LMFDB - Embedded newform 8001.2.a.a.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:	$2$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$

$=$

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

$q-2.00000 q^{2} +2.00000 q^{4} -3.00000 q^{5} -1.00000 q^{7} +6.00000 q^{10} -2.00000 q^{11} +5.00000 q^{13} +2.00000 q^{14} -4.00000 q^{16} -6.00000 q^{17} -8.00000 q^{19} -6.00000 q^{20} +4.00000 q^{22} -5.00000 q^{23} +4.00000 q^{25} -10.0000 q^{26} -2.00000 q^{28} +3.00000 q^{29} -1.00000 q^{31} +8.00000 q^{32} +12.0000 q^{34} +3.00000 q^{35} -11.0000 q^{37} +16.0000 q^{38} -6.00000 q^{41} -8.00000 q^{43} -4.00000 q^{44} +10.0000 q^{46} +2.00000 q^{47} +1.00000 q^{49} -8.00000 q^{50} +10.0000 q^{52} -3.00000 q^{53} +6.00000 q^{55} -6.00000 q^{58} +3.00000 q^{59} -1.00000 q^{61} +2.00000 q^{62} -8.00000 q^{64} -15.0000 q^{65} +2.00000 q^{67} -12.0000 q^{68} -6.00000 q^{70} -10.0000 q^{71} +11.0000 q^{73} +22.0000 q^{74} -16.0000 q^{76} +2.00000 q^{77} +12.0000 q^{80} +12.0000 q^{82} -17.0000 q^{83} +18.0000 q^{85} +16.0000 q^{86} +1.00000 q^{89} -5.00000 q^{91} -10.0000 q^{92} -4.00000 q^{94} +24.0000 q^{95} -6.00000 q^{97} -2.00000 q^{98} +O(q^{100})$

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	$-0.750000\pi$
$2$	−2.00000	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$3$	0	0
$3$	0	0
$4$	2.00000	1.00000
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	6.00000	1.89737
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	−4.00000	−1.00000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	0	0
$19$	−8.00000	−1.83533	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−8.00000	−1.83533	−0.917663	+	0.397360i	$0.869927\pi$
$20$	−6.00000	−1.34164
$21$	0	0
$22$	4.00000	0.852803
$23$	−5.00000	−1.04257	−0.521286	−	0.853382i	$-0.674548\pi$
$23$	−5.00000	−1.04257	−0.521286	+	0.853382i	$0.674548\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	−10.0000	−1.96116
$27$	0	0
$28$	−2.00000	−0.377964
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	−1.00000	−0.179605	−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000	−0.179605	−0.0898027	+	0.995960i	$0.528624\pi$
$32$	8.00000	1.41421
$33$	0	0
$34$	12.0000	2.05798
$35$	3.00000	0.507093
$36$	0	0
$37$	−11.0000	−1.80839	−0.904194	−	0.427121i	$-0.859528\pi$
$37$	−11.0000	−1.80839	−0.904194	+	0.427121i	$0.859528\pi$
$38$	16.0000	2.59554
$39$	0	0
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	10.0000	1.47442
$47$	2.00000	0.291730	0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000	0.291730	0.145865	+	0.989305i	$0.453403\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−8.00000	−1.13137
$51$	0	0
$52$	10.0000	1.38675
$53$	−3.00000	−0.412082	−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000	−0.412082	−0.206041	+	0.978543i	$0.566058\pi$
$54$	0	0
$55$	6.00000	0.809040
$56$	0	0
$57$	0	0
$58$	−6.00000	−0.787839
$59$	3.00000	0.390567	0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000	0.390567	0.195283	+	0.980747i	$0.437437\pi$
$60$	0	0
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	2.00000	0.254000
$63$	0	0
$64$	−8.00000	−1.00000
$65$	−15.0000	−1.86052
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	−12.0000	−1.45521
$69$	0	0
$70$	−6.00000	−0.717137
$71$	−10.0000	−1.18678	−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000	−1.18678	−0.593391	+	0.804914i	$0.702211\pi$
$72$	0	0
$73$	11.0000	1.28745	0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000	1.28745	0.643726	+	0.765256i	$0.277388\pi$
$74$	22.0000	2.55745
$75$	0	0
$76$	−16.0000	−1.83533
$77$	2.00000	0.227921
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	12.0000	1.34164
$81$	0	0
$82$	12.0000	1.32518
$83$	−17.0000	−1.86599	−0.932996	−	0.359886i	$-0.882816\pi$
$83$	−17.0000	−1.86599	−0.932996	+	0.359886i	$0.882816\pi$
$84$	0	0
$85$	18.0000	1.95237
$86$	16.0000	1.72532
$87$	0	0
$88$	0	0
$89$	1.00000	0.106000	0.0529999	−	0.998595i	$-0.483122\pi$
$89$	1.00000	0.106000	0.0529999	+	0.998595i	$0.483122\pi$
$90$	0	0
$91$	−5.00000	−0.524142
