LMFDB - Embedded newform 222.6.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [222,6,Mod(1,222)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(222, 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("222.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 222 = 2 \cdot 3 \cdot 37 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	222.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:	$35.6052079985$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	222.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} -60.0000 q^{5} -36.0000 q^{6} -112.000 q^{7} -64.0000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q-4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} -60.0000 q^{5} -36.0000 q^{6} -112.000 q^{7} -64.0000 q^{8} +81.0000 q^{9} +240.000 q^{10} +336.000 q^{11} +144.000 q^{12} +1106.00 q^{13} +448.000 q^{14} -540.000 q^{15} +256.000 q^{16} +192.000 q^{17} -324.000 q^{18} -952.000 q^{19} -960.000 q^{20} -1008.00 q^{21} -1344.00 q^{22} +114.000 q^{23} -576.000 q^{24} +475.000 q^{25} -4424.00 q^{26} +729.000 q^{27} -1792.00 q^{28} -8484.00 q^{29} +2160.00 q^{30} +2024.00 q^{31} -1024.00 q^{32} +3024.00 q^{33} -768.000 q^{34} +6720.00 q^{35} +1296.00 q^{36} +1369.00 q^{37} +3808.00 q^{38} +9954.00 q^{39} +3840.00 q^{40} -15630.0 q^{41} +4032.00 q^{42} -3748.00 q^{43} +5376.00 q^{44} -4860.00 q^{45} -456.000 q^{46} -1176.00 q^{47} +2304.00 q^{48} -4263.00 q^{49} -1900.00 q^{50} +1728.00 q^{51} +17696.0 q^{52} -4686.00 q^{53} -2916.00 q^{54} -20160.0 q^{55} +7168.00 q^{56} -8568.00 q^{57} +33936.0 q^{58} -26370.0 q^{59} -8640.00 q^{60} +14342.0 q^{61} -8096.00 q^{62} -9072.00 q^{63} +4096.00 q^{64} -66360.0 q^{65} -12096.0 q^{66} -11188.0 q^{67} +3072.00 q^{68} +1026.00 q^{69} -26880.0 q^{70} -31080.0 q^{71} -5184.00 q^{72} +45542.0 q^{73} -5476.00 q^{74} +4275.00 q^{75} -15232.0 q^{76} -37632.0 q^{77} -39816.0 q^{78} -45796.0 q^{79} -15360.0 q^{80} +6561.00 q^{81} +62520.0 q^{82} -76296.0 q^{83} -16128.0 q^{84} -11520.0 q^{85} +14992.0 q^{86} -76356.0 q^{87} -21504.0 q^{88} -4308.00 q^{89} +19440.0 q^{90} -123872. q^{91} +1824.00 q^{92} +18216.0 q^{93} +4704.00 q^{94} +57120.0 q^{95} -9216.00 q^{96} -119650. q^{97} +17052.0 q^{98} +27216.0 q^{99} +O(q^{100})\)

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
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	16.0000	0.500000
$5$	−60.0000	−1.07331	−0.536656	−	0.843801i	$-0.680313\pi$
$5$	−60.0000	−1.07331	−0.536656	+	0.843801i	$0.680313\pi$
$6$	−36.0000	−0.408248
$7$	−112.000	−0.863919	−0.431959	−	0.901893i	$-0.642178\pi$
$7$	−112.000	−0.863919	−0.431959	+	0.901893i	$0.642178\pi$
$8$	−64.0000	−0.353553
$9$	81.0000	0.333333
$10$	240.000	0.758947
$11$	336.000	0.837255	0.418627	−	0.908158i	$-0.362511\pi$
$11$	336.000	0.837255	0.418627	+	0.908158i	$0.362511\pi$
$12$	144.000	0.288675
$13$	1106.00	1.81508	0.907542	−	0.419961i	$-0.137956\pi$
$13$	1106.00	1.81508	0.907542	+	0.419961i	$0.137956\pi$
$14$	448.000	0.610883
$15$	−540.000	−0.619677
$16$	256.000	0.250000
$17$	192.000	0.161131	0.0805655	−	0.996749i	$-0.474327\pi$
$17$	192.000	0.161131	0.0805655	+	0.996749i	$0.474327\pi$
$18$	−324.000	−0.235702
$19$	−952.000	−0.604997	−0.302498	−	0.953150i	$-0.597821\pi$
$19$	−952.000	−0.604997	−0.302498	+	0.953150i	$0.597821\pi$
$20$	−960.000	−0.536656
$21$	−1008.00	−0.498784
$22$	−1344.00	−0.592028
$23$	114.000	0.0449351	0.0224675	−	0.999748i	$-0.492848\pi$
$23$	114.000	0.0449351	0.0224675	+	0.999748i	$0.492848\pi$
$24$	−576.000	−0.204124
$25$	475.000	0.152000
$26$	−4424.00	−1.28346
$27$	729.000	0.192450
$28$	−1792.00	−0.431959
$29$	−8484.00	−1.87329	−0.936646	−	0.350276i	$-0.886088\pi$
$29$	−8484.00	−1.87329	−0.936646	+	0.350276i	$0.886088\pi$
$30$	2160.00	0.438178
$31$	2024.00	0.378274	0.189137	−	0.981951i	$-0.439431\pi$
$31$	2024.00	0.378274	0.189137	+	0.981951i	$0.439431\pi$
$32$	−1024.00	−0.176777
$33$	3024.00	0.483389
$34$	−768.000	−0.113937
$35$	6720.00	0.927255
$36$	1296.00	0.166667
$37$	1369.00	0.164399
$37$	1369.00	0.164399
$38$	3808.00	0.427797
$39$	9954.00	1.04794
$40$	3840.00	0.379473
$41$	−15630.0	−1.45211	−0.726055	−	0.687637i	$-0.758648\pi$
$41$	−15630.0	−1.45211	−0.726055	+	0.687637i	$0.758648\pi$
$42$	4032.00	0.352693
$43$	−3748.00	−0.309121	−0.154560	−	0.987983i	$-0.549396\pi$
$43$	−3748.00	−0.309121	−0.154560	+	0.987983i	$0.549396\pi$
$44$	5376.00	0.418627
$45$	−4860.00	−0.357771
$46$	−456.000	−0.0317739
$47$	−1176.00	−0.0776538	−0.0388269	−	0.999246i	$-0.512362\pi$
$47$	−1176.00	−0.0776538	−0.0388269	+	0.999246i	$0.512362\pi$
$48$	2304.00	0.144338
$49$	−4263.00	−0.253644
$50$	−1900.00	−0.107480
$51$	1728.00	0.0930290
$52$	17696.0	0.907542
$53$	−4686.00	−0.229146	−0.114573	−	0.993415i	$-0.536550\pi$
$53$	−4686.00	−0.229146	−0.114573	+	0.993415i	$0.536550\pi$
$54$	−2916.00	−0.136083
$55$	−20160.0	−0.898636
$56$	7168.00	0.305441
$57$	−8568.00	−0.349295
$58$	33936.0	1.32462
$59$	−26370.0	−0.986234	−0.493117	−	0.869963i	$-0.664143\pi$
$59$	−26370.0	−0.986234	−0.493117	+	0.869963i	$0.664143\pi$
$60$	−8640.00	−0.309839
$61$	14342.0	0.493498	0.246749	−	0.969079i	$-0.420638\pi$
$61$	14342.0	0.493498	0.246749	+	0.969079i	$0.420638\pi$
$62$	−8096.00	−0.267480
$63$	−9072.00	−0.287973
$64$	4096.00	0.125000
$65$	−66360.0	−1.94815
$66$	−12096.0	−0.341808
$67$	−11188.0	−0.304485	−0.152242	−	0.988343i	$-0.548649\pi$
$67$	−11188.0	−0.304485	−0.152242	+	0.988343i	$0.548649\pi$
$68$	3072.00	0.0805655
$69$	1026.00	0.0259433
$70$	−26880.0	−0.655668
$71$	−31080.0	−0.731704	−0.365852	−	0.930673i	$-0.619222\pi$
