LMFDB - Embedded newform 8001.2.a.r.1.13

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:	$16$
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} - 5 x^{15} - 13 x^{14} + 98 x^{13} + 9 x^{12} - 712 x^{11} + 565 x^{10} + 2282 x^{9} - 3082 x^{8} + \cdots + 4 $ x^16 - 5x^15 - 13x^14 + 98x^13 + 9x^12 - 712x^11 + 565x^10 + 2282x^9 - 3082x^8 - 2747x^7 + 5821x^6 - 158x^5 - 3341x^4 + 1002x^3 + 416x^2 - 148*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$-1.82937$ 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)

$f(q)$	$=$	$q+1.82937 q^{2} +1.34661 q^{4} +1.45514 q^{5} -1.00000 q^{7} -1.19530 q^{8} +O(q^{10})$	\(q+1.82937 q^{2} +1.34661 q^{4} +1.45514 q^{5} -1.00000 q^{7} -1.19530 q^{8} +2.66200 q^{10} -2.19029 q^{11} +6.28967 q^{13} -1.82937 q^{14} -4.87986 q^{16} -6.17655 q^{17} +7.62323 q^{19} +1.95951 q^{20} -4.00686 q^{22} -5.57657 q^{23} -2.88256 q^{25} +11.5062 q^{26} -1.34661 q^{28} -3.99429 q^{29} -3.97226 q^{31} -6.53650 q^{32} -11.2992 q^{34} -1.45514 q^{35} -1.95957 q^{37} +13.9457 q^{38} -1.73933 q^{40} -9.92690 q^{41} -7.53303 q^{43} -2.94947 q^{44} -10.2016 q^{46} +1.82436 q^{47} +1.00000 q^{49} -5.27329 q^{50} +8.46972 q^{52} +2.70508 q^{53} -3.18719 q^{55} +1.19530 q^{56} -7.30704 q^{58} +5.00355 q^{59} +4.52270 q^{61} -7.26675 q^{62} -2.19797 q^{64} +9.15236 q^{65} +4.03056 q^{67} -8.31740 q^{68} -2.66200 q^{70} -13.6861 q^{71} -0.355776 q^{73} -3.58478 q^{74} +10.2655 q^{76} +2.19029 q^{77} -5.46858 q^{79} -7.10089 q^{80} -18.1600 q^{82} +0.444405 q^{83} -8.98776 q^{85} -13.7807 q^{86} +2.61805 q^{88} +6.92589 q^{89} -6.28967 q^{91} -7.50946 q^{92} +3.33744 q^{94} +11.0929 q^{95} +2.38835 q^{97} +1.82937 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8}+O(q^{10}) $ 16 * q - 5 * q^2 + 19 * q^4 + q^5 - 16 * q^7 - 6 * q^8	$ 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8} - 12 q^{10} - 11 q^{11} + 18 q^{13} + 5 q^{14} + 25 q^{16} + 5 q^{17} - 11 q^{19} + q^{20} + q^{22} - 13 q^{23} + 33 q^{25} - 8 q^{26} - 19 q^{28} - 24 q^{29} - 42 q^{31} - 42 q^{32} + 9 q^{34} - q^{35} + 40 q^{37} - 38 q^{38} - 61 q^{40} - 9 q^{41} + 7 q^{43} - 3 q^{44} + 24 q^{46} - 31 q^{47} + 16 q^{49} - 6 q^{50} + 52 q^{52} - 66 q^{53} - 36 q^{55} + 6 q^{56} + 19 q^{58} + 7 q^{59} + 6 q^{61} - 52 q^{62} + 10 q^{64} - 51 q^{65} + 16 q^{67} - 14 q^{68} + 12 q^{70} - 46 q^{71} + 39 q^{73} - 72 q^{74} + 24 q^{76} + 11 q^{77} + 4 q^{79} + 2 q^{80} - 18 q^{82} - 15 q^{83} - 4 q^{85} - 14 q^{86} + 58 q^{88} + q^{89} - 18 q^{91} - 26 q^{92} + 5 q^{94} - 44 q^{95} + 41 q^{97} - 5 q^{98}+O(q^{100}) $ 16 * q - 5 * q^2 + 19 * q^4 + q^5 - 16 * q^7 - 6 * q^8 - 12 * q^10 - 11 * q^11 + 18 * q^13 + 5 * q^14 + 25 * q^16 + 5 * q^17 - 11 * q^19 + q^20 + q^22 - 13 * q^23 + 33 * q^25 - 8 * q^26 - 19 * q^28 - 24 * q^29 - 42 * q^31 - 42 * q^32 + 9 * q^34 - q^35 + 40 * q^37 - 38 * q^38 - 61 * q^40 - 9 * q^41 + 7 * q^43 - 3 * q^44 + 24 * q^46 - 31 * q^47 + 16 * q^49 - 6 * q^50 + 52 * q^52 - 66 * q^53 - 36 * q^55 + 6 * q^56 + 19 * q^58 + 7 * q^59 + 6 * q^61 - 52 * q^62 + 10 * q^64 - 51 * q^65 + 16 * q^67 - 14 * q^68 + 12 * q^70 - 46 * q^71 + 39 * q^73 - 72 * q^74 + 24 * q^76 + 11 * q^77 + 4 * q^79 + 2 * q^80 - 18 * q^82 - 15 * q^83 - 4 * q^85 - 14 * q^86 + 58 * q^88 + q^89 - 18 * q^91 - 26 * q^92 + 5 * q^94 - 44 * q^95 + 41 * q^97 - 5 * 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 $

$2$	1.82937	1.29356	0.646781	−	0.762676i	$-0.276115\pi$
$2$	1.82937	1.29356	0.646781	+	0.762676i	$0.276115\pi$
$3$	0	0
$3$	0	0
$4$	1.34661	0.673304
$5$	1.45514	0.650759	0.325380	−	0.945584i	$-0.394508\pi$
$5$	1.45514	0.650759	0.325380	+	0.945584i	$0.394508\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.19530	−0.422602
$9$	0	0
$10$	2.66200	0.841798
$11$	−2.19029	−0.660398	−0.330199	−	0.943911i	$-0.607116\pi$
$11$	−2.19029	−0.660398	−0.330199	+	0.943911i	$0.607116\pi$
$12$	0	0
$13$	6.28967	1.74444	0.872221	−	0.489113i	$-0.162679\pi$
$13$	6.28967	1.74444	0.872221	+	0.489113i	$0.162679\pi$
$14$	−1.82937	−0.488921
$15$	0	0
$16$	−4.87986	−1.21997
$17$	−6.17655	−1.49803	−0.749017	−	0.662550i	$-0.769474\pi$
$17$	−6.17655	−1.49803	−0.749017	+	0.662550i	$0.769474\pi$
$18$	0	0
$19$	7.62323	1.74889	0.874445	−	0.485125i	$-0.161226\pi$
$19$	7.62323	1.74889	0.874445	+	0.485125i	$0.161226\pi$
$20$	1.95951	0.438159
$21$	0	0
$22$	−4.00686	−0.854266
$23$	−5.57657	−1.16280	−0.581398	−	0.813619i	$-0.697494\pi$
$23$	−5.57657	−1.16280	−0.581398	+	0.813619i	$0.697494\pi$
$24$	0	0
$25$	−2.88256	−0.576513
$26$	11.5062	2.25654
$27$	0	0
$28$	−1.34661	−0.254485
$29$	−3.99429	−0.741721	−0.370860	−	0.928689i	$-0.620937\pi$
$29$	−3.99429	−0.741721	−0.370860	+	0.928689i	$0.620937\pi$
$30$	0	0
$31$	−3.97226	−0.713439	−0.356720	−	0.934211i	$-0.616105\pi$
$31$	−3.97226	−0.713439	−0.356720	+	0.934211i	$0.616105\pi$
$32$	−6.53650	−1.15550
$33$	0	0
$34$	−11.2992	−1.93780
$35$	−1.45514	−0.245964
$36$	0	0
$37$	−1.95957	−0.322151	−0.161075	−	0.986942i	$-0.551496\pi$
$37$	−1.95957	−0.322151	−0.161075	+	0.986942i	$0.551496\pi$
$38$	13.9457	2.26230
$39$	0	0
$40$	−1.73933	−0.275012
$41$	−9.92690	−1.55032	−0.775161	−	0.631764i	$-0.782331\pi$
$41$	−9.92690	−1.55032	−0.775161	+	0.631764i	$0.782331\pi$
$42$	0	0
