LMFDB - Embedded newform 4002.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4002,2,Mod(1,4002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4002, 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("4002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4002 = 2 \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4002.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:	$31.9561308889$
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	$\chi$	$=$	4002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -4.00000 q^{10} +1.00000 q^{12} -2.00000 q^{13} +4.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} +4.00000 q^{20} -1.00000 q^{23} -1.00000 q^{24} +11.0000 q^{25} +2.00000 q^{26} +1.00000 q^{27} -1.00000 q^{29} -4.00000 q^{30} -1.00000 q^{32} -4.00000 q^{34} +1.00000 q^{36} +2.00000 q^{37} -4.00000 q^{38} -2.00000 q^{39} -4.00000 q^{40} -6.00000 q^{41} +4.00000 q^{43} +4.00000 q^{45} +1.00000 q^{46} +1.00000 q^{48} -7.00000 q^{49} -11.0000 q^{50} +4.00000 q^{51} -2.00000 q^{52} -4.00000 q^{53} -1.00000 q^{54} +4.00000 q^{57} +1.00000 q^{58} +8.00000 q^{59} +4.00000 q^{60} +6.00000 q^{61} +1.00000 q^{64} -8.00000 q^{65} +2.00000 q^{67} +4.00000 q^{68} -1.00000 q^{69} -1.00000 q^{72} +6.00000 q^{73} -2.00000 q^{74} +11.0000 q^{75} +4.00000 q^{76} +2.00000 q^{78} -2.00000 q^{79} +4.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +14.0000 q^{83} +16.0000 q^{85} -4.00000 q^{86} -1.00000 q^{87} -4.00000 q^{90} -1.00000 q^{92} +16.0000 q^{95} -1.00000 q^{96} -12.0000 q^{97} +7.00000 q^{98} +O(q^{100})\)

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
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	4.00000	1.78885	0.894427	−	0.447214i	$-0.147584\pi$
$5$	4.00000	1.78885	0.894427	+	0.447214i	$0.147584\pi$
$6$	−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
$9$	1.00000	0.333333
$10$	−4.00000	−1.26491
$11$	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	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	4.00000	1.03280
$16$	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	$0.338794\pi$
$18$	−1.00000	−0.235702
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	4.00000	0.894427
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−1.00000	−0.204124
$25$	11.0000	2.20000
$26$	2.00000	0.392232
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−4.00000	−0.730297
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−4.00000	−0.685994
$35$	0	0
$36$	1.00000	0.166667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−4.00000	−0.648886
$39$	−2.00000	−0.320256
$40$	−4.00000	−0.632456
$41$	−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
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	4.00000	0.596285
$46$	1.00000	0.147442
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	1.00000	0.144338
$49$	−7.00000	−1.00000
$50$	−11.0000	−1.55563
$51$	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	$0.588586\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0	0
$57$	4.00000	0.529813
$58$	1.00000	0.131306
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	4.00000	0.516398
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	0	0
$64$	1.00000	0.125000
$65$	−8.00000	−0.992278
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	4.00000	0.485071
$69$	−1.00000	−0.120386
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−1.00000	−0.117851
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	−2.00000	−0.232495
$75$	11.0000	1.27017
$76$	4.00000	0.458831
$77$	0	0
$78$	2.00000	0.226455
$79$	−2.00000	−0.225018	−0.112509	−	0.993651i	$-0.535889\pi$
$79$	−2.00000	−0.225018	−0.112509	+	0.993651i	$0.535889\pi$
$80$	4.00000	0.447214
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	14.0000	1.53670	0.768350	−	0.640030i	$-0.221078\pi$
$83$	14.0000	1.53670	0.768350	+	0.640030i	$0.221078\pi$
$84$	0	0
$85$	16.0000	1.73544
$86$	−4.00000	−0.431331
$87$	−1.00000	−0.107211
