LMFDB - Embedded newform 22.8.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [22,8,Mod(1,22)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(22, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("22.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 22 = 2 \cdot 11 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	22.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:	$6.87247056065$
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	$\chi$	$=$	22.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+8.00000 q^{2} -21.0000 q^{3} +64.0000 q^{4} -551.000 q^{5} -168.000 q^{6} +62.0000 q^{7} +512.000 q^{8} -1746.00 q^{9} +O(q^{10})$

\(q+8.00000 q^{2} -21.0000 q^{3} +64.0000 q^{4} -551.000 q^{5} -168.000 q^{6} +62.0000 q^{7} +512.000 q^{8} -1746.00 q^{9} -4408.00 q^{10} -1331.00 q^{11} -1344.00 q^{12} +1500.00 q^{13} +496.000 q^{14} +11571.0 q^{15} +4096.00 q^{16} -29930.0 q^{17} -13968.0 q^{18} +29512.0 q^{19} -35264.0 q^{20} -1302.00 q^{21} -10648.0 q^{22} +31499.0 q^{23} -10752.0 q^{24} +225476. q^{25} +12000.0 q^{26} +82593.0 q^{27} +3968.00 q^{28} -75168.0 q^{29} +92568.0 q^{30} -235845. q^{31} +32768.0 q^{32} +27951.0 q^{33} -239440. q^{34} -34162.0 q^{35} -111744. q^{36} +75507.0 q^{37} +236096. q^{38} -31500.0 q^{39} -282112. q^{40} -270288. q^{41} -10416.0 q^{42} -1.02803e6 q^{43} -85184.0 q^{44} +962046. q^{45} +251992. q^{46} -771840. q^{47} -86016.0 q^{48} -819699. q^{49} +1.80381e6 q^{50} +628530. q^{51} +96000.0 q^{52} +765778. q^{53} +660744. q^{54} +733381. q^{55} +31744.0 q^{56} -619752. q^{57} -601344. q^{58} -392007. q^{59} +740544. q^{60} +1.24846e6 q^{61} -1.88676e6 q^{62} -108252. q^{63} +262144. q^{64} -826500. q^{65} +223608. q^{66} +3.49813e6 q^{67} -1.91552e6 q^{68} -661479. q^{69} -273296. q^{70} +1.10175e6 q^{71} -893952. q^{72} -1.12300e6 q^{73} +604056. q^{74} -4.73500e6 q^{75} +1.88877e6 q^{76} -82522.0 q^{77} -252000. q^{78} -4.36295e6 q^{79} -2.25690e6 q^{80} +2.08405e6 q^{81} -2.16230e6 q^{82} -4.43779e6 q^{83} -83328.0 q^{84} +1.64914e7 q^{85} -8.22424e6 q^{86} +1.57853e6 q^{87} -681472. q^{88} -521233. q^{89} +7.69637e6 q^{90} +93000.0 q^{91} +2.01594e6 q^{92} +4.95275e6 q^{93} -6.17472e6 q^{94} -1.62611e7 q^{95} -688128. q^{96} -2.12983e6 q^{97} -6.55759e6 q^{98} +2.32393e6 q^{99} +O(q^{100})\)

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 $

$2$	8.00000	0.707107
$2$	8.00000	0.707107
$3$	−21.0000	−0.449050	−0.224525	−	0.974468i	$-0.572083\pi$
$3$	−21.0000	−0.449050	−0.224525	+	0.974468i	$0.572083\pi$
$4$	64.0000	0.500000
$5$	−551.000	−1.97132	−0.985659	−	0.168751i	$-0.946027\pi$
$5$	−551.000	−1.97132	−0.985659	+	0.168751i	$0.946027\pi$
$6$	−168.000	−0.317526
$7$	62.0000	0.0683201	0.0341601	−	0.999416i	$-0.489124\pi$
$7$	62.0000	0.0683201	0.0341601	+	0.999416i	$0.489124\pi$
$8$	512.000	0.353553
$9$	−1746.00	−0.798354
$10$	−4408.00	−1.39393
$11$	−1331.00	−0.301511
$11$	−1331.00	−0.301511
$12$	−1344.00	−0.224525
$13$	1500.00	0.189361	0.0946803	−	0.995508i	$-0.469817\pi$
$13$	1500.00	0.189361	0.0946803	+	0.995508i	$0.469817\pi$
$14$	496.000	0.0483096
$15$	11571.0	0.885221
$16$	4096.00	0.250000
$17$	−29930.0	−1.47753	−0.738764	−	0.673965i	$-0.764590\pi$
$17$	−29930.0	−1.47753	−0.738764	+	0.673965i	$0.764590\pi$
$18$	−13968.0	−0.564521
$19$	29512.0	0.987100	0.493550	−	0.869717i	$-0.335699\pi$
$19$	29512.0	0.987100	0.493550	+	0.869717i	$0.335699\pi$
$20$	−35264.0	−0.985659
$21$	−1302.00	−0.0306792
$22$	−10648.0	−0.213201
$23$	31499.0	0.539820	0.269910	−	0.962885i	$-0.413006\pi$
$23$	31499.0	0.539820	0.269910	+	0.962885i	$0.413006\pi$
$24$	−10752.0	−0.158763
$25$	225476.	2.88609
$26$	12000.0	0.133898
$27$	82593.0	0.807551
$28$	3968.00	0.0341601
$29$	−75168.0	−0.572321	−0.286161	−	0.958182i	$-0.592379\pi$
$29$	−75168.0	−0.572321	−0.286161	+	0.958182i	$0.592379\pi$
$30$	92568.0	0.625945
$31$	−235845.	−1.42187	−0.710936	−	0.703256i	$-0.751729\pi$
$31$	−235845.	−1.42187	−0.710936	+	0.703256i	$0.751729\pi$
$32$	32768.0	0.176777
$33$	27951.0	0.135394
$34$	−239440.	−1.04477
$35$	−34162.0	−0.134681
$36$	−111744.	−0.399177
$37$	75507.0	0.245065	0.122532	−	0.992465i	$-0.460898\pi$
$37$	75507.0	0.245065	0.122532	+	0.992465i	$0.460898\pi$
$38$	236096.	0.697985
$39$	−31500.0	−0.0850324
$40$	−282112.	−0.696966
$41$	−270288.	−0.612468	−0.306234	−	0.951956i	$-0.599069\pi$
$41$	−270288.	−0.612468	−0.306234	+	0.951956i	$0.599069\pi$
$42$	−10416.0	−0.0216934
$43$	−1.02803e6	−1.97182	−0.985908	−	0.167291i	$-0.946498\pi$
$43$	−1.02803e6	−1.97182	−0.985908	+	0.167291i	$0.946498\pi$
$44$	−85184.0	−0.150756
$45$	962046.	1.57381
$46$	251992.	0.381711
$47$	−771840.	−1.08439	−0.542194	−	0.840253i	$-0.682406\pi$
$47$	−771840.	−1.08439	−0.542194	+	0.840253i	$0.682406\pi$
$48$	−86016.0	−0.112263
$49$	−819699.	−0.995332
$50$	1.80381e6	2.04078
$51$	628530.	0.663484
$52$	96000.0	0.0946803
$53$	765778.	0.706541	0.353270	−	0.935521i	$-0.385070\pi$
$53$	765778.	0.706541	0.353270	+	0.935521i	$0.385070\pi$
$54$	660744.	0.571025
$55$	733381.	0.594375
$56$	31744.0	0.0241548
$57$	−619752.	−0.443257
$58$	−601344.	−0.404692
$59$	−392007.	−0.248492	−0.124246	−	0.992251i	$-0.539651\pi$
$59$	−392007.	−0.248492	−0.124246	+	0.992251i	$0.539651\pi$
$60$	740544.	0.442610
$61$	1.24846e6	0.704239	0.352120	−	0.935955i	$-0.385461\pi$
$61$	1.24846e6	0.704239	0.352120	+	0.935955i	$0.385461\pi$
$62$	−1.88676e6	−1.00542
$63$	−108252.	−0.0545436
$64$	262144.	0.125000
$65$	−826500.	−0.373290
$66$	223608.	0.0957378
$67$	3.49813e6	1.42094	0.710468	−	0.703730i	$-0.248483\pi$
$67$	3.49813e6	1.42094	0.710468	+	0.703730i	$0.248483\pi$
$68$	−1.91552e6	−0.738764
$69$	−661479.	−0.242406
$70$	−273296.	−0.0952336
