LMFDB - Embedded newform 54.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [54,2,Mod(1,54)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(54, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("54.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 54 = 2 \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	54.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:	$0.431192170915$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	54.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$

$q+1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} -3.00000 q^{10} +3.00000 q^{11} -4.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +2.00000 q^{19} -3.00000 q^{20} +3.00000 q^{22} +6.00000 q^{23} +4.00000 q^{25} -4.00000 q^{26} -1.00000 q^{28} -6.00000 q^{29} +5.00000 q^{31} +1.00000 q^{32} +3.00000 q^{35} +2.00000 q^{37} +2.00000 q^{38} -3.00000 q^{40} +6.00000 q^{41} -10.0000 q^{43} +3.00000 q^{44} +6.00000 q^{46} -6.00000 q^{47} -6.00000 q^{49} +4.00000 q^{50} -4.00000 q^{52} -9.00000 q^{53} -9.00000 q^{55} -1.00000 q^{56} -6.00000 q^{58} -12.0000 q^{59} +8.00000 q^{61} +5.00000 q^{62} +1.00000 q^{64} +12.0000 q^{65} +14.0000 q^{67} +3.00000 q^{70} -7.00000 q^{73} +2.00000 q^{74} +2.00000 q^{76} -3.00000 q^{77} +8.00000 q^{79} -3.00000 q^{80} +6.00000 q^{82} +3.00000 q^{83} -10.0000 q^{86} +3.00000 q^{88} +18.0000 q^{89} +4.00000 q^{91} +6.00000 q^{92} -6.00000 q^{94} -6.00000 q^{95} -1.00000 q^{97} -6.00000 q^{98} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	0	0
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−3.00000	−0.948683
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	−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		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	−3.00000	−0.670820
$21$	0	0
$22$	3.00000	0.639602
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	−4.00000	−0.784465
$27$	0	0
$28$	−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
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0	0
$35$	3.00000	0.507093
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	2.00000	0.324443
$39$	0	0
$40$	−3.00000	−0.474342
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	3.00000	0.452267
$45$	0	0
$46$	6.00000	0.884652
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	4.00000	0.565685
$51$	0	0
$52$	−4.00000	−0.554700
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	0	0
$55$	−9.00000	−1.21356
$56$	−1.00000	−0.133631
$57$	0	0
$58$	−6.00000	−0.787839
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	5.00000	0.635001
$63$	0	0
$64$	1.00000	0.125000
$65$	12.0000	1.48842
$66$	0	0
$67$	14.0000	1.71037	0.855186	−	0.518321i	$-0.173443\pi$
$67$	14.0000	1.71037	0.855186	+	0.518321i	$0.173443\pi$
$68$	0	0
$69$	0	0
$70$	3.00000	0.358569
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−7.00000	−0.819288	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000	−0.819288	−0.409644	+	0.912245i	$0.634347\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	2.00000	0.229416
$77$	−3.00000	−0.341882
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	−3.00000	−0.335410
$81$	0	0
$82$	6.00000	0.662589
$83$	3.00000	0.329293	0.164646	−	0.986353i	$-0.447352\pi$
$83$	3.00000	0.329293	0.164646	+	0.986353i	$0.447352\pi$
$84$	0	0
$85$	0	0
$86$	−10.0000	−1.07833
$87$	0	0
$88$	3.00000	0.319801
$89$	18.0000	1.90800	0.953998	−	0.299813i	$-0.0969242\pi$
$89$	18.0000	1.90800	0.953998	+	0.299813i	$0.0969242\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	6.00000	0.625543
$93$	0	0
$94$	−6.00000	−0.618853
$95$	−6.00000	−0.615587
$96$	0	0
$97$	−1.00000	−0.101535	−0.0507673	−	0.998711i	$-0.516167\pi$
$97$	−1.00000	−0.101535	−0.0507673	+	0.998711i	$0.516167\pi$
$98$	−6.00000	−0.606092
$99$	0	0
$100$	4.00000	0.400000
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	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
$106$	−9.00000	−0.874157
$107$	−9.00000	−0.870063	−0.435031	−	0.900415i	$-0.643263\pi$
