LMFDB - Embedded newform 6003.2.a.t.1.11

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:	$1$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
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)

$f(q)$	$=$	$q-0.230149 q^{2} -1.94703 q^{4} +2.59479 q^{5} -4.77655 q^{7} +0.908406 q^{8} +O(q^{10})$	\(q-0.230149 q^{2} -1.94703 q^{4} +2.59479 q^{5} -4.77655 q^{7} +0.908406 q^{8} -0.597188 q^{10} -3.70131 q^{11} -3.06512 q^{13} +1.09932 q^{14} +3.68499 q^{16} +3.40309 q^{17} +5.00538 q^{19} -5.05213 q^{20} +0.851854 q^{22} +1.00000 q^{23} +1.73291 q^{25} +0.705435 q^{26} +9.30009 q^{28} +1.00000 q^{29} +6.71761 q^{31} -2.66491 q^{32} -0.783218 q^{34} -12.3941 q^{35} +1.21966 q^{37} -1.15198 q^{38} +2.35712 q^{40} -8.51699 q^{41} +8.34577 q^{43} +7.20657 q^{44} -0.230149 q^{46} +3.13922 q^{47} +15.8154 q^{49} -0.398828 q^{50} +5.96789 q^{52} +6.13173 q^{53} -9.60411 q^{55} -4.33904 q^{56} -0.230149 q^{58} -3.83602 q^{59} +3.39205 q^{61} -1.54605 q^{62} -6.75666 q^{64} -7.95333 q^{65} +1.42070 q^{67} -6.62592 q^{68} +2.85250 q^{70} -1.21273 q^{71} -8.66204 q^{73} -0.280704 q^{74} -9.74564 q^{76} +17.6795 q^{77} -12.2141 q^{79} +9.56177 q^{80} +1.96018 q^{82} +14.6713 q^{83} +8.83029 q^{85} -1.92077 q^{86} -3.36229 q^{88} -10.8906 q^{89} +14.6407 q^{91} -1.94703 q^{92} -0.722489 q^{94} +12.9879 q^{95} -1.64503 q^{97} -3.63990 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8}+O(q^{10}) $ 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8	$ 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8} - 12 q^{10} - 28 q^{13} - q^{14} + 3 q^{16} - 10 q^{17} - 8 q^{19} - 11 q^{22} + 22 q^{23} + 11 q^{26} - 21 q^{28} + 22 q^{29} - 18 q^{31} + 5 q^{32} - 33 q^{34} + 2 q^{35} - 28 q^{37} + 14 q^{38} - 30 q^{40} - 10 q^{41} - 14 q^{43} + 37 q^{44} - 3 q^{46} - 18 q^{47} + 2 q^{49} + 7 q^{50} - 57 q^{52} + 20 q^{53} - 42 q^{55} - 2 q^{56} - 3 q^{58} - 20 q^{59} - 38 q^{61} + 4 q^{62} - 24 q^{64} + 12 q^{65} - 50 q^{67} + 11 q^{68} - 48 q^{70} + 12 q^{71} - 46 q^{73} - 6 q^{74} - 16 q^{76} - 14 q^{77} - 20 q^{79} - 58 q^{80} - 42 q^{82} + 22 q^{83} - 66 q^{85} + 22 q^{86} - 68 q^{88} - 14 q^{89} - 16 q^{91} + 17 q^{92} - 27 q^{94} - 20 q^{95} - 48 q^{97} - 28 q^{98}+O(q^{100}) $ 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8 - 12 * q^10 - 28 * q^13 - q^14 + 3 * q^16 - 10 * q^17 - 8 * q^19 - 11 * q^22 + 22 * q^23 + 11 * q^26 - 21 * q^28 + 22 * q^29 - 18 * q^31 + 5 * q^32 - 33 * q^34 + 2 * q^35 - 28 * q^37 + 14 * q^38 - 30 * q^40 - 10 * q^41 - 14 * q^43 + 37 * q^44 - 3 * q^46 - 18 * q^47 + 2 * q^49 + 7 * q^50 - 57 * q^52 + 20 * q^53 - 42 * q^55 - 2 * q^56 - 3 * q^58 - 20 * q^59 - 38 * q^61 + 4 * q^62 - 24 * q^64 + 12 * q^65 - 50 * q^67 + 11 * q^68 - 48 * q^70 + 12 * q^71 - 46 * q^73 - 6 * q^74 - 16 * q^76 - 14 * q^77 - 20 * q^79 - 58 * q^80 - 42 * q^82 + 22 * q^83 - 66 * q^85 + 22 * q^86 - 68 * q^88 - 14 * q^89 - 16 * q^91 + 17 * q^92 - 27 * q^94 - 20 * q^95 - 48 * q^97 - 28 * 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 $

$2$	−0.230149	−0.162740	−0.0813700	−	0.996684i	$-0.525930\pi$
$2$	−0.230149	−0.162740	−0.0813700	+	0.996684i	$0.525930\pi$
$3$	0	0
$3$	0	0
$4$	−1.94703	−0.973516
$5$	2.59479	1.16042	0.580212	−	0.814466i	$-0.302970\pi$
$5$	2.59479	1.16042	0.580212	+	0.814466i	$0.302970\pi$
$6$	0	0
$7$	−4.77655	−1.80537	−0.902683	−	0.430307i	$-0.858405\pi$
$7$	−4.77655	−1.80537	−0.902683	+	0.430307i	$0.858405\pi$
$8$	0.908406	0.321170
$9$	0	0
$10$	−0.597188	−0.188847
$11$	−3.70131	−1.11599	−0.557994	−	0.829845i	$-0.688429\pi$
$11$	−3.70131	−1.11599	−0.557994	+	0.829845i	$0.688429\pi$
$12$	0	0
$13$	−3.06512	−0.850112	−0.425056	−	0.905167i	$-0.639746\pi$
$13$	−3.06512	−0.850112	−0.425056	+	0.905167i	$0.639746\pi$
$14$	1.09932	0.293805
$15$	0	0
$16$	3.68499	0.921248
$17$	3.40309	0.825370	0.412685	−	0.910874i	$-0.364591\pi$
$17$	3.40309	0.825370	0.412685	+	0.910874i	$0.364591\pi$
$18$	0	0
$19$	5.00538	1.14831	0.574157	−	0.818745i	$-0.305330\pi$
$19$	5.00538	1.14831	0.574157	+	0.818745i	$0.305330\pi$
$20$	−5.05213	−1.12969
$21$	0	0
$22$	0.851854	0.181616
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.73291	0.346582
$26$	0.705435	0.138347
$27$	0	0
$28$	9.30009	1.75755
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	6.71761	1.20652	0.603259	−	0.797545i	$-0.293869\pi$
$31$	6.71761	1.20652	0.603259	+	0.797545i	$0.293869\pi$
$32$	−2.66491	−0.471094
$33$	0	0
$34$	−0.783218	−0.134321
$35$	−12.3941	−2.09499
$36$	0	0
$37$	1.21966	0.200511	0.100256	−	0.994962i	$-0.468034\pi$
$37$	1.21966	0.200511	0.100256	+	0.994962i	$0.468034\pi$
$38$	−1.15198	−0.186877
$39$	0	0
$40$	2.35712	0.372693
$41$	−8.51699	−1.33013	−0.665065	−	0.746785i	$-0.731596\pi$
$41$	−8.51699	−1.33013	−0.665065	+	0.746785i	$0.731596\pi$
$42$	0	0
$43$	8.34577	1.27272	0.636359	−	0.771393i	$-0.280440\pi$
$43$	8.34577	1.27272	0.636359	+	0.771393i	$0.280440\pi$
$44$	7.20657	1.08643
$45$	0	0
$46$	−0.230149	−0.0339336
$47$	3.13922	0.457903	0.228951	−	0.973438i	$-0.426470\pi$
$47$	3.13922	0.457903	0.228951	+	0.973438i	$0.426470\pi$
$48$	0	0
$49$	15.8154	2.25934
$50$	−0.398828	−0.0564028
$51$	0	0
$52$	5.96789	0.827597
$53$	6.13173	0.842258	0.421129	−	0.907001i	$-0.361634\pi$
