LMFDB - Embedded newform 855.2.a.b.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:	$1$
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)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{7} +3.00000 q^{8} +O(q^{10})$

$q-1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{7} +3.00000 q^{8} -1.00000 q^{10} +2.00000 q^{11} -4.00000 q^{13} +2.00000 q^{14} -1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{19} -1.00000 q^{20} -2.00000 q^{22} +4.00000 q^{23} +1.00000 q^{25} +4.00000 q^{26} +2.00000 q^{28} -4.00000 q^{29} -5.00000 q^{32} +2.00000 q^{34} -2.00000 q^{35} +1.00000 q^{38} +3.00000 q^{40} -10.0000 q^{43} -2.00000 q^{44} -4.00000 q^{46} -12.0000 q^{47} -3.00000 q^{49} -1.00000 q^{50} +4.00000 q^{52} +2.00000 q^{53} +2.00000 q^{55} -6.00000 q^{56} +4.00000 q^{58} -4.00000 q^{59} +2.00000 q^{61} +7.00000 q^{64} -4.00000 q^{65} -16.0000 q^{67} +2.00000 q^{68} +2.00000 q^{70} -2.00000 q^{73} +1.00000 q^{76} -4.00000 q^{77} -8.00000 q^{79} -1.00000 q^{80} +12.0000 q^{83} -2.00000 q^{85} +10.0000 q^{86} +6.00000 q^{88} +8.00000 q^{91} -4.00000 q^{92} +12.0000 q^{94} -1.00000 q^{95} -16.0000 q^{97} +3.00000 q^{98} +O(q^{100})$

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
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	3.00000	1.06066
$9$	0	0
$10$	−1.00000	−0.316228
$11$	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
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	2.00000	0.534522
$15$	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
$22$	−2.00000	−0.426401
$23$	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
$26$	4.00000	0.784465
$27$	0	0
$28$	2.00000	0.377964
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	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
$35$	−2.00000	−0.338062
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	3.00000	0.474342
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	−4.00000	−0.589768
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	−1.00000	−0.141421
$51$	0	0
$52$	4.00000	0.554700
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	−6.00000	−0.801784
$57$	0	0
$58$	4.00000	0.525226
$59$	−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
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	0	0
$64$	7.00000	0.875000
$65$	−4.00000	−0.496139
$66$	0	0
$67$	−16.0000	−1.95471	−0.977356	−	0.211604i	$-0.932131\pi$
$67$	−16.0000	−1.95471	−0.977356	+	0.211604i	$0.932131\pi$
$68$	2.00000	0.242536
$69$	0	0
$70$	2.00000	0.239046
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	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
$75$	0	0
$76$	1.00000	0.114708
$77$	−4.00000	−0.455842
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	0	0
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	10.0000	1.07833
$87$	0	0
$88$	6.00000	0.639602
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	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
$97$	−16.0000	−1.62455	−0.812277	−	0.583272i	$-0.801772\pi$
$97$	−16.0000	−1.62455	−0.812277	+	0.583272i	$0.801772\pi$
$98$	3.00000	0.303046
$99$	0	0
$100$	−1.00000	−0.100000
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	−12.0000	−1.17670
$105$	0	0
$106$	−2.00000	−0.194257
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	−2.00000	−0.190693
$111$	0	0
$112$	2.00000	0.188982
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	4.00000	0.371391
$117$	0	0
$118$	4.00000	0.368230
$119$	4.00000	0.366679
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−2.00000	−0.181071
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	3.00000	0.265165
$129$	0	0
$130$	4.00000	0.350823
