LMFDB - Embedded newform 8001.2.a.t.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, 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("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} - 20 x^{14} + 38 x^{13} + 155 x^{12} - 275 x^{11} - 593 x^{10} + 957 x^{9} + 1177 x^{8} + \cdots + 1 $ x^16 - 2x^15 - 20x^14 + 38x^13 + 155x^12 - 275x^11 - 593x^10 + 957x^9 + 1177x^8 - 1655x^7 - 1150x^6 + 1279x^5 + 474x^4 - 280x^3 - 83x^2 + x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$2.18919$ of defining polynomial
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.18919 q^{2} +2.79257 q^{4} -0.868150 q^{5} -1.00000 q^{7} +1.73509 q^{8} +O(q^{10})$	\(q+2.18919 q^{2} +2.79257 q^{4} -0.868150 q^{5} -1.00000 q^{7} +1.73509 q^{8} -1.90055 q^{10} -0.280542 q^{11} +3.69064 q^{13} -2.18919 q^{14} -1.78669 q^{16} +1.92048 q^{17} +1.42421 q^{19} -2.42437 q^{20} -0.614161 q^{22} +6.52129 q^{23} -4.24632 q^{25} +8.07954 q^{26} -2.79257 q^{28} +3.07387 q^{29} -0.0163952 q^{31} -7.38160 q^{32} +4.20429 q^{34} +0.868150 q^{35} +0.359357 q^{37} +3.11787 q^{38} -1.50632 q^{40} -8.75364 q^{41} +4.09079 q^{43} -0.783433 q^{44} +14.2764 q^{46} -4.32210 q^{47} +1.00000 q^{49} -9.29601 q^{50} +10.3064 q^{52} +14.1066 q^{53} +0.243552 q^{55} -1.73509 q^{56} +6.72929 q^{58} +13.7905 q^{59} +0.486241 q^{61} -0.0358923 q^{62} -12.5864 q^{64} -3.20403 q^{65} +0.701192 q^{67} +5.36306 q^{68} +1.90055 q^{70} +9.14768 q^{71} -0.554740 q^{73} +0.786702 q^{74} +3.97721 q^{76} +0.280542 q^{77} -14.4508 q^{79} +1.55112 q^{80} -19.1634 q^{82} +11.1517 q^{83} -1.66726 q^{85} +8.95554 q^{86} -0.486765 q^{88} +16.6821 q^{89} -3.69064 q^{91} +18.2111 q^{92} -9.46192 q^{94} -1.23643 q^{95} +7.95594 q^{97} +2.18919 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8}+O(q^{10}) $ 16 * q + 2 * q^2 + 12 * q^4 + 9 * q^5 - 16 * q^7 + 6 * q^8	$ 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8} - 2 q^{10} + 22 q^{11} - 4 q^{13} - 2 q^{14} + 12 q^{16} + 18 q^{17} - 15 q^{19} + 40 q^{20} - 11 q^{22} + 5 q^{23} + 15 q^{25} + 24 q^{26} - 12 q^{28} + 12 q^{29} - 32 q^{31} + 9 q^{32} - 14 q^{34} - 9 q^{35} - 2 q^{37} - 3 q^{38} - 14 q^{40} + 45 q^{41} - 3 q^{43} + 54 q^{44} + 49 q^{47} + 16 q^{49} + 6 q^{50} + 38 q^{52} - 16 q^{53} + 7 q^{55} - 6 q^{56} + 16 q^{58} + 35 q^{59} - 11 q^{61} - 17 q^{62} - 2 q^{64} - 14 q^{65} + 17 q^{67} + 71 q^{68} + 2 q^{70} + 81 q^{71} - 15 q^{73} - 13 q^{74} + 14 q^{76} - 22 q^{77} - 34 q^{79} + 33 q^{80} - 14 q^{82} + 39 q^{83} - 17 q^{85} - 36 q^{86} + 61 q^{88} + 32 q^{89} + 4 q^{91} - 37 q^{92} + 13 q^{94} + 33 q^{95} - 4 q^{97} + 2 q^{98}+O(q^{100}) $ 16 * q + 2 * q^2 + 12 * q^4 + 9 * q^5 - 16 * q^7 + 6 * q^8 - 2 * q^10 + 22 * q^11 - 4 * q^13 - 2 * q^14 + 12 * q^16 + 18 * q^17 - 15 * q^19 + 40 * q^20 - 11 * q^22 + 5 * q^23 + 15 * q^25 + 24 * q^26 - 12 * q^28 + 12 * q^29 - 32 * q^31 + 9 * q^32 - 14 * q^34 - 9 * q^35 - 2 * q^37 - 3 * q^38 - 14 * q^40 + 45 * q^41 - 3 * q^43 + 54 * q^44 + 49 * q^47 + 16 * q^49 + 6 * q^50 + 38 * q^52 - 16 * q^53 + 7 * q^55 - 6 * q^56 + 16 * q^58 + 35 * q^59 - 11 * q^61 - 17 * q^62 - 2 * q^64 - 14 * q^65 + 17 * q^67 + 71 * q^68 + 2 * q^70 + 81 * q^71 - 15 * q^73 - 13 * q^74 + 14 * q^76 - 22 * q^77 - 34 * q^79 + 33 * q^80 - 14 * q^82 + 39 * q^83 - 17 * q^85 - 36 * q^86 + 61 * q^88 + 32 * q^89 + 4 * q^91 - 37 * q^92 + 13 * q^94 + 33 * q^95 - 4 * q^97 + 2 * 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 $

$2$	2.18919	1.54799	0.773997	−	0.633189i	$-0.218255\pi$
$2$	2.18919	1.54799	0.773997	+	0.633189i	$0.218255\pi$
$3$	0	0
$3$	0	0
$4$	2.79257	1.39629
$5$	−0.868150	−0.388249	−0.194124	−	0.980977i	$-0.562186\pi$
$5$	−0.868150	−0.388249	−0.194124	+	0.980977i	$0.562186\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.73509	0.613447
$9$	0	0
$10$	−1.90055	−0.601006
$11$	−0.280542	−0.0845866	−0.0422933	−	0.999105i	$-0.513466\pi$
$11$	−0.280542	−0.0845866	−0.0422933	+	0.999105i	$0.513466\pi$
$12$	0	0
$13$	3.69064	1.02360	0.511800	−	0.859104i	$-0.328979\pi$
$13$	3.69064	1.02360	0.511800	+	0.859104i	$0.328979\pi$
$14$	−2.18919	−0.585087
$15$	0	0
$16$	−1.78669	−0.446673
$17$	1.92048	0.465784	0.232892	−	0.972503i	$-0.425181\pi$
$17$	1.92048	0.465784	0.232892	+	0.972503i	$0.425181\pi$
$18$	0	0
$19$	1.42421	0.326736	0.163368	−	0.986565i	$-0.447764\pi$
$19$	1.42421	0.326736	0.163368	+	0.986565i	$0.447764\pi$
$20$	−2.42437	−0.542106
$21$	0	0
$22$	−0.614161	−0.130939
$23$	6.52129	1.35978	0.679891	−	0.733313i	$-0.262027\pi$
$23$	6.52129	1.35978	0.679891	+	0.733313i	$0.262027\pi$
$24$	0	0
$25$	−4.24632	−0.849263
$26$	8.07954	1.58453
$27$	0	0
$28$	−2.79257	−0.527746
$29$	3.07387	0.570803	0.285402	−	0.958408i	$-0.407873\pi$
$29$	3.07387	0.570803	0.285402	+	0.958408i	$0.407873\pi$
$30$	0	0
$31$	−0.0163952	−0.00294467	−0.00147234	−	0.999999i	$-0.500469\pi$
$31$	−0.0163952	−0.00294467	−0.00147234	+	0.999999i	$0.500469\pi$
$32$	−7.38160	−1.30489
$33$	0	0
$34$	4.20429	0.721031
$35$	0.868150	0.146744
$36$	0	0
$37$	0.359357	0.0590779	0.0295389	−	0.999564i	$-0.490596\pi$
$37$	0.359357	0.0590779	0.0295389	+	0.999564i	$0.490596\pi$
$38$	3.11787	0.505786
$39$	0	0
$40$	−1.50632	−0.238170
$41$	−8.75364	−1.36709	−0.683544	−	0.729909i	$-0.739562\pi$
$41$	−8.75364	−1.36709	−0.683544	+	0.729909i	$0.739562\pi$
$42$	0	0
