LMFDB - Embedded newform 6003.2.a.r.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$47.9341963334$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{15} - 28 x^{14} + 27 x^{13} + 316 x^{12} - 295 x^{11} - 1835 x^{10} + 1665 x^{9} + \cdots - 192 $ x^16 - x^15 - 28x^14 + 27x^13 + 316x^12 - 295x^11 - 1835x^10 + 1665x^9 + 5778x^8 - 5124x^7 - 9405x^6 + 8288x^5 + 6405x^4 - 6032x^3 - 400x^2 + 1088x - 192
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$1.78406$ 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)

$f(q)$	$=$	$q+1.78406 q^{2} +1.18288 q^{4} -3.42296 q^{5} +3.39829 q^{7} -1.45779 q^{8} +O(q^{10})$	\(q+1.78406 q^{2} +1.18288 q^{4} -3.42296 q^{5} +3.39829 q^{7} -1.45779 q^{8} -6.10677 q^{10} -1.48855 q^{11} +1.54078 q^{13} +6.06276 q^{14} -4.96655 q^{16} -1.71252 q^{17} +5.94186 q^{19} -4.04896 q^{20} -2.65566 q^{22} +1.00000 q^{23} +6.71663 q^{25} +2.74885 q^{26} +4.01978 q^{28} +1.00000 q^{29} -2.03180 q^{31} -5.94507 q^{32} -3.05524 q^{34} -11.6322 q^{35} -2.86662 q^{37} +10.6006 q^{38} +4.98994 q^{40} -2.74906 q^{41} +2.40752 q^{43} -1.76078 q^{44} +1.78406 q^{46} -0.626003 q^{47} +4.54837 q^{49} +11.9829 q^{50} +1.82256 q^{52} +1.96409 q^{53} +5.09523 q^{55} -4.95398 q^{56} +1.78406 q^{58} +10.3160 q^{59} -6.34964 q^{61} -3.62487 q^{62} -0.673284 q^{64} -5.27402 q^{65} -3.34937 q^{67} -2.02571 q^{68} -20.7526 q^{70} +15.8477 q^{71} -8.02067 q^{73} -5.11424 q^{74} +7.02852 q^{76} -5.05851 q^{77} +2.84758 q^{79} +17.0003 q^{80} -4.90450 q^{82} +16.8010 q^{83} +5.86188 q^{85} +4.29516 q^{86} +2.16999 q^{88} +14.5368 q^{89} +5.23601 q^{91} +1.18288 q^{92} -1.11683 q^{94} -20.3387 q^{95} +6.93709 q^{97} +8.11457 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + q^{2} + 25 q^{4} - 3 q^{5} + 13 q^{7}+O(q^{10}) $ 16 * q + q^2 + 25 * q^4 - 3 * q^5 + 13 * q^7	$ 16 q + q^{2} + 25 q^{4} - 3 q^{5} + 13 q^{7} + 11 q^{10} - 8 q^{11} + 19 q^{13} - 16 q^{14} + 31 q^{16} + 4 q^{17} + 19 q^{19} - 16 q^{20} + 6 q^{22} + 16 q^{23} + 23 q^{25} + 15 q^{26} + 18 q^{28} + 16 q^{29} + 24 q^{31} + 21 q^{32} - 9 q^{34} + 13 q^{35} + 26 q^{37} - 22 q^{40} + 15 q^{41} + 33 q^{43} - 6 q^{44} + q^{46} - 13 q^{47} + 41 q^{49} - 13 q^{50} - 26 q^{52} - 5 q^{53} + 9 q^{55} - 40 q^{56} + q^{58} - 2 q^{59} + 29 q^{61} + 32 q^{62} + 28 q^{64} - 18 q^{65} + 32 q^{67} + 26 q^{68} + 18 q^{70} - 29 q^{71} + 19 q^{73} + 16 q^{74} + 64 q^{76} + 21 q^{77} + 56 q^{79} + 14 q^{82} - 5 q^{83} + 16 q^{85} + 20 q^{86} + q^{88} - 7 q^{89} - 6 q^{91} + 25 q^{92} - 11 q^{94} - 39 q^{95} + 35 q^{97} + 109 q^{98}+O(q^{100}) $ 16 * q + q^2 + 25 * q^4 - 3 * q^5 + 13 * q^7 + 11 * q^10 - 8 * q^11 + 19 * q^13 - 16 * q^14 + 31 * q^16 + 4 * q^17 + 19 * q^19 - 16 * q^20 + 6 * q^22 + 16 * q^23 + 23 * q^25 + 15 * q^26 + 18 * q^28 + 16 * q^29 + 24 * q^31 + 21 * q^32 - 9 * q^34 + 13 * q^35 + 26 * q^37 - 22 * q^40 + 15 * q^41 + 33 * q^43 - 6 * q^44 + q^46 - 13 * q^47 + 41 * q^49 - 13 * q^50 - 26 * q^52 - 5 * q^53 + 9 * q^55 - 40 * q^56 + q^58 - 2 * q^59 + 29 * q^61 + 32 * q^62 + 28 * q^64 - 18 * q^65 + 32 * q^67 + 26 * q^68 + 18 * q^70 - 29 * q^71 + 19 * q^73 + 16 * q^74 + 64 * q^76 + 21 * q^77 + 56 * q^79 + 14 * q^82 - 5 * q^83 + 16 * q^85 + 20 * q^86 + q^88 - 7 * q^89 - 6 * q^91 + 25 * q^92 - 11 * q^94 - 39 * q^95 + 35 * q^97 + 109 * 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 $

$2$	1.78406	1.26152	0.630762	−	0.775976i	$-0.282742\pi$
$2$	1.78406	1.26152	0.630762	+	0.775976i	$0.282742\pi$
$3$	0	0
$3$	0	0
$4$	1.18288	0.591442
$5$	−3.42296	−1.53079	−0.765396	−	0.643559i	$-0.777457\pi$
$5$	−3.42296	−1.53079	−0.765396	+	0.643559i	$0.777457\pi$
$6$	0	0
$7$	3.39829	1.28443	0.642216	−	0.766524i	$-0.278015\pi$
$7$	3.39829	1.28443	0.642216	+	0.766524i	$0.278015\pi$
$8$	−1.45779	−0.515406
$9$	0	0
$10$	−6.10677	−1.93113
$11$	−1.48855	−0.448814	−0.224407	−	0.974496i	$-0.572044\pi$
$11$	−1.48855	−0.448814	−0.224407	+	0.974496i	$0.572044\pi$
$12$	0	0
$13$	1.54078	0.427335	0.213668	−	0.976906i	$-0.431459\pi$
$13$	1.54078	0.427335	0.213668	+	0.976906i	$0.431459\pi$
$14$	6.06276	1.62034
$15$	0	0
$16$	−4.96655	−1.24164
$17$	−1.71252	−0.415347	−0.207673	−	0.978198i	$-0.566589\pi$
$17$	−1.71252	−0.415347	−0.207673	+	0.978198i	$0.566589\pi$
$18$	0	0
$19$	5.94186	1.36316	0.681578	−	0.731746i	$-0.261294\pi$
$19$	5.94186	1.36316	0.681578	+	0.731746i	$0.261294\pi$
$20$	−4.04896	−0.905375
$21$	0	0
$22$	−2.65566	−0.566189
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	6.71663	1.34333
$26$	2.74885	0.539094
$27$	0	0
$28$	4.01978	0.759667
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−2.03180	−0.364923	−0.182461	−	0.983213i	$-0.558406\pi$
$31$	−2.03180	−0.364923	−0.182461	+	0.983213i	$0.558406\pi$
$32$	−5.94507	−1.05095
$33$	0	0
$34$	−3.05524	−0.523970
$35$	−11.6322	−1.96620
$36$	0	0
$37$	−2.86662	−0.471270	−0.235635	−	0.971842i	$-0.575717\pi$
$37$	−2.86662	−0.471270	−0.235635	+	0.971842i	$0.575717\pi$
$38$	10.6006	1.71965
$39$	0	0
$40$	4.98994	0.788979
$41$	−2.74906	−0.429331	−0.214665	−	0.976688i	$-0.568866\pi$
$41$	−2.74906	−0.429331	−0.214665	+	0.976688i	$0.568866\pi$
$42$	0	0
$43$	2.40752	0.367143	0.183571	−	0.983006i	$-0.441234\pi$
$43$	2.40752	0.367143	0.183571	+	0.983006i	$0.441234\pi$
$44$	−1.76078	−0.265447