$107$	−9.00000	−0.870063	−0.435031	+	0.900415i	$0.643263\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	−9.00000	−0.858116
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	−18.0000	−1.67851
$116$	−6.00000	−0.557086
$117$	0	0
$118$	−12.0000	−1.10469
$119$	0	0
$120$	0	0
$121$	−2.00000	−0.181818
$122$	8.00000	0.724286
$123$	0	0
$124$	5.00000	0.449013
$125$	3.00000	0.268328
$126$	0	0
$127$	−7.00000	−0.621150	−0.310575	−	0.950549i	$-0.600522\pi$
$127$	−7.00000	−0.621150	−0.310575	+	0.950549i	$0.600522\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	12.0000	1.05247
$131$	15.0000	1.31056	0.655278	−	0.755388i	$-0.272551\pi$
$131$	15.0000	1.31056	0.655278	+	0.755388i	$0.272551\pi$
$132$	0	0
$133$	−2.00000	−0.173422
$134$	14.0000	1.20942
$135$	0	0
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	3.00000	0.253546
$141$	0	0
$142$	0	0
$143$	−12.0000	−1.00349
$144$	0	0
$145$	18.0000	1.49482
$146$	−7.00000	−0.579324
$147$	0	0
$148$	2.00000	0.164399
$149$	−3.00000	−0.245770	−0.122885	−	0.992421i	$-0.539215\pi$
$149$	−3.00000	−0.245770	−0.122885	+	0.992421i	$0.539215\pi$
$150$	0	0
$151$	17.0000	1.38344	0.691720	−	0.722166i	$-0.256853\pi$
$151$	17.0000	1.38344	0.691720	+	0.722166i	$0.256853\pi$
$152$	2.00000	0.162221
$153$	0	0
$154$	−3.00000	−0.241747
$155$	−15.0000	−1.20483
$156$	0	0
$157$	−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
$159$	0	0
$160$	−3.00000	−0.237171
$161$	−6.00000	−0.472866
$162$	0	0
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	3.00000	0.232845
$167$	6.00000	0.464294	0.232147	−	0.972681i	$-0.425425\pi$
$167$	6.00000	0.464294	0.232147	+	0.972681i	$0.425425\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	0	0
$172$	−10.0000	−0.762493
$173$	−15.0000	−1.14043	−0.570214	−	0.821496i	$-0.693140\pi$
$173$	−15.0000	−1.14043	−0.570214	+	0.821496i	$0.693140\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	3.00000	0.226134
$177$	0	0
$178$	18.0000	1.34916
$179$	9.00000	0.672692	0.336346	−	0.941739i	$-0.390809\pi$
$179$	9.00000	0.672692	0.336346	+	0.941739i	$0.390809\pi$
$180$	0	0
$181$	−16.0000	−1.18927	−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000	−1.18927	−0.594635	+	0.803996i	$0.702704\pi$
$182$	4.00000	0.296500
$183$	0	0
$184$	6.00000	0.442326
$185$	−6.00000	−0.441129
$186$	0	0
$187$	0	0
$188$	−6.00000	−0.437595
$189$	0	0
$190$	−6.00000	−0.435286
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	5.00000	0.359908	0.179954	−	0.983675i	$-0.442405\pi$
$193$	5.00000	0.359908	0.179954	+	0.983675i	$0.442405\pi$
$194$	−1.00000	−0.0717958
$195$	0	0
$196$	−6.00000	−0.428571
$197$	−9.00000	−0.641223	−0.320612	−	0.947211i	$-0.603888\pi$
$197$	−9.00000	−0.641223	−0.320612	+	0.947211i	$0.603888\pi$
$198$	0	0
$199$	−7.00000	−0.496217	−0.248108	−	0.968732i	$-0.579809\pi$
$199$	−7.00000	−0.496217	−0.248108	+	0.968732i	$0.579809\pi$
$200$	4.00000	0.282843
$201$	0	0
$202$	3.00000	0.211079
$203$	6.00000	0.421117
$204$	0	0
$205$	−18.0000	−1.25717
$206$	−4.00000	−0.278693
$207$	0	0
$208$	−4.00000	−0.277350
$209$	6.00000	0.415029
$210$	0	0
$211$	−22.0000	−1.51454	−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000	−1.51454	−0.757271	+	0.653101i	$0.773468\pi$
$212$	−9.00000	−0.618123
$213$	0	0
$214$	−9.00000	−0.615227
$215$	30.0000	2.04598
$216$	0	0
$217$	−5.00000	−0.339422
$218$	2.00000	0.135457
$219$	0	0
$220$	−9.00000	−0.606780
$221$	0	0
$222$	0	0
$223$	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
$225$	0	0
$226$	6.00000	0.399114
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	0	0
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	−18.0000	−1.18688
$231$	0	0
$232$	−6.00000	−0.393919
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	18.0000	1.17419
$236$	−12.0000	−0.781133
$237$	0	0
$238$	0	0
$239$	−30.0000	−1.94054	−0.970269	−	0.242028i	$-0.922188\pi$
$239$	−30.0000	−1.94054	−0.970269	+	0.242028i	$0.922188\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	−2.00000	−0.128565
$243$	0	0
$244$	8.00000	0.512148
$245$	18.0000	1.14998
$246$	0	0
$247$	−8.00000	−0.509028
$248$	5.00000	0.317500
$249$	0	0
