LMFDB - Embedded newform 5070.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5070,2,Mod(1,5070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5070, 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("5070.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5070.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:	$40.4841538248$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 390)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5070.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -3.00000 q^{11} +1.00000 q^{12} -2.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{18} +2.00000 q^{19} -1.00000 q^{20} +2.00000 q^{21} +3.00000 q^{22} +3.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} +2.00000 q^{28} +3.00000 q^{29} +1.00000 q^{30} +5.00000 q^{31} -1.00000 q^{32} -3.00000 q^{33} -6.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} -7.00000 q^{37} -2.00000 q^{38} +1.00000 q^{40} +6.00000 q^{41} -2.00000 q^{42} -1.00000 q^{43} -3.00000 q^{44} -1.00000 q^{45} -3.00000 q^{46} -3.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} +6.00000 q^{51} -6.00000 q^{53} -1.00000 q^{54} +3.00000 q^{55} -2.00000 q^{56} +2.00000 q^{57} -3.00000 q^{58} -9.00000 q^{59} -1.00000 q^{60} +2.00000 q^{61} -5.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +3.00000 q^{66} +8.00000 q^{67} +6.00000 q^{68} +3.00000 q^{69} +2.00000 q^{70} -12.0000 q^{71} -1.00000 q^{72} +14.0000 q^{73} +7.00000 q^{74} +1.00000 q^{75} +2.00000 q^{76} -6.00000 q^{77} +5.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -6.00000 q^{83} +2.00000 q^{84} -6.00000 q^{85} +1.00000 q^{86} +3.00000 q^{87} +3.00000 q^{88} -18.0000 q^{89} +1.00000 q^{90} +3.00000 q^{92} +5.00000 q^{93} +3.00000 q^{94} -2.00000 q^{95} -1.00000 q^{96} +14.0000 q^{97} +3.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.00000	−0.408248
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	−2.00000	−0.534522
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	−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	$0.426318\pi$
$20$	−1.00000	−0.223607
$21$	2.00000	0.436436
$22$	3.00000	0.639602
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	2.00000	0.377964
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	1.00000	0.182574
$31$	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
$35$	−2.00000	−0.338062
$36$	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
$39$	0	0
$40$	1.00000	0.158114
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	−2.00000	−0.308607
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	−3.00000	−0.452267
$45$	−1.00000	−0.149071
$46$	−3.00000	−0.442326
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	1.00000	0.144338
$49$	−3.00000	−0.428571
$50$	−1.00000	−0.141421
$51$	6.00000	0.840168
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	−1.00000	−0.136083
$55$	3.00000	0.404520
$56$	−2.00000	−0.267261
$57$	2.00000	0.264906
$58$	−3.00000	−0.393919
$59$	−9.00000	−1.17170	−0.585850	−	0.810419i	$-0.699239\pi$
$59$	−9.00000	−1.17170	−0.585850	+	0.810419i	$0.699239\pi$
$60$	−1.00000	−0.129099
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−5.00000	−0.635001
$63$	2.00000	0.251976
$64$	1.00000	0.125000
$65$	0	0
$66$	3.00000	0.369274
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	6.00000	0.727607
$69$	3.00000	0.361158
$70$	2.00000	0.239046
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	−1.00000	−0.117851
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	7.00000	0.813733
$75$	1.00000	0.115470
$76$	2.00000	0.229416
$77$	−6.00000	−0.683763
$78$	0	0
$79$	5.00000	0.562544	0.281272	−	0.959628i	$-0.409244\pi$
$79$	5.00000	0.562544	0.281272	+	0.959628i	$0.409244\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	2.00000	0.218218
$85$	−6.00000	−0.650791
$86$	1.00000	0.107833
$87$	3.00000	0.321634
$88$	3.00000	0.319801
$89$	−18.0000	−1.90800	−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−18.0000	−1.90800	−0.953998	+	0.299813i	$0.903076\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	3.00000	0.312772
$93$	5.00000	0.518476
$94$	3.00000	0.309426
$95$	−2.00000	−0.205196
$96$	−1.00000	−0.102062
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	3.00000	0.303046
$99$	−3.00000	−0.301511
$100$	1.00000	0.100000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	−6.00000	−0.594089
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	6.00000	0.582772
$107$	−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
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	−3.00000	−0.286039
$111$	−7.00000	−0.664411
