LMFDB - Embedded newform 6003.2.a.s.1.4

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			1.4
Root			$2.44527$ 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)

$f(q)$	$=$	$q-2.44527 q^{2} +3.97935 q^{4} +1.74520 q^{5} +5.10283 q^{7} -4.84005 q^{8} +O(q^{10})$	\(q-2.44527 q^{2} +3.97935 q^{4} +1.74520 q^{5} +5.10283 q^{7} -4.84005 q^{8} -4.26748 q^{10} +5.64276 q^{11} +6.72828 q^{13} -12.4778 q^{14} +3.87652 q^{16} +0.427470 q^{17} -1.22783 q^{19} +6.94475 q^{20} -13.7981 q^{22} -1.00000 q^{23} -1.95428 q^{25} -16.4525 q^{26} +20.3059 q^{28} -1.00000 q^{29} +5.44908 q^{31} +0.200940 q^{32} -1.04528 q^{34} +8.90544 q^{35} -10.5826 q^{37} +3.00237 q^{38} -8.44684 q^{40} -6.52884 q^{41} -4.88914 q^{43} +22.4545 q^{44} +2.44527 q^{46} +8.92503 q^{47} +19.0388 q^{49} +4.77875 q^{50} +26.7742 q^{52} -5.99381 q^{53} +9.84773 q^{55} -24.6979 q^{56} +2.44527 q^{58} -0.440433 q^{59} -6.08220 q^{61} -13.3245 q^{62} -8.24440 q^{64} +11.7422 q^{65} +6.28479 q^{67} +1.70105 q^{68} -21.7762 q^{70} -3.56305 q^{71} -7.59201 q^{73} +25.8774 q^{74} -4.88596 q^{76} +28.7940 q^{77} +14.5546 q^{79} +6.76530 q^{80} +15.9648 q^{82} +8.12019 q^{83} +0.746019 q^{85} +11.9553 q^{86} -27.3112 q^{88} -4.45271 q^{89} +34.3333 q^{91} -3.97935 q^{92} -21.8241 q^{94} -2.14280 q^{95} -8.18389 q^{97} -46.5551 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.44527	−1.72907	−0.864534	−	0.502575i	$-0.832386\pi$
$2$	−2.44527	−1.72907	−0.864534	+	0.502575i	$0.832386\pi$
$3$	0	0
$3$	0	0
$4$	3.97935	1.98967
$5$	1.74520	0.780476	0.390238	−	0.920714i	$-0.372393\pi$
$5$	1.74520	0.780476	0.390238	+	0.920714i	$0.372393\pi$
$6$	0	0
$7$	5.10283	1.92869	0.964344	−	0.264654i	$-0.0852576\pi$
$7$	5.10283	1.92869	0.964344	+	0.264654i	$0.0852576\pi$
$8$	−4.84005	−1.71121
$9$	0	0
$10$	−4.26748	−1.34950
$11$	5.64276	1.70136	0.850678	−	0.525688i	$-0.176192\pi$
$11$	5.64276	1.70136	0.850678	+	0.525688i	$0.176192\pi$
$12$	0	0
$13$	6.72828	1.86609	0.933045	−	0.359759i	$-0.117141\pi$
$13$	6.72828	1.86609	0.933045	+	0.359759i	$0.117141\pi$
$14$	−12.4778	−3.33483
$15$	0	0
$16$	3.87652	0.969131
$17$	0.427470	0.103677	0.0518383	−	0.998655i	$-0.483492\pi$
$17$	0.427470	0.103677	0.0518383	+	0.998655i	$0.483492\pi$
$18$	0	0
$19$	−1.22783	−0.281683	−0.140842	−	0.990032i	$-0.544981\pi$
$19$	−1.22783	−0.281683	−0.140842	+	0.990032i	$0.544981\pi$
$20$	6.94475	1.55289
$21$	0	0
$22$	−13.7981	−2.94176
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−1.95428	−0.390857
$26$	−16.4525	−3.22660
$27$	0	0
$28$	20.3059	3.83746
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	5.44908	0.978684	0.489342	−	0.872092i	$-0.337237\pi$
$31$	5.44908	0.978684	0.489342	+	0.872092i	$0.337237\pi$
$32$	0.200940	0.0355216
$33$	0	0
$34$	−1.04528	−0.179264
$35$	8.90544	1.50529
$36$	0	0
$37$	−10.5826	−1.73977	−0.869887	−	0.493250i	$-0.835809\pi$
$37$	−10.5826	−1.73977	−0.869887	+	0.493250i	$0.835809\pi$
$38$	3.00237	0.487049
$39$	0	0
$40$	−8.44684	−1.33556
$41$	−6.52884	−1.01963	−0.509817	−	0.860283i	$-0.670287\pi$
$41$	−6.52884	−1.01963	−0.509817	+	0.860283i	$0.670287\pi$
$42$	0	0
$43$	−4.88914	−0.745586	−0.372793	−	0.927914i	$-0.621600\pi$
$43$	−4.88914	−0.745586	−0.372793	+	0.927914i	$0.621600\pi$
$44$	22.4545	3.38514
$45$	0	0
$46$	2.44527	0.360536
$47$	8.92503	1.30185	0.650925	−	0.759142i	$-0.274381\pi$
$47$	8.92503	1.30185	0.650925	+	0.759142i	$0.274381\pi$
$48$	0	0
$49$	19.0388	2.71983
$50$	4.77875	0.675818
$51$	0	0
$52$	26.7742	3.71291
$53$	−5.99381	−0.823313	−0.411657	−	0.911339i	$-0.635050\pi$
$53$	−5.99381	−0.823313	−0.411657	+	0.911339i	$0.635050\pi$
$54$	0	0
$55$	9.84773	1.32787
$56$	−24.6979	−3.30040
$57$	0	0
$58$	2.44527	0.321080
$59$	−0.440433	−0.0573395	−0.0286697	−	0.999589i	$-0.509127\pi$
$59$	−0.440433	−0.0573395	−0.0286697	+	0.999589i	$0.509127\pi$
$60$	0	0
$61$	−6.08220	−0.778746	−0.389373	−	0.921080i	$-0.627308\pi$
$61$	−6.08220	−0.778746	−0.389373	+	0.921080i	$0.627308\pi$
$62$	−13.3245	−1.69221
$63$	0	0
$64$	−8.24440	−1.03055
$65$	11.7422	1.45644
$66$	0	0
$67$	6.28479	0.767810	0.383905	−	0.923373i	$-0.374579\pi$
$67$	6.28479	0.767810	0.383905	+	0.923373i	$0.374579\pi$
$68$	1.70105	0.206283
$69$	0	0
$70$	−21.7762	−2.60276
$71$	−3.56305	−0.422856	−0.211428	−	0.977394i	$-0.567811\pi$
$71$	−3.56305	−0.422856	−0.211428	+	0.977394i	$0.567811\pi$
$72$	0	0
$73$	−7.59201	−0.888577	−0.444289	−	0.895884i	$-0.646544\pi$
$73$	−7.59201	−0.888577	−0.444289	+	0.895884i	$0.646544\pi$
$74$	25.8774	3.00819
$75$	0	0
$76$	−4.88596	−0.560458
$77$	28.7940	3.28138
$78$	0	0
$79$	14.5546	1.63752	0.818758	−	0.574139i	$-0.194663\pi$
$79$	14.5546	1.63752	0.818758	+	0.574139i	$0.194663\pi$
$80$	6.76530	0.756384
$81$	0	0
$82$	15.9648	1.76302
$83$	8.12019	0.891306	0.445653	−	0.895206i	$-0.352971\pi$
