LMFDB - Embedded newform 1.78.a.a.1.1

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			1.1
Root			$3.81583e9$ 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)

$f(q)$	$=$	$q-6.88520e11 q^{2} +1.25704e18 q^{3} +3.22944e23 q^{4} +7.49066e26 q^{5} -8.65494e29 q^{6} +4.37098e32 q^{7} -1.18307e35 q^{8} -3.89426e36 q^{9} +O(q^{10})$	\(q-6.88520e11 q^{2} +1.25704e18 q^{3} +3.22944e23 q^{4} +7.49066e26 q^{5} -8.65494e29 q^{6} +4.37098e32 q^{7} -1.18307e35 q^{8} -3.89426e36 q^{9} -5.15747e38 q^{10} -2.13283e40 q^{11} +4.05952e41 q^{12} +3.45913e42 q^{13} -3.00950e44 q^{14} +9.41603e44 q^{15} +3.26547e46 q^{16} -1.55573e47 q^{17} +2.68128e48 q^{18} +1.76144e49 q^{19} +2.41906e50 q^{20} +5.49447e50 q^{21} +1.46850e52 q^{22} -4.81185e52 q^{23} -1.48716e53 q^{24} -1.00645e53 q^{25} -2.38168e54 q^{26} -1.17767e55 q^{27} +1.41158e56 q^{28} +1.07750e56 q^{29} -6.48312e56 q^{30} -2.03054e57 q^{31} -4.60540e57 q^{32} -2.68105e58 q^{33} +1.07115e59 q^{34} +3.27415e59 q^{35} -1.25763e60 q^{36} -2.31877e60 q^{37} -1.21279e61 q^{38} +4.34825e60 q^{39} -8.86196e61 q^{40} +8.70352e61 q^{41} -3.78305e62 q^{42} +1.34272e62 q^{43} -6.88784e63 q^{44} -2.91706e63 q^{45} +3.31305e64 q^{46} -1.39186e64 q^{47} +4.10482e64 q^{48} +7.28729e64 q^{49} +6.92958e64 q^{50} -1.95561e65 q^{51} +1.11710e66 q^{52} +6.15735e65 q^{53} +8.10852e66 q^{54} -1.59763e67 q^{55} -5.17116e67 q^{56} +2.21419e67 q^{57} -7.41882e67 q^{58} -1.10779e67 q^{59} +3.04085e68 q^{60} +1.77613e68 q^{61} +1.39806e69 q^{62} -1.70217e69 q^{63} -1.76373e69 q^{64} +2.59112e69 q^{65} +1.84595e70 q^{66} +1.11673e70 q^{67} -5.02413e70 q^{68} -6.04867e70 q^{69} -2.25432e71 q^{70} +8.55276e70 q^{71} +4.60718e71 q^{72} -5.87097e71 q^{73} +1.59652e72 q^{74} -1.26514e71 q^{75} +5.68846e72 q^{76} -9.32255e72 q^{77} -2.99386e72 q^{78} -1.64901e72 q^{79} +2.44606e73 q^{80} +6.51496e72 q^{81} -5.99255e73 q^{82} -1.48423e74 q^{83} +1.77440e74 q^{84} -1.16534e74 q^{85} -9.24487e73 q^{86} +1.35446e74 q^{87} +2.52329e75 q^{88} -1.76014e75 q^{89} +2.00845e75 q^{90} +1.51198e75 q^{91} -1.55396e76 q^{92} -2.55246e75 q^{93} +9.58322e75 q^{94} +1.31944e76 q^{95} -5.78916e75 q^{96} -3.86061e75 q^{97} -5.01744e76 q^{98} +8.30580e76 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−6.88520e11	−1.77118	−0.885588	−	0.464472i	$-0.846244\pi$
$2$	−6.88520e11	−1.77118	−0.885588	+	0.464472i	$0.846244\pi$
$3$	1.25704e18	0.537254	0.268627	−	0.963244i	$-0.413430\pi$
$3$	1.25704e18	0.537254	0.268627	+	0.963244i	$0.413430\pi$
$4$	3.22944e23	2.13706
$5$	7.49066e26	0.920820	0.460410	−	0.887706i	$-0.347702\pi$
$5$	7.49066e26	0.920820	0.460410	+	0.887706i	$0.347702\pi$
$6$	−8.65494e29	−0.951570
$7$	4.37098e32	1.27146	0.635732	−	0.771910i	$-0.280698\pi$
$7$	4.37098e32	1.27146	0.635732	+	0.771910i	$0.280698\pi$
$8$	−1.18307e35	−2.01393
$9$	−3.89426e36	−0.711359
$10$	−5.15747e38	−1.63093
$11$	−2.13283e40	−1.71925	−0.859623	−	0.510928i	$-0.829302\pi$
$11$	−2.13283e40	−1.71925	−0.859623	+	0.510928i	$0.829302\pi$
$12$	4.05952e41	1.14814
$13$	3.45913e42	0.448898	0.224449	−	0.974486i	$-0.427942\pi$
$13$	3.45913e42	0.448898	0.224449	+	0.974486i	$0.427942\pi$
$14$	−3.00950e44	−2.25198
$15$	9.41603e44	0.494714
$16$	3.26547e46	1.42997
$17$	−1.55573e47	−0.660162	−0.330081	−	0.943953i	$-0.607076\pi$
$17$	−1.55573e47	−0.660162	−0.330081	+	0.943953i	$0.607076\pi$
$18$	2.68128e48	1.25994
$19$	1.76144e49	1.03242	0.516208	−	0.856463i	$-0.327343\pi$
$19$	1.76144e49	1.03242	0.516208	+	0.856463i	$0.327343\pi$
$20$	2.41906e50	1.96785
$21$	5.49447e50	0.683098
$22$	1.46850e52	3.04509
$23$	−4.81185e52	−1.80215	−0.901074	−	0.433666i	$-0.857220\pi$
$23$	−4.81185e52	−1.80215	−0.901074	+	0.433666i	$0.857220\pi$
$24$	−1.48716e53	−1.08199
$25$	−1.00645e53	−0.152090
$26$	−2.38168e54	−0.795077
$27$	−1.17767e55	−0.919434
$28$	1.41158e56	2.71720
$29$	1.07750e56	0.537150	0.268575	−	0.963259i	$-0.413447\pi$
$29$	1.07750e56	0.537150	0.268575	+	0.963259i	$0.413447\pi$
$30$	−6.48312e56	−0.876225
$31$	−2.03054e57	−0.776579	−0.388290	−	0.921537i	$-0.626934\pi$
$31$	−2.03054e57	−0.776579	−0.388290	+	0.921537i	$0.626934\pi$
$32$	−4.60540e57	−0.518792
$33$	−2.68105e58	−0.923671
$34$	1.07115e59	1.16926
$35$	3.27415e59	1.17079
$36$	−1.25763e60	−1.52022
$37$	−2.31877e60	−0.976094	−0.488047	−	0.872817i	$-0.662290\pi$
$37$	−2.31877e60	−0.976094	−0.488047	+	0.872817i	$0.662290\pi$
$38$	−1.21279e61	−1.82859
$39$	4.34825e60	0.241172
$40$	−8.86196e61	−1.85447
$41$	8.70352e61	0.703909	0.351955	−	0.936017i	$-0.385517\pi$
$41$	8.70352e61	0.703909	0.351955	+	0.936017i	$0.385517\pi$
$42$	−3.78305e62	−1.20989
