LMFDB - Embedded newform 882.6.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,6,Mod(1,882)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("882.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	882.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:	$141.458529075$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 294)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	882.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{2} +16.0000 q^{4} -26.0000 q^{5} +64.0000 q^{8} +O(q^{10})$

$q+4.00000 q^{2} +16.0000 q^{4} -26.0000 q^{5} +64.0000 q^{8} -104.000 q^{10} +358.000 q^{11} +332.000 q^{13} +256.000 q^{16} -126.000 q^{17} -2200.00 q^{19} -416.000 q^{20} +1432.00 q^{22} +2142.00 q^{23} -2449.00 q^{25} +1328.00 q^{26} +3610.00 q^{29} +5668.00 q^{31} +1024.00 q^{32} -504.000 q^{34} -2922.00 q^{37} -8800.00 q^{38} -1664.00 q^{40} +2142.00 q^{41} +6388.00 q^{43} +5728.00 q^{44} +8568.00 q^{46} +6520.00 q^{47} -9796.00 q^{50} +5312.00 q^{52} +10702.0 q^{53} -9308.00 q^{55} +14440.0 q^{58} -42524.0 q^{59} -44840.0 q^{61} +22672.0 q^{62} +4096.00 q^{64} -8632.00 q^{65} -1448.00 q^{67} -2016.00 q^{68} +4402.00 q^{71} +20500.0 q^{73} -11688.0 q^{74} -35200.0 q^{76} +65236.0 q^{79} -6656.00 q^{80} +8568.00 q^{82} +102804. q^{83} +3276.00 q^{85} +25552.0 q^{86} +22912.0 q^{88} +128006. q^{89} +34272.0 q^{92} +26080.0 q^{94} +57200.0 q^{95} -113324. q^{97} +O(q^{100})$

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 $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	−26.0000	−0.465102	−0.232551	−	0.972584i	$-0.574707\pi$
$5$	−26.0000	−0.465102	−0.232551	+	0.972584i	$0.574707\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	64.0000	0.353553
$9$	0	0
$10$	−104.000	−0.328877
$11$	358.000	0.892075	0.446037	−	0.895014i	$-0.352835\pi$
$11$	358.000	0.892075	0.446037	+	0.895014i	$0.352835\pi$
$12$	0	0
$13$	332.000	0.544853	0.272427	−	0.962177i	$-0.412174\pi$
$13$	332.000	0.544853	0.272427	+	0.962177i	$0.412174\pi$
$14$	0	0
$15$	0	0
$16$	256.000	0.250000
$17$	−126.000	−0.105742	−0.0528711	−	0.998601i	$-0.516837\pi$
$17$	−126.000	−0.105742	−0.0528711	+	0.998601i	$0.516837\pi$
$18$	0	0
$19$	−2200.00	−1.39810	−0.699051	−	0.715072i	$-0.746394\pi$
$19$	−2200.00	−1.39810	−0.699051	+	0.715072i	$0.746394\pi$
$20$	−416.000	−0.232551
$21$	0	0
$22$	1432.00	0.630792
$23$	2142.00	0.844306	0.422153	−	0.906525i	$-0.361274\pi$
$23$	2142.00	0.844306	0.422153	+	0.906525i	$0.361274\pi$
$24$	0	0
$25$	−2449.00	−0.783680
$26$	1328.00	0.385270
$27$	0	0
$28$	0	0
$29$	3610.00	0.797099	0.398549	−	0.917147i	$-0.369514\pi$
$29$	3610.00	0.797099	0.398549	+	0.917147i	$0.369514\pi$
$30$	0	0
$31$	5668.00	1.05932	0.529658	−	0.848211i	$-0.322320\pi$
$31$	5668.00	1.05932	0.529658	+	0.848211i	$0.322320\pi$
$32$	1024.00	0.176777
$33$	0	0
$34$	−504.000	−0.0747710
$35$	0	0
$36$	0	0
$37$	−2922.00	−0.350894	−0.175447	−	0.984489i	$-0.556137\pi$
$37$	−2922.00	−0.350894	−0.175447	+	0.984489i	$0.556137\pi$
$38$	−8800.00	−0.988607
$39$	0	0
$40$	−1664.00	−0.164438
$41$	2142.00	0.199003	0.0995015	−	0.995037i	$-0.468275\pi$
$41$	2142.00	0.199003	0.0995015	+	0.995037i	$0.468275\pi$
$42$	0	0
$43$	6388.00	0.526858	0.263429	−	0.964679i	$-0.415146\pi$
$43$	6388.00	0.526858	0.263429	+	0.964679i	$0.415146\pi$
$44$	5728.00	0.446037
$45$	0	0
$46$	8568.00	0.597014
$47$	6520.00	0.430530	0.215265	−	0.976556i	$-0.430939\pi$
$47$	6520.00	0.430530	0.215265	+	0.976556i	$0.430939\pi$
$48$	0	0
$49$	0	0
$50$	−9796.00	−0.554145
$51$	0	0
$52$	5312.00	0.272427
$53$	10702.0	0.523330	0.261665	−	0.965159i	$-0.415729\pi$
$53$	10702.0	0.523330	0.261665	+	0.965159i	$0.415729\pi$
$54$	0	0
$55$	−9308.00	−0.414906
$56$	0	0
$57$	0	0
$58$	14440.0	0.563634
$59$	−42524.0	−1.59039	−0.795196	−	0.606353i	$-0.792632\pi$
$59$	−42524.0	−1.59039	−0.795196	+	0.606353i	$0.792632\pi$
$60$	0	0
$61$	−44840.0	−1.54291	−0.771456	−	0.636283i	$-0.780471\pi$
$61$	−44840.0	−1.54291	−0.771456	+	0.636283i	$0.780471\pi$
$62$	22672.0	0.749050
$63$	0	0
$64$	4096.00	0.125000
$65$	−8632.00	−0.253413
$66$	0	0
$67$	−1448.00	−0.0394077	−0.0197039	−	0.999806i	$-0.506272\pi$
$67$	−1448.00	−0.0394077	−0.0197039	+	0.999806i	$0.506272\pi$
$68$	−2016.00	−0.0528711
$69$	0	0
$70$	0	0
$71$	4402.00	0.103634	0.0518172	−	0.998657i	$-0.483499\pi$
$71$	4402.00	0.103634	0.0518172	+	0.998657i	$0.483499\pi$
$72$	0	0
$73$	20500.0	0.450243	0.225121	−	0.974331i	$-0.427722\pi$
$73$	20500.0	0.450243	0.225121	+	0.974331i	$0.427722\pi$
$74$	−11688.0	−0.248120
$75$	0	0
$76$	−35200.0	−0.699051
$77$	0	0
$78$	0	0
$79$	65236.0	1.17603	0.588017	−	0.808849i	$-0.299909\pi$
$79$	65236.0	1.17603	0.588017	+	0.808849i	$0.299909\pi$
