LMFDB - Embedded newform 832.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [832,2,Mod(1,832)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(832, 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("832.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 832 = 2^{6} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	832.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:	$6.64355344817$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 416)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	832.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{3} -1.00000 q^{5} -3.00000 q^{7} -2.00000 q^{9} +O(q^{10})$

$q+1.00000 q^{3} -1.00000 q^{5} -3.00000 q^{7} -2.00000 q^{9} +2.00000 q^{11} -1.00000 q^{13} -1.00000 q^{15} -3.00000 q^{17} +2.00000 q^{19} -3.00000 q^{21} -4.00000 q^{23} -4.00000 q^{25} -5.00000 q^{27} -2.00000 q^{29} -4.00000 q^{31} +2.00000 q^{33} +3.00000 q^{35} -5.00000 q^{37} -1.00000 q^{39} -12.0000 q^{41} +7.00000 q^{43} +2.00000 q^{45} +9.00000 q^{47} +2.00000 q^{49} -3.00000 q^{51} -4.00000 q^{53} -2.00000 q^{55} +2.00000 q^{57} +6.00000 q^{59} +4.00000 q^{61} +6.00000 q^{63} +1.00000 q^{65} -10.0000 q^{67} -4.00000 q^{69} +15.0000 q^{71} -2.00000 q^{73} -4.00000 q^{75} -6.00000 q^{77} +8.00000 q^{79} +1.00000 q^{81} -4.00000 q^{83} +3.00000 q^{85} -2.00000 q^{87} +2.00000 q^{89} +3.00000 q^{91} -4.00000 q^{93} -2.00000 q^{95} +10.0000 q^{97} -4.00000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	0	0
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	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
$21$	−3.00000	−0.654654
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	3.00000	0.507093
$36$	0	0
$37$	−5.00000	−0.821995	−0.410997	−	0.911636i	$-0.634819\pi$
$37$	−5.00000	−0.821995	−0.410997	+	0.911636i	$0.634819\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−12.0000	−1.87409	−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000	−1.87409	−0.937043	+	0.349215i	$0.886448\pi$
$42$	0	0
$43$	7.00000	1.06749	0.533745	−	0.845645i	$-0.320784\pi$
$43$	7.00000	1.06749	0.533745	+	0.845645i	$0.320784\pi$
$44$	0	0
$45$	2.00000	0.298142
$46$	0	0
$47$	9.00000	1.31278	0.656392	−	0.754420i	$-0.272082\pi$
$47$	9.00000	1.31278	0.656392	+	0.754420i	$0.272082\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	−3.00000	−0.420084
$52$	0	0
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	0	0
$57$	2.00000	0.264906
$58$	0	0
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	0	0
$63$	6.00000	0.755929
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	15.0000	1.78017	0.890086	−	0.455792i	$-0.150644\pi$
$71$	15.0000	1.78017	0.890086	+	0.455792i	$0.150644\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	0	0
$75$	−4.00000	−0.461880
$76$	0	0
$77$	−6.00000	−0.683763
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	3.00000	0.325396
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	3.00000	0.314485
$92$	0	0
$93$	−4.00000	−0.414781
$94$	0	0
$95$	−2.00000	−0.205196
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	−4.00000	−0.402015
$100$	0	0
$101$	−4.00000	−0.398015	−0.199007	−	0.979998i	$-0.563772\pi$
$101$	−4.00000	−0.398015	−0.199007	+	0.979998i	$0.563772\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	3.00000	0.292770
$106$	0	0
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	19.0000	1.81987	0.909935	−	0.414751i	$-0.136131\pi$
$109$	19.0000	1.81987	0.909935	+	0.414751i	$0.136131\pi$
$110$	0	0
$111$	−5.00000	−0.474579
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	9.00000	0.825029
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−12.0000	−1.08200
$124$	0	0
$125$	9.00000	0.804984
$126$	0	0
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	0	0
$129$	7.00000	0.616316
$130$	0	0
$131$	−21.0000	−1.83478	−0.917389	−	0.397991i	$-0.869707\pi$
