LMFDB - Embedded newform 1.78.a.a.1.6

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			1.6
Root			$-3.42124e9$ 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)

$f(q)$	$=$	$q+7.00999e11 q^{2} +9.61452e17 q^{3} +3.40284e23 q^{4} -1.50068e27 q^{5} +6.73977e29 q^{6} -2.07773e32 q^{7} +1.32607e35 q^{8} -4.55001e36 q^{9} +O(q^{10})$	\(q+7.00999e11 q^{2} +9.61452e17 q^{3} +3.40284e23 q^{4} -1.50068e27 q^{5} +6.73977e29 q^{6} -2.07773e32 q^{7} +1.32607e35 q^{8} -4.55001e36 q^{9} -1.05197e39 q^{10} -5.71890e39 q^{11} +3.27166e41 q^{12} +4.10741e40 q^{13} -1.45648e44 q^{14} -1.44283e45 q^{15} +4.15348e46 q^{16} -3.37382e47 q^{17} -3.18955e48 q^{18} +1.24104e49 q^{19} -5.10656e50 q^{20} -1.99763e50 q^{21} -4.00894e51 q^{22} -2.06967e52 q^{23} +1.27495e53 q^{24} +1.59029e54 q^{25} +2.87929e52 q^{26} -9.63799e54 q^{27} -7.07017e55 q^{28} +8.42524e55 q^{29} -1.01142e57 q^{30} +4.46777e57 q^{31} +9.07693e57 q^{32} -5.49844e57 q^{33} -2.36504e59 q^{34} +3.11800e59 q^{35} -1.54829e60 q^{36} -7.21183e59 q^{37} +8.69970e60 q^{38} +3.94908e58 q^{39} -1.99000e62 q^{40} -7.96800e61 q^{41} -1.40034e62 q^{42} -2.46355e62 q^{43} -1.94605e63 q^{44} +6.82810e63 q^{45} -1.45083e64 q^{46} -7.65815e63 q^{47} +3.99337e64 q^{48} -7.50119e64 q^{49} +1.11479e66 q^{50} -3.24376e65 q^{51} +1.39768e64 q^{52} -1.05807e66 q^{53} -6.75622e66 q^{54} +8.58222e66 q^{55} -2.75520e67 q^{56} +1.19320e67 q^{57} +5.90609e67 q^{58} -2.11868e68 q^{59} -4.90972e68 q^{60} +1.33545e68 q^{61} +3.13190e69 q^{62} +9.45368e68 q^{63} +8.63510e67 q^{64} -6.16390e67 q^{65} -3.85440e69 q^{66} +1.91437e70 q^{67} -1.14805e71 q^{68} -1.98988e70 q^{69} +2.18571e71 q^{70} +7.67990e70 q^{71} -6.03361e71 q^{72} +1.11746e71 q^{73} -5.05549e71 q^{74} +1.52899e72 q^{75} +4.22307e72 q^{76} +1.18823e72 q^{77} +2.76830e70 q^{78} +1.26080e73 q^{79} -6.23304e73 q^{80} +1.56421e73 q^{81} -5.58556e73 q^{82} +2.46787e73 q^{83} -6.79763e73 q^{84} +5.06301e74 q^{85} -1.72694e74 q^{86} +8.10047e73 q^{87} -7.58363e74 q^{88} -5.29212e74 q^{89} +4.78649e75 q^{90} -8.53408e72 q^{91} -7.04273e75 q^{92} +4.29555e75 q^{93} -5.36836e75 q^{94} -1.86241e76 q^{95} +8.72703e75 q^{96} -1.16975e76 q^{97} -5.25833e76 q^{98} +2.60210e76 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	7.00999e11	1.80328	0.901639	−	0.432490i	$-0.142365\pi$
$2$	7.00999e11	1.80328	0.901639	+	0.432490i	$0.142365\pi$
$3$	9.61452e17	0.410922	0.205461	−	0.978665i	$-0.434131\pi$
$3$	9.61452e17	0.410922	0.205461	+	0.978665i	$0.434131\pi$
$4$	3.40284e23	2.25181
$5$	−1.50068e27	−1.84477	−0.922386	−	0.386270i	$-0.873763\pi$
$5$	−1.50068e27	−1.84477	−0.922386	+	0.386270i	$0.873763\pi$
$6$	6.73977e29	0.741006
$7$	−2.07773e32	−0.604385	−0.302193	−	0.953247i	$-0.597719\pi$
$7$	−2.07773e32	−0.604385	−0.302193	+	0.953247i	$0.597719\pi$
$8$	1.32607e35	2.25736
$9$	−4.55001e36	−0.831143
$10$	−1.05197e39	−3.32663
$11$	−5.71890e39	−0.460992	−0.230496	−	0.973073i	$-0.574035\pi$
$11$	−5.71890e39	−0.460992	−0.230496	+	0.973073i	$0.574035\pi$
$12$	3.27166e41	0.925318
$13$	4.10741e40	0.00533027	0.00266513	−	0.999996i	$-0.499152\pi$
$13$	4.10741e40	0.00533027	0.00266513	+	0.999996i	$0.499152\pi$
$14$	−1.45648e44	−1.08987
$15$	−1.44283e45	−0.758057
$16$	4.15348e46	1.81883
$17$	−3.37382e47	−1.43165	−0.715827	−	0.698278i	$-0.753950\pi$
$17$	−3.37382e47	−1.43165	−0.715827	+	0.698278i	$0.753950\pi$
$18$	−3.18955e48	−1.49878
$19$	1.24104e49	0.727400	0.363700	−	0.931516i	$-0.381513\pi$
$19$	1.24104e49	0.727400	0.363700	+	0.931516i	$0.381513\pi$
$20$	−5.10656e50	−4.15407
$21$	−1.99763e50	−0.248355
$22$	−4.00894e51	−0.831297
$23$	−2.06967e52	−0.775137	−0.387569	−	0.921841i	$-0.626685\pi$
$23$	−2.06967e52	−0.775137	−0.387569	+	0.921841i	$0.626685\pi$
$24$	1.27495e53	0.927598
$25$	1.59029e54	2.40318
$26$	2.87929e52	0.00961195
$27$	−9.63799e54	−0.752457
$28$	−7.07017e55	−1.36096
$29$	8.42524e55	0.420010	0.210005	−	0.977700i	$-0.432652\pi$
$29$	8.42524e55	0.420010	0.210005	+	0.977700i	$0.432652\pi$
$30$	−1.01142e57	−1.36699
$31$	4.46777e57	1.70870	0.854350	−	0.519698i	$-0.173955\pi$
$31$	4.46777e57	1.70870	0.854350	+	0.519698i	$0.173955\pi$
$32$	9.07693e57	1.02250
$33$	−5.49844e57	−0.189432
$34$	−2.36504e59	−2.58167
$35$	3.11800e59	1.11495
$36$	−1.54829e60	−1.87158
$37$	−7.21183e59	−0.303584	−0.151792	−	0.988412i	$-0.548504\pi$
$37$	−7.21183e59	−0.303584	−0.151792	+	0.988412i	$0.548504\pi$
$38$	8.69970e60	1.31170
$39$	3.94908e58	0.00219032
$40$	−1.99000e62	−4.16431
$41$	−7.96800e61	−0.644422	−0.322211	−	0.946668i	$-0.604426\pi$
$41$	−7.96800e61	−0.644422	−0.322211	+	0.946668i	$0.604426\pi$
$42$	−1.40034e62	−0.447853
$43$	−2.46355e62	−0.318438	−0.159219	−	0.987243i	$-0.550898\pi$
$43$	−2.46355e62	−0.318438	−0.159219	+	0.987243i	$0.550898\pi$
$44$	−1.94605e63	−1.03807
$45$	6.82810e63	1.53327
$46$	−1.45083e64	−1.39779
$47$	−7.65815e63	−0.322371	−0.161186	−	0.986924i	$-0.551532\pi$
$47$	−7.65815e63	−0.322371	−0.161186	+	0.986924i	$0.551532\pi$
$48$	3.99337e64	0.747399
$49$	−7.50119e64	−0.634718
$50$	1.11479e66	4.33360
$51$	−3.24376e65	−0.588298
$52$	1.39768e64	0.0120027
$53$	−1.05807e66	−0.436410	−0.218205	−	0.975903i	$-0.570020\pi$
$53$	−1.05807e66	−0.436410	−0.218205	+	0.975903i	$0.570020\pi$
$54$	−6.75622e66	−1.35689
$55$	8.58222e66	0.850426
$56$	−2.75520e67	−1.36431
$57$	1.19320e67	0.298904
$58$	5.90609e67	0.757394
$59$	−2.11868e68	−1.40690	−0.703449	−	0.710745i	$-0.748358\pi$
$59$	−2.11868e68	−1.40690	−0.703449	+	0.710745i	$0.748358\pi$
$60$	−4.90972e68	−1.70700
$61$	1.33545e68	0.245713	0.122857	−	0.992424i	$-0.460794\pi$
$61$	1.33545e68	0.245713	0.122857	+	0.992424i	$0.460794\pi$
$62$	3.13190e69	3.08126
$63$	9.45368e68	0.502331
$64$	8.63510e67	0.0250229
$65$	−6.16390e67	−0.00983312
$66$	−3.85440e69	−0.341598
$67$	1.91437e70	0.950923	0.475461	−	0.879737i	$-0.342281\pi$
$67$	1.91437e70	0.950923	0.475461	+	0.879737i	$0.342281\pi$
$68$	−1.14805e71	−3.22381
$69$	−1.98988e70	−0.318521
$70$	2.18571e71	2.01057
$71$	7.67990e70	0.409175	0.204587	−	0.978848i	$-0.434415\pi$
$71$	7.67990e70	0.409175	0.204587	+	0.978848i	$0.434415\pi$
$72$	−6.03361e71	−1.87619
$73$	1.11746e71	0.204316	0.102158	−	0.994768i	$-0.467425\pi$
