LMFDB - Embedded newform 1000.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1000,2,Mod(1,1000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1000, 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("1000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1000 = 2^{3} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1000.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:	$7.98504020213$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	1000.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.61803 q^{3} +4.23607 q^{7} -0.381966 q^{9} +O(q^{10})$	$q-1.61803 q^{3} +4.23607 q^{7} -0.381966 q^{9} -6.23607 q^{11} -2.38197 q^{13} -2.23607 q^{17} +4.09017 q^{19} -6.85410 q^{21} +5.23607 q^{23} +5.47214 q^{27} -10.2361 q^{29} -5.85410 q^{31} +10.0902 q^{33} +1.23607 q^{37} +3.85410 q^{39} -3.00000 q^{41} -2.70820 q^{43} -6.32624 q^{47} +10.9443 q^{49} +3.61803 q^{51} -9.85410 q^{53} -6.61803 q^{57} -3.38197 q^{59} -12.5623 q^{61} -1.61803 q^{63} +2.61803 q^{67} -8.47214 q^{69} +0.236068 q^{71} -6.14590 q^{73} -26.4164 q^{77} -1.76393 q^{79} -7.70820 q^{81} -6.94427 q^{83} +16.5623 q^{87} +0.472136 q^{89} -10.0902 q^{91} +9.47214 q^{93} -11.5623 q^{97} +2.38197 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} + 4 q^{7} - 3 q^{9}+O(q^{10}) $ 2 * q - q^3 + 4 * q^7 - 3 * q^9	$ 2 q - q^{3} + 4 q^{7} - 3 q^{9} - 8 q^{11} - 7 q^{13} - 3 q^{19} - 7 q^{21} + 6 q^{23} + 2 q^{27} - 16 q^{29} - 5 q^{31} + 9 q^{33} - 2 q^{37} + q^{39} - 6 q^{41} + 8 q^{43} + 3 q^{47} + 4 q^{49} + 5 q^{51} - 13 q^{53} - 11 q^{57} - 9 q^{59} - 5 q^{61} - q^{63} + 3 q^{67} - 8 q^{69} - 4 q^{71} - 19 q^{73} - 26 q^{77} - 8 q^{79} - 2 q^{81} + 4 q^{83} + 13 q^{87} - 8 q^{89} - 9 q^{91} + 10 q^{93} - 3 q^{97} + 7 q^{99}+O(q^{100}) $ 2 * q - q^3 + 4 * q^7 - 3 * q^9 - 8 * q^11 - 7 * q^13 - 3 * q^19 - 7 * q^21 + 6 * q^23 + 2 * q^27 - 16 * q^29 - 5 * q^31 + 9 * q^33 - 2 * q^37 + q^39 - 6 * q^41 + 8 * q^43 + 3 * q^47 + 4 * q^49 + 5 * q^51 - 13 * q^53 - 11 * q^57 - 9 * q^59 - 5 * q^61 - q^63 + 3 * q^67 - 8 * q^69 - 4 * q^71 - 19 * q^73 - 26 * q^77 - 8 * q^79 - 2 * q^81 + 4 * q^83 + 13 * q^87 - 8 * q^89 - 9 * q^91 + 10 * q^93 - 3 * q^97 + 7 * q^99

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	0
$2$	0	0
$3$	−1.61803	−0.934172	−0.467086	−	0.884212i	$-0.654696\pi$
$3$	−1.61803	−0.934172	−0.467086	+	0.884212i	$0.654696\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.23607	1.60108	0.800542	−	0.599277i	$-0.204545\pi$
$7$	4.23607	1.60108	0.800542	+	0.599277i	$0.204545\pi$
$8$	0	0
$9$	−0.381966	−0.127322
$10$	0	0
$11$	−6.23607	−1.88025	−0.940123	−	0.340836i	$-0.889290\pi$
$11$	−6.23607	−1.88025	−0.940123	+	0.340836i	$0.889290\pi$
$12$	0	0
$13$	−2.38197	−0.660639	−0.330319	−	0.943869i	$-0.607156\pi$
$13$	−2.38197	−0.660639	−0.330319	+	0.943869i	$0.607156\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.23607	−0.542326	−0.271163	−	0.962533i	$-0.587408\pi$
$17$	−2.23607	−0.542326	−0.271163	+	0.962533i	$0.587408\pi$
$18$	0	0
$19$	4.09017	0.938349	0.469175	−	0.883105i	$-0.344551\pi$
$19$	4.09017	0.938349	0.469175	+	0.883105i	$0.344551\pi$
$20$	0	0
$21$	−6.85410	−1.49569
$22$	0	0
$23$	5.23607	1.09180	0.545898	−	0.837852i	$-0.316189\pi$
$23$	5.23607	1.09180	0.545898	+	0.837852i	$0.316189\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.47214	1.05311
$28$	0	0
$29$	−10.2361	−1.90079	−0.950395	−	0.311045i	$-0.899321\pi$
$29$	−10.2361	−1.90079	−0.950395	+	0.311045i	$0.899321\pi$
$30$	0	0
$31$	−5.85410	−1.05143	−0.525714	−	0.850661i	$-0.676202\pi$
$31$	−5.85410	−1.05143	−0.525714	+	0.850661i	$0.676202\pi$
$32$	0	0
$33$	10.0902	1.75647
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.23607	0.203208	0.101604	−	0.994825i	$-0.467602\pi$
$37$	1.23607	0.203208	0.101604	+	0.994825i	$0.467602\pi$
$38$	0	0
$39$	3.85410	0.617150
$40$	0	0
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	0	0
$43$	−2.70820	−0.412997	−0.206499	−	0.978447i	$-0.566207\pi$
$43$	−2.70820	−0.412997	−0.206499	+	0.978447i	$0.566207\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.32624	−0.922777	−0.461388	−	0.887198i	$-0.652648\pi$
$47$	−6.32624	−0.922777	−0.461388	+	0.887198i	$0.652648\pi$
$48$	0	0
$49$	10.9443	1.56347
$50$	0	0
$51$	3.61803	0.506626
$52$	0	0
$53$	−9.85410	−1.35357	−0.676783	−	0.736183i	$-0.736626\pi$
$53$	−9.85410	−1.35357	−0.676783	+	0.736183i	$0.736626\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−6.61803	−0.876580
$58$	0	0
$59$	−3.38197	−0.440294	−0.220147	−	0.975467i	$-0.570654\pi$
$59$	−3.38197	−0.440294	−0.220147	+	0.975467i	$0.570654\pi$
$60$	0	0
$61$	−12.5623	−1.60844	−0.804219	−	0.594333i	$-0.797416\pi$
$61$	−12.5623	−1.60844	−0.804219	+	0.594333i	$0.797416\pi$
$62$	0	0