$131$	−21.0000	−1.83478	−0.917389	+	0.397991i	$0.869707\pi$
$132$	0	0
$133$	−6.00000	−0.520266
$134$	0	0
$135$	5.00000	0.430331
$136$	0	0
$137$	−4.00000	−0.341743	−0.170872	−	0.985293i	$-0.554658\pi$
$137$	−4.00000	−0.341743	−0.170872	+	0.985293i	$0.554658\pi$
$138$	0	0
$139$	19.0000	1.61156	0.805779	−	0.592216i	$-0.201747\pi$
$139$	19.0000	1.61156	0.805779	+	0.592216i	$0.201747\pi$
$140$	0	0
$141$	9.00000	0.757937
$142$	0	0
$143$	−2.00000	−0.167248
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	2.00000	0.164957
$148$	0	0
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−5.00000	−0.406894	−0.203447	−	0.979086i	$-0.565214\pi$
$151$	−5.00000	−0.406894	−0.203447	+	0.979086i	$0.565214\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	0	0
$155$	4.00000	0.321288
$156$	0	0
$157$	22.0000	1.75579	0.877896	−	0.478852i	$-0.158947\pi$
$157$	22.0000	1.75579	0.877896	+	0.478852i	$0.158947\pi$
$158$	0	0
$159$	−4.00000	−0.317221
$160$	0	0
$161$	12.0000	0.945732
$162$	0	0
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	0	0
$165$	−2.00000	−0.155700
$166$	0	0
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−4.00000	−0.305888
$172$	0	0
$173$	−16.0000	−1.21646	−0.608229	−	0.793762i	$-0.708120\pi$
$173$	−16.0000	−1.21646	−0.608229	+	0.793762i	$0.708120\pi$
$174$	0	0
$175$	12.0000	0.907115
$176$	0	0
$177$	6.00000	0.450988
$178$	0	0
$179$	11.0000	0.822179	0.411089	−	0.911595i	$-0.365148\pi$
$179$	11.0000	0.822179	0.411089	+	0.911595i	$0.365148\pi$
$180$	0	0
$181$	−20.0000	−1.48659	−0.743294	−	0.668965i	$-0.766738\pi$
$181$	−20.0000	−1.48659	−0.743294	+	0.668965i	$0.766738\pi$
$182$	0	0
$183$	4.00000	0.295689
$184$	0	0
$185$	5.00000	0.367607
$186$	0	0
$187$	−6.00000	−0.438763
$188$	0	0
$189$	15.0000	1.09109
$190$	0	0
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	0	0
$193$	12.0000	0.863779	0.431889	−	0.901927i	$-0.357847\pi$
$193$	12.0000	0.863779	0.431889	+	0.901927i	$0.357847\pi$
$194$	0	0
$195$	1.00000	0.0716115
$196$	0	0
$197$	−15.0000	−1.06871	−0.534353	−	0.845262i	$-0.679445\pi$
$197$	−15.0000	−1.06871	−0.534353	+	0.845262i	$0.679445\pi$
$198$	0	0
$199$	−26.0000	−1.84309	−0.921546	−	0.388270i	$-0.873073\pi$
$199$	−26.0000	−1.84309	−0.921546	+	0.388270i	$0.873073\pi$
$200$	0	0
$201$	−10.0000	−0.705346
$202$	0	0
$203$	6.00000	0.421117
$204$	0	0
$205$	12.0000	0.838116
$206$	0	0
$207$	8.00000	0.556038
$208$	0	0
$209$	4.00000	0.276686
$210$	0	0
$211$	27.0000	1.85876	0.929378	−	0.369129i	$-0.120344\pi$
$211$	27.0000	1.85876	0.929378	+	0.369129i	$0.120344\pi$
$212$	0	0
$213$	15.0000	1.02778
$214$	0	0
$215$	−7.00000	−0.477396
$216$	0	0
$217$	12.0000	0.814613
$218$	0	0
$219$	−2.00000	−0.135147
$220$	0	0
$221$	3.00000	0.201802
$222$	0	0
$223$	15.0000	1.00447	0.502237	−	0.864730i	$-0.332510\pi$
$223$	15.0000	1.00447	0.502237	+	0.864730i	$0.332510\pi$
$224$	0	0
$225$	8.00000	0.533333
$226$	0	0
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	0	0
$229$	9.00000	0.594737	0.297368	−	0.954763i	$-0.403891\pi$
$229$	9.00000	0.594737	0.297368	+	0.954763i	$0.403891\pi$
$230$	0	0
$231$	−6.00000	−0.394771
$232$	0	0
$233$	−27.0000	−1.76883	−0.884414	−	0.466702i	$-0.845442\pi$
$233$	−27.0000	−1.76883	−0.884414	+	0.466702i	$0.845442\pi$
$234$	0	0
$235$	−9.00000	−0.587095
$236$	0	0
$237$	8.00000	0.519656
$238$	0	0
$239$	−27.0000	−1.74648	−0.873242	−	0.487286i	$-0.837987\pi$
$239$	−27.0000	−1.74648	−0.873242	+	0.487286i	$0.837987\pi$
$240$	0	0
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	0	0
$245$	−2.00000	−0.127775
$246$	0	0
$247$	−2.00000	−0.127257
$248$	0	0
$249$	−4.00000	−0.253490
$250$	0	0
$251$	−8.00000	−0.504956	−0.252478	−	0.967603i	$-0.581245\pi$
$251$	−8.00000	−0.504956	−0.252478	+	0.967603i	$0.581245\pi$
$252$	0	0
$253$	−8.00000	−0.502956
$254$	0	0
$255$	3.00000	0.187867
$256$	0	0
