LMFDB - Embedded newform 8043.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8043,2,Mod(1,8043)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8043, 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("8043.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8043 = 3 \cdot 7 \cdot 383 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8043.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:	$64.2236783457$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8043.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} +3.00000 q^{5} -2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

\(q+2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} +3.00000 q^{5} -2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{9} +6.00000 q^{10} +1.00000 q^{11} -2.00000 q^{12} -4.00000 q^{13} +2.00000 q^{14} -3.00000 q^{15} -4.00000 q^{16} +4.00000 q^{17} +2.00000 q^{18} -5.00000 q^{19} +6.00000 q^{20} -1.00000 q^{21} +2.00000 q^{22} +6.00000 q^{23} +4.00000 q^{25} -8.00000 q^{26} -1.00000 q^{27} +2.00000 q^{28} -6.00000 q^{30} +7.00000 q^{31} -8.00000 q^{32} -1.00000 q^{33} +8.00000 q^{34} +3.00000 q^{35} +2.00000 q^{36} +4.00000 q^{37} -10.0000 q^{38} +4.00000 q^{39} -3.00000 q^{41} -2.00000 q^{42} +5.00000 q^{43} +2.00000 q^{44} +3.00000 q^{45} +12.0000 q^{46} -3.00000 q^{47} +4.00000 q^{48} +1.00000 q^{49} +8.00000 q^{50} -4.00000 q^{51} -8.00000 q^{52} +10.0000 q^{53} -2.00000 q^{54} +3.00000 q^{55} +5.00000 q^{57} +13.0000 q^{59} -6.00000 q^{60} -10.0000 q^{61} +14.0000 q^{62} +1.00000 q^{63} -8.00000 q^{64} -12.0000 q^{65} -2.00000 q^{66} +13.0000 q^{67} +8.00000 q^{68} -6.00000 q^{69} +6.00000 q^{70} +8.00000 q^{71} +11.0000 q^{73} +8.00000 q^{74} -4.00000 q^{75} -10.0000 q^{76} +1.00000 q^{77} +8.00000 q^{78} -8.00000 q^{79} -12.0000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +4.00000 q^{83} -2.00000 q^{84} +12.0000 q^{85} +10.0000 q^{86} +9.00000 q^{89} +6.00000 q^{90} -4.00000 q^{91} +12.0000 q^{92} -7.00000 q^{93} -6.00000 q^{94} -15.0000 q^{95} +8.00000 q^{96} -4.00000 q^{97} +2.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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 $

$2$	2.00000	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$2$	2.00000	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.00000	1.00000
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	−2.00000	−0.816497
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	6.00000	1.89737
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	−2.00000	−0.577350
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	2.00000	0.534522
$15$	−3.00000	−0.774597
$16$	−4.00000	−1.00000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	2.00000	0.471405
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	6.00000	1.34164
$21$	−1.00000	−0.218218
$22$	2.00000	0.426401
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	−8.00000	−1.56893
$27$	−1.00000	−0.192450
$28$	2.00000	0.377964
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	−6.00000	−1.09545
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	−8.00000	−1.41421
$33$	−1.00000	−0.174078
$34$	8.00000	1.37199
$35$	3.00000	0.507093
$36$	2.00000	0.333333
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	−10.0000	−1.62221
$39$	4.00000	0.640513
$40$	0	0
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	−2.00000	−0.308607
$43$	5.00000	0.762493	0.381246	−	0.924473i	$-0.375495\pi$
$43$	5.00000	0.762493	0.381246	+	0.924473i	$0.375495\pi$
$44$	2.00000	0.301511
$45$	3.00000	0.447214
$46$	12.0000	1.76930
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	4.00000	0.577350
$49$	1.00000	0.142857
$50$	8.00000	1.13137
$51$	−4.00000	−0.560112
$52$	−8.00000	−1.10940
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	−2.00000	−0.272166
$55$	3.00000	0.404520
$56$	0	0
$57$	5.00000	0.662266
$58$	0	0
$59$	13.0000	1.69246	0.846228	−	0.532821i	$-0.178868\pi$
$59$	13.0000	1.69246	0.846228	+	0.532821i	$0.178868\pi$
$60$	−6.00000	−0.774597
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	14.0000	1.77800
$63$	1.00000	0.125988
$64$	−8.00000	−1.00000
$65$	−12.0000	−1.48842
$66$	−2.00000	−0.246183
$67$	13.0000	1.58820	0.794101	−	0.607785i	$-0.207942\pi$
$67$	13.0000	1.58820	0.794101	+	0.607785i	$0.207942\pi$
$68$	8.00000	0.970143
$69$	−6.00000	−0.722315
$70$	6.00000	0.717137
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	11.0000	1.28745	0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000	1.28745	0.643726	+	0.765256i	$0.277388\pi$
$74$	8.00000	0.929981
$75$	−4.00000	−0.461880
$76$	−10.0000	−1.14708
$77$	1.00000	0.113961
$78$	8.00000	0.905822