$112$	2.00000	0.188982
$113$	15.0000	1.41108	0.705541	−	0.708669i	$-0.250704\pi$
$113$	15.0000	1.41108	0.705541	+	0.708669i	$0.250704\pi$
$114$	−2.00000	−0.187317
$115$	−3.00000	−0.279751
$116$	3.00000	0.278543
$117$	0	0
$118$	9.00000	0.828517
$119$	12.0000	1.10004
$120$	1.00000	0.0912871
$121$	−2.00000	−0.181818
$122$	−2.00000	−0.181071
$123$	6.00000	0.541002
$124$	5.00000	0.449013
$125$	−1.00000	−0.0894427
$126$	−2.00000	−0.178174
$127$	14.0000	1.24230	0.621150	−	0.783692i	$-0.286666\pi$
$127$	14.0000	1.24230	0.621150	+	0.783692i	$0.286666\pi$
$128$	−1.00000	−0.0883883
$129$	−1.00000	−0.0880451
$130$	0	0
$131$	9.00000	0.786334	0.393167	−	0.919467i	$-0.371379\pi$
$131$	9.00000	0.786334	0.393167	+	0.919467i	$0.371379\pi$
$132$	−3.00000	−0.261116
$133$	4.00000	0.346844
$134$	−8.00000	−0.691095
$135$	−1.00000	−0.0860663
$136$	−6.00000	−0.514496
$137$	9.00000	0.768922	0.384461	−	0.923141i	$-0.374387\pi$
$137$	9.00000	0.768922	0.384461	+	0.923141i	$0.374387\pi$
$138$	−3.00000	−0.255377
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	−2.00000	−0.169031
$141$	−3.00000	−0.252646
$142$	12.0000	1.00702
$143$	0	0
$144$	1.00000	0.0833333
$145$	−3.00000	−0.249136
$146$	−14.0000	−1.15865
$147$	−3.00000	−0.247436
$148$	−7.00000	−0.575396
$149$	−9.00000	−0.737309	−0.368654	−	0.929567i	$-0.620181\pi$
$149$	−9.00000	−0.737309	−0.368654	+	0.929567i	$0.620181\pi$
$150$	−1.00000	−0.0816497
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	−2.00000	−0.162221
$153$	6.00000	0.485071
$154$	6.00000	0.483494
$155$	−5.00000	−0.401610
$156$	0	0
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	−5.00000	−0.397779
$159$	−6.00000	−0.475831
$160$	1.00000	0.0790569
$161$	6.00000	0.472866
$162$	−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	$0.670029\pi$
$164$	6.00000	0.468521
$165$	3.00000	0.233550
$166$	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	$0.613214\pi$
$168$	−2.00000	−0.154303
$169$	0	0
$170$	6.00000	0.460179
$171$	2.00000	0.152944
$172$	−1.00000	−0.0762493
$173$	12.0000	0.912343	0.456172	−	0.889892i	$-0.349220\pi$
$173$	12.0000	0.912343	0.456172	+	0.889892i	$0.349220\pi$
$174$	−3.00000	−0.227429
$175$	2.00000	0.151186
$176$	−3.00000	−0.226134
$177$	−9.00000	−0.676481
$178$	18.0000	1.34916
$179$	−3.00000	−0.224231	−0.112115	−	0.993695i	$-0.535763\pi$
$179$	−3.00000	−0.224231	−0.112115	+	0.993695i	$0.535763\pi$
$180$	−1.00000	−0.0745356
$181$	−16.0000	−1.18927	−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000	−1.18927	−0.594635	+	0.803996i	$0.702704\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	−3.00000	−0.221163
$185$	7.00000	0.514650
$186$	−5.00000	−0.366618
$187$	−18.0000	−1.31629
$188$	−3.00000	−0.218797
$189$	2.00000	0.145479
$190$	2.00000	0.145095
$191$	−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
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	−3.00000	−0.214286
$197$	24.0000	1.70993	0.854965	−	0.518686i	$-0.173579\pi$
$197$	24.0000	1.70993	0.854965	+	0.518686i	$0.173579\pi$
$198$	3.00000	0.213201
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	−1.00000	−0.0707107
$201$	8.00000	0.564276
$202$	−6.00000	−0.422159
$203$	6.00000	0.421117
$204$	6.00000	0.420084
$205$	−6.00000	−0.419058
$206$	−14.0000	−0.975426
$207$	3.00000	0.208514
$208$	0	0
$209$	−6.00000	−0.415029
$210$	2.00000	0.138013
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	−6.00000	−0.412082
$213$	−12.0000	−0.822226
$214$	6.00000	0.410152
$215$	1.00000	0.0681994
$216$	−1.00000	−0.0680414
$217$	10.0000	0.678844
$218$	−14.0000	−0.948200
$219$	14.0000	0.946032
$220$	3.00000	0.202260
$221$	0	0
$222$	7.00000	0.469809
$223$	−10.0000	−0.669650	−0.334825	−	0.942280i	$-0.608677\pi$
$223$	−10.0000	−0.669650	−0.334825	+	0.942280i	$0.608677\pi$
$224$	−2.00000	−0.133631
$225$	1.00000	0.0666667
$226$	−15.0000	−0.997785
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	2.00000	0.132453
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	3.00000	0.197814
$231$	−6.00000	−0.394771
$232$	−3.00000	−0.196960
$233$	21.0000	1.37576	0.687878	−	0.725826i	$-0.258542\pi$
$233$	21.0000	1.37576	0.687878	+	0.725826i	$0.258542\pi$
$234$	0	0
$235$	3.00000	0.195698
$236$	−9.00000	−0.585850
$237$	5.00000	0.324785
$238$	−12.0000	−0.777844
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	−1.00000	−0.0645497
$241$	17.0000	1.09507	0.547533	−	0.836784i	$-0.315567\pi$
$241$	17.0000	1.09507	0.547533	+	0.836784i	$0.315567\pi$
$242$	2.00000	0.128565
$243$	1.00000	0.0641500
$244$	2.00000	0.128037
$245$	3.00000	0.191663
$246$	−6.00000	−0.382546
$247$	0	0
