LMFDB - Embedded newform 1470.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1470,2,Mod(1,1470)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1470, 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("1470.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1470.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:	$11.7380090971$
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	$\chi$	$=$	1470.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +2.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{18} -1.00000 q^{20} +2.00000 q^{22} +8.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -1.00000 q^{30} +2.00000 q^{31} +1.00000 q^{32} +2.00000 q^{33} +4.00000 q^{34} +1.00000 q^{36} +8.00000 q^{37} -2.00000 q^{39} -1.00000 q^{40} +2.00000 q^{41} -2.00000 q^{43} +2.00000 q^{44} -1.00000 q^{45} +8.00000 q^{46} -10.0000 q^{47} +1.00000 q^{48} +1.00000 q^{50} +4.00000 q^{51} -2.00000 q^{52} -2.00000 q^{53} +1.00000 q^{54} -2.00000 q^{55} -4.00000 q^{59} -1.00000 q^{60} +10.0000 q^{61} +2.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} +2.00000 q^{66} +2.00000 q^{67} +4.00000 q^{68} +8.00000 q^{69} -12.0000 q^{71} +1.00000 q^{72} -10.0000 q^{73} +8.00000 q^{74} +1.00000 q^{75} -2.00000 q^{78} +16.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} -16.0000 q^{83} -4.00000 q^{85} -2.00000 q^{86} +2.00000 q^{88} -14.0000 q^{89} -1.00000 q^{90} +8.00000 q^{92} +2.00000 q^{93} -10.0000 q^{94} +1.00000 q^{96} -6.00000 q^{97} +2.00000 q^{99} +O(q^{100})

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
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	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
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	1.00000	0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	2.00000	0.426401
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	−2.00000	−0.392232
$27$	1.00000	0.192450
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	−1.00000	−0.182574
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	1.00000	0.176777
$33$	2.00000	0.348155
$34$	4.00000	0.685994
$35$	0	0
$36$	1.00000	0.166667
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	−1.00000	−0.158114
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	2.00000	0.301511
$45$	−1.00000	−0.149071
$46$	8.00000	1.17954
$47$	−10.0000	−1.45865	−0.729325	−	0.684167i	$-0.760166\pi$
$47$	−10.0000	−1.45865	−0.729325	+	0.684167i	$0.760166\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	1.00000	0.141421
$51$	4.00000	0.560112
$52$	−2.00000	−0.277350
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	1.00000	0.136083
$55$	−2.00000	−0.269680
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	−1.00000	−0.129099
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	2.00000	0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	2.00000	0.246183
$67$	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
$69$	8.00000	0.963087
$70$	0	0
$71$	−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
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	8.00000	0.929981
$75$	1.00000	0.115470
$76$	0	0
$77$	0	0
$78$	−2.00000	−0.226455
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	−16.0000	−1.75623	−0.878114	−	0.478451i	$-0.841198\pi$
$83$	−16.0000	−1.75623	−0.878114	+	0.478451i	$0.841198\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	−2.00000	−0.215666
$87$	0	0
$88$	2.00000	0.213201
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	8.00000	0.834058
$93$	2.00000	0.207390
$94$	−10.0000	−1.03142
$95$	0	0
$96$	1.00000	0.102062
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	2.00000	0.201008
$100$	1.00000	0.100000
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	4.00000	0.396059
$103$	20.0000	1.97066	0.985329	−	0.170664i	$-0.0545913\pi$
$103$	20.0000	1.97066	0.985329	+	0.170664i	$0.0545913\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	−2.00000	−0.194257
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	1.00000	0.0962250
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	−2.00000	−0.190693
$111$	8.00000	0.759326
$112$	0	0