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	−4.00000	−0.421637
$91$	0	0
$92$	−1.00000	−0.104257
$93$	0	0
$94$	0	0
$95$	16.0000	1.64157
$96$	−1.00000	−0.102062
$97$	−12.0000	−1.21842	−0.609208	−	0.793011i	$-0.708512\pi$
$97$	−12.0000	−1.21842	−0.609208	+	0.793011i	$0.708512\pi$
$98$	7.00000	0.707107
$99$	0	0
$100$	11.0000	1.10000
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	−4.00000	−0.396059
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	4.00000	0.388514
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	1.00000	0.0962250
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	−4.00000	−0.376288	−0.188144	−	0.982141i	$-0.560247\pi$
$113$	−4.00000	−0.376288	−0.188144	+	0.982141i	$0.560247\pi$
$114$	−4.00000	−0.374634
$115$	−4.00000	−0.373002
$116$	−1.00000	−0.0928477
$117$	−2.00000	−0.184900
$118$	−8.00000	−0.736460
$119$	0	0
$120$	−4.00000	−0.365148
$121$	−11.0000	−1.00000
$122$	−6.00000	−0.543214
$123$	−6.00000	−0.541002
$124$	0	0
$125$	24.0000	2.14663
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	−1.00000	−0.0883883
$129$	4.00000	0.352180
$130$	8.00000	0.701646
$131$	20.0000	1.74741	0.873704	−	0.486458i	$-0.161711\pi$
$131$	20.0000	1.74741	0.873704	+	0.486458i	$0.161711\pi$
$132$	0	0
$133$	0	0
$134$	−2.00000	−0.172774
$135$	4.00000	0.344265
$136$	−4.00000	−0.342997
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	1.00000	0.0851257
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	1.00000	0.0833333
$145$	−4.00000	−0.332182
$146$	−6.00000	−0.496564
$147$	−7.00000	−0.577350
$148$	2.00000	0.164399
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	−11.0000	−0.898146
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	−4.00000	−0.324443
$153$	4.00000	0.323381
$154$	0	0
$155$	0	0
$156$	−2.00000	−0.160128
$157$	−6.00000	−0.478852	−0.239426	−	0.970915i	$-0.576959\pi$
$157$	−6.00000	−0.478852	−0.239426	+	0.970915i	$0.576959\pi$
$158$	2.00000	0.159111
$159$	−4.00000	−0.317221
$160$	−4.00000	−0.316228
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	−14.0000	−1.08661
$167$	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
$169$	−9.00000	−0.692308
$170$	−16.0000	−1.22714
$171$	4.00000	0.305888
$172$	4.00000	0.304997
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	1.00000	0.0758098
$175$	0	0
$176$	0	0
$177$	8.00000	0.601317
$178$	0	0
$179$	−8.00000	−0.597948	−0.298974	−	0.954261i	$-0.596644\pi$
$179$	−8.00000	−0.597948	−0.298974	+	0.954261i	$0.596644\pi$
$180$	4.00000	0.298142
$181$	8.00000	0.594635	0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000	0.594635	0.297318	+	0.954779i	$0.403908\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	1.00000	0.0737210
$185$	8.00000	0.588172
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	−16.0000	−1.16076
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	1.00000	0.0721688
$193$	−26.0000	−1.87152	−0.935760	−	0.352636i	$-0.885285\pi$
$193$	−26.0000	−1.87152	−0.935760	+	0.352636i	$0.885285\pi$
$194$	12.0000	0.861550
$195$	−8.00000	−0.572892
$196$	−7.00000	−0.500000
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	−11.0000	−0.777817
$201$	2.00000	0.141069
$202$	−10.0000	−0.703598
$203$	0	0
$204$	4.00000	0.280056
$205$	−24.0000	−1.67623
$206$	−8.00000	−0.557386
$207$	−1.00000	−0.0695048
$208$	−2.00000	−0.138675
$209$	0	0
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	−4.00000	−0.274721
$213$	0	0
$214$	6.00000	0.410152
$215$	16.0000	1.09119
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	4.00000	0.270914
$219$	6.00000	0.405442
$220$	0	0
$221$	−8.00000	−0.538138
$222$	−2.00000	−0.134231
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	0	0
$225$	11.0000	0.733333
$226$	4.00000	0.266076
$227$	−6.00000	−0.398234	−0.199117	−	0.979976i	$-0.563807\pi$
$227$	−6.00000	−0.398234	−0.199117	+	0.979976i	$0.563807\pi$
$228$	4.00000	0.264906
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	4.00000	0.263752
$231$	0	0
$232$	1.00000	0.0656532
