LMFDB - Embedded newform 1.78.a.a.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1,78,Mod(1,1)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1, base_ring=CyclotomicField(1))

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

N = Newforms(chi, 78, names="a")

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

chi := DirichletCharacter("1.1");

S:= CuspForms(chi, 78);

N := Newforms(S);

Level:	$ N $	$=$	$ 1 $
Weight:	$ k $	$=$	$ 78 $
Character orbit:	$[\chi]$	$=$	1.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:	$37.5479417817$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + \cdots - 44\!\cdots\!16 $ x^6 - 17761277524319496093x^4 - 5418653098480012458842128200x^3 + 63613941732383091007079230349679850008x^2 + 25593744940652395086700970560658253564764671200x - 44271959822073232588337476834309941911954845745332334416
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	multiple of $ 2^{64}\cdot 3^{20}\cdot 5^{8}\cdot 7^{3}\cdot 11^{2}\cdot 13^{2}\cdot 19 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.47555e9$ of defining polynomial
Character	$\chi$	$=$	1.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.27426e11 q^{2} -3.10225e18 q^{3} -4.39082e22 q^{4} -2.60453e25 q^{5} -1.01576e30 q^{6} +6.63145e32 q^{7} -6.38558e34 q^{8} +4.14958e36 q^{9} +O(q^{10})$	\(q+3.27426e11 q^{2} -3.10225e18 q^{3} -4.39082e22 q^{4} -2.60453e25 q^{5} -1.01576e30 q^{6} +6.63145e32 q^{7} -6.38558e34 q^{8} +4.14958e36 q^{9} -8.52790e36 q^{10} +6.48617e39 q^{11} +1.36214e41 q^{12} +7.32735e42 q^{13} +2.17131e44 q^{14} +8.07991e43 q^{15} -1.42728e46 q^{16} -3.12387e47 q^{17} +1.35868e48 q^{18} -1.10609e49 q^{19} +1.14360e48 q^{20} -2.05724e51 q^{21} +2.12374e51 q^{22} -1.83256e51 q^{23} +1.98097e53 q^{24} -6.61066e53 q^{25} +2.39916e54 q^{26} +4.10994e54 q^{27} -2.91175e55 q^{28} -2.13393e56 q^{29} +2.64557e55 q^{30} +2.56966e57 q^{31} +4.97633e57 q^{32} -2.01218e58 q^{33} -1.02284e59 q^{34} -1.72718e58 q^{35} -1.82200e59 q^{36} -1.15746e60 q^{37} -3.62163e60 q^{38} -2.27313e61 q^{39} +1.66314e60 q^{40} -2.18823e62 q^{41} -6.73594e62 q^{42} +2.71146e62 q^{43} -2.84796e62 q^{44} -1.08077e62 q^{45} -6.00029e62 q^{46} -3.14518e64 q^{47} +4.42779e64 q^{48} +3.21580e65 q^{49} -2.16450e65 q^{50} +9.69104e65 q^{51} -3.21731e65 q^{52} -1.58400e66 q^{53} +1.34570e66 q^{54} -1.68934e65 q^{55} -4.23457e67 q^{56} +3.43138e67 q^{57} -6.98703e67 q^{58} +5.89775e66 q^{59} -3.54774e66 q^{60} -3.43144e68 q^{61} +8.41373e68 q^{62} +2.75177e69 q^{63} +3.78623e69 q^{64} -1.90843e68 q^{65} -6.58838e69 q^{66} -3.29495e70 q^{67} +1.37164e70 q^{68} +5.68508e69 q^{69} -5.65523e69 q^{70} -1.52686e71 q^{71} -2.64975e71 q^{72} +6.42182e71 q^{73} -3.78981e71 q^{74} +2.05079e72 q^{75} +4.85665e71 q^{76} +4.30127e72 q^{77} -7.44281e72 q^{78} +5.72645e72 q^{79} +3.71740e71 q^{80} -3.54665e73 q^{81} -7.16481e73 q^{82} +1.49967e73 q^{83} +9.03298e73 q^{84} +8.13622e72 q^{85} +8.87801e73 q^{86} +6.61999e74 q^{87} -4.14180e74 q^{88} -7.74510e74 q^{89} -3.53872e73 q^{90} +4.85910e75 q^{91} +8.04646e73 q^{92} -7.97174e75 q^{93} -1.02981e76 q^{94} +2.88085e74 q^{95} -1.54378e76 q^{96} -2.13933e76 q^{97} +1.05293e77 q^{98} +2.69149e76 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 264721893120 q^{2} + 14\!\cdots\!80 q^{3}+ \cdots - 48\!\cdots\!42 q^{9}+O(q^{10}) $ 6 * q + 264721893120 * q^2 + 1441611375788089080 * q^3 + 414488718051562044137472 * q^4 - 262034655809276597802623100 * q^5 - 366806373876514422355130201088 * q^6 + 270857839709680313679329082207600 * q^7 - 21222715175771442181432808272035840 * q^8 - 4895011902698601362943838014820938642 * q^9	$ 6 q + 264721893120 q^{2} + 14\!\cdots\!80 q^{3}+ \cdots + 22\!\cdots\!76 q^{99}+O(q^{100}) $ 6 * q + 264721893120 * q^2 + 1441611375788089080 * q^3 + 414488718051562044137472 * q^4 - 262034655809276597802623100 * q^5 - 366806373876514422355130201088 * q^6 + 270857839709680313679329082207600 * q^7 - 21222715175771442181432808272035840 * q^8 - 4895011902698601362943838014820938642 * q^9 - 943981839595570481933588950406110963200 * q^10 - 18259523505888985403727472716999830396568 * q^11 + 464880745107061653664227300592441229475840 * q^12 + 2380194429688668511174277092390661204777460 * q^13 - 294635885851315312797761966406968268616083456 * q^14 - 205398878502504702020984224024402427189799600 * q^15 + 44678683808767880535489561508977888302859288576 * q^16 - 354360527898668112383938884095051354700497739540 * q^17 - 579271368199620858231236048886957435939670283520 * q^18 + 5609349588345922508653624528305080863470217945560 * q^19 - 203683258723050328883517820082887345024612662067200 * q^20 - 540436528239602082418470626392633746858753885327168 * q^21 + 6114758837406209570140637531269380807939078825354240 * q^22 - 47704848518150781083381010364499925248017731105789360 * q^23 + 54816594321451146263884987133438313725918160963502080 * q^24 + 1218179184124654235834607721614209488740550081397516250 * q^25 - 550866287661514215594856318864157268196554716662599168 * q^26 - 18043016639677384483149377528753631298041104958650825040 * q^27 + 27673374074418016612675778175139670616136146452210319360 * q^28 + 16280759001892520996902191080778000816917007177111344340 * q^29 - 1011764162094599483190536262352170796094588963959467571200 * q^30 + 1790157695657501252076176026171679301117057950393828071872 * q^31 + 2181738275733175796144863923114890278726253762313317253120 * q^32 - 43587817236406175039900898992317145479130853281029781929440 * q^33 - 194953896897775283488395124499645480734536214093757337125376 * q^34 - 154692900477132731635374066357063627778968619353419533615200 * q^35 - 3492826558014976862445701719551945606786728595490426565320704 * q^36 - 7407081044970614504852172741178628092123742757389002078546620 * q^37 - 23437112252897896880014527036750794366537612969976459081384960 * q^38 - 70639623729776089938517469701883961219469901500376562016990704 * q^39 - 405188969845671269189619479481322249890553020909404426600448000 * q^40 - 622334267497818741826318492502909006845617873220194725690266308 * q^41 - 1785314244915654620512096109946411089204707510656231457419632640 * q^42 - 2618736058252833489926178823787488683637632009899828264542685400 * q^43 - 10432822146524346418364354026489166880919966717673650220195414016 * q^44 - 4261664397684963408032474279116855329416026736411628094706748300 * q^45 + 15015668129695292865283752907788076308022505321300082329335498752 * q^46 + 10456714010948157967838849947567752123265493969429352826408225440 * q^47 + 195413572872589369177356962560292275025549393856976542715493744640 * q^48 + 471540371160970162947042524598603402842322365532951363545112427542 * q^49 + 1643157939326752820476079797938119435570111533434962615482069920000 * q^50 + 1437480019590559075427205620912951154862381268364307425321563068272 * q^51 + 2497724095217025746860726558921243156289473926034163020782517862400 * q^52 - 1351874258191312033682238211039504811418206289179811594973968454620 * q^53 - 5802743153283665491865673044971704841325735354018068579171943024640 * q^54 - 31869417904451543700720919226087498810908552969327767789062308093200 * q^55 - 108662999456853131644146391920304784213919375663746888603843313008640 * q^56 - 80192344792625061999550741265686291327620988057496835084561604104480 * q^57 - 198994309984504890955955717028136272430515707470032642085684388912640 * q^58 - 314181202100222481576760882531528728694115253385468703103388053744120 * q^59 + 31757517058237454003450287009369504103827834377957826095111285964800 * q^60 + 1205268428250020260156089514176723127199311484106453915524972341789332 * q^61 + 5847595327663334724128627418101253733235456067016746715116600210186240 * q^62 + 5797157259817187838172511312525532039665297350177795293712349527116720 * q^63 + 7434996765286658306744699729117001037072361082977441755818145446100992 * q^64 + 8871205916648888694302367444130318424721491973331706441245421034929400 * q^65 + 2619665127058305678398408811495791734563636436120923156834123240689664 * q^66 - 20237777137309439154263683707086560559031166786916812035021461385807240 * q^67 - 192282119728407192675118903436134256417865682110612543314367901015326720 * q^68 - 