LMFDB - Embedded newform 3.68.a.a.1.2

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			1.2
Root			$1.48809e8$ 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)

$f(q)$	$=$	$q-1.03939e10 q^{2} +5.55906e15 q^{3} -3.95417e19 q^{4} +1.67944e23 q^{5} -5.77801e25 q^{6} -3.85597e28 q^{7} +1.94485e30 q^{8} +3.09032e31 q^{9} +O(q^{10})$	\(q-1.03939e10 q^{2} +5.55906e15 q^{3} -3.95417e19 q^{4} +1.67944e23 q^{5} -5.77801e25 q^{6} -3.85597e28 q^{7} +1.94485e30 q^{8} +3.09032e31 q^{9} -1.74559e33 q^{10} -2.10881e34 q^{11} -2.19815e35 q^{12} +2.64496e37 q^{13} +4.00784e38 q^{14} +9.33612e38 q^{15} -1.43792e40 q^{16} -1.55302e41 q^{17} -3.21203e41 q^{18} +4.02214e42 q^{19} -6.64081e42 q^{20} -2.14356e44 q^{21} +2.19187e44 q^{22} +5.77103e45 q^{23} +1.08116e46 q^{24} -3.95574e46 q^{25} -2.74914e47 q^{26} +1.71793e47 q^{27} +1.52472e48 q^{28} -8.56944e48 q^{29} -9.70383e48 q^{30} -2.43118e49 q^{31} -1.37554e50 q^{32} -1.17230e50 q^{33} +1.61419e51 q^{34} -6.47588e51 q^{35} -1.22196e51 q^{36} +5.12588e52 q^{37} -4.18056e52 q^{38} +1.47035e53 q^{39} +3.26627e53 q^{40} -1.27508e54 q^{41} +2.22798e54 q^{42} +6.40780e54 q^{43} +8.33861e53 q^{44} +5.19000e54 q^{45} -5.99833e55 q^{46} +1.57548e56 q^{47} -7.99348e55 q^{48} +1.06847e57 q^{49} +4.11154e56 q^{50} -8.63333e56 q^{51} -1.04586e57 q^{52} +4.51622e57 q^{53} -1.78559e57 q^{54} -3.54163e57 q^{55} -7.49929e58 q^{56} +2.23593e58 q^{57} +8.90696e58 q^{58} -1.96190e59 q^{59} -3.69166e58 q^{60} -2.54683e59 q^{61} +2.52693e59 q^{62} -1.19162e60 q^{63} +3.55172e60 q^{64} +4.44206e60 q^{65} +1.21847e60 q^{66} -2.16510e61 q^{67} +6.14091e60 q^{68} +3.20815e61 q^{69} +6.73093e61 q^{70} -6.59771e61 q^{71} +6.01021e61 q^{72} -1.31948e62 q^{73} -5.32777e62 q^{74} -2.19902e62 q^{75} -1.59043e62 q^{76} +8.13151e62 q^{77} -1.52826e63 q^{78} -3.95091e63 q^{79} -2.41490e63 q^{80} +9.55005e62 q^{81} +1.32530e64 q^{82} +3.05217e63 q^{83} +8.47600e63 q^{84} -2.60821e64 q^{85} -6.66017e64 q^{86} -4.76381e64 q^{87} -4.10133e64 q^{88} -1.74764e65 q^{89} -5.39442e64 q^{90} -1.01989e66 q^{91} -2.28197e65 q^{92} -1.35151e65 q^{93} -1.63753e66 q^{94} +6.75496e65 q^{95} -7.64674e65 q^{96} +2.23298e66 q^{97} -1.11055e67 q^{98} -6.51689e65 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.03939e10	−0.855602	−0.427801	−	0.903873i	$-0.640712\pi$
$2$	−1.03939e10	−0.855602	−0.427801	+	0.903873i	$0.640712\pi$
$3$	5.55906e15	0.577350
$3$	5.55906e15	0.577350
$4$	−3.95417e19	−0.267945
$5$	1.67944e23	0.645164	0.322582	−	0.946542i	$-0.395449\pi$
$5$	1.67944e23	0.645164	0.322582	+	0.946542i	$0.395449\pi$
$6$	−5.77801e25	−0.493982
$7$	−3.85597e28	−1.88516	−0.942582	−	0.333974i	$-0.891610\pi$
$7$	−3.85597e28	−1.88516	−0.942582	+	0.333974i	$0.891610\pi$
$8$	1.94485e30	1.08486
$9$	3.09032e31	0.333333
$10$	−1.74559e33	−0.552003
$11$	−2.10881e34	−0.273768	−0.136884	−	0.990587i	$-0.543709\pi$
$11$	−2.10881e34	−0.273768	−0.136884	+	0.990587i	$0.543709\pi$
$12$	−2.19815e35	−0.154698
$13$	2.64496e37	1.27443	0.637217	−	0.770684i	$-0.280085\pi$
$13$	2.64496e37	1.27443	0.637217	+	0.770684i	$0.280085\pi$
$14$	4.00784e38	1.61295
$15$	9.33612e38	0.372485
$16$	−1.43792e40	−0.660260
$17$	−1.55302e41	−0.935703	−0.467851	−	0.883807i	$-0.654972\pi$
$17$	−1.55302e41	−0.935703	−0.467851	+	0.883807i	$0.654972\pi$
$18$	−3.21203e41	−0.285201
$19$	4.02214e42	0.583730	0.291865	−	0.956460i	$-0.405724\pi$
$19$	4.02214e42	0.583730	0.291865	+	0.956460i	$0.405724\pi$
$20$	−6.64081e42	−0.172869
$21$	−2.14356e44	−1.08840
$22$	2.19187e44	0.234237
$23$	5.77103e45	1.39114	0.695570	−	0.718458i	$-0.255152\pi$
$23$	5.77103e45	1.39114	0.695570	+	0.718458i	$0.255152\pi$
$24$	1.08116e46	0.626342
$25$	−3.95574e46	−0.583764
$26$	−2.74914e47	−1.09041
$27$	1.71793e47	0.192450
$28$	1.52472e48	0.505121
$29$	−8.56944e48	−0.876233	−0.438117	−	0.898918i	$-0.644354\pi$
$29$	−8.56944e48	−0.876233	−0.438117	+	0.898918i	$0.644354\pi$
$30$	−9.70383e48	−0.318699
$31$	−2.43118e49	−0.266195	−0.133097	−	0.991103i	$-0.542492\pi$
$31$	−2.43118e49	−0.266195	−0.133097	+	0.991103i	$0.542492\pi$
$32$	−1.37554e50	−0.519937
$33$	−1.17230e50	−0.158060
$34$	1.61419e51	0.800589
$35$	−6.47588e51	−1.21624
$36$	−1.22196e51	−0.0893151
$37$	5.12588e52	1.49627	0.748137	−	0.663545i	$-0.230949\pi$
$37$	5.12588e52	1.49627	0.748137	+	0.663545i	$0.230949\pi$
$38$	−4.18056e52	−0.499440
$39$	1.47035e53	0.735795
$40$	3.26627e53	0.699910
$41$	−1.27508e54	−1.19475	−0.597376	−	0.801962i	$-0.703790\pi$
$41$	−1.27508e54	−1.19475	−0.597376	+	0.801962i	$0.703790\pi$
$42$	2.22798e54	0.931237
$43$	6.40780e54	1.21763	0.608815	−	0.793312i	$-0.291645\pi$
$43$	6.40780e54	1.21763	0.608815	+	0.793312i	$0.291645\pi$
$44$	8.33861e53	0.0733549
$45$	5.19000e54	0.215055
$46$	−5.99833e55	−1.19026
$47$	1.57548e56	1.52102	0.760508	−	0.649328i	$-0.224950\pi$
$47$	1.57548e56	1.52102	0.760508	+	0.649328i	$0.224950\pi$
$48$	−7.99348e55	−0.381201
$49$	1.06847e57	2.55384
$50$	4.11154e56	0.499470
$51$	−8.63333e56	−0.540228
$52$	−1.04586e57	−0.341479
$53$	4.51622e57	0.778992	0.389496	−	0.921028i	$-0.372649\pi$
$53$	4.51622e57	0.778992	0.389496	+	0.921028i	$0.372649\pi$
$54$	−1.78559e57	−0.164661
$55$	−3.54163e57	−0.176625
$56$	−7.49929e58	−2.04513
$57$	2.23593e58	0.337017
$58$	8.90696e58	0.749707
$59$	−1.96190e59	−0.931396	−0.465698	−	0.884944i	$-0.654197\pi$
$59$	−1.96190e59	−0.931396	−0.465698	+	0.884944i	$0.654197\pi$
$60$	−3.69166e58	−0.0998057
$61$	−2.54683e59	−0.395776	−0.197888	−	0.980225i	$-0.563408\pi$
$61$	−2.54683e59	−0.395776	−0.197888	+	0.980225i	$0.563408\pi$
$62$	2.52693e59	0.227757
$63$	−1.19162e60	−0.628388
$64$	3.55172e60	1.10512
$65$	4.44206e60	0.822218
$66$	1.21847e60	0.135237
$67$	−2.16510e61	−1.45202	−0.726008	−	0.687686i	$-0.758626\pi$
$67$	−2.16510e61	−1.45202	−0.726008	+	0.687686i	$0.758626\pi$
$68$	6.14091e60	0.250717
$69$	3.20815e61	0.803175
$70$	6.73093e61	1.04062
$71$	−6.59771e61	−0.634218	−0.317109	−	0.948389i	$-0.602712\pi$
$71$	−6.59771e61	−0.634218	−0.317109	+	0.948389i	$0.602712\pi$
$72$	6.01021e61	0.361619
$73$	−1.31948e62	−0.500133	−0.250067	−	0.968229i	$-0.580453\pi$