$250$	3.00000	0.189737
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	18.0000	1.13165
$254$	−7.00000	−0.439219
$255$	0	0
$256$	1.00000	0.0625000
$257$	−12.0000	−0.748539	−0.374270	−	0.927320i	$-0.622107\pi$
$257$	−12.0000	−0.748539	−0.374270	+	0.927320i	$0.622107\pi$
$258$	0	0
$259$	−2.00000	−0.124274
$260$	12.0000	0.744208
$261$	0	0
$262$	15.0000	0.926703
$263$	30.0000	1.84988	0.924940	−	0.380114i	$-0.124115\pi$
$263$	30.0000	1.84988	0.924940	+	0.380114i	$0.124115\pi$
$264$	0	0
$265$	27.0000	1.65860
$266$	−2.00000	−0.122628
$267$	0	0
$268$	14.0000	0.855186
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	−25.0000	−1.51864	−0.759321	−	0.650716i	$-0.774469\pi$
$271$	−25.0000	−1.51864	−0.759321	+	0.650716i	$0.774469\pi$
$272$	0	0
$273$	0	0
$274$	−6.00000	−0.362473
$275$	12.0000	0.723627
$276$	0	0
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	−4.00000	−0.239904
$279$	0	0
$280$	3.00000	0.179284
$281$	−24.0000	−1.43172	−0.715860	−	0.698244i	$-0.753965\pi$
$281$	−24.0000	−1.43172	−0.715860	+	0.698244i	$0.753965\pi$
$282$	0	0
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	0	0
$285$	0	0
$286$	−12.0000	−0.709575
$287$	−6.00000	−0.354169
$288$	0	0
$289$	−17.0000	−1.00000
$290$	18.0000	1.05700
$291$	0	0
$292$	−7.00000	−0.409644
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	36.0000	2.09600
$296$	2.00000	0.116248
$297$	0	0
$298$	−3.00000	−0.173785
$299$	−24.0000	−1.38796
$300$	0	0
$301$	10.0000	0.576390
$302$	17.0000	0.978240
$303$	0	0
$304$	2.00000	0.114708
$305$	−24.0000	−1.37424
$306$	0	0
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	−3.00000	−0.170941
$309$	0	0
$310$	−15.0000	−0.851943
$311$	6.00000	0.340229	0.170114	−	0.985424i	$-0.445586\pi$
$311$	6.00000	0.340229	0.170114	+	0.985424i	$0.445586\pi$
$312$	0	0
$313$	−19.0000	−1.07394	−0.536972	−	0.843600i	$-0.680432\pi$
$313$	−19.0000	−1.07394	−0.536972	+	0.843600i	$0.680432\pi$
$314$	−4.00000	−0.225733
$315$	0	0
$316$	8.00000	0.450035
$317$	3.00000	0.168497	0.0842484	−	0.996445i	$-0.473151\pi$
$317$	3.00000	0.168497	0.0842484	+	0.996445i	$0.473151\pi$
$318$	0	0
$319$	−18.0000	−1.00781
$320$	−3.00000	−0.167705
$321$	0	0
$322$	−6.00000	−0.334367
$323$	0	0
$324$	0	0
$325$	−16.0000	−0.887520
$326$	20.0000	1.10770
$327$	0	0
$328$	6.00000	0.331295
$329$	6.00000	0.330791
$330$	0	0
$331$	−10.0000	−0.549650	−0.274825	−	0.961494i	$-0.588620\pi$
$331$	−10.0000	−0.549650	−0.274825	+	0.961494i	$0.588620\pi$
$332$	3.00000	0.164646
$333$	0	0
$334$	6.00000	0.328305
$335$	−42.0000	−2.29471
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	3.00000	0.163178
$339$	0	0
$340$	0	0
$341$	15.0000	0.812296
$342$	0	0
$343$	13.0000	0.701934
$344$	−10.0000	−0.539164
$345$	0	0
$346$	−15.0000	−0.806405
$347$	−3.00000	−0.161048	−0.0805242	−	0.996753i	$-0.525659\pi$
$347$	−3.00000	−0.161048	−0.0805242	+	0.996753i	$0.525659\pi$
$348$	0	0
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	−4.00000	−0.213809
$351$	0	0
$352$	3.00000	0.159901
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	0	0
$356$	18.0000	0.953998
$357$	0	0
$358$	9.00000	0.475665
$359$	−18.0000	−0.950004	−0.475002	−	0.879985i	$-0.657553\pi$
$359$	−18.0000	−0.950004	−0.475002	+	0.879985i	$0.657553\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	−16.0000	−0.840941
$363$	0	0
$364$	4.00000	0.209657
$365$	21.0000	1.09919
$366$	0	0
$367$	17.0000	0.887393	0.443696	−	0.896177i	$-0.353667\pi$
$367$	17.0000	0.887393	0.443696	+	0.896177i	$0.353667\pi$
$368$	6.00000	0.312772
$369$	0	0
$370$	−6.00000	−0.311925
$371$	9.00000	0.467257
$372$	0	0
$373$	32.0000	1.65690	0.828449	−	0.560065i	$-0.189224\pi$
$373$	32.0000	1.65690	0.828449	+	0.560065i	$0.189224\pi$
$374$	0	0
$375$	0	0
$376$	−6.00000	−0.309426
$377$	24.0000	1.23606
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	−6.00000	−0.307794
$381$	0	0
$382$	12.0000	0.613973
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	0	0
$385$	9.00000	0.458682
$386$	5.00000	0.254493
$387$	0	0
$388$	−1.00000	−0.0507673
$389$	21.0000	1.06474	0.532371	−	0.846511i	$-0.321301\pi$