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	−12.0000	−1.34164
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	−2.00000	−0.218218
$85$	12.0000	1.30158
$86$	10.0000	1.07833
$87$	0	0
$88$	0	0
$89$	9.00000	0.953998	0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000	0.953998	0.476999	+	0.878904i	$0.341725\pi$
$90$	6.00000	0.632456
$91$	−4.00000	−0.419314
$92$	12.0000	1.25109
$93$	−7.00000	−0.725866
$94$	−6.00000	−0.618853
$95$	−15.0000	−1.53897
$96$	8.00000	0.816497
$97$	−4.00000	−0.406138	−0.203069	−	0.979164i	$-0.565092\pi$
$97$	−4.00000	−0.406138	−0.203069	+	0.979164i	$0.565092\pi$
$98$	2.00000	0.202031
$99$	1.00000	0.100504
$100$	8.00000	0.800000
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	−8.00000	−0.792118
$103$	5.00000	0.492665	0.246332	−	0.969185i	$-0.420775\pi$
$103$	5.00000	0.492665	0.246332	+	0.969185i	$0.420775\pi$
$104$	0	0
$105$	−3.00000	−0.292770
$106$	20.0000	1.94257
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	−2.00000	−0.192450
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	6.00000	0.572078
$111$	−4.00000	−0.379663
$112$	−4.00000	−0.377964
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	10.0000	0.936586
$115$	18.0000	1.67851
$116$	0	0
$117$	−4.00000	−0.369800
$118$	26.0000	2.39349
$119$	4.00000	0.366679
$120$	0	0
$121$	−10.0000	−0.909091
$122$	−20.0000	−1.81071
$123$	3.00000	0.270501
$124$	14.0000	1.25724
$125$	−3.00000	−0.268328
$126$	2.00000	0.178174
$127$	−18.0000	−1.59724	−0.798621	−	0.601834i	$-0.794437\pi$
$127$	−18.0000	−1.59724	−0.798621	+	0.601834i	$0.794437\pi$
$128$	0	0
$129$	−5.00000	−0.440225
$130$	−24.0000	−2.10494
$131$	1.00000	0.0873704	0.0436852	−	0.999045i	$-0.486090\pi$
$131$	1.00000	0.0873704	0.0436852	+	0.999045i	$0.486090\pi$
$132$	−2.00000	−0.174078
$133$	−5.00000	−0.433555
$134$	26.0000	2.24606
$135$	−3.00000	−0.258199
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	−12.0000	−1.02151
$139$	−9.00000	−0.763370	−0.381685	−	0.924292i	$-0.624656\pi$
$139$	−9.00000	−0.763370	−0.381685	+	0.924292i	$0.624656\pi$
$140$	6.00000	0.507093
$141$	3.00000	0.252646
$142$	16.0000	1.34269
$143$	−4.00000	−0.334497
$144$	−4.00000	−0.333333
$145$	0	0
$146$	22.0000	1.82073
$147$	−1.00000	−0.0824786
$148$	8.00000	0.657596
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	−8.00000	−0.653197
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	0	0
$153$	4.00000	0.323381
$154$	2.00000	0.161165
$155$	21.0000	1.68676
$156$	8.00000	0.640513
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	−16.0000	−1.27289
$159$	−10.0000	−0.793052
$160$	−24.0000	−1.89737
$161$	6.00000	0.472866
$162$	2.00000	0.157135
$163$	−20.0000	−1.56652	−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000	−1.56652	−0.783260	+	0.621694i	$0.786445\pi$
$164$	−6.00000	−0.468521
$165$	−3.00000	−0.233550
$166$	8.00000	0.620920
$167$	−4.00000	−0.309529	−0.154765	−	0.987951i	$-0.549462\pi$
$167$	−4.00000	−0.309529	−0.154765	+	0.987951i	$0.549462\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	24.0000	1.84072
$171$	−5.00000	−0.382360
$172$	10.0000	0.762493
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	−4.00000	−0.301511
$177$	−13.0000	−0.977140
$178$	18.0000	1.34916
$179$	15.0000	1.12115	0.560576	−	0.828103i	$-0.310580\pi$
$179$	15.0000	1.12115	0.560576	+	0.828103i	$0.310580\pi$
$180$	6.00000	0.447214
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	−8.00000	−0.592999
$183$	10.0000	0.739221
$184$	0	0
$185$	12.0000	0.882258
$186$	−14.0000	−1.02653
$187$	4.00000	0.292509
$188$	−6.00000	−0.437595
$189$	−1.00000	−0.0727393
$190$	−30.0000	−2.17643
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	8.00000	0.577350
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	−8.00000	−0.574367
$195$	12.0000	0.859338
$196$	2.00000	0.142857
$197$	−1.00000	−0.0712470	−0.0356235	−	0.999365i	$-0.511342\pi$
$197$	−1.00000	−0.0712470	−0.0356235	+	0.999365i	$0.511342\pi$
$198$	2.00000	0.142134
$199$	−2.00000	−0.141776	−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000	−0.141776	−0.0708881	+	0.997484i	$0.522583\pi$
$200$	0	0
$201$	−13.0000	−0.916949
$202$	36.0000	2.53295
$203$	0	0
$204$	−8.00000	−0.560112
$205$	−9.00000	−0.628587
$206$	10.0000	0.696733
$207$	6.00000	0.417029
$208$	16.0000	1.10940
$209$	−5.00000	−0.345857
$210$	−6.00000	−0.414039
$211$	−22.0000	−1.51454	−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000	−1.51454	−0.757271	+	0.653101i	$0.773468\pi$
$212$	20.0000	1.37361
$213$	−8.00000	−0.548151
$214$	−16.0000	−1.09374
$215$	15.0000	1.02299
$216$	0	0
$217$	7.00000	0.475191
$218$	8.00000	0.541828
$219$	−11.0000	−0.743311
$220$	6.00000	0.404520
