LMFDB - Embedded newform 4002.2.a.m.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:	$1$
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	$\chi$	$=$	4002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{10} +3.00000 q^{11} +1.00000 q^{12} +5.00000 q^{13} -4.00000 q^{14} -3.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} +2.00000 q^{19} -3.00000 q^{20} -4.00000 q^{21} +3.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} +4.00000 q^{25} +5.00000 q^{26} +1.00000 q^{27} -4.00000 q^{28} -1.00000 q^{29} -3.00000 q^{30} +5.00000 q^{31} +1.00000 q^{32} +3.00000 q^{33} -6.00000 q^{34} +12.0000 q^{35} +1.00000 q^{36} -7.00000 q^{37} +2.00000 q^{38} +5.00000 q^{39} -3.00000 q^{40} -9.00000 q^{41} -4.00000 q^{42} -10.0000 q^{43} +3.00000 q^{44} -3.00000 q^{45} -1.00000 q^{46} -12.0000 q^{47} +1.00000 q^{48} +9.00000 q^{49} +4.00000 q^{50} -6.00000 q^{51} +5.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} -9.00000 q^{55} -4.00000 q^{56} +2.00000 q^{57} -1.00000 q^{58} -3.00000 q^{59} -3.00000 q^{60} +5.00000 q^{61} +5.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} -15.0000 q^{65} +3.00000 q^{66} -13.0000 q^{67} -6.00000 q^{68} -1.00000 q^{69} +12.0000 q^{70} -3.00000 q^{71} +1.00000 q^{72} +2.00000 q^{73} -7.00000 q^{74} +4.00000 q^{75} +2.00000 q^{76} -12.0000 q^{77} +5.00000 q^{78} +8.00000 q^{79} -3.00000 q^{80} +1.00000 q^{81} -9.00000 q^{82} -6.00000 q^{83} -4.00000 q^{84} +18.0000 q^{85} -10.0000 q^{86} -1.00000 q^{87} +3.00000 q^{88} -12.0000 q^{89} -3.00000 q^{90} -20.0000 q^{91} -1.00000 q^{92} +5.00000 q^{93} -12.0000 q^{94} -6.00000 q^{95} +1.00000 q^{96} -10.0000 q^{97} +9.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	1.00000	0.408248
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−3.00000	−0.948683
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	1.00000	0.288675
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	−4.00000	−1.06904
$15$	−3.00000	−0.774597
$16$	1.00000	0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	1.00000	0.235702
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	−3.00000	−0.670820
$21$	−4.00000	−0.872872
$22$	3.00000	0.639602
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	1.00000	0.204124
$25$	4.00000	0.800000
$26$	5.00000	0.980581
$27$	1.00000	0.192450
$28$	−4.00000	−0.755929
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−3.00000	−0.547723
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	1.00000	0.176777
$33$	3.00000	0.522233
$34$	−6.00000	−1.02899
$35$	12.0000	2.02837
$36$	1.00000	0.166667
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	2.00000	0.324443
$39$	5.00000	0.800641
$40$	−3.00000	−0.474342
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	−4.00000	−0.617213
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	3.00000	0.452267
$45$	−3.00000	−0.447214
$46$	−1.00000	−0.147442
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	1.00000	0.144338
$49$	9.00000	1.28571
$50$	4.00000	0.565685
$51$	−6.00000	−0.840168
$52$	5.00000	0.693375
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	1.00000	0.136083
$55$	−9.00000	−1.21356
$56$	−4.00000	−0.534522
$57$	2.00000	0.264906
$58$	−1.00000	−0.131306
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	−3.00000	−0.387298
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	5.00000	0.635001
$63$	−4.00000	−0.503953
$64$	1.00000	0.125000
$65$	−15.0000	−1.86052
$66$	3.00000	0.369274
$67$	−13.0000	−1.58820	−0.794101	−	0.607785i	$-0.792058\pi$
$67$	−13.0000	−1.58820	−0.794101	+	0.607785i	$0.792058\pi$
$68$	−6.00000	−0.727607
$69$	−1.00000	−0.120386
$70$	12.0000	1.43427
$71$	−3.00000	−0.356034	−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000	−0.356034	−0.178017	+	0.984027i	$0.556968\pi$
$72$	1.00000	0.117851
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	−7.00000	−0.813733
$75$	4.00000	0.461880
$76$	2.00000	0.229416
$77$	−12.0000	−1.36753
$78$	5.00000	0.566139
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	−3.00000	−0.335410
$81$	1.00000	0.111111
$82$	−9.00000	−0.993884
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	−4.00000	−0.436436
$85$	18.0000	1.95237
$86$	−10.0000	−1.07833
$87$	−1.00000	−0.107211
$88$	3.00000	0.319801
$89$	−12.0000	−1.27200	−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	−3.00000	−0.316228
$91$	−20.0000	−2.09657
$92$	−1.00000	−0.104257
$93$	5.00000	0.518476
$94$	−12.0000	−1.23771
$95$	−6.00000	−0.615587
$96$	1.00000	0.102062
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	9.00000	0.909137
$99$	3.00000	0.301511
$100$	4.00000	0.400000
