LMFDB - Embedded newform 8002.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8002,2,Mod(1,8002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8002, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8002 = 2 \cdot 4001 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8002.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:	$63.8962916974$
Analytic rank:	$1$
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	$\chi$	$=$	8002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} -2.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} -2.00000 q^{9} +1.00000 q^{10} -1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} -2.00000 q^{18} -4.00000 q^{19} +1.00000 q^{20} +1.00000 q^{23} -1.00000 q^{24} -4.00000 q^{25} +2.00000 q^{26} +5.00000 q^{27} -6.00000 q^{29} -1.00000 q^{30} -6.00000 q^{31} +1.00000 q^{32} +4.00000 q^{34} -2.00000 q^{36} -6.00000 q^{37} -4.00000 q^{38} -2.00000 q^{39} +1.00000 q^{40} +10.0000 q^{41} -9.00000 q^{43} -2.00000 q^{45} +1.00000 q^{46} +6.00000 q^{47} -1.00000 q^{48} -7.00000 q^{49} -4.00000 q^{50} -4.00000 q^{51} +2.00000 q^{52} -4.00000 q^{53} +5.00000 q^{54} +4.00000 q^{57} -6.00000 q^{58} -1.00000 q^{60} -9.00000 q^{61} -6.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} -1.00000 q^{67} +4.00000 q^{68} -1.00000 q^{69} -2.00000 q^{71} -2.00000 q^{72} +4.00000 q^{73} -6.00000 q^{74} +4.00000 q^{75} -4.00000 q^{76} -2.00000 q^{78} -10.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} +12.0000 q^{83} +4.00000 q^{85} -9.00000 q^{86} +6.00000 q^{87} +7.00000 q^{89} -2.00000 q^{90} +1.00000 q^{92} +6.00000 q^{93} +6.00000 q^{94} -4.00000 q^{95} -1.00000 q^{96} -1.00000 q^{97} -7.00000 q^{98} +O(q^{100})\)

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	−1.00000	−0.408248
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	1.00000	0.353553
$9$	−2.00000	−0.666667
$10$	1.00000	0.316228
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	−1.00000	−0.288675
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	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
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514	0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000	0.208514	0.104257	+	0.994550i	$0.466753\pi$
$24$	−1.00000	−0.204124
$25$	−4.00000	−0.800000
$26$	2.00000	0.392232
$27$	5.00000	0.962250
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	−1.00000	−0.182574
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	4.00000	0.685994
$35$	0	0
$36$	−2.00000	−0.333333
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	−4.00000	−0.648886
$39$	−2.00000	−0.320256
$40$	1.00000	0.158114
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	−9.00000	−1.37249	−0.686244	−	0.727372i	$-0.740742\pi$
$43$	−9.00000	−1.37249	−0.686244	+	0.727372i	$0.740742\pi$
$44$	0	0
$45$	−2.00000	−0.298142
$46$	1.00000	0.147442
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	−1.00000	−0.144338
$49$	−7.00000	−1.00000
$50$	−4.00000	−0.565685
$51$	−4.00000	−0.560112
$52$	2.00000	0.277350
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	5.00000	0.680414
$55$	0	0
$56$	0	0
$57$	4.00000	0.529813
$58$	−6.00000	−0.787839
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	−1.00000	−0.129099
$61$	−9.00000	−1.15233	−0.576166	−	0.817333i	$-0.695452\pi$
$61$	−9.00000	−1.15233	−0.576166	+	0.817333i	$0.695452\pi$
$62$	−6.00000	−0.762001
$63$	0	0
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	0	0
$67$	−1.00000	−0.122169	−0.0610847	−	0.998133i	$-0.519456\pi$
$67$	−1.00000	−0.122169	−0.0610847	+	0.998133i	$0.519456\pi$
$68$	4.00000	0.485071
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	−2.00000	−0.235702
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	−6.00000	−0.697486
$75$	4.00000	0.461880
$76$	−4.00000	−0.458831
$77$	0	0
$78$	−2.00000	−0.226455
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	10.0000	1.10432
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	−9.00000	−0.970495
$87$	6.00000	0.643268
$88$	0	0
$89$	7.00000	0.741999	0.370999	−	0.928633i	$-0.379015\pi$
$89$	7.00000	0.741999	0.370999	+	0.928633i	$0.379015\pi$
$90$	−2.00000	−0.210819
$91$	0	0
$92$	1.00000	0.104257
$93$	6.00000	0.622171
$94$	6.00000	0.618853
$95$	−4.00000	−0.410391
$96$	−1.00000	−0.102062
$97$	−1.00000	−0.101535	−0.0507673	−	0.998711i	$-0.516167\pi$
$97$	−1.00000	−0.101535	−0.0507673	+	0.998711i	$0.516167\pi$
$98$	−7.00000	−0.707107
$99$	0	0