$233$	22.0000	1.44127	0.720634	−	0.693316i	$-0.243851\pi$
$233$	22.0000	1.44127	0.720634	+	0.693316i	$0.243851\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	8.00000	0.520756
$237$	−2.00000	−0.129914
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	4.00000	0.258199
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	11.0000	0.707107
$243$	1.00000	0.0641500
$244$	6.00000	0.384111
$245$	−28.0000	−1.78885
$246$	6.00000	0.382546
$247$	−8.00000	−0.509028
$248$	0	0
$249$	14.0000	0.887214
$250$	−24.0000	−1.51789
$251$	−28.0000	−1.76734	−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000	−1.76734	−0.883672	+	0.468106i	$0.844936\pi$
$252$	0	0
$253$	0	0
$254$	−4.00000	−0.250982
$255$	16.0000	1.00196
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	−4.00000	−0.249029
$259$	0	0
$260$	−8.00000	−0.496139
$261$	−1.00000	−0.0618984
$262$	−20.0000	−1.23560
$263$	26.0000	1.60323	0.801614	−	0.597841i	$-0.203975\pi$
$263$	26.0000	1.60323	0.801614	+	0.597841i	$0.203975\pi$
$264$	0	0
$265$	−16.0000	−0.982872
$266$	0	0
$267$	0	0
$268$	2.00000	0.122169
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	−4.00000	−0.243432
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	12.0000	0.724947
$275$	0	0
$276$	−1.00000	−0.0601929
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	−8.00000	−0.479808
$279$	0	0
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	−14.0000	−0.832214	−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000	−0.832214	−0.416107	+	0.909316i	$0.636606\pi$
$284$	0	0
$285$	16.0000	0.947758
$286$	0	0
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	−1.00000	−0.0588235
$290$	4.00000	0.234888
$291$	−12.0000	−0.703452
$292$	6.00000	0.351123
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	7.00000	0.408248
$295$	32.0000	1.86311
$296$	−2.00000	−0.116248
$297$	0	0
$298$	0	0
$299$	2.00000	0.115663
$300$	11.0000	0.635085
$301$	0	0
$302$	0	0
$303$	10.0000	0.574485
$304$	4.00000	0.229416
$305$	24.0000	1.37424
$306$	−4.00000	−0.228665
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	−4.00000	−0.226819	−0.113410	−	0.993548i	$-0.536177\pi$
$311$	−4.00000	−0.226819	−0.113410	+	0.993548i	$0.536177\pi$
$312$	2.00000	0.113228
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	6.00000	0.338600
$315$	0	0
$316$	−2.00000	−0.112509
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	4.00000	0.224309
$319$	0	0
$320$	4.00000	0.223607
$321$	−6.00000	−0.334887
$322$	0	0
$323$	16.0000	0.890264
$324$	1.00000	0.0555556
$325$	−22.0000	−1.22034
$326$	12.0000	0.664619
$327$	−4.00000	−0.221201
$328$	6.00000	0.331295
$329$	0	0
$330$	0	0
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	14.0000	0.768350
$333$	2.00000	0.109599
$334$	−8.00000	−0.437741
$335$	8.00000	0.437087
$336$	0	0
$337$	−32.0000	−1.74315	−0.871576	−	0.490261i	$-0.836901\pi$
$337$	−32.0000	−1.74315	−0.871576	+	0.490261i	$0.836901\pi$
$338$	9.00000	0.489535
$339$	−4.00000	−0.217250
$340$	16.0000	0.867722
$341$	0	0
$342$	−4.00000	−0.216295
$343$	0	0
$344$	−4.00000	−0.215666
$345$	−4.00000	−0.215353
$346$	6.00000	0.322562
$347$	−20.0000	−1.07366	−0.536828	−	0.843692i	$-0.680378\pi$
$347$	−20.0000	−1.07366	−0.536828	+	0.843692i	$0.680378\pi$
$348$	−1.00000	−0.0536056
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	0	0
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	−8.00000	−0.425195
$355$	0	0
$356$	0	0
$357$	0	0
$358$	8.00000	0.422813
$359$	−14.0000	−0.738892	−0.369446	−	0.929252i	$-0.620452\pi$
$359$	−14.0000	−0.738892	−0.369446	+	0.929252i	$0.620452\pi$
$360$	−4.00000	−0.210819
$361$	−3.00000	−0.157895
$362$	−8.00000	−0.420471
$363$	−11.0000	−0.577350
$364$	0	0
$365$	24.0000	1.25622
$366$	−6.00000	−0.313625
$367$	14.0000	0.730794	0.365397	−	0.930852i	$-0.380933\pi$
$367$	14.0000	0.730794	0.365397	+	0.930852i	$0.380933\pi$
$368$	−1.00000	−0.0521286
$369$	−6.00000	−0.312348
$370$	−8.00000	−0.415900
$371$	0	0
$372$	0	0
$373$	12.0000	0.621336	0.310668	−	0.950518i	$-0.399447\pi$