$80$	−6656.00	−0.116276
$81$	0	0
$82$	8568.00	0.140716
$83$	102804.	1.63800	0.819002	−	0.573791i	$-0.194528\pi$
$83$	102804.	1.63800	0.819002	+	0.573791i	$0.194528\pi$
$84$	0	0
$85$	3276.00	0.0491809
$86$	25552.0	0.372545
$87$	0	0
$88$	22912.0	0.315396
$89$	128006.	1.71299	0.856496	−	0.516154i	$-0.172637\pi$
$89$	128006.	1.71299	0.856496	+	0.516154i	$0.172637\pi$
$90$	0	0
$91$	0	0
$92$	34272.0	0.422153
$93$	0	0
$94$	26080.0	0.304430
$95$	57200.0	0.650260
$96$	0	0
$97$	−113324.	−1.22290	−0.611452	−	0.791281i	$-0.709414\pi$
$97$	−113324.	−1.22290	−0.611452	+	0.791281i	$0.709414\pi$
$98$	0	0
$99$	0	0
$100$	−39184.0	−0.391840
$101$	139714.	1.36281	0.681407	−	0.731905i	$-0.261368\pi$
$101$	139714.	1.36281	0.681407	+	0.731905i	$0.261368\pi$
$102$	0	0
$103$	−142180.	−1.32052	−0.660261	−	0.751036i	$-0.729554\pi$
$103$	−142180.	−1.32052	−0.660261	+	0.751036i	$0.729554\pi$
$104$	21248.0	0.192635
$105$	0	0
$106$	42808.0	0.370050
$107$	198518.	1.67626	0.838128	−	0.545473i	$-0.183650\pi$
$107$	198518.	1.67626	0.838128	+	0.545473i	$0.183650\pi$
$108$	0	0
$109$	132538.	1.06850	0.534250	−	0.845327i	$-0.320594\pi$
$109$	132538.	1.06850	0.534250	+	0.845327i	$0.320594\pi$
$110$	−37232.0	−0.293383
$111$	0	0
$112$	0	0
$113$	−47026.0	−0.346451	−0.173226	−	0.984882i	$-0.555419\pi$
$113$	−47026.0	−0.346451	−0.173226	+	0.984882i	$0.555419\pi$
$114$	0	0
$115$	−55692.0	−0.392689
$116$	57760.0	0.398549
$117$	0	0
$118$	−170096.	−1.12458
$119$	0	0
$120$	0	0
$121$	−32887.0	−0.204202
$122$	−179360.	−1.09100
$123$	0	0
$124$	90688.0	0.529658
$125$	144924.	0.829593
$126$	0	0
$127$	165548.	0.910782	0.455391	−	0.890291i	$-0.349499\pi$
$127$	165548.	0.910782	0.455391	+	0.890291i	$0.349499\pi$
$128$	16384.0	0.0883883
$129$	0	0
$130$	−34528.0	−0.179190
$131$	139308.	0.709248	0.354624	−	0.935009i	$-0.384609\pi$
$131$	139308.	0.709248	0.354624	+	0.935009i	$0.384609\pi$
$132$	0	0
$133$	0	0
$134$	−5792.00	−0.0278655
$135$	0	0
$136$	−8064.00	−0.0373855
$137$	332842.	1.51508	0.757542	−	0.652786i	$-0.226400\pi$
$137$	332842.	1.51508	0.757542	+	0.652786i	$0.226400\pi$
$138$	0	0
$139$	8556.00	0.0375607	0.0187804	−	0.999824i	$-0.494022\pi$
$139$	8556.00	0.0375607	0.0187804	+	0.999824i	$0.494022\pi$
$140$	0	0
$141$	0	0
$142$	17608.0	0.0732806
$143$	118856.	0.486050
$144$	0	0
$145$	−93860.0	−0.370732
$146$	82000.0	0.318370
$147$	0	0
$148$	−46752.0	−0.175447
$149$	−69554.0	−0.256659	−0.128329	−	0.991732i	$-0.540962\pi$
$149$	−69554.0	−0.256659	−0.128329	+	0.991732i	$0.540962\pi$
$150$	0	0
$151$	529240.	1.88891	0.944453	−	0.328647i	$-0.106593\pi$
$151$	529240.	1.88891	0.944453	+	0.328647i	$0.106593\pi$
$152$	−140800.	−0.494303
$153$	0	0
$154$	0	0
$155$	−147368.	−0.492690
$156$	0	0
$157$	13040.0	0.0422210	0.0211105	−	0.999777i	$-0.493280\pi$
$157$	13040.0	0.0422210	0.0211105	+	0.999777i	$0.493280\pi$
$158$	260944.	0.831581
$159$	0	0
$160$	−26624.0	−0.0822192
$161$	0	0
$162$	0	0
$163$	−351240.	−1.03546	−0.517732	−	0.855543i	$-0.673224\pi$
$163$	−351240.	−1.03546	−0.517732	+	0.855543i	$0.673224\pi$
$164$	34272.0	0.0995015
$165$	0	0
$166$	411216.	1.15824
$167$	−626128.	−1.73729	−0.868644	−	0.495436i	$-0.835008\pi$
$167$	−626128.	−1.73729	−0.868644	+	0.495436i	$0.835008\pi$
$168$	0	0
$169$	−261069.	−0.703135
$170$	13104.0	0.0347762
$171$	0	0
$172$	102208.	0.263429
$173$	184826.	0.469513	0.234757	−	0.972054i	$-0.424571\pi$
$173$	184826.	0.469513	0.234757	+	0.972054i	$0.424571\pi$
$174$	0	0
$175$	0	0
$176$	91648.0	0.223019
$177$	0	0
$178$	512024.	1.21127
$179$	357522.	0.834008	0.417004	−	0.908905i	$-0.363080\pi$
$179$	357522.	0.834008	0.417004	+	0.908905i	$0.363080\pi$
$180$	0	0
$181$	696508.	1.58026	0.790132	−	0.612937i	$-0.210012\pi$
$181$	696508.	1.58026	0.790132	+	0.612937i	$0.210012\pi$
$182$	0	0
$183$	0	0
$184$	137088.	0.298507
$185$	75972.0	0.163202
$186$	0	0
$187$	−45108.0	−0.0943299
$188$	104320.	0.215265
$189$	0	0
$190$	228800.	0.459803
$191$	−68670.0	−0.136202	−0.0681010	−	0.997678i	$-0.521694\pi$
$191$	−68670.0	−0.136202	−0.0681010	+	0.997678i	$0.521694\pi$
$192$	0	0
$193$	827222.	1.59856	0.799280	−	0.600959i	$-0.205215\pi$
$193$	827222.	1.59856	0.799280	+	0.600959i	$0.205215\pi$
$194$	−453296.	−0.864724
$195$	0	0
$196$	0	0
$197$	143382.	0.263226	0.131613	−	0.991301i	$-0.457984\pi$
$197$	143382.	0.263226	0.131613	+	0.991301i	$0.457984\pi$
$198$	0	0
$199$	−542600.	−0.971286	−0.485643	−	0.874157i	$-0.661414\pi$
$199$	−542600.	−0.971286	−0.485643	+	0.874157i	$0.661414\pi$
$200$	−156736.	−0.277073
$201$	0	0
$202$	558856.	0.963655
$203$	0	0
$204$	0	0
$205$	−55692.0	−0.0925568
$206$	−568720.	−0.933750
$207$	0	0
$208$	84992.0	0.136213
$209$	−787600.	−1.24721