$100$	−4.00000	−0.400000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	−4.00000	−0.396059
$103$	13.0000	1.28093	0.640464	−	0.767988i	$-0.278742\pi$
$103$	13.0000	1.28093	0.640464	+	0.767988i	$0.278742\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	−4.00000	−0.388514
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	5.00000	0.481125
$109$	−12.0000	−1.14939	−0.574696	−	0.818367i	$-0.694880\pi$
$109$	−12.0000	−1.14939	−0.574696	+	0.818367i	$0.694880\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	0	0
$113$	−16.0000	−1.50515	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	−16.0000	−1.50515	−0.752577	+	0.658505i	$0.771189\pi$
$114$	4.00000	0.374634
$115$	1.00000	0.0932505
$116$	−6.00000	−0.557086
$117$	−4.00000	−0.369800
$118$	0	0
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	−11.0000	−1.00000
$122$	−9.00000	−0.814822
$123$	−10.0000	−0.901670
$124$	−6.00000	−0.538816
$125$	−9.00000	−0.804984
$126$	0	0
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	1.00000	0.0883883
$129$	9.00000	0.792406
$130$	2.00000	0.175412
$131$	−7.00000	−0.611593	−0.305796	−	0.952097i	$-0.598923\pi$
$131$	−7.00000	−0.611593	−0.305796	+	0.952097i	$0.598923\pi$
$132$	0	0
$133$	0	0
$134$	−1.00000	−0.0863868
$135$	5.00000	0.430331
$136$	4.00000	0.342997
$137$	−3.00000	−0.256307	−0.128154	−	0.991754i	$-0.540905\pi$
$137$	−3.00000	−0.256307	−0.128154	+	0.991754i	$0.540905\pi$
$138$	−1.00000	−0.0851257
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	−2.00000	−0.167836
$143$	0	0
$144$	−2.00000	−0.166667
$145$	−6.00000	−0.498273
$146$	4.00000	0.331042
$147$	7.00000	0.577350
$148$	−6.00000	−0.493197
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	4.00000	0.326599
$151$	3.00000	0.244137	0.122068	−	0.992522i	$-0.461047\pi$
$151$	3.00000	0.244137	0.122068	+	0.992522i	$0.461047\pi$
$152$	−4.00000	−0.324443
$153$	−8.00000	−0.646762
$154$	0	0
$155$	−6.00000	−0.481932
$156$	−2.00000	−0.160128
$157$	−22.0000	−1.75579	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000	−1.75579	−0.877896	+	0.478852i	$0.841053\pi$
$158$	−10.0000	−0.795557
$159$	4.00000	0.317221
$160$	1.00000	0.0790569
$161$	0	0
$162$	1.00000	0.0785674
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	10.0000	0.780869
$165$	0	0
$166$	12.0000	0.931381
$167$	25.0000	1.93456	0.967279	−	0.253715i	$-0.0816525\pi$
$167$	25.0000	1.93456	0.967279	+	0.253715i	$0.0816525\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	4.00000	0.306786
$171$	8.00000	0.611775
$172$	−9.00000	−0.686244
$173$	21.0000	1.59660	0.798300	−	0.602260i	$-0.205733\pi$
$173$	21.0000	1.59660	0.798300	+	0.602260i	$0.205733\pi$
$174$	6.00000	0.454859
$175$	0	0
$176$	0	0
$177$	0	0
$178$	7.00000	0.524672
$179$	9.00000	0.672692	0.336346	−	0.941739i	$-0.390809\pi$
$179$	9.00000	0.672692	0.336346	+	0.941739i	$0.390809\pi$
$180$	−2.00000	−0.149071
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	9.00000	0.665299
$184$	1.00000	0.0737210
$185$	−6.00000	−0.441129
$186$	6.00000	0.439941
$187$	0	0
$188$	6.00000	0.437595
$189$	0	0
$190$	−4.00000	−0.290191
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	−1.00000	−0.0721688
$193$	−12.0000	−0.863779	−0.431889	−	0.901927i	$-0.642153\pi$
$193$	−12.0000	−0.863779	−0.431889	+	0.901927i	$0.642153\pi$
$194$	−1.00000	−0.0717958
$195$	−2.00000	−0.143223
$196$	−7.00000	−0.500000
$197$	9.00000	0.641223	0.320612	−	0.947211i	$-0.396112\pi$
$197$	9.00000	0.641223	0.320612	+	0.947211i	$0.396112\pi$
$198$	0	0
$199$	−11.0000	−0.779769	−0.389885	−	0.920864i	$-0.627485\pi$
$199$	−11.0000	−0.779769	−0.389885	+	0.920864i	$0.627485\pi$
$200$	−4.00000	−0.282843
$201$	1.00000	0.0705346
$202$	6.00000	0.422159
$203$	0	0
$204$	−4.00000	−0.280056
$205$	10.0000	0.698430
$206$	13.0000	0.905753
$207$	−2.00000	−0.139010
$208$	2.00000	0.138675
$209$	0	0
$210$	0	0
$211$	−22.0000	−1.51454	−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000	−1.51454	−0.757271	+	0.653101i	$0.773468\pi$
$212$	−4.00000	−0.274721
$213$	2.00000	0.137038
$214$	6.00000	0.410152
$215$	−9.00000	−0.613795
$216$	5.00000	0.340207
$217$	0	0
$218$	−12.0000	−0.812743
$219$	−4.00000	−0.270295
$220$	0	0
$221$	8.00000	0.538138
$222$	6.00000	0.402694
$223$	−12.0000	−0.803579	−0.401790	−	0.915732i	$-0.631612\pi$
$223$	−12.0000	−0.803579	−0.401790	+	0.915732i	$0.631612\pi$
$224$	0	0
$225$	8.00000	0.533333
$226$	−16.0000	−1.06430
$227$	−3.00000	−0.199117	−0.0995585	−	0.995032i	$-0.531743\pi$