$389$	21.0000	1.06474	0.532371	+	0.846511i	$0.321301\pi$
$390$	0	0
$391$	0	0
$392$	−6.00000	−0.303046
$393$	0	0
$394$	−9.00000	−0.453413
$395$	−24.0000	−1.20757
$396$	0	0
$397$	20.0000	1.00377	0.501886	−	0.864934i	$-0.332640\pi$
$397$	20.0000	1.00377	0.501886	+	0.864934i	$0.332640\pi$
$398$	−7.00000	−0.350878
$399$	0	0
$400$	4.00000	0.200000
$401$	−12.0000	−0.599251	−0.299626	−	0.954057i	$-0.596862\pi$
$401$	−12.0000	−0.599251	−0.299626	+	0.954057i	$0.596862\pi$
$402$	0	0
$403$	−20.0000	−0.996271
$404$	3.00000	0.149256
$405$	0	0
$406$	6.00000	0.297775
$407$	6.00000	0.297409
$408$	0	0
$409$	23.0000	1.13728	0.568638	−	0.822588i	$-0.307470\pi$
$409$	23.0000	1.13728	0.568638	+	0.822588i	$0.307470\pi$
$410$	−18.0000	−0.888957
$411$	0	0
$412$	−4.00000	−0.197066
$413$	12.0000	0.590481
$414$	0	0
$415$	−9.00000	−0.441793
$416$	−4.00000	−0.196116
$417$	0	0
$418$	6.00000	0.293470
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	8.00000	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	−22.0000	−1.07094
$423$	0	0
$424$	−9.00000	−0.437079
$425$	0	0
$426$	0	0
$427$	−8.00000	−0.387147
$428$	−9.00000	−0.435031
$429$	0	0
$430$	30.0000	1.44673
$431$	−18.0000	−0.867029	−0.433515	−	0.901146i	$-0.642727\pi$
$431$	−18.0000	−0.867029	−0.433515	+	0.901146i	$0.642727\pi$
$432$	0	0
$433$	29.0000	1.39365	0.696826	−	0.717241i	$-0.254595\pi$
$433$	29.0000	1.39365	0.696826	+	0.717241i	$0.254595\pi$
$434$	−5.00000	−0.240008
$435$	0	0
$436$	2.00000	0.0957826
$437$	12.0000	0.574038
$438$	0	0
$439$	−19.0000	−0.906821	−0.453410	−	0.891302i	$-0.649793\pi$
$439$	−19.0000	−0.906821	−0.453410	+	0.891302i	$0.649793\pi$
$440$	−9.00000	−0.429058
$441$	0	0
$442$	0	0
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	0	0
$445$	−54.0000	−2.55985
$446$	8.00000	0.378811
$447$	0	0
$448$	−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	$0.360367\pi$
$450$	0	0
$451$	18.0000	0.847587
$452$	6.00000	0.282216
$453$	0	0
$454$	12.0000	0.563188
$455$	−12.0000	−0.562569
$456$	0	0
$457$	−1.00000	−0.0467780	−0.0233890	−	0.999726i	$-0.507446\pi$
$457$	−1.00000	−0.0467780	−0.0233890	+	0.999726i	$0.507446\pi$
$458$	14.0000	0.654177
$459$	0	0
$460$	−18.0000	−0.839254
$461$	21.0000	0.978068	0.489034	−	0.872265i	$-0.337349\pi$
$461$	21.0000	0.978068	0.489034	+	0.872265i	$0.337349\pi$
$462$	0	0
$463$	−13.0000	−0.604161	−0.302081	−	0.953282i	$-0.597681\pi$
$463$	−13.0000	−0.604161	−0.302081	+	0.953282i	$0.597681\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	−18.0000	−0.833834
$467$	27.0000	1.24941	0.624705	−	0.780860i	$-0.285219\pi$
$467$	27.0000	1.24941	0.624705	+	0.780860i	$0.285219\pi$
$468$	0	0
$469$	−14.0000	−0.646460
$470$	18.0000	0.830278
$471$	0	0
$472$	−12.0000	−0.552345
$473$	−30.0000	−1.37940
$474$	0	0
$475$	8.00000	0.367065
$476$	0	0
$477$	0	0
$478$	−30.0000	−1.37217
$479$	−6.00000	−0.274147	−0.137073	−	0.990561i	$-0.543770\pi$
$479$	−6.00000	−0.274147	−0.137073	+	0.990561i	$0.543770\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	−10.0000	−0.455488
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	3.00000	0.136223
$486$	0	0
$487$	−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
$489$	0	0
$490$	18.0000	0.813157
$491$	−39.0000	−1.76005	−0.880023	−	0.474932i	$-0.842473\pi$
$491$	−39.0000	−1.76005	−0.880023	+	0.474932i	$0.842473\pi$
$492$	0	0
$493$	0	0
$494$	−8.00000	−0.359937
$495$	0	0
$496$	5.00000	0.224507
$497$	0	0
$498$	0	0
$499$	14.0000	0.626726	0.313363	−	0.949633i	$-0.398544\pi$
$499$	14.0000	0.626726	0.313363	+	0.949633i	$0.398544\pi$
$500$	3.00000	0.134164
$501$	0	0
$502$	0	0
$503$	18.0000	0.802580	0.401290	−	0.915951i	$-0.368562\pi$
$503$	18.0000	0.802580	0.401290	+	0.915951i	$0.368562\pi$
$504$	0	0
$505$	−9.00000	−0.400495
$506$	18.0000	0.800198
$507$	0	0
$508$	−7.00000	−0.310575
$509$	15.0000	0.664863	0.332432	−	0.943127i	$-0.392131\pi$
$509$	15.0000	0.664863	0.332432	+	0.943127i	$0.392131\pi$
$510$	0	0
$511$	7.00000	0.309662
$512$	1.00000	0.0441942
$513$	0	0
$514$	−12.0000	−0.529297
$515$	12.0000	0.528783
