LMFDB - Embedded newform 8001.2.a.q.1.5

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:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 22 x^{13} + 186 x^{11} - 763 x^{9} - 7 x^{8} + 1588 x^{7} + 64 x^{6} - 1625 x^{5} - 185 x^{4} + \cdots - 13 $ x^15 - 22x^13 + 186x^11 - 763x^9 - 7x^8 + 1588x^7 + 64x^6 - 1625x^5 - 185x^4 + 726x^3 + 145x^2 - 83*x - 13
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			1.5
Root			$-0.955258$ 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)

$f(q)$	$=$	$q-0.955258 q^{2} -1.08748 q^{4} +2.56419 q^{5} -1.00000 q^{7} +2.94934 q^{8} +O(q^{10})$	\(q-0.955258 q^{2} -1.08748 q^{4} +2.56419 q^{5} -1.00000 q^{7} +2.94934 q^{8} -2.44946 q^{10} -0.200367 q^{11} +6.35172 q^{13} +0.955258 q^{14} -0.642416 q^{16} -2.09762 q^{17} -4.55157 q^{19} -2.78851 q^{20} +0.191402 q^{22} -6.44422 q^{23} +1.57505 q^{25} -6.06753 q^{26} +1.08748 q^{28} +5.01546 q^{29} +2.16363 q^{31} -5.28501 q^{32} +2.00377 q^{34} -2.56419 q^{35} +1.92730 q^{37} +4.34792 q^{38} +7.56266 q^{40} -2.64628 q^{41} +3.25300 q^{43} +0.217896 q^{44} +6.15589 q^{46} -7.34897 q^{47} +1.00000 q^{49} -1.50458 q^{50} -6.90739 q^{52} +4.05023 q^{53} -0.513779 q^{55} -2.94934 q^{56} -4.79105 q^{58} -7.20824 q^{59} -8.95485 q^{61} -2.06682 q^{62} +6.33338 q^{64} +16.2870 q^{65} -8.84866 q^{67} +2.28112 q^{68} +2.44946 q^{70} -0.482080 q^{71} -1.76693 q^{73} -1.84106 q^{74} +4.94975 q^{76} +0.200367 q^{77} -16.5418 q^{79} -1.64727 q^{80} +2.52788 q^{82} -14.8359 q^{83} -5.37869 q^{85} -3.10745 q^{86} -0.590951 q^{88} -1.52543 q^{89} -6.35172 q^{91} +7.00798 q^{92} +7.02016 q^{94} -11.6711 q^{95} +13.7114 q^{97} -0.955258 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7}+O(q^{10}) $ 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7	$ 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7} + 10 q^{10} - 14 q^{11} + 6 q^{13} + 20 q^{16} - 10 q^{17} + 13 q^{19} - 8 q^{20} - 11 q^{22} - 15 q^{23} - 22 q^{26} - 14 q^{28} - 16 q^{29} + 22 q^{31} - 15 q^{34} + 7 q^{35} - 14 q^{37} + 6 q^{38} + 22 q^{40} - 19 q^{41} - q^{43} - 25 q^{44} - 28 q^{46} - 49 q^{47} + 15 q^{49} - 24 q^{50} - 17 q^{52} + 28 q^{53} + 39 q^{55} - 10 q^{58} - 43 q^{59} + 27 q^{61} - 14 q^{62} + 18 q^{64} + 8 q^{65} + 3 q^{67} - 13 q^{68} - 10 q^{70} - 55 q^{71} - 3 q^{73} + 12 q^{74} - 20 q^{76} + 14 q^{77} + 18 q^{79} - 29 q^{80} + 14 q^{82} - 17 q^{83} + 7 q^{85} - 4 q^{86} - 114 q^{88} - 36 q^{89} - 6 q^{91} - 45 q^{92} - 15 q^{94} - 59 q^{95} - 2 q^{97}+O(q^{100}) $ 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7 + 10 * q^10 - 14 * q^11 + 6 * q^13 + 20 * q^16 - 10 * q^17 + 13 * q^19 - 8 * q^20 - 11 * q^22 - 15 * q^23 - 22 * q^26 - 14 * q^28 - 16 * q^29 + 22 * q^31 - 15 * q^34 + 7 * q^35 - 14 * q^37 + 6 * q^38 + 22 * q^40 - 19 * q^41 - q^43 - 25 * q^44 - 28 * q^46 - 49 * q^47 + 15 * q^49 - 24 * q^50 - 17 * q^52 + 28 * q^53 + 39 * q^55 - 10 * q^58 - 43 * q^59 + 27 * q^61 - 14 * q^62 + 18 * q^64 + 8 * q^65 + 3 * q^67 - 13 * q^68 - 10 * q^70 - 55 * q^71 - 3 * q^73 + 12 * q^74 - 20 * q^76 + 14 * q^77 + 18 * q^79 - 29 * q^80 + 14 * q^82 - 17 * q^83 + 7 * q^85 - 4 * q^86 - 114 * q^88 - 36 * q^89 - 6 * q^91 - 45 * q^92 - 15 * q^94 - 59 * q^95 - 2 * q^97

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 $

$2$	−0.955258	−0.675469	−0.337735	−	0.941241i	$-0.609661\pi$
$2$	−0.955258	−0.675469	−0.337735	+	0.941241i	$0.609661\pi$
$3$	0	0
$3$	0	0
$4$	−1.08748	−0.543741
$5$	2.56419	1.14674	0.573370	−	0.819297i	$-0.305636\pi$
$5$	2.56419	1.14674	0.573370	+	0.819297i	$0.305636\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.94934	1.04275
$9$	0	0
$10$	−2.44946	−0.774587
$11$	−0.200367	−0.0604130	−0.0302065	−	0.999544i	$-0.509616\pi$
$11$	−0.200367	−0.0604130	−0.0302065	+	0.999544i	$0.509616\pi$
$12$	0	0
$13$	6.35172	1.76165	0.880825	−	0.473441i	$-0.156988\pi$
$13$	6.35172	1.76165	0.880825	+	0.473441i	$0.156988\pi$
$14$	0.955258	0.255303
$15$	0	0
$16$	−0.642416	−0.160604
$17$	−2.09762	−0.508747	−0.254374	−	0.967106i	$-0.581869\pi$
$17$	−2.09762	−0.508747	−0.254374	+	0.967106i	$0.581869\pi$
$18$	0	0
$19$	−4.55157	−1.04420	−0.522101	−	0.852884i	$-0.674851\pi$
$19$	−4.55157	−1.04420	−0.522101	+	0.852884i	$0.674851\pi$
$20$	−2.78851	−0.623530
$21$	0	0
$22$	0.191402	0.0408071
$23$	−6.44422	−1.34371	−0.671856	−	0.740682i	$-0.734503\pi$
$23$	−6.44422	−1.34371	−0.671856	+	0.740682i	$0.734503\pi$
$24$	0	0
$25$	1.57505	0.315011
$26$	−6.06753	−1.18994
$27$	0	0
$28$	1.08748	0.205515
$29$	5.01546	0.931347	0.465673	−	0.884957i	$-0.345812\pi$
$29$	5.01546	0.931347	0.465673	+	0.884957i	$0.345812\pi$
$30$	0	0
$31$	2.16363	0.388599	0.194300	−	0.980942i	$-0.437757\pi$
$31$	2.16363	0.388599	0.194300	+	0.980942i	$0.437757\pi$
$32$	−5.28501	−0.934267
$33$	0	0
$34$	2.00377	0.343643
$35$	−2.56419	−0.433427
$36$	0	0
$37$	1.92730	0.316845	0.158423	−	0.987371i	$-0.449359\pi$
$37$	1.92730	0.316845	0.158423	+	0.987371i	$0.449359\pi$
$38$	4.34792	0.705326
$39$	0	0
$40$	7.56266	1.19576
$41$	−2.64628	−0.413280	−0.206640	−	0.978417i	$-0.566253\pi$
$41$	−2.64628	−0.413280	−0.206640	+	0.978417i	$0.566253\pi$
$42$	0	0
$43$	3.25300	0.496078	0.248039	−	0.968750i	$-0.420214\pi$
$43$	3.25300	0.496078	0.248039	+	0.968750i	$0.420214\pi$
$44$	0.217896	0.0328490
