LMFDB - Embedded newform 8019.2.a.k.1.15

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			1.15
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)

$f(q)$	$=$	$q-1.31194 q^{2} -0.278801 q^{4} +0.542226 q^{5} -2.66849 q^{7} +2.98966 q^{8} +O(q^{10})$	\(q-1.31194 q^{2} -0.278801 q^{4} +0.542226 q^{5} -2.66849 q^{7} +2.98966 q^{8} -0.711370 q^{10} -1.00000 q^{11} +0.353276 q^{13} +3.50092 q^{14} -3.36467 q^{16} +0.396851 q^{17} -6.23129 q^{19} -0.151173 q^{20} +1.31194 q^{22} -2.34418 q^{23} -4.70599 q^{25} -0.463479 q^{26} +0.743980 q^{28} -3.28095 q^{29} -8.28785 q^{31} -1.56507 q^{32} -0.520646 q^{34} -1.44693 q^{35} +5.54028 q^{37} +8.17510 q^{38} +1.62107 q^{40} +8.05719 q^{41} +3.04207 q^{43} +0.278801 q^{44} +3.07543 q^{46} -10.0983 q^{47} +0.120863 q^{49} +6.17400 q^{50} -0.0984940 q^{52} -3.72978 q^{53} -0.542226 q^{55} -7.97789 q^{56} +4.30443 q^{58} +1.74607 q^{59} -10.2846 q^{61} +10.8732 q^{62} +8.78261 q^{64} +0.191556 q^{65} +12.3450 q^{67} -0.110642 q^{68} +1.89829 q^{70} +7.92922 q^{71} -4.52866 q^{73} -7.26854 q^{74} +1.73729 q^{76} +2.66849 q^{77} +2.17303 q^{79} -1.82441 q^{80} -10.5706 q^{82} -10.4678 q^{83} +0.215183 q^{85} -3.99103 q^{86} -2.98966 q^{88} -10.0498 q^{89} -0.942716 q^{91} +0.653560 q^{92} +13.2484 q^{94} -3.37876 q^{95} +11.2727 q^{97} -0.158566 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.31194	−0.927685	−0.463842	−	0.885918i	$-0.653530\pi$
$2$	−1.31194	−0.927685	−0.463842	+	0.885918i	$0.653530\pi$
$3$	0	0
$3$	0	0
$4$	−0.278801	−0.139401
$5$	0.542226	0.242491	0.121245	−	0.992623i	$-0.461311\pi$
$5$	0.542226	0.242491	0.121245	+	0.992623i	$0.461311\pi$
$6$	0	0
$7$	−2.66849	−1.00860	−0.504298	−	0.863530i	$-0.668249\pi$
$7$	−2.66849	−1.00860	−0.504298	+	0.863530i	$0.668249\pi$
$8$	2.98966	1.05700
$9$	0	0
$10$	−0.711370	−0.224955
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	0.353276	0.0979813	0.0489906	−	0.998799i	$-0.484400\pi$
$13$	0.353276	0.0979813	0.0489906	+	0.998799i	$0.484400\pi$
$14$	3.50092	0.935659
$15$	0	0
$16$	−3.36467	−0.841167
$17$	0.396851	0.0962504	0.0481252	−	0.998841i	$-0.484675\pi$
$17$	0.396851	0.0962504	0.0481252	+	0.998841i	$0.484675\pi$
$18$	0	0
$19$	−6.23129	−1.42955	−0.714777	−	0.699352i	$-0.753472\pi$
$19$	−6.23129	−1.42955	−0.714777	+	0.699352i	$0.753472\pi$
$20$	−0.151173	−0.0338034
$21$	0	0
$22$	1.31194	0.279708
$23$	−2.34418	−0.488794	−0.244397	−	0.969675i	$-0.578590\pi$
$23$	−2.34418	−0.488794	−0.244397	+	0.969675i	$0.578590\pi$
$24$	0	0
$25$	−4.70599	−0.941198
$26$	−0.463479	−0.0908957
$27$	0	0
$28$	0.743980	0.140599
$29$	−3.28095	−0.609258	−0.304629	−	0.952471i	$-0.598532\pi$
$29$	−3.28095	−0.609258	−0.304629	+	0.952471i	$0.598532\pi$
$30$	0	0
$31$	−8.28785	−1.48854	−0.744271	−	0.667878i	$-0.767203\pi$
$31$	−8.28785	−1.48854	−0.744271	+	0.667878i	$0.767203\pi$
$32$	−1.56507	−0.276667
$33$	0	0
$34$	−0.520646	−0.0892900
$35$	−1.44693	−0.244575
$36$	0	0
$37$	5.54028	0.910816	0.455408	−	0.890283i	$-0.349493\pi$
$37$	5.54028	0.910816	0.455408	+	0.890283i	$0.349493\pi$
$38$	8.17510	1.32618
$39$	0	0
$40$	1.62107	0.256314
$41$	8.05719	1.25832	0.629161	−	0.777275i	$-0.283399\pi$
$41$	8.05719	1.25832	0.629161	+	0.777275i	$0.283399\pi$
$42$	0	0
$43$	3.04207	0.463911	0.231956	−	0.972726i	$-0.425488\pi$
$43$	3.04207	0.463911	0.231956	+	0.972726i	$0.425488\pi$
$44$	0.278801	0.0420309
$45$	0	0
$46$	3.07543	0.453447
$47$	−10.0983	−1.47298	−0.736491	−	0.676447i	$-0.763519\pi$
$47$	−10.0983	−1.47298	−0.736491	+	0.676447i	$0.763519\pi$
$48$	0	0
$49$	0.120863	0.0172662
$50$	6.17400	0.873135
$51$	0	0
$52$	−0.0984940	−0.0136587
$53$	−3.72978	−0.512324	−0.256162	−	0.966634i	$-0.582458\pi$
$53$	−3.72978	−0.512324	−0.256162	+	0.966634i	$0.582458\pi$
$54$	0	0
$55$	−0.542226	−0.0731137
$56$	−7.97789	−1.06609
$57$	0	0
$58$	4.30443	0.565199
$59$	1.74607	0.227319	0.113659	−	0.993520i	$-0.463743\pi$
$59$	1.74607	0.227319	0.113659	+	0.993520i	$0.463743\pi$
$60$	0	0
$61$	−10.2846	−1.31681	−0.658404	−	0.752665i	$-0.728768\pi$
$61$	−10.2846	−1.31681	−0.658404	+	0.752665i	$0.728768\pi$
$62$	10.8732	1.38090
$63$	0	0
$64$	8.78261	1.09783
$65$	0.191556	0.0237596
$66$	0	0
$67$	12.3450	1.50818	0.754092	−	0.656769i	$-0.228077\pi$
$67$	12.3450	1.50818	0.754092	+	0.656769i	$0.228077\pi$
$68$	−0.110642	−0.0134174
$69$	0	0
$70$	1.89829	0.226889
$71$	7.92922	0.941025	0.470513	−	0.882393i	$-0.344069\pi$
$71$	7.92922	0.941025	0.470513	+	0.882393i	$0.344069\pi$
$72$	0	0
$73$	−4.52866	−0.530039	−0.265020	−	0.964243i	$-0.585378\pi$
$73$	−4.52866	−0.530039	−0.265020	+	0.964243i	$0.585378\pi$
$74$	−7.26854	−0.844950
$75$	0	0
$76$	1.73729	0.199281
$77$	2.66849	0.304103
$78$	0	0
$79$	2.17303	0.244484	0.122242	−	0.992500i	$-0.460992\pi$
$79$	2.17303	0.244484	0.122242	+	0.992500i	$0.460992\pi$
$80$	−1.82441	−0.203975
$81$	0	0
$82$	−10.5706	−1.16733
$83$	−10.4678	−1.14899	−0.574495	−	0.818508i	$-0.694801\pi$
