LMFDB - Embedded newform 8001.2.a.v.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:	$0$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 4 x^{18} - 22 x^{17} + 101 x^{16} + 178 x^{15} - 1035 x^{14} - 583 x^{13} + 5572 x^{12} + \cdots + 210 $ x^19 - 4x^18 - 22x^17 + 101x^16 + 178x^15 - 1035x^14 - 583x^13 + 5572x^12 + 21x^11 - 17032x^10 + 4985x^9 + 29792x^8 - 13249x^7 - 28600x^6 + 14000x^5 + 13725x^4 - 5723x^3 - 2913x^2 + 608x + 210
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.88777$ 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)

$f(q)$	$=$	$q-1.88777 q^{2} +1.56369 q^{4} +3.95777 q^{5} +1.00000 q^{7} +0.823654 q^{8} +O(q^{10})$	\(q-1.88777 q^{2} +1.56369 q^{4} +3.95777 q^{5} +1.00000 q^{7} +0.823654 q^{8} -7.47138 q^{10} -2.14742 q^{11} +4.63999 q^{13} -1.88777 q^{14} -4.68225 q^{16} +2.02948 q^{17} +5.69266 q^{19} +6.18873 q^{20} +4.05384 q^{22} +1.45406 q^{23} +10.6640 q^{25} -8.75925 q^{26} +1.56369 q^{28} +7.67597 q^{29} +9.00443 q^{31} +7.19173 q^{32} -3.83120 q^{34} +3.95777 q^{35} +7.50700 q^{37} -10.7465 q^{38} +3.25984 q^{40} -1.70524 q^{41} +11.2877 q^{43} -3.35790 q^{44} -2.74493 q^{46} -10.1989 q^{47} +1.00000 q^{49} -20.1312 q^{50} +7.25551 q^{52} -7.66310 q^{53} -8.49900 q^{55} +0.823654 q^{56} -14.4905 q^{58} +3.85596 q^{59} -7.14847 q^{61} -16.9983 q^{62} -4.21185 q^{64} +18.3640 q^{65} -5.67445 q^{67} +3.17348 q^{68} -7.47138 q^{70} +6.26208 q^{71} -9.32720 q^{73} -14.1715 q^{74} +8.90157 q^{76} -2.14742 q^{77} +8.47237 q^{79} -18.5313 q^{80} +3.21911 q^{82} -6.28388 q^{83} +8.03222 q^{85} -21.3087 q^{86} -1.76873 q^{88} -10.4894 q^{89} +4.63999 q^{91} +2.27370 q^{92} +19.2533 q^{94} +22.5303 q^{95} -1.36340 q^{97} -1.88777 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8}+O(q^{10}) $ 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8	$ 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8} + 9 q^{11} + 24 q^{13} - 4 q^{14} + 20 q^{16} - 17 q^{17} + 23 q^{19} - 5 q^{20} - 3 q^{22} + 17 q^{23} + 38 q^{25} - 28 q^{26} + 22 q^{28} - 2 q^{29} + 16 q^{31} - 17 q^{32} + 29 q^{34} - 5 q^{35} + 56 q^{37} - 2 q^{38} - 13 q^{40} + 7 q^{41} + 19 q^{43} + 29 q^{44} + 10 q^{46} - 25 q^{47} + 19 q^{49} + 9 q^{50} + 16 q^{52} - 18 q^{53} + 10 q^{55} - 9 q^{56} + 31 q^{58} - 11 q^{59} + 26 q^{61} - 26 q^{62} + 45 q^{64} - 27 q^{65} + 24 q^{67} - 14 q^{68} + 32 q^{71} + 51 q^{73} + 12 q^{76} + 9 q^{77} + 30 q^{79} + 30 q^{80} - 52 q^{82} - q^{83} + 44 q^{85} + 24 q^{86} - 30 q^{88} - 5 q^{89} + 24 q^{91} + 88 q^{92} + 7 q^{94} + 24 q^{95} + 5 q^{97} - 4 q^{98}+O(q^{100}) $ 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8 + 9 * q^11 + 24 * q^13 - 4 * q^14 + 20 * q^16 - 17 * q^17 + 23 * q^19 - 5 * q^20 - 3 * q^22 + 17 * q^23 + 38 * q^25 - 28 * q^26 + 22 * q^28 - 2 * q^29 + 16 * q^31 - 17 * q^32 + 29 * q^34 - 5 * q^35 + 56 * q^37 - 2 * q^38 - 13 * q^40 + 7 * q^41 + 19 * q^43 + 29 * q^44 + 10 * q^46 - 25 * q^47 + 19 * q^49 + 9 * q^50 + 16 * q^52 - 18 * q^53 + 10 * q^55 - 9 * q^56 + 31 * q^58 - 11 * q^59 + 26 * q^61 - 26 * q^62 + 45 * q^64 - 27 * q^65 + 24 * q^67 - 14 * q^68 + 32 * q^71 + 51 * q^73 + 12 * q^76 + 9 * q^77 + 30 * q^79 + 30 * q^80 - 52 * q^82 - q^83 + 44 * q^85 + 24 * q^86 - 30 * q^88 - 5 * q^89 + 24 * q^91 + 88 * q^92 + 7 * q^94 + 24 * q^95 + 5 * q^97 - 4 * 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 $

$2$	−1.88777	−1.33486	−0.667429	−	0.744674i	$-0.732605\pi$
$2$	−1.88777	−1.33486	−0.667429	+	0.744674i	$0.732605\pi$
$3$	0	0
$3$	0	0
$4$	1.56369	0.781845
$5$	3.95777	1.76997	0.884985	−	0.465619i	$-0.154168\pi$
$5$	3.95777	1.76997	0.884985	+	0.465619i	$0.154168\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0.823654	0.291206
$9$	0	0
$10$	−7.47138	−2.36266
$11$	−2.14742	−0.647471	−0.323736	−	0.946148i	$-0.604939\pi$
$11$	−2.14742	−0.647471	−0.323736	+	0.946148i	$0.604939\pi$
$12$	0	0
$13$	4.63999	1.28690	0.643451	−	0.765488i	$-0.277502\pi$
$13$	4.63999	1.28690	0.643451	+	0.765488i	$0.277502\pi$
$14$	−1.88777	−0.504529
$15$	0	0
$16$	−4.68225	−1.17056
$17$	2.02948	0.492221	0.246111	−	0.969242i	$-0.420847\pi$
$17$	2.02948	0.492221	0.246111	+	0.969242i	$0.420847\pi$
$18$	0	0
$19$	5.69266	1.30599	0.652993	−	0.757364i	$-0.273513\pi$
$19$	5.69266	1.30599	0.652993	+	0.757364i	$0.273513\pi$
$20$	6.18873	1.38384
$21$	0	0
$22$	4.05384	0.864282
$23$	1.45406	0.303192	0.151596	−	0.988443i	$-0.451559\pi$
$23$	1.45406	0.303192	0.151596	+	0.988443i	$0.451559\pi$
$24$	0	0
$25$	10.6640	2.13280
$26$	−8.75925	−1.71783
$27$	0	0
$28$	1.56369	0.295510
$29$	7.67597	1.42539	0.712696	−	0.701473i	$-0.247474\pi$
$29$	7.67597	1.42539	0.712696	+	0.701473i	$0.247474\pi$
$30$	0	0
$31$	9.00443	1.61724	0.808622	−	0.588329i	$-0.200214\pi$
$31$	9.00443	1.61724	0.808622	+	0.588329i	$0.200214\pi$
$32$	7.19173	1.27133
$33$	0	0
$34$	−3.83120	−0.657045
$35$	3.95777	0.668986
$36$	0	0
$37$	7.50700	1.23414	0.617071	−	0.786907i	$-0.288319\pi$
$37$	7.50700	1.23414	0.617071	+	0.786907i	$0.288319\pi$
$38$	−10.7465	−1.74331
$39$	0	0
$40$	3.25984	0.515425
$41$	−1.70524	−0.266314	−0.133157	−	0.991095i	$-0.542511\pi$
$41$	−1.70524	−0.266314	−0.133157	+	0.991095i	$0.542511\pi$
$42$	0	0
$43$	11.2877	1.72136	0.860681	−	0.509144i	$-0.170038\pi$
