LMFDB - Embedded newform 855.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [855,2,Mod(1,855)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(855, 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("855.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 855 = 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	855.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:	$6.82720937282$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 285)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	855.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{4} -1.00000 q^{5} +4.00000 q^{7} +3.00000 q^{8} +O(q^{10})$

$q-1.00000 q^{2} -1.00000 q^{4} -1.00000 q^{5} +4.00000 q^{7} +3.00000 q^{8} +1.00000 q^{10} -4.00000 q^{11} +2.00000 q^{13} -4.00000 q^{14} -1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{19} +1.00000 q^{20} +4.00000 q^{22} +4.00000 q^{23} +1.00000 q^{25} -2.00000 q^{26} -4.00000 q^{28} +2.00000 q^{29} -5.00000 q^{32} +2.00000 q^{34} -4.00000 q^{35} -6.00000 q^{37} +1.00000 q^{38} -3.00000 q^{40} +6.00000 q^{41} +8.00000 q^{43} +4.00000 q^{44} -4.00000 q^{46} +12.0000 q^{47} +9.00000 q^{49} -1.00000 q^{50} -2.00000 q^{52} +14.0000 q^{53} +4.00000 q^{55} +12.0000 q^{56} -2.00000 q^{58} -4.00000 q^{59} +14.0000 q^{61} +7.00000 q^{64} -2.00000 q^{65} -4.00000 q^{67} +2.00000 q^{68} +4.00000 q^{70} -14.0000 q^{73} +6.00000 q^{74} +1.00000 q^{76} -16.0000 q^{77} +16.0000 q^{79} +1.00000 q^{80} -6.00000 q^{82} +2.00000 q^{85} -8.00000 q^{86} -12.0000 q^{88} +6.00000 q^{89} +8.00000 q^{91} -4.00000 q^{92} -12.0000 q^{94} +1.00000 q^{95} -10.0000 q^{97} -9.00000 q^{98} +O(q^{100})$

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	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	3.00000	1.06066
$9$	0	0
$10$	1.00000	0.316228
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−4.00000	−1.06904
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	1.00000	0.223607
$21$	0	0
$22$	4.00000	0.852803
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−2.00000	−0.392232
$27$	0	0
$28$	−4.00000	−0.755929
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−5.00000	−0.883883
$33$	0	0
$34$	2.00000	0.342997
$35$	−4.00000	−0.676123
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	1.00000	0.162221
$39$	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
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	−4.00000	−0.589768
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−2.00000	−0.277350
$53$	14.0000	1.92305	0.961524	−	0.274721i	$-0.0885855\pi$
$53$	14.0000	1.92305	0.961524	+	0.274721i	$0.0885855\pi$
$54$	0	0
$55$	4.00000	0.539360
$56$	12.0000	1.60357
$57$	0	0
$58$	−2.00000	−0.262613
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	0	0
$63$	0	0
$64$	7.00000	0.875000
$65$	−2.00000	−0.248069
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	2.00000	0.242536
$69$	0	0
$70$	4.00000	0.478091
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	1.00000	0.114708
$77$	−16.0000	−1.82337
$78$	0	0
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−6.00000	−0.662589
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	2.00000	0.216930
$86$	−8.00000	−0.862662
$87$	0	0
$88$	−12.0000	−1.27920
$89$	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
$91$	8.00000	0.838628
$92$	−4.00000	−0.417029
$93$	0	0
$94$	−12.0000	−1.23771
$95$	1.00000	0.102598
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	−9.00000	−0.909137
$99$	0	0
$100$	−1.00000	−0.100000
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	−14.0000	−1.35980