$131$	14.0000	1.22319	0.611593	−	0.791173i	$-0.290529\pi$
$131$	14.0000	1.22319	0.611593	+	0.791173i	$0.290529\pi$
$132$	0	0
$133$	2.00000	0.173422
$134$	16.0000	1.38219
$135$	0	0
$136$	−6.00000	−0.514496
$137$	−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
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	2.00000	0.169031
$141$	0	0
$142$	0	0
$143$	−8.00000	−0.668994
$144$	0	0
$145$	−4.00000	−0.332182
$146$	2.00000	0.165521
$147$	0	0
$148$	0	0
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	24.0000	1.95309	0.976546	−	0.215308i	$-0.0690756\pi$
$151$	24.0000	1.95309	0.976546	+	0.215308i	$0.0690756\pi$
$152$	−3.00000	−0.243332
$153$	0	0
$154$	4.00000	0.322329
$155$	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	$0.755069\pi$
$158$	8.00000	0.636446
$159$	0	0
$160$	−5.00000	−0.395285
$161$	−8.00000	−0.630488
$162$	0	0
$163$	−6.00000	−0.469956	−0.234978	−	0.972001i	$-0.575502\pi$
$163$	−6.00000	−0.469956	−0.234978	+	0.972001i	$0.575502\pi$
$164$	0	0
$165$	0	0
$166$	−12.0000	−0.931381
$167$	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
$169$	3.00000	0.230769
$170$	2.00000	0.153393
$171$	0	0
$172$	10.0000	0.762493
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	−2.00000	−0.150756
$177$	0	0
$178$	0	0
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	−8.00000	−0.592999
$183$	0	0
$184$	12.0000	0.884652
$185$	0	0
$186$	0	0
$187$	−4.00000	−0.292509
$188$	12.0000	0.875190
$189$	0	0
$190$	1.00000	0.0725476
$191$	18.0000	1.30243	0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000	1.30243	0.651217	+	0.758891i	$0.274259\pi$
$192$	0	0
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	16.0000	1.14873
$195$	0	0
$196$	3.00000	0.214286
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	0	0
$199$	12.0000	0.850657	0.425329	−	0.905039i	$-0.360158\pi$
$199$	12.0000	0.850657	0.425329	+	0.905039i	$0.360158\pi$
$200$	3.00000	0.212132
$201$	0	0
$202$	14.0000	0.985037
$203$	8.00000	0.561490
$204$	0	0
$205$	0	0
$206$	8.00000	0.557386
$207$	0	0
$208$	4.00000	0.277350
$209$	−2.00000	−0.138343
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	−2.00000	−0.137361
$213$	0	0
$214$	−12.0000	−0.820303
$215$	−10.0000	−0.681994
$216$	0	0
$217$	0	0
$218$	−10.0000	−0.677285
$219$	0	0
$220$	−2.00000	−0.134840
$221$	8.00000	0.538138
$222$	0	0
$223$	24.0000	1.60716	0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000	1.60716	0.803579	+	0.595198i	$0.202926\pi$
$224$	10.0000	0.668153
$225$	0	0
$226$	14.0000	0.931266
$227$	−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
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	−4.00000	−0.263752
$231$	0	0
$232$	−12.0000	−0.787839
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	−12.0000	−0.782794
$236$	4.00000	0.260378
$237$	0	0
$238$	−4.00000	−0.259281
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	7.00000	0.449977
$243$	0	0
$244$	−2.00000	−0.128037
$245$	−3.00000	−0.191663
$246$	0	0
$247$	4.00000	0.254514
$248$	0	0
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	−22.0000	−1.38863	−0.694314	−	0.719672i	$-0.744292\pi$
$251$	−22.0000	−1.38863	−0.694314	+	0.719672i	$0.744292\pi$
$252$	0	0
$253$	8.00000	0.502956
$254$	−4.00000	−0.250982
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−22.0000	−1.37232	−0.686161	−	0.727450i	$-0.740706\pi$
$257$	−22.0000	−1.37232	−0.686161	+	0.727450i	$0.740706\pi$
$258$	0	0
$259$	0	0
$260$	4.00000	0.248069
$261$	0	0
$262$	−14.0000	−0.864923
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	0	0
$265$	2.00000	0.122859
$266$	−2.00000	−0.122628
$267$	0	0
$268$	16.0000	0.977356
$269$	−4.00000	−0.243884	−0.121942	−	0.992537i	$-0.538912\pi$