200313232582951000798678573547728345115884560145782233425701325968721344 * q^69 - 275296905202040079140538321149869121020079259159855285360874363784294400 * q^70 - 84566701931998894021025479230103599698793740396961643247253155987263248 * q^71 - 111057066523302390867411925011041674769499898042748718262699840917995520 * q^72 + 314847243747931153403355639258701372803638975422668152047464555954073340 * q^73 + 2440978652597431277933746515377916372937009497503279937672013012437072384 * q^74 + 5406848712313524713947990348797970278787563587939911155459007261158885000 * q^75 + 12583857788866874491493997574394644552563831648352611876045224548363960320 * q^76 + 3503704566884313110653172829033905566483467863121365108497302013892404800 * q^77 - 3205965675068953356081325009194962500817273160296149320647178104103475200 * q^78 - 4478934474840817947707335445643189482501134177625819638140993021754592160 * q^79 - 71161053770974020381924657869896203411328788664145358716969422377556377600 * q^80 - 62119692812757584111204435712960193411658364945985590227036022119327600874 * q^81 - 215870435463876846311226667757904584387222051304478591060278478661208783360 * q^82 - 148140974166416303075124558783351182994193920545724368826080939980520858280 * q^83 + 51356650106390838431654087814077196024066314243448648781044119034071875584 * q^84 + 491597676589966787491574950515749558236649703170109339586274361878505865800 * q^85 + 282408262347204426247008847369759267182197911237017110679724533137839752192 * q^86 + 1363181037080164755262169098102975505194220445144287377094744451132352801680 * q^87 + 2657681751590091680961785563694793344274050442909114832295757477808460267520 * q^88 + 244875270415762051982571278168423166555270869887140541520680077076967187420 * q^89 + 4978403011748231687089157306213454731057762280800383772471249777474918822400 * q^90 + 637560104295539679774601429928812188462565548465392346037583647743398360352 * q^91 - 20568828732565451743944414428223922568301505472542794858122155367508257341440 * q^92 - 14490478249669505799006967900921922007746327667753988409512015425992979344640 * q^93 - 17973701798427203443369711151367318601937605119380706693990709867752930471936 * q^94 - 32941519370397986858775692207480112776415936672941584218885058160279506566000 * q^95 - 18725276161952705842933385524361085962413728935875751155361188816688573841408 * q^96 - 12282501490438986599941765679523755451099259695347356745999502279014355079860 * q^97 + 34498422662482146891923464099753801985751726005296311870693615309501356240640 * q^98 + 225826616422430389876328344183536014760241704676820880296433754570070950840776 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	3.27426e11	0.842283	0.421141	−	0.906995i	$-0.361630\pi$
$2$	3.27426e11	0.842283	0.421141	+	0.906995i	$0.361630\pi$
$3$	−3.10225e18	−1.32589	−0.662947	−	0.748666i	$-0.730695\pi$
$3$	−3.10225e18	−1.32589	−0.662947	+	0.748666i	$0.730695\pi$
$4$	−4.39082e22	−0.290560
$5$	−2.60453e25	−0.0320173	−0.0160086	−	0.999872i	$-0.505096\pi$
$5$	−2.60453e25	−0.0320173	−0.0160086	+	0.999872i	$0.505096\pi$
$6$	−1.01576e30	−1.11678
$7$	6.63145e32	1.92901	0.964503	−	0.264070i	$-0.0850650\pi$
$7$	6.63145e32	1.92901	0.964503	+	0.264070i	$0.0850650\pi$
$8$	−6.38558e34	−1.08702
$9$	4.14958e36	0.757996
$10$	−8.52790e36	−0.0269676
$11$	6.48617e39	0.522842	0.261421	−	0.965225i	$-0.415809\pi$
$11$	6.48617e39	0.522842	0.261421	+	0.965225i	$0.415809\pi$
$12$	1.36214e41	0.385252
$13$	7.32735e42	0.950885	0.475443	−	0.879747i	$-0.342288\pi$
$13$	7.32735e42	0.950885	0.475443	+	0.879747i	$0.342288\pi$
$14$	2.17131e44	1.62477
$15$	8.07991e43	0.0424515
$16$	−1.42728e46	−0.625015
$17$	−3.12387e47	−1.32559	−0.662796	−	0.748800i	$-0.730630\pi$
$17$	−3.12387e47	−1.32559	−0.662796	+	0.748800i	$0.730630\pi$
$18$	1.35868e48	0.638447
$19$	−1.10609e49	−0.648303	−0.324151	−	0.946005i	$-0.605079\pi$
$19$	−1.10609e49	−0.648303	−0.324151	+	0.946005i	$0.605079\pi$
$20$	1.14360e48	0.00930293
$21$	−2.05724e51	−2.55766
$22$	2.12374e51	0.440380
$23$	−1.83256e51	−0.0686337	−0.0343169	−	0.999411i	$-0.510926\pi$
$23$	−1.83256e51	−0.0686337	−0.0343169	+	0.999411i	$0.510926\pi$
$24$	1.98097e53	1.44127
$25$	−6.61066e53	−0.998975
$26$	2.39916e54	0.800914
$27$	4.10994e54	0.320871
$28$	−2.91175e55	−0.560492
$29$	−2.13393e56	−1.06379	−0.531897	−	0.846809i	$-0.678521\pi$
$29$	−2.13393e56	−1.06379	−0.531897	+	0.846809i	$0.678521\pi$
$30$	2.64557e55	0.0357562
$31$	2.56966e57	0.982767	0.491384	−	0.870943i	$-0.336491\pi$
$31$	2.56966e57	0.982767	0.491384	+	0.870943i	$0.336491\pi$
$32$	4.97633e57	0.560577
$33$	−2.01218e58	−0.693233
$34$	−1.02284e59	−1.11652
$35$	−1.72718e58	−0.0617615
$36$	−1.82200e59	−0.220243
$37$	−1.15746e60	−0.487236	−0.243618	−	0.969871i	$-0.578334\pi$
$37$	−1.15746e60	−0.487236	−0.243618	+	0.969871i	$0.578334\pi$
$38$	−3.62163e60	−0.546054
$39$	−2.27313e61	−1.26077
$40$	1.66314e60	0.0348033
$41$	−2.18823e62	−1.76976	−0.884878	−	0.465822i	$-0.845759\pi$
$41$	−2.18823e62	−1.76976	−0.884878	+	0.465822i	$0.845759\pi$
$42$	−6.73594e62	−2.15427
$43$	2.71146e62	0.350483	0.175242	−	0.984525i	$-0.443929\pi$
$43$	2.71146e62	0.350483	0.175242	+	0.984525i	$0.443929\pi$
$44$	−2.84796e62	−0.151917
$45$	−1.08077e62	−0.0242690
$46$	−6.00029e62	−0.0578090
$47$	−3.14518e64	−1.32397	−0.661984	−	0.749518i	$-0.730286\pi$
$47$	−3.14518e64	−1.32397	−0.661984	+	0.749518i	$0.730286\pi$
$48$	4.42779e64	0.828704
$49$	3.21580e65	2.72107
$50$	−2.16450e65	−0.841419
$51$	9.69104e65	1.75759
$52$	−3.21731e65	−0.276289
$53$	−1.58400e66	−0.653334	−0.326667	−	0.945140i	$-0.605925\pi$
$53$	−1.58400e66	−0.653334	−0.326667	+	0.945140i	$0.605925\pi$
$54$	1.34570e66	0.270264
$55$	−1.68934e65	−0.0167400
$56$	−4.23457e67	−2.09686
$57$	3.43138e67	0.859581
$58$	−6.98703e67	−0.896015
$59$	5.89775e66	0.0391637	0.0195819	−	0.999808i	$-0.493767\pi$
$59$	5.89775e66	0.0391637	0.0195819	+	0.999808i	$0.493767\pi$
$60$	−3.54774e66	−0.0123347
$61$	−3.43144e68	−0.631362	−0.315681	−	0.948865i	$-0.602233\pi$
$61$	−3.43144e68	−0.631362	−0.315681	+	0.948865i	$0.602233\pi$
$62$	8.41373e68	0.827768
$63$	2.75177e69	1.46218
$64$	3.78623e69	1.09718
$65$	−1.90843e68	−0.0304447
$66$	−6.58838e69	−0.583898
$67$	−3.29495e70	−1.63670	−0.818349	−	0.574722i	$-0.805110\pi$
$67$	−3.29495e70	−1.63670	−0.818349	+	0.574722i	$0.805110\pi$
$68$	1.37164e70	0.385164
$69$	5.68508e69	0.0910011
$70$	−5.65523e69	−0.0520207
$71$	−1.52686e71	−0.813490	−0.406745	−	0.913542i	$-0.633336\pi$
$71$	−1.52686e71	−0.813490	−0.406745	+	0.913542i	$0.633336\pi$
$72$	−2.64975e71	−0.823954
$73$	6.42182e71	1.17416	0.587081	−	0.809528i	$-0.300277\pi$
$73$	6.42182e71	1.17416	0.587081	+	0.809528i	$0.300277\pi$
$74$	−3.78981e71	−0.410390
$75$	2.05079e72	1.32454
$76$	4.85665e71	0.188371
$77$	4.30127e72	1.00857
$78$	−7.44281e72	−1.06193
$79$	5.72645e72	0.500314	0.250157	−	0.968205i	$-0.419518\pi$
$79$	5.72645e72	0.500314	0.250157	+	0.968205i	$0.419518\pi$
$80$	3.71740e71	0.0200113
$81$	−3.54665e73	−1.18344
$82$	−7.16481e73	−1.49064
$83$	1.49967e73	0.195655	0.0978273	−	0.995203i	$-0.468811\pi$
$83$	1.49967e73	0.195655	0.0978273	+	0.995203i	$0.468811\pi$
$84$	9.03298e73	0.743154
$85$	8.13622e72	0.0424418
$86$	8.87801e73	0.295206
$87$	6.61999e74	1.41048
$88$	−4.14180e74	−0.568337
$89$	−7.74510e74	−0.687878	−0.343939	−	0.938992i	$-0.611761\pi$
$89$	−7.74510e74	−0.687878	−0.343939	+	0.938992i	$0.611761\pi$
$90$	−3.53872e73	−0.0204413
$91$	4.85910e75	1.83426
$92$	8.04646e73	0.0199422
$93$	−7.97174e75	−1.30305
$94$	−1.02981e76	−1.11516
$95$	2.88085e74	0.0207569
$96$	−1.54378e76	−0.743266
$97$	−2.13933e76	−0.691139	−0.345569	−	0.938393i	$-0.612314\pi$
$97$	−2.13933e76	−0.691139	−0.345569	+	0.938393i	$0.612314\pi$
$98$	1.05293e77	2.29191
$99$	2.69149e76	0.396312
$100$	2.90262e76	0.290262
$101$	−1.49463e77	−1.01897	−0.509485	−	0.860480i	$-0.670164\pi$
$101$	−1.49463e77	−1.01897	−0.509485	+	0.860480i	$0.670164\pi$
$102$	3.17310e77	1.48039
$103$	3.05945e76	0.0980417	0.0490208	−	0.998798i	$-0.484390\pi$
$103$	3.05945e76	0.0980417	0.0490208	+	0.998798i	$0.484390\pi$
$104$	−4.67894e77	−1.03363
$105$	5.35815e76	0.0818893
$106$	−5.18643e77	−0.550292
$107$	−1.16397e78	−0.860332	−0.430166	−	0.902750i	$-0.641545\pi$