$45$	0	0
$46$	1.78406	0.263046
$47$	−0.626003	−0.0913120	−0.0456560	−	0.998957i	$-0.514538\pi$
$47$	−0.626003	−0.0913120	−0.0456560	+	0.998957i	$0.514538\pi$
$48$	0	0
$49$	4.54837	0.649766
$50$	11.9829	1.69464
$51$	0	0
$52$	1.82256	0.252744
$53$	1.96409	0.269789	0.134894	−	0.990860i	$-0.456930\pi$
$53$	1.96409	0.269789	0.134894	+	0.990860i	$0.456930\pi$
$54$	0	0
$55$	5.09523	0.687041
$56$	−4.95398	−0.662004
$57$	0	0
$58$	1.78406	0.234259
$59$	10.3160	1.34303	0.671514	−	0.740992i	$-0.265645\pi$
$59$	10.3160	1.34303	0.671514	+	0.740992i	$0.265645\pi$
$60$	0	0
$61$	−6.34964	−0.812988	−0.406494	−	0.913653i	$-0.633249\pi$
$61$	−6.34964	−0.812988	−0.406494	+	0.913653i	$0.633249\pi$
$62$	−3.62487	−0.460359
$63$	0	0
$64$	−0.673284	−0.0841605
$65$	−5.27402	−0.654162
$66$	0	0
$67$	−3.34937	−0.409190	−0.204595	−	0.978847i	$-0.565588\pi$
$67$	−3.34937	−0.409190	−0.204595	+	0.978847i	$0.565588\pi$
$68$	−2.02571	−0.245654
$69$	0	0
$70$	−20.7526	−2.48041
$71$	15.8477	1.88078	0.940388	−	0.340104i	$-0.110462\pi$
$71$	15.8477	1.88078	0.940388	+	0.340104i	$0.110462\pi$
$72$	0	0
$73$	−8.02067	−0.938748	−0.469374	−	0.882999i	$-0.655520\pi$
$73$	−8.02067	−0.938748	−0.469374	+	0.882999i	$0.655520\pi$
$74$	−5.11424	−0.594518
$75$	0	0
$76$	7.02852	0.806227
$77$	−5.05851	−0.576471
$78$	0	0
$79$	2.84758	0.320378	0.160189	−	0.987086i	$-0.448790\pi$
$79$	2.84758	0.320378	0.160189	+	0.987086i	$0.448790\pi$
$80$	17.0003	1.90069
$81$	0	0
$82$	−4.90450	−0.541611
$83$	16.8010	1.84415	0.922077	−	0.387006i	$-0.126491\pi$
$83$	16.8010	1.84415	0.922077	+	0.387006i	$0.126491\pi$
$84$	0	0
$85$	5.86188	0.635810
$86$	4.29516	0.463159
$87$	0	0
$88$	2.16999	0.231321
$89$	14.5368	1.54090	0.770449	−	0.637502i	$-0.220032\pi$
$89$	14.5368	1.54090	0.770449	+	0.637502i	$0.220032\pi$
$90$	0	0
$91$	5.23601	0.548883
$92$	1.18288	0.123324
$93$	0	0
$94$	−1.11683	−0.115192
$95$	−20.3387	−2.08671
$96$	0	0
$97$	6.93709	0.704355	0.352177	−	0.935933i	$-0.385441\pi$
$97$	6.93709	0.704355	0.352177	+	0.935933i	$0.385441\pi$
$98$	8.11457	0.819696
$99$	0	0
$100$	7.94500	0.794500
$101$	17.0941	1.70093	0.850465	−	0.526031i	$-0.176320\pi$
$101$	17.0941	1.70093	0.850465	+	0.526031i	$0.176320\pi$
$102$	0	0
$103$	0.593783	0.0585071	0.0292536	−	0.999572i	$-0.490687\pi$
$103$	0.593783	0.0585071	0.0292536	+	0.999572i	$0.490687\pi$
$104$	−2.24613	−0.220251
$105$	0	0
$106$	3.50407	0.340345
$107$	6.00249	0.580283	0.290141	−	0.956984i	$-0.406298\pi$
$107$	6.00249	0.580283	0.290141	+	0.956984i	$0.406298\pi$
$108$	0	0
$109$	16.3993	1.57077	0.785385	−	0.619008i	$-0.212465\pi$
$109$	16.3993	1.57077	0.785385	+	0.619008i	$0.212465\pi$
$110$	9.09022	0.866718
$111$	0	0
$112$	−16.8778	−1.59480
$113$	−6.64814	−0.625404	−0.312702	−	0.949851i	$-0.601234\pi$
$113$	−6.64814	−0.625404	−0.312702	+	0.949851i	$0.601234\pi$
$114$	0	0
$115$	−3.42296	−0.319192
$116$	1.18288	0.109828
$117$	0	0
$118$	18.4044	1.69426
$119$	−5.81963	−0.533485
$120$	0	0
$121$	−8.78423	−0.798566
$122$	−11.3282	−1.02560
$123$	0	0
$124$	−2.40339	−0.215831
$125$	−5.87596	−0.525562
$126$	0	0
$127$	2.85194	0.253069	0.126535	−	0.991962i	$-0.459615\pi$
$127$	2.85194	0.253069	0.126535	+	0.991962i	$0.459615\pi$
$128$	10.6890	0.944780
$129$	0	0
$130$	−9.40919	−0.825241
$131$	1.25044	0.109251	0.0546256	−	0.998507i	$-0.482603\pi$
$131$	1.25044	0.109251	0.0546256	+	0.998507i	$0.482603\pi$
$132$	0	0
$133$	20.1921	1.75088
$134$	−5.97549	−0.516203
$135$	0	0
$136$	2.49649	0.214072
$137$	17.0060	1.45292	0.726459	−	0.687210i	$-0.241165\pi$
$137$	17.0060	1.45292	0.726459	+	0.687210i	$0.241165\pi$
$138$	0	0
$139$	−14.2040	−1.20477	−0.602386	−	0.798205i	$-0.705783\pi$
$139$	−14.2040	−1.20477	−0.602386	+	0.798205i	$0.705783\pi$
$140$	−13.7595	−1.16289
$141$	0	0
$142$	28.2733	2.37264
$143$	−2.29352	−0.191794
$144$	0	0
$145$	−3.42296	−0.284261
$146$	−14.3094	−1.18425
$147$	0	0
$148$	−3.39088	−0.278729
$149$	−4.72788	−0.387323	−0.193661	−	0.981068i	$-0.562036\pi$
$149$	−4.72788	−0.387323	−0.193661	+	0.981068i	$0.562036\pi$
$150$	0	0
$151$	13.9866	1.13822	0.569108	−	0.822263i	$-0.307289\pi$
$151$	13.9866	1.13822	0.569108	+	0.822263i	$0.307289\pi$
$152$	−8.66196	−0.702578
$153$	0	0
$154$	−9.02471	−0.727232
$155$	6.95478	0.558621
$156$	0	0
$157$	−3.91044	−0.312087	−0.156044	−	0.987750i	$-0.549874\pi$
$157$	−3.91044	−0.312087	−0.156044	+	0.987750i	$0.549874\pi$
$158$	5.08026	0.404164
$159$	0	0
$160$	20.3497	1.60879
$161$	3.39829	0.267823
$162$	0	0
$163$	24.1757	1.89359	0.946794	−	0.321840i	$-0.104301\pi$
$163$	24.1757	1.89359	0.946794	+	0.321840i	$0.104301\pi$
$164$	−3.25182	−0.253924
$165$	0	0
$166$	29.9741	2.32644
$167$	8.39892	0.649928	0.324964	−	0.945726i	$-0.394648\pi$
$167$	8.39892	0.649928	0.324964	+	0.945726i	$0.394648\pi$
$168$	0	0
$169$	−10.6260	−0.817385
$170$	10.4580	0.802089
$171$	0	0
$172$	2.84781	0.217144
$173$	−5.63120	−0.428132	−0.214066	−	0.976819i	$-0.568671\pi$
$173$	−5.63120	−0.428132	−0.214066	+	0.976819i	$0.568671\pi$
$174$	0	0
$175$	22.8251	1.72541
$176$	7.39295	0.557264
$177$	0	0
$178$	25.9346	1.94388
$179$	14.2510	1.06517	0.532584	−	0.846377i	$-0.321221\pi$
$179$	14.2510	1.06517	0.532584	+	0.846377i	$0.321221\pi$
$180$	0	0
$181$	−3.49672	−0.259909	−0.129955	−	0.991520i	$-0.541483\pi$