$248$	−5.00000	−0.317500
$249$	−6.00000	−0.380235
$250$	1.00000	0.0632456
$251$	15.0000	0.946792	0.473396	−	0.880850i	$-0.343028\pi$
$251$	15.0000	0.946792	0.473396	+	0.880850i	$0.343028\pi$
$252$	2.00000	0.125988
$253$	−9.00000	−0.565825
$254$	−14.0000	−0.878438
$255$	−6.00000	−0.375735
$256$	1.00000	0.0625000
$257$	−21.0000	−1.30994	−0.654972	−	0.755653i	$-0.727320\pi$
$257$	−21.0000	−1.30994	−0.654972	+	0.755653i	$0.727320\pi$
$258$	1.00000	0.0622573
$259$	−14.0000	−0.869918
$260$	0	0
$261$	3.00000	0.185695
$262$	−9.00000	−0.556022
$263$	−15.0000	−0.924940	−0.462470	−	0.886635i	$-0.653037\pi$
$263$	−15.0000	−0.924940	−0.462470	+	0.886635i	$0.653037\pi$
$264$	3.00000	0.184637
$265$	6.00000	0.368577
$266$	−4.00000	−0.245256
$267$	−18.0000	−1.10158
$268$	8.00000	0.488678
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	1.00000	0.0608581
$271$	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
$273$	0	0
$274$	−9.00000	−0.543710
$275$	−3.00000	−0.180907
$276$	3.00000	0.180579
$277$	−1.00000	−0.0600842	−0.0300421	−	0.999549i	$-0.509564\pi$
$277$	−1.00000	−0.0600842	−0.0300421	+	0.999549i	$0.509564\pi$
$278$	−14.0000	−0.839664
$279$	5.00000	0.299342
$280$	2.00000	0.119523
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	3.00000	0.178647
$283$	−31.0000	−1.84276	−0.921379	−	0.388664i	$-0.872937\pi$
$283$	−31.0000	−1.84276	−0.921379	+	0.388664i	$0.872937\pi$
$284$	−12.0000	−0.712069
$285$	−2.00000	−0.118470
$286$	0	0
$287$	12.0000	0.708338
$288$	−1.00000	−0.0589256
$289$	19.0000	1.11765
$290$	3.00000	0.176166
$291$	14.0000	0.820695
$292$	14.0000	0.819288
$293$	−30.0000	−1.75262	−0.876309	−	0.481749i	$-0.840002\pi$
$293$	−30.0000	−1.75262	−0.876309	+	0.481749i	$0.840002\pi$
$294$	3.00000	0.174964
$295$	9.00000	0.524000
$296$	7.00000	0.406867
$297$	−3.00000	−0.174078
$298$	9.00000	0.521356
$299$	0	0
$300$	1.00000	0.0577350
$301$	−2.00000	−0.115278
$302$	−8.00000	−0.460348
$303$	6.00000	0.344691
$304$	2.00000	0.114708
$305$	−2.00000	−0.114520
$306$	−6.00000	−0.342997
$307$	8.00000	0.456584	0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000	0.456584	0.228292	+	0.973593i	$0.426686\pi$
$308$	−6.00000	−0.341882
$309$	14.0000	0.796432
$310$	5.00000	0.283981
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	8.00000	0.452187	0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000	0.452187	0.226093	+	0.974106i	$0.427405\pi$
$314$	13.0000	0.733632
$315$	−2.00000	−0.112687
$316$	5.00000	0.281272
$317$	12.0000	0.673987	0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000	0.673987	0.336994	+	0.941507i	$0.390590\pi$
$318$	6.00000	0.336463
$319$	−9.00000	−0.503903
$320$	−1.00000	−0.0559017
$321$	−6.00000	−0.334887
$322$	−6.00000	−0.334367
$323$	12.0000	0.667698
$324$	1.00000	0.0555556
$325$	0	0
$326$	13.0000	0.720003
$327$	14.0000	0.774202
$328$	−6.00000	−0.331295
$329$	−6.00000	−0.330791
$330$	−3.00000	−0.165145
$331$	32.0000	1.75888	0.879440	−	0.476011i	$-0.157918\pi$
$331$	32.0000	1.75888	0.879440	+	0.476011i	$0.157918\pi$
$332$	−6.00000	−0.329293
$333$	−7.00000	−0.383598
$334$	9.00000	0.492458
$335$	−8.00000	−0.437087
$336$	2.00000	0.109109
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	0	0
$339$	15.0000	0.814688
$340$	−6.00000	−0.325396
$341$	−15.0000	−0.812296
$342$	−2.00000	−0.108148
$343$	−20.0000	−1.07990
$344$	1.00000	0.0539164
$345$	−3.00000	−0.161515
$346$	−12.0000	−0.645124
$347$	−30.0000	−1.61048	−0.805242	−	0.592946i	$-0.797965\pi$
$347$	−30.0000	−1.61048	−0.805242	+	0.592946i	$0.797965\pi$
$348$	3.00000	0.160817
$349$	8.00000	0.428230	0.214115	−	0.976808i	$-0.431313\pi$
$349$	8.00000	0.428230	0.214115	+	0.976808i	$0.431313\pi$
$350$	−2.00000	−0.106904
$351$	0	0
$352$	3.00000	0.159901
$353$	30.0000	1.59674	0.798369	−	0.602168i	$-0.205696\pi$
$353$	30.0000	1.59674	0.798369	+	0.602168i	$0.205696\pi$
$354$	9.00000	0.478345
$355$	12.0000	0.636894
$356$	−18.0000	−0.953998
$357$	12.0000	0.635107
$358$	3.00000	0.158555
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	1.00000	0.0527046
$361$	−15.0000	−0.789474
$362$	16.0000	0.840941
$363$	−2.00000	−0.104973
$364$	0	0
$365$	−14.0000	−0.732793
$366$	−2.00000	−0.104542
$367$	20.0000	1.04399	0.521996	−	0.852948i	$-0.325188\pi$
$367$	20.0000	1.04399	0.521996	+	0.852948i	$0.325188\pi$
$368$	3.00000	0.156386
$369$	6.00000	0.312348
$370$	−7.00000	−0.363913
$371$	−12.0000	−0.623009
$372$	5.00000	0.259238
$373$	−25.0000	−1.29445	−0.647225	−	0.762299i	$-0.724071\pi$
$373$	−25.0000	−1.29445	−0.647225	+	0.762299i	$0.724071\pi$
$374$	18.0000	0.930758
$375$	−1.00000	−0.0516398