$71$	1.10175e6	0.365326	0.182663	−	0.983176i	$-0.441528\pi$
$71$	1.10175e6	0.365326	0.182663	+	0.983176i	$0.441528\pi$
$72$	−893952.	−0.282261
$73$	−1.12300e6	−0.337869	−0.168934	−	0.985627i	$-0.554033\pi$
$73$	−1.12300e6	−0.337869	−0.168934	+	0.985627i	$0.554033\pi$
$74$	604056.	0.173287
$75$	−4.73500e6	−1.29600
$76$	1.88877e6	0.493550
$77$	−82522.0	−0.0205993
$78$	−252000.	−0.0601270
$79$	−4.36295e6	−0.995600	−0.497800	−	0.867292i	$-0.665859\pi$
$79$	−4.36295e6	−0.995600	−0.497800	+	0.867292i	$0.665859\pi$
$80$	−2.25690e6	−0.492829
$81$	2.08405e6	0.435723
$82$	−2.16230e6	−0.433080
$83$	−4.43779e6	−0.851909	−0.425955	−	0.904744i	$-0.640062\pi$
$83$	−4.43779e6	−0.851909	−0.425955	+	0.904744i	$0.640062\pi$
$84$	−83328.0	−0.0153396
$85$	1.64914e7	2.91268
$86$	−8.22424e6	−1.39428
$87$	1.57853e6	0.257001
$88$	−681472.	−0.106600
$89$	−521233.	−0.0783731	−0.0391865	−	0.999232i	$-0.512477\pi$
$89$	−521233.	−0.0783731	−0.0391865	+	0.999232i	$0.512477\pi$
$90$	7.69637e6	1.11285
$91$	93000.0	0.0129371
$92$	2.01594e6	0.269910
$93$	4.95275e6	0.638492
$94$	−6.17472e6	−0.766778
$95$	−1.62611e7	−1.94589
$96$	−688128.	−0.0793816
$97$	−2.12983e6	−0.236943	−0.118472	−	0.992957i	$-0.537799\pi$
$97$	−2.12983e6	−0.236943	−0.118472	+	0.992957i	$0.537799\pi$
$98$	−6.55759e6	−0.703806
$99$	2.32393e6	0.240713
$100$	1.44305e7	1.44305
$101$	1.24266e7	1.20013	0.600063	−	0.799953i	$-0.295142\pi$
$101$	1.24266e7	1.20013	0.600063	+	0.799953i	$0.295142\pi$
$102$	5.02824e6	0.469154
$103$	−9.36718e6	−0.844653	−0.422327	−	0.906444i	$-0.638786\pi$
$103$	−9.36718e6	−0.844653	−0.422327	+	0.906444i	$0.638786\pi$
$104$	768000.	0.0669491
$105$	717402.	0.0604784
$106$	6.12622e6	0.499600
$107$	1.10211e7	0.869724	0.434862	−	0.900497i	$-0.356797\pi$
$107$	1.10211e7	0.869724	0.434862	+	0.900497i	$0.356797\pi$
$108$	5.28595e6	0.403776
$109$	1.97072e7	1.45758	0.728790	−	0.684737i	$-0.240083\pi$
$109$	1.97072e7	1.45758	0.728790	+	0.684737i	$0.240083\pi$
$110$	5.86705e6	0.420286
$111$	−1.58565e6	−0.110046
$112$	253952.	0.0170800
$113$	−1.23988e7	−0.808359	−0.404180	−	0.914680i	$-0.632443\pi$
$113$	−1.23988e7	−0.808359	−0.404180	+	0.914680i	$0.632443\pi$
$114$	−4.95802e6	−0.313430
$115$	−1.73559e7	−1.06416
$116$	−4.81075e6	−0.286161
$117$	−2.61900e6	−0.151177
$118$	−3.13606e6	−0.175710
$119$	−1.85566e6	−0.100945
$120$	5.92435e6	0.312973
$121$	1.77156e6	0.0909091
$122$	9.98768e6	0.497972
$123$	5.67605e6	0.275029
$124$	−1.50941e7	−0.710936
$125$	−8.11904e7	−3.71809
$126$	−866016.	−0.0385682
$127$	3.23235e7	1.40025	0.700124	−	0.714021i	$-0.253128\pi$
$127$	3.23235e7	1.40025	0.700124	+	0.714021i	$0.253128\pi$
$128$	2.09715e6	0.0883883
$129$	2.15886e7	0.885444
$130$	−6.61200e6	−0.263956
$131$	−3.72281e7	−1.44684	−0.723422	−	0.690406i	$-0.757432\pi$
$131$	−3.72281e7	−1.44684	−0.723422	+	0.690406i	$0.757432\pi$
$132$	1.78886e6	0.0676969
$133$	1.82974e6	0.0674388
$134$	2.79851e7	1.00475
$135$	−4.55087e7	−1.59194
$136$	−1.53242e7	−0.522385
$137$	7.60704e6	0.252752	0.126376	−	0.991982i	$-0.459665\pi$
$137$	7.60704e6	0.252752	0.126376	+	0.991982i	$0.459665\pi$
$138$	−5.29183e6	−0.171407
$139$	1.51140e7	0.477339	0.238670	−	0.971101i	$-0.423289\pi$
$139$	1.51140e7	0.477339	0.238670	+	0.971101i	$0.423289\pi$
$140$	−2.18637e6	−0.0673403
$141$	1.62086e7	0.486945
$142$	8.81402e6	0.258324
$143$	−1.99650e6	−0.0570944
$144$	−7.15162e6	−0.199588
$145$	4.14176e7	1.12823
$146$	−8.98397e6	−0.238909
$147$	1.72137e7	0.446954
$148$	4.83245e6	0.122532
$149$	−3.96300e7	−0.981458	−0.490729	−	0.871312i	$-0.663269\pi$
$149$	−3.96300e7	−0.981458	−0.490729	+	0.871312i	$0.663269\pi$
$150$	−3.78800e7	−0.916411
$151$	7.40579e7	1.75046	0.875230	−	0.483707i	$-0.160710\pi$
$151$	7.40579e7	1.75046	0.875230	+	0.483707i	$0.160710\pi$
$152$	1.51101e7	0.348992
$153$	5.22578e7	1.17959
$154$	−660176.	−0.0145659
$155$	1.29951e8	2.80296
$156$	−2.01600e6	−0.0425162
$157$	4.73609e7	0.976722	0.488361	−	0.872642i	$-0.337595\pi$
$157$	4.73609e7	0.976722	0.488361	+	0.872642i	$0.337595\pi$
$158$	−3.49036e7	−0.703996
$159$	−1.60813e7	−0.317272
$160$	−1.80552e7	−0.348483
$161$	1.95294e6	0.0368806
$162$	1.66724e7	0.308103
$163$	−8.40724e7	−1.52054	−0.760268	−	0.649609i	$-0.774933\pi$
$163$	−8.40724e7	−1.52054	−0.760268	+	0.649609i	$0.774933\pi$
$164$	−1.72984e7	−0.306234
$165$	−1.54010e7	−0.266904
$166$	−3.55023e7	−0.602391
$167$	−6.05363e7	−1.00579	−0.502896	−	0.864347i	$-0.667732\pi$
$167$	−6.05363e7	−1.00579	−0.502896	+	0.864347i	$0.667732\pi$
$168$	−666624.	−0.0108467
$169$	−6.04985e7	−0.964143
$170$	1.31931e8	2.05957
$171$	−5.15280e7	−0.788055
$172$	−6.57939e7	−0.985908
$173$	−6.54378e6	−0.0960876	−0.0480438	−	0.998845i	$-0.515299\pi$
$173$	−6.54378e6	−0.0960876	−0.0480438	+	0.998845i	$0.515299\pi$
$174$	1.26282e7	0.181727
$175$	1.39795e7	0.197178
$176$	−5.45178e6	−0.0753778
$177$	8.23215e6	0.111585
$178$	−4.16986e6	−0.0554181
$179$	1.16987e8	1.52458	0.762290	−	0.647235i	$-0.224075\pi$
$179$	1.16987e8	1.52458	0.762290	+	0.647235i	$0.224075\pi$
$180$	6.15709e7	0.786905
$181$	−8.22779e7	−1.03136	−0.515678	−	0.856783i	$-0.672460\pi$
$181$	−8.22779e7	−1.03136	−0.515678	+	0.856783i	$0.672460\pi$
$182$	744000.	0.00914794
$183$	−2.62177e7	−0.316239
$184$	1.61275e7	0.190855
$185$	−4.16044e7	−0.483101
$186$	3.96220e7	0.451482
$187$	3.98368e7	0.445491
$188$	−4.93978e7	−0.542194
$189$	5.12077e6	0.0551720
$190$	−1.30089e8	−1.37595
$191$	5.14775e7	0.534565	0.267283	−	0.963618i	$-0.413874\pi$
$191$	5.14775e7	0.534565	0.267283	+	0.963618i	$0.413874\pi$
$192$	−5.50502e6	−0.0561313
$193$	−1.07593e8	−1.07729	−0.538645	−	0.842533i	$-0.681064\pi$
$193$	−1.07593e8	−1.07729	−0.538645	+	0.842533i	$0.681064\pi$
$194$	−1.70386e7	−0.167544
$195$	1.73565e7	0.167626
$196$	−5.24607e7	−0.497666