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	−8.00000	−0.746004
$116$	0	0
$117$	−2.00000	−0.184900
$118$	−4.00000	−0.368230
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	−7.00000	−0.636364
$122$	10.0000	0.905357
$123$	2.00000	0.180334
$124$	2.00000	0.179605
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−12.0000	−1.06483	−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000	−1.06483	−0.532414	+	0.846484i	$0.678715\pi$
$128$	1.00000	0.0883883
$129$	−2.00000	−0.176090
$130$	2.00000	0.175412
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	2.00000	0.174078
$133$	0	0
$134$	2.00000	0.172774
$135$	−1.00000	−0.0860663
$136$	4.00000	0.342997
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	8.00000	0.681005
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	−10.0000	−0.842152
$142$	−12.0000	−1.00702
$143$	−4.00000	−0.334497
$144$	1.00000	0.0833333
$145$	0	0
$146$	−10.0000	−0.827606
$147$	0	0
$148$	8.00000	0.657596
$149$	−16.0000	−1.31077	−0.655386	−	0.755295i	$-0.727494\pi$
$149$	−16.0000	−1.31077	−0.655386	+	0.755295i	$0.727494\pi$
$150$	1.00000	0.0816497
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	4.00000	0.323381
$154$	0	0
$155$	−2.00000	−0.160644
$156$	−2.00000	−0.160128
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	16.0000	1.27289
$159$	−2.00000	−0.158610
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	1.00000	0.0785674
$163$	−10.0000	−0.783260	−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000	−0.783260	−0.391630	+	0.920123i	$0.628089\pi$
$164$	2.00000	0.156174
$165$	−2.00000	−0.155700
$166$	−16.0000	−1.24184
$167$	−18.0000	−1.39288	−0.696441	−	0.717614i	$-0.745234\pi$
$167$	−18.0000	−1.39288	−0.696441	+	0.717614i	$0.745234\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−4.00000	−0.306786
$171$	0	0
$172$	−2.00000	−0.152499
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	0	0
$176$	2.00000	0.150756
$177$	−4.00000	−0.300658
$178$	−14.0000	−1.04934
$179$	−2.00000	−0.149487	−0.0747435	−	0.997203i	$-0.523814\pi$
$179$	−2.00000	−0.149487	−0.0747435	+	0.997203i	$0.523814\pi$
$180$	−1.00000	−0.0745356
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	0	0
$183$	10.0000	0.739221
$184$	8.00000	0.589768
$185$	−8.00000	−0.588172
$186$	2.00000	0.146647
$187$	8.00000	0.585018
$188$	−10.0000	−0.729325
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	1.00000	0.0721688
$193$	−18.0000	−1.29567	−0.647834	−	0.761781i	$-0.724325\pi$
$193$	−18.0000	−1.29567	−0.647834	+	0.761781i	$0.724325\pi$
$194$	−6.00000	−0.430775
$195$	2.00000	0.143223
$196$	0	0
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	2.00000	0.142134
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	1.00000	0.0707107
$201$	2.00000	0.141069
$202$	14.0000	0.985037
$203$	0	0
$204$	4.00000	0.280056
$205$	−2.00000	−0.139686
$206$	20.0000	1.39347
$207$	8.00000	0.556038
$208$	−2.00000	−0.138675
$209$	0	0
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	−2.00000	−0.137361
$213$	−12.0000	−0.822226
$214$	12.0000	0.820303
$215$	2.00000	0.136399
$216$	1.00000	0.0680414
$217$	0	0
$218$	−2.00000	−0.135457
$219$	−10.0000	−0.675737
$220$	−2.00000	−0.134840
$221$	−8.00000	−0.538138
$222$	8.00000	0.536925
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	−14.0000	−0.931266
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	−8.00000	−0.527504
$231$	0	0
$232$	0	0
$233$	−14.0000	−0.917170	−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000	−0.917170	−0.458585	+	0.888650i	$0.651644\pi$
$234$	−2.00000	−0.130744
$235$	10.0000	0.652328
$236$	−4.00000	−0.260378
$237$	16.0000	1.03931
$238$	0	0
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	−1.00000	−0.0645497
$241$	20.0000	1.28831	0.644157	−	0.764894i	$-0.277208\pi$
$241$	20.0000	1.28831	0.644157	+	0.764894i	$0.277208\pi$
$242$	−7.00000	−0.449977
$243$	1.00000	0.0641500
$244$	10.0000	0.640184
$245$	0	0
$246$	2.00000	0.127515
$247$	0	0
$248$	2.00000	0.127000
$249$	−16.0000	−1.01396
$250$	−1.00000	−0.0632456
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	0	0
$253$	16.0000	1.00591
$254$	−12.0000	−0.752947
