LMFDB - Embedded newform 3.68.a.a.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3,68,Mod(1,3)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 68, names="a")

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

chi := DirichletCharacter("3.1");

S:= CuspForms(chi, 68);

N := Newforms(S);

Level:	$ N $	$=$	$ 3 $
Weight:	$ k $	$=$	$ 68 $
Character orbit:	$[\chi]$	$=$	3.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:	$85.2871055790$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} + \cdots - 17\!\cdots\!00 $ x^5 - x^4 - 189398708752260728x^3 + 10367208713972597963236992x^2 + 3357934964760116559921767608965120*x - 178120650251916997056936485095832786534400
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{40}\cdot 3^{20}\cdot 5^{3}\cdot 7^{2}\cdot 11\cdot 17 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.37573e8$ of defining polynomial
Character	$\chi$	$=$	3.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.35246e9 q^{2} +5.55906e15 q^{3} -1.36335e20 q^{4} -3.80674e23 q^{5} +1.86366e25 q^{6} +2.28291e27 q^{7} -9.51794e29 q^{8} +3.09032e31 q^{9} +O(q^{10})$	\(q+3.35246e9 q^{2} +5.55906e15 q^{3} -1.36335e20 q^{4} -3.80674e23 q^{5} +1.86366e25 q^{6} +2.28291e27 q^{7} -9.51794e29 q^{8} +3.09032e31 q^{9} -1.27619e33 q^{10} +6.32880e34 q^{11} -7.57894e35 q^{12} +1.06832e37 q^{13} +7.65339e36 q^{14} -2.11619e39 q^{15} +1.69286e40 q^{16} +1.88759e41 q^{17} +1.03602e41 q^{18} +7.47789e42 q^{19} +5.18991e43 q^{20} +1.26909e43 q^{21} +2.12171e44 q^{22} -9.84296e44 q^{23} -5.29108e45 q^{24} +7.71498e46 q^{25} +3.58149e46 q^{26} +1.71793e47 q^{27} -3.11241e47 q^{28} -1.61330e49 q^{29} -7.09444e48 q^{30} -1.42506e50 q^{31} +1.97213e50 q^{32} +3.51822e50 q^{33} +6.32809e50 q^{34} -8.69045e50 q^{35} -4.21318e51 q^{36} -3.12191e52 q^{37} +2.50694e52 q^{38} +5.93883e52 q^{39} +3.62323e53 q^{40} +3.16803e53 q^{41} +4.25456e52 q^{42} -4.14652e54 q^{43} -8.62837e54 q^{44} -1.17640e55 q^{45} -3.29982e54 q^{46} +7.48599e54 q^{47} +9.41073e55 q^{48} -4.13166e56 q^{49} +2.58642e56 q^{50} +1.04933e57 q^{51} -1.45649e57 q^{52} +2.48975e57 q^{53} +5.75928e56 q^{54} -2.40921e58 q^{55} -2.17286e57 q^{56} +4.15701e58 q^{57} -5.40852e58 q^{58} +2.25769e59 q^{59} +2.88510e59 q^{60} -1.12737e59 q^{61} -4.77747e59 q^{62} +7.05492e58 q^{63} -1.83708e60 q^{64} -4.06680e60 q^{65} +1.17947e60 q^{66} +2.53292e61 q^{67} -2.57345e61 q^{68} -5.47176e60 q^{69} -2.91344e60 q^{70} -1.31821e62 q^{71} -2.94134e61 q^{72} +4.80052e62 q^{73} -1.04661e62 q^{74} +4.28880e62 q^{75} -1.01950e63 q^{76} +1.44481e62 q^{77} +1.99097e62 q^{78} -4.20019e63 q^{79} -6.44428e63 q^{80} +9.55005e62 q^{81} +1.06207e63 q^{82} +2.02548e64 q^{83} -1.73021e63 q^{84} -7.18557e64 q^{85} -1.39011e64 q^{86} -8.96842e64 q^{87} -6.02372e64 q^{88} -2.92734e65 q^{89} -3.94384e64 q^{90} +2.43887e64 q^{91} +1.34194e65 q^{92} -7.92201e65 q^{93} +2.50965e64 q^{94} -2.84664e66 q^{95} +1.09632e66 q^{96} -7.09139e66 q^{97} -1.38512e66 q^{98} +1.95580e66 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 16255223088 q^{2} + 27\!\cdots\!15 q^{3}+ \cdots + 15\!\cdots\!45 q^{9}+O(q^{10}) $ 5 * q - 16255223088 * q^2 + 27795302832777615 * q^3 + 187725942510165022976 * q^4 - 78587975159033653413210 * q^5 - 90363769669063698103515024 * q^6 + 2897133779647323835395360808 * q^7 - 446560396075670111196535861248 * q^8 + 154515771913163061809603209017645 * q^9	$ 5 q - 16255223088 q^{2} + 27\!\cdots\!15 q^{3}+ \cdots - 33\!\cdots\!28 q^{99}+O(q^{100}) $ 5 * q - 16255223088 * q^2 + 27795302832777615 * q^3 + 187725942510165022976 * q^4 - 78587975159033653413210 * q^5 - 90363769669063698103515024 * q^6 + 2897133779647323835395360808 * q^7 - 446560396075670111196535861248 * q^8 + 154515771913163061809603209017645 * q^9 - 3359687674843328387731353730265760 * q^10 - 107218677940017498877929099247513932 * q^11 + 1043579884327727512139439968474696448 * q^12 + 18720658700128270654310089005018371278 * q^13 - 61671302560002915461244055855418861952 * q^14 - 436875313712028989080558866721926658830 * q^15 + 19828810558474760279626207936836822695936 * q^16 + 155634077783406813100849154582687160437850 * q^17 - 502337668612598026687246628668502560677552 * q^18 - 6330823499631802877762269921902501094783988 * q^19 + 8806685958461200090592049790099809534328320 * q^20 + 16105342150473395769449245566105034350142584 * q^21 - 1285334407460495219610552944494458560199940800 * q^22 - 3096470811979061341218583695224599825546419864 * q^23 - 2482456288409673438086642069250837537616072704 * q^24 - 128510195883326013861116838537875526813134026525 * q^25 - 309893301421656093037055250012280159780714544160 * q^26 + 858962534553352218394101882942702121170179203335 * q^27 - 1758216414458764947454106710991446824456120252416 * q^28 - 17195448123638013107201791699494703244624614194066 * q^29 - 18676707269164160844617717039655071934318585792480 * q^30 - 167299848682287535534976349940903215242774636120144 * q^31 - 661581347508711830434048681313730640500855508303872 * q^32 - 596035124534567832987517244399918779918082194046436 * q^33 - 3668629124130223548817939535033459308602451741777248 * q^34 - 597313366964957875633721976544849058693729215396560 * q^35 + 5801323783016844061209039192798206302504038362882304 * q^36 + 30043361009726020519419208243072073319518814491934358 * q^37 + 260485037699007543111228693866771891318790849526359744 * q^38 + 104069275559827645019221591022442107451411644015468394 * q^39 + 1115094440991577594559320173651217718577849903091507200 * q^40 - 808437303878393630827363118778237447937170861730235454 * q^41 - 342834506149426883197674031813537074530930837280160896 * q^42 - 4736429276246859157617871173658532851277246462721149228 * q^43 + 7015258532306255548162192439376864244003799076792468480 * q^44 - 2428616328958113717945826873336511479963953561073218090 * q^45 + 12527540911364294799696045411504819449186256978468080256 * q^46 + 108290076402229329642983829719434835232002632660130515056 * q^47 + 110229558857316837304648760016604589737407028314290454528 * q^48 + 880181676573802609519730686079181363021552872382740232333 * q^49 + 1052494889081109131658159592164188937755980711801220679600 * q^50 + 865179264617971813637134219389277404805805348251015745550 * q^51 + 2308777474667263284772763232859451620790190619345360422400 * q^52 + 3245226878182367959789303611224433348480161481518744370966 * q^53 - 2792525524679729749647164642126896980230414112629707719696 * q^54 - 51262912349629136715335621059095130466245073679837843274920 * q^55 - 47772133165991162191839606887952190261990284108568121671680 * q^56 - 35193431270626188960239511075559223006610539094694893365724 * q^57 - 84493666334602655307554841072065967874913257934429532582432 * q^58 - 179624529165245332659094970865033289643789158511842592792412 * q^59 + 48956900633719888068348132839178395661941003500967791311360 * q^60 + 584907690374744349731719712847344294312321913116060488567102 * q^61 + 3025719874056683992201295546200663430948598802434153503920896 * q^62 + 89530572459581180641513960803016595914572442189036802691432 * q^63 - 4499392031332841330291967248952523929085218266739063232921600 * q^64 - 4475515698809998982184518632813763939217510489440113220535420 * q^65 - 7145251819350648004204211437078044590913023079552444513038400 * q^66 + 1344914323542783268647242614869504266340689501787079942327420 * q^67 + 19365780359678204058695683499114787965920302224535027287671296 * q^68 - 17213468786362961074546343099951406673447837437328506826108872 * q^69 + 31668988367649764945620440396543964676121485136652812185168640 * q^70 + 208099792629824800554470944953685171609425458107698538896812280 * q^71 - 13800124861096000027311301435875177780070484438004484020744192 * q^72 - 252414514587179259177941988046481265295589404121206374617470334 * q^73 - 358084904219721857079108717596560064061662918994206334990793632 * q^74 - 714395962335323550124996935032662148018445244009738166967247575 * q^75 - 3167576238289455647607470438405921376289304779045740803124282368 * q^76 - 2715616264867565296962962180641800513550454260043685733361085024 * q^77 - 1722715631772832981701583674878761695633721340944383480275395680 * q^78 - 3431503850832591202644393930474874375763697022794649483222945280 * q^79 - 8322155504340769679256255462767373452465836703993138541717749760 * q^80 + 4775024753984126184465953508872070059599675694871715649184269205 * q^81 - 6362169783346643199240774371522649930233919884701663896646628064 * q^82 - 2753559226275970537111940225815871118779223599597864793376534836 * q^83 - 9774031537088362109664563191733058527530044006822089118438893568 * q^84 - 100223085545462292798863671541625307722645907180923010395975039060 * q^85 - 287767761108017143223082018540844348985876033069110036187064944832 * q^86 - 95590537588367238050824370796559187863343340269759771352286126518 * q^87 - 475509956271309169302126186902650458005506404316097446988140167168 * q^88 - 953231813381767774308665680609247777419192406201469535932206722638 * q^89 - 103824946893111374784083486264977336654984096886031891078875867040 * q^90 - 1826343975172985917715752368227201077015187248090890361406237190544 * q^91 - 3132766731772277225542233484759817592417648594742828753496091850752 * q^92 - 930029991600410615305341159302088161126938972921803642822874755312 * q^93 - 4377709041417318550457656800988375516480152699177767581662212342016 * q^94 - 3334812729598028343033336500168508357868459521335461166424246211800 * q^95 - 3677770780504345932935680810193675103543675932078897134851043885056 * q^96 - 11968150959482375162440074477550294838351889417758320270285978711190 * q^97 - 17297597363568888846780657367452226513119832175100344485603054876848 * q^98 - 3313395357082126364699797809873297365206219515646735115126034266028 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	3.35246e9	0.275968	0.137984	−	0.990434i	$-0.455938\pi$
$2$	3.35246e9	0.275968	0.137984	+	0.990434i	$0.455938\pi$
$3$	5.55906e15	0.577350
$3$	5.55906e15	0.577350
$4$	−1.36335e20	−0.923841
$5$	−3.80674e23	−1.46237	−0.731186	−	0.682178i	$-0.761033\pi$
$5$	−3.80674e23	−1.46237	−0.731186	+	0.682178i	$0.761033\pi$
$6$	1.86366e25	0.159330
$7$	2.28291e27	0.111611	0.0558053	−	0.998442i	$-0.482227\pi$
$7$	2.28291e27	0.111611	0.0558053	+	0.998442i	$0.482227\pi$
$8$	−9.51794e29	−0.530919
$9$	3.09032e31	0.333333
$10$	−1.27619e33	−0.403568
$11$	6.32880e34	0.821611	0.410806	−	0.911723i	$-0.365247\pi$
$11$	6.32880e34	0.821611	0.410806	+	0.911723i	$0.365247\pi$
$12$	−7.57894e35	−0.533380
$13$	1.06832e37	0.514751	0.257376	−	0.966311i	$-0.417142\pi$
$13$	1.06832e37	0.514751	0.257376	+	0.966311i	$0.417142\pi$
$14$	7.65339e36	0.0308010
$15$	−2.11619e39	−0.844301
$16$	1.69286e40	0.777325
$17$	1.88759e41	1.13729	0.568643	−	0.822584i	$-0.307469\pi$
$17$	1.88759e41	1.13729	0.568643	+	0.822584i	$0.307469\pi$
$18$	1.03602e41	0.0919894
$19$	7.47789e42	1.08526	0.542630	−	0.839972i	$-0.317429\pi$
$19$	7.47789e42	1.08526	0.542630	+	0.839972i	$0.317429\pi$
$20$	5.18991e43	1.35100
$21$	1.26909e43	0.0644384
$22$	2.12171e44	0.226739
$23$	−9.84296e44	−0.237270	−0.118635	−	0.992938i	$-0.537852\pi$
$23$	−9.84296e44	−0.237270	−0.118635	+	0.992938i	$0.537852\pi$
$24$	−5.29108e45	−0.306526
$25$	7.71498e46	1.13853
$26$	3.58149e46	0.142055
$27$	1.71793e47	0.192450
$28$	−3.11241e47	−0.103110
$29$	−1.61330e49	−1.64961	−0.824805	−	0.565417i	$-0.808715\pi$
$29$	−1.61330e49	−1.64961	−0.824805	+	0.565417i	$0.808715\pi$
$30$	−7.09444e48	−0.233000
$31$	−1.42506e50	−1.56033	−0.780167	−	0.625572i	$-0.784866\pi$
$31$	−1.42506e50	−1.56033	−0.780167	+	0.625572i	$0.784866\pi$
$32$	1.97213e50	0.745436
$33$	3.51822e50	0.474358
$34$	6.32809e50	0.313855
$35$	−8.69045e50	−0.163216
$36$	−4.21318e51	−0.307947
$37$	−3.12191e52	−0.911301	−0.455651	−	0.890159i	$-0.650593\pi$
$37$	−3.12191e52	−0.911301	−0.455651	+	0.890159i	$0.650593\pi$
$38$	2.50694e52	0.299497
$39$	5.93883e52	0.297192
$40$	3.62323e53	0.776401
$41$	3.16803e53	0.296845	0.148423	−	0.988924i	$-0.452580\pi$
$41$	3.16803e53	0.296845	0.148423	+	0.988924i	$0.452580\pi$
$42$	4.25456e52	0.0177830
$43$	−4.14652e54	−0.787936	−0.393968	−	0.919124i	$-0.628898\pi$
$43$	−4.14652e54	−0.787936	−0.393968	+	0.919124i	$0.628898\pi$
$44$	−8.62837e54	−0.759039
$45$	−1.17640e55	−0.487457
$46$	−3.29982e54	−0.0654791
$47$	7.48599e54	0.0722721	0.0361361	−	0.999347i	$-0.488495\pi$
$47$	7.48599e54	0.0722721	0.0361361	+	0.999347i	$0.488495\pi$
$48$	9.41073e55	0.448789
$49$	−4.13166e56	−0.987543
$50$	2.58642e56	0.314198
$51$	1.04933e57	0.656613
$52$	−1.45649e57	−0.475548
$53$	2.48975e57	0.429451	0.214725	−	0.976674i	$-0.431114\pi$
$53$	2.48975e57	0.429451	0.214725	+	0.976674i	$0.431114\pi$
$54$	5.75928e56	0.0531101
$55$	−2.40921e58	−1.20150
$56$	−2.17286e57	−0.0592562
$57$	4.15701e58	0.626575
$58$	−5.40852e58	−0.455240
$59$	2.25769e59	1.07182	0.535909	−	0.844276i	$-0.319969\pi$
$59$	2.25769e59	1.07182	0.535909	+	0.844276i	$0.319969\pi$
$60$	2.88510e59	0.780000
$61$	−1.12737e59	−0.175193	−0.0875966	−	0.996156i	$-0.527919\pi$
$61$	−1.12737e59	−0.175193	−0.0875966	+	0.996156i	$0.527919\pi$
$62$	−4.77747e59	−0.430603
$63$	7.05492e58	0.0372035
$64$	−1.83708e60	−0.571608
$65$	−4.06680e60	−0.752757
$66$	1.17947e60	0.130908
$67$	2.53292e61	1.69870	0.849348	−	0.527834i	$-0.176996\pi$
$67$	2.53292e61	1.69870	0.849348	+	0.527834i	$0.176996\pi$
$68$	−2.57345e61	−1.05067
$69$	−5.47176e60	−0.136988
$70$	−2.91344e60	−0.0450425
$71$	−1.31821e62	−1.26715	−0.633576	−	0.773681i	$-0.718413\pi$
$71$	−1.31821e62	−1.26715	−0.633576	+	0.773681i	$0.718413\pi$
$72$	−2.94134e61	−0.176973
$73$	4.80052e62	1.81959	0.909793	−	0.415062i	$-0.136240\pi$
$73$	4.80052e62	1.81959	0.909793	+	0.415062i	$0.136240\pi$
$74$	−1.04661e62	−0.251490
$75$	4.28880e62	0.657331
$76$	−1.01950e63	−1.00261
$77$	1.44481e62	0.0917005
$78$	1.99097e62	0.0820155
$79$	−4.20019e63	−1.12918	−0.564588	−	0.825373i	$-0.690965\pi$
$79$	−4.20019e63	−1.12918	−0.564588	+	0.825373i	$0.690965\pi$
$80$	−6.44428e63	−1.13674
$81$	9.55005e62	0.111111
$82$	1.06207e63	0.0819199
$83$	2.02548e64	1.04091	0.520455	−	0.853889i	$-0.325763\pi$
$83$	2.02548e64	1.04091	0.520455	+	0.853889i	$0.325763\pi$
$84$	−1.73021e63	−0.0595308
$85$	−7.18557e64	−1.66313
$86$	−1.39011e64	−0.217445
$87$	−8.96842e64	−0.952403
$88$	−6.02372e64	−0.436209
$89$	−2.92734e65	−1.45181	−0.725903	−	0.687797i	$-0.758578\pi$
$89$	−2.92734e65	−1.45181	−0.725903	+	0.687797i	$0.758578\pi$
$90$	−3.94384e64	−0.134523
$91$	2.43887e64	0.0574517
$92$	1.34194e65	0.219200
$93$	−7.92201e65	−0.900859
$94$	2.50965e64	0.0199448
$95$	−2.84664e66	−1.58705
$96$	1.09632e66	0.430378
$97$	−7.09139e66	−1.96733	−0.983667	−	0.179997i	$-0.942391\pi$
$97$	−7.09139e66	−1.96733	−0.983667	+	0.179997i	$0.942391\pi$
$98$	−1.38512e66	−0.272531
$99$	1.95580e66	0.273870
$100$	−1.05182e67	−1.05182
$101$	−1.94342e67	−1.39252	−0.696259	−	0.717791i	$-0.745153\pi$
$101$	−1.94342e67	−1.39252	−0.696259	+	0.717791i	$0.745153\pi$
$102$	3.51782e66	0.181204
$103$	3.79390e66	0.140942	0.0704708	−	0.997514i	$-0.477550\pi$
$103$	3.79390e66	0.140942	0.0704708	+	0.997514i	$0.477550\pi$
$104$	−1.01682e67	−0.273291
$105$	−4.83108e66	−0.0942328
$106$	8.34679e66	0.118515
$107$	6.84871e67	0.709990	0.354995	−	0.934868i	$-0.384483\pi$
$107$	6.84871e67	0.709990	0.354995	+	0.934868i	$0.384483\pi$
$108$	−2.34213e67	−0.177793
$109$	2.03645e68	1.13524	0.567619	−	0.823291i	$-0.307865\pi$
$109$	2.03645e68	1.13524	0.567619	+	0.823291i	$0.307865\pi$
$110$	−8.07678e67	−0.331576
$111$	−1.73549e68	−0.526140
$112$	3.86466e67	0.0867576
$113$	5.28465e68	0.880823	0.440412	−	0.897796i	$-0.354833\pi$
$113$	5.28465e68	0.880823	0.440412	+	0.897796i	$0.354833\pi$
$114$	1.39362e68	0.172915
$115$	3.74696e68	0.346977
$116$	2.19949e69	1.52398
$117$	3.30143e68	0.171584