$257$	−7.00000	−0.436648	−0.218324	−	0.975876i	$-0.570059\pi$
$257$	−7.00000	−0.436648	−0.218324	+	0.975876i	$0.570059\pi$
$258$	0	0
$259$	15.0000	0.932055
$260$	0	0
$261$	4.00000	0.247594
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	4.00000	0.245718
$266$	0	0
$267$	2.00000	0.122398
$268$	0	0
$269$	4.00000	0.243884	0.121942	−	0.992537i	$-0.461088\pi$
$269$	4.00000	0.243884	0.121942	+	0.992537i	$0.461088\pi$
$270$	0	0
$271$	−15.0000	−0.911185	−0.455593	−	0.890188i	$-0.650573\pi$
$271$	−15.0000	−0.911185	−0.455593	+	0.890188i	$0.650573\pi$
$272$	0	0
$273$	3.00000	0.181568
$274$	0	0
$275$	−8.00000	−0.482418
$276$	0	0
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	0	0
$279$	8.00000	0.478947
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	20.0000	1.18888	0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	0	0
$285$	−2.00000	−0.118470
$286$	0	0
$287$	36.0000	2.12501
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	10.0000	0.586210
$292$	0	0
$293$	7.00000	0.408944	0.204472	−	0.978872i	$-0.434452\pi$
$293$	7.00000	0.408944	0.204472	+	0.978872i	$0.434452\pi$
$294$	0	0
$295$	−6.00000	−0.349334
$296$	0	0
$297$	−10.0000	−0.580259
$298$	0	0
$299$	4.00000	0.231326
$300$	0	0
$301$	−21.0000	−1.21042
$302$	0	0
$303$	−4.00000	−0.229794
$304$	0	0
$305$	−4.00000	−0.229039
$306$	0	0
$307$	−26.0000	−1.48390	−0.741949	−	0.670456i	$-0.766098\pi$
$307$	−26.0000	−1.48390	−0.741949	+	0.670456i	$0.766098\pi$
$308$	0	0
$309$	−4.00000	−0.227552
$310$	0	0
$311$	−2.00000	−0.113410	−0.0567048	−	0.998391i	$-0.518059\pi$
$311$	−2.00000	−0.113410	−0.0567048	+	0.998391i	$0.518059\pi$
$312$	0	0
$313$	23.0000	1.30004	0.650018	−	0.759918i	$-0.274761\pi$
$313$	23.0000	1.30004	0.650018	+	0.759918i	$0.274761\pi$
$314$	0	0
$315$	−6.00000	−0.338062
$316$	0	0
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	−4.00000	−0.223957
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	−6.00000	−0.333849
$324$	0	0
$325$	4.00000	0.221880
$326$	0	0
$327$	19.0000	1.05070
$328$	0	0
$329$	−27.0000	−1.48856
$330$	0	0
$331$	−16.0000	−0.879440	−0.439720	−	0.898135i	$-0.644922\pi$
$331$	−16.0000	−0.879440	−0.439720	+	0.898135i	$0.644922\pi$
$332$	0	0
$333$	10.0000	0.547997
$334$	0	0
$335$	10.0000	0.546358
$336$	0	0
$337$	−25.0000	−1.36184	−0.680918	−	0.732359i	$-0.738419\pi$
$337$	−25.0000	−1.36184	−0.680918	+	0.732359i	$0.738419\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	0	0
$341$	−8.00000	−0.433224
$342$	0	0
$343$	15.0000	0.809924
$344$	0	0
$345$	4.00000	0.215353
$346$	0	0
$347$	11.0000	0.590511	0.295255	−	0.955418i	$-0.404595\pi$
$347$	11.0000	0.590511	0.295255	+	0.955418i	$0.404595\pi$
$348$	0	0
$349$	7.00000	0.374701	0.187351	−	0.982293i	$-0.440010\pi$
$349$	7.00000	0.374701	0.187351	+	0.982293i	$0.440010\pi$
$350$	0	0
$351$	5.00000	0.266880
$352$	0	0
$353$	−24.0000	−1.27739	−0.638696	−	0.769460i	$-0.720526\pi$
$353$	−24.0000	−1.27739	−0.638696	+	0.769460i	$0.720526\pi$
$354$	0	0
$355$	−15.0000	−0.796117
$356$	0	0
$357$	9.00000	0.476331
$358$	0	0
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	−7.00000	−0.367405
$364$	0	0
$365$	2.00000	0.104685
$366$	0	0
$367$	−10.0000	−0.521996	−0.260998	−	0.965339i	$-0.584052\pi$
$367$	−10.0000	−0.521996	−0.260998	+	0.965339i	$0.584052\pi$
$368$	0	0
$369$	24.0000	1.24939
$370$	0	0
$371$	12.0000	0.623009
$372$	0	0
$373$	−36.0000	−1.86401	−0.932005	−	0.362446i	$-0.881942\pi$
$373$	−36.0000	−1.86401	−0.932005	+	0.362446i	$0.881942\pi$
$374$	0	0
$375$	9.00000	0.464758
$376$	0	0
$377$	2.00000	0.103005
$378$	0	0
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	0	0
$383$	7.00000	0.357683	0.178842	−	0.983878i	$-0.442765\pi$
$383$	7.00000	0.357683	0.178842	+	0.983878i	$0.442765\pi$
$384$	0	0
$385$	6.00000	0.305788
$386$	0	0
$387$	−14.0000	−0.711660
$388$	0	0
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	0	0