$221$	−16.0000	−1.07628
$222$	−8.00000	−0.536925
$223$	−9.00000	−0.602685	−0.301342	−	0.953516i	$-0.597435\pi$
$223$	−9.00000	−0.602685	−0.301342	+	0.953516i	$0.597435\pi$
$224$	−8.00000	−0.534522
$225$	4.00000	0.266667
$226$	12.0000	0.798228
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	10.0000	0.662266
$229$	7.00000	0.462573	0.231287	−	0.972886i	$-0.425707\pi$
$229$	7.00000	0.462573	0.231287	+	0.972886i	$0.425707\pi$
$230$	36.0000	2.37377
$231$	−1.00000	−0.0657952
$232$	0	0
$233$	27.0000	1.76883	0.884414	−	0.466702i	$-0.154558\pi$
$233$	27.0000	1.76883	0.884414	+	0.466702i	$0.154558\pi$
$234$	−8.00000	−0.522976
$235$	−9.00000	−0.587095
$236$	26.0000	1.69246
$237$	8.00000	0.519656
$238$	8.00000	0.518563
$239$	−5.00000	−0.323423	−0.161712	−	0.986838i	$-0.551701\pi$
$239$	−5.00000	−0.323423	−0.161712	+	0.986838i	$0.551701\pi$
$240$	12.0000	0.774597
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	−20.0000	−1.28565
$243$	−1.00000	−0.0641500
$244$	−20.0000	−1.28037
$245$	3.00000	0.191663
$246$	6.00000	0.382546
$247$	20.0000	1.27257
$248$	0	0
$249$	−4.00000	−0.253490
$250$	−6.00000	−0.379473
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	2.00000	0.125988
$253$	6.00000	0.377217
$254$	−36.0000	−2.25884
$255$	−12.0000	−0.751469
$256$	16.0000	1.00000
$257$	−27.0000	−1.68421	−0.842107	−	0.539311i	$-0.818685\pi$
$257$	−27.0000	−1.68421	−0.842107	+	0.539311i	$0.818685\pi$
$258$	−10.0000	−0.622573
$259$	4.00000	0.248548
$260$	−24.0000	−1.48842
$261$	0	0
$262$	2.00000	0.123560
$263$	30.0000	1.84988	0.924940	−	0.380114i	$-0.124115\pi$
$263$	30.0000	1.84988	0.924940	+	0.380114i	$0.124115\pi$
$264$	0	0
$265$	30.0000	1.84289
$266$	−10.0000	−0.613139
$267$	−9.00000	−0.550791
$268$	26.0000	1.58820
$269$	3.00000	0.182913	0.0914566	−	0.995809i	$-0.470848\pi$
$269$	3.00000	0.182913	0.0914566	+	0.995809i	$0.470848\pi$
$270$	−6.00000	−0.365148
$271$	−2.00000	−0.121491	−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000	−0.121491	−0.0607457	+	0.998153i	$0.519348\pi$
$272$	−16.0000	−0.970143
$273$	4.00000	0.242091
$274$	−4.00000	−0.241649
$275$	4.00000	0.241209
$276$	−12.0000	−0.722315
$277$	−11.0000	−0.660926	−0.330463	−	0.943819i	$-0.607205\pi$
$277$	−11.0000	−0.660926	−0.330463	+	0.943819i	$0.607205\pi$
$278$	−18.0000	−1.07957
$279$	7.00000	0.419079
$280$	0	0
$281$	27.0000	1.61068	0.805342	−	0.592810i	$-0.201981\pi$
$281$	27.0000	1.61068	0.805342	+	0.592810i	$0.201981\pi$
$282$	6.00000	0.357295
$283$	−24.0000	−1.42665	−0.713326	−	0.700832i	$-0.752812\pi$
$283$	−24.0000	−1.42665	−0.713326	+	0.700832i	$0.752812\pi$
$284$	16.0000	0.949425
$285$	15.0000	0.888523
$286$	−8.00000	−0.473050
$287$	−3.00000	−0.177084
$288$	−8.00000	−0.471405
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	4.00000	0.234484
$292$	22.0000	1.28745
$293$	−30.0000	−1.75262	−0.876309	−	0.481749i	$-0.840002\pi$
$293$	−30.0000	−1.75262	−0.876309	+	0.481749i	$0.840002\pi$
$294$	−2.00000	−0.116642
$295$	39.0000	2.27067
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	20.0000	1.15857
$299$	−24.0000	−1.38796
$300$	−8.00000	−0.461880
$301$	5.00000	0.288195
$302$	4.00000	0.230174
$303$	−18.0000	−1.03407
$304$	20.0000	1.14708
$305$	−30.0000	−1.71780
$306$	8.00000	0.457330
$307$	2.00000	0.114146	0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000	0.114146	0.0570730	+	0.998370i	$0.481823\pi$
$308$	2.00000	0.113961
$309$	−5.00000	−0.284440
$310$	42.0000	2.38544
$311$	−15.0000	−0.850572	−0.425286	−	0.905059i	$-0.639826\pi$
$311$	−15.0000	−0.850572	−0.425286	+	0.905059i	$0.639826\pi$
$312$	0	0
$313$	−27.0000	−1.52613	−0.763065	−	0.646322i	$-0.776306\pi$
$313$	−27.0000	−1.52613	−0.763065	+	0.646322i	$0.776306\pi$
$314$	−8.00000	−0.451466
$315$	3.00000	0.169031
$316$	−16.0000	−0.900070
$317$	32.0000	1.79730	0.898650	−	0.438667i	$-0.144549\pi$
$317$	32.0000	1.79730	0.898650	+	0.438667i	$0.144549\pi$
$318$	−20.0000	−1.12154
$319$	0	0
$320$	−24.0000	−1.34164
$321$	8.00000	0.446516
$322$	12.0000	0.668734
$323$	−20.0000	−1.11283
$324$	2.00000	0.111111
$325$	−16.0000	−0.887520
$326$	−40.0000	−2.21540
$327$	−4.00000	−0.221201
$328$	0	0
$329$	−3.00000	−0.165395
$330$	−6.00000	−0.330289
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	8.00000	0.439057
$333$	4.00000	0.219199
$334$	−8.00000	−0.437741
$335$	39.0000	2.13080
$336$	4.00000	0.218218
$337$	−20.0000	−1.08947	−0.544735	−	0.838608i	$-0.683370\pi$
$337$	−20.0000	−1.08947	−0.544735	+	0.838608i	$0.683370\pi$
$338$	6.00000	0.326357
$339$	−6.00000	−0.325875
$340$	24.0000	1.30158
$341$	7.00000	0.379071
$342$	−10.0000	−0.540738
$343$	1.00000	0.0539949
$344$	0	0
$345$	−18.0000	−0.969087
$346$	12.0000	0.645124
