LMFDB - Embedded newform 8019.2.a.i.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8019,2,Mod(1,8019)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8019, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8019.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.0320373809$
Analytic rank:	$1$
Dimension:	$48$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
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)

$f(q)$	$=$	$q-0.892531 q^{2} -1.20339 q^{4} -2.80076 q^{5} -1.37622 q^{7} +2.85912 q^{8} +O(q^{10})$	\(q-0.892531 q^{2} -1.20339 q^{4} -2.80076 q^{5} -1.37622 q^{7} +2.85912 q^{8} +2.49977 q^{10} -1.00000 q^{11} +6.00471 q^{13} +1.22831 q^{14} -0.145080 q^{16} -0.780297 q^{17} -8.40346 q^{19} +3.37040 q^{20} +0.892531 q^{22} -0.00142148 q^{23} +2.84427 q^{25} -5.35939 q^{26} +1.65612 q^{28} -7.14418 q^{29} +6.41205 q^{31} -5.58876 q^{32} +0.696439 q^{34} +3.85445 q^{35} +9.98419 q^{37} +7.50035 q^{38} -8.00773 q^{40} +0.111180 q^{41} -10.1621 q^{43} +1.20339 q^{44} +0.00126871 q^{46} +2.31635 q^{47} -5.10603 q^{49} -2.53860 q^{50} -7.22600 q^{52} +3.14045 q^{53} +2.80076 q^{55} -3.93477 q^{56} +6.37640 q^{58} +5.54194 q^{59} +2.76901 q^{61} -5.72295 q^{62} +5.27830 q^{64} -16.8178 q^{65} +7.59953 q^{67} +0.939000 q^{68} -3.44022 q^{70} +7.74768 q^{71} +8.20093 q^{73} -8.91120 q^{74} +10.1126 q^{76} +1.37622 q^{77} +3.44213 q^{79} +0.406335 q^{80} -0.0992316 q^{82} -10.5916 q^{83} +2.18543 q^{85} +9.07003 q^{86} -2.85912 q^{88} +15.1612 q^{89} -8.26377 q^{91} +0.00171059 q^{92} -2.06742 q^{94} +23.5361 q^{95} +0.717491 q^{97} +4.55729 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 6 q^{2} + 54 q^{4} - 24 q^{5} - 18 q^{8}+O(q^{10}) $ 48 * q - 6 * q^2 + 54 * q^4 - 24 * q^5 - 18 * q^8	$ 48 q - 6 q^{2} + 54 q^{4} - 24 q^{5} - 18 q^{8} - 48 q^{11} - 24 q^{14} + 66 q^{16} - 24 q^{17} - 48 q^{20} + 6 q^{22} - 12 q^{23} + 60 q^{25} - 36 q^{26} - 18 q^{28} - 60 q^{29} + 36 q^{31} - 42 q^{32} + 12 q^{34} - 24 q^{35} + 6 q^{37} - 24 q^{38} - 72 q^{41} - 12 q^{43} - 54 q^{44} - 30 q^{46} - 36 q^{47} + 60 q^{49} - 42 q^{50} - 48 q^{53} + 24 q^{55} - 72 q^{56} + 12 q^{58} - 60 q^{59} - 24 q^{61} - 36 q^{62} + 90 q^{64} - 48 q^{65} - 60 q^{68} - 30 q^{70} - 60 q^{71} - 18 q^{73} - 36 q^{74} - 42 q^{76} - 12 q^{79} - 96 q^{80} + 12 q^{82} - 36 q^{83} + 18 q^{85} - 48 q^{86} + 18 q^{88} - 96 q^{89} + 30 q^{91} - 36 q^{92} - 48 q^{94} - 48 q^{95} + 30 q^{97} - 54 q^{98}+O(q^{100}) $ 48 * q - 6 * q^2 + 54 * q^4 - 24 * q^5 - 18 * q^8 - 48 * q^11 - 24 * q^14 + 66 * q^16 - 24 * q^17 - 48 * q^20 + 6 * q^22 - 12 * q^23 + 60 * q^25 - 36 * q^26 - 18 * q^28 - 60 * q^29 + 36 * q^31 - 42 * q^32 + 12 * q^34 - 24 * q^35 + 6 * q^37 - 24 * q^38 - 72 * q^41 - 12 * q^43 - 54 * q^44 - 30 * q^46 - 36 * q^47 + 60 * q^49 - 42 * q^50 - 48 * q^53 + 24 * q^55 - 72 * q^56 + 12 * q^58 - 60 * q^59 - 24 * q^61 - 36 * q^62 + 90 * q^64 - 48 * q^65 - 60 * q^68 - 30 * q^70 - 60 * q^71 - 18 * q^73 - 36 * q^74 - 42 * q^76 - 12 * q^79 - 96 * q^80 + 12 * q^82 - 36 * q^83 + 18 * q^85 - 48 * q^86 + 18 * q^88 - 96 * q^89 + 30 * q^91 - 36 * q^92 - 48 * q^94 - 48 * q^95 + 30 * q^97 - 54 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.892531	−0.631115	−0.315557	−	0.948906i	$-0.602191\pi$
$2$	−0.892531	−0.631115	−0.315557	+	0.948906i	$0.602191\pi$
$3$	0	0
$3$	0	0
$4$	−1.20339	−0.601694
$5$	−2.80076	−1.25254	−0.626270	−	0.779607i	$-0.715419\pi$
$5$	−2.80076	−1.25254	−0.626270	+	0.779607i	$0.715419\pi$
$6$	0	0
$7$	−1.37622	−0.520160	−0.260080	−	0.965587i	$-0.583749\pi$
$7$	−1.37622	−0.520160	−0.260080	+	0.965587i	$0.583749\pi$
$8$	2.85912	1.01085
$9$	0	0
$10$	2.49977	0.790496
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	6.00471	1.66541	0.832703	−	0.553719i	$-0.186792\pi$
$13$	6.00471	1.66541	0.832703	+	0.553719i	$0.186792\pi$
$14$	1.22831	0.328281
$15$	0	0
$16$	−0.145080	−0.0362701
$17$	−0.780297	−0.189250	−0.0946249	−	0.995513i	$-0.530165\pi$
$17$	−0.780297	−0.189250	−0.0946249	+	0.995513i	$0.530165\pi$
$18$	0	0
$19$	−8.40346	−1.92788	−0.963942	−	0.266111i	$-0.914261\pi$
$19$	−8.40346	−1.92788	−0.963942	+	0.266111i	$0.914261\pi$
$20$	3.37040	0.753645
$21$	0	0
$22$	0.892531	0.190288
$23$	−0.00142148	−0.000296399 0	−0.000148200	−	1.00000i	$-0.500047\pi$
$23$	−0.00142148	−0.000296399 0	−0.000148200		1.00000i	$0.500047\pi$
$24$	0	0
$25$	2.84427	0.568854
$26$	−5.35939	−1.05106
$27$	0	0
$28$	1.65612	0.312977
$29$	−7.14418	−1.32664	−0.663320	−	0.748336i	$-0.730853\pi$
$29$	−7.14418	−1.32664	−0.663320	+	0.748336i	$0.730853\pi$
$30$	0	0
$31$	6.41205	1.15164	0.575819	−	0.817577i	$-0.304683\pi$
$31$	6.41205	1.15164	0.575819	+	0.817577i	$0.304683\pi$
$32$	−5.58876	−0.987962
$33$	0	0
$34$	0.696439	0.119438
$35$	3.85445	0.651521
$36$	0	0
$37$	9.98419	1.64139	0.820696	−	0.571366i	$-0.193586\pi$
$37$	9.98419	1.64139	0.820696	+	0.571366i	$0.193586\pi$
$38$	7.50035	1.21672
$39$	0	0
$40$	−8.00773	−1.26613
$41$	0.111180	0.0173634	0.00868170	−	0.999962i	$-0.497236\pi$
$41$	0.111180	0.0173634	0.00868170	+	0.999962i	$0.497236\pi$
$42$	0	0
$43$	−10.1621	−1.54971	−0.774857	−	0.632137i	$-0.782178\pi$
$43$	−10.1621	−1.54971	−0.774857	+	0.632137i	$0.782178\pi$
$44$	1.20339	0.181418
$45$	0	0
$46$	0.00126871	0.000187062 0
