LMFDB - Embedded newform 6003.2.a.l.1.1

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 3x^{9} - 10x^{8} + 32x^{7} + 32x^{6} - 118x^{5} - 29x^{4} + 182x^{3} - 28x^{2} - 101x + 43 $ x^10 - 3x^9 - 10x^8 + 32x^7 + 32x^6 - 118x^5 - 29x^4 + 182x^3 - 28x^2 - 101*x + 43
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.17460$ 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)

$f(q)$	$=$	$q-2.17460 q^{2} +2.72887 q^{4} +1.26068 q^{5} +1.97284 q^{7} -1.58501 q^{8} +O(q^{10})$	\(q-2.17460 q^{2} +2.72887 q^{4} +1.26068 q^{5} +1.97284 q^{7} -1.58501 q^{8} -2.74147 q^{10} +2.80559 q^{11} -4.91313 q^{13} -4.29013 q^{14} -2.01100 q^{16} +7.61789 q^{17} +0.948384 q^{19} +3.44024 q^{20} -6.10102 q^{22} -1.00000 q^{23} -3.41068 q^{25} +10.6841 q^{26} +5.38363 q^{28} -1.00000 q^{29} +0.780558 q^{31} +7.54312 q^{32} -16.5658 q^{34} +2.48712 q^{35} +2.59391 q^{37} -2.06235 q^{38} -1.99819 q^{40} -1.16041 q^{41} -3.06328 q^{43} +7.65610 q^{44} +2.17460 q^{46} -5.32731 q^{47} -3.10791 q^{49} +7.41686 q^{50} -13.4073 q^{52} +9.00013 q^{53} +3.53695 q^{55} -3.12696 q^{56} +2.17460 q^{58} +8.64531 q^{59} +10.1026 q^{61} -1.69740 q^{62} -12.3813 q^{64} -6.19389 q^{65} +2.24279 q^{67} +20.7883 q^{68} -5.40848 q^{70} -13.3724 q^{71} +1.58151 q^{73} -5.64071 q^{74} +2.58802 q^{76} +5.53497 q^{77} +5.39042 q^{79} -2.53522 q^{80} +2.52343 q^{82} -10.5871 q^{83} +9.60373 q^{85} +6.66141 q^{86} -4.44687 q^{88} +9.44598 q^{89} -9.69281 q^{91} -2.72887 q^{92} +11.5848 q^{94} +1.19561 q^{95} -8.45927 q^{97} +6.75845 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 3 q^{2} + 9 q^{4} + 10 q^{5} + q^{7} + 9 q^{8}+O(q^{10}) $ 10 * q + 3 * q^2 + 9 * q^4 + 10 * q^5 + q^7 + 9 * q^8	$ 10 q + 3 q^{2} + 9 q^{4} + 10 q^{5} + q^{7} + 9 q^{8} - 6 q^{10} - 13 q^{13} + 12 q^{14} - 5 q^{16} + 22 q^{17} - 2 q^{19} - 3 q^{20} + 3 q^{22} - 10 q^{23} + 10 q^{25} + 25 q^{26} + 19 q^{28} - 10 q^{29} - 22 q^{31} + 31 q^{32} + 13 q^{34} + 15 q^{35} - 9 q^{37} + 10 q^{38} - 6 q^{40} + 25 q^{41} + 3 q^{43} + 27 q^{44} - 3 q^{46} + 17 q^{47} + 17 q^{49} - 2 q^{50} - 18 q^{52} + 43 q^{53} - 11 q^{55} + 7 q^{56} - 3 q^{58} + 7 q^{59} - 6 q^{61} - 3 q^{62} + 33 q^{64} - 11 q^{65} + 11 q^{67} + 51 q^{68} + 34 q^{70} + 17 q^{71} - 44 q^{73} - 9 q^{74} + 24 q^{76} + 71 q^{77} + 5 q^{79} - 38 q^{80} + 33 q^{82} + 32 q^{83} + 16 q^{85} + 9 q^{86} + 18 q^{88} + 10 q^{89} - 3 q^{91} - 9 q^{92} + 47 q^{94} + 8 q^{95} + 6 q^{97} + 73 q^{98}+O(q^{100}) $ 10 * q + 3 * q^2 + 9 * q^4 + 10 * q^5 + q^7 + 9 * q^8 - 6 * q^10 - 13 * q^13 + 12 * q^14 - 5 * q^16 + 22 * q^17 - 2 * q^19 - 3 * q^20 + 3 * q^22 - 10 * q^23 + 10 * q^25 + 25 * q^26 + 19 * q^28 - 10 * q^29 - 22 * q^31 + 31 * q^32 + 13 * q^34 + 15 * q^35 - 9 * q^37 + 10 * q^38 - 6 * q^40 + 25 * q^41 + 3 * q^43 + 27 * q^44 - 3 * q^46 + 17 * q^47 + 17 * q^49 - 2 * q^50 - 18 * q^52 + 43 * q^53 - 11 * q^55 + 7 * q^56 - 3 * q^58 + 7 * q^59 - 6 * q^61 - 3 * q^62 + 33 * q^64 - 11 * q^65 + 11 * q^67 + 51 * q^68 + 34 * q^70 + 17 * q^71 - 44 * q^73 - 9 * q^74 + 24 * q^76 + 71 * q^77 + 5 * q^79 - 38 * q^80 + 33 * q^82 + 32 * q^83 + 16 * q^85 + 9 * q^86 + 18 * q^88 + 10 * q^89 - 3 * q^91 - 9 * q^92 + 47 * q^94 + 8 * q^95 + 6 * q^97 + 73 * 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 $

$2$	−2.17460	−1.53767	−0.768836	−	0.639446i	$-0.779164\pi$
$2$	−2.17460	−1.53767	−0.768836	+	0.639446i	$0.779164\pi$
$3$	0	0
$3$	0	0
$4$	2.72887	1.36444
$5$	1.26068	0.563794	0.281897	−	0.959445i	$-0.409036\pi$
$5$	1.26068	0.563794	0.281897	+	0.959445i	$0.409036\pi$
$6$	0	0
$7$	1.97284	0.745663	0.372831	−	0.927899i	$-0.378387\pi$
$7$	1.97284	0.745663	0.372831	+	0.927899i	$0.378387\pi$
$8$	−1.58501	−0.560384
$9$	0	0
$10$	−2.74147	−0.866930
$11$	2.80559	0.845917	0.422958	−	0.906149i	$-0.360992\pi$
$11$	2.80559	0.845917	0.422958	+	0.906149i	$0.360992\pi$
$12$	0	0
$13$	−4.91313	−1.36266	−0.681329	−	0.731977i	$-0.738598\pi$
$13$	−4.91313	−1.36266	−0.681329	+	0.731977i	$0.738598\pi$
$14$	−4.29013	−1.14659
$15$	0	0
$16$	−2.01100	−0.502749
$17$	7.61789	1.84761	0.923805	−	0.382863i	$-0.125062\pi$
$17$	7.61789	1.84761	0.923805	+	0.382863i	$0.125062\pi$
$18$	0	0
$19$	0.948384	0.217574	0.108787	−	0.994065i	$-0.465303\pi$
$19$	0.948384	0.217574	0.108787	+	0.994065i	$0.465303\pi$
$20$	3.44024	0.769261
$21$	0	0
$22$	−6.10102	−1.30074
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−3.41068	−0.682137
$26$	10.6841	2.09532
$27$	0	0
$28$	5.38363	1.01741
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	0.780558	0.140192	0.0700962	−	0.997540i	$-0.477669\pi$
$31$	0.780558	0.140192	0.0700962	+	0.997540i	$0.477669\pi$
$32$	7.54312	1.33345
$33$	0	0
$34$	−16.5658	−2.84102
$35$	2.48712	0.420400
$36$	0	0
$37$	2.59391	0.426436	0.213218	−	0.977005i	$-0.431605\pi$
$37$	2.59391	0.426436	0.213218	+	0.977005i	$0.431605\pi$
$38$	−2.06235	−0.334558
$39$	0	0
$40$	−1.99819	−0.315941
$41$	−1.16041	−0.181226	−0.0906132	−	0.995886i	$-0.528883\pi$
$41$	−1.16041	−0.181226	−0.0906132	+	0.995886i	$0.528883\pi$
$42$	0	0
$43$	−3.06328	−0.467147	−0.233573	−	0.972339i	$-0.575042\pi$
$43$	−3.06328	−0.467147	−0.233573	+	0.972339i	$0.575042\pi$
$44$	7.65610	1.15420
$45$	0	0
$46$	2.17460	0.320627
$47$	−5.32731	−0.777068	−0.388534	−	0.921434i	$-0.627018\pi$