$43$	11.2877	1.72136	0.860681	+	0.509144i	$0.170038\pi$
$44$	−3.35790	−0.506222
$45$	0	0
$46$	−2.74493	−0.404719
$47$	−10.1989	−1.48767	−0.743835	−	0.668363i	$-0.766995\pi$
$47$	−10.1989	−1.48767	−0.743835	+	0.668363i	$0.766995\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−20.1312	−2.84698
$51$	0	0
$52$	7.25551	1.00616
$53$	−7.66310	−1.05261	−0.526304	−	0.850297i	$-0.676423\pi$
$53$	−7.66310	−1.05261	−0.526304	+	0.850297i	$0.676423\pi$
$54$	0	0
$55$	−8.49900	−1.14601
$56$	0.823654	0.110065
$57$	0	0
$58$	−14.4905	−1.90270
$59$	3.85596	0.502004	0.251002	−	0.967987i	$-0.419240\pi$
$59$	3.85596	0.502004	0.251002	+	0.967987i	$0.419240\pi$
$60$	0	0
$61$	−7.14847	−0.915268	−0.457634	−	0.889141i	$-0.651303\pi$
$61$	−7.14847	−0.915268	−0.457634	+	0.889141i	$0.651303\pi$
$62$	−16.9983	−2.15879
$63$	0	0
$64$	−4.21185	−0.526481
$65$	18.3640	2.27778
$66$	0	0
$67$	−5.67445	−0.693245	−0.346622	−	0.938005i	$-0.612671\pi$
$67$	−5.67445	−0.693245	−0.346622	+	0.938005i	$0.612671\pi$
$68$	3.17348	0.384841
$69$	0	0
$70$	−7.47138	−0.893001
$71$	6.26208	0.743172	0.371586	−	0.928398i	$-0.378814\pi$
$71$	6.26208	0.743172	0.371586	+	0.928398i	$0.378814\pi$
$72$	0	0
$73$	−9.32720	−1.09167	−0.545833	−	0.837894i	$-0.683787\pi$
$73$	−9.32720	−1.09167	−0.545833	+	0.837894i	$0.683787\pi$
$74$	−14.1715	−1.64741
$75$	0	0
$76$	8.90157	1.02108
$77$	−2.14742	−0.244721
$78$	0	0
$79$	8.47237	0.953216	0.476608	−	0.879116i	$-0.341866\pi$
$79$	8.47237	0.953216	0.476608	+	0.879116i	$0.341866\pi$
$80$	−18.5313	−2.07186
$81$	0	0
$82$	3.21911	0.355491
$83$	−6.28388	−0.689746	−0.344873	−	0.938649i	$-0.612078\pi$
$83$	−6.28388	−0.689746	−0.344873	+	0.938649i	$0.612078\pi$
$84$	0	0
$85$	8.03222	0.871217
$86$	−21.3087	−2.29777
$87$	0	0
$88$	−1.76873	−0.188547
$89$	−10.4894	−1.11187	−0.555935	−	0.831226i	$-0.687640\pi$
$89$	−10.4894	−1.11187	−0.555935	+	0.831226i	$0.687640\pi$
$90$	0	0
$91$	4.63999	0.486403
$92$	2.27370	0.237049
$93$	0	0
$94$	19.2533	1.98583
$95$	22.5303	2.31156
$96$	0	0
$97$	−1.36340	−0.138432	−0.0692159	−	0.997602i	$-0.522050\pi$
$97$	−1.36340	−0.138432	−0.0692159	+	0.997602i	$0.522050\pi$
$98$	−1.88777	−0.190694
$99$	0	0
$100$	16.6752	1.66752
$101$	−11.4803	−1.14233	−0.571165	−	0.820835i	$-0.693508\pi$
$101$	−11.4803	−1.14233	−0.571165	+	0.820835i	$0.693508\pi$
$102$	0	0
$103$	8.90018	0.876961	0.438480	−	0.898741i	$-0.355517\pi$
$103$	8.90018	0.876961	0.438480	+	0.898741i	$0.355517\pi$
$104$	3.82174	0.374753
$105$	0	0
$106$	14.4662	1.40508
$107$	19.2178	1.85786	0.928929	−	0.370258i	$-0.120731\pi$
$107$	19.2178	1.85786	0.928929	+	0.370258i	$0.120731\pi$
$108$	0	0
$109$	−16.3725	−1.56820	−0.784101	−	0.620633i	$-0.786876\pi$
$109$	−16.3725	−1.56820	−0.784101	+	0.620633i	$0.786876\pi$
$110$	16.0442	1.52975
$111$	0	0
$112$	−4.68225	−0.442431
$113$	11.4751	1.07948	0.539742	−	0.841830i	$-0.318522\pi$
$113$	11.4751	1.07948	0.539742	+	0.841830i	$0.318522\pi$
$114$	0	0
$115$	5.75484	0.536641
$116$	12.0028	1.11444
$117$	0	0
$118$	−7.27919	−0.670104
$119$	2.02948	0.186042
$120$	0	0
$121$	−6.38859	−0.580781
$122$	13.4947	1.22175
$123$	0	0
$124$	14.0801	1.26443
$125$	22.4168	2.00502
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−6.43243	−0.568552
$129$	0	0
$130$	−34.6671	−3.04051
$131$	−16.8610	−1.47315	−0.736577	−	0.676354i	$-0.763559\pi$
$131$	−16.8610	−1.47315	−0.736577	+	0.676354i	$0.763559\pi$
$132$	0	0
$133$	5.69266	0.493617
$134$	10.7121	0.925383
$135$	0	0
$136$	1.67159	0.143338
$137$	19.1796	1.63862	0.819311	−	0.573350i	$-0.194356\pi$
$137$	19.1796	1.63862	0.819311	+	0.573350i	$0.194356\pi$
$138$	0	0
$139$	−8.01183	−0.679554	−0.339777	−	0.940506i	$-0.610352\pi$
$139$	−8.01183	−0.679554	−0.339777	+	0.940506i	$0.610352\pi$
$140$	6.18873	0.523044
$141$	0	0
$142$	−11.8214	−0.992029
$143$	−9.96400	−0.833232
$144$	0	0
$145$	30.3798	2.52290
$146$	17.6076	1.45722
$147$	0	0
$148$	11.7386	0.964909
$149$	−13.1891	−1.08049	−0.540246	−	0.841507i	$-0.681669\pi$
$149$	−13.1891	−1.08049	−0.540246	+	0.841507i	$0.681669\pi$
$150$	0	0
$151$	13.0804	1.06447	0.532233	−	0.846598i	$-0.321353\pi$
$151$	13.0804	1.06447	0.532233	+	0.846598i	$0.321353\pi$
$152$	4.68878	0.380311
$153$	0	0
$154$	4.05384	0.326668
$155$	35.6375	2.86247
$156$	0	0
$157$	−4.19203	−0.334560	−0.167280	−	0.985909i	$-0.553498\pi$
$157$	−4.19203	−0.334560	−0.167280	+	0.985909i	$0.553498\pi$
$158$	−15.9939	−1.27241
$159$	0	0
$160$	28.4632	2.25022
$161$	1.45406	0.114596
$162$	0	0
$163$	−15.0370	−1.17779	−0.588895	−	0.808210i	$-0.700437\pi$
$163$	−15.0370	−1.17779	−0.588895	+	0.808210i	$0.700437\pi$
$164$	−2.66647	−0.208216
$165$	0	0
$166$	11.8625	0.920712
$167$	−9.25065	−0.715837	−0.357919	−	0.933753i	$-0.616513\pi$
$167$	−9.25065	−0.715837	−0.357919	+	0.933753i	$0.616513\pi$
$168$	0	0
$169$	8.52949	0.656115
$170$	−15.1630	−1.16295
$171$	0	0
$172$	17.6505	1.34584
$173$	−7.57346	−0.575800	−0.287900	−	0.957661i	$-0.592957\pi$
$173$	−7.57346	−0.575800	−0.287900	+	0.957661i	$0.592957\pi$
$174$	0	0
$175$	10.6640	0.806121
$176$	10.0548	0.757906
$177$	0	0
$178$	19.8015	1.48419
$179$	−7.28211	−0.544290	−0.272145	−	0.962256i	$-0.587733\pi$
$179$	−7.28211	−0.544290	−0.272145	+	0.962256i	$0.587733\pi$
$180$	0	0
$181$	−23.4346	−1.74188	−0.870941	−	0.491388i	$-0.836490\pi$