$269$	−4.00000	−0.243884	−0.121942	+	0.992537i	$0.538912\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	2.00000	0.121268
$273$	0	0
$274$	6.00000	0.362473
$275$	2.00000	0.120605
$276$	0	0
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	−8.00000	−0.479808
$279$	0	0
$280$	−6.00000	−0.358569
$281$	−28.0000	−1.67034	−0.835170	−	0.549992i	$-0.814631\pi$
$281$	−28.0000	−1.67034	−0.835170	+	0.549992i	$0.814631\pi$
$282$	0	0
$283$	26.0000	1.54554	0.772770	−	0.634686i	$-0.218871\pi$
$283$	26.0000	1.54554	0.772770	+	0.634686i	$0.218871\pi$
$284$	0	0
$285$	0	0
$286$	8.00000	0.473050
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	4.00000	0.234888
$291$	0	0
$292$	2.00000	0.117041
$293$	−26.0000	−1.51894	−0.759468	−	0.650545i	$-0.774541\pi$
$293$	−26.0000	−1.51894	−0.759468	+	0.650545i	$0.774541\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	0	0
$297$	0	0
$298$	−18.0000	−1.04271
$299$	−16.0000	−0.925304
$300$	0	0
$301$	20.0000	1.15278
$302$	−24.0000	−1.38104
$303$	0	0
$304$	1.00000	0.0573539
$305$	2.00000	0.114520
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	4.00000	0.227921
$309$	0	0
$310$	0	0
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	0	0
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	18.0000	1.01580
$315$	0	0
$316$	8.00000	0.450035
$317$	−14.0000	−0.786318	−0.393159	−	0.919470i	$-0.628618\pi$
$317$	−14.0000	−0.786318	−0.393159	+	0.919470i	$0.628618\pi$
$318$	0	0
$319$	−8.00000	−0.447914
$320$	7.00000	0.391312
$321$	0	0
$322$	8.00000	0.445823
$323$	2.00000	0.111283
$324$	0	0
$325$	−4.00000	−0.221880
$326$	6.00000	0.332309
$327$	0	0
$328$	0	0
$329$	24.0000	1.32316
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	−12.0000	−0.658586
$333$	0	0
$334$	−8.00000	−0.437741
$335$	−16.0000	−0.874173
$336$	0	0
$337$	16.0000	0.871576	0.435788	−	0.900049i	$-0.356470\pi$
$337$	16.0000	0.871576	0.435788	+	0.900049i	$0.356470\pi$
$338$	−3.00000	−0.163178
$339$	0	0
$340$	2.00000	0.108465
$341$	0	0
$342$	0	0
$343$	20.0000	1.07990
$344$	−30.0000	−1.61749
$345$	0	0
$346$	18.0000	0.967686
$347$	28.0000	1.50312	0.751559	−	0.659665i	$-0.229302\pi$
$347$	28.0000	1.50312	0.751559	+	0.659665i	$0.229302\pi$
$348$	0	0
$349$	−18.0000	−0.963518	−0.481759	−	0.876304i	$-0.660002\pi$
$349$	−18.0000	−0.963518	−0.481759	+	0.876304i	$0.660002\pi$
$350$	2.00000	0.106904
$351$	0	0
$352$	−10.0000	−0.533002
$353$	2.00000	0.106449	0.0532246	−	0.998583i	$-0.483050\pi$
$353$	2.00000	0.106449	0.0532246	+	0.998583i	$0.483050\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	−12.0000	−0.634220
$359$	2.00000	0.105556	0.0527780	−	0.998606i	$-0.483192\pi$
$359$	2.00000	0.105556	0.0527780	+	0.998606i	$0.483192\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−10.0000	−0.525588
$363$	0	0
$364$	−8.00000	−0.419314
$365$	−2.00000	−0.104685
$366$	0	0
$367$	10.0000	0.521996	0.260998	−	0.965339i	$-0.415948\pi$
$367$	10.0000	0.521996	0.260998	+	0.965339i	$0.415948\pi$
$368$	−4.00000	−0.208514
$369$	0	0
$370$	0	0
$371$	−4.00000	−0.207670
$372$	0	0
$373$	4.00000	0.207112	0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000	0.207112	0.103556	+	0.994624i	$0.466978\pi$
$374$	4.00000	0.206835
$375$	0	0
$376$	−36.0000	−1.85656
$377$	16.0000	0.824042
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	1.00000	0.0512989
$381$	0	0
$382$	−18.0000	−0.920960
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	0	0
$385$	−4.00000	−0.203859
$386$	−4.00000	−0.203595
$387$	0	0
$388$	16.0000	0.812277
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	−9.00000	−0.454569
$393$	0	0
$394$	−2.00000	−0.100759
$395$	−8.00000	−0.402524