$73$	1.11746e71	0.204316	0.102158	+	0.994768i	$0.467425\pi$
$74$	−5.05549e71	−0.547447
$75$	1.52899e72	0.987520
$76$	4.22307e72	1.63796
$77$	1.18823e72	0.278617
$78$	2.76830e70	0.00394976
$79$	1.26080e73	1.10154	0.550772	−	0.834656i	$-0.314333\pi$
$79$	1.26080e73	1.10154	0.550772	+	0.834656i	$0.314333\pi$
$80$	−6.23304e73	−3.35533
$81$	1.56421e73	0.521942
$82$	−5.58556e73	−1.16207
$83$	2.46787e73	0.321970	0.160985	−	0.986957i	$-0.448533\pi$
$83$	2.46787e73	0.321970	0.160985	+	0.986957i	$0.448533\pi$
$84$	−6.79763e73	−0.559248
$85$	5.06301e74	2.64107
$86$	−1.72694e74	−0.574232
$87$	8.10047e73	0.172591
$88$	−7.58363e74	−1.04062
$89$	−5.29212e74	−0.470018	−0.235009	−	0.971993i	$-0.575512\pi$
$89$	−5.29212e74	−0.470018	−0.235009	+	0.971993i	$0.575512\pi$
$90$	4.78649e75	2.76491
$91$	−8.53408e72	−0.00322154
$92$	−7.04273e75	−1.74546
$93$	4.29555e75	0.702142
$94$	−5.36836e75	−0.581325
$95$	−1.86241e76	−1.34189
$96$	8.72703e75	0.420169
$97$	−1.16975e76	−0.377903	−0.188952	−	0.981986i	$-0.560509\pi$
$97$	−1.16975e76	−0.377903	−0.188952	+	0.981986i	$0.560509\pi$
$98$	−5.25833e76	−1.14457
$99$	2.60210e76	0.383151
$100$	5.41150e77	5.41150
$101$	−1.86675e77	−1.27267	−0.636333	−	0.771415i	$-0.719549\pi$
$101$	−1.86675e77	−1.27267	−0.636333	+	0.771415i	$0.719549\pi$
$102$	−2.27387e77	−1.06086
$103$	−4.72504e77	−1.51416	−0.757081	−	0.653321i	$-0.773375\pi$
$103$	−4.72504e77	−1.51416	−0.757081	+	0.653321i	$0.773375\pi$
$104$	5.44669e75	0.0120323
$105$	2.99781e77	0.458159
$106$	−7.41707e77	−0.786967
$107$	2.11792e78	1.56543	0.782715	−	0.622380i	$-0.213834\pi$
$107$	2.11792e78	1.56543	0.782715	+	0.622380i	$0.213834\pi$
$108$	−3.27965e78	−1.69439
$109$	−1.51300e78	−0.548175	−0.274088	−	0.961705i	$-0.588376\pi$
$109$	−1.51300e78	−0.548175	−0.274088	+	0.961705i	$0.588376\pi$
$110$	6.01613e78	1.53355
$111$	−6.93383e77	−0.124749
$112$	−8.62980e78	−1.09928
$113$	3.41772e78	0.309184	0.154592	−	0.987978i	$-0.450594\pi$
$113$	3.41772e78	0.309184	0.154592	+	0.987978i	$0.450594\pi$
$114$	8.36434e78	0.539008
$115$	3.10590e79	1.42995
$116$	2.86697e79	0.945782
$117$	−1.86888e77	−0.00443021
$118$	−1.48519e80	−2.53703
$119$	7.00987e79	0.865270
$120$	−1.91329e80	−1.71121
$121$	−1.21194e80	−0.787486
$122$	9.36148e79	0.443089
$123$	−7.66085e79	−0.264807
$124$	1.52031e81	3.84767
$125$	−1.39345e81	−2.58855
$126$	6.62702e80	0.905842
$127$	1.21899e81	1.22901	0.614507	−	0.788911i	$-0.289355\pi$
$127$	1.21899e81	1.22901	0.614507	+	0.788911i	$0.289355\pi$
$128$	−1.31113e81	−0.977380
$129$	−2.36858e80	−0.130853
$130$	−4.32089e79	−0.0177318
$131$	−1.92243e81	−0.587361	−0.293681	−	0.955904i	$-0.594880\pi$
$131$	−1.92243e81	−0.587361	−0.293681	+	0.955904i	$0.594880\pi$
$132$	−1.87103e81	−0.426564
$133$	−2.57855e81	−0.439630
$134$	1.34197e82	1.71478
$135$	1.44635e82	1.38811
$136$	−4.47390e82	−3.23175
$137$	−1.55543e82	−0.847441	−0.423721	−	0.905793i	$-0.639276\pi$
$137$	−1.55543e82	−0.847441	−0.423721	+	0.905793i	$0.639276\pi$
$138$	−1.39491e82	−0.574381
$139$	4.50035e82	1.40338	0.701692	−	0.712481i	$-0.252428\pi$
$139$	4.50035e82	1.40338	0.701692	+	0.712481i	$0.252428\pi$
$140$	1.06100e83	2.51066
$141$	−7.36295e81	−0.132469
$142$	5.38360e82	0.737856
$143$	−2.34898e80	−0.00245721
$144$	−1.88984e83	−1.51171
$145$	−1.26436e83	−0.774822
$146$	7.83340e82	0.368439
$147$	−7.21203e82	−0.260820
$148$	−2.45407e83	−0.683614
$149$	−8.28787e83	−1.78144	−0.890720	−	0.454551i	$-0.849800\pi$
$149$	−8.28787e83	−1.78144	−0.890720	+	0.454551i	$0.849800\pi$
$150$	1.07182e84	1.78077
$151$	−1.13735e83	−0.146312	−0.0731562	−	0.997320i	$-0.523307\pi$
$151$	−1.13735e83	−0.146312	−0.0731562	+	0.997320i	$0.523307\pi$
$152$	1.64570e84	1.64200
$153$	1.53509e84	1.18991
$154$	8.32948e83	0.502424
$155$	−6.70469e84	−3.15216
$156$	1.34381e82	0.00493219
$157$	1.30643e84	0.374928	0.187464	−	0.982271i	$-0.439973\pi$
$157$	1.30643e84	0.374928	0.187464	+	0.982271i	$0.439973\pi$
$158$	8.83817e84	1.98639
$159$	−1.01729e84	−0.179330
$160$	−1.36215e85	−1.88628
$161$	4.30020e84	0.468481
$162$	1.09651e85	0.941206
$163$	−1.58574e85	−1.07401	−0.537005	−	0.843579i	$-0.680444\pi$
$163$	−1.58574e85	−1.07401	−0.537005	+	0.843579i	$0.680444\pi$
$164$	−2.71138e85	−1.45112
$165$	8.25140e84	0.349459
$166$	1.72997e85	0.580601
$167$	2.09070e85	0.556812	0.278406	−	0.960464i	$-0.410194\pi$
$167$	2.09070e85	0.556812	0.278406	+	0.960464i	$0.410194\pi$
$168$	−2.64899e85	−0.560627
$169$	−5.93780e85	−0.999972
$170$	3.54917e86	4.76259
$171$	−5.64676e85	−0.604573
$172$	−8.38304e85	−0.717062
$173$	−1.56213e86	−1.06891	−0.534456	−	0.845196i	$-0.679484\pi$
$173$	−1.56213e86	−1.06891	−0.534456	+	0.845196i	$0.679484\pi$
$174$	5.67842e85	0.311230
$175$	−3.30419e86	−1.45245
$176$	−2.37533e86	−0.838469
$177$	−2.03701e86	−0.578126
$178$	−3.70977e86	−0.847573
$179$	8.32421e86	1.53286	0.766428	−	0.642330i	$-0.222032\pi$
$179$	8.32421e86	1.53286	0.766428	+	0.642330i	$0.222032\pi$
$180$	2.32349e87	3.45263
$181$	−1.34805e87	−1.61838	−0.809191	−	0.587545i	$-0.800094\pi$
$181$	−1.34805e87	−1.61838	−0.809191	+	0.587545i	$0.800094\pi$
$182$	−5.98238e84	−0.00580932
$183$	1.28397e86	0.100969
$184$	−2.74451e87	−1.74976
$185$	1.08226e87	0.560044
$186$	3.01118e87	1.26616
$187$	1.92945e87	0.659981
$188$	−2.60594e87	−0.725919
$189$	2.00251e87	0.454774
$190$	−1.30554e88	−2.41979
$191$	7.79558e87	1.18049	0.590247	−	0.807223i	$-0.299030\pi$
$191$	7.79558e87	1.18049	0.590247	+	0.807223i	$0.299030\pi$
$192$	8.30224e85	0.0102825
$193$	4.27720e87	0.433714	0.216857	−	0.976203i	$-0.430419\pi$
$193$	4.27720e87	0.433714	0.216857	+	0.976203i	$0.430419\pi$
$194$	−8.19992e87	−0.681464
$195$	−5.92630e85	−0.00404065
$196$	−2.55253e88	−1.42926
$197$	−1.13472e87	−0.0522325	−0.0261162	−	0.999659i	$-0.508314\pi$
$197$	−1.13472e87	−0.0522325	−0.0261162	+	0.999659i	$0.508314\pi$
$198$	1.82407e88	0.690927
$199$	−3.21718e88	−1.00377	−0.501883	−	0.864936i	$-0.667359\pi$
$199$	−3.21718e88	−1.00377	−0.501883	+	0.864936i	$0.667359\pi$
$200$	2.10883e89	5.42484
$201$	1.84057e88	0.390755
$202$	−1.30859e89	−2.29497
$203$	−1.75054e88	−0.253848
$204$	−1.10380e89	−1.32473
$205$	1.19574e89	1.18881
$206$	−3.31225e89	−2.73045
