LMFDB - Embedded newform 6003.2.a.w.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6003,2,Mod(1,6003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6003, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6003 = 3^{2} \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6003.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:	$47.9341963334$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	6003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.809520 q^{2} -1.34468 q^{4} +2.87436 q^{5} +0.883619 q^{7} -2.70758 q^{8} +O(q^{10})$	\(q+0.809520 q^{2} -1.34468 q^{4} +2.87436 q^{5} +0.883619 q^{7} -2.70758 q^{8} +2.32685 q^{10} +3.13012 q^{11} -3.71846 q^{13} +0.715307 q^{14} +0.497513 q^{16} -7.58717 q^{17} +2.80564 q^{19} -3.86508 q^{20} +2.53390 q^{22} -1.00000 q^{23} +3.26192 q^{25} -3.01017 q^{26} -1.18818 q^{28} +1.00000 q^{29} +0.437909 q^{31} +5.81791 q^{32} -6.14196 q^{34} +2.53984 q^{35} +7.43453 q^{37} +2.27122 q^{38} -7.78256 q^{40} +6.99969 q^{41} -1.35447 q^{43} -4.20901 q^{44} -0.809520 q^{46} +8.71212 q^{47} -6.21922 q^{49} +2.64059 q^{50} +5.00013 q^{52} +13.0730 q^{53} +8.99709 q^{55} -2.39247 q^{56} +0.809520 q^{58} -1.55333 q^{59} +4.35753 q^{61} +0.354496 q^{62} +3.71469 q^{64} -10.6882 q^{65} +6.46860 q^{67} +10.2023 q^{68} +2.05605 q^{70} +13.8126 q^{71} -10.4979 q^{73} +6.01840 q^{74} -3.77268 q^{76} +2.76584 q^{77} +4.47449 q^{79} +1.43003 q^{80} +5.66639 q^{82} +4.00053 q^{83} -21.8082 q^{85} -1.09647 q^{86} -8.47507 q^{88} +9.36855 q^{89} -3.28571 q^{91} +1.34468 q^{92} +7.05263 q^{94} +8.06441 q^{95} +4.35023 q^{97} -5.03458 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + q^{2} + 37 q^{4} + 10 q^{7} + 6 q^{8}+O(q^{10}) $ 30 * q + q^2 + 37 * q^4 + 10 * q^7 + 6 * q^8	$ 30 q + q^{2} + 37 q^{4} + 10 q^{7} + 6 q^{8} + 8 q^{10} + 36 q^{13} + 7 q^{14} + 47 q^{16} + 18 q^{17} + 16 q^{19} + 25 q^{22} - 30 q^{23} + 56 q^{25} + 11 q^{26} + 27 q^{28} + 30 q^{29} + 14 q^{31} - 7 q^{32} + 3 q^{34} - 22 q^{35} + 40 q^{37} + 6 q^{38} + 30 q^{40} + 14 q^{41} + 34 q^{43} + 5 q^{44} - q^{46} - 2 q^{47} + 74 q^{49} - 21 q^{50} + 71 q^{52} + 16 q^{53} + 22 q^{55} + 14 q^{56} + q^{58} - 32 q^{59} + 46 q^{61} + 20 q^{62} + 68 q^{64} + 12 q^{65} + 14 q^{67} + 27 q^{68} - 32 q^{71} + 50 q^{73} - 26 q^{74} + 56 q^{76} + 34 q^{77} + 16 q^{79} + 2 q^{80} + 38 q^{82} - 14 q^{83} + 38 q^{85} + 10 q^{86} + 40 q^{88} - 2 q^{89} + 32 q^{91} - 37 q^{92} + 29 q^{94} - 28 q^{95} + 56 q^{97} + 8 q^{98}+O(q^{100}) $ 30 * q + q^2 + 37 * q^4 + 10 * q^7 + 6 * q^8 + 8 * q^10 + 36 * q^13 + 7 * q^14 + 47 * q^16 + 18 * q^17 + 16 * q^19 + 25 * q^22 - 30 * q^23 + 56 * q^25 + 11 * q^26 + 27 * q^28 + 30 * q^29 + 14 * q^31 - 7 * q^32 + 3 * q^34 - 22 * q^35 + 40 * q^37 + 6 * q^38 + 30 * q^40 + 14 * q^41 + 34 * q^43 + 5 * q^44 - q^46 - 2 * q^47 + 74 * q^49 - 21 * q^50 + 71 * q^52 + 16 * q^53 + 22 * q^55 + 14 * q^56 + q^58 - 32 * q^59 + 46 * q^61 + 20 * q^62 + 68 * q^64 + 12 * q^65 + 14 * q^67 + 27 * q^68 - 32 * q^71 + 50 * q^73 - 26 * q^74 + 56 * q^76 + 34 * q^77 + 16 * q^79 + 2 * q^80 + 38 * q^82 - 14 * q^83 + 38 * q^85 + 10 * q^86 + 40 * q^88 - 2 * q^89 + 32 * q^91 - 37 * q^92 + 29 * q^94 - 28 * q^95 + 56 * q^97 + 8 * 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 $

$2$	0.809520	0.572417	0.286208	−	0.958167i	$-0.407605\pi$
$2$	0.809520	0.572417	0.286208	+	0.958167i	$0.407605\pi$
$3$	0	0
$3$	0	0
$4$	−1.34468	−0.672339
$5$	2.87436	1.28545	0.642726	−	0.766096i	$-0.277804\pi$
$5$	2.87436	1.28545	0.642726	+	0.766096i	$0.277804\pi$
$6$	0	0
$7$	0.883619	0.333977	0.166988	−	0.985959i	$-0.446596\pi$
$7$	0.883619	0.333977	0.166988	+	0.985959i	$0.446596\pi$
$8$	−2.70758	−0.957275
$9$	0	0
$10$	2.32685	0.735814
$11$	3.13012	0.943768	0.471884	−	0.881661i	$-0.343574\pi$
$11$	3.13012	0.943768	0.471884	+	0.881661i	$0.343574\pi$
$12$	0	0
$13$	−3.71846	−1.03132	−0.515658	−	0.856794i	$-0.672453\pi$
$13$	−3.71846	−1.03132	−0.515658	+	0.856794i	$0.672453\pi$
$14$	0.715307	0.191174
$15$	0	0
$16$	0.497513	0.124378
$17$	−7.58717	−1.84016	−0.920079	−	0.391733i	$-0.871876\pi$
$17$	−7.58717	−1.84016	−0.920079	+	0.391733i	$0.871876\pi$
$18$	0	0
$19$	2.80564	0.643658	0.321829	−	0.946798i	$-0.395702\pi$
$19$	2.80564	0.643658	0.321829	+	0.946798i	$0.395702\pi$
$20$	−3.86508	−0.864259
$21$	0	0
$22$	2.53390	0.540229
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	3.26192	0.652385
$26$	−3.01017	−0.590343
$27$	0	0
$28$	−1.18818	−0.224545
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	0.437909	0.0786508	0.0393254	−	0.999226i	$-0.487479\pi$
$31$	0.437909	0.0786508	0.0393254	+	0.999226i	$0.487479\pi$
$32$	5.81791	1.02847
$33$	0	0
$34$	−6.14196	−1.05334
$35$	2.53984	0.429311
$36$	0	0
$37$	7.43453	1.22223	0.611115	−	0.791542i	$-0.290721\pi$
$37$	7.43453	1.22223	0.611115	+	0.791542i	$0.290721\pi$
$38$	2.27122	0.368441
$39$	0	0
$40$	−7.78256	−1.23053
$41$	6.99969	1.09317	0.546584	−	0.837404i	$-0.315928\pi$
$41$	6.99969	1.09317	0.546584	+	0.837404i	$0.315928\pi$
$42$	0	0
$43$	−1.35447	−0.206555	−0.103277	−	0.994653i	$-0.532933\pi$
$43$	−1.35447	−0.206555	−0.103277	+	0.994653i	$0.532933\pi$
$44$	−4.20901	−0.634532
$45$	0	0
$46$	−0.809520	−0.119357
$47$	8.71212	1.27079	0.635396	−	0.772186i	$-0.280837\pi$