$43$	1.34272e62	0.173560	0.0867799	−	0.996228i	$-0.472342\pi$
$43$	1.34272e62	0.173560	0.0867799	+	0.996228i	$0.472342\pi$
$44$	−6.88784e63	−3.67414
$45$	−2.91706e63	−0.655033
$46$	3.31305e64	3.19192
$47$	−1.39186e64	−0.585905	−0.292953	−	0.956127i	$-0.594638\pi$
$47$	−1.39186e64	−0.585905	−0.292953	+	0.956127i	$0.594638\pi$
$48$	4.10482e64	0.768257
$49$	7.28729e64	0.616619
$50$	6.92958e64	0.269378
$51$	−1.95561e65	−0.354675
$52$	1.11710e66	0.959323
$53$	6.15735e65	0.253964	0.126982	−	0.991905i	$-0.459471\pi$
$53$	6.15735e65	0.253964	0.126982	+	0.991905i	$0.459471\pi$
$54$	8.10852e66	1.62848
$55$	−1.59763e67	−1.58312
$56$	−5.17116e67	−2.56064
$57$	2.21419e67	0.554669
$58$	−7.41882e67	−0.951387
$59$	−1.10779e67	−0.0735624	−0.0367812	−	0.999323i	$-0.511710\pi$
$59$	−1.10779e67	−0.0735624	−0.0367812	+	0.999323i	$0.511710\pi$
$60$	3.04085e68	1.05723
$61$	1.77613e68	0.326796	0.163398	−	0.986560i	$-0.447755\pi$
$61$	1.77613e68	0.326796	0.163398	+	0.986560i	$0.447755\pi$
$62$	1.39806e69	1.37546
$63$	−1.70217e69	−0.904466
$64$	−1.76373e69	−0.511098
$65$	2.59112e69	0.413354
$66$	1.84595e70	1.63598
$67$	1.11673e70	0.554711	0.277356	−	0.960767i	$-0.410542\pi$
$67$	1.11673e70	0.554711	0.277356	+	0.960767i	$0.410542\pi$
$68$	−5.02413e70	−1.41081
$69$	−6.04867e70	−0.968210
$70$	−2.25432e71	−2.07367
$71$	8.55276e70	0.455680	0.227840	−	0.973699i	$-0.426834\pi$
$71$	8.55276e70	0.455680	0.227840	+	0.973699i	$0.426834\pi$
$72$	4.60718e71	1.43263
$73$	−5.87097e71	−1.07345	−0.536723	−	0.843759i	$-0.680338\pi$
$73$	−5.87097e71	−1.07345	−0.536723	+	0.843759i	$0.680338\pi$
$74$	1.59652e72	1.72883
$75$	−1.26514e71	−0.0817108
$76$	5.68846e72	2.20633
$77$	−9.32255e72	−2.18596
$78$	−2.99386e72	−0.427158
$79$	−1.64901e72	−0.144072	−0.0720361	−	0.997402i	$-0.522950\pi$
$79$	−1.64901e72	−0.144072	−0.0720361	+	0.997402i	$0.522950\pi$
$80$	2.44606e73	1.31675
$81$	6.51496e72	0.217389
$82$	−5.99255e73	−1.24675
$83$	−1.48423e74	−1.93640	−0.968201	−	0.250172i	$-0.919513\pi$
$83$	−1.48423e74	−1.93640	−0.968201	+	0.250172i	$0.919513\pi$
$84$	1.77440e74	1.45982
$85$	−1.16534e74	−0.607891
$86$	−9.24487e73	−0.307405
$87$	1.35446e74	0.288586
$88$	2.52329e75	3.46245
$89$	−1.76014e75	−1.56327	−0.781633	−	0.623738i	$-0.785613\pi$
$89$	−1.76014e75	−1.56327	−0.781633	+	0.623738i	$0.785613\pi$
$90$	2.00845e75	1.16018
$91$	1.51198e75	0.570757
$92$	−1.55396e76	−3.85130
$93$	−2.55246e75	−0.417220
$94$	9.58322e75	1.03774
$95$	1.31944e76	0.950669
$96$	−5.78916e75	−0.278723
$97$	−3.86061e75	−0.124722	−0.0623612	−	0.998054i	$-0.519863\pi$
$97$	−3.86061e75	−0.124722	−0.0623612	+	0.998054i	$0.519863\pi$
$98$	−5.01744e76	−1.09214
$99$	8.30580e76	1.22300
$100$	−3.25025e76	−0.325025
$101$	8.69379e76	0.592702	0.296351	−	0.955079i	$-0.404230\pi$
$101$	8.69379e76	0.592702	0.296351	+	0.955079i	$0.404230\pi$
$102$	1.34647e77	0.628191
$103$	1.73809e77	0.556979	0.278490	−	0.960439i	$-0.410166\pi$
$103$	1.73809e77	0.556979	0.278490	+	0.960439i	$0.410166\pi$
$104$	−4.09239e77	−0.904051
$105$	4.11572e77	0.629011
$106$	−4.23946e77	−0.449816
$107$	−8.25803e77	−0.610381	−0.305190	−	0.952291i	$-0.598720\pi$
$107$	−8.25803e77	−0.610381	−0.305190	+	0.952291i	$0.598720\pi$
$108$	−3.80322e78	−1.96489
$109$	−3.51483e78	−1.27346	−0.636729	−	0.771088i	$-0.719713\pi$
$109$	−3.51483e78	−1.27346	−0.636729	+	0.771088i	$0.719713\pi$
$110$	1.10000e79	2.80398
$111$	−2.91478e78	−0.524410
$112$	1.42733e79	1.81815
$113$	1.32843e79	1.20176	0.600879	−	0.799340i	$-0.294817\pi$
$113$	1.32843e79	1.20176	0.600879	+	0.799340i	$0.294817\pi$
$114$	−1.52452e79	−0.982416
$115$	−3.60439e79	−1.65945
$116$	3.47973e79	1.14792
$117$	−1.34708e79	−0.319327
$118$	7.62736e78	0.130292
$119$	−6.80006e79	−0.839372
$120$	−1.11398e80	−0.996322
$121$	3.00998e80	1.95581
$122$	−1.22290e80	−0.578812
$123$	1.09406e80	0.378178
$124$	−6.55749e80	−1.65960
$125$	−5.71080e80	−1.06087
$126$	1.17198e81	1.60197
$127$	1.42117e81	1.43286	0.716428	−	0.697661i	$-0.245776\pi$
$127$	1.42117e81	1.43286	0.716428	+	0.697661i	$0.245776\pi$
$128$	1.91031e81	1.42404
$129$	1.68784e80	0.0932456
$130$	−1.78403e81	−0.732123
$131$	−2.86301e81	−0.874738	−0.437369	−	0.899282i	$-0.644090\pi$
$131$	−2.86301e81	−0.874738	−0.437369	+	0.899282i	$0.644090\pi$
$132$	−8.65826e81	−1.97394
$133$	7.69921e81	1.31268
$134$	−7.68888e81	−0.982491
$135$	−8.82156e81	−0.846633
$136$	1.84054e82	1.32952
$137$	−7.83571e81	−0.426910	−0.213455	−	0.976953i	$-0.568472\pi$
$137$	−7.83571e81	−0.426910	−0.213455	+	0.976953i	$0.568472\pi$
$138$	4.16463e82	1.71487
$139$	−4.52889e82	−1.41228	−0.706141	−	0.708071i	$-0.749566\pi$
$139$	−4.52889e82	−1.41228	−0.706141	+	0.708071i	$0.749566\pi$