$83$	8.12019	0.891306	0.445653	+	0.895206i	$0.352971\pi$
$84$	0	0
$85$	0.746019	0.0809172
$86$	11.9553	1.28917
$87$	0	0
$88$	−27.3112	−2.91138
$89$	−4.45271	−0.471986	−0.235993	−	0.971755i	$-0.575834\pi$
$89$	−4.45271	−0.471986	−0.235993	+	0.971755i	$0.575834\pi$
$90$	0	0
$91$	34.3333	3.59910
$92$	−3.97935	−0.414876
$93$	0	0
$94$	−21.8241	−2.25099
$95$	−2.14280	−0.219847
$96$	0	0
$97$	−8.18389	−0.830948	−0.415474	−	0.909605i	$-0.636384\pi$
$97$	−8.18389	−0.830948	−0.415474	+	0.909605i	$0.636384\pi$
$98$	−46.5551	−4.70278
$99$	0	0
$100$	−7.77678	−0.777678
$101$	17.5954	1.75081	0.875406	−	0.483388i	$-0.160594\pi$
$101$	17.5954	1.75081	0.875406	+	0.483388i	$0.160594\pi$
$102$	0	0
$103$	11.7113	1.15395	0.576973	−	0.816763i	$-0.304234\pi$
$103$	11.7113	1.15395	0.576973	+	0.816763i	$0.304234\pi$
$104$	−32.5652	−3.19328
$105$	0	0
$106$	14.6565	1.42356
$107$	11.2667	1.08919	0.544597	−	0.838698i	$-0.316683\pi$
$107$	11.2667	1.08919	0.544597	+	0.838698i	$0.316683\pi$
$108$	0	0
$109$	−15.2337	−1.45913	−0.729563	−	0.683913i	$-0.760277\pi$
$109$	−15.2337	−1.45913	−0.729563	+	0.683913i	$0.760277\pi$
$110$	−24.0804	−2.29597
$111$	0	0
$112$	19.7812	1.86915
$113$	−8.40919	−0.791070	−0.395535	−	0.918451i	$-0.629441\pi$
$113$	−8.40919	−0.791070	−0.395535	+	0.918451i	$0.629441\pi$
$114$	0	0
$115$	−1.74520	−0.162741
$116$	−3.97935	−0.369473
$117$	0	0
$118$	1.07698	0.0991439
$119$	2.18130	0.199960
$120$	0	0
$121$	20.8407	1.89461
$122$	14.8726	1.34651
$123$	0	0
$124$	21.6838	1.94726
$125$	−12.1366	−1.08553
$126$	0	0
$127$	−22.3171	−1.98032	−0.990159	−	0.139947i	$-0.955307\pi$
$127$	−22.3171	−1.98032	−0.990159	+	0.139947i	$0.955307\pi$
$128$	19.7579	1.74637
$129$	0	0
$130$	−28.7128	−2.51828
$131$	−1.92039	−0.167785	−0.0838925	−	0.996475i	$-0.526735\pi$
$131$	−1.92039	−0.167785	−0.0838925	+	0.996475i	$0.526735\pi$
$132$	0	0
$133$	−6.26539	−0.543278
$134$	−15.3680	−1.32759
$135$	0	0
$136$	−2.06897	−0.177413
$137$	3.76984	0.322079	0.161040	−	0.986948i	$-0.448515\pi$
$137$	3.76984	0.322079	0.161040	+	0.986948i	$0.448515\pi$
$138$	0	0
$139$	1.93727	0.164317	0.0821585	−	0.996619i	$-0.473819\pi$
$139$	1.93727	0.164317	0.0821585	+	0.996619i	$0.473819\pi$
$140$	35.4379	2.99505
$141$	0	0
$142$	8.71262	0.731146
$143$	37.9661	3.17488
$144$	0	0
$145$	−1.74520	−0.144931
$146$	18.5645	1.53641
$147$	0	0
$148$	−42.1120	−3.46159
$149$	13.6999	1.12234	0.561171	−	0.827700i	$-0.310351\pi$
$149$	13.6999	1.12234	0.561171	+	0.827700i	$0.310351\pi$
$150$	0	0
$151$	−5.39735	−0.439230	−0.219615	−	0.975587i	$-0.570480\pi$
$151$	−5.39735	−0.439230	−0.219615	+	0.975587i	$0.570480\pi$
$152$	5.94274	0.482020
$153$	0	0
$154$	−70.4091	−5.67373
$155$	9.50972	0.763839
$156$	0	0
$157$	0.0239922	0.00191479	0.000957394	−	1.00000i	$-0.499695\pi$
$157$	0.0239922	0.00191479	0.000957394		1.00000i	$0.499695\pi$
$158$	−35.5898	−2.83138
$159$	0	0
$160$	0.350681	0.0277238
$161$	−5.10283	−0.402159
$162$	0	0
$163$	0.0252041	0.00197414	0.000987068	−	1.00000i	$-0.499686\pi$
$163$	0.0252041	0.00197414	0.000987068		1.00000i	$0.499686\pi$
$164$	−25.9806	−2.02874
$165$	0	0
$166$	−19.8561	−1.54113
$167$	−7.32969	−0.567188	−0.283594	−	0.958944i	$-0.591527\pi$
$167$	−7.32969	−0.567188	−0.283594	+	0.958944i	$0.591527\pi$
$168$	0	0
$169$	32.2698	2.48229
$170$	−1.82422	−0.139911
$171$	0	0
$172$	−19.4556	−1.48347
$173$	19.5370	1.48537	0.742685	−	0.669641i	$-0.233552\pi$
$173$	19.5370	1.48537	0.742685	+	0.669641i	$0.233552\pi$
$174$	0	0
$175$	−9.97237	−0.753840
$176$	21.8743	1.64884
$177$	0	0
$178$	10.8881	0.816096
$179$	−6.09592	−0.455631	−0.227815	−	0.973704i	$-0.573158\pi$
$179$	−6.09592	−0.455631	−0.227815	+	0.973704i	$0.573158\pi$
$180$	0	0
$181$	6.02909	0.448139	0.224069	−	0.974573i	$-0.428066\pi$
$181$	6.02909	0.448139	0.224069	+	0.974573i	$0.428066\pi$
$182$	−83.9541	−6.22309
$183$	0	0
$184$	4.84005	0.356813
$185$	−18.4688	−1.35785
$186$	0	0
$187$	2.41211	0.176391
$188$	35.5158	2.59026
$189$	0	0
$190$	5.23973	0.380130
$191$	−12.5256	−0.906318	−0.453159	−	0.891430i	$-0.649703\pi$
$191$	−12.5256	−0.906318	−0.453159	+	0.891430i	$0.649703\pi$
$192$	0	0
$193$	9.90192	0.712756	0.356378	−	0.934342i	$-0.384012\pi$
$193$	9.90192	0.712756	0.356378	+	0.934342i	$0.384012\pi$
$194$	20.0118	1.43676
$195$	0	0
$196$	75.7622	5.41158
$197$	6.13949	0.437421	0.218710	−	0.975790i	$-0.429815\pi$
$197$	6.13949	0.437421	0.218710	+	0.975790i	$0.429815\pi$
$198$	0	0
$199$	14.2857	1.01268	0.506342	−	0.862333i	$-0.330997\pi$
$199$	14.2857	1.01268	0.506342	+	0.862333i	$0.330997\pi$
$200$	9.45882	0.668840
$201$	0	0
$202$	−43.0256	−3.02727
$203$	−5.10283	−0.358148
$204$	0	0
$205$	−11.3941	−0.795800
$206$	−28.6372	−1.99525
$207$	0	0
$208$	26.0824	1.80849
$209$	−6.92833	−0.479243
$210$	0	0
$211$	8.67262	0.597048	0.298524	−	0.954402i	$-0.403506\pi$