$83$	−10.4678	−1.14899	−0.574495	+	0.818508i	$0.694801\pi$
$84$	0	0
$85$	0.215183	0.0233398
$86$	−3.99103	−0.430363
$87$	0	0
$88$	−2.98966	−0.318699
$89$	−10.0498	−1.06528	−0.532640	−	0.846342i	$-0.678800\pi$
$89$	−10.0498	−1.06528	−0.532640	+	0.846342i	$0.678800\pi$
$90$	0	0
$91$	−0.942716	−0.0988235
$92$	0.653560	0.0681383
$93$	0	0
$94$	13.2484	1.36646
$95$	−3.37876	−0.346654
$96$	0	0
$97$	11.2727	1.14456	0.572282	−	0.820057i	$-0.306058\pi$
$97$	11.2727	1.14456	0.572282	+	0.820057i	$0.306058\pi$
$98$	−0.158566	−0.0160176
$99$	0	0
$100$	1.31204	0.131204
$101$	12.1582	1.20979	0.604895	−	0.796305i	$-0.293215\pi$
$101$	12.1582	1.20979	0.604895	+	0.796305i	$0.293215\pi$
$102$	0	0
$103$	16.2611	1.60226	0.801129	−	0.598492i	$-0.204233\pi$
$103$	16.2611	1.60226	0.801129	+	0.598492i	$0.204233\pi$
$104$	1.05618	0.103567
$105$	0	0
$106$	4.89326	0.475275
$107$	13.4508	1.30033	0.650167	−	0.759791i	$-0.274699\pi$
$107$	13.4508	1.30033	0.650167	+	0.759791i	$0.274699\pi$
$108$	0	0
$109$	−0.977953	−0.0936710	−0.0468355	−	0.998903i	$-0.514914\pi$
$109$	−0.977953	−0.0936710	−0.0468355	+	0.998903i	$0.514914\pi$
$110$	0.711370	0.0678265
$111$	0	0
$112$	8.97860	0.848398
$113$	−5.86056	−0.551315	−0.275658	−	0.961256i	$-0.588896\pi$
$113$	−5.86056	−0.551315	−0.275658	+	0.961256i	$0.588896\pi$
$114$	0	0
$115$	−1.27107	−0.118528
$116$	0.914735	0.0849310
$117$	0	0
$118$	−2.29074	−0.210880
$119$	−1.05899	−0.0970778
$120$	0	0
$121$	1.00000	0.0909091
$122$	13.4928	1.22158
$123$	0	0
$124$	2.31066	0.207504
$125$	−5.26284	−0.470723
$126$	0	0
$127$	−5.30900	−0.471097	−0.235549	−	0.971863i	$-0.575689\pi$
$127$	−5.30900	−0.471097	−0.235549	+	0.971863i	$0.575689\pi$
$128$	−8.39217	−0.741770
$129$	0	0
$130$	−0.251310	−0.0220414
$131$	−22.4076	−1.95776	−0.978880	−	0.204435i	$-0.934464\pi$
$131$	−22.4076	−1.95776	−0.978880	+	0.204435i	$0.934464\pi$
$132$	0	0
$133$	16.6282	1.44184
$134$	−16.1960	−1.39912
$135$	0	0
$136$	1.18645	0.101737
$137$	−6.78495	−0.579678	−0.289839	−	0.957075i	$-0.593602\pi$
$137$	−6.78495	−0.579678	−0.289839	+	0.957075i	$0.593602\pi$
$138$	0	0
$139$	3.17970	0.269698	0.134849	−	0.990866i	$-0.456945\pi$
$139$	3.17970	0.269698	0.134849	+	0.990866i	$0.456945\pi$
$140$	0.403405	0.0340940
$141$	0	0
$142$	−10.4027	−0.872975
$143$	−0.353276	−0.0295425
$144$	0	0
$145$	−1.77902	−0.147739
$146$	5.94135	0.491709
$147$	0	0
$148$	−1.54464	−0.126968
$149$	−2.52807	−0.207108	−0.103554	−	0.994624i	$-0.533021\pi$
$149$	−2.52807	−0.207108	−0.103554	+	0.994624i	$0.533021\pi$
$150$	0	0
$151$	−9.30273	−0.757045	−0.378523	−	0.925592i	$-0.623568\pi$
$151$	−9.30273	−0.757045	−0.378523	+	0.925592i	$0.623568\pi$
$152$	−18.6294	−1.51105
$153$	0	0
$154$	−3.50092	−0.282112
$155$	−4.49389	−0.360958
$156$	0	0
$157$	−10.8241	−0.863854	−0.431927	−	0.901909i	$-0.642166\pi$
$157$	−10.8241	−0.863854	−0.431927	+	0.901909i	$0.642166\pi$
$158$	−2.85089	−0.226805
$159$	0	0
$160$	−0.848619	−0.0670892
$161$	6.25542	0.492996
$162$	0	0
$163$	−11.1189	−0.870902	−0.435451	−	0.900212i	$-0.643411\pi$
$163$	−11.1189	−0.870902	−0.435451	+	0.900212i	$0.643411\pi$
$164$	−2.24636	−0.175411
$165$	0	0
$166$	13.7332	1.06590
$167$	−0.530462	−0.0410484	−0.0205242	−	0.999789i	$-0.506534\pi$
$167$	−0.530462	−0.0410484	−0.0205242	+	0.999789i	$0.506534\pi$
$168$	0	0
$169$	−12.8752	−0.990400
$170$	−0.282308	−0.0216520
$171$	0	0
$172$	−0.848133	−0.0646695
$173$	−18.2778	−1.38964	−0.694818	−	0.719185i	$-0.744515\pi$
$173$	−18.2778	−1.38964	−0.694818	+	0.719185i	$0.744515\pi$
$174$	0	0
$175$	12.5579	0.949289
$176$	3.36467	0.253621
$177$	0	0
$178$	13.1848	0.988245
$179$	−8.97918	−0.671136	−0.335568	−	0.942016i	$-0.608928\pi$
$179$	−8.97918	−0.671136	−0.335568	+	0.942016i	$0.608928\pi$
$180$	0	0
$181$	10.1909	0.757483	0.378742	−	0.925502i	$-0.376357\pi$
$181$	10.1909	0.757483	0.378742	+	0.925502i	$0.376357\pi$
$182$	1.23679	0.0916771
$183$	0	0
$184$	−7.00829	−0.516658
$185$	3.00408	0.220864
$186$	0	0
$187$	−0.396851	−0.0290206
$188$	2.81541	0.205335
$189$	0	0
$190$	4.43275	0.321586
$191$	−9.51263	−0.688310	−0.344155	−	0.938913i	$-0.611834\pi$
$191$	−9.51263	−0.688310	−0.344155	+	0.938913i	$0.611834\pi$
$192$	0	0
$193$	18.2166	1.31126	0.655630	−	0.755082i	$-0.272403\pi$
$193$	18.2166	1.31126	0.655630	+	0.755082i	$0.272403\pi$
$194$	−14.7891	−1.06180
$195$	0	0
$196$	−0.0336969	−0.00240692
$197$	15.4739	1.10247	0.551236	−	0.834349i	$-0.314156\pi$
$197$	15.4739	1.10247	0.551236	+	0.834349i	$0.314156\pi$
$198$	0	0
$199$	0.254448	0.0180374	0.00901868	−	0.999959i	$-0.497129\pi$
$199$	0.254448	0.0180374	0.00901868	+	0.999959i	$0.497129\pi$
$200$	−14.0693	−0.994851
$201$	0	0
$202$	−15.9509	−1.12230
$203$	8.75521	0.614495
$204$	0	0
$205$	4.36882	0.305132
$206$	−21.3337	−1.48639
$207$	0	0
$208$	−1.18866	−0.0824186
$209$	6.23129	0.431027
$210$	0	0
$211$	0.407250	0.0280363	0.0140181	−	0.999902i	$-0.495538\pi$
