LMFDB - Embedded newform 7406.2.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7406,2,Mod(1,7406)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7406.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7406, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$N$	$=$	$7406 = 2 \cdot 7 \cdot 23^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	7406.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-1,-2,1,0,2,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$59.1372077370$
Analytic rank:	$2$
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	$\chi$	$=$	7406.1

$q$ -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} -6.00000 q^{17} -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} +6.00000 q^{34} +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} +12.0000 q^{51} -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} -6.00000 q^{68} -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	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	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
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	−1.00000	−0.235702
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	0	0
$23$	0	0
$24$	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	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	6.00000	1.02899
$35$	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	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	−2.00000	−0.308607
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	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
$51$	12.0000	1.68034
$52$	−4.00000	−0.554700
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	−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	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	−6.00000	−0.727607
$69$	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	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	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	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	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	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	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	$0.5\pi$
$102$	−12.0000	−1.18818
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	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
$119$	6.00000	0.550019
$120$	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	$-0.211977\pi$
$131$	18.0000	1.57267	0.786334	+	0.617802i	$0.211977\pi$
$132$	0	0
$133$	2.00000	0.173422
$134$	−4.00000	−0.345547
$135$	0	0
$136$	6.00000	0.514496
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	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	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	−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
$153$	−6.00000	−0.485071
$154$	0	0
$155$	0	0
$156$	8.00000	0.640513
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	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	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	−6.00000	−0.465690
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	−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	$-0.650780\pi$
$173$	−12.0000	−0.912343	−0.456172	+	0.889892i	$0.650780\pi$
$174$	−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	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	−2.00000	−0.144338
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	1.00000	0.0714286
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	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
$204$	12.0000	0.840168
$205$	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	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	−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
$221$	24.0000	1.61441
$222$	−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	$-0.457808\pi$
$229$	4.00000	0.264327	0.132164	+	0.991228i	$0.457808\pi$
$230$	0	0
$231$	0	0
$232$	6.00000	0.393919
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	4.00000	0.261488
$235$	0	0
$236$	−6.00000	−0.390567
$237$	16.0000	1.03931
$238$	−6.00000	−0.388922
$239$	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	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	−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	$-0.619214\pi$
$269$	−12.0000	−0.731653	−0.365826	+	0.930683i	$0.619214\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	−6.00000	−0.363803
$273$	−8.00000	−0.484182
$274$	18.0000	1.08742
$275$	0	0