$373$	12.0000	0.621336	0.310668	+	0.950518i	$0.399447\pi$
$374$	0	0
$375$	24.0000	1.23935
$376$	0	0
$377$	2.00000	0.103005
$378$	0	0
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	16.0000	0.820783
$381$	4.00000	0.204926
$382$	18.0000	0.920960
$383$	20.0000	1.02195	0.510976	−	0.859595i	$-0.329284\pi$
$383$	20.0000	1.02195	0.510976	+	0.859595i	$0.329284\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	26.0000	1.32337
$387$	4.00000	0.203331
$388$	−12.0000	−0.609208
$389$	−22.0000	−1.11544	−0.557722	−	0.830028i	$-0.688325\pi$
$389$	−22.0000	−1.11544	−0.557722	+	0.830028i	$0.688325\pi$
$390$	8.00000	0.405096
$391$	−4.00000	−0.202289
$392$	7.00000	0.353553
$393$	20.0000	1.00887
$394$	−18.0000	−0.906827
$395$	−8.00000	−0.402524
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	−4.00000	−0.200502
$399$	0	0
$400$	11.0000	0.550000
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	−2.00000	−0.0997509
$403$	0	0
$404$	10.0000	0.497519
$405$	4.00000	0.198762
$406$	0	0
$407$	0	0
$408$	−4.00000	−0.198030
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	24.0000	1.18528
$411$	−12.0000	−0.591916
$412$	8.00000	0.394132
$413$	0	0
$414$	1.00000	0.0491473
$415$	56.0000	2.74893
$416$	2.00000	0.0980581
$417$	8.00000	0.391762
$418$	0	0
$419$	6.00000	0.293119	0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000	0.293119	0.146560	+	0.989202i	$0.453180\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	12.0000	0.584151
$423$	0	0
$424$	4.00000	0.194257
$425$	44.0000	2.13431
$426$	0	0
$427$	0	0
$428$	−6.00000	−0.290021
$429$	0	0
$430$	−16.0000	−0.771589
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	1.00000	0.0481125
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	0	0
$435$	−4.00000	−0.191785
$436$	−4.00000	−0.191565
$437$	−4.00000	−0.191346
$438$	−6.00000	−0.286691
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	8.00000	0.380521
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	2.00000	0.0949158
$445$	0	0
$446$	−8.00000	−0.378811
$447$	0	0
$448$	0	0
$449$	−22.0000	−1.03824	−0.519122	−	0.854700i	$-0.673741\pi$
$449$	−22.0000	−1.03824	−0.519122	+	0.854700i	$0.673741\pi$
$450$	−11.0000	−0.518545
$451$	0	0
$452$	−4.00000	−0.188144
$453$	0	0
$454$	6.00000	0.281594
$455$	0	0
$456$	−4.00000	−0.187317
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	6.00000	0.280362
$459$	4.00000	0.186704
$460$	−4.00000	−0.186501
$461$	38.0000	1.76984	0.884918	−	0.465746i	$-0.154214\pi$
$461$	38.0000	1.76984	0.884918	+	0.465746i	$0.154214\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	−1.00000	−0.0464238
$465$	0	0
$466$	−22.0000	−1.01913
$467$	32.0000	1.48078	0.740392	−	0.672176i	$-0.234640\pi$
$467$	32.0000	1.48078	0.740392	+	0.672176i	$0.234640\pi$
$468$	−2.00000	−0.0924500
$469$	0	0
$470$	0	0
$471$	−6.00000	−0.276465
$472$	−8.00000	−0.368230
$473$	0	0
$474$	2.00000	0.0918630
$475$	44.0000	2.01886
$476$	0	0
$477$	−4.00000	−0.183147
$478$	−24.0000	−1.09773
$479$	30.0000	1.37073	0.685367	−	0.728197i	$-0.259642\pi$
$479$	30.0000	1.37073	0.685367	+	0.728197i	$0.259642\pi$
$480$	−4.00000	−0.182574
$481$	−4.00000	−0.182384
$482$	14.0000	0.637683
$483$	0	0
$484$	−11.0000	−0.500000
$485$	−48.0000	−2.17957
$486$	−1.00000	−0.0453609
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	−6.00000	−0.271607
$489$	−12.0000	−0.542659
$490$	28.0000	1.26491
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	−6.00000	−0.270501
$493$	−4.00000	−0.180151
$494$	8.00000	0.359937
$495$	0	0
$496$	0	0
$497$	0	0
$498$	−14.0000	−0.627355
$499$	12.0000	0.537194	0.268597	−	0.963253i	$-0.413440\pi$
$499$	12.0000	0.537194	0.268597	+	0.963253i	$0.413440\pi$
$500$	24.0000	1.07331
$501$	8.00000	0.357414
$502$	28.0000	1.24970
$503$	−34.0000	−1.51599	−0.757993	−	0.652263i	$-0.773820\pi$
$503$	−34.0000	−1.51599	−0.757993	+	0.652263i	$0.773820\pi$
$504$	0	0
$505$	40.0000	1.77998
$506$	0	0
$507$	−9.00000	−0.399704
$508$	4.00000	0.177471
