LMFDB - Embedded newform 6003.2.a.s.1.2

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:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} - 33 x^{18} + 64 x^{17} + 453 x^{16} - 846 x^{15} - 3353 x^{14} + 5985 x^{13} + \cdots - 256 $ x^20 - 2x^19 - 33x^18 + 64x^17 + 453x^16 - 846x^15 - 3353x^14 + 5985x^13 + 14484x^12 - 24566x^11 - 36791x^10 + 59410x^9 + 52109x^8 - 82362x^7 - 34967x^6 + 60661x^5 + 5201x^4 - 19624x^3 + 2768x^2 + 1408*x - 256
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.62822$ of defining polynomial
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)

$f(q)$	$=$	$q-2.62822 q^{2} +4.90753 q^{4} +4.03060 q^{5} -1.30880 q^{7} -7.64164 q^{8} +O(q^{10})$	\(q-2.62822 q^{2} +4.90753 q^{4} +4.03060 q^{5} -1.30880 q^{7} -7.64164 q^{8} -10.5933 q^{10} -3.92402 q^{11} +4.45983 q^{13} +3.43982 q^{14} +10.2688 q^{16} -5.05684 q^{17} -4.27409 q^{19} +19.7803 q^{20} +10.3132 q^{22} -1.00000 q^{23} +11.2457 q^{25} -11.7214 q^{26} -6.42299 q^{28} -1.00000 q^{29} +3.94960 q^{31} -11.7055 q^{32} +13.2905 q^{34} -5.27526 q^{35} +10.9136 q^{37} +11.2332 q^{38} -30.8004 q^{40} +10.8994 q^{41} -4.04647 q^{43} -19.2573 q^{44} +2.62822 q^{46} -9.36211 q^{47} -5.28704 q^{49} -29.5563 q^{50} +21.8868 q^{52} +11.0423 q^{53} -15.8162 q^{55} +10.0014 q^{56} +2.62822 q^{58} +1.85620 q^{59} -6.11452 q^{61} -10.3804 q^{62} +10.2268 q^{64} +17.9758 q^{65} -10.5649 q^{67} -24.8166 q^{68} +13.8645 q^{70} +0.234097 q^{71} -9.40763 q^{73} -28.6833 q^{74} -20.9752 q^{76} +5.13577 q^{77} +5.42650 q^{79} +41.3895 q^{80} -28.6459 q^{82} +12.1730 q^{83} -20.3821 q^{85} +10.6350 q^{86} +29.9860 q^{88} +3.98074 q^{89} -5.83704 q^{91} -4.90753 q^{92} +24.6057 q^{94} -17.2271 q^{95} +17.7045 q^{97} +13.8955 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8}+O(q^{10}) $ 20 * q - 2 * q^2 + 30 * q^4 + q^5 + 9 * q^7 - 6 * q^8	$ 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8} + 7 q^{10} + 21 q^{13} + q^{14} + 58 q^{16} + 4 q^{17} + 7 q^{19} + 20 q^{20} + 7 q^{22} - 20 q^{23} + 47 q^{25} - 8 q^{26} + 11 q^{28} - 20 q^{29} + 28 q^{31} - 14 q^{32} + 16 q^{34} - 9 q^{35} + 14 q^{37} + 20 q^{38} + 34 q^{40} - 7 q^{41} + 3 q^{43} + q^{44} + 2 q^{46} - 3 q^{47} + 35 q^{49} + 24 q^{50} + 73 q^{52} + 19 q^{53} + 29 q^{55} + 30 q^{56} + 2 q^{58} - 20 q^{59} + 15 q^{61} - 12 q^{62} + 82 q^{64} + 28 q^{65} + 20 q^{67} + 23 q^{68} - 24 q^{70} - 63 q^{71} + 19 q^{73} - 16 q^{74} - 44 q^{76} + 7 q^{77} + 32 q^{79} + 56 q^{80} - 20 q^{82} + 21 q^{83} + 4 q^{85} + 6 q^{86} + 55 q^{88} + 13 q^{89} + 70 q^{91} - 30 q^{92} - 12 q^{94} - 9 q^{95} - 9 q^{97} - 31 q^{98}+O(q^{100}) $ 20 * q - 2 * q^2 + 30 * q^4 + q^5 + 9 * q^7 - 6 * q^8 + 7 * q^10 + 21 * q^13 + q^14 + 58 * q^16 + 4 * q^17 + 7 * q^19 + 20 * q^20 + 7 * q^22 - 20 * q^23 + 47 * q^25 - 8 * q^26 + 11 * q^28 - 20 * q^29 + 28 * q^31 - 14 * q^32 + 16 * q^34 - 9 * q^35 + 14 * q^37 + 20 * q^38 + 34 * q^40 - 7 * q^41 + 3 * q^43 + q^44 + 2 * q^46 - 3 * q^47 + 35 * q^49 + 24 * q^50 + 73 * q^52 + 19 * q^53 + 29 * q^55 + 30 * q^56 + 2 * q^58 - 20 * q^59 + 15 * q^61 - 12 * q^62 + 82 * q^64 + 28 * q^65 + 20 * q^67 + 23 * q^68 - 24 * q^70 - 63 * q^71 + 19 * q^73 - 16 * q^74 - 44 * q^76 + 7 * q^77 + 32 * q^79 + 56 * q^80 - 20 * q^82 + 21 * q^83 + 4 * q^85 + 6 * q^86 + 55 * q^88 + 13 * q^89 + 70 * q^91 - 30 * q^92 - 12 * q^94 - 9 * q^95 - 9 * q^97 - 31 * 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 $

$2$	−2.62822	−1.85843	−0.929216	−	0.369538i	$-0.879516\pi$
$2$	−2.62822	−1.85843	−0.929216	+	0.369538i	$0.879516\pi$
$3$	0	0
$3$	0	0
$4$	4.90753	2.45377
$5$	4.03060	1.80254	0.901270	−	0.433259i	$-0.142636\pi$
$5$	4.03060	1.80254	0.901270	+	0.433259i	$0.142636\pi$
$6$	0	0
$7$	−1.30880	−0.494681	−0.247340	−	0.968929i	$-0.579557\pi$
$7$	−1.30880	−0.494681	−0.247340	+	0.968929i	$0.579557\pi$
$8$	−7.64164	−2.70173
$9$	0	0
$10$	−10.5933	−3.34990
$11$	−3.92402	−1.18314	−0.591569	−	0.806255i	$-0.701491\pi$
$11$	−3.92402	−1.18314	−0.591569	+	0.806255i	$0.701491\pi$
$12$	0	0
$13$	4.45983	1.23694	0.618468	−	0.785810i	$-0.287754\pi$
$13$	4.45983	1.23694	0.618468	+	0.785810i	$0.287754\pi$
$14$	3.43982	0.919330
$15$	0	0
$16$	10.2688	2.56721
$17$	−5.05684	−1.22646	−0.613232	−	0.789903i	$-0.710131\pi$
$17$	−5.05684	−1.22646	−0.613232	+	0.789903i	$0.710131\pi$
$18$	0	0
$19$	−4.27409	−0.980543	−0.490271	−	0.871570i	$-0.663102\pi$
$19$	−4.27409	−0.980543	−0.490271	+	0.871570i	$0.663102\pi$
$20$	19.7803	4.42301
$21$	0	0
$22$	10.3132	2.19878
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	11.2457	2.24915
$26$	−11.7214	−2.29876
$27$	0	0
$28$	−6.42299	−1.21383
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	3.94960	0.709368	0.354684	−	0.934986i	$-0.384588\pi$
$31$	3.94960	0.709368	0.354684	+	0.934986i	$0.384588\pi$
$32$	−11.7055	−2.06925
$33$	0	0
$34$	13.2905	2.27930
$35$	−5.27526	−0.891681
$36$	0	0
$37$	10.9136	1.79418	0.897091	−	0.441845i	$-0.145676\pi$
$37$	10.9136	1.79418	0.897091	+	0.441845i	$0.145676\pi$
$38$	11.2332	1.82227
$39$	0	0
$40$	−30.8004	−4.86997
$41$	10.8994	1.70219	0.851097	−	0.525009i	$-0.175938\pi$
$41$	10.8994	1.70219	0.851097	+	0.525009i	$0.175938\pi$
$42$	0	0
$43$	−4.04647	−0.617080	−0.308540	−	0.951211i	$-0.599840\pi$