$516$	0	0
$517$	−18.0000	−0.791639
$518$	−2.00000	−0.0878750
$519$	0	0
$520$	12.0000	0.526235
$521$	−36.0000	−1.57719	−0.788594	−	0.614914i	$-0.789191\pi$
$521$	−36.0000	−1.57719	−0.788594	+	0.614914i	$0.789191\pi$
$522$	0	0
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	15.0000	0.655278
$525$	0	0
$526$	30.0000	1.30806
$527$	0	0
$528$	0	0
$529$	13.0000	0.565217
$530$	27.0000	1.17281
$531$	0	0
$532$	−2.00000	−0.0867110
$533$	−24.0000	−1.03956
$534$	0	0
$535$	27.0000	1.16731
$536$	14.0000	0.604708
$537$	0	0
$538$	18.0000	0.776035
$539$	−18.0000	−0.775315
$540$	0	0
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	−25.0000	−1.07384
$543$	0	0
$544$	0	0
$545$	−6.00000	−0.257012
$546$	0	0
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	−6.00000	−0.256307
$549$	0	0
$550$	12.0000	0.511682
$551$	−12.0000	−0.511217
$552$	0	0
$553$	−8.00000	−0.340195
$554$	8.00000	0.339887
$555$	0	0
$556$	−4.00000	−0.169638
$557$	−27.0000	−1.14403	−0.572013	−	0.820244i	$-0.693837\pi$
$557$	−27.0000	−1.14403	−0.572013	+	0.820244i	$0.693837\pi$
$558$	0	0
$559$	40.0000	1.69182
$560$	3.00000	0.126773
$561$	0	0
$562$	−24.0000	−1.01238
$563$	−3.00000	−0.126435	−0.0632175	−	0.998000i	$-0.520136\pi$
$563$	−3.00000	−0.126435	−0.0632175	+	0.998000i	$0.520136\pi$
$564$	0	0
$565$	−18.0000	−0.757266
$566$	14.0000	0.588464
$567$	0	0
$568$	0	0
$569$	12.0000	0.503066	0.251533	−	0.967849i	$-0.419065\pi$
$569$	12.0000	0.503066	0.251533	+	0.967849i	$0.419065\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	−12.0000	−0.501745
$573$	0	0
$574$	−6.00000	−0.250435
$575$	24.0000	1.00087
$576$	0	0
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	−17.0000	−0.707107
$579$	0	0
$580$	18.0000	0.747409
$581$	−3.00000	−0.124461
$582$	0	0
$583$	−27.0000	−1.11823
$584$	−7.00000	−0.289662
$585$	0	0
$586$	6.00000	0.247858
$587$	3.00000	0.123823	0.0619116	−	0.998082i	$-0.480280\pi$
$587$	3.00000	0.123823	0.0619116	+	0.998082i	$0.480280\pi$
$588$	0	0
$589$	10.0000	0.412043
$590$	36.0000	1.48210
$591$	0	0
$592$	2.00000	0.0821995
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	0	0
$595$	0	0
$596$	−3.00000	−0.122885
$597$	0	0
$598$	−24.0000	−0.981433
$599$	42.0000	1.71607	0.858037	−	0.513588i	$-0.171684\pi$
$599$	42.0000	1.71607	0.858037	+	0.513588i	$0.171684\pi$
$600$	0	0
$601$	35.0000	1.42768	0.713840	−	0.700309i	$-0.246954\pi$
$601$	35.0000	1.42768	0.713840	+	0.700309i	$0.246954\pi$
$602$	10.0000	0.407570
$603$	0	0
$604$	17.0000	0.691720
$605$	6.00000	0.243935
$606$	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
$609$	0	0
$610$	−24.0000	−0.971732
$611$	24.0000	0.970936
$612$	0	0
$613$	−34.0000	−1.37325	−0.686624	−	0.727013i	$-0.740908\pi$
$613$	−34.0000	−1.37325	−0.686624	+	0.727013i	$0.740908\pi$
$614$	−16.0000	−0.645707
$615$	0	0
$616$	−3.00000	−0.120873
$617$	42.0000	1.69086	0.845428	−	0.534089i	$-0.179345\pi$
$617$	42.0000	1.69086	0.845428	+	0.534089i	$0.179345\pi$
$618$	0	0
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	−15.0000	−0.602414
$621$	0	0
$622$	6.00000	0.240578
$623$	−18.0000	−0.721155
$624$	0	0
$625$	−29.0000	−1.16000
$626$	−19.0000	−0.759393
$627$	0	0
$628$	−4.00000	−0.159617
$629$	0	0
$630$	0	0
$631$	−25.0000	−0.995234	−0.497617	−	0.867397i	$-0.665792\pi$
$631$	−25.0000	−0.995234	−0.497617	+	0.867397i	$0.665792\pi$
$632$	8.00000	0.318223
$633$	0	0
$634$	3.00000	0.119145
$635$	21.0000	0.833360
$636$	0	0
$637$	24.0000	0.950915
$638$	−18.0000	−0.712627
$639$	0	0
$640$	−3.00000	−0.118585
$641$	−42.0000	−1.65890	−0.829450	−	0.558581i	$-0.811346\pi$
$641$	−42.0000	−1.65890	−0.829450	+	0.558581i	$0.811346\pi$
$642$	0	0
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	−6.00000	−0.236433
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	−36.0000	−1.41312
$650$	−16.0000	−0.627572
$651$	0	0
$652$	20.0000	0.783260
$653$	−39.0000	−1.52619	−0.763094	−	0.646288i	$-0.776321\pi$
$653$	−39.0000	−1.52619	−0.763094	+	0.646288i	$0.776321\pi$
$654$	0	0
$655$	−45.0000	−1.75830
$656$	6.00000	0.234261
$657$	0	0
$658$	6.00000	0.233904
