LMFDB - Embedded newform 1856.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1856,2,Mod(1,1856)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1856, 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("1856.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1856 = 2^{6} \cdot 29$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1856.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:	$14.8202346151$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 232)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1856.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q+1.00000 q^{3} +3.00000 q^{5} +2.00000 q^{7} -2.00000 q^{9} +3.00000 q^{11} +5.00000 q^{13} +3.00000 q^{15} -4.00000 q^{17} +2.00000 q^{21} +4.00000 q^{25} -5.00000 q^{27} +1.00000 q^{29} +9.00000 q^{31} +3.00000 q^{33} +6.00000 q^{35} -8.00000 q^{37} +5.00000 q^{39} -2.00000 q^{41} +11.0000 q^{43} -6.00000 q^{45} -7.00000 q^{47} -3.00000 q^{49} -4.00000 q^{51} -9.00000 q^{53} +9.00000 q^{55} -4.00000 q^{59} +12.0000 q^{61} -4.00000 q^{63} +15.0000 q^{65} -12.0000 q^{67} +2.00000 q^{71} -4.00000 q^{73} +4.00000 q^{75} +6.00000 q^{77} +3.00000 q^{79} +1.00000 q^{81} +16.0000 q^{83} -12.0000 q^{85} +1.00000 q^{87} +2.00000 q^{89} +10.0000 q^{91} +9.00000 q^{93} -14.0000 q^{97} -6.00000 q^{99} +O(q^{100})

Coefficient data

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

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	0	0
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	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
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	0	0
$15$	3.00000	0.774597
$16$	0	0
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	9.00000	1.61645	0.808224	−	0.588875i	$-0.200429\pi$
$31$	9.00000	1.61645	0.808224	+	0.588875i	$0.200429\pi$
$32$	0	0
$33$	3.00000	0.522233
$34$	0	0
$35$	6.00000	1.01419
$36$	0	0
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	0	0
$39$	5.00000	0.800641
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	11.0000	1.67748	0.838742	−	0.544529i	$-0.183292\pi$
$43$	11.0000	1.67748	0.838742	+	0.544529i	$0.183292\pi$
$44$	0	0
$45$	−6.00000	−0.894427
$46$	0	0
$47$	−7.00000	−1.02105	−0.510527	−	0.859861i	$-0.670550\pi$
$47$	−7.00000	−1.02105	−0.510527	+	0.859861i	$0.670550\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	−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
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−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
$61$	12.0000	1.53644	0.768221	−	0.640184i	$-0.221142\pi$
$61$	12.0000	1.53644	0.768221	+	0.640184i	$0.221142\pi$
$62$	0	0
$63$	−4.00000	−0.503953
$64$	0	0
$65$	15.0000	1.86052
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	0	0
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	0	0
$75$	4.00000	0.461880
$76$	0	0
$77$	6.00000	0.683763
$78$	0	0
$79$	3.00000	0.337526	0.168763	−	0.985657i	$-0.446023\pi$
$79$	3.00000	0.337526	0.168763	+	0.985657i	$0.446023\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	16.0000	1.75623	0.878114	−	0.478451i	$-0.158802\pi$
$83$	16.0000	1.75623	0.878114	+	0.478451i	$0.158802\pi$
$84$	0	0
$85$	−12.0000	−1.30158
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	10.0000	1.04828
$92$	0	0
$93$	9.00000	0.933257
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	0	0
$99$	−6.00000	−0.603023
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	−10.0000	−0.985329	−0.492665	−	0.870219i	$-0.663977\pi$
$103$	−10.0000	−0.985329	−0.492665	+	0.870219i	$0.663977\pi$
$104$	0	0
$105$	6.00000	0.585540
$106$	0	0
$107$	14.0000	1.35343	0.676716	−	0.736245i	$-0.263403\pi$
$107$	14.0000	1.35343	0.676716	+	0.736245i	$0.263403\pi$
$108$	0	0
$109$	−1.00000	−0.0957826	−0.0478913	−	0.998853i	$-0.515250\pi$
