LMFDB - Embedded newform 8037.2.a.j.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8037,2,Mod(1,8037)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8037, 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("8037.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8037 = 3^{2} \cdot 19 \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8037.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:	$64.1757681045$
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} - 2x^{6} - 7x^{5} + 16x^{4} + x^{3} - 13x^{2} + 2x + 1 $ x^7 - 2x^6 - 7x^5 + 16x^4 + x^3 - 13x^2 + 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2679)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.423829$ of defining polynomial
Character	$\chi$	$=$	8037.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.423829 q^{2} -1.82037 q^{4} -3.15942 q^{5} -2.93562 q^{7} -1.61918 q^{8} +O(q^{10})$	\(q+0.423829 q^{2} -1.82037 q^{4} -3.15942 q^{5} -2.93562 q^{7} -1.61918 q^{8} -1.33905 q^{10} +0.303664 q^{11} -2.46309 q^{13} -1.24420 q^{14} +2.95448 q^{16} +0.539325 q^{17} +1.00000 q^{19} +5.75132 q^{20} +0.128701 q^{22} -5.82494 q^{23} +4.98195 q^{25} -1.04393 q^{26} +5.34391 q^{28} +8.39258 q^{29} +2.68919 q^{31} +4.49056 q^{32} +0.228581 q^{34} +9.27485 q^{35} -1.96859 q^{37} +0.423829 q^{38} +5.11568 q^{40} +2.99841 q^{41} +8.07617 q^{43} -0.552780 q^{44} -2.46878 q^{46} -1.00000 q^{47} +1.61784 q^{49} +2.11149 q^{50} +4.48373 q^{52} +5.24045 q^{53} -0.959402 q^{55} +4.75330 q^{56} +3.55701 q^{58} +0.592694 q^{59} +4.40512 q^{61} +1.13976 q^{62} -4.00574 q^{64} +7.78193 q^{65} -8.48521 q^{67} -0.981771 q^{68} +3.93095 q^{70} +4.36730 q^{71} +0.0473605 q^{73} -0.834347 q^{74} -1.82037 q^{76} -0.891440 q^{77} +10.3711 q^{79} -9.33446 q^{80} +1.27081 q^{82} -6.09672 q^{83} -1.70396 q^{85} +3.42291 q^{86} -0.491686 q^{88} +8.02474 q^{89} +7.23068 q^{91} +10.6035 q^{92} -0.423829 q^{94} -3.15942 q^{95} -7.95225 q^{97} +0.685687 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 2 q^{2} + 4 q^{4} - 6 q^{5} - 3 q^{7} - 6 q^{8}+O(q^{10}) $ 7 * q + 2 * q^2 + 4 * q^4 - 6 * q^5 - 3 * q^7 - 6 * q^8	$ 7 q + 2 q^{2} + 4 q^{4} - 6 q^{5} - 3 q^{7} - 6 q^{8} - 10 q^{10} + 9 q^{11} - 8 q^{13} + 9 q^{14} + 14 q^{16} - 7 q^{17} + 7 q^{19} + 4 q^{20} - 7 q^{22} + 7 q^{23} - 9 q^{25} - q^{26} - 12 q^{28} + 12 q^{29} - 13 q^{31} - 15 q^{32} + 2 q^{34} - 4 q^{35} - 10 q^{37} + 2 q^{38} - 26 q^{40} + 8 q^{41} + 12 q^{43} + 14 q^{44} - 7 q^{46} - 7 q^{47} - 16 q^{49} + 12 q^{50} - 6 q^{52} + 17 q^{53} + 7 q^{55} + 38 q^{56} - 12 q^{58} - 12 q^{59} - 23 q^{61} - 23 q^{62} + 12 q^{64} + 13 q^{65} + q^{67} - 15 q^{68} + 8 q^{70} - 29 q^{73} - 5 q^{74} + 4 q^{76} - 4 q^{77} - 3 q^{79} - 15 q^{80} - 7 q^{82} - 19 q^{83} - 5 q^{85} + 23 q^{86} - 33 q^{88} + 14 q^{89} - 3 q^{91} + 5 q^{92} - 2 q^{94} - 6 q^{95} - 17 q^{97} - 38 q^{98}+O(q^{100}) $ 7 * q + 2 * q^2 + 4 * q^4 - 6 * q^5 - 3 * q^7 - 6 * q^8 - 10 * q^10 + 9 * q^11 - 8 * q^13 + 9 * q^14 + 14 * q^16 - 7 * q^17 + 7 * q^19 + 4 * q^20 - 7 * q^22 + 7 * q^23 - 9 * q^25 - q^26 - 12 * q^28 + 12 * q^29 - 13 * q^31 - 15 * q^32 + 2 * q^34 - 4 * q^35 - 10 * q^37 + 2 * q^38 - 26 * q^40 + 8 * q^41 + 12 * q^43 + 14 * q^44 - 7 * q^46 - 7 * q^47 - 16 * q^49 + 12 * q^50 - 6 * q^52 + 17 * q^53 + 7 * q^55 + 38 * q^56 - 12 * q^58 - 12 * q^59 - 23 * q^61 - 23 * q^62 + 12 * q^64 + 13 * q^65 + q^67 - 15 * q^68 + 8 * q^70 - 29 * q^73 - 5 * q^74 + 4 * q^76 - 4 * q^77 - 3 * q^79 - 15 * q^80 - 7 * q^82 - 19 * q^83 - 5 * q^85 + 23 * q^86 - 33 * q^88 + 14 * q^89 - 3 * q^91 + 5 * q^92 - 2 * q^94 - 6 * q^95 - 17 * q^97 - 38 * 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 $

$2$	0.423829	0.299692	0.149846	−	0.988709i	$-0.452122\pi$
$2$	0.423829	0.299692	0.149846	+	0.988709i	$0.452122\pi$
$3$	0	0
$3$	0	0
$4$	−1.82037	−0.910185
$5$	−3.15942	−1.41294	−0.706468	−	0.707745i	$-0.749713\pi$
$5$	−3.15942	−1.41294	−0.706468	+	0.707745i	$0.749713\pi$
$6$	0	0
$7$	−2.93562	−1.10956	−0.554779	−	0.831998i	$-0.687197\pi$
$7$	−2.93562	−1.10956	−0.554779	+	0.831998i	$0.687197\pi$
$8$	−1.61918	−0.572467
$9$	0	0
$10$	−1.33905	−0.423446
$11$	0.303664	0.0915580	0.0457790	−	0.998952i	$-0.485423\pi$
$11$	0.303664	0.0915580	0.0457790	+	0.998952i	$0.485423\pi$
$12$	0	0
$13$	−2.46309	−0.683137	−0.341569	−	0.939857i	$-0.610958\pi$
$13$	−2.46309	−0.683137	−0.341569	+	0.939857i	$0.610958\pi$
$14$	−1.24420	−0.332526
$15$	0	0
$16$	2.95448	0.738621
$17$	0.539325	0.130806	0.0654028	−	0.997859i	$-0.479167\pi$
$17$	0.539325	0.130806	0.0654028	+	0.997859i	$0.479167\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	5.75132	1.28603
$21$	0	0
$22$	0.128701	0.0274392
$23$	−5.82494	−1.21458	−0.607292	−	0.794478i	$-0.707744\pi$
$23$	−5.82494	−1.21458	−0.607292	+	0.794478i	$0.707744\pi$
$24$	0	0
$25$	4.98195	0.996391
$26$	−1.04393	−0.204731
$27$	0	0
$28$	5.34391	1.00990
$29$	8.39258	1.55846	0.779231	−	0.626736i	$-0.215610\pi$
$29$	8.39258	1.55846	0.779231	+	0.626736i	$0.215610\pi$
$30$	0	0
$31$	2.68919	0.482993	0.241496	−	0.970402i	$-0.422362\pi$
$31$	2.68919	0.482993	0.241496	+	0.970402i	$0.422362\pi$
$32$	4.49056	0.793826
$33$	0	0
$34$	0.228581	0.0392014
$35$	9.27485	1.56774
$36$	0	0
$37$	−1.96859	−0.323635	−0.161817	−	0.986821i	$-0.551736\pi$
$37$	−1.96859	−0.323635	−0.161817	+	0.986821i	$0.551736\pi$