$107$	−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
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	−4.00000	−0.381385
$111$	0	0
$112$	−4.00000	−0.377964
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	0	0
$115$	−4.00000	−0.373002
$116$	−2.00000	−0.185695
$117$	0	0
$118$	4.00000	0.368230
$119$	−8.00000	−0.733359
$120$	0	0
$121$	5.00000	0.454545
$122$	−14.0000	−1.26750
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	3.00000	0.265165
$129$	0	0
$130$	2.00000	0.175412
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	−4.00000	−0.346844
$134$	4.00000	0.345547
$135$	0	0
$136$	−6.00000	−0.514496
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	4.00000	0.338062
$141$	0	0
$142$	0	0
$143$	−8.00000	−0.668994
$144$	0	0
$145$	−2.00000	−0.166091
$146$	14.0000	1.15865
$147$	0	0
$148$	6.00000	0.493197
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	−3.00000	−0.243332
$153$	0	0
$154$	16.0000	1.28932
$155$	0	0
$156$	0	0
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	−16.0000	−1.27289
$159$	0	0
$160$	5.00000	0.395285
$161$	16.0000	1.26098
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	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
$169$	−9.00000	−0.692308
$170$	−2.00000	−0.153393
$171$	0	0
$172$	−8.00000	−0.609994
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	4.00000	0.301511
$177$	0	0
$178$	−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
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	−8.00000	−0.592999
$183$	0	0
$184$	12.0000	0.884652
$185$	6.00000	0.441129
$186$	0	0
$187$	8.00000	0.585018
$188$	−12.0000	−0.875190
$189$	0	0
$190$	−1.00000	−0.0725476
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	22.0000	1.58359	0.791797	−	0.610784i	$-0.209146\pi$
$193$	22.0000	1.58359	0.791797	+	0.610784i	$0.209146\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	−9.00000	−0.642857
$197$	−22.0000	−1.56744	−0.783718	−	0.621117i	$-0.786679\pi$
$197$	−22.0000	−1.56744	−0.783718	+	0.621117i	$0.786679\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	3.00000	0.212132
$201$	0	0
$202$	−10.0000	−0.703598
$203$	8.00000	0.561490
$204$	0	0
$205$	−6.00000	−0.419058
$206$	−16.0000	−1.11477
$207$	0	0
$208$	−2.00000	−0.138675
$209$	4.00000	0.276686
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	−14.0000	−0.961524
$213$	0	0
$214$	12.0000	0.820303
$215$	−8.00000	−0.545595
$216$	0	0
$217$	0	0
$218$	2.00000	0.135457
$219$	0	0
$220$	−4.00000	−0.269680
$221$	−4.00000	−0.269069
$222$	0	0
$223$	−24.0000	−1.60716	−0.803579	−	0.595198i	$-0.797074\pi$
$223$	−24.0000	−1.60716	−0.803579	+	0.595198i	$0.797074\pi$
$224$	−20.0000	−1.33631
$225$	0	0
$226$	−10.0000	−0.665190
$227$	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
$229$	−26.0000	−1.71813	−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000	−1.71813	−0.859064	+	0.511868i	$0.828954\pi$
$230$	4.00000	0.263752
$231$	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
$235$	−12.0000	−0.782794
$236$	4.00000	0.260378
$237$	0	0
$238$	8.00000	0.518563
$239$	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
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	−5.00000	−0.321412
$243$	0	0
$244$	−14.0000	−0.896258
$245$	−9.00000	−0.574989
$246$	0	0
$247$	−2.00000	−0.127257
$248$	0	0
$249$	0	0
$250$	1.00000	0.0632456
$251$	−28.0000	−1.76734	−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000	−1.76734	−0.883672	+	0.468106i	$0.844936\pi$
$252$	0	0
$253$	−16.0000	−1.00591
$254$	8.00000	0.501965