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	−16.0000	−0.708492
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	4.00000	0.176604
$514$	−6.00000	−0.264649
$515$	32.0000	1.41009
$516$	4.00000	0.176090
$517$	0	0
$518$	0	0
$519$	−6.00000	−0.263371
$520$	8.00000	0.350823
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	1.00000	0.0437688
$523$	−14.0000	−0.612177	−0.306089	−	0.952003i	$-0.599020\pi$
$523$	−14.0000	−0.612177	−0.306089	+	0.952003i	$0.599020\pi$
$524$	20.0000	0.873704
$525$	0	0
$526$	−26.0000	−1.13365
$527$	0	0
$528$	0	0
$529$	1.00000	0.0434783
$530$	16.0000	0.694996
$531$	8.00000	0.347170
$532$	0	0
$533$	12.0000	0.519778
$534$	0	0
$535$	−24.0000	−1.03761
$536$	−2.00000	−0.0863868
$537$	−8.00000	−0.345225
$538$	−6.00000	−0.258678
$539$	0	0
$540$	4.00000	0.172133
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	28.0000	1.20270
$543$	8.00000	0.343313
$544$	−4.00000	−0.171499
$545$	−16.0000	−0.685365
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	−12.0000	−0.512615
$549$	6.00000	0.256074
$550$	0	0
$551$	−4.00000	−0.170406
$552$	1.00000	0.0425628
$553$	0	0
$554$	−2.00000	−0.0849719
$555$	8.00000	0.339581
$556$	8.00000	0.339276
$557$	−40.0000	−1.69485	−0.847427	−	0.530912i	$-0.821850\pi$
$557$	−40.0000	−1.69485	−0.847427	+	0.530912i	$0.821850\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	0	0
$561$	0	0
$562$	−6.00000	−0.253095
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	0	0
$565$	−16.0000	−0.673125
$566$	14.0000	0.588464
$567$	0	0
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	−16.0000	−0.670166
$571$	14.0000	0.585882	0.292941	−	0.956131i	$-0.405366\pi$
$571$	14.0000	0.585882	0.292941	+	0.956131i	$0.405366\pi$
$572$	0	0
$573$	−18.0000	−0.751961
$574$	0	0
$575$	−11.0000	−0.458732
$576$	1.00000	0.0416667
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	1.00000	0.0415945
$579$	−26.0000	−1.08052
$580$	−4.00000	−0.166091
$581$	0	0
$582$	12.0000	0.497416
$583$	0	0
$584$	−6.00000	−0.248282
$585$	−8.00000	−0.330759
$586$	−6.00000	−0.247858
$587$	4.00000	0.165098	0.0825488	−	0.996587i	$-0.473694\pi$
$587$	4.00000	0.165098	0.0825488	+	0.996587i	$0.473694\pi$
$588$	−7.00000	−0.288675
$589$	0	0
$590$	−32.0000	−1.31742
$591$	18.0000	0.740421
$592$	2.00000	0.0821995
$593$	34.0000	1.39621	0.698106	−	0.715994i	$-0.254026\pi$
$593$	34.0000	1.39621	0.698106	+	0.715994i	$0.254026\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4.00000	0.163709
$598$	−2.00000	−0.0817861
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	−11.0000	−0.449073
$601$	−42.0000	−1.71322	−0.856608	−	0.515968i	$-0.827432\pi$
$601$	−42.0000	−1.71322	−0.856608	+	0.515968i	$0.827432\pi$
$602$	0	0
$603$	2.00000	0.0814463
$604$	0	0
$605$	−44.0000	−1.78885
$606$	−10.0000	−0.406222
$607$	20.0000	0.811775	0.405887	−	0.913923i	$-0.366962\pi$
$607$	20.0000	0.811775	0.405887	+	0.913923i	$0.366962\pi$
$608$	−4.00000	−0.162221
$609$	0	0
$610$	−24.0000	−0.971732
$611$	0	0
$612$	4.00000	0.161690
$613$	−28.0000	−1.13091	−0.565455	−	0.824779i	$-0.691299\pi$
$613$	−28.0000	−1.13091	−0.565455	+	0.824779i	$0.691299\pi$
$614$	−12.0000	−0.484281
$615$	−24.0000	−0.967773
$616$	0	0
$617$	−12.0000	−0.483102	−0.241551	−	0.970388i	$-0.577656\pi$
$617$	−12.0000	−0.483102	−0.241551	+	0.970388i	$0.577656\pi$
$618$	−8.00000	−0.321807
$619$	−24.0000	−0.964641	−0.482321	−	0.875995i	$-0.660206\pi$
$619$	−24.0000	−0.964641	−0.482321	+	0.875995i	$0.660206\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	4.00000	0.160385
$623$	0	0
$624$	−2.00000	−0.0800641
$625$	41.0000	1.64000
$626$	−10.0000	−0.399680
$627$	0	0
$628$	−6.00000	−0.239426
$629$	8.00000	0.318981
$630$	0	0
$631$	−32.0000	−1.27390	−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000	−1.27390	−0.636950	+	0.770905i	$0.719804\pi$
$632$	2.00000	0.0795557
$633$	−12.0000	−0.476957
$634$	2.00000	0.0794301
$635$	16.0000	0.634941
$636$	−4.00000	−0.158610
$637$	14.0000	0.554700
$638$	0	0
$639$	0	0
$640$	−4.00000	−0.158114