$118$	7.56881e68	0.295788
$119$	4.30922e68	0.126933
$120$	2.01418e69	0.448255
$121$	−1.92811e69	−0.324955
$122$	−3.77947e68	−0.0483478
$123$	1.76113e69	0.171384
$124$	1.94286e70	1.44150
$125$	−3.57344e69	−0.202582
$126$	2.36514e68	0.0102670
$127$	−6.05630e69	−0.201736	−0.100868	−	0.994900i	$-0.532162\pi$
$127$	−6.05630e69	−0.201736	−0.100868	+	0.994900i	$0.532162\pi$
$128$	−3.52622e70	−0.903182
$129$	−2.30508e70	−0.454915
$130$	−1.36338e70	−0.207737
$131$	−9.53303e70	−1.12368	−0.561840	−	0.827246i	$-0.689906\pi$
$131$	−9.53303e70	−1.12368	−0.561840	+	0.827246i	$0.689906\pi$
$132$	−4.79656e70	−0.438231
$133$	1.70714e70	0.121126
$134$	8.49152e70	0.468786
$135$	−6.53969e70	−0.281434
$136$	−1.79660e71	−0.603807
$137$	−2.03132e71	−0.534121	−0.267061	−	0.963680i	$-0.586052\pi$
$137$	−2.03132e71	−0.534121	−0.267061	+	0.963680i	$0.586052\pi$
$138$	−1.83439e70	−0.0378044
$139$	−8.33421e71	−1.34855	−0.674277	−	0.738478i	$-0.735545\pi$
$139$	−8.33421e71	−1.34855	−0.674277	+	0.738478i	$0.735545\pi$
$140$	1.18481e71	0.150786
$141$	4.16151e70	0.0417263
$142$	−4.41924e71	−0.349694
$143$	6.76116e71	0.422925
$144$	5.23148e71	0.259108
$145$	6.14140e72	2.41234
$146$	1.60936e72	0.502148
$147$	−2.29682e72	−0.570158
$148$	4.25625e72	0.841898
$149$	−8.26372e72	−1.30447	−0.652237	−	0.758015i	$-0.726169\pi$
$149$	−8.26372e72	−1.30447	−0.652237	+	0.758015i	$0.726169\pi$
$150$	1.43781e72	0.181402
$151$	4.29912e72	0.434162	0.217081	−	0.976154i	$-0.430346\pi$
$151$	4.29912e72	0.434162	0.217081	+	0.976154i	$0.430346\pi$
$152$	−7.11742e72	−0.576185
$153$	5.83326e72	0.379095
$154$	4.84368e71	0.0253064
$155$	5.42484e73	2.28179
$156$	−8.09670e72	−0.274558
$157$	1.94458e73	0.532338	0.266169	−	0.963926i	$-0.414242\pi$
$157$	1.94458e73	0.532338	0.266169	+	0.963926i	$0.414242\pi$
$158$	−1.40810e73	−0.311617
$159$	1.38407e73	0.247944
$160$	−7.50737e73	−1.09010
$161$	−2.24706e72	−0.0264819
$162$	3.20162e72	0.0306631
$163$	−1.41942e74	−1.10618	−0.553089	−	0.833122i	$-0.686551\pi$
$163$	−1.41942e74	−1.10618	−0.553089	+	0.833122i	$0.686551\pi$
$164$	−4.31913e73	−0.274238
$165$	−1.33929e74	−0.693687
$166$	6.79036e73	0.287258
$167$	−1.52804e74	−0.528606	−0.264303	−	0.964440i	$-0.585142\pi$
$167$	−1.52804e74	−0.528606	−0.264303	+	0.964440i	$0.585142\pi$
$168$	−1.20791e73	−0.0342116
$169$	−3.16600e74	−0.735031
$170$	−2.40894e74	−0.458973
$171$	2.31090e74	0.361753
$172$	5.65316e74	0.727928
$173$	−9.24294e74	−0.980090	−0.490045	−	0.871697i	$-0.663020\pi$
$173$	−9.24294e74	−0.980090	−0.490045	+	0.871697i	$0.663020\pi$
$174$	−3.00663e74	−0.262833
$175$	1.76126e74	0.127072
$176$	1.07138e75	0.638659
$177$	1.25506e75	0.618814
$178$	−9.81382e74	−0.400652
$179$	3.33858e75	1.12976	0.564878	−	0.825174i	$-0.308923\pi$
$179$	3.33858e75	1.12976	0.564878	+	0.825174i	$0.308923\pi$
$180$	1.60385e75	0.450333
$181$	−8.81641e74	−0.205617	−0.102809	−	0.994701i	$-0.532783\pi$
$181$	−8.81641e74	−0.205617	−0.102809	+	0.994701i	$0.532783\pi$
$182$	8.17623e73	0.0158548
$183$	−6.26712e74	−0.101148
$184$	9.36847e74	0.125971
$185$	1.18843e76	1.33266
$186$	−2.65583e75	−0.248609
$187$	1.19462e76	0.934407
$188$	−1.02060e75	−0.0667680
$189$	3.92188e74	0.0214795
$190$	−9.54325e75	−0.437976
$191$	−1.49891e76	−0.576975	−0.288487	−	0.957484i	$-0.593152\pi$
$191$	−1.49891e76	−0.576975	−0.288487	+	0.957484i	$0.593152\pi$
$192$	−1.02124e76	−0.330018
$193$	9.85154e74	0.0267506	0.0133753	−	0.999911i	$-0.495742\pi$
$193$	9.85154e74	0.0267506	0.0133753	+	0.999911i	$0.495742\pi$
$194$	−2.37736e76	−0.542922
$195$	−2.26076e76	−0.434605
$196$	5.63290e76	0.912333
$197$	−3.04616e76	−0.416039	−0.208020	−	0.978125i	$-0.566702\pi$
$197$	−3.04616e76	−0.416039	−0.208020	+	0.978125i	$0.566702\pi$
$198$	6.55675e75	0.0755796
$199$	−1.34434e77	−1.30897	−0.654487	−	0.756073i	$-0.727115\pi$
$199$	−1.34434e77	−1.30897	−0.654487	+	0.756073i	$0.727115\pi$
$200$	−7.34307e76	−0.604468
$201$	1.40807e77	0.980742
$202$	−6.51524e76	−0.384291
$203$	−3.68302e76	−0.184114
$204$	−1.43060e77	−0.606606
$205$	−1.20599e77	−0.434098
$206$	1.27189e76	0.0388954
$207$	−3.04179e76	−0.0790901
$208$	1.80851e77	0.400129
$209$	4.73261e77	0.891662
$210$	−1.61960e76	−0.0260053
$211$	8.41155e77	1.15190	0.575948	−	0.817486i	$-0.304633\pi$
$211$	8.41155e77	1.15190	0.575948	+	0.817486i	$0.304633\pi$
$212$	−3.39439e77	−0.396745
$213$	−7.32798e77	−0.731590
$214$	2.29601e77	0.195935
$215$	1.57847e78	1.15225
$216$	−1.63511e77	−0.102175
$217$	−3.25330e77	−0.174150
$218$	6.82714e77	0.313290
$219$	2.66864e78	1.05054
$220$	3.28459e78	1.11000
$221$	2.01655e78	0.585419
$222$	−5.81815e77	−0.145198
$223$	−8.19828e78	−1.75999	−0.879994	−	0.474985i	$-0.842453\pi$
$223$	−8.19828e78	−1.75999	−0.879994	+	0.474985i	$0.842453\pi$
$224$	4.50220e77	0.0831986
$225$	2.38417e78	0.379510
$226$	1.77166e78	0.243079
$227$	−1.56505e79	−1.85209	−0.926045	−	0.377414i	$-0.876813\pi$
$227$	−1.56505e79	−1.85209	−0.926045	+	0.377414i	$0.876813\pi$
$228$	−5.66745e78	−0.578856
$229$	−2.10680e79	−1.85838	−0.929191	−	0.369601i	$-0.879494\pi$
$229$	−2.10680e79	−1.85838	−0.929191	+	0.369601i	$0.879494\pi$
$230$	1.25615e78	0.0957547
$231$	8.03179e77	0.0529433
$232$	1.53553e79	0.875810
$233$	2.97821e79	1.47073	0.735364	−	0.677672i	$-0.237011\pi$
$233$	2.97821e79	1.47073	0.735364	+	0.677672i	$0.237011\pi$
$234$	1.10679e78	0.0473517
$235$	−2.84972e78	−0.105689
$236$	−3.07801e79	−0.990190
$237$	−2.33491e79	−0.651930
$238$	1.44465e78	0.0350295
$239$	3.43905e79	0.724618	0.362309	−	0.932058i	$-0.381989\pi$
$239$	3.43905e79	0.724618	0.362309	+	0.932058i	$0.381989\pi$
$240$	−3.58242e79	−0.656295
$241$	−4.23443e79	−0.674876	−0.337438	−	0.941348i	$-0.609560\pi$
$241$	−4.23443e79	−0.674876	−0.337438	+	0.941348i	$0.609560\pi$
$242$	−6.46394e78	−0.0896772
$243$	5.30893e78	0.0641500
$244$	1.53700e79	0.161851
$245$	1.57281e80	1.44415
$246$	5.90412e78	0.0472965
$247$	7.98875e79	0.558639
$248$	1.35637e80	0.828411
$249$	1.12598e80	0.600969
$250$	−1.19798e79	−0.0559063
$251$	3.93091e79	0.160481	0.0802405	−	0.996776i	$-0.474431\pi$
$251$	3.93091e79	0.160481	0.0802405	+	0.996776i	$0.474431\pi$
$252$	−9.61833e78	−0.0343701
$253$	−6.22941e79	−0.194944
$254$	−2.03035e79	−0.0556727
$255$	−3.99450e80	−0.960211
$256$	1.52889e80	0.322358
$257$	1.25034e80	0.231349	0.115675	−	0.993287i	$-0.463097\pi$
$257$	1.25034e80	0.231349	0.115675	+	0.993287i	$0.463097\pi$
$258$	−7.72769e79	−0.125542
$259$	−7.12704e79	−0.101711
$260$	5.54446e80	0.695429
$261$	−4.98560e80	−0.549870
$262$	−3.19592e80	−0.310100
$263$	−4.53381e79	−0.0387209	−0.0193605	−	0.999813i	$-0.506163\pi$
$263$	−4.53381e79	−0.0387209	−0.0193605	+	0.999813i	$0.506163\pi$
$264$	−3.34862e80	−0.251846
$265$	−9.47781e80	−0.628017
$266$	5.72312e79	0.0334271
$267$	−1.62733e81	−0.838200
$268$	−3.45325e81	−1.56933
$269$	3.78943e81	1.52010	0.760050	−	0.649864i	$-0.225174\pi$
$269$	3.78943e81	1.52010	0.760050	+	0.649864i	$0.225174\pi$
$270$	−2.19241e80	−0.0776667
$271$	−4.35153e81	−1.36198	−0.680990	−	0.732292i	$-0.738450\pi$
$271$	−4.35153e81	−1.36198	−0.680990	+	0.732292i	$0.738450\pi$
$272$	3.19544e81	0.884041
$273$	1.35578e80	0.0331697
$274$	−6.80994e80	−0.147401
$275$	4.88266e81	0.935429
$276$	7.45992e80	0.126555
$277$	1.35111e81	0.203057	0.101529	−	0.994833i	$-0.467627\pi$
$277$	1.35111e81	0.203057	0.101529	+	0.994833i	$0.467627\pi$
$278$	−2.79401e81	−0.372158
$279$	−4.40390e81	−0.520111
$280$	8.27152e80	0.0866546
$281$	1.65152e82	1.53540	0.767699	−	0.640811i	$-0.221402\pi$
$281$	1.65152e82	1.53540	0.767699	+	0.640811i	$0.221402\pi$
$282$	1.39513e80	0.0115151
$283$	1.17492e82	0.861318	0.430659	−	0.902515i	$-0.358281\pi$
$283$	1.17492e82	0.861318	0.430659	+	0.902515i	$0.358281\pi$
$284$	1.79717e82	1.17065
$285$	−1.58246e82	−0.916285
$286$	2.26665e81	0.116714
$287$	7.23234e80	0.0331311
$288$	6.09449e81	0.248479
$289$	8.08291e81	0.293420
$290$	2.05888e82	0.665730
$291$	−3.94215e82	−1.13584
