LMFDB - Embedded newform 832.4.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [832,4,Mod(1,832)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(832, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("832.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$832 = 2^{6} \cdot 13$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	832.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$49.0895891248$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 52)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

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

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q+3.00000 q^{3} +13.0000 q^{5} -11.0000 q^{7} -18.0000 q^{9} +2.00000 q^{11} +13.0000 q^{13} +39.0000 q^{15} -51.0000 q^{17} -150.000 q^{19} -33.0000 q^{21} -4.00000 q^{23} +44.0000 q^{25} -135.000 q^{27} +118.000 q^{29} -116.000 q^{31} +6.00000 q^{33} -143.000 q^{35} -63.0000 q^{37} +39.0000 q^{39} -288.000 q^{41} +293.000 q^{43} -234.000 q^{45} -335.000 q^{47} -222.000 q^{49} -153.000 q^{51} +708.000 q^{53} +26.0000 q^{55} -450.000 q^{57} -566.000 q^{59} -904.000 q^{61} +198.000 q^{63} +169.000 q^{65} -382.000 q^{67} -12.0000 q^{69} +7.00000 q^{71} +518.000 q^{73} +132.000 q^{75} -22.0000 q^{77} -100.000 q^{79} +81.0000 q^{81} +1440.00 q^{83} -663.000 q^{85} +354.000 q^{87} +1254.00 q^{89} -143.000 q^{91} -348.000 q^{93} -1950.00 q^{95} +1262.00 q^{97} -36.0000 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
$3$	3.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	3.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	0	0
$5$	13.0000	1.16276	0.581378	−	0.813634i	$-0.302514\pi$
$5$	13.0000	1.16276	0.581378	+	0.813634i	$0.302514\pi$
$6$	0	0
$7$	−11.0000	−0.593944	−0.296972	−	0.954886i	$-0.595977\pi$
$7$	−11.0000	−0.593944	−0.296972	+	0.954886i	$0.595977\pi$
$8$	0	0
$9$	−18.0000	−0.666667
$10$	0	0
$11$	2.00000	0.0548202	0.0274101	−	0.999624i	$-0.491274\pi$
$11$	2.00000	0.0548202	0.0274101	+	0.999624i	$0.491274\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	39.0000	0.671317
$16$	0	0
$17$	−51.0000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−51.0000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	−150.000	−1.81118	−0.905588	−	0.424158i	$-0.860570\pi$
$19$	−150.000	−1.81118	−0.905588	+	0.424158i	$0.860570\pi$
$20$	0	0
$21$	−33.0000	−0.342914
$22$	0	0
$23$	−4.00000	−0.0362634	−0.0181317	−	0.999836i	$-0.505772\pi$
$23$	−4.00000	−0.0362634	−0.0181317	+	0.999836i	$0.505772\pi$
$24$	0	0
$25$	44.0000	0.352000
$26$	0	0
$27$	−135.000	−0.962250
$28$	0	0
$29$	118.000	0.755588	0.377794	−	0.925890i	$-0.376683\pi$
$29$	118.000	0.755588	0.377794	+	0.925890i	$0.376683\pi$
$30$	0	0
$31$	−116.000	−0.672071	−0.336036	−	0.941849i	$-0.609086\pi$
$31$	−116.000	−0.672071	−0.336036	+	0.941849i	$0.609086\pi$
$32$	0	0
$33$	6.00000	0.0316505
$34$	0	0
$35$	−143.000	−0.690612
$36$	0	0
$37$	−63.0000	−0.279923	−0.139961	−	0.990157i	$-0.544698\pi$
$37$	−63.0000	−0.279923	−0.139961	+	0.990157i	$0.544698\pi$
$38$	0	0
$39$	39.0000	0.160128
$40$	0	0
$41$	−288.000	−1.09703	−0.548513	−	0.836142i	$-0.684806\pi$
$41$	−288.000	−1.09703	−0.548513	+	0.836142i	$0.684806\pi$
$42$	0	0
$43$	293.000	1.03912	0.519559	−	0.854435i	$-0.326096\pi$
$43$	293.000	1.03912	0.519559	+	0.854435i	$0.326096\pi$
$44$	0	0
$45$	−234.000	−0.775170
$46$	0	0
$47$	−335.000	−1.03968	−0.519838	−	0.854265i	$-0.674008\pi$
$47$	−335.000	−1.03968	−0.519838	+	0.854265i	$0.674008\pi$
$48$	0	0
$49$	−222.000	−0.647230
$50$	0	0
$51$	−153.000	−0.420084
$52$	0	0
$53$	708.000	1.83493	0.917465	−	0.397817i	$-0.130232\pi$
$53$	708.000	1.83493	0.917465	+	0.397817i	$0.130232\pi$
$54$	0	0
$55$	26.0000	0.0637425
$56$	0	0
$57$	−450.000	−1.04568
$58$	0	0
$59$	−566.000	−1.24893	−0.624465	−	0.781052i	$-0.714683\pi$
$59$	−566.000	−1.24893	−0.624465	+	0.781052i	$0.714683\pi$
$60$	0	0
$61$	−904.000	−1.89746	−0.948732	−	0.316081i	$-0.897633\pi$
$61$	−904.000	−1.89746	−0.948732	+	0.316081i	$0.897633\pi$
$62$	0	0
$63$	198.000	0.395963
$64$	0	0
$65$	169.000	0.322490
$66$	0	0
$67$	−382.000	−0.696548	−0.348274	−	0.937393i	$-0.613232\pi$
$67$	−382.000	−0.696548	−0.348274	+	0.937393i	$0.613232\pi$
$68$	0	0
$69$	−12.0000	−0.0209367
$70$	0	0
$71$	7.00000	0.0117007	0.00585033	−	0.999983i	$-0.498138\pi$
$71$	7.00000	0.0117007	0.00585033	+	0.999983i	$0.498138\pi$
$72$	0	0
$73$	518.000	0.830511	0.415256	−	0.909705i	$-0.363692\pi$
$73$	518.000	0.830511	0.415256	+	0.909705i	$0.363692\pi$
$74$	0	0
$75$	132.000	0.203227
$76$	0	0
$77$	−22.0000	−0.0325602
$78$	0	0