$73$	−1.31948e62	−0.500133	−0.250067	+	0.968229i	$0.580453\pi$
$74$	−5.32777e62	−1.28021
$75$	−2.19902e62	−0.337036
$76$	−1.59043e62	−0.156408
$77$	8.13151e62	0.516098
$78$	−1.52826e63	−0.629548
$79$	−3.95091e63	−1.06216	−0.531079	−	0.847322i	$-0.678213\pi$
$79$	−3.95091e63	−1.06216	−0.531079	+	0.847322i	$0.678213\pi$
$80$	−2.41490e63	−0.425976
$81$	9.55005e62	0.111111
$82$	1.32530e64	1.02223
$83$	3.05217e63	0.156853	0.0784264	−	0.996920i	$-0.475010\pi$
$83$	3.05217e63	0.156853	0.0784264	+	0.996920i	$0.475010\pi$
$84$	8.47600e63	0.291632
$85$	−2.60821e64	−0.603681
$86$	−6.66017e64	−1.04181
$87$	−4.76381e64	−0.505894
$88$	−4.10133e64	−0.296999
$89$	−1.74764e65	−0.866734	−0.433367	−	0.901217i	$-0.642675\pi$
$89$	−1.74764e65	−0.866734	−0.433367	+	0.901217i	$0.642675\pi$
$90$	−5.39442e64	−0.184001
$91$	−1.01989e66	−2.40252
$92$	−2.28197e65	−0.372749
$93$	−1.35151e65	−0.153688
$94$	−1.63753e66	−1.30138
$95$	6.75496e65	0.376601
$96$	−7.64674e65	−0.300186
$97$	2.23298e66	0.619488	0.309744	−	0.950820i	$-0.399757\pi$
$97$	2.23298e66	0.619488	0.309744	+	0.950820i	$0.399757\pi$
$98$	−1.11055e67	−2.18507
$99$	−6.51689e65	−0.0912561
$100$	1.56417e66	0.156417
$101$	−1.97014e67	−1.41166	−0.705831	−	0.708380i	$-0.749426\pi$
$101$	−1.97014e67	−1.41166	−0.705831	+	0.708380i	$0.749426\pi$
$102$	8.97336e66	0.462220
$103$	−1.45814e67	−0.541691	−0.270845	−	0.962623i	$-0.587303\pi$
$103$	−1.45814e67	−0.541691	−0.270845	+	0.962623i	$0.587303\pi$
$104$	5.14407e67	1.38258
$105$	−3.59998e67	−0.702196
$106$	−4.69409e67	−0.666507
$107$	−1.53387e67	−0.159013	−0.0795066	−	0.996834i	$-0.525334\pi$
$107$	−1.53387e67	−0.159013	−0.0795066	+	0.996834i	$0.525334\pi$
$108$	−6.79298e66	−0.0515661
$109$	−1.61115e68	−0.898149	−0.449074	−	0.893494i	$-0.648246\pi$
$109$	−1.61115e68	−0.898149	−0.449074	+	0.893494i	$0.648246\pi$
$110$	3.68112e67	0.151121
$111$	2.84951e68	0.863874
$112$	5.54457e68	1.24470
$113$	3.93638e68	0.656098	0.328049	−	0.944661i	$-0.393609\pi$
$113$	3.93638e68	0.656098	0.328049	+	0.944661i	$0.393609\pi$
$114$	−2.32400e68	−0.288352
$115$	9.69211e68	0.897513
$116$	3.38851e68	0.234783
$117$	8.17377e68	0.424811
$118$	2.03917e69	0.796904
$119$	5.98839e69	1.76395
$120$	1.81574e69	0.404093
$121$	−5.48878e69	−0.925051
$122$	2.64713e69	0.338627
$123$	−7.08823e69	−0.689790
$124$	9.61329e68	0.0713256
$125$	−1.80238e70	−1.02179
$126$	1.23855e70	0.537650
$127$	−1.98306e70	−0.660557	−0.330278	−	0.943884i	$-0.607143\pi$
$127$	−1.98306e70	−0.660557	−0.330278	+	0.943884i	$0.607143\pi$
$128$	−1.66166e70	−0.425605
$129$	3.56213e70	0.703000
$130$	−4.61702e70	−0.703492
$131$	3.11780e69	0.0367502	0.0183751	−	0.999831i	$-0.494151\pi$
$131$	3.11780e69	0.0367502	0.0183751	+	0.999831i	$0.494151\pi$
$132$	4.63548e69	0.0423515
$133$	−1.55093e71	−1.10043
$134$	2.25037e71	1.24235
$135$	2.88516e70	0.124162
$136$	−3.02039e71	−1.01510
$137$	−3.78607e71	−0.995518	−0.497759	−	0.867315i	$-0.665844\pi$
$137$	−3.78607e71	−0.995518	−0.497759	+	0.867315i	$0.665844\pi$
$138$	−3.33451e71	−0.687198
$139$	−8.76617e71	−1.41845	−0.709225	−	0.704982i	$-0.750955\pi$
$139$	−8.76617e71	−1.41845	−0.709225	+	0.704982i	$0.750955\pi$
$140$	2.56067e71	0.325886
$141$	8.75817e71	0.878159
$142$	6.85756e71	0.542638
$143$	−5.57773e71	−0.348900
$144$	−4.44362e71	−0.220087
$145$	−1.43919e72	−0.565314
$146$	1.37144e72	0.427915
$147$	5.93970e72	1.47446
$148$	−2.02686e72	−0.400919
$149$	1.21287e73	1.91458	0.957289	−	0.289131i	$-0.0933665\pi$
$149$	1.21287e73	1.91458	0.957289	+	0.289131i	$0.0933665\pi$
$150$	2.28563e72	0.288369
$151$	3.99913e72	0.403867	0.201933	−	0.979399i	$-0.435278\pi$
$151$	3.99913e72	0.403867	0.201933	+	0.979399i	$0.435278\pi$
$152$	7.82248e72	0.633263
$153$	−4.79932e72	−0.311901
$154$	−8.45178e72	−0.441574
$155$	−4.08302e72	−0.171739
$156$	−5.81403e72	−0.197153
$157$	2.62592e73	0.718858	0.359429	−	0.933172i	$-0.382971\pi$
$157$	2.62592e73	0.718858	0.359429	+	0.933172i	$0.382971\pi$
$158$	4.10652e73	0.908785
$159$	2.51059e73	0.449751
$160$	−2.31015e73	−0.335444
$161$	−2.22529e74	−2.62253
$162$	−9.92618e72	−0.0950669
$163$	−1.03139e74	−0.803784	−0.401892	−	0.915687i	$-0.631647\pi$
$163$	−1.03139e74	−0.803784	−0.401892	+	0.915687i	$0.631647\pi$
$164$	5.04188e73	0.320128
$165$	−1.96881e73	−0.101975
$166$	−3.17238e73	−0.134204
$167$	5.86661e73	0.202948	0.101474	−	0.994838i	$-0.467644\pi$
$167$	5.86661e73	0.202948	0.101474	+	0.994838i	$0.467644\pi$
$168$	−4.16890e74	−1.18076
$169$	2.68854e74	0.624182
$170$	2.71093e74	0.516511
$171$	1.24297e74	0.194577
$172$	−2.53376e74	−0.326258
$173$	−1.50339e75	−1.59415	−0.797074	−	0.603882i	$-0.793620\pi$
$173$	−1.50339e75	−1.59415	−0.797074	+	0.603882i	$0.793620\pi$
$174$	4.95143e74	0.432843
$175$	1.52532e75	1.10049
$176$	3.03230e74	0.180758
$177$	−1.09063e75	−0.537742
$178$	1.81647e75	0.741579
$179$	−4.91440e75	−1.66301	−0.831503	−	0.555520i	$-0.812519\pi$
$179$	−4.91440e75	−1.66301	−0.831503	+	0.555520i	$0.812519\pi$
$180$	−2.05222e74	−0.0576228
$181$	−5.79651e74	−0.135187	−0.0675934	−	0.997713i	$-0.521532\pi$
$181$	−5.79651e74	−0.135187	−0.0675934	+	0.997713i	$0.521532\pi$
$182$	1.06006e76	2.05560
$183$	−1.41580e75	−0.228502
$184$	1.12238e76	1.50919
$185$	8.60862e75	0.965341
$186$	1.40473e75	0.131495
$187$	3.27503e75	0.256166
$188$	−6.22971e75	−0.407549
$189$	−6.62427e75	−0.362800
$190$	−7.02100e75	−0.322221
$191$	1.62221e76	0.624436	0.312218	−	0.950010i	$-0.398928\pi$
$191$	1.62221e76	0.624436	0.312218	+	0.950010i	$0.398928\pi$
$192$	1.97442e76	0.638041
$193$	−4.49053e76	−1.21935	−0.609674	−	0.792652i	$-0.708700\pi$
$193$	−4.49053e76	−1.21935	−0.609674	+	0.792652i	$0.708700\pi$
$194$	−2.32093e76	−0.530035
$195$	2.46937e76	0.474708
$196$	−4.22492e76	−0.684291
$197$	−9.61352e76	−1.31300	−0.656498	−	0.754327i	$-0.727963\pi$
$197$	−9.61352e76	−1.31300	−0.656498	+	0.754327i	$0.727963\pi$
$198$	6.77357e75	0.0780789
$199$	−1.55638e77	−1.51544	−0.757720	−	0.652580i	$-0.773687\pi$
$199$	−1.55638e77	−1.51544	−0.757720	+	0.652580i	$0.773687\pi$
$200$	−7.69333e76	−0.633300
$201$	−1.20359e77	−0.838322
$202$	2.04773e77	1.20782