$92$	−10.0000	−1.04257
$93$	0	0
$94$	−4.00000	−0.412568
$95$	24.0000	2.46235
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	−2.00000	−0.202031
$99$	0	0
$100$	8.00000	0.800000
$101$	−7.00000	−0.696526	−0.348263	−	0.937397i	$-0.613228\pi$
$101$	−7.00000	−0.696526	−0.348263	+	0.937397i	$0.613228\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	0	0
$106$	6.00000	0.582772
$107$	18.0000	1.74013	0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000	1.74013	0.870063	+	0.492941i	$0.164078\pi$
$108$	0	0
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	−12.0000	−1.14416
$111$	0	0
$112$	4.00000	0.377964
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	15.0000	1.39876
$116$	6.00000	0.557086
$117$	0	0
$118$	−6.00000	−0.552345
$119$	6.00000	0.550019
$120$	0	0
$121$	−7.00000	−0.636364
$122$	2.00000	0.181071
$123$	0	0
$124$	−2.00000	−0.179605
$125$	3.00000	0.268328
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	0	0
$129$	0	0
$130$	30.0000	2.63117
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	8.00000	0.693688
$134$	−4.00000	−0.345547
$135$	0	0
$136$	0	0
$137$	7.00000	0.598050	0.299025	−	0.954245i	$-0.403339\pi$
$137$	7.00000	0.598050	0.299025	+	0.954245i	$0.403339\pi$
$138$	0	0
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	6.00000	0.507093
$141$	0	0
$142$	20.0000	1.67836
$143$	−10.0000	−0.836242
$144$	0	0
$145$	−9.00000	−0.747409
$146$	−22.0000	−1.82073
$147$	0	0
$148$	−22.0000	−1.80839
$149$	−16.0000	−1.31077	−0.655386	−	0.755295i	$-0.727494\pi$
$149$	−16.0000	−1.31077	−0.655386	+	0.755295i	$0.727494\pi$
$150$	0	0
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	0	0
$153$	0	0
$154$	−4.00000	−0.322329
$155$	3.00000	0.240966
$156$	0	0
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	0	0
$159$	0	0
$160$	−24.0000	−1.89737
$161$	5.00000	0.394055
$162$	0	0
$163$	−1.00000	−0.0783260	−0.0391630	−	0.999233i	$-0.512469\pi$
$163$	−1.00000	−0.0783260	−0.0391630	+	0.999233i	$0.512469\pi$
$164$	−12.0000	−0.937043
$165$	0	0
$166$	34.0000	2.63891
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	−36.0000	−2.76107
$171$	0	0
$172$	−16.0000	−1.21999
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	8.00000	0.603023
$177$	0	0
$178$	−2.00000	−0.149906
$179$	−26.0000	−1.94333	−0.971666	−	0.236360i	$-0.924046\pi$
$179$	−26.0000	−1.94333	−0.971666	+	0.236360i	$0.924046\pi$
$180$	0	0
$181$	−16.0000	−1.18927	−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000	−1.18927	−0.594635	+	0.803996i	$0.702704\pi$
$182$	10.0000	0.741249
$183$	0	0
$184$	0	0
$185$	33.0000	2.42621
$186$	0	0
$187$	12.0000	0.877527
$188$	4.00000	0.291730
$189$	0	0
$190$	−48.0000	−3.48229
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	−18.0000	−1.29567	−0.647834	−	0.761781i	$-0.724325\pi$
$193$	−18.0000	−1.29567	−0.647834	+	0.761781i	$0.724325\pi$
$194$	12.0000	0.861550
$195$	0	0
$196$	2.00000	0.142857
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	0	0
$199$	−5.00000	−0.354441	−0.177220	−	0.984171i	$-0.556711\pi$
$199$	−5.00000	−0.354441	−0.177220	+	0.984171i	$0.556711\pi$
$200$	0	0
$201$	0	0
$202$	14.0000	0.985037
$203$	−3.00000	−0.210559
$204$	0	0
$205$	18.0000	1.25717
$206$	16.0000	1.11477
$207$	0	0
$208$	−20.0000	−1.38675
$209$	16.0000	1.10674
$210$	0	0
$211$	−25.0000	−1.72107	−0.860535	−	0.509390i	$-0.829871\pi$
$211$	−25.0000	−1.72107	−0.860535	+	0.509390i	$0.829871\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	−36.0000	−2.46091
$215$	24.0000	1.63679
$216$	0	0
$217$	1.00000	0.0678844
$218$	8.00000	0.541828
$219$	0	0
$220$	12.0000	0.809040
$221$	−30.0000	−2.01802
$222$	0	0
$223$	−10.0000	−0.669650	−0.334825	−	0.942280i	$-0.608677\pi$
$223$	−10.0000	−0.669650	−0.334825	+	0.942280i	$0.608677\pi$
$224$	−8.00000	−0.534522
$225$	0	0
$226$	4.00000	0.266076
$227$	28.0000	1.85843	0.929213	−	0.369546i	$-0.120487\pi$
$227$	28.0000	1.85843	0.929213	+	0.369546i	$0.120487\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	−30.0000	−1.97814
$231$	0	0
$232$	0	0
$233$	−15.0000	−0.982683	−0.491341	−	0.870967i	$-0.663493\pi$