$43$	4.09079	0.623840	0.311920	−	0.950108i	$-0.399028\pi$
$43$	4.09079	0.623840	0.311920	+	0.950108i	$0.399028\pi$
$44$	−0.783433	−0.118107
$45$	0	0
$46$	14.2764	2.10493
$47$	−4.32210	−0.630443	−0.315222	−	0.949018i	$-0.602079\pi$
$47$	−4.32210	−0.630443	−0.315222	+	0.949018i	$0.602079\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−9.29601	−1.31465
$51$	0	0
$52$	10.3064	1.42924
$53$	14.1066	1.93769	0.968846	−	0.247662i	$-0.0796623\pi$
$53$	14.1066	1.93769	0.968846	+	0.247662i	$0.0796623\pi$
$54$	0	0
$55$	0.243552	0.0328406
$56$	−1.73509	−0.231861
$57$	0	0
$58$	6.72929	0.883600
$59$	13.7905	1.79537	0.897685	−	0.440638i	$-0.145248\pi$
$59$	13.7905	1.79537	0.897685	+	0.440638i	$0.145248\pi$
$60$	0	0
$61$	0.486241	0.0622568	0.0311284	−	0.999515i	$-0.490090\pi$
$61$	0.486241	0.0622568	0.0311284	+	0.999515i	$0.490090\pi$
$62$	−0.0358923	−0.00455833
$63$	0	0
$64$	−12.5864	−1.57329
$65$	−3.20403	−0.397411
$66$	0	0
$67$	0.701192	0.0856643	0.0428321	−	0.999082i	$-0.486362\pi$
$67$	0.701192	0.0856643	0.0428321	+	0.999082i	$0.486362\pi$
$68$	5.36306	0.650367
$69$	0	0
$70$	1.90055	0.227159
$71$	9.14768	1.08563	0.542815	−	0.839852i	$-0.317358\pi$
$71$	9.14768	1.08563	0.542815	+	0.839852i	$0.317358\pi$
$72$	0	0
$73$	−0.554740	−0.0649273	−0.0324637	−	0.999473i	$-0.510335\pi$
$73$	−0.554740	−0.0649273	−0.0324637	+	0.999473i	$0.510335\pi$
$74$	0.786702	0.0914522
$75$	0	0
$76$	3.97721	0.456217
$77$	0.280542	0.0319707
$78$	0	0
$79$	−14.4508	−1.62584	−0.812921	−	0.582374i	$-0.802124\pi$
$79$	−14.4508	−1.62584	−0.812921	+	0.582374i	$0.802124\pi$
$80$	1.55112	0.173420
$81$	0	0
$82$	−19.1634	−2.11624
$83$	11.1517	1.22405	0.612027	−	0.790837i	$-0.290354\pi$
$83$	11.1517	1.22405	0.612027	+	0.790837i	$0.290354\pi$
$84$	0	0
$85$	−1.66726	−0.180840
$86$	8.95554	0.965700
$87$	0	0
$88$	−0.486765	−0.0518894
$89$	16.6821	1.76830	0.884149	−	0.467205i	$-0.154739\pi$
$89$	16.6821	1.76830	0.884149	+	0.467205i	$0.154739\pi$
$90$	0	0
$91$	−3.69064	−0.386885
$92$	18.2111	1.89864
$93$	0	0
$94$	−9.46192	−0.975922
$95$	−1.23643	−0.126855
$96$	0	0
$97$	7.95594	0.807803	0.403902	−	0.914802i	$-0.367654\pi$
$97$	7.95594	0.807803	0.403902	+	0.914802i	$0.367654\pi$
$98$	2.18919	0.221142
$99$	0	0
$100$	−11.8581	−1.18581
$101$	−5.03438	−0.500939	−0.250470	−	0.968124i	$-0.580585\pi$
$101$	−5.03438	−0.500939	−0.250470	+	0.968124i	$0.580585\pi$
$102$	0	0
$103$	2.57322	0.253547	0.126774	−	0.991932i	$-0.459538\pi$
$103$	2.57322	0.253547	0.126774	+	0.991932i	$0.459538\pi$
$104$	6.40360	0.627925
$105$	0	0
$106$	30.8821	2.99954
$107$	6.71132	0.648808	0.324404	−	0.945919i	$-0.394836\pi$
$107$	6.71132	0.648808	0.324404	+	0.945919i	$0.394836\pi$
$108$	0	0
$109$	1.76219	0.168787	0.0843935	−	0.996433i	$-0.473105\pi$
$109$	1.76219	0.168787	0.0843935	+	0.996433i	$0.473105\pi$
$110$	0.533184	0.0508371
$111$	0	0
$112$	1.78669	0.168827
$113$	8.28904	0.779768	0.389884	−	0.920864i	$-0.372515\pi$
$113$	8.28904	0.779768	0.389884	+	0.920864i	$0.372515\pi$
$114$	0	0
$115$	−5.66145	−0.527933
$116$	8.58399	0.797004
$117$	0	0
$118$	30.1901	2.77922
$119$	−1.92048	−0.176050
$120$	0	0
$121$	−10.9213	−0.992845
$122$	1.06448	0.0963732
$123$	0	0
$124$	−0.0457848	−0.00411160
$125$	8.02719	0.717974
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−12.7908	−1.13056
$129$	0	0
$130$	−7.01425	−0.615190
$131$	15.0169	1.31203	0.656015	−	0.754748i	$-0.272241\pi$
$131$	15.0169	1.31203	0.656015	+	0.754748i	$0.272241\pi$
$132$	0	0
$133$	−1.42421	−0.123495
$134$	1.53505	0.132608
$135$	0	0
$136$	3.33220	0.285734
$137$	−16.5669	−1.41540	−0.707702	−	0.706511i	$-0.750268\pi$
$137$	−16.5669	−1.41540	−0.707702	+	0.706511i	$0.750268\pi$
$138$	0	0
$139$	−14.5682	−1.23566	−0.617830	−	0.786312i	$-0.711988\pi$
$139$	−14.5682	−1.23566	−0.617830	+	0.786312i	$0.711988\pi$
$140$	2.42437	0.204897
$141$	0	0
$142$	20.0261	1.68055
$143$	−1.03538	−0.0865828
$144$	0	0
$145$	−2.66858	−0.221613
$146$	−1.21443	−0.100507
$147$	0	0
$148$	1.00353	0.0824896
$149$	11.3687	0.931364	0.465682	−	0.884952i	$-0.345809\pi$
$149$	11.3687	0.931364	0.465682	+	0.884952i	$0.345809\pi$
$150$	0	0
$151$	−9.59570	−0.780887	−0.390443	−	0.920627i	$-0.627678\pi$
$151$	−9.59570	−0.780887	−0.390443	+	0.920627i	$0.627678\pi$
$152$	2.47113	0.200435
$153$	0	0
$154$	0.614161	0.0494905
$155$	0.0142335	0.00114326
$156$	0	0
$157$	10.8227	0.863743	0.431871	−	0.901935i	$-0.357853\pi$
$157$	10.8227	0.863743	0.431871	+	0.901935i	$0.357853\pi$
$158$	−31.6356	−2.51679
$159$	0	0
$160$	6.40833	0.506623
$161$	−6.52129	−0.513949
$162$	0	0
$163$	24.3493	1.90718	0.953590	−	0.301107i	$-0.0973561\pi$
$163$	24.3493	1.90718	0.953590	+	0.301107i	$0.0973561\pi$
$164$	−24.4451	−1.90885
$165$	0	0
$166$	24.4131	1.89483
$167$	−14.3860	−1.11322	−0.556611	−	0.830773i	$-0.687899\pi$
$167$	−14.3860	−1.11322	−0.556611	+	0.830773i	$0.687899\pi$
$168$	0	0
$169$	0.620853	0.0477579
$170$	−3.64996	−0.279939
$171$	0	0
$172$	11.4238	0.871058
$173$	20.4335	1.55353	0.776765	−	0.629790i	$-0.216859\pi$
$173$	20.4335	1.55353	0.776765	+	0.629790i	$0.216859\pi$
$174$	0	0
$175$	4.24632	0.320991
$176$	0.501242	0.0377825
$177$	0	0
$178$	36.5203	2.73731
$179$	1.77187	0.132436	0.0662178	−	0.997805i	$-0.478907\pi$
$179$	1.77187	0.132436	0.0662178	+	0.997805i	$0.478907\pi$
$180$	0	0
$181$	13.7325	1.02073	0.510365	−	0.859958i	$-0.329510\pi$