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	−12.0000	−0.601506
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	−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
$403$	0	0
$404$	14.0000	0.696526
$405$	0	0
$406$	−8.00000	−0.397033
$407$	0	0
$408$	0	0
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	0	0
$411$	0	0
$412$	8.00000	0.394132
$413$	8.00000	0.393654
$414$	0	0
$415$	12.0000	0.589057
$416$	20.0000	0.980581
$417$	0	0
$418$	2.00000	0.0978232
$419$	26.0000	1.27018	0.635092	−	0.772437i	$-0.280962\pi$
$419$	26.0000	1.27018	0.635092	+	0.772437i	$0.280962\pi$
$420$	0	0
$421$	−34.0000	−1.65706	−0.828529	−	0.559946i	$-0.810822\pi$
$421$	−34.0000	−1.65706	−0.828529	+	0.559946i	$0.810822\pi$
$422$	12.0000	0.584151
$423$	0	0
$424$	6.00000	0.291386
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	−4.00000	−0.193574
$428$	−12.0000	−0.580042
$429$	0	0
$430$	10.0000	0.482243
$431$	4.00000	0.192673	0.0963366	−	0.995349i	$-0.469287\pi$
$431$	4.00000	0.192673	0.0963366	+	0.995349i	$0.469287\pi$
$432$	0	0
$433$	−36.0000	−1.73005	−0.865025	−	0.501729i	$-0.832697\pi$
$433$	−36.0000	−1.73005	−0.865025	+	0.501729i	$0.832697\pi$
$434$	0	0
$435$	0	0
$436$	−10.0000	−0.478913
$437$	−4.00000	−0.191346
$438$	0	0
$439$	−40.0000	−1.90910	−0.954548	−	0.298057i	$-0.903661\pi$
$439$	−40.0000	−1.90910	−0.954548	+	0.298057i	$0.903661\pi$
$440$	6.00000	0.286039
$441$	0	0
$442$	−8.00000	−0.380521
$443$	−16.0000	−0.760183	−0.380091	−	0.924949i	$-0.624107\pi$
$443$	−16.0000	−0.760183	−0.380091	+	0.924949i	$0.624107\pi$
$444$	0	0
$445$	0	0
$446$	−24.0000	−1.13643
$447$	0	0
$448$	−14.0000	−0.661438
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	0	0
$452$	14.0000	0.658505
$453$	0	0
$454$	12.0000	0.563188
$455$	8.00000	0.375046
$456$	0	0
$457$	−14.0000	−0.654892	−0.327446	−	0.944870i	$-0.606188\pi$
$457$	−14.0000	−0.654892	−0.327446	+	0.944870i	$0.606188\pi$
$458$	14.0000	0.654177
$459$	0	0
$460$	−4.00000	−0.186501
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	−22.0000	−1.02243	−0.511213	−	0.859454i	$-0.670804\pi$
$463$	−22.0000	−1.02243	−0.511213	+	0.859454i	$0.670804\pi$
$464$	4.00000	0.185695
$465$	0	0
$466$	−18.0000	−0.833834
$467$	−16.0000	−0.740392	−0.370196	−	0.928954i	$-0.620709\pi$
$467$	−16.0000	−0.740392	−0.370196	+	0.928954i	$0.620709\pi$
$468$	0	0
$469$	32.0000	1.47762
$470$	12.0000	0.553519
$471$	0	0
$472$	−12.0000	−0.552345
$473$	−20.0000	−0.919601
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	−4.00000	−0.183340
$477$	0	0
$478$	−6.00000	−0.274434
$479$	−38.0000	−1.73626	−0.868132	−	0.496333i	$-0.834679\pi$
$479$	−38.0000	−1.73626	−0.868132	+	0.496333i	$0.834679\pi$
$480$	0	0
$481$	0	0
$482$	−26.0000	−1.18427
$483$	0	0
$484$	7.00000	0.318182
$485$	−16.0000	−0.726523
$486$	0	0
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	6.00000	0.271607
$489$	0	0
$490$	3.00000	0.135526
$491$	6.00000	0.270776	0.135388	−	0.990793i	$-0.456772\pi$
$491$	6.00000	0.270776	0.135388	+	0.990793i	$0.456772\pi$
$492$	0	0
$493$	8.00000	0.360302
$494$	−4.00000	−0.179969
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	32.0000	1.43252	0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000	1.43252	0.716258	+	0.697835i	$0.245853\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	22.0000	0.981908
$503$	8.00000	0.356702	0.178351	−	0.983967i	$-0.442924\pi$
$503$	8.00000	0.356702	0.178351	+	0.983967i	$0.442924\pi$
$504$	0	0
$505$	−14.0000	−0.622992
$506$	−8.00000	−0.355643
$507$	0	0
$508$	−4.00000	−0.177471
$509$	24.0000	1.06378	0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000	1.06378	0.531891	+	0.846813i	$0.321482\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	11.0000	0.486136