$227$	−3.00000	−0.199117	−0.0995585	+	0.995032i	$0.531743\pi$
$228$	4.00000	0.264906
$229$	−9.00000	−0.594737	−0.297368	−	0.954763i	$-0.596109\pi$
$229$	−9.00000	−0.594737	−0.297368	+	0.954763i	$0.596109\pi$
$230$	1.00000	0.0659380
$231$	0	0
$232$	−6.00000	−0.393919
$233$	−8.00000	−0.524097	−0.262049	−	0.965055i	$-0.584398\pi$
$233$	−8.00000	−0.524097	−0.262049	+	0.965055i	$0.584398\pi$
$234$	−4.00000	−0.261488
$235$	6.00000	0.391397
$236$	0	0
$237$	10.0000	0.649570
$238$	0	0
$239$	−15.0000	−0.970269	−0.485135	−	0.874439i	$-0.661229\pi$
$239$	−15.0000	−0.970269	−0.485135	+	0.874439i	$0.661229\pi$
$240$	−1.00000	−0.0645497
$241$	23.0000	1.48156	0.740780	−	0.671748i	$-0.234456\pi$
$241$	23.0000	1.48156	0.740780	+	0.671748i	$0.234456\pi$
$242$	−11.0000	−0.707107
$243$	−16.0000	−1.02640
$244$	−9.00000	−0.576166
$245$	−7.00000	−0.447214
$246$	−10.0000	−0.637577
$247$	−8.00000	−0.509028
$248$	−6.00000	−0.381000
$249$	−12.0000	−0.760469
$250$	−9.00000	−0.569210
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	0	0
$253$	0	0
$254$	12.0000	0.752947
$255$	−4.00000	−0.250490
$256$	1.00000	0.0625000
$257$	19.0000	1.18519	0.592594	−	0.805502i	$-0.298104\pi$
$257$	19.0000	1.18519	0.592594	+	0.805502i	$0.298104\pi$
$258$	9.00000	0.560316
$259$	0	0
$260$	2.00000	0.124035
$261$	12.0000	0.742781
$262$	−7.00000	−0.432461
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	0	0
$265$	−4.00000	−0.245718
$266$	0	0
$267$	−7.00000	−0.428393
$268$	−1.00000	−0.0610847
$269$	−27.0000	−1.64622	−0.823110	−	0.567883i	$-0.807763\pi$
$269$	−27.0000	−1.64622	−0.823110	+	0.567883i	$0.807763\pi$
$270$	5.00000	0.304290
$271$	25.0000	1.51864	0.759321	−	0.650716i	$-0.225531\pi$
$271$	25.0000	1.51864	0.759321	+	0.650716i	$0.225531\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	−3.00000	−0.181237
$275$	0	0
$276$	−1.00000	−0.0601929
$277$	19.0000	1.14160	0.570800	−	0.821089i	$-0.306633\pi$
$277$	19.0000	1.14160	0.570800	+	0.821089i	$0.306633\pi$
$278$	−12.0000	−0.719712
$279$	12.0000	0.718421
$280$	0	0
$281$	−16.0000	−0.954480	−0.477240	−	0.878773i	$-0.658363\pi$
$281$	−16.0000	−0.954480	−0.477240	+	0.878773i	$0.658363\pi$
$282$	−6.00000	−0.357295
$283$	33.0000	1.96165	0.980823	−	0.194900i	$-0.0624381\pi$
$283$	33.0000	1.96165	0.980823	+	0.194900i	$0.0624381\pi$
$284$	−2.00000	−0.118678
$285$	4.00000	0.236940
$286$	0	0
$287$	0	0
$288$	−2.00000	−0.117851
$289$	−1.00000	−0.0588235
$290$	−6.00000	−0.352332
$291$	1.00000	0.0586210
$292$	4.00000	0.234082
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	7.00000	0.408248
$295$	0	0
$296$	−6.00000	−0.348743
$297$	0	0
$298$	−6.00000	−0.347571
$299$	2.00000	0.115663
$300$	4.00000	0.230940
$301$	0	0
$302$	3.00000	0.172631
$303$	−6.00000	−0.344691
$304$	−4.00000	−0.229416
$305$	−9.00000	−0.515339
$306$	−8.00000	−0.457330
$307$	−22.0000	−1.25561	−0.627803	−	0.778372i	$-0.716046\pi$
$307$	−22.0000	−1.25561	−0.627803	+	0.778372i	$0.716046\pi$
$308$	0	0
$309$	−13.0000	−0.739544
$310$	−6.00000	−0.340777
$311$	4.00000	0.226819	0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000	0.226819	0.113410	+	0.993548i	$0.463823\pi$
$312$	−2.00000	−0.113228
$313$	−16.0000	−0.904373	−0.452187	−	0.891923i	$-0.649356\pi$
$313$	−16.0000	−0.904373	−0.452187	+	0.891923i	$0.649356\pi$
$314$	−22.0000	−1.24153
$315$	0	0
$316$	−10.0000	−0.562544
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	4.00000	0.224309
$319$	0	0
$320$	1.00000	0.0559017
$321$	−6.00000	−0.334887
$322$	0	0
$323$	−16.0000	−0.890264
$324$	1.00000	0.0555556
$325$	−8.00000	−0.443760
$326$	−16.0000	−0.886158
$327$	12.0000	0.663602
$328$	10.0000	0.552158
$329$	0	0
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	12.0000	0.658586
$333$	12.0000	0.657596
$334$	25.0000	1.36794
$335$	−1.00000	−0.0546358
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	−9.00000	−0.489535
$339$	16.0000	0.869001
$340$	4.00000	0.216930
$341$	0	0
$342$	8.00000	0.432590
$343$	0	0
$344$	−9.00000	−0.485247
$345$	−1.00000	−0.0538382
$346$	21.0000	1.12897
$347$	8.00000	0.429463	0.214731	−	0.976673i	$-0.431112\pi$
$347$	8.00000	0.429463	0.214731	+	0.976673i	$0.431112\pi$
$348$	6.00000	0.321634
$349$	16.0000	0.856460	0.428230	−	0.903670i	$-0.359137\pi$
$349$	16.0000	0.856460	0.428230	+	0.903670i	$0.359137\pi$
$350$	0	0
$351$	10.0000	0.533761
$352$	0	0
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	−2.00000	−0.106149
$356$	7.00000	0.370999
$357$	0	0