$347$	−3.00000	−0.161048	−0.0805242	−	0.996753i	$-0.525659\pi$
$347$	−3.00000	−0.161048	−0.0805242	+	0.996753i	$0.525659\pi$
$348$	0	0
$349$	28.0000	1.49881	0.749403	−	0.662114i	$-0.230341\pi$
$349$	28.0000	1.49881	0.749403	+	0.662114i	$0.230341\pi$
$350$	8.00000	0.427618
$351$	4.00000	0.213504
$352$	−8.00000	−0.426401
$353$	−30.0000	−1.59674	−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000	−1.59674	−0.798369	+	0.602168i	$0.794304\pi$
$354$	−26.0000	−1.38188
$355$	24.0000	1.27379
$356$	18.0000	0.953998
$357$	−4.00000	−0.211702
$358$	30.0000	1.58555
$359$	−3.00000	−0.158334	−0.0791670	−	0.996861i	$-0.525226\pi$
$359$	−3.00000	−0.158334	−0.0791670	+	0.996861i	$0.525226\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	−28.0000	−1.47165
$363$	10.0000	0.524864
$364$	−8.00000	−0.419314
$365$	33.0000	1.72730
$366$	20.0000	1.04542
$367$	−34.0000	−1.77479	−0.887393	−	0.461014i	$-0.847486\pi$
$367$	−34.0000	−1.77479	−0.887393	+	0.461014i	$0.847486\pi$
$368$	−24.0000	−1.25109
$369$	−3.00000	−0.156174
$370$	24.0000	1.24770
$371$	10.0000	0.519174
$372$	−14.0000	−0.725866
$373$	15.0000	0.776671	0.388335	−	0.921518i	$-0.373050\pi$
$373$	15.0000	0.776671	0.388335	+	0.921518i	$0.373050\pi$
$374$	8.00000	0.413670
$375$	3.00000	0.154919
$376$	0	0
$377$	0	0
$378$	−2.00000	−0.102869
$379$	22.0000	1.13006	0.565032	−	0.825069i	$-0.308864\pi$
$379$	22.0000	1.13006	0.565032	+	0.825069i	$0.308864\pi$
$380$	−30.0000	−1.53897
$381$	18.0000	0.922168
$382$	24.0000	1.22795
$383$	−1.00000	−0.0510976
$383$	−1.00000	−0.0510976
$384$	0	0
$385$	3.00000	0.152894
$386$	−20.0000	−1.01797
$387$	5.00000	0.254164
$388$	−8.00000	−0.406138
$389$	−16.0000	−0.811232	−0.405616	−	0.914044i	$-0.632943\pi$
$389$	−16.0000	−0.811232	−0.405616	+	0.914044i	$0.632943\pi$
$390$	24.0000	1.21529
$391$	24.0000	1.21373
$392$	0	0
$393$	−1.00000	−0.0504433
$394$	−2.00000	−0.100759
$395$	−24.0000	−1.20757
$396$	2.00000	0.100504
$397$	25.0000	1.25471	0.627357	−	0.778732i	$-0.284137\pi$
$397$	25.0000	1.25471	0.627357	+	0.778732i	$0.284137\pi$
$398$	−4.00000	−0.200502
$399$	5.00000	0.250313
$400$	−16.0000	−0.800000
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	−26.0000	−1.29676
$403$	−28.0000	−1.39478
$404$	36.0000	1.79107
$405$	3.00000	0.149071
$406$	0	0
$407$	4.00000	0.198273
$408$	0	0
$409$	2.00000	0.0988936	0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000	0.0988936	0.0494468	+	0.998777i	$0.484254\pi$
$410$	−18.0000	−0.888957
$411$	2.00000	0.0986527
$412$	10.0000	0.492665
$413$	13.0000	0.639688
$414$	12.0000	0.589768
$415$	12.0000	0.589057
$416$	32.0000	1.56893
$417$	9.00000	0.440732
$418$	−10.0000	−0.489116
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	−6.00000	−0.292770
$421$	−30.0000	−1.46211	−0.731055	−	0.682318i	$-0.760972\pi$
$421$	−30.0000	−1.46211	−0.731055	+	0.682318i	$0.760972\pi$
$422$	−44.0000	−2.14189
$423$	−3.00000	−0.145865
$424$	0	0
$425$	16.0000	0.776114
$426$	−16.0000	−0.775203
$427$	−10.0000	−0.483934
$428$	−16.0000	−0.773389
$429$	4.00000	0.193122
$430$	30.0000	1.44673
$431$	−32.0000	−1.54139	−0.770693	−	0.637207i	$-0.780090\pi$
$431$	−32.0000	−1.54139	−0.770693	+	0.637207i	$0.780090\pi$
$432$	4.00000	0.192450
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	14.0000	0.672022
$435$	0	0
$436$	8.00000	0.383131
$437$	−30.0000	−1.43509
$438$	−22.0000	−1.05120
$439$	−15.0000	−0.715911	−0.357955	−	0.933739i	$-0.616526\pi$
$439$	−15.0000	−0.715911	−0.357955	+	0.933739i	$0.616526\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	−32.0000	−1.52208
$443$	21.0000	0.997740	0.498870	−	0.866677i	$-0.333748\pi$
$443$	21.0000	0.997740	0.498870	+	0.866677i	$0.333748\pi$
$444$	−8.00000	−0.379663
$445$	27.0000	1.27992
$446$	−18.0000	−0.852325
$447$	−10.0000	−0.472984
$448$	−8.00000	−0.377964
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	8.00000	0.377124
$451$	−3.00000	−0.141264
$452$	12.0000	0.564433
$453$	−2.00000	−0.0939682
$454$	0	0
$455$	−12.0000	−0.562569
$456$	0	0
$457$	16.0000	0.748448	0.374224	−	0.927338i	$-0.377909\pi$
$457$	16.0000	0.748448	0.374224	+	0.927338i	$0.377909\pi$
$458$	14.0000	0.654177
$459$	−4.00000	−0.186704
$460$	36.0000	1.67851
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	−2.00000	−0.0930484
$463$	−20.0000	−0.929479	−0.464739	−	0.885448i	$-0.653852\pi$
$463$	−20.0000	−0.929479	−0.464739	+	0.885448i	$0.653852\pi$
$464$	0	0
$465$	−21.0000	−0.973852
$466$	54.0000	2.50150
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	−8.00000	−0.369800
$469$	13.0000	0.600284
$470$	−18.0000	−0.830278
$471$	4.00000	0.184310
$472$	0	0
$473$	5.00000	0.229900
$474$	16.0000	0.734904