$47$	2.31635	0.337875	0.168937	−	0.985627i	$-0.445966\pi$
$47$	2.31635	0.337875	0.168937	+	0.985627i	$0.445966\pi$
$48$	0	0
$49$	−5.10603	−0.729433
$50$	−2.53860	−0.359012
$51$	0	0
$52$	−7.22600	−1.00207
$53$	3.14045	0.431374	0.215687	−	0.976463i	$-0.430801\pi$
$53$	3.14045	0.431374	0.215687	+	0.976463i	$0.430801\pi$
$54$	0	0
$55$	2.80076	0.377655
$56$	−3.93477	−0.525806
$57$	0	0
$58$	6.37640	0.837263
$59$	5.54194	0.721499	0.360750	−	0.932663i	$-0.382521\pi$
$59$	5.54194	0.721499	0.360750	+	0.932663i	$0.382521\pi$
$60$	0	0
$61$	2.76901	0.354536	0.177268	−	0.984163i	$-0.443274\pi$
$61$	2.76901	0.354536	0.177268	+	0.984163i	$0.443274\pi$
$62$	−5.72295	−0.726816
$63$	0	0
$64$	5.27830	0.659788
$65$	−16.8178	−2.08599
$66$	0	0
$67$	7.59953	0.928431	0.464215	−	0.885722i	$-0.346336\pi$
$67$	7.59953	0.928431	0.464215	+	0.885722i	$0.346336\pi$
$68$	0.939000	0.113870
$69$	0	0
$70$	−3.44022	−0.411185
$71$	7.74768	0.919481	0.459740	−	0.888053i	$-0.347942\pi$
$71$	7.74768	0.919481	0.459740	+	0.888053i	$0.347942\pi$
$72$	0	0
$73$	8.20093	0.959846	0.479923	−	0.877311i	$-0.340665\pi$
$73$	8.20093	0.959846	0.479923	+	0.877311i	$0.340665\pi$
$74$	−8.91120	−1.03591
$75$	0	0
$76$	10.1126	1.16000
$77$	1.37622	0.156834
$78$	0	0
$79$	3.44213	0.387270	0.193635	−	0.981074i	$-0.437972\pi$
$79$	3.44213	0.387270	0.193635	+	0.981074i	$0.437972\pi$
$80$	0.406335	0.0454297
$81$	0	0
$82$	−0.0992316	−0.0109583
$83$	−10.5916	−1.16258	−0.581288	−	0.813698i	$-0.697451\pi$
$83$	−10.5916	−1.16258	−0.581288	+	0.813698i	$0.697451\pi$
$84$	0	0
$85$	2.18543	0.237043
$86$	9.07003	0.978047
$87$	0	0
$88$	−2.85912	−0.304784
$89$	15.1612	1.60709	0.803543	−	0.595247i	$-0.202946\pi$
$89$	15.1612	1.60709	0.803543	+	0.595247i	$0.202946\pi$
$90$	0	0
$91$	−8.26377	−0.866279
$92$	0.00171059	0.000178342 0
$93$	0	0
$94$	−2.06742	−0.213238
$95$	23.5361	2.41475
$96$	0	0
$97$	0.717491	0.0728502	0.0364251	−	0.999336i	$-0.488403\pi$
$97$	0.717491	0.0728502	0.0364251	+	0.999336i	$0.488403\pi$
$98$	4.55729	0.460356
$99$	0	0
$100$	−3.42276	−0.342276
$101$	−13.5671	−1.34998	−0.674989	−	0.737827i	$-0.735852\pi$
$101$	−13.5671	−1.34998	−0.674989	+	0.737827i	$0.735852\pi$
$102$	0	0
$103$	12.7102	1.25237	0.626185	−	0.779674i	$-0.284615\pi$
$103$	12.7102	1.25237	0.626185	+	0.779674i	$0.284615\pi$
$104$	17.1682	1.68348
$105$	0	0
$106$	−2.80295	−0.272246
$107$	−17.3314	−1.67549	−0.837745	−	0.546062i	$-0.816126\pi$
$107$	−17.3314	−1.67549	−0.837745	+	0.546062i	$0.816126\pi$
$108$	0	0
$109$	−4.36047	−0.417657	−0.208829	−	0.977952i	$-0.566965\pi$
$109$	−4.36047	−0.417657	−0.208829	+	0.977952i	$0.566965\pi$
$110$	−2.49977	−0.238343
$111$	0	0
$112$	0.199662	0.0188663
$113$	−17.1679	−1.61502	−0.807509	−	0.589855i	$-0.799185\pi$
$113$	−17.1679	−1.61502	−0.807509	+	0.589855i	$0.799185\pi$
$114$	0	0
$115$	0.00398123	0.000371251 0
$116$	8.59722	0.798232
$117$	0	0
$118$	−4.94635	−0.455349
$119$	1.07386	0.0984402
$120$	0	0
$121$	1.00000	0.0909091
$122$	−2.47143	−0.223753
$123$	0	0
$124$	−7.71619	−0.692934
$125$	6.03769	0.540027
$126$	0	0
$127$	15.9246	1.41308	0.706539	−	0.707674i	$-0.250255\pi$
$127$	15.9246	1.41308	0.706539	+	0.707674i	$0.250255\pi$
$128$	6.46647	0.571561
$129$	0	0
$130$	15.0104	1.31650
$131$	9.86434	0.861851	0.430926	−	0.902387i	$-0.358187\pi$
$131$	9.86434	0.861851	0.430926	+	0.902387i	$0.358187\pi$
$132$	0	0
$133$	11.5650	1.00281
$134$	−6.78282	−0.585946
$135$	0	0
$136$	−2.23096	−0.191304
$137$	2.53507	0.216586	0.108293	−	0.994119i	$-0.465462\pi$
$137$	2.53507	0.216586	0.108293	+	0.994119i	$0.465462\pi$
$138$	0	0
$139$	−17.5833	−1.49140	−0.745698	−	0.666284i	$-0.767884\pi$
$139$	−17.5833	−1.49140	−0.745698	+	0.666284i	$0.767884\pi$
$140$	−4.63840	−0.392016
$141$	0	0
$142$	−6.91505	−0.580298
$143$	−6.00471	−0.502139
$144$	0	0
$145$	20.0091	1.66167
$146$	−7.31958	−0.605773
$147$	0	0
$148$	−12.0149	−0.987615
$149$	11.6413	0.953691	0.476845	−	0.878987i	$-0.341780\pi$
$149$	11.6413	0.953691	0.476845	+	0.878987i	$0.341780\pi$
$150$	0	0
$151$	5.59542	0.455349	0.227674	−	0.973737i	$-0.426888\pi$
$151$	5.59542	0.455349	0.227674	+	0.973737i	$0.426888\pi$
$152$	−24.0265	−1.94881
$153$	0	0
$154$	−1.22831	−0.0989804
$155$	−17.9586	−1.44247
$156$	0	0
$157$	13.9638	1.11443	0.557217	−	0.830367i	$-0.311869\pi$
$157$	13.9638	1.11443	0.557217	+	0.830367i	$0.311869\pi$
$158$	−3.07221	−0.244412
$159$	0	0
$160$	15.6528	1.23746
$161$	0.00195626	0.000154175 0
$162$	0	0
$163$	17.8634	1.39917	0.699585	−	0.714549i	$-0.253368\pi$
$163$	17.8634	1.39917	0.699585	+	0.714549i	$0.253368\pi$
$164$	−0.133793	−0.0104475
$165$	0	0
$166$	9.45331	0.733719
$167$	1.81119	0.140154	0.0700770	−	0.997542i	$-0.477676\pi$
$167$	1.81119	0.140154	0.0700770	+	0.997542i	$0.477676\pi$
$168$	0	0
$169$	23.0565	1.77358
$170$	−1.95056	−0.149601
$171$	0	0
$172$	12.2290	0.932453
$173$	−15.4998	−1.17843	−0.589215	−	0.807976i	$-0.700563\pi$
$173$	−15.4998	−1.17843	−0.589215	+	0.807976i	$0.700563\pi$
$174$	0	0
$175$	−3.91433	−0.295895
$176$	0.145080	0.0109358
$177$	0	0
$178$	−13.5319	−1.01426
$179$	2.40914	0.180068	0.0900339	−	0.995939i	$-0.471302\pi$
$179$	2.40914	0.180068	0.0900339	+	0.995939i	$0.471302\pi$
$180$	0	0
$181$	−1.12996	−0.0839892	−0.0419946	−	0.999118i	$-0.513371\pi$
$181$	−1.12996	−0.0839892	−0.0419946	+	0.999118i	$0.513371\pi$
