LMFDB - Embedded newform 1089.4.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,4,Mod(1,1089)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1089.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1089.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:	$64.2530799963$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 363)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1089.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{2} +1.00000 q^{4} +12.0000 q^{5} -12.0000 q^{7} -21.0000 q^{8} +O(q^{10})$

$q+3.00000 q^{2} +1.00000 q^{4} +12.0000 q^{5} -12.0000 q^{7} -21.0000 q^{8} +36.0000 q^{10} +66.0000 q^{13} -36.0000 q^{14} -71.0000 q^{16} -114.000 q^{17} -42.0000 q^{19} +12.0000 q^{20} -18.0000 q^{23} +19.0000 q^{25} +198.000 q^{26} -12.0000 q^{28} +186.000 q^{29} -308.000 q^{31} -45.0000 q^{32} -342.000 q^{34} -144.000 q^{35} -146.000 q^{37} -126.000 q^{38} -252.000 q^{40} +42.0000 q^{41} +366.000 q^{43} -54.0000 q^{46} -618.000 q^{47} -199.000 q^{49} +57.0000 q^{50} +66.0000 q^{52} +408.000 q^{53} +252.000 q^{56} +558.000 q^{58} +132.000 q^{59} -630.000 q^{61} -924.000 q^{62} +433.000 q^{64} +792.000 q^{65} -452.000 q^{67} -114.000 q^{68} -432.000 q^{70} +282.000 q^{71} -684.000 q^{73} -438.000 q^{74} -42.0000 q^{76} -1272.00 q^{79} -852.000 q^{80} +126.000 q^{82} -432.000 q^{83} -1368.00 q^{85} +1098.00 q^{86} -954.000 q^{89} -792.000 q^{91} -18.0000 q^{92} -1854.00 q^{94} -504.000 q^{95} +326.000 q^{97} -597.000 q^{98} +O(q^{100})$

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 $

$2$	3.00000	1.06066	0.530330	−	0.847791i	$-0.322068\pi$
$2$	3.00000	1.06066	0.530330	+	0.847791i	$0.322068\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.125000
$5$	12.0000	1.07331	0.536656	−	0.843801i	$-0.319687\pi$
$5$	12.0000	1.07331	0.536656	+	0.843801i	$0.319687\pi$
$6$	0	0
$7$	−12.0000	−0.647939	−0.323970	−	0.946068i	$-0.605018\pi$
$7$	−12.0000	−0.647939	−0.323970	+	0.946068i	$0.605018\pi$
$8$	−21.0000	−0.928078
$9$	0	0
$10$	36.0000	1.13842
$11$	0	0
$11$	0	0
$12$	0	0
$13$	66.0000	1.40809	0.704043	−	0.710158i	$-0.251376\pi$
$13$	66.0000	1.40809	0.704043	+	0.710158i	$0.251376\pi$
$14$	−36.0000	−0.687243
$15$	0	0
$16$	−71.0000	−1.10938
$17$	−114.000	−1.62642	−0.813208	−	0.581974i	$-0.802281\pi$
$17$	−114.000	−1.62642	−0.813208	+	0.581974i	$0.802281\pi$
$18$	0	0
$19$	−42.0000	−0.507130	−0.253565	−	0.967318i	$-0.581603\pi$
$19$	−42.0000	−0.507130	−0.253565	+	0.967318i	$0.581603\pi$
$20$	12.0000	0.134164
$21$	0	0
$22$	0	0
$23$	−18.0000	−0.163185	−0.0815926	−	0.996666i	$-0.526001\pi$
$23$	−18.0000	−0.163185	−0.0815926	+	0.996666i	$0.526001\pi$
$24$	0	0
$25$	19.0000	0.152000
$26$	198.000	1.49350
$27$	0	0
$28$	−12.0000	−0.0809924
$29$	186.000	1.19101	0.595506	−	0.803351i	$-0.296952\pi$
$29$	186.000	1.19101	0.595506	+	0.803351i	$0.296952\pi$
$30$	0	0
$31$	−308.000	−1.78447	−0.892233	−	0.451576i	$-0.850862\pi$
$31$	−308.000	−1.78447	−0.892233	+	0.451576i	$0.850862\pi$
$32$	−45.0000	−0.248592
$33$	0	0
$34$	−342.000	−1.72507
$35$	−144.000	−0.695441
$36$	0	0
$37$	−146.000	−0.648710	−0.324355	−	0.945936i	$-0.605147\pi$
$37$	−146.000	−0.648710	−0.324355	+	0.945936i	$0.605147\pi$
$38$	−126.000	−0.537892
$39$	0	0
$40$	−252.000	−0.996117
$41$	42.0000	0.159983	0.0799914	−	0.996796i	$-0.474511\pi$
$41$	42.0000	0.159983	0.0799914	+	0.996796i	$0.474511\pi$
$42$	0	0
$43$	366.000	1.29801	0.649006	−	0.760784i	$-0.275185\pi$
$43$	366.000	1.29801	0.649006	+	0.760784i	$0.275185\pi$
$44$	0	0
$45$	0	0
$46$	−54.0000	−0.173084
$47$	−618.000	−1.91797	−0.958985	−	0.283458i	$-0.908518\pi$
$47$	−618.000	−1.91797	−0.958985	+	0.283458i	$0.908518\pi$
$48$	0	0
$49$	−199.000	−0.580175
$50$	57.0000	0.161220
$51$	0	0
$52$	66.0000	0.176011
$53$	408.000	1.05742	0.528709	−	0.848803i	$-0.322676\pi$
$53$	408.000	1.05742	0.528709	+	0.848803i	$0.322676\pi$
$54$	0	0
$55$	0	0
$56$	252.000	0.601338
$57$	0	0
$58$	558.000	1.26326
$59$	132.000	0.291270	0.145635	−	0.989338i	$-0.453477\pi$
$59$	132.000	0.291270	0.145635	+	0.989338i	$0.453477\pi$
$60$	0	0
$61$	−630.000	−1.32235	−0.661174	−	0.750233i	$-0.729942\pi$
$61$	−630.000	−1.32235	−0.661174	+	0.750233i	$0.729942\pi$
$62$	−924.000	−1.89271
$63$	0	0
$64$	433.000	0.845703
$65$	792.000	1.51132
$66$	0	0
$67$	−452.000	−0.824188	−0.412094	−	0.911141i	$-0.635202\pi$
$67$	−452.000	−0.824188	−0.412094	+	0.911141i	$0.635202\pi$
$68$	−114.000	−0.203302
$69$	0	0
$70$	−432.000	−0.737627
$71$	282.000	0.471370	0.235685	−	0.971830i	$-0.424267\pi$
$71$	282.000	0.471370	0.235685	+	0.971830i	$0.424267\pi$
$72$	0	0