$376$	3.00000	0.154713
$377$	0	0
$378$	−2.00000	−0.102869
$379$	38.0000	1.95193	0.975964	−	0.217930i	$-0.0699304\pi$
$379$	38.0000	1.95193	0.975964	+	0.217930i	$0.0699304\pi$
$380$	−2.00000	−0.102598
$381$	14.0000	0.717242
$382$	12.0000	0.613973
$383$	−21.0000	−1.07305	−0.536525	−	0.843884i	$-0.680263\pi$
$383$	−21.0000	−1.07305	−0.536525	+	0.843884i	$0.680263\pi$
$384$	−1.00000	−0.0510310
$385$	6.00000	0.305788
$386$	4.00000	0.203595
$387$	−1.00000	−0.0508329
$388$	14.0000	0.710742
$389$	3.00000	0.152106	0.0760530	−	0.997104i	$-0.475768\pi$
$389$	3.00000	0.152106	0.0760530	+	0.997104i	$0.475768\pi$
$390$	0	0
$391$	18.0000	0.910299
$392$	3.00000	0.151523
$393$	9.00000	0.453990
$394$	−24.0000	−1.20910
$395$	−5.00000	−0.251577
$396$	−3.00000	−0.150756
$397$	−31.0000	−1.55585	−0.777923	−	0.628360i	$-0.783727\pi$
$397$	−31.0000	−1.55585	−0.777923	+	0.628360i	$0.783727\pi$
$398$	−8.00000	−0.401004
$399$	4.00000	0.200250
$400$	1.00000	0.0500000
$401$	−12.0000	−0.599251	−0.299626	−	0.954057i	$-0.596862\pi$
$401$	−12.0000	−0.599251	−0.299626	+	0.954057i	$0.596862\pi$
$402$	−8.00000	−0.399004
$403$	0	0
$404$	6.00000	0.298511
$405$	−1.00000	−0.0496904
$406$	−6.00000	−0.297775
$407$	21.0000	1.04093
$408$	−6.00000	−0.297044
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	6.00000	0.296319
$411$	9.00000	0.443937
$412$	14.0000	0.689730
$413$	−18.0000	−0.885722
$414$	−3.00000	−0.147442
$415$	6.00000	0.294528
$416$	0	0
$417$	14.0000	0.685583
$418$	6.00000	0.293470
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	−2.00000	−0.0975900
$421$	−16.0000	−0.779792	−0.389896	−	0.920859i	$-0.627489\pi$
$421$	−16.0000	−0.779792	−0.389896	+	0.920859i	$0.627489\pi$
$422$	−20.0000	−0.973585
$423$	−3.00000	−0.145865
$424$	6.00000	0.291386
$425$	6.00000	0.291043
$426$	12.0000	0.581402
$427$	4.00000	0.193574
$428$	−6.00000	−0.290021
$429$	0	0
$430$	−1.00000	−0.0482243
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	1.00000	0.0481125
$433$	−40.0000	−1.92228	−0.961139	−	0.276066i	$-0.910969\pi$
$433$	−40.0000	−1.92228	−0.961139	+	0.276066i	$0.910969\pi$
$434$	−10.0000	−0.480015
$435$	−3.00000	−0.143839
$436$	14.0000	0.670478
$437$	6.00000	0.287019
$438$	−14.0000	−0.668946
$439$	−4.00000	−0.190910	−0.0954548	−	0.995434i	$-0.530431\pi$
$439$	−4.00000	−0.190910	−0.0954548	+	0.995434i	$0.530431\pi$
$440$	−3.00000	−0.143019
$441$	−3.00000	−0.142857
$442$	0	0
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	−7.00000	−0.332205
$445$	18.0000	0.853282
$446$	10.0000	0.473514
$447$	−9.00000	−0.425685
$448$	2.00000	0.0944911
$449$	36.0000	1.69895	0.849473	−	0.527633i	$-0.176920\pi$
$449$	36.0000	1.69895	0.849473	+	0.527633i	$0.176920\pi$
$450$	−1.00000	−0.0471405
$451$	−18.0000	−0.847587
$452$	15.0000	0.705541
$453$	8.00000	0.375873
$454$	0	0
$455$	0	0
$456$	−2.00000	−0.0936586
$457$	2.00000	0.0935561	0.0467780	−	0.998905i	$-0.485105\pi$
$457$	2.00000	0.0935561	0.0467780	+	0.998905i	$0.485105\pi$
$458$	−14.0000	−0.654177
$459$	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	$0.386416\pi$
$462$	6.00000	0.279145
$463$	−34.0000	−1.58011	−0.790057	−	0.613033i	$-0.789949\pi$
$463$	−34.0000	−1.58011	−0.790057	+	0.613033i	$0.789949\pi$
$464$	3.00000	0.139272
$465$	−5.00000	−0.231869
$466$	−21.0000	−0.972806
$467$	18.0000	0.832941	0.416470	−	0.909149i	$-0.363267\pi$
$467$	18.0000	0.832941	0.416470	+	0.909149i	$0.363267\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	−3.00000	−0.138380
$471$	−13.0000	−0.599008
$472$	9.00000	0.414259
$473$	3.00000	0.137940
$474$	−5.00000	−0.229658
$475$	2.00000	0.0917663
$476$	12.0000	0.550019
$477$	−6.00000	−0.274721
$478$	−24.0000	−1.09773
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	−17.0000	−0.774329
$483$	6.00000	0.273009
$484$	−2.00000	−0.0909091
$485$	−14.0000	−0.635707
$486$	−1.00000	−0.0453609
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	−2.00000	−0.0905357
$489$	−13.0000	−0.587880
$490$	−3.00000	−0.135526
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	6.00000	0.270501
$493$	18.0000	0.810679
$494$	0	0
$495$	3.00000	0.134840
$496$	5.00000	0.224507
$497$	−24.0000	−1.07655
$498$	6.00000	0.268866
$499$	32.0000	1.43252	0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000	1.43252	0.716258	+	0.697835i	$0.245853\pi$
$500$	−1.00000	−0.0447214
$501$	−9.00000	−0.402090
$502$	−15.0000	−0.669483
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	−2.00000	−0.0890871
$505$	−6.00000	−0.266996
$506$	9.00000	0.400099
$507$	0	0
$508$	14.0000	0.621150
$509$	3.00000	0.132973	0.0664863	−	0.997787i	$-0.478821\pi$
