LMFDB - Embedded newform 3.68.a.a.1.1

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			1.1
Root			$3.76085e8$ 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)

$f(q)$	$=$	$q-2.13031e10 q^{2} +5.55906e15 q^{3} +3.06250e20 q^{4} -4.34254e22 q^{5} -1.18425e26 q^{6} +7.31825e27 q^{7} -3.38029e30 q^{8} +3.09032e31 q^{9} +O(q^{10})$	\(q-2.13031e10 q^{2} +5.55906e15 q^{3} +3.06250e20 q^{4} -4.34254e22 q^{5} -1.18425e26 q^{6} +7.31825e27 q^{7} -3.38029e30 q^{8} +3.09032e31 q^{9} +9.25096e32 q^{10} +4.75664e34 q^{11} +1.70246e36 q^{12} +7.97368e36 q^{13} -1.55902e38 q^{14} -2.41404e38 q^{15} +2.68163e40 q^{16} +1.90565e41 q^{17} -6.58334e41 q^{18} -1.00826e43 q^{19} -1.32990e43 q^{20} +4.06826e43 q^{21} -1.01331e45 q^{22} -7.07884e45 q^{23} -1.87912e46 q^{24} -6.58769e46 q^{25} -1.69864e47 q^{26} +1.71793e47 q^{27} +2.24121e48 q^{28} +1.16251e49 q^{29} +5.14267e48 q^{30} -1.43085e50 q^{31} -7.24284e49 q^{32} +2.64424e50 q^{33} -4.05964e51 q^{34} -3.17798e50 q^{35} +9.46408e51 q^{36} +6.93252e51 q^{37} +2.14791e53 q^{38} +4.43262e52 q^{39} +1.46790e53 q^{40} -4.79867e53 q^{41} -8.66667e53 q^{42} +5.19896e54 q^{43} +1.45672e55 q^{44} -1.34198e54 q^{45} +1.50801e56 q^{46} +1.06751e56 q^{47} +1.49073e56 q^{48} -3.64821e56 q^{49} +1.40338e57 q^{50} +1.05936e57 q^{51} +2.44194e57 q^{52} +2.07016e56 q^{53} -3.65972e57 q^{54} -2.06559e57 q^{55} -2.47378e58 q^{56} -5.60498e58 q^{57} -2.47652e59 q^{58} -2.69298e59 q^{59} -7.39300e58 q^{60} -1.30079e59 q^{61} +3.04815e60 q^{62} +2.26157e59 q^{63} -2.41443e60 q^{64} -3.46260e59 q^{65} -5.63307e60 q^{66} +1.00707e61 q^{67} +5.83606e61 q^{68} -3.93517e61 q^{69} +6.77009e60 q^{70} +2.01874e62 q^{71} -1.04462e62 q^{72} -3.79880e62 q^{73} -1.47684e62 q^{74} -3.66214e62 q^{75} -3.08779e63 q^{76} +3.48102e62 q^{77} -9.44287e62 q^{78} +4.95942e63 q^{79} -1.16451e63 q^{80} +9.55005e62 q^{81} +1.02227e64 q^{82} -3.63568e63 q^{83} +1.24590e64 q^{84} -8.27537e63 q^{85} -1.10754e65 q^{86} +6.46249e64 q^{87} -1.60788e65 q^{88} -2.05203e65 q^{89} +2.85884e64 q^{90} +5.83534e64 q^{91} -2.16789e66 q^{92} -7.95416e65 q^{93} -2.27413e66 q^{94} +4.37840e65 q^{95} -4.02634e65 q^{96} +1.12980e66 q^{97} +7.77183e66 q^{98} +1.46995e66 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.13031e10	−1.75363	−0.876816	−	0.480825i	$-0.840337\pi$
$2$	−2.13031e10	−1.75363	−0.876816	+	0.480825i	$0.840337\pi$
$3$	5.55906e15	0.577350
$3$	5.55906e15	0.577350
$4$	3.06250e20	2.07523
$5$	−4.34254e22	−0.166820	−0.0834100	−	0.996515i	$-0.526581\pi$
$5$	−4.34254e22	−0.166820	−0.0834100	+	0.996515i	$0.526581\pi$
$6$	−1.18425e26	−1.01246
$7$	7.31825e27	0.357786	0.178893	−	0.983869i	$-0.442748\pi$
$7$	7.31825e27	0.357786	0.178893	+	0.983869i	$0.442748\pi$
$8$	−3.38029e30	−1.88556
$9$	3.09032e31	0.333333
$10$	9.25096e32	0.292541
$11$	4.75664e34	0.617511	0.308756	−	0.951141i	$-0.400087\pi$
$11$	4.75664e34	0.617511	0.308756	+	0.951141i	$0.400087\pi$
$12$	1.70246e36	1.19813
$13$	7.97368e36	0.384199	0.192100	−	0.981375i	$-0.438470\pi$
$13$	7.97368e36	0.384199	0.192100	+	0.981375i	$0.438470\pi$
$14$	−1.55902e38	−0.627425
$15$	−2.41404e38	−0.0963136
$16$	2.68163e40	1.23134
$17$	1.90565e41	1.14817	0.574083	−	0.818797i	$-0.305359\pi$
$17$	1.90565e41	1.14817	0.574083	+	0.818797i	$0.305359\pi$
$18$	−6.58334e41	−0.584544
$19$	−1.00826e43	−1.46328	−0.731639	−	0.681693i	$-0.761244\pi$
$19$	−1.00826e43	−1.46328	−0.731639	+	0.681693i	$0.761244\pi$
$20$	−1.32990e43	−0.346190
$21$	4.06826e43	0.206568
$22$	−1.01331e45	−1.08289
$23$	−7.07884e45	−1.70639	−0.853197	−	0.521589i	$-0.825340\pi$
$23$	−7.07884e45	−1.70639	−0.853197	+	0.521589i	$0.825340\pi$
$24$	−1.87912e46	−1.08863
$25$	−6.58769e46	−0.972171
$26$	−1.69864e47	−0.673745
$27$	1.71793e47	0.192450
$28$	2.24121e48	0.742487
$29$	1.16251e49	1.18868	0.594341	−	0.804213i	$-0.297413\pi$
$29$	1.16251e49	1.18868	0.594341	+	0.804213i	$0.297413\pi$
$30$	5.14267e48	0.168899
$31$	−1.43085e50	−1.56667	−0.783333	−	0.621603i	$-0.786482\pi$
$31$	−1.43085e50	−1.56667	−0.783333	+	0.621603i	$0.786482\pi$
$32$	−7.24284e49	−0.273769
$33$	2.64424e50	0.356520
$34$	−4.05964e51	−2.01346
$35$	−3.17798e50	−0.0596858
$36$	9.46408e51	0.691743
$37$	6.93252e51	0.202364	0.101182	−	0.994868i	$-0.467738\pi$
$37$	6.93252e51	0.202364	0.101182	+	0.994868i	$0.467738\pi$
$38$	2.14791e53	2.56605
$39$	4.43262e52	0.221818
$40$	1.46790e53	0.314549
$41$	−4.79867e53	−0.449637	−0.224818	−	0.974401i	$-0.572179\pi$
$41$	−4.79867e53	−0.449637	−0.224818	+	0.974401i	$0.572179\pi$
$42$	−8.66667e53	−0.362244
$43$	5.19896e54	0.987923	0.493962	−	0.869484i	$-0.335548\pi$
$43$	5.19896e54	0.987923	0.493962	+	0.869484i	$0.335548\pi$
$44$	1.45672e55	1.28148
$45$	−1.34198e54	−0.0556067
$46$	1.50801e56	2.99239
$47$	1.06751e56	1.03061	0.515304	−	0.857007i	$-0.327679\pi$
$47$	1.06751e56	1.03061	0.515304	+	0.857007i	$0.327679\pi$
$48$	1.49073e56	0.710917
$49$	−3.64821e56	−0.871989
$50$	1.40338e57	1.70483
$51$	1.05936e57	0.662894
$52$	2.44194e57	0.797301
$53$	2.07016e56	0.0357077	0.0178539	−	0.999841i	$-0.494317\pi$
$53$	2.07016e56	0.0357077	0.0178539	+	0.999841i	$0.494317\pi$
$54$	−3.65972e57	−0.337487
$55$	−2.06559e57	−0.103013
$56$	−2.47378e58	−0.674625
$57$	−5.60498e58	−0.844824
$58$	−2.47652e59	−2.08451
$59$	−2.69298e59	−1.27847	−0.639236	−	0.769011i	$-0.720749\pi$
$59$	−2.69298e59	−1.27847	−0.639236	+	0.769011i	$0.720749\pi$
$60$	−7.39300e58	−0.199873
$61$	−1.30079e59	−0.202142	−0.101071	−	0.994879i	$-0.532227\pi$
$61$	−1.30079e59	−0.202142	−0.101071	+	0.994879i	$0.532227\pi$
$62$	3.04815e60	2.74736
$63$	2.26157e59	0.119262
$64$	−2.41443e60	−0.751253
$65$	−3.46260e59	−0.0640922
$66$	−5.63307e60	−0.625206
$67$	1.00707e61	0.675391	0.337695	−	0.941255i	$-0.390353\pi$
$67$	1.00707e61	0.675391	0.337695	+	0.941255i	$0.390353\pi$
$68$	5.83606e61	2.38271
$69$	−3.93517e61	−0.985187
$70$	6.77009e60	0.104667
$71$	2.01874e62	1.94055	0.970276	−	0.242002i	$-0.0778039\pi$
$71$	2.01874e62	1.94055	0.970276	+	0.242002i	$0.0778039\pi$
$72$	−1.04462e62	−0.628518
$73$	−3.79880e62	−1.43989	−0.719947	−	0.694029i	$-0.755834\pi$