$181$	−23.4346	−1.74188	−0.870941	+	0.491388i	$0.836490\pi$
$182$	−8.75925	−0.649279
$183$	0	0
$184$	1.19764	0.0882913
$185$	29.7110	2.18440
$186$	0	0
$187$	−4.35814	−0.318699
$188$	−15.9480	−1.16313
$189$	0	0
$190$	−42.5321	−3.08560
$191$	−15.0684	−1.09031	−0.545157	−	0.838334i	$-0.683530\pi$
$191$	−15.0684	−1.09031	−0.545157	+	0.838334i	$0.683530\pi$
$192$	0	0
$193$	−22.0680	−1.58849	−0.794243	−	0.607600i	$-0.792133\pi$
$193$	−22.0680	−1.58849	−0.794243	+	0.607600i	$0.792133\pi$
$194$	2.57378	0.184787
$195$	0	0
$196$	1.56369	0.111692
$197$	−5.41669	−0.385923	−0.192962	−	0.981206i	$-0.561809\pi$
$197$	−5.41669	−0.385923	−0.192962	+	0.981206i	$0.561809\pi$
$198$	0	0
$199$	11.5150	0.816279	0.408140	−	0.912919i	$-0.366178\pi$
$199$	11.5150	0.816279	0.408140	+	0.912919i	$0.366178\pi$
$200$	8.78343	0.621082
$201$	0	0
$202$	21.6722	1.52485
$203$	7.67597	0.538748
$204$	0	0
$205$	−6.74897	−0.471368
$206$	−16.8015	−1.17062
$207$	0	0
$208$	−21.7256	−1.50640
$209$	−12.2245	−0.845589
$210$	0	0
$211$	5.20139	0.358079	0.179039	−	0.983842i	$-0.442701\pi$
$211$	5.20139	0.358079	0.179039	+	0.983842i	$0.442701\pi$
$212$	−11.9827	−0.822976
$213$	0	0
$214$	−36.2789	−2.47998
$215$	44.6743	3.04676
$216$	0	0
$217$	9.00443	0.611261
$218$	30.9076	2.09333
$219$	0	0
$220$	−13.2898	−0.895999
$221$	9.41676	0.633440
$222$	0	0
$223$	21.3493	1.42965	0.714827	−	0.699301i	$-0.246505\pi$
$223$	21.3493	1.42965	0.714827	+	0.699301i	$0.246505\pi$
$224$	7.19173	0.480518
$225$	0	0
$226$	−21.6623	−1.44096
$227$	6.08058	0.403582	0.201791	−	0.979429i	$-0.435324\pi$
$227$	6.08058	0.403582	0.201791	+	0.979429i	$0.435324\pi$
$228$	0	0
$229$	21.0455	1.39073	0.695364	−	0.718658i	$-0.255243\pi$
$229$	21.0455	1.39073	0.695364	+	0.718658i	$0.255243\pi$
$230$	−10.8638	−0.716340
$231$	0	0
$232$	6.32234	0.415082
$233$	−22.1764	−1.45282	−0.726411	−	0.687260i	$-0.758813\pi$
$233$	−22.1764	−1.45282	−0.726411	+	0.687260i	$0.758813\pi$
$234$	0	0
$235$	−40.3651	−2.63313
$236$	6.02953	0.392489
$237$	0	0
$238$	−3.83120	−0.248340
$239$	25.5967	1.65572	0.827858	−	0.560938i	$-0.189559\pi$
$239$	25.5967	1.65572	0.827858	+	0.560938i	$0.189559\pi$
$240$	0	0
$241$	−19.9316	−1.28391	−0.641954	−	0.766743i	$-0.721876\pi$
$241$	−19.9316	−1.28391	−0.641954	+	0.766743i	$0.721876\pi$
$242$	12.0602	0.775260
$243$	0	0
$244$	−11.1780	−0.715598
$245$	3.95777	0.252853
$246$	0	0
$247$	26.4139	1.68068
$248$	7.41653	0.470950
$249$	0	0
$250$	−42.3178	−2.67641
$251$	−10.1150	−0.638453	−0.319226	−	0.947679i	$-0.603423\pi$
$251$	−10.1150	−0.638453	−0.319226	+	0.947679i	$0.603423\pi$
$252$	0	0
$253$	−3.12248	−0.196308
$254$	−1.88777	−0.118449
$255$	0	0
$256$	20.5667	1.28542
$257$	−11.7107	−0.730492	−0.365246	−	0.930911i	$-0.619015\pi$
$257$	−11.7107	−0.730492	−0.365246	+	0.930911i	$0.619015\pi$
$258$	0	0
$259$	7.50700	0.466462
$260$	28.7157	1.78087
$261$	0	0
$262$	31.8298	1.96645
$263$	11.8920	0.733292	0.366646	−	0.930360i	$-0.380506\pi$
$263$	11.8920	0.733292	0.366646	+	0.930360i	$0.380506\pi$
$264$	0	0
$265$	−30.3288	−1.86308
$266$	−10.7465	−0.658908
$267$	0	0
$268$	−8.87309	−0.542010
$269$	3.47050	0.211600	0.105800	−	0.994387i	$-0.466260\pi$
$269$	3.47050	0.211600	0.105800	+	0.994387i	$0.466260\pi$
$270$	0	0
$271$	−30.0603	−1.82603	−0.913015	−	0.407926i	$-0.866252\pi$
$271$	−30.0603	−1.82603	−0.913015	+	0.407926i	$0.866252\pi$
$272$	−9.50254	−0.576176
$273$	0	0
$274$	−36.2067	−2.18733
$275$	−22.9000	−1.38092
$276$	0	0
$277$	−27.2744	−1.63876	−0.819380	−	0.573251i	$-0.805682\pi$
$277$	−27.2744	−1.63876	−0.819380	+	0.573251i	$0.805682\pi$
$278$	15.1245	0.907108
$279$	0	0
$280$	3.25984	0.194812
$281$	17.7977	1.06172	0.530862	−	0.847459i	$-0.321868\pi$
$281$	17.7977	1.06172	0.530862	+	0.847459i	$0.321868\pi$
$282$	0	0
$283$	4.85009	0.288308	0.144154	−	0.989555i	$-0.453954\pi$
$283$	4.85009	0.288308	0.144154	+	0.989555i	$0.453954\pi$
$284$	9.79196	0.581046
$285$	0	0
$286$	18.8098	1.11225
$287$	−1.70524	−0.100657
$288$	0	0
$289$	−12.8812	−0.757718
$290$	−57.3501	−3.36772
$291$	0	0
$292$	−14.5849	−0.853514
$293$	14.8955	0.870207	0.435103	−	0.900381i	$-0.356712\pi$
$293$	14.8955	0.870207	0.435103	+	0.900381i	$0.356712\pi$
$294$	0	0
$295$	15.2610	0.888532
$296$	6.18317	0.359389
$297$	0	0
$298$	24.8980	1.44230
$299$	6.74682	0.390179
$300$	0	0
$301$	11.2877	0.650614
$302$	−24.6928	−1.42091
$303$	0	0
$304$	−26.6545	−1.52874
$305$	−28.2920	−1.62000
$306$	0	0
$307$	−1.06872	−0.0609948	−0.0304974	−	0.999535i	$-0.509709\pi$
$307$	−1.06872	−0.0609948	−0.0304974	+	0.999535i	$0.509709\pi$
$308$	−3.35790	−0.191334
$309$	0	0
$310$	−67.2756	−3.82100
$311$	0.583290	0.0330754	0.0165377	−	0.999863i	$-0.494736\pi$
$311$	0.583290	0.0330754	0.0165377	+	0.999863i	$0.494736\pi$
$312$	0	0
$313$	−26.9052	−1.52077	−0.760385	−	0.649472i	$-0.774990\pi$
$313$	−26.9052	−1.52077	−0.760385	+	0.649472i	$0.774990\pi$
$314$	7.91360	0.446590
$315$	0	0
$316$	13.2482	0.745267
$317$	−9.34430	−0.524828	−0.262414	−	0.964955i	$-0.584519\pi$
$317$	−9.34430	−0.524828	−0.262414	+	0.964955i	$0.584519\pi$
$318$	0	0
$319$	−16.4835	−0.922901
$320$	−16.6696	−0.931856
$321$	0	0
$322$	−2.74493	−0.152969
$323$	11.5531	0.642834
$324$	0	0
$325$	49.4807	2.74470
$326$	28.3865	1.57218
$327$	0	0