$255$	0	0
$256$	−17.0000	−1.06250
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	0	0
$259$	−24.0000	−1.49129
$260$	2.00000	0.124035
$261$	0	0
$262$	4.00000	0.247121
$263$	4.00000	0.246651	0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000	0.246651	0.123325	+	0.992366i	$0.460644\pi$
$264$	0	0
$265$	−14.0000	−0.860013
$266$	4.00000	0.245256
$267$	0	0
$268$	4.00000	0.244339
$269$	26.0000	1.58525	0.792624	−	0.609711i	$-0.208714\pi$
$269$	26.0000	1.58525	0.792624	+	0.609711i	$0.208714\pi$
$270$	0	0
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	2.00000	0.121268
$273$	0	0
$274$	−6.00000	−0.362473
$275$	−4.00000	−0.241209
$276$	0	0
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	4.00000	0.239904
$279$	0	0
$280$	−12.0000	−0.717137
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	−16.0000	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$284$	0	0
$285$	0	0
$286$	8.00000	0.473050
$287$	24.0000	1.41668
$288$	0	0
$289$	−13.0000	−0.764706
$290$	2.00000	0.117444
$291$	0	0
$292$	14.0000	0.819288
$293$	−2.00000	−0.116841	−0.0584206	−	0.998292i	$-0.518606\pi$
$293$	−2.00000	−0.116841	−0.0584206	+	0.998292i	$0.518606\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	−18.0000	−1.04623
$297$	0	0
$298$	6.00000	0.347571
$299$	8.00000	0.462652
$300$	0	0
$301$	32.0000	1.84445
$302$	0	0
$303$	0	0
$304$	1.00000	0.0573539
$305$	−14.0000	−0.801638
$306$	0	0
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	16.0000	0.911685
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	18.0000	1.01580
$315$	0	0
$316$	−16.0000	−0.900070
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	0	0
$319$	−8.00000	−0.447914
$320$	−7.00000	−0.391312
$321$	0	0
$322$	−16.0000	−0.891645
$323$	2.00000	0.111283
$324$	0	0
$325$	2.00000	0.110940
$326$	0	0
$327$	0	0
$328$	18.0000	0.993884
$329$	48.0000	2.64633
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	0	0
$333$	0	0
$334$	−8.00000	−0.437741
$335$	4.00000	0.218543
$336$	0	0
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	9.00000	0.489535
$339$	0	0
$340$	−2.00000	−0.108465
$341$	0	0
$342$	0	0
$343$	8.00000	0.431959
$344$	24.0000	1.29399
$345$	0	0
$346$	−6.00000	−0.322562
$347$	−32.0000	−1.71785	−0.858925	−	0.512101i	$-0.828867\pi$
$347$	−32.0000	−1.71785	−0.858925	+	0.512101i	$0.828867\pi$
$348$	0	0
$349$	30.0000	1.60586	0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000	1.60586	0.802932	+	0.596071i	$0.203272\pi$
$350$	−4.00000	−0.213809
$351$	0	0
$352$	20.0000	1.06600
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	0	0
$355$	0	0
$356$	−6.00000	−0.317999
$357$	0	0
$358$	12.0000	0.634220
$359$	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
$361$	1.00000	0.0526316
$362$	−22.0000	−1.15629
$363$	0	0
$364$	−8.00000	−0.419314
$365$	14.0000	0.732793
$366$	0	0
$367$	−20.0000	−1.04399	−0.521996	−	0.852948i	$-0.674812\pi$
$367$	−20.0000	−1.04399	−0.521996	+	0.852948i	$0.674812\pi$
$368$	−4.00000	−0.208514
$369$	0	0
$370$	−6.00000	−0.311925
$371$	56.0000	2.90738
$372$	0	0
$373$	34.0000	1.76045	0.880227	−	0.474554i	$-0.157390\pi$
$373$	34.0000	1.76045	0.880227	+	0.474554i	$0.157390\pi$
$374$	−8.00000	−0.413670
$375$	0	0
$376$	36.0000	1.85656
$377$	4.00000	0.206010
$378$	0	0
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	−1.00000	−0.0512989
$381$	0	0
$382$	0	0
$383$	−32.0000	−1.63512	−0.817562	−	0.575841i	$-0.804675\pi$
$383$	−32.0000	−1.63512	−0.817562	+	0.575841i	$0.804675\pi$
$384$	0	0
$385$	16.0000	0.815436
$386$	−22.0000	−1.11977
$387$	0	0