$73$	−3.79880e62	−1.43989	−0.719947	+	0.694029i	$0.755834\pi$
$74$	−1.47684e62	−0.354872
$75$	−3.66214e62	−0.561283
$76$	−3.08779e63	−3.03663
$77$	3.48102e62	0.220937
$78$	−9.44287e62	−0.388987
$79$	4.95942e63	1.33329	0.666643	−	0.745377i	$-0.267731\pi$
$79$	4.95942e63	1.33329	0.666643	+	0.745377i	$0.267731\pi$
$80$	−1.16451e63	−0.205413
$81$	9.55005e62	0.111111
$82$	1.02227e64	0.788498
$83$	−3.63568e63	−0.186840	−0.0934198	−	0.995627i	$-0.529780\pi$
$83$	−3.63568e63	−0.186840	−0.0934198	+	0.995627i	$0.529780\pi$
$84$	1.24590e64	0.428675
$85$	−8.27537e63	−0.191537
$86$	−1.10754e65	−1.73246
$87$	6.46249e64	0.686286
$88$	−1.60788e65	−1.16435
$89$	−2.05203e65	−1.01770	−0.508848	−	0.860856i	$-0.669929\pi$
$89$	−2.05203e65	−1.01770	−0.508848	+	0.860856i	$0.669929\pi$
$90$	2.85884e64	0.0975137
$91$	5.83534e64	0.137461
$92$	−2.16789e66	−3.54116
$93$	−7.95416e65	−0.904515
$94$	−2.27413e66	−1.80731
$95$	4.37840e65	0.244104
$96$	−4.02634e65	−0.158061
$97$	1.12980e66	0.313435	0.156717	−	0.987643i	$-0.449909\pi$
$97$	1.12980e66	0.313435	0.156717	+	0.987643i	$0.449909\pi$
$98$	7.77183e66	1.52915
$99$	1.46995e66	0.205837
$100$	−2.01748e67	−2.01748
$101$	1.24250e67	0.890291	0.445146	−	0.895458i	$-0.353152\pi$
$101$	1.24250e67	0.890291	0.445146	+	0.895458i	$0.353152\pi$
$102$	−2.25678e67	−1.16247
$103$	−8.53843e66	−0.317198	−0.158599	−	0.987343i	$-0.550698\pi$
$103$	−8.53843e66	−0.317198	−0.158599	+	0.987343i	$0.550698\pi$
$104$	−2.69534e67	−0.724429
$105$	−1.76666e66	−0.0344596
$106$	−4.41009e66	−0.0626182
$107$	−7.62710e67	−0.790684	−0.395342	−	0.918534i	$-0.629374\pi$
$107$	−7.62710e67	−0.790684	−0.395342	+	0.918534i	$0.629374\pi$
$108$	5.26114e67	0.399378
$109$	2.26896e67	0.126485	0.0632426	−	0.997998i	$-0.479856\pi$
$109$	2.26896e67	0.126485	0.0632426	+	0.997998i	$0.479856\pi$
$110$	4.40035e67	0.180647
$111$	3.85383e67	0.116835
$112$	1.96248e68	0.440557
$113$	−6.63308e68	−1.10557	−0.552786	−	0.833323i	$-0.686435\pi$
$113$	−6.63308e68	−1.10557	−0.552786	+	0.833323i	$0.686435\pi$
$114$	1.19404e69	1.48151
$115$	3.07401e68	0.284661
$116$	3.56020e69	2.46679
$117$	2.46412e68	0.128066
$118$	5.73690e69	2.24197
$119$	1.39460e69	0.410798
$120$	8.16016e68	0.181605
$121$	−3.67093e69	−0.618680
$122$	2.77109e69	0.354483
$123$	−2.66761e69	−0.259598
$124$	−4.38196e70	−3.25119
$125$	5.80334e69	0.328998
$126$	−4.81785e69	−0.209142
$127$	−4.12385e70	−1.37366	−0.686828	−	0.726820i	$-0.740997\pi$
$127$	−4.12385e70	−1.37366	−0.686828	+	0.726820i	$0.740997\pi$
$128$	6.21236e70	1.59119
$129$	2.89013e70	0.570378
$130$	7.37643e69	0.112394
$131$	1.66773e70	0.196580	0.0982898	−	0.995158i	$-0.468663\pi$
$131$	1.66773e70	0.196580	0.0982898	+	0.995158i	$0.468663\pi$
$132$	8.09798e70	0.739861
$133$	−7.37870e70	−0.523540
$134$	−2.14538e71	−1.18439
$135$	−7.46015e69	−0.0321045
$136$	−6.44166e71	−2.16493
$137$	−1.33284e71	−0.350461	−0.175231	−	0.984527i	$-0.556067\pi$
$137$	−1.33284e71	−0.350461	−0.175231	+	0.984527i	$0.556067\pi$
$138$	8.38314e71	1.72766
$139$	−2.69374e71	−0.435873	−0.217936	−	0.975963i	$-0.569933\pi$
$139$	−2.69374e71	−0.435873	−0.217936	+	0.975963i	$0.569933\pi$
$140$	−9.73254e70	−0.123862
$141$	5.93435e71	0.595022
$142$	−4.30054e72	−3.40302
$143$	3.79279e71	0.237247
$144$	8.28708e71	0.410448
$145$	−5.04826e71	−0.198296
$146$	8.09263e72	2.52505
$147$	−2.02806e72	−0.503443
$148$	2.12308e72	0.419951
$149$	5.38162e72	0.849519	0.424760	−	0.905306i	$-0.360359\pi$
$149$	5.38162e72	0.849519	0.424760	+	0.905306i	$0.360359\pi$
$150$	7.80150e72	0.984285
$151$	−5.82617e72	−0.588377	−0.294189	−	0.955747i	$-0.595049\pi$
$151$	−5.82617e72	−0.588377	−0.294189	+	0.955747i	$0.595049\pi$
$152$	3.40821e73	2.75909
$153$	5.88907e72	0.382722
$154$	−7.41567e72	−0.387442
$155$	6.21350e72	0.261351
$156$	1.35749e73	0.460322
$157$	−4.60245e73	−1.25994	−0.629972	−	0.776618i	$-0.716933\pi$
$157$	−4.60245e73	−1.25994	−0.629972	+	0.776618i	$0.716933\pi$
$158$	−1.05651e74	−2.33809
$159$	1.15081e72	0.0206159
$160$	3.14523e72	0.0456702
$161$	−5.18047e73	−0.610523
$162$	−2.03446e73	−0.194848
$163$	1.78024e74	1.38737	0.693686	−	0.720277i	$-0.255985\pi$
$163$	1.78024e74	1.38737	0.693686	+	0.720277i	$0.255985\pi$
$164$	−1.46959e74	−0.933099
$165$	−1.14827e73	−0.0594747
$166$	7.74513e73	0.327648
$167$	−1.63814e74	−0.566694	−0.283347	−	0.959018i	$-0.591445\pi$
$167$	−1.63814e74	−0.566694	−0.283347	+	0.959018i	$0.591445\pi$
$168$	−1.37519e74	−0.389495
$169$	−3.67150e74	−0.852391
$170$	1.76291e74	0.335886
$171$	−3.11584e74	−0.487759
$172$	1.59218e75	2.05017
$173$	−3.21576e72	−0.00340989	−0.00170494	−	0.999999i	$-0.500543\pi$
$173$	−3.21576e72	−0.00340989	−0.00170494	+	0.999999i	$0.500543\pi$
$174$	−1.37671e75	−1.20349
$175$	−4.82103e74	−0.347829
$176$	1.27555e75	0.760369
$177$	−1.49705e75	−0.738126
$178$	4.37146e75	1.78467
$179$	−4.79857e75	−1.62381	−0.811905	−	0.583789i	$-0.801570\pi$
$179$	−4.79857e75	−1.62381	−0.811905	+	0.583789i	$0.801570\pi$
$180$	−4.10981e74	−0.115397
$181$	−5.30992e75	−1.23838	−0.619192	−	0.785239i	$-0.712540\pi$
$181$	−5.30992e75	−1.23838	−0.619192	+	0.785239i	$0.712540\pi$
$182$	−1.24311e75	−0.241056
$183$	−7.23116e74	−0.116707
$184$	2.39285e76	3.21750
$185$	−3.01047e74	−0.0337584
$186$	1.69449e76	1.58619
$187$	9.06450e75	0.709006
$188$	3.26925e76	2.13875
$189$	1.25722e75	0.0688559
$190$	−9.32737e75	−0.428069
$191$	−4.33840e76	−1.66998	−0.834991	−	0.550264i	$-0.814527\pi$
$191$	−4.33840e76	−1.66998	−0.834991	+	0.550264i	$0.814527\pi$
$192$	−1.34220e76	−0.433736
$193$	−4.35247e76	−1.18186	−0.590929	−	0.806723i	$-0.701239\pi$
$193$	−4.35247e76	−1.18186	−0.590929	+	0.806723i	$0.701239\pi$
$194$	−2.40682e76	−0.549649
$195$	−1.92488e75	−0.0370036
$196$	−1.11726e77	−1.80958
$197$	1.41121e76	0.192741	0.0963704	−	0.995346i	$-0.469277\pi$
$197$	1.41121e76	0.192741	0.0963704	+	0.995346i	$0.469277\pi$
$198$	−3.13146e76	−0.360963
$199$	−1.91712e77	−1.86669	−0.933345	−	0.358982i	$-0.883124\pi$
$199$	−1.91712e77	−1.86669	−0.933345	+	0.358982i	$0.883124\pi$
$200$	2.22683e77	1.83308
$201$	5.59838e76	0.389937
$202$	−2.64692e77	−1.56124
$203$	8.50758e76	0.425293