$210$	0	0
$211$	1.12776e6	1.74385	0.871925	−	0.489640i	$-0.162872\pi$
$211$	1.12776e6	1.74385	0.871925	+	0.489640i	$0.162872\pi$
$212$	171232.	0.261665
$213$	0	0
$214$	794072.	1.18529
$215$	−166088.	−0.245043
$216$	0	0
$217$	0	0
$218$	530152.	0.755543
$219$	0	0
$220$	−148928.	−0.207453
$221$	−41832.0	−0.0576140
$222$	0	0
$223$	897976.	1.20921	0.604606	−	0.796525i	$-0.293331\pi$
$223$	897976.	1.20921	0.604606	+	0.796525i	$0.293331\pi$
$224$	0	0
$225$	0	0
$226$	−188104.	−0.244978
$227$	467612.	0.602311	0.301156	−	0.953575i	$-0.402628\pi$
$227$	467612.	0.602311	0.301156	+	0.953575i	$0.402628\pi$
$228$	0	0
$229$	−446140.	−0.562189	−0.281095	−	0.959680i	$-0.590697\pi$
$229$	−446140.	−0.562189	−0.281095	+	0.959680i	$0.590697\pi$
$230$	−222768.	−0.277673
$231$	0	0
$232$	231040.	0.281817
$233$	−701486.	−0.846504	−0.423252	−	0.906012i	$-0.639112\pi$
$233$	−701486.	−0.846504	−0.423252	+	0.906012i	$0.639112\pi$
$234$	0	0
$235$	−169520.	−0.200240
$236$	−680384.	−0.795196
$237$	0	0
$238$	0	0
$239$	384198.	0.435071	0.217536	−	0.976052i	$-0.430198\pi$
$239$	384198.	0.435071	0.217536	+	0.976052i	$0.430198\pi$
$240$	0	0
$241$	−953780.	−1.05780	−0.528902	−	0.848683i	$-0.677396\pi$
$241$	−953780.	−1.05780	−0.528902	+	0.848683i	$0.677396\pi$
$242$	−131548.	−0.144393
$243$	0	0
$244$	−717440.	−0.771456
$245$	0	0
$246$	0	0
$247$	−730400.	−0.761760
$248$	362752.	0.374525
$249$	0	0
$250$	579696.	0.586611
$251$	−569540.	−0.570611	−0.285305	−	0.958437i	$-0.592095\pi$
$251$	−569540.	−0.570611	−0.285305	+	0.958437i	$0.592095\pi$
$252$	0	0
$253$	766836.	0.753184
$254$	662192.	0.644020
$255$	0	0
$256$	65536.0	0.0625000
$257$	−1.06664e6	−1.00736	−0.503681	−	0.863890i	$-0.668021\pi$
$257$	−1.06664e6	−1.00736	−0.503681	+	0.863890i	$0.668021\pi$
$258$	0	0
$259$	0	0
$260$	−138112.	−0.126706
$261$	0	0
$262$	557232.	0.501514
$263$	1.48243e6	1.32155	0.660777	−	0.750582i	$-0.270227\pi$
$263$	1.48243e6	1.32155	0.660777	+	0.750582i	$0.270227\pi$
$264$	0	0
$265$	−278252.	−0.243402
$266$	0	0
$267$	0	0
$268$	−23168.0	−0.0197039
$269$	215110.	0.181251	0.0906254	−	0.995885i	$-0.471113\pi$
$269$	215110.	0.181251	0.0906254	+	0.995885i	$0.471113\pi$
$270$	0	0
$271$	1.93104e6	1.59723	0.798614	−	0.601843i	$-0.205567\pi$
$271$	1.93104e6	1.59723	0.798614	+	0.601843i	$0.205567\pi$
$272$	−32256.0	−0.0264355
$273$	0	0
$274$	1.33137e6	1.07133
$275$	−876742.	−0.699101
$276$	0	0
$277$	2.03756e6	1.59555	0.797777	−	0.602953i	$-0.206009\pi$
$277$	2.03756e6	1.59555	0.797777	+	0.602953i	$0.206009\pi$
$278$	34224.0	0.0265594
$279$	0	0
$280$	0	0
$281$	639066.	0.482814	0.241407	−	0.970424i	$-0.422391\pi$
$281$	639066.	0.482814	0.241407	+	0.970424i	$0.422391\pi$
$282$	0	0
$283$	37744.0	0.0280144	0.0140072	−	0.999902i	$-0.495541\pi$
$283$	37744.0	0.0280144	0.0140072	+	0.999902i	$0.495541\pi$
$284$	70432.0	0.0518172
$285$	0	0
$286$	475424.	0.343689
$287$	0	0
$288$	0	0
$289$	−1.40398e6	−0.988819
$290$	−375440.	−0.262147
$291$	0	0
$292$	328000.	0.225121
$293$	1.83921e6	1.25159	0.625795	−	0.779987i	$-0.284775\pi$
$293$	1.83921e6	1.25159	0.625795	+	0.779987i	$0.284775\pi$
$294$	0	0
$295$	1.10562e6	0.739695
$296$	−187008.	−0.124060
$297$	0	0
$298$	−278216.	−0.181485
$299$	711144.	0.460023
$300$	0	0
$301$	0	0
$302$	2.11696e6	1.33566
$303$	0	0
$304$	−563200.	−0.349525
$305$	1.16584e6	0.717611
$306$	0	0
$307$	−1.06472e6	−0.644747	−0.322374	−	0.946613i	$-0.604481\pi$
$307$	−1.06472e6	−0.644747	−0.322374	+	0.946613i	$0.604481\pi$
$308$	0	0
$309$	0	0
$310$	−589472.	−0.348385
$311$	1.00952e6	0.591853	0.295927	−	0.955211i	$-0.404372\pi$
$311$	1.00952e6	0.591853	0.295927	+	0.955211i	$0.404372\pi$
$312$	0	0
$313$	−1.44910e6	−0.836058	−0.418029	−	0.908434i	$-0.637279\pi$
$313$	−1.44910e6	−0.836058	−0.418029	+	0.908434i	$0.637279\pi$
$314$	52160.0	0.0298548
$315$	0	0
$316$	1.04378e6	0.588017
$317$	−2.72311e6	−1.52201	−0.761003	−	0.648748i	$-0.775293\pi$
$317$	−2.72311e6	−1.52201	−0.761003	+	0.648748i	$0.775293\pi$
$318$	0	0
$319$	1.29238e6	0.711072
$320$	−106496.	−0.0581378
$321$	0	0
$322$	0	0
$323$	277200.	0.147838
$324$	0	0
$325$	−813068.	−0.426991
$326$	−1.40496e6	−0.732184
$327$	0	0
$328$	137088.	0.0703582
$329$	0	0
$330$	0	0
$331$	−1.10040e6	−0.552055	−0.276027	−	0.961150i	$-0.589018\pi$
$331$	−1.10040e6	−0.552055	−0.276027	+	0.961150i	$0.589018\pi$
$332$	1.64486e6	0.819002
$333$	0	0
$334$	−2.50451e6	−1.22845
$335$	37648.0	0.0183286
$336$	0	0
$337$	1.73512e6	0.832251	0.416125	−	0.909307i	$-0.363388\pi$
$337$	1.73512e6	0.832251	0.416125	+	0.909307i	$0.363388\pi$
$338$	−1.04428e6	−0.497191
$339$	0	0
$340$	52416.0	0.0245905
$341$	2.02914e6	0.944989
$342$	0	0
$343$	0	0