$181$	13.7325	1.02073	0.510365	+	0.859958i	$0.329510\pi$
$182$	−8.07954	−0.598895
$183$	0	0
$184$	11.3150	0.834154
$185$	−0.311976	−0.0229369
$186$	0	0
$187$	−0.538774	−0.0393991
$188$	−12.0698	−0.880279
$189$	0	0
$190$	−2.70678	−0.196371
$191$	−7.22638	−0.522883	−0.261441	−	0.965219i	$-0.584198\pi$
$191$	−7.22638	−0.522883	−0.261441	+	0.965219i	$0.584198\pi$
$192$	0	0
$193$	13.0486	0.939255	0.469628	−	0.882865i	$-0.344388\pi$
$193$	13.0486	0.939255	0.469628	+	0.882865i	$0.344388\pi$
$194$	17.4171	1.25047
$195$	0	0
$196$	2.79257	0.199469
$197$	3.12406	0.222580	0.111290	−	0.993788i	$-0.464502\pi$
$197$	3.12406	0.222580	0.111290	+	0.993788i	$0.464502\pi$
$198$	0	0
$199$	−10.9671	−0.777438	−0.388719	−	0.921356i	$-0.627082\pi$
$199$	−10.9671	−0.777438	−0.388719	+	0.921356i	$0.627082\pi$
$200$	−7.36774	−0.520978
$201$	0	0
$202$	−11.0212	−0.775451
$203$	−3.07387	−0.215743
$204$	0	0
$205$	7.59947	0.530770
$206$	5.63328	0.392489
$207$	0	0
$208$	−6.59404	−0.457215
$209$	−0.399551	−0.0276375
$210$	0	0
$211$	−13.8674	−0.954668	−0.477334	−	0.878722i	$-0.658397\pi$
$211$	−13.8674	−0.954668	−0.477334	+	0.878722i	$0.658397\pi$
$212$	39.3937	2.70557
$213$	0	0
$214$	14.6924	1.00435
$215$	−3.55142	−0.242205
$216$	0	0
$217$	0.0163952	0.00111298
$218$	3.85777	0.261281
$219$	0	0
$220$	0.680137	0.0458548
$221$	7.08779	0.476777
$222$	0	0
$223$	9.77971	0.654898	0.327449	−	0.944869i	$-0.393811\pi$
$223$	9.77971	0.654898	0.327449	+	0.944869i	$0.393811\pi$
$224$	7.38160	0.493204
$225$	0	0
$226$	18.1463	1.20708
$227$	12.3932	0.822563	0.411282	−	0.911508i	$-0.365081\pi$
$227$	12.3932	0.822563	0.411282	+	0.911508i	$0.365081\pi$
$228$	0	0
$229$	24.5147	1.61998	0.809988	−	0.586447i	$-0.199474\pi$
$229$	24.5147	1.61998	0.809988	+	0.586447i	$0.199474\pi$
$230$	−12.3940	−0.817238
$231$	0	0
$232$	5.33344	0.350157
$233$	7.75695	0.508175	0.254087	−	0.967181i	$-0.418225\pi$
$233$	7.75695	0.508175	0.254087	+	0.967181i	$0.418225\pi$
$234$	0	0
$235$	3.75223	0.244769
$236$	38.5109	2.50685
$237$	0	0
$238$	−4.20429	−0.272524
$239$	−12.1161	−0.783723	−0.391862	−	0.920024i	$-0.628169\pi$
$239$	−12.1161	−0.783723	−0.391862	+	0.920024i	$0.628169\pi$
$240$	0	0
$241$	3.36131	0.216521	0.108261	−	0.994123i	$-0.465472\pi$
$241$	3.36131	0.216521	0.108261	+	0.994123i	$0.465472\pi$
$242$	−23.9088	−1.53692
$243$	0	0
$244$	1.35786	0.0869283
$245$	−0.868150	−0.0554641
$246$	0	0
$247$	5.25626	0.334448
$248$	−0.0284472	−0.00180640
$249$	0	0
$250$	17.5731	1.11142
$251$	23.2358	1.46663	0.733314	−	0.679890i	$-0.237973\pi$
$251$	23.2358	1.46663	0.733314	+	0.679890i	$0.237973\pi$
$252$	0	0
$253$	−1.82949	−0.115019
$254$	−2.18919	−0.137362
$255$	0	0
$256$	−2.82880	−0.176800
$257$	−19.4430	−1.21282	−0.606411	−	0.795151i	$-0.707392\pi$
$257$	−19.4430	−1.21282	−0.606411	+	0.795151i	$0.707392\pi$
$258$	0	0
$259$	−0.359357	−0.0223293
$260$	−8.94749	−0.554900
$261$	0	0
$262$	32.8749	2.03102
$263$	4.25647	0.262465	0.131233	−	0.991352i	$-0.458106\pi$
$263$	4.25647	0.262465	0.131233	+	0.991352i	$0.458106\pi$
$264$	0	0
$265$	−12.2467	−0.752306
$266$	−3.11787	−0.191169
$267$	0	0
$268$	1.95813	0.119612
$269$	−4.00443	−0.244154	−0.122077	−	0.992521i	$-0.538956\pi$
$269$	−4.00443	−0.244154	−0.122077	+	0.992521i	$0.538956\pi$
$270$	0	0
$271$	0.684262	0.0415659	0.0207830	−	0.999784i	$-0.493384\pi$
$271$	0.684262	0.0415659	0.0207830	+	0.999784i	$0.493384\pi$
$272$	−3.43130	−0.208053
$273$	0	0
$274$	−36.2681	−2.19104
$275$	1.19127	0.0718362
$276$	0	0
$277$	5.19540	0.312162	0.156081	−	0.987744i	$-0.450114\pi$
$277$	5.19540	0.312162	0.156081	+	0.987744i	$0.450114\pi$
$278$	−31.8926	−1.91279
$279$	0	0
$280$	1.50632	0.0900197
$281$	−23.1569	−1.38143	−0.690713	−	0.723129i	$-0.742703\pi$
$281$	−23.1569	−1.38143	−0.690713	+	0.723129i	$0.742703\pi$
$282$	0	0
$283$	−0.107190	−0.00637180	−0.00318590	−	0.999995i	$-0.501014\pi$
$283$	−0.107190	−0.00637180	−0.00318590	+	0.999995i	$0.501014\pi$
$284$	25.5455	1.51585
$285$	0	0
$286$	−2.26665	−0.134030
$287$	8.75364	0.516711
$288$	0	0
$289$	−13.3118	−0.783045
$290$	−5.84204	−0.343056
$291$	0	0
$292$	−1.54915	−0.0906571
$293$	−10.5671	−0.617337	−0.308668	−	0.951170i	$-0.599883\pi$
$293$	−10.5671	−0.617337	−0.308668	+	0.951170i	$0.599883\pi$
$294$	0	0
$295$	−11.9722	−0.697050
$296$	0.623516	0.0362411
$297$	0	0
$298$	24.8884	1.44175
$299$	24.0677	1.39187
$300$	0	0
$301$	−4.09079	−0.235789
$302$	−21.0068	−1.20881
$303$	0	0
$304$	−2.54463	−0.145944
$305$	−0.422130	−0.0241711
$306$	0	0
$307$	−3.41074	−0.194661	−0.0973307	−	0.995252i	$-0.531030\pi$
$307$	−3.41074	−0.194661	−0.0973307	+	0.995252i	$0.531030\pi$
$308$	0.783433	0.0446402
$309$	0	0
$310$	0.0311599	0.00176977
$311$	−29.7332	−1.68602	−0.843008	−	0.537901i	$-0.819218\pi$
$311$	−29.7332	−1.68602	−0.843008	+	0.537901i	$0.819218\pi$
$312$	0	0
$313$	21.5328	1.21710	0.608552	−	0.793514i	$-0.291751\pi$
$313$	21.5328	1.21710	0.608552	+	0.793514i	$0.291751\pi$
$314$	23.6929	1.33707
$315$	0	0
$316$	−40.3549	−2.27014
$317$	−28.3682	−1.59332	−0.796660	−	0.604428i	$-0.793402\pi$
$317$	−28.3682	−1.59332	−0.796660	+	0.604428i	$0.793402\pi$
$318$	0	0
$319$	−0.862349	−0.0482823
$320$	10.9268	0.610829
$321$	0	0
$322$	−14.2764	−0.795590
$323$	2.73516	0.152189
$324$	0	0
$325$	−15.6716	−0.869306
$326$	53.3052	2.95230
$327$	0	0