$328$	−1.40453	−0.0775522
$329$	−10.1989	−0.562286
$330$	0	0
$331$	−1.03557	−0.0569200	−0.0284600	−	0.999595i	$-0.509060\pi$
$331$	−1.03557	−0.0569200	−0.0284600	+	0.999595i	$0.509060\pi$
$332$	−9.82605	−0.539274
$333$	0	0
$334$	17.4631	0.955541
$335$	−22.4582	−1.22702
$336$	0	0
$337$	−14.1889	−0.772918	−0.386459	−	0.922307i	$-0.626302\pi$
$337$	−14.1889	−0.772918	−0.386459	+	0.922307i	$0.626302\pi$
$338$	−16.1018	−0.875820
$339$	0	0
$340$	12.5599	0.681157
$341$	−19.3363	−1.04712
$342$	0	0
$343$	1.00000	0.0539949
$344$	9.29718	0.501270
$345$	0	0
$346$	14.2970	0.768611
$347$	−17.6484	−0.947416	−0.473708	−	0.880682i	$-0.657085\pi$
$347$	−17.6484	−0.947416	−0.473708	+	0.880682i	$0.657085\pi$
$348$	0	0
$349$	11.4412	0.612435	0.306218	−	0.951962i	$-0.400936\pi$
$349$	11.4412	0.612435	0.306218	+	0.951962i	$0.400936\pi$
$350$	−20.1312	−1.07606
$351$	0	0
$352$	−15.4437	−0.823150
$353$	20.3642	1.08388	0.541939	−	0.840418i	$-0.317691\pi$
$353$	20.3642	1.08388	0.541939	+	0.840418i	$0.317691\pi$
$354$	0	0
$355$	24.7839	1.31539
$356$	−16.4021	−0.869310
$357$	0	0
$358$	13.7470	0.726550
$359$	−3.45273	−0.182228	−0.0911140	−	0.995840i	$-0.529043\pi$
$359$	−3.45273	−0.182228	−0.0911140	+	0.995840i	$0.529043\pi$
$360$	0	0
$361$	13.4064	0.705602
$362$	44.2393	2.32516
$363$	0	0
$364$	7.25551	0.380292
$365$	−36.9150	−1.93222
$366$	0	0
$367$	34.3928	1.79529	0.897645	−	0.440718i	$-0.145276\pi$
$367$	34.3928	1.79529	0.897645	+	0.440718i	$0.145276\pi$
$368$	−6.80827	−0.354906
$369$	0	0
$370$	−56.0877	−2.91586
$371$	−7.66310	−0.397848
$372$	0	0
$373$	−1.96560	−0.101775	−0.0508874	−	0.998704i	$-0.516205\pi$
$373$	−1.96560	−0.101775	−0.0508874	+	0.998704i	$0.516205\pi$
$374$	8.22719	0.425418
$375$	0	0
$376$	−8.40040	−0.433218
$377$	35.6164	1.83434
$378$	0	0
$379$	−8.58920	−0.441197	−0.220599	−	0.975365i	$-0.570801\pi$
$379$	−8.58920	−0.441197	−0.220599	+	0.975365i	$0.570801\pi$
$380$	35.2304	1.80728
$381$	0	0
$382$	28.4458	1.45541
$383$	−10.3226	−0.527460	−0.263730	−	0.964597i	$-0.584953\pi$
$383$	−10.3226	−0.527460	−0.263730	+	0.964597i	$0.584953\pi$
$384$	0	0
$385$	−8.49900	−0.433149
$386$	41.6593	2.12040
$387$	0	0
$388$	−2.13193	−0.108232
$389$	−19.6783	−0.997730	−0.498865	−	0.866680i	$-0.666250\pi$
$389$	−19.6783	−0.997730	−0.498865	+	0.866680i	$0.666250\pi$
$390$	0	0
$391$	2.95098	0.149238
$392$	0.823654	0.0416008
$393$	0	0
$394$	10.2255	0.515153
$395$	33.5317	1.68716
$396$	0	0
$397$	−16.9533	−0.850863	−0.425432	−	0.904991i	$-0.639878\pi$
$397$	−16.9533	−0.850863	−0.425432	+	0.904991i	$0.639878\pi$
$398$	−21.7378	−1.08962
$399$	0	0
$400$	−49.9314	−2.49657
$401$	14.7737	0.737762	0.368881	−	0.929477i	$-0.379741\pi$
$401$	14.7737	0.737762	0.368881	+	0.929477i	$0.379741\pi$
$402$	0	0
$403$	41.7805	2.08123
$404$	−17.9516	−0.893126
$405$	0	0
$406$	−14.4905	−0.719152
$407$	−16.1207	−0.799072
$408$	0	0
$409$	−22.3107	−1.10319	−0.551596	−	0.834111i	$-0.685981\pi$
$409$	−22.3107	−1.10319	−0.551596	+	0.834111i	$0.685981\pi$
$410$	12.7405	0.629209
$411$	0	0
$412$	13.9171	0.685647
$413$	3.85596	0.189740
$414$	0	0
$415$	−24.8702	−1.22083
$416$	33.3695	1.63608
$417$	0	0
$418$	23.0772	1.12874
$419$	−3.91746	−0.191380	−0.0956901	−	0.995411i	$-0.530506\pi$
$419$	−3.91746	−0.191380	−0.0956901	+	0.995411i	$0.530506\pi$
$420$	0	0
$421$	−9.93242	−0.484076	−0.242038	−	0.970267i	$-0.577816\pi$
$421$	−9.93242	−0.484076	−0.242038	+	0.970267i	$0.577816\pi$
$422$	−9.81906	−0.477984
$423$	0	0
$424$	−6.31174	−0.306525
$425$	21.6423	1.04981
$426$	0	0
$427$	−7.14847	−0.345939
$428$	30.0507	1.45256
$429$	0	0
$430$	−84.3349	−4.06699
$431$	−4.01227	−0.193264	−0.0966320	−	0.995320i	$-0.530807\pi$
$431$	−4.01227	−0.193264	−0.0966320	+	0.995320i	$0.530807\pi$
$432$	0	0
$433$	−16.6134	−0.798388	−0.399194	−	0.916867i	$-0.630710\pi$
$433$	−16.6134	−0.798388	−0.399194	+	0.916867i	$0.630710\pi$
$434$	−16.9983	−0.815946
$435$	0	0
$436$	−25.6015	−1.22609
$437$	8.27747	0.395965
$438$	0	0
$439$	−1.61324	−0.0769956	−0.0384978	−	0.999259i	$-0.512257\pi$
$439$	−1.61324	−0.0769956	−0.0384978	+	0.999259i	$0.512257\pi$
$440$	−7.00024	−0.333723
$441$	0	0
$442$	−17.7767	−0.845552
$443$	−15.3314	−0.728416	−0.364208	−	0.931318i	$-0.618660\pi$
$443$	−15.3314	−0.728416	−0.364208	+	0.931318i	$0.618660\pi$
$444$	0	0
$445$	−41.5145	−1.96798
$446$	−40.3026	−1.90838
$447$	0	0
$448$	−4.21185	−0.198991
$449$	−25.6227	−1.20921	−0.604604	−	0.796526i	$-0.706669\pi$
$449$	−25.6227	−1.20921	−0.604604	+	0.796526i	$0.706669\pi$
$450$	0	0
$451$	3.66187	0.172431
$452$	17.9435	0.843989
$453$	0	0
$454$	−11.4788	−0.538725
$455$	18.3640	0.860919
$456$	0	0
$457$	10.3360	0.483496	0.241748	−	0.970339i	$-0.422279\pi$
$457$	10.3360	0.483496	0.241748	+	0.970339i	$0.422279\pi$
$458$	−39.7292	−1.85642
$459$	0	0
$460$	8.99879	0.419571
$461$	17.7774	0.827976	0.413988	−	0.910282i	$-0.364135\pi$
$461$	17.7774	0.827976	0.413988	+	0.910282i	$0.364135\pi$
$462$	0	0
$463$	−17.3660	−0.807065	−0.403532	−	0.914965i	$-0.632218\pi$
$463$	−17.3660	−0.807065	−0.403532	+	0.914965i	$0.632218\pi$
$464$	−35.9408	−1.66851
$465$	0	0
$466$	41.8640	1.93931
$467$	26.1427	1.20974	0.604869	−	0.796325i	$-0.293226\pi$
$467$	26.1427	1.20974	0.604869	+	0.796325i	$0.293226\pi$
$468$	0	0