$47$	−5.32731	−0.777068	−0.388534	+	0.921434i	$0.627018\pi$
$48$	0	0
$49$	−3.10791	−0.443987
$50$	7.41686	1.04890
$51$	0	0
$52$	−13.4073	−1.85926
$53$	9.00013	1.23626	0.618132	−	0.786074i	$-0.287890\pi$
$53$	9.00013	1.23626	0.618132	+	0.786074i	$0.287890\pi$
$54$	0	0
$55$	3.53695	0.476922
$56$	−3.12696	−0.417858
$57$	0	0
$58$	2.17460	0.285539
$59$	8.64531	1.12552	0.562762	−	0.826619i	$-0.309739\pi$
$59$	8.64531	1.12552	0.562762	+	0.826619i	$0.309739\pi$
$60$	0	0
$61$	10.1026	1.29351	0.646756	−	0.762697i	$-0.276125\pi$
$61$	10.1026	1.29351	0.646756	+	0.762697i	$0.276125\pi$
$62$	−1.69740	−0.215570
$63$	0	0
$64$	−12.3813	−1.54766
$65$	−6.19389	−0.768258
$66$	0	0
$67$	2.24279	0.274000	0.137000	−	0.990571i	$-0.456254\pi$
$67$	2.24279	0.274000	0.137000	+	0.990571i	$0.456254\pi$
$68$	20.7883	2.52095
$69$	0	0
$70$	−5.40848	−0.646437
$71$	−13.3724	−1.58701	−0.793506	−	0.608562i	$-0.791747\pi$
$71$	−13.3724	−1.58701	−0.793506	+	0.608562i	$0.791747\pi$
$72$	0	0
$73$	1.58151	0.185102	0.0925511	−	0.995708i	$-0.470498\pi$
$73$	1.58151	0.185102	0.0925511	+	0.995708i	$0.470498\pi$
$74$	−5.64071	−0.655720
$75$	0	0
$76$	2.58802	0.296866
$77$	5.53497	0.630769
$78$	0	0
$79$	5.39042	0.606470	0.303235	−	0.952916i	$-0.401933\pi$
$79$	5.39042	0.606470	0.303235	+	0.952916i	$0.401933\pi$
$80$	−2.53522	−0.283447
$81$	0	0
$82$	2.52343	0.278667
$83$	−10.5871	−1.16208	−0.581041	−	0.813875i	$-0.697354\pi$
$83$	−10.5871	−1.16208	−0.581041	+	0.813875i	$0.697354\pi$
$84$	0	0
$85$	9.60373	1.04167
$86$	6.66141	0.718318
$87$	0	0
$88$	−4.44687	−0.474038
$89$	9.44598	1.00127	0.500636	−	0.865658i	$-0.333100\pi$
$89$	9.44598	1.00127	0.500636	+	0.865658i	$0.333100\pi$
$90$	0	0
$91$	−9.69281	−1.01608
$92$	−2.72887	−0.284505
$93$	0	0
$94$	11.5848	1.19488
$95$	1.19561	0.122667
$96$	0	0
$97$	−8.45927	−0.858908	−0.429454	−	0.903089i	$-0.641294\pi$
$97$	−8.45927	−0.858908	−0.429454	+	0.903089i	$0.641294\pi$
$98$	6.75845	0.682707
$99$	0	0
$100$	−9.30732	−0.930732
$101$	8.79445	0.875081	0.437540	−	0.899199i	$-0.355850\pi$
$101$	8.79445	0.875081	0.437540	+	0.899199i	$0.355850\pi$
$102$	0	0
$103$	18.0439	1.77792	0.888958	−	0.457989i	$-0.151430\pi$
$103$	18.0439	1.77792	0.888958	+	0.457989i	$0.151430\pi$
$104$	7.78734	0.763612
$105$	0	0
$106$	−19.5717	−1.90097
$107$	8.70700	0.841738	0.420869	−	0.907122i	$-0.361725\pi$
$107$	8.70700	0.841738	0.420869	+	0.907122i	$0.361725\pi$
$108$	0	0
$109$	17.1911	1.64661	0.823303	−	0.567602i	$-0.192129\pi$
$109$	17.1911	1.64661	0.823303	+	0.567602i	$0.192129\pi$
$110$	−7.69144	−0.733350
$111$	0	0
$112$	−3.96737	−0.374881
$113$	9.32014	0.876765	0.438383	−	0.898788i	$-0.355551\pi$
$113$	9.32014	0.876765	0.438383	+	0.898788i	$0.355551\pi$
$114$	0	0
$115$	−1.26068	−0.117559
$116$	−2.72887	−0.253370
$117$	0	0
$118$	−18.8001	−1.73069
$119$	15.0289	1.37769
$120$	0	0
$121$	−3.12867	−0.284425
$122$	−21.9692	−1.98900
$123$	0	0
$124$	2.13004	0.191284
$125$	−10.6032	−0.948378
$126$	0	0
$127$	−0.277678	−0.0246399	−0.0123199	−	0.999924i	$-0.503922\pi$
$127$	−0.277678	−0.0246399	−0.0123199	+	0.999924i	$0.503922\pi$
$128$	11.8380	1.04634
$129$	0	0
$130$	13.4692	1.18133
$131$	14.1746	1.23844	0.619222	−	0.785216i	$-0.287448\pi$
$131$	14.1746	1.23844	0.619222	+	0.785216i	$0.287448\pi$
$132$	0	0
$133$	1.87101	0.162237
$134$	−4.87716	−0.421322
$135$	0	0
$136$	−12.0744	−1.03537
$137$	18.5400	1.58398	0.791991	−	0.610533i	$-0.209045\pi$
$137$	18.5400	1.58398	0.791991	+	0.610533i	$0.209045\pi$
$138$	0	0
$139$	2.45921	0.208587	0.104294	−	0.994547i	$-0.466742\pi$
$139$	2.45921	0.208587	0.104294	+	0.994547i	$0.466742\pi$
$140$	6.78703	0.573609
$141$	0	0
$142$	29.0796	2.44030
$143$	−13.7842	−1.15269
$144$	0	0
$145$	−1.26068	−0.104694
$146$	−3.43916	−0.284627
$147$	0	0
$148$	7.07846	0.581846
$149$	−1.49012	−0.122076	−0.0610378	−	0.998135i	$-0.519441\pi$
$149$	−1.49012	−0.122076	−0.0610378	+	0.998135i	$0.519441\pi$
$150$	0	0
$151$	1.26392	0.102856	0.0514280	−	0.998677i	$-0.483623\pi$
$151$	1.26392	0.102856	0.0514280	+	0.998677i	$0.483623\pi$
$152$	−1.50320	−0.121925
$153$	0	0
$154$	−12.0363	−0.969915
$155$	0.984035	0.0790396
$156$	0	0
$157$	−22.6734	−1.80954	−0.904769	−	0.425903i	$-0.859957\pi$
$157$	−22.6734	−1.80954	−0.904769	+	0.425903i	$0.859957\pi$
$158$	−11.7220	−0.932552
$159$	0	0
$160$	9.50947	0.751789
$161$	−1.97284	−0.155481
$162$	0	0
$163$	10.3552	0.811084	0.405542	−	0.914076i	$-0.367083\pi$
$163$	10.3552	0.811084	0.405542	+	0.914076i	$0.367083\pi$
$164$	−3.16663	−0.247272
$165$	0	0
$166$	23.0226	1.78690
$167$	2.41321	0.186740	0.0933700	−	0.995631i	$-0.470236\pi$
$167$	2.41321	0.186740	0.0933700	+	0.995631i	$0.470236\pi$
$168$	0	0
$169$	11.1389	0.856836
$170$	−20.8842	−1.60175
$171$	0	0
$172$	−8.35932	−0.637392
$173$	−6.21692	−0.472664	−0.236332	−	0.971672i	$-0.575945\pi$
$173$	−6.21692	−0.472664	−0.236332	+	0.971672i	$0.575945\pi$
$174$	0	0
$175$	−6.72873	−0.508644
$176$	−5.64203	−0.425284
$177$	0	0
$178$	−20.5412	−1.53963
$179$	−6.15265	−0.459871	−0.229935	−	0.973206i	$-0.573851\pi$
$179$	−6.15265	−0.459871	−0.229935	+	0.973206i	$0.573851\pi$
$180$	0	0
$181$	−11.6477	−0.865766	−0.432883	−	0.901450i	$-0.642504\pi$
$181$	−11.6477	−0.865766	−0.432883	+	0.901450i	$0.642504\pi$