$73$	−684.000	−1.09666	−0.548330	−	0.836262i	$-0.684736\pi$
$73$	−684.000	−1.09666	−0.548330	+	0.836262i	$0.684736\pi$
$74$	−438.000	−0.688060
$75$	0	0
$76$	−42.0000	−0.0633912
$77$	0	0
$78$	0	0
$79$	−1272.00	−1.81153	−0.905767	−	0.423776i	$-0.860704\pi$
$79$	−1272.00	−1.81153	−0.905767	+	0.423776i	$0.860704\pi$
$80$	−852.000	−1.19071
$81$	0	0
$82$	126.000	0.169687
$83$	−432.000	−0.571303	−0.285652	−	0.958334i	$-0.592210\pi$
$83$	−432.000	−0.571303	−0.285652	+	0.958334i	$0.592210\pi$
$84$	0	0
$85$	−1368.00	−1.74565
$86$	1098.00	1.37675
$87$	0	0
$88$	0	0
$89$	−954.000	−1.13622	−0.568111	−	0.822952i	$-0.692326\pi$
$89$	−954.000	−1.13622	−0.568111	+	0.822952i	$0.692326\pi$
$90$	0	0
$91$	−792.000	−0.912353
$92$	−18.0000	−0.0203981
$93$	0	0
$94$	−1854.00	−2.03431
$95$	−504.000	−0.544309
$96$	0	0
$97$	326.000	0.341240	0.170620	−	0.985337i	$-0.445423\pi$
$97$	326.000	0.341240	0.170620	+	0.985337i	$0.445423\pi$
$98$	−597.000	−0.615368
$99$	0	0
$100$	19.0000	0.0190000
$101$	1326.00	1.30636	0.653178	−	0.757205i	$-0.273435\pi$
$101$	1326.00	1.30636	0.653178	+	0.757205i	$0.273435\pi$
$102$	0	0
$103$	−1564.00	−1.49617	−0.748085	−	0.663603i	$-0.769026\pi$
$103$	−1564.00	−1.49617	−0.748085	+	0.663603i	$0.769026\pi$
$104$	−1386.00	−1.30681
$105$	0	0
$106$	1224.00	1.12156
$107$	−84.0000	−0.0758933	−0.0379467	−	0.999280i	$-0.512082\pi$
$107$	−84.0000	−0.0758933	−0.0379467	+	0.999280i	$0.512082\pi$
$108$	0	0
$109$	1122.00	0.985946	0.492973	−	0.870045i	$-0.335910\pi$
$109$	1122.00	0.985946	0.492973	+	0.870045i	$0.335910\pi$
$110$	0	0
$111$	0	0
$112$	852.000	0.718807
$113$	906.000	0.754242	0.377121	−	0.926164i	$-0.376914\pi$
$113$	906.000	0.754242	0.377121	+	0.926164i	$0.376914\pi$
$114$	0	0
$115$	−216.000	−0.175149
$116$	186.000	0.148876
$117$	0	0
$118$	396.000	0.308939
$119$	1368.00	1.05382
$120$	0	0
$121$	0	0
$122$	−1890.00	−1.40256
$123$	0	0
$124$	−308.000	−0.223058
$125$	−1272.00	−0.910169
$126$	0	0
$127$	1992.00	1.39182	0.695911	−	0.718128i	$-0.255001\pi$
$127$	1992.00	1.39182	0.695911	+	0.718128i	$0.255001\pi$
$128$	1659.00	1.14560
$129$	0	0
$130$	2376.00	1.60299
$131$	1068.00	0.712302	0.356151	−	0.934428i	$-0.384089\pi$
$131$	1068.00	0.712302	0.356151	+	0.934428i	$0.384089\pi$
$132$	0	0
$133$	504.000	0.328589
$134$	−1356.00	−0.874183
$135$	0	0
$136$	2394.00	1.50944
$137$	462.000	0.288112	0.144056	−	0.989570i	$-0.453985\pi$
$137$	462.000	0.288112	0.144056	+	0.989570i	$0.453985\pi$
$138$	0	0
$139$	−2274.00	−1.38761	−0.693806	−	0.720162i	$-0.744068\pi$
$139$	−2274.00	−1.38761	−0.693806	+	0.720162i	$0.744068\pi$
$140$	−144.000	−0.0869302
$141$	0	0
$142$	846.000	0.499963
$143$	0	0
$144$	0	0
$145$	2232.00	1.27833
$146$	−2052.00	−1.16318
$147$	0	0
$148$	−146.000	−0.0810887
$149$	−750.000	−0.412365	−0.206183	−	0.978514i	$-0.566104\pi$
$149$	−750.000	−0.412365	−0.206183	+	0.978514i	$0.566104\pi$
$150$	0	0
$151$	720.000	0.388032	0.194016	−	0.980998i	$-0.437849\pi$
$151$	720.000	0.388032	0.194016	+	0.980998i	$0.437849\pi$
$152$	882.000	0.470656
$153$	0	0
$154$	0	0
$155$	−3696.00	−1.91529
$156$	0	0
$157$	−3206.00	−1.62972	−0.814862	−	0.579655i	$-0.803187\pi$
$157$	−3206.00	−1.62972	−0.814862	+	0.579655i	$0.803187\pi$
$158$	−3816.00	−1.92142
$159$	0	0
$160$	−540.000	−0.266817
$161$	216.000	0.105734
$162$	0	0
$163$	740.000	0.355591	0.177795	−	0.984067i	$-0.443104\pi$
$163$	740.000	0.355591	0.177795	+	0.984067i	$0.443104\pi$
$164$	42.0000	0.0199979
$165$	0	0
$166$	−1296.00	−0.605958
$167$	−1968.00	−0.911907	−0.455953	−	0.890004i	$-0.650702\pi$
$167$	−1968.00	−0.911907	−0.455953	+	0.890004i	$0.650702\pi$
$168$	0	0
$169$	2159.00	0.982704
$170$	−4104.00	−1.85154
$171$	0	0
$172$	366.000	0.162251
$173$	2898.00	1.27359	0.636794	−	0.771034i	$-0.280260\pi$
$173$	2898.00	1.27359	0.636794	+	0.771034i	$0.280260\pi$
$174$	0	0
$175$	−228.000	−0.0984867
$176$	0	0
$177$	0	0
$178$	−2862.00	−1.20515
$179$	−120.000	−0.0501074	−0.0250537	−	0.999686i	$-0.507976\pi$
$179$	−120.000	−0.0501074	−0.0250537	+	0.999686i	$0.507976\pi$
$180$	0	0
$181$	−1150.00	−0.472259	−0.236129	−	0.971722i	$-0.575879\pi$
$181$	−1150.00	−0.472259	−0.236129	+	0.971722i	$0.575879\pi$
$182$	−2376.00	−0.967697
$183$	0	0
$184$	378.000	0.151449
$185$	−1752.00	−0.696268
$186$	0	0
$187$	0	0
$188$	−618.000	−0.239746
$189$	0	0
$190$	−1512.00	−0.577326
$191$	1518.00	0.575071	0.287536	−	0.957770i	$-0.407164\pi$
$191$	1518.00	0.575071	0.287536	+	0.957770i	$0.407164\pi$
$192$	0	0
$193$	2052.00	0.765317	0.382659	−	0.923890i	$-0.375009\pi$
$193$	2052.00	0.765317	0.382659	+	0.923890i	$0.375009\pi$
$194$	978.000	0.361940
$195$	0	0
$196$	−199.000	−0.0725219