$43$	−4.04647	−0.617080	−0.308540	+	0.951211i	$0.599840\pi$
$44$	−19.2573	−2.90314
$45$	0	0
$46$	2.62822	0.387510
$47$	−9.36211	−1.36560	−0.682802	−	0.730604i	$-0.739239\pi$
$47$	−9.36211	−1.36560	−0.682802	+	0.730604i	$0.739239\pi$
$48$	0	0
$49$	−5.28704	−0.755291
$50$	−29.5563	−4.17989
$51$	0	0
$52$	21.8868	3.03515
$53$	11.0423	1.51678	0.758388	−	0.651804i	$-0.225987\pi$
$53$	11.0423	1.51678	0.758388	+	0.651804i	$0.225987\pi$
$54$	0	0
$55$	−15.8162	−2.13265
$56$	10.0014	1.33649
$57$	0	0
$58$	2.62822	0.345102
$59$	1.85620	0.241656	0.120828	−	0.992673i	$-0.461445\pi$
$59$	1.85620	0.241656	0.120828	+	0.992673i	$0.461445\pi$
$60$	0	0
$61$	−6.11452	−0.782884	−0.391442	−	0.920203i	$-0.628024\pi$
$61$	−6.11452	−0.782884	−0.391442	+	0.920203i	$0.628024\pi$
$62$	−10.3804	−1.31831
$63$	0	0
$64$	10.2268	1.27835
$65$	17.9758	2.22962
$66$	0	0
$67$	−10.5649	−1.29071	−0.645354	−	0.763883i	$-0.723290\pi$
$67$	−10.5649	−1.29071	−0.645354	+	0.763883i	$0.723290\pi$
$68$	−24.8166	−3.00946
$69$	0	0
$70$	13.8645	1.65713
$71$	0.234097	0.0277822	0.0138911	−	0.999904i	$-0.495578\pi$
$71$	0.234097	0.0277822	0.0138911	+	0.999904i	$0.495578\pi$
$72$	0	0
$73$	−9.40763	−1.10108	−0.550540	−	0.834809i	$-0.685578\pi$
$73$	−9.40763	−1.10108	−0.550540	+	0.834809i	$0.685578\pi$
$74$	−28.6833	−3.33436
$75$	0	0
$76$	−20.9752	−2.40602
$77$	5.13577	0.585275
$78$	0	0
$79$	5.42650	0.610529	0.305264	−	0.952268i	$-0.401255\pi$
$79$	5.42650	0.610529	0.305264	+	0.952268i	$0.401255\pi$
$80$	41.3895	4.62749
$81$	0	0
$82$	−28.6459	−3.16341
$83$	12.1730	1.33616	0.668082	−	0.744087i	$-0.267115\pi$
$83$	12.1730	1.33616	0.668082	+	0.744087i	$0.267115\pi$
$84$	0	0
$85$	−20.3821	−2.21075
$86$	10.6350	1.14680
$87$	0	0
$88$	29.9860	3.19651
$89$	3.98074	0.421957	0.210979	−	0.977491i	$-0.432335\pi$
$89$	3.98074	0.421957	0.210979	+	0.977491i	$0.432335\pi$
$90$	0	0
$91$	−5.83704	−0.611888
$92$	−4.90753	−0.511646
$93$	0	0
$94$	24.6057	2.53788
$95$	−17.2271	−1.76747
$96$	0	0
$97$	17.7045	1.79761	0.898807	−	0.438344i	$-0.144435\pi$
$97$	17.7045	1.79761	0.898807	+	0.438344i	$0.144435\pi$
$98$	13.8955	1.40366
$99$	0	0
$100$	55.1888	5.51888
$101$	4.47540	0.445319	0.222660	−	0.974896i	$-0.428526\pi$
$101$	4.47540	0.445319	0.222660	+	0.974896i	$0.428526\pi$
$102$	0	0
$103$	2.13558	0.210425	0.105212	−	0.994450i	$-0.466448\pi$
$103$	2.13558	0.210425	0.105212	+	0.994450i	$0.466448\pi$
$104$	−34.0804	−3.34186
$105$	0	0
$106$	−29.0216	−2.81882
$107$	−0.364469	−0.0352346	−0.0176173	−	0.999845i	$-0.505608\pi$
$107$	−0.364469	−0.0352346	−0.0176173	+	0.999845i	$0.505608\pi$
$108$	0	0
$109$	5.86304	0.561578	0.280789	−	0.959770i	$-0.409404\pi$
$109$	5.86304	0.561578	0.280789	+	0.959770i	$0.409404\pi$
$110$	41.5684	3.96339
$111$	0	0
$112$	−13.4399	−1.26995
$113$	−4.04672	−0.380683	−0.190341	−	0.981718i	$-0.560960\pi$
$113$	−4.04672	−0.380683	−0.190341	+	0.981718i	$0.560960\pi$
$114$	0	0
$115$	−4.03060	−0.375855
$116$	−4.90753	−0.455653
$117$	0	0
$118$	−4.87849	−0.449101
$119$	6.61840	0.606708
$120$	0	0
$121$	4.39796	0.399814
$122$	16.0703	1.45494
$123$	0	0
$124$	19.3828	1.74062
$125$	25.1741	2.25164
$126$	0	0
$127$	12.1075	1.07437	0.537186	−	0.843464i	$-0.319487\pi$
$127$	12.1075	1.07437	0.537186	+	0.843464i	$0.319487\pi$
$128$	−3.46746	−0.306483
$129$	0	0
$130$	−47.2444	−4.14360
$131$	2.52075	0.220239	0.110119	−	0.993918i	$-0.464877\pi$
$131$	2.52075	0.220239	0.110119	+	0.993918i	$0.464877\pi$
$132$	0	0
$133$	5.59393	0.485055
$134$	27.7669	2.39869
$135$	0	0
$136$	38.6426	3.31357
$137$	8.87993	0.758663	0.379332	−	0.925261i	$-0.376154\pi$
$137$	8.87993	0.758663	0.379332	+	0.925261i	$0.376154\pi$
$138$	0	0
$139$	−13.0214	−1.10446	−0.552229	−	0.833692i	$-0.686223\pi$
$139$	−13.0214	−1.10446	−0.552229	+	0.833692i	$0.686223\pi$
$140$	−25.8885	−2.18798
$141$	0	0
$142$	−0.615258	−0.0516313
$143$	−17.5005	−1.46346
$144$	0	0
$145$	−4.03060	−0.334723
$146$	24.7253	2.04628
$147$	0	0
$148$	53.5588	4.40251
$149$	−2.45763	−0.201337	−0.100668	−	0.994920i	$-0.532098\pi$
$149$	−2.45763	−0.201337	−0.100668	+	0.994920i	$0.532098\pi$
$150$	0	0
$151$	6.21818	0.506029	0.253014	−	0.967463i	$-0.418578\pi$
$151$	6.21818	0.506029	0.253014	+	0.967463i	$0.418578\pi$
$152$	32.6610	2.64916
$153$	0	0
$154$	−13.4979	−1.08769
$155$	15.9192	1.27866
$156$	0	0
$157$	−13.6785	−1.09166	−0.545830	−	0.837896i	$-0.683786\pi$
$157$	−13.6785	−1.09166	−0.545830	+	0.837896i	$0.683786\pi$
$158$	−14.2620	−1.13463
$159$	0	0
$160$	−47.1800	−3.72991
$161$	1.30880	0.103148
$162$	0	0
$163$	22.3241	1.74856	0.874280	−	0.485422i	$-0.161334\pi$
$163$	22.3241	1.74856	0.874280	+	0.485422i	$0.161334\pi$
$164$	53.4890	4.17679
$165$	0	0
$166$	−31.9934	−2.48317
$167$	15.0420	1.16398	0.581992	−	0.813195i	$-0.302274\pi$
$167$	15.0420	1.16398	0.581992	+	0.813195i	$0.302274\pi$
$168$	0	0
$169$	6.89012	0.530009
$170$	53.5686	4.10853
$171$	0	0
$172$	−19.8582	−1.51417
$173$	17.7084	1.34634	0.673171	−	0.739487i	$-0.264932\pi$
$173$	17.7084	1.34634	0.673171	+	0.739487i	$0.264932\pi$
$174$	0	0
$175$	−14.7184	−1.11261
$176$	−40.2951	−3.03736
$177$	0	0
$178$	−10.4622	−0.784179
$179$	−14.3858	−1.07524	−0.537622	−	0.843186i	$-0.680677\pi$
$179$	−14.3858	−1.07524	−0.537622	+	0.843186i	$0.680677\pi$
