LMFDB - Embedded newform 6003.2.a.j.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:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 5x^{5} + 18x^{4} + 4x^{3} - 26x^{2} + x + 8 $ x^7 - 3x^6 - 5x^5 + 18x^4 + 4x^3 - 26*x^2 + x + 8
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.21072$ 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)

$f(q)$	$=$	$q-1.21072 q^{2} -0.534159 q^{4} -3.03795 q^{5} -3.40847 q^{7} +3.06816 q^{8} +O(q^{10})$	\(q-1.21072 q^{2} -0.534159 q^{4} -3.03795 q^{5} -3.40847 q^{7} +3.06816 q^{8} +3.67810 q^{10} +3.85707 q^{11} -5.15691 q^{13} +4.12670 q^{14} -2.64635 q^{16} +2.07488 q^{17} -0.0953170 q^{19} +1.62275 q^{20} -4.66983 q^{22} -1.00000 q^{23} +4.22914 q^{25} +6.24357 q^{26} +1.82067 q^{28} +1.00000 q^{29} -3.70527 q^{31} -2.93232 q^{32} -2.51210 q^{34} +10.3548 q^{35} -8.24097 q^{37} +0.115402 q^{38} -9.32090 q^{40} -4.31059 q^{41} +2.81904 q^{43} -2.06029 q^{44} +1.21072 q^{46} +2.48043 q^{47} +4.61770 q^{49} -5.12030 q^{50} +2.75461 q^{52} +6.40448 q^{53} -11.7176 q^{55} -10.4577 q^{56} -1.21072 q^{58} +10.8564 q^{59} +5.67822 q^{61} +4.48604 q^{62} +8.84292 q^{64} +15.6664 q^{65} +12.8177 q^{67} -1.10832 q^{68} -12.5367 q^{70} +2.25971 q^{71} +14.0533 q^{73} +9.97750 q^{74} +0.0509145 q^{76} -13.1467 q^{77} +0.679730 q^{79} +8.03950 q^{80} +5.21891 q^{82} -4.99221 q^{83} -6.30339 q^{85} -3.41307 q^{86} +11.8341 q^{88} -8.87404 q^{89} +17.5772 q^{91} +0.534159 q^{92} -3.00310 q^{94} +0.289568 q^{95} +12.0257 q^{97} -5.59073 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 3 q^{2} + 5 q^{4} + 3 q^{5} - 5 q^{7} + 6 q^{8}+O(q^{10}) $ 7 * q + 3 * q^2 + 5 * q^4 + 3 * q^5 - 5 * q^7 + 6 * q^8	$ 7 q + 3 q^{2} + 5 q^{4} + 3 q^{5} - 5 q^{7} + 6 q^{8} + 3 q^{10} + 4 q^{11} - 18 q^{13} + 2 q^{14} - 7 q^{16} + 3 q^{17} - 4 q^{19} + 2 q^{20} - 26 q^{22} - 7 q^{23} - 8 q^{25} + 7 q^{26} - 6 q^{28} + 7 q^{29} - 22 q^{31} - 5 q^{32} + 9 q^{34} - 3 q^{35} - 25 q^{37} - 14 q^{38} - 10 q^{40} + 13 q^{41} - 2 q^{43} - 4 q^{44} - 3 q^{46} + 25 q^{47} - 8 q^{49} - 19 q^{50} - 12 q^{52} + 5 q^{53} - 15 q^{55} - 18 q^{56} + 3 q^{58} - 11 q^{59} - 33 q^{61} - 28 q^{62} - 14 q^{64} + 2 q^{65} + 8 q^{67} - 12 q^{68} - 22 q^{70} + 6 q^{71} + 15 q^{73} - 34 q^{74} - 28 q^{76} + q^{77} - 15 q^{79} + 12 q^{80} - 14 q^{82} - 21 q^{83} - 28 q^{85} + 12 q^{86} - 13 q^{88} - 8 q^{89} + 6 q^{91} - 5 q^{92} - 35 q^{94} + 25 q^{95} + 13 q^{97} - q^{98}+O(q^{100}) $ 7 * q + 3 * q^2 + 5 * q^4 + 3 * q^5 - 5 * q^7 + 6 * q^8 + 3 * q^10 + 4 * q^11 - 18 * q^13 + 2 * q^14 - 7 * q^16 + 3 * q^17 - 4 * q^19 + 2 * q^20 - 26 * q^22 - 7 * q^23 - 8 * q^25 + 7 * q^26 - 6 * q^28 + 7 * q^29 - 22 * q^31 - 5 * q^32 + 9 * q^34 - 3 * q^35 - 25 * q^37 - 14 * q^38 - 10 * q^40 + 13 * q^41 - 2 * q^43 - 4 * q^44 - 3 * q^46 + 25 * q^47 - 8 * q^49 - 19 * q^50 - 12 * q^52 + 5 * q^53 - 15 * q^55 - 18 * q^56 + 3 * q^58 - 11 * q^59 - 33 * q^61 - 28 * q^62 - 14 * q^64 + 2 * q^65 + 8 * q^67 - 12 * q^68 - 22 * q^70 + 6 * q^71 + 15 * q^73 - 34 * q^74 - 28 * q^76 + q^77 - 15 * q^79 + 12 * q^80 - 14 * q^82 - 21 * q^83 - 28 * q^85 + 12 * q^86 - 13 * q^88 - 8 * q^89 + 6 * q^91 - 5 * q^92 - 35 * q^94 + 25 * q^95 + 13 * q^97 - 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 $

$2$	−1.21072	−0.856108	−0.428054	−	0.903753i	$-0.640801\pi$
$2$	−1.21072	−0.856108	−0.428054	+	0.903753i	$0.640801\pi$
$3$	0	0
$3$	0	0
$4$	−0.534159	−0.267080
$5$	−3.03795	−1.35861	−0.679306	−	0.733855i	$-0.737719\pi$
$5$	−3.03795	−1.35861	−0.679306	+	0.733855i	$0.737719\pi$
$6$	0	0
$7$	−3.40847	−1.28828	−0.644141	−	0.764907i	$-0.722785\pi$
$7$	−3.40847	−1.28828	−0.644141	+	0.764907i	$0.722785\pi$
$8$	3.06816	1.08476
$9$	0	0
$10$	3.67810	1.16312
$11$	3.85707	1.16295	0.581476	−	0.813564i	$-0.302476\pi$
$11$	3.85707	1.16295	0.581476	+	0.813564i	$0.302476\pi$
$12$	0	0
$13$	−5.15691	−1.43027	−0.715135	−	0.698987i	$-0.753635\pi$
$13$	−5.15691	−1.43027	−0.715135	+	0.698987i	$0.753635\pi$
$14$	4.12670	1.10291
$15$	0	0
$16$	−2.64635	−0.661589
$17$	2.07488	0.503233	0.251616	−	0.967827i	$-0.419038\pi$
$17$	2.07488	0.503233	0.251616	+	0.967827i	$0.419038\pi$
$18$	0	0
$19$	−0.0953170	−0.0218672	−0.0109336	−	0.999940i	$-0.503480\pi$
$19$	−0.0953170	−0.0218672	−0.0109336	+	0.999940i	$0.503480\pi$
$20$	1.62275	0.362858
$21$	0	0
$22$	−4.66983	−0.995612
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	4.22914	0.845828
$26$	6.24357	1.22446
$27$	0	0
$28$	1.82067	0.344074
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−3.70527	−0.665485	−0.332743	−	0.943018i	$-0.607974\pi$
$31$	−3.70527	−0.665485	−0.332743	+	0.943018i	$0.607974\pi$
$32$	−2.93232	−0.518365
$33$	0	0
$34$	−2.51210	−0.430821
$35$	10.3548	1.75028
$36$	0	0
$37$	−8.24097	−1.35481	−0.677404	−	0.735612i	$-0.736895\pi$
$37$	−8.24097	−1.35481	−0.677404	+	0.735612i	$0.736895\pi$
$38$	0.115402	0.0187207
$39$	0	0
$40$	−9.32090	−1.47376
$41$	−4.31059	−0.673201	−0.336600	−	0.941648i	$-0.609277\pi$
$41$	−4.31059	−0.673201	−0.336600	+	0.941648i	$0.609277\pi$
$42$	0	0
$43$	2.81904	0.429900	0.214950	−	0.976625i	$-0.431041\pi$
$43$	2.81904	0.429900	0.214950	+	0.976625i	$0.431041\pi$
$44$	−2.06029	−0.310601
$45$	0	0
$46$	1.21072	0.178511
$47$	2.48043	0.361808	0.180904	−	0.983501i	$-0.442098\pi$