$393$	−21.0000	−1.05931
$394$	0	0
$395$	−8.00000	−0.402524
$396$	0	0
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	0	0
$399$	−6.00000	−0.300376
$400$	0	0
$401$	−24.0000	−1.19850	−0.599251	−	0.800561i	$-0.704535\pi$
$401$	−24.0000	−1.19850	−0.599251	+	0.800561i	$0.704535\pi$
$402$	0	0
$403$	4.00000	0.199254
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−10.0000	−0.495682
$408$	0	0
$409$	12.0000	0.593362	0.296681	−	0.954977i	$-0.404120\pi$
$409$	12.0000	0.593362	0.296681	+	0.954977i	$0.404120\pi$
$410$	0	0
$411$	−4.00000	−0.197305
$412$	0	0
$413$	−18.0000	−0.885722
$414$	0	0
$415$	4.00000	0.196352
$416$	0	0
$417$	19.0000	0.930434
$418$	0	0
$419$	1.00000	0.0488532	0.0244266	−	0.999702i	$-0.492224\pi$
$419$	1.00000	0.0488532	0.0244266	+	0.999702i	$0.492224\pi$
$420$	0	0
$421$	3.00000	0.146211	0.0731055	−	0.997324i	$-0.476709\pi$
$421$	3.00000	0.146211	0.0731055	+	0.997324i	$0.476709\pi$
$422$	0	0
$423$	−18.0000	−0.875190
$424$	0	0
$425$	12.0000	0.582086
$426$	0	0
$427$	−12.0000	−0.580721
$428$	0	0
$429$	−2.00000	−0.0965609
$430$	0	0
$431$	−3.00000	−0.144505	−0.0722525	−	0.997386i	$-0.523019\pi$
$431$	−3.00000	−0.144505	−0.0722525	+	0.997386i	$0.523019\pi$
$432$	0	0
$433$	−25.0000	−1.20142	−0.600712	−	0.799466i	$-0.705116\pi$
$433$	−25.0000	−1.20142	−0.600712	+	0.799466i	$0.705116\pi$
$434$	0	0
$435$	2.00000	0.0958927
$436$	0	0
$437$	−8.00000	−0.382692
$438$	0	0
$439$	2.00000	0.0954548	0.0477274	−	0.998860i	$-0.484802\pi$
$439$	2.00000	0.0954548	0.0477274	+	0.998860i	$0.484802\pi$
$440$	0	0
$441$	−4.00000	−0.190476
$442$	0	0
$443$	−11.0000	−0.522626	−0.261313	−	0.965254i	$-0.584155\pi$
$443$	−11.0000	−0.522626	−0.261313	+	0.965254i	$0.584155\pi$
$444$	0	0
$445$	−2.00000	−0.0948091
$446$	0	0
$447$	6.00000	0.283790
$448$	0	0
$449$	26.0000	1.22702	0.613508	−	0.789689i	$-0.289758\pi$
$449$	26.0000	1.22702	0.613508	+	0.789689i	$0.289758\pi$
$450$	0	0
$451$	−24.0000	−1.13012
$452$	0	0
$453$	−5.00000	−0.234920
$454$	0	0
$455$	−3.00000	−0.140642
$456$	0	0
$457$	−34.0000	−1.59045	−0.795226	−	0.606313i	$-0.792648\pi$
$457$	−34.0000	−1.59045	−0.795226	+	0.606313i	$0.792648\pi$
$458$	0	0
$459$	15.0000	0.700140
$460$	0	0
$461$	−21.0000	−0.978068	−0.489034	−	0.872265i	$-0.662651\pi$
$461$	−21.0000	−0.978068	−0.489034	+	0.872265i	$0.662651\pi$
$462$	0	0
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	0	0
$465$	4.00000	0.185496
$466$	0	0
$467$	−36.0000	−1.66588	−0.832941	−	0.553362i	$-0.813345\pi$
$467$	−36.0000	−1.66588	−0.832941	+	0.553362i	$0.813345\pi$
$468$	0	0
$469$	30.0000	1.38527
$470$	0	0
$471$	22.0000	1.01371
$472$	0	0
$473$	14.0000	0.643721
$474$	0	0
$475$	−8.00000	−0.367065
$476$	0	0
$477$	8.00000	0.366295
$478$	0	0
$479$	−15.0000	−0.685367	−0.342684	−	0.939451i	$-0.611336\pi$
$479$	−15.0000	−0.685367	−0.342684	+	0.939451i	$0.611336\pi$
$480$	0	0
$481$	5.00000	0.227980
$482$	0	0
$483$	12.0000	0.546019
$484$	0	0
$485$	−10.0000	−0.454077
$486$	0	0
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	0	0
$489$	20.0000	0.904431
$490$	0	0
$491$	31.0000	1.39901	0.699505	−	0.714628i	$-0.253404\pi$
$491$	31.0000	1.39901	0.699505	+	0.714628i	$0.253404\pi$
$492$	0	0
$493$	6.00000	0.270226
$494$	0	0
$495$	4.00000	0.179787
$496$	0	0
$497$	−45.0000	−2.01853
$498$	0	0
$499$	−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
$501$	8.00000	0.357414
$502$	0	0
$503$	−10.0000	−0.445878	−0.222939	−	0.974832i	$-0.571565\pi$
$503$	−10.0000	−0.445878	−0.222939	+	0.974832i	$0.571565\pi$
$504$	0	0
$505$	4.00000	0.177998
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	0	0
$511$	6.00000	0.265424
$512$	0	0
$513$	−10.0000	−0.441511
$514$	0	0
$515$	4.00000	0.176261
$516$	0	0
$517$	18.0000	0.791639
$518$	0	0
$519$	−16.0000	−0.702322
$520$	0	0
$521$	−1.00000	−0.0438108	−0.0219054	−	0.999760i	$-0.506973\pi$