$211$	8.67262	0.597048	0.298524	+	0.954402i	$0.403506\pi$
$212$	−23.8515	−1.63813
$213$	0	0
$214$	−27.5502	−1.88329
$215$	−8.53251	−0.581912
$216$	0	0
$217$	27.8057	1.88757
$218$	37.2506	2.52293
$219$	0	0
$220$	39.1875	2.64202
$221$	2.87614	0.193470
$222$	0	0
$223$	−6.91512	−0.463071	−0.231535	−	0.972827i	$-0.574375\pi$
$223$	−6.91512	−0.463071	−0.231535	+	0.972827i	$0.574375\pi$
$224$	1.02536	0.0685100
$225$	0	0
$226$	20.5627	1.36781
$227$	10.0823	0.669187	0.334593	−	0.942363i	$-0.391401\pi$
$227$	10.0823	0.669187	0.334593	+	0.942363i	$0.391401\pi$
$228$	0	0
$229$	−10.2947	−0.680292	−0.340146	−	0.940373i	$-0.610476\pi$
$229$	−10.2947	−0.680292	−0.340146	+	0.940373i	$0.610476\pi$
$230$	4.26748	0.281389
$231$	0	0
$232$	4.84005	0.317765
$233$	−3.95614	−0.259175	−0.129588	−	0.991568i	$-0.541365\pi$
$233$	−3.95614	−0.259175	−0.129588	+	0.991568i	$0.541365\pi$
$234$	0	0
$235$	15.5759	1.01606
$236$	−1.75264	−0.114087
$237$	0	0
$238$	−5.33388	−0.345744
$239$	29.3893	1.90104	0.950518	−	0.310670i	$-0.100553\pi$
$239$	29.3893	1.90104	0.950518	+	0.310670i	$0.100553\pi$
$240$	0	0
$241$	−13.4864	−0.868737	−0.434369	−	0.900735i	$-0.643028\pi$
$241$	−13.4864	−0.868737	−0.434369	+	0.900735i	$0.643028\pi$
$242$	−50.9612	−3.27591
$243$	0	0
$244$	−24.2032	−1.54945
$245$	33.2265	2.12277
$246$	0	0
$247$	−8.26117	−0.525646
$248$	−26.3738	−1.67474
$249$	0	0
$250$	29.6773	1.87696
$251$	−8.57329	−0.541141	−0.270571	−	0.962700i	$-0.587212\pi$
$251$	−8.57329	−0.541141	−0.270571	+	0.962700i	$0.587212\pi$
$252$	0	0
$253$	−5.64276	−0.354757
$254$	54.5712	3.42410
$255$	0	0
$256$	−31.8247	−1.98904
$257$	−28.3655	−1.76939	−0.884695	−	0.466169i	$-0.845634\pi$
$257$	−28.3655	−1.76939	−0.884695	+	0.466169i	$0.845634\pi$
$258$	0	0
$259$	−54.0014	−3.35548
$260$	46.7263	2.89784
$261$	0	0
$262$	4.69587	0.290112
$263$	12.4022	0.764750	0.382375	−	0.924007i	$-0.375106\pi$
$263$	12.4022	0.764750	0.382375	+	0.924007i	$0.375106\pi$
$264$	0	0
$265$	−10.4604	−0.642576
$266$	15.3206	0.939365
$267$	0	0
$268$	25.0094	1.52769
$269$	−26.8677	−1.63815	−0.819075	−	0.573686i	$-0.805513\pi$
$269$	−26.8677	−1.63815	−0.819075	+	0.573686i	$0.805513\pi$
$270$	0	0
$271$	2.24407	0.136317	0.0681587	−	0.997674i	$-0.478288\pi$
$271$	2.24407	0.136317	0.0681587	+	0.997674i	$0.478288\pi$
$272$	1.65710	0.100476
$273$	0	0
$274$	−9.21828	−0.556896
$275$	−11.0275	−0.664986
$276$	0	0
$277$	−1.42115	−0.0853887	−0.0426944	−	0.999088i	$-0.513594\pi$
$277$	−1.42115	−0.0853887	−0.0426944	+	0.999088i	$0.513594\pi$
$278$	−4.73715	−0.284115
$279$	0	0
$280$	−43.1027	−2.57588
$281$	−24.2843	−1.44868	−0.724339	−	0.689444i	$-0.757855\pi$
$281$	−24.2843	−1.44868	−0.724339	+	0.689444i	$0.757855\pi$
$282$	0	0
$283$	−19.9648	−1.18678	−0.593392	−	0.804914i	$-0.702212\pi$
$283$	−19.9648	−1.18678	−0.593392	+	0.804914i	$0.702212\pi$
$284$	−14.1786	−0.841346
$285$	0	0
$286$	−92.8373	−5.48959
$287$	−33.3156	−1.96656
$288$	0	0
$289$	−16.8173	−0.989251
$290$	4.26748	0.250595
$291$	0	0
$292$	−30.2113	−1.76798
$293$	11.9515	0.698216	0.349108	−	0.937083i	$-0.386485\pi$
$293$	11.9515	0.698216	0.349108	+	0.937083i	$0.386485\pi$
$294$	0	0
$295$	−0.768643	−0.0447521
$296$	51.2204	2.97713
$297$	0	0
$298$	−33.5000	−1.94060
$299$	−6.72828	−0.389107
$300$	0	0
$301$	−24.9484	−1.43800
$302$	13.1980	0.759458
$303$	0	0
$304$	−4.75970	−0.272988
$305$	−10.6146	−0.607793
$306$	0	0
$307$	−12.3877	−0.707006	−0.353503	−	0.935433i	$-0.615010\pi$
$307$	−12.3877	−0.707006	−0.353503	+	0.935433i	$0.615010\pi$
$308$	114.581	6.52888
$309$	0	0
$310$	−23.2539	−1.32073
$311$	−11.8535	−0.672150	−0.336075	−	0.941835i	$-0.609100\pi$
$311$	−11.8535	−0.672150	−0.336075	+	0.941835i	$0.609100\pi$
$312$	0	0
$313$	−19.8879	−1.12413	−0.562065	−	0.827093i	$-0.689993\pi$
$313$	−19.8879	−1.12413	−0.562065	+	0.827093i	$0.689993\pi$
$314$	−0.0586675	−0.00331080
$315$	0	0
$316$	57.9177	3.25812
$317$	−9.90945	−0.556570	−0.278285	−	0.960499i	$-0.589766\pi$
$317$	−9.90945	−0.556570	−0.278285	+	0.960499i	$0.589766\pi$
$318$	0	0
$319$	−5.64276	−0.315934
$320$	−14.3881	−0.804320
$321$	0	0
$322$	12.4778	0.695360
$323$	−0.524859	−0.0292040
$324$	0	0
$325$	−13.1490	−0.729374
$326$	−0.0616308	−0.00341341
$327$	0	0
$328$	31.5999	1.74481
$329$	45.5429	2.51086
$330$	0	0
$331$	−10.0660	−0.553276	−0.276638	−	0.960974i	$-0.589220\pi$
$331$	−10.0660	−0.553276	−0.276638	+	0.960974i	$0.589220\pi$
$332$	32.3131	1.77341
$333$	0	0
$334$	17.9231	0.980707
$335$	10.9682	0.599257
$336$	0	0
$337$	8.52111	0.464175	0.232087	−	0.972695i	$-0.425444\pi$
$337$	8.52111	0.464175	0.232087	+	0.972695i	$0.425444\pi$
$338$	−78.9084	−4.29205
$339$	0	0
$340$	2.96867	0.160999
$341$	30.7478	1.66509
$342$	0	0
$343$	61.4321	3.31702
$344$	23.6636	1.27586
$345$	0	0
$346$	−47.7732	−2.56830
$347$	−36.6769	−1.96892	−0.984461	−	0.175605i	$-0.943812\pi$