$140$	1.05737e83	2.50205
$141$	−1.74962e82	−0.314780
$142$	−5.88875e82	−0.807089
$143$	−7.37774e82	−0.771766
$144$	−1.27166e83	−1.01722
$145$	8.07121e82	0.494619
$146$	4.04228e83	1.90126
$147$	9.16038e82	0.331281
$148$	−7.48832e83	−2.08597
$149$	−2.81315e83	−0.604675	−0.302338	−	0.953201i	$-0.597767\pi$
$149$	−2.81315e83	−0.604675	−0.302338	+	0.953201i	$0.597767\pi$
$150$	8.71073e82	0.144724
$151$	4.59517e83	0.591138	0.295569	−	0.955321i	$-0.404491\pi$
$151$	4.59517e83	0.591138	0.295569	+	0.955321i	$0.404491\pi$
$152$	−2.08391e84	−2.07922
$153$	6.05842e83	0.469612
$154$	6.41876e84	3.87172
$155$	−1.52101e84	−0.715090
$156$	1.40424e84	0.515400
$157$	−3.14465e84	−0.902477	−0.451239	−	0.892403i	$-0.649018\pi$
$157$	−3.14465e84	−0.902477	−0.451239	+	0.892403i	$0.649018\pi$
$158$	1.13538e84	0.255177
$159$	7.74001e83	0.136443
$160$	−3.44975e84	−0.477715
$161$	−2.10325e85	−2.29136
$162$	−4.48568e84	−0.385035
$163$	−2.85867e84	−0.193616	−0.0968080	−	0.995303i	$-0.530863\pi$
$163$	−2.85867e84	−0.193616	−0.0968080	+	0.995303i	$0.530863\pi$
$164$	2.81075e85	1.50430
$165$	−2.00828e85	−0.850535
$166$	1.02192e86	3.42971
$167$	−1.60028e85	−0.426198	−0.213099	−	0.977031i	$-0.568356\pi$
$167$	−1.60028e85	−0.426198	−0.213099	+	0.977031i	$0.568356\pi$
$168$	−6.50034e85	−1.37572
$169$	−4.74141e85	−0.798491
$170$	8.02362e85	1.07668
$171$	−6.85951e85	−0.734417
$172$	4.33622e85	0.370908
$173$	2.22223e86	1.52060	0.760299	−	0.649573i	$-0.225052\pi$
$173$	2.22223e86	1.52060	0.760299	+	0.649573i	$0.225052\pi$
$174$	−9.32572e85	−0.511136
$175$	−4.39915e85	−0.193377
$176$	−6.96471e86	−2.45847
$177$	−1.39253e85	−0.0395217
$178$	1.21189e87	2.76882
$179$	−5.98273e86	−1.10169	−0.550843	−	0.834609i	$-0.685694\pi$
$179$	−5.98273e86	−1.10169	−0.550843	+	0.834609i	$0.685694\pi$
$180$	−9.42046e86	−1.39985
$181$	3.68947e86	0.442934	0.221467	−	0.975168i	$-0.428915\pi$
$181$	3.68947e86	0.442934	0.221467	+	0.975168i	$0.428915\pi$
$182$	−1.04103e87	−1.01091
$183$	2.23266e86	0.175572
$184$	5.69275e87	3.62941
$185$	−1.73691e87	−0.898807
$186$	1.75742e87	0.738970
$187$	3.31811e87	1.13498
$188$	−4.49492e87	−1.25212
$189$	−5.14759e87	−1.16903
$190$	−9.08457e87	−1.68380
$191$	−3.56104e87	−0.539252	−0.269626	−	0.962965i	$-0.586900\pi$
$191$	−3.56104e87	−0.539252	−0.269626	+	0.962965i	$0.586900\pi$
$192$	−2.21708e87	−0.274589
$193$	1.09426e88	1.10960	0.554799	−	0.831984i	$-0.312795\pi$
$193$	1.09426e88	1.10960	0.554799	+	0.831984i	$0.312795\pi$
$194$	2.65811e87	0.220905
$195$	3.25713e87	0.222076
$196$	2.35338e88	1.31775
$197$	9.41777e87	0.433509	0.216755	−	0.976226i	$-0.430453\pi$
$197$	9.41777e87	0.433509	0.216755	+	0.976226i	$0.430453\pi$
$198$	−5.71871e88	−2.16615
$199$	4.03707e88	1.25957	0.629786	−	0.776769i	$-0.283143\pi$
$199$	4.03707e88	1.25957	0.629786	+	0.776769i	$0.283143\pi$
$200$	1.19070e88	0.306299
$201$	1.40377e88	0.298021
$202$	−5.98585e88	−1.04978
$203$	4.70974e88	0.682966
$204$	−6.31551e88	−0.757961
$205$	6.51951e88	0.648174
$206$	−1.19671e89	−0.986508
$207$	1.87386e89	1.28197
$208$	1.12957e89	0.641911
$209$	−3.75686e89	−1.77498
$210$	−2.83376e89	−1.11409
$211$	−1.68858e89	−0.552902	−0.276451	−	0.961028i	$-0.589158\pi$
$211$	−1.68858e89	−0.552902	−0.276451	+	0.961028i	$0.589158\pi$
$212$	1.98848e89	0.542738
$213$	1.07511e89	0.244816
$214$	5.68582e89	1.08109
$215$	1.00578e89	0.159817
$216$	1.39327e90	1.85168
$217$	−8.87543e89	−0.987392
$218$	2.42003e90	2.25552
$219$	−7.38002e89	−0.576712
$220$	−5.15945e90	−3.38322
$221$	−5.38147e89	−0.296346
$222$	2.00688e90	0.928822
$223$	5.40084e89	0.210244	0.105122	−	0.994459i	$-0.466477\pi$
$223$	5.40084e89	0.210244	0.105122	+	0.994459i	$0.466477\pi$
$224$	−2.01301e90	−0.659626
$225$	3.91937e89	0.108190
$226$	−9.14647e90	−2.12852
$227$	3.77136e90	0.740462	0.370231	−	0.928940i	$-0.379279\pi$
$227$	3.77136e90	0.740462	0.370231	+	0.928940i	$0.379279\pi$
$228$	7.15060e90	1.18536
$229$	6.10644e90	0.855307	0.427654	−	0.903943i	$-0.359340\pi$
$229$	6.10644e90	0.855307	0.427654	+	0.903943i	$0.359340\pi$
$230$	2.48170e91	2.93918
$231$	−1.17188e91	−1.17441
$232$	−1.27476e91	−1.08179
$233$	−1.75471e91	−1.26183	−0.630917	−	0.775851i	$-0.717321\pi$
$233$	−1.75471e91	−1.26183	−0.630917	+	0.775851i	$0.717321\pi$
$234$	9.27488e90	0.565585
$235$	−1.04259e91	−0.539514
$236$	−3.57754e90	−0.157207
$237$	−2.07286e90	−0.0774033
$238$	4.68197e91	1.48668
$239$	2.74160e91	0.740773	0.370387	−	0.928878i	$-0.379225\pi$
$239$	2.74160e91	0.740773	0.370387	+	0.928878i	$0.379225\pi$
$240$	3.07478e91	0.707426
$241$	−8.34823e91	−1.63658	−0.818292	−	0.574803i	$-0.805079\pi$
$241$	−8.34823e91	−1.63658	−0.818292	+	0.574803i	$0.805079\pi$