$181$	−3.49672	−0.259909	−0.129955	+	0.991520i	$0.541483\pi$
$182$	9.34138	0.692429
$183$	0	0
$184$	−1.45779	−0.107470
$185$	9.81232	0.721416
$186$	0	0
$187$	2.54917	0.186413
$188$	−0.740489	−0.0540057
$189$	0	0
$190$	−36.2856	−2.63243
$191$	−8.41875	−0.609159	−0.304580	−	0.952487i	$-0.598516\pi$
$191$	−8.41875	−0.609159	−0.304580	+	0.952487i	$0.598516\pi$
$192$	0	0
$193$	22.9526	1.65217	0.826083	−	0.563548i	$-0.190564\pi$
$193$	22.9526	1.65217	0.826083	+	0.563548i	$0.190564\pi$
$194$	12.3762	0.888560
$195$	0	0
$196$	5.38019	0.384299
$197$	−6.02811	−0.429485	−0.214743	−	0.976671i	$-0.568891\pi$
$197$	−6.02811	−0.429485	−0.214743	+	0.976671i	$0.568891\pi$
$198$	0	0
$199$	−2.91164	−0.206401	−0.103200	−	0.994661i	$-0.532908\pi$
$199$	−2.91164	−0.206401	−0.103200	+	0.994661i	$0.532908\pi$
$200$	−9.79143	−0.692358
$201$	0	0
$202$	30.4970	2.14576
$203$	3.39829	0.238513
$204$	0	0
$205$	9.40991	0.657217
$206$	1.05935	0.0738081
$207$	0	0
$208$	−7.65236	−0.530596
$209$	−8.84473	−0.611803
$210$	0	0
$211$	−8.02009	−0.552126	−0.276063	−	0.961140i	$-0.589030\pi$
$211$	−8.02009	−0.552126	−0.276063	+	0.961140i	$0.589030\pi$
$212$	2.32329	0.159564
$213$	0	0
$214$	10.7088	0.732040
$215$	−8.24083	−0.562020
$216$	0	0
$217$	−6.90466	−0.468719
$218$	29.2574	1.98156
$219$	0	0
$220$	6.02707	0.406345
$221$	−2.63861	−0.177492
$222$	0	0
$223$	18.6751	1.25058	0.625288	−	0.780394i	$-0.284981\pi$
$223$	18.6751	1.25058	0.625288	+	0.780394i	$0.284981\pi$
$224$	−20.2031	−1.34987
$225$	0	0
$226$	−11.8607	−0.788962
$227$	−7.54846	−0.501009	−0.250504	−	0.968115i	$-0.580596\pi$
$227$	−7.54846	−0.501009	−0.250504	+	0.968115i	$0.580596\pi$
$228$	0	0
$229$	−6.17163	−0.407833	−0.203916	−	0.978988i	$-0.565367\pi$
$229$	−6.17163	−0.407833	−0.203916	+	0.978988i	$0.565367\pi$
$230$	−6.10677	−0.402669
$231$	0	0
$232$	−1.45779	−0.0957084
$233$	−1.49889	−0.0981957	−0.0490979	−	0.998794i	$-0.515635\pi$
$233$	−1.49889	−0.0981957	−0.0490979	+	0.998794i	$0.515635\pi$
$234$	0	0
$235$	2.14278	0.139780
$236$	12.2026	0.794322
$237$	0	0
$238$	−10.3826	−0.673004
$239$	−2.45089	−0.158535	−0.0792675	−	0.996853i	$-0.525258\pi$
$239$	−2.45089	−0.158535	−0.0792675	+	0.996853i	$0.525258\pi$
$240$	0	0
$241$	−3.23042	−0.208089	−0.104045	−	0.994573i	$-0.533179\pi$
$241$	−3.23042	−0.208089	−0.104045	+	0.994573i	$0.533179\pi$
$242$	−15.6716	−1.00741
$243$	0	0
$244$	−7.51088	−0.480835
$245$	−15.5689	−0.994658
$246$	0	0
$247$	9.15509	0.582524
$248$	2.96194	0.188083
$249$	0	0
$250$	−10.4831	−0.663009
$251$	−27.8680	−1.75902	−0.879508	−	0.475885i	$-0.842128\pi$
$251$	−27.8680	−1.75902	−0.879508	+	0.475885i	$0.842128\pi$
$252$	0	0
$253$	−1.48855	−0.0935842
$254$	5.08805	0.319253
$255$	0	0
$256$	20.4164	1.27602
$257$	−11.8216	−0.737409	−0.368704	−	0.929547i	$-0.620199\pi$
$257$	−11.8216	−0.737409	−0.368704	+	0.929547i	$0.620199\pi$
$258$	0	0
$259$	−9.74161	−0.605314
$260$	−6.23855	−0.386899
$261$	0	0
$262$	2.23086	0.137823
$263$	11.6260	0.716890	0.358445	−	0.933551i	$-0.383307\pi$
$263$	11.6260	0.716890	0.358445	+	0.933551i	$0.383307\pi$
$264$	0	0
$265$	−6.72300	−0.412991
$266$	36.0241	2.20878
$267$	0	0
$268$	−3.96191	−0.242012
$269$	−23.7915	−1.45060	−0.725298	−	0.688435i	$-0.758298\pi$
$269$	−23.7915	−1.45060	−0.725298	+	0.688435i	$0.758298\pi$
$270$	0	0
$271$	19.3046	1.17267	0.586335	−	0.810069i	$-0.300570\pi$
$271$	19.3046	1.17267	0.586335	+	0.810069i	$0.300570\pi$
$272$	8.50532	0.515711
$273$	0	0
$274$	30.3397	1.83289
$275$	−9.99803	−0.602904
$276$	0	0
$277$	−5.11059	−0.307066	−0.153533	−	0.988144i	$-0.549065\pi$
$277$	−5.11059	−0.307066	−0.153533	+	0.988144i	$0.549065\pi$
$278$	−25.3409	−1.51985
$279$	0	0
$280$	16.9573	1.01339
$281$	−28.6341	−1.70817	−0.854084	−	0.520135i	$-0.825882\pi$
$281$	−28.6341	−1.70817	−0.854084	+	0.520135i	$0.825882\pi$
$282$	0	0
$283$	−16.4299	−0.976659	−0.488330	−	0.872659i	$-0.662394\pi$
$283$	−16.4299	−0.976659	−0.488330	+	0.872659i	$0.662394\pi$
$284$	18.7460	1.11237
$285$	0	0
$286$	−4.09179	−0.241953
$287$	−9.34210	−0.551447
$288$	0	0
$289$	−14.0673	−0.827487
$290$	−6.10677	−0.358602
$291$	0	0
$292$	−9.48752	−0.555215
$293$	5.69142	0.332496	0.166248	−	0.986084i	$-0.446835\pi$
$293$	5.69142	0.332496	0.166248	+	0.986084i	$0.446835\pi$
$294$	0	0
$295$	−35.3112	−2.05590
$296$	4.17893	0.242895
$297$	0	0
$298$	−8.43484	−0.488617
$299$	1.54078	0.0891056
$300$	0	0
$301$	8.18144	0.471570
$302$	24.9530	1.43589
$303$	0	0
$304$	−29.5105	−1.69255
$305$	21.7345	1.24452
$306$	0	0
$307$	34.0596	1.94389	0.971943	−	0.235218i	$-0.0755803\pi$
$307$	34.0596	1.94389	0.971943	+	0.235218i	$0.0755803\pi$
$308$	−5.98363	−0.340949
$309$	0	0
$310$	12.4078	0.704714
$311$	3.80385	0.215696	0.107848	−	0.994167i	$-0.465604\pi$
$311$	3.80385	0.215696	0.107848	+	0.994167i	$0.465604\pi$
$312$	0	0
$313$	−6.17586	−0.349080	−0.174540	−	0.984650i	$-0.555844\pi$
$313$	−6.17586	−0.349080	−0.174540	+	0.984650i	$0.555844\pi$
$314$	−6.97648	−0.393706
$315$	0	0
$316$	3.36836	0.189485
$317$	0.190758	0.0107140	0.00535701	−	0.999986i	$-0.498295\pi$
$317$	0.190758	0.0107140	0.00535701	+	0.999986i	$0.498295\pi$
$318$	0	0
$319$	−1.48855	−0.0833426
$320$	2.30462	0.128832
$321$	0	0
$322$	6.06276	0.337865
$323$	−10.1755	−0.566182
$324$	0	0
$325$	10.3489	0.574051