$203$	3.30435e77	1.65184
$204$	3.41377e76	0.144752
$205$	−2.14142e77	−0.770810
$206$	1.51557e77	0.463472
$207$	1.78343e77	0.463713
$208$	−3.80324e77	−0.841458
$209$	−8.48194e76	−0.159807
$210$	3.74177e77	0.600800
$211$	5.91131e77	0.809509	0.404754	−	0.914425i	$-0.367357\pi$
$211$	5.91131e77	0.809509	0.404754	+	0.914425i	$0.367357\pi$
$212$	−1.78579e77	−0.208727
$213$	−3.66771e77	−0.366166
$214$	1.59429e77	0.136052
$215$	1.07615e78	0.785571
$216$	3.34111e77	0.208781
$217$	9.37454e77	0.501821
$218$	1.67461e78	0.768458
$219$	−7.33505e77	−0.288752
$220$	1.40042e77	0.0473259
$221$	−4.10768e78	−1.19249
$222$	−2.96174e78	−0.739132
$223$	5.78857e78	1.24268	0.621339	−	0.783542i	$-0.286589\pi$
$223$	5.78857e78	1.24268	0.621339	+	0.783542i	$0.286589\pi$
$224$	5.30406e78	0.980166
$225$	−1.22245e78	−0.194588
$226$	−4.09141e78	−0.561359
$227$	−4.22273e78	−0.499720	−0.249860	−	0.968282i	$-0.580385\pi$
$227$	−4.22273e78	−0.499720	−0.249860	+	0.968282i	$0.580385\pi$
$228$	−8.84127e77	−0.0903020
$229$	4.58937e78	0.404822	0.202411	−	0.979301i	$-0.435122\pi$
$229$	4.58937e78	0.404822	0.202411	+	0.979301i	$0.435122\pi$
$230$	−1.00738e79	−0.767914
$231$	4.52036e78	0.297969
$232$	−1.66663e79	−0.950587
$233$	−1.72317e79	−0.850950	−0.425475	−	0.904970i	$-0.639893\pi$
$233$	−1.72317e79	−0.850950	−0.425475	+	0.904970i	$0.639893\pi$
$234$	−8.49570e78	−0.363469
$235$	2.64592e79	0.981304
$236$	7.75769e78	0.249563
$237$	−2.19633e79	−0.613238
$238$	−6.22425e79	−1.50924
$239$	4.25827e79	0.897229	0.448614	−	0.893725i	$-0.351918\pi$
$239$	4.25827e79	0.897229	0.448614	+	0.893725i	$0.351918\pi$
$240$	−1.34246e79	−0.245937
$241$	−6.62697e79	−1.05620	−0.528098	−	0.849184i	$-0.677095\pi$
$241$	−6.62697e79	−1.05620	−0.528098	+	0.849184i	$0.677095\pi$
$242$	5.70495e79	0.791475
$243$	5.30893e78	0.0641500
$244$	1.00706e79	0.106046
$245$	1.79444e80	1.64765
$246$	7.36741e79	0.590186
$247$	1.06384e80	0.743925
$248$	−4.72828e79	−0.288783
$249$	1.69672e79	0.0905590
$250$	1.87336e80	0.874243
$251$	1.15208e80	0.470340	0.235170	−	0.971954i	$-0.424435\pi$
$251$	1.15208e80	0.470340	0.235170	+	0.971954i	$0.424435\pi$
$252$	4.71186e79	0.168374
$253$	−1.21700e80	−0.380850
$254$	2.06116e80	0.565174
$255$	−1.44992e80	−0.348536
$256$	−3.51431e80	−0.740970
$257$	7.44854e80	1.37820	0.689099	−	0.724667i	$-0.258007\pi$
$257$	7.44854e80	1.37820	0.689099	+	0.724667i	$0.258007\pi$
$258$	−3.70243e80	−0.601488
$259$	−1.97652e81	−2.82072
$260$	−1.75647e80	−0.220310
$261$	−2.64823e80	−0.292078
$262$	−3.24059e79	−0.0314435
$263$	8.44350e79	0.0721115	0.0360558	−	0.999350i	$-0.488521\pi$
$263$	8.44350e79	0.0721115	0.0360558	+	0.999350i	$0.488521\pi$
$264$	−2.27995e80	−0.171473
$265$	7.58472e80	0.502577
$266$	1.61201e81	0.941527
$267$	−9.71522e80	−0.500409
$268$	8.56117e80	0.389061
$269$	−1.30791e81	−0.524658	−0.262329	−	0.964978i	$-0.584491\pi$
$269$	−1.30791e81	−0.524658	−0.262329	+	0.964978i	$0.584491\pi$
$270$	−2.99879e80	−0.106233
$271$	−2.76858e81	−0.866533	−0.433267	−	0.901266i	$-0.642639\pi$
$271$	−2.76858e81	−0.866533	−0.433267	+	0.901266i	$0.642639\pi$
$272$	2.23312e81	0.617807
$273$	−5.66963e81	−1.38709
$274$	3.93518e81	0.851767
$275$	8.34191e80	0.159816
$276$	−1.26856e81	−0.215207
$277$	4.18239e81	0.628569	0.314284	−	0.949329i	$-0.398235\pi$
$277$	4.18239e81	0.628569	0.314284	+	0.949329i	$0.398235\pi$
$278$	9.11143e81	1.21363
$279$	−7.51310e80	−0.0887316
$280$	−1.25946e82	−1.31945
$281$	1.50887e82	1.40278	0.701392	−	0.712776i	$-0.252562\pi$
$281$	1.50887e82	1.40278	0.701392	+	0.712776i	$0.252562\pi$
$282$	−9.10312e81	−0.751355
$283$	−5.73170e81	−0.420182	−0.210091	−	0.977682i	$-0.567376\pi$
$283$	−5.73170e81	−0.420182	−0.210091	+	0.977682i	$0.567376\pi$
$284$	2.60885e81	0.169936
$285$	3.75512e81	0.217431
$286$	5.79741e81	0.298519
$287$	4.91666e82	2.25230
$288$	−4.25087e81	−0.173312
$289$	−3.42853e81	−0.124460
$290$	1.49587e82	0.483684
$291$	1.24133e82	0.357661
$292$	5.21744e81	0.134008
$293$	−6.47811e82	−1.48383	−0.741913	−	0.670496i	$-0.766081\pi$
$293$	−6.47811e82	−1.48383	−0.741913	+	0.670496i	$0.766081\pi$
$294$	−6.17364e82	−1.26155
$295$	−3.29490e82	−0.600903
$296$	9.96909e82	1.62324
$297$	−3.62278e81	−0.0526867
$298$	−1.26064e83	−1.63812
$299$	1.52642e83	1.77292
$300$	8.69531e81	0.0903073
$301$	−2.47083e83	−2.29543
$302$	−4.15664e82	−0.345549
$303$	−1.09521e83	−0.815023
$304$	−5.78352e82	−0.385413
$305$	−4.27725e82	−0.255340
$306$	4.98834e82	0.266863
$307$	2.02311e83	0.970249	0.485124	−	0.874445i	$-0.338774\pi$
$307$	2.02311e83	0.970249	0.485124	+	0.874445i	$0.338774\pi$
$308$	−3.21534e82	−0.138286
$309$	−8.10587e82	−0.312745
$310$	4.24383e82	0.146940
$311$	3.12987e83	0.972868	0.486434	−	0.873717i	$-0.338297\pi$
$311$	3.12987e83	0.972868	0.486434	+	0.873717i	$0.338297\pi$
$312$	2.85962e83	0.798232
$313$	7.95394e82	0.199456	0.0997279	−	0.995015i	$-0.468203\pi$
$313$	7.95394e82	0.199456	0.0997279	+	0.995015i	$0.468203\pi$
$314$	−2.72934e83	−0.615056
$315$	−2.00125e83	−0.405413
$316$	1.56226e83	0.284600
$317$	7.58586e83	1.24314	0.621569	−	0.783359i	$-0.286495\pi$
$317$	7.58586e83	1.24314	0.621569	+	0.783359i	$0.286495\pi$
$318$	−2.60947e83	−0.384808
$319$	1.80713e83	0.239885
$320$	5.96490e83	0.712982
$321$	−8.52690e82	−0.0918063
$322$	2.31294e84	2.24384
$323$	−6.24647e83	−0.546198
$324$	−3.77626e82	−0.0297717
$325$	−1.04628e84	−0.743969
$326$	1.07201e84	0.687719
$327$	−8.95648e83	−0.518546
$328$	−2.47984e84	−1.29613
$329$	−6.07499e84	−2.86737
$330$	2.04635e83	0.0872497
$331$	3.80638e84	1.46647	0.733236	−	0.679974i	$-0.238009\pi$
$331$	3.80638e84	1.46647	0.733236	+	0.679974i	$0.238009\pi$
$332$	−1.20688e83	−0.0420280
$333$	1.58406e84	0.498758
$334$	−6.09767e83	−0.173643
$335$	−3.63615e84	−0.936787
$336$	3.08226e84	0.718627
$337$	1.36745e84	0.288608	0.144304	−	0.989533i	$-0.453906\pi$
$337$	1.36745e84	0.288608	0.144304	+	0.989533i	$0.453906\pi$
$338$	−2.79443e84	−0.534052
$339$	2.18826e84	0.378799
$340$	1.03133e84	0.161754
$341$	5.12689e83	0.0728757
$342$	−1.29192e84	−0.166480
$343$	−2.50674e85	−2.92925
$344$	1.24622e85	1.32095