$255$	−4.00000	−0.250490
$256$	1.00000	0.0625000
$257$	−12.0000	−0.748539	−0.374270	−	0.927320i	$-0.622107\pi$
$257$	−12.0000	−0.748539	−0.374270	+	0.927320i	$0.622107\pi$
$258$	−2.00000	−0.124515
$259$	0	0
$260$	2.00000	0.124035
$261$	0	0
$262$	−12.0000	−0.741362
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	2.00000	0.123091
$265$	2.00000	0.122859
$266$	0	0
$267$	−14.0000	−0.856786
$268$	2.00000	0.122169
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	−1.00000	−0.0608581
$271$	14.0000	0.850439	0.425220	−	0.905090i	$-0.360197\pi$
$271$	14.0000	0.850439	0.425220	+	0.905090i	$0.360197\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	−2.00000	−0.120824
$275$	2.00000	0.120605
$276$	8.00000	0.481543
$277$	−28.0000	−1.68236	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	−28.0000	−1.68236	−0.841178	+	0.540758i	$0.818138\pi$
$278$	4.00000	0.239904
$279$	2.00000	0.119737
$280$	0	0
$281$	14.0000	0.835170	0.417585	−	0.908638i	$-0.362877\pi$
$281$	14.0000	0.835170	0.417585	+	0.908638i	$0.362877\pi$
$282$	−10.0000	−0.595491
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	−4.00000	−0.236525
$287$	0	0
$288$	1.00000	0.0589256
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−6.00000	−0.351726
$292$	−10.0000	−0.585206
$293$	−30.0000	−1.75262	−0.876309	−	0.481749i	$-0.840002\pi$
$293$	−30.0000	−1.75262	−0.876309	+	0.481749i	$0.840002\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	8.00000	0.464991
$297$	2.00000	0.116052
$298$	−16.0000	−0.926855
$299$	−16.0000	−0.925304
$300$	1.00000	0.0577350
$301$	0	0
$302$	0	0
$303$	14.0000	0.804279
$304$	0	0
$305$	−10.0000	−0.572598
$306$	4.00000	0.228665
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	0	0
$309$	20.0000	1.13776
$310$	−2.00000	−0.113592
$311$	20.0000	1.13410	0.567048	−	0.823685i	$-0.308085\pi$
$311$	20.0000	1.13410	0.567048	+	0.823685i	$0.308085\pi$
$312$	−2.00000	−0.113228
$313$	26.0000	1.46961	0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	10.0000	0.564333
$315$	0	0
$316$	16.0000	0.900070
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	−2.00000	−0.112154
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	12.0000	0.669775
$322$	0	0
$323$	0	0
$324$	1.00000	0.0555556
$325$	−2.00000	−0.110940
$326$	−10.0000	−0.553849
$327$	−2.00000	−0.110600
$328$	2.00000	0.110432
$329$	0	0
$330$	−2.00000	−0.110096
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	−16.0000	−0.878114
$333$	8.00000	0.438397
$334$	−18.0000	−0.984916
$335$	−2.00000	−0.109272
$336$	0	0
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	−9.00000	−0.489535
$339$	−14.0000	−0.760376
$340$	−4.00000	−0.216930
$341$	4.00000	0.216612
$342$	0	0
$343$	0	0
$344$	−2.00000	−0.107833
$345$	−8.00000	−0.430706
$346$	6.00000	0.322562
$347$	−4.00000	−0.214731	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000	−0.214731	−0.107366	+	0.994220i	$0.534242\pi$
$348$	0	0
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	2.00000	0.106600
$353$	−24.0000	−1.27739	−0.638696	−	0.769460i	$-0.720526\pi$
$353$	−24.0000	−1.27739	−0.638696	+	0.769460i	$0.720526\pi$
$354$	−4.00000	−0.212598
$355$	12.0000	0.636894
$356$	−14.0000	−0.741999
$357$	0	0
$358$	−2.00000	−0.105703
$359$	−20.0000	−1.05556	−0.527780	−	0.849381i	$-0.676975\pi$
$359$	−20.0000	−1.05556	−0.527780	+	0.849381i	$0.676975\pi$
$360$	−1.00000	−0.0527046
$361$	−19.0000	−1.00000
$362$	22.0000	1.15629
$363$	−7.00000	−0.367405
$364$	0	0
$365$	10.0000	0.523424
$366$	10.0000	0.522708
$367$	28.0000	1.46159	0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000	1.46159	0.730794	+	0.682598i	$0.239150\pi$
$368$	8.00000	0.417029
$369$	2.00000	0.104116
$370$	−8.00000	−0.415900
$371$	0	0
$372$	2.00000	0.103695
$373$	−36.0000	−1.86401	−0.932005	−	0.362446i	$-0.881942\pi$
$373$	−36.0000	−1.86401	−0.932005	+	0.362446i	$0.881942\pi$
$374$	8.00000	0.413670
$375$	−1.00000	−0.0516398
$376$	−10.0000	−0.515711
$377$	0	0
$378$	0	0
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	0	0
$381$	−12.0000	−0.614779
$382$	0	0
$383$	−14.0000	−0.715367	−0.357683	−	0.933843i	$-0.616433\pi$