$347$	−36.6769	−1.96892	−0.984461	+	0.175605i	$0.943812\pi$
$348$	0	0
$349$	16.0444	0.858835	0.429417	−	0.903106i	$-0.358719\pi$
$349$	16.0444	0.858835	0.429417	+	0.903106i	$0.358719\pi$
$350$	24.3851	1.30344
$351$	0	0
$352$	1.13386	0.0604348
$353$	16.0098	0.852117	0.426059	−	0.904696i	$-0.359902\pi$
$353$	16.0098	0.852117	0.426059	+	0.904696i	$0.359902\pi$
$354$	0	0
$355$	−6.21822	−0.330029
$356$	−17.7189	−0.939099
$357$	0	0
$358$	14.9062	0.787816
$359$	22.6647	1.19620	0.598098	−	0.801423i	$-0.295923\pi$
$359$	22.6647	1.19620	0.598098	+	0.801423i	$0.295923\pi$
$360$	0	0
$361$	−17.4924	−0.920655
$362$	−14.7428	−0.774862
$363$	0	0
$364$	136.624	7.16105
$365$	−13.2496	−0.693514
$366$	0	0
$367$	−21.6748	−1.13142	−0.565708	−	0.824605i	$-0.691397\pi$
$367$	−21.6748	−1.13142	−0.565708	+	0.824605i	$0.691397\pi$
$368$	−3.87652	−0.202078
$369$	0	0
$370$	45.1612	2.34782
$371$	−30.5854	−1.58791
$372$	0	0
$373$	−0.141606	−0.00733210	−0.00366605	−	0.999993i	$-0.501167\pi$
$373$	−0.141606	−0.00733210	−0.00366605	+	0.999993i	$0.501167\pi$
$374$	−5.89826	−0.304992
$375$	0	0
$376$	−43.1976	−2.22774
$377$	−6.72828	−0.346524
$378$	0	0
$379$	20.6660	1.06154	0.530771	−	0.847515i	$-0.321902\pi$
$379$	20.6660	1.06154	0.530771	+	0.847515i	$0.321902\pi$
$380$	−8.52696	−0.437424
$381$	0	0
$382$	30.6284	1.56709
$383$	12.1786	0.622298	0.311149	−	0.950361i	$-0.399286\pi$
$383$	12.1786	0.622298	0.311149	+	0.950361i	$0.399286\pi$
$384$	0	0
$385$	50.2512	2.56104
$386$	−24.2129	−1.23240
$387$	0	0
$388$	−32.5665	−1.65332
$389$	−36.0697	−1.82880	−0.914402	−	0.404807i	$-0.867339\pi$
$389$	−36.0697	−1.82880	−0.914402	+	0.404807i	$0.867339\pi$
$390$	0	0
$391$	−0.427470	−0.0216181
$392$	−92.1488	−4.65422
$393$	0	0
$394$	−15.0127	−0.756330
$395$	25.4006	1.27804
$396$	0	0
$397$	29.7914	1.49519	0.747595	−	0.664155i	$-0.231209\pi$
$397$	29.7914	1.49519	0.747595	+	0.664155i	$0.231209\pi$
$398$	−34.9323	−1.75100
$399$	0	0
$400$	−7.57583	−0.378791
$401$	17.1118	0.854521	0.427261	−	0.904129i	$-0.359479\pi$
$401$	17.1118	0.854521	0.427261	+	0.904129i	$0.359479\pi$
$402$	0	0
$403$	36.6630	1.82631
$404$	70.0184	3.48355
$405$	0	0
$406$	12.4778	0.619262
$407$	−59.7152	−2.95997
$408$	0	0
$409$	−2.30574	−0.114011	−0.0570057	−	0.998374i	$-0.518155\pi$
$409$	−2.30574	−0.114011	−0.0570057	+	0.998374i	$0.518155\pi$
$410$	27.8617	1.37599
$411$	0	0
$412$	46.6033	2.29598
$413$	−2.24745	−0.110590
$414$	0	0
$415$	14.1713	0.695643
$416$	1.35198	0.0662865
$417$	0	0
$418$	16.9417	0.828643
$419$	−13.5344	−0.661199	−0.330600	−	0.943771i	$-0.607251\pi$
$419$	−13.5344	−0.661199	−0.330600	+	0.943771i	$0.607251\pi$
$420$	0	0
$421$	29.2502	1.42557	0.712783	−	0.701385i	$-0.247435\pi$
$421$	29.2502	1.42557	0.712783	+	0.701385i	$0.247435\pi$
$422$	−21.2069	−1.03234
$423$	0	0
$424$	29.0103	1.40887
$425$	−0.835397	−0.0405227
$426$	0	0
$427$	−31.0364	−1.50196
$428$	44.8342	2.16714
$429$	0	0
$430$	20.8643	1.00617
$431$	−21.8598	−1.05295	−0.526476	−	0.850190i	$-0.676487\pi$
$431$	−21.8598	−1.05295	−0.526476	+	0.850190i	$0.676487\pi$
$432$	0	0
$433$	−7.10245	−0.341322	−0.170661	−	0.985330i	$-0.554590\pi$
$433$	−7.10245	−0.341322	−0.170661	+	0.985330i	$0.554590\pi$
$434$	−67.9925	−3.26374
$435$	0	0
$436$	−60.6203	−2.90319
$437$	1.22783	0.0587350
$438$	0	0
$439$	−28.6168	−1.36581	−0.682903	−	0.730509i	$-0.739283\pi$
$439$	−28.6168	−1.36581	−0.682903	+	0.730509i	$0.739283\pi$
$440$	−47.6635	−2.27227
$441$	0	0
$442$	−7.03294	−0.334523
$443$	−17.1209	−0.813440	−0.406720	−	0.913553i	$-0.633328\pi$
$443$	−17.1209	−0.813440	−0.406720	+	0.913553i	$0.633328\pi$
$444$	0	0
$445$	−7.77085	−0.368374
$446$	16.9093	0.800680
$447$	0	0
$448$	−42.0698	−1.98761
$449$	−14.4814	−0.683419	−0.341710	−	0.939806i	$-0.611006\pi$
$449$	−14.4814	−0.683419	−0.341710	+	0.939806i	$0.611006\pi$
$450$	0	0
$451$	−36.8407	−1.73476
$452$	−33.4631	−1.57397
$453$	0	0
$454$	−24.6540	−1.15707
$455$	59.9183	2.80902
$456$	0	0
$457$	27.8152	1.30114	0.650569	−	0.759447i	$-0.274530\pi$
$457$	27.8152	1.30114	0.650569	+	0.759447i	$0.274530\pi$
$458$	25.1733	1.17627
$459$	0	0
$460$	−6.94475	−0.323801
$461$	−15.2303	−0.709346	−0.354673	−	0.934990i	$-0.615408\pi$
$461$	−15.2303	−0.709346	−0.354673	+	0.934990i	$0.615408\pi$
$462$	0	0
$463$	26.1514	1.21536	0.607680	−	0.794182i	$-0.292100\pi$
$463$	26.1514	1.21536	0.607680	+	0.794182i	$0.292100\pi$
$464$	−3.87652	−0.179963
$465$	0	0
$466$	9.67384	0.448132
$467$	17.6426	0.816402	0.408201	−	0.912892i	$-0.366156\pi$
$467$	17.6426	0.816402	0.408201	+	0.912892i	$0.366156\pi$
$468$	0	0
$469$	32.0702	1.48086
$470$	−38.0874	−1.75684
$471$	0	0
$472$	2.13172	0.0981202
$473$	−27.5882	−1.26851
$474$	0	0
$475$	2.39952	0.110098
$476$	8.68017	0.397855
$477$	0	0
$478$	−71.8648	−3.28702