$513$	0	0
$514$	22.0000	0.970378
$515$	−8.00000	−0.352522
$516$	0	0
$517$	−24.0000	−1.05552
$518$	0	0
$519$	0	0
$520$	−12.0000	−0.526235
$521$	12.0000	0.525730	0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000	0.525730	0.262865	+	0.964833i	$0.415333\pi$
$522$	0	0
$523$	−28.0000	−1.22435	−0.612177	−	0.790721i	$-0.709706\pi$
$523$	−28.0000	−1.22435	−0.612177	+	0.790721i	$0.709706\pi$
$524$	−14.0000	−0.611593
$525$	0	0
$526$	−16.0000	−0.697633
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	−2.00000	−0.0868744
$531$	0	0
$532$	−2.00000	−0.0867110
$533$	0	0
$534$	0	0
$535$	12.0000	0.518805
$536$	−48.0000	−2.07328
$537$	0	0
$538$	4.00000	0.172452
$539$	−6.00000	−0.258438
$540$	0	0
$541$	−42.0000	−1.80572	−0.902861	−	0.429934i	$-0.858537\pi$
$541$	−42.0000	−1.80572	−0.902861	+	0.429934i	$0.858537\pi$
$542$	−12.0000	−0.515444
$543$	0	0
$544$	10.0000	0.428746
$545$	10.0000	0.428353
$546$	0	0
$547$	16.0000	0.684111	0.342055	−	0.939680i	$-0.388877\pi$
$547$	16.0000	0.684111	0.342055	+	0.939680i	$0.388877\pi$
$548$	6.00000	0.256307
$549$	0	0
$550$	−2.00000	−0.0852803
$551$	4.00000	0.170406
$552$	0	0
$553$	16.0000	0.680389
$554$	−22.0000	−0.934690
$555$	0	0
$556$	−8.00000	−0.339276
$557$	38.0000	1.61011	0.805056	−	0.593199i	$-0.202135\pi$
$557$	38.0000	1.61011	0.805056	+	0.593199i	$0.202135\pi$
$558$	0	0
$559$	40.0000	1.69182
$560$	2.00000	0.0845154
$561$	0	0
$562$	28.0000	1.18111
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	0	0
$565$	−14.0000	−0.588984
$566$	−26.0000	−1.09286
$567$	0	0
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	−24.0000	−1.00437	−0.502184	−	0.864761i	$-0.667470\pi$
$571$	−24.0000	−1.00437	−0.502184	+	0.864761i	$0.667470\pi$
$572$	8.00000	0.334497
$573$	0	0
$574$	0	0
$575$	4.00000	0.166812
$576$	0	0
$577$	42.0000	1.74848	0.874241	−	0.485491i	$-0.161359\pi$
$577$	42.0000	1.74848	0.874241	+	0.485491i	$0.161359\pi$
$578$	13.0000	0.540729
$579$	0	0
$580$	4.00000	0.166091
$581$	−24.0000	−0.995688
$582$	0	0
$583$	4.00000	0.165663
$584$	−6.00000	−0.248282
$585$	0	0
$586$	26.0000	1.07405
$587$	32.0000	1.32078	0.660391	−	0.750922i	$-0.270391\pi$
$587$	32.0000	1.32078	0.660391	+	0.750922i	$0.270391\pi$
$588$	0	0
$589$	0	0
$590$	4.00000	0.164677
$591$	0	0
$592$	0	0
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	0	0
$595$	4.00000	0.163984
$596$	−18.0000	−0.737309
$597$	0	0
$598$	16.0000	0.654289
$599$	−36.0000	−1.47092	−0.735460	−	0.677568i	$-0.763034\pi$
$599$	−36.0000	−1.47092	−0.735460	+	0.677568i	$0.763034\pi$
$600$	0	0
$601$	30.0000	1.22373	0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000	1.22373	0.611863	+	0.790964i	$0.290420\pi$
$602$	−20.0000	−0.815139
$603$	0	0
$604$	−24.0000	−0.976546
$605$	−7.00000	−0.284590
$606$	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
$610$	−2.00000	−0.0809776
$611$	48.0000	1.94187
$612$	0	0
$613$	−18.0000	−0.727013	−0.363507	−	0.931592i	$-0.618421\pi$
$613$	−18.0000	−0.727013	−0.363507	+	0.931592i	$0.618421\pi$
$614$	0	0
$615$	0	0
$616$	−12.0000	−0.483494
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	0	0
$619$	16.0000	0.643094	0.321547	−	0.946894i	$-0.395797\pi$
$619$	16.0000	0.643094	0.321547	+	0.946894i	$0.395797\pi$
$620$	0	0
$621$	0	0
$622$	−18.0000	−0.721734
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	14.0000	0.559553
$627$	0	0
$628$	18.0000	0.718278
$629$	0	0
$630$	0	0
$631$	48.0000	1.91085	0.955425	−	0.295234i	$-0.0953977\pi$
$631$	48.0000	1.91085	0.955425	+	0.295234i	$0.0953977\pi$
$632$	−24.0000	−0.954669
$633$	0	0
$634$	14.0000	0.556011
$635$	4.00000	0.158735
$636$	0	0
$637$	12.0000	0.475457
$638$	8.00000	0.316723