$383$	−14.0000	−0.715367	−0.357683	+	0.933843i	$0.616433\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−18.0000	−0.916176
$387$	−2.00000	−0.101666
$388$	−6.00000	−0.304604
$389$	−24.0000	−1.21685	−0.608424	−	0.793612i	$-0.708198\pi$
$389$	−24.0000	−1.21685	−0.608424	+	0.793612i	$0.708198\pi$
$390$	2.00000	0.101274
$391$	32.0000	1.61831
$392$	0	0
$393$	−12.0000	−0.605320
$394$	18.0000	0.906827
$395$	−16.0000	−0.805047
$396$	2.00000	0.100504
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	−10.0000	−0.501255
$399$	0	0
$400$	1.00000	0.0500000
$401$	−14.0000	−0.699127	−0.349563	−	0.936913i	$-0.613670\pi$
$401$	−14.0000	−0.699127	−0.349563	+	0.936913i	$0.613670\pi$
$402$	2.00000	0.0997509
$403$	−4.00000	−0.199254
$404$	14.0000	0.696526
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	16.0000	0.793091
$408$	4.00000	0.198030
$409$	32.0000	1.58230	0.791149	−	0.611623i	$-0.209483\pi$
$409$	32.0000	1.58230	0.791149	+	0.611623i	$0.209483\pi$
$410$	−2.00000	−0.0987730
$411$	−2.00000	−0.0986527
$412$	20.0000	0.985329
$413$	0	0
$414$	8.00000	0.393179
$415$	16.0000	0.785409
$416$	−2.00000	−0.0980581
$417$	4.00000	0.195881
$418$	0	0
$419$	−36.0000	−1.75872	−0.879358	−	0.476162i	$-0.842028\pi$
$419$	−36.0000	−1.75872	−0.879358	+	0.476162i	$0.842028\pi$
$420$	0	0
$421$	38.0000	1.85201	0.926003	−	0.377515i	$-0.123221\pi$
$421$	38.0000	1.85201	0.926003	+	0.377515i	$0.123221\pi$
$422$	−4.00000	−0.194717
$423$	−10.0000	−0.486217
$424$	−2.00000	−0.0971286
$425$	4.00000	0.194029
$426$	−12.0000	−0.581402
$427$	0	0
$428$	12.0000	0.580042
$429$	−4.00000	−0.193122
$430$	2.00000	0.0964486
$431$	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
$433$	−38.0000	−1.82616	−0.913082	−	0.407777i	$-0.866304\pi$
$433$	−38.0000	−1.82616	−0.913082	+	0.407777i	$0.866304\pi$
$434$	0	0
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	0	0
$438$	−10.0000	−0.477818
$439$	26.0000	1.24091	0.620456	−	0.784241i	$-0.286947\pi$
$439$	26.0000	1.24091	0.620456	+	0.784241i	$0.286947\pi$
$440$	−2.00000	−0.0953463
$441$	0	0
$442$	−8.00000	−0.380521
$443$	28.0000	1.33032	0.665160	−	0.746701i	$-0.268363\pi$
$443$	28.0000	1.33032	0.665160	+	0.746701i	$0.268363\pi$
$444$	8.00000	0.379663
$445$	14.0000	0.663664
$446$	−16.0000	−0.757622
$447$	−16.0000	−0.756774
$448$	0	0
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	1.00000	0.0471405
$451$	4.00000	0.188353
$452$	−14.0000	−0.658505
$453$	0	0
$454$	12.0000	0.563188
$455$	0	0
$456$	0	0
$457$	−42.0000	−1.96468	−0.982339	−	0.187112i	$-0.940087\pi$
$457$	−42.0000	−1.96468	−0.982339	+	0.187112i	$0.940087\pi$
$458$	−10.0000	−0.467269
$459$	4.00000	0.186704
$460$	−8.00000	−0.373002
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	0	0
$465$	−2.00000	−0.0927478
$466$	−14.0000	−0.648537
$467$	−8.00000	−0.370196	−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000	−0.370196	−0.185098	+	0.982720i	$0.559260\pi$
$468$	−2.00000	−0.0924500
$469$	0	0
$470$	10.0000	0.461266
$471$	10.0000	0.460776
$472$	−4.00000	−0.184115
$473$	−4.00000	−0.183920
$474$	16.0000	0.734904
$475$	0	0
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	−8.00000	−0.365911
$479$	−4.00000	−0.182765	−0.0913823	−	0.995816i	$-0.529129\pi$
$479$	−4.00000	−0.182765	−0.0913823	+	0.995816i	$0.529129\pi$
$480$	−1.00000	−0.0456435
$481$	−16.0000	−0.729537
$482$	20.0000	0.910975
$483$	0	0
$484$	−7.00000	−0.318182
$485$	6.00000	0.272446
$486$	1.00000	0.0453609
$487$	28.0000	1.26880	0.634401	−	0.773004i	$-0.281247\pi$
$487$	28.0000	1.26880	0.634401	+	0.773004i	$0.281247\pi$
$488$	10.0000	0.452679
$489$	−10.0000	−0.452216
$490$	0	0
$491$	6.00000	0.270776	0.135388	−	0.990793i	$-0.456772\pi$
$491$	6.00000	0.270776	0.135388	+	0.990793i	$0.456772\pi$
$492$	2.00000	0.0901670
$493$	0	0
$494$	0	0
$495$	−2.00000	−0.0898933
$496$	2.00000	0.0898027
$497$	0	0
$498$	−16.0000	−0.716977
$499$	40.0000	1.79065	0.895323	−	0.445418i	$-0.146945\pi$
$499$	40.0000	1.79065	0.895323	+	0.445418i	$0.146945\pi$
$500$	−1.00000	−0.0447214
$501$	−18.0000	−0.804181
$502$	20.0000	0.892644
$503$	6.00000	0.267527	0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000	0.267527	0.133763	+	0.991013i	$0.457294\pi$