$211$	0.407250	0.0280363	0.0140181	+	0.999902i	$0.495538\pi$
$212$	1.03987	0.0714183
$213$	0	0
$214$	−17.6467	−1.20630
$215$	1.64949	0.112494
$216$	0	0
$217$	22.1161	1.50134
$218$	1.28302	0.0868971
$219$	0	0
$220$	0.151173	0.0101921
$221$	0.140198	0.00943074
$222$	0	0
$223$	16.5484	1.10816	0.554081	−	0.832462i	$-0.313070\pi$
$223$	16.5484	1.10816	0.554081	+	0.832462i	$0.313070\pi$
$224$	4.17637	0.279045
$225$	0	0
$226$	7.68873	0.511447
$227$	−12.4087	−0.823597	−0.411798	−	0.911275i	$-0.635099\pi$
$227$	−12.4087	−0.823597	−0.411798	+	0.911275i	$0.635099\pi$
$228$	0	0
$229$	25.9009	1.71158	0.855788	−	0.517326i	$-0.173073\pi$
$229$	25.9009	1.71158	0.855788	+	0.517326i	$0.173073\pi$
$230$	1.66758	0.109957
$231$	0	0
$232$	−9.80894	−0.643989
$233$	18.0586	1.18306	0.591528	−	0.806284i	$-0.298525\pi$
$233$	18.0586	1.18306	0.591528	+	0.806284i	$0.298525\pi$
$234$	0	0
$235$	−5.47554	−0.357185
$236$	−0.486806	−0.0316884
$237$	0	0
$238$	1.38934	0.0900576
$239$	17.8979	1.15772	0.578861	−	0.815426i	$-0.303497\pi$
$239$	17.8979	1.15772	0.578861	+	0.815426i	$0.303497\pi$
$240$	0	0
$241$	25.4244	1.63773	0.818863	−	0.573988i	$-0.194605\pi$
$241$	25.4244	1.63773	0.818863	+	0.573988i	$0.194605\pi$
$242$	−1.31194	−0.0843350
$243$	0	0
$244$	2.86736	0.183564
$245$	0.0655353	0.00418690
$246$	0	0
$247$	−2.20137	−0.140070
$248$	−24.7779	−1.57340
$249$	0	0
$250$	6.90455	0.436682
$251$	7.35656	0.464342	0.232171	−	0.972675i	$-0.425417\pi$
$251$	7.35656	0.464342	0.232171	+	0.972675i	$0.425417\pi$
$252$	0	0
$253$	2.34418	0.147377
$254$	6.96511	0.437030
$255$	0	0
$256$	−6.55516	−0.409698
$257$	−17.7738	−1.10870	−0.554351	−	0.832283i	$-0.687033\pi$
$257$	−17.7738	−1.10870	−0.554351	+	0.832283i	$0.687033\pi$
$258$	0	0
$259$	−14.7842	−0.918645
$260$	−0.0534060	−0.00331210
$261$	0	0
$262$	29.3975	1.81618
$263$	−4.53708	−0.279768	−0.139884	−	0.990168i	$-0.544673\pi$
$263$	−4.53708	−0.279768	−0.139884	+	0.990168i	$0.544673\pi$
$264$	0	0
$265$	−2.02238	−0.124234
$266$	−21.8152	−1.33758
$267$	0	0
$268$	−3.44181	−0.210242
$269$	24.6753	1.50448	0.752241	−	0.658888i	$-0.228973\pi$
$269$	24.6753	1.50448	0.752241	+	0.658888i	$0.228973\pi$
$270$	0	0
$271$	−28.3025	−1.71926	−0.859628	−	0.510920i	$-0.829305\pi$
$271$	−28.3025	−1.71926	−0.859628	+	0.510920i	$0.829305\pi$
$272$	−1.33527	−0.0809626
$273$	0	0
$274$	8.90148	0.537758
$275$	4.70599	0.283782
$276$	0	0
$277$	20.6983	1.24364	0.621820	−	0.783160i	$-0.286393\pi$
$277$	20.6983	1.24364	0.621820	+	0.783160i	$0.286393\pi$
$278$	−4.17159	−0.250195
$279$	0	0
$280$	−4.32582	−0.258517
$281$	−11.4988	−0.685964	−0.342982	−	0.939342i	$-0.611437\pi$
$281$	−11.4988	−0.685964	−0.342982	+	0.939342i	$0.611437\pi$
$282$	0	0
$283$	18.3493	1.09075	0.545375	−	0.838192i	$-0.316387\pi$
$283$	18.3493	1.09075	0.545375	+	0.838192i	$0.316387\pi$
$284$	−2.21068	−0.131180
$285$	0	0
$286$	0.463479	0.0274061
$287$	−21.5006	−1.26914
$288$	0	0
$289$	−16.8425	−0.990736
$290$	2.33397	0.137056
$291$	0	0
$292$	1.26260	0.0738878
$293$	−8.90221	−0.520073	−0.260036	−	0.965599i	$-0.583735\pi$
$293$	−8.90221	−0.520073	−0.260036	+	0.965599i	$0.583735\pi$
$294$	0	0
$295$	0.946763	0.0551226
$296$	16.5635	0.962737
$297$	0	0
$298$	3.31669	0.192131
$299$	−0.828142	−0.0478927
$300$	0	0
$301$	−8.11774	−0.467899
$302$	12.2047	0.702299
$303$	0	0
$304$	20.9662	1.20249
$305$	−5.57658	−0.319314
$306$	0	0
$307$	−8.64895	−0.493622	−0.246811	−	0.969064i	$-0.579383\pi$
$307$	−8.64895	−0.493622	−0.246811	+	0.969064i	$0.579383\pi$
$308$	−0.743980	−0.0423922
$309$	0	0
$310$	5.89573	0.334855
$311$	−33.0438	−1.87374	−0.936870	−	0.349679i	$-0.886291\pi$
$311$	−33.0438	−1.87374	−0.936870	+	0.349679i	$0.886291\pi$
$312$	0	0
$313$	16.1697	0.913968	0.456984	−	0.889475i	$-0.348930\pi$
$313$	16.1697	0.913968	0.456984	+	0.889475i	$0.348930\pi$
$314$	14.2006	0.801385
$315$	0	0
$316$	−0.605842	−0.0340813
$317$	−20.0311	−1.12506	−0.562529	−	0.826778i	$-0.690172\pi$
$317$	−20.0311	−1.12506	−0.562529	+	0.826778i	$0.690172\pi$
$318$	0	0
$319$	3.28095	0.183698
$320$	4.76216	0.266213
$321$	0	0
$322$	−8.20677	−0.457345
$323$	−2.47289	−0.137595
$324$	0	0
$325$	−1.66252	−0.0922198
$326$	14.5874	0.807923
$327$	0	0
$328$	24.0883	1.33005
$329$	26.9471	1.48564
$330$	0	0
$331$	−16.1003	−0.884950	−0.442475	−	0.896781i	$-0.645899\pi$
$331$	−16.1003	−0.884950	−0.442475	+	0.896781i	$0.645899\pi$
$332$	2.91844	0.160170
$333$	0	0
$334$	0.695937	0.0380800
$335$	6.69379	0.365721
$336$	0	0
$337$	21.4313	1.16743	0.583717	−	0.811957i	$-0.301598\pi$
$337$	21.4313	1.16743	0.583717	+	0.811957i	$0.301598\pi$
$338$	16.8915	0.918779
$339$	0	0
$340$	−0.0599932	−0.00325359
$341$	8.28785	0.448812
$342$	0	0
$343$	18.3569	0.991182
$344$	9.09476	0.490356
$345$	0	0
$346$	23.9795	1.28915
$347$	28.5122	1.53061	0.765307	−	0.643666i	$-0.222587\pi$
$347$	28.5122	1.53061	0.765307	+	0.643666i	$0.222587\pi$
$348$	0	0