$182$	7.37567	0.546721
$183$	0	0
$184$	−0.00406419	−0.000299616 0
$185$	−27.9634	−2.05591
$186$	0	0
$187$	0.780297	0.0570609
$188$	−2.78747	−0.203297
$189$	0	0
$190$	−21.0067	−1.52399
$191$	20.9686	1.51723	0.758616	−	0.651538i	$-0.225876\pi$
$191$	20.9686	1.51723	0.758616	+	0.651538i	$0.225876\pi$
$192$	0	0
$193$	−14.9764	−1.07802	−0.539012	−	0.842298i	$-0.681202\pi$
$193$	−14.9764	−1.07802	−0.539012	+	0.842298i	$0.681202\pi$
$194$	−0.640383	−0.0459768
$195$	0	0
$196$	6.14454	0.438896
$197$	−11.7333	−0.835963	−0.417982	−	0.908455i	$-0.637262\pi$
$197$	−11.7333	−0.835963	−0.417982	+	0.908455i	$0.637262\pi$
$198$	0	0
$199$	7.22675	0.512291	0.256145	−	0.966638i	$-0.417547\pi$
$199$	7.22675	0.512291	0.256145	+	0.966638i	$0.417547\pi$
$200$	8.13212	0.575028
$201$	0	0
$202$	12.1091	0.851992
$203$	9.83193	0.690066
$204$	0	0
$205$	−0.311389	−0.0217483
$206$	−11.3442	−0.790390
$207$	0	0
$208$	−0.871165	−0.0604044
$209$	8.40346	0.581279
$210$	0	0
$211$	−3.37690	−0.232475	−0.116238	−	0.993221i	$-0.537083\pi$
$211$	−3.37690	−0.232475	−0.116238	+	0.993221i	$0.537083\pi$
$212$	−3.77918	−0.259555
$213$	0	0
$214$	15.4688	1.05743
$215$	28.4618	1.94108
$216$	0	0
$217$	−8.82436	−0.599037
$218$	3.89185	0.263590
$219$	0	0
$220$	−3.37040	−0.227233
$221$	−4.68545	−0.315178
$222$	0	0
$223$	24.2545	1.62420	0.812100	−	0.583519i	$-0.198324\pi$
$223$	24.2545	1.62420	0.812100	+	0.583519i	$0.198324\pi$
$224$	7.69133	0.513899
$225$	0	0
$226$	15.3229	1.01926
$227$	−6.01188	−0.399023	−0.199511	−	0.979896i	$-0.563935\pi$
$227$	−6.01188	−0.399023	−0.199511	+	0.979896i	$0.563935\pi$
$228$	0	0
$229$	5.44847	0.360045	0.180022	−	0.983662i	$-0.442383\pi$
$229$	5.44847	0.360045	0.180022	+	0.983662i	$0.442383\pi$
$230$	−0.00355337	−0.000234302 0
$231$	0	0
$232$	−20.4261	−1.34104
$233$	−1.58978	−0.104150	−0.0520749	−	0.998643i	$-0.516583\pi$
$233$	−1.58978	−0.104150	−0.0520749	+	0.998643i	$0.516583\pi$
$234$	0	0
$235$	−6.48755	−0.423201
$236$	−6.66911	−0.434122
$237$	0	0
$238$	−0.958450	−0.0621271
$239$	21.0601	1.36226	0.681131	−	0.732161i	$-0.261488\pi$
$239$	21.0601	1.36226	0.681131	+	0.732161i	$0.261488\pi$
$240$	0	0
$241$	20.1237	1.29628	0.648142	−	0.761520i	$-0.275546\pi$
$241$	20.1237	1.29628	0.648142	+	0.761520i	$0.275546\pi$
$242$	−0.892531	−0.0573741
$243$	0	0
$244$	−3.33220	−0.213322
$245$	14.3008	0.913644
$246$	0	0
$247$	−50.4603	−3.21071
$248$	18.3328	1.16414
$249$	0	0
$250$	−5.38882	−0.340819
$251$	−14.3364	−0.904903	−0.452452	−	0.891789i	$-0.649450\pi$
$251$	−14.3364	−0.904903	−0.452452	+	0.891789i	$0.649450\pi$
$252$	0	0
$253$	0.00142148	8.93677e−5 0
$254$	−14.2132	−0.891815
$255$	0	0
$256$	−16.3281	−1.02051
$257$	−18.5744	−1.15864	−0.579318	−	0.815101i	$-0.696681\pi$
$257$	−18.5744	−1.15864	−0.579318	+	0.815101i	$0.696681\pi$
$258$	0	0
$259$	−13.7404	−0.853787
$260$	20.2383	1.25513
$261$	0	0
$262$	−8.80423	−0.543927
$263$	11.3434	0.699463	0.349731	−	0.936850i	$-0.386273\pi$
$263$	11.3434	0.699463	0.349731	+	0.936850i	$0.386273\pi$
$264$	0	0
$265$	−8.79565	−0.540312
$266$	−10.3221	−0.632888
$267$	0	0
$268$	−9.14519	−0.558631
$269$	−28.5513	−1.74081	−0.870403	−	0.492340i	$-0.836142\pi$
$269$	−28.5513	−1.74081	−0.870403	+	0.492340i	$0.836142\pi$
$270$	0	0
$271$	−0.0383753	−0.00233113	−0.00116557	−	0.999999i	$-0.500371\pi$
$271$	−0.0383753	−0.00233113	−0.00116557	+	0.999999i	$0.500371\pi$
$272$	0.113206	0.00686410
$273$	0	0
$274$	−2.26263	−0.136690
$275$	−2.84427	−0.171516
$276$	0	0
$277$	−3.00400	−0.180493	−0.0902465	−	0.995919i	$-0.528765\pi$
$277$	−3.00400	−0.180493	−0.0902465	+	0.995919i	$0.528765\pi$
$278$	15.6936	0.941242
$279$	0	0
$280$	11.0204	0.658592
$281$	−30.1801	−1.80039	−0.900197	−	0.435482i	$-0.856578\pi$
$281$	−30.1801	−1.80039	−0.900197	+	0.435482i	$0.856578\pi$
$282$	0	0
$283$	−6.49877	−0.386312	−0.193156	−	0.981168i	$-0.561872\pi$
$283$	−6.49877	−0.386312	−0.193156	+	0.981168i	$0.561872\pi$
$284$	−9.32347	−0.553246
$285$	0	0
$286$	5.35939	0.316907
$287$	−0.153008	−0.00903175
$288$	0	0
$289$	−16.3911	−0.964185
$290$	−17.8588	−1.04870
$291$	0	0
$292$	−9.86890	−0.577534
$293$	19.0961	1.11561	0.557803	−	0.829973i	$-0.311644\pi$
$293$	19.0961	1.11561	0.557803	+	0.829973i	$0.311644\pi$
$294$	0	0
$295$	−15.5217	−0.903706
$296$	28.5460	1.65920
$297$	0	0
$298$	−10.3902	−0.601888
$299$	−0.00853557	−0.000493625 0
$300$	0	0
$301$	13.9853	0.806100
$302$	−4.99409	−0.287377
$303$	0	0
$304$	1.21918	0.0699245
$305$	−7.75535	−0.444070
$306$	0	0
$307$	−20.3399	−1.16086	−0.580430	−	0.814310i	$-0.697116\pi$
$307$	−20.3399	−1.16086	−0.580430	+	0.814310i	$0.697116\pi$
$308$	−1.65612	−0.0943663
$309$	0	0
$310$	16.0286	0.910365
$311$	3.70845	0.210287	0.105144	−	0.994457i	$-0.466470\pi$
$311$	3.70845	0.210287	0.105144	+	0.994457i	$0.466470\pi$
$312$	0	0
$313$	−8.74948	−0.494550	−0.247275	−	0.968945i	$-0.579535\pi$
$313$	−8.74948	−0.494550	−0.247275	+	0.968945i	$0.579535\pi$
$314$	−12.4631	−0.703335
$315$	0	0
$316$	−4.14222	−0.233018
$317$	−28.6516	−1.60924	−0.804618	−	0.593792i	$-0.797630\pi$
$317$	−28.6516	−1.60924	−0.804618	+	0.593792i	$0.797630\pi$
$318$	0	0
$319$	7.14418	0.399997
$320$	−14.7833	−0.826410
$321$	0	0
$322$	−0.00174602	−9.73021e−5 0
$323$	6.55719	0.364852
$324$	0	0
$325$	17.0790	0.947373
$326$	−15.9436	−0.883037