$79$	−100.000	−0.142416	−0.0712081	−	0.997461i	$-0.522685\pi$
$79$	−100.000	−0.142416	−0.0712081	+	0.997461i	$0.522685\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	1440.00	1.90434	0.952172	−	0.305563i	$-0.0988446\pi$
$83$	1440.00	1.90434	0.952172	+	0.305563i	$0.0988446\pi$
$84$	0	0
$85$	−663.000	−0.846029
$86$	0	0
$87$	354.000	0.436239
$88$	0	0
$89$	1254.00	1.49353	0.746763	−	0.665091i	$-0.231607\pi$
$89$	1254.00	1.49353	0.746763	+	0.665091i	$0.231607\pi$
$90$	0	0
$91$	−143.000	−0.164730
$92$	0	0
$93$	−348.000	−0.388021
$94$	0	0
$95$	−1950.00	−2.10596
$96$	0	0
$97$	1262.00	1.32100	0.660498	−	0.750827i	$-0.270345\pi$
$97$	1262.00	1.32100	0.660498	+	0.750827i	$0.270345\pi$
$98$	0	0
$99$	−36.0000	−0.0365468
$100$	0	0
$101$	1772.00	1.74575	0.872874	−	0.487945i	$-0.162253\pi$
$101$	1772.00	1.74575	0.872874	+	0.487945i	$0.162253\pi$
$102$	0	0
$103$	−1160.00	−1.10969	−0.554846	−	0.831953i	$-0.687223\pi$
$103$	−1160.00	−1.10969	−0.554846	+	0.831953i	$0.687223\pi$
$104$	0	0
$105$	−429.000	−0.398725
$106$	0	0
$107$	−500.000	−0.451746	−0.225873	−	0.974157i	$-0.572523\pi$
$107$	−500.000	−0.451746	−0.225873	+	0.974157i	$0.572523\pi$
$108$	0	0
$109$	329.000	0.289105	0.144553	−	0.989497i	$-0.453826\pi$
$109$	329.000	0.289105	0.144553	+	0.989497i	$0.453826\pi$
$110$	0	0
$111$	−189.000	−0.161613
$112$	0	0
$113$	−766.000	−0.637692	−0.318846	−	0.947807i	$-0.603295\pi$
$113$	−766.000	−0.637692	−0.318846	+	0.947807i	$0.603295\pi$
$114$	0	0
$115$	−52.0000	−0.0421654
$116$	0	0
$117$	−234.000	−0.184900
$118$	0	0
$119$	561.000	0.432158
$120$	0	0
$121$	−1327.00	−0.996995
$122$	0	0
$123$	−864.000	−0.633368
$124$	0	0
$125$	−1053.00	−0.753465
$126$	0	0
$127$	−2144.00	−1.49803	−0.749013	−	0.662556i	$-0.769472\pi$
$127$	−2144.00	−1.49803	−0.749013	+	0.662556i	$0.769472\pi$
$128$	0	0
$129$	879.000	0.599935
$130$	0	0
$131$	−1847.00	−1.23186	−0.615928	−	0.787802i	$-0.711219\pi$
$131$	−1847.00	−1.23186	−0.615928	+	0.787802i	$0.711219\pi$
$132$	0	0
$133$	1650.00	1.07574
$134$	0	0
$135$	−1755.00	−1.11886
$136$	0	0
$137$	−564.000	−0.351721	−0.175860	−	0.984415i	$-0.556271\pi$
$137$	−564.000	−0.351721	−0.175860	+	0.984415i	$0.556271\pi$
$138$	0	0
$139$	−2719.00	−1.65916	−0.829578	−	0.558391i	$-0.811419\pi$
$139$	−2719.00	−1.65916	−0.829578	+	0.558391i	$0.811419\pi$
$140$	0	0
$141$	−1005.00	−0.600257
$142$	0	0
$143$	26.0000	0.0152044
$144$	0	0
$145$	1534.00	0.878564
$146$	0	0
$147$	−666.000	−0.373679
$148$	0	0
$149$	−1254.00	−0.689474	−0.344737	−	0.938699i	$-0.612032\pi$
$149$	−1254.00	−0.689474	−0.344737	+	0.938699i	$0.612032\pi$
$150$	0	0
$151$	−1173.00	−0.632168	−0.316084	−	0.948731i	$-0.602368\pi$
$151$	−1173.00	−0.632168	−0.316084	+	0.948731i	$0.602368\pi$
$152$	0	0
$153$	918.000	0.485071
$154$	0	0
$155$	−1508.00	−0.781455
$156$	0	0
$157$	1714.00	0.871287	0.435644	−	0.900119i	$-0.356521\pi$
$157$	1714.00	0.871287	0.435644	+	0.900119i	$0.356521\pi$
$158$	0	0
$159$	2124.00	1.05940
$160$	0	0
$161$	44.0000	0.0215384
$162$	0	0
$163$	−2300.00	−1.10521	−0.552607	−	0.833442i	$-0.686367\pi$
$163$	−2300.00	−1.10521	−0.552607	+	0.833442i	$0.686367\pi$
$164$	0	0
$165$	78.0000	0.0368018
$166$	0	0
$167$	1176.00	0.544920	0.272460	−	0.962167i	$-0.412163\pi$
$167$	1176.00	0.544920	0.272460	+	0.962167i	$0.412163\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	2700.00	1.20745
$172$	0	0
$173$	844.000	0.370914	0.185457	−	0.982652i	$-0.440623\pi$
$173$	844.000	0.370914	0.185457	+	0.982652i	$0.440623\pi$
$174$	0	0
$175$	−484.000	−0.209068
$176$	0	0
$177$	−1698.00	−0.721071
$178$	0	0
$179$	−1527.00	−0.637616	−0.318808	−	0.947819i	$-0.603283\pi$
$179$	−1527.00	−0.637616	−0.318808	+	0.947819i	$0.603283\pi$
$180$	0	0
$181$	960.000	0.394233	0.197117	−	0.980380i	$-0.436842\pi$
$181$	960.000	0.394233	0.197117	+	0.980380i	$0.436842\pi$
$182$	0	0
$183$	−2712.00	−1.09550
$184$	0	0
$185$	−819.000	−0.325482
$186$	0	0
$187$	−102.000	−0.0398876
$188$	0	0
$189$	1485.00	0.571523
$190$	0	0
$191$	−110.000	−0.0416718	−0.0208359	−	0.999783i	$-0.506633\pi$
$191$	−110.000	−0.0416718	−0.0208359	+	0.999783i	$0.506633\pi$
$192$	0	0
$193$	−4096.00	−1.52765	−0.763825	−	0.645423i	$-0.776681\pi$
$193$	−4096.00	−1.52765	−0.763825	+	0.645423i	$0.776681\pi$
$194$	0	0
$195$	507.000	0.186190
$196$	0	0
$197$	1635.00	0.591314	0.295657	−	0.955294i	$-0.404461\pi$
$197$	1635.00	0.591314	0.295657	+	0.955294i	$0.404461\pi$
$198$	0	0
$199$	2414.00	0.859919	0.429960	−	0.902848i	$-0.358528\pi$
$199$	2414.00	0.859919	0.429960	+	0.902848i	$0.358528\pi$
$200$	0	0
$201$	−1146.00	−0.402152
$202$	0	0