$45$	0	0
$46$	6.15589	0.907636
$47$	−7.34897	−1.07196	−0.535978	−	0.844232i	$-0.680057\pi$
$47$	−7.34897	−1.07196	−0.535978	+	0.844232i	$0.680057\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.50458	−0.212780
$51$	0	0
$52$	−6.90739	−0.957882
$53$	4.05023	0.556342	0.278171	−	0.960532i	$-0.410272\pi$
$53$	4.05023	0.556342	0.278171	+	0.960532i	$0.410272\pi$
$54$	0	0
$55$	−0.513779	−0.0692779
$56$	−2.94934	−0.394122
$57$	0	0
$58$	−4.79105	−0.629096
$59$	−7.20824	−0.938432	−0.469216	−	0.883083i	$-0.655463\pi$
$59$	−7.20824	−0.938432	−0.469216	+	0.883083i	$0.655463\pi$
$60$	0	0
$61$	−8.95485	−1.14655	−0.573275	−	0.819363i	$-0.694328\pi$
$61$	−8.95485	−1.14655	−0.573275	+	0.819363i	$0.694328\pi$
$62$	−2.06682	−0.262487
$63$	0	0
$64$	6.33338	0.791672
$65$	16.2870	2.02015
$66$	0	0
$67$	−8.84866	−1.08104	−0.540518	−	0.841332i	$-0.681772\pi$
$67$	−8.84866	−1.08104	−0.540518	+	0.841332i	$0.681772\pi$
$68$	2.28112	0.276627
$69$	0	0
$70$	2.44946	0.292766
$71$	−0.482080	−0.0572124	−0.0286062	−	0.999591i	$-0.509107\pi$
$71$	−0.482080	−0.0572124	−0.0286062	+	0.999591i	$0.509107\pi$
$72$	0	0
$73$	−1.76693	−0.206804	−0.103402	−	0.994640i	$-0.532973\pi$
$73$	−1.76693	−0.206804	−0.103402	+	0.994640i	$0.532973\pi$
$74$	−1.84106	−0.214019
$75$	0	0
$76$	4.94975	0.567775
$77$	0.200367	0.0228340
$78$	0	0
$79$	−16.5418	−1.86110	−0.930549	−	0.366167i	$-0.880670\pi$
$79$	−16.5418	−1.86110	−0.930549	+	0.366167i	$0.880670\pi$
$80$	−1.64727	−0.184171
$81$	0	0
$82$	2.52788	0.279158
$83$	−14.8359	−1.62845	−0.814227	−	0.580547i	$-0.802839\pi$
$83$	−14.8359	−1.62845	−0.814227	+	0.580547i	$0.802839\pi$
$84$	0	0
$85$	−5.37869	−0.583401
$86$	−3.10745	−0.335085
$87$	0	0
$88$	−0.590951	−0.0629956
$89$	−1.52543	−0.161695	−0.0808475	−	0.996726i	$-0.525763\pi$
$89$	−1.52543	−0.161695	−0.0808475	+	0.996726i	$0.525763\pi$
$90$	0	0
$91$	−6.35172	−0.665841
$92$	7.00798	0.730632
$93$	0	0
$94$	7.02016	0.724074
$95$	−11.6711	−1.19743
$96$	0	0
$97$	13.7114	1.39218	0.696091	−	0.717953i	$-0.254921\pi$
$97$	13.7114	1.39218	0.696091	+	0.717953i	$0.254921\pi$
$98$	−0.955258	−0.0964956
$99$	0	0
$100$	−1.71284	−0.171284
$101$	−12.8402	−1.27764	−0.638822	−	0.769355i	$-0.720578\pi$
$101$	−12.8402	−1.27764	−0.638822	+	0.769355i	$0.720578\pi$
$102$	0	0
$103$	8.70087	0.857322	0.428661	−	0.903465i	$-0.358985\pi$
$103$	8.70087	0.857322	0.428661	+	0.903465i	$0.358985\pi$
$104$	18.7334	1.83696
$105$	0	0
$106$	−3.86902	−0.375792
$107$	−4.80585	−0.464599	−0.232299	−	0.972644i	$-0.574625\pi$
$107$	−4.80585	−0.464599	−0.232299	+	0.972644i	$0.574625\pi$
$108$	0	0
$109$	12.4500	1.19249	0.596247	−	0.802801i	$-0.296658\pi$
$109$	12.4500	1.19249	0.596247	+	0.802801i	$0.296658\pi$
$110$	0.490791	0.0467951
$111$	0	0
$112$	0.642416	0.0607026
$113$	−11.6903	−1.09973	−0.549866	−	0.835253i	$-0.685321\pi$
$113$	−11.6903	−1.09973	−0.549866	+	0.835253i	$0.685321\pi$
$114$	0	0
$115$	−16.5242	−1.54089
$116$	−5.45422	−0.506412
$117$	0	0
$118$	6.88572	0.633882
$119$	2.09762	0.192288
$120$	0	0
$121$	−10.9599	−0.996350
$122$	8.55419	0.774460
$123$	0	0
$124$	−2.35291	−0.211297
$125$	−8.78220	−0.785504
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	4.52001	0.399517
$129$	0	0
$130$	−15.5583	−1.36455
$131$	−5.86721	−0.512621	−0.256310	−	0.966595i	$-0.582507\pi$
$131$	−5.86721	−0.512621	−0.256310	+	0.966595i	$0.582507\pi$
$132$	0	0
$133$	4.55157	0.394671
$134$	8.45275	0.730207
$135$	0	0
$136$	−6.18660	−0.530496
$137$	6.00082	0.512684	0.256342	−	0.966586i	$-0.417483\pi$
$137$	6.00082	0.512684	0.256342	+	0.966586i	$0.417483\pi$
$138$	0	0
$139$	8.07811	0.685177	0.342588	−	0.939486i	$-0.388696\pi$
$139$	8.07811	0.685177	0.342588	+	0.939486i	$0.388696\pi$
$140$	2.78851	0.235672
$141$	0	0
$142$	0.460511	0.0386452
$143$	−1.27268	−0.106427
$144$	0	0
$145$	12.8606	1.06801
$146$	1.68787	0.139690
$147$	0	0
$148$	−2.09590	−0.172282
$149$	18.9326	1.55102	0.775509	−	0.631336i	$-0.217493\pi$
$149$	18.9326	1.55102	0.775509	+	0.631336i	$0.217493\pi$
$150$	0	0
$151$	−1.10912	−0.0902586	−0.0451293	−	0.998981i	$-0.514370\pi$
$151$	−1.10912	−0.0902586	−0.0451293	+	0.998981i	$0.514370\pi$
$152$	−13.4241	−1.08884
$153$	0	0
$154$	−0.191402	−0.0154236
$155$	5.54795	0.445622
$156$	0	0
$157$	−16.8696	−1.34634	−0.673171	−	0.739487i	$-0.735068\pi$
$157$	−16.8696	−1.34634	−0.673171	+	0.739487i	$0.735068\pi$
$158$	15.8017	1.25711
$159$	0	0
$160$	−13.5518	−1.07136
$161$	6.44422	0.507876
$162$	0	0
$163$	1.09884	0.0860677	0.0430339	−	0.999074i	$-0.486298\pi$
$163$	1.09884	0.0860677	0.0430339	+	0.999074i	$0.486298\pi$
$164$	2.87779	0.224718
$165$	0	0
$166$	14.1721	1.09997
$167$	−7.70901	−0.596541	−0.298270	−	0.954481i	$-0.596410\pi$
$167$	−7.70901	−0.596541	−0.298270	+	0.954481i	$0.596410\pi$
$168$	0	0
$169$	27.3444	2.10341
$170$	5.13803	0.394069
$171$	0	0
$172$	−3.53758	−0.269738
$173$	−2.60380	−0.197963	−0.0989816	−	0.995089i	$-0.531559\pi$
$173$	−2.60380	−0.197963	−0.0989816	+	0.995089i	$0.531559\pi$
$174$	0	0
$175$	−1.57505	−0.119063
$176$	0.128719	0.00970256
$177$	0	0
$178$	1.45718	0.109220
$179$	6.77637	0.506490	0.253245	−	0.967402i	$-0.418502\pi$
$179$	6.77637	0.506490	0.253245	+	0.967402i	$0.418502\pi$
$180$	0	0
$181$	10.6343	0.790444	0.395222	−	0.918586i	$-0.370668\pi$