$349$	10.5277	0.563534	0.281767	−	0.959483i	$-0.409080\pi$
$349$	10.5277	0.563534	0.281767	+	0.959483i	$0.409080\pi$
$350$	−16.4753	−0.880641
$351$	0	0
$352$	1.56507	0.0834183
$353$	3.11563	0.165828	0.0829140	−	0.996557i	$-0.473577\pi$
$353$	3.11563	0.165828	0.0829140	+	0.996557i	$0.473577\pi$
$354$	0	0
$355$	4.29943	0.228190
$356$	2.80191	0.148501
$357$	0	0
$358$	11.7802	0.622602
$359$	10.7623	0.568014	0.284007	−	0.958822i	$-0.408336\pi$
$359$	10.7623	0.568014	0.284007	+	0.958822i	$0.408336\pi$
$360$	0	0
$361$	19.8289	1.04363
$362$	−13.3699	−0.702706
$363$	0	0
$364$	0.262831	0.0137761
$365$	−2.45555	−0.128530
$366$	0	0
$367$	4.44818	0.232193	0.116097	−	0.993238i	$-0.462962\pi$
$367$	4.44818	0.232193	0.116097	+	0.993238i	$0.462962\pi$
$368$	7.88737	0.411158
$369$	0	0
$370$	−3.94119	−0.204893
$371$	9.95289	0.516728
$372$	0	0
$373$	−0.0568600	−0.00294410	−0.00147205	−	0.999999i	$-0.500469\pi$
$373$	−0.0568600	−0.00294410	−0.00147205	+	0.999999i	$0.500469\pi$
$374$	0.520646	0.0269220
$375$	0	0
$376$	−30.1904	−1.55695
$377$	−1.15908	−0.0596959
$378$	0	0
$379$	−32.6774	−1.67852	−0.839262	−	0.543727i	$-0.817013\pi$
$379$	−32.6774	−1.67852	−0.839262	+	0.543727i	$0.817013\pi$
$380$	0.942004	0.0483238
$381$	0	0
$382$	12.4800	0.638534
$383$	20.5092	1.04797	0.523986	−	0.851727i	$-0.324444\pi$
$383$	20.5092	1.04797	0.523986	+	0.851727i	$0.324444\pi$
$384$	0	0
$385$	1.44693	0.0737422
$386$	−23.8992	−1.21644
$387$	0	0
$388$	−3.14283	−0.159553
$389$	26.4011	1.33859	0.669294	−	0.742997i	$-0.266597\pi$
$389$	26.4011	1.33859	0.669294	+	0.742997i	$0.266597\pi$
$390$	0	0
$391$	−0.930288	−0.0470467
$392$	0.361341	0.0182505
$393$	0	0
$394$	−20.3010	−1.02275
$395$	1.17827	0.0592852
$396$	0	0
$397$	23.4392	1.17638	0.588190	−	0.808723i	$-0.299841\pi$
$397$	23.4392	1.17638	0.588190	+	0.808723i	$0.299841\pi$
$398$	−0.333822	−0.0167330
$399$	0	0
$400$	15.8341	0.791705
$401$	−21.3484	−1.06609	−0.533043	−	0.846088i	$-0.678952\pi$
$401$	−21.3484	−1.06609	−0.533043	+	0.846088i	$0.678952\pi$
$402$	0	0
$403$	−2.92790	−0.145849
$404$	−3.38974	−0.168646
$405$	0	0
$406$	−11.4864	−0.570058
$407$	−5.54028	−0.274621
$408$	0	0
$409$	33.7617	1.66941	0.834705	−	0.550698i	$-0.185638\pi$
$409$	33.7617	1.66941	0.834705	+	0.550698i	$0.185638\pi$
$410$	−5.73165	−0.283066
$411$	0	0
$412$	−4.53363	−0.223356
$413$	−4.65937	−0.229273
$414$	0	0
$415$	−5.67591	−0.278619
$416$	−0.552901	−0.0271082
$417$	0	0
$418$	−8.17510	−0.399857
$419$	3.30472	0.161446	0.0807232	−	0.996737i	$-0.474277\pi$
$419$	3.30472	0.161446	0.0807232	+	0.996737i	$0.474277\pi$
$420$	0	0
$421$	3.01525	0.146954	0.0734770	−	0.997297i	$-0.476590\pi$
$421$	3.01525	0.146954	0.0734770	+	0.997297i	$0.476590\pi$
$422$	−0.534290	−0.0260088
$423$	0	0
$424$	−11.1508	−0.541529
$425$	−1.86758	−0.0905907
$426$	0	0
$427$	27.4444	1.32813
$428$	−3.75009	−0.181267
$429$	0	0
$430$	−2.16404	−0.104359
$431$	−33.6827	−1.62244	−0.811220	−	0.584741i	$-0.801196\pi$
$431$	−33.6827	−1.62244	−0.811220	+	0.584741i	$0.801196\pi$
$432$	0	0
$433$	19.8183	0.952409	0.476205	−	0.879335i	$-0.342012\pi$
$433$	19.8183	0.952409	0.476205	+	0.879335i	$0.342012\pi$
$434$	−29.0151	−1.39277
$435$	0	0
$436$	0.272655	0.0130578
$437$	14.6072	0.698759
$438$	0	0
$439$	25.9161	1.23691	0.618453	−	0.785822i	$-0.287760\pi$
$439$	25.9161	1.23691	0.618453	+	0.785822i	$0.287760\pi$
$440$	−1.62107	−0.0772816
$441$	0	0
$442$	−0.183932	−0.00874875
$443$	−31.0382	−1.47467	−0.737334	−	0.675529i	$-0.763915\pi$
$443$	−31.0382	−1.47467	−0.737334	+	0.675529i	$0.763915\pi$
$444$	0	0
$445$	−5.44928	−0.258321
$446$	−21.7106	−1.02803
$447$	0	0
$448$	−23.4364	−1.10726
$449$	19.6949	0.929461	0.464731	−	0.885452i	$-0.346151\pi$
$449$	19.6949	0.929461	0.464731	+	0.885452i	$0.346151\pi$
$450$	0	0
$451$	−8.05719	−0.379398
$452$	1.63393	0.0768537
$453$	0	0
$454$	16.2796	0.764038
$455$	−0.511165	−0.0239638
$456$	0	0
$457$	8.38941	0.392440	0.196220	−	0.980560i	$-0.437133\pi$
$457$	8.38941	0.392440	0.196220	+	0.980560i	$0.437133\pi$
$458$	−33.9805	−1.58780
$459$	0	0
$460$	0.354377	0.0165229
$461$	5.74580	0.267609	0.133804	−	0.991008i	$-0.457281\pi$
$461$	5.74580	0.267609	0.133804	+	0.991008i	$0.457281\pi$
$462$	0	0
$463$	−39.2557	−1.82437	−0.912184	−	0.409781i	$-0.865605\pi$
$463$	−39.2557	−1.82437	−0.912184	+	0.409781i	$0.865605\pi$
$464$	11.0393	0.512488
$465$	0	0
$466$	−23.6919	−1.09750
$467$	−9.31678	−0.431129	−0.215565	−	0.976490i	$-0.569159\pi$
$467$	−9.31678	−0.431129	−0.215565	+	0.976490i	$0.569159\pi$
$468$	0	0
$469$	−32.9426	−1.52115
$470$	7.18360	0.331355
$471$	0	0
$472$	5.22015	0.240277
$473$	−3.04207	−0.139874
$474$	0	0
$475$	29.3244	1.34549
$476$	0.295249	0.0135327
$477$	0	0
$478$	−23.4811	−1.07400
$479$	−7.38099	−0.337246	−0.168623	−	0.985681i	$-0.553932\pi$
$479$	−7.38099	−0.337246	−0.168623	+	0.985681i	$0.553932\pi$
$480$	0	0
$481$	1.95725	0.0892429
$482$	−33.3553	−1.51929