$71$	−31080.0	−0.731704	−0.365852	+	0.930673i	$0.619222\pi$
$72$	−5184.00	−0.117851
$73$	45542.0	1.00024	0.500121	−	0.865956i	$-0.333289\pi$
$73$	45542.0	1.00024	0.500121	+	0.865956i	$0.333289\pi$
$74$	−5476.00	−0.116248
$75$	4275.00	0.0877572
$76$	−15232.0	−0.302498
$77$	−37632.0	−0.723320
$78$	−39816.0	−0.741005
$79$	−45796.0	−0.825581	−0.412791	−	0.910826i	$-0.635446\pi$
$79$	−45796.0	−0.825581	−0.412791	+	0.910826i	$0.635446\pi$
$80$	−15360.0	−0.268328
$81$	6561.00	0.111111
$82$	62520.0	1.02680
$83$	−76296.0	−1.21564	−0.607822	−	0.794073i	$-0.707957\pi$
$83$	−76296.0	−1.21564	−0.607822	+	0.794073i	$0.707957\pi$
$84$	−16128.0	−0.249392
$85$	−11520.0	−0.172944
$86$	14992.0	0.218582
$87$	−76356.0	−1.08155
$88$	−21504.0	−0.296014
$89$	−4308.00	−0.0576502	−0.0288251	−	0.999584i	$-0.509177\pi$
$89$	−4308.00	−0.0576502	−0.0288251	+	0.999584i	$0.509177\pi$
$90$	19440.0	0.252982
$91$	−123872.	−1.56809
$92$	1824.00	0.0224675
$93$	18216.0	0.218396
$94$	4704.00	0.0549095
$95$	57120.0	0.649351
$96$	−9216.00	−0.102062
$97$	−119650.	−1.29117	−0.645585	−	0.763688i	$-0.723386\pi$
$97$	−119650.	−1.29117	−0.645585	+	0.763688i	$0.723386\pi$
$98$	17052.0	0.179354
$99$	27216.0	0.279085
$100$	7600.00	0.0760000
$101$	−104586.	−1.02016	−0.510082	−	0.860126i	$-0.670385\pi$
$101$	−104586.	−1.02016	−0.510082	+	0.860126i	$0.670385\pi$
$102$	−6912.00	−0.0657814
$103$	3788.00	0.0351817	0.0175909	−	0.999845i	$-0.494400\pi$
$103$	3788.00	0.0351817	0.0175909	+	0.999845i	$0.494400\pi$
$104$	−70784.0	−0.641729
$105$	60480.0	0.535351
$106$	18744.0	0.162031
$107$	−14628.0	−0.123517	−0.0617583	−	0.998091i	$-0.519671\pi$
$107$	−14628.0	−0.123517	−0.0617583	+	0.998091i	$0.519671\pi$
$108$	11664.0	0.0962250
$109$	199226.	1.60613	0.803063	−	0.595894i	$-0.203202\pi$
$109$	199226.	1.60613	0.803063	+	0.595894i	$0.203202\pi$
$110$	80640.0	0.635432
$111$	12321.0	0.0949158
$112$	−28672.0	−0.215980
$113$	−28692.0	−0.211380	−0.105690	−	0.994399i	$-0.533705\pi$
$113$	−28692.0	−0.211380	−0.105690	+	0.994399i	$0.533705\pi$
$114$	34272.0	0.246989
$115$	−6840.00	−0.0482294
$116$	−135744.	−0.936646
$117$	89586.0	0.605028
$118$	105480.	0.697373
$119$	−21504.0	−0.139204
$120$	34560.0	0.219089
$121$	−48155.0	−0.299005
$122$	−57368.0	−0.348956
$123$	−140670.	−0.838376
$124$	32384.0	0.189137
$125$	159000.	0.910169
$126$	36288.0	0.203628
$127$	−122632.	−0.674675	−0.337337	−	0.941384i	$-0.609526\pi$
$127$	−122632.	−0.674675	−0.337337	+	0.941384i	$0.609526\pi$
$128$	−16384.0	−0.0883883
$129$	−33732.0	−0.178471
$130$	265440.	1.37755
$131$	207450.	1.05617	0.528087	−	0.849190i	$-0.322910\pi$
$131$	207450.	1.05617	0.528087	+	0.849190i	$0.322910\pi$
$132$	48384.0	0.241695
$133$	106624.	0.522668
$134$	44752.0	0.215303
$135$	−43740.0	−0.206559
$136$	−12288.0	−0.0569684
$137$	−11490.0	−0.0523020	−0.0261510	−	0.999658i	$-0.508325\pi$
$137$	−11490.0	−0.0523020	−0.0261510	+	0.999658i	$0.508325\pi$
$138$	−4104.00	−0.0183447
$139$	−73540.0	−0.322839	−0.161420	−	0.986886i	$-0.551607\pi$
$139$	−73540.0	−0.322839	−0.161420	+	0.986886i	$0.551607\pi$
$140$	107520.	0.463627
$141$	−10584.0	−0.0448334
$142$	124320.	0.517393
$143$	371616.	1.51969
$144$	20736.0	0.0833333
$145$	509040.	2.01063
$146$	−182168.	−0.707278
$147$	−38367.0	−0.146442
$148$	21904.0	0.0821995
$149$	−82002.0	−0.302593	−0.151296	−	0.988488i	$-0.548345\pi$
$149$	−82002.0	−0.302593	−0.151296	+	0.988488i	$0.548345\pi$
$150$	−17100.0	−0.0620537
$151$	540536.	1.92922	0.964611	−	0.263677i	$-0.0849352\pi$
$151$	540536.	1.92922	0.964611	+	0.263677i	$0.0849352\pi$
$152$	60928.0	0.213899
$153$	15552.0	0.0537103
$154$	150528.	0.511464
$155$	−121440.	−0.406006
$156$	159264.	0.523970
$157$	−551494.	−1.78563	−0.892815	−	0.450423i	$-0.851273\pi$
$157$	−551494.	−1.78563	−0.892815	+	0.450423i	$0.851273\pi$
$158$	183184.	0.583774
$159$	−42174.0	−0.132298
$160$	61440.0	0.189737
$161$	−12768.0	−0.0388202
$162$	−26244.0	−0.0785674
$163$	−194308.	−0.572825	−0.286412	−	0.958106i	$-0.592463\pi$
$163$	−194308.	−0.572825	−0.286412	+	0.958106i	$0.592463\pi$
$164$	−250080.	−0.726055
$165$	−181440.	−0.518828
$166$	305184.	0.859590
$167$	−198114.	−0.549698	−0.274849	−	0.961487i	$-0.588628\pi$
$167$	−198114.	−0.549698	−0.274849	+	0.961487i	$0.588628\pi$
$168$	64512.0	0.176347
$169$	851943.	2.29453
$170$	46080.0	0.122290
$171$	−77112.0	−0.201666
$172$	−59968.0	−0.154560
$173$	−623334.	−1.58345	−0.791727	−	0.610875i	$-0.790818\pi$
$173$	−623334.	−1.58345	−0.791727	+	0.610875i	$0.790818\pi$
$174$	305424.	0.764769
$175$	−53200.0	−0.131316
$176$	86016.0	0.209314
$177$	−237330.	−0.569403
$178$	17232.0	0.0407648
$179$	−547074.	−1.27618	−0.638092	−	0.769960i	$-0.720276\pi$
$179$	−547074.	−1.27618	−0.638092	+	0.769960i	$0.720276\pi$
$180$	−77760.0	−0.178885
$181$	362810.	0.823157	0.411579	−	0.911374i	$-0.364978\pi$
$181$	362810.	0.823157	0.411579	+	0.911374i	$0.364978\pi$
$182$	495488.	1.10880
$183$	129078.	0.284921
$184$	−7296.00	−0.0158869
$185$	−82140.0	−0.176452
$186$	−72864.0	−0.154430
$187$	64512.0	0.134908
$188$	−18816.0	−0.0388269
$189$	−81648.0	−0.166261
$190$	−228480.	−0.459160
$191$	6438.00	0.0127693	0.00638466	−	0.999980i	$-0.497968\pi$
$191$	6438.00	0.0127693	0.00638466	+	0.999980i	$0.497968\pi$
$192$	36864.0	0.0721688
$193$	745802.	1.44122	0.720610	−	0.693341i	$-0.243862\pi$
$193$	745802.	1.44122	0.720610	+	0.693341i	$0.243862\pi$
$194$	478600.	0.912995
$195$	−597240.	−1.12477
$196$	−68208.0	−0.126822
$197$	−194802.	−0.357625	−0.178812	−	0.983883i	$-0.557226\pi$