$344$	408832.	0.186273
$345$	0	0
$346$	739304.	0.331996
$347$	−1.59145e6	−0.709526	−0.354763	−	0.934956i	$-0.615438\pi$
$347$	−1.59145e6	−0.709526	−0.354763	+	0.934956i	$0.615438\pi$
$348$	0	0
$349$	−2.33376e6	−1.02563	−0.512817	−	0.858498i	$-0.671398\pi$
$349$	−2.33376e6	−1.02563	−0.512817	+	0.858498i	$0.671398\pi$
$350$	0	0
$351$	0	0
$352$	366592.	0.157698
$353$	−2.81081e6	−1.20059	−0.600296	−	0.799778i	$-0.704951\pi$
$353$	−2.81081e6	−1.20059	−0.600296	+	0.799778i	$0.704951\pi$
$354$	0	0
$355$	−114452.	−0.0482006
$356$	2.04810e6	0.856496
$357$	0	0
$358$	1.43009e6	0.589733
$359$	−939310.	−0.384656	−0.192328	−	0.981331i	$-0.561604\pi$
$359$	−939310.	−0.384656	−0.192328	+	0.981331i	$0.561604\pi$
$360$	0	0
$361$	2.36390e6	0.954688
$362$	2.78603e6	1.11742
$363$	0	0
$364$	0	0
$365$	−533000.	−0.209409
$366$	0	0
$367$	3.09851e6	1.20085	0.600424	−	0.799682i	$-0.294999\pi$
$367$	3.09851e6	1.20085	0.600424	+	0.799682i	$0.294999\pi$
$368$	548352.	0.211077
$369$	0	0
$370$	303888.	0.115401
$371$	0	0
$372$	0	0
$373$	−228266.	−0.0849511	−0.0424756	−	0.999098i	$-0.513524\pi$
$373$	−228266.	−0.0849511	−0.0424756	+	0.999098i	$0.513524\pi$
$374$	−180432.	−0.0667013
$375$	0	0
$376$	417280.	0.152215
$377$	1.19852e6	0.434302
$378$	0	0
$379$	−1.03669e6	−0.370725	−0.185362	−	0.982670i	$-0.559346\pi$
$379$	−1.03669e6	−0.370725	−0.185362	+	0.982670i	$0.559346\pi$
$380$	915200.	0.325130
$381$	0	0
$382$	−274680.	−0.0963094
$383$	211776.	0.0737700	0.0368850	−	0.999320i	$-0.488256\pi$
$383$	211776.	0.0737700	0.0368850	+	0.999320i	$0.488256\pi$
$384$	0	0
$385$	0	0
$386$	3.30889e6	1.13035
$387$	0	0
$388$	−1.81318e6	−0.611452
$389$	−1.41325e6	−0.473526	−0.236763	−	0.971567i	$-0.576086\pi$
$389$	−1.41325e6	−0.473526	−0.236763	+	0.971567i	$0.576086\pi$
$390$	0	0
$391$	−269892.	−0.0892788
$392$	0	0
$393$	0	0
$394$	573528.	0.186129
$395$	−1.69614e6	−0.546976
$396$	0	0
$397$	−1.09034e6	−0.347203	−0.173602	−	0.984816i	$-0.555541\pi$
$397$	−1.09034e6	−0.347203	−0.173602	+	0.984816i	$0.555541\pi$
$398$	−2.17040e6	−0.686803
$399$	0	0
$400$	−626944.	−0.195920
$401$	−2.64253e6	−0.820651	−0.410325	−	0.911939i	$-0.634585\pi$
$401$	−2.64253e6	−0.820651	−0.410325	+	0.911939i	$0.634585\pi$
$402$	0	0
$403$	1.88178e6	0.577172
$404$	2.23542e6	0.681407
$405$	0	0
$406$	0	0
$407$	−1.04608e6	−0.313024
$408$	0	0
$409$	6.25427e6	1.84871	0.924354	−	0.381536i	$-0.124605\pi$
$409$	6.25427e6	1.84871	0.924354	+	0.381536i	$0.124605\pi$
$410$	−222768.	−0.0654475
$411$	0	0
$412$	−2.27488e6	−0.660261
$413$	0	0
$414$	0	0
$415$	−2.67290e6	−0.761839
$416$	339968.	0.0963174
$417$	0	0
$418$	−3.15040e6	−0.881911
$419$	973924.	0.271013	0.135506	−	0.990776i	$-0.456734\pi$
$419$	973924.	0.271013	0.135506	+	0.990776i	$0.456734\pi$
$420$	0	0
$421$	864618.	0.237749	0.118875	−	0.992909i	$-0.462071\pi$
$421$	864618.	0.237749	0.118875	+	0.992909i	$0.462071\pi$
$422$	4.51102e6	1.23309
$423$	0	0
$424$	684928.	0.185025
$425$	308574.	0.0828680
$426$	0	0
$427$	0	0
$428$	3.17629e6	0.838128
$429$	0	0
$430$	−664352.	−0.173271
$431$	3.66046e6	0.949166	0.474583	−	0.880211i	$-0.342599\pi$
$431$	3.66046e6	0.949166	0.474583	+	0.880211i	$0.342599\pi$
$432$	0	0
$433$	−4.93667e6	−1.26536	−0.632681	−	0.774413i	$-0.718045\pi$
$433$	−4.93667e6	−1.26536	−0.632681	+	0.774413i	$0.718045\pi$
$434$	0	0
$435$	0	0
$436$	2.12061e6	0.534250
$437$	−4.71240e6	−1.18043
$438$	0	0
$439$	731304.	0.181108	0.0905538	−	0.995892i	$-0.471136\pi$
$439$	731304.	0.181108	0.0905538	+	0.995892i	$0.471136\pi$
$440$	−595712.	−0.146691
$441$	0	0
$442$	−167328.	−0.0407392
$443$	−4.86620e6	−1.17810	−0.589048	−	0.808098i	$-0.700497\pi$
$443$	−4.86620e6	−1.17810	−0.589048	+	0.808098i	$0.700497\pi$
$444$	0	0
$445$	−3.32816e6	−0.796716
$446$	3.59190e6	0.855042
$447$	0	0
$448$	0	0
$449$	−5.71987e6	−1.33897	−0.669484	−	0.742827i	$-0.733485\pi$
$449$	−5.71987e6	−1.33897	−0.669484	+	0.742827i	$0.733485\pi$
$450$	0	0
$451$	766836.	0.177526
$452$	−752416.	−0.173226
$453$	0	0
$454$	1.87045e6	0.425898
$455$	0	0
$456$	0	0
$457$	−6.82034e6	−1.52762	−0.763811	−	0.645440i	$-0.776674\pi$
$457$	−6.82034e6	−1.52762	−0.763811	+	0.645440i	$0.776674\pi$
$458$	−1.78456e6	−0.397528
$459$	0	0
$460$	−891072.	−0.196344
$461$	7.45934e6	1.63474	0.817369	−	0.576115i	$-0.195432\pi$
$461$	7.45934e6	1.63474	0.817369	+	0.576115i	$0.195432\pi$
$462$	0	0
$463$	−5.23848e6	−1.13567	−0.567836	−	0.823142i	$-0.692219\pi$
$463$	−5.23848e6	−1.13567	−0.567836	+	0.823142i	$0.692219\pi$
$464$	924160.	0.199275
$465$	0	0
$466$	−2.80594e6	−0.598569
$467$	−8.95995e6	−1.90114	−0.950568	−	0.310516i	$-0.899498\pi$
$467$	−8.95995e6	−1.90114	−0.950568	+	0.310516i	$0.899498\pi$
$468$	0	0
