LMFDB - Embedded newform 8001.2.a.n.1.2

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 3 x^{10} + 41 x^{9} - 11 x^{8} - 123 x^{7} + 44 x^{6} + 159 x^{5} - 39 x^{4} + \cdots - 1 $ x^12 - 5x^11 - 3x^10 + 41x^9 - 11x^8 - 123x^7 + 44x^6 + 159x^5 - 39x^4 - 71x^3 + 16x^2 + 7*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			1.2
Root			$2.59078$ 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)

$f(q)$	$=$	$q-1.59078 q^{2} +0.530574 q^{4} +1.30400 q^{5} +1.00000 q^{7} +2.33753 q^{8} +O(q^{10})$	\(q-1.59078 q^{2} +0.530574 q^{4} +1.30400 q^{5} +1.00000 q^{7} +2.33753 q^{8} -2.07438 q^{10} -0.521308 q^{11} -5.72032 q^{13} -1.59078 q^{14} -4.77964 q^{16} +0.651574 q^{17} +4.34946 q^{19} +0.691871 q^{20} +0.829285 q^{22} -0.286695 q^{23} -3.29957 q^{25} +9.09975 q^{26} +0.530574 q^{28} +9.23886 q^{29} -3.19814 q^{31} +2.92828 q^{32} -1.03651 q^{34} +1.30400 q^{35} -6.41494 q^{37} -6.91903 q^{38} +3.04815 q^{40} -7.05179 q^{41} +6.08025 q^{43} -0.276592 q^{44} +0.456069 q^{46} -4.36677 q^{47} +1.00000 q^{49} +5.24889 q^{50} -3.03505 q^{52} +4.71280 q^{53} -0.679788 q^{55} +2.33753 q^{56} -14.6970 q^{58} -6.14848 q^{59} -0.825841 q^{61} +5.08753 q^{62} +4.90103 q^{64} -7.45932 q^{65} +5.94124 q^{67} +0.345708 q^{68} -2.07438 q^{70} +11.6748 q^{71} -12.7724 q^{73} +10.2047 q^{74} +2.30771 q^{76} -0.521308 q^{77} +2.01464 q^{79} -6.23267 q^{80} +11.2178 q^{82} -11.6550 q^{83} +0.849656 q^{85} -9.67232 q^{86} -1.21857 q^{88} +7.39692 q^{89} -5.72032 q^{91} -0.152113 q^{92} +6.94656 q^{94} +5.67172 q^{95} +13.1868 q^{97} -1.59078 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 7 q^{2} + 9 q^{4} + 7 q^{5} + 12 q^{7} + 15 q^{8}+O(q^{10}) $ 12 * q + 7 * q^2 + 9 * q^4 + 7 * q^5 + 12 * q^7 + 15 * q^8	$ 12 q + 7 q^{2} + 9 q^{4} + 7 q^{5} + 12 q^{7} + 15 q^{8} - 2 q^{10} + 22 q^{11} + 7 q^{14} + 7 q^{16} + 6 q^{17} - 7 q^{19} + 8 q^{20} + 13 q^{22} + 29 q^{23} + 3 q^{25} + 9 q^{28} + 22 q^{29} - 16 q^{31} + 27 q^{32} - 5 q^{34} + 7 q^{35} - 4 q^{37} - 2 q^{38} + 16 q^{40} + 21 q^{41} + 11 q^{43} + 11 q^{44} + 31 q^{47} + 12 q^{49} + 21 q^{50} + 3 q^{52} + 38 q^{53} - 11 q^{55} + 15 q^{56} + 20 q^{58} + 15 q^{59} - 3 q^{61} + 4 q^{62} + 29 q^{64} + 32 q^{65} - q^{67} - 17 q^{68} - 2 q^{70} + 57 q^{71} - 7 q^{73} + 42 q^{74} - 44 q^{76} + 22 q^{77} - 18 q^{79} - q^{80} + 56 q^{82} + 21 q^{83} - 5 q^{85} + 32 q^{86} - 10 q^{88} - 6 q^{89} + 15 q^{92} + 35 q^{94} + 57 q^{95} + 4 q^{97} + 7 q^{98}+O(q^{100}) $ 12 * q + 7 * q^2 + 9 * q^4 + 7 * q^5 + 12 * q^7 + 15 * q^8 - 2 * q^10 + 22 * q^11 + 7 * q^14 + 7 * q^16 + 6 * q^17 - 7 * q^19 + 8 * q^20 + 13 * q^22 + 29 * q^23 + 3 * q^25 + 9 * q^28 + 22 * q^29 - 16 * q^31 + 27 * q^32 - 5 * q^34 + 7 * q^35 - 4 * q^37 - 2 * q^38 + 16 * q^40 + 21 * q^41 + 11 * q^43 + 11 * q^44 + 31 * q^47 + 12 * q^49 + 21 * q^50 + 3 * q^52 + 38 * q^53 - 11 * q^55 + 15 * q^56 + 20 * q^58 + 15 * q^59 - 3 * q^61 + 4 * q^62 + 29 * q^64 + 32 * q^65 - q^67 - 17 * q^68 - 2 * q^70 + 57 * q^71 - 7 * q^73 + 42 * q^74 - 44 * q^76 + 22 * q^77 - 18 * q^79 - q^80 + 56 * q^82 + 21 * q^83 - 5 * q^85 + 32 * q^86 - 10 * q^88 - 6 * q^89 + 15 * q^92 + 35 * q^94 + 57 * q^95 + 4 * q^97 + 7 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.59078	−1.12485	−0.562425	−	0.826848i	$-0.690131\pi$
$2$	−1.59078	−1.12485	−0.562425	+	0.826848i	$0.690131\pi$
$3$	0	0
$3$	0	0
$4$	0.530574	0.265287
$5$	1.30400	0.583168	0.291584	−	0.956545i	$-0.405818\pi$
$5$	1.30400	0.583168	0.291584	+	0.956545i	$0.405818\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.33753	0.826442
$9$	0	0
$10$	−2.07438	−0.655977
$11$	−0.521308	−0.157180	−0.0785902	−	0.996907i	$-0.525042\pi$
$11$	−0.521308	−0.157180	−0.0785902	+	0.996907i	$0.525042\pi$
$12$	0	0
$13$	−5.72032	−1.58653	−0.793265	−	0.608877i	$-0.791620\pi$
$13$	−5.72032	−1.58653	−0.793265	+	0.608877i	$0.791620\pi$
$14$	−1.59078	−0.425153
$15$	0	0
$16$	−4.77964	−1.19491
$17$	0.651574	0.158030	0.0790150	−	0.996873i	$-0.474823\pi$
$17$	0.651574	0.158030	0.0790150	+	0.996873i	$0.474823\pi$
$18$	0	0
$19$	4.34946	0.997835	0.498918	−	0.866649i	$-0.333731\pi$
$19$	4.34946	0.997835	0.498918	+	0.866649i	$0.333731\pi$
$20$	0.691871	0.154707
$21$	0	0
$22$	0.829285	0.176804
$23$	−0.286695	−0.0597801	−0.0298901	−	0.999553i	$-0.509516\pi$
$23$	−0.286695	−0.0597801	−0.0298901	+	0.999553i	$0.509516\pi$
$24$	0	0
$25$	−3.29957	−0.659915
$26$	9.09975	1.78461
$27$	0	0
$28$	0.530574	0.100269
$29$	9.23886	1.71561	0.857807	−	0.513972i	$-0.171826\pi$
$29$	9.23886	1.71561	0.857807	+	0.513972i	$0.171826\pi$
$30$	0	0
$31$	−3.19814	−0.574403	−0.287201	−	0.957870i	$-0.592725\pi$
$31$	−3.19814	−0.574403	−0.287201	+	0.957870i	$0.592725\pi$
$32$	2.92828	0.517652
$33$	0	0
$34$	−1.03651	−0.177760
$35$	1.30400	0.220417
$36$	0	0
$37$	−6.41494	−1.05461	−0.527305	−	0.849676i	$-0.676797\pi$
$37$	−6.41494	−1.05461	−0.527305	+	0.849676i	$0.676797\pi$
$38$	−6.91903	−1.12241
$39$	0	0
$40$	3.04815	0.481955
$41$	−7.05179	−1.10130	−0.550652	−	0.834735i	$-0.685621\pi$
$41$	−7.05179	−1.10130	−0.550652	+	0.834735i	$0.685621\pi$
$42$	0	0
$43$	6.08025	0.927229	0.463615	−	0.886037i	$-0.346552\pi$
$43$	6.08025	0.927229	0.463615	+	0.886037i	$0.346552\pi$
$44$	−0.276592	−0.0416979
$45$	0	0
$46$	0.456069	0.0672437