$233$	−15.0000	−0.982683	−0.491341	+	0.870967i	$0.663493\pi$
$234$	0	0
$235$	−6.00000	−0.391397
$236$	6.00000	0.390567
$237$	0	0
$238$	−12.0000	−0.777844
$239$	19.0000	1.22901	0.614504	−	0.788914i	$-0.289356\pi$
$239$	19.0000	1.22901	0.614504	+	0.788914i	$0.289356\pi$
$240$	0	0
$241$	−8.00000	−0.515325	−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000	−0.515325	−0.257663	+	0.966235i	$0.582952\pi$
$242$	14.0000	0.899954
$243$	0	0
$244$	−2.00000	−0.128037
$245$	−3.00000	−0.191663
$246$	0	0
$247$	−40.0000	−2.54514
$248$	0	0
$249$	0	0
$250$	−6.00000	−0.379473
$251$	16.0000	1.00991	0.504956	−	0.863145i	$-0.331509\pi$
$251$	16.0000	1.00991	0.504956	+	0.863145i	$0.331509\pi$
$252$	0	0
$253$	10.0000	0.628695
$254$	2.00000	0.125491
$255$	0	0
$256$	16.0000	1.00000
$257$	−5.00000	−0.311891	−0.155946	−	0.987766i	$-0.549842\pi$
$257$	−5.00000	−0.311891	−0.155946	+	0.987766i	$0.549842\pi$
$258$	0	0
$259$	11.0000	0.683507
$260$	−30.0000	−1.86052
$261$	0	0
$262$	−8.00000	−0.494242
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	0	0
$265$	9.00000	0.552866
$266$	−16.0000	−0.981023
$267$	0	0
$268$	4.00000	0.244339
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	13.0000	0.789694	0.394847	−	0.918747i	$-0.370798\pi$
$271$	13.0000	0.789694	0.394847	+	0.918747i	$0.370798\pi$
$272$	24.0000	1.45521
$273$	0	0
$274$	−14.0000	−0.845771
$275$	−8.00000	−0.482418
$276$	0	0
$277$	30.0000	1.80253	0.901263	−	0.433273i	$-0.142641\pi$
$277$	30.0000	1.80253	0.901263	+	0.433273i	$0.142641\pi$
$278$	28.0000	1.67933
$279$	0	0
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	−20.0000	−1.18678
$285$	0	0
$286$	20.0000	1.18262
$287$	6.00000	0.354169
$288$	0	0
$289$	19.0000	1.11765
$290$	18.0000	1.05700
$291$	0	0
$292$	22.0000	1.28745
$293$	18.0000	1.05157	0.525786	−	0.850617i	$-0.323771\pi$
$293$	18.0000	1.05157	0.525786	+	0.850617i	$0.323771\pi$
$294$	0	0
$295$	−9.00000	−0.524000
$296$	0	0
$297$	0	0
$298$	32.0000	1.85371
$299$	−25.0000	−1.44579
$300$	0	0
$301$	8.00000	0.461112
$302$	4.00000	0.230174
$303$	0	0
$304$	32.0000	1.83533
$305$	3.00000	0.171780
$306$	0	0
$307$	−2.00000	−0.114146	−0.0570730	−	0.998370i	$-0.518177\pi$
$307$	−2.00000	−0.114146	−0.0570730	+	0.998370i	$0.518177\pi$
$308$	4.00000	0.227921
$309$	0	0
$310$	−6.00000	−0.340777
$311$	7.00000	0.396934	0.198467	−	0.980108i	$-0.436404\pi$
$311$	7.00000	0.396934	0.198467	+	0.980108i	$0.436404\pi$
$312$	0	0
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	28.0000	1.58013
$315$	0	0
$316$	0	0
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	0	0
$319$	−6.00000	−0.335936
$320$	24.0000	1.34164
$321$	0	0
$322$	−10.0000	−0.557278
$323$	48.0000	2.67079
$324$	0	0
$325$	20.0000	1.10940
$326$	2.00000	0.110770
$327$	0	0
$328$	0	0
$329$	−2.00000	−0.110264
$330$	0	0
$331$	18.0000	0.989369	0.494685	−	0.869072i	$-0.335284\pi$
$331$	18.0000	0.989369	0.494685	+	0.869072i	$0.335284\pi$
$332$	−34.0000	−1.86599
$333$	0	0
$334$	−16.0000	−0.875481
$335$	−6.00000	−0.327815
$336$	0	0
$337$	−28.0000	−1.52526	−0.762629	−	0.646837i	$-0.776092\pi$
$337$	−28.0000	−1.52526	−0.762629	+	0.646837i	$0.776092\pi$
$338$	−24.0000	−1.30543
$339$	0	0
$340$	36.0000	1.95237
$341$	2.00000	0.108306
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	36.0000	1.93537
$347$	−37.0000	−1.98626	−0.993132	−	0.116999i	$-0.962673\pi$
$347$	−37.0000	−1.98626	−0.993132	+	0.116999i	$0.962673\pi$
$348$	0	0
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	8.00000	0.427618
$351$	0	0
$352$	−16.0000	−0.852803
$353$	24.0000	1.27739	0.638696	−	0.769460i	$-0.279474\pi$
$353$	24.0000	1.27739	0.638696	+	0.769460i	$0.279474\pi$
$354$	0	0
$355$	30.0000	1.59223
$356$	2.00000	0.106000
$357$	0	0
$358$	52.0000	2.74829
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	32.0000	1.68188
$363$	0	0
$364$	−10.0000	−0.524142
$365$	−33.0000	−1.72730
$366$	0	0
$367$	−5.00000	−0.260998	−0.130499	−	0.991448i	$-0.541658\pi$
$367$	−5.00000	−0.260998	−0.130499	+	0.991448i	$0.541658\pi$
$368$	20.0000	1.04257
$369$	0	0
$370$	−66.0000	−3.43118