$358$	9.00000	0.475665
$359$	9.00000	0.475002	0.237501	−	0.971387i	$-0.423672\pi$
$359$	9.00000	0.475002	0.237501	+	0.971387i	$0.423672\pi$
$360$	−2.00000	−0.105409
$361$	−3.00000	−0.157895
$362$	−10.0000	−0.525588
$363$	11.0000	0.577350
$364$	0	0
$365$	4.00000	0.209370
$366$	9.00000	0.470438
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	1.00000	0.0521286
$369$	−20.0000	−1.04116
$370$	−6.00000	−0.311925
$371$	0	0
$372$	6.00000	0.311086
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	0	0
$375$	9.00000	0.464758
$376$	6.00000	0.309426
$377$	−12.0000	−0.618031
$378$	0	0
$379$	2.00000	0.102733	0.0513665	−	0.998680i	$-0.483642\pi$
$379$	2.00000	0.102733	0.0513665	+	0.998680i	$0.483642\pi$
$380$	−4.00000	−0.205196
$381$	−12.0000	−0.614779
$382$	−12.0000	−0.613973
$383$	−4.00000	−0.204390	−0.102195	−	0.994764i	$-0.532587\pi$
$383$	−4.00000	−0.204390	−0.102195	+	0.994764i	$0.532587\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−12.0000	−0.610784
$387$	18.0000	0.914991
$388$	−1.00000	−0.0507673
$389$	−9.00000	−0.456318	−0.228159	−	0.973624i	$-0.573271\pi$
$389$	−9.00000	−0.456318	−0.228159	+	0.973624i	$0.573271\pi$
$390$	−2.00000	−0.101274
$391$	4.00000	0.202289
$392$	−7.00000	−0.353553
$393$	7.00000	0.353103
$394$	9.00000	0.453413
$395$	−10.0000	−0.503155
$396$	0	0
$397$	−3.00000	−0.150566	−0.0752828	−	0.997162i	$-0.523986\pi$
$397$	−3.00000	−0.150566	−0.0752828	+	0.997162i	$0.523986\pi$
$398$	−11.0000	−0.551380
$399$	0	0
$400$	−4.00000	−0.200000
$401$	−27.0000	−1.34832	−0.674158	−	0.738587i	$-0.735493\pi$
$401$	−27.0000	−1.34832	−0.674158	+	0.738587i	$0.735493\pi$
$402$	1.00000	0.0498755
$403$	−12.0000	−0.597763
$404$	6.00000	0.298511
$405$	1.00000	0.0496904
$406$	0	0
$407$	0	0
$408$	−4.00000	−0.198030
$409$	35.0000	1.73064	0.865319	−	0.501221i	$-0.167116\pi$
$409$	35.0000	1.73064	0.865319	+	0.501221i	$0.167116\pi$
$410$	10.0000	0.493865
$411$	3.00000	0.147979
$412$	13.0000	0.640464
$413$	0	0
$414$	−2.00000	−0.0982946
$415$	12.0000	0.589057
$416$	2.00000	0.0980581
$417$	12.0000	0.587643
$418$	0	0
$419$	5.00000	0.244266	0.122133	−	0.992514i	$-0.461027\pi$
$419$	5.00000	0.244266	0.122133	+	0.992514i	$0.461027\pi$
$420$	0	0
$421$	−32.0000	−1.55958	−0.779792	−	0.626038i	$-0.784675\pi$
$421$	−32.0000	−1.55958	−0.779792	+	0.626038i	$0.784675\pi$
$422$	−22.0000	−1.07094
$423$	−12.0000	−0.583460
$424$	−4.00000	−0.194257
$425$	−16.0000	−0.776114
$426$	2.00000	0.0969003
$427$	0	0
$428$	6.00000	0.290021
$429$	0	0
$430$	−9.00000	−0.434019
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	5.00000	0.240563
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	0	0
$435$	6.00000	0.287678
$436$	−12.0000	−0.574696
$437$	−4.00000	−0.191346
$438$	−4.00000	−0.191127
$439$	−2.00000	−0.0954548	−0.0477274	−	0.998860i	$-0.515198\pi$
$439$	−2.00000	−0.0954548	−0.0477274	+	0.998860i	$0.515198\pi$
$440$	0	0
$441$	14.0000	0.666667
$442$	8.00000	0.380521
$443$	−20.0000	−0.950229	−0.475114	−	0.879924i	$-0.657593\pi$
$443$	−20.0000	−0.950229	−0.475114	+	0.879924i	$0.657593\pi$
$444$	6.00000	0.284747
$445$	7.00000	0.331832
$446$	−12.0000	−0.568216
$447$	6.00000	0.283790
$448$	0	0
$449$	−23.0000	−1.08544	−0.542719	−	0.839915i	$-0.682605\pi$
$449$	−23.0000	−1.08544	−0.542719	+	0.839915i	$0.682605\pi$
$450$	8.00000	0.377124
$451$	0	0
$452$	−16.0000	−0.752577
$453$	−3.00000	−0.140952
$454$	−3.00000	−0.140797
$455$	0	0
$456$	4.00000	0.187317
$457$	35.0000	1.63723	0.818615	−	0.574342i	$-0.194742\pi$
$457$	35.0000	1.63723	0.818615	+	0.574342i	$0.194742\pi$
$458$	−9.00000	−0.420542
$459$	20.0000	0.933520
$460$	1.00000	0.0466252
$461$	−36.0000	−1.67669	−0.838344	−	0.545142i	$-0.816476\pi$
$461$	−36.0000	−1.67669	−0.838344	+	0.545142i	$0.816476\pi$
$462$	0	0
$463$	28.0000	1.30127	0.650635	−	0.759390i	$-0.274503\pi$
$463$	28.0000	1.30127	0.650635	+	0.759390i	$0.274503\pi$
$464$	−6.00000	−0.278543
$465$	6.00000	0.278243
$466$	−8.00000	−0.370593
$467$	15.0000	0.694117	0.347059	−	0.937843i	$-0.387180\pi$
$467$	15.0000	0.694117	0.347059	+	0.937843i	$0.387180\pi$
$468$	−4.00000	−0.184900
$469$	0	0
$470$	6.00000	0.276759
$471$	22.0000	1.01371
$472$	0	0
$473$	0	0
$474$	10.0000	0.459315
$475$	16.0000	0.734130
$476$	0	0
$477$	8.00000	0.366295
$478$	−15.0000	−0.686084
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	−1.00000	−0.0456435
$481$	−12.0000	−0.547153
$482$	23.0000	1.04762
$483$	0	0
$484$	−11.0000	−0.500000
$485$	−1.00000	−0.0454077