$659$	21.0000	0.818044	0.409022	−	0.912525i	$-0.365870\pi$
$659$	21.0000	0.818044	0.409022	+	0.912525i	$0.365870\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	−10.0000	−0.388661
$663$	0	0
$664$	3.00000	0.116423
$665$	6.00000	0.232670
$666$	0	0
$667$	−36.0000	−1.39393
$668$	6.00000	0.232147
$669$	0	0
$670$	−42.0000	−1.62260
$671$	24.0000	0.926510
$672$	0	0
$673$	−19.0000	−0.732396	−0.366198	−	0.930537i	$-0.619341\pi$
$673$	−19.0000	−0.732396	−0.366198	+	0.930537i	$0.619341\pi$
$674$	−22.0000	−0.847408
$675$	0	0
$676$	3.00000	0.115385
$677$	−42.0000	−1.61419	−0.807096	−	0.590421i	$-0.798962\pi$
$677$	−42.0000	−1.61419	−0.807096	+	0.590421i	$0.798962\pi$
$678$	0	0
$679$	1.00000	0.0383765
$680$	0	0
$681$	0	0
$682$	15.0000	0.574380
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	0	0
$685$	18.0000	0.687745
$686$	13.0000	0.496342
$687$	0	0
$688$	−10.0000	−0.381246
$689$	36.0000	1.37149
$690$	0	0
$691$	44.0000	1.67384	0.836919	−	0.547326i	$-0.184354\pi$
$691$	44.0000	1.67384	0.836919	+	0.547326i	$0.184354\pi$
$692$	−15.0000	−0.570214
$693$	0	0
$694$	−3.00000	−0.113878
$695$	12.0000	0.455186
$696$	0	0
$697$	0	0
$698$	−10.0000	−0.378506
$699$	0	0
$700$	−4.00000	−0.151186
$701$	−9.00000	−0.339925	−0.169963	−	0.985451i	$-0.554365\pi$
$701$	−9.00000	−0.339925	−0.169963	+	0.985451i	$0.554365\pi$
$702$	0	0
$703$	4.00000	0.150863
$704$	3.00000	0.113067
$705$	0	0
$706$	−6.00000	−0.225813
$707$	−3.00000	−0.112827
$708$	0	0
$709$	44.0000	1.65245	0.826227	−	0.563337i	$-0.190483\pi$
$709$	44.0000	1.65245	0.826227	+	0.563337i	$0.190483\pi$
$710$	0	0
$711$	0	0
$712$	18.0000	0.674579
$713$	30.0000	1.12351
$714$	0	0
$715$	36.0000	1.34632
$716$	9.00000	0.336346
$717$	0	0
$718$	−18.0000	−0.671754
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	−15.0000	−0.558242
$723$	0	0
$724$	−16.0000	−0.594635
$725$	−24.0000	−0.891338
$726$	0	0
$727$	−1.00000	−0.0370879	−0.0185440	−	0.999828i	$-0.505903\pi$
$727$	−1.00000	−0.0370879	−0.0185440	+	0.999828i	$0.505903\pi$
$728$	4.00000	0.148250
$729$	0	0
$730$	21.0000	0.777245
$731$	0	0
$732$	0	0
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	17.0000	0.627481
$735$	0	0
$736$	6.00000	0.221163
$737$	42.0000	1.54709
$738$	0	0
$739$	−16.0000	−0.588570	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000	−0.588570	−0.294285	+	0.955718i	$0.595081\pi$
$740$	−6.00000	−0.220564
$741$	0	0
$742$	9.00000	0.330400
$743$	−12.0000	−0.440237	−0.220119	−	0.975473i	$-0.570644\pi$
$743$	−12.0000	−0.440237	−0.220119	+	0.975473i	$0.570644\pi$
$744$	0	0
$745$	9.00000	0.329734
$746$	32.0000	1.17160
$747$	0	0
$748$	0	0
$749$	9.00000	0.328853
$750$	0	0
$751$	41.0000	1.49611	0.748056	−	0.663636i	$-0.230988\pi$
$751$	41.0000	1.49611	0.748056	+	0.663636i	$0.230988\pi$
$752$	−6.00000	−0.218797
$753$	0	0
$754$	24.0000	0.874028
$755$	−51.0000	−1.85608
$756$	0	0
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	20.0000	0.726433
$759$	0	0
$760$	−6.00000	−0.217643
$761$	−48.0000	−1.74000	−0.869999	−	0.493053i	$-0.835881\pi$
$761$	−48.0000	−1.74000	−0.869999	+	0.493053i	$0.835881\pi$
$762$	0	0
$763$	−2.00000	−0.0724049
$764$	12.0000	0.434145
$765$	0	0
$766$	24.0000	0.867155
$767$	48.0000	1.73318
$768$	0	0
$769$	−31.0000	−1.11789	−0.558944	−	0.829205i	$-0.688793\pi$
$769$	−31.0000	−1.11789	−0.558944	+	0.829205i	$0.688793\pi$
$770$	9.00000	0.324337
$771$	0	0
$772$	5.00000	0.179954
$773$	18.0000	0.647415	0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000	0.647415	0.323708	+	0.946157i	$0.395071\pi$
$774$	0	0
$775$	20.0000	0.718421
$776$	−1.00000	−0.0358979
$777$	0	0
$778$	21.0000	0.752886
$779$	12.0000	0.429945
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−6.00000	−0.214286
$785$	12.0000	0.428298
$786$	0	0
$787$	32.0000	1.14068	0.570338	−	0.821410i	$-0.306812\pi$
$787$	32.0000	1.14068	0.570338	+	0.821410i	$0.306812\pi$
$788$	−9.00000	−0.320612
$789$	0	0
$790$	−24.0000	−0.853882
$791$	−6.00000	−0.213335
$792$	0	0
$793$	−32.0000	−1.13635
$794$	20.0000	0.709773
$795$	0	0
$796$	−7.00000	−0.248108