$47$	2.48043	0.361808	0.180904	+	0.983501i	$0.442098\pi$
$48$	0	0
$49$	4.61770	0.659671
$50$	−5.12030	−0.724120
$51$	0	0
$52$	2.75461	0.381996
$53$	6.40448	0.879723	0.439861	−	0.898066i	$-0.355028\pi$
$53$	6.40448	0.879723	0.439861	+	0.898066i	$0.355028\pi$
$54$	0	0
$55$	−11.7176	−1.58000
$56$	−10.4577	−1.39747
$57$	0	0
$58$	−1.21072	−0.158975
$59$	10.8564	1.41338	0.706692	−	0.707521i	$-0.250187\pi$
$59$	10.8564	1.41338	0.706692	+	0.707521i	$0.250187\pi$
$60$	0	0
$61$	5.67822	0.727021	0.363510	−	0.931590i	$-0.381578\pi$
$61$	5.67822	0.727021	0.363510	+	0.931590i	$0.381578\pi$
$62$	4.48604	0.569727
$63$	0	0
$64$	8.84292	1.10537
$65$	15.6664	1.94318
$66$	0	0
$67$	12.8177	1.56593	0.782963	−	0.622068i	$-0.213707\pi$
$67$	12.8177	1.56593	0.782963	+	0.622068i	$0.213707\pi$
$68$	−1.10832	−0.134403
$69$	0	0
$70$	−12.5367	−1.49843
$71$	2.25971	0.268178	0.134089	−	0.990969i	$-0.457189\pi$
$71$	2.25971	0.268178	0.134089	+	0.990969i	$0.457189\pi$
$72$	0	0
$73$	14.0533	1.64482	0.822408	−	0.568898i	$-0.192630\pi$
$73$	14.0533	1.64482	0.822408	+	0.568898i	$0.192630\pi$
$74$	9.97750	1.15986
$75$	0	0
$76$	0.0509145	0.00584029
$77$	−13.1467	−1.49821
$78$	0	0
$79$	0.679730	0.0764756	0.0382378	−	0.999269i	$-0.487826\pi$
$79$	0.679730	0.0764756	0.0382378	+	0.999269i	$0.487826\pi$
$80$	8.03950	0.898843
$81$	0	0
$82$	5.21891	0.576332
$83$	−4.99221	−0.547967	−0.273983	−	0.961734i	$-0.588341\pi$
$83$	−4.99221	−0.547967	−0.273983	+	0.961734i	$0.588341\pi$
$84$	0	0
$85$	−6.30339	−0.683698
$86$	−3.41307	−0.368041
$87$	0	0
$88$	11.8341	1.26152
$89$	−8.87404	−0.940646	−0.470323	−	0.882494i	$-0.655863\pi$
$89$	−8.87404	−0.940646	−0.470323	+	0.882494i	$0.655863\pi$
$90$	0	0
$91$	17.5772	1.84259
$92$	0.534159	0.0556900
$93$	0	0
$94$	−3.00310	−0.309747
$95$	0.289568	0.0297091
$96$	0	0
$97$	12.0257	1.22102	0.610511	−	0.792007i	$-0.290964\pi$
$97$	12.0257	1.22102	0.610511	+	0.792007i	$0.290964\pi$
$98$	−5.59073	−0.564749
$99$	0	0
$100$	−2.25904	−0.225904
$101$	12.8874	1.28234	0.641172	−	0.767397i	$-0.278448\pi$
$101$	12.8874	1.28234	0.641172	+	0.767397i	$0.278448\pi$
$102$	0	0
$103$	−16.6608	−1.64164	−0.820818	−	0.571190i	$-0.806482\pi$
$103$	−16.6608	−1.64164	−0.820818	+	0.571190i	$0.806482\pi$
$104$	−15.8222	−1.55149
$105$	0	0
$106$	−7.75402	−0.753137
$107$	3.34894	0.323754	0.161877	−	0.986811i	$-0.448245\pi$
$107$	3.34894	0.323754	0.161877	+	0.986811i	$0.448245\pi$
$108$	0	0
$109$	1.99607	0.191189	0.0955944	−	0.995420i	$-0.469525\pi$
$109$	1.99607	0.191189	0.0955944	+	0.995420i	$0.469525\pi$
$110$	14.1867	1.35265
$111$	0	0
$112$	9.02003	0.852313
$113$	7.18948	0.676330	0.338165	−	0.941087i	$-0.390194\pi$
$113$	7.18948	0.676330	0.338165	+	0.941087i	$0.390194\pi$
$114$	0	0
$115$	3.03795	0.283290
$116$	−0.534159	−0.0495955
$117$	0	0
$118$	−13.1441	−1.21001
$119$	−7.07218	−0.648306
$120$	0	0
$121$	3.87702	0.352456
$122$	−6.87472	−0.622408
$123$	0	0
$124$	1.97920	0.177738
$125$	2.34183	0.209459
$126$	0	0
$127$	3.54176	0.314280	0.157140	−	0.987576i	$-0.449773\pi$
$127$	3.54176	0.314280	0.157140	+	0.987576i	$0.449773\pi$
$128$	−4.84166	−0.427946
$129$	0	0
$130$	−18.9676	−1.66357
$131$	4.63906	0.405316	0.202658	−	0.979250i	$-0.435042\pi$
$131$	4.63906	0.405316	0.202658	+	0.979250i	$0.435042\pi$
$132$	0	0
$133$	0.324885	0.0281711
$134$	−15.5186	−1.34060
$135$	0	0
$136$	6.36606	0.545885
$137$	16.9239	1.44591	0.722954	−	0.690897i	$-0.242784\pi$
$137$	16.9239	1.44591	0.722954	+	0.690897i	$0.242784\pi$
$138$	0	0
$139$	6.59644	0.559503	0.279751	−	0.960072i	$-0.409748\pi$
$139$	6.59644	0.559503	0.279751	+	0.960072i	$0.409748\pi$
$140$	−5.53110	−0.467463
$141$	0	0
$142$	−2.73587	−0.229589
$143$	−19.8906	−1.66333
$144$	0	0
$145$	−3.03795	−0.252288
$146$	−17.0146	−1.40814
$147$	0	0
$148$	4.40199	0.361841
$149$	−3.08255	−0.252532	−0.126266	−	0.991996i	$-0.540299\pi$
$149$	−3.08255	−0.252532	−0.126266	+	0.991996i	$0.540299\pi$
$150$	0	0
$151$	−11.3917	−0.927046	−0.463523	−	0.886085i	$-0.653415\pi$
$151$	−11.3917	−0.927046	−0.463523	+	0.886085i	$0.653415\pi$
$152$	−0.292447	−0.0237206
$153$	0	0
$154$	15.9170	1.28263
$155$	11.2564	0.904137
$156$	0	0
$157$	−7.45670	−0.595109	−0.297555	−	0.954705i	$-0.596171\pi$
$157$	−7.45670	−0.595109	−0.297555	+	0.954705i	$0.596171\pi$
$158$	−0.822962	−0.0654713
$159$	0	0
$160$	8.90824	0.704258
$161$	3.40847	0.268625
$162$	0	0
$163$	−16.9258	−1.32573	−0.662865	−	0.748739i	$-0.730660\pi$
$163$	−16.9258	−1.32573	−0.662865	+	0.748739i	$0.730660\pi$
$164$	2.30254	0.179798
$165$	0	0
$166$	6.04417	0.469118
$167$	−5.32428	−0.412005	−0.206002	−	0.978551i	$-0.566045\pi$
$167$	−5.32428	−0.412005	−0.206002	+	0.978551i	$0.566045\pi$
$168$	0	0
$169$	13.5937	1.04567
$170$	7.63163	0.585319
$171$	0	0
$172$	−1.50582	−0.114818
$173$	−24.7597	−1.88244	−0.941222	−	0.337789i	$-0.890321\pi$
$173$	−24.7597	−1.88244	−0.941222	+	0.337789i	$0.890321\pi$
$174$	0	0
$175$	−14.4149	−1.08967
$176$	−10.2072	−0.769396
$177$	0	0
$178$	10.7440	0.805294
$179$	6.67712	0.499071	0.249536	−	0.968366i	$-0.419722\pi$
$179$	6.67712	0.499071	0.249536	+	0.968366i	$0.419722\pi$
$180$	0	0
$181$	10.7524	0.799219	0.399609	−	0.916686i	$-0.369146\pi$
$181$	10.7524	0.799219	0.399609	+	0.916686i	$0.369146\pi$
$182$	−21.2810	−1.57746
$183$	0	0