$53$	6.13173	0.842258	0.421129	+	0.907001i	$0.361634\pi$
$54$	0	0
$55$	−9.60411	−1.29502
$56$	−4.33904	−0.579829
$57$	0	0
$58$	−0.230149	−0.0302201
$59$	−3.83602	−0.499408	−0.249704	−	0.968322i	$-0.580333\pi$
$59$	−3.83602	−0.499408	−0.249704	+	0.968322i	$0.580333\pi$
$60$	0	0
$61$	3.39205	0.434307	0.217154	−	0.976137i	$-0.430323\pi$
$61$	3.39205	0.434307	0.217154	+	0.976137i	$0.430323\pi$
$62$	−1.54605	−0.196349
$63$	0	0
$64$	−6.75666	−0.844583
$65$	−7.95333	−0.986490
$66$	0	0
$67$	1.42070	0.173566	0.0867830	−	0.996227i	$-0.472341\pi$
$67$	1.42070	0.173566	0.0867830	+	0.996227i	$0.472341\pi$
$68$	−6.62592	−0.803511
$69$	0	0
$70$	2.85250	0.340938
$71$	−1.21273	−0.143925	−0.0719626	−	0.997407i	$-0.522926\pi$
$71$	−1.21273	−0.143925	−0.0719626	+	0.997407i	$0.522926\pi$
$72$	0	0
$73$	−8.66204	−1.01382	−0.506908	−	0.862000i	$-0.669212\pi$
$73$	−8.66204	−1.01382	−0.506908	+	0.862000i	$0.669212\pi$
$74$	−0.280704	−0.0326312
$75$	0	0
$76$	−9.74564	−1.11790
$77$	17.6795	2.01477
$78$	0	0
$79$	−12.2141	−1.37419	−0.687095	−	0.726567i	$-0.741115\pi$
$79$	−12.2141	−1.37419	−0.687095	+	0.726567i	$0.741115\pi$
$80$	9.56177	1.06904
$81$	0	0
$82$	1.96018	0.216465
$83$	14.6713	1.61038	0.805192	−	0.593014i	$-0.202062\pi$
$83$	14.6713	1.61038	0.805192	+	0.593014i	$0.202062\pi$
$84$	0	0
$85$	8.83029	0.957779
$86$	−1.92077	−0.207122
$87$	0	0
$88$	−3.36229	−0.358422
$89$	−10.8906	−1.15440	−0.577198	−	0.816604i	$-0.695854\pi$
$89$	−10.8906	−1.15440	−0.577198	+	0.816604i	$0.695854\pi$
$90$	0	0
$91$	14.6407	1.53476
$92$	−1.94703	−0.202992
$93$	0	0
$94$	−0.722489	−0.0745191
$95$	12.9879	1.33253
$96$	0	0
$97$	−1.64503	−0.167027	−0.0835137	−	0.996507i	$-0.526614\pi$
$97$	−1.64503	−0.167027	−0.0835137	+	0.996507i	$0.526614\pi$
$98$	−3.63990	−0.367686
$99$	0	0
$100$	−3.37403	−0.337403
$101$	−18.2082	−1.81178	−0.905890	−	0.423512i	$-0.860797\pi$
$101$	−18.2082	−1.81178	−0.905890	+	0.423512i	$0.860797\pi$
$102$	0	0
$103$	−7.91947	−0.780329	−0.390164	−	0.920745i	$-0.627582\pi$
$103$	−7.91947	−0.780329	−0.390164	+	0.920745i	$0.627582\pi$
$104$	−2.78437	−0.273030
$105$	0	0
$106$	−1.41121	−0.137069
$107$	10.1399	0.980259	0.490130	−	0.871650i	$-0.336949\pi$
$107$	10.1399	0.980259	0.490130	+	0.871650i	$0.336949\pi$
$108$	0	0
$109$	−18.2734	−1.75027	−0.875137	−	0.483874i	$-0.839229\pi$
$109$	−18.2734	−1.75027	−0.875137	+	0.483874i	$0.839229\pi$
$110$	2.21038	0.210751
$111$	0	0
$112$	−17.6015	−1.66319
$113$	−8.88638	−0.835961	−0.417980	−	0.908456i	$-0.637262\pi$
$113$	−8.88638	−0.835961	−0.417980	+	0.908456i	$0.637262\pi$
$114$	0	0
$115$	2.59479	0.241965
$116$	−1.94703	−0.180777
$117$	0	0
$118$	0.882858	0.0812737
$119$	−16.2550	−1.49009
$120$	0	0
$121$	2.69971	0.245428
$122$	−0.780677	−0.0706791
$123$	0	0
$124$	−13.0794	−1.17456
$125$	−8.47739	−0.758241
$126$	0	0
$127$	9.25291	0.821063	0.410532	−	0.911846i	$-0.365343\pi$
$127$	9.25291	0.821063	0.410532	+	0.911846i	$0.365343\pi$
$128$	6.88486	0.608541
$129$	0	0
$130$	1.83045	0.160541
$131$	3.01107	0.263078	0.131539	−	0.991311i	$-0.458008\pi$
$131$	3.01107	0.263078	0.131539	+	0.991311i	$0.458008\pi$
$132$	0	0
$133$	−23.9085	−2.07313
$134$	−0.326972	−0.0282461
$135$	0	0
$136$	3.09139	0.265084
$137$	−14.1358	−1.20770	−0.603850	−	0.797098i	$-0.706367\pi$
$137$	−14.1358	−1.20770	−0.603850	+	0.797098i	$0.706367\pi$
$138$	0	0
$139$	7.05151	0.598101	0.299051	−	0.954237i	$-0.403330\pi$
$139$	7.05151	0.598101	0.299051	+	0.954237i	$0.403330\pi$
$140$	24.1317	2.03950
$141$	0	0
$142$	0.279110	0.0234224
$143$	11.3450	0.948714
$144$	0	0
$145$	2.59479	0.215485
$146$	1.99356	0.164988
$147$	0	0
$148$	−2.37472	−0.195201
$149$	−8.33724	−0.683013	−0.341507	−	0.939879i	$-0.610937\pi$
$149$	−8.33724	−0.683013	−0.341507	+	0.939879i	$0.610937\pi$
$150$	0	0
$151$	−10.0416	−0.817170	−0.408585	−	0.912720i	$-0.633978\pi$
$151$	−10.0416	−0.817170	−0.408585	+	0.912720i	$0.633978\pi$
$152$	4.54692	0.368804
$153$	0	0
$154$	−4.06892	−0.327883
$155$	17.4308	1.40007
$156$	0	0
$157$	−3.30825	−0.264027	−0.132014	−	0.991248i	$-0.542144\pi$
$157$	−3.30825	−0.264027	−0.132014	+	0.991248i	$0.542144\pi$
$158$	2.81106	0.223636
$159$	0	0
$160$	−6.91487	−0.546668
$161$	−4.77655	−0.376445
$162$	0	0
$163$	−1.69369	−0.132660	−0.0663301	−	0.997798i	$-0.521129\pi$
$163$	−1.69369	−0.132660	−0.0663301	+	0.997798i	$0.521129\pi$
$164$	16.5828	1.29490
$165$	0	0
$166$	−3.37659	−0.262074
$167$	−4.53024	−0.350561	−0.175280	−	0.984519i	$-0.556083\pi$
$167$	−4.53024	−0.350561	−0.175280	+	0.984519i	$0.556083\pi$
$168$	0	0
$169$	−3.60503	−0.277310
$170$	−2.03228	−0.155869
$171$	0	0
$172$	−16.2495	−1.23901
$173$	−21.7611	−1.65447	−0.827234	−	0.561858i	$-0.810087\pi$
$173$	−21.7611	−1.65447	−0.827234	+	0.561858i	$0.810087\pi$
$174$	0	0
$175$	−8.27734	−0.625708
$176$	−13.6393	−1.02810
$177$	0	0
$178$	2.50645	0.187866
$179$	20.8274	1.55671	0.778357	−	0.627822i	$-0.216053\pi$
$179$	20.8274	1.55671	0.778357	+	0.627822i	$0.216053\pi$
$180$	0	0
$181$	−6.54612	−0.486569	−0.243285	−	0.969955i	$-0.578225\pi$
$181$	−6.54612	−0.486569	−0.243285	+	0.969955i	$0.578225\pi$
$182$	−3.36954	−0.249767
$183$	0	0
$184$	0.908406	0.0669686
$185$	3.16476	0.232678
$186$	0	0
$187$	−12.5959	−0.921103