$641$	−24.0000	−0.947943	−0.473972	−	0.880540i	$-0.657180\pi$
$641$	−24.0000	−0.947943	−0.473972	+	0.880540i	$0.657180\pi$
$642$	6.00000	0.236801
$643$	−26.0000	−1.02534	−0.512670	−	0.858586i	$-0.671344\pi$
$643$	−26.0000	−1.02534	−0.512670	+	0.858586i	$0.671344\pi$
$644$	0	0
$645$	16.0000	0.629999
$646$	−16.0000	−0.629512
$647$	8.00000	0.314512	0.157256	−	0.987558i	$-0.449735\pi$
$647$	8.00000	0.314512	0.157256	+	0.987558i	$0.449735\pi$
$648$	−1.00000	−0.0392837
$649$	0	0
$650$	22.0000	0.862911
$651$	0	0
$652$	−12.0000	−0.469956
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	4.00000	0.156412
$655$	80.0000	3.12586
$656$	−6.00000	−0.234261
$657$	6.00000	0.234082
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	20.0000	0.777910	0.388955	−	0.921257i	$-0.372836\pi$
$661$	20.0000	0.777910	0.388955	+	0.921257i	$0.372836\pi$
$662$	−28.0000	−1.08825
$663$	−8.00000	−0.310694
$664$	−14.0000	−0.543305
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	1.00000	0.0387202
$668$	8.00000	0.309529
$669$	8.00000	0.309298
$670$	−8.00000	−0.309067
$671$	0	0
$672$	0	0
$673$	6.00000	0.231283	0.115642	−	0.993291i	$-0.463108\pi$
$673$	6.00000	0.231283	0.115642	+	0.993291i	$0.463108\pi$
$674$	32.0000	1.23259
$675$	11.0000	0.423390
$676$	−9.00000	−0.346154
$677$	38.0000	1.46046	0.730229	−	0.683202i	$-0.239413\pi$
$677$	38.0000	1.46046	0.730229	+	0.683202i	$0.239413\pi$
$678$	4.00000	0.153619
$679$	0	0
$680$	−16.0000	−0.613572
$681$	−6.00000	−0.229920
$682$	0	0
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	4.00000	0.152944
$685$	−48.0000	−1.83399
$686$	0	0
$687$	−6.00000	−0.228914
$688$	4.00000	0.152499
$689$	8.00000	0.304776
$690$	4.00000	0.152277
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	20.0000	0.759190
$695$	32.0000	1.21383
$696$	1.00000	0.0379049
$697$	−24.0000	−0.909065
$698$	14.0000	0.529908
$699$	22.0000	0.832116
$700$	0	0
$701$	44.0000	1.66186	0.830929	−	0.556379i	$-0.187810\pi$
$701$	44.0000	1.66186	0.830929	+	0.556379i	$0.187810\pi$
$702$	2.00000	0.0754851
$703$	8.00000	0.301726
$704$	0	0
$705$	0	0
$706$	−14.0000	−0.526897
$707$	0	0
$708$	8.00000	0.300658
$709$	28.0000	1.05156	0.525781	−	0.850620i	$-0.323773\pi$
$709$	28.0000	1.05156	0.525781	+	0.850620i	$0.323773\pi$
$710$	0	0
$711$	−2.00000	−0.0750059
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−8.00000	−0.298974
$717$	24.0000	0.896296
$718$	14.0000	0.522475
$719$	16.0000	0.596699	0.298350	−	0.954457i	$-0.403564\pi$
$719$	16.0000	0.596699	0.298350	+	0.954457i	$0.403564\pi$
$720$	4.00000	0.149071
$721$	0	0
$722$	3.00000	0.111648
$723$	−14.0000	−0.520666
$724$	8.00000	0.297318
$725$	−11.0000	−0.408530
$726$	11.0000	0.408248
$727$	2.00000	0.0741759	0.0370879	−	0.999312i	$-0.488192\pi$
$727$	2.00000	0.0741759	0.0370879	+	0.999312i	$0.488192\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−24.0000	−0.888280
$731$	16.0000	0.591781
$732$	6.00000	0.221766
$733$	22.0000	0.812589	0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000	0.812589	0.406294	+	0.913742i	$0.366821\pi$
$734$	−14.0000	−0.516749
$735$	−28.0000	−1.03280
$736$	1.00000	0.0368605
$737$	0	0
$738$	6.00000	0.220863
$739$	−36.0000	−1.32428	−0.662141	−	0.749380i	$-0.730352\pi$
$739$	−36.0000	−1.32428	−0.662141	+	0.749380i	$0.730352\pi$
$740$	8.00000	0.294086
$741$	−8.00000	−0.293887
$742$	0	0
$743$	−6.00000	−0.220119	−0.110059	−	0.993925i	$-0.535104\pi$
$743$	−6.00000	−0.220119	−0.110059	+	0.993925i	$0.535104\pi$
$744$	0	0
$745$	0	0
$746$	−12.0000	−0.439351
$747$	14.0000	0.512233
$748$	0	0
$749$	0	0
$750$	−24.0000	−0.876356
$751$	−14.0000	−0.510867	−0.255434	−	0.966827i	$-0.582218\pi$
$751$	−14.0000	−0.510867	−0.255434	+	0.966827i	$0.582218\pi$
$752$	0	0
$753$	−28.0000	−1.02038
$754$	−2.00000	−0.0728357
$755$	0	0
$756$	0	0
$757$	−38.0000	−1.38113	−0.690567	−	0.723269i	$-0.742639\pi$
$757$	−38.0000	−1.38113	−0.690567	+	0.723269i	$0.742639\pi$
$758$	4.00000	0.145287
$759$	0	0
$760$	−16.0000	−0.580381