$184$	−3.06816	−0.226187
$185$	25.0357	1.84066
$186$	0	0
$187$	8.00297	0.585235
$188$	−1.32494	−0.0966315
$189$	0	0
$190$	−0.350586	−0.0254342
$191$	9.96489	0.721034	0.360517	−	0.932753i	$-0.382600\pi$
$191$	9.96489	0.721034	0.360517	+	0.932753i	$0.382600\pi$
$192$	0	0
$193$	5.72373	0.412003	0.206002	−	0.978552i	$-0.433955\pi$
$193$	5.72373	0.412003	0.206002	+	0.978552i	$0.433955\pi$
$194$	−14.5597	−1.04533
$195$	0	0
$196$	−2.46659	−0.176185
$197$	7.92245	0.564451	0.282226	−	0.959348i	$-0.408927\pi$
$197$	7.92245	0.564451	0.282226	+	0.959348i	$0.408927\pi$
$198$	0	0
$199$	−16.2857	−1.15446	−0.577230	−	0.816581i	$-0.695866\pi$
$199$	−16.2857	−1.15446	−0.577230	+	0.816581i	$0.695866\pi$
$200$	12.9757	0.917518
$201$	0	0
$202$	−15.6030	−1.09782
$203$	−3.40847	−0.239228
$204$	0	0
$205$	13.0953	0.914619
$206$	20.1715	1.40542
$207$	0	0
$208$	13.6470	0.946250
$209$	−0.367645	−0.0254305
$210$	0	0
$211$	−8.49660	−0.584930	−0.292465	−	0.956276i	$-0.594476\pi$
$211$	−8.49660	−0.584930	−0.292465	+	0.956276i	$0.594476\pi$
$212$	−3.42101	−0.234956
$213$	0	0
$214$	−4.05462	−0.277168
$215$	−8.56411	−0.584068
$216$	0	0
$217$	12.6293	0.857333
$218$	−2.41668	−0.163678
$219$	0	0
$220$	6.25907	0.421986
$221$	−10.7000	−0.719758
$222$	0	0
$223$	−22.8578	−1.53067	−0.765334	−	0.643633i	$-0.777426\pi$
$223$	−22.8578	−1.53067	−0.765334	+	0.643633i	$0.777426\pi$
$224$	9.99473	0.667801
$225$	0	0
$226$	−8.70444	−0.579011
$227$	7.74967	0.514363	0.257182	−	0.966363i	$-0.417206\pi$
$227$	7.74967	0.514363	0.257182	+	0.966363i	$0.417206\pi$
$228$	0	0
$229$	−22.2253	−1.46869	−0.734344	−	0.678778i	$-0.762510\pi$
$229$	−22.2253	−1.46869	−0.734344	+	0.678778i	$0.762510\pi$
$230$	−3.67810	−0.242527
$231$	0	0
$232$	3.06816	0.201434
$233$	−10.7454	−0.703954	−0.351977	−	0.936009i	$-0.614490\pi$
$233$	−10.7454	−0.703954	−0.351977	+	0.936009i	$0.614490\pi$
$234$	0	0
$235$	−7.53542	−0.491557
$236$	−5.79905	−0.377486
$237$	0	0
$238$	8.56242	0.555019
$239$	13.8541	0.896146	0.448073	−	0.893997i	$-0.352110\pi$
$239$	13.8541	0.896146	0.448073	+	0.893997i	$0.352110\pi$
$240$	0	0
$241$	10.4272	0.671676	0.335838	−	0.941920i	$-0.390980\pi$
$241$	10.4272	0.671676	0.335838	+	0.941920i	$0.390980\pi$
$242$	−4.69398	−0.301741
$243$	0	0
$244$	−3.03307	−0.194173
$245$	−14.0283	−0.896237
$246$	0	0
$247$	0.491541	0.0312760
$248$	−11.3683	−0.721890
$249$	0	0
$250$	−2.83529	−0.179320
$251$	−1.35842	−0.0857428	−0.0428714	−	0.999081i	$-0.513651\pi$
$251$	−1.35842	−0.0857428	−0.0428714	+	0.999081i	$0.513651\pi$
$252$	0	0
$253$	−3.85707	−0.242492
$254$	−4.28807	−0.269058
$255$	0	0
$256$	−11.8240	−0.738997
$257$	−5.57472	−0.347741	−0.173871	−	0.984768i	$-0.555627\pi$
$257$	−5.57472	−0.347741	−0.173871	+	0.984768i	$0.555627\pi$
$258$	0	0
$259$	28.0891	1.74537
$260$	−8.36837	−0.518984
$261$	0	0
$262$	−5.61660	−0.346994
$263$	−31.2075	−1.92433	−0.962167	−	0.272460i	$-0.912163\pi$
$263$	−31.2075	−1.92433	−0.962167	+	0.272460i	$0.912163\pi$
$264$	0	0
$265$	−19.4565	−1.19520
$266$	−0.393345	−0.0241175
$267$	0	0
$268$	−6.84667	−0.418227
$269$	−15.0114	−0.915259	−0.457629	−	0.889143i	$-0.651301\pi$
$269$	−15.0114	−0.915259	−0.457629	+	0.889143i	$0.651301\pi$
$270$	0	0
$271$	−16.3564	−0.993584	−0.496792	−	0.867870i	$-0.665489\pi$
$271$	−16.3564	−0.993584	−0.496792	+	0.867870i	$0.665489\pi$
$272$	−5.49087	−0.332933
$273$	0	0
$274$	−20.4901	−1.23785
$275$	16.3121	0.983658
$276$	0	0
$277$	21.8350	1.31194	0.655969	−	0.754788i	$-0.272260\pi$
$277$	21.8350	1.31194	0.655969	+	0.754788i	$0.272260\pi$
$278$	−7.98644	−0.478995
$279$	0	0
$280$	31.7701	1.89862
$281$	−24.1432	−1.44026	−0.720130	−	0.693839i	$-0.755918\pi$
$281$	−24.1432	−1.44026	−0.720130	+	0.693839i	$0.755918\pi$
$282$	0	0
$283$	−26.8035	−1.59330	−0.796651	−	0.604439i	$-0.793397\pi$
$283$	−26.8035	−1.59330	−0.796651	+	0.604439i	$0.793397\pi$
$284$	−1.20704	−0.0716249
$285$	0	0
$286$	24.0819	1.42399
$287$	14.6925	0.867272
$288$	0	0
$289$	−12.6949	−0.746757
$290$	3.67810	0.215986
$291$	0	0
$292$	−7.50671	−0.439297
$293$	−13.5767	−0.793156	−0.396578	−	0.918001i	$-0.629802\pi$
$293$	−13.5767	−0.793156	−0.396578	+	0.918001i	$0.629802\pi$
$294$	0	0
$295$	−32.9812	−1.92024
$296$	−25.2846	−1.46964
$297$	0	0
$298$	3.73210	0.216195
$299$	5.15691	0.298232
$300$	0	0
$301$	−9.60863	−0.553833
$302$	13.7922	0.793651
$303$	0	0
$304$	0.252243	0.0144671
$305$	−17.2501	−0.987740
$306$	0	0
$307$	−19.6964	−1.12414	−0.562068	−	0.827091i	$-0.689994\pi$
$307$	−19.6964	−1.12414	−0.562068	+	0.827091i	$0.689994\pi$
$308$	7.02245	0.400141
$309$	0	0
$310$	−13.6284	−0.774039
$311$	29.6322	1.68029	0.840143	−	0.542364i	$-0.182471\pi$
$311$	29.6322	1.68029	0.840143	+	0.542364i	$0.182471\pi$
$312$	0	0
$313$	−23.8630	−1.34882	−0.674408	−	0.738359i	$-0.735601\pi$
$313$	−23.8630	−1.34882	−0.674408	+	0.738359i	$0.735601\pi$
$314$	9.02797	0.509478
$315$	0	0
$316$	−0.363084	−0.0204251
$317$	−27.3388	−1.53550	−0.767751	−	0.640748i	$-0.778624\pi$
$317$	−27.3388	−1.53550	−0.767751	+	0.640748i	$0.778624\pi$
$318$	0	0
$319$	3.85707	0.215955
$320$	−26.8644	−1.50176
$321$	0	0
$322$	−4.12670	−0.229972
$323$	−0.197771	−0.0110043
$324$	0	0
$325$	−21.8093	−1.20976
$326$	20.4924	1.13497
$327$	0	0
$328$	−13.2255	−0.730259
$329$	−8.45448	−0.466111
$330$	0	0
