LMFDB - Embedded newform 4046.2.a.f.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4046,2,Mod(1,4046)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4046, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4046.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$4046 = 2 \cdot 7 \cdot 17^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4046.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-1,2,1,0,-2,-1,-1,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$32.3074726578$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 14)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4046.1

$q$ -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} +2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{12} -4.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -1.00000 q^{18} +2.00000 q^{19} -2.00000 q^{21} -2.00000 q^{24} -5.00000 q^{25} +4.00000 q^{26} -4.00000 q^{27} -1.00000 q^{28} +6.00000 q^{29} +4.00000 q^{31} -1.00000 q^{32} +1.00000 q^{36} -2.00000 q^{37} -2.00000 q^{38} -8.00000 q^{39} -6.00000 q^{41} +2.00000 q^{42} +8.00000 q^{43} -12.0000 q^{47} +2.00000 q^{48} +1.00000 q^{49} +5.00000 q^{50} -4.00000 q^{52} +6.00000 q^{53} +4.00000 q^{54} +1.00000 q^{56} +4.00000 q^{57} -6.00000 q^{58} -6.00000 q^{59} -8.00000 q^{61} -4.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -4.00000 q^{67} -1.00000 q^{72} -2.00000 q^{73} +2.00000 q^{74} -10.0000 q^{75} +2.00000 q^{76} +8.00000 q^{78} -8.00000 q^{79} -11.0000 q^{81} +6.00000 q^{82} -6.00000 q^{83} -2.00000 q^{84} -8.00000 q^{86} +12.0000 q^{87} -6.00000 q^{89} +4.00000 q^{91} +8.00000 q^{93} +12.0000 q^{94} -2.00000 q^{96} +10.0000 q^{97} -1.00000 q^{98} +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
$3$	2.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	1.00000	0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	−2.00000	−0.816497
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	2.00000	0.577350
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	−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	$0.426318\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−2.00000	−0.408248
$25$	−5.00000	−1.00000
$26$	4.00000	0.784465
$27$	−4.00000	−0.769800
$28$	−1.00000	−0.188982
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	0	0
$35$	0	0
$36$	1.00000	0.166667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−2.00000	−0.324443
$39$	−8.00000	−1.28103
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	2.00000	0.308607
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	2.00000	0.288675
$49$	1.00000	0.142857
$50$	5.00000	0.707107
$51$	0	0
$52$	−4.00000	−0.554700
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	4.00000	0.544331
$55$	0	0
$56$	1.00000	0.133631
$57$	4.00000	0.529813
$58$	−6.00000	−0.787839
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	−4.00000	−0.508001
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−1.00000	−0.117851
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	2.00000	0.232495
$75$	−10.0000	−1.15470
$76$	2.00000	0.229416
$77$	0	0
$78$	8.00000	0.905822
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	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
$85$	0	0
$86$	−8.00000	−0.862662
$87$	12.0000	1.28654
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	0	0
$93$	8.00000	0.829561
$94$	12.0000	1.23771
$95$	0	0
$96$	−2.00000	−0.204124
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	−5.00000	−0.500000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	4.00000	0.392232
$105$	0	0
$106$	−6.00000	−0.582772
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	−4.00000	−0.384900
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	−1.00000	−0.0944911
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	6.00000	0.557086
$117$	−4.00000	−0.369800
$118$	6.00000	0.552345
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	8.00000	0.724286
$123$	−12.0000	−1.08200