$197$	−3282.00	−1.18697	−0.593484	−	0.804846i	$-0.702248\pi$
$197$	−3282.00	−1.18697	−0.593484	+	0.804846i	$0.702248\pi$
$198$	0	0
$199$	−704.000	−0.250780	−0.125390	−	0.992108i	$-0.540018\pi$
$199$	−704.000	−0.250780	−0.125390	+	0.992108i	$0.540018\pi$
$200$	−399.000	−0.141068
$201$	0	0
$202$	3978.00	1.38560
$203$	−2232.00	−0.771703
$204$	0	0
$205$	504.000	0.171712
$206$	−4692.00	−1.58693
$207$	0	0
$208$	−4686.00	−1.56209
$209$	0	0
$210$	0	0
$211$	1986.00	0.647971	0.323985	−	0.946062i	$-0.394977\pi$
$211$	1986.00	0.647971	0.323985	+	0.946062i	$0.394977\pi$
$212$	408.000	0.132177
$213$	0	0
$214$	−252.000	−0.0804970
$215$	4392.00	1.39317
$216$	0	0
$217$	3696.00	1.15623
$218$	3366.00	1.04575
$219$	0	0
$220$	0	0
$221$	−7524.00	−2.29013
$222$	0	0
$223$	−2248.00	−0.675055	−0.337527	−	0.941316i	$-0.609591\pi$
$223$	−2248.00	−0.675055	−0.337527	+	0.941316i	$0.609591\pi$
$224$	540.000	0.161073
$225$	0	0
$226$	2718.00	0.799994
$227$	−2796.00	−0.817520	−0.408760	−	0.912642i	$-0.634039\pi$
$227$	−2796.00	−0.817520	−0.408760	+	0.912642i	$0.634039\pi$
$228$	0	0
$229$	−2198.00	−0.634270	−0.317135	−	0.948380i	$-0.602721\pi$
$229$	−2198.00	−0.634270	−0.317135	+	0.948380i	$0.602721\pi$
$230$	−648.000	−0.185773
$231$	0	0
$232$	−3906.00	−1.10535
$233$	5802.00	1.63134	0.815669	−	0.578519i	$-0.196369\pi$
$233$	5802.00	1.63134	0.815669	+	0.578519i	$0.196369\pi$
$234$	0	0
$235$	−7416.00	−2.05858
$236$	132.000	0.0364088
$237$	0	0
$238$	4104.00	1.11774
$239$	1104.00	0.298794	0.149397	−	0.988777i	$-0.452267\pi$
$239$	1104.00	0.298794	0.149397	+	0.988777i	$0.452267\pi$
$240$	0	0
$241$	−5208.00	−1.39202	−0.696010	−	0.718032i	$-0.745043\pi$
$241$	−5208.00	−1.39202	−0.696010	+	0.718032i	$0.745043\pi$
$242$	0	0
$243$	0	0
$244$	−630.000	−0.165294
$245$	−2388.00	−0.622709
$246$	0	0
$247$	−2772.00	−0.714082
$248$	6468.00	1.65612
$249$	0	0
$250$	−3816.00	−0.965380
$251$	2880.00	0.724239	0.362119	−	0.932132i	$-0.382053\pi$
$251$	2880.00	0.724239	0.362119	+	0.932132i	$0.382053\pi$
$252$	0	0
$253$	0	0
$254$	5976.00	1.47625
$255$	0	0
$256$	1513.00	0.369385
$257$	−3762.00	−0.913102	−0.456551	−	0.889697i	$-0.650915\pi$
$257$	−3762.00	−0.913102	−0.456551	+	0.889697i	$0.650915\pi$
$258$	0	0
$259$	1752.00	0.420324
$260$	792.000	0.188914
$261$	0	0
$262$	3204.00	0.755511
$263$	−1920.00	−0.450161	−0.225080	−	0.974340i	$-0.572264\pi$
$263$	−1920.00	−0.450161	−0.225080	+	0.974340i	$0.572264\pi$
$264$	0	0
$265$	4896.00	1.13494
$266$	1512.00	0.348521
$267$	0	0
$268$	−452.000	−0.103023
$269$	6144.00	1.39259	0.696294	−	0.717756i	$-0.254831\pi$
$269$	6144.00	1.39259	0.696294	+	0.717756i	$0.254831\pi$
$270$	0	0
$271$	5904.00	1.32340	0.661702	−	0.749767i	$-0.269834\pi$
$271$	5904.00	1.32340	0.661702	+	0.749767i	$0.269834\pi$
$272$	8094.00	1.80430
$273$	0	0
$274$	1386.00	0.305589
$275$	0	0
$276$	0	0
$277$	3630.00	0.787385	0.393692	−	0.919242i	$-0.371198\pi$
$277$	3630.00	0.787385	0.393692	+	0.919242i	$0.371198\pi$
$278$	−6822.00	−1.47179
$279$	0	0
$280$	3024.00	0.645423
$281$	1842.00	0.391048	0.195524	−	0.980699i	$-0.437359\pi$
$281$	1842.00	0.391048	0.195524	+	0.980699i	$0.437359\pi$
$282$	0	0
$283$	2382.00	0.500336	0.250168	−	0.968202i	$-0.419514\pi$
$283$	2382.00	0.500336	0.250168	+	0.968202i	$0.419514\pi$
$284$	282.000	0.0589212
$285$	0	0
$286$	0	0
$287$	−504.000	−0.103659
$288$	0	0
$289$	8083.00	1.64523
$290$	6696.00	1.35587
$291$	0	0
$292$	−684.000	−0.137082
$293$	2238.00	0.446230	0.223115	−	0.974792i	$-0.428377\pi$
$293$	2238.00	0.446230	0.223115	+	0.974792i	$0.428377\pi$
$294$	0	0
$295$	1584.00	0.312624
$296$	3066.00	0.602053
$297$	0	0
$298$	−2250.00	−0.437379
$299$	−1188.00	−0.229779
$300$	0	0
$301$	−4392.00	−0.841032
$302$	2160.00	0.411570
$303$	0	0
$304$	2982.00	0.562597
$305$	−7560.00	−1.41929
$306$	0	0
$307$	558.000	0.103735	0.0518677	−	0.998654i	$-0.483483\pi$
$307$	558.000	0.103735	0.0518677	+	0.998654i	$0.483483\pi$
$308$	0	0
$309$	0	0
$310$	−11088.0	−2.03147
$311$	4062.00	0.740627	0.370313	−	0.928907i	$-0.379250\pi$
$311$	4062.00	0.740627	0.370313	+	0.928907i	$0.379250\pi$
$312$	0	0
$313$	−3098.00	−0.559455	−0.279727	−	0.960079i	$-0.590244\pi$
$313$	−3098.00	−0.559455	−0.279727	+	0.960079i	$0.590244\pi$
$314$	−9618.00	−1.72858
$315$	0	0
$316$	−1272.00	−0.226442
$317$	3600.00	0.637843	0.318921	−	0.947781i	$-0.396679\pi$
$317$	3600.00	0.637843	0.318921	+	0.947781i	$0.396679\pi$
$318$	0	0
$319$	0	0
$320$	5196.00	0.907704
$321$	0	0
$322$	648.000	0.112148
$323$	4788.00	0.824803
$324$	0	0
$325$	1254.00	0.214029
$326$	2220.00	0.377161
$327$	0	0
$328$	−882.000	−0.148477
$329$	7416.00	1.24273