$197$	−1.12130e8	−1.04493	−0.522467	−	0.852659i	$-0.674988\pi$
$197$	−1.12130e8	−1.04493	−0.522467	+	0.852659i	$0.674988\pi$
$198$	1.85914e7	0.170210
$199$	−2.35053e7	−0.211436	−0.105718	−	0.994396i	$-0.533714\pi$
$199$	−2.35053e7	−0.211436	−0.105718	+	0.994396i	$0.533714\pi$
$200$	1.15444e8	1.02039
$201$	−7.34608e7	−0.638072
$202$	9.94127e7	0.848618
$203$	−4.66042e6	−0.0391011
$204$	4.02259e7	0.331742
$205$	1.48929e8	1.20737
$206$	−7.49374e7	−0.597260
$207$	−5.49973e7	−0.430968
$208$	6.14400e6	0.0473401
$209$	−3.92805e7	−0.297622
$210$	5.73922e6	0.0427647
$211$	−1.86758e8	−1.36864	−0.684322	−	0.729180i	$-0.739902\pi$
$211$	−1.86758e8	−1.36864	−0.684322	+	0.729180i	$0.739902\pi$
$212$	4.90098e7	0.353270
$213$	−2.31368e7	−0.164050
$214$	8.81687e7	0.614988
$215$	5.66445e8	3.88707
$216$	4.22876e7	0.285512
$217$	−1.46224e7	−0.0971425
$218$	1.57658e8	1.03066
$219$	2.35829e7	0.151720
$220$	4.69364e7	0.297187
$221$	−4.48950e7	−0.279785
$222$	−1.26852e7	−0.0778146
$223$	−1.38809e8	−0.838208	−0.419104	−	0.907938i	$-0.637656\pi$
$223$	−1.38809e8	−0.838208	−0.419104	+	0.907938i	$0.637656\pi$
$224$	2.03162e6	0.0120774
$225$	−3.93681e8	−2.30412
$226$	−9.91903e7	−0.571596
$227$	−6.51335e7	−0.369585	−0.184792	−	0.982778i	$-0.559161\pi$
$227$	−6.51335e7	−0.369585	−0.184792	+	0.982778i	$0.559161\pi$
$228$	−3.96641e7	−0.221629
$229$	2.90274e8	1.59729	0.798646	−	0.601801i	$-0.205550\pi$
$229$	2.90274e8	1.59729	0.798646	+	0.601801i	$0.205550\pi$
$230$	−1.38848e8	−0.752473
$231$	1.73296e6	0.00925011
$232$	−3.84860e7	−0.202346
$233$	−6.53560e7	−0.338485	−0.169243	−	0.985574i	$-0.554132\pi$
$233$	−6.53560e7	−0.338485	−0.169243	+	0.985574i	$0.554132\pi$
$234$	−2.09520e7	−0.106898
$235$	4.25284e8	2.13767
$236$	−2.50884e7	−0.124246
$237$	9.16219e7	0.447075
$238$	−1.48453e7	−0.0713788
$239$	−7.03014e7	−0.333098	−0.166549	−	0.986033i	$-0.553262\pi$
$239$	−7.03014e7	−0.333098	−0.166549	+	0.986033i	$0.553262\pi$
$240$	4.73948e7	0.221305
$241$	1.63087e8	0.750514	0.375257	−	0.926921i	$-0.377554\pi$
$241$	1.63087e8	0.750514	0.375257	+	0.926921i	$0.377554\pi$
$242$	1.41725e7	0.0642824
$243$	−2.24396e8	−1.00321
$244$	7.99014e7	0.352120
$245$	4.51654e8	1.96212
$246$	4.54084e7	0.194475
$247$	4.42680e7	0.186918
$248$	−1.20753e8	−0.502708
$249$	9.31936e7	0.382550
$250$	−6.49523e8	−2.62909
$251$	−8.53709e7	−0.340762	−0.170381	−	0.985378i	$-0.554500\pi$
$251$	−8.53709e7	−0.340762	−0.170381	+	0.985378i	$0.554500\pi$
$252$	−6.92813e6	−0.0272718
$253$	−4.19252e7	−0.162762
$254$	2.58588e8	0.990125
$255$	−3.46320e8	−1.30794
$256$	1.67772e7	0.0625000
$257$	−3.06745e8	−1.12723	−0.563613	−	0.826039i	$-0.690589\pi$
$257$	−3.06745e8	−1.12723	−0.563613	+	0.826039i	$0.690589\pi$
$258$	1.72709e8	0.626103
$259$	4.68143e6	0.0167429
$260$	−5.28960e7	−0.186645
$261$	1.31243e8	0.456915
$262$	−2.97825e8	−1.02307
$263$	−3.65645e8	−1.23941	−0.619704	−	0.784835i	$-0.712747\pi$
$263$	−3.65645e8	−1.23941	−0.619704	+	0.784835i	$0.712747\pi$
$264$	1.43109e7	0.0478689
$265$	−4.21944e8	−1.39282
$266$	1.46380e7	0.0476864
$267$	1.09459e7	0.0351934
$268$	2.23881e8	0.710468
$269$	3.74620e8	1.17343	0.586716	−	0.809793i	$-0.300420\pi$
$269$	3.74620e8	1.17343	0.586716	+	0.809793i	$0.300420\pi$
$270$	−3.64070e8	−1.12567
$271$	2.75299e8	0.840258	0.420129	−	0.907464i	$-0.361985\pi$
$271$	2.75299e8	0.840258	0.420129	+	0.907464i	$0.361985\pi$
$272$	−1.22593e8	−0.369382
$273$	−1.95300e6	−0.00580942
$274$	6.08563e7	0.178722
$275$	−3.00109e8	−0.870190
$276$	−4.23347e7	−0.121203
$277$	−2.63367e8	−0.744531	−0.372265	−	0.928126i	$-0.621419\pi$
$277$	−2.63367e8	−0.744531	−0.372265	+	0.928126i	$0.621419\pi$
$278$	1.20912e8	0.337530
$279$	4.11785e8	1.13516
$280$	−1.74909e7	−0.0476168
$281$	−1.77369e8	−0.476877	−0.238439	−	0.971158i	$-0.576636\pi$
$281$	−1.77369e8	−0.476877	−0.238439	+	0.971158i	$0.576636\pi$
$282$	1.29669e8	0.344322
$283$	2.21037e8	0.579713	0.289856	−	0.957070i	$-0.406392\pi$
$283$	2.21037e8	0.579713	0.289856	+	0.957070i	$0.406392\pi$
$284$	7.05122e7	0.182663
$285$	3.41483e8	0.873801
$286$	−1.59720e7	−0.0403718
$287$	−1.67579e7	−0.0418439
$288$	−5.72129e7	−0.141130
$289$	4.85466e8	1.18309
$290$	3.31341e8	0.797777
$291$	4.47265e7	0.106399
$292$	−7.18717e7	−0.168934
$293$	1.57114e8	0.364904	0.182452	−	0.983215i	$-0.441597\pi$
$293$	1.57114e8	0.364904	0.182452	+	0.983215i	$0.441597\pi$
$294$	1.37709e8	0.316044
$295$	2.15996e8	0.489856
$296$	3.86596e7	0.0866435
$297$	−1.09931e8	−0.243486
$298$	−3.17040e8	−0.693995
$299$	4.72485e7	0.102221
$300$	−3.03040e8	−0.648000
$301$	−6.37379e7	−0.134715
$302$	5.92463e8	1.23776
$303$	−2.60958e8	−0.538917
$304$	1.20881e8	0.246775
$305$	−6.87901e8	−1.38828
$306$	4.18062e8	0.834096
$307$	−3.35074e8	−0.660932	−0.330466	−	0.943818i	$-0.607206\pi$
$307$	−3.35074e8	−0.660932	−0.330466	+	0.943818i	$0.607206\pi$
$308$	−5.28141e6	−0.0102996
$309$	1.96711e8	0.379292
$310$	1.03960e9	1.98199
$311$	5.58672e8	1.05316	0.526581	−	0.850125i	$-0.323474\pi$
$311$	5.58672e8	1.05316	0.526581	+	0.850125i	$0.323474\pi$
$312$	−1.61280e7	−0.0300635
$313$	−3.69424e8	−0.680957	−0.340479	−	0.940252i	$-0.610589\pi$
$313$	−3.69424e8	−0.680957	−0.340479	+	0.940252i	$0.610589\pi$
$314$	3.78887e8	0.690647
$315$	5.96469e7	0.107523
$316$	−2.79229e8	−0.497800
$317$	5.27905e8	0.930783	0.465392	−	0.885105i	$-0.345913\pi$
$317$	5.27905e8	0.930783	0.465392	+	0.885105i	$0.345913\pi$
$318$	−1.28651e8	−0.224345
$319$	1.00049e8	0.172561
$320$	−1.44441e8	−0.246415
$321$	−2.31443e8	−0.390550
$322$	1.56235e7	0.0260785
$323$	−8.83294e8	−1.45847
$324$	1.33379e8	0.217861
$325$	3.38214e8	0.546512
$326$	−6.72579e8	−1.07518
$327$	−4.13851e8	−0.654527
$328$	−1.38387e8	−0.216540
$329$	−4.78541e7	−0.0740855