$371$	3.00000	0.155752
$372$	0	0
$373$	−38.0000	−1.96757	−0.983783	−	0.179364i	$-0.942596\pi$
$373$	−38.0000	−1.96757	−0.983783	+	0.179364i	$0.942596\pi$
$374$	−24.0000	−1.24101
$375$	0	0
$376$	0	0
$377$	15.0000	0.772539
$378$	0	0
$379$	32.0000	1.64373	0.821865	−	0.569683i	$-0.192934\pi$
$379$	32.0000	1.64373	0.821865	+	0.569683i	$0.192934\pi$
$380$	48.0000	2.46235
$381$	0	0
$382$	−24.0000	−1.22795
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	0	0
$385$	−6.00000	−0.305788
$386$	36.0000	1.83235
$387$	0	0
$388$	−12.0000	−0.609208
$389$	−14.0000	−0.709828	−0.354914	−	0.934899i	$-0.615490\pi$
$389$	−14.0000	−0.709828	−0.354914	+	0.934899i	$0.615490\pi$
$390$	0	0
$391$	30.0000	1.51717
$392$	0	0
$393$	0	0
$394$	36.0000	1.81365
$395$	0	0
$396$	0	0
$397$	17.0000	0.853206	0.426603	−	0.904439i	$-0.359710\pi$
$397$	17.0000	0.853206	0.426603	+	0.904439i	$0.359710\pi$
$398$	10.0000	0.501255
$399$	0	0
$400$	−16.0000	−0.800000
$401$	−7.00000	−0.349563	−0.174782	−	0.984607i	$-0.555922\pi$
$401$	−7.00000	−0.349563	−0.174782	+	0.984607i	$0.555922\pi$
$402$	0	0
$403$	−5.00000	−0.249068
$404$	−14.0000	−0.696526
$405$	0	0
$406$	6.00000	0.297775
$407$	22.0000	1.09050
$408$	0	0
$409$	−8.00000	−0.395575	−0.197787	−	0.980245i	$-0.563376\pi$
$409$	−8.00000	−0.395575	−0.197787	+	0.980245i	$0.563376\pi$
$410$	−36.0000	−1.77791
$411$	0	0
$412$	−16.0000	−0.788263
$413$	−3.00000	−0.147620
$414$	0	0
$415$	51.0000	2.50349
$416$	40.0000	1.96116
$417$	0	0
$418$	−32.0000	−1.56517
$419$	−16.0000	−0.781651	−0.390826	−	0.920465i	$-0.627810\pi$
$419$	−16.0000	−0.781651	−0.390826	+	0.920465i	$0.627810\pi$
$420$	0	0
$421$	38.0000	1.85201	0.926003	−	0.377515i	$-0.123221\pi$
$421$	38.0000	1.85201	0.926003	+	0.377515i	$0.123221\pi$
$422$	50.0000	2.43396
$423$	0	0
$424$	0	0
$425$	−24.0000	−1.16417
$426$	0	0
$427$	1.00000	0.0483934
$428$	36.0000	1.74013
$429$	0	0
$430$	−48.0000	−2.31477
$431$	−18.0000	−0.867029	−0.433515	−	0.901146i	$-0.642727\pi$
$431$	−18.0000	−0.867029	−0.433515	+	0.901146i	$0.642727\pi$
$432$	0	0
$433$	26.0000	1.24948	0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000	1.24948	0.624740	+	0.780833i	$0.285205\pi$
$434$	−2.00000	−0.0960031
$435$	0	0
$436$	−8.00000	−0.383131
$437$	40.0000	1.91346
$438$	0	0
$439$	2.00000	0.0954548	0.0477274	−	0.998860i	$-0.484802\pi$
$439$	2.00000	0.0954548	0.0477274	+	0.998860i	$0.484802\pi$
$440$	0	0
$441$	0	0
$442$	60.0000	2.85391
$443$	−2.00000	−0.0950229	−0.0475114	−	0.998871i	$-0.515129\pi$
$443$	−2.00000	−0.0950229	−0.0475114	+	0.998871i	$0.515129\pi$
$444$	0	0
$445$	−3.00000	−0.142214
$446$	20.0000	0.947027
$447$	0	0
$448$	8.00000	0.377964
$449$	−8.00000	−0.377543	−0.188772	−	0.982021i	$-0.560451\pi$
$449$	−8.00000	−0.377543	−0.188772	+	0.982021i	$0.560451\pi$
$450$	0	0
$451$	12.0000	0.565058
$452$	−4.00000	−0.188144
$453$	0	0
$454$	−56.0000	−2.62821
$455$	15.0000	0.703211
$456$	0	0
$457$	11.0000	0.514558	0.257279	−	0.966337i	$-0.417174\pi$
$457$	11.0000	0.514558	0.257279	+	0.966337i	$0.417174\pi$
$458$	20.0000	0.934539
$459$	0	0
$460$	30.0000	1.39876
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	0	0
$463$	−35.0000	−1.62659	−0.813294	−	0.581853i	$-0.802328\pi$
$463$	−35.0000	−1.62659	−0.813294	+	0.581853i	$0.802328\pi$
$464$	−12.0000	−0.557086
$465$	0	0
$466$	30.0000	1.38972
$467$	−7.00000	−0.323921	−0.161961	−	0.986797i	$-0.551782\pi$
$467$	−7.00000	−0.323921	−0.161961	+	0.986797i	$0.551782\pi$
$468$	0	0
$469$	−2.00000	−0.0923514
$470$	12.0000	0.553519
$471$	0	0
$472$	0	0
$473$	16.0000	0.735681
$474$	0	0
$475$	−32.0000	−1.46826
$476$	12.0000	0.550019
$477$	0	0
$478$	−38.0000	−1.73808
$479$	14.0000	0.639676	0.319838	−	0.947472i	$-0.396371\pi$
$479$	14.0000	0.639676	0.319838	+	0.947472i	$0.396371\pi$
$480$	0	0
$481$	−55.0000	−2.50778
$482$	16.0000	0.728780
$483$	0	0
$484$	−14.0000	−0.636364
$485$	18.0000	0.817338
$486$	0	0
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	0	0
$489$	0	0
$490$	6.00000	0.271052
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	0	0