$182$	21.0780	1.56240
$183$	0	0
$184$	1.58501	0.116848
$185$	3.27009	0.240422
$186$	0	0
$187$	21.3727	1.56292
$188$	−14.5376	−1.06026
$189$	0	0
$190$	−2.59997	−0.188622
$191$	2.34372	0.169585	0.0847927	−	0.996399i	$-0.472977\pi$
$191$	2.34372	0.169585	0.0847927	+	0.996399i	$0.472977\pi$
$192$	0	0
$193$	−6.54014	−0.470770	−0.235385	−	0.971902i	$-0.575635\pi$
$193$	−6.54014	−0.470770	−0.235385	+	0.971902i	$0.575635\pi$
$194$	18.3955	1.32072
$195$	0	0
$196$	−8.48109	−0.605792
$197$	21.5991	1.53887	0.769435	−	0.638725i	$-0.220538\pi$
$197$	21.5991	1.53887	0.769435	+	0.638725i	$0.220538\pi$
$198$	0	0
$199$	19.8789	1.40918	0.704588	−	0.709617i	$-0.251132\pi$
$199$	19.8789	1.40918	0.704588	+	0.709617i	$0.251132\pi$
$200$	5.40595	0.382259
$201$	0	0
$202$	−19.1244	−1.34559
$203$	−1.97284	−0.138466
$204$	0	0
$205$	−1.46291	−0.102174
$206$	−39.2382	−2.73385
$207$	0	0
$208$	9.88029	0.685075
$209$	2.66078	0.184050
$210$	0	0
$211$	−25.3034	−1.74196	−0.870979	−	0.491320i	$-0.836515\pi$
$211$	−25.3034	−1.74196	−0.870979	+	0.491320i	$0.836515\pi$
$212$	24.5602	1.68680
$213$	0	0
$214$	−18.9342	−1.29432
$215$	−3.86182	−0.263374
$216$	0	0
$217$	1.53991	0.104536
$218$	−37.3837	−2.53194
$219$	0	0
$220$	9.65189	0.650730
$221$	−37.4277	−2.51766
$222$	0	0
$223$	2.52194	0.168881	0.0844407	−	0.996429i	$-0.473090\pi$
$223$	2.52194	0.168881	0.0844407	+	0.996429i	$0.473090\pi$
$224$	14.8814	0.994302
$225$	0	0
$226$	−20.2676	−1.34818
$227$	2.47395	0.164202	0.0821010	−	0.996624i	$-0.473837\pi$
$227$	2.47395	0.164202	0.0821010	+	0.996624i	$0.473837\pi$
$228$	0	0
$229$	−1.98196	−0.130971	−0.0654857	−	0.997854i	$-0.520860\pi$
$229$	−1.98196	−0.130971	−0.0654857	+	0.997854i	$0.520860\pi$
$230$	2.74147	0.180767
$231$	0	0
$232$	1.58501	0.104061
$233$	−11.1016	−0.727288	−0.363644	−	0.931538i	$-0.618468\pi$
$233$	−11.1016	−0.727288	−0.363644	+	0.931538i	$0.618468\pi$
$234$	0	0
$235$	−6.71604	−0.438106
$236$	23.5920	1.53571
$237$	0	0
$238$	−32.6817	−2.11844
$239$	12.5734	0.813306	0.406653	−	0.913583i	$-0.366696\pi$
$239$	12.5734	0.813306	0.406653	+	0.913583i	$0.366696\pi$
$240$	0	0
$241$	−2.28503	−0.147192	−0.0735960	−	0.997288i	$-0.523448\pi$
$241$	−2.28503	−0.147192	−0.0735960	+	0.997288i	$0.523448\pi$
$242$	6.80361	0.437352
$243$	0	0
$244$	27.5688	1.76491
$245$	−3.91808	−0.250317
$246$	0	0
$247$	−4.65954	−0.296479
$248$	−1.23719	−0.0785616
$249$	0	0
$250$	23.0577	1.45829
$251$	−11.2547	−0.710390	−0.355195	−	0.934792i	$-0.615586\pi$
$251$	−11.2547	−0.710390	−0.355195	+	0.934792i	$0.615586\pi$
$252$	0	0
$253$	−2.80559	−0.176386
$254$	0.603837	0.0378881
$255$	0	0
$256$	−0.980381	−0.0612738
$257$	−8.39977	−0.523963	−0.261982	−	0.965073i	$-0.584376\pi$
$257$	−8.39977	−0.523963	−0.261982	+	0.965073i	$0.584376\pi$
$258$	0	0
$259$	5.11737	0.317978
$260$	−16.9023	−1.04824
$261$	0	0
$262$	−30.8241	−1.90432
$263$	4.20481	0.259280	0.129640	−	0.991561i	$-0.458618\pi$
$263$	4.20481	0.259280	0.129640	+	0.991561i	$0.458618\pi$
$264$	0	0
$265$	11.3463	0.696997
$266$	−4.06869	−0.249467
$267$	0	0
$268$	6.12028	0.373856
$269$	−9.25812	−0.564478	−0.282239	−	0.959344i	$-0.591077\pi$
$269$	−9.25812	−0.564478	−0.282239	+	0.959344i	$0.591077\pi$
$270$	0	0
$271$	−2.26843	−0.137797	−0.0688986	−	0.997624i	$-0.521948\pi$
$271$	−2.26843	−0.137797	−0.0688986	+	0.997624i	$0.521948\pi$
$272$	−15.3196	−0.928885
$273$	0	0
$274$	−40.3171	−2.43564
$275$	−9.56898	−0.577031
$276$	0	0
$277$	−11.6491	−0.699929	−0.349965	−	0.936763i	$-0.613806\pi$
$277$	−11.6491	−0.699929	−0.349965	+	0.936763i	$0.613806\pi$
$278$	−5.34779	−0.320739
$279$	0	0
$280$	−3.94210	−0.235585
$281$	−17.3301	−1.03383	−0.516914	−	0.856037i	$-0.672919\pi$
$281$	−17.3301	−1.03383	−0.516914	+	0.856037i	$0.672919\pi$
$282$	0	0
$283$	23.1480	1.37601	0.688004	−	0.725707i	$-0.258487\pi$
$283$	23.1480	1.37601	0.688004	+	0.725707i	$0.258487\pi$
$284$	−36.4916	−2.16538
$285$	0	0
$286$	29.9751	1.77247
$287$	−2.28931	−0.135134
$288$	0	0
$289$	41.0323	2.41366
$290$	2.74147	0.160985
$291$	0	0
$292$	4.31575	0.252560
$293$	−3.48973	−0.203872	−0.101936	−	0.994791i	$-0.532504\pi$
$293$	−3.48973	−0.203872	−0.101936	+	0.994791i	$0.532504\pi$
$294$	0	0
$295$	10.8990	0.634563
$296$	−4.11137	−0.238968
$297$	0	0
$298$	3.24042	0.187712
$299$	4.91313	0.284134
$300$	0	0
$301$	−6.04336	−0.348334
$302$	−2.74851	−0.158159
$303$	0	0
$304$	−1.90720	−0.109385
$305$	12.7362	0.729273
$306$	0	0
$307$	10.5471	0.601956	0.300978	−	0.953631i	$-0.402687\pi$
$307$	10.5471	0.601956	0.300978	+	0.953631i	$0.402687\pi$
$308$	15.1042	0.860644
$309$	0	0
$310$	−2.13988	−0.121537
$311$	25.6998	1.45730	0.728650	−	0.684886i	$-0.240148\pi$
$311$	25.6998	1.45730	0.728650	+	0.684886i	$0.240148\pi$
$312$	0	0
$313$	7.52351	0.425254	0.212627	−	0.977133i	$-0.431798\pi$
$313$	7.52351	0.425254	0.212627	+	0.977133i	$0.431798\pi$
$314$	49.3056	2.78248
$315$	0	0
$316$	14.7098	0.827489
$317$	−32.9266	−1.84934	−0.924670	−	0.380770i	$-0.875659\pi$
$317$	−32.9266	−1.84934	−0.924670	+	0.380770i	$0.875659\pi$
$318$	0	0
$319$	−2.80559	−0.157083
$320$	−15.6088	−0.872559
$321$	0	0
$322$	4.29013	0.239080
$323$	7.22469	0.401993
$324$	0	0
$325$	16.7571	0.929519
$326$	−22.5184	−1.24718
$327$	0	0
$328$	1.83926	0.101556
$329$	−10.5099	−0.579431