$330$	−1.23208e8	−0.188730
$331$	−6.01784e8	−0.912101	−0.456050	−	0.889954i	$-0.650736\pi$
$331$	−6.01784e8	−0.912101	−0.456050	+	0.889954i	$0.650736\pi$
$332$	−2.84019e8	−0.425955
$333$	−1.31835e8	−0.195649
$334$	−4.84290e8	−0.711202
$335$	−1.92747e9	−2.80112
$336$	−5.33299e6	−0.00766979
$337$	8.14101e8	1.15871	0.579354	−	0.815076i	$-0.303305\pi$
$337$	8.14101e8	1.15871	0.579354	+	0.815076i	$0.303305\pi$
$338$	−4.83988e8	−0.681752
$339$	2.60374e8	0.362994
$340$	1.05545e9	1.45634
$341$	3.13910e8	0.428711
$342$	−4.12224e8	−0.557239
$343$	−1.01881e8	−0.136321
$344$	−5.26351e8	−0.697142
$345$	3.64475e8	0.477860
$346$	−5.23503e7	−0.0679442
$347$	−1.93524e8	−0.248646	−0.124323	−	0.992242i	$-0.539676\pi$
$347$	−1.93524e8	−0.248646	−0.124323	+	0.992242i	$0.539676\pi$
$348$	1.01026e8	0.128501
$349$	7.35639e8	0.926351	0.463176	−	0.886267i	$-0.346710\pi$
$349$	7.35639e8	0.926351	0.463176	+	0.886267i	$0.346710\pi$
$350$	1.11836e8	0.139426
$351$	1.23890e8	0.152918
$352$	−4.36142e7	−0.0533002
$353$	2.46985e8	0.298854	0.149427	−	0.988773i	$-0.452257\pi$
$353$	2.46985e8	0.298854	0.149427	+	0.988773i	$0.452257\pi$
$354$	6.58572e7	0.0789027
$355$	−6.07066e8	−0.720173
$356$	−3.33589e7	−0.0391865
$357$	3.89689e7	0.0453293
$358$	9.35893e8	1.07804
$359$	1.08186e9	1.23407	0.617036	−	0.786935i	$-0.288333\pi$
$359$	1.08186e9	1.23407	0.617036	+	0.786935i	$0.288333\pi$
$360$	4.92568e8	0.556426
$361$	−2.29136e7	−0.0256341
$362$	−6.58224e8	−0.729279
$363$	−3.72028e7	−0.0408227
$364$	5.95200e6	0.00646857
$365$	6.18771e8	0.666047
$366$	−2.09741e8	−0.223615
$367$	−1.00475e9	−1.06103	−0.530513	−	0.847677i	$-0.678001\pi$
$367$	−1.00475e9	−1.06103	−0.530513	+	0.847677i	$0.678001\pi$
$368$	1.29020e8	0.134955
$369$	4.71923e8	0.488966
$370$	−3.32835e8	−0.341604
$371$	4.74782e7	0.0482710
$372$	3.16976e8	0.319246
$373$	−1.06441e9	−1.06201	−0.531007	−	0.847368i	$-0.678186\pi$
$373$	−1.06441e9	−1.06201	−0.531007	+	0.847368i	$0.678186\pi$
$374$	3.18695e8	0.315010
$375$	1.70500e9	1.66961
$376$	−3.95182e8	−0.383389
$377$	−1.12752e8	−0.108375
$378$	4.09661e7	0.0390125
$379$	9.57022e8	0.902994	0.451497	−	0.892273i	$-0.350890\pi$
$379$	9.57022e8	0.902994	0.451497	+	0.892273i	$0.350890\pi$
$380$	−1.04071e9	−0.972944
$381$	−6.78793e8	−0.628782
$382$	4.11820e8	0.377995
$383$	−1.52159e9	−1.38389	−0.691943	−	0.721952i	$-0.743245\pi$
$383$	−1.52159e9	−1.38389	−0.691943	+	0.721952i	$0.743245\pi$
$384$	−4.40402e7	−0.0396908
$385$	4.54696e7	0.0406077
$386$	−8.60742e8	−0.761759
$387$	1.79494e9	1.57421
$388$	−1.36309e8	−0.118472
$389$	1.69855e9	1.46304	0.731518	−	0.681822i	$-0.238812\pi$
$389$	1.69855e9	1.46304	0.731518	+	0.681822i	$0.238812\pi$
$390$	1.38852e8	0.118529
$391$	−9.42765e8	−0.797599
$392$	−4.19686e8	−0.351903
$393$	7.81790e8	0.649705
$394$	−8.97038e8	−0.738880
$395$	2.40398e9	1.96264
$396$	1.48731e8	0.120356
$397$	1.61745e8	0.129737	0.0648686	−	0.997894i	$-0.479337\pi$
$397$	1.61745e8	0.129737	0.0648686	+	0.997894i	$0.479337\pi$
$398$	−1.88042e8	−0.149508
$399$	−3.84246e7	−0.0302834
$400$	9.23550e8	0.721523
$401$	5.18083e8	0.401231	0.200615	−	0.979670i	$-0.435706\pi$
$401$	5.18083e8	0.401231	0.200615	+	0.979670i	$0.435706\pi$
$402$	−5.87686e8	−0.451185
$403$	−3.53768e8	−0.269247
$404$	7.95302e8	0.600063
$405$	−1.14831e9	−0.858948
$406$	−3.72833e7	−0.0276486
$407$	−1.00500e8	−0.0738899
$408$	3.21807e8	0.234577
$409$	−2.31946e9	−1.67631	−0.838157	−	0.545429i	$-0.816367\pi$
$409$	−2.31946e9	−1.67631	−0.838157	+	0.545429i	$0.816367\pi$
$410$	1.19143e9	0.853738
$411$	−1.59748e8	−0.113498
$412$	−5.99499e8	−0.422327
$413$	−2.43044e7	−0.0169770
$414$	−4.39978e8	−0.304740
$415$	2.44522e9	1.67938
$416$	4.91520e7	0.0334745
$417$	−3.17394e8	−0.214349
$418$	−3.14244e8	−0.210450
$419$	−1.63260e8	−0.108425	−0.0542127	−	0.998529i	$-0.517265\pi$
$419$	−1.63260e8	−0.108425	−0.0542127	+	0.998529i	$0.517265\pi$
$420$	4.59137e7	0.0302392
$421$	−1.70927e8	−0.111641	−0.0558203	−	0.998441i	$-0.517777\pi$
$421$	−1.70927e8	−0.111641	−0.0558203	+	0.998441i	$0.517777\pi$
$422$	−1.49406e9	−0.967777
$423$	1.34763e9	0.865726
$424$	3.92078e8	0.249800
$425$	−6.74850e9	−4.26428
$426$	−1.85095e8	−0.116001
$427$	7.74045e7	0.0481137
$428$	7.05350e8	0.434862
$429$	4.19265e7	0.0256382
$430$	4.53156e9	2.74858
$431$	−2.53433e7	−0.0152473	−0.00762365	−	0.999971i	$-0.502427\pi$
$431$	−2.53433e7	−0.0152473	−0.00762365	+	0.999971i	$0.502427\pi$
$432$	3.38301e8	0.201888
$433$	−1.55797e9	−0.922259	−0.461129	−	0.887333i	$-0.652556\pi$
$433$	−1.55797e9	−0.922259	−0.461129	+	0.887333i	$0.652556\pi$
$434$	−1.16979e8	−0.0686901
$435$	−8.69769e8	−0.506631
$436$	1.26126e9	0.728790
$437$	9.29598e8	0.532857
$438$	1.88663e8	0.107282
$439$	−1.51204e9	−0.852978	−0.426489	−	0.904493i	$-0.640250\pi$
$439$	−1.51204e9	−0.852978	−0.426489	+	0.904493i	$0.640250\pi$
$440$	3.75491e8	0.210143
$441$	1.43119e9	0.794627
$442$	−3.59160e8	−0.197838
$443$	−1.90710e9	−1.04222	−0.521110	−	0.853490i	$-0.674482\pi$
$443$	−1.90710e9	−1.04222	−0.521110	+	0.853490i	$0.674482\pi$
$444$	−1.01481e8	−0.0550232
$445$	2.87199e8	0.154498
$446$	−1.11047e9	−0.592702
$447$	8.32229e8	0.440724
$448$	1.62529e7	0.00854001
$449$	−1.75105e9	−0.912926	−0.456463	−	0.889742i	$-0.650884\pi$
$449$	−1.75105e9	−0.912926	−0.456463	+	0.889742i	$0.650884\pi$
$450$	−3.14945e9	−1.62926
$451$	3.59753e8	0.184666
$452$	−7.93522e8	−0.404180
$453$	−1.55522e9	−0.786044
$454$	−5.21068e8	−0.261336
$455$	−5.12430e7	−0.0255032
$456$	−3.17313e8	−0.156715
$457$	2.91724e9	1.42977	0.714883	−	0.699244i	$-0.246480\pi$
$457$	2.91724e9	1.42977	0.714883	+	0.699244i	$0.246480\pi$
$458$	2.32219e9	1.12946
$459$	−2.47201e9	−1.19318
$460$	−1.11078e9	−0.532079
$461$	3.09362e9	1.47066	0.735331	−	0.677708i	$-0.237026\pi$
