LMFDB - Embedded newform 8001.2.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

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

Embedding invariants

Embedding label			1.1
Root			$-2.57541$ of defining polynomial
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.57541 q^{2} +4.63271 q^{4} +3.61540 q^{5} -1.00000 q^{7} -6.78030 q^{8} +O(q^{10})$	\(q-2.57541 q^{2} +4.63271 q^{4} +3.61540 q^{5} -1.00000 q^{7} -6.78030 q^{8} -9.31112 q^{10} +4.80505 q^{11} -3.02226 q^{13} +2.57541 q^{14} +8.19660 q^{16} +5.99473 q^{17} +5.27314 q^{19} +16.7491 q^{20} -12.3750 q^{22} -5.63109 q^{23} +8.07110 q^{25} +7.78354 q^{26} -4.63271 q^{28} +5.62713 q^{29} +4.07939 q^{31} -7.54897 q^{32} -15.4389 q^{34} -3.61540 q^{35} +3.58935 q^{37} -13.5805 q^{38} -24.5135 q^{40} +9.98794 q^{41} +4.00758 q^{43} +22.2604 q^{44} +14.5023 q^{46} +9.98363 q^{47} +1.00000 q^{49} -20.7864 q^{50} -14.0013 q^{52} -1.00415 q^{53} +17.3722 q^{55} +6.78030 q^{56} -14.4922 q^{58} -2.21412 q^{59} +6.57832 q^{61} -10.5061 q^{62} +3.04845 q^{64} -10.9267 q^{65} -5.13122 q^{67} +27.7719 q^{68} +9.31112 q^{70} +13.8883 q^{71} -10.0400 q^{73} -9.24403 q^{74} +24.4290 q^{76} -4.80505 q^{77} -17.1340 q^{79} +29.6340 q^{80} -25.7230 q^{82} +5.98037 q^{83} +21.6733 q^{85} -10.3211 q^{86} -32.5797 q^{88} -15.9290 q^{89} +3.02226 q^{91} -26.0872 q^{92} -25.7119 q^{94} +19.0645 q^{95} -16.1200 q^{97} -2.57541 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8}+O(q^{10}) $ 16 * q + 2 * q^2 + 12 * q^4 + 9 * q^5 - 16 * q^7 + 6 * q^8	$ 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8} - 2 q^{10} + 22 q^{11} - 4 q^{13} - 2 q^{14} + 12 q^{16} + 18 q^{17} - 15 q^{19} + 40 q^{20} - 11 q^{22} + 5 q^{23} + 15 q^{25} + 24 q^{26} - 12 q^{28} + 12 q^{29} - 32 q^{31} + 9 q^{32} - 14 q^{34} - 9 q^{35} - 2 q^{37} - 3 q^{38} - 14 q^{40} + 45 q^{41} - 3 q^{43} + 54 q^{44} + 49 q^{47} + 16 q^{49} + 6 q^{50} + 38 q^{52} - 16 q^{53} + 7 q^{55} - 6 q^{56} + 16 q^{58} + 35 q^{59} - 11 q^{61} - 17 q^{62} - 2 q^{64} - 14 q^{65} + 17 q^{67} + 71 q^{68} + 2 q^{70} + 81 q^{71} - 15 q^{73} - 13 q^{74} + 14 q^{76} - 22 q^{77} - 34 q^{79} + 33 q^{80} - 14 q^{82} + 39 q^{83} - 17 q^{85} - 36 q^{86} + 61 q^{88} + 32 q^{89} + 4 q^{91} - 37 q^{92} + 13 q^{94} + 33 q^{95} - 4 q^{97} + 2 q^{98}+O(q^{100}) $ 16 * q + 2 * q^2 + 12 * q^4 + 9 * q^5 - 16 * q^7 + 6 * q^8 - 2 * q^10 + 22 * q^11 - 4 * q^13 - 2 * q^14 + 12 * q^16 + 18 * q^17 - 15 * q^19 + 40 * q^20 - 11 * q^22 + 5 * q^23 + 15 * q^25 + 24 * q^26 - 12 * q^28 + 12 * q^29 - 32 * q^31 + 9 * q^32 - 14 * q^34 - 9 * q^35 - 2 * q^37 - 3 * q^38 - 14 * q^40 + 45 * q^41 - 3 * q^43 + 54 * q^44 + 49 * q^47 + 16 * q^49 + 6 * q^50 + 38 * q^52 - 16 * q^53 + 7 * q^55 - 6 * q^56 + 16 * q^58 + 35 * q^59 - 11 * q^61 - 17 * q^62 - 2 * q^64 - 14 * q^65 + 17 * q^67 + 71 * q^68 + 2 * q^70 + 81 * q^71 - 15 * q^73 - 13 * q^74 + 14 * q^76 - 22 * q^77 - 34 * q^79 + 33 * q^80 - 14 * q^82 + 39 * q^83 - 17 * q^85 - 36 * q^86 + 61 * q^88 + 32 * q^89 + 4 * q^91 - 37 * q^92 + 13 * q^94 + 33 * q^95 - 4 * q^97 + 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.57541	−1.82109	−0.910543	−	0.413414i	$-0.864336\pi$
$2$	−2.57541	−1.82109	−0.910543	+	0.413414i	$0.864336\pi$
$3$	0	0
$3$	0	0
$4$	4.63271	2.31636
$5$	3.61540	1.61686	0.808428	−	0.588596i	$-0.200319\pi$
$5$	3.61540	1.61686	0.808428	+	0.588596i	$0.200319\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−6.78030	−2.39720
$9$	0	0
$10$	−9.31112	−2.94443
$11$	4.80505	1.44878	0.724389	−	0.689392i	$-0.242122\pi$
$11$	4.80505	1.44878	0.724389	+	0.689392i	$0.242122\pi$
$12$	0	0
$13$	−3.02226	−0.838224	−0.419112	−	0.907935i	$-0.637659\pi$
$13$	−3.02226	−0.838224	−0.419112	+	0.907935i	$0.637659\pi$
$14$	2.57541	0.688306
$15$	0	0
$16$	8.19660	2.04915
$17$	5.99473	1.45394	0.726968	−	0.686671i	$-0.240929\pi$
$17$	5.99473	1.45394	0.726968	+	0.686671i	$0.240929\pi$
$18$	0	0
$19$	5.27314	1.20974	0.604871	−	0.796324i	$-0.293225\pi$
$19$	5.27314	1.20974	0.604871	+	0.796324i	$0.293225\pi$
$20$	16.7491	3.74521
$21$	0	0
$22$	−12.3750	−2.63835
$23$	−5.63109	−1.17416	−0.587082	−	0.809528i	$-0.699723\pi$
$23$	−5.63109	−1.17416	−0.587082	+	0.809528i	$0.699723\pi$
$24$	0	0
$25$	8.07110	1.61422
$26$	7.78354	1.52648
$27$	0	0
$28$	−4.63271	−0.875500
$29$	5.62713	1.04493	0.522466	−	0.852660i	$-0.325012\pi$
$29$	5.62713	1.04493	0.522466	+	0.852660i	$0.325012\pi$
$30$	0	0
$31$	4.07939	0.732680	0.366340	−	0.930481i	$-0.380611\pi$
$31$	4.07939	0.732680	0.366340	+	0.930481i	$0.380611\pi$
$32$	−7.54897	−1.33448
$33$	0	0
$34$	−15.4389	−2.64774
$35$	−3.61540	−0.611114
$36$	0	0
$37$	3.58935	0.590085	0.295043	−	0.955484i	$-0.404666\pi$
$37$	3.58935	0.590085	0.295043	+	0.955484i	$0.404666\pi$
$38$	−13.5805	−2.20304
$39$	0	0
$40$	−24.5135	−3.87592
$41$	9.98794	1.55985	0.779927	−	0.625870i	$-0.215256\pi$
$41$	9.98794	1.55985	0.779927	+	0.625870i	$0.215256\pi$
$42$	0	0
$43$	4.00758	0.611150	0.305575	−	0.952168i	$-0.401151\pi$
$43$	4.00758	0.611150	0.305575	+	0.952168i	$0.401151\pi$
$44$	22.2604	3.35588
$45$	0	0
$46$	14.5023	2.13825
$47$	9.98363	1.45626	0.728131	−	0.685438i	$-0.240389\pi$
