LMFDB - Embedded newform 3.68.a.a.1.5

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			1.5
Root			$-4.39911e8$ 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)

$f(q)$	$=$	$q+1.78647e10 q^{2} +5.55906e15 q^{3} +1.71572e20 q^{4} -1.00461e22 q^{5} +9.93107e25 q^{6} -5.50883e27 q^{7} +4.28720e29 q^{8} +3.09032e31 q^{9} +O(q^{10})$	\(q+1.78647e10 q^{2} +5.55906e15 q^{3} +1.71572e20 q^{4} -1.00461e22 q^{5} +9.93107e25 q^{6} -5.50883e27 q^{7} +4.28720e29 q^{8} +3.09032e31 q^{9} -1.79470e32 q^{10} -7.78780e34 q^{11} +9.53780e35 q^{12} -2.25540e36 q^{13} -9.84134e37 q^{14} -5.58468e37 q^{15} -1.76606e40 q^{16} -9.52180e40 q^{17} +5.52074e41 q^{18} +7.47941e41 q^{19} -1.72363e42 q^{20} -3.06239e43 q^{21} -1.39126e45 q^{22} -3.36871e45 q^{23} +2.38328e45 q^{24} -6.76617e46 q^{25} -4.02919e46 q^{26} +1.71793e47 q^{27} -9.45163e47 q^{28} +4.41582e48 q^{29} -9.97684e47 q^{30} +4.34097e49 q^{31} -3.78769e50 q^{32} -4.32929e50 q^{33} -1.70104e51 q^{34} +5.53422e49 q^{35} +5.30212e51 q^{36} +1.88144e52 q^{37} +1.33617e52 q^{38} -1.25379e52 q^{39} -4.30696e51 q^{40} -1.15326e54 q^{41} -5.47086e53 q^{42} -7.06222e54 q^{43} -1.33617e55 q^{44} -3.10456e53 q^{45} -6.01808e55 q^{46} -6.07185e55 q^{47} -9.81766e55 q^{48} -3.88031e56 q^{49} -1.20875e57 q^{50} -5.29323e56 q^{51} -3.86964e56 q^{52} +5.87139e57 q^{53} +3.06902e57 q^{54} +7.82369e56 q^{55} -2.36175e57 q^{56} +4.15785e57 q^{57} +7.88872e58 q^{58} +2.78418e59 q^{59} -9.58175e57 q^{60} +5.66402e59 q^{61} +7.75499e59 q^{62} -1.70240e59 q^{63} -4.16034e60 q^{64} +2.26579e58 q^{65} -7.73412e60 q^{66} -2.08739e61 q^{67} -1.63368e61 q^{68} -1.87268e61 q^{69} +9.88670e59 q^{70} +1.83580e62 q^{71} +1.32488e61 q^{72} -4.46947e61 q^{73} +3.36113e62 q^{74} -3.76136e62 q^{75} +1.28326e62 q^{76} +4.29017e62 q^{77} -2.23985e62 q^{78} -1.25916e62 q^{79} +1.77420e62 q^{80} +9.55005e62 q^{81} -2.06026e64 q^{82} -3.26210e64 q^{83} -5.25422e63 q^{84} +9.56568e62 q^{85} -1.26164e65 q^{86} +2.45478e64 q^{87} -3.33879e64 q^{88} +2.61799e63 q^{89} -5.54619e63 q^{90} +1.24246e64 q^{91} -5.77977e65 q^{92} +2.41317e65 q^{93} -1.08472e66 q^{94} -7.51388e63 q^{95} -2.10560e66 q^{96} -3.95806e66 q^{97} -6.93204e66 q^{98} -2.40668e66 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.78647e10	1.47058	0.735292	−	0.677750i	$-0.237045\pi$
$2$	1.78647e10	1.47058	0.735292	+	0.677750i	$0.237045\pi$
$3$	5.55906e15	0.577350
$3$	5.55906e15	0.577350
$4$	1.71572e20	1.16262
$5$	−1.00461e22	−0.0385924	−0.0192962	−	0.999814i	$-0.506143\pi$
$5$	−1.00461e22	−0.0385924	−0.0192962	+	0.999814i	$0.506143\pi$
$6$	9.93107e25	0.849042
$7$	−5.50883e27	−0.269324	−0.134662	−	0.990892i	$-0.542995\pi$
$7$	−5.50883e27	−0.269324	−0.134662	+	0.990892i	$0.542995\pi$
$8$	4.28720e29	0.239144
$9$	3.09032e31	0.333333
$10$	−1.79470e32	−0.0567534
$11$	−7.78780e34	−1.01102	−0.505510	−	0.862821i	$-0.668696\pi$
$11$	−7.78780e34	−1.01102	−0.505510	+	0.862821i	$0.668696\pi$
$12$	9.53780e35	0.671238
$13$	−2.25540e36	−0.108673	−0.0543364	−	0.998523i	$-0.517304\pi$
$13$	−2.25540e36	−0.108673	−0.0543364	+	0.998523i	$0.517304\pi$
$14$	−9.84134e37	−0.396064
$15$	−5.58468e37	−0.0222813
$16$	−1.76606e40	−0.810937
$17$	−9.52180e40	−0.573694	−0.286847	−	0.957976i	$-0.592607\pi$
$17$	−9.52180e40	−0.573694	−0.286847	+	0.957976i	$0.592607\pi$
$18$	5.52074e41	0.490195
$19$	7.47941e41	0.108548	0.0542740	−	0.998526i	$-0.482716\pi$
$19$	7.47941e41	0.108548	0.0542740	+	0.998526i	$0.482716\pi$
$20$	−1.72363e42	−0.0448682
$21$	−3.06239e43	−0.155494
$22$	−1.39126e45	−1.48679
$23$	−3.36871e45	−0.812046	−0.406023	−	0.913863i	$-0.633085\pi$
$23$	−3.36871e45	−0.812046	−0.406023	+	0.913863i	$0.633085\pi$
$24$	2.38328e45	0.138070
$25$	−6.76617e46	−0.998511
$26$	−4.02919e46	−0.159813
$27$	1.71793e47	0.192450
$28$	−9.45163e47	−0.313121
$29$	4.41582e48	0.451522	0.225761	−	0.974183i	$-0.427513\pi$
$29$	4.41582e48	0.451522	0.225761	+	0.974183i	$0.427513\pi$
$30$	−9.97684e47	−0.0327666
$31$	4.34097e49	0.475302	0.237651	−	0.971351i	$-0.423622\pi$
$31$	4.34097e49	0.475302	0.237651	+	0.971351i	$0.423622\pi$
$32$	−3.78769e50	−1.43170
$33$	−4.32929e50	−0.583713
$34$	−1.70104e51	−0.843665
$35$	5.53422e49	0.0103939
$36$	5.30212e51	0.387539
$37$	1.88144e52	0.549203	0.274602	−	0.961558i	$-0.411454\pi$
$37$	1.88144e52	0.549203	0.274602	+	0.961558i	$0.411454\pi$
$38$	1.33617e52	0.159629
$39$	−1.25379e52	−0.0627423
$40$	−4.30696e51	−0.00922914
$41$	−1.15326e54	−1.08061	−0.540304	−	0.841470i	$-0.681691\pi$
$41$	−1.15326e54	−1.08061	−0.540304	+	0.841470i	$0.681691\pi$
$42$	−5.47086e53	−0.228668
$43$	−7.06222e54	−1.34199	−0.670993	−	0.741464i	$-0.734132\pi$
$43$	−7.06222e54	−1.34199	−0.670993	+	0.741464i	$0.734132\pi$
$44$	−1.33617e55	−1.17543
$45$	−3.10456e53	−0.0128641
$46$	−6.01808e55	−1.19418
$47$	−6.07185e55	−0.586196	−0.293098	−	0.956082i	$-0.594686\pi$
$47$	−6.07185e55	−0.586196	−0.293098	+	0.956082i	$0.594686\pi$
$48$	−9.81766e55	−0.468195
$49$	−3.88031e56	−0.927465
$50$	−1.20875e57	−1.46839
$51$	−5.29323e56	−0.331222
$52$	−3.86964e56	−0.126345
$53$	5.87139e57	1.01274	0.506372	−	0.862315i	$-0.330986\pi$
$53$	5.87139e57	1.01274	0.506372	+	0.862315i	$0.330986\pi$
$54$	3.06902e57	0.283014
$55$	7.82369e56	0.0390177
$56$	−2.36175e57	−0.0644073
$57$	4.15785e57	0.0626702
$58$	7.88872e58	0.664001
$59$	2.78418e59	1.32177	0.660884	−	0.750488i	$-0.270182\pi$
$59$	2.78418e59	1.32177	0.660884	+	0.750488i	$0.270182\pi$
$60$	−9.58175e57	−0.0259047
$61$	5.66402e59	0.880188	0.440094	−	0.897952i	$-0.354945\pi$
$61$	5.66402e59	0.880188	0.440094	+	0.897952i	$0.354945\pi$
$62$	7.75499e59	0.698972
$63$	−1.70240e59	−0.0897747
$64$	−4.16034e60	−1.29449
$65$	2.26579e58	0.00419395
$66$	−7.73412e60	−0.858399
$67$	−2.08739e61	−1.39990	−0.699949	−	0.714192i	$-0.746794\pi$
$67$	−2.08739e61	−1.39990	−0.699949	+	0.714192i	$0.746794\pi$
$68$	−1.63368e61	−0.666987
$69$	−1.87268e61	−0.468835
$70$	9.88670e59	0.0152850
$71$	1.83580e62	1.76470	0.882352	−	0.470590i	$-0.155959\pi$
$71$	1.83580e62	1.76470	0.882352	+	0.470590i	$0.155959\pi$
$72$	1.32488e61	0.0797147