$493$	−18.0000	−0.810679
$494$	80.0000	3.59937
$495$	0	0
$496$	4.00000	0.179605
$497$	10.0000	0.448561
$498$	0	0
$499$	28.0000	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$500$	6.00000	0.268328
$501$	0	0
$502$	−32.0000	−1.42823
$503$	26.0000	1.15928	0.579641	−	0.814872i	$-0.303193\pi$
$503$	26.0000	1.15928	0.579641	+	0.814872i	$0.303193\pi$
$504$	0	0
$505$	21.0000	0.934488
$506$	−20.0000	−0.889108
$507$	0	0
$508$	−2.00000	−0.0887357
$509$	−12.0000	−0.531891	−0.265945	−	0.963988i	$-0.585684\pi$
$509$	−12.0000	−0.531891	−0.265945	+	0.963988i	$0.585684\pi$
$510$	0	0
$511$	−11.0000	−0.486611
$512$	−32.0000	−1.41421
$513$	0	0
$514$	10.0000	0.441081
$515$	24.0000	1.05757
$516$	0	0
$517$	−4.00000	−0.175920
$518$	−22.0000	−0.966625
$519$	0	0
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	0	0
$523$	11.0000	0.480996	0.240498	−	0.970650i	$-0.422689\pi$
$523$	11.0000	0.480996	0.240498	+	0.970650i	$0.422689\pi$
$524$	8.00000	0.349482
$525$	0	0
$526$	−24.0000	−1.04645
$527$	6.00000	0.261364
$528$	0	0
$529$	2.00000	0.0869565
$530$	−18.0000	−0.781870
$531$	0	0
$532$	16.0000	0.693688
$533$	−30.0000	−1.29944
$534$	0	0
$535$	−54.0000	−2.33462
$536$	0	0
$537$	0	0
$538$	36.0000	1.55207
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	−26.0000	−1.11680
$543$	0	0
$544$	−48.0000	−2.05798
$545$	12.0000	0.514024
$546$	0	0
$547$	36.0000	1.53925	0.769624	−	0.638497i	$-0.220443\pi$
$547$	36.0000	1.53925	0.769624	+	0.638497i	$0.220443\pi$
$548$	14.0000	0.598050
$549$	0	0
$550$	16.0000	0.682242
$551$	−24.0000	−1.02243
$552$	0	0
$553$	0	0
$554$	−60.0000	−2.54916
$555$	0	0
$556$	−28.0000	−1.18746
$557$	10.0000	0.423714	0.211857	−	0.977301i	$-0.432049\pi$
$557$	10.0000	0.423714	0.211857	+	0.977301i	$0.432049\pi$
$558$	0	0
$559$	−40.0000	−1.69182
$560$	−12.0000	−0.507093
$561$	0	0
$562$	12.0000	0.506189
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	0	0
$565$	6.00000	0.252422
$566$	−28.0000	−1.17693
$567$	0	0
$568$	0	0
$569$	−34.0000	−1.42535	−0.712677	−	0.701492i	$-0.752517\pi$
$569$	−34.0000	−1.42535	−0.712677	+	0.701492i	$0.752517\pi$
$570$	0	0
$571$	18.0000	0.753277	0.376638	−	0.926360i	$-0.377080\pi$
$571$	18.0000	0.753277	0.376638	+	0.926360i	$0.377080\pi$
$572$	−20.0000	−0.836242
$573$	0	0
$574$	−12.0000	−0.500870
$575$	−20.0000	−0.834058
$576$	0	0
$577$	−35.0000	−1.45707	−0.728535	−	0.685009i	$-0.759798\pi$
$577$	−35.0000	−1.45707	−0.728535	+	0.685009i	$0.759798\pi$
$578$	−38.0000	−1.58059
$579$	0	0
$580$	−18.0000	−0.747409
$581$	17.0000	0.705279
$582$	0	0
$583$	6.00000	0.248495
$584$	0	0
$585$	0	0
$586$	−36.0000	−1.48715
$587$	42.0000	1.73353	0.866763	−	0.498721i	$-0.166197\pi$
$587$	42.0000	1.73353	0.866763	+	0.498721i	$0.166197\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	18.0000	0.741048
$591$	0	0
$592$	44.0000	1.80839
$593$	−5.00000	−0.205325	−0.102663	−	0.994716i	$-0.532736\pi$
$593$	−5.00000	−0.205325	−0.102663	+	0.994716i	$0.532736\pi$
$594$	0	0
$595$	−18.0000	−0.737928
$596$	−32.0000	−1.31077
$597$	0	0
$598$	50.0000	2.04465
$599$	−15.0000	−0.612883	−0.306442	−	0.951889i	$-0.599138\pi$
$599$	−15.0000	−0.612883	−0.306442	+	0.951889i	$0.599138\pi$
$600$	0	0
$601$	34.0000	1.38689	0.693444	−	0.720510i	$-0.256092\pi$
$601$	34.0000	1.38689	0.693444	+	0.720510i	$0.256092\pi$
$602$	−16.0000	−0.652111
$603$	0	0
$604$	−4.00000	−0.162758
$605$	21.0000	0.853771
$606$	0	0
$607$	−29.0000	−1.17707	−0.588537	−	0.808470i	$-0.700296\pi$
$607$	−29.0000	−1.17707	−0.588537	+	0.808470i	$0.700296\pi$
$608$	−64.0000	−2.59554
$609$	0	0
$610$	−6.00000	−0.242933
$611$	10.0000	0.404557
$612$	0	0
$613$	8.00000	0.323117	0.161558	−	0.986863i	$-0.448348\pi$
$613$	8.00000	0.323117	0.161558	+	0.986863i	$0.448348\pi$
$614$	4.00000	0.161427
$615$	0	0
$616$	0	0
$617$	7.00000	0.281809	0.140905	−	0.990023i	$-0.454999\pi$
$617$	7.00000	0.281809	0.140905	+	0.990023i	$0.454999\pi$
$618$	0	0
$619$	−8.00000	−0.321547	−0.160774	−	0.986991i	$-0.551399\pi$
$619$	−8.00000	−0.321547	−0.160774	+	0.986991i	$0.551399\pi$
$620$	6.00000	0.240966
$621$	0	0