$475$	−20.0000	−0.917663
$476$	8.00000	0.366679
$477$	10.0000	0.457869
$478$	−10.0000	−0.457389
$479$	−6.00000	−0.274147	−0.137073	−	0.990561i	$-0.543770\pi$
$479$	−6.00000	−0.274147	−0.137073	+	0.990561i	$0.543770\pi$
$480$	24.0000	1.09545
$481$	−16.0000	−0.729537
$482$	52.0000	2.36854
$483$	−6.00000	−0.273009
$484$	−20.0000	−0.909091
$485$	−12.0000	−0.544892
$486$	−2.00000	−0.0907218
$487$	20.0000	0.906287	0.453143	−	0.891438i	$-0.350303\pi$
$487$	20.0000	0.906287	0.453143	+	0.891438i	$0.350303\pi$
$488$	0	0
$489$	20.0000	0.904431
$490$	6.00000	0.271052
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	6.00000	0.270501
$493$	0	0
$494$	40.0000	1.79969
$495$	3.00000	0.134840
$496$	−28.0000	−1.25724
$497$	8.00000	0.358849
$498$	−8.00000	−0.358489
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	−6.00000	−0.268328
$501$	4.00000	0.178707
$502$	−24.0000	−1.07117
$503$	−37.0000	−1.64975	−0.824874	−	0.565316i	$-0.808754\pi$
$503$	−37.0000	−1.64975	−0.824874	+	0.565316i	$0.808754\pi$
$504$	0	0
$505$	54.0000	2.40297
$506$	12.0000	0.533465
$507$	−3.00000	−0.133235
$508$	−36.0000	−1.59724
$509$	−16.0000	−0.709188	−0.354594	−	0.935020i	$-0.615381\pi$
$509$	−16.0000	−0.709188	−0.354594	+	0.935020i	$0.615381\pi$
$510$	−24.0000	−1.06274
$511$	11.0000	0.486611
$512$	32.0000	1.41421
$513$	5.00000	0.220755
$514$	−54.0000	−2.38184
$515$	15.0000	0.660979
$516$	−10.0000	−0.440225
$517$	−3.00000	−0.131940
$518$	8.00000	0.351500
$519$	−6.00000	−0.263371
$520$	0	0
$521$	12.0000	0.525730	0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000	0.525730	0.262865	+	0.964833i	$0.415333\pi$
$522$	0	0
$523$	−24.0000	−1.04945	−0.524723	−	0.851273i	$-0.675831\pi$
$523$	−24.0000	−1.04945	−0.524723	+	0.851273i	$0.675831\pi$
$524$	2.00000	0.0873704
$525$	−4.00000	−0.174574
$526$	60.0000	2.61612
$527$	28.0000	1.21970
$528$	4.00000	0.174078
$529$	13.0000	0.565217
$530$	60.0000	2.60623
$531$	13.0000	0.564152
$532$	−10.0000	−0.433555
$533$	12.0000	0.519778
$534$	−18.0000	−0.778936
$535$	−24.0000	−1.03761
$536$	0	0
$537$	−15.0000	−0.647298
$538$	6.00000	0.258678
$539$	1.00000	0.0430730
$540$	−6.00000	−0.258199
$541$	32.0000	1.37579	0.687894	−	0.725811i	$-0.258536\pi$
$541$	32.0000	1.37579	0.687894	+	0.725811i	$0.258536\pi$
$542$	−4.00000	−0.171815
$543$	14.0000	0.600798
$544$	−32.0000	−1.37199
$545$	12.0000	0.514024
$546$	8.00000	0.342368
$547$	−14.0000	−0.598597	−0.299298	−	0.954160i	$-0.596753\pi$
$547$	−14.0000	−0.598597	−0.299298	+	0.954160i	$0.596753\pi$
$548$	−4.00000	−0.170872
$549$	−10.0000	−0.426790
$550$	8.00000	0.341121
$551$	0	0
$552$	0	0
$553$	−8.00000	−0.340195
$554$	−22.0000	−0.934690
$555$	−12.0000	−0.509372
$556$	−18.0000	−0.763370
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	14.0000	0.592667
$559$	−20.0000	−0.845910
$560$	−12.0000	−0.507093
$561$	−4.00000	−0.168880
$562$	54.0000	2.27785
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	6.00000	0.252646
$565$	18.0000	0.757266
$566$	−48.0000	−2.01759
$567$	1.00000	0.0419961
$568$	0	0
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	30.0000	1.25656
$571$	36.0000	1.50655	0.753277	−	0.657704i	$-0.228472\pi$
$571$	36.0000	1.50655	0.753277	+	0.657704i	$0.228472\pi$
$572$	−8.00000	−0.334497
$573$	−12.0000	−0.501307
$574$	−6.00000	−0.250435
$575$	24.0000	1.00087
$576$	−8.00000	−0.333333
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	−2.00000	−0.0831890
$579$	10.0000	0.415586
$580$	0	0
$581$	4.00000	0.165948
$582$	8.00000	0.331611
$583$	10.0000	0.414158
$584$	0	0
$585$	−12.0000	−0.496139
$586$	−60.0000	−2.47858
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	−2.00000	−0.0824786
$589$	−35.0000	−1.44215
$590$	78.0000	3.21121
$591$	1.00000	0.0411345
$592$	−16.0000	−0.657596
$593$	17.0000	0.698106	0.349053	−	0.937103i	$-0.386503\pi$
$593$	17.0000	0.698106	0.349053	+	0.937103i	$0.386503\pi$
$594$	−2.00000	−0.0820610
$595$	12.0000	0.491952
$596$	20.0000	0.819232
$597$	2.00000	0.0818546
$598$	−48.0000	−1.96287
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	48.0000	1.95796	0.978980	−	0.203954i	$-0.0653794\pi$
$601$	48.0000	1.95796	0.978980	+	0.203954i	$0.0653794\pi$
$602$	10.0000	0.407570
$603$	13.0000	0.529401
$604$	4.00000	0.162758
$605$	−30.0000	−1.21967
$606$	−36.0000	−1.46240
$607$	23.0000	0.933541	0.466771	−	0.884378i	$-0.345417\pi$
$607$	23.0000	0.933541	0.466771	+	0.884378i	$0.345417\pi$
$608$	40.0000	1.62221
$609$	0	0
$610$	−60.0000	−2.42933
$611$	12.0000	0.485468
$612$	8.00000	0.323381
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	4.00000	0.161427
$615$	9.00000	0.362915
$616$	0	0
$617$	43.0000	1.73111	0.865557	−	0.500810i	$-0.166964\pi$