$63$	−1.61803	−0.203853
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.61803	0.319844	0.159922	−	0.987130i	$-0.448876\pi$
$67$	2.61803	0.319844	0.159922	+	0.987130i	$0.448876\pi$
$68$	0	0
$69$	−8.47214	−1.01993
$70$	0	0
$71$	0.236068	0.0280161	0.0140081	−	0.999902i	$-0.495541\pi$
$71$	0.236068	0.0280161	0.0140081	+	0.999902i	$0.495541\pi$
$72$	0	0
$73$	−6.14590	−0.719323	−0.359661	−	0.933083i	$-0.617108\pi$
$73$	−6.14590	−0.719323	−0.359661	+	0.933083i	$0.617108\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−26.4164	−3.01043
$78$	0	0
$79$	−1.76393	−0.198458	−0.0992289	−	0.995065i	$-0.531638\pi$
$79$	−1.76393	−0.198458	−0.0992289	+	0.995065i	$0.531638\pi$
$80$	0	0
$81$	−7.70820	−0.856467
$82$	0	0
$83$	−6.94427	−0.762233	−0.381116	−	0.924527i	$-0.624460\pi$
$83$	−6.94427	−0.762233	−0.381116	+	0.924527i	$0.624460\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	16.5623	1.77567
$88$	0	0
$89$	0.472136	0.0500463	0.0250232	−	0.999687i	$-0.492034\pi$
$89$	0.472136	0.0500463	0.0250232	+	0.999687i	$0.492034\pi$
$90$	0	0
$91$	−10.0902	−1.05774
$92$	0	0
$93$	9.47214	0.982215
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−11.5623	−1.17397	−0.586987	−	0.809596i	$-0.699686\pi$
$97$	−11.5623	−1.17397	−0.586987	+	0.809596i	$0.699686\pi$
$98$	0	0
$99$	2.38197	0.239397
$100$	0	0
$101$	−4.52786	−0.450539	−0.225270	−	0.974296i	$-0.572326\pi$
$101$	−4.52786	−0.450539	−0.225270	+	0.974296i	$0.572326\pi$
$102$	0	0
$103$	4.52786	0.446144	0.223072	−	0.974802i	$-0.428392\pi$
$103$	4.52786	0.446144	0.223072	+	0.974802i	$0.428392\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.2705	1.37958	0.689791	−	0.724008i	$-0.257702\pi$
$107$	14.2705	1.37958	0.689791	+	0.724008i	$0.257702\pi$
$108$	0	0
$109$	14.8541	1.42276	0.711382	−	0.702805i	$-0.248069\pi$
$109$	14.8541	1.42276	0.711382	+	0.702805i	$0.248069\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	−9.00000	−0.846649	−0.423324	−	0.905978i	$-0.639137\pi$
$113$	−9.00000	−0.846649	−0.423324	+	0.905978i	$0.639137\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0.909830	0.0841138
$118$	0	0
$119$	−9.47214	−0.868309
$120$	0	0
$121$	27.8885	2.53532
$122$	0	0
$123$	4.85410	0.437680
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2.29180	0.203364	0.101682	−	0.994817i	$-0.467578\pi$
$127$	2.29180	0.203364	0.101682	+	0.994817i	$0.467578\pi$
$128$	0	0
$129$	4.38197	0.385811
$130$	0	0
$131$	13.3262	1.16432	0.582159	−	0.813075i	$-0.302208\pi$
$131$	13.3262	1.16432	0.582159	+	0.813075i	$0.302208\pi$
$132$	0	0
$133$	17.3262	1.50238
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10.5623	0.902399	0.451199	−	0.892423i	$-0.350996\pi$
$137$	10.5623	0.902399	0.451199	+	0.892423i	$0.350996\pi$
$138$	0	0
$139$	−18.8885	−1.60211	−0.801053	−	0.598594i	$-0.795726\pi$
$139$	−18.8885	−1.60211	−0.801053	+	0.598594i	$0.795726\pi$
$140$	0	0
$141$	10.2361	0.862032
$142$	0	0
$143$	14.8541	1.24216
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−17.7082	−1.46055
$148$	0	0
$149$	−1.47214	−0.120602	−0.0603010	−	0.998180i	$-0.519206\pi$
$149$	−1.47214	−0.120602	−0.0603010	+	0.998180i	$0.519206\pi$
$150$	0	0
$151$	8.90983	0.725072	0.362536	−	0.931970i	$-0.381911\pi$
$151$	8.90983	0.725072	0.362536	+	0.931970i	$0.381911\pi$
$152$	0	0
$153$	0.854102	0.0690501
$154$	0	0
$155$	0	0
$156$	0	0
$157$	13.0344	1.04026	0.520131	−	0.854087i	$-0.325883\pi$
$157$	13.0344	1.04026	0.520131	+	0.854087i	$0.325883\pi$
$158$	0	0
$159$	15.9443	1.26446
$160$	0	0
$161$	22.1803	1.74806
$162$	0	0
$163$	−19.3820	−1.51811	−0.759056	−	0.651025i	$-0.774339\pi$
$163$	−19.3820	−1.51811	−0.759056	+	0.651025i	$0.774339\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.7639	0.832938	0.416469	−	0.909150i	$-0.363267\pi$
$167$	10.7639	0.832938	0.416469	+	0.909150i	$0.363267\pi$
$168$	0	0
$169$	−7.32624	−0.563557
$170$	0	0
$171$	−1.56231	−0.119473
$172$	0	0
$173$	6.41641	0.487830	0.243915	−	0.969797i	$-0.421568\pi$
$173$	6.41641	0.487830	0.243915	+	0.969797i	$0.421568\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	5.47214	0.411311
$178$	0	0
$179$	6.52786	0.487915	0.243958	−	0.969786i	$-0.421554\pi$
$179$	6.52786	0.487915	0.243958	+	0.969786i	$0.421554\pi$
$180$	0	0
$181$	−5.23607	−0.389194	−0.194597	−	0.980883i	$-0.562340\pi$
$181$	−5.23607	−0.389194	−0.194597	+	0.980883i	$0.562340\pi$
$182$	0	0
$183$	20.3262	1.50256
$184$	0	0
$185$	0	0
$186$	0	0
$187$	13.9443	1.01971
$188$	0	0
$189$	23.1803	1.68612
$190$	0	0
$191$	−20.0000	−1.44715	−0.723575	−	0.690246i	$-0.757502\pi$
$191$	−20.0000	−1.44715	−0.723575	+	0.690246i	$0.757502\pi$
$192$	0	0
$193$	−26.9787	−1.94197	−0.970985	−	0.239140i	$-0.923135\pi$
$193$	−26.9787	−1.94197	−0.970985	+	0.239140i	$0.923135\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.52786	0.607585	0.303793	−	0.952738i	$-0.401747\pi$