$486$	−16.0000	−0.725775
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	−9.00000	−0.407411
$489$	16.0000	0.723545
$490$	−7.00000	−0.316228
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	−10.0000	−0.450835
$493$	−24.0000	−1.08091
$494$	−8.00000	−0.359937
$495$	0	0
$496$	−6.00000	−0.269408
$497$	0	0
$498$	−12.0000	−0.537733
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	−9.00000	−0.402492
$501$	−25.0000	−1.11692
$502$	−20.0000	−0.892644
$503$	−7.00000	−0.312115	−0.156057	−	0.987748i	$-0.549878\pi$
$503$	−7.00000	−0.312115	−0.156057	+	0.987748i	$0.549878\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	0	0
$507$	9.00000	0.399704
$508$	12.0000	0.532414
$509$	21.0000	0.930809	0.465404	−	0.885098i	$-0.345909\pi$
$509$	21.0000	0.930809	0.465404	+	0.885098i	$0.345909\pi$
$510$	−4.00000	−0.177123
$511$	0	0
$512$	1.00000	0.0441942
$513$	−20.0000	−0.883022
$514$	19.0000	0.838054
$515$	13.0000	0.572848
$516$	9.00000	0.396203
$517$	0	0
$518$	0	0
$519$	−21.0000	−0.921798
$520$	2.00000	0.0877058
$521$	−24.0000	−1.05146	−0.525730	−	0.850652i	$-0.676208\pi$
$521$	−24.0000	−1.05146	−0.525730	+	0.850652i	$0.676208\pi$
$522$	12.0000	0.525226
$523$	40.0000	1.74908	0.874539	−	0.484955i	$-0.161164\pi$
$523$	40.0000	1.74908	0.874539	+	0.484955i	$0.161164\pi$
$524$	−7.00000	−0.305796
$525$	0	0
$526$	8.00000	0.348817
$527$	−24.0000	−1.04546
$528$	0	0
$529$	−22.0000	−0.956522
$530$	−4.00000	−0.173749
$531$	0	0
$532$	0	0
$533$	20.0000	0.866296
$534$	−7.00000	−0.302920
$535$	6.00000	0.259403
$536$	−1.00000	−0.0431934
$537$	−9.00000	−0.388379
$538$	−27.0000	−1.16405
$539$	0	0
$540$	5.00000	0.215166
$541$	13.0000	0.558914	0.279457	−	0.960158i	$-0.409846\pi$
$541$	13.0000	0.558914	0.279457	+	0.960158i	$0.409846\pi$
$542$	25.0000	1.07384
$543$	10.0000	0.429141
$544$	4.00000	0.171499
$545$	−12.0000	−0.514024
$546$	0	0
$547$	−31.0000	−1.32546	−0.662732	−	0.748857i	$-0.730603\pi$
$547$	−31.0000	−1.32546	−0.662732	+	0.748857i	$0.730603\pi$
$548$	−3.00000	−0.128154
$549$	18.0000	0.768221
$550$	0	0
$551$	24.0000	1.02243
$552$	−1.00000	−0.0425628
$553$	0	0
$554$	19.0000	0.807233
$555$	6.00000	0.254686
$556$	−12.0000	−0.508913
$557$	19.0000	0.805056	0.402528	−	0.915408i	$-0.368132\pi$
$557$	19.0000	0.805056	0.402528	+	0.915408i	$0.368132\pi$
$558$	12.0000	0.508001
$559$	−18.0000	−0.761319
$560$	0	0
$561$	0	0
$562$	−16.0000	−0.674919
$563$	−3.00000	−0.126435	−0.0632175	−	0.998000i	$-0.520136\pi$
$563$	−3.00000	−0.126435	−0.0632175	+	0.998000i	$0.520136\pi$
$564$	−6.00000	−0.252646
$565$	−16.0000	−0.673125
$566$	33.0000	1.38709
$567$	0	0
$568$	−2.00000	−0.0839181
$569$	15.0000	0.628833	0.314416	−	0.949285i	$-0.398191\pi$
$569$	15.0000	0.628833	0.314416	+	0.949285i	$0.398191\pi$
$570$	4.00000	0.167542
$571$	18.0000	0.753277	0.376638	−	0.926360i	$-0.377080\pi$
$571$	18.0000	0.753277	0.376638	+	0.926360i	$0.377080\pi$
$572$	0	0
$573$	12.0000	0.501307
$574$	0	0
$575$	−4.00000	−0.166812
$576$	−2.00000	−0.0833333
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	−1.00000	−0.0415945
$579$	12.0000	0.498703
$580$	−6.00000	−0.249136
$581$	0	0
$582$	1.00000	0.0414513
$583$	0	0
$584$	4.00000	0.165521
$585$	−4.00000	−0.165380
$586$	−18.0000	−0.743573
$587$	−13.0000	−0.536567	−0.268284	−	0.963340i	$-0.586456\pi$
$587$	−13.0000	−0.536567	−0.268284	+	0.963340i	$0.586456\pi$
$588$	7.00000	0.288675
$589$	24.0000	0.988903
$590$	0	0
$591$	−9.00000	−0.370211
$592$	−6.00000	−0.246598
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	0	0
$595$	0	0
$596$	−6.00000	−0.245770
$597$	11.0000	0.450200
$598$	2.00000	0.0817861
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	4.00000	0.163299
$601$	−16.0000	−0.652654	−0.326327	−	0.945257i	$-0.605811\pi$
$601$	−16.0000	−0.652654	−0.326327	+	0.945257i	$0.605811\pi$
$602$	0	0
$603$	2.00000	0.0814463
$604$	3.00000	0.122068
$605$	−11.0000	−0.447214
$606$	−6.00000	−0.243733
$607$	−38.0000	−1.54237	−0.771186	−	0.636610i	$-0.780336\pi$
$607$	−38.0000	−1.54237	−0.771186	+	0.636610i	$0.780336\pi$
$608$	−4.00000	−0.162221
$609$	0	0
$610$	−9.00000	−0.364399
$611$	12.0000	0.485468
$612$	−8.00000	−0.323381
$613$	−23.0000	−0.928961	−0.464481	−	0.885583i	$-0.653759\pi$
$613$	−23.0000	−0.928961	−0.464481	+	0.885583i	$0.653759\pi$
$614$	−22.0000	−0.887848
$615$	−10.0000	−0.403239
$616$	0	0
$617$	21.0000	0.845428	0.422714	−	0.906263i	$-0.361077\pi$
$617$	21.0000	0.845428	0.422714	+	0.906263i	$0.361077\pi$