$483$	0	0
$484$	−0.278801	−0.0126728
$485$	6.11232	0.277546
$486$	0	0
$487$	22.3296	1.01185	0.505925	−	0.862578i	$-0.331151\pi$
$487$	22.3296	1.01185	0.505925	+	0.862578i	$0.331151\pi$
$488$	−30.7475	−1.39187
$489$	0	0
$490$	−0.0859787	−0.00388412
$491$	10.0829	0.455037	0.227518	−	0.973774i	$-0.426939\pi$
$491$	10.0829	0.455037	0.227518	+	0.973774i	$0.426939\pi$
$492$	0	0
$493$	−1.30205	−0.0586413
$494$	2.88807	0.129940
$495$	0	0
$496$	27.8859	1.25211
$497$	−21.1591	−0.949115
$498$	0	0
$499$	−20.5621	−0.920484	−0.460242	−	0.887794i	$-0.652237\pi$
$499$	−20.5621	−0.920484	−0.460242	+	0.887794i	$0.652237\pi$
$500$	1.46729	0.0656191
$501$	0	0
$502$	−9.65140	−0.430763
$503$	36.4192	1.62385	0.811926	−	0.583761i	$-0.198419\pi$
$503$	36.4192	1.62385	0.811926	+	0.583761i	$0.198419\pi$
$504$	0	0
$505$	6.59252	0.293363
$506$	−3.07543	−0.136719
$507$	0	0
$508$	1.48016	0.0656713
$509$	23.7320	1.05190	0.525952	−	0.850514i	$-0.323709\pi$
$509$	23.7320	1.05190	0.525952	+	0.850514i	$0.323709\pi$
$510$	0	0
$511$	12.0847	0.534595
$512$	25.3844	1.12184
$513$	0	0
$514$	23.3183	1.02853
$515$	8.81721	0.388533
$516$	0	0
$517$	10.0983	0.444121
$518$	19.3960	0.852213
$519$	0	0
$520$	0.572687	0.0251140
$521$	−32.8659	−1.43988	−0.719941	−	0.694035i	$-0.755831\pi$
$521$	−32.8659	−1.43988	−0.719941	+	0.694035i	$0.755831\pi$
$522$	0	0
$523$	−20.0614	−0.877226	−0.438613	−	0.898676i	$-0.644530\pi$
$523$	−20.0614	−0.877226	−0.438613	+	0.898676i	$0.644530\pi$
$524$	6.24727	0.272913
$525$	0	0
$526$	5.95239	0.259537
$527$	−3.28904	−0.143273
$528$	0	0
$529$	−17.5048	−0.761080
$530$	2.65325	0.115250
$531$	0	0
$532$	−4.63595	−0.200994
$533$	2.84642	0.123292
$534$	0	0
$535$	7.29335	0.315319
$536$	36.9074	1.59416
$537$	0	0
$538$	−32.3727	−1.39569
$539$	−0.120863	−0.00520596
$540$	0	0
$541$	11.0319	0.474300	0.237150	−	0.971473i	$-0.423787\pi$
$541$	11.0319	0.474300	0.237150	+	0.971473i	$0.423787\pi$
$542$	37.1314	1.59493
$543$	0	0
$544$	−0.621097	−0.0266293
$545$	−0.530272	−0.0227143
$546$	0	0
$547$	−11.1216	−0.475527	−0.237763	−	0.971323i	$-0.576414\pi$
$547$	−11.1216	−0.475527	−0.237763	+	0.971323i	$0.576414\pi$
$548$	1.89165	0.0808075
$549$	0	0
$550$	−6.17400	−0.263260
$551$	20.4446	0.870968
$552$	0	0
$553$	−5.79871	−0.246586
$554$	−27.1550	−1.15371
$555$	0	0
$556$	−0.886504	−0.0375961
$557$	−18.0105	−0.763130	−0.381565	−	0.924342i	$-0.624615\pi$
$557$	−18.0105	−0.763130	−0.381565	+	0.924342i	$0.624615\pi$
$558$	0	0
$559$	1.07469	0.0454546
$560$	4.86843	0.205729
$561$	0	0
$562$	15.0859	0.636358
$563$	32.6488	1.37598	0.687992	−	0.725718i	$-0.258492\pi$
$563$	32.6488	1.37598	0.687992	+	0.725718i	$0.258492\pi$
$564$	0	0
$565$	−3.17775	−0.133689
$566$	−24.0732	−1.01187
$567$	0	0
$568$	23.7057	0.994668
$569$	46.9407	1.96786	0.983928	−	0.178568i	$-0.0571464\pi$
$569$	46.9407	1.96786	0.983928	+	0.178568i	$0.0571464\pi$
$570$	0	0
$571$	−25.1154	−1.05105	−0.525524	−	0.850779i	$-0.676131\pi$
$571$	−25.1154	−1.05105	−0.525524	+	0.850779i	$0.676131\pi$
$572$	0.0984940	0.00411824
$573$	0	0
$574$	28.2076	1.17736
$575$	11.0317	0.460052
$576$	0	0
$577$	0.0825991	0.00343864	0.00171932	−	0.999999i	$-0.499453\pi$
$577$	0.0825991	0.00343864	0.00171932	+	0.999999i	$0.499453\pi$
$578$	22.0964	0.919091
$579$	0	0
$580$	0.495993	0.0205950
$581$	27.9333	1.15887
$582$	0	0
$583$	3.72978	0.154472
$584$	−13.5391	−0.560254
$585$	0	0
$586$	11.6792	0.482464
$587$	2.35042	0.0970123	0.0485061	−	0.998823i	$-0.484554\pi$
$587$	2.35042	0.0970123	0.0485061	+	0.998823i	$0.484554\pi$
$588$	0	0
$589$	51.6440	2.12795
$590$	−1.24210	−0.0511365
$591$	0	0
$592$	−18.6412	−0.766148
$593$	4.78627	0.196549	0.0982743	−	0.995159i	$-0.468668\pi$
$593$	4.78627	0.196549	0.0982743	+	0.995159i	$0.468668\pi$
$594$	0	0
$595$	−0.574214	−0.0235405
$596$	0.704829	0.0288709
$597$	0	0
$598$	1.08648	0.0444293
$599$	13.9806	0.571233	0.285617	−	0.958344i	$-0.407802\pi$
$599$	13.9806	0.571233	0.285617	+	0.958344i	$0.407802\pi$
$600$	0	0
$601$	16.1009	0.656770	0.328385	−	0.944544i	$-0.393496\pi$
$601$	16.1009	0.656770	0.328385	+	0.944544i	$0.393496\pi$
$602$	10.6500	0.434063
$603$	0	0
$604$	2.59361	0.105533
$605$	0.542226	0.0220446
$606$	0	0
$607$	−22.7675	−0.924103	−0.462051	−	0.886853i	$-0.652886\pi$
$607$	−22.7675	−0.924103	−0.462051	+	0.886853i	$0.652886\pi$
$608$	9.75237	0.395511
$609$	0	0
$610$	7.31616	0.296223
$611$	−3.56748	−0.144325
$612$	0	0
$613$	−0.605814	−0.0244686	−0.0122343	−	0.999925i	$-0.503894\pi$
$613$	−0.605814	−0.0244686	−0.0122343	+	0.999925i	$0.503894\pi$
$614$	11.3469	0.457926
$615$	0	0
$616$	7.97789	0.321439
$617$	6.86635	0.276429	0.138215	−	0.990402i	$-0.455864\pi$
$617$	6.86635	0.276429	0.138215	+	0.990402i	$0.455864\pi$
$618$	0	0
$619$	4.75439	0.191095	0.0955476	−	0.995425i	$-0.469540\pi$
$619$	4.75439	0.191095	0.0955476	+	0.995425i	$0.469540\pi$
$620$	1.25290	0.0503178
$621$	0	0
$622$	43.3516	1.73824