$330$	0	0
$331$	−26.0679	−1.43282	−0.716410	−	0.697679i	$-0.754216\pi$
$331$	−26.0679	−1.43282	−0.716410	+	0.697679i	$0.754216\pi$
$332$	−28.8908	−1.58559
$333$	0	0
$334$	−5.24776	−0.287145
$335$	2.82744	0.154479
$336$	0	0
$337$	14.5956	0.795071	0.397536	−	0.917587i	$-0.369866\pi$
$337$	14.5956	0.795071	0.397536	+	0.917587i	$0.369866\pi$
$338$	−24.2226	−1.31753
$339$	0	0
$340$	26.2074	1.42129
$341$	2.18992	0.118591
$342$	0	0
$343$	−19.9413	−1.07673
$344$	4.85532	0.261782
$345$	0	0
$346$	13.5193	0.726802
$347$	25.8181	1.38599	0.692993	−	0.720944i	$-0.256292\pi$
$347$	25.8181	1.38599	0.692993	+	0.720944i	$0.256292\pi$
$348$	0	0
$349$	−15.3388	−0.821065	−0.410533	−	0.911846i	$-0.634657\pi$
$349$	−15.3388	−0.821065	−0.410533	+	0.911846i	$0.634657\pi$
$350$	14.6323	0.782128
$351$	0	0
$352$	21.1629	1.12799
$353$	27.1672	1.44596	0.722981	−	0.690867i	$-0.242771\pi$
$353$	27.1672	1.44596	0.722981	+	0.690867i	$0.242771\pi$
$354$	0	0
$355$	−16.8583	−0.894747
$356$	25.7769	1.36617
$357$	0	0
$358$	13.3795	0.707130
$359$	27.0423	1.42724	0.713619	−	0.700534i	$-0.247055\pi$
$359$	27.0423	1.42724	0.713619	+	0.700534i	$0.247055\pi$
$360$	0	0
$361$	−18.1006	−0.952661
$362$	25.3290	1.33126
$363$	0	0
$364$	−26.4505	−1.38638
$365$	1.99378	0.104359
$366$	0	0
$367$	10.5334	0.549838	0.274919	−	0.961467i	$-0.411349\pi$
$367$	10.5334	0.549838	0.274919	+	0.961467i	$0.411349\pi$
$368$	2.01100	0.104830
$369$	0	0
$370$	−7.11114	−0.369690
$371$	17.7558	0.921836
$372$	0	0
$373$	0.938818	0.0486101	0.0243051	−	0.999705i	$-0.492263\pi$
$373$	0.938818	0.0486101	0.0243051	+	0.999705i	$0.492263\pi$
$374$	−46.4770	−2.40327
$375$	0	0
$376$	8.44382	0.435457
$377$	4.91313	0.253039
$378$	0	0
$379$	−7.94239	−0.407973	−0.203987	−	0.978974i	$-0.565390\pi$
$379$	−7.94239	−0.407973	−0.203987	+	0.978974i	$0.565390\pi$
$380$	3.26267	0.167371
$381$	0	0
$382$	−5.09664	−0.260767
$383$	−20.6050	−1.05287	−0.526433	−	0.850217i	$-0.676471\pi$
$383$	−20.6050	−1.05287	−0.526433	+	0.850217i	$0.676471\pi$
$384$	0	0
$385$	6.97783	0.355623
$386$	14.2222	0.723890
$387$	0	0
$388$	−23.0843	−1.17193
$389$	−20.6799	−1.04851	−0.524257	−	0.851560i	$-0.675657\pi$
$389$	−20.6799	−1.04851	−0.524257	+	0.851560i	$0.675657\pi$
$390$	0	0
$391$	−7.61789	−0.385253
$392$	4.92606	0.248803
$393$	0	0
$394$	−46.9693	−2.36628
$395$	6.79560	0.341924
$396$	0	0
$397$	9.70884	0.487273	0.243636	−	0.969867i	$-0.421660\pi$
$397$	9.70884	0.487273	0.243636	+	0.969867i	$0.421660\pi$
$398$	−43.2285	−2.16685
$399$	0	0
$400$	6.85887	0.342944
$401$	0.722519	0.0360809	0.0180404	−	0.999837i	$-0.494257\pi$
$401$	0.722519	0.0360809	0.0180404	+	0.999837i	$0.494257\pi$
$402$	0	0
$403$	−3.83499	−0.191034
$404$	23.9989	1.19399
$405$	0	0
$406$	4.29013	0.212915
$407$	7.27745	0.360730
$408$	0	0
$409$	−27.8520	−1.37719	−0.688596	−	0.725145i	$-0.741773\pi$
$409$	−27.8520	−1.37719	−0.688596	+	0.725145i	$0.741773\pi$
$410$	3.18125	0.157111
$411$	0	0
$412$	49.2394	2.42585
$413$	17.0558	0.839261
$414$	0	0
$415$	−13.3469	−0.655174
$416$	−37.0603	−1.81703
$417$	0	0
$418$	−5.78612	−0.283008
$419$	−31.9366	−1.56021	−0.780103	−	0.625651i	$-0.784833\pi$
$419$	−31.9366	−1.56021	−0.780103	+	0.625651i	$0.784833\pi$
$420$	0	0
$421$	12.4853	0.608497	0.304249	−	0.952593i	$-0.401595\pi$
$421$	12.4853	0.608497	0.304249	+	0.952593i	$0.401595\pi$
$422$	55.0247	2.67856
$423$	0	0
$424$	−14.2653	−0.692783
$425$	−25.9822	−1.26032
$426$	0	0
$427$	19.9309	0.964523
$428$	23.7603	1.14850
$429$	0	0
$430$	8.39791	0.404983
$431$	19.2481	0.927146	0.463573	−	0.886059i	$-0.346567\pi$
$431$	19.2481	0.927146	0.463573	+	0.886059i	$0.346567\pi$
$432$	0	0
$433$	15.0631	0.723889	0.361944	−	0.932200i	$-0.382113\pi$
$433$	15.0631	0.723889	0.361944	+	0.932200i	$0.382113\pi$
$434$	−3.34869	−0.160742
$435$	0	0
$436$	46.9123	2.24669
$437$	−0.948384	−0.0453674
$438$	0	0
$439$	16.2718	0.776609	0.388304	−	0.921531i	$-0.373061\pi$
$439$	16.2718	0.776609	0.388304	+	0.921531i	$0.373061\pi$
$440$	−5.60609	−0.267260
$441$	0	0
$442$	81.3902	3.87134
$443$	33.6668	1.59956	0.799780	−	0.600294i	$-0.204950\pi$
$443$	33.6668	1.59956	0.799780	+	0.600294i	$0.204950\pi$
$444$	0	0
$445$	11.9084	0.564511
$446$	−5.48419	−0.259684
$447$	0	0
$448$	−24.4262	−1.15403
$449$	10.2329	0.482919	0.241460	−	0.970411i	$-0.422374\pi$
$449$	10.2329	0.482919	0.241460	+	0.970411i	$0.422374\pi$
$450$	0	0
$451$	−3.25565	−0.153302
$452$	25.4335	1.19629
$453$	0	0
$454$	−5.37985	−0.252489
$455$	−12.2195	−0.572861
$456$	0	0
$457$	−1.11474	−0.0521454	−0.0260727	−	0.999660i	$-0.508300\pi$
$457$	−1.11474	−0.0521454	−0.0260727	+	0.999660i	$0.508300\pi$
$458$	4.30996	0.201391
$459$	0	0
$460$	−3.44024	−0.160402
$461$	10.2123	0.475633	0.237817	−	0.971310i	$-0.423568\pi$
$461$	10.2123	0.475633	0.237817	+	0.971310i	$0.423568\pi$
$462$	0	0
$463$	−23.9211	−1.11171	−0.555854	−	0.831280i	$-0.687609\pi$
$463$	−23.9211	−1.11171	−0.555854	+	0.831280i	$0.687609\pi$
$464$	2.01100	0.0933582
$465$	0	0
$466$	24.1414	1.11833
$467$	5.43565	0.251532	0.125766	−	0.992060i	$-0.459861\pi$
$467$	5.43565	0.251532	0.125766	+	0.992060i	$0.459861\pi$
$468$	0	0
$469$	4.42465	0.204312
$470$	14.6047	0.673664
$471$	0	0
$472$	−13.7029	−0.630726
$473$	−8.59432	−0.395167
$474$	0	0