$618$	−13.0000	−0.522937
$619$	−14.0000	−0.562708	−0.281354	−	0.959604i	$-0.590783\pi$
$619$	−14.0000	−0.562708	−0.281354	+	0.959604i	$0.590783\pi$
$620$	−6.00000	−0.240966
$621$	5.00000	0.200643
$622$	4.00000	0.160385
$623$	0	0
$624$	−2.00000	−0.0800641
$625$	11.0000	0.440000
$626$	−16.0000	−0.639489
$627$	0	0
$628$	−22.0000	−0.877896
$629$	−24.0000	−0.956943
$630$	0	0
$631$	28.0000	1.11466	0.557331	−	0.830290i	$-0.311825\pi$
$631$	28.0000	1.11466	0.557331	+	0.830290i	$0.311825\pi$
$632$	−10.0000	−0.397779
$633$	22.0000	0.874421
$634$	18.0000	0.714871
$635$	12.0000	0.476205
$636$	4.00000	0.158610
$637$	−14.0000	−0.554700
$638$	0	0
$639$	4.00000	0.158238
$640$	1.00000	0.0395285
$641$	22.0000	0.868948	0.434474	−	0.900684i	$-0.356934\pi$
$641$	22.0000	0.868948	0.434474	+	0.900684i	$0.356934\pi$
$642$	−6.00000	−0.236801
$643$	14.0000	0.552106	0.276053	−	0.961142i	$-0.410973\pi$
$643$	14.0000	0.552106	0.276053	+	0.961142i	$0.410973\pi$
$644$	0	0
$645$	9.00000	0.354375
$646$	−16.0000	−0.629512
$647$	−36.0000	−1.41531	−0.707653	−	0.706560i	$-0.750246\pi$
$647$	−36.0000	−1.41531	−0.707653	+	0.706560i	$0.750246\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	−8.00000	−0.313786
$651$	0	0
$652$	−16.0000	−0.626608
$653$	−40.0000	−1.56532	−0.782660	−	0.622449i	$-0.786138\pi$
$653$	−40.0000	−1.56532	−0.782660	+	0.622449i	$0.786138\pi$
$654$	12.0000	0.469237
$655$	−7.00000	−0.273513
$656$	10.0000	0.390434
$657$	−8.00000	−0.312110
$658$	0	0
$659$	−25.0000	−0.973862	−0.486931	−	0.873441i	$-0.661884\pi$
$659$	−25.0000	−0.973862	−0.486931	+	0.873441i	$0.661884\pi$
$660$	0	0
$661$	−42.0000	−1.63361	−0.816805	−	0.576913i	$-0.804257\pi$
$661$	−42.0000	−1.63361	−0.816805	+	0.576913i	$0.804257\pi$
$662$	−28.0000	−1.08825
$663$	−8.00000	−0.310694
$664$	12.0000	0.465690
$665$	0	0
$666$	12.0000	0.464991
$667$	−6.00000	−0.232321
$668$	25.0000	0.967279
$669$	12.0000	0.463947
$670$	−1.00000	−0.0386334
$671$	0	0
$672$	0	0
$673$	−50.0000	−1.92736	−0.963679	−	0.267063i	$-0.913947\pi$
$673$	−50.0000	−1.92736	−0.963679	+	0.267063i	$0.913947\pi$
$674$	2.00000	0.0770371
$675$	−20.0000	−0.769800
$676$	−9.00000	−0.346154
$677$	26.0000	0.999261	0.499631	−	0.866239i	$-0.333469\pi$
$677$	26.0000	0.999261	0.499631	+	0.866239i	$0.333469\pi$
$678$	16.0000	0.614476
$679$	0	0
$680$	4.00000	0.153393
$681$	3.00000	0.114960
$682$	0	0
$683$	20.0000	0.765279	0.382639	−	0.923898i	$-0.375015\pi$
$683$	20.0000	0.765279	0.382639	+	0.923898i	$0.375015\pi$
$684$	8.00000	0.305888
$685$	−3.00000	−0.114624
$686$	0	0
$687$	9.00000	0.343371
$688$	−9.00000	−0.343122
$689$	−8.00000	−0.304776
$690$	−1.00000	−0.0380693
$691$	32.0000	1.21734	0.608669	−	0.793424i	$-0.291704\pi$
$691$	32.0000	1.21734	0.608669	+	0.793424i	$0.291704\pi$
$692$	21.0000	0.798300
$693$	0	0
$694$	8.00000	0.303676
$695$	−12.0000	−0.455186
$696$	6.00000	0.227429
$697$	40.0000	1.51511
$698$	16.0000	0.605609
$699$	8.00000	0.302588
$700$	0	0
$701$	31.0000	1.17085	0.585427	−	0.810725i	$-0.300927\pi$
$701$	31.0000	1.17085	0.585427	+	0.810725i	$0.300927\pi$
$702$	10.0000	0.377426
$703$	24.0000	0.905177
$704$	0	0
$705$	−6.00000	−0.225973
$706$	−6.00000	−0.225813
$707$	0	0
$708$	0	0
$709$	6.00000	0.225335	0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000	0.225335	0.112667	+	0.993633i	$0.464061\pi$
$710$	−2.00000	−0.0750587
$711$	20.0000	0.750059
$712$	7.00000	0.262336
$713$	−6.00000	−0.224702
$714$	0	0
$715$	0	0
$716$	9.00000	0.336346
$717$	15.0000	0.560185
$718$	9.00000	0.335877
$719$	6.00000	0.223762	0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	−2.00000	−0.0745356
$721$	0	0
$722$	−3.00000	−0.111648
$723$	−23.0000	−0.855379
$724$	−10.0000	−0.371647
$725$	24.0000	0.891338
$726$	11.0000	0.408248
$727$	23.0000	0.853023	0.426511	−	0.904482i	$-0.359742\pi$
$727$	23.0000	0.853023	0.426511	+	0.904482i	$0.359742\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	4.00000	0.148047
$731$	−36.0000	−1.33151
$732$	9.00000	0.332650
$733$	−4.00000	−0.147743	−0.0738717	−	0.997268i	$-0.523536\pi$
$733$	−4.00000	−0.147743	−0.0738717	+	0.997268i	$0.523536\pi$
$734$	0	0
$735$	7.00000	0.258199
$736$	1.00000	0.0368605
$737$	0	0
$738$	−20.0000	−0.736210
$739$	−9.00000	−0.331070	−0.165535	−	0.986204i	$-0.552935\pi$
$739$	−9.00000	−0.331070	−0.165535	+	0.986204i	$0.552935\pi$
$740$	−6.00000	−0.220564
$741$	8.00000	0.293887
$742$	0	0
$743$	1.00000	0.0366864	0.0183432	−	0.999832i	$-0.494161\pi$