$388$	10.0000	0.507673
$389$	−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
$391$	−8.00000	−0.404577
$392$	27.0000	1.36371
$393$	0	0
$394$	22.0000	1.10834
$395$	−16.0000	−0.805047
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	0	0
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	0	0
$403$	0	0
$404$	−10.0000	−0.497519
$405$	0	0
$406$	−8.00000	−0.397033
$407$	24.0000	1.18964
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	6.00000	0.296319
$411$	0	0
$412$	−16.0000	−0.788263
$413$	−16.0000	−0.787309
$414$	0	0
$415$	0	0
$416$	−10.0000	−0.490290
$417$	0	0
$418$	−4.00000	−0.195646
$419$	−28.0000	−1.36789	−0.683945	−	0.729534i	$-0.739737\pi$
$419$	−28.0000	−1.36789	−0.683945	+	0.729534i	$0.739737\pi$
$420$	0	0
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	−12.0000	−0.584151
$423$	0	0
$424$	42.0000	2.03970
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	56.0000	2.71003
$428$	12.0000	0.580042
$429$	0	0
$430$	8.00000	0.385794
$431$	−32.0000	−1.54139	−0.770693	−	0.637207i	$-0.780090\pi$
$431$	−32.0000	−1.54139	−0.770693	+	0.637207i	$0.780090\pi$
$432$	0	0
$433$	6.00000	0.288342	0.144171	−	0.989553i	$-0.453949\pi$
$433$	6.00000	0.288342	0.144171	+	0.989553i	$0.453949\pi$
$434$	0	0
$435$	0	0
$436$	2.00000	0.0957826
$437$	−4.00000	−0.191346
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	12.0000	0.572078
$441$	0	0
$442$	4.00000	0.190261
$443$	−40.0000	−1.90046	−0.950229	−	0.311553i	$-0.899151\pi$
$443$	−40.0000	−1.90046	−0.950229	+	0.311553i	$0.899151\pi$
$444$	0	0
$445$	−6.00000	−0.284427
$446$	24.0000	1.13643
$447$	0	0
$448$	28.0000	1.32288
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	−24.0000	−1.13012
$452$	−10.0000	−0.470360
$453$	0	0
$454$	−12.0000	−0.563188
$455$	−8.00000	−0.375046
$456$	0	0
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	26.0000	1.21490
$459$	0	0
$460$	4.00000	0.186501
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	18.0000	0.833834
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	12.0000	0.553519
$471$	0	0
$472$	−12.0000	−0.552345
$473$	−32.0000	−1.47136
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	8.00000	0.366679
$477$	0	0
$478$	−24.0000	−1.09773
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	22.0000	1.00207
$483$	0	0
$484$	−5.00000	−0.227273
$485$	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	$0.441983\pi$
$488$	42.0000	1.90125
$489$	0	0
$490$	9.00000	0.406579
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	0	0
$493$	−4.00000	−0.180151
$494$	2.00000	0.0899843
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	28.0000	1.24970
$503$	20.0000	0.891756	0.445878	−	0.895094i	$-0.352892\pi$
$503$	20.0000	0.891756	0.445878	+	0.895094i	$0.352892\pi$
$504$	0	0
$505$	−10.0000	−0.444994
$506$	16.0000	0.711287
$507$	0	0
$508$	8.00000	0.354943
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	0	0
$511$	−56.0000	−2.47729
$512$	11.0000	0.486136
$513$	0	0
$514$	−2.00000	−0.0882162
$515$	−16.0000	−0.705044
$516$	0	0
$517$	−48.0000	−2.11104
$518$	24.0000	1.05450
$519$	0	0
$520$	−6.00000	−0.263117
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	0	0
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	4.00000	0.174741
$525$	0	0
$526$	−4.00000	−0.174408
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	14.0000	0.608121
$531$	0	0
$532$	4.00000	0.173422
$533$	12.0000	0.519778
$534$	0	0
$535$	12.0000	0.518805
$536$	−12.0000	−0.518321
$537$	0	0
$538$	−26.0000	−1.12094
$539$	−36.0000	−1.55063
$540$	0	0