$479$	5.12689	0.234253	0.117127	−	0.993117i	$-0.462632\pi$
$479$	5.12689	0.234253	0.117127	+	0.993117i	$0.462632\pi$
$480$	0	0
$481$	−71.2030	−3.24658
$482$	32.9780	1.50211
$483$	0	0
$484$	82.9324	3.76966
$485$	−14.2825	−0.648535
$486$	0	0
$487$	6.10506	0.276647	0.138323	−	0.990387i	$-0.455829\pi$
$487$	6.10506	0.276647	0.138323	+	0.990387i	$0.455829\pi$
$488$	29.4381	1.33260
$489$	0	0
$490$	−81.2479	−3.67041
$491$	−5.01074	−0.226131	−0.113066	−	0.993588i	$-0.536067\pi$
$491$	−5.01074	−0.226131	−0.113066	+	0.993588i	$0.536067\pi$
$492$	0	0
$493$	−0.427470	−0.0192523
$494$	20.2008	0.908877
$495$	0	0
$496$	21.1235	0.948473
$497$	−18.1816	−0.815557
$498$	0	0
$499$	26.9336	1.20571	0.602856	−	0.797850i	$-0.294029\pi$
$499$	26.9336	1.20571	0.602856	+	0.797850i	$0.294029\pi$
$500$	−48.2958	−2.15985
$501$	0	0
$502$	20.9640	0.935670
$503$	−2.34501	−0.104559	−0.0522794	−	0.998632i	$-0.516649\pi$
$503$	−2.34501	−0.104559	−0.0522794	+	0.998632i	$0.516649\pi$
$504$	0	0
$505$	30.7075	1.36647
$506$	13.7981	0.613399
$507$	0	0
$508$	−88.8073	−3.94019
$509$	−25.6358	−1.13629	−0.568143	−	0.822930i	$-0.692338\pi$
$509$	−25.6358	−1.13629	−0.568143	+	0.822930i	$0.692338\pi$
$510$	0	0
$511$	−38.7407	−1.71379
$512$	38.3041	1.69282
$513$	0	0
$514$	69.3613	3.05940
$515$	20.4385	0.900628
$516$	0	0
$517$	50.3618	2.21491
$518$	132.048	5.80185
$519$	0	0
$520$	−56.8327	−2.49228
$521$	−0.166481	−0.00729366	−0.00364683	−	0.999993i	$-0.501161\pi$
$521$	−0.166481	−0.00729366	−0.00364683	+	0.999993i	$0.501161\pi$
$522$	0	0
$523$	44.8249	1.96005	0.980027	−	0.198864i	$-0.0637253\pi$
$523$	44.8249	1.96005	0.980027	+	0.198864i	$0.0637253\pi$
$524$	−7.64189	−0.333838
$525$	0	0
$526$	−30.3266	−1.32230
$527$	2.32932	0.101467
$528$	0	0
$529$	1.00000	0.0434783
$530$	25.5785	1.11106
$531$	0	0
$532$	−24.9322	−1.08095
$533$	−43.9279	−1.90273
$534$	0	0
$535$	19.6626	0.850090
$536$	−30.4187	−1.31389
$537$	0	0
$538$	65.6987	2.83247
$539$	107.432	4.62740
$540$	0	0
$541$	−8.32708	−0.358009	−0.179005	−	0.983848i	$-0.557288\pi$
$541$	−8.32708	−0.358009	−0.179005	+	0.983848i	$0.557288\pi$
$542$	−5.48735	−0.235702
$543$	0	0
$544$	0.0858960	0.00368276
$545$	−26.5859	−1.13881
$546$	0	0
$547$	32.0556	1.37060	0.685299	−	0.728262i	$-0.259671\pi$
$547$	32.0556	1.37060	0.685299	+	0.728262i	$0.259671\pi$
$548$	15.0015	0.640833
$549$	0	0
$550$	26.9653	1.14981
$551$	1.22783	0.0523072
$552$	0	0
$553$	74.2694	3.15826
$554$	3.47510	0.147643
$555$	0	0
$556$	7.70907	0.326937
$557$	−15.4539	−0.654803	−0.327402	−	0.944885i	$-0.606173\pi$
$557$	−15.4539	−0.654803	−0.327402	+	0.944885i	$0.606173\pi$
$558$	0	0
$559$	−32.8955	−1.39133
$560$	34.5222	1.45883
$561$	0	0
$562$	59.3816	2.50486
$563$	10.5091	0.442905	0.221453	−	0.975171i	$-0.428920\pi$
$563$	10.5091	0.442905	0.221453	+	0.975171i	$0.428920\pi$
$564$	0	0
$565$	−14.6757	−0.617411
$566$	48.8193	2.05203
$567$	0	0
$568$	17.2453	0.723597
$569$	−2.83273	−0.118754	−0.0593770	−	0.998236i	$-0.518911\pi$
$569$	−2.83273	−0.118754	−0.0593770	+	0.998236i	$0.518911\pi$
$570$	0	0
$571$	−18.5624	−0.776812	−0.388406	−	0.921488i	$-0.626974\pi$
$571$	−18.5624	−0.776812	−0.388406	+	0.921488i	$0.626974\pi$
$572$	151.080	6.31698
$573$	0	0
$574$	81.4656	3.40031
$575$	1.95428	0.0814993
$576$	0	0
$577$	−17.3886	−0.723897	−0.361949	−	0.932198i	$-0.617888\pi$
$577$	−17.3886	−0.723897	−0.361949	+	0.932198i	$0.617888\pi$
$578$	41.1228	1.71048
$579$	0	0
$580$	−6.94475	−0.288365
$581$	41.4359	1.71905
$582$	0	0
$583$	−33.8216	−1.40075
$584$	36.7457	1.52055
$585$	0	0
$586$	−29.2247	−1.20726
$587$	5.05110	0.208481	0.104241	−	0.994552i	$-0.466759\pi$
$587$	5.05110	0.208481	0.104241	+	0.994552i	$0.466759\pi$
$588$	0	0
$589$	−6.69053	−0.275679
$590$	1.87954	0.0773794
$591$	0	0
$592$	−41.0238	−1.68607
$593$	13.2024	0.542158	0.271079	−	0.962557i	$-0.412619\pi$
$593$	13.2024	0.542158	0.271079	+	0.962557i	$0.412619\pi$
$594$	0	0
$595$	3.80681	0.156064
$596$	54.5168	2.23310
$597$	0	0
$598$	16.4525	0.672792
$599$	−19.1932	−0.784212	−0.392106	−	0.919920i	$-0.628253\pi$
$599$	−19.1932	−0.784212	−0.392106	+	0.919920i	$0.628253\pi$
$600$	0	0
$601$	7.62036	0.310841	0.155421	−	0.987848i	$-0.450327\pi$
$601$	7.62036	0.310841	0.155421	+	0.987848i	$0.450327\pi$
$602$	61.0056	2.48640
$603$	0	0
$604$	−21.4779	−0.873924
$605$	36.3711	1.47870
$606$	0	0
$607$	−19.0718	−0.774101	−0.387051	−	0.922059i	$-0.626506\pi$
$607$	−19.0718	−0.774101	−0.387051	+	0.922059i	$0.626506\pi$
$608$	−0.246720	−0.0100058
$609$	0	0
$610$	25.9557	1.05092
$611$	60.0501	2.42937
$612$	0	0
$613$	−20.5471	−0.829891	−0.414945	−	0.909846i	$-0.636199\pi$
$613$	−20.5471	−0.829891	−0.414945	+	0.909846i	$0.636199\pi$
$614$	30.2914	1.22246
$615$	0	0
$616$	−139.364	−5.61515
$617$	13.7223	0.552438	0.276219	−	0.961095i	$-0.410918\pi$
$617$	13.7223	0.552438	0.276219	+	0.961095i	$0.410918\pi$