$124$	4.00000	0.359211
$125$	0	0
$126$	1.00000	0.0890871
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	−1.00000	−0.0883883
$129$	16.0000	1.40872
$130$	0	0
$131$	−18.0000	−1.57267	−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000	−1.57267	−0.786334	+	0.617802i	$0.788023\pi$
$132$	0	0
$133$	−2.00000	−0.173422
$134$	4.00000	0.345547
$135$	0	0
$136$	0	0
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	0	0
$141$	−24.0000	−2.02116
$142$	0	0
$143$	0	0
$144$	1.00000	0.0833333
$145$	0	0
$146$	2.00000	0.165521
$147$	2.00000	0.164957
$148$	−2.00000	−0.164399
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	10.0000	0.816497
$151$	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
$153$	0	0
$154$	0	0
$155$	0	0
$156$	−8.00000	−0.640513
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	8.00000	0.636446
$159$	12.0000	0.951662
$160$	0	0
$161$	0	0
$162$	11.0000	0.864242
$163$	16.0000	1.25322	0.626608	−	0.779334i	$-0.284443\pi$
$163$	16.0000	1.25322	0.626608	+	0.779334i	$0.284443\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	6.00000	0.465690
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	2.00000	0.154303
$169$	3.00000	0.230769
$170$	0	0
$171$	2.00000	0.152944
$172$	8.00000	0.609994
$173$	12.0000	0.912343	0.456172	−	0.889892i	$-0.349220\pi$
$173$	12.0000	0.912343	0.456172	+	0.889892i	$0.349220\pi$
$174$	−12.0000	−0.909718
$175$	5.00000	0.377964
$176$	0	0
$177$	−12.0000	−0.901975
$178$	6.00000	0.449719
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−20.0000	−1.48659	−0.743294	−	0.668965i	$-0.766738\pi$
$181$	−20.0000	−1.48659	−0.743294	+	0.668965i	$0.766738\pi$
$182$	−4.00000	−0.296500
$183$	−16.0000	−1.18275
$184$	0	0
$185$	0	0
$186$	−8.00000	−0.586588
$187$	0	0
$188$	−12.0000	−0.875190
$189$	4.00000	0.290957
$190$	0	0
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	2.00000	0.144338
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	1.00000	0.0714286
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	5.00000	0.353553
$201$	−8.00000	−0.564276
$202$	0	0
$203$	−6.00000	−0.421117
$204$	0	0
$205$	0	0
$206$	4.00000	0.278693
$207$	0	0
$208$	−4.00000	−0.277350
$209$	0	0
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	12.0000	0.820303
$215$	0	0
$216$	4.00000	0.272166
$217$	−4.00000	−0.271538
$218$	2.00000	0.135457
$219$	−4.00000	−0.270295
$220$	0	0
$221$	0	0
$222$	4.00000	0.268462
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	1.00000	0.0668153
$225$	−5.00000	−0.333333
$226$	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	$0.703780\pi$
$228$	4.00000	0.264906
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	0	0
$231$	0	0
$232$	−6.00000	−0.393919
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	4.00000	0.261488
$235$	0	0
$236$	−6.00000	−0.390567
$237$	−16.0000	−1.03931
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	11.0000	0.707107
$243$	−10.0000	−0.641500
$244$	−8.00000	−0.512148
$245$	0	0
$246$	12.0000	0.765092
$247$	−8.00000	−0.509028
$248$	−4.00000	−0.254000
$249$	−12.0000	−0.760469
$250$	0	0
$251$	−18.0000	−1.13615	−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000	−1.13615	−0.568075	+	0.822977i	$0.692312\pi$
$252$	−1.00000	−0.0629941
$253$	0	0
$254$	16.0000	1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	−16.0000	−0.996116
$259$	2.00000	0.124274
$260$	0	0
$261$	6.00000	0.371391
$262$	18.0000	1.11204
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	2.00000	0.122628
$267$	−12.0000	−0.734388
$268$	−4.00000	−0.244339
$269$	12.0000	0.731653	0.365826	−	0.930683i	$-0.380786\pi$
$269$	12.0000	0.731653	0.365826	+	0.930683i	$0.380786\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	0	0
$273$	8.00000	0.484182
$274$	−18.0000	−1.08742
$275$	0	0
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	14.0000	0.839664
$279$	4.00000	0.239474
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	24.0000	1.42918
$283$	22.0000	1.30776	0.653882	−	0.756596i	$-0.273139\pi$