$180$	0	0
$181$	17.9508	1.33427	0.667135	−	0.744937i	$-0.267520\pi$
$181$	17.9508	1.33427	0.667135	+	0.744937i	$0.267520\pi$
$182$	15.3410	1.13715
$183$	0	0
$184$	7.64164	0.563349
$185$	43.9883	3.23408
$186$	0	0
$187$	19.8432	1.45108
$188$	−45.9449	−3.35087
$189$	0	0
$190$	45.2767	3.28472
$191$	2.96160	0.214294	0.107147	−	0.994243i	$-0.465829\pi$
$191$	2.96160	0.214294	0.107147	+	0.994243i	$0.465829\pi$
$192$	0	0
$193$	6.26171	0.450727	0.225364	−	0.974275i	$-0.427643\pi$
$193$	6.26171	0.450727	0.225364	+	0.974275i	$0.427643\pi$
$194$	−46.5312	−3.34074
$195$	0	0
$196$	−25.9463	−1.85331
$197$	23.1754	1.65118	0.825589	−	0.564272i	$-0.190843\pi$
$197$	23.1754	1.65118	0.825589	+	0.564272i	$0.190843\pi$
$198$	0	0
$199$	1.65529	0.117340	0.0586702	−	0.998277i	$-0.481314\pi$
$199$	1.65529	0.117340	0.0586702	+	0.998277i	$0.481314\pi$
$200$	−85.9358	−6.07658
$201$	0	0
$202$	−11.7623	−0.827595
$203$	1.30880	0.0918599
$204$	0	0
$205$	43.9309	3.06827
$206$	−5.61276	−0.391060
$207$	0	0
$208$	45.7973	3.17547
$209$	16.7716	1.16012
$210$	0	0
$211$	−0.335765	−0.0231150	−0.0115575	−	0.999933i	$-0.503679\pi$
$211$	−0.335765	−0.0231150	−0.0115575	+	0.999933i	$0.503679\pi$
$212$	54.1904	3.72181
$213$	0	0
$214$	0.957905	0.0654810
$215$	−16.3097	−1.11231
$216$	0	0
$217$	−5.16924	−0.350911
$218$	−15.4094	−1.04365
$219$	0	0
$220$	−77.6184	−5.23303
$221$	−22.5527	−1.51706
$222$	0	0
$223$	11.4430	0.766283	0.383142	−	0.923690i	$-0.374842\pi$
$223$	11.4430	0.766283	0.383142	+	0.923690i	$0.374842\pi$
$224$	15.3201	1.02362
$225$	0	0
$226$	10.6357	0.707473
$227$	−13.0515	−0.866259	−0.433129	−	0.901332i	$-0.642591\pi$
$227$	−13.0515	−0.866259	−0.433129	+	0.901332i	$0.642591\pi$
$228$	0	0
$229$	17.2421	1.13939	0.569695	−	0.821856i	$-0.307061\pi$
$229$	17.2421	1.13939	0.569695	+	0.821856i	$0.307061\pi$
$230$	10.5933	0.698501
$231$	0	0
$232$	7.64164	0.501698
$233$	−23.7833	−1.55809	−0.779046	−	0.626966i	$-0.784296\pi$
$233$	−23.7833	−1.55809	−0.779046	+	0.626966i	$0.784296\pi$
$234$	0	0
$235$	−37.7349	−2.46155
$236$	9.10935	0.592968
$237$	0	0
$238$	−17.3946	−1.12753
$239$	17.1777	1.11113	0.555566	−	0.831472i	$-0.312501\pi$
$239$	17.1777	1.11113	0.555566	+	0.831472i	$0.312501\pi$
$240$	0	0
$241$	−26.8413	−1.72900	−0.864501	−	0.502631i	$-0.832365\pi$
$241$	−26.8413	−1.72900	−0.864501	+	0.502631i	$0.832365\pi$
$242$	−11.5588	−0.743028
$243$	0	0
$244$	−30.0072	−1.92102
$245$	−21.3099	−1.36144
$246$	0	0
$247$	−19.0617	−1.21287
$248$	−30.1814	−1.91652
$249$	0	0
$250$	−66.1630	−4.18451
$251$	23.8599	1.50603	0.753013	−	0.658006i	$-0.228600\pi$
$251$	23.8599	1.50603	0.753013	+	0.658006i	$0.228600\pi$
$252$	0	0
$253$	3.92402	0.246701
$254$	−31.8213	−1.99665
$255$	0	0
$256$	−11.3404	−0.708777
$257$	−6.38623	−0.398362	−0.199181	−	0.979963i	$-0.563828\pi$
$257$	−6.38623	−0.398362	−0.199181	+	0.979963i	$0.563828\pi$
$258$	0	0
$259$	−14.2837	−0.887547
$260$	88.2169	5.47098
$261$	0	0
$262$	−6.62507	−0.409298
$263$	14.5150	0.895035	0.447518	−	0.894275i	$-0.352308\pi$
$263$	14.5150	0.895035	0.447518	+	0.894275i	$0.352308\pi$
$264$	0	0
$265$	44.5071	2.73405
$266$	−14.7021	−0.901442
$267$	0	0
$268$	−51.8477	−3.16710
$269$	−8.53374	−0.520311	−0.260156	−	0.965567i	$-0.583774\pi$
$269$	−8.53374	−0.520311	−0.260156	+	0.965567i	$0.583774\pi$
$270$	0	0
$271$	−32.1862	−1.95517	−0.977585	−	0.210540i	$-0.932478\pi$
$271$	−32.1862	−1.95517	−0.977585	+	0.210540i	$0.932478\pi$
$272$	−51.9278	−3.14859
$273$	0	0
$274$	−23.3384	−1.40992
$275$	−44.1285	−2.66105
$276$	0	0
$277$	23.6051	1.41829	0.709147	−	0.705060i	$-0.249080\pi$
$277$	23.6051	1.41829	0.709147	+	0.705060i	$0.249080\pi$
$278$	34.2230	2.05256
$279$	0	0
$280$	40.3116	2.40908
$281$	13.4076	0.799831	0.399915	−	0.916552i	$-0.369039\pi$
$281$	13.4076	0.799831	0.399915	+	0.916552i	$0.369039\pi$
$282$	0	0
$283$	9.07775	0.539616	0.269808	−	0.962914i	$-0.413040\pi$
$283$	9.07775	0.539616	0.269808	+	0.962914i	$0.413040\pi$
$284$	1.14884	0.0681710
$285$	0	0
$286$	45.9951	2.71975
$287$	−14.2651	−0.842042
$288$	0	0
$289$	8.57165	0.504215
$290$	10.5933	0.622060
$291$	0	0
$292$	−46.1683	−2.70179
$293$	−17.4273	−1.01811	−0.509057	−	0.860733i	$-0.670006\pi$
$293$	−17.4273	−1.01811	−0.509057	+	0.860733i	$0.670006\pi$
$294$	0	0
$295$	7.48158	0.435595
$296$	−83.3977	−4.84739
$297$	0	0
$298$	6.45919	0.374171
$299$	−4.45983	−0.257919
$300$	0	0
$301$	5.29602	0.305258
$302$	−16.3427	−0.940419
$303$	0	0
$304$	−43.8899	−2.51726
$305$	−24.6452	−1.41118
$306$	0	0
$307$	−18.3100	−1.04501	−0.522504	−	0.852637i	$-0.675002\pi$
$307$	−18.3100	−1.04501	−0.522504	+	0.852637i	$0.675002\pi$
$308$	25.2040	1.43613
$309$	0	0
$310$	−41.8392	−2.37631
$311$	25.1692	1.42722	0.713609	−	0.700545i	$-0.247060\pi$
$311$	25.1692	1.42722	0.713609	+	0.700545i	$0.247060\pi$
$312$	0	0
$313$	5.07476	0.286843	0.143421	−	0.989662i	$-0.454190\pi$
$313$	5.07476	0.286843	0.143421	+	0.989662i	$0.454190\pi$
$314$	35.9500	2.02878
$315$	0	0
$316$	26.6307	1.49810
$317$	9.98799	0.560981	0.280491	−	0.959857i	$-0.409503\pi$
$317$	9.98799	0.560981	0.280491	+	0.959857i	$0.409503\pi$
$318$	0	0
$319$	3.92402	0.219703
$320$	41.2203	2.30428
$321$	0	0
$322$	−3.43982	−0.191694
$323$	21.6134	1.20260
$324$	0	0
$325$	50.1541	2.78205