$109$	−1.00000	−0.0957826	−0.0478913	+	0.998853i	$0.515250\pi$
$110$	0	0
$111$	−8.00000	−0.759326
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−10.0000	−0.924500
$118$	0	0
$119$	−8.00000	−0.733359
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	−2.00000	−0.180334
$124$	0	0
$125$	−3.00000	−0.268328
$126$	0	0
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	0	0
$129$	11.0000	0.968496
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−15.0000	−1.29099
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	−7.00000	−0.589506
$142$	0	0
$143$	15.0000	1.25436
$144$	0	0
$145$	3.00000	0.249136
$146$	0	0
$147$	−3.00000	−0.247436
$148$	0	0
$149$	−11.0000	−0.901155	−0.450578	−	0.892737i	$-0.648782\pi$
$149$	−11.0000	−0.901155	−0.450578	+	0.892737i	$0.648782\pi$
$150$	0	0
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	0	0
$153$	8.00000	0.646762
$154$	0	0
$155$	27.0000	2.16869
$156$	0	0
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	0	0
$159$	−9.00000	−0.713746
$160$	0	0
$161$	0	0
$162$	0	0
$163$	15.0000	1.17489	0.587445	−	0.809264i	$-0.300134\pi$
$163$	15.0000	1.17489	0.587445	+	0.809264i	$0.300134\pi$
$164$	0	0
$165$	9.00000	0.700649
$166$	0	0
$167$	−18.0000	−1.39288	−0.696441	−	0.717614i	$-0.745234\pi$
$167$	−18.0000	−1.39288	−0.696441	+	0.717614i	$0.745234\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	0	0
$175$	8.00000	0.604743
$176$	0	0
$177$	−4.00000	−0.300658
$178$	0	0
$179$	−22.0000	−1.64436	−0.822179	−	0.569230i	$-0.807242\pi$
$179$	−22.0000	−1.64436	−0.822179	+	0.569230i	$0.807242\pi$
$180$	0	0
$181$	−19.0000	−1.41226	−0.706129	−	0.708083i	$-0.749560\pi$
$181$	−19.0000	−1.41226	−0.706129	+	0.708083i	$0.749560\pi$
$182$	0	0
$183$	12.0000	0.887066
$184$	0	0
$185$	−24.0000	−1.76452
$186$	0	0
$187$	−12.0000	−0.877527
$188$	0	0
$189$	−10.0000	−0.727393
$190$	0	0
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	0	0
$195$	15.0000	1.07417
$196$	0	0
$197$	14.0000	0.997459	0.498729	−	0.866758i	$-0.333800\pi$
$197$	14.0000	0.997459	0.498729	+	0.866758i	$0.333800\pi$
$198$	0	0
$199$	26.0000	1.84309	0.921546	−	0.388270i	$-0.126927\pi$
$199$	26.0000	1.84309	0.921546	+	0.388270i	$0.126927\pi$
$200$	0	0
$201$	−12.0000	−0.846415
$202$	0	0
$203$	2.00000	0.140372
$204$	0	0
$205$	−6.00000	−0.419058
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	3.00000	0.206529	0.103264	−	0.994654i	$-0.467071\pi$
$211$	3.00000	0.206529	0.103264	+	0.994654i	$0.467071\pi$
$212$	0	0
$213$	2.00000	0.137038
$214$	0	0
$215$	33.0000	2.25058
$216$	0	0
$217$	18.0000	1.22192
$218$	0	0
$219$	−4.00000	−0.270295
$220$	0	0
$221$	−20.0000	−1.34535
$222$	0	0
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	0	0
$225$	−8.00000	−0.533333
$226$	0	0
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	0	0
$229$	26.0000	1.71813	0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000	1.71813	0.859064	+	0.511868i	$0.171046\pi$
$230$	0	0
$231$	6.00000	0.394771
$232$	0	0
$233$	−9.00000	−0.589610	−0.294805	−	0.955557i	$-0.595255\pi$
$233$	−9.00000	−0.589610	−0.294805	+	0.955557i	$0.595255\pi$
$234$	0	0
$235$	−21.0000	−1.36989
$236$	0	0
$237$	3.00000	0.194871
$238$	0	0
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	0	0
$241$	1.00000	0.0644157	0.0322078	−	0.999481i	$-0.489746\pi$
$241$	1.00000	0.0644157	0.0322078	+	0.999481i	$0.489746\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	0	0
$245$	−9.00000	−0.574989
$246$	0	0
$247$	0	0
$248$	0	0
$249$	16.0000	1.01396
$250$	0	0
$251$	−3.00000	−0.189358	−0.0946792	−	0.995508i	$-0.530183\pi$
$251$	−3.00000	−0.189358	−0.0946792	+	0.995508i	$0.530183\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−12.0000	−0.751469
$256$	0	0
$257$	29.0000	1.80897	0.904485	−	0.426505i	$-0.140255\pi$
$257$	29.0000	1.80897	0.904485	+	0.426505i	$0.140255\pi$
$258$	0	0