$204$	3.24430e77	1.37566
$205$	2.08384e76	0.0750084
$206$	1.81895e77	0.556249
$207$	−2.18758e77	−0.568798
$208$	2.13825e77	0.473081
$209$	−4.79592e77	−0.903590
$210$	3.76353e76	0.0604295
$211$	4.33960e77	0.594275	0.297137	−	0.954835i	$-0.403968\pi$
$211$	4.33960e77	0.594275	0.297137	+	0.954835i	$0.403968\pi$
$212$	6.33985e76	0.0741016
$213$	1.12223e78	1.12038
$214$	1.62481e78	1.38657
$215$	−2.25767e77	−0.164805
$216$	−5.80708e77	−0.362875
$217$	−1.04713e78	−0.560530
$218$	−4.83360e77	−0.221809
$219$	−2.11178e78	−0.831323
$220$	−6.32585e77	−0.213776
$221$	1.51951e78	0.441125
$222$	−8.20986e77	−0.204885
$223$	2.41287e78	0.517990	0.258995	−	0.965879i	$-0.416609\pi$
$223$	2.41287e78	0.517990	0.258995	+	0.965879i	$0.416609\pi$
$224$	−5.30049e77	−0.0979507
$225$	−2.03580e78	−0.324057
$226$	1.41305e79	1.93877
$227$	8.94151e78	1.05814	0.529072	−	0.848577i	$-0.322540\pi$
$227$	8.94151e78	1.05814	0.529072	+	0.848577i	$0.322540\pi$
$228$	−1.71652e79	−1.75320
$229$	1.27524e79	1.12487	0.562436	−	0.826841i	$-0.309864\pi$
$229$	1.27524e79	1.12487	0.562436	+	0.826841i	$0.309864\pi$
$230$	−6.54861e78	−0.499191
$231$	1.93512e78	0.127558
$232$	−3.92964e79	−2.24133
$233$	−2.98312e78	−0.147315	−0.0736577	−	0.997284i	$-0.523467\pi$
$233$	−2.98312e78	−0.147315	−0.0736577	+	0.997284i	$0.523467\pi$
$234$	−5.24935e78	−0.224582
$235$	−4.63570e78	−0.171926
$236$	−8.24725e79	−2.65312
$237$	2.75697e79	0.769773
$238$	−2.97094e79	−0.720388
$239$	2.27590e79	0.479538	0.239769	−	0.970830i	$-0.422928\pi$
$239$	2.27590e79	0.479538	0.239769	+	0.970830i	$0.422928\pi$
$240$	−6.47357e78	−0.118595
$241$	7.38899e79	1.17764	0.588822	−	0.808262i	$-0.299592\pi$
$241$	7.38899e79	1.17764	0.588822	+	0.808262i	$0.299592\pi$
$242$	7.82023e79	1.08494
$243$	5.30893e78	0.0641500
$244$	−3.98366e79	−0.419491
$245$	1.58425e79	0.145465
$246$	5.68285e79	0.455239
$247$	−8.03954e79	−0.562190
$248$	4.83667e80	2.95403
$249$	−2.02109e79	−0.107872
$250$	−1.23629e80	−0.576941
$251$	3.53345e80	1.44254	0.721272	−	0.692652i	$-0.243558\pi$
$251$	3.53345e80	1.44254	0.721272	+	0.692652i	$0.243558\pi$
$252$	6.92605e79	0.247496
$253$	−3.36714e80	−1.05372
$254$	8.78508e80	2.40889
$255$	−4.60033e79	−0.110584
$256$	−9.67119e80	−2.03911
$257$	−6.61020e80	−1.22308	−0.611540	−	0.791213i	$-0.709450\pi$
$257$	−6.61020e80	−1.22308	−0.611540	+	0.791213i	$0.709450\pi$
$258$	−6.15689e80	−1.00023
$259$	5.07339e79	0.0724029
$260$	−1.06042e80	−0.133006
$261$	3.59254e80	0.396227
$262$	−3.55280e80	−0.344728
$263$	−2.01456e81	−1.72053	−0.860265	−	0.509846i	$-0.829702\pi$
$263$	−2.01456e81	−1.72053	−0.860265	+	0.509846i	$0.829702\pi$
$264$	−8.93830e80	−0.672239
$265$	−8.98974e78	−0.00595676
$266$	1.57189e81	0.918096
$267$	−1.14074e81	−0.587567
$268$	3.08416e81	1.40159
$269$	1.06749e81	0.428214	0.214107	−	0.976810i	$-0.431316\pi$
$269$	1.06749e81	0.428214	0.214107	+	0.976810i	$0.431316\pi$
$270$	1.58925e80	0.0562996
$271$	8.68000e80	0.271674	0.135837	−	0.990731i	$-0.456628\pi$
$271$	8.68000e80	0.271674	0.135837	+	0.990731i	$0.456628\pi$
$272$	5.11025e81	1.41379
$273$	3.24390e80	0.0793632
$274$	2.83937e81	0.614580
$275$	−3.13352e81	−0.600327
$276$	−1.20514e82	−2.04449
$277$	−1.27622e82	−1.91802	−0.959012	−	0.283366i	$-0.908549\pi$
$277$	−1.27622e82	−1.91802	−0.959012	+	0.283366i	$0.908549\pi$
$278$	5.73851e81	0.764361
$279$	−4.42177e81	−0.522222
$280$	1.07425e81	0.112541
$281$	−7.55125e80	−0.0702032	−0.0351016	−	0.999384i	$-0.511175\pi$
$281$	−7.55125e80	−0.0702032	−0.0351016	+	0.999384i	$0.511175\pi$
$282$	−1.26420e82	−1.04345
$283$	2.02202e82	1.48231	0.741157	−	0.671332i	$-0.234277\pi$
$283$	2.02202e82	1.48231	0.741157	+	0.671332i	$0.234277\pi$
$284$	6.18238e82	4.02709
$285$	2.43398e81	0.140934
$286$	−8.07983e81	−0.416045
$287$	−3.51179e81	−0.160874
$288$	−2.23827e81	−0.0912564
$289$	8.76792e81	0.318287
$290$	1.07544e82	0.347738
$291$	6.28061e81	0.180962
$292$	−1.16338e83	−2.98811
$293$	6.41486e82	1.46934	0.734669	−	0.678426i	$-0.237337\pi$
$293$	6.41486e82	1.46934	0.734669	+	0.678426i	$0.237337\pi$
$294$	4.32041e82	0.882855
$295$	1.16944e82	0.213275
$296$	−2.34339e82	−0.381568
$297$	8.17154e81	0.118840
$298$	−1.14645e83	−1.48974
$299$	−5.64444e82	−0.655596
$300$	−1.12153e83	−1.16479
$301$	3.80473e82	0.353465
$302$	1.24116e83	1.03180
$303$	6.90716e82	0.514010
$304$	−2.70378e83	−1.80180
$305$	5.64872e81	0.0337214
$306$	−1.25456e83	−0.671154
$307$	−3.75315e83	−1.79995	−0.899974	−	0.435943i	$-0.856415\pi$
$307$	−3.75315e83	−1.79995	−0.899974	+	0.435943i	$0.856415\pi$
$308$	1.06606e83	0.458494
$309$	−4.74656e82	−0.183135
$310$	−1.32367e83	−0.458314
$311$	5.73120e83	1.78145	0.890723	−	0.454547i	$-0.150199\pi$
$311$	5.73120e83	1.78145	0.890723	+	0.454547i	$0.150199\pi$
$312$	−1.49835e83	−0.418249
$313$	4.23883e83	1.06295	0.531473	−	0.847075i	$-0.321639\pi$
$313$	4.23883e83	1.06295	0.531473	+	0.847075i	$0.321639\pi$
$314$	9.80467e83	2.20948
$315$	−9.82095e81	−0.0198953
$316$	1.51882e84	2.76687
$317$	−9.09685e83	−1.49075	−0.745376	−	0.666644i	$-0.767730\pi$
$317$	−9.09685e83	−1.49075	−0.745376	+	0.666644i	$0.767730\pi$
$318$	−2.45159e82	−0.0361526
$319$	5.52966e83	0.734024
$320$	1.04848e83	0.125324
$321$	−4.23995e83	−0.456502
$322$	1.10360e84	1.07063
$323$	−1.92139e84	−1.68009
$324$	2.92470e83	0.230581
$325$	−5.25281e83	−0.373507
$326$	−3.79246e84	−2.43294
$327$	1.26133e83	0.0730262
$328$	1.62209e84	0.847815
$329$	7.81231e83	0.368737
$330$	2.44618e83	0.104297
$331$	−3.95258e84	−1.52280	−0.761400	−	0.648283i	$-0.775487\pi$
$331$	−3.95258e84	−1.52280	−0.761400	+	0.648283i	$0.775487\pi$
$332$	−1.11342e84	−0.387735
$333$	2.14237e83	0.0674546
$334$	3.48974e84	0.993773
$335$	−4.37325e83	−0.112669
$336$	1.09096e84	0.254356
$337$	2.28534e83	0.0482336	0.0241168	−	0.999709i	$-0.492323\pi$
$337$	2.28534e83	0.0482336	0.0241168	+	0.999709i	$0.492323\pi$
$338$	7.82144e84	1.49478
$339$	−3.68737e84	−0.638302
$340$	−2.53433e84	−0.397484
$341$	−6.80601e84	−0.967434
$342$	6.63772e84	0.855351
$343$	−5.73165e84	−0.669771
$344$	−1.75740e85	−1.86278
$345$	1.70886e84	0.164349
$346$	6.85059e82	0.00597969