$504$	0	0
$505$	−14.0000	−0.622992
$506$	16.0000	0.711287
$507$	−9.00000	−0.399704
$508$	−12.0000	−0.532414
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	−4.00000	−0.177123
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	−12.0000	−0.529297
$515$	−20.0000	−0.881305
$516$	−2.00000	−0.0880451
$517$	−20.0000	−0.879599
$518$	0	0
$519$	6.00000	0.263371
$520$	2.00000	0.0877058
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	0	0
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	0	0
$527$	8.00000	0.348485
$528$	2.00000	0.0870388
$529$	41.0000	1.78261
$530$	2.00000	0.0868744
$531$	−4.00000	−0.173585
$532$	0	0
$533$	−4.00000	−0.173259
$534$	−14.0000	−0.605839
$535$	−12.0000	−0.518805
$536$	2.00000	0.0863868
$537$	−2.00000	−0.0863064
$538$	10.0000	0.431131
$539$	0	0
$540$	−1.00000	−0.0430331
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	14.0000	0.601351
$543$	22.0000	0.944110
$544$	4.00000	0.171499
$545$	2.00000	0.0856706
$546$	0	0
$547$	−14.0000	−0.598597	−0.299298	−	0.954160i	$-0.596753\pi$
$547$	−14.0000	−0.598597	−0.299298	+	0.954160i	$0.596753\pi$
$548$	−2.00000	−0.0854358
$549$	10.0000	0.426790
$550$	2.00000	0.0852803
$551$	0	0
$552$	8.00000	0.340503
$553$	0	0
$554$	−28.0000	−1.18961
$555$	−8.00000	−0.339581
$556$	4.00000	0.169638
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	2.00000	0.0846668
$559$	4.00000	0.169182
$560$	0	0
$561$	8.00000	0.337760
$562$	14.0000	0.590554
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	−10.0000	−0.421076
$565$	14.0000	0.588984
$566$	−4.00000	−0.168133
$567$	0	0
$568$	−12.0000	−0.503509
$569$	−38.0000	−1.59304	−0.796521	−	0.604610i	$-0.793329\pi$
$569$	−38.0000	−1.59304	−0.796521	+	0.604610i	$0.793329\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	−4.00000	−0.167248
$573$	0	0
$574$	0	0
$575$	8.00000	0.333623
$576$	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	$0.486745\pi$
$578$	−1.00000	−0.0415945
$579$	−18.0000	−0.748054
$580$	0	0
$581$	0	0
$582$	−6.00000	−0.248708
$583$	−4.00000	−0.165663
$584$	−10.0000	−0.413803
$585$	2.00000	0.0826898
$586$	−30.0000	−1.23929
$587$	16.0000	0.660391	0.330195	−	0.943913i	$-0.392885\pi$
$587$	16.0000	0.660391	0.330195	+	0.943913i	$0.392885\pi$
$588$	0	0
$589$	0	0
$590$	4.00000	0.164677
$591$	18.0000	0.740421
$592$	8.00000	0.328798
$593$	20.0000	0.821302	0.410651	−	0.911793i	$-0.365302\pi$
$593$	20.0000	0.821302	0.410651	+	0.911793i	$0.365302\pi$
$594$	2.00000	0.0820610
$595$	0	0
$596$	−16.0000	−0.655386
$597$	−10.0000	−0.409273
$598$	−16.0000	−0.654289
$599$	32.0000	1.30748	0.653742	−	0.756717i	$-0.273198\pi$
$599$	32.0000	1.30748	0.653742	+	0.756717i	$0.273198\pi$
$600$	1.00000	0.0408248
$601$	−4.00000	−0.163163	−0.0815817	−	0.996667i	$-0.525997\pi$
$601$	−4.00000	−0.163163	−0.0815817	+	0.996667i	$0.525997\pi$
$602$	0	0
$603$	2.00000	0.0814463
$604$	0	0
$605$	7.00000	0.284590
$606$	14.0000	0.568711
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	0	0
$609$	0	0
$610$	−10.0000	−0.404888
$611$	20.0000	0.809113
$612$	4.00000	0.161690
$613$	−8.00000	−0.323117	−0.161558	−	0.986863i	$-0.551652\pi$
$613$	−8.00000	−0.323117	−0.161558	+	0.986863i	$0.551652\pi$
$614$	20.0000	0.807134
$615$	−2.00000	−0.0806478
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	20.0000	0.804518
$619$	24.0000	0.964641	0.482321	−	0.875995i	$-0.339794\pi$
$619$	24.0000	0.964641	0.482321	+	0.875995i	$0.339794\pi$
$620$	−2.00000	−0.0803219
$621$	8.00000	0.321029
$622$	20.0000	0.801927
$623$	0	0
$624$	−2.00000	−0.0800641
$625$	1.00000	0.0400000
$626$	26.0000	1.03917
$627$	0	0
$628$	10.0000	0.399043
$629$	32.0000	1.27592
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	16.0000	0.636446
$633$	−4.00000	−0.158986
$634$	6.00000	0.238290
$635$	12.0000	0.476205
$636$	−2.00000	−0.0793052
$637$	0	0
$638$	0	0
$639$	−12.0000	−0.474713
$640$	−1.00000	−0.0395285
$641$	42.0000	1.65890	0.829450	−	0.558581i	$-0.188654\pi$
$641$	42.0000	1.65890	0.829450	+	0.558581i	$0.188654\pi$
$642$	12.0000	0.473602
$643$	36.0000	1.41970	0.709851	−	0.704352i	$-0.248762\pi$