$38$	0.423829	0.0687541
$39$	0	0
$40$	5.11568	0.808860
$41$	2.99841	0.468273	0.234136	−	0.972204i	$-0.424774\pi$
$41$	2.99841	0.468273	0.234136	+	0.972204i	$0.424774\pi$
$42$	0	0
$43$	8.07617	1.23160	0.615802	−	0.787901i	$-0.288832\pi$
$43$	8.07617	1.23160	0.615802	+	0.787901i	$0.288832\pi$
$44$	−0.552780	−0.0833347
$45$	0	0
$46$	−2.46878	−0.364001
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	1.61784	0.231120
$50$	2.11149	0.298610
$51$	0	0
$52$	4.48373	0.621781
$53$	5.24045	0.719831	0.359915	−	0.932985i	$-0.382806\pi$
$53$	5.24045	0.719831	0.359915	+	0.932985i	$0.382806\pi$
$54$	0	0
$55$	−0.959402	−0.129366
$56$	4.75330	0.635186
$57$	0	0
$58$	3.55701	0.467059
$59$	0.592694	0.0771622	0.0385811	−	0.999255i	$-0.487716\pi$
$59$	0.592694	0.0771622	0.0385811	+	0.999255i	$0.487716\pi$
$60$	0	0
$61$	4.40512	0.564017	0.282009	−	0.959412i	$-0.408999\pi$
$61$	4.40512	0.564017	0.282009	+	0.959412i	$0.408999\pi$
$62$	1.13976	0.144749
$63$	0	0
$64$	−4.00574	−0.500717
$65$	7.78193	0.965230
$66$	0	0
$67$	−8.48521	−1.03663	−0.518316	−	0.855189i	$-0.673441\pi$
$67$	−8.48521	−1.03663	−0.518316	+	0.855189i	$0.673441\pi$
$68$	−0.981771	−0.119057
$69$	0	0
$70$	3.93095	0.469838
$71$	4.36730	0.518303	0.259151	−	0.965837i	$-0.416557\pi$
$71$	4.36730	0.518303	0.259151	+	0.965837i	$0.416557\pi$
$72$	0	0
$73$	0.0473605	0.00554313	0.00277156	−	0.999996i	$-0.499118\pi$
$73$	0.0473605	0.00554313	0.00277156	+	0.999996i	$0.499118\pi$
$74$	−0.834347	−0.0969908
$75$	0	0
$76$	−1.82037	−0.208811
$77$	−0.891440	−0.101589
$78$	0	0
$79$	10.3711	1.16684	0.583420	−	0.812171i	$-0.301714\pi$
$79$	10.3711	1.16684	0.583420	+	0.812171i	$0.301714\pi$
$80$	−9.33446	−1.04362
$81$	0	0
$82$	1.27081	0.140338
$83$	−6.09672	−0.669202	−0.334601	−	0.942360i	$-0.608602\pi$
$83$	−6.09672	−0.669202	−0.334601	+	0.942360i	$0.608602\pi$
$84$	0	0
$85$	−1.70396	−0.184820
$86$	3.42291	0.369102
$87$	0	0
$88$	−0.491686	−0.0524140
$89$	8.02474	0.850621	0.425310	−	0.905048i	$-0.360165\pi$
$89$	8.02474	0.850621	0.425310	+	0.905048i	$0.360165\pi$
$90$	0	0
$91$	7.23068	0.757981
$92$	10.6035	1.10550
$93$	0	0
$94$	−0.423829	−0.0437146
$95$	−3.15942	−0.324150
$96$	0	0
$97$	−7.95225	−0.807429	−0.403715	−	0.914885i	$-0.632281\pi$
$97$	−7.95225	−0.807429	−0.403715	+	0.914885i	$0.632281\pi$
$98$	0.685687	0.0692649
$99$	0	0
$100$	−9.06900	−0.906900
$101$	11.8851	1.18262	0.591308	−	0.806446i	$-0.298612\pi$
$101$	11.8851	1.18262	0.591308	+	0.806446i	$0.298612\pi$
$102$	0	0
$103$	−14.8767	−1.46585	−0.732924	−	0.680310i	$-0.761845\pi$
$103$	−14.8767	−1.46585	−0.732924	+	0.680310i	$0.761845\pi$
$104$	3.98818	0.391074
$105$	0	0
$106$	2.22105	0.215728
$107$	−7.91293	−0.764972	−0.382486	−	0.923961i	$-0.624932\pi$
$107$	−7.91293	−0.764972	−0.382486	+	0.923961i	$0.624932\pi$
$108$	0	0
$109$	12.8823	1.23390	0.616951	−	0.787001i	$-0.288368\pi$
$109$	12.8823	1.23390	0.616951	+	0.787001i	$0.288368\pi$
$110$	−0.406622	−0.0387699
$111$	0	0
$112$	−8.67323	−0.819543
$113$	−4.47790	−0.421246	−0.210623	−	0.977567i	$-0.567549\pi$
$113$	−4.47790	−0.421246	−0.210623	+	0.977567i	$0.567549\pi$
$114$	0	0
$115$	18.4035	1.71613
$116$	−15.2776	−1.41849
$117$	0	0
$118$	0.251201	0.0231249
$119$	−1.58325	−0.145136
$120$	0	0
$121$	−10.9078	−0.991617
$122$	1.86701	0.169032
$123$	0	0
$124$	−4.89532	−0.439613
$125$	0.0570139	0.00509948
$126$	0	0
$127$	−10.2232	−0.907166	−0.453583	−	0.891214i	$-0.649854\pi$
$127$	−10.2232	−0.907166	−0.453583	+	0.891214i	$0.649854\pi$
$128$	−10.6789	−0.943887
$129$	0	0
$130$	3.29821	0.289272
$131$	−0.531543	−0.0464411	−0.0232206	−	0.999730i	$-0.507392\pi$
$131$	−0.531543	−0.0464411	−0.0232206	+	0.999730i	$0.507392\pi$
$132$	0	0
$133$	−2.93562	−0.254550
$134$	−3.59627	−0.310671
$135$	0	0
$136$	−0.873265	−0.0748819
$137$	−0.429653	−0.0367078	−0.0183539	−	0.999832i	$-0.505843\pi$
$137$	−0.429653	−0.0367078	−0.0183539	+	0.999832i	$0.505843\pi$
$138$	0	0
$139$	−8.58919	−0.728526	−0.364263	−	0.931296i	$-0.618679\pi$
$139$	−8.58919	−0.728526	−0.364263	+	0.931296i	$0.618679\pi$
$140$	−16.8837	−1.42693
$141$	0	0
$142$	1.85099	0.155331
$143$	−0.747950	−0.0625467
$144$	0	0
$145$	−26.5157	−2.20201
$146$	0.0200727	0.00166123
$147$	0	0
$148$	3.58357	0.294568
$149$	16.6382	1.36305	0.681527	−	0.731793i	$-0.261316\pi$
$149$	16.6382	1.36305	0.681527	+	0.731793i	$0.261316\pi$
$150$	0	0
$151$	1.33598	0.108720	0.0543601	−	0.998521i	$-0.482688\pi$
$151$	1.33598	0.108720	0.0543601	+	0.998521i	$0.482688\pi$
$152$	−1.61918	−0.131333
$153$	0	0
$154$	−0.377818	−0.0304454
$155$	−8.49629	−0.682438
$156$	0	0
$157$	−4.44203	−0.354513	−0.177256	−	0.984165i	$-0.556722\pi$
$157$	−4.44203	−0.354513	−0.177256	+	0.984165i	$0.556722\pi$
$158$	4.39557	0.349693
$159$	0	0
$160$	−14.1876	−1.12163
$161$	17.0998	1.34765
$162$	0	0
$163$	−11.1318	−0.871907	−0.435954	−	0.899969i	$-0.643589\pi$
$163$	−11.1318	−0.871907	−0.435954	+	0.899969i	$0.643589\pi$
$164$	−5.45821	−0.426215
$165$	0	0
$166$	−2.58396	−0.200554
$167$	11.6491	0.901435	0.450718	−	0.892667i	$-0.351168\pi$
$167$	11.6491	0.901435	0.450718	+	0.892667i	$0.351168\pi$
$168$	0	0
$169$	−6.93320	−0.533323
$170$	−0.722185	−0.0553891
$171$	0	0
$172$	−14.7016	−1.12099
$173$	−5.12665	−0.389772	−0.194886	−	0.980826i	$-0.562434\pi$
$173$	−5.12665	−0.389772	−0.194886	+	0.980826i	$0.562434\pi$
$174$	0	0
$175$	−14.6251	−1.10555