$197$	8.52786	0.607585	0.303793	+	0.952738i	$0.401747\pi$
$198$	0	0
$199$	−18.6525	−1.32224	−0.661119	−	0.750281i	$-0.729918\pi$
$199$	−18.6525	−1.32224	−0.661119	+	0.750281i	$0.729918\pi$
$200$	0	0
$201$	−4.23607	−0.298789
$202$	0	0
$203$	−43.3607	−3.04332
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−2.00000	−0.139010
$208$	0	0
$209$	−25.5066	−1.76433
$210$	0	0
$211$	14.6525	1.00872	0.504359	−	0.863494i	$-0.331729\pi$
$211$	14.6525	1.00872	0.504359	+	0.863494i	$0.331729\pi$
$212$	0	0
$213$	−0.381966	−0.0261719
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−24.7984	−1.68342
$218$	0	0
$219$	9.94427	0.671972
$220$	0	0
$221$	5.32624	0.358282
$222$	0	0
$223$	20.7984	1.39276	0.696381	−	0.717672i	$-0.254792\pi$
$223$	20.7984	1.39276	0.696381	+	0.717672i	$0.254792\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.79837	0.185735	0.0928673	−	0.995678i	$-0.470397\pi$
$227$	2.79837	0.185735	0.0928673	+	0.995678i	$0.470397\pi$
$228$	0	0
$229$	17.0902	1.12935	0.564675	−	0.825313i	$-0.309001\pi$
$229$	17.0902	1.12935	0.564675	+	0.825313i	$0.309001\pi$
$230$	0	0
$231$	42.7426	2.81226
$232$	0	0
$233$	11.0000	0.720634	0.360317	−	0.932830i	$-0.382669\pi$
$233$	11.0000	0.720634	0.360317	+	0.932830i	$0.382669\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	2.85410	0.185394
$238$	0	0
$239$	−12.5066	−0.808983	−0.404492	−	0.914542i	$-0.632552\pi$
$239$	−12.5066	−0.808983	−0.404492	+	0.914542i	$0.632552\pi$
$240$	0	0
$241$	20.2361	1.30352	0.651760	−	0.758425i	$-0.274031\pi$
$241$	20.2361	1.30352	0.651760	+	0.758425i	$0.274031\pi$
$242$	0	0
$243$	−3.94427	−0.253025
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−9.74265	−0.619910
$248$	0	0
$249$	11.2361	0.712057
$250$	0	0
$251$	17.0344	1.07520	0.537602	−	0.843199i	$-0.319330\pi$
$251$	17.0344	1.07520	0.537602	+	0.843199i	$0.319330\pi$
$252$	0	0
$253$	−32.6525	−2.05284
$254$	0	0
$255$	0	0
$256$	0	0
$257$	28.4164	1.77257	0.886283	−	0.463143i	$-0.153278\pi$
$257$	28.4164	1.77257	0.886283	+	0.463143i	$0.153278\pi$
$258$	0	0
$259$	5.23607	0.325353
$260$	0	0
$261$	3.90983	0.242012
$262$	0	0
$263$	−4.14590	−0.255647	−0.127824	−	0.991797i	$-0.540799\pi$
$263$	−4.14590	−0.255647	−0.127824	+	0.991797i	$0.540799\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−0.763932	−0.0467519
$268$	0	0
$269$	−14.7426	−0.898875	−0.449437	−	0.893312i	$-0.648376\pi$
$269$	−14.7426	−0.898875	−0.449437	+	0.893312i	$0.648376\pi$
$270$	0	0
$271$	5.90983	0.358997	0.179498	−	0.983758i	$-0.442553\pi$
$271$	5.90983	0.358997	0.179498	+	0.983758i	$0.442553\pi$
$272$	0	0
$273$	16.3262	0.988109
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−9.76393	−0.586658	−0.293329	−	0.956012i	$-0.594763\pi$
$277$	−9.76393	−0.586658	−0.293329	+	0.956012i	$0.594763\pi$
$278$	0	0
$279$	2.23607	0.133870
$280$	0	0
$281$	−12.0000	−0.715860	−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000	−0.715860	−0.357930	+	0.933748i	$0.616517\pi$
$282$	0	0
$283$	24.6525	1.46544	0.732719	−	0.680532i	$-0.238251\pi$
$283$	24.6525	1.46544	0.732719	+	0.680532i	$0.238251\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−12.7082	−0.750142
$288$	0	0
$289$	−12.0000	−0.705882
$290$	0	0
$291$	18.7082	1.09669
$292$	0	0
$293$	23.0000	1.34367	0.671837	−	0.740699i	$-0.265505\pi$
$293$	23.0000	1.34367	0.671837	+	0.740699i	$0.265505\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−34.1246	−1.98011
$298$	0	0
$299$	−12.4721	−0.721282
$300$	0	0
$301$	−11.4721	−0.661243
$302$	0	0
$303$	7.32624	0.420881
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−18.3820	−1.04911	−0.524557	−	0.851375i	$-0.675769\pi$
$307$	−18.3820	−1.04911	−0.524557	+	0.851375i	$0.675769\pi$
$308$	0	0
$309$	−7.32624	−0.416775
$310$	0	0
$311$	8.79837	0.498910	0.249455	−	0.968386i	$-0.419749\pi$
$311$	8.79837	0.498910	0.249455	+	0.968386i	$0.419749\pi$
$312$	0	0
$313$	11.4164	0.645294	0.322647	−	0.946519i	$-0.395427\pi$
$313$	11.4164	0.645294	0.322647	+	0.946519i	$0.395427\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6.94427	−0.390029	−0.195015	−	0.980800i	$-0.562475\pi$
$317$	−6.94427	−0.390029	−0.195015	+	0.980800i	$0.562475\pi$
$318$	0	0
$319$	63.8328	3.57395
$320$	0	0
$321$	−23.0902	−1.28877
$322$	0	0
$323$	−9.14590	−0.508891
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−24.0344	−1.32911
$328$	0	0
$329$	−26.7984	−1.47744
$330$	0	0
$331$	−24.0344	−1.32105	−0.660526	−	0.750803i	$-0.729667\pi$
$331$	−24.0344	−1.32105	−0.660526	+	0.750803i	$0.729667\pi$
$332$	0	0
$333$	−0.472136	−0.0258729
$334$	0	0
$335$	0	0
$336$	0	0
$337$	26.8885	1.46471	0.732356	−	0.680922i	$-0.238421\pi$
$337$	26.8885	1.46471	0.732356	+	0.680922i	$0.238421\pi$