$328$	−15.1883	−0.838636
$329$	4.32210	0.238285
$330$	0	0
$331$	−7.53693	−0.414267	−0.207134	−	0.978313i	$-0.566414\pi$
$331$	−7.53693	−0.414267	−0.207134	+	0.978313i	$0.566414\pi$
$332$	31.1418	1.70913
$333$	0	0
$334$	−31.4938	−1.72326
$335$	−0.608740	−0.0332590
$336$	0	0
$337$	11.9671	0.651888	0.325944	−	0.945389i	$-0.394318\pi$
$337$	11.9671	0.651888	0.325944	+	0.945389i	$0.394318\pi$
$338$	1.35917	0.0739289
$339$	0	0
$340$	−4.65594	−0.252504
$341$	0.00459955	0.000249080 0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	7.09789	0.382693
$345$	0	0
$346$	44.7329	2.40486
$347$	−4.07527	−0.218772	−0.109386	−	0.993999i	$-0.534888\pi$
$347$	−4.07527	−0.218772	−0.109386	+	0.993999i	$0.534888\pi$
$348$	0	0
$349$	−2.38630	−0.127736	−0.0638679	−	0.997958i	$-0.520344\pi$
$349$	−2.38630	−0.127736	−0.0638679	+	0.997958i	$0.520344\pi$
$350$	9.29601	0.496893
$351$	0	0
$352$	2.07085	0.110376
$353$	−19.0735	−1.01518	−0.507591	−	0.861598i	$-0.669464\pi$
$353$	−19.0735	−1.01518	−0.507591	+	0.861598i	$0.669464\pi$
$354$	0	0
$355$	−7.94156	−0.421494
$356$	46.5859	2.46905
$357$	0	0
$358$	3.87896	0.205010
$359$	−1.49830	−0.0790775	−0.0395387	−	0.999218i	$-0.512589\pi$
$359$	−1.49830	−0.0790775	−0.0395387	+	0.999218i	$0.512589\pi$
$360$	0	0
$361$	−16.9716	−0.893243
$362$	30.0631	1.58008
$363$	0	0
$364$	−10.3064	−0.540201
$365$	0.481597	0.0252079
$366$	0	0
$367$	10.1206	0.528291	0.264146	−	0.964483i	$-0.414910\pi$
$367$	10.1206	0.528291	0.264146	+	0.964483i	$0.414910\pi$
$368$	−11.6515	−0.607378
$369$	0	0
$370$	−0.682975	−0.0355062
$371$	−14.1066	−0.732379
$372$	0	0
$373$	−25.6071	−1.32588	−0.662942	−	0.748671i	$-0.730693\pi$
$373$	−25.6071	−1.32588	−0.662942	+	0.748671i	$0.730693\pi$
$374$	−1.17948	−0.0609895
$375$	0	0
$376$	−7.49923	−0.386743
$377$	11.3446	0.584274
$378$	0	0
$379$	−13.9735	−0.717771	−0.358886	−	0.933382i	$-0.616843\pi$
$379$	−13.9735	−0.717771	−0.358886	+	0.933382i	$0.616843\pi$
$380$	−3.45281	−0.177126
$381$	0	0
$382$	−15.8199	−0.809419
$383$	25.7039	1.31341	0.656705	−	0.754147i	$-0.271950\pi$
$383$	25.7039	1.31341	0.656705	+	0.754147i	$0.271950\pi$
$384$	0	0
$385$	−0.243552	−0.0124126
$386$	28.5658	1.45396
$387$	0	0
$388$	22.2175	1.12792
$389$	−4.06415	−0.206061	−0.103030	−	0.994678i	$-0.532854\pi$
$389$	−4.06415	−0.206061	−0.103030	+	0.994678i	$0.532854\pi$
$390$	0	0
$391$	12.5240	0.633365
$392$	1.73509	0.0876353
$393$	0	0
$394$	6.83918	0.344553
$395$	12.5455	0.631231
$396$	0	0
$397$	1.03102	0.0517453	0.0258726	−	0.999665i	$-0.491764\pi$
$397$	1.03102	0.0517453	0.0258726	+	0.999665i	$0.491764\pi$
$398$	−24.0091	−1.20347
$399$	0	0
$400$	7.58686	0.379343
$401$	3.59965	0.179758	0.0898789	−	0.995953i	$-0.471352\pi$
$401$	3.59965	0.179758	0.0898789	+	0.995953i	$0.471352\pi$
$402$	0	0
$403$	−0.0605090	−0.00301417
$404$	−14.0588	−0.699454
$405$	0	0
$406$	−6.72929	−0.333969
$407$	−0.100815	−0.00499720
$408$	0	0
$409$	−21.8694	−1.08137	−0.540685	−	0.841225i	$-0.681835\pi$
$409$	−21.8694	−1.08137	−0.540685	+	0.841225i	$0.681835\pi$
$410$	16.6367	0.821629
$411$	0	0
$412$	7.18590	0.354024
$413$	−13.7905	−0.678586
$414$	0	0
$415$	−9.68131	−0.475237
$416$	−27.2428	−1.33569
$417$	0	0
$418$	−0.874694	−0.0427827
$419$	14.5188	0.709288	0.354644	−	0.935001i	$-0.384602\pi$
$419$	14.5188	0.709288	0.354644	+	0.935001i	$0.384602\pi$
$420$	0	0
$421$	1.74372	0.0849836	0.0424918	−	0.999097i	$-0.486470\pi$
$421$	1.74372	0.0849836	0.0424918	+	0.999097i	$0.486470\pi$
$422$	−30.3583	−1.47782
$423$	0	0
$424$	24.4762	1.18867
$425$	−8.15495	−0.395573
$426$	0	0
$427$	−0.486241	−0.0235309
$428$	18.7418	0.905920
$429$	0	0
$430$	−7.77475	−0.374932
$431$	−24.6850	−1.18904	−0.594518	−	0.804082i	$-0.702657\pi$
$431$	−24.6850	−1.18904	−0.594518	+	0.804082i	$0.702657\pi$
$432$	0	0
$433$	−16.9930	−0.816633	−0.408316	−	0.912840i	$-0.633884\pi$
$433$	−16.9930	−0.816633	−0.408316	+	0.912840i	$0.633884\pi$
$434$	0.0358923	0.00172289
$435$	0	0
$436$	4.92103	0.235675
$437$	9.28769	0.444290
$438$	0	0
$439$	9.88431	0.471752	0.235876	−	0.971783i	$-0.424204\pi$
$439$	9.88431	0.471752	0.235876	+	0.971783i	$0.424204\pi$
$440$	0.422585	0.0201460
$441$	0	0
$442$	15.5166	0.738047
$443$	−39.8440	−1.89305	−0.946523	−	0.322636i	$-0.895431\pi$
$443$	−39.8440	−1.89305	−0.946523	+	0.322636i	$0.895431\pi$
$444$	0	0
$445$	−14.4826	−0.686539
$446$	21.4097	1.01378
$447$	0	0
$448$	12.5864	0.594650
$449$	−24.0250	−1.13381	−0.566904	−	0.823784i	$-0.691859\pi$
$449$	−24.0250	−1.13381	−0.566904	+	0.823784i	$0.691859\pi$
$450$	0	0
$451$	2.45576	0.115637
$452$	23.1477	1.08878
$453$	0	0
$454$	27.1310	1.27332
$455$	3.20403	0.150207
$456$	0	0
$457$	3.81846	0.178620	0.0893101	−	0.996004i	$-0.471534\pi$
$457$	3.81846	0.178620	0.0893101	+	0.996004i	$0.471534\pi$
$458$	53.6674	2.50771
$459$	0	0
$460$	−15.8100	−0.737145
$461$	27.3775	1.27510	0.637549	−	0.770409i	$-0.279948\pi$
$461$	27.3775	1.27510	0.637549	+	0.770409i	$0.279948\pi$
$462$	0	0
$463$	−6.11153	−0.284027	−0.142013	−	0.989865i	$-0.545358\pi$
$463$	−6.11153	−0.284027	−0.142013	+	0.989865i	$0.545358\pi$
$464$	−5.49206	−0.254962
$465$	0	0
$466$	16.9815	0.786652
$467$	−23.4738	−1.08624	−0.543118	−	0.839656i	$-0.682756\pi$
$467$	−23.4738	−1.08624	−0.543118	+	0.839656i	$0.682756\pi$
$468$	0	0
$469$	−0.701192	−0.0323781
$470$	8.21437	0.378900
$471$	0	0