$259$	−16.0000	−0.994192
$260$	0	0
$261$	−2.00000	−0.123797
$262$	0	0
$263$	−19.0000	−1.17159	−0.585795	−	0.810459i	$-0.699218\pi$
$263$	−19.0000	−1.17159	−0.585795	+	0.810459i	$0.699218\pi$
$264$	0	0
$265$	−27.0000	−1.65860
$266$	0	0
$267$	2.00000	0.122398
$268$	0	0
$269$	12.0000	0.731653	0.365826	−	0.930683i	$-0.380786\pi$
$269$	12.0000	0.731653	0.365826	+	0.930683i	$0.380786\pi$
$270$	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
$273$	10.0000	0.605228
$274$	0	0
$275$	12.0000	0.723627
$276$	0	0
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	0	0
$279$	−18.0000	−1.07763
$280$	0	0
$281$	−29.0000	−1.72999	−0.864997	−	0.501776i	$-0.832680\pi$
$281$	−29.0000	−1.72999	−0.864997	+	0.501776i	$0.832680\pi$
$282$	0	0
$283$	−24.0000	−1.42665	−0.713326	−	0.700832i	$-0.752812\pi$
$283$	−24.0000	−1.42665	−0.713326	+	0.700832i	$0.752812\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4.00000	−0.236113
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−14.0000	−0.820695
$292$	0	0
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	0	0
$295$	−12.0000	−0.698667
$296$	0	0
$297$	−15.0000	−0.870388
$298$	0	0
$299$	0	0
$300$	0	0
$301$	22.0000	1.26806
$302$	0	0
$303$	0	0
$304$	0	0
$305$	36.0000	2.06135
$306$	0	0
$307$	7.00000	0.399511	0.199756	−	0.979846i	$-0.435985\pi$
$307$	7.00000	0.399511	0.199756	+	0.979846i	$0.435985\pi$
$308$	0	0
$309$	−10.0000	−0.568880
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	−15.0000	−0.847850	−0.423925	−	0.905697i	$-0.639348\pi$
$313$	−15.0000	−0.847850	−0.423925	+	0.905697i	$0.639348\pi$
$314$	0	0
$315$	−12.0000	−0.676123
$316$	0	0
$317$	−8.00000	−0.449325	−0.224662	−	0.974437i	$-0.572128\pi$
$317$	−8.00000	−0.449325	−0.224662	+	0.974437i	$0.572128\pi$
$318$	0	0
$319$	3.00000	0.167968
$320$	0	0
$321$	14.0000	0.781404
$322$	0	0
$323$	0	0
$324$	0	0
$325$	20.0000	1.10940
$326$	0	0
$327$	−1.00000	−0.0553001
$328$	0	0
$329$	−14.0000	−0.771845
$330$	0	0
$331$	−17.0000	−0.934405	−0.467202	−	0.884150i	$-0.654738\pi$
$331$	−17.0000	−0.934405	−0.467202	+	0.884150i	$0.654738\pi$
$332$	0	0
$333$	16.0000	0.876795
$334$	0	0
$335$	−36.0000	−1.96689
$336$	0	0
$337$	−12.0000	−0.653682	−0.326841	−	0.945079i	$-0.605984\pi$
$337$	−12.0000	−0.653682	−0.326841	+	0.945079i	$0.605984\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	0	0
$341$	27.0000	1.46213
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−30.0000	−1.61048	−0.805242	−	0.592946i	$-0.797965\pi$
$347$	−30.0000	−1.61048	−0.805242	+	0.592946i	$0.797965\pi$
$348$	0	0
$349$	−21.0000	−1.12410	−0.562052	−	0.827102i	$-0.689988\pi$
$349$	−21.0000	−1.12410	−0.562052	+	0.827102i	$0.689988\pi$
$350$	0	0
$351$	−25.0000	−1.33440
$352$	0	0
$353$	30.0000	1.59674	0.798369	−	0.602168i	$-0.205696\pi$
$353$	30.0000	1.59674	0.798369	+	0.602168i	$0.205696\pi$
$354$	0	0
$355$	6.00000	0.318447
$356$	0	0
$357$	−8.00000	−0.423405
$358$	0	0
$359$	3.00000	0.158334	0.0791670	−	0.996861i	$-0.474774\pi$
$359$	3.00000	0.158334	0.0791670	+	0.996861i	$0.474774\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	−2.00000	−0.104973
$364$	0	0
$365$	−12.0000	−0.628109
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	0	0
$369$	4.00000	0.208232
$370$	0	0
$371$	−18.0000	−0.934513
$372$	0	0
$373$	−15.0000	−0.776671	−0.388335	−	0.921518i	$-0.626950\pi$
$373$	−15.0000	−0.776671	−0.388335	+	0.921518i	$0.626950\pi$
$374$	0	0
$375$	−3.00000	−0.154919
$376$	0	0
$377$	5.00000	0.257513
$378$	0	0
$379$	28.0000	1.43826	0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	0	0
$381$	16.0000	0.819705
$382$	0	0
$383$	−6.00000	−0.306586	−0.153293	−	0.988181i	$-0.548988\pi$
$383$	−6.00000	−0.306586	−0.153293	+	0.988181i	$0.548988\pi$
$384$	0	0
$385$	18.0000	0.917365
$386$	0	0
$387$	−22.0000	−1.11832
$388$	0	0