$345$	5.38790e84	0.518179
$346$	1.56261e85	1.36396
$347$	−1.91444e85	−1.51707	−0.758533	−	0.651634i	$-0.774084\pi$
$347$	−1.91444e85	−1.51707	−0.758533	+	0.651634i	$0.774084\pi$
$348$	1.88369e84	0.135552
$349$	2.32098e85	1.51712	0.758560	−	0.651603i	$-0.225903\pi$
$349$	2.32098e85	1.51712	0.758560	+	0.651603i	$0.225903\pi$
$350$	−1.58540e85	−0.941582
$351$	4.54385e84	0.245265
$352$	2.90077e84	0.142342
$353$	−2.89966e85	−1.29388	−0.646941	−	0.762540i	$-0.723952\pi$
$353$	−2.89966e85	−1.29388	−0.646941	+	0.762540i	$0.723952\pi$
$354$	1.13359e85	0.460093
$355$	−1.10805e85	−0.409174
$356$	6.91046e84	0.232237
$357$	3.32898e85	1.01842
$358$	5.10795e85	1.42287
$359$	5.20267e85	1.31996	0.659982	−	0.751281i	$-0.270564\pi$
$359$	5.20267e85	1.31996	0.659982	+	0.751281i	$0.270564\pi$
$360$	1.00938e85	0.233303
$361$	−3.13002e85	−0.659259
$362$	6.02481e84	0.115666
$363$	−3.05124e85	−0.534078
$364$	4.03282e85	0.643743
$365$	−2.21598e85	−0.322668
$366$	1.47156e85	0.195506
$367$	4.99070e85	0.605129	0.302565	−	0.953129i	$-0.402157\pi$
$367$	4.99070e85	0.605129	0.302565	+	0.953129i	$0.402157\pi$
$368$	−8.29827e85	−0.918514
$369$	−3.94039e85	−0.398250
$370$	−8.94768e85	−0.825948
$371$	−1.74144e86	−1.46853
$372$	5.34409e84	0.0411799
$373$	−6.82399e85	−0.480609	−0.240305	−	0.970698i	$-0.577247\pi$
$373$	−6.82399e85	−0.480609	−0.240305	+	0.970698i	$0.577247\pi$
$374$	−3.40401e85	−0.219176
$375$	−1.00195e86	−0.589929
$376$	3.06407e86	1.65008
$377$	−2.26659e86	−1.11670
$378$	6.88517e85	0.310412
$379$	−1.31157e86	−0.541225	−0.270613	−	0.962688i	$-0.587226\pi$
$379$	−1.31157e86	−0.541225	−0.270613	+	0.962688i	$0.587226\pi$
$380$	−2.67103e85	−0.100909
$381$	−1.10239e86	−0.381373
$382$	−1.68610e86	−0.534269
$383$	4.70284e86	1.36522	0.682608	−	0.730785i	$-0.260846\pi$
$383$	4.70284e86	1.36522	0.682608	+	0.730785i	$0.260846\pi$
$384$	−9.23724e85	−0.245723
$385$	1.36564e86	0.332968
$386$	4.66739e86	1.04328
$387$	1.98021e86	0.405877
$388$	−8.82961e85	−0.165989
$389$	1.41077e86	0.243301	0.121650	−	0.992573i	$-0.461181\pi$
$389$	1.41077e86	0.243301	0.121650	+	0.992573i	$0.461181\pi$
$390$	−2.56663e86	−0.406161
$391$	−8.96252e86	−1.30169
$392$	2.07802e87	2.77055
$393$	1.73320e85	0.0212177
$394$	9.99215e86	1.12340
$395$	−6.63532e86	−0.685266
$396$	2.57689e85	0.0244516
$397$	−1.34451e87	−1.17241	−0.586207	−	0.810161i	$-0.699380\pi$
$397$	−1.34451e87	−1.17241	−0.586207	+	0.810161i	$0.699380\pi$
$398$	1.61768e87	1.29661
$399$	−8.62169e86	−0.635332
$400$	5.68803e86	0.385436
$401$	−1.23685e87	−0.770870	−0.385435	−	0.922735i	$-0.625949\pi$
$401$	−1.23685e87	−0.770870	−0.385435	+	0.922735i	$0.625949\pi$
$402$	1.25099e87	0.717270
$403$	−6.43037e86	−0.339248
$404$	7.79027e86	0.378248
$405$	1.60388e86	0.0716848
$406$	−3.43449e87	−1.41332
$407$	−1.08095e87	−0.409632
$408$	−1.67906e87	−0.586070
$409$	−1.43261e87	−0.460678	−0.230339	−	0.973110i	$-0.573984\pi$
$409$	−1.43261e87	−0.460678	−0.230339	+	0.973110i	$0.573984\pi$
$410$	2.22576e87	0.659506
$411$	−2.10470e87	−0.574763
$412$	5.76573e86	0.145143
$413$	7.56502e87	1.75583
$414$	−1.85367e87	−0.396754
$415$	5.12594e86	0.101196
$416$	−3.63827e87	−0.662625
$417$	−4.87317e87	−0.818942
$418$	8.81601e86	0.136731
$419$	−3.24220e87	−0.464162	−0.232081	−	0.972696i	$-0.574553\pi$
$419$	−3.24220e87	−0.464162	−0.232081	+	0.972696i	$0.574553\pi$
$420$	1.42349e87	0.188150
$421$	−5.60594e87	−0.684224	−0.342112	−	0.939659i	$-0.611142\pi$
$421$	−5.60594e87	−0.684224	−0.342112	+	0.939659i	$0.611142\pi$
$422$	−6.14413e87	−0.692617
$423$	4.86872e87	0.507006
$424$	8.78338e87	0.845095
$425$	6.14334e87	0.546230
$426$	3.81216e87	0.313292
$427$	9.82048e87	0.746103
$428$	6.06521e86	0.0426069
$429$	−3.10069e87	−0.201437
$430$	−1.11854e88	−0.672136
$431$	−1.10616e88	−0.614934	−0.307467	−	0.951559i	$-0.599481\pi$
$431$	−1.10616e88	−0.614934	−0.307467	+	0.951559i	$0.599481\pi$
$432$	−2.47024e87	−0.127067
$433$	3.08620e88	1.46919	0.734596	−	0.678504i	$-0.237372\pi$
$433$	3.08620e88	1.46919	0.734596	+	0.678504i	$0.237372\pi$
$434$	−9.74376e87	−0.429359
$435$	−8.00053e87	−0.326384
$436$	6.37077e87	0.240655
$437$	2.32119e88	0.812050
$438$	7.62394e87	0.247057
$439$	1.34418e88	0.403548	0.201774	−	0.979432i	$-0.435329\pi$
$439$	1.34418e88	0.403548	0.201774	+	0.979432i	$0.435329\pi$
$440$	−6.88795e87	−0.191613
$441$	3.30192e88	0.851281
$442$	4.26946e88	1.02030
$443$	2.87068e88	0.636005	0.318002	−	0.948090i	$-0.396988\pi$
$443$	2.87068e88	0.636005	0.318002	+	0.948090i	$0.396988\pi$
$444$	−1.12675e88	−0.231471
$445$	−2.93505e88	−0.559185
$446$	−6.01655e88	−1.06324
$447$	6.74240e88	1.10538
$448$	−1.36953e89	−2.08333
$449$	−1.13335e89	−1.59997	−0.799987	−	0.600018i	$-0.795160\pi$
$449$	−1.13335e89	−1.59997	−0.799987	+	0.600018i	$0.795160\pi$
$450$	1.27059e88	0.166490
$451$	2.68890e88	0.327085
$452$	−1.55651e88	−0.175798
$453$	2.22314e88	0.233173
$454$	4.38904e88	0.427561
$455$	−1.71285e89	−1.55002
$456$	4.34856e88	0.365615
$457$	−2.05288e89	−1.60388	−0.801939	−	0.597406i	$-0.796198\pi$
$457$	−2.05288e89	−1.60388	−0.801939	+	0.597406i	$0.796198\pi$
$458$	−4.77012e88	−0.346367
$459$	−2.66797e88	−0.180076
$460$	−3.83243e88	−0.240484
$461$	1.13552e89	0.662544	0.331272	−	0.943535i	$-0.392522\pi$
$461$	1.13552e89	0.662544	0.331272	+	0.943535i	$0.392522\pi$
$462$	−4.69839e88	−0.254943
$463$	2.34175e89	1.18189	0.590944	−	0.806712i	$-0.298755\pi$
$463$	2.34175e89	1.18189	0.590944	+	0.806712i	$0.298755\pi$
$464$	1.23222e89	0.578542
$465$	−2.26977e88	−0.0991537
$466$	1.79103e89	0.728074
$467$	1.26348e89	0.478027	0.239013	−	0.971016i	$-0.423176\pi$
$467$	1.26348e89	0.478027	0.239013	+	0.971016i	$0.423176\pi$
$468$	−3.23205e88	−0.113826
$469$	8.34854e89	2.73729
$470$	−2.75013e89	−0.839606
$471$	1.45976e89	0.415033
$472$	−3.81561e89	−1.01043
$473$	−1.35128e89	−0.333349
$474$	2.28284e89	0.524687
$475$	−1.59105e89	−0.340760
$476$	−2.36792e89	−0.472643
$477$	1.39565e89	0.259664
$478$	−4.42598e89	−0.767671
$479$	2.55611e89	0.413371	0.206686	−	0.978407i	$-0.433732\pi$
$479$	2.55611e89	0.413371	0.206686	+	0.978407i	$0.433732\pi$
$480$	−1.28422e89	−0.193669