$639$	0	0
$640$	3.00000	0.118585
$641$	24.0000	0.947943	0.473972	−	0.880540i	$-0.342820\pi$
$641$	24.0000	0.947943	0.473972	+	0.880540i	$0.342820\pi$
$642$	0	0
$643$	26.0000	1.02534	0.512670	−	0.858586i	$-0.328656\pi$
$643$	26.0000	1.02534	0.512670	+	0.858586i	$0.328656\pi$
$644$	8.00000	0.315244
$645$	0	0
$646$	−2.00000	−0.0786889
$647$	24.0000	0.943537	0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000	0.943537	0.471769	+	0.881722i	$0.343616\pi$
$648$	0	0
$649$	−8.00000	−0.314027
$650$	4.00000	0.156893
$651$	0	0
$652$	6.00000	0.234978
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	0	0
$655$	14.0000	0.547025
$656$	0	0
$657$	0	0
$658$	−24.0000	−0.935617
$659$	−16.0000	−0.623272	−0.311636	−	0.950202i	$-0.600877\pi$
$659$	−16.0000	−0.623272	−0.311636	+	0.950202i	$0.600877\pi$
$660$	0	0
$661$	18.0000	0.700119	0.350059	−	0.936727i	$-0.386161\pi$
$661$	18.0000	0.700119	0.350059	+	0.936727i	$0.386161\pi$
$662$	28.0000	1.08825
$663$	0	0
$664$	36.0000	1.39707
$665$	2.00000	0.0775567
$666$	0	0
$667$	−16.0000	−0.619522
$668$	−8.00000	−0.309529
$669$	0	0
$670$	16.0000	0.618134
$671$	4.00000	0.154418
$672$	0	0
$673$	24.0000	0.925132	0.462566	−	0.886585i	$-0.346929\pi$
$673$	24.0000	0.925132	0.462566	+	0.886585i	$0.346929\pi$
$674$	−16.0000	−0.616297
$675$	0	0
$676$	−3.00000	−0.115385
$677$	50.0000	1.92166	0.960828	−	0.277145i	$-0.0893883\pi$
$677$	50.0000	1.92166	0.960828	+	0.277145i	$0.0893883\pi$
$678$	0	0
$679$	32.0000	1.22805
$680$	−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
$686$	−20.0000	−0.763604
$687$	0	0
$688$	10.0000	0.381246
$689$	−8.00000	−0.304776
$690$	0	0
$691$	−16.0000	−0.608669	−0.304334	−	0.952565i	$-0.598434\pi$
$691$	−16.0000	−0.608669	−0.304334	+	0.952565i	$0.598434\pi$
$692$	18.0000	0.684257
$693$	0	0
$694$	−28.0000	−1.06287
$695$	8.00000	0.303457
$696$	0	0
$697$	0	0
$698$	18.0000	0.681310
$699$	0	0
$700$	2.00000	0.0755929
$701$	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
$703$	0	0
$704$	14.0000	0.527645
$705$	0	0
$706$	−2.00000	−0.0752710
$707$	28.0000	1.05305
$708$	0	0
$709$	18.0000	0.676004	0.338002	−	0.941145i	$-0.390249\pi$
$709$	18.0000	0.676004	0.338002	+	0.941145i	$0.390249\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−8.00000	−0.299183
$716$	−12.0000	−0.448461
$717$	0	0
$718$	−2.00000	−0.0746393
$719$	−42.0000	−1.56634	−0.783168	−	0.621810i	$-0.786397\pi$
$719$	−42.0000	−1.56634	−0.783168	+	0.621810i	$0.786397\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	−1.00000	−0.0372161
$723$	0	0
$724$	−10.0000	−0.371647
$725$	−4.00000	−0.148556
$726$	0	0
$727$	−14.0000	−0.519231	−0.259616	−	0.965712i	$-0.583596\pi$
$727$	−14.0000	−0.519231	−0.259616	+	0.965712i	$0.583596\pi$
$728$	24.0000	0.889499
$729$	0	0
$730$	2.00000	0.0740233
$731$	20.0000	0.739727
$732$	0	0
$733$	34.0000	1.25582	0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000	1.25582	0.627909	+	0.778287i	$0.283911\pi$
$734$	−10.0000	−0.369107
$735$	0	0
$736$	−20.0000	−0.737210
$737$	−32.0000	−1.17874
$738$	0	0
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	0	0
$741$	0	0
$742$	4.00000	0.146845
$743$	8.00000	0.293492	0.146746	−	0.989174i	$-0.453120\pi$
$743$	8.00000	0.293492	0.146746	+	0.989174i	$0.453120\pi$
$744$	0	0
$745$	18.0000	0.659469
$746$	−4.00000	−0.146450
$747$	0	0
$748$	4.00000	0.146254
$749$	−24.0000	−0.876941
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	12.0000	0.437595
$753$	0	0
$754$	−16.0000	−0.582686
$755$	24.0000	0.873449
$756$	0	0
$757$	10.0000	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$758$	20.0000	0.726433
$759$	0	0
$760$	−3.00000	−0.108821