$461$	3.09362e9	1.47066	0.735331	+	0.677708i	$0.237026\pi$
$462$	1.38637e7	0.00654082
$463$	−2.80698e8	−0.131434	−0.0657168	−	0.997838i	$-0.520933\pi$
$463$	−2.80698e8	−0.131434	−0.0657168	+	0.997838i	$0.520933\pi$
$464$	−3.07888e8	−0.143080
$465$	−2.72896e9	−1.25867
$466$	−5.22848e8	−0.239345
$467$	5.75352e8	0.261411	0.130706	−	0.991421i	$-0.458276\pi$
$467$	5.75352e8	0.261411	0.130706	+	0.991421i	$0.458276\pi$
$468$	−1.67616e8	−0.0755884
$469$	2.16884e8	0.0970785
$470$	3.40227e9	1.51156
$471$	−9.94578e8	−0.438597
$472$	−2.00708e8	−0.0878551
$473$	1.36831e9	0.594525
$474$	7.32975e8	0.316129
$475$	6.65425e9	2.84886
$476$	−1.18762e8	−0.0504724
$477$	−1.33705e9	−0.564070
$478$	−5.62411e8	−0.235536
$479$	2.31069e9	0.960654	0.480327	−	0.877090i	$-0.340518\pi$
$479$	2.31069e9	0.960654	0.480327	+	0.877090i	$0.340518\pi$
$480$	3.79159e8	0.156486
$481$	1.13260e8	0.0464056
$482$	1.30469e9	0.530694
$483$	−4.10117e7	−0.0165612
$484$	1.13380e8	0.0454545
$485$	1.17354e9	0.467090
$486$	−1.79517e9	−0.709378
$487$	−1.34577e9	−0.527983	−0.263991	−	0.964525i	$-0.585039\pi$
$487$	−1.34577e9	−0.527983	−0.263991	+	0.964525i	$0.585039\pi$
$488$	6.39212e8	0.248986
$489$	1.76552e9	0.682797
$490$	3.61323e9	1.38743
$491$	8.80452e8	0.335676	0.167838	−	0.985815i	$-0.446321\pi$
$491$	8.80452e8	0.335676	0.167838	+	0.985815i	$0.446321\pi$
$492$	3.63267e8	0.137514
$493$	2.24978e9	0.845620
$494$	3.54144e8	0.132171
$495$	−1.28048e9	−0.474521
$496$	−9.66021e8	−0.355468
$497$	6.83087e7	0.0249591
$498$	7.45549e8	0.270504
$499$	−3.60240e9	−1.29790	−0.648949	−	0.760832i	$-0.724791\pi$
$499$	−3.60240e9	−1.29790	−0.648949	+	0.760832i	$0.724791\pi$
$500$	−5.19619e9	−1.85904
$501$	1.27126e9	0.451651
$502$	−6.82967e8	−0.240955
$503$	9.72776e8	0.340820	0.170410	−	0.985373i	$-0.445491\pi$
$503$	9.72776e8	0.340820	0.170410	+	0.985373i	$0.445491\pi$
$504$	−5.54250e7	−0.0192841
$505$	−6.84705e9	−2.36583
$506$	−3.35401e8	−0.115090
$507$	1.27047e9	0.432948
$508$	2.06870e9	0.700124
$509$	1.01644e9	0.341642	0.170821	−	0.985302i	$-0.445358\pi$
$509$	1.01644e9	0.341642	0.170821	+	0.985302i	$0.445358\pi$
$510$	−2.77056e9	−0.924851
$511$	−6.96258e7	−0.0230832
$512$	1.34218e8	0.0441942
$513$	2.43748e9	0.797134
$514$	−2.45396e9	−0.797070
$515$	5.16131e9	1.66508
$516$	1.38167e9	0.442722
$517$	1.02732e9	0.326955
$518$	3.74515e7	0.0118390
$519$	1.37419e8	0.0431482
$520$	−4.23168e8	−0.131978
$521$	1.80128e9	0.558021	0.279010	−	0.960288i	$-0.409994\pi$
$521$	1.80128e9	0.558021	0.279010	+	0.960288i	$0.409994\pi$
$522$	1.04995e9	0.323088
$523$	−4.60802e9	−1.40851	−0.704253	−	0.709949i	$-0.748718\pi$
$523$	−4.60802e9	−1.40851	−0.704253	+	0.709949i	$0.748718\pi$
$524$	−2.38260e9	−0.723422
$525$	−2.93570e8	−0.0885429
$526$	−2.92516e9	−0.876394
$527$	7.05884e9	2.10086
$528$	1.14487e8	0.0338484
$529$	−2.41264e9	−0.708594
$530$	−3.37555e9	−0.984870
$531$	6.84444e8	0.198384
$532$	1.17104e8	0.0337194
$533$	−4.05432e8	−0.115977
$534$	8.75671e7	0.0248855
$535$	−6.07262e9	−1.71450
$536$	1.79104e9	0.502377
$537$	−2.45672e9	−0.684613
$538$	2.99696e9	0.829742
$539$	1.09102e9	0.300104
$540$	−2.91256e9	−0.795970
$541$	3.01371e9	0.818296	0.409148	−	0.912468i	$-0.365826\pi$
$541$	3.01371e9	0.818296	0.409148	+	0.912468i	$0.365826\pi$
$542$	2.20239e9	0.594152
$543$	1.72784e9	0.463130
$544$	−9.80746e8	−0.261192
$545$	−1.08587e10	−2.87335
$546$	−1.56240e7	−0.00410788
$547$	−6.06759e9	−1.58511	−0.792557	−	0.609797i	$-0.791251\pi$
$547$	−6.06759e9	−1.58511	−0.792557	+	0.609797i	$0.791251\pi$
$548$	4.86851e8	0.126376
$549$	−2.17981e9	−0.562232
$550$	−2.40087e9	−0.615317
$551$	−2.21836e9	−0.564938
$552$	−3.38677e8	−0.0857036
$553$	−2.70503e8	−0.0680195
$554$	−2.10694e9	−0.526463
$555$	8.73691e8	0.216937
$556$	9.67295e8	0.238670
$557$	4.17725e9	1.02423	0.512114	−	0.858917i	$-0.328862\pi$
$557$	4.17725e9	1.02423	0.512114	+	0.858917i	$0.328862\pi$
$558$	3.29428e9	0.802678
$559$	−1.54204e9	−0.373384
$560$	−1.39928e8	−0.0336702
$561$	−8.36573e8	−0.200048
$562$	−1.41896e9	−0.337203
$563$	2.32616e9	0.549365	0.274682	−	0.961535i	$-0.411427\pi$
$563$	2.32616e9	0.549365	0.274682	+	0.961535i	$0.411427\pi$
$564$	1.03735e9	0.243472
$565$	6.83173e9	1.59353
$566$	1.76830e9	0.409919
$567$	1.29211e8	0.0297686
$568$	5.64098e8	0.129162
$569$	−1.78134e9	−0.405373	−0.202687	−	0.979244i	$-0.564967\pi$
$569$	−1.78134e9	−0.405373	−0.202687	+	0.979244i	$0.564967\pi$
$570$	2.73187e9	0.617871
$571$	−1.52713e9	−0.343281	−0.171641	−	0.985160i	$-0.554907\pi$
$571$	−1.52713e9	−0.343281	−0.171641	+	0.985160i	$0.554907\pi$
$572$	−1.27776e8	−0.0285472
$573$	−1.08103e9	−0.240047
$574$	−1.34063e8	−0.0295881
$575$	7.10227e9	1.55797
$576$	−4.57703e8	−0.0997942
$577$	5.18847e9	1.12441	0.562204	−	0.826998i	$-0.309953\pi$
$577$	5.18847e9	1.12441	0.562204	+	0.826998i	$0.309953\pi$
$578$	3.88373e9	0.836569
$579$	2.25945e9	0.483757
$580$	2.65072e9	0.564114
$581$	−2.75143e8	−0.0582025
$582$	3.57812e8	0.0752357
$583$	−1.01925e9	−0.213030
$584$	−5.74974e8	−0.119455
$585$	1.44307e9	0.298017
$586$	1.25691e9	0.258026
$587$	3.52006e9	0.718319	0.359159	−	0.933276i	$-0.383063\pi$
$587$	3.52006e9	0.718319	0.359159	+	0.933276i	$0.383063\pi$
$588$	1.10168e9	0.223477
$589$	−6.96026e9	−1.40353
$590$	1.72797e9	0.346380
$591$	2.35472e9	0.469228
$592$	3.09277e8	0.0612662
$593$	3.15165e9	0.620650	0.310325	−	0.950631i	$-0.399562\pi$
$593$	3.15165e9	0.620650	0.310325	+	0.950631i	$0.399562\pi$
$594$	−8.79450e8	−0.172170
$595$	1.02247e9	0.198994
$596$	−2.53632e9	−0.490729
$597$	4.93610e8	0.0949454
$598$	3.77988e8	0.0722810
$599$	−4.06814e9	−0.773397	−0.386699	−	0.922206i	$-0.626385\pi$
$599$	−4.06814e9	−0.773397	−0.386699	+	0.922206i	$0.626385\pi$
$600$	−2.42432e9	−0.458205