$330$	0	0
$331$	−4156.00	−0.690134	−0.345067	−	0.938578i	$-0.612144\pi$
$331$	−4156.00	−0.690134	−0.345067	+	0.938578i	$0.612144\pi$
$332$	−432.000	−0.0714129
$333$	0	0
$334$	−5904.00	−0.967223
$335$	−5424.00	−0.884611
$336$	0	0
$337$	−2748.00	−0.444193	−0.222097	−	0.975025i	$-0.571290\pi$
$337$	−2748.00	−0.444193	−0.222097	+	0.975025i	$0.571290\pi$
$338$	6477.00	1.04231
$339$	0	0
$340$	−1368.00	−0.218207
$341$	0	0
$342$	0	0
$343$	6504.00	1.02386
$344$	−7686.00	−1.20466
$345$	0	0
$346$	8694.00	1.35084
$347$	696.000	0.107675	0.0538375	−	0.998550i	$-0.482855\pi$
$347$	696.000	0.107675	0.0538375	+	0.998550i	$0.482855\pi$
$348$	0	0
$349$	3246.00	0.497864	0.248932	−	0.968521i	$-0.419920\pi$
$349$	3246.00	0.497864	0.248932	+	0.968521i	$0.419920\pi$
$350$	−684.000	−0.104461
$351$	0	0
$352$	0	0
$353$	10926.0	1.64740	0.823700	−	0.567026i	$-0.191906\pi$
$353$	10926.0	1.64740	0.823700	+	0.567026i	$0.191906\pi$
$354$	0	0
$355$	3384.00	0.505927
$356$	−954.000	−0.142028
$357$	0	0
$358$	−360.000	−0.0531469
$359$	−1584.00	−0.232870	−0.116435	−	0.993198i	$-0.537147\pi$
$359$	−1584.00	−0.232870	−0.116435	+	0.993198i	$0.537147\pi$
$360$	0	0
$361$	−5095.00	−0.742820
$362$	−3450.00	−0.500906
$363$	0	0
$364$	−792.000	−0.114044
$365$	−8208.00	−1.17706
$366$	0	0
$367$	−232.000	−0.0329981	−0.0164990	−	0.999864i	$-0.505252\pi$
$367$	−232.000	−0.0329981	−0.0164990	+	0.999864i	$0.505252\pi$
$368$	1278.00	0.181034
$369$	0	0
$370$	−5256.00	−0.738504
$371$	−4896.00	−0.685142
$372$	0	0
$373$	10806.0	1.50004	0.750018	−	0.661417i	$-0.230045\pi$
$373$	10806.0	1.50004	0.750018	+	0.661417i	$0.230045\pi$
$374$	0	0
$375$	0	0
$376$	12978.0	1.78002
$377$	12276.0	1.67705
$378$	0	0
$379$	164.000	0.0222272	0.0111136	−	0.999938i	$-0.496462\pi$
$379$	164.000	0.0222272	0.0111136	+	0.999938i	$0.496462\pi$
$380$	−504.000	−0.0680386
$381$	0	0
$382$	4554.00	0.609955
$383$	−8058.00	−1.07505	−0.537526	−	0.843247i	$-0.680641\pi$
$383$	−8058.00	−1.07505	−0.537526	+	0.843247i	$0.680641\pi$
$384$	0	0
$385$	0	0
$386$	6156.00	0.811741
$387$	0	0
$388$	326.000	0.0426550
$389$	708.000	0.0922803	0.0461401	−	0.998935i	$-0.485308\pi$
$389$	708.000	0.0922803	0.0461401	+	0.998935i	$0.485308\pi$
$390$	0	0
$391$	2052.00	0.265407
$392$	4179.00	0.538447
$393$	0	0
$394$	−9846.00	−1.25897
$395$	−15264.0	−1.94434
$396$	0	0
$397$	−286.000	−0.0361560	−0.0180780	−	0.999837i	$-0.505755\pi$
$397$	−286.000	−0.0361560	−0.0180780	+	0.999837i	$0.505755\pi$
$398$	−2112.00	−0.265992
$399$	0	0
$400$	−1349.00	−0.168625
$401$	5262.00	0.655291	0.327646	−	0.944801i	$-0.393745\pi$
$401$	5262.00	0.655291	0.327646	+	0.944801i	$0.393745\pi$
$402$	0	0
$403$	−20328.0	−2.51268
$404$	1326.00	0.163294
$405$	0	0
$406$	−6696.00	−0.818515
$407$	0	0
$408$	0	0
$409$	2280.00	0.275645	0.137822	−	0.990457i	$-0.455990\pi$
$409$	2280.00	0.275645	0.137822	+	0.990457i	$0.455990\pi$
$410$	1512.00	0.182128
$411$	0	0
$412$	−1564.00	−0.187021
$413$	−1584.00	−0.188725
$414$	0	0
$415$	−5184.00	−0.613187
$416$	−2970.00	−0.350039
$417$	0	0
$418$	0	0
$419$	−4272.00	−0.498093	−0.249046	−	0.968492i	$-0.580117\pi$
$419$	−4272.00	−0.498093	−0.249046	+	0.968492i	$0.580117\pi$
$420$	0	0
$421$	10150.0	1.17501	0.587507	−	0.809219i	$-0.300110\pi$
$421$	10150.0	1.17501	0.587507	+	0.809219i	$0.300110\pi$
$422$	5958.00	0.687277
$423$	0	0
$424$	−8568.00	−0.981365
$425$	−2166.00	−0.247215
$426$	0	0
$427$	7560.00	0.856801
$428$	−84.0000	−0.00948667
$429$	0	0
$430$	13176.0	1.47768
$431$	13056.0	1.45913	0.729565	−	0.683911i	$-0.239722\pi$
$431$	13056.0	1.45913	0.729565	+	0.683911i	$0.239722\pi$
$432$	0	0
$433$	13682.0	1.51851	0.759255	−	0.650793i	$-0.225563\pi$
$433$	13682.0	1.51851	0.759255	+	0.650793i	$0.225563\pi$
$434$	11088.0	1.22636
$435$	0	0
$436$	1122.00	0.123243
$437$	756.000	0.0827560
$438$	0	0
$439$	7020.00	0.763203	0.381602	−	0.924327i	$-0.375373\pi$
$439$	7020.00	0.763203	0.381602	+	0.924327i	$0.375373\pi$
$440$	0	0
$441$	0	0
$442$	−22572.0	−2.42905
$443$	8916.00	0.956235	0.478117	−	0.878296i	$-0.341319\pi$
$443$	8916.00	0.956235	0.478117	+	0.878296i	$0.341319\pi$
$444$	0	0
$445$	−11448.0	−1.21952
$446$	−6744.00	−0.716004
$447$	0	0
$448$	−5196.00	−0.547964
$449$	−10110.0	−1.06263	−0.531314	−	0.847175i	$-0.678302\pi$
$449$	−10110.0	−1.06263	−0.531314	+	0.847175i	$0.678302\pi$
$450$	0	0
$451$	0	0
$452$	906.000	0.0942802
$453$	0	0
$454$	−8388.00	−0.867111
$455$	−9504.00	−0.979240
$456$	0	0
$457$	−1824.00	−0.186703	−0.0933513	−	0.995633i	$-0.529758\pi$
$457$	−1824.00	−0.186703	−0.0933513	+	0.995633i	$0.529758\pi$
$458$	−6594.00	−0.672745
$459$	0	0