$326$	−58.6727	−3.24958
$327$	0	0
$328$	−83.2889	−4.59886
$329$	12.2531	0.675538
$330$	0	0
$331$	11.8792	0.652941	0.326471	−	0.945207i	$-0.394141\pi$
$331$	11.8792	0.652941	0.326471	+	0.945207i	$0.394141\pi$
$332$	59.7396	3.27864
$333$	0	0
$334$	−39.5336	−2.16318
$335$	−42.5829	−2.32655
$336$	0	0
$337$	4.20337	0.228972	0.114486	−	0.993425i	$-0.463478\pi$
$337$	4.20337	0.228972	0.114486	+	0.993425i	$0.463478\pi$
$338$	−18.1087	−0.984985
$339$	0	0
$340$	−100.026	−5.42467
$341$	−15.4983	−0.839280
$342$	0	0
$343$	16.0813	0.868308
$344$	30.9216	1.66718
$345$	0	0
$346$	−46.5415	−2.50208
$347$	13.6858	0.734691	0.367346	−	0.930084i	$-0.380267\pi$
$347$	13.6858	0.734691	0.367346	+	0.930084i	$0.380267\pi$
$348$	0	0
$349$	−25.6384	−1.37239	−0.686195	−	0.727418i	$-0.740720\pi$
$349$	−25.6384	−1.37239	−0.686195	+	0.727418i	$0.740720\pi$
$350$	38.6833	2.06771
$351$	0	0
$352$	45.9325	2.44821
$353$	−24.7065	−1.31499	−0.657497	−	0.753457i	$-0.728385\pi$
$353$	−24.7065	−1.31499	−0.657497	+	0.753457i	$0.728385\pi$
$354$	0	0
$355$	0.943551	0.0500785
$356$	19.5356	1.03538
$357$	0	0
$358$	37.8090	1.99827
$359$	−20.2553	−1.06903	−0.534517	−	0.845158i	$-0.679507\pi$
$359$	−20.2553	−1.06903	−0.534517	+	0.845158i	$0.679507\pi$
$360$	0	0
$361$	−0.732187	−0.0385362
$362$	−47.1786	−2.47965
$363$	0	0
$364$	−28.6455	−1.50143
$365$	−37.9184	−1.98474
$366$	0	0
$367$	33.6637	1.75723	0.878615	−	0.477530i	$-0.158468\pi$
$367$	33.6637	1.75723	0.878615	+	0.477530i	$0.158468\pi$
$368$	−10.2688	−0.535300
$369$	0	0
$370$	−115.611	−6.01032
$371$	−14.4522	−0.750319
$372$	0	0
$373$	−18.8819	−0.977667	−0.488833	−	0.872377i	$-0.662577\pi$
$373$	−18.8819	−0.977667	−0.488833	+	0.872377i	$0.662577\pi$
$374$	−52.1522	−2.69672
$375$	0	0
$376$	71.5418	3.68949
$377$	−4.45983	−0.229693
$378$	0	0
$379$	6.30502	0.323867	0.161933	−	0.986802i	$-0.448227\pi$
$379$	6.30502	0.323867	0.161933	+	0.986802i	$0.448227\pi$
$380$	−84.5427	−4.33695
$381$	0	0
$382$	−7.78373	−0.398250
$383$	2.87741	0.147029	0.0735145	−	0.997294i	$-0.476578\pi$
$383$	2.87741	0.147029	0.0735145	+	0.997294i	$0.476578\pi$
$384$	0	0
$385$	20.7002	1.05498
$386$	−16.4571	−0.837646
$387$	0	0
$388$	86.8852	4.41093
$389$	−36.8649	−1.86913	−0.934563	−	0.355797i	$-0.884209\pi$
$389$	−36.8649	−1.86913	−0.934563	+	0.355797i	$0.884209\pi$
$390$	0	0
$391$	5.05684	0.255735
$392$	40.4016	2.04059
$393$	0	0
$394$	−60.9100	−3.06860
$395$	21.8720	1.10050
$396$	0	0
$397$	−0.604130	−0.0303204	−0.0151602	−	0.999885i	$-0.504826\pi$
$397$	−0.604130	−0.0303204	−0.0151602	+	0.999885i	$0.504826\pi$
$398$	−4.35047	−0.218069
$399$	0	0
$400$	115.481	5.77403
$401$	28.4282	1.41964	0.709818	−	0.704385i	$-0.248777\pi$
$401$	28.4282	1.41964	0.709818	+	0.704385i	$0.248777\pi$
$402$	0	0
$403$	17.6145	0.877443
$404$	21.9632	1.09271
$405$	0	0
$406$	−3.43982	−0.170715
$407$	−42.8252	−2.12276
$408$	0	0
$409$	−28.2203	−1.39540	−0.697702	−	0.716388i	$-0.745794\pi$
$409$	−28.2203	−1.39540	−0.697702	+	0.716388i	$0.745794\pi$
$410$	−115.460	−5.70217
$411$	0	0
$412$	10.4804	0.516333
$413$	−2.42939	−0.119543
$414$	0	0
$415$	49.0646	2.40849
$416$	−52.2044	−2.55953
$417$	0	0
$418$	−44.0795	−2.15600
$419$	7.93616	0.387707	0.193853	−	0.981031i	$-0.437901\pi$
$419$	7.93616	0.387707	0.193853	+	0.981031i	$0.437901\pi$
$420$	0	0
$421$	17.1419	0.835445	0.417722	−	0.908575i	$-0.362828\pi$
$421$	17.1419	0.835445	0.417722	+	0.908575i	$0.362828\pi$
$422$	0.882464	0.0429577
$423$	0	0
$424$	−84.3812	−4.09791
$425$	−56.8679	−2.75850
$426$	0	0
$427$	8.00270	0.387278
$428$	−1.78865	−0.0864574
$429$	0	0
$430$	42.8654	2.06715
$431$	20.8868	1.00608	0.503040	−	0.864263i	$-0.332215\pi$
$431$	20.8868	1.00608	0.503040	+	0.864263i	$0.332215\pi$
$432$	0	0
$433$	17.3374	0.833181	0.416590	−	0.909094i	$-0.363225\pi$
$433$	17.3374	0.833181	0.416590	+	0.909094i	$0.363225\pi$
$434$	13.5859	0.652143
$435$	0	0
$436$	28.7731	1.37798
$437$	4.27409	0.204457
$438$	0	0
$439$	3.80024	0.181375	0.0906877	−	0.995879i	$-0.471093\pi$
$439$	3.80024	0.181375	0.0906877	+	0.995879i	$0.471093\pi$
$440$	120.861	5.76184
$441$	0	0
$442$	59.2734	2.81935
$443$	28.3193	1.34549	0.672745	−	0.739875i	$-0.265115\pi$
$443$	28.3193	1.34549	0.672745	+	0.739875i	$0.265115\pi$
$444$	0	0
$445$	16.0448	0.760594
$446$	−30.0748	−1.42408
$447$	0	0
$448$	−13.3849	−0.632377
$449$	−32.2036	−1.51978	−0.759892	−	0.650049i	$-0.774748\pi$
$449$	−32.2036	−1.51978	−0.759892	+	0.650049i	$0.774748\pi$
$450$	0	0
$451$	−42.7693	−2.01393
$452$	−19.8594	−0.934107
$453$	0	0
$454$	34.3022	1.60988
$455$	−23.5268	−1.10295
$456$	0	0
$457$	−20.5248	−0.960110	−0.480055	−	0.877238i	$-0.659383\pi$
$457$	−20.5248	−0.960110	−0.480055	+	0.877238i	$0.659383\pi$
$458$	−45.3160	−2.11748
$459$	0	0
$460$	−19.7803	−0.922262
$461$	12.9526	0.603263	0.301632	−	0.953425i	$-0.402469\pi$
$461$	12.9526	0.603263	0.301632	+	0.953425i	$0.402469\pi$
$462$	0	0
$463$	−3.31491	−0.154057	−0.0770285	−	0.997029i	$-0.524543\pi$
$463$	−3.31491	−0.154057	−0.0770285	+	0.997029i	$0.524543\pi$
$464$	−10.2688	−0.476718
$465$	0	0
$466$	62.5076	2.89561
$467$	12.3457	0.571292	0.285646	−	0.958335i	$-0.407792\pi$
$467$	12.3457	0.571292	0.285646	+	0.958335i	$0.407792\pi$
$468$	0	0
$469$	13.8274	0.638489