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	−6.00000	−0.594089
$103$	−1.00000	−0.0985329	−0.0492665	−	0.998786i	$-0.515688\pi$
$103$	−1.00000	−0.0985329	−0.0492665	+	0.998786i	$0.515688\pi$
$104$	5.00000	0.490290
$105$	12.0000	1.17108
$106$	6.00000	0.582772
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	1.00000	0.0962250
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	−9.00000	−0.858116
$111$	−7.00000	−0.664411
$112$	−4.00000	−0.377964
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	2.00000	0.187317
$115$	3.00000	0.279751
$116$	−1.00000	−0.0928477
$117$	5.00000	0.462250
$118$	−3.00000	−0.276172
$119$	24.0000	2.20008
$120$	−3.00000	−0.273861
$121$	−2.00000	−0.181818
$122$	5.00000	0.452679
$123$	−9.00000	−0.811503
$124$	5.00000	0.449013
$125$	3.00000	0.268328
$126$	−4.00000	−0.356348
$127$	−7.00000	−0.621150	−0.310575	−	0.950549i	$-0.600522\pi$
$127$	−7.00000	−0.621150	−0.310575	+	0.950549i	$0.600522\pi$
$128$	1.00000	0.0883883
$129$	−10.0000	−0.880451
$130$	−15.0000	−1.31559
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	3.00000	0.261116
$133$	−8.00000	−0.693688
$134$	−13.0000	−1.12303
$135$	−3.00000	−0.258199
$136$	−6.00000	−0.514496
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	−1.00000	−0.0851257
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	12.0000	1.01419
$141$	−12.0000	−1.01058
$142$	−3.00000	−0.251754
$143$	15.0000	1.25436
$144$	1.00000	0.0833333
$145$	3.00000	0.249136
$146$	2.00000	0.165521
$147$	9.00000	0.742307
$148$	−7.00000	−0.575396
$149$	15.0000	1.22885	0.614424	−	0.788976i	$-0.289388\pi$
$149$	15.0000	1.22885	0.614424	+	0.788976i	$0.289388\pi$
$150$	4.00000	0.326599
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	2.00000	0.162221
$153$	−6.00000	−0.485071
$154$	−12.0000	−0.966988
$155$	−15.0000	−1.20483
$156$	5.00000	0.400320
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	8.00000	0.636446
$159$	6.00000	0.475831
$160$	−3.00000	−0.237171
$161$	4.00000	0.315244
$162$	1.00000	0.0785674
$163$	−13.0000	−1.01824	−0.509119	−	0.860696i	$-0.670029\pi$
$163$	−13.0000	−1.01824	−0.509119	+	0.860696i	$0.670029\pi$
$164$	−9.00000	−0.702782
$165$	−9.00000	−0.700649
$166$	−6.00000	−0.465690
$167$	−9.00000	−0.696441	−0.348220	−	0.937413i	$-0.613214\pi$
$167$	−9.00000	−0.696441	−0.348220	+	0.937413i	$0.613214\pi$
$168$	−4.00000	−0.308607
$169$	12.0000	0.923077
$170$	18.0000	1.38054
$171$	2.00000	0.152944
$172$	−10.0000	−0.762493
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	−1.00000	−0.0758098
$175$	−16.0000	−1.20949
$176$	3.00000	0.226134
$177$	−3.00000	−0.225494
$178$	−12.0000	−0.899438
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	−3.00000	−0.223607
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	−20.0000	−1.48250
$183$	5.00000	0.369611
$184$	−1.00000	−0.0737210
$185$	21.0000	1.54395
$186$	5.00000	0.366618
$187$	−18.0000	−1.31629
$188$	−12.0000	−0.875190
$189$	−4.00000	−0.290957
$190$	−6.00000	−0.435286
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	1.00000	0.0721688
$193$	−22.0000	−1.58359	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	−22.0000	−1.58359	−0.791797	+	0.610784i	$0.790854\pi$
$194$	−10.0000	−0.717958
$195$	−15.0000	−1.07417
$196$	9.00000	0.642857
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	3.00000	0.213201
$199$	−25.0000	−1.77220	−0.886102	−	0.463491i	$-0.846597\pi$
$199$	−25.0000	−1.77220	−0.886102	+	0.463491i	$0.846597\pi$
$200$	4.00000	0.282843
$201$	−13.0000	−0.916949
$202$	3.00000	0.211079
$203$	4.00000	0.280745
$204$	−6.00000	−0.420084
$205$	27.0000	1.88576
$206$	−1.00000	−0.0696733
$207$	−1.00000	−0.0695048
$208$	5.00000	0.346688
$209$	6.00000	0.415029
$210$	12.0000	0.828079
$211$	5.00000	0.344214	0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000	0.344214	0.172107	+	0.985078i	$0.444942\pi$
$212$	6.00000	0.412082
$213$	−3.00000	−0.205557
$214$	6.00000	0.410152
$215$	30.0000	2.04598
$216$	1.00000	0.0680414
$217$	−20.0000	−1.35769
$218$	−10.0000	−0.677285
$219$	2.00000	0.135147
$220$	−9.00000	−0.606780
$221$	−30.0000	−2.01802
$222$	−7.00000	−0.469809
$223$	26.0000	1.74109	0.870544	−	0.492090i	$-0.163767\pi$
$223$	26.0000	1.74109	0.870544	+	0.492090i	$0.163767\pi$
$224$	−4.00000	−0.267261
$225$	4.00000	0.266667
$226$	−6.00000	−0.399114
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	2.00000	0.132453
$229$	17.0000	1.12339	0.561696	−	0.827344i	$-0.310149\pi$
$229$	17.0000	1.12339	0.561696	+	0.827344i	$0.310149\pi$
$230$	3.00000	0.197814
$231$	−12.0000	−0.789542
$232$	−1.00000	−0.0656532