$472$	23.9278	1.10136
$473$	−1.14764	−0.0527685
$474$	0	0
$475$	−6.04765	−0.277485
$476$	−5.36306	−0.245816
$477$	0	0
$478$	−26.5244	−1.21320
$479$	30.4428	1.39097	0.695484	−	0.718541i	$-0.255190\pi$
$479$	30.4428	1.39097	0.695484	+	0.718541i	$0.255190\pi$
$480$	0	0
$481$	1.32626	0.0604722
$482$	7.35857	0.335173
$483$	0	0
$484$	−30.4985	−1.38629
$485$	−6.90695	−0.313628
$486$	0	0
$487$	−43.9485	−1.99150	−0.995749	−	0.0921094i	$-0.970639\pi$
$487$	−43.9485	−1.99150	−0.995749	+	0.0921094i	$0.970639\pi$
$488$	0.843672	0.0381912
$489$	0	0
$490$	−1.90055	−0.0858580
$491$	4.76508	0.215045	0.107522	−	0.994203i	$-0.465708\pi$
$491$	4.76508	0.215045	0.107522	+	0.994203i	$0.465708\pi$
$492$	0	0
$493$	5.90329	0.265871
$494$	11.5070	0.517723
$495$	0	0
$496$	0.0292932	0.00131531
$497$	−9.14768	−0.410330
$498$	0	0
$499$	−9.87038	−0.441859	−0.220929	−	0.975290i	$-0.570909\pi$
$499$	−9.87038	−0.441859	−0.220929	+	0.975290i	$0.570909\pi$
$500$	22.4165	1.00250
$501$	0	0
$502$	50.8676	2.27033
$503$	29.9723	1.33640	0.668200	−	0.743982i	$-0.267065\pi$
$503$	29.9723	1.33640	0.668200	+	0.743982i	$0.267065\pi$
$504$	0	0
$505$	4.37059	0.194489
$506$	−4.00512	−0.178049
$507$	0	0
$508$	−2.79257	−0.123900
$509$	−33.7727	−1.49695	−0.748474	−	0.663165i	$-0.769213\pi$
$509$	−33.7727	−1.49695	−0.748474	+	0.663165i	$0.769213\pi$
$510$	0	0
$511$	0.554740	0.0245402
$512$	19.3888	0.856871
$513$	0	0
$514$	−42.5646	−1.87744
$515$	−2.23394	−0.0984393
$516$	0	0
$517$	1.21253	0.0533270
$518$	−0.786702	−0.0345657
$519$	0	0
$520$	−5.55928	−0.243791
$521$	−23.4277	−1.02639	−0.513194	−	0.858273i	$-0.671538\pi$
$521$	−23.4277	−1.02639	−0.513194	+	0.858273i	$0.671538\pi$
$522$	0	0
$523$	0.337038	0.0147376	0.00736882	−	0.999973i	$-0.497654\pi$
$523$	0.337038	0.0147376	0.00736882	+	0.999973i	$0.497654\pi$
$524$	41.9357	1.83197
$525$	0	0
$526$	9.31825	0.406295
$527$	−0.0314867	−0.00137158
$528$	0	0
$529$	19.5272	0.849007
$530$	−26.8103	−1.16457
$531$	0	0
$532$	−3.97721	−0.172434
$533$	−32.3066	−1.39935
$534$	0	0
$535$	−5.82643	−0.251899
$536$	1.21663	0.0525505
$537$	0	0
$538$	−8.76646	−0.377949
$539$	−0.280542	−0.0120838
$540$	0	0
$541$	5.85948	0.251919	0.125959	−	0.992035i	$-0.459799\pi$
$541$	5.85948	0.251919	0.125959	+	0.992035i	$0.459799\pi$
$542$	1.49798	0.0643438
$543$	0	0
$544$	−14.1762	−0.607799
$545$	−1.52984	−0.0655313
$546$	0	0
$547$	31.8131	1.36023	0.680114	−	0.733106i	$-0.261930\pi$
$547$	31.8131	1.36023	0.680114	+	0.733106i	$0.261930\pi$
$548$	−46.2642	−1.97631
$549$	0	0
$550$	2.60792	0.111202
$551$	4.37784	0.186502
$552$	0	0
$553$	14.4508	0.614510
$554$	11.3737	0.483224
$555$	0	0
$556$	−40.6827	−1.72533
$557$	7.19289	0.304773	0.152386	−	0.988321i	$-0.451304\pi$
$557$	7.19289	0.304773	0.152386	+	0.988321i	$0.451304\pi$
$558$	0	0
$559$	15.0977	0.638563
$560$	−1.55112	−0.0655467
$561$	0	0
$562$	−50.6950	−2.13844
$563$	−28.3208	−1.19358	−0.596789	−	0.802398i	$-0.703557\pi$
$563$	−28.3208	−1.19358	−0.596789	+	0.802398i	$0.703557\pi$
$564$	0	0
$565$	−7.19613	−0.302744
$566$	−0.234660	−0.00986351
$567$	0	0
$568$	15.8721	0.665977
$569$	−26.6346	−1.11658	−0.558290	−	0.829646i	$-0.688542\pi$
$569$	−26.6346	−1.11658	−0.558290	+	0.829646i	$0.688542\pi$
$570$	0	0
$571$	15.4515	0.646625	0.323313	−	0.946292i	$-0.395203\pi$
$571$	15.4515	0.646625	0.323313	+	0.946292i	$0.395203\pi$
$572$	−2.89137	−0.120894
$573$	0	0
$574$	19.1634	0.799865
$575$	−27.6914	−1.15481
$576$	0	0
$577$	−24.0575	−1.00153	−0.500763	−	0.865585i	$-0.666947\pi$
$577$	−24.0575	−1.00153	−0.500763	+	0.865585i	$0.666947\pi$
$578$	−29.1420	−1.21215
$579$	0	0
$580$	−7.45219	−0.309436
$581$	−11.1517	−0.462649
$582$	0	0
$583$	−3.95750	−0.163903
$584$	−0.962523	−0.0398295
$585$	0	0
$586$	−23.1334	−0.955634
$587$	38.3645	1.58347	0.791736	−	0.610863i	$-0.209177\pi$
$587$	38.3645	1.58347	0.791736	+	0.610863i	$0.209177\pi$
$588$	0	0
$589$	−0.0233503	−0.000962131 0
$590$	−26.2095	−1.07903
$591$	0	0
$592$	−0.642060	−0.0263885
$593$	39.1039	1.60580	0.802902	−	0.596111i	$-0.203288\pi$
$593$	39.1039	1.60580	0.802902	+	0.596111i	$0.203288\pi$
$594$	0	0
$595$	1.66726	0.0683511
$596$	31.7480	1.30045
$597$	0	0
$598$	52.6890	2.15461
$599$	34.6581	1.41609	0.708045	−	0.706167i	$-0.249577\pi$
$599$	34.6581	1.41609	0.708045	+	0.706167i	$0.249577\pi$
$600$	0	0
$601$	31.9773	1.30438	0.652190	−	0.758056i	$-0.273850\pi$
$601$	31.9773	1.30438	0.652190	+	0.758056i	$0.273850\pi$
$602$	−8.95554	−0.365000
$603$	0	0
$604$	−26.7967	−1.09034
$605$	9.48132	0.385471
$606$	0	0
$607$	−24.7650	−1.00518	−0.502591	−	0.864525i	$-0.667620\pi$
$607$	−24.7650	−1.00518	−0.502591	+	0.864525i	$0.667620\pi$
$608$	−10.5129	−0.426356
$609$	0	0
$610$	−0.924125	−0.0374167
$611$	−15.9513	−0.645322
$612$	0	0
$613$	28.3938	1.14681	0.573407	−	0.819271i	$-0.305621\pi$
$613$	28.3938	1.14681	0.573407	+	0.819271i	$0.305621\pi$
$614$	−7.46678	−0.301335
$615$	0	0
$616$	0.486765	0.0196123
$617$	−5.98355	−0.240889	−0.120444	−	0.992720i	$-0.538432\pi$
$617$	−5.98355	−0.240889	−0.120444	+	0.992720i	$0.538432\pi$
$618$	0	0
$619$	4.70457	0.189093	0.0945464	−	0.995520i	$-0.469860\pi$
$619$	4.70457	0.189093	0.0945464	+	0.995520i	$0.469860\pi$
$620$	0.0397481	0.00159632
$621$	0	0
$622$	−65.0918	−2.60994
$623$	−16.6821	−0.668354
$624$	0	0
$625$	14.2628	0.570511
$626$	47.1394	1.88407