$73$	−4.46947e61	−0.169410	−0.0847052	−	0.996406i	$-0.526995\pi$
$73$	−4.46947e61	−0.169410	−0.0847052	+	0.996406i	$0.526995\pi$
$74$	3.36113e62	0.807650
$75$	−3.76136e62	−0.576490
$76$	1.28326e62	0.126200
$77$	4.29017e62	0.272292
$78$	−2.23985e62	−0.0922679
$79$	−1.25916e62	−0.0338512	−0.0169256	−	0.999857i	$-0.505388\pi$
$79$	−1.25916e62	−0.0338512	−0.0169256	+	0.999857i	$0.505388\pi$
$80$	1.77420e62	0.0312960
$81$	9.55005e62	0.111111
$82$	−2.06026e64	−1.58913
$83$	−3.26210e64	−1.67642	−0.838208	−	0.545351i	$-0.816397\pi$
$83$	−3.26210e64	−1.67642	−0.838208	+	0.545351i	$0.816397\pi$
$84$	−5.25422e63	−0.180781
$85$	9.56568e62	0.0221402
$86$	−1.26164e65	−1.97350
$87$	2.45478e64	0.260686
$88$	−3.33879e64	−0.241780
$89$	2.61799e63	0.0129838	0.00649192	−	0.999979i	$-0.497934\pi$
$89$	2.61799e63	0.0129838	0.00649192	+	0.999979i	$0.497934\pi$
$90$	−5.54619e63	−0.0189178
$91$	1.24246e64	0.0292682
$92$	−5.77977e65	−0.944100
$93$	2.41317e65	0.274416
$94$	−1.08472e66	−0.862051
$95$	−7.51388e63	−0.00418912
$96$	−2.10560e66	−0.826590
$97$	−3.95806e66	−1.09807	−0.549033	−	0.835800i	$-0.685004\pi$
$97$	−3.95806e66	−1.09807	−0.549033	+	0.835800i	$0.685004\pi$
$98$	−6.93204e66	−1.36391
$99$	−2.40668e66	−0.337007
$100$	−1.16089e67	−1.16089
$101$	−3.43037e66	−0.245796	−0.122898	−	0.992419i	$-0.539219\pi$
$101$	−3.43037e66	−0.245796	−0.122898	+	0.992419i	$0.539219\pi$
$102$	−9.45617e66	−0.487090
$103$	−3.78657e67	−1.40669	−0.703345	−	0.710849i	$-0.748311\pi$
$103$	−3.78657e67	−1.40669	−0.703345	+	0.710849i	$0.748311\pi$
$104$	−9.66936e65	−0.0259885
$105$	3.07651e65	0.00600090
$106$	1.04890e68	1.48932
$107$	1.07484e68	1.11426	0.557129	−	0.830426i	$-0.311903\pi$
$107$	1.07484e68	1.11426	0.557129	+	0.830426i	$0.311903\pi$
$108$	2.94748e67	0.223746
$109$	1.96698e68	1.09651	0.548254	−	0.836312i	$-0.315293\pi$
$109$	1.96698e68	1.09651	0.548254	+	0.836312i	$0.315293\pi$
$110$	1.39768e67	0.0573788
$111$	1.04591e68	0.317083
$112$	9.72895e67	0.218405
$113$	−8.47224e68	−1.41212	−0.706058	−	0.708154i	$-0.749528\pi$
$113$	−8.47224e68	−1.41212	−0.706058	+	0.708154i	$0.749528\pi$
$114$	7.42786e67	0.0921618
$115$	3.38423e67	0.0313388
$116$	7.57632e68	0.524947
$117$	−6.96990e67	−0.0362243
$118$	4.97384e69	1.94377
$119$	5.24540e68	0.154510
$120$	−2.39427e67	−0.00532845
$121$	1.31502e68	0.0221626
$122$	1.01186e70	1.29439
$123$	−6.41105e69	−0.623890
$124$	7.44790e69	0.552595
$125$	1.36048e69	0.0771273
$126$	−3.04129e69	−0.132021
$127$	2.84972e70	0.949242	0.474621	−	0.880190i	$-0.342585\pi$
$127$	2.84972e70	0.949242	0.474621	+	0.880190i	$0.342585\pi$
$128$	−1.84265e70	−0.471964
$129$	−3.92593e70	−0.774795
$130$	4.04776e68	0.00616755
$131$	−9.59898e70	−1.13145	−0.565727	−	0.824593i	$-0.691404\pi$
$131$	−9.59898e70	−1.13145	−0.565727	+	0.824593i	$0.691404\pi$
$132$	−7.42785e70	−0.678635
$133$	−4.12028e69	−0.0292346
$134$	−3.72904e71	−2.05867
$135$	−1.72584e69	−0.00742711
$136$	−4.08219e70	−0.137195
$137$	−5.01638e71	−1.31902	−0.659510	−	0.751696i	$-0.729236\pi$
$137$	−5.01638e71	−1.31902	−0.659510	+	0.751696i	$0.729236\pi$
$138$	−3.34549e71	−0.689462
$139$	1.09442e72	1.77088	0.885442	−	0.464750i	$-0.153856\pi$
$139$	1.09442e72	1.77088	0.885442	+	0.464750i	$0.153856\pi$
$140$	9.49518e69	0.0120841
$141$	−3.37538e71	−0.338440
$142$	3.27960e72	2.59515
$143$	1.75646e71	0.109871
$144$	−5.45770e71	−0.270312
$145$	−4.43617e70	−0.0174253
$146$	−7.98455e71	−0.249132
$147$	−2.15709e72	−0.535472
$148$	3.22803e72	0.638514
$149$	2.24981e72	0.355145	0.177572	−	0.984108i	$-0.443176\pi$
$149$	2.24981e72	0.355145	0.177572	+	0.984108i	$0.443176\pi$
$150$	−6.71953e72	−0.847778
$151$	1.22480e73	1.23690	0.618452	−	0.785823i	$-0.287760\pi$
$151$	1.22480e73	1.23690	0.618452	+	0.785823i	$0.287760\pi$
$152$	3.20657e71	0.0259586
$153$	−2.94254e72	−0.191231
$154$	7.66425e72	0.400429
$155$	−4.36097e71	−0.0183431
$156$	−2.15116e72	−0.0729454
$157$	3.47174e73	0.950406	0.475203	−	0.879876i	$-0.342375\pi$
$157$	3.47174e73	0.950406	0.475203	+	0.879876i	$0.342375\pi$
$158$	−2.24945e72	−0.0497810
$159$	3.26394e73	0.584708
$160$	3.80515e72	0.0552525
$161$	1.85576e73	0.218704
$162$	1.70608e73	0.163398
$163$	2.83191e73	0.220696	0.110348	−	0.993893i	$-0.464804\pi$
$163$	2.83191e73	0.220696	0.110348	+	0.993893i	$0.464804\pi$
$164$	−1.97868e74	−1.25634
$165$	4.34924e72	0.0225269
$166$	−5.82764e74	−2.46531
$167$	−1.83227e74	−0.633852	−0.316926	−	0.948450i	$-0.602651\pi$
$167$	−1.83227e74	−0.633852	−0.316926	+	0.948450i	$0.602651\pi$
$168$	−1.31291e73	−0.0371855
$169$	−4.25643e74	−0.988190
$170$	1.70888e73	0.0325590
$171$	2.31137e73	0.0361827
$172$	−1.21168e75	−1.56022
$173$	1.55406e75	1.64788	0.823938	−	0.566680i	$-0.191772\pi$
$173$	1.55406e75	1.64788	0.823938	+	0.566680i	$0.191772\pi$
$174$	4.38539e74	0.383361
$175$	3.72737e74	0.268923
$176$	1.37538e75	0.819874
$177$	1.54774e75	0.763123
$178$	4.67696e73	0.0190938
$179$	3.01440e75	1.02006	0.510028	−	0.860158i	$-0.329635\pi$
$179$	3.01440e75	1.02006	0.510028	+	0.860158i	$0.329635\pi$
$180$	−5.32656e73	−0.0149561
$181$	−1.34884e75	−0.314577	−0.157289	−	0.987553i	$-0.550275\pi$
$181$	−1.34884e75	−0.314577	−0.157289	+	0.987553i	$0.550275\pi$
$182$	2.21962e74	0.0430414
$183$	3.14867e75	0.508177
$184$	−1.44423e75	−0.194196
$185$	−1.89011e74	−0.0211951
$186$	4.31105e75	0.403552
$187$	7.41539e75	0.580016
$188$	−1.04176e76	−0.681522
$189$	−9.46376e74	−0.0518315
$190$	−1.34233e74	−0.00616046
$191$	7.42759e75	0.285910	0.142955	−	0.989729i	$-0.454340\pi$
$191$	7.42759e75	0.285910	0.142955	+	0.989729i	$0.454340\pi$
$192$	−2.31276e76	−0.747375
$193$	3.03921e76	0.825261	0.412631	−	0.910898i	$-0.364610\pi$
$193$	3.03921e76	0.825261	0.412631	+	0.910898i	$0.364610\pi$
$194$	−7.07093e76	−1.61480
$195$	1.25957e74	0.00242138
$196$	−6.65753e76	−1.07829
$197$	8.15855e76	1.11428	0.557140	−	0.830419i	$-0.311899\pi$
$197$	8.15855e76	1.11428	0.557140	+	0.830419i	$0.311899\pi$
$198$	−4.29945e76	−0.495597
$199$	1.98025e77	1.92815	0.964075	−	0.265629i	$-0.0855795\pi$
$199$	1.98025e77	1.92815	0.964075	+	0.265629i	$0.0855795\pi$
$200$	−2.90080e76	−0.238788
$201$	−1.16039e77	−0.808232