$743$	1.00000	0.0366864	0.0183432	+	0.999832i	$0.494161\pi$
$744$	6.00000	0.219971
$745$	−6.00000	−0.219823
$746$	−4.00000	−0.146450
$747$	−24.0000	−0.878114
$748$	0	0
$749$	0	0
$750$	9.00000	0.328634
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	6.00000	0.218797
$753$	20.0000	0.728841
$754$	−12.0000	−0.437014
$755$	3.00000	0.109181
$756$	0	0
$757$	22.0000	0.799604	0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000	0.799604	0.399802	+	0.916602i	$0.369079\pi$
$758$	2.00000	0.0726433
$759$	0	0
$760$	−4.00000	−0.145095
$761$	9.00000	0.326250	0.163125	−	0.986605i	$-0.447843\pi$
$761$	9.00000	0.326250	0.163125	+	0.986605i	$0.447843\pi$
$762$	−12.0000	−0.434714
$763$	0	0
$764$	−12.0000	−0.434145
$765$	−8.00000	−0.289241
$766$	−4.00000	−0.144526
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	−36.0000	−1.29819	−0.649097	−	0.760706i	$-0.724853\pi$
$769$	−36.0000	−1.29819	−0.649097	+	0.760706i	$0.724853\pi$
$770$	0	0
$771$	−19.0000	−0.684268
$772$	−12.0000	−0.431889
$773$	38.0000	1.36677	0.683383	−	0.730061i	$-0.260508\pi$
$773$	38.0000	1.36677	0.683383	+	0.730061i	$0.260508\pi$
$774$	18.0000	0.646997
$775$	24.0000	0.862105
$776$	−1.00000	−0.0358979
$777$	0	0
$778$	−9.00000	−0.322666
$779$	−40.0000	−1.43315
$780$	−2.00000	−0.0716115
$781$	0	0
$782$	4.00000	0.143040
$783$	−30.0000	−1.07211
$784$	−7.00000	−0.250000
$785$	−22.0000	−0.785214
$786$	7.00000	0.249682
$787$	18.0000	0.641631	0.320815	−	0.947142i	$-0.396043\pi$
$787$	18.0000	0.641631	0.320815	+	0.947142i	$0.396043\pi$
$788$	9.00000	0.320612
$789$	−8.00000	−0.284808
$790$	−10.0000	−0.355784
$791$	0	0
$792$	0	0
$793$	−18.0000	−0.639199
$794$	−3.00000	−0.106466
$795$	4.00000	0.141865
$796$	−11.0000	−0.389885
$797$	−15.0000	−0.531327	−0.265664	−	0.964066i	$-0.585591\pi$
$797$	−15.0000	−0.531327	−0.265664	+	0.964066i	$0.585591\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	−4.00000	−0.141421
$801$	−14.0000	−0.494666
$802$	−27.0000	−0.953403
$803$	0	0
$804$	1.00000	0.0352673
$805$	0	0
$806$	−12.0000	−0.422682
$807$	27.0000	0.950445
$808$	6.00000	0.211079
$809$	−50.0000	−1.75791	−0.878953	−	0.476908i	$-0.841757\pi$
$809$	−50.0000	−1.75791	−0.878953	+	0.476908i	$0.841757\pi$
$810$	1.00000	0.0351364
$811$	39.0000	1.36948	0.684738	−	0.728790i	$-0.259917\pi$
$811$	39.0000	1.36948	0.684738	+	0.728790i	$0.259917\pi$
$812$	0	0
$813$	−25.0000	−0.876788
$814$	0	0
$815$	−16.0000	−0.560456
$816$	−4.00000	−0.140028
$817$	36.0000	1.25948
$818$	35.0000	1.22375
$819$	0	0
$820$	10.0000	0.349215
$821$	−3.00000	−0.104701	−0.0523504	−	0.998629i	$-0.516671\pi$
$821$	−3.00000	−0.104701	−0.0523504	+	0.998629i	$0.516671\pi$
$822$	3.00000	0.104637
$823$	9.00000	0.313720	0.156860	−	0.987621i	$-0.449863\pi$
$823$	9.00000	0.313720	0.156860	+	0.987621i	$0.449863\pi$
$824$	13.0000	0.452876
$825$	0	0
$826$	0	0
$827$	5.00000	0.173867	0.0869335	−	0.996214i	$-0.472293\pi$
$827$	5.00000	0.173867	0.0869335	+	0.996214i	$0.472293\pi$
$828$	−2.00000	−0.0695048
$829$	11.0000	0.382046	0.191023	−	0.981586i	$-0.438820\pi$
$829$	11.0000	0.382046	0.191023	+	0.981586i	$0.438820\pi$
$830$	12.0000	0.416526
$831$	−19.0000	−0.659103
$832$	2.00000	0.0693375
$833$	−28.0000	−0.970143
$834$	12.0000	0.415526
$835$	25.0000	0.865161
$836$	0	0
$837$	−30.0000	−1.03695
$838$	5.00000	0.172722
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−32.0000	−1.10279
$843$	16.0000	0.551069
$844$	−22.0000	−0.757271
$845$	−9.00000	−0.309609
$846$	−12.0000	−0.412568
$847$	0	0
$848$	−4.00000	−0.137361
$849$	−33.0000	−1.13256
$850$	−16.0000	−0.548795
$851$	−6.00000	−0.205677
$852$	2.00000	0.0685189
$853$	18.0000	0.616308	0.308154	−	0.951336i	$-0.400289\pi$
$853$	18.0000	0.616308	0.308154	+	0.951336i	$0.400289\pi$
$854$	0	0
$855$	8.00000	0.273594
$856$	6.00000	0.205076
$857$	7.00000	0.239115	0.119558	−	0.992827i	$-0.461852\pi$
$857$	7.00000	0.239115	0.119558	+	0.992827i	$0.461852\pi$
$858$	0	0
$859$	21.0000	0.716511	0.358255	−	0.933624i	$-0.383372\pi$
$859$	21.0000	0.716511	0.358255	+	0.933624i	$0.383372\pi$
$860$	−9.00000	−0.306897
$861$	0	0
$862$	24.0000	0.817443
$863$	−4.00000	−0.136162	−0.0680808	−	0.997680i	$-0.521688\pi$
$863$	−4.00000	−0.136162	−0.0680808	+	0.997680i	$0.521688\pi$
$864$	5.00000	0.170103
$865$	21.0000	0.714021
$866$	−14.0000	−0.475739
$867$	1.00000	0.0339618
$868$	0	0
$869$	0	0
$870$	6.00000	0.203419
$871$	−2.00000	−0.0677674
$872$	−12.0000	−0.406371
$873$	2.00000	0.0676897
$874$	−4.00000	−0.135302