$338$	0	0
$339$	14.5623	0.790916
$340$	0	0
$341$	36.5066	1.97694
$342$	0	0
$343$	16.7082	0.902158
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−14.7082	−0.789578	−0.394789	−	0.918772i	$-0.629182\pi$
$347$	−14.7082	−0.789578	−0.394789	+	0.918772i	$0.629182\pi$
$348$	0	0
$349$	4.88854	0.261678	0.130839	−	0.991404i	$-0.458233\pi$
$349$	4.88854	0.261678	0.130839	+	0.991404i	$0.458233\pi$
$350$	0	0
$351$	−13.0344	−0.695727
$352$	0	0
$353$	30.0000	1.59674	0.798369	−	0.602168i	$-0.205696\pi$
$353$	30.0000	1.59674	0.798369	+	0.602168i	$0.205696\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	15.3262	0.811151
$358$	0	0
$359$	−30.5623	−1.61302	−0.806508	−	0.591223i	$-0.798645\pi$
$359$	−30.5623	−1.61302	−0.806508	+	0.591223i	$0.798645\pi$
$360$	0	0
$361$	−2.27051	−0.119501
$362$	0	0
$363$	−45.1246	−2.36843
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−2.56231	−0.133751	−0.0668756	−	0.997761i	$-0.521303\pi$
$367$	−2.56231	−0.133751	−0.0668756	+	0.997761i	$0.521303\pi$
$368$	0	0
$369$	1.14590	0.0596531
$370$	0	0
$371$	−41.7426	−2.16717
$372$	0	0
$373$	−29.9230	−1.54935	−0.774677	−	0.632357i	$-0.782087\pi$
$373$	−29.9230	−1.54935	−0.774677	+	0.632357i	$0.782087\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	24.3820	1.25574
$378$	0	0
$379$	20.3262	1.04409	0.522044	−	0.852918i	$-0.325170\pi$
$379$	20.3262	1.04409	0.522044	+	0.852918i	$0.325170\pi$
$380$	0	0
$381$	−3.70820	−0.189977
$382$	0	0
$383$	2.70820	0.138383	0.0691914	−	0.997603i	$-0.477958\pi$
$383$	2.70820	0.138383	0.0691914	+	0.997603i	$0.477958\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.03444	0.0525836
$388$	0	0
$389$	2.88854	0.146455	0.0732275	−	0.997315i	$-0.476670\pi$
$389$	2.88854	0.146455	0.0732275	+	0.997315i	$0.476670\pi$
$390$	0	0
$391$	−11.7082	−0.592109
$392$	0	0
$393$	−21.5623	−1.08767
$394$	0	0
$395$	0	0
$396$	0	0
$397$	7.94427	0.398712	0.199356	−	0.979927i	$-0.436115\pi$
$397$	7.94427	0.398712	0.199356	+	0.979927i	$0.436115\pi$
$398$	0	0
$399$	−28.0344	−1.40348
$400$	0	0
$401$	−15.0557	−0.751847	−0.375924	−	0.926651i	$-0.622674\pi$
$401$	−15.0557	−0.751847	−0.375924	+	0.926651i	$0.622674\pi$
$402$	0	0
$403$	13.9443	0.694614
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−7.70820	−0.382081
$408$	0	0
$409$	6.76393	0.334455	0.167227	−	0.985918i	$-0.446519\pi$
$409$	6.76393	0.334455	0.167227	+	0.985918i	$0.446519\pi$
$410$	0	0
$411$	−17.0902	−0.842996
$412$	0	0
$413$	−14.3262	−0.704948
$414$	0	0
$415$	0	0
$416$	0	0
$417$	30.5623	1.49664
$418$	0	0
$419$	−9.58359	−0.468189	−0.234095	−	0.972214i	$-0.575213\pi$
$419$	−9.58359	−0.468189	−0.234095	+	0.972214i	$0.575213\pi$
$420$	0	0
$421$	−0.618034	−0.0301211	−0.0150606	−	0.999887i	$-0.504794\pi$
$421$	−0.618034	−0.0301211	−0.0150606	+	0.999887i	$0.504794\pi$
$422$	0	0
$423$	2.41641	0.117490
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−53.2148	−2.57524
$428$	0	0
$429$	−24.0344	−1.16039
$430$	0	0
$431$	4.74265	0.228445	0.114223	−	0.993455i	$-0.463562\pi$
$431$	4.74265	0.228445	0.114223	+	0.993455i	$0.463562\pi$
$432$	0	0
$433$	−16.1803	−0.777578	−0.388789	−	0.921327i	$-0.627106\pi$
$433$	−16.1803	−0.777578	−0.388789	+	0.921327i	$0.627106\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	21.4164	1.02449
$438$	0	0
$439$	24.4164	1.16533	0.582666	−	0.812712i	$-0.302010\pi$
$439$	24.4164	1.16533	0.582666	+	0.812712i	$0.302010\pi$
$440$	0	0
$441$	−4.18034	−0.199064
$442$	0	0
$443$	−6.52786	−0.310148	−0.155074	−	0.987903i	$-0.549562\pi$
$443$	−6.52786	−0.310148	−0.155074	+	0.987903i	$0.549562\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	2.38197	0.112663
$448$	0	0
$449$	3.56231	0.168116	0.0840578	−	0.996461i	$-0.473212\pi$
$449$	3.56231	0.168116	0.0840578	+	0.996461i	$0.473212\pi$
$450$	0	0
$451$	18.7082	0.880935
$452$	0	0
$453$	−14.4164	−0.677342
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3.88854	0.181898	0.0909492	−	0.995856i	$-0.471010\pi$
$457$	3.88854	0.181898	0.0909492	+	0.995856i	$0.471010\pi$
$458$	0	0
$459$	−12.2361	−0.571131
$460$	0	0
$461$	16.2705	0.757793	0.378897	−	0.925439i	$-0.376304\pi$
$461$	16.2705	0.757793	0.378897	+	0.925439i	$0.376304\pi$
$462$	0	0
$463$	31.3607	1.45745	0.728727	−	0.684804i	$-0.240112\pi$
$463$	31.3607	1.45745	0.728727	+	0.684804i	$0.240112\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.18034	0.378541	0.189270	−	0.981925i	$-0.439388\pi$
$467$	8.18034	0.378541	0.189270	+	0.981925i	$0.439388\pi$
$468$	0	0
$469$	11.0902	0.512096
$470$	0	0
$471$	−21.0902	−0.971784
$472$	0	0
$473$	16.8885	0.776536
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3.76393	0.172339
$478$	0	0
$479$	17.9787	0.821468	0.410734	−	0.911755i	$-0.365272\pi$
$479$	17.9787	0.821468	0.410734	+	0.911755i	$0.365272\pi$