$207$	9.41700e88	0.644250
$208$	1.70601e87	0.00969487
$209$	−7.09739e88	−0.335326
$210$	2.10146e89	0.826187
$211$	1.63247e89	0.534530	0.267265	−	0.963623i	$-0.413880\pi$
$211$	1.63247e89	0.534530	0.267265	+	0.963623i	$0.413880\pi$
$212$	−3.60045e89	−0.982711
$213$	7.38385e88	0.168139
$214$	1.48466e90	2.82291
$215$	3.69699e89	0.587446
$216$	−1.27806e90	−1.69856
$217$	−9.28281e89	−1.03271
$218$	−1.06061e90	−0.988512
$219$	1.07439e89	0.0839580
$220$	2.92039e90	1.91500
$221$	−1.38576e88	−0.00763110
$222$	−4.86061e89	−0.224958
$223$	1.81597e90	0.706923	0.353461	−	0.935449i	$-0.385005\pi$
$223$	1.81597e90	0.706923	0.353461	+	0.935449i	$0.385005\pi$
$224$	−1.88594e90	−0.617986
$225$	−7.23585e90	−1.99739
$226$	2.39582e90	0.557544
$227$	−4.35527e90	−0.855106	−0.427553	−	0.903990i	$-0.640624\pi$
$227$	−4.35527e90	−0.855106	−0.427553	+	0.903990i	$0.640624\pi$
$228$	4.06028e90	0.673076
$229$	−1.22938e91	−1.72195	−0.860976	−	0.508645i	$-0.830146\pi$
$229$	−1.22938e91	−1.72195	−0.860976	+	0.508645i	$0.830146\pi$
$230$	2.17723e91	2.57860
$231$	1.14243e90	0.114490
$232$	1.11724e91	0.948113
$233$	1.35618e91	0.975248	0.487624	−	0.873054i	$-0.337864\pi$
$233$	1.35618e91	0.975248	0.487624	+	0.873054i	$0.337864\pi$
$234$	−1.31008e89	−0.00798891
$235$	1.14924e91	0.594702
$236$	−7.20952e91	−3.16807
$237$	1.21219e91	0.452649
$238$	4.91391e91	1.56032
$239$	−5.38232e91	−1.45429	−0.727144	−	0.686485i	$-0.759153\pi$
$239$	−5.38232e91	−1.45429	−0.727144	+	0.686485i	$0.759153\pi$
$240$	−5.99277e91	−1.37878
$241$	2.19486e91	0.430279	0.215139	−	0.976583i	$-0.430979\pi$
$241$	2.19486e91	0.430279	0.215139	+	0.976583i	$0.430979\pi$
$242$	−8.49566e91	−1.42006
$243$	6.78014e91	0.966934
$244$	4.54432e91	0.553299
$245$	1.12569e92	1.17091
$246$	−5.37024e91	−0.477521
$247$	5.09747e89	0.00387723
$248$	5.92456e92	3.85715
$249$	2.37274e91	0.132305
$250$	−9.76808e92	−4.66787
$251$	−3.41401e92	−1.39903	−0.699515	−	0.714618i	$-0.746600\pi$
$251$	−3.41401e92	−1.39903	−0.699515	+	0.714618i	$0.746600\pi$
$252$	3.21693e92	1.13115
$253$	1.18362e92	0.357332
$254$	8.54511e92	2.21625
$255$	4.86784e92	1.08527
$256$	−9.32153e92	−1.78751
$257$	−7.13632e92	−1.17774	−0.588870	−	0.808228i	$-0.700427\pi$
$257$	−7.13632e92	−1.17774	−0.588870	+	0.808228i	$0.700427\pi$
$258$	−1.66037e92	−0.235965
$259$	1.49842e92	0.183482
$260$	−2.09748e91	−0.0221423
$261$	−3.83350e92	−0.349088
$262$	−1.34762e93	−1.05917
$263$	2.71215e93	1.84085	0.920425	−	0.390920i	$-0.127843\pi$
$263$	2.71215e93	1.84085	0.920425	+	0.390920i	$0.127843\pi$
$264$	−7.29130e92	−0.427616
$265$	1.58783e93	0.805076
$266$	−1.80756e93	−0.792774
$267$	−5.08812e92	−0.193141
$268$	6.51428e93	2.14130
$269$	−6.33988e92	−0.180559	−0.0902793	−	0.995916i	$-0.528776\pi$
$269$	−6.33988e92	−0.180559	−0.0902793	+	0.995916i	$0.528776\pi$
$270$	1.01389e94	2.50315
$271$	−4.24496e93	−0.908983	−0.454492	−	0.890751i	$-0.650179\pi$
$271$	−4.24496e93	−0.908983	−0.454492	+	0.890751i	$0.650179\pi$
$272$	−1.40131e94	−2.60394
$273$	−8.20510e90	−0.00132380
$274$	−1.09036e94	−1.52817
$275$	−9.09471e93	−1.10785
$276$	−6.77125e93	−0.717248
$277$	9.69194e93	0.893181	0.446591	−	0.894738i	$-0.352638\pi$
$277$	9.69194e93	0.893181	0.446591	+	0.894738i	$0.352638\pi$
$278$	3.15474e94	2.53069
$279$	−2.03284e94	−1.42017
$280$	4.13467e94	2.51685
$281$	−1.23325e94	−0.654421	−0.327210	−	0.944952i	$-0.606109\pi$
$281$	−1.23325e94	−0.654421	−0.327210	+	0.944952i	$0.606109\pi$
$282$	−5.16142e93	−0.238879
$283$	−9.83998e93	−0.397389	−0.198695	−	0.980061i	$-0.563670\pi$
$283$	−9.83998e93	−0.397389	−0.198695	+	0.980061i	$0.563670\pi$
$284$	2.61334e94	0.921383
$285$	−1.79061e94	−0.551410
$286$	−1.64664e92	−0.00443104
$287$	1.65553e94	0.389479
$288$	−4.13001e94	−0.849846
$289$	5.82913e94	1.04963
$290$	−8.86314e94	−1.39722
$291$	−1.12466e94	−0.155289
$292$	3.80254e94	0.460081
$293$	−8.03099e94	−0.851855	−0.425927	−	0.904757i	$-0.640052\pi$
$293$	−8.03099e94	−0.851855	−0.425927	+	0.904757i	$0.640052\pi$
$294$	−5.05563e94	−0.470330
$295$	3.17946e95	2.59541
$296$	−9.56336e94	−0.685299
$297$	5.51187e94	0.346877
$298$	−5.80979e95	−3.21243
$299$	−8.50096e92	−0.00413169
$300$	5.20290e95	2.22371
$301$	5.11858e94	0.192459
$302$	−7.97280e94	−0.263842
$303$	−1.79480e95	−0.522966
$304$	5.15465e95	1.32302
$305$	−2.00408e95	−0.453285
$306$	1.07610e96	2.14574
$307$	2.58511e95	0.454622	0.227311	−	0.973822i	$-0.427007\pi$
$307$	2.58511e95	0.454622	0.227311	+	0.973822i	$0.427007\pi$
$308$	4.04335e95	0.627392
$309$	−4.54290e95	−0.622202
$310$	−4.69998e96	−5.68422
$311$	9.78183e95	1.04507	0.522535	−	0.852618i	$-0.324987\pi$
$311$	9.78183e95	1.04507	0.522535	+	0.852618i	$0.324987\pi$
$312$	5.23673e93	0.00494434
$313$	−3.84011e95	−0.320544	−0.160272	−	0.987073i	$-0.551237\pi$
$313$	−3.84011e95	−0.320544	−0.160272	+	0.987073i	$0.551237\pi$
$314$	9.15803e95	0.676100
$315$	−1.41869e96	−0.926685
$316$	4.29028e96	2.48047
$317$	1.31244e96	0.671889	0.335945	−	0.941882i	$-0.390944\pi$
$317$	1.31244e96	0.671889	0.335945	+	0.941882i	$0.390944\pi$
$318$	−7.13116e95	−0.323382
$319$	−4.81831e95	−0.193621
$320$	−1.29585e95	−0.0461616
$321$	2.03628e96	0.643270
$322$	3.01444e96	0.844802
$323$	−4.18705e96	−1.04138
$324$	5.32276e96	1.17531
$325$	6.53198e94	0.0128096
$326$	−1.11160e97	−1.93674
$327$	−1.45468e96	−0.225257
$328$	−1.05661e97	−1.45469
$329$	1.59116e96	0.194837
$330$	5.78422e96	0.630171
$331$	8.09044e96	0.784503	0.392251	−	0.919858i	$-0.371696\pi$
$331$	8.09044e96	0.784503	0.392251	+	0.919858i	$0.371696\pi$
$332$	8.39776e96	0.725015
$333$	3.28139e96	0.252322
$334$	1.46558e97	1.00409
$335$	−2.87285e97	−1.75423
$336$	−8.29714e96	−0.451717
$337$	−2.59358e97	−1.25936	−0.629678	−	0.776856i	$-0.716813\pi$
$337$	−2.59358e97	−1.25936	−0.629678	+	0.776856i	$0.716813\pi$
$338$	−4.16239e97	−1.80323
$339$	3.28598e96	0.127050
$340$	1.72286e98	5.94719
$341$	−2.55507e97	−0.787698
$342$	−3.95837e97	−1.09021
$343$	4.01403e97	0.988000
$344$	−3.26682e97	−0.718829
$345$	2.98618e97	0.587598
$346$	−1.09505e98	−1.92755
$347$	8.86133e97	1.39577	0.697886	−	0.716209i	$-0.254124\pi$
$347$	8.86133e97	1.39577	0.697886	+	0.716209i	$0.254124\pi$
$348$	2.75646e97	0.388643