$181$	10.6343	0.790444	0.395222	+	0.918586i	$0.370668\pi$
$182$	6.06753	0.449755
$183$	0	0
$184$	−19.0062	−1.40116
$185$	4.94195	0.363339
$186$	0	0
$187$	0.420294	0.0307349
$188$	7.99187	0.582867
$189$	0	0
$190$	11.1489	0.808825
$191$	−17.2669	−1.24939	−0.624696	−	0.780868i	$-0.714777\pi$
$191$	−17.2669	−1.24939	−0.624696	+	0.780868i	$0.714777\pi$
$192$	0	0
$193$	10.8484	0.780886	0.390443	−	0.920627i	$-0.372322\pi$
$193$	10.8484	0.780886	0.390443	+	0.920627i	$0.372322\pi$
$194$	−13.0979	−0.940376
$195$	0	0
$196$	−1.08748	−0.0776773
$197$	19.6418	1.39942	0.699711	−	0.714426i	$-0.253312\pi$
$197$	19.6418	1.39942	0.699711	+	0.714426i	$0.253312\pi$
$198$	0	0
$199$	13.3086	0.943420	0.471710	−	0.881754i	$-0.343637\pi$
$199$	13.3086	0.943420	0.471710	+	0.881754i	$0.343637\pi$
$200$	4.64537	0.328478
$201$	0	0
$202$	12.2657	0.863009
$203$	−5.01546	−0.352016
$204$	0	0
$205$	−6.78557	−0.473925
$206$	−8.31157	−0.579095
$207$	0	0
$208$	−4.08045	−0.282928
$209$	0.911985	0.0630833
$210$	0	0
$211$	23.3178	1.60526	0.802632	−	0.596474i	$-0.203432\pi$
$211$	23.3178	1.60526	0.802632	+	0.596474i	$0.203432\pi$
$212$	−4.40456	−0.302506
$213$	0	0
$214$	4.59082	0.313822
$215$	8.34130	0.568872
$216$	0	0
$217$	−2.16363	−0.146877
$218$	−11.8930	−0.805493
$219$	0	0
$220$	0.558726	0.0376693
$221$	−13.3235	−0.896235
$222$	0	0
$223$	−20.5555	−1.37650	−0.688250	−	0.725474i	$-0.741621\pi$
$223$	−20.5555	−1.37650	−0.688250	+	0.725474i	$0.741621\pi$
$224$	5.28501	0.353120
$225$	0	0
$226$	11.1673	0.742835
$227$	20.1494	1.33736	0.668680	−	0.743550i	$-0.266860\pi$
$227$	20.1494	1.33736	0.668680	+	0.743550i	$0.266860\pi$
$228$	0	0
$229$	−9.82303	−0.649124	−0.324562	−	0.945864i	$-0.605217\pi$
$229$	−9.82303	−0.649124	−0.324562	+	0.945864i	$0.605217\pi$
$230$	15.7849	1.04082
$231$	0	0
$232$	14.7923	0.971162
$233$	−19.3189	−1.26562	−0.632811	−	0.774306i	$-0.718099\pi$
$233$	−19.3189	−1.26562	−0.632811	+	0.774306i	$0.718099\pi$
$234$	0	0
$235$	−18.8441	−1.22925
$236$	7.83883	0.510264
$237$	0	0
$238$	−2.00377	−0.129885
$239$	−6.25451	−0.404571	−0.202285	−	0.979327i	$-0.564837\pi$
$239$	−6.25451	−0.404571	−0.202285	+	0.979327i	$0.564837\pi$
$240$	0	0
$241$	−9.13401	−0.588374	−0.294187	−	0.955748i	$-0.595049\pi$
$241$	−9.13401	−0.588374	−0.294187	+	0.955748i	$0.595049\pi$
$242$	10.4695	0.673004
$243$	0	0
$244$	9.73824	0.623427
$245$	2.56419	0.163820
$246$	0	0
$247$	−28.9103	−1.83952
$248$	6.38128	0.405212
$249$	0	0
$250$	8.38926	0.530584
$251$	−12.4238	−0.784182	−0.392091	−	0.919926i	$-0.628248\pi$
$251$	−12.4238	−0.784182	−0.392091	+	0.919926i	$0.628248\pi$
$252$	0	0
$253$	1.29121	0.0811777
$254$	−0.955258	−0.0599382
$255$	0	0
$256$	−16.9845	−1.06153
$257$	0.894303	0.0557851	0.0278926	−	0.999611i	$-0.491120\pi$
$257$	0.894303	0.0557851	0.0278926	+	0.999611i	$0.491120\pi$
$258$	0	0
$259$	−1.92730	−0.119756
$260$	−17.7118	−1.09844
$261$	0	0
$262$	5.60470	0.346260
$263$	−10.4415	−0.643848	−0.321924	−	0.946765i	$-0.604330\pi$
$263$	−10.4415	−0.643848	−0.321924	+	0.946765i	$0.604330\pi$
$264$	0	0
$265$	10.3856	0.637980
$266$	−4.34792	−0.266588
$267$	0	0
$268$	9.62277	0.587804
$269$	15.5131	0.945852	0.472926	−	0.881102i	$-0.343198\pi$
$269$	15.5131	0.945852	0.472926	+	0.881102i	$0.343198\pi$
$270$	0	0
$271$	25.9192	1.57448	0.787240	−	0.616646i	$-0.211509\pi$
$271$	25.9192	1.57448	0.787240	+	0.616646i	$0.211509\pi$
$272$	1.34754	0.0817068
$273$	0	0
$274$	−5.73233	−0.346302
$275$	−0.315589	−0.0190307
$276$	0	0
$277$	22.8649	1.37382	0.686911	−	0.726742i	$-0.258966\pi$
$277$	22.8649	1.37382	0.686911	+	0.726742i	$0.258966\pi$
$278$	−7.71668	−0.462816
$279$	0	0
$280$	−7.56266	−0.451956
$281$	−30.2457	−1.80431	−0.902155	−	0.431412i	$-0.858015\pi$
$281$	−30.2457	−1.80431	−0.902155	+	0.431412i	$0.858015\pi$
$282$	0	0
$283$	21.9540	1.30503	0.652515	−	0.757776i	$-0.273714\pi$
$283$	21.9540	1.30503	0.652515	+	0.757776i	$0.273714\pi$
$284$	0.524254	0.0311088
$285$	0	0
$286$	1.21573	0.0718879
$287$	2.64628	0.156205
$288$	0	0
$289$	−12.6000	−0.741176
$290$	−12.2852	−0.721409
$291$	0	0
$292$	1.92151	0.112448
$293$	20.9256	1.22248	0.611242	−	0.791443i	$-0.290670\pi$
$293$	20.9256	1.22248	0.611242	+	0.791443i	$0.290670\pi$
$294$	0	0
$295$	−18.4833	−1.07614
$296$	5.68425	0.330391
$297$	0	0
$298$	−18.0855	−1.04767
$299$	−40.9319	−2.36715
$300$	0	0
$301$	−3.25300	−0.187500
$302$	1.05949	0.0609669
$303$	0	0
$304$	2.92400	0.167703
$305$	−22.9619	−1.31479
$306$	0	0
$307$	−12.5547	−0.716534	−0.358267	−	0.933619i	$-0.616632\pi$
$307$	−12.5547	−0.716534	−0.358267	+	0.933619i	$0.616632\pi$
$308$	−0.217896	−0.0124158
$309$	0	0
$310$	−5.29972	−0.301004
$311$	−2.75470	−0.156205	−0.0781023	−	0.996945i	$-0.524886\pi$
$311$	−2.75470	−0.156205	−0.0781023	+	0.996945i	$0.524886\pi$
$312$	0	0
$313$	−24.2096	−1.36841	−0.684204	−	0.729291i	$-0.739850\pi$
$313$	−24.2096	−1.36841	−0.684204	+	0.729291i	$0.739850\pi$
$314$	16.1148	0.909412
$315$	0	0
$316$	17.9889	1.01196
$317$	−22.3393	−1.25470	−0.627351	−	0.778737i	$-0.715861\pi$
$317$	−22.3393	−1.25470	−0.627351	+	0.778737i	$0.715861\pi$
$318$	0	0
$319$	−1.00493	−0.0562654
$320$	16.2400	0.907842
$321$	0	0
$322$	−6.15589	−0.343054
$323$	9.54746	0.531235
$324$	0	0
$325$	10.0043	0.554939
$326$	−1.04967	−0.0581361
$327$	0	0
$328$	−7.80480	−0.430948