$47$	−4.36677	−0.636959	−0.318479	−	0.947930i	$-0.603172\pi$
$47$	−4.36677	−0.636959	−0.318479	+	0.947930i	$0.603172\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	5.24889	0.742305
$51$	0	0
$52$	−3.03505	−0.420886
$53$	4.71280	0.647353	0.323676	−	0.946168i	$-0.395081\pi$
$53$	4.71280	0.647353	0.323676	+	0.946168i	$0.395081\pi$
$54$	0	0
$55$	−0.679788	−0.0916626
$56$	2.33753	0.312366
$57$	0	0
$58$	−14.6970	−1.92981
$59$	−6.14848	−0.800464	−0.400232	−	0.916414i	$-0.631070\pi$
$59$	−6.14848	−0.800464	−0.400232	+	0.916414i	$0.631070\pi$
$60$	0	0
$61$	−0.825841	−0.105738	−0.0528690	−	0.998601i	$-0.516837\pi$
$61$	−0.825841	−0.105738	−0.0528690	+	0.998601i	$0.516837\pi$
$62$	5.08753	0.646117
$63$	0	0
$64$	4.90103	0.612629
$65$	−7.45932	−0.925214
$66$	0	0
$67$	5.94124	0.725839	0.362919	−	0.931821i	$-0.381780\pi$
$67$	5.94124	0.725839	0.362919	+	0.931821i	$0.381780\pi$
$68$	0.345708	0.0419233
$69$	0	0
$70$	−2.07438	−0.247936
$71$	11.6748	1.38554	0.692771	−	0.721157i	$-0.256389\pi$
$71$	11.6748	1.38554	0.692771	+	0.721157i	$0.256389\pi$
$72$	0	0
$73$	−12.7724	−1.49489	−0.747447	−	0.664321i	$-0.768721\pi$
$73$	−12.7724	−1.49489	−0.747447	+	0.664321i	$0.768721\pi$
$74$	10.2047	1.18628
$75$	0	0
$76$	2.30771	0.264713
$77$	−0.521308	−0.0594086
$78$	0	0
$79$	2.01464	0.226665	0.113332	−	0.993557i	$-0.463847\pi$
$79$	2.01464	0.226665	0.113332	+	0.993557i	$0.463847\pi$
$80$	−6.23267	−0.696834
$81$	0	0
$82$	11.2178	1.23880
$83$	−11.6550	−1.27931	−0.639653	−	0.768664i	$-0.720922\pi$
$83$	−11.6550	−1.27931	−0.639653	+	0.768664i	$0.720922\pi$
$84$	0	0
$85$	0.849656	0.0921581
$86$	−9.67232	−1.04299
$87$	0	0
$88$	−1.21857	−0.129900
$89$	7.39692	0.784072	0.392036	−	0.919950i	$-0.371771\pi$
$89$	7.39692	0.784072	0.392036	+	0.919950i	$0.371771\pi$
$90$	0	0
$91$	−5.72032	−0.599652
$92$	−0.152113	−0.0158589
$93$	0	0
$94$	6.94656	0.716483
$95$	5.67172	0.581906
$96$	0	0
$97$	13.1868	1.33892	0.669459	−	0.742849i	$-0.266526\pi$
$97$	13.1868	1.33892	0.669459	+	0.742849i	$0.266526\pi$
$98$	−1.59078	−0.160693
$99$	0	0
$100$	−1.75067	−0.175067
$101$	15.7463	1.56682	0.783410	−	0.621505i	$-0.213478\pi$
$101$	15.7463	1.56682	0.783410	+	0.621505i	$0.213478\pi$
$102$	0	0
$103$	−10.0660	−0.991830	−0.495915	−	0.868371i	$-0.665167\pi$
$103$	−10.0660	−0.991830	−0.495915	+	0.868371i	$0.665167\pi$
$104$	−13.3714	−1.31117
$105$	0	0
$106$	−7.49702	−0.728175
$107$	6.30388	0.609419	0.304709	−	0.952445i	$-0.401441\pi$
$107$	6.30388	0.609419	0.304709	+	0.952445i	$0.401441\pi$
$108$	0	0
$109$	16.6139	1.59132	0.795662	−	0.605741i	$-0.207123\pi$
$109$	16.6139	1.59132	0.795662	+	0.605741i	$0.207123\pi$
$110$	1.08139	0.103107
$111$	0	0
$112$	−4.77964	−0.451633
$113$	8.17414	0.768959	0.384479	−	0.923134i	$-0.374381\pi$
$113$	8.17414	0.768959	0.384479	+	0.923134i	$0.374381\pi$
$114$	0	0
$115$	−0.373852	−0.0348619
$116$	4.90190	0.455130
$117$	0	0
$118$	9.78086	0.900401
$119$	0.651574	0.0597297
$120$	0	0
$121$	−10.7282	−0.975294
$122$	1.31373	0.118939
$123$	0	0
$124$	−1.69685	−0.152382
$125$	−10.8227	−0.968010
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−13.6530	−1.20677
$129$	0	0
$130$	11.8661	1.04073
$131$	−6.39666	−0.558878	−0.279439	−	0.960163i	$-0.590149\pi$
$131$	−6.39666	−0.558878	−0.279439	+	0.960163i	$0.590149\pi$
$132$	0	0
$133$	4.34946	0.377146
$134$	−9.45120	−0.816459
$135$	0	0
$136$	1.52307	0.130603
$137$	11.6456	0.994948	0.497474	−	0.867479i	$-0.334261\pi$
$137$	11.6456	0.994948	0.497474	+	0.867479i	$0.334261\pi$
$138$	0	0
$139$	22.7750	1.93175	0.965876	−	0.259007i	$-0.0833951\pi$
$139$	22.7750	1.93175	0.965876	+	0.259007i	$0.0833951\pi$
$140$	0.691871	0.0584737
$141$	0	0
$142$	−18.5720	−1.55853
$143$	2.98205	0.249371
$144$	0	0
$145$	12.0475	1.00049
$146$	20.3180	1.68153
$147$	0	0
$148$	−3.40360	−0.279774
$149$	18.3038	1.49951	0.749754	−	0.661717i	$-0.230172\pi$
$149$	18.3038	1.49951	0.749754	+	0.661717i	$0.230172\pi$
$150$	0	0
$151$	−8.81417	−0.717287	−0.358644	−	0.933475i	$-0.616761\pi$
$151$	−8.81417	−0.717287	−0.358644	+	0.933475i	$0.616761\pi$
$152$	10.1670	0.824653
$153$	0	0
$154$	0.829285	0.0668257
$155$	−4.17039	−0.334974
$156$	0	0
$157$	19.6305	1.56669	0.783343	−	0.621590i	$-0.213513\pi$
$157$	19.6305	1.56669	0.783343	+	0.621590i	$0.213513\pi$
$158$	−3.20485	−0.254964
$159$	0	0
$160$	3.81849	0.301878
$161$	−0.286695	−0.0225948
$162$	0	0
$163$	4.64101	0.363512	0.181756	−	0.983344i	$-0.441822\pi$
$163$	4.64101	0.363512	0.181756	+	0.983344i	$0.441822\pi$
$164$	−3.74150	−0.292162
$165$	0	0
$166$	18.5406	1.43903
$167$	−15.6648	−1.21218	−0.606091	−	0.795395i	$-0.707263\pi$
$167$	−15.6648	−1.21218	−0.606091	+	0.795395i	$0.707263\pi$
$168$	0	0
$169$	19.7220	1.51708
$170$	−1.35161	−0.103664
$171$	0	0
$172$	3.22602	0.245982
$173$	5.54841	0.421838	0.210919	−	0.977504i	$-0.432354\pi$
$173$	5.54841	0.421838	0.210919	+	0.977504i	$0.432354\pi$
$174$	0	0
$175$	−3.29957	−0.249424
$176$	2.49166	0.187816
$177$	0	0
$178$	−11.7669	−0.881963
$179$	10.7678	0.804822	0.402411	−	0.915459i	$-0.368172\pi$
$179$	10.7678	0.804822	0.402411	+	0.915459i	$0.368172\pi$
$180$	0	0
$181$	−4.32593	−0.321544	−0.160772	−	0.986992i	$-0.551398\pi$
$181$	−4.32593	−0.321544	−0.160772	+	0.986992i	$0.551398\pi$
$182$	9.09975	0.674518