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	−24.0000	−1.03089
$543$	0	0
$544$	10.0000	0.428746
$545$	2.00000	0.0856706
$546$	0	0
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	−6.00000	−0.256307
$549$	0	0
$550$	4.00000	0.170561
$551$	−2.00000	−0.0852029
$552$	0	0
$553$	64.0000	2.72156
$554$	−22.0000	−0.934690
$555$	0	0
$556$	4.00000	0.169638
$557$	26.0000	1.10166	0.550828	−	0.834619i	$-0.314312\pi$
$557$	26.0000	1.10166	0.550828	+	0.834619i	$0.314312\pi$
$558$	0	0
$559$	16.0000	0.676728
$560$	4.00000	0.169031
$561$	0	0
$562$	10.0000	0.421825
$563$	−36.0000	−1.51722	−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000	−1.51722	−0.758610	+	0.651546i	$0.774121\pi$
$564$	0	0
$565$	−10.0000	−0.420703
$566$	16.0000	0.672530
$567$	0	0
$568$	0	0
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	0	0
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	8.00000	0.334497
$573$	0	0
$574$	−24.0000	−1.00174
$575$	4.00000	0.166812
$576$	0	0
$577$	18.0000	0.749350	0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000	0.749350	0.374675	+	0.927156i	$0.377754\pi$
$578$	13.0000	0.540729
$579$	0	0
$580$	2.00000	0.0830455
$581$	0	0
$582$	0	0
$583$	−56.0000	−2.31928
$584$	−42.0000	−1.73797
$585$	0	0
$586$	2.00000	0.0826192
$587$	−16.0000	−0.660391	−0.330195	−	0.943913i	$-0.607115\pi$
$587$	−16.0000	−0.660391	−0.330195	+	0.943913i	$0.607115\pi$
$588$	0	0
$589$	0	0
$590$	−4.00000	−0.164677
$591$	0	0
$592$	6.00000	0.246598
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	8.00000	0.327968
$596$	6.00000	0.245770
$597$	0	0
$598$	−8.00000	−0.327144
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	18.0000	0.734235	0.367118	−	0.930175i	$-0.380345\pi$
$601$	18.0000	0.734235	0.367118	+	0.930175i	$0.380345\pi$
$602$	−32.0000	−1.30422
$603$	0	0
$604$	0	0
$605$	−5.00000	−0.203279
$606$	0	0
$607$	−16.0000	−0.649420	−0.324710	−	0.945814i	$-0.605267\pi$
$607$	−16.0000	−0.649420	−0.324710	+	0.945814i	$0.605267\pi$
$608$	5.00000	0.202777
$609$	0	0
$610$	14.0000	0.566843
$611$	24.0000	0.970936
$612$	0	0
$613$	−42.0000	−1.69636	−0.848182	−	0.529705i	$-0.822303\pi$
$613$	−42.0000	−1.69636	−0.848182	+	0.529705i	$0.822303\pi$
$614$	12.0000	0.484281
$615$	0	0
$616$	−48.0000	−1.93398
$617$	−42.0000	−1.69086	−0.845428	−	0.534089i	$-0.820655\pi$
$617$	−42.0000	−1.69086	−0.845428	+	0.534089i	$0.820655\pi$
$618$	0	0
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	24.0000	0.961540
$624$	0	0
$625$	1.00000	0.0400000
$626$	−10.0000	−0.399680
$627$	0	0
$628$	18.0000	0.718278
$629$	12.0000	0.478471
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	48.0000	1.90934
$633$	0	0
$634$	2.00000	0.0794301
$635$	8.00000	0.317470
$636$	0	0
$637$	18.0000	0.713186
$638$	8.00000	0.316723
$639$	0	0
$640$	−3.00000	−0.118585
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	0	0
$643$	−40.0000	−1.57745	−0.788723	−	0.614749i	$-0.789257\pi$
$643$	−40.0000	−1.57745	−0.788723	+	0.614749i	$0.789257\pi$
$644$	−16.0000	−0.630488
$645$	0	0
$646$	−2.00000	−0.0786889
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	0	0
$649$	16.0000	0.628055
$650$	−2.00000	−0.0784465
$651$	0	0
$652$	0	0
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	0	0
$655$	4.00000	0.156293
$656$	−6.00000	−0.234261
$657$	0	0
$658$	−48.0000	−1.87123
$659$	−28.0000	−1.09073	−0.545363	−	0.838200i	$-0.683608\pi$
$659$	−28.0000	−1.09073	−0.545363	+	0.838200i	$0.683608\pi$
$660$	0	0
$661$	−18.0000	−0.700119	−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000	−0.700119	−0.350059	+	0.936727i	$0.613839\pi$