$460$	−216.000	−0.0218936
$461$	17118.0	1.72942	0.864712	−	0.502267i	$-0.167501\pi$
$461$	17118.0	1.72942	0.864712	+	0.502267i	$0.167501\pi$
$462$	0	0
$463$	596.000	0.0598239	0.0299120	−	0.999553i	$-0.490477\pi$
$463$	596.000	0.0598239	0.0299120	+	0.999553i	$0.490477\pi$
$464$	−13206.0	−1.32128
$465$	0	0
$466$	17406.0	1.73029
$467$	−5856.00	−0.580264	−0.290132	−	0.956987i	$-0.593699\pi$
$467$	−5856.00	−0.580264	−0.290132	+	0.956987i	$0.593699\pi$
$468$	0	0
$469$	5424.00	0.534024
$470$	−22248.0	−2.18345
$471$	0	0
$472$	−2772.00	−0.270321
$473$	0	0
$474$	0	0
$475$	−798.000	−0.0770837
$476$	1368.00	0.131727
$477$	0	0
$478$	3312.00	0.316919
$479$	3264.00	0.311349	0.155674	−	0.987808i	$-0.450245\pi$
$479$	3264.00	0.311349	0.155674	+	0.987808i	$0.450245\pi$
$480$	0	0
$481$	−9636.00	−0.913438
$482$	−15624.0	−1.47646
$483$	0	0
$484$	0	0
$485$	3912.00	0.366257
$486$	0	0
$487$	−9920.00	−0.923035	−0.461518	−	0.887131i	$-0.652695\pi$
$487$	−9920.00	−0.923035	−0.461518	+	0.887131i	$0.652695\pi$
$488$	13230.0	1.22724
$489$	0	0
$490$	−7164.00	−0.660483
$491$	−10620.0	−0.976118	−0.488059	−	0.872811i	$-0.662295\pi$
$491$	−10620.0	−0.976118	−0.488059	+	0.872811i	$0.662295\pi$
$492$	0	0
$493$	−21204.0	−1.93708
$494$	−8316.00	−0.757398
$495$	0	0
$496$	21868.0	1.97964
$497$	−3384.00	−0.305419
$498$	0	0
$499$	52.0000	0.00466501	0.00233250	−	0.999997i	$-0.499258\pi$
$499$	52.0000	0.00466501	0.00233250	+	0.999997i	$0.499258\pi$
$500$	−1272.00	−0.113771
$501$	0	0
$502$	8640.00	0.768171
$503$	8568.00	0.759499	0.379750	−	0.925089i	$-0.376010\pi$
$503$	8568.00	0.759499	0.379750	+	0.925089i	$0.376010\pi$
$504$	0	0
$505$	15912.0	1.40213
$506$	0	0
$507$	0	0
$508$	1992.00	0.173978
$509$	21096.0	1.83706	0.918530	−	0.395351i	$-0.129377\pi$
$509$	21096.0	1.83706	0.918530	+	0.395351i	$0.129377\pi$
$510$	0	0
$511$	8208.00	0.710569
$512$	−8733.00	−0.753804
$513$	0	0
$514$	−11286.0	−0.968491
$515$	−18768.0	−1.60586
$516$	0	0
$517$	0	0
$518$	5256.00	0.445821
$519$	0	0
$520$	−16632.0	−1.40262
$521$	−15762.0	−1.32542	−0.662712	−	0.748874i	$-0.730595\pi$
$521$	−15762.0	−1.32542	−0.662712	+	0.748874i	$0.730595\pi$
$522$	0	0
$523$	−23778.0	−1.98803	−0.994015	−	0.109247i	$-0.965156\pi$
$523$	−23778.0	−1.98803	−0.994015	+	0.109247i	$0.965156\pi$
$524$	1068.00	0.0890378
$525$	0	0
$526$	−5760.00	−0.477468
$527$	35112.0	2.90228
$528$	0	0
$529$	−11843.0	−0.973371
$530$	14688.0	1.20378
$531$	0	0
$532$	504.000	0.0410736
$533$	2772.00	0.225270
$534$	0	0
$535$	−1008.00	−0.0814573
$536$	9492.00	0.764910
$537$	0	0
$538$	18432.0	1.47706
$539$	0	0
$540$	0	0
$541$	−5478.00	−0.435338	−0.217669	−	0.976023i	$-0.569845\pi$
$541$	−5478.00	−0.435338	−0.217669	+	0.976023i	$0.569845\pi$
$542$	17712.0	1.40368
$543$	0	0
$544$	5130.00	0.404314
$545$	13464.0	1.05823
$546$	0	0
$547$	6714.00	0.524808	0.262404	−	0.964958i	$-0.415485\pi$
$547$	6714.00	0.524808	0.262404	+	0.964958i	$0.415485\pi$
$548$	462.000	0.0360140
$549$	0	0
$550$	0	0
$551$	−7812.00	−0.603997
$552$	0	0
$553$	15264.0	1.17376
$554$	10890.0	0.835148
$555$	0	0
$556$	−2274.00	−0.173452
$557$	−7218.00	−0.549078	−0.274539	−	0.961576i	$-0.588525\pi$
$557$	−7218.00	−0.549078	−0.274539	+	0.961576i	$0.588525\pi$
$558$	0	0
$559$	24156.0	1.82771
$560$	10224.0	0.771505
$561$	0	0
$562$	5526.00	0.414769
$563$	552.000	0.0413215	0.0206608	−	0.999787i	$-0.493423\pi$
$563$	552.000	0.0413215	0.0206608	+	0.999787i	$0.493423\pi$
$564$	0	0
$565$	10872.0	0.809537
$566$	7146.00	0.530687
$567$	0	0
$568$	−5922.00	−0.437468
$569$	5766.00	0.424821	0.212411	−	0.977180i	$-0.431869\pi$
$569$	5766.00	0.424821	0.212411	+	0.977180i	$0.431869\pi$
$570$	0	0
$571$	−17070.0	−1.25106	−0.625532	−	0.780199i	$-0.715118\pi$
$571$	−17070.0	−1.25106	−0.625532	+	0.780199i	$0.715118\pi$
$572$	0	0
$573$	0	0
$574$	−1512.00	−0.109947
$575$	−342.000	−0.0248041
$576$	0	0
$577$	26170.0	1.88817	0.944083	−	0.329709i	$-0.106951\pi$
$577$	26170.0	1.88817	0.944083	+	0.329709i	$0.106951\pi$
$578$	24249.0	1.74503
$579$	0	0
$580$	2232.00	0.159791
$581$	5184.00	0.370170
$582$	0	0
$583$	0	0
$584$	14364.0	1.01779
$585$	0	0
$586$	6714.00	0.473298
$587$	−27996.0	−1.96852	−0.984258	−	0.176739i	$-0.943445\pi$
$587$	−27996.0	−1.96852	−0.984258	+	0.176739i	$0.943445\pi$
$588$	0	0
$589$	12936.0	0.904955
$590$	4752.00	0.331588
$591$	0	0
$592$	10366.0	0.719662
$593$	−28398.0	−1.96655	−0.983277	−	0.182118i	$-0.941705\pi$
$593$	−28398.0	−1.96655	−0.983277	+	0.182118i	$0.941705\pi$
$594$	0	0
$595$	16416.0	1.13108
$596$	−750.000	−0.0515456
$597$	0	0
$598$	−3564.00	−0.243717
$599$	−13482.0	−0.919632	−0.459816	−	0.888014i	$-0.652085\pi$