$389$	−24.0000	−1.21685	−0.608424	−	0.793612i	$-0.708198\pi$
$389$	−24.0000	−1.21685	−0.608424	+	0.793612i	$0.708198\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	4.00000	0.201773
$394$	0	0
$395$	9.00000	0.452839
$396$	0	0
$397$	13.0000	0.652451	0.326226	−	0.945292i	$-0.394223\pi$
$397$	13.0000	0.652451	0.326226	+	0.945292i	$0.394223\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	11.0000	0.549314	0.274657	−	0.961542i	$-0.411436\pi$
$401$	11.0000	0.549314	0.274657	+	0.961542i	$0.411436\pi$
$402$	0	0
$403$	45.0000	2.24161
$404$	0	0
$405$	3.00000	0.149071
$406$	0	0
$407$	−24.0000	−1.18964
$408$	0	0
$409$	−18.0000	−0.890043	−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000	−0.890043	−0.445021	+	0.895520i	$0.646804\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	0	0
$413$	−8.00000	−0.393654
$414$	0	0
$415$	48.0000	2.35623
$416$	0	0
$417$	−8.00000	−0.391762
$418$	0	0
$419$	−6.00000	−0.293119	−0.146560	−	0.989202i	$-0.546820\pi$
$419$	−6.00000	−0.293119	−0.146560	+	0.989202i	$0.546820\pi$
$420$	0	0
$421$	20.0000	0.974740	0.487370	−	0.873195i	$-0.337956\pi$
$421$	20.0000	0.974740	0.487370	+	0.873195i	$0.337956\pi$
$422$	0	0
$423$	14.0000	0.680703
$424$	0	0
$425$	−16.0000	−0.776114
$426$	0	0
$427$	24.0000	1.16144
$428$	0	0
$429$	15.0000	0.724207
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	0	0
$433$	8.00000	0.384455	0.192228	−	0.981350i	$-0.438429\pi$
$433$	8.00000	0.384455	0.192228	+	0.981350i	$0.438429\pi$
$434$	0	0
$435$	3.00000	0.143839
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	6.00000	0.285714
$442$	0	0
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	0	0
$445$	6.00000	0.284427
$446$	0	0
$447$	−11.0000	−0.520282
$448$	0	0
$449$	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
$451$	−6.00000	−0.282529
$452$	0	0
$453$	−2.00000	−0.0939682
$454$	0	0
$455$	30.0000	1.40642
$456$	0	0
$457$	38.0000	1.77757	0.888783	−	0.458329i	$-0.151552\pi$
$457$	38.0000	1.77757	0.888783	+	0.458329i	$0.151552\pi$
$458$	0	0
$459$	20.0000	0.933520
$460$	0	0
$461$	−10.0000	−0.465746	−0.232873	−	0.972507i	$-0.574813\pi$
$461$	−10.0000	−0.465746	−0.232873	+	0.972507i	$0.574813\pi$
$462$	0	0
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	0	0
$465$	27.0000	1.25210
$466$	0	0
$467$	−21.0000	−0.971764	−0.485882	−	0.874024i	$-0.661502\pi$
$467$	−21.0000	−0.971764	−0.485882	+	0.874024i	$0.661502\pi$
$468$	0	0
$469$	−24.0000	−1.10822
$470$	0	0
$471$	18.0000	0.829396
$472$	0	0
$473$	33.0000	1.51734
$474$	0	0
$475$	0	0
$476$	0	0
$477$	18.0000	0.824163
$478$	0	0
$479$	−17.0000	−0.776750	−0.388375	−	0.921501i	$-0.626963\pi$
$479$	−17.0000	−0.776750	−0.388375	+	0.921501i	$0.626963\pi$
$480$	0	0
$481$	−40.0000	−1.82384
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−42.0000	−1.90712
$486$	0	0
$487$	14.0000	0.634401	0.317200	−	0.948359i	$-0.397257\pi$
$487$	14.0000	0.634401	0.317200	+	0.948359i	$0.397257\pi$
$488$	0	0
$489$	15.0000	0.678323
$490$	0	0
$491$	−15.0000	−0.676941	−0.338470	−	0.940977i	$-0.609909\pi$
$491$	−15.0000	−0.676941	−0.338470	+	0.940977i	$0.609909\pi$
$492$	0	0
$493$	−4.00000	−0.180151
$494$	0	0
$495$	−18.0000	−0.809040
$496$	0	0
$497$	4.00000	0.179425
$498$	0	0
$499$	16.0000	0.716258	0.358129	−	0.933672i	$-0.383415\pi$
$499$	16.0000	0.716258	0.358129	+	0.933672i	$0.383415\pi$
$500$	0	0
$501$	−18.0000	−0.804181
$502$	0	0
$503$	−9.00000	−0.401290	−0.200645	−	0.979664i	$-0.564304\pi$
$503$	−9.00000	−0.401290	−0.200645	+	0.979664i	$0.564304\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	12.0000	0.532939
$508$	0	0
$509$	−29.0000	−1.28540	−0.642701	−	0.766117i	$-0.722186\pi$
$509$	−29.0000	−1.28540	−0.642701	+	0.766117i	$0.722186\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−30.0000	−1.32196
$516$	0	0
$517$	−21.0000	−0.923579
$518$	0	0
$519$	14.0000	0.614532