$242$	−2.07243e92	−3.46408
$243$	7.26602e91	1.03623
$244$	5.73590e91	0.698382
$245$	5.45866e91	0.567795
$246$	−7.53285e91	−0.669819
$247$	6.09305e91	0.463449
$248$	2.40227e92	1.56398
$249$	−1.86573e92	−1.04034
$250$	3.93200e92	1.87898
$251$	1.09937e92	0.450510	0.225255	−	0.974300i	$-0.427678\pi$
$251$	1.09937e92	0.450510	0.225255	+	0.974300i	$0.427678\pi$
$252$	−5.49706e92	−1.93290
$253$	1.02629e93	3.09834
$254$	−9.78503e92	−2.53784
$255$	−1.46488e92	−0.326592
$256$	−1.04876e93	−2.01112
$257$	4.57693e92	0.755352	0.377676	−	0.925938i	$-0.376723\pi$
$257$	4.57693e92	0.755352	0.377676	+	0.925938i	$0.376723\pi$
$258$	−1.16211e92	−0.165154
$259$	−1.01353e93	−1.24107
$260$	8.36784e92	0.883364
$261$	−4.19608e92	−0.382106
$262$	1.97124e93	1.54931
$263$	7.85101e92	0.532880	0.266440	−	0.963852i	$-0.414153\pi$
$263$	7.85101e92	0.532880	0.266440	+	0.963852i	$0.414153\pi$
$264$	3.17186e93	1.86021
$265$	4.61226e92	0.233856
$266$	−5.30106e93	−2.32498
$267$	−2.21256e93	−0.839870
$268$	3.60640e93	1.18545
$269$	−2.66601e93	−0.759274	−0.379637	−	0.925136i	$-0.623951\pi$
$269$	−2.66601e93	−0.759274	−0.379637	+	0.925136i	$0.623951\pi$
$270$	6.07382e93	1.49954
$271$	−3.20437e93	−0.686158	−0.343079	−	0.939306i	$-0.611470\pi$
$271$	−3.20437e93	−0.686158	−0.343079	+	0.939306i	$0.611470\pi$
$272$	−5.08020e93	−0.944012
$273$	1.90061e93	0.306641
$274$	5.39504e93	0.756133
$275$	2.14658e93	0.261480
$276$	−1.95338e94	−2.06912
$277$	−9.20537e93	−0.848340	−0.424170	−	0.905583i	$-0.639434\pi$
$277$	−9.20537e93	−0.848340	−0.424170	+	0.905583i	$0.639434\pi$
$278$	3.11823e94	2.50140
$279$	7.90744e93	0.552426
$280$	−3.87354e94	−2.35789
$281$	4.99518e93	0.265068	0.132534	−	0.991178i	$-0.457689\pi$
$281$	4.99518e93	0.265068	0.132534	+	0.991178i	$0.457689\pi$
$282$	1.20465e94	0.557530
$283$	−2.26252e94	−0.913723	−0.456861	−	0.889538i	$-0.651026\pi$
$283$	−2.26252e94	−0.913723	−0.456861	+	0.889538i	$0.651026\pi$
$284$	2.76206e94	0.973816
$285$	1.65858e94	0.510750
$286$	5.07972e94	1.36693
$287$	3.80429e94	0.894995
$288$	1.79347e94	0.369047
$289$	−3.13321e94	−0.564186
$290$	−5.55719e94	−0.876056
$291$	−4.85293e93	−0.0670076
$292$	−1.89599e95	−2.29402
$293$	−1.33456e95	−1.41558	−0.707791	−	0.706422i	$-0.750308\pi$
$293$	−1.33456e95	−1.41558	−0.707791	+	0.706422i	$0.750308\pi$
$294$	−6.30710e94	−0.586756
$295$	−8.29809e93	−0.0677377
$296$	2.74327e95	1.96579
$297$	2.51178e95	1.58073
$298$	1.93691e95	1.07099
$299$	−1.66448e95	−0.808980
$300$	−4.08569e94	−0.174621
$301$	5.86899e94	0.220675
$302$	−3.16386e95	−1.04701
$303$	1.09284e95	0.318431
$304$	5.75194e95	1.47632
$305$	1.33044e95	0.300920
$306$	−4.17134e95	−0.831765
$307$	8.74081e95	1.53718	0.768589	−	0.639743i	$-0.220959\pi$
$307$	8.74081e95	1.53718	0.768589	+	0.639743i	$0.220959\pi$
$308$	−3.01066e96	−4.67153
$309$	2.18484e95	0.299239
$310$	1.04724e96	1.26655
$311$	1.47925e96	1.58040	0.790201	−	0.612847i	$-0.209976\pi$
$311$	1.47925e96	1.58040	0.790201	+	0.612847i	$0.209976\pi$
$312$	−5.14428e95	−0.485705
$313$	−1.70225e96	−1.42091	−0.710453	−	0.703745i	$-0.751510\pi$
$313$	−1.70225e96	−1.42091	−0.710453	+	0.703745i	$0.751510\pi$
$314$	2.16516e96	1.59845
$315$	−1.27504e96	−0.832851
$316$	−5.32537e95	−0.307891
$317$	9.09001e95	0.465353	0.232677	−	0.972554i	$-0.425252\pi$
$317$	9.09001e95	0.465353	0.232677	+	0.972554i	$0.425252\pi$
$318$	−5.32915e95	−0.241665
$319$	−2.29813e96	−0.923493
$320$	−1.32115e96	−0.470629
$321$	−1.03806e96	−0.327929
$322$	1.44813e97	4.05841
$323$	−2.74033e96	−0.681562
$324$	2.10396e96	0.464575
$325$	−3.48143e95	−0.0682729
$326$	1.96825e96	0.342928
$327$	−4.41827e96	−0.684170
$328$	−1.02969e97	−1.41763
$329$	−6.08378e96	−0.744957
$330$	1.38274e97	1.50645
$331$	1.23246e96	0.119508	0.0597539	−	0.998213i	$-0.480968\pi$
$331$	1.23246e96	0.119508	0.0597539	+	0.998213i	$0.480968\pi$
$332$	−4.79324e97	−4.13821
$333$	9.02990e96	0.694353
$334$	1.10182e97	0.754872
$335$	8.36502e96	0.510789
$336$	1.79421e97	0.976810
$337$	−7.75639e96	−0.376624	−0.188312	−	0.982109i	$-0.560302\pi$
$337$	−7.75639e96	−0.376624	−0.188312	+	0.982109i	$0.560302\pi$
$338$	3.26456e97	1.41427
$339$	1.66988e97	0.645649
$340$	−3.76340e97	−1.29910
$341$	4.33079e97	1.33513
$342$	4.72291e97	1.30078
$343$	−1.98042e97	−0.487455
$344$	−1.58853e97	−0.349538
$345$	−4.53085e97	−0.891548
$346$	−1.53005e98	−2.69325
$347$	−4.82241e97	−0.759590	−0.379795	−	0.925071i	$-0.624005\pi$
$347$	−4.82241e97	−0.759590	−0.379795	+	0.925071i	$0.624005\pi$
$348$	4.37414e97	0.616726
$349$	1.39007e98	1.75492	0.877458	−	0.479653i	$-0.159237\pi$
$349$	1.39007e98	1.75492	0.877458	+	0.479653i	$0.159237\pi$