$347$	1.75200e85	1.38834	0.694170	−	0.719811i	$-0.255772\pi$
$347$	1.75200e85	1.38834	0.694170	+	0.719811i	$0.255772\pi$
$348$	1.97914e85	1.42420
$349$	−1.78094e85	−1.16412	−0.582059	−	0.813147i	$-0.697753\pi$
$349$	−1.78094e85	−1.16412	−0.582059	+	0.813147i	$0.697753\pi$
$350$	1.02703e85	0.609964
$351$	1.36982e84	0.0739392
$352$	−3.44516e84	−0.169056
$353$	−3.42212e85	−1.52701	−0.763507	−	0.645800i	$-0.776524\pi$
$353$	−3.42212e85	−1.52701	−0.763507	+	0.645800i	$0.776524\pi$
$354$	3.18918e85	1.29440
$355$	−8.76644e84	−0.323723
$356$	−6.28433e85	−2.11195
$357$	7.75269e84	0.237174
$358$	1.02225e86	2.84757
$359$	−1.74464e85	−0.442631	−0.221315	−	0.975202i	$-0.571035\pi$
$359$	−1.74464e85	−0.442631	−0.221315	+	0.975202i	$0.571035\pi$
$360$	4.53628e84	0.104850
$361$	5.41809e85	1.14118
$362$	1.13118e86	2.17167
$363$	−2.04069e85	−0.357195
$364$	1.78707e85	0.285263
$365$	1.64964e85	0.240203
$366$	1.54046e85	0.204661
$367$	2.04155e85	0.247541	0.123771	−	0.992311i	$-0.460501\pi$
$367$	2.04155e85	0.247541	0.123771	+	0.992311i	$0.460501\pi$
$368$	−1.89828e86	−2.10116
$369$	−1.48294e85	−0.149879
$370$	6.41325e84	0.0591998
$371$	1.51499e84	0.0127757
$372$	−2.43596e86	−1.87707
$373$	−1.23583e86	−0.870387	−0.435193	−	0.900337i	$-0.643320\pi$
$373$	−1.23583e86	−0.870387	−0.435193	+	0.900337i	$0.643320\pi$
$374$	−1.93102e86	−1.24334
$375$	3.22611e85	0.189947
$376$	−3.60849e86	−1.94327
$377$	9.26953e85	0.456691
$378$	−2.67827e85	−0.120748
$379$	1.09067e86	0.450069	0.225035	−	0.974351i	$-0.427750\pi$
$379$	1.09067e86	0.450069	0.225035	+	0.974351i	$0.427750\pi$
$380$	1.34088e86	0.506572
$381$	−2.29247e86	−0.793080
$382$	9.24215e86	2.92854
$383$	−6.27334e86	−1.82113	−0.910563	−	0.413370i	$-0.864352\pi$
$383$	−6.27334e86	−1.82113	−0.910563	+	0.413370i	$0.864352\pi$
$384$	3.45349e86	0.918675
$385$	−1.51165e85	−0.0368567
$386$	9.27212e86	2.07255
$387$	1.60664e86	0.329308
$388$	3.46000e86	0.650448
$389$	−1.96530e86	−0.338935	−0.169467	−	0.985536i	$-0.554205\pi$
$389$	−1.96530e86	−0.338935	−0.169467	+	0.985536i	$0.554205\pi$
$390$	4.10060e85	0.0648908
$391$	−1.34898e87	−1.95923
$392$	1.23320e87	1.64418
$393$	9.27103e85	0.113495
$394$	−3.00632e86	−0.337997
$395$	−2.15365e86	−0.222419
$396$	4.50172e86	0.427159
$397$	−1.11201e87	−0.969680	−0.484840	−	0.874603i	$-0.661122\pi$
$397$	−1.11201e87	−0.969680	−0.484840	+	0.874603i	$0.661122\pi$
$398$	4.08407e87	3.27349
$399$	−4.10186e86	−0.302266
$400$	−1.76657e87	−1.19708
$401$	−1.53616e87	−0.957414	−0.478707	−	0.877975i	$-0.658894\pi$
$401$	−1.53616e87	−0.957414	−0.478707	+	0.877975i	$0.658894\pi$
$402$	−1.19263e87	−0.683806
$403$	−1.14091e87	−0.601912
$404$	3.80517e87	1.84756
$405$	−4.14714e85	−0.0185356
$406$	−1.81238e87	−0.745808
$407$	3.29755e86	0.124962
$408$	−3.58096e87	−1.24992
$409$	−5.68351e86	−0.182762	−0.0913810	−	0.995816i	$-0.529128\pi$
$409$	−5.68351e86	−0.182762	−0.0913810	+	0.995816i	$0.529128\pi$
$410$	−4.43923e86	−0.131537
$411$	−7.40935e86	−0.202339
$412$	−2.61489e87	−0.658259
$413$	−1.97079e87	−0.457419
$414$	4.66024e87	0.997463
$415$	1.57881e86	0.0311686
$416$	−5.77521e86	−0.105182
$417$	−1.49747e87	−0.251651
$418$	1.02168e88	1.58457
$419$	3.57728e87	0.512133	0.256067	−	0.966659i	$-0.417573\pi$
$419$	3.57728e87	0.512133	0.256067	+	0.966659i	$0.417573\pi$
$420$	−5.41038e86	−0.0715116
$421$	1.17240e88	1.43096	0.715479	−	0.698634i	$-0.246209\pi$
$421$	1.17240e88	1.43096	0.715479	+	0.698634i	$0.246209\pi$
$422$	−9.24471e87	−1.04214
$423$	3.29894e87	0.343536
$424$	−6.99773e86	−0.0673289
$425$	−1.25538e88	−1.11621
$426$	−2.39070e88	−1.96473
$427$	−9.51950e86	−0.0723236
$428$	−2.33580e88	−1.64085
$429$	2.10843e87	0.136975
$430$	4.80954e87	0.289008
$431$	1.11815e88	0.621599	0.310799	−	0.950476i	$-0.399403\pi$
$431$	1.11815e88	0.621599	0.310799	+	0.950476i	$0.399403\pi$
$432$	4.60684e87	0.236972
$433$	1.96971e88	0.937686	0.468843	−	0.883282i	$-0.344671\pi$
$433$	1.96971e88	0.937686	0.468843	+	0.883282i	$0.344671\pi$
$434$	2.23071e88	0.982965
$435$	−2.80636e87	−0.114486
$436$	6.94869e87	0.262486
$437$	7.13730e88	2.49693
$438$	4.49874e88	1.45784
$439$	−5.01948e86	−0.0150695	−0.00753473	−	0.999972i	$-0.502398\pi$
$439$	−5.01948e86	−0.0150695	−0.00753473	+	0.999972i	$0.502398\pi$
$440$	6.98228e87	0.194237
$441$	−1.12741e88	−0.290663
$442$	−3.23703e88	−0.773571
$443$	−5.26560e88	−1.16660	−0.583302	−	0.812256i	$-0.698239\pi$
$443$	−5.26560e88	−1.16660	−0.583302	+	0.812256i	$0.698239\pi$
$444$	1.18023e88	0.242459
$445$	8.91101e87	0.169772
$446$	−5.14018e88	−0.908365
$447$	2.99168e88	0.490470
$448$	−1.76694e88	−0.268787
$449$	−3.02115e88	−0.426502	−0.213251	−	0.976998i	$-0.568405\pi$
$449$	−3.02115e88	−0.426502	−0.213251	+	0.976998i	$0.568405\pi$
$450$	4.33690e88	0.568277
$451$	−2.28255e88	−0.277656
$452$	−2.03138e89	−2.29431
$453$	−3.23881e88	−0.339700
$454$	−1.90482e89	−1.85560
$455$	−2.53402e87	−0.0229313
$456$	1.89464e89	1.59296
$457$	3.75835e88	0.293633	0.146816	−	0.989164i	$-0.453097\pi$
$457$	3.75835e88	0.293633	0.146816	+	0.989164i	$0.453097\pi$
$458$	−2.71666e89	−1.97261
$459$	3.27377e88	0.220965
$460$	9.41415e88	0.590736
$461$	2.48216e88	0.144827	0.0724134	−	0.997375i	$-0.476930\pi$
$461$	2.48216e88	0.144827	0.0724134	+	0.997375i	$0.476930\pi$
$462$	−4.12242e88	−0.223690
$463$	−3.41579e88	−0.172396	−0.0861980	−	0.996278i	$-0.527472\pi$
$463$	−3.41579e88	−0.172396	−0.0861980	+	0.996278i	$0.527472\pi$
$464$	3.11743e89	1.46368
$465$	3.45412e88	0.150891
$466$	6.35499e88	0.258337
$467$	1.02120e89	0.386364	0.193182	−	0.981163i	$-0.438119\pi$
$467$	1.02120e89	0.386364	0.193182	+	0.981163i	$0.438119\pi$
$468$	7.54636e88	0.265767
$469$	7.37001e88	0.241645
$470$	9.87550e88	0.301495
$471$	−2.55853e89	−0.727429
$472$	9.10306e89	2.41063
$473$	2.47296e89	0.610054
$474$	−5.87321e89	−1.34990
$475$	6.64210e89	1.42256
$476$	4.27097e89	0.852499
$477$	6.39744e87	0.0119026
$478$	−4.84837e89	−0.840933
$479$	−3.72929e89	−0.603096	−0.301548	−	0.953451i	$-0.597503\pi$
$479$	−3.72929e89	−0.603096	−0.301548	+	0.953451i	$0.597503\pi$
$480$	1.74845e88	0.0263677
$481$	5.52777e88	0.0777481