$618$	0	0
$619$	27.1177	1.08995	0.544976	−	0.838452i	$-0.316539\pi$
$619$	27.1177	1.08995	0.544976	+	0.838452i	$0.316539\pi$
$620$	37.8425	1.51979
$621$	0	0
$622$	28.9850	1.16219
$623$	−22.7214	−0.910313
$624$	0	0
$625$	−11.4094	−0.456374
$626$	48.6313	1.94370
$627$	0	0
$628$	0.0954734	0.00380980
$629$	−4.52376	−0.180374
$630$	0	0
$631$	−8.00036	−0.318489	−0.159245	−	0.987239i	$-0.550906\pi$
$631$	−8.00036	−0.318489	−0.159245	+	0.987239i	$0.550906\pi$
$632$	−70.4447	−2.80214
$633$	0	0
$634$	24.2313	0.962348
$635$	−38.9477	−1.54559
$636$	0	0
$637$	128.099	5.07546
$638$	13.7981	0.546271
$639$	0	0
$640$	34.4815	1.36300
$641$	21.8605	0.863436	0.431718	−	0.902009i	$-0.357908\pi$
$641$	21.8605	0.863436	0.431718	+	0.902009i	$0.357908\pi$
$642$	0	0
$643$	29.0946	1.14738	0.573689	−	0.819073i	$-0.305512\pi$
$643$	29.0946	1.14738	0.573689	+	0.819073i	$0.305512\pi$
$644$	−20.3059	−0.800166
$645$	0	0
$646$	1.28342	0.0504956
$647$	4.13054	0.162388	0.0811941	−	0.996698i	$-0.474127\pi$
$647$	4.13054	0.162388	0.0811941	+	0.996698i	$0.474127\pi$
$648$	0	0
$649$	−2.48526	−0.0975548
$650$	32.1528	1.26114
$651$	0	0
$652$	0.100296	0.00392789
$653$	−14.6791	−0.574436	−0.287218	−	0.957865i	$-0.592730\pi$
$653$	−14.6791	−0.574436	−0.287218	+	0.957865i	$0.592730\pi$
$654$	0	0
$655$	−3.35146	−0.130952
$656$	−25.3092	−0.988159
$657$	0	0
$658$	−111.365	−4.34145
$659$	8.51122	0.331550	0.165775	−	0.986164i	$-0.446987\pi$
$659$	8.51122	0.331550	0.165775	+	0.986164i	$0.446987\pi$
$660$	0	0
$661$	3.66553	0.142573	0.0712864	−	0.997456i	$-0.477290\pi$
$661$	3.66553	0.142573	0.0712864	+	0.997456i	$0.477290\pi$
$662$	24.6140	0.956651
$663$	0	0
$664$	−39.3021	−1.52522
$665$	−10.9343	−0.424016
$666$	0	0
$667$	1.00000	0.0387202
$668$	−29.1674	−1.12852
$669$	0	0
$670$	−26.8202	−1.03616
$671$	−34.3204	−1.32492
$672$	0	0
$673$	41.8994	1.61510	0.807551	−	0.589798i	$-0.200793\pi$
$673$	41.8994	1.61510	0.807551	+	0.589798i	$0.200793\pi$
$674$	−20.8364	−0.802589
$675$	0	0
$676$	128.413	4.93896
$677$	−25.4243	−0.977133	−0.488567	−	0.872527i	$-0.662480\pi$
$677$	−25.4243	−0.977133	−0.488567	+	0.872527i	$0.662480\pi$
$678$	0	0
$679$	−41.7609	−1.60264
$680$	−3.61077	−0.138467
$681$	0	0
$682$	−75.1868	−2.87905
$683$	−0.993670	−0.0380217	−0.0190109	−	0.999819i	$-0.506052\pi$
$683$	−0.993670	−0.0380217	−0.0190109	+	0.999819i	$0.506052\pi$
$684$	0	0
$685$	6.57911	0.251375
$686$	−150.218	−5.73535
$687$	0	0
$688$	−18.9529	−0.722571
$689$	−40.3281	−1.53638
$690$	0	0
$691$	−2.93531	−0.111664	−0.0558322	−	0.998440i	$-0.517781\pi$
$691$	−2.93531	−0.111664	−0.0558322	+	0.998440i	$0.517781\pi$
$692$	77.7445	2.95540
$693$	0	0
$694$	89.6851	3.40440
$695$	3.38092	0.128246
$696$	0	0
$697$	−2.79088	−0.105712
$698$	−39.2328	−1.48498
$699$	0	0
$700$	−39.6836	−1.49990
$701$	−26.7695	−1.01107	−0.505536	−	0.862806i	$-0.668705\pi$
$701$	−26.7695	−1.01107	−0.505536	+	0.862806i	$0.668705\pi$
$702$	0	0
$703$	12.9937	0.490065
$704$	−46.5212	−1.75333
$705$	0	0
$706$	−39.1484	−1.47337
$707$	89.7865	3.37677
$708$	0	0
$709$	−38.4277	−1.44318	−0.721591	−	0.692319i	$-0.756589\pi$
$709$	−38.4277	−1.44318	−0.721591	+	0.692319i	$0.756589\pi$
$710$	15.2052	0.570642
$711$	0	0
$712$	21.5513	0.807669
$713$	−5.44908	−0.204070
$714$	0	0
$715$	66.2583	2.47792
$716$	−24.2578	−0.906557
$717$	0	0
$718$	−55.4213	−2.06830
$719$	−9.36859	−0.349390	−0.174695	−	0.984623i	$-0.555894\pi$
$719$	−9.36859	−0.349390	−0.174695	+	0.984623i	$0.555894\pi$
$720$	0	0
$721$	59.7606	2.22560
$722$	42.7738	1.59187
$723$	0	0
$724$	23.9919	0.891650
$725$	1.95428	0.0725803
$726$	0	0
$727$	−3.00591	−0.111483	−0.0557415	−	0.998445i	$-0.517752\pi$
$727$	−3.00591	−0.111483	−0.0557415	+	0.998445i	$0.517752\pi$
$728$	−166.175	−6.15884
$729$	0	0
$730$	32.3988	1.19913
$731$	−2.08996	−0.0772999
$732$	0	0
$733$	3.57332	0.131984	0.0659919	−	0.997820i	$-0.478979\pi$
$733$	3.57332	0.131984	0.0659919	+	0.997820i	$0.478979\pi$
$734$	53.0008	1.95630
$735$	0	0
$736$	−0.200940	−0.00740676
$737$	35.4636	1.30632
$738$	0	0
$739$	29.3307	1.07895	0.539474	−	0.842003i	$-0.318623\pi$
$739$	29.3307	1.07895	0.539474	+	0.842003i	$0.318623\pi$
$740$	−73.4938	−2.70169
$741$	0	0
$742$	74.7895	2.74561
$743$	−11.5099	−0.422259	−0.211129	−	0.977458i	$-0.567714\pi$
$743$	−11.5099	−0.422259	−0.211129	+	0.977458i	$0.567714\pi$
$744$	0	0
$745$	23.9091	0.875961
$746$	0.346266	0.0126777
$747$	0	0
$748$	9.59862	0.350960
$749$	57.4921	2.10071
$750$	0	0
$751$	14.4430	0.527033	0.263517	−	0.964655i	$-0.415118\pi$
$751$	14.4430	0.527033	0.263517	+	0.964655i	$0.415118\pi$
$752$	34.5981	1.26166
$753$	0	0
$754$	16.4525	0.599164
$755$	−9.41944	−0.342808
$756$	0	0
$757$	23.0783	0.838796	0.419398	−	0.907803i	$-0.362241\pi$
$757$	23.0783	0.838796	0.419398	+	0.907803i	$0.362241\pi$