$202$	−6.12824e76	−0.361464
$203$	−2.43260e76	−0.121606
$204$	−9.08170e76	−0.385085
$205$	1.15858e76	0.0417033
$206$	−6.76457e77	−2.06866
$207$	−1.04104e77	−0.270682
$208$	3.98318e76	0.0881268
$209$	−5.82482e76	−0.109744
$210$	5.49607e75	0.00882483
$211$	1.97602e77	0.270601	0.135300	−	0.990805i	$-0.456800\pi$
$211$	1.97602e77	0.270601	0.135300	+	0.990805i	$0.456800\pi$
$212$	1.00737e78	1.17743
$213$	1.02053e78	1.01885
$214$	1.92016e78	1.63861
$215$	7.09476e76	0.0517904
$216$	7.36509e76	0.0460233
$217$	−2.39137e77	−0.128010
$218$	3.51394e78	1.61251
$219$	−2.48460e77	−0.0978091
$220$	1.34233e77	0.0453627
$221$	2.14755e77	0.0623449
$222$	1.86847e78	0.466297
$223$	−8.33686e78	−1.78974	−0.894869	−	0.446329i	$-0.852731\pi$
$223$	−8.33686e78	−1.78974	−0.894869	+	0.446329i	$0.852731\pi$
$224$	2.08658e78	0.385590
$225$	−2.09096e78	−0.332837
$226$	−1.51354e79	−2.07664
$227$	3.01004e78	0.356210	0.178105	−	0.984012i	$-0.443003\pi$
$227$	3.01004e78	0.356210	0.178105	+	0.984012i	$0.443003\pi$
$228$	7.13371e77	0.0728615
$229$	−1.15420e79	−1.01811	−0.509054	−	0.860734i	$-0.670005\pi$
$229$	−1.15420e79	−1.01811	−0.509054	+	0.860734i	$0.670005\pi$
$230$	6.04582e77	0.0460864
$231$	2.38493e78	0.157208
$232$	1.89315e78	0.107979
$233$	−2.61072e79	−1.28925	−0.644625	−	0.764499i	$-0.722987\pi$
$233$	−2.61072e79	−1.28925	−0.644625	+	0.764499i	$0.722987\pi$
$234$	−1.24515e78	−0.0532709
$235$	6.09983e77	0.0226227
$236$	4.77688e79	1.53671
$237$	−6.99976e77	−0.0195440
$238$	9.37073e78	0.227219
$239$	−1.36380e79	−0.287356	−0.143678	−	0.989625i	$-0.545893\pi$
$239$	−1.36380e79	−0.287356	−0.143678	+	0.989625i	$0.545893\pi$
$240$	9.86290e77	0.0180687
$241$	1.46176e79	0.232972	0.116486	−	0.993192i	$-0.462837\pi$
$241$	1.46176e79	0.232972	0.116486	+	0.993192i	$0.462837\pi$
$242$	2.34923e78	0.0325920
$243$	5.30893e78	0.0641500
$244$	9.71789e79	1.02332
$245$	3.89819e78	0.0357931
$246$	−1.14531e80	−0.917482
$247$	−1.68691e78	−0.0117962
$248$	1.86106e79	0.113666
$249$	−1.81342e80	−0.967879
$250$	2.43046e79	0.113422
$251$	1.45431e80	0.593728	0.296864	−	0.954920i	$-0.404059\pi$
$251$	1.45431e80	0.593728	0.296864	+	0.954920i	$0.404059\pi$
$252$	−2.92085e79	−0.104374
$253$	2.62348e80	0.820996
$254$	5.09092e80	1.39594
$255$	5.31762e78	0.0127827
$256$	2.84774e80	0.600429
$257$	4.07925e80	0.754780	0.377390	−	0.926054i	$-0.376822\pi$
$257$	4.07925e80	0.754780	0.377390	+	0.926054i	$0.376822\pi$
$258$	−7.01354e80	−1.13940
$259$	−1.03646e80	−0.147914
$260$	3.88747e78	0.00487596
$261$	1.36463e80	0.150507
$262$	−1.71483e81	−1.66390
$263$	−4.77486e80	−0.407796	−0.203898	−	0.978992i	$-0.565361\pi$
$263$	−4.77486e80	−0.407796	−0.203898	+	0.978992i	$0.565361\pi$
$264$	−1.85605e80	−0.139591
$265$	−5.89845e79	−0.0390842
$266$	−7.36074e79	−0.0429919
$267$	1.45536e79	0.00749623
$268$	−3.58137e81	−1.62755
$269$	1.39609e81	0.560032	0.280016	−	0.959995i	$-0.409660\pi$
$269$	1.39609e81	0.560032	0.280016	+	0.959995i	$0.409660\pi$
$270$	−3.08316e79	−0.0109222
$271$	−1.94630e81	−0.609171	−0.304585	−	0.952485i	$-0.598518\pi$
$271$	−1.94630e81	−0.609171	−0.304585	+	0.952485i	$0.598518\pi$
$272$	1.68161e81	0.465229
$273$	6.90692e79	0.0168980
$274$	−8.96159e81	−1.93973
$275$	5.26936e81	1.00951
$276$	−3.21301e81	−0.545076
$277$	1.06361e81	0.159849	0.0799243	−	0.996801i	$-0.474532\pi$
$277$	1.06361e81	0.159849	0.0799243	+	0.996801i	$0.474532\pi$
$278$	1.95515e82	2.60423
$279$	1.34150e81	0.158434
$280$	2.37263e79	0.00248563
$281$	1.26260e82	1.17383	0.586913	−	0.809650i	$-0.300343\pi$
$281$	1.26260e82	1.17383	0.586913	+	0.809650i	$0.300343\pi$
$282$	−6.03000e81	−0.497705
$283$	8.69837e80	0.0637664	0.0318832	−	0.999492i	$-0.489850\pi$
$283$	8.69837e80	0.0637664	0.0318832	+	0.999492i	$0.489850\pi$
$284$	3.14973e82	2.05168
$285$	−4.17701e79	−0.00241859
$286$	3.13786e81	0.161574
$287$	6.35312e81	0.291034
$288$	−1.17052e82	−0.477232
$289$	−1.84808e82	−0.670875
$290$	−7.92507e80	−0.0256254
$291$	−2.20031e82	−0.633969
$292$	−7.66836e81	−0.196960
$293$	−7.44032e82	−1.70422	−0.852112	−	0.523360i	$-0.824678\pi$
$293$	−7.44032e82	−1.70422	−0.852112	+	0.523360i	$0.824678\pi$
$294$	−3.85356e82	−0.787457
$295$	−2.79701e81	−0.0510102
$296$	8.06613e81	0.131339
$297$	−1.33789e82	−0.194571
$298$	4.01920e82	0.522270
$299$	7.59778e81	0.0882474
$300$	−6.45344e82	−0.670238
$301$	3.89046e82	0.361429
$302$	2.18806e83	1.81897
$303$	−1.90696e82	−0.141910
$304$	−1.32091e82	−0.0880255
$305$	−5.69013e81	−0.0339686
$306$	−5.25674e82	−0.281222
$307$	−1.22726e83	−0.588575	−0.294288	−	0.955717i	$-0.595082\pi$
$307$	−1.22726e83	−0.588575	−0.294288	+	0.955717i	$0.595082\pi$
$308$	7.36074e82	0.316572
$309$	−2.10497e83	−0.812153
$310$	−7.79073e81	−0.0269750
$311$	−2.22891e83	−0.692819	−0.346409	−	0.938083i	$-0.612599\pi$
$311$	−2.22891e83	−0.692819	−0.346409	+	0.938083i	$0.612599\pi$
$312$	−5.37525e81	−0.0150044
$313$	−3.50226e83	−0.878241	−0.439120	−	0.898428i	$-0.644710\pi$
$313$	−3.50226e83	−0.878241	−0.439120	+	0.898428i	$0.644710\pi$
$314$	6.20215e83	1.39765
$315$	1.71025e81	0.00346462
$316$	−2.16037e82	−0.0393560
$317$	5.66883e83	0.928983	0.464492	−	0.885577i	$-0.346237\pi$
$317$	5.66883e83	0.928983	0.464492	+	0.885577i	$0.346237\pi$
$318$	5.83092e83	0.859862
$319$	−3.43896e83	−0.456498
$320$	4.17951e82	0.0499575
$321$	5.97508e83	0.643318
$322$	3.31526e83	0.321622
$323$	−7.12174e82	−0.0622733
$324$	1.63852e83	0.129180
$325$	1.52604e83	0.108511
$326$	5.05910e83	0.324552
$327$	1.09345e84	0.633069
$328$	−4.94426e83	−0.258421
$329$	3.34488e83	0.157877
$330$	7.76977e82	0.0331277
$331$	1.88884e84	0.727710	0.363855	−	0.931456i	$-0.381460\pi$
$331$	1.88884e84	0.727710	0.363855	+	0.931456i	$0.381460\pi$
$332$	−5.59686e84	−1.94903
$333$	5.81425e83	0.183068
$334$	−3.27329e84	−0.932133
$335$	2.09700e83	0.0540254
$336$	5.40838e83	0.126096
$337$	−4.04709e84	−0.854164	−0.427082	−	0.904213i	$-0.640458\pi$
$337$	−4.04709e84	−0.854164	−0.427082	+	0.904213i	$0.640458\pi$
$338$	−7.60396e84	−1.45322
$339$	−4.70977e84	−0.815285
$340$	1.64120e83	0.0257406
$341$	−3.38066e84	−0.480540
$342$	4.12919e83	0.0532096
$343$	4.44237e84	0.519113