$292$	−6.54479e82	−1.68101
$293$	6.21701e82	1.42402	0.712010	−	0.702169i	$-0.247785\pi$
$293$	6.21701e82	1.42402	0.712010	+	0.702169i	$0.247785\pi$
$294$	−7.69999e81	−0.157346
$295$	−8.59441e82	−1.56740
$296$	2.97141e82	0.483827
$297$	1.08724e82	0.158119
$298$	−2.77038e82	−0.359994
$299$	−1.05154e82	−0.122135
$300$	−5.84714e82	−0.607269
$301$	−9.46616e81	−0.0879419
$302$	1.44127e82	0.119815
$303$	−1.08036e83	−0.803970
$304$	1.26590e83	0.843599
$305$	4.29160e82	0.256198
$306$	1.95558e82	0.104618
$307$	−2.65112e83	−1.27143	−0.635716	−	0.771923i	$-0.719295\pi$
$307$	−2.65112e83	−1.27143	−0.635716	+	0.771923i	$0.719295\pi$
$308$	−1.96978e82	−0.0847167
$309$	2.10905e82	0.0813727
$310$	1.81866e83	0.629701
$311$	5.62904e82	0.174969	0.0874846	−	0.996166i	$-0.472117\pi$
$311$	5.62904e82	0.174969	0.0874846	+	0.996166i	$0.472117\pi$
$312$	−5.65255e82	−0.157785
$313$	1.23760e83	0.310346	0.155173	−	0.987887i	$-0.450407\pi$
$313$	1.23760e83	0.310346	0.155173	+	0.987887i	$0.450407\pi$
$314$	6.51913e82	0.146908
$315$	−2.68562e82	−0.0544054
$316$	5.72633e83	1.04318
$317$	9.23522e82	0.151343	0.0756714	−	0.997133i	$-0.475890\pi$
$317$	9.23522e82	0.151343	0.0756714	+	0.997133i	$0.475890\pi$
$318$	4.64003e82	0.0684246
$319$	−1.02102e84	−1.35534
$320$	6.99327e83	0.835903
$321$	3.80724e83	0.409913
$322$	−7.53320e81	−0.00730815
$323$	1.41152e84	1.23425
$324$	−1.30201e83	−0.102649
$325$	8.24203e83	0.586060
$326$	−4.75854e83	−0.305270
$327$	1.13208e84	0.655430
$328$	−3.01531e83	−0.157601
$329$	1.70899e82	0.00806633
$330$	−4.48993e83	−0.191436
$331$	−2.42406e84	−0.933911	−0.466956	−	0.884281i	$-0.654649\pi$
$331$	−2.42406e84	−0.933911	−0.466956	+	0.884281i	$0.654649\pi$
$332$	−2.76144e84	−0.961635
$333$	−9.64767e83	−0.303767
$334$	−5.12269e83	−0.145879
$335$	−9.64216e84	−2.48412
$336$	2.14839e83	0.0500895
$337$	4.59459e84	0.969717	0.484858	−	0.874593i	$-0.338871\pi$
$337$	4.59459e84	0.969717	0.484858	+	0.874593i	$0.338871\pi$
$338$	−1.06139e84	−0.202845
$339$	2.93777e84	0.508543
$340$	9.79645e84	1.53647
$341$	−9.01894e84	−1.28199
$342$	7.74722e83	0.0998324
$343$	−1.89834e84	−0.221831
$344$	3.94664e84	0.418330
$345$	2.08296e84	0.200327
$346$	−3.09866e84	−0.270474
$347$	−9.65988e84	−0.765481	−0.382740	−	0.923856i	$-0.625020\pi$
$347$	−9.65988e84	−0.765481	−0.382740	+	0.923856i	$0.625020\pi$
$348$	1.22271e85	0.879869
$349$	−1.13452e85	−0.741586	−0.370793	−	0.928715i	$-0.620914\pi$
$349$	−1.13452e85	−0.741586	−0.370793	+	0.928715i	$0.620914\pi$
$350$	5.90457e83	0.0350678
$351$	1.83529e84	0.0990639
$352$	1.24812e85	0.612459
$353$	−1.87681e85	−0.837467	−0.418733	−	0.908109i	$-0.637526\pi$
$353$	−1.87681e85	−0.837467	−0.418733	+	0.908109i	$0.637526\pi$
$354$	4.20755e84	0.170773
$355$	5.01806e85	1.85305
$356$	3.99099e85	1.34124
$357$	2.39552e84	0.0732849
$358$	1.11925e85	0.311777
$359$	6.21320e83	0.0157634	0.00788172	−	0.999969i	$-0.497491\pi$
$359$	6.21320e83	0.0157634	0.00788172	+	0.999969i	$0.497491\pi$
$360$	1.11969e85	0.258800
$361$	8.44101e84	0.177788
$362$	−2.95567e84	−0.0567439
$363$	−1.07185e85	−0.187613
$364$	−3.32504e84	−0.0530762
$365$	−1.82743e86	−2.66091
$366$	−2.10103e84	−0.0279136
$367$	−1.24848e85	−0.151379	−0.0756897	−	0.997131i	$-0.524116\pi$
$367$	−1.24848e85	−0.151379	−0.0756897	+	0.997131i	$0.524116\pi$
$368$	−1.66628e85	−0.184436
$369$	9.79021e84	0.0989484
$370$	3.98416e85	0.367772
$371$	5.68388e84	0.0479313
$372$	1.08005e86	0.832251
$373$	−1.31895e86	−0.928929	−0.464464	−	0.885592i	$-0.653753\pi$
$373$	−1.31895e86	−0.928929	−0.464464	+	0.885592i	$0.653753\pi$
$374$	4.00492e85	0.257867
$375$	−1.98650e85	−0.116961
$376$	−7.12512e84	−0.0383707
$377$	−1.72351e86	−0.849139
$378$	1.31479e84	0.00592765
$379$	−3.61053e86	−1.48990	−0.744950	−	0.667120i	$-0.767527\pi$
$379$	−3.61053e86	−1.48990	−0.744950	+	0.667120i	$0.767527\pi$
$380$	3.88096e86	1.46618
$381$	−3.36674e85	−0.116472
$382$	−5.02503e85	−0.159227
$383$	3.87398e85	0.112460	0.0562301	−	0.998418i	$-0.482092\pi$
$383$	3.87398e85	0.112460	0.0562301	+	0.998418i	$0.482092\pi$
$384$	−1.96025e86	−0.521452
$385$	−5.50001e85	−0.134100
$386$	3.30269e84	0.00738233
$387$	−1.28141e86	−0.262645
$388$	9.66804e86	1.81750
$389$	4.23652e86	0.730631	0.365315	−	0.930884i	$-0.380961\pi$
$389$	4.23652e86	0.730631	0.365315	+	0.930884i	$0.380961\pi$
$390$	−7.57911e85	−0.119937
$391$	−1.85795e86	−0.269844
$392$	3.93249e86	0.524306
$393$	−5.29947e86	−0.648757
$394$	−1.02121e86	−0.114814
$395$	1.59890e87	1.65127
$396$	−2.66644e86	−0.253013
$397$	−4.37543e86	−0.381539	−0.190769	−	0.981635i	$-0.561098\pi$
$397$	−4.37543e86	−0.381539	−0.190769	+	0.981635i	$0.561098\pi$
$398$	−4.50685e86	−0.361235
$399$	9.49009e85	0.0699324
$400$	1.30604e87	0.885007
$401$	3.53648e86	0.220412	0.110206	−	0.993909i	$-0.464849\pi$
$401$	3.53648e86	0.220412	0.110206	+	0.993909i	$0.464849\pi$
$402$	4.72049e86	0.270654
$403$	−1.52242e87	−0.803184
$404$	2.64956e87	1.28646
$405$	−3.63545e86	−0.162486
$406$	−1.23472e86	−0.0508096
$407$	−1.97579e87	−0.748735
$408$	−9.98742e86	−0.348608
$409$	−3.87688e87	−1.24667	−0.623336	−	0.781954i	$-0.714223\pi$
$409$	−3.87688e87	−1.24667	−0.623336	+	0.781954i	$0.714223\pi$
$410$	−4.04302e86	−0.119797
$411$	−1.12922e87	−0.308375
$412$	−5.17242e86	−0.130208
$413$	5.15410e86	0.119626
$414$	−1.01975e86	−0.0218264
$415$	−7.71049e87	−1.52220
$416$	2.10685e87	0.383714
$417$	−4.63304e87	−0.778588
$418$	1.58659e87	0.246070
$419$	−1.26351e87	−0.180888	−0.0904440	−	0.995902i	$-0.528829\pi$
$419$	−1.26351e87	−0.180888	−0.0904440	+	0.995902i	$0.528829\pi$
$420$	6.58644e86	0.0870562
$421$	8.72843e87	1.06533	0.532667	−	0.846325i	$-0.321190\pi$
$421$	8.72843e87	1.06533	0.532667	+	0.846325i	$0.321190\pi$
$422$	2.81994e87	0.317887
$423$	2.31341e86	0.0240907
$424$	−2.36973e87	−0.228004
$425$	1.45628e88	1.29483
$426$	−2.45668e87	−0.201896
$427$	−2.57369e86	−0.0195534
$428$	−9.33718e87	−0.655918
$429$	3.75857e87	0.244176
$430$	5.29177e87	0.317986
$431$	−3.03682e87	−0.168823	−0.0844114	−	0.996431i	$-0.526901\pi$
$431$	−3.03682e87	−0.168823	−0.0844114	+	0.996431i	$0.526901\pi$
$432$	2.90821e87	0.149596
$433$	2.79557e88	1.33084	0.665420	−	0.746469i	$-0.268252\pi$
$433$	2.79557e88	1.33084	0.665420	+	0.746469i	$0.268252\pi$
$434$	−1.09066e87	−0.0480598
$435$	3.41404e88	1.39277
$436$	−2.77640e88	−1.04878
$437$	−7.36046e87	−0.257500
$438$	8.94652e87	0.289915
$439$	−1.60107e88	−0.480673	−0.240336	−	0.970690i	$-0.577258\pi$
$439$	−1.60107e88	−0.480673	−0.240336	+	0.970690i	$0.577258\pi$
$440$	2.29307e88	0.637900
$441$	−1.27681e88	−0.329181
$442$	6.76040e87	0.161557
$443$	−8.51451e88	−1.88641	−0.943203	−	0.332218i	$-0.892203\pi$
$443$	−8.51451e88	−1.88641	−0.943203	+	0.332218i	$0.892203\pi$
$444$	2.36607e88	0.486070
$445$	1.11436e89	2.12308
$446$	−2.74844e88	−0.485701
$447$	−4.59385e88	−0.753138
$448$	−4.19389e87	−0.0637974
$449$	−5.20598e88	−0.734936	−0.367468	−	0.930036i	$-0.619775\pi$
$449$	−5.20598e88	−0.734936	−0.367468	+	0.930036i	$0.619775\pi$
$450$	7.99285e87	0.104733
$451$	2.00498e88	0.243891
$452$	−7.20483e88	−0.813741
$453$	2.38991e88	0.250664
$454$	−5.24677e88	−0.511118
$455$	−9.28415e87	−0.0840157
$456$	−3.95661e88	−0.332661
$457$	−6.92145e88	−0.540759	−0.270380	−	0.962754i	$-0.587149\pi$
$457$	−6.92145e88	−0.540759	−0.270380	+	0.962754i	$0.587149\pi$
$458$	−7.06297e88	−0.512854
$459$	3.24275e88	0.218871
$460$	−5.10841e88	−0.320552
$461$	9.18444e87	0.0535885	0.0267943	−	0.999641i	$-0.491470\pi$
$461$	9.18444e87	0.0535885	0.0267943	+	0.999641i	$0.491470\pi$
$462$	2.69263e87	0.0146107
$463$	1.12588e89	0.568235	0.284117	−	0.958790i	$-0.408299\pi$
$463$	1.12588e89	0.568235	0.284117	+	0.958790i	$0.408299\pi$
$464$	−2.73109e89	−1.28228
$465$	3.01570e89	1.31739
$466$	9.98435e88	0.405874
$467$	4.76530e89	1.80291	0.901456	−	0.432872i	$-0.142500\pi$
$467$	4.76530e89	1.80291	0.901456	+	0.432872i	$0.142500\pi$
$468$	−4.50101e88	−0.158516
$469$	5.78244e88	0.189592