$521$	−1.00000	−0.0438108	−0.0219054	+	0.999760i	$0.506973\pi$
$522$	0	0
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	0	0
$525$	12.0000	0.523723
$526$	0	0
$527$	12.0000	0.522728
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−12.0000	−0.520756
$532$	0	0
$533$	12.0000	0.519778
$534$	0	0
$535$	12.0000	0.518805
$536$	0	0
$537$	11.0000	0.474685
$538$	0	0
$539$	4.00000	0.172292
$540$	0	0
$541$	17.0000	0.730887	0.365444	−	0.930834i	$-0.380917\pi$
$541$	17.0000	0.730887	0.365444	+	0.930834i	$0.380917\pi$
$542$	0	0
$543$	−20.0000	−0.858282
$544$	0	0
$545$	−19.0000	−0.813871
$546$	0	0
$547$	17.0000	0.726868	0.363434	−	0.931620i	$-0.381604\pi$
$547$	17.0000	0.726868	0.363434	+	0.931620i	$0.381604\pi$
$548$	0	0
$549$	−8.00000	−0.341432
$550$	0	0
$551$	−4.00000	−0.170406
$552$	0	0
$553$	−24.0000	−1.02058
$554$	0	0
$555$	5.00000	0.212238
$556$	0	0
$557$	−7.00000	−0.296600	−0.148300	−	0.988942i	$-0.547380\pi$
$557$	−7.00000	−0.296600	−0.148300	+	0.988942i	$0.547380\pi$
$558$	0	0
$559$	−7.00000	−0.296068
$560$	0	0
$561$	−6.00000	−0.253320
$562$	0	0
$563$	39.0000	1.64365	0.821827	−	0.569737i	$-0.192955\pi$
$563$	39.0000	1.64365	0.821827	+	0.569737i	$0.192955\pi$
$564$	0	0
$565$	6.00000	0.252422
$566$	0	0
$567$	−3.00000	−0.125988
$568$	0	0
$569$	15.0000	0.628833	0.314416	−	0.949285i	$-0.398191\pi$
$569$	15.0000	0.628833	0.314416	+	0.949285i	$0.398191\pi$
$570$	0	0
$571$	29.0000	1.21361	0.606806	−	0.794850i	$-0.292450\pi$
$571$	29.0000	1.21361	0.606806	+	0.794850i	$0.292450\pi$
$572$	0	0
$573$	−18.0000	−0.751961
$574$	0	0
$575$	16.0000	0.667246
$576$	0	0
$577$	10.0000	0.416305	0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000	0.416305	0.208153	+	0.978096i	$0.433255\pi$
$578$	0	0
$579$	12.0000	0.498703
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	−8.00000	−0.331326
$584$	0	0
$585$	−2.00000	−0.0826898
$586$	0	0
$587$	20.0000	0.825488	0.412744	−	0.910847i	$-0.364570\pi$
$587$	20.0000	0.825488	0.412744	+	0.910847i	$0.364570\pi$
$588$	0	0
$589$	−8.00000	−0.329634
$590$	0	0
$591$	−15.0000	−0.617018
$592$	0	0
$593$	−34.0000	−1.39621	−0.698106	−	0.715994i	$-0.745974\pi$
$593$	−34.0000	−1.39621	−0.698106	+	0.715994i	$0.745974\pi$
$594$	0	0
$595$	−9.00000	−0.368964
$596$	0	0
$597$	−26.0000	−1.06411
$598$	0	0
$599$	−42.0000	−1.71607	−0.858037	−	0.513588i	$-0.828316\pi$
$599$	−42.0000	−1.71607	−0.858037	+	0.513588i	$0.828316\pi$
$600$	0	0
$601$	21.0000	0.856608	0.428304	−	0.903635i	$-0.359111\pi$
$601$	21.0000	0.856608	0.428304	+	0.903635i	$0.359111\pi$
$602$	0	0
$603$	20.0000	0.814463
$604$	0	0
$605$	7.00000	0.284590
$606$	0	0
$607$	38.0000	1.54237	0.771186	−	0.636610i	$-0.219664\pi$
$607$	38.0000	1.54237	0.771186	+	0.636610i	$0.219664\pi$
$608$	0	0
$609$	6.00000	0.243132
$610$	0	0
$611$	−9.00000	−0.364101
$612$	0	0
$613$	−22.0000	−0.888572	−0.444286	−	0.895885i	$-0.646543\pi$
$613$	−22.0000	−0.888572	−0.444286	+	0.895885i	$0.646543\pi$
$614$	0	0
$615$	12.0000	0.483887
$616$	0	0
$617$	8.00000	0.322068	0.161034	−	0.986949i	$-0.448517\pi$
$617$	8.00000	0.322068	0.161034	+	0.986949i	$0.448517\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	20.0000	0.802572
$622$	0	0
$623$	−6.00000	−0.240385
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	4.00000	0.159745
$628$	0	0
$629$	15.0000	0.598089
$630$	0	0
$631$	−25.0000	−0.995234	−0.497617	−	0.867397i	$-0.665792\pi$
$631$	−25.0000	−0.995234	−0.497617	+	0.867397i	$0.665792\pi$
$632$	0	0
$633$	27.0000	1.07315
$634$	0	0
$635$	16.0000	0.634941
$636$	0	0
$637$	−2.00000	−0.0792429
$638$	0	0
$639$	−30.0000	−1.18678
$640$	0	0
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	0	0
$643$	−2.00000	−0.0788723	−0.0394362	−	0.999222i	$-0.512556\pi$
$643$	−2.00000	−0.0788723	−0.0394362	+	0.999222i	$0.512556\pi$
$644$	0	0
$645$	−7.00000	−0.275625
$646$	0	0
$647$	26.0000	1.02217	0.511083	−	0.859532i	$-0.329245\pi$