$326$	43.1310	2.38881
$327$	0	0
$328$	4.00754	0.221280
$329$	−2.12734	−0.117284
$330$	0	0
$331$	27.8480	1.53066	0.765331	−	0.643637i	$-0.222575\pi$
$331$	27.8480	1.53066	0.765331	+	0.643637i	$0.222575\pi$
$332$	19.8737	1.09071
$333$	0	0
$334$	14.9842	0.819899
$335$	11.4647	0.626386
$336$	0	0
$337$	−26.8796	−1.46423	−0.732113	−	0.681183i	$-0.761466\pi$
$337$	−26.8796	−1.46423	−0.732113	+	0.681183i	$0.761466\pi$
$338$	−18.9575	−1.03115
$339$	0	0
$340$	6.93392	0.376045
$341$	3.02444	0.163782
$342$	0	0
$343$	−8.33136	−0.449851
$344$	−3.50965	−0.189228
$345$	0	0
$346$	−10.0464	−0.540099
$347$	−19.4468	−1.04396	−0.521979	−	0.852958i	$-0.674806\pi$
$347$	−19.4468	−1.04396	−0.521979	+	0.852958i	$0.674806\pi$
$348$	0	0
$349$	7.49870	0.401396	0.200698	−	0.979653i	$-0.435679\pi$
$349$	7.49870	0.401396	0.200698	+	0.979653i	$0.435679\pi$
$350$	40.7214	2.17665
$351$	0	0
$352$	8.84952	0.471681
$353$	13.1539	0.700110	0.350055	−	0.936729i	$-0.386163\pi$
$353$	13.1539	0.700110	0.350055	+	0.936729i	$0.386163\pi$
$354$	0	0
$355$	−54.2460	−2.87908
$356$	17.1953	0.911351
$357$	0	0
$358$	25.4246	1.34373
$359$	−4.78965	−0.252788	−0.126394	−	0.991980i	$-0.540340\pi$
$359$	−4.78965	−0.252788	−0.126394	+	0.991980i	$0.540340\pi$
$360$	0	0
$361$	16.3056	0.858192
$362$	−6.23837	−0.327881
$363$	0	0
$364$	6.19359	0.324633
$365$	27.4544	1.43703
$366$	0	0
$367$	−5.18576	−0.270695	−0.135347	−	0.990798i	$-0.543215\pi$
$367$	−5.18576	−0.270695	−0.135347	+	0.990798i	$0.543215\pi$
$368$	−4.96655	−0.258899
$369$	0	0
$370$	17.5058	0.910084
$371$	6.67455	0.346526
$372$	0	0
$373$	−25.4860	−1.31962	−0.659808	−	0.751434i	$-0.729362\pi$
$373$	−25.4860	−1.31962	−0.659808	+	0.751434i	$0.729362\pi$
$374$	4.54787	0.235165
$375$	0	0
$376$	0.912580	0.0470627
$377$	1.54078	0.0793542
$378$	0	0
$379$	19.7633	1.01517	0.507586	−	0.861601i	$-0.330538\pi$
$379$	19.7633	1.01517	0.507586	+	0.861601i	$0.330538\pi$
$380$	−24.0583	−1.23417
$381$	0	0
$382$	−15.0196	−0.768469
$383$	0.518090	0.0264732	0.0132366	−	0.999912i	$-0.495787\pi$
$383$	0.518090	0.0264732	0.0132366	+	0.999912i	$0.495787\pi$
$384$	0	0
$385$	17.3151	0.882458
$386$	40.9490	2.08425
$387$	0	0
$388$	8.20577	0.416585
$389$	16.2237	0.822574	0.411287	−	0.911506i	$-0.365079\pi$
$389$	16.2237	0.822574	0.411287	+	0.911506i	$0.365079\pi$
$390$	0	0
$391$	−1.71252	−0.0866058
$392$	−6.63055	−0.334893
$393$	0	0
$394$	−10.7545	−0.541806
$395$	−9.74714	−0.490432
$396$	0	0
$397$	−11.7502	−0.589727	−0.294863	−	0.955539i	$-0.595274\pi$
$397$	−11.7502	−0.589727	−0.294863	+	0.955539i	$0.595274\pi$
$398$	−5.19455	−0.260379
$399$	0	0
$400$	−33.3585	−1.66793
$401$	1.11434	0.0556473	0.0278236	−	0.999613i	$-0.491142\pi$
$401$	1.11434	0.0556473	0.0278236	+	0.999613i	$0.491142\pi$
$402$	0	0
$403$	−3.13056	−0.155944
$404$	20.2204	1.00600
$405$	0	0
$406$	6.06276	0.300890
$407$	4.26710	0.211512
$408$	0	0
$409$	−11.1891	−0.553267	−0.276634	−	0.960975i	$-0.589219\pi$
$409$	−11.1891	−0.553267	−0.276634	+	0.960975i	$0.589219\pi$
$410$	16.7879	0.829095
$411$	0	0
$412$	0.702376	0.0346036
$413$	35.0567	1.72503
$414$	0	0
$415$	−57.5093	−2.82302
$416$	−9.16005	−0.449108
$417$	0	0
$418$	−15.7796	−0.771804
$419$	28.0240	1.36906	0.684531	−	0.728983i	$-0.260007\pi$
$419$	28.0240	1.36906	0.684531	+	0.728983i	$0.260007\pi$
$420$	0	0
$421$	5.58458	0.272176	0.136088	−	0.990697i	$-0.456547\pi$
$421$	5.58458	0.272176	0.136088	+	0.990697i	$0.456547\pi$
$422$	−14.3083	−0.696519
$423$	0	0
$424$	−2.86323	−0.139051
$425$	−11.5024	−0.557947
$426$	0	0
$427$	−21.5779	−1.04423
$428$	7.10025	0.343203
$429$	0	0
$430$	−14.7022	−0.709001
$431$	−10.7167	−0.516205	−0.258102	−	0.966118i	$-0.583097\pi$
$431$	−10.7167	−0.516205	−0.258102	+	0.966118i	$0.583097\pi$
$432$	0	0
$433$	17.9099	0.860697	0.430349	−	0.902663i	$-0.358391\pi$
$433$	17.9099	0.860697	0.430349	+	0.902663i	$0.358391\pi$
$434$	−12.3183	−0.591300
$435$	0	0
$436$	19.3985	0.929019
$437$	5.94186	0.284237
$438$	0	0
$439$	−15.0813	−0.719792	−0.359896	−	0.932992i	$-0.617188\pi$
$439$	−15.0813	−0.719792	−0.359896	+	0.932992i	$0.617188\pi$
$440$	−7.42777	−0.354105
$441$	0	0
$442$	−4.70746	−0.223911
$443$	−21.6438	−1.02833	−0.514164	−	0.857692i	$-0.671898\pi$
$443$	−21.6438	−1.02833	−0.514164	+	0.857692i	$0.671898\pi$
$444$	0	0
$445$	−49.7588	−2.35880
$446$	33.3176	1.57763
$447$	0	0
$448$	−2.28801	−0.108098
$449$	−24.8955	−1.17489	−0.587445	−	0.809264i	$-0.699866\pi$
$449$	−24.8955	−1.17489	−0.587445	+	0.809264i	$0.699866\pi$
$450$	0	0
$451$	4.09210	0.192690
$452$	−7.86398	−0.369890
$453$	0	0
$454$	−13.4669	−0.632035
$455$	−17.9226	−0.840227
$456$	0	0
$457$	13.6695	0.639433	0.319717	−	0.947513i	$-0.396412\pi$
$457$	13.6695	0.639433	0.319717	+	0.947513i	$0.396412\pi$
$458$	−11.0106	−0.514491
$459$	0	0
$460$	−4.04896	−0.188784
$461$	−30.3073	−1.41155	−0.705776	−	0.708435i	$-0.749402\pi$
$461$	−30.3073	−1.41155	−0.705776	+	0.708435i	$0.749402\pi$
$462$	0	0
$463$	21.9221	1.01881	0.509403	−	0.860528i	$-0.329866\pi$
$463$	21.9221	1.01881	0.509403	+	0.860528i	$0.329866\pi$
$464$	−4.96655	−0.230566
$465$	0	0
$466$	−2.67412	−0.123876
$467$	−7.29438	−0.337544	−0.168772	−	0.985655i	$-0.553980\pi$
$467$	−7.29438	−0.337544	−0.168772	+	0.985655i	$0.553980\pi$
$468$	0	0
$469$	−11.3821	−0.525577