$480$	0	0
$481$	−2.94427	−0.134247
$482$	0	0
$483$	−35.8885	−1.63299
$484$	0	0
$485$	0	0
$486$	0	0
$487$	11.3475	0.514205	0.257103	−	0.966384i	$-0.417232\pi$
$487$	11.3475	0.514205	0.257103	+	0.966384i	$0.417232\pi$
$488$	0	0
$489$	31.3607	1.41818
$490$	0	0
$491$	−14.8541	−0.670356	−0.335178	−	0.942155i	$-0.608796\pi$
$491$	−14.8541	−0.670356	−0.335178	+	0.942155i	$0.608796\pi$
$492$	0	0
$493$	22.8885	1.03085
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.00000	0.0448561
$498$	0	0
$499$	6.00000	0.268597	0.134298	−	0.990941i	$-0.457122\pi$
$499$	6.00000	0.268597	0.134298	+	0.990941i	$0.457122\pi$
$500$	0	0
$501$	−17.4164	−0.778108
$502$	0	0
$503$	−11.4377	−0.509982	−0.254991	−	0.966943i	$-0.582072\pi$
$503$	−11.4377	−0.509982	−0.254991	+	0.966943i	$0.582072\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	11.8541	0.526459
$508$	0	0
$509$	−11.5279	−0.510964	−0.255482	−	0.966814i	$-0.582234\pi$
$509$	−11.5279	−0.510964	−0.255482	+	0.966814i	$0.582234\pi$
$510$	0	0
$511$	−26.0344	−1.15170
$512$	0	0
$513$	22.3820	0.988188
$514$	0	0
$515$	0	0
$516$	0	0
$517$	39.4508	1.73505
$518$	0	0
$519$	−10.3820	−0.455718
$520$	0	0
$521$	−27.0344	−1.18440	−0.592200	−	0.805791i	$-0.701741\pi$
$521$	−27.0344	−1.18440	−0.592200	+	0.805791i	$0.701741\pi$
$522$	0	0
$523$	−18.9443	−0.828375	−0.414188	−	0.910192i	$-0.635934\pi$
$523$	−18.9443	−0.828375	−0.414188	+	0.910192i	$0.635934\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	13.0902	0.570217
$528$	0	0
$529$	4.41641	0.192018
$530$	0	0
$531$	1.29180	0.0560592
$532$	0	0
$533$	7.14590	0.309523
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−10.5623	−0.455797
$538$	0	0
$539$	−68.2492	−2.93970
$540$	0	0
$541$	−23.7082	−1.01930	−0.509648	−	0.860383i	$-0.670224\pi$
$541$	−23.7082	−1.01930	−0.509648	+	0.860383i	$0.670224\pi$
$542$	0	0
$543$	8.47214	0.363574
$544$	0	0
$545$	0	0
$546$	0	0
$547$	35.2705	1.50806	0.754029	−	0.656841i	$-0.228108\pi$
$547$	35.2705	1.50806	0.754029	+	0.656841i	$0.228108\pi$
$548$	0	0
$549$	4.79837	0.204790
$550$	0	0
$551$	−41.8673	−1.78361
$552$	0	0
$553$	−7.47214	−0.317748
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−40.3607	−1.71014	−0.855068	−	0.518515i	$-0.826485\pi$
$557$	−40.3607	−1.71014	−0.855068	+	0.518515i	$0.826485\pi$
$558$	0	0
$559$	6.45085	0.272842
$560$	0	0
$561$	−22.5623	−0.952581
$562$	0	0
$563$	28.0000	1.18006	0.590030	−	0.807382i	$-0.299116\pi$
$563$	28.0000	1.18006	0.590030	+	0.807382i	$0.299116\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−32.6525	−1.37128
$568$	0	0
$569$	−16.9443	−0.710341	−0.355170	−	0.934802i	$-0.615577\pi$
$569$	−16.9443	−0.710341	−0.355170	+	0.934802i	$0.615577\pi$
$570$	0	0
$571$	4.34752	0.181938	0.0909691	−	0.995854i	$-0.471004\pi$
$571$	4.34752	0.181938	0.0909691	+	0.995854i	$0.471004\pi$
$572$	0	0
$573$	32.3607	1.35189
$574$	0	0
$575$	0	0
$576$	0	0
$577$	18.5066	0.770439	0.385219	−	0.922825i	$-0.374126\pi$
$577$	18.5066	0.770439	0.385219	+	0.922825i	$0.374126\pi$
$578$	0	0
$579$	43.6525	1.81413
$580$	0	0
$581$	−29.4164	−1.22040
$582$	0	0
$583$	61.4508	2.54503
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−8.58359	−0.354283	−0.177141	−	0.984185i	$-0.556685\pi$
$587$	−8.58359	−0.354283	−0.177141	+	0.984185i	$0.556685\pi$
$588$	0	0
$589$	−23.9443	−0.986607
$590$	0	0
$591$	−13.7984	−0.567589
$592$	0	0
$593$	−4.41641	−0.181360	−0.0906801	−	0.995880i	$-0.528904\pi$
$593$	−4.41641	−0.181360	−0.0906801	+	0.995880i	$0.528904\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	30.1803	1.23520
$598$	0	0
$599$	−27.0344	−1.10460	−0.552299	−	0.833646i	$-0.686249\pi$
$599$	−27.0344	−1.10460	−0.552299	+	0.833646i	$0.686249\pi$
$600$	0	0
$601$	−31.4721	−1.28378	−0.641888	−	0.766799i	$-0.721848\pi$
$601$	−31.4721	−1.28378	−0.641888	+	0.766799i	$0.721848\pi$
$602$	0	0
$603$	−1.00000	−0.0407231
$604$	0	0
$605$	0	0
$606$	0	0
$607$	17.2361	0.699590	0.349795	−	0.936826i	$-0.386251\pi$
$607$	17.2361	0.699590	0.349795	+	0.936826i	$0.386251\pi$
$608$	0	0
$609$	70.1591	2.84299
$610$	0	0
$611$	15.0689	0.609622
$612$	0	0
$613$	−32.8885	−1.32836	−0.664178	−	0.747575i	$-0.731218\pi$
$613$	−32.8885	−1.32836	−0.664178	+	0.747575i	$0.731218\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−29.1803	−1.17476	−0.587378	−	0.809313i	$-0.699840\pi$
$617$	−29.1803	−1.17476	−0.587378	+	0.809313i	$0.699840\pi$
$618$	0	0
$619$	23.0689	0.927217	0.463608	−	0.886040i	$-0.346554\pi$
$619$	23.0689	0.927217	0.463608	+	0.886040i	$0.346554\pi$
$620$	0	0
$621$	28.6525	1.14978
$622$	0	0
$623$	2.00000	0.0801283
$624$	0	0
$625$	0	0
$626$	0	0
$627$	41.2705	1.64819
$628$	0	0
$629$	−2.76393	−0.110205
$630$	0	0
$631$	4.47214	0.178033	0.0890165	−	0.996030i	$-0.471628\pi$