$349$	−1.77054e97	−0.223525	−0.111762	−	0.993735i	$-0.535650\pi$
$349$	−1.77054e97	−0.223525	−0.111762	+	0.993735i	$0.535650\pi$
$350$	−2.31624e98	−2.61917
$351$	−3.95872e95	−0.00401080
$352$	−5.19100e97	−0.471366
$353$	2.31183e98	1.88204	0.941020	−	0.338351i	$-0.109869\pi$
$353$	2.31183e98	1.88204	0.941020	+	0.338351i	$0.109869\pi$
$354$	−1.42794e98	−1.04252
$355$	−1.15251e98	−0.754834
$356$	−1.80082e98	−1.05839
$357$	6.73965e97	0.355559
$358$	5.83526e98	2.76416
$359$	1.40084e98	0.596008	0.298004	−	0.954565i	$-0.403679\pi$
$359$	1.40084e98	0.596008	0.298004	+	0.954565i	$0.403679\pi$
$360$	9.05451e98	3.46114
$361$	−1.37071e98	−0.470890
$362$	−9.44982e98	−2.91839
$363$	−1.16522e98	−0.323595
$364$	−2.90401e96	−0.00725428
$365$	−1.67695e98	−0.376917
$366$	9.00062e97	0.182075
$367$	−6.37378e98	−1.16079	−0.580395	−	0.814335i	$-0.697102\pi$
$367$	−6.37378e98	−1.16079	−0.580395	+	0.814335i	$0.697102\pi$
$368$	−8.59632e98	−1.40985
$369$	3.62545e98	0.535607
$370$	7.58666e98	1.00991
$371$	2.19839e98	0.263760
$372$	1.46171e99	1.58109
$373$	−1.61161e99	−1.57206	−0.786031	−	0.618187i	$-0.787867\pi$
$373$	−1.61161e99	−1.57206	−0.786031	+	0.618187i	$0.787867\pi$
$374$	1.35254e99	1.19013
$375$	−1.33974e99	−1.06369
$376$	−1.01552e99	−0.727708
$377$	3.46059e96	0.00223877
$378$	1.40376e99	0.820083
$379$	2.57828e99	1.36057	0.680285	−	0.732948i	$-0.261856\pi$
$379$	2.57828e99	1.36057	0.680285	+	0.732948i	$0.261856\pi$
$380$	−6.33746e99	−3.02167
$381$	1.17200e99	0.505029
$382$	5.46469e99	2.12876
$383$	1.88845e99	0.665200	0.332600	−	0.943068i	$-0.392074\pi$
$383$	1.88845e99	0.665200	0.332600	+	0.943068i	$0.392074\pi$
$384$	−1.26059e99	−0.401627
$385$	−1.78315e99	−0.513985
$386$	2.99831e99	0.782107
$387$	1.12092e99	0.264668
$388$	−3.98046e99	−0.850966
$389$	−5.21133e99	−1.00900	−0.504498	−	0.863413i	$-0.668322\pi$
$389$	−5.21133e99	−1.00900	−0.504498	+	0.863413i	$0.668322\pi$
$390$	−4.15433e97	−0.00728640
$391$	6.98267e99	1.10973
$392$	−9.94707e99	−1.43279
$393$	−1.84832e99	−0.241360
$394$	−7.95440e98	−0.0941896
$395$	−1.89205e100	−2.03210
$396$	8.85454e99	0.862782
$397$	1.31714e100	1.16466	0.582328	−	0.812954i	$-0.302142\pi$
$397$	1.31714e100	1.16466	0.582328	+	0.812954i	$0.302142\pi$
$398$	−2.25524e100	−1.81007
$399$	−2.47915e99	−0.180653
$400$	6.60525e100	4.37099
$401$	2.97570e100	1.78867	0.894335	−	0.447397i	$-0.147649\pi$
$401$	2.97570e100	1.78867	0.894335	+	0.447397i	$0.147649\pi$
$402$	1.29024e100	0.704640
$403$	1.83510e98	0.00910783
$404$	−6.35226e100	−2.86580
$405$	−2.34738e100	−0.962864
$406$	−1.22712e100	−0.457758
$407$	4.12437e99	0.139950
$408$	−4.30144e100	−1.32800
$409$	2.39595e100	0.673179	0.336589	−	0.941652i	$-0.390727\pi$
$409$	2.39595e100	0.673179	0.336589	+	0.941652i	$0.390727\pi$
$410$	8.38212e100	2.14376
$411$	−1.49547e100	−0.348232
$412$	−1.60785e101	−3.40960
$413$	4.40204e100	0.850309
$414$	6.60131e100	1.16176
$415$	−3.70348e100	−0.593961
$416$	3.72827e98	0.00545021
$417$	4.32687e100	0.576681
$418$	−4.97527e100	−0.604685
$419$	1.17757e101	1.30541	0.652705	−	0.757612i	$-0.273634\pi$
$419$	1.17757e101	1.30541	0.652705	+	0.757612i	$0.273634\pi$
$420$	1.02011e101	1.03169
$421$	−5.43826e100	−0.501880	−0.250940	−	0.968003i	$-0.580740\pi$
$421$	−5.43826e100	−0.501880	−0.250940	+	0.968003i	$0.580740\pi$
$422$	1.14436e101	0.963905
$423$	3.48447e100	0.267937
$424$	−1.40307e101	−0.985133
$425$	−5.36535e101	−3.44052
$426$	5.17607e100	0.303201
$427$	−2.77470e100	−0.148506
$428$	7.20693e101	3.52505
$429$	−2.25844e98	−0.00100972
$430$	2.59159e101	1.05933
$431$	−1.05349e100	−0.0393780	−0.0196890	−	0.999806i	$-0.506268\pi$
$431$	−1.05349e100	−0.0393780	−0.0196890	+	0.999806i	$0.506268\pi$
$432$	−4.00312e101	−1.36859
$433$	2.09962e101	0.656683	0.328342	−	0.944559i	$-0.393510\pi$
$433$	2.09962e101	0.656683	0.328342	+	0.944559i	$0.393510\pi$
$434$	−6.50724e101	−1.86227
$435$	−1.21562e101	−0.318391
$436$	−5.14850e101	−1.23439
$437$	−2.56854e101	−0.563834
$438$	7.53144e100	0.151400
$439$	−3.80683e101	−0.700938	−0.350469	−	0.936574i	$-0.613978\pi$
$439$	−3.80683e101	−0.700938	−0.350469	+	0.936574i	$0.613978\pi$
$440$	1.13806e102	1.91972
$441$	3.41305e101	0.527542
$442$	−9.71419e99	−0.0137610
$443$	9.14564e101	1.18760	0.593801	−	0.804612i	$-0.297627\pi$
$443$	9.14564e101	1.18760	0.593801	+	0.804612i	$0.297627\pi$
$444$	−2.35947e101	−0.280912
$445$	7.94177e101	0.867076
$446$	1.27299e102	1.27478
$447$	−7.96839e101	−0.732033
$448$	−1.79414e100	−0.0151235
$449$	1.67233e102	1.29371	0.646856	−	0.762612i	$-0.276083\pi$
$449$	1.67233e102	1.29371	0.646856	+	0.762612i	$0.276083\pi$
$450$	−5.07232e102	−3.60184
$451$	4.55681e101	0.297074
$452$	1.16300e102	0.696223
$453$	−1.09351e101	−0.0601230
$454$	−3.05304e102	−1.54199
$455$	1.28069e100	0.00594300
$456$	1.58227e102	0.674734
$457$	−1.61789e102	−0.634127	−0.317063	−	0.948404i	$-0.602697\pi$
$457$	−1.61789e102	−0.634127	−0.317063	+	0.948404i	$0.602697\pi$
$458$	−8.61796e102	−3.10516
$459$	3.25168e102	1.07726
$460$	1.05689e103	3.21997
$461$	3.17423e102	0.889513	0.444756	−	0.895652i	$-0.353290\pi$
$461$	3.17423e102	0.889513	0.444756	+	0.895652i	$0.353290\pi$
$462$	8.00840e101	0.206457
$463$	−3.62306e102	−0.859425	−0.429712	−	0.902966i	$-0.641385\pi$
$463$	−3.62306e102	−0.859425	−0.429712	+	0.902966i	$0.641385\pi$
$464$	3.49941e102	0.763928
$465$	−6.44624e102	−1.29529
$466$	9.50682e102	1.75864
$467$	−5.79239e102	−0.986638	−0.493319	−	0.869849i	$-0.664216\pi$
$467$	−5.79239e102	−0.986638	−0.493319	+	0.869849i	$0.664216\pi$
$468$	−6.35948e100	−0.00997600
$469$	−3.97753e102	−0.574724
$470$	8.05618e102	1.07241
$471$	1.25607e102	0.154066
$472$	−2.80951e103	−3.17587
$473$	1.40888e102	0.146798
$474$	8.49747e102	0.816251
$475$	1.97362e103	1.74807
$476$	2.38534e103	1.94842
$477$	4.81424e102	0.362719
$478$	−3.77300e103	−2.62248
$479$	−1.90795e103	−1.22363	−0.611814	−	0.791001i	$-0.709560\pi$
$479$	−1.90795e103	−1.22363	−0.611814	+	0.791001i	$0.709560\pi$
$480$	−1.30965e103	−0.775116
$481$	−2.96220e100	−0.00161819
$482$	1.53859e103	0.775912
$483$	4.13444e102	0.192509
$484$	−4.12402e103	−1.77327
$485$	1.75542e103	0.697145
$486$	4.75287e103	1.74365