$329$	7.34897	0.405162
$330$	0	0
$331$	4.12792	0.226891	0.113445	−	0.993544i	$-0.463811\pi$
$331$	4.12792	0.226891	0.113445	+	0.993544i	$0.463811\pi$
$332$	16.1338	0.885458
$333$	0	0
$334$	7.36409	0.402945
$335$	−22.6896	−1.23967
$336$	0	0
$337$	5.28852	0.288084	0.144042	−	0.989572i	$-0.453990\pi$
$337$	5.28852	0.288084	0.144042	+	0.989572i	$0.453990\pi$
$338$	−26.1209	−1.42079
$339$	0	0
$340$	5.84923	0.317219
$341$	−0.433520	−0.0234764
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	9.59421	0.517285
$345$	0	0
$346$	2.48730	0.133718
$347$	−12.8368	−0.689117	−0.344559	−	0.938765i	$-0.611971\pi$
$347$	−12.8368	−0.689117	−0.344559	+	0.938765i	$0.611971\pi$
$348$	0	0
$349$	−20.3867	−1.09127	−0.545637	−	0.838022i	$-0.683712\pi$
$349$	−20.3867	−1.09127	−0.545637	+	0.838022i	$0.683712\pi$
$350$	1.50458	0.0804233
$351$	0	0
$352$	1.05894	0.0564418
$353$	−28.0556	−1.49325	−0.746626	−	0.665244i	$-0.768327\pi$
$353$	−28.0556	−1.49325	−0.746626	+	0.665244i	$0.768327\pi$
$354$	0	0
$355$	−1.23614	−0.0656077
$356$	1.65888	0.0879203
$357$	0	0
$358$	−6.47318	−0.342118
$359$	−26.4042	−1.39356	−0.696780	−	0.717285i	$-0.745384\pi$
$359$	−26.4042	−1.39356	−0.696780	+	0.717285i	$0.745384\pi$
$360$	0	0
$361$	1.71677	0.0903561
$362$	−10.1585	−0.533921
$363$	0	0
$364$	6.90739	0.362046
$365$	−4.53074	−0.237150
$366$	0	0
$367$	10.2803	0.536627	0.268314	−	0.963332i	$-0.413534\pi$
$367$	10.2803	0.536627	0.268314	+	0.963332i	$0.413534\pi$
$368$	4.13987	0.215805
$369$	0	0
$370$	−4.72083	−0.245424
$371$	−4.05023	−0.210278
$372$	0	0
$373$	−30.9604	−1.60307	−0.801535	−	0.597948i	$-0.795983\pi$
$373$	−30.9604	−1.60307	−0.801535	+	0.597948i	$0.795983\pi$
$374$	−0.401489	−0.0207605
$375$	0	0
$376$	−21.6746	−1.11778
$377$	31.8568	1.64071
$378$	0	0
$379$	−16.7108	−0.858378	−0.429189	−	0.903215i	$-0.641201\pi$
$379$	−16.7108	−0.858378	−0.429189	+	0.903215i	$0.641201\pi$
$380$	12.6921	0.651090
$381$	0	0
$382$	16.4944	0.843925
$383$	4.61926	0.236033	0.118017	−	0.993012i	$-0.462346\pi$
$383$	4.61926	0.236033	0.118017	+	0.993012i	$0.462346\pi$
$384$	0	0
$385$	0.513779	0.0261846
$386$	−10.3630	−0.527464
$387$	0	0
$388$	−14.9109	−0.756987
$389$	−4.68771	−0.237676	−0.118838	−	0.992914i	$-0.537917\pi$
$389$	−4.68771	−0.237676	−0.118838	+	0.992914i	$0.537917\pi$
$390$	0	0
$391$	13.5175	0.683610
$392$	2.94934	0.148964
$393$	0	0
$394$	−18.7630	−0.945266
$395$	−42.4163	−2.13419
$396$	0	0
$397$	17.2229	0.864393	0.432197	−	0.901779i	$-0.357739\pi$
$397$	17.2229	0.864393	0.432197	+	0.901779i	$0.357739\pi$
$398$	−12.7131	−0.637251
$399$	0	0
$400$	−1.01184	−0.0505920
$401$	9.90101	0.494433	0.247216	−	0.968960i	$-0.420484\pi$
$401$	9.90101	0.494433	0.247216	+	0.968960i	$0.420484\pi$
$402$	0	0
$403$	13.7428	0.684576
$404$	13.9634	0.694708
$405$	0	0
$406$	4.79105	0.237776
$407$	−0.386167	−0.0191416
$408$	0	0
$409$	4.81954	0.238311	0.119155	−	0.992876i	$-0.461981\pi$
$409$	4.81954	0.238311	0.119155	+	0.992876i	$0.461981\pi$
$410$	6.48197	0.320121
$411$	0	0
$412$	−9.46204	−0.466161
$413$	7.20824	0.354694
$414$	0	0
$415$	−38.0421	−1.86741
$416$	−33.5689	−1.64585
$417$	0	0
$418$	−0.871180	−0.0426108
$419$	20.0860	0.981264	0.490632	−	0.871367i	$-0.336766\pi$
$419$	20.0860	0.981264	0.490632	+	0.871367i	$0.336766\pi$
$420$	0	0
$421$	10.0258	0.488626	0.244313	−	0.969696i	$-0.421438\pi$
$421$	10.0258	0.488626	0.244313	+	0.969696i	$0.421438\pi$
$422$	−22.2745	−1.08431
$423$	0	0
$424$	11.9455	0.580126
$425$	−3.30386	−0.160261
$426$	0	0
$427$	8.95485	0.433356
$428$	5.22627	0.252621
$429$	0	0
$430$	−7.96809	−0.384255
$431$	−18.4714	−0.889737	−0.444869	−	0.895596i	$-0.646750\pi$
$431$	−18.4714	−0.889737	−0.444869	+	0.895596i	$0.646750\pi$
$432$	0	0
$433$	−32.0478	−1.54012	−0.770061	−	0.637971i	$-0.779774\pi$
$433$	−32.0478	−1.54012	−0.770061	+	0.637971i	$0.779774\pi$
$434$	2.06682	0.0992106
$435$	0	0
$436$	−13.5392	−0.648409
$437$	29.3313	1.40311
$438$	0	0
$439$	23.6580	1.12914	0.564568	−	0.825387i	$-0.309043\pi$
$439$	23.6580	1.12914	0.564568	+	0.825387i	$0.309043\pi$
$440$	−1.51531	−0.0722395
$441$	0	0
$442$	12.7274	0.605379
$443$	2.51077	0.119290	0.0596452	−	0.998220i	$-0.481003\pi$
$443$	2.51077	0.119290	0.0596452	+	0.998220i	$0.481003\pi$
$444$	0	0
$445$	−3.91148	−0.185422
$446$	19.6358	0.929783
$447$	0	0
$448$	−6.33338	−0.299224
$449$	32.2341	1.52122	0.760611	−	0.649208i	$-0.224899\pi$
$449$	32.2341	1.52122	0.760611	+	0.649208i	$0.224899\pi$
$450$	0	0
$451$	0.530228	0.0249675
$452$	12.7130	0.597970
$453$	0	0
$454$	−19.2478	−0.903346
$455$	−16.2870	−0.763546
$456$	0	0
$457$	−8.53005	−0.399019	−0.199510	−	0.979896i	$-0.563935\pi$
$457$	−8.53005	−0.399019	−0.199510	+	0.979896i	$0.563935\pi$
$458$	9.38352	0.438463
$459$	0	0
$460$	17.9698	0.837844
$461$	15.1973	0.707807	0.353904	−	0.935282i	$-0.384854\pi$
$461$	15.1973	0.707807	0.353904	+	0.935282i	$0.384854\pi$
$462$	0	0
$463$	32.5301	1.51180	0.755901	−	0.654686i	$-0.227199\pi$
$463$	32.5301	1.51180	0.755901	+	0.654686i	$0.227199\pi$
$464$	−3.22201	−0.149578
$465$	0	0
$466$	18.4545	0.854889
$467$	−30.6691	−1.41919	−0.709597	−	0.704607i	$-0.751123\pi$
$467$	−30.6691	−1.41919	−0.709597	+	0.704607i	$0.751123\pi$
$468$	0	0
$469$	8.84866	0.408593
$470$	18.0010	0.830324
$471$	0	0
$472$	−21.2595	−0.978550