$197$	−194802.	−0.357625	−0.178812	+	0.983883i	$0.557226\pi$
$198$	−108864.	−0.197343
$199$	−244816.	−0.438235	−0.219118	−	0.975698i	$-0.570318\pi$
$199$	−244816.	−0.438235	−0.219118	+	0.975698i	$0.570318\pi$
$200$	−30400.0	−0.0537401
$201$	−100692.	−0.175794
$202$	418344.	0.721365
$203$	950208.	1.61837
$204$	27648.0	0.0465145
$205$	937800.	1.55857
$206$	−15152.0	−0.0248772
$207$	9234.00	0.0149784
$208$	283136.	0.453771
$209$	−319872.	−0.506536
$210$	−241920.	−0.378550
$211$	117332.	0.181431	0.0907153	−	0.995877i	$-0.471085\pi$
$211$	117332.	0.181431	0.0907153	+	0.995877i	$0.471085\pi$
$212$	−74976.0	−0.114573
$213$	−279720.	−0.422449
$214$	58512.0	0.0873395
$215$	224880.	0.331783
$216$	−46656.0	−0.0680414
$217$	−226688.	−0.326798
$218$	−796904.	−1.13570
$219$	409878.	0.577490
$220$	−322560.	−0.449318
$221$	212352.	0.292466
$222$	−49284.0	−0.0671156
$223$	62096.0	0.0836183	0.0418092	−	0.999126i	$-0.486688\pi$
$223$	62096.0	0.0836183	0.0418092	+	0.999126i	$0.486688\pi$
$224$	114688.	0.152721
$225$	38475.0	0.0506667
$226$	114768.	0.149469
$227$	−1.53538e6	−1.97766	−0.988830	−	0.149045i	$-0.952380\pi$
$227$	−1.53538e6	−1.97766	−0.988830	+	0.149045i	$0.952380\pi$
$228$	−137088.	−0.174647
$229$	−565930.	−0.713139	−0.356569	−	0.934269i	$-0.616054\pi$
$229$	−565930.	−0.713139	−0.356569	+	0.934269i	$0.616054\pi$
$230$	27360.0	0.0341033
$231$	−338688.	−0.417609
$232$	542976.	0.662309
$233$	−1.07257e6	−1.29430	−0.647149	−	0.762364i	$-0.724039\pi$
$233$	−1.07257e6	−1.29430	−0.647149	+	0.762364i	$0.724039\pi$
$234$	−358344.	−0.427819
$235$	70560.0	0.0833468
$236$	−421920.	−0.493117
$237$	−412164.	−0.476650
$238$	86016.0	0.0984321
$239$	1.18283e6	1.33946	0.669729	−	0.742606i	$-0.266410\pi$
$239$	1.18283e6	1.33946	0.669729	+	0.742606i	$0.266410\pi$
$240$	−138240.	−0.154919
$241$	552398.	0.612646	0.306323	−	0.951928i	$-0.400901\pi$
$241$	552398.	0.612646	0.306323	+	0.951928i	$0.400901\pi$
$242$	192620.	0.211428
$243$	59049.0	0.0641500
$244$	229472.	0.246749
$245$	255780.	0.272240
$246$	562680.	0.592821
$247$	−1.05291e6	−1.09812
$248$	−129536.	−0.133740
$249$	−686664.	−0.701853
$250$	−636000.	−0.643587
$251$	−1.02998e6	−1.03191	−0.515957	−	0.856614i	$-0.672564\pi$
$251$	−1.02998e6	−1.03191	−0.515957	+	0.856614i	$0.672564\pi$
$252$	−145152.	−0.143986
$253$	38304.0	0.0376221
$254$	490528.	0.477067
$255$	−103680.	−0.0998492
$256$	65536.0	0.0625000
$257$	575244.	0.543274	0.271637	−	0.962400i	$-0.412435\pi$
$257$	575244.	0.543274	0.271637	+	0.962400i	$0.412435\pi$
$258$	134928.	0.126198
$259$	−153328.	−0.142027
$260$	−1.06176e6	−0.974076
$261$	−687204.	−0.624431
$262$	−829800.	−0.746827
$263$	1.91596e6	1.70803	0.854016	−	0.520246i	$-0.174160\pi$
$263$	1.91596e6	1.70803	0.854016	+	0.520246i	$0.174160\pi$
$264$	−193536.	−0.170904
$265$	281160.	0.245945
$266$	−426496.	−0.369582
$267$	−38772.0	−0.0332843
$268$	−179008.	−0.152242
$269$	921438.	0.776400	0.388200	−	0.921575i	$-0.373097\pi$
$269$	921438.	0.776400	0.388200	+	0.921575i	$0.373097\pi$
$270$	174960.	0.146059
$271$	9272.00	0.00766920	0.00383460	−	0.999993i	$-0.498779\pi$
$271$	9272.00	0.00766920	0.00383460	+	0.999993i	$0.498779\pi$
$272$	49152.0	0.0402827
$273$	−1.11485e6	−0.905334
$274$	45960.0	0.0369831
$275$	159600.	0.127263
$276$	16416.0	0.0129716
$277$	−1.15649e6	−0.905609	−0.452805	−	0.891610i	$-0.649577\pi$
$277$	−1.15649e6	−0.905609	−0.452805	+	0.891610i	$0.649577\pi$
$278$	294160.	0.228282
$279$	163944.	0.126091
$280$	−430080.	−0.327834
$281$	97764.0	0.0738607	0.0369303	−	0.999318i	$-0.488242\pi$
$281$	97764.0	0.0738607	0.0369303	+	0.999318i	$0.488242\pi$
$282$	42336.0	0.0317020
$283$	37448.0	0.0277947	0.0138974	−	0.999903i	$-0.495576\pi$
$283$	37448.0	0.0277947	0.0138974	+	0.999903i	$0.495576\pi$
$284$	−497280.	−0.365852
$285$	514080.	0.374903
$286$	−1.48646e6	−1.07458
$287$	1.75056e6	1.25450
$288$	−82944.0	−0.0589256
$289$	−1.38299e6	−0.974037
$290$	−2.03616e6	−1.42173
$291$	−1.07685e6	−0.745457
$292$	728672.	0.500121
$293$	410826.	0.279569	0.139784	−	0.990182i	$-0.455359\pi$
$293$	410826.	0.279569	0.139784	+	0.990182i	$0.455359\pi$
$294$	153468.	0.103550
$295$	1.58220e6	1.05854
$296$	−87616.0	−0.0581238
$297$	244944.	0.161130
$298$	328008.	0.213966
$299$	126084.	0.0815609
$300$	68400.0	0.0438786
$301$	419776.	0.267055
$302$	−2.16214e6	−1.36417
$303$	−941274.	−0.588992
$304$	−243712.	−0.151249
$305$	−860520.	−0.529677
$306$	−62208.0	−0.0379789
$307$	883100.	0.534766	0.267383	−	0.963590i	$-0.413841\pi$
$307$	883100.	0.534766	0.267383	+	0.963590i	$0.413841\pi$
$308$	−602112.	−0.361660
$309$	34092.0	0.0203122
$310$	485760.	0.287090
$311$	3.24866e6	1.90460	0.952298	−	0.305169i	$-0.0987128\pi$
$311$	3.24866e6	1.90460	0.952298	+	0.305169i	$0.0987128\pi$
$312$	−637056.	−0.370502
$313$	−1.02163e6	−0.589431	−0.294715	−	0.955585i	$-0.595225\pi$
$313$	−1.02163e6	−0.589431	−0.294715	+	0.955585i	$0.595225\pi$
$314$	2.20598e6	1.26263
$315$	544320.	0.309085
$316$	−732736.	−0.412791
$317$	2.03714e6	1.13860	0.569301	−	0.822129i	$-0.307214\pi$
$317$	2.03714e6	1.13860	0.569301	+	0.822129i	$0.307214\pi$
$318$	168696.	0.0935485
$319$	−2.85062e6	−1.56842
$320$	−245760.	−0.134164
$321$	−131652.	−0.0713124
$322$	51072.0	0.0274501
$323$	−182784.	−0.0974837
$324$	104976.	0.0555556
$325$	525350.	0.275893
$326$	777232.	0.405048
$327$	1.79303e6	0.927298
$328$	1.00032e6	0.513398
$329$	131712.	0.0670866
$330$	725760.	0.366867
$331$	3.67045e6	1.84140	0.920702	−	0.390266i	$-0.127617\pi$
$331$	3.67045e6	1.84140	0.920702	+	0.390266i	$0.127617\pi$
$332$	−1.22074e6	−0.607822