$43$	−7.53303	−1.14878	−0.574388	−	0.818583i	$-0.694760\pi$
$43$	−7.53303	−1.14878	−0.574388	+	0.818583i	$0.694760\pi$
$44$	−2.94947	−0.444649
$45$	0	0
$46$	−10.2016	−1.50415
$47$	1.82436	0.266111	0.133055	−	0.991109i	$-0.457521\pi$
$47$	1.82436	0.266111	0.133055	+	0.991109i	$0.457521\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−5.27329	−0.745755
$51$	0	0
$52$	8.46972	1.17454
$53$	2.70508	0.371571	0.185785	−	0.982590i	$-0.440517\pi$
$53$	2.70508	0.371571	0.185785	+	0.982590i	$0.440517\pi$
$54$	0	0
$55$	−3.18719	−0.429760
$56$	1.19530	0.159728
$57$	0	0
$58$	−7.30704	−0.959462
$59$	5.00355	0.651406	0.325703	−	0.945472i	$-0.394399\pi$
$59$	5.00355	0.651406	0.325703	+	0.945472i	$0.394399\pi$
$60$	0	0
$61$	4.52270	0.579072	0.289536	−	0.957167i	$-0.406499\pi$
$61$	4.52270	0.579072	0.289536	+	0.957167i	$0.406499\pi$
$62$	−7.26675	−0.922879
$63$	0	0
$64$	−2.19797	−0.274746
$65$	9.15236	1.13521
$66$	0	0
$67$	4.03056	0.492412	0.246206	−	0.969218i	$-0.420816\pi$
$67$	4.03056	0.492412	0.246206	+	0.969218i	$0.420816\pi$
$68$	−8.31740	−1.00863
$69$	0	0
$70$	−2.66200	−0.318170
$71$	−13.6861	−1.62425	−0.812123	−	0.583486i	$-0.801688\pi$
$71$	−13.6861	−1.62425	−0.812123	+	0.583486i	$0.801688\pi$
$72$	0	0
$73$	−0.355776	−0.0416404	−0.0208202	−	0.999783i	$-0.506628\pi$
$73$	−0.355776	−0.0416404	−0.0208202	+	0.999783i	$0.506628\pi$
$74$	−3.58478	−0.416722
$75$	0	0
$76$	10.2655	1.17753
$77$	2.19029	0.249607
$78$	0	0
$79$	−5.46858	−0.615264	−0.307632	−	0.951505i	$-0.599537\pi$
$79$	−5.46858	−0.615264	−0.307632	+	0.951505i	$0.599537\pi$
$80$	−7.10089	−0.793904
$81$	0	0
$82$	−18.1600	−2.00544
$83$	0.444405	0.0487798	0.0243899	−	0.999703i	$-0.492236\pi$
$83$	0.444405	0.0487798	0.0243899	+	0.999703i	$0.492236\pi$
$84$	0	0
$85$	−8.98776	−0.974860
$86$	−13.7807	−1.48601
$87$	0	0
$88$	2.61805	0.279085
$89$	6.92589	0.734142	0.367071	−	0.930193i	$-0.380361\pi$
$89$	6.92589	0.734142	0.367071	+	0.930193i	$0.380361\pi$
$90$	0	0
$91$	−6.28967	−0.659337
$92$	−7.50946	−0.782915
$93$	0	0
$94$	3.33744	0.344231
$95$	11.0929	1.13811
$96$	0	0
$97$	2.38835	0.242500	0.121250	−	0.992622i	$-0.461310\pi$
$97$	2.38835	0.242500	0.121250	+	0.992622i	$0.461310\pi$
$98$	1.82937	0.184795
$99$	0	0
$100$	−3.88168	−0.388168
$101$	3.75467	0.373604	0.186802	−	0.982398i	$-0.440188\pi$
$101$	3.75467	0.373604	0.186802	+	0.982398i	$0.440188\pi$
$102$	0	0
$103$	1.91280	0.188473	0.0942367	−	0.995550i	$-0.469959\pi$
$103$	1.91280	0.188473	0.0942367	+	0.995550i	$0.469959\pi$
$104$	−7.51803	−0.737204
$105$	0	0
$106$	4.94860	0.480650
$107$	−9.22188	−0.891512	−0.445756	−	0.895154i	$-0.647065\pi$
$107$	−9.22188	−0.891512	−0.445756	+	0.895154i	$0.647065\pi$
$108$	0	0
$109$	−0.780448	−0.0747533	−0.0373767	−	0.999301i	$-0.511900\pi$
$109$	−0.780448	−0.0747533	−0.0373767	+	0.999301i	$0.511900\pi$
$110$	−5.83055	−0.555922
$111$	0	0
$112$	4.87986	0.461104
$113$	−13.1911	−1.24092	−0.620458	−	0.784240i	$-0.713053\pi$
$113$	−13.1911	−1.24092	−0.620458	+	0.784240i	$0.713053\pi$
$114$	0	0
$115$	−8.11471	−0.756700
$116$	−5.37874	−0.499404
$117$	0	0
$118$	9.15336	0.842635
$119$	6.17655	0.566204
$120$	0	0
$121$	−6.20262	−0.563874
$122$	8.27370	0.749066
$123$	0	0
$124$	−5.34908	−0.480362
$125$	−11.4702	−1.02593
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	9.05209	0.800099
$129$	0	0
$130$	16.7431	1.46847
$131$	20.8994	1.82599	0.912995	−	0.407972i	$-0.133764\pi$
$131$	20.8994	1.82599	0.912995	+	0.407972i	$0.133764\pi$
$132$	0	0
$133$	−7.62323	−0.661018
$134$	7.37341	0.636966
$135$	0	0
$136$	7.38282	0.633072
$137$	11.4265	0.976236	0.488118	−	0.872778i	$-0.337684\pi$
$137$	11.4265	0.976236	0.488118	+	0.872778i	$0.337684\pi$
$138$	0	0
$139$	18.1243	1.53728	0.768642	−	0.639679i	$-0.220933\pi$
$139$	18.1243	1.53728	0.768642	+	0.639679i	$0.220933\pi$
$140$	−1.95951	−0.165608
$141$	0	0
$142$	−25.0371	−2.10106
$143$	−13.7762	−1.15203
$144$	0	0
$145$	−5.81225	−0.482681
$146$	−0.650847	−0.0538645
$147$	0	0
$148$	−2.63877	−0.216906
$149$	−18.3159	−1.50049	−0.750247	−	0.661157i	$-0.770066\pi$
$149$	−18.3159	−1.50049	−0.750247	+	0.661157i	$0.770066\pi$
$150$	0	0
$151$	−21.9311	−1.78473	−0.892365	−	0.451314i	$-0.850955\pi$
$151$	−21.9311	−1.78473	−0.892365	+	0.451314i	$0.850955\pi$
$152$	−9.11203	−0.739084
$153$	0	0
$154$	4.00686	0.322882
$155$	−5.78020	−0.464277
$156$	0	0
$157$	12.2689	0.979167	0.489583	−	0.871957i	$-0.337149\pi$
$157$	12.2689	0.979167	0.489583	+	0.871957i	$0.337149\pi$
$158$	−10.0041	−0.795882
$159$	0	0
$160$	−9.51153	−0.751952
$161$	5.57657	0.439496
$162$	0	0
$163$	−23.5755	−1.84657	−0.923287	−	0.384111i	$-0.874508\pi$
$163$	−23.5755	−1.84657	−0.923287	+	0.384111i	$0.874508\pi$
$164$	−13.3676	−1.04384
$165$	0	0
$166$	0.812983	0.0630997
$167$	0.575768	0.0445543	0.0222772	−	0.999752i	$-0.492908\pi$
$167$	0.575768	0.0445543	0.0222772	+	0.999752i	$0.492908\pi$
$168$	0	0
$169$	26.5600	2.04308
$170$	−16.4420	−1.26104
$171$	0	0
$172$	−10.1440	−0.773476
$173$	15.7831	1.19997	0.599984	−	0.800012i	$-0.295174\pi$
$173$	15.7831	1.19997	0.599984	+	0.800012i	$0.295174\pi$
$174$	0	0
$175$	2.88256	0.217901
$176$	10.6883	0.805663
$177$	0	0
$178$	12.6700	0.949659
$179$	2.70842	0.202437	0.101218	−	0.994864i	$-0.467726\pi$
$179$	2.70842	0.202437	0.101218	+	0.994864i	$0.467726\pi$