$643$	36.0000	1.41970	0.709851	+	0.704352i	$0.248762\pi$
$644$	0	0
$645$	2.00000	0.0787499
$646$	0	0
$647$	2.00000	0.0786281	0.0393141	−	0.999227i	$-0.487483\pi$
$647$	2.00000	0.0786281	0.0393141	+	0.999227i	$0.487483\pi$
$648$	1.00000	0.0392837
$649$	−8.00000	−0.314027
$650$	−2.00000	−0.0784465
$651$	0	0
$652$	−10.0000	−0.391630
$653$	2.00000	0.0782660	0.0391330	−	0.999234i	$-0.487540\pi$
$653$	2.00000	0.0782660	0.0391330	+	0.999234i	$0.487540\pi$
$654$	−2.00000	−0.0782062
$655$	12.0000	0.468879
$656$	2.00000	0.0780869
$657$	−10.0000	−0.390137
$658$	0	0
$659$	−34.0000	−1.32445	−0.662226	−	0.749304i	$-0.730388\pi$
$659$	−34.0000	−1.32445	−0.662226	+	0.749304i	$0.730388\pi$
$660$	−2.00000	−0.0778499
$661$	26.0000	1.01128	0.505641	−	0.862744i	$-0.331256\pi$
$661$	26.0000	1.01128	0.505641	+	0.862744i	$0.331256\pi$
$662$	−4.00000	−0.155464
$663$	−8.00000	−0.310694
$664$	−16.0000	−0.620920
$665$	0	0
$666$	8.00000	0.309994
$667$	0	0
$668$	−18.0000	−0.696441
$669$	−16.0000	−0.618596
$670$	−2.00000	−0.0772667
$671$	20.0000	0.772091
$672$	0	0
$673$	10.0000	0.385472	0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000	0.385472	0.192736	+	0.981251i	$0.438264\pi$
$674$	−2.00000	−0.0770371
$675$	1.00000	0.0384900
$676$	−9.00000	−0.346154
$677$	−26.0000	−0.999261	−0.499631	−	0.866239i	$-0.666531\pi$
$677$	−26.0000	−0.999261	−0.499631	+	0.866239i	$0.666531\pi$
$678$	−14.0000	−0.537667
$679$	0	0
$680$	−4.00000	−0.153393
$681$	12.0000	0.459841
$682$	4.00000	0.153168
$683$	24.0000	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$684$	0	0
$685$	2.00000	0.0764161
$686$	0	0
$687$	−10.0000	−0.381524
$688$	−2.00000	−0.0762493
$689$	4.00000	0.152388
$690$	−8.00000	−0.304555
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	−4.00000	−0.151838
$695$	−4.00000	−0.151729
$696$	0	0
$697$	8.00000	0.303022
$698$	−10.0000	−0.378506
$699$	−14.0000	−0.529529
$700$	0	0
$701$	16.0000	0.604312	0.302156	−	0.953259i	$-0.402294\pi$
$701$	16.0000	0.604312	0.302156	+	0.953259i	$0.402294\pi$
$702$	−2.00000	−0.0754851
$703$	0	0
$704$	2.00000	0.0753778
$705$	10.0000	0.376622
$706$	−24.0000	−0.903252
$707$	0	0
$708$	−4.00000	−0.150329
$709$	42.0000	1.57734	0.788672	−	0.614815i	$-0.210769\pi$
$709$	42.0000	1.57734	0.788672	+	0.614815i	$0.210769\pi$
$710$	12.0000	0.450352
$711$	16.0000	0.600047
$712$	−14.0000	−0.524672
$713$	16.0000	0.599205
$714$	0	0
$715$	4.00000	0.149592
$716$	−2.00000	−0.0747435
$717$	−8.00000	−0.298765
$718$	−20.0000	−0.746393
$719$	48.0000	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$720$	−1.00000	−0.0372678
$721$	0	0
$722$	−19.0000	−0.707107
$723$	20.0000	0.743808
$724$	22.0000	0.817624
$725$	0	0
$726$	−7.00000	−0.259794
$727$	−32.0000	−1.18681	−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000	−1.18681	−0.593407	+	0.804902i	$0.702218\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	10.0000	0.370117
$731$	−8.00000	−0.295891
$732$	10.0000	0.369611
$733$	−30.0000	−1.10808	−0.554038	−	0.832492i	$-0.686914\pi$
$733$	−30.0000	−1.10808	−0.554038	+	0.832492i	$0.686914\pi$
$734$	28.0000	1.03350
$735$	0	0
$736$	8.00000	0.294884
$737$	4.00000	0.147342
$738$	2.00000	0.0736210
$739$	48.0000	1.76571	0.882854	−	0.469647i	$-0.155619\pi$
$739$	48.0000	1.76571	0.882854	+	0.469647i	$0.155619\pi$
$740$	−8.00000	−0.294086
$741$	0	0
$742$	0	0
$743$	−48.0000	−1.76095	−0.880475	−	0.474093i	$-0.842776\pi$
$743$	−48.0000	−1.76095	−0.880475	+	0.474093i	$0.842776\pi$
$744$	2.00000	0.0733236
$745$	16.0000	0.586195
$746$	−36.0000	−1.31805
$747$	−16.0000	−0.585409
$748$	8.00000	0.292509
$749$	0	0
$750$	−1.00000	−0.0365148
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	−10.0000	−0.364662
$753$	20.0000	0.728841
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−20.0000	−0.726912	−0.363456	−	0.931611i	$-0.618403\pi$
$757$	−20.0000	−0.726912	−0.363456	+	0.931611i	$0.618403\pi$
$758$	8.00000	0.290573
$759$	16.0000	0.580763
$760$	0	0
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	−12.0000	−0.434714
$763$	0	0
$764$	0	0
$765$	−4.00000	−0.144620
$766$	−14.0000	−0.505841
$767$	8.00000	0.288863
$768$	1.00000	0.0360844