$333$	110889.	0.0547997
$334$	792456.	0.388695
$335$	671280.	0.326807
$336$	−258048.	−0.124696
$337$	−3.30225e6	−1.58393	−0.791965	−	0.610567i	$-0.790942\pi$
$337$	−3.30225e6	−1.58393	−0.791965	+	0.610567i	$0.790942\pi$
$338$	−3.40777e6	−1.62248
$339$	−258228.	−0.122041
$340$	−184320.	−0.0864719
$341$	680064.	0.316712
$342$	308448.	0.142599
$343$	2.35984e6	1.08305
$344$	239872.	0.109291
$345$	−61560.0	−0.0278452
$346$	2.49334e6	1.11967
$347$	544902.	0.242938	0.121469	−	0.992595i	$-0.461240\pi$
$347$	544902.	0.242938	0.121469	+	0.992595i	$0.461240\pi$
$348$	−1.22170e6	−0.540773
$349$	−1.83595e6	−0.806858	−0.403429	−	0.915011i	$-0.632182\pi$
$349$	−1.83595e6	−0.806858	−0.403429	+	0.915011i	$0.632182\pi$
$350$	212800.	0.0928542
$351$	806274.	0.349313
$352$	−344064.	−0.148007
$353$	−3.13151e6	−1.33757	−0.668785	−	0.743456i	$-0.733185\pi$
$353$	−3.13151e6	−1.33757	−0.668785	+	0.743456i	$0.733185\pi$
$354$	949320.	0.402628
$355$	1.86480e6	0.785347
$356$	−68928.0	−0.0288251
$357$	−193536.	−0.0803695
$358$	2.18830e6	0.902399
$359$	−1.73394e6	−0.710065	−0.355032	−	0.934854i	$-0.615530\pi$
$359$	−1.73394e6	−0.710065	−0.355032	+	0.934854i	$0.615530\pi$
$360$	311040.	0.126491
$361$	−1.56980e6	−0.633979
$362$	−1.45124e6	−0.582060
$363$	−433395.	−0.172630
$364$	−1.98195e6	−0.784043
$365$	−2.73252e6	−1.07357
$366$	−516312.	−0.201470
$367$	−843016.	−0.326716	−0.163358	−	0.986567i	$-0.552233\pi$
$367$	−843016.	−0.326716	−0.163358	+	0.986567i	$0.552233\pi$
$368$	29184.0	0.0112338
$369$	−1.26603e6	−0.484036
$370$	328560.	0.124770
$371$	524832.	0.197964
$372$	291456.	0.109198
$373$	3.29849e6	1.22756	0.613781	−	0.789477i	$-0.289648\pi$
$373$	3.29849e6	1.22756	0.613781	+	0.789477i	$0.289648\pi$
$374$	−258048.	−0.0953941
$375$	1.43100e6	0.525486
$376$	75264.0	0.0274548
$377$	−9.38330e6	−3.40018
$378$	326592.	0.117564
$379$	−5.08476e6	−1.81833	−0.909165	−	0.416436i	$-0.863279\pi$
$379$	−5.08476e6	−1.81833	−0.909165	+	0.416436i	$0.863279\pi$
$380$	913920.	0.324675
$381$	−1.10369e6	−0.389524
$382$	−25752.0	−0.00902927
$383$	1.40767e6	0.490346	0.245173	−	0.969479i	$-0.421155\pi$
$383$	1.40767e6	0.490346	0.245173	+	0.969479i	$0.421155\pi$
$384$	−147456.	−0.0510310
$385$	2.25792e6	0.776349
$386$	−2.98321e6	−1.01910
$387$	−303588.	−0.103040
$388$	−1.91440e6	−0.645585
$389$	2.92068e6	0.978611	0.489305	−	0.872113i	$-0.337250\pi$
$389$	2.92068e6	0.978611	0.489305	+	0.872113i	$0.337250\pi$
$390$	2.38896e6	0.795330
$391$	21888.0	0.00724043
$392$	272832.	0.0896768
$393$	1.86705e6	0.609782
$394$	779208.	0.252879
$395$	2.74776e6	0.886107
$396$	435456.	0.139542
$397$	−43702.0	−0.0139163	−0.00695817	−	0.999976i	$-0.502215\pi$
$397$	−43702.0	−0.0139163	−0.00695817	+	0.999976i	$0.502215\pi$
$398$	979264.	0.309879
$399$	959616.	0.301762
$400$	121600.	0.0380000
$401$	4.08998e6	1.27017	0.635083	−	0.772444i	$-0.280966\pi$
$401$	4.08998e6	1.27017	0.635083	+	0.772444i	$0.280966\pi$
$402$	402768.	0.124305
$403$	2.23854e6	0.686599
$404$	−1.67338e6	−0.510082
$405$	−393660.	−0.119257
$406$	−3.80083e6	−1.14436
$407$	459984.	0.137644
$408$	−110592.	−0.0328907
$409$	−4.57223e6	−1.35151	−0.675755	−	0.737126i	$-0.736182\pi$
$409$	−4.57223e6	−1.35151	−0.675755	+	0.737126i	$0.736182\pi$
$410$	−3.75120e6	−1.10207
$411$	−103410.	−0.0301966
$412$	60608.0	0.0175909
$413$	2.95344e6	0.852026
$414$	−36936.0	−0.0105913
$415$	4.57776e6	1.30477
$416$	−1.13254e6	−0.320865
$417$	−661860.	−0.186391
$418$	1.27949e6	0.358175
$419$	4.11206e6	1.14426	0.572130	−	0.820163i	$-0.306117\pi$
$419$	4.11206e6	1.14426	0.572130	+	0.820163i	$0.306117\pi$
$420$	967680.	0.267675
$421$	2.90691e6	0.799329	0.399665	−	0.916661i	$-0.369127\pi$
$421$	2.90691e6	0.799329	0.399665	+	0.916661i	$0.369127\pi$
$422$	−469328.	−0.128291
$423$	−95256.0	−0.0258846
$424$	299904.	0.0810154
$425$	91200.0	0.0244919
$426$	1.11888e6	0.298717
$427$	−1.60630e6	−0.426342
$428$	−234048.	−0.0617583
$429$	3.34454e6	0.877392
$430$	−899520.	−0.234606
$431$	−649110.	−0.168316	−0.0841579	−	0.996452i	$-0.526820\pi$
$431$	−649110.	−0.168316	−0.0841579	+	0.996452i	$0.526820\pi$
$432$	186624.	0.0481125
$433$	−4.17474e6	−1.07006	−0.535032	−	0.844832i	$-0.679700\pi$
$433$	−4.17474e6	−1.07006	−0.535032	+	0.844832i	$0.679700\pi$
$434$	906752.	0.231081
$435$	4.58136e6	1.16084
$436$	3.18762e6	0.803063
$437$	−108528.	−0.0271856
$438$	−1.63951e6	−0.408347
$439$	−4.97114e6	−1.23110	−0.615551	−	0.788097i	$-0.711067\pi$
$439$	−4.97114e6	−1.23110	−0.615551	+	0.788097i	$0.711067\pi$
$440$	1.29024e6	0.317716
$441$	−345303.	−0.0845481
$442$	−849408.	−0.206805
$443$	−3.68156e6	−0.891298	−0.445649	−	0.895208i	$-0.647027\pi$
$443$	−3.68156e6	−0.891298	−0.445649	+	0.895208i	$0.647027\pi$
$444$	197136.	0.0474579
$445$	258480.	0.0618767
$446$	−248384.	−0.0591271
$447$	−738018.	−0.174702
$448$	−458752.	−0.107990
$449$	8.18128e6	1.91516	0.957580	−	0.288167i	$-0.0930457\pi$
$449$	8.18128e6	1.91516	0.957580	+	0.288167i	$0.0930457\pi$
$450$	−153900.	−0.0358267
$451$	−5.25168e6	−1.21579
$452$	−459072.	−0.105690
$453$	4.86482e6	1.11384
$454$	6.14153e6	1.39842
$455$	7.43232e6	1.68305
$456$	548352.	0.123494
$457$	−1.61653e6	−0.362071	−0.181035	−	0.983477i	$-0.557945\pi$
$457$	−1.61653e6	−0.362071	−0.181035	+	0.983477i	$0.557945\pi$
$458$	2.26372e6	0.504265
$459$	139968.	0.0310097
$460$	−109440.	−0.0241147
$461$	−168960.	−0.0370281	−0.0185141	−	0.999829i	$-0.505894\pi$
$461$	−168960.	−0.0370281	−0.0185141	+	0.999829i	$0.505894\pi$
$462$	1.35475e6	0.295294
$463$	4.55098e6	0.986627	0.493313	−	0.869852i	$-0.335786\pi$