$470$	99.1756	4.57463
$471$	0	0
$472$	−14.1844	−0.652889
$473$	15.8784	0.730091
$474$	0	0
$475$	−48.0652	−2.20538
$476$	32.4800	1.48872
$477$	0	0
$478$	−45.1467	−2.06496
$479$	34.2771	1.56616	0.783081	−	0.621920i	$-0.213647\pi$
$479$	34.2771	1.56616	0.783081	+	0.621920i	$0.213647\pi$
$480$	0	0
$481$	48.6728	2.21929
$482$	70.5449	3.21323
$483$	0	0
$484$	21.5831	0.981051
$485$	71.3596	3.24027
$486$	0	0
$487$	23.2630	1.05415	0.527073	−	0.849820i	$-0.323289\pi$
$487$	23.2630	1.05415	0.527073	+	0.849820i	$0.323289\pi$
$488$	46.7250	2.11514
$489$	0	0
$490$	56.0072	2.53015
$491$	−9.25429	−0.417640	−0.208820	−	0.977954i	$-0.566962\pi$
$491$	−9.25429	−0.417640	−0.208820	+	0.977954i	$0.566962\pi$
$492$	0	0
$493$	5.05684	0.227749
$494$	50.0984	2.25403
$495$	0	0
$496$	40.5577	1.82110
$497$	−0.306386	−0.0137433
$498$	0	0
$499$	−9.98925	−0.447180	−0.223590	−	0.974683i	$-0.571778\pi$
$499$	−9.98925	−0.447180	−0.223590	+	0.974683i	$0.571778\pi$
$500$	123.543	5.52499
$501$	0	0
$502$	−62.7091	−2.79885
$503$	−30.4391	−1.35721	−0.678607	−	0.734502i	$-0.737416\pi$
$503$	−30.4391	−1.35721	−0.678607	+	0.734502i	$0.737416\pi$
$504$	0	0
$505$	18.0386	0.802706
$506$	−10.3132	−0.458477
$507$	0	0
$508$	59.4182	2.63626
$509$	−17.7690	−0.787599	−0.393799	−	0.919196i	$-0.628840\pi$
$509$	−17.7690	−0.787599	−0.393799	+	0.919196i	$0.628840\pi$
$510$	0	0
$511$	12.3127	0.544683
$512$	36.7401	1.62370
$513$	0	0
$514$	16.7844	0.740328
$515$	8.60766	0.379299
$516$	0	0
$517$	36.7371	1.61570
$518$	37.5407	1.64945
$519$	0	0
$520$	−137.365	−6.02384
$521$	26.3429	1.15410	0.577052	−	0.816708i	$-0.304203\pi$
$521$	26.3429	1.15410	0.577052	+	0.816708i	$0.304203\pi$
$522$	0	0
$523$	16.3553	0.715169	0.357585	−	0.933881i	$-0.383600\pi$
$523$	16.3553	0.715169	0.357585	+	0.933881i	$0.383600\pi$
$524$	12.3706	0.540414
$525$	0	0
$526$	−38.1487	−1.66336
$527$	−19.9725	−0.870015
$528$	0	0
$529$	1.00000	0.0434783
$530$	−116.974	−5.08104
$531$	0	0
$532$	27.4524	1.19021
$533$	48.6093	2.10550
$534$	0	0
$535$	−1.46903	−0.0635117
$536$	80.7332	3.48714
$537$	0	0
$538$	22.4285	0.966963
$539$	20.7465	0.893613
$540$	0	0
$541$	−35.8863	−1.54287	−0.771436	−	0.636307i	$-0.780461\pi$
$541$	−35.8863	−1.54287	−0.771436	+	0.636307i	$0.780461\pi$
$542$	84.5923	3.63355
$543$	0	0
$544$	59.1926	2.53786
$545$	23.6316	1.01227
$546$	0	0
$547$	7.69359	0.328954	0.164477	−	0.986381i	$-0.447406\pi$
$547$	7.69359	0.328954	0.164477	+	0.986381i	$0.447406\pi$
$548$	43.5785	1.86158
$549$	0	0
$550$	115.979	4.94538
$551$	4.27409	0.182082
$552$	0	0
$553$	−7.10221	−0.302017
$554$	−62.0394	−2.63580
$555$	0	0
$556$	−63.9029	−2.71009
$557$	−9.35667	−0.396455	−0.198227	−	0.980156i	$-0.563518\pi$
$557$	−9.35667	−0.396455	−0.198227	+	0.980156i	$0.563518\pi$
$558$	0	0
$559$	−18.0466	−0.763288
$560$	−54.1707	−2.28913
$561$	0	0
$562$	−35.2381	−1.48643
$563$	36.0938	1.52117	0.760587	−	0.649236i	$-0.224911\pi$
$563$	36.0938	1.52117	0.760587	+	0.649236i	$0.224911\pi$
$564$	0	0
$565$	−16.3107	−0.686196
$566$	−23.8583	−1.00284
$567$	0	0
$568$	−1.78888	−0.0750599
$569$	−3.60296	−0.151044	−0.0755219	−	0.997144i	$-0.524062\pi$
$569$	−3.60296	−0.151044	−0.0755219	+	0.997144i	$0.524062\pi$
$570$	0	0
$571$	8.79729	0.368155	0.184078	−	0.982912i	$-0.441070\pi$
$571$	8.79729	0.368155	0.184078	+	0.982912i	$0.441070\pi$
$572$	−85.8843	−3.59100
$573$	0	0
$574$	37.4918	1.56488
$575$	−11.2457	−0.468980
$576$	0	0
$577$	8.19811	0.341292	0.170646	−	0.985332i	$-0.445415\pi$
$577$	8.19811	0.341292	0.170646	+	0.985332i	$0.445415\pi$
$578$	−22.5282	−0.937048
$579$	0	0
$580$	−19.7803	−0.821333
$581$	−15.9321	−0.660975
$582$	0	0
$583$	−43.3302	−1.79455
$584$	71.8897	2.97482
$585$	0	0
$586$	45.8028	1.89210
$587$	−5.43093	−0.224158	−0.112079	−	0.993699i	$-0.535751\pi$
$587$	−5.43093	−0.224158	−0.112079	+	0.993699i	$0.535751\pi$
$588$	0	0
$589$	−16.8809	−0.695566
$590$	−19.6632	−0.809523
$591$	0	0
$592$	112.070	4.60604
$593$	13.1806	0.541263	0.270631	−	0.962683i	$-0.412768\pi$
$593$	13.1806	0.541263	0.270631	+	0.962683i	$0.412768\pi$
$594$	0	0
$595$	26.6761	1.09361
$596$	−12.0609	−0.494034
$597$	0	0
$598$	11.7214	0.479324
$599$	−2.52635	−0.103224	−0.0516119	−	0.998667i	$-0.516436\pi$
$599$	−2.52635	−0.103224	−0.0516119	+	0.998667i	$0.516436\pi$
$600$	0	0
$601$	14.8703	0.606571	0.303285	−	0.952900i	$-0.401916\pi$
$601$	14.8703	0.606571	0.303285	+	0.952900i	$0.401916\pi$
$602$	−13.9191	−0.567300
$603$	0	0
$604$	30.5160	1.24168
$605$	17.7264	0.720681
$606$	0	0
$607$	34.5555	1.40256	0.701282	−	0.712884i	$-0.252611\pi$
$607$	34.5555	1.40256	0.701282	+	0.712884i	$0.252611\pi$
$608$	50.0301	2.02899
$609$	0	0
$610$	64.7730	2.62258
$611$	−41.7534	−1.68916
$612$	0	0
$613$	6.55533	0.264767	0.132384	−	0.991199i	$-0.457737\pi$
$613$	6.55533	0.264767	0.132384	+	0.991199i	$0.457737\pi$
$614$	48.1227	1.94208
$615$	0	0
$616$	−39.2457	−1.58125
$617$	24.5329	0.987658	0.493829	−	0.869559i	$-0.335597\pi$
$617$	24.5329	0.987658	0.493829	+	0.869559i	$0.335597\pi$
$618$	0	0
$619$	25.3807	1.02014	0.510068	−	0.860134i	$-0.329620\pi$
$619$	25.3807	1.02014	0.510068	+	0.860134i	$0.329620\pi$
$620$	78.1242	3.13754
$621$	0	0
$622$	−66.1503	−2.65239
$623$	−5.20999	−0.208734
$624$	0	0
$625$	45.2379	1.80952
$626$	−13.3376	−0.533077