$331$	33.9104	1.86388	0.931942	−	0.362607i	$-0.118113\pi$
$331$	33.9104	1.86388	0.931942	+	0.362607i	$0.118113\pi$
$332$	2.66664	0.146351
$333$	0	0
$334$	6.44620	0.352721
$335$	−38.9394	−2.12749
$336$	0	0
$337$	7.35469	0.400636	0.200318	−	0.979731i	$-0.435803\pi$
$337$	7.35469	0.400636	0.200318	+	0.979731i	$0.435803\pi$
$338$	−16.4582	−0.895206
$339$	0	0
$340$	3.36701	0.182602
$341$	−14.2915	−0.773927
$342$	0	0
$343$	8.12002	0.438440
$344$	8.64926	0.466337
$345$	0	0
$346$	29.9770	1.61157
$347$	−11.9665	−0.642394	−0.321197	−	0.947012i	$-0.604085\pi$
$347$	−11.9665	−0.642394	−0.321197	+	0.947012i	$0.604085\pi$
$348$	0	0
$349$	21.1613	1.13274	0.566368	−	0.824152i	$-0.308348\pi$
$349$	21.1613	1.13274	0.566368	+	0.824152i	$0.308348\pi$
$350$	17.4524	0.932871
$351$	0	0
$352$	−11.3102	−0.602834
$353$	29.9515	1.59416	0.797080	−	0.603874i	$-0.206377\pi$
$353$	29.9515	1.59416	0.797080	+	0.603874i	$0.206377\pi$
$354$	0	0
$355$	−6.86488	−0.364350
$356$	4.74015	0.251227
$357$	0	0
$358$	−8.08411	−0.427259
$359$	33.8970	1.78901	0.894507	−	0.447053i	$-0.147527\pi$
$359$	33.8970	1.78901	0.894507	+	0.447053i	$0.147527\pi$
$360$	0	0
$361$	−18.9909	−0.999522
$362$	−13.0181	−0.684217
$363$	0	0
$364$	−9.38902	−0.492118
$365$	−42.6933	−2.23467
$366$	0	0
$367$	−1.12298	−0.0586189	−0.0293094	−	0.999570i	$-0.509331\pi$
$367$	−1.12298	−0.0586189	−0.0293094	+	0.999570i	$0.509331\pi$
$368$	2.64635	0.137951
$369$	0	0
$370$	−30.3111	−1.57580
$371$	−21.8295	−1.13333
$372$	0	0
$373$	11.0715	0.573261	0.286630	−	0.958041i	$-0.407465\pi$
$373$	11.0715	0.573261	0.286630	+	0.958041i	$0.407465\pi$
$374$	−9.68935	−0.501024
$375$	0	0
$376$	7.61034	0.392474
$377$	−5.15691	−0.265594
$378$	0	0
$379$	13.9430	0.716203	0.358101	−	0.933683i	$-0.383424\pi$
$379$	13.9430	0.716203	0.358101	+	0.933683i	$0.383424\pi$
$380$	−0.154676	−0.00793469
$381$	0	0
$382$	−12.0647	−0.617283
$383$	−15.0622	−0.769644	−0.384822	−	0.922991i	$-0.625737\pi$
$383$	−15.0622	−0.769644	−0.384822	+	0.922991i	$0.625737\pi$
$384$	0	0
$385$	39.9391	2.03549
$386$	−6.92983	−0.352719
$387$	0	0
$388$	−6.42363	−0.326110
$389$	−10.4235	−0.528491	−0.264245	−	0.964456i	$-0.585123\pi$
$389$	−10.4235	−0.528491	−0.264245	+	0.964456i	$0.585123\pi$
$390$	0	0
$391$	−2.07488	−0.104931
$392$	14.1678	0.715582
$393$	0	0
$394$	−9.59186	−0.483231
$395$	−2.06499	−0.103901
$396$	0	0
$397$	26.3518	1.32256	0.661280	−	0.750139i	$-0.270013\pi$
$397$	26.3518	1.32256	0.661280	+	0.750139i	$0.270013\pi$
$398$	19.7174	0.988342
$399$	0	0
$400$	−11.1918	−0.559591
$401$	−16.3597	−0.816964	−0.408482	−	0.912766i	$-0.633942\pi$
$401$	−16.3597	−0.816964	−0.408482	+	0.912766i	$0.633942\pi$
$402$	0	0
$403$	19.1077	0.951823
$404$	−6.88393	−0.342488
$405$	0	0
$406$	4.12670	0.204805
$407$	−31.7860	−1.57558
$408$	0	0
$409$	−16.8579	−0.833568	−0.416784	−	0.909005i	$-0.636843\pi$
$409$	−16.8579	−0.833568	−0.416784	+	0.909005i	$0.636843\pi$
$410$	−15.8548	−0.783012
$411$	0	0
$412$	8.89951	0.438447
$413$	−37.0038	−1.82084
$414$	0	0
$415$	15.1661	0.744474
$416$	15.1217	0.741402
$417$	0	0
$418$	0.445114	0.0217713
$419$	0.578582	0.0282656	0.0141328	−	0.999900i	$-0.495501\pi$
$419$	0.578582	0.0282656	0.0141328	+	0.999900i	$0.495501\pi$
$420$	0	0
$421$	21.3261	1.03937	0.519686	−	0.854357i	$-0.326049\pi$
$421$	21.3261	1.03937	0.519686	+	0.854357i	$0.326049\pi$
$422$	10.2870	0.500763
$423$	0	0
$424$	19.6499	0.954285
$425$	8.77497	0.425649
$426$	0	0
$427$	−19.3540	−0.936608
$428$	−1.78887	−0.0864681
$429$	0	0
$430$	10.3687	0.500025
$431$	1.62870	0.0784518	0.0392259	−	0.999230i	$-0.487511\pi$
$431$	1.62870	0.0784518	0.0392259	+	0.999230i	$0.487511\pi$
$432$	0	0
$433$	6.74749	0.324264	0.162132	−	0.986769i	$-0.448163\pi$
$433$	6.74749	0.324264	0.162132	+	0.986769i	$0.448163\pi$
$434$	−15.2905	−0.733969
$435$	0	0
$436$	−1.06622	−0.0510626
$437$	0.0953170	0.00455963
$438$	0	0
$439$	−7.43797	−0.354995	−0.177498	−	0.984121i	$-0.556800\pi$
$439$	−7.43797	−0.354995	−0.177498	+	0.984121i	$0.556800\pi$
$440$	−35.9514	−1.71392
$441$	0	0
$442$	12.9547	0.616191
$443$	−2.20485	−0.104755	−0.0523777	−	0.998627i	$-0.516680\pi$
$443$	−2.20485	−0.104755	−0.0523777	+	0.998627i	$0.516680\pi$
$444$	0	0
$445$	26.9589	1.27797
$446$	27.6743	1.31042
$447$	0	0
$448$	−30.1409	−1.42402
$449$	37.0431	1.74817	0.874087	−	0.485770i	$-0.161461\pi$
$449$	37.0431	1.74817	0.874087	+	0.485770i	$0.161461\pi$
$450$	0	0
$451$	−16.6263	−0.782900
$452$	−3.84033	−0.180634
$453$	0	0
$454$	−9.38267	−0.440351
$455$	−53.3986	−2.50337
$456$	0	0
$457$	28.8088	1.34762	0.673810	−	0.738904i	$-0.264657\pi$
$457$	28.8088	1.34762	0.673810	+	0.738904i	$0.264657\pi$
$458$	26.9086	1.25735
$459$	0	0
$460$	−1.62275	−0.0756611
$461$	22.4332	1.04482	0.522409	−	0.852695i	$-0.325034\pi$
$461$	22.4332	1.04482	0.522409	+	0.852695i	$0.325034\pi$
$462$	0	0
$463$	−0.887793	−0.0412593	−0.0206296	−	0.999787i	$-0.506567\pi$
$463$	−0.887793	−0.0412593	−0.0206296	+	0.999787i	$0.506567\pi$
$464$	−2.64635	−0.122854
$465$	0	0
$466$	13.0096	0.602660
$467$	−21.3965	−0.990113	−0.495056	−	0.868861i	$-0.664853\pi$
$467$	−21.3965	−0.990113	−0.495056	+	0.868861i	$0.664853\pi$
$468$	0	0
$469$	−43.6886	−2.01735
$470$	9.12328	0.420826
$471$	0	0
$472$	33.3091	1.53318
$473$	10.8733	0.499953
$474$	0	0