$47$	8.71212	1.27079	0.635396	+	0.772186i	$0.280837\pi$
$48$	0	0
$49$	−6.21922	−0.888460
$50$	2.64059	0.373436
$51$	0	0
$52$	5.00013	0.693394
$53$	13.0730	1.79572	0.897860	−	0.440280i	$-0.145121\pi$
$53$	13.0730	1.79572	0.897860	+	0.440280i	$0.145121\pi$
$54$	0	0
$55$	8.99709	1.21317
$56$	−2.39247	−0.319708
$57$	0	0
$58$	0.809520	0.106295
$59$	−1.55333	−0.202227	−0.101113	−	0.994875i	$-0.532240\pi$
$59$	−1.55333	−0.202227	−0.101113	+	0.994875i	$0.532240\pi$
$60$	0	0
$61$	4.35753	0.557925	0.278962	−	0.960302i	$-0.410010\pi$
$61$	4.35753	0.557925	0.278962	+	0.960302i	$0.410010\pi$
$62$	0.354496	0.0450210
$63$	0	0
$64$	3.71469	0.464336
$65$	−10.6882	−1.32571
$66$	0	0
$67$	6.46860	0.790266	0.395133	−	0.918624i	$-0.370699\pi$
$67$	6.46860	0.790266	0.395133	+	0.918624i	$0.370699\pi$
$68$	10.2023	1.23721
$69$	0	0
$70$	2.05605	0.245745
$71$	13.8126	1.63925	0.819627	−	0.572897i	$-0.194181\pi$
$71$	13.8126	1.63925	0.819627	+	0.572897i	$0.194181\pi$
$72$	0	0
$73$	−10.4979	−1.22868	−0.614342	−	0.789040i	$-0.710578\pi$
$73$	−10.4979	−1.22868	−0.614342	+	0.789040i	$0.710578\pi$
$74$	6.01840	0.699625
$75$	0	0
$76$	−3.77268	−0.432756
$77$	2.76584	0.315196
$78$	0	0
$79$	4.47449	0.503420	0.251710	−	0.967803i	$-0.419007\pi$
$79$	4.47449	0.503420	0.251710	+	0.967803i	$0.419007\pi$
$80$	1.43003	0.159882
$81$	0	0
$82$	5.66639	0.625748
$83$	4.00053	0.439115	0.219558	−	0.975600i	$-0.429539\pi$
$83$	4.00053	0.439115	0.219558	+	0.975600i	$0.429539\pi$
$84$	0	0
$85$	−21.8082	−2.36543
$86$	−1.09647	−0.118235
$87$	0	0
$88$	−8.47507	−0.903445
$89$	9.36855	0.993064	0.496532	−	0.868018i	$-0.334607\pi$
$89$	9.36855	0.993064	0.496532	+	0.868018i	$0.334607\pi$
$90$	0	0
$91$	−3.28571	−0.344436
$92$	1.34468	0.140192
$93$	0	0
$94$	7.05263	0.727423
$95$	8.06441	0.827391
$96$	0	0
$97$	4.35023	0.441699	0.220849	−	0.975308i	$-0.429117\pi$
$97$	4.35023	0.441699	0.220849	+	0.975308i	$0.429117\pi$
$98$	−5.03458	−0.508569
$99$	0	0
$100$	−4.38624	−0.438624
$101$	−17.3662	−1.72800	−0.864001	−	0.503490i	$-0.832049\pi$
$101$	−17.3662	−1.72800	−0.864001	+	0.503490i	$0.832049\pi$
$102$	0	0
$103$	5.02647	0.495273	0.247636	−	0.968853i	$-0.420346\pi$
$103$	5.02647	0.495273	0.247636	+	0.968853i	$0.420346\pi$
$104$	10.0680	0.987253
$105$	0	0
$106$	10.5829	1.02790
$107$	−4.57065	−0.441862	−0.220931	−	0.975289i	$-0.570910\pi$
$107$	−4.57065	−0.441862	−0.220931	+	0.975289i	$0.570910\pi$
$108$	0	0
$109$	1.72662	0.165380	0.0826902	−	0.996575i	$-0.473649\pi$
$109$	1.72662	0.165380	0.0826902	+	0.996575i	$0.473649\pi$
$110$	7.28332	0.694437
$111$	0	0
$112$	0.439612	0.0415394
$113$	7.22055	0.679252	0.339626	−	0.940561i	$-0.389699\pi$
$113$	7.22055	0.679252	0.339626	+	0.940561i	$0.389699\pi$
$114$	0	0
$115$	−2.87436	−0.268035
$116$	−1.34468	−0.124850
$117$	0	0
$118$	−1.25745	−0.115758
$119$	−6.70417	−0.614570
$120$	0	0
$121$	−1.20233	−0.109303
$122$	3.52751	0.319366
$123$	0	0
$124$	−0.588846	−0.0528800
$125$	−4.99585	−0.446842
$126$	0	0
$127$	14.0287	1.24484	0.622422	−	0.782682i	$-0.286149\pi$
$127$	14.0287	1.24484	0.622422	+	0.782682i	$0.286149\pi$
$128$	−8.62871	−0.762677
$129$	0	0
$130$	−8.65230	−0.758857
$131$	5.02385	0.438936	0.219468	−	0.975620i	$-0.429568\pi$
$131$	5.02385	0.438936	0.219468	+	0.975620i	$0.429568\pi$
$132$	0	0
$133$	2.47912	0.214967
$134$	5.23646	0.452361
$135$	0	0
$136$	20.5429	1.76154
$137$	−11.5563	−0.987325	−0.493662	−	0.869654i	$-0.664342\pi$
$137$	−11.5563	−0.987325	−0.493662	+	0.869654i	$0.664342\pi$
$138$	0	0
$139$	13.0949	1.11070	0.555349	−	0.831618i	$-0.312585\pi$
$139$	13.0949	1.11070	0.555349	+	0.831618i	$0.312585\pi$
$140$	−3.41526	−0.288642
$141$	0	0
$142$	11.1816	0.938337
$143$	−11.6392	−0.973323
$144$	0	0
$145$	2.87436	0.238702
$146$	−8.49824	−0.703320
$147$	0	0
$148$	−9.99705	−0.821752
$149$	−0.0991848	−0.00812553	−0.00406277	−	0.999992i	$-0.501293\pi$
$149$	−0.0991848	−0.00812553	−0.00406277	+	0.999992i	$0.501293\pi$
$150$	0	0
$151$	−0.619231	−0.0503923	−0.0251962	−	0.999683i	$-0.508021\pi$
$151$	−0.619231	−0.0503923	−0.0251962	+	0.999683i	$0.508021\pi$
$152$	−7.59650	−0.616158
$153$	0	0
$154$	2.23900	0.180424
$155$	1.25871	0.101102
$156$	0	0
$157$	7.83176	0.625043	0.312521	−	0.949911i	$-0.398826\pi$
$157$	7.83176	0.625043	0.312521	+	0.949911i	$0.398826\pi$
$158$	3.62219	0.288166
$159$	0	0
$160$	16.7228	1.32205
$161$	−0.883619	−0.0696390
$162$	0	0
$163$	−4.29528	−0.336432	−0.168216	−	0.985750i	$-0.553801\pi$
$163$	−4.29528	−0.336432	−0.168216	+	0.985750i	$0.553801\pi$
$164$	−9.41233	−0.734980
$165$	0	0
$166$	3.23851	0.251357
$167$	12.4381	0.962489	0.481244	−	0.876586i	$-0.340185\pi$
$167$	12.4381	0.962489	0.481244	+	0.876586i	$0.340185\pi$
$168$	0	0
$169$	0.826970	0.0636131
$170$	−17.6542	−1.35401
$171$	0	0
$172$	1.82132	0.138875
$173$	−3.20942	−0.244008	−0.122004	−	0.992530i	$-0.538932\pi$
$173$	−3.20942	−0.244008	−0.122004	+	0.992530i	$0.538932\pi$
$174$	0	0
$175$	2.88230	0.217881
$176$	1.55728	0.117384
$177$	0	0
$178$	7.58402	0.568447
$179$	11.2810	0.843180	0.421590	−	0.906787i	$-0.361472\pi$
$179$	11.2810	0.843180	0.421590	+	0.906787i	$0.361472\pi$
$180$	0	0
$181$	6.10430	0.453729	0.226864	−	0.973926i	$-0.427153\pi$
$181$	6.10430	0.453729	0.226864	+	0.973926i	$0.427153\pi$
$182$	−2.65984	−0.197161