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	−14.0000	−0.839664
$279$	−4.00000	−0.239474
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	−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
$289$	19.0000	1.11765
$290$	0	0
$291$	−20.0000	−1.17242
$292$	2.00000	0.117041
$293$	−24.0000	−1.40209	−0.701047	−	0.713115i	$-0.747284\pi$
$293$	−24.0000	−1.40209	−0.701047	+	0.713115i	$0.747284\pi$
$294$	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
$306$	6.00000	0.342997
$307$	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	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	−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	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	−12.0000	−0.672927
$319$	0	0
$320$	0	0
$321$	24.0000	1.33955
$322$	0	0
$323$	12.0000	0.667698
$324$	−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	$-0.722807\pi$
$347$	−24.0000	−1.28839	−0.644194	+	0.764862i	$0.722807\pi$
$348$	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
$357$	−12.0000	−0.635107
$358$	12.0000	0.634220
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	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	$-0.618058\pi$
$373$	−14.0000	−0.724893	−0.362446	+	0.932005i	$0.618058\pi$
$374$	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	$-0.871614\pi$
$383$	−36.0000	−1.83951	−0.919757	+	0.392488i	$0.871614\pi$
$384$	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	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	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	$-0.332640\pi$
$397$	20.0000	1.00377	0.501886	+	0.864934i	$0.332640\pi$
$398$	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
$408$	−12.0000	−0.594089
$409$	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	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	4.00000	0.194717
$423$	−12.0000	−0.583460
$424$	6.00000	0.291386
$425$	30.0000	1.45521
$426$	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	$-0.195653\pi$
$433$	34.0000	1.63394	0.816968	+	0.576683i	$0.195653\pi$
$434$	−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	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	−24.0000	−1.14156
$443$	−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	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	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	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	−4.00000	−0.186908
$459$	−24.0000	−1.12022
$460$	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	$-0.455668\pi$
$467$	6.00000	0.277647	0.138823	+	0.990317i	$0.455668\pi$
$468$	−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
$476$	6.00000	0.275010
$477$	−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	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	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
$493$	36.0000	1.62136
$494$	−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	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	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	$-0.513923\pi$
$523$	−2.00000	−0.0874539	−0.0437269	+	0.999044i	$0.513923\pi$
$524$	18.0000	0.786334
$525$	−10.0000	−0.436436
$526$	0	0
$527$	24.0000	1.04546
$528$	0	0
$529$	0	0
$530$	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	$-0.195705\pi$
$541$	38.0000	1.63375	0.816874	+	0.576816i	$0.195705\pi$
$542$	16.0000	0.687259
$543$	40.0000	1.71656
$544$	6.00000	0.257248
$545$	0	0
$546$	8.00000	0.342368
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	−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	$-0.540571\pi$
$557$	−6.00000	−0.254228	−0.127114	+	0.991888i	$0.540571\pi$
$558$	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	$-0.717837\pi$
$563$	−30.0000	−1.26435	−0.632175	+	0.774826i	$0.717837\pi$
$564$	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	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	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	$0.486745\pi$
$578$	−19.0000	−0.790296
$579$	−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	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	−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	$-0.275012\pi$
$607$	32.0000	1.29884	0.649420	+	0.760430i	$0.275012\pi$
$608$	2.00000	0.0811107
$609$	−12.0000	−0.486265
$610$	0	0
$611$	48.0000	1.94187
$612$	−6.00000	−0.242536
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	−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
$629$	12.0000	0.478471