$627$	0	0
$628$	30.2231	1.20603
$629$	0.690136	0.0275175
$630$	0	0
$631$	−21.1503	−0.841978	−0.420989	−	0.907066i	$-0.638317\pi$
$631$	−21.1503	−0.841978	−0.420989	+	0.907066i	$0.638317\pi$
$632$	−25.0734	−0.997367
$633$	0	0
$634$	−62.1036	−2.46645
$635$	0.868150	0.0344515
$636$	0	0
$637$	3.69064	0.146229
$638$	−1.88785	−0.0747406
$639$	0	0
$640$	11.1043	0.438937
$641$	−8.70143	−0.343686	−0.171843	−	0.985124i	$-0.554972\pi$
$641$	−8.70143	−0.343686	−0.171843	+	0.985124i	$0.554972\pi$
$642$	0	0
$643$	−12.9855	−0.512100	−0.256050	−	0.966664i	$-0.582421\pi$
$643$	−12.9855	−0.512100	−0.256050	+	0.966664i	$0.582421\pi$
$644$	−18.2111	−0.717620
$645$	0	0
$646$	5.98780	0.235587
$647$	29.0762	1.14310	0.571552	−	0.820566i	$-0.306342\pi$
$647$	29.0762	1.14310	0.571552	+	0.820566i	$0.306342\pi$
$648$	0	0
$649$	−3.86881	−0.151864
$650$	−34.3083	−1.34568
$651$	0	0
$652$	67.9970	2.66297
$653$	−0.0994735	−0.00389270	−0.00194635	−	0.999998i	$-0.500620\pi$
$653$	−0.0994735	−0.00389270	−0.00194635	+	0.999998i	$0.500620\pi$
$654$	0	0
$655$	−13.0369	−0.509394
$656$	15.6401	0.610642
$657$	0	0
$658$	9.46192	0.368864
$659$	−31.4715	−1.22596	−0.612978	−	0.790100i	$-0.710029\pi$
$659$	−31.4715	−1.22596	−0.612978	+	0.790100i	$0.710029\pi$
$660$	0	0
$661$	−41.6056	−1.61827	−0.809135	−	0.587623i	$-0.800064\pi$
$661$	−41.6056	−1.61827	−0.809135	+	0.587623i	$0.800064\pi$
$662$	−16.4998	−0.641283
$663$	0	0
$664$	19.3491	0.750892
$665$	1.23643	0.0479467
$666$	0	0
$667$	20.0456	0.776168
$668$	−40.1739	−1.55438
$669$	0	0
$670$	−1.33265	−0.0514848
$671$	−0.136411	−0.00526609
$672$	0	0
$673$	2.99578	0.115479	0.0577394	−	0.998332i	$-0.481611\pi$
$673$	2.99578	0.115479	0.0577394	+	0.998332i	$0.481611\pi$
$674$	26.1982	1.00912
$675$	0	0
$676$	1.73377	0.0666837
$677$	−48.2569	−1.85466	−0.927332	−	0.374239i	$-0.877904\pi$
$677$	−48.2569	−1.85466	−0.927332	+	0.374239i	$0.877904\pi$
$678$	0	0
$679$	−7.95594	−0.305321
$680$	−2.89285	−0.110936
$681$	0	0
$682$	0.0100693	0.000385574 0
$683$	−12.9192	−0.494338	−0.247169	−	0.968972i	$-0.579500\pi$
$683$	−12.9192	−0.494338	−0.247169	+	0.968972i	$0.579500\pi$
$684$	0	0
$685$	14.3825	0.549529
$686$	−2.18919	−0.0835838
$687$	0	0
$688$	−7.30899	−0.278652
$689$	52.0625	1.98342
$690$	0	0
$691$	20.8880	0.794616	0.397308	−	0.917685i	$-0.369945\pi$
$691$	20.8880	0.794616	0.397308	+	0.917685i	$0.369945\pi$
$692$	57.0620	2.16917
$693$	0	0
$694$	−8.92155	−0.338657
$695$	12.6474	0.479743
$696$	0	0
$697$	−16.8111	−0.636768
$698$	−5.22407	−0.197734
$699$	0	0
$700$	11.8581	0.448195
$701$	−46.8697	−1.77024	−0.885122	−	0.465358i	$-0.845926\pi$
$701$	−46.8697	−1.77024	−0.885122	+	0.465358i	$0.845926\pi$
$702$	0	0
$703$	0.511800	0.0193029
$704$	3.53100	0.133080
$705$	0	0
$706$	−41.7557	−1.57150
$707$	5.03438	0.189337
$708$	0	0
$709$	−33.3937	−1.25413	−0.627064	−	0.778968i	$-0.715744\pi$
$709$	−33.3937	−1.25413	−0.627064	+	0.778968i	$0.715744\pi$
$710$	−17.3856	−0.652471
$711$	0	0
$712$	28.9449	1.08476
$713$	−0.106918	−0.00400411
$714$	0	0
$715$	0.898865	0.0336157
$716$	4.94807	0.184918
$717$	0	0
$718$	−3.28008	−0.122411
$719$	14.6697	0.547088	0.273544	−	0.961860i	$-0.411804\pi$
$719$	14.6697	0.547088	0.273544	+	0.961860i	$0.411804\pi$
$720$	0	0
$721$	−2.57322	−0.0958318
$722$	−37.1542	−1.38274
$723$	0	0
$724$	38.3490	1.42523
$725$	−13.0526	−0.484762
$726$	0	0
$727$	−39.9187	−1.48050	−0.740250	−	0.672331i	$-0.765293\pi$
$727$	−39.9187	−1.48050	−0.740250	+	0.672331i	$0.765293\pi$
$728$	−6.40360	−0.237333
$729$	0	0
$730$	1.05431	0.0390217
$731$	7.85627	0.290575
$732$	0	0
$733$	44.3086	1.63657	0.818287	−	0.574810i	$-0.194924\pi$
$733$	44.3086	1.63657	0.818287	+	0.574810i	$0.194924\pi$
$734$	22.1560	0.817792
$735$	0	0
$736$	−48.1375	−1.77437
$737$	−0.196714	−0.00724605
$738$	0	0
$739$	−9.90080	−0.364207	−0.182103	−	0.983279i	$-0.558291\pi$
$739$	−9.90080	−0.364207	−0.182103	+	0.983279i	$0.558291\pi$
$740$	−0.871214	−0.0320265
$741$	0	0
$742$	−30.8821	−1.13372
$743$	−43.7246	−1.60410	−0.802049	−	0.597258i	$-0.796257\pi$
$743$	−43.7246	−1.60410	−0.802049	+	0.597258i	$0.796257\pi$
$744$	0	0
$745$	−9.86977	−0.361601
$746$	−56.0589	−2.05246
$747$	0	0
$748$	−1.50456	−0.0550123
$749$	−6.71132	−0.245226
$750$	0	0
$751$	−1.71759	−0.0626756	−0.0313378	−	0.999509i	$-0.509977\pi$
$751$	−1.71759	−0.0626756	−0.0313378	+	0.999509i	$0.509977\pi$
$752$	7.72227	0.281602
$753$	0	0
$754$	24.8354	0.904453
$755$	8.33050	0.303178
$756$	0	0
$757$	33.7461	1.22652	0.613261	−	0.789880i	$-0.289857\pi$
$757$	33.7461	1.22652	0.613261	+	0.789880i	$0.289857\pi$
$758$	−30.5907	−1.11111
$759$	0	0
$760$	−2.14532	−0.0778188
$761$	5.95307	0.215799	0.107899	−	0.994162i	$-0.465588\pi$
$761$	5.95307	0.215799	0.107899	+	0.994162i	$0.465588\pi$
$762$	0	0
$763$	−1.76219	−0.0637955
$764$	−20.1802	−0.730093
$765$	0	0
$766$	56.2709	2.03315
$767$	50.8958	1.83774
$768$	0	0
$769$	19.4719	0.702173	0.351087	−	0.936343i	$-0.385812\pi$
$769$	19.4719	0.702173	0.351087	+	0.936343i	$0.385812\pi$
$770$	−0.533184	−0.0192146
$771$	0	0
$772$	36.4390	1.31147
$773$	34.4291	1.23833	0.619165	−	0.785261i	$-0.287471\pi$
$773$	34.4291	1.23833	0.619165	+	0.785261i	$0.287471\pi$
$774$	0	0
$775$	0.0696193	0.00250080
$776$	13.8043	0.495544
$777$	0	0
$778$	−8.89721	−0.318981
$779$	−12.4670	−0.446678
$780$	0	0