$622$	−14.0000	−0.561349
$623$	−1.00000	−0.0400642
$624$	0	0
$625$	−29.0000	−1.16000
$626$	28.0000	1.11911
$627$	0	0
$628$	−28.0000	−1.11732
$629$	66.0000	2.63159
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	0	0
$633$	0	0
$634$	12.0000	0.476581
$635$	3.00000	0.119051
$636$	0	0
$637$	5.00000	0.198107
$638$	12.0000	0.475085
$639$	0	0
$640$	0	0
$641$	14.0000	0.552967	0.276483	−	0.961019i	$-0.410831\pi$
$641$	14.0000	0.552967	0.276483	+	0.961019i	$0.410831\pi$
$642$	0	0
$643$	28.0000	1.10421	0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000	1.10421	0.552106	+	0.833774i	$0.313824\pi$
$644$	10.0000	0.394055
$645$	0	0
$646$	−96.0000	−3.77707
$647$	3.00000	0.117942	0.0589711	−	0.998260i	$-0.481218\pi$
$647$	3.00000	0.117942	0.0589711	+	0.998260i	$0.481218\pi$
$648$	0	0
$649$	−6.00000	−0.235521
$650$	−40.0000	−1.56893
$651$	0	0
$652$	−2.00000	−0.0783260
$653$	−14.0000	−0.547862	−0.273931	−	0.961749i	$-0.588324\pi$
$653$	−14.0000	−0.547862	−0.273931	+	0.961749i	$0.588324\pi$
$654$	0	0
$655$	−12.0000	−0.468879
$656$	24.0000	0.937043
$657$	0	0
$658$	4.00000	0.155936
$659$	−11.0000	−0.428499	−0.214250	−	0.976779i	$-0.568731\pi$
$659$	−11.0000	−0.428499	−0.214250	+	0.976779i	$0.568731\pi$
$660$	0	0
$661$	7.00000	0.272268	0.136134	−	0.990690i	$-0.456532\pi$
$661$	7.00000	0.272268	0.136134	+	0.990690i	$0.456532\pi$
$662$	−36.0000	−1.39918
$663$	0	0
$664$	0	0
$665$	−24.0000	−0.930680
$666$	0	0
$667$	−15.0000	−0.580802
$668$	16.0000	0.619059
$669$	0	0
$670$	12.0000	0.463600
$671$	2.00000	0.0772091
$672$	0	0
$673$	19.0000	0.732396	0.366198	−	0.930537i	$-0.380659\pi$
$673$	19.0000	0.732396	0.366198	+	0.930537i	$0.380659\pi$
$674$	56.0000	2.15704
$675$	0	0
$676$	24.0000	0.923077
$677$	−28.0000	−1.07613	−0.538064	−	0.842904i	$-0.680844\pi$
$677$	−28.0000	−1.07613	−0.538064	+	0.842904i	$0.680844\pi$
$678$	0	0
$679$	6.00000	0.230259
$680$	0	0
$681$	0	0
$682$	−4.00000	−0.153168
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	0	0
$685$	−21.0000	−0.802369
$686$	2.00000	0.0763604
$687$	0	0
$688$	32.0000	1.21999
$689$	−15.0000	−0.571454
$690$	0	0
$691$	42.0000	1.59776	0.798878	−	0.601494i	$-0.205427\pi$
$691$	42.0000	1.59776	0.798878	+	0.601494i	$0.205427\pi$
$692$	−36.0000	−1.36851
$693$	0	0
$694$	74.0000	2.80900
$695$	42.0000	1.59315
$696$	0	0
$697$	36.0000	1.36360
$698$	28.0000	1.05982
$699$	0	0
$700$	−8.00000	−0.302372
$701$	−33.0000	−1.24639	−0.623196	−	0.782065i	$-0.714166\pi$
$701$	−33.0000	−1.24639	−0.623196	+	0.782065i	$0.714166\pi$
$702$	0	0
$703$	88.0000	3.31898
$704$	16.0000	0.603023
$705$	0	0
$706$	−48.0000	−1.80650
$707$	7.00000	0.263262
$708$	0	0
$709$	38.0000	1.42712	0.713560	−	0.700594i	$-0.247082\pi$
$709$	38.0000	1.42712	0.713560	+	0.700594i	$0.247082\pi$
$710$	−60.0000	−2.25176
$711$	0	0
$712$	0	0
$713$	5.00000	0.187251
$714$	0	0
$715$	30.0000	1.12194
$716$	−52.0000	−1.94333
$717$	0	0
$718$	−16.0000	−0.597115
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	−90.0000	−3.34945
$723$	0	0
$724$	−32.0000	−1.18927
$725$	12.0000	0.445669
$726$	0	0
$727$	30.0000	1.11264	0.556319	−	0.830969i	$-0.312213\pi$
$727$	30.0000	1.11264	0.556319	+	0.830969i	$0.312213\pi$
$728$	0	0
$729$	0	0
$730$	66.0000	2.44277
$731$	48.0000	1.77534
$732$	0	0
$733$	−23.0000	−0.849524	−0.424762	−	0.905305i	$-0.639642\pi$
$733$	−23.0000	−0.849524	−0.424762	+	0.905305i	$0.639642\pi$
$734$	10.0000	0.369107
$735$	0	0
$736$	−40.0000	−1.47442
$737$	−4.00000	−0.147342
$738$	0	0
$739$	−15.0000	−0.551784	−0.275892	−	0.961189i	$-0.588973\pi$
$739$	−15.0000	−0.551784	−0.275892	+	0.961189i	$0.588973\pi$
$740$	66.0000	2.42621
$741$	0	0
$742$	−6.00000	−0.220267
$743$	−27.0000	−0.990534	−0.495267	−	0.868741i	$-0.664930\pi$
$743$	−27.0000	−0.990534	−0.495267	+	0.868741i	$0.664930\pi$
$744$	0	0
$745$	48.0000	1.75858
$746$	76.0000	2.78256
$747$	0	0
$748$	24.0000	0.877527
$749$	−18.0000	−0.657706
$750$	0	0
$751$	−22.0000	−0.802791	−0.401396	−	0.915905i	$-0.631475\pi$
$751$	−22.0000	−0.802791	−0.401396	+	0.915905i	$0.631475\pi$
$752$	−8.00000	−0.291730
$753$	0	0