$463$	4.55098e6	0.986627	0.493313	+	0.869852i	$0.335786\pi$
$464$	−2.17190e6	−0.468323
$465$	−1.09296e6	−0.234408
$466$	4.29026e6	0.915207
$467$	7.12121e6	1.51099	0.755495	−	0.655154i	$-0.227396\pi$
$467$	7.12121e6	1.51099	0.755495	+	0.655154i	$0.227396\pi$
$468$	1.43338e6	0.302514
$469$	1.25306e6	0.263050
$470$	−282240.	−0.0589351
$471$	−4.96345e6	−1.03093
$472$	1.68768e6	0.348686
$473$	−1.25933e6	−0.258813
$474$	1.64866e6	0.337042
$475$	−452200.	−0.0919595
$476$	−344064.	−0.0696020
$477$	−379566.	−0.0763821
$478$	−4.73134e6	−0.947140
$479$	732606.	0.145892	0.0729460	−	0.997336i	$-0.476760\pi$
$479$	732606.	0.145892	0.0729460	+	0.997336i	$0.476760\pi$
$480$	552960.	0.109545
$481$	1.51411e6	0.298398
$482$	−2.20959e6	−0.433206
$483$	−114912.	−0.0224129
$484$	−770480.	−0.149502
$485$	7.17900e6	1.38583
$486$	−236196.	−0.0453609
$487$	−7.58144e6	−1.44854	−0.724268	−	0.689519i	$-0.757822\pi$
$487$	−7.58144e6	−1.44854	−0.724268	+	0.689519i	$0.757822\pi$
$488$	−917888.	−0.174478
$489$	−1.74877e6	−0.330720
$490$	−1.02312e6	−0.192503
$491$	−9.03742e6	−1.69177	−0.845883	−	0.533368i	$-0.820926\pi$
$491$	−9.03742e6	−1.69177	−0.845883	+	0.533368i	$0.820926\pi$
$492$	−2.25072e6	−0.419188
$493$	−1.62893e6	−0.301845
$494$	4.21165e6	0.776488
$495$	−1.63296e6	−0.299545
$496$	518144.	0.0945685
$497$	3.48096e6	0.632132
$498$	2.74666e6	0.496285
$499$	3.90554e6	0.702150	0.351075	−	0.936347i	$-0.385816\pi$
$499$	3.90554e6	0.702150	0.351075	+	0.936347i	$0.385816\pi$
$500$	2.54400e6	0.455085
$501$	−1.78303e6	−0.317368
$502$	4.11991e6	0.729674
$503$	7.75771e6	1.36714	0.683571	−	0.729884i	$-0.260426\pi$
$503$	7.75771e6	1.36714	0.683571	+	0.729884i	$0.260426\pi$
$504$	580608.	0.101814
$505$	6.27516e6	1.09496
$506$	−153216.	−0.0266028
$507$	7.66749e6	1.32475
$508$	−1.96211e6	−0.337337
$509$	1.52519e6	0.260934	0.130467	−	0.991453i	$-0.458352\pi$
$509$	1.52519e6	0.260934	0.130467	+	0.991453i	$0.458352\pi$
$510$	414720.	0.0706040
$511$	−5.10070e6	−0.864128
$512$	−262144.	−0.0441942
$513$	−694008.	−0.116432
$514$	−2.30098e6	−0.384153
$515$	−227280.	−0.0377610
$516$	−539712.	−0.0892355
$517$	−395136.	−0.0650160
$518$	613312.	0.100429
$519$	−5.61001e6	−0.914208
$520$	4.24704e6	0.688776
$521$	−439230.	−0.0708921	−0.0354460	−	0.999372i	$-0.511285\pi$
$521$	−439230.	−0.0708921	−0.0354460	+	0.999372i	$0.511285\pi$
$522$	2.74882e6	0.441539
$523$	7.85565e6	1.25582	0.627911	−	0.778285i	$-0.283910\pi$
$523$	7.85565e6	1.25582	0.627911	+	0.778285i	$0.283910\pi$
$524$	3.31920e6	0.528087
$525$	−478800.	−0.0758151
$526$	−7.66382e6	−1.20776
$527$	388608.	0.0609516
$528$	774144.	0.120847
$529$	−6.42335e6	−0.997981
$530$	−1.12464e6	−0.173910
$531$	−2.13597e6	−0.328745
$532$	1.70598e6	0.261334
$533$	−1.72868e7	−2.63570
$534$	155088.	0.0235356
$535$	877680.	0.132572
$536$	716032.	0.107652
$537$	−4.92367e6	−0.736806
$538$	−3.68575e6	−0.548998
$539$	−1.43237e6	−0.212365
$540$	−699840.	−0.103280
$541$	−1.02990e7	−1.51287	−0.756437	−	0.654066i	$-0.773062\pi$
$541$	−1.02990e7	−1.51287	−0.756437	+	0.654066i	$0.773062\pi$
$542$	−37088.0	−0.00542294
$543$	3.26529e6	0.475250
$544$	−196608.	−0.0284842
$545$	−1.19536e7	−1.72388
$546$	4.45939e6	0.640168
$547$	−1.48515e6	−0.212227	−0.106114	−	0.994354i	$-0.533841\pi$
$547$	−1.48515e6	−0.212227	−0.106114	+	0.994354i	$0.533841\pi$
$548$	−183840.	−0.0261510
$549$	1.16170e6	0.164499
$550$	−638400.	−0.0899883
$551$	8.07677e6	1.13334
$552$	−65664.0	−0.00917233
$553$	5.12915e6	0.713235
$554$	4.62594e6	0.640363
$555$	−739260.	−0.101874
$556$	−1.17664e6	−0.161420
$557$	−1.29462e7	−1.76809	−0.884045	−	0.467402i	$-0.845190\pi$
$557$	−1.29462e7	−1.76809	−0.884045	+	0.467402i	$0.845190\pi$
$558$	−655776.	−0.0891600
$559$	−4.14529e6	−0.561081
$560$	1.72032e6	0.231814
$561$	580608.	0.0778890
$562$	−391056.	−0.0522274
$563$	3.78879e6	0.503767	0.251883	−	0.967758i	$-0.418950\pi$
$563$	3.78879e6	0.503767	0.251883	+	0.967758i	$0.418950\pi$
$564$	−169344.	−0.0224167
$565$	1.72152e6	0.226877
$566$	−149792.	−0.0196538
$567$	−734832.	−0.0959910
$568$	1.98912e6	0.258696
$569$	−2.40760e6	−0.311747	−0.155874	−	0.987777i	$-0.549819\pi$
$569$	−2.40760e6	−0.311747	−0.155874	+	0.987777i	$0.549819\pi$
$570$	−2.05632e6	−0.265096
$571$	−7.06827e6	−0.907241	−0.453621	−	0.891195i	$-0.649868\pi$
$571$	−7.06827e6	−0.907241	−0.453621	+	0.891195i	$0.649868\pi$
$572$	5.94586e6	0.759844
$573$	57942.0	0.00737237
$574$	−7.00224e6	−0.887069
$575$	54150.0	0.00683013
$576$	331776.	0.0416667
$577$	−2.14574e6	−0.268311	−0.134155	−	0.990960i	$-0.542832\pi$
$577$	−2.14574e6	−0.268311	−0.134155	+	0.990960i	$0.542832\pi$
$578$	5.53197e6	0.688748
$579$	6.71222e6	0.832089
$580$	8.14464e6	1.00531
$581$	8.54515e6	1.05022
$582$	4.30740e6	0.527118
$583$	−1.57450e6	−0.191854
$584$	−2.91469e6	−0.353639
$585$	−5.37516e6	−0.649384
$586$	−1.64330e6	−0.197685
$587$	−1.11706e7	−1.33808	−0.669039	−	0.743227i	$-0.733294\pi$
$587$	−1.11706e7	−1.33808	−0.669039	+	0.743227i	$0.733294\pi$
$588$	−613872.	−0.0732208
$589$	−1.92685e6	−0.228854
$590$	−6.32880e6	−0.748499
$591$	−1.75322e6	−0.206475
$592$	350464.	0.0410997
$593$	1.08359e7	1.26540	0.632699	−	0.774398i	$-0.281947\pi$
$593$	1.08359e7	1.26540	0.632699	+	0.774398i	$0.281947\pi$
$594$	−979776.	−0.113936
$595$	1.29024e6	0.149409
$596$	−1.31203e6	−0.151296
$597$	−2.20334e6	−0.253015
$598$	−504336.	−0.0576723
$599$	−575016.	−0.0654806	−0.0327403	−	0.999464i	$-0.510423\pi$
$599$	−575016.	−0.0654806	−0.0327403	+	0.999464i	$0.510423\pi$
$600$	−273600.	−0.0310269
$601$	−9.79631e6	−1.10631	−0.553154	−	0.833079i	$-0.686576\pi$