$599$	−13482.0	−0.919632	−0.459816	+	0.888014i	$0.652085\pi$
$600$	0	0
$601$	−4488.00	−0.304608	−0.152304	−	0.988334i	$-0.548669\pi$
$601$	−4488.00	−0.304608	−0.152304	+	0.988334i	$0.548669\pi$
$602$	−13176.0	−0.892049
$603$	0	0
$604$	720.000	0.0485039
$605$	0	0
$606$	0	0
$607$	600.000	0.0401207	0.0200603	−	0.999799i	$-0.493614\pi$
$607$	600.000	0.0401207	0.0200603	+	0.999799i	$0.493614\pi$
$608$	1890.00	0.126068
$609$	0	0
$610$	−22680.0	−1.50539
$611$	−40788.0	−2.70066
$612$	0	0
$613$	1674.00	0.110297	0.0551486	−	0.998478i	$-0.482437\pi$
$613$	1674.00	0.110297	0.0551486	+	0.998478i	$0.482437\pi$
$614$	1674.00	0.110028
$615$	0	0
$616$	0	0
$617$	18834.0	1.22890	0.614448	−	0.788958i	$-0.289379\pi$
$617$	18834.0	1.22890	0.614448	+	0.788958i	$0.289379\pi$
$618$	0	0
$619$	7436.00	0.482840	0.241420	−	0.970421i	$-0.422387\pi$
$619$	7436.00	0.482840	0.241420	+	0.970421i	$0.422387\pi$
$620$	−3696.00	−0.239411
$621$	0	0
$622$	12186.0	0.785553
$623$	11448.0	0.736203
$624$	0	0
$625$	−17639.0	−1.12890
$626$	−9294.00	−0.593391
$627$	0	0
$628$	−3206.00	−0.203715
$629$	16644.0	1.05507
$630$	0	0
$631$	−23852.0	−1.50481	−0.752403	−	0.658703i	$-0.771106\pi$
$631$	−23852.0	−1.50481	−0.752403	+	0.658703i	$0.771106\pi$
$632$	26712.0	1.68124
$633$	0	0
$634$	10800.0	0.676534
$635$	23904.0	1.49386
$636$	0	0
$637$	−13134.0	−0.816936
$638$	0	0
$639$	0	0
$640$	19908.0	1.22958
$641$	−10998.0	−0.677683	−0.338842	−	0.940843i	$-0.610035\pi$
$641$	−10998.0	−0.677683	−0.338842	+	0.940843i	$0.610035\pi$
$642$	0	0
$643$	16724.0	1.02571	0.512854	−	0.858476i	$-0.328588\pi$
$643$	16724.0	1.02571	0.512854	+	0.858476i	$0.328588\pi$
$644$	216.000	0.0132168
$645$	0	0
$646$	14364.0	0.874836
$647$	−26226.0	−1.59359	−0.796793	−	0.604252i	$-0.793472\pi$
$647$	−26226.0	−1.59359	−0.796793	+	0.604252i	$0.793472\pi$
$648$	0	0
$649$	0	0
$650$	3762.00	0.227012
$651$	0	0
$652$	740.000	0.0444488
$653$	1884.00	0.112904	0.0564522	−	0.998405i	$-0.482021\pi$
$653$	1884.00	0.112904	0.0564522	+	0.998405i	$0.482021\pi$
$654$	0	0
$655$	12816.0	0.764523
$656$	−2982.00	−0.177481
$657$	0	0
$658$	22248.0	1.31811
$659$	−22584.0	−1.33497	−0.667487	−	0.744622i	$-0.732630\pi$
$659$	−22584.0	−1.33497	−0.667487	+	0.744622i	$0.732630\pi$
$660$	0	0
$661$	2266.00	0.133339	0.0666696	−	0.997775i	$-0.478763\pi$
$661$	2266.00	0.133339	0.0666696	+	0.997775i	$0.478763\pi$
$662$	−12468.0	−0.731998
$663$	0	0
$664$	9072.00	0.530214
$665$	6048.00	0.352679
$666$	0	0
$667$	−3348.00	−0.194355
$668$	−1968.00	−0.113988
$669$	0	0
$670$	−16272.0	−0.938272
$671$	0	0
$672$	0	0
$673$	−23604.0	−1.35196	−0.675979	−	0.736921i	$-0.736279\pi$
$673$	−23604.0	−1.35196	−0.675979	+	0.736921i	$0.736279\pi$
$674$	−8244.00	−0.471138
$675$	0	0
$676$	2159.00	0.122838
$677$	−6594.00	−0.374340	−0.187170	−	0.982328i	$-0.559931\pi$
$677$	−6594.00	−0.374340	−0.187170	+	0.982328i	$0.559931\pi$
$678$	0	0
$679$	−3912.00	−0.221103
$680$	28728.0	1.62010
$681$	0	0
$682$	0	0
$683$	−16368.0	−0.916990	−0.458495	−	0.888697i	$-0.651611\pi$
$683$	−16368.0	−0.916990	−0.458495	+	0.888697i	$0.651611\pi$
$684$	0	0
$685$	5544.00	0.309234
$686$	19512.0	1.08596
$687$	0	0
$688$	−25986.0	−1.43998
$689$	26928.0	1.48893
$690$	0	0
$691$	−11860.0	−0.652931	−0.326466	−	0.945209i	$-0.605858\pi$
$691$	−11860.0	−0.652931	−0.326466	+	0.945209i	$0.605858\pi$
$692$	2898.00	0.159199
$693$	0	0
$694$	2088.00	0.114207
$695$	−27288.0	−1.48934
$696$	0	0
$697$	−4788.00	−0.260199
$698$	9738.00	0.528064
$699$	0	0
$700$	−228.000	−0.0123108
$701$	−28698.0	−1.54623	−0.773116	−	0.634265i	$-0.781303\pi$
$701$	−28698.0	−1.54623	−0.773116	+	0.634265i	$0.781303\pi$
$702$	0	0
$703$	6132.00	0.328980
$704$	0	0
$705$	0	0
$706$	32778.0	1.74733
$707$	−15912.0	−0.846439
$708$	0	0
$709$	−10766.0	−0.570276	−0.285138	−	0.958486i	$-0.592039\pi$
$709$	−10766.0	−0.570276	−0.285138	+	0.958486i	$0.592039\pi$
$710$	10152.0	0.536617
$711$	0	0
$712$	20034.0	1.05450
$713$	5544.00	0.291198
$714$	0	0
$715$	0	0
$716$	−120.000	−0.00626342
$717$	0	0
$718$	−4752.00	−0.246996
$719$	20226.0	1.04910	0.524550	−	0.851380i	$-0.324234\pi$
$719$	20226.0	1.04910	0.524550	+	0.851380i	$0.324234\pi$
$720$	0	0
$721$	18768.0	0.969427
$722$	−15285.0	−0.787879
$723$	0	0
$724$	−1150.00	−0.0590323
$725$	3534.00	0.181034
$726$	0	0
$727$	−29576.0	−1.50882	−0.754411	−	0.656403i	$-0.772077\pi$
$727$	−29576.0	−1.50882	−0.754411	+	0.656403i	$0.772077\pi$
$728$	16632.0	0.846735
$729$	0	0
$730$	−24624.0	−1.24846
$731$	−41724.0	−2.11111
$732$	0	0
$733$	−16878.0	−0.850482	−0.425241	−	0.905080i	$-0.639811\pi$
$733$	−16878.0	−0.850482	−0.425241	+	0.905080i	$0.639811\pi$