$469$	0	0
$470$	−678080.	−0.141591
$471$	0	0
$472$	−2.72154e6	−0.562288
$473$	2.28690e6	0.469997
$474$	0	0
$475$	5.38780e6	1.09566
$476$	0	0
$477$	0	0
$478$	1.53679e6	0.307642
$479$	1.75354e6	0.349201	0.174601	−	0.984639i	$-0.444137\pi$
$479$	1.75354e6	0.349201	0.174601	+	0.984639i	$0.444137\pi$
$480$	0	0
$481$	−970104.	−0.191186
$482$	−3.81512e6	−0.747981
$483$	0	0
$484$	−526192.	−0.102101
$485$	2.94642e6	0.568776
$486$	0	0
$487$	927568.	0.177224	0.0886122	−	0.996066i	$-0.471757\pi$
$487$	927568.	0.177224	0.0886122	+	0.996066i	$0.471757\pi$
$488$	−2.86976e6	−0.545502
$489$	0	0
$490$	0	0
$491$	−8.43733e6	−1.57943	−0.789716	−	0.613472i	$-0.789772\pi$
$491$	−8.43733e6	−1.57943	−0.789716	+	0.613472i	$0.789772\pi$
$492$	0	0
$493$	−454860.	−0.0842870
$494$	−2.92160e6	−0.538646
$495$	0	0
$496$	1.45101e6	0.264829
$497$	0	0
$498$	0	0
$499$	1.33278e6	0.239611	0.119806	−	0.992797i	$-0.461773\pi$
$499$	1.33278e6	0.239611	0.119806	+	0.992797i	$0.461773\pi$
$500$	2.31878e6	0.414797
$501$	0	0
$502$	−2.27816e6	−0.403483
$503$	−3.64494e6	−0.642349	−0.321174	−	0.947020i	$-0.604078\pi$
$503$	−3.64494e6	−0.642349	−0.321174	+	0.947020i	$0.604078\pi$
$504$	0	0
$505$	−3.63256e6	−0.633848
$506$	3.06734e6	0.532582
$507$	0	0
$508$	2.64877e6	0.455391
$509$	3.26166e6	0.558013	0.279007	−	0.960289i	$-0.409995\pi$
$509$	3.26166e6	0.558013	0.279007	+	0.960289i	$0.409995\pi$
$510$	0	0
$511$	0	0
$512$	262144.	0.0441942
$513$	0	0
$514$	−4.26657e6	−0.712313
$515$	3.69668e6	0.614177
$516$	0	0
$517$	2.33416e6	0.384065
$518$	0	0
$519$	0	0
$520$	−552448.	−0.0895948
$521$	2.18741e6	0.353050	0.176525	−	0.984296i	$-0.443514\pi$
$521$	2.18741e6	0.353050	0.176525	+	0.984296i	$0.443514\pi$
$522$	0	0
$523$	1.03890e7	1.66081	0.830406	−	0.557159i	$-0.188109\pi$
$523$	1.03890e7	1.66081	0.830406	+	0.557159i	$0.188109\pi$
$524$	2.22893e6	0.354624
$525$	0	0
$526$	5.92972e6	0.934480
$527$	−714168.	−0.112014
$528$	0	0
$529$	−1.84818e6	−0.287147
$530$	−1.11301e6	−0.172111
$531$	0	0
$532$	0	0
$533$	711144.	0.108428
$534$	0	0
$535$	−5.16147e6	−0.779630
$536$	−92672.0	−0.0139327
$537$	0	0
$538$	860440.	0.128164
$539$	0	0
$540$	0	0
$541$	1.27724e7	1.87620	0.938101	−	0.346363i	$-0.112583\pi$
$541$	1.27724e7	1.87620	0.938101	+	0.346363i	$0.112583\pi$
$542$	7.72414e6	1.12941
$543$	0	0
$544$	−129024.	−0.0186928
$545$	−3.44599e6	−0.496961
$546$	0	0
$547$	−5.22238e6	−0.746278	−0.373139	−	0.927776i	$-0.621719\pi$
$547$	−5.22238e6	−0.746278	−0.373139	+	0.927776i	$0.621719\pi$
$548$	5.32547e6	0.757542
$549$	0	0
$550$	−3.50697e6	−0.494339
$551$	−7.94200e6	−1.11443
$552$	0	0
$553$	0	0
$554$	8.15025e6	1.12823
$555$	0	0
$556$	136896.	0.0187804
$557$	5.74047e6	0.783988	0.391994	−	0.919968i	$-0.371785\pi$
$557$	5.74047e6	0.783988	0.391994	+	0.919968i	$0.371785\pi$
$558$	0	0
$559$	2.12082e6	0.287061
$560$	0	0
$561$	0	0
$562$	2.55626e6	0.341401
$563$	2.30448e6	0.306409	0.153204	−	0.988195i	$-0.451041\pi$
$563$	2.30448e6	0.306409	0.153204	+	0.988195i	$0.451041\pi$
$564$	0	0
$565$	1.22268e6	0.161135
$566$	150976.	0.0198092
$567$	0	0
$568$	281728.	0.0366403
$569$	5.12150e6	0.663157	0.331578	−	0.943428i	$-0.392419\pi$
$569$	5.12150e6	0.663157	0.331578	+	0.943428i	$0.392419\pi$
$570$	0	0
$571$	−2.38637e6	−0.306300	−0.153150	−	0.988203i	$-0.548942\pi$
$571$	−2.38637e6	−0.306300	−0.153150	+	0.988203i	$0.548942\pi$
$572$	1.90170e6	0.243025
$573$	0	0
$574$	0	0
$575$	−5.24576e6	−0.661666
$576$	0	0
$577$	−5.24151e6	−0.655416	−0.327708	−	0.944779i	$-0.606276\pi$
$577$	−5.24151e6	−0.655416	−0.327708	+	0.944779i	$0.606276\pi$
$578$	−5.61592e6	−0.699200
$579$	0	0
$580$	−1.50176e6	−0.185366
$581$	0	0
$582$	0	0
$583$	3.83132e6	0.466849
$584$	1.31200e6	0.159185
$585$	0	0
$586$	7.35684e6	0.885008
$587$	−9.11548e6	−1.09190	−0.545952	−	0.837816i	$-0.683832\pi$
$587$	−9.11548e6	−1.09190	−0.545952	+	0.837816i	$0.683832\pi$
$588$	0	0
$589$	−1.24696e7	−1.48103
$590$	4.42250e6	0.523043
$591$	0	0
$592$	−748032.	−0.0877235
$593$	−3.05043e6	−0.356225	−0.178112	−	0.984010i	$-0.556999\pi$
$593$	−3.05043e6	−0.356225	−0.178112	+	0.984010i	$0.556999\pi$
$594$	0	0
$595$	0	0
$596$	−1.11286e6	−0.128329
$597$	0	0
$598$	2.84458e6	0.325285
$599$	−1.43408e7	−1.63308	−0.816539	−	0.577290i	$-0.804110\pi$
$599$	−1.43408e7	−1.63308	−0.816539	+	0.577290i	$0.804110\pi$
$600$	0	0
$601$	−3.12662e6	−0.353092	−0.176546	−	0.984292i	$-0.556492\pi$
$601$	−3.12662e6	−0.353092	−0.176546	+	0.984292i	$0.556492\pi$
$602$	0	0
$603$	0	0
$604$	8.46784e6	0.944453
$605$	855062.	0.0949750
$606$	0	0
$607$	1.15098e7	1.26794	0.633969	−	0.773359i	$-0.281425\pi$
$607$	1.15098e7	1.26794	0.633969	+	0.773359i	$0.281425\pi$
$608$	−2.25280e6	−0.247152