$475$	−0.403109	−0.0184959
$476$	3.77767	0.173149
$477$	0	0
$478$	−16.7734	−0.767198
$479$	27.8301	1.27159	0.635795	−	0.771858i	$-0.280673\pi$
$479$	27.8301	1.27159	0.635795	+	0.771858i	$0.280673\pi$
$480$	0	0
$481$	42.4979	1.93774
$482$	−12.6244	−0.575027
$483$	0	0
$484$	−2.07095	−0.0941339
$485$	−36.5334	−1.65890
$486$	0	0
$487$	−6.27093	−0.284163	−0.142081	−	0.989855i	$-0.545379\pi$
$487$	−6.27093	−0.284163	−0.142081	+	0.989855i	$0.545379\pi$
$488$	17.4216	0.788641
$489$	0	0
$490$	16.9844	0.767275
$491$	23.6758	1.06847	0.534236	−	0.845335i	$-0.320599\pi$
$491$	23.6758	1.06847	0.534236	+	0.845335i	$0.320599\pi$
$492$	0	0
$493$	2.07488	0.0934480
$494$	−0.595118	−0.0267756
$495$	0	0
$496$	9.80545	0.440278
$497$	−7.70216	−0.345489
$498$	0	0
$499$	−16.9094	−0.756969	−0.378484	−	0.925608i	$-0.623555\pi$
$499$	−16.9094	−0.756969	−0.378484	+	0.925608i	$0.623555\pi$
$500$	−1.25091	−0.0559424
$501$	0	0
$502$	1.64467	0.0734050
$503$	−4.36684	−0.194708	−0.0973539	−	0.995250i	$-0.531038\pi$
$503$	−4.36684	−0.194708	−0.0973539	+	0.995250i	$0.531038\pi$
$504$	0	0
$505$	−39.1513	−1.74221
$506$	4.66983	0.207599
$507$	0	0
$508$	−1.89186	−0.0839378
$509$	23.8848	1.05868	0.529338	−	0.848411i	$-0.322441\pi$
$509$	23.8848	1.05868	0.529338	+	0.848411i	$0.322441\pi$
$510$	0	0
$511$	−47.9004	−2.11899
$512$	23.9988	1.06061
$513$	0	0
$514$	6.74942	0.297704
$515$	50.6146	2.23035
$516$	0	0
$517$	9.56720	0.420765
$518$	−34.0080	−1.49423
$519$	0	0
$520$	48.0670	2.10788
$521$	−11.2081	−0.491037	−0.245519	−	0.969392i	$-0.578958\pi$
$521$	−11.2081	−0.491037	−0.245519	+	0.969392i	$0.578958\pi$
$522$	0	0
$523$	2.67524	0.116980	0.0584901	−	0.998288i	$-0.481371\pi$
$523$	2.67524	0.116980	0.0584901	+	0.998288i	$0.481371\pi$
$524$	−2.47800	−0.108252
$525$	0	0
$526$	37.7835	1.64744
$527$	−7.68799	−0.334894
$528$	0	0
$529$	1.00000	0.0434783
$530$	23.5563	1.02322
$531$	0	0
$532$	−0.173541	−0.00752394
$533$	22.2293	0.962858
$534$	0	0
$535$	−10.1739	−0.439856
$536$	39.3266	1.69865
$537$	0	0
$538$	18.1745	0.783560
$539$	17.8108	0.767165
$540$	0	0
$541$	−4.28908	−0.184402	−0.0922010	−	0.995740i	$-0.529390\pi$
$541$	−4.28908	−0.184402	−0.0922010	+	0.995740i	$0.529390\pi$
$542$	19.8031	0.850614
$543$	0	0
$544$	−6.08421	−0.260858
$545$	−6.06396	−0.259752
$546$	0	0
$547$	−35.7338	−1.52787	−0.763933	−	0.645296i	$-0.776734\pi$
$547$	−35.7338	−1.52787	−0.763933	+	0.645296i	$0.776734\pi$
$548$	−9.04006	−0.386172
$549$	0	0
$550$	−19.7494	−0.842117
$551$	−0.0953170	−0.00406064
$552$	0	0
$553$	−2.31684	−0.0985221
$554$	−26.4361	−1.12316
$555$	0	0
$556$	−3.52355	−0.149432
$557$	3.81954	0.161839	0.0809195	−	0.996721i	$-0.474214\pi$
$557$	3.81954	0.161839	0.0809195	+	0.996721i	$0.474214\pi$
$558$	0	0
$559$	−14.5375	−0.614873
$560$	−27.4024	−1.15796
$561$	0	0
$562$	29.2306	1.23302
$563$	20.4663	0.862554	0.431277	−	0.902220i	$-0.358063\pi$
$563$	20.4663	0.862554	0.431277	+	0.902220i	$0.358063\pi$
$564$	0	0
$565$	−21.8413	−0.918870
$566$	32.4515	1.36404
$567$	0	0
$568$	6.93314	0.290908
$569$	−10.7702	−0.451510	−0.225755	−	0.974184i	$-0.572485\pi$
$569$	−10.7702	−0.451510	−0.225755	+	0.974184i	$0.572485\pi$
$570$	0	0
$571$	−19.2253	−0.804556	−0.402278	−	0.915518i	$-0.631781\pi$
$571$	−19.2253	−0.804556	−0.402278	+	0.915518i	$0.631781\pi$
$572$	10.6247	0.444243
$573$	0	0
$574$	−17.7885	−0.742478
$575$	−4.22914	−0.176367
$576$	0	0
$577$	−16.6266	−0.692174	−0.346087	−	0.938203i	$-0.612490\pi$
$577$	−16.6266	−0.692174	−0.346087	+	0.938203i	$0.612490\pi$
$578$	15.3699	0.639304
$579$	0	0
$580$	1.62275	0.0673810
$581$	17.0158	0.705936
$582$	0	0
$583$	24.7025	1.02308
$584$	43.1177	1.78423
$585$	0	0
$586$	16.4375	0.679027
$587$	−38.1799	−1.57585	−0.787926	−	0.615769i	$-0.788845\pi$
$587$	−38.1799	−1.57585	−0.787926	+	0.615769i	$0.788845\pi$
$588$	0	0
$589$	0.353175	0.0145523
$590$	39.9310	1.64393
$591$	0	0
$592$	21.8085	0.896325
$593$	−24.9127	−1.02304	−0.511522	−	0.859270i	$-0.670918\pi$
$593$	−24.9127	−1.02304	−0.511522	+	0.859270i	$0.670918\pi$
$594$	0	0
$595$	21.4849	0.880796
$596$	1.64657	0.0674462
$597$	0	0
$598$	−6.24357	−0.255318
$599$	41.2275	1.68451	0.842255	−	0.539079i	$-0.181228\pi$
$599$	41.2275	1.68451	0.842255	+	0.539079i	$0.181228\pi$
$600$	0	0
$601$	20.1084	0.820241	0.410120	−	0.912031i	$-0.365487\pi$
$601$	20.1084	0.820241	0.410120	+	0.912031i	$0.365487\pi$
$602$	11.6334	0.474140
$603$	0	0
$604$	6.08500	0.247595
$605$	−11.7782	−0.478852
$606$	0	0
$607$	−15.0972	−0.612776	−0.306388	−	0.951907i	$-0.599121\pi$
$607$	−15.0972	−0.612776	−0.306388	+	0.951907i	$0.599121\pi$
$608$	0.279500	0.0113352
$609$	0	0
$610$	20.8851	0.845612
$611$	−12.7914	−0.517483
$612$	0	0
$613$	−26.9957	−1.09035	−0.545174	−	0.838323i	$-0.683536\pi$
$613$	−26.9957	−1.09035	−0.545174	+	0.838323i	$0.683536\pi$
$614$	23.8469	0.962381
$615$	0	0
$616$	−40.3362	−1.62519
$617$	−8.06126	−0.324534	−0.162267	−	0.986747i	$-0.551881\pi$
$617$	−8.06126	−0.324534	−0.162267	+	0.986747i	$0.551881\pi$
$618$	0	0
$619$	−38.6945	−1.55526	−0.777631	−	0.628721i	$-0.783579\pi$
$619$	−38.6945	−1.55526	−0.777631	+	0.628721i	$0.783579\pi$
$620$	−6.01272	−0.241477
$621$	0	0
$622$	−35.8762	−1.43851
$623$	30.2469	1.21182
$624$	0	0
$625$	−28.2601	−1.13040
$626$	28.8914	1.15473
$627$	0	0