$470$	−9.55358e87	−0.0291667
$471$	1.08100e89	0.307345
$472$	−2.14885e89	−0.569049
$473$	−2.62425e89	−0.647377
$474$	−7.82771e88	−0.179912
$475$	5.76918e89	1.23560
$476$	−5.87497e88	−0.117266
$477$	7.69410e88	0.143150
$478$	1.15293e89	0.199972
$479$	6.71576e88	0.108606	0.0543032	−	0.998524i	$-0.482706\pi$
$479$	6.71576e88	0.108606	0.0543032	+	0.998524i	$0.482706\pi$
$480$	−4.17339e89	−0.629372
$481$	−3.33518e89	−0.469093
$482$	−1.41958e89	−0.186244
$483$	−1.24916e88	−0.0152893
$484$	2.62869e89	0.300207
$485$	2.69950e90	2.87697
$486$	1.77980e88	0.0177034
$487$	−6.01760e89	−0.558732	−0.279366	−	0.960185i	$-0.590124\pi$
$487$	−6.01760e89	−0.558732	−0.279366	+	0.960185i	$0.590124\pi$
$488$	1.07303e89	0.0930135
$489$	−7.89062e89	−0.638652
$490$	5.27280e89	0.398541
$491$	−1.54202e90	−1.08858	−0.544289	−	0.838898i	$-0.683200\pi$
$491$	−1.54202e90	−1.08858	−0.544289	+	0.838898i	$0.683200\pi$
$492$	−2.40103e89	−0.158331
$493$	−3.04525e90	−1.87608
$494$	2.67820e89	0.154167
$495$	−7.44521e89	−0.400500
$496$	−2.41244e90	−1.21289
$497$	−3.00935e89	−0.141427
$498$	3.77480e89	0.165848
$499$	−2.50059e90	−1.02724	−0.513622	−	0.858017i	$-0.671697\pi$
$499$	−2.50059e90	−1.02724	−0.513622	+	0.858017i	$0.671697\pi$
$500$	4.87185e89	0.187154
$501$	−8.49446e89	−0.305191
$502$	1.31782e89	0.0442877
$503$	5.51357e90	1.73343	0.866713	−	0.498807i	$-0.166228\pi$
$503$	5.51357e90	1.73343	0.866713	+	0.498807i	$0.166228\pi$
$504$	−6.71484e88	−0.0197521
$505$	7.39809e90	2.03638
$506$	−2.08839e89	−0.0537983
$507$	−1.76000e90	−0.424370
$508$	8.25686e89	0.186372
$509$	−4.80271e90	−1.01494	−0.507470	−	0.861669i	$-0.669419\pi$
$509$	−4.80271e90	−1.01494	−0.507470	+	0.861669i	$0.669419\pi$
$510$	−1.33914e90	−0.264988
$511$	1.09592e90	0.203085
$512$	5.71634e90	0.992143
$513$	1.28465e90	0.208858
$514$	4.19172e89	0.0638451
$515$	−1.44424e90	−0.206109
$516$	3.14263e90	0.420269
$517$	4.73773e89	0.0593796
$518$	−2.38931e89	−0.0280690
$519$	−5.13821e90	−0.565855
$520$	3.87075e90	0.399653
$521$	1.73864e91	1.68324	0.841618	−	0.540073i	$-0.181603\pi$
$521$	1.73864e91	1.68324	0.841618	+	0.540073i	$0.181603\pi$
$522$	−1.67140e90	−0.151747
$523$	−5.15385e90	−0.438859	−0.219430	−	0.975628i	$-0.570420\pi$
$523$	−5.15385e90	−0.438859	−0.219430	+	0.975628i	$0.570420\pi$
$524$	1.29969e91	1.03810
$525$	9.79097e89	0.0733650
$526$	−1.51994e89	−0.0106857
$527$	−2.68994e91	−1.77455
$528$	5.95586e90	0.368730
$529$	−1.62405e91	−0.943703
$530$	−3.17740e90	−0.173313
$531$	6.97696e90	0.357273
$532$	−2.32743e90	−0.111902
$533$	3.38446e90	0.152801
$534$	−5.45556e90	−0.231317
$535$	−2.60712e91	−1.03827
$536$	−2.41082e91	−0.901870
$537$	1.85593e91	0.652265
$538$	1.27039e91	0.419500
$539$	−2.61485e91	−0.811377
$540$	8.91588e90	0.260000
$541$	4.02418e91	1.10298	0.551492	−	0.834180i	$-0.314059\pi$
$541$	4.02418e91	1.10298	0.551492	+	0.834180i	$0.314059\pi$
$542$	−1.45884e91	−0.375864
$543$	−4.90110e90	−0.118713
$544$	3.72258e91	0.847775
$545$	−7.75225e91	−1.66014
$546$	4.54522e89	0.00915380
$547$	−3.43638e90	−0.0650917	−0.0325459	−	0.999470i	$-0.510361\pi$
$547$	−3.43638e90	−0.0650917	−0.0325459	+	0.999470i	$0.510361\pi$
$548$	2.76940e91	0.493443
$549$	−3.48393e90	−0.0583978
$550$	1.63689e91	0.258149
$551$	−1.20641e92	−1.79026
$552$	5.20799e90	0.0727296
$553$	−9.58868e90	−0.126028
$554$	4.52955e90	0.0560374
$555$	6.60654e91	0.769412
$556$	1.13624e92	1.24585
$557$	1.44119e92	1.48790	0.743949	−	0.668237i	$-0.232951\pi$
$557$	1.44119e92	1.48790	0.743949	+	0.668237i	$0.232951\pi$
$558$	−1.47639e91	−0.143534
$559$	−4.42980e91	−0.405591
$560$	−1.47117e91	−0.126872
$561$	6.64097e91	0.539480
$562$	5.53665e91	0.423721
$563$	1.95881e92	1.41241	0.706206	−	0.708007i	$-0.250405\pi$
$563$	1.95881e92	1.41241	0.706206	+	0.708007i	$0.250405\pi$
$564$	−5.67359e90	−0.0385485
$565$	−2.01173e92	−1.28809
$566$	3.93889e91	0.237697
$567$	2.18019e90	0.0124012
$568$	1.25466e92	0.672755
$569$	−2.35622e92	−1.19112	−0.595560	−	0.803311i	$-0.703070\pi$
$569$	−2.35622e92	−1.19112	−0.595560	+	0.803311i	$0.703070\pi$
$570$	−5.30515e91	−0.252866
$571$	1.13360e92	0.509508	0.254754	−	0.967006i	$-0.418006\pi$
$571$	1.13360e92	0.509508	0.254754	+	0.967006i	$0.418006\pi$
$572$	−9.21782e91	−0.390716
$573$	−8.33251e91	−0.333116
$574$	2.42462e90	0.00914313
$575$	−7.59382e91	−0.270139
$576$	−5.67715e91	−0.190536
$577$	3.53984e92	1.12097	0.560486	−	0.828164i	$-0.310615\pi$
$577$	3.53984e92	1.12097	0.560486	+	0.828164i	$0.310615\pi$
$578$	2.70977e91	0.0809747
$579$	5.47653e90	0.0154445
$580$	−8.37287e92	−2.22862
$581$	4.62401e91	0.116176
$582$	−1.32159e92	−0.313456
$583$	1.57571e92	0.352842
$584$	−4.56911e92	−0.966054
$585$	−1.25677e92	−0.250919
$586$	2.08423e92	0.392985
$587$	7.72129e92	1.37503	0.687517	−	0.726168i	$-0.258701\pi$
$587$	7.72129e92	1.37503	0.687517	+	0.726168i	$0.258701\pi$
$588$	3.13136e92	0.526736
$589$	−1.06565e93	−1.69337
$590$	−2.88125e92	−0.432552
$591$	−1.69338e92	−0.240200
$592$	−5.28496e92	−0.708377
$593$	5.37898e92	0.681346	0.340673	−	0.940182i	$-0.389345\pi$
$593$	5.37898e92	0.681346	0.340673	+	0.940182i	$0.389345\pi$
$594$	3.64493e91	0.0436359
$595$	−1.64040e92	−0.185623
$596$	1.12663e93	1.20513
$597$	−7.47327e92	−0.755737
$598$	−3.52525e91	−0.0337054
$599$	1.64953e93	1.49129	0.745645	−	0.666344i	$-0.232142\pi$
$599$	1.64953e93	1.49129	0.745645	+	0.666344i	$0.232142\pi$
$600$	−4.08206e92	−0.348990
$601$	6.82596e92	0.551911	0.275955	−	0.961170i	$-0.411006\pi$
$601$	6.82596e92	0.551911	0.275955	+	0.961170i	$0.411006\pi$
$602$	−3.17349e91	−0.0242692
$603$	7.82752e92	0.566232
$604$	−5.86121e92	−0.401097
$605$	7.33982e92	0.475205
$606$	−3.62186e92	−0.221870
$607$	8.74935e92	0.507171	0.253586	−	0.967313i	$-0.418390\pi$
$607$	8.74935e92	0.507171	0.253586	+	0.967313i	$0.418390\pi$
$608$	1.47474e93	0.808992
$609$	−2.04741e92	−0.106298
$610$	1.43874e92	0.0707024
$611$	7.99740e91	0.0372022
$612$	−7.95277e92	−0.350224
$613$	3.67026e93	1.53028	0.765141	−	0.643863i	$-0.222669\pi$
$613$	3.67026e93	1.53028	0.765141	+	0.643863i	$0.222669\pi$
$614$	−8.88777e92	−0.350875
$615$	−6.70415e92	−0.250627
$616$	−1.37516e92	−0.0486856
$617$	5.28402e92	0.177179	0.0885894	−	0.996068i	$-0.471764\pi$
$617$	5.28402e92	0.177179	0.0885894	+	0.996068i	$0.471764\pi$
$618$	7.07053e91	0.0224563
$619$	−2.63583e93	−0.793015	−0.396507	−	0.918032i	$-0.629778\pi$
$619$	−2.63583e93	−0.793015	−0.396507	+	0.918032i	$0.629778\pi$
$620$	−7.39595e93	−2.10801
$621$	−1.69095e92	−0.0456627
$622$	1.88712e92	0.0482860
$623$	−6.68287e92	−0.162037
$624$	1.00536e93	0.231014
$625$	−3.86756e93	−0.842279
$626$	4.14902e92	0.0856456
$627$	2.63089e93	0.514801
$628$	−2.65114e93	−0.491796
$629$	−5.89289e93	−1.03641
$630$	−9.00346e91	−0.0150142
$631$	5.00934e93	0.792131	0.396065	−	0.918222i	$-0.370375\pi$
$631$	5.00934e93	0.792131	0.396065	+	0.918222i	$0.370375\pi$
$632$	3.99772e93	0.599501
$633$	4.67603e93	0.665048
$634$	3.09607e92	0.0417658
$635$	2.30548e93	0.295013
$636$	−1.88696e93	−0.229061
$637$	−4.41392e93	−0.508339
$638$	−3.42294e93	−0.374031
$639$	−4.07367e93	−0.422384
$640$	1.34234e94	1.32079
$641$	−6.57435e93	−0.613916	−0.306958	−	0.951723i	$-0.599311\pi$
$641$	−6.57435e93	−0.613916	−0.306958	+	0.951723i	$0.599311\pi$
$642$	1.27636e93	0.113123
$643$	−1.69643e94	−1.42715	−0.713575	−	0.700579i	$-0.752925\pi$
$643$	−1.69643e94	−1.42715	−0.713575	+	0.700579i	$0.752925\pi$
$644$	3.06353e92	0.0244650
$645$	8.77482e93	0.665255
$646$	4.73208e93	0.340614
$647$	−1.70171e94	−1.16304	−0.581518	−	0.813534i	$-0.697541\pi$
$647$	−1.70171e94	−1.16304	−0.581518	+	0.813534i	$0.697541\pi$
$648$	−9.08968e92	−0.0589910
$649$	1.42884e94	0.880618
$650$	2.76311e93	0.161734
$651$	−1.80853e93	−0.100545
$652$	1.93516e94	1.02193
$653$	1.60857e94	0.806954	0.403477	−	0.914990i	$-0.367802\pi$
$653$	1.60857e94	0.806954	0.403477	+	0.914990i	$0.367802\pi$
$654$	3.79525e93	0.180878
$655$	3.62897e94	1.64324
$656$	5.36304e93	0.230745