$188$	−6.11216	−0.445775
$189$	0	0
$190$	−2.98915	−0.216856
$191$	−21.0972	−1.52654	−0.763271	−	0.646078i	$-0.776408\pi$
$191$	−21.0972	−1.52654	−0.763271	+	0.646078i	$0.776408\pi$
$192$	0	0
$193$	5.55042	0.399528	0.199764	−	0.979844i	$-0.435982\pi$
$193$	5.55042	0.399528	0.199764	+	0.979844i	$0.435982\pi$
$194$	0.378602	0.0271820
$195$	0	0
$196$	−30.7931	−2.19951
$197$	20.1657	1.43674	0.718372	−	0.695659i	$-0.244888\pi$
$197$	20.1657	1.43674	0.718372	+	0.695659i	$0.244888\pi$
$198$	0	0
$199$	−23.4093	−1.65944	−0.829719	−	0.558181i	$-0.811500\pi$
$199$	−23.4093	−1.65944	−0.829719	+	0.558181i	$0.811500\pi$
$200$	1.57419	0.111312
$201$	0	0
$202$	4.19060	0.294849
$203$	−4.77655	−0.335248
$204$	0	0
$205$	−22.0998	−1.54351
$206$	1.82266	0.126991
$207$	0	0
$208$	−11.2950	−0.783164
$209$	−18.5265	−1.28150
$210$	0	0
$211$	26.1869	1.80278	0.901391	−	0.433007i	$-0.142547\pi$
$211$	26.1869	1.80278	0.901391	+	0.433007i	$0.142547\pi$
$212$	−11.9387	−0.819952
$213$	0	0
$214$	−2.33369	−0.159527
$215$	21.6555	1.47689
$216$	0	0
$217$	−32.0870	−2.17821
$218$	4.20561	0.284840
$219$	0	0
$220$	18.6995	1.26072
$221$	−10.4309	−0.701657
$222$	0	0
$223$	−14.4965	−0.970755	−0.485378	−	0.874305i	$-0.661318\pi$
$223$	−14.4965	−0.970755	−0.485378	+	0.874305i	$0.661318\pi$
$224$	12.7291	0.850497
$225$	0	0
$226$	2.04519	0.136044
$227$	−13.7819	−0.914738	−0.457369	−	0.889277i	$-0.651208\pi$
$227$	−13.7819	−0.914738	−0.457369	+	0.889277i	$0.651208\pi$
$228$	0	0
$229$	27.3057	1.80441	0.902207	−	0.431303i	$-0.141946\pi$
$229$	27.3057	1.80441	0.902207	+	0.431303i	$0.141946\pi$
$230$	−0.597188	−0.0393774
$231$	0	0
$232$	0.908406	0.0596398
$233$	−8.33427	−0.545996	−0.272998	−	0.962015i	$-0.588015\pi$
$233$	−8.33427	−0.545996	−0.272998	+	0.962015i	$0.588015\pi$
$234$	0	0
$235$	8.14561	0.531361
$236$	7.46886	0.486181
$237$	0	0
$238$	3.74108	0.242498
$239$	21.1215	1.36624	0.683119	−	0.730307i	$-0.260623\pi$
$239$	21.1215	1.36624	0.683119	+	0.730307i	$0.260623\pi$
$240$	0	0
$241$	−6.71214	−0.432367	−0.216184	−	0.976353i	$-0.569361\pi$
$241$	−6.71214	−0.432367	−0.216184	+	0.976353i	$0.569361\pi$
$242$	−0.621336	−0.0399410
$243$	0	0
$244$	−6.60442	−0.422805
$245$	41.0376	2.62180
$246$	0	0
$247$	−15.3421	−0.976195
$248$	6.10231	0.387497
$249$	0	0
$250$	1.95106	0.123396
$251$	23.6023	1.48976	0.744882	−	0.667197i	$-0.232506\pi$
$251$	23.6023	1.48976	0.744882	+	0.667197i	$0.232506\pi$
$252$	0	0
$253$	−3.70131	−0.232699
$254$	−2.12955	−0.133620
$255$	0	0
$256$	11.9288	0.745549
$257$	−11.8463	−0.738955	−0.369477	−	0.929240i	$-0.620463\pi$
$257$	−11.8463	−0.738955	−0.369477	+	0.929240i	$0.620463\pi$
$258$	0	0
$259$	−5.82578	−0.361996
$260$	15.4854	0.960363
$261$	0	0
$262$	−0.692995	−0.0428134
$263$	3.24225	0.199926	0.0999630	−	0.994991i	$-0.468128\pi$
$263$	3.24225	0.199926	0.0999630	+	0.994991i	$0.468128\pi$
$264$	0	0
$265$	15.9105	0.977376
$266$	5.50251	0.337381
$267$	0	0
$268$	−2.76614	−0.168969
$269$	3.56842	0.217570	0.108785	−	0.994065i	$-0.465304\pi$
$269$	3.56842	0.217570	0.108785	+	0.994065i	$0.465304\pi$
$270$	0	0
$271$	6.17978	0.375395	0.187697	−	0.982227i	$-0.439898\pi$
$271$	6.17978	0.375395	0.187697	+	0.982227i	$0.439898\pi$
$272$	12.5404	0.760371
$273$	0	0
$274$	3.25333	0.196541
$275$	−6.41405	−0.386782
$276$	0	0
$277$	20.3224	1.22106	0.610529	−	0.791994i	$-0.290957\pi$
$277$	20.3224	1.22106	0.610529	+	0.791994i	$0.290957\pi$
$278$	−1.62290	−0.0973350
$279$	0	0
$280$	−11.2589	−0.672847
$281$	2.07151	0.123576	0.0617880	−	0.998089i	$-0.480320\pi$
$281$	2.07151	0.123576	0.0617880	+	0.998089i	$0.480320\pi$
$282$	0	0
$283$	−5.58757	−0.332147	−0.166073	−	0.986113i	$-0.553109\pi$
$283$	−5.58757	−0.332147	−0.166073	+	0.986113i	$0.553109\pi$
$284$	2.36123	0.140113
$285$	0	0
$286$	−2.61104	−0.154394
$287$	40.6818	2.40137
$288$	0	0
$289$	−5.41898	−0.318764
$290$	−0.597188	−0.0350681
$291$	0	0
$292$	16.8653	0.986965
$293$	17.7965	1.03969	0.519843	−	0.854262i	$-0.325991\pi$
$293$	17.7965	1.03969	0.519843	+	0.854262i	$0.325991\pi$
$294$	0	0
$295$	−9.95366	−0.579525
$296$	1.10795	0.0643982
$297$	0	0
$298$	1.91881	0.111154
$299$	−3.06512	−0.177261
$300$	0	0
$301$	−39.8640	−2.29772
$302$	2.31105	0.132986
$303$	0	0
$304$	18.4448	1.05788
$305$	8.80163	0.503980
$306$	0	0
$307$	−13.7399	−0.784180	−0.392090	−	0.919927i	$-0.628248\pi$
$307$	−13.7399	−0.784180	−0.392090	+	0.919927i	$0.628248\pi$
$308$	−34.4225	−1.96141
$309$	0	0
$310$	−4.01167	−0.227848
$311$	5.10837	0.289669	0.144835	−	0.989456i	$-0.453735\pi$
$311$	5.10837	0.289669	0.144835	+	0.989456i	$0.453735\pi$
$312$	0	0
$313$	−31.5536	−1.78351	−0.891756	−	0.452516i	$-0.850527\pi$
$313$	−31.5536	−1.78351	−0.891756	+	0.452516i	$0.850527\pi$
$314$	0.761391	0.0429678
$315$	0	0
$316$	23.7812	1.33780
$317$	6.59462	0.370391	0.185195	−	0.982702i	$-0.440708\pi$
$317$	6.59462	0.370391	0.185195	+	0.982702i	$0.440708\pi$
$318$	0	0
$319$	−3.70131	−0.207234
$320$	−17.5321	−0.980073
$321$	0	0
$322$	1.09932	0.0612626
$323$	17.0338	0.947784
$324$	0	0
$325$	−5.31159	−0.294634
$326$	0.389802	0.0215891
$327$	0	0
$328$	−7.73688	−0.427198
$329$	−14.9946	−0.826681
$330$	0	0
$331$	−16.9553	−0.931947	−0.465974	−	0.884799i	$-0.654296\pi$
$331$	−16.9553	−0.931947	−0.465974	+	0.884799i	$0.654296\pi$