$350$	3.02890e97	0.342504
$351$	−4.07373e97	−0.412732
$352$	9.82255e97	0.891932
$353$	−8.25635e96	−0.0672144	−0.0336072	−	0.999435i	$-0.510700\pi$
$353$	−8.25635e96	−0.0672144	−0.0336072	+	0.999435i	$0.510700\pi$
$354$	9.58787e96	0.0699998
$355$	6.40658e97	0.419599
$356$	−5.68427e98	−3.34080
$357$	−8.54792e97	−0.450956
$358$	4.11923e98	1.95128
$359$	3.37229e97	0.143479	0.0717395	−	0.997423i	$-0.477145\pi$
$359$	3.37229e97	0.143479	0.0717395	+	0.997423i	$0.477145\pi$
$360$	3.45108e98	1.31919
$361$	1.91774e97	0.0658813
$362$	−2.54028e98	−0.784514
$363$	3.78365e98	1.05077
$364$	4.88283e98	1.21974
$365$	−4.39775e98	−0.988450
$366$	−1.53723e98	−0.310969
$367$	4.53760e98	0.826385	0.413192	−	0.910644i	$-0.364414\pi$
$367$	4.53760e98	0.826385	0.413192	+	0.910644i	$0.364414\pi$
$368$	−1.57130e99	−2.57702
$369$	−3.38938e98	−0.500732
$370$	1.19590e99	1.59194
$371$	2.69136e98	0.322907
$372$	−8.24300e98	−0.891625
$373$	−2.82254e98	−0.275328	−0.137664	−	0.990479i	$-0.543959\pi$
$373$	−2.82254e98	−0.275328	−0.137664	+	0.990479i	$0.543959\pi$
$374$	−2.28458e99	−2.01025
$375$	−7.17868e98	−0.569955
$376$	1.64666e99	1.17998
$377$	3.72722e98	0.241126
$378$	3.54421e99	2.07055
$379$	−1.22113e99	−0.644396	−0.322198	−	0.946672i	$-0.604422\pi$
$379$	−1.22113e99	−0.644396	−0.322198	+	0.946672i	$0.604422\pi$
$380$	4.26103e99	2.03164
$381$	1.78646e99	0.769807
$382$	2.45184e99	0.955110
$383$	−4.84912e99	−1.70809	−0.854043	−	0.520202i	$-0.825857\pi$
$383$	−4.84912e99	−1.70809	−0.854043	+	0.520202i	$0.825857\pi$
$384$	2.40133e99	0.765069
$385$	−6.98321e99	−2.01288
$386$	−7.53422e99	−1.96529
$387$	−5.22889e98	−0.123463
$388$	−1.24676e99	−0.266539
$389$	9.27222e99	1.79525	0.897623	−	0.440764i	$-0.145293\pi$
$389$	9.27222e99	1.79525	0.897623	+	0.440764i	$0.145293\pi$
$390$	−2.24260e99	−0.393336
$391$	7.48594e99	1.18971
$392$	−8.62136e99	−1.24183
$393$	−3.59890e99	−0.469956
$394$	−6.48432e99	−0.767821
$395$	−1.23522e99	−0.132665
$396$	2.68231e100	2.61363
$397$	1.27500e100	1.12739	0.563696	−	0.825983i	$-0.309379\pi$
$397$	1.27500e100	1.12739	0.563696	+	0.825983i	$0.309379\pi$
$398$	−2.77960e100	−2.23092
$399$	9.67819e99	0.705241
$400$	−3.28653e99	−0.217484
$401$	−6.24513e98	−0.0375390	−0.0187695	−	0.999824i	$-0.505975\pi$
$401$	−6.24513e98	−0.0375390	−0.0187695	+	0.999824i	$0.505975\pi$
$402$	−9.66520e99	−0.527847
$403$	−7.02389e99	−0.348605
$404$	2.80761e100	1.26664
$405$	4.88014e99	0.200177
$406$	−3.24275e100	−1.20965
$407$	4.94555e100	1.67815
$408$	2.31362e100	0.714292
$409$	−1.43353e100	−0.402772	−0.201386	−	0.979512i	$-0.564545\pi$
$409$	−1.43353e100	−0.402772	−0.201386	+	0.979512i	$0.564545\pi$
$410$	−4.48881e100	−1.14803
$411$	−9.84977e99	−0.229359
$412$	5.61305e100	1.19030
$413$	−4.84213e99	−0.0935319
$414$	−1.29019e101	−2.27060
$415$	−1.11179e101	−1.78308
$416$	−1.59307e100	−0.232885
$417$	−5.69297e100	−0.758754
$418$	2.58667e101	3.14379
$419$	−6.66038e100	−0.738345	−0.369172	−	0.929361i	$-0.620359\pi$
$419$	−6.66038e100	−0.738345	−0.369172	+	0.929361i	$0.620359\pi$
$420$	1.32915e101	1.34423
$421$	9.78282e100	0.902824	0.451412	−	0.892316i	$-0.350920\pi$
$421$	9.78282e100	0.902824	0.451412	+	0.892316i	$0.350920\pi$
$422$	1.16262e101	0.979286
$423$	5.42026e100	0.416789
$424$	−7.28457e100	−0.511468
$425$	1.56576e100	0.100404
$426$	−7.40236e100	−0.433612
$427$	7.76342e100	0.415509
$428$	−2.66688e101	−1.30442
$429$	−9.27408e100	−0.414634
$430$	−6.92502e100	−0.283065
$431$	−1.74766e101	−0.653255	−0.326627	−	0.945153i	$-0.605912\pi$
$431$	−1.74766e101	−0.653255	−0.326627	+	0.945153i	$0.605912\pi$
$432$	−3.84567e101	−1.31476
$433$	4.19340e101	1.31154	0.655771	−	0.754960i	$-0.272344\pi$
$433$	4.19340e101	1.31154	0.655771	+	0.754960i	$0.272344\pi$
$434$	6.11091e101	1.74884
$435$	1.01458e101	0.265736
$436$	−1.13509e102	−2.72146
$437$	−8.47579e101	−1.86056
$438$	5.08129e101	1.02146
$439$	4.21020e101	0.775208	0.387604	−	0.921826i	$-0.373303\pi$
$439$	4.21020e101	0.775208	0.387604	+	0.921826i	$0.373303\pi$
$440$	1.89011e102	3.18829
$441$	−2.83786e101	−0.438637
$442$	3.70525e101	0.524880
$443$	7.09163e101	0.920879	0.460440	−	0.887691i	$-0.347692\pi$
$443$	7.09163e101	0.920879	0.460440	+	0.887691i	$0.347692\pi$
$444$	−9.41309e101	−1.12070
$445$	−1.31846e102	−1.43949
$446$	−3.71859e101	−0.372379
$447$	−3.53623e101	−0.324864
$448$	−7.70924e101	−0.649842
$449$	2.32998e102	1.80247	0.901235	−	0.433331i	$-0.142662\pi$
$449$	2.32998e102	1.80247	0.901235	+	0.433331i	$0.142662\pi$
$450$	−2.69856e101	−0.191624
$451$	−1.85631e102	−1.21019
$452$	4.29007e102	2.56823
$453$	5.77629e101	0.317591
$454$	−2.59666e102	−1.31149
$455$	1.13257e102	0.525565