$609$	0	0
$610$	4.66336e6	0.507428
$611$	2.16464e6	0.234576
$612$	0	0
$613$	−1.21782e7	−1.30898	−0.654488	−	0.756072i	$-0.727116\pi$
$613$	−1.21782e7	−1.30898	−0.654488	+	0.756072i	$0.727116\pi$
$614$	−4.25888e6	−0.455905
$615$	0	0
$616$	0	0
$617$	−1.77629e6	−0.187845	−0.0939226	−	0.995580i	$-0.529941\pi$
$617$	−1.77629e6	−0.187845	−0.0939226	+	0.995580i	$0.529941\pi$
$618$	0	0
$619$	−5.95516e6	−0.624694	−0.312347	−	0.949968i	$-0.601115\pi$
$619$	−5.95516e6	−0.624694	−0.312347	+	0.949968i	$0.601115\pi$
$620$	−2.35789e6	−0.246345
$621$	0	0
$622$	4.03808e6	0.418503
$623$	0	0
$624$	0	0
$625$	3.88510e6	0.397834
$626$	−5.79638e6	−0.591182
$627$	0	0
$628$	208640.	0.0211105
$629$	368172.	0.0371043
$630$	0	0
$631$	−1.45351e7	−1.45327	−0.726633	−	0.687026i	$-0.758916\pi$
$631$	−1.45351e7	−1.45327	−0.726633	+	0.687026i	$0.758916\pi$
$632$	4.17510e6	0.415791
$633$	0	0
$634$	−1.08924e7	−1.07622
$635$	−4.30425e6	−0.423607
$636$	0	0
$637$	0	0
$638$	5.16952e6	0.502804
$639$	0	0
$640$	−425984.	−0.0411096
$641$	1.07349e7	1.03194	0.515970	−	0.856607i	$-0.327432\pi$
$641$	1.07349e7	1.03194	0.515970	+	0.856607i	$0.327432\pi$
$642$	0	0
$643$	1.62815e7	1.55298	0.776492	−	0.630127i	$-0.216997\pi$
$643$	1.62815e7	1.55298	0.776492	+	0.630127i	$0.216997\pi$
$644$	0	0
$645$	0	0
$646$	1.10880e6	0.104537
$647$	7.91947e6	0.743765	0.371882	−	0.928280i	$-0.378712\pi$
$647$	7.91947e6	0.743765	0.371882	+	0.928280i	$0.378712\pi$
$648$	0	0
$649$	−1.52236e7	−1.41875
$650$	−3.25227e6	−0.301928
$651$	0	0
$652$	−5.61984e6	−0.517732
$653$	−1.34478e6	−0.123415	−0.0617076	−	0.998094i	$-0.519655\pi$
$653$	−1.34478e6	−0.123415	−0.0617076	+	0.998094i	$0.519655\pi$
$654$	0	0
$655$	−3.62201e6	−0.329873
$656$	548352.	0.0497508
$657$	0	0
$658$	0	0
$659$	−2.02235e7	−1.81402	−0.907010	−	0.421109i	$-0.861641\pi$
$659$	−2.02235e7	−1.81402	−0.907010	+	0.421109i	$0.861641\pi$
$660$	0	0
$661$	7.17802e6	0.639001	0.319500	−	0.947586i	$-0.396485\pi$
$661$	7.17802e6	0.639001	0.319500	+	0.947586i	$0.396485\pi$
$662$	−4.40162e6	−0.390362
$663$	0	0
$664$	6.57946e6	0.579122
$665$	0	0
$666$	0	0
$667$	7.73262e6	0.672995
$668$	−1.00180e7	−0.868644
$669$	0	0
$670$	150592.	0.0129603
$671$	−1.60527e7	−1.37639
$672$	0	0
$673$	9.61217e6	0.818057	0.409029	−	0.912522i	$-0.365868\pi$
$673$	9.61217e6	0.818057	0.409029	+	0.912522i	$0.365868\pi$
$674$	6.94047e6	0.588490
$675$	0	0
$676$	−4.17710e6	−0.351567
$677$	9.66815e6	0.810722	0.405361	−	0.914157i	$-0.367146\pi$
$677$	9.66815e6	0.810722	0.405361	+	0.914157i	$0.367146\pi$
$678$	0	0
$679$	0	0
$680$	209664.	0.0173881
$681$	0	0
$682$	8.11658e6	0.668208
$683$	−389854.	−0.0319779	−0.0159890	−	0.999872i	$-0.505090\pi$
$683$	−389854.	−0.0319779	−0.0159890	+	0.999872i	$0.505090\pi$
$684$	0	0
$685$	−8.65389e6	−0.704669
$686$	0	0
$687$	0	0
$688$	1.63533e6	0.131715
$689$	3.55306e6	0.285138
$690$	0	0
$691$	3.73985e6	0.297961	0.148980	−	0.988840i	$-0.452401\pi$
$691$	3.73985e6	0.297961	0.148980	+	0.988840i	$0.452401\pi$
$692$	2.95722e6	0.234757
$693$	0	0
$694$	−6.36578e6	−0.501711
$695$	−222456.	−0.0174696
$696$	0	0
$697$	−269892.	−0.0210430
$698$	−9.33504e6	−0.725233
$699$	0	0
$700$	0	0
$701$	−2.49886e7	−1.92064	−0.960322	−	0.278893i	$-0.910033\pi$
$701$	−2.49886e7	−1.92064	−0.960322	+	0.278893i	$0.910033\pi$
$702$	0	0
$703$	6.42840e6	0.490585
$704$	1.46637e6	0.111509
$705$	0	0
$706$	−1.12433e7	−0.848947
$707$	0	0
$708$	0	0
$709$	−9.83584e6	−0.734845	−0.367423	−	0.930054i	$-0.619760\pi$
$709$	−9.83584e6	−0.734845	−0.367423	+	0.930054i	$0.619760\pi$
$710$	−457808.	−0.0340830
$711$	0	0
$712$	8.19238e6	0.605634
$713$	1.21409e7	0.894387
$714$	0	0
$715$	−3.09026e6	−0.226063
$716$	5.72035e6	0.417004
$717$	0	0
$718$	−3.75724e6	−0.271993
$719$	2.13624e7	1.54109	0.770546	−	0.637384i	$-0.219983\pi$
$719$	2.13624e7	1.54109	0.770546	+	0.637384i	$0.219983\pi$
$720$	0	0
$721$	0	0
$722$	9.45560e6	0.675066
$723$	0	0
$724$	1.11441e7	0.790132
$725$	−8.84089e6	−0.624670
$726$	0	0
$727$	6.53025e6	0.458241	0.229120	−	0.973398i	$-0.426415\pi$
$727$	6.53025e6	0.458241	0.229120	+	0.973398i	$0.426415\pi$
$728$	0	0
$729$	0	0
$730$	−2.13200e6	−0.148074
$731$	−804888.	−0.0557111
$732$	0	0
$733$	−1.66571e7	−1.14509	−0.572545	−	0.819873i	$-0.694044\pi$
$733$	−1.66571e7	−1.14509	−0.572545	+	0.819873i	$0.694044\pi$
$734$	1.23940e7	0.849128
$735$	0	0
$736$	2.19341e6	0.149254
$737$	−518384.	−0.0351547
$738$	0	0
$739$	−2.39536e7	−1.61347	−0.806733	−	0.590917i	$-0.798766\pi$
$739$	−2.39536e7	−1.61347	−0.806733	+	0.590917i	$0.798766\pi$
$740$	1.21555e6	0.0816008
$741$	0	0
$742$	0	0
$743$	7.48982e6	0.497736	0.248868	−	0.968537i	$-0.419941\pi$
$743$	7.48982e6	0.497736	0.248868	+	0.968537i	$0.419941\pi$