$47$	9.98363	1.45626	0.728131	+	0.685438i	$0.240389\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−20.7864	−2.93964
$51$	0	0
$52$	−14.0013	−1.94163
$53$	−1.00415	−0.137931	−0.0689655	−	0.997619i	$-0.521970\pi$
$53$	−1.00415	−0.137931	−0.0689655	+	0.997619i	$0.521970\pi$
$54$	0	0
$55$	17.3722	2.34246
$56$	6.78030	0.906056
$57$	0	0
$58$	−14.4922	−1.90291
$59$	−2.21412	−0.288254	−0.144127	−	0.989559i	$-0.546037\pi$
$59$	−2.21412	−0.288254	−0.144127	+	0.989559i	$0.546037\pi$
$60$	0	0
$61$	6.57832	0.842267	0.421133	−	0.906999i	$-0.361632\pi$
$61$	6.57832	0.842267	0.421133	+	0.906999i	$0.361632\pi$
$62$	−10.5061	−1.33427
$63$	0	0
$64$	3.04845	0.381056
$65$	−10.9267	−1.35529
$66$	0	0
$67$	−5.13122	−0.626879	−0.313439	−	0.949608i	$-0.601481\pi$
$67$	−5.13122	−0.626879	−0.313439	+	0.949608i	$0.601481\pi$
$68$	27.7719	3.36784
$69$	0	0
$70$	9.31112	1.11289
$71$	13.8883	1.64824	0.824121	−	0.566414i	$-0.191670\pi$
$71$	13.8883	1.64824	0.824121	+	0.566414i	$0.191670\pi$
$72$	0	0
$73$	−10.0400	−1.17509	−0.587544	−	0.809192i	$-0.699905\pi$
$73$	−10.0400	−1.17509	−0.587544	+	0.809192i	$0.699905\pi$
$74$	−9.24403	−1.07460
$75$	0	0
$76$	24.4290	2.80219
$77$	−4.80505	−0.547586
$78$	0	0
$79$	−17.1340	−1.92772	−0.963861	−	0.266405i	$-0.914164\pi$
$79$	−17.1340	−1.92772	−0.963861	+	0.266405i	$0.914164\pi$
$80$	29.6340	3.31318
$81$	0	0
$82$	−25.7230	−2.84063
$83$	5.98037	0.656431	0.328216	−	0.944603i	$-0.393553\pi$
$83$	5.98037	0.656431	0.328216	+	0.944603i	$0.393553\pi$
$84$	0	0
$85$	21.6733	2.35080
$86$	−10.3211	−1.11296
$87$	0	0
$88$	−32.5797	−3.47301
$89$	−15.9290	−1.68847	−0.844237	−	0.535970i	$-0.819946\pi$
$89$	−15.9290	−1.68847	−0.844237	+	0.535970i	$0.819946\pi$
$90$	0	0
$91$	3.02226	0.316819
$92$	−26.0872	−2.71978
$93$	0	0
$94$	−25.7119	−2.65198
$95$	19.0645	1.95598
$96$	0	0
$97$	−16.1200	−1.63673	−0.818367	−	0.574695i	$-0.805121\pi$
$97$	−16.1200	−1.63673	−0.818367	+	0.574695i	$0.805121\pi$
$98$	−2.57541	−0.260155
$99$	0	0
$100$	37.3911	3.73911
$101$	−15.0825	−1.50077	−0.750385	−	0.661001i	$-0.770132\pi$
$101$	−15.0825	−1.50077	−0.750385	+	0.661001i	$0.770132\pi$
$102$	0	0
$103$	13.8383	1.36353	0.681764	−	0.731572i	$-0.261213\pi$
$103$	13.8383	1.36353	0.681764	+	0.731572i	$0.261213\pi$
$104$	20.4918	2.00939
$105$	0	0
$106$	2.58610	0.251184
$107$	−14.4768	−1.39952	−0.699761	−	0.714377i	$-0.746710\pi$
$107$	−14.4768	−1.39952	−0.699761	+	0.714377i	$0.746710\pi$
$108$	0	0
$109$	5.17787	0.495950	0.247975	−	0.968766i	$-0.420235\pi$
$109$	5.17787	0.495950	0.247975	+	0.968766i	$0.420235\pi$
$110$	−44.7404	−4.26583
$111$	0	0
$112$	−8.19660	−0.774506
$113$	2.17656	0.204753	0.102377	−	0.994746i	$-0.467355\pi$
$113$	2.17656	0.204753	0.102377	+	0.994746i	$0.467355\pi$
$114$	0	0
$115$	−20.3586	−1.89845
$116$	26.0689	2.42044
$117$	0	0
$118$	5.70227	0.524936
$119$	−5.99473	−0.549536
$120$	0	0
$121$	12.0885	1.09896
$122$	−16.9418	−1.53384
$123$	0	0
$124$	18.8986	1.69715
$125$	11.1033	0.993105
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	7.24694	0.640545
$129$	0	0
$130$	28.1406	2.46809
$131$	−1.35652	−0.118520	−0.0592598	−	0.998243i	$-0.518874\pi$
$131$	−1.35652	−0.118520	−0.0592598	+	0.998243i	$0.518874\pi$
$132$	0	0
$133$	−5.27314	−0.457239
$134$	13.2150	1.14160
$135$	0	0
$136$	−40.6461	−3.48538
$137$	−7.96241	−0.680274	−0.340137	−	0.940376i	$-0.610474\pi$
$137$	−7.96241	−0.680274	−0.340137	+	0.940376i	$0.610474\pi$
$138$	0	0
$139$	3.78746	0.321248	0.160624	−	0.987016i	$-0.448649\pi$
$139$	3.78746	0.321248	0.160624	+	0.987016i	$0.448649\pi$
$140$	−16.7491	−1.41556
$141$	0	0
$142$	−35.7681	−3.00159
$143$	−14.5221	−1.21440
$144$	0	0
$145$	20.3443	1.68950
$146$	25.8570	2.13994
$147$	0	0
$148$	16.6284	1.36685
$149$	10.5322	0.862835	0.431417	−	0.902152i	$-0.358014\pi$
$149$	10.5322	0.862835	0.431417	+	0.902152i	$0.358014\pi$
$150$	0	0
$151$	−8.21168	−0.668257	−0.334128	−	0.942528i	$-0.608442\pi$
$151$	−8.21168	−0.668257	−0.334128	+	0.942528i	$0.608442\pi$
$152$	−35.7535	−2.89999
$153$	0	0
$154$	12.3750	0.997202
$155$	14.7486	1.18464
$156$	0	0
$157$	4.22460	0.337160	0.168580	−	0.985688i	$-0.446082\pi$
$157$	4.22460	0.337160	0.168580	+	0.985688i	$0.446082\pi$
$158$	44.1269	3.51055
$159$	0	0
$160$	−27.2925	−2.15766
$161$	5.63109	0.443792
$162$	0	0
$163$	0.350024	0.0274160	0.0137080	−	0.999906i	$-0.495636\pi$
$163$	0.350024	0.0274160	0.0137080	+	0.999906i	$0.495636\pi$
$164$	46.2713	3.61318
$165$	0	0
$166$	−15.4019	−1.19542
$167$	−0.955076	−0.0739060	−0.0369530	−	0.999317i	$-0.511765\pi$
$167$	−0.955076	−0.0739060	−0.0369530	+	0.999317i	$0.511765\pi$
$168$	0	0
$169$	−3.86595	−0.297381
$170$	−55.8177	−4.28102
$171$	0	0
$172$	18.5660	1.41564
$173$	6.62247	0.503497	0.251749	−	0.967793i	$-0.418994\pi$
$173$	6.62247	0.503497	0.251749	+	0.967793i	$0.418994\pi$
$174$	0	0
$175$	−8.07110	−0.610118
$176$	39.3851	2.96876
$177$	0	0
$178$	41.0237	3.07486
$179$	12.6159	0.942958	0.471479	−	0.881877i	$-0.343720\pi$
$179$	12.6159	0.942958	0.471479	+	0.881877i	$0.343720\pi$
$180$	0	0
$181$	−11.9530	−0.888460	−0.444230	−	0.895913i	$-0.646523\pi$
$181$	−11.9530	−0.888460	−0.444230	+	0.895913i	$0.646523\pi$
$182$	−7.78354	−0.576955
$183$	0	0
$184$	38.1805	2.81470
$185$	12.9769	0.954082
$186$	0	0
$187$	28.8050	2.10643