$662$	28.0000	1.08825
$663$	0	0
$664$	0	0
$665$	4.00000	0.155113
$666$	0	0
$667$	8.00000	0.309761
$668$	−8.00000	−0.309529
$669$	0	0
$670$	−4.00000	−0.154533
$671$	−56.0000	−2.16186
$672$	0	0
$673$	30.0000	1.15642	0.578208	−	0.815890i	$-0.303752\pi$
$673$	30.0000	1.15642	0.578208	+	0.815890i	$0.303752\pi$
$674$	−22.0000	−0.847408
$675$	0	0
$676$	9.00000	0.346154
$677$	14.0000	0.538064	0.269032	−	0.963131i	$-0.413296\pi$
$677$	14.0000	0.538064	0.269032	+	0.963131i	$0.413296\pi$
$678$	0	0
$679$	−40.0000	−1.53506
$680$	6.00000	0.230089
$681$	0	0
$682$	0	0
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	0	0
$685$	−6.00000	−0.229248
$686$	−8.00000	−0.305441
$687$	0	0
$688$	−8.00000	−0.304997
$689$	28.0000	1.06672
$690$	0	0
$691$	−4.00000	−0.152167	−0.0760836	−	0.997101i	$-0.524242\pi$
$691$	−4.00000	−0.152167	−0.0760836	+	0.997101i	$0.524242\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	32.0000	1.21470
$695$	4.00000	0.151729
$696$	0	0
$697$	−12.0000	−0.454532
$698$	−30.0000	−1.13552
$699$	0	0
$700$	−4.00000	−0.151186
$701$	34.0000	1.28416	0.642081	−	0.766637i	$-0.278071\pi$
$701$	34.0000	1.28416	0.642081	+	0.766637i	$0.278071\pi$
$702$	0	0
$703$	6.00000	0.226294
$704$	−28.0000	−1.05529
$705$	0	0
$706$	−14.0000	−0.526897
$707$	40.0000	1.50435
$708$	0	0
$709$	6.00000	0.225335	0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000	0.225335	0.112667	+	0.993633i	$0.464061\pi$
$710$	0	0
$711$	0	0
$712$	18.0000	0.674579
$713$	0	0
$714$	0	0
$715$	8.00000	0.299183
$716$	12.0000	0.448461
$717$	0	0
$718$	−8.00000	−0.298557
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	0	0
$721$	64.0000	2.38348
$722$	−1.00000	−0.0372161
$723$	0	0
$724$	−22.0000	−0.817624
$725$	2.00000	0.0742781
$726$	0	0
$727$	4.00000	0.148352	0.0741759	−	0.997245i	$-0.476367\pi$
$727$	4.00000	0.148352	0.0741759	+	0.997245i	$0.476367\pi$
$728$	24.0000	0.889499
$729$	0	0
$730$	−14.0000	−0.518163
$731$	−16.0000	−0.591781
$732$	0	0
$733$	−26.0000	−0.960332	−0.480166	−	0.877178i	$-0.659424\pi$
$733$	−26.0000	−0.960332	−0.480166	+	0.877178i	$0.659424\pi$
$734$	20.0000	0.738213
$735$	0	0
$736$	−20.0000	−0.737210
$737$	16.0000	0.589368
$738$	0	0
$739$	28.0000	1.03000	0.514998	−	0.857191i	$-0.327793\pi$
$739$	28.0000	1.03000	0.514998	+	0.857191i	$0.327793\pi$
$740$	−6.00000	−0.220564
$741$	0	0
$742$	−56.0000	−2.05582
$743$	−40.0000	−1.46746	−0.733729	−	0.679442i	$-0.762222\pi$
$743$	−40.0000	−1.46746	−0.733729	+	0.679442i	$0.762222\pi$
$744$	0	0
$745$	6.00000	0.219823
$746$	−34.0000	−1.24483
$747$	0	0
$748$	−8.00000	−0.292509
$749$	−48.0000	−1.75388
$750$	0	0
$751$	−24.0000	−0.875772	−0.437886	−	0.899030i	$-0.644273\pi$
$751$	−24.0000	−0.875772	−0.437886	+	0.899030i	$0.644273\pi$
$752$	−12.0000	−0.437595
$753$	0	0
$754$	−4.00000	−0.145671
$755$	0	0
$756$	0	0
$757$	−50.0000	−1.81728	−0.908640	−	0.417579i	$-0.862879\pi$
$757$	−50.0000	−1.81728	−0.908640	+	0.417579i	$0.862879\pi$
$758$	−4.00000	−0.145287
$759$	0	0
$760$	3.00000	0.108821
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	0	0
$763$	−8.00000	−0.289619
$764$	0	0
$765$	0	0
$766$	32.0000	1.15621
$767$	−8.00000	−0.288863
$768$	0	0
$769$	50.0000	1.80305	0.901523	−	0.432731i	$-0.142450\pi$
$769$	50.0000	1.80305	0.901523	+	0.432731i	$0.142450\pi$
$770$	−16.0000	−0.576600
$771$	0	0
$772$	−22.0000	−0.791797
$773$	−26.0000	−0.935155	−0.467578	−	0.883952i	$-0.654873\pi$
$773$	−26.0000	−0.935155	−0.467578	+	0.883952i	$0.654873\pi$
$774$	0	0
$775$	0	0
$776$	−30.0000	−1.07694
$777$	0	0
$778$	6.00000	0.215110