$183$	0	0
$184$	−0.670159	−0.0494048
$185$	−8.36511	−0.615015
$186$	0	0
$187$	−0.339671	−0.0248392
$188$	−2.31689	−0.168977
$189$	0	0
$190$	−9.02244	−0.654557
$191$	17.8341	1.29043	0.645217	−	0.764000i	$-0.276767\pi$
$191$	17.8341	1.29043	0.645217	+	0.764000i	$0.276767\pi$
$192$	0	0
$193$	−3.23135	−0.232598	−0.116299	−	0.993214i	$-0.537103\pi$
$193$	−3.23135	−0.232598	−0.116299	+	0.993214i	$0.537103\pi$
$194$	−20.9773	−1.50608
$195$	0	0
$196$	0.530574	0.0378981
$197$	−18.7509	−1.33595	−0.667974	−	0.744184i	$-0.732838\pi$
$197$	−18.7509	−1.33595	−0.667974	+	0.744184i	$0.732838\pi$
$198$	0	0
$199$	−15.7318	−1.11520	−0.557599	−	0.830110i	$-0.688277\pi$
$199$	−15.7318	−1.11520	−0.557599	+	0.830110i	$0.688277\pi$
$200$	−7.71285	−0.545381
$201$	0	0
$202$	−25.0489	−1.76244
$203$	9.23886	0.648441
$204$	0	0
$205$	−9.19557	−0.642246
$206$	16.0127	1.11566
$207$	0	0
$208$	27.3410	1.89576
$209$	−2.26741	−0.156840
$210$	0	0
$211$	−14.6365	−1.00762	−0.503808	−	0.863816i	$-0.668068\pi$
$211$	−14.6365	−1.00762	−0.503808	+	0.863816i	$0.668068\pi$
$212$	2.50049	0.171734
$213$	0	0
$214$	−10.0281	−0.685505
$215$	7.92867	0.540731
$216$	0	0
$217$	−3.19814	−0.217104
$218$	−26.4290	−1.79000
$219$	0	0
$220$	−0.360678	−0.0243169
$221$	−3.72721	−0.250719
$222$	0	0
$223$	12.3058	0.824056	0.412028	−	0.911171i	$-0.364821\pi$
$223$	12.3058	0.824056	0.412028	+	0.911171i	$0.364821\pi$
$224$	2.92828	0.195654
$225$	0	0
$226$	−13.0032	−0.864963
$227$	4.72697	0.313740	0.156870	−	0.987619i	$-0.449860\pi$
$227$	4.72697	0.313740	0.156870	+	0.987619i	$0.449860\pi$
$228$	0	0
$229$	−22.5953	−1.49314	−0.746571	−	0.665306i	$-0.768301\pi$
$229$	−22.5953	−1.49314	−0.746571	+	0.665306i	$0.768301\pi$
$230$	0.594716	0.0392144
$231$	0	0
$232$	21.5961	1.41786
$233$	13.3642	0.875517	0.437759	−	0.899093i	$-0.355772\pi$
$233$	13.3642	0.875517	0.437759	+	0.899093i	$0.355772\pi$
$234$	0	0
$235$	−5.69429	−0.371454
$236$	−3.26222	−0.212352
$237$	0	0
$238$	−1.03651	−0.0671870
$239$	11.7947	0.762934	0.381467	−	0.924382i	$-0.375419\pi$
$239$	11.7947	0.762934	0.381467	+	0.924382i	$0.375419\pi$
$240$	0	0
$241$	9.52367	0.613473	0.306737	−	0.951794i	$-0.400763\pi$
$241$	9.52367	0.613473	0.306737	+	0.951794i	$0.400763\pi$
$242$	17.0662	1.09706
$243$	0	0
$244$	−0.438169	−0.0280509
$245$	1.30400	0.0833098
$246$	0	0
$247$	−24.8803	−1.58310
$248$	−7.47575	−0.474710
$249$	0	0
$250$	17.2165	1.08887
$251$	6.71535	0.423869	0.211935	−	0.977284i	$-0.432024\pi$
$251$	6.71535	0.423869	0.211935	+	0.977284i	$0.432024\pi$
$252$	0	0
$253$	0.149457	0.00939626
$254$	−1.59078	−0.0998143
$255$	0	0
$256$	11.9169	0.744803
$257$	12.9374	0.807015	0.403508	−	0.914976i	$-0.367791\pi$
$257$	12.9374	0.807015	0.403508	+	0.914976i	$0.367791\pi$
$258$	0	0
$259$	−6.41494	−0.398605
$260$	−3.95772	−0.245447
$261$	0	0
$262$	10.1757	0.628654
$263$	−24.9192	−1.53658	−0.768291	−	0.640101i	$-0.778893\pi$
$263$	−24.9192	−1.53658	−0.768291	+	0.640101i	$0.778893\pi$
$264$	0	0
$265$	6.14551	0.377516
$266$	−6.91903	−0.424233
$267$	0	0
$268$	3.15227	0.192555
$269$	−4.07243	−0.248300	−0.124150	−	0.992263i	$-0.539620\pi$
$269$	−4.07243	−0.248300	−0.124150	+	0.992263i	$0.539620\pi$
$270$	0	0
$271$	8.19755	0.497966	0.248983	−	0.968508i	$-0.419904\pi$
$271$	8.19755	0.497966	0.248983	+	0.968508i	$0.419904\pi$
$272$	−3.11429	−0.188832
$273$	0	0
$274$	−18.5255	−1.11917
$275$	1.72009	0.103726
$276$	0	0
$277$	−10.1528	−0.610024	−0.305012	−	0.952349i	$-0.598660\pi$
$277$	−10.1528	−0.610024	−0.305012	+	0.952349i	$0.598660\pi$
$278$	−36.2300	−2.17293
$279$	0	0
$280$	3.04815	0.182162
$281$	−1.96821	−0.117413	−0.0587067	−	0.998275i	$-0.518698\pi$
$281$	−1.96821	−0.117413	−0.0587067	+	0.998275i	$0.518698\pi$
$282$	0	0
$283$	−30.8457	−1.83359	−0.916795	−	0.399359i	$-0.869233\pi$
$283$	−30.8457	−1.83359	−0.916795	+	0.399359i	$0.869233\pi$
$284$	6.19434	0.367566
$285$	0	0
$286$	−4.74377	−0.280505
$287$	−7.05179	−0.416254
$288$	0	0
$289$	−16.5755	−0.975027
$290$	−19.1649	−1.12540
$291$	0	0
$292$	−6.77669	−0.396576
$293$	15.6537	0.914498	0.457249	−	0.889339i	$-0.348835\pi$
$293$	15.6537	0.914498	0.457249	+	0.889339i	$0.348835\pi$
$294$	0	0
$295$	−8.01764	−0.466805
$296$	−14.9951	−0.871573
$297$	0	0
$298$	−29.1173	−1.68672
$299$	1.63999	0.0948430
$300$	0	0
$301$	6.08025	0.350460
$302$	14.0214	0.806840
$303$	0	0
$304$	−20.7889	−1.19232
$305$	−1.07690	−0.0616631
$306$	0	0
$307$	−12.8702	−0.734542	−0.367271	−	0.930114i	$-0.619708\pi$
$307$	−12.8702	−0.734542	−0.367271	+	0.930114i	$0.619708\pi$
$308$	−0.276592	−0.0157603
$309$	0	0
$310$	6.63416	0.376795
$311$	−13.3375	−0.756300	−0.378150	−	0.925744i	$-0.623440\pi$
$311$	−13.3375	−0.756300	−0.378150	+	0.925744i	$0.623440\pi$
$312$	0	0
$313$	−12.5300	−0.708237	−0.354119	−	0.935200i	$-0.615219\pi$
$313$	−12.5300	−0.708237	−0.354119	+	0.935200i	$0.615219\pi$
$314$	−31.2278	−1.76229
$315$	0	0
$316$	1.06892	0.0601312
$317$	28.7700	1.61589	0.807943	−	0.589260i	$-0.200581\pi$
$317$	28.7700	1.61589	0.807943	+	0.589260i	$0.200581\pi$
$318$	0	0
$319$	−4.81629	−0.269661
$320$	6.39097	0.357266
$321$	0	0
$322$	0.456069	0.0254157
$323$	2.83400	0.157688
$324$	0	0
$325$	18.8746	1.04697
$326$	−7.38281	−0.408896
$327$	0	0
$328$	−16.4838	−0.910164
$329$	−4.36677	−0.240748
$330$	0	0