$188$	46.2513	3.37322
$189$	0	0
$190$	−49.0988	−3.56200
$191$	21.1585	1.53098	0.765489	−	0.643449i	$-0.222497\pi$
$191$	21.1585	1.53098	0.765489	+	0.643449i	$0.222497\pi$
$192$	0	0
$193$	−20.2465	−1.45737	−0.728686	−	0.684848i	$-0.759869\pi$
$193$	−20.2465	−1.45737	−0.728686	+	0.684848i	$0.759869\pi$
$194$	41.5155	2.98064
$195$	0	0
$196$	4.63271	0.330908
$197$	−7.09872	−0.505763	−0.252881	−	0.967497i	$-0.581378\pi$
$197$	−7.09872	−0.505763	−0.252881	+	0.967497i	$0.581378\pi$
$198$	0	0
$199$	−16.6871	−1.18292	−0.591458	−	0.806336i	$-0.701447\pi$
$199$	−16.6871	−1.18292	−0.591458	+	0.806336i	$0.701447\pi$
$200$	−54.7245	−3.86961
$201$	0	0
$202$	38.8437	2.73303
$203$	−5.62713	−0.394947
$204$	0	0
$205$	36.1104	2.52206
$206$	−35.6392	−2.48310
$207$	0	0
$208$	−24.7723	−1.71765
$209$	25.3377	1.75265
$210$	0	0
$211$	17.6748	1.21678	0.608391	−	0.793637i	$-0.291815\pi$
$211$	17.6748	1.21678	0.608391	+	0.793637i	$0.291815\pi$
$212$	−4.65195	−0.319497
$213$	0	0
$214$	37.2835	2.54865
$215$	14.4890	0.988141
$216$	0	0
$217$	−4.07939	−0.276927
$218$	−13.3351	−0.903167
$219$	0	0
$220$	80.4803	5.42598
$221$	−18.1176	−1.21872
$222$	0	0
$223$	2.34873	0.157283	0.0786414	−	0.996903i	$-0.474942\pi$
$223$	2.34873	0.157283	0.0786414	+	0.996903i	$0.474942\pi$
$224$	7.54897	0.504387
$225$	0	0
$226$	−5.60551	−0.372873
$227$	−8.80210	−0.584216	−0.292108	−	0.956385i	$-0.594357\pi$
$227$	−8.80210	−0.584216	−0.292108	+	0.956385i	$0.594357\pi$
$228$	0	0
$229$	−0.995190	−0.0657640	−0.0328820	−	0.999459i	$-0.510469\pi$
$229$	−0.995190	−0.0657640	−0.0328820	+	0.999459i	$0.510469\pi$
$230$	52.4317	3.45724
$231$	0	0
$232$	−38.1537	−2.50491
$233$	16.8582	1.10442	0.552209	−	0.833706i	$-0.313785\pi$
$233$	16.8582	1.10442	0.552209	+	0.833706i	$0.313785\pi$
$234$	0	0
$235$	36.0948	2.35456
$236$	−10.2574	−0.667700
$237$	0	0
$238$	15.4389	1.00075
$239$	5.12666	0.331616	0.165808	−	0.986158i	$-0.446977\pi$
$239$	5.12666	0.331616	0.165808	+	0.986158i	$0.446977\pi$
$240$	0	0
$241$	8.86855	0.571273	0.285637	−	0.958338i	$-0.407795\pi$
$241$	8.86855	0.571273	0.285637	+	0.958338i	$0.407795\pi$
$242$	−31.1328	−2.00129
$243$	0	0
$244$	30.4754	1.95099
$245$	3.61540	0.230979
$246$	0	0
$247$	−15.9368	−1.01403
$248$	−27.6595	−1.75638
$249$	0	0
$250$	−28.5954	−1.80853
$251$	8.12236	0.512679	0.256339	−	0.966587i	$-0.417484\pi$
$251$	8.12236	0.512679	0.256339	+	0.966587i	$0.417484\pi$
$252$	0	0
$253$	−27.0577	−1.70110
$254$	2.57541	0.161595
$255$	0	0
$256$	−24.7607	−1.54754
$257$	−23.4193	−1.46086	−0.730429	−	0.682989i	$-0.760680\pi$
$257$	−23.4193	−1.46086	−0.730429	+	0.682989i	$0.760680\pi$
$258$	0	0
$259$	−3.58935	−0.223031
$260$	−50.6201	−3.13933
$261$	0	0
$262$	3.49358	0.215834
$263$	8.96856	0.553025	0.276513	−	0.961010i	$-0.410821\pi$
$263$	8.96856	0.553025	0.276513	+	0.961010i	$0.410821\pi$
$264$	0	0
$265$	−3.63041	−0.223014
$266$	13.5805	0.832673
$267$	0	0
$268$	−23.7715	−1.45207
$269$	−12.9224	−0.787892	−0.393946	−	0.919134i	$-0.628890\pi$
$269$	−12.9224	−0.787892	−0.393946	+	0.919134i	$0.628890\pi$
$270$	0	0
$271$	−7.70820	−0.468240	−0.234120	−	0.972208i	$-0.575221\pi$
$271$	−7.70820	−0.468240	−0.234120	+	0.972208i	$0.575221\pi$
$272$	49.1365	2.97934
$273$	0	0
$274$	20.5064	1.23884
$275$	38.7821	2.33865
$276$	0	0
$277$	−8.26154	−0.496388	−0.248194	−	0.968710i	$-0.579837\pi$
$277$	−8.26154	−0.496388	−0.248194	+	0.968710i	$0.579837\pi$
$278$	−9.75425	−0.585021
$279$	0	0
$280$	24.5135	1.46496
$281$	11.9817	0.714766	0.357383	−	0.933958i	$-0.383669\pi$
$281$	11.9817	0.714766	0.357383	+	0.933958i	$0.383669\pi$
$282$	0	0
$283$	−5.98073	−0.355517	−0.177759	−	0.984074i	$-0.556885\pi$
$283$	−5.98073	−0.355517	−0.177759	+	0.984074i	$0.556885\pi$
$284$	64.3407	3.81792
$285$	0	0
$286$	37.4003	2.21153
$287$	−9.98794	−0.589570
$288$	0	0
$289$	18.9368	1.11393
$290$	−52.3949	−3.07673
$291$	0	0
$292$	−46.5123	−2.72192
$293$	−9.49311	−0.554593	−0.277297	−	0.960784i	$-0.589439\pi$
$293$	−9.49311	−0.554593	−0.277297	+	0.960784i	$0.589439\pi$
$294$	0	0
$295$	−8.00494	−0.466066
$296$	−24.3369	−1.41455
$297$	0	0
$298$	−27.1248	−1.57130
$299$	17.0186	0.984212
$300$	0	0
$301$	−4.00758	−0.230993
$302$	21.1484	1.21695
$303$	0	0
$304$	43.2219	2.47894
$305$	23.7832	1.36182
$306$	0	0
$307$	−20.3886	−1.16364	−0.581820	−	0.813318i	$-0.697659\pi$
$307$	−20.3886	−1.16364	−0.581820	+	0.813318i	$0.697659\pi$
$308$	−22.2604	−1.26841
$309$	0	0
$310$	−37.9837	−2.15733
$311$	−13.4256	−0.761298	−0.380649	−	0.924720i	$-0.624299\pi$
$311$	−13.4256	−0.761298	−0.380649	+	0.924720i	$0.624299\pi$
$312$	0	0
$313$	−8.23860	−0.465673	−0.232837	−	0.972516i	$-0.574801\pi$
$313$	−8.23860	−0.465673	−0.232837	+	0.972516i	$0.574801\pi$
$314$	−10.8801	−0.613997
$315$	0	0
$316$	−79.3768	−4.46529
$317$	−18.9347	−1.06348	−0.531739	−	0.846908i	$-0.678461\pi$
$317$	−18.9347	−1.06348	−0.531739	+	0.846908i	$0.678461\pi$
$318$	0	0
$319$	27.0387	1.51387
$320$	11.0214	0.616113
$321$	0	0
$322$	−14.5023	−0.808183
$323$	31.6111	1.75889
$324$	0	0
$325$	−24.3930	−1.35308
$326$	−0.901455	−0.0499270
$327$	0	0
$328$	−67.7213	−3.73928
$329$	−9.98363	−0.550415
$330$	0	0
$331$	10.9260	0.600547	0.300274	−	0.953853i	$-0.402922\pi$
