LMFDB - Embedded newform 6003.2.a.v.1.12

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			1.12
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	$-0.592395\pi$
$2$	−0.809520	−0.572417	−0.286208	+	0.958167i	$0.592395\pi$
$3$	0	0
$3$	0	0
$4$	−1.34468	−0.672339
$5$	−2.87436	−1.28545	−0.642726	−	0.766096i	$-0.722196\pi$
$5$	−2.87436	−1.28545	−0.642726	+	0.766096i	$0.722196\pi$
$6$	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	$-0.656426\pi$
$11$	−3.13012	−0.943768	−0.471884	+	0.881661i	$0.656426\pi$
$12$	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	$-0.128124\pi$
$17$	7.58717	1.84016	0.920079	+	0.391733i	$0.128124\pi$
$18$	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	$-0.684072\pi$
$41$	−6.99969	−1.09317	−0.546584	+	0.837404i	$0.684072\pi$
$42$	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	$-0.719163\pi$
$47$	−8.71212	−1.27079	−0.635396	+	0.772186i	$0.719163\pi$
$48$	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	$-0.854879\pi$
$53$	−13.0730	−1.79572	−0.897860	+	0.440280i	$0.854879\pi$
$54$	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	$-0.467760\pi$
$59$	1.55333	0.202227	0.101113	+	0.994875i	$0.467760\pi$
$60$	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	$-0.805819\pi$
$71$	−13.8126	−1.63925	−0.819627	+	0.572897i	$0.805819\pi$
$72$	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	$-0.570461\pi$
$83$	−4.00053	−0.439115	−0.219558	+	0.975600i	$0.570461\pi$
$84$	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	$-0.665393\pi$
$89$	−9.36855	−0.993064	−0.496532	+	0.868018i	$0.665393\pi$
$90$	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	$-0.167951\pi$
$101$	17.3662	1.72800	0.864001	+	0.503490i	$0.167951\pi$
$102$	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	$-0.429090\pi$
$107$	4.57065	0.441862	0.220931	+	0.975289i	$0.429090\pi$
$108$	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	$-0.610301\pi$
$113$	−7.22055	−0.679252	−0.339626	+	0.940561i	$0.610301\pi$
$114$	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	$-0.570432\pi$
$131$	−5.02385	−0.438936	−0.219468	+	0.975620i	$0.570432\pi$
$132$	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	$-0.335658\pi$
$137$	11.5563	0.987325	0.493662	+	0.869654i	$0.335658\pi$
$138$	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	$-0.498707\pi$
$149$	0.0991848	0.00812553	0.00406277	+	0.999992i	$0.498707\pi$
$150$	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	$-0.659815\pi$
$167$	−12.4381	−0.962489	−0.481244	+	0.876586i	$0.659815\pi$
$168$	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	$-0.461068\pi$
$173$	3.20942	0.244008	0.122004	+	0.992530i	$0.461068\pi$
$174$	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	$-0.638528\pi$
$179$	−11.2810	−0.843180	−0.421590	+	0.906787i	$0.638528\pi$
$180$	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	$-0.360519\pi$
$191$	11.7280	0.848606	0.424303	+	0.905520i	$0.360519\pi$
$192$	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	$-0.449619\pi$
$197$	4.42451	0.315233	0.157617	+	0.987500i	$0.449619\pi$
$198$	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	$-0.605759\pi$
$227$	−9.82855	−0.652344	−0.326172	+	0.945311i	$0.605759\pi$
$228$	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	$-0.524161\pi$
$233$	−2.31506	−0.151664	−0.0758322	+	0.997121i	$0.524161\pi$
$234$	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	$-0.478486\pi$
$239$	2.08815	0.135071	0.0675355	+	0.997717i	$0.478486\pi$
$240$	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	$-0.548224\pi$
$251$	−4.78209	−0.301843	−0.150921	+	0.988546i	$0.548224\pi$
$252$	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	$-0.929195\pi$
$257$	−31.2725	−1.95072	−0.975362	+	0.220612i	$0.929195\pi$
$258$	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	$-0.431577\pi$
$263$	6.91848	0.426612	0.213306	+	0.976985i	$0.431577\pi$
$264$	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	$-0.362715\pi$
$269$	13.7129	0.836091	0.418046	+	0.908426i	$0.362715\pi$