$475$	−3.23464	−0.148415
$476$	41.0119	1.87978
$477$	0	0
$478$	−27.3421	−1.25060
$479$	4.24526	0.193971	0.0969855	−	0.995286i	$-0.469080\pi$
$479$	4.24526	0.193971	0.0969855	+	0.995286i	$0.469080\pi$
$480$	0	0
$481$	−12.7442	−0.581087
$482$	4.96903	0.226333
$483$	0	0
$484$	−8.53776	−0.388080
$485$	−10.6644	−0.484247
$486$	0	0
$487$	−4.65303	−0.210849	−0.105424	−	0.994427i	$-0.533620\pi$
$487$	−4.65303	−0.210849	−0.105424	+	0.994427i	$0.533620\pi$
$488$	−16.0128	−0.724863
$489$	0	0
$490$	8.52025	0.384906
$491$	−14.4447	−0.651878	−0.325939	−	0.945391i	$-0.605680\pi$
$491$	−14.4447	−0.651878	−0.325939	+	0.945391i	$0.605680\pi$
$492$	0	0
$493$	−7.61789	−0.343093
$494$	10.1326	0.455888
$495$	0	0
$496$	−1.56970	−0.0704816
$497$	−26.3816	−1.18338
$498$	0	0
$499$	−37.4369	−1.67591	−0.837953	−	0.545743i	$-0.816247\pi$
$499$	−37.4369	−1.67591	−0.837953	+	0.545743i	$0.816247\pi$
$500$	−28.9348	−1.29400
$501$	0	0
$502$	24.4744	1.09235
$503$	−7.62692	−0.340067	−0.170034	−	0.985438i	$-0.554388\pi$
$503$	−7.62692	−0.340067	−0.170034	+	0.985438i	$0.554388\pi$
$504$	0	0
$505$	11.0870	0.493365
$506$	6.10102	0.271224
$507$	0	0
$508$	−0.757747	−0.0336196
$509$	21.3713	0.947267	0.473634	−	0.880722i	$-0.342942\pi$
$509$	21.3713	0.947267	0.473634	+	0.880722i	$0.342942\pi$
$510$	0	0
$511$	3.12007	0.138024
$512$	−21.5441	−0.952123
$513$	0	0
$514$	18.2661	0.805684
$515$	22.7476	1.00238
$516$	0	0
$517$	−14.9462	−0.657335
$518$	−11.1282	−0.488946
$519$	0	0
$520$	9.81736	0.430520
$521$	21.6716	0.949452	0.474726	−	0.880134i	$-0.342547\pi$
$521$	21.6716	0.949452	0.474726	+	0.880134i	$0.342547\pi$
$522$	0	0
$523$	−23.4720	−1.02636	−0.513179	−	0.858282i	$-0.671532\pi$
$523$	−23.4720	−1.02636	−0.513179	+	0.858282i	$0.671532\pi$
$524$	38.6808	1.68978
$525$	0	0
$526$	−9.14378	−0.398688
$527$	5.94621	0.259021
$528$	0	0
$529$	1.00000	0.0434783
$530$	−24.6736	−1.07175
$531$	0	0
$532$	5.10575	0.221362
$533$	5.70127	0.246949
$534$	0	0
$535$	10.9767	0.474566
$536$	−3.55483	−0.153545
$537$	0	0
$538$	20.1327	0.867982
$539$	−8.71952	−0.375576
$540$	0	0
$541$	−4.20268	−0.180687	−0.0903436	−	0.995911i	$-0.528797\pi$
$541$	−4.20268	−0.180687	−0.0903436	+	0.995911i	$0.528797\pi$
$542$	4.93292	0.211887
$543$	0	0
$544$	57.4627	2.46369
$545$	21.6725	0.928346
$546$	0	0
$547$	33.2643	1.42228	0.711140	−	0.703050i	$-0.248179\pi$
$547$	33.2643	1.42228	0.711140	+	0.703050i	$0.248179\pi$
$548$	50.5934	2.16124
$549$	0	0
$550$	20.8087	0.887285
$551$	−0.948384	−0.0404025
$552$	0	0
$553$	10.6344	0.452222
$554$	25.3322	1.07626
$555$	0	0
$556$	6.71087	0.284604
$557$	29.2119	1.23775	0.618875	−	0.785489i	$-0.287589\pi$
$557$	29.2119	1.23775	0.618875	+	0.785489i	$0.287589\pi$
$558$	0	0
$559$	15.0503	0.636561
$560$	−5.00159	−0.211356
$561$	0	0
$562$	37.6860	1.58969
$563$	−24.8028	−1.04531	−0.522656	−	0.852544i	$-0.675059\pi$
$563$	−24.8028	−1.04531	−0.522656	+	0.852544i	$0.675059\pi$
$564$	0	0
$565$	11.7497	0.494315
$566$	−50.3377	−2.11585
$567$	0	0
$568$	21.1953	0.889337
$569$	35.8760	1.50400	0.752000	−	0.659163i	$-0.229089\pi$
$569$	35.8760	1.50400	0.752000	+	0.659163i	$0.229089\pi$
$570$	0	0
$571$	2.29056	0.0958568	0.0479284	−	0.998851i	$-0.484738\pi$
$571$	2.29056	0.0958568	0.0479284	+	0.998851i	$0.484738\pi$
$572$	−37.6154	−1.57278
$573$	0	0
$574$	4.97833	0.207791
$575$	3.41068	0.142235
$576$	0	0
$577$	−16.5189	−0.687692	−0.343846	−	0.939026i	$-0.611730\pi$
$577$	−16.5189	−0.687692	−0.343846	+	0.939026i	$0.611730\pi$
$578$	−89.2287	−3.71143
$579$	0	0
$580$	−3.44024	−0.142848
$581$	−20.8866	−0.866520
$582$	0	0
$583$	25.2507	1.04578
$584$	−2.50671	−0.103728
$585$	0	0
$586$	7.58876	0.313489
$587$	40.8024	1.68409	0.842047	−	0.539404i	$-0.181351\pi$
$587$	40.8024	1.68409	0.842047	+	0.539404i	$0.181351\pi$
$588$	0	0
$589$	0.740269	0.0305023
$590$	−23.7009	−0.975750
$591$	0	0
$592$	−5.21635	−0.214391
$593$	31.5907	1.29728	0.648638	−	0.761097i	$-0.275339\pi$
$593$	31.5907	1.29728	0.648638	+	0.761097i	$0.275339\pi$
$594$	0	0
$595$	18.9466	0.776735
$596$	−4.06636	−0.166564
$597$	0	0
$598$	−10.6841	−0.436905
$599$	3.17573	0.129757	0.0648784	−	0.997893i	$-0.479334\pi$
$599$	3.17573	0.129757	0.0648784	+	0.997893i	$0.479334\pi$
$600$	0	0
$601$	7.76111	0.316582	0.158291	−	0.987392i	$-0.449402\pi$
$601$	7.76111	0.316582	0.158291	+	0.987392i	$0.449402\pi$
$602$	13.1419	0.535623
$603$	0	0
$604$	3.44907	0.140341
$605$	−3.94426	−0.160357
$606$	0	0
$607$	−6.99598	−0.283958	−0.141979	−	0.989870i	$-0.545347\pi$
$607$	−6.99598	−0.283958	−0.141979	+	0.989870i	$0.545347\pi$
$608$	7.15378	0.290124
$609$	0	0
$610$	−27.6961	−1.12138
$611$	26.1738	1.05888
$612$	0	0
$613$	22.6656	0.915454	0.457727	−	0.889093i	$-0.348664\pi$
$613$	22.6656	0.915454	0.457727	+	0.889093i	$0.348664\pi$
$614$	−22.9357	−0.925611
$615$	0	0
$616$	−8.77296	−0.353473
$617$	39.8319	1.60357	0.801786	−	0.597611i	$-0.203883\pi$
$617$	39.8319	1.60357	0.801786	+	0.597611i	$0.203883\pi$
$618$	0	0
$619$	0.945926	0.0380200	0.0190100	−	0.999819i	$-0.493949\pi$
$619$	0.945926	0.0380200	0.0190100	+	0.999819i	$0.493949\pi$
$620$	2.68531	0.107844
$621$	0	0
$622$	−55.8867	−2.24085
$623$	18.6354	0.746611
$624$	0	0
$625$	3.68619	0.147447
$626$	−16.3606	−0.653901
$627$	0	0
$628$	−61.8730	−2.46900