$761$	22.0000	0.797499	0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000	0.797499	0.398750	+	0.917060i	$0.369444\pi$
$762$	−4.00000	−0.144905
$763$	0	0
$764$	−18.0000	−0.651217
$765$	16.0000	0.578481
$766$	−20.0000	−0.722629
$767$	−16.0000	−0.577727
$768$	1.00000	0.0360844
$769$	−40.0000	−1.44244	−0.721218	−	0.692708i	$-0.756418\pi$
$769$	−40.0000	−1.44244	−0.721218	+	0.692708i	$0.756418\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	−26.0000	−0.935760
$773$	−42.0000	−1.51064	−0.755318	−	0.655359i	$-0.772517\pi$
$773$	−42.0000	−1.51064	−0.755318	+	0.655359i	$0.772517\pi$
$774$	−4.00000	−0.143777
$775$	0	0
$776$	12.0000	0.430775
$777$	0	0
$778$	22.0000	0.788738
$779$	−24.0000	−0.859889
$780$	−8.00000	−0.286446
$781$	0	0
$782$	4.00000	0.143040
$783$	−1.00000	−0.0357371
$784$	−7.00000	−0.250000
$785$	−24.0000	−0.856597
$786$	−20.0000	−0.713376
$787$	−50.0000	−1.78231	−0.891154	−	0.453701i	$-0.850103\pi$
$787$	−50.0000	−1.78231	−0.891154	+	0.453701i	$0.850103\pi$
$788$	18.0000	0.641223
$789$	26.0000	0.925625
$790$	8.00000	0.284627
$791$	0	0
$792$	0	0
$793$	−12.0000	−0.426132
$794$	2.00000	0.0709773
$795$	−16.0000	−0.567462
$796$	4.00000	0.141776
$797$	−34.0000	−1.20434	−0.602171	−	0.798367i	$-0.705697\pi$
$797$	−34.0000	−1.20434	−0.602171	+	0.798367i	$0.705697\pi$
$798$	0	0
$799$	0	0
$800$	−11.0000	−0.388909
$801$	0	0
$802$	−2.00000	−0.0706225
$803$	0	0
$804$	2.00000	0.0705346
$805$	0	0
$806$	0	0
$807$	6.00000	0.211210
$808$	−10.0000	−0.351799
$809$	22.0000	0.773479	0.386739	−	0.922189i	$-0.373601\pi$
$809$	22.0000	0.773479	0.386739	+	0.922189i	$0.373601\pi$
$810$	−4.00000	−0.140546
$811$	44.0000	1.54505	0.772524	−	0.634985i	$-0.218994\pi$
$811$	44.0000	1.54505	0.772524	+	0.634985i	$0.218994\pi$
$812$	0	0
$813$	−28.0000	−0.982003
$814$	0	0
$815$	−48.0000	−1.68137
$816$	4.00000	0.140028
$817$	16.0000	0.559769
$818$	26.0000	0.909069
$819$	0	0
$820$	−24.0000	−0.838116
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	12.0000	0.418548
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	−8.00000	−0.278693
$825$	0	0
$826$	0	0
$827$	48.0000	1.66912	0.834562	−	0.550914i	$-0.185721\pi$
$827$	48.0000	1.66912	0.834562	+	0.550914i	$0.185721\pi$
$828$	−1.00000	−0.0347524
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	−56.0000	−1.94379
$831$	2.00000	0.0693792
$832$	−2.00000	−0.0693375
$833$	−28.0000	−0.970143
$834$	−8.00000	−0.277017
$835$	32.0000	1.10741
$836$	0	0
$837$	0	0
$838$	−6.00000	−0.207267
$839$	14.0000	0.483334	0.241667	−	0.970359i	$-0.422306\pi$
$839$	14.0000	0.483334	0.241667	+	0.970359i	$0.422306\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−10.0000	−0.344623
$843$	6.00000	0.206651
$844$	−12.0000	−0.413057
$845$	−36.0000	−1.23844
$846$	0	0
$847$	0	0
$848$	−4.00000	−0.137361
$849$	−14.0000	−0.480479
$850$	−44.0000	−1.50919
$851$	−2.00000	−0.0685591
$852$	0	0
$853$	22.0000	0.753266	0.376633	−	0.926363i	$-0.377082\pi$
$853$	22.0000	0.753266	0.376633	+	0.926363i	$0.377082\pi$
$854$	0	0
$855$	16.0000	0.547188
$856$	6.00000	0.205076
$857$	22.0000	0.751506	0.375753	−	0.926720i	$-0.377384\pi$
$857$	22.0000	0.751506	0.375753	+	0.926720i	$0.377384\pi$
$858$	0	0
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	16.0000	0.545595
$861$	0	0
$862$	12.0000	0.408722
$863$	−32.0000	−1.08929	−0.544646	−	0.838666i	$-0.683336\pi$
$863$	−32.0000	−1.08929	−0.544646	+	0.838666i	$0.683336\pi$
$864$	−1.00000	−0.0340207
$865$	−24.0000	−0.816024
$866$	16.0000	0.543702
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	0	0
$870$	4.00000	0.135613
$871$	−4.00000	−0.135535
$872$	4.00000	0.135457
$873$	−12.0000	−0.406138
$874$	4.00000	0.135302
$875$	0	0
$876$	6.00000	0.202721
$877$	−38.0000	−1.28317	−0.641584	−	0.767052i	$-0.721723\pi$
$877$	−38.0000	−1.28317	−0.641584	+	0.767052i	$0.721723\pi$
$878$	0	0
$879$	6.00000	0.202375
$880$	0	0
$881$	16.0000	0.539054	0.269527	−	0.962993i	$-0.413133\pi$
$881$	16.0000	0.539054	0.269527	+	0.962993i	$0.413133\pi$