$332$	−28.5655	−1.56773
$333$	0	0
$334$	1.04263	0.0570502
$335$	3.68641	0.201410
$336$	0	0
$337$	27.3791	1.49143	0.745717	−	0.666263i	$-0.232107\pi$
$337$	27.3791	1.49143	0.745717	+	0.666263i	$0.232107\pi$
$338$	0.829694	0.0451294
$339$	0	0
$340$	−17.1928	−0.932413
$341$	−24.8640	−1.34646
$342$	0	0
$343$	−42.1072	−2.27358
$344$	7.58135	0.408759
$345$	0	0
$346$	5.00830	0.269248
$347$	−2.62438	−0.140884	−0.0704421	−	0.997516i	$-0.522441\pi$
$347$	−2.62438	−0.140884	−0.0704421	+	0.997516i	$0.522441\pi$
$348$	0	0
$349$	6.65979	0.356490	0.178245	−	0.983986i	$-0.442958\pi$
$349$	6.65979	0.356490	0.178245	+	0.983986i	$0.442958\pi$
$350$	1.90502	0.101828
$351$	0	0
$352$	9.86366	0.525735
$353$	−3.89509	−0.207315	−0.103657	−	0.994613i	$-0.533055\pi$
$353$	−3.89509	−0.207315	−0.103657	+	0.994613i	$0.533055\pi$
$354$	0	0
$355$	−3.14679	−0.167014
$356$	21.2042	1.12382
$357$	0	0
$358$	−4.79341	−0.253340
$359$	13.3307	0.703565	0.351783	−	0.936082i	$-0.385576\pi$
$359$	13.3307	0.703565	0.351783	+	0.936082i	$0.385576\pi$
$360$	0	0
$361$	6.05387	0.318625
$362$	1.50658	0.0791843
$363$	0	0
$364$	−28.5059	−1.49412
$365$	−22.4761	−1.17646
$366$	0	0
$367$	−1.83274	−0.0956684	−0.0478342	−	0.998855i	$-0.515232\pi$
$367$	−1.83274	−0.0956684	−0.0478342	+	0.998855i	$0.515232\pi$
$368$	3.68499	0.192094
$369$	0	0
$370$	−0.728368	−0.0378660
$371$	−29.2885	−1.52058
$372$	0	0
$373$	−37.0251	−1.91709	−0.958544	−	0.284945i	$-0.908025\pi$
$373$	−37.0251	−1.91709	−0.958544	+	0.284945i	$0.908025\pi$
$374$	2.89893	0.149900
$375$	0	0
$376$	2.85169	0.147065
$377$	−3.06512	−0.157862
$378$	0	0
$379$	8.53785	0.438560	0.219280	−	0.975662i	$-0.429629\pi$
$379$	8.53785	0.438560	0.219280	+	0.975662i	$0.429629\pi$
$380$	−25.2878	−1.29724
$381$	0	0
$382$	4.85551	0.248430
$383$	−18.2560	−0.932841	−0.466420	−	0.884563i	$-0.654457\pi$
$383$	−18.2560	−0.932841	−0.466420	+	0.884563i	$0.654457\pi$
$384$	0	0
$385$	45.8745	2.33798
$386$	−1.27743	−0.0650192
$387$	0	0
$388$	3.20292	0.162604
$389$	−19.6829	−0.997965	−0.498983	−	0.866612i	$-0.666293\pi$
$389$	−19.6829	−0.997965	−0.498983	+	0.866612i	$0.666293\pi$
$390$	0	0
$391$	3.40309	0.172102
$392$	14.3668	0.725633
$393$	0	0
$394$	−4.64111	−0.233816
$395$	−31.6929	−1.59464
$396$	0	0
$397$	−1.39049	−0.0697869	−0.0348934	−	0.999391i	$-0.511109\pi$
$397$	−1.39049	−0.0697869	−0.0348934	+	0.999391i	$0.511109\pi$
$398$	5.38762	0.270057
$399$	0	0
$400$	6.38577	0.319289
$401$	5.28138	0.263739	0.131870	−	0.991267i	$-0.457902\pi$
$401$	5.28138	0.263739	0.131870	+	0.991267i	$0.457902\pi$
$402$	0	0
$403$	−20.5903	−1.02568
$404$	35.4519	1.76380
$405$	0	0
$406$	1.09932	0.0545583
$407$	−4.51435	−0.223768
$408$	0	0
$409$	3.95248	0.195438	0.0977188	−	0.995214i	$-0.468845\pi$
$409$	3.95248	0.195438	0.0977188	+	0.995214i	$0.468845\pi$
$410$	5.08624	0.251192
$411$	0	0
$412$	15.4195	0.759662
$413$	18.3230	0.901614
$414$	0	0
$415$	38.0689	1.86873
$416$	8.16827	0.400483
$417$	0	0
$418$	4.26386	0.208552
$419$	−6.29691	−0.307624	−0.153812	−	0.988100i	$-0.549155\pi$
$419$	−6.29691	−0.307624	−0.153812	+	0.988100i	$0.549155\pi$
$420$	0	0
$421$	4.87154	0.237424	0.118712	−	0.992929i	$-0.462123\pi$
$421$	4.87154	0.237424	0.118712	+	0.992929i	$0.462123\pi$
$422$	−6.02690	−0.293385
$423$	0	0
$424$	5.57010	0.270508
$425$	5.89725	0.286059
$426$	0	0
$427$	−16.2023	−0.784083
$428$	−19.7427	−0.954298
$429$	0	0
$430$	−4.98399	−0.240349
$431$	−24.0540	−1.15864	−0.579321	−	0.815100i	$-0.696682\pi$
$431$	−24.0540	−1.15864	−0.579321	+	0.815100i	$0.696682\pi$
$432$	0	0
$433$	−20.1836	−0.969961	−0.484980	−	0.874525i	$-0.661173\pi$
$433$	−20.1836	−0.969961	−0.484980	+	0.874525i	$0.661173\pi$
$434$	7.38479	0.354481
$435$	0	0
$436$	35.5789	1.70392
$437$	5.00538	0.239440
$438$	0	0
$439$	−26.4320	−1.26153	−0.630766	−	0.775973i	$-0.717259\pi$
$439$	−26.4320	−1.26153	−0.630766	+	0.775973i	$0.717259\pi$
$440$	−8.72443	−0.415921
$441$	0	0
$442$	2.40066	0.114188
$443$	−31.8533	−1.51339	−0.756697	−	0.653765i	$-0.773188\pi$
$443$	−31.8533	−1.51339	−0.756697	+	0.653765i	$0.773188\pi$
$444$	0	0
$445$	−28.2586	−1.33959
$446$	3.33635	0.157981
$447$	0	0
$448$	32.2735	1.52478
$449$	29.9464	1.41326	0.706629	−	0.707584i	$-0.250215\pi$
$449$	29.9464	1.41326	0.706629	+	0.707584i	$0.250215\pi$
$450$	0	0
$451$	31.5240	1.48441
$452$	17.3021	0.813821
$453$	0	0
$454$	3.17190	0.148864
$455$	37.9895	1.78097
$456$	0	0
$457$	−23.3812	−1.09373	−0.546864	−	0.837221i	$-0.684179\pi$
$457$	−23.3812	−1.09373	−0.546864	+	0.837221i	$0.684179\pi$
$458$	−6.28439	−0.293650
$459$	0	0
$460$	−5.05213	−0.235557
$461$	3.01774	0.140550	0.0702751	−	0.997528i	$-0.477612\pi$
$461$	3.01774	0.140550	0.0702751	+	0.997528i	$0.477612\pi$
$462$	0	0
$463$	−2.43464	−0.113147	−0.0565736	−	0.998398i	$-0.518018\pi$
$463$	−2.43464	−0.113147	−0.0565736	+	0.998398i	$0.518018\pi$
$464$	3.68499	0.171072
$465$	0	0
$466$	1.91812	0.0888554
$467$	6.71105	0.310551	0.155275	−	0.987871i	$-0.450374\pi$
$467$	6.71105	0.310551	0.155275	+	0.987871i	$0.450374\pi$
$468$	0	0
$469$	−6.78603	−0.313350
$470$	−1.87470	−0.0864737
$471$	0	0
$472$	−3.48467	−0.160395
$473$	−30.8903	−1.42034
$474$	0	0
$475$	8.67389	0.397985
$476$	31.6490	1.45063
$477$	0	0