$797$	−39.0000	−1.38145	−0.690725	−	0.723117i	$-0.742709\pi$
$797$	−39.0000	−1.38145	−0.690725	+	0.723117i	$0.742709\pi$
$798$	0	0
$799$	0	0
$800$	4.00000	0.141421
$801$	0	0
$802$	−12.0000	−0.423735
$803$	−21.0000	−0.741074
$804$	0	0
$805$	18.0000	0.634417
$806$	−20.0000	−0.704470
$807$	0	0
$808$	3.00000	0.105540
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	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
$813$	0	0
$814$	6.00000	0.210300
$815$	−60.0000	−2.10171
$816$	0	0
$817$	−20.0000	−0.699711
$818$	23.0000	0.804176
$819$	0	0
$820$	−18.0000	−0.628587
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	0	0
$823$	−31.0000	−1.08059	−0.540296	−	0.841475i	$-0.681688\pi$
$823$	−31.0000	−1.08059	−0.540296	+	0.841475i	$0.681688\pi$
$824$	−4.00000	−0.139347
$825$	0	0
$826$	12.0000	0.417533
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	−9.00000	−0.312395
$831$	0	0
$832$	−4.00000	−0.138675
$833$	0	0
$834$	0	0
$835$	−18.0000	−0.622916
$836$	6.00000	0.207514
$837$	0	0
$838$	−12.0000	−0.414533
$839$	−24.0000	−0.828572	−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000	−0.828572	−0.414286	+	0.910147i	$0.635969\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	8.00000	0.275698
$843$	0	0
$844$	−22.0000	−0.757271
$845$	−9.00000	−0.309609
$846$	0	0
$847$	2.00000	0.0687208
$848$	−9.00000	−0.309061
$849$	0	0
$850$	0	0
$851$	12.0000	0.411355
$852$	0	0
$853$	−10.0000	−0.342393	−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000	−0.342393	−0.171197	+	0.985237i	$0.554763\pi$
$854$	−8.00000	−0.273754
$855$	0	0
$856$	−9.00000	−0.307614
$857$	48.0000	1.63965	0.819824	−	0.572615i	$-0.194071\pi$
$857$	48.0000	1.63965	0.819824	+	0.572615i	$0.194071\pi$
$858$	0	0
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	30.0000	1.02299
$861$	0	0
$862$	−18.0000	−0.613082
$863$	36.0000	1.22545	0.612727	−	0.790295i	$-0.290072\pi$
$863$	36.0000	1.22545	0.612727	+	0.790295i	$0.290072\pi$
$864$	0	0
$865$	45.0000	1.53005
$866$	29.0000	0.985460
$867$	0	0
$868$	−5.00000	−0.169711
$869$	24.0000	0.814144
$870$	0	0
$871$	−56.0000	−1.89749
$872$	2.00000	0.0677285
$873$	0	0
$874$	12.0000	0.405906
$875$	−3.00000	−0.101419
$876$	0	0
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	−19.0000	−0.641219
$879$	0	0
$880$	−9.00000	−0.303390
$881$	36.0000	1.21287	0.606435	−	0.795133i	$-0.292599\pi$
$881$	36.0000	1.21287	0.606435	+	0.795133i	$0.292599\pi$
$882$	0	0
$883$	2.00000	0.0673054	0.0336527	−	0.999434i	$-0.489286\pi$
$883$	2.00000	0.0673054	0.0336527	+	0.999434i	$0.489286\pi$
$884$	0	0
$885$	0	0
$886$	12.0000	0.403148
$887$	−12.0000	−0.402921	−0.201460	−	0.979497i	$-0.564569\pi$
$887$	−12.0000	−0.402921	−0.201460	+	0.979497i	$0.564569\pi$
$888$	0	0
$889$	7.00000	0.234772
$890$	−54.0000	−1.81008
$891$	0	0
$892$	8.00000	0.267860
$893$	−12.0000	−0.401565
$894$	0	0
$895$	−27.0000	−0.902510
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	18.0000	0.600668
$899$	−30.0000	−1.00056
$900$	0	0
$901$	0	0
$902$	18.0000	0.599334
$903$	0	0
$904$	6.00000	0.199557
$905$	48.0000	1.59557
$906$	0	0
$907$	26.0000	0.863316	0.431658	−	0.902037i	$-0.357929\pi$
$907$	26.0000	0.863316	0.431658	+	0.902037i	$0.357929\pi$
$908$	12.0000	0.398234
$909$	0	0
$910$	−12.0000	−0.397796
$911$	−6.00000	−0.198789	−0.0993944	−	0.995048i	$-0.531691\pi$
$911$	−6.00000	−0.198789	−0.0993944	+	0.995048i	$0.531691\pi$
$912$	0	0
$913$	9.00000	0.297857
$914$	−1.00000	−0.0330771
$915$	0	0
$916$	14.0000	0.462573
$917$	−15.0000	−0.495344
$918$	0	0
$919$	29.0000	0.956622	0.478311	−	0.878191i	$-0.341249\pi$
$919$	29.0000	0.956622	0.478311	+	0.878191i	$0.341249\pi$
$920$	−18.0000	−0.593442
$921$	0	0
$922$	21.0000	0.691598
$923$	0	0
$924$	0	0
$925$	8.00000	0.263038
$926$	−13.0000	−0.427207
$927$	0	0
$928$	−6.00000	−0.196960
$929$	12.0000	0.393707	0.196854	−	0.980433i	$-0.436928\pi$
$929$	12.0000	0.393707	0.196854	+	0.980433i	$0.436928\pi$
$930$	0	0
$931$	−12.0000	−0.393284
$932$	−18.0000	−0.589610
$933$	0	0
$934$	27.0000	0.883467
$935$	0	0
$936$	0	0