$183$	0	0
$184$	2.70758	0.199606
$185$	21.3695	1.57112
$186$	0	0
$187$	−23.7488	−1.73668
$188$	−11.7150	−0.854403
$189$	0	0
$190$	6.52830	0.473612
$191$	−11.7280	−0.848606	−0.424303	−	0.905520i	$-0.639481\pi$
$191$	−11.7280	−0.848606	−0.424303	+	0.905520i	$0.639481\pi$
$192$	0	0
$193$	−2.83076	−0.203762	−0.101881	−	0.994797i	$-0.532486\pi$
$193$	−2.83076	−0.203762	−0.101881	+	0.994797i	$0.532486\pi$
$194$	3.52160	0.252836
$195$	0	0
$196$	8.36284	0.597346
$197$	−4.42451	−0.315233	−0.157617	−	0.987500i	$-0.550381\pi$
$197$	−4.42451	−0.315233	−0.157617	+	0.987500i	$0.550381\pi$
$198$	0	0
$199$	−22.4160	−1.58903	−0.794513	−	0.607248i	$-0.792274\pi$
$199$	−22.4160	−1.58903	−0.794513	+	0.607248i	$0.792274\pi$
$200$	−8.83193	−0.624512
$201$	0	0
$202$	−14.0583	−0.989137
$203$	0.883619	0.0620179
$204$	0	0
$205$	20.1196	1.40521
$206$	4.06903	0.283502
$207$	0	0
$208$	−1.84998	−0.128273
$209$	8.78199	0.607463
$210$	0	0
$211$	8.31524	0.572445	0.286222	−	0.958163i	$-0.407600\pi$
$211$	8.31524	0.572445	0.286222	+	0.958163i	$0.407600\pi$
$212$	−17.5790	−1.20733
$213$	0	0
$214$	−3.70004	−0.252929
$215$	−3.89323	−0.265516
$216$	0	0
$217$	0.386945	0.0262675
$218$	1.39773	0.0946665
$219$	0	0
$220$	−12.0982	−0.815659
$221$	28.2126	1.89778
$222$	0	0
$223$	−22.6285	−1.51532	−0.757659	−	0.652651i	$-0.773657\pi$
$223$	−22.6285	−1.51532	−0.757659	+	0.652651i	$0.773657\pi$
$224$	5.14082	0.343485
$225$	0	0
$226$	5.84518	0.388815
$227$	9.82855	0.652344	0.326172	−	0.945311i	$-0.394241\pi$
$227$	9.82855	0.652344	0.326172	+	0.945311i	$0.394241\pi$
$228$	0	0
$229$	2.91799	0.192826	0.0964130	−	0.995341i	$-0.469263\pi$
$229$	2.91799	0.192826	0.0964130	+	0.995341i	$0.469263\pi$
$230$	−2.32685	−0.153428
$231$	0	0
$232$	−2.70758	−0.177762
$233$	2.31506	0.151664	0.0758322	−	0.997121i	$-0.475839\pi$
$233$	2.31506	0.151664	0.0758322	+	0.997121i	$0.475839\pi$
$234$	0	0
$235$	25.0417	1.63354
$236$	2.08873	0.135965
$237$	0	0
$238$	−5.42716	−0.351790
$239$	−2.08815	−0.135071	−0.0675355	−	0.997717i	$-0.521514\pi$
$239$	−2.08815	−0.135071	−0.0675355	+	0.997717i	$0.521514\pi$
$240$	0	0
$241$	7.15468	0.460874	0.230437	−	0.973087i	$-0.425985\pi$
$241$	7.15468	0.460874	0.230437	+	0.973087i	$0.425985\pi$
$242$	−0.973311	−0.0625668
$243$	0	0
$244$	−5.85948	−0.375114
$245$	−17.8762	−1.14207
$246$	0	0
$247$	−10.4327	−0.663815
$248$	−1.18567	−0.0752904
$249$	0	0
$250$	−4.04424	−0.255780
$251$	4.78209	0.301843	0.150921	−	0.988546i	$-0.451776\pi$
$251$	4.78209	0.301843	0.150921	+	0.988546i	$0.451776\pi$
$252$	0	0
$253$	−3.13012	−0.196789
$254$	11.3565	0.712570
$255$	0	0
$256$	−14.4145	−0.900906
$257$	31.2725	1.95072	0.975362	−	0.220612i	$-0.0708053\pi$
$257$	31.2725	1.95072	0.975362	+	0.220612i	$0.0708053\pi$
$258$	0	0
$259$	6.56930	0.408196
$260$	14.3722	0.891324
$261$	0	0
$262$	4.06691	0.251254
$263$	−6.91848	−0.426612	−0.213306	−	0.976985i	$-0.568423\pi$
$263$	−6.91848	−0.426612	−0.213306	+	0.976985i	$0.568423\pi$
$264$	0	0
$265$	37.5766	2.30831
$266$	2.00689	0.123051
$267$	0	0
$268$	−8.69819	−0.531326
$269$	−13.7129	−0.836091	−0.418046	−	0.908426i	$-0.637285\pi$
$269$	−13.7129	−0.836091	−0.418046	+	0.908426i	$0.637285\pi$
$270$	0	0
$271$	3.14468	0.191026	0.0955128	−	0.995428i	$-0.469551\pi$
$271$	3.14468	0.191026	0.0955128	+	0.995428i	$0.469551\pi$
$272$	−3.77471	−0.228876
$273$	0	0
$274$	−9.35509	−0.565162
$275$	10.2102	0.615700
$276$	0	0
$277$	21.6186	1.29894	0.649468	−	0.760389i	$-0.274992\pi$
$277$	21.6186	1.29894	0.649468	+	0.760389i	$0.274992\pi$
$278$	10.6006	0.635782
$279$	0	0
$280$	−6.87682	−0.410968
$281$	−16.8548	−1.00548	−0.502738	−	0.864439i	$-0.667674\pi$
$281$	−16.8548	−1.00548	−0.502738	+	0.864439i	$0.667674\pi$
$282$	0	0
$283$	21.7015	1.29002	0.645011	−	0.764174i	$-0.276853\pi$
$283$	21.7015	1.29002	0.645011	+	0.764174i	$0.276853\pi$
$284$	−18.5735	−1.10213
$285$	0	0
$286$	−9.42220	−0.557146
$287$	6.18507	0.365093
$288$	0	0
$289$	40.5651	2.38618
$290$	2.32685	0.136637
$291$	0	0
$292$	14.1163	0.826092
$293$	−3.09119	−0.180589	−0.0902947	−	0.995915i	$-0.528781\pi$
$293$	−3.09119	−0.180589	−0.0902947	+	0.995915i	$0.528781\pi$
$294$	0	0
$295$	−4.46483	−0.259952
$296$	−20.1296	−1.17001
$297$	0	0
$298$	−0.0802920	−0.00465119
$299$	3.71846	0.215044
$300$	0	0
$301$	−1.19683	−0.0689844
$302$	−0.501280	−0.0288454
$303$	0	0
$304$	1.39584	0.0800570
$305$	12.5251	0.717185
$306$	0	0
$307$	29.2889	1.67161	0.835803	−	0.549030i	$-0.185003\pi$
$307$	29.2889	1.67161	0.835803	+	0.549030i	$0.185003\pi$
$308$	−3.71916	−0.211919
$309$	0	0
$310$	1.01895	0.0578723
$311$	−25.0994	−1.42326	−0.711629	−	0.702556i	$-0.752042\pi$
$311$	−25.0994	−1.42326	−0.711629	+	0.702556i	$0.752042\pi$
$312$	0	0
$313$	31.9053	1.80340	0.901698	−	0.432367i	$-0.142322\pi$
$313$	31.9053	1.80340	0.901698	+	0.432367i	$0.142322\pi$
$314$	6.33997	0.357785
$315$	0	0
$316$	−6.01675	−0.338469
$317$	−2.41067	−0.135397	−0.0676985	−	0.997706i	$-0.521566\pi$
$317$	−2.41067	−0.135397	−0.0676985	+	0.997706i	$0.521566\pi$
$318$	0	0
$319$	3.13012	0.175253
$320$	10.6773	0.596881
$321$	0	0
$322$	−0.715307	−0.0398625
$323$	−21.2868	−1.18443
$324$	0	0
$325$	−12.1293	−0.672815
$326$	−3.47711	−0.192580
$327$	0	0
$328$	−18.9523	−1.04646
$329$	7.69819	0.424415
$330$	0	0