$478$	−4.86110	−0.222342
$479$	33.6174	1.53602	0.768008	−	0.640440i	$-0.221248\pi$
$479$	33.6174	1.53602	0.768008	+	0.640440i	$0.221248\pi$
$480$	0	0
$481$	−3.73842	−0.170457
$482$	1.54479	0.0703634
$483$	0	0
$484$	−5.25642	−0.238928
$485$	−4.26850	−0.193822
$486$	0	0
$487$	−22.3802	−1.01414	−0.507072	−	0.861904i	$-0.669272\pi$
$487$	−22.3802	−1.01414	−0.507072	+	0.861904i	$0.669272\pi$
$488$	3.08135	0.139486
$489$	0	0
$490$	−9.44477	−0.426671
$491$	13.3390	0.601983	0.300991	−	0.953627i	$-0.402682\pi$
$491$	13.3390	0.601983	0.300991	+	0.953627i	$0.402682\pi$
$492$	0	0
$493$	3.40309	0.153267
$494$	3.53097	0.158866
$495$	0	0
$496$	24.7543	1.11150
$497$	5.79269	0.259837
$498$	0	0
$499$	−15.5475	−0.696001	−0.348000	−	0.937494i	$-0.613139\pi$
$499$	−15.5475	−0.696001	−0.348000	+	0.937494i	$0.613139\pi$
$500$	16.5057	0.738160
$501$	0	0
$502$	−5.43205	−0.242444
$503$	−25.5361	−1.13860	−0.569300	−	0.822130i	$-0.692786\pi$
$503$	−25.5361	−1.13860	−0.569300	+	0.822130i	$0.692786\pi$
$504$	0	0
$505$	−47.2463	−2.10243
$506$	0.851854	0.0378695
$507$	0	0
$508$	−18.0157	−0.799318
$509$	10.4831	0.464653	0.232327	−	0.972638i	$-0.425366\pi$
$509$	10.4831	0.464653	0.232327	+	0.972638i	$0.425366\pi$
$510$	0	0
$511$	41.3747	1.83031
$512$	−16.5151	−0.729872
$513$	0	0
$514$	2.72643	0.120257
$515$	−20.5493	−0.905512
$516$	0	0
$517$	−11.6192	−0.511013
$518$	1.34080	0.0589113
$519$	0	0
$520$	−7.22486	−0.316831
$521$	−13.6878	−0.599673	−0.299836	−	0.953991i	$-0.596932\pi$
$521$	−13.6878	−0.599673	−0.299836	+	0.953991i	$0.596932\pi$
$522$	0	0
$523$	6.86730	0.300286	0.150143	−	0.988664i	$-0.452027\pi$
$523$	6.86730	0.300286	0.150143	+	0.988664i	$0.452027\pi$
$524$	−5.86265	−0.256111
$525$	0	0
$526$	−0.746202	−0.0325360
$527$	22.8606	0.995824
$528$	0	0
$529$	1.00000	0.0434783
$530$	−3.66180	−0.159058
$531$	0	0
$532$	46.5505	2.01822
$533$	26.1056	1.13076
$534$	0	0
$535$	26.3108	1.13752
$536$	1.29057	0.0557442
$537$	0	0
$538$	−0.821268	−0.0354074
$539$	−58.5377	−2.52140
$540$	0	0
$541$	12.8009	0.550355	0.275178	−	0.961393i	$-0.411263\pi$
$541$	12.8009	0.550355	0.275178	+	0.961393i	$0.411263\pi$
$542$	−1.42227	−0.0610917
$543$	0	0
$544$	−9.06893	−0.388827
$545$	−47.4156	−2.03106
$546$	0	0
$547$	37.3728	1.59794	0.798972	−	0.601368i	$-0.205377\pi$
$547$	37.3728	1.59794	0.798972	+	0.601368i	$0.205377\pi$
$548$	27.5228	1.17571
$549$	0	0
$550$	1.47619	0.0629449
$551$	5.00538	0.213237
$552$	0	0
$553$	58.3411	2.48092
$554$	−4.67719	−0.198715
$555$	0	0
$556$	−13.7295	−0.582261
$557$	−35.1787	−1.49057	−0.745286	−	0.666745i	$-0.767687\pi$
$557$	−35.1787	−1.49057	−0.745286	+	0.666745i	$0.767687\pi$
$558$	0	0
$559$	−25.5808	−1.08195
$560$	−45.6722	−1.93000
$561$	0	0
$562$	−0.476757	−0.0201108
$563$	−26.0407	−1.09749	−0.548743	−	0.835991i	$-0.684894\pi$
$563$	−26.0407	−1.09749	−0.548743	+	0.835991i	$0.684894\pi$
$564$	0	0
$565$	−23.0583	−0.970068
$566$	1.28597	0.0540535
$567$	0	0
$568$	−1.10166	−0.0462244
$569$	−17.4792	−0.732767	−0.366383	−	0.930464i	$-0.619404\pi$
$569$	−17.4792	−0.732767	−0.366383	+	0.930464i	$0.619404\pi$
$570$	0	0
$571$	8.05298	0.337007	0.168503	−	0.985701i	$-0.446107\pi$
$571$	8.05298	0.337007	0.168503	+	0.985701i	$0.446107\pi$
$572$	−22.0890	−0.923588
$573$	0	0
$574$	−9.36288	−0.390799
$575$	1.73291	0.0722674
$576$	0	0
$577$	−38.1194	−1.58693	−0.793466	−	0.608614i	$-0.791726\pi$
$577$	−38.1194	−1.58693	−0.793466	+	0.608614i	$0.791726\pi$
$578$	1.24717	0.0518756
$579$	0	0
$580$	−5.05213	−0.209778
$581$	−70.0782	−2.90733
$582$	0	0
$583$	−22.6955	−0.939950
$584$	−7.86865	−0.325607
$585$	0	0
$586$	−4.09586	−0.169198
$587$	−31.8205	−1.31337	−0.656687	−	0.754163i	$-0.728043\pi$
$587$	−31.8205	−1.31337	−0.656687	+	0.754163i	$0.728043\pi$
$588$	0	0
$589$	33.6242	1.38546
$590$	2.29083	0.0943118
$591$	0	0
$592$	4.49445	0.184721
$593$	33.9571	1.39445	0.697226	−	0.716852i	$-0.254418\pi$
$593$	33.9571	1.39445	0.697226	+	0.716852i	$0.254418\pi$
$594$	0	0
$595$	−42.1783	−1.72914
$596$	16.2329	0.664924
$597$	0	0
$598$	0.705435	0.0288474
$599$	4.88042	0.199408	0.0997042	−	0.995017i	$-0.468210\pi$
$599$	4.88042	0.199408	0.0997042	+	0.995017i	$0.468210\pi$
$600$	0	0
$601$	3.53581	0.144229	0.0721143	−	0.997396i	$-0.477025\pi$
$601$	3.53581	0.144229	0.0721143	+	0.997396i	$0.477025\pi$
$602$	9.17466	0.373931
$603$	0	0
$604$	19.5512	0.795528
$605$	7.00517	0.284800
$606$	0	0
$607$	−23.9417	−0.971764	−0.485882	−	0.874024i	$-0.661502\pi$
$607$	−23.9417	−0.971764	−0.485882	+	0.874024i	$0.661502\pi$
$608$	−13.3389	−0.540964
$609$	0	0
$610$	−2.02569	−0.0820177
$611$	−9.62210	−0.389268
$612$	0	0
$613$	24.8422	1.00337	0.501684	−	0.865051i	$-0.332714\pi$
$613$	24.8422	1.00337	0.501684	+	0.865051i	$0.332714\pi$
$614$	3.16223	0.127617
$615$	0	0
$616$	16.0602	0.647082
$617$	−14.1565	−0.569919	−0.284960	−	0.958539i	$-0.591980\pi$
$617$	−14.1565	−0.569919	−0.284960	+	0.958539i	$0.591980\pi$
$618$	0	0
$619$	15.7115	0.631500	0.315750	−	0.948842i	$-0.397744\pi$
$619$	15.7115	0.631500	0.315750	+	0.948842i	$0.397744\pi$
$620$	−33.9382	−1.36299
$621$	0	0
$622$	−1.17569	−0.0471408
$623$	52.0192	2.08411
$624$	0	0
$625$	−30.6616	−1.22646
$626$	7.26203	0.290249
$627$	0	0
$628$	6.44127	0.257035