$601$	−9.79631e6	−1.10631	−0.553154	+	0.833079i	$0.686576\pi$
$602$	−1.67910e6	−0.188837
$603$	−906228.	−0.101495
$604$	8.64858e6	0.964611
$605$	2.88930e6	0.320925
$606$	3.76510e6	0.416480
$607$	−1.97369e6	−0.217424	−0.108712	−	0.994073i	$-0.534673\pi$
$607$	−1.97369e6	−0.217424	−0.108712	+	0.994073i	$0.534673\pi$
$608$	974848.	0.106949
$609$	8.55187e6	0.934368
$610$	3.44208e6	0.374538
$611$	−1.30066e6	−0.140948
$612$	248832.	0.0268552
$613$	−2.91495e6	−0.313315	−0.156657	−	0.987653i	$-0.550072\pi$
$613$	−2.91495e6	−0.313315	−0.156657	+	0.987653i	$0.550072\pi$
$614$	−3.53240e6	−0.378137
$615$	8.44020e6	0.899839
$616$	2.40845e6	0.255732
$617$	−7.75024e6	−0.819601	−0.409800	−	0.912175i	$-0.634402\pi$
$617$	−7.75024e6	−0.819601	−0.409800	+	0.912175i	$0.634402\pi$
$618$	−136368.	−0.0143629
$619$	−1.24929e7	−1.31050	−0.655248	−	0.755414i	$-0.727436\pi$
$619$	−1.24929e7	−1.31050	−0.655248	+	0.755414i	$0.727436\pi$
$620$	−1.94304e6	−0.203003
$621$	83106.0	0.00864776
$622$	−1.29946e7	−1.34675
$623$	482496.	0.0498051
$624$	2.54822e6	0.261985
$625$	−1.10244e7	−1.12890
$626$	4.08652e6	0.416790
$627$	−2.87885e6	−0.292449
$628$	−8.82390e6	−0.892815
$629$	262848.	0.0264898
$630$	−2.17728e6	−0.218556
$631$	1.08052e7	1.08034	0.540169	−	0.841557i	$-0.318360\pi$
$631$	1.08052e7	1.08034	0.540169	+	0.841557i	$0.318360\pi$
$632$	2.93094e6	0.291887
$633$	1.05599e6	0.104749
$634$	−8.14855e6	−0.805114
$635$	7.35792e6	0.724137
$636$	−674784.	−0.0661488
$637$	−4.71488e6	−0.460386
$638$	1.14025e7	1.10904
$639$	−2.51748e6	−0.243901
$640$	983040.	0.0948683
$641$	−1.30548e7	−1.25494	−0.627472	−	0.778639i	$-0.715910\pi$
$641$	−1.30548e7	−1.25494	−0.627472	+	0.778639i	$0.715910\pi$
$642$	526608.	0.0504255
$643$	1.21503e7	1.15894	0.579468	−	0.814995i	$-0.303260\pi$
$643$	1.21503e7	1.15894	0.579468	+	0.814995i	$0.303260\pi$
$644$	−204288.	−0.0194101
$645$	2.02392e6	0.191555
$646$	731136.	0.0689314
$647$	1.67384e7	1.57200	0.786001	−	0.618226i	$-0.212148\pi$
$647$	1.67384e7	1.57200	0.786001	+	0.618226i	$0.212148\pi$
$648$	−419904.	−0.0392837
$649$	−8.86032e6	−0.825729
$650$	−2.10140e6	−0.195086
$651$	−2.04019e6	−0.188677
$652$	−3.10893e6	−0.286412
$653$	−2.31641e6	−0.212585	−0.106292	−	0.994335i	$-0.533898\pi$
$653$	−2.31641e6	−0.212585	−0.106292	+	0.994335i	$0.533898\pi$
$654$	−7.17214e6	−0.655698
$655$	−1.24470e7	−1.13360
$656$	−4.00128e6	−0.363027
$657$	3.68890e6	0.333414
$658$	−526848.	−0.0474374
$659$	1.12373e7	1.00797	0.503987	−	0.863711i	$-0.331866\pi$
$659$	1.12373e7	1.00797	0.503987	+	0.863711i	$0.331866\pi$
$660$	−2.90304e6	−0.259414
$661$	−4.09515e6	−0.364558	−0.182279	−	0.983247i	$-0.558347\pi$
$661$	−4.09515e6	−0.364558	−0.182279	+	0.983247i	$0.558347\pi$
$662$	−1.46818e7	−1.30207
$663$	1.91117e6	0.168855
$664$	4.88294e6	0.429795
$665$	−6.39744e6	−0.560986
$666$	−443556.	−0.0387492
$667$	−967176.	−0.0841765
$668$	−3.16982e6	−0.274849
$669$	558864.	0.0482771
$670$	−2.68512e6	−0.231088
$671$	4.81891e6	0.413183
$672$	1.03219e6	0.0881733
$673$	−1.24815e7	−1.06226	−0.531130	−	0.847290i	$-0.678232\pi$
$673$	−1.24815e7	−1.06226	−0.531130	+	0.847290i	$0.678232\pi$
$674$	1.32090e7	1.12001
$675$	346275.	0.0292524
$676$	1.36311e7	1.14727
$677$	30018.0	0.00251716	0.00125858	−	0.999999i	$-0.499599\pi$
$677$	30018.0	0.00251716	0.00125858	+	0.999999i	$0.499599\pi$
$678$	1.03291e6	0.0862957
$679$	1.34008e7	1.11547
$680$	737280.	0.0611449
$681$	−1.38184e7	−1.14180
$682$	−2.72026e6	−0.223949
$683$	−1.46873e7	−1.20473	−0.602367	−	0.798220i	$-0.705775\pi$
$683$	−1.46873e7	−1.20473	−0.602367	+	0.798220i	$0.705775\pi$
$684$	−1.23379e6	−0.100833
$685$	689400.	0.0561364
$686$	−9.43936e6	−0.765830
$687$	−5.09337e6	−0.411731
$688$	−959488.	−0.0772802
$689$	−5.18272e6	−0.415920
$690$	246240.	0.0196896
$691$	1.71028e7	1.36261	0.681306	−	0.731999i	$-0.261412\pi$
$691$	1.71028e7	1.36261	0.681306	+	0.731999i	$0.261412\pi$
$692$	−9.97334e6	−0.791727
$693$	−3.04819e6	−0.241107
$694$	−2.17961e6	−0.171783
$695$	4.41240e6	0.346508
$696$	4.88678e6	0.382384
$697$	−3.00096e6	−0.233980
$698$	7.34380e6	0.570535
$699$	−9.65309e6	−0.747263
$700$	−851200.	−0.0656578
$701$	−1.09524e7	−0.841808	−0.420904	−	0.907105i	$-0.638287\pi$
$701$	−1.09524e7	−0.841808	−0.420904	+	0.907105i	$0.638287\pi$
$702$	−3.22510e6	−0.247002
$703$	−1.30329e6	−0.0994608
$704$	1.37626e6	0.104657
$705$	635040.	0.0481203
$706$	1.25260e7	0.945805
$707$	1.17136e7	0.881339
$708$	−3.79728e6	−0.284701
$709$	7.22957e6	0.540128	0.270064	−	0.962842i	$-0.412955\pi$
$709$	7.22957e6	0.540128	0.270064	+	0.962842i	$0.412955\pi$
$710$	−7.45920e6	−0.555324
$711$	−3.70948e6	−0.275194
$712$	275712.	0.0203824
$713$	230736.	0.0169978
$714$	774144.	0.0568298
$715$	−2.22970e7	−1.63110
$716$	−8.75318e6	−0.638092
$717$	1.06455e7	0.773336
$718$	6.93576e6	0.502092
$719$	507972.	0.0366452	0.0183226	−	0.999832i	$-0.494167\pi$
$719$	507972.	0.0366452	0.0183226	+	0.999832i	$0.494167\pi$
$720$	−1.24416e6	−0.0894427
$721$	−424256.	−0.0303941
$722$	6.27918e6	0.448291
$723$	4.97158e6	0.353711
$724$	5.80496e6	0.411579
$725$	−4.02990e6	−0.284741
$726$	1.73358e6	0.122068
$727$	5.77458e6	0.405214	0.202607	−	0.979260i	$-0.435059\pi$
$727$	5.77458e6	0.405214	0.202607	+	0.979260i	$0.435059\pi$
$728$	7.92781e6	0.554402
$729$	531441.	0.0370370
$730$	1.09301e7	0.759130
$731$	−719616.	−0.0498089
$732$	2.06525e6	0.142461
$733$	−2.32028e7	−1.59507	−0.797536	−	0.603271i	$-0.793864\pi$
$733$	−2.32028e7	−1.59507	−0.797536	+	0.603271i	$0.793864\pi$
$734$	3.37206e6	0.231023
$735$	2.30202e6	0.157178