$331$	2.42750	0.133428	0.0667138	−	0.997772i	$-0.478749\pi$
$331$	2.42750	0.133428	0.0667138	+	0.997772i	$0.478749\pi$
$332$	−6.18385	−0.339383
$333$	0	0
$334$	24.9193	1.36352
$335$	7.74741	0.423286
$336$	0	0
$337$	8.54479	0.465464	0.232732	−	0.972541i	$-0.425233\pi$
$337$	8.54479	0.465464	0.232732	+	0.972541i	$0.425233\pi$
$338$	−31.3733	−1.70648
$339$	0	0
$340$	0.450805	0.0244483
$341$	1.66722	0.0902848
$342$	0	0
$343$	1.00000	0.0539949
$344$	14.2128	0.766301
$345$	0	0
$346$	−8.82629	−0.474504
$347$	9.82829	0.527610	0.263805	−	0.964576i	$-0.415022\pi$
$347$	9.82829	0.527610	0.263805	+	0.964576i	$0.415022\pi$
$348$	0	0
$349$	21.3449	1.14257	0.571284	−	0.820753i	$-0.306446\pi$
$349$	21.3449	1.14257	0.571284	+	0.820753i	$0.306446\pi$
$350$	5.24889	0.280565
$351$	0	0
$352$	−1.52654	−0.0813647
$353$	22.9885	1.22356	0.611778	−	0.791030i	$-0.290455\pi$
$353$	22.9885	1.22356	0.611778	+	0.791030i	$0.290455\pi$
$354$	0	0
$355$	15.2240	0.808005
$356$	3.92461	0.208004
$357$	0	0
$358$	−17.1292	−0.905304
$359$	10.8117	0.570622	0.285311	−	0.958435i	$-0.407903\pi$
$359$	10.8117	0.570622	0.285311	+	0.958435i	$0.407903\pi$
$360$	0	0
$361$	−0.0821779	−0.00432515
$362$	6.88159	0.361688
$363$	0	0
$364$	−3.03505	−0.159080
$365$	−16.6552	−0.871775
$366$	0	0
$367$	−35.5314	−1.85473	−0.927363	−	0.374163i	$-0.877930\pi$
$367$	−35.5314	−1.85473	−0.927363	+	0.374163i	$0.877930\pi$
$368$	1.37030	0.0714319
$369$	0	0
$370$	13.3070	0.691799
$371$	4.71280	0.244676
$372$	0	0
$373$	−0.714018	−0.0369705	−0.0184852	−	0.999829i	$-0.505884\pi$
$373$	−0.714018	−0.0369705	−0.0184852	+	0.999829i	$0.505884\pi$
$374$	0.540341	0.0279404
$375$	0	0
$376$	−10.2075	−0.526409
$377$	−52.8492	−2.72187
$378$	0	0
$379$	25.0091	1.28463	0.642316	−	0.766440i	$-0.277974\pi$
$379$	25.0091	1.28463	0.642316	+	0.766440i	$0.277974\pi$
$380$	3.00927	0.154372
$381$	0	0
$382$	−28.3702	−1.45154
$383$	0.276102	0.0141082	0.00705408	−	0.999975i	$-0.497755\pi$
$383$	0.276102	0.0141082	0.00705408	+	0.999975i	$0.497755\pi$
$384$	0	0
$385$	−0.679788	−0.0346452
$386$	5.14036	0.261638
$387$	0	0
$388$	6.99658	0.355198
$389$	−9.51463	−0.482411	−0.241205	−	0.970474i	$-0.577543\pi$
$389$	−9.51463	−0.482411	−0.241205	+	0.970474i	$0.577543\pi$
$390$	0	0
$391$	−0.186803	−0.00944705
$392$	2.33753	0.118063
$393$	0	0
$394$	29.8286	1.50274
$395$	2.62710	0.132184
$396$	0	0
$397$	−4.38744	−0.220199	−0.110100	−	0.993921i	$-0.535117\pi$
$397$	−4.38744	−0.220199	−0.110100	+	0.993921i	$0.535117\pi$
$398$	25.0258	1.25443
$399$	0	0
$400$	15.7708	0.788538
$401$	−10.9136	−0.544999	−0.272500	−	0.962156i	$-0.587850\pi$
$401$	−10.9136	−0.544999	−0.272500	+	0.962156i	$0.587850\pi$
$402$	0	0
$403$	18.2944	0.911307
$404$	8.35460	0.415657
$405$	0	0
$406$	−14.6970	−0.729399
$407$	3.34416	0.165764
$408$	0	0
$409$	20.7763	1.02732	0.513660	−	0.857994i	$-0.328289\pi$
$409$	20.7763	1.02732	0.513660	+	0.857994i	$0.328289\pi$
$410$	14.6281	0.722431
$411$	0	0
$412$	−5.34074	−0.263120
$413$	−6.14848	−0.302547
$414$	0	0
$415$	−15.1982	−0.746051
$416$	−16.7507	−0.821271
$417$	0	0
$418$	3.60695	0.176421
$419$	15.7009	0.767038	0.383519	−	0.923533i	$-0.374712\pi$
$419$	15.7009	0.767038	0.383519	+	0.923533i	$0.374712\pi$
$420$	0	0
$421$	−4.25717	−0.207482	−0.103741	−	0.994604i	$-0.533081\pi$
$421$	−4.25717	−0.207482	−0.103741	+	0.994604i	$0.533081\pi$
$422$	23.2834	1.13342
$423$	0	0
$424$	11.0163	0.535000
$425$	−2.14992	−0.104286
$426$	0	0
$427$	−0.825841	−0.0399652
$428$	3.34467	0.161671
$429$	0	0
$430$	−12.6128	−0.608241
$431$	30.6100	1.47443	0.737216	−	0.675657i	$-0.236140\pi$
$431$	30.6100	1.47443	0.737216	+	0.675657i	$0.236140\pi$
$432$	0	0
$433$	−3.87570	−0.186254	−0.0931272	−	0.995654i	$-0.529686\pi$
$433$	−3.87570	−0.186254	−0.0931272	+	0.995654i	$0.529686\pi$
$434$	5.08753	0.244209
$435$	0	0
$436$	8.81491	0.422157
$437$	−1.24697	−0.0596507
$438$	0	0
$439$	2.39641	0.114374	0.0571872	−	0.998363i	$-0.481787\pi$
$439$	2.39641	0.114374	0.0571872	+	0.998363i	$0.481787\pi$
$440$	−1.58903	−0.0757538
$441$	0	0
$442$	5.92916	0.282022
$443$	−0.331964	−0.0157721	−0.00788605	−	0.999969i	$-0.502510\pi$
$443$	−0.331964	−0.0157721	−0.00788605	+	0.999969i	$0.502510\pi$
$444$	0	0
$445$	9.64561	0.457246
$446$	−19.5758	−0.926940
$447$	0	0
$448$	4.90103	0.231552
$449$	3.41097	0.160974	0.0804868	−	0.996756i	$-0.474353\pi$
$449$	3.41097	0.160974	0.0804868	+	0.996756i	$0.474353\pi$
$450$	0	0
$451$	3.67616	0.173103
$452$	4.33699	0.203995
$453$	0	0
$454$	−7.51955	−0.352910
$455$	−7.45932	−0.349698
$456$	0	0
$457$	12.6408	0.591311	0.295655	−	0.955295i	$-0.404462\pi$
$457$	12.6408	0.591311	0.295655	+	0.955295i	$0.404462\pi$
$458$	35.9442	1.67956
$459$	0	0
$460$	−0.198356	−0.00924840
$461$	24.0115	1.11833	0.559163	−	0.829058i	$-0.311123\pi$
$461$	24.0115	1.11833	0.559163	+	0.829058i	$0.311123\pi$
$462$	0	0
$463$	40.4054	1.87780	0.938900	−	0.344191i	$-0.111847\pi$
$463$	40.4054	1.87780	0.938900	+	0.344191i	$0.111847\pi$
$464$	−44.1584	−2.05000
$465$	0	0
$466$	−21.2595	−0.984825
$467$	0.185985	0.00860637	0.00430318	−	0.999991i	$-0.498630\pi$
$467$	0.185985	0.00860637	0.00430318	+	0.999991i	$0.498630\pi$
$468$	0	0
$469$	5.94124	0.274341
$470$	9.05834	0.417830
$471$	0	0
$472$	−14.3723	−0.661537
$473$	−3.16968	−0.145742