$107$	−1.16397e78	−0.860332	−0.430166	+	0.902750i	$0.641545\pi$
$108$	−1.80460e77	−0.0932324
$109$	1.84440e78	0.668245	0.334123	−	0.942530i	$-0.391560\pi$
$109$	1.84440e78	0.668245	0.334123	+	0.942530i	$0.391560\pi$
$110$	−5.53134e76	−0.0140998
$111$	3.59073e78	0.646023
$112$	−9.46494e78	−1.20566
$113$	1.00123e79	0.905765	0.452883	−	0.891570i	$-0.350396\pi$
$113$	1.00123e79	0.905765	0.452883	+	0.891570i	$0.350396\pi$
$114$	1.12352e79	0.724010
$115$	4.77297e76	0.00219746
$116$	9.36970e78	0.309096
$117$	3.04054e79	0.720767
$118$	1.93108e78	0.0329869
$119$	−2.07158e80	−2.55708
$120$	−5.15949e78	−0.0461455
$121$	−1.11829e80	−0.726637
$122$	−1.12354e80	−0.531785
$123$	6.78843e80	2.34651
$124$	−1.12829e80	−0.285553
$125$	3.44530e79	0.0640017
$126$	9.01000e80	1.23157
$127$	−8.72357e80	−0.879531	−0.439765	−	0.898113i	$-0.644938\pi$
$127$	−8.72357e80	−0.879531	−0.439765	+	0.898113i	$0.644938\pi$
$128$	4.87705e80	0.363558
$129$	−8.41163e80	−0.464704
$130$	−6.24869e79	−0.0256431
$131$	2.40701e81	0.735416	0.367708	−	0.929941i	$-0.380143\pi$
$131$	2.40701e81	0.735416	0.367708	+	0.929941i	$0.380143\pi$
$132$	8.83510e80	0.201426
$133$	−7.33500e81	−1.25058
$134$	−1.07885e82	−1.37856
$135$	−1.07045e80	−0.0102734
$136$	1.99477e82	1.44094
$137$	−8.39121e81	−0.457175	−0.228588	−	0.973523i	$-0.573411\pi$
$137$	−8.39121e81	−0.457175	−0.228588	+	0.973523i	$0.573411\pi$
$138$	1.86144e81	0.0766486
$139$	2.71804e81	0.0847590	0.0423795	−	0.999102i	$-0.486506\pi$
$139$	2.71804e81	0.0847590	0.0423795	+	0.999102i	$0.486506\pi$
$140$	7.58373e80	0.0179454
$141$	9.75714e82	1.75544
$142$	−4.99932e82	−0.685188
$143$	4.75265e82	0.497162
$144$	−5.92261e82	−0.473759
$145$	5.55788e81	0.0340597
$146$	2.10267e83	0.988976
$147$	−9.97621e83	−3.60785
$148$	5.08219e82	0.141571
$149$	−1.49810e83	−0.322010	−0.161005	−	0.986954i	$-0.551474\pi$
$149$	−1.49810e83	−0.322010	−0.161005	+	0.986954i	$0.551474\pi$
$150$	6.71483e83	1.11563
$151$	8.08802e83	1.04047	0.520235	−	0.854023i	$-0.325844\pi$
$151$	8.08802e83	1.04047	0.520235	+	0.854023i	$0.325844\pi$
$152$	7.06305e83	0.704716
$153$	−1.29627e84	−1.00479
$154$	1.40835e84	0.849497
$155$	−6.69276e82	−0.0314655
$156$	9.98090e83	0.366330
$157$	−5.04447e84	−1.44770	−0.723850	−	0.689957i	$-0.757629\pi$
$157$	−5.04447e84	−1.44770	−0.723850	+	0.689957i	$0.757629\pi$
$158$	1.87499e84	0.421406
$159$	4.91398e84	0.866252
$160$	−1.29610e83	−0.0179481
$161$	−1.21526e84	−0.132395
$162$	−1.16127e85	−0.996789
$163$	−1.89276e85	−1.28196	−0.640978	−	0.767559i	$-0.721471\pi$
$163$	−1.89276e85	−1.28196	−0.640978	+	0.767559i	$0.721471\pi$
$164$	9.60810e84	0.514220
$165$	5.24077e83	0.0221954
$166$	4.91031e84	0.164796
$167$	−2.38288e85	−0.634627	−0.317313	−	0.948321i	$-0.602781\pi$
$167$	−2.38288e85	−0.634627	−0.317313	+	0.948321i	$0.602781\pi$
$168$	1.31367e86	2.78022
$169$	−5.68961e84	−0.0958173
$170$	2.66401e84	0.0357480
$171$	−4.58982e85	−0.491411
$172$	−1.19055e85	−0.101836
$173$	−4.08427e85	−0.279472	−0.139736	−	0.990189i	$-0.544625\pi$
$173$	−4.08427e85	−0.279472	−0.139736	+	0.990189i	$0.544625\pi$
$174$	2.16756e86	1.18802
$175$	−4.38383e86	−1.92703
$176$	−9.25760e85	−0.326784
$177$	−1.82963e85	−0.0519270
$178$	−2.53594e86	−0.579388
$179$	2.63570e86	0.485349	0.242674	−	0.970108i	$-0.421975\pi$
$179$	2.63570e86	0.485349	0.242674	+	0.970108i	$0.421975\pi$
$180$	4.74546e84	0.00705159
$181$	−2.67150e86	−0.320723	−0.160361	−	0.987058i	$-0.551266\pi$
$181$	−2.67150e86	−0.320723	−0.160361	+	0.987058i	$0.551266\pi$
$182$	1.59099e87	1.54497
$183$	1.06452e87	0.837119
$184$	1.17020e86	0.0746060
$185$	3.01463e85	0.0155999
$186$	−2.61015e87	−1.09753
$187$	−2.02620e87	−0.693075
$188$	1.38099e87	0.384692
$189$	2.72549e87	0.618963
$190$	9.43265e85	0.0174832
$191$	3.41040e87	0.516441	0.258221	−	0.966086i	$-0.416864\pi$
$191$	3.41040e87	0.516441	0.258221	+	0.966086i	$0.416864\pi$
$192$	−1.17458e88	−1.45474
$193$	−7.35171e86	−0.0745474	−0.0372737	−	0.999305i	$-0.511867\pi$
$193$	−7.35171e86	−0.0745474	−0.0372737	+	0.999305i	$0.511867\pi$
$194$	−7.00470e87	−0.582134
$195$	5.92044e86	0.0403665
$196$	−1.41200e88	−0.790633
$197$	2.65806e88	1.22353	0.611766	−	0.791039i	$-0.290459\pi$
$197$	2.65806e88	1.22353	0.611766	+	0.791039i	$0.290459\pi$
$198$	8.81262e87	0.333807
$199$	3.09853e87	0.0966746	0.0483373	−	0.998831i	$-0.484608\pi$
$199$	3.09853e87	0.0966746	0.0483373	+	0.998831i	$0.484608\pi$
$200$	4.22129e88	1.08590
$201$	1.02218e89	2.17009
$202$	−4.89381e88	−0.858261
$203$	−1.41510e89	−2.05206
$204$	−4.25516e88	−0.510687
$205$	5.69930e87	0.0566628
$206$	1.00174e88	0.0825788
$207$	−7.60437e87	−0.0520241
$208$	−1.04582e89	−0.594317
$209$	−7.17431e88	−0.338960
$210$	1.75440e88	0.0689739
$211$	1.72113e89	0.563560	0.281780	−	0.959479i	$-0.409075\pi$
$211$	1.72113e89	0.563560	0.281780	+	0.959479i	$0.409075\pi$
$212$	6.95507e88	0.189833
$213$	4.73670e89	1.07860
$214$	−3.81113e89	−0.724643
$215$	−7.06207e87	−0.0112215
$216$	−2.62444e89	−0.348792
$217$	1.70406e90	1.89577
$218$	6.03905e89	0.562851
$219$	−1.99221e90	−1.55682
$220$	7.41760e87	0.00486396
$221$	−2.28897e90	−1.26049
$222$	1.17570e90	0.544134
$223$	−1.57808e90	−0.614315	−0.307157	−	0.951659i	$-0.599378\pi$
$223$	−1.57808e90	−0.614315	−0.307157	+	0.951659i	$0.599378\pi$
$224$	3.30003e90	1.08136
$225$	−2.74314e90	−0.757219
$226$	3.27830e90	0.762910
$227$	−8.66916e90	−1.70209	−0.851043	−	0.525095i	$-0.824030\pi$
$227$	−8.66916e90	−1.70209	−0.851043	+	0.525095i	$0.824030\pi$
$228$	−1.50666e90	−0.249760
$229$	1.20017e91	1.68103	0.840514	−	0.541789i	$-0.182253\pi$
$229$	1.20017e91	1.68103	0.840514	+	0.541789i	$0.182253\pi$
$230$	1.56279e88	0.00185089
$231$	−1.33436e91	−1.33725
$232$	1.36264e91	1.15636
$233$	2.04433e91	1.47010	0.735051	−	0.678012i	$-0.237158\pi$
$233$	2.04433e91	1.47010	0.735051	+	0.678012i	$0.237158\pi$
$234$	9.95551e90	0.607090
$235$	8.19171e89	0.0423899
$236$	−2.58960e89	−0.0113794
$237$	−1.77649e91	−0.663364
$238$	−6.78288e91	−2.15378
$239$	−1.38414e91	−0.373992	−0.186996	−	0.982361i	$-0.559875\pi$
$239$	−1.38414e91	−0.373992	−0.186996	+	0.982361i	$0.559875\pi$
$240$	−1.15323e90	−0.0265328
$241$	3.26079e90	0.0639244	0.0319622	−	0.999489i	$-0.489824\pi$
$241$	3.26079e90	0.0639244	0.0319622	+	0.999489i	$0.489824\pi$
$242$	−3.66156e91	−0.612033
$243$	8.75267e91	1.24824
$244$	1.50668e91	0.183448
$245$	−8.37564e90	−0.0871211
$246$	2.22271e92	1.97643
$247$	−8.10473e91	−0.616461
$248$	−1.64088e92	−1.06828
$249$	−4.65237e91	−0.259417
$250$	1.12808e91	0.0539075
$251$	4.58790e92	1.88008	0.940038	−	0.341071i	$-0.110790\pi$
$251$	4.58790e92	1.88008	0.940038	+	0.341071i	$0.110790\pi$
$252$	−1.20825e92	−0.424851
$253$	−1.18863e91	−0.0358846
$254$	−2.85632e92	−0.740814
$255$	−2.52406e91	−0.0562734
$256$	−4.12471e92	−0.790961
$257$	5.19296e92	0.857018	0.428509	−	0.903538i	$-0.359039\pi$
$257$	5.19296e92	0.857018	0.428509	+	0.903538i	$0.359039\pi$
$258$	−2.75418e92	−0.391412
$259$	−7.67562e92	−0.939881
$260$	8.37957e90	0.00884602
$261$	−8.85490e92	−0.806351
$262$	7.88117e92	0.619428
$263$	1.62122e93	1.10038	0.550192	−	0.835038i	$-0.314554\pi$
$263$	1.62122e93	1.10038	0.550192	+	0.835038i	$0.314554\pi$
$264$	1.28489e93	0.753555
$265$	4.12558e91	0.0209180
$266$	−2.40167e93	−1.05334
$267$	2.40273e93	0.912054
$268$	1.44675e93	0.475559
$269$	−7.65097e92	−0.217898	−0.108949	−	0.994047i	$-0.534749\pi$
$269$	−7.65097e92	−0.217898	−0.108949	+	0.994047i	$0.534749\pi$
$270$	−3.50492e91	−0.00865313
$271$	2.71370e93	0.581091	0.290546	−	0.956861i	$-0.406163\pi$
$271$	2.71370e93	0.581091	0.290546	+	0.956861i	$0.406163\pi$
$272$	4.45865e93	0.828515
$273$	−1.50741e94	−2.43204
$274$	−2.74750e93	−0.385071
$275$	−4.28779e93	−0.522306
$276$	−2.49622e92	−0.0264413
$277$	−4.92011e93	−0.453423	−0.226712	−	0.973962i	$-0.572798\pi$
$277$	−4.92011e93	−0.453423	−0.226712	+	0.973962i	$0.572798\pi$
$278$	8.89956e92	0.0713911
$279$	1.06630e94	0.744934
$280$	1.10291e93	0.0671358
$281$	9.10473e91	0.00483141	0.00241570	−	0.999997i	$-0.499231\pi$
$281$	9.10473e91	0.00483141	0.00241570	+	0.999997i	$0.499231\pi$
$282$	3.19474e94	1.47858
$283$	1.70979e94	0.690503	0.345251	−	0.938510i	$-0.387794\pi$
$283$	1.70979e94	0.690503	0.345251	+	0.938510i	$0.387794\pi$
$284$	6.70415e93	0.236368