$758$	−50.5340	−1.83548
$759$	0	0
$760$	10.3713	0.376205
$761$	42.2443	1.53135	0.765677	−	0.643226i	$-0.222404\pi$
$761$	42.2443	1.53135	0.765677	+	0.643226i	$0.222404\pi$
$762$	0	0
$763$	−77.7351	−2.81420
$764$	−49.8436	−1.80328
$765$	0	0
$766$	−29.7800	−1.07600
$767$	−2.96336	−0.107001
$768$	0	0
$769$	44.6402	1.60977	0.804884	−	0.593433i	$-0.202228\pi$
$769$	44.6402	1.60977	0.804884	+	0.593433i	$0.202228\pi$
$770$	−122.878	−4.42821
$771$	0	0
$772$	39.4032	1.41815
$773$	29.2734	1.05289	0.526445	−	0.850209i	$-0.323525\pi$
$773$	29.2734	1.05289	0.526445	+	0.850209i	$0.323525\pi$
$774$	0	0
$775$	−10.6490	−0.382525
$776$	39.6104	1.42193
$777$	0	0
$778$	88.2001	3.16213
$779$	8.01630	0.287214
$780$	0	0
$781$	−20.1054	−0.719428
$782$	1.04528	0.0373791
$783$	0	0
$784$	73.8045	2.63588
$785$	0.0418712	0.00149445
$786$	0	0
$787$	−2.63680	−0.0939917	−0.0469959	−	0.998895i	$-0.514965\pi$
$787$	−2.63680	−0.0939917	−0.0469959	+	0.998895i	$0.514965\pi$
$788$	24.4312	0.870325
$789$	0	0
$790$	−62.1113	−2.20982
$791$	−42.9106	−1.52573
$792$	0	0
$793$	−40.9228	−1.45321
$794$	−72.8481	−2.58528
$795$	0	0
$796$	56.8476	2.01491
$797$	2.62808	0.0930912	0.0465456	−	0.998916i	$-0.485179\pi$
$797$	2.62808	0.0930912	0.0465456	+	0.998916i	$0.485179\pi$
$798$	0	0
$799$	3.81518	0.134971
$800$	−0.392695	−0.0138839
$801$	0	0
$802$	−41.8429	−1.47752
$803$	−42.8399	−1.51179
$804$	0	0
$805$	−8.90544	−0.313876
$806$	−89.6509	−3.15782
$807$	0	0
$808$	−85.1628	−2.99602
$809$	43.6439	1.53444	0.767219	−	0.641385i	$-0.221640\pi$
$809$	43.6439	1.53444	0.767219	+	0.641385i	$0.221640\pi$
$810$	0	0
$811$	−39.6254	−1.39143	−0.695717	−	0.718316i	$-0.744913\pi$
$811$	−39.6254	−1.39143	−0.695717	+	0.718316i	$0.744913\pi$
$812$	−20.3059	−0.712598
$813$	0	0
$814$	146.020	5.11800
$815$	0.0439861	0.00154077
$816$	0	0
$817$	6.00302	0.210019
$818$	5.63815	0.197133
$819$	0	0
$820$	−45.3412	−1.58338
$821$	7.02625	0.245218	0.122609	−	0.992455i	$-0.460874\pi$
$821$	7.02625	0.245218	0.122609	+	0.992455i	$0.460874\pi$
$822$	0	0
$823$	10.0297	0.349614	0.174807	−	0.984603i	$-0.444070\pi$
$823$	10.0297	0.349614	0.174807	+	0.984603i	$0.444070\pi$
$824$	−56.6831	−1.97465
$825$	0	0
$826$	5.49563	0.191217
$827$	−9.94286	−0.345747	−0.172874	−	0.984944i	$-0.555305\pi$
$827$	−9.94286	−0.345747	−0.172874	+	0.984944i	$0.555305\pi$
$828$	0	0
$829$	−34.6153	−1.20224	−0.601119	−	0.799159i	$-0.705278\pi$
$829$	−34.6153	−1.20224	−0.601119	+	0.799159i	$0.705278\pi$
$830$	−34.6527	−1.20281
$831$	0	0
$832$	−55.4707	−1.92310
$833$	8.13853	0.281983
$834$	0	0
$835$	−12.7918	−0.442677
$836$	−27.5703	−0.953537
$837$	0	0
$838$	33.0953	1.14326
$839$	−36.0858	−1.24582	−0.622909	−	0.782294i	$-0.714049\pi$
$839$	−36.0858	−1.24582	−0.622909	+	0.782294i	$0.714049\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−71.5246	−2.46490
$843$	0	0
$844$	34.5114	1.18793
$845$	56.3172	1.93737
$846$	0	0
$847$	106.346	3.65411
$848$	−23.2352	−0.797898
$849$	0	0
$850$	2.04277	0.0700665
$851$	10.5826	0.362768
$852$	0	0
$853$	−20.5219	−0.702656	−0.351328	−	0.936253i	$-0.614270\pi$
$853$	−20.5219	−0.702656	−0.351328	+	0.936253i	$0.614270\pi$
$854$	75.8925	2.59699
$855$	0	0
$856$	−54.5314	−1.86384
$857$	−40.8417	−1.39513	−0.697563	−	0.716523i	$-0.745732\pi$
$857$	−40.8417	−1.39513	−0.697563	+	0.716523i	$0.745732\pi$
$858$	0	0
$859$	−26.1071	−0.890762	−0.445381	−	0.895341i	$-0.646932\pi$
$859$	−26.1071	−0.890762	−0.445381	+	0.895341i	$0.646932\pi$
$860$	−33.9538	−1.15782
$861$	0	0
$862$	53.4532	1.82062
$863$	24.0875	0.819949	0.409974	−	0.912097i	$-0.365538\pi$
$863$	24.0875	0.819949	0.409974	+	0.912097i	$0.365538\pi$
$864$	0	0
$865$	34.0959	1.15930
$866$	17.3674	0.590169
$867$	0	0
$868$	110.649	3.75566
$869$	82.1278	2.78600
$870$	0	0
$871$	42.2859	1.43280
$872$	73.7319	2.49688
$873$	0	0
$874$	−3.00237	−0.101557
$875$	−61.9310	−2.09365
$876$	0	0
$877$	−22.3908	−0.756085	−0.378043	−	0.925788i	$-0.623403\pi$
$877$	−22.3908	−0.756085	−0.378043	+	0.925788i	$0.623403\pi$
$878$	69.9759	2.36157
$879$	0	0
$880$	38.1750	1.28688
$881$	−5.28714	−0.178128	−0.0890642	−	0.996026i	$-0.528388\pi$
$881$	−5.28714	−0.178128	−0.0890642	+	0.996026i	$0.528388\pi$
$882$	0	0
$883$	−15.6717	−0.527395	−0.263697	−	0.964605i	$-0.584942\pi$
$883$	−15.6717	−0.527395	−0.263697	+	0.964605i	$0.584942\pi$
$884$	11.4452	0.384942
$885$	0	0
$886$	41.8653	1.40649
$887$	9.06847	0.304489	0.152245	−	0.988343i	$-0.451350\pi$
$887$	9.06847	0.304489	0.152245	+	0.988343i	$0.451350\pi$
$888$	0	0
$889$	−113.880	−3.81941
$890$	19.0018	0.636943
$891$	0	0
$892$	−27.5177	−0.921360
$893$	−10.9584	−0.366709
$894$	0	0
$895$	−10.6386	−0.355609
$896$	100.821	3.36820
$897$	0	0
$898$	35.4109	1.18168
$899$	−5.44908	−0.181737
$900$	0	0
$901$	−2.56217	−0.0853584
$902$	90.0854	2.99952
$903$	0	0