$331$	10.9260	0.600547	0.300274	+	0.953853i	$0.402922\pi$
$332$	27.7054	1.52053
$333$	0	0
$334$	2.45971	0.134589
$335$	−18.5514	−1.01357
$336$	0	0
$337$	19.2079	1.04632	0.523162	−	0.852233i	$-0.324752\pi$
$337$	19.2079	1.04632	0.523162	+	0.852233i	$0.324752\pi$
$338$	9.95638	0.541556
$339$	0	0
$340$	100.406	5.44530
$341$	19.6017	1.06149
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−27.1726	−1.46505
$345$	0	0
$346$	−17.0555	−0.916912
$347$	−2.30435	−0.123704	−0.0618519	−	0.998085i	$-0.519701\pi$
$347$	−2.30435	−0.123704	−0.0618519	+	0.998085i	$0.519701\pi$
$348$	0	0
$349$	−10.0945	−0.540347	−0.270173	−	0.962812i	$-0.587081\pi$
$349$	−10.0945	−0.540347	−0.270173	+	0.962812i	$0.587081\pi$
$350$	20.7864	1.11108
$351$	0	0
$352$	−36.2732	−1.93337
$353$	20.0098	1.06501	0.532506	−	0.846426i	$-0.321250\pi$
$353$	20.0098	1.06501	0.532506	+	0.846426i	$0.321250\pi$
$354$	0	0
$355$	50.2118	2.66497
$356$	−73.7946	−3.91111
$357$	0	0
$358$	−32.4911	−1.71721
$359$	−24.9914	−1.31899	−0.659497	−	0.751707i	$-0.729231\pi$
$359$	−24.9914	−1.31899	−0.659497	+	0.751707i	$0.729231\pi$
$360$	0	0
$361$	8.80603	0.463475
$362$	30.7838	1.61796
$363$	0	0
$364$	14.0013	0.733865
$365$	−36.2985	−1.89995
$366$	0	0
$367$	37.5916	1.96226	0.981132	−	0.193340i	$-0.0619321\pi$
$367$	37.5916	1.96226	0.981132	+	0.193340i	$0.0619321\pi$
$368$	−46.1558	−2.40604
$369$	0	0
$370$	−33.4208	−1.73747
$371$	1.00415	0.0521330
$372$	0	0
$373$	21.3793	1.10698	0.553489	−	0.832856i	$-0.313296\pi$
$373$	21.3793	1.10698	0.553489	+	0.832856i	$0.313296\pi$
$374$	−74.1846	−3.83599
$375$	0	0
$376$	−67.6920	−3.49095
$377$	−17.0067	−0.875887
$378$	0	0
$379$	−24.9688	−1.28256	−0.641279	−	0.767308i	$-0.721596\pi$
$379$	−24.9688	−1.28256	−0.641279	+	0.767308i	$0.721596\pi$
$380$	88.3204	4.53074
$381$	0	0
$382$	−54.4918	−2.78804
$383$	−21.9425	−1.12121	−0.560605	−	0.828084i	$-0.689431\pi$
$383$	−21.9425	−1.12121	−0.560605	+	0.828084i	$0.689431\pi$
$384$	0	0
$385$	−17.3722	−0.885368
$386$	52.1428	2.65400
$387$	0	0
$388$	−74.6792	−3.79126
$389$	8.14442	0.412938	0.206469	−	0.978453i	$-0.433803\pi$
$389$	8.14442	0.412938	0.206469	+	0.978453i	$0.433803\pi$
$390$	0	0
$391$	−33.7569	−1.70716
$392$	−6.78030	−0.342457
$393$	0	0
$394$	18.2821	0.921038
$395$	−61.9461	−3.11685
$396$	0	0
$397$	−24.5214	−1.23069	−0.615347	−	0.788257i	$-0.710984\pi$
$397$	−24.5214	−1.23069	−0.615347	+	0.788257i	$0.710984\pi$
$398$	42.9760	2.15419
$399$	0	0
$400$	66.1556	3.30778
$401$	−17.6974	−0.883763	−0.441882	−	0.897073i	$-0.645689\pi$
$401$	−17.6974	−0.883763	−0.441882	+	0.897073i	$0.645689\pi$
$402$	0	0
$403$	−12.3290	−0.614150
$404$	−69.8731	−3.47632
$405$	0	0
$406$	14.4922	0.719233
$407$	17.2470	0.854902
$408$	0	0
$409$	−3.61653	−0.178826	−0.0894130	−	0.995995i	$-0.528499\pi$
$409$	−3.61653	−0.178826	−0.0894130	+	0.995995i	$0.528499\pi$
$410$	−92.9989	−4.59289
$411$	0	0
$412$	64.1088	3.15842
$413$	2.21412	0.108950
$414$	0	0
$415$	21.6214	1.06135
$416$	22.8149	1.11859
$417$	0	0
$418$	−65.2549	−3.19172
$419$	7.39479	0.361259	0.180630	−	0.983551i	$-0.442186\pi$
$419$	7.39479	0.361259	0.180630	+	0.983551i	$0.442186\pi$
$420$	0	0
$421$	−14.6420	−0.713608	−0.356804	−	0.934179i	$-0.616134\pi$
$421$	−14.6420	−0.713608	−0.356804	+	0.934179i	$0.616134\pi$
$422$	−45.5197	−2.21587
$423$	0	0
$424$	6.80846	0.330648
$425$	48.3841	2.34697
$426$	0	0
$427$	−6.57832	−0.318347
$428$	−67.0667	−3.24179
$429$	0	0
$430$	−37.3150	−1.79949
$431$	20.5154	0.988194	0.494097	−	0.869407i	$-0.335499\pi$
$431$	20.5154	0.988194	0.494097	+	0.869407i	$0.335499\pi$
$432$	0	0
$433$	15.2805	0.734333	0.367167	−	0.930155i	$-0.380328\pi$
$433$	15.2805	0.734333	0.367167	+	0.930155i	$0.380328\pi$
$434$	10.5061	0.504308
$435$	0	0
$436$	23.9876	1.14880
$437$	−29.6935	−1.42043
$438$	0	0
$439$	−12.3526	−0.589556	−0.294778	−	0.955566i	$-0.595246\pi$
$439$	−12.3526	−0.589556	−0.294778	+	0.955566i	$0.595246\pi$
$440$	−117.789	−5.61535
$441$	0	0
$442$	46.6603	2.21940
$443$	−23.3013	−1.10708	−0.553540	−	0.832823i	$-0.686723\pi$
$443$	−23.3013	−1.10708	−0.553540	+	0.832823i	$0.686723\pi$
$444$	0	0
$445$	−57.5898	−2.73002
$446$	−6.04894	−0.286426
$447$	0	0
$448$	−3.04845	−0.144026
$449$	−41.9961	−1.98192	−0.990959	−	0.134165i	$-0.957165\pi$
$449$	−41.9961	−1.98192	−0.990959	+	0.134165i	$0.957165\pi$
$450$	0	0
$451$	47.9926	2.25988
$452$	10.0834	0.474281
$453$	0	0
$454$	22.6690	1.06391
$455$	10.9267	0.512250
$456$	0	0
$457$	16.9573	0.793231	0.396615	−	0.917985i	$-0.370185\pi$
$457$	16.9573	0.793231	0.396615	+	0.917985i	$0.370185\pi$
$458$	2.56302	0.119762
$459$	0	0
$460$	−94.3157	−4.39749
$461$	22.4259	1.04448	0.522239	−	0.852799i	$-0.325097\pi$
$461$	22.4259	1.04448	0.522239	+	0.852799i	$0.325097\pi$
$462$	0	0
$463$	4.90815	0.228101	0.114051	−	0.993475i	$-0.463617\pi$
$463$	4.90815	0.228101	0.114051	+	0.993475i	$0.463617\pi$
$464$	46.1234	2.14122
$465$	0	0
$466$	−43.4167	−2.01124
$467$	−10.0412	−0.464654	−0.232327	−	0.972638i	$-0.574634\pi$
$467$	−10.0412	−0.464654	−0.232327	+	0.972638i	$0.574634\pi$
$468$	0	0
$469$	5.13122	0.236938
$470$	−92.9587	−4.28787
$471$	0	0
$472$	15.0124	0.691003
$473$	19.2566	0.885420
$474$	0	0
$475$	42.5601	1.95279
$476$	−27.7719	−1.27292