$176$	0.897169	0.0676266
$177$	0	0
$178$	3.40111	0.254924
$179$	26.2295	1.96048	0.980241	−	0.197809i	$-0.0633824\pi$
$179$	26.2295	1.96048	0.980241	+	0.197809i	$0.0633824\pi$
$180$	0	0
$181$	1.21898	0.0906059	0.0453029	−	0.998973i	$-0.485575\pi$
$181$	1.21898	0.0906059	0.0453029	+	0.998973i	$0.485575\pi$
$182$	3.06457	0.227161
$183$	0	0
$184$	9.43164	0.695310
$185$	6.21962	0.457276
$186$	0	0
$187$	0.163773	0.0119763
$188$	1.82037	0.132764
$189$	0	0
$190$	−1.33905	−0.0971452
$191$	4.42752	0.320364	0.160182	−	0.987087i	$-0.448792\pi$
$191$	4.42752	0.320364	0.160182	+	0.987087i	$0.448792\pi$
$192$	0	0
$193$	−4.98825	−0.359062	−0.179531	−	0.983752i	$-0.557458\pi$
$193$	−4.98825	−0.359062	−0.179531	+	0.983752i	$0.557458\pi$
$194$	−3.37039	−0.241980
$195$	0	0
$196$	−2.94507	−0.210362
$197$	12.0190	0.856315	0.428158	−	0.903704i	$-0.359163\pi$
$197$	12.0190	0.856315	0.428158	+	0.903704i	$0.359163\pi$
$198$	0	0
$199$	2.39396	0.169703	0.0848515	−	0.996394i	$-0.472958\pi$
$199$	2.39396	0.169703	0.0848515	+	0.996394i	$0.472958\pi$
$200$	−8.06669	−0.570401
$201$	0	0
$202$	5.03726	0.354421
$203$	−24.6374	−1.72921
$204$	0	0
$205$	−9.47324	−0.661640
$206$	−6.30519	−0.439303
$207$	0	0
$208$	−7.27715	−0.504579
$209$	0.303664	0.0210048
$210$	0	0
$211$	−14.0379	−0.966409	−0.483205	−	0.875507i	$-0.660527\pi$
$211$	−14.0379	−0.966409	−0.483205	+	0.875507i	$0.660527\pi$
$212$	−9.53955	−0.655179
$213$	0	0
$214$	−3.35373	−0.229256
$215$	−25.5160	−1.74018
$216$	0	0
$217$	−7.89443	−0.535909
$218$	5.45989	0.369791
$219$	0	0
$220$	1.74647	0.117747
$221$	−1.32840	−0.0893582
$222$	0	0
$223$	8.10997	0.543084	0.271542	−	0.962427i	$-0.412466\pi$
$223$	8.10997	0.543084	0.271542	+	0.962427i	$0.412466\pi$
$224$	−13.1826	−0.880796
$225$	0	0
$226$	−1.89786	−0.126244
$227$	−4.67918	−0.310568	−0.155284	−	0.987870i	$-0.549629\pi$
$227$	−4.67918	−0.310568	−0.155284	+	0.987870i	$0.549629\pi$
$228$	0	0
$229$	−17.1478	−1.13316	−0.566579	−	0.824007i	$-0.691733\pi$
$229$	−17.1478	−1.13316	−0.566579	+	0.824007i	$0.691733\pi$
$230$	7.79991	0.514311
$231$	0	0
$232$	−13.5891	−0.892169
$233$	−8.86484	−0.580755	−0.290378	−	0.956912i	$-0.593781\pi$
$233$	−8.86484	−0.580755	−0.290378	+	0.956912i	$0.593781\pi$
$234$	0	0
$235$	3.15942	0.206098
$236$	−1.07892	−0.0702319
$237$	0	0
$238$	−0.671027	−0.0434962
$239$	19.7348	1.27654	0.638270	−	0.769813i	$-0.279651\pi$
$239$	19.7348	1.27654	0.638270	+	0.769813i	$0.279651\pi$
$240$	0	0
$241$	−1.97256	−0.127064	−0.0635318	−	0.997980i	$-0.520236\pi$
$241$	−1.97256	−0.127064	−0.0635318	+	0.997980i	$0.520236\pi$
$242$	−4.62303	−0.297180
$243$	0	0
$244$	−8.01894	−0.513360
$245$	−5.11145	−0.326558
$246$	0	0
$247$	−2.46309	−0.156722
$248$	−4.35429	−0.276497
$249$	0	0
$250$	0.0241641	0.00152827
$251$	−1.98734	−0.125439	−0.0627197	−	0.998031i	$-0.519977\pi$
$251$	−1.98734	−0.125439	−0.0627197	+	0.998031i	$0.519977\pi$
$252$	0	0
$253$	−1.76882	−0.111205
$254$	−4.33290	−0.271870
$255$	0	0
$256$	3.48547	0.217842
$257$	21.0580	1.31356	0.656782	−	0.754081i	$-0.271917\pi$
$257$	21.0580	1.31356	0.656782	+	0.754081i	$0.271917\pi$
$258$	0	0
$259$	5.77904	0.359092
$260$	−14.1660	−0.878538
$261$	0	0
$262$	−0.225283	−0.0139180
$263$	2.35245	0.145058	0.0725292	−	0.997366i	$-0.476893\pi$
$263$	2.35245	0.145058	0.0725292	+	0.997366i	$0.476893\pi$
$264$	0	0
$265$	−16.5568	−1.01708
$266$	−1.24420	−0.0762867
$267$	0	0
$268$	15.4462	0.943527
$269$	−21.6571	−1.32046	−0.660229	−	0.751065i	$-0.729541\pi$
$269$	−21.6571	−1.32046	−0.660229	+	0.751065i	$0.729541\pi$
$270$	0	0
$271$	−14.3914	−0.874214	−0.437107	−	0.899410i	$-0.643997\pi$
$271$	−14.3914	−0.874214	−0.437107	+	0.899410i	$0.643997\pi$
$272$	1.59343	0.0966157
$273$	0	0
$274$	−0.182099	−0.0110010
$275$	1.51284	0.0912276
$276$	0	0
$277$	9.17199	0.551092	0.275546	−	0.961288i	$-0.411141\pi$
$277$	9.17199	0.551092	0.275546	+	0.961288i	$0.411141\pi$
$278$	−3.64034	−0.218333
$279$	0	0
$280$	−15.0177	−0.897478
$281$	9.61795	0.573759	0.286879	−	0.957967i	$-0.407382\pi$
$281$	9.61795	0.573759	0.286879	+	0.957967i	$0.407382\pi$
$282$	0	0
$283$	8.28086	0.492246	0.246123	−	0.969239i	$-0.420843\pi$
$283$	8.28086	0.492246	0.246123	+	0.969239i	$0.420843\pi$
$284$	−7.95009	−0.471751
$285$	0	0
$286$	−0.317002	−0.0187447
$287$	−8.80218	−0.519576
$288$	0	0
$289$	−16.7091	−0.982890
$290$	−11.2381	−0.659925
$291$	0	0
$292$	−0.0862136	−0.00504527
$293$	14.5507	0.850061	0.425031	−	0.905179i	$-0.360263\pi$
$293$	14.5507	0.850061	0.425031	+	0.905179i	$0.360263\pi$
$294$	0	0
$295$	−1.87257	−0.109025
$296$	3.18751	0.185270
$297$	0	0
$298$	7.05174	0.408496
$299$	14.3473	0.829728
$300$	0	0
$301$	−23.7085	−1.36654
$302$	0.566225	0.0325826
$303$	0	0
$304$	2.95448	0.169451
$305$	−13.9176	−0.796921
$306$	0	0
$307$	5.47148	0.312274	0.156137	−	0.987735i	$-0.450096\pi$
$307$	5.47148	0.312274	0.156137	+	0.987735i	$0.450096\pi$
$308$	1.62275	0.0924647
$309$	0	0
$310$	−3.60097	−0.204521
$311$	−4.16596	−0.236230	−0.118115	−	0.993000i	$-0.537685\pi$
$311$	−4.16596	−0.236230	−0.118115	+	0.993000i	$0.537685\pi$
$312$	0	0
$313$	−7.54040	−0.426209	−0.213104	−	0.977029i	$-0.568357\pi$
$313$	−7.54040	−0.426209	−0.213104	+	0.977029i	$0.568357\pi$
$314$	−1.88266	−0.106245
$315$	0	0
$316$	−18.8792	−1.06204
$317$	−4.11782	−0.231280	−0.115640	−	0.993291i	$-0.536892\pi$