$628$	3.98306	0.158942
$629$	−17.0990	−0.681783
$630$	0	0
$631$	18.5647	0.739048	0.369524	−	0.929221i	$-0.379521\pi$
$631$	18.5647	0.739048	0.369524	+	0.929221i	$0.379521\pi$
$632$	2.08552	0.0829574
$633$	0	0
$634$	33.0997	1.31456
$635$	−10.7597	−0.426985
$636$	0	0
$637$	−23.8130	−0.943507
$638$	−4.66983	−0.184880
$639$	0	0
$640$	14.7087	0.581413
$641$	3.84832	0.151999	0.0759997	−	0.997108i	$-0.475785\pi$
$641$	3.84832	0.151999	0.0759997	+	0.997108i	$0.475785\pi$
$642$	0	0
$643$	19.0776	0.752348	0.376174	−	0.926549i	$-0.377239\pi$
$643$	19.0776	0.752348	0.376174	+	0.926549i	$0.377239\pi$
$644$	−1.82067	−0.0717444
$645$	0	0
$646$	0.239446	0.00942086
$647$	−35.1122	−1.38040	−0.690202	−	0.723617i	$-0.742478\pi$
$647$	−35.1122	−1.38040	−0.690202	+	0.723617i	$0.742478\pi$
$648$	0	0
$649$	41.8740	1.64370
$650$	26.4049	1.03569
$651$	0	0
$652$	9.04106	0.354075
$653$	12.2338	0.478745	0.239372	−	0.970928i	$-0.423058\pi$
$653$	12.2338	0.478745	0.239372	+	0.970928i	$0.423058\pi$
$654$	0	0
$655$	−14.0932	−0.550668
$656$	11.4073	0.445382
$657$	0	0
$658$	10.2360	0.399041
$659$	2.80788	0.109380	0.0546898	−	0.998503i	$-0.482583\pi$
$659$	2.80788	0.109380	0.0546898	+	0.998503i	$0.482583\pi$
$660$	0	0
$661$	−37.6385	−1.46397	−0.731984	−	0.681322i	$-0.761405\pi$
$661$	−37.6385	−1.46397	−0.731984	+	0.681322i	$0.761405\pi$
$662$	−41.0560	−1.59569
$663$	0	0
$664$	−15.3169	−0.594410
$665$	−0.986986	−0.0382737
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	2.84401	0.110038
$669$	0	0
$670$	47.1447	1.82136
$671$	21.9013	0.845490
$672$	0	0
$673$	−50.4621	−1.94517	−0.972586	−	0.232543i	$-0.925295\pi$
$673$	−50.4621	−1.94517	−0.972586	+	0.232543i	$0.925295\pi$
$674$	−8.90447	−0.342987
$675$	0	0
$676$	−7.26121	−0.279277
$677$	−18.5792	−0.714055	−0.357028	−	0.934094i	$-0.616210\pi$
$677$	−18.5792	−0.714055	−0.357028	+	0.934094i	$0.616210\pi$
$678$	0	0
$679$	−40.9892	−1.57302
$680$	−19.3398	−0.741646
$681$	0	0
$682$	17.3030	0.662565
$683$	−49.7804	−1.90480	−0.952398	−	0.304858i	$-0.901391\pi$
$683$	−49.7804	−1.90480	−0.952398	+	0.304858i	$0.901391\pi$
$684$	0	0
$685$	−51.4140	−1.96443
$686$	−9.83107	−0.375352
$687$	0	0
$688$	−7.46019	−0.284417
$689$	−33.0273	−1.25824
$690$	0	0
$691$	45.4174	1.72776	0.863880	−	0.503698i	$-0.168027\pi$
$691$	45.4174	1.72776	0.863880	+	0.503698i	$0.168027\pi$
$692$	13.2256	0.502763
$693$	0	0
$694$	14.4880	0.549959
$695$	−20.0397	−0.760148
$696$	0	0
$697$	−8.94396	−0.338777
$698$	−25.6203	−0.969744
$699$	0	0
$700$	7.69987	0.291028
$701$	−16.6480	−0.628788	−0.314394	−	0.949293i	$-0.601801\pi$
$701$	−16.6480	−0.628788	−0.314394	+	0.949293i	$0.601801\pi$
$702$	0	0
$703$	0.785504	0.0296259
$704$	34.1078	1.28549
$705$	0	0
$706$	−36.2629	−1.36477
$707$	−43.9264	−1.65202
$708$	0	0
$709$	−8.65867	−0.325183	−0.162592	−	0.986693i	$-0.551985\pi$
$709$	−8.65867	−0.325183	−0.162592	+	0.986693i	$0.551985\pi$
$710$	8.31145	0.311923
$711$	0	0
$712$	−27.2269	−1.02037
$713$	3.70527	0.138763
$714$	0	0
$715$	60.4266	2.25983
$716$	−3.56664	−0.133292
$717$	0	0
$718$	−41.0397	−1.53159
$719$	19.1222	0.713138	0.356569	−	0.934269i	$-0.383946\pi$
$719$	19.1222	0.713138	0.356569	+	0.934269i	$0.383946\pi$
$720$	0	0
$721$	56.7878	2.11489
$722$	22.9927	0.855698
$723$	0	0
$724$	−5.74349	−0.213455
$725$	4.22914	0.157066
$726$	0	0
$727$	−9.85298	−0.365427	−0.182713	−	0.983166i	$-0.558488\pi$
$727$	−9.85298	−0.365427	−0.182713	+	0.983166i	$0.558488\pi$
$728$	53.9295	1.99876
$729$	0	0
$730$	51.6896	1.91312
$731$	5.84918	0.216340
$732$	0	0
$733$	29.0623	1.07344	0.536721	−	0.843760i	$-0.319663\pi$
$733$	29.0623	1.07344	0.536721	+	0.843760i	$0.319663\pi$
$734$	1.35961	0.0501841
$735$	0	0
$736$	2.93232	0.108087
$737$	49.4386	1.82110
$738$	0	0
$739$	−18.5216	−0.681327	−0.340663	−	0.940185i	$-0.610652\pi$
$739$	−18.5216	−0.681327	−0.340663	+	0.940185i	$0.610652\pi$
$740$	−13.3730	−0.491602
$741$	0	0
$742$	26.4294	0.970253
$743$	−33.6795	−1.23558	−0.617790	−	0.786343i	$-0.711972\pi$
$743$	−33.6795	−1.23558	−0.617790	+	0.786343i	$0.711972\pi$
$744$	0	0
$745$	9.36462	0.343093
$746$	−13.4045	−0.490773
$747$	0	0
$748$	−4.27486	−0.156304
$749$	−11.4148	−0.417086
$750$	0	0
$751$	51.9526	1.89578	0.947889	−	0.318600i	$-0.103213\pi$
$751$	51.9526	1.89578	0.947889	+	0.318600i	$0.103213\pi$
$752$	−6.56410	−0.239368
$753$	0	0
$754$	6.24357	0.227377
$755$	34.6075	1.25950
$756$	0	0
$757$	33.8994	1.23210	0.616048	−	0.787709i	$-0.288733\pi$
$757$	33.8994	1.23210	0.616048	+	0.787709i	$0.288733\pi$
$758$	−16.8810	−0.613147
$759$	0	0
$760$	0.888440	0.0322271
$761$	−47.1104	−1.70775	−0.853875	−	0.520478i	$-0.825754\pi$
$761$	−47.1104	−1.70775	−0.853875	+	0.520478i	$0.825754\pi$
$762$	0	0
$763$	−6.80355	−0.246305
$764$	−5.32284	−0.192574
$765$	0	0
$766$	18.2361	0.658898
$767$	−55.9855	−2.02152
$768$	0	0
$769$	−23.6926	−0.854377	−0.427188	−	0.904163i	$-0.640496\pi$
$769$	−23.6926	−0.854377	−0.427188	+	0.904163i	$0.640496\pi$
$770$	−48.3551	−1.74260
$771$	0	0
$772$	−3.05738	−0.110038
$773$	20.8577	0.750201	0.375100	−	0.926984i	$-0.377608\pi$
$773$	20.8577	0.750201	0.375100	+	0.926984i	$0.377608\pi$
$774$	0	0
$775$	−15.6701	−0.562887
$776$	36.8967	1.32451
$777$	0	0
$778$	12.6199	0.452445
$779$	0.410872	0.0147210
$780$	0	0
$781$	8.71586	0.311878