$769$	−16.0000	−0.576975	−0.288487	−	0.957484i	$-0.593152\pi$
$769$	−16.0000	−0.576975	−0.288487	+	0.957484i	$0.593152\pi$
$770$	0	0
$771$	−12.0000	−0.432169
$772$	−18.0000	−0.647834
$773$	−30.0000	−1.07903	−0.539513	−	0.841978i	$-0.681391\pi$
$773$	−30.0000	−1.07903	−0.539513	+	0.841978i	$0.681391\pi$
$774$	−2.00000	−0.0718885
$775$	2.00000	0.0718421
$776$	−6.00000	−0.215387
$777$	0	0
$778$	−24.0000	−0.860442
$779$	0	0
$780$	2.00000	0.0716115
$781$	−24.0000	−0.858788
$782$	32.0000	1.14432
$783$	0	0
$784$	0	0
$785$	−10.0000	−0.356915
$786$	−12.0000	−0.428026
$787$	12.0000	0.427754	0.213877	−	0.976861i	$-0.431391\pi$
$787$	12.0000	0.427754	0.213877	+	0.976861i	$0.431391\pi$
$788$	18.0000	0.641223
$789$	0	0
$790$	−16.0000	−0.569254
$791$	0	0
$792$	2.00000	0.0710669
$793$	−20.0000	−0.710221
$794$	−14.0000	−0.496841
$795$	2.00000	0.0709327
$796$	−10.0000	−0.354441
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	0	0
$799$	−40.0000	−1.41510
$800$	1.00000	0.0353553
$801$	−14.0000	−0.494666
$802$	−14.0000	−0.494357
$803$	−20.0000	−0.705785
$804$	2.00000	0.0705346
$805$	0	0
$806$	−4.00000	−0.140894
$807$	10.0000	0.352017
$808$	14.0000	0.492518
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	−1.00000	−0.0351364
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	0	0
$813$	14.0000	0.491001
$814$	16.0000	0.560800
$815$	10.0000	0.350285
$816$	4.00000	0.140028
$817$	0	0
$818$	32.0000	1.11885
$819$	0	0
$820$	−2.00000	−0.0698430
$821$	24.0000	0.837606	0.418803	−	0.908077i	$-0.362450\pi$
$821$	24.0000	0.837606	0.418803	+	0.908077i	$0.362450\pi$
$822$	−2.00000	−0.0697580
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	20.0000	0.696733
$825$	2.00000	0.0696311
$826$	0	0
$827$	8.00000	0.278187	0.139094	−	0.990279i	$-0.455581\pi$
$827$	8.00000	0.278187	0.139094	+	0.990279i	$0.455581\pi$
$828$	8.00000	0.278019
$829$	10.0000	0.347314	0.173657	−	0.984806i	$-0.444442\pi$
$829$	10.0000	0.347314	0.173657	+	0.984806i	$0.444442\pi$
$830$	16.0000	0.555368
$831$	−28.0000	−0.971309
$832$	−2.00000	−0.0693375
$833$	0	0
$834$	4.00000	0.138509
$835$	18.0000	0.622916
$836$	0	0
$837$	2.00000	0.0691301
$838$	−36.0000	−1.24360
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	38.0000	1.30957
$843$	14.0000	0.482186
$844$	−4.00000	−0.137686
$845$	9.00000	0.309609
$846$	−10.0000	−0.343807
$847$	0	0
$848$	−2.00000	−0.0686803
$849$	−4.00000	−0.137280
$850$	4.00000	0.137199
$851$	64.0000	2.19389
$852$	−12.0000	−0.411113
$853$	−38.0000	−1.30110	−0.650548	−	0.759465i	$-0.725461\pi$
$853$	−38.0000	−1.30110	−0.650548	+	0.759465i	$0.725461\pi$
$854$	0	0
$855$	0	0
$856$	12.0000	0.410152
$857$	12.0000	0.409912	0.204956	−	0.978771i	$-0.434295\pi$
$857$	12.0000	0.409912	0.204956	+	0.978771i	$0.434295\pi$
$858$	−4.00000	−0.136558
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	2.00000	0.0681994
$861$	0	0
$862$	12.0000	0.408722
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	1.00000	0.0340207
$865$	−6.00000	−0.204006
$866$	−38.0000	−1.29129
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	32.0000	1.08553
$870$	0	0
$871$	−4.00000	−0.135535
$872$	−2.00000	−0.0677285
$873$	−6.00000	−0.203069
$874$	0	0
$875$	0	0
$876$	−10.0000	−0.337869
$877$	12.0000	0.405211	0.202606	−	0.979260i	$-0.435059\pi$
$877$	12.0000	0.405211	0.202606	+	0.979260i	$0.435059\pi$
$878$	26.0000	0.877457
$879$	−30.0000	−1.01187
$880$	−2.00000	−0.0674200
$881$	46.0000	1.54978	0.774890	−	0.632096i	$-0.217805\pi$
$881$	46.0000	1.54978	0.774890	+	0.632096i	$0.217805\pi$
$882$	0	0
$883$	34.0000	1.14419	0.572096	−	0.820187i	$-0.306131\pi$
$883$	34.0000	1.14419	0.572096	+	0.820187i	$0.306131\pi$
$884$	−8.00000	−0.269069
$885$	4.00000	0.134459
$886$	28.0000	0.940678
$887$	2.00000	0.0671534	0.0335767	−	0.999436i	$-0.489310\pi$
$887$	2.00000	0.0671534	0.0335767	+	0.999436i	$0.489310\pi$
$888$	8.00000	0.268462
$889$	0	0
$890$	14.0000	0.469281
$891$	2.00000	0.0670025
$892$	−16.0000	−0.535720
$893$	0	0
$894$	−16.0000	−0.535120
$895$	2.00000	0.0668526
$896$	0	0
$897$	−16.0000	−0.534224
$898$	6.00000	0.200223
$899$	0	0
$900$	1.00000	0.0333333
$901$	−8.00000	−0.266519