$629$	4.15062	0.165496
$630$	0	0
$631$	26.8165	1.06755	0.533774	−	0.845627i	$-0.320773\pi$
$631$	26.8165	1.06755	0.533774	+	0.845627i	$0.320773\pi$
$632$	−11.0953	−0.441349
$633$	0	0
$634$	−1.51775	−0.0602774
$635$	24.0093	0.952781
$636$	0	0
$637$	−48.4761	−1.92069
$638$	0.851854	0.0337252
$639$	0	0
$640$	17.8647	0.706166
$641$	−9.40758	−0.371577	−0.185789	−	0.982590i	$-0.559484\pi$
$641$	−9.40758	−0.371577	−0.185789	+	0.982590i	$0.559484\pi$
$642$	0	0
$643$	−16.8530	−0.664616	−0.332308	−	0.943171i	$-0.607827\pi$
$643$	−16.8530	−0.664616	−0.332308	+	0.943171i	$0.607827\pi$
$644$	9.30009	0.366475
$645$	0	0
$646$	−3.92031	−0.154242
$647$	7.52867	0.295983	0.147991	−	0.988989i	$-0.452719\pi$
$647$	7.52867	0.295983	0.147991	+	0.988989i	$0.452719\pi$
$648$	0	0
$649$	14.1983	0.557333
$650$	1.22246	0.0479487
$651$	0	0
$652$	3.29767	0.129147
$653$	−8.35597	−0.326995	−0.163497	−	0.986544i	$-0.552277\pi$
$653$	−8.35597	−0.326995	−0.163497	+	0.986544i	$0.552277\pi$
$654$	0	0
$655$	7.81308	0.305282
$656$	−31.3851	−1.22538
$657$	0	0
$658$	3.45100	0.134534
$659$	−47.0599	−1.83319	−0.916596	−	0.399815i	$-0.869074\pi$
$659$	−47.0599	−1.83319	−0.916596	+	0.399815i	$0.869074\pi$
$660$	0	0
$661$	−19.4971	−0.758350	−0.379175	−	0.925325i	$-0.623792\pi$
$661$	−19.4971	−0.758350	−0.379175	+	0.925325i	$0.623792\pi$
$662$	3.90225	0.151665
$663$	0	0
$664$	13.3275	0.517207
$665$	−62.0373	−2.40570
$666$	0	0
$667$	1.00000	0.0387202
$668$	8.82052	0.341276
$669$	0	0
$670$	−0.848423	−0.0327775
$671$	−12.5550	−0.484681
$672$	0	0
$673$	−25.3526	−0.977271	−0.488636	−	0.872488i	$-0.662505\pi$
$673$	−25.3526	−0.977271	−0.488636	+	0.872488i	$0.662505\pi$
$674$	−6.30127	−0.242716
$675$	0	0
$676$	7.01910	0.269965
$677$	10.0166	0.384969	0.192485	−	0.981300i	$-0.438345\pi$
$677$	10.0166	0.384969	0.192485	+	0.981300i	$0.438345\pi$
$678$	0	0
$679$	7.85756	0.301545
$680$	8.02148	0.307610
$681$	0	0
$682$	5.72242	0.219123
$683$	−22.0140	−0.842342	−0.421171	−	0.906981i	$-0.638381\pi$
$683$	−22.0140	−0.842342	−0.421171	+	0.906981i	$0.638381\pi$
$684$	0	0
$685$	−36.6793	−1.40144
$686$	9.69094	0.370002
$687$	0	0
$688$	30.7541	1.17249
$689$	−18.7945	−0.716014
$690$	0	0
$691$	−23.6597	−0.900059	−0.450029	−	0.893014i	$-0.648586\pi$
$691$	−23.6597	−0.900059	−0.450029	+	0.893014i	$0.648586\pi$
$692$	42.3696	1.61065
$693$	0	0
$694$	0.603999	0.0229275
$695$	18.2972	0.694051
$696$	0	0
$697$	−28.9841	−1.09785
$698$	−1.53275	−0.0580153
$699$	0	0
$700$	16.1162	0.609136
$701$	44.6815	1.68760	0.843799	−	0.536659i	$-0.180314\pi$
$701$	44.6815	1.68760	0.843799	+	0.536659i	$0.180314\pi$
$702$	0	0
$703$	6.10488	0.230250
$704$	25.0085	0.942544
$705$	0	0
$706$	0.896452	0.0337384
$707$	86.9722	3.27093
$708$	0	0
$709$	−15.8352	−0.594705	−0.297353	−	0.954768i	$-0.596104\pi$
$709$	−15.8352	−0.594705	−0.297353	+	0.954768i	$0.596104\pi$
$710$	0.724230	0.0271799
$711$	0	0
$712$	−9.89304	−0.370757
$713$	6.71761	0.251576
$714$	0	0
$715$	29.4378	1.10091
$716$	−40.5516	−1.51549
$717$	0	0
$718$	−3.06804	−0.114498
$719$	−52.1881	−1.94629	−0.973144	−	0.230196i	$-0.926063\pi$
$719$	−52.1881	−1.94629	−0.973144	+	0.230196i	$0.926063\pi$
$720$	0	0
$721$	37.8277	1.40878
$722$	−1.39329	−0.0518530
$723$	0	0
$724$	12.7455	0.473683
$725$	1.73291	0.0643587
$726$	0	0
$727$	−14.0872	−0.522467	−0.261233	−	0.965276i	$-0.584129\pi$
$727$	−14.0872	−0.522467	−0.261233	+	0.965276i	$0.584129\pi$
$728$	13.2997	0.492920
$729$	0	0
$730$	5.17287	0.191456
$731$	28.4014	1.05046
$732$	0	0
$733$	−32.7271	−1.20880	−0.604401	−	0.796680i	$-0.706588\pi$
$733$	−32.7271	−1.20880	−0.604401	+	0.796680i	$0.706588\pi$
$734$	0.421804	0.0155691
$735$	0	0
$736$	−2.66491	−0.0982299
$737$	−5.25845	−0.193697
$738$	0	0
$739$	27.7120	1.01940	0.509701	−	0.860351i	$-0.329756\pi$
$739$	27.7120	1.01940	0.509701	+	0.860351i	$0.329756\pi$
$740$	−6.16189	−0.226516
$741$	0	0
$742$	6.74073	0.247460
$743$	21.1573	0.776186	0.388093	−	0.921620i	$-0.373134\pi$
$743$	21.1573	0.776186	0.388093	+	0.921620i	$0.373134\pi$
$744$	0	0
$745$	−21.6334	−0.792585
$746$	8.52130	0.311987
$747$	0	0
$748$	24.5246	0.896708
$749$	−48.4336	−1.76973
$750$	0	0
$751$	29.3387	1.07058	0.535291	−	0.844668i	$-0.320202\pi$
$751$	29.3387	1.07058	0.535291	+	0.844668i	$0.320202\pi$
$752$	11.5680	0.421842
$753$	0	0
$754$	0.705435	0.0256904
$755$	−26.0557	−0.948263
$756$	0	0
$757$	−38.3555	−1.39405	−0.697027	−	0.717045i	$-0.745494\pi$
$757$	−38.3555	−1.39405	−0.697027	+	0.717045i	$0.745494\pi$
$758$	−1.96498	−0.0713712
$759$	0	0
$760$	11.7983	0.427969
$761$	5.41281	0.196214	0.0981071	−	0.995176i	$-0.468721\pi$
$761$	5.41281	0.196214	0.0981071	+	0.995176i	$0.468721\pi$
$762$	0	0
$763$	87.2838	3.15989
$764$	41.0770	1.48611
$765$	0	0
$766$	4.20161	0.151810
$767$	11.7579	0.424553
$768$	0	0
$769$	−15.7821	−0.569119	−0.284559	−	0.958658i	$-0.591847\pi$
$769$	−15.7821	−0.569119	−0.284559	+	0.958658i	$0.591847\pi$
$770$	−10.5580	−0.380483
$771$	0	0
$772$	−10.8068	−0.388947
$773$	35.5162	1.27743	0.638715	−	0.769443i	$-0.279466\pi$
$773$	35.5162	1.27743	0.638715	+	0.769443i	$0.279466\pi$
$774$	0	0
$775$	11.6410	0.418158
$776$	−1.49435	−0.0536442
$777$	0	0
$778$	4.53001	0.162409
$779$	−42.6308	−1.52741
$780$	0	0
$781$	4.48871	0.160619