$734$	−696.000	−0.0349998
$735$	0	0
$736$	810.000	0.0405666
$737$	0	0
$738$	0	0
$739$	6726.00	0.334804	0.167402	−	0.985889i	$-0.446462\pi$
$739$	6726.00	0.334804	0.167402	+	0.985889i	$0.446462\pi$
$740$	−1752.00	−0.0870335
$741$	0	0
$742$	−14688.0	−0.726703
$743$	20592.0	1.01675	0.508376	−	0.861135i	$-0.330246\pi$
$743$	20592.0	1.01675	0.508376	+	0.861135i	$0.330246\pi$
$744$	0	0
$745$	−9000.00	−0.442597
$746$	32418.0	1.59103
$747$	0	0
$748$	0	0
$749$	1008.00	0.0491743
$750$	0	0
$751$	8764.00	0.425836	0.212918	−	0.977070i	$-0.431703\pi$
$751$	8764.00	0.425836	0.212918	+	0.977070i	$0.431703\pi$
$752$	43878.0	2.12775
$753$	0	0
$754$	36828.0	1.77878
$755$	8640.00	0.416479
$756$	0	0
$757$	−11810.0	−0.567030	−0.283515	−	0.958968i	$-0.591501\pi$
$757$	−11810.0	−0.567030	−0.283515	+	0.958968i	$0.591501\pi$
$758$	492.000	0.0235755
$759$	0	0
$760$	10584.0	0.505161
$761$	−29346.0	−1.39789	−0.698943	−	0.715177i	$-0.746346\pi$
$761$	−29346.0	−1.39789	−0.698943	+	0.715177i	$0.746346\pi$
$762$	0	0
$763$	−13464.0	−0.638833
$764$	1518.00	0.0718839
$765$	0	0
$766$	−24174.0	−1.14026
$767$	8712.00	0.410133
$768$	0	0
$769$	−15684.0	−0.735474	−0.367737	−	0.929930i	$-0.619867\pi$
$769$	−15684.0	−0.735474	−0.367737	+	0.929930i	$0.619867\pi$
$770$	0	0
$771$	0	0
$772$	2052.00	0.0956646
$773$	25632.0	1.19265	0.596325	−	0.802743i	$-0.296627\pi$
$773$	25632.0	1.19265	0.596325	+	0.802743i	$0.296627\pi$
$774$	0	0
$775$	−5852.00	−0.271239
$776$	−6846.00	−0.316697
$777$	0	0
$778$	2124.00	0.0978780
$779$	−1764.00	−0.0811320
$780$	0	0
$781$	0	0
$782$	6156.00	0.281507
$783$	0	0
$784$	14129.0	0.643632
$785$	−38472.0	−1.74920
$786$	0	0
$787$	37854.0	1.71455	0.857274	−	0.514860i	$-0.172156\pi$
$787$	37854.0	1.71455	0.857274	+	0.514860i	$0.172156\pi$
$788$	−3282.00	−0.148371
$789$	0	0
$790$	−45792.0	−2.06229
$791$	−10872.0	−0.488703
$792$	0	0
$793$	−41580.0	−1.86198
$794$	−858.000	−0.0383492
$795$	0	0
$796$	−704.000	−0.0313475
$797$	23484.0	1.04372	0.521861	−	0.853031i	$-0.325238\pi$
$797$	23484.0	1.04372	0.521861	+	0.853031i	$0.325238\pi$
$798$	0	0
$799$	70452.0	3.11942
$800$	−855.000	−0.0377860
$801$	0	0
$802$	15786.0	0.695041
$803$	0	0
$804$	0	0
$805$	2592.00	0.113486
$806$	−60984.0	−2.66510
$807$	0	0
$808$	−27846.0	−1.21240
$809$	9054.00	0.393476	0.196738	−	0.980456i	$-0.436965\pi$
$809$	9054.00	0.393476	0.196738	+	0.980456i	$0.436965\pi$
$810$	0	0
$811$	9858.00	0.426833	0.213416	−	0.976961i	$-0.431541\pi$
$811$	9858.00	0.426833	0.213416	+	0.976961i	$0.431541\pi$
$812$	−2232.00	−0.0964629
$813$	0	0
$814$	0	0
$815$	8880.00	0.381660
$816$	0	0
$817$	−15372.0	−0.658260
$818$	6840.00	0.292366
$819$	0	0
$820$	504.000	0.0214640
$821$	−18210.0	−0.774097	−0.387048	−	0.922059i	$-0.626505\pi$
$821$	−18210.0	−0.774097	−0.387048	+	0.922059i	$0.626505\pi$
$822$	0	0
$823$	41240.0	1.74670	0.873351	−	0.487091i	$-0.161942\pi$
$823$	41240.0	1.74670	0.873351	+	0.487091i	$0.161942\pi$
$824$	32844.0	1.38856
$825$	0	0
$826$	−4752.00	−0.200173
$827$	37680.0	1.58436	0.792178	−	0.610290i	$-0.208947\pi$
$827$	37680.0	1.58436	0.792178	+	0.610290i	$0.208947\pi$
$828$	0	0
$829$	−11626.0	−0.487078	−0.243539	−	0.969891i	$-0.578308\pi$
$829$	−11626.0	−0.487078	−0.243539	+	0.969891i	$0.578308\pi$
$830$	−15552.0	−0.650383
$831$	0	0
$832$	28578.0	1.19082
$833$	22686.0	0.943605
$834$	0	0
$835$	−23616.0	−0.978761
$836$	0	0
$837$	0	0
$838$	−12816.0	−0.528307
$839$	−12150.0	−0.499958	−0.249979	−	0.968251i	$-0.580424\pi$
$839$	−12150.0	−0.499958	−0.249979	+	0.968251i	$0.580424\pi$
$840$	0	0
$841$	10207.0	0.418508
$842$	30450.0	1.24629
$843$	0	0
$844$	1986.00	0.0809964
$845$	25908.0	1.05475
$846$	0	0
$847$	0	0
$848$	−28968.0	−1.17307
$849$	0	0
$850$	−6498.00	−0.262211
$851$	2628.00	0.105860
$852$	0	0
$853$	−41022.0	−1.64662	−0.823310	−	0.567592i	$-0.807875\pi$
$853$	−41022.0	−1.64662	−0.823310	+	0.567592i	$0.807875\pi$
$854$	22680.0	0.908775
$855$	0	0
$856$	1764.00	0.0704349
$857$	−9234.00	−0.368060	−0.184030	−	0.982921i	$-0.558914\pi$
$857$	−9234.00	−0.368060	−0.184030	+	0.982921i	$0.558914\pi$
$858$	0	0
$859$	−30004.0	−1.19176	−0.595881	−	0.803073i	$-0.703197\pi$
$859$	−30004.0	−1.19176	−0.595881	+	0.803073i	$0.703197\pi$
$860$	4392.00	0.174146
$861$	0	0
$862$	39168.0	1.54764
$863$	−14958.0	−0.590007	−0.295004	−	0.955496i	$-0.595321\pi$
$863$	−14958.0	−0.590007	−0.295004	+	0.955496i	$0.595321\pi$
$864$	0	0
$865$	34776.0	1.36696
$866$	41046.0	1.61062
$867$	0	0
$868$	3696.00	0.144528
$869$	0	0
$870$	0	0
$871$	−29832.0	−1.16053
$872$	−23562.0	−0.915034
$873$	0	0
$874$	2268.00	0.0877760