$331$	−5.27304	−0.289833	−0.144916	−	0.989444i	$-0.546291\pi$
$331$	−5.27304	−0.289833	−0.144916	+	0.989444i	$0.546291\pi$
$332$	−5.37942	−0.295234
$333$	0	0
$334$	10.0689	0.550945
$335$	18.5931	1.01585
$336$	0	0
$337$	−21.9350	−1.19488	−0.597438	−	0.801915i	$-0.703815\pi$
$337$	−21.9350	−1.19488	−0.597438	+	0.801915i	$0.703815\pi$
$338$	0.669448	0.0364132
$339$	0	0
$340$	29.3250	1.59037
$341$	1.37071	0.0742281
$342$	0	0
$343$	−11.6808	−0.630701
$344$	3.66734	0.197730
$345$	0	0
$346$	−2.59809	−0.139674
$347$	10.3430	0.555244	0.277622	−	0.960690i	$-0.410454\pi$
$347$	10.3430	0.555244	0.277622	+	0.960690i	$0.410454\pi$
$348$	0	0
$349$	−12.0168	−0.643247	−0.321624	−	0.946868i	$-0.604229\pi$
$349$	−12.0168	−0.643247	−0.321624	+	0.946868i	$0.604229\pi$
$350$	2.33328	0.124719
$351$	0	0
$352$	18.2108	0.970638
$353$	17.3683	0.924421	0.462210	−	0.886770i	$-0.347056\pi$
$353$	17.3683	0.924421	0.462210	+	0.886770i	$0.347056\pi$
$354$	0	0
$355$	39.7024	2.10718
$356$	−12.5977	−0.667675
$357$	0	0
$358$	9.13217	0.482650
$359$	−9.90317	−0.522669	−0.261335	−	0.965248i	$-0.584163\pi$
$359$	−9.90317	−0.522669	−0.261335	+	0.965248i	$0.584163\pi$
$360$	0	0
$361$	−11.1284	−0.585705
$362$	4.94155	0.259722
$363$	0	0
$364$	4.41822	0.231577
$365$	−30.1746	−1.57941
$366$	0	0
$367$	10.0885	0.526614	0.263307	−	0.964712i	$-0.415187\pi$
$367$	10.0885	0.526614	0.263307	+	0.964712i	$0.415187\pi$
$368$	−0.497513	−0.0259347
$369$	0	0
$370$	17.2990	0.899334
$371$	11.5516	0.599729
$372$	0	0
$373$	−7.12698	−0.369021	−0.184511	−	0.982831i	$-0.559070\pi$
$373$	−7.12698	−0.369021	−0.184511	+	0.982831i	$0.559070\pi$
$374$	−19.2251	−0.994106
$375$	0	0
$376$	−23.5888	−1.21650
$377$	−3.71846	−0.191511
$378$	0	0
$379$	28.9649	1.48783	0.743914	−	0.668276i	$-0.232967\pi$
$379$	28.9649	1.48783	0.743914	+	0.668276i	$0.232967\pi$
$380$	−10.8440	−0.556287
$381$	0	0
$382$	−9.49403	−0.485757
$383$	32.2072	1.64571	0.822857	−	0.568249i	$-0.192379\pi$
$383$	32.2072	1.64571	0.822857	+	0.568249i	$0.192379\pi$
$384$	0	0
$385$	7.95000	0.405170
$386$	−2.29156	−0.116637
$387$	0	0
$388$	−5.84965	−0.296971
$389$	−18.8698	−0.956738	−0.478369	−	0.878159i	$-0.658772\pi$
$389$	−18.8698	−0.956738	−0.478369	+	0.878159i	$0.658772\pi$
$390$	0	0
$391$	7.58717	0.383699
$392$	16.8390	0.850500
$393$	0	0
$394$	−3.58173	−0.180445
$395$	12.8613	0.647122
$396$	0	0
$397$	−2.26705	−0.113780	−0.0568900	−	0.998380i	$-0.518118\pi$
$397$	−2.26705	−0.113780	−0.0568900	+	0.998380i	$0.518118\pi$
$398$	−18.1462	−0.909585
$399$	0	0
$400$	1.62285	0.0811425
$401$	−19.4241	−0.969992	−0.484996	−	0.874517i	$-0.661179\pi$
$401$	−19.4241	−0.969992	−0.484996	+	0.874517i	$0.661179\pi$
$402$	0	0
$403$	−1.62835	−0.0811138
$404$	23.3519	1.16180
$405$	0	0
$406$	0.715307	0.0355001
$407$	23.2710	1.15350
$408$	0	0
$409$	4.96080	0.245296	0.122648	−	0.992450i	$-0.460861\pi$
$409$	4.96080	0.245296	0.122648	+	0.992450i	$0.460861\pi$
$410$	16.2872	0.804369
$411$	0	0
$412$	−6.75898	−0.332991
$413$	−1.37255	−0.0675390
$414$	0	0
$415$	11.4989	0.564461
$416$	−21.6337	−1.06068
$417$	0	0
$418$	7.10920	0.347722
$419$	−34.2462	−1.67304	−0.836518	−	0.547939i	$-0.815412\pi$
$419$	−34.2462	−1.67304	−0.836518	+	0.547939i	$0.815412\pi$
$420$	0	0
$421$	−0.614100	−0.0299294	−0.0149647	−	0.999888i	$-0.504764\pi$
$421$	−0.614100	−0.0299294	−0.0149647	+	0.999888i	$0.504764\pi$
$422$	6.73135	0.327677
$423$	0	0
$424$	−35.3963	−1.71900
$425$	−24.7488	−1.20049
$426$	0	0
$427$	3.85040	0.186334
$428$	6.14606	0.297081
$429$	0	0
$430$	−3.15164	−0.151986
$431$	−24.4445	−1.17745	−0.588726	−	0.808333i	$-0.700370\pi$
$431$	−24.4445	−1.17745	−0.588726	+	0.808333i	$0.700370\pi$
$432$	0	0
$433$	32.8240	1.57742	0.788710	−	0.614766i	$-0.210749\pi$
$433$	32.8240	1.57742	0.788710	+	0.614766i	$0.210749\pi$
$434$	0.313240	0.0150360
$435$	0	0
$436$	−2.32175	−0.111192
$437$	−2.80564	−0.134212
$438$	0	0
$439$	30.8312	1.47149	0.735746	−	0.677257i	$-0.236832\pi$
$439$	30.8312	1.47149	0.735746	+	0.677257i	$0.236832\pi$
$440$	−24.3604	−1.16133
$441$	0	0
$442$	22.8387	1.08632
$443$	−3.00139	−0.142601	−0.0713003	−	0.997455i	$-0.522715\pi$
$443$	−3.00139	−0.142601	−0.0713003	+	0.997455i	$0.522715\pi$
$444$	0	0
$445$	26.9285	1.27654
$446$	−18.3182	−0.867394
$447$	0	0
$448$	3.28237	0.155077
$449$	−19.2743	−0.909609	−0.454805	−	0.890591i	$-0.650291\pi$
$449$	−19.2743	−0.909609	−0.454805	+	0.890591i	$0.650291\pi$
$450$	0	0
$451$	21.9099	1.03170
$452$	−9.70931	−0.456688
$453$	0	0
$454$	7.95640	0.373413
$455$	−9.44429	−0.442755
$456$	0	0
$457$	−6.01229	−0.281243	−0.140622	−	0.990063i	$-0.544910\pi$
$457$	−6.01229	−0.281243	−0.140622	+	0.990063i	$0.544910\pi$
$458$	2.36217	0.110377
$459$	0	0
$460$	3.86508	0.180210
$461$	−8.56537	−0.398929	−0.199465	−	0.979905i	$-0.563920\pi$
$461$	−8.56537	−0.398929	−0.199465	+	0.979905i	$0.563920\pi$
$462$	0	0
$463$	5.85107	0.271922	0.135961	−	0.990714i	$-0.456588\pi$
$463$	5.85107	0.271922	0.135961	+	0.990714i	$0.456588\pi$
$464$	0.497513	0.0230965
$465$	0	0
$466$	1.87409	0.0868153
$467$	−1.80231	−0.0834008	−0.0417004	−	0.999130i	$-0.513278\pi$
$467$	−1.80231	−0.0834008	−0.0417004	+	0.999130i	$0.513278\pi$
$468$	0	0
$469$	5.71578	0.263930
$470$	20.2718	0.935067
$471$	0	0
$472$	4.20577	0.193586
$473$	−4.23965	−0.194939
$474$	0	0