$469$	−5.67445	−0.262022
$470$	76.2003	3.51486
$471$	0	0
$472$	3.17598	0.146186
$473$	−24.2395	−1.11453
$474$	0	0
$475$	60.7065	2.78540
$476$	3.17348	0.145456
$477$	0	0
$478$	−48.3209	−2.21014
$479$	−24.4009	−1.11490	−0.557452	−	0.830209i	$-0.688221\pi$
$479$	−24.4009	−1.11490	−0.557452	+	0.830209i	$0.688221\pi$
$480$	0	0
$481$	34.8324	1.58822
$482$	37.6264	1.71384
$483$	0	0
$484$	−9.98977	−0.454081
$485$	−5.39601	−0.245020
$486$	0	0
$487$	4.27526	0.193731	0.0968653	−	0.995298i	$-0.469118\pi$
$487$	4.27526	0.193731	0.0968653	+	0.995298i	$0.469118\pi$
$488$	−5.88786	−0.266531
$489$	0	0
$490$	−7.47138	−0.337523
$491$	−19.5487	−0.882220	−0.441110	−	0.897453i	$-0.645415\pi$
$491$	−19.5487	−0.882220	−0.441110	+	0.897453i	$0.645415\pi$
$492$	0	0
$493$	15.5782	0.701608
$494$	−49.8635	−2.24346
$495$	0	0
$496$	−42.1610	−1.89309
$497$	6.26208	0.280893
$498$	0	0
$499$	15.2354	0.682029	0.341015	−	0.940058i	$-0.389229\pi$
$499$	15.2354	0.682029	0.341015	+	0.940058i	$0.389229\pi$
$500$	35.0529	1.56761
$501$	0	0
$502$	19.0948	0.852243
$503$	−34.0607	−1.51869	−0.759346	−	0.650687i	$-0.774481\pi$
$503$	−34.0607	−1.51869	−0.759346	+	0.650687i	$0.774481\pi$
$504$	0	0
$505$	−45.4364	−2.02189
$506$	5.89453	0.262044
$507$	0	0
$508$	1.56369	0.0693775
$509$	−9.48321	−0.420336	−0.210168	−	0.977665i	$-0.567401\pi$
$509$	−9.48321	−0.420336	−0.210168	+	0.977665i	$0.567401\pi$
$510$	0	0
$511$	−9.32720	−0.412611
$512$	−25.9604	−1.14730
$513$	0	0
$514$	22.1071	0.975103
$515$	35.2249	1.55219
$516$	0	0
$517$	21.9014	0.963224
$518$	−14.1715	−0.622661
$519$	0	0
$520$	15.1256	0.663301
$521$	−14.5732	−0.638464	−0.319232	−	0.947677i	$-0.603425\pi$
$521$	−14.5732	−0.638464	−0.319232	+	0.947677i	$0.603425\pi$
$522$	0	0
$523$	12.4195	0.543068	0.271534	−	0.962429i	$-0.412469\pi$
$523$	12.4195	0.543068	0.271534	+	0.962429i	$0.412469\pi$
$524$	−26.3654	−1.15178
$525$	0	0
$526$	−22.4494	−0.978841
$527$	18.2743	0.796041
$528$	0	0
$529$	−20.8857	−0.908074
$530$	57.2540	2.48695
$531$	0	0
$532$	8.90157	0.385932
$533$	−7.91231	−0.342720
$534$	0	0
$535$	76.0598	3.28835
$536$	−4.67379	−0.201877
$537$	0	0
$538$	−6.55152	−0.282456
$539$	−2.14742	−0.0924959
$540$	0	0
$541$	−0.384270	−0.0165211	−0.00826053	−	0.999966i	$-0.502629\pi$
$541$	−0.384270	−0.0165211	−0.00826053	+	0.999966i	$0.502629\pi$
$542$	56.7470	2.43749
$543$	0	0
$544$	14.5955	0.625775
$545$	−64.7987	−2.77567
$546$	0	0
$547$	21.1127	0.902715	0.451358	−	0.892343i	$-0.350940\pi$
$547$	21.1127	0.902715	0.451358	+	0.892343i	$0.350940\pi$
$548$	29.9909	1.28115
$549$	0	0
$550$	43.2301	1.84334
$551$	43.6967	1.86154
$552$	0	0
$553$	8.47237	0.360282
$554$	51.4879	2.18751
$555$	0	0
$556$	−12.5280	−0.531306
$557$	7.73413	0.327706	0.163853	−	0.986485i	$-0.447608\pi$
$557$	7.73413	0.327706	0.163853	+	0.986485i	$0.447608\pi$
$558$	0	0
$559$	52.3749	2.21522
$560$	−18.5313	−0.783090
$561$	0	0
$562$	−33.5981	−1.41725
$563$	−36.6656	−1.54527	−0.772636	−	0.634849i	$-0.781062\pi$
$563$	−36.6656	−1.54527	−0.772636	+	0.634849i	$0.781062\pi$
$564$	0	0
$565$	45.4158	1.91066
$566$	−9.15588	−0.384850
$567$	0	0
$568$	5.15779	0.216416
$569$	18.2910	0.766798	0.383399	−	0.923583i	$-0.374753\pi$
$569$	18.2910	0.766798	0.383399	+	0.923583i	$0.374753\pi$
$570$	0	0
$571$	19.4223	0.812800	0.406400	−	0.913695i	$-0.366784\pi$
$571$	19.4223	0.812800	0.406400	+	0.913695i	$0.366784\pi$
$572$	−15.5806	−0.651458
$573$	0	0
$574$	3.21911	0.134363
$575$	15.5061	0.646647
$576$	0	0
$577$	−39.4979	−1.64432	−0.822160	−	0.569257i	$-0.807231\pi$
$577$	−39.4979	−1.64432	−0.822160	+	0.569257i	$0.807231\pi$
$578$	24.3168	1.01145
$579$	0	0
$580$	47.5046	1.97252
$581$	−6.28388	−0.260699
$582$	0	0
$583$	16.4559	0.681533
$584$	−7.68239	−0.317899
$585$	0	0
$586$	−28.1194	−1.16160
$587$	6.56836	0.271105	0.135553	−	0.990770i	$-0.456719\pi$
$587$	6.56836	0.271105	0.135553	+	0.990770i	$0.456719\pi$
$588$	0	0
$589$	51.2592	2.11210
$590$	−28.8094	−1.18606
$591$	0	0
$592$	−35.1497	−1.44464
$593$	−36.5594	−1.50131	−0.750657	−	0.660693i	$-0.770263\pi$
$593$	−36.5594	−1.50131	−0.750657	+	0.660693i	$0.770263\pi$
$594$	0	0
$595$	8.03222	0.329289
$596$	−20.6236	−0.844778
$597$	0	0
$598$	−12.7365	−0.520833
$599$	−10.1789	−0.415897	−0.207948	−	0.978140i	$-0.566679\pi$
$599$	−10.1789	−0.415897	−0.207948	+	0.978140i	$0.566679\pi$
$600$	0	0
$601$	20.3312	0.829328	0.414664	−	0.909975i	$-0.363899\pi$
$601$	20.3312	0.829328	0.414664	+	0.909975i	$0.363899\pi$
$602$	−21.3087	−0.868477
$603$	0	0
$604$	20.4537	0.832247
$605$	−25.2846	−1.02796
$606$	0	0
$607$	27.5219	1.11708	0.558540	−	0.829478i	$-0.311362\pi$
$607$	27.5219	1.11708	0.558540	+	0.829478i	$0.311362\pi$
$608$	40.9401	1.66034
$609$	0	0
$610$	53.4089	2.16246
$611$	−47.3230	−1.91448
$612$	0	0
$613$	0.970338	0.0391916	0.0195958	−	0.999808i	$-0.493762\pi$
$613$	0.970338	0.0391916	0.0195958	+	0.999808i	$0.493762\pi$
$614$	2.01749	0.0814194
$615$	0	0
$616$	−1.76873	−0.0712642
$617$	6.80882	0.274113	0.137056	−	0.990563i	$-0.456236\pi$
$617$	6.80882	0.274113	0.137056	+	0.990563i	$0.456236\pi$
$618$	0	0
$619$	−24.7537	−0.994936	−0.497468	−	0.867482i	$-0.665737\pi$
$619$	−24.7537	−0.994936	−0.497468	+	0.867482i	$0.665737\pi$
$620$	55.7260	2.23801
$621$	0	0
$622$	−1.10112	−0.0441509