$875$	15264.0	0.589734
$876$	0	0
$877$	−3714.00	−0.143002	−0.0715011	−	0.997441i	$-0.522779\pi$
$877$	−3714.00	−0.143002	−0.0715011	+	0.997441i	$0.522779\pi$
$878$	21060.0	0.809500
$879$	0	0
$880$	0	0
$881$	−45390.0	−1.73579	−0.867893	−	0.496751i	$-0.834526\pi$
$881$	−45390.0	−1.73579	−0.867893	+	0.496751i	$0.834526\pi$
$882$	0	0
$883$	23452.0	0.893797	0.446898	−	0.894585i	$-0.352529\pi$
$883$	23452.0	0.893797	0.446898	+	0.894585i	$0.352529\pi$
$884$	−7524.00	−0.286266
$885$	0	0
$886$	26748.0	1.01424
$887$	27072.0	1.02479	0.512395	−	0.858750i	$-0.328758\pi$
$887$	27072.0	1.02479	0.512395	+	0.858750i	$0.328758\pi$
$888$	0	0
$889$	−23904.0	−0.901816
$890$	−34344.0	−1.29350
$891$	0	0
$892$	−2248.00	−0.0843818
$893$	25956.0	0.972659
$894$	0	0
$895$	−1440.00	−0.0537809
$896$	−19908.0	−0.742276
$897$	0	0
$898$	−30330.0	−1.12709
$899$	−57288.0	−2.12532
$900$	0	0
$901$	−46512.0	−1.71980
$902$	0	0
$903$	0	0
$904$	−19026.0	−0.699995
$905$	−13800.0	−0.506881
$906$	0	0
$907$	−29716.0	−1.08788	−0.543938	−	0.839125i	$-0.683067\pi$
$907$	−29716.0	−1.08788	−0.543938	+	0.839125i	$0.683067\pi$
$908$	−2796.00	−0.102190
$909$	0	0
$910$	−28512.0	−1.03864
$911$	−17094.0	−0.621679	−0.310839	−	0.950462i	$-0.600610\pi$
$911$	−17094.0	−0.621679	−0.310839	+	0.950462i	$0.600610\pi$
$912$	0	0
$913$	0	0
$914$	−5472.00	−0.198028
$915$	0	0
$916$	−2198.00	−0.0792838
$917$	−12816.0	−0.461528
$918$	0	0
$919$	−24840.0	−0.891617	−0.445808	−	0.895128i	$-0.647084\pi$
$919$	−24840.0	−0.891617	−0.445808	+	0.895128i	$0.647084\pi$
$920$	4536.00	0.162552
$921$	0	0
$922$	51354.0	1.83433
$923$	18612.0	0.663729
$924$	0	0
$925$	−2774.00	−0.0986038
$926$	1788.00	0.0634528
$927$	0	0
$928$	−8370.00	−0.296076
$929$	−10578.0	−0.373577	−0.186788	−	0.982400i	$-0.559808\pi$
$929$	−10578.0	−0.373577	−0.186788	+	0.982400i	$0.559808\pi$
$930$	0	0
$931$	8358.00	0.294224
$932$	5802.00	0.203917
$933$	0	0
$934$	−17568.0	−0.615463
$935$	0	0
$936$	0	0
$937$	38208.0	1.33212	0.666062	−	0.745896i	$-0.267978\pi$
$937$	38208.0	1.33212	0.666062	+	0.745896i	$0.267978\pi$
$938$	16272.0	0.566418
$939$	0	0
$940$	−7416.00	−0.257323
$941$	−2154.00	−0.0746210	−0.0373105	−	0.999304i	$-0.511879\pi$
$941$	−2154.00	−0.0746210	−0.0373105	+	0.999304i	$0.511879\pi$
$942$	0	0
$943$	−756.000	−0.0261068
$944$	−9372.00	−0.323128
$945$	0	0
$946$	0	0
$947$	−6564.00	−0.225239	−0.112620	−	0.993638i	$-0.535924\pi$
$947$	−6564.00	−0.225239	−0.112620	+	0.993638i	$0.535924\pi$
$948$	0	0
$949$	−45144.0	−1.54419
$950$	−2394.00	−0.0817596
$951$	0	0
$952$	−28728.0	−0.978025
$953$	−36162.0	−1.22917	−0.614587	−	0.788849i	$-0.710677\pi$
$953$	−36162.0	−1.22917	−0.614587	+	0.788849i	$0.710677\pi$
$954$	0	0
$955$	18216.0	0.617231
$956$	1104.00	0.0373493
$957$	0	0
$958$	9792.00	0.330235
$959$	−5544.00	−0.186679
$960$	0	0
$961$	65073.0	2.18432
$962$	−28908.0	−0.968848
$963$	0	0
$964$	−5208.00	−0.174002
$965$	24624.0	0.821424
$966$	0	0
$967$	37728.0	1.25465	0.627327	−	0.778756i	$-0.284149\pi$
$967$	37728.0	1.25465	0.627327	+	0.778756i	$0.284149\pi$
$968$	0	0
$969$	0	0
$970$	11736.0	0.388474
$971$	−24156.0	−0.798355	−0.399178	−	0.916874i	$-0.630704\pi$
$971$	−24156.0	−0.798355	−0.399178	+	0.916874i	$0.630704\pi$
$972$	0	0
$973$	27288.0	0.899089
$974$	−29760.0	−0.979027
$975$	0	0
$976$	44730.0	1.46698
$977$	23742.0	0.777455	0.388728	−	0.921353i	$-0.372915\pi$
$977$	23742.0	0.777455	0.388728	+	0.921353i	$0.372915\pi$
$978$	0	0
$979$	0	0
$980$	−2388.00	−0.0778386
$981$	0	0
$982$	−31860.0	−1.03533
$983$	−11562.0	−0.375148	−0.187574	−	0.982250i	$-0.560062\pi$
$983$	−11562.0	−0.375148	−0.187574	+	0.982250i	$0.560062\pi$
$984$	0	0
$985$	−39384.0	−1.27399
$986$	−63612.0	−2.05458
$987$	0	0
$988$	−2772.00	−0.0892602
$989$	−6588.00	−0.211816
$990$	0	0
$991$	52180.0	1.67261	0.836303	−	0.548268i	$-0.184713\pi$
$991$	52180.0	1.67261	0.836303	+	0.548268i	$0.184713\pi$
$992$	13860.0	0.443604
$993$	0	0
$994$	−10152.0	−0.323946
$995$	−8448.00	−0.269165
$996$	0	0
$997$	−15714.0	−0.499165	−0.249582	−	0.968354i	$-0.580293\pi$
$997$	−15714.0	−0.499165	−0.249582	+	0.968354i	$0.580293\pi$
$998$	156.000	0.00494799
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.4.a.i.1.1		1
3.2	odd	2		363.4.a.b.1.1	✓	1
11.10	odd	2		1089.4.a.c.1.1		1
33.32	even	2		363.4.a.f.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
363.4.a.b.1.1	✓	1	3.2	odd	2
363.4.a.f.1.1	yes	1	33.32	even	2
1089.4.a.c.1.1		1	11.10	odd	2
1089.4.a.i.1.1		1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1089.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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