$180$	0	0
$181$	−14.6608	−1.08973	−0.544864	−	0.838525i	$-0.683419\pi$
$181$	−14.6608	−1.08973	−0.544864	+	0.838525i	$0.683419\pi$
$182$	−11.5062	−0.852893
$183$	0	0
$184$	6.66567	0.491400
$185$	−2.85145	−0.209643
$186$	0	0
$187$	13.5285	0.989299
$188$	2.45670	0.179173
$189$	0	0
$190$	20.2930	1.47221
$191$	−26.7099	−1.93266	−0.966330	−	0.257305i	$-0.917165\pi$
$191$	−26.7099	−1.93266	−0.966330	+	0.257305i	$0.917165\pi$
$192$	0	0
$193$	4.29894	0.309445	0.154722	−	0.987958i	$-0.450552\pi$
$193$	4.29894	0.309445	0.154722	+	0.987958i	$0.450552\pi$
$194$	4.36918	0.313689
$195$	0	0
$196$	1.34661	0.0961863
$197$	−12.3520	−0.880041	−0.440020	−	0.897988i	$-0.645029\pi$
$197$	−12.3520	−0.880041	−0.440020	+	0.897988i	$0.645029\pi$
$198$	0	0
$199$	1.45985	0.103486	0.0517431	−	0.998660i	$-0.483522\pi$
$199$	1.45985	0.103486	0.0517431	+	0.998660i	$0.483522\pi$
$200$	3.44552	0.243635
$201$	0	0
$202$	6.86870	0.483280
$203$	3.99429	0.280344
$204$	0	0
$205$	−14.4450	−1.00889
$206$	3.49922	0.243802
$207$	0	0
$208$	−30.6927	−2.12816
$209$	−16.6971	−1.15496
$210$	0	0
$211$	−23.2602	−1.60130	−0.800649	−	0.599133i	$-0.795512\pi$
$211$	−23.2602	−1.60130	−0.800649	+	0.599133i	$0.795512\pi$
$212$	3.64268	0.250180
$213$	0	0
$214$	−16.8703	−1.15323
$215$	−10.9616	−0.747576
$216$	0	0
$217$	3.97226	0.269655
$218$	−1.42773	−0.0966981
$219$	0	0
$220$	−4.29189	−0.289359
$221$	−38.8485	−2.61323
$222$	0	0
$223$	−4.08720	−0.273699	−0.136850	−	0.990592i	$-0.543698\pi$
$223$	−4.08720	−0.273699	−0.136850	+	0.990592i	$0.543698\pi$
$224$	6.53650	0.436738
$225$	0	0
$226$	−24.1315	−1.60520
$227$	−23.9736	−1.59119	−0.795593	−	0.605832i	$-0.792841\pi$
$227$	−23.9736	−1.59119	−0.795593	+	0.605832i	$0.792841\pi$
$228$	0	0
$229$	−17.2205	−1.13796	−0.568981	−	0.822351i	$-0.692662\pi$
$229$	−17.2205	−1.13796	−0.568981	+	0.822351i	$0.692662\pi$
$230$	−14.8448	−0.978839
$231$	0	0
$232$	4.77436	0.313452
$233$	−15.1867	−0.994914	−0.497457	−	0.867489i	$-0.665733\pi$
$233$	−15.1867	−0.994914	−0.497457	+	0.867489i	$0.665733\pi$
$234$	0	0
$235$	2.65471	0.173174
$236$	6.73782	0.438594
$237$	0	0
$238$	11.2992	0.732420
$239$	12.0366	0.778580	0.389290	−	0.921115i	$-0.372720\pi$
$239$	12.0366	0.778580	0.389290	+	0.921115i	$0.372720\pi$
$240$	0	0
$241$	−5.91405	−0.380958	−0.190479	−	0.981691i	$-0.561004\pi$
$241$	−5.91405	−0.380958	−0.190479	+	0.981691i	$0.561004\pi$
$242$	−11.3469	−0.729407
$243$	0	0
$244$	6.09030	0.389891
$245$	1.45514	0.0929656
$246$	0	0
$247$	47.9476	3.05084
$248$	4.74804	0.301501
$249$	0	0
$250$	−20.9834	−1.32710
$251$	−4.09811	−0.258670	−0.129335	−	0.991601i	$-0.541284\pi$
$251$	−4.09811	−0.258670	−0.129335	+	0.991601i	$0.541284\pi$
$252$	0	0
$253$	12.2143	0.767908
$254$	1.82937	0.114785
$255$	0	0
$256$	20.9556	1.30972
$257$	0.462923	0.0288764	0.0144382	−	0.999896i	$-0.495404\pi$
$257$	0.462923	0.0288764	0.0144382	+	0.999896i	$0.495404\pi$
$258$	0	0
$259$	1.95957	0.121762
$260$	12.3246	0.764342
$261$	0	0
$262$	38.2328	2.36203
$263$	−27.1526	−1.67430	−0.837151	−	0.546972i	$-0.815780\pi$
$263$	−27.1526	−1.67430	−0.837151	+	0.546972i	$0.815780\pi$
$264$	0	0
$265$	3.93627	0.241803
$266$	−13.9457	−0.855068
$267$	0	0
$268$	5.42759	0.331543
$269$	19.8660	1.21125	0.605626	−	0.795749i	$-0.292923\pi$
$269$	19.8660	1.21125	0.605626	+	0.795749i	$0.292923\pi$
$270$	0	0
$271$	−18.8698	−1.14626	−0.573130	−	0.819464i	$-0.694271\pi$
$271$	−18.8698	−1.14626	−0.573130	+	0.819464i	$0.694271\pi$
$272$	30.1407	1.82755
$273$	0	0
$274$	20.9034	1.26282
$275$	6.31366	0.380728
$276$	0	0
$277$	29.8172	1.79154	0.895771	−	0.444515i	$-0.146624\pi$
$277$	29.8172	1.79154	0.895771	+	0.444515i	$0.146624\pi$
$278$	33.1561	1.98857
$279$	0	0
$280$	1.73933	0.103945
$281$	30.2631	1.80534	0.902672	−	0.430329i	$-0.141603\pi$
$281$	30.2631	1.80534	0.902672	+	0.430329i	$0.141603\pi$
$282$	0	0
$283$	8.97112	0.533278	0.266639	−	0.963796i	$-0.414087\pi$
$283$	8.97112	0.533278	0.266639	+	0.963796i	$0.414087\pi$
$284$	−18.4299	−1.09361
$285$	0	0
$286$	−25.2019	−1.49022
$287$	9.92690	0.585966
$288$	0	0
$289$	21.1498	1.24411
$290$	−10.6328	−0.624379
$291$	0	0
$292$	−0.479091	−0.0280367
$293$	−18.2970	−1.06892	−0.534461	−	0.845193i	$-0.679485\pi$
$293$	−18.2970	−1.06892	−0.534461	+	0.845193i	$0.679485\pi$
$294$	0	0
$295$	7.28087	0.423908
$296$	2.34227	0.136142
$297$	0	0
$298$	−33.5066	−1.94098
$299$	−35.0748	−2.02843
$300$	0	0
$301$	7.53303	0.434196
$302$	−40.1202	−2.30866
$303$	0	0
$304$	−37.2003	−2.13359
$305$	6.58116	0.376836
$306$	0	0
$307$	−28.2136	−1.61024	−0.805118	−	0.593115i	$-0.797898\pi$
$307$	−28.2136	−1.61024	−0.805118	+	0.593115i	$0.797898\pi$
$308$	2.94947	0.168061
$309$	0	0
$310$	−10.5742	−0.600572
$311$	8.42306	0.477628	0.238814	−	0.971065i	$-0.423241\pi$
$311$	8.42306	0.477628	0.238814	+	0.971065i	$0.423241\pi$
$312$	0	0
$313$	−32.5732	−1.84114	−0.920572	−	0.390573i	$-0.872277\pi$
$313$	−32.5732	−1.84114	−0.920572	+	0.390573i	$0.872277\pi$
$314$	22.4444	1.26661
$315$	0	0
$316$	−7.36404	−0.414260
$317$	−18.6701	−1.04862	−0.524309	−	0.851528i	$-0.675676\pi$
$317$	−18.6701	−1.04862	−0.524309	+	0.851528i	$0.675676\pi$
$318$	0	0
$319$	8.74866	0.489831
$320$	−3.19836	−0.178794
$321$	0	0
$322$	10.2016	0.568515
$323$	−47.0853	−2.61990
$324$	0	0