$623$	26.8179	1.07444
$624$	0	0
$625$	20.6763	0.827052
$626$	−21.2138	−0.847874
$627$	0	0
$628$	3.01776	0.120422
$629$	2.19866	0.0876664
$630$	0	0
$631$	37.0844	1.47631	0.738154	−	0.674632i	$-0.235698\pi$
$631$	37.0844	1.47631	0.738154	+	0.674632i	$0.235698\pi$
$632$	6.49661	0.258421
$633$	0	0
$634$	26.2797	1.04370
$635$	−2.87868	−0.114237
$636$	0	0
$637$	0.0426982	0.00169176
$638$	−4.30443	−0.170414
$639$	0	0
$640$	−4.55045	−0.179872
$641$	10.6165	0.419328	0.209664	−	0.977774i	$-0.432763\pi$
$641$	10.6165	0.419328	0.209664	+	0.977774i	$0.432763\pi$
$642$	0	0
$643$	−9.84898	−0.388406	−0.194203	−	0.980961i	$-0.562212\pi$
$643$	−9.84898	−0.388406	−0.194203	+	0.980961i	$0.562212\pi$
$644$	−1.74402	−0.0687240
$645$	0	0
$646$	3.24429	0.127645
$647$	41.9013	1.64731	0.823655	−	0.567092i	$-0.191931\pi$
$647$	41.9013	1.64731	0.823655	+	0.567092i	$0.191931\pi$
$648$	0	0
$649$	−1.74607	−0.0685391
$650$	2.18113	0.0855509
$651$	0	0
$652$	3.09997	0.121404
$653$	−48.0212	−1.87922	−0.939608	−	0.342253i	$-0.888810\pi$
$653$	−48.0212	−1.87922	−0.939608	+	0.342253i	$0.888810\pi$
$654$	0	0
$655$	−12.1500	−0.474739
$656$	−27.1098	−1.05846
$657$	0	0
$658$	−35.3532	−1.37821
$659$	33.9833	1.32380	0.661900	−	0.749592i	$-0.269750\pi$
$659$	33.9833	1.32380	0.661900	+	0.749592i	$0.269750\pi$
$660$	0	0
$661$	−6.04058	−0.234951	−0.117476	−	0.993076i	$-0.537480\pi$
$661$	−6.04058	−0.234951	−0.117476	+	0.993076i	$0.537480\pi$
$662$	21.1226	0.820955
$663$	0	0
$664$	−31.2952	−1.21449
$665$	9.01622	0.349634
$666$	0	0
$667$	7.69114	0.297802
$668$	0.147894	0.00572217
$669$	0	0
$670$	−8.78188	−0.339274
$671$	10.2846	0.397032
$672$	0	0
$673$	−50.4214	−1.94360	−0.971801	−	0.235804i	$-0.924228\pi$
$673$	−50.4214	−1.94360	−0.971801	+	0.235804i	$0.924228\pi$
$674$	−28.1166	−1.08301
$675$	0	0
$676$	3.58962	0.138062
$677$	30.8067	1.18400	0.592000	−	0.805938i	$-0.298339\pi$
$677$	30.8067	1.18400	0.592000	+	0.805938i	$0.298339\pi$
$678$	0	0
$679$	−30.0810	−1.15440
$680$	0.643323	0.0246703
$681$	0	0
$682$	−10.8732	−0.416356
$683$	−6.73059	−0.257539	−0.128769	−	0.991675i	$-0.541103\pi$
$683$	−6.73059	−0.257539	−0.128769	+	0.991675i	$0.541103\pi$
$684$	0	0
$685$	−3.67898	−0.140566
$686$	−24.0833	−0.919504
$687$	0	0
$688$	−10.2355	−0.390227
$689$	−1.31764	−0.0501982
$690$	0	0
$691$	−2.69755	−0.102620	−0.0513099	−	0.998683i	$-0.516340\pi$
$691$	−2.69755	−0.102620	−0.0513099	+	0.998683i	$0.516340\pi$
$692$	5.09588	0.193716
$693$	0	0
$694$	−37.4064	−1.41993
$695$	1.72411	0.0653994
$696$	0	0
$697$	3.19750	0.121114
$698$	−13.8117	−0.522782
$699$	0	0
$700$	−3.50116	−0.132332
$701$	−2.01048	−0.0759346	−0.0379673	−	0.999279i	$-0.512088\pi$
$701$	−2.01048	−0.0759346	−0.0379673	+	0.999279i	$0.512088\pi$
$702$	0	0
$703$	−34.5230	−1.30206
$704$	−8.78261	−0.331007
$705$	0	0
$706$	−4.08753	−0.153836
$707$	−32.4442	−1.22019
$708$	0	0
$709$	−13.7176	−0.515175	−0.257587	−	0.966255i	$-0.582927\pi$
$709$	−13.7176	−0.515175	−0.257587	+	0.966255i	$0.582927\pi$
$710$	−5.64061	−0.211688
$711$	0	0
$712$	−30.0456	−1.12601
$713$	19.4282	0.727591
$714$	0	0
$715$	−0.191556	−0.00716378
$716$	2.50341	0.0935568
$717$	0	0
$718$	−14.1196	−0.526938
$719$	48.5479	1.81053	0.905265	−	0.424848i	$-0.139672\pi$
$719$	48.5479	1.81053	0.905265	+	0.424848i	$0.139672\pi$
$720$	0	0
$721$	−43.3928	−1.61603
$722$	−26.0144	−0.968157
$723$	0	0
$724$	−2.84124	−0.105594
$725$	15.4401	0.573433
$726$	0	0
$727$	42.0867	1.56091	0.780455	−	0.625212i	$-0.214988\pi$
$727$	42.0867	1.56091	0.780455	+	0.625212i	$0.214988\pi$
$728$	−2.81840	−0.104457
$729$	0	0
$730$	3.22155	0.119235
$731$	1.20725	0.0446516
$732$	0	0
$733$	41.6263	1.53750	0.768751	−	0.639548i	$-0.220879\pi$
$733$	41.6263	1.53750	0.768751	+	0.639548i	$0.220879\pi$
$734$	−5.83577	−0.215402
$735$	0	0
$736$	3.66879	0.135233
$737$	−12.3450	−0.454735
$738$	0	0
$739$	0.815436	0.0299963	0.0149982	−	0.999888i	$-0.495226\pi$
$739$	0.815436	0.0299963	0.0149982	+	0.999888i	$0.495226\pi$
$740$	−0.837542	−0.0307887
$741$	0	0
$742$	−13.0576	−0.479361
$743$	−23.3040	−0.854940	−0.427470	−	0.904030i	$-0.640595\pi$
$743$	−23.3040	−0.854940	−0.427470	+	0.904030i	$0.640595\pi$
$744$	0	0
$745$	−1.37078	−0.0502217
$746$	0.0745972	0.00273120
$747$	0	0
$748$	0.110642	0.00404549
$749$	−35.8933	−1.31151
$750$	0	0
$751$	−39.4154	−1.43829	−0.719144	−	0.694861i	$-0.755466\pi$
$751$	−39.4154	−1.43829	−0.719144	+	0.694861i	$0.755466\pi$
$752$	33.9773	1.23902
$753$	0	0
$754$	1.52065	0.0553790
$755$	−5.04418	−0.183576
$756$	0	0
$757$	22.3542	0.812476	0.406238	−	0.913767i	$-0.366840\pi$
$757$	22.3542	0.812476	0.406238	+	0.913767i	$0.366840\pi$
$758$	42.8709	1.55714
$759$	0	0
$760$	−10.1014	−0.366415
$761$	24.3127	0.881336	0.440668	−	0.897670i	$-0.354742\pi$
$761$	24.3127	0.881336	0.440668	+	0.897670i	$0.354742\pi$
$762$	0	0
$763$	2.60966	0.0944762
$764$	2.65213	0.0959508
$765$	0	0
$766$	−26.9070	−0.972189