$285$	−8.93713e92	−0.0275214
$286$	1.55614e94	0.418751
$287$	−1.45111e95	−3.41387
$288$	2.06497e94	0.424915
$289$	4.20507e94	0.757193
$290$	1.81979e93	0.0286879
$291$	6.63673e94	0.916377
$292$	−2.81971e94	−0.341165
$293$	1.24342e95	1.31891	0.659455	−	0.751744i	$-0.270787\pi$
$293$	1.24342e95	1.31891	0.659455	+	0.751744i	$0.270787\pi$
$294$	−3.26647e95	−3.03883
$295$	−1.53609e92	−0.00125392
$296$	7.39104e94	0.529633
$297$	2.66578e94	0.167765
$298$	−4.90517e94	−0.271224
$299$	−1.34279e94	−0.0652628
$300$	−9.00467e94	−0.384857
$301$	1.79809e95	0.676084
$302$	2.64822e95	0.876370
$303$	4.63673e95	1.35105
$304$	1.57871e95	0.405199
$305$	8.93730e93	0.0202145
$306$	−4.24433e95	−0.846320
$307$	−4.07119e95	−0.715969	−0.357984	−	0.933728i	$-0.616536\pi$
$307$	−4.07119e95	−0.715969	−0.357984	+	0.933728i	$0.616536\pi$
$308$	−1.88861e95	−0.293049
$309$	−9.49120e94	−0.129993
$310$	−2.19138e94	−0.0265029
$311$	−1.17503e96	−1.25537	−0.627687	−	0.778466i	$-0.715998\pi$
$311$	−1.17503e96	−1.25537	−0.627687	+	0.778466i	$0.715998\pi$
$312$	1.45153e96	1.37048
$313$	−5.18358e95	−0.432686	−0.216343	−	0.976317i	$-0.569413\pi$
$313$	−5.18358e95	−0.432686	−0.216343	+	0.976317i	$0.569413\pi$
$314$	−1.65169e96	−1.21937
$315$	−7.16707e94	−0.0468150
$316$	−2.51438e95	−0.145371
$317$	2.78357e96	1.42502	0.712510	−	0.701662i	$-0.247559\pi$
$317$	2.78357e96	1.42502	0.712510	+	0.701662i	$0.247559\pi$
$318$	1.60896e96	0.729629
$319$	−1.38410e96	−0.556195
$320$	−9.86134e94	−0.0351287
$321$	3.61093e96	1.14071
$322$	−3.97906e95	−0.111514
$323$	3.45529e96	0.859385
$324$	1.55727e96	0.343860
$325$	−4.84387e96	−0.949910
$326$	−6.19739e96	−1.07977
$327$	−5.72181e96	−0.886023
$328$	1.39731e97	1.92375
$329$	−2.08571e97	−2.55395
$330$	1.71596e95	0.0186948
$331$	1.48132e97	1.43639	0.718193	−	0.695844i	$-0.244970\pi$
$331$	1.48132e97	1.43639	0.718193	+	0.695844i	$0.244970\pi$
$332$	−6.58479e95	−0.0568494
$333$	−4.80296e96	−0.369323
$334$	−7.80217e96	−0.534535
$335$	8.58179e95	0.0524026
$336$	2.93627e97	1.59858
$337$	−7.94818e96	−0.385937	−0.192969	−	0.981205i	$-0.561812\pi$
$337$	−7.94818e96	−0.385937	−0.192969	+	0.981205i	$0.561812\pi$
$338$	−1.86292e96	−0.0807053
$339$	−3.10608e97	−1.20095
$340$	−3.57247e95	−0.0123319
$341$	1.66673e97	0.513832
$342$	−1.50282e97	−0.413907
$343$	1.34882e98	3.31995
$344$	−1.73142e97	−0.380981
$345$	−1.48070e95	−0.00291361
$346$	−1.33729e97	−0.235395
$347$	−7.41782e97	−1.16840	−0.584200	−	0.811610i	$-0.698592\pi$
$347$	−7.41782e97	−1.16840	−0.584200	+	0.811610i	$0.698592\pi$
$348$	−2.90672e97	−0.409828
$349$	−5.44739e96	−0.0687717	−0.0343858	−	0.999409i	$-0.510948\pi$
$349$	−5.44739e96	−0.0687717	−0.0343858	+	0.999409i	$0.510948\pi$
$350$	−1.43538e98	−1.62310
$351$	3.01150e97	0.305112
$352$	3.22774e97	0.293093
$353$	−1.22962e98	−1.00103	−0.500514	−	0.865729i	$-0.666856\pi$
$353$	−1.22962e98	−1.00103	−0.500514	+	0.865729i	$0.666856\pi$
$354$	−5.99068e96	−0.0437372
$355$	3.97675e96	0.0260457
$356$	3.40073e97	0.199870
$357$	6.42656e98	3.39041
$358$	8.62995e97	0.408801
$359$	−2.91491e98	−1.24019	−0.620095	−	0.784526i	$-0.712906\pi$
$359$	−2.91491e98	−1.24019	−0.620095	+	0.784526i	$0.712906\pi$
$360$	6.90134e96	0.0263808
$361$	−1.68746e98	−0.579704
$362$	−8.74718e97	−0.270139
$363$	3.46922e98	0.963444
$364$	−2.13354e98	−0.532964
$365$	−1.67258e97	−0.0375935
$366$	3.48551e98	0.705091
$367$	−2.26762e98	−0.412978	−0.206489	−	0.978449i	$-0.566204\pi$
$367$	−2.26762e98	−0.412978	−0.206489	+	0.978449i	$0.566204\pi$
$368$	2.61559e97	0.0428971
$369$	−9.08021e98	−1.34147
$370$	9.87069e96	0.0131396
$371$	−1.05042e99	−1.26029
$372$	3.50025e98	0.378613
$373$	1.18489e99	1.15581	0.577905	−	0.816104i	$-0.303871\pi$
$373$	1.18489e99	1.15581	0.577905	+	0.816104i	$0.303871\pi$
$374$	−6.63429e98	−0.583765
$375$	−1.06882e98	−0.0848595
$376$	2.00838e99	1.43918
$377$	−1.56361e99	−1.01155
$378$	8.92394e98	0.521342
$379$	−7.03274e97	−0.0371120	−0.0185560	−	0.999828i	$-0.505907\pi$
$379$	−7.03274e97	−0.0371120	−0.0185560	+	0.999828i	$0.505907\pi$
$380$	−1.26493e97	−0.00603112
$381$	2.70627e99	1.16617
$382$	1.11665e99	0.434989
$383$	−2.46939e99	−0.869836	−0.434918	−	0.900470i	$-0.643223\pi$
$383$	−2.46939e99	−0.869836	−0.434918	+	0.900470i	$0.643223\pi$
$384$	−1.51299e99	−0.482040
$385$	−1.12028e98	−0.0322915
$386$	−2.40714e98	−0.0627899
$387$	1.12514e99	0.265665
$388$	9.39339e98	0.200817
$389$	3.53826e99	0.685063	0.342531	−	0.939506i	$-0.388716\pi$
$389$	3.53826e99	0.685063	0.342531	+	0.939506i	$0.388716\pi$
$390$	1.93850e98	0.0340000
$391$	5.72470e98	0.0909803
$392$	−2.05347e100	−2.95784
$393$	−7.46715e99	−0.975084
$394$	8.70318e99	1.03056
$395$	−1.49147e98	−0.0160187
$396$	−1.18178e99	−0.115152
$397$	5.06702e99	0.448041	0.224021	−	0.974584i	$-0.428082\pi$
$397$	5.06702e99	0.448041	0.224021	+	0.974584i	$0.428082\pi$
$398$	1.01454e99	0.0814273
$399$	2.27550e100	1.65814
$400$	9.43528e99	0.624374
$401$	−2.80080e100	−1.68354	−0.841771	−	0.539836i	$-0.818486\pi$
$401$	−2.80080e100	−1.68354	−0.841771	+	0.539836i	$0.818486\pi$
$402$	3.34687e100	1.82783
$403$	1.88288e100	0.934499
$404$	6.56266e99	0.296072
$405$	9.23736e98	0.0378904
$406$	−4.63341e100	−1.72842
$407$	−7.50747e99	−0.254747
$408$	−6.18830e100	−1.91053
$409$	4.11993e99	0.115756	0.0578779	−	0.998324i	$-0.481567\pi$
$409$	4.11993e99	0.115756	0.0578779	+	0.998324i	$0.481567\pi$
$410$	1.86610e99	0.0477261
$411$	2.60317e100	0.606166
$412$	−1.34335e99	−0.0284870
$413$	3.91106e99	0.0755471
$414$	−2.48986e99	−0.0438190
$415$	−3.90594e98	−0.00626432
$416$	3.64634e100	0.533044
$417$	−8.43205e99	−0.112382
$418$	−2.34905e100	−0.285500
$419$	7.71096e97	0.000854808 0	0.000427404	−	1.00000i	$-0.499864\pi$
$419$	7.71096e97	0.000854808 0	0.000427404		1.00000i	$0.499864\pi$
$420$	−2.35267e99	−0.0237937
$421$	3.26838e100	0.301628	0.150814	−	0.988562i	$-0.451811\pi$
$421$	3.26838e100	0.301628	0.150814	+	0.988562i	$0.451811\pi$
$422$	5.63544e100	0.474677
$423$	−1.30512e101	−1.00356
$424$	1.01148e101	0.710184
$425$	2.06509e101	1.32423
$426$	1.55092e101	0.908488
$427$	−2.27554e101	−1.21790
$428$	5.11078e100	0.249978
$429$	−1.47439e101	−0.659185
$430$	−2.31230e99	−0.00945168
$431$	−1.81193e101	−0.677278	−0.338639	−	0.940916i	$-0.609966\pi$
$431$	−1.81193e101	−0.677278	−0.338639	+	0.940916i	$0.609966\pi$
$432$	−5.86605e100	−0.200549
$433$	−5.04854e101	−1.57900	−0.789499	−	0.613752i	$-0.789659\pi$
$433$	−5.04854e101	−1.57900	−0.789499	+	0.613752i	$0.789659\pi$
$434$	5.57952e101	1.59677
$435$	−1.72420e100	−0.0451596
$436$	−8.09844e100	−0.194165
$437$	2.02699e100	0.0444954
$438$	−6.52301e101	−1.31128
$439$	1.73808e101	0.320026	0.160013	−	0.987115i	$-0.448846\pi$
$439$	1.73808e101	0.320026	0.160013	+	0.987115i	$0.448846\pi$
$440$	1.07874e100	0.0181966
$441$	1.33442e102	2.06256
$442$	−7.49468e101	−1.06169
$443$	6.39517e101	0.830441	0.415221	−	0.909721i	$-0.363704\pi$
$443$	6.39517e101	0.830441	0.415221	+	0.909721i	$0.363704\pi$
$444$	−1.57662e101	−0.187708
$445$	2.01723e100	0.0220240
$446$	−5.16703e101	−0.517427
$447$	4.64749e101	0.426952
$448$	2.51082e102	2.11647
$449$	8.45392e101	0.653994	0.326997	−	0.945025i	$-0.393963\pi$
$449$	8.45392e101	0.653994	0.326997	+	0.945025i	$0.393963\pi$
$450$	−8.98176e101	−0.637793
$451$	−1.41932e102	−0.925302
$452$	−4.39624e101	−0.263179
$453$	−2.50911e102	−1.37955
$454$	−2.83851e102	−1.43364
$455$	−1.26557e101	−0.0587281
$456$	−2.19114e102	−0.934379
$457$	2.25251e102	0.882861	0.441430	−	0.897296i	$-0.354471\pi$
$457$	2.25251e102	0.882861	0.441430	+	0.897296i	$0.354471\pi$
$458$	3.92965e102	1.41590
$459$	−1.28389e102	−0.425344
$460$	−2.09572e99	−0.000638495 0
$461$	3.74843e102	1.05042	0.525210	−	0.850972i	$-0.323987\pi$
$461$	3.74843e102	1.05042	0.525210	+	0.850972i	$0.323987\pi$
$462$	−4.36905e102	−1.12634
$463$	3.03537e102	0.720019	0.360010	−	0.932949i	$-0.382773\pi$
$463$	3.03537e102	0.720019	0.360010	+	0.932949i	$0.382773\pi$
$464$	3.04572e102	0.664887
$465$	2.07626e101	0.0417200
$466$	6.69365e102	1.23824
$467$	8.84453e101	0.150652	0.0753260	−	0.997159i	$-0.476000\pi$
$467$	8.84453e101	0.150652	0.0753260	+	0.997159i	$0.476000\pi$
$468$	−1.33505e102	−0.209426
$469$	−2.18503e103	−3.15720
$470$	2.68218e101	0.0357042
$471$	1.56492e103	1.91950
$472$	−3.76606e101	−0.0425716