$317$	−4.11782	−0.231280	−0.115640	+	0.993291i	$0.536892\pi$
$318$	0	0
$319$	2.54852	0.142690
$320$	12.6558	0.707482
$321$	0	0
$322$	7.24738	0.403881
$323$	0.539325	0.0300089
$324$	0	0
$325$	−12.2710	−0.680672
$326$	−4.71796	−0.261304
$327$	0	0
$328$	−4.85497	−0.268071
$329$	2.93562	0.161846
$330$	0	0
$331$	−13.0543	−0.717531	−0.358765	−	0.933428i	$-0.616802\pi$
$331$	−13.0543	−0.717531	−0.358765	+	0.933428i	$0.616802\pi$
$332$	11.0983	0.609097
$333$	0	0
$334$	4.93723	0.270153
$335$	26.8084	1.46470
$336$	0	0
$337$	−30.0222	−1.63541	−0.817706	−	0.575636i	$-0.804754\pi$
$337$	−30.0222	−1.63541	−0.817706	+	0.575636i	$0.804754\pi$
$338$	−2.93849	−0.159833
$339$	0	0
$340$	3.10183	0.168220
$341$	0.816609	0.0442218
$342$	0	0
$343$	15.8000	0.853117
$344$	−13.0768	−0.705053
$345$	0	0
$346$	−2.17282	−0.116812
$347$	−33.2049	−1.78253	−0.891267	−	0.453478i	$-0.850183\pi$
$347$	−33.2049	−1.78253	−0.891267	+	0.453478i	$0.850183\pi$
$348$	0	0
$349$	−17.1537	−0.918215	−0.459108	−	0.888381i	$-0.651831\pi$
$349$	−17.1537	−0.918215	−0.459108	+	0.888381i	$0.651831\pi$
$350$	−6.19854	−0.331326
$351$	0	0
$352$	1.36362	0.0726811
$353$	−10.1530	−0.540388	−0.270194	−	0.962806i	$-0.587088\pi$
$353$	−10.1530	−0.540388	−0.270194	+	0.962806i	$0.587088\pi$
$354$	0	0
$355$	−13.7981	−0.732329
$356$	−14.6080	−0.774222
$357$	0	0
$358$	11.1168	0.587541
$359$	−10.0967	−0.532883	−0.266442	−	0.963851i	$-0.585848\pi$
$359$	−10.0967	−0.532883	−0.266442	+	0.963851i	$0.585848\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0.516637	0.0271539
$363$	0	0
$364$	−13.1625	−0.689903
$365$	−0.149632	−0.00783209
$366$	0	0
$367$	3.97202	0.207338	0.103669	−	0.994612i	$-0.466942\pi$
$367$	3.97202	0.207338	0.103669	+	0.994612i	$0.466942\pi$
$368$	−17.2097	−0.897117
$369$	0	0
$370$	2.63605	0.137042
$371$	−15.3839	−0.798694
$372$	0	0
$373$	8.91937	0.461828	0.230914	−	0.972974i	$-0.425828\pi$
$373$	8.91937	0.461828	0.230914	+	0.972974i	$0.425828\pi$
$374$	0.0694118	0.00358920
$375$	0	0
$376$	1.61918	0.0835029
$377$	−20.6716	−1.06464
$378$	0	0
$379$	19.0337	0.977696	0.488848	−	0.872369i	$-0.337417\pi$
$379$	19.0337	0.977696	0.488848	+	0.872369i	$0.337417\pi$
$380$	5.75132	0.295036
$381$	0	0
$382$	1.87651	0.0960106
$383$	−22.0810	−1.12829	−0.564143	−	0.825677i	$-0.690793\pi$
$383$	−22.0810	−1.12829	−0.564143	+	0.825677i	$0.690793\pi$
$384$	0	0
$385$	2.81643	0.143539
$386$	−2.11416	−0.107608
$387$	0	0
$388$	14.4760	0.734910
$389$	19.1175	0.969297	0.484648	−	0.874709i	$-0.338948\pi$
$389$	19.1175	0.969297	0.484648	+	0.874709i	$0.338948\pi$
$390$	0	0
$391$	−3.14154	−0.158874
$392$	−2.61958	−0.132309
$393$	0	0
$394$	5.09398	0.256631
$395$	−32.7667	−1.64867
$396$	0	0
$397$	23.1416	1.16144	0.580722	−	0.814102i	$-0.302770\pi$
$397$	23.1416	1.16144	0.580722	+	0.814102i	$0.302770\pi$
$398$	1.01463	0.0508587
$399$	0	0
$400$	14.7191	0.735955
$401$	29.4016	1.46825	0.734124	−	0.679016i	$-0.237593\pi$
$401$	29.4016	1.46825	0.734124	+	0.679016i	$0.237593\pi$
$402$	0	0
$403$	−6.62371	−0.329950
$404$	−21.6353	−1.07640
$405$	0	0
$406$	−10.4420	−0.518229
$407$	−0.597790	−0.0296314
$408$	0	0
$409$	15.4262	0.762776	0.381388	−	0.924415i	$-0.375446\pi$
$409$	15.4262	0.762776	0.381388	+	0.924415i	$0.375446\pi$
$410$	−4.01503	−0.198288
$411$	0	0
$412$	27.0812	1.33419
$413$	−1.73992	−0.0856160
$414$	0	0
$415$	19.2621	0.945540
$416$	−11.0606	−0.542292
$417$	0	0
$418$	0.128701	0.00629499
$419$	−20.2923	−0.991346	−0.495673	−	0.868509i	$-0.665079\pi$
$419$	−20.2923	−0.991346	−0.495673	+	0.868509i	$0.665079\pi$
$420$	0	0
$421$	−4.01359	−0.195611	−0.0978053	−	0.995206i	$-0.531182\pi$
$421$	−4.01359	−0.195611	−0.0978053	+	0.995206i	$0.531182\pi$
$422$	−5.94967	−0.289625
$423$	0	0
$424$	−8.48523	−0.412079
$425$	2.68689	0.130333
$426$	0	0
$427$	−12.9317	−0.625810
$428$	14.4045	0.696266
$429$	0	0
$430$	−10.8144	−0.521518
$431$	−18.3874	−0.885689	−0.442845	−	0.896598i	$-0.646031\pi$
$431$	−18.3874	−0.885689	−0.442845	+	0.896598i	$0.646031\pi$
$432$	0	0
$433$	−0.340913	−0.0163832	−0.00819161	−	0.999966i	$-0.502608\pi$
$433$	−0.340913	−0.0163832	−0.00819161	+	0.999966i	$0.502608\pi$
$434$	−3.34588	−0.160608
$435$	0	0
$436$	−23.4506	−1.12308
$437$	−5.82494	−0.278645
$438$	0	0
$439$	−13.8559	−0.661308	−0.330654	−	0.943752i	$-0.607269\pi$
$439$	−13.8559	−0.661308	−0.330654	+	0.943752i	$0.607269\pi$
$440$	1.55345	0.0740576
$441$	0	0
$442$	−0.563016	−0.0267799
$443$	−14.8394	−0.705042	−0.352521	−	0.935804i	$-0.614676\pi$
$443$	−14.8394	−0.705042	−0.352521	+	0.935804i	$0.614676\pi$
$444$	0	0
$445$	−25.3536	−1.20187
$446$	3.43724	0.162758
$447$	0	0
$448$	11.7593	0.555575
$449$	−22.4486	−1.05941	−0.529707	−	0.848181i	$-0.677698\pi$
$449$	−22.4486	−1.05941	−0.529707	+	0.848181i	$0.677698\pi$
$450$	0	0
$451$	0.910508	0.0428741
$452$	8.15144	0.383411
$453$	0	0
$454$	−1.98317	−0.0930748
$455$	−22.8448	−1.07098
$456$	0	0
$457$	17.4062	0.814228	0.407114	−	0.913377i	$-0.366535\pi$
$457$	17.4062	0.814228	0.407114	+	0.913377i	$0.366535\pi$
$458$	−7.26772	−0.339598
$459$	0	0
$460$	−33.5011	−1.56200
$461$	−5.12733	−0.238804	−0.119402	−	0.992846i	$-0.538098\pi$
$461$	−5.12733	−0.238804	−0.119402	+	0.992846i	$0.538098\pi$
$462$	0	0
$463$	−23.0707	−1.07219	−0.536094	−	0.844158i	$-0.680101\pi$
$463$	−23.0707	−1.07219	−0.536094	+	0.844158i	$0.680101\pi$
$464$	24.7957	1.15111
$465$	0	0