$203$	−1298.00	−0.448777
$204$	0	0
$205$	−3744.00	−1.27557
$206$	0	0
$207$	72.0000	0.0241756
$208$	0	0
$209$	−300.000	−0.0992892
$210$	0	0
$211$	3697.00	1.20622	0.603109	−	0.797659i	$-0.293928\pi$
$211$	3697.00	1.20622	0.603109	+	0.797659i	$0.293928\pi$
$212$	0	0
$213$	21.0000	0.00675538
$214$	0	0
$215$	3809.00	1.20824
$216$	0	0
$217$	1276.00	0.399173
$218$	0	0
$219$	1554.00	0.479496
$220$	0	0
$221$	−663.000	−0.201802
$222$	0	0
$223$	−2841.00	−0.853127	−0.426564	−	0.904457i	$-0.640276\pi$
$223$	−2841.00	−0.853127	−0.426564	+	0.904457i	$0.640276\pi$
$224$	0	0
$225$	−792.000	−0.234667
$226$	0	0
$227$	2424.00	0.708751	0.354376	−	0.935103i	$-0.384693\pi$
$227$	2424.00	0.708751	0.354376	+	0.935103i	$0.384693\pi$
$228$	0	0
$229$	−429.000	−0.123795	−0.0618976	−	0.998083i	$-0.519715\pi$
$229$	−429.000	−0.123795	−0.0618976	+	0.998083i	$0.519715\pi$
$230$	0	0
$231$	−66.0000	−0.0187986
$232$	0	0
$233$	−683.000	−0.192038	−0.0960189	−	0.995380i	$-0.530611\pi$
$233$	−683.000	−0.192038	−0.0960189	+	0.995380i	$0.530611\pi$
$234$	0	0
$235$	−4355.00	−1.20889
$236$	0	0
$237$	−300.000	−0.0822240
$238$	0	0
$239$	4053.00	1.09693	0.548466	−	0.836173i	$-0.315212\pi$
$239$	4053.00	1.09693	0.548466	+	0.836173i	$0.315212\pi$
$240$	0	0
$241$	−4206.00	−1.12420	−0.562100	−	0.827069i	$-0.690006\pi$
$241$	−4206.00	−1.12420	−0.562100	+	0.827069i	$0.690006\pi$
$242$	0	0
$243$	3888.00	1.02640
$244$	0	0
$245$	−2886.00	−0.752571
$246$	0	0
$247$	−1950.00	−0.502330
$248$	0	0
$249$	4320.00	1.09947
$250$	0	0
$251$	−6312.00	−1.58729	−0.793645	−	0.608381i	$-0.791819\pi$
$251$	−6312.00	−1.58729	−0.793645	+	0.608381i	$0.791819\pi$
$252$	0	0
$253$	−8.00000	−0.00198797
$254$	0	0
$255$	−1989.00	−0.488455
$256$	0	0
$257$	4065.00	0.986645	0.493322	−	0.869847i	$-0.335782\pi$
$257$	4065.00	0.986645	0.493322	+	0.869847i	$0.335782\pi$
$258$	0	0
$259$	693.000	0.166258
$260$	0	0
$261$	−2124.00	−0.503725
$262$	0	0
$263$	7308.00	1.71342	0.856712	−	0.515795i	$-0.172503\pi$
$263$	7308.00	1.71342	0.856712	+	0.515795i	$0.172503\pi$
$264$	0	0
$265$	9204.00	2.13357
$266$	0	0
$267$	3762.00	0.862287
$268$	0	0
$269$	4488.00	1.01724	0.508621	−	0.860990i	$-0.330155\pi$
$269$	4488.00	1.01724	0.508621	+	0.860990i	$0.330155\pi$
$270$	0	0
$271$	−3935.00	−0.882045	−0.441023	−	0.897496i	$-0.645384\pi$
$271$	−3935.00	−0.882045	−0.441023	+	0.897496i	$0.645384\pi$
$272$	0	0
$273$	−429.000	−0.0951072
$274$	0	0
$275$	88.0000	0.0192967
$276$	0	0
$277$	1140.00	0.247278	0.123639	−	0.992327i	$-0.460544\pi$
$277$	1140.00	0.247278	0.123639	+	0.992327i	$0.460544\pi$
$278$	0	0
$279$	2088.00	0.448048
$280$	0	0
$281$	838.000	0.177904	0.0889518	−	0.996036i	$-0.471648\pi$
$281$	838.000	0.177904	0.0889518	+	0.996036i	$0.471648\pi$
$282$	0	0
$283$	5972.00	1.25441	0.627206	−	0.778853i	$-0.284198\pi$
$283$	5972.00	1.25441	0.627206	+	0.778853i	$0.284198\pi$
$284$	0	0
$285$	−5850.00	−1.21587
$286$	0	0
$287$	3168.00	0.651572
$288$	0	0
$289$	−2312.00	−0.470588
$290$	0	0
$291$	3786.00	0.762678
$292$	0	0
$293$	8093.00	1.61365	0.806823	−	0.590794i	$-0.201185\pi$
$293$	8093.00	1.61365	0.806823	+	0.590794i	$0.201185\pi$
$294$	0	0
$295$	−7358.00	−1.45220
$296$	0	0
$297$	−270.000	−0.0527508
$298$	0	0
$299$	−52.0000	−0.0100577
$300$	0	0
$301$	−3223.00	−0.617178
$302$	0	0
$303$	5316.00	1.00791
$304$	0	0
$305$	−11752.0	−2.20629
$306$	0	0
$307$	−2894.00	−0.538011	−0.269005	−	0.963139i	$-0.586695\pi$
$307$	−2894.00	−0.538011	−0.269005	+	0.963139i	$0.586695\pi$
$308$	0	0
$309$	−3480.00	−0.640681
$310$	0	0
$311$	−6006.00	−1.09508	−0.547539	−	0.836780i	$-0.684435\pi$
$311$	−6006.00	−1.09508	−0.547539	+	0.836780i	$0.684435\pi$
$312$	0	0
$313$	2063.00	0.372548	0.186274	−	0.982498i	$-0.440359\pi$
$313$	2063.00	0.372548	0.186274	+	0.982498i	$0.440359\pi$
$314$	0	0
$315$	2574.00	0.460408
$316$	0	0
$317$	−5622.00	−0.996098	−0.498049	−	0.867149i	$-0.665950\pi$
$317$	−5622.00	−0.996098	−0.498049	+	0.867149i	$0.665950\pi$
$318$	0	0
$319$	236.000	0.0414215
$320$	0	0
$321$	−1500.00	−0.260816
$322$	0	0
$323$	7650.00	1.31782
$324$	0	0
$325$	572.000	0.0976272
$326$	0	0
$327$	987.000	0.166915
$328$	0	0
$329$	3685.00	0.617510
$330$	0	0
$331$	5188.00	0.861505	0.430753	−	0.902470i	$-0.358248\pi$
$331$	5188.00	0.861505	0.430753	+	0.902470i	$0.358248\pi$
$332$	0	0
$333$	1134.00	0.186615
$334$	0	0
$335$	−4966.00	−0.809915
$336$	0	0
$337$	−5761.00	−0.931222	−0.465611	−	0.884990i	$-0.654165\pi$
$337$	−5761.00	−0.931222	−0.465611	+	0.884990i	$0.654165\pi$
$338$	0	0
$339$	−2298.00	−0.368172
$340$	0	0
$341$	−232.000	−0.0368431
$342$	0	0
$343$	6215.00	0.978363