$623$	−10.4894	−0.420247
$624$	0	0
$625$	35.4006	1.41602
$626$	50.7909	2.03001
$627$	0	0
$628$	−6.55503	−0.261574
$629$	15.2353	0.607471
$630$	0	0
$631$	−4.67050	−0.185930	−0.0929648	−	0.995669i	$-0.529634\pi$
$631$	−4.67050	−0.185930	−0.0929648	+	0.995669i	$0.529634\pi$
$632$	6.97830	0.277582
$633$	0	0
$634$	17.6399	0.700571
$635$	3.95777	0.157059
$636$	0	0
$637$	4.63999	0.183843
$638$	31.1172	1.23194
$639$	0	0
$640$	−25.4581	−1.00632
$641$	25.9544	1.02514	0.512569	−	0.858646i	$-0.328694\pi$
$641$	25.9544	1.02514	0.512569	+	0.858646i	$0.328694\pi$
$642$	0	0
$643$	26.4794	1.04425	0.522123	−	0.852870i	$-0.325140\pi$
$643$	26.4794	1.04425	0.522123	+	0.852870i	$0.325140\pi$
$644$	2.27370	0.0895963
$645$	0	0
$646$	−21.8097	−0.858092
$647$	30.6460	1.20482	0.602409	−	0.798187i	$-0.294208\pi$
$647$	30.6460	1.20482	0.602409	+	0.798187i	$0.294208\pi$
$648$	0	0
$649$	−8.28037	−0.325033
$650$	−93.4085	−3.66378
$651$	0	0
$652$	−23.5132	−0.920849
$653$	46.1775	1.80706	0.903532	−	0.428521i	$-0.140965\pi$
$653$	46.1775	1.80706	0.903532	+	0.428521i	$0.140965\pi$
$654$	0	0
$655$	−66.7321	−2.60744
$656$	7.98438	0.311738
$657$	0	0
$658$	19.2533	0.750572
$659$	33.6935	1.31251	0.656255	−	0.754539i	$-0.272139\pi$
$659$	33.6935	1.31251	0.656255	+	0.754539i	$0.272139\pi$
$660$	0	0
$661$	−5.86710	−0.228204	−0.114102	−	0.993469i	$-0.536399\pi$
$661$	−5.86710	−0.228204	−0.114102	+	0.993469i	$0.536399\pi$
$662$	1.95492	0.0759801
$663$	0	0
$664$	−5.17574	−0.200858
$665$	22.5303	0.873687
$666$	0	0
$667$	11.1613	0.432168
$668$	−14.4652	−0.559674
$669$	0	0
$670$	42.3960	1.63790
$671$	15.3508	0.592610
$672$	0	0
$673$	42.5282	1.63934	0.819672	−	0.572833i	$-0.194156\pi$
$673$	42.5282	1.63934	0.819672	+	0.572833i	$0.194156\pi$
$674$	26.7854	1.03174
$675$	0	0
$676$	13.3375	0.512980
$677$	31.9041	1.22618	0.613088	−	0.790015i	$-0.289927\pi$
$677$	31.9041	1.22618	0.613088	+	0.790015i	$0.289927\pi$
$678$	0	0
$679$	−1.36340	−0.0523223
$680$	6.61577	0.253703
$681$	0	0
$682$	36.5026	1.39776
$683$	20.7623	0.794445	0.397223	−	0.917722i	$-0.369974\pi$
$683$	20.7623	0.794445	0.397223	+	0.917722i	$0.369974\pi$
$684$	0	0
$685$	75.9084	2.90031
$686$	−1.88777	−0.0720755
$687$	0	0
$688$	−52.8520	−2.01496
$689$	−35.5567	−1.35460
$690$	0	0
$691$	4.89130	0.186074	0.0930370	−	0.995663i	$-0.470343\pi$
$691$	4.89130	0.186074	0.0930370	+	0.995663i	$0.470343\pi$
$692$	−11.8426	−0.450186
$693$	0	0
$694$	33.3162	1.26467
$695$	−31.7090	−1.20279
$696$	0	0
$697$	−3.46075	−0.131085
$698$	−21.5985	−0.817514
$699$	0	0
$700$	16.6752	0.630262
$701$	−46.2580	−1.74714	−0.873570	−	0.486699i	$-0.838201\pi$
$701$	−46.2580	−1.74714	−0.873570	+	0.486699i	$0.838201\pi$
$702$	0	0
$703$	42.7348	1.61177
$704$	9.04461	0.340882
$705$	0	0
$706$	−38.4430	−1.44682
$707$	−11.4803	−0.431760
$708$	0	0
$709$	6.50323	0.244234	0.122117	−	0.992516i	$-0.461032\pi$
$709$	6.50323	0.244234	0.122117	+	0.992516i	$0.461032\pi$
$710$	−46.7864	−1.75586
$711$	0	0
$712$	−8.63960	−0.323783
$713$	13.0930	0.490336
$714$	0	0
$715$	−39.4353	−1.47480
$716$	−11.3870	−0.425551
$717$	0	0
$718$	6.51797	0.243248
$719$	−24.1984	−0.902449	−0.451224	−	0.892411i	$-0.649013\pi$
$719$	−24.1984	−0.902449	−0.451224	+	0.892411i	$0.649013\pi$
$720$	0	0
$721$	8.90018	0.331460
$722$	−25.3083	−0.941878
$723$	0	0
$724$	−36.6445	−1.36188
$725$	81.8564	3.04007
$726$	0	0
$727$	−38.4129	−1.42466	−0.712328	−	0.701846i	$-0.752359\pi$
$727$	−38.4129	−1.42466	−0.712328	+	0.701846i	$0.752359\pi$
$728$	3.82174	0.141643
$729$	0	0
$730$	69.6871	2.57924
$731$	22.9082	0.847291
$732$	0	0
$733$	1.75404	0.0647869	0.0323934	−	0.999475i	$-0.489687\pi$
$733$	1.75404	0.0647869	0.0323934	+	0.999475i	$0.489687\pi$
$734$	−64.9259	−2.39646
$735$	0	0
$736$	10.4572	0.385457
$737$	12.1854	0.448856
$738$	0	0
$739$	51.1599	1.88195	0.940973	−	0.338480i	$-0.109913\pi$
$739$	51.1599	1.88195	0.940973	+	0.338480i	$0.109913\pi$
$740$	46.4588	1.70786
$741$	0	0
$742$	14.4662	0.531071
$743$	−40.5974	−1.48937	−0.744686	−	0.667415i	$-0.767401\pi$
$743$	−40.5974	−1.48937	−0.744686	+	0.667415i	$0.767401\pi$
$744$	0	0
$745$	−52.1994	−1.91244
$746$	3.71060	0.135855
$747$	0	0
$748$	−6.81479	−0.249173
$749$	19.2178	0.702204
$750$	0	0
$751$	29.4652	1.07520	0.537601	−	0.843199i	$-0.319331\pi$
$751$	29.4652	1.07520	0.537601	+	0.843199i	$0.319331\pi$
$752$	47.7541	1.74141
$753$	0	0
$754$	−67.2358	−2.44858
$755$	51.7692	1.88407
$756$	0	0
$757$	24.4676	0.889290	0.444645	−	0.895707i	$-0.353330\pi$
$757$	24.4676	0.889290	0.444645	+	0.895707i	$0.353330\pi$
$758$	16.2145	0.588936
$759$	0	0
$760$	18.5572	0.673139
$761$	9.90305	0.358985	0.179493	−	0.983759i	$-0.442554\pi$
$761$	9.90305	0.358985	0.179493	+	0.983759i	$0.442554\pi$
$762$	0	0
$763$	−16.3725	−0.592725
$764$	−23.5624	−0.852457
$765$	0	0
$766$	19.4867	0.704084
$767$	17.8916	0.646029
$768$	0	0
$769$	24.5030	0.883599	0.441800	−	0.897114i	$-0.354340\pi$
$769$	24.5030	0.883599	0.441800	+	0.897114i	$0.354340\pi$
$770$	16.0442	0.578193
$771$	0	0
$772$	−34.5075	−1.24195
$773$	5.48889	0.197422	0.0987109	−	0.995116i	$-0.468528\pi$
$773$	5.48889	0.197422	0.0987109	+	0.995116i	$0.468528\pi$
$774$	0	0
$775$	96.0231	3.44925
$776$	−1.12297	−0.0403121
$777$	0	0
$778$	37.1482	1.33183
$779$	−9.70737	−0.347803