$327$	0	0
$328$	0.317877	0.0175518
$329$	−3.18780	−0.175749
$330$	0	0
$331$	11.4276	0.628117	0.314059	−	0.949404i	$-0.398311\pi$
$331$	11.4276	0.628117	0.314059	+	0.949404i	$0.398311\pi$
$332$	12.7458	0.699515
$333$	0	0
$334$	−1.61654	−0.0884532
$335$	−21.2845	−1.16290
$336$	0	0
$337$	−1.24204	−0.0676581	−0.0338290	−	0.999428i	$-0.510770\pi$
$337$	−1.24204	−0.0676581	−0.0338290	+	0.999428i	$0.510770\pi$
$338$	−20.5787	−1.11933
$339$	0	0
$340$	−2.62992	−0.142627
$341$	−6.41205	−0.347232
$342$	0	0
$343$	16.6605	0.899583
$344$	−29.0548	−1.56653
$345$	0	0
$346$	13.8341	0.743725
$347$	−27.3589	−1.46870	−0.734351	−	0.678770i	$-0.762514\pi$
$347$	−27.3589	−1.46870	−0.734351	+	0.678770i	$0.762514\pi$
$348$	0	0
$349$	−18.1763	−0.972957	−0.486479	−	0.873693i	$-0.661719\pi$
$349$	−18.1763	−0.972957	−0.486479	+	0.873693i	$0.661719\pi$
$350$	3.49366	0.186744
$351$	0	0
$352$	5.58876	0.297882
$353$	−19.1167	−1.01748	−0.508740	−	0.860920i	$-0.669888\pi$
$353$	−19.1167	−1.01748	−0.508740	+	0.860920i	$0.669888\pi$
$354$	0	0
$355$	−21.6994	−1.15169
$356$	−18.2448	−0.966974
$357$	0	0
$358$	−2.15024	−0.113643
$359$	−26.1659	−1.38098	−0.690491	−	0.723341i	$-0.742605\pi$
$359$	−26.1659	−1.38098	−0.690491	+	0.723341i	$0.742605\pi$
$360$	0	0
$361$	51.6181	2.71674
$362$	1.00852	0.0530069
$363$	0	0
$364$	9.94453	0.521235
$365$	−22.9688	−1.20224
$366$	0	0
$367$	4.32109	0.225559	0.112780	−	0.993620i	$-0.464025\pi$
$367$	4.32109	0.225559	0.112780	+	0.993620i	$0.464025\pi$
$368$	0.000206229 0	1.07504e−5 0
$369$	0	0
$370$	24.9582	1.29751
$371$	−4.32193	−0.224383
$372$	0	0
$373$	−34.0363	−1.76233	−0.881167	−	0.472805i	$-0.843242\pi$
$373$	−34.0363	−1.76233	−0.881167	+	0.472805i	$0.843242\pi$
$374$	−0.696439	−0.0360120
$375$	0	0
$376$	6.62274	0.341542
$377$	−42.8987	−2.20940
$378$	0	0
$379$	−7.19795	−0.369734	−0.184867	−	0.982764i	$-0.559185\pi$
$379$	−7.19795	−0.369734	−0.184867	+	0.982764i	$0.559185\pi$
$380$	−28.3230	−1.45294
$381$	0	0
$382$	−18.7151	−0.957548
$383$	−12.3452	−0.630809	−0.315404	−	0.948957i	$-0.602140\pi$
$383$	−12.3452	−0.630809	−0.315404	+	0.948957i	$0.602140\pi$
$384$	0	0
$385$	−3.85445	−0.196441
$386$	13.3669	0.680357
$387$	0	0
$388$	−0.863420	−0.0438335
$389$	−8.27488	−0.419553	−0.209777	−	0.977749i	$-0.567274\pi$
$389$	−8.27488	−0.419553	−0.209777	+	0.977749i	$0.567274\pi$
$390$	0	0
$391$	0.00110918	5.60934e−5 0
$392$	−14.5988	−0.737350
$393$	0	0
$394$	10.4723	0.527589
$395$	−9.64059	−0.485071
$396$	0	0
$397$	7.32917	0.367841	0.183920	−	0.982941i	$-0.441121\pi$
$397$	7.32917	0.367841	0.183920	+	0.982941i	$0.441121\pi$
$398$	−6.45010	−0.323314
$399$	0	0
$400$	−0.412648	−0.0206324
$401$	−6.51948	−0.325567	−0.162784	−	0.986662i	$-0.552047\pi$
$401$	−6.51948	−0.325567	−0.162784	+	0.986662i	$0.552047\pi$
$402$	0	0
$403$	38.5025	1.91795
$404$	16.3265	0.812274
$405$	0	0
$406$	−8.77530	−0.435511
$407$	−9.98419	−0.494898
$408$	0	0
$409$	3.04727	0.150678	0.0753389	−	0.997158i	$-0.475996\pi$
$409$	3.04727	0.150678	0.0753389	+	0.997158i	$0.475996\pi$
$410$	0.277924	0.0137257
$411$	0	0
$412$	−15.2953	−0.753544
$413$	−7.62690	−0.375295
$414$	0	0
$415$	29.6645	1.45617
$416$	−33.5589	−1.64536
$417$	0	0
$418$	−7.50035	−0.366854
$419$	11.3292	0.553468	0.276734	−	0.960947i	$-0.410748\pi$
$419$	11.3292	0.553468	0.276734	+	0.960947i	$0.410748\pi$
$420$	0	0
$421$	0.530082	0.0258346	0.0129173	−	0.999917i	$-0.495888\pi$
$421$	0.530082	0.0258346	0.0129173	+	0.999917i	$0.495888\pi$
$422$	3.01399	0.146719
$423$	0	0
$424$	8.97893	0.436055
$425$	−2.21937	−0.107655
$426$	0	0
$427$	−3.81076	−0.184415
$428$	20.8564	1.00813
$429$	0	0
$430$	−25.4030	−1.22504
$431$	37.7914	1.82035	0.910173	−	0.414229i	$-0.135949\pi$
$431$	37.7914	1.82035	0.910173	+	0.414229i	$0.135949\pi$
$432$	0	0
$433$	−27.1269	−1.30363	−0.651817	−	0.758377i	$-0.725993\pi$
$433$	−27.1269	−1.30363	−0.651817	+	0.758377i	$0.725993\pi$
$434$	7.87602	0.378061
$435$	0	0
$436$	5.24734	0.251302
$437$	0.0119453	0.000571423 0
$438$	0	0
$439$	11.7215	0.559435	0.279718	−	0.960082i	$-0.409759\pi$
$439$	11.7215	0.559435	0.279718	+	0.960082i	$0.409759\pi$
$440$	8.00773	0.381753
$441$	0	0
$442$	4.18191	0.198913
$443$	−4.27600	−0.203159	−0.101580	−	0.994827i	$-0.532390\pi$
$443$	−4.27600	−0.203159	−0.101580	+	0.994827i	$0.532390\pi$
$444$	0	0
$445$	−42.4630	−2.01294
$446$	−21.6479	−1.02506
$447$	0	0
$448$	−7.26408	−0.343195
$449$	−36.5313	−1.72402	−0.862009	−	0.506893i	$-0.830794\pi$
$449$	−36.5313	−1.72402	−0.862009	+	0.506893i	$0.830794\pi$
$450$	0	0
$451$	−0.111180	−0.00523526
$452$	20.6596	0.971747
$453$	0	0
$454$	5.36579	0.251829
$455$	23.1449	1.08505
$456$	0	0
$457$	−17.6602	−0.826109	−0.413054	−	0.910706i	$-0.635538\pi$
$457$	−17.6602	−0.826109	−0.413054	+	0.910706i	$0.635538\pi$
$458$	−4.86293	−0.227230
$459$	0	0
$460$	−0.00479096	−0.000223380 0
$461$	2.37433	0.110584	0.0552918	−	0.998470i	$-0.482391\pi$
$461$	2.37433	0.110584	0.0552918	+	0.998470i	$0.482391\pi$
$462$	0	0
$463$	−19.1691	−0.890865	−0.445432	−	0.895316i	$-0.646950\pi$
$463$	−19.1691	−0.890865	−0.445432	+	0.895316i	$0.646950\pi$
$464$	1.03648	0.0481173
$465$	0	0
$466$	1.41893	0.0657305
$467$	6.93428	0.320880	0.160440	−	0.987046i	$-0.448709\pi$
$467$	6.93428	0.320880	0.160440	+	0.987046i	$0.448709\pi$
$468$	0	0