$767$	0.616844	0.0222730
$768$	0	0
$769$	17.0398	0.614470	0.307235	−	0.951634i	$-0.400596\pi$
$769$	17.0398	0.614470	0.307235	+	0.951634i	$0.400596\pi$
$770$	−1.89829	−0.0684096
$771$	0	0
$772$	−5.07882	−0.182791
$773$	34.3989	1.23724	0.618622	−	0.785689i	$-0.287691\pi$
$773$	34.3989	1.23724	0.618622	+	0.785689i	$0.287691\pi$
$774$	0	0
$775$	39.0025	1.40101
$776$	33.7014	1.20981
$777$	0	0
$778$	−34.6368	−1.24179
$779$	−50.2067	−1.79884
$780$	0	0
$781$	−7.92922	−0.283730
$782$	1.22049	0.0436445
$783$	0	0
$784$	−0.406665	−0.0145238
$785$	−5.86909	−0.209477
$786$	0	0
$787$	26.7595	0.953872	0.476936	−	0.878938i	$-0.341747\pi$
$787$	26.7595	0.953872	0.476936	+	0.878938i	$0.341747\pi$
$788$	−4.31416	−0.153685
$789$	0	0
$790$	−1.54583	−0.0549980
$791$	15.6389	0.556054
$792$	0	0
$793$	−3.63331	−0.129022
$794$	−30.7509	−1.09131
$795$	0	0
$796$	−0.0709405	−0.00251442
$797$	−24.4344	−0.865512	−0.432756	−	0.901511i	$-0.642459\pi$
$797$	−24.4344	−0.865512	−0.432756	+	0.901511i	$0.642459\pi$
$798$	0	0
$799$	−4.00750	−0.141775
$800$	7.36519	0.260399
$801$	0	0
$802$	28.0079	0.988992
$803$	4.52866	0.159813
$804$	0	0
$805$	3.39185	0.119547
$806$	3.84125	0.135302
$807$	0	0
$808$	36.3490	1.27875
$809$	44.8214	1.57583	0.787917	−	0.615781i	$-0.211159\pi$
$809$	44.8214	1.57583	0.787917	+	0.615781i	$0.211159\pi$
$810$	0	0
$811$	−45.1908	−1.58686	−0.793431	−	0.608660i	$-0.791708\pi$
$811$	−45.1908	−1.58686	−0.793431	+	0.608660i	$0.791708\pi$
$812$	−2.44096	−0.0856611
$813$	0	0
$814$	7.26854	0.254762
$815$	−6.02897	−0.211186
$816$	0	0
$817$	−18.9560	−0.663187
$818$	−44.2935	−1.54869
$819$	0	0
$820$	−1.21803	−0.0425355
$821$	−16.0677	−0.560766	−0.280383	−	0.959888i	$-0.590461\pi$
$821$	−16.0677	−0.560766	−0.280383	+	0.959888i	$0.590461\pi$
$822$	0	0
$823$	41.2618	1.43829	0.719147	−	0.694858i	$-0.244533\pi$
$823$	41.2618	1.43829	0.719147	+	0.694858i	$0.244533\pi$
$824$	48.6153	1.69359
$825$	0	0
$826$	6.11283	0.212693
$827$	25.9356	0.901868	0.450934	−	0.892557i	$-0.351091\pi$
$827$	25.9356	0.901868	0.450934	+	0.892557i	$0.351091\pi$
$828$	0	0
$829$	4.75429	0.165123	0.0825617	−	0.996586i	$-0.473690\pi$
$829$	4.75429	0.165123	0.0825617	+	0.996586i	$0.473690\pi$
$830$	7.44648	0.258471
$831$	0	0
$832$	3.10269	0.107566
$833$	0.0479647	0.00166188
$834$	0	0
$835$	−0.287630	−0.00995386
$836$	−1.73729	−0.0600855
$837$	0	0
$838$	−4.33561	−0.149771
$839$	23.1850	0.800436	0.400218	−	0.916420i	$-0.368934\pi$
$839$	23.1850	0.800436	0.400218	+	0.916420i	$0.368934\pi$
$840$	0	0
$841$	−18.2353	−0.628805
$842$	−3.95583	−0.136327
$843$	0	0
$844$	−0.113542	−0.00390827
$845$	−6.98127	−0.240163
$846$	0	0
$847$	−2.66849	−0.0916906
$848$	12.5495	0.430950
$849$	0	0
$850$	2.45016	0.0840396
$851$	−12.9874	−0.445202
$852$	0	0
$853$	25.3754	0.868836	0.434418	−	0.900711i	$-0.356954\pi$
$853$	25.3754	0.868836	0.434418	+	0.900711i	$0.356954\pi$
$854$	−36.0055	−1.23208
$855$	0	0
$856$	40.2132	1.37446
$857$	−35.5796	−1.21537	−0.607687	−	0.794176i	$-0.707903\pi$
$857$	−35.5796	−1.21537	−0.607687	+	0.794176i	$0.707903\pi$
$858$	0	0
$859$	49.2064	1.67890	0.839450	−	0.543437i	$-0.182877\pi$
$859$	49.2064	1.67890	0.839450	+	0.543437i	$0.182877\pi$
$860$	−0.459880	−0.0156818
$861$	0	0
$862$	44.1899	1.50511
$863$	−5.74431	−0.195539	−0.0977693	−	0.995209i	$-0.531171\pi$
$863$	−5.74431	−0.195539	−0.0977693	+	0.995209i	$0.531171\pi$
$864$	0	0
$865$	−9.91071	−0.336974
$866$	−26.0006	−0.883536
$867$	0	0
$868$	−6.16599	−0.209287
$869$	−2.17303	−0.0737148
$870$	0	0
$871$	4.36121	0.147774
$872$	−2.92375	−0.0990107
$873$	0	0
$874$	−19.1639	−0.648228
$875$	14.0439	0.474769
$876$	0	0
$877$	−40.8520	−1.37947	−0.689737	−	0.724060i	$-0.742274\pi$
$877$	−40.8520	−1.37947	−0.689737	+	0.724060i	$0.742274\pi$
$878$	−34.0004	−1.14746
$879$	0	0
$880$	1.82441	0.0615008
$881$	−11.0783	−0.373238	−0.186619	−	0.982432i	$-0.559753\pi$
$881$	−11.0783	−0.373238	−0.186619	+	0.982432i	$0.559753\pi$
$882$	0	0
$883$	−27.1634	−0.914122	−0.457061	−	0.889435i	$-0.651098\pi$
$883$	−27.1634	−0.914122	−0.457061	+	0.889435i	$0.651098\pi$
$884$	−0.0390874	−0.00131465
$885$	0	0
$886$	40.7203	1.36803
$887$	−39.6725	−1.33207	−0.666037	−	0.745919i	$-0.732011\pi$
$887$	−39.6725	−1.33207	−0.666037	+	0.745919i	$0.732011\pi$
$888$	0	0
$889$	14.1670	0.475147
$890$	7.14916	0.239640
$891$	0	0
$892$	−4.61372	−0.154479
$893$	62.9251	2.10571
$894$	0	0
$895$	−4.86875	−0.162744
$896$	22.3945	0.748146
$897$	0	0
$898$	−25.8387	−0.862247
$899$	27.1921	0.906906
$900$	0	0
$901$	−1.48016	−0.0493114
$902$	10.5706	0.351962
$903$	0	0
$904$	−17.5211	−0.582743
$905$	5.52577	0.183683
$906$	0	0
$907$	−15.5945	−0.517807	−0.258903	−	0.965903i	$-0.583361\pi$
$907$	−15.5945	−0.517807	−0.258903	+	0.965903i	$0.583361\pi$
$908$	3.45957	0.114810
$909$	0	0
$910$	0.670620	0.0222309
$911$	14.7406	0.488378	0.244189	−	0.969728i	$-0.421478\pi$