$520$	0	0
$521$	−29.0000	−1.27051	−0.635257	−	0.772301i	$-0.719106\pi$
$521$	−29.0000	−1.27051	−0.635257	+	0.772301i	$0.719106\pi$
$522$	0	0
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	0	0
$525$	8.00000	0.349149
$526$	0	0
$527$	−36.0000	−1.56818
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	8.00000	0.347170
$532$	0	0
$533$	−10.0000	−0.433148
$534$	0	0
$535$	42.0000	1.81582
$536$	0	0
$537$	−22.0000	−0.949370
$538$	0	0
$539$	−9.00000	−0.387657
$540$	0	0
$541$	8.00000	0.343947	0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000	0.343947	0.171973	+	0.985102i	$0.444986\pi$
$542$	0	0
$543$	−19.0000	−0.815368
$544$	0	0
$545$	−3.00000	−0.128506
$546$	0	0
$547$	22.0000	0.940652	0.470326	−	0.882493i	$-0.344136\pi$
$547$	22.0000	0.940652	0.470326	+	0.882493i	$0.344136\pi$
$548$	0	0
$549$	−24.0000	−1.02430
$550$	0	0
$551$	0	0
$552$	0	0
$553$	6.00000	0.255146
$554$	0	0
$555$	−24.0000	−1.01874
$556$	0	0
$557$	34.0000	1.44063	0.720313	−	0.693649i	$-0.243998\pi$
$557$	34.0000	1.44063	0.720313	+	0.693649i	$0.243998\pi$
$558$	0	0
$559$	55.0000	2.32625
$560$	0	0
$561$	−12.0000	−0.506640
$562$	0	0
$563$	−13.0000	−0.547885	−0.273942	−	0.961746i	$-0.588328\pi$
$563$	−13.0000	−0.547885	−0.273942	+	0.961746i	$0.588328\pi$
$564$	0	0
$565$	−18.0000	−0.757266
$566$	0	0
$567$	2.00000	0.0839921
$568$	0	0
$569$	26.0000	1.08998	0.544988	−	0.838444i	$-0.316534\pi$
$569$	26.0000	1.08998	0.544988	+	0.838444i	$0.316534\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	24.0000	1.00261
$574$	0	0
$575$	0	0
$576$	0	0
$577$	28.0000	1.16566	0.582828	−	0.812596i	$-0.301946\pi$
$577$	28.0000	1.16566	0.582828	+	0.812596i	$0.301946\pi$
$578$	0	0
$579$	2.00000	0.0831172
$580$	0	0
$581$	32.0000	1.32758
$582$	0	0
$583$	−27.0000	−1.11823
$584$	0	0
$585$	−30.0000	−1.24035
$586$	0	0
$587$	−32.0000	−1.32078	−0.660391	−	0.750922i	$-0.729609\pi$
$587$	−32.0000	−1.32078	−0.660391	+	0.750922i	$0.729609\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	14.0000	0.575883
$592$	0	0
$593$	−25.0000	−1.02663	−0.513313	−	0.858201i	$-0.671582\pi$
$593$	−25.0000	−1.02663	−0.513313	+	0.858201i	$0.671582\pi$
$594$	0	0
$595$	−24.0000	−0.983904
$596$	0	0
$597$	26.0000	1.06411
$598$	0	0
$599$	−1.00000	−0.0408589	−0.0204294	−	0.999791i	$-0.506503\pi$
$599$	−1.00000	−0.0408589	−0.0204294	+	0.999791i	$0.506503\pi$
$600$	0	0
$601$	22.0000	0.897399	0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000	0.897399	0.448699	+	0.893683i	$0.351887\pi$
$602$	0	0
$603$	24.0000	0.977356
$604$	0	0
$605$	−6.00000	−0.243935
$606$	0	0
$607$	−9.00000	−0.365299	−0.182649	−	0.983178i	$-0.558467\pi$
$607$	−9.00000	−0.365299	−0.182649	+	0.983178i	$0.558467\pi$
$608$	0	0
$609$	2.00000	0.0810441
$610$	0	0
$611$	−35.0000	−1.41595
$612$	0	0
$613$	−37.0000	−1.49442	−0.747208	−	0.664590i	$-0.768606\pi$
$613$	−37.0000	−1.49442	−0.747208	+	0.664590i	$0.768606\pi$
$614$	0	0
$615$	−6.00000	−0.241943
$616$	0	0
$617$	−28.0000	−1.12724	−0.563619	−	0.826035i	$-0.690591\pi$
$617$	−28.0000	−1.12724	−0.563619	+	0.826035i	$0.690591\pi$
$618$	0	0
$619$	−5.00000	−0.200967	−0.100483	−	0.994939i	$-0.532039\pi$
$619$	−5.00000	−0.200967	−0.100483	+	0.994939i	$0.532039\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4.00000	0.160257
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	32.0000	1.27592
$630$	0	0
$631$	18.0000	0.716569	0.358284	−	0.933613i	$-0.383362\pi$
$631$	18.0000	0.716569	0.358284	+	0.933613i	$0.383362\pi$
$632$	0	0
$633$	3.00000	0.119239
$634$	0	0
$635$	48.0000	1.90482
$636$	0	0
$637$	−15.0000	−0.594322
$638$	0	0
$639$	−4.00000	−0.158238
$640$	0	0
$641$	28.0000	1.10593	0.552967	−	0.833203i	$-0.313496\pi$
$641$	28.0000	1.10593	0.552967	+	0.833203i	$0.313496\pi$
$642$	0	0
$643$	34.0000	1.34083	0.670415	−	0.741987i	$-0.266116\pi$