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	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
$646$	−12.0000	−0.472134
$647$	−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	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	−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	$-0.345171\pi$
$659$	24.0000	0.934907	0.467454	+	0.884018i	$0.345171\pi$
$660$	0	0
$661$	40.0000	1.55582	0.777910	−	0.628376i	$-0.216280\pi$
$661$	40.0000	1.55582	0.777910	+	0.628376i	$0.216280\pi$
$662$	−8.00000	−0.310929
$663$	−48.0000	−1.86417
$664$	−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	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	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	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	−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	$-0.839113\pi$
$691$	−46.0000	−1.74992	−0.874961	+	0.484193i	$0.839113\pi$
$692$	−12.0000	−0.456172
$693$	0	0
$694$	24.0000	0.911028
$695$	0	0
$696$	−12.0000	−0.454859
$697$	−36.0000	−1.36360
$698$	28.0000	1.05982
$699$	12.0000	0.453882
$700$	5.00000	0.188982
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	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
$714$	12.0000	0.449089
$715$	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	$-0.428166\pi$
$719$	12.0000	0.447524	0.223762	+	0.974644i	$0.428166\pi$
$720$	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	$-0.803777\pi$
$727$	−44.0000	−1.63187	−0.815935	+	0.578144i	$0.803777\pi$
$728$	−4.00000	−0.148250
$729$	13.0000	0.481481
$730$	0	0
$731$	48.0000	1.77534
$732$	16.0000	0.591377
$733$	40.0000	1.47743	0.738717	−	0.674016i	$-0.235432\pi$
$733$	40.0000	1.47743	0.738717	+	0.674016i	$0.235432\pi$
$734$	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	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	−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	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	−36.0000	−1.29651
$772$	14.0000	0.503871
$773$	−24.0000	−0.863220	−0.431610	−	0.902060i	$-0.642054\pi$
$773$	−24.0000	−0.863220	−0.431610	+	0.902060i	$0.642054\pi$
$774$	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	$-0.431829\pi$
$797$	12.0000	0.425062	0.212531	+	0.977154i	$0.431829\pi$
$798$	4.00000	0.141598
$799$	72.0000	2.54718
$800$	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	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	6.00000	0.210559
$813$	32.0000	1.12229
$814$	0	0
$815$	0	0
$816$	12.0000	0.420084
$817$	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	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	−36.0000	−1.25564
$823$	−40.0000	−1.39431	−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000	−1.39431	−0.697156	+	0.716919i	$0.745552\pi$
$824$	−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
$833$	−6.00000	−0.207888
$834$	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
$850$	−30.0000	−1.02899
$851$	0	0
$852$	0	0
$853$	44.0000	1.50653	0.753266	−	0.657716i	$-0.228477\pi$
$853$	44.0000	1.50653	0.753266	+	0.657716i	$0.228477\pi$
$854$	−8.00000	−0.273754
$855$	0	0
$856$	12.0000	0.410152
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	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
$867$	−38.0000	−1.29055
$868$	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	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	−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	$0.390748\pi$
$884$	24.0000	0.807207
$885$	0	0
$886$	12.0000	0.403148
$887$	−36.0000	−1.20876	−0.604381	−	0.796696i	$-0.706579\pi$
$887$	−36.0000	−1.20876	−0.604381	+	0.796696i	$0.706579\pi$
$888$	−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
$901$	36.0000	1.19933
$902$	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
$918$	24.0000	0.792118
$919$	−56.0000	−1.84727	−0.923635	−	0.383274i	$-0.874797\pi$
$919$	−56.0000	−1.84727	−0.923635	+	0.383274i	$0.874797\pi$
$920$	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	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	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	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	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	$-0.372493\pi$
$947$	24.0000	0.779895	0.389948	+	0.920837i	$0.372493\pi$
$948$	16.0000	0.519656
$949$	−8.00000	−0.259691
$950$	−10.0000	−0.324443
$951$	−12.0000	−0.389127
$952$	−6.00000	−0.194461
$953$	54.0000	1.74923	0.874616	−	0.484817i	$-0.161114\pi$
$953$	54.0000	1.74923	0.874616	+	0.484817i	$0.161114\pi$
$954$	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