$473$	1.75870e102	0.183247
$474$	−5.81669e102	−0.558740
$475$	7.31200e102	0.647638
$476$	9.09593e102	0.742984
$477$	−6.57294e102	−0.495224
$478$	−4.53204e102	−0.315007
$479$	−2.84087e103	−1.82194	−0.910970	−	0.412474i	$-0.864665\pi$
$479$	−2.84087e103	−1.82194	−0.910970	+	0.412474i	$0.864665\pi$
$480$	4.02083e101	0.0237973
$481$	−8.48111e102	−0.463305
$482$	1.06767e102	0.0538424
$483$	3.77003e102	0.175542
$484$	4.91020e102	0.211132
$485$	5.57194e101	0.0221284
$486$	2.86585e103	1.05137
$487$	−3.06397e103	−1.03853	−0.519265	−	0.854613i	$-0.673794\pi$
$487$	−3.06397e103	−1.03853	−0.519265	+	0.854613i	$0.673794\pi$
$488$	2.19118e103	0.686300
$489$	5.87183e103	1.69974
$490$	−2.74240e102	−0.0733806
$491$	9.53453e102	0.235864	0.117932	−	0.993022i	$-0.462373\pi$
$491$	9.53453e102	0.235864	0.117932	+	0.993022i	$0.462373\pi$
$492$	−2.98068e103	−0.681802
$493$	6.66612e103	1.41016
$494$	−2.65370e103	−0.519235
$495$	−7.01006e101	−0.0126888
$496$	−3.66763e103	−0.614244
$497$	−1.01253e104	−1.56923
$498$	−1.52330e103	−0.218503
$499$	−1.66577e103	−0.221180	−0.110590	−	0.993866i	$-0.535274\pi$
$499$	−1.66577e103	−0.221180	−0.110590	+	0.993866i	$0.535274\pi$
$500$	−1.51277e102	−0.0185963
$501$	7.39231e103	0.841448
$502$	1.50219e104	1.58355
$503$	−1.70452e104	−1.66431	−0.832154	−	0.554544i	$-0.812893\pi$
$503$	−1.70452e104	−1.66431	−0.832154	+	0.554544i	$0.812893\pi$
$504$	−1.75717e104	−1.58941
$505$	3.89281e102	0.0326246
$506$	−3.89189e102	−0.0302250
$507$	1.76506e103	0.127044
$508$	3.83036e103	0.255556
$509$	2.02452e104	1.25224	0.626121	−	0.779726i	$-0.284642\pi$
$509$	2.02452e104	1.25224	0.626121	+	0.779726i	$0.284642\pi$
$510$	−8.26442e102	−0.0473981
$511$	4.25860e104	2.26497
$512$	−2.08754e104	−1.02977
$513$	−4.54598e103	−0.208022
$514$	1.70031e104	0.721851
$515$	−7.96844e101	−0.00313903
$516$	3.69339e103	0.135024
$517$	−2.04002e104	−0.692226
$518$	−2.51320e104	−0.791645
$519$	1.26704e104	0.370551
$520$	1.21864e103	0.0330939
$521$	−2.71157e104	−0.683861	−0.341931	−	0.939725i	$-0.611081\pi$
$521$	−2.71157e104	−0.683861	−0.341931	+	0.939725i	$0.611081\pi$
$522$	−2.89932e104	−0.679176
$523$	−5.54459e104	−1.20658	−0.603288	−	0.797523i	$-0.706143\pi$
$523$	−5.54459e104	−1.20658	−0.603288	+	0.797523i	$0.706143\pi$
$524$	−1.05687e104	−0.213682
$525$	1.35997e105	2.55504
$526$	5.30827e104	0.926835
$527$	−8.02729e104	−1.30275
$528$	2.87194e104	0.433281
$529$	−7.09566e104	−0.995289
$530$	1.35082e103	0.0176188
$531$	2.44732e103	0.0296860
$532$	3.22066e104	0.363369
$533$	−1.60339e105	−1.68284
$534$	7.86714e104	0.768207
$535$	3.03159e103	0.0275455
$536$	2.10402e105	1.77912
$537$	−8.17661e104	−0.643521
$538$	−2.50512e104	−0.183532
$539$	2.08582e105	1.42269
$540$	4.70014e102	0.00298505
$541$	5.06493e104	0.299556	0.149778	−	0.988720i	$-0.452144\pi$
$541$	5.06493e104	0.299556	0.149778	+	0.988720i	$0.452144\pi$
$542$	8.88535e104	0.489443
$543$	8.28767e104	0.425245
$544$	−1.55454e105	−0.743096
$545$	−4.80380e103	−0.0213954
$546$	−4.93566e105	−2.04847
$547$	−2.62820e105	−1.01659	−0.508295	−	0.861183i	$-0.669724\pi$
$547$	−2.62820e105	−1.01659	−0.508295	+	0.861183i	$0.669724\pi$
$548$	3.68443e104	0.132837
$549$	−1.42390e105	−0.478570
$550$	−1.40393e105	−0.439929
$551$	2.36032e105	0.689660
$552$	−3.63026e104	−0.0989197
$553$	3.79747e105	0.965109
$554$	−1.61097e105	−0.381911
$555$	−9.35216e103	−0.0206839
$556$	−1.19344e104	−0.0246276
$557$	4.09238e105	0.788046	0.394023	−	0.919101i	$-0.371083\pi$
$557$	4.09238e105	0.788046	0.394023	+	0.919101i	$0.371083\pi$
$558$	3.49134e105	0.627445
$559$	1.98678e105	0.333269
$560$	2.46517e104	0.0386019
$561$	6.28578e105	0.918944
$562$	2.98112e103	0.00406941
$563$	−5.26421e105	−0.671058	−0.335529	−	0.942030i	$-0.608915\pi$
$563$	−5.26421e105	−0.671058	−0.335529	+	0.942030i	$0.608915\pi$
$564$	−4.28418e105	−0.510062
$565$	−2.60774e104	−0.0290001
$566$	5.59830e105	0.581599
$567$	−2.35194e106	−2.28286
$568$	9.74988e105	0.884277
$569$	1.74428e106	1.47841	0.739203	−	0.673482i	$-0.235202\pi$
$569$	1.74428e106	1.47841	0.739203	+	0.673482i	$0.235202\pi$
$570$	−2.92625e104	−0.0231808
$571$	−3.16647e105	−0.234469	−0.117234	−	0.993104i	$-0.537403\pi$
$571$	−3.16647e105	−0.234469	−0.117234	+	0.993104i	$0.537403\pi$
$572$	−2.08680e105	−0.144455
$573$	−1.05799e106	−0.684747
$574$	−4.75131e106	−2.87545
$575$	1.21145e105	0.0685634
$576$	1.57112e106	0.831658
$577$	1.82733e106	0.904790	0.452395	−	0.891818i	$-0.350570\pi$
$577$	1.82733e106	0.904790	0.452395	+	0.891818i	$0.350570\pi$
$578$	1.37685e106	0.637771
$579$	2.28069e105	0.0988419
$580$	−2.44037e104	−0.00989640
$581$	9.94500e105	0.377419
$582$	2.17304e106	0.771849
$583$	−1.02741e106	−0.341590
$584$	−4.10071e106	−1.27633
$585$	−7.91918e104	−0.0230770
$586$	4.07128e106	1.11090
$587$	1.56831e106	0.400742	0.200371	−	0.979720i	$-0.435785\pi$
$587$	1.56831e106	0.400742	0.200371	+	0.979720i	$0.435785\pi$
$588$	4.38037e106	1.04830
$589$	−2.84228e106	−0.637131
$590$	−5.02954e103	−0.00105615
$591$	−8.24598e106	−1.62228
$592$	1.65202e106	0.304529
$593$	3.39053e106	0.585683	0.292841	−	0.956161i	$-0.405399\pi$
$593$	3.39053e106	0.585683	0.292841	+	0.956161i	$0.405399\pi$
$594$	8.72845e105	0.141305
$595$	5.39549e105	0.0818706
$596$	6.57789e105	0.0935633
$597$	−9.61243e105	−0.128180
$598$	−4.39662e105	−0.0549697
$599$	5.53238e106	0.648604	0.324302	−	0.945954i	$-0.394871\pi$
$599$	5.53238e106	0.648604	0.324302	+	0.945954i	$0.394871\pi$
$600$	−1.30955e107	−1.43979
$601$	−1.69106e107	−1.74378	−0.871888	−	0.489705i	$-0.837105\pi$
$601$	−1.69106e107	−1.74378	−0.871888	+	0.489705i	$0.837105\pi$
$602$	5.88740e106	0.569454
$603$	−1.36726e107	−1.24061
$604$	−3.55130e106	−0.302319
$605$	2.91262e105	0.0232649
$606$	1.51818e107	1.13796
$607$	2.07566e107	1.46013	0.730064	−	0.683379i	$-0.239490\pi$
$607$	2.07566e107	1.46013	0.730064	+	0.683379i	$0.239490\pi$
$608$	−5.50429e106	−0.363424
$609$	4.39001e107	2.72082
$610$	2.92630e105	0.0170263
$611$	−2.30458e107	−1.25894
$612$	5.69171e106	0.291953
$613$	3.51234e107	1.69187	0.845937	−	0.533283i	$-0.179042\pi$
$613$	3.51234e107	1.69187	0.845937	+	0.533283i	$0.179042\pi$
$614$	−1.33301e107	−0.603048
$615$	−1.76807e106	−0.0751288
$616$	−2.74661e107	−1.09633
$617$	−3.03502e107	−1.13810	−0.569052	−	0.822301i	$-0.692690\pi$
$617$	−3.03502e107	−1.13810	−0.569052	+	0.822301i	$0.692690\pi$
$618$	−3.10766e106	−0.109491
$619$	8.67369e106	0.287153	0.143577	−	0.989639i	$-0.454140\pi$
$619$	8.67369e106	0.287153	0.143577	+	0.989639i	$0.454140\pi$
$620$	2.93867e105	0.00914262
$621$	−7.53174e105	−0.0220226
$622$	−3.84734e107	−1.05738
$623$	−5.13612e107	−1.32692
$624$	3.24440e107	0.788002
$625$	4.36560e107	0.996926
$626$	−1.69724e107	−0.364444
$627$	2.22565e107	0.449425
$628$	2.21493e107	0.420644
$629$	3.61575e107	0.645875
$630$	−2.34668e106	−0.0394315
$631$	−5.87884e107	−0.929311	−0.464656	−	0.885492i	$-0.653822\pi$
$631$	−5.87884e107	−0.929311	−0.464656	+	0.885492i	$0.653822\pi$
$632$	−3.65667e107	−0.543850
$633$	−5.33940e107	−0.747221
$634$	9.11412e107	1.20027
$635$	2.27208e106	0.0281602
$636$	−2.15764e107	−0.251698
$637$	2.35633e108	2.58742
$638$	−4.53191e107	−0.468474
$639$	−6.33581e107	−0.616622
$640$	−1.27024e106	−0.0116401
$641$	2.27779e107	0.196553	0.0982766	−	0.995159i	$-0.468667\pi$
$641$	2.27779e107	0.196553	0.0982766	+	0.995159i	$0.468667\pi$
$642$	1.18231e108	0.960800
$643$	−2.18232e108	−1.67031	−0.835154	−	0.550015i	$-0.814622\pi$
$643$	−2.18232e108	−1.67031	−0.835154	+	0.550015i	$0.814622\pi$
$644$	5.33597e106	0.0384687
$645$	2.19083e106	0.0148785
$646$	1.13135e108	0.723845
$647$	−2.75178e108	−1.65882	−0.829411	−	0.558639i	$-0.811324\pi$
$647$	−2.75178e108	−1.65882	−0.829411	+	0.558639i	$0.811324\pi$
$648$	2.26474e108	1.28642
$649$	3.82538e106	0.0204764
$650$	−1.58601e108	−0.800093
$651$	−5.28642e108	−2.51358
$652$	8.31078e107	0.372485
$653$	4.26985e108	1.80408	0.902038	−	0.431656i	$-0.142071\pi$
$653$	4.26985e108	1.80408	0.902038	+	0.431656i	$0.142071\pi$
$654$	−1.87347e108	−0.746281
$655$	−6.26913e106	−0.0235460
$656$	3.12321e108	1.10612
$657$	2.66478e108	0.890011
$658$	−6.82915e108	−2.15114
$659$	1.74595e108	0.518733	0.259366	−	0.965779i	$-0.416486\pi$
$659$	1.74595e108	0.518733	0.259366	+	0.965779i	$0.416486\pi$