$509$	3.00000	0.132973	0.0664863	+	0.997787i	$0.478821\pi$
$510$	6.00000	0.265684
$511$	28.0000	1.23865
$512$	−1.00000	−0.0441942
$513$	2.00000	0.0883022
$514$	21.0000	0.926270
$515$	−14.0000	−0.616914
$516$	−1.00000	−0.0440225
$517$	9.00000	0.395820
$518$	14.0000	0.615125
$519$	12.0000	0.526742
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	−3.00000	−0.131306
$523$	11.0000	0.480996	0.240498	−	0.970650i	$-0.422689\pi$
$523$	11.0000	0.480996	0.240498	+	0.970650i	$0.422689\pi$
$524$	9.00000	0.393167
$525$	2.00000	0.0872872
$526$	15.0000	0.654031
$527$	30.0000	1.30682
$528$	−3.00000	−0.130558
$529$	−14.0000	−0.608696
$530$	−6.00000	−0.260623
$531$	−9.00000	−0.390567
$532$	4.00000	0.173422
$533$	0	0
$534$	18.0000	0.778936
$535$	6.00000	0.259403
$536$	−8.00000	−0.345547
$537$	−3.00000	−0.129460
$538$	−18.0000	−0.776035
$539$	9.00000	0.387657
$540$	−1.00000	−0.0430331
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	−11.0000	−0.472490
$543$	−16.0000	−0.686626
$544$	−6.00000	−0.257248
$545$	−14.0000	−0.599694
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	9.00000	0.384461
$549$	2.00000	0.0853579
$550$	3.00000	0.127920
$551$	6.00000	0.255609
$552$	−3.00000	−0.127688
$553$	10.0000	0.425243
$554$	1.00000	0.0424859
$555$	7.00000	0.297133
$556$	14.0000	0.593732
$557$	−6.00000	−0.254228	−0.127114	−	0.991888i	$-0.540571\pi$
$557$	−6.00000	−0.254228	−0.127114	+	0.991888i	$0.540571\pi$
$558$	−5.00000	−0.211667
$559$	0	0
$560$	−2.00000	−0.0845154
$561$	−18.0000	−0.759961
$562$	−18.0000	−0.759284
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	−3.00000	−0.126323
$565$	−15.0000	−0.631055
$566$	31.0000	1.30303
$567$	2.00000	0.0839921
$568$	12.0000	0.503509
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	2.00000	0.0837708
$571$	−40.0000	−1.67395	−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000	−1.67395	−0.836974	+	0.547243i	$0.815677\pi$
$572$	0	0
$573$	−12.0000	−0.501307
$574$	−12.0000	−0.500870
$575$	3.00000	0.125109
$576$	1.00000	0.0416667
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	−19.0000	−0.790296
$579$	−4.00000	−0.166234
$580$	−3.00000	−0.124568
$581$	−12.0000	−0.497844
$582$	−14.0000	−0.580319
$583$	18.0000	0.745484
$584$	−14.0000	−0.579324
$585$	0	0
$586$	30.0000	1.23929
$587$	18.0000	0.742940	0.371470	−	0.928445i	$-0.378854\pi$
$587$	18.0000	0.742940	0.371470	+	0.928445i	$0.378854\pi$
$588$	−3.00000	−0.123718
$589$	10.0000	0.412043
$590$	−9.00000	−0.370524
$591$	24.0000	0.987228
$592$	−7.00000	−0.287698
$593$	27.0000	1.10876	0.554379	−	0.832265i	$-0.312956\pi$
$593$	27.0000	1.10876	0.554379	+	0.832265i	$0.312956\pi$
$594$	3.00000	0.123091
$595$	−12.0000	−0.491952
$596$	−9.00000	−0.368654
$597$	8.00000	0.327418
$598$	0	0
$599$	6.00000	0.245153	0.122577	−	0.992459i	$-0.460884\pi$
$599$	6.00000	0.245153	0.122577	+	0.992459i	$0.460884\pi$
$600$	−1.00000	−0.0408248
$601$	−19.0000	−0.775026	−0.387513	−	0.921864i	$-0.626666\pi$
$601$	−19.0000	−0.775026	−0.387513	+	0.921864i	$0.626666\pi$
$602$	2.00000	0.0815139
$603$	8.00000	0.325785
$604$	8.00000	0.325515
$605$	2.00000	0.0813116
$606$	−6.00000	−0.243733
$607$	−22.0000	−0.892952	−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000	−0.892952	−0.446476	+	0.894795i	$0.647321\pi$
$608$	−2.00000	−0.0811107
$609$	6.00000	0.243132
$610$	2.00000	0.0809776
$611$	0	0
$612$	6.00000	0.242536
$613$	−31.0000	−1.25208	−0.626039	−	0.779792i	$-0.715325\pi$
$613$	−31.0000	−1.25208	−0.626039	+	0.779792i	$0.715325\pi$
$614$	−8.00000	−0.322854
$615$	−6.00000	−0.241943
$616$	6.00000	0.241747
$617$	21.0000	0.845428	0.422714	−	0.906263i	$-0.361077\pi$
$617$	21.0000	0.845428	0.422714	+	0.906263i	$0.361077\pi$
$618$	−14.0000	−0.563163
$619$	−46.0000	−1.84890	−0.924448	−	0.381308i	$-0.875474\pi$
$619$	−46.0000	−1.84890	−0.924448	+	0.381308i	$0.875474\pi$
$620$	−5.00000	−0.200805
$621$	3.00000	0.120386
$622$	12.0000	0.481156
$623$	−36.0000	−1.44231
$624$	0	0
$625$	1.00000	0.0400000
$626$	−8.00000	−0.319744
$627$	−6.00000	−0.239617
$628$	−13.0000	−0.518756
$629$	−42.0000	−1.67465
$630$	2.00000	0.0796819
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	−5.00000	−0.198889
$633$	20.0000	0.794929
$634$	−12.0000	−0.476581
$635$	−14.0000	−0.555573
$636$	−6.00000	−0.237915
$637$	0	0
$638$	9.00000	0.356313
$639$	−12.0000	−0.474713
$640$	1.00000	0.0395285
$641$	6.00000	0.236986	0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000	0.236986	0.118493	+	0.992955i	$0.462194\pi$
$642$	6.00000	0.236801
$643$	−16.0000	−0.630978	−0.315489	−	0.948929i	$-0.602169\pi$
$643$	−16.0000	−0.630978	−0.315489	+	0.948929i	$0.602169\pi$