$270$	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	$-0.332326\pi$
$281$	16.8548	1.00548	0.502738	+	0.864439i	$0.332326\pi$
$282$	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	$-0.471219\pi$
$293$	3.09119	0.180589	0.0902947	+	0.995915i	$0.471219\pi$
$294$	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	$-0.247958\pi$
$311$	25.0994	1.42326	0.711629	+	0.702556i	$0.247958\pi$
$312$	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	$-0.478434\pi$
$317$	2.41067	0.135397	0.0676985	+	0.997706i	$0.478434\pi$
$318$	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	$-0.589546\pi$
$347$	−10.3430	−0.555244	−0.277622	+	0.960690i	$0.589546\pi$
$348$	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	$-0.652944\pi$
$353$	−17.3683	−0.924421	−0.462210	+	0.886770i	$0.652944\pi$
$354$	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	$-0.415837\pi$
$359$	9.90317	0.522669	0.261335	+	0.965248i	$0.415837\pi$
$360$	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	$-0.807621\pi$
$383$	−32.2072	−1.64571	−0.822857	+	0.568249i	$0.807621\pi$
$384$	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	$-0.341228\pi$
$389$	18.8698	0.956738	0.478369	+	0.878159i	$0.341228\pi$
$390$	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	$-0.338821\pi$
$401$	19.4241	0.969992	0.484996	+	0.874517i	$0.338821\pi$
$402$	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	$-0.184588\pi$
$419$	34.2462	1.67304	0.836518	+	0.547939i	$0.184588\pi$
$420$	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	$-0.299630\pi$
$431$	24.4445	1.17745	0.588726	+	0.808333i	$0.299630\pi$
$432$	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	$-0.477285\pi$
$443$	3.00139	0.142601	0.0713003	+	0.997455i	$0.477285\pi$
$444$	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	$-0.349709\pi$
$449$	19.2743	0.909609	0.454805	+	0.890591i	$0.349709\pi$
$450$	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	$-0.436080\pi$
$461$	8.56537	0.398929	0.199465	+	0.979905i	$0.436080\pi$
$462$	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	$-0.486722\pi$
$467$	1.80231	0.0834008	0.0417004	+	0.999130i	$0.486722\pi$
$468$	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	$-0.313871\pi$
$479$	24.1615	1.10397	0.551984	+	0.833854i	$0.313871\pi$
$480$	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	$-0.556091\pi$
$491$	−7.76902	−0.350611	−0.175305	+	0.984514i	$0.556091\pi$
$492$	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	$-0.249931\pi$
$503$	31.7244	1.41452	0.707261	+	0.706953i	$0.249931\pi$
$504$	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	$-0.368580\pi$
$509$	18.1047	0.802477	0.401238	+	0.915974i	$0.368580\pi$
$510$	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	$-0.599215\pi$
$521$	−13.9997	−0.613339	−0.306670	+	0.951816i	$0.599215\pi$
$522$	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	$-0.216245\pi$
$557$	36.7219	1.55596	0.777978	+	0.628291i	$0.216245\pi$
$558$	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	$-0.380597\pi$
$563$	17.3866	0.732759	0.366379	+	0.930466i	$0.380597\pi$
$564$	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	$-0.657298\pi$
$569$	−22.6275	−0.948594	−0.474297	+	0.880365i	$0.657298\pi$
$570$	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	$-0.562447\pi$
$587$	−9.44544	−0.389855	−0.194927	+	0.980818i	$0.562447\pi$
$588$	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	$-0.851661\pi$
$593$	−43.5097	−1.78673	−0.893364	+	0.449334i	$0.851661\pi$
$594$	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	$-0.225302\pi$
$599$	37.1909	1.51958	0.759790	+	0.650169i	$0.225302\pi$
$600$	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	$-0.636727\pi$
$617$	−20.6889	−0.832906	−0.416453	+	0.909157i	$0.636727\pi$
$618$	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	$-0.647117\pi$
$641$	−22.5787	−0.891805	−0.445902	+	0.895082i	$0.647117\pi$
$642$	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	$-0.606290\pi$