$474$	0	0
$475$	−14.3514	−0.658486
$476$	0.345708	0.0158455
$477$	0	0
$478$	−18.7627	−0.858187
$479$	5.64544	0.257947	0.128973	−	0.991648i	$-0.458832\pi$
$479$	5.64544	0.257947	0.128973	+	0.991648i	$0.458832\pi$
$480$	0	0
$481$	36.6955	1.67317
$482$	−15.1500	−0.690065
$483$	0	0
$484$	−5.69212	−0.258733
$485$	17.1957	0.780815
$486$	0	0
$487$	37.8492	1.71511	0.857554	−	0.514393i	$-0.171983\pi$
$487$	37.8492	1.71511	0.857554	+	0.514393i	$0.171983\pi$
$488$	−1.93043	−0.0873863
$489$	0	0
$490$	−2.07438	−0.0937110
$491$	−27.8633	−1.25745	−0.628727	−	0.777626i	$-0.716424\pi$
$491$	−27.8633	−1.25745	−0.628727	+	0.777626i	$0.716424\pi$
$492$	0	0
$493$	6.01981	0.271118
$494$	39.5790	1.78074
$495$	0	0
$496$	15.2859	0.686359
$497$	11.6748	0.523686
$498$	0	0
$499$	14.5737	0.652410	0.326205	−	0.945299i	$-0.394230\pi$
$499$	14.5737	0.652410	0.326205	+	0.945299i	$0.394230\pi$
$500$	−5.74223	−0.256800
$501$	0	0
$502$	−10.6826	−0.476789
$503$	29.6802	1.32338	0.661688	−	0.749780i	$-0.269841\pi$
$503$	29.6802	1.32338	0.661688	+	0.749780i	$0.269841\pi$
$504$	0	0
$505$	20.5333	0.913720
$506$	−0.237752	−0.0105694
$507$	0	0
$508$	0.530574	0.0235404
$509$	−18.2788	−0.810193	−0.405096	−	0.914274i	$-0.632762\pi$
$509$	−18.2788	−0.810193	−0.405096	+	0.914274i	$0.632762\pi$
$510$	0	0
$511$	−12.7724	−0.565017
$512$	8.34897	0.368976
$513$	0	0
$514$	−20.5806	−0.907771
$515$	−13.1261	−0.578404
$516$	0	0
$517$	2.27643	0.100117
$518$	10.2047	0.448370
$519$	0	0
$520$	−17.4364	−0.764636
$521$	0.897038	0.0393000	0.0196500	−	0.999807i	$-0.493745\pi$
$521$	0.897038	0.0393000	0.0196500	+	0.999807i	$0.493745\pi$
$522$	0	0
$523$	21.0586	0.920829	0.460414	−	0.887704i	$-0.347701\pi$
$523$	21.0586	0.920829	0.460414	+	0.887704i	$0.347701\pi$
$524$	−3.39390	−0.148263
$525$	0	0
$526$	39.6409	1.72842
$527$	−2.08382	−0.0907728
$528$	0	0
$529$	−22.9178	−0.996426
$530$	−9.77614	−0.424649
$531$	0	0
$532$	2.30771	0.100052
$533$	40.3385	1.74725
$534$	0	0
$535$	8.22028	0.355394
$536$	13.8878	0.599863
$537$	0	0
$538$	6.47833	0.279301
$539$	−0.521308	−0.0224543
$540$	0	0
$541$	22.9058	0.984797	0.492398	−	0.870370i	$-0.336120\pi$
$541$	22.9058	0.984797	0.492398	+	0.870370i	$0.336120\pi$
$542$	−13.0405	−0.560136
$543$	0	0
$544$	1.90799	0.0818046
$545$	21.6646	0.928010
$546$	0	0
$547$	41.6847	1.78231	0.891156	−	0.453698i	$-0.149895\pi$
$547$	41.6847	1.78231	0.891156	+	0.453698i	$0.149895\pi$
$548$	6.17883	0.263947
$549$	0	0
$550$	−2.73629	−0.116676
$551$	40.1841	1.71190
$552$	0	0
$553$	2.01464	0.0856713
$554$	16.1509	0.686185
$555$	0	0
$556$	12.0838	0.512468
$557$	−41.5137	−1.75899	−0.879495	−	0.475908i	$-0.842120\pi$
$557$	−41.5137	−1.75899	−0.879495	+	0.475908i	$0.842120\pi$
$558$	0	0
$559$	−34.7809	−1.47108
$560$	−6.23267	−0.263378
$561$	0	0
$562$	3.13098	0.132072
$563$	38.7879	1.63472	0.817358	−	0.576130i	$-0.195438\pi$
$563$	38.7879	1.63472	0.817358	+	0.576130i	$0.195438\pi$
$564$	0	0
$565$	10.6591	0.448432
$566$	49.0687	2.06251
$567$	0	0
$568$	27.2902	1.14507
$569$	−17.7510	−0.744160	−0.372080	−	0.928201i	$-0.621355\pi$
$569$	−17.7510	−0.744160	−0.372080	+	0.928201i	$0.621355\pi$
$570$	0	0
$571$	−12.7110	−0.531937	−0.265968	−	0.963982i	$-0.585692\pi$
$571$	−12.7110	−0.531937	−0.265968	+	0.963982i	$0.585692\pi$
$572$	1.58220	0.0661549
$573$	0	0
$574$	11.2178	0.468223
$575$	0.945972	0.0394498
$576$	0	0
$577$	−22.2884	−0.927877	−0.463939	−	0.885867i	$-0.653564\pi$
$577$	−22.2884	−0.927877	−0.463939	+	0.885867i	$0.653564\pi$
$578$	26.3679	1.09676
$579$	0	0
$580$	6.39210	0.265417
$581$	−11.6550	−0.483532
$582$	0	0
$583$	−2.45682	−0.101751
$584$	−29.8558	−1.23544
$585$	0	0
$586$	−24.9015	−1.02867
$587$	25.5532	1.05469	0.527347	−	0.849650i	$-0.323187\pi$
$587$	25.5532	1.05469	0.527347	+	0.849650i	$0.323187\pi$
$588$	0	0
$589$	−13.9102	−0.573159
$590$	12.7543	0.525086
$591$	0	0
$592$	30.6611	1.26016
$593$	0.108404	0.00445161	0.00222580	−	0.999998i	$-0.499292\pi$
$593$	0.108404	0.00445161	0.00222580	+	0.999998i	$0.499292\pi$
$594$	0	0
$595$	0.849656	0.0348325
$596$	9.71153	0.397800
$597$	0	0
$598$	−2.60886	−0.106684
$599$	−10.4057	−0.425165	−0.212582	−	0.977143i	$-0.568187\pi$
$599$	−10.4057	−0.425165	−0.212582	+	0.977143i	$0.568187\pi$
$600$	0	0
$601$	−18.0611	−0.736727	−0.368364	−	0.929682i	$-0.620082\pi$
$601$	−18.0611	−0.736727	−0.368364	+	0.929682i	$0.620082\pi$
$602$	−9.67232	−0.394214
$603$	0	0
$604$	−4.67657	−0.190287
$605$	−13.9897	−0.568761
$606$	0	0
$607$	−21.5961	−0.876559	−0.438279	−	0.898839i	$-0.644412\pi$
$607$	−21.5961	−0.876559	−0.438279	+	0.898839i	$0.644412\pi$
$608$	12.7365	0.516531
$609$	0	0
$610$	1.71311	0.0693617
$611$	24.9793	1.01055
$612$	0	0
$613$	−32.4946	−1.31245	−0.656223	−	0.754567i	$-0.727847\pi$
$613$	−32.4946	−1.31245	−0.656223	+	0.754567i	$0.727847\pi$
$614$	20.4736	0.826249
$615$	0	0
$616$	−1.21857	−0.0490977
$617$	13.8499	0.557575	0.278787	−	0.960353i	$-0.410068\pi$
$617$	13.8499	0.557575	0.278787	+	0.960353i	$0.410068\pi$
$618$	0	0
$619$	−1.84964	−0.0743433	−0.0371716	−	0.999309i	$-0.511835\pi$
$619$	−1.84964	−0.0743433	−0.0371716	+	0.999309i	$0.511835\pi$
$620$	−2.21270	−0.0888641
$621$	0	0
$622$	21.2170	0.850724
$623$	7.39692	0.296351
$624$	0	0
$625$	2.38504	0.0954017
$626$	19.9324	0.796661
$627$	0	0