$660$	−2.30113e106	−0.00644910
$661$	−2.31209e108	−0.611291	−0.305645	−	0.952145i	$-0.598872\pi$
$661$	−2.31209e108	−0.611291	−0.305645	+	0.952145i	$0.598872\pi$
$662$	4.85022e108	1.20984
$663$	7.10097e108	1.67127
$664$	−9.57628e107	−0.212680
$665$	1.91042e107	0.0400402
$666$	−1.57261e108	−0.311074
$667$	3.91057e107	0.0730121
$668$	1.04628e108	0.184397
$669$	4.89560e108	0.814517
$670$	2.80990e107	0.0441378
$671$	−2.22569e108	−0.330102
$672$	−1.02375e109	−1.43377
$673$	3.40355e107	0.0450145	0.0225073	−	0.999747i	$-0.492835\pi$
$673$	3.40355e107	0.0450145	0.0225073	+	0.999747i	$0.492835\pi$
$674$	−2.60244e108	−0.325068
$675$	−2.71694e108	−0.320542
$676$	2.49820e107	0.0278407
$677$	−1.04002e109	−1.09491	−0.547455	−	0.836835i	$-0.684403\pi$
$677$	−1.04002e109	−1.09491	−0.547455	+	0.836835i	$0.684403\pi$
$678$	−1.01701e109	−1.01154
$679$	−1.41868e109	−1.33321
$680$	−5.19545e107	−0.0461349
$681$	2.68939e109	2.25679
$682$	5.45729e108	0.432791
$683$	1.75962e109	1.31893	0.659464	−	0.751736i	$-0.270783\pi$
$683$	1.75962e109	1.31893	0.659464	+	0.751736i	$0.270783\pi$
$684$	2.01530e108	0.142784
$685$	2.18552e107	0.0146375
$686$	4.41640e109	2.79634
$687$	−3.72322e109	−2.22887
$688$	−3.87001e108	−0.219057
$689$	−1.16066e109	−0.621245
$690$	−4.84818e106	−0.00245408
$691$	7.52732e108	0.360359	0.180180	−	0.983634i	$-0.442332\pi$
$691$	7.52732e108	0.360359	0.180180	+	0.983634i	$0.442332\pi$
$692$	1.79333e108	0.0812035
$693$	1.78485e109	0.764489
$694$	−2.42879e109	−0.984123
$695$	−7.07922e106	−0.00271375
$696$	−4.22725e109	−1.53321
$697$	6.83574e109	2.34597
$698$	−1.78362e108	−0.0579252
$699$	−6.34202e109	−1.94920
$700$	1.92486e109	0.559918
$701$	1.08140e109	0.297744	0.148872	−	0.988857i	$-0.452436\pi$
$701$	1.08140e109	0.297744	0.148872	+	0.988857i	$0.452436\pi$
$702$	9.86043e108	0.256990
$703$	1.28026e109	0.315876
$704$	2.45581e109	0.573651
$705$	−2.54128e108	−0.0562045
$706$	−4.02610e109	−0.843148
$707$	−9.91157e109	−1.96560
$708$	8.03358e107	0.0150879
$709$	−5.35255e109	−0.952096	−0.476048	−	0.879419i	$-0.657931\pi$
$709$	−5.35255e109	−0.952096	−0.476048	+	0.879419i	$0.657931\pi$
$710$	1.30209e108	0.0219379
$711$	2.37624e109	0.379236
$712$	4.94570e109	0.747735
$713$	−4.70907e108	−0.0674510
$714$	2.10422e110	2.85569
$715$	−1.23784e108	−0.0159178
$716$	−1.15729e109	−0.141023
$717$	4.29397e109	0.495874
$718$	−9.54415e109	−1.04459
$719$	3.68655e109	0.382435	0.191218	−	0.981548i	$-0.438756\pi$
$719$	3.68655e109	0.382435	0.191218	+	0.981548i	$0.438756\pi$
$720$	1.54256e108	0.0151685
$721$	2.02886e109	0.189123
$722$	−5.52517e109	−0.488274
$723$	−1.01158e109	−0.0847569
$724$	1.17301e109	0.0931892
$725$	1.41067e110	1.06270
$726$	1.13591e110	0.811492
$727$	−1.94342e110	−1.31671	−0.658356	−	0.752707i	$-0.728748\pi$
$727$	−1.94342e110	−1.31671	−0.658356	+	0.752707i	$0.728748\pi$
$728$	−3.10282e110	−1.99387
$729$	−7.73720e109	−0.471600
$730$	−5.47646e108	−0.0316643
$731$	−8.47024e109	−0.464598
$732$	−4.67412e109	−0.243233
$733$	3.36660e110	1.66222	0.831110	−	0.556108i	$-0.187706\pi$
$733$	3.36660e110	1.66222	0.831110	+	0.556108i	$0.187706\pi$
$734$	−7.42477e109	−0.347844
$735$	2.59833e109	0.115513
$736$	−9.11945e108	−0.0384745
$737$	−2.13716e110	−0.855734
$738$	−2.97309e110	−1.12990
$739$	5.02422e110	1.81241	0.906207	−	0.422834i	$-0.138965\pi$
$739$	5.02422e110	1.81241	0.906207	+	0.422834i	$0.138965\pi$
$740$	−1.32367e108	−0.00453272
$741$	2.51429e110	0.817363
$742$	−3.43936e110	−1.06152
$743$	2.72652e110	0.798986	0.399493	−	0.916736i	$-0.369186\pi$
$743$	2.72652e110	0.798986	0.399493	+	0.916736i	$0.369186\pi$
$744$	5.09042e110	1.41643
$745$	3.90185e108	0.0103099
$746$	3.87962e110	0.973518
$747$	6.22301e109	0.148305
$748$	8.89666e109	0.201380
$749$	−7.71880e110	−1.65959
$750$	−3.49959e109	−0.0714757
$751$	1.97983e110	0.384141	0.192071	−	0.981381i	$-0.438480\pi$
$751$	1.97983e110	0.384141	0.192071	+	0.981381i	$0.438480\pi$
$752$	4.48906e110	0.827500
$753$	−1.42328e111	−2.49278
$754$	−5.11965e110	−0.852007
$755$	−2.10655e109	−0.0333130
$756$	−1.19671e110	−0.179846
$757$	8.36086e110	1.19415	0.597076	−	0.802185i	$-0.296329\pi$
$757$	8.36086e110	1.19415	0.597076	+	0.802185i	$0.296329\pi$
$758$	−2.30270e109	−0.0312588
$759$	3.68744e109	0.0475792
$760$	−1.83959e109	−0.0225631
$761$	−1.51180e111	−1.76273	−0.881366	−	0.472435i	$-0.843375\pi$
$761$	−1.51180e111	−1.76273	−0.881366	+	0.472435i	$0.843375\pi$
$762$	8.86103e110	0.982241
$763$	1.22311e111	1.28905
$764$	−1.49745e110	−0.150057
$765$	3.37619e109	0.0321707
$766$	−8.08543e110	−0.732648
$767$	4.32149e109	0.0372402
$768$	1.27959e111	1.04873
$769$	−9.96596e109	−0.0776882	−0.0388441	−	0.999245i	$-0.512368\pi$
$769$	−9.96596e109	−0.0776882	−0.0388441	+	0.999245i	$0.512368\pi$
$770$	−3.66808e109	−0.0271986
$771$	−1.61099e111	−1.13632
$772$	3.22800e109	0.0216605
$773$	6.62170e110	0.422726	0.211363	−	0.977408i	$-0.432210\pi$
$773$	6.62170e110	0.422726	0.211363	+	0.977408i	$0.432210\pi$
$774$	3.68400e110	0.223765
$775$	−1.69872e111	−0.981760
$776$	1.36608e111	0.751279
$777$	2.38117e111	1.24618
$778$	1.15852e111	0.577017
$779$	2.42038e111	1.14734
$780$	−2.59956e109	−0.0117289
$781$	−9.90346e110	−0.425326
$782$	1.87441e110	0.0766311
$783$	−8.77033e110	−0.341341
$784$	−4.58985e111	−1.70071
$785$	1.31385e110	0.0463514
$786$	−2.44494e111	−0.821296
$787$	−9.28283e110	−0.296930	−0.148465	−	0.988918i	$-0.547433\pi$
$787$	−9.28283e110	−0.296930	−0.148465	+	0.988918i	$0.547433\pi$
$788$	−1.16711e111	−0.355510
$789$	−5.02942e111	−1.45899
$790$	−4.88346e109	−0.0134923
$791$	6.63963e111	1.74723
$792$	−1.71867e111	−0.430798
$793$	−2.51434e111	−0.600352
$794$	1.65907e111	0.377377
$795$	−1.27986e110	−0.0277350
$796$	−1.36051e110	−0.0280898
$797$	6.02404e111	1.18507	0.592533	−	0.805546i	$-0.298128\pi$
$797$	6.02404e111	1.18507	0.592533	+	0.805546i	$0.298128\pi$
$798$	7.45058e111	1.39662
$799$	9.82514e111	1.75504
$800$	−3.28969e111	−0.560002
$801$	−3.21389e111	−0.521409
$802$	−9.17054e111	−1.41802
$803$	4.16530e111	0.613901
$804$	−4.48819e111	−0.630541
$805$	3.16517e109	0.00423892
$806$	6.16504e111	0.787112
$807$	2.37352e111	0.288910
$808$	9.54409e111	1.10764
$809$	−1.45095e112	−1.60559	−0.802794	−	0.596257i	$-0.796654\pi$
$809$	−1.45095e112	−1.60559	−0.802794	+	0.596257i	$0.796654\pi$
$810$	3.02455e110	0.0319145
$811$	−6.97257e111	−0.701601	−0.350800	−	0.936450i	$-0.614090\pi$
$811$	−6.97257e111	−0.701601	−0.350800	+	0.936450i	$0.614090\pi$
$812$	6.21347e111	0.596248
$813$	−8.41859e111	−0.770465
$814$	−2.45814e111	−0.214569
$815$	4.92976e110	0.0410447
$816$	−1.38318e112	−1.09852
$817$	−2.99912e111	−0.227219
$818$	1.34897e111	0.0974991
$819$	2.01632e112	1.39037
$820$	−2.50246e110	−0.0164639
$821$	−1.99056e112	−1.24958	−0.624791	−	0.780792i	$-0.714816\pi$
$821$	−1.99056e112	−1.24958	−0.624791	+	0.780792i	$0.714816\pi$
$822$	8.52343e111	0.510563
$823$	−1.12656e112	−0.643965	−0.321983	−	0.946746i	$-0.604349\pi$
$823$	−1.12656e112	−0.643965	−0.321983	+	0.946746i	$0.604349\pi$
$824$	−1.95364e111	−0.106573
$825$	1.33018e112	0.692522
$826$	1.28058e111	0.0636320
$827$	1.68258e112	0.798021	0.399011	−	0.916946i	$-0.369354\pi$
$827$	1.68258e112	0.798021	0.399011	+	0.916946i	$0.369354\pi$
$828$	3.33894e110	0.0151161
$829$	3.98139e112	1.72062	0.860310	−	0.509772i	$-0.170270\pi$
$829$	3.98139e112	1.72062	0.860310	+	0.509772i	$0.170270\pi$
$830$	−1.27891e110	−0.00527633
$831$	1.52634e112	0.601192
$832$	2.77430e112	1.04329
$833$	−1.00457e113	−3.60702
$834$	−2.76087e111	−0.0946570
$835$	6.20629e110	0.0203190
$836$	3.15011e111	0.0984881
$837$	1.05612e112	0.315342
$838$	2.52477e109	0.000719990 0
$839$	2.50194e112	0.681461	0.340730	−	0.940161i	$-0.389326\pi$
$839$	2.50194e112	0.681461	0.340730	+	0.940161i	$0.389326\pi$
$840$	−3.42149e111	−0.0890150
$841$	5.29770e111	0.131656
$842$	1.07015e112	0.254056
$843$	−2.82452e110	−0.00640594
$844$	−7.55719e111	−0.163748
$845$	1.48187e110	0.00306781
$846$	−4.27328e112	−0.845284
$847$	−7.41587e112	−1.40169
$848$	2.26082e112	0.408343
$849$	−5.30421e112	−0.915534
$850$	6.76162e112	1.11538
$851$	2.12112e111	0.0334408
$852$	−2.07980e112	−0.313398
$853$	4.01222e112	0.577893	0.288946	−	0.957345i	$-0.406695\pi$
$853$	4.01222e112	0.577893	0.288946	+	0.957345i	$0.406695\pi$