$647$	26.0000	1.02217	0.511083	+	0.859532i	$0.329245\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	12.0000	0.470317
$652$	0	0
$653$	−32.0000	−1.25226	−0.626128	−	0.779720i	$-0.715361\pi$
$653$	−32.0000	−1.25226	−0.626128	+	0.779720i	$0.715361\pi$
$654$	0	0
$655$	21.0000	0.820538
$656$	0	0
$657$	4.00000	0.156055
$658$	0	0
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	0	0
$663$	3.00000	0.116510
$664$	0	0
$665$	6.00000	0.232670
$666$	0	0
$667$	8.00000	0.309761
$668$	0	0
$669$	15.0000	0.579934
$670$	0	0
$671$	8.00000	0.308837
$672$	0	0
$673$	13.0000	0.501113	0.250557	−	0.968102i	$-0.419386\pi$
$673$	13.0000	0.501113	0.250557	+	0.968102i	$0.419386\pi$
$674$	0	0
$675$	20.0000	0.769800
$676$	0	0
$677$	40.0000	1.53732	0.768662	−	0.639655i	$-0.220923\pi$
$677$	40.0000	1.53732	0.768662	+	0.639655i	$0.220923\pi$
$678$	0	0
$679$	−30.0000	−1.15129
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	−16.0000	−0.612223	−0.306111	−	0.951996i	$-0.599028\pi$
$683$	−16.0000	−0.612223	−0.306111	+	0.951996i	$0.599028\pi$
$684$	0	0
$685$	4.00000	0.152832
$686$	0	0
$687$	9.00000	0.343371
$688$	0	0
$689$	4.00000	0.152388
$690$	0	0
$691$	−32.0000	−1.21734	−0.608669	−	0.793424i	$-0.708296\pi$
$691$	−32.0000	−1.21734	−0.608669	+	0.793424i	$0.708296\pi$
$692$	0	0
$693$	12.0000	0.455842
$694$	0	0
$695$	−19.0000	−0.720711
$696$	0	0
$697$	36.0000	1.36360
$698$	0	0
$699$	−27.0000	−1.02123
$700$	0	0
$701$	−20.0000	−0.755390	−0.377695	−	0.925930i	$-0.623283\pi$
$701$	−20.0000	−0.755390	−0.377695	+	0.925930i	$0.623283\pi$
$702$	0	0
$703$	−10.0000	−0.377157
$704$	0	0
$705$	−9.00000	−0.338960
$706$	0	0
$707$	12.0000	0.451306
$708$	0	0
$709$	30.0000	1.12667	0.563337	−	0.826227i	$-0.309517\pi$
$709$	30.0000	1.12667	0.563337	+	0.826227i	$0.309517\pi$
$710$	0	0
$711$	−16.0000	−0.600047
$712$	0	0
$713$	16.0000	0.599205
$714$	0	0
$715$	2.00000	0.0747958
$716$	0	0
$717$	−27.0000	−1.00833
$718$	0	0
$719$	18.0000	0.671287	0.335643	−	0.941989i	$-0.391046\pi$
$719$	18.0000	0.671287	0.335643	+	0.941989i	$0.391046\pi$
$720$	0	0
$721$	12.0000	0.446903
$722$	0	0
$723$	22.0000	0.818189
$724$	0	0
$725$	8.00000	0.297113
$726$	0	0
$727$	14.0000	0.519231	0.259616	−	0.965712i	$-0.416404\pi$
$727$	14.0000	0.519231	0.259616	+	0.965712i	$0.416404\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−21.0000	−0.776713
$732$	0	0
$733$	29.0000	1.07114	0.535570	−	0.844491i	$-0.320097\pi$
$733$	29.0000	1.07114	0.535570	+	0.844491i	$0.320097\pi$
$734$	0	0
$735$	−2.00000	−0.0737711
$736$	0	0
$737$	−20.0000	−0.736709
$738$	0	0
$739$	16.0000	0.588570	0.294285	−	0.955718i	$-0.404919\pi$
$739$	16.0000	0.588570	0.294285	+	0.955718i	$0.404919\pi$
$740$	0	0
$741$	−2.00000	−0.0734718
$742$	0	0
$743$	−51.0000	−1.87101	−0.935504	−	0.353315i	$-0.885054\pi$
$743$	−51.0000	−1.87101	−0.935504	+	0.353315i	$0.885054\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	0	0
$747$	8.00000	0.292705
$748$	0	0
$749$	36.0000	1.31541
$750$	0	0
$751$	4.00000	0.145962	0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000	0.145962	0.0729810	+	0.997333i	$0.476749\pi$
$752$	0	0
$753$	−8.00000	−0.291536
$754$	0	0
$755$	5.00000	0.181969
$756$	0	0
$757$	−20.0000	−0.726912	−0.363456	−	0.931611i	$-0.618403\pi$
$757$	−20.0000	−0.726912	−0.363456	+	0.931611i	$0.618403\pi$
$758$	0	0
$759$	−8.00000	−0.290382
$760$	0	0
$761$	14.0000	0.507500	0.253750	−	0.967270i	$-0.418336\pi$
$761$	14.0000	0.507500	0.253750	+	0.967270i	$0.418336\pi$
$762$	0	0
$763$	−57.0000	−2.06354
$764$	0	0
$765$	−6.00000	−0.216930
$766$	0	0
$767$	−6.00000	−0.216647
$768$	0	0
$769$	8.00000	0.288487	0.144244	−	0.989542i	$-0.453925\pi$
$769$	8.00000	0.288487	0.144244	+	0.989542i	$0.453925\pi$
$770$	0	0
$771$	−7.00000	−0.252099
$772$	0	0
$773$	51.0000	1.83434	0.917171	−	0.398493i	$-0.130467\pi$
$773$	51.0000	1.83434	0.917171	+	0.398493i	$0.130467\pi$