$628$	10.4154	0.415621
$629$	−4.17981	−0.166660
$630$	0	0
$631$	−38.3507	−1.52672	−0.763358	−	0.645976i	$-0.776451\pi$
$631$	−38.3507	−1.52672	−0.763358	+	0.645976i	$0.776451\pi$
$632$	4.70929	0.187325
$633$	0	0
$634$	−45.7667	−1.81763
$635$	1.30400	0.0517478
$636$	0	0
$637$	−5.72032	−0.226647
$638$	7.66165	0.303328
$639$	0	0
$640$	−17.8036	−0.703749
$641$	22.8906	0.904124	0.452062	−	0.891986i	$-0.350689\pi$
$641$	22.8906	0.904124	0.452062	+	0.891986i	$0.350689\pi$
$642$	0	0
$643$	40.5697	1.59991	0.799957	−	0.600058i	$-0.204856\pi$
$643$	40.5697	1.59991	0.799957	+	0.600058i	$0.204856\pi$
$644$	−0.152113	−0.00599410
$645$	0	0
$646$	−4.50826	−0.177375
$647$	3.41956	0.134437	0.0672184	−	0.997738i	$-0.478588\pi$
$647$	3.41956	0.134437	0.0672184	+	0.997738i	$0.478588\pi$
$648$	0	0
$649$	3.20525	0.125817
$650$	−30.0253	−1.17769
$651$	0	0
$652$	2.46240	0.0964349
$653$	20.9029	0.817994	0.408997	−	0.912536i	$-0.365879\pi$
$653$	20.9029	0.817994	0.408997	+	0.912536i	$0.365879\pi$
$654$	0	0
$655$	−8.34127	−0.325920
$656$	33.7050	1.31596
$657$	0	0
$658$	6.94656	0.270805
$659$	47.2056	1.83887	0.919435	−	0.393243i	$-0.128647\pi$
$659$	47.2056	1.83887	0.919435	+	0.393243i	$0.128647\pi$
$660$	0	0
$661$	12.3951	0.482113	0.241056	−	0.970511i	$-0.422506\pi$
$661$	12.3951	0.482113	0.241056	+	0.970511i	$0.422506\pi$
$662$	−3.86162	−0.150086
$663$	0	0
$664$	−27.2440	−1.05727
$665$	5.67172	0.219940
$666$	0	0
$667$	−2.64874	−0.102560
$668$	−8.31136	−0.321576
$669$	0	0
$670$	−12.3244	−0.476133
$671$	0.430517	0.0166199
$672$	0	0
$673$	−10.5636	−0.407199	−0.203599	−	0.979054i	$-0.565264\pi$
$673$	−10.5636	−0.407199	−0.203599	+	0.979054i	$0.565264\pi$
$674$	−13.5929	−0.523577
$675$	0	0
$676$	10.4640	0.402461
$677$	−31.7648	−1.22082	−0.610411	−	0.792085i	$-0.708996\pi$
$677$	−31.7648	−1.22082	−0.610411	+	0.792085i	$0.708996\pi$
$678$	0	0
$679$	13.1868	0.506064
$680$	1.98610	0.0761633
$681$	0	0
$682$	−2.65217	−0.101557
$683$	−47.1501	−1.80415	−0.902073	−	0.431582i	$-0.857955\pi$
$683$	−47.1501	−1.80415	−0.902073	+	0.431582i	$0.857955\pi$
$684$	0	0
$685$	15.1859	0.580222
$686$	−1.59078	−0.0607362
$687$	0	0
$688$	−29.0614	−1.10796
$689$	−26.9587	−1.02704
$690$	0	0
$691$	30.4238	1.15737	0.578687	−	0.815550i	$-0.303565\pi$
$691$	30.4238	1.15737	0.578687	+	0.815550i	$0.303565\pi$
$692$	2.94384	0.111908
$693$	0	0
$694$	−15.6346	−0.593482
$695$	29.6987	1.12654
$696$	0	0
$697$	−4.59477	−0.174039
$698$	−33.9550	−1.28522
$699$	0	0
$700$	−1.75067	−0.0661690
$701$	42.2439	1.59553	0.797766	−	0.602968i	$-0.206015\pi$
$701$	42.2439	1.59553	0.797766	+	0.602968i	$0.206015\pi$
$702$	0	0
$703$	−27.9015	−1.05233
$704$	−2.55495	−0.0962932
$705$	0	0
$706$	−36.5696	−1.37632
$707$	15.7463	0.592202
$708$	0	0
$709$	21.1609	0.794716	0.397358	−	0.917664i	$-0.369927\pi$
$709$	21.1609	0.794716	0.397358	+	0.917664i	$0.369927\pi$
$710$	−24.2180	−0.908884
$711$	0	0
$712$	17.2905	0.647990
$713$	0.916892	0.0343379
$714$	0	0
$715$	3.88860	0.145425
$716$	5.71311	0.213509
$717$	0	0
$718$	−17.1991	−0.641864
$719$	−10.0753	−0.375746	−0.187873	−	0.982193i	$-0.560159\pi$
$719$	−10.0753	−0.375746	−0.187873	+	0.982193i	$0.560159\pi$
$720$	0	0
$721$	−10.0660	−0.374877
$722$	0.130727	0.00486515
$723$	0	0
$724$	−2.29522	−0.0853013
$725$	−30.4843	−1.13216
$726$	0	0
$727$	40.8339	1.51445	0.757223	−	0.653157i	$-0.226556\pi$
$727$	40.8339	1.51445	0.757223	+	0.653157i	$0.226556\pi$
$728$	−13.3714	−0.495577
$729$	0	0
$730$	26.4948	0.980616
$731$	3.96173	0.146530
$732$	0	0
$733$	32.7845	1.21092	0.605461	−	0.795875i	$-0.292989\pi$
$733$	32.7845	1.21092	0.605461	+	0.795875i	$0.292989\pi$
$734$	56.5226	2.08629
$735$	0	0
$736$	−0.839525	−0.0309453
$737$	−3.09722	−0.114088
$738$	0	0
$739$	30.1571	1.10935	0.554673	−	0.832068i	$-0.312843\pi$
$739$	30.1571	1.10935	0.554673	+	0.832068i	$0.312843\pi$
$740$	−4.43831	−0.163155
$741$	0	0
$742$	−7.49702	−0.275224
$743$	−48.1245	−1.76552	−0.882758	−	0.469827i	$-0.844316\pi$
$743$	−48.1245	−1.76552	−0.882758	+	0.469827i	$0.844316\pi$
$744$	0	0
$745$	23.8683	0.874465
$746$	1.13584	0.0415862
$747$	0	0
$748$	−0.180221	−0.00658951
$749$	6.30388	0.230339
$750$	0	0
$751$	42.2740	1.54260	0.771301	−	0.636471i	$-0.219606\pi$
$751$	42.2740	1.54260	0.771301	+	0.636471i	$0.219606\pi$
$752$	20.8716	0.761108
$753$	0	0
$754$	84.0714	3.06170
$755$	−11.4937	−0.418299
$756$	0	0
$757$	−13.1036	−0.476258	−0.238129	−	0.971234i	$-0.576534\pi$
$757$	−13.1036	−0.476258	−0.238129	+	0.971234i	$0.576534\pi$
$758$	−39.7840	−1.44502
$759$	0	0
$760$	13.2578	0.480911
$761$	−44.3506	−1.60771	−0.803853	−	0.594828i	$-0.797220\pi$
$761$	−44.3506	−1.60771	−0.803853	+	0.594828i	$0.797220\pi$
$762$	0	0
$763$	16.6139	0.601464
$764$	9.46233	0.342335
$765$	0	0
$766$	−0.439217	−0.0158696
$767$	35.1712	1.26996
$768$	0	0
$769$	−5.95989	−0.214919	−0.107460	−	0.994209i	$-0.534272\pi$
$769$	−5.95989	−0.214919	−0.107460	+	0.994209i	$0.534272\pi$
$770$	1.08139	0.0389707
$771$	0	0
$772$	−1.71447	−0.0617052
$773$	−50.5340	−1.81758	−0.908791	−	0.417252i	$-0.862993\pi$
$773$	−50.5340	−1.81758	−0.908791	+	0.417252i	$0.862993\pi$
$774$	0	0
$775$	10.5525	0.379057
$776$	30.8246	1.10654
$777$	0	0
$778$	15.1357	0.542640
$779$	−30.6715	−1.09892
$780$	0	0
$781$	−6.08616	−0.217780