$456$	−2.61954e102	−1.11707
$457$	1.91172e102	0.749290	0.374645	−	0.927168i	$-0.377765\pi$
$457$	1.91172e102	0.749290	0.374645	+	0.927168i	$0.377765\pi$
$458$	−4.20441e102	−1.51490
$459$	1.83214e102	0.606975
$460$	−1.16402e103	−3.54636
$461$	4.26059e101	0.119394	0.0596972	−	0.998217i	$-0.480986\pi$
$461$	4.26059e101	0.119394	0.0596972	+	0.998217i	$0.480986\pi$
$462$	8.06861e102	2.08009
$463$	2.74566e102	0.651297	0.325649	−	0.945491i	$-0.394417\pi$
$463$	2.74566e102	0.651297	0.325649	+	0.945491i	$0.394417\pi$
$464$	3.51856e102	0.768108
$465$	−1.91196e102	−0.384185
$466$	1.20815e103	2.23493
$467$	−4.27291e102	−0.727820	−0.363910	−	0.931434i	$-0.618558\pi$
$467$	−4.27291e102	−0.727820	−0.363910	+	0.931434i	$0.618558\pi$
$468$	−4.35029e102	−0.682422
$469$	4.88118e102	0.705295
$470$	7.17846e102	0.955573
$471$	−3.95294e102	−0.484859
$472$	1.31059e102	0.148150
$473$	−2.86379e102	−0.298392
$474$	1.42721e102	0.137095
$475$	−1.77280e102	−0.157020
$476$	−2.19603e103	−1.79379
$477$	−2.39783e102	−0.180660
$478$	−1.88765e103	−1.31204
$479$	−1.51774e103	−0.973375	−0.486687	−	0.873576i	$-0.661795\pi$
$479$	−1.51774e103	−0.973375	−0.486687	+	0.873576i	$0.661795\pi$
$480$	−4.33646e102	−0.256654
$481$	−8.02093e102	−0.438167
$482$	5.74792e103	2.89868
$483$	−2.64386e103	−1.23104
$484$	9.72053e103	4.17968
$485$	−2.89185e102	−0.114847
$486$	−5.00280e103	−1.83534
$487$	3.72781e103	1.26354	0.631769	−	0.775157i	$-0.282329\pi$
$487$	3.72781e103	1.26354	0.631769	+	0.775157i	$0.282329\pi$
$488$	−2.10128e103	−0.658145
$489$	−3.59345e102	−0.104021
$490$	−3.75839e103	−1.00566
$491$	2.54355e102	0.0629220	0.0314610	−	0.999505i	$-0.489984\pi$
$491$	2.54355e102	0.0629220	0.0314610	+	0.999505i	$0.489984\pi$
$492$	3.53321e103	0.808189
$493$	−1.67630e103	−0.354606
$494$	−4.19519e103	−0.820850
$495$	6.22160e103	1.12616
$496$	−6.63067e103	−1.11049
$497$	3.73839e103	0.579380
$498$	1.28460e104	1.84262
$499$	−4.02749e103	−0.534765	−0.267383	−	0.963590i	$-0.586159\pi$
$499$	−4.02749e103	−0.534765	−0.267383	+	0.963590i	$0.586159\pi$
$500$	−1.84427e104	−2.26714
$501$	−2.01161e103	−0.228977
$502$	−7.56936e103	−0.797933
$503$	1.76399e104	1.72238	0.861190	−	0.508283i	$-0.169720\pi$
$503$	1.76399e104	1.72238	0.861190	+	0.508283i	$0.169720\pi$
$504$	2.01379e104	1.82154
$505$	6.51223e103	0.545772
$506$	−7.06618e104	−5.48769
$507$	−5.96013e103	−0.428992
$508$	4.58957e104	3.06210
$509$	9.53375e103	0.589698	0.294849	−	0.955544i	$-0.404731\pi$
$509$	9.53375e103	0.589698	0.294849	+	0.955544i	$0.404731\pi$
$510$	1.00860e104	0.578451
$511$	−2.56619e104	−1.36485
$512$	4.33414e104	2.13801
$513$	−2.07440e104	−0.949237
$514$	−3.15131e104	−1.33786
$515$	1.30194e104	0.512878
$516$	5.45078e103	0.199272
$517$	2.96860e104	1.00732
$518$	6.97835e104	2.19815
$519$	2.79343e104	0.816947
$520$	−3.06547e104	−0.832469
$521$	−6.64948e104	−1.67701	−0.838504	−	0.544895i	$-0.816570\pi$
$521$	−6.64948e104	−1.67701	−0.838504	+	0.544895i	$0.816570\pi$
$522$	2.88908e104	0.676777
$523$	−8.67549e104	−1.88790	−0.943950	−	0.330087i	$-0.892922\pi$
$523$	−8.67549e104	−1.88790	−0.943950	+	0.330087i	$0.892922\pi$
$524$	−9.24590e104	−1.86937
$525$	−5.52989e103	−0.103892
$526$	−5.40557e104	−0.943823
$527$	3.15897e104	0.512668
$528$	−8.75489e104	−1.32082
$529$	1.60246e105	2.24774
$530$	−3.17563e104	−0.414199
$531$	4.31403e103	0.0523292
$532$	2.48641e105	2.80527
$533$	3.01066e104	0.315983
$534$	1.52339e105	1.48756
$535$	−6.18581e104	−0.562051
$536$	−1.32116e105	−1.11715
$537$	−7.52051e104	−0.591885
$538$	1.83560e105	1.34481
$539$	−1.55426e105	−1.06012
$540$	−2.84887e105	−1.80931
$541$	6.82949e104	0.403918	0.201959	−	0.979394i	$-0.435269\pi$
$541$	6.82949e104	0.403918	0.201959	+	0.979394i	$0.435269\pi$
$542$	2.20627e105	1.21531
$543$	4.63780e104	0.237968
$544$	7.16477e104	0.342487
$545$	−2.63284e105	−1.17263
$546$	−1.30861e105	−0.543116
$547$	7.55745e104	0.292323	0.146162	−	0.989261i	$-0.453308\pi$
$547$	7.55745e104	0.292323	0.146162	+	0.989261i	$0.453308\pi$
$548$	−2.53049e105	−0.912333
$549$	−6.91672e104	−0.232469
$550$	−1.47796e105	−0.463127
$551$	1.89796e105	0.554562
$552$	7.15599e105	1.94991
$553$	−7.20778e104	−0.183182
$554$	6.33808e105	1.50256
$555$	−2.18336e105	−0.482887
$556$	−1.46257e106	−3.01813
$557$	3.10589e105	0.598082	0.299041	−	0.954240i	$-0.403333\pi$
$557$	3.10589e105	0.598082	0.299041	+	0.954240i	$0.403333\pi$
$558$	−5.44443e105	−0.978444
$559$	4.64463e104	0.0779107
$560$	1.06916e106	1.67419
$561$	4.17098e105	0.609773
$562$	−3.43928e105	−0.469482
$563$	1.57841e105	0.201209	0.100604	−	0.994926i	$-0.467922\pi$
$563$	1.57841e105	0.201209	0.100604	+	0.994926i	$0.467922\pi$