$774$	0	0
$775$	16.0000	0.574737
$776$	0	0
$777$	15.0000	0.538122
$778$	0	0
$779$	−24.0000	−0.859889
$780$	0	0
$781$	30.0000	1.07348
$782$	0	0
$783$	10.0000	0.357371
$784$	0	0
$785$	−22.0000	−0.785214
$786$	0	0
$787$	−52.0000	−1.85360	−0.926800	−	0.375555i	$-0.877452\pi$
$787$	−52.0000	−1.85360	−0.926800	+	0.375555i	$0.877452\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	18.0000	0.640006
$792$	0	0
$793$	−4.00000	−0.142044
$794$	0	0
$795$	4.00000	0.141865
$796$	0	0
$797$	−54.0000	−1.91278	−0.956389	−	0.292096i	$-0.905647\pi$
$797$	−54.0000	−1.91278	−0.956389	+	0.292096i	$0.905647\pi$
$798$	0	0
$799$	−27.0000	−0.955191
$800$	0	0
$801$	−4.00000	−0.141333
$802$	0	0
$803$	−4.00000	−0.141157
$804$	0	0
$805$	−12.0000	−0.422944
$806$	0	0
$807$	4.00000	0.140807
$808$	0	0
$809$	47.0000	1.65243	0.826216	−	0.563353i	$-0.190489\pi$
$809$	47.0000	1.65243	0.826216	+	0.563353i	$0.190489\pi$
$810$	0	0
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	0	0
$813$	−15.0000	−0.526073
$814$	0	0
$815$	−20.0000	−0.700569
$816$	0	0
$817$	14.0000	0.489798
$818$	0	0
$819$	−6.00000	−0.209657
$820$	0	0
$821$	15.0000	0.523504	0.261752	−	0.965135i	$-0.415700\pi$
$821$	15.0000	0.523504	0.261752	+	0.965135i	$0.415700\pi$
$822$	0	0
$823$	46.0000	1.60346	0.801730	−	0.597687i	$-0.203913\pi$
$823$	46.0000	1.60346	0.801730	+	0.597687i	$0.203913\pi$
$824$	0	0
$825$	−8.00000	−0.278524
$826$	0	0
$827$	18.0000	0.625921	0.312961	−	0.949766i	$-0.398679\pi$
$827$	18.0000	0.625921	0.312961	+	0.949766i	$0.398679\pi$
$828$	0	0
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	0	0
$831$	8.00000	0.277517
$832$	0	0
$833$	−6.00000	−0.207888
$834$	0	0
$835$	−8.00000	−0.276851
$836$	0	0
$837$	20.0000	0.691301
$838$	0	0
$839$	40.0000	1.38095	0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	18.0000	0.619953
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	21.0000	0.721569
$848$	0	0
$849$	20.0000	0.686398
$850$	0	0
$851$	20.0000	0.685591
$852$	0	0
$853$	9.00000	0.308154	0.154077	−	0.988059i	$-0.450760\pi$
$853$	9.00000	0.308154	0.154077	+	0.988059i	$0.450760\pi$
$854$	0	0
$855$	4.00000	0.136797
$856$	0	0
$857$	30.0000	1.02478	0.512390	−	0.858753i	$-0.328760\pi$
$857$	30.0000	1.02478	0.512390	+	0.858753i	$0.328760\pi$
$858$	0	0
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	0	0
$861$	36.0000	1.22688
$862$	0	0
$863$	9.00000	0.306364	0.153182	−	0.988198i	$-0.451048\pi$
$863$	9.00000	0.306364	0.153182	+	0.988198i	$0.451048\pi$
$864$	0	0
$865$	16.0000	0.544016
$866$	0	0
$867$	−8.00000	−0.271694
$868$	0	0
$869$	16.0000	0.542763
$870$	0	0
$871$	10.0000	0.338837
$872$	0	0
$873$	−20.0000	−0.676897
$874$	0	0
$875$	−27.0000	−0.912767
$876$	0	0
$877$	−23.0000	−0.776655	−0.388327	−	0.921521i	$-0.626947\pi$
$877$	−23.0000	−0.776655	−0.388327	+	0.921521i	$0.626947\pi$
$878$	0	0
$879$	7.00000	0.236104
$880$	0	0
$881$	−35.0000	−1.17918	−0.589590	−	0.807703i	$-0.700711\pi$
$881$	−35.0000	−1.17918	−0.589590	+	0.807703i	$0.700711\pi$
$882$	0	0
$883$	−11.0000	−0.370179	−0.185090	−	0.982722i	$-0.559258\pi$
$883$	−11.0000	−0.370179	−0.185090	+	0.982722i	$0.559258\pi$
$884$	0	0
$885$	−6.00000	−0.201688
$886$	0	0
$887$	−20.0000	−0.671534	−0.335767	−	0.941945i	$-0.608996\pi$
$887$	−20.0000	−0.671534	−0.335767	+	0.941945i	$0.608996\pi$
$888$	0	0
$889$	48.0000	1.60987
$890$	0	0
$891$	2.00000	0.0670025
$892$	0	0
$893$	18.0000	0.602347
$894$	0	0
$895$	−11.0000	−0.367689
$896$	0	0
$897$	4.00000	0.133556
$898$	0	0
$899$	8.00000	0.266815
$900$	0	0
$901$	12.0000	0.399778
$902$	0	0
$903$	−21.0000	−0.698836
$904$	0	0
$905$	20.0000	0.664822
$906$	0	0
$907$	−5.00000	−0.166022	−0.0830111	−	0.996549i	$-0.526454\pi$
$907$	−5.00000	−0.166022	−0.0830111	+	0.996549i	$0.526454\pi$
$908$	0	0
$909$	8.00000	0.265343
$910$	0	0
$911$	−26.0000	−0.861418	−0.430709	−	0.902491i	$-0.641737\pi$