$473$	−0.651794	−0.0299695
$474$	0	0
$475$	−7.16897	−0.328935
$476$	−2.28112	−0.104555
$477$	0	0
$478$	5.97467	0.273275
$479$	22.7841	1.04103	0.520515	−	0.853853i	$-0.325740\pi$
$479$	22.7841	1.04103	0.520515	+	0.853853i	$0.325740\pi$
$480$	0	0
$481$	12.2416	0.558171
$482$	8.72534	0.397428
$483$	0	0
$484$	11.9187	0.541757
$485$	35.1586	1.59647
$486$	0	0
$487$	21.5900	0.978338	0.489169	−	0.872189i	$-0.337300\pi$
$487$	21.5900	0.978338	0.489169	+	0.872189i	$0.337300\pi$
$488$	−26.4109	−1.19557
$489$	0	0
$490$	−2.44946	−0.110655
$491$	−2.24210	−0.101185	−0.0505924	−	0.998719i	$-0.516111\pi$
$491$	−2.24210	−0.101185	−0.0505924	+	0.998719i	$0.516111\pi$
$492$	0	0
$493$	−10.5205	−0.473820
$494$	27.6168	1.24254
$495$	0	0
$496$	−1.38995	−0.0624105
$497$	0.482080	0.0216243
$498$	0	0
$499$	5.88464	0.263433	0.131716	−	0.991287i	$-0.457951\pi$
$499$	5.88464	0.263433	0.131716	+	0.991287i	$0.457951\pi$
$500$	9.55049	0.427111
$501$	0	0
$502$	11.8679	0.529691
$503$	−40.7930	−1.81887	−0.909434	−	0.415848i	$-0.863485\pi$
$503$	−40.7930	−1.81887	−0.909434	+	0.415848i	$0.863485\pi$
$504$	0	0
$505$	−32.9246	−1.46512
$506$	−1.23344	−0.0548330
$507$	0	0
$508$	−1.08748	−0.0482492
$509$	−31.5813	−1.39982	−0.699908	−	0.714233i	$-0.746776\pi$
$509$	−31.5813	−1.39982	−0.699908	+	0.714233i	$0.746776\pi$
$510$	0	0
$511$	1.76693	0.0781644
$512$	7.18458	0.317517
$513$	0	0
$514$	−0.854290	−0.0376811
$515$	22.3107	0.983125
$516$	0	0
$517$	1.47249	0.0647601
$518$	1.84106	0.0808917
$519$	0	0
$520$	48.0359	2.10652
$521$	−8.82300	−0.386543	−0.193271	−	0.981145i	$-0.561910\pi$
$521$	−8.82300	−0.386543	−0.193271	+	0.981145i	$0.561910\pi$
$522$	0	0
$523$	−15.1383	−0.661950	−0.330975	−	0.943640i	$-0.607378\pi$
$523$	−15.1383	−0.661950	−0.330975	+	0.943640i	$0.607378\pi$
$524$	6.38049	0.278733
$525$	0	0
$526$	9.97429	0.434900
$527$	−4.53847	−0.197699
$528$	0	0
$529$	18.5280	0.805563
$530$	−9.92088	−0.430936
$531$	0	0
$532$	−4.94975	−0.214599
$533$	−16.8085	−0.728055
$534$	0	0
$535$	−12.3231	−0.532773
$536$	−26.0977	−1.12725
$537$	0	0
$538$	−14.8190	−0.638894
$539$	−0.200367	−0.00863042
$540$	0	0
$541$	−13.4317	−0.577472	−0.288736	−	0.957409i	$-0.593235\pi$
$541$	−13.4317	−0.577472	−0.288736	+	0.957409i	$0.593235\pi$
$542$	−24.7595	−1.06351
$543$	0	0
$544$	11.0859	0.475306
$545$	31.9242	1.36748
$546$	0	0
$547$	−44.2889	−1.89366	−0.946828	−	0.321739i	$-0.895733\pi$
$547$	−44.2889	−1.89366	−0.946828	+	0.321739i	$0.895733\pi$
$548$	−6.52578	−0.278768
$549$	0	0
$550$	0.301469	0.0128547
$551$	−22.8282	−0.972514
$552$	0	0
$553$	16.5418	0.703429
$554$	−21.8419	−0.927974
$555$	0	0
$556$	−8.78481	−0.372559
$557$	28.3796	1.20248	0.601241	−	0.799068i	$-0.294673\pi$
$557$	28.3796	1.20248	0.601241	+	0.799068i	$0.294673\pi$
$558$	0	0
$559$	20.6622	0.873916
$560$	1.64727	0.0696100
$561$	0	0
$562$	28.8925	1.21876
$563$	19.3286	0.814602	0.407301	−	0.913294i	$-0.366470\pi$
$563$	19.3286	0.814602	0.407301	+	0.913294i	$0.366470\pi$
$564$	0	0
$565$	−29.9761	−1.26111
$566$	−20.9717	−0.881508
$567$	0	0
$568$	−1.42182	−0.0596582
$569$	−20.5554	−0.861725	−0.430863	−	0.902418i	$-0.641791\pi$
$569$	−20.5554	−0.861725	−0.430863	+	0.902418i	$0.641791\pi$
$570$	0	0
$571$	−4.88437	−0.204404	−0.102202	−	0.994764i	$-0.532589\pi$
$571$	−4.88437	−0.204404	−0.102202	+	0.994764i	$0.532589\pi$
$572$	1.38401	0.0578685
$573$	0	0
$574$	−2.52788	−0.105512
$575$	−10.1500	−0.423284
$576$	0	0
$577$	−23.4647	−0.976847	−0.488423	−	0.872607i	$-0.662428\pi$
$577$	−23.4647	−0.976847	−0.488423	+	0.872607i	$0.662428\pi$
$578$	12.0362	0.500642
$579$	0	0
$580$	−13.9856	−0.580722
$581$	14.8359	0.615498
$582$	0	0
$583$	−0.811534	−0.0336103
$584$	−5.21128	−0.215644
$585$	0	0
$586$	−19.9893	−0.825751
$587$	−29.5065	−1.21786	−0.608932	−	0.793223i	$-0.708402\pi$
$587$	−29.5065	−1.21786	−0.608932	+	0.793223i	$0.708402\pi$
$588$	0	0
$589$	−9.84790	−0.405776
$590$	17.6563	0.726897
$591$	0	0
$592$	−1.23812	−0.0508866
$593$	−28.2398	−1.15967	−0.579836	−	0.814733i	$-0.696883\pi$
$593$	−28.2398	−1.15967	−0.579836	+	0.814733i	$0.696883\pi$
$594$	0	0
$595$	5.37869	0.220505
$596$	−20.5889	−0.843353
$597$	0	0
$598$	39.1005	1.59894
$599$	−6.80523	−0.278054	−0.139027	−	0.990289i	$-0.544398\pi$
$599$	−6.80523	−0.278054	−0.139027	+	0.990289i	$0.544398\pi$
$600$	0	0
$601$	−27.9026	−1.13817	−0.569086	−	0.822278i	$-0.692703\pi$
$601$	−27.9026	−1.13817	−0.569086	+	0.822278i	$0.692703\pi$
$602$	3.10745	0.126650
$603$	0	0
$604$	1.20614	0.0490773
$605$	−28.1031	−1.14255
$606$	0	0
$607$	42.7418	1.73484	0.867418	−	0.497581i	$-0.165778\pi$
$607$	42.7418	1.73484	0.867418	+	0.497581i	$0.165778\pi$
$608$	24.0551	0.975562
$609$	0	0
$610$	21.9345	0.888103
$611$	−46.6786	−1.88841
$612$	0	0
$613$	2.42264	0.0978496	0.0489248	−	0.998802i	$-0.484421\pi$
$613$	2.42264	0.0978496	0.0489248	+	0.998802i	$0.484421\pi$
$614$	11.9930	0.483996
$615$	0	0
$616$	0.590951	0.0238101
$617$	34.6980	1.39689	0.698445	−	0.715664i	$-0.253876\pi$
$617$	34.6980	1.39689	0.698445	+	0.715664i	$0.253876\pi$
$618$	0	0
$619$	10.6845	0.429448	0.214724	−	0.976675i	$-0.431115\pi$
$619$	10.6845	0.429448	0.214724	+	0.976675i	$0.431115\pi$
$620$	−6.03330	−0.242303
$621$	0	0
$622$	2.63145	0.105511
$623$	1.52543	0.0611150
$624$	0	0
$625$	−30.3945	−1.21578