$233$	−24.0000	−1.57229	−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000	−1.57229	−0.786146	+	0.618041i	$0.787927\pi$
$234$	5.00000	0.326860
$235$	36.0000	2.34838
$236$	−3.00000	−0.195283
$237$	8.00000	0.519656
$238$	24.0000	1.55569
$239$	9.00000	0.582162	0.291081	−	0.956698i	$-0.405985\pi$
$239$	9.00000	0.582162	0.291081	+	0.956698i	$0.405985\pi$
$240$	−3.00000	−0.193649
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	−2.00000	−0.128565
$243$	1.00000	0.0641500
$244$	5.00000	0.320092
$245$	−27.0000	−1.72497
$246$	−9.00000	−0.573819
$247$	10.0000	0.636285
$248$	5.00000	0.317500
$249$	−6.00000	−0.380235
$250$	3.00000	0.189737
$251$	15.0000	0.946792	0.473396	−	0.880850i	$-0.343028\pi$
$251$	15.0000	0.946792	0.473396	+	0.880850i	$0.343028\pi$
$252$	−4.00000	−0.251976
$253$	−3.00000	−0.188608
$254$	−7.00000	−0.439219
$255$	18.0000	1.12720
$256$	1.00000	0.0625000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	−10.0000	−0.622573
$259$	28.0000	1.73984
$260$	−15.0000	−0.930261
$261$	−1.00000	−0.0618984
$262$	6.00000	0.370681
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	3.00000	0.184637
$265$	−18.0000	−1.10573
$266$	−8.00000	−0.490511
$267$	−12.0000	−0.734388
$268$	−13.0000	−0.794101
$269$	−3.00000	−0.182913	−0.0914566	−	0.995809i	$-0.529152\pi$
$269$	−3.00000	−0.182913	−0.0914566	+	0.995809i	$0.529152\pi$
$270$	−3.00000	−0.182574
$271$	11.0000	0.668202	0.334101	−	0.942537i	$-0.391567\pi$
$271$	11.0000	0.668202	0.334101	+	0.942537i	$0.391567\pi$
$272$	−6.00000	−0.363803
$273$	−20.0000	−1.21046
$274$	−6.00000	−0.362473
$275$	12.0000	0.723627
$276$	−1.00000	−0.0601929
$277$	17.0000	1.02143	0.510716	−	0.859750i	$-0.329381\pi$
$277$	17.0000	1.02143	0.510716	+	0.859750i	$0.329381\pi$
$278$	20.0000	1.19952
$279$	5.00000	0.299342
$280$	12.0000	0.717137
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	−12.0000	−0.714590
$283$	11.0000	0.653882	0.326941	−	0.945045i	$-0.393982\pi$
$283$	11.0000	0.653882	0.326941	+	0.945045i	$0.393982\pi$
$284$	−3.00000	−0.178017
$285$	−6.00000	−0.355409
$286$	15.0000	0.886969
$287$	36.0000	2.12501
$288$	1.00000	0.0589256
$289$	19.0000	1.11765
$290$	3.00000	0.176166
$291$	−10.0000	−0.586210
$292$	2.00000	0.117041
$293$	12.0000	0.701047	0.350524	−	0.936554i	$-0.386004\pi$
$293$	12.0000	0.701047	0.350524	+	0.936554i	$0.386004\pi$
$294$	9.00000	0.524891
$295$	9.00000	0.524000
$296$	−7.00000	−0.406867
$297$	3.00000	0.174078
$298$	15.0000	0.868927
$299$	−5.00000	−0.289157
$300$	4.00000	0.230940
$301$	40.0000	2.30556
$302$	−16.0000	−0.920697
$303$	3.00000	0.172345
$304$	2.00000	0.114708
$305$	−15.0000	−0.858898
$306$	−6.00000	−0.342997
$307$	29.0000	1.65512	0.827559	−	0.561379i	$-0.189729\pi$
$307$	29.0000	1.65512	0.827559	+	0.561379i	$0.189729\pi$
$308$	−12.0000	−0.683763
$309$	−1.00000	−0.0568880
$310$	−15.0000	−0.851943
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	5.00000	0.283069
$313$	20.0000	1.13047	0.565233	−	0.824931i	$-0.308786\pi$
$313$	20.0000	1.13047	0.565233	+	0.824931i	$0.308786\pi$
$314$	14.0000	0.790066
$315$	12.0000	0.676123
$316$	8.00000	0.450035
$317$	3.00000	0.168497	0.0842484	−	0.996445i	$-0.473151\pi$
$317$	3.00000	0.168497	0.0842484	+	0.996445i	$0.473151\pi$
$318$	6.00000	0.336463
$319$	−3.00000	−0.167968
$320$	−3.00000	−0.167705
$321$	6.00000	0.334887
$322$	4.00000	0.222911
$323$	−12.0000	−0.667698
$324$	1.00000	0.0555556
$325$	20.0000	1.10940
$326$	−13.0000	−0.720003
$327$	−10.0000	−0.553001
$328$	−9.00000	−0.496942
$329$	48.0000	2.64633
$330$	−9.00000	−0.495434
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	−6.00000	−0.329293
$333$	−7.00000	−0.383598
$334$	−9.00000	−0.492458
$335$	39.0000	2.13080
$336$	−4.00000	−0.218218
$337$	−13.0000	−0.708155	−0.354078	−	0.935216i	$-0.615205\pi$
$337$	−13.0000	−0.708155	−0.354078	+	0.935216i	$0.615205\pi$
$338$	12.0000	0.652714
$339$	−6.00000	−0.325875
$340$	18.0000	0.976187
$341$	15.0000	0.812296
$342$	2.00000	0.108148
$343$	−8.00000	−0.431959
$344$	−10.0000	−0.539164
$345$	3.00000	0.161515
$346$	0	0
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	−1.00000	−0.0536056
$349$	35.0000	1.87351	0.936754	−	0.349990i	$-0.113815\pi$
$349$	35.0000	1.87351	0.936754	+	0.349990i	$0.113815\pi$
$350$	−16.0000	−0.855236
$351$	5.00000	0.266880
$352$	3.00000	0.159901
$353$	−24.0000	−1.27739	−0.638696	−	0.769460i	$-0.720526\pi$
$353$	−24.0000	−1.27739	−0.638696	+	0.769460i	$0.720526\pi$
$354$	−3.00000	−0.159448
$355$	9.00000	0.477670
$356$	−12.0000	−0.635999
$357$	24.0000	1.27021
$358$	−12.0000	−0.634220
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	−3.00000	−0.158114
$361$	−15.0000	−0.789474