$782$	0.297163	0.0106265
$783$	0	0
$784$	−4.77964	−0.170701
$785$	25.5983	0.913642
$786$	0	0
$787$	−5.68901	−0.202791	−0.101396	−	0.994846i	$-0.532331\pi$
$787$	−5.68901	−0.202791	−0.101396	+	0.994846i	$0.532331\pi$
$788$	−9.94875	−0.354410
$789$	0	0
$790$	−4.17913	−0.148687
$791$	8.17414	0.290639
$792$	0	0
$793$	4.72407	0.167757
$794$	6.97945	0.247691
$795$	0	0
$796$	−8.34689	−0.295848
$797$	0.814967	0.0288676	0.0144338	−	0.999896i	$-0.495405\pi$
$797$	0.814967	0.0288676	0.0144338	+	0.999896i	$0.495405\pi$
$798$	0	0
$799$	−2.84527	−0.100659
$800$	−9.66208	−0.341606
$801$	0	0
$802$	17.3611	0.613042
$803$	6.65835	0.234968
$804$	0	0
$805$	−0.373852	−0.0131766
$806$	−29.1023	−1.02508
$807$	0	0
$808$	36.8076	1.29489
$809$	22.8522	0.803441	0.401721	−	0.915762i	$-0.368412\pi$
$809$	22.8522	0.803441	0.401721	+	0.915762i	$0.368412\pi$
$810$	0	0
$811$	25.0839	0.880817	0.440408	−	0.897798i	$-0.354834\pi$
$811$	25.0839	0.880817	0.440408	+	0.897798i	$0.354834\pi$
$812$	4.90190	0.172023
$813$	0	0
$814$	−5.31981	−0.186459
$815$	6.05189	0.211989
$816$	0	0
$817$	26.4458	0.925222
$818$	−33.0504	−1.15558
$819$	0	0
$820$	−4.87893	−0.170380
$821$	32.7422	1.14271	0.571356	−	0.820702i	$-0.306417\pi$
$821$	32.7422	1.14271	0.571356	+	0.820702i	$0.306417\pi$
$822$	0	0
$823$	39.8068	1.38758	0.693789	−	0.720178i	$-0.255940\pi$
$823$	39.8068	1.38758	0.693789	+	0.720178i	$0.255940\pi$
$824$	−23.5295	−0.819690
$825$	0	0
$826$	9.78086	0.340320
$827$	−12.9976	−0.451969	−0.225985	−	0.974131i	$-0.572560\pi$
$827$	−12.9976	−0.451969	−0.225985	+	0.974131i	$0.572560\pi$
$828$	0	0
$829$	17.0717	0.592925	0.296462	−	0.955045i	$-0.404193\pi$
$829$	17.0717	0.592925	0.296462	+	0.955045i	$0.404193\pi$
$830$	24.1770	0.839195
$831$	0	0
$832$	−28.0354	−0.971954
$833$	0.651574	0.0225757
$834$	0	0
$835$	−20.4270	−0.706907
$836$	−1.20303	−0.0416076
$837$	0	0
$838$	−24.9766	−0.862802
$839$	−9.78228	−0.337722	−0.168861	−	0.985640i	$-0.554009\pi$
$839$	−9.78228	−0.337722	−0.168861	+	0.985640i	$0.554009\pi$
$840$	0	0
$841$	56.3566	1.94333
$842$	6.77222	0.233386
$843$	0	0
$844$	−7.76573	−0.267307
$845$	25.7176	0.884712
$846$	0	0
$847$	−10.7282	−0.368627
$848$	−22.5255	−0.773528
$849$	0	0
$850$	3.42004	0.117306
$851$	1.83913	0.0630447
$852$	0	0
$853$	−10.0916	−0.345531	−0.172766	−	0.984963i	$-0.555270\pi$
$853$	−10.0916	−0.345531	−0.172766	+	0.984963i	$0.555270\pi$
$854$	1.31373	0.0449549
$855$	0	0
$856$	14.7355	0.503649
$857$	22.2950	0.761584	0.380792	−	0.924661i	$-0.375651\pi$
$857$	22.2950	0.761584	0.380792	+	0.924661i	$0.375651\pi$
$858$	0	0
$859$	−1.53754	−0.0524602	−0.0262301	−	0.999656i	$-0.508350\pi$
$859$	−1.53754	−0.0524602	−0.0262301	+	0.999656i	$0.508350\pi$
$860$	4.20674	0.143449
$861$	0	0
$862$	−48.6937	−1.65851
$863$	−11.6494	−0.396550	−0.198275	−	0.980146i	$-0.563534\pi$
$863$	−11.6494	−0.396550	−0.198275	+	0.980146i	$0.563534\pi$
$864$	0	0
$865$	7.23515	0.246003
$866$	6.16538	0.209508
$867$	0	0
$868$	−1.69685	−0.0575948
$869$	−1.05025	−0.0356273
$870$	0	0
$871$	−33.9858	−1.15156
$872$	38.8355	1.31514
$873$	0	0
$874$	1.98365	0.0670981
$875$	−10.8227	−0.365873
$876$	0	0
$877$	48.7347	1.64565	0.822827	−	0.568292i	$-0.192396\pi$
$877$	48.7347	1.64565	0.822827	+	0.568292i	$0.192396\pi$
$878$	−3.81216	−0.128654
$879$	0	0
$880$	3.24914	0.109529
$881$	23.0422	0.776313	0.388157	−	0.921593i	$-0.373112\pi$
$881$	23.0422	0.776313	0.388157	+	0.921593i	$0.373112\pi$
$882$	0	0
$883$	36.8282	1.23937	0.619683	−	0.784852i	$-0.287261\pi$
$883$	36.8282	1.23937	0.619683	+	0.784852i	$0.287261\pi$
$884$	−1.97756	−0.0665125
$885$	0	0
$886$	0.528081	0.0177412
$887$	−6.37999	−0.214219	−0.107110	−	0.994247i	$-0.534160\pi$
$887$	−6.37999	−0.214219	−0.107110	+	0.994247i	$0.534160\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−15.3440	−0.514333
$891$	0	0
$892$	6.52913	0.218611
$893$	−18.9931	−0.635580
$894$	0	0
$895$	14.0412	0.469347
$896$	−13.6530	−0.456115
$897$	0	0
$898$	−5.42609	−0.181071
$899$	−29.5472	−0.985453
$900$	0	0
$901$	3.07074	0.102301
$902$	−5.84795	−0.194715
$903$	0	0
$904$	19.1073	0.635500
$905$	−5.64103	−0.187514
$906$	0	0
$907$	12.2607	0.407111	0.203556	−	0.979063i	$-0.434750\pi$
$907$	12.2607	0.407111	0.203556	+	0.979063i	$0.434750\pi$
$908$	2.50800	0.0832311
$909$	0	0
$910$	11.8661	0.393358
$911$	30.1637	0.999368	0.499684	−	0.866208i	$-0.333449\pi$
$911$	30.1637	0.999368	0.499684	+	0.866208i	$0.333449\pi$
$912$	0	0
$913$	6.07586	0.201082
$914$	−20.1087	−0.665136
$915$	0	0
$916$	−11.9885	−0.396111
$917$	−6.39666	−0.211236
$918$	0	0
$919$	−56.7967	−1.87355	−0.936775	−	0.349934i	$-0.886204\pi$
$919$	−56.7967	−1.87355	−0.936775	+	0.349934i	$0.886204\pi$
$920$	−0.873891	−0.0288113
$921$	0	0
$922$	−38.1969	−1.25795
$923$	−66.7835	−2.19821
$924$	0	0
$925$	21.1665	0.695952
$926$	−64.2760	−2.11224
$927$	0	0
$928$	27.0540	0.888091
$929$	13.9800	0.458670	0.229335	−	0.973348i	$-0.426345\pi$
$929$	13.9800	0.458670	0.229335	+	0.973348i	$0.426345\pi$
$930$	0	0
$931$	4.34946	0.142548
$932$	7.09069	0.232263
$933$	0	0
$934$	−0.295861	−0.00968087
$935$	−0.442932	−0.0144854
$936$	0	0
$937$	3.51614	0.114867	0.0574337	−	0.998349i	$-0.481708\pi$
$937$	3.51614	0.114867	0.0574337	+	0.998349i	$0.481708\pi$
$938$	−9.45120	−0.308593
$939$	0	0