$911$	−26.0000	−0.861418	−0.430709	+	0.902491i	$0.641737\pi$
$912$	0	0
$913$	−8.00000	−0.264761
$914$	0	0
$915$	−4.00000	−0.132236
$916$	0	0
$917$	63.0000	2.08044
$918$	0	0
$919$	48.0000	1.58337	0.791687	−	0.610927i	$-0.209203\pi$
$919$	48.0000	1.58337	0.791687	+	0.610927i	$0.209203\pi$
$920$	0	0
$921$	−26.0000	−0.856729
$922$	0	0
$923$	−15.0000	−0.493731
$924$	0	0
$925$	20.0000	0.657596
$926$	0	0
$927$	8.00000	0.262754
$928$	0	0
$929$	36.0000	1.18112	0.590561	−	0.806993i	$-0.298907\pi$
$929$	36.0000	1.18112	0.590561	+	0.806993i	$0.298907\pi$
$930$	0	0
$931$	4.00000	0.131095
$932$	0	0
$933$	−2.00000	−0.0654771
$934$	0	0
$935$	6.00000	0.196221
$936$	0	0
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	0	0
$939$	23.0000	0.750577
$940$	0	0
$941$	1.00000	0.0325991	0.0162995	−	0.999867i	$-0.494811\pi$
$941$	1.00000	0.0325991	0.0162995	+	0.999867i	$0.494811\pi$
$942$	0	0
$943$	48.0000	1.56310
$944$	0	0
$945$	−15.0000	−0.487950
$946$	0	0
$947$	−46.0000	−1.49480	−0.747400	−	0.664375i	$-0.768698\pi$
$947$	−46.0000	−1.49480	−0.747400	+	0.664375i	$0.768698\pi$
$948$	0	0
$949$	2.00000	0.0649227
$950$	0	0
$951$	−18.0000	−0.583690
$952$	0	0
$953$	−33.0000	−1.06897	−0.534487	−	0.845176i	$-0.679495\pi$
$953$	−33.0000	−1.06897	−0.534487	+	0.845176i	$0.679495\pi$
$954$	0	0
$955$	18.0000	0.582466
$956$	0	0
$957$	−4.00000	−0.129302
$958$	0	0
$959$	12.0000	0.387500
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	24.0000	0.773389
$964$	0	0
$965$	−12.0000	−0.386294
$966$	0	0
$967$	43.0000	1.38279	0.691393	−	0.722478i	$-0.256997\pi$
$967$	43.0000	1.38279	0.691393	+	0.722478i	$0.256997\pi$
$968$	0	0
$969$	−6.00000	−0.192748
$970$	0	0
$971$	−51.0000	−1.63667	−0.818334	−	0.574743i	$-0.805102\pi$
$971$	−51.0000	−1.63667	−0.818334	+	0.574743i	$0.805102\pi$
$972$	0	0
$973$	−57.0000	−1.82734
$974$	0	0
$975$	4.00000	0.128103
$976$	0	0
$977$	−14.0000	−0.447900	−0.223950	−	0.974601i	$-0.571895\pi$
$977$	−14.0000	−0.447900	−0.223950	+	0.974601i	$0.571895\pi$
$978$	0	0
$979$	4.00000	0.127841
$980$	0	0
$981$	−38.0000	−1.21325
$982$	0	0
$983$	−27.0000	−0.861166	−0.430583	−	0.902551i	$-0.641692\pi$
$983$	−27.0000	−0.861166	−0.430583	+	0.902551i	$0.641692\pi$
$984$	0	0
$985$	15.0000	0.477940
$986$	0	0
$987$	−27.0000	−0.859419
$988$	0	0
$989$	−28.0000	−0.890348
$990$	0	0
$991$	−2.00000	−0.0635321	−0.0317660	−	0.999495i	$-0.510113\pi$
$991$	−2.00000	−0.0635321	−0.0317660	+	0.999495i	$0.510113\pi$
$992$	0	0
$993$	−16.0000	−0.507745
$994$	0	0
$995$	26.0000	0.824255
$996$	0	0
$997$	10.0000	0.316703	0.158352	−	0.987383i	$-0.449382\pi$
$997$	10.0000	0.316703	0.158352	+	0.987383i	$0.449382\pi$
$998$	0	0
$999$	25.0000	0.790965

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	832.2.a.g.1.1		1
3.2	odd	2		7488.2.a.bi.1.1		1
4.3	odd	2		832.2.a.b.1.1		1
8.3	odd	2		416.2.a.b.1.1	yes	1
8.5	even	2		416.2.a.a.1.1	✓	1
12.11	even	2		7488.2.a.bj.1.1		1
16.3	odd	4		3328.2.b.c.1665.1		2
16.5	even	4		3328.2.b.r.1665.1		2
16.11	odd	4		3328.2.b.c.1665.2		2
16.13	even	4		3328.2.b.r.1665.2		2
24.5	odd	2		3744.2.a.e.1.1		1
24.11	even	2		3744.2.a.f.1.1		1
104.51	odd	2		5408.2.a.i.1.1		1
104.77	even	2		5408.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
416.2.a.a.1.1	✓	1	8.5	even	2
416.2.a.b.1.1	yes	1	8.3	odd	2
832.2.a.b.1.1		1	4.3	odd	2
832.2.a.g.1.1		1	1.1	even	1	trivial
3328.2.b.c.1665.1		2	16.3	odd	4
3328.2.b.c.1665.2		2	16.11	odd	4
3328.2.b.r.1665.1		2	16.5	even	4
3328.2.b.r.1665.2		2	16.13	even	4
3744.2.a.e.1.1		1	24.5	odd	2
3744.2.a.f.1.1		1	24.11	even	2
5408.2.a.e.1.1		1	104.77	even	2
5408.2.a.i.1.1		1	104.51	odd	2
7488.2.a.bi.1.1		1	3.2	odd	2
7488.2.a.bj.1.1		1	12.11	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 832 = 2^{6} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	832.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more