$780$	0	0
$781$	−13.4473	−0.481183
$782$	−5.57079	−0.199211
$783$	0	0
$784$	−4.68225	−0.167223
$785$	−16.5911	−0.592161
$786$	0	0
$787$	20.1270	0.717448	0.358724	−	0.933444i	$-0.383212\pi$
$787$	20.1270	0.717448	0.358724	+	0.933444i	$0.383212\pi$
$788$	−8.47003	−0.301732
$789$	0	0
$790$	−63.3003	−2.25212
$791$	11.4751	0.408007
$792$	0	0
$793$	−33.1688	−1.17786
$794$	32.0041	1.13578
$795$	0	0
$796$	18.0060	0.638204
$797$	37.5515	1.33014	0.665071	−	0.746781i	$-0.268401\pi$
$797$	37.5515	1.33014	0.665071	+	0.746781i	$0.268401\pi$
$798$	0	0
$799$	−20.6986	−0.732262
$800$	76.6924	2.71149
$801$	0	0
$802$	−27.8894	−0.984808
$803$	20.0294	0.706823
$804$	0	0
$805$	5.75484	0.202831
$806$	−78.8721	−2.77815
$807$	0	0
$808$	−9.45577	−0.332653
$809$	15.5218	0.545719	0.272859	−	0.962054i	$-0.412031\pi$
$809$	15.5218	0.545719	0.272859	+	0.962054i	$0.412031\pi$
$810$	0	0
$811$	−36.0347	−1.26535	−0.632675	−	0.774417i	$-0.718043\pi$
$811$	−36.0347	−1.26535	−0.632675	+	0.774417i	$0.718043\pi$
$812$	12.0028	0.421217
$813$	0	0
$814$	30.4322	1.06665
$815$	−59.5131	−2.08465
$816$	0	0
$817$	64.2572	2.24808
$818$	42.1175	1.47261
$819$	0	0
$820$	−10.5533	−0.368537
$821$	36.1102	1.26025	0.630127	−	0.776492i	$-0.283003\pi$
$821$	36.1102	1.26025	0.630127	+	0.776492i	$0.283003\pi$
$822$	0	0
$823$	−40.1465	−1.39942	−0.699709	−	0.714428i	$-0.746687\pi$
$823$	−40.1465	−1.39942	−0.699709	+	0.714428i	$0.746687\pi$
$824$	7.33066	0.255376
$825$	0	0
$826$	−7.27919	−0.253275
$827$	17.4221	0.605827	0.302914	−	0.953018i	$-0.402041\pi$
$827$	17.4221	0.605827	0.302914	+	0.953018i	$0.402041\pi$
$828$	0	0
$829$	−52.1085	−1.80980	−0.904902	−	0.425620i	$-0.860056\pi$
$829$	−52.1085	−1.80980	−0.904902	+	0.425620i	$0.860056\pi$
$830$	46.9493	1.62963
$831$	0	0
$832$	−19.5429	−0.677529
$833$	2.02948	0.0703173
$834$	0	0
$835$	−36.6120	−1.26701
$836$	−19.1154	−0.661120
$837$	0	0
$838$	7.39527	0.255465
$839$	−11.3620	−0.392260	−0.196130	−	0.980578i	$-0.562837\pi$
$839$	−11.3620	−0.392260	−0.196130	+	0.980578i	$0.562837\pi$
$840$	0	0
$841$	29.9206	1.03174
$842$	18.7502	0.646173
$843$	0	0
$844$	8.13337	0.279962
$845$	33.7578	1.16130
$846$	0	0
$847$	−6.38859	−0.219514
$848$	35.8806	1.23214
$849$	0	0
$850$	−40.8558	−1.40134
$851$	10.9156	0.374183
$852$	0	0
$853$	−11.4023	−0.390406	−0.195203	−	0.980763i	$-0.562537\pi$
$853$	−11.4023	−0.390406	−0.195203	+	0.980763i	$0.562537\pi$
$854$	13.4947	0.461779
$855$	0	0
$856$	15.8288	0.541019
$857$	0.950674	0.0324744	0.0162372	−	0.999868i	$-0.494831\pi$
$857$	0.950674	0.0324744	0.0162372	+	0.999868i	$0.494831\pi$
$858$	0	0
$859$	0.210633	0.00718671	0.00359336	−	0.999994i	$-0.498856\pi$
$859$	0.210633	0.00718671	0.00359336	+	0.999994i	$0.498856\pi$
$860$	69.8567	2.38210
$861$	0	0
$862$	7.57425	0.257980
$863$	−10.9776	−0.373681	−0.186841	−	0.982390i	$-0.559825\pi$
$863$	−10.9776	−0.373681	−0.186841	+	0.982390i	$0.559825\pi$
$864$	0	0
$865$	−29.9741	−1.01915
$866$	31.3623	1.06573
$867$	0	0
$868$	14.0801	0.477911
$869$	−18.1937	−0.617180
$870$	0	0
$871$	−26.3294	−0.892138
$872$	−13.4853	−0.456669
$873$	0	0
$874$	−15.6260	−0.528557
$875$	22.4168	0.757825
$876$	0	0
$877$	0.159264	0.00537796	0.00268898	−	0.999996i	$-0.499144\pi$
$877$	0.159264	0.00537796	0.00268898	+	0.999996i	$0.499144\pi$
$878$	3.04542	0.102778
$879$	0	0
$880$	39.7945	1.34147
$881$	23.8224	0.802598	0.401299	−	0.915947i	$-0.368559\pi$
$881$	23.8224	0.802598	0.401299	+	0.915947i	$0.368559\pi$
$882$	0	0
$883$	−13.0665	−0.439722	−0.219861	−	0.975531i	$-0.570560\pi$
$883$	−13.0665	−0.439722	−0.219861	+	0.975531i	$0.570560\pi$
$884$	14.7249	0.495252
$885$	0	0
$886$	28.9422	0.972332
$887$	−30.3390	−1.01868	−0.509341	−	0.860565i	$-0.670111\pi$
$887$	−30.3390	−1.01868	−0.509341	+	0.860565i	$0.670111\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	78.3700	2.62697
$891$	0	0
$892$	33.3837	1.11777
$893$	−58.0592	−1.94288
$894$	0	0
$895$	−28.8209	−0.963378
$896$	−6.43243	−0.214893
$897$	0	0
$898$	48.3698	1.61412
$899$	69.1178	2.30521
$900$	0	0
$901$	−15.5521	−0.518116
$902$	−6.91279	−0.230171
$903$	0	0
$904$	9.45149	0.314352
$905$	−92.7490	−3.08308
$906$	0	0
$907$	−29.7564	−0.988044	−0.494022	−	0.869449i	$-0.664474\pi$
$907$	−29.7564	−0.988044	−0.494022	+	0.869449i	$0.664474\pi$
$908$	9.50814	0.315539
$909$	0	0
$910$	−34.6671	−1.14920
$911$	60.0700	1.99021	0.995103	−	0.0988416i	$-0.0315137\pi$
$911$	60.0700	1.99021	0.995103	+	0.0988416i	$0.0315137\pi$
$912$	0	0
$913$	13.4941	0.446591
$914$	−19.5119	−0.645398
$915$	0	0
$916$	32.9087	1.08733
$917$	−16.8610	−0.556800
$918$	0	0
$919$	25.7936	0.850852	0.425426	−	0.904993i	$-0.360124\pi$
$919$	25.7936	0.850852	0.425426	+	0.904993i	$0.360124\pi$
$920$	4.73999	0.156273
$921$	0	0
$922$	−33.5597	−1.10523
$923$	29.0560	0.956389
$924$	0	0
$925$	80.0545	2.63217
$926$	32.7830	1.07732
$927$	0	0
$928$	55.2035	1.81214
$929$	46.1737	1.51491	0.757455	−	0.652887i	$-0.226442\pi$
$929$	46.1737	1.51491	0.757455	+	0.652887i	$0.226442\pi$
$930$	0	0
$931$	5.69266	0.186570
$932$	−34.6770	−1.13588
$933$	0	0
$934$	−49.3514	−1.61483
$935$	−17.2486	−0.564088
$936$	0	0
$937$	−54.6743	−1.78613	−0.893066	−	0.449926i	$-0.851450\pi$
$937$	−54.6743	−1.78613	−0.893066	+	0.449926i	$0.851450\pi$
$938$	10.7121	0.349762