$940$	−3.02124	−0.0985420
$941$	−46.0497	−1.50118	−0.750589	−	0.660769i	$-0.770230\pi$
$941$	−46.0497	−1.50118	−0.750589	+	0.660769i	$0.770230\pi$
$942$	0	0
$943$	2.02172	0.0658361
$944$	29.3875	0.956482
$945$	0	0
$946$	5.04226	0.163938
$947$	32.6336	1.06045	0.530225	−	0.847857i	$-0.322107\pi$
$947$	32.6336	1.06045	0.530225	+	0.847857i	$0.322107\pi$
$948$	0	0
$949$	73.0620	2.37169
$950$	22.8298	0.740698
$951$	0	0
$952$	1.52307	0.0493631
$953$	−48.3871	−1.56741	−0.783705	−	0.621133i	$-0.786673\pi$
$953$	−48.3871	−1.56741	−0.783705	+	0.621133i	$0.786673\pi$
$954$	0	0
$955$	23.2558	0.752540
$956$	6.25795	0.202397
$957$	0	0
$958$	−8.98064	−0.290151
$959$	11.6456	0.376055
$960$	0	0
$961$	−20.7719	−0.670062
$962$	−58.3743	−1.88206
$963$	0	0
$964$	5.05301	0.162746
$965$	−4.21370	−0.135644
$966$	0	0
$967$	2.97146	0.0955556	0.0477778	−	0.998858i	$-0.484786\pi$
$967$	2.97146	0.0955556	0.0477778	+	0.998858i	$0.484786\pi$
$968$	−25.0776	−0.806024
$969$	0	0
$970$	−27.3545	−0.878300
$971$	−10.9273	−0.350675	−0.175337	−	0.984508i	$-0.556102\pi$
$971$	−10.9273	−0.350675	−0.175337	+	0.984508i	$0.556102\pi$
$972$	0	0
$973$	22.7750	0.730133
$974$	−60.2096	−1.92924
$975$	0	0
$976$	3.94722	0.126347
$977$	−11.8977	−0.380641	−0.190321	−	0.981722i	$-0.560953\pi$
$977$	−11.8977	−0.380641	−0.190321	+	0.981722i	$0.560953\pi$
$978$	0	0
$979$	−3.85607	−0.123241
$980$	0.691871	0.0221010
$981$	0	0
$982$	44.3244	1.41445
$983$	48.5821	1.54953	0.774765	−	0.632250i	$-0.217868\pi$
$983$	48.5821	1.54953	0.774765	+	0.632250i	$0.217868\pi$
$984$	0	0
$985$	−24.4513	−0.779083
$986$	−9.57617	−0.304967
$987$	0	0
$988$	−13.2008	−0.419974
$989$	−1.74318	−0.0554299
$990$	0	0
$991$	−54.4168	−1.72861	−0.864303	−	0.502972i	$-0.832240\pi$
$991$	−54.4168	−1.72861	−0.864303	+	0.502972i	$0.832240\pi$
$992$	−9.36505	−0.297341
$993$	0	0
$994$	−18.5720	−0.589068
$995$	−20.5144	−0.650349
$996$	0	0
$997$	19.3110	0.611584	0.305792	−	0.952098i	$-0.401079\pi$
$997$	19.3110	0.611584	0.305792	+	0.952098i	$0.401079\pi$
$998$	−23.1836	−0.733863
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.n.1.2		12
3.2	odd	2		889.2.a.a.1.11	✓	12
21.20	even	2		6223.2.a.i.1.11		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.a.1.11	✓	12	3.2	odd	2
6223.2.a.i.1.11		12	21.20	even	2
8001.2.a.n.1.2		12	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)

\(f(q)\)	\(=\)	\(q-1.59078 q^{2} +0.530574 q^{4} +1.30400 q^{5} +1.00000 q^{7} +2.33753 q^{8} +O(q^{10})\)	\(q-1.59078 q^{2} +0.530574 q^{4} +1.30400 q^{5} +1.00000 q^{7} +2.33753 q^{8} -2.07438 q^{10} -0.521308 q^{11} -5.72032 q^{13} -1.59078 q^{14} -4.77964 q^{16} +0.651574 q^{17} +4.34946 q^{19} +0.691871 q^{20} +0.829285 q^{22} -0.286695 q^{23} -3.29957 q^{25} +9.09975 q^{26} +0.530574 q^{28} +9.23886 q^{29} -3.19814 q^{31} +2.92828 q^{32} -1.03651 q^{34} +1.30400 q^{35} -6.41494 q^{37} -6.91903 q^{38} +3.04815 q^{40} -7.05179 q^{41} +6.08025 q^{43} -0.276592 q^{44} +0.456069 q^{46} -4.36677 q^{47} +1.00000 q^{49} +5.24889 q^{50} -3.03505 q^{52} +4.71280 q^{53} -0.679788 q^{55} +2.33753 q^{56} -14.6970 q^{58} -6.14848 q^{59} -0.825841 q^{61} +5.08753 q^{62} +4.90103 q^{64} -7.45932 q^{65} +5.94124 q^{67} +0.345708 q^{68} -2.07438 q^{70} +11.6748 q^{71} -12.7724 q^{73} +10.2047 q^{74} +2.30771 q^{76} -0.521308 q^{77} +2.01464 q^{79} -6.23267 q^{80} +11.2178 q^{82} -11.6550 q^{83} +0.849656 q^{85} -9.67232 q^{86} -1.21857 q^{88} +7.39692 q^{89} -5.72032 q^{91} -0.152113 q^{92} +6.94656 q^{94} +5.67172 q^{95} +13.1868 q^{97} -1.59078 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 7 q^{2} + 9 q^{4} + 7 q^{5} + 12 q^{7} + 15 q^{8}+O(q^{10}) \) 12 * q + 7 * q^2 + 9 * q^4 + 7 * q^5 + 12 * q^7 + 15 * q^8	\( 12 q + 7 q^{2} + 9 q^{4} + 7 q^{5} + 12 q^{7} + 15 q^{8} - 2 q^{10} + 22 q^{11} + 7 q^{14} + 7 q^{16} + 6 q^{17} - 7 q^{19} + 8 q^{20} + 13 q^{22} + 29 q^{23} + 3 q^{25} + 9 q^{28} + 22 q^{29} - 16 q^{31} + 27 q^{32} - 5 q^{34} + 7 q^{35} - 4 q^{37} - 2 q^{38} + 16 q^{40} + 21 q^{41} + 11 q^{43} + 11 q^{44} + 31 q^{47} + 12 q^{49} + 21 q^{50} + 3 q^{52} + 38 q^{53} - 11 q^{55} + 15 q^{56} + 20 q^{58} + 15 q^{59} - 3 q^{61} + 4 q^{62} + 29 q^{64} + 32 q^{65} - q^{67} - 17 q^{68} - 2 q^{70} + 57 q^{71} - 7 q^{73} + 42 q^{74} - 44 q^{76} + 22 q^{77} - 18 q^{79} - q^{80} + 56 q^{82} + 21 q^{83} - 5 q^{85} + 32 q^{86} - 10 q^{88} - 6 q^{89} + 15 q^{92} + 35 q^{94} + 57 q^{95} + 4 q^{97} + 7 q^{98}+O(q^{100}) \) 12 * q + 7 * q^2 + 9 * q^4 + 7 * q^5 + 12 * q^7 + 15 * q^8 - 2 * q^10 + 22 * q^11 + 7 * q^14 + 7 * q^16 + 6 * q^17 - 7 * q^19 + 8 * q^20 + 13 * q^22 + 29 * q^23 + 3 * q^25 + 9 * q^28 + 22 * q^29 - 16 * q^31 + 27 * q^32 - 5 * q^34 + 7 * q^35 - 4 * q^37 - 2 * q^38 + 16 * q^40 + 21 * q^41 + 11 * q^43 + 11 * q^44 + 31 * q^47 + 12 * q^49 + 21 * q^50 + 3 * q^52 + 38 * q^53 - 11 * q^55 + 15 * q^56 + 20 * q^58 + 15 * q^59 - 3 * q^61 + 4 * q^62 + 29 * q^64 + 32 * q^65 - q^67 - 17 * q^68 - 2 * q^70 + 57 * q^71 - 7 * q^73 + 42 * q^74 - 44 * q^76 + 22 * q^77 - 18 * q^79 - q^80 + 56 * q^82 + 21 * q^83 - 5 * q^85 + 32 * q^86 - 10 * q^88 - 6 * q^89 + 15 * q^92 + 35 * q^94 + 57 * q^95 + 4 * q^97 + 7 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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