$362$	14.0000	0.735824
$363$	−2.00000	−0.104973
$364$	−20.0000	−1.04828
$365$	−6.00000	−0.314054
$366$	5.00000	0.261354
$367$	−22.0000	−1.14839	−0.574195	−	0.818718i	$-0.694685\pi$
$367$	−22.0000	−1.14839	−0.574195	+	0.818718i	$0.694685\pi$
$368$	−1.00000	−0.0521286
$369$	−9.00000	−0.468521
$370$	21.0000	1.09174
$371$	−24.0000	−1.24602
$372$	5.00000	0.259238
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	−18.0000	−0.930758
$375$	3.00000	0.154919
$376$	−12.0000	−0.618853
$377$	−5.00000	−0.257513
$378$	−4.00000	−0.205738
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	−6.00000	−0.307794
$381$	−7.00000	−0.358621
$382$	−3.00000	−0.153493
$383$	12.0000	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$384$	1.00000	0.0510310
$385$	36.0000	1.83473
$386$	−22.0000	−1.11977
$387$	−10.0000	−0.508329
$388$	−10.0000	−0.507673
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	−15.0000	−0.759555
$391$	6.00000	0.303433
$392$	9.00000	0.454569
$393$	6.00000	0.302660
$394$	−18.0000	−0.906827
$395$	−24.0000	−1.20757
$396$	3.00000	0.150756
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	−25.0000	−1.25314
$399$	−8.00000	−0.400501
$400$	4.00000	0.200000
$401$	21.0000	1.04869	0.524345	−	0.851506i	$-0.324310\pi$
$401$	21.0000	1.04869	0.524345	+	0.851506i	$0.324310\pi$
$402$	−13.0000	−0.648381
$403$	25.0000	1.24534
$404$	3.00000	0.149256
$405$	−3.00000	−0.149071
$406$	4.00000	0.198517
$407$	−21.0000	−1.04093
$408$	−6.00000	−0.297044
$409$	32.0000	1.58230	0.791149	−	0.611623i	$-0.209483\pi$
$409$	32.0000	1.58230	0.791149	+	0.611623i	$0.209483\pi$
$410$	27.0000	1.33343
$411$	−6.00000	−0.295958
$412$	−1.00000	−0.0492665
$413$	12.0000	0.590481
$414$	−1.00000	−0.0491473
$415$	18.0000	0.883585
$416$	5.00000	0.245145
$417$	20.0000	0.979404
$418$	6.00000	0.293470
$419$	−30.0000	−1.46560	−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000	−1.46560	−0.732798	+	0.680446i	$0.761786\pi$
$420$	12.0000	0.585540
$421$	23.0000	1.12095	0.560476	−	0.828171i	$-0.310618\pi$
$421$	23.0000	1.12095	0.560476	+	0.828171i	$0.310618\pi$
$422$	5.00000	0.243396
$423$	−12.0000	−0.583460
$424$	6.00000	0.291386
$425$	−24.0000	−1.16417
$426$	−3.00000	−0.145350
$427$	−20.0000	−0.967868
$428$	6.00000	0.290021
$429$	15.0000	0.724207
$430$	30.0000	1.44673
$431$	−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
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	−20.0000	−0.960031
$435$	3.00000	0.143839
$436$	−10.0000	−0.478913
$437$	−2.00000	−0.0956730
$438$	2.00000	0.0955637
$439$	−10.0000	−0.477274	−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000	−0.477274	−0.238637	+	0.971109i	$0.576701\pi$
$440$	−9.00000	−0.429058
$441$	9.00000	0.428571
$442$	−30.0000	−1.42695
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	−7.00000	−0.332205
$445$	36.0000	1.70656
$446$	26.0000	1.23114
$447$	15.0000	0.709476
$448$	−4.00000	−0.188982
$449$	3.00000	0.141579	0.0707894	−	0.997491i	$-0.477448\pi$
$449$	3.00000	0.141579	0.0707894	+	0.997491i	$0.477448\pi$
$450$	4.00000	0.188562
$451$	−27.0000	−1.27138
$452$	−6.00000	−0.282216
$453$	−16.0000	−0.751746
$454$	−18.0000	−0.844782
$455$	60.0000	2.81284
$456$	2.00000	0.0936586
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	17.0000	0.794358
$459$	−6.00000	−0.280056
$460$	3.00000	0.139876
$461$	15.0000	0.698620	0.349310	−	0.937007i	$-0.386416\pi$
$461$	15.0000	0.698620	0.349310	+	0.937007i	$0.386416\pi$
$462$	−12.0000	−0.558291
$463$	−22.0000	−1.02243	−0.511213	−	0.859454i	$-0.670804\pi$
$463$	−22.0000	−1.02243	−0.511213	+	0.859454i	$0.670804\pi$
$464$	−1.00000	−0.0464238
$465$	−15.0000	−0.695608
$466$	−24.0000	−1.11178
$467$	3.00000	0.138823	0.0694117	−	0.997588i	$-0.477888\pi$
$467$	3.00000	0.138823	0.0694117	+	0.997588i	$0.477888\pi$
$468$	5.00000	0.231125
$469$	52.0000	2.40114
$470$	36.0000	1.66056
$471$	14.0000	0.645086
$472$	−3.00000	−0.138086
$473$	−30.0000	−1.37940
$474$	8.00000	0.367452
$475$	8.00000	0.367065
$476$	24.0000	1.10004
$477$	6.00000	0.274721
$478$	9.00000	0.411650
$479$	−21.0000	−0.959514	−0.479757	−	0.877401i	$-0.659275\pi$
$479$	−21.0000	−0.959514	−0.479757	+	0.877401i	$0.659275\pi$
$480$	−3.00000	−0.136931
$481$	−35.0000	−1.59586
$482$	−22.0000	−1.00207
$483$	4.00000	0.182006
$484$	−2.00000	−0.0909091
$485$	30.0000	1.36223
$486$	1.00000	0.0453609
$487$	38.0000	1.72194	0.860972	−	0.508652i	$-0.169856\pi$
$487$	38.0000	1.72194	0.860972	+	0.508652i	$0.169856\pi$
$488$	5.00000	0.226339
$489$	−13.0000	−0.587880
$490$	−27.0000	−1.21974
$491$	−24.0000	−1.08310	−0.541552	−	0.840667i	$-0.682163\pi$