$487$	−2.65108e103	−0.898582	−0.449291	−	0.893385i	$-0.648323\pi$
$487$	−2.65108e103	−0.898582	−0.449291	+	0.893385i	$0.648323\pi$
$488$	1.77089e103	0.554663
$489$	−1.52461e103	−0.441334
$490$	7.89106e103	2.11148
$491$	1.36377e103	0.337367	0.168684	−	0.985670i	$-0.446048\pi$
$491$	1.36377e103	0.337367	0.168684	+	0.985670i	$0.446048\pi$
$492$	−2.60686e103	−0.596295
$493$	−2.84252e103	−0.601309
$494$	3.57332e101	0.00699173
$495$	−3.90492e103	−0.706825
$496$	1.85568e104	3.10784
$497$	−1.59567e103	−0.247299
$498$	1.66329e103	0.238582
$499$	3.85441e103	0.511784	0.255892	−	0.966705i	$-0.417631\pi$
$499$	3.85441e103	0.511784	0.255892	+	0.966705i	$0.417631\pi$
$500$	−4.74169e104	−5.82892
$501$	2.01011e103	0.228806
$502$	−2.39322e104	−2.52284
$503$	5.63268e102	0.0549981	0.0274990	−	0.999622i	$-0.491246\pi$
$503$	5.63268e102	0.0549981	0.0274990	+	0.999622i	$0.491246\pi$
$504$	1.25362e104	1.13394
$505$	2.80140e104	2.34778
$506$	8.29716e103	0.644369
$507$	−5.70891e103	−0.410910
$508$	4.14802e104	2.76750
$509$	−1.43689e104	−0.888767	−0.444384	−	0.895837i	$-0.646577\pi$
$509$	−1.43689e104	−0.888767	−0.444384	+	0.895837i	$0.646577\pi$
$510$	3.41235e104	1.95705
$511$	−2.32178e103	−0.123486
$512$	−4.55305e104	−2.24600
$513$	−1.19612e104	−0.547337
$514$	−5.00256e104	−2.12379
$515$	7.09077e104	2.79328
$516$	−8.05990e103	−0.294656
$517$	4.37962e103	0.148611
$518$	1.05039e104	0.330869
$519$	−1.50191e104	−0.439240
$520$	−8.17374e102	−0.0221969
$521$	−6.52397e103	−0.164536	−0.0822678	−	0.996610i	$-0.526216\pi$
$521$	−6.52397e103	−0.164536	−0.0822678	+	0.996610i	$0.526216\pi$
$522$	−2.68728e104	−0.629503
$523$	2.90203e104	0.631520	0.315760	−	0.948839i	$-0.397740\pi$
$523$	2.90203e104	0.631520	0.315760	+	0.948839i	$0.397740\pi$
$524$	−6.54171e104	−1.32262
$525$	−3.17682e104	−0.596843
$526$	1.90122e105	3.31956
$527$	−1.50734e105	−2.44627
$528$	−2.28377e104	−0.344545
$529$	−2.84573e104	−0.399163
$530$	1.11306e105	1.45178
$531$	9.64002e104	1.16933
$532$	−8.77438e104	−0.989962
$533$	−3.27278e102	−0.00343494
$534$	−3.56677e104	−0.348286
$535$	−3.17832e105	−2.88786
$536$	2.53858e105	2.14657
$537$	8.00333e104	0.629884
$538$	−4.44425e104	−0.325597
$539$	4.28985e104	0.292600
$540$	4.92170e105	3.12576
$541$	1.85470e105	1.09693	0.548466	−	0.836173i	$-0.315212\pi$
$541$	1.85470e105	1.09693	0.548466	+	0.836173i	$0.315212\pi$
$542$	−2.97571e105	−1.63915
$543$	−1.29609e105	−0.665029
$544$	−3.06239e105	−1.46387
$545$	2.27053e105	1.01126
$546$	−5.75177e102	−0.00238718
$547$	1.19425e105	0.461937	0.230969	−	0.972961i	$-0.425811\pi$
$547$	1.19425e105	0.461937	0.230969	+	0.972961i	$0.425811\pi$
$548$	−5.29289e105	−1.90828
$549$	−6.07631e104	−0.204223
$550$	−6.37538e105	−1.99776
$551$	1.04561e105	0.305515
$552$	−2.63872e105	−0.719015
$553$	−2.61959e105	−0.665757
$554$	6.79404e105	1.61065
$555$	1.04055e105	0.230134
$556$	1.53140e106	3.16015
$557$	4.64256e105	0.893989	0.446995	−	0.894537i	$-0.352494\pi$
$557$	4.64256e105	0.893989	0.446995	+	0.894537i	$0.352494\pi$
$558$	−1.42502e106	−2.56097
$559$	−1.01188e103	−0.00169736
$560$	1.29506e106	2.02791
$561$	1.85507e105	0.271201
$562$	−8.64505e105	−1.18010
$563$	9.98047e105	1.27226	0.636132	−	0.771580i	$-0.280533\pi$
$563$	9.98047e105	1.27226	0.636132	+	0.771580i	$0.280533\pi$
$564$	−2.50549e105	−0.298296
$565$	−5.12890e105	−0.570373
$566$	−6.89781e105	−0.716603
$567$	−3.25001e105	−0.315454
$568$	1.01840e106	0.923654
$569$	−1.92340e106	−1.63022	−0.815112	−	0.579303i	$-0.803325\pi$
$569$	−1.92340e106	−1.63022	−0.815112	+	0.579303i	$0.803325\pi$
$570$	−1.25522e106	−0.994346
$571$	3.85546e105	0.285487	0.142744	−	0.989760i	$-0.454408\pi$
$571$	3.85546e105	0.285487	0.142744	+	0.989760i	$0.454408\pi$
$572$	−7.99321e103	−0.00553317
$573$	7.49507e105	0.485091
$574$	1.16053e106	0.702339
$575$	−3.29137e106	−1.86279
$576$	−3.92898e104	−0.0207977
$577$	3.48767e106	1.72690	0.863448	−	0.504439i	$-0.168301\pi$
$577$	3.48767e106	1.72690	0.863448	+	0.504439i	$0.168301\pi$
$578$	4.08622e106	1.89278
$579$	4.11233e105	0.178223
$580$	−4.30241e106	−1.74475
$581$	−5.12756e105	−0.194594
$582$	−7.88383e105	−0.280029
$583$	6.05100e105	0.201182
$584$	1.48183e106	0.461215
$585$	2.80458e104	0.00817273
$586$	−5.62971e106	−1.53613
$587$	−3.89434e106	−0.995101	−0.497551	−	0.867435i	$-0.665767\pi$
$587$	−3.89434e106	−0.995101	−0.497551	+	0.867435i	$0.665767\pi$
$588$	−2.45414e106	−0.587316
$589$	5.54470e106	1.24291
$590$	2.22880e107	4.68024
$591$	−1.09098e105	−0.0214635
$592$	−2.99542e106	−0.552169
$593$	4.93367e106	0.852245	0.426122	−	0.904665i	$-0.359879\pi$
$593$	4.93367e106	0.852245	0.426122	+	0.904665i	$0.359879\pi$
$594$	3.86381e106	0.625515
$595$	−1.05196e107	−1.59623
$596$	−2.82023e107	−4.01146
$597$	−3.09317e106	−0.412469
$598$	−5.95917e104	−0.00745058
$599$	−6.01308e106	−0.704960	−0.352480	−	0.935819i	$-0.614662\pi$
$599$	−6.01308e106	−0.704960	−0.352480	+	0.935819i	$0.614662\pi$
$600$	2.02754e107	2.22919
$601$	−5.26712e106	−0.543133	−0.271566	−	0.962420i	$-0.587542\pi$
$601$	−5.26712e106	−0.543133	−0.271566	+	0.962420i	$0.587542\pi$
$602$	3.58812e106	0.347058
$603$	−8.71039e106	−0.790353
$604$	−3.87021e106	−0.329467
$605$	1.81873e107	1.45273
$606$	−1.25815e107	−0.943053
$607$	9.90399e106	0.696700	0.348350	−	0.937365i	$-0.386742\pi$
$607$	9.90399e106	0.696700	0.348350	+	0.937365i	$0.386742\pi$
$608$	1.12649e107	0.743768
$609$	−1.68306e106	−0.104312
$610$	−1.40486e107	−0.817398
$611$	−3.14552e104	−0.00171833
$612$	5.22366e107	2.67945
$613$	−8.64866e106	−0.416601	−0.208301	−	0.978065i	$-0.566793\pi$
$613$	−8.64866e106	−0.416601	−0.208301	+	0.978065i	$0.566793\pi$
$614$	1.81216e107	0.819810
$615$	1.14965e107	0.488509
$616$	1.57567e107	0.628939
$617$	6.48537e106	0.243196	0.121598	−	0.992579i	$-0.461198\pi$
$617$	6.48537e106	0.243196	0.121598	+	0.992579i	$0.461198\pi$
$618$	−3.18457e107	−1.12200
$619$	−9.31205e106	−0.308287	−0.154144	−	0.988048i	$-0.549262\pi$
$619$	−9.31205e106	−0.308287	−0.154144	+	0.988048i	$0.549262\pi$
$620$	−2.28150e108	−7.09806
$621$	1.99474e107	0.583257
$622$	6.85705e107	1.88455
$623$	1.09956e107	0.284072
$624$	1.64024e105	0.00398383
$625$	1.03876e108	2.37210
$626$	−2.69191e107	−0.578029