$854$	−7.45071e112	−1.02582
$855$	1.19543e111	0.0157336
$856$	7.43262e112	0.935195
$857$	−6.28976e112	−0.756610	−0.378305	−	0.925681i	$-0.623493\pi$
$857$	−6.28976e112	−0.756610	−0.378305	+	0.925681i	$0.623493\pi$
$858$	−4.82754e112	−0.555220
$859$	−1.77939e112	−0.195675	−0.0978375	−	0.995202i	$-0.531193\pi$
$859$	−1.77939e112	−0.195675	−0.0978375	+	0.995202i	$0.531193\pi$
$860$	3.10083e110	0.00326052
$861$	4.50171e113	4.52644
$862$	−5.93273e112	−0.570459
$863$	1.76886e113	1.62659	0.813295	−	0.581852i	$-0.197672\pi$
$863$	1.76886e113	1.62659	0.813295	+	0.581852i	$0.197672\pi$
$864$	2.04524e112	0.179873
$865$	1.06376e111	0.00894794
$866$	−1.65302e113	−1.32996
$867$	−1.30452e113	−1.00396
$868$	−7.48221e112	−0.550833
$869$	3.71428e112	0.261585
$870$	−5.64546e111	−0.0380372
$871$	−2.41432e113	−1.55631
$872$	−1.17776e113	−0.726393
$873$	−8.87730e112	−0.523881
$874$	6.63687e111	0.0374777
$875$	2.28473e112	0.123460
$876$	8.74744e112	0.452348
$877$	1.80338e113	0.892491	0.446245	−	0.894911i	$-0.352761\pi$
$877$	1.80338e113	0.892491	0.446245	+	0.894911i	$0.352761\pi$
$878$	5.69091e112	0.269552
$879$	−3.85741e113	−1.74874
$880$	2.41117e111	0.0104627
$881$	−1.57058e113	−0.652358	−0.326179	−	0.945308i	$-0.605761\pi$
$881$	−1.57058e113	−0.652358	−0.326179	+	0.945308i	$0.605761\pi$
$882$	4.36923e113	1.73726
$883$	7.68944e112	0.292690	0.146345	−	0.989234i	$-0.453249\pi$
$883$	7.68944e112	0.292690	0.146345	+	0.989234i	$0.453249\pi$
$884$	1.00505e113	0.366247
$885$	4.76533e110	0.00166256
$886$	2.09394e113	0.699466
$887$	4.53412e113	1.45022	0.725110	−	0.688633i	$-0.241789\pi$
$887$	4.53412e113	1.45022	0.725110	+	0.688633i	$0.241789\pi$
$888$	−2.29289e113	−0.702237
$889$	−5.78499e113	−1.69662
$890$	6.60494e111	0.0185504
$891$	−2.30042e113	−0.618751
$892$	6.92905e112	0.178495
$893$	3.47886e113	0.858333
$894$	1.52171e113	0.359614
$895$	−6.86476e111	−0.0155395
$896$	3.23419e113	0.701306
$897$	4.16566e112	0.0865316
$898$	2.76803e113	0.550848
$899$	−5.48348e113	−1.04546
$900$	1.20446e113	0.220018
$901$	4.94822e113	0.866054
$902$	−4.64722e113	−0.779366
$903$	−5.57813e113	−0.896417
$904$	−6.39346e113	−0.984581
$905$	6.95800e111	0.0102687
$906$	−8.21546e113	−1.16197
$907$	5.84563e113	0.792411	0.396206	−	0.918162i	$-0.370327\pi$
$907$	5.84563e113	0.792411	0.396206	+	0.918162i	$0.370327\pi$
$908$	3.80647e113	0.494558
$909$	−6.20209e113	−0.772375
$910$	−4.14379e112	−0.0494657
$911$	7.27518e113	0.832504	0.416252	−	0.909249i	$-0.363343\pi$
$911$	7.27518e113	0.832504	0.416252	+	0.909249i	$0.363343\pi$
$912$	−4.89755e113	−0.537251
$913$	9.72714e112	0.102296
$914$	7.37528e113	0.743618
$915$	−2.77258e112	−0.0268023
$916$	−5.26971e113	−0.488440
$917$	1.59620e114	1.41862
$918$	−4.20380e113	−0.358260
$919$	−1.53423e114	−1.25384	−0.626920	−	0.779083i	$-0.715685\pi$
$919$	−1.53423e114	−1.25384	−0.626920	+	0.779083i	$0.715685\pi$
$920$	−3.04782e111	−0.00238868
$921$	1.26299e114	0.949299
$922$	1.22733e114	0.884751
$923$	−1.11878e114	−0.773535
$924$	5.85895e113	0.388552
$925$	7.65156e113	0.486736
$926$	9.93859e113	0.606460
$927$	1.26954e113	0.0743152
$928$	−1.06191e114	−0.596338
$929$	1.80192e114	0.970803	0.485401	−	0.874291i	$-0.338674\pi$
$929$	1.80192e114	0.970803	0.485401	+	0.874291i	$0.338674\pi$
$930$	6.79822e112	0.0351400
$931$	−3.55697e114	−1.76408
$932$	−8.97627e113	−0.427153
$933$	3.64523e114	1.66449
$934$	2.89593e113	0.126892
$935$	5.27729e112	0.0221903
$936$	−1.94156e114	−0.783486
$937$	−2.54013e114	−0.983741	−0.491871	−	0.870668i	$-0.663687\pi$
$937$	−2.54013e114	−0.983741	−0.491871	+	0.870668i	$0.663687\pi$
$938$	−7.15434e114	−2.65926
$939$	1.60808e114	0.573696
$940$	−3.59683e112	−0.0123168
$941$	−3.99277e114	−1.31242	−0.656211	−	0.754577i	$-0.727842\pi$
$941$	−3.99277e114	−1.31242	−0.656211	+	0.754577i	$0.727842\pi$
$942$	5.12396e114	1.61676
$943$	4.01006e113	0.121465
$944$	−8.41775e112	−0.0244779
$945$	−7.09861e112	−0.0198175
$946$	5.75843e113	0.154346
$947$	5.77836e114	1.48707	0.743533	−	0.668699i	$-0.233149\pi$
$947$	5.77836e114	1.48707	0.743533	+	0.668699i	$0.233149\pi$
$948$	7.80025e113	0.192747
$949$	4.70550e114	1.11649
$950$	2.39414e114	0.545494
$951$	−8.63534e114	−1.88942
$952$	1.32282e115	2.77958
$953$	−2.87043e114	−0.579256	−0.289628	−	0.957139i	$-0.593532\pi$
$953$	−2.87043e114	−0.579256	−0.289628	+	0.957139i	$0.593532\pi$
$954$	−2.15215e114	−0.417119
$955$	−8.88249e112	−0.0165350
$956$	6.07753e113	0.108667
$957$	4.29384e114	0.737456
$958$	−9.30173e114	−1.53459
$959$	−5.56458e114	−0.881894
$960$	3.05924e113	0.0465769
$961$	−2.33601e113	−0.0341684
$962$	−2.77693e114	−0.390234
$963$	−4.82998e114	−0.652129
$964$	−1.43175e113	−0.0185739
$965$	1.91478e112	0.00238680
$966$	1.23440e114	0.147856
$967$	1.48323e115	1.70722	0.853610	−	0.520913i	$-0.174408\pi$
$967$	1.48323e115	1.70722	0.853610	+	0.520913i	$0.174408\pi$
$968$	7.14093e114	0.789866
$969$	−1.07192e115	−1.13945
$970$	1.82440e113	0.0186383
$971$	−1.14318e115	−1.12247	−0.561234	−	0.827657i	$-0.689673\pi$
$971$	−1.14318e115	−1.12247	−0.561234	+	0.827657i	$0.689673\pi$
$972$	−3.84314e114	−0.362689
$973$	1.80245e114	0.163501
$974$	−1.00322e115	−0.874736
$975$	1.50269e115	1.25948
$976$	4.89764e114	0.394610
$977$	−2.13312e115	−1.65224	−0.826122	−	0.563492i	$-0.809458\pi$
$977$	−2.13312e115	−1.65224	−0.826122	+	0.563492i	$0.809458\pi$
$978$	1.92259e115	1.43166
$979$	−5.02360e114	−0.359651
$980$	3.67759e113	0.0253139
$981$	7.65349e114	0.506527
$982$	3.12185e114	0.198664
$983$	4.42075e114	0.270511	0.135256	−	0.990811i	$-0.456814\pi$
$983$	4.42075e114	0.270511	0.135256	+	0.990811i	$0.456814\pi$
$984$	−4.33481e115	−2.55070
$985$	−6.92300e113	−0.0391742
$986$	2.18266e115	1.18775
$987$	6.47040e115	3.38626
$988$	3.55864e114	0.179119
$989$	−4.96892e113	−0.0240550
$990$	−2.29527e113	−0.0106876
$991$	1.01787e115	0.455887	0.227943	−	0.973674i	$-0.426800\pi$
$991$	1.01787e115	0.455887	0.227943	+	0.973674i	$0.426800\pi$
$992$	1.27875e115	0.550917
$993$	−4.59543e115	−1.90450
$994$	−3.31528e115	−1.32173
$995$	−8.07022e112	−0.00309526
$996$	2.04277e114	0.0753763
$997$	−3.84476e115	−1.36492	−0.682458	−	0.730925i	$-0.739089\pi$
$997$	−3.84476e115	−1.36492	−0.682458	+	0.730925i	$0.739089\pi$
$998$	−5.45417e114	−0.186296
$999$	−4.75709e114	−0.156340

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1.78.a.a.1.4	✓	6
3.2	odd	2		9.78.a.a.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.78.a.a.1.4	✓	6	1.1	even	1	trivial
9.78.a.a.1.3		6	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:	\( N \)	\(=\)	\( 1 \)
Weight:	\( k \)	\(=\)	\( 78 \)
Character orbit:	\([\chi]\)	\(=\)	1.a (trivial)

\(f(q)\)	\(=\)	\(q+3.27426e11 q^{2} -3.10225e18 q^{3} -4.39082e22 q^{4} -2.60453e25 q^{5} -1.01576e30 q^{6} +6.63145e32 q^{7} -6.38558e34 q^{8} +4.14958e36 q^{9} +O(q^{10})\)	\(q+3.27426e11 q^{2} -3.10225e18 q^{3} -4.39082e22 q^{4} -2.60453e25 q^{5} -1.01576e30 q^{6} +6.63145e32 q^{7} -6.38558e34 q^{8} +4.14958e36 q^{9} -8.52790e36 q^{10} +6.48617e39 q^{11} +1.36214e41 q^{12} +7.32735e42 q^{13} +2.17131e44 q^{14} +8.07991e43 q^{15} -1.42728e46 q^{16} -3.12387e47 q^{17} +1.35868e48 q^{18} -1.10609e49 q^{19} +1.14360e48 q^{20} -2.05724e51 q^{21} +2.12374e51 q^{22} -1.83256e51 q^{23} +1.98097e53 q^{24} -6.61066e53 q^{25} +2.39916e54 q^{26} +4.10994e54 q^{27} -2.91175e55 q^{28} -2.13393e56 q^{29} +2.64557e55 q^{30} +2.56966e57 q^{31} +4.97633e57 q^{32} -2.01218e58 q^{33} -1.02284e59 q^{34} -1.72718e58 q^{35} -1.82200e59 q^{36} -1.15746e60 q^{37} -3.62163e60 q^{38} -2.27313e61 q^{39} +1.66314e60 q^{40} -2.18823e62 q^{41} -6.73594e62 q^{42} +2.71146e62 q^{43} -2.84796e62 q^{44} -1.08077e62 q^{45} -6.00029e62 q^{46} -3.14518e64 q^{47} +4.42779e64 q^{48} +3.21580e65 q^{49} -2.16450e65 q^{50} +9.69104e65 q^{51} -3.21731e65 q^{52} -1.58400e66 q^{53} +1.34570e66 q^{54} -1.68934e65 q^{55} -4.23457e67 q^{56} +3.43138e67 q^{57} -6.98703e67 q^{58} +5.89775e66 q^{59} -3.54774e66 q^{60} -3.43144e68 q^{61} +8.41373e68 q^{62} +2.75177e69 q^{63} +3.78623e69 q^{64} -1.90843e68 q^{65} -6.58838e69 q^{66} -3.29495e70 q^{67} +1.37164e70 q^{68} +5.68508e69 q^{69} -5.65523e69 q^{70} -1.52686e71 q^{71} -2.64975e71 q^{72} +6.42182e71 q^{73} -3.78981e71 q^{74} +2.05079e72 q^{75} +4.85665e71 q^{76} +4.30127e72 q^{77} -7.44281e72 q^{78} +5.72645e72 q^{79} +3.71740e71 q^{80} -3.54665e73 q^{81} -7.16481e73 q^{82} +1.49967e73 q^{83} +9.03298e73 q^{84} +8.13622e72 q^{85} +8.87801e73 q^{86} +6.61999e74 q^{87} -4.14180e74 q^{88} -7.74510e74 q^{89} -3.53872e73 q^{90} +4.85910e75 q^{91} +8.04646e73 q^{92} -7.97174e75 q^{93} -1.02981e76 q^{94} +2.88085e74 q^{95} -1.54378e76 q^{96} -2.13933e76 q^{97} +1.05293e77 q^{98} +2.69149e76 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 264721893120 q^{2} + 14\!