$782$	2.51210	0.0898325
$783$	0	0
$784$	−12.2201	−0.436431
$785$	22.6531	0.808523
$786$	0	0
$787$	15.5046	0.552680	0.276340	−	0.961060i	$-0.410878\pi$
$787$	15.5046	0.552680	0.276340	+	0.961060i	$0.410878\pi$
$788$	−4.23185	−0.150753
$789$	0	0
$790$	2.50012	0.0889502
$791$	−24.5052	−0.871303
$792$	0	0
$793$	−29.2820	−1.03984
$794$	−31.9047	−1.13225
$795$	0	0
$796$	8.69914	0.308333
$797$	18.8889	0.669080	0.334540	−	0.942382i	$-0.391419\pi$
$797$	18.8889	0.669080	0.334540	+	0.942382i	$0.391419\pi$
$798$	0	0
$799$	5.14660	0.182074
$800$	−12.4012	−0.438448
$801$	0	0
$802$	19.8070	0.699409
$803$	54.2047	1.91284
$804$	0	0
$805$	−10.3548	−0.364958
$806$	−23.1341	−0.814863
$807$	0	0
$808$	39.5405	1.39103
$809$	−10.3408	−0.363563	−0.181781	−	0.983339i	$-0.558186\pi$
$809$	−10.3408	−0.363563	−0.181781	+	0.983339i	$0.558186\pi$
$810$	0	0
$811$	−31.1199	−1.09277	−0.546383	−	0.837535i	$-0.683996\pi$
$811$	−31.1199	−1.09277	−0.546383	+	0.837535i	$0.683996\pi$
$812$	1.82067	0.0638929
$813$	0	0
$814$	38.4840	1.34886
$815$	51.4197	1.80115
$816$	0	0
$817$	−0.268703	−0.00940072
$818$	20.4102	0.713624
$819$	0	0
$820$	−6.99500	−0.244276
$821$	20.8286	0.726922	0.363461	−	0.931609i	$-0.381595\pi$
$821$	20.8286	0.726922	0.363461	+	0.931609i	$0.381595\pi$
$822$	0	0
$823$	23.2091	0.809018	0.404509	−	0.914534i	$-0.367443\pi$
$823$	23.2091	0.809018	0.404509	+	0.914534i	$0.367443\pi$
$824$	−51.1179	−1.78077
$825$	0	0
$826$	44.8012	1.55883
$827$	18.4758	0.642468	0.321234	−	0.947000i	$-0.395902\pi$
$827$	18.4758	0.642468	0.321234	+	0.947000i	$0.395902\pi$
$828$	0	0
$829$	20.8312	0.723497	0.361749	−	0.932276i	$-0.382180\pi$
$829$	20.8312	0.723497	0.361749	+	0.932276i	$0.382180\pi$
$830$	−18.3619	−0.637350
$831$	0	0
$832$	−45.6021	−1.58097
$833$	9.58117	0.331968
$834$	0	0
$835$	16.1749	0.559755
$836$	0.196381	0.00679197
$837$	0	0
$838$	−0.700501	−0.0241984
$839$	7.97701	0.275397	0.137698	−	0.990474i	$-0.456030\pi$
$839$	7.97701	0.275397	0.137698	+	0.990474i	$0.456030\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−25.8200	−0.889815
$843$	0	0
$844$	4.53854	0.156223
$845$	−41.2970	−1.42066
$846$	0	0
$847$	−13.2147	−0.454063
$848$	−16.9485	−0.582015
$849$	0	0
$850$	−10.6240	−0.364401
$851$	8.24097	0.282497
$852$	0	0
$853$	−28.3893	−0.972030	−0.486015	−	0.873950i	$-0.661550\pi$
$853$	−28.3893	−0.972030	−0.486015	+	0.873950i	$0.661550\pi$
$854$	23.4323	0.801837
$855$	0	0
$856$	10.2751	0.351194
$857$	−1.06086	−0.0362384	−0.0181192	−	0.999836i	$-0.505768\pi$
$857$	−1.06086	−0.0362384	−0.0181192	+	0.999836i	$0.505768\pi$
$858$	0	0
$859$	−34.9191	−1.19142	−0.595711	−	0.803199i	$-0.703130\pi$
$859$	−34.9191	−1.19142	−0.595711	+	0.803199i	$0.703130\pi$
$860$	4.57460	0.155993
$861$	0	0
$862$	−1.97190	−0.0671632
$863$	31.8477	1.08411	0.542054	−	0.840343i	$-0.317647\pi$
$863$	31.8477	1.08411	0.542054	+	0.840343i	$0.317647\pi$
$864$	0	0
$865$	75.2187	2.55751
$866$	−8.16932	−0.277605
$867$	0	0
$868$	−6.74606	−0.228976
$869$	2.62177	0.0889374
$870$	0	0
$871$	−66.0995	−2.23970
$872$	6.12425	0.207393
$873$	0	0
$874$	−0.115402	−0.00390353
$875$	−7.98206	−0.269843
$876$	0	0
$877$	−28.7892	−0.972141	−0.486071	−	0.873919i	$-0.661570\pi$
$877$	−28.7892	−0.972141	−0.486071	+	0.873919i	$0.661570\pi$
$878$	9.00530	0.303914
$879$	0	0
$880$	31.0089	1.04531
$881$	−6.20651	−0.209103	−0.104551	−	0.994519i	$-0.533341\pi$
$881$	−6.20651	−0.209103	−0.104551	+	0.994519i	$0.533341\pi$
$882$	0	0
$883$	−38.2460	−1.28708	−0.643540	−	0.765413i	$-0.722535\pi$
$883$	−38.2460	−1.28708	−0.643540	+	0.765413i	$0.722535\pi$
$884$	5.71549	0.192233
$885$	0	0
$886$	2.66945	0.0896819
$887$	13.5713	0.455681	0.227841	−	0.973698i	$-0.426834\pi$
$887$	13.5713	0.455681	0.227841	+	0.973698i	$0.426834\pi$
$888$	0	0
$889$	−12.0720	−0.404881
$890$	−32.6396	−1.09408
$891$	0	0
$892$	12.2097	0.408810
$893$	−0.236427	−0.00791173
$894$	0	0
$895$	−20.2847	−0.678044
$896$	16.5027	0.551316
$897$	0	0
$898$	−44.8488	−1.49662
$899$	−3.70527	−0.123578
$900$	0	0
$901$	13.2885	0.442705
$902$	20.1297	0.670246
$903$	0	0
$904$	22.0584	0.733653
$905$	−32.6652	−1.08583
$906$	0	0
$907$	−8.32618	−0.276466	−0.138233	−	0.990400i	$-0.544142\pi$
$907$	−8.32618	−0.276466	−0.138233	+	0.990400i	$0.544142\pi$
$908$	−4.13956	−0.137376
$909$	0	0
$910$	64.6507	2.14315
$911$	−35.8689	−1.18839	−0.594194	−	0.804321i	$-0.702529\pi$
$911$	−35.8689	−1.18839	−0.594194	+	0.804321i	$0.702529\pi$
$912$	0	0
$913$	−19.2553	−0.637259
$914$	−34.8794	−1.15371
$915$	0	0
$916$	11.8718	0.392256
$917$	−15.8121	−0.522162
$918$	0	0
$919$	38.8879	1.28279	0.641397	−	0.767209i	$-0.278355\pi$
$919$	38.8879	1.28279	0.641397	+	0.767209i	$0.278355\pi$
$920$	9.32090	0.307301
$921$	0	0
$922$	−27.1603	−0.894476
$923$	−11.6531	−0.383567
$924$	0	0
$925$	−34.8522	−1.14593
$926$	1.07487	0.0353224
$927$	0	0
$928$	−2.93232	−0.0962580
$929$	6.58929	0.216188	0.108094	−	0.994141i	$-0.465525\pi$
$929$	6.58929	0.216188	0.108094	+	0.994141i	$0.465525\pi$
$930$	0	0
$931$	−0.440145	−0.0144252
$932$	5.73975	0.188012
$933$	0	0
$934$	25.9052	0.847643
$935$	−24.3126	−0.795108
$936$	0	0
$937$	4.77562	0.156013	0.0780063	−	0.996953i	$-0.475145\pi$
$937$	4.77562	0.156013	0.0780063	+	0.996953i	$0.475145\pi$
$938$	52.8947	1.72707
$939$	0	0