$283$	22.0000	1.30776	0.653882	+	0.756596i	$0.273139\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6.00000	0.354169
$288$	−1.00000	−0.0589256
$289$	0	0
$290$	0	0
$291$	20.0000	1.17242
$292$	−2.00000	−0.117041
$293$	24.0000	1.40209	0.701047	−	0.713115i	$-0.252716\pi$
$293$	24.0000	1.40209	0.701047	+	0.713115i	$0.252716\pi$
$294$	−2.00000	−0.116642
$295$	0	0
$296$	2.00000	0.116248
$297$	0	0
$298$	18.0000	1.04271
$299$	0	0
$300$	−10.0000	−0.577350
$301$	−8.00000	−0.461112
$302$	−8.00000	−0.460348
$303$	0	0
$304$	2.00000	0.114708
$305$	0	0
$306$	0	0
$307$	2.00000	0.114146	0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000	0.114146	0.0570730	+	0.998370i	$0.481823\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	8.00000	0.452911
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	4.00000	0.225733
$315$	0	0
$316$	−8.00000	−0.450035
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	−12.0000	−0.672927
$319$	0	0
$320$	0	0
$321$	−24.0000	−1.33955
$322$	0	0
$323$	0	0
$324$	−11.0000	−0.611111
$325$	20.0000	1.10940
$326$	−16.0000	−0.886158
$327$	−4.00000	−0.221201
$328$	6.00000	0.331295
$329$	12.0000	0.661581
$330$	0	0
$331$	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
$333$	−2.00000	−0.109599
$334$	−12.0000	−0.656611
$335$	0	0
$336$	−2.00000	−0.109109
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	−3.00000	−0.163178
$339$	−12.0000	−0.651751
$340$	0	0
$341$	0	0
$342$	−2.00000	−0.108148
$343$	−1.00000	−0.0539949
$344$	−8.00000	−0.431331
$345$	0	0
$346$	−12.0000	−0.645124
$347$	24.0000	1.28839	0.644194	−	0.764862i	$-0.277193\pi$
$347$	24.0000	1.28839	0.644194	+	0.764862i	$0.277193\pi$
$348$	12.0000	0.643268
$349$	−28.0000	−1.49881	−0.749403	−	0.662114i	$-0.769659\pi$
$349$	−28.0000	−1.49881	−0.749403	+	0.662114i	$0.769659\pi$
$350$	−5.00000	−0.267261
$351$	16.0000	0.854017
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	12.0000	0.637793
$355$	0	0
$356$	−6.00000	−0.317999
$357$	0	0
$358$	12.0000	0.634220
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	20.0000	1.05118
$363$	−22.0000	−1.15470
$364$	4.00000	0.209657
$365$	0	0
$366$	16.0000	0.836333
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	−6.00000	−0.311504
$372$	8.00000	0.414781
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	0	0
$375$	0	0
$376$	12.0000	0.618853
$377$	−24.0000	−1.23606
$378$	−4.00000	−0.205738
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	0	0
$381$	−32.0000	−1.63941
$382$	−24.0000	−1.22795
$383$	36.0000	1.83951	0.919757	−	0.392488i	$-0.128386\pi$
$383$	36.0000	1.83951	0.919757	+	0.392488i	$0.128386\pi$
$384$	−2.00000	−0.102062
$385$	0	0
$386$	14.0000	0.712581
$387$	8.00000	0.406663
$388$	10.0000	0.507673
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	0	0
$392$	−1.00000	−0.0505076
$393$	−36.0000	−1.81596
$394$	−18.0000	−0.906827
$395$	0	0
$396$	0	0
$397$	−20.0000	−1.00377	−0.501886	−	0.864934i	$-0.667360\pi$
$397$	−20.0000	−1.00377	−0.501886	+	0.864934i	$0.667360\pi$
$398$	20.0000	1.00251
$399$	−4.00000	−0.200250
$400$	−5.00000	−0.250000
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	8.00000	0.399004
$403$	−16.0000	−0.797017
$404$	0	0
$405$	0	0
$406$	6.00000	0.297775
$407$	0	0
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	36.0000	1.77575
$412$	−4.00000	−0.197066
$413$	6.00000	0.295241
$414$	0	0
$415$	0	0
$416$	4.00000	0.196116
$417$	−28.0000	−1.37117
$418$	0	0
$419$	−6.00000	−0.293119	−0.146560	−	0.989202i	$-0.546820\pi$
$419$	−6.00000	−0.293119	−0.146560	+	0.989202i	$0.546820\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	−4.00000	−0.194717
$423$	−12.0000	−0.583460
$424$	−6.00000	−0.291386
$425$	0	0
$426$	0	0
$427$	8.00000	0.387147
$428$	−12.0000	−0.580042
$429$	0	0
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	−4.00000	−0.192450
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	4.00000	0.192006
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	0	0