$601$	9.12077e9	1.71384	0.856921	−	0.515448i	$-0.172374\pi$
$601$	9.12077e9	1.71384	0.856921	+	0.515448i	$0.172374\pi$
$602$	−5.09903e8	−0.0952576
$603$	−6.10774e9	−1.13441
$604$	4.73971e9	0.875230
$605$	−9.76130e8	−0.179211
$606$	−2.08767e9	−0.381072
$607$	−9.66953e9	−1.75487	−0.877435	−	0.479696i	$-0.840747\pi$
$607$	−9.66953e9	−1.75487	−0.877435	+	0.479696i	$0.840747\pi$
$608$	9.67049e8	0.174496
$609$	9.78687e7	0.0175583
$610$	−5.50321e9	−0.981662
$611$	−1.15776e9	−0.205340
$612$	3.34450e9	0.589795
$613$	−5.80960e9	−1.01867	−0.509336	−	0.860568i	$-0.670109\pi$
$613$	−5.80960e9	−1.01867	−0.509336	+	0.860568i	$0.670109\pi$
$614$	−2.68059e9	−0.467349
$615$	−3.12750e9	−0.542169
$616$	−4.22513e7	−0.00728295
$617$	−5.67349e9	−0.972417	−0.486208	−	0.873843i	$-0.661620\pi$
$617$	−5.67349e9	−0.972417	−0.486208	+	0.873843i	$0.661620\pi$
$618$	1.57369e9	0.268200
$619$	−1.02063e10	−1.72962	−0.864810	−	0.502100i	$-0.832561\pi$
$619$	−1.02063e10	−1.72962	−0.864810	+	0.502100i	$0.832561\pi$
$620$	8.31684e9	1.40148
$621$	2.60160e9	0.435933
$622$	4.46937e9	0.744698
$623$	−3.23164e7	−0.00535446
$624$	−1.29024e8	−0.0212581
$625$	2.71206e10	4.44344
$626$	−2.95539e9	−0.481509
$627$	8.24890e8	0.133647
$628$	3.03110e9	0.488361
$629$	−2.25992e9	−0.362090
$630$	4.77175e8	0.0760301
$631$	9.17510e8	0.145381	0.0726906	−	0.997355i	$-0.476841\pi$
$631$	9.17510e8	0.145381	0.0726906	+	0.997355i	$0.476841\pi$
$632$	−2.23383e9	−0.351998
$633$	3.92192e9	0.614590
$634$	4.22324e9	0.658163
$635$	−1.78102e10	−2.76033
$636$	−1.02921e9	−0.158636
$637$	−1.22955e9	−0.188477
$638$	8.00389e8	0.122019
$639$	−1.92366e9	−0.291659
$640$	−1.15553e9	−0.174241
$641$	7.27488e9	1.09099	0.545497	−	0.838113i	$-0.316341\pi$
$641$	7.27488e9	1.09099	0.545497	+	0.838113i	$0.316341\pi$
$642$	−1.85154e9	−0.276160
$643$	3.77263e9	0.559636	0.279818	−	0.960053i	$-0.409726\pi$
$643$	3.77263e9	0.559636	0.279818	+	0.960053i	$0.409726\pi$
$644$	1.24988e8	0.0184403
$645$	−1.18953e10	−1.74549
$646$	−7.06635e9	−1.03129
$647$	−4.77280e9	−0.692800	−0.346400	−	0.938087i	$-0.612596\pi$
$647$	−4.77280e9	−0.692800	−0.346400	+	0.938087i	$0.612596\pi$
$648$	1.06703e9	0.154051
$649$	5.21761e8	0.0749230
$650$	2.70571e9	0.386442
$651$	3.07070e8	0.0436219
$652$	−5.38064e9	−0.760268
$653$	−1.12692e10	−1.58379	−0.791895	−	0.610657i	$-0.790905\pi$
$653$	−1.12692e10	−1.58379	−0.791895	+	0.610657i	$0.790905\pi$
$654$	−3.31081e9	−0.462820
$655$	2.05127e10	2.85219
$656$	−1.10710e9	−0.153117
$657$	1.96075e9	0.269739
$658$	−3.82833e8	−0.0523864
$659$	1.30260e9	0.177301	0.0886506	−	0.996063i	$-0.471745\pi$
$659$	1.30260e9	0.177301	0.0886506	+	0.996063i	$0.471745\pi$
$660$	−9.85664e8	−0.133452
$661$	−4.90256e9	−0.660265	−0.330132	−	0.943935i	$-0.607093\pi$
$661$	−4.90256e9	−0.660265	−0.330132	+	0.943935i	$0.607093\pi$
$662$	−4.81427e9	−0.644952
$663$	9.42795e8	0.125638
$664$	−2.27215e9	−0.301195
$665$	−1.00819e9	−0.132943
$666$	−1.05468e9	−0.138344
$667$	−2.36772e9	−0.308951
$668$	−3.87432e9	−0.502896
$669$	2.91500e9	0.376397
$670$	−1.54198e10	−1.98069
$671$	−1.66170e9	−0.212336
$672$	−4.26639e7	−0.00542336
$673$	6.92999e9	0.876355	0.438177	−	0.898889i	$-0.355624\pi$
$673$	6.92999e9	0.876355	0.438177	+	0.898889i	$0.355624\pi$
$674$	6.51281e9	0.819330
$675$	1.86227e10	2.33067
$676$	−3.87191e9	−0.482071
$677$	7.87483e9	0.975396	0.487698	−	0.873012i	$-0.337837\pi$
$677$	7.87483e9	0.975396	0.487698	+	0.873012i	$0.337837\pi$
$678$	2.08300e9	0.256675
$679$	−1.32050e8	−0.0161880
$680$	8.44361e9	1.02979
$681$	1.36780e9	0.165962
$682$	2.51128e9	0.303144
$683$	−2.48171e9	−0.298042	−0.149021	−	0.988834i	$-0.547612\pi$
$683$	−2.48171e9	−0.298042	−0.149021	+	0.988834i	$0.547612\pi$
$684$	−3.29779e9	−0.394027
$685$	−4.19148e9	−0.498253
$686$	−8.15048e8	−0.0963937
$687$	−6.09576e9	−0.717264
$688$	−4.21081e9	−0.492954
$689$	1.14867e9	0.133791
$690$	2.91580e9	0.337898
$691$	−6.43186e9	−0.741589	−0.370795	−	0.928715i	$-0.620915\pi$
$691$	−6.43186e9	−0.741589	−0.370795	+	0.928715i	$0.620915\pi$
$692$	−4.18802e8	−0.0480438
$693$	1.44083e8	0.0164455
$694$	−1.54819e9	−0.175819
$695$	−8.32780e9	−0.940987
$696$	8.08206e8	0.0908636
$697$	8.08972e9	0.904938
$698$	5.88511e9	0.655029
$699$	1.37248e9	0.151997
$700$	8.94689e8	0.0985891
$701$	2.48956e9	0.272966	0.136483	−	0.990642i	$-0.456420\pi$
$701$	2.48956e9	0.272966	0.136483	+	0.990642i	$0.456420\pi$
$702$	9.91116e8	0.108130
$703$	2.22836e9	0.241904
$704$	−3.48914e8	−0.0376889
$705$	−8.93096e9	−0.959923
$706$	1.97588e9	0.211322
$707$	7.70448e8	0.0819928
$708$	5.26857e8	0.0557926
$709$	−2.63056e8	−0.0277196	−0.0138598	−	0.999904i	$-0.504412\pi$
$709$	−2.63056e8	−0.0277196	−0.0138598	+	0.999904i	$0.504412\pi$
$710$	−4.85653e9	−0.509239
$711$	7.61770e9	0.794841
$712$	−2.66871e8	−0.0277091
$713$	−7.42888e9	−0.767556
$714$	3.11751e8	0.0320527
$715$	1.10007e9	0.112551
$716$	7.48714e9	0.762290
$717$	1.47633e9	0.149578
$718$	8.65488e9	0.872621
$719$	3.66923e9	0.368149	0.184074	−	0.982912i	$-0.441071\pi$
$719$	3.66923e9	0.368149	0.184074	+	0.982912i	$0.441071\pi$
$720$	3.94054e9	0.393452
$721$	−5.80765e8	−0.0577068
$722$	−1.83309e8	−0.0181260
$723$	−3.42482e9	−0.337019
$724$	−5.26579e9	−0.515678
$725$	−1.69486e10	−1.65177
$726$	−2.97622e8	−0.0288660
$727$	8.53327e9	0.823654	0.411827	−	0.911262i	$-0.364891\pi$
$727$	8.53327e9	0.823654	0.411827	+	0.911262i	$0.364891\pi$
$728$	4.76160e7	0.00457397
$729$	1.54499e8	0.0147700
$730$	4.95017e9	0.470966
$731$	3.07689e10	2.91341
$732$	−1.67793e9	−0.158119
$733$	1.52031e10	1.42583	0.712915	−	0.701251i	$-0.247375\pi$
$733$	1.52031e10	1.42583	0.712915	+	0.701251i	$0.247375\pi$
$734$	−8.03798e9	−0.750258
$735$	−9.48474e9	−0.881089
$736$	1.03216e9	0.0954277