$626$	23.1264	0.924317
$627$	0	0
$628$	18.3454	0.732062
$629$	−4.04273	−0.161194
$630$	0	0
$631$	−37.1731	−1.47984	−0.739918	−	0.672697i	$-0.765136\pi$
$631$	−37.1731	−1.47984	−0.739918	+	0.672697i	$0.765136\pi$
$632$	−48.7874	−1.94066
$633$	0	0
$634$	21.3398	0.847513
$635$	2.56419	0.101757
$636$	0	0
$637$	6.35172	0.251664
$638$	0.959970	0.0380056
$639$	0	0
$640$	11.5902	0.458141
$641$	38.6068	1.52488	0.762438	−	0.647061i	$-0.224002\pi$
$641$	38.6068	1.52488	0.762438	+	0.647061i	$0.224002\pi$
$642$	0	0
$643$	0.235030	0.00926867	0.00463434	−	0.999989i	$-0.498525\pi$
$643$	0.235030	0.00926867	0.00463434	+	0.999989i	$0.498525\pi$
$644$	−7.00798	−0.276153
$645$	0	0
$646$	−9.12028	−0.358833
$647$	−8.35481	−0.328462	−0.164231	−	0.986422i	$-0.552514\pi$
$647$	−8.35481	−0.328462	−0.164231	+	0.986422i	$0.552514\pi$
$648$	0	0
$649$	1.44429	0.0566935
$650$	−9.55669	−0.374844
$651$	0	0
$652$	−1.19497	−0.0467986
$653$	−7.96069	−0.311526	−0.155763	−	0.987794i	$-0.549784\pi$
$653$	−7.96069	−0.311526	−0.155763	+	0.987794i	$0.549784\pi$
$654$	0	0
$655$	−15.0446	−0.587843
$656$	1.70001	0.0663744
$657$	0	0
$658$	−7.02016	−0.273674
$659$	−13.9270	−0.542518	−0.271259	−	0.962506i	$-0.587440\pi$
$659$	−13.9270	−0.542518	−0.271259	+	0.962506i	$0.587440\pi$
$660$	0	0
$661$	−1.66745	−0.0648564	−0.0324282	−	0.999474i	$-0.510324\pi$
$661$	−1.66745	−0.0648564	−0.0324282	+	0.999474i	$0.510324\pi$
$662$	−3.94322	−0.153258
$663$	0	0
$664$	−43.7562	−1.69807
$665$	11.6711	0.452585
$666$	0	0
$667$	−32.3207	−1.25146
$668$	8.38341	0.324364
$669$	0	0
$670$	21.6744	0.837357
$671$	1.79426	0.0692665
$672$	0	0
$673$	11.3630	0.438010	0.219005	−	0.975724i	$-0.429719\pi$
$673$	11.3630	0.438010	0.219005	+	0.975724i	$0.429719\pi$
$674$	−5.05190	−0.194592
$675$	0	0
$676$	−29.7365	−1.14371
$677$	20.8605	0.801733	0.400867	−	0.916136i	$-0.368709\pi$
$677$	20.8605	0.801733	0.400867	+	0.916136i	$0.368709\pi$
$678$	0	0
$679$	−13.7114	−0.526195
$680$	−15.8636	−0.608341
$681$	0	0
$682$	0.414123	0.0158576
$683$	−3.20522	−0.122644	−0.0613221	−	0.998118i	$-0.519532\pi$
$683$	−3.20522	−0.122644	−0.0613221	+	0.998118i	$0.519532\pi$
$684$	0	0
$685$	15.3872	0.587915
$686$	0.955258	0.0364719
$687$	0	0
$688$	−2.08978	−0.0796720
$689$	25.7260	0.980081
$690$	0	0
$691$	11.1189	0.422984	0.211492	−	0.977380i	$-0.432168\pi$
$691$	11.1189	0.422984	0.211492	+	0.977380i	$0.432168\pi$
$692$	2.83159	0.107641
$693$	0	0
$694$	12.2625	0.465477
$695$	20.7138	0.785719
$696$	0	0
$697$	5.55090	0.210255
$698$	19.4745	0.737122
$699$	0	0
$700$	1.71284	0.0647394
$701$	−3.00205	−0.113386	−0.0566930	−	0.998392i	$-0.518056\pi$
$701$	−3.00205	−0.113386	−0.0566930	+	0.998392i	$0.518056\pi$
$702$	0	0
$703$	−8.77222	−0.330850
$704$	−1.26900	−0.0478273
$705$	0	0
$706$	26.8004	1.00865
$707$	12.8402	0.482904
$708$	0	0
$709$	30.1487	1.13226	0.566129	−	0.824316i	$-0.308440\pi$
$709$	30.1487	1.13226	0.566129	+	0.824316i	$0.308440\pi$
$710$	1.18084	0.0443160
$711$	0	0
$712$	−4.49901	−0.168607
$713$	−13.9429	−0.522165
$714$	0	0
$715$	−3.26338	−0.122043
$716$	−7.36919	−0.275399
$717$	0	0
$718$	25.2228	0.941306
$719$	48.1784	1.79675	0.898376	−	0.439226i	$-0.144747\pi$
$719$	48.1784	1.79675	0.898376	+	0.439226i	$0.144747\pi$
$720$	0	0
$721$	−8.70087	−0.324037
$722$	−1.63995	−0.0610327
$723$	0	0
$724$	−11.5647	−0.429797
$725$	7.89962	0.293384
$726$	0	0
$727$	42.9298	1.59218	0.796089	−	0.605180i	$-0.206899\pi$
$727$	42.9298	1.59218	0.796089	+	0.605180i	$0.206899\pi$
$728$	−18.7334	−0.694306
$729$	0	0
$730$	4.32803	0.160187
$731$	−6.82356	−0.252378
$732$	0	0
$733$	−4.92095	−0.181759	−0.0908797	−	0.995862i	$-0.528968\pi$
$733$	−4.92095	−0.181759	−0.0908797	+	0.995862i	$0.528968\pi$
$734$	−9.82033	−0.362475
$735$	0	0
$736$	34.0578	1.25539
$737$	1.77298	0.0653086
$738$	0	0
$739$	−36.3828	−1.33836	−0.669181	−	0.743099i	$-0.733355\pi$
$739$	−36.3828	−1.33836	−0.669181	+	0.743099i	$0.733355\pi$
$740$	−5.37428	−0.197563
$741$	0	0
$742$	3.86902	0.142036
$743$	2.76072	0.101281	0.0506406	−	0.998717i	$-0.483874\pi$
$743$	2.76072	0.101281	0.0506406	+	0.998717i	$0.483874\pi$
$744$	0	0
$745$	48.5467	1.77861
$746$	29.5752	1.08282
$747$	0	0
$748$	−0.457062	−0.0167119
$749$	4.80585	0.175602
$750$	0	0
$751$	−20.9105	−0.763035	−0.381517	−	0.924362i	$-0.624598\pi$
$751$	−20.9105	−0.763035	−0.381517	+	0.924362i	$0.624598\pi$
$752$	4.72109	0.172160
$753$	0	0
$754$	−30.4314	−1.10825
$755$	−2.84398	−0.103503
$756$	0	0
$757$	−38.6655	−1.40532	−0.702661	−	0.711525i	$-0.748005\pi$
$757$	−38.6655	−1.40532	−0.702661	+	0.711525i	$0.748005\pi$
$758$	15.9632	0.579808
$759$	0	0
$760$	−34.4220	−1.24862
$761$	−19.6524	−0.712399	−0.356200	−	0.934410i	$-0.615928\pi$
$761$	−19.6524	−0.712399	−0.356200	+	0.934410i	$0.615928\pi$
$762$	0	0
$763$	−12.4500	−0.450721
$764$	18.7775	0.679346
$765$	0	0
$766$	−4.41258	−0.159433
$767$	−45.7847	−1.65319
$768$	0	0
$769$	−18.8783	−0.680768	−0.340384	−	0.940287i	$-0.610557\pi$
$769$	−18.8783	−0.680768	−0.340384	+	0.940287i	$0.610557\pi$
$770$	−0.490791	−0.0176869
$771$	0	0
$772$	−11.7975	−0.424600
$773$	−6.35473	−0.228564	−0.114282	−	0.993448i	$-0.536457\pi$
$773$	−6.35473	−0.228564	−0.114282	+	0.993448i	$0.536457\pi$
$774$	0	0
$775$	3.40783	0.122413
$776$	40.4396	1.45170
$777$	0	0
$778$	4.47797	0.160543