$325$	−18.1304	−1.00569
$326$	−43.1283	−2.38866
$327$	0	0
$328$	11.8656	0.655168
$329$	−1.82436	−0.100580
$330$	0	0
$331$	21.2362	1.16725	0.583623	−	0.812025i	$-0.301635\pi$
$331$	21.2362	1.16725	0.583623	+	0.812025i	$0.301635\pi$
$332$	0.598440	0.0328437
$333$	0	0
$334$	1.05330	0.0576338
$335$	5.86504	0.320441
$336$	0	0
$337$	19.7374	1.07516	0.537582	−	0.843211i	$-0.319338\pi$
$337$	19.7374	1.07516	0.537582	+	0.843211i	$0.319338\pi$
$338$	48.5881	2.64285
$339$	0	0
$340$	−12.1030	−0.656377
$341$	8.70042	0.471154
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	9.00421	0.485475
$345$	0	0
$346$	28.8732	1.55223
$347$	27.9381	1.49980	0.749898	−	0.661554i	$-0.230103\pi$
$347$	27.9381	1.49980	0.749898	+	0.661554i	$0.230103\pi$
$348$	0	0
$349$	−16.0730	−0.860366	−0.430183	−	0.902742i	$-0.641551\pi$
$349$	−16.0730	−0.860366	−0.430183	+	0.902742i	$0.641551\pi$
$350$	5.27329	0.281869
$351$	0	0
$352$	14.3168	0.763090
$353$	−11.5576	−0.615148	−0.307574	−	0.951524i	$-0.599517\pi$
$353$	−11.5576	−0.615148	−0.307574	+	0.951524i	$0.599517\pi$
$354$	0	0
$355$	−19.9153	−1.05699
$356$	9.32645	0.494301
$357$	0	0
$358$	4.95471	0.261865
$359$	−9.21020	−0.486096	−0.243048	−	0.970014i	$-0.578147\pi$
$359$	−9.21020	−0.486096	−0.243048	+	0.970014i	$0.578147\pi$
$360$	0	0
$361$	39.1137	2.05862
$362$	−26.8201	−1.40963
$363$	0	0
$364$	−8.46972	−0.443934
$365$	−0.517704	−0.0270979
$366$	0	0
$367$	23.3937	1.22114	0.610571	−	0.791962i	$-0.290940\pi$
$367$	23.3937	1.22114	0.610571	+	0.791962i	$0.290940\pi$
$368$	27.2129	1.41857
$369$	0	0
$370$	−5.21636	−0.271186
$371$	−2.70508	−0.140441
$372$	0	0
$373$	20.2084	1.04635	0.523175	−	0.852225i	$-0.324747\pi$
$373$	20.2084	1.04635	0.523175	+	0.852225i	$0.324747\pi$
$374$	24.7486	1.27972
$375$	0	0
$376$	−2.18066	−0.112459
$377$	−25.1228	−1.29389
$378$	0	0
$379$	25.5931	1.31463	0.657314	−	0.753617i	$-0.271693\pi$
$379$	25.5931	1.31463	0.657314	+	0.753617i	$0.271693\pi$
$380$	14.9378	0.766291
$381$	0	0
$382$	−48.8624	−2.50002
$383$	−10.1116	−0.516681	−0.258340	−	0.966054i	$-0.583176\pi$
$383$	−10.1116	−0.516681	−0.258340	+	0.966054i	$0.583176\pi$
$384$	0	0
$385$	3.18719	0.162434
$386$	7.86437	0.400286
$387$	0	0
$388$	3.21617	0.163276
$389$	−4.17187	−0.211522	−0.105761	−	0.994392i	$-0.533728\pi$
$389$	−4.17187	−0.211522	−0.105761	+	0.994392i	$0.533728\pi$
$390$	0	0
$391$	34.4440	1.74191
$392$	−1.19530	−0.0603717
$393$	0	0
$394$	−22.5963	−1.13839
$395$	−7.95756	−0.400388
$396$	0	0
$397$	3.54478	0.177907	0.0889537	−	0.996036i	$-0.471648\pi$
$397$	3.54478	0.177907	0.0889537	+	0.996036i	$0.471648\pi$
$398$	2.67062	0.133866
$399$	0	0
$400$	14.0665	0.703326
$401$	−7.16011	−0.357559	−0.178779	−	0.983889i	$-0.557215\pi$
$401$	−7.16011	−0.357559	−0.178779	+	0.983889i	$0.557215\pi$
$402$	0	0
$403$	−24.9842	−1.24455
$404$	5.05607	0.251549
$405$	0	0
$406$	7.30704	0.362643
$407$	4.29203	0.212748
$408$	0	0
$409$	−20.8877	−1.03283	−0.516416	−	0.856338i	$-0.672734\pi$
$409$	−20.8877	−1.03283	−0.516416	+	0.856338i	$0.672734\pi$
$410$	−26.4254	−1.30506
$411$	0	0
$412$	2.57579	0.126900
$413$	−5.00355	−0.246208
$414$	0	0
$415$	0.646673	0.0317439
$416$	−41.1124	−2.01570
$417$	0	0
$418$	−30.5453	−1.49402
$419$	3.23628	0.158102	0.0790512	−	0.996871i	$-0.474811\pi$
$419$	3.23628	0.158102	0.0790512	+	0.996871i	$0.474811\pi$
$420$	0	0
$421$	24.3455	1.18653	0.593264	−	0.805008i	$-0.297839\pi$
$421$	24.3455	1.18653	0.593264	+	0.805008i	$0.297839\pi$
$422$	−42.5516	−2.07138
$423$	0	0
$424$	−3.23337	−0.157026
$425$	17.8043	0.863636
$426$	0	0
$427$	−4.52270	−0.218869
$428$	−12.4183	−0.600259
$429$	0	0
$430$	−20.0529	−0.967037
$431$	38.0138	1.83106	0.915531	−	0.402248i	$-0.131771\pi$
$431$	38.0138	1.83106	0.915531	+	0.402248i	$0.131771\pi$
$432$	0	0
$433$	3.75990	0.180689	0.0903445	−	0.995911i	$-0.471203\pi$
$433$	3.75990	0.180689	0.0903445	+	0.995911i	$0.471203\pi$
$434$	7.26675	0.348815
$435$	0	0
$436$	−1.05096	−0.0503317
$437$	−42.5115	−2.03360
$438$	0	0
$439$	3.62013	0.172779	0.0863897	−	0.996261i	$-0.472467\pi$
$439$	3.62013	0.172779	0.0863897	+	0.996261i	$0.472467\pi$
$440$	3.80964	0.181617
$441$	0	0
$442$	−71.0684	−3.38038
$443$	22.0624	1.04821	0.524107	−	0.851652i	$-0.324399\pi$
$443$	22.0624	1.04821	0.524107	+	0.851652i	$0.324399\pi$
$444$	0	0
$445$	10.0781	0.477750
$446$	−7.47701	−0.354047
$447$	0	0
$448$	2.19797	0.103844
$449$	5.21757	0.246232	0.123116	−	0.992392i	$-0.460711\pi$
$449$	5.21757	0.246232	0.123116	+	0.992392i	$0.460711\pi$
$450$	0	0
$451$	21.7428	1.02383
$452$	−17.7633	−0.835513
$453$	0	0
$454$	−43.8567	−2.05830
$455$	−9.15236	−0.429069
$456$	0	0
$457$	33.6570	1.57441	0.787204	−	0.616692i	$-0.211528\pi$
$457$	33.6570	1.57441	0.787204	+	0.616692i	$0.211528\pi$
$458$	−31.5027	−1.47202
$459$	0	0
$460$	−10.9273	−0.509489
$461$	19.9884	0.930951	0.465476	−	0.885061i	$-0.345883\pi$
$461$	19.9884	0.930951	0.465476	+	0.885061i	$0.345883\pi$
$462$	0	0
$463$	9.71478	0.451484	0.225742	−	0.974187i	$-0.427519\pi$
$463$	9.71478	0.451484	0.225742	+	0.974187i	$0.427519\pi$
$464$	19.4916	0.904874
$465$	0	0
$466$	−27.7822	−1.28698
$467$	16.8882	0.781490	0.390745	−	0.920499i	$-0.372217\pi$
$467$	16.8882	0.781490	0.390745	+	0.920499i	$0.372217\pi$
$468$	0	0
$469$	−4.03056	−0.186114