$564$	−5.65027e105	−0.672704
$565$	9.95078e105	1.10660
$566$	1.55779e106	1.61836
$567$	2.84767e105	0.276403
$568$	−1.01185e106	−0.917710
$569$	1.98176e106	1.67969	0.839845	−	0.542826i	$-0.182646\pi$
$569$	1.98176e106	1.67969	0.839845	+	0.542826i	$0.182646\pi$
$570$	−1.14196e106	−0.904628
$571$	−2.42412e106	−1.79500	−0.897499	−	0.441017i	$-0.854618\pi$
$571$	−2.42412e106	−1.79500	−0.897499	+	0.441017i	$0.854618\pi$
$572$	−2.38259e106	−1.64931
$573$	−4.47635e105	−0.289715
$574$	−2.61933e106	−1.58519
$575$	4.84287e105	0.274088
$576$	6.86844e105	0.363574
$577$	1.24532e106	0.616612	0.308306	−	0.951287i	$-0.400238\pi$
$577$	1.24532e106	0.616612	0.308306	+	0.951287i	$0.400238\pi$
$578$	2.15727e106	0.999272
$579$	1.37553e106	0.596136
$580$	2.60655e106	1.05703
$581$	−6.48755e106	−2.46207
$582$	3.34134e105	0.118682
$583$	−1.31326e106	−0.436628
$584$	6.94576e106	2.16185
$585$	−1.00905e106	−0.294043
$586$	9.18871e106	2.50724
$587$	−2.48954e106	−0.636138	−0.318069	−	0.948068i	$-0.603034\pi$
$587$	−2.48954e106	−0.636138	−0.318069	+	0.948068i	$0.603034\pi$
$588$	2.95829e106	0.707967
$589$	−3.57667e106	−0.801752
$590$	5.71340e105	0.119975
$591$	1.18385e106	0.232905
$592$	−7.57189e106	−1.39578
$593$	5.97288e106	1.03176	0.515880	−	0.856661i	$-0.327465\pi$
$593$	5.97288e106	1.03176	0.515880	+	0.856661i	$0.327465\pi$
$594$	−1.72941e107	−2.79975
$595$	−5.09369e106	−0.772911
$596$	−9.08490e106	−1.29223
$597$	5.07474e106	0.676709
$598$	1.14603e107	1.43285
$599$	−3.11667e106	−0.365391	−0.182695	−	0.983170i	$-0.558482\pi$
$599$	−3.11667e106	−0.365391	−0.182695	+	0.983170i	$0.558482\pi$
$600$	1.49675e106	0.164560
$601$	1.07889e107	1.11252	0.556261	−	0.831008i	$-0.312236\pi$
$601$	1.07889e107	1.11252	0.556261	+	0.831008i	$0.312236\pi$
$602$	−4.04091e106	−0.390854
$603$	−4.34883e106	−0.394598
$604$	1.48398e107	1.26330
$605$	2.25467e107	1.80095
$606$	−7.52443e106	−0.563998
$607$	−2.43291e107	−1.71144	−0.855721	−	0.517438i	$-0.826886\pi$
$607$	−2.43291e107	−1.71144	−0.855721	+	0.517438i	$0.826886\pi$
$608$	−8.11215e106	−0.535609
$609$	5.92031e106	0.366926
$610$	−9.16033e106	−0.532982
$611$	−4.81462e106	−0.263012
$612$	1.95653e107	1.00359
$613$	1.09299e107	0.526489	0.263244	−	0.964729i	$-0.415207\pi$
$613$	1.09299e107	0.526489	0.263244	+	0.964729i	$0.415207\pi$
$614$	−6.01822e107	−2.72261
$615$	8.19526e106	0.348234
$616$	1.10292e108	4.40238
$617$	−1.17702e107	−0.441371	−0.220686	−	0.975345i	$-0.570830\pi$
$617$	−1.17702e107	−0.441371	−0.220686	+	0.975345i	$0.570830\pi$
$618$	−1.50431e107	−0.530005
$619$	3.82718e107	1.26704	0.633518	−	0.773728i	$-0.281610\pi$
$619$	3.82718e107	1.26704	0.633518	+	0.773728i	$0.281610\pi$
$620$	−4.91199e107	−1.52819
$621$	5.66679e107	1.65695
$622$	−1.01850e108	−2.79917
$623$	−7.69355e107	−1.98764
$624$	1.41991e107	0.344869
$625$	−3.61175e107	−0.824779
$626$	1.17203e108	2.51667
$627$	−4.72250e107	−0.953612
$628$	−1.01555e108	−1.92865
$629$	3.60738e107	0.644380
$630$	8.77890e107	1.47512
$631$	−2.26558e107	−0.358137	−0.179068	−	0.983837i	$-0.557308\pi$
$631$	−2.26558e107	−0.358137	−0.179068	+	0.983837i	$0.557308\pi$
$632$	1.95089e107	0.290152
$633$	−2.12261e107	−0.297049
$634$	−6.25865e107	−0.824222
$635$	1.06455e108	1.31940
$636$	2.49959e107	0.291588
$637$	2.52077e107	0.276799
$638$	1.58231e108	1.63567
$639$	−3.33067e107	−0.324152
$640$	1.43095e108	1.31128
$641$	9.09145e107	0.784511	0.392255	−	0.919856i	$-0.371695\pi$
$641$	9.09145e107	0.784511	0.392255	+	0.919856i	$0.371695\pi$
$642$	7.14727e107	0.580820
$643$	−7.27867e107	−0.557095	−0.278548	−	0.960422i	$-0.589853\pi$
$643$	−7.27867e107	−0.557095	−0.278548	+	0.960422i	$0.589853\pi$
$644$	−6.79230e108	−4.89679
$645$	1.26431e107	0.0858625
$646$	1.88677e108	1.20717
$647$	−2.33270e108	−1.40619	−0.703094	−	0.711097i	$-0.748199\pi$
$647$	−2.33270e108	−1.40619	−0.703094	+	0.711097i	$0.748199\pi$
$648$	−7.70765e107	−0.437808
$649$	2.36273e107	0.126472
$650$	2.39703e107	0.120923
$651$	−1.11567e108	−0.530480
$652$	−9.23189e107	−0.413769
$653$	3.26494e108	1.37949	0.689745	−	0.724053i	$-0.257723\pi$
$653$	3.26494e108	1.37949	0.689745	+	0.724053i	$0.257723\pi$
$654$	3.04207e108	1.21178
$655$	−2.14458e108	−0.805476
$656$	2.84211e108	1.00657
$657$	2.28631e108	0.763604
$658$	4.18880e108	1.31945
$659$	−3.11478e108	−0.925419	−0.462710	−	0.886510i	$-0.653123\pi$
$659$	−3.11478e108	−0.925419	−0.462710	+	0.886510i	$0.653123\pi$
$660$	−6.48561e108	−1.81765
$661$	−3.98965e108	−1.05482	−0.527411	−	0.849610i	$-0.676837\pi$
$661$	−3.98965e108	−1.05482	−0.527411	+	0.849610i	$0.676837\pi$
$662$	−8.48575e107	−0.211669
$663$	−6.76470e107	−0.159213
$664$	1.75595e109	3.89979