$470$	3.82286	0.176335
$471$	0	0
$472$	−15.0385	−0.692204
$473$	−3.58370	−0.164779
$474$	0	0
$475$	39.9093	1.83116
$476$	−6.88395	−0.315525
$477$	0	0
$478$	−4.37255	−0.199996
$479$	26.8328	1.22602	0.613012	−	0.790074i	$-0.289958\pi$
$479$	26.8328	1.22602	0.613012	+	0.790074i	$0.289958\pi$
$480$	0	0
$481$	−4.41683	−0.201390
$482$	−5.76327	−0.262510
$483$	0	0
$484$	−10.3907	−0.472305
$485$	−23.7454	−1.07822
$486$	0	0
$487$	11.3571	0.514638	0.257319	−	0.966326i	$-0.417161\pi$
$487$	11.3571	0.514638	0.257319	+	0.966326i	$0.417161\pi$
$488$	9.25642	0.419019
$489$	0	0
$490$	−27.7758	−1.25478
$491$	21.7557	0.981821	0.490911	−	0.871210i	$-0.336664\pi$
$491$	21.7557	0.981821	0.490911	+	0.871210i	$0.336664\pi$
$492$	0	0
$493$	−1.71252	−0.0771280
$494$	16.3333	0.734868
$495$	0	0
$496$	10.0911	0.453102
$497$	53.8551	2.41573
$498$	0	0
$499$	5.25062	0.235050	0.117525	−	0.993070i	$-0.462504\pi$
$499$	5.25062	0.235050	0.117525	+	0.993070i	$0.462504\pi$
$500$	−6.95058	−0.310840
$501$	0	0
$502$	−49.7184	−2.21904
$503$	29.2770	1.30540	0.652698	−	0.757618i	$-0.273637\pi$
$503$	29.2770	1.30540	0.652698	+	0.757618i	$0.273637\pi$
$504$	0	0
$505$	−58.5125	−2.60377
$506$	−2.65566	−0.118059
$507$	0	0
$508$	3.37352	0.149676
$509$	39.0644	1.73150	0.865750	−	0.500477i	$-0.166842\pi$
$509$	39.0644	1.73150	0.865750	+	0.500477i	$0.166842\pi$
$510$	0	0
$511$	−27.2565	−1.20576
$512$	15.0462	0.664953
$513$	0	0
$514$	−21.0904	−0.930259
$515$	−2.03249	−0.0895623
$516$	0	0
$517$	0.931835	0.0409821
$518$	−17.3797	−0.763618
$519$	0	0
$520$	7.68840	0.337159
$521$	7.75144	0.339597	0.169798	−	0.985479i	$-0.445688\pi$
$521$	7.75144	0.339597	0.169798	+	0.985479i	$0.445688\pi$
$522$	0	0
$523$	28.2941	1.23721	0.618607	−	0.785701i	$-0.287697\pi$
$523$	28.2941	1.23721	0.618607	+	0.785701i	$0.287697\pi$
$524$	1.47912	0.0646157
$525$	0	0
$526$	20.7415	0.904373
$527$	3.47950	0.151570
$528$	0	0
$529$	1.00000	0.0434783
$530$	−11.9943	−0.520998
$531$	0	0
$532$	23.8850	1.03554
$533$	−4.23569	−0.183468
$534$	0	0
$535$	−20.5463	−0.888293
$536$	4.88267	0.210899
$537$	0	0
$538$	−42.4456	−1.82996
$539$	−6.77046	−0.291624
$540$	0	0
$541$	14.5626	0.626093	0.313047	−	0.949738i	$-0.398650\pi$
$541$	14.5626	0.626093	0.313047	+	0.949738i	$0.398650\pi$
$542$	34.4406	1.47935
$543$	0	0
$544$	10.1811	0.436509
$545$	−56.1342	−2.40452
$546$	0	0
$547$	−19.0407	−0.814120	−0.407060	−	0.913401i	$-0.633446\pi$
$547$	−19.0407	−0.814120	−0.407060	+	0.913401i	$0.633446\pi$
$548$	20.1161	0.859317
$549$	0	0
$550$	−17.8371	−0.760577
$551$	5.94186	0.253132
$552$	0	0
$553$	9.67690	0.411504
$554$	−9.11763	−0.387371
$555$	0	0
$556$	−16.8017	−0.712552
$557$	−23.4385	−0.993121	−0.496561	−	0.868002i	$-0.665404\pi$
$557$	−23.4385	−0.993121	−0.496561	+	0.868002i	$0.665404\pi$
$558$	0	0
$559$	3.70945	0.156893
$560$	57.7719	2.44131
$561$	0	0
$562$	−51.0851	−2.15489
$563$	−24.2755	−1.02309	−0.511546	−	0.859256i	$-0.670927\pi$
$563$	−24.2755	−1.02309	−0.511546	+	0.859256i	$0.670927\pi$
$564$	0	0
$565$	22.7563	0.957364
$566$	−29.3121	−1.23208
$567$	0	0
$568$	−23.1026	−0.969363
$569$	−3.68449	−0.154462	−0.0772309	−	0.997013i	$-0.524608\pi$
$569$	−3.68449	−0.154462	−0.0772309	+	0.997013i	$0.524608\pi$
$570$	0	0
$571$	8.23061	0.344440	0.172220	−	0.985058i	$-0.444906\pi$
$571$	8.23061	0.344440	0.172220	+	0.985058i	$0.444906\pi$
$572$	−2.71297	−0.113435
$573$	0	0
$574$	−16.6669	−0.695663
$575$	6.71663	0.280103
$576$	0	0
$577$	−14.9312	−0.621595	−0.310798	−	0.950476i	$-0.600596\pi$
$577$	−14.9312	−0.621595	−0.310798	+	0.950476i	$0.600596\pi$
$578$	−25.0969	−1.04389
$579$	0	0
$580$	−4.04896	−0.168124
$581$	57.0948	2.36869
$582$	0	0
$583$	−2.92364	−0.121085
$584$	11.6924	0.483836
$585$	0	0
$586$	10.1539	0.419452
$587$	32.3414	1.33487	0.667436	−	0.744667i	$-0.267392\pi$
$587$	32.3414	1.33487	0.667436	+	0.744667i	$0.267392\pi$
$588$	0	0
$589$	−12.0727	−0.497446
$590$	−62.9974	−2.59356
$591$	0	0
$592$	14.2372	0.585147
$593$	40.7038	1.67150	0.835752	−	0.549107i	$-0.185032\pi$
$593$	40.7038	1.67150	0.835752	+	0.549107i	$0.185032\pi$
$594$	0	0
$595$	19.9204	0.816655
$596$	−5.59253	−0.229079
$597$	0	0
$598$	2.74885	0.112409
$599$	−8.42483	−0.344229	−0.172115	−	0.985077i	$-0.555060\pi$
$599$	−8.42483	−0.344229	−0.172115	+	0.985077i	$0.555060\pi$
$600$	0	0
$601$	−5.26509	−0.214768	−0.107384	−	0.994218i	$-0.534247\pi$
$601$	−5.26509	−0.214768	−0.107384	+	0.994218i	$0.534247\pi$
$602$	14.5962	0.594897
$603$	0	0
$604$	16.5445	0.673188
$605$	30.0680	1.22244
$606$	0	0
$607$	24.9444	1.01246	0.506231	−	0.862398i	$-0.331038\pi$
$607$	24.9444	1.01246	0.506231	+	0.862398i	$0.331038\pi$
$608$	−35.3248	−1.43261
$609$	0	0
$610$	38.7758	1.56999
$611$	−0.964533	−0.0390208
$612$	0	0
$613$	−47.3822	−1.91375	−0.956874	−	0.290504i	$-0.906177\pi$
$613$	−47.3822	−1.91375	−0.956874	+	0.290504i	$0.906177\pi$
$614$	60.7645	2.45226
$615$	0	0
$616$	7.37424	0.297116
$617$	−18.4036	−0.740900	−0.370450	−	0.928852i	$-0.620797\pi$
$617$	−18.4036	−0.740900	−0.370450	+	0.928852i	$0.620797\pi$
$618$	0	0
$619$	26.9757	1.08424	0.542122	−	0.840299i	$-0.317621\pi$
$619$	26.9757	1.08424	0.542122	+	0.840299i	$0.317621\pi$
$620$	8.22669	0.330392
$621$	0	0
$622$	6.78631	0.272106
$623$	49.4002	1.97918
$624$	0	0
$625$	−13.4700	−0.538800