$470$	4.85645	0.224011
$471$	0	0
$472$	−5.98073	−0.275285
$473$	16.4995	0.758650
$474$	0	0
$475$	−21.9745	−1.00826
$476$	8.31740	0.381227
$477$	0	0
$478$	22.0194	1.00714
$479$	23.8468	1.08959	0.544795	−	0.838569i	$-0.316607\pi$
$479$	23.8468	1.08959	0.544795	+	0.838569i	$0.316607\pi$
$480$	0	0
$481$	−12.3250	−0.561973
$482$	−10.8190	−0.492793
$483$	0	0
$484$	−8.35249	−0.379659
$485$	3.47538	0.157809
$486$	0	0
$487$	−8.13005	−0.368408	−0.184204	−	0.982888i	$-0.558971\pi$
$487$	−8.13005	−0.368408	−0.184204	+	0.982888i	$0.558971\pi$
$488$	−5.40597	−0.244717
$489$	0	0
$490$	2.66200	0.120257
$491$	−15.5782	−0.703035	−0.351518	−	0.936181i	$-0.614334\pi$
$491$	−15.5782	−0.703035	−0.351518	+	0.936181i	$0.614334\pi$
$492$	0	0
$493$	24.6709	1.11112
$494$	87.7142	3.94645
$495$	0	0
$496$	19.3841	0.870372
$497$	13.6861	0.613907
$498$	0	0
$499$	−12.3396	−0.552395	−0.276197	−	0.961101i	$-0.589074\pi$
$499$	−12.3396	−0.552395	−0.276197	+	0.961101i	$0.589074\pi$
$500$	−15.4459	−0.690763
$501$	0	0
$502$	−7.49698	−0.334606
$503$	15.5250	0.692226	0.346113	−	0.938193i	$-0.387501\pi$
$503$	15.5250	0.692226	0.346113	+	0.938193i	$0.387501\pi$
$504$	0	0
$505$	5.46358	0.243126
$506$	22.3446	0.993338
$507$	0	0
$508$	1.34661	0.0597461
$509$	9.59742	0.425398	0.212699	−	0.977118i	$-0.431775\pi$
$509$	9.59742	0.425398	0.212699	+	0.977118i	$0.431775\pi$
$510$	0	0
$511$	0.355776	0.0157386
$512$	20.2314	0.894111
$513$	0	0
$514$	0.846860	0.0373534
$515$	2.78339	0.122651
$516$	0	0
$517$	−3.99589	−0.175739
$518$	3.58478	0.157506
$519$	0	0
$520$	−10.9398	−0.479742
$521$	−6.14341	−0.269148	−0.134574	−	0.990904i	$-0.542967\pi$
$521$	−6.14341	−0.269148	−0.134574	+	0.990904i	$0.542967\pi$
$522$	0	0
$523$	15.6084	0.682509	0.341255	−	0.939971i	$-0.389148\pi$
$523$	15.6084	0.682509	0.341255	+	0.939971i	$0.389148\pi$
$524$	28.1433	1.22945
$525$	0	0
$526$	−49.6723	−2.16581
$527$	24.5349	1.06876
$528$	0	0
$529$	8.09819	0.352095
$530$	7.20091	0.312787
$531$	0	0
$532$	−10.2655	−0.445066
$533$	−62.4369	−2.70444
$534$	0	0
$535$	−13.4191	−0.580160
$536$	−4.81773	−0.208094
$537$	0	0
$538$	36.3424	1.56683
$539$	−2.19029	−0.0943426
$540$	0	0
$541$	−23.7248	−1.02001	−0.510005	−	0.860171i	$-0.670356\pi$
$541$	−23.7248	−1.02001	−0.510005	+	0.860171i	$0.670356\pi$
$542$	−34.5200	−1.48276
$543$	0	0
$544$	40.3730	1.73098
$545$	−1.13566	−0.0486464
$546$	0	0
$547$	−12.7750	−0.546219	−0.273109	−	0.961983i	$-0.588052\pi$
$547$	−12.7750	−0.546219	−0.273109	+	0.961983i	$0.588052\pi$
$548$	15.3871	0.657304
$549$	0	0
$550$	11.5500	0.492495
$551$	−30.4494	−1.29719
$552$	0	0
$553$	5.46858	0.232548
$554$	54.5468	2.31747
$555$	0	0
$556$	24.4063	1.03506
$557$	−3.20615	−0.135849	−0.0679245	−	0.997690i	$-0.521638\pi$
$557$	−3.20615	−0.135849	−0.0679245	+	0.997690i	$0.521638\pi$
$558$	0	0
$559$	−47.3803	−2.00397
$560$	7.10089	0.300067
$561$	0	0
$562$	55.3625	2.33533
$563$	21.0118	0.885541	0.442770	−	0.896635i	$-0.353996\pi$
$563$	21.0118	0.885541	0.442770	+	0.896635i	$0.353996\pi$
$564$	0	0
$565$	−19.1949	−0.807537
$566$	16.4115	0.689828
$567$	0	0
$568$	16.3590	0.686409
$569$	24.6207	1.03215	0.516076	−	0.856543i	$-0.327392\pi$
$569$	24.6207	1.03215	0.516076	+	0.856543i	$0.327392\pi$
$570$	0	0
$571$	−40.6375	−1.70063	−0.850313	−	0.526277i	$-0.823587\pi$
$571$	−40.6375	−1.70063	−0.850313	+	0.526277i	$0.823587\pi$
$572$	−18.5512	−0.775664
$573$	0	0
$574$	18.1600	0.757984
$575$	16.0748	0.670367
$576$	0	0
$577$	31.7031	1.31982	0.659909	−	0.751346i	$-0.270595\pi$
$577$	31.7031	1.31982	0.659909	+	0.751346i	$0.270595\pi$
$578$	38.6909	1.60933
$579$	0	0
$580$	−7.82683	−0.324991
$581$	−0.444405	−0.0184370
$582$	0	0
$583$	−5.92491	−0.245385
$584$	0.425258	0.0175973
$585$	0	0
$586$	−33.4721	−1.38272
$587$	−6.03061	−0.248910	−0.124455	−	0.992225i	$-0.539718\pi$
$587$	−6.03061	−0.248910	−0.124455	+	0.992225i	$0.539718\pi$
$588$	0	0
$589$	−30.2815	−1.24773
$590$	13.3194	0.548352
$591$	0	0
$592$	9.56242	0.393013
$593$	43.0369	1.76731	0.883656	−	0.468136i	$-0.155074\pi$
$593$	43.0369	1.76731	0.883656	+	0.468136i	$0.155074\pi$
$594$	0	0
$595$	8.98776	0.368462
$596$	−24.6643	−1.01029
$597$	0	0
$598$	−64.1650	−2.62390
$599$	28.5562	1.16678	0.583388	−	0.812193i	$-0.301727\pi$
$599$	28.5562	1.16678	0.583388	+	0.812193i	$0.301727\pi$
$600$	0	0
$601$	16.6010	0.677171	0.338585	−	0.940936i	$-0.390052\pi$
$601$	16.6010	0.677171	0.338585	+	0.940936i	$0.390052\pi$
$602$	13.7807	0.561660
$603$	0	0
$604$	−29.5326	−1.20167
$605$	−9.02569	−0.366946
$606$	0	0
$607$	−6.36767	−0.258456	−0.129228	−	0.991615i	$-0.541250\pi$
$607$	−6.36767	−0.258456	−0.129228	+	0.991615i	$0.541250\pi$
$608$	−49.8292	−2.02084
$609$	0	0
$610$	12.0394	0.487461
$611$	11.4746	0.464215
$612$	0	0
$613$	47.6729	1.92549	0.962744	−	0.270413i	$-0.0871602\pi$
$613$	47.6729	1.92549	0.962744	+	0.270413i	$0.0871602\pi$
$614$	−51.6132	−2.08294
$615$	0	0
$616$	−2.61805	−0.105484
$617$	−17.1461	−0.690276	−0.345138	−	0.938552i	$-0.612168\pi$
$617$	−17.1461	−0.690276	−0.345138	+	0.938552i	$0.612168\pi$
$618$	0	0
$619$	−4.28642	−0.172286	−0.0861429	−	0.996283i	$-0.527454\pi$
$619$	−4.28642	−0.172286	−0.0861429	+	0.996283i	$0.527454\pi$
$620$	−7.78367	−0.312600
$621$	0	0
$622$	15.4089	0.617841
$623$	−6.92589	−0.277480