$481$	1.35578e90	1.90690
$482$	6.88798e89	0.903683
$483$	−1.23705e90	−1.51412
$484$	2.17036e89	0.247863
$485$	3.75017e89	0.399671
$486$	−5.51802e88	−0.0548869
$487$	−1.83594e90	−1.70466	−0.852329	−	0.523005i	$-0.824811\pi$
$487$	−1.83594e90	−1.70466	−0.852329	+	0.523005i	$0.824811\pi$
$488$	−4.95320e89	−0.429360
$489$	−5.73357e89	−0.464065
$490$	−1.86511e90	−1.40973
$491$	1.54963e90	1.09395	0.546976	−	0.837148i	$-0.315779\pi$
$491$	1.54963e90	1.09395	0.546976	+	0.837148i	$0.315779\pi$
$492$	2.80281e89	0.184826
$493$	1.33085e90	0.819894
$494$	−1.10574e90	−0.636504
$495$	−1.09447e89	−0.0588751
$496$	3.49583e89	0.175758
$497$	2.54406e90	1.19561
$498$	−1.76354e89	−0.0774825
$499$	−3.16417e90	−1.29985	−0.649923	−	0.760000i	$-0.725199\pi$
$499$	−3.16417e90	−1.29985	−0.649923	+	0.760000i	$0.725199\pi$
$500$	7.12691e89	0.273783
$501$	3.26128e89	0.117172
$502$	−1.19745e90	−0.402424
$503$	−2.63171e90	−0.827391	−0.413696	−	0.910415i	$-0.635762\pi$
$503$	−2.63171e90	−0.827391	−0.413696	+	0.910415i	$0.635762\pi$
$504$	−2.31752e90	−0.681711
$505$	−3.30873e90	−0.910753
$506$	1.26493e90	0.325856
$507$	1.49457e90	0.360372
$508$	7.84135e89	0.176993
$509$	5.29884e90	1.11979	0.559893	−	0.828565i	$-0.310842\pi$
$509$	5.29884e90	1.11979	0.559893	+	0.828565i	$0.310842\pi$
$510$	1.50702e90	0.298208
$511$	5.08786e90	0.942834
$512$	6.10489e90	1.05958
$513$	6.90974e89	0.112339
$514$	−7.74190e90	−1.17919
$515$	−2.44886e90	−0.349479
$516$	−1.40853e90	−0.188365
$517$	−3.32239e90	−0.416406
$518$	2.05437e91	2.41341
$519$	−8.35746e90	−0.920381
$520$	8.63916e90	0.891989
$521$	−1.77125e91	−1.71481	−0.857403	−	0.514646i	$-0.827923\pi$
$521$	−1.77125e91	−1.71481	−0.857403	+	0.514646i	$0.827923\pi$
$522$	2.75253e90	0.249902
$523$	9.54877e90	0.813094	0.406547	−	0.913630i	$-0.366733\pi$
$523$	9.54877e90	0.813094	0.406547	+	0.913630i	$0.366733\pi$
$524$	−1.23283e89	−0.00984704
$525$	8.47935e90	0.635369
$526$	−8.77605e89	−0.0616988
$527$	3.77566e90	0.249079
$528$	1.68567e90	0.104361
$529$	1.60954e91	0.935271
$530$	−7.88345e90	−0.430006
$531$	−6.06289e90	−0.310465
$532$	6.13263e90	0.294854
$533$	−3.37253e91	−1.52263
$534$	1.00979e91	0.428151
$535$	−2.57605e90	−0.102590
$536$	−4.21079e91	−1.57523
$537$	−2.73194e91	−0.960137
$538$	1.35942e91	0.448899
$539$	−2.25321e91	−0.699161
$540$	−1.14084e90	−0.0332686
$541$	−3.95124e91	−1.08299	−0.541495	−	0.840704i	$-0.682142\pi$
$541$	−3.95124e91	−1.08299	−0.541495	+	0.840704i	$0.682142\pi$
$542$	2.87762e91	0.741407
$543$	−3.22231e90	−0.0780501
$544$	2.13625e91	0.486506
$545$	−2.70583e91	−0.579453
$546$	5.89293e91	1.18680
$547$	−7.26253e91	−1.37566	−0.687832	−	0.725870i	$-0.741437\pi$
$547$	−7.26253e91	−1.37566	−0.687832	+	0.725870i	$0.741437\pi$
$548$	1.49708e91	0.266744
$549$	−7.87050e90	−0.131925
$550$	−8.67046e90	−0.136739
$551$	−3.44675e91	−0.511484
$552$	6.23938e91	0.871330
$553$	1.52346e92	2.00234
$554$	−4.34712e91	−0.537804
$555$	4.78559e91	0.557340
$556$	3.46630e91	0.380067
$557$	−1.39421e91	−0.143940	−0.0719698	−	0.997407i	$-0.522929\pi$
$557$	−1.39421e91	−0.143940	−0.0719698	+	0.997407i	$0.522929\pi$
$558$	7.80901e90	0.0759189
$559$	1.69484e92	1.55179
$560$	9.31178e91	0.803034
$561$	1.82061e91	0.147897
$562$	−1.56830e92	−1.20022
$563$	−1.57237e91	−0.113377	−0.0566884	−	0.998392i	$-0.518054\pi$
$563$	−1.57237e91	−0.113377	−0.0566884	+	0.998392i	$0.518054\pi$
$564$	−3.46313e91	−0.235299
$565$	6.61092e91	0.423291
$566$	5.95745e91	0.359509
$567$	−3.68247e91	−0.209463
$568$	−1.28316e92	−0.688036
$569$	1.12945e92	0.570962	0.285481	−	0.958384i	$-0.407847\pi$
$569$	1.12945e92	0.570962	0.285481	+	0.958384i	$0.407847\pi$
$570$	−3.90302e91	−0.186034
$571$	−1.66161e92	−0.746824	−0.373412	−	0.927666i	$-0.621812\pi$
$571$	−1.66161e92	−0.746824	−0.373412	+	0.927666i	$0.621812\pi$
$572$	2.20553e91	0.0934860
$573$	9.01794e91	0.360518
$574$	−5.11031e92	−1.92707
$575$	−2.28287e92	−0.812098
$576$	1.09759e92	0.368373
$577$	3.72309e92	1.17900	0.589500	−	0.807769i	$-0.299325\pi$
$577$	3.72309e92	1.17900	0.589500	+	0.807769i	$0.299325\pi$
$578$	3.56356e91	0.106488
$579$	−2.49631e92	−0.703991
$580$	5.69080e91	0.151473
$581$	−1.17691e92	−0.295693
$582$	−1.29022e92	−0.306016
$583$	−9.52385e91	−0.213263
$584$	−2.56619e92	−0.542573
$585$	1.37274e92	0.274073
$586$	6.73325e92	1.26956
$587$	4.35735e92	0.775973	0.387986	−	0.921665i	$-0.373171\pi$
$587$	4.35735e92	0.775973	0.387986	+	0.921665i	$0.373171\pi$
$588$	−2.34866e92	−0.395075
$589$	−9.77854e91	−0.155386
$590$	3.42467e92	0.514134
$591$	−5.34421e92	−0.758059
$592$	−7.37061e92	−0.987930
$593$	−1.52013e92	−0.192553	−0.0962763	−	0.995355i	$-0.530693\pi$
$593$	−1.52013e92	−0.192553	−0.0962763	+	0.995355i	$0.530693\pi$
$594$	3.76547e91	0.0450788
$595$	1.00572e93	1.13804
$596$	−4.79589e92	−0.513002
$597$	−8.65203e92	−0.874940
$598$	−1.58654e93	−1.51691
$599$	1.85461e93	1.67669	0.838345	−	0.545139i	$-0.183523\pi$
$599$	1.85461e93	1.67669	0.838345	+	0.545139i	$0.183523\pi$
$600$	−4.27677e92	−0.365636
$601$	−1.29011e91	−0.0104311	−0.00521556	−	0.999986i	$-0.501660\pi$
$601$	−1.29011e91	−0.0104311	−0.00521556	+	0.999986i	$0.501660\pi$
$602$	2.56814e93	1.96398
$603$	−6.69083e92	−0.484005
$604$	−1.58133e92	−0.108214
$605$	−9.21808e92	−0.596809
$606$	1.13835e93	0.697336
$607$	1.54230e93	0.894020	0.447010	−	0.894529i	$-0.352489\pi$
$607$	1.54230e93	0.894020	0.447010	+	0.894529i	$0.352489\pi$
$608$	−5.53264e92	−0.303503
$609$	1.83691e93	0.953692
$610$	4.44571e92	0.218470
$611$	4.16708e93	1.93844
$612$	1.89773e92	0.0835724
$613$	−1.38764e93	−0.578565	−0.289283	−	0.957244i	$-0.593417\pi$
$613$	−1.38764e93	−0.578565	−0.289283	+	0.957244i	$0.593417\pi$
$614$	−2.10279e93	−0.830147
$615$	−1.19043e93	−0.445027
$616$	1.58146e93	0.559892
$617$	−3.43145e93	−1.15060	−0.575300	−	0.817942i	$-0.695115\pi$
$617$	−3.43145e93	−1.15060	−0.575300	+	0.817942i	$0.695115\pi$
$618$	8.42513e92	0.267585
$619$	−2.77920e93	−0.836148	−0.418074	−	0.908413i	$-0.637295\pi$
$619$	−2.77920e93	−0.836148	−0.418074	+	0.908413i	$0.637295\pi$
$620$	1.61450e92	0.0460167
$621$	9.91420e92	0.267725