$782$	−0.783218	−0.0280078
$783$	0	0
$784$	58.2797	2.08142
$785$	−8.58420	−0.306383
$786$	0	0
$787$	45.4908	1.62157	0.810785	−	0.585344i	$-0.199040\pi$
$787$	45.4908	1.62157	0.810785	+	0.585344i	$0.199040\pi$
$788$	−39.2632	−1.39869
$789$	0	0
$790$	7.29409	0.259512
$791$	42.4462	1.50921
$792$	0	0
$793$	−10.3970	−0.369210
$794$	0.320021	0.0113571
$795$	0	0
$796$	45.5786	1.61549
$797$	−30.5490	−1.08210	−0.541051	−	0.840990i	$-0.681973\pi$
$797$	−30.5490	−1.08210	−0.541051	+	0.840990i	$0.681973\pi$
$798$	0	0
$799$	10.6831	0.377939
$800$	−4.61806	−0.163273
$801$	0	0
$802$	−1.21550	−0.0429210
$803$	32.0609	1.13141
$804$	0	0
$805$	−12.3941	−0.436835
$806$	4.73884	0.166918
$807$	0	0
$808$	−16.5404	−0.581890
$809$	−25.5128	−0.896981	−0.448491	−	0.893788i	$-0.648038\pi$
$809$	−25.5128	−0.896981	−0.448491	+	0.893788i	$0.648038\pi$
$810$	0	0
$811$	50.4381	1.77112	0.885560	−	0.464524i	$-0.153775\pi$
$811$	50.4381	1.77112	0.885560	+	0.464524i	$0.153775\pi$
$812$	9.30009	0.326369
$813$	0	0
$814$	1.03897	0.0364160
$815$	−4.39477	−0.153942
$816$	0	0
$817$	41.7738	1.46148
$818$	−0.909660	−0.0318055
$819$	0	0
$820$	43.0289	1.50264
$821$	−2.96844	−0.103599	−0.0517996	−	0.998657i	$-0.516496\pi$
$821$	−2.96844	−0.103599	−0.0517996	+	0.998657i	$0.516496\pi$
$822$	0	0
$823$	18.1118	0.631336	0.315668	−	0.948870i	$-0.397771\pi$
$823$	18.1118	0.631336	0.315668	+	0.948870i	$0.397771\pi$
$824$	−7.19410	−0.250618
$825$	0	0
$826$	−4.21701	−0.146729
$827$	40.2128	1.39834	0.699168	−	0.714957i	$-0.253554\pi$
$827$	40.2128	1.39834	0.699168	+	0.714957i	$0.253554\pi$
$828$	0	0
$829$	−40.4489	−1.40485	−0.702424	−	0.711759i	$-0.747899\pi$
$829$	−40.4489	−1.40485	−0.702424	+	0.711759i	$0.747899\pi$
$830$	−8.76152	−0.304117
$831$	0	0
$832$	20.7100	0.717990
$833$	53.8212	1.86480
$834$	0	0
$835$	−11.7550	−0.406799
$836$	36.0717	1.24756
$837$	0	0
$838$	1.44923	0.0500627
$839$	33.9645	1.17259	0.586293	−	0.810099i	$-0.300587\pi$
$839$	33.9645	1.17259	0.586293	+	0.810099i	$0.300587\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−1.12118	−0.0386384
$843$	0	0
$844$	−50.9867	−1.75504
$845$	−9.35427	−0.321797
$846$	0	0
$847$	−12.8953	−0.443087
$848$	22.5954	0.775929
$849$	0	0
$850$	−1.35725	−0.0465532
$851$	1.21966	0.0418095
$852$	0	0
$853$	21.1443	0.723967	0.361983	−	0.932185i	$-0.382100\pi$
$853$	21.1443	0.723967	0.361983	+	0.932185i	$0.382100\pi$
$854$	3.72894	0.127602
$855$	0	0
$856$	9.21113	0.314830
$857$	20.8537	0.712350	0.356175	−	0.934419i	$-0.384081\pi$
$857$	20.8537	0.712350	0.356175	+	0.934419i	$0.384081\pi$
$858$	0	0
$859$	−39.2514	−1.33924	−0.669621	−	0.742703i	$-0.733543\pi$
$859$	−39.2514	−1.33924	−0.669621	+	0.742703i	$0.733543\pi$
$860$	−42.1639	−1.43778
$861$	0	0
$862$	5.53601	0.188557
$863$	−57.0716	−1.94274	−0.971371	−	0.237569i	$-0.923649\pi$
$863$	−57.0716	−1.94274	−0.971371	+	0.237569i	$0.923649\pi$
$864$	0	0
$865$	−56.4654	−1.91988
$866$	4.64523	0.157851
$867$	0	0
$868$	62.4743	2.12052
$869$	45.2081	1.53358
$870$	0	0
$871$	−4.35461	−0.147550
$872$	−16.5997	−0.562136
$873$	0	0
$874$	−1.15198	−0.0389665
$875$	40.4927	1.36890
$876$	0	0
$877$	1.49604	0.0505176	0.0252588	−	0.999681i	$-0.491959\pi$
$877$	1.49604	0.0505176	0.0252588	+	0.999681i	$0.491959\pi$
$878$	6.08331	0.205302
$879$	0	0
$880$	−35.3911	−1.19303
$881$	34.7534	1.17087	0.585435	−	0.810719i	$-0.300924\pi$
$881$	34.7534	1.17087	0.585435	+	0.810719i	$0.300924\pi$
$882$	0	0
$883$	−21.5140	−0.724002	−0.362001	−	0.932178i	$-0.617906\pi$
$883$	−21.5140	−0.724002	−0.362001	+	0.932178i	$0.617906\pi$
$884$	20.3093	0.683074
$885$	0	0
$886$	7.33100	0.246290
$887$	10.3028	0.345934	0.172967	−	0.984928i	$-0.444665\pi$
$887$	10.3028	0.345934	0.172967	+	0.984928i	$0.444665\pi$
$888$	0	0
$889$	−44.1970	−1.48232
$890$	6.50370	0.218005
$891$	0	0
$892$	28.2251	0.945045
$893$	15.7130	0.525816
$894$	0	0
$895$	54.0427	1.80645
$896$	−32.8859	−1.09864
$897$	0	0
$898$	−6.89214	−0.229994
$899$	6.71761	0.224045
$900$	0	0
$901$	20.8668	0.695175
$902$	−7.25523	−0.241573
$903$	0	0
$904$	−8.07244	−0.268485
$905$	−16.9858	−0.564626
$906$	0	0
$907$	−17.4319	−0.578818	−0.289409	−	0.957206i	$-0.593459\pi$
$907$	−17.4319	−0.578818	−0.289409	+	0.957206i	$0.593459\pi$
$908$	26.8338	0.890511
$909$	0	0
$910$	−8.74325	−0.289836
$911$	33.5503	1.11157	0.555785	−	0.831326i	$-0.312418\pi$
$911$	33.5503	1.11157	0.555785	+	0.831326i	$0.312418\pi$
$912$	0	0
$913$	−54.3030	−1.79717
$914$	5.38117	0.177993
$915$	0	0
$916$	−53.1651	−1.75663
$917$	−14.3825	−0.474953
$918$	0	0
$919$	−24.8480	−0.819661	−0.409831	−	0.912162i	$-0.634412\pi$
$919$	−24.8480	−0.819661	−0.409831	+	0.912162i	$0.634412\pi$
$920$	2.35712	0.0777119
$921$	0	0
$922$	−0.694531	−0.0228732
$923$	3.71718	0.122352
$924$	0	0
$925$	2.11357	0.0694937
$926$	0.560330	0.0184136
$927$	0	0
$928$	−2.66491	−0.0874800
$929$	30.1456	0.989044	0.494522	−	0.869165i	$-0.335343\pi$
$929$	30.1456	0.989044	0.494522	+	0.869165i	$0.335343\pi$
$930$	0	0
$931$	79.1622	2.59444
$932$	16.2271	0.531536
$933$	0	0
$934$	−1.54454	−0.0505390
$935$	−32.6836	−1.06887
$936$	0	0
$937$	−50.8757	−1.66204	−0.831019	−	0.556244i	$-0.812242\pi$
$937$	−50.8757	−1.66204	−0.831019	+	0.556244i	$0.812242\pi$
$938$	1.56180	0.0509946