$627$	−6.82380e106	−0.137793
$628$	4.44555e107	0.844267
$629$	2.43314e107	0.434628
$630$	−9.94503e107	−1.67107
$631$	−4.89967e107	−0.774527	−0.387264	−	0.921969i	$-0.626580\pi$
$631$	−4.89967e107	−0.774527	−0.387264	+	0.921969i	$0.626580\pi$
$632$	1.67190e108	2.48658
$633$	1.56955e107	0.219650
$634$	9.20019e107	1.21160
$635$	−1.82931e108	−2.26725
$636$	−3.46166e107	−0.403817
$637$	−3.08105e105	−0.00338322
$638$	−3.37763e107	−0.349153
$639$	−3.49436e107	−0.340083
$640$	1.96759e108	1.80304
$641$	3.07516e105	0.00265359	0.00132680	−	0.999999i	$-0.499578\pi$
$641$	3.07516e105	0.00265359	0.00132680	+	0.999999i	$0.499578\pi$
$642$	1.42743e108	1.15999
$643$	−2.52847e108	−1.93524	−0.967620	−	0.252412i	$-0.918776\pi$
$643$	−2.52847e108	−1.93524	−0.967620	+	0.252412i	$0.918776\pi$
$644$	1.46329e108	1.05493
$645$	3.55448e107	0.241394
$646$	−2.93512e108	−1.87790
$647$	1.75204e107	0.105616	0.0528080	−	0.998605i	$-0.483183\pi$
$647$	1.75204e107	0.105616	0.0528080	+	0.998605i	$0.483183\pi$
$648$	2.07425e108	1.17821
$649$	1.21165e108	0.648570
$650$	4.57891e106	0.0230993
$651$	−8.92498e107	−0.424365
$652$	−5.39600e108	−2.41846
$653$	6.10854e107	0.258095	0.129048	−	0.991638i	$-0.458808\pi$
$653$	6.10854e107	0.258095	0.129048	+	0.991638i	$0.458808\pi$
$654$	−1.01973e108	−0.406201
$655$	2.88495e108	1.08355
$656$	−3.30949e108	−1.17210
$657$	−5.08447e107	−0.169816
$658$	1.11540e108	0.351344
$659$	−2.35889e108	−0.700839	−0.350420	−	0.936593i	$-0.613961\pi$
$659$	−2.35889e108	−0.700839	−0.350420	+	0.936593i	$0.613961\pi$
$660$	2.80782e108	0.786914
$661$	2.17627e108	0.575383	0.287691	−	0.957723i	$-0.407112\pi$
$661$	2.17627e108	0.575383	0.287691	+	0.957723i	$0.407112\pi$
$662$	5.67139e108	1.41468
$663$	−1.33235e106	−0.00313578
$664$	3.27256e108	0.726802
$665$	3.86957e108	0.811016
$666$	2.30025e108	0.455007
$667$	−1.74374e108	−0.325565
$668$	7.11432e108	1.25383
$669$	1.74597e108	0.290490
$670$	−2.01386e109	−3.16337
$671$	−7.63729e107	−0.113272
$672$	−1.81324e108	−0.253944
$673$	1.15770e109	1.53114	0.765569	−	0.643354i	$-0.222458\pi$
$673$	1.15770e109	1.53114	0.765569	+	0.643354i	$0.222458\pi$
$674$	−1.81810e109	−2.27097
$675$	−1.53272e109	−1.80829
$676$	−2.02054e109	−2.25174
$677$	8.28409e108	0.872130	0.436065	−	0.899915i	$-0.356372\pi$
$677$	8.28409e108	0.872130	0.436065	+	0.899915i	$0.356372\pi$
$678$	2.30347e108	0.229107
$679$	2.43042e108	0.228399
$680$	6.71389e109	5.96185
$681$	−4.18739e108	−0.351382
$682$	−1.79110e109	−1.42044
$683$	−2.14317e109	−1.60642	−0.803211	−	0.595695i	$-0.796877\pi$
$683$	−2.14317e109	−1.60642	−0.803211	+	0.595695i	$0.796877\pi$
$684$	−1.92150e109	−1.36138
$685$	2.33421e109	1.56334
$686$	2.81383e109	1.78164
$687$	−1.18199e109	−0.707588
$688$	−1.02323e109	−0.579186
$689$	−4.34594e106	−0.00232618
$690$	2.09331e109	1.05960
$691$	2.98797e109	1.43045	0.715223	−	0.698897i	$-0.246325\pi$
$691$	2.98797e109	1.43045	0.715223	+	0.698897i	$0.246325\pi$
$692$	−5.31567e109	−2.40699
$693$	−5.40646e108	−0.231571
$694$	6.21178e109	2.51696
$695$	−6.75358e109	−2.58892
$696$	1.07418e109	0.389600
$697$	2.68826e109	0.922590
$698$	−1.24114e109	−0.403077
$699$	1.30390e109	0.400751
$700$	−1.12436e110	−3.27063
$701$	3.76374e109	1.03628	0.518138	−	0.855297i	$-0.326625\pi$
$701$	3.76374e109	1.03628	0.518138	+	0.855297i	$0.326625\pi$
$702$	−2.77506e107	−0.00723258
$703$	−8.95019e108	−0.220827
$704$	−4.93833e107	−0.0115354
$705$	1.10494e109	0.244376
$706$	1.62059e110	3.39384
$707$	3.87861e109	0.769181
$708$	−6.93161e109	−1.30183
$709$	−4.95782e109	−0.881883	−0.440941	−	0.897536i	$-0.645355\pi$
$709$	−4.95782e109	−0.881883	−0.440941	+	0.897536i	$0.645355\pi$
$710$	−8.07905e109	−1.36117
$711$	−5.73664e109	−0.915541
$712$	−7.01770e109	−1.06100
$713$	−9.24680e109	−1.32448
$714$	4.72449e109	0.641171
$715$	3.52507e107	0.00453300
$716$	2.83259e110	3.45170
$717$	−5.17484e109	−0.597599
$718$	9.81987e109	1.07477
$719$	−9.06361e108	−0.0940240	−0.0470120	−	0.998894i	$-0.514970\pi$
$719$	−9.06361e108	−0.0940240	−0.0470120	+	0.998894i	$0.514970\pi$
$720$	2.83604e110	2.78876
$721$	9.81735e109	0.915138
$722$	−9.60868e109	−0.849145
$723$	2.11025e109	0.176811
$724$	−4.58720e110	−3.64429
$725$	1.33986e110	1.00936
$726$	−8.16817e109	−0.583532
$727$	−1.41473e110	−0.958513	−0.479256	−	0.877675i	$-0.659094\pi$
$727$	−1.41473e110	−0.958513	−0.479256	+	0.877675i	$0.659094\pi$
$728$	−1.13167e108	−0.00727216
$729$	−2.04435e109	−0.124608
$730$	−1.17554e110	−0.679685
$731$	8.31155e109	0.455893
$732$	4.36914e109	0.227363
$733$	7.07465e109	0.349303	0.174651	−	0.984630i	$-0.444120\pi$
$733$	7.07465e109	0.349303	0.174651	+	0.984630i	$0.444120\pi$
$734$	−4.46801e110	−2.09323
$735$	1.08229e110	0.481153
$736$	−1.87862e110	−0.792580
$737$	−1.09481e110	−0.438368
$738$	2.54143e110	0.965848
$739$	4.30955e110	1.55461	0.777304	−	0.629125i	$-0.216587\pi$
$739$	4.30955e110	1.55461	0.777304	+	0.629125i	$0.216587\pi$
$740$	3.68277e110	1.26111
$741$	4.90097e107	0.00159324
$742$	1.54107e110	0.475632
$743$	−4.90436e110	−1.43719	−0.718593	−	0.695430i	$-0.755214\pi$
$743$	−4.90436e110	−1.43719	−0.718593	+	0.695430i	$0.755214\pi$
$744$	5.69618e110	1.58499
$745$	1.24374e111	3.28635
$746$	−1.12974e111	−2.83486
$747$	−1.12288e110	−0.267603
$748$	6.56560e110	1.48615
$749$	−4.40046e110	−0.946124
$750$	−9.39154e110	−1.91813
$751$	−3.57628e110	−0.693895	−0.346948	−	0.937884i	$-0.612782\pi$
$751$	−3.57628e110	−0.693895	−0.346948	+	0.937884i	$0.612782\pi$
$752$	−3.18080e110	−0.586340
$753$	−3.28241e110	−0.574892
$754$	2.42587e108	0.00403711
$755$	1.70679e110	0.269913
$756$	6.81422e110	1.02406
$757$	−1.11237e110	−0.158876	−0.0794379	−	0.996840i	$-0.525313\pi$
$757$	−1.11237e110	−0.158876	−0.0794379	+	0.996840i	$0.525313\pi$
$758$	1.80737e111	2.45348
$759$	1.13799e110	0.146836
$760$	−2.46967e111	−3.02912
$761$	−9.15339e110	−1.06727	−0.533633	−	0.845716i	$-0.679174\pi$
$761$	−9.15339e110	−1.06727	−0.533633	+	0.845716i	$0.679174\pi$
$762$	8.21571e110	0.910707
$763$	3.14361e110	0.331309
$764$	2.65271e111	2.65825
$765$	−2.30368e111	−2.19511
$766$	1.32380e111	1.19954
$767$	−8.70229e108	−0.00749915
$768$	−8.96220e110	−0.734527