$626$	−11.0181	−0.440373
$627$	0	0
$628$	−4.62560	−0.184582
$629$	4.90915	0.195740
$630$	0	0
$631$	−20.0249	−0.797179	−0.398589	−	0.917130i	$-0.630500\pi$
$631$	−20.0249	−0.797179	−0.398589	+	0.917130i	$0.630500\pi$
$632$	−4.15117	−0.165125
$633$	0	0
$634$	0.340324	0.0135160
$635$	−9.76208	−0.387396
$636$	0	0
$637$	7.00803	0.277668
$638$	−2.65566	−0.105139
$639$	0	0
$640$	−36.5879	−1.44626
$641$	0.303011	0.0119682	0.00598410	−	0.999982i	$-0.498095\pi$
$641$	0.303011	0.0119682	0.00598410	+	0.999982i	$0.498095\pi$
$642$	0	0
$643$	−13.8559	−0.546425	−0.273212	−	0.961954i	$-0.588086\pi$
$643$	−13.8559	−0.546425	−0.273212	+	0.961954i	$0.588086\pi$
$644$	4.01978	0.158402
$645$	0	0
$646$	−18.1538	−0.714252
$647$	−2.92725	−0.115082	−0.0575411	−	0.998343i	$-0.518326\pi$
$647$	−2.92725	−0.115082	−0.0575411	+	0.998343i	$0.518326\pi$
$648$	0	0
$649$	−15.3558	−0.602769
$650$	18.4630	0.724179
$651$	0	0
$652$	28.5971	1.11995
$653$	11.4806	0.449270	0.224635	−	0.974443i	$-0.427881\pi$
$653$	11.4806	0.449270	0.224635	+	0.974443i	$0.427881\pi$
$654$	0	0
$655$	−4.28019	−0.167241
$656$	13.6534	0.533074
$657$	0	0
$658$	−3.79531	−0.147957
$659$	−16.8223	−0.655304	−0.327652	−	0.944798i	$-0.606257\pi$
$659$	−16.8223	−0.655304	−0.327652	+	0.944798i	$0.606257\pi$
$660$	0	0
$661$	42.3254	1.64627	0.823134	−	0.567847i	$-0.192223\pi$
$661$	42.3254	1.64627	0.823134	+	0.567847i	$0.192223\pi$
$662$	49.6825	1.93097
$663$	0	0
$664$	−24.4924	−0.950488
$665$	−69.1168	−2.68024
$666$	0	0
$667$	1.00000	0.0387202
$668$	9.93494	0.384394
$669$	0	0
$670$	20.4538	0.790200
$671$	9.45174	0.364880
$672$	0	0
$673$	−42.8345	−1.65115	−0.825575	−	0.564292i	$-0.809149\pi$
$673$	−42.8345	−1.65115	−0.825575	+	0.564292i	$0.809149\pi$
$674$	−47.9550	−1.84716
$675$	0	0
$676$	−12.5693	−0.483435
$677$	−0.731629	−0.0281188	−0.0140594	−	0.999901i	$-0.504475\pi$
$677$	−0.731629	−0.0281188	−0.0140594	+	0.999901i	$0.504475\pi$
$678$	0	0
$679$	23.5742	0.904696
$680$	−8.54537	−0.327700
$681$	0	0
$682$	5.39579	0.206615
$683$	−32.6387	−1.24888	−0.624442	−	0.781071i	$-0.714674\pi$
$683$	−32.6387	−1.24888	−0.624442	+	0.781071i	$0.714674\pi$
$684$	0	0
$685$	−58.2107	−2.22412
$686$	−14.8637	−0.567498
$687$	0	0
$688$	−11.9571	−0.455859
$689$	3.02623	0.115290
$690$	0	0
$691$	34.7545	1.32212	0.661062	−	0.750331i	$-0.270106\pi$
$691$	34.7545	1.32212	0.661062	+	0.750331i	$0.270106\pi$
$692$	−6.66105	−0.253215
$693$	0	0
$694$	−34.6943	−1.31698
$695$	48.6198	1.84426
$696$	0	0
$697$	4.70782	0.178321
$698$	13.3782	0.506371
$699$	0	0
$700$	26.9994	1.02048
$701$	−36.0044	−1.35987	−0.679934	−	0.733273i	$-0.737992\pi$
$701$	−36.0044	−1.35987	−0.679934	+	0.733273i	$0.737992\pi$
$702$	0	0
$703$	−17.0331	−0.642414
$704$	1.00221	0.0377724
$705$	0	0
$706$	23.4674	0.883206
$707$	58.0908	2.18473
$708$	0	0
$709$	−36.0363	−1.35337	−0.676687	−	0.736271i	$-0.736585\pi$
$709$	−36.0363	−1.35337	−0.676687	+	0.736271i	$0.736585\pi$
$710$	−96.7783	−3.63203
$711$	0	0
$712$	−21.1916	−0.794187
$713$	−2.03180	−0.0760916
$714$	0	0
$715$	7.85063	0.293597
$716$	16.8572	0.629985
$717$	0	0
$718$	−8.54504	−0.318898
$719$	−39.4319	−1.47056	−0.735281	−	0.677763i	$-0.762950\pi$
$719$	−39.4319	−1.47056	−0.735281	+	0.677763i	$0.762950\pi$
$720$	0	0
$721$	2.01784	0.0751485
$722$	29.0903	1.08263
$723$	0	0
$724$	−4.13621	−0.153721
$725$	6.71663	0.249450
$726$	0	0
$727$	27.5541	1.02193	0.510963	−	0.859603i	$-0.329289\pi$
$727$	27.5541	1.02193	0.510963	+	0.859603i	$0.329289\pi$
$728$	−7.63299	−0.282898
$729$	0	0
$730$	48.9804	1.81285
$731$	−4.12292	−0.152492
$732$	0	0
$733$	−9.01914	−0.333129	−0.166565	−	0.986031i	$-0.553267\pi$
$733$	−9.01914	−0.333129	−0.166565	+	0.986031i	$0.553267\pi$
$734$	−9.25174	−0.341488
$735$	0	0
$736$	−5.94507	−0.219138
$737$	4.98569	0.183650
$738$	0	0
$739$	31.5605	1.16097	0.580486	−	0.814271i	$-0.302863\pi$
$739$	31.5605	1.16097	0.580486	+	0.814271i	$0.302863\pi$
$740$	11.6068	0.426676
$741$	0	0
$742$	11.9078	0.437150
$743$	−8.31854	−0.305177	−0.152589	−	0.988290i	$-0.548761\pi$
$743$	−8.31854	−0.305177	−0.152589	+	0.988290i	$0.548761\pi$
$744$	0	0
$745$	16.1833	0.592911
$746$	−45.4687	−1.66473
$747$	0	0
$748$	3.01537	0.110253
$749$	20.3982	0.745334
$750$	0	0
$751$	23.4686	0.856380	0.428190	−	0.903689i	$-0.359151\pi$
$751$	23.4686	0.856380	0.428190	+	0.903689i	$0.359151\pi$
$752$	3.10908	0.113376
$753$	0	0
$754$	2.74885	0.100107
$755$	−47.8756	−1.74237
$756$	0	0
$757$	19.5805	0.711665	0.355832	−	0.934550i	$-0.384197\pi$
$757$	19.5805	0.711665	0.355832	+	0.934550i	$0.384197\pi$
$758$	35.2590	1.28066
$759$	0	0
$760$	29.6495	1.07550
$761$	−37.1441	−1.34647	−0.673236	−	0.739427i	$-0.735096\pi$
$761$	−37.1441	−1.34647	−0.673236	+	0.739427i	$0.735096\pi$
$762$	0	0
$763$	55.7296	2.01755
$764$	−9.95840	−0.360282
$765$	0	0
$766$	0.924306	0.0333965
$767$	15.8947	0.573923
$768$	0	0
$769$	−32.0627	−1.15621	−0.578106	−	0.815962i	$-0.696208\pi$
$769$	−32.0627	−1.15621	−0.578106	+	0.815962i	$0.696208\pi$
$770$	30.8912	1.11324
$771$	0	0
$772$	27.1503	0.977161
$773$	15.9858	0.574971	0.287486	−	0.957785i	$-0.407181\pi$
$773$	15.9858	0.574971	0.287486	+	0.957785i	$0.407181\pi$
$774$	0	0
$775$	−13.6469	−0.490210
$776$	−10.1128	−0.363028
$777$	0	0
$778$	28.9441	1.03770