$631$	4.47214	0.178033	0.0890165	+	0.996030i	$0.471628\pi$
$632$	0	0
$633$	−23.7082	−0.942317
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−26.0689	−1.03289
$638$	0	0
$639$	−0.0901699	−0.00356707
$640$	0	0
$641$	−20.1459	−0.795715	−0.397858	−	0.917447i	$-0.630246\pi$
$641$	−20.1459	−0.795715	−0.397858	+	0.917447i	$0.630246\pi$
$642$	0	0
$643$	−40.0902	−1.58100	−0.790501	−	0.612461i	$-0.790180\pi$
$643$	−40.0902	−1.58100	−0.790501	+	0.612461i	$0.790180\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−47.4721	−1.86632	−0.933161	−	0.359458i	$-0.882961\pi$
$647$	−47.4721	−1.86632	−0.933161	+	0.359458i	$0.882961\pi$
$648$	0	0
$649$	21.0902	0.827862
$650$	0	0
$651$	40.1246	1.57261
$652$	0	0
$653$	25.5967	1.00168	0.500839	−	0.865540i	$-0.333025\pi$
$653$	25.5967	1.00168	0.500839	+	0.865540i	$0.333025\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2.34752	0.0915856
$658$	0	0
$659$	−0.798374	−0.0311002	−0.0155501	−	0.999879i	$-0.504950\pi$
$659$	−0.798374	−0.0311002	−0.0155501	+	0.999879i	$0.504950\pi$
$660$	0	0
$661$	19.5279	0.759546	0.379773	−	0.925080i	$-0.376002\pi$
$661$	19.5279	0.759546	0.379773	+	0.925080i	$0.376002\pi$
$662$	0	0
$663$	−8.61803	−0.334697
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−53.5967	−2.07527
$668$	0	0
$669$	−33.6525	−1.30108
$670$	0	0
$671$	78.3394	3.02426
$672$	0	0
$673$	−23.6869	−0.913064	−0.456532	−	0.889707i	$-0.650909\pi$
$673$	−23.6869	−0.913064	−0.456532	+	0.889707i	$0.650909\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6.94427	0.266890	0.133445	−	0.991056i	$-0.457396\pi$
$677$	6.94427	0.266890	0.133445	+	0.991056i	$0.457396\pi$
$678$	0	0
$679$	−48.9787	−1.87963
$680$	0	0
$681$	−4.52786	−0.173508
$682$	0	0
$683$	−21.9443	−0.839674	−0.419837	−	0.907599i	$-0.637913\pi$
$683$	−21.9443	−0.839674	−0.419837	+	0.907599i	$0.637913\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−27.6525	−1.05501
$688$	0	0
$689$	23.4721	0.894217
$690$	0	0
$691$	−44.6525	−1.69866	−0.849330	−	0.527862i	$-0.822994\pi$
$691$	−44.6525	−1.69866	−0.849330	+	0.527862i	$0.822994\pi$
$692$	0	0
$693$	10.0902	0.383294
$694$	0	0
$695$	0	0
$696$	0	0
$697$	6.70820	0.254091
$698$	0	0
$699$	−17.7984	−0.673196
$700$	0	0
$701$	18.5066	0.698984	0.349492	−	0.936939i	$-0.386354\pi$
$701$	18.5066	0.698984	0.349492	+	0.936939i	$0.386354\pi$
$702$	0	0
$703$	5.05573	0.190680
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−19.1803	−0.721351
$708$	0	0
$709$	−33.9787	−1.27610	−0.638049	−	0.769996i	$-0.720258\pi$
$709$	−33.9787	−1.27610	−0.638049	+	0.769996i	$0.720258\pi$
$710$	0	0
$711$	0.673762	0.0252681
$712$	0	0
$713$	−30.6525	−1.14794
$714$	0	0
$715$	0	0
$716$	0	0
$717$	20.2361	0.755730
$718$	0	0
$719$	−18.7984	−0.701061	−0.350531	−	0.936551i	$-0.613999\pi$
$719$	−18.7984	−0.701061	−0.350531	+	0.936551i	$0.613999\pi$
$720$	0	0
$721$	19.1803	0.714313
$722$	0	0
$723$	−32.7426	−1.21771
$724$	0	0
$725$	0	0
$726$	0	0
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	0	0
$729$	29.5066	1.09284
$730$	0	0
$731$	6.05573	0.223979
$732$	0	0
$733$	12.5066	0.461941	0.230970	−	0.972961i	$-0.425810\pi$
$733$	12.5066	0.461941	0.230970	+	0.972961i	$0.425810\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−16.3262	−0.601385
$738$	0	0
$739$	−8.52786	−0.313703	−0.156851	−	0.987622i	$-0.550134\pi$
$739$	−8.52786	−0.313703	−0.156851	+	0.987622i	$0.550134\pi$
$740$	0	0
$741$	15.7639	0.579103
$742$	0	0
$743$	−31.5066	−1.15586	−0.577932	−	0.816085i	$-0.696140\pi$
$743$	−31.5066	−1.15586	−0.577932	+	0.816085i	$0.696140\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2.65248	0.0970490
$748$	0	0
$749$	60.4508	2.20883
$750$	0	0
$751$	19.6525	0.717129	0.358565	−	0.933505i	$-0.383266\pi$
$751$	19.6525	0.717129	0.358565	+	0.933505i	$0.383266\pi$
$752$	0	0
$753$	−27.5623	−1.00443
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.47214	0.0535057	0.0267528	−	0.999642i	$-0.491483\pi$
$757$	1.47214	0.0535057	0.0267528	+	0.999642i	$0.491483\pi$
$758$	0	0
$759$	52.8328	1.91771
$760$	0	0
$761$	−33.7771	−1.22442	−0.612209	−	0.790696i	$-0.709719\pi$
$761$	−33.7771	−1.22442	−0.612209	+	0.790696i	$0.709719\pi$
$762$	0	0
$763$	62.9230	2.27797
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.05573	0.290875
$768$	0	0
$769$	−51.7214	−1.86512	−0.932560	−	0.361015i	$-0.882430\pi$
$769$	−51.7214	−1.86512	−0.932560	+	0.361015i	$0.882430\pi$
$770$	0	0
$771$	−45.9787	−1.65588
$772$	0	0
$773$	37.2361	1.33929	0.669644	−	0.742682i	$-0.266447\pi$
$773$	37.2361	1.33929	0.669644	+	0.742682i	$0.266447\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−8.47214	−0.303936
$778$	0	0
$779$	−12.2705	−0.439637
$780$	0	0
$781$	−1.47214	−0.0526772
$782$	0	0
$783$	−56.0132	−2.00175