$344$	−3.02772e84	−0.320928
$345$	1.88131e83	0.0180935
$346$	2.77628e85	2.42334
$347$	−1.19841e85	−0.949661	−0.474830	−	0.880077i	$-0.657491\pi$
$347$	−1.19841e85	−0.949661	−0.474830	+	0.880077i	$0.657491\pi$
$348$	4.21172e84	0.303079
$349$	1.14656e85	0.749451	0.374726	−	0.927136i	$-0.377737\pi$
$349$	1.14656e85	0.749451	0.374726	+	0.927136i	$0.377737\pi$
$350$	6.65882e84	0.395474
$351$	−3.87461e83	−0.0209141
$352$	2.94978e85	1.44747
$353$	−2.45579e85	−1.09582	−0.547908	−	0.836538i	$-0.684576\pi$
$353$	−2.45579e85	−1.09582	−0.547908	+	0.836538i	$0.684576\pi$
$354$	2.76499e85	1.12224
$355$	−1.84426e84	−0.0681041
$356$	4.49175e83	0.0150953
$357$	2.91595e84	0.0892061
$358$	5.38512e85	1.50008
$359$	−4.48164e85	−1.13703	−0.568516	−	0.822672i	$-0.692482\pi$
$359$	−4.48164e85	−1.13703	−0.568516	+	0.822672i	$0.692482\pi$
$360$	−1.33099e83	−0.00307638
$361$	−4.69185e85	−0.988217
$362$	−2.40965e85	−0.462613
$363$	7.31025e83	0.0127956
$364$	2.13172e84	0.0340278
$365$	4.49006e83	0.00653795
$366$	5.62498e85	0.747317
$367$	−1.26316e86	−1.53159	−0.765797	−	0.643083i	$-0.777655\pi$
$367$	−1.26316e86	−1.53159	−0.765797	+	0.643083i	$0.777655\pi$
$368$	5.94935e85	0.658518
$369$	−3.56394e85	−0.360203
$370$	−3.37662e84	−0.0311691
$371$	−3.23445e85	−0.272756
$372$	4.14033e85	0.319041
$373$	−2.24552e86	−1.58150	−0.790752	−	0.612136i	$-0.790310\pi$
$373$	−2.24552e86	−1.58150	−0.790752	+	0.612136i	$0.790310\pi$
$374$	1.32473e86	0.852963
$375$	7.56301e84	0.0445295
$376$	−2.60313e85	−0.140185
$377$	−9.95944e84	−0.0490682
$378$	−1.69067e85	−0.0762225
$379$	1.61057e85	0.0664610	0.0332305	−	0.999448i	$-0.489420\pi$
$379$	1.61057e85	0.0664610	0.0332305	+	0.999448i	$0.489420\pi$
$380$	−1.28917e84	−0.00487035
$381$	1.58418e86	0.548045
$382$	1.32691e86	0.420455
$383$	−9.46487e85	−0.274761	−0.137381	−	0.990518i	$-0.543868\pi$
$383$	−9.46487e85	−0.274761	−0.137381	+	0.990518i	$0.543868\pi$
$384$	−1.02434e86	−0.272489
$385$	−4.30994e84	−0.0105084
$386$	5.42945e86	1.21362
$387$	−2.18245e86	−0.447328
$388$	−6.79092e86	−1.27663
$389$	−5.61275e86	−0.967975	−0.483988	−	0.875075i	$-0.660812\pi$
$389$	−5.61275e86	−0.967975	−0.483988	+	0.875075i	$0.660812\pi$
$390$	2.25018e84	0.00356084
$391$	3.20762e86	0.465866
$392$	−1.66357e86	−0.221798
$393$	−5.33613e86	−0.653245
$394$	1.45750e87	1.63864
$395$	1.26496e84	0.00130640
$396$	−4.12919e86	−0.391810
$397$	5.63115e86	0.491038	0.245519	−	0.969392i	$-0.421042\pi$
$397$	5.63115e86	0.491038	0.245519	+	0.969392i	$0.421042\pi$
$398$	3.53764e87	2.83551
$399$	−2.29049e85	−0.0168786
$400$	1.19495e87	0.809729
$401$	2.09444e87	1.30537	0.652683	−	0.757632i	$-0.273644\pi$
$401$	2.09444e87	1.30537	0.652683	+	0.757632i	$0.273644\pi$
$402$	−2.07300e87	−1.18857
$403$	−9.79062e85	−0.0516525
$404$	−5.88556e86	−0.285767
$405$	−9.59406e84	−0.00428804
$406$	−4.34576e86	−0.178831
$407$	−1.46523e87	−0.555256
$408$	−2.26931e86	−0.0792098
$409$	−5.40921e87	−1.73942	−0.869708	−	0.493567i	$-0.835693\pi$
$409$	−5.40921e87	−1.73942	−0.869708	+	0.493567i	$0.835693\pi$
$410$	2.06976e86	0.0613282
$411$	−2.78864e87	−0.761537
$412$	−6.49669e87	−1.63544
$413$	−1.53376e87	−0.355984
$414$	−1.85978e87	−0.398061
$415$	3.27714e86	0.0646969
$416$	8.54276e86	0.155586
$417$	6.08397e87	1.02242
$418$	−1.04058e87	−0.161388
$419$	−2.59511e87	−0.371523	−0.185761	−	0.982595i	$-0.559475\pi$
$419$	−2.59511e87	−0.371523	−0.185761	+	0.982595i	$0.559475\pi$
$420$	5.27843e85	0.00697675
$421$	−1.59011e87	−0.194078	−0.0970389	−	0.995281i	$-0.530937\pi$
$421$	−1.59011e87	−0.194078	−0.0970389	+	0.995281i	$0.530937\pi$
$422$	3.53009e87	0.397941
$423$	−1.87639e87	−0.195399
$424$	2.51718e87	0.242192
$425$	6.44261e87	0.572839
$426$	1.82315e88	1.49831
$427$	−3.12022e87	−0.237056
$428$	1.84412e88	1.29546
$429$	9.76427e86	0.0634338
$430$	1.26745e87	0.0761622
$431$	4.83628e86	0.0268858	0.0134429	−	0.999910i	$-0.495721\pi$
$431$	4.83628e86	0.0268858	0.0134429	+	0.999910i	$0.495721\pi$
$432$	−3.03397e87	−0.156065
$433$	−1.26114e88	−0.600372	−0.300186	−	0.953881i	$-0.597049\pi$
$433$	−1.26114e88	−0.600372	−0.300186	+	0.953881i	$0.597049\pi$
$434$	−4.27210e87	−0.188250
$435$	−2.46609e86	−0.0100605
$436$	3.37479e88	1.27482
$437$	−2.51959e87	−0.0881460
$438$	−4.43866e87	−0.143837
$439$	6.91031e87	0.207461	0.103730	−	0.994605i	$-0.466922\pi$
$439$	6.91031e87	0.207461	0.103730	+	0.994605i	$0.466922\pi$
$440$	3.35418e86	0.00933085
$441$	−1.19914e88	−0.309155
$442$	3.83652e87	0.0916835
$443$	−3.01713e88	−0.668451	−0.334226	−	0.942493i	$-0.608475\pi$
$443$	−3.01713e88	−0.668451	−0.334226	+	0.942493i	$0.608475\pi$
$444$	1.79448e88	0.368646
$445$	−2.63006e85	−0.000501078 0
$446$	−1.48935e89	−2.63196
$447$	1.25068e88	0.205043
$448$	2.29186e88	0.348638
$449$	2.56507e88	0.362116	0.181058	−	0.983472i	$-0.442048\pi$
$449$	2.56507e88	0.362116	0.181058	+	0.983472i	$0.442048\pi$
$450$	−3.73543e88	−0.489465
$451$	8.98137e88	1.09252
$452$	−1.45360e89	−1.64175
$453$	6.80871e88	0.714127
$454$	5.37733e88	0.523836
$455$	−1.24819e86	−0.00112953
$456$	1.78255e87	0.0149872
$457$	1.96286e89	1.53355	0.766773	−	0.641918i	$-0.221861\pi$
$457$	1.96286e89	1.53355	0.766773	+	0.641918i	$0.221861\pi$
$458$	−2.06195e89	−1.49721
$459$	−1.63577e88	−0.110407
$460$	5.80640e87	0.0364351
$461$	3.41992e88	0.199543	0.0997713	−	0.995010i	$-0.468189\pi$
$461$	3.41992e88	0.199543	0.0997713	+	0.995010i	$0.468189\pi$
$462$	4.26060e88	0.231188
$463$	−9.70950e88	−0.490042	−0.245021	−	0.969518i	$-0.578795\pi$
$463$	−9.70950e88	−0.490042	−0.245021	+	0.969518i	$0.578795\pi$
$464$	−7.79863e88	−0.366156
$465$	−2.42429e87	−0.0105904
$466$	−4.66397e89	−1.89595
$467$	1.41296e89	0.534582	0.267291	−	0.963616i	$-0.413872\pi$
$467$	1.41296e89	0.534582	0.267291	+	0.963616i	$0.413872\pi$
$468$	−1.19584e88	−0.0421150
$469$	1.14991e89	0.377027
$470$	1.08971e88	0.0332686
$471$	1.92996e89	0.548717
$472$	1.19363e89	0.316093
$473$	5.49991e89	1.35677
$474$	−1.25048e88	−0.0287411
$475$	−5.06070e88	−0.108386
$476$	8.99965e88	0.179636
$477$	1.81445e89	0.337581
$478$	−2.43638e89	−0.422581
$479$	7.92571e89	1.28174	0.640868	−	0.767651i	$-0.278575\pi$