$781$	−2.56631	−0.0918297
$782$	27.4174	0.980444
$783$	0	0
$784$	−1.78669	−0.0638104
$785$	−9.39570	−0.335347
$786$	0	0
$787$	−26.7540	−0.953676	−0.476838	−	0.878991i	$-0.658217\pi$
$787$	−26.7540	−0.953676	−0.476838	+	0.878991i	$0.658217\pi$
$788$	8.72416	0.310785
$789$	0	0
$790$	27.4644	0.977141
$791$	−8.28904	−0.294724
$792$	0	0
$793$	1.79454	0.0637261
$794$	2.25710	0.0801014
$795$	0	0
$796$	−30.6264	−1.08552
$797$	−4.82222	−0.170812	−0.0854058	−	0.996346i	$-0.527219\pi$
$797$	−4.82222	−0.170812	−0.0854058	+	0.996346i	$0.527219\pi$
$798$	0	0
$799$	−8.30049	−0.293650
$800$	31.3446	1.10820
$801$	0	0
$802$	7.88033	0.278264
$803$	0.155628	0.00549198
$804$	0	0
$805$	5.66145	0.199540
$806$	−0.132466	−0.00466591
$807$	0	0
$808$	−8.73509	−0.307300
$809$	−7.80400	−0.274374	−0.137187	−	0.990545i	$-0.543806\pi$
$809$	−7.80400	−0.274374	−0.137187	+	0.990545i	$0.543806\pi$
$810$	0	0
$811$	26.2261	0.920924	0.460462	−	0.887679i	$-0.347684\pi$
$811$	26.2261	0.920924	0.460462	+	0.887679i	$0.347684\pi$
$812$	−8.58399	−0.301239
$813$	0	0
$814$	−0.220703	−0.00773563
$815$	−21.1388	−0.740460
$816$	0	0
$817$	5.82615	0.203831
$818$	−47.8762	−1.67395
$819$	0	0
$820$	21.2221	0.741106
$821$	−2.81989	−0.0984148	−0.0492074	−	0.998789i	$-0.515670\pi$
$821$	−2.81989	−0.0984148	−0.0492074	+	0.998789i	$0.515670\pi$
$822$	0	0
$823$	36.8534	1.28463	0.642314	−	0.766441i	$-0.277974\pi$
$823$	36.8534	1.28463	0.642314	+	0.766441i	$0.277974\pi$
$824$	4.46477	0.155538
$825$	0	0
$826$	−30.1901	−1.05045
$827$	17.0927	0.594371	0.297186	−	0.954820i	$-0.403952\pi$
$827$	17.0927	0.594371	0.297186	+	0.954820i	$0.403952\pi$
$828$	0	0
$829$	−34.5735	−1.20079	−0.600394	−	0.799704i	$-0.704990\pi$
$829$	−34.5735	−1.20079	−0.600394	+	0.799704i	$0.704990\pi$
$830$	−21.1943	−0.735664
$831$	0	0
$832$	−46.4518	−1.61043
$833$	1.92048	0.0665406
$834$	0	0
$835$	12.4892	0.432207
$836$	−1.11577	−0.0385898
$837$	0	0
$838$	31.7844	1.09797
$839$	17.1104	0.590718	0.295359	−	0.955386i	$-0.404561\pi$
$839$	17.1104	0.590718	0.295359	+	0.955386i	$0.404561\pi$
$840$	0	0
$841$	−19.5513	−0.674184
$842$	3.81733	0.131554
$843$	0	0
$844$	−38.7256	−1.33299
$845$	−0.538993	−0.0185419
$846$	0	0
$847$	10.9213	0.375260
$848$	−25.2042	−0.865515
$849$	0	0
$850$	−17.8528	−0.612345
$851$	2.34347	0.0803331
$852$	0	0
$853$	−45.5681	−1.56022	−0.780111	−	0.625642i	$-0.784837\pi$
$853$	−45.5681	−1.56022	−0.780111	+	0.625642i	$0.784837\pi$
$854$	−1.06448	−0.0364256
$855$	0	0
$856$	11.6447	0.398009
$857$	30.0586	1.02678	0.513391	−	0.858155i	$-0.328389\pi$
$857$	30.0586	1.02678	0.513391	+	0.858155i	$0.328389\pi$
$858$	0	0
$859$	24.8645	0.848365	0.424183	−	0.905577i	$-0.360561\pi$
$859$	24.8645	0.848365	0.424183	+	0.905577i	$0.360561\pi$
$860$	−9.91759	−0.338187
$861$	0	0
$862$	−54.0403	−1.84062
$863$	14.4612	0.492264	0.246132	−	0.969236i	$-0.420840\pi$
$863$	14.4612	0.492264	0.246132	+	0.969236i	$0.420840\pi$
$864$	0	0
$865$	−17.7393	−0.603156
$866$	−37.2010	−1.26414
$867$	0	0
$868$	0.0457848	0.00155404
$869$	4.05405	0.137524
$870$	0	0
$871$	2.58785	0.0876860
$872$	3.05755	0.103542
$873$	0	0
$874$	20.3325	0.687759
$875$	−8.02719	−0.271369
$876$	0	0
$877$	−32.6606	−1.10287	−0.551435	−	0.834218i	$-0.685920\pi$
$877$	−32.6606	−1.10287	−0.551435	+	0.834218i	$0.685920\pi$
$878$	21.6387	0.730269
$879$	0	0
$880$	−0.435153	−0.0146690
$881$	−11.2108	−0.377700	−0.188850	−	0.982006i	$-0.560476\pi$
$881$	−11.2108	−0.377700	−0.188850	+	0.982006i	$0.560476\pi$
$882$	0	0
$883$	27.6874	0.931754	0.465877	−	0.884849i	$-0.345739\pi$
$883$	27.6874	0.931754	0.465877	+	0.884849i	$0.345739\pi$
$884$	19.7932	0.665716
$885$	0	0
$886$	−87.2263	−2.93042
$887$	−9.88366	−0.331861	−0.165930	−	0.986137i	$-0.553063\pi$
$887$	−9.88366	−0.331861	−0.165930	+	0.986137i	$0.553063\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−31.7051	−1.06276
$891$	0	0
$892$	27.3105	0.914424
$893$	−6.15558	−0.205989
$894$	0	0
$895$	−1.53825	−0.0514179
$896$	12.7908	0.427310
$897$	0	0
$898$	−52.5953	−1.75513
$899$	−0.0503968	−0.00168083
$900$	0	0
$901$	27.0914	0.902546
$902$	5.37614	0.179006
$903$	0	0
$904$	14.3822	0.478346
$905$	−11.9219	−0.396297
$906$	0	0
$907$	−2.03008	−0.0674076	−0.0337038	−	0.999432i	$-0.510730\pi$
$907$	−2.03008	−0.0674076	−0.0337038	+	0.999432i	$0.510730\pi$
$908$	34.6088	1.14853
$909$	0	0
$910$	7.01425	0.232520
$911$	29.2269	0.968330	0.484165	−	0.874977i	$-0.339123\pi$
$911$	29.2269	0.968330	0.484165	+	0.874977i	$0.339123\pi$
$912$	0	0
$913$	−3.12851	−0.103538
$914$	8.35936	0.276503
$915$	0	0
$916$	68.4590	2.26195
$917$	−15.0169	−0.495901
$918$	0	0
$919$	15.5165	0.511842	0.255921	−	0.966698i	$-0.417621\pi$
$919$	15.5165	0.511842	0.255921	+	0.966698i	$0.417621\pi$
$920$	−9.82313	−0.323859
$921$	0	0
$922$	59.9347	1.97385
$923$	33.7608	1.11125
$924$	0	0
$925$	−1.52594	−0.0501727
$926$	−13.3793	−0.439672
$927$	0	0
$928$	−22.6901	−0.744838
$929$	−53.4586	−1.75392	−0.876960	−	0.480562i	$-0.840433\pi$
$929$	−53.4586	−1.75392	−0.876960	+	0.480562i	$0.840433\pi$
$930$	0	0
$931$	1.42421	0.0466766
$932$	21.6618	0.709557
$933$	0	0
$934$	−51.3886	−1.68149
$935$	0.467737	0.0152966
$936$	0	0
$937$	−25.9970	−0.849285	−0.424642	−	0.905361i	$-0.639600\pi$
$937$	−25.9970	−0.849285	−0.424642	+	0.905361i	$0.639600\pi$
$938$	−1.53505	−0.0501210
$939$	0	0
$940$	10.4784	0.341767