$466$	−3.75717	−0.174048
$467$	16.4367	0.760599	0.380300	−	0.924863i	$-0.375821\pi$
$467$	16.4367	0.760599	0.380300	+	0.924863i	$0.375821\pi$
$468$	0	0
$469$	24.9093	1.15020
$470$	1.33905	0.0617659
$471$	0	0
$472$	−0.959680	−0.0441728
$473$	2.45244	0.112763
$474$	0	0
$475$	4.98195	0.228588
$476$	2.88210	0.132101
$477$	0	0
$478$	8.36418	0.382569
$479$	−8.48465	−0.387673	−0.193837	−	0.981034i	$-0.562093\pi$
$479$	−8.48465	−0.387673	−0.193837	+	0.981034i	$0.562093\pi$
$480$	0	0
$481$	4.84882	0.221087
$482$	−0.836026	−0.0380799
$483$	0	0
$484$	19.8562	0.902555
$485$	25.1245	1.14085
$486$	0	0
$487$	8.13090	0.368446	0.184223	−	0.982884i	$-0.441023\pi$
$487$	8.13090	0.368446	0.184223	+	0.982884i	$0.441023\pi$
$488$	−7.13268	−0.322881
$489$	0	0
$490$	−2.16638	−0.0978669
$491$	2.71349	0.122458	0.0612290	−	0.998124i	$-0.480498\pi$
$491$	2.71349	0.122458	0.0612290	+	0.998124i	$0.480498\pi$
$492$	0	0
$493$	4.52633	0.203856
$494$	−1.04393	−0.0469685
$495$	0	0
$496$	7.94517	0.356748
$497$	−12.8207	−0.575087
$498$	0	0
$499$	39.6466	1.77482	0.887412	−	0.460977i	$-0.152501\pi$
$499$	39.6466	1.77482	0.887412	+	0.460977i	$0.152501\pi$
$500$	−0.103786	−0.00464147
$501$	0	0
$502$	−0.842289	−0.0375932
$503$	−35.1740	−1.56833	−0.784165	−	0.620552i	$-0.786908\pi$
$503$	−35.1740	−1.56833	−0.784165	+	0.620552i	$0.786908\pi$
$504$	0	0
$505$	−37.5502	−1.67096
$506$	−0.749678	−0.0333272
$507$	0	0
$508$	18.6101	0.825688
$509$	18.8708	0.836432	0.418216	−	0.908348i	$-0.362655\pi$
$509$	18.8708	0.836432	0.418216	+	0.908348i	$0.362655\pi$
$510$	0	0
$511$	−0.139032	−0.00615043
$512$	22.8350	1.00917
$513$	0	0
$514$	8.92500	0.393665
$515$	47.0019	2.07115
$516$	0	0
$517$	−0.303664	−0.0133551
$518$	2.44932	0.107617
$519$	0	0
$520$	−12.6004	−0.552562
$521$	7.63632	0.334553	0.167277	−	0.985910i	$-0.446503\pi$
$521$	7.63632	0.334553	0.167277	+	0.985910i	$0.446503\pi$
$522$	0	0
$523$	−28.0258	−1.22548	−0.612742	−	0.790283i	$-0.709934\pi$
$523$	−28.0258	−1.22548	−0.612742	+	0.790283i	$0.709934\pi$
$524$	0.967605	0.0422700
$525$	0	0
$526$	0.997037	0.0434729
$527$	1.45035	0.0631781
$528$	0	0
$529$	10.9300	0.475216
$530$	−7.01724	−0.304809
$531$	0	0
$532$	5.34391	0.231688
$533$	−7.38534	−0.319895
$534$	0	0
$535$	25.0003	1.08086
$536$	13.7391	0.593438
$537$	0	0
$538$	−9.17890	−0.395731
$539$	0.491279	0.0211609
$540$	0	0
$541$	28.4455	1.22297	0.611483	−	0.791257i	$-0.290573\pi$
$541$	28.4455	1.22297	0.611483	+	0.791257i	$0.290573\pi$
$542$	−6.09948	−0.261995
$543$	0	0
$544$	2.42187	0.103837
$545$	−40.7007	−1.74343
$546$	0	0
$547$	19.7488	0.844399	0.422200	−	0.906503i	$-0.361258\pi$
$547$	19.7488	0.844399	0.422200	+	0.906503i	$0.361258\pi$
$548$	0.782128	0.0334108
$549$	0	0
$550$	0.641184	0.0273402
$551$	8.39258	0.357536
$552$	0	0
$553$	−30.4456	−1.29468
$554$	3.88735	0.165158
$555$	0	0
$556$	15.6355	0.663093
$557$	−25.2662	−1.07056	−0.535281	−	0.844674i	$-0.679794\pi$
$557$	−25.2662	−1.07056	−0.535281	+	0.844674i	$0.679794\pi$
$558$	0	0
$559$	−19.8923	−0.841355
$560$	27.4024	1.15796
$561$	0	0
$562$	4.07636	0.171951
$563$	13.3230	0.561497	0.280748	−	0.959781i	$-0.409417\pi$
$563$	13.3230	0.561497	0.280748	+	0.959781i	$0.409417\pi$
$564$	0	0
$565$	14.1476	0.595194
$566$	3.50967	0.147522
$567$	0	0
$568$	−7.07145	−0.296711
$569$	−13.7317	−0.575661	−0.287831	−	0.957681i	$-0.592934\pi$
$569$	−13.7317	−0.575661	−0.287831	+	0.957681i	$0.592934\pi$
$570$	0	0
$571$	14.3677	0.601268	0.300634	−	0.953740i	$-0.402802\pi$
$571$	14.3677	0.601268	0.300634	+	0.953740i	$0.402802\pi$
$572$	1.36154	0.0569290
$573$	0	0
$574$	−3.73062	−0.155713
$575$	−29.0196	−1.21020
$576$	0	0
$577$	−24.1672	−1.00609	−0.503047	−	0.864259i	$-0.667788\pi$
$577$	−24.1672	−1.00609	−0.503047	+	0.864259i	$0.667788\pi$
$578$	−7.08181	−0.294564
$579$	0	0
$580$	48.2684	2.00424
$581$	17.8976	0.742519
$582$	0	0
$583$	1.59133	0.0659063
$584$	−0.0766853	−0.00317326
$585$	0	0
$586$	6.16701	0.254757
$587$	−23.8264	−0.983422	−0.491711	−	0.870758i	$-0.663628\pi$
$587$	−23.8264	−0.983422	−0.491711	+	0.870758i	$0.663628\pi$
$588$	0	0
$589$	2.68919	0.110806
$590$	−0.793650	−0.0326740
$591$	0	0
$592$	−5.81618	−0.239043
$593$	−41.3865	−1.69954	−0.849770	−	0.527154i	$-0.823259\pi$
$593$	−41.3865	−1.69954	−0.849770	+	0.527154i	$0.823259\pi$
$594$	0	0
$595$	5.00216	0.205069
$596$	−30.2877	−1.24063
$597$	0	0
$598$	6.08081	0.248663
$599$	2.37581	0.0970728	0.0485364	−	0.998821i	$-0.484544\pi$
$599$	2.37581	0.0970728	0.0485364	+	0.998821i	$0.484544\pi$
$600$	0	0
$601$	−4.32033	−0.176230	−0.0881149	−	0.996110i	$-0.528084\pi$
$601$	−4.32033	−0.176230	−0.0881149	+	0.996110i	$0.528084\pi$
$602$	−10.0484	−0.409540
$603$	0	0
$604$	−2.43197	−0.0989554
$605$	34.4623	1.40109
$606$	0	0
$607$	36.8357	1.49511	0.747557	−	0.664197i	$-0.231226\pi$
$607$	36.8357	1.49511	0.747557	+	0.664197i	$0.231226\pi$
$608$	4.49056	0.182116
$609$	0	0
$610$	−5.89869	−0.238831
$611$	2.46309	0.0996458
$612$	0	0
$613$	19.5563	0.789873	0.394936	−	0.918709i	$-0.370767\pi$
$613$	19.5563	0.789873	0.394936	+	0.918709i	$0.370767\pi$
$614$	2.31897	0.0935860
$615$	0	0
$616$	1.44340	0.0581563
$617$	35.6342	1.43458	0.717290	−	0.696775i	$-0.245382\pi$
$617$	35.6342	1.43458	0.717290	+	0.696775i	$0.245382\pi$
$618$	0	0
$619$	5.13166	0.206259	0.103129	−	0.994668i	$-0.467114\pi$
$619$	5.13166	0.206259	0.103129	+	0.994668i	$0.467114\pi$