$438$	4.00000	0.191127
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	−4.00000	−0.189832
$445$	0	0
$446$	−8.00000	−0.378811
$447$	−36.0000	−1.70274
$448$	−1.00000	−0.0472456
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	5.00000	0.235702
$451$	0	0
$452$	−6.00000	−0.282216
$453$	16.0000	0.751746
$454$	18.0000	0.844782
$455$	0	0
$456$	−4.00000	−0.187317
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	4.00000	0.186908
$459$	0	0
$460$	0	0
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	0	0
$463$	32.0000	1.48717	0.743583	−	0.668644i	$-0.233125\pi$
$463$	32.0000	1.48717	0.743583	+	0.668644i	$0.233125\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	−6.00000	−0.277945
$467$	−6.00000	−0.277647	−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000	−0.277647	−0.138823	+	0.990317i	$0.544332\pi$
$468$	−4.00000	−0.184900
$469$	4.00000	0.184703
$470$	0	0
$471$	−8.00000	−0.368621
$472$	6.00000	0.276172
$473$	0	0
$474$	16.0000	0.734904
$475$	−10.0000	−0.458831
$476$	0	0
$477$	6.00000	0.274721
$478$	−24.0000	−1.09773
$479$	36.0000	1.64488	0.822441	−	0.568850i	$-0.192612\pi$
$479$	36.0000	1.64488	0.822441	+	0.568850i	$0.192612\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	−10.0000	−0.455488
$483$	0	0
$484$	−11.0000	−0.500000
$485$	0	0
$486$	10.0000	0.453609
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	8.00000	0.362143
$489$	32.0000	1.44709
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	−12.0000	−0.541002
$493$	0	0
$494$	8.00000	0.359937
$495$	0	0
$496$	4.00000	0.179605
$497$	0	0
$498$	12.0000	0.537733
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	24.0000	1.07224
$502$	18.0000	0.803379
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	1.00000	0.0445435
$505$	0	0
$506$	0	0
$507$	6.00000	0.266469
$508$	−16.0000	−0.709885
$509$	36.0000	1.59567	0.797836	−	0.602875i	$-0.205978\pi$
$509$	36.0000	1.59567	0.797836	+	0.602875i	$0.205978\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	−1.00000	−0.0441942
$513$	−8.00000	−0.353209
$514$	−18.0000	−0.793946
$515$	0	0
$516$	16.0000	0.704361
$517$	0	0
$518$	−2.00000	−0.0878750
$519$	24.0000	1.05348
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	−6.00000	−0.262613
$523$	2.00000	0.0874539	0.0437269	−	0.999044i	$-0.486077\pi$
$523$	2.00000	0.0874539	0.0437269	+	0.999044i	$0.486077\pi$
$524$	−18.0000	−0.786334
$525$	10.0000	0.436436
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	−6.00000	−0.260378
$532$	−2.00000	−0.0867110
$533$	24.0000	1.03956
$534$	12.0000	0.519291
$535$	0	0
$536$	4.00000	0.172774
$537$	−24.0000	−1.03568
$538$	−12.0000	−0.517357
$539$	0	0
$540$	0	0
$541$	−38.0000	−1.63375	−0.816874	−	0.576816i	$-0.804295\pi$
$541$	−38.0000	−1.63375	−0.816874	+	0.576816i	$0.804295\pi$
$542$	16.0000	0.687259
$543$	−40.0000	−1.71656
$544$	0	0
$545$	0	0
$546$	−8.00000	−0.342368
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	18.0000	0.768922
$549$	−8.00000	−0.341432
$550$	0	0
$551$	12.0000	0.511217
$552$	0	0
$553$	8.00000	0.340195
$554$	−10.0000	−0.424859
$555$	0	0
$556$	−14.0000	−0.593732
$557$	6.00000	0.254228	0.127114	−	0.991888i	$-0.459429\pi$
$557$	6.00000	0.254228	0.127114	+	0.991888i	$0.459429\pi$
$558$	−4.00000	−0.169334
$559$	−32.0000	−1.35346
$560$	0	0
$561$	0	0
$562$	6.00000	0.253095
$563$	30.0000	1.26435	0.632175	−	0.774826i	$-0.282163\pi$
$563$	30.0000	1.26435	0.632175	+	0.774826i	$0.282163\pi$
$564$	−24.0000	−1.01058
$565$	0	0
$566$	−22.0000	−0.924729
$567$	11.0000	0.461957
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	−32.0000	−1.33916	−0.669579	−	0.742741i	$-0.733526\pi$
$571$	−32.0000	−1.33916	−0.669579	+	0.742741i	$0.733526\pi$
$572$	0	0
$573$	48.0000	2.00523
$574$	−6.00000	−0.250435
$575$	0	0
$576$	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	$0.486745\pi$
$578$	0	0
$579$	−28.0000	−1.16364
$580$	0	0
$581$	6.00000	0.248922
$582$	−20.0000	−0.829027
$583$	0	0
$584$	2.00000	0.0827606
$585$	0	0