$622$	−3.25315e93	−0.832388
$623$	6.73883e93	1.63394
$624$	−2.11425e93	−0.485816
$625$	−3.46475e92	−0.0754555
$626$	−8.26721e92	−0.170655
$627$	−4.71516e92	−0.0922644
$628$	−1.03833e93	−0.192615
$629$	−7.96060e93	−1.40007
$630$	2.08007e93	0.346872
$631$	−5.80746e93	−0.918338	−0.459169	−	0.888349i	$-0.651853\pi$
$631$	−5.80746e93	−0.918338	−0.459169	+	0.888349i	$0.651853\pi$
$632$	−7.68394e93	−1.15229
$633$	3.28613e93	0.467370
$634$	−7.88463e93	−1.06363
$635$	−3.33043e93	−0.426167
$636$	−9.92732e92	−0.120509
$637$	2.82607e94	3.25471
$638$	−1.87831e93	−0.205246
$639$	−2.03890e93	−0.211406
$640$	−2.79065e93	−0.274585
$641$	−1.55793e94	−1.45480	−0.727402	−	0.686212i	$-0.759272\pi$
$641$	−1.55793e94	−1.45480	−0.727402	+	0.686212i	$0.759272\pi$
$642$	8.86274e92	0.0785497
$643$	9.20388e93	0.774289	0.387145	−	0.922019i	$-0.373461\pi$
$643$	9.20388e93	0.774289	0.387145	+	0.922019i	$0.373461\pi$
$644$	8.79919e93	0.702694
$645$	5.98240e93	0.453550
$646$	6.49249e93	0.467328
$647$	−3.77051e93	−0.257695	−0.128848	−	0.991664i	$-0.541128\pi$
$647$	−3.77051e93	−0.257695	−0.128848	+	0.991664i	$0.541128\pi$
$648$	1.85734e93	0.120540
$649$	4.13728e93	0.254987
$650$	1.08749e94	0.636541
$651$	5.21136e93	0.289726
$652$	4.07831e93	0.215370
$653$	−1.24838e94	−0.626261	−0.313130	−	0.949710i	$-0.601378\pi$
$653$	−1.24838e94	−0.626261	−0.313130	+	0.949710i	$0.601378\pi$
$654$	9.30924e93	0.443669
$655$	5.23616e92	0.0237099
$656$	1.83346e94	0.788846
$657$	−4.07760e93	−0.166711
$658$	6.31426e94	2.45332
$659$	7.87386e93	0.290754	0.145377	−	0.989376i	$-0.453560\pi$
$659$	7.87386e93	0.290754	0.145377	+	0.989376i	$0.453560\pi$
$660$	7.78503e92	0.0273236
$661$	−4.77483e94	−1.59297	−0.796487	−	0.604656i	$-0.793311\pi$
$661$	−4.77483e94	−1.59297	−0.796487	+	0.604656i	$0.793311\pi$
$662$	−3.95629e94	−1.25472
$663$	−2.28348e94	−0.688485
$664$	5.93602e93	0.170163
$665$	−2.60469e94	−0.709955
$666$	−1.64645e94	−0.426738
$667$	−4.94545e94	−1.21896
$668$	−2.31976e93	−0.0543790
$669$	3.21790e94	0.717460
$670$	3.77936e94	0.801517
$671$	5.37078e93	0.108351
$672$	2.94856e94	0.565899
$673$	−3.75749e94	−0.686109	−0.343054	−	0.939316i	$-0.611462\pi$
$673$	−3.75749e94	−0.686109	−0.343054	+	0.939316i	$0.611462\pi$
$674$	−1.42130e94	−0.246934
$675$	−6.79566e93	−0.112345
$676$	−1.06309e94	−0.167247
$677$	−7.42048e93	−0.111099	−0.0555497	−	0.998456i	$-0.517691\pi$
$677$	−7.42048e93	−0.111099	−0.0555497	+	0.998456i	$0.517691\pi$
$678$	−2.27444e94	−0.324101
$679$	−8.61032e94	−1.16784
$680$	−5.07258e94	−0.654908
$681$	−2.34744e94	−0.288513
$682$	−5.32882e93	−0.0623526
$683$	1.53091e95	1.70552	0.852759	−	0.522304i	$-0.174927\pi$
$683$	1.53091e95	1.70552	0.852759	+	0.522304i	$0.174927\pi$
$684$	−4.91492e93	−0.0521359
$685$	−6.35848e94	−0.642272
$686$	2.60547e95	2.50627
$687$	2.55126e94	0.233724
$688$	−9.21390e94	−0.803953
$689$	1.19452e95	0.992774
$690$	−5.60011e94	−0.443355
$691$	−1.70611e95	−1.28675	−0.643373	−	0.765553i	$-0.722466\pi$
$691$	−1.70611e95	−1.28675	−0.643373	+	0.765553i	$0.722466\pi$
$692$	5.94468e94	0.427144
$693$	2.51289e94	0.172033
$694$	1.98984e95	1.29800
$695$	−1.47223e95	−0.915132
$696$	−9.26490e94	−0.548822
$697$	1.98022e95	1.11793
$698$	−2.41239e95	−1.29805
$699$	−9.57919e94	−0.491296
$700$	−6.03138e94	−0.294871
$701$	−6.03623e94	−0.281327	−0.140664	−	0.990057i	$-0.544924\pi$
$701$	−6.03623e94	−0.281327	−0.140664	+	0.990057i	$0.544924\pi$
$702$	−4.72281e94	−0.209849
$703$	2.06170e95	0.873419
$704$	−7.48990e94	−0.302546
$705$	1.47088e95	0.566556
$706$	3.01387e95	1.10705
$707$	7.59679e95	2.66121
$708$	4.31255e94	0.144085
$709$	1.74967e94	0.0557579	0.0278789	−	0.999611i	$-0.491125\pi$
$709$	1.74967e94	0.0557579	0.0278789	+	0.999611i	$0.491125\pi$
$710$	1.15169e95	0.350090
$711$	−1.22096e95	−0.354053
$712$	−3.39890e95	−0.940282
$713$	−1.40304e95	−0.370314
$714$	−3.46010e95	−0.871361
$715$	−9.36748e94	−0.225097
$716$	1.94324e95	0.445595
$717$	2.36720e95	0.518015
$718$	−5.40758e95	−1.12936
$719$	−5.94055e95	−1.18416	−0.592078	−	0.805881i	$-0.701692\pi$
$719$	−5.94055e95	−1.18416	−0.592078	+	0.805881i	$0.701692\pi$
$720$	−7.46281e94	−0.141992
$721$	5.62253e95	1.02118
$722$	3.25330e95	0.564064
$723$	−3.68397e95	−0.609795
$724$	2.29204e94	0.0362227
$725$	3.38985e95	0.511513
$726$	3.17142e95	0.456959
$727$	−1.88760e95	−0.259721	−0.129861	−	0.991532i	$-0.541453\pi$
$727$	−1.88760e95	−0.259721	−0.129861	+	0.991532i	$0.541453\pi$
$728$	−1.98354e96	−2.60639
$729$	2.95127e94	0.0370370
$730$	2.30326e95	0.276075
$731$	−9.95144e95	−1.13934
$732$	5.59830e94	0.0612259
$733$	−9.47346e95	−0.989750	−0.494875	−	0.868964i	$-0.664786\pi$
$733$	−9.47346e95	−0.989750	−0.494875	+	0.868964i	$0.664786\pi$
$734$	−5.18726e95	−0.517750
$735$	9.97538e95	0.951270
$736$	−7.93831e95	−0.723305
$737$	4.56578e95	0.397516
$738$	4.09559e95	0.340744
$739$	4.24915e95	0.337841	0.168921	−	0.985630i	$-0.445972\pi$
$739$	4.24915e95	0.337841	0.168921	+	0.985630i	$0.445972\pi$
$740$	−3.40400e95	−0.258659
$741$	5.91396e95	0.429505
$742$	1.81003e96	1.25648
$743$	1.87178e96	1.24202	0.621010	−	0.783803i	$-0.286723\pi$
$743$	1.87178e96	1.24202	0.621010	+	0.783803i	$0.286723\pi$
$744$	−2.62848e95	−0.166729
$745$	2.03694e96	1.23522
$746$	7.09275e95	0.411210
$747$	9.43216e94	0.0522843
$748$	−1.29500e95	−0.0686384
$749$	5.91457e95	0.299766
$750$	1.04141e96	0.504744
$751$	−1.84305e96	−0.854278	−0.427139	−	0.904186i	$-0.640478\pi$
$751$	−1.84305e96	−0.854278	−0.427139	+	0.904186i	$0.640478\pi$
$752$	−2.26541e96	−1.00427
$753$	6.40447e95	0.271551
$754$	2.35586e96	0.955452
$755$	6.71631e95	0.260560
$756$	2.61935e95	0.0972106
$757$	−2.99008e96	−1.06162	−0.530812	−	0.847489i	$-0.678113\pi$
$757$	−2.99008e96	−1.06162	−0.530812	+	0.847489i	$0.678113\pi$
$758$	1.36323e96	0.463073
$759$	−6.76539e95	−0.219884
$760$	1.31374e96	0.408558
$761$	−3.97451e96	−1.18276	−0.591381	−	0.806392i	$-0.701417\pi$
$761$	−3.97451e96	−1.18276	−0.591381	+	0.806392i	$0.701417\pi$
$762$	1.14581e96	0.326303
$763$	6.21254e96	1.69316
$764$	−6.41448e95	−0.167315
$765$	−8.06018e95	−0.201227
$766$	−4.88806e96	−1.16808