$479$	7.92571e89	1.28174	0.640868	+	0.767651i	$0.278575\pi$
$480$	2.11530e88	0.0319001
$481$	−4.24341e88	−0.0596835
$482$	2.61138e89	0.342605
$483$	1.03163e89	0.126269
$484$	2.25620e88	0.0257667
$485$	3.97630e88	0.0423770
$486$	9.48422e88	0.0943380
$487$	−8.88468e89	−0.824938	−0.412469	−	0.910972i	$-0.635334\pi$
$487$	−8.88468e89	−0.824938	−0.412469	+	0.910972i	$0.635334\pi$
$488$	2.42828e89	0.210492
$489$	1.57427e89	0.127419
$490$	6.96398e88	0.0526367
$491$	−1.17958e90	−0.832714	−0.416357	−	0.909201i	$-0.636693\pi$
$491$	−1.17958e90	−0.832714	−0.416357	+	0.909201i	$0.636693\pi$
$492$	−1.09996e90	−0.725346
$493$	−4.20466e89	−0.259035
$494$	−3.01360e88	−0.0173473
$495$	2.41777e88	0.0130059
$496$	−7.66643e89	−0.385440
$497$	−1.01131e90	−0.475277
$498$	−3.23962e90	−1.42335
$499$	−2.98688e90	−1.22701	−0.613506	−	0.789690i	$-0.710241\pi$
$499$	−2.98688e90	−1.22701	−0.613506	+	0.789690i	$0.710241\pi$
$500$	2.33421e89	0.0896696
$501$	−1.01857e90	−0.365955
$502$	2.59808e90	0.873128
$503$	−1.00459e90	−0.315834	−0.157917	−	0.987452i	$-0.550478\pi$
$503$	−1.00459e90	−0.315834	−0.157917	+	0.987452i	$0.550478\pi$
$504$	−7.29855e88	−0.0214691
$505$	3.44618e88	0.00948586
$506$	4.68676e90	1.20734
$507$	−2.36617e90	−0.570532
$508$	4.88932e90	1.10361
$509$	6.42651e90	1.35809	0.679047	−	0.734095i	$-0.262393\pi$
$509$	6.42651e90	1.35809	0.679047	+	0.734095i	$0.262393\pi$
$510$	9.49974e88	0.0187980
$511$	2.46215e89	0.0456263
$512$	7.80666e90	1.35495
$513$	1.28491e89	0.0208901
$514$	7.28744e90	1.10997
$515$	3.80401e89	0.0542875
$516$	−6.73580e90	−0.900791
$517$	4.72864e90	0.592656
$518$	−1.85159e90	−0.217520
$519$	8.63914e90	0.951402
$520$	9.71391e87	0.00100296
$521$	−1.83965e90	−0.178103	−0.0890514	−	0.996027i	$-0.528384\pi$
$521$	−1.83965e90	−0.178103	−0.0890514	+	0.996027i	$0.528384\pi$
$522$	2.43786e90	0.221334
$523$	−2.21396e91	−1.88522	−0.942612	−	0.333890i	$-0.891639\pi$
$523$	−2.21396e91	−1.88522	−0.942612	+	0.333890i	$0.891639\pi$
$524$	−1.64692e91	−1.31545
$525$	2.07207e90	0.155263
$526$	−8.53012e90	−0.599698
$527$	−4.13338e90	−0.272678
$528$	7.64580e90	0.473354
$529$	−5.86118e90	−0.340581
$530$	−1.05374e90	−0.0574766
$531$	8.60400e90	0.440589
$532$	−7.06926e89	−0.0339887
$533$	2.60106e90	0.117433
$534$	2.59995e89	0.0110238
$535$	−1.07979e90	−0.0430019
$536$	−8.94904e90	−0.334777
$537$	1.67572e91	0.588929
$538$	2.49407e91	0.823574
$539$	3.02191e91	0.937686
$540$	−2.96106e89	−0.00863489
$541$	−5.03634e91	−1.38040	−0.690202	−	0.723616i	$-0.742478\pi$
$541$	−5.03634e91	−1.38040	−0.690202	+	0.723616i	$0.742478\pi$
$542$	−3.47700e91	−0.895837
$543$	−7.49827e90	−0.181621
$544$	3.60657e91	0.821355
$545$	−1.97604e90	−0.0423168
$546$	1.23390e90	0.0248500
$547$	−8.79660e91	−1.66625	−0.833123	−	0.553087i	$-0.813450\pi$
$547$	−8.79660e91	−1.66625	−0.833123	+	0.553087i	$0.813450\pi$
$548$	−8.60671e91	−1.53352
$549$	1.75036e91	0.293396
$550$	9.41353e91	1.48458
$551$	3.30277e90	0.0490118
$552$	−8.02858e90	−0.112119
$553$	6.93651e89	0.00911694
$554$	1.90010e91	0.235071
$555$	−1.05073e90	−0.0122370
$556$	1.87773e92	2.05886
$557$	−1.05456e92	−1.08874	−0.544368	−	0.838847i	$-0.683230\pi$
$557$	−1.05456e92	−1.08874	−0.544368	+	0.838847i	$0.683230\pi$
$558$	2.39654e91	0.232991
$559$	1.59281e91	0.145837
$560$	−9.77379e89	−0.00842877
$561$	4.12226e91	0.334872
$562$	2.25559e92	1.72621
$563$	−2.32894e92	−1.67930	−0.839648	−	0.543131i	$-0.817239\pi$
$563$	−2.32894e92	−1.67930	−0.839648	+	0.543131i	$0.817239\pi$
$564$	−5.79121e91	−0.393477
$565$	8.51128e90	0.0544969
$566$	1.55393e91	0.0937739
$567$	−5.26096e90	−0.0299249
$568$	7.87047e91	0.422018
$569$	2.80447e92	1.41772	0.708858	−	0.705351i	$-0.249211\pi$
$569$	2.80447e92	1.41772	0.708858	+	0.705351i	$0.249211\pi$
$570$	−7.46209e89	−0.00355674
$571$	−1.85514e91	−0.0833810	−0.0416905	−	0.999131i	$-0.513274\pi$
$571$	−1.85514e91	−0.0833810	−0.0416905	+	0.999131i	$0.513274\pi$
$572$	3.01360e91	0.127737
$573$	4.12904e91	0.165070
$574$	1.13496e92	0.427990
$575$	2.27933e92	0.810837
$576$	−1.28567e92	−0.431497
$577$	−5.73771e92	−1.81698	−0.908488	−	0.417912i	$-0.862762\pi$
$577$	−5.73771e92	−1.81698	−0.908488	+	0.417912i	$0.862762\pi$
$578$	−3.30152e92	−0.986579
$579$	1.68952e92	0.476465
$580$	−7.61124e90	−0.0202590
$581$	1.79704e92	0.451499
$582$	−3.93077e92	−0.932305
$583$	−4.57252e92	−1.02390
$584$	−1.91615e91	−0.0405135
$585$	7.00201e89	0.00139798
$586$	−1.32919e93	−2.50620
$587$	4.40351e92	0.784193	0.392097	−	0.919924i	$-0.371750\pi$
$587$	4.40351e92	0.784193	0.392097	+	0.919924i	$0.371750\pi$
$588$	−3.70096e92	−0.622549
$589$	3.24679e91	0.0515931
$590$	−4.99677e91	−0.0750147
$591$	4.53539e92	0.643330
$592$	−3.32275e92	−0.445369
$593$	4.24255e92	0.537397	0.268698	−	0.963224i	$-0.413407\pi$
$593$	4.24255e92	0.537397	0.268698	+	0.963224i	$0.413407\pi$
$594$	−2.39009e92	−0.286133
$595$	−5.26957e90	−0.00596289
$596$	3.86004e92	0.412898
$597$	1.10083e93	1.11322
$598$	1.35732e92	0.129775
$599$	2.10853e93	1.90626	0.953128	−	0.302568i	$-0.0978440\pi$
$599$	2.10853e93	1.90626	0.953128	+	0.302568i	$0.0978440\pi$
$600$	−1.61257e92	−0.137864
$601$	−2.59701e92	−0.209981	−0.104990	−	0.994473i	$-0.533481\pi$
$601$	−2.59701e92	−0.209981	−0.104990	+	0.994473i	$0.533481\pi$
$602$	6.95017e92	0.531512
$603$	−6.45068e92	−0.466633
$604$	2.10141e93	1.43805
$605$	−1.32108e90	−0.000855308 0
$606$	−3.40673e92	−0.208691
$607$	2.20940e93	1.28072	0.640358	−	0.768076i	$-0.278786\pi$
$607$	2.20940e93	1.28072	0.640358	+	0.768076i	$0.278786\pi$
$608$	−2.83297e92	−0.155408
$609$	−1.35230e92	−0.0702091
$610$	−1.01652e92	−0.0499536
$611$	1.36944e92	0.0637036
$612$	−5.04857e92	−0.222329
$613$	−3.18272e93	−1.32701	−0.663503	−	0.748174i	$-0.730931\pi$
$613$	−3.18272e93	−1.32701	−0.663503	+	0.748174i	$0.730931\pi$
$614$	−2.19246e93	−0.865550
$615$	6.44059e91	0.0240774
$616$	1.83928e92	0.0651171
$617$	7.51462e92	0.251973	0.125986	−	0.992032i	$-0.459790\pi$
$617$	7.51462e92	0.251973	0.125986	+	0.992032i	$0.459790\pi$
$618$	−3.76047e93	−1.19434
$619$	2.64778e93	0.796608	0.398304	−	0.917254i	$-0.369599\pi$