$779$	−16.3345	−0.585245
$780$	0	0
$781$	−23.5900	−0.844118
$782$	−3.05524	−0.109255
$783$	0	0
$784$	−22.5897	−0.806775
$785$	13.3853	0.477741
$786$	0	0
$787$	16.2988	0.580990	0.290495	−	0.956876i	$-0.406180\pi$
$787$	16.2988	0.580990	0.290495	+	0.956876i	$0.406180\pi$
$788$	−7.13056	−0.254016
$789$	0	0
$790$	−17.3895	−0.618691
$791$	−22.5923	−0.803290
$792$	0	0
$793$	−9.78339	−0.347418
$794$	−20.9632	−0.743954
$795$	0	0
$796$	−3.44413	−0.122074
$797$	−2.65493	−0.0940424	−0.0470212	−	0.998894i	$-0.514973\pi$
$797$	−2.65493	−0.0940424	−0.0470212	+	0.998894i	$0.514973\pi$
$798$	0	0
$799$	1.07204	0.0379261
$800$	−39.9309	−1.41177
$801$	0	0
$802$	1.98805	0.0702003
$803$	11.9391	0.421323
$804$	0	0
$805$	−11.6322	−0.409981
$806$	−5.58512	−0.196727
$807$	0	0
$808$	−24.9196	−0.876669
$809$	−40.5385	−1.42526	−0.712628	−	0.701542i	$-0.752495\pi$
$809$	−40.5385	−1.42526	−0.712628	+	0.701542i	$0.752495\pi$
$810$	0	0
$811$	39.5091	1.38735	0.693677	−	0.720287i	$-0.255990\pi$
$811$	39.5091	1.38735	0.693677	+	0.720287i	$0.255990\pi$
$812$	4.01978	0.141067
$813$	0	0
$814$	7.61278	0.266828
$815$	−82.7524	−2.89869
$816$	0	0
$817$	14.3051	0.500473
$818$	−19.9621	−0.697960
$819$	0	0
$820$	11.1308	0.388706
$821$	−0.156877	−0.00547506	−0.00273753	−	0.999996i	$-0.500871\pi$
$821$	−0.156877	−0.00547506	−0.00273753	+	0.999996i	$0.500871\pi$
$822$	0	0
$823$	27.8701	0.971490	0.485745	−	0.874101i	$-0.338548\pi$
$823$	27.8701	0.971490	0.485745	+	0.874101i	$0.338548\pi$
$824$	−0.865609	−0.0301549
$825$	0	0
$826$	62.5434	2.17616
$827$	36.9020	1.28321	0.641605	−	0.767035i	$-0.278269\pi$
$827$	36.9020	1.28321	0.641605	+	0.767035i	$0.278269\pi$
$828$	0	0
$829$	2.26800	0.0787709	0.0393855	−	0.999224i	$-0.487460\pi$
$829$	2.26800	0.0787709	0.0393855	+	0.999224i	$0.487460\pi$
$830$	−102.600	−3.56131
$831$	0	0
$832$	−1.03738	−0.0359647
$833$	−7.78916	−0.269879
$834$	0	0
$835$	−28.7491	−0.994905
$836$	−10.4623	−0.361846
$837$	0	0
$838$	49.9966	1.72711
$839$	−20.0828	−0.693334	−0.346667	−	0.937988i	$-0.612687\pi$
$839$	−20.0828	−0.693334	−0.346667	+	0.937988i	$0.612687\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	9.96324	0.343356
$843$	0	0
$844$	−9.48683	−0.326550
$845$	36.3723	1.25125
$846$	0	0
$847$	−29.8513	−1.02570
$848$	−9.75477	−0.334980
$849$	0	0
$850$	−20.5210	−0.703863
$851$	−2.86662	−0.0982665
$852$	0	0
$853$	−24.1153	−0.825694	−0.412847	−	0.910800i	$-0.635466\pi$
$853$	−24.1153	−0.825694	−0.412847	+	0.910800i	$0.635466\pi$
$854$	−38.4964	−1.31732
$855$	0	0
$856$	−8.75036	−0.299081
$857$	10.3653	0.354071	0.177036	−	0.984204i	$-0.443349\pi$
$857$	10.3653	0.354071	0.177036	+	0.984204i	$0.443349\pi$
$858$	0	0
$859$	17.5952	0.600339	0.300170	−	0.953886i	$-0.402957\pi$
$859$	17.5952	0.600339	0.300170	+	0.953886i	$0.402957\pi$
$860$	−9.74794	−0.332402
$861$	0	0
$862$	−19.1193	−0.651205
$863$	−47.9193	−1.63119	−0.815596	−	0.578622i	$-0.803591\pi$
$863$	−47.9193	−1.63119	−0.815596	+	0.578622i	$0.803591\pi$
$864$	0	0
$865$	19.2753	0.655382
$866$	31.9525	1.08579
$867$	0	0
$868$	−8.16740	−0.277220
$869$	−4.23876	−0.143790
$870$	0	0
$871$	−5.16064	−0.174861
$872$	−23.9067	−0.809584
$873$	0	0
$874$	10.6006	0.358572
$875$	−19.9682	−0.675049
$876$	0	0
$877$	−42.4419	−1.43316	−0.716580	−	0.697505i	$-0.754293\pi$
$877$	−42.4419	−1.43316	−0.716580	+	0.697505i	$0.754293\pi$
$878$	−26.9060	−0.908035
$879$	0	0
$880$	−25.3057	−0.853057
$881$	−38.4672	−1.29599	−0.647996	−	0.761644i	$-0.724393\pi$
$881$	−38.4672	−1.29599	−0.647996	+	0.761644i	$0.724393\pi$
$882$	0	0
$883$	−20.5029	−0.689977	−0.344989	−	0.938607i	$-0.612117\pi$
$883$	−20.5029	−0.689977	−0.344989	+	0.938607i	$0.612117\pi$
$884$	−3.12117	−0.104976
$885$	0	0
$886$	−38.6139	−1.29726
$887$	16.1194	0.541236	0.270618	−	0.962687i	$-0.412772\pi$
$887$	16.1194	0.541236	0.270618	+	0.962687i	$0.412772\pi$
$888$	0	0
$889$	9.69173	0.325050
$890$	−88.7729	−2.97568
$891$	0	0
$892$	22.0905	0.739643
$893$	−3.71962	−0.124472
$894$	0	0
$895$	−48.7804	−1.63055
$896$	36.3242	1.21351
$897$	0	0
$898$	−44.4151	−1.48215
$899$	−2.03180	−0.0677644
$900$	0	0
$901$	−3.36355	−0.112056
$902$	7.30058	0.243083
$903$	0	0
$904$	9.69157	0.322337
$905$	11.9691	0.397867
$906$	0	0
$907$	−31.6215	−1.04997	−0.524987	−	0.851110i	$-0.675930\pi$
$907$	−31.6215	−1.04997	−0.524987	+	0.851110i	$0.675930\pi$
$908$	−8.92895	−0.296318
$909$	0	0
$910$	−31.9751	−1.05997
$911$	−38.6264	−1.27975	−0.639874	−	0.768480i	$-0.721014\pi$
$911$	−38.6264	−1.27975	−0.639874	+	0.768480i	$0.721014\pi$
$912$	0	0
$913$	−25.0092	−0.827682
$914$	24.3873	0.806660
$915$	0	0
$916$	−7.30032	−0.241209
$917$	4.24934	0.140326
$918$	0	0
$919$	−12.5300	−0.413326	−0.206663	−	0.978412i	$-0.566260\pi$
$919$	−12.5300	−0.413326	−0.206663	+	0.978412i	$0.566260\pi$
$920$	4.98994	0.164514
$921$	0	0
$922$	−54.0702	−1.78071
$923$	24.4178	0.803722
$924$	0	0
$925$	−19.2541	−0.633069
$926$	39.1104	1.28525
$927$	0	0
$928$	−5.94507	−0.195157
$929$	−29.9489	−0.982592	−0.491296	−	0.870993i	$-0.663477\pi$
$929$	−29.9489	−0.982592	−0.491296	+	0.870993i	$0.663477\pi$
$930$	0	0
$931$	27.0257	0.885732
$932$	−1.77302	−0.0580771
$933$	0	0
$934$	−13.0136	−0.425819
$935$	−8.72568	−0.285360
$936$	0	0
$937$	−38.4575	−1.25635	−0.628177	−	0.778071i	$-0.716198\pi$