$643$	34.0000	1.34083	0.670415	+	0.741987i	$0.266116\pi$
$644$	0	0
$645$	33.0000	1.29937
$646$	0	0
$647$	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
$649$	−12.0000	−0.471041
$650$	0	0
$651$	18.0000	0.705476
$652$	0	0
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	0	0
$655$	12.0000	0.468879
$656$	0	0
$657$	8.00000	0.312110
$658$	0	0
$659$	25.0000	0.973862	0.486931	−	0.873441i	$-0.338116\pi$
$659$	25.0000	0.973862	0.486931	+	0.873441i	$0.338116\pi$
$660$	0	0
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	0	0
$663$	−20.0000	−0.776736
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−2.00000	−0.0773245
$670$	0	0
$671$	36.0000	1.38976
$672$	0	0
$673$	1.00000	0.0385472	0.0192736	−	0.999814i	$-0.493865\pi$
$673$	1.00000	0.0385472	0.0192736	+	0.999814i	$0.493865\pi$
$674$	0	0
$675$	−20.0000	−0.769800
$676$	0	0
$677$	−30.0000	−1.15299	−0.576497	−	0.817099i	$-0.695581\pi$
$677$	−30.0000	−1.15299	−0.576497	+	0.817099i	$0.695581\pi$
$678$	0	0
$679$	−28.0000	−1.07454
$680$	0	0
$681$	−18.0000	−0.689761
$682$	0	0
$683$	8.00000	0.306111	0.153056	−	0.988218i	$-0.451089\pi$
$683$	8.00000	0.306111	0.153056	+	0.988218i	$0.451089\pi$
$684$	0	0
$685$	−36.0000	−1.37549
$686$	0	0
$687$	26.0000	0.991962
$688$	0	0
$689$	−45.0000	−1.71436
$690$	0	0
$691$	4.00000	0.152167	0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000	0.152167	0.0760836	+	0.997101i	$0.475758\pi$
$692$	0	0
$693$	−12.0000	−0.455842
$694$	0	0
$695$	−24.0000	−0.910372
$696$	0	0
$697$	8.00000	0.303022
$698$	0	0
$699$	−9.00000	−0.340411
$700$	0	0
$701$	−7.00000	−0.264386	−0.132193	−	0.991224i	$-0.542202\pi$
$701$	−7.00000	−0.264386	−0.132193	+	0.991224i	$0.542202\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−21.0000	−0.790906
$706$	0	0
$707$	0	0
$708$	0	0
$709$	37.0000	1.38956	0.694782	−	0.719220i	$-0.255501\pi$
$709$	37.0000	1.38956	0.694782	+	0.719220i	$0.255501\pi$
$710$	0	0
$711$	−6.00000	−0.225018
$712$	0	0
$713$	0	0
$714$	0	0
$715$	45.0000	1.68290
$716$	0	0
$717$	20.0000	0.746914
$718$	0	0
$719$	−26.0000	−0.969636	−0.484818	−	0.874615i	$-0.661114\pi$
$719$	−26.0000	−0.969636	−0.484818	+	0.874615i	$0.661114\pi$
$720$	0	0
$721$	−20.0000	−0.744839
$722$	0	0
$723$	1.00000	0.0371904
$724$	0	0
$725$	4.00000	0.148556
$726$	0	0
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−44.0000	−1.62740
$732$	0	0
$733$	24.0000	0.886460	0.443230	−	0.896408i	$-0.353832\pi$
$733$	24.0000	0.886460	0.443230	+	0.896408i	$0.353832\pi$
$734$	0	0
$735$	−9.00000	−0.331970
$736$	0	0
$737$	−36.0000	−1.32608
$738$	0	0
$739$	37.0000	1.36107	0.680534	−	0.732717i	$-0.261748\pi$
$739$	37.0000	1.36107	0.680534	+	0.732717i	$0.261748\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	20.0000	0.733729	0.366864	−	0.930274i	$-0.380431\pi$
$743$	20.0000	0.733729	0.366864	+	0.930274i	$0.380431\pi$
$744$	0	0
$745$	−33.0000	−1.20903
$746$	0	0
$747$	−32.0000	−1.17082
$748$	0	0
$749$	28.0000	1.02310
$750$	0	0
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	0	0
$753$	−3.00000	−0.109326
$754$	0	0
$755$	−6.00000	−0.218362
$756$	0	0
$757$	20.0000	0.726912	0.363456	−	0.931611i	$-0.381597\pi$
$757$	20.0000	0.726912	0.363456	+	0.931611i	$0.381597\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	50.0000	1.81250	0.906249	−	0.422744i	$-0.138933\pi$
$761$	50.0000	1.81250	0.906249	+	0.422744i	$0.138933\pi$
$762$	0	0
$763$	−2.00000	−0.0724049
$764$	0	0
$765$	24.0000	0.867722
$766$	0	0
$767$	−20.0000	−0.722158
$768$	0	0
$769$	−32.0000	−1.15395	−0.576975	−	0.816762i	$-0.695767\pi$
$769$	−32.0000	−1.15395	−0.576975	+	0.816762i	$0.695767\pi$
$770$	0	0
$771$	29.0000	1.04441
$772$	0	0
$773$	14.0000	0.503545	0.251773	−	0.967786i	$-0.418987\pi$
$773$	14.0000	0.503545	0.251773	+	0.967786i	$0.418987\pi$
$774$	0	0
$775$	36.0000	1.29316
$776$	0	0