$627$	0	0
$628$	−67.1275	−2.67868
$629$	−55.1883	−2.20050
$630$	0	0
$631$	−4.56564	−0.181755	−0.0908776	−	0.995862i	$-0.528967\pi$
$631$	−4.56564	−0.181755	−0.0908776	+	0.995862i	$0.528967\pi$
$632$	−41.4673	−1.64948
$633$	0	0
$634$	−26.2506	−1.04255
$635$	48.8007	1.93660
$636$	0	0
$637$	−23.5793	−0.934246
$638$	−10.3132	−0.408303
$639$	0	0
$640$	−13.9759	−0.552447
$641$	26.4626	1.04521	0.522604	−	0.852575i	$-0.324961\pi$
$641$	26.4626	1.04521	0.522604	+	0.852575i	$0.324961\pi$
$642$	0	0
$643$	19.6322	0.774220	0.387110	−	0.922034i	$-0.373473\pi$
$643$	19.6322	0.774220	0.387110	+	0.922034i	$0.373473\pi$
$644$	6.42299	0.253101
$645$	0	0
$646$	−56.8047	−2.23495
$647$	−36.3256	−1.42811	−0.714054	−	0.700091i	$-0.753143\pi$
$647$	−36.3256	−1.42811	−0.714054	+	0.700091i	$0.753143\pi$
$648$	0	0
$649$	−7.28376	−0.285912
$650$	−131.816	−5.17025
$651$	0	0
$652$	109.556	4.29056
$653$	−22.3417	−0.874297	−0.437149	−	0.899389i	$-0.644012\pi$
$653$	−22.3417	−0.874297	−0.437149	+	0.899389i	$0.644012\pi$
$654$	0	0
$655$	10.1601	0.396989
$656$	111.924	4.36988
$657$	0	0
$658$	−32.2039	−1.25544
$659$	−13.3343	−0.519431	−0.259716	−	0.965685i	$-0.583629\pi$
$659$	−13.3343	−0.519431	−0.259716	+	0.965685i	$0.583629\pi$
$660$	0	0
$661$	−1.80412	−0.0701720	−0.0350860	−	0.999384i	$-0.511171\pi$
$661$	−1.80412	−0.0701720	−0.0350860	+	0.999384i	$0.511171\pi$
$662$	−31.2212	−1.21345
$663$	0	0
$664$	−93.0219	−3.60995
$665$	22.5469	0.874331
$666$	0	0
$667$	1.00000	0.0387202
$668$	73.8191	2.85615
$669$	0	0
$670$	111.917	4.32374
$671$	23.9935	0.926260
$672$	0	0
$673$	−35.3630	−1.36314	−0.681572	−	0.731751i	$-0.738703\pi$
$673$	−35.3630	−1.36314	−0.681572	+	0.731751i	$0.738703\pi$
$674$	−11.0474	−0.425529
$675$	0	0
$676$	33.8135	1.30052
$677$	5.78388	0.222293	0.111146	−	0.993804i	$-0.464548\pi$
$677$	5.78388	0.222293	0.111146	+	0.993804i	$0.464548\pi$
$678$	0	0
$679$	−23.1716	−0.889245
$680$	155.753	5.97284
$681$	0	0
$682$	40.7329	1.55974
$683$	41.1493	1.57453	0.787267	−	0.616612i	$-0.211495\pi$
$683$	41.1493	1.57453	0.787267	+	0.616612i	$0.211495\pi$
$684$	0	0
$685$	35.7914	1.36752
$686$	−42.2652	−1.61369
$687$	0	0
$688$	−41.5525	−1.58417
$689$	49.2468	1.87615
$690$	0	0
$691$	32.8290	1.24887	0.624436	−	0.781076i	$-0.285329\pi$
$691$	32.8290	1.24887	0.624436	+	0.781076i	$0.285329\pi$
$692$	86.9044	3.30361
$693$	0	0
$694$	−35.9692	−1.36537
$695$	−52.4840	−1.99083
$696$	0	0
$697$	−55.1163	−2.08768
$698$	67.3832	2.55049
$699$	0	0
$700$	−72.2312	−2.73008
$701$	−1.79732	−0.0678838	−0.0339419	−	0.999424i	$-0.510806\pi$
$701$	−1.79732	−0.0678838	−0.0339419	+	0.999424i	$0.510806\pi$
$702$	0	0
$703$	−46.6456	−1.75927
$704$	−40.1303	−1.51247
$705$	0	0
$706$	64.9341	2.44383
$707$	−5.85742	−0.220291
$708$	0	0
$709$	−6.32493	−0.237538	−0.118769	−	0.992922i	$-0.537895\pi$
$709$	−6.32493	−0.237538	−0.118769	+	0.992922i	$0.537895\pi$
$710$	−2.47986	−0.0930674
$711$	0	0
$712$	−30.4193	−1.14001
$713$	−3.94960	−0.147914
$714$	0	0
$715$	−70.5375	−2.63795
$716$	−70.5987	−2.63840
$717$	0	0
$718$	53.2354	1.98673
$719$	−32.7107	−1.21990	−0.609952	−	0.792438i	$-0.708811\pi$
$719$	−32.7107	−1.21990	−0.609952	+	0.792438i	$0.708811\pi$
$720$	0	0
$721$	−2.79505	−0.104093
$722$	1.92435	0.0716168
$723$	0	0
$724$	88.0940	3.27399
$725$	−11.2457	−0.417656
$726$	0	0
$727$	21.3139	0.790489	0.395244	−	0.918576i	$-0.370660\pi$
$727$	21.3139	0.790489	0.395244	+	0.918576i	$0.370660\pi$
$728$	44.6045	1.65315
$729$	0	0
$730$	99.6579	3.68850
$731$	20.4623	0.756827
$732$	0	0
$733$	−1.91080	−0.0705769	−0.0352884	−	0.999377i	$-0.511235\pi$
$733$	−1.91080	−0.0705769	−0.0352884	+	0.999377i	$0.511235\pi$
$734$	−88.4756	−3.26569
$735$	0	0
$736$	11.7055	0.431469
$737$	41.4569	1.52709
$738$	0	0
$739$	−35.8192	−1.31763	−0.658816	−	0.752304i	$-0.728942\pi$
$739$	−35.8192	−1.31763	−0.658816	+	0.752304i	$0.728942\pi$
$740$	215.874	7.93569
$741$	0	0
$742$	37.9835	1.39442
$743$	−44.0312	−1.61535	−0.807673	−	0.589630i	$-0.799273\pi$
$743$	−44.0312	−1.61535	−0.807673	+	0.589630i	$0.799273\pi$
$744$	0	0
$745$	−9.90573	−0.362918
$746$	49.6257	1.81693
$747$	0	0
$748$	97.3810	3.56060
$749$	0.477018	0.0174299
$750$	0	0
$751$	27.3898	0.999467	0.499734	−	0.866179i	$-0.333431\pi$
$751$	27.3898	0.999467	0.499734	+	0.866179i	$0.333431\pi$
$752$	−96.1379	−3.50579
$753$	0	0
$754$	11.7214	0.426869
$755$	25.0630	0.912136
$756$	0	0
$757$	21.5043	0.781588	0.390794	−	0.920478i	$-0.372200\pi$
$757$	21.5043	0.781588	0.390794	+	0.920478i	$0.372200\pi$
$758$	−16.5710	−0.601885
$759$	0	0
$760$	131.644	4.77521
$761$	−11.1035	−0.402503	−0.201251	−	0.979540i	$-0.564501\pi$
$761$	−11.1035	−0.402503	−0.201251	+	0.979540i	$0.564501\pi$
$762$	0	0
$763$	−7.67356	−0.277802
$764$	14.5341	0.525827
$765$	0	0
$766$	−7.56247	−0.273243
$767$	8.27833	0.298913
$768$	0	0
$769$	25.5509	0.921390	0.460695	−	0.887558i	$-0.347600\pi$
$769$	25.5509	0.921390	0.460695	+	0.887558i	$0.347600\pi$
$770$	−54.4047	−1.96061
$771$	0	0
$772$	30.7295	1.10598
$773$	7.81970	0.281255	0.140628	−	0.990063i	$-0.455088\pi$
$773$	7.81970	0.281255	0.140628	+	0.990063i	$0.455088\pi$
$774$	0	0
$775$	44.4161	1.59547
$776$	−135.291	−4.85666
$777$	0	0
$778$	96.8891	3.47364
$779$	−46.5848	−1.66907
$780$	0	0