$657$	1.48351e94	0.606529
$658$	5.72932e91	0.00222605
$659$	−3.12702e94	−1.15470	−0.577351	−	0.816496i	$-0.695913\pi$
$659$	−3.12702e94	−1.15470	−0.577351	+	0.816496i	$0.695913\pi$
$660$	1.82592e94	0.640857
$661$	−3.91430e94	−1.30589	−0.652943	−	0.757407i	$-0.726466\pi$
$661$	−3.91430e94	−1.30589	−0.652943	+	0.757407i	$0.726466\pi$
$662$	−8.12657e93	−0.257730
$663$	1.12101e94	0.337992
$664$	−1.92784e94	−0.552639
$665$	−6.49863e93	−0.177132
$666$	−3.23435e93	−0.0838301
$667$	1.58796e94	0.391403
$668$	2.08325e94	0.488349
$669$	−4.55747e94	−1.01613
$670$	−3.23250e94	−0.685540
$671$	−7.13491e93	−0.143941
$672$	2.50280e93	0.0480347
$673$	−5.60089e94	−1.02271	−0.511355	−	0.859370i	$-0.670856\pi$
$673$	−5.60089e94	−1.02271	−0.511355	+	0.859370i	$0.670856\pi$
$674$	1.54032e94	0.267611
$675$	1.32538e94	0.219110
$676$	4.31636e94	0.679052
$677$	−3.87098e94	−0.579563	−0.289781	−	0.957093i	$-0.593583\pi$
$677$	−3.87098e94	−0.579563	−0.289781	+	0.957093i	$0.593583\pi$
$678$	9.84877e93	0.140342
$679$	−1.61890e94	−0.219575
$680$	6.83919e94	0.882991
$681$	−8.70021e94	−1.06930
$682$	−3.02357e94	−0.353788
$683$	−4.48374e94	−0.499513	−0.249757	−	0.968309i	$-0.580351\pi$
$683$	−4.48374e94	−0.499513	−0.249757	+	0.968309i	$0.580351\pi$
$684$	−3.15057e94	−0.334203
$685$	7.73271e94	0.781084
$686$	−6.36413e93	−0.0612183
$687$	−1.17118e95	−1.07294
$688$	−7.01950e94	−0.612482
$689$	2.65984e94	0.221060
$690$	6.98303e93	0.0552840
$691$	4.77869e94	0.360408	0.180204	−	0.983629i	$-0.442324\pi$
$691$	4.77869e94	0.360408	0.180204	+	0.983629i	$0.442324\pi$
$692$	1.26014e95	0.905447
$693$	4.46492e93	0.0305668
$694$	−3.23844e94	−0.211248
$695$	3.17261e95	1.97209
$696$	8.53609e94	0.505649
$697$	5.97996e94	0.337598
$698$	−3.80345e94	−0.204654
$699$	1.65561e95	0.849125
$700$	−2.40122e94	−0.117394
$701$	2.06236e95	0.961193	0.480596	−	0.876942i	$-0.340420\pi$
$701$	2.06236e95	0.961193	0.480596	+	0.876942i	$0.340420\pi$
$702$	6.15273e93	0.0273385
$703$	−2.33453e95	−0.988998
$704$	−1.16265e95	−0.469639
$705$	−1.58418e94	−0.0610194
$706$	−6.29194e94	−0.231114
$707$	−4.43666e94	−0.155420
$708$	−1.71109e95	−0.571686
$709$	7.46676e94	0.237949	0.118974	−	0.992897i	$-0.462039\pi$
$709$	7.46676e94	0.237949	0.118974	+	0.992897i	$0.462039\pi$
$710$	1.68229e95	0.511382
$711$	−1.29799e95	−0.376392
$712$	2.78623e95	0.770792
$713$	1.40268e95	0.370221
$714$	8.03089e93	0.0202243
$715$	−2.57379e95	−0.618474
$716$	−4.55165e95	−1.04372
$717$	1.91179e95	0.418358
$718$	2.08295e93	0.00435021
$719$	4.88213e95	0.973176	0.486588	−	0.873632i	$-0.338241\pi$
$719$	4.88213e95	0.973176	0.486588	+	0.873632i	$0.338241\pi$
$720$	−1.99149e95	−0.378912
$721$	8.66116e93	0.0157306
$722$	2.82982e94	0.0490639
$723$	−2.35395e95	−0.389640
$724$	1.20199e95	0.189958
$725$	−1.24466e96	−1.87813
$726$	−3.59334e94	−0.0517752
$727$	−6.79529e95	−0.934986	−0.467493	−	0.883997i	$-0.654843\pi$
$727$	−6.79529e95	−0.934986	−0.467493	+	0.883997i	$0.654843\pi$
$728$	−2.32131e94	−0.0305022
$729$	2.95127e94	0.0370370
$730$	−6.12640e95	−0.734327
$731$	−7.82695e95	−0.896109
$732$	8.54428e94	0.0934446
$733$	9.40814e95	0.982926	0.491463	−	0.870899i	$-0.336462\pi$
$733$	9.40814e95	0.982926	0.491463	+	0.870899i	$0.336462\pi$
$734$	−4.18547e94	−0.0417759
$735$	8.74337e95	0.833783
$736$	−1.94116e95	−0.176870
$737$	1.60303e96	1.39567
$738$	3.28213e94	0.0273066
$739$	1.08587e96	0.863355	0.431678	−	0.902028i	$-0.357922\pi$
$739$	1.08587e96	0.863355	0.431678	+	0.902028i	$0.357922\pi$
$740$	−1.62024e96	−1.23117
$741$	4.44100e95	0.322530
$742$	1.90550e94	0.0132275
$743$	−1.14387e96	−0.759019	−0.379509	−	0.925188i	$-0.623907\pi$
$743$	−1.14387e96	−0.759019	−0.379509	+	0.925188i	$0.623907\pi$
$744$	7.54013e95	0.478284
$745$	3.14578e96	1.90763
$746$	−4.42173e95	−0.256355
$747$	6.25939e95	0.346970
$748$	−1.62869e96	−0.863244
$749$	1.56350e95	0.0792424
$750$	−6.65967e94	−0.0322775
$751$	1.14463e96	0.530551	0.265275	−	0.964173i	$-0.414537\pi$
$751$	1.14463e96	0.530551	0.265275	+	0.964173i	$0.414537\pi$
$752$	1.26728e95	0.0561789
$753$	2.18522e95	0.0926538
$754$	−5.77801e95	−0.234335
$755$	−1.63656e96	−0.634907
$756$	−5.34689e94	−0.0198436
$757$	1.59430e96	0.566056	0.283028	−	0.959112i	$-0.408661\pi$
$757$	1.59430e96	0.566056	0.283028	+	0.959112i	$0.408661\pi$
$758$	−1.21042e96	−0.411165
$759$	−3.46297e95	−0.112551
$760$	2.70941e96	0.842597
$761$	6.24206e95	0.185756	0.0928778	−	0.995678i	$-0.470393\pi$
$761$	6.24206e95	0.185756	0.0928778	+	0.995678i	$0.470393\pi$
$762$	−1.12869e95	−0.0321426
$763$	4.64905e95	0.126705
$764$	2.04353e96	0.533033
$765$	−2.22057e96	−0.554378
$766$	1.29874e95	0.0310355
$767$	2.41192e96	0.551720
$768$	8.49921e95	0.186114
$769$	−5.72458e96	−1.20008	−0.600041	−	0.799969i	$-0.704849\pi$
$769$	−5.72458e96	−1.20008	−0.600041	+	0.799969i	$0.704849\pi$
$770$	−1.84386e95	−0.0370074
$771$	6.95071e95	0.133570
$772$	−1.34311e95	−0.0247134
$773$	−7.07720e96	−1.24695	−0.623474	−	0.781844i	$-0.714279\pi$
$773$	−7.07720e96	−1.24695	−0.623474	+	0.781844i	$0.714279\pi$
$774$	−4.29587e95	−0.0724818
$775$	−1.09943e97	−1.77649
$776$	6.74954e96	1.04450
$777$	−3.96197e95	−0.0587228
$778$	1.42028e96	0.201631
$779$	2.36902e96	0.322154
$780$	3.08220e96	0.401506
$781$	−8.34266e96	−1.04111
$782$	−6.22872e95	−0.0744684
$783$	−2.77152e96	−0.317468
$784$	−6.99434e96	−0.767641
$785$	−7.40250e96	−0.778476
$786$	−1.77663e96	−0.179036
$787$	1.00852e97	0.973934	0.486967	−	0.873420i	$-0.338103\pi$
$787$	1.00852e97	0.973934	0.486967	+	0.873420i	$0.338103\pi$
$788$	4.15298e96	0.384354
$789$	−2.52037e95	−0.0223555
$790$	5.36026e96	0.455700
$791$	1.20644e96	0.0983091
$792$	−1.86152e96	−0.145403
$793$	−1.20439e96	−0.0901809
$794$	−1.46685e96	−0.105293
$795$	−5.26877e96	−0.362586
$796$	1.83281e97	1.20928
$797$	2.93518e97	1.85687	0.928434	−	0.371498i	$-0.121156\pi$
$797$	2.93518e97	1.85687	0.928434	+	0.371498i	$0.121156\pi$
$798$	3.18152e95	0.0192991
$799$	1.41305e96	0.0821941
$800$	1.52149e97	0.848702
$801$	−9.04642e96	−0.483935
$802$	1.18559e96	0.0608266
$803$	3.03815e97	1.49499
$804$	−1.91969e97	−0.906050
$805$	8.55398e95	0.0387263
$806$	−5.10385e96	−0.221653
$807$	2.10657e97	0.877630
$808$	1.84974e97	0.739314
$809$	−4.15643e97	−1.59384	−0.796921	−	0.604083i	$-0.793539\pi$
$809$	−4.15643e97	−1.59384	−0.796921	+	0.604083i	$0.793539\pi$
$810$	−1.21877e96	−0.0448409
$811$	−4.69239e97	−1.65652	−0.828258	−	0.560347i	$-0.810668\pi$
$811$	−4.69239e97	−1.65652	−0.828258	+	0.560347i	$0.810668\pi$
$812$	5.02124e96	0.170092
$813$	−2.41904e97	−0.786340
$814$	−6.62377e96	−0.206627
$815$	5.40334e97	1.61764
$816$	1.77636e97	0.510401
$817$	−3.10073e97	−0.855115
$818$	−1.29971e97	−0.344042
$819$	7.53689e95	0.0191506
$820$	1.64418e97	0.401038
$821$	−7.54708e96	−0.176719	−0.0883595	−	0.996089i	$-0.528162\pi$
$821$	−7.54708e96	−0.176719	−0.0883595	+	0.996089i	$0.528162\pi$
$822$	−3.78568e96	−0.0851018
$823$	−2.81141e97	−0.606778	−0.303389	−	0.952867i	$-0.598118\pi$
$823$	−2.81141e97	−0.606778	−0.303389	+	0.952867i	$0.598118\pi$
$824$	−3.61102e96	−0.0748286
$825$	2.71430e97	0.540070
$826$	1.72789e96	0.0330130
$827$	−5.42093e97	−0.994579	−0.497290	−	0.867585i	$-0.665671\pi$
$827$	−5.42093e97	−0.994579	−0.497290	+	0.867585i	$0.665671\pi$
$828$	4.14702e96	0.0730667
$829$	3.18721e97	0.539304	0.269652	−	0.962958i	$-0.413091\pi$
$829$	3.18721e97	0.539304	0.269652	+	0.962958i	$0.413091\pi$
$830$	−2.58491e97	−0.420078
$831$	7.51091e96	0.117235
$832$	−1.96258e97	−0.294236
$833$	−7.79890e97	−1.12312
$834$	−1.55321e97	−0.214866
$835$	5.81684e97	0.773019
$836$	−6.45220e97	−0.823754
$837$	−2.44815e97	−0.300286
$838$	−4.23588e96	−0.0499193
$839$	−1.06669e98	−1.20785	−0.603925	−	0.797041i	$-0.706398\pi$
$839$	−1.06669e98	−1.20785	−0.603925	+	0.797041i	$0.706398\pi$
$840$	4.59819e96	0.0500300
$841$	1.64627e98	1.72121
$842$	2.92617e97	0.293998
$843$	9.18087e97	0.886462
$844$	−1.14679e98	−1.06417
$845$	1.20521e98	1.07489