$779$	−6.00000	−0.214972
$780$	0	0
$781$	0	0
$782$	8.00000	0.286079
$783$	0	0
$784$	−9.00000	−0.321429
$785$	18.0000	0.642448
$786$	0	0
$787$	−44.0000	−1.56843	−0.784215	−	0.620489i	$-0.786934\pi$
$787$	−44.0000	−1.56843	−0.784215	+	0.620489i	$0.786934\pi$
$788$	22.0000	0.783718
$789$	0	0
$790$	16.0000	0.569254
$791$	40.0000	1.42224
$792$	0	0
$793$	28.0000	0.994309
$794$	2.00000	0.0709773
$795$	0	0
$796$	0	0
$797$	38.0000	1.34603	0.673015	−	0.739629i	$-0.264999\pi$
$797$	38.0000	1.34603	0.673015	+	0.739629i	$0.264999\pi$
$798$	0	0
$799$	−24.0000	−0.849059
$800$	−5.00000	−0.176777
$801$	0	0
$802$	−6.00000	−0.211867
$803$	56.0000	1.97620
$804$	0	0
$805$	−16.0000	−0.563926
$806$	0	0
$807$	0	0
$808$	30.0000	1.05540
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	0	0
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	−8.00000	−0.280745
$813$	0	0
$814$	−24.0000	−0.841200
$815$	0	0
$816$	0	0
$817$	−8.00000	−0.279885
$818$	−10.0000	−0.349642
$819$	0	0
$820$	6.00000	0.209529
$821$	−54.0000	−1.88461	−0.942306	−	0.334751i	$-0.891348\pi$
$821$	−54.0000	−1.88461	−0.942306	+	0.334751i	$0.891348\pi$
$822$	0	0
$823$	−20.0000	−0.697156	−0.348578	−	0.937280i	$-0.613335\pi$
$823$	−20.0000	−0.697156	−0.348578	+	0.937280i	$0.613335\pi$
$824$	48.0000	1.67216
$825$	0	0
$826$	16.0000	0.556711
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	0	0
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	0	0
$831$	0	0
$832$	14.0000	0.485363
$833$	−18.0000	−0.623663
$834$	0	0
$835$	−8.00000	−0.276851
$836$	−4.00000	−0.138343
$837$	0	0
$838$	28.0000	0.967244
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−14.0000	−0.482472
$843$	0	0
$844$	−12.0000	−0.413057
$845$	9.00000	0.309609
$846$	0	0
$847$	20.0000	0.687208
$848$	−14.0000	−0.480762
$849$	0	0
$850$	2.00000	0.0685994
$851$	−24.0000	−0.822709
$852$	0	0
$853$	−2.00000	−0.0684787	−0.0342393	−	0.999414i	$-0.510901\pi$
$853$	−2.00000	−0.0684787	−0.0342393	+	0.999414i	$0.510901\pi$
$854$	−56.0000	−1.91628
$855$	0	0
$856$	−36.0000	−1.23045
$857$	26.0000	0.888143	0.444072	−	0.895991i	$-0.353534\pi$
$857$	26.0000	0.888143	0.444072	+	0.895991i	$0.353534\pi$
$858$	0	0
$859$	36.0000	1.22830	0.614152	−	0.789188i	$-0.289498\pi$
$859$	36.0000	1.22830	0.614152	+	0.789188i	$0.289498\pi$
$860$	8.00000	0.272798
$861$	0	0
$862$	32.0000	1.08992
$863$	32.0000	1.08929	0.544646	−	0.838666i	$-0.316664\pi$
$863$	32.0000	1.08929	0.544646	+	0.838666i	$0.316664\pi$
$864$	0	0
$865$	−6.00000	−0.204006
$866$	−6.00000	−0.203888
$867$	0	0
$868$	0	0
$869$	−64.0000	−2.17105
$870$	0	0
$871$	−8.00000	−0.271070
$872$	−6.00000	−0.203186
$873$	0	0
$874$	4.00000	0.135302
$875$	−4.00000	−0.135225
$876$	0	0
$877$	42.0000	1.41824	0.709120	−	0.705088i	$-0.249093\pi$
$877$	42.0000	1.41824	0.709120	+	0.705088i	$0.249093\pi$
$878$	−8.00000	−0.269987
$879$	0	0
$880$	−4.00000	−0.134840
$881$	−2.00000	−0.0673817	−0.0336909	−	0.999432i	$-0.510726\pi$
$881$	−2.00000	−0.0673817	−0.0336909	+	0.999432i	$0.510726\pi$
$882$	0	0
$883$	−16.0000	−0.538443	−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000	−0.538443	−0.269221	+	0.963078i	$0.586766\pi$
$884$	4.00000	0.134535
$885$	0	0
$886$	40.0000	1.34383
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	−32.0000	−1.07325
$890$	6.00000	0.201120
$891$	0	0
$892$	24.0000	0.803579
$893$	−12.0000	−0.401565
$894$	0	0
$895$	12.0000	0.401116
$896$	12.0000	0.400892
$897$	0	0
$898$	−6.00000	−0.200223
$899$	0	0
$900$	0	0
$901$	−28.0000	−0.932815
$902$	24.0000	0.799113
$903$	0	0
$904$	30.0000	0.997785