$586$	−24.0000	−0.991431
$587$	−42.0000	−1.73353	−0.866763	−	0.498721i	$-0.833803\pi$
$587$	−42.0000	−1.73353	−0.866763	+	0.498721i	$0.833803\pi$
$588$	2.00000	0.0824786
$589$	8.00000	0.329634
$590$	0	0
$591$	36.0000	1.48084
$592$	−2.00000	−0.0821995
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	0	0
$595$	0	0
$596$	−18.0000	−0.737309
$597$	−40.0000	−1.63709
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	10.0000	0.408248
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	8.00000	0.326056
$603$	−4.00000	−0.162893
$604$	8.00000	0.325515
$605$	0	0
$606$	0	0
$607$	−32.0000	−1.29884	−0.649420	−	0.760430i	$-0.724988\pi$
$607$	−32.0000	−1.29884	−0.649420	+	0.760430i	$0.724988\pi$
$608$	−2.00000	−0.0811107
$609$	−12.0000	−0.486265
$610$	0	0
$611$	48.0000	1.94187
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	−2.00000	−0.0807134
$615$	0	0
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	8.00000	0.321807
$619$	−26.0000	−1.04503	−0.522514	−	0.852631i	$-0.675006\pi$
$619$	−26.0000	−1.04503	−0.522514	+	0.852631i	$0.675006\pi$
$620$	0	0
$621$	0	0
$622$	−24.0000	−0.962312
$623$	6.00000	0.240385
$624$	−8.00000	−0.320256
$625$	25.0000	1.00000
$626$	−10.0000	−0.399680
$627$	0	0
$628$	−4.00000	−0.159617
$629$	0	0
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	8.00000	0.318223
$633$	8.00000	0.317971
$634$	6.00000	0.238290
$635$	0	0
$636$	12.0000	0.475831
$637$	−4.00000	−0.158486
$638$	0	0
$639$	0	0
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	24.0000	0.947204
$643$	−14.0000	−0.552106	−0.276053	−	0.961142i	$-0.589027\pi$
$643$	−14.0000	−0.552106	−0.276053	+	0.961142i	$0.589027\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	11.0000	0.432121
$649$	0	0
$650$	−20.0000	−0.784465
$651$	−8.00000	−0.313545
$652$	16.0000	0.626608
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	4.00000	0.156412
$655$	0	0
$656$	−6.00000	−0.234261
$657$	−2.00000	−0.0780274
$658$	−12.0000	−0.467809
$659$	−24.0000	−0.934907	−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000	−0.934907	−0.467454	+	0.884018i	$0.654829\pi$
$660$	0	0
$661$	−40.0000	−1.55582	−0.777910	−	0.628376i	$-0.783720\pi$
$661$	−40.0000	−1.55582	−0.777910	+	0.628376i	$0.783720\pi$
$662$	−8.00000	−0.310929
$663$	0	0
$664$	6.00000	0.232845
$665$	0	0
$666$	2.00000	0.0774984
$667$	0	0
$668$	12.0000	0.464294
$669$	16.0000	0.618596
$670$	0	0
$671$	0	0
$672$	2.00000	0.0771517
$673$	−26.0000	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$674$	14.0000	0.539260
$675$	20.0000	0.769800
$676$	3.00000	0.115385
$677$	12.0000	0.461197	0.230599	−	0.973049i	$-0.425932\pi$
$677$	12.0000	0.461197	0.230599	+	0.973049i	$0.425932\pi$
$678$	12.0000	0.460857
$679$	−10.0000	−0.383765
$680$	0	0
$681$	−36.0000	−1.37952
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	2.00000	0.0764719
$685$	0	0
$686$	1.00000	0.0381802
$687$	−8.00000	−0.305219
$688$	8.00000	0.304997
$689$	−24.0000	−0.914327
$690$	0	0
$691$	46.0000	1.74992	0.874961	−	0.484193i	$-0.160887\pi$
$691$	46.0000	1.74992	0.874961	+	0.484193i	$0.160887\pi$
$692$	12.0000	0.456172
$693$	0	0
$694$	−24.0000	−0.911028
$695$	0	0
$696$	−12.0000	−0.454859
$697$	0	0
$698$	28.0000	1.05982
$699$	12.0000	0.453882
$700$	5.00000	0.188982
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	−16.0000	−0.603881
$703$	−4.00000	−0.150863
$704$	0	0
$705$	0	0
$706$	−18.0000	−0.677439
$707$	0	0
$708$	−12.0000	−0.450988
$709$	46.0000	1.72757	0.863783	−	0.503864i	$-0.168089\pi$
$709$	46.0000	1.72757	0.863783	+	0.503864i	$0.168089\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	6.00000	0.224860
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	48.0000	1.79259
$718$	24.0000	0.895672
$719$	−12.0000	−0.447524	−0.223762	−	0.974644i	$-0.571834\pi$
$719$	−12.0000	−0.447524	−0.223762	+	0.974644i	$0.571834\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	15.0000	0.558242
$723$	20.0000	0.743808
$724$	−20.0000	−0.743294
$725$	−30.0000	−1.11417