$937$	11.0000	0.359354	0.179677	−	0.983726i	$-0.442495\pi$
$937$	11.0000	0.359354	0.179677	+	0.983726i	$0.442495\pi$
$938$	−14.0000	−0.457116
$939$	0	0
$940$	18.0000	0.587095
$941$	33.0000	1.07577	0.537885	−	0.843018i	$-0.319224\pi$
$941$	33.0000	1.07577	0.537885	+	0.843018i	$0.319224\pi$
$942$	0	0
$943$	36.0000	1.17232
$944$	−12.0000	−0.390567
$945$	0	0
$946$	−30.0000	−0.975384
$947$	−51.0000	−1.65728	−0.828639	−	0.559784i	$-0.810884\pi$
$947$	−51.0000	−1.65728	−0.828639	+	0.559784i	$0.810884\pi$
$948$	0	0
$949$	28.0000	0.908918
$950$	8.00000	0.259554
$951$	0	0
$952$	0	0
$953$	36.0000	1.16615	0.583077	−	0.812417i	$-0.301849\pi$
$953$	36.0000	1.16615	0.583077	+	0.812417i	$0.301849\pi$
$954$	0	0
$955$	−36.0000	−1.16493
$956$	−30.0000	−0.970269
$957$	0	0
$958$	−6.00000	−0.193851
$959$	6.00000	0.193750
$960$	0	0
$961$	−6.00000	−0.193548
$962$	−8.00000	−0.257930
$963$	0	0
$964$	−10.0000	−0.322078
$965$	−15.0000	−0.482867
$966$	0	0
$967$	23.0000	0.739630	0.369815	−	0.929105i	$-0.379421\pi$
$967$	23.0000	0.739630	0.369815	+	0.929105i	$0.379421\pi$
$968$	−2.00000	−0.0642824
$969$	0	0
$970$	3.00000	0.0963242
$971$	27.0000	0.866471	0.433236	−	0.901281i	$-0.357372\pi$
$971$	27.0000	0.866471	0.433236	+	0.901281i	$0.357372\pi$
$972$	0	0
$973$	4.00000	0.128234
$974$	−16.0000	−0.512673
$975$	0	0
$976$	8.00000	0.256074
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	0	0
$979$	54.0000	1.72585
$980$	18.0000	0.574989
$981$	0	0
$982$	−39.0000	−1.24454
$983$	−6.00000	−0.191370	−0.0956851	−	0.995412i	$-0.530504\pi$
$983$	−6.00000	−0.191370	−0.0956851	+	0.995412i	$0.530504\pi$
$984$	0	0
$985$	27.0000	0.860292
$986$	0	0
$987$	0	0
$988$	−8.00000	−0.254514
$989$	−60.0000	−1.90789
$990$	0	0
$991$	47.0000	1.49300	0.746502	−	0.665383i	$-0.231732\pi$
$991$	47.0000	1.49300	0.746502	+	0.665383i	$0.231732\pi$
$992$	5.00000	0.158750
$993$	0	0
$994$	0	0
$995$	21.0000	0.665745
$996$	0	0
$997$	−28.0000	−0.886769	−0.443384	−	0.896332i	$-0.646222\pi$
$997$	−28.0000	−0.886769	−0.443384	+	0.896332i	$0.646222\pi$
$998$	14.0000	0.443162
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	54.2.a.b.1.1	yes	1
3.2	odd	2		54.2.a.a.1.1	✓	1
4.3	odd	2		432.2.a.b.1.1		1
5.2	odd	4		1350.2.c.k.649.2		2
5.3	odd	4		1350.2.c.k.649.1		2
5.4	even	2		1350.2.a.h.1.1		1
7.6	odd	2		2646.2.a.bd.1.1		1
8.3	odd	2		1728.2.a.z.1.1		1
8.5	even	2		1728.2.a.y.1.1		1
9.2	odd	6		162.2.c.c.109.1		2
9.4	even	3		162.2.c.b.55.1		2
9.5	odd	6		162.2.c.c.55.1		2
9.7	even	3		162.2.c.b.109.1		2
11.10	odd	2		6534.2.a.b.1.1		1
12.11	even	2		432.2.a.g.1.1		1
13.12	even	2		9126.2.a.r.1.1		1
15.2	even	4		1350.2.c.b.649.1		2
15.8	even	4		1350.2.c.b.649.2		2
15.14	odd	2		1350.2.a.r.1.1		1
21.20	even	2		2646.2.a.a.1.1		1
24.5	odd	2		1728.2.a.c.1.1		1
24.11	even	2		1728.2.a.d.1.1		1
33.32	even	2		6534.2.a.bc.1.1		1
36.7	odd	6		1296.2.i.o.433.1		2
36.11	even	6		1296.2.i.c.433.1		2
36.23	even	6		1296.2.i.c.865.1		2
36.31	odd	6		1296.2.i.o.865.1		2
39.38	odd	2		9126.2.a.u.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.2.a.a.1.1	✓	1	3.2	odd	2
54.2.a.b.1.1	yes	1	1.1	even	1	trivial
162.2.c.b.55.1		2	9.4	even	3
162.2.c.b.109.1		2	9.7	even	3
162.2.c.c.55.1		2	9.5	odd	6
162.2.c.c.109.1		2	9.2	odd	6
432.2.a.b.1.1		1	4.3	odd	2
432.2.a.g.1.1		1	12.11	even	2
1296.2.i.c.433.1		2	36.11	even	6
1296.2.i.c.865.1		2	36.23	even	6
1296.2.i.o.433.1		2	36.7	odd	6
1296.2.i.o.865.1		2	36.31	odd	6
1350.2.a.h.1.1		1	5.4	even	2
1350.2.a.r.1.1		1	15.14	odd	2
1350.2.c.b.649.1		2	15.2	even	4
1350.2.c.b.649.2		2	15.8	even	4
1350.2.c.k.649.1		2	5.3	odd	4
1350.2.c.k.649.2		2	5.2	odd	4
1728.2.a.c.1.1		1	24.5	odd	2
1728.2.a.d.1.1		1	24.11	even	2
1728.2.a.y.1.1		1	8.5	even	2
1728.2.a.z.1.1		1	8.3	odd	2
2646.2.a.a.1.1		1	21.20	even	2
2646.2.a.bd.1.1		1	7.6	odd	2
6534.2.a.b.1.1		1	11.10	odd	2
6534.2.a.bc.1.1		1	33.32	even	2
9126.2.a.r.1.1		1	13.12	even	2
9126.2.a.u.1.1		1	39.38	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 54 = 2 \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	54.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more