$905$	−22.0000	−0.731305
$906$	0	0
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	−12.0000	−0.398234
$909$	0	0
$910$	8.00000	0.265197
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	0	0
$913$	0	0
$914$	−10.0000	−0.330771
$915$	0	0
$916$	26.0000	0.859064
$917$	−16.0000	−0.528367
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	−12.0000	−0.395628
$921$	0	0
$922$	30.0000	0.987997
$923$	0	0
$924$	0	0
$925$	−6.00000	−0.197279
$926$	4.00000	0.131448
$927$	0	0
$928$	−10.0000	−0.328266
$929$	14.0000	0.459325	0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000	0.459325	0.229663	+	0.973270i	$0.426238\pi$
$930$	0	0
$931$	−9.00000	−0.294963
$932$	18.0000	0.589610
$933$	0	0
$934$	−8.00000	−0.261768
$935$	−8.00000	−0.261628
$936$	0	0
$937$	18.0000	0.588034	0.294017	−	0.955800i	$-0.405008\pi$
$937$	18.0000	0.588034	0.294017	+	0.955800i	$0.405008\pi$
$938$	16.0000	0.522419
$939$	0	0
$940$	12.0000	0.391397
$941$	42.0000	1.36916	0.684580	−	0.728937i	$-0.259985\pi$
$941$	42.0000	1.36916	0.684580	+	0.728937i	$0.259985\pi$
$942$	0	0
$943$	24.0000	0.781548
$944$	4.00000	0.130189
$945$	0	0
$946$	32.0000	1.04041
$947$	40.0000	1.29983	0.649913	−	0.760009i	$-0.274805\pi$
$947$	40.0000	1.29983	0.649913	+	0.760009i	$0.274805\pi$
$948$	0	0
$949$	−28.0000	−0.908918
$950$	1.00000	0.0324443
$951$	0	0
$952$	−24.0000	−0.777844
$953$	26.0000	0.842223	0.421111	−	0.907009i	$-0.361640\pi$
$953$	26.0000	0.842223	0.421111	+	0.907009i	$0.361640\pi$
$954$	0	0
$955$	0	0
$956$	−24.0000	−0.776215
$957$	0	0
$958$	−16.0000	−0.516937
$959$	24.0000	0.775000
$960$	0	0
$961$	−31.0000	−1.00000
$962$	12.0000	0.386896
$963$	0	0
$964$	22.0000	0.708572
$965$	−22.0000	−0.708205
$966$	0	0
$967$	−20.0000	−0.643157	−0.321578	−	0.946883i	$-0.604213\pi$
$967$	−20.0000	−0.643157	−0.321578	+	0.946883i	$0.604213\pi$
$968$	15.0000	0.482118
$969$	0	0
$970$	−10.0000	−0.321081
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	0	0
$973$	−16.0000	−0.512936
$974$	−8.00000	−0.256337
$975$	0	0
$976$	−14.0000	−0.448129
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	0	0
$979$	−24.0000	−0.767043
$980$	9.00000	0.287494
$981$	0	0
$982$	12.0000	0.382935
$983$	56.0000	1.78612	0.893061	−	0.449935i	$-0.148553\pi$
$983$	56.0000	1.78612	0.893061	+	0.449935i	$0.148553\pi$
$984$	0	0
$985$	22.0000	0.700978
$986$	4.00000	0.127386
$987$	0	0
$988$	2.00000	0.0636285
$989$	32.0000	1.01754
$990$	0	0
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−26.0000	−0.823428	−0.411714	−	0.911313i	$-0.635070\pi$
$997$	−26.0000	−0.823428	−0.411714	+	0.911313i	$0.635070\pi$
$998$	−20.0000	−0.633089
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	855.2.a.a.1.1		1
3.2	odd	2		285.2.a.c.1.1	✓	1
5.4	even	2		4275.2.a.j.1.1		1
12.11	even	2		4560.2.a.w.1.1		1
15.2	even	4		1425.2.c.f.799.2		2
15.8	even	4		1425.2.c.f.799.1		2
15.14	odd	2		1425.2.a.c.1.1		1
57.56	even	2		5415.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.a.c.1.1	✓	1	3.2	odd	2
855.2.a.a.1.1		1	1.1	even	1	trivial
1425.2.a.c.1.1		1	15.14	odd	2
1425.2.c.f.799.1		2	15.8	even	4
1425.2.c.f.799.2		2	15.2	even	4
4275.2.a.j.1.1		1	5.4	even	2
4560.2.a.w.1.1		1	12.11	even	2
5415.2.a.e.1.1		1	57.56	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:	\( N \)	\(=\)	\( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	855.a (trivial)

Introduction

L-functions

Modular forms

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Fields

Representations

Groups

Database

Properties

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