$744$	0	0
$745$	1.80840e6	0.119373
$746$	−913064.	−0.0600695
$747$	0	0
$748$	−721728.	−0.0471650
$749$	0	0
$750$	0	0
$751$	−4.71845e6	−0.305281	−0.152640	−	0.988282i	$-0.548778\pi$
$751$	−4.71845e6	−0.305281	−0.152640	+	0.988282i	$0.548778\pi$
$752$	1.66912e6	0.107632
$753$	0	0
$754$	4.79408e6	0.307098
$755$	−1.37602e7	−0.878534
$756$	0	0
$757$	−2.67397e7	−1.69597	−0.847983	−	0.530024i	$-0.822183\pi$
$757$	−2.67397e7	−1.69597	−0.847983	+	0.530024i	$0.822183\pi$
$758$	−4.14677e6	−0.262142
$759$	0	0
$760$	3.66080e6	0.229902
$761$	1.44331e7	0.903435	0.451718	−	0.892161i	$-0.350811\pi$
$761$	1.44331e7	0.903435	0.451718	+	0.892161i	$0.350811\pi$
$762$	0	0
$763$	0	0
$764$	−1.09872e6	−0.0681010
$765$	0	0
$766$	847104.	0.0521633
$767$	−1.41180e7	−0.866530
$768$	0	0
$769$	−8.55510e6	−0.521686	−0.260843	−	0.965381i	$-0.584001\pi$
$769$	−8.55510e6	−0.521686	−0.260843	+	0.965381i	$0.584001\pi$
$770$	0	0
$771$	0	0
$772$	1.32356e7	0.799280
$773$	−1.92272e7	−1.15735	−0.578677	−	0.815557i	$-0.696431\pi$
$773$	−1.92272e7	−1.15735	−0.578677	+	0.815557i	$0.696431\pi$
$774$	0	0
$775$	−1.38809e7	−0.830165
$776$	−7.25274e6	−0.432362
$777$	0	0
$778$	−5.65298e6	−0.334833
$779$	−4.71240e6	−0.278227
$780$	0	0
$781$	1.57592e6	0.0924497
$782$	−1.07957e6	−0.0631296
$783$	0	0
$784$	0	0
$785$	−339040.	−0.0196371
$786$	0	0
$787$	−2.53316e7	−1.45789	−0.728947	−	0.684570i	$-0.759990\pi$
$787$	−2.53316e7	−1.45789	−0.728947	+	0.684570i	$0.759990\pi$
$788$	2.29411e6	0.131613
$789$	0	0
$790$	−6.78454e6	−0.386770
$791$	0	0
$792$	0	0
$793$	−1.48869e7	−0.840661
$794$	−4.36134e6	−0.245510
$795$	0	0
$796$	−8.68160e6	−0.485643
$797$	3.13162e7	1.74632	0.873158	−	0.487437i	$-0.162068\pi$
$797$	3.13162e7	1.74632	0.873158	+	0.487437i	$0.162068\pi$
$798$	0	0
$799$	−821520.	−0.0455251
$800$	−2.50778e6	−0.138536
$801$	0	0
$802$	−1.05701e7	−0.580288
$803$	7.33900e6	0.401650
$804$	0	0
$805$	0	0
$806$	7.52710e6	0.408122
$807$	0	0
$808$	8.94170e6	0.481827
$809$	−484890.	−0.0260479	−0.0130239	−	0.999915i	$-0.504146\pi$
$809$	−484890.	−0.0260479	−0.0130239	+	0.999915i	$0.504146\pi$
$810$	0	0
$811$	−5.32623e6	−0.284359	−0.142180	−	0.989841i	$-0.545411\pi$
$811$	−5.32623e6	−0.284359	−0.142180	+	0.989841i	$0.545411\pi$
$812$	0	0
$813$	0	0
$814$	−4.18430e6	−0.221341
$815$	9.13224e6	0.481596
$816$	0	0
$817$	−1.40536e7	−0.736601
$818$	2.50171e7	1.30723
$819$	0	0
$820$	−891072.	−0.0462784
$821$	−3.21777e7	−1.66609	−0.833043	−	0.553209i	$-0.813403\pi$
$821$	−3.21777e7	−1.66609	−0.833043	+	0.553209i	$0.813403\pi$
$822$	0	0
$823$	−8.07408e6	−0.415521	−0.207761	−	0.978180i	$-0.566618\pi$
$823$	−8.07408e6	−0.415521	−0.207761	+	0.978180i	$0.566618\pi$
$824$	−9.09952e6	−0.466875
$825$	0	0
$826$	0	0
$827$	8.04922e6	0.409251	0.204626	−	0.978840i	$-0.434402\pi$
$827$	8.04922e6	0.409251	0.204626	+	0.978840i	$0.434402\pi$
$828$	0	0
$829$	−1.35889e7	−0.686751	−0.343375	−	0.939198i	$-0.611570\pi$
$829$	−1.35889e7	−0.686751	−0.343375	+	0.939198i	$0.611570\pi$
$830$	−1.06916e7	−0.538701
$831$	0	0
$832$	1.35987e6	0.0681067
$833$	0	0
$834$	0	0
$835$	1.62793e7	0.808017
$836$	−1.26016e7	−0.623606
$837$	0	0
$838$	3.89570e6	0.191635
$839$	3.67721e6	0.180349	0.0901744	−	0.995926i	$-0.471258\pi$
$839$	3.67721e6	0.180349	0.0901744	+	0.995926i	$0.471258\pi$
$840$	0	0
$841$	−7.47905e6	−0.364633
$842$	3.45847e6	0.168114
$843$	0	0
$844$	1.80441e7	0.871925
$845$	6.78779e6	0.327029
$846$	0	0
$847$	0	0
$848$	2.73971e6	0.130832
$849$	0	0
$850$	1.23430e6	0.0585965
$851$	−6.25892e6	−0.296262
$852$	0	0
$853$	−3.25379e7	−1.53115	−0.765573	−	0.643349i	$-0.777544\pi$
$853$	−3.25379e7	−1.53115	−0.765573	+	0.643349i	$0.777544\pi$
$854$	0	0
$855$	0	0
$856$	1.27052e7	0.592646
$857$	−1.12723e7	−0.524278	−0.262139	−	0.965030i	$-0.584428\pi$
$857$	−1.12723e7	−0.524278	−0.262139	+	0.965030i	$0.584428\pi$
$858$	0	0
$859$	−7.69694e6	−0.355906	−0.177953	−	0.984039i	$-0.556948\pi$
$859$	−7.69694e6	−0.355906	−0.177953	+	0.984039i	$0.556948\pi$
$860$	−2.65741e6	−0.122521
$861$	0	0
$862$	1.46418e7	0.671162
$863$	4.58785e6	0.209692	0.104846	−	0.994488i	$-0.466565\pi$
$863$	4.58785e6	0.209692	0.104846	+	0.994488i	$0.466565\pi$
$864$	0	0
$865$	−4.80548e6	−0.218372
$866$	−1.97467e7	−0.894746
$867$	0	0
$868$	0	0
$869$	2.33545e7	1.04911
$870$	0	0
$871$	−480736.	−0.0214714
$872$	8.48243e6	0.377771
$873$	0	0
$874$	−1.88496e7	−0.834687
$875$	0	0
$876$	0	0
$877$	−1.14666e7	−0.503424	−0.251712	−	0.967802i	$-0.580994\pi$
$877$	−1.14666e7	−0.503424	−0.251712	+	0.967802i	$0.580994\pi$
$878$	2.92522e6	0.128062
$879$	0	0
$880$	−2.38285e6	−0.103726
$881$	3.02550e7	1.31328	0.656640	−	0.754204i	$-0.271977\pi$