$491$	−24.0000	−1.08310	−0.541552	+	0.840667i	$0.682163\pi$
$492$	−9.00000	−0.405751
$493$	6.00000	0.270226
$494$	10.0000	0.449921
$495$	−9.00000	−0.404520
$496$	5.00000	0.224507
$497$	12.0000	0.538274
$498$	−6.00000	−0.268866
$499$	−22.0000	−0.984855	−0.492428	−	0.870353i	$-0.663890\pi$
$499$	−22.0000	−0.984855	−0.492428	+	0.870353i	$0.663890\pi$
$500$	3.00000	0.134164
$501$	−9.00000	−0.402090
$502$	15.0000	0.669483
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	−4.00000	−0.178174
$505$	−9.00000	−0.400495
$506$	−3.00000	−0.133366
$507$	12.0000	0.532939
$508$	−7.00000	−0.310575
$509$	−36.0000	−1.59567	−0.797836	−	0.602875i	$-0.794022\pi$
$509$	−36.0000	−1.59567	−0.797836	+	0.602875i	$0.794022\pi$
$510$	18.0000	0.797053
$511$	−8.00000	−0.353899
$512$	1.00000	0.0441942
$513$	2.00000	0.0883022
$514$	0	0
$515$	3.00000	0.132196
$516$	−10.0000	−0.440225
$517$	−36.0000	−1.58328
$518$	28.0000	1.23025
$519$	0	0
$520$	−15.0000	−0.657794
$521$	−42.0000	−1.84005	−0.920027	−	0.391856i	$-0.871833\pi$
$521$	−42.0000	−1.84005	−0.920027	+	0.391856i	$0.871833\pi$
$522$	−1.00000	−0.0437688
$523$	23.0000	1.00572	0.502860	−	0.864368i	$-0.332281\pi$
$523$	23.0000	1.00572	0.502860	+	0.864368i	$0.332281\pi$
$524$	6.00000	0.262111
$525$	−16.0000	−0.698297
$526$	12.0000	0.523225
$527$	−30.0000	−1.30682
$528$	3.00000	0.130558
$529$	1.00000	0.0434783
$530$	−18.0000	−0.781870
$531$	−3.00000	−0.130189
$532$	−8.00000	−0.346844
$533$	−45.0000	−1.94917
$534$	−12.0000	−0.519291
$535$	−18.0000	−0.778208
$536$	−13.0000	−0.561514
$537$	−12.0000	−0.517838
$538$	−3.00000	−0.129339
$539$	27.0000	1.16297
$540$	−3.00000	−0.129099
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	11.0000	0.472490
$543$	14.0000	0.600798
$544$	−6.00000	−0.257248
$545$	30.0000	1.28506
$546$	−20.0000	−0.855921
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	−6.00000	−0.256307
$549$	5.00000	0.213395
$550$	12.0000	0.511682
$551$	−2.00000	−0.0852029
$552$	−1.00000	−0.0425628
$553$	−32.0000	−1.36078
$554$	17.0000	0.722261
$555$	21.0000	0.891400
$556$	20.0000	0.848189
$557$	−9.00000	−0.381342	−0.190671	−	0.981654i	$-0.561066\pi$
$557$	−9.00000	−0.381342	−0.190671	+	0.981654i	$0.561066\pi$
$558$	5.00000	0.211667
$559$	−50.0000	−2.11477
$560$	12.0000	0.507093
$561$	−18.0000	−0.759961
$562$	−6.00000	−0.253095
$563$	21.0000	0.885044	0.442522	−	0.896758i	$-0.354084\pi$
$563$	21.0000	0.885044	0.442522	+	0.896758i	$0.354084\pi$
$564$	−12.0000	−0.505291
$565$	18.0000	0.757266
$566$	11.0000	0.462364
$567$	−4.00000	−0.167984
$568$	−3.00000	−0.125877
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	−6.00000	−0.251312
$571$	−7.00000	−0.292941	−0.146470	−	0.989215i	$-0.546791\pi$
$571$	−7.00000	−0.292941	−0.146470	+	0.989215i	$0.546791\pi$
$572$	15.0000	0.627182
$573$	−3.00000	−0.125327
$574$	36.0000	1.50261
$575$	−4.00000	−0.166812
$576$	1.00000	0.0416667
$577$	−10.0000	−0.416305	−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000	−0.416305	−0.208153	+	0.978096i	$0.566745\pi$
$578$	19.0000	0.790296
$579$	−22.0000	−0.914289
$580$	3.00000	0.124568
$581$	24.0000	0.995688
$582$	−10.0000	−0.414513
$583$	18.0000	0.745484
$584$	2.00000	0.0827606
$585$	−15.0000	−0.620174
$586$	12.0000	0.495715
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	9.00000	0.371154
$589$	10.0000	0.412043
$590$	9.00000	0.370524
$591$	−18.0000	−0.740421
$592$	−7.00000	−0.287698
$593$	24.0000	0.985562	0.492781	−	0.870153i	$-0.335980\pi$
$593$	24.0000	0.985562	0.492781	+	0.870153i	$0.335980\pi$
$594$	3.00000	0.123091
$595$	−72.0000	−2.95171
$596$	15.0000	0.614424
$597$	−25.0000	−1.02318
$598$	−5.00000	−0.204465
$599$	−6.00000	−0.245153	−0.122577	−	0.992459i	$-0.539116\pi$
$599$	−6.00000	−0.245153	−0.122577	+	0.992459i	$0.539116\pi$
$600$	4.00000	0.163299
$601$	20.0000	0.815817	0.407909	−	0.913023i	$-0.366258\pi$
$601$	20.0000	0.815817	0.407909	+	0.913023i	$0.366258\pi$
$602$	40.0000	1.63028
$603$	−13.0000	−0.529401
$604$	−16.0000	−0.651031
$605$	6.00000	0.243935
$606$	3.00000	0.121867
$607$	−40.0000	−1.62355	−0.811775	−	0.583970i	$-0.801498\pi$
$607$	−40.0000	−1.62355	−0.811775	+	0.583970i	$0.801498\pi$
$608$	2.00000	0.0811107
$609$	4.00000	0.162088
$610$	−15.0000	−0.607332
$611$	−60.0000	−2.42734
$612$	−6.00000	−0.242536
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	29.0000	1.17034
$615$	27.0000	1.08875
$616$	−12.0000	−0.483494
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	−1.00000	−0.0402259
$619$	−16.0000	−0.643094	−0.321547	−	0.946894i	$-0.604203\pi$
$619$	−16.0000	−0.643094	−0.321547	+	0.946894i	$0.604203\pi$
$620$	−15.0000	−0.602414