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−23.8328	−0.849548	−0.424774	−	0.905299i	$-0.639646\pi$
$787$	−23.8328	−0.849548	−0.424774	+	0.905299i	$0.639646\pi$
$788$	0	0
$789$	6.70820	0.238818
$790$	0	0
$791$	−38.1246	−1.35556
$792$	0	0
$793$	29.9230	1.06260
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−40.5623	−1.43679	−0.718395	−	0.695635i	$-0.755123\pi$
$797$	−40.5623	−1.43679	−0.718395	+	0.695635i	$0.755123\pi$
$798$	0	0
$799$	14.1459	0.500446
$800$	0	0
$801$	−0.180340	−0.00637200
$802$	0	0
$803$	38.3262	1.35250
$804$	0	0
$805$	0	0
$806$	0	0
$807$	23.8541	0.839704
$808$	0	0
$809$	−22.6525	−0.796419	−0.398209	−	0.917295i	$-0.630368\pi$
$809$	−22.6525	−0.796419	−0.398209	+	0.917295i	$0.630368\pi$
$810$	0	0
$811$	0.180340	0.00633259	0.00316629	−	0.999995i	$-0.498992\pi$
$811$	0.180340	0.00633259	0.00316629	+	0.999995i	$0.498992\pi$
$812$	0	0
$813$	−9.56231	−0.335365
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−11.0770	−0.387536
$818$	0	0
$819$	3.85410	0.134673
$820$	0	0
$821$	48.1033	1.67882	0.839409	−	0.543501i	$-0.182901\pi$
$821$	48.1033	1.67882	0.839409	+	0.543501i	$0.182901\pi$
$822$	0	0
$823$	7.87539	0.274519	0.137259	−	0.990535i	$-0.456171\pi$
$823$	7.87539	0.274519	0.137259	+	0.990535i	$0.456171\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.61803	0.0562646	0.0281323	−	0.999604i	$-0.491044\pi$
$827$	1.61803	0.0562646	0.0281323	+	0.999604i	$0.491044\pi$
$828$	0	0
$829$	13.0902	0.454640	0.227320	−	0.973820i	$-0.427004\pi$
$829$	13.0902	0.454640	0.227320	+	0.973820i	$0.427004\pi$
$830$	0	0
$831$	15.7984	0.548040
$832$	0	0
$833$	−24.4721	−0.847909
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−32.0344	−1.10727
$838$	0	0
$839$	25.9098	0.894507	0.447253	−	0.894407i	$-0.352402\pi$
$839$	25.9098	0.894507	0.447253	+	0.894407i	$0.352402\pi$
$840$	0	0
$841$	75.7771	2.61300
$842$	0	0
$843$	19.4164	0.668737
$844$	0	0
$845$	0	0
$846$	0	0
$847$	118.138	4.05926
$848$	0	0
$849$	−39.8885	−1.36897
$850$	0	0
$851$	6.47214	0.221862
$852$	0	0
$853$	27.4721	0.940628	0.470314	−	0.882499i	$-0.344141\pi$
$853$	27.4721	0.940628	0.470314	+	0.882499i	$0.344141\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−35.4853	−1.21215	−0.606077	−	0.795406i	$-0.707258\pi$
$857$	−35.4853	−1.21215	−0.606077	+	0.795406i	$0.707258\pi$
$858$	0	0
$859$	−0.180340	−0.00615312	−0.00307656	−	0.999995i	$-0.500979\pi$
$859$	−0.180340	−0.00615312	−0.00307656	+	0.999995i	$0.500979\pi$
$860$	0	0
$861$	20.5623	0.700762
$862$	0	0
$863$	39.7082	1.35168	0.675841	−	0.737047i	$-0.263780\pi$
$863$	39.7082	1.35168	0.675841	+	0.737047i	$0.263780\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	19.4164	0.659416
$868$	0	0
$869$	11.0000	0.373149
$870$	0	0
$871$	−6.23607	−0.211301
$872$	0	0
$873$	4.41641	0.149473
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−32.5066	−1.09767	−0.548835	−	0.835931i	$-0.684928\pi$
$877$	−32.5066	−1.09767	−0.548835	+	0.835931i	$0.684928\pi$
$878$	0	0
$879$	−37.2148	−1.25522
$880$	0	0
$881$	3.29180	0.110903	0.0554517	−	0.998461i	$-0.482340\pi$
$881$	3.29180	0.110903	0.0554517	+	0.998461i	$0.482340\pi$
$882$	0	0
$883$	0.124612	0.00419352	0.00209676	−	0.999998i	$-0.499333\pi$
$883$	0.124612	0.00419352	0.00209676	+	0.999998i	$0.499333\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	11.4164	0.383325	0.191663	−	0.981461i	$-0.438612\pi$
$887$	11.4164	0.383325	0.191663	+	0.981461i	$0.438612\pi$
$888$	0	0
$889$	9.70820	0.325603
$890$	0	0
$891$	48.0689	1.61037
$892$	0	0
$893$	−25.8754	−0.865887
$894$	0	0
$895$	0	0
$896$	0	0
$897$	20.1803	0.673802
$898$	0	0
$899$	59.9230	1.99854
$900$	0	0
$901$	22.0344	0.734074
$902$	0	0
$903$	18.5623	0.617715
$904$	0	0
$905$	0	0
$906$	0	0
$907$	23.9098	0.793913	0.396956	−	0.917837i	$-0.370066\pi$
$907$	23.9098	0.793913	0.396956	+	0.917837i	$0.370066\pi$
$908$	0	0
$909$	1.72949	0.0573636
$910$	0	0
$911$	6.29180	0.208457	0.104228	−	0.994553i	$-0.466763\pi$
$911$	6.29180	0.208457	0.104228	+	0.994553i	$0.466763\pi$
$912$	0	0
$913$	43.3050	1.43318
$914$	0	0
$915$	0	0
$916$	0	0
$917$	56.4508	1.86417
$918$	0	0
$919$	−22.5410	−0.743560	−0.371780	−	0.928321i	$-0.621252\pi$
$919$	−22.5410	−0.743560	−0.371780	+	0.928321i	$0.621252\pi$
$920$	0	0
$921$	29.7426	0.980053
$922$	0	0
$923$	−0.562306	−0.0185085
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−1.72949	−0.0568039
$928$	0	0
$929$	−54.4853	−1.78760	−0.893802	−	0.448461i	$-0.851972\pi$
$929$	−54.4853	−1.78760	−0.893802	+	0.448461i	$0.851972\pi$
$930$	0	0
$931$	44.7639	1.46708
$932$	0	0
$933$	−14.2361	−0.466068
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−40.4508	−1.32147	−0.660736	−	0.750619i	$-0.729756\pi$
$937$	−40.4508	−1.32147	−0.660736	+	0.750619i	$0.729756\pi$
$938$	0	0