$475$	9.15178	0.419912
$476$	9.01494	0.413199
$477$	0	0
$478$	−1.69040	−0.0773170
$479$	−24.1615	−1.10397	−0.551984	−	0.833854i	$-0.686129\pi$
$479$	−24.1615	−1.10397	−0.551984	+	0.833854i	$0.686129\pi$
$480$	0	0
$481$	−27.6450	−1.26051
$482$	5.79186	0.263812
$483$	0	0
$484$	1.61675	0.0734885
$485$	12.5041	0.567782
$486$	0	0
$487$	21.3798	0.968810	0.484405	−	0.874844i	$-0.339036\pi$
$487$	21.3798	0.968810	0.484405	+	0.874844i	$0.339036\pi$
$488$	−11.7984	−0.534088
$489$	0	0
$490$	−14.4712	−0.653741
$491$	7.76902	0.350611	0.175305	−	0.984514i	$-0.443909\pi$
$491$	7.76902	0.350611	0.175305	+	0.984514i	$0.443909\pi$
$492$	0	0
$493$	−7.58717	−0.341709
$494$	−8.44545	−0.379979
$495$	0	0
$496$	0.217865	0.00978245
$497$	12.2051	0.547473
$498$	0	0
$499$	−17.2323	−0.771425	−0.385712	−	0.922619i	$-0.626044\pi$
$499$	−17.2323	−0.771425	−0.385712	+	0.922619i	$0.626044\pi$
$500$	6.71781	0.300430
$501$	0	0
$502$	3.87120	0.172780
$503$	−31.7244	−1.41452	−0.707261	−	0.706953i	$-0.750069\pi$
$503$	−31.7244	−1.41452	−0.707261	+	0.706953i	$0.750069\pi$
$504$	0	0
$505$	−49.9166	−2.22126
$506$	−2.53390	−0.112645
$507$	0	0
$508$	−18.8640	−0.836957
$509$	−18.1047	−0.802477	−0.401238	−	0.915974i	$-0.631420\pi$
$509$	−18.1047	−0.802477	−0.401238	+	0.915974i	$0.631420\pi$
$510$	0	0
$511$	−9.27613	−0.410352
$512$	5.58860	0.246984
$513$	0	0
$514$	25.3157	1.11663
$515$	14.4479	0.636649
$516$	0	0
$517$	27.2700	1.19933
$518$	5.31798	0.233658
$519$	0	0
$520$	28.9392	1.26907
$521$	13.9997	0.613339	0.306670	−	0.951816i	$-0.400785\pi$
$521$	13.9997	0.613339	0.306670	+	0.951816i	$0.400785\pi$
$522$	0	0
$523$	25.2533	1.10425	0.552125	−	0.833761i	$-0.313817\pi$
$523$	25.2533	1.10425	0.552125	+	0.833761i	$0.313817\pi$
$524$	−6.75546	−0.295114
$525$	0	0
$526$	−5.60065	−0.244200
$527$	−3.32249	−0.144730
$528$	0	0
$529$	1.00000	0.0434783
$530$	30.4190	1.32132
$531$	0	0
$532$	−3.33361	−0.144530
$533$	−26.0281	−1.12740
$534$	0	0
$535$	−13.1377	−0.567992
$536$	−17.5143	−0.756502
$537$	0	0
$538$	−11.1009	−0.478593
$539$	−19.4669	−0.838499
$540$	0	0
$541$	23.1425	0.994973	0.497486	−	0.867472i	$-0.334256\pi$
$541$	23.1425	0.994973	0.497486	+	0.867472i	$0.334256\pi$
$542$	2.54568	0.109346
$543$	0	0
$544$	−44.1415	−1.89255
$545$	4.96293	0.212588
$546$	0	0
$547$	−36.9010	−1.57777	−0.788887	−	0.614538i	$-0.789342\pi$
$547$	−36.9010	−1.57777	−0.788887	+	0.614538i	$0.789342\pi$
$548$	15.5396	0.663817
$549$	0	0
$550$	8.26538	0.352437
$551$	2.80564	0.119524
$552$	0	0
$553$	3.95375	0.168130
$554$	17.5007	0.743533
$555$	0	0
$556$	−17.6085	−0.746765
$557$	−36.7219	−1.55596	−0.777978	−	0.628291i	$-0.783755\pi$
$557$	−36.7219	−1.55596	−0.777978	+	0.628291i	$0.783755\pi$
$558$	0	0
$559$	5.03654	0.213023
$560$	1.26360	0.0533969
$561$	0	0
$562$	−13.6443	−0.575551
$563$	−17.3866	−0.732759	−0.366379	−	0.930466i	$-0.619403\pi$
$563$	−17.3866	−0.732759	−0.366379	+	0.930466i	$0.619403\pi$
$564$	0	0
$565$	20.7544	0.873145
$566$	17.5678	0.738430
$567$	0	0
$568$	−37.3988	−1.56922
$569$	22.6275	0.948594	0.474297	−	0.880365i	$-0.342702\pi$
$569$	22.6275	0.948594	0.474297	+	0.880365i	$0.342702\pi$
$570$	0	0
$571$	−39.1499	−1.63837	−0.819186	−	0.573529i	$-0.805574\pi$
$571$	−39.1499	−1.63837	−0.819186	+	0.573529i	$0.805574\pi$
$572$	15.6510	0.654403
$573$	0	0
$574$	5.00693	0.208985
$575$	−3.26192	−0.136032
$576$	0	0
$577$	−4.97090	−0.206941	−0.103471	−	0.994633i	$-0.532995\pi$
$577$	−4.97090	−0.206941	−0.103471	+	0.994633i	$0.532995\pi$
$578$	32.8382	1.36589
$579$	0	0
$580$	−3.86508	−0.160489
$581$	3.53494	0.146654
$582$	0	0
$583$	40.9202	1.69474
$584$	28.4239	1.17619
$585$	0	0
$586$	−2.50238	−0.103372
$587$	9.44544	0.389855	0.194927	−	0.980818i	$-0.437553\pi$
$587$	9.44544	0.389855	0.194927	+	0.980818i	$0.437553\pi$
$588$	0	0
$589$	1.22861	0.0506242
$590$	−3.61437	−0.148801
$591$	0	0
$592$	3.69878	0.152019
$593$	43.5097	1.78673	0.893364	−	0.449334i	$-0.148339\pi$
$593$	43.5097	1.78673	0.893364	+	0.449334i	$0.148339\pi$
$594$	0	0
$595$	−19.2702	−0.790000
$596$	0.133372	0.00546311
$597$	0	0
$598$	3.01017	0.123095
$599$	−37.1909	−1.51958	−0.759790	−	0.650169i	$-0.774698\pi$
$599$	−37.1909	−1.51958	−0.759790	+	0.650169i	$0.774698\pi$
$600$	0	0
$601$	22.6491	0.923877	0.461938	−	0.886912i	$-0.347154\pi$
$601$	22.6491	0.923877	0.461938	+	0.886912i	$0.347154\pi$
$602$	−0.968861	−0.0394878
$603$	0	0
$604$	0.832666	0.0338807
$605$	−3.45593	−0.140503
$606$	0	0
$607$	30.7230	1.24701	0.623504	−	0.781821i	$-0.285709\pi$
$607$	30.7230	1.24701	0.623504	+	0.781821i	$0.285709\pi$
$608$	16.3230	0.661984
$609$	0	0
$610$	10.1393	0.410529
$611$	−32.3957	−1.31059
$612$	0	0
$613$	1.20728	0.0487616	0.0243808	−	0.999703i	$-0.492239\pi$
$613$	1.20728	0.0487616	0.0243808	+	0.999703i	$0.492239\pi$
$614$	23.7099	0.956856
$615$	0	0
$616$	−7.48873	−0.301730
$617$	20.6889	0.832906	0.416453	−	0.909157i	$-0.363273\pi$
$617$	20.6889	0.832906	0.416453	+	0.909157i	$0.363273\pi$
$618$	0	0
$619$	−24.6432	−0.990496	−0.495248	−	0.868752i	$-0.664923\pi$
$619$	−24.6432	−0.990496	−0.495248	+	0.868752i	$0.664923\pi$
$620$	−1.69255	−0.0679746
$621$	0	0
$622$	−20.3185	−0.814697
$623$	8.27823	0.331660
$624$	0	0
$625$	−30.6695	−1.22678
$626$	25.8280	1.03229
$627$	0	0
$628$	−10.5312	−0.420241
$629$	−56.4070	−2.24910