$969$	−24.0000	−0.770991
$970$	0	0
$971$	6.00000	0.192549	0.0962746	−	0.995355i	$-0.469307\pi$
$971$	6.00000	0.192549	0.0962746	+	0.995355i	$0.469307\pi$
$972$	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	$-0.469402\pi$
$977$	6.00000	0.191957	0.0959785	+	0.995383i	$0.469402\pi$
$978$	−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
$986$	−36.0000	−1.14647
$987$	−24.0000	−0.763928
$988$	8.00000	0.254514
$989$	0	0
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	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	$-0.459567\pi$
$997$	8.00000	0.253363	0.126681	+	0.991943i	$0.459567\pi$
$998$	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	7406.2.a.a.1.1		1
23.22	odd	2		14.2.a.a.1.1	✓	1
69.68	even	2		126.2.a.b.1.1		1
92.91	even	2		112.2.a.c.1.1		1
115.22	even	4		350.2.c.d.99.1		2
115.68	even	4		350.2.c.d.99.2		2
115.114	odd	2		350.2.a.f.1.1		1
161.45	even	6		98.2.c.a.79.1		2
161.68	even	6		98.2.c.a.67.1		2
161.114	odd	6		98.2.c.b.67.1		2
161.137	odd	6		98.2.c.b.79.1		2
161.160	even	2		98.2.a.a.1.1		1
184.45	odd	2		448.2.a.g.1.1		1
184.91	even	2		448.2.a.a.1.1		1
207.22	odd	6		1134.2.f.l.379.1		2
207.68	even	6		1134.2.f.f.379.1		2
207.137	even	6		1134.2.f.f.757.1		2
207.160	odd	6		1134.2.f.l.757.1		2
253.252	even	2		1694.2.a.e.1.1		1
276.275	odd	2		1008.2.a.h.1.1		1
299.229	even	4		2366.2.d.b.337.1		2
299.252	even	4		2366.2.d.b.337.2		2
299.298	odd	2		2366.2.a.j.1.1		1
345.68	odd	4		3150.2.g.j.2899.1		2
345.137	odd	4		3150.2.g.j.2899.2		2
345.344	even	2		3150.2.a.i.1.1		1
368.45	odd	4		1792.2.b.c.897.1		2
368.91	even	4		1792.2.b.g.897.1		2
368.229	odd	4		1792.2.b.c.897.2		2
368.275	even	4		1792.2.b.g.897.2		2
391.390	odd	2		4046.2.a.f.1.1		1
437.436	even	2		5054.2.a.c.1.1		1
460.183	odd	4		2800.2.g.h.449.2		2
460.367	odd	4		2800.2.g.h.449.1		2
460.459	even	2		2800.2.a.g.1.1		1
483.68	odd	6		882.2.g.d.361.1		2
483.137	even	6		882.2.g.c.667.1		2
483.206	odd	6		882.2.g.d.667.1		2
483.275	even	6		882.2.g.c.361.1		2
483.482	odd	2		882.2.a.i.1.1		1
552.275	odd	2		4032.2.a.r.1.1		1
552.413	even	2		4032.2.a.w.1.1		1
644.275	even	6		784.2.i.c.753.1		2
644.367	odd	6		784.2.i.i.177.1		2
644.459	even	6		784.2.i.c.177.1		2
644.551	odd	6		784.2.i.i.753.1		2
644.643	odd	2		784.2.a.b.1.1		1
805.482	odd	4		2450.2.c.c.99.1		2
805.643	odd	4		2450.2.c.c.99.2		2
805.804	even	2		2450.2.a.t.1.1		1
1288.643	odd	2		3136.2.a.z.1.1		1
1288.965	even	2		3136.2.a.e.1.1		1
1932.1931	even	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	23.22	odd	2
98.2.a.a.1.1		1	161.160	even	2
98.2.c.a.67.1		2	161.68	even	6
98.2.c.a.79.1		2	161.45	even	6
98.2.c.b.67.1		2	161.114	odd	6
98.2.c.b.79.1		2	161.137	odd	6
112.2.a.c.1.1		1	92.91	even	2
126.2.a.b.1.1		1	69.68	even	2
350.2.a.f.1.1		1	115.114	odd	2
350.2.c.d.99.1		2	115.22	even	4
350.2.c.d.99.2		2	115.68	even	4
448.2.a.a.1.1		1	184.91	even	2
448.2.a.g.1.1		1	184.45	odd	2
784.2.a.b.1.1		1	644.643	odd	2
784.2.i.c.177.1		2	644.459	even	6
784.2.i.c.753.1		2	644.275	even	6
784.2.i.i.177.1		2	644.367	odd	6
784.2.i.i.753.1		2	644.551	odd	6
882.2.a.i.1.1		1	483.482	odd	2
882.2.g.c.361.1		2	483.275	even	6
882.2.g.c.667.1		2	483.137	even	6
882.2.g.d.361.1		2	483.68	odd	6
882.2.g.d.667.1		2	483.206	odd	6
1008.2.a.h.1.1		1	276.275	odd	2
1134.2.f.f.379.1		2	207.68	even	6
1134.2.f.f.757.1		2	207.137	even	6
1134.2.f.l.379.1		2	207.22	odd	6
1134.2.f.l.757.1		2	207.160	odd	6
1694.2.a.e.1.1		1	253.252	even	2
1792.2.b.c.897.1		2	368.45	odd	4
1792.2.b.c.897.2		2	368.229	odd	4
1792.2.b.g.897.1		2	368.91	even	4
1792.2.b.g.897.2		2	368.275	even	4
2366.2.a.j.1.1		1	299.298	odd	2
2366.2.d.b.337.1		2	299.229	even	4
2366.2.d.b.337.2		2	299.252	even	4
2450.2.a.t.1.1		1	805.804	even	2
2450.2.c.c.99.1		2	805.482	odd	4
2450.2.c.c.99.2		2	805.643	odd	4
2800.2.a.g.1.1		1	460.459	even	2
2800.2.g.h.449.1		2	460.367	odd	4
2800.2.g.h.449.2		2	460.183	odd	4
3136.2.a.e.1.1		1	1288.965	even	2
3136.2.a.z.1.1		1	1288.643	odd	2
3150.2.a.i.1.1		1	345.344	even	2
3150.2.g.j.2899.1		2	345.68	odd	4
3150.2.g.j.2899.2		2	345.137	odd	4
4032.2.a.r.1.1		1	552.275	odd	2
4032.2.a.w.1.1		1	552.413	even	2
4046.2.a.f.1.1		1	391.390	odd	2
5054.2.a.c.1.1		1	437.436	even	2
7056.2.a.bd.1.1		1	1932.1931	even	2
7406.2.a.a.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
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	7406.2.a.a.1.1

Level	$7406$
Weight	$2$
Character	7406.1
Self dual	yes
Analytic conductor	$59.137$
Analytic rank	$2$
Dimension	$1$
CM	no
Inner twists	$1$