$477$	0	0
$478$	−13.2032	−0.603901
$479$	25.4444	1.16258	0.581291	−	0.813695i	$-0.302548\pi$
$479$	25.4444	1.16258	0.581291	+	0.813695i	$0.302548\pi$
$480$	0	0
$481$	−10.8479	−0.494624
$482$	−22.8401	−1.04034
$483$	0	0
$484$	56.0026	2.54557
$485$	−58.2801	−2.64636
$486$	0	0
$487$	16.0841	0.728839	0.364419	−	0.931235i	$-0.381267\pi$
$487$	16.0841	0.728839	0.364419	+	0.931235i	$0.381267\pi$
$488$	−44.6030	−2.01908
$489$	0	0
$490$	−9.31112	−0.420633
$491$	28.4648	1.28460	0.642300	−	0.766453i	$-0.277980\pi$
$491$	28.4648	1.28460	0.642300	+	0.766453i	$0.277980\pi$
$492$	0	0
$493$	33.7332	1.51927
$494$	41.0437	1.84664
$495$	0	0
$496$	33.4372	1.50137
$497$	−13.8883	−0.622977
$498$	0	0
$499$	13.5099	0.604786	0.302393	−	0.953183i	$-0.402214\pi$
$499$	13.5099	0.604786	0.302393	+	0.953183i	$0.402214\pi$
$500$	51.4382	2.30039
$501$	0	0
$502$	−20.9184	−0.933633
$503$	20.6323	0.919949	0.459975	−	0.887932i	$-0.347859\pi$
$503$	20.6323	0.919949	0.459975	+	0.887932i	$0.347859\pi$
$504$	0	0
$505$	−54.5294	−2.42653
$506$	69.6845	3.09785
$507$	0	0
$508$	−4.63271	−0.205543
$509$	1.46524	0.0649456	0.0324728	−	0.999473i	$-0.489662\pi$
$509$	1.46524	0.0649456	0.0324728	+	0.999473i	$0.489662\pi$
$510$	0	0
$511$	10.0400	0.444142
$512$	49.2750	2.17767
$513$	0	0
$514$	60.3143	2.66035
$515$	50.0309	2.20463
$516$	0	0
$517$	47.9718	2.10980
$518$	9.24403	0.406159
$519$	0	0
$520$	74.0861	3.24889
$521$	15.1230	0.662549	0.331275	−	0.943534i	$-0.392521\pi$
$521$	15.1230	0.662549	0.331275	+	0.943534i	$0.392521\pi$
$522$	0	0
$523$	−29.7942	−1.30281	−0.651404	−	0.758731i	$-0.725820\pi$
$523$	−29.7942	−1.30281	−0.651404	+	0.758731i	$0.725820\pi$
$524$	−6.28436	−0.274533
$525$	0	0
$526$	−23.0977	−1.00711
$527$	24.4549	1.06527
$528$	0	0
$529$	8.70915	0.378659
$530$	9.34978	0.406128
$531$	0	0
$532$	−24.4290	−1.05913
$533$	−30.1862	−1.30751
$534$	0	0
$535$	−52.3392	−2.26282
$536$	34.7912	1.50275
$537$	0	0
$538$	33.2804	1.43482
$539$	4.80505	0.206968
$540$	0	0
$541$	−9.71401	−0.417638	−0.208819	−	0.977954i	$-0.566962\pi$
$541$	−9.71401	−0.417638	−0.208819	+	0.977954i	$0.566962\pi$
$542$	19.8517	0.852705
$543$	0	0
$544$	−45.2541	−1.94025
$545$	18.7200	0.801879
$546$	0	0
$547$	10.6353	0.454732	0.227366	−	0.973809i	$-0.426989\pi$
$547$	10.6353	0.454732	0.227366	+	0.973809i	$0.426989\pi$
$548$	−36.8876	−1.57576
$549$	0	0
$550$	−99.8795	−4.25888
$551$	29.6727	1.26410
$552$	0	0
$553$	17.1340	0.728611
$554$	21.2768	0.903966
$555$	0	0
$556$	17.5462	0.744125
$557$	−8.26750	−0.350305	−0.175153	−	0.984541i	$-0.556042\pi$
$557$	−8.26750	−0.350305	−0.175153	+	0.984541i	$0.556042\pi$
$558$	0	0
$559$	−12.1119	−0.512280
$560$	−29.6340	−1.25226
$561$	0	0
$562$	−30.8576	−1.30165
$563$	19.6229	0.827008	0.413504	−	0.910502i	$-0.364305\pi$
$563$	19.6229	0.827008	0.413504	+	0.910502i	$0.364305\pi$
$564$	0	0
$565$	7.86912	0.331056
$566$	15.4028	0.647428
$567$	0	0
$568$	−94.1671	−3.95116
$569$	−36.0303	−1.51047	−0.755235	−	0.655454i	$-0.772477\pi$
$569$	−36.0303	−1.51047	−0.755235	+	0.655454i	$0.772477\pi$
$570$	0	0
$571$	21.8466	0.914250	0.457125	−	0.889402i	$-0.348879\pi$
$571$	21.8466	0.914250	0.457125	+	0.889402i	$0.348879\pi$
$572$	−67.2768	−2.81298
$573$	0	0
$574$	25.7230	1.07366
$575$	−45.4491	−1.89536
$576$	0	0
$577$	−19.8478	−0.826274	−0.413137	−	0.910669i	$-0.635567\pi$
$577$	−19.8478	−0.826274	−0.413137	+	0.910669i	$0.635567\pi$
$578$	−48.7700	−2.02857
$579$	0	0
$580$	94.2494	3.91349
$581$	−5.98037	−0.248108
$582$	0	0
$583$	−4.82500	−0.199831
$584$	68.0740	2.81692
$585$	0	0
$586$	24.4486	1.00996
$587$	−2.95082	−0.121793	−0.0608966	−	0.998144i	$-0.519396\pi$
$587$	−2.95082	−0.121793	−0.0608966	+	0.998144i	$0.519396\pi$
$588$	0	0
$589$	21.5112	0.886354
$590$	20.6160	0.848746
$591$	0	0
$592$	29.4205	1.20917
$593$	16.6183	0.682432	0.341216	−	0.939985i	$-0.389161\pi$
$593$	16.6183	0.682432	0.341216	+	0.939985i	$0.389161\pi$
$594$	0	0
$595$	−21.6733	−0.888521
$596$	48.7929	1.99863
$597$	0	0
$598$	−43.8298	−1.79233
$599$	38.1190	1.55750	0.778749	−	0.627335i	$-0.215854\pi$
$599$	38.1190	1.55750	0.778749	+	0.627335i	$0.215854\pi$
$600$	0	0
$601$	23.5289	0.959763	0.479881	−	0.877333i	$-0.340680\pi$
$601$	23.5289	0.959763	0.479881	+	0.877333i	$0.340680\pi$
$602$	10.3211	0.420658
$603$	0	0
$604$	−38.0423	−1.54792
$605$	43.7048	1.77685
$606$	0	0
$607$	−24.9940	−1.01448	−0.507239	−	0.861806i	$-0.669334\pi$
$607$	−24.9940	−1.01448	−0.507239	+	0.861806i	$0.669334\pi$
$608$	−39.8068	−1.61438
$609$	0	0
$610$	−61.2514	−2.48000
$611$	−30.1731	−1.22067
$612$	0	0
$613$	−24.4873	−0.989033	−0.494516	−	0.869168i	$-0.664655\pi$
$613$	−24.4873	−0.989033	−0.494516	+	0.869168i	$0.664655\pi$
$614$	52.5089	2.11909
$615$	0	0
$616$	32.5797	1.31267
$617$	−25.5183	−1.02733	−0.513664	−	0.857992i	$-0.671712\pi$
$617$	−25.5183	−1.02733	−0.513664	+	0.857992i	$0.671712\pi$
$618$	0	0
$619$	−8.02886	−0.322707	−0.161354	−	0.986897i	$-0.551586\pi$
$619$	−8.02886	−0.322707	−0.161354	+	0.986897i	$0.551586\pi$
$620$	68.3261	2.74404
$621$	0	0
$622$	34.5764	1.38639
$623$	15.9290	0.638183
$624$	0	0
$625$	−0.212826	−0.00851305
$626$	21.2177	0.848031
$627$	0	0
$628$	19.5714	0.780983
$629$	21.5172	0.857947
$630$	0	0