$846$	7.75561e95	0.00664827
$847$	−4.40172e96	−0.0362684
$848$	4.21480e97	0.333823
$849$	6.53146e97	0.497282
$850$	4.88211e97	0.357333
$851$	3.07288e97	0.216225
$852$	9.99060e97	0.675873
$853$	1.55010e98	1.00825	0.504123	−	0.863632i	$-0.331816\pi$
$853$	1.55010e98	1.00825	0.504123	+	0.863632i	$0.331816\pi$
$854$	−8.62821e95	−0.00539612
$855$	−8.79701e97	−0.529017
$856$	−6.51856e97	−0.376947
$857$	1.00718e98	0.560080	0.280040	−	0.959988i	$-0.409652\pi$
$857$	1.00718e98	0.560080	0.280040	+	0.959988i	$0.409652\pi$
$858$	1.26005e97	0.0673849
$859$	−2.94332e98	−1.51380	−0.756898	−	0.653534i	$-0.773286\pi$
$859$	−2.94332e98	−1.51380	−0.756898	+	0.653534i	$0.773286\pi$
$860$	−2.15201e98	−1.06450
$861$	4.02050e96	0.0191282
$862$	−1.01808e97	−0.0465897
$863$	4.46451e97	0.196523	0.0982613	−	0.995161i	$-0.468672\pi$
$863$	4.46451e97	0.196523	0.0982613	+	0.995161i	$0.468672\pi$
$864$	3.38797e97	0.143459
$865$	3.51854e98	1.43326
$866$	9.37205e97	0.367270
$867$	4.49334e97	0.169406
$868$	4.43538e97	0.160887
$869$	−2.65822e98	−0.927744
$870$	1.14454e98	0.384360
$871$	2.70596e98	0.874406
$872$	−1.93829e98	−0.602720
$873$	−2.19146e98	−0.655778
$874$	−2.46757e97	−0.0710618
$875$	−8.15787e96	−0.0226103
$876$	−3.63829e98	−0.970531
$877$	1.22553e98	0.314657	0.157329	−	0.987546i	$-0.449712\pi$
$877$	1.22553e98	0.314657	0.157329	+	0.987546i	$0.449712\pi$
$878$	−5.36754e97	−0.132650
$879$	3.45607e98	0.822159
$880$	−4.07846e98	−0.933956
$881$	−2.44838e98	−0.539741	−0.269870	−	0.962897i	$-0.586981\pi$
$881$	−2.44838e98	−0.539741	−0.269870	+	0.962897i	$0.586981\pi$
$882$	−4.28047e97	−0.0908435
$883$	−5.43205e98	−1.10989	−0.554946	−	0.831887i	$-0.687261\pi$
$883$	−5.43205e98	−1.10989	−0.554946	+	0.831887i	$0.687261\pi$
$884$	−2.74926e98	−0.540835
$885$	−4.77769e98	−0.904937
$886$	−2.85446e98	−0.520588
$887$	5.54794e98	0.974295	0.487148	−	0.873320i	$-0.338037\pi$
$887$	5.54794e98	0.974295	0.487148	+	0.873320i	$0.338037\pi$
$888$	1.65183e98	0.279338
$889$	−1.38260e97	−0.0225158
$890$	3.73586e98	0.585903
$891$	6.04404e97	0.0912901
$892$	1.11771e99	1.62595
$893$	5.59794e97	0.0784340
$894$	−1.54007e98	−0.207842
$895$	−1.27091e99	−1.65212
$896$	−8.05005e97	−0.100805
$897$	−5.84557e97	−0.0705147
$898$	−1.74528e98	−0.202819
$899$	2.29905e99	2.57394
$900$	−3.25046e98	−0.350607
$901$	4.69963e98	0.488409
$902$	6.72163e97	0.0673063
$903$	−5.26229e97	−0.0507733
$904$	−5.02990e98	−0.467646
$905$	3.35618e98	0.300689
$906$	8.01208e97	0.0691753
$907$	−2.34440e99	−1.95069	−0.975345	−	0.220686i	$-0.929171\pi$
$907$	−2.34440e99	−1.95069	−0.975345	+	0.220686i	$0.929171\pi$
$908$	2.13371e99	1.71104
$909$	−6.00578e98	−0.464172
$910$	−3.11248e97	−0.0231857
$911$	−3.63118e98	−0.260725	−0.130362	−	0.991466i	$-0.541614\pi$
$911$	−3.63118e98	−0.260725	−0.130362	+	0.991466i	$0.541614\pi$
$912$	7.03724e98	0.487052
$913$	1.28189e99	0.855223
$914$	−2.32039e98	−0.149232
$915$	2.38573e98	0.147916
$916$	2.87231e99	1.71685
$917$	−2.17631e98	−0.125415
$918$	1.08712e98	0.0604014
$919$	3.23150e99	1.73115	0.865574	−	0.500780i	$-0.166954\pi$
$919$	3.23150e99	1.73115	0.865574	+	0.500780i	$0.166954\pi$
$920$	−3.56633e98	−0.184217
$921$	−1.47377e99	−0.734062
$922$	3.07905e97	0.0147887
$923$	−1.40826e99	−0.652268
$924$	−1.09501e98	−0.0489112
$925$	−2.40854e99	−1.03754
$926$	3.77446e98	0.156815
$927$	1.17244e98	0.0469805
$928$	−3.18163e99	−1.22968
$929$	−2.67496e98	−0.0997221	−0.0498611	−	0.998756i	$-0.515878\pi$
$929$	−2.67496e98	−0.0997221	−0.0498611	+	0.998756i	$0.515878\pi$
$930$	1.01100e99	0.363558
$931$	−3.08961e99	−1.07174
$932$	−4.06035e99	−1.35872
$933$	3.12922e98	0.101019
$934$	1.59755e99	0.497546
$935$	−4.54761e99	−1.36645
$936$	−3.14229e98	−0.0910971
$937$	−7.86079e98	−0.219882	−0.109941	−	0.993938i	$-0.535066\pi$
$937$	−7.86079e98	−0.219882	−0.109941	+	0.993938i	$0.535066\pi$
$938$	1.93854e98	0.0523215
$939$	6.87991e98	0.179178
$940$	3.88516e98	0.0976396
$941$	9.21226e98	0.223416	0.111708	−	0.993741i	$-0.464368\pi$
$941$	9.21226e98	0.223416	0.111708	+	0.993741i	$0.464368\pi$
$942$	3.62403e98	0.0848176
$943$	−3.11828e98	−0.0704325
$944$	3.82195e99	0.833150
$945$	−1.49295e98	−0.0314109
$946$	−8.79771e98	−0.178656
$947$	2.58483e99	0.506650	0.253325	−	0.967381i	$-0.418476\pi$
$947$	2.58483e99	0.506650	0.253325	+	0.967381i	$0.418476\pi$
$948$	3.18330e99	0.602280
$949$	5.12847e99	0.936634
$950$	1.93410e99	0.340987
$951$	5.13392e98	0.0873779
$952$	−4.10149e98	−0.0673913
$953$	−1.07901e100	−1.71165	−0.855823	−	0.517269i	$-0.826949\pi$
$953$	−1.07901e100	−1.71165	−0.855823	+	0.517269i	$0.826949\pi$
$954$	2.57942e98	0.0395050
$955$	5.70594e99	0.843751
$956$	−4.68863e99	−0.669432
$957$	−5.67593e99	−0.782505
$958$	2.25144e98	0.0299719
$959$	−4.63734e98	−0.0596136
$960$	3.88760e99	0.482609
$961$	1.19668e100	1.43464
$962$	−1.11811e99	−0.129455
$963$	2.11647e99	0.236663
$964$	5.77301e99	0.623479
$965$	−3.75022e98	−0.0391194
$966$	−4.18775e97	−0.00421936
$967$	−7.85337e99	−0.764309	−0.382155	−	0.924098i	$-0.624818\pi$
$967$	−7.85337e99	−0.764309	−0.382155	+	0.924098i	$0.624818\pi$
$968$	1.83517e99	0.172525
$969$	7.84674e99	0.712595
$970$	9.04999e99	0.793954
$971$	−3.25437e99	−0.275818	−0.137909	−	0.990445i	$-0.544038\pi$
$971$	−3.25437e99	−0.275818	−0.137909	+	0.990445i	$0.544038\pi$
$972$	−7.23793e98	−0.0592645
$973$	−1.90263e99	−0.150513
$974$	−2.01738e99	−0.154192
$975$	4.58180e99	0.338362
$976$	−1.90848e99	−0.136182
$977$	1.05571e100	0.727907	0.363953	−	0.931417i	$-0.381427\pi$
$977$	1.05571e100	0.727907	0.363953	+	0.931417i	$0.381427\pi$
$978$	−2.64530e99	−0.176248
$979$	−1.85266e100	−1.19282
$980$	−2.14430e100	−1.33417
$981$	6.29329e99	0.378413
$982$	−5.16956e99	−0.300413
$983$	3.00556e100	1.68804	0.844020	−	0.536311i	$-0.180183\pi$
$983$	3.00556e100	1.68804	0.844020	+	0.536311i	$0.180183\pi$
$984$	−1.67623e99	−0.0909909
$985$	1.15959e100	0.608404
$986$	−1.02091e100	−0.517739
$987$	9.50036e97	0.00465710
$988$	−1.08915e100	−0.516094
$989$	4.08141e99	0.186954
$990$	−2.49598e99	−0.110525
$991$	−3.79483e100	−1.62452	−0.812261	−	0.583295i	$-0.801763\pi$
$991$	−3.79483e100	−1.62452	−0.812261	+	0.583295i	$0.801763\pi$
$992$	−2.81041e100	−1.16313
$993$	−1.34755e100	−0.539194
$994$	−1.00887e99	−0.0390295
$995$	5.11755e100	1.91421
$996$	−1.53510e100	−0.555200
$997$	−2.22986e100	−0.779812	−0.389906	−	0.920855i	$-0.627492\pi$
$997$	−2.22986e100	−0.779812	−0.389906	+	0.920855i	$0.627492\pi$
$998$	−8.38313e99	−0.283487
$999$	−5.36320e99	−0.175380

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3.68.a.a.1.4	✓	5
3.2	odd	2		9.68.a.b.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.68.a.a.1.4	✓	5	1.1	even	1	trivial
9.68.a.b.1.2		5	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:	\( N \)	\(=\)	\( 3 \)
Weight:	\( k \)	\(=\)	\( 68 \)
Character orbit:	\([\chi]\)	\(=\)	3.a (trivial)