$647$	−16.6734	−0.655501	−0.327750	+	0.944764i	$0.606290\pi$
$648$	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	$-0.578404\pi$
$653$	−12.4616	−0.487662	−0.243831	+	0.969818i	$0.578404\pi$
$654$	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	$-0.395817\pi$
$659$	16.5058	0.642973	0.321487	+	0.946914i	$0.395817\pi$
$660$	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	$-0.442713\pi$
$677$	9.31505	0.358006	0.179003	+	0.983848i	$0.442713\pi$
$678$	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	$-0.478391\pi$
$683$	3.54564	0.135670	0.0678350	+	0.997697i	$0.478391\pi$
$684$	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	$-0.449505\pi$
$701$	8.36492	0.315939	0.157969	+	0.987444i	$0.449505\pi$
$702$	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	$-0.636288\pi$
$719$	−22.2665	−0.830399	−0.415199	+	0.909730i	$0.636288\pi$
$720$	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	$-0.292090\pi$
$743$	33.1298	1.21541	0.607707	+	0.794162i	$0.292090\pi$
$744$	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	$-0.500283\pi$
$761$	−0.0490802	−0.00177916	−0.000889578		1.00000i	$0.500283\pi$
$762$	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	$-0.485791\pi$
$773$	2.48136	0.0892484	0.0446242	+	0.999004i	$0.485791\pi$
$774$	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	$-0.901771\pi$
$797$	−53.7952	−1.90552	−0.952761	+	0.303721i	$0.901771\pi$
$798$	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	$-0.314833\pi$
$809$	31.2565	1.09892	0.549460	+	0.835520i	$0.314833\pi$
$810$	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	$-0.420458\pi$
$821$	14.1716	0.494594	0.247297	+	0.968940i	$0.420458\pi$
$822$	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	$-0.187532\pi$
$827$	47.8189	1.66283	0.831413	+	0.555655i	$0.187532\pi$
$828$	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	$-0.402331\pi$
$839$	17.4978	0.604091	0.302046	+	0.953294i	$0.402331\pi$
$840$	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	$-0.517155\pi$
$857$	−3.15401	−0.107739	−0.0538695	+	0.998548i	$0.517155\pi$
$858$	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	$-0.732225\pi$
$863$	−39.1616	−1.33308	−0.666538	+	0.745471i	$0.732225\pi$
$864$	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	$-0.623486\pi$
$881$	−22.4562	−0.756568	−0.378284	+	0.925690i	$0.623486\pi$
$882$	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	$-0.660462\pi$
$887$	−28.7714	−0.966048	−0.483024	+	0.875607i	$0.660462\pi$
$888$	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	$-0.508801\pi$
$911$	−1.66879	−0.0552894	−0.0276447	+	0.999618i	$0.508801\pi$
$912$	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	$-0.510893\pi$
$929$	−2.08561	−0.0684265	−0.0342133	+	0.999415i	$0.510893\pi$
$930$	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	$-0.442375\pi$
$941$	11.0462	0.360096	0.180048	+	0.983658i	$0.442375\pi$
$942$	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	$-0.539574\pi$
$947$	−7.63217	−0.248012	−0.124006	+	0.992281i	$0.539574\pi$
$948$	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	$-0.247472\pi$
$953$	44.0031	1.42540	0.712700	+	0.701469i	$0.247472\pi$
$954$	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	$-0.208560\pi$
$971$	49.4161	1.58584	0.792919	+	0.609327i	$0.208560\pi$
$972$	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	$-0.277241\pi$
$977$	40.2639	1.28816	0.644078	+	0.764959i	$0.277241\pi$
$978$	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	$-0.465441\pi$
$983$	6.79464	0.216715	0.108358	+	0.994112i	$0.465441\pi$
$984$	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.v.1.12	✓	30
3.2	odd	2		6003.2.a.w.1.19	yes	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.v.1.12	✓	30	1.1	even	1	trivial
6003.2.a.w.1.19	yes	30	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:	\( N \)	\(=\)	\( 6003 = 3^{2} \cdot 23 \cdot 29 \)
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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