$781$	−0.918602	−0.0328702
$782$	−13.2905	−0.475267
$783$	0	0
$784$	−54.2917	−1.93899
$785$	−55.1324	−1.96776
$786$	0	0
$787$	−20.8943	−0.744801	−0.372400	−	0.928072i	$-0.621465\pi$
$787$	−20.8943	−0.744801	−0.372400	+	0.928072i	$0.621465\pi$
$788$	113.734	4.05161
$789$	0	0
$790$	−57.4845	−2.04521
$791$	5.29635	0.188316
$792$	0	0
$793$	−27.2698	−0.968377
$794$	1.58779	0.0563484
$795$	0	0
$796$	8.12340	0.287926
$797$	−17.0846	−0.605167	−0.302584	−	0.953123i	$-0.597849\pi$
$797$	−17.0846	−0.605167	−0.302584	+	0.953123i	$0.597849\pi$
$798$	0	0
$799$	47.3427	1.67486
$800$	−131.636	−4.65405
$801$	0	0
$802$	−74.7155	−2.63830
$803$	36.9158	1.30273
$804$	0	0
$805$	5.27526	0.185928
$806$	−46.2949	−1.63067
$807$	0	0
$808$	−34.1994	−1.20313
$809$	−37.5299	−1.31948	−0.659741	−	0.751493i	$-0.729334\pi$
$809$	−37.5299	−1.31948	−0.659741	+	0.751493i	$0.729334\pi$
$810$	0	0
$811$	−44.9728	−1.57921	−0.789604	−	0.613617i	$-0.789714\pi$
$811$	−44.9728	−1.57921	−0.789604	+	0.613617i	$0.789714\pi$
$812$	6.42299	0.225403
$813$	0	0
$814$	112.554	3.94501
$815$	89.9796	3.15185
$816$	0	0
$817$	17.2949	0.605073
$818$	74.1691	2.59326
$819$	0	0
$820$	215.593	7.52882
$821$	−7.99805	−0.279134	−0.139567	−	0.990213i	$-0.544571\pi$
$821$	−7.99805	−0.279134	−0.139567	+	0.990213i	$0.544571\pi$
$822$	0	0
$823$	−8.07635	−0.281524	−0.140762	−	0.990043i	$-0.544955\pi$
$823$	−8.07635	−0.281524	−0.140762	+	0.990043i	$0.544955\pi$
$824$	−16.3193	−0.568510
$825$	0	0
$826$	6.38498	0.222162
$827$	−6.72715	−0.233926	−0.116963	−	0.993136i	$-0.537316\pi$
$827$	−6.72715	−0.233926	−0.116963	+	0.993136i	$0.537316\pi$
$828$	0	0
$829$	−19.6878	−0.683787	−0.341894	−	0.939739i	$-0.611068\pi$
$829$	−19.6878	−0.683787	−0.341894	+	0.939739i	$0.611068\pi$
$830$	−128.953	−4.47601
$831$	0	0
$832$	45.6100	1.58124
$833$	26.7357	0.926338
$834$	0	0
$835$	60.6282	2.09813
$836$	82.3073	2.84666
$837$	0	0
$838$	−20.8580	−0.720527
$839$	−34.4713	−1.19008	−0.595041	−	0.803695i	$-0.702864\pi$
$839$	−34.4713	−1.19008	−0.595041	+	0.803695i	$0.702864\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−45.0526	−1.55262
$843$	0	0
$844$	−1.64778	−0.0567189
$845$	27.7713	0.955362
$846$	0	0
$847$	−5.75606	−0.197780
$848$	113.391	3.89388
$849$	0	0
$850$	149.461	5.12648
$851$	−10.9136	−0.374113
$852$	0	0
$853$	27.6682	0.947341	0.473670	−	0.880702i	$-0.342929\pi$
$853$	27.6682	0.947341	0.473670	+	0.880702i	$0.342929\pi$
$854$	−21.0328	−0.719729
$855$	0	0
$856$	2.78514	0.0951942
$857$	54.9024	1.87543	0.937715	−	0.347406i	$-0.112937\pi$
$857$	54.9024	1.87543	0.937715	+	0.347406i	$0.112937\pi$
$858$	0	0
$859$	46.1147	1.57341	0.786706	−	0.617328i	$-0.211785\pi$
$859$	46.1147	1.57341	0.786706	+	0.617328i	$0.211785\pi$
$860$	−80.0403	−2.72935
$861$	0	0
$862$	−54.8950	−1.86973
$863$	23.9805	0.816305	0.408152	−	0.912914i	$-0.366173\pi$
$863$	23.9805	0.816305	0.408152	+	0.912914i	$0.366173\pi$
$864$	0	0
$865$	71.3754	2.42683
$866$	−45.5664	−1.54841
$867$	0	0
$868$	−25.3682	−0.861053
$869$	−21.2937	−0.722339
$870$	0	0
$871$	−47.1177	−1.59652
$872$	−44.8032	−1.51723
$873$	0	0
$874$	−11.2332	−0.379970
$875$	−32.9479	−1.11384
$876$	0	0
$877$	−43.0874	−1.45496	−0.727478	−	0.686131i	$-0.759308\pi$
$877$	−43.0874	−1.45496	−0.727478	+	0.686131i	$0.759308\pi$
$878$	−9.98786	−0.337074
$879$	0	0
$880$	−162.414	−5.47496
$881$	−11.7949	−0.397382	−0.198691	−	0.980062i	$-0.563669\pi$
$881$	−11.7949	−0.397382	−0.198691	+	0.980062i	$0.563669\pi$
$882$	0	0
$883$	−10.4990	−0.353320	−0.176660	−	0.984272i	$-0.556529\pi$
$883$	−10.4990	−0.353320	−0.176660	+	0.984272i	$0.556529\pi$
$884$	−110.678	−3.72250
$885$	0	0
$886$	−74.4292	−2.50050
$887$	7.04493	0.236546	0.118273	−	0.992981i	$-0.462264\pi$
$887$	7.04493	0.236546	0.118273	+	0.992981i	$0.462264\pi$
$888$	0	0
$889$	−15.8464	−0.531470
$890$	−42.1691	−1.41351
$891$	0	0
$892$	56.1572	1.88028
$893$	40.0145	1.33903
$894$	0	0
$895$	−57.9833	−1.93817
$896$	4.53821	0.151611
$897$	0	0
$898$	84.6382	2.82441
$899$	−3.94960	−0.131726
$900$	0	0
$901$	−55.8391	−1.86027
$902$	112.407	3.74275
$903$	0	0
$904$	30.9235	1.02850
$905$	72.3524	2.40507
$906$	0	0
$907$	36.2555	1.20384	0.601922	−	0.798555i	$-0.294402\pi$
$907$	36.2555	1.20384	0.601922	+	0.798555i	$0.294402\pi$
$908$	−64.0507	−2.12560
$909$	0	0
$910$	61.8335	2.04976
$911$	−20.2954	−0.672417	−0.336209	−	0.941788i	$-0.609145\pi$
$911$	−20.2954	−0.672417	−0.336209	+	0.941788i	$0.609145\pi$
$912$	0	0
$913$	−47.7673	−1.58087
$914$	53.9437	1.78430
$915$	0	0
$916$	84.6162	2.79580
$917$	−3.29916	−0.108948
$918$	0	0
$919$	−49.1613	−1.62168	−0.810841	−	0.585266i	$-0.800990\pi$
$919$	−49.1613	−1.62168	−0.810841	+	0.585266i	$0.800990\pi$
$920$	30.8004	1.01546
$921$	0	0
$922$	−34.0423	−1.12112
$923$	1.04403	0.0343648
$924$	0	0
$925$	122.731	4.03538
$926$	8.71231	0.286304
$927$	0	0
$928$	11.7055	0.384250
$929$	−39.8349	−1.30694	−0.653471	−	0.756952i	$-0.726688\pi$
$929$	−39.8349	−1.30694	−0.653471	+	0.756952i	$0.726688\pi$
$930$	0	0
$931$	22.5973	0.740595
$932$	−116.717	−3.82320
$933$	0	0
$934$	−32.4473	−1.06171
$935$	79.9799	2.61562
$936$	0	0
$937$	−16.1779	−0.528510	−0.264255	−	0.964453i	$-0.585126\pi$
$937$	−16.1779	−0.528510	−0.264255	+	0.964453i	$0.585126\pi$