$767$	−5.18915e96	−1.18700
$768$	−1.95362e96	−0.427799
$769$	−2.62342e96	−0.549964	−0.274982	−	0.961449i	$-0.588672\pi$
$769$	−2.62342e96	−0.549964	−0.274982	+	0.961449i	$0.588672\pi$
$770$	−1.41943e96	−0.284888
$771$	4.14069e96	0.795703
$772$	1.77563e96	0.326719
$773$	1.63182e96	0.287515	0.143757	−	0.989613i	$-0.454082\pi$
$773$	1.63182e96	0.287515	0.143757	+	0.989613i	$0.454082\pi$
$774$	−2.05820e96	−0.347269
$775$	9.61709e95	0.155395
$776$	4.34283e96	0.672055
$777$	−1.09876e97	−1.62854
$778$	−1.46633e96	−0.208169
$779$	−5.12855e96	−0.697412
$780$	−9.76432e95	−0.127196
$781$	1.39133e96	0.173629
$782$	9.31551e96	1.11373
$783$	−1.47217e96	−0.168631
$784$	−1.53638e97	−1.68620
$785$	4.41008e96	0.463781
$786$	−1.80147e95	−0.0181539
$787$	1.52024e97	1.46811	0.734053	−	0.679092i	$-0.237626\pi$
$787$	1.52024e97	1.46811	0.734053	+	0.679092i	$0.237626\pi$
$788$	3.80135e96	0.351811
$789$	4.69379e95	0.0416336
$790$	6.89666e96	0.586315
$791$	−1.51786e97	−1.23685
$792$	−1.26744e96	−0.0989997
$793$	−6.73626e96	−0.504391
$794$	1.39746e97	1.00312
$795$	4.21639e96	0.290163
$796$	6.15421e96	0.406055
$797$	2.51897e97	1.59356	0.796782	−	0.604266i	$-0.206534\pi$
$797$	2.51897e97	1.59356	0.796782	+	0.604266i	$0.206534\pi$
$798$	8.96126e96	0.543591
$799$	−2.44675e97	−1.42322
$800$	5.44130e96	0.303520
$801$	−5.40075e96	−0.288911
$802$	1.28557e97	0.659558
$803$	2.78253e96	0.136921
$804$	4.75920e96	0.224624
$805$	−3.73725e97	−1.69196
$806$	6.68363e96	0.290261
$807$	−7.27075e96	−0.302912
$808$	−3.83163e97	−1.53145
$809$	4.33159e97	1.66101	0.830505	−	0.557011i	$-0.188052\pi$
$809$	4.33159e97	1.66101	0.830505	+	0.557011i	$0.188052\pi$
$810$	−1.66704e96	−0.0613337
$811$	4.13849e97	1.46098	0.730488	−	0.682926i	$-0.239293\pi$
$811$	4.13849e97	1.46098	0.730488	+	0.682926i	$0.239293\pi$
$812$	−1.30660e97	−0.442604
$813$	−1.53907e97	−0.500293
$814$	1.12353e97	0.350482
$815$	−1.73216e97	−0.518572
$816$	1.24140e97	0.356691
$817$	2.57731e97	0.710767
$818$	1.48904e97	0.394157
$819$	−3.15178e97	−0.800839
$820$	8.46754e96	0.206535
$821$	1.13510e97	0.265791	0.132895	−	0.991130i	$-0.457573\pi$
$821$	1.13510e97	0.265791	0.132895	+	0.991130i	$0.457573\pi$
$822$	2.18759e97	0.491768
$823$	−8.60119e96	−0.185637	−0.0928184	−	0.995683i	$-0.529588\pi$
$823$	−8.60119e96	−0.185637	−0.0928184	+	0.995683i	$0.529588\pi$
$824$	−2.83586e97	−0.587657
$825$	4.63732e96	0.0922698
$826$	−7.86298e97	−1.50230
$827$	−3.87968e97	−0.711805	−0.355902	−	0.934523i	$-0.615826\pi$
$827$	−3.87968e97	−0.711805	−0.355902	+	0.934523i	$0.615826\pi$
$828$	−7.05200e96	−0.124250
$829$	−2.96648e97	−0.501955	−0.250977	−	0.967993i	$-0.580752\pi$
$829$	−2.96648e97	−0.501955	−0.250977	+	0.967993i	$0.580752\pi$
$830$	−5.32782e96	−0.0865832
$831$	2.32502e97	0.362904
$832$	9.39416e97	1.40840
$833$	−1.65936e98	−2.38964
$834$	5.06510e97	0.700689
$835$	9.85263e96	0.130935
$836$	3.35391e96	0.0428194
$837$	−4.17658e96	−0.0512292
$838$	3.36989e97	0.397138
$839$	−1.34167e98	−1.51922	−0.759608	−	0.650381i	$-0.774609\pi$
$839$	−1.34167e98	−1.51922	−0.759608	+	0.650381i	$0.774609\pi$
$840$	−7.00143e97	−0.761782
$841$	−2.22104e97	−0.232215
$842$	5.82673e97	0.585423
$843$	8.38791e97	0.809897
$844$	−2.33744e97	−0.216904
$845$	4.51524e97	0.402700
$846$	−5.06048e97	−0.433795
$847$	2.11646e98	1.74387
$848$	−6.49395e97	−0.514337
$849$	−3.18629e97	−0.242592
$850$	−6.38530e97	−0.467355
$851$	2.95816e98	2.08153
$852$	1.45028e97	0.0981125
$853$	2.26854e98	1.47555	0.737774	−	0.675048i	$-0.235877\pi$
$853$	2.26854e98	1.47555	0.737774	+	0.675048i	$0.235877\pi$
$854$	−1.02073e98	−0.638367
$855$	2.08749e97	0.125534
$856$	−2.98316e97	−0.172507
$857$	2.17739e97	0.121082	0.0605409	−	0.998166i	$-0.480717\pi$
$857$	2.17739e97	0.121082	0.0605409	+	0.998166i	$0.480717\pi$
$858$	3.22282e97	0.172350
$859$	2.55887e98	1.31606	0.658032	−	0.752990i	$-0.271389\pi$
$859$	2.55887e98	1.31606	0.658032	+	0.752990i	$0.271389\pi$
$860$	−4.25530e97	−0.210490
$861$	2.73320e98	1.30037
$862$	1.14972e98	0.526139
$863$	−1.48428e98	−0.653364	−0.326682	−	0.945134i	$-0.605931\pi$
$863$	−1.48428e98	−0.653364	−0.326682	+	0.945134i	$0.605931\pi$
$864$	−2.36308e97	−0.100062
$865$	−2.52486e98	−1.02849
$866$	−3.20775e98	−1.25704
$867$	−1.90594e97	−0.0718570
$868$	−3.70686e97	−0.134461
$869$	8.33172e97	0.290785
$870$	8.31564e97	0.279255
$871$	−5.72660e98	−1.85050
$872$	−3.13345e98	−0.974363
$873$	6.90063e97	0.206496
$874$	−2.41261e98	−0.694792
$875$	6.94991e98	1.92624
$876$	2.90041e97	0.0773698
$877$	−4.00787e98	−1.02903	−0.514514	−	0.857482i	$-0.672028\pi$
$877$	−4.00787e98	−1.02903	−0.514514	+	0.857482i	$0.672028\pi$
$878$	−1.39712e98	−0.345277
$879$	−3.60122e98	−0.856687
$880$	5.09257e97	0.116619
$881$	2.58772e98	0.570459	0.285230	−	0.958459i	$-0.407930\pi$
$881$	2.58772e98	0.570459	0.285230	+	0.958459i	$0.407930\pi$
$882$	−3.43196e98	−0.728358
$883$	6.62436e97	0.135351	0.0676754	−	0.997707i	$-0.478442\pi$
$883$	6.62436e97	0.135351	0.0676754	+	0.997707i	$0.478442\pi$
$884$	1.62425e98	0.319523
$885$	−1.83165e98	−0.346931
$886$	−2.98375e98	−0.544167
$887$	−5.91704e98	−1.03911	−0.519557	−	0.854436i	$-0.673903\pi$
$887$	−5.91704e98	−1.03911	−0.519557	+	0.854436i	$0.673903\pi$
$888$	5.54188e98	0.937179
$889$	7.64660e98	1.24526
$890$	3.05065e98	0.478440
$891$	−2.01393e97	−0.0304187
$892$	−2.28890e98	−0.332969
$893$	6.33680e98	0.887863
$894$	−7.00795e98	−0.945767
$895$	−8.25345e98	−1.07291
$896$	6.40729e98	0.802336
$897$	8.48544e98	1.02359
$898$	1.17799e99	1.36894
$899$	2.08338e98	0.233249
$900$	4.83377e97	0.0521389
$901$	−7.01377e98	−0.728905
$902$	−2.79480e98	−0.279854
$903$	−1.37355e99	−1.32527
$904$	7.65568e98	0.711773
$905$	−9.73490e97	−0.0872176
$906$	−2.31070e98	−0.199503
$907$	2.08456e98	0.173449	0.0867244	−	0.996232i	$-0.472360\pi$
$907$	2.08456e98	0.173449	0.0867244	+	0.996232i	$0.472360\pi$
$908$	1.66974e98	0.133898
$909$	−6.08835e98	−0.470554
$910$	1.78031e99	1.32620
$911$	5.32981e98	0.382689	0.191345	−	0.981523i	$-0.438715\pi$
$911$	5.32981e98	0.382689	0.191345	+	0.981523i	$0.438715\pi$
$912$	−3.21509e98	−0.222519