$621$	−1.00000	−0.0401286
$622$	−12.0000	−0.481156
$623$	48.0000	1.92308
$624$	5.00000	0.200160
$625$	−29.0000	−1.16000
$626$	20.0000	0.799361
$627$	6.00000	0.239617
$628$	14.0000	0.558661
$629$	42.0000	1.67465
$630$	12.0000	0.478091
$631$	−43.0000	−1.71180	−0.855901	−	0.517139i	$-0.826997\pi$
$631$	−43.0000	−1.71180	−0.855901	+	0.517139i	$0.826997\pi$
$632$	8.00000	0.318223
$633$	5.00000	0.198732
$634$	3.00000	0.119145
$635$	21.0000	0.833360
$636$	6.00000	0.237915
$637$	45.0000	1.78296
$638$	−3.00000	−0.118771
$639$	−3.00000	−0.118678
$640$	−3.00000	−0.118585
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	6.00000	0.236801
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	4.00000	0.157622
$645$	30.0000	1.18125
$646$	−12.0000	−0.472134
$647$	−3.00000	−0.117942	−0.0589711	−	0.998260i	$-0.518782\pi$
$647$	−3.00000	−0.117942	−0.0589711	+	0.998260i	$0.518782\pi$
$648$	1.00000	0.0392837
$649$	−9.00000	−0.353281
$650$	20.0000	0.784465
$651$	−20.0000	−0.783862
$652$	−13.0000	−0.509119
$653$	−27.0000	−1.05659	−0.528296	−	0.849060i	$-0.677169\pi$
$653$	−27.0000	−1.05659	−0.528296	+	0.849060i	$0.677169\pi$
$654$	−10.0000	−0.391031
$655$	−18.0000	−0.703318
$656$	−9.00000	−0.351391
$657$	2.00000	0.0780274
$658$	48.0000	1.87123
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	−9.00000	−0.350325
$661$	8.00000	0.311164	0.155582	−	0.987823i	$-0.450275\pi$
$661$	8.00000	0.311164	0.155582	+	0.987823i	$0.450275\pi$
$662$	8.00000	0.310929
$663$	−30.0000	−1.16510
$664$	−6.00000	−0.232845
$665$	24.0000	0.930680
$666$	−7.00000	−0.271244
$667$	1.00000	0.0387202
$668$	−9.00000	−0.348220
$669$	26.0000	1.00522
$670$	39.0000	1.50670
$671$	15.0000	0.579069
$672$	−4.00000	−0.154303
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	−13.0000	−0.500741
$675$	4.00000	0.153960
$676$	12.0000	0.461538
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	−6.00000	−0.230429
$679$	40.0000	1.53506
$680$	18.0000	0.690268
$681$	−18.0000	−0.689761
$682$	15.0000	0.574380
$683$	9.00000	0.344375	0.172188	−	0.985064i	$-0.444916\pi$
$683$	9.00000	0.344375	0.172188	+	0.985064i	$0.444916\pi$
$684$	2.00000	0.0764719
$685$	18.0000	0.687745
$686$	−8.00000	−0.305441
$687$	17.0000	0.648590
$688$	−10.0000	−0.381246
$689$	30.0000	1.14291
$690$	3.00000	0.114208
$691$	50.0000	1.90209	0.951045	−	0.309053i	$-0.100012\pi$
$691$	50.0000	1.90209	0.951045	+	0.309053i	$0.100012\pi$
$692$	0	0
$693$	−12.0000	−0.455842
$694$	12.0000	0.455514
$695$	−60.0000	−2.27593
$696$	−1.00000	−0.0379049
$697$	54.0000	2.04540
$698$	35.0000	1.32477
$699$	−24.0000	−0.907763
$700$	−16.0000	−0.604743
$701$	−33.0000	−1.24639	−0.623196	−	0.782065i	$-0.714166\pi$
$701$	−33.0000	−1.24639	−0.623196	+	0.782065i	$0.714166\pi$
$702$	5.00000	0.188713
$703$	−14.0000	−0.528020
$704$	3.00000	0.113067
$705$	36.0000	1.35584
$706$	−24.0000	−0.903252
$707$	−12.0000	−0.451306
$708$	−3.00000	−0.112747
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	9.00000	0.337764
$711$	8.00000	0.300023
$712$	−12.0000	−0.449719
$713$	−5.00000	−0.187251
$714$	24.0000	0.898177
$715$	−45.0000	−1.68290
$716$	−12.0000	−0.448461
$717$	9.00000	0.336111
$718$	24.0000	0.895672
$719$	15.0000	0.559406	0.279703	−	0.960087i	$-0.409764\pi$
$719$	15.0000	0.559406	0.279703	+	0.960087i	$0.409764\pi$
$720$	−3.00000	−0.111803
$721$	4.00000	0.148968
$722$	−15.0000	−0.558242
$723$	−22.0000	−0.818189
$724$	14.0000	0.520306
$725$	−4.00000	−0.148556
$726$	−2.00000	−0.0742270
$727$	−46.0000	−1.70605	−0.853023	−	0.521874i	$-0.825233\pi$
$727$	−46.0000	−1.70605	−0.853023	+	0.521874i	$0.825233\pi$
$728$	−20.0000	−0.741249
$729$	1.00000	0.0370370
$730$	−6.00000	−0.222070
$731$	60.0000	2.21918
$732$	5.00000	0.184805
$733$	−7.00000	−0.258551	−0.129275	−	0.991609i	$-0.541265\pi$
$733$	−7.00000	−0.258551	−0.129275	+	0.991609i	$0.541265\pi$
$734$	−22.0000	−0.812035
$735$	−27.0000	−0.995910
$736$	−1.00000	−0.0368605
$737$	−39.0000	−1.43658
$738$	−9.00000	−0.331295
$739$	5.00000	0.183928	0.0919640	−	0.995762i	$-0.470686\pi$
$739$	5.00000	0.183928	0.0919640	+	0.995762i	$0.470686\pi$
$740$	21.0000	0.771975
$741$	10.0000	0.367359
$742$	−24.0000	−0.881068
$743$	−15.0000	−0.550297	−0.275148	−	0.961402i	$-0.588727\pi$
$743$	−15.0000	−0.550297	−0.275148	+	0.961402i	$0.588727\pi$
$744$	5.00000	0.183309
$745$	−45.0000	−1.64867
$746$	−4.00000	−0.146450
$747$	−6.00000	−0.219529
$748$	−18.0000	−0.658145
$749$	−24.0000	−0.876941
$750$	3.00000	0.109545
$751$	38.0000	1.38664	0.693320	−	0.720630i	$-0.256147\pi$
$751$	38.0000	1.38664	0.693320	+	0.720630i	$0.256147\pi$
$752$	−12.0000	−0.437595
$753$	15.0000	0.546630