$630$	0	0
$631$	3.58591	0.142753	0.0713764	−	0.997449i	$-0.477261\pi$
$631$	3.58591	0.142753	0.0713764	+	0.997449i	$0.477261\pi$
$632$	−12.1151	−0.481911
$633$	0	0
$634$	−1.95149	−0.0775035
$635$	40.3234	1.60019
$636$	0	0
$637$	23.1259	0.916283
$638$	2.53390	0.100318
$639$	0	0
$640$	−24.8020	−0.980385
$641$	22.5787	0.891805	0.445902	−	0.895082i	$-0.352883\pi$
$641$	22.5787	0.891805	0.445902	+	0.895082i	$0.352883\pi$
$642$	0	0
$643$	−18.5402	−0.731153	−0.365577	−	0.930781i	$-0.619128\pi$
$643$	−18.5402	−0.731153	−0.365577	+	0.930781i	$0.619128\pi$
$644$	1.18818	0.0468210
$645$	0	0
$646$	−17.2321	−0.677989
$647$	16.6734	0.655501	0.327750	−	0.944764i	$-0.393710\pi$
$647$	16.6734	0.655501	0.327750	+	0.944764i	$0.393710\pi$
$648$	0	0
$649$	−4.86212	−0.190855
$650$	−9.81894	−0.385131
$651$	0	0
$652$	5.77577	0.226196
$653$	12.4616	0.487662	0.243831	−	0.969818i	$-0.421596\pi$
$653$	12.4616	0.487662	0.243831	+	0.969818i	$0.421596\pi$
$654$	0	0
$655$	14.4403	0.564231
$656$	3.48244	0.135966
$657$	0	0
$658$	6.23184	0.242942
$659$	−16.5058	−0.642973	−0.321487	−	0.946914i	$-0.604183\pi$
$659$	−16.5058	−0.642973	−0.321487	+	0.946914i	$0.604183\pi$
$660$	0	0
$661$	−33.0840	−1.28682	−0.643408	−	0.765523i	$-0.722480\pi$
$661$	−33.0840	−1.28682	−0.643408	+	0.765523i	$0.722480\pi$
$662$	−4.26863	−0.165905
$663$	0	0
$664$	−10.8318	−0.420354
$665$	7.12586	0.276329
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	−16.7252	−0.647119
$669$	0	0
$670$	15.0515	0.581489
$671$	13.6396	0.526551
$672$	0	0
$673$	−31.1514	−1.20080	−0.600399	−	0.799700i	$-0.704992\pi$
$673$	−31.1514	−1.20080	−0.600399	+	0.799700i	$0.704992\pi$
$674$	−17.7568	−0.683968
$675$	0	0
$676$	−1.11201	−0.0427695
$677$	−9.31505	−0.358006	−0.179003	−	0.983848i	$-0.557287\pi$
$677$	−9.31505	−0.358006	−0.179003	+	0.983848i	$0.557287\pi$
$678$	0	0
$679$	3.84394	0.147517
$680$	59.0476	2.26437
$681$	0	0
$682$	1.10962	0.0424894
$683$	−3.54564	−0.135670	−0.0678350	−	0.997697i	$-0.521609\pi$
$683$	−3.54564	−0.135670	−0.0678350	+	0.997697i	$0.521609\pi$
$684$	0	0
$685$	−33.2170	−1.26916
$686$	−9.45580	−0.361024
$687$	0	0
$688$	−0.673866	−0.0256909
$689$	−48.6116	−1.85196
$690$	0	0
$691$	28.1377	1.07041	0.535204	−	0.844723i	$-0.320235\pi$
$691$	28.1377	1.07041	0.535204	+	0.844723i	$0.320235\pi$
$692$	4.31563	0.164056
$693$	0	0
$694$	8.37290	0.317831
$695$	37.6395	1.42775
$696$	0	0
$697$	−53.1078	−2.01160
$698$	−9.72788	−0.368206
$699$	0	0
$700$	−3.87576	−0.146490
$701$	−8.36492	−0.315939	−0.157969	−	0.987444i	$-0.550495\pi$
$701$	−8.36492	−0.315939	−0.157969	+	0.987444i	$0.550495\pi$
$702$	0	0
$703$	20.8586	0.786697
$704$	11.6274	0.438225
$705$	0	0
$706$	14.0600	0.529154
$707$	−15.3451	−0.577112
$708$	0	0
$709$	−34.2293	−1.28551	−0.642754	−	0.766073i	$-0.722208\pi$
$709$	−34.2293	−1.28551	−0.642754	+	0.766073i	$0.722208\pi$
$710$	32.1398	1.20619
$711$	0	0
$712$	−25.3661	−0.950635
$713$	−0.437909	−0.0163998
$714$	0	0
$715$	−33.4553	−1.25116
$716$	−15.1693	−0.566902
$717$	0	0
$718$	−8.01681	−0.299185
$719$	22.2665	0.830399	0.415199	−	0.909730i	$-0.363712\pi$
$719$	22.2665	0.830399	0.415199	+	0.909730i	$0.363712\pi$
$720$	0	0
$721$	4.44148	0.165410
$722$	−9.00865	−0.335267
$723$	0	0
$724$	−8.20831	−0.305060
$725$	3.26192	0.121145
$726$	0	0
$727$	24.3621	0.903541	0.451770	−	0.892134i	$-0.350793\pi$
$727$	24.3621	0.903541	0.451770	+	0.892134i	$0.350793\pi$
$728$	8.89632	0.329720
$729$	0	0
$730$	−24.4270	−0.904083
$731$	10.2766	0.380093
$732$	0	0
$733$	−27.6426	−1.02100	−0.510502	−	0.859876i	$-0.670540\pi$
$733$	−27.6426	−1.02100	−0.510502	+	0.859876i	$0.670540\pi$
$734$	8.16682	0.301443
$735$	0	0
$736$	−5.81791	−0.214451
$737$	20.2475	0.745827
$738$	0	0
$739$	−33.2693	−1.22383	−0.611916	−	0.790922i	$-0.709601\pi$
$739$	−33.2693	−1.22383	−0.611916	+	0.790922i	$0.709601\pi$
$740$	−28.7351	−1.05632
$741$	0	0
$742$	9.35124	0.343295
$743$	−33.1298	−1.21541	−0.607707	−	0.794162i	$-0.707910\pi$
$743$	−33.1298	−1.21541	−0.607707	+	0.794162i	$0.707910\pi$
$744$	0	0
$745$	−0.285092	−0.0104450
$746$	−5.76943	−0.211234
$747$	0	0
$748$	31.9344	1.16764
$749$	−4.03872	−0.147572
$750$	0	0
$751$	2.19627	0.0801429	0.0400714	−	0.999197i	$-0.487241\pi$
$751$	2.19627	0.0801429	0.0400714	+	0.999197i	$0.487241\pi$
$752$	4.33439	0.158059
$753$	0	0
$754$	−3.01017	−0.109624
$755$	−1.77989	−0.0647769
$756$	0	0
$757$	51.4768	1.87096	0.935478	−	0.353386i	$-0.114970\pi$
$757$	51.4768	1.87096	0.935478	+	0.353386i	$0.114970\pi$
$758$	23.4477	0.851658
$759$	0	0
$760$	−21.8350	−0.792040
$761$	0.0490802	0.00177916	0.000889578	−	1.00000i	$-0.499717\pi$
$761$	0.0490802	0.00177916	0.000889578		1.00000i	$0.499717\pi$
$762$	0	0
$763$	1.52568	0.0552332
$764$	15.7703	0.570551
$765$	0	0
$766$	26.0724	0.942034
$767$	5.77601	0.208560
$768$	0	0
$769$	28.0182	1.01036	0.505181	−	0.863013i	$-0.331426\pi$
$769$	28.0182	1.01036	0.505181	+	0.863013i	$0.331426\pi$
$770$	6.43568	0.231926
$771$	0	0
$772$	3.80646	0.136997
$773$	−2.48136	−0.0892484	−0.0446242	−	0.999004i	$-0.514209\pi$
$773$	−2.48136	−0.0892484	−0.0446242	+	0.999004i	$0.514209\pi$
$774$	0	0
$775$	1.42843	0.0513106
$776$	−11.7786	−0.422827
$777$	0	0
$778$	−15.2755	−0.547653
$779$	19.6386	0.703626
$780$	0	0
$781$	43.2352	1.54708