$344$	0	0
$345$	−156.000	−0.0243442
$346$	0	0
$347$	4161.00	0.643730	0.321865	−	0.946786i	$-0.395690\pi$
$347$	4161.00	0.643730	0.321865	+	0.946786i	$0.395690\pi$
$348$	0	0
$349$	−6211.00	−0.952628	−0.476314	−	0.879275i	$-0.658027\pi$
$349$	−6211.00	−0.952628	−0.476314	+	0.879275i	$0.658027\pi$
$350$	0	0
$351$	−1755.00	−0.266880
$352$	0	0
$353$	−332.000	−0.0500583	−0.0250291	−	0.999687i	$-0.507968\pi$
$353$	−332.000	−0.0500583	−0.0250291	+	0.999687i	$0.507968\pi$
$354$	0	0
$355$	91.0000	0.0136050
$356$	0	0
$357$	1683.00	0.249506
$358$	0	0
$359$	1296.00	0.190530	0.0952650	−	0.995452i	$-0.469630\pi$
$359$	1296.00	0.190530	0.0952650	+	0.995452i	$0.469630\pi$
$360$	0	0
$361$	15641.0	2.28036
$362$	0	0
$363$	−3981.00	−0.575615
$364$	0	0
$365$	6734.00	0.965681
$366$	0	0
$367$	11726.0	1.66783	0.833913	−	0.551896i	$-0.186095\pi$
$367$	11726.0	1.66783	0.833913	+	0.551896i	$0.186095\pi$
$368$	0	0
$369$	5184.00	0.731350
$370$	0	0
$371$	−7788.00	−1.08985
$372$	0	0
$373$	−2780.00	−0.385906	−0.192953	−	0.981208i	$-0.561806\pi$
$373$	−2780.00	−0.385906	−0.192953	+	0.981208i	$0.561806\pi$
$374$	0	0
$375$	−3159.00	−0.435013
$376$	0	0
$377$	1534.00	0.209562
$378$	0	0
$379$	9920.00	1.34448	0.672238	−	0.740335i	$-0.265333\pi$
$379$	9920.00	1.34448	0.672238	+	0.740335i	$0.265333\pi$
$380$	0	0
$381$	−6432.00	−0.864885
$382$	0	0
$383$	−3033.00	−0.404645	−0.202323	−	0.979319i	$-0.564849\pi$
$383$	−3033.00	−0.404645	−0.202323	+	0.979319i	$0.564849\pi$
$384$	0	0
$385$	−286.000	−0.0378595
$386$	0	0
$387$	−5274.00	−0.692745
$388$	0	0
$389$	12630.0	1.64619	0.823093	−	0.567906i	$-0.192246\pi$
$389$	12630.0	1.64619	0.823093	+	0.567906i	$0.192246\pi$
$390$	0	0
$391$	204.000	0.0263855
$392$	0	0
$393$	−5541.00	−0.711212
$394$	0	0
$395$	−1300.00	−0.165595
$396$	0	0
$397$	−674.000	−0.0852068	−0.0426034	−	0.999092i	$-0.513565\pi$
$397$	−674.000	−0.0852068	−0.0426034	+	0.999092i	$0.513565\pi$
$398$	0	0
$399$	4950.00	0.621078
$400$	0	0
$401$	3144.00	0.391531	0.195765	−	0.980651i	$-0.437281\pi$
$401$	3144.00	0.391531	0.195765	+	0.980651i	$0.437281\pi$
$402$	0	0
$403$	−1508.00	−0.186399
$404$	0	0
$405$	1053.00	0.129195
$406$	0	0
$407$	−126.000	−0.0153454
$408$	0	0
$409$	1108.00	0.133954	0.0669769	−	0.997755i	$-0.478665\pi$
$409$	1108.00	0.133954	0.0669769	+	0.997755i	$0.478665\pi$
$410$	0	0
$411$	−1692.00	−0.203066
$412$	0	0
$413$	6226.00	0.741795
$414$	0	0
$415$	18720.0	2.21429
$416$	0	0
$417$	−8157.00	−0.957914
$418$	0	0
$419$	−2277.00	−0.265486	−0.132743	−	0.991150i	$-0.542379\pi$
$419$	−2277.00	−0.265486	−0.132743	+	0.991150i	$0.542379\pi$
$420$	0	0
$421$	8225.00	0.952166	0.476083	−	0.879400i	$-0.342056\pi$
$421$	8225.00	0.952166	0.476083	+	0.879400i	$0.342056\pi$
$422$	0	0
$423$	6030.00	0.693117
$424$	0	0
$425$	−2244.00	−0.256118
$426$	0	0
$427$	9944.00	1.12699
$428$	0	0
$429$	78.0000	0.00877826
$430$	0	0
$431$	7005.00	0.782875	0.391437	−	0.920205i	$-0.371978\pi$
$431$	7005.00	0.782875	0.391437	+	0.920205i	$0.371978\pi$
$432$	0	0
$433$	5215.00	0.578792	0.289396	−	0.957209i	$-0.406546\pi$
$433$	5215.00	0.578792	0.289396	+	0.957209i	$0.406546\pi$
$434$	0	0
$435$	4602.00	0.507239
$436$	0	0
$437$	600.000	0.0656794
$438$	0	0
$439$	4658.00	0.506411	0.253205	−	0.967413i	$-0.418515\pi$
$439$	4658.00	0.506411	0.253205	+	0.967413i	$0.418515\pi$
$440$	0	0
$441$	3996.00	0.431487
$442$	0	0
$443$	−8217.00	−0.881267	−0.440634	−	0.897687i	$-0.645246\pi$
$443$	−8217.00	−0.881267	−0.440634	+	0.897687i	$0.645246\pi$
$444$	0	0
$445$	16302.0	1.73660
$446$	0	0
$447$	−3762.00	−0.398068
$448$	0	0
$449$	−16634.0	−1.74835	−0.874173	−	0.485615i	$-0.838596\pi$
$449$	−16634.0	−1.74835	−0.874173	+	0.485615i	$0.838596\pi$
$450$	0	0
$451$	−576.000	−0.0601392
$452$	0	0
$453$	−3519.00	−0.364982
$454$	0	0
$455$	−1859.00	−0.191541
$456$	0	0
$457$	−9254.00	−0.947229	−0.473615	−	0.880732i	$-0.657051\pi$
$457$	−9254.00	−0.947229	−0.473615	+	0.880732i	$0.657051\pi$
$458$	0	0
$459$	6885.00	0.700140
$460$	0	0
$461$	−5295.00	−0.534952	−0.267476	−	0.963565i	$-0.586190\pi$
$461$	−5295.00	−0.534952	−0.267476	+	0.963565i	$0.586190\pi$
$462$	0	0
$463$	−14984.0	−1.50403	−0.752015	−	0.659146i	$-0.770918\pi$
$463$	−14984.0	−1.50403	−0.752015	+	0.659146i	$0.770918\pi$
$464$	0	0
$465$	−4524.00	−0.451173
$466$	0	0
$467$	9076.00	0.899330	0.449665	−	0.893197i	$-0.351543\pi$
$467$	9076.00	0.899330	0.449665	+	0.893197i	$0.351543\pi$
$468$	0	0
$469$	4202.00	0.413711
$470$	0	0
$471$	5142.00	0.503038
$472$	0	0
$473$	586.000	0.0569647
$474$	0	0
$475$	−6600.00	−0.637534
$476$	0	0
$477$	−12744.0	−1.22329
$478$	0	0