$624$	0	0
$625$	−2.27801	−0.0911204
$626$	−59.5885	−2.38164
$627$	0	0
$628$	16.5214	0.659277
$629$	12.1034	0.482593
$630$	0	0
$631$	−10.9843	−0.437279	−0.218640	−	0.975806i	$-0.570162\pi$
$631$	−10.9843	−0.437279	−0.218640	+	0.975806i	$0.570162\pi$
$632$	6.53659	0.260011
$633$	0	0
$634$	−34.1546	−1.35645
$635$	1.45514	0.0577455
$636$	0	0
$637$	6.28967	0.249206
$638$	16.0046	0.633627
$639$	0	0
$640$	13.1721	0.520672
$641$	6.52496	0.257720	0.128860	−	0.991663i	$-0.458868\pi$
$641$	6.52496	0.257720	0.128860	+	0.991663i	$0.458868\pi$
$642$	0	0
$643$	−9.45870	−0.373015	−0.186507	−	0.982454i	$-0.559717\pi$
$643$	−9.45870	−0.373015	−0.186507	+	0.982454i	$0.559717\pi$
$644$	7.50946	0.295914
$645$	0	0
$646$	−86.1366	−3.38900
$647$	−32.6574	−1.28389	−0.641947	−	0.766749i	$-0.721873\pi$
$647$	−32.6574	−1.28389	−0.641947	+	0.766749i	$0.721873\pi$
$648$	0	0
$649$	−10.9592	−0.430187
$650$	−33.1672	−1.30093
$651$	0	0
$652$	−31.7469	−1.24331
$653$	29.6322	1.15960	0.579800	−	0.814759i	$-0.303131\pi$
$653$	29.6322	1.15960	0.579800	+	0.814759i	$0.303131\pi$
$654$	0	0
$655$	30.4116	1.18828
$656$	48.4419	1.89134
$657$	0	0
$658$	−3.33744	−0.130107
$659$	−9.36429	−0.364781	−0.182391	−	0.983226i	$-0.558384\pi$
$659$	−9.36429	−0.364781	−0.182391	+	0.983226i	$0.558384\pi$
$660$	0	0
$661$	15.4094	0.599356	0.299678	−	0.954040i	$-0.403121\pi$
$661$	15.4094	0.599356	0.299678	+	0.954040i	$0.403121\pi$
$662$	38.8489	1.50991
$663$	0	0
$664$	−0.531197	−0.0206144
$665$	−11.0929	−0.430164
$666$	0	0
$667$	22.2744	0.862470
$668$	0.775335	0.0299986
$669$	0	0
$670$	10.7294	0.414511
$671$	−9.90603	−0.382418
$672$	0	0
$673$	−48.0244	−1.85121	−0.925603	−	0.378496i	$-0.876442\pi$
$673$	−48.0244	−1.85121	−0.925603	+	0.378496i	$0.876442\pi$
$674$	36.1071	1.39079
$675$	0	0
$676$	35.7659	1.37561
$677$	27.0870	1.04104	0.520518	−	0.853850i	$-0.325739\pi$
$677$	27.0870	1.04104	0.520518	+	0.853850i	$0.325739\pi$
$678$	0	0
$679$	−2.38835	−0.0916563
$680$	10.7431	0.411977
$681$	0	0
$682$	15.9163	0.609467
$683$	4.87002	0.186346	0.0931730	−	0.995650i	$-0.470299\pi$
$683$	4.87002	0.186346	0.0931730	+	0.995650i	$0.470299\pi$
$684$	0	0
$685$	16.6272	0.635294
$686$	−1.82937	−0.0698458
$687$	0	0
$688$	36.7601	1.40147
$689$	17.0140	0.648183
$690$	0	0
$691$	−29.2548	−1.11290	−0.556452	−	0.830880i	$-0.687838\pi$
$691$	−29.2548	−1.11290	−0.556452	+	0.830880i	$0.687838\pi$
$692$	21.2537	0.807944
$693$	0	0
$694$	51.1092	1.94008
$695$	26.3734	1.00040
$696$	0	0
$697$	61.3140	2.32243
$698$	−29.4035	−1.11294
$699$	0	0
$700$	3.88168	0.146714
$701$	−41.8736	−1.58154	−0.790771	−	0.612112i	$-0.790320\pi$
$701$	−41.8736	−1.58154	−0.790771	+	0.612112i	$0.790320\pi$
$702$	0	0
$703$	−14.9382	−0.563407
$704$	4.81420	0.181442
$705$	0	0
$706$	−21.1431	−0.795732
$707$	−3.75467	−0.141209
$708$	0	0
$709$	−1.82760	−0.0686368	−0.0343184	−	0.999411i	$-0.510926\pi$
$709$	−1.82760	−0.0686368	−0.0343184	+	0.999411i	$0.510926\pi$
$710$	−36.4325	−1.36729
$711$	0	0
$712$	−8.27850	−0.310250
$713$	22.1516	0.829585
$714$	0	0
$715$	−20.0464	−0.749691
$716$	3.64718	0.136302
$717$	0	0
$718$	−16.8489	−0.628795
$719$	18.6331	0.694897	0.347449	−	0.937699i	$-0.387048\pi$
$719$	18.6331	0.694897	0.347449	+	0.937699i	$0.387048\pi$
$720$	0	0
$721$	−1.91280	−0.0712363
$722$	71.5536	2.66295
$723$	0	0
$724$	−19.7423	−0.733718
$725$	11.5138	0.427611
$726$	0	0
$727$	23.2446	0.862095	0.431047	−	0.902329i	$-0.358144\pi$
$727$	23.2446	0.862095	0.431047	+	0.902329i	$0.358144\pi$
$728$	7.51803	0.278637
$729$	0	0
$730$	−0.947074	−0.0350528
$731$	46.5282	1.72091
$732$	0	0
$733$	37.1964	1.37388	0.686940	−	0.726714i	$-0.258954\pi$
$733$	37.1964	1.37388	0.686940	+	0.726714i	$0.258954\pi$
$734$	42.7958	1.57962
$735$	0	0
$736$	36.4513	1.34361
$737$	−8.82812	−0.325188
$738$	0	0
$739$	−7.04930	−0.259313	−0.129656	−	0.991559i	$-0.541387\pi$
$739$	−7.04930	−0.259313	−0.129656	+	0.991559i	$0.541387\pi$
$740$	−3.83978	−0.141153
$741$	0	0
$742$	−4.94860	−0.181669
$743$	−42.0471	−1.54256	−0.771280	−	0.636496i	$-0.780383\pi$
$743$	−42.0471	−1.54256	−0.771280	+	0.636496i	$0.780383\pi$
$744$	0	0
$745$	−26.6522	−0.976461
$746$	36.9687	1.35352
$747$	0	0
$748$	18.2175	0.666099
$749$	9.22188	0.336960
$750$	0	0
$751$	50.3652	1.83785	0.918926	−	0.394430i	$-0.129058\pi$
$751$	50.3652	1.83785	0.918926	+	0.394430i	$0.129058\pi$
$752$	−8.90264	−0.324646
$753$	0	0
$754$	−45.9589	−1.67373
$755$	−31.9129	−1.16143
$756$	0	0
$757$	−10.0813	−0.366411	−0.183206	−	0.983075i	$-0.558647\pi$
$757$	−10.0813	−0.366411	−0.183206	+	0.983075i	$0.558647\pi$
$758$	46.8193	1.70055
$759$	0	0
$760$	−13.2593	−0.480965
$761$	−30.8685	−1.11898	−0.559491	−	0.828837i	$-0.689003\pi$
$761$	−30.8685	−1.11898	−0.559491	+	0.828837i	$0.689003\pi$
$762$	0	0
$763$	0.780448	0.0282541
$764$	−35.9678	−1.30127
$765$	0	0
$766$	−18.4980	−0.668359
$767$	31.4707	1.13634
$768$	0	0
$769$	−7.84894	−0.283040	−0.141520	−	0.989935i	$-0.545199\pi$
$769$	−7.84894	−0.283040	−0.141520	+	0.989935i	$0.545199\pi$
$770$	5.83055	0.210119
$771$	0	0
$772$	5.78899	0.208350
$773$	−1.48989	−0.0535877	−0.0267939	−	0.999641i	$-0.508530\pi$
$773$	−1.48989	−0.0535877	−0.0267939	+	0.999641i	$0.508530\pi$
$774$	0	0
$775$	11.4503	0.411307
$776$	−2.85479	−0.102481
$777$	0	0