$782$	6.14196	0.219636
$783$	0	0
$784$	−3.09414	−0.110505
$785$	22.5113	0.803462
$786$	0	0
$787$	−13.8187	−0.492583	−0.246292	−	0.969196i	$-0.579212\pi$
$787$	−13.8187	−0.492583	−0.246292	+	0.969196i	$0.579212\pi$
$788$	5.94954	0.211943
$789$	0	0
$790$	10.4115	0.370423
$791$	6.38022	0.226854
$792$	0	0
$793$	−16.2033	−0.575397
$794$	−1.83522	−0.0651296
$795$	0	0
$796$	30.1422	1.06836
$797$	53.7952	1.90552	0.952761	−	0.303721i	$-0.0982290\pi$
$797$	53.7952	1.90552	0.952761	+	0.303721i	$0.0982290\pi$
$798$	0	0
$799$	−66.1003	−2.33846
$800$	18.9776	0.670959
$801$	0	0
$802$	−15.7242	−0.555240
$803$	−32.8597	−1.15959
$804$	0	0
$805$	−2.53984	−0.0895175
$806$	−1.31818	−0.0464309
$807$	0	0
$808$	47.0204	1.65417
$809$	−31.2565	−1.09892	−0.549460	−	0.835520i	$-0.685167\pi$
$809$	−31.2565	−1.09892	−0.549460	+	0.835520i	$0.685167\pi$
$810$	0	0
$811$	9.48433	0.333040	0.166520	−	0.986038i	$-0.446747\pi$
$811$	9.48433	0.333040	0.166520	+	0.986038i	$0.446747\pi$
$812$	−1.18818	−0.0416971
$813$	0	0
$814$	18.8383	0.660283
$815$	−12.3462	−0.432467
$816$	0	0
$817$	−3.80015	−0.132950
$818$	4.01586	0.140411
$819$	0	0
$820$	−27.0544	−0.944781
$821$	−14.1716	−0.494594	−0.247297	−	0.968940i	$-0.579542\pi$
$821$	−14.1716	−0.494594	−0.247297	+	0.968940i	$0.579542\pi$
$822$	0	0
$823$	3.50100	0.122037	0.0610186	−	0.998137i	$-0.480565\pi$
$823$	3.50100	0.122037	0.0610186	+	0.998137i	$0.480565\pi$
$824$	−13.6096	−0.474112
$825$	0	0
$826$	−1.11111	−0.0386604
$827$	−47.8189	−1.66283	−0.831413	−	0.555655i	$-0.812468\pi$
$827$	−47.8189	−1.66283	−0.831413	+	0.555655i	$0.812468\pi$
$828$	0	0
$829$	22.6429	0.786421	0.393211	−	0.919448i	$-0.371364\pi$
$829$	22.6429	0.786421	0.393211	+	0.919448i	$0.371364\pi$
$830$	9.30862	0.323107
$831$	0	0
$832$	−13.8129	−0.478877
$833$	47.1862	1.63491
$834$	0	0
$835$	35.7515	1.23723
$836$	−11.8090	−0.408421
$837$	0	0
$838$	−27.7230	−0.957674
$839$	−17.4978	−0.604091	−0.302046	−	0.953294i	$-0.597669\pi$
$839$	−17.4978	−0.604091	−0.302046	+	0.953294i	$0.597669\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−0.497126	−0.0171321
$843$	0	0
$844$	−11.1813	−0.384877
$845$	2.37701	0.0817715
$846$	0	0
$847$	−1.06240	−0.0365046
$848$	6.50401	0.223349
$849$	0	0
$850$	−20.0346	−0.687181
$851$	−7.43453	−0.254852
$852$	0	0
$853$	−17.4760	−0.598368	−0.299184	−	0.954195i	$-0.596715\pi$
$853$	−17.4760	−0.598368	−0.299184	+	0.954195i	$0.596715\pi$
$854$	3.11697	0.106661
$855$	0	0
$856$	12.3754	0.422983
$857$	3.15401	0.107739	0.0538695	−	0.998548i	$-0.482845\pi$
$857$	3.15401	0.107739	0.0538695	+	0.998548i	$0.482845\pi$
$858$	0	0
$859$	10.3857	0.354355	0.177177	−	0.984179i	$-0.443303\pi$
$859$	10.3857	0.354355	0.177177	+	0.984179i	$0.443303\pi$
$860$	5.23513	0.178517
$861$	0	0
$862$	−19.7883	−0.673993
$863$	39.1616	1.33308	0.666538	−	0.745471i	$-0.267775\pi$
$863$	39.1616	1.33308	0.666538	+	0.745471i	$0.267775\pi$
$864$	0	0
$865$	−9.22502	−0.313660
$866$	26.5717	0.902942
$867$	0	0
$868$	−0.520316	−0.0176607
$869$	14.0057	0.475111
$870$	0	0
$871$	−24.0533	−0.815014
$872$	−4.67497	−0.158314
$873$	0	0
$874$	−2.27122	−0.0768252
$875$	−4.41443	−0.149235
$876$	0	0
$877$	34.0623	1.15020	0.575102	−	0.818082i	$-0.304962\pi$
$877$	34.0623	1.15020	0.575102	+	0.818082i	$0.304962\pi$
$878$	24.9585	0.842307
$879$	0	0
$880$	4.47617	0.150892
$881$	22.4562	0.756568	0.378284	−	0.925690i	$-0.376514\pi$
$881$	22.4562	0.756568	0.378284	+	0.925690i	$0.376514\pi$
$882$	0	0
$883$	23.5167	0.791401	0.395701	−	0.918380i	$-0.370502\pi$
$883$	23.5167	0.791401	0.395701	+	0.918380i	$0.370502\pi$
$884$	−37.9368	−1.27595
$885$	0	0
$886$	−2.42969	−0.0816270
$887$	28.7714	0.966048	0.483024	−	0.875607i	$-0.339538\pi$
$887$	28.7714	0.966048	0.483024	+	0.875607i	$0.339538\pi$
$888$	0	0
$889$	12.3960	0.415749
$890$	21.7992	0.730710
$891$	0	0
$892$	30.4281	1.01881
$893$	24.4431	0.817956
$894$	0	0
$895$	32.4255	1.08387
$896$	−7.62449	−0.254716
$897$	0	0
$898$	−15.6029	−0.520676
$899$	0.437909	0.0146051
$900$	0	0
$901$	−99.1873	−3.30441
$902$	17.7365	0.590561
$903$	0	0
$904$	−19.5502	−0.650231
$905$	17.5459	0.583246
$906$	0	0
$907$	−43.9992	−1.46097	−0.730484	−	0.682929i	$-0.760706\pi$
$907$	−43.9992	−1.46097	−0.730484	+	0.682929i	$0.760706\pi$
$908$	−13.2162	−0.438596
$909$	0	0
$910$	−7.64534	−0.253441
$911$	1.66879	0.0552894	0.0276447	−	0.999618i	$-0.491199\pi$
$911$	1.66879	0.0552894	0.0276447	+	0.999618i	$0.491199\pi$
$912$	0	0
$913$	12.5221	0.414423
$914$	−4.86707	−0.160988
$915$	0	0
$916$	−3.92375	−0.129644
$917$	4.43917	0.146594
$918$	0	0
$919$	−35.9710	−1.18657	−0.593287	−	0.804991i	$-0.702170\pi$
$919$	−35.9710	−1.18657	−0.593287	+	0.804991i	$0.702170\pi$
$920$	7.78256	0.256583
$921$	0	0
$922$	−6.93384	−0.228354
$923$	−51.3617	−1.69059
$924$	0	0
$925$	24.2509	0.797364
$926$	4.73655	0.155653
$927$	0	0
$928$	5.81791	0.190982
$929$	2.08561	0.0684265	0.0342133	−	0.999415i	$-0.489107\pi$
$929$	2.08561	0.0684265	0.0342133	+	0.999415i	$0.489107\pi$
$930$	0	0
$931$	−17.4489	−0.571864
$932$	−3.11301	−0.101970
$933$	0	0
$934$	−1.45900	−0.0477401
$935$	−68.2624	−2.23242
$936$	0	0
$937$	28.7301	0.938572	0.469286	−	0.883046i	$-0.344511\pi$
$937$	28.7301	0.938572	0.469286	+	0.883046i	$0.344511\pi$
$938$	4.62704	0.151078
$939$	0	0
$940$	−33.6730	−1.09829