$911$	14.7406	0.488378	0.244189	+	0.969728i	$0.421478\pi$
$912$	0	0
$913$	10.4678	0.346433
$914$	−11.0064	−0.364061
$915$	0	0
$916$	−7.22119	−0.238595
$917$	59.7945	1.97459
$918$	0	0
$919$	57.4744	1.89591	0.947953	−	0.318409i	$-0.103148\pi$
$919$	57.4744	1.89591	0.947953	+	0.318409i	$0.103148\pi$
$920$	−3.80008	−0.125285
$921$	0	0
$922$	−7.53817	−0.248256
$923$	2.80121	0.0922029
$924$	0	0
$925$	−26.0725	−0.857258
$926$	51.5013	1.69244
$927$	0	0
$928$	5.13491	0.168562
$929$	−16.3393	−0.536074	−0.268037	−	0.963409i	$-0.586375\pi$
$929$	−16.3393	−0.536074	−0.268037	+	0.963409i	$0.586375\pi$
$930$	0	0
$931$	−0.753135	−0.0246830
$932$	−5.03476	−0.164919
$933$	0	0
$934$	12.2231	0.399952
$935$	−0.215183	−0.00703723
$936$	0	0
$937$	35.8302	1.17052	0.585262	−	0.810845i	$-0.300992\pi$
$937$	35.8302	1.17052	0.585262	+	0.810845i	$0.300992\pi$
$938$	43.2189	1.41115
$939$	0	0
$940$	1.52659	0.0497918
$941$	27.4328	0.894284	0.447142	−	0.894463i	$-0.352442\pi$
$941$	27.4328	0.894284	0.447142	+	0.894463i	$0.352442\pi$
$942$	0	0
$943$	−18.8875	−0.615061
$944$	−5.87493	−0.191213
$945$	0	0
$946$	3.99103	0.129759
$947$	−16.2274	−0.527320	−0.263660	−	0.964616i	$-0.584930\pi$
$947$	−16.2274	−0.527320	−0.263660	+	0.964616i	$0.584930\pi$
$948$	0	0
$949$	−1.59987	−0.0519339
$950$	−38.4720	−1.24819
$951$	0	0
$952$	−3.16603	−0.102612
$953$	−16.2333	−0.525849	−0.262925	−	0.964816i	$-0.584687\pi$
$953$	−16.2333	−0.525849	−0.262925	+	0.964816i	$0.584687\pi$
$954$	0	0
$955$	−5.15799	−0.166909
$956$	−4.98997	−0.161387
$957$	0	0
$958$	9.68345	0.312858
$959$	18.1056	0.584661
$960$	0	0
$961$	37.6884	1.21576
$962$	−2.56780	−0.0827893
$963$	0	0
$964$	−7.08835	−0.228300
$965$	9.87752	0.317969
$966$	0	0
$967$	−27.0741	−0.870645	−0.435323	−	0.900275i	$-0.643366\pi$
$967$	−27.0741	−0.870645	−0.435323	+	0.900275i	$0.643366\pi$
$968$	2.98966	0.0960913
$969$	0	0
$970$	−8.01903	−0.257476
$971$	−43.8900	−1.40850	−0.704248	−	0.709954i	$-0.748716\pi$
$971$	−43.8900	−1.40850	−0.704248	+	0.709954i	$0.748716\pi$
$972$	0	0
$973$	−8.48501	−0.272017
$974$	−29.2952	−0.938678
$975$	0	0
$976$	34.6042	1.10765
$977$	3.75071	0.119996	0.0599979	−	0.998199i	$-0.480891\pi$
$977$	3.75071	0.119996	0.0599979	+	0.998199i	$0.480891\pi$
$978$	0	0
$979$	10.0498	0.321194
$980$	−0.0182713	−0.000583656 0
$981$	0	0
$982$	−13.2283	−0.422131
$983$	44.5684	1.42151	0.710756	−	0.703439i	$-0.248353\pi$
$983$	44.5684	1.42151	0.710756	+	0.703439i	$0.248353\pi$
$984$	0	0
$985$	8.39037	0.267339
$986$	1.70822	0.0544007
$987$	0	0
$988$	0.613744	0.0195258
$989$	−7.13114	−0.226757
$990$	0	0
$991$	10.1581	0.322684	0.161342	−	0.986899i	$-0.448418\pi$
$991$	10.1581	0.322684	0.161342	+	0.986899i	$0.448418\pi$
$992$	12.9710	0.411831
$993$	0	0
$994$	27.7595	0.880479
$995$	0.137968	0.00437389
$996$	0	0
$997$	−12.8463	−0.406846	−0.203423	−	0.979091i	$-0.565207\pi$
$997$	−12.8463	−0.406846	−0.203423	+	0.979091i	$0.565207\pi$
$998$	26.9763	0.853919
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8019.2.a.k.1.15		51
3.2	odd	2		8019.2.a.l.1.37		51
27.4	even	9		891.2.j.c.694.6		102
27.7	even	9		891.2.j.c.199.6		102
27.20	odd	18		297.2.j.c.265.12	yes	102
27.23	odd	18		297.2.j.c.232.12	✓	102

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
297.2.j.c.232.12	✓	102	27.23	odd	18
297.2.j.c.265.12	yes	102	27.20	odd	18
891.2.j.c.199.6		102	27.7	even	9
891.2.j.c.694.6		102	27.4	even	9
8019.2.a.k.1.15		51	1.1	even	1	trivial
8019.2.a.l.1.37		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)

\(f(q)\)	\(=\)	\(q-1.31194 q^{2} -0.278801 q^{4} +0.542226 q^{5} -2.66849 q^{7} +2.98966 q^{8} +O(q^{10})\)	\(q-1.31194 q^{2} -0.278801 q^{4} +0.542226 q^{5} -2.66849 q^{7} +2.98966 q^{8} -0.711370 q^{10} -1.00000 q^{11} +0.353276 q^{13} +3.50092 q^{14} -3.36467 q^{16} +0.396851 q^{17} -6.23129 q^{19} -0.151173 q^{20} +1.31194 q^{22} -2.34418 q^{23} -4.70599 q^{25} -0.463479 q^{26} +0.743980 q^{28} -3.28095 q^{29} -8.28785 q^{31} -1.56507 q^{32} -0.520646 q^{34} -1.44693 q^{35} +5.54028 q^{37} +8.17510 q^{38} +1.62107 q^{40} +8.05719 q^{41} +3.04207 q^{43} +0.278801 q^{44} +3.07543 q^{46} -10.0983 q^{47} +0.120863 q^{49} +6.17400 q^{50} -0.0984940 q^{52} -3.72978 q^{53} -0.542226 q^{55} -7.97789 q^{56} +4.30443 q^{58} +1.74607 q^{59} -10.2846 q^{61} +10.8732 q^{62} +8.78261 q^{64} +0.191556 q^{65} +12.3450 q^{67} -0.110642 q^{68} +1.89829 q^{70} +7.92922 q^{71} -4.52866 q^{73} -7.26854 q^{74} +1.73729 q^{76} +2.66849 q^{77} +2.17303 q^{79} -1.82441 q^{80} -10.5706 q^{82} -10.4678 q^{83} +0.215183 q^{85} -3.99103 q^{86} -2.98966 q^{88} -10.0498 q^{89} -0.942716 q^{91} +0.653560 q^{92} +13.2484 q^{94} -3.37876 q^{95} +11.2727 q^{97} -0.158566 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 51 q + 60 q^{4} - 6 q^{5} + 12 q^{7}+O(q^{10}) \) 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

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more