$479$	−4575.00	−0.436403	−0.218202	−	0.975904i	$-0.570019\pi$
$479$	−4575.00	−0.436403	−0.218202	+	0.975904i	$0.570019\pi$
$480$	0	0
$481$	−819.000	−0.0776366
$482$	0	0
$483$	132.000	0.0124352
$484$	0	0
$485$	16406.0	1.53600
$486$	0	0
$487$	20504.0	1.90785	0.953927	−	0.300039i	$-0.0969996\pi$
$487$	20504.0	1.90785	0.953927	+	0.300039i	$0.0969996\pi$
$488$	0	0
$489$	−6900.00	−0.638096
$490$	0	0
$491$	16949.0	1.55784	0.778918	−	0.627126i	$-0.215769\pi$
$491$	16949.0	1.55784	0.778918	+	0.627126i	$0.215769\pi$
$492$	0	0
$493$	−6018.00	−0.549771
$494$	0	0
$495$	−468.000	−0.0424950
$496$	0	0
$497$	−77.0000	−0.00694954
$498$	0	0
$499$	11072.0	0.993288	0.496644	−	0.867954i	$-0.334565\pi$
$499$	11072.0	0.993288	0.496644	+	0.867954i	$0.334565\pi$
$500$	0	0
$501$	3528.00	0.314610
$502$	0	0
$503$	5194.00	0.460416	0.230208	−	0.973142i	$-0.426059\pi$
$503$	5194.00	0.460416	0.230208	+	0.973142i	$0.426059\pi$
$504$	0	0
$505$	23036.0	2.02988
$506$	0	0
$507$	507.000	0.0444116
$508$	0	0
$509$	−16186.0	−1.40949	−0.704746	−	0.709459i	$-0.748939\pi$
$509$	−16186.0	−1.40949	−0.704746	+	0.709459i	$0.748939\pi$
$510$	0	0
$511$	−5698.00	−0.493277
$512$	0	0
$513$	20250.0	1.74281
$514$	0	0
$515$	−15080.0	−1.29030
$516$	0	0
$517$	−670.000	−0.0569953
$518$	0	0
$519$	2532.00	0.214147
$520$	0	0
$521$	8055.00	0.677343	0.338672	−	0.940905i	$-0.390022\pi$
$521$	8055.00	0.677343	0.338672	+	0.940905i	$0.390022\pi$
$522$	0	0
$523$	−7092.00	−0.592947	−0.296474	−	0.955041i	$-0.595811\pi$
$523$	−7092.00	−0.592947	−0.296474	+	0.955041i	$0.595811\pi$
$524$	0	0
$525$	−1452.00	−0.120706
$526$	0	0
$527$	5916.00	0.489004
$528$	0	0
$529$	−12151.0	−0.998685
$530$	0	0
$531$	10188.0	0.832621
$532$	0	0
$533$	−3744.00	−0.304260
$534$	0	0
$535$	−6500.00	−0.525270
$536$	0	0
$537$	−4581.00	−0.368128
$538$	0	0
$539$	−444.000	−0.0354813
$540$	0	0
$541$	−11741.0	−0.933059	−0.466530	−	0.884506i	$-0.654496\pi$
$541$	−11741.0	−0.933059	−0.466530	+	0.884506i	$0.654496\pi$
$542$	0	0
$543$	2880.00	0.227611
$544$	0	0
$545$	4277.00	0.336159
$546$	0	0
$547$	−11605.0	−0.907119	−0.453559	−	0.891226i	$-0.649846\pi$
$547$	−11605.0	−0.907119	−0.453559	+	0.891226i	$0.649846\pi$
$548$	0	0
$549$	16272.0	1.26498
$550$	0	0
$551$	−17700.0	−1.36850
$552$	0	0
$553$	1100.00	0.0845873
$554$	0	0
$555$	−2457.00	−0.187917
$556$	0	0
$557$	−13773.0	−1.04772	−0.523861	−	0.851804i	$-0.675509\pi$
$557$	−13773.0	−1.04772	−0.523861	+	0.851804i	$0.675509\pi$
$558$	0	0
$559$	3809.00	0.288200
$560$	0	0
$561$	−306.000	−0.0230291
$562$	0	0
$563$	4165.00	0.311783	0.155891	−	0.987774i	$-0.450175\pi$
$563$	4165.00	0.311783	0.155891	+	0.987774i	$0.450175\pi$
$564$	0	0
$565$	−9958.00	−0.741480
$566$	0	0
$567$	−891.000	−0.0659938
$568$	0	0
$569$	−10505.0	−0.773976	−0.386988	−	0.922085i	$-0.626485\pi$
$569$	−10505.0	−0.773976	−0.386988	+	0.922085i	$0.626485\pi$
$570$	0	0
$571$	3831.00	0.280775	0.140387	−	0.990097i	$-0.455165\pi$
$571$	3831.00	0.280775	0.140387	+	0.990097i	$0.455165\pi$
$572$	0	0
$573$	−330.000	−0.0240592
$574$	0	0
$575$	−176.000	−0.0127647
$576$	0	0
$577$	−13566.0	−0.978787	−0.489393	−	0.872063i	$-0.662782\pi$
$577$	−13566.0	−0.978787	−0.489393	+	0.872063i	$0.662782\pi$
$578$	0	0
$579$	−12288.0	−0.881989
$580$	0	0
$581$	−15840.0	−1.13107
$582$	0	0
$583$	1416.00	0.100591
$584$	0	0
$585$	−3042.00	−0.214994
$586$	0	0
$587$	−6932.00	−0.487418	−0.243709	−	0.969848i	$-0.578364\pi$
$587$	−6932.00	−0.487418	−0.243709	+	0.969848i	$0.578364\pi$
$588$	0	0
$589$	17400.0	1.21724
$590$	0	0
$591$	4905.00	0.341395
$592$	0	0
$593$	14762.0	1.02226	0.511132	−	0.859502i	$-0.329226\pi$
$593$	14762.0	1.02226	0.511132	+	0.859502i	$0.329226\pi$
$594$	0	0
$595$	7293.00	0.502494
$596$	0	0
$597$	7242.00	0.496475
$598$	0	0
$599$	−27962.0	−1.90734	−0.953670	−	0.300855i	$-0.902728\pi$
$599$	−27962.0	−1.90734	−0.953670	+	0.300855i	$0.902728\pi$
$600$	0	0
$601$	4165.00	0.282685	0.141343	−	0.989961i	$-0.454858\pi$
$601$	4165.00	0.282685	0.141343	+	0.989961i	$0.454858\pi$
$602$	0	0
$603$	6876.00	0.464365
$604$	0	0
$605$	−17251.0	−1.15926
$606$	0	0
$607$	8286.00	0.554066	0.277033	−	0.960860i	$-0.410649\pi$
$607$	8286.00	0.554066	0.277033	+	0.960860i	$0.410649\pi$
$608$	0	0
$609$	−3894.00	−0.259102
$610$	0	0
$611$	−4355.00	−0.288354
$612$	0	0
$613$	1534.00	0.101073	0.0505364	−	0.998722i	$-0.483907\pi$
$613$	1534.00	0.101073	0.0505364	+	0.998722i	$0.483907\pi$
$614$	0	0
$615$	−11232.0	−0.736452
$616$	0	0
$617$	−21056.0	−1.37388	−0.686939	−	0.726715i	$-0.741046\pi$
$617$	−21056.0	−1.37388	−0.686939	+	0.726715i	$0.741046\pi$
$618$	0	0