$777$	−16.0000	−0.573997
$778$	0	0
$779$	0	0
$780$	0	0
$781$	6.00000	0.214697
$782$	0	0
$783$	−5.00000	−0.178685
$784$	0	0
$785$	54.0000	1.92734
$786$	0	0
$787$	30.0000	1.06938	0.534692	−	0.845047i	$-0.320428\pi$
$787$	30.0000	1.06938	0.534692	+	0.845047i	$0.320428\pi$
$788$	0	0
$789$	−19.0000	−0.676418
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	60.0000	2.13066
$794$	0	0
$795$	−27.0000	−0.957591
$796$	0	0
$797$	20.0000	0.708436	0.354218	−	0.935163i	$-0.384747\pi$
$797$	20.0000	0.708436	0.354218	+	0.935163i	$0.384747\pi$
$798$	0	0
$799$	28.0000	0.990569
$800$	0	0
$801$	−4.00000	−0.141333
$802$	0	0
$803$	−12.0000	−0.423471
$804$	0	0
$805$	0	0
$806$	0	0
$807$	12.0000	0.422420
$808$	0	0
$809$	8.00000	0.281265	0.140633	−	0.990062i	$-0.455086\pi$
$809$	8.00000	0.281265	0.140633	+	0.990062i	$0.455086\pi$
$810$	0	0
$811$	−2.00000	−0.0702295	−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000	−0.0702295	−0.0351147	+	0.999383i	$0.511180\pi$
$812$	0	0
$813$	−25.0000	−0.876788
$814$	0	0
$815$	45.0000	1.57628
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−20.0000	−0.698857
$820$	0	0
$821$	45.0000	1.57051	0.785255	−	0.619172i	$-0.212532\pi$
$821$	45.0000	1.57051	0.785255	+	0.619172i	$0.212532\pi$
$822$	0	0
$823$	40.0000	1.39431	0.697156	−	0.716919i	$-0.254448\pi$
$823$	40.0000	1.39431	0.697156	+	0.716919i	$0.254448\pi$
$824$	0	0
$825$	12.0000	0.417786
$826$	0	0
$827$	−45.0000	−1.56480	−0.782402	−	0.622774i	$-0.786006\pi$
$827$	−45.0000	−1.56480	−0.782402	+	0.622774i	$0.786006\pi$
$828$	0	0
$829$	−28.0000	−0.972480	−0.486240	−	0.873825i	$-0.661632\pi$
$829$	−28.0000	−0.972480	−0.486240	+	0.873825i	$0.661632\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	0	0
$833$	12.0000	0.415775
$834$	0	0
$835$	−54.0000	−1.86875
$836$	0	0
$837$	−45.0000	−1.55543
$838$	0	0
$839$	17.0000	0.586905	0.293453	−	0.955974i	$-0.405196\pi$
$839$	17.0000	0.586905	0.293453	+	0.955974i	$0.405196\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−29.0000	−0.998813
$844$	0	0
$845$	36.0000	1.23844
$846$	0	0
$847$	−4.00000	−0.137442
$848$	0	0
$849$	−24.0000	−0.823678
$850$	0	0
$851$	0	0
$852$	0	0
$853$	2.00000	0.0684787	0.0342393	−	0.999414i	$-0.489099\pi$
$853$	2.00000	0.0684787	0.0342393	+	0.999414i	$0.489099\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3.00000	−0.102478	−0.0512390	−	0.998686i	$-0.516317\pi$
$857$	−3.00000	−0.102478	−0.0512390	+	0.998686i	$0.516317\pi$
$858$	0	0
$859$	41.0000	1.39890	0.699451	−	0.714681i	$-0.253428\pi$
$859$	41.0000	1.39890	0.699451	+	0.714681i	$0.253428\pi$
$860$	0	0
$861$	−4.00000	−0.136320
$862$	0	0
$863$	−46.0000	−1.56586	−0.782929	−	0.622111i	$-0.786275\pi$
$863$	−46.0000	−1.56586	−0.782929	+	0.622111i	$0.786275\pi$
$864$	0	0
$865$	42.0000	1.42804
$866$	0	0
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	9.00000	0.305304
$870$	0	0
$871$	−60.0000	−2.03302
$872$	0	0
$873$	28.0000	0.947656
$874$	0	0
$875$	−6.00000	−0.202837
$876$	0	0
$877$	−17.0000	−0.574049	−0.287025	−	0.957923i	$-0.592666\pi$
$877$	−17.0000	−0.574049	−0.287025	+	0.957923i	$0.592666\pi$
$878$	0	0
$879$	−6.00000	−0.202375
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	22.0000	0.740359	0.370179	−	0.928960i	$-0.379296\pi$
$883$	22.0000	0.740359	0.370179	+	0.928960i	$0.379296\pi$
$884$	0	0
$885$	−12.0000	−0.403376
$886$	0	0
$887$	−3.00000	−0.100730	−0.0503651	−	0.998731i	$-0.516038\pi$
$887$	−3.00000	−0.100730	−0.0503651	+	0.998731i	$0.516038\pi$
$888$	0	0
$889$	32.0000	1.07325
$890$	0	0
$891$	3.00000	0.100504
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−66.0000	−2.20614
$896$	0	0
$897$	0	0
$898$	0	0
$899$	9.00000	0.300167
$900$	0	0
$901$	36.0000	1.19933
$902$	0	0
$903$	22.0000	0.732114
$904$	0	0
$905$	−57.0000	−1.89474
$906$	0	0
$907$	44.0000	1.46100	0.730498	−	0.682915i	$-0.239288\pi$