$941$	45.2741	1.47589	0.737947	−	0.674858i	$-0.235795\pi$
$941$	45.2741	1.47589	0.737947	+	0.674858i	$0.235795\pi$
$942$	0	0
$943$	−57.0850	−1.85894
$944$	−24.6394	−0.801943
$945$	0	0
$946$	−2.51240	−0.0816853
$947$	24.4990	0.796111	0.398056	−	0.917361i	$-0.369685\pi$
$947$	24.4990	0.796111	0.398056	+	0.917361i	$0.369685\pi$
$948$	0	0
$949$	−2.04735	−0.0664597
$950$	−13.2395	−0.429545
$951$	0	0
$952$	−3.33220	−0.107997
$953$	−11.2283	−0.363720	−0.181860	−	0.983324i	$-0.558212\pi$
$953$	−11.2283	−0.363720	−0.181860	+	0.983324i	$0.558212\pi$
$954$	0	0
$955$	6.27358	0.203008
$956$	−33.8350	−1.09430
$957$	0	0
$958$	66.6453	2.15321
$959$	16.5669	0.534973
$960$	0	0
$961$	−30.9997	−0.999991
$962$	2.90344	0.0936105
$963$	0	0
$964$	9.38670	0.302325
$965$	−11.3281	−0.364665
$966$	0	0
$967$	4.25392	0.136797	0.0683984	−	0.997658i	$-0.478211\pi$
$967$	4.25392	0.136797	0.0683984	+	0.997658i	$0.478211\pi$
$968$	−18.9494	−0.609058
$969$	0	0
$970$	−15.1206	−0.485495
$971$	33.0193	1.05964	0.529821	−	0.848110i	$-0.322259\pi$
$971$	33.0193	1.05964	0.529821	+	0.848110i	$0.322259\pi$
$972$	0	0
$973$	14.5682	0.467035
$974$	−96.2118	−3.08283
$975$	0	0
$976$	−0.868763	−0.0278084
$977$	−28.1942	−0.902013	−0.451006	−	0.892521i	$-0.648935\pi$
$977$	−28.1942	−0.902013	−0.451006	+	0.892521i	$0.648935\pi$
$978$	0	0
$979$	−4.68002	−0.149574
$980$	−2.42437	−0.0774437
$981$	0	0
$982$	10.4317	0.332888
$983$	40.8708	1.30358	0.651788	−	0.758401i	$-0.274019\pi$
$983$	40.8708	1.30358	0.651788	+	0.758401i	$0.274019\pi$
$984$	0	0
$985$	−2.71215	−0.0864164
$986$	12.9234	0.411566
$987$	0	0
$988$	14.6785	0.466984
$989$	26.6772	0.848286
$990$	0	0
$991$	−2.56939	−0.0816194	−0.0408097	−	0.999167i	$-0.512994\pi$
$991$	−2.56939	−0.0816194	−0.0408097	+	0.999167i	$0.512994\pi$
$992$	0.121023	0.00384248
$993$	0	0
$994$	−20.0261	−0.635188
$995$	9.52110	0.301839
$996$	0	0
$997$	−36.9676	−1.17077	−0.585387	−	0.810754i	$-0.699057\pi$
$997$	−36.9676	−1.17077	−0.585387	+	0.810754i	$0.699057\pi$
$998$	−21.6082	−0.683995
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.t.1.14		16
3.2	odd	2		889.2.a.c.1.3	✓	16
21.20	even	2		6223.2.a.k.1.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.c.1.3	✓	16	3.2	odd	2
6223.2.a.k.1.3		16	21.20	even	2
8001.2.a.t.1.14		16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q+2.18919 q^{2} +2.79257 q^{4} -0.868150 q^{5} -1.00000 q^{7} +1.73509 q^{8} +O(q^{10})\)	\(q+2.18919 q^{2} +2.79257 q^{4} -0.868150 q^{5} -1.00000 q^{7} +1.73509 q^{8} -1.90055 q^{10} -0.280542 q^{11} +3.69064 q^{13} -2.18919 q^{14} -1.78669 q^{16} +1.92048 q^{17} +1.42421 q^{19} -2.42437 q^{20} -0.614161 q^{22} +6.52129 q^{23} -4.24632 q^{25} +8.07954 q^{26} -2.79257 q^{28} +3.07387 q^{29} -0.0163952 q^{31} -7.38160 q^{32} +4.20429 q^{34} +0.868150 q^{35} +0.359357 q^{37} +3.11787 q^{38} -1.50632 q^{40} -8.75364 q^{41} +4.09079 q^{43} -0.783433 q^{44} +14.2764 q^{46} -4.32210 q^{47} +1.00000 q^{49} -9.29601 q^{50} +10.3064 q^{52} +14.1066 q^{53} +0.243552 q^{55} -1.73509 q^{56} +6.72929 q^{58} +13.7905 q^{59} +0.486241 q^{61} -0.0358923 q^{62} -12.5864 q^{64} -3.20403 q^{65} +0.701192 q^{67} +5.36306 q^{68} +1.90055 q^{70} +9.14768 q^{71} -0.554740 q^{73} +0.786702 q^{74} +3.97721 q^{76} +0.280542 q^{77} -14.4508 q^{79} +1.55112 q^{80} -19.1634 q^{82} +11.1517 q^{83} -1.66726 q^{85} +8.95554 q^{86} -0.486765 q^{88} +16.6821 q^{89} -3.69064 q^{91} +18.2111 q^{92} -9.46192 q^{94} -1.23643 q^{95} +7.95594 q^{97} +2.18919 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8}+O(q^{10}) \) 16 * q + 2 * q^2 + 12 * q^4 + 9 * q^5 - 16 * q^7 + 6 * q^8	\( 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8} - 2 q^{10} + 22 q^{11} - 4 q^{13} - 2 q^{14} + 12 q^{16} + 18 q^{17} - 15 q^{19} + 40 q^{20} - 11 q^{22} + 5 q^{23} + 15 q^{25} + 24 q^{26} - 12 q^{28} + 12 q^{29} - 32 q^{31} + 9 q^{32} - 14 q^{34} - 9 q^{35} - 2 q^{37} - 3 q^{38} - 14 q^{40} + 45 q^{41} - 3 q^{43} + 54 q^{44} + 49 q^{47} + 16 q^{49} + 6 q^{50} + 38 q^{52} - 16 q^{53} + 7 q^{55} - 6 q^{56} + 16 q^{58} + 35 q^{59} - 11 q^{61} - 17 q^{62} - 2 q^{64} - 14 q^{65} + 17 q^{67} + 71 q^{68} + 2 q^{70} + 81 q^{71} - 15 q^{73} - 13 q^{74} + 14 q^{76} - 22 q^{77} - 34 q^{79} + 33 q^{80} - 14 q^{82} + 39 q^{83} - 17 q^{85} - 36 q^{86} + 61 q^{88} + 32 q^{89} + 4 q^{91} - 37 q^{92} + 13 q^{94} + 33 q^{95} - 4 q^{97} + 2 q^{98}+O(q^{100}) \) 16 * q + 2 * q^2 + 12 * q^4 + 9 * q^5 - 16 * q^7 + 6 * q^8 - 2 * q^10 + 22 * q^11 - 4 * q^13 - 2 * q^14 + 12 * q^16 + 18 * q^17 - 15 * q^19 + 40 * q^20 - 11 * q^22 + 5 * q^23 + 15 * q^25 + 24 * q^26 - 12 * q^28 + 12 * q^29 - 32 * q^31 + 9 * q^32 - 14 * q^34 - 9 * q^35 - 2 * q^37 - 3 * q^38 - 14 * q^40 + 45 * q^41 - 3 * q^43 + 54 * q^44 + 49 * q^47 + 16 * q^49 + 6 * q^50 + 38 * q^52 - 16 * q^53 + 7 * q^55 - 6 * q^56 + 16 * q^58 + 35 * q^59 - 11 * q^61 - 17 * q^62 - 2 * q^64 - 14 * q^65 + 17 * q^67 + 71 * q^68 + 2 * q^70 + 81 * q^71 - 15 * q^73 - 13 * q^74 + 14 * q^76 - 22 * q^77 - 34 * q^79 + 33 * q^80 - 14 * q^82 + 39 * q^83 - 17 * q^85 - 36 * q^86 + 61 * q^88 + 32 * q^89 + 4 * q^91 - 37 * q^92 + 13 * q^94 + 33 * q^95 - 4 * q^97 + 2 * q^98

Introduction

L-functions

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Database

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