$619$	2.64778e93	0.796608	0.398304	+	0.917254i	$0.369599\pi$
$620$	−7.48222e91	−0.0213260
$621$	−5.78719e92	−0.156278
$622$	−3.98187e93	−1.01885
$623$	−1.44221e91	−0.00349686
$624$	2.21427e92	0.0508801
$625$	4.57127e93	0.995534
$626$	−6.25668e93	−1.29153
$627$	−3.23805e92	−0.0633609
$628$	5.95655e93	1.10496
$629$	−1.79147e93	−0.315075
$630$	3.05530e91	0.00509502
$631$	−1.11722e94	−1.76667	−0.883336	−	0.468741i	$-0.844708\pi$
$631$	−1.11722e94	−1.76667	−0.883336	+	0.468741i	$0.844708\pi$
$632$	−5.39828e91	−0.00809531
$633$	1.09848e93	0.156231
$634$	1.01272e94	1.36615
$635$	−2.86285e92	−0.0366335
$636$	5.60002e93	0.679792
$637$	8.75164e92	0.100790
$638$	−6.14358e93	−0.671318
$639$	5.67322e93	0.588235
$640$	1.85114e92	0.0182142
$641$	−7.47110e93	−0.697655	−0.348827	−	0.937187i	$-0.613420\pi$
$641$	−7.47110e93	−0.697655	−0.348827	+	0.937187i	$0.613420\pi$
$642$	1.06743e94	0.946053
$643$	−5.86512e92	−0.0493412	−0.0246706	−	0.999696i	$-0.507854\pi$
$643$	−5.86512e92	−0.0493412	−0.0246706	+	0.999696i	$0.507854\pi$
$644$	3.18398e93	0.254269
$645$	3.94402e92	0.0299012
$646$	−1.27228e93	−0.0915781
$647$	−1.16155e94	−0.793861	−0.396930	−	0.917849i	$-0.629925\pi$
$647$	−1.16155e94	−0.793861	−0.396930	+	0.917849i	$0.629925\pi$
$648$	4.09430e92	0.0265716
$649$	−2.16827e94	−1.33633
$650$	2.72622e93	0.159575
$651$	−1.32938e93	−0.0739068
$652$	4.85876e93	0.256585
$653$	−1.79722e94	−0.901590	−0.450795	−	0.892627i	$-0.648859\pi$
$653$	−1.79722e94	−0.901590	−0.450795	+	0.892627i	$0.648859\pi$
$654$	1.95342e94	0.930981
$655$	9.64322e92	0.0436655
$656$	2.03673e94	0.876305
$657$	−1.38121e93	−0.0564701
$658$	5.97552e93	0.232171
$659$	4.02543e94	1.48645	0.743227	−	0.669040i	$-0.233294\pi$
$659$	4.02543e94	1.48645	0.743227	+	0.669040i	$0.233294\pi$
$660$	7.46208e92	0.0261902
$661$	3.58187e94	1.19498	0.597491	−	0.801875i	$-0.296164\pi$
$661$	3.58187e94	1.19498	0.597491	+	0.801875i	$0.296164\pi$
$662$	3.37436e94	1.07016
$663$	1.19383e93	0.0359949
$664$	−1.39853e94	−0.400905
$665$	4.13927e91	0.00112823
$666$	1.03870e94	0.269217
$667$	−1.48756e94	−0.366657
$668$	−3.14367e94	−0.736928
$669$	−4.63451e94	−1.03331
$670$	3.74623e93	0.0794490
$671$	−4.41103e94	−0.889889
$672$	1.15994e94	0.222621
$673$	6.33303e94	1.15640	0.578199	−	0.815896i	$-0.303756\pi$
$673$	6.33303e94	1.15640	0.578199	+	0.815896i	$0.303756\pi$
$674$	−7.22999e94	−1.25612
$675$	−1.16238e94	−0.192163
$676$	−7.30284e94	−1.14889
$677$	−6.35836e94	−0.951973	−0.475987	−	0.879452i	$-0.657909\pi$
$677$	−6.35836e94	−0.951973	−0.475987	+	0.879452i	$0.657909\pi$
$678$	−8.41384e94	−1.19895
$679$	2.18043e94	0.295736
$680$	4.10100e92	0.00529470
$681$	1.67330e94	0.205658
$682$	−6.03944e94	−0.706675
$683$	−5.42313e93	−0.0604167	−0.0302083	−	0.999544i	$-0.509617\pi$
$683$	−5.42313e93	−0.0604167	−0.0302083	+	0.999544i	$0.509617\pi$
$684$	3.96567e93	0.0420666
$685$	5.03950e93	0.0509041
$686$	7.93614e94	0.763399
$687$	−6.41629e94	−0.587805
$688$	1.24723e95	1.08827
$689$	−1.32423e94	−0.110058
$690$	3.36091e93	0.0266080
$691$	7.80197e94	0.588423	0.294212	−	0.955740i	$-0.404943\pi$
$691$	7.80197e94	0.588423	0.294212	+	0.955740i	$0.404943\pi$
$692$	2.66634e95	1.91585
$693$	1.32580e94	0.0907641
$694$	−2.14092e95	−1.39656
$695$	−1.09947e94	−0.0683426
$696$	1.05242e94	0.0623415
$697$	1.09811e95	0.619938
$698$	2.04828e95	1.10213
$699$	−1.45132e95	−0.744349
$700$	6.39513e94	0.312655
$701$	2.56876e95	1.19721	0.598605	−	0.801044i	$-0.295722\pi$
$701$	2.56876e95	1.19721	0.598605	+	0.801044i	$0.295722\pi$
$702$	−6.92185e93	−0.0307560
$703$	1.40721e94	0.0596149
$704$	3.23999e95	1.30876
$705$	3.39093e93	0.0130612
$706$	−4.38718e95	−1.61149
$707$	1.88973e94	0.0661988
$708$	2.65550e95	0.887221
$709$	−2.85394e95	−0.909485	−0.454742	−	0.890623i	$-0.650269\pi$
$709$	−2.85394e95	−0.909485	−0.454742	+	0.890623i	$0.650269\pi$
$710$	−3.29472e94	−0.100153
$711$	−3.89121e93	−0.0112837
$712$	1.12239e93	0.00310501
$713$	−1.46235e95	−0.385968
$714$	5.20925e94	0.131185
$715$	−1.76455e93	−0.00424016
$716$	5.17187e95	1.18594
$717$	−7.58143e94	−0.165905
$718$	−8.00629e95	−1.67210
$719$	−8.93677e95	−1.78141	−0.890703	−	0.454586i	$-0.849787\pi$
$719$	−8.93677e95	−1.78141	−0.890703	+	0.454586i	$0.849787\pi$
$720$	5.48285e93	0.0104320
$721$	2.08596e95	0.378856
$722$	−8.38182e95	−1.45326
$723$	8.12600e94	0.134507
$724$	−2.31423e95	−0.365734
$725$	−2.98782e95	−0.450849
$726$	1.30595e94	0.0188170
$727$	7.00552e95	0.963912	0.481956	−	0.876195i	$-0.339926\pi$
$727$	7.00552e95	0.963912	0.481956	+	0.876195i	$0.339926\pi$
$728$	5.32669e93	0.00699932
$729$	2.95127e94	0.0370370
$730$	8.02135e93	0.00961461
$731$	6.72450e95	0.769888
$732$	5.40223e95	0.590816
$733$	1.25555e96	1.31175	0.655877	−	0.754868i	$-0.272299\pi$
$733$	1.25555e96	1.31175	0.655877	+	0.754868i	$0.272299\pi$
$734$	−2.25658e96	−2.25234
$735$	2.16703e94	0.0206651
$736$	1.27596e96	1.16260
$737$	1.62561e96	1.41533
$738$	−6.36686e95	−0.529709
$739$	4.63498e95	0.368519	0.184259	−	0.982878i	$-0.441011\pi$
$739$	4.63498e95	0.368519	0.184259	+	0.982878i	$0.441011\pi$
$740$	−3.24291e94	−0.0246418
$741$	−9.37761e93	−0.00681055
$742$	−5.77824e95	−0.401111
$743$	5.89585e95	0.391220	0.195610	−	0.980682i	$-0.437331\pi$
$743$	5.89585e95	0.391220	0.195610	+	0.980682i	$0.437331\pi$
$744$	1.03458e95	0.0656249
$745$	−2.26017e94	−0.0137059
$746$	−4.01154e96	−2.32574
$747$	−1.00809e96	−0.558805
$748$	1.27227e96	0.674337
$749$	−5.92110e95	−0.300097
$750$	1.35111e95	0.0654843
$751$	8.12036e95	0.376390	0.188195	−	0.982132i	$-0.439736\pi$
$751$	8.12036e95	0.376390	0.188195	+	0.982132i	$0.439736\pi$
$752$	1.07233e96	0.475368
$753$	8.08461e95	0.342789
$754$	−1.77922e95	−0.0721589
$755$	−1.23044e95	−0.0477351
$756$	−1.62372e95	−0.0602602
$757$	4.48960e96	1.59403	0.797014	−	0.603961i	$-0.206412\pi$
$757$	4.48960e96	1.59403	0.797014	+	0.603961i	$0.206412\pi$
$758$	2.87724e95	0.0977365
$759$	1.45841e96	0.474002
$760$	−3.22135e93	−0.00100180
$761$	−1.04777e96	−0.311804	−0.155902	−	0.987773i	$-0.549828\pi$
$761$	−1.04777e96	−0.311804	−0.155902	+	0.987773i	$0.549828\pi$
$762$	2.83008e96	0.805947
$763$	−1.08357e96	−0.295316