$617$	43.0000	1.73111	0.865557	+	0.500810i	$0.166964\pi$
$618$	−10.0000	−0.402259
$619$	14.0000	0.562708	0.281354	−	0.959604i	$-0.409217\pi$
$619$	14.0000	0.562708	0.281354	+	0.959604i	$0.409217\pi$
$620$	42.0000	1.68676
$621$	−6.00000	−0.240772
$622$	−30.0000	−1.20289
$623$	9.00000	0.360577
$624$	−16.0000	−0.640513
$625$	−29.0000	−1.16000
$626$	−54.0000	−2.15827
$627$	5.00000	0.199681
$628$	−8.00000	−0.319235
$629$	16.0000	0.637962
$630$	6.00000	0.239046
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	0	0
$633$	22.0000	0.874421
$634$	64.0000	2.54176
$635$	−54.0000	−2.14292
$636$	−20.0000	−0.793052
$637$	−4.00000	−0.158486
$638$	0	0
$639$	8.00000	0.316475
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	16.0000	0.631470
$643$	−9.00000	−0.354925	−0.177463	−	0.984128i	$-0.556789\pi$
$643$	−9.00000	−0.354925	−0.177463	+	0.984128i	$0.556789\pi$
$644$	12.0000	0.472866
$645$	−15.0000	−0.590624
$646$	−40.0000	−1.57378
$647$	−41.0000	−1.61188	−0.805938	−	0.592000i	$-0.798339\pi$
$647$	−41.0000	−1.61188	−0.805938	+	0.592000i	$0.798339\pi$
$648$	0	0
$649$	13.0000	0.510295
$650$	−32.0000	−1.25514
$651$	−7.00000	−0.274352
$652$	−40.0000	−1.56652
$653$	34.0000	1.33052	0.665261	−	0.746611i	$-0.268320\pi$
$653$	34.0000	1.33052	0.665261	+	0.746611i	$0.268320\pi$
$654$	−8.00000	−0.312825
$655$	3.00000	0.117220
$656$	12.0000	0.468521
$657$	11.0000	0.429151
$658$	−6.00000	−0.233904
$659$	−32.0000	−1.24654	−0.623272	−	0.782006i	$-0.714197\pi$
$659$	−32.0000	−1.24654	−0.623272	+	0.782006i	$0.714197\pi$
$660$	−6.00000	−0.233550
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	−16.0000	−0.621858
$663$	16.0000	0.621389
$664$	0	0
$665$	−15.0000	−0.581675
$666$	8.00000	0.309994
$667$	0	0
$668$	−8.00000	−0.309529
$669$	9.00000	0.347960
$670$	78.0000	3.01340
$671$	−10.0000	−0.386046
$672$	8.00000	0.308607
$673$	−42.0000	−1.61898	−0.809491	−	0.587133i	$-0.800257\pi$
$673$	−42.0000	−1.61898	−0.809491	+	0.587133i	$0.800257\pi$
$674$	−40.0000	−1.54074
$675$	−4.00000	−0.153960
$676$	6.00000	0.230769
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	−12.0000	−0.460857
$679$	−4.00000	−0.153506
$680$	0	0
$681$	0	0
$682$	14.0000	0.536088
$683$	−6.00000	−0.229584	−0.114792	−	0.993390i	$-0.536620\pi$
$683$	−6.00000	−0.229584	−0.114792	+	0.993390i	$0.536620\pi$
$684$	−10.0000	−0.382360
$685$	−6.00000	−0.229248
$686$	2.00000	0.0763604
$687$	−7.00000	−0.267067
$688$	−20.0000	−0.762493
$689$	−40.0000	−1.52388
$690$	−36.0000	−1.37050
$691$	−30.0000	−1.14125	−0.570627	−	0.821209i	$-0.693300\pi$
$691$	−30.0000	−1.14125	−0.570627	+	0.821209i	$0.693300\pi$
$692$	12.0000	0.456172
$693$	1.00000	0.0379869
$694$	−6.00000	−0.227757
$695$	−27.0000	−1.02417
$696$	0	0
$697$	−12.0000	−0.454532
$698$	56.0000	2.11963
$699$	−27.0000	−1.02123
$700$	8.00000	0.302372
$701$	−15.0000	−0.566542	−0.283271	−	0.959040i	$-0.591420\pi$
$701$	−15.0000	−0.566542	−0.283271	+	0.959040i	$0.591420\pi$
$702$	8.00000	0.301941
$703$	−20.0000	−0.754314
$704$	−8.00000	−0.301511
$705$	9.00000	0.338960
$706$	−60.0000	−2.25813
$707$	18.0000	0.676960
$708$	−26.0000	−0.977140
$709$	−16.0000	−0.600893	−0.300446	−	0.953799i	$-0.597136\pi$
$709$	−16.0000	−0.600893	−0.300446	+	0.953799i	$0.597136\pi$
$710$	48.0000	1.80141
$711$	−8.00000	−0.300023
$712$	0	0
$713$	42.0000	1.57291
$714$	−8.00000	−0.299392
$715$	−12.0000	−0.448775
$716$	30.0000	1.12115
$717$	5.00000	0.186728
$718$	−6.00000	−0.223918
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	−12.0000	−0.447214
$721$	5.00000	0.186210
$722$	12.0000	0.446594
$723$	−26.0000	−0.966950
$724$	−28.0000	−1.04061
$725$	0	0
$726$	20.0000	0.742270
$727$	−31.0000	−1.14973	−0.574863	−	0.818250i	$-0.694945\pi$
$727$	−31.0000	−1.14973	−0.574863	+	0.818250i	$0.694945\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	66.0000	2.44277
$731$	20.0000	0.739727
$732$	20.0000	0.739221
$733$	−38.0000	−1.40356	−0.701781	−	0.712393i	$-0.747612\pi$
$733$	−38.0000	−1.40356	−0.701781	+	0.712393i	$0.747612\pi$
$734$	−68.0000	−2.50993
$735$	−3.00000	−0.110657
$736$	−48.0000	−1.76930
$737$	13.0000	0.478861
$738$	−6.00000	−0.220863
$739$	−12.0000	−0.441427	−0.220714	−	0.975339i	$-0.570839\pi$
$739$	−12.0000	−0.441427	−0.220714	+	0.975339i	$0.570839\pi$
$740$	24.0000	0.882258
$741$	−20.0000	−0.734718
$742$	20.0000	0.734223
$743$	−5.00000	−0.183432	−0.0917161	−	0.995785i	$-0.529235\pi$
$743$	−5.00000	−0.183432	−0.0917161	+	0.995785i	$0.529235\pi$
$744$	0	0
$745$	30.0000	1.09911
$746$	30.0000	1.09838
$747$	4.00000	0.146352
$748$	8.00000	0.292509
$749$	−8.00000	−0.292314
$750$	6.00000	0.219089
$751$	1.00000	0.0364905	0.0182453	−	0.999834i	$-0.494192\pi$