$644$	6.00000	0.236433
$645$	1.00000	0.0393750
$646$	−12.0000	−0.472134
$647$	24.0000	0.943537	0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000	0.943537	0.471769	+	0.881722i	$0.343616\pi$
$648$	−1.00000	−0.0392837
$649$	27.0000	1.05984
$650$	0	0
$651$	10.0000	0.391931
$652$	−13.0000	−0.509119
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	−14.0000	−0.547443
$655$	−9.00000	−0.351659
$656$	6.00000	0.234261
$657$	14.0000	0.546192
$658$	6.00000	0.233904
$659$	15.0000	0.584317	0.292159	−	0.956370i	$-0.405627\pi$
$659$	15.0000	0.584317	0.292159	+	0.956370i	$0.405627\pi$
$660$	3.00000	0.116775
$661$	32.0000	1.24466	0.622328	−	0.782757i	$-0.286187\pi$
$661$	32.0000	1.24466	0.622328	+	0.782757i	$0.286187\pi$
$662$	−32.0000	−1.24372
$663$	0	0
$664$	6.00000	0.232845
$665$	−4.00000	−0.155113
$666$	7.00000	0.271244
$667$	9.00000	0.348481
$668$	−9.00000	−0.348220
$669$	−10.0000	−0.386622
$670$	8.00000	0.309067
$671$	−6.00000	−0.231627
$672$	−2.00000	−0.0771517
$673$	−4.00000	−0.154189	−0.0770943	−	0.997024i	$-0.524564\pi$
$673$	−4.00000	−0.154189	−0.0770943	+	0.997024i	$0.524564\pi$
$674$	−14.0000	−0.539260
$675$	1.00000	0.0384900
$676$	0	0
$677$	−36.0000	−1.38359	−0.691796	−	0.722093i	$-0.743180\pi$
$677$	−36.0000	−1.38359	−0.691796	+	0.722093i	$0.743180\pi$
$678$	−15.0000	−0.576072
$679$	28.0000	1.07454
$680$	6.00000	0.230089
$681$	0	0
$682$	15.0000	0.574380
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	2.00000	0.0764719
$685$	−9.00000	−0.343872
$686$	20.0000	0.763604
$687$	14.0000	0.534133
$688$	−1.00000	−0.0381246
$689$	0	0
$690$	3.00000	0.114208
$691$	−46.0000	−1.74992	−0.874961	−	0.484193i	$-0.839113\pi$
$691$	−46.0000	−1.74992	−0.874961	+	0.484193i	$0.839113\pi$
$692$	12.0000	0.456172
$693$	−6.00000	−0.227921
$694$	30.0000	1.13878
$695$	−14.0000	−0.531050
$696$	−3.00000	−0.113715
$697$	36.0000	1.36360
$698$	−8.00000	−0.302804
$699$	21.0000	0.794293
$700$	2.00000	0.0755929
$701$	−21.0000	−0.793159	−0.396580	−	0.918000i	$-0.629803\pi$
$701$	−21.0000	−0.793159	−0.396580	+	0.918000i	$0.629803\pi$
$702$	0	0
$703$	−14.0000	−0.528020
$704$	−3.00000	−0.113067
$705$	3.00000	0.112987
$706$	−30.0000	−1.12906
$707$	12.0000	0.451306
$708$	−9.00000	−0.338241
$709$	32.0000	1.20179	0.600893	−	0.799330i	$-0.294812\pi$
$709$	32.0000	1.20179	0.600893	+	0.799330i	$0.294812\pi$
$710$	−12.0000	−0.450352
$711$	5.00000	0.187515
$712$	18.0000	0.674579
$713$	15.0000	0.561754
$714$	−12.0000	−0.449089
$715$	0	0
$716$	−3.00000	−0.112115
$717$	24.0000	0.896296
$718$	−24.0000	−0.895672
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	−1.00000	−0.0372678
$721$	28.0000	1.04277
$722$	15.0000	0.558242
$723$	17.0000	0.632237
$724$	−16.0000	−0.594635
$725$	3.00000	0.111417
$726$	2.00000	0.0742270
$727$	−4.00000	−0.148352	−0.0741759	−	0.997245i	$-0.523633\pi$
$727$	−4.00000	−0.148352	−0.0741759	+	0.997245i	$0.523633\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	14.0000	0.518163
$731$	−6.00000	−0.221918
$732$	2.00000	0.0739221
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	−20.0000	−0.738213
$735$	3.00000	0.110657
$736$	−3.00000	−0.110581
$737$	−24.0000	−0.884051
$738$	−6.00000	−0.220863
$739$	−16.0000	−0.588570	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000	−0.588570	−0.294285	+	0.955718i	$0.595081\pi$
$740$	7.00000	0.257325
$741$	0	0
$742$	12.0000	0.440534
$743$	−9.00000	−0.330178	−0.165089	−	0.986279i	$-0.552791\pi$
$743$	−9.00000	−0.330178	−0.165089	+	0.986279i	$0.552791\pi$
$744$	−5.00000	−0.183309
$745$	9.00000	0.329734
$746$	25.0000	0.915315
$747$	−6.00000	−0.219529
$748$	−18.0000	−0.658145
$749$	−12.0000	−0.438470
$750$	1.00000	0.0365148
$751$	41.0000	1.49611	0.748056	−	0.663636i	$-0.230988\pi$
$751$	41.0000	1.49611	0.748056	+	0.663636i	$0.230988\pi$
$752$	−3.00000	−0.109399
$753$	15.0000	0.546630
$754$	0	0
$755$	−8.00000	−0.291150
$756$	2.00000	0.0727393
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	−38.0000	−1.38022
$759$	−9.00000	−0.326679
$760$	2.00000	0.0725476
$761$	−12.0000	−0.435000	−0.217500	−	0.976060i	$-0.569790\pi$
$761$	−12.0000	−0.435000	−0.217500	+	0.976060i	$0.569790\pi$
$762$	−14.0000	−0.507166
$763$	28.0000	1.01367
$764$	−12.0000	−0.434145
$765$	−6.00000	−0.216930
$766$	21.0000	0.758761
$767$	0	0
$768$	1.00000	0.0360844
$769$	−13.0000	−0.468792	−0.234396	−	0.972141i	$-0.575311\pi$
$769$	−13.0000	−0.468792	−0.234396	+	0.972141i	$0.575311\pi$
$770$	−6.00000	−0.216225
$771$	−21.0000	−0.756297
$772$	−4.00000	−0.143963
$773$	−48.0000	−1.72644	−0.863220	−	0.504828i	$-0.831556\pi$