$726$	22.0000	0.816497
$727$	44.0000	1.63187	0.815935	−	0.578144i	$-0.196223\pi$
$727$	44.0000	1.63187	0.815935	+	0.578144i	$0.196223\pi$
$728$	−4.00000	−0.148250
$729$	13.0000	0.481481
$730$	0	0
$731$	0	0
$732$	−16.0000	−0.591377
$733$	−40.0000	−1.47743	−0.738717	−	0.674016i	$-0.764568\pi$
$733$	−40.0000	−1.47743	−0.738717	+	0.674016i	$0.764568\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	0	0
$737$	0	0
$738$	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	$0.595081\pi$
$740$	0	0
$741$	−16.0000	−0.587775
$742$	6.00000	0.220267
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	−8.00000	−0.293294
$745$	0	0
$746$	−14.0000	−0.512576
$747$	−6.00000	−0.219529
$748$	0	0
$749$	12.0000	0.438470
$750$	0	0
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	−12.0000	−0.437595
$753$	−36.0000	−1.31191
$754$	24.0000	0.874028
$755$	0	0
$756$	4.00000	0.145479
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	−16.0000	−0.581146
$759$	0	0
$760$	0	0
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	32.0000	1.15924
$763$	2.00000	0.0724049
$764$	24.0000	0.868290
$765$	0	0
$766$	−36.0000	−1.30073
$767$	24.0000	0.866590
$768$	2.00000	0.0721688
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	36.0000	1.29651
$772$	−14.0000	−0.503871
$773$	24.0000	0.863220	0.431610	−	0.902060i	$-0.357946\pi$
$773$	24.0000	0.863220	0.431610	+	0.902060i	$0.357946\pi$
$774$	−8.00000	−0.287554
$775$	−20.0000	−0.718421
$776$	−10.0000	−0.358979
$777$	4.00000	0.143499
$778$	−18.0000	−0.645331
$779$	−12.0000	−0.429945
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−24.0000	−0.857690
$784$	1.00000	0.0357143
$785$	0	0
$786$	36.0000	1.28408
$787$	22.0000	0.784215	0.392108	−	0.919919i	$-0.371746\pi$
$787$	22.0000	0.784215	0.392108	+	0.919919i	$0.371746\pi$
$788$	18.0000	0.641223
$789$	0	0
$790$	0	0
$791$	6.00000	0.213335
$792$	0	0
$793$	32.0000	1.13635
$794$	20.0000	0.709773
$795$	0	0
$796$	−20.0000	−0.708881
$797$	−12.0000	−0.425062	−0.212531	−	0.977154i	$-0.568171\pi$
$797$	−12.0000	−0.425062	−0.212531	+	0.977154i	$0.568171\pi$
$798$	4.00000	0.141598
$799$	0	0
$800$	5.00000	0.176777
$801$	−6.00000	−0.212000
$802$	−18.0000	−0.635602
$803$	0	0
$804$	−8.00000	−0.282138
$805$	0	0
$806$	16.0000	0.563576
$807$	24.0000	0.844840
$808$	0	0
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	−2.00000	−0.0702295	−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000	−0.0702295	−0.0351147	+	0.999383i	$0.511180\pi$
$812$	−6.00000	−0.210559
$813$	−32.0000	−1.12229
$814$	0	0
$815$	0	0
$816$	0	0
$817$	16.0000	0.559769
$818$	−14.0000	−0.489499
$819$	4.00000	0.139771
$820$	0	0
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	−36.0000	−1.25564
$823$	40.0000	1.39431	0.697156	−	0.716919i	$-0.254448\pi$
$823$	40.0000	1.39431	0.697156	+	0.716919i	$0.254448\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	−6.00000	−0.208767
$827$	36.0000	1.25184	0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000	1.25184	0.625921	+	0.779886i	$0.284723\pi$
$828$	0	0
$829$	56.0000	1.94496	0.972480	−	0.232986i	$-0.0748495\pi$
$829$	56.0000	1.94496	0.972480	+	0.232986i	$0.0748495\pi$
$830$	0	0
$831$	20.0000	0.693792
$832$	−4.00000	−0.138675
$833$	0	0
$834$	28.0000	0.969561
$835$	0	0
$836$	0	0
$837$	−16.0000	−0.553041
$838$	6.00000	0.207267
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	10.0000	0.344623
$843$	−12.0000	−0.413302
$844$	4.00000	0.137686
$845$	0	0
$846$	12.0000	0.412568
$847$	11.0000	0.377964
$848$	6.00000	0.206041
$849$	44.0000	1.51008
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−44.0000	−1.50653	−0.753266	−	0.657716i	$-0.771523\pi$
$853$	−44.0000	−1.50653	−0.753266	+	0.657716i	$0.771523\pi$
$854$	−8.00000	−0.273754
$855$	0	0
$856$	12.0000	0.410152
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	0	0
$859$	14.0000	0.477674	0.238837	−	0.971060i	$-0.423234\pi$
$859$	14.0000	0.477674	0.238837	+	0.971060i	$0.423234\pi$
$860$	0	0