$764$	1.27437e96	0.332405
$765$	2.95610e94	0.00738007
$766$	−1.69087e96	−0.404060
$767$	−6.27944e95	−0.143640
$768$	1.58308e96	0.346658
$769$	−8.44815e96	−1.77104	−0.885522	−	0.464597i	$-0.846199\pi$
$769$	−8.44815e96	−1.77104	−0.885522	+	0.464597i	$0.846199\pi$
$770$	−7.69956e94	−0.0154535
$771$	2.26768e96	0.435772
$772$	5.21444e96	0.959464
$773$	8.12199e95	0.143103	0.0715516	−	0.997437i	$-0.477205\pi$
$773$	8.12199e95	0.143103	0.0715516	+	0.997437i	$0.477205\pi$
$774$	−3.89887e96	−0.657834
$775$	−2.93717e96	−0.474594
$776$	−1.69690e96	−0.262596
$777$	−5.76172e95	−0.0853981
$778$	−1.00270e97	−1.42349
$779$	−8.62571e95	−0.117298
$780$	2.16107e94	0.00281514
$781$	−1.42969e97	−1.78415
$782$	5.73030e96	0.685095
$783$	7.58605e95	0.0868954
$784$	6.85287e96	0.752115
$785$	−3.48774e95	−0.0366784
$786$	−9.53282e96	−0.960652
$787$	−1.69912e97	−1.64085	−0.820426	−	0.571752i	$-0.806264\pi$
$787$	−1.69912e97	−1.64085	−0.820426	+	0.571752i	$0.806264\pi$
$788$	1.39978e97	1.29548
$789$	−2.65437e96	−0.235441
$790$	2.25982e94	0.00192117
$791$	4.66721e96	0.380317
$792$	−1.03179e96	−0.0805932
$793$	−1.27746e96	−0.0956526
$794$	1.00599e97	0.722113
$795$	−3.27898e95	−0.0225653
$796$	3.39755e97	2.24170
$797$	−4.96633e96	−0.314182	−0.157091	−	0.987584i	$-0.550212\pi$
$797$	−4.96633e96	−0.314182	−0.157091	+	0.987584i	$0.550212\pi$
$798$	−4.09188e95	−0.0248214
$799$	5.78149e96	0.336297
$800$	2.56282e97	1.42956
$801$	8.09043e94	0.00432795
$802$	3.74165e97	1.91965
$803$	3.48073e96	0.171277
$804$	−1.99091e97	−0.939665
$805$	−1.86432e95	−0.00844030
$806$	−1.74906e96	−0.0759593
$807$	7.76096e96	0.323335
$808$	−1.47067e96	−0.0587807
$809$	−4.25651e96	−0.163222	−0.0816108	−	0.996664i	$-0.526006\pi$
$809$	−4.25651e96	−0.163222	−0.0816108	+	0.996664i	$0.526006\pi$
$810$	−1.71395e95	−0.00630593
$811$	9.79815e96	0.345896	0.172948	−	0.984931i	$-0.444671\pi$
$811$	9.79815e96	0.345896	0.172948	+	0.984931i	$0.444671\pi$
$812$	−4.17367e96	−0.141381
$813$	−1.08196e97	−0.351705
$814$	−2.61759e97	−0.816551
$815$	−2.84496e95	−0.00851717
$816$	9.34818e96	0.268600
$817$	−5.28212e96	−0.145670
$818$	−9.66337e97	−2.55796
$819$	3.83960e95	0.00975608
$820$	1.98779e96	0.0484850
$821$	1.36596e97	0.319846	0.159923	−	0.987129i	$-0.448875\pi$
$821$	1.36596e97	0.319846	0.159923	+	0.987129i	$0.448875\pi$
$822$	−4.98180e97	−1.11990
$823$	−4.91029e97	−1.05977	−0.529886	−	0.848069i	$-0.677765\pi$
$823$	−4.91029e97	−1.05977	−0.529886	+	0.848069i	$0.677765\pi$
$824$	−1.62338e97	−0.336402
$825$	2.92927e97	0.582844
$826$	−2.74001e97	−0.523504
$827$	4.47243e97	0.820557	0.410279	−	0.911960i	$-0.365431\pi$
$827$	4.47243e97	0.820557	0.410279	+	0.911960i	$0.365431\pi$
$828$	−1.78613e97	−0.314700
$829$	1.56789e97	0.265301	0.132650	−	0.991163i	$-0.457651\pi$
$829$	1.56789e97	0.265301	0.132650	+	0.991163i	$0.457651\pi$
$830$	5.85450e96	0.0951422
$831$	5.91266e96	0.0922886
$832$	9.38322e96	0.140676
$833$	3.69475e97	0.532081
$834$	1.08688e98	1.50356
$835$	1.84071e96	0.0244619
$836$	−9.99376e96	−0.127591
$837$	7.45746e96	0.0914720
$838$	−4.63607e97	−0.546356
$839$	1.62162e97	0.183622	0.0918108	−	0.995776i	$-0.470734\pi$
$839$	1.62162e97	0.183622	0.0918108	+	0.995776i	$0.470734\pi$
$840$	1.31896e95	0.00143508
$841$	−7.61463e97	−0.796128
$842$	−2.84067e97	−0.285408
$843$	7.01886e97	0.677709
$844$	3.39030e97	0.314605
$845$	4.27604e96	0.0381366
$846$	−3.35211e97	−0.287350
$847$	−7.24420e95	−0.00596893
$848$	−1.03693e98	−0.821271
$849$	4.83548e96	0.0368156
$850$	1.15095e98	0.842409
$851$	−6.33803e97	−0.445979
$852$	1.75095e98	1.18454
$853$	4.40380e97	0.286441	0.143220	−	0.989691i	$-0.454254\pi$
$853$	4.40380e97	0.286441	0.143220	+	0.989691i	$0.454254\pi$
$854$	−5.57416e97	−0.348611
$855$	−2.32202e95	−0.00139637
$856$	4.60804e97	0.266468
$857$	−3.89106e97	−0.216377	−0.108188	−	0.994130i	$-0.534505\pi$
$857$	−3.89106e97	−0.216377	−0.108188	+	0.994130i	$0.534505\pi$
$858$	1.74435e97	0.0932847
$859$	2.41506e98	1.24210	0.621051	−	0.783771i	$-0.286706\pi$
$859$	2.41506e98	1.24210	0.621051	+	0.783771i	$0.286706\pi$
$860$	1.21726e97	0.0602125
$861$	3.53174e97	0.168029
$862$	8.63985e96	0.0395378
$863$	−5.51839e97	−0.242913	−0.121456	−	0.992597i	$-0.538756\pi$
$863$	−5.51839e97	−0.242913	−0.121456	+	0.992597i	$0.538756\pi$
$864$	−6.50697e97	−0.275530
$865$	−1.56123e97	−0.0635955
$866$	−2.25299e98	−0.882898
$867$	−1.02736e98	−0.387330
$868$	−4.10292e97	−0.148827
$869$	9.80610e96	0.0342243
$870$	−4.40559e96	−0.0147948
$871$	4.70789e97	0.152131
$872$	8.43283e97	0.262223
$873$	−1.22316e98	−0.366022
$874$	−4.50117e97	−0.129626
$875$	−7.49468e96	−0.0207722
$876$	−4.26289e97	−0.113715
$877$	5.44375e98	1.39769	0.698846	−	0.715272i	$-0.253697\pi$
$877$	5.44375e98	1.39769	0.698846	+	0.715272i	$0.253697\pi$
$878$	1.23450e98	0.305089
$879$	−4.13612e98	−0.983934
$880$	−1.38171e97	−0.0316409
$881$	3.46521e98	0.763899	0.381950	−	0.924183i	$-0.375253\pi$
$881$	3.46521e98	0.763899	0.381950	+	0.924183i	$0.375253\pi$
$882$	−2.14222e98	−0.454638
$883$	2.92506e97	0.0597655	0.0298828	−	0.999553i	$-0.490487\pi$
$883$	2.92506e97	0.0597655	0.0298828	+	0.999553i	$0.490487\pi$
$884$	3.68459e97	0.0724834
$885$	−1.55488e97	−0.0294507
$886$	−5.39001e98	−0.983014
$887$	−4.59178e98	−0.806380	−0.403190	−	0.915116i	$-0.632099\pi$
$887$	−4.59178e98	−0.806380	−0.403190	+	0.915116i	$0.632099\pi$
$888$	4.48401e97	0.0758285
$889$	−1.56986e98	−0.255654
$890$	−4.69851e95	−0.000736877 0
$891$	−7.43739e97	−0.112336
$892$	−1.43037e99	−2.08078
$893$	−4.54138e97	−0.0636304
$894$	2.23430e98	0.301533
$895$	−3.02829e97	−0.0393664
$896$	1.01509e98	0.127111
$897$	4.22365e97	0.0509497
$898$	4.58242e98	0.532522
$899$	1.91689e98	0.214609
$900$	−3.58751e98	−0.386962
$901$	−5.59062e98	−0.581005
$902$	1.60449e99	1.60664
$903$	2.16273e98	0.208671
$904$	−3.63222e98	−0.337699
$905$	1.35505e97	0.0121403
$906$	1.21635e99	1.05018
$907$	9.42261e98	0.784020	0.392010	−	0.919961i	$-0.371780\pi$
$907$	9.42261e98	0.784020	0.392010	+	0.919961i	$0.371780\pi$
$908$	5.16439e98	0.414136
$909$	−1.06009e98	−0.0819321
$910$	−2.22984e96	−0.00166107