$779$	12.0447	0.431548
$780$	0	0
$781$	0.0965930	0.00345637
$782$	−12.9127	−0.461758
$783$	0	0
$784$	−0.642416	−0.0229434
$785$	−43.2568	−1.54390
$786$	0	0
$787$	−46.4731	−1.65659	−0.828294	−	0.560294i	$-0.810688\pi$
$787$	−46.4731	−1.65659	−0.828294	+	0.560294i	$0.810688\pi$
$788$	−21.3601	−0.760923
$789$	0	0
$790$	40.5185	1.44158
$791$	11.6903	0.415659
$792$	0	0
$793$	−56.8787	−2.01982
$794$	−16.4523	−0.583871
$795$	0	0
$796$	−14.4728	−0.512976
$797$	−9.85474	−0.349073	−0.174536	−	0.984651i	$-0.555843\pi$
$797$	−9.85474	−0.349073	−0.174536	+	0.984651i	$0.555843\pi$
$798$	0	0
$799$	15.4153	0.545355
$800$	−8.32418	−0.294304
$801$	0	0
$802$	−9.45802	−0.333974
$803$	0.354035	0.0124936
$804$	0	0
$805$	16.5242	0.582401
$806$	−13.1279	−0.462410
$807$	0	0
$808$	−37.8700	−1.33226
$809$	−0.0782396	−0.00275076	−0.00137538	−	0.999999i	$-0.500438\pi$
$809$	−0.0782396	−0.00275076	−0.00137538	+	0.999999i	$0.500438\pi$
$810$	0	0
$811$	−41.9787	−1.47407	−0.737036	−	0.675854i	$-0.763775\pi$
$811$	−41.9787	−1.47407	−0.737036	+	0.675854i	$0.763775\pi$
$812$	5.45422	0.191406
$813$	0	0
$814$	0.368889	0.0129295
$815$	2.81763	0.0986972
$816$	0	0
$817$	−14.8062	−0.518005
$818$	−4.60390	−0.160972
$819$	0	0
$820$	7.37919	0.257692
$821$	−45.1684	−1.57639	−0.788194	−	0.615427i	$-0.788984\pi$
$821$	−45.1684	−1.57639	−0.788194	+	0.615427i	$0.788984\pi$
$822$	0	0
$823$	6.36935	0.222022	0.111011	−	0.993819i	$-0.464591\pi$
$823$	6.36935	0.222022	0.111011	+	0.993819i	$0.464591\pi$
$824$	25.6618	0.893972
$825$	0	0
$826$	−6.88572	−0.239585
$827$	−31.4505	−1.09364	−0.546821	−	0.837250i	$-0.684162\pi$
$827$	−31.4505	−1.09364	−0.546821	+	0.837250i	$0.684162\pi$
$828$	0	0
$829$	−48.3801	−1.68031	−0.840156	−	0.542345i	$-0.817536\pi$
$829$	−48.3801	−1.68031	−0.840156	+	0.542345i	$0.817536\pi$
$830$	36.3400	1.26138
$831$	0	0
$832$	40.2279	1.39465
$833$	−2.09762	−0.0726782
$834$	0	0
$835$	−19.7673	−0.684077
$836$	−0.991768	−0.0343010
$837$	0	0
$838$	−19.1873	−0.662813
$839$	−22.8918	−0.790313	−0.395157	−	0.918614i	$-0.629310\pi$
$839$	−22.8918	−0.790313	−0.395157	+	0.918614i	$0.629310\pi$
$840$	0	0
$841$	−3.84519	−0.132593
$842$	−9.57719	−0.330052
$843$	0	0
$844$	−25.3577	−0.872849
$845$	70.1161	2.41207
$846$	0	0
$847$	10.9599	0.376585
$848$	−2.60193	−0.0893508
$849$	0	0
$850$	3.15604	0.108251
$851$	−12.4199	−0.425749
$852$	0	0
$853$	−9.61146	−0.329090	−0.164545	−	0.986370i	$-0.552616\pi$
$853$	−9.61146	−0.329090	−0.164545	+	0.986370i	$0.552616\pi$
$854$	−8.55419	−0.292718
$855$	0	0
$856$	−14.1741	−0.484460
$857$	−43.7361	−1.49400	−0.746998	−	0.664826i	$-0.768506\pi$
$857$	−43.7361	−1.49400	−0.746998	+	0.664826i	$0.768506\pi$
$858$	0	0
$859$	8.40862	0.286899	0.143449	−	0.989658i	$-0.454181\pi$
$859$	8.40862	0.286899	0.143449	+	0.989658i	$0.454181\pi$
$860$	−9.07102	−0.309319
$861$	0	0
$862$	17.6450	0.600990
$863$	2.92301	0.0995006	0.0497503	−	0.998762i	$-0.484157\pi$
$863$	2.92301	0.0995006	0.0497503	+	0.998762i	$0.484157\pi$
$864$	0	0
$865$	−6.67663	−0.227012
$866$	30.6140	1.04030
$867$	0	0
$868$	2.35291	0.0798629
$869$	3.31443	0.112434
$870$	0	0
$871$	−56.2043	−1.90441
$872$	36.7193	1.24347
$873$	0	0
$874$	−28.0189	−0.947755
$875$	8.78220	0.296893
$876$	0	0
$877$	−7.69264	−0.259762	−0.129881	−	0.991530i	$-0.541460\pi$
$877$	−7.69264	−0.259762	−0.129881	+	0.991530i	$0.541460\pi$
$878$	−22.5995	−0.762697
$879$	0	0
$880$	0.330060	0.0111263
$881$	34.9805	1.17852	0.589261	−	0.807943i	$-0.299419\pi$
$881$	34.9805	1.17852	0.589261	+	0.807943i	$0.299419\pi$
$882$	0	0
$883$	−52.4407	−1.76477	−0.882386	−	0.470527i	$-0.844064\pi$
$883$	−52.4407	−1.76477	−0.882386	+	0.470527i	$0.844064\pi$
$884$	14.4891	0.487320
$885$	0	0
$886$	−2.39843	−0.0805769
$887$	4.77380	0.160289	0.0801443	−	0.996783i	$-0.474462\pi$
$887$	4.77380	0.160289	0.0801443	+	0.996783i	$0.474462\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	3.73647	0.125247
$891$	0	0
$892$	22.3538	0.748460
$893$	33.4493	1.11934
$894$	0	0
$895$	17.3759	0.580812
$896$	−4.52001	−0.151003
$897$	0	0
$898$	−30.7919	−1.02754
$899$	10.8516	0.361921
$900$	0	0
$901$	−8.49585	−0.283038
$902$	−0.506505	−0.0168648
$903$	0	0
$904$	−34.4787	−1.14674
$905$	27.2684	0.906433
$906$	0	0
$907$	−29.5559	−0.981389	−0.490695	−	0.871332i	$-0.663257\pi$
$907$	−29.5559	−0.981389	−0.490695	+	0.871332i	$0.663257\pi$
$908$	−21.9121	−0.727178
$909$	0	0
$910$	15.5583	0.515752
$911$	−9.78643	−0.324239	−0.162119	−	0.986771i	$-0.551833\pi$
$911$	−9.78643	−0.324239	−0.162119	+	0.986771i	$0.551833\pi$
$912$	0	0
$913$	2.97263	0.0983797
$914$	8.14840	0.269525
$915$	0	0
$916$	10.6824	0.352956
$917$	5.86721	0.193752
$918$	0	0
$919$	16.8745	0.556638	0.278319	−	0.960489i	$-0.410223\pi$
$919$	16.8745	0.556638	0.278319	+	0.960489i	$0.410223\pi$
$920$	−48.7355	−1.60676
$921$	0	0
$922$	−14.5173	−0.478102
$923$	−3.06204	−0.100788
$924$	0	0
$925$	3.03560	0.0998098
$926$	−31.0746	−1.02118
$927$	0	0
$928$	−26.5067	−0.870127
$929$	51.9812	1.70545	0.852724	−	0.522362i	$-0.174949\pi$
$929$	51.9812	1.70545	0.852724	+	0.522362i	$0.174949\pi$
$930$	0	0
$931$	−4.55157	−0.149172
$932$	21.0090	0.688171
$933$	0	0
$934$	29.2969	0.958622
$935$	1.07771	0.0352450
$936$	0	0
$937$	35.9690	1.17506	0.587528	−	0.809204i	$-0.300101\pi$