$754$	−5.00000	−0.182089
$755$	48.0000	1.74690
$756$	−4.00000	−0.145479
$757$	−19.0000	−0.690567	−0.345283	−	0.938498i	$-0.612217\pi$
$757$	−19.0000	−0.690567	−0.345283	+	0.938498i	$0.612217\pi$
$758$	20.0000	0.726433
$759$	−3.00000	−0.108893
$760$	−6.00000	−0.217643
$761$	−42.0000	−1.52250	−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000	−1.52250	−0.761249	+	0.648459i	$0.775414\pi$
$762$	−7.00000	−0.253583
$763$	40.0000	1.44810
$764$	−3.00000	−0.108536
$765$	18.0000	0.650791
$766$	12.0000	0.433578
$767$	−15.0000	−0.541619
$768$	1.00000	0.0360844
$769$	23.0000	0.829401	0.414701	−	0.909958i	$-0.363886\pi$
$769$	23.0000	0.829401	0.414701	+	0.909958i	$0.363886\pi$
$770$	36.0000	1.29735
$771$	0	0
$772$	−22.0000	−0.791797
$773$	36.0000	1.29483	0.647415	−	0.762138i	$-0.275850\pi$
$773$	36.0000	1.29483	0.647415	+	0.762138i	$0.275850\pi$
$774$	−10.0000	−0.359443
$775$	20.0000	0.718421
$776$	−10.0000	−0.358979
$777$	28.0000	1.00449
$778$	6.00000	0.215110
$779$	−18.0000	−0.644917
$780$	−15.0000	−0.537086
$781$	−9.00000	−0.322045
$782$	6.00000	0.214560
$783$	−1.00000	−0.0357371
$784$	9.00000	0.321429
$785$	−42.0000	−1.49904
$786$	6.00000	0.214013
$787$	53.0000	1.88925	0.944623	−	0.328158i	$-0.106428\pi$
$787$	53.0000	1.88925	0.944623	+	0.328158i	$0.106428\pi$
$788$	−18.0000	−0.641223
$789$	12.0000	0.427211
$790$	−24.0000	−0.853882
$791$	24.0000	0.853342
$792$	3.00000	0.106600
$793$	25.0000	0.887776
$794$	14.0000	0.496841
$795$	−18.0000	−0.638394
$796$	−25.0000	−0.886102
$797$	−24.0000	−0.850124	−0.425062	−	0.905164i	$-0.639748\pi$
$797$	−24.0000	−0.850124	−0.425062	+	0.905164i	$0.639748\pi$
$798$	−8.00000	−0.283197
$799$	72.0000	2.54718
$800$	4.00000	0.141421
$801$	−12.0000	−0.423999
$802$	21.0000	0.741536
$803$	6.00000	0.211735
$804$	−13.0000	−0.458475
$805$	−12.0000	−0.422944
$806$	25.0000	0.880587
$807$	−3.00000	−0.105605
$808$	3.00000	0.105540
$809$	−39.0000	−1.37117	−0.685583	−	0.727994i	$-0.740453\pi$
$809$	−39.0000	−1.37117	−0.685583	+	0.727994i	$0.740453\pi$
$810$	−3.00000	−0.105409
$811$	−22.0000	−0.772524	−0.386262	−	0.922389i	$-0.626234\pi$
$811$	−22.0000	−0.772524	−0.386262	+	0.922389i	$0.626234\pi$
$812$	4.00000	0.140372
$813$	11.0000	0.385787
$814$	−21.0000	−0.736050
$815$	39.0000	1.36611
$816$	−6.00000	−0.210042
$817$	−20.0000	−0.699711
$818$	32.0000	1.11885
$819$	−20.0000	−0.698857
$820$	27.0000	0.942881
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	−6.00000	−0.209274
$823$	−31.0000	−1.08059	−0.540296	−	0.841475i	$-0.681688\pi$
$823$	−31.0000	−1.08059	−0.540296	+	0.841475i	$0.681688\pi$
$824$	−1.00000	−0.0348367
$825$	12.0000	0.417786
$826$	12.0000	0.417533
$827$	21.0000	0.730242	0.365121	−	0.930960i	$-0.381028\pi$
$827$	21.0000	0.730242	0.365121	+	0.930960i	$0.381028\pi$
$828$	−1.00000	−0.0347524
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	18.0000	0.624789
$831$	17.0000	0.589723
$832$	5.00000	0.173344
$833$	−54.0000	−1.87099
$834$	20.0000	0.692543
$835$	27.0000	0.934374
$836$	6.00000	0.207514
$837$	5.00000	0.172825
$838$	−30.0000	−1.03633
$839$	3.00000	0.103572	0.0517858	−	0.998658i	$-0.483509\pi$
$839$	3.00000	0.103572	0.0517858	+	0.998658i	$0.483509\pi$
$840$	12.0000	0.414039
$841$	1.00000	0.0344828
$842$	23.0000	0.792632
$843$	−6.00000	−0.206651
$844$	5.00000	0.172107
$845$	−36.0000	−1.23844
$846$	−12.0000	−0.412568
$847$	8.00000	0.274883
$848$	6.00000	0.206041
$849$	11.0000	0.377519
$850$	−24.0000	−0.823193
$851$	7.00000	0.239957
$852$	−3.00000	−0.102778
$853$	−22.0000	−0.753266	−0.376633	−	0.926363i	$-0.622918\pi$
$853$	−22.0000	−0.753266	−0.376633	+	0.926363i	$0.622918\pi$
$854$	−20.0000	−0.684386
$855$	−6.00000	−0.205196
$856$	6.00000	0.205076
$857$	−54.0000	−1.84460	−0.922302	−	0.386469i	$-0.873695\pi$
$857$	−54.0000	−1.84460	−0.922302	+	0.386469i	$0.873695\pi$
$858$	15.0000	0.512092
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	30.0000	1.02299
$861$	36.0000	1.22688
$862$	−12.0000	−0.408722
$863$	−33.0000	−1.12333	−0.561667	−	0.827364i	$-0.689840\pi$
$863$	−33.0000	−1.12333	−0.561667	+	0.827364i	$0.689840\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	2.00000	0.0679628
$867$	19.0000	0.645274
$868$	−20.0000	−0.678844
$869$	24.0000	0.814144
$870$	3.00000	0.101710
$871$	−65.0000	−2.20244
$872$	−10.0000	−0.338643
$873$	−10.0000	−0.338449
$874$	−2.00000	−0.0676510
$875$	−12.0000	−0.405674
$876$	2.00000	0.0675737
$877$	−46.0000	−1.55331	−0.776655	−	0.629926i	$-0.783085\pi$
$877$	−46.0000	−1.55331	−0.776655	+	0.629926i	$0.783085\pi$
$878$	−10.0000	−0.337484
$879$	12.0000	0.404750
$880$	−9.00000	−0.303390