$754$	−30.0000	−1.09254
$755$	6.00000	0.218362
$756$	0	0
$757$	−31.0000	−1.12671	−0.563357	−	0.826214i	$-0.690490\pi$
$757$	−31.0000	−1.12671	−0.563357	+	0.826214i	$0.690490\pi$
$758$	−64.0000	−2.32458
$759$	0	0
$760$	0	0
$761$	15.0000	0.543750	0.271875	−	0.962333i	$-0.412356\pi$
$761$	15.0000	0.543750	0.271875	+	0.962333i	$0.412356\pi$
$762$	0	0
$763$	4.00000	0.144810
$764$	24.0000	0.868290
$765$	0	0
$766$	48.0000	1.73431
$767$	15.0000	0.541619
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	12.0000	0.432450
$771$	0	0
$772$	−36.0000	−1.29567
$773$	−34.0000	−1.22290	−0.611448	−	0.791285i	$-0.709412\pi$
$773$	−34.0000	−1.22290	−0.611448	+	0.791285i	$0.709412\pi$
$774$	0	0
$775$	−4.00000	−0.143684
$776$	0	0
$777$	0	0
$778$	28.0000	1.00385
$779$	48.0000	1.71978
$780$	0	0
$781$	20.0000	0.715656
$782$	−60.0000	−2.14560
$783$	0	0
$784$	−4.00000	−0.142857
$785$	42.0000	1.49904
$786$	0	0
$787$	−7.00000	−0.249523	−0.124762	−	0.992187i	$-0.539817\pi$
$787$	−7.00000	−0.249523	−0.124762	+	0.992187i	$0.539817\pi$
$788$	−36.0000	−1.28245
$789$	0	0
$790$	0	0
$791$	2.00000	0.0711118
$792$	0	0
$793$	−5.00000	−0.177555
$794$	−34.0000	−1.20661
$795$	0	0
$796$	−10.0000	−0.354441
$797$	16.0000	0.566749	0.283375	−	0.959009i	$-0.408546\pi$
$797$	16.0000	0.566749	0.283375	+	0.959009i	$0.408546\pi$
$798$	0	0
$799$	−12.0000	−0.424529
$800$	32.0000	1.13137
$801$	0	0
$802$	14.0000	0.494357
$803$	−22.0000	−0.776363
$804$	0	0
$805$	−15.0000	−0.528681
$806$	10.0000	0.352235
$807$	0	0
$808$	0	0
$809$	16.0000	0.562530	0.281265	−	0.959630i	$-0.409246\pi$
$809$	16.0000	0.562530	0.281265	+	0.959630i	$0.409246\pi$
$810$	0	0
$811$	29.0000	1.01833	0.509164	−	0.860670i	$-0.329955\pi$
$811$	29.0000	1.01833	0.509164	+	0.860670i	$0.329955\pi$
$812$	−6.00000	−0.210559
$813$	0	0
$814$	−44.0000	−1.54220
$815$	3.00000	0.105085
$816$	0	0
$817$	64.0000	2.23908
$818$	16.0000	0.559427
$819$	0	0
$820$	36.0000	1.25717
$821$	29.0000	1.01211	0.506053	−	0.862502i	$-0.331104\pi$
$821$	29.0000	1.01211	0.506053	+	0.862502i	$0.331104\pi$
$822$	0	0
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	0	0
$825$	0	0
$826$	6.00000	0.208767
$827$	21.0000	0.730242	0.365121	−	0.930960i	$-0.381028\pi$
$827$	21.0000	0.730242	0.365121	+	0.930960i	$0.381028\pi$
$828$	0	0
$829$	10.0000	0.347314	0.173657	−	0.984806i	$-0.444442\pi$
$829$	10.0000	0.347314	0.173657	+	0.984806i	$0.444442\pi$
$830$	−102.000	−3.54047
$831$	0	0
$832$	−40.0000	−1.38675
$833$	−6.00000	−0.207888
$834$	0	0
$835$	−24.0000	−0.830554
$836$	32.0000	1.10674
$837$	0	0
$838$	32.0000	1.10542
$839$	−29.0000	−1.00119	−0.500596	−	0.865681i	$-0.666886\pi$
$839$	−29.0000	−1.00119	−0.500596	+	0.865681i	$0.666886\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	−76.0000	−2.61913
$843$	0	0
$844$	−50.0000	−1.72107
$845$	−36.0000	−1.23844
$846$	0	0
$847$	7.00000	0.240523
$848$	12.0000	0.412082
$849$	0	0
$850$	48.0000	1.64639
$851$	55.0000	1.88538
$852$	0	0
$853$	14.0000	0.479351	0.239675	−	0.970853i	$-0.422959\pi$
$853$	14.0000	0.479351	0.239675	+	0.970853i	$0.422959\pi$
$854$	−2.00000	−0.0684386
$855$	0	0
$856$	0	0
$857$	3.00000	0.102478	0.0512390	−	0.998686i	$-0.483683\pi$
$857$	3.00000	0.102478	0.0512390	+	0.998686i	$0.483683\pi$
$858$	0	0
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	48.0000	1.63679
$861$	0	0
$862$	36.0000	1.22616
$863$	53.0000	1.80414	0.902070	−	0.431589i	$-0.142047\pi$
$863$	53.0000	1.80414	0.902070	+	0.431589i	$0.142047\pi$
$864$	0	0
$865$	54.0000	1.83606
$866$	−52.0000	−1.76703
$867$	0	0
$868$	2.00000	0.0678844
$869$	0	0
$870$	0	0
$871$	10.0000	0.338837
$872$	0	0
$873$	0	0
$874$	−80.0000	−2.70604
$875$	−3.00000	−0.101419
$876$	0	0
$877$	46.0000	1.55331	0.776655	−	0.629926i	$-0.216915\pi$
$877$	46.0000	1.55331	0.776655	+	0.629926i	$0.216915\pi$
$878$	−4.00000	−0.134993
$879$	0	0
$880$	−24.0000	−0.809040
$881$	47.0000	1.58347	0.791735	−	0.610865i	$-0.209178\pi$
$881$	47.0000	1.58347	0.791735	+	0.610865i	$0.209178\pi$
$882$	0	0
$883$	−19.0000	−0.639401	−0.319700	−	0.947519i	$-0.603582\pi$