$736$	−116736.	−0.00794347
$737$	−3.75917e6	−0.254931
$738$	5.06412e6	0.342265
$739$	1.17221e7	0.789578	0.394789	−	0.918772i	$-0.370818\pi$
$739$	1.17221e7	0.789578	0.394789	+	0.918772i	$0.370818\pi$
$740$	−1.31424e6	−0.0882258
$741$	−9.47621e6	−0.634000
$742$	−2.09933e6	−0.139981
$743$	1.29736e7	0.862163	0.431081	−	0.902313i	$-0.358132\pi$
$743$	1.29736e7	0.862163	0.431081	+	0.902313i	$0.358132\pi$
$744$	−1.16582e6	−0.0772148
$745$	4.92012e6	0.324777
$746$	−1.31940e7	−0.868017
$747$	−6.17998e6	−0.405215
$748$	1.03219e6	0.0674538
$749$	1.63834e6	0.106708
$750$	−5.72400e6	−0.371575
$751$	2.32702e7	1.50557	0.752784	−	0.658268i	$-0.228711\pi$
$751$	2.32702e7	1.50557	0.752784	+	0.658268i	$0.228711\pi$
$752$	−301056.	−0.0194134
$753$	−9.26980e6	−0.595776
$754$	3.75332e7	2.40429
$755$	−3.24322e7	−2.07066
$756$	−1.30637e6	−0.0831306
$757$	−2.10731e7	−1.33656	−0.668280	−	0.743909i	$-0.732969\pi$
$757$	−2.10731e7	−1.33656	−0.668280	+	0.743909i	$0.732969\pi$
$758$	2.03391e7	1.28575
$759$	344736.	0.0217211
$760$	−3.65568e6	−0.229580
$761$	8.42463e6	0.527338	0.263669	−	0.964613i	$-0.415067\pi$
$761$	8.42463e6	0.527338	0.263669	+	0.964613i	$0.415067\pi$
$762$	4.41475e6	0.275435
$763$	−2.23133e7	−1.38756
$764$	103008.	0.00638466
$765$	−933120.	−0.0576480
$766$	−5.63066e6	−0.346727
$767$	−2.91652e7	−1.79010
$768$	589824.	0.0360844
$769$	2.04689e7	1.24818	0.624092	−	0.781351i	$-0.285469\pi$
$769$	2.04689e7	1.24818	0.624092	+	0.781351i	$0.285469\pi$
$770$	−9.03168e6	−0.548961
$771$	5.17720e6	0.313660
$772$	1.19328e7	0.720610
$773$	1.08012e7	0.650166	0.325083	−	0.945685i	$-0.394608\pi$
$773$	1.08012e7	0.650166	0.325083	+	0.945685i	$0.394608\pi$
$774$	1.21435e6	0.0728605
$775$	961400.	0.0574976
$776$	7.65760e6	0.456497
$777$	−1.37995e6	−0.0819995
$778$	−1.16827e7	−0.691982
$779$	1.48798e7	0.878521
$780$	−9.55584e6	−0.562383
$781$	−1.04429e7	−0.612622
$782$	−87552.0	−0.00511976
$783$	−6.18484e6	−0.360515
$784$	−1.09133e6	−0.0634111
$785$	3.30896e7	1.91654
$786$	−7.46820e6	−0.431181
$787$	1.82811e7	1.05212	0.526061	−	0.850447i	$-0.323668\pi$
$787$	1.82811e7	1.05212	0.526061	+	0.850447i	$0.323668\pi$
$788$	−3.11683e6	−0.178812
$789$	1.72436e7	0.986133
$790$	−1.09910e7	−0.626572
$791$	3.21350e6	0.182615
$792$	−1.74182e6	−0.0986714
$793$	1.58623e7	0.895740
$794$	174808.	0.00984034
$795$	2.53044e6	0.141997
$796$	−3.91706e6	−0.219118
$797$	2.36553e7	1.31911	0.659557	−	0.751655i	$-0.270744\pi$
$797$	2.36553e7	1.31911	0.659557	+	0.751655i	$0.270744\pi$
$798$	−3.83846e6	−0.213378
$799$	−225792.	−0.0125124
$800$	−486400.	−0.0268701
$801$	−348948.	−0.0192167
$802$	−1.63599e7	−0.898143
$803$	1.53021e7	0.837457
$804$	−1.61107e6	−0.0878972
$805$	766080.	0.0416663
$806$	−8.95418e6	−0.485499
$807$	8.29294e6	0.448255
$808$	6.69350e6	0.360683
$809$	3.04745e7	1.63706	0.818530	−	0.574464i	$-0.194789\pi$
$809$	3.04745e7	1.63706	0.818530	+	0.574464i	$0.194789\pi$
$810$	1.57464e6	0.0843274
$811$	2.23010e7	1.19062	0.595309	−	0.803497i	$-0.297030\pi$
$811$	2.23010e7	1.19062	0.595309	+	0.803497i	$0.297030\pi$
$812$	1.52033e7	0.809186
$813$	83448.0	0.00442782
$814$	−1.83994e6	−0.0973289
$815$	1.16585e7	0.614820
$816$	442368.	0.0232572
$817$	3.56810e6	0.187017
$818$	1.82889e7	0.955662
$819$	−1.00336e7	−0.522695
$820$	1.50048e7	0.779284
$821$	−1.26099e6	−0.0652911	−0.0326455	−	0.999467i	$-0.510393\pi$
$821$	−1.26099e6	−0.0652911	−0.0326455	+	0.999467i	$0.510393\pi$
$822$	413640.	0.0213522
$823$	−1.57113e7	−0.808558	−0.404279	−	0.914636i	$-0.632478\pi$
$823$	−1.57113e7	−0.808558	−0.404279	+	0.914636i	$0.632478\pi$
$824$	−242432.	−0.0124386
$825$	1.43640e6	0.0734752
$826$	−1.18138e7	−0.602474
$827$	1.09705e7	0.557780	0.278890	−	0.960323i	$-0.410034\pi$
$827$	1.09705e7	0.557780	0.278890	+	0.960323i	$0.410034\pi$
$828$	147744.	0.00748918
$829$	2.21798e7	1.12091	0.560456	−	0.828184i	$-0.310626\pi$
$829$	2.21798e7	1.12091	0.560456	+	0.828184i	$0.310626\pi$
$830$	−1.83110e7	−0.922609
$831$	−1.04084e7	−0.522854
$832$	4.53018e6	0.226886
$833$	−818496.	−0.0408699
$834$	2.64744e6	0.131799
$835$	1.18868e7	0.589998
$836$	−5.11795e6	−0.253268
$837$	1.47550e6	0.0727988
$838$	−1.64483e7	−0.809114
$839$	1.53519e7	0.752935	0.376467	−	0.926430i	$-0.377139\pi$
$839$	1.53519e7	0.752935	0.376467	+	0.926430i	$0.377139\pi$
$840$	−3.87072e6	−0.189275
$841$	5.14671e7	2.50923
$842$	−1.16276e7	−0.565211
$843$	879876.	0.0426435
$844$	1.87731e6	0.0907153
$845$	−5.11166e7	−2.46275
$846$	381024.	0.0183032
$847$	5.39336e6	0.258316
$848$	−1.19962e6	−0.0572865
$849$	337032.	0.0160473
$850$	−364800.	−0.0173184
$851$	156066.	0.00738728
$852$	−4.47552e6	−0.211225
$853$	1.81072e7	0.852074	0.426037	−	0.904706i	$-0.359909\pi$
$853$	1.81072e7	0.852074	0.426037	+	0.904706i	$0.359909\pi$
$854$	6.42522e6	0.301469
$855$	4.62672e6	0.216450
$856$	936192.	0.0436697
$857$	1.02686e7	0.477593	0.238797	−	0.971070i	$-0.423247\pi$
$857$	1.02686e7	0.477593	0.238797	+	0.971070i	$0.423247\pi$
$858$	−1.33782e7	−0.620410
$859$	3.64904e7	1.68731	0.843656	−	0.536884i	$-0.180399\pi$
$859$	3.64904e7	1.68731	0.843656	+	0.536884i	$0.180399\pi$
$860$	3.59808e6	0.165892
$861$	1.57550e7	0.724289
$862$	2.59644e6	0.119017
$863$	−3.49319e7	−1.59660	−0.798298	−	0.602263i	$-0.794266\pi$
$863$	−3.49319e7	−1.59660	−0.798298	+	0.602263i	$0.794266\pi$
$864$	−746496.	−0.0340207
$865$	3.74000e7	1.69954
$866$	1.66990e7	0.756649
$867$	−1.24469e7	−0.562360
$868$	−3.62701e6	−0.163399
$869$	−1.53875e7	−0.691222
$870$	−1.83254e7	−0.820836
$871$	−1.23739e7	−0.552665
$872$	−1.27505e7	−0.567851