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	0	0
$763$	−20.0000	−0.724049
$764$	−18.0000	−0.651217
$765$	0	0
$766$	8.00000	0.289052
$767$	16.0000	0.577727
$768$	0	0
$769$	−46.0000	−1.65880	−0.829401	−	0.558653i	$-0.811318\pi$
$769$	−46.0000	−1.65880	−0.829401	+	0.558653i	$0.811318\pi$
$770$	4.00000	0.144150
$771$	0	0
$772$	−4.00000	−0.143963
$773$	−14.0000	−0.503545	−0.251773	−	0.967786i	$-0.581013\pi$
$773$	−14.0000	−0.503545	−0.251773	+	0.967786i	$0.581013\pi$
$774$	0	0
$775$	0	0
$776$	−48.0000	−1.72310
$777$	0	0
$778$	−30.0000	−1.07555
$779$	0	0
$780$	0	0
$781$	0	0
$782$	8.00000	0.286079
$783$	0	0
$784$	3.00000	0.107143
$785$	−18.0000	−0.642448
$786$	0	0
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	−2.00000	−0.0712470
$789$	0	0
$790$	8.00000	0.284627
$791$	28.0000	0.995565
$792$	0	0
$793$	−8.00000	−0.284088
$794$	2.00000	0.0709773
$795$	0	0
$796$	−12.0000	−0.425329
$797$	14.0000	0.495905	0.247953	−	0.968772i	$-0.420242\pi$
$797$	14.0000	0.495905	0.247953	+	0.968772i	$0.420242\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	−5.00000	−0.176777
$801$	0	0
$802$	12.0000	0.423735
$803$	−4.00000	−0.141157
$804$	0	0
$805$	−8.00000	−0.281963
$806$	0	0
$807$	0	0
$808$	−42.0000	−1.47755
$809$	−2.00000	−0.0703163	−0.0351581	−	0.999382i	$-0.511193\pi$
$809$	−2.00000	−0.0703163	−0.0351581	+	0.999382i	$0.511193\pi$
$810$	0	0
$811$	4.00000	0.140459	0.0702295	−	0.997531i	$-0.477627\pi$
$811$	4.00000	0.140459	0.0702295	+	0.997531i	$0.477627\pi$
$812$	−8.00000	−0.280745
$813$	0	0
$814$	0	0
$815$	−6.00000	−0.210171
$816$	0	0
$817$	10.0000	0.349856
$818$	26.0000	0.909069
$819$	0	0
$820$	0	0
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	0	0
$823$	−14.0000	−0.488009	−0.244005	−	0.969774i	$-0.578461\pi$
$823$	−14.0000	−0.488009	−0.244005	+	0.969774i	$0.578461\pi$
$824$	−24.0000	−0.836080
$825$	0	0
$826$	−8.00000	−0.278356
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	0	0
$829$	−38.0000	−1.31979	−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000	−1.31979	−0.659897	+	0.751356i	$0.729400\pi$
$830$	−12.0000	−0.416526
$831$	0	0
$832$	−28.0000	−0.970725
$833$	6.00000	0.207888
$834$	0	0
$835$	8.00000	0.276851
$836$	2.00000	0.0691714
$837$	0	0
$838$	−26.0000	−0.898155
$839$	−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
$841$	−13.0000	−0.448276
$842$	34.0000	1.17172
$843$	0	0
$844$	12.0000	0.413057
$845$	3.00000	0.103203
$846$	0	0
$847$	14.0000	0.481046
$848$	−2.00000	−0.0686803
$849$	0	0
$850$	2.00000	0.0685994
$851$	0	0
$852$	0	0
$853$	10.0000	0.342393	0.171197	−	0.985237i	$-0.445237\pi$
$853$	10.0000	0.342393	0.171197	+	0.985237i	$0.445237\pi$
$854$	4.00000	0.136877
$855$	0	0
$856$	36.0000	1.23045
$857$	−10.0000	−0.341593	−0.170797	−	0.985306i	$-0.554634\pi$
$857$	−10.0000	−0.341593	−0.170797	+	0.985306i	$0.554634\pi$
$858$	0	0
$859$	−12.0000	−0.409435	−0.204717	−	0.978821i	$-0.565628\pi$
$859$	−12.0000	−0.409435	−0.204717	+	0.978821i	$0.565628\pi$
$860$	10.0000	0.340997
$861$	0	0
$862$	−4.00000	−0.136241
$863$	−40.0000	−1.36162	−0.680808	−	0.732462i	$-0.738371\pi$
$863$	−40.0000	−1.36162	−0.680808	+	0.732462i	$0.738371\pi$
$864$	0	0
$865$	−18.0000	−0.612018
$866$	36.0000	1.22333
$867$	0	0
$868$	0	0
$869$	−16.0000	−0.542763
$870$	0	0
$871$	64.0000	2.16856
$872$	30.0000	1.01593
$873$	0	0
$874$	4.00000	0.135302
$875$	−2.00000	−0.0676123
$876$	0	0
$877$	12.0000	0.405211	0.202606	−	0.979260i	$-0.435059\pi$
$877$	12.0000	0.405211	0.202606	+	0.979260i	$0.435059\pi$
$878$	40.0000	1.34993
$879$	0	0
$880$	−2.00000	−0.0674200
$881$	10.0000	0.336909	0.168454	−	0.985709i	$-0.446122\pi$
$881$	10.0000	0.336909	0.168454	+	0.985709i	$0.446122\pi$
$882$	0	0