$941$	−11.0462	−0.360096	−0.180048	−	0.983658i	$-0.557625\pi$
$941$	−11.0462	−0.360096	−0.180048	+	0.983658i	$0.557625\pi$
$942$	0	0
$943$	−6.99969	−0.227941
$944$	−0.772803	−0.0251526
$945$	0	0
$946$	−3.43208	−0.111587
$947$	7.63217	0.248012	0.124006	−	0.992281i	$-0.460426\pi$
$947$	7.63217	0.248012	0.124006	+	0.992281i	$0.460426\pi$
$948$	0	0
$949$	39.0360	1.26716
$950$	7.40855	0.240365
$951$	0	0
$952$	18.1521	0.588312
$953$	−44.0031	−1.42540	−0.712700	−	0.701469i	$-0.752528\pi$
$953$	−44.0031	−1.42540	−0.712700	+	0.701469i	$0.752528\pi$
$954$	0	0
$955$	−33.7104	−1.09084
$956$	2.80789	0.0908135
$957$	0	0
$958$	−19.5592	−0.631931
$959$	−10.2114	−0.329744
$960$	0	0
$961$	−30.8082	−0.993814
$962$	−22.3792	−0.721535
$963$	0	0
$964$	−9.62074	−0.309863
$965$	−8.13661	−0.261927
$966$	0	0
$967$	14.2064	0.456846	0.228423	−	0.973562i	$-0.426643\pi$
$967$	14.2064	0.456846	0.228423	+	0.973562i	$0.426643\pi$
$968$	3.25541	0.104633
$969$	0	0
$970$	10.1223	0.325008
$971$	−49.4161	−1.58584	−0.792919	−	0.609327i	$-0.791440\pi$
$971$	−49.4161	−1.58584	−0.792919	+	0.609327i	$0.791440\pi$
$972$	0	0
$973$	11.5709	0.370947
$974$	17.3074	0.554564
$975$	0	0
$976$	2.16793	0.0693937
$977$	−40.2639	−1.28816	−0.644078	−	0.764959i	$-0.722759\pi$
$977$	−40.2639	−1.28816	−0.644078	+	0.764959i	$0.722759\pi$
$978$	0	0
$979$	29.3247	0.937221
$980$	24.0378	0.767859
$981$	0	0
$982$	6.28917	0.200696
$983$	−6.79464	−0.216715	−0.108358	−	0.994112i	$-0.534559\pi$
$983$	−6.79464	−0.216715	−0.108358	+	0.994112i	$0.534559\pi$
$984$	0	0
$985$	−12.7176	−0.405217
$986$	−6.14196	−0.195600
$987$	0	0
$988$	14.0286	0.446308
$989$	1.35447	0.0430696
$990$	0	0
$991$	−7.70480	−0.244751	−0.122375	−	0.992484i	$-0.539051\pi$
$991$	−7.70480	−0.244751	−0.122375	+	0.992484i	$0.539051\pi$
$992$	2.54772	0.0808901
$993$	0	0
$994$	9.88026	0.313383
$995$	−64.4315	−2.04261
$996$	0	0
$997$	−7.70589	−0.244048	−0.122024	−	0.992527i	$-0.538938\pi$
$997$	−7.70589	−0.244048	−0.122024	+	0.992527i	$0.538938\pi$
$998$	−13.9499	−0.441577
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.w.1.19	yes	30
3.2	odd	2		6003.2.a.v.1.12	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.v.1.12	✓	30	3.2	odd	2
6003.2.a.w.1.19	yes	30	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:	\( N \)	\(=\)	\( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6003.a (trivial)

\(f(q)\)	\(=\)	\(q+0.809520 q^{2} -1.34468 q^{4} +2.87436 q^{5} +0.883619 q^{7} -2.70758 q^{8} +O(q^{10})\)	\(q+0.809520 q^{2} -1.34468 q^{4} +2.87436 q^{5} +0.883619 q^{7} -2.70758 q^{8} +2.32685 q^{10} +3.13012 q^{11} -3.71846 q^{13} +0.715307 q^{14} +0.497513 q^{16} -7.58717 q^{17} +2.80564 q^{19} -3.86508 q^{20} +2.53390 q^{22} -1.00000 q^{23} +3.26192 q^{25} -3.01017 q^{26} -1.18818 q^{28} +1.00000 q^{29} +0.437909 q^{31} +5.81791 q^{32} -6.14196 q^{34} +2.53984 q^{35} +7.43453 q^{37} +2.27122 q^{38} -7.78256 q^{40} +6.99969 q^{41} -1.35447 q^{43} -4.20901 q^{44} -0.809520 q^{46} +8.71212 q^{47} -6.21922 q^{49} +2.64059 q^{50} +5.00013 q^{52} +13.0730 q^{53} +8.99709 q^{55} -2.39247 q^{56} +0.809520 q^{58} -1.55333 q^{59} +4.35753 q^{61} +0.354496 q^{62} +3.71469 q^{64} -10.6882 q^{65} +6.46860 q^{67} +10.2023 q^{68} +2.05605 q^{70} +13.8126 q^{71} -10.4979 q^{73} +6.01840 q^{74} -3.77268 q^{76} +2.76584 q^{77} +4.47449 q^{79} +1.43003 q^{80} +5.66639 q^{82} +4.00053 q^{83} -21.8082 q^{85} -1.09647 q^{86} -8.47507 q^{88} +9.36855 q^{89} -3.28571 q^{91} +1.34468 q^{92} +7.05263 q^{94} +8.06441 q^{95} +4.35023 q^{97} -5.03458 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + q^{2} + 37 q^{4} + 10 q^{7} + 6 q^{8}+O(q^{10}) \) 30 * q + q^2 + 37 * q^4 + 10 * q^7 + 6 * q^8	\( 30 q + q^{2} + 37 q^{4} + 10 q^{7} + 6 q^{8} + 8 q^{10} + 36 q^{13} + 7 q^{14} + 47 q^{16} + 18 q^{17} + 16 q^{19} + 25 q^{22} - 30 q^{23} + 56 q^{25} + 11 q^{26} + 27 q^{28} + 30 q^{29} + 14 q^{31} - 7 q^{32} + 3 q^{34} - 22 q^{35} + 40 q^{37} + 6 q^{38} + 30 q^{40} + 14 q^{41} + 34 q^{43} + 5 q^{44} - q^{46} - 2 q^{47} + 74 q^{49} - 21 q^{50} + 71 q^{52} + 16 q^{53} + 22 q^{55} + 14 q^{56} + q^{58} - 32 q^{59} + 46 q^{61} + 20 q^{62} + 68 q^{64} + 12 q^{65} + 14 q^{67} + 27 q^{68} - 32 q^{71} + 50 q^{73} - 26 q^{74} + 56 q^{76} + 34 q^{77} + 16 q^{79} + 2 q^{80} + 38 q^{82} - 14 q^{83} + 38 q^{85} + 10 q^{86} + 40 q^{88} - 2 q^{89} + 32 q^{91} - 37 q^{92} + 29 q^{94} - 28 q^{95} + 56 q^{97} + 8 q^{98}+O(q^{100}) \) 30 * q + q^2 + 37 * q^4 + 10 * q^7 + 6 * q^8 + 8 * q^10 + 36 * q^13 + 7 * q^14 + 47 * q^16 + 18 * q^17 + 16 * q^19 + 25 * q^22 - 30 * q^23 + 56 * q^25 + 11 * q^26 + 27 * q^28 + 30 * q^29 + 14 * q^31 - 7 * q^32 + 3 * q^34 - 22 * q^35 + 40 * q^37 + 6 * q^38 + 30 * q^40 + 14 * q^41 + 34 * q^43 + 5 * q^44 - q^46 - 2 * q^47 + 74 * q^49 - 21 * q^50 + 71 * q^52 + 16 * q^53 + 22 * q^55 + 14 * q^56 + q^58 - 32 * q^59 + 46 * q^61 + 20 * q^62 + 68 * q^64 + 12 * q^65 + 14 * q^67 + 27 * q^68 - 32 * q^71 + 50 * q^73 - 26 * q^74 + 56 * q^76 + 34 * q^77 + 16 * q^79 + 2 * q^80 + 38 * q^82 - 14 * q^83 + 38 * q^85 + 10 * q^86 + 40 * q^88 - 2 * q^89 + 32 * q^91 - 37 * q^92 + 29 * q^94 - 28 * q^95 + 56 * q^97 + 8 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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