$482$	−1.57409e90	−2.06516
$483$	−2.87985e89	−0.352486
$484$	−1.12422e90	−1.28390
$485$	−4.90618e88	−0.0522872
$486$	−1.13097e89	−0.112496
$487$	3.95617e89	0.367328	0.183664	−	0.982989i	$-0.441204\pi$
$487$	3.95617e89	0.367328	0.183664	+	0.982989i	$0.441204\pi$
$488$	4.39704e89	0.381151
$489$	9.89645e89	0.801000
$490$	−3.37495e89	−0.255093
$491$	−2.74760e89	−0.193965	−0.0969826	−	0.995286i	$-0.530919\pi$
$491$	−2.74760e89	−0.193965	−0.0969826	+	0.995286i	$0.530919\pi$
$492$	−8.16955e89	−0.538725
$493$	2.21535e90	1.36480
$494$	1.71267e90	0.985875
$495$	−6.38331e88	−0.0343378
$496$	−3.83700e90	−1.92910
$497$	1.47736e90	0.694302
$498$	4.30556e89	0.189168
$499$	1.56862e90	0.644392	0.322196	−	0.946673i	$-0.395579\pi$
$499$	1.56862e90	0.644392	0.322196	+	0.946673i	$0.395579\pi$
$500$	1.77727e90	0.682745
$501$	−9.10650e89	−0.327181
$502$	−7.52736e90	−2.52969
$503$	3.52944e89	0.110963	0.0554815	−	0.998460i	$-0.482331\pi$
$503$	3.52944e89	0.110963	0.0554815	+	0.998460i	$0.482331\pi$
$504$	−7.64476e89	−0.224875
$505$	−5.39562e89	−0.148518
$506$	7.17307e90	1.84783
$507$	−2.04101e90	−0.492128
$508$	−1.26293e91	−2.85065
$509$	−5.44378e89	−0.115042	−0.0575208	−	0.998344i	$-0.518320\pi$
$509$	−5.44378e89	−0.115042	−0.0575208	+	0.998344i	$0.518320\pi$
$510$	9.80014e89	0.193924
$511$	−2.78006e90	−0.515173
$512$	1.14348e91	1.98466
$513$	−1.73211e90	−0.281608
$514$	1.40818e91	2.14483
$515$	3.70784e89	0.0529151
$516$	8.85102e90	1.18366
$517$	5.07776e90	0.636412
$518$	−1.08079e90	−0.126968
$519$	−1.78766e88	−0.00196870
$520$	1.17046e90	0.120849
$521$	−1.26908e91	−1.22864	−0.614321	−	0.789056i	$-0.710570\pi$
$521$	−1.26908e91	−1.22864	−0.614321	+	0.789056i	$0.710570\pi$
$522$	−7.65323e90	−0.694837
$523$	−1.03677e91	−0.882829	−0.441414	−	0.897303i	$-0.645523\pi$
$523$	−1.03677e91	−0.882829	−0.441414	+	0.897303i	$0.645523\pi$
$524$	5.10743e90	0.407947
$525$	−2.68004e90	−0.200819
$526$	4.29164e91	3.01718
$527$	−2.72670e91	−1.79879
$528$	7.09088e90	0.438999
$529$	3.29006e91	1.91178
$530$	1.91510e89	0.0104460
$531$	−8.32216e90	−0.426157
$532$	−2.25972e91	−1.08646
$533$	−3.82631e90	−0.172750
$534$	2.43012e91	1.03038
$535$	3.31210e90	0.131902
$536$	−3.40420e91	−1.27349
$537$	−2.66755e91	−0.937508
$538$	−2.27408e91	−0.750931
$539$	−1.73532e91	−0.538463
$540$	−2.28467e90	−0.0666242
$541$	−5.03047e91	−1.37880	−0.689398	−	0.724382i	$-0.742125\pi$
$541$	−5.03047e91	−1.37880	−0.689398	+	0.724382i	$0.742125\pi$
$542$	−1.84911e91	−0.476417
$543$	−2.95182e91	−0.714982
$544$	−1.38023e91	−0.314333
$545$	−9.85305e89	−0.0211003
$546$	−6.91053e90	−0.139174
$547$	7.39209e91	1.40021	0.700103	−	0.714042i	$-0.253137\pi$
$547$	7.39209e91	1.40021	0.700103	+	0.714042i	$0.253137\pi$
$548$	−4.08183e91	−0.727287
$549$	−4.01985e90	−0.0673808
$550$	6.67539e91	1.05275
$551$	−1.17212e92	−1.73937
$552$	1.33020e92	1.85762
$553$	3.62943e91	0.477030
$554$	2.71875e92	3.36351
$555$	−1.67354e90	−0.0194904
$556$	−8.24956e91	−0.904535
$557$	1.01395e92	1.04681	0.523406	−	0.852083i	$-0.324661\pi$
$557$	1.01395e92	1.04681	0.523406	+	0.852083i	$0.324661\pi$
$558$	9.41975e91	0.915785
$559$	4.14549e91	0.379560
$560$	−8.52216e90	−0.0734938
$561$	5.03901e91	0.409345
$562$	1.60865e91	0.123111
$563$	−6.32573e91	−0.456121	−0.228060	−	0.973647i	$-0.573238\pi$
$563$	−6.32573e91	−0.456121	−0.228060	+	0.973647i	$0.573238\pi$
$564$	1.81739e92	1.23481
$565$	2.88044e91	0.184432
$566$	−4.30754e92	−2.59943
$567$	6.98897e90	0.0397540
$568$	−6.82392e92	−3.65902
$569$	−1.14738e92	−0.580022	−0.290011	−	0.957023i	$-0.593659\pi$
$569$	−1.14738e92	−0.580022	−0.290011	+	0.957023i	$0.593659\pi$
$570$	−5.18514e91	−0.247146
$571$	3.63351e92	1.63311	0.816557	−	0.577264i	$-0.195880\pi$
$571$	3.63351e92	1.63311	0.816557	+	0.577264i	$0.195880\pi$
$572$	1.16154e92	0.492343
$573$	−2.41174e92	−0.964165
$574$	7.48121e91	0.282113
$575$	4.66332e92	1.65891
$576$	−7.46136e91	−0.250418
$577$	2.56354e92	0.811804	0.405902	−	0.913917i	$-0.366958\pi$
$577$	2.56354e92	0.811804	0.405902	+	0.913917i	$0.366958\pi$
$578$	−1.86784e92	−0.558158
$579$	−2.41956e92	−0.682347
$580$	−1.54603e92	−0.411509
$581$	−2.66068e91	−0.0668485
$582$	−1.33797e92	−0.317340
$583$	9.84699e90	0.0220499
$584$	1.28410e93	2.71500
$585$	−1.07005e91	−0.0213641
$586$	−1.36657e93	−2.57668
$587$	−1.99505e92	−0.355285	−0.177642	−	0.984095i	$-0.556847\pi$
$587$	−1.99505e92	−0.355285	−0.177642	+	0.984095i	$0.556847\pi$
$588$	−6.21093e92	−1.04476
$589$	1.44266e93	2.29247
$590$	−2.49127e92	−0.374006
$591$	7.84501e91	0.111279
$592$	1.85904e92	0.249180
$593$	−1.52207e93	−1.92798	−0.963991	−	0.265936i	$-0.914319\pi$
$593$	−1.52207e93	−1.92798	−0.963991	+	0.265936i	$0.914319\pi$
$594$	−1.74079e92	−0.208402
$595$	−6.05612e91	−0.0685293
$596$	1.64812e93	1.76295
$597$	−1.06574e93	−1.07773
$598$	1.20244e93	1.14967
$599$	−4.67709e92	−0.422841	−0.211420	−	0.977395i	$-0.567809\pi$
$599$	−4.67709e92	−0.422841	−0.211420	+	0.977395i	$0.567809\pi$
$600$	1.23791e93	1.05833
$601$	7.64797e92	0.618375	0.309187	−	0.951001i	$-0.399943\pi$
$601$	7.64797e92	0.618375	0.309187	+	0.951001i	$0.399943\pi$
$602$	−8.10527e92	−0.619848
$603$	3.11217e92	0.225130
$604$	−1.78426e93	−1.22102
$605$	1.59411e92	0.103208
$606$	−1.47144e93	−0.901384
$607$	2.89487e92	0.167806	0.0839031	−	0.996474i	$-0.473261\pi$
$607$	2.89487e92	0.167806	0.0839031	+	0.996474i	$0.473261\pi$
$608$	7.30266e92	0.400600
$609$	4.72941e92	0.245543
$610$	−1.20335e92	−0.0591350
$611$	8.51199e92	0.395959
$612$	1.80353e93	0.794236
$613$	−3.38496e93	−1.41133	−0.705664	−	0.708546i	$-0.749351\pi$
$613$	−3.38496e93	−1.41133	−0.705664	+	0.708546i	$0.749351\pi$
$614$	7.99538e93	3.15645
$615$	1.15842e92	0.0433061
$616$	−1.17669e93	−0.416588
$617$	2.14446e93	0.719061	0.359531	−	0.933133i	$-0.382937\pi$
$617$	2.14446e93	0.719061	0.359531	+	0.933133i	$0.382937\pi$
$618$	1.01117e93	0.321151
$619$	6.09133e93	1.83263	0.916316	−	0.400456i	$-0.131148\pi$
$619$	6.09133e93	1.83263	0.916316	+	0.400456i	$0.131148\pi$
$620$	1.90288e93	0.542363
$621$	−1.21609e93	−0.328396
$622$	−1.22093e94	−3.12400
$623$	−1.50173e93	−0.364117