$778$	−7.63191	−0.273617
$779$	−75.6751	−2.71134
$780$	0	0
$781$	29.9767	1.07265
$782$	63.0110	2.25327
$783$	0	0
$784$	−4.87986	−0.174281
$785$	17.8530	0.637202
$786$	0	0
$787$	50.1610	1.78805	0.894023	−	0.448022i	$-0.147871\pi$
$787$	50.1610	1.78805	0.894023	+	0.448022i	$0.147871\pi$
$788$	−16.6332	−0.592535
$789$	0	0
$790$	−14.5574	−0.517927
$791$	13.1911	0.469022
$792$	0	0
$793$	28.4463	1.01016
$794$	6.48473	0.230134
$795$	0	0
$796$	1.96585	0.0696777
$797$	20.0436	0.709979	0.354989	−	0.934870i	$-0.384484\pi$
$797$	20.0436	0.709979	0.354989	+	0.934870i	$0.384484\pi$
$798$	0	0
$799$	−11.2683	−0.398643
$800$	18.8419	0.666161
$801$	0	0
$802$	−13.0985	−0.462524
$803$	0.779253	0.0274992
$804$	0	0
$805$	8.11471	0.286006
$806$	−45.7055	−1.60991
$807$	0	0
$808$	−4.48795	−0.157886
$809$	20.6385	0.725612	0.362806	−	0.931865i	$-0.381819\pi$
$809$	20.6385	0.725612	0.362806	+	0.931865i	$0.381819\pi$
$810$	0	0
$811$	−47.5447	−1.66952	−0.834760	−	0.550615i	$-0.814393\pi$
$811$	−47.5447	−1.66952	−0.834760	+	0.550615i	$0.814393\pi$
$812$	5.37874	0.188757
$813$	0	0
$814$	7.85172	0.275203
$815$	−34.3056	−1.20167
$816$	0	0
$817$	−57.4260	−2.00908
$818$	−38.2115	−1.33603
$819$	0	0
$820$	−19.4518	−0.679287
$821$	27.2365	0.950559	0.475280	−	0.879835i	$-0.342347\pi$
$821$	27.2365	0.950559	0.475280	+	0.879835i	$0.342347\pi$
$822$	0	0
$823$	−0.662707	−0.0231005	−0.0115503	−	0.999933i	$-0.503677\pi$
$823$	−0.662707	−0.0231005	−0.0115503	+	0.999933i	$0.503677\pi$
$824$	−2.28636	−0.0796492
$825$	0	0
$826$	−9.15336	−0.318486
$827$	−15.3219	−0.532794	−0.266397	−	0.963863i	$-0.585833\pi$
$827$	−15.3219	−0.532794	−0.266397	+	0.963863i	$0.585833\pi$
$828$	0	0
$829$	9.05283	0.314418	0.157209	−	0.987565i	$-0.449750\pi$
$829$	9.05283	0.314418	0.157209	+	0.987565i	$0.449750\pi$
$830$	1.18301	0.0410627
$831$	0	0
$832$	−13.8245	−0.479279
$833$	−6.17655	−0.214005
$834$	0	0
$835$	0.837825	0.0289941
$836$	−22.4845	−0.777642
$837$	0	0
$838$	5.92036	0.204515
$839$	27.5086	0.949703	0.474851	−	0.880066i	$-0.342502\pi$
$839$	27.5086	0.949703	0.474851	+	0.880066i	$0.342502\pi$
$840$	0	0
$841$	−13.0457	−0.449851
$842$	44.5370	1.53485
$843$	0	0
$844$	−31.3224	−1.07816
$845$	38.6485	1.32955
$846$	0	0
$847$	6.20262	0.213124
$848$	−13.2004	−0.453304
$849$	0	0
$850$	32.5707	1.11717
$851$	10.9277	0.374596
$852$	0	0
$853$	25.0679	0.858309	0.429155	−	0.903231i	$-0.358812\pi$
$853$	25.0679	0.858309	0.429155	+	0.903231i	$0.358812\pi$
$854$	−8.27370	−0.283120
$855$	0	0
$856$	11.0229	0.376755
$857$	−31.5604	−1.07808	−0.539041	−	0.842279i	$-0.681213\pi$
$857$	−31.5604	−1.07808	−0.539041	+	0.842279i	$0.681213\pi$
$858$	0	0
$859$	−23.9525	−0.817247	−0.408624	−	0.912703i	$-0.633991\pi$
$859$	−23.9525	−0.817247	−0.408624	+	0.912703i	$0.633991\pi$
$860$	−14.7610	−0.503346
$861$	0	0
$862$	69.5415	2.36859
$863$	−17.6675	−0.601409	−0.300705	−	0.953717i	$-0.597222\pi$
$863$	−17.6675	−0.601409	−0.300705	+	0.953717i	$0.597222\pi$
$864$	0	0
$865$	22.9667	0.780890
$866$	6.87826	0.233733
$867$	0	0
$868$	5.34908	0.181560
$869$	11.9778	0.406319
$870$	0	0
$871$	25.3509	0.858984
$872$	0.932868	0.0315909
$873$	0	0
$874$	−77.7695	−2.63059
$875$	11.4702	0.387765
$876$	0	0
$877$	−38.4371	−1.29793	−0.648964	−	0.760819i	$-0.724797\pi$
$877$	−38.4371	−1.29793	−0.648964	+	0.760819i	$0.724797\pi$
$878$	6.62257	0.223501
$879$	0	0
$880$	15.5530	0.524293
$881$	−53.2171	−1.79293	−0.896465	−	0.443115i	$-0.853873\pi$
$881$	−53.2171	−1.79293	−0.896465	+	0.443115i	$0.853873\pi$
$882$	0	0
$883$	−30.2549	−1.01816	−0.509079	−	0.860720i	$-0.670014\pi$
$883$	−30.2549	−1.01816	−0.509079	+	0.860720i	$0.670014\pi$
$884$	−52.3137	−1.75950
$885$	0	0
$886$	40.3603	1.35593
$887$	23.1393	0.776943	0.388472	−	0.921461i	$-0.373003\pi$
$887$	23.1393	0.776943	0.388472	+	0.921461i	$0.373003\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	18.4367	0.617999
$891$	0	0
$892$	−5.50386	−0.184283
$893$	13.9075	0.465398
$894$	0	0
$895$	3.94114	0.131738
$896$	−9.05209	−0.302409
$897$	0	0
$898$	9.54489	0.318517
$899$	15.8664	0.529173
$900$	0	0
$901$	−16.7081	−0.556626
$902$	39.7757	1.32439
$903$	0	0
$904$	15.7673	0.524413
$905$	−21.3335	−0.709150
$906$	0	0
$907$	−51.2332	−1.70117	−0.850585	−	0.525838i	$-0.823752\pi$
$907$	−51.2332	−1.70117	−0.850585	+	0.525838i	$0.823752\pi$
$908$	−32.2831	−1.07135
$909$	0	0
$910$	−16.7431	−0.555028
$911$	−42.9037	−1.42146	−0.710731	−	0.703464i	$-0.751636\pi$
$911$	−42.9037	−1.42146	−0.710731	+	0.703464i	$0.751636\pi$
$912$	0	0
$913$	−0.973378	−0.0322141
$914$	61.5712	2.03660
$915$	0	0
$916$	−23.1892	−0.766194
$917$	−20.8994	−0.690159
$918$	0	0
$919$	19.9703	0.658759	0.329379	−	0.944198i	$-0.393160\pi$
$919$	19.9703	0.658759	0.329379	+	0.944198i	$0.393160\pi$
$920$	9.69949	0.319783
$921$	0	0
$922$	36.5662	1.20424
$923$	−86.0813	−2.83340
$924$	0	0
$925$	5.64858	0.185724
$926$	17.7720	0.584023
$927$	0	0
$928$	26.1086	0.857058
$929$	−5.81386	−0.190747	−0.0953734	−	0.995442i	$-0.530404\pi$
$929$	−5.81386	−0.190747	−0.0953734	+	0.995442i	$0.530404\pi$
$930$	0	0
$931$	7.62323	0.249841
$932$	−20.4505	−0.669880
$933$	0	0
$934$	30.8947	1.01091
$935$	19.6858	0.643795
$936$	0	0
$937$	32.4003	1.05847	0.529236	−	0.848474i	$-0.322478\pi$
$937$	32.4003	1.05847	0.529236	+	0.848474i	$0.322478\pi$