$769$	−1.68751e111	−1.31548	−0.657738	−	0.753247i	$-0.728487\pi$
$769$	−1.68751e111	−1.31548	−0.657738	+	0.753247i	$0.728487\pi$
$770$	−1.24999e111	−0.926857
$771$	−6.86123e110	−0.483959
$772$	1.45546e111	0.976641
$773$	8.92222e110	0.569590	0.284795	−	0.958588i	$-0.408074\pi$
$773$	8.92222e110	0.569590	0.284795	+	0.958588i	$0.408074\pi$
$774$	7.85761e110	0.477269
$775$	7.10506e111	4.10632
$776$	−1.55116e111	−0.853063
$777$	1.44066e110	0.0753967
$778$	−3.65314e111	−1.81950
$779$	−9.88862e110	−0.468753
$780$	−2.01662e109	−0.00909876
$781$	−4.39205e110	−0.188627
$782$	4.89484e111	2.00115
$783$	−8.12024e110	−0.316039
$784$	−3.11561e111	−1.15445
$785$	−1.96053e111	−0.691657
$786$	−1.29567e111	−0.435238
$787$	1.78649e111	0.571444	0.285722	−	0.958313i	$-0.407767\pi$
$787$	1.78649e111	0.571444	0.285722	+	0.958313i	$0.407767\pi$
$788$	−3.86128e110	−0.117618
$789$	2.60761e111	0.756445
$790$	−1.32632e112	−3.66443
$791$	−7.10109e110	−0.186866
$792$	3.45056e111	0.864908
$793$	5.48524e108	0.00130972
$794$	9.23314e111	2.10020
$795$	1.52662e111	0.330823
$796$	−1.09476e112	−2.26029
$797$	−2.70232e111	−0.531608	−0.265804	−	0.964027i	$-0.585637\pi$
$797$	−2.70232e111	−0.531608	−0.265804	+	0.964027i	$0.585637\pi$
$798$	−1.73788e111	−0.325768
$799$	2.58372e111	0.461524
$800$	1.44350e112	2.45726
$801$	2.40792e111	0.390652
$802$	2.08596e112	3.22547
$803$	−6.39065e110	−0.0941883
$804$	6.26317e111	0.879905
$805$	−6.45322e111	−0.864241
$806$	1.28640e110	0.0164239
$807$	−6.09549e110	−0.0741955
$808$	−2.47544e112	−2.87286
$809$	−4.18836e111	−0.463474	−0.231737	−	0.972778i	$-0.574441\pi$
$809$	−4.18836e111	−0.463474	−0.231737	+	0.972778i	$0.574441\pi$
$810$	−1.64551e112	−1.73631
$811$	−1.34948e111	−0.135788	−0.0678942	−	0.997693i	$-0.521628\pi$
$811$	−1.34948e111	−0.135788	−0.0678942	+	0.997693i	$0.521628\pi$
$812$	−5.95679e111	−0.571617
$813$	−4.08133e111	−0.373521
$814$	2.89118e111	0.252369
$815$	2.37968e112	1.98130
$816$	−1.34729e112	−1.07002
$817$	−3.05737e111	−0.231632
$818$	1.67956e112	1.21393
$819$	3.88301e109	0.00267756
$820$	4.06891e112	2.67698
$821$	−2.16182e112	−1.35709	−0.678545	−	0.734559i	$-0.737389\pi$
$821$	−2.16182e112	−1.35709	−0.678545	+	0.734559i	$0.737389\pi$
$822$	−1.04833e112	−0.627959
$823$	2.34088e112	1.33809	0.669046	−	0.743221i	$-0.266703\pi$
$823$	2.34088e112	1.33809	0.669046	+	0.743221i	$0.266703\pi$
$824$	−6.26571e112	−3.41801
$825$	−8.74413e111	−0.455239
$826$	3.08582e112	1.53334
$827$	2.51771e111	0.119411	0.0597054	−	0.998216i	$-0.480984\pi$
$827$	2.51771e111	0.119411	0.0597054	+	0.998216i	$0.480984\pi$
$828$	3.20445e112	1.45073
$829$	−1.18267e110	−0.00511109	−0.00255554	−	0.999997i	$-0.500813\pi$
$829$	−1.18267e110	−0.00511109	−0.00255554	+	0.999997i	$0.500813\pi$
$830$	−2.59613e112	−1.07108
$831$	9.31833e111	0.367028
$832$	3.54679e108	0.000133379 0
$833$	2.53076e112	0.908697
$834$	3.03313e112	1.03992
$835$	−3.13747e112	−1.02719
$836$	−2.41513e112	−0.755089
$837$	−4.30604e112	−1.28572
$838$	8.25476e112	2.35402
$839$	−5.15958e112	−1.40533	−0.702666	−	0.711520i	$-0.748007\pi$
$839$	−5.15958e112	−1.40533	−0.702666	+	0.711520i	$0.748007\pi$
$840$	3.97529e112	1.03423
$841$	−3.31404e112	−0.823592
$842$	−3.81222e112	−0.905028
$843$	−1.18571e112	−0.268916
$844$	5.55504e112	1.20366
$845$	8.91073e112	1.84472
$846$	2.44261e112	0.483164
$847$	2.51807e112	0.475945
$848$	−4.39468e112	−0.793756
$849$	−9.46067e111	−0.163296
$850$	−3.76111e113	−6.20422
$851$	1.49261e112	0.235319
$852$	2.51260e112	0.378617
$853$	−4.77308e112	−0.687481	−0.343741	−	0.939065i	$-0.611694\pi$
$853$	−4.77308e112	−0.687481	−0.343741	+	0.939065i	$0.611694\pi$
$854$	−1.94506e112	−0.267797
$855$	8.47397e112	1.11530
$856$	2.80850e113	3.53374
$857$	4.59078e112	0.552236	0.276118	−	0.961124i	$-0.410952\pi$
$857$	4.59078e112	0.552236	0.276118	+	0.961124i	$0.410952\pi$
$858$	−1.58316e110	−0.00182081
$859$	1.06981e112	0.117644	0.0588219	−	0.998268i	$-0.481266\pi$
$859$	1.06981e112	0.117644	0.0588219	+	0.998268i	$0.481266\pi$
$860$	1.25803e113	1.32281
$861$	1.59171e112	0.160046
$862$	−7.38493e111	−0.0710095
$863$	−1.51588e113	−1.39396	−0.696980	−	0.717091i	$-0.745473\pi$
$863$	−1.51588e113	−1.39396	−0.696980	+	0.717091i	$0.745473\pi$
$864$	−8.74833e112	−0.769390
$865$	2.34426e113	1.97190
$866$	1.47183e113	1.18418
$867$	5.60443e112	0.431317
$868$	−3.15879e113	−2.32547
$869$	−7.21036e112	−0.507803
$870$	−8.52148e112	−0.574148
$871$	7.86309e110	0.00506867
$872$	−2.00634e113	−1.23743
$873$	5.32237e112	0.314092
$874$	−1.80055e113	−1.01675
$875$	2.89521e113	1.56448
$876$	3.65596e112	0.189057
$877$	−3.28325e113	−1.62487	−0.812437	−	0.583049i	$-0.801860\pi$
$877$	−3.28325e113	−1.62487	−0.812437	+	0.583049i	$0.801860\pi$
$878$	−2.66858e113	−1.26399
$879$	−7.72141e112	−0.350046
$880$	3.56461e113	1.54678
$881$	2.32865e112	0.0967235	0.0483618	−	0.998830i	$-0.484600\pi$
$881$	2.32865e112	0.0967235	0.0483618	+	0.998830i	$0.484600\pi$
$882$	2.39254e113	0.951304
$883$	−8.28592e112	−0.315394	−0.157697	−	0.987488i	$-0.550407\pi$
$883$	−8.28592e112	−0.315394	−0.157697	+	0.987488i	$0.550407\pi$
$884$	−4.71553e111	−0.0171838
$885$	3.05690e113	1.06651
$886$	6.41108e113	2.14157
$887$	−4.10233e113	−1.31211	−0.656056	−	0.754712i	$-0.727776\pi$
$887$	−4.10233e113	−1.31211	−0.656056	+	0.754712i	$0.727776\pi$
$888$	−9.19472e112	−0.281604
$889$	−2.53273e113	−0.742798
$890$	5.56717e113	1.56358
$891$	−8.94556e112	−0.240611
$892$	6.17946e113	1.59185
$893$	−9.50410e112	−0.234493
$894$	−5.58583e113	−1.32006
$895$	−1.24920e114	−2.82777
$896$	2.72418e113	0.590714
$897$	−8.17327e110	−0.00169780
$898$	1.17230e114	2.33292
$899$	3.76421e113	0.717671
$900$	−2.46224e114	−4.49773
$901$	3.56974e113	0.624787
$902$	3.19432e113	0.535706
$903$	4.92126e112	0.0790858
$904$	4.53212e113	0.697938
$905$	2.02299e114	2.98555
$906$	−7.66546e112	−0.108418
$907$	−6.38293e113	−0.865245	−0.432623	−	0.901575i	$-0.642412\pi$
$907$	−6.38293e113	−0.865245	−0.432623	+	0.901575i	$0.642412\pi$
$908$	−1.48203e114	−1.92554
$909$	8.49375e113	1.05777
$910$	8.97763e111	0.0107169
$911$	−1.00011e114	−1.14443	−0.572215	−	0.820104i	$-0.693916\pi$
$911$	−1.00011e114	−1.14443	−0.572215	+	0.820104i	$0.693916\pi$