$938$	−36.3414	−1.18659
$939$	0	0
$940$	−185.185	−6.04008
$941$	59.5192	1.94027	0.970136	−	0.242562i	$-0.0779878\pi$
$941$	59.5192	1.94027	0.970136	+	0.242562i	$0.0779878\pi$
$942$	0	0
$943$	−10.8994	−0.354932
$944$	19.0610	0.620381
$945$	0	0
$946$	−41.7320	−1.35682
$947$	−31.7387	−1.03137	−0.515684	−	0.856779i	$-0.672462\pi$
$947$	−31.7387	−1.03137	−0.515684	+	0.856779i	$0.672462\pi$
$948$	0	0
$949$	−41.9565	−1.36196
$950$	126.326	4.09856
$951$	0	0
$952$	−50.5754	−1.63916
$953$	32.5881	1.05563	0.527817	−	0.849358i	$-0.323011\pi$
$953$	32.5881	1.05563	0.527817	+	0.849358i	$0.323011\pi$
$954$	0	0
$955$	11.9370	0.386273
$956$	84.3001	2.72646
$957$	0	0
$958$	−90.0877	−2.91060
$959$	−11.6221	−0.375296
$960$	0	0
$961$	−15.4007	−0.496797
$962$	−127.923	−4.12439
$963$	0	0
$964$	−131.725	−4.24257
$965$	25.2384	0.812454
$966$	0	0
$967$	6.36939	0.204826	0.102413	−	0.994742i	$-0.467344\pi$
$967$	6.36939	0.204826	0.102413	+	0.994742i	$0.467344\pi$
$968$	−33.6076	−1.08019
$969$	0	0
$970$	−187.549	−6.02182
$971$	−44.5308	−1.42906	−0.714530	−	0.699605i	$-0.753360\pi$
$971$	−44.5308	−1.42906	−0.714530	+	0.699605i	$0.753360\pi$
$972$	0	0
$973$	17.0424	0.546354
$974$	−61.1402	−1.95906
$975$	0	0
$976$	−62.7890	−2.00983
$977$	52.8522	1.69089	0.845445	−	0.534062i	$-0.179335\pi$
$977$	52.8522	1.69089	0.845445	+	0.534062i	$0.179335\pi$
$978$	0	0
$979$	−15.6205	−0.499233
$980$	−104.579	−3.34066
$981$	0	0
$982$	24.3223	0.776156
$983$	−28.6636	−0.914228	−0.457114	−	0.889408i	$-0.651117\pi$
$983$	−28.6636	−0.914228	−0.457114	+	0.889408i	$0.651117\pi$
$984$	0	0
$985$	93.4107	2.97631
$986$	−13.2905	−0.423255
$987$	0	0
$988$	−93.5460	−2.97610
$989$	4.04647	0.128670
$990$	0	0
$991$	−0.248409	−0.00789096	−0.00394548	−	0.999992i	$-0.501256\pi$
$991$	−0.248409	−0.00789096	−0.00394548	+	0.999992i	$0.501256\pi$
$992$	−46.2318	−1.46786
$993$	0	0
$994$	0.805251	0.0255410
$995$	6.67182	0.211511
$996$	0	0
$997$	−11.9581	−0.378716	−0.189358	−	0.981908i	$-0.560641\pi$
$997$	−11.9581	−0.378716	−0.189358	+	0.981908i	$0.560641\pi$
$998$	26.2539	0.831054
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.s.1.2		20
3.2	odd	2		2001.2.a.o.1.19	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.o.1.19	✓	20	3.2	odd	2
6003.2.a.s.1.2		20	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6003.a (trivial)

\(f(q)\)	\(=\)	\(q-2.62822 q^{2} +4.90753 q^{4} +4.03060 q^{5} -1.30880 q^{7} -7.64164 q^{8} +O(q^{10})\)	\(q-2.62822 q^{2} +4.90753 q^{4} +4.03060 q^{5} -1.30880 q^{7} -7.64164 q^{8} -10.5933 q^{10} -3.92402 q^{11} +4.45983 q^{13} +3.43982 q^{14} +10.2688 q^{16} -5.05684 q^{17} -4.27409 q^{19} +19.7803 q^{20} +10.3132 q^{22} -1.00000 q^{23} +11.2457 q^{25} -11.7214 q^{26} -6.42299 q^{28} -1.00000 q^{29} +3.94960 q^{31} -11.7055 q^{32} +13.2905 q^{34} -5.27526 q^{35} +10.9136 q^{37} +11.2332 q^{38} -30.8004 q^{40} +10.8994 q^{41} -4.04647 q^{43} -19.2573 q^{44} +2.62822 q^{46} -9.36211 q^{47} -5.28704 q^{49} -29.5563 q^{50} +21.8868 q^{52} +11.0423 q^{53} -15.8162 q^{55} +10.0014 q^{56} +2.62822 q^{58} +1.85620 q^{59} -6.11452 q^{61} -10.3804 q^{62} +10.2268 q^{64} +17.9758 q^{65} -10.5649 q^{67} -24.8166 q^{68} +13.8645 q^{70} +0.234097 q^{71} -9.40763 q^{73} -28.6833 q^{74} -20.9752 q^{76} +5.13577 q^{77} +5.42650 q^{79} +41.3895 q^{80} -28.6459 q^{82} +12.1730 q^{83} -20.3821 q^{85} +10.6350 q^{86} +29.9860 q^{88} +3.98074 q^{89} -5.83704 q^{91} -4.90753 q^{92} +24.6057 q^{94} -17.2271 q^{95} +17.7045 q^{97} +13.8955 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8}+O(q^{10}) \) 20 * q - 2 * q^2 + 30 * q^4 + q^5 + 9 * q^7 - 6 * q^8	\( 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8} + 7 q^{10} + 21 q^{13} + q^{14} + 58 q^{16} + 4 q^{17} + 7 q^{19} + 20 q^{20} + 7 q^{22} - 20 q^{23} + 47 q^{25} - 8 q^{26} + 11 q^{28} - 20 q^{29} + 28 q^{31} - 14 q^{32} + 16 q^{34} - 9 q^{35} + 14 q^{37} + 20 q^{38} + 34 q^{40} - 7 q^{41} + 3 q^{43} + q^{44} + 2 q^{46} - 3 q^{47} + 35 q^{49} + 24 q^{50} + 73 q^{52} + 19 q^{53} + 29 q^{55} + 30 q^{56} + 2 q^{58} - 20 q^{59} + 15 q^{61} - 12 q^{62} + 82 q^{64} + 28 q^{65} + 20 q^{67} + 23 q^{68} - 24 q^{70} - 63 q^{71} + 19 q^{73} - 16 q^{74} - 44 q^{76} + 7 q^{77} + 32 q^{79} + 56 q^{80} - 20 q^{82} + 21 q^{83} + 4 q^{85} + 6 q^{86} + 55 q^{88} + 13 q^{89} + 70 q^{91} - 30 q^{92} - 12 q^{94} - 9 q^{95} - 9 q^{97} - 31 q^{98}+O(q^{100}) \) 20 * q - 2 * q^2 + 30 * q^4 + q^5 + 9 * q^7 - 6 * q^8 + 7 * q^10 + 21 * q^13 + q^14 + 58 * q^16 + 4 * q^17 + 7 * q^19 + 20 * q^20 + 7 * q^22 - 20 * q^23 + 47 * q^25 - 8 * q^26 + 11 * q^28 - 20 * q^29 + 28 * q^31 - 14 * q^32 + 16 * q^34 - 9 * q^35 + 14 * q^37 + 20 * q^38 + 34 * q^40 - 7 * q^41 + 3 * q^43 + q^44 + 2 * q^46 - 3 * q^47 + 35 * q^49 + 24 * q^50 + 73 * q^52 + 19 * q^53 + 29 * q^55 + 30 * q^56 + 2 * q^58 - 20 * q^59 + 15 * q^61 - 12 * q^62 + 82 * q^64 + 28 * q^65 + 20 * q^67 + 23 * q^68 - 24 * q^70 - 63 * q^71 + 19 * q^73 - 16 * q^74 - 44 * q^76 + 7 * q^77 + 32 * q^79 + 56 * q^80 - 20 * q^82 + 21 * q^83 + 4 * q^85 + 6 * q^86 + 55 * q^88 + 13 * q^89 + 70 * q^91 - 30 * q^92 - 12 * q^94 - 9 * q^95 - 9 * q^97 - 31 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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