$861$	12.0000	0.408959
$862$	24.0000	0.817443
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	4.00000	0.136083
$865$	0	0
$866$	34.0000	1.15537
$867$	0	0
$868$	−4.00000	−0.135769
$869$	0	0
$870$	0	0
$871$	16.0000	0.542139
$872$	2.00000	0.0677285
$873$	10.0000	0.338449
$874$	0	0
$875$	0	0
$876$	−4.00000	−0.135147
$877$	22.0000	0.742887	0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000	0.742887	0.371444	+	0.928456i	$0.378863\pi$
$878$	8.00000	0.269987
$879$	48.0000	1.61900
$880$	0	0
$881$	54.0000	1.81931	0.909653	−	0.415369i	$-0.136347\pi$
$881$	54.0000	1.81931	0.909653	+	0.415369i	$0.136347\pi$
$882$	−1.00000	−0.0336718
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	0	0
$885$	0	0
$886$	12.0000	0.403148
$887$	36.0000	1.20876	0.604381	−	0.796696i	$-0.293421\pi$
$887$	36.0000	1.20876	0.604381	+	0.796696i	$0.293421\pi$
$888$	4.00000	0.134231
$889$	16.0000	0.536623
$890$	0	0
$891$	0	0
$892$	8.00000	0.267860
$893$	−24.0000	−0.803129
$894$	36.0000	1.20402
$895$	0	0
$896$	1.00000	0.0334077
$897$	0	0
$898$	18.0000	0.600668
$899$	24.0000	0.800445
$900$	−5.00000	−0.166667
$901$	0	0
$902$	0	0
$903$	−16.0000	−0.532447
$904$	6.00000	0.199557
$905$	0	0
$906$	−16.0000	−0.531564
$907$	−44.0000	−1.46100	−0.730498	−	0.682915i	$-0.760712\pi$
$907$	−44.0000	−1.46100	−0.730498	+	0.682915i	$0.760712\pi$
$908$	−18.0000	−0.597351
$909$	0	0
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	4.00000	0.132453
$913$	0	0
$914$	10.0000	0.330771
$915$	0	0
$916$	−4.00000	−0.132164
$917$	18.0000	0.594412
$918$	0	0
$919$	56.0000	1.84727	0.923635	−	0.383274i	$-0.125203\pi$
$919$	56.0000	1.84727	0.923635	+	0.383274i	$0.125203\pi$
$920$	0	0
$921$	4.00000	0.131804
$922$	−12.0000	−0.395199
$923$	0	0
$924$	0	0
$925$	10.0000	0.328798
$926$	−32.0000	−1.05159
$927$	−4.00000	−0.131377
$928$	−6.00000	−0.196960
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	6.00000	0.196537
$933$	48.0000	1.57145
$934$	6.00000	0.196326
$935$	0	0
$936$	4.00000	0.130744
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	−4.00000	−0.130605
$939$	20.0000	0.652675
$940$	0	0
$941$	24.0000	0.782378	0.391189	−	0.920310i	$-0.372064\pi$
$941$	24.0000	0.782378	0.391189	+	0.920310i	$0.372064\pi$
$942$	8.00000	0.260654
$943$	0	0
$944$	−6.00000	−0.195283
$945$	0	0
$946$	0	0
$947$	−24.0000	−0.779895	−0.389948	−	0.920837i	$-0.627507\pi$
$947$	−24.0000	−0.779895	−0.389948	+	0.920837i	$0.627507\pi$
$948$	−16.0000	−0.519656
$949$	8.00000	0.259691
$950$	10.0000	0.324443
$951$	−12.0000	−0.389127
$952$	0	0
$953$	−54.0000	−1.74923	−0.874616	−	0.484817i	$-0.838886\pi$
$953$	−54.0000	−1.74923	−0.874616	+	0.484817i	$0.838886\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	24.0000	0.776215
$957$	0	0
$958$	−36.0000	−1.16311
$959$	−18.0000	−0.581250
$960$	0	0
$961$	−15.0000	−0.483871
$962$	−8.00000	−0.257930
$963$	−12.0000	−0.386695
$964$	10.0000	0.322078
$965$	0	0
$966$	0	0
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	11.0000	0.353553
$969$	0	0
$970$	0	0
$971$	−6.00000	−0.192549	−0.0962746	−	0.995355i	$-0.530693\pi$
$971$	−6.00000	−0.192549	−0.0962746	+	0.995355i	$0.530693\pi$
$972$	−10.0000	−0.320750
$973$	14.0000	0.448819
$974$	−16.0000	−0.512673
$975$	40.0000	1.28103
$976$	−8.00000	−0.256074
$977$	−6.00000	−0.191957	−0.0959785	−	0.995383i	$-0.530598\pi$
$977$	−6.00000	−0.191957	−0.0959785	+	0.995383i	$0.530598\pi$
$978$	−32.0000	−1.02325
$979$	0	0
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	12.0000	0.382935
$983$	36.0000	1.14822	0.574111	−	0.818778i	$-0.305348\pi$
$983$	36.0000	1.14822	0.574111	+	0.818778i	$0.305348\pi$
$984$	12.0000	0.382546
$985$	0	0
$986$	0	0
$987$	24.0000	0.763928
$988$	−8.00000	−0.254514
$989$	0	0
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	−4.00000	−0.127000
$993$	16.0000	0.507745
$994$	0	0
$995$	0	0
$996$	−12.0000	−0.380235
$997$	−8.00000	−0.253363	−0.126681	−	0.991943i	$-0.540433\pi$
$997$	−8.00000	−0.253363	−0.126681	+	0.991943i	$0.540433\pi$