$469$	−10.4586	−0.482933
$470$	5.79034	0.267089
$471$	0	0
$472$	15.8451	0.729330
$473$	10.1621	0.467256
$474$	0	0
$475$	−23.9017	−1.09669
$476$	−1.29227	−0.0592309
$477$	0	0
$478$	−18.7968	−0.859744
$479$	−30.4688	−1.39216	−0.696078	−	0.717967i	$-0.745073\pi$
$479$	−30.4688	−1.39216	−0.696078	+	0.717967i	$0.745073\pi$
$480$	0	0
$481$	59.9522	2.73358
$482$	−17.9611	−0.818104
$483$	0	0
$484$	−1.20339	−0.0546995
$485$	−2.00952	−0.0912477
$486$	0	0
$487$	−30.6270	−1.38784	−0.693921	−	0.720051i	$-0.744118\pi$
$487$	−30.6270	−1.38784	−0.693921	+	0.720051i	$0.744118\pi$
$488$	7.91695	0.358383
$489$	0	0
$490$	−12.7639	−0.576614
$491$	6.03597	0.272399	0.136200	−	0.990681i	$-0.456511\pi$
$491$	6.03597	0.272399	0.136200	+	0.990681i	$0.456511\pi$
$492$	0	0
$493$	5.57458	0.251066
$494$	45.0374	2.02633
$495$	0	0
$496$	−0.930262	−0.0417700
$497$	−10.6625	−0.478278
$498$	0	0
$499$	15.9577	0.714363	0.357182	−	0.934035i	$-0.383738\pi$
$499$	15.9577	0.714363	0.357182	+	0.934035i	$0.383738\pi$
$500$	−7.26568	−0.324931
$501$	0	0
$502$	12.7956	0.571098
$503$	−9.49862	−0.423523	−0.211761	−	0.977321i	$-0.567920\pi$
$503$	−9.49862	−0.423523	−0.211761	+	0.977321i	$0.567920\pi$
$504$	0	0
$505$	37.9983	1.69090
$506$	−0.00126871	−5.64013e−5 0
$507$	0	0
$508$	−19.1635	−0.850241
$509$	3.17756	0.140843	0.0704215	−	0.997517i	$-0.477566\pi$
$509$	3.17756	0.140843	0.0704215	+	0.997517i	$0.477566\pi$
$510$	0	0
$511$	−11.2862	−0.499274
$512$	1.64042	0.0724972
$513$	0	0
$514$	16.5782	0.731233
$515$	−35.5982	−1.56864
$516$	0	0
$517$	−2.31635	−0.101873
$518$	12.2637	0.538837
$519$	0	0
$520$	−48.0841	−2.10863
$521$	−21.9108	−0.959928	−0.479964	−	0.877288i	$-0.659350\pi$
$521$	−21.9108	−0.959928	−0.479964	+	0.877288i	$0.659350\pi$
$522$	0	0
$523$	−3.76433	−0.164603	−0.0823014	−	0.996607i	$-0.526227\pi$
$523$	−3.76433	−0.164603	−0.0823014	+	0.996607i	$0.526227\pi$
$524$	−11.8706	−0.518571
$525$	0	0
$526$	−10.1243	−0.441441
$527$	−5.00330	−0.217947
$528$	0	0
$529$	−23.0000	−1.00000
$530$	7.85039	0.340999
$531$	0	0
$532$	−13.9171	−0.603385
$533$	0.667603	0.0289171
$534$	0	0
$535$	48.5411	2.09862
$536$	21.7280	0.938507
$537$	0	0
$538$	25.4830	1.09865
$539$	5.10603	0.219932
$540$	0	0
$541$	−9.71991	−0.417892	−0.208946	−	0.977927i	$-0.567003\pi$
$541$	−9.71991	−0.417892	−0.208946	+	0.977927i	$0.567003\pi$
$542$	0.0342511	0.00147121
$543$	0	0
$544$	4.36089	0.186972
$545$	12.2126	0.523132
$546$	0	0
$547$	23.1495	0.989799	0.494900	−	0.868950i	$-0.335205\pi$
$547$	23.1495	0.989799	0.494900	+	0.868950i	$0.335205\pi$
$548$	−3.05068	−0.130318
$549$	0	0
$550$	2.53860	0.108246
$551$	60.0358	2.55761
$552$	0	0
$553$	−4.73711	−0.201442
$554$	2.68117	0.113912
$555$	0	0
$556$	21.1595	0.897364
$557$	13.5001	0.572019	0.286009	−	0.958227i	$-0.407671\pi$
$557$	13.5001	0.572019	0.286009	+	0.958227i	$0.407671\pi$
$558$	0	0
$559$	−61.0208	−2.58090
$560$	−0.559205	−0.0236307
$561$	0	0
$562$	26.9367	1.13626
$563$	7.01823	0.295783	0.147892	−	0.989004i	$-0.452751\pi$
$563$	7.01823	0.295783	0.147892	+	0.989004i	$0.452751\pi$
$564$	0	0
$565$	48.0832	2.02287
$566$	5.80035	0.243807
$567$	0	0
$568$	22.1516	0.929460
$569$	2.85242	0.119580	0.0597899	−	0.998211i	$-0.480957\pi$
$569$	2.85242	0.119580	0.0597899	+	0.998211i	$0.480957\pi$
$570$	0	0
$571$	−8.09215	−0.338646	−0.169323	−	0.985561i	$-0.554158\pi$
$571$	−8.09215	−0.338646	−0.169323	+	0.985561i	$0.554158\pi$
$572$	7.22600	0.302134
$573$	0	0
$574$	0.136564	0.00570007
$575$	−0.00404307	−0.000168608 0
$576$	0	0
$577$	−35.7905	−1.48998	−0.744989	−	0.667076i	$-0.767545\pi$
$577$	−35.7905	−1.48998	−0.744989	+	0.667076i	$0.767545\pi$
$578$	14.6296	0.608511
$579$	0	0
$580$	−24.0788	−0.999817
$581$	14.5763	0.604726
$582$	0	0
$583$	−3.14045	−0.130064
$584$	23.4475	0.970263
$585$	0	0
$586$	−17.0439	−0.704076
$587$	−11.6244	−0.479792	−0.239896	−	0.970799i	$-0.577113\pi$
$587$	−11.6244	−0.479792	−0.239896	+	0.970799i	$0.577113\pi$
$588$	0	0
$589$	−53.8834	−2.22023
$590$	13.8536	0.570342
$591$	0	0
$592$	−1.44851	−0.0595334
$593$	19.5576	0.803134	0.401567	−	0.915830i	$-0.368466\pi$
$593$	19.5576	0.803134	0.401567	+	0.915830i	$0.368466\pi$
$594$	0	0
$595$	−3.00762	−0.123300
$596$	−14.0090	−0.573830
$597$	0	0
$598$	0.00761826	0.000311534 0
$599$	46.7336	1.90948	0.954742	−	0.297434i	$-0.0961307\pi$
$599$	46.7336	1.90948	0.954742	+	0.297434i	$0.0961307\pi$
$600$	0	0
$601$	−44.6894	−1.82292	−0.911459	−	0.411392i	$-0.865043\pi$
$601$	−44.6894	−1.82292	−0.911459	+	0.411392i	$0.865043\pi$
$602$	−12.4823	−0.508741
$603$	0	0
$604$	−6.73346	−0.273981
$605$	−2.80076	−0.113867
$606$	0	0
$607$	8.70264	0.353229	0.176615	−	0.984280i	$-0.443485\pi$
$607$	8.70264	0.353229	0.176615	+	0.984280i	$0.443485\pi$
$608$	46.9649	1.90468
$609$	0	0
$610$	6.92189	0.280259
$611$	13.9090	0.562699
$612$	0	0
$613$	3.11952	0.125996	0.0629981	−	0.998014i	$-0.479934\pi$
$613$	3.11952	0.125996	0.0629981	+	0.998014i	$0.479934\pi$
$614$	18.1540	0.732636
$615$	0	0
$616$	3.93477	0.158536
$617$	−24.3068	−0.978555	−0.489277	−	0.872128i	$-0.662739\pi$
$617$	−24.3068	−0.978555	−0.489277	+	0.872128i	$0.662739\pi$
$618$	0	0
$619$	−5.70193	−0.229180	−0.114590	−	0.993413i	$-0.536555\pi$
$619$	−5.70193	−0.229180	−0.114590	+	0.993413i	$0.536555\pi$
$620$	21.6112	0.867927
$621$	0	0