$751$	1.00000	0.0364905	0.0182453	+	0.999834i	$0.494192\pi$
$752$	12.0000	0.437595
$753$	12.0000	0.437304
$754$	0	0
$755$	6.00000	0.218362
$756$	−2.00000	−0.0727393
$757$	16.0000	0.581530	0.290765	−	0.956795i	$-0.406090\pi$
$757$	16.0000	0.581530	0.290765	+	0.956795i	$0.406090\pi$
$758$	44.0000	1.59815
$759$	−6.00000	−0.217786
$760$	0	0
$761$	−42.0000	−1.52250	−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000	−1.52250	−0.761249	+	0.648459i	$0.775414\pi$
$762$	36.0000	1.30414
$763$	4.00000	0.144810
$764$	24.0000	0.868290
$765$	12.0000	0.433861
$766$	−2.00000	−0.0722629
$767$	−52.0000	−1.87761
$768$	−16.0000	−0.577350
$769$	−25.0000	−0.901523	−0.450762	−	0.892644i	$-0.648848\pi$
$769$	−25.0000	−0.901523	−0.450762	+	0.892644i	$0.648848\pi$
$770$	6.00000	0.216225
$771$	27.0000	0.972381
$772$	−20.0000	−0.719816
$773$	46.0000	1.65451	0.827253	−	0.561830i	$-0.189903\pi$
$773$	46.0000	1.65451	0.827253	+	0.561830i	$0.189903\pi$
$774$	10.0000	0.359443
$775$	28.0000	1.00579
$776$	0	0
$777$	−4.00000	−0.143499
$778$	−32.0000	−1.14726
$779$	15.0000	0.537431
$780$	24.0000	0.859338
$781$	8.00000	0.286263
$782$	48.0000	1.71648
$783$	0	0
$784$	−4.00000	−0.142857
$785$	−12.0000	−0.428298
$786$	−2.00000	−0.0713376
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	−2.00000	−0.0712470
$789$	−30.0000	−1.06803
$790$	−48.0000	−1.70776
$791$	6.00000	0.213335
$792$	0	0
$793$	40.0000	1.42044
$794$	50.0000	1.77443
$795$	−30.0000	−1.06399
$796$	−4.00000	−0.141776
$797$	−12.0000	−0.425062	−0.212531	−	0.977154i	$-0.568171\pi$
$797$	−12.0000	−0.425062	−0.212531	+	0.977154i	$0.568171\pi$
$798$	10.0000	0.353996
$799$	−12.0000	−0.424529
$800$	−32.0000	−1.13137
$801$	9.00000	0.317999
$802$	12.0000	0.423735
$803$	11.0000	0.388182
$804$	−26.0000	−0.916949
$805$	18.0000	0.634417
$806$	−56.0000	−1.97252
$807$	−3.00000	−0.105605
$808$	0	0
$809$	−24.0000	−0.843795	−0.421898	−	0.906644i	$-0.638636\pi$
$809$	−24.0000	−0.843795	−0.421898	+	0.906644i	$0.638636\pi$
$810$	6.00000	0.210819
$811$	−40.0000	−1.40459	−0.702295	−	0.711886i	$-0.747841\pi$
$811$	−40.0000	−1.40459	−0.702295	+	0.711886i	$0.747841\pi$
$812$	0	0
$813$	2.00000	0.0701431
$814$	8.00000	0.280400
$815$	−60.0000	−2.10171
$816$	16.0000	0.560112
$817$	−25.0000	−0.874639
$818$	4.00000	0.139857
$819$	−4.00000	−0.139771
$820$	−18.0000	−0.628587
$821$	26.0000	0.907406	0.453703	−	0.891153i	$-0.350103\pi$
$821$	26.0000	0.907406	0.453703	+	0.891153i	$0.350103\pi$
$822$	4.00000	0.139516
$823$	−51.0000	−1.77775	−0.888874	−	0.458151i	$-0.848512\pi$
$823$	−51.0000	−1.77775	−0.888874	+	0.458151i	$0.848512\pi$
$824$	0	0
$825$	−4.00000	−0.139262
$826$	26.0000	0.904656
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	12.0000	0.417029
$829$	−29.0000	−1.00721	−0.503606	−	0.863934i	$-0.667994\pi$
$829$	−29.0000	−1.00721	−0.503606	+	0.863934i	$0.667994\pi$
$830$	24.0000	0.833052
$831$	11.0000	0.381586
$832$	32.0000	1.10940
$833$	4.00000	0.138592
$834$	18.0000	0.623289
$835$	−12.0000	−0.415277
$836$	−10.0000	−0.345857
$837$	−7.00000	−0.241955
$838$	24.0000	0.829066
$839$	−30.0000	−1.03572	−0.517858	−	0.855467i	$-0.673270\pi$
$839$	−30.0000	−1.03572	−0.517858	+	0.855467i	$0.673270\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	−60.0000	−2.06774
$843$	−27.0000	−0.929929
$844$	−44.0000	−1.51454
$845$	9.00000	0.309609
$846$	−6.00000	−0.206284
$847$	−10.0000	−0.343604
$848$	−40.0000	−1.37361
$849$	24.0000	0.823678
$850$	32.0000	1.09759
$851$	24.0000	0.822709
$852$	−16.0000	−0.548151
$853$	−3.00000	−0.102718	−0.0513590	−	0.998680i	$-0.516355\pi$
$853$	−3.00000	−0.102718	−0.0513590	+	0.998680i	$0.516355\pi$
$854$	−20.0000	−0.684386
$855$	−15.0000	−0.512989
$856$	0	0
$857$	−10.0000	−0.341593	−0.170797	−	0.985306i	$-0.554634\pi$
$857$	−10.0000	−0.341593	−0.170797	+	0.985306i	$0.554634\pi$
$858$	8.00000	0.273115
$859$	−36.0000	−1.22830	−0.614152	−	0.789188i	$-0.710502\pi$
$859$	−36.0000	−1.22830	−0.614152	+	0.789188i	$0.710502\pi$
$860$	30.0000	1.02299
$861$	3.00000	0.102240
$862$	−64.0000	−2.17985
$863$	25.0000	0.851010	0.425505	−	0.904956i	$-0.360097\pi$
$863$	25.0000	0.851010	0.425505	+	0.904956i	$0.360097\pi$
$864$	8.00000	0.272166
$865$	18.0000	0.612018
$866$	−28.0000	−0.951479
$867$	1.00000	0.0339618
$868$	14.0000	0.475191
$869$	−8.00000	−0.271381
$870$	0	0
$871$	−52.0000	−1.76195
$872$	0	0
$873$	−4.00000	−0.135379
$874$	−60.0000	−2.02953
$875$	−3.00000	−0.101419
$876$	−22.0000	−0.743311
$877$	4.00000	0.135070	0.0675352	−	0.997717i	$-0.478487\pi$
$877$	4.00000	0.135070	0.0675352	+	0.997717i	$0.478487\pi$
$878$	−30.0000	−1.01245
$879$	30.0000	1.01187
$880$	−12.0000	−0.404520