$911$	1.29846e99	0.932314	0.466157	−	0.884702i	$-0.345638\pi$
$911$	1.29846e99	0.932314	0.466157	+	0.884702i	$0.345638\pi$
$912$	−7.34303e97	−0.0508216
$913$	2.54046e99	1.69489
$914$	3.50659e99	2.25521
$915$	−3.16318e97	−0.0196118
$916$	−1.98029e99	−1.18367
$917$	5.28792e98	0.304728
$918$	−2.92225e98	−0.162363
$919$	−3.15353e99	−1.68938	−0.844690	−	0.535256i	$-0.820215\pi$
$919$	−3.15353e99	−1.68938	−0.844690	+	0.535256i	$0.820215\pi$
$920$	1.45089e97	0.00749449
$921$	−6.82243e98	−0.339814
$922$	6.10958e98	0.293444
$923$	−4.14047e98	−0.191775
$924$	4.09188e98	0.182773
$925$	−1.27302e99	−0.548386
$926$	−1.73457e99	−0.720649
$927$	−1.17017e99	−0.468897
$928$	−1.67258e99	−0.646441
$929$	−4.16321e99	−1.55204	−0.776019	−	0.630710i	$-0.782764\pi$
$929$	−4.16321e99	−1.55204	−0.776019	+	0.630710i	$0.782764\pi$
$930$	−4.33092e97	−0.0155740
$931$	−2.90224e98	−0.100674
$932$	−4.47927e99	−1.49891
$933$	−1.23906e99	−0.399999
$934$	2.52420e99	0.786148
$935$	−7.44956e97	−0.0223842
$936$	−2.98814e97	−0.00866282
$937$	1.48127e99	0.414342	0.207171	−	0.978305i	$-0.433574\pi$
$937$	1.48127e99	0.414342	0.207171	+	0.978305i	$0.433574\pi$
$938$	2.05427e99	0.554449
$939$	−1.94693e99	−0.507053
$940$	1.04656e98	0.0263016
$941$	−1.91465e99	−0.464341	−0.232171	−	0.972675i	$-0.574583\pi$
$941$	−1.91465e99	−0.464341	−0.232171	+	0.972675i	$0.574583\pi$
$942$	3.44781e99	0.806935
$943$	3.88500e99	0.877504
$944$	−4.91704e99	−1.07187
$945$	9.50737e96	0.00200030
$946$	9.82541e99	1.99525
$947$	−5.82478e99	−1.14171	−0.570854	−	0.821051i	$-0.693388\pi$
$947$	−5.82478e99	−1.14171	−0.570854	+	0.821051i	$0.693388\pi$
$948$	−1.20096e98	−0.0227222
$949$	1.00804e98	0.0184103
$950$	−9.04076e98	−0.159391
$951$	3.15134e99	0.536349
$952$	2.24881e98	0.0369500
$953$	2.65980e99	0.421927	0.210964	−	0.977494i	$-0.432340\pi$
$953$	2.65980e99	0.421927	0.210964	+	0.977494i	$0.432340\pi$
$954$	3.24144e99	0.496442
$955$	−7.46181e97	−0.0110340
$956$	−2.33990e99	−0.334085
$957$	−1.91174e99	−0.263559
$958$	1.41590e100	1.88490
$959$	2.76344e99	0.355244
$960$	2.32341e98	0.0288430
$961$	−6.45689e99	−0.774088
$962$	−7.58070e98	−0.0877697
$963$	3.32159e99	0.371420
$964$	2.50797e99	0.270858
$965$	−3.05322e98	−0.0318488
$966$	1.84297e99	0.185689
$967$	−1.41710e100	−1.37916	−0.689580	−	0.724209i	$-0.742205\pi$
$967$	−1.41710e100	−1.37916	−0.689580	+	0.724209i	$0.742205\pi$
$968$	5.63774e97	0.00530006
$969$	−3.95902e98	−0.0359535
$970$	7.10352e98	0.0623190
$971$	1.28291e100	1.08731	0.543654	−	0.839309i	$-0.317040\pi$
$971$	1.28291e100	1.08731	0.543654	+	0.839309i	$0.317040\pi$
$972$	9.10865e98	0.0745820
$973$	−6.02900e99	−0.476942
$974$	−1.58722e100	−1.21314
$975$	8.48336e98	0.0626489
$976$	−1.00030e100	−0.713777
$977$	5.76405e99	0.397429	0.198714	−	0.980057i	$-0.436323\pi$
$977$	5.76405e99	0.397429	0.198714	+	0.980057i	$0.436323\pi$
$978$	2.81239e99	0.187380
$979$	−2.03884e98	−0.0131269
$980$	6.68821e98	0.0416137
$981$	6.07858e99	0.365503
$982$	−2.10727e100	−1.22458
$983$	2.74127e100	1.53960	0.769802	−	0.638282i	$-0.220355\pi$
$983$	2.74127e100	1.53960	0.769802	+	0.638282i	$0.220355\pi$
$984$	−2.74855e99	−0.149199
$985$	−8.19615e98	−0.0430027
$986$	−7.51148e99	−0.380933
$987$	1.85944e99	0.0911502
$988$	−2.89426e98	−0.0137145
$989$	2.37905e100	1.08975
$990$	4.31926e98	0.0191263
$991$	3.14760e100	1.34745	0.673724	−	0.738983i	$-0.264693\pi$
$991$	3.14760e100	1.34745	0.673724	+	0.738983i	$0.264693\pi$
$992$	−1.64423e100	−0.680488
$993$	1.05002e100	0.420144
$994$	−1.80668e100	−0.698935
$995$	−1.98937e99	−0.0744119
$996$	−3.11133e100	−1.12527
$997$	−9.78770e99	−0.342289	−0.171144	−	0.985246i	$-0.554746\pi$
$997$	−9.78770e99	−0.342289	−0.171144	+	0.985246i	$0.554746\pi$
$998$	−5.33595e100	−1.80443
$999$	3.23218e99	0.105694

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.78647e10 q^{2} +5.55906e15 q^{3} +1.71572e20 q^{4} -1.00461e22 q^{5} +9.93107e25 q^{6} -5.50883e27 q^{7} +4.28720e29 q^{8} +3.09032e31 q^{9} +O(q^{10})\)	\(q+1.78647e10 q^{2} +5.55906e15 q^{3} +1.71572e20 q^{4} -1.00461e22 q^{5} +9.93107e25 q^{6} -5.50883e27 q^{7} +4.28720e29 q^{8} +3.09032e31 q^{9} -1.79470e32 q^{10} -7.78780e34 q^{11} +9.53780e35 q^{12} -2.25540e36 q^{13} -9.84134e37 q^{14} -5.58468e37 q^{15} -1.76606e40 q^{16} -9.52180e40 q^{17} +5.52074e41 q^{18} +7.47941e41 q^{19} -1.72363e42 q^{20} -3.06239e43 q^{21} -1.39126e45 q^{22} -3.36871e45 q^{23} +2.38328e45 q^{24} -6.76617e46 q^{25} -4.02919e46 q^{26} +1.71793e47 q^{27} -9.45163e47 q^{28} +4.41582e48 q^{29} -9.97684e47 q^{30} +4.34097e49 q^{31} -3.78769e50 q^{32} -4.32929e50 q^{33} -1.70104e51 q^{34} +5.53422e49 q^{35} +5.30212e51 q^{36} +1.88144e52 q^{37} +1.33617e52 q^{38} -1.25379e52 q^{39} -4.30696e51 q^{40} -1.15326e54 q^{41} -5.47086e53 q^{42} -7.06222e54 q^{43} -1.33617e55 q^{44} -3.10456e53 q^{45} -6.01808e55 q^{46} -6.07185e55 q^{47} -9.81766e55 q^{48} -3.88031e56 q^{49} -1.20875e57 q^{50} -5.29323e56 q^{51} -3.86964e56 q^{52} +5.87139e57 q^{53} +3.06902e57 q^{54} +7.82369e56 q^{55} -2.36175e57 q^{56} +4.15785e57 q^{57} +7.88872e58 q^{58} +2.78418e59 q^{59} -9.58175e57 q^{60} +5.66402e59 q^{61} +7.75499e59 q^{62} -1.70240e59 q^{63} -4.16034e60 q^{64} +2.26579e58 q^{65} -7.73412e60 q^{66} -2.08739e61 q^{67} -1.63368e61 q^{68} -1.87268e61 q^{69} +9.88670e59 q^{70} +1.83580e62 q^{71} +1.32488e61 q^{72} -4.46947e61 q^{73} +3.36113e62 q^{74} -3.76136e62 q^{75} +1.28326e62 q^{76} +4.29017e62 q^{77} -2.23985e62 q^{78} -1.25916e62 q^{79} +1.77420e62 q^{80} +9.55005e62 q^{81} -2.06026e64 q^{82} -3.26210e64 q^{83} -5.25422e63 q^{84} +9.56568e62 q^{85} -1.26164e65 q^{86} +2.45478e64 q^{87} -3.33879e64 q^{88} +2.61799e63 q^{89} -5.54619e63 q^{90} +1.24246e64 q^{91} -5.77977e65 q^{92} +2.41317e65 q^{93} -1.08472e66 q^{94} -7.51388e63 q^{95} -2.10560e66 q^{96} -3.95806e66 q^{97} -6.93204e66 q^{98} -2.40668e66 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 16255223088 q^{2} + 27\!\cdots\!15 q^{3}+ \cdots + 15\!\cdots\!45 q^{9}+O(q^{10}) \) 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

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more