$913$	−6.43645e97	−0.0429413
$914$	2.13373e99	1.37228
$915$	−2.37775e98	−0.147421
$916$	−1.81472e98	−0.108470
$917$	−1.20221e98	−0.0692801
$918$	2.77305e98	0.154073
$919$	−1.92351e99	−1.03044	−0.515221	−	0.857057i	$-0.672290\pi$
$919$	−1.92351e99	−1.03044	−0.515221	+	0.857057i	$0.672290\pi$
$920$	1.88497e99	0.973673
$921$	1.12466e99	0.560173
$922$	−1.18025e99	−0.566874
$923$	−1.74507e99	−0.808269
$924$	−1.78743e98	−0.0798395
$925$	−2.02767e99	−0.873471
$926$	−2.43398e99	−1.01123
$927$	−4.50610e98	−0.180564
$928$	1.17877e99	0.455586
$929$	−1.75066e98	−0.0652644	−0.0326322	−	0.999467i	$-0.510389\pi$
$929$	−1.75066e98	−0.0652644	−0.0326322	+	0.999467i	$0.510389\pi$
$930$	2.35917e98	0.0848361
$931$	4.29755e99	1.49076
$932$	6.81370e98	0.228008
$933$	1.73992e99	0.561686
$934$	−1.31324e99	−0.409001
$935$	5.50022e98	0.165269
$936$	1.58968e99	0.460859
$937$	−2.21598e98	−0.0619856	−0.0309928	−	0.999520i	$-0.509867\pi$
$937$	−2.21598e98	−0.0619856	−0.0309928	+	0.999520i	$0.509867\pi$
$938$	−8.67735e99	−2.34203
$939$	4.42164e98	0.115156
$940$	−1.04624e99	−0.262936
$941$	−4.01175e99	−0.972929	−0.486464	−	0.873700i	$-0.661714\pi$
$941$	−4.01175e99	−0.972929	−0.486464	+	0.873700i	$0.661714\pi$
$942$	−1.51726e99	−0.355103
$943$	−7.35851e99	−1.66207
$944$	2.82105e99	0.614964
$945$	−1.11251e99	−0.234065
$946$	1.40451e99	0.285214
$947$	3.46437e99	0.679048	0.339524	−	0.940597i	$-0.389734\pi$
$947$	3.46437e99	0.679048	0.339524	+	0.940597i	$0.389734\pi$
$948$	8.68469e98	0.164314
$949$	−3.48997e99	−0.637387
$950$	1.65372e99	0.291555
$951$	4.21702e99	0.717726
$952$	1.16465e100	1.91364
$953$	−3.05518e99	−0.484645	−0.242323	−	0.970196i	$-0.577909\pi$
$953$	−3.05518e99	−0.484645	−0.242323	+	0.970196i	$0.577909\pi$
$954$	−1.45062e99	−0.222169
$955$	2.72440e99	0.402864
$956$	−1.68379e99	−0.240408
$957$	1.00460e99	0.138498
$958$	−2.65679e99	−0.353681
$959$	1.45989e100	1.87671
$960$	3.31592e99	0.411641
$961$	−7.75023e99	−0.929140
$962$	−1.40918e100	−1.63155
$963$	−4.74016e98	−0.0530044
$964$	2.62042e99	0.283003
$965$	−7.54159e99	−0.786679
$966$	1.28577e100	1.29548
$967$	1.53190e100	1.49088	0.745441	−	0.666572i	$-0.232239\pi$
$967$	1.53190e100	1.49088	0.745441	+	0.666572i	$0.232239\pi$
$968$	−1.06749e100	−1.00355
$969$	−3.47245e99	−0.315347
$970$	−3.89787e99	−0.341959
$971$	6.24686e99	0.529441	0.264720	−	0.964325i	$-0.414720\pi$
$971$	6.24686e99	0.529441	0.264720	+	0.964325i	$0.414720\pi$
$972$	−2.09924e98	−0.0171887
$973$	3.38021e100	2.67401
$974$	1.90825e100	1.45851
$975$	−5.81633e99	−0.429531
$976$	3.66213e99	0.261315
$977$	−4.20953e99	−0.290246	−0.145123	−	0.989414i	$-0.546358\pi$
$977$	−4.20953e99	−0.290246	−0.145123	+	0.989414i	$0.546358\pi$
$978$	5.95939e99	0.397055
$979$	3.68544e99	0.237284
$980$	−7.09551e99	−0.441479
$981$	−4.97896e99	−0.299383
$982$	−1.61066e100	−0.935988
$983$	−2.77525e100	−1.55869	−0.779344	−	0.626596i	$-0.784447\pi$
$983$	−2.77525e100	−1.55869	−0.779344	+	0.626596i	$0.784447\pi$
$984$	−1.37856e100	−0.748323
$985$	−1.61453e100	−0.847098
$986$	−1.38327e100	−0.701503
$987$	−3.37712e100	−1.65547
$988$	−4.20662e99	−0.199331
$989$	3.69796e100	1.69390
$990$	1.13758e99	0.0503736
$991$	−8.60539e99	−0.368386	−0.184193	−	0.982890i	$-0.558967\pi$
$991$	−8.60539e99	−0.368386	−0.184193	+	0.982890i	$0.558967\pi$
$992$	3.34419e99	0.138404
$993$	2.11599e100	0.846668
$994$	−2.64426e100	−1.02296
$995$	−2.61386e100	−0.977707
$996$	−6.70912e98	−0.0242649
$997$	−2.15408e100	−0.753310	−0.376655	−	0.926354i	$-0.622926\pi$
$997$	−2.15408e100	−0.753310	−0.376655	+	0.926354i	$0.622926\pi$
$998$	3.28880e100	1.11215
$999$	8.80588e99	0.287958

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.68.a.a.1.2	✓	5	1.1	even	1	trivial
9.68.a.b.1.4		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)

\(f(q)\)	\(=\)	\(q-1.03939e10 q^{2} +5.55906e15 q^{3} -3.95417e19 q^{4} +1.67944e23 q^{5} -5.77801e25 q^{6} -3.85597e28 q^{7} +1.94485e30 q^{8} +3.09032e31 q^{9} +O(q^{10})\)	\(q-1.03939e10 q^{2} +5.55906e15 q^{3} -3.95417e19 q^{4} +1.67944e23 q^{5} -5.77801e25 q^{6} -3.85597e28 q^{7} +1.94485e30 q^{8} +3.09032e31 q^{9} -1.74559e33 q^{10} -2.10881e34 q^{11} -2.19815e35 q^{12} +2.64496e37 q^{13} +4.00784e38 q^{14} +9.33612e38 q^{15} -1.43792e40 q^{16} -1.55302e41 q^{17} -3.21203e41 q^{18} +4.02214e42 q^{19} -6.64081e42 q^{20} -2.14356e44 q^{21} +2.19187e44 q^{22} +5.77103e45 q^{23} +1.08116e46 q^{24} -3.95574e46 q^{25} -2.74914e47 q^{26} +1.71793e47 q^{27} +1.52472e48 q^{28} -8.56944e48 q^{29} -9.70383e48 q^{30} -2.43118e49 q^{31} -1.37554e50 q^{32} -1.17230e50 q^{33} +1.61419e51 q^{34} -6.47588e51 q^{35} -1.22196e51 q^{36} +5.12588e52 q^{37} -4.18056e52 q^{38} +1.47035e53 q^{39} +3.26627e53 q^{40} -1.27508e54 q^{41} +2.22798e54 q^{42} +6.40780e54 q^{43} +8.33861e53 q^{44} +5.19000e54 q^{45} -5.99833e55 q^{46} +1.57548e56 q^{47} -7.99348e55 q^{48} +1.06847e57 q^{49} +4.11154e56 q^{50} -8.63333e56 q^{51} -1.04586e57 q^{52} +4.51622e57 q^{53} -1.78559e57 q^{54} -3.54163e57 q^{55} -7.49929e58 q^{56} +2.23593e58 q^{57} +8.90696e58 q^{58} -1.96190e59 q^{59} -3.69166e58 q^{60} -2.54683e59 q^{61} +2.52693e59 q^{62} -1.19162e60 q^{63} +3.55172e60 q^{64} +4.44206e60 q^{65} +1.21847e60 q^{66} -2.16510e61 q^{67} +6.14091e60 q^{68} +3.20815e61 q^{69} +6.73093e61 q^{70} -6.59771e61 q^{71} +6.01021e61 q^{72} -1.31948e62 q^{73} -5.32777e62 q^{74} -2.19902e62 q^{75} -1.59043e62 q^{76} +8.13151e62 q^{77} -1.52826e63 q^{78} -3.95091e63 q^{79} -2.41490e63 q^{80} +9.55005e62 q^{81} +1.32530e64 q^{82} +3.05217e63 q^{83} +8.47600e63 q^{84} -2.60821e64 q^{85} -6.66017e64 q^{86} -4.76381e64 q^{87} -4.10133e64 q^{88} -1.74764e65 q^{89} -5.39442e64 q^{90} -1.01989e66 q^{91} -2.28197e65 q^{92} -1.35151e65 q^{93} -1.63753e66 q^{94} +6.75496e65 q^{95} -7.64674e65 q^{96} +2.23298e66 q^{97} -1.11055e67 q^{98} -6.51689e65 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 16255223088 q^{2} + 27\!\cdots\!15 q^{3}+ \cdots + 15\!\cdots\!45 q^{9}+O(q^{10}) \) 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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