$620$	15.4664	0.621145
$621$	0	0
$622$	−1.76565	−0.0707962
$623$	−23.5576	−0.943814
$624$	0	0
$625$	−25.0899	−1.00360
$626$	−3.19584	−0.127731
$627$	0	0
$628$	8.08614	0.322672
$629$	−1.06171	−0.0423333
$630$	0	0
$631$	−27.5260	−1.09579	−0.547896	−	0.836546i	$-0.684571\pi$
$631$	−27.5260	−1.09579	−0.547896	+	0.836546i	$0.684571\pi$
$632$	−16.7927	−0.667977
$633$	0	0
$634$	−1.74525	−0.0693127
$635$	32.2995	1.28177
$636$	0	0
$637$	−3.98488	−0.157887
$638$	1.08014	0.0427630
$639$	0	0
$640$	33.7390	1.33365
$641$	25.2374	0.996816	0.498408	−	0.866943i	$-0.333918\pi$
$641$	25.2374	0.996816	0.498408	+	0.866943i	$0.333918\pi$
$642$	0	0
$643$	−21.9362	−0.865080	−0.432540	−	0.901615i	$-0.642383\pi$
$643$	−21.9362	−0.865080	−0.432540	+	0.901615i	$0.642383\pi$
$644$	−31.1279	−1.22661
$645$	0	0
$646$	0.228581	0.00899342
$647$	32.5525	1.27977	0.639886	−	0.768470i	$-0.278982\pi$
$647$	32.5525	1.27977	0.639886	+	0.768470i	$0.278982\pi$
$648$	0	0
$649$	0.179980	0.00706482
$650$	−5.20079	−0.203992
$651$	0	0
$652$	20.2639	0.793597
$653$	−10.3970	−0.406866	−0.203433	−	0.979089i	$-0.565210\pi$
$653$	−10.3970	−0.406866	−0.203433	+	0.979089i	$0.565210\pi$
$654$	0	0
$655$	1.67937	0.0656184
$656$	8.85875	0.345876
$657$	0	0
$658$	1.24420	0.0485039
$659$	−33.0884	−1.28894	−0.644470	−	0.764630i	$-0.722922\pi$
$659$	−33.0884	−1.28894	−0.644470	+	0.764630i	$0.722922\pi$
$660$	0	0
$661$	−17.0452	−0.662981	−0.331491	−	0.943458i	$-0.607552\pi$
$661$	−17.0452	−0.662981	−0.331491	+	0.943458i	$0.607552\pi$
$662$	−5.53280	−0.215038
$663$	0	0
$664$	9.87169	0.383096
$665$	9.27485	0.359663
$666$	0	0
$667$	−48.8863	−1.89288
$668$	−21.2057	−0.820473
$669$	0	0
$670$	11.3621	0.438958
$671$	1.33767	0.0516403
$672$	0	0
$673$	−30.6437	−1.18123	−0.590614	−	0.806954i	$-0.701114\pi$
$673$	−30.6437	−1.18123	−0.590614	+	0.806954i	$0.701114\pi$
$674$	−12.7243	−0.490120
$675$	0	0
$676$	12.6210	0.485423
$677$	−32.1432	−1.23537	−0.617683	−	0.786428i	$-0.711928\pi$
$677$	−32.1432	−1.23537	−0.617683	+	0.786428i	$0.711928\pi$
$678$	0	0
$679$	23.3448	0.895890
$680$	2.75902	0.105803
$681$	0	0
$682$	0.346102	0.0132529
$683$	40.3626	1.54443	0.772215	−	0.635361i	$-0.219149\pi$
$683$	40.3626	1.54443	0.772215	+	0.635361i	$0.219149\pi$
$684$	0	0
$685$	1.35746	0.0518658
$686$	6.69647	0.255672
$687$	0	0
$688$	23.8609	0.909689
$689$	−12.9077	−0.491743
$690$	0	0
$691$	3.81465	0.145116	0.0725581	−	0.997364i	$-0.476884\pi$
$691$	3.81465	0.145116	0.0725581	+	0.997364i	$0.476884\pi$
$692$	9.33240	0.354765
$693$	0	0
$694$	−14.0732	−0.534211
$695$	27.1369	1.02936
$696$	0	0
$697$	1.61712	0.0612527
$698$	−7.27022	−0.275182
$699$	0	0
$700$	26.6231	1.00626
$701$	12.6399	0.477403	0.238702	−	0.971093i	$-0.423278\pi$
$701$	12.6399	0.477403	0.238702	+	0.971093i	$0.423278\pi$
$702$	0	0
$703$	−1.96859	−0.0742469
$704$	−1.21640	−0.0458447
$705$	0	0
$706$	−4.30312	−0.161950
$707$	−34.8902	−1.31218
$708$	0	0
$709$	−5.36339	−0.201426	−0.100713	−	0.994916i	$-0.532112\pi$
$709$	−5.36339	−0.201426	−0.100713	+	0.994916i	$0.532112\pi$
$710$	−5.84805	−0.219473
$711$	0	0
$712$	−12.9935	−0.486953
$713$	−15.6644	−0.586635
$714$	0	0
$715$	2.36309	0.0883745
$716$	−47.7473	−1.78440
$717$	0	0
$718$	−4.27927	−0.159701
$719$	50.2371	1.87353	0.936763	−	0.349963i	$-0.113806\pi$
$719$	50.2371	1.87353	0.936763	+	0.349963i	$0.113806\pi$
$720$	0	0
$721$	43.6724	1.62644
$722$	0.423829	0.0157733
$723$	0	0
$724$	−2.21899	−0.0824681
$725$	41.8114	1.55284
$726$	0	0
$727$	47.0531	1.74510	0.872552	−	0.488521i	$-0.162463\pi$
$727$	47.0531	1.74510	0.872552	+	0.488521i	$0.162463\pi$
$728$	−11.7078	−0.433919
$729$	0	0
$730$	−0.0634183	−0.00234722
$731$	4.35568	0.161101
$732$	0	0
$733$	−3.52075	−0.130042	−0.0650210	−	0.997884i	$-0.520711\pi$
$733$	−3.52075	−0.130042	−0.0650210	+	0.997884i	$0.520711\pi$
$734$	1.68345	0.0621374
$735$	0	0
$736$	−26.1572	−0.964169
$737$	−2.57665	−0.0949120
$738$	0	0
$739$	−45.3020	−1.66646	−0.833231	−	0.552926i	$-0.813511\pi$
$739$	−45.3020	−1.66646	−0.833231	+	0.552926i	$0.813511\pi$
$740$	−11.3220	−0.416205
$741$	0	0
$742$	−6.52015	−0.239362
$743$	8.69214	0.318884	0.159442	−	0.987207i	$-0.449031\pi$
$743$	8.69214	0.318884	0.159442	+	0.987207i	$0.449031\pi$
$744$	0	0
$745$	−52.5671	−1.92591
$746$	3.78029	0.138406
$747$	0	0
$748$	−0.298128	−0.0109006
$749$	23.2293	0.848781
$750$	0	0
$751$	22.3249	0.814646	0.407323	−	0.913284i	$-0.366462\pi$
$751$	22.3249	0.814646	0.407323	+	0.913284i	$0.366462\pi$
$752$	−2.95448	−0.107739
$753$	0	0
$754$	−8.76123	−0.319065
$755$	−4.22091	−0.153615
$756$	0	0
$757$	−38.7360	−1.40788	−0.703942	−	0.710257i	$-0.748579\pi$
$757$	−38.7360	−1.40788	−0.703942	+	0.710257i	$0.748579\pi$
$758$	8.06703	0.293008
$759$	0	0
$760$	5.11568	0.185565
$761$	−28.3758	−1.02862	−0.514311	−	0.857604i	$-0.671952\pi$
$761$	−28.3758	−1.02862	−0.514311	+	0.857604i	$0.671952\pi$
$762$	0	0
$763$	−37.8175	−1.36909
$764$	−8.05973	−0.291591
$765$	0	0
$766$	−9.35856	−0.338138
$767$	−1.45986	−0.0527124
$768$	0	0
$769$	37.9346	1.36796	0.683979	−	0.729502i	$-0.260248\pi$
$769$	37.9346	1.36796	0.683979	+	0.729502i	$0.260248\pi$
$770$	1.19369	0.0430174
$771$	0	0
$772$	9.08046	0.326813
$773$	−6.24992	−0.224794	−0.112397	−	0.993663i	$-0.535853\pi$
$773$	−6.24992	−0.224794	−0.112397	+	0.993663i	$0.535853\pi$
$774$	0	0
$775$	13.3974	0.481250