$737$	−4.65602e9	−0.428428
$738$	3.77538e9	0.345751
$739$	1.00321e9	0.0914397	0.0457199	−	0.998954i	$-0.485442\pi$
$739$	1.00321e9	0.0914397	0.0457199	+	0.998954i	$0.485442\pi$
$740$	−2.66268e9	−0.241550
$741$	−9.29628e8	−0.0839355
$742$	3.79826e8	0.0341327
$743$	1.63631e9	0.146354	0.0731771	−	0.997319i	$-0.476686\pi$
$743$	1.63631e9	0.146354	0.0731771	+	0.997319i	$0.476686\pi$
$744$	2.53581e9	0.225741
$745$	2.18361e10	1.93476
$746$	−8.51532e9	−0.750957
$747$	7.74838e9	0.680125
$748$	2.54956e9	0.222746
$749$	6.83308e8	0.0594196
$750$	1.36400e10	1.18059
$751$	1.16289e10	1.00184	0.500919	−	0.865494i	$-0.332996\pi$
$751$	1.16289e10	1.00184	0.500919	+	0.865494i	$0.332996\pi$
$752$	−3.16146e9	−0.271097
$753$	1.79279e9	0.153019
$754$	−9.02016e8	−0.0766328
$755$	−4.08059e10	−3.45071
$756$	3.27729e8	0.0275860
$757$	−4.81507e9	−0.403429	−0.201714	−	0.979444i	$-0.564651\pi$
$757$	−4.81507e9	−0.403429	−0.201714	+	0.979444i	$0.564651\pi$
$758$	7.65618e9	0.638513
$759$	8.80429e8	0.0730883
$760$	−8.32569e9	−0.687975
$761$	−9.54754e9	−0.785318	−0.392659	−	0.919684i	$-0.628445\pi$
$761$	−9.54754e9	−0.785318	−0.392659	+	0.919684i	$0.628445\pi$
$762$	−5.43035e9	−0.444616
$763$	1.22185e9	0.0995820
$764$	3.29456e9	0.267283
$765$	−2.87940e10	−2.32535
$766$	−1.21727e10	−0.978556
$767$	−5.88010e8	−0.0470545
$768$	−3.52322e8	−0.0280656
$769$	7.36001e9	0.583628	0.291814	−	0.956475i	$-0.405741\pi$
$769$	7.36001e9	0.583628	0.291814	+	0.956475i	$0.405741\pi$
$770$	3.63757e8	0.0287140
$771$	6.44164e9	0.506181
$772$	−6.88594e9	−0.538645
$773$	3.64678e9	0.283976	0.141988	−	0.989868i	$-0.454651\pi$
$773$	3.64678e9	0.283976	0.141988	+	0.989868i	$0.454651\pi$
$774$	1.43595e10	1.11313
$775$	−5.31774e10	−4.10366
$776$	−1.09047e9	−0.0837720
$777$	−9.83101e7	−0.00751839
$778$	1.35884e10	1.03452
$779$	−7.97674e9	−0.604567
$780$	1.11082e9	0.0838129
$781$	−1.46643e9	−0.110150
$782$	−7.54212e9	−0.563988
$783$	−6.20835e9	−0.462179
$784$	−3.35749e9	−0.248833
$785$	−2.60958e10	−1.92543
$786$	6.25432e9	0.459411
$787$	4.89669e9	0.358089	0.179045	−	0.983841i	$-0.442699\pi$
$787$	4.89669e9	0.358089	0.179045	+	0.983841i	$0.442699\pi$
$788$	−7.17630e9	−0.522467
$789$	7.67855e9	0.556557
$790$	1.92319e10	1.38780
$791$	−7.68725e8	−0.0552272
$792$	1.18985e9	0.0851048
$793$	1.87269e9	0.133355
$794$	1.29396e9	0.0917381
$795$	8.86082e9	0.625445
$796$	−1.50434e9	−0.105718
$797$	−6.18871e8	−0.0433008	−0.0216504	−	0.999766i	$-0.506892\pi$
$797$	−6.18871e8	−0.0433008	−0.0216504	+	0.999766i	$0.506892\pi$
$798$	−3.07397e8	−0.0214136
$799$	2.31012e10	1.60221
$800$	7.38840e9	0.510194
$801$	9.10073e8	0.0625694
$802$	4.14467e9	0.283713
$803$	1.49471e9	0.101871
$804$	−4.70149e9	−0.319036
$805$	−1.07607e9	−0.0727034
$806$	−2.83014e9	−0.190386
$807$	−7.86702e9	−0.526930
$808$	6.36241e9	0.424309
$809$	−1.90876e9	−0.126745	−0.0633726	−	0.997990i	$-0.520186\pi$
$809$	−1.90876e9	−0.126745	−0.0633726	+	0.997990i	$0.520186\pi$
$810$	−9.18649e9	−0.607368
$811$	−1.42262e10	−0.936518	−0.468259	−	0.883591i	$-0.655119\pi$
$811$	−1.42262e10	−0.936518	−0.468259	+	0.883591i	$0.655119\pi$
$812$	−2.98267e8	−0.0195505
$813$	−5.78129e9	−0.377318
$814$	−8.03999e8	−0.0522480
$815$	4.63239e10	2.99746
$816$	2.57446e9	0.165871
$817$	−3.03392e10	−1.94638
$818$	−1.85557e10	−1.18533
$819$	−1.62378e8	−0.0103284
$820$	9.53144e9	0.603684
$821$	1.03473e10	0.652570	0.326285	−	0.945271i	$-0.394203\pi$
$821$	1.03473e10	0.652570	0.326285	+	0.945271i	$0.394203\pi$
$822$	−1.27798e9	−0.0802553
$823$	2.72568e9	0.170441	0.0852207	−	0.996362i	$-0.472840\pi$
$823$	2.72568e9	0.170441	0.0852207	+	0.996362i	$0.472840\pi$
$824$	−4.79599e9	−0.298630
$825$	6.30228e9	0.390759
$826$	−1.94435e8	−0.0120045
$827$	2.76749e10	1.70144	0.850722	−	0.525616i	$-0.176165\pi$
$827$	2.76749e10	1.70144	0.850722	+	0.525616i	$0.176165\pi$
$828$	−3.51982e9	−0.215484
$829$	−2.24970e10	−1.37146	−0.685731	−	0.727855i	$-0.740517\pi$
$829$	−2.24970e10	−1.37146	−0.685731	+	0.727855i	$0.740517\pi$
$830$	1.95618e10	1.18750
$831$	5.53071e9	0.334332
$832$	3.93216e8	0.0236701
$833$	2.45336e10	1.47063
$834$	−2.53915e9	−0.151568
$835$	3.33555e10	1.98274
$836$	−2.51395e9	−0.148811
$837$	−1.94791e10	−1.14824
$838$	−1.30608e9	−0.0766683
$839$	−3.24344e9	−0.189601	−0.0948003	−	0.995496i	$-0.530221\pi$
$839$	−3.24344e9	−0.189601	−0.0948003	+	0.995496i	$0.530221\pi$
$840$	3.67310e8	0.0213823
$841$	−1.15996e10	−0.672448
$842$	−1.36741e9	−0.0789419
$843$	3.72476e9	0.214142
$844$	−1.19525e10	−0.684322
$845$	3.33347e10	1.90063
$846$	1.07811e10	0.612160
$847$	1.09837e8	0.00621092
$848$	3.13663e9	0.176635
$849$	−4.64178e9	−0.260320
$850$	−5.39880e10	−3.01530
$851$	2.37839e9	0.132291
$852$	−1.48076e9	−0.0820248
$853$	7.37362e9	0.406779	0.203390	−	0.979098i	$-0.434804\pi$
$853$	7.37362e9	0.406779	0.203390	+	0.979098i	$0.434804\pi$
$854$	6.19236e8	0.0340215
$855$	2.83919e10	1.55351
$856$	5.64280e9	0.307494
$857$	−9.67049e9	−0.524826	−0.262413	−	0.964956i	$-0.584518\pi$
$857$	−9.67049e9	−0.524826	−0.262413	+	0.964956i	$0.584518\pi$
$858$	3.35412e8	0.0181290
$859$	1.64804e10	0.887140	0.443570	−	0.896240i	$-0.353712\pi$
$859$	1.64804e10	0.887140	0.443570	+	0.896240i	$0.353712\pi$
$860$	3.62524e10	1.94354
$861$	3.51915e8	0.0187900
$862$	−2.02746e8	−0.0107815
$863$	−1.50741e10	−0.798352	−0.399176	−	0.916874i	$-0.630704\pi$
$863$	−1.50741e10	−0.798352	−0.399176	+	0.916874i	$0.630704\pi$
$864$	2.70641e9	0.142756
$865$	3.60562e9	0.189419
$866$	−1.24638e10	−0.652135
$867$	−1.01948e10	−0.531265
$868$	−9.35833e8	−0.0485713
$869$	5.80708e9	0.300185
$870$	−6.95815e9	−0.358242
$871$	5.24720e9	0.269069
$872$	1.00901e10	0.515332
$873$	3.71868e9	0.189164
$874$	7.43679e9	0.376787
$875$	−5.03380e9	−0.254020