$873$	−9.69165e6	−0.430390
$874$	434112.	0.0192231
$875$	−1.78080e7	−0.786312
$876$	6.55805e6	0.288745
$877$	3.54393e7	1.55592	0.777958	−	0.628317i	$-0.216256\pi$
$877$	3.54393e7	1.55592	0.777958	+	0.628317i	$0.216256\pi$
$878$	1.98845e7	0.870521
$879$	3.69743e6	0.161409
$880$	−5.16096e6	−0.224659
$881$	−6.63184e6	−0.287869	−0.143934	−	0.989587i	$-0.545975\pi$
$881$	−6.63184e6	−0.287869	−0.143934	+	0.989587i	$0.545975\pi$
$882$	1.38121e6	0.0597845
$883$	2.96209e7	1.27849	0.639243	−	0.769005i	$-0.279248\pi$
$883$	2.96209e7	1.27849	0.639243	+	0.769005i	$0.279248\pi$
$884$	3.39763e6	0.146233
$885$	1.42398e7	0.611147
$886$	1.47263e7	0.630243
$887$	1.33538e7	0.569897	0.284949	−	0.958543i	$-0.408023\pi$
$887$	1.33538e7	0.569897	0.284949	+	0.958543i	$0.408023\pi$
$888$	−788544.	−0.0335578
$889$	1.37348e7	0.582864
$890$	−1.03392e6	−0.0437534
$891$	2.20450e6	0.0930283
$892$	993536.	0.0418092
$893$	1.11955e6	0.0469803
$894$	2.95207e6	0.123533
$895$	3.28244e7	1.36975
$896$	1.83501e6	0.0763604
$897$	1.13476e6	0.0470892
$898$	−3.27251e7	−1.35422
$899$	−1.71716e7	−0.708618
$900$	615600.	0.0253333
$901$	−899712.	−0.0369225
$902$	2.10067e7	0.859690
$903$	3.77798e6	0.154185
$904$	1.83629e6	0.0747343
$905$	−2.17686e7	−0.883505
$906$	−1.94593e7	−0.787602
$907$	−2.49718e6	−0.100793	−0.0503966	−	0.998729i	$-0.516049\pi$
$907$	−2.49718e6	−0.100793	−0.0503966	+	0.998729i	$0.516049\pi$
$908$	−2.45661e7	−0.988830
$909$	−8.47147e6	−0.340055
$910$	−2.97293e7	−1.19009
$911$	−2.58450e7	−1.03176	−0.515882	−	0.856660i	$-0.672536\pi$
$911$	−2.58450e7	−1.03176	−0.515882	+	0.856660i	$0.672536\pi$
$912$	−2.19341e6	−0.0873237
$913$	−2.56355e7	−1.01780
$914$	6.46612e6	0.256023
$915$	−7.74468e6	−0.305809
$916$	−9.05488e6	−0.356569
$917$	−2.32344e7	−0.912448
$918$	−559872.	−0.0219271
$919$	3.15894e7	1.23382	0.616911	−	0.787033i	$-0.288384\pi$
$919$	3.15894e7	1.23382	0.616911	+	0.787033i	$0.288384\pi$
$920$	437760.	0.0170517
$921$	7.94790e6	0.308747
$922$	675840.	0.0261828
$923$	−3.43745e7	−1.32810
$924$	−5.41901e6	−0.208805
$925$	650275.	0.0249886
$926$	−1.82039e7	−0.697650
$927$	306828.	0.0117272
$928$	8.68762e6	0.331155
$929$	2.80142e7	1.06497	0.532487	−	0.846438i	$-0.321257\pi$
$929$	2.80142e7	1.06497	0.532487	+	0.846438i	$0.321257\pi$
$930$	4.37184e6	0.165751
$931$	4.05838e6	0.153454
$932$	−1.71611e7	−0.647149
$933$	2.92379e7	1.09962
$934$	−2.84849e7	−1.06843
$935$	−3.87072e6	−0.144798
$936$	−5.73350e6	−0.213910
$937$	−3.07827e7	−1.14540	−0.572700	−	0.819765i	$-0.694104\pi$
$937$	−3.07827e7	−1.14540	−0.572700	+	0.819765i	$0.694104\pi$
$938$	−5.01222e6	−0.186004
$939$	−9.19467e6	−0.340308
$940$	1.12896e6	0.0416734
$941$	1.39029e7	0.511838	0.255919	−	0.966698i	$-0.417622\pi$
$941$	1.39029e7	0.511838	0.255919	+	0.966698i	$0.417622\pi$
$942$	1.98538e7	0.728981
$943$	−1.78182e6	−0.0652506
$944$	−6.75072e6	−0.246559
$945$	4.89888e6	0.178450
$946$	5.03731e6	0.183008
$947$	−2.30409e7	−0.834882	−0.417441	−	0.908704i	$-0.637073\pi$
$947$	−2.30409e7	−0.834882	−0.417441	+	0.908704i	$0.637073\pi$
$948$	−6.59462e6	−0.238325
$949$	5.03695e7	1.81552
$950$	1.80880e6	0.0650252
$951$	1.83342e7	0.657373
$952$	1.37626e6	0.0492161
$953$	−4.79253e7	−1.70935	−0.854677	−	0.519159i	$-0.826245\pi$
$953$	−4.79253e7	−1.70935	−0.854677	+	0.519159i	$0.826245\pi$
$954$	1.51826e6	0.0540103
$955$	−386280.	−0.0137055
$956$	1.89253e7	0.669729
$957$	−2.56556e7	−0.905529
$958$	−2.93042e6	−0.103161
$959$	1.28688e6	0.0451847
$960$	−2.21184e6	−0.0774597
$961$	−2.45326e7	−0.856909
$962$	−6.05646e6	−0.210999
$963$	−1.18487e6	−0.0411722
$964$	8.83837e6	0.306323
$965$	−4.47481e7	−1.54688
$966$	459648.	0.0158483
$967$	4.48357e7	1.54191	0.770953	−	0.636892i	$-0.219780\pi$
$967$	4.48357e7	1.54191	0.770953	+	0.636892i	$0.219780\pi$
$968$	3.08192e6	0.105714
$969$	−1.64506e6	−0.0562822
$970$	−2.87160e7	−0.979929
$971$	5.45714e6	0.185745	0.0928725	−	0.995678i	$-0.470395\pi$
$971$	5.45714e6	0.185745	0.0928725	+	0.995678i	$0.470395\pi$
$972$	944784.	0.0320750
$973$	8.23648e6	0.278907
$974$	3.03257e7	1.02427
$975$	4.72815e6	0.159287
$976$	3.67155e6	0.123374
$977$	−4.32190e7	−1.44857	−0.724283	−	0.689503i	$-0.757829\pi$
$977$	−4.32190e7	−1.44857	−0.724283	+	0.689503i	$0.757829\pi$
$978$	6.99509e6	0.233855
$979$	−1.44749e6	−0.0482679
$980$	4.09248e6	0.136120
$981$	1.61373e7	0.535376
$982$	3.61497e7	1.19626
$983$	6.92920e6	0.228717	0.114359	−	0.993440i	$-0.463519\pi$
$983$	6.92920e6	0.228717	0.114359	+	0.993440i	$0.463519\pi$
$984$	9.00288e6	0.296411
$985$	1.16881e7	0.383843
$986$	6.51571e6	0.213437
$987$	1.18541e6	0.0387325
$988$	−1.68466e7	−0.549060
$989$	−427272.	−0.0138904
$990$	6.53184e6	0.211811
$991$	−3.77607e7	−1.22139	−0.610697	−	0.791864i	$-0.709111\pi$
$991$	−3.77607e7	−1.22139	−0.610697	+	0.791864i	$0.709111\pi$
$992$	−2.07258e6	−0.0668700
$993$	3.30340e7	1.06314
$994$	−1.39238e7	−0.446985
$995$	1.46890e7	0.470363
$996$	−1.09866e7	−0.350926
$997$	−3.58129e7	−1.14104	−0.570520	−	0.821284i	$-0.693258\pi$
$997$	−3.58129e7	−1.14104	−0.570520	+	0.821284i	$0.693258\pi$
$998$	−1.56222e7	−0.496495
$999$	998001.	0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	222.6.a.a.1.1	✓	1
3.2	odd	2		666.6.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
222.6.a.a.1.1	✓	1	1.1	even	1	trivial
666.6.a.d.1.1		1	3.2	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:	\( N \)	\(=\)	\( 222 = 2 \cdot 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	222.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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