$881$	3.02550e7	1.31328	0.656640	+	0.754204i	$0.271977\pi$
$882$	0	0
$883$	9.83052e6	0.424302	0.212151	−	0.977237i	$-0.431953\pi$
$883$	9.83052e6	0.424302	0.212151	+	0.977237i	$0.431953\pi$
$884$	−669312.	−0.0288070
$885$	0	0
$886$	−1.94648e7	−0.833039
$887$	−2.32272e7	−0.991263	−0.495631	−	0.868533i	$-0.665063\pi$
$887$	−2.32272e7	−0.991263	−0.495631	+	0.868533i	$0.665063\pi$
$888$	0	0
$889$	0	0
$890$	−1.33126e7	−0.563363
$891$	0	0
$892$	1.43676e7	0.604606
$893$	−1.43440e7	−0.601924
$894$	0	0
$895$	−9.29557e6	−0.387899
$896$	0	0
$897$	0	0
$898$	−2.28795e7	−0.946793
$899$	2.04615e7	0.844380
$900$	0	0
$901$	−1.34845e6	−0.0553380
$902$	3.06734e6	0.125530
$903$	0	0
$904$	−3.00966e6	−0.122489
$905$	−1.81092e7	−0.734984
$906$	0	0
$907$	3.05501e7	1.23309	0.616544	−	0.787321i	$-0.288532\pi$
$907$	3.05501e7	1.23309	0.616544	+	0.787321i	$0.288532\pi$
$908$	7.48179e6	0.301156
$909$	0	0
$910$	0	0
$911$	−2.21502e7	−0.884265	−0.442133	−	0.896950i	$-0.645778\pi$
$911$	−2.21502e7	−0.884265	−0.442133	+	0.896950i	$0.645778\pi$
$912$	0	0
$913$	3.68038e7	1.46122
$914$	−2.72814e7	−1.08019
$915$	0	0
$916$	−7.13824e6	−0.281095
$917$	0	0
$918$	0	0
$919$	−1.26723e7	−0.494955	−0.247477	−	0.968894i	$-0.579602\pi$
$919$	−1.26723e7	−0.494955	−0.247477	+	0.968894i	$0.579602\pi$
$920$	−3.56429e6	−0.138836
$921$	0	0
$922$	2.98374e7	1.15593
$923$	1.46146e6	0.0564656
$924$	0	0
$925$	7.15598e6	0.274989
$926$	−2.09539e7	−0.803042
$927$	0	0
$928$	3.69664e6	0.140909
$929$	4.02840e7	1.53142	0.765709	−	0.643187i	$-0.222388\pi$
$929$	4.02840e7	1.53142	0.765709	+	0.643187i	$0.222388\pi$
$930$	0	0
$931$	0	0
$932$	−1.12238e7	−0.423252
$933$	0	0
$934$	−3.58398e7	−1.34431
$935$	1.17281e6	0.0438731
$936$	0	0
$937$	1.34104e7	0.498992	0.249496	−	0.968376i	$-0.419735\pi$
$937$	1.34104e7	0.498992	0.249496	+	0.968376i	$0.419735\pi$
$938$	0	0
$939$	0	0
$940$	−2.71232e6	−0.100120
$941$	2.73213e7	1.00584	0.502918	−	0.864334i	$-0.332260\pi$
$941$	2.73213e7	1.00584	0.502918	+	0.864334i	$0.332260\pi$
$942$	0	0
$943$	4.58816e6	0.168020
$944$	−1.08861e7	−0.397598
$945$	0	0
$946$	9.14762e6	0.332338
$947$	−8.71745e6	−0.315874	−0.157937	−	0.987449i	$-0.550484\pi$
$947$	−8.71745e6	−0.315874	−0.157937	+	0.987449i	$0.550484\pi$
$948$	0	0
$949$	6.80600e6	0.245316
$950$	2.15512e7	0.774752
$951$	0	0
$952$	0	0
$953$	1.62984e7	0.581315	0.290658	−	0.956827i	$-0.406126\pi$
$953$	1.62984e7	0.581315	0.290658	+	0.956827i	$0.406126\pi$
$954$	0	0
$955$	1.78542e6	0.0633479
$956$	6.14717e6	0.217536
$957$	0	0
$958$	7.01414e6	0.246923
$959$	0	0
$960$	0	0
$961$	3.49707e6	0.122151
$962$	−3.88042e6	−0.135189
$963$	0	0
$964$	−1.52605e7	−0.528902
$965$	−2.15078e7	−0.743493
$966$	0	0
$967$	−5.49067e6	−0.188825	−0.0944124	−	0.995533i	$-0.530097\pi$
$967$	−5.49067e6	−0.188825	−0.0944124	+	0.995533i	$0.530097\pi$
$968$	−2.10477e6	−0.0721964
$969$	0	0
$970$	1.17857e7	0.402185
$971$	−4.51675e7	−1.53737	−0.768685	−	0.639628i	$-0.779089\pi$
$971$	−4.51675e7	−1.53737	−0.768685	+	0.639628i	$0.779089\pi$
$972$	0	0
$973$	0	0
$974$	3.71027e6	0.125317
$975$	0	0
$976$	−1.14790e7	−0.385728
$977$	2.38010e7	0.797737	0.398868	−	0.917008i	$-0.369403\pi$
$977$	2.38010e7	0.797737	0.398868	+	0.917008i	$0.369403\pi$
$978$	0	0
$979$	4.58261e7	1.52812
$980$	0	0
$981$	0	0
$982$	−3.37493e7	−1.11683
$983$	−9.36478e6	−0.309111	−0.154555	−	0.987984i	$-0.549394\pi$
$983$	−9.36478e6	−0.309111	−0.154555	+	0.987984i	$0.549394\pi$
$984$	0	0
$985$	−3.72793e6	−0.122427
$986$	−1.81944e6	−0.0595999
$987$	0	0
$988$	−1.16864e7	−0.380880
$989$	1.36831e7	0.444830
$990$	0	0
$991$	−4.33916e7	−1.40353	−0.701764	−	0.712409i	$-0.747604\pi$
$991$	−4.33916e7	−1.40353	−0.701764	+	0.712409i	$0.747604\pi$
$992$	5.80403e6	0.187262
$993$	0	0
$994$	0	0
$995$	1.41076e7	0.451747
$996$	0	0
$997$	4.35294e6	0.138690	0.0693449	−	0.997593i	$-0.477909\pi$
$997$	4.35294e6	0.138690	0.0693449	+	0.997593i	$0.477909\pi$
$998$	5.33112e6	0.169431
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.6.a.p.1.1		1
3.2	odd	2		294.6.a.g.1.1	yes	1
7.6	odd	2		882.6.a.t.1.1		1
21.2	odd	6		294.6.e.j.67.1		2
21.5	even	6		294.6.e.q.67.1		2
21.11	odd	6		294.6.e.j.79.1		2
21.17	even	6		294.6.e.q.79.1		2
21.20	even	2		294.6.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
294.6.a.a.1.1	✓	1	21.20	even	2
294.6.a.g.1.1	yes	1	3.2	odd	2
294.6.e.j.67.1		2	21.2	odd	6
294.6.e.j.79.1		2	21.11	odd	6
294.6.e.q.67.1		2	21.5	even	6
294.6.e.q.79.1		2	21.17	even	6
882.6.a.p.1.1		1	1.1	even	1	trivial
882.6.a.t.1.1		1	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	882.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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