$629$	19.7601	0.787888
$630$	0	0
$631$	48.0868	1.91430	0.957152	−	0.289585i	$-0.0935173\pi$
$631$	48.0868	1.91430	0.957152	+	0.289585i	$0.0935173\pi$
$632$	−8.54385	−0.339856
$633$	0	0
$634$	71.6020	2.84368
$635$	−0.350063	−0.0138918
$636$	0	0
$637$	15.2696	0.605003
$638$	6.10102	0.241542
$639$	0	0
$640$	14.9239	0.589921
$641$	−25.1911	−0.994988	−0.497494	−	0.867468i	$-0.665746\pi$
$641$	−25.1911	−0.994988	−0.497494	+	0.867468i	$0.665746\pi$
$642$	0	0
$643$	−19.7165	−0.777544	−0.388772	−	0.921334i	$-0.627101\pi$
$643$	−19.7165	−0.777544	−0.388772	+	0.921334i	$0.627101\pi$
$644$	−5.38363	−0.212145
$645$	0	0
$646$	−15.7108	−0.618133
$647$	−7.35803	−0.289274	−0.144637	−	0.989485i	$-0.546201\pi$
$647$	−7.35803	−0.289274	−0.144637	+	0.989485i	$0.546201\pi$
$648$	0	0
$649$	24.2552	0.952099
$650$	−36.4400	−1.42930
$651$	0	0
$652$	28.2581	1.10667
$653$	39.4879	1.54528	0.772641	−	0.634843i	$-0.218935\pi$
$653$	39.4879	1.54528	0.772641	+	0.634843i	$0.218935\pi$
$654$	0	0
$655$	17.8697	0.698226
$656$	2.33359	0.0911114
$657$	0	0
$658$	22.8549	0.890975
$659$	−42.0600	−1.63842	−0.819212	−	0.573491i	$-0.805589\pi$
$659$	−42.0600	−1.63842	−0.819212	+	0.573491i	$0.805589\pi$
$660$	0	0
$661$	5.95136	0.231481	0.115741	−	0.993279i	$-0.463076\pi$
$661$	5.95136	0.231481	0.115741	+	0.993279i	$0.463076\pi$
$662$	56.6872	2.20321
$663$	0	0
$664$	16.7806	0.651212
$665$	2.35875	0.0914682
$666$	0	0
$667$	1.00000	0.0387202
$668$	6.58535	0.254795
$669$	0	0
$670$	−6.14854	−0.237539
$671$	28.3439	1.09420
$672$	0	0
$673$	−9.35345	−0.360549	−0.180274	−	0.983616i	$-0.557699\pi$
$673$	−9.35345	−0.360549	−0.180274	+	0.983616i	$0.557699\pi$
$674$	−31.7395	−1.22256
$675$	0	0
$676$	30.3966	1.16910
$677$	32.4221	1.24608	0.623042	−	0.782188i	$-0.285897\pi$
$677$	32.4221	1.24608	0.623042	+	0.782188i	$0.285897\pi$
$678$	0	0
$679$	−16.6888	−0.640456
$680$	−15.2220	−0.583736
$681$	0	0
$682$	−4.76220	−0.182354
$683$	34.0491	1.30285	0.651427	−	0.758711i	$-0.274171\pi$
$683$	34.0491	1.30285	0.651427	+	0.758711i	$0.274171\pi$
$684$	0	0
$685$	23.3731	0.893038
$686$	43.3642	1.65565
$687$	0	0
$688$	6.16026	0.234858
$689$	−44.2189	−1.68460
$690$	0	0
$691$	−40.1018	−1.52554	−0.762772	−	0.646667i	$-0.776162\pi$
$691$	−40.1018	−1.52554	−0.762772	+	0.646667i	$0.776162\pi$
$692$	−16.9652	−0.644919
$693$	0	0
$694$	−56.1439	−2.13119
$695$	3.10028	0.117600
$696$	0	0
$697$	−8.83992	−0.334836
$698$	33.3556	1.26253
$699$	0	0
$700$	−18.3618	−0.694012
$701$	−16.9770	−0.641211	−0.320605	−	0.947213i	$-0.603886\pi$
$701$	−16.9770	−0.641211	−0.320605	+	0.947213i	$0.603886\pi$
$702$	0	0
$703$	2.46003	0.0927816
$704$	−34.7367	−1.30919
$705$	0	0
$706$	−59.0777	−2.22342
$707$	17.3500	0.652515
$708$	0	0
$709$	26.7087	1.00306	0.501532	−	0.865139i	$-0.332770\pi$
$709$	26.7087	1.00306	0.501532	+	0.865139i	$0.332770\pi$
$710$	36.6601	1.37583
$711$	0	0
$712$	−14.9719	−0.561097
$713$	−0.780558	−0.0292321
$714$	0	0
$715$	−17.3775	−0.649882
$716$	−16.7898	−0.627464
$717$	0	0
$718$	−58.8061	−2.19463
$719$	14.3165	0.533916	0.266958	−	0.963708i	$-0.413981\pi$
$719$	14.3165	0.533916	0.266958	+	0.963708i	$0.413981\pi$
$720$	0	0
$721$	35.5976	1.32573
$722$	39.3614	1.46488
$723$	0	0
$724$	−31.7851	−1.18128
$725$	3.41068	0.126670
$726$	0	0
$727$	−43.6185	−1.61772	−0.808860	−	0.588001i	$-0.799915\pi$
$727$	−43.6185	−1.61772	−0.808860	+	0.588001i	$0.799915\pi$
$728$	15.3632	0.569397
$729$	0	0
$730$	−4.33568	−0.160471
$731$	−23.3358	−0.863105
$732$	0	0
$733$	20.6163	0.761481	0.380740	−	0.924682i	$-0.375669\pi$
$733$	20.6163	0.761481	0.380740	+	0.924682i	$0.375669\pi$
$734$	−22.9059	−0.845471
$735$	0	0
$736$	−7.54312	−0.278043
$737$	6.29233	0.231781
$738$	0	0
$739$	−14.2345	−0.523626	−0.261813	−	0.965119i	$-0.584320\pi$
$739$	−14.2345	−0.523626	−0.261813	+	0.965119i	$0.584320\pi$
$740$	8.92367	0.328041
$741$	0	0
$742$	−38.6117	−1.41748
$743$	50.2563	1.84372	0.921862	−	0.387519i	$-0.126668\pi$
$743$	50.2563	1.84372	0.921862	+	0.387519i	$0.126668\pi$
$744$	0	0
$745$	−1.87857	−0.0688254
$746$	−2.04155	−0.0747465
$747$	0	0
$748$	58.3233	2.13251
$749$	17.1775	0.627652
$750$	0	0
$751$	12.8396	0.468523	0.234261	−	0.972174i	$-0.424733\pi$
$751$	12.8396	0.468523	0.234261	+	0.972174i	$0.424733\pi$
$752$	10.7132	0.390671
$753$	0	0
$754$	−10.6841	−0.389091
$755$	1.59339	0.0579896
$756$	0	0
$757$	0.0409374	0.00148789	0.000743947	−	1.00000i	$-0.499763\pi$
$757$	0.0409374	0.00148789	0.000743947		1.00000i	$0.499763\pi$
$758$	17.2715	0.627329
$759$	0	0
$760$	−1.89505	−0.0687407
$761$	−14.0206	−0.508248	−0.254124	−	0.967172i	$-0.581787\pi$
$761$	−14.0206	−0.508248	−0.254124	+	0.967172i	$0.581787\pi$
$762$	0	0
$763$	33.9152	1.22781
$764$	6.39570	0.231388
$765$	0	0
$766$	44.8075	1.61896
$767$	−42.4756	−1.53370
$768$	0	0
$769$	19.5698	0.705706	0.352853	−	0.935679i	$-0.385212\pi$
$769$	19.5698	0.705706	0.352853	+	0.935679i	$0.385212\pi$
$770$	−15.1740	−0.546832
$771$	0	0
$772$	−17.8472	−0.642336
$773$	19.5272	0.702346	0.351173	−	0.936311i	$-0.385783\pi$
$773$	19.5272	0.702346	0.351173	+	0.936311i	$0.385783\pi$
$774$	0	0
$775$	−2.66224	−0.0956304
$776$	13.4080	0.481319
$777$	0	0
$778$	44.9705	1.61227
$779$	−1.10052	−0.0394302
$780$	0	0
$781$	−37.5175	−1.34248
$782$	16.5658	0.592394