$773$	−48.0000	−1.72644	−0.863220	+	0.504828i	$0.831556\pi$
$774$	1.00000	0.0359443
$775$	5.00000	0.179605
$776$	−14.0000	−0.502571
$777$	−14.0000	−0.502247
$778$	−3.00000	−0.107555
$779$	12.0000	0.429945
$780$	0	0
$781$	36.0000	1.28818
$782$	−18.0000	−0.643679
$783$	3.00000	0.107211
$784$	−3.00000	−0.107143
$785$	13.0000	0.463990
$786$	−9.00000	−0.321019
$787$	35.0000	1.24762	0.623808	−	0.781578i	$-0.285585\pi$
$787$	35.0000	1.24762	0.623808	+	0.781578i	$0.285585\pi$
$788$	24.0000	0.854965
$789$	−15.0000	−0.534014
$790$	5.00000	0.177892
$791$	30.0000	1.06668
$792$	3.00000	0.106600
$793$	0	0
$794$	31.0000	1.10015
$795$	6.00000	0.212798
$796$	8.00000	0.283552
$797$	−24.0000	−0.850124	−0.425062	−	0.905164i	$-0.639748\pi$
$797$	−24.0000	−0.850124	−0.425062	+	0.905164i	$0.639748\pi$
$798$	−4.00000	−0.141598
$799$	−18.0000	−0.636794
$800$	−1.00000	−0.0353553
$801$	−18.0000	−0.635999
$802$	12.0000	0.423735
$803$	−42.0000	−1.48215
$804$	8.00000	0.282138
$805$	−6.00000	−0.211472
$806$	0	0
$807$	18.0000	0.633630
$808$	−6.00000	−0.211079
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	1.00000	0.0351364
$811$	32.0000	1.12367	0.561836	−	0.827249i	$-0.310095\pi$
$811$	32.0000	1.12367	0.561836	+	0.827249i	$0.310095\pi$
$812$	6.00000	0.210559
$813$	11.0000	0.385787
$814$	−21.0000	−0.736050
$815$	13.0000	0.455370
$816$	6.00000	0.210042
$817$	−2.00000	−0.0699711
$818$	10.0000	0.349642
$819$	0	0
$820$	−6.00000	−0.209529
$821$	27.0000	0.942306	0.471153	−	0.882051i	$-0.343838\pi$
$821$	27.0000	0.942306	0.471153	+	0.882051i	$0.343838\pi$
$822$	−9.00000	−0.313911
$823$	14.0000	0.488009	0.244005	−	0.969774i	$-0.421539\pi$
$823$	14.0000	0.488009	0.244005	+	0.969774i	$0.421539\pi$
$824$	−14.0000	−0.487713
$825$	−3.00000	−0.104447
$826$	18.0000	0.626300
$827$	6.00000	0.208640	0.104320	−	0.994544i	$-0.466733\pi$
$827$	6.00000	0.208640	0.104320	+	0.994544i	$0.466733\pi$
$828$	3.00000	0.104257
$829$	26.0000	0.903017	0.451509	−	0.892267i	$-0.350886\pi$
$829$	26.0000	0.903017	0.451509	+	0.892267i	$0.350886\pi$
$830$	−6.00000	−0.208263
$831$	−1.00000	−0.0346896
$832$	0	0
$833$	−18.0000	−0.623663
$834$	−14.0000	−0.484780
$835$	9.00000	0.311458
$836$	−6.00000	−0.207514
$837$	5.00000	0.172825
$838$	12.0000	0.414533
$839$	42.0000	1.45000	0.725001	−	0.688748i	$-0.241839\pi$
$839$	42.0000	1.45000	0.725001	+	0.688748i	$0.241839\pi$
$840$	2.00000	0.0690066
$841$	−20.0000	−0.689655
$842$	16.0000	0.551396
$843$	18.0000	0.619953
$844$	20.0000	0.688428
$845$	0	0
$846$	3.00000	0.103142
$847$	−4.00000	−0.137442
$848$	−6.00000	−0.206041
$849$	−31.0000	−1.06392
$850$	−6.00000	−0.205798
$851$	−21.0000	−0.719871
$852$	−12.0000	−0.411113
$853$	−19.0000	−0.650548	−0.325274	−	0.945620i	$-0.605456\pi$
$853$	−19.0000	−0.650548	−0.325274	+	0.945620i	$0.605456\pi$
$854$	−4.00000	−0.136877
$855$	−2.00000	−0.0683986
$856$	6.00000	0.205076
$857$	3.00000	0.102478	0.0512390	−	0.998686i	$-0.483683\pi$
$857$	3.00000	0.102478	0.0512390	+	0.998686i	$0.483683\pi$
$858$	0	0
$859$	−46.0000	−1.56950	−0.784750	−	0.619813i	$-0.787209\pi$
$859$	−46.0000	−1.56950	−0.784750	+	0.619813i	$0.787209\pi$
$860$	1.00000	0.0340997
$861$	12.0000	0.408959
$862$	−12.0000	−0.408722
$863$	−45.0000	−1.53182	−0.765909	−	0.642949i	$-0.777711\pi$
$863$	−45.0000	−1.53182	−0.765909	+	0.642949i	$0.777711\pi$
$864$	−1.00000	−0.0340207
$865$	−12.0000	−0.408012
$866$	40.0000	1.35926
$867$	19.0000	0.645274
$868$	10.0000	0.339422
$869$	−15.0000	−0.508840
$870$	3.00000	0.101710
$871$	0	0
$872$	−14.0000	−0.474100
$873$	14.0000	0.473828
$874$	−6.00000	−0.202953
$875$	−2.00000	−0.0676123
$876$	14.0000	0.473016
$877$	23.0000	0.776655	0.388327	−	0.921521i	$-0.373053\pi$
$877$	23.0000	0.776655	0.388327	+	0.921521i	$0.373053\pi$
$878$	4.00000	0.134993
$879$	−30.0000	−1.01187
$880$	3.00000	0.101130
$881$	6.00000	0.202145	0.101073	−	0.994879i	$-0.467773\pi$
$881$	6.00000	0.202145	0.101073	+	0.994879i	$0.467773\pi$
$882$	3.00000	0.101015
$883$	−19.0000	−0.639401	−0.319700	−	0.947519i	$-0.603582\pi$
$883$	−19.0000	−0.639401	−0.319700	+	0.947519i	$0.603582\pi$
$884$	0	0
$885$	9.00000	0.302532
$886$	36.0000	1.20944
$887$	−57.0000	−1.91387	−0.956936	−	0.290298i	$-0.906246\pi$
$887$	−57.0000	−1.91387	−0.956936	+	0.290298i	$0.906246\pi$
$888$	7.00000	0.234905
$889$	28.0000	0.939090
$890$	−18.0000	−0.603361
$891$	−3.00000	−0.100504
$892$	−10.0000	−0.334825
$893$	−6.00000	−0.200782
$894$	9.00000	0.301005
$895$	3.00000	0.100279
$896$	−2.00000	−0.0668153
$897$	0	0
$898$	−36.0000	−1.20134
$899$	15.0000	0.500278
$900$	1.00000	0.0333333
$901$	−36.0000	−1.19933