$622$	−3.30991	−0.132715
$623$	−20.8651	−0.835942
$624$	0	0
$625$	−31.1315	−1.24526
$626$	7.80918	0.312118
$627$	0	0
$628$	−16.8039	−0.670548
$629$	−7.79063	−0.310633
$630$	0	0
$631$	−8.49898	−0.338339	−0.169170	−	0.985587i	$-0.554109\pi$
$631$	−8.49898	−0.338339	−0.169170	+	0.985587i	$0.554109\pi$
$632$	9.84147	0.391473
$633$	0	0
$634$	25.5725	1.01561
$635$	−44.6010	−1.76994
$636$	0	0
$637$	−30.6602	−1.21480
$638$	−6.37640	−0.252444
$639$	0	0
$640$	−18.1110	−0.715902
$641$	22.1081	0.873218	0.436609	−	0.899651i	$-0.356179\pi$
$641$	22.1081	0.873218	0.436609	+	0.899651i	$0.356179\pi$
$642$	0	0
$643$	43.3117	1.70805	0.854024	−	0.520234i	$-0.174155\pi$
$643$	43.3117	1.70805	0.854024	+	0.520234i	$0.174155\pi$
$644$	−0.00235414	−9.27662e−5 0
$645$	0	0
$646$	−5.85249	−0.230263
$647$	−40.0358	−1.57397	−0.786986	−	0.616971i	$-0.788360\pi$
$647$	−40.0358	−1.57397	−0.786986	+	0.616971i	$0.788360\pi$
$648$	0	0
$649$	−5.54194	−0.217540
$650$	−15.2436	−0.597901
$651$	0	0
$652$	−21.4966	−0.841872
$653$	−39.7450	−1.55534	−0.777671	−	0.628672i	$-0.783599\pi$
$653$	−39.7450	−1.55534	−0.777671	+	0.628672i	$0.783599\pi$
$654$	0	0
$655$	−27.6277	−1.07950
$656$	−0.0161300	−0.000629772 0
$657$	0	0
$658$	2.84521	0.110918
$659$	−16.8219	−0.655287	−0.327643	−	0.944801i	$-0.606254\pi$
$659$	−16.8219	−0.655287	−0.327643	+	0.944801i	$0.606254\pi$
$660$	0	0
$661$	8.84274	0.343943	0.171971	−	0.985102i	$-0.444986\pi$
$661$	8.84274	0.343943	0.171971	+	0.985102i	$0.444986\pi$
$662$	−10.1995	−0.396414
$663$	0	0
$664$	−30.2826	−1.17519
$665$	−32.3907	−1.25606
$666$	0	0
$667$	0.0101553	0.000393215 0
$668$	−2.17956	−0.0843298
$669$	0	0
$670$	18.9971	0.733921
$671$	−2.76901	−0.106897
$672$	0	0
$673$	3.75521	0.144753	0.0723764	−	0.997377i	$-0.476942\pi$
$673$	3.75521	0.144753	0.0723764	+	0.997377i	$0.476942\pi$
$674$	1.10856	0.0427000
$675$	0	0
$676$	−27.7460	−1.06715
$677$	30.0479	1.15483	0.577416	−	0.816450i	$-0.304061\pi$
$677$	30.0479	1.15483	0.577416	+	0.816450i	$0.304061\pi$
$678$	0	0
$679$	−0.987422	−0.0378938
$680$	6.24840	0.239615
$681$	0	0
$682$	5.72295	0.219143
$683$	15.5319	0.594310	0.297155	−	0.954829i	$-0.403962\pi$
$683$	15.5319	0.594310	0.297155	+	0.954829i	$0.403962\pi$
$684$	0	0
$685$	−7.10013	−0.271282
$686$	−14.8700	−0.567740
$687$	0	0
$688$	1.47433	0.0562082
$689$	18.8575	0.718413
$690$	0	0
$691$	15.6215	0.594271	0.297135	−	0.954835i	$-0.403969\pi$
$691$	15.6215	0.594271	0.297135	+	0.954835i	$0.403969\pi$
$692$	18.6523	0.709055
$693$	0	0
$694$	24.4187	0.926920
$695$	49.2466	1.86803
$696$	0	0
$697$	−0.0867534	−0.00328602
$698$	16.2229	0.614048
$699$	0	0
$700$	4.71046	0.178039
$701$	−34.7312	−1.31178	−0.655890	−	0.754857i	$-0.727706\pi$
$701$	−34.7312	−1.31178	−0.655890	+	0.754857i	$0.727706\pi$
$702$	0	0
$703$	−83.9017	−3.16441
$704$	−5.27830	−0.198933
$705$	0	0
$706$	17.0622	0.642146
$707$	18.6713	0.702206
$708$	0	0
$709$	9.53665	0.358156	0.179078	−	0.983835i	$-0.442689\pi$
$709$	9.53665	0.358156	0.179078	+	0.983835i	$0.442689\pi$
$710$	19.3674	0.726846
$711$	0	0
$712$	43.3478	1.62453
$713$	−0.00911460	−0.000341344 0
$714$	0	0
$715$	16.8178	0.628949
$716$	−2.89914	−0.108346
$717$	0	0
$718$	23.3538	0.871558
$719$	39.8527	1.48625	0.743127	−	0.669151i	$-0.233342\pi$
$719$	39.8527	1.48625	0.743127	+	0.669151i	$0.233342\pi$
$720$	0	0
$721$	−17.4919	−0.651434
$722$	−46.0707	−1.71457
$723$	0	0
$724$	1.35978	0.0505358
$725$	−20.3200	−0.754665
$726$	0	0
$727$	4.76079	0.176568	0.0882840	−	0.996095i	$-0.471862\pi$
$727$	4.76079	0.176568	0.0882840	+	0.996095i	$0.471862\pi$
$728$	−23.6271	−0.875680
$729$	0	0
$730$	20.5004	0.758754
$731$	7.92949	0.293283
$732$	0	0
$733$	22.6196	0.835473	0.417736	−	0.908568i	$-0.362824\pi$
$733$	22.6196	0.835473	0.417736	+	0.908568i	$0.362824\pi$
$734$	−3.85671	−0.142354
$735$	0	0
$736$	0.00794431	0.000292831 0
$737$	−7.59953	−0.279932
$738$	0	0
$739$	−19.6502	−0.722845	−0.361422	−	0.932402i	$-0.617709\pi$
$739$	−19.6502	−0.722845	−0.361422	+	0.932402i	$0.617709\pi$
$740$	33.6508	1.23703
$741$	0	0
$742$	3.85746	0.141612
$743$	−10.7530	−0.394488	−0.197244	−	0.980354i	$-0.563199\pi$
$743$	−10.7530	−0.394488	−0.197244	+	0.980354i	$0.563199\pi$
$744$	0	0
$745$	−32.6045	−1.19453
$746$	30.3785	1.11224
$747$	0	0
$748$	−0.939000	−0.0343332
$749$	23.8517	0.871524
$750$	0	0
$751$	−5.88610	−0.214787	−0.107393	−	0.994217i	$-0.534250\pi$
$751$	−5.88610	−0.214787	−0.107393	+	0.994217i	$0.534250\pi$
$752$	−0.336057	−0.0122547
$753$	0	0
$754$	38.2884	1.39438
$755$	−15.6714	−0.570342
$756$	0	0
$757$	19.4228	0.705934	0.352967	−	0.935636i	$-0.385173\pi$
$757$	19.4228	0.705934	0.352967	+	0.935636i	$0.385173\pi$
$758$	6.42440	0.233345
$759$	0	0
$760$	67.2926	2.44096
$761$	−31.8911	−1.15605	−0.578025	−	0.816019i	$-0.696177\pi$
$761$	−31.8911	−1.15605	−0.578025	+	0.816019i	$0.696177\pi$
$762$	0	0
$763$	6.00094	0.217249
$764$	−25.2333	−0.912910
$765$	0	0
$766$	11.0184	0.398113
$767$	33.2777	1.20159
$768$	0	0
$769$	23.2712	0.839180	0.419590	−	0.907714i	$-0.362174\pi$
$769$	23.2712	0.839180	0.419590	+	0.907714i	$0.362174\pi$
$770$	3.44022	0.123977
$771$	0	0
$772$	18.0224	0.648641
$773$	15.4051	0.554084	0.277042	−	0.960858i	$-0.410646\pi$
$773$	15.4051	0.554084	0.277042	+	0.960858i	$0.410646\pi$
$774$	0	0
$775$	18.2376	0.655114