$937$	−38.4575	−1.25635	−0.628177	+	0.778071i	$0.716198\pi$
$938$	−20.3064	−0.663028
$939$	0	0
$940$	2.53466	0.0826716
$941$	−50.0232	−1.63071	−0.815354	−	0.578962i	$-0.803458\pi$
$941$	−50.0232	−1.63071	−0.815354	+	0.578962i	$0.803458\pi$
$942$	0	0
$943$	−2.74906	−0.0895217
$944$	−51.2349	−1.66755
$945$	0	0
$946$	−6.39355	−0.207872
$947$	46.4362	1.50897	0.754486	−	0.656316i	$-0.227886\pi$
$947$	46.4362	1.50897	0.754486	+	0.656316i	$0.227886\pi$
$948$	0	0
$949$	−12.3581	−0.401160
$950$	71.2007	2.31006
$951$	0	0
$952$	8.48379	0.274961
$953$	−19.8670	−0.643556	−0.321778	−	0.946815i	$-0.604280\pi$
$953$	−19.8670	−0.643556	−0.321778	+	0.946815i	$0.604280\pi$
$954$	0	0
$955$	28.8170	0.932497
$956$	−2.89912	−0.0937642
$957$	0	0
$958$	47.8715	1.54666
$959$	57.7912	1.86618
$960$	0	0
$961$	−26.8718	−0.866831
$962$	−7.87991	−0.254058
$963$	0	0
$964$	−3.82121	−0.123073
$965$	−78.5659	−2.52913
$966$	0	0
$967$	−46.8673	−1.50715	−0.753575	−	0.657362i	$-0.771672\pi$
$967$	−46.8673	−1.50715	−0.753575	+	0.657362i	$0.771672\pi$
$968$	12.8055	0.411586
$969$	0	0
$970$	−42.3632	−1.36020
$971$	32.0173	1.02748	0.513742	−	0.857945i	$-0.328259\pi$
$971$	32.0173	1.02748	0.513742	+	0.857945i	$0.328259\pi$
$972$	0	0
$973$	−48.2695	−1.54745
$974$	20.2618	0.649228
$975$	0	0
$976$	31.5358	1.00944
$977$	6.57514	0.210357	0.105179	−	0.994453i	$-0.466459\pi$
$977$	6.57514	0.210357	0.105179	+	0.994453i	$0.466459\pi$
$978$	0	0
$979$	−21.6387	−0.691576
$980$	−18.4161	−0.588282
$981$	0	0
$982$	38.8136	1.23859
$983$	39.0158	1.24441	0.622205	−	0.782854i	$-0.286237\pi$
$983$	39.0158	1.24441	0.622205	+	0.782854i	$0.286237\pi$
$984$	0	0
$985$	20.6340	0.657453
$986$	−3.05524	−0.0972988
$987$	0	0
$988$	10.8294	0.344529
$989$	2.40752	0.0765546
$990$	0	0
$991$	49.9715	1.58740	0.793699	−	0.608311i	$-0.208153\pi$
$991$	49.9715	1.58740	0.793699	+	0.608311i	$0.208153\pi$
$992$	12.0792	0.383516
$993$	0	0
$994$	96.0809	3.04750
$995$	9.96641	0.315957
$996$	0	0
$997$	30.3821	0.962212	0.481106	−	0.876662i	$-0.340235\pi$
$997$	30.3821	0.962212	0.481106	+	0.876662i	$0.340235\pi$
$998$	9.36744	0.296521
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.r.1.12		16
3.2	odd	2		2001.2.a.n.1.5	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.n.1.5	✓	16	3.2	odd	2
6003.2.a.r.1.12		16	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)

\(f(q)\)	\(=\)	\(q+1.78406 q^{2} +1.18288 q^{4} -3.42296 q^{5} +3.39829 q^{7} -1.45779 q^{8} +O(q^{10})\)	\(q+1.78406 q^{2} +1.18288 q^{4} -3.42296 q^{5} +3.39829 q^{7} -1.45779 q^{8} -6.10677 q^{10} -1.48855 q^{11} +1.54078 q^{13} +6.06276 q^{14} -4.96655 q^{16} -1.71252 q^{17} +5.94186 q^{19} -4.04896 q^{20} -2.65566 q^{22} +1.00000 q^{23} +6.71663 q^{25} +2.74885 q^{26} +4.01978 q^{28} +1.00000 q^{29} -2.03180 q^{31} -5.94507 q^{32} -3.05524 q^{34} -11.6322 q^{35} -2.86662 q^{37} +10.6006 q^{38} +4.98994 q^{40} -2.74906 q^{41} +2.40752 q^{43} -1.76078 q^{44} +1.78406 q^{46} -0.626003 q^{47} +4.54837 q^{49} +11.9829 q^{50} +1.82256 q^{52} +1.96409 q^{53} +5.09523 q^{55} -4.95398 q^{56} +1.78406 q^{58} +10.3160 q^{59} -6.34964 q^{61} -3.62487 q^{62} -0.673284 q^{64} -5.27402 q^{65} -3.34937 q^{67} -2.02571 q^{68} -20.7526 q^{70} +15.8477 q^{71} -8.02067 q^{73} -5.11424 q^{74} +7.02852 q^{76} -5.05851 q^{77} +2.84758 q^{79} +17.0003 q^{80} -4.90450 q^{82} +16.8010 q^{83} +5.86188 q^{85} +4.29516 q^{86} +2.16999 q^{88} +14.5368 q^{89} +5.23601 q^{91} +1.18288 q^{92} -1.11683 q^{94} -20.3387 q^{95} +6.93709 q^{97} +8.11457 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + q^{2} + 25 q^{4} - 3 q^{5} + 13 q^{7}+O(q^{10}) \) 16 * q + q^2 + 25 * q^4 - 3 * q^5 + 13 * q^7	\( 16 q + q^{2} + 25 q^{4} - 3 q^{5} + 13 q^{7} + 11 q^{10} - 8 q^{11} + 19 q^{13} - 16 q^{14} + 31 q^{16} + 4 q^{17} + 19 q^{19} - 16 q^{20} + 6 q^{22} + 16 q^{23} + 23 q^{25} + 15 q^{26} + 18 q^{28} + 16 q^{29} + 24 q^{31} + 21 q^{32} - 9 q^{34} + 13 q^{35} + 26 q^{37} - 22 q^{40} + 15 q^{41} + 33 q^{43} - 6 q^{44} + q^{46} - 13 q^{47} + 41 q^{49} - 13 q^{50} - 26 q^{52} - 5 q^{53} + 9 q^{55} - 40 q^{56} + q^{58} - 2 q^{59} + 29 q^{61} + 32 q^{62} + 28 q^{64} - 18 q^{65} + 32 q^{67} + 26 q^{68} + 18 q^{70} - 29 q^{71} + 19 q^{73} + 16 q^{74} + 64 q^{76} + 21 q^{77} + 56 q^{79} + 14 q^{82} - 5 q^{83} + 16 q^{85} + 20 q^{86} + q^{88} - 7 q^{89} - 6 q^{91} + 25 q^{92} - 11 q^{94} - 39 q^{95} + 35 q^{97} + 109 q^{98}+O(q^{100}) \) 16 * q + q^2 + 25 * q^4 - 3 * q^5 + 13 * q^7 + 11 * q^10 - 8 * q^11 + 19 * q^13 - 16 * q^14 + 31 * q^16 + 4 * q^17 + 19 * q^19 - 16 * q^20 + 6 * q^22 + 16 * q^23 + 23 * q^25 + 15 * q^26 + 18 * q^28 + 16 * q^29 + 24 * q^31 + 21 * q^32 - 9 * q^34 + 13 * q^35 + 26 * q^37 - 22 * q^40 + 15 * q^41 + 33 * q^43 - 6 * q^44 + q^46 - 13 * q^47 + 41 * q^49 - 13 * q^50 - 26 * q^52 - 5 * q^53 + 9 * q^55 - 40 * q^56 + q^58 - 2 * q^59 + 29 * q^61 + 32 * q^62 + 28 * q^64 - 18 * q^65 + 32 * q^67 + 26 * q^68 + 18 * q^70 - 29 * q^71 + 19 * q^73 + 16 * q^74 + 64 * q^76 + 21 * q^77 + 56 * q^79 + 14 * q^82 - 5 * q^83 + 16 * q^85 + 20 * q^86 + q^88 - 7 * q^89 - 6 * q^91 + 25 * q^92 - 11 * q^94 - 39 * q^95 + 35 * q^97 + 109 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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