$938$	−7.37341	−0.240750
$939$	0	0
$940$	3.57485	0.116599
$941$	−11.9726	−0.390295	−0.195147	−	0.980774i	$-0.562519\pi$
$941$	−11.9726	−0.390295	−0.195147	+	0.980774i	$0.562519\pi$
$942$	0	0
$943$	55.3581	1.80271
$944$	−24.4166	−0.794693
$945$	0	0
$946$	30.1838	0.981361
$947$	−42.5720	−1.38341	−0.691703	−	0.722183i	$-0.743139\pi$
$947$	−42.5720	−1.38341	−0.691703	+	0.722183i	$0.743139\pi$
$948$	0	0
$949$	−2.23771	−0.0726392
$950$	−40.1995	−1.30424
$951$	0	0
$952$	−7.38282	−0.239279
$953$	29.9119	0.968940	0.484470	−	0.874808i	$-0.339012\pi$
$953$	29.9119	0.968940	0.484470	+	0.874808i	$0.339012\pi$
$954$	0	0
$955$	−38.8667	−1.25770
$956$	16.2085	0.524221
$957$	0	0
$958$	43.6248	1.40945
$959$	−11.4265	−0.368982
$960$	0	0
$961$	−15.2211	−0.491004
$962$	−22.5471	−0.726948
$963$	0	0
$964$	−7.96391	−0.256500
$965$	6.25557	0.201374
$966$	0	0
$967$	6.13602	0.197321	0.0986606	−	0.995121i	$-0.468544\pi$
$967$	6.13602	0.197321	0.0986606	+	0.995121i	$0.468544\pi$
$968$	7.41397	0.238294
$969$	0	0
$970$	6.35777	0.204136
$971$	−33.4056	−1.07204	−0.536018	−	0.844207i	$-0.680072\pi$
$971$	−33.4056	−1.07204	−0.536018	+	0.844207i	$0.680072\pi$
$972$	0	0
$973$	−18.1243	−0.581039
$974$	−14.8729	−0.476558
$975$	0	0
$976$	−22.0701	−0.706448
$977$	24.2129	0.774639	0.387319	−	0.921946i	$-0.373401\pi$
$977$	24.2129	0.774639	0.387319	+	0.921946i	$0.373401\pi$
$978$	0	0
$979$	−15.1697	−0.484826
$980$	1.95951	0.0625941
$981$	0	0
$982$	−28.4984	−0.909420
$983$	10.3817	0.331125	0.165563	−	0.986199i	$-0.447056\pi$
$983$	10.3817	0.331125	0.165563	+	0.986199i	$0.447056\pi$
$984$	0	0
$985$	−17.9738	−0.572694
$986$	45.1324	1.43731
$987$	0	0
$988$	64.5667	2.05414
$989$	42.0085	1.33579
$990$	0	0
$991$	34.2360	1.08754	0.543771	−	0.839233i	$-0.316996\pi$
$991$	34.2360	1.08754	0.543771	+	0.839233i	$0.316996\pi$
$992$	25.9647	0.824380
$993$	0	0
$994$	25.0371	0.794127
$995$	2.12429	0.0673446
$996$	0	0
$997$	−16.6391	−0.526966	−0.263483	−	0.964664i	$-0.584871\pi$
$997$	−16.6391	−0.526966	−0.263483	+	0.964664i	$0.584871\pi$
$998$	−22.5737	−0.714557
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.r.1.13		16
3.2	odd	2		2667.2.a.o.1.4	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.o.1.4	✓	16	3.2	odd	2
8001.2.a.r.1.13		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)

\(f(q)\)	\(=\)	\(q+1.82937 q^{2} +1.34661 q^{4} +1.45514 q^{5} -1.00000 q^{7} -1.19530 q^{8} +O(q^{10})\)	\(q+1.82937 q^{2} +1.34661 q^{4} +1.45514 q^{5} -1.00000 q^{7} -1.19530 q^{8} +2.66200 q^{10} -2.19029 q^{11} +6.28967 q^{13} -1.82937 q^{14} -4.87986 q^{16} -6.17655 q^{17} +7.62323 q^{19} +1.95951 q^{20} -4.00686 q^{22} -5.57657 q^{23} -2.88256 q^{25} +11.5062 q^{26} -1.34661 q^{28} -3.99429 q^{29} -3.97226 q^{31} -6.53650 q^{32} -11.2992 q^{34} -1.45514 q^{35} -1.95957 q^{37} +13.9457 q^{38} -1.73933 q^{40} -9.92690 q^{41} -7.53303 q^{43} -2.94947 q^{44} -10.2016 q^{46} +1.82436 q^{47} +1.00000 q^{49} -5.27329 q^{50} +8.46972 q^{52} +2.70508 q^{53} -3.18719 q^{55} +1.19530 q^{56} -7.30704 q^{58} +5.00355 q^{59} +4.52270 q^{61} -7.26675 q^{62} -2.19797 q^{64} +9.15236 q^{65} +4.03056 q^{67} -8.31740 q^{68} -2.66200 q^{70} -13.6861 q^{71} -0.355776 q^{73} -3.58478 q^{74} +10.2655 q^{76} +2.19029 q^{77} -5.46858 q^{79} -7.10089 q^{80} -18.1600 q^{82} +0.444405 q^{83} -8.98776 q^{85} -13.7807 q^{86} +2.61805 q^{88} +6.92589 q^{89} -6.28967 q^{91} -7.50946 q^{92} +3.33744 q^{94} +11.0929 q^{95} +2.38835 q^{97} +1.82937 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8}+O(q^{10}) \) 16 * q - 5 * q^2 + 19 * q^4 + q^5 - 16 * q^7 - 6 * q^8	\( 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8} - 12 q^{10} - 11 q^{11} + 18 q^{13} + 5 q^{14} + 25 q^{16} + 5 q^{17} - 11 q^{19} + q^{20} + q^{22} - 13 q^{23} + 33 q^{25} - 8 q^{26} - 19 q^{28} - 24 q^{29} - 42 q^{31} - 42 q^{32} + 9 q^{34} - q^{35} + 40 q^{37} - 38 q^{38} - 61 q^{40} - 9 q^{41} + 7 q^{43} - 3 q^{44} + 24 q^{46} - 31 q^{47} + 16 q^{49} - 6 q^{50} + 52 q^{52} - 66 q^{53} - 36 q^{55} + 6 q^{56} + 19 q^{58} + 7 q^{59} + 6 q^{61} - 52 q^{62} + 10 q^{64} - 51 q^{65} + 16 q^{67} - 14 q^{68} + 12 q^{70} - 46 q^{71} + 39 q^{73} - 72 q^{74} + 24 q^{76} + 11 q^{77} + 4 q^{79} + 2 q^{80} - 18 q^{82} - 15 q^{83} - 4 q^{85} - 14 q^{86} + 58 q^{88} + q^{89} - 18 q^{91} - 26 q^{92} + 5 q^{94} - 44 q^{95} + 41 q^{97} - 5 q^{98}+O(q^{100}) \) 16 * q - 5 * q^2 + 19 * q^4 + q^5 - 16 * q^7 - 6 * q^8 - 12 * q^10 - 11 * q^11 + 18 * q^13 + 5 * q^14 + 25 * q^16 + 5 * q^17 - 11 * q^19 + q^20 + q^22 - 13 * q^23 + 33 * q^25 - 8 * q^26 - 19 * q^28 - 24 * q^29 - 42 * q^31 - 42 * q^32 + 9 * q^34 - q^35 + 40 * q^37 - 38 * q^38 - 61 * q^40 - 9 * q^41 + 7 * q^43 - 3 * q^44 + 24 * q^46 - 31 * q^47 + 16 * q^49 - 6 * q^50 + 52 * q^52 - 66 * q^53 - 36 * q^55 + 6 * q^56 + 19 * q^58 + 7 * q^59 + 6 * q^61 - 52 * q^62 + 10 * q^64 - 51 * q^65 + 16 * q^67 - 14 * q^68 + 12 * q^70 - 46 * q^71 + 39 * q^73 - 72 * q^74 + 24 * q^76 + 11 * q^77 + 4 * q^79 + 2 * q^80 - 18 * q^82 - 15 * q^83 - 4 * q^85 - 14 * q^86 + 58 * q^88 + q^89 - 18 * q^91 - 26 * q^92 + 5 * q^94 - 44 * q^95 + 41 * q^97 - 5 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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