$902$	18.0000	0.599334
$903$	−2.00000	−0.0665558
$904$	−15.0000	−0.498893
$905$	16.0000	0.531858
$906$	−8.00000	−0.265782
$907$	17.0000	0.564476	0.282238	−	0.959344i	$-0.408923\pi$
$907$	17.0000	0.564476	0.282238	+	0.959344i	$0.408923\pi$
$908$	0	0
$909$	6.00000	0.199007
$910$	0	0
$911$	−6.00000	−0.198789	−0.0993944	−	0.995048i	$-0.531691\pi$
$911$	−6.00000	−0.198789	−0.0993944	+	0.995048i	$0.531691\pi$
$912$	2.00000	0.0662266
$913$	18.0000	0.595713
$914$	−2.00000	−0.0661541
$915$	−2.00000	−0.0661180
$916$	14.0000	0.462573
$917$	18.0000	0.594412
$918$	−6.00000	−0.198030
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	3.00000	0.0989071
$921$	8.00000	0.263609
$922$	−15.0000	−0.493999
$923$	0	0
$924$	−6.00000	−0.197386
$925$	−7.00000	−0.230159
$926$	34.0000	1.11731
$927$	14.0000	0.459820
$928$	−3.00000	−0.0984798
$929$	−12.0000	−0.393707	−0.196854	−	0.980433i	$-0.563072\pi$
$929$	−12.0000	−0.393707	−0.196854	+	0.980433i	$0.563072\pi$
$930$	5.00000	0.163956
$931$	−6.00000	−0.196642
$932$	21.0000	0.687878
$933$	−12.0000	−0.392862
$934$	−18.0000	−0.588978
$935$	18.0000	0.588663
$936$	0	0
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	−16.0000	−0.522419
$939$	8.00000	0.261070
$940$	3.00000	0.0978492
$941$	42.0000	1.36916	0.684580	−	0.728937i	$-0.259985\pi$
$941$	42.0000	1.36916	0.684580	+	0.728937i	$0.259985\pi$
$942$	13.0000	0.423563
$943$	18.0000	0.586161
$944$	−9.00000	−0.292925
$945$	−2.00000	−0.0650600
$946$	−3.00000	−0.0975384
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	5.00000	0.162392
$949$	0	0
$950$	−2.00000	−0.0648886
$951$	12.0000	0.389127
$952$	−12.0000	−0.388922
$953$	9.00000	0.291539	0.145769	−	0.989319i	$-0.453434\pi$
$953$	9.00000	0.291539	0.145769	+	0.989319i	$0.453434\pi$
$954$	6.00000	0.194257
$955$	12.0000	0.388311
$956$	24.0000	0.776215
$957$	−9.00000	−0.290929
$958$	0	0
$959$	18.0000	0.581250
$960$	−1.00000	−0.0322749
$961$	−6.00000	−0.193548
$962$	0	0
$963$	−6.00000	−0.193347
$964$	17.0000	0.547533
$965$	4.00000	0.128765
$966$	−6.00000	−0.193047
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	2.00000	0.0642824
$969$	12.0000	0.385496
$970$	14.0000	0.449513
$971$	−48.0000	−1.54039	−0.770197	−	0.637806i	$-0.779842\pi$
$971$	−48.0000	−1.54039	−0.770197	+	0.637806i	$0.779842\pi$
$972$	1.00000	0.0320750
$973$	28.0000	0.897639
$974$	−2.00000	−0.0640841
$975$	0	0
$976$	2.00000	0.0640184
$977$	−57.0000	−1.82359	−0.911796	−	0.410644i	$-0.865304\pi$
$977$	−57.0000	−1.82359	−0.911796	+	0.410644i	$0.865304\pi$
$978$	13.0000	0.415694
$979$	54.0000	1.72585
$980$	3.00000	0.0958315
$981$	14.0000	0.446986
$982$	0	0
$983$	9.00000	0.287055	0.143528	−	0.989646i	$-0.454155\pi$
$983$	9.00000	0.287055	0.143528	+	0.989646i	$0.454155\pi$
$984$	−6.00000	−0.191273
$985$	−24.0000	−0.764704
$986$	−18.0000	−0.573237
$987$	−6.00000	−0.190982
$988$	0	0
$989$	−3.00000	−0.0953945
$990$	−3.00000	−0.0953463
$991$	−25.0000	−0.794151	−0.397076	−	0.917786i	$-0.629975\pi$
$991$	−25.0000	−0.794151	−0.397076	+	0.917786i	$0.629975\pi$
$992$	−5.00000	−0.158750
$993$	32.0000	1.01549
$994$	24.0000	0.761234
$995$	−8.00000	−0.253617
$996$	−6.00000	−0.190117
$997$	−22.0000	−0.696747	−0.348373	−	0.937356i	$-0.613266\pi$
$997$	−22.0000	−0.696747	−0.348373	+	0.937356i	$0.613266\pi$
$998$	−32.0000	−1.01294
$999$	−7.00000	−0.221470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.a.j.1.1		1
13.3	even	3		390.2.i.c.61.1	✓	2
13.5	odd	4		5070.2.b.j.1351.2		2
13.8	odd	4		5070.2.b.j.1351.1		2
13.9	even	3		390.2.i.c.211.1	yes	2
13.12	even	2		5070.2.a.v.1.1		1
39.29	odd	6		1170.2.i.d.451.1		2
39.35	odd	6		1170.2.i.d.991.1		2
65.3	odd	12		1950.2.z.k.1699.1		4
65.9	even	6		1950.2.i.n.601.1		2
65.22	odd	12		1950.2.z.k.1849.1		4
65.29	even	6		1950.2.i.n.451.1		2
65.42	odd	12		1950.2.z.k.1699.2		4
65.48	odd	12		1950.2.z.k.1849.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.i.c.61.1	✓	2	13.3	even	3
390.2.i.c.211.1	yes	2	13.9	even	3
1170.2.i.d.451.1		2	39.29	odd	6
1170.2.i.d.991.1		2	39.35	odd	6
1950.2.i.n.451.1		2	65.29	even	6
1950.2.i.n.601.1		2	65.9	even	6
1950.2.z.k.1699.1		4	65.3	odd	12
1950.2.z.k.1699.2		4	65.42	odd	12
1950.2.z.k.1849.1		4	65.22	odd	12
1950.2.z.k.1849.2		4	65.48	odd	12
5070.2.a.j.1.1		1	1.1	even	1	trivial
5070.2.a.v.1.1		1	13.12	even	2
5070.2.b.j.1351.1		2	13.8	odd	4
5070.2.b.j.1351.2		2	13.5	odd	4

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:	\( N \)	\(=\)	\( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5070.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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