\(f(q)\)	\(=\)	\(q+3.35246e9 q^{2} +5.55906e15 q^{3} -1.36335e20 q^{4} -3.80674e23 q^{5} +1.86366e25 q^{6} +2.28291e27 q^{7} -9.51794e29 q^{8} +3.09032e31 q^{9} +O(q^{10})\)	\(q+3.35246e9 q^{2} +5.55906e15 q^{3} -1.36335e20 q^{4} -3.80674e23 q^{5} +1.86366e25 q^{6} +2.28291e27 q^{7} -9.51794e29 q^{8} +3.09032e31 q^{9} -1.27619e33 q^{10} +6.32880e34 q^{11} -7.57894e35 q^{12} +1.06832e37 q^{13} +7.65339e36 q^{14} -2.11619e39 q^{15} +1.69286e40 q^{16} +1.88759e41 q^{17} +1.03602e41 q^{18} +7.47789e42 q^{19} +5.18991e43 q^{20} +1.26909e43 q^{21} +2.12171e44 q^{22} -9.84296e44 q^{23} -5.29108e45 q^{24} +7.71498e46 q^{25} +3.58149e46 q^{26} +1.71793e47 q^{27} -3.11241e47 q^{28} -1.61330e49 q^{29} -7.09444e48 q^{30} -1.42506e50 q^{31} +1.97213e50 q^{32} +3.51822e50 q^{33} +6.32809e50 q^{34} -8.69045e50 q^{35} -4.21318e51 q^{36} -3.12191e52 q^{37} +2.50694e52 q^{38} +5.93883e52 q^{39} +3.62323e53 q^{40} +3.16803e53 q^{41} +4.25456e52 q^{42} -4.14652e54 q^{43} -8.62837e54 q^{44} -1.17640e55 q^{45} -3.29982e54 q^{46} +7.48599e54 q^{47} +9.41073e55 q^{48} -4.13166e56 q^{49} +2.58642e56 q^{50} +1.04933e57 q^{51} -1.45649e57 q^{52} +2.48975e57 q^{53} +5.75928e56 q^{54} -2.40921e58 q^{55} -2.17286e57 q^{56} +4.15701e58 q^{57} -5.40852e58 q^{58} +2.25769e59 q^{59} +2.88510e59 q^{60} -1.12737e59 q^{61} -4.77747e59 q^{62} +7.05492e58 q^{63} -1.83708e60 q^{64} -4.06680e60 q^{65} +1.17947e60 q^{66} +2.53292e61 q^{67} -2.57345e61 q^{68} -5.47176e60 q^{69} -2.91344e60 q^{70} -1.31821e62 q^{71} -2.94134e61 q^{72} +4.80052e62 q^{73} -1.04661e62 q^{74} +4.28880e62 q^{75} -1.01950e63 q^{76} +1.44481e62 q^{77} +1.99097e62 q^{78} -4.20019e63 q^{79} -6.44428e63 q^{80} +9.55005e62 q^{81} +1.06207e63 q^{82} +2.02548e64 q^{83} -1.73021e63 q^{84} -7.18557e64 q^{85} -1.39011e64 q^{86} -8.96842e64 q^{87} -6.02372e64 q^{88} -2.92734e65 q^{89} -3.94384e64 q^{90} +2.43887e64 q^{91} +1.34194e65 q^{92} -7.92201e65 q^{93} +2.50965e64 q^{94} -2.84664e66 q^{95} +1.09632e66 q^{96} -7.09139e66 q^{97} -1.38512e66 q^{98} +1.95580e66 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 16255223088 q^{2} + 27\!\cdots\!15 q^{3}+ \cdots + 15\!\cdots\!45 q^{9}+O(q^{10}) \) 5 * q - 16255223088 * q^2 + 27795302832777615 * q^3 + 187725942510165022976 * q^4 - 78587975159033653413210 * q^5 - 90363769669063698103515024 * q^6 + 2897133779647323835395360808 * q^7 - 446560396075670111196535861248 * q^8 + 154515771913163061809603209017645 * q^9	\( 5 q - 16255223088 q^{2} + 27\!\cdots\!15 q^{3}+ \cdots - 33\!\cdots\!28 q^{99}+O(q^{100}) \) 5 * q - 16255223088 * q^2 + 27795302832777615 * q^3 + 187725942510165022976 * q^4 - 78587975159033653413210 * q^5 - 90363769669063698103515024 * q^6 + 2897133779647323835395360808 * q^7 - 446560396075670111196535861248 * q^8 + 154515771913163061809603209017645 * q^9 - 3359687674843328387731353730265760 * q^10 - 107218677940017498877929099247513932 * q^11 + 1043579884327727512139439968474696448 * q^12 + 18720658700128270654310089005018371278 * q^13 - 61671302560002915461244055855418861952 * q^14 - 436875313712028989080558866721926658830 * q^15 + 19828810558474760279626207936836822695936 * q^16 + 155634077783406813100849154582687160437850 * q^17 - 502337668612598026687246628668502560677552 * q^18 - 6330823499631802877762269921902501094783988 * q^19 + 8806685958461200090592049790099809534328320 * q^20 + 16105342150473395769449245566105034350142584 * q^21 - 1285334407460495219610552944494458560199940800 * q^22 - 3096470811979061341218583695224599825546419864 * q^23 - 2482456288409673438086642069250837537616072704 * q^24 - 128510195883326013861116838537875526813134026525 * q^25 - 309893301421656093037055250012280159780714544160 * q^26 + 858962534553352218394101882942702121170179203335 * q^27 - 1758216414458764947454106710991446824456120252416 * q^28 - 17195448123638013107201791699494703244624614194066 * q^29 - 18676707269164160844617717039655071934318585792480 * q^30 - 167299848682287535534976349940903215242774636120144 * q^31 - 661581347508711830434048681313730640500855508303872 * q^32 - 596035124534567832987517244399918779918082194046436 * q^33 - 3668629124130223548817939535033459308602451741777248 * q^34 - 597313366964957875633721976544849058693729215396560 * q^35 + 5801323783016844061209039192798206302504038362882304 * q^36 + 30043361009726020519419208243072073319518814491934358 * q^37 + 260485037699007543111228693866771891318790849526359744 * q^38 + 104069275559827645019221591022442107451411644015468394 * q^39 + 1115094440991577594559320173651217718577849903091507200 * q^40 - 808437303878393630827363118778237447937170861730235454 * q^41 - 342834506149426883197674031813537074530930837280160896 * q^42 - 4736429276246859157617871173658532851277246462721149228 * q^43 + 7015258532306255548162192439376864244003799076792468480 * q^44 - 2428616328958113717945826873336511479963953561073218090 * q^45 + 12527540911364294799696045411504819449186256978468080256 * q^46 + 108290076402229329642983829719434835232002632660130515056 * q^47 + 110229558857316837304648760016604589737407028314290454528 * q^48 + 880181676573802609519730686079181363021552872382740232333 * q^49 + 1052494889081109131658159592164188937755980711801220679600 * q^50 + 865179264617971813637134219389277404805805348251015745550 * q^51 + 2308777474667263284772763232859451620790190619345360422400 * q^52 + 3245226878182367959789303611224433348480161481518744370966 * q^53 - 2792525524679729749647164642126896980230414112629707719696 * q^54 - 51262912349629136715335621059095130466245073679837843274920 * q^55 - 47772133165991162191839606887952190261990284108568121671680 * q^56 - 35193431270626188960239511075559223006610539094694893365724 * q^57 - 84493666334602655307554841072065967874913257934429532582432 * q^58 - 179624529165245332659094970865033289643789158511842592792412 * q^59 + 48956900633719888068348132839178395661941003500967791311360 * q^60 + 584907690374744349731719712847344294312321913116060488567102 * q^61 + 3025719874056683992201295546200663430948598802434153503920896 * q^62 + 89530572459581180641513960803016595914572442189036802691432 * q^63 - 4499392031332841330291967248952523929085218266739063232921600 * q^64 - 4475515698809998982184518632813763939217510489440113220535420 * q^65 - 7145251819350648004204211437078044590913023079552444513038400 * q^66 + 1344914323542783268647242614869504266340689501787079942327420 * q^67 + 19365780359678204058695683499114787965920302224535027287671296 * q^68 - 17213468786362961074546343099951406673447837437328506826108872 * q^69 + 31668988367649764945620440396543964676121485136652812185168640 * q^70 + 208099792629824800554470944953685171609425458107698538896812280 * q^71 - 13800124861096000027311301435875177780070484438004484020744192 * q^72 - 252414514587179259177941988046481265295589404121206374617470334 * q^73 - 358084904219721857079108717596560064061662918994206334990793632 * q^74 - 714395962335323550124996935032662148018445244009738166967247575 * q^75 - 3167576238289455647607470438405921376289304779045740803124282368 * q^76 - 2715616264867565296962962180641800513550454260043685733361085024 * q^77 - 1722715631772832981701583674878761695633721340944383480275395680 * q^78 - 3431503850832591202644393930474874375763697022794649483222945280 * q^79 - 8322155504340769679256255462767373452465836703993138541717749760 * q^80 + 4775024753984126184465953508872070059599675694871715649184269205 * q^81 - 6362169783346643199240774371522649930233919884701663896646628064 * q^82 - 2753559226275970537111940225815871118779223599597864793376534836 * q^83 - 9774031537088362109664563191733058527530044006822089118438893568 * q^84 - 100223085545462292798863671541625307722645907180923010395975039060 * q^85 - 287767761108017143223082018540844348985876033069110036187064944832 * q^86 - 95590537588367238050824370796559187863343340269759771352286126518 * q^87 - 475509956271309169302126186902650458005506404316097446988140167168 * q^88 - 953231813381767774308665680609247777419192406201469535932206722638 * q^89 - 103824946893111374784083486264977336654984096886031891078875867040 * q^90 - 1826343975172985917715752368227201077015187248090890361406237190544 * q^91 - 3132766731772277225542233484759817592417648594742828753496091850752 * q^92 - 930029991600410615305341159302088161126938972921803642822874755312 * q^93 - 4377709041417318550457656800988375516480152699177767581662212342016 * q^94 - 3334812729598028343033336500168508357868459521335461166424246211800 * q^95 - 3677770780504345932935680810193675103543675932078897134851043885056 * q^96 - 11968150959482375162440074477550294838351889417758320270285978711190 * q^97 - 17297597363568888846780657367452226513119832175100344485603054876848 * q^98 - 3313395357082126364699797809873297365206219515646735115126034266028 * q^99

Introduction

L-functions

Modular forms

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