$940$	4.02512	0.131285
$941$	48.1961	1.57115	0.785573	−	0.618769i	$-0.212368\pi$
$941$	48.1961	1.57115	0.785573	+	0.618769i	$0.212368\pi$
$942$	0	0
$943$	4.31059	0.140372
$944$	−28.7299	−0.935079
$945$	0	0
$946$	−13.1645	−0.428014
$947$	−35.1573	−1.14246	−0.571229	−	0.820790i	$-0.693533\pi$
$947$	−35.1573	−1.14246	−0.571229	+	0.820790i	$0.693533\pi$
$948$	0	0
$949$	−72.4717	−2.35253
$950$	0.488052	0.0158345
$951$	0	0
$952$	−21.6985	−0.703254
$953$	24.9046	0.806740	0.403370	−	0.915037i	$-0.367839\pi$
$953$	24.9046	0.806740	0.403370	+	0.915037i	$0.367839\pi$
$954$	0	0
$955$	−30.2728	−0.979606
$956$	−7.40029	−0.239342
$957$	0	0
$958$	−33.6944	−1.08862
$959$	−57.6847	−1.86274
$960$	0	0
$961$	−17.2710	−0.557129
$962$	−51.4531	−1.65891
$963$	0	0
$964$	−5.56980	−0.179391
$965$	−17.3884	−0.559753
$966$	0	0
$967$	−10.6116	−0.341245	−0.170622	−	0.985337i	$-0.554578\pi$
$967$	−10.6116	−0.341245	−0.170622	+	0.985337i	$0.554578\pi$
$968$	11.8953	0.382329
$969$	0	0
$970$	44.2317	1.42019
$971$	−25.5705	−0.820595	−0.410298	−	0.911952i	$-0.634575\pi$
$971$	−25.5705	−0.820595	−0.410298	+	0.911952i	$0.634575\pi$
$972$	0	0
$973$	−22.4838	−0.720798
$974$	7.59233	0.243274
$975$	0	0
$976$	−15.0266	−0.480989
$977$	−48.7768	−1.56051	−0.780255	−	0.625462i	$-0.784911\pi$
$977$	−48.7768	−1.56051	−0.780255	+	0.625462i	$0.784911\pi$
$978$	0	0
$979$	−34.2278	−1.09393
$980$	7.49336	0.239367
$981$	0	0
$982$	−28.6647	−0.914727
$983$	42.2885	1.34879	0.674397	−	0.738369i	$-0.264404\pi$
$983$	42.2885	1.34879	0.674397	+	0.738369i	$0.264404\pi$
$984$	0	0
$985$	−24.0680	−0.766871
$986$	−2.51210	−0.0800015
$987$	0	0
$988$	−0.262561	−0.00835319
$989$	−2.81904	−0.0896404
$990$	0	0
$991$	−2.03129	−0.0645259	−0.0322630	−	0.999479i	$-0.510271\pi$
$991$	−2.03129	−0.0645259	−0.0322630	+	0.999479i	$0.510271\pi$
$992$	10.8650	0.344965
$993$	0	0
$994$	9.32515	0.295776
$995$	49.4750	1.56846
$996$	0	0
$997$	4.13490	0.130954	0.0654768	−	0.997854i	$-0.479143\pi$
$997$	4.13490	0.130954	0.0654768	+	0.997854i	$0.479143\pi$
$998$	20.4725	0.648047
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.j.1.2		7
3.2	odd	2		2001.2.a.i.1.6	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.i.1.6	✓	7	3.2	odd	2
6003.2.a.j.1.2		7	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)

\(f(q)\)	\(=\)	\(q-1.21072 q^{2} -0.534159 q^{4} -3.03795 q^{5} -3.40847 q^{7} +3.06816 q^{8} +O(q^{10})\)	\(q-1.21072 q^{2} -0.534159 q^{4} -3.03795 q^{5} -3.40847 q^{7} +3.06816 q^{8} +3.67810 q^{10} +3.85707 q^{11} -5.15691 q^{13} +4.12670 q^{14} -2.64635 q^{16} +2.07488 q^{17} -0.0953170 q^{19} +1.62275 q^{20} -4.66983 q^{22} -1.00000 q^{23} +4.22914 q^{25} +6.24357 q^{26} +1.82067 q^{28} +1.00000 q^{29} -3.70527 q^{31} -2.93232 q^{32} -2.51210 q^{34} +10.3548 q^{35} -8.24097 q^{37} +0.115402 q^{38} -9.32090 q^{40} -4.31059 q^{41} +2.81904 q^{43} -2.06029 q^{44} +1.21072 q^{46} +2.48043 q^{47} +4.61770 q^{49} -5.12030 q^{50} +2.75461 q^{52} +6.40448 q^{53} -11.7176 q^{55} -10.4577 q^{56} -1.21072 q^{58} +10.8564 q^{59} +5.67822 q^{61} +4.48604 q^{62} +8.84292 q^{64} +15.6664 q^{65} +12.8177 q^{67} -1.10832 q^{68} -12.5367 q^{70} +2.25971 q^{71} +14.0533 q^{73} +9.97750 q^{74} +0.0509145 q^{76} -13.1467 q^{77} +0.679730 q^{79} +8.03950 q^{80} +5.21891 q^{82} -4.99221 q^{83} -6.30339 q^{85} -3.41307 q^{86} +11.8341 q^{88} -8.87404 q^{89} +17.5772 q^{91} +0.534159 q^{92} -3.00310 q^{94} +0.289568 q^{95} +12.0257 q^{97} -5.59073 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 3 q^{2} + 5 q^{4} + 3 q^{5} - 5 q^{7} + 6 q^{8}+O(q^{10}) \) 7 * q + 3 * q^2 + 5 * q^4 + 3 * q^5 - 5 * q^7 + 6 * q^8	\( 7 q + 3 q^{2} + 5 q^{4} + 3 q^{5} - 5 q^{7} + 6 q^{8} + 3 q^{10} + 4 q^{11} - 18 q^{13} + 2 q^{14} - 7 q^{16} + 3 q^{17} - 4 q^{19} + 2 q^{20} - 26 q^{22} - 7 q^{23} - 8 q^{25} + 7 q^{26} - 6 q^{28} + 7 q^{29} - 22 q^{31} - 5 q^{32} + 9 q^{34} - 3 q^{35} - 25 q^{37} - 14 q^{38} - 10 q^{40} + 13 q^{41} - 2 q^{43} - 4 q^{44} - 3 q^{46} + 25 q^{47} - 8 q^{49} - 19 q^{50} - 12 q^{52} + 5 q^{53} - 15 q^{55} - 18 q^{56} + 3 q^{58} - 11 q^{59} - 33 q^{61} - 28 q^{62} - 14 q^{64} + 2 q^{65} + 8 q^{67} - 12 q^{68} - 22 q^{70} + 6 q^{71} + 15 q^{73} - 34 q^{74} - 28 q^{76} + q^{77} - 15 q^{79} + 12 q^{80} - 14 q^{82} - 21 q^{83} - 28 q^{85} + 12 q^{86} - 13 q^{88} - 8 q^{89} + 6 q^{91} - 5 q^{92} - 35 q^{94} + 25 q^{95} + 13 q^{97} - q^{98}+O(q^{100}) \) 7 * q + 3 * q^2 + 5 * q^4 + 3 * q^5 - 5 * q^7 + 6 * q^8 + 3 * q^10 + 4 * q^11 - 18 * q^13 + 2 * q^14 - 7 * q^16 + 3 * q^17 - 4 * q^19 + 2 * q^20 - 26 * q^22 - 7 * q^23 - 8 * q^25 + 7 * q^26 - 6 * q^28 + 7 * q^29 - 22 * q^31 - 5 * q^32 + 9 * q^34 - 3 * q^35 - 25 * q^37 - 14 * q^38 - 10 * q^40 + 13 * q^41 - 2 * q^43 - 4 * q^44 - 3 * q^46 + 25 * q^47 - 8 * q^49 - 19 * q^50 - 12 * q^52 + 5 * q^53 - 15 * q^55 - 18 * q^56 + 3 * q^58 - 11 * q^59 - 33 * q^61 - 28 * q^62 - 14 * q^64 + 2 * q^65 + 8 * q^67 - 12 * q^68 - 22 * q^70 + 6 * q^71 + 15 * q^73 - 34 * q^74 - 28 * q^76 + q^77 - 15 * q^79 + 12 * q^80 - 14 * q^82 - 21 * q^83 - 28 * q^85 + 12 * q^86 - 13 * q^88 - 8 * q^89 + 6 * q^91 - 5 * q^92 - 35 * q^94 + 25 * q^95 + 13 * q^97 - q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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