +0.0168813 q^{76} +5.22882 q^{77} -13.0021 q^{79} +14.5152 q^{80} +3.35959 q^{82} -1.86629 q^{83} +17.0879 q^{85} +0.0370216 q^{86} +2.75716 q^{88} -16.4859 q^{89} -9.77964 q^{91} -0.402989 q^{92} -17.4780 q^{94} -0.615593 q^{95} -2.08477 q^{97} -29.4425 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 51 q + 60 q^{4} + 6 q^{5} + 12 q^{7}+O(q^{10}) \) 51 * q + 60 * q^4 + 6 * q^5 + 12 * q^7	\( 51 q + 60 q^{4} + 6 q^{5} + 12 q^{7} + 18 q^{10} + 51 q^{11} + 30 q^{13} + 12 q^{14} + 78 q^{16} + 30 q^{19} + 18 q^{20} + 3 q^{23} + 75 q^{25} + 9 q^{26} + 36 q^{28} + 42 q^{31} - 15 q^{32} + 42 q^{34} - 9 q^{35} + 48 q^{37} - 3 q^{38} + 54 q^{40} + 18 q^{43} + 60 q^{44} + 42 q^{46} + 30 q^{47} + 99 q^{49} - 30 q^{50} + 60 q^{52} + 18 q^{53} + 6 q^{55} + 21 q^{56} + 30 q^{58} + 24 q^{59} + 99 q^{61} + 114 q^{64} - 15 q^{65} + 39 q^{67} - 39 q^{68} + 48 q^{70} + 30 q^{71} + 69 q^{73} + 90 q^{76} + 12 q^{77} + 48 q^{79} + 42 q^{80} + 42 q^{82} - 21 q^{83} + 84 q^{85} + 24 q^{86} + 15 q^{89} + 69 q^{91} - 66 q^{92} + 66 q^{94} - 12 q^{95} + 72 q^{97} - 57 q^{98}+O(q^{100}) \) 51 * q + 60 * q^4 + 6 * q^5 + 12 * q^7 + 18 * q^10 + 51 * q^11 + 30 * q^13 + 12 * q^14 + 78 * q^16 + 30 * q^19 + 18 * q^20 + 3 * q^23 + 75 * q^25 + 9 * q^26 + 36 * q^28 + 42 * q^31 - 15 * q^32 + 42 * q^34 - 9 * q^35 + 48 * q^37 - 3 * q^38 + 54 * q^40 + 18 * q^43 + 60 * q^44 + 42 * q^46 + 30 * q^47 + 99 * q^49 - 30 * q^50 + 60 * q^52 + 18 * q^53 + 6 * q^55 + 21 * q^56 + 30 * q^58 + 24 * q^59 + 99 * q^61 + 114 * q^64 - 15 * q^65 + 39 * q^67 - 39 * q^68 + 48 * q^70 + 30 * q^71 + 69 * q^73 + 90 * q^76 + 12 * q^77 + 48 * q^79 + 42 * q^80 + 42 * q^82 - 21 * q^83 + 84 * q^85 + 24 * q^86 + 15 * q^89 + 69 * q^91 - 66 * q^92 + 66 * q^94 - 12 * q^95 + 72 * q^97 - 57 * q^98

Introduction

L-functions

Modular forms

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Database

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