$875$	0	0
$876$	−4.00000	−0.135147
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	−2.00000	−0.0674967
$879$	18.0000	0.607125
$880$	0	0
$881$	−4.00000	−0.134763	−0.0673817	−	0.997727i	$-0.521465\pi$
$881$	−4.00000	−0.134763	−0.0673817	+	0.997727i	$0.521465\pi$
$882$	14.0000	0.471405
$883$	24.0000	0.807664	0.403832	−	0.914833i	$-0.367678\pi$
$883$	24.0000	0.807664	0.403832	+	0.914833i	$0.367678\pi$
$884$	8.00000	0.269069
$885$	0	0
$886$	−20.0000	−0.671913
$887$	−14.0000	−0.470074	−0.235037	−	0.971986i	$-0.575521\pi$
$887$	−14.0000	−0.470074	−0.235037	+	0.971986i	$0.575521\pi$
$888$	6.00000	0.201347
$889$	0	0
$890$	7.00000	0.234641
$891$	0	0
$892$	−12.0000	−0.401790
$893$	−24.0000	−0.803129
$894$	6.00000	0.200670
$895$	9.00000	0.300837
$896$	0	0
$897$	−2.00000	−0.0667781
$898$	−23.0000	−0.767520
$899$	36.0000	1.20067
$900$	8.00000	0.266667
$901$	−16.0000	−0.533037
$902$	0	0
$903$	0	0
$904$	−16.0000	−0.532152
$905$	−10.0000	−0.332411
$906$	−3.00000	−0.0996683
$907$	31.0000	1.02934	0.514669	−	0.857389i	$-0.327915\pi$
$907$	31.0000	1.02934	0.514669	+	0.857389i	$0.327915\pi$
$908$	−3.00000	−0.0995585
$909$	−12.0000	−0.398015
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	4.00000	0.132453
$913$	0	0
$914$	35.0000	1.15770
$915$	9.00000	0.297531
$916$	−9.00000	−0.297368
$917$	0	0
$918$	20.0000	0.660098
$919$	−4.00000	−0.131948	−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000	−0.131948	−0.0659739	+	0.997821i	$0.521015\pi$
$920$	1.00000	0.0329690
$921$	22.0000	0.724925
$922$	−36.0000	−1.18560
$923$	−4.00000	−0.131662
$924$	0	0
$925$	24.0000	0.789115
$926$	28.0000	0.920137
$927$	−26.0000	−0.853952
$928$	−6.00000	−0.196960
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	6.00000	0.196748
$931$	28.0000	0.917663
$932$	−8.00000	−0.262049
$933$	−4.00000	−0.130954
$934$	15.0000	0.490815
$935$	0	0
$936$	−4.00000	−0.130744
$937$	9.00000	0.294017	0.147009	−	0.989135i	$-0.453036\pi$
$937$	9.00000	0.294017	0.147009	+	0.989135i	$0.453036\pi$
$938$	0	0
$939$	16.0000	0.522140
$940$	6.00000	0.195698
$941$	−36.0000	−1.17357	−0.586783	−	0.809744i	$-0.699606\pi$
$941$	−36.0000	−1.17357	−0.586783	+	0.809744i	$0.699606\pi$
$942$	22.0000	0.716799
$943$	10.0000	0.325645
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.00000	0.0974869	0.0487435	−	0.998811i	$-0.484478\pi$
$947$	3.00000	0.0974869	0.0487435	+	0.998811i	$0.484478\pi$
$948$	10.0000	0.324785
$949$	8.00000	0.259691
$950$	16.0000	0.519109
$951$	−18.0000	−0.583690
$952$	0	0
$953$	24.0000	0.777436	0.388718	−	0.921357i	$-0.372918\pi$
$953$	24.0000	0.777436	0.388718	+	0.921357i	$0.372918\pi$
$954$	8.00000	0.259010
$955$	−12.0000	−0.388311
$956$	−15.0000	−0.485135
$957$	0	0
$958$	0	0
$959$	0	0
$960$	−1.00000	−0.0322749
$961$	5.00000	0.161290
$962$	−12.0000	−0.386896
$963$	−12.0000	−0.386695
$964$	23.0000	0.740780
$965$	−12.0000	−0.386294
$966$	0	0
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	−11.0000	−0.353553
$969$	16.0000	0.513994
$970$	−1.00000	−0.0321081
$971$	50.0000	1.60458	0.802288	−	0.596937i	$-0.203616\pi$
$971$	50.0000	1.60458	0.802288	+	0.596937i	$0.203616\pi$
$972$	−16.0000	−0.513200
$973$	0	0
$974$	−16.0000	−0.512673
$975$	8.00000	0.256205
$976$	−9.00000	−0.288083
$977$	54.0000	1.72761	0.863807	−	0.503824i	$-0.168074\pi$
$977$	54.0000	1.72761	0.863807	+	0.503824i	$0.168074\pi$
$978$	16.0000	0.511624
$979$	0	0
$980$	−7.00000	−0.223607
$981$	24.0000	0.766261
$982$	12.0000	0.382935
$983$	−6.00000	−0.191370	−0.0956851	−	0.995412i	$-0.530504\pi$
$983$	−6.00000	−0.191370	−0.0956851	+	0.995412i	$0.530504\pi$
$984$	−10.0000	−0.318788
$985$	9.00000	0.286764
$986$	−24.0000	−0.764316
$987$	0	0
$988$	−8.00000	−0.254514
$989$	−9.00000	−0.286183
$990$	0	0
$991$	−7.00000	−0.222362	−0.111181	−	0.993800i	$-0.535463\pi$
$991$	−7.00000	−0.222362	−0.111181	+	0.993800i	$0.535463\pi$
$992$	−6.00000	−0.190500
$993$	28.0000	0.888553
$994$	0	0
$995$	−11.0000	−0.348723
$996$	−12.0000	−0.380235
$997$	−47.0000	−1.48850	−0.744252	−	0.667898i	$-0.767194\pi$
$997$	−47.0000	−1.48850	−0.744252	+	0.667898i	$0.767194\pi$
$998$	4.00000	0.126618
$999$	−30.0000	−0.949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.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:	\( N \)	\(=\)	\( 8002 = 2 \cdot 4001 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8002.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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