$631$	14.9596	0.595534	0.297767	−	0.954639i	$-0.403758\pi$
$631$	14.9596	0.595534	0.297767	+	0.954639i	$0.403758\pi$
$632$	116.174	4.62113
$633$	0	0
$634$	48.7645	1.93669
$635$	−3.61540	−0.143473
$636$	0	0
$637$	−3.02226	−0.119746
$638$	−69.6355	−2.75690
$639$	0	0
$640$	26.2006	1.03567
$641$	−40.7640	−1.61008	−0.805040	−	0.593220i	$-0.797856\pi$
$641$	−40.7640	−1.61008	−0.805040	+	0.593220i	$0.797856\pi$
$642$	0	0
$643$	36.2589	1.42991	0.714956	−	0.699170i	$-0.246447\pi$
$643$	36.2589	1.42991	0.714956	+	0.699170i	$0.246447\pi$
$644$	26.0872	1.02798
$645$	0	0
$646$	−81.4114	−3.20309
$647$	−2.06513	−0.0811887	−0.0405943	−	0.999176i	$-0.512925\pi$
$647$	−2.06513	−0.0811887	−0.0405943	+	0.999176i	$0.512925\pi$
$648$	0	0
$649$	−10.6390	−0.417617
$650$	62.8218	2.46407
$651$	0	0
$652$	1.62156	0.0635053
$653$	−23.6696	−0.926263	−0.463131	−	0.886290i	$-0.653274\pi$
$653$	−23.6696	−0.926263	−0.463131	+	0.886290i	$0.653274\pi$
$654$	0	0
$655$	−4.90435	−0.191629
$656$	81.8672	3.19638
$657$	0	0
$658$	25.7119	1.00235
$659$	−50.5108	−1.96762	−0.983812	−	0.179206i	$-0.942647\pi$
$659$	−50.5108	−1.96762	−0.983812	+	0.179206i	$0.942647\pi$
$660$	0	0
$661$	−9.85995	−0.383508	−0.191754	−	0.981443i	$-0.561418\pi$
$661$	−9.85995	−0.383508	−0.191754	+	0.981443i	$0.561418\pi$
$662$	−28.1389	−1.09365
$663$	0	0
$664$	−40.5488	−1.57360
$665$	−19.0645	−0.739290
$666$	0	0
$667$	−31.6869	−1.22692
$668$	−4.42460	−0.171193
$669$	0	0
$670$	47.7774	1.84580
$671$	31.6091	1.22026
$672$	0	0
$673$	33.6849	1.29846	0.649229	−	0.760593i	$-0.275092\pi$
$673$	33.6849	1.29846	0.649229	+	0.760593i	$0.275092\pi$
$674$	−49.4683	−1.90545
$675$	0	0
$676$	−17.9098	−0.688840
$677$	13.9184	0.534927	0.267464	−	0.963568i	$-0.413814\pi$
$677$	13.9184	0.534927	0.267464	+	0.963568i	$0.413814\pi$
$678$	0	0
$679$	16.1200	0.618628
$680$	−146.952	−5.63535
$681$	0	0
$682$	−50.4823	−1.93307
$683$	−0.478159	−0.0182963	−0.00914813	−	0.999958i	$-0.502912\pi$
$683$	−0.478159	−0.0182963	−0.00914813	+	0.999958i	$0.502912\pi$
$684$	0	0
$685$	−28.7873	−1.09991
$686$	2.57541	0.0983294
$687$	0	0
$688$	32.8485	1.25234
$689$	3.03481	0.115617
$690$	0	0
$691$	−23.8010	−0.905435	−0.452717	−	0.891654i	$-0.649545\pi$
$691$	−23.8010	−0.905435	−0.452717	+	0.891654i	$0.649545\pi$
$692$	30.6800	1.16628
$693$	0	0
$694$	5.93463	0.225275
$695$	13.6932	0.519412
$696$	0	0
$697$	59.8751	2.26793
$698$	25.9974	0.984018
$699$	0	0
$700$	−37.3911	−1.41325
$701$	−12.3729	−0.467320	−0.233660	−	0.972318i	$-0.575070\pi$
$701$	−12.3729	−0.467320	−0.233660	+	0.972318i	$0.575070\pi$
$702$	0	0
$703$	18.9271	0.713851
$704$	14.6480	0.552066
$705$	0	0
$706$	−51.5333	−1.93948
$707$	15.0825	0.567237
$708$	0	0
$709$	−6.03901	−0.226800	−0.113400	−	0.993549i	$-0.536174\pi$
$709$	−6.03901	−0.226800	−0.113400	+	0.993549i	$0.536174\pi$
$710$	−129.316	−4.85314
$711$	0	0
$712$	108.004	4.04761
$713$	−22.9714	−0.860286
$714$	0	0
$715$	−52.5032	−1.96351
$716$	58.4459	2.18423
$717$	0	0
$718$	64.3629	2.40200
$719$	−38.8243	−1.44790	−0.723951	−	0.689852i	$-0.757676\pi$
$719$	−38.8243	−1.44790	−0.723951	+	0.689852i	$0.757676\pi$
$720$	0	0
$721$	−13.8383	−0.515365
$722$	−22.6791	−0.844029
$723$	0	0
$724$	−55.3748	−2.05799
$725$	45.4172	1.68675
$726$	0	0
$727$	−45.7266	−1.69591	−0.847953	−	0.530071i	$-0.822165\pi$
$727$	−45.7266	−1.69591	−0.847953	+	0.530071i	$0.822165\pi$
$728$	−20.4918	−0.759478
$729$	0	0
$730$	93.4832	3.45997
$731$	24.0244	0.888573
$732$	0	0
$733$	26.3784	0.974309	0.487155	−	0.873316i	$-0.338035\pi$
$733$	26.3784	0.974309	0.487155	+	0.873316i	$0.338035\pi$
$734$	−96.8135	−3.57345
$735$	0	0
$736$	42.5089	1.56690
$737$	−24.6558	−0.908208
$738$	0	0
$739$	−44.0186	−1.61925	−0.809625	−	0.586948i	$-0.800329\pi$
$739$	−44.0186	−1.61925	−0.809625	+	0.586948i	$0.800329\pi$
$740$	60.1184	2.20999
$741$	0	0
$742$	−2.58610	−0.0949387
$743$	39.6817	1.45578	0.727891	−	0.685693i	$-0.240501\pi$
$743$	39.6817	1.45578	0.727891	+	0.685693i	$0.240501\pi$
$744$	0	0
$745$	38.0782	1.39508
$746$	−55.0604	−2.01590
$747$	0	0
$748$	133.445	4.87924
$749$	14.4768	0.528969
$750$	0	0
$751$	−40.3374	−1.47193	−0.735965	−	0.677019i	$-0.763271\pi$
$751$	−40.3374	−1.47193	−0.735965	+	0.677019i	$0.763271\pi$
$752$	81.8318	2.98410
$753$	0	0
$754$	43.7990	1.59507
$755$	−29.6885	−1.08047
$756$	0	0
$757$	31.1435	1.13193	0.565965	−	0.824429i	$-0.308504\pi$
$757$	31.1435	1.13193	0.565965	+	0.824429i	$0.308504\pi$
$758$	64.3047	2.33565
$759$	0	0
$760$	−129.263	−4.68887
$761$	−1.19795	−0.0434255	−0.0217128	−	0.999764i	$-0.506912\pi$
$761$	−1.19795	−0.0434255	−0.0217128	+	0.999764i	$0.506912\pi$
$762$	0	0
$763$	−5.17787	−0.187451
$764$	98.0214	3.54629
$765$	0	0
$766$	56.5108	2.04182
$767$	6.69166	0.241622
$768$	0	0
$769$	9.53802	0.343950	0.171975	−	0.985101i	$-0.444985\pi$
$769$	9.53802	0.343950	0.171975	+	0.985101i	$0.444985\pi$
$770$	44.7404	1.61233
$771$	0	0
$772$	−93.7960	−3.37579
$773$	27.3573	0.983974	0.491987	−	0.870603i	$-0.336271\pi$
$773$	27.3573	0.983974	0.491987	+	0.870603i	$0.336271\pi$
$774$	0	0
$775$	32.9252	1.18271
$776$	109.298	3.92358
$777$	0	0
$778$	−20.9752	−0.751996
$779$	52.6678	1.88702
$780$	0	0
$781$	66.7341	2.38794
$782$	86.9376	3.10888
$783$	0	0