$619$	19916.0	1.29320	0.646601	−	0.762829i	$-0.276190\pi$
$619$	19916.0	1.29320	0.646601	+	0.762829i	$0.276190\pi$
$620$	0	0
$621$	540.000	0.0348945
$622$	0	0
$623$	−13794.0	−0.887071
$624$	0	0
$625$	−19189.0	−1.22810
$626$	0	0
$627$	−900.000	−0.0573246
$628$	0	0
$629$	3213.00	0.203674
$630$	0	0
$631$	−2993.00	−0.188826	−0.0944132	−	0.995533i	$-0.530097\pi$
$631$	−2993.00	−0.188826	−0.0944132	+	0.995533i	$0.530097\pi$
$632$	0	0
$633$	11091.0	0.696410
$634$	0	0
$635$	−27872.0	−1.74184
$636$	0	0
$637$	−2886.00	−0.179509
$638$	0	0
$639$	−126.000	−0.00780044
$640$	0	0
$641$	5950.00	0.366632	0.183316	−	0.983054i	$-0.441317\pi$
$641$	5950.00	0.366632	0.183316	+	0.983054i	$0.441317\pi$
$642$	0	0
$643$	−5198.00	−0.318801	−0.159401	−	0.987214i	$-0.550956\pi$
$643$	−5198.00	−0.318801	−0.159401	+	0.987214i	$0.550956\pi$
$644$	0	0
$645$	11427.0	0.697578
$646$	0	0
$647$	−25982.0	−1.57876	−0.789380	−	0.613905i	$-0.789598\pi$
$647$	−25982.0	−1.57876	−0.789380	+	0.613905i	$0.789598\pi$
$648$	0	0
$649$	−1132.00	−0.0684667
$650$	0	0
$651$	3828.00	0.230463
$652$	0	0
$653$	−28824.0	−1.72737	−0.863683	−	0.504035i	$-0.831848\pi$
$653$	−28824.0	−1.72737	−0.863683	+	0.504035i	$0.831848\pi$
$654$	0	0
$655$	−24011.0	−1.43235
$656$	0	0
$657$	−9324.00	−0.553674
$658$	0	0
$659$	−8220.00	−0.485896	−0.242948	−	0.970039i	$-0.578115\pi$
$659$	−8220.00	−0.485896	−0.242948	+	0.970039i	$0.578115\pi$
$660$	0	0
$661$	8242.00	0.484987	0.242494	−	0.970153i	$-0.422035\pi$
$661$	8242.00	0.484987	0.242494	+	0.970153i	$0.422035\pi$
$662$	0	0
$663$	−1989.00	−0.116510
$664$	0	0
$665$	21450.0	1.25082
$666$	0	0
$667$	−472.000	−0.0274002
$668$	0	0
$669$	−8523.00	−0.492553
$670$	0	0
$671$	−1808.00	−0.104019
$672$	0	0
$673$	9013.00	0.516234	0.258117	−	0.966114i	$-0.416898\pi$
$673$	9013.00	0.516234	0.258117	+	0.966114i	$0.416898\pi$
$674$	0	0
$675$	−5940.00	−0.338712
$676$	0	0
$677$	14100.0	0.800454	0.400227	−	0.916416i	$-0.368931\pi$
$677$	14100.0	0.800454	0.400227	+	0.916416i	$0.368931\pi$
$678$	0	0
$679$	−13882.0	−0.784598
$680$	0	0
$681$	7272.00	0.409198
$682$	0	0
$683$	−32260.0	−1.80731	−0.903656	−	0.428258i	$-0.859127\pi$
$683$	−32260.0	−1.80731	−0.903656	+	0.428258i	$0.859127\pi$
$684$	0	0
$685$	−7332.00	−0.408965
$686$	0	0
$687$	−1287.00	−0.0714732
$688$	0	0
$689$	9204.00	0.508918
$690$	0	0
$691$	20840.0	1.14731	0.573655	−	0.819097i	$-0.305525\pi$
$691$	20840.0	1.14731	0.573655	+	0.819097i	$0.305525\pi$
$692$	0	0
$693$	396.000	0.0217068
$694$	0	0
$695$	−35347.0	−1.92919
$696$	0	0
$697$	14688.0	0.798203
$698$	0	0
$699$	−2049.00	−0.110873
$700$	0	0
$701$	−15492.0	−0.834700	−0.417350	−	0.908746i	$-0.637041\pi$
$701$	−15492.0	−0.834700	−0.417350	+	0.908746i	$0.637041\pi$
$702$	0	0
$703$	9450.00	0.506989
$704$	0	0
$705$	−13065.0	−0.697952
$706$	0	0
$707$	−19492.0	−1.03688
$708$	0	0
$709$	8770.00	0.464548	0.232274	−	0.972650i	$-0.425383\pi$
$709$	8770.00	0.464548	0.232274	+	0.972650i	$0.425383\pi$
$710$	0	0
$711$	1800.00	0.0949441
$712$	0	0
$713$	464.000	0.0243716
$714$	0	0
$715$	338.000	0.0176790
$716$	0	0
$717$	12159.0	0.633314
$718$	0	0
$719$	24626.0	1.27732	0.638661	−	0.769488i	$-0.279489\pi$
$719$	24626.0	1.27732	0.638661	+	0.769488i	$0.279489\pi$
$720$	0	0
$721$	12760.0	0.659095
$722$	0	0
$723$	−12618.0	−0.649057
$724$	0	0
$725$	5192.00	0.265967
$726$	0	0
$727$	31162.0	1.58973	0.794866	−	0.606786i	$-0.207541\pi$
$727$	31162.0	1.58973	0.794866	+	0.606786i	$0.207541\pi$
$728$	0	0
$729$	9477.00	0.481481
$730$	0	0
$731$	−14943.0	−0.756070
$732$	0	0
$733$	21031.0	1.05975	0.529876	−	0.848075i	$-0.322239\pi$
$733$	21031.0	1.05975	0.529876	+	0.848075i	$0.322239\pi$
$734$	0	0
$735$	−8658.00	−0.434497
$736$	0	0
$737$	−764.000	−0.0381849
$738$	0	0
$739$	−19116.0	−0.951547	−0.475774	−	0.879568i	$-0.657832\pi$
$739$	−19116.0	−0.951547	−0.475774	+	0.879568i	$0.657832\pi$
$740$	0	0
$741$	−5850.00	−0.290020
$742$	0	0
$743$	25309.0	1.24966	0.624830	−	0.780761i	$-0.285168\pi$
$743$	25309.0	1.24966	0.624830	+	0.780761i	$0.285168\pi$
$744$	0	0
$745$	−16302.0	−0.801690
$746$	0	0
$747$	−25920.0	−1.26956
$748$	0	0
$749$	5500.00	0.268312
$750$	0	0
$751$	−6552.00	−0.318357	−0.159178	−	0.987250i	$-0.550884\pi$
$751$	−6552.00	−0.318357	−0.159178	+	0.987250i	$0.550884\pi$
$752$	0	0
$753$	−18936.0	−0.916423
$754$	0	0
$755$	−15249.0	−0.735057
$756$	0	0
$757$	−9684.00	−0.464955	−0.232478	−	0.972602i	$-0.574683\pi$
$757$	−9684.00	−0.464955	−0.232478	+	0.972602i	$0.574683\pi$
$758$	0	0
$759$	−24.0000	−0.00114775
$760$	0	0
$761$	8118.00	0.386698	0.193349	−	0.981130i	$-0.438065\pi$