$912$	4.95595e113	0.543657
$913$	−1.41135e113	−0.148426
$914$	−1.13414e114	−1.14351
$915$	−1.92683e113	−0.186265
$916$	−4.18339e114	−3.87751
$917$	3.99428e113	0.354992
$918$	2.27942e114	1.94259
$919$	7.23363e113	0.591166	0.295583	−	0.955317i	$-0.404486\pi$
$919$	7.23363e113	0.591166	0.295583	+	0.955317i	$0.404486\pi$
$920$	4.11863e114	3.22791
$921$	2.48546e113	0.186814
$922$	2.22513e114	1.60404
$923$	3.15445e111	0.00218101
$924$	3.88749e113	0.257809
$925$	−1.14689e114	−0.729568
$926$	−2.53976e114	−1.54978
$927$	2.14990e114	1.25849
$928$	7.64753e113	0.429461
$929$	1.89352e113	0.102015	0.0510076	−	0.998698i	$-0.483757\pi$
$929$	1.89352e113	0.102015	0.0510076	+	0.998698i	$0.483757\pi$
$930$	−4.51881e114	−2.33577
$931$	−9.30930e113	−0.461694
$932$	4.61486e114	2.19607
$933$	9.40476e113	0.429442
$934$	−4.06046e114	−1.77918
$935$	−2.89548e114	−1.21751
$936$	−2.47825e112	−0.0100006
$937$	−1.10387e114	−0.427506	−0.213753	−	0.976888i	$-0.568569\pi$
$937$	−1.10387e114	−0.427506	−0.213753	+	0.976888i	$0.568569\pi$
$938$	−2.78824e114	−1.03639
$939$	−3.69208e113	−0.131718
$940$	3.91069e114	1.33915
$941$	1.79108e113	0.0588727	0.0294363	−	0.999567i	$-0.490629\pi$
$941$	1.79108e113	0.0588727	0.0294363	+	0.999567i	$0.490629\pi$
$942$	8.80501e113	0.277824
$943$	1.64911e114	0.499516
$944$	−8.79990e114	−2.55891
$945$	−3.00513e114	−0.838954
$946$	9.87621e113	0.264717
$947$	4.55062e114	1.17111	0.585553	−	0.810634i	$-0.300877\pi$
$947$	4.55062e114	1.17111	0.585553	+	0.810634i	$0.300877\pi$
$948$	4.12490e114	1.01928
$949$	4.58988e111	0.00108906
$950$	1.38351e115	3.15226
$951$	1.26185e114	0.276094
$952$	9.29554e114	1.95323
$953$	3.48766e114	0.703814	0.351907	−	0.936035i	$-0.385533\pi$
$953$	3.48766e114	0.703814	0.351907	+	0.936035i	$0.385533\pi$
$954$	3.37478e114	0.654083
$955$	−1.16987e115	−2.17774
$956$	−1.83152e115	−3.27478
$957$	−4.63257e113	−0.0795633
$958$	−1.33747e115	−2.20654
$959$	3.23177e114	0.512181
$960$	−1.24590e113	−0.0189688
$961$	1.31242e115	1.91966
$962$	−2.07650e112	−0.00291804
$963$	−9.63655e114	−1.30110
$964$	7.46874e114	0.968905
$965$	−6.41871e114	−0.800103
$966$	2.89823e114	0.347148
$967$	1.67001e115	1.92221	0.961104	−	0.276186i	$-0.0890707\pi$
$967$	1.67001e115	1.92221	0.961104	+	0.276186i	$0.0890707\pi$
$968$	−1.60711e115	−1.77764
$969$	−4.02565e114	−0.427928
$970$	1.23054e115	1.25715
$971$	−1.23824e115	−1.21581	−0.607904	−	0.794011i	$-0.707989\pi$
$971$	−1.23824e115	−1.21581	−0.607904	+	0.794011i	$0.707989\pi$
$972$	2.30717e115	2.17735
$973$	−9.35050e114	−0.848184
$974$	−1.85840e115	−1.62039
$975$	6.28019e112	0.00526374
$976$	5.54676e114	0.446912
$977$	−1.52087e115	−1.17801	−0.589007	−	0.808128i	$-0.700481\pi$
$977$	−1.52087e115	−1.17801	−0.589007	+	0.808128i	$0.700481\pi$
$978$	−1.06875e115	−0.795848
$979$	3.02651e114	0.216675
$980$	3.83053e115	2.63667
$981$	6.88418e114	0.455612
$982$	9.55999e114	0.608366
$983$	−2.27613e114	−0.139280	−0.0696398	−	0.997572i	$-0.522185\pi$
$983$	−2.27613e114	−0.139280	−0.0696398	+	0.997572i	$0.522185\pi$
$984$	−1.01588e115	−0.597765
$985$	1.70286e114	0.0963570
$986$	−1.99261e115	−1.08433
$987$	1.52982e114	0.0800626
$988$	1.73459e113	0.00873079
$989$	5.09872e114	0.246833
$990$	−2.73735e115	−1.27460
$991$	1.76625e115	0.791072	0.395536	−	0.918451i	$-0.370559\pi$
$991$	1.76625e115	0.791072	0.395536	+	0.918451i	$0.370559\pi$
$992$	4.05536e115	1.74715
$993$	7.77857e114	0.322369
$994$	−1.11856e115	−0.445949
$995$	4.82796e115	1.85172
$996$	8.07404e114	0.297925
$997$	−1.95818e115	−0.695168	−0.347584	−	0.937649i	$-0.612998\pi$
$997$	−1.95818e115	−0.695168	−0.347584	+	0.937649i	$0.612998\pi$
$998$	2.70194e115	0.922889
$999$	6.95076e114	0.228434

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1.78.a.a.1.6	✓	6
3.2	odd	2		9.78.a.a.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.78.a.a.1.6	✓	6	1.1	even	1	trivial
9.78.a.a.1.1		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)

\(f(q)\)	\(=\)	\(q+7.00999e11 q^{2} +9.61452e17 q^{3} +3.40284e23 q^{4} -1.50068e27 q^{5} +6.73977e29 q^{6} -2.07773e32 q^{7} +1.32607e35 q^{8} -4.55001e36 q^{9} +O(q^{10})\)	\(q+7.00999e11 q^{2} +9.61452e17 q^{3} +3.40284e23 q^{4} -1.50068e27 q^{5} +6.73977e29 q^{6} -2.07773e32 q^{7} +1.32607e35 q^{8} -4.55001e36 q^{9} -1.05197e39 q^{10} -5.71890e39 q^{11} +3.27166e41 q^{12} +4.10741e40 q^{13} -1.45648e44 q^{14} -1.44283e45 q^{15} +4.15348e46 q^{16} -3.37382e47 q^{17} -3.18955e48 q^{18} +1.24104e49 q^{19} -5.10656e50 q^{20} -1.99763e50 q^{21} -4.00894e51 q^{22} -2.06967e52 q^{23} +1.27495e53 q^{24} +1.59029e54 q^{25} +2.87929e52 q^{26} -9.63799e54 q^{27} -7.07017e55 q^{28} +8.42524e55 q^{29} -1.01142e57 q^{30} +4.46777e57 q^{31} +9.07693e57 q^{32} -5.49844e57 q^{33} -2.36504e59 q^{34} +3.11800e59 q^{35} -1.54829e60 q^{36} -7.21183e59 q^{37} +8.69970e60 q^{38} +3.94908e58 q^{39} -1.99000e62 q^{40} -7.96800e61 q^{41} -1.40034e62 q^{42} -2.46355e62 q^{43} -1.94605e63 q^{44} +6.82810e63 q^{45} -1.45083e64 q^{46} -7.65815e63 q^{47} +3.99337e64 q^{48} -7.50119e64 q^{49} +1.11479e66 q^{50} -3.24376e65 q^{51} +1.39768e64 q^{52} -1.05807e66 q^{53} -6.75622e66 q^{54} +8.58222e66 q^{55} -2.75520e67 q^{56} +1.19320e67 q^{57} +5.90609e67 q^{58} -2.11868e68 q^{59} -4.90972e68 q^{60} +1.33545e68 q^{61} +3.13190e69 q^{62} +9.45368e68 q^{63} +8.63510e67 q^{64} -6.16390e67 q^{65} -3.85440e69 q^{66} +1.91437e70 q^{67} -1.14805e71 q^{68} -1.98988e70 q^{69} +2.18571e71 q^{70} +7.67990e70 q^{71} -6.03361e71 q^{72} +1.11746e71 q^{73} -5.05549e71 q^{74} +1.52899e72 q^{75} +4.22307e72 q^{76} +1.18823e72 q^{77} +2.76830e70 q^{78} +1.26080e73 q^{79} -6.23304e73 q^{80} +1.56421e73 q^{81} -5.58556e73 q^{82} +2.46787e73 q^{83} -6.79763e73 q^{84} +5.06301e74 q^{85} -1.72694e74 q^{86} +8.10047e73 q^{87} -7.58363e74 q^{88} -5.29212e74 q^{89} +4.78649e75 q^{90} -8.53408e72 q^{91} -7.04273e75 q^{92} +4.29555e75 q^{93} -5.36836e75 q^{94} -1.86241e76 q^{95} +8.72703e75 q^{96} -1.16975e76 q^{97} -5.25833e76 q^{98} +2.60210e76 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 264721893120 q^{2} + 14\!\cdots\!80 q^{3}+ \cdots - 48\!\cdots\!42 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more