$783$	0	0
$784$	6.25000	0.223214
$785$	−28.5840	−1.02021
$786$	0	0
$787$	−1.88331	−0.0671327	−0.0335663	−	0.999436i	$-0.510687\pi$
$787$	−1.88331	−0.0671327	−0.0335663	+	0.999436i	$0.510687\pi$
$788$	58.9412	2.09969
$789$	0	0
$790$	−14.7777	−0.525767
$791$	18.3871	0.653771
$792$	0	0
$793$	−49.6356	−1.76261
$794$	−21.1128	−0.749266
$795$	0	0
$796$	54.2469	1.92273
$797$	−41.4554	−1.46843	−0.734213	−	0.678919i	$-0.762449\pi$
$797$	−41.4554	−1.46843	−0.734213	+	0.678919i	$0.762449\pi$
$798$	0	0
$799$	−40.5829	−1.43572
$800$	−25.7272	−0.909594
$801$	0	0
$802$	−1.57119	−0.0554805
$803$	4.43708	0.156581
$804$	0	0
$805$	−2.48712	−0.0876594
$806$	8.33955	0.293748
$807$	0	0
$808$	−13.9393	−0.490382
$809$	−0.539019	−0.0189509	−0.00947545	−	0.999955i	$-0.503016\pi$
$809$	−0.539019	−0.0189509	−0.00947545	+	0.999955i	$0.503016\pi$
$810$	0	0
$811$	−9.11311	−0.320005	−0.160002	−	0.987117i	$-0.551150\pi$
$811$	−9.11311	−0.320005	−0.160002	+	0.987117i	$0.551150\pi$
$812$	−5.38363	−0.188928
$813$	0	0
$814$	−15.8255	−0.554684
$815$	13.0546	0.457284
$816$	0	0
$817$	−2.90517	−0.101639
$818$	60.5669	2.11767
$819$	0	0
$820$	−3.99210	−0.139410
$821$	−6.54944	−0.228577	−0.114288	−	0.993448i	$-0.536459\pi$
$821$	−6.54944	−0.228577	−0.114288	+	0.993448i	$0.536459\pi$
$822$	0	0
$823$	24.6041	0.857646	0.428823	−	0.903389i	$-0.358928\pi$
$823$	24.6041	0.857646	0.428823	+	0.903389i	$0.358928\pi$
$824$	−28.5996	−0.996316
$825$	0	0
$826$	−37.0895	−1.29051
$827$	−10.1294	−0.352235	−0.176118	−	0.984369i	$-0.556354\pi$
$827$	−10.1294	−0.352235	−0.176118	+	0.984369i	$0.556354\pi$
$828$	0	0
$829$	44.1771	1.53433	0.767166	−	0.641448i	$-0.221666\pi$
$829$	44.1771	1.53433	0.767166	+	0.641448i	$0.221666\pi$
$830$	29.0241	1.00744
$831$	0	0
$832$	60.8307	2.10893
$833$	−23.6757	−0.820315
$834$	0	0
$835$	3.04229	0.105283
$836$	7.26092	0.251124
$837$	0	0
$838$	69.4493	2.39909
$839$	−43.0106	−1.48489	−0.742445	−	0.669907i	$-0.766334\pi$
$839$	−43.0106	−1.48489	−0.742445	+	0.669907i	$0.766334\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−27.1505	−0.935669
$843$	0	0
$844$	−69.0498	−2.37679
$845$	14.0426	0.483079
$846$	0	0
$847$	−6.17237	−0.212085
$848$	−18.0992	−0.621531
$849$	0	0
$850$	56.5009	1.93796
$851$	−2.59391	−0.0889181
$852$	0	0
$853$	40.8126	1.39740	0.698698	−	0.715417i	$-0.253763\pi$
$853$	40.8126	1.39740	0.698698	+	0.715417i	$0.253763\pi$
$854$	−43.3416	−1.48312
$855$	0	0
$856$	−13.8007	−0.471697
$857$	−13.3169	−0.454897	−0.227448	−	0.973790i	$-0.573038\pi$
$857$	−13.3169	−0.454897	−0.227448	+	0.973790i	$0.573038\pi$
$858$	0	0
$859$	20.1413	0.687211	0.343606	−	0.939114i	$-0.388352\pi$
$859$	20.1413	0.687211	0.343606	+	0.939114i	$0.388352\pi$
$860$	−10.5384	−0.359357
$861$	0	0
$862$	−41.8568	−1.42565
$863$	3.71870	0.126586	0.0632930	−	0.997995i	$-0.479840\pi$
$863$	3.71870	0.126586	0.0632930	+	0.997995i	$0.479840\pi$
$864$	0	0
$865$	−7.83755	−0.266485
$866$	−32.7563	−1.11310
$867$	0	0
$868$	4.20223	0.142633
$869$	15.1233	0.513023
$870$	0	0
$871$	−11.0191	−0.373368
$872$	−27.2480	−0.922732
$873$	0	0
$874$	2.06235	0.0697602
$875$	−20.9184	−0.707170
$876$	0	0
$877$	22.4550	0.758251	0.379126	−	0.925345i	$-0.376225\pi$
$877$	22.4550	0.758251	0.379126	+	0.925345i	$0.376225\pi$
$878$	−35.3845	−1.19417
$879$	0	0
$880$	−7.11280	−0.239772
$881$	26.9030	0.906386	0.453193	−	0.891412i	$-0.350285\pi$
$881$	26.9030	0.906386	0.453193	+	0.891412i	$0.350285\pi$
$882$	0	0
$883$	28.4011	0.955775	0.477887	−	0.878421i	$-0.341403\pi$
$883$	28.4011	0.955775	0.477887	+	0.878421i	$0.341403\pi$
$884$	−102.135	−3.43519
$885$	0	0
$886$	−73.2118	−2.45960
$887$	14.0374	0.471329	0.235665	−	0.971834i	$-0.424273\pi$
$887$	14.0374	0.471329	0.235665	+	0.971834i	$0.424273\pi$
$888$	0	0
$889$	−0.547813	−0.0183731
$890$	−25.8959	−0.868032
$891$	0	0
$892$	6.88204	0.230428
$893$	−5.05234	−0.169070
$894$	0	0
$895$	−7.75653	−0.259272
$896$	23.3545	0.780218
$897$	0	0
$898$	−22.2524	−0.742572
$899$	−0.780558	−0.0260331
$900$	0	0
$901$	68.5621	2.28413
$902$	7.07972	0.235729
$903$	0	0
$904$	−14.7725	−0.491326
$905$	−14.6840	−0.488113
$906$	0	0
$907$	23.7647	0.789095	0.394548	−	0.918875i	$-0.370901\pi$
$907$	23.7647	0.789095	0.394548	+	0.918875i	$0.370901\pi$
$908$	6.75110	0.224043
$909$	0	0
$910$	26.5726	0.880873
$911$	−33.8905	−1.12284	−0.561421	−	0.827530i	$-0.689745\pi$
$911$	−33.8905	−1.12284	−0.561421	+	0.827530i	$0.689745\pi$
$912$	0	0
$913$	−29.7029	−0.983024
$914$	2.42411	0.0801826
$915$	0	0
$916$	−5.40851	−0.178702
$917$	27.9642	0.923461
$918$	0	0
$919$	−8.74138	−0.288352	−0.144176	−	0.989552i	$-0.546053\pi$
$919$	−8.74138	−0.288352	−0.144176	+	0.989552i	$0.546053\pi$
$920$	1.99819	0.0658783
$921$	0	0
$922$	−22.2076	−0.731368
$923$	65.7004	2.16255
$924$	0	0
$925$	−8.84701	−0.290888
$926$	52.0188	1.70944
$927$	0	0
$928$	−7.54312	−0.247615
$929$	23.5070	0.771239	0.385620	−	0.922658i	$-0.373988\pi$
$929$	23.5070	0.771239	0.385620	+	0.922658i	$0.373988\pi$
$930$	0	0
$931$	−2.94749	−0.0966002
$932$	−30.2948	−0.992338
$933$	0	0
$934$	−11.8203	−0.386774
$935$	26.9441	0.881167
$936$	0	0
$937$	−9.26029	−0.302520	−0.151260	−	0.988494i	$-0.548333\pi$
$937$	−9.26029	−0.302520	−0.151260	+	0.988494i	$0.548333\pi$
$938$	−9.62184	−0.314164