$761$	8118.00	0.386698	0.193349	+	0.981130i	$0.438065\pi$
$762$	0	0
$763$	−3619.00	−0.171712
$764$	0	0
$765$	11934.0	0.564019
$766$	0	0
$767$	−7358.00	−0.346391
$768$	0	0
$769$	768.000	0.0360140	0.0180070	−	0.999838i	$-0.494268\pi$
$769$	768.000	0.0360140	0.0180070	+	0.999838i	$0.494268\pi$
$770$	0	0
$771$	12195.0	0.569640
$772$	0	0
$773$	11137.0	0.518202	0.259101	−	0.965850i	$-0.416574\pi$
$773$	11137.0	0.518202	0.259101	+	0.965850i	$0.416574\pi$
$774$	0	0
$775$	−5104.00	−0.236569
$776$	0	0
$777$	2079.00	0.0959893
$778$	0	0
$779$	43200.0	1.98691
$780$	0	0
$781$	14.0000	0.000641433 0
$782$	0	0
$783$	−15930.0	−0.727065
$784$	0	0
$785$	22282.0	1.01309
$786$	0	0
$787$	28000.0	1.26822	0.634112	−	0.773241i	$-0.281366\pi$
$787$	28000.0	1.26822	0.634112	+	0.773241i	$0.281366\pi$
$788$	0	0
$789$	21924.0	0.989246
$790$	0	0
$791$	8426.00	0.378754
$792$	0	0
$793$	−11752.0	−0.526262
$794$	0	0
$795$	27612.0	1.23182
$796$	0	0
$797$	26082.0	1.15919	0.579593	−	0.814906i	$-0.303211\pi$
$797$	26082.0	1.15919	0.579593	+	0.814906i	$0.303211\pi$
$798$	0	0
$799$	17085.0	0.756475
$800$	0	0
$801$	−22572.0	−0.995683
$802$	0	0
$803$	1036.00	0.0455288
$804$	0	0
$805$	572.000	0.0250439
$806$	0	0
$807$	13464.0	0.587305
$808$	0	0
$809$	−8409.00	−0.365445	−0.182722	−	0.983165i	$-0.558491\pi$
$809$	−8409.00	−0.365445	−0.182722	+	0.983165i	$0.558491\pi$
$810$	0	0
$811$	−41444.0	−1.79445	−0.897223	−	0.441578i	$-0.854419\pi$
$811$	−41444.0	−1.79445	−0.897223	+	0.441578i	$0.854419\pi$
$812$	0	0
$813$	−11805.0	−0.509249
$814$	0	0
$815$	−29900.0	−1.28509
$816$	0	0
$817$	−43950.0	−1.88203
$818$	0	0
$819$	2574.00	0.109820
$820$	0	0
$821$	−10715.0	−0.455489	−0.227744	−	0.973721i	$-0.573135\pi$
$821$	−10715.0	−0.455489	−0.227744	+	0.973721i	$0.573135\pi$
$822$	0	0
$823$	32622.0	1.38169	0.690845	−	0.723003i	$-0.257239\pi$
$823$	32622.0	1.38169	0.690845	+	0.723003i	$0.257239\pi$
$824$	0	0
$825$	264.000	0.0111410
$826$	0	0
$827$	−26718.0	−1.12343	−0.561715	−	0.827331i	$-0.689858\pi$
$827$	−26718.0	−1.12343	−0.561715	+	0.827331i	$0.689858\pi$
$828$	0	0
$829$	−23266.0	−0.974743	−0.487371	−	0.873195i	$-0.662044\pi$
$829$	−23266.0	−0.974743	−0.487371	+	0.873195i	$0.662044\pi$
$830$	0	0
$831$	3420.00	0.142766
$832$	0	0
$833$	11322.0	0.470929
$834$	0	0
$835$	15288.0	0.633608
$836$	0	0
$837$	15660.0	0.646701
$838$	0	0
$839$	−44080.0	−1.81384	−0.906919	−	0.421304i	$-0.861572\pi$
$839$	−44080.0	−1.81384	−0.906919	+	0.421304i	$0.861572\pi$
$840$	0	0
$841$	−10465.0	−0.429087
$842$	0	0
$843$	2514.00	0.102713
$844$	0	0
$845$	2197.00	0.0894427
$846$	0	0
$847$	14597.0	0.592159
$848$	0	0
$849$	17916.0	0.724235
$850$	0	0
$851$	252.000	0.0101509
$852$	0	0
$853$	16787.0	0.673829	0.336914	−	0.941535i	$-0.390617\pi$
$853$	16787.0	0.673829	0.336914	+	0.941535i	$0.390617\pi$
$854$	0	0
$855$	35100.0	1.40397
$856$	0	0
$857$	28030.0	1.11725	0.558627	−	0.829419i	$-0.311328\pi$
$857$	28030.0	1.11725	0.558627	+	0.829419i	$0.311328\pi$
$858$	0	0
$859$	1684.00	0.0668886	0.0334443	−	0.999441i	$-0.489352\pi$
$859$	1684.00	0.0668886	0.0334443	+	0.999441i	$0.489352\pi$
$860$	0	0
$861$	9504.00	0.376185
$862$	0	0
$863$	−41135.0	−1.62254	−0.811270	−	0.584672i	$-0.801223\pi$
$863$	−41135.0	−1.62254	−0.811270	+	0.584672i	$0.801223\pi$
$864$	0	0
$865$	10972.0	0.431282
$866$	0	0
$867$	−6936.00	−0.271694
$868$	0	0
$869$	−200.000	−0.00780729
$870$	0	0
$871$	−4966.00	−0.193188
$872$	0	0
$873$	−22716.0	−0.880665
$874$	0	0
$875$	11583.0	0.447516
$876$	0	0
$877$	12483.0	0.480640	0.240320	−	0.970694i	$-0.422748\pi$
$877$	12483.0	0.480640	0.240320	+	0.970694i	$0.422748\pi$
$878$	0	0
$879$	24279.0	0.931639
$880$	0	0
$881$	−22299.0	−0.852750	−0.426375	−	0.904547i	$-0.640210\pi$
$881$	−22299.0	−0.852750	−0.426375	+	0.904547i	$0.640210\pi$
$882$	0	0
$883$	2663.00	0.101492	0.0507458	−	0.998712i	$-0.483840\pi$
$883$	2663.00	0.101492	0.0507458	+	0.998712i	$0.483840\pi$
$884$	0	0
$885$	−22074.0	−0.838429
$886$	0	0
$887$	14488.0	0.548432	0.274216	−	0.961668i	$-0.411582\pi$
$887$	14488.0	0.548432	0.274216	+	0.961668i	$0.411582\pi$
$888$	0	0
$889$	23584.0	0.889744
$890$	0	0
$891$	162.000	0.00609114
$892$	0	0
$893$	50250.0	1.88304
$894$	0	0
$895$	−19851.0	−0.741392
$896$	0	0
$897$	−156.000	−0.00580679
$898$	0	0
$899$	−13688.0	−0.507809
$900$	0	0
$901$	−36108.0	−1.33511
$902$	0	0
$903$	−9669.00	−0.356328
$904$	0	0
$905$	12480.0	0.458397
$906$	0	0
$907$	−43863.0	−1.60579	−0.802893	−	0.596124i	$-0.796707\pi$
$907$	−43863.0	−1.60579	−0.802893	+	0.596124i	$0.796707\pi$
$908$	0	0
$909$	−31896.0	−1.16383
$910$	0	0