$776$	12.8761	0.462227
$777$	0	0
$778$	8.10255	0.290491
$779$	2.99841	0.107429
$780$	0	0
$781$	1.32619	0.0474548
$782$	−1.33147	−0.0476134
$783$	0	0
$784$	4.77989	0.170710
$785$	14.0343	0.500904
$786$	0	0
$787$	−16.2637	−0.579737	−0.289868	−	0.957067i	$-0.593612\pi$
$787$	−16.2637	−0.579737	−0.289868	+	0.957067i	$0.593612\pi$
$788$	−21.8789	−0.779405
$789$	0	0
$790$	−13.8875	−0.494094
$791$	13.1454	0.467397
$792$	0	0
$793$	−10.8502	−0.385301
$794$	9.80808	0.348076
$795$	0	0
$796$	−4.35788	−0.154461
$797$	31.1810	1.10449	0.552244	−	0.833683i	$-0.313772\pi$
$797$	31.1810	1.10449	0.552244	+	0.833683i	$0.313772\pi$
$798$	0	0
$799$	−0.539325	−0.0190800
$800$	22.3718	0.790961
$801$	0	0
$802$	12.4612	0.440022
$803$	0.0143817	0.000507518 0
$804$	0	0
$805$	−54.0255	−1.90415
$806$	−2.80732	−0.0988835
$807$	0	0
$808$	−19.2442	−0.677009
$809$	−2.14307	−0.0753463	−0.0376732	−	0.999290i	$-0.511995\pi$
$809$	−2.14307	−0.0753463	−0.0376732	+	0.999290i	$0.511995\pi$
$810$	0	0
$811$	−28.3140	−0.994239	−0.497120	−	0.867682i	$-0.665609\pi$
$811$	−28.3140	−0.994239	−0.497120	+	0.867682i	$0.665609\pi$
$812$	44.8491	1.57390
$813$	0	0
$814$	−0.253361	−0.00888029
$815$	35.1700	1.23195
$816$	0	0
$817$	8.07617	0.282549
$818$	6.53806	0.228598
$819$	0	0
$820$	17.2448	0.602215
$821$	−0.753844	−0.0263093	−0.0131547	−	0.999913i	$-0.504187\pi$
$821$	−0.753844	−0.0263093	−0.0131547	+	0.999913i	$0.504187\pi$
$822$	0	0
$823$	47.9651	1.67196	0.835979	−	0.548761i	$-0.184900\pi$
$823$	47.9651	1.67196	0.835979	+	0.548761i	$0.184900\pi$
$824$	24.0881	0.839150
$825$	0	0
$826$	−0.737429	−0.0256584
$827$	−14.7060	−0.511377	−0.255688	−	0.966759i	$-0.582302\pi$
$827$	−14.7060	−0.511377	−0.255688	+	0.966759i	$0.582302\pi$
$828$	0	0
$829$	49.0188	1.70249	0.851247	−	0.524765i	$-0.175847\pi$
$829$	49.0188	1.70249	0.851247	+	0.524765i	$0.175847\pi$
$830$	8.16383	0.283371
$831$	0	0
$832$	9.86648	0.342059
$833$	0.872543	0.0302318
$834$	0	0
$835$	−36.8045	−1.27367
$836$	−0.552780	−0.0191183
$837$	0	0
$838$	−8.60047	−0.297098
$839$	−31.2986	−1.08055	−0.540274	−	0.841489i	$-0.681679\pi$
$839$	−31.2986	−1.08055	−0.540274	+	0.841489i	$0.681679\pi$
$840$	0	0
$841$	41.4354	1.42881
$842$	−1.70108	−0.0586229
$843$	0	0
$844$	25.5542	0.879611
$845$	21.9049	0.753552
$846$	0	0
$847$	32.0211	1.10026
$848$	15.4828	0.531682
$849$	0	0
$850$	1.13878	0.0390599
$851$	11.4670	0.393082
$852$	0	0
$853$	−4.17396	−0.142914	−0.0714569	−	0.997444i	$-0.522765\pi$
$853$	−4.17396	−0.142914	−0.0714569	+	0.997444i	$0.522765\pi$
$854$	−5.48084	−0.187550
$855$	0	0
$856$	12.8125	0.437921
$857$	14.7419	0.503575	0.251788	−	0.967783i	$-0.418982\pi$
$857$	14.7419	0.503575	0.251788	+	0.967783i	$0.418982\pi$
$858$	0	0
$859$	−41.7172	−1.42337	−0.711686	−	0.702498i	$-0.752068\pi$
$859$	−41.7172	−1.42337	−0.711686	+	0.702498i	$0.752068\pi$
$860$	46.4486	1.58388
$861$	0	0
$862$	−7.79310	−0.265434
$863$	−30.3462	−1.03300	−0.516498	−	0.856288i	$-0.672765\pi$
$863$	−30.3462	−1.03300	−0.516498	+	0.856288i	$0.672765\pi$
$864$	0	0
$865$	16.1973	0.550723
$866$	−0.144489	−0.00490992
$867$	0	0
$868$	14.3708	0.487776
$869$	3.14932	0.106834
$870$	0	0
$871$	20.8998	0.708163
$872$	−20.8588	−0.706369
$873$	0	0
$874$	−2.46878	−0.0835076
$875$	−0.167371	−0.00565817
$876$	0	0
$877$	−31.3243	−1.05775	−0.528873	−	0.848701i	$-0.677385\pi$
$877$	−31.3243	−1.05775	−0.528873	+	0.848701i	$0.677385\pi$
$878$	−5.87254	−0.198189
$879$	0	0
$880$	−2.83454	−0.0955522
$881$	26.3561	0.887960	0.443980	−	0.896037i	$-0.353566\pi$
$881$	26.3561	0.887960	0.443980	+	0.896037i	$0.353566\pi$
$882$	0	0
$883$	45.3804	1.52717	0.763587	−	0.645705i	$-0.223436\pi$
$883$	45.3804	1.52717	0.763587	+	0.645705i	$0.223436\pi$
$884$	2.41819	0.0813324
$885$	0	0
$886$	−6.28937	−0.211296
$887$	15.1318	0.508076	0.254038	−	0.967194i	$-0.418241\pi$
$887$	15.1318	0.508076	0.254038	+	0.967194i	$0.418241\pi$
$888$	0	0
$889$	30.0115	1.00655
$890$	−10.7456	−0.360192
$891$	0	0
$892$	−14.7631	−0.494307
$893$	−1.00000	−0.0334637
$894$	0	0
$895$	−82.8699	−2.77004
$896$	31.3490	1.04730
$897$	0	0
$898$	−9.51435	−0.317498
$899$	22.5692	0.752726
$900$	0	0
$901$	2.82630	0.0941579
$902$	0.385899	0.0128490
$903$	0	0
$904$	7.25054	0.241149
$905$	−3.85127	−0.128020
$906$	0	0
$907$	−42.7640	−1.41995	−0.709977	−	0.704225i	$-0.751295\pi$
$907$	−42.7640	−1.41995	−0.709977	+	0.704225i	$0.751295\pi$
$908$	8.51784	0.282674
$909$	0	0
$910$	−9.68226	−0.320964
$911$	11.0779	0.367028	0.183514	−	0.983017i	$-0.441253\pi$
$911$	11.0779	0.367028	0.183514	+	0.983017i	$0.441253\pi$
$912$	0	0
$913$	−1.85135	−0.0612708
$914$	7.37724	0.244018
$915$	0	0
$916$	31.2153	1.03138
$917$	1.56041	0.0515291
$918$	0	0
$919$	11.8875	0.392133	0.196067	−	0.980591i	$-0.437183\pi$
$919$	11.8875	0.392133	0.196067	+	0.980591i	$0.437183\pi$
$920$	−29.7985	−0.982429
$921$	0	0
$922$	−2.17311	−0.0715676
$923$	−10.7570	−0.354072
$924$	0	0
$925$	−9.80745	−0.322467
$926$	−9.77804	−0.321326
$927$	0	0
$928$	37.6874	1.23715
$929$	36.1678	1.18663	0.593313	−	0.804972i	$-0.297820\pi$
$929$	36.1678	1.18663	0.593313	+	0.804972i	$0.297820\pi$
$930$	0	0
$931$	1.61784	0.0530226
$932$	16.1373	0.528595
$933$	0	0
$934$	6.96634	0.227946
$935$	−0.517429	−0.0169218
$936$	0	0
$937$	12.7308	0.415898	0.207949	−	0.978140i	$-0.433321\pi$
$937$	12.7308	0.415898	0.207949	+	0.978140i	$0.433321\pi$