$939$	−18.4721	−0.602815
$940$	0	0
$941$	−0.798374	−0.0260262	−0.0130131	−	0.999915i	$-0.504142\pi$
$941$	−0.798374	−0.0260262	−0.0130131	+	0.999915i	$0.504142\pi$
$942$	0	0
$943$	−15.7082	−0.511529
$944$	0	0
$945$	0	0
$946$	0	0
$947$	42.8885	1.39369	0.696845	−	0.717222i	$-0.254586\pi$
$947$	42.8885	1.39369	0.696845	+	0.717222i	$0.254586\pi$
$948$	0	0
$949$	14.6393	0.475212
$950$	0	0
$951$	11.2361	0.364354
$952$	0	0
$953$	51.0476	1.65359	0.826797	−	0.562501i	$-0.190161\pi$
$953$	51.0476	1.65359	0.826797	+	0.562501i	$0.190161\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−103.284	−3.33869
$958$	0	0
$959$	44.7426	1.44482
$960$	0	0
$961$	3.27051	0.105500
$962$	0	0
$963$	−5.45085	−0.175651
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0.978714	0.0314733	0.0157367	−	0.999876i	$-0.494991\pi$
$967$	0.978714	0.0314733	0.0157367	+	0.999876i	$0.494991\pi$
$968$	0	0
$969$	14.7984	0.475392
$970$	0	0
$971$	17.9443	0.575859	0.287930	−	0.957652i	$-0.407033\pi$
$971$	17.9443	0.575859	0.287930	+	0.957652i	$0.407033\pi$
$972$	0	0
$973$	−80.0132	−2.56510
$974$	0	0
$975$	0	0
$976$	0	0
$977$	41.2705	1.32036	0.660180	−	0.751107i	$-0.270480\pi$
$977$	41.2705	1.32036	0.660180	+	0.751107i	$0.270480\pi$
$978$	0	0
$979$	−2.94427	−0.0940993
$980$	0	0
$981$	−5.67376	−0.181149
$982$	0	0
$983$	−23.7984	−0.759050	−0.379525	−	0.925181i	$-0.623913\pi$
$983$	−23.7984	−0.759050	−0.379525	+	0.925181i	$0.623913\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	43.3607	1.38019
$988$	0	0
$989$	−14.1803	−0.450909
$990$	0	0
$991$	25.3607	0.805609	0.402804	−	0.915286i	$-0.368036\pi$
$991$	25.3607	0.805609	0.402804	+	0.915286i	$0.368036\pi$
$992$	0	0
$993$	38.8885	1.23409
$994$	0	0
$995$	0	0
$996$	0	0
$997$	12.2705	0.388611	0.194305	−	0.980941i	$-0.437755\pi$
$997$	12.2705	0.388611	0.194305	+	0.980941i	$0.437755\pi$
$998$	0	0
$999$	6.76393	0.214001

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1000.2.a.b.1.1	✓	2
3.2	odd	2		9000.2.a.o.1.2		2
4.3	odd	2		2000.2.a.g.1.2		2
5.2	odd	4		1000.2.c.a.249.4		4
5.3	odd	4		1000.2.c.a.249.1		4
5.4	even	2		1000.2.a.c.1.2	yes	2
8.3	odd	2		8000.2.a.g.1.1		2
8.5	even	2		8000.2.a.r.1.2		2
15.14	odd	2		9000.2.a.c.1.1		2
20.3	even	4		2000.2.c.h.1249.4		4
20.7	even	4		2000.2.c.h.1249.1		4
20.19	odd	2		2000.2.a.f.1.1		2
40.19	odd	2		8000.2.a.q.1.2		2
40.29	even	2		8000.2.a.h.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1000.2.a.b.1.1	✓	2	1.1	even	1	trivial
1000.2.a.c.1.2	yes	2	5.4	even	2
1000.2.c.a.249.1		4	5.3	odd	4
1000.2.c.a.249.4		4	5.2	odd	4
2000.2.a.f.1.1		2	20.19	odd	2
2000.2.a.g.1.2		2	4.3	odd	2
2000.2.c.h.1249.1		4	20.7	even	4
2000.2.c.h.1249.4		4	20.3	even	4
8000.2.a.g.1.1		2	8.3	odd	2
8000.2.a.h.1.1		2	40.29	even	2
8000.2.a.q.1.2		2	40.19	odd	2
8000.2.a.r.1.2		2	8.5	even	2
9000.2.a.c.1.1		2	15.14	odd	2
9000.2.a.o.1.2		2	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:	\( N \)	\(=\)	\( 1000 = 2^{3} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1000.a (trivial)

\(f(q)\)	\(=\)	\(q-1.61803 q^{3} +4.23607 q^{7} -0.381966 q^{9} +O(q^{10})\)	\(q-1.61803 q^{3} +4.23607 q^{7} -0.381966 q^{9} -6.23607 q^{11} -2.38197 q^{13} -2.23607 q^{17} +4.09017 q^{19} -6.85410 q^{21} +5.23607 q^{23} +5.47214 q^{27} -10.2361 q^{29} -5.85410 q^{31} +10.0902 q^{33} +1.23607 q^{37} +3.85410 q^{39} -3.00000 q^{41} -2.70820 q^{43} -6.32624 q^{47} +10.9443 q^{49} +3.61803 q^{51} -9.85410 q^{53} -6.61803 q^{57} -3.38197 q^{59} -12.5623 q^{61} -1.61803 q^{63} +2.61803 q^{67} -8.47214 q^{69} +0.236068 q^{71} -6.14590 q^{73} -26.4164 q^{77} -1.76393 q^{79} -7.70820 q^{81} -6.94427 q^{83} +16.5623 q^{87} +0.472136 q^{89} -10.0902 q^{91} +9.47214 q^{93} -11.5623 q^{97} +2.38197 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + 4 q^{7} - 3 q^{9}+O(q^{10}) \) 2 * q - q^3 + 4 * q^7 - 3 * q^9	\( 2 q - q^{3} + 4 q^{7} - 3 q^{9} - 8 q^{11} - 7 q^{13} - 3 q^{19} - 7 q^{21} + 6 q^{23} + 2 q^{27} - 16 q^{29} - 5 q^{31} + 9 q^{33} - 2 q^{37} + q^{39} - 6 q^{41} + 8 q^{43} + 3 q^{47} + 4 q^{49} + 5 q^{51} - 13 q^{53} - 11 q^{57} - 9 q^{59} - 5 q^{61} - q^{63} + 3 q^{67} - 8 q^{69} - 4 q^{71} - 19 q^{73} - 26 q^{77} - 8 q^{79} - 2 q^{81} + 4 q^{83} + 13 q^{87} - 8 q^{89} - 9 q^{91} + 10 q^{93} - 3 q^{97} + 7 q^{99}+O(q^{100}) \) 2 * q - q^3 + 4 * q^7 - 3 * q^9 - 8 * q^11 - 7 * q^13 - 3 * q^19 - 7 * q^21 + 6 * q^23 + 2 * q^27 - 16 * q^29 - 5 * q^31 + 9 * q^33 - 2 * q^37 + q^39 - 6 * q^41 + 8 * q^43 + 3 * q^47 + 4 * q^49 + 5 * q^51 - 13 * q^53 - 11 * q^57 - 9 * q^59 - 5 * q^61 - q^63 + 3 * q^67 - 8 * q^69 - 4 * q^71 - 19 * q^73 - 26 * q^77 - 8 * q^79 - 2 * q^81 + 4 * q^83 + 13 * q^87 - 8 * q^89 - 9 * q^91 + 10 * q^93 - 3 * q^97 + 7 * q^99

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