$998$	−4.00000	−0.126618
$999$	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	4046.2.a.f.1.1		1
17.16	even	2		14.2.a.a.1.1	✓	1
51.50	odd	2		126.2.a.b.1.1		1
68.67	odd	2		112.2.a.c.1.1		1
85.33	odd	4		350.2.c.d.99.2		2
85.67	odd	4		350.2.c.d.99.1		2
85.84	even	2		350.2.a.f.1.1		1
119.16	even	6		98.2.c.b.67.1		2
119.33	odd	6		98.2.c.a.67.1		2
119.67	even	6		98.2.c.b.79.1		2
119.101	odd	6		98.2.c.a.79.1		2
119.118	odd	2		98.2.a.a.1.1		1
136.67	odd	2		448.2.a.a.1.1		1
136.101	even	2		448.2.a.g.1.1		1
153.16	even	6		1134.2.f.l.757.1		2
153.50	odd	6		1134.2.f.f.379.1		2
153.67	even	6		1134.2.f.l.379.1		2
153.101	odd	6		1134.2.f.f.757.1		2
187.186	odd	2		1694.2.a.e.1.1		1
204.203	even	2		1008.2.a.h.1.1		1
221.135	odd	4		2366.2.d.b.337.2		2
221.203	odd	4		2366.2.d.b.337.1		2
221.220	even	2		2366.2.a.j.1.1		1
255.152	even	4		3150.2.g.j.2899.2		2
255.203	even	4		3150.2.g.j.2899.1		2
255.254	odd	2		3150.2.a.i.1.1		1
272.67	odd	4		1792.2.b.g.897.2		2
272.101	even	4		1792.2.b.c.897.2		2
272.203	odd	4		1792.2.b.g.897.1		2
272.237	even	4		1792.2.b.c.897.1		2
323.322	odd	2		5054.2.a.c.1.1		1
340.67	even	4		2800.2.g.h.449.1		2
340.203	even	4		2800.2.g.h.449.2		2
340.339	odd	2		2800.2.a.g.1.1		1
357.101	even	6		882.2.g.d.667.1		2
357.152	even	6		882.2.g.d.361.1		2
357.254	odd	6		882.2.g.c.361.1		2
357.305	odd	6		882.2.g.c.667.1		2
357.356	even	2		882.2.a.i.1.1		1
391.390	odd	2		7406.2.a.a.1.1		1
408.101	odd	2		4032.2.a.w.1.1		1
408.203	even	2		4032.2.a.r.1.1		1
476.67	odd	6		784.2.i.c.177.1		2
476.135	odd	6		784.2.i.c.753.1		2
476.271	even	6		784.2.i.i.753.1		2
476.339	even	6		784.2.i.i.177.1		2
476.475	even	2		784.2.a.b.1.1		1
595.118	even	4		2450.2.c.c.99.2		2
595.237	even	4		2450.2.c.c.99.1		2
595.594	odd	2		2450.2.a.t.1.1		1
952.237	odd	2		3136.2.a.e.1.1		1
952.475	even	2		3136.2.a.z.1.1		1
1428.1427	odd	2		7056.2.a.bd.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.2.a.a.1.1	✓	1	17.16	even	2
98.2.a.a.1.1		1	119.118	odd	2
98.2.c.a.67.1		2	119.33	odd	6
98.2.c.a.79.1		2	119.101	odd	6
98.2.c.b.67.1		2	119.16	even	6
98.2.c.b.79.1		2	119.67	even	6
112.2.a.c.1.1		1	68.67	odd	2
126.2.a.b.1.1		1	51.50	odd	2
350.2.a.f.1.1		1	85.84	even	2
350.2.c.d.99.1		2	85.67	odd	4
350.2.c.d.99.2		2	85.33	odd	4
448.2.a.a.1.1		1	136.67	odd	2
448.2.a.g.1.1		1	136.101	even	2
784.2.a.b.1.1		1	476.475	even	2
784.2.i.c.177.1		2	476.67	odd	6
784.2.i.c.753.1		2	476.135	odd	6
784.2.i.i.177.1		2	476.339	even	6
784.2.i.i.753.1		2	476.271	even	6
882.2.a.i.1.1		1	357.356	even	2
882.2.g.c.361.1		2	357.254	odd	6
882.2.g.c.667.1		2	357.305	odd	6
882.2.g.d.361.1		2	357.152	even	6
882.2.g.d.667.1		2	357.101	even	6
1008.2.a.h.1.1		1	204.203	even	2
1134.2.f.f.379.1		2	153.50	odd	6
1134.2.f.f.757.1		2	153.101	odd	6
1134.2.f.l.379.1		2	153.67	even	6
1134.2.f.l.757.1		2	153.16	even	6
1694.2.a.e.1.1		1	187.186	odd	2
1792.2.b.c.897.1		2	272.237	even	4
1792.2.b.c.897.2		2	272.101	even	4
1792.2.b.g.897.1		2	272.203	odd	4
1792.2.b.g.897.2		2	272.67	odd	4
2366.2.a.j.1.1		1	221.220	even	2
2366.2.d.b.337.1		2	221.203	odd	4
2366.2.d.b.337.2		2	221.135	odd	4
2450.2.a.t.1.1		1	595.594	odd	2
2450.2.c.c.99.1		2	595.237	even	4
2450.2.c.c.99.2		2	595.118	even	4
2800.2.a.g.1.1		1	340.339	odd	2
2800.2.g.h.449.1		2	340.67	even	4
2800.2.g.h.449.2		2	340.203	even	4
3136.2.a.e.1.1		1	952.237	odd	2
3136.2.a.z.1.1		1	952.475	even	2
3150.2.a.i.1.1		1	255.254	odd	2
3150.2.g.j.2899.1		2	255.203	even	4
3150.2.g.j.2899.2		2	255.152	even	4
4032.2.a.r.1.1		1	408.203	even	2
4032.2.a.w.1.1		1	408.101	odd	2
4046.2.a.f.1.1		1	1.1	even	1	trivial
5054.2.a.c.1.1		1	323.322	odd	2
7056.2.a.bd.1.1		1	1428.1427	odd	2
7406.2.a.a.1.1		1	391.390	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 $\F_{q}$
Belyi maps

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	4046.2.a.f.1.1

Level	$4046$
Weight	$2$
Character	4046.1
Self dual	yes
Analytic conductor	$32.307$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$