$938$	10.5573	0.344707
$939$	0	0
$940$	−5.75132	−0.187587
$941$	−5.19767	−0.169439	−0.0847196	−	0.996405i	$-0.526999\pi$
$941$	−5.19767	−0.169439	−0.0847196	+	0.996405i	$0.526999\pi$
$942$	0	0
$943$	−17.4656	−0.568757
$944$	1.75111	0.0569936
$945$	0	0
$946$	1.03941	0.0337943
$947$	16.2309	0.527433	0.263717	−	0.964600i	$-0.415052\pi$
$947$	16.2309	0.527433	0.263717	+	0.964600i	$0.415052\pi$
$948$	0	0
$949$	−0.116653	−0.00378672
$950$	2.11149	0.0685059
$951$	0	0
$952$	2.56357	0.0830858
$953$	−26.4349	−0.856311	−0.428156	−	0.903705i	$-0.640837\pi$
$953$	−26.4349	−0.856311	−0.428156	+	0.903705i	$0.640837\pi$
$954$	0	0
$955$	−13.9884	−0.452655
$956$	−35.9247	−1.16189
$957$	0	0
$958$	−3.59604	−0.116183
$959$	1.26130	0.0407294
$960$	0	0
$961$	−23.7683	−0.766718
$962$	2.05507	0.0662580
$963$	0	0
$964$	3.59078	0.115651
$965$	15.7600	0.507332
$966$	0	0
$967$	31.7951	1.02246	0.511231	−	0.859443i	$-0.329190\pi$
$967$	31.7951	1.02246	0.511231	+	0.859443i	$0.329190\pi$
$968$	17.6617	0.567668
$969$	0	0
$970$	10.6485	0.341903
$971$	−8.13555	−0.261082	−0.130541	−	0.991443i	$-0.541671\pi$
$971$	−8.13555	−0.261082	−0.130541	+	0.991443i	$0.541671\pi$
$972$	0	0
$973$	25.2146	0.808342
$974$	3.44611	0.110420
$975$	0	0
$976$	13.0148	0.416595
$977$	10.6826	0.341765	0.170883	−	0.985291i	$-0.445338\pi$
$977$	10.6826	0.341765	0.170883	+	0.985291i	$0.445338\pi$
$978$	0	0
$979$	2.43682	0.0778811
$980$	9.30472	0.297228
$981$	0	0
$982$	1.15005	0.0366997
$983$	−25.9332	−0.827142	−0.413571	−	0.910472i	$-0.635719\pi$
$983$	−25.9332	−0.827142	−0.413571	+	0.910472i	$0.635719\pi$
$984$	0	0
$985$	−37.9730	−1.20992
$986$	1.91839	0.0610939
$987$	0	0
$988$	4.48373	0.142646
$989$	−47.0432	−1.49589
$990$	0	0
$991$	−30.9546	−0.983305	−0.491652	−	0.870792i	$-0.663607\pi$
$991$	−30.9546	−0.983305	−0.491652	+	0.870792i	$0.663607\pi$
$992$	12.0760	0.383412
$993$	0	0
$994$	−5.43378	−0.172349
$995$	−7.56352	−0.239780
$996$	0	0
$997$	39.5365	1.25213	0.626067	−	0.779769i	$-0.284664\pi$
$997$	39.5365	1.25213	0.626067	+	0.779769i	$0.284664\pi$
$998$	16.8033	0.531901
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.j.1.4		7
3.2	odd	2		2679.2.a.k.1.4	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2679.2.a.k.1.4	✓	7	3.2	odd	2
8037.2.a.j.1.4		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:	\( N \)	\(=\)	\( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8037.a (trivial)

\(f(q)\)	\(=\)	\(q+0.423829 q^{2} -1.82037 q^{4} -3.15942 q^{5} -2.93562 q^{7} -1.61918 q^{8} +O(q^{10})\)	\(q+0.423829 q^{2} -1.82037 q^{4} -3.15942 q^{5} -2.93562 q^{7} -1.61918 q^{8} -1.33905 q^{10} +0.303664 q^{11} -2.46309 q^{13} -1.24420 q^{14} +2.95448 q^{16} +0.539325 q^{17} +1.00000 q^{19} +5.75132 q^{20} +0.128701 q^{22} -5.82494 q^{23} +4.98195 q^{25} -1.04393 q^{26} +5.34391 q^{28} +8.39258 q^{29} +2.68919 q^{31} +4.49056 q^{32} +0.228581 q^{34} +9.27485 q^{35} -1.96859 q^{37} +0.423829 q^{38} +5.11568 q^{40} +2.99841 q^{41} +8.07617 q^{43} -0.552780 q^{44} -2.46878 q^{46} -1.00000 q^{47} +1.61784 q^{49} +2.11149 q^{50} +4.48373 q^{52} +5.24045 q^{53} -0.959402 q^{55} +4.75330 q^{56} +3.55701 q^{58} +0.592694 q^{59} +4.40512 q^{61} +1.13976 q^{62} -4.00574 q^{64} +7.78193 q^{65} -8.48521 q^{67} -0.981771 q^{68} +3.93095 q^{70} +4.36730 q^{71} +0.0473605 q^{73} -0.834347 q^{74} -1.82037 q^{76} -0.891440 q^{77} +10.3711 q^{79} -9.33446 q^{80} +1.27081 q^{82} -6.09672 q^{83} -1.70396 q^{85} +3.42291 q^{86} -0.491686 q^{88} +8.02474 q^{89} +7.23068 q^{91} +10.6035 q^{92} -0.423829 q^{94} -3.15942 q^{95} -7.95225 q^{97} +0.685687 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{2} + 4 q^{4} - 6 q^{5} - 3 q^{7} - 6 q^{8}+O(q^{10}) \) 7 * q + 2 * q^2 + 4 * q^4 - 6 * q^5 - 3 * q^7 - 6 * q^8	\( 7 q + 2 q^{2} + 4 q^{4} - 6 q^{5} - 3 q^{7} - 6 q^{8} - 10 q^{10} + 9 q^{11} - 8 q^{13} + 9 q^{14} + 14 q^{16} - 7 q^{17} + 7 q^{19} + 4 q^{20} - 7 q^{22} + 7 q^{23} - 9 q^{25} - q^{26} - 12 q^{28} + 12 q^{29} - 13 q^{31} - 15 q^{32} + 2 q^{34} - 4 q^{35} - 10 q^{37} + 2 q^{38} - 26 q^{40} + 8 q^{41} + 12 q^{43} + 14 q^{44} - 7 q^{46} - 7 q^{47} - 16 q^{49} + 12 q^{50} - 6 q^{52} + 17 q^{53} + 7 q^{55} + 38 q^{56} - 12 q^{58} - 12 q^{59} - 23 q^{61} - 23 q^{62} + 12 q^{64} + 13 q^{65} + q^{67} - 15 q^{68} + 8 q^{70} - 29 q^{73} - 5 q^{74} + 4 q^{76} - 4 q^{77} - 3 q^{79} - 15 q^{80} - 7 q^{82} - 19 q^{83} - 5 q^{85} + 23 q^{86} - 33 q^{88} + 14 q^{89} - 3 q^{91} + 5 q^{92} - 2 q^{94} - 6 q^{95} - 17 q^{97} - 38 q^{98}+O(q^{100}) \) 7 * q + 2 * q^2 + 4 * q^4 - 6 * q^5 - 3 * q^7 - 6 * q^8 - 10 * q^10 + 9 * q^11 - 8 * q^13 + 9 * q^14 + 14 * q^16 - 7 * q^17 + 7 * q^19 + 4 * q^20 - 7 * q^22 + 7 * q^23 - 9 * q^25 - q^26 - 12 * q^28 + 12 * q^29 - 13 * q^31 - 15 * q^32 + 2 * q^34 - 4 * q^35 - 10 * q^37 + 2 * q^38 - 26 * q^40 + 8 * q^41 + 12 * q^43 + 14 * q^44 - 7 * q^46 - 7 * q^47 - 16 * q^49 + 12 * q^50 - 6 * q^52 + 17 * q^53 + 7 * q^55 + 38 * q^56 - 12 * q^58 - 12 * q^59 - 23 * q^61 - 23 * q^62 + 12 * q^64 + 13 * q^65 + q^67 - 15 * q^68 + 8 * q^70 - 29 * q^73 - 5 * q^74 + 4 * q^76 - 4 * q^77 - 3 * q^79 - 15 * q^80 - 7 * q^82 - 19 * q^83 - 5 * q^85 + 23 * q^86 - 33 * q^88 + 14 * q^89 - 3 * q^91 + 5 * q^92 - 2 * q^94 - 6 * q^95 - 17 * q^97 - 38 * q^98

Introduction

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