$876$	1.50931e9	0.0758600
$877$	3.62952e9	0.181698	0.0908491	−	0.995865i	$-0.471042\pi$
$877$	3.62952e9	0.181698	0.0908491	+	0.995865i	$0.471042\pi$
$878$	−1.20963e10	−0.603147
$879$	−3.29940e9	−0.163860
$880$	3.00393e9	0.148594
$881$	2.97881e9	0.146766	0.0733832	−	0.997304i	$-0.476620\pi$
$881$	2.97881e9	0.146766	0.0733832	+	0.997304i	$0.476620\pi$
$882$	1.14496e10	0.561886
$883$	−2.96787e10	−1.45072	−0.725359	−	0.688371i	$-0.758326\pi$
$883$	−2.96787e10	−1.45072	−0.725359	+	0.688371i	$0.758326\pi$
$884$	−2.87328e9	−0.139893
$885$	−4.53591e9	−0.219970
$886$	−1.52568e10	−0.736961
$887$	2.58852e10	1.24543	0.622715	−	0.782449i	$-0.286030\pi$
$887$	2.58852e10	1.24543	0.622715	+	0.782449i	$0.286030\pi$
$888$	−8.11851e8	−0.0389073
$889$	2.00406e9	0.0956651
$890$	2.29760e9	0.109247
$891$	−2.77387e9	−0.131375
$892$	−8.88380e9	−0.419104
$893$	−2.27785e10	−1.07040
$894$	6.65783e9	0.311639
$895$	−6.44596e10	−3.00543
$896$	1.30023e8	0.00603870
$897$	−9.92219e8	−0.0459022
$898$	−1.40084e10	−0.645536
$899$	1.77280e10	0.813768
$900$	−2.51956e10	−1.15206
$901$	−2.29197e10	−1.04393
$902$	2.87803e9	0.130579
$903$	1.33850e9	0.0604936
$904$	−6.34818e9	−0.285798
$905$	4.53351e10	2.03313
$906$	−1.24417e10	−0.555817
$907$	7.12409e9	0.317033	0.158516	−	0.987356i	$-0.449329\pi$
$907$	7.12409e9	0.317033	0.158516	+	0.987356i	$0.449329\pi$
$908$	−4.16854e9	−0.184792
$909$	−2.16968e10	−0.958126
$910$	−4.09944e8	−0.0180335
$911$	−2.75887e10	−1.20897	−0.604487	−	0.796615i	$-0.706622\pi$
$911$	−2.75887e10	−1.20897	−0.604487	+	0.796615i	$0.706622\pi$
$912$	−2.53850e9	−0.110814
$913$	5.90670e9	0.256860
$914$	2.33379e10	1.01100
$915$	1.44459e10	0.623407
$916$	1.85776e10	0.798646
$917$	−2.30814e9	−0.0988485
$918$	−1.97761e10	−0.843705
$919$	1.77159e10	0.752936	0.376468	−	0.926430i	$-0.377138\pi$
$919$	1.77159e10	0.752936	0.376468	+	0.926430i	$0.377138\pi$
$920$	−8.88625e9	−0.376237
$921$	7.03656e9	0.296792
$922$	2.47489e10	1.03992
$923$	1.65263e9	0.0691782
$924$	1.10910e8	0.00462506
$925$	1.70250e10	0.707280
$926$	−2.24559e9	−0.0929376
$927$	1.63551e10	0.674332
$928$	−2.46311e9	−0.101173
$929$	2.36999e10	0.969821	0.484911	−	0.874564i	$-0.338852\pi$
$929$	2.36999e10	0.969821	0.484911	+	0.874564i	$0.338852\pi$
$930$	−2.18317e10	−0.890015
$931$	−2.41910e10	−0.982492
$932$	−4.18279e9	−0.169243
$933$	−1.17321e10	−0.472923
$934$	4.60282e9	0.184846
$935$	−2.19501e10	−0.878205
$936$	−1.34093e9	−0.0534490
$937$	6.84304e9	0.271744	0.135872	−	0.990726i	$-0.456616\pi$
$937$	6.84304e9	0.271744	0.135872	+	0.990726i	$0.456616\pi$
$938$	1.73507e9	0.0686449
$939$	7.75790e9	0.305784
$940$	2.72182e10	1.06884
$941$	−2.77504e10	−1.08569	−0.542845	−	0.839833i	$-0.682653\pi$
$941$	−2.77504e10	−1.08569	−0.542845	+	0.839833i	$0.682653\pi$
$942$	−7.95663e9	−0.310135
$943$	−8.51380e9	−0.330623
$944$	−1.60566e9	−0.0621229
$945$	−2.82154e9	−0.108762
$946$	1.09465e10	0.420392
$947$	−4.22715e10	−1.61742	−0.808710	−	0.588208i	$-0.799834\pi$
$947$	−4.22715e10	−1.61742	−0.808710	+	0.588208i	$0.799834\pi$
$948$	5.86380e9	0.223537
$949$	−1.68449e9	−0.0639790
$950$	5.32340e10	2.01445
$951$	−1.10860e10	−0.417968
$952$	−9.50098e8	−0.0356894
$953$	−3.87038e10	−1.44853	−0.724267	−	0.689520i	$-0.757822\pi$
$953$	−3.87038e10	−1.44853	−0.724267	+	0.689520i	$0.757822\pi$
$954$	−1.06964e10	−0.398858
$955$	−2.83641e10	−1.05380
$956$	−4.49929e9	−0.166549
$957$	−2.10102e9	−0.0774887
$958$	1.84855e10	0.679285
$959$	4.71637e8	0.0172680
$960$	3.03327e9	0.110653
$961$	2.81102e10	1.02172
$962$	9.06084e8	0.0328137
$963$	−1.92428e10	−0.694347
$964$	1.04376e10	0.375257
$965$	5.92836e10	2.12368
$966$	−3.28094e8	−0.0117106
$967$	3.63884e10	1.29411	0.647054	−	0.762444i	$-0.276001\pi$
$967$	3.63884e10	1.29411	0.647054	+	0.762444i	$0.276001\pi$
$968$	9.07039e8	0.0321412
$969$	1.85492e10	0.654925
$970$	9.38830e9	0.330283
$971$	−2.83770e9	−0.0994717	−0.0497359	−	0.998762i	$-0.515838\pi$
$971$	−2.83770e9	−0.0994717	−0.0497359	+	0.998762i	$0.515838\pi$
$972$	−1.43613e10	−0.501606
$973$	9.37067e8	0.0326119
$974$	−1.07662e10	−0.373340
$975$	−7.10249e9	−0.245411
$976$	5.11369e9	0.176060
$977$	2.20171e10	0.755316	0.377658	−	0.925945i	$-0.376729\pi$
$977$	2.20171e10	0.755316	0.377658	+	0.925945i	$0.376729\pi$
$978$	1.41242e10	0.482811
$979$	6.93761e8	0.0236304
$980$	2.89059e10	0.981058
$981$	−3.44088e10	−1.16366
$982$	7.04362e9	0.237359
$983$	−1.29252e10	−0.434011	−0.217006	−	0.976170i	$-0.569629\pi$
$983$	−1.29252e10	−0.434011	−0.217006	+	0.976170i	$0.569629\pi$
$984$	2.90614e9	0.0972374
$985$	6.17835e10	2.05990
$986$	1.79982e10	0.597944
$987$	1.00494e9	0.0332681
$988$	2.83315e9	0.0934589
$989$	−3.23819e10	−1.06443
$990$	−1.02439e10	−0.335537
$991$	−5.37746e10	−1.75517	−0.877585	−	0.479421i	$-0.840847\pi$
$991$	−5.37746e10	−1.75517	−0.877585	+	0.479421i	$0.840847\pi$
$992$	−7.72817e9	−0.251354
$993$	1.26375e10	0.409579
$994$	5.46469e8	0.0176487
$995$	1.29514e10	0.416808
$996$	5.96439e9	0.191275
$997$	1.18188e10	0.377693	0.188847	−	0.982007i	$-0.439525\pi$
$997$	1.18188e10	0.377693	0.188847	+	0.982007i	$0.439525\pi$
$998$	−2.88192e10	−0.917752
$999$	6.23635e9	0.197902

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	22.8.a.c.1.1	✓	1
3.2	odd	2		198.8.a.b.1.1		1
4.3	odd	2		176.8.a.c.1.1		1
5.4	even	2		550.8.a.a.1.1		1
11.10	odd	2		242.8.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.8.a.c.1.1	✓	1	1.1	even	1	trivial
176.8.a.c.1.1		1	4.3	odd	2
198.8.a.b.1.1		1	3.2	odd	2
242.8.a.b.1.1		1	11.10	odd	2
550.8.a.a.1.1		1	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 22 = 2 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	22.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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