$624$	1.18866e93	0.273134
$625$	4.21198e93	0.917288
$626$	−9.03004e93	−1.86402
$627$	−2.66608e93	−0.521688
$628$	−1.40950e94	−2.61467
$629$	1.32110e93	0.232348
$630$	2.09217e92	0.0348890
$631$	−1.71132e93	−0.270612	−0.135306	−	0.990804i	$-0.543202\pi$
$631$	−1.71132e93	−0.270612	−0.135306	+	0.990804i	$0.543202\pi$
$632$	−1.67643e94	−2.51398
$633$	2.41241e93	0.343105
$634$	1.93791e94	2.61423
$635$	1.79080e93	0.229153
$636$	3.52436e92	0.0427826
$637$	−2.90897e93	−0.335018
$638$	−1.17799e94	−1.28721
$639$	6.23854e93	0.646851
$640$	−2.69774e93	−0.265443
$641$	3.19909e93	0.298732	0.149366	−	0.988782i	$-0.452277\pi$
$641$	3.19909e93	0.298732	0.149366	+	0.988782i	$0.452277\pi$
$642$	9.03243e93	0.800536
$643$	1.40320e94	1.18046	0.590231	−	0.807235i	$-0.299037\pi$
$643$	1.40320e94	1.18046	0.590231	+	0.807235i	$0.299037\pi$
$644$	−1.58652e94	−1.26698
$645$	−1.25505e93	−0.0951505
$646$	4.09317e94	2.94626
$647$	3.61307e93	0.246935	0.123468	−	0.992349i	$-0.460598\pi$
$647$	3.61307e93	0.246935	0.123468	+	0.992349i	$0.460598\pi$
$648$	−3.22819e93	−0.209506
$649$	−1.28095e94	−0.789471
$650$	1.11901e94	0.654995
$651$	−5.82105e93	−0.323622
$652$	5.45197e94	2.87912
$653$	−7.71068e93	−0.386813	−0.193407	−	0.981119i	$-0.561954\pi$
$653$	−7.71068e93	−0.386813	−0.193407	+	0.981119i	$0.561954\pi$
$654$	−2.68703e93	−0.128061
$655$	−7.24220e92	−0.0327934
$656$	−1.28683e94	−0.553657
$657$	−1.17395e94	−0.479965
$658$	−1.66427e94	−0.646629
$659$	−3.19576e94	−1.18008	−0.590041	−	0.807373i	$-0.700888\pi$
$659$	−3.19576e94	−1.18008	−0.590041	+	0.807373i	$0.700888\pi$
$660$	−3.51658e93	−0.123424
$661$	−1.29579e94	−0.432299	−0.216150	−	0.976360i	$-0.569350\pi$
$661$	−1.29579e94	−0.432299	−0.216150	+	0.976360i	$0.569350\pi$
$662$	8.42023e94	2.67043
$663$	8.44703e93	0.254684
$664$	1.22896e94	0.352296
$665$	3.20423e93	0.0873369
$666$	−4.56391e93	−0.118291
$667$	−8.22925e94	−2.02836
$668$	−5.01679e94	−1.17602
$669$	1.34133e94	0.299062
$670$	9.31640e93	0.197580
$671$	−6.18738e93	−0.124825
$672$	−2.94658e93	−0.0565519
$673$	−8.59582e94	−1.56958	−0.784789	−	0.619764i	$-0.787229\pi$
$673$	−8.59582e94	−1.56958	−0.784789	+	0.619764i	$0.787229\pi$
$674$	−4.86850e93	−0.0845841
$675$	−1.13172e94	−0.187094
$676$	−1.12440e95	−1.76891
$677$	−8.59790e93	−0.128728	−0.0643639	−	0.997926i	$-0.520502\pi$
$677$	−8.59790e93	−0.128728	−0.0643639	+	0.997926i	$0.520502\pi$
$678$	7.85525e94	1.11935
$679$	8.26813e93	0.112142
$680$	2.79731e94	0.361154
$681$	4.97064e94	0.610920
$682$	1.44989e95	1.69652
$683$	2.78866e94	0.310672	0.155336	−	0.987862i	$-0.450354\pi$
$683$	2.78866e94	0.310672	0.155336	+	0.987862i	$0.450354\pi$
$684$	−9.54225e94	−1.01221
$685$	5.78792e93	0.0584640
$686$	1.22102e95	1.17453
$687$	7.08914e94	0.649445
$688$	1.39417e95	1.21647
$689$	1.65068e93	0.0137189
$690$	−3.64041e94	−0.288208
$691$	−1.06729e95	−0.804946	−0.402473	−	0.915432i	$-0.631849\pi$
$691$	−1.06729e95	−0.804946	−0.402473	+	0.915432i	$0.631849\pi$
$692$	−9.84827e92	−0.00707629
$693$	1.07575e94	0.0736456
$694$	−3.73230e95	−2.43464
$695$	1.16977e94	0.0727123
$696$	−2.18451e95	−1.29403
$697$	−9.14460e94	−0.516258
$698$	3.79395e95	2.04143
$699$	−1.65834e94	−0.0850525
$700$	−1.47644e95	−0.721824
$701$	9.12662e94	0.425360	0.212680	−	0.977122i	$-0.431781\pi$
$701$	9.12662e94	0.425360	0.212680	+	0.977122i	$0.431781\pi$
$702$	−2.91814e94	−0.129662
$703$	−6.98978e94	−0.296115
$704$	−1.14846e95	−0.463907
$705$	−2.57701e94	−0.0992616
$706$	7.29019e95	2.67782
$707$	9.09296e94	0.318533
$708$	−4.58469e95	−1.53178
$709$	2.76217e95	0.880240	0.440120	−	0.897939i	$-0.354936\pi$
$709$	2.76217e95	0.880240	0.440120	+	0.897939i	$0.354936\pi$
$710$	1.86753e95	0.567691
$711$	1.53262e95	0.444428
$712$	6.93645e95	1.91892
$713$	1.01287e96	2.67335
$714$	−1.65157e95	−0.415916
$715$	−1.64703e94	−0.0395776
$716$	−1.46956e96	−3.36978
$717$	1.26518e95	0.276861
$718$	3.71663e95	0.776212
$719$	−2.73294e95	−0.544769	−0.272385	−	0.962188i	$-0.587812\pi$
$719$	−2.73294e95	−0.544769	−0.272385	+	0.962188i	$0.587812\pi$
$720$	−3.59869e94	−0.0684710
$721$	−6.24864e94	−0.113489
$722$	−1.15422e96	−2.00121
$723$	4.10759e95	0.679914
$724$	−1.62616e96	−2.56993
$725$	−7.65829e95	−1.15560
$726$	4.34731e95	0.626389
$727$	−1.45037e95	−0.199561	−0.0997805	−	0.995009i	$-0.531814\pi$
$727$	−1.45037e95	−0.199561	−0.0997805	+	0.995009i	$0.531814\pi$
$728$	−1.97251e95	−0.259190
$729$	2.95127e94	0.0370370
$730$	−3.51426e95	−0.421228
$731$	9.90742e95	1.13430
$732$	−2.21454e95	−0.242193
$733$	−7.02719e94	−0.0734174	−0.0367087	−	0.999326i	$-0.511687\pi$
$733$	−7.02719e94	−0.0734174	−0.0367087	+	0.999326i	$0.511687\pi$
$734$	−4.34915e95	−0.434097
$735$	8.80694e94	0.0839845
$736$	5.12709e95	0.467158
$737$	4.79028e95	0.417061
$738$	3.15913e95	0.262833
$739$	−9.72139e95	−0.772930	−0.386465	−	0.922304i	$-0.626304\pi$
$739$	−9.72139e95	−0.772930	−0.386465	+	0.922304i	$0.626304\pi$
$740$	−9.21955e94	−0.0700563
$741$	−4.46923e95	−0.324581
$742$	−3.22741e94	−0.0224039
$743$	−3.87379e94	−0.0257046	−0.0128523	−	0.999917i	$-0.504091\pi$
$743$	−3.87379e94	−0.0257046	−0.0128523	+	0.999917i	$0.504091\pi$
$744$	2.68874e96	1.70551
$745$	−2.33699e95	−0.141717
$746$	2.63270e96	1.52634
$747$	−1.12354e95	−0.0622799
$748$	2.77600e96	1.47135
$749$	−5.58170e95	−0.282895
$750$	−6.87264e95	−0.333097
$751$	−4.77793e95	−0.221463	−0.110732	−	0.993850i	$-0.535319\pi$
$751$	−4.77793e95	−0.221463	−0.110732	+	0.993850i	$0.535319\pi$
$752$	2.86267e96	1.26903
$753$	1.96427e96	0.832853
$754$	−1.97470e96	−0.800868
$755$	2.53004e95	0.0981531
$756$	3.85023e95	0.142892
$757$	2.53359e96	0.899548	0.449774	−	0.893142i	$-0.351505\pi$
$757$	2.53359e96	0.899548	0.449774	+	0.893142i	$0.351505\pi$
$758$	−2.32347e96	−0.789256
$759$	−1.87182e96	−0.608364
$760$	−1.48003e96	−0.460272
$761$	−4.58277e95	−0.136377	−0.0681886	−	0.997672i	$-0.521722\pi$
$761$	−4.58277e95	−0.136377	−0.0681886	+	0.997672i	$0.521722\pi$
$762$	4.88368e96	1.39077
$763$	1.66048e95	0.0452546
$764$	−1.32863e97	−3.46559
$765$	−2.55735e95	−0.0638458
$766$	1.33642e97	3.19359
$767$	−2.14730e96	−0.491188