\cdots\!80 q^{3}+ \cdots - 48\!\cdots\!42 q^{9}+O(q^{10}) \) 6 * q + 264721893120 * q^2 + 1441611375788089080 * q^3 + 414488718051562044137472 * q^4 - 262034655809276597802623100 * q^5 - 366806373876514422355130201088 * q^6 + 270857839709680313679329082207600 * q^7 - 21222715175771442181432808272035840 * q^8 - 4895011902698601362943838014820938642 * q^9	\( 6 q + 264721893120 q^{2} + 14\!\cdots\!80 q^{3}+ \cdots + 22\!\cdots\!76 q^{99}+O(q^{100}) \) 6 * q + 264721893120 * q^2 + 1441611375788089080 * q^3 + 414488718051562044137472 * q^4 - 262034655809276597802623100 * q^5 - 366806373876514422355130201088 * q^6 + 270857839709680313679329082207600 * q^7 - 21222715175771442181432808272035840 * q^8 - 4895011902698601362943838014820938642 * q^9 - 943981839595570481933588950406110963200 * q^10 - 18259523505888985403727472716999830396568 * q^11 + 464880745107061653664227300592441229475840 * q^12 + 2380194429688668511174277092390661204777460 * q^13 - 294635885851315312797761966406968268616083456 * q^14 - 205398878502504702020984224024402427189799600 * q^15 + 44678683808767880535489561508977888302859288576 * q^16 - 354360527898668112383938884095051354700497739540 * q^17 - 579271368199620858231236048886957435939670283520 * q^18 + 5609349588345922508653624528305080863470217945560 * q^19 - 203683258723050328883517820082887345024612662067200 * q^20 - 540436528239602082418470626392633746858753885327168 * q^21 + 6114758837406209570140637531269380807939078825354240 * q^22 - 47704848518150781083381010364499925248017731105789360 * q^23 + 54816594321451146263884987133438313725918160963502080 * q^24 + 1218179184124654235834607721614209488740550081397516250 * q^25 - 550866287661514215594856318864157268196554716662599168 * q^26 - 18043016639677384483149377528753631298041104958650825040 * q^27 + 27673374074418016612675778175139670616136146452210319360 * q^28 + 16280759001892520996902191080778000816917007177111344340 * q^29 - 1011764162094599483190536262352170796094588963959467571200 * q^30 + 1790157695657501252076176026171679301117057950393828071872 * q^31 + 2181738275733175796144863923114890278726253762313317253120 * q^32 - 43587817236406175039900898992317145479130853281029781929440 * q^33 - 194953896897775283488395124499645480734536214093757337125376 * q^34 - 154692900477132731635374066357063627778968619353419533615200 * q^35 - 3492826558014976862445701719551945606786728595490426565320704 * q^36 - 7407081044970614504852172741178628092123742757389002078546620 * q^37 - 23437112252897896880014527036750794366537612969976459081384960 * q^38 - 70639623729776089938517469701883961219469901500376562016990704 * q^39 - 405188969845671269189619479481322249890553020909404426600448000 * q^40 - 622334267497818741826318492502909006845617873220194725690266308 * q^41 - 1785314244915654620512096109946411089204707510656231457419632640 * q^42 - 2618736058252833489926178823787488683637632009899828264542685400 * q^43 - 10432822146524346418364354026489166880919966717673650220195414016 * q^44 - 4261664397684963408032474279116855329416026736411628094706748300 * q^45 + 15015668129695292865283752907788076308022505321300082329335498752 * q^46 + 10456714010948157967838849947567752123265493969429352826408225440 * q^47 + 195413572872589369177356962560292275025549393856976542715493744640 * q^48 + 471540371160970162947042524598603402842322365532951363545112427542 * q^49 + 1643157939326752820476079797938119435570111533434962615482069920000 * q^50 + 1437480019590559075427205620912951154862381268364307425321563068272 * q^51 + 2497724095217025746860726558921243156289473926034163020782517862400 * q^52 - 1351874258191312033682238211039504811418206289179811594973968454620 * q^53 - 5802743153283665491865673044971704841325735354018068579171943024640 * q^54 - 31869417904451543700720919226087498810908552969327767789062308093200 * q^55 - 108662999456853131644146391920304784213919375663746888603843313008640 * q^56 - 80192344792625061999550741265686291327620988057496835084561604104480 * q^57 - 198994309984504890955955717028136272430515707470032642085684388912640 * q^58 - 314181202100222481576760882531528728694115253385468703103388053744120 * q^59 + 31757517058237454003450287009369504103827834377957826095111285964800 * q^60 + 1205268428250020260156089514176723127199311484106453915524972341789332 * q^61 + 5847595327663334724128627418101253733235456067016746715116600210186240 * q^62 + 5797157259817187838172511312525532039665297350177795293712349527116720 * q^63 + 7434996765286658306744699729117001037072361082977441755818145446100992 * q^64 + 8871205916648888694302367444130318424721491973331706441245421034929400 * q^65 + 2619665127058305678398408811495791734563636436120923156834123240689664 * q^66 - 20237777137309439154263683707086560559031166786916812035021461385807240 * q^67 - 192282119728407192675118903436134256417865682110612543314367901015326720 * q^68 - 200313232582951000798678573547728345115884560145782233425701325968721344 * q^69 - 275296905202040079140538321149869121020079259159855285360874363784294400 * q^70 - 84566701931998894021025479230103599698793740396961643247253155987263248 * q^71 - 111057066523302390867411925011041674769499898042748718262699840917995520 * q^72 + 314847243747931153403355639258701372803638975422668152047464555954073340 * q^73 + 2440978652597431277933746515377916372937009497503279937672013012437072384 * q^74 + 5406848712313524713947990348797970278787563587939911155459007261158885000 * q^75 + 12583857788866874491493997574394644552563831648352611876045224548363960320 * q^76 + 3503704566884313110653172829033905566483467863121365108497302013892404800 * q^77 - 3205965675068953356081325009194962500817273160296149320647178104103475200 * q^78 - 4478934474840817947707335445643189482501134177625819638140993021754592160 * q^79 - 71161053770974020381924657869896203411328788664145358716969422377556377600 * q^80 - 62119692812757584111204435712960193411658364945985590227036022119327600874 * q^81 - 215870435463876846311226667757904584387222051304478591060278478661208783360 * q^82 - 148140974166416303075124558783351182994193920545724368826080939980520858280 * q^83 + 51356650106390838431654087814077196024066314243448648781044119034071875584 * q^84 + 491597676589966787491574950515749558236649703170109339586274361878505865800 * q^85 + 282408262347204426247008847369759267182197911237017110679724533137839752192 * q^86 + 1363181037080164755262169098102975505194220445144287377094744451132352801680 * q^87 + 2657681751590091680961785563694793344274050442909114832295757477808460267520 * q^88 + 244875270415762051982571278168423166555270869887140541520680077076967187420 * q^89 + 4978403011748231687089157306213454731057762280800383772471249777474918822400 * q^90 + 637560104295539679774601429928812188462565548465392346037583647743398360352 * q^91 - 20568828732565451743944414428223922568301505472542794858122155367508257341440 * q^92 - 14490478249669505799006967900921922007746327667753988409512015425992979344640 * q^93 - 17973701798427203443369711151367318601937605119380706693990709867752930471936 * q^94 - 32941519370397986858775692207480112776415936672941584218885058160279506566000 * q^95 - 18725276161952705842933385524361085962413728935875751155361188816688573841408 * q^96 - 12282501490438986599941765679523755451099259695347356745999502279014355079860 * q^97 + 34498422662482146891923464099753801985751726005296311870693615309501356240640 * q^98 + 225826616422430389876328344183536014760241704676820880296433754570070950840776 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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