$784$	8.19660	0.292736
$785$	15.2736	0.545139
$786$	0	0
$787$	19.7641	0.704514	0.352257	−	0.935903i	$-0.385414\pi$
$787$	19.7641	0.704514	0.352257	+	0.935903i	$0.385414\pi$
$788$	−32.8863	−1.17153
$789$	0	0
$790$	159.536	5.67605
$791$	−2.17656	−0.0773894
$792$	0	0
$793$	−19.8814	−0.706008
$794$	63.1525	2.24120
$795$	0	0
$796$	−77.3065	−2.74005
$797$	48.1120	1.70422	0.852108	−	0.523366i	$-0.175324\pi$
$797$	48.1120	1.70422	0.852108	+	0.523366i	$0.175324\pi$
$798$	0	0
$799$	59.8492	2.11731
$800$	−60.9285	−2.15415
$801$	0	0
$802$	45.5779	1.60941
$803$	−48.2425	−1.70244
$804$	0	0
$805$	20.3586	0.717547
$806$	31.7521	1.11842
$807$	0	0
$808$	102.264	3.59764
$809$	45.1368	1.58693	0.793463	−	0.608618i	$-0.208276\pi$
$809$	45.1368	1.58693	0.793463	+	0.608618i	$0.208276\pi$
$810$	0	0
$811$	−44.9022	−1.57673	−0.788366	−	0.615207i	$-0.789072\pi$
$811$	−44.9022	−1.57673	−0.788366	+	0.615207i	$0.789072\pi$
$812$	−26.0689	−0.914839
$813$	0	0
$814$	−44.4180	−1.55685
$815$	1.26548	0.0443277
$816$	0	0
$817$	21.1325	0.739333
$818$	9.31403	0.325658
$819$	0	0
$820$	167.289	5.84199
$821$	47.1163	1.64437	0.822185	−	0.569221i	$-0.192755\pi$
$821$	47.1163	1.64437	0.822185	+	0.569221i	$0.192755\pi$
$822$	0	0
$823$	−3.85102	−0.134238	−0.0671191	−	0.997745i	$-0.521381\pi$
$823$	−3.85102	−0.134238	−0.0671191	+	0.997745i	$0.521381\pi$
$824$	−93.8278	−3.26865
$825$	0	0
$826$	−5.70227	−0.198407
$827$	3.78077	0.131470	0.0657351	−	0.997837i	$-0.479061\pi$
$827$	3.78077	0.131470	0.0657351	+	0.997837i	$0.479061\pi$
$828$	0	0
$829$	−23.5958	−0.819518	−0.409759	−	0.912194i	$-0.634387\pi$
$829$	−23.5958	−0.819518	−0.409759	+	0.912194i	$0.634387\pi$
$830$	−55.6840	−1.93282
$831$	0	0
$832$	−9.21321	−0.319411
$833$	5.99473	0.207705
$834$	0	0
$835$	−3.45298	−0.119495
$836$	117.382	4.05975
$837$	0	0
$838$	−19.0446	−0.657884
$839$	25.2441	0.871522	0.435761	−	0.900062i	$-0.356479\pi$
$839$	25.2441	0.871522	0.435761	+	0.900062i	$0.356479\pi$
$840$	0	0
$841$	2.66464	0.0918840
$842$	37.7091	1.29954
$843$	0	0
$844$	81.8822	2.81850
$845$	−13.9769	−0.480821
$846$	0	0
$847$	−12.0885	−0.415366
$848$	−8.23064	−0.282641
$849$	0	0
$850$	−124.609	−4.27404
$851$	−20.2119	−0.692856
$852$	0	0
$853$	−1.23491	−0.0422826	−0.0211413	−	0.999776i	$-0.506730\pi$
$853$	−1.23491	−0.0422826	−0.0211413	+	0.999776i	$0.506730\pi$
$854$	16.9418	0.579737
$855$	0	0
$856$	98.1568	3.35493
$857$	−21.9609	−0.750169	−0.375085	−	0.926991i	$-0.622386\pi$
$857$	−21.9609	−0.750169	−0.375085	+	0.926991i	$0.622386\pi$
$858$	0	0
$859$	17.3623	0.592395	0.296197	−	0.955127i	$-0.404281\pi$
$859$	17.3623	0.592395	0.296197	+	0.955127i	$0.404281\pi$
$860$	67.1233	2.28889
$861$	0	0
$862$	−52.8356	−1.79959
$863$	−47.6552	−1.62220	−0.811102	−	0.584905i	$-0.801132\pi$
$863$	−47.6552	−1.62220	−0.811102	+	0.584905i	$0.801132\pi$
$864$	0	0
$865$	23.9429	0.814082
$866$	−39.3535	−1.33728
$867$	0	0
$868$	−18.8986	−0.641462
$869$	−82.3296	−2.79284
$870$	0	0
$871$	15.5079	0.525465
$872$	−35.1075	−1.18889
$873$	0	0
$874$	76.4729	2.58673
$875$	−11.1033	−0.375359
$876$	0	0
$877$	38.2044	1.29007	0.645035	−	0.764153i	$-0.276843\pi$
$877$	38.2044	1.29007	0.645035	+	0.764153i	$0.276843\pi$
$878$	31.8129	1.07363
$879$	0	0
$880$	142.393	4.80006
$881$	−19.8672	−0.669342	−0.334671	−	0.942335i	$-0.608625\pi$
$881$	−19.8672	−0.669342	−0.334671	+	0.942335i	$0.608625\pi$
$882$	0	0
$883$	9.70573	0.326624	0.163312	−	0.986574i	$-0.447782\pi$
$883$	9.70573	0.326624	0.163312	+	0.986574i	$0.447782\pi$
$884$	−83.9338	−2.82300
$885$	0	0
$886$	60.0104	2.01609
$887$	44.4765	1.49338	0.746688	−	0.665175i	$-0.231643\pi$
$887$	44.4765	1.49338	0.746688	+	0.665175i	$0.231643\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	148.317	4.97160
$891$	0	0
$892$	10.8810	0.364323
$893$	52.6451	1.76170
$894$	0	0
$895$	45.6115	1.52463
$896$	−7.24694	−0.242103
$897$	0	0
$898$	108.157	3.60924
$899$	22.9553	0.765602
$900$	0	0
$901$	−6.01963	−0.200543
$902$	−123.600	−4.11544
$903$	0	0
$904$	−14.7577	−0.490834
$905$	−43.2149	−1.43651
$906$	0	0
$907$	−24.0803	−0.799572	−0.399786	−	0.916608i	$-0.630916\pi$
$907$	−24.0803	−0.799572	−0.399786	+	0.916608i	$0.630916\pi$
$908$	−40.7776	−1.35325
$909$	0	0
$910$	−28.1406	−0.932852
$911$	−1.68964	−0.0559802	−0.0279901	−	0.999608i	$-0.508911\pi$
$911$	−1.68964	−0.0559802	−0.0279901	+	0.999608i	$0.508911\pi$
$912$	0	0
$913$	28.7360	0.951023
$914$	−43.6720	−1.44454
$915$	0	0
$916$	−4.61043	−0.152333
$917$	1.35652	0.0447962
$918$	0	0
$919$	40.5462	1.33750	0.668748	−	0.743489i	$-0.266831\pi$
$919$	40.5462	1.33750	0.668748	+	0.743489i	$0.266831\pi$
$920$	138.038	4.55097
$921$	0	0
$922$	−57.7558	−1.90208
$923$	−41.9741	−1.38160
$924$	0	0
$925$	28.9700	0.952528
$926$	−12.6405	−0.415392
$927$	0	0
$928$	−42.4791	−1.39444
$929$	33.1179	1.08656	0.543281	−	0.839551i	$-0.317182\pi$
$929$	33.1179	1.08656	0.543281	+	0.839551i	$0.317182\pi$
$930$	0	0
$931$	5.27314	0.172820
$932$	78.0992	2.55822
$933$	0	0
$934$	25.8603	0.846174
$935$	104.142	3.40579
$936$	0	0
$937$	29.0490	0.948991	0.474495	−	0.880258i	$-0.342631\pi$
$937$	29.0490	0.948991	0.474495	+	0.880258i	$0.342631\pi$
$938$	−13.2150	−0.431484
$939$	0	0
$940$	167.217	5.45401