$768$	−5.37627e96	−1.17728
$769$	−4.53561e96	−0.950832	−0.475416	−	0.879761i	$-0.657702\pi$
$769$	−4.53561e96	−0.950832	−0.475416	+	0.879761i	$0.657702\pi$
$770$	3.22028e95	0.0646331
$771$	−3.67465e96	−0.706146
$772$	−1.33294e97	−2.45263
$773$	−6.51508e96	−1.14791	−0.573953	−	0.818888i	$-0.694591\pi$
$773$	−6.51508e96	−1.14791	−0.573953	+	0.818888i	$0.694591\pi$
$774$	−3.42265e96	−0.577485
$775$	9.42597e96	1.52307
$776$	−3.81904e96	−0.590998
$777$	2.82033e95	0.0418018
$778$	4.18670e96	0.594367
$779$	4.83831e96	0.657943
$780$	−5.89494e95	−0.0767910
$781$	9.60240e96	1.19831
$782$	2.87375e97	3.43576
$783$	1.99711e96	0.228762
$784$	−9.78315e96	−1.07372
$785$	1.99863e96	0.210184
$786$	−1.97502e96	−0.199029
$787$	−7.16665e96	−0.692091	−0.346045	−	0.938218i	$-0.612476\pi$
$787$	−7.16665e96	−0.692091	−0.346045	+	0.938218i	$0.612476\pi$
$788$	4.32183e96	0.399981
$789$	−1.11991e97	−0.993349
$790$	4.58794e96	0.390041
$791$	−4.85425e96	−0.395558
$792$	−4.96886e96	−0.388117
$793$	−1.03721e96	−0.0776629
$794$	2.36894e97	1.70046
$795$	−4.99745e94	−0.00343914
$796$	−5.87118e97	−3.87381
$797$	3.88056e96	0.245494	0.122747	−	0.992438i	$-0.460830\pi$
$797$	3.88056e96	0.245494	0.122747	+	0.992438i	$0.460830\pi$
$798$	8.73825e96	0.530063
$799$	2.03430e97	1.18331
$800$	4.77136e96	0.266151
$801$	−6.34142e96	−0.339232
$802$	3.27250e97	1.67895
$803$	−1.80695e97	−0.889151
$804$	1.71450e97	0.809208
$805$	2.24964e96	0.101848
$806$	2.43050e97	1.05553
$807$	5.93423e96	0.247230
$808$	−4.20002e97	−1.67869
$809$	−1.01809e97	−0.390400	−0.195200	−	0.980763i	$-0.562536\pi$
$809$	−1.01809e97	−0.390400	−0.195200	+	0.980763i	$0.562536\pi$
$810$	8.83472e95	0.0325046
$811$	1.57707e97	0.556740	0.278370	−	0.960474i	$-0.410206\pi$
$811$	1.57707e97	0.556740	0.278370	+	0.960474i	$0.410206\pi$
$812$	2.60544e97	0.882581
$813$	4.82527e96	0.156851
$814$	−7.02480e96	−0.219137
$815$	−7.73075e96	−0.231442
$816$	2.84082e97	0.816251
$817$	−5.24190e97	−1.44561
$818$	1.21077e97	0.320498
$819$	1.80330e96	0.0458203
$820$	6.38175e96	0.155660
$821$	−4.92393e97	−1.15296	−0.576482	−	0.817110i	$-0.695575\pi$
$821$	−4.92393e97	−1.15296	−0.576482	+	0.817110i	$0.695575\pi$
$822$	1.57842e97	0.354828
$823$	−2.72469e97	−0.588061	−0.294030	−	0.955796i	$-0.594997\pi$
$823$	−2.72469e97	−0.588061	−0.294030	+	0.955796i	$0.594997\pi$
$824$	2.88624e97	0.598095
$825$	−1.74194e97	−0.346599
$826$	4.19840e97	0.802145
$827$	3.34382e97	0.613492	0.306746	−	0.951791i	$-0.400760\pi$
$827$	3.34382e97	0.613492	0.306746	+	0.951791i	$0.400760\pi$
$828$	−6.69947e97	−1.18039
$829$	9.06340e97	1.53361	0.766804	−	0.641881i	$-0.221846\pi$
$829$	9.06340e97	1.53361	0.766804	+	0.641881i	$0.221846\pi$
$830$	−3.36335e96	−0.0546583
$831$	−7.09460e97	−1.10737
$832$	−1.92519e97	−0.288631
$833$	−6.95222e97	−1.00119
$834$	3.19007e97	0.441304
$835$	7.11367e96	0.0945359
$836$	−1.46875e98	−1.87516
$837$	−2.45809e97	−0.301505
$838$	−7.62073e97	−0.898094
$839$	7.10400e97	0.804410	0.402205	−	0.915550i	$-0.368244\pi$
$839$	7.10400e97	0.804410	0.402205	+	0.915550i	$0.368244\pi$
$840$	5.97181e96	0.0649755
$841$	3.94983e97	0.412964
$842$	−2.49759e98	−2.50937
$843$	−4.19778e96	−0.0405318
$844$	1.32900e98	1.23326
$845$	1.59436e97	0.142196
$846$	−7.02778e97	−0.602436
$847$	−2.68648e97	−0.221355
$848$	5.55140e96	0.0439685
$849$	1.12405e98	0.855814
$850$	2.67436e98	1.95743
$851$	−4.90741e97	−0.345313
$852$	3.43682e98	2.32504
$853$	1.18361e98	0.769868	0.384934	−	0.922944i	$-0.374224\pi$
$853$	1.18361e98	0.769868	0.384934	+	0.922944i	$0.374224\pi$
$854$	2.02795e97	0.126829
$855$	1.35306e97	0.0813680
$856$	2.57818e98	1.49088
$857$	2.38779e98	1.32782	0.663908	−	0.747814i	$-0.268897\pi$
$857$	2.38779e98	1.32782	0.663908	+	0.747814i	$0.268897\pi$
$858$	−4.49163e97	−0.240204
$859$	9.85017e97	0.506609	0.253305	−	0.967387i	$-0.418483\pi$
$859$	9.85017e97	0.506609	0.253305	+	0.967387i	$0.418483\pi$
$860$	−6.91410e97	−0.342009
$861$	−1.95222e97	−0.0928804
$862$	−2.38200e98	−1.09006
$863$	1.65327e97	0.0727749	0.0363875	−	0.999338i	$-0.488415\pi$
$863$	1.65327e97	0.0727749	0.0363875	+	0.999338i	$0.488415\pi$
$864$	−1.24427e97	−0.0526869
$865$	1.39646e95	0.000568837 0
$866$	−4.19610e98	−1.64436
$867$	4.87414e97	0.183763
$868$	−3.20683e98	−1.16323
$869$	2.35901e98	0.823319
$870$	5.97843e97	0.200767
$871$	8.03008e97	0.259485
$872$	−7.66975e97	−0.238495
$873$	3.49143e97	0.104478
$874$	−1.52047e99	−4.37870
$875$	4.24703e97	0.117711
$876$	−6.46730e98	−1.72519
$877$	−1.90622e98	−0.489425	−0.244713	−	0.969596i	$-0.578694\pi$
$877$	−1.90622e98	−0.489425	−0.244713	+	0.969596i	$0.578694\pi$
$878$	1.06931e97	0.0264263
$879$	3.56606e98	0.848323
$880$	−5.53914e97	−0.126845
$881$	4.94325e98	1.08973	0.544866	−	0.838523i	$-0.316580\pi$
$881$	4.94325e98	1.08973	0.544866	+	0.838523i	$0.316580\pi$
$882$	2.40174e98	0.509716
$883$	−7.14153e98	−1.45918	−0.729588	−	0.683887i	$-0.760288\pi$
$883$	−7.14153e98	−1.45918	−0.729588	+	0.683887i	$0.760288\pi$
$884$	4.65348e98	0.915435
$885$	6.50097e97	0.123134
$886$	1.12174e99	2.04579
$887$	1.08574e99	1.90671	0.953356	−	0.301847i	$-0.0976032\pi$
$887$	1.08574e99	1.90671	0.953356	+	0.301847i	$0.0976032\pi$
$888$	−1.30271e98	−0.220299
$889$	−3.01793e98	−0.491474
$890$	−1.89832e98	−0.297718
$891$	4.54261e97	0.0686124
$892$	7.38942e98	1.07495
$893$	−1.07633e99	−1.50807
$894$	−6.37321e98	−0.860105
$895$	2.08380e98	0.270884
$896$	4.54636e98	0.569305
$897$	−3.13778e98	−0.378508
$898$	6.43601e98	0.747927
$899$	−1.66338e99	−1.86227
$900$	−6.23464e98	−0.672492
$901$	3.94500e97	0.0409984
$902$	4.86255e98	0.486906
$903$	2.11507e98	0.204073
$904$	2.24217e99	2.08462
$905$	2.30585e98	0.206587
$906$	6.89967e98	0.595708
$907$	−1.96225e99	−1.63271	−0.816356	−	0.577549i	$-0.804009\pi$
$907$	−1.96225e99	−1.63271	−0.816356	+	0.577549i	$0.804009\pi$
$908$	2.73834e99	2.19589
$909$	3.83973e98	0.296764
$910$	5.39825e97	0.0402130
$911$	−6.27026e98	−0.450215	−0.225107	−	0.974334i	$-0.572273\pi$
$911$	−6.27026e98	−0.450215	−0.225107	+	0.974334i	$0.572273\pi$
$912$	−1.50305e99	−1.04027
$913$	−1.72936e98	−0.115376
$914$	−8.00646e98	−0.514924