$881$	33.0000	1.11180	0.555899	−	0.831250i	$-0.312374\pi$
$881$	33.0000	1.11180	0.555899	+	0.831250i	$0.312374\pi$
$882$	2.00000	0.0673435
$883$	16.0000	0.538443	0.269221	−	0.963078i	$-0.413234\pi$
$883$	16.0000	0.538443	0.269221	+	0.963078i	$0.413234\pi$
$884$	−32.0000	−1.07628
$885$	−39.0000	−1.31097
$886$	42.0000	1.41102
$887$	−12.0000	−0.402921	−0.201460	−	0.979497i	$-0.564569\pi$
$887$	−12.0000	−0.402921	−0.201460	+	0.979497i	$0.564569\pi$
$888$	0	0
$889$	−18.0000	−0.603701
$890$	54.0000	1.81008
$891$	1.00000	0.0335013
$892$	−18.0000	−0.602685
$893$	15.0000	0.501956
$894$	−20.0000	−0.668900
$895$	45.0000	1.50418
$896$	0	0
$897$	24.0000	0.801337
$898$	36.0000	1.20134
$899$	0	0
$900$	8.00000	0.266667
$901$	40.0000	1.33259
$902$	−6.00000	−0.199778
$903$	−5.00000	−0.166390
$904$	0	0
$905$	−42.0000	−1.39613
$906$	−4.00000	−0.132891
$907$	8.00000	0.265636	0.132818	−	0.991140i	$-0.457597\pi$
$907$	8.00000	0.265636	0.132818	+	0.991140i	$0.457597\pi$
$908$	0	0
$909$	18.0000	0.597022
$910$	−24.0000	−0.795592
$911$	−29.0000	−0.960813	−0.480406	−	0.877046i	$-0.659511\pi$
$911$	−29.0000	−0.960813	−0.480406	+	0.877046i	$0.659511\pi$
$912$	−20.0000	−0.662266
$913$	4.00000	0.132381
$914$	32.0000	1.05847
$915$	30.0000	0.991769
$916$	14.0000	0.462573
$917$	1.00000	0.0330229
$918$	−8.00000	−0.264039
$919$	5.00000	0.164935	0.0824674	−	0.996594i	$-0.473720\pi$
$919$	5.00000	0.164935	0.0824674	+	0.996594i	$0.473720\pi$
$920$	0	0
$921$	−2.00000	−0.0659022
$922$	60.0000	1.97599
$923$	−32.0000	−1.05329
$924$	−2.00000	−0.0657952
$925$	16.0000	0.526077
$926$	−40.0000	−1.31448
$927$	5.00000	0.164222
$928$	0	0
$929$	−34.0000	−1.11550	−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000	−1.11550	−0.557752	+	0.830008i	$0.688336\pi$
$930$	−42.0000	−1.37723
$931$	−5.00000	−0.163868
$932$	54.0000	1.76883
$933$	15.0000	0.491078
$934$	−24.0000	−0.785304
$935$	12.0000	0.392442
$936$	0	0
$937$	38.0000	1.24141	0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000	1.24141	0.620703	+	0.784046i	$0.286847\pi$
$938$	26.0000	0.848930
$939$	27.0000	0.881112
$940$	−18.0000	−0.587095
$941$	−8.00000	−0.260793	−0.130396	−	0.991462i	$-0.541625\pi$
$941$	−8.00000	−0.260793	−0.130396	+	0.991462i	$0.541625\pi$
$942$	8.00000	0.260654
$943$	−18.0000	−0.586161
$944$	−52.0000	−1.69246
$945$	−3.00000	−0.0975900
$946$	10.0000	0.325128
$947$	−3.00000	−0.0974869	−0.0487435	−	0.998811i	$-0.515522\pi$
$947$	−3.00000	−0.0974869	−0.0487435	+	0.998811i	$0.515522\pi$
$948$	16.0000	0.519656
$949$	−44.0000	−1.42830
$950$	−40.0000	−1.29777
$951$	−32.0000	−1.03767
$952$	0	0
$953$	−42.0000	−1.36051	−0.680257	−	0.732974i	$-0.738132\pi$
$953$	−42.0000	−1.36051	−0.680257	+	0.732974i	$0.738132\pi$
$954$	20.0000	0.647524
$955$	36.0000	1.16493
$956$	−10.0000	−0.323423
$957$	0	0
$958$	−12.0000	−0.387702
$959$	−2.00000	−0.0645834
$960$	24.0000	0.774597
$961$	18.0000	0.580645
$962$	−32.0000	−1.03172
$963$	−8.00000	−0.257796
$964$	52.0000	1.67481
$965$	−30.0000	−0.965734
$966$	−12.0000	−0.386094
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	0	0
$969$	20.0000	0.642493
$970$	−24.0000	−0.770594
$971$	34.0000	1.09111	0.545556	−	0.838074i	$-0.316319\pi$
$971$	34.0000	1.09111	0.545556	+	0.838074i	$0.316319\pi$
$972$	−2.00000	−0.0641500
$973$	−9.00000	−0.288527
$974$	40.0000	1.28168
$975$	16.0000	0.512410
$976$	40.0000	1.28037
$977$	−45.0000	−1.43968	−0.719839	−	0.694141i	$-0.755784\pi$
$977$	−45.0000	−1.43968	−0.719839	+	0.694141i	$0.755784\pi$
$978$	40.0000	1.27906
$979$	9.00000	0.287641
$980$	6.00000	0.191663
$981$	4.00000	0.127710
$982$	72.0000	2.29761
$983$	−18.0000	−0.574111	−0.287055	−	0.957914i	$-0.592676\pi$
$983$	−18.0000	−0.574111	−0.287055	+	0.957914i	$0.592676\pi$
$984$	0	0
$985$	−3.00000	−0.0955879
$986$	0	0
$987$	3.00000	0.0954911
$988$	40.0000	1.27257
$989$	30.0000	0.953945
$990$	6.00000	0.190693
$991$	−20.0000	−0.635321	−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000	−0.635321	−0.317660	+	0.948205i	$0.602897\pi$
$992$	−56.0000	−1.77800
$993$	8.00000	0.253872
$994$	16.0000	0.507489
$995$	−6.00000	−0.190213
$996$	−8.00000	−0.253490
$997$	14.0000	0.443384	0.221692	−	0.975117i	$-0.428842\pi$
$997$	14.0000	0.443384	0.221692	+	0.975117i	$0.428842\pi$
$998$	−8.00000	−0.253236
$999$	−4.00000	−0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8043.2.a.i.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8043.2.a.i.1.1	✓	1	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:	\( N \)	\(=\)	\( 8043 = 3 \cdot 7 \cdot 383 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8043.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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