$883$	−46.0000	−1.54802	−0.774012	−	0.633171i	$-0.781753\pi$
$883$	−46.0000	−1.54802	−0.774012	+	0.633171i	$0.781753\pi$
$884$	−8.00000	−0.269069
$885$	0	0
$886$	16.0000	0.537531
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	0	0
$891$	0	0
$892$	−24.0000	−0.803579
$893$	12.0000	0.401565
$894$	0	0
$895$	12.0000	0.401116
$896$	−6.00000	−0.200446
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−4.00000	−0.133259
$902$	0	0
$903$	0	0
$904$	−42.0000	−1.39690
$905$	10.0000	0.332411
$906$	0	0
$907$	52.0000	1.72663	0.863316	−	0.504664i	$-0.168384\pi$
$907$	52.0000	1.72663	0.863316	+	0.504664i	$0.168384\pi$
$908$	12.0000	0.398234
$909$	0	0
$910$	−8.00000	−0.265197
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	24.0000	0.794284
$914$	14.0000	0.463079
$915$	0	0
$916$	14.0000	0.462573
$917$	−28.0000	−0.924641
$918$	0	0
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	12.0000	0.395628
$921$	0	0
$922$	6.00000	0.197599
$923$	0	0
$924$	0	0
$925$	0	0
$926$	22.0000	0.722965
$927$	0	0
$928$	20.0000	0.656532
$929$	−10.0000	−0.328089	−0.164045	−	0.986453i	$-0.552454\pi$
$929$	−10.0000	−0.328089	−0.164045	+	0.986453i	$0.552454\pi$
$930$	0	0
$931$	3.00000	0.0983210
$932$	−18.0000	−0.589610
$933$	0	0
$934$	16.0000	0.523536
$935$	−4.00000	−0.130814
$936$	0	0
$937$	−42.0000	−1.37208	−0.686040	−	0.727564i	$-0.740653\pi$
$937$	−42.0000	−1.37208	−0.686040	+	0.727564i	$0.740653\pi$
$938$	−32.0000	−1.04484
$939$	0	0
$940$	12.0000	0.391397
$941$	60.0000	1.95594	0.977972	−	0.208736i	$-0.0669349\pi$
$941$	60.0000	1.95594	0.977972	+	0.208736i	$0.0669349\pi$
$942$	0	0
$943$	0	0
$944$	4.00000	0.130189
$945$	0	0
$946$	20.0000	0.650256
$947$	52.0000	1.68977	0.844886	−	0.534946i	$-0.179668\pi$
$947$	52.0000	1.68977	0.844886	+	0.534946i	$0.179668\pi$
$948$	0	0
$949$	8.00000	0.259691
$950$	1.00000	0.0324443
$951$	0	0
$952$	12.0000	0.388922
$953$	−46.0000	−1.49009	−0.745043	−	0.667016i	$-0.767571\pi$
$953$	−46.0000	−1.49009	−0.745043	+	0.667016i	$0.767571\pi$
$954$	0	0
$955$	18.0000	0.582466
$956$	−6.00000	−0.194054
$957$	0	0
$958$	38.0000	1.22772
$959$	12.0000	0.387500
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	0	0
$964$	−26.0000	−0.837404
$965$	4.00000	0.128765
$966$	0	0
$967$	34.0000	1.09337	0.546683	−	0.837340i	$-0.315890\pi$
$967$	34.0000	1.09337	0.546683	+	0.837340i	$0.315890\pi$
$968$	−21.0000	−0.674966
$969$	0	0
$970$	16.0000	0.513729
$971$	24.0000	0.770197	0.385098	−	0.922876i	$-0.374168\pi$
$971$	24.0000	0.770197	0.385098	+	0.922876i	$0.374168\pi$
$972$	0	0
$973$	−16.0000	−0.512936
$974$	−8.00000	−0.256337
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	0	0
$979$	0	0
$980$	3.00000	0.0958315
$981$	0	0
$982$	−6.00000	−0.191468
$983$	−16.0000	−0.510321	−0.255160	−	0.966899i	$-0.582128\pi$
$983$	−16.0000	−0.510321	−0.255160	+	0.966899i	$0.582128\pi$
$984$	0	0
$985$	2.00000	0.0637253
$986$	−8.00000	−0.254772
$987$	0	0
$988$	−4.00000	−0.127257
$989$	−40.0000	−1.27193
$990$	0	0
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	12.0000	0.380426
$996$	0	0
$997$	−2.00000	−0.0633406	−0.0316703	−	0.999498i	$-0.510083\pi$
$997$	−2.00000	−0.0633406	−0.0316703	+	0.999498i	$0.510083\pi$
$998$	−32.0000	−1.01294
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.a.b.1.1	✓	1	3.2	odd	2
855.2.a.b.1.1		1	1.1	even	1	trivial
1425.2.a.d.1.1		1	15.14	odd	2
1425.2.c.d.799.1		2	15.8	even	4
1425.2.c.d.799.2		2	15.2	even	4
4275.2.a.o.1.1		1	5.4	even	2
4560.2.a.v.1.1		1	12.11	even	2
5415.2.a.c.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

Varieties

Fields

Representations

Groups

Database

Properties

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