$915$	3.14016e97	0.0194691
$916$	3.90542e99	2.33437
$917$	1.22049e98	0.0703334
$918$	−6.97415e98	−0.387491
$919$	−3.46918e99	−1.85848	−0.929238	−	0.369482i	$-0.879535\pi$
$919$	−3.46918e99	−1.85848	−0.929238	+	0.369482i	$0.879535\pi$
$920$	−1.03910e99	−0.536744
$921$	−2.08640e99	−1.03920
$922$	−5.28778e98	−0.253973
$923$	1.60968e99	0.745559
$924$	5.92631e98	0.264712
$925$	−4.56692e98	−0.196732
$926$	7.27669e98	0.302319
$927$	−2.63864e98	−0.105733
$928$	−8.41991e98	−0.325425
$929$	4.11167e99	1.53282	0.766411	−	0.642351i	$-0.222041\pi$
$929$	4.11167e99	1.53282	0.766411	+	0.642351i	$0.222041\pi$
$930$	−7.35837e98	−0.264608
$931$	3.67834e99	1.27596
$932$	−9.13580e98	−0.305713
$933$	3.18601e99	1.02852
$934$	−2.17548e99	−0.677540
$935$	−3.93629e98	−0.118276
$936$	−8.32944e98	−0.241476
$937$	−5.99760e99	−1.67765	−0.838825	−	0.544401i	$-0.816757\pi$
$937$	−5.99760e99	−1.67765	−0.838825	+	0.544401i	$0.816757\pi$
$938$	−1.57004e99	−0.423757
$939$	2.35639e99	0.613692
$940$	−1.41968e99	−0.356786
$941$	−2.70800e99	−0.656745	−0.328372	−	0.944548i	$-0.606500\pi$
$941$	−2.70800e99	−0.656745	−0.328372	+	0.944548i	$0.606500\pi$
$942$	5.45047e99	1.27564
$943$	3.39690e99	0.767257
$944$	−7.22158e99	−1.57424
$945$	−5.45953e97	−0.0114865
$946$	−5.26817e99	−1.06981
$947$	8.76769e98	0.171855	0.0859273	−	0.996301i	$-0.472615\pi$
$947$	8.76769e98	0.171855	0.0859273	+	0.996301i	$0.472615\pi$
$948$	8.44321e99	1.59745
$949$	−3.02904e99	−0.553206
$950$	−1.41498e100	−2.49464
$951$	−5.05699e99	−0.860686
$952$	−4.71417e99	−0.774582
$953$	6.23220e98	0.0988619	0.0494310	−	0.998778i	$-0.484259\pi$
$953$	6.23220e98	0.0988619	0.0494310	+	0.998778i	$0.484259\pi$
$954$	−1.36286e98	−0.0208727
$955$	1.88397e99	0.278587
$956$	6.96993e99	0.995151
$957$	3.07397e99	0.423789
$958$	7.94456e99	1.05761
$959$	−9.75408e98	−0.125390
$960$	5.82854e98	0.0723559
$961$	1.21319e100	1.45444
$962$	−1.17759e99	−0.136342
$963$	−2.35702e99	−0.263561
$964$	2.26288e100	2.44388
$965$	1.89007e99	0.197158
$966$	6.13499e99	0.618131
$967$	1.87788e99	0.182760	0.0913801	−	0.995816i	$-0.470872\pi$
$967$	1.87788e99	0.182760	0.0913801	+	0.995816i	$0.470872\pi$
$968$	1.24088e100	1.16656
$969$	−1.06811e100	−0.969998
$970$	1.04517e99	0.0916925
$971$	8.38931e99	0.711020	0.355510	−	0.934672i	$-0.384307\pi$
$971$	8.38931e99	0.711020	0.355510	+	0.934672i	$0.384307\pi$
$972$	1.62586e99	0.133126
$973$	−1.97135e99	−0.155949
$974$	−8.42788e99	−0.644159
$975$	−2.92007e99	−0.215645
$976$	−3.48823e99	−0.248907
$977$	5.30698e99	0.365915	0.182957	−	0.983121i	$-0.441433\pi$
$977$	5.30698e99	0.365915	0.182957	+	0.983121i	$0.441433\pi$
$978$	−2.10825e100	−1.40466
$979$	−9.76075e99	−0.628439
$980$	4.85176e99	0.301874
$981$	7.01181e98	0.0421617
$982$	5.85325e99	0.340144
$983$	1.59598e99	0.0896365	0.0448182	−	0.998995i	$-0.485729\pi$
$983$	1.59598e99	0.0896365	0.0448182	+	0.998995i	$0.485729\pi$
$984$	9.01729e99	0.489486
$985$	−6.12824e98	−0.0321530
$986$	−4.71939e100	−2.39337
$987$	4.34291e99	0.212890
$988$	−2.46211e100	−1.16667
$989$	−3.68026e100	−1.68579
$990$	1.35985e99	0.0602158
$991$	4.16191e100	1.78166	0.890831	−	0.454334i	$-0.150123\pi$
$991$	4.16191e100	1.78166	0.890831	+	0.454334i	$0.150123\pi$
$992$	1.03634e100	0.428905
$993$	−2.19726e100	−0.879189
$994$	−3.14725e100	−1.21755
$995$	8.32518e99	0.311401
$996$	−6.18959e99	−0.223859
$997$	2.29671e100	0.803189	0.401594	−	0.915818i	$-0.368456\pi$
$997$	2.29671e100	0.803189	0.401594	+	0.915818i	$0.368456\pi$
$998$	−3.34166e100	−1.13003
$999$	1.19095e99	0.0389450

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3.68.a.a.1.1	✓	5
3.2	odd	2		9.68.a.b.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.68.a.a.1.1	✓	5	1.1	even	1	trivial
9.68.a.b.1.5		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)

\(f(q)\)	\(=\)	\(q-2.13031e10 q^{2} +5.55906e15 q^{3} +3.06250e20 q^{4} -4.34254e22 q^{5} -1.18425e26 q^{6} +7.31825e27 q^{7} -3.38029e30 q^{8} +3.09032e31 q^{9} +O(q^{10})\)	\(q-2.13031e10 q^{2} +5.55906e15 q^{3} +3.06250e20 q^{4} -4.34254e22 q^{5} -1.18425e26 q^{6} +7.31825e27 q^{7} -3.38029e30 q^{8} +3.09032e31 q^{9} +9.25096e32 q^{10} +4.75664e34 q^{11} +1.70246e36 q^{12} +7.97368e36 q^{13} -1.55902e38 q^{14} -2.41404e38 q^{15} +2.68163e40 q^{16} +1.90565e41 q^{17} -6.58334e41 q^{18} -1.00826e43 q^{19} -1.32990e43 q^{20} +4.06826e43 q^{21} -1.01331e45 q^{22} -7.07884e45 q^{23} -1.87912e46 q^{24} -6.58769e46 q^{25} -1.69864e47 q^{26} +1.71793e47 q^{27} +2.24121e48 q^{28} +1.16251e49 q^{29} +5.14267e48 q^{30} -1.43085e50 q^{31} -7.24284e49 q^{32} +2.64424e50 q^{33} -4.05964e51 q^{34} -3.17798e50 q^{35} +9.46408e51 q^{36} +6.93252e51 q^{37} +2.14791e53 q^{38} +4.43262e52 q^{39} +1.46790e53 q^{40} -4.79867e53 q^{41} -8.66667e53 q^{42} +5.19896e54 q^{43} +1.45672e55 q^{44} -1.34198e54 q^{45} +1.50801e56 q^{46} +1.06751e56 q^{47} +1.49073e56 q^{48} -3.64821e56 q^{49} +1.40338e57 q^{50} +1.05936e57 q^{51} +2.44194e57 q^{52} +2.07016e56 q^{53} -3.65972e57 q^{54} -2.06559e57 q^{55} -2.47378e58 q^{56} -5.60498e58 q^{57} -2.47652e59 q^{58} -2.69298e59 q^{59} -7.39300e58 q^{60} -1.30079e59 q^{61} +3.04815e60 q^{62} +2.26157e59 q^{63} -2.41443e60 q^{64} -3.46260e59 q^{65} -5.63307e60 q^{66} +1.00707e61 q^{67} +5.83606e61 q^{68} -3.93517e61 q^{69} +6.77009e60 q^{70} +2.01874e62 q^{71} -1.04462e62 q^{72} -3.79880e62 q^{73} -1.47684e62 q^{74} -3.66214e62 q^{75} -3.08779e63 q^{76} +3.48102e62 q^{77} -9.44287e62 q^{78} +4.95942e63 q^{79} -1.16451e63 q^{80} +9.55005e62 q^{81} +1.02227e64 q^{82} -3.63568e63 q^{83} +1.24590e64 q^{84} -8.27537e63 q^{85} -1.10754e65 q^{86} +6.46249e64 q^{87} -1.60788e65 q^{88} -2.05203e65 q^{89} +2.85884e64 q^{90} +5.83534e64 q^{91} -2.16789e66 q^{92} -7.95416e65 q^{93} -2.27413e66 q^{94} +4.37840e65 q^{95} -4.02634e65 q^{96} +1.12980e66 q^{97} +7.77183e66 q^{98} +1.46995e66 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 16255223088 q^{2} + 27\!\cdots\!15 q^{3}+ \cdots + 15\!\cdots\!45 q^{9}+O(q^{10}) \) 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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