$937$	35.9690	1.17506	0.587528	+	0.809204i	$0.300101\pi$
$938$	−8.45275	−0.275992
$939$	0	0
$940$	20.4927	0.668397
$941$	−50.3657	−1.64188	−0.820938	−	0.571018i	$-0.806549\pi$
$941$	−50.3657	−1.64188	−0.820938	+	0.571018i	$0.806549\pi$
$942$	0	0
$943$	17.0532	0.555330
$944$	4.63068	0.150716
$945$	0	0
$946$	0.622632	0.0202435
$947$	−49.8917	−1.62126	−0.810631	−	0.585557i	$-0.800876\pi$
$947$	−49.8917	−1.62126	−0.810631	+	0.585557i	$0.800876\pi$
$948$	0	0
$949$	−11.2231	−0.364316
$950$	6.84821	0.222185
$951$	0	0
$952$	6.18660	0.200509
$953$	23.1081	0.748544	0.374272	−	0.927319i	$-0.377893\pi$
$953$	23.1081	0.748544	0.374272	+	0.927319i	$0.377893\pi$
$954$	0	0
$955$	−44.2756	−1.43273
$956$	6.80167	0.219982
$957$	0	0
$958$	−21.7646	−0.703184
$959$	−6.00082	−0.193776
$960$	0	0
$961$	−26.3187	−0.848991
$962$	−11.6939	−0.377027
$963$	0	0
$964$	9.93308	0.319923
$965$	27.8174	0.895472
$966$	0	0
$967$	−33.9010	−1.09018	−0.545091	−	0.838377i	$-0.683505\pi$
$967$	−33.9010	−1.09018	−0.545091	+	0.838377i	$0.683505\pi$
$968$	−32.3244	−1.03894
$969$	0	0
$970$	−33.5855	−1.07837
$971$	37.1869	1.19338	0.596692	−	0.802471i	$-0.296482\pi$
$971$	37.1869	1.19338	0.596692	+	0.802471i	$0.296482\pi$
$972$	0	0
$973$	−8.07811	−0.258972
$974$	−20.6240	−0.660837
$975$	0	0
$976$	5.75273	0.184141
$977$	7.75983	0.248259	0.124129	−	0.992266i	$-0.460386\pi$
$977$	7.75983	0.248259	0.124129	+	0.992266i	$0.460386\pi$
$978$	0	0
$979$	0.305646	0.00976848
$980$	−2.78851	−0.0890757
$981$	0	0
$982$	2.14179	0.0683471
$983$	1.49388	0.0476475	0.0238237	−	0.999716i	$-0.492416\pi$
$983$	1.49388	0.0476475	0.0238237	+	0.999716i	$0.492416\pi$
$984$	0	0
$985$	50.3653	1.60477
$986$	10.0498	0.320051
$987$	0	0
$988$	31.4394	1.00022
$989$	−20.9630	−0.666586
$990$	0	0
$991$	56.9824	1.81010	0.905052	−	0.425301i	$-0.139832\pi$
$991$	56.9824	1.81010	0.905052	+	0.425301i	$0.139832\pi$
$992$	−11.4348	−0.363055
$993$	0	0
$994$	−0.460511	−0.0146065
$995$	34.1257	1.08186
$996$	0	0
$997$	−7.29689	−0.231095	−0.115547	−	0.993302i	$-0.536862\pi$
$997$	−7.29689	−0.231095	−0.115547	+	0.993302i	$0.536862\pi$
$998$	−5.62135	−0.177941
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.q.1.5		15
3.2	odd	2		889.2.a.b.1.11	✓	15
21.20	even	2		6223.2.a.j.1.11		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.b.1.11	✓	15	3.2	odd	2
6223.2.a.j.1.11		15	21.20	even	2
8001.2.a.q.1.5		15	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)

\(f(q)\)	\(=\)	\(q-0.955258 q^{2} -1.08748 q^{4} +2.56419 q^{5} -1.00000 q^{7} +2.94934 q^{8} +O(q^{10})\)	\(q-0.955258 q^{2} -1.08748 q^{4} +2.56419 q^{5} -1.00000 q^{7} +2.94934 q^{8} -2.44946 q^{10} -0.200367 q^{11} +6.35172 q^{13} +0.955258 q^{14} -0.642416 q^{16} -2.09762 q^{17} -4.55157 q^{19} -2.78851 q^{20} +0.191402 q^{22} -6.44422 q^{23} +1.57505 q^{25} -6.06753 q^{26} +1.08748 q^{28} +5.01546 q^{29} +2.16363 q^{31} -5.28501 q^{32} +2.00377 q^{34} -2.56419 q^{35} +1.92730 q^{37} +4.34792 q^{38} +7.56266 q^{40} -2.64628 q^{41} +3.25300 q^{43} +0.217896 q^{44} +6.15589 q^{46} -7.34897 q^{47} +1.00000 q^{49} -1.50458 q^{50} -6.90739 q^{52} +4.05023 q^{53} -0.513779 q^{55} -2.94934 q^{56} -4.79105 q^{58} -7.20824 q^{59} -8.95485 q^{61} -2.06682 q^{62} +6.33338 q^{64} +16.2870 q^{65} -8.84866 q^{67} +2.28112 q^{68} +2.44946 q^{70} -0.482080 q^{71} -1.76693 q^{73} -1.84106 q^{74} +4.94975 q^{76} +0.200367 q^{77} -16.5418 q^{79} -1.64727 q^{80} +2.52788 q^{82} -14.8359 q^{83} -5.37869 q^{85} -3.10745 q^{86} -0.590951 q^{88} -1.52543 q^{89} -6.35172 q^{91} +7.00798 q^{92} +7.02016 q^{94} -11.6711 q^{95} +13.7114 q^{97} -0.955258 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7}+O(q^{10}) \) 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7	\( 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7} + 10 q^{10} - 14 q^{11} + 6 q^{13} + 20 q^{16} - 10 q^{17} + 13 q^{19} - 8 q^{20} - 11 q^{22} - 15 q^{23} - 22 q^{26} - 14 q^{28} - 16 q^{29} + 22 q^{31} - 15 q^{34} + 7 q^{35} - 14 q^{37} + 6 q^{38} + 22 q^{40} - 19 q^{41} - q^{43} - 25 q^{44} - 28 q^{46} - 49 q^{47} + 15 q^{49} - 24 q^{50} - 17 q^{52} + 28 q^{53} + 39 q^{55} - 10 q^{58} - 43 q^{59} + 27 q^{61} - 14 q^{62} + 18 q^{64} + 8 q^{65} + 3 q^{67} - 13 q^{68} - 10 q^{70} - 55 q^{71} - 3 q^{73} + 12 q^{74} - 20 q^{76} + 14 q^{77} + 18 q^{79} - 29 q^{80} + 14 q^{82} - 17 q^{83} + 7 q^{85} - 4 q^{86} - 114 q^{88} - 36 q^{89} - 6 q^{91} - 45 q^{92} - 15 q^{94} - 59 q^{95} - 2 q^{97}+O(q^{100}) \) 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7 + 10 * q^10 - 14 * q^11 + 6 * q^13 + 20 * q^16 - 10 * q^17 + 13 * q^19 - 8 * q^20 - 11 * q^22 - 15 * q^23 - 22 * q^26 - 14 * q^28 - 16 * q^29 + 22 * q^31 - 15 * q^34 + 7 * q^35 - 14 * q^37 + 6 * q^38 + 22 * q^40 - 19 * q^41 - q^43 - 25 * q^44 - 28 * q^46 - 49 * q^47 + 15 * q^49 - 24 * q^50 - 17 * q^52 + 28 * q^53 + 39 * q^55 - 10 * q^58 - 43 * q^59 + 27 * q^61 - 14 * q^62 + 18 * q^64 + 8 * q^65 + 3 * q^67 - 13 * q^68 - 10 * q^70 - 55 * q^71 - 3 * q^73 + 12 * q^74 - 20 * q^76 + 14 * q^77 + 18 * q^79 - 29 * q^80 + 14 * q^82 - 17 * q^83 + 7 * q^85 - 4 * q^86 - 114 * q^88 - 36 * q^89 - 6 * q^91 - 45 * q^92 - 15 * q^94 - 59 * q^95 - 2 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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