$939$	0	0
$940$	−18.3272	−0.597768
$941$	−9.23069	−0.300912	−0.150456	−	0.988617i	$-0.548074\pi$
$941$	−9.23069	−0.300912	−0.150456	+	0.988617i	$0.548074\pi$
$942$	0	0
$943$	1.16041	0.0377883
$944$	−17.3857	−0.565856
$945$	0	0
$946$	18.6892	0.607638
$947$	−28.3587	−0.921534	−0.460767	−	0.887521i	$-0.652426\pi$
$947$	−28.3587	−0.921534	−0.460767	+	0.887521i	$0.652426\pi$
$948$	0	0
$949$	−7.77019	−0.252231
$950$	7.03404	0.228214
$951$	0	0
$952$	−23.8208	−0.772038
$953$	−39.7690	−1.28824	−0.644122	−	0.764922i	$-0.722777\pi$
$953$	−39.7690	−1.28824	−0.644122	+	0.764922i	$0.722777\pi$
$954$	0	0
$955$	2.95468	0.0956111
$956$	34.3112	1.10970
$957$	0	0
$958$	−9.23174	−0.298264
$959$	36.5765	1.18112
$960$	0	0
$961$	−30.3907	−0.980346
$962$	27.7136	0.893521
$963$	0	0
$964$	−6.23557	−0.200834
$965$	−8.24503	−0.265417
$966$	0	0
$967$	−24.2808	−0.780818	−0.390409	−	0.920642i	$-0.627666\pi$
$967$	−24.2808	−0.780818	−0.390409	+	0.920642i	$0.627666\pi$
$968$	4.95897	0.159387
$969$	0	0
$970$	23.1908	0.744613
$971$	17.8792	0.573770	0.286885	−	0.957965i	$-0.407380\pi$
$971$	17.8792	0.573770	0.286885	+	0.957965i	$0.407380\pi$
$972$	0	0
$973$	4.85162	0.155536
$974$	10.1185	0.324216
$975$	0	0
$976$	−20.3164	−0.650312
$977$	47.0835	1.50633	0.753167	−	0.657829i	$-0.228525\pi$
$977$	47.0835	1.50633	0.753167	+	0.657829i	$0.228525\pi$
$978$	0	0
$979$	26.5015	0.846993
$980$	−10.6920	−0.341542
$981$	0	0
$982$	31.4113	1.00238
$983$	−35.3151	−1.12638	−0.563189	−	0.826328i	$-0.690426\pi$
$983$	−35.3151	−1.12638	−0.563189	+	0.826328i	$0.690426\pi$
$984$	0	0
$985$	27.2295	0.867606
$986$	16.5658	0.527564
$987$	0	0
$988$	−12.7153	−0.404527
$989$	3.06328	0.0974068
$990$	0	0
$991$	16.6079	0.527569	0.263784	−	0.964582i	$-0.415029\pi$
$991$	16.6079	0.527569	0.263784	+	0.964582i	$0.415029\pi$
$992$	5.88784	0.186939
$993$	0	0
$994$	57.3693	1.81964
$995$	25.0609	0.794484
$996$	0	0
$997$	21.3164	0.675097	0.337548	−	0.941308i	$-0.390402\pi$
$997$	21.3164	0.675097	0.337548	+	0.941308i	$0.390402\pi$
$998$	81.4101	2.57699
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.l.1.1		10
3.2	odd	2		667.2.a.a.1.10	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.a.a.1.10	✓	10	3.2	odd	2
6003.2.a.l.1.1		10	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)

\(f(q)\)	\(=\)	\(q-2.17460 q^{2} +2.72887 q^{4} +1.26068 q^{5} +1.97284 q^{7} -1.58501 q^{8} +O(q^{10})\)	\(q-2.17460 q^{2} +2.72887 q^{4} +1.26068 q^{5} +1.97284 q^{7} -1.58501 q^{8} -2.74147 q^{10} +2.80559 q^{11} -4.91313 q^{13} -4.29013 q^{14} -2.01100 q^{16} +7.61789 q^{17} +0.948384 q^{19} +3.44024 q^{20} -6.10102 q^{22} -1.00000 q^{23} -3.41068 q^{25} +10.6841 q^{26} +5.38363 q^{28} -1.00000 q^{29} +0.780558 q^{31} +7.54312 q^{32} -16.5658 q^{34} +2.48712 q^{35} +2.59391 q^{37} -2.06235 q^{38} -1.99819 q^{40} -1.16041 q^{41} -3.06328 q^{43} +7.65610 q^{44} +2.17460 q^{46} -5.32731 q^{47} -3.10791 q^{49} +7.41686 q^{50} -13.4073 q^{52} +9.00013 q^{53} +3.53695 q^{55} -3.12696 q^{56} +2.17460 q^{58} +8.64531 q^{59} +10.1026 q^{61} -1.69740 q^{62} -12.3813 q^{64} -6.19389 q^{65} +2.24279 q^{67} +20.7883 q^{68} -5.40848 q^{70} -13.3724 q^{71} +1.58151 q^{73} -5.64071 q^{74} +2.58802 q^{76} +5.53497 q^{77} +5.39042 q^{79} -2.53522 q^{80} +2.52343 q^{82} -10.5871 q^{83} +9.60373 q^{85} +6.66141 q^{86} -4.44687 q^{88} +9.44598 q^{89} -9.69281 q^{91} -2.72887 q^{92} +11.5848 q^{94} +1.19561 q^{95} -8.45927 q^{97} +6.75845 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 3 q^{2} + 9 q^{4} + 10 q^{5} + q^{7} + 9 q^{8}+O(q^{10}) \) 10 * q + 3 * q^2 + 9 * q^4 + 10 * q^5 + q^7 + 9 * q^8	\( 10 q + 3 q^{2} + 9 q^{4} + 10 q^{5} + q^{7} + 9 q^{8} - 6 q^{10} - 13 q^{13} + 12 q^{14} - 5 q^{16} + 22 q^{17} - 2 q^{19} - 3 q^{20} + 3 q^{22} - 10 q^{23} + 10 q^{25} + 25 q^{26} + 19 q^{28} - 10 q^{29} - 22 q^{31} + 31 q^{32} + 13 q^{34} + 15 q^{35} - 9 q^{37} + 10 q^{38} - 6 q^{40} + 25 q^{41} + 3 q^{43} + 27 q^{44} - 3 q^{46} + 17 q^{47} + 17 q^{49} - 2 q^{50} - 18 q^{52} + 43 q^{53} - 11 q^{55} + 7 q^{56} - 3 q^{58} + 7 q^{59} - 6 q^{61} - 3 q^{62} + 33 q^{64} - 11 q^{65} + 11 q^{67} + 51 q^{68} + 34 q^{70} + 17 q^{71} - 44 q^{73} - 9 q^{74} + 24 q^{76} + 71 q^{77} + 5 q^{79} - 38 q^{80} + 33 q^{82} + 32 q^{83} + 16 q^{85} + 9 q^{86} + 18 q^{88} + 10 q^{89} - 3 q^{91} - 9 q^{92} + 47 q^{94} + 8 q^{95} + 6 q^{97} + 73 q^{98}+O(q^{100}) \) 10 * q + 3 * q^2 + 9 * q^4 + 10 * q^5 + q^7 + 9 * q^8 - 6 * q^10 - 13 * q^13 + 12 * q^14 - 5 * q^16 + 22 * q^17 - 2 * q^19 - 3 * q^20 + 3 * q^22 - 10 * q^23 + 10 * q^25 + 25 * q^26 + 19 * q^28 - 10 * q^29 - 22 * q^31 + 31 * q^32 + 13 * q^34 + 15 * q^35 - 9 * q^37 + 10 * q^38 - 6 * q^40 + 25 * q^41 + 3 * q^43 + 27 * q^44 - 3 * q^46 + 17 * q^47 + 17 * q^49 - 2 * q^50 - 18 * q^52 + 43 * q^53 - 11 * q^55 + 7 * q^56 - 3 * q^58 + 7 * q^59 - 6 * q^61 - 3 * q^62 + 33 * q^64 - 11 * q^65 + 11 * q^67 + 51 * q^68 + 34 * q^70 + 17 * q^71 - 44 * q^73 - 9 * q^74 + 24 * q^76 + 71 * q^77 + 5 * q^79 - 38 * q^80 + 33 * q^82 + 32 * q^83 + 16 * q^85 + 9 * q^86 + 18 * q^88 + 10 * q^89 - 3 * q^91 - 9 * q^92 + 47 * q^94 + 8 * q^95 + 6 * q^97 + 73 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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