$882$	7.00000	0.235702
$883$	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
$885$	32.0000	1.07567
$886$	12.0000	0.403148
$887$	−36.0000	−1.20876	−0.604381	−	0.796696i	$-0.706579\pi$
$887$	−36.0000	−1.20876	−0.604381	+	0.796696i	$0.706579\pi$
$888$	−2.00000	−0.0671156
$889$	0	0
$890$	0	0
$891$	0	0
$892$	8.00000	0.267860
$893$	0	0
$894$	0	0
$895$	−32.0000	−1.06964
$896$	0	0
$897$	2.00000	0.0667781
$898$	22.0000	0.734150
$899$	0	0
$900$	11.0000	0.366667
$901$	−16.0000	−0.533037
$902$	0	0
$903$	0	0
$904$	4.00000	0.133038
$905$	32.0000	1.06372
$906$	0	0
$907$	4.00000	0.132818	0.0664089	−	0.997792i	$-0.478846\pi$
$907$	4.00000	0.132818	0.0664089	+	0.997792i	$0.478846\pi$
$908$	−6.00000	−0.199117
$909$	10.0000	0.331679
$910$	0	0
$911$	−10.0000	−0.331315	−0.165657	−	0.986183i	$-0.552975\pi$
$911$	−10.0000	−0.331315	−0.165657	+	0.986183i	$0.552975\pi$
$912$	4.00000	0.132453
$913$	0	0
$914$	−22.0000	−0.727695
$915$	24.0000	0.793416
$916$	−6.00000	−0.198246
$917$	0	0
$918$	−4.00000	−0.132020
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	4.00000	0.131876
$921$	12.0000	0.395413
$922$	−38.0000	−1.25146
$923$	0	0
$924$	0	0
$925$	22.0000	0.723356
$926$	0	0
$927$	8.00000	0.262754
$928$	1.00000	0.0328266
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	−28.0000	−0.917663
$932$	22.0000	0.720634
$933$	−4.00000	−0.130954
$934$	−32.0000	−1.04707
$935$	0	0
$936$	2.00000	0.0653720
$937$	−6.00000	−0.196011	−0.0980057	−	0.995186i	$-0.531246\pi$
$937$	−6.00000	−0.196011	−0.0980057	+	0.995186i	$0.531246\pi$
$938$	0	0
$939$	10.0000	0.326338
$940$	0	0
$941$	−28.0000	−0.912774	−0.456387	−	0.889781i	$-0.650857\pi$
$941$	−28.0000	−0.912774	−0.456387	+	0.889781i	$0.650857\pi$
$942$	6.00000	0.195491
$943$	6.00000	0.195387
$944$	8.00000	0.260378
$945$	0	0
$946$	0	0
$947$	−36.0000	−1.16984	−0.584921	−	0.811090i	$-0.698875\pi$
$947$	−36.0000	−1.16984	−0.584921	+	0.811090i	$0.698875\pi$
$948$	−2.00000	−0.0649570
$949$	−12.0000	−0.389536
$950$	−44.0000	−1.42755
$951$	−2.00000	−0.0648544
$952$	0	0
$953$	−30.0000	−0.971795	−0.485898	−	0.874016i	$-0.661507\pi$
$953$	−30.0000	−0.971795	−0.485898	+	0.874016i	$0.661507\pi$
$954$	4.00000	0.129505
$955$	−72.0000	−2.32987
$956$	24.0000	0.776215
$957$	0	0
$958$	−30.0000	−0.969256
$959$	0	0
$960$	4.00000	0.129099
$961$	−31.0000	−1.00000
$962$	4.00000	0.128965
$963$	−6.00000	−0.193347
$964$	−14.0000	−0.450910
$965$	−104.000	−3.34788
$966$	0	0
$967$	28.0000	0.900419	0.450210	−	0.892923i	$-0.351349\pi$
$967$	28.0000	0.900419	0.450210	+	0.892923i	$0.351349\pi$
$968$	11.0000	0.353553
$969$	16.0000	0.513994
$970$	48.0000	1.54119
$971$	−28.0000	−0.898563	−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000	−0.898563	−0.449281	+	0.893390i	$0.648320\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	0	0
$975$	−22.0000	−0.704564
$976$	6.00000	0.192055
$977$	38.0000	1.21573	0.607864	−	0.794041i	$-0.292027\pi$
$977$	38.0000	1.21573	0.607864	+	0.794041i	$0.292027\pi$
$978$	12.0000	0.383718
$979$	0	0
$980$	−28.0000	−0.894427
$981$	−4.00000	−0.127710
$982$	−20.0000	−0.638226
$983$	2.00000	0.0637901	0.0318950	−	0.999491i	$-0.489846\pi$
$983$	2.00000	0.0637901	0.0318950	+	0.999491i	$0.489846\pi$
$984$	6.00000	0.191273
$985$	72.0000	2.29411
$986$	4.00000	0.127386
$987$	0	0
$988$	−8.00000	−0.254514
$989$	−4.00000	−0.127193
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	0	0
$993$	28.0000	0.888553
$994$	0	0
$995$	16.0000	0.507234
$996$	14.0000	0.443607
$997$	42.0000	1.33015	0.665077	−	0.746775i	$-0.268399\pi$
$997$	42.0000	1.33015	0.665077	+	0.746775i	$0.268399\pi$
$998$	−12.0000	−0.379853
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.g.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.g.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:	\( N \)	\(=\)	\( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4002.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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