$776$	2.05140	0.0736408
$777$	0	0
$778$	7.38559	0.264786
$779$	−0.934296	−0.0334746
$780$	0	0
$781$	−7.74768	−0.277234
$782$	−0.000989974 0	−3.54014e−5 0
$783$	0	0
$784$	0.740784	0.0264566
$785$	−39.1093	−1.39587
$786$	0	0
$787$	−9.89293	−0.352645	−0.176322	−	0.984332i	$-0.556420\pi$
$787$	−9.89293	−0.352645	−0.176322	+	0.984332i	$0.556420\pi$
$788$	14.1197	0.502994
$789$	0	0
$790$	8.60452	0.306135
$791$	23.6267	0.840069
$792$	0	0
$793$	16.6271	0.590446
$794$	−6.54151	−0.232150
$795$	0	0
$796$	−8.69659	−0.308242
$797$	22.0892	0.782440	0.391220	−	0.920297i	$-0.372053\pi$
$797$	22.0892	0.782440	0.391220	+	0.920297i	$0.372053\pi$
$798$	0	0
$799$	−1.80744	−0.0639427
$800$	−15.8959	−0.562006
$801$	0	0
$802$	5.81884	0.205470
$803$	−8.20093	−0.289404
$804$	0	0
$805$	−0.00547903	−0.000193110 0
$806$	−34.3647	−1.21044
$807$	0	0
$808$	−38.7901	−1.36463
$809$	34.4037	1.20957	0.604785	−	0.796389i	$-0.293259\pi$
$809$	34.4037	1.20957	0.604785	+	0.796389i	$0.293259\pi$
$810$	0	0
$811$	−12.4360	−0.436688	−0.218344	−	0.975872i	$-0.570065\pi$
$811$	−12.4360	−0.436688	−0.218344	+	0.975872i	$0.570065\pi$
$812$	−11.8316	−0.415209
$813$	0	0
$814$	8.91120	0.312337
$815$	−50.0312	−1.75252
$816$	0	0
$817$	85.3972	2.98767
$818$	−2.71978	−0.0950950
$819$	0	0
$820$	0.374722	0.0130858
$821$	36.1453	1.26148	0.630740	−	0.775994i	$-0.282751\pi$
$821$	36.1453	1.26148	0.630740	+	0.775994i	$0.282751\pi$
$822$	0	0
$823$	0.650224	0.0226654	0.0113327	−	0.999936i	$-0.496393\pi$
$823$	0.650224	0.0226654	0.0113327	+	0.999936i	$0.496393\pi$
$824$	36.3400	1.26596
$825$	0	0
$826$	6.80725	0.236854
$827$	20.2222	0.703194	0.351597	−	0.936151i	$-0.385639\pi$
$827$	20.2222	0.703194	0.351597	+	0.936151i	$0.385639\pi$
$828$	0	0
$829$	40.8313	1.41813	0.709064	−	0.705144i	$-0.249118\pi$
$829$	40.8313	1.41813	0.709064	+	0.705144i	$0.249118\pi$
$830$	−26.4765	−0.919012
$831$	0	0
$832$	31.6947	1.09881
$833$	3.98422	0.138045
$834$	0	0
$835$	−5.07271	−0.175548
$836$	−10.1126	−0.349752
$837$	0	0
$838$	−10.1117	−0.349302
$839$	−16.0811	−0.555182	−0.277591	−	0.960699i	$-0.589536\pi$
$839$	−16.0811	−0.555182	−0.277591	+	0.960699i	$0.589536\pi$
$840$	0	0
$841$	22.0393	0.759976
$842$	−0.473114	−0.0163046
$843$	0	0
$844$	4.06372	0.139879
$845$	−64.5759	−2.22148
$846$	0	0
$847$	−1.37622	−0.0472873
$848$	−0.455617	−0.0156459
$849$	0	0
$850$	1.98086	0.0679430
$851$	−0.0141923	−0.000486507 0
$852$	0	0
$853$	−31.0010	−1.06146	−0.530728	−	0.847542i	$-0.678081\pi$
$853$	−31.0010	−1.06146	−0.530728	+	0.847542i	$0.678081\pi$
$854$	3.40122	0.116387
$855$	0	0
$856$	−49.5526	−1.69367
$857$	−6.06516	−0.207182	−0.103591	−	0.994620i	$-0.533033\pi$
$857$	−6.06516	−0.207182	−0.103591	+	0.994620i	$0.533033\pi$
$858$	0	0
$859$	−50.4213	−1.72035	−0.860176	−	0.509998i	$-0.829646\pi$
$859$	−50.4213	−1.72035	−0.860176	+	0.509998i	$0.829646\pi$
$860$	−34.2506	−1.16793
$861$	0	0
$862$	−33.7300	−1.14885
$863$	−3.92700	−0.133677	−0.0668384	−	0.997764i	$-0.521291\pi$
$863$	−3.92700	−0.133677	−0.0668384	+	0.997764i	$0.521291\pi$
$864$	0	0
$865$	43.4114	1.47603
$866$	24.2116	0.822742
$867$	0	0
$868$	10.6191	0.360437
$869$	−3.44213	−0.116766
$870$	0	0
$871$	45.6330	1.54621
$872$	−12.4671	−0.422190
$873$	0	0
$874$	−0.0106616	−0.000360634 0
$875$	−8.30915	−0.280901
$876$	0	0
$877$	−53.6348	−1.81112	−0.905559	−	0.424220i	$-0.860548\pi$
$877$	−53.6348	−1.81112	−0.905559	+	0.424220i	$0.860548\pi$
$878$	−10.4618	−0.353068
$879$	0	0
$880$	−0.406335	−0.0136976
$881$	29.7861	1.00352	0.501760	−	0.865007i	$-0.332686\pi$
$881$	29.7861	1.00352	0.501760	+	0.865007i	$0.332686\pi$
$882$	0	0
$883$	4.16244	0.140077	0.0700387	−	0.997544i	$-0.477688\pi$
$883$	4.16244	0.140077	0.0700387	+	0.997544i	$0.477688\pi$
$884$	5.63842	0.189641
$885$	0	0
$886$	3.81647	0.128217
$887$	−36.7078	−1.23253	−0.616264	−	0.787540i	$-0.711354\pi$
$887$	−36.7078	−1.23253	−0.616264	+	0.787540i	$0.711354\pi$
$888$	0	0
$889$	−21.9157	−0.735028
$890$	37.8995	1.27039
$891$	0	0
$892$	−29.1875	−0.977271
$893$	−19.4654	−0.651383
$894$	0	0
$895$	−6.74744	−0.225542
$896$	−8.89925	−0.297303
$897$	0	0
$898$	32.6053	1.08805
$899$	−45.8088	−1.52781
$900$	0	0
$901$	−2.45048	−0.0816373
$902$	0.0992316	0.00330405
$903$	0	0
$904$	−49.0851	−1.63255
$905$	3.16475	0.105200
$906$	0	0
$907$	39.7421	1.31961	0.659807	−	0.751435i	$-0.270638\pi$
$907$	39.7421	1.31961	0.659807	+	0.751435i	$0.270638\pi$
$908$	7.23463	0.240090
$909$	0	0
$910$	−20.6575	−0.684790
$911$	4.82078	0.159720	0.0798598	−	0.996806i	$-0.474553\pi$
$911$	4.82078	0.159720	0.0798598	+	0.996806i	$0.474553\pi$
$912$	0	0
$913$	10.5916	0.350530
$914$	15.7623	0.521370
$915$	0	0
$916$	−6.55662	−0.216637
$917$	−13.5755	−0.448301
$918$	0	0
$919$	25.6775	0.847023	0.423511	−	0.905891i	$-0.360797\pi$
$919$	25.6775	0.847023	0.423511	+	0.905891i	$0.360797\pi$
$920$	0.0113828	0.000375280 0
$921$	0	0
$922$	−2.11916	−0.0697909
$923$	46.5226	1.53131
$924$	0	0
$925$	28.3977	0.933712
$926$	17.1090	0.562238
$927$	0	0
$928$	39.9271	1.31067
$929$	26.6969	0.875897	0.437949	−	0.899000i	$-0.355705\pi$
$929$	26.6969	0.875897	0.437949	+	0.899000i	$0.355705\pi$
$930$	0	0
$931$	42.9083	1.40626
$932$	1.91312	0.0626663
$933$	0	0
$934$	−6.18906	−0.202512
$935$	−2.18543	−0.0714710
$936$	0	0