$883$	−19.0000	−0.639401	−0.319700	+	0.947519i	$0.603582\pi$
$884$	−60.0000	−2.01802
$885$	0	0
$886$	4.00000	0.134383
$887$	−29.0000	−0.973725	−0.486862	−	0.873479i	$-0.661859\pi$
$887$	−29.0000	−0.973725	−0.486862	+	0.873479i	$0.661859\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	6.00000	0.201120
$891$	0	0
$892$	−20.0000	−0.669650
$893$	−16.0000	−0.535420
$894$	0	0
$895$	78.0000	2.60725
$896$	0	0
$897$	0	0
$898$	16.0000	0.533927
$899$	−3.00000	−0.100056
$900$	0	0
$901$	18.0000	0.599667
$902$	−24.0000	−0.799113
$903$	0	0
$904$	0	0
$905$	48.0000	1.59557
$906$	0	0
$907$	−37.0000	−1.22856	−0.614282	−	0.789086i	$-0.710554\pi$
$907$	−37.0000	−1.22856	−0.614282	+	0.789086i	$0.710554\pi$
$908$	56.0000	1.85843
$909$	0	0
$910$	−30.0000	−0.994490
$911$	−26.0000	−0.861418	−0.430709	−	0.902491i	$-0.641737\pi$
$911$	−26.0000	−0.861418	−0.430709	+	0.902491i	$0.641737\pi$
$912$	0	0
$913$	34.0000	1.12524
$914$	−22.0000	−0.727695
$915$	0	0
$916$	−20.0000	−0.660819
$917$	−4.00000	−0.132092
$918$	0	0
$919$	−19.0000	−0.626752	−0.313376	−	0.949629i	$-0.601460\pi$
$919$	−19.0000	−0.626752	−0.313376	+	0.949629i	$0.601460\pi$
$920$	0	0
$921$	0	0
$922$	−36.0000	−1.18560
$923$	−50.0000	−1.64577
$924$	0	0
$925$	−44.0000	−1.44671
$926$	70.0000	2.30034
$927$	0	0
$928$	24.0000	0.787839
$929$	−14.0000	−0.459325	−0.229663	−	0.973270i	$-0.573762\pi$
$929$	−14.0000	−0.459325	−0.229663	+	0.973270i	$0.573762\pi$
$930$	0	0
$931$	−8.00000	−0.262189
$932$	−30.0000	−0.982683
$933$	0	0
$934$	14.0000	0.458094
$935$	−36.0000	−1.17733
$936$	0	0
$937$	−40.0000	−1.30674	−0.653372	−	0.757037i	$-0.726646\pi$
$937$	−40.0000	−1.30674	−0.653372	+	0.757037i	$0.726646\pi$
$938$	4.00000	0.130605
$939$	0	0
$940$	−12.0000	−0.391397
$941$	−12.0000	−0.391189	−0.195594	−	0.980685i	$-0.562664\pi$
$941$	−12.0000	−0.391189	−0.195594	+	0.980685i	$0.562664\pi$
$942$	0	0
$943$	30.0000	0.976934
$944$	−12.0000	−0.390567
$945$	0	0
$946$	−32.0000	−1.04041
$947$	36.0000	1.16984	0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000	1.16984	0.584921	+	0.811090i	$0.301125\pi$
$948$	0	0
$949$	55.0000	1.78538
$950$	64.0000	2.07643
$951$	0	0
$952$	0	0
$953$	−44.0000	−1.42530	−0.712650	−	0.701520i	$-0.752505\pi$
$953$	−44.0000	−1.42530	−0.712650	+	0.701520i	$0.752505\pi$
$954$	0	0
$955$	−36.0000	−1.16493
$956$	38.0000	1.22901
$957$	0	0
$958$	−28.0000	−0.904639
$959$	−7.00000	−0.226042
$960$	0	0
$961$	−30.0000	−0.967742
$962$	110.000	3.54654
$963$	0	0
$964$	−16.0000	−0.515325
$965$	54.0000	1.73832
$966$	0	0
$967$	−6.00000	−0.192947	−0.0964735	−	0.995336i	$-0.530756\pi$
$967$	−6.00000	−0.192947	−0.0964735	+	0.995336i	$0.530756\pi$
$968$	0	0
$969$	0	0
$970$	−36.0000	−1.15589
$971$	−16.0000	−0.513464	−0.256732	−	0.966483i	$-0.582646\pi$
$971$	−16.0000	−0.513464	−0.256732	+	0.966483i	$0.582646\pi$
$972$	0	0
$973$	14.0000	0.448819
$974$	4.00000	0.128168
$975$	0	0
$976$	4.00000	0.128037
$977$	−28.0000	−0.895799	−0.447900	−	0.894084i	$-0.647828\pi$
$977$	−28.0000	−0.895799	−0.447900	+	0.894084i	$0.647828\pi$
$978$	0	0
$979$	−2.00000	−0.0639203
$980$	−6.00000	−0.191663
$981$	0	0
$982$	−40.0000	−1.27645
$983$	42.0000	1.33959	0.669796	−	0.742545i	$-0.266382\pi$
$983$	42.0000	1.33959	0.669796	+	0.742545i	$0.266382\pi$
$984$	0	0
$985$	54.0000	1.72058
$986$	36.0000	1.14647
$987$	0	0
$988$	−80.0000	−2.54514
$989$	40.0000	1.27193
$990$	0	0
$991$	62.0000	1.96949	0.984747	−	0.173990i	$-0.0556660\pi$
$991$	62.0000	1.96949	0.984747	+	0.173990i	$0.0556660\pi$
$992$	−8.00000	−0.254000
$993$	0	0
$994$	−20.0000	−0.634361
$995$	15.0000	0.475532
$996$	0	0
$997$	24.0000	0.760088	0.380044	−	0.924968i	$-0.375909\pi$
$997$	24.0000	0.760088	0.380044	+	0.924968i	$0.375909\pi$
$998$	−56.0000	−1.77265
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.a.1.1		1
3.2	odd	2		2667.2.a.e.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.e.1.1	✓	1	3.2	odd	2
8001.2.a.a.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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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