$881$	36.0000	1.21287	0.606435	−	0.795133i	$-0.292599\pi$
$881$	36.0000	1.21287	0.606435	+	0.795133i	$0.292599\pi$
$882$	9.00000	0.303046
$883$	2.00000	0.0673054	0.0336527	−	0.999434i	$-0.489286\pi$
$883$	2.00000	0.0673054	0.0336527	+	0.999434i	$0.489286\pi$
$884$	−30.0000	−1.00901
$885$	9.00000	0.302532
$886$	24.0000	0.806296
$887$	−36.0000	−1.20876	−0.604381	−	0.796696i	$-0.706579\pi$
$887$	−36.0000	−1.20876	−0.604381	+	0.796696i	$0.706579\pi$
$888$	−7.00000	−0.234905
$889$	28.0000	0.939090
$890$	36.0000	1.20672
$891$	3.00000	0.100504
$892$	26.0000	0.870544
$893$	−24.0000	−0.803129
$894$	15.0000	0.501675
$895$	36.0000	1.20335
$896$	−4.00000	−0.133631
$897$	−5.00000	−0.166945
$898$	3.00000	0.100111
$899$	−5.00000	−0.166759
$900$	4.00000	0.133333
$901$	−36.0000	−1.19933
$902$	−27.0000	−0.899002
$903$	40.0000	1.33112
$904$	−6.00000	−0.199557
$905$	−42.0000	−1.39613
$906$	−16.0000	−0.531564
$907$	−10.0000	−0.332045	−0.166022	−	0.986122i	$-0.553092\pi$
$907$	−10.0000	−0.332045	−0.166022	+	0.986122i	$0.553092\pi$
$908$	−18.0000	−0.597351
$909$	3.00000	0.0995037
$910$	60.0000	1.98898
$911$	−15.0000	−0.496972	−0.248486	−	0.968635i	$-0.579933\pi$
$911$	−15.0000	−0.496972	−0.248486	+	0.968635i	$0.579933\pi$
$912$	2.00000	0.0662266
$913$	−18.0000	−0.595713
$914$	8.00000	0.264616
$915$	−15.0000	−0.495885
$916$	17.0000	0.561696
$917$	−24.0000	−0.792550
$918$	−6.00000	−0.198030
$919$	−25.0000	−0.824674	−0.412337	−	0.911031i	$-0.635287\pi$
$919$	−25.0000	−0.824674	−0.412337	+	0.911031i	$0.635287\pi$
$920$	3.00000	0.0989071
$921$	29.0000	0.955582
$922$	15.0000	0.493999
$923$	−15.0000	−0.493731
$924$	−12.0000	−0.394771
$925$	−28.0000	−0.920634
$926$	−22.0000	−0.722965
$927$	−1.00000	−0.0328443
$928$	−1.00000	−0.0328266
$929$	24.0000	0.787414	0.393707	−	0.919236i	$-0.371192\pi$
$929$	24.0000	0.787414	0.393707	+	0.919236i	$0.371192\pi$
$930$	−15.0000	−0.491869
$931$	18.0000	0.589926
$932$	−24.0000	−0.786146
$933$	−12.0000	−0.392862
$934$	3.00000	0.0981630
$935$	54.0000	1.76599
$936$	5.00000	0.163430
$937$	32.0000	1.04539	0.522697	−	0.852518i	$-0.324926\pi$
$937$	32.0000	1.04539	0.522697	+	0.852518i	$0.324926\pi$
$938$	52.0000	1.69786
$939$	20.0000	0.652675
$940$	36.0000	1.17419
$941$	−27.0000	−0.880175	−0.440087	−	0.897955i	$-0.645053\pi$
$941$	−27.0000	−0.880175	−0.440087	+	0.897955i	$0.645053\pi$
$942$	14.0000	0.456145
$943$	9.00000	0.293080
$944$	−3.00000	−0.0976417
$945$	12.0000	0.390360
$946$	−30.0000	−0.975384
$947$	42.0000	1.36482	0.682408	−	0.730971i	$-0.260933\pi$
$947$	42.0000	1.36482	0.682408	+	0.730971i	$0.260933\pi$
$948$	8.00000	0.259828
$949$	10.0000	0.324614
$950$	8.00000	0.259554
$951$	3.00000	0.0972817
$952$	24.0000	0.777844
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	6.00000	0.194257
$955$	9.00000	0.291233
$956$	9.00000	0.291081
$957$	−3.00000	−0.0969762
$958$	−21.0000	−0.678479
$959$	24.0000	0.775000
$960$	−3.00000	−0.0968246
$961$	−6.00000	−0.193548
$962$	−35.0000	−1.12845
$963$	6.00000	0.193347
$964$	−22.0000	−0.708572
$965$	66.0000	2.12462
$966$	4.00000	0.128698
$967$	−4.00000	−0.128631	−0.0643157	−	0.997930i	$-0.520486\pi$
$967$	−4.00000	−0.128631	−0.0643157	+	0.997930i	$0.520486\pi$
$968$	−2.00000	−0.0642824
$969$	−12.0000	−0.385496
$970$	30.0000	0.963242
$971$	21.0000	0.673922	0.336961	−	0.941519i	$-0.390601\pi$
$971$	21.0000	0.673922	0.336961	+	0.941519i	$0.390601\pi$
$972$	1.00000	0.0320750
$973$	−80.0000	−2.56468
$974$	38.0000	1.21760
$975$	20.0000	0.640513
$976$	5.00000	0.160046
$977$	−3.00000	−0.0959785	−0.0479893	−	0.998848i	$-0.515281\pi$
$977$	−3.00000	−0.0959785	−0.0479893	+	0.998848i	$0.515281\pi$
$978$	−13.0000	−0.415694
$979$	−36.0000	−1.15056
$980$	−27.0000	−0.862483
$981$	−10.0000	−0.319275
$982$	−24.0000	−0.765871
$983$	33.0000	1.05254	0.526268	−	0.850319i	$-0.323591\pi$
$983$	33.0000	1.05254	0.526268	+	0.850319i	$0.323591\pi$
$984$	−9.00000	−0.286910
$985$	54.0000	1.72058
$986$	6.00000	0.191079
$987$	48.0000	1.52786
$988$	10.0000	0.318142
$989$	10.0000	0.317982
$990$	−9.00000	−0.286039
$991$	44.0000	1.39771	0.698853	−	0.715265i	$-0.253694\pi$
$991$	44.0000	1.39771	0.698853	+	0.715265i	$0.253694\pi$
$992$	5.00000	0.158750
$993$	8.00000	0.253872
$994$	12.0000	0.380617
$995$	75.0000	2.37766
$996$	−6.00000	−0.190117
$997$	8.00000	0.253363	0.126681	−	0.991943i	$-0.459567\pi$
$997$	8.00000	0.253363	0.126681	+	0.991943i	$0.459567\pi$
$998$	−22.0000	−0.696398
$999$	−7.00000	−0.221470

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.m.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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