$902$	4.00000	0.133185
$903$	0	0
$904$	−14.0000	−0.465633
$905$	−22.0000	−0.731305
$906$	0	0
$907$	10.0000	0.332045	0.166022	−	0.986122i	$-0.446908\pi$
$907$	10.0000	0.332045	0.166022	+	0.986122i	$0.446908\pi$
$908$	12.0000	0.398234
$909$	14.0000	0.464351
$910$	0	0
$911$	16.0000	0.530104	0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000	0.530104	0.265052	+	0.964234i	$0.414611\pi$
$912$	0	0
$913$	−32.0000	−1.05905
$914$	−42.0000	−1.38924
$915$	−10.0000	−0.330590
$916$	−10.0000	−0.330409
$917$	0	0
$918$	4.00000	0.132020
$919$	56.0000	1.84727	0.923635	−	0.383274i	$-0.125203\pi$
$919$	56.0000	1.84727	0.923635	+	0.383274i	$0.125203\pi$
$920$	−8.00000	−0.263752
$921$	20.0000	0.659022
$922$	18.0000	0.592798
$923$	24.0000	0.789970
$924$	0	0
$925$	8.00000	0.263038
$926$	16.0000	0.525793
$927$	20.0000	0.656886
$928$	0	0
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	−2.00000	−0.0655826
$931$	0	0
$932$	−14.0000	−0.458585
$933$	20.0000	0.654771
$934$	−8.00000	−0.261768
$935$	−8.00000	−0.261628
$936$	−2.00000	−0.0653720
$937$	−42.0000	−1.37208	−0.686040	−	0.727564i	$-0.740653\pi$
$937$	−42.0000	−1.37208	−0.686040	+	0.727564i	$0.740653\pi$
$938$	0	0
$939$	26.0000	0.848478
$940$	10.0000	0.326164
$941$	−50.0000	−1.62995	−0.814977	−	0.579494i	$-0.803250\pi$
$941$	−50.0000	−1.62995	−0.814977	+	0.579494i	$0.803250\pi$
$942$	10.0000	0.325818
$943$	16.0000	0.521032
$944$	−4.00000	−0.130189
$945$	0	0
$946$	−4.00000	−0.130051
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	16.0000	0.519656
$949$	20.0000	0.649227
$950$	0	0
$951$	6.00000	0.194563
$952$	0	0
$953$	−58.0000	−1.87880	−0.939402	−	0.342817i	$-0.888619\pi$
$953$	−58.0000	−1.87880	−0.939402	+	0.342817i	$0.888619\pi$
$954$	−2.00000	−0.0647524
$955$	0	0
$956$	−8.00000	−0.258738
$957$	0	0
$958$	−4.00000	−0.129234
$959$	0	0
$960$	−1.00000	−0.0322749
$961$	−27.0000	−0.870968
$962$	−16.0000	−0.515861
$963$	12.0000	0.386695
$964$	20.0000	0.644157
$965$	18.0000	0.579441
$966$	0	0
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	−7.00000	−0.224989
$969$	0	0
$970$	6.00000	0.192648
$971$	20.0000	0.641831	0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000	0.641831	0.320915	+	0.947108i	$0.396010\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	28.0000	0.897178
$975$	−2.00000	−0.0640513
$976$	10.0000	0.320092
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	−10.0000	−0.319765
$979$	−28.0000	−0.894884
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	6.00000	0.191468
$983$	46.0000	1.46717	0.733586	−	0.679597i	$-0.237845\pi$
$983$	46.0000	1.46717	0.733586	+	0.679597i	$0.237845\pi$
$984$	2.00000	0.0637577
$985$	−18.0000	−0.573528
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−16.0000	−0.508770
$990$	−2.00000	−0.0635642
$991$	−24.0000	−0.762385	−0.381193	−	0.924496i	$-0.624487\pi$
$991$	−24.0000	−0.762385	−0.381193	+	0.924496i	$0.624487\pi$
$992$	2.00000	0.0635001
$993$	−4.00000	−0.126936
$994$	0	0
$995$	10.0000	0.317021
$996$	−16.0000	−0.506979
$997$	−6.00000	−0.190022	−0.0950110	−	0.995476i	$-0.530289\pi$
$997$	−6.00000	−0.190022	−0.0950110	+	0.995476i	$0.530289\pi$
$998$	40.0000	1.26618
$999$	8.00000	0.253109

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1470.2.a.p.1.1	yes	1
3.2	odd	2		4410.2.a.n.1.1		1
5.4	even	2		7350.2.a.o.1.1		1
7.2	even	3		1470.2.i.c.361.1		2
7.3	odd	6		1470.2.i.g.961.1		2
7.4	even	3		1470.2.i.c.961.1		2
7.5	odd	6		1470.2.i.g.361.1		2
7.6	odd	2		1470.2.a.n.1.1	✓	1
21.20	even	2		4410.2.a.e.1.1		1
35.34	odd	2		7350.2.a.bh.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1470.2.a.n.1.1	✓	1	7.6	odd	2
1470.2.a.p.1.1	yes	1	1.1	even	1	trivial
1470.2.i.c.361.1		2	7.2	even	3
1470.2.i.c.961.1		2	7.4	even	3
1470.2.i.g.361.1		2	7.5	odd	6
1470.2.i.g.961.1		2	7.3	odd	6
4410.2.a.e.1.1		1	21.20	even	2
4410.2.a.n.1.1		1	3.2	odd	2
7350.2.a.o.1.1		1	5.4	even	2
7350.2.a.bh.1.1		1	35.34	odd	2

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1470.2.a.p.1.1

Level	$1470$
Weight	$2$
Character	1470.1
Self dual	yes
Analytic conductor	$11.738$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$