$904$	40.7009	1.35369
$905$	10.5220	0.349762
$906$	0	0
$907$	31.3343	1.04044	0.520219	−	0.854033i	$-0.325850\pi$
$907$	31.3343	1.04044	0.520219	+	0.854033i	$0.325850\pi$
$908$	40.1211	1.33146
$909$	0	0
$910$	−146.517	−4.85698
$911$	−20.8240	−0.689929	−0.344965	−	0.938616i	$-0.612109\pi$
$911$	−20.8240	−0.689929	−0.344965	+	0.938616i	$0.612109\pi$
$912$	0	0
$913$	45.8202	1.51643
$914$	−68.0156	−2.24976
$915$	0	0
$916$	−40.9661	−1.35356
$917$	−9.79940	−0.323605
$918$	0	0
$919$	48.3596	1.59524	0.797618	−	0.603163i	$-0.206093\pi$
$919$	48.3596	1.59524	0.797618	+	0.603163i	$0.206093\pi$
$920$	8.44684	0.278484
$921$	0	0
$922$	37.2422	1.22651
$923$	−23.9732	−0.789087
$924$	0	0
$925$	20.6815	0.680003
$926$	−63.9473	−2.10144
$927$	0	0
$928$	−0.200940	−0.00659619
$929$	−37.0018	−1.21399	−0.606995	−	0.794706i	$-0.707625\pi$
$929$	−37.0018	−1.21399	−0.606995	+	0.794706i	$0.707625\pi$
$930$	0	0
$931$	−23.3764	−0.766131
$932$	−15.7429	−0.515675
$933$	0	0
$934$	−43.1409	−1.41161
$935$	4.20961	0.137669
$936$	0	0
$937$	−45.0999	−1.47335	−0.736675	−	0.676247i	$-0.763605\pi$
$937$	−45.0999	−1.47335	−0.736675	+	0.676247i	$0.763605\pi$
$938$	−78.4203	−2.56051
$939$	0	0
$940$	61.9821	2.02163
$941$	−2.74512	−0.0894883	−0.0447442	−	0.998998i	$-0.514247\pi$
$941$	−2.74512	−0.0894883	−0.0447442	+	0.998998i	$0.514247\pi$
$942$	0	0
$943$	6.52884	0.212608
$944$	−1.70735	−0.0555695
$945$	0	0
$946$	67.4606	2.19333
$947$	22.5531	0.732876	0.366438	−	0.930442i	$-0.380577\pi$
$947$	22.5531	0.732876	0.366438	+	0.930442i	$0.380577\pi$
$948$	0	0
$949$	−51.0812	−1.65817
$950$	−5.86749	−0.190366
$951$	0	0
$952$	−10.5576	−0.342174
$953$	46.4313	1.50406	0.752029	−	0.659130i	$-0.229075\pi$
$953$	46.4313	1.50406	0.752029	+	0.659130i	$0.229075\pi$
$954$	0	0
$955$	−21.8596	−0.707360
$956$	116.950	3.78244
$957$	0	0
$958$	−12.5366	−0.405040
$959$	19.2368	0.621190
$960$	0	0
$961$	−1.30753	−0.0421783
$962$	174.111	5.61355
$963$	0	0
$964$	−53.6672	−1.72850
$965$	17.2808	0.556289
$966$	0	0
$967$	40.8729	1.31439	0.657193	−	0.753723i	$-0.271744\pi$
$967$	40.8729	1.31439	0.657193	+	0.753723i	$0.271744\pi$
$968$	−100.870	−3.24208
$969$	0	0
$970$	34.9246	1.12136
$971$	−41.7944	−1.34124	−0.670622	−	0.741799i	$-0.733973\pi$
$971$	−41.7944	−1.34124	−0.670622	+	0.741799i	$0.733973\pi$
$972$	0	0
$973$	9.88555	0.316916
$974$	−14.9285	−0.478341
$975$	0	0
$976$	−23.5778	−0.754707
$977$	−17.6158	−0.563581	−0.281790	−	0.959476i	$-0.590928\pi$
$977$	−17.6158	−0.563581	−0.281790	+	0.959476i	$0.590928\pi$
$978$	0	0
$979$	−25.1255	−0.803016
$980$	132.220	4.22361
$981$	0	0
$982$	12.2526	0.390996
$983$	3.62572	0.115642	0.0578212	−	0.998327i	$-0.481585\pi$
$983$	3.62572	0.115642	0.0578212	+	0.998327i	$0.481585\pi$
$984$	0	0
$985$	10.7146	0.341397
$986$	1.04528	0.0332885
$987$	0	0
$988$	−32.8741	−1.04586
$989$	4.88914	0.155465
$990$	0	0
$991$	27.8298	0.884044	0.442022	−	0.897004i	$-0.354261\pi$
$991$	27.8298	0.884044	0.442022	+	0.897004i	$0.354261\pi$
$992$	1.09494	0.0347644
$993$	0	0
$994$	44.4590	1.41015
$995$	24.9313	0.790375
$996$	0	0
$997$	−54.2859	−1.71925	−0.859626	−	0.510923i	$-0.829304\pi$
$997$	−54.2859	−1.71925	−0.859626	+	0.510923i	$0.829304\pi$
$998$	−65.8599	−2.08476
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.s.1.4		20
3.2	odd	2		2001.2.a.o.1.17	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.o.1.17	✓	20	3.2	odd	2
6003.2.a.s.1.4		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)

\(f(q)\)	\(=\)	\(q-2.44527 q^{2} +3.97935 q^{4} +1.74520 q^{5} +5.10283 q^{7} -4.84005 q^{8} +O(q^{10})\)	\(q-2.44527 q^{2} +3.97935 q^{4} +1.74520 q^{5} +5.10283 q^{7} -4.84005 q^{8} -4.26748 q^{10} +5.64276 q^{11} +6.72828 q^{13} -12.4778 q^{14} +3.87652 q^{16} +0.427470 q^{17} -1.22783 q^{19} +6.94475 q^{20} -13.7981 q^{22} -1.00000 q^{23} -1.95428 q^{25} -16.4525 q^{26} +20.3059 q^{28} -1.00000 q^{29} +5.44908 q^{31} +0.200940 q^{32} -1.04528 q^{34} +8.90544 q^{35} -10.5826 q^{37} +3.00237 q^{38} -8.44684 q^{40} -6.52884 q^{41} -4.88914 q^{43} +22.4545 q^{44} +2.44527 q^{46} +8.92503 q^{47} +19.0388 q^{49} +4.77875 q^{50} +26.7742 q^{52} -5.99381 q^{53} +9.84773 q^{55} -24.6979 q^{56} +2.44527 q^{58} -0.440433 q^{59} -6.08220 q^{61} -13.3245 q^{62} -8.24440 q^{64} +11.7422 q^{65} +6.28479 q^{67} +1.70105 q^{68} -21.7762 q^{70} -3.56305 q^{71} -7.59201 q^{73} +25.8774 q^{74} -4.88596 q^{76} +28.7940 q^{77} +14.5546 q^{79} +6.76530 q^{80} +15.9648 q^{82} +8.12019 q^{83} +0.746019 q^{85} +11.9553 q^{86} -27.3112 q^{88} -4.45271 q^{89} +34.3333 q^{91} -3.97935 q^{92} -21.8241 q^{94} -2.14280 q^{95} -8.18389 q^{97} -46.5551 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8}+O(q^{10}) \) 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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