$939$	0	0
$940$	−63.1186	−2.05870
$941$	−16.7357	−0.545569	−0.272785	−	0.962075i	$-0.587945\pi$
$941$	−16.7357	−0.545569	−0.272785	+	0.962075i	$0.587945\pi$
$942$	0	0
$943$	−2.47952	−0.0807444
$944$	−18.0546	−0.587627
$945$	0	0
$946$	45.7587	1.48774
$947$	−60.2549	−1.95802	−0.979011	−	0.203808i	$-0.934668\pi$
$947$	−60.2549	−1.95802	−0.979011	+	0.203808i	$0.934668\pi$
$948$	0	0
$949$	−43.2781	−1.40487
$950$	−114.600	−3.71812
$951$	0	0
$952$	1.67159	0.0541765
$953$	−8.73051	−0.282809	−0.141405	−	0.989952i	$-0.545162\pi$
$953$	−8.73051	−0.282809	−0.141405	+	0.989952i	$0.545162\pi$
$954$	0	0
$955$	−59.6375	−1.92982
$956$	40.0254	1.29451
$957$	0	0
$958$	46.0634	1.48824
$959$	19.1796	0.619341
$960$	0	0
$961$	50.0798	1.61548
$962$	−65.7557	−2.12005
$963$	0	0
$964$	−31.1669	−1.00382
$965$	−87.3400	−2.81157
$966$	0	0
$967$	59.2643	1.90581	0.952905	−	0.303268i	$-0.0980777\pi$
$967$	59.2643	1.90581	0.952905	+	0.303268i	$0.0980777\pi$
$968$	−5.26198	−0.169127
$969$	0	0
$970$	10.1865	0.327067
$971$	15.3180	0.491577	0.245789	−	0.969323i	$-0.420953\pi$
$971$	15.3180	0.491577	0.245789	+	0.969323i	$0.420953\pi$
$972$	0	0
$973$	−8.01183	−0.256847
$974$	−8.07073	−0.258603
$975$	0	0
$976$	33.4709	1.07138
$977$	55.5903	1.77849	0.889245	−	0.457431i	$-0.151230\pi$
$977$	55.5903	1.77849	0.889245	+	0.457431i	$0.151230\pi$
$978$	0	0
$979$	22.5251	0.719904
$980$	6.18873	0.197692
$981$	0	0
$982$	36.9035	1.17764
$983$	−2.43713	−0.0777323	−0.0388662	−	0.999244i	$-0.512375\pi$
$983$	−2.43713	−0.0777323	−0.0388662	+	0.999244i	$0.512375\pi$
$984$	0	0
$985$	−21.4380	−0.683073
$986$	−29.4082	−0.936547
$987$	0	0
$988$	41.3032	1.31403
$989$	16.4130	0.521904
$990$	0	0
$991$	−8.15365	−0.259009	−0.129505	−	0.991579i	$-0.541339\pi$
$991$	−8.15365	−0.259009	−0.129505	+	0.991579i	$0.541339\pi$
$992$	64.7574	2.05605
$993$	0	0
$994$	−11.8214	−0.374952
$995$	45.5739	1.44479
$996$	0	0
$997$	−33.5398	−1.06222	−0.531109	−	0.847304i	$-0.678224\pi$
$997$	−33.5398	−1.06222	−0.531109	+	0.847304i	$0.678224\pi$
$998$	−28.7609	−0.910412
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.v.1.5		19
3.2	odd	2		2667.2.a.q.1.15	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.q.1.15	✓	19	3.2	odd	2
8001.2.a.v.1.5		19	1.1	even	1	trivial

Overview	Random
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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)

\(f(q)\)	\(=\)	\(q-1.88777 q^{2} +1.56369 q^{4} +3.95777 q^{5} +1.00000 q^{7} +0.823654 q^{8} +O(q^{10})\)	\(q-1.88777 q^{2} +1.56369 q^{4} +3.95777 q^{5} +1.00000 q^{7} +0.823654 q^{8} -7.47138 q^{10} -2.14742 q^{11} +4.63999 q^{13} -1.88777 q^{14} -4.68225 q^{16} +2.02948 q^{17} +5.69266 q^{19} +6.18873 q^{20} +4.05384 q^{22} +1.45406 q^{23} +10.6640 q^{25} -8.75925 q^{26} +1.56369 q^{28} +7.67597 q^{29} +9.00443 q^{31} +7.19173 q^{32} -3.83120 q^{34} +3.95777 q^{35} +7.50700 q^{37} -10.7465 q^{38} +3.25984 q^{40} -1.70524 q^{41} +11.2877 q^{43} -3.35790 q^{44} -2.74493 q^{46} -10.1989 q^{47} +1.00000 q^{49} -20.1312 q^{50} +7.25551 q^{52} -7.66310 q^{53} -8.49900 q^{55} +0.823654 q^{56} -14.4905 q^{58} +3.85596 q^{59} -7.14847 q^{61} -16.9983 q^{62} -4.21185 q^{64} +18.3640 q^{65} -5.67445 q^{67} +3.17348 q^{68} -7.47138 q^{70} +6.26208 q^{71} -9.32720 q^{73} -14.1715 q^{74} +8.90157 q^{76} -2.14742 q^{77} +8.47237 q^{79} -18.5313 q^{80} +3.21911 q^{82} -6.28388 q^{83} +8.03222 q^{85} -21.3087 q^{86} -1.76873 q^{88} -10.4894 q^{89} +4.63999 q^{91} +2.27370 q^{92} +19.2533 q^{94} +22.5303 q^{95} -1.36340 q^{97} -1.88777 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8}+O(q^{10}) \) 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8	\( 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8} + 9 q^{11} + 24 q^{13} - 4 q^{14} + 20 q^{16} - 17 q^{17} + 23 q^{19} - 5 q^{20} - 3 q^{22} + 17 q^{23} + 38 q^{25} - 28 q^{26} + 22 q^{28} - 2 q^{29} + 16 q^{31} - 17 q^{32} + 29 q^{34} - 5 q^{35} + 56 q^{37} - 2 q^{38} - 13 q^{40} + 7 q^{41} + 19 q^{43} + 29 q^{44} + 10 q^{46} - 25 q^{47} + 19 q^{49} + 9 q^{50} + 16 q^{52} - 18 q^{53} + 10 q^{55} - 9 q^{56} + 31 q^{58} - 11 q^{59} + 26 q^{61} - 26 q^{62} + 45 q^{64} - 27 q^{65} + 24 q^{67} - 14 q^{68} + 32 q^{71} + 51 q^{73} + 12 q^{76} + 9 q^{77} + 30 q^{79} + 30 q^{80} - 52 q^{82} - q^{83} + 44 q^{85} + 24 q^{86} - 30 q^{88} - 5 q^{89} + 24 q^{91} + 88 q^{92} + 7 q^{94} + 24 q^{95} + 5 q^{97} - 4 q^{98}+O(q^{100}) \) 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8 + 9 * q^11 + 24 * q^13 - 4 * q^14 + 20 * q^16 - 17 * q^17 + 23 * q^19 - 5 * q^20 - 3 * q^22 + 17 * q^23 + 38 * q^25 - 28 * q^26 + 22 * q^28 - 2 * q^29 + 16 * q^31 - 17 * q^32 + 29 * q^34 - 5 * q^35 + 56 * q^37 - 2 * q^38 - 13 * q^40 + 7 * q^41 + 19 * q^43 + 29 * q^44 + 10 * q^46 - 25 * q^47 + 19 * q^49 + 9 * q^50 + 16 * q^52 - 18 * q^53 + 10 * q^55 - 9 * q^56 + 31 * q^58 - 11 * q^59 + 26 * q^61 - 26 * q^62 + 45 * q^64 - 27 * q^65 + 24 * q^67 - 14 * q^68 + 32 * q^71 + 51 * q^73 + 12 * q^76 + 9 * q^77 + 30 * q^79 + 30 * q^80 - 52 * q^82 - q^83 + 44 * q^85 + 24 * q^86 - 30 * q^88 - 5 * q^89 + 24 * q^91 + 88 * q^92 + 7 * q^94 + 24 * q^95 + 5 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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