$939$	0	0
$940$	−15.8598	−0.517288
$941$	55.5956	1.81236	0.906182	−	0.422888i	$-0.138984\pi$
$941$	55.5956	1.81236	0.906182	+	0.422888i	$0.138984\pi$
$942$	0	0
$943$	−8.51699	−0.277351
$944$	−14.1357	−0.460079
$945$	0	0
$946$	7.10938	0.231146
$947$	−54.0049	−1.75492	−0.877462	−	0.479647i	$-0.840765\pi$
$947$	−54.0049	−1.75492	−0.877462	+	0.479647i	$0.840765\pi$
$948$	0	0
$949$	26.5502	0.861857
$950$	−1.99629	−0.0647682
$951$	0	0
$952$	−14.7662	−0.478574
$953$	51.8326	1.67902	0.839511	−	0.543342i	$-0.182841\pi$
$953$	51.8326	1.67902	0.839511	+	0.543342i	$0.182841\pi$
$954$	0	0
$955$	−54.7428	−1.77144
$956$	−41.1243	−1.33005
$957$	0	0
$958$	−7.73701	−0.249971
$959$	67.5202	2.18034
$960$	0	0
$961$	14.1263	0.455686
$962$	0.860393	0.0277402
$963$	0	0
$964$	13.0688	0.420916
$965$	14.4022	0.463622
$966$	0	0
$967$	−37.8809	−1.21817	−0.609083	−	0.793106i	$-0.708462\pi$
$967$	−37.8809	−1.21817	−0.609083	+	0.793106i	$0.708462\pi$
$968$	2.45243	0.0788241
$969$	0	0
$970$	0.982391	0.0315427
$971$	−2.62752	−0.0843210	−0.0421605	−	0.999111i	$-0.513424\pi$
$971$	−2.62752	−0.0843210	−0.0421605	+	0.999111i	$0.513424\pi$
$972$	0	0
$973$	−33.6819	−1.07979
$974$	5.15078	0.165042
$975$	0	0
$976$	12.4997	0.400105
$977$	−29.4200	−0.941229	−0.470615	−	0.882339i	$-0.655968\pi$
$977$	−29.4200	−0.941229	−0.470615	+	0.882339i	$0.655968\pi$
$978$	0	0
$979$	40.3093	1.28829
$980$	−79.9015	−2.55236
$981$	0	0
$982$	−3.06997	−0.0979667
$983$	23.9487	0.763845	0.381923	−	0.924194i	$-0.375262\pi$
$983$	23.9487	0.763845	0.381923	+	0.924194i	$0.375262\pi$
$984$	0	0
$985$	52.3256	1.66723
$986$	−0.783218	−0.0249427
$987$	0	0
$988$	29.8716	0.950341
$989$	8.34577	0.265380
$990$	0	0
$991$	0.962968	0.0305897	0.0152948	−	0.999883i	$-0.495131\pi$
$991$	0.962968	0.0305897	0.0152948	+	0.999883i	$0.495131\pi$
$992$	−17.9018	−0.568383
$993$	0	0
$994$	−1.33318	−0.0422860
$995$	−60.7420	−1.92565
$996$	0	0
$997$	−50.5602	−1.60126	−0.800629	−	0.599161i	$-0.795501\pi$
$997$	−50.5602	−1.60126	−0.800629	+	0.599161i	$0.795501\pi$
$998$	3.57824	0.113267
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.t.1.11	✓	22
3.2	odd	2		6003.2.a.u.1.12	yes	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.t.1.11	✓	22	1.1	even	1	trivial
6003.2.a.u.1.12	yes	22	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)

\(f(q)\)	\(=\)	\(q-0.230149 q^{2} -1.94703 q^{4} +2.59479 q^{5} -4.77655 q^{7} +0.908406 q^{8} +O(q^{10})\)	\(q-0.230149 q^{2} -1.94703 q^{4} +2.59479 q^{5} -4.77655 q^{7} +0.908406 q^{8} -0.597188 q^{10} -3.70131 q^{11} -3.06512 q^{13} +1.09932 q^{14} +3.68499 q^{16} +3.40309 q^{17} +5.00538 q^{19} -5.05213 q^{20} +0.851854 q^{22} +1.00000 q^{23} +1.73291 q^{25} +0.705435 q^{26} +9.30009 q^{28} +1.00000 q^{29} +6.71761 q^{31} -2.66491 q^{32} -0.783218 q^{34} -12.3941 q^{35} +1.21966 q^{37} -1.15198 q^{38} +2.35712 q^{40} -8.51699 q^{41} +8.34577 q^{43} +7.20657 q^{44} -0.230149 q^{46} +3.13922 q^{47} +15.8154 q^{49} -0.398828 q^{50} +5.96789 q^{52} +6.13173 q^{53} -9.60411 q^{55} -4.33904 q^{56} -0.230149 q^{58} -3.83602 q^{59} +3.39205 q^{61} -1.54605 q^{62} -6.75666 q^{64} -7.95333 q^{65} +1.42070 q^{67} -6.62592 q^{68} +2.85250 q^{70} -1.21273 q^{71} -8.66204 q^{73} -0.280704 q^{74} -9.74564 q^{76} +17.6795 q^{77} -12.2141 q^{79} +9.56177 q^{80} +1.96018 q^{82} +14.6713 q^{83} +8.83029 q^{85} -1.92077 q^{86} -3.36229 q^{88} -10.8906 q^{89} +14.6407 q^{91} -1.94703 q^{92} -0.722489 q^{94} +12.9879 q^{95} -1.64503 q^{97} -3.63990 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8}+O(q^{10}) \) 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8	\( 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8} - 12 q^{10} - 28 q^{13} - q^{14} + 3 q^{16} - 10 q^{17} - 8 q^{19} - 11 q^{22} + 22 q^{23} + 11 q^{26} - 21 q^{28} + 22 q^{29} - 18 q^{31} + 5 q^{32} - 33 q^{34} + 2 q^{35} - 28 q^{37} + 14 q^{38} - 30 q^{40} - 10 q^{41} - 14 q^{43} + 37 q^{44} - 3 q^{46} - 18 q^{47} + 2 q^{49} + 7 q^{50} - 57 q^{52} + 20 q^{53} - 42 q^{55} - 2 q^{56} - 3 q^{58} - 20 q^{59} - 38 q^{61} + 4 q^{62} - 24 q^{64} + 12 q^{65} - 50 q^{67} + 11 q^{68} - 48 q^{70} + 12 q^{71} - 46 q^{73} - 6 q^{74} - 16 q^{76} - 14 q^{77} - 20 q^{79} - 58 q^{80} - 42 q^{82} + 22 q^{83} - 66 q^{85} + 22 q^{86} - 68 q^{88} - 14 q^{89} - 16 q^{91} + 17 q^{92} - 27 q^{94} - 20 q^{95} - 48 q^{97} - 28 q^{98}+O(q^{100}) \) 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8 - 12 * q^10 - 28 * q^13 - q^14 + 3 * q^16 - 10 * q^17 - 8 * q^19 - 11 * q^22 + 22 * q^23 + 11 * q^26 - 21 * q^28 + 22 * q^29 - 18 * q^31 + 5 * q^32 - 33 * q^34 + 2 * q^35 - 28 * q^37 + 14 * q^38 - 30 * q^40 - 10 * q^41 - 14 * q^43 + 37 * q^44 - 3 * q^46 - 18 * q^47 + 2 * q^49 + 7 * q^50 - 57 * q^52 + 20 * q^53 - 42 * q^55 - 2 * q^56 - 3 * q^58 - 20 * q^59 - 38 * q^61 + 4 * q^62 - 24 * q^64 + 12 * q^65 - 50 * q^67 + 11 * q^68 - 48 * q^70 + 12 * q^71 - 46 * q^73 - 6 * q^74 - 16 * q^76 - 14 * q^77 - 20 * q^79 - 58 * q^80 - 42 * q^82 + 22 * q^83 - 66 * q^85 + 22 * q^86 - 68 * q^88 - 14 * q^89 - 16 * q^91 + 17 * q^92 - 27 * q^94 - 20 * q^95 - 48 * q^97 - 28 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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