$937$	32.2348	1.05307	0.526533	−	0.850155i	$-0.323492\pi$
$937$	32.2348	1.05307	0.526533	+	0.850155i	$0.323492\pi$
$938$	9.33462	0.304786
$939$	0	0
$940$	7.80704	0.254638
$941$	55.5758	1.81172	0.905860	−	0.423578i	$-0.139226\pi$
$941$	55.5758	1.81172	0.905860	+	0.423578i	$0.139226\pi$
$942$	0	0
$943$	−0.000158040 0	−5.14649e−6 0
$944$	−0.804026	−0.0261688
$945$	0	0
$946$	−9.07003	−0.294892
$947$	−53.1747	−1.72795	−0.863973	−	0.503537i	$-0.832032\pi$
$947$	−53.1747	−1.72795	−0.863973	+	0.503537i	$0.832032\pi$
$948$	0	0
$949$	49.2442	1.59853
$950$	21.3330	0.692134
$951$	0	0
$952$	3.07029	0.0995086
$953$	19.2233	0.622704	0.311352	−	0.950295i	$-0.399218\pi$
$953$	19.2233	0.622704	0.311352	+	0.950295i	$0.399218\pi$
$954$	0	0
$955$	−58.7280	−1.90039
$956$	−25.3434	−0.819665
$957$	0	0
$958$	27.1944	0.878610
$959$	−3.48880	−0.112659
$960$	0	0
$961$	10.1144	0.326270
$962$	−53.5092	−1.72521
$963$	0	0
$964$	−24.2167	−0.779966
$965$	41.9453	1.35027
$966$	0	0
$967$	−39.9307	−1.28408	−0.642042	−	0.766669i	$-0.721913\pi$
$967$	−39.9307	−1.28408	−0.642042	+	0.766669i	$0.721913\pi$
$968$	2.85912	0.0918957
$969$	0	0
$970$	1.79356	0.0575878
$971$	37.0225	1.18811	0.594055	−	0.804425i	$-0.297526\pi$
$971$	37.0225	1.18811	0.594055	+	0.804425i	$0.297526\pi$
$972$	0	0
$973$	24.1984	0.775765
$974$	27.3355	0.875887
$975$	0	0
$976$	−0.401729	−0.0128590
$977$	−46.2148	−1.47854	−0.739272	−	0.673407i	$-0.764830\pi$
$977$	−46.2148	−1.47854	−0.739272	+	0.673407i	$0.764830\pi$
$978$	0	0
$979$	−15.1612	−0.484555
$980$	−17.2094	−0.549734
$981$	0	0
$982$	−5.38729	−0.171915
$983$	19.7236	0.629087	0.314543	−	0.949243i	$-0.398149\pi$
$983$	19.7236	0.629087	0.314543	+	0.949243i	$0.398149\pi$
$984$	0	0
$985$	32.8622	1.04708
$986$	−4.97548	−0.158452
$987$	0	0
$988$	60.7233	1.93187
$989$	0.0144453	0.000459334 0
$990$	0	0
$991$	−36.5126	−1.15986	−0.579930	−	0.814666i	$-0.696920\pi$
$991$	−36.5126	−1.15986	−0.579930	+	0.814666i	$0.696920\pi$
$992$	−35.8354	−1.13778
$993$	0	0
$994$	9.51659	0.301848
$995$	−20.2404	−0.641664
$996$	0	0
$997$	0.824508	0.0261124	0.0130562	−	0.999915i	$-0.495844\pi$
$997$	0.824508	0.0261124	0.0130562	+	0.999915i	$0.495844\pi$
$998$	−14.2427	−0.450845
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8019.2.a.i.1.19	✓	48
3.2	odd	2		8019.2.a.j.1.30	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8019.2.a.i.1.19	✓	48	1.1	even	1	trivial
8019.2.a.j.1.30	yes	48	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.892531 q^{2} -1.20339 q^{4} -2.80076 q^{5} -1.37622 q^{7} +2.85912 q^{8} +O(q^{10})\)	\(q-0.892531 q^{2} -1.20339 q^{4} -2.80076 q^{5} -1.37622 q^{7} +2.85912 q^{8} +2.49977 q^{10} -1.00000 q^{11} +6.00471 q^{13} +1.22831 q^{14} -0.145080 q^{16} -0.780297 q^{17} -8.40346 q^{19} +3.37040 q^{20} +0.892531 q^{22} -0.00142148 q^{23} +2.84427 q^{25} -5.35939 q^{26} +1.65612 q^{28} -7.14418 q^{29} +6.41205 q^{31} -5.58876 q^{32} +0.696439 q^{34} +3.85445 q^{35} +9.98419 q^{37} +7.50035 q^{38} -8.00773 q^{40} +0.111180 q^{41} -10.1621 q^{43} +1.20339 q^{44} +0.00126871 q^{46} +2.31635 q^{47} -5.10603 q^{49} -2.53860 q^{50} -7.22600 q^{52} +3.14045 q^{53} +2.80076 q^{55} -3.93477 q^{56} +6.37640 q^{58} +5.54194 q^{59} +2.76901 q^{61} -5.72295 q^{62} +5.27830 q^{64} -16.8178 q^{65} +7.59953 q^{67} +0.939000 q^{68} -3.44022 q^{70} +7.74768 q^{71} +8.20093 q^{73} -8.91120 q^{74} +10.1126 q^{76} +1.37622 q^{77} +3.44213 q^{79} +0.406335 q^{80} -0.0992316 q^{82} -10.5916 q^{83} +2.18543 q^{85} +9.07003 q^{86} -2.85912 q^{88} +15.1612 q^{89} -8.26377 q^{91} +0.00171059 q^{92} -2.06742 q^{94} +23.5361 q^{95} +0.717491 q^{97} +4.55729 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 6 q^{2} + 54 q^{4} - 24 q^{5} - 18 q^{8}+O(q^{10}) \) 48 * q - 6 * q^2 + 54 * q^4 - 24 * q^5 - 18 * q^8	\( 48 q - 6 q^{2} + 54 q^{4} - 24 q^{5} - 18 q^{8} - 48 q^{11} - 24 q^{14} + 66 q^{16} - 24 q^{17} - 48 q^{20} + 6 q^{22} - 12 q^{23} + 60 q^{25} - 36 q^{26} - 18 q^{28} - 60 q^{29} + 36 q^{31} - 42 q^{32} + 12 q^{34} - 24 q^{35} + 6 q^{37} - 24 q^{38} - 72 q^{41} - 12 q^{43} - 54 q^{44} - 30 q^{46} - 36 q^{47} + 60 q^{49} - 42 q^{50} - 48 q^{53} + 24 q^{55} - 72 q^{56} + 12 q^{58} - 60 q^{59} - 24 q^{61} - 36 q^{62} + 90 q^{64} - 48 q^{65} - 60 q^{68} - 30 q^{70} - 60 q^{71} - 18 q^{73} - 36 q^{74} - 42 q^{76} - 12 q^{79} - 96 q^{80} + 12 q^{82} - 36 q^{83} + 18 q^{85} - 48 q^{86} + 18 q^{88} - 96 q^{89} + 30 q^{91} - 36 q^{92} - 48 q^{94} - 48 q^{95} + 30 q^{97} - 54 q^{98}+O(q^{100}) \) 48 * q - 6 * q^2 + 54 * q^4 - 24 * q^5 - 18 * q^8 - 48 * q^11 - 24 * q^14 + 66 * q^16 - 24 * q^17 - 48 * q^20 + 6 * q^22 - 12 * q^23 + 60 * q^25 - 36 * q^26 - 18 * q^28 - 60 * q^29 + 36 * q^31 - 42 * q^32 + 12 * q^34 - 24 * q^35 + 6 * q^37 - 24 * q^38 - 72 * q^41 - 12 * q^43 - 54 * q^44 - 30 * q^46 - 36 * q^47 + 60 * q^49 - 42 * q^50 - 48 * q^53 + 24 * q^55 - 72 * q^56 + 12 * q^58 - 60 * q^59 - 24 * q^61 - 36 * q^62 + 90 * q^64 - 48 * q^65 - 60 * q^68 - 30 * q^70 - 60 * q^71 - 18 * q^73 - 36 * q^74 - 42 * q^76 - 12 * q^79 - 96 * q^80 + 12 * q^82 - 36 * q^83 + 18 * q^85 - 48 * q^86 + 18 * q^88 - 96 * q^89 + 30 * q^91 - 36 * q^92 - 48 * q^94 - 48 * q^95 + 30 * q^97 - 54 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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