$907$	44.0000	1.46100	0.730498	+	0.682915i	$0.239288\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	39.0000	1.29213	0.646064	−	0.763283i	$-0.276414\pi$
$911$	39.0000	1.29213	0.646064	+	0.763283i	$0.276414\pi$
$912$	0	0
$913$	48.0000	1.58857
$914$	0	0
$915$	36.0000	1.19012
$916$	0	0
$917$	8.00000	0.264183
$918$	0	0
$919$	54.0000	1.78130	0.890648	−	0.454694i	$-0.150251\pi$
$919$	54.0000	1.78130	0.890648	+	0.454694i	$0.150251\pi$
$920$	0	0
$921$	7.00000	0.230658
$922$	0	0
$923$	10.0000	0.329154
$924$	0	0
$925$	−32.0000	−1.05215
$926$	0	0
$927$	20.0000	0.656886
$928$	0	0
$929$	−14.0000	−0.459325	−0.229663	−	0.973270i	$-0.573762\pi$
$929$	−14.0000	−0.459325	−0.229663	+	0.973270i	$0.573762\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−24.0000	−0.785725
$934$	0	0
$935$	−36.0000	−1.17733
$936$	0	0
$937$	30.0000	0.980057	0.490029	−	0.871706i	$-0.336986\pi$
$937$	30.0000	0.980057	0.490029	+	0.871706i	$0.336986\pi$
$938$	0	0
$939$	−15.0000	−0.489506
$940$	0	0
$941$	−33.0000	−1.07577	−0.537885	−	0.843018i	$-0.680776\pi$
$941$	−33.0000	−1.07577	−0.537885	+	0.843018i	$0.680776\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−30.0000	−0.975900
$946$	0	0
$947$	−9.00000	−0.292461	−0.146230	−	0.989251i	$-0.546714\pi$
$947$	−9.00000	−0.292461	−0.146230	+	0.989251i	$0.546714\pi$
$948$	0	0
$949$	−20.0000	−0.649227
$950$	0	0
$951$	−8.00000	−0.259418
$952$	0	0
$953$	−33.0000	−1.06897	−0.534487	−	0.845176i	$-0.679495\pi$
$953$	−33.0000	−1.06897	−0.534487	+	0.845176i	$0.679495\pi$
$954$	0	0
$955$	72.0000	2.32987
$956$	0	0
$957$	3.00000	0.0969762
$958$	0	0
$959$	−24.0000	−0.775000
$960$	0	0
$961$	50.0000	1.61290
$962$	0	0
$963$	−28.0000	−0.902287
$964$	0	0
$965$	6.00000	0.193147
$966$	0	0
$967$	−23.0000	−0.739630	−0.369815	−	0.929105i	$-0.620579\pi$
$967$	−23.0000	−0.739630	−0.369815	+	0.929105i	$0.620579\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	20.0000	0.641831	0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000	0.641831	0.320915	+	0.947108i	$0.396010\pi$
$972$	0	0
$973$	−16.0000	−0.512936
$974$	0	0
$975$	20.0000	0.640513
$976$	0	0
$977$	−3.00000	−0.0959785	−0.0479893	−	0.998848i	$-0.515281\pi$
$977$	−3.00000	−0.0959785	−0.0479893	+	0.998848i	$0.515281\pi$
$978$	0	0
$979$	6.00000	0.191761
$980$	0	0
$981$	2.00000	0.0638551
$982$	0	0
$983$	−3.00000	−0.0956851	−0.0478426	−	0.998855i	$-0.515235\pi$
$983$	−3.00000	−0.0956851	−0.0478426	+	0.998855i	$0.515235\pi$
$984$	0	0
$985$	42.0000	1.33823
$986$	0	0
$987$	−14.0000	−0.445625
$988$	0	0
$989$	0	0
$990$	0	0
$991$	46.0000	1.46124	0.730619	−	0.682785i	$-0.239232\pi$
$991$	46.0000	1.46124	0.730619	+	0.682785i	$0.239232\pi$
$992$	0	0
$993$	−17.0000	−0.539479
$994$	0	0
$995$	78.0000	2.47277
$996$	0	0
$997$	−8.00000	−0.253363	−0.126681	−	0.991943i	$-0.540433\pi$
$997$	−8.00000	−0.253363	−0.126681	+	0.991943i	$0.540433\pi$
$998$	0	0
$999$	40.0000	1.26554

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.2.a.m.1.1		1
4.3	odd	2		1856.2.a.h.1.1		1
8.3	odd	2		464.2.a.d.1.1		1
8.5	even	2		232.2.a.a.1.1	✓	1
24.5	odd	2		2088.2.a.m.1.1		1
24.11	even	2		4176.2.a.be.1.1		1
40.29	even	2		5800.2.a.j.1.1		1
232.173	even	2		6728.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
232.2.a.a.1.1	✓	1	8.5	even	2
464.2.a.d.1.1		1	8.3	odd	2
1856.2.a.h.1.1		1	4.3	odd	2
1856.2.a.m.1.1		1	1.1	even	1	trivial
2088.2.a.m.1.1		1	24.5	odd	2
4176.2.a.be.1.1		1	24.11	even	2
5800.2.a.j.1.1		1	40.29	even	2
6728.2.a.c.1.1		1	232.173	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 $\F_{q}$

Introduction

L-functions

Modular forms

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1856.2.a.m.1.1

Level	$1856$
Weight	$2$
Character	1856.1
Self dual	yes
Analytic conductor	$14.820$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$