$941$	−9.45338	−0.308171	−0.154086	−	0.988057i	$-0.549243\pi$
$941$	−9.45338	−0.308171	−0.154086	+	0.988057i	$0.549243\pi$
$942$	0	0
$943$	−56.2430	−1.83152
$944$	−18.1483	−0.590677
$945$	0	0
$946$	−49.5936	−1.61243
$947$	−22.3098	−0.724970	−0.362485	−	0.931990i	$-0.618072\pi$
$947$	−22.3098	−0.724970	−0.362485	+	0.931990i	$0.618072\pi$
$948$	0	0
$949$	30.3434	0.984987
$950$	−109.609	−3.55620
$951$	0	0
$952$	40.6461	1.31735
$953$	49.6079	1.60696	0.803478	−	0.595334i	$-0.202980\pi$
$953$	49.6079	1.60696	0.803478	+	0.595334i	$0.202980\pi$
$954$	0	0
$955$	76.4965	2.47537
$956$	23.7503	0.768141
$957$	0	0
$958$	−65.5296	−2.11716
$959$	7.96241	0.257120
$960$	0	0
$961$	−14.3586	−0.463180
$962$	27.9378	0.900752
$963$	0	0
$964$	41.0854	1.32327
$965$	−73.1990	−2.35636
$966$	0	0
$967$	44.5415	1.43236	0.716179	−	0.697916i	$-0.245889\pi$
$967$	44.5415	1.43236	0.716179	+	0.697916i	$0.245889\pi$
$968$	−81.9638	−2.63442
$969$	0	0
$970$	150.095	4.81926
$971$	−10.9162	−0.350318	−0.175159	−	0.984540i	$-0.556044\pi$
$971$	−10.9162	−0.350318	−0.175159	+	0.984540i	$0.556044\pi$
$972$	0	0
$973$	−3.78746	−0.121420
$974$	−41.4230	−1.32728
$975$	0	0
$976$	53.9198	1.72593
$977$	−47.4189	−1.51706	−0.758532	−	0.651636i	$-0.774083\pi$
$977$	−47.4189	−1.51706	−0.758532	+	0.651636i	$0.774083\pi$
$978$	0	0
$979$	−76.5398	−2.44622
$980$	16.7491	0.535030
$981$	0	0
$982$	−73.3085	−2.33937
$983$	27.6546	0.882046	0.441023	−	0.897496i	$-0.354616\pi$
$983$	27.6546	0.882046	0.441023	+	0.897496i	$0.354616\pi$
$984$	0	0
$985$	−25.6647	−0.817745
$986$	−86.8766	−2.76671
$987$	0	0
$988$	−73.8306	−2.34887
$989$	−22.5670	−0.717589
$990$	0	0
$991$	−9.83870	−0.312537	−0.156268	−	0.987715i	$-0.549946\pi$
$991$	−9.83870	−0.312537	−0.156268	+	0.987715i	$0.549946\pi$
$992$	−30.7952	−0.977749
$993$	0	0
$994$	35.7681	1.13449
$995$	−60.3304	−1.91260
$996$	0	0
$997$	52.0999	1.65002	0.825010	−	0.565119i	$-0.191170\pi$
$997$	52.0999	1.65002	0.825010	+	0.565119i	$0.191170\pi$
$998$	−34.7935	−1.10137
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.t.1.1		16
3.2	odd	2		889.2.a.c.1.16	✓	16
21.20	even	2		6223.2.a.k.1.16		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.c.1.16	✓	16	3.2	odd	2
6223.2.a.k.1.16		16	21.20	even	2
8001.2.a.t.1.1		16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.57541 q^{2} +4.63271 q^{4} +3.61540 q^{5} -1.00000 q^{7} -6.78030 q^{8} +O(q^{10})\)	\(q-2.57541 q^{2} +4.63271 q^{4} +3.61540 q^{5} -1.00000 q^{7} -6.78030 q^{8} -9.31112 q^{10} +4.80505 q^{11} -3.02226 q^{13} +2.57541 q^{14} +8.19660 q^{16} +5.99473 q^{17} +5.27314 q^{19} +16.7491 q^{20} -12.3750 q^{22} -5.63109 q^{23} +8.07110 q^{25} +7.78354 q^{26} -4.63271 q^{28} +5.62713 q^{29} +4.07939 q^{31} -7.54897 q^{32} -15.4389 q^{34} -3.61540 q^{35} +3.58935 q^{37} -13.5805 q^{38} -24.5135 q^{40} +9.98794 q^{41} +4.00758 q^{43} +22.2604 q^{44} +14.5023 q^{46} +9.98363 q^{47} +1.00000 q^{49} -20.7864 q^{50} -14.0013 q^{52} -1.00415 q^{53} +17.3722 q^{55} +6.78030 q^{56} -14.4922 q^{58} -2.21412 q^{59} +6.57832 q^{61} -10.5061 q^{62} +3.04845 q^{64} -10.9267 q^{65} -5.13122 q^{67} +27.7719 q^{68} +9.31112 q^{70} +13.8883 q^{71} -10.0400 q^{73} -9.24403 q^{74} +24.4290 q^{76} -4.80505 q^{77} -17.1340 q^{79} +29.6340 q^{80} -25.7230 q^{82} +5.98037 q^{83} +21.6733 q^{85} -10.3211 q^{86} -32.5797 q^{88} -15.9290 q^{89} +3.02226 q^{91} -26.0872 q^{92} -25.7119 q^{94} +19.0645 q^{95} -16.1200 q^{97} -2.57541 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8}+O(q^{10}) \) 16 * q + 2 * q^2 + 12 * q^4 + 9 * q^5 - 16 * q^7 + 6 * q^8	\( 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8} - 2 q^{10} + 22 q^{11} - 4 q^{13} - 2 q^{14} + 12 q^{16} + 18 q^{17} - 15 q^{19} + 40 q^{20} - 11 q^{22} + 5 q^{23} + 15 q^{25} + 24 q^{26} - 12 q^{28} + 12 q^{29} - 32 q^{31} + 9 q^{32} - 14 q^{34} - 9 q^{35} - 2 q^{37} - 3 q^{38} - 14 q^{40} + 45 q^{41} - 3 q^{43} + 54 q^{44} + 49 q^{47} + 16 q^{49} + 6 q^{50} + 38 q^{52} - 16 q^{53} + 7 q^{55} - 6 q^{56} + 16 q^{58} + 35 q^{59} - 11 q^{61} - 17 q^{62} - 2 q^{64} - 14 q^{65} + 17 q^{67} + 71 q^{68} + 2 q^{70} + 81 q^{71} - 15 q^{73} - 13 q^{74} + 14 q^{76} - 22 q^{77} - 34 q^{79} + 33 q^{80} - 14 q^{82} + 39 q^{83} - 17 q^{85} - 36 q^{86} + 61 q^{88} + 32 q^{89} + 4 q^{91} - 37 q^{92} + 13 q^{94} + 33 q^{95} - 4 q^{97} + 2 q^{98}+O(q^{100}) \) 16 * q + 2 * q^2 + 12 * q^4 + 9 * q^5 - 16 * q^7 + 6 * q^8 - 2 * q^10 + 22 * q^11 - 4 * q^13 - 2 * q^14 + 12 * q^16 + 18 * q^17 - 15 * q^19 + 40 * q^20 - 11 * q^22 + 5 * q^23 + 15 * q^25 + 24 * q^26 - 12 * q^28 + 12 * q^29 - 32 * q^31 + 9 * q^32 - 14 * q^34 - 9 * q^35 - 2 * q^37 - 3 * q^38 - 14 * q^40 + 45 * q^41 - 3 * q^43 + 54 * q^44 + 49 * q^47 + 16 * q^49 + 6 * q^50 + 38 * q^52 - 16 * q^53 + 7 * q^55 - 6 * q^56 + 16 * q^58 + 35 * q^59 - 11 * q^61 - 17 * q^62 - 2 * q^64 - 14 * q^65 + 17 * q^67 + 71 * q^68 + 2 * q^70 + 81 * q^71 - 15 * q^73 - 13 * q^74 + 14 * q^76 - 22 * q^77 - 34 * q^79 + 33 * q^80 - 14 * q^82 + 39 * q^83 - 17 * q^85 - 36 * q^86 + 61 * q^88 + 32 * q^89 + 4 * q^91 - 37 * q^92 + 13 * q^94 + 33 * q^95 - 4 * q^97 + 2 * q^98

Introduction

L-functions

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