$911$	−8142.00	−0.296110	−0.148055	−	0.988979i	$-0.547301\pi$
$911$	−8142.00	−0.296110	−0.148055	+	0.988979i	$0.547301\pi$
$912$	0	0
$913$	2880.00	0.104397
$914$	0	0
$915$	−35256.0	−1.27380
$916$	0	0
$917$	20317.0	0.731654
$918$	0	0
$919$	18504.0	0.664190	0.332095	−	0.943246i	$-0.392245\pi$
$919$	18504.0	0.664190	0.332095	+	0.943246i	$0.392245\pi$
$920$	0	0
$921$	−8682.00	−0.310621
$922$	0	0
$923$	91.0000	0.00324518
$924$	0	0
$925$	−2772.00	−0.0985328
$926$	0	0
$927$	20880.0	0.739794
$928$	0	0
$929$	−26484.0	−0.935320	−0.467660	−	0.883909i	$-0.654903\pi$
$929$	−26484.0	−0.935320	−0.467660	+	0.883909i	$0.654903\pi$
$930$	0	0
$931$	33300.0	1.17225
$932$	0	0
$933$	−18018.0	−0.632243
$934$	0	0
$935$	−1326.00	−0.0463795
$936$	0	0
$937$	−4890.00	−0.170490	−0.0852451	−	0.996360i	$-0.527167\pi$
$937$	−4890.00	−0.170490	−0.0852451	+	0.996360i	$0.527167\pi$
$938$	0	0
$939$	6189.00	0.215091
$940$	0	0
$941$	−39709.0	−1.37564	−0.687820	−	0.725882i	$-0.741432\pi$
$941$	−39709.0	−1.37564	−0.687820	+	0.725882i	$0.741432\pi$
$942$	0	0
$943$	1152.00	0.0397818
$944$	0	0
$945$	19305.0	0.664541
$946$	0	0
$947$	−15582.0	−0.534685	−0.267343	−	0.963602i	$-0.586146\pi$
$947$	−15582.0	−0.534685	−0.267343	+	0.963602i	$0.586146\pi$
$948$	0	0
$949$	6734.00	0.230342
$950$	0	0
$951$	−16866.0	−0.575097
$952$	0	0
$953$	45887.0	1.55973	0.779867	−	0.625946i	$-0.215287\pi$
$953$	45887.0	1.55973	0.779867	+	0.625946i	$0.215287\pi$
$954$	0	0
$955$	−1430.00	−0.0484542
$956$	0	0
$957$	708.000	0.0239147
$958$	0	0
$959$	6204.00	0.208903
$960$	0	0
$961$	−16335.0	−0.548320
$962$	0	0
$963$	9000.00	0.301164
$964$	0	0
$965$	−53248.0	−1.77628
$966$	0	0
$967$	25139.0	0.836004	0.418002	−	0.908446i	$-0.362731\pi$
$967$	25139.0	0.836004	0.418002	+	0.908446i	$0.362731\pi$
$968$	0	0
$969$	22950.0	0.760846
$970$	0	0
$971$	36591.0	1.20933	0.604666	−	0.796479i	$-0.293307\pi$
$971$	36591.0	1.20933	0.604666	+	0.796479i	$0.293307\pi$
$972$	0	0
$973$	29909.0	0.985446
$974$	0	0
$975$	1716.00	0.0563651
$976$	0	0
$977$	−19182.0	−0.628134	−0.314067	−	0.949401i	$-0.601692\pi$
$977$	−19182.0	−0.628134	−0.314067	+	0.949401i	$0.601692\pi$
$978$	0	0
$979$	2508.00	0.0818754
$980$	0	0
$981$	−5922.00	−0.192737
$982$	0	0
$983$	−39835.0	−1.29251	−0.646256	−	0.763121i	$-0.723666\pi$
$983$	−39835.0	−1.29251	−0.646256	+	0.763121i	$0.723666\pi$
$984$	0	0
$985$	21255.0	0.687554
$986$	0	0
$987$	11055.0	0.356519
$988$	0	0
$989$	−1172.00	−0.0376819
$990$	0	0
$991$	−28662.0	−0.918747	−0.459374	−	0.888243i	$-0.651926\pi$
$991$	−28662.0	−0.918747	−0.459374	+	0.888243i	$0.651926\pi$
$992$	0	0
$993$	15564.0	0.497390
$994$	0	0
$995$	31382.0	0.999876
$996$	0	0
$997$	−39182.0	−1.24464	−0.622320	−	0.782763i	$-0.713810\pi$
$997$	−39182.0	−1.24464	−0.622320	+	0.782763i	$0.713810\pi$
$998$	0	0
$999$	8505.00	0.269356

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	832.4.a.n.1.1		1
4.3	odd	2		832.4.a.f.1.1		1
8.3	odd	2		208.4.a.f.1.1		1
8.5	even	2		52.4.a.a.1.1	✓	1
24.5	odd	2		468.4.a.c.1.1		1
24.11	even	2		1872.4.a.n.1.1		1
40.13	odd	4		1300.4.c.b.1249.1		2
40.29	even	2		1300.4.a.d.1.1		1
40.37	odd	4		1300.4.c.b.1249.2		2
104.5	odd	4		676.4.d.a.337.2		2
104.21	odd	4		676.4.d.a.337.1		2
104.29	even	6		676.4.e.a.529.1		2
104.37	odd	12		676.4.h.d.485.1		4
104.45	odd	12		676.4.h.d.361.2		4
104.61	even	6		676.4.e.a.653.1		2
104.69	even	6		676.4.e.b.653.1		2
104.77	even	2		676.4.a.a.1.1		1
104.85	odd	12		676.4.h.d.361.1		4
104.93	odd	12		676.4.h.d.485.2		4
104.101	even	6		676.4.e.b.529.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
52.4.a.a.1.1	✓	1	8.5	even	2
208.4.a.f.1.1		1	8.3	odd	2
468.4.a.c.1.1		1	24.5	odd	2
676.4.a.a.1.1		1	104.77	even	2
676.4.d.a.337.1		2	104.21	odd	4
676.4.d.a.337.2		2	104.5	odd	4
676.4.e.a.529.1		2	104.29	even	6
676.4.e.a.653.1		2	104.61	even	6
676.4.e.b.529.1		2	104.101	even	6
676.4.e.b.653.1		2	104.69	even	6
676.4.h.d.361.1		4	104.85	odd	12
676.4.h.d.361.2		4	104.45	odd	12
676.4.h.d.485.1		4	104.37	odd	12
676.4.h.d.485.2		4	104.93	odd	12
832.4.a.f.1.1		1	4.3	odd	2
832.4.a.n.1.1		1	1.1	even	1	trivial
1300.4.a.d.1.1		1	40.29	even	2
1300.4.c.b.1249.1		2	40.13	odd	4
1300.4.c.b.1249.2		2	40.37	odd	4
1872.4.a.n.1.1		1	24.11	even	2

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Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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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	832.4.a.n.1.1

Level	$832$
Weight	$4$
Character	832.1
Self dual	yes
Analytic conductor	$49.090$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$