$665$	5.76722e108	1.20874
$666$	−6.21726e108	−1.22982
$667$	−5.18478e108	−0.968023
$668$	−5.16800e108	−0.910812
$669$	6.78905e107	0.112954
$670$	−5.75948e108	−0.904697
$671$	−3.78819e108	−0.561842
$672$	−2.53043e108	−0.354386
$673$	1.40437e109	1.85738	0.928691	−	0.370854i	$-0.120935\pi$
$673$	1.40437e109	1.85738	0.928691	+	0.370854i	$0.120935\pi$
$674$	5.34043e108	0.667068
$675$	1.18527e108	0.139837
$676$	−1.53121e109	−1.70642
$677$	1.32329e109	1.39313	0.696563	−	0.717496i	$-0.254712\pi$
$677$	1.32329e109	1.39313	0.696563	+	0.717496i	$0.254712\pi$
$678$	−1.14974e109	−1.14356
$679$	−1.68746e108	−0.158580
$680$	1.37868e109	1.22425
$681$	4.74074e108	0.397816
$682$	−2.98184e109	−2.36475
$683$	4.72910e108	0.354472	0.177236	−	0.984168i	$-0.443284\pi$
$683$	4.72910e108	0.354472	0.177236	+	0.984168i	$0.443284\pi$
$684$	−2.21524e109	−1.56950
$685$	−5.86946e108	−0.393108
$686$	1.36356e109	0.863368
$687$	7.67602e108	0.459517
$688$	4.38461e108	0.248185
$689$	2.12991e108	0.114004
$690$	3.11958e109	1.57909
$691$	−3.11698e109	−1.49221	−0.746103	−	0.665831i	$-0.768077\pi$
$691$	−3.11698e109	−1.49221	−0.746103	+	0.665831i	$0.768077\pi$
$692$	7.17656e109	3.24961
$693$	3.63045e109	1.55500
$694$	3.32032e109	1.34537
$695$	−3.39243e109	−1.30046
$696$	−1.60242e109	−0.581193
$697$	−1.35403e109	−0.464694
$698$	−9.57088e109	−3.10826
$699$	−2.20573e109	−0.677924
$700$	−1.42068e109	−0.413258
$701$	−1.30468e109	−0.359220	−0.179610	−	0.983738i	$-0.557484\pi$
$701$	−1.30468e109	−0.359220	−0.179610	+	0.983738i	$0.557484\pi$
$702$	2.80484e109	0.731021
$703$	−4.08438e109	−1.00773
$704$	3.76175e109	0.878703
$705$	−1.31058e109	−0.289856
$706$	5.68466e108	0.119048
$707$	3.80004e109	0.753599
$708$	−4.49710e108	−0.0844602
$709$	−6.04389e109	−1.07507	−0.537535	−	0.843242i	$-0.680644\pi$
$709$	−6.04389e109	−1.07507	−0.537535	+	0.843242i	$0.680644\pi$
$710$	−4.41106e109	−0.743184
$711$	6.42167e108	0.102487
$712$	2.08237e110	3.14832
$713$	9.77064e109	1.39951
$714$	5.88541e109	0.798722
$715$	−5.52641e109	−0.710658
$716$	−1.93209e110	−2.35437
$717$	3.44629e109	0.397983
$718$	−2.32189e109	−0.254127
$719$	1.43672e110	1.49042	0.745211	−	0.666829i	$-0.232349\pi$
$719$	1.43672e110	1.49042	0.745211	+	0.666829i	$0.232349\pi$
$720$	−9.52558e109	−0.936678
$721$	7.59715e109	0.708179
$722$	−1.32040e109	−0.116687
$723$	−1.04940e110	−0.879260
$724$	1.19149e110	0.946578
$725$	−1.08445e109	−0.0816951
$726$	−2.60512e110	−1.86109
$727$	−5.19727e109	−0.352128	−0.176064	−	0.984379i	$-0.556337\pi$
$727$	−5.19727e109	−0.352128	−0.176064	+	0.984379i	$0.556337\pi$
$728$	−1.78877e110	−1.14947
$729$	5.56709e109	0.339327
$730$	3.02793e110	1.75072
$731$	−2.08891e109	−0.114578
$732$	7.21023e109	0.375208
$733$	2.05509e110	1.01468	0.507338	−	0.861747i	$-0.330629\pi$
$733$	2.05509e110	1.01468	0.507338	+	0.861747i	$0.330629\pi$
$734$	−3.12422e110	−1.46367
$735$	6.86173e109	0.305050
$736$	2.21605e110	0.934940
$737$	−2.38179e110	−0.953685
$738$	2.33365e110	0.886884
$739$	−3.15476e110	−1.13803	−0.569017	−	0.822326i	$-0.692676\pi$
$739$	−3.15476e110	−1.13803	−0.569017	+	0.822326i	$0.692676\pi$
$740$	−5.60925e110	−1.92081
$741$	7.65918e109	0.248990
$742$	−1.85306e110	−0.571924
$743$	4.29386e110	1.25828	0.629142	−	0.777291i	$-0.283407\pi$
$743$	4.29386e110	1.25828	0.629142	+	0.777291i	$0.283407\pi$
$744$	3.01973e110	0.840254
$745$	−2.10724e110	−0.556797
$746$	1.94337e110	0.487654
$747$	5.77999e110	1.37748
$748$	1.07156e111	2.42553
$749$	−3.60956e110	−0.776077
$750$	4.94266e110	1.00949
$751$	−8.77821e110	−1.70321	−0.851606	−	0.524183i	$-0.824371\pi$
$751$	−8.77821e110	−1.70321	−0.851606	+	0.524183i	$0.824371\pi$
$752$	−4.54508e110	−0.837827
$753$	1.38194e110	0.242038
$754$	−2.56627e110	−0.427076
$755$	3.44208e110	0.544332
$756$	−1.66238e111	−2.49828
$757$	−2.92955e110	−0.418417	−0.209208	−	0.977871i	$-0.567089\pi$
$757$	−2.92955e110	−0.418417	−0.209208	+	0.977871i	$0.567089\pi$
$758$	8.40773e110	1.14134
$759$	1.29008e111	1.66459
$760$	−1.56098e111	−1.91459
$761$	7.47929e110	0.872070	0.436035	−	0.899930i	$-0.356382\pi$
$761$	7.47929e110	0.872070	0.436035	+	0.899930i	$0.356382\pi$
$762$	−1.23001e111	−1.36346
$763$	−1.53632e111	−1.61915
$764$	−1.15001e111	−1.15242
$765$	4.53816e110	0.432428
$766$	3.33871e111	3.02532
$767$	−3.83200e109	−0.0330220
$768$	−1.31833e111	−1.08048
$769$	−2.13743e111	−1.66620	−0.833101	−	0.553121i	$-0.813437\pi$
$769$	−2.13743e111	−1.66620	−0.833101	+	0.553121i	$0.813437\pi$
$770$	4.80808e111	3.56515
$771$	5.75337e110	0.405816
$772$	3.53386e111	2.37128
$773$	−3.29736e110	−0.210502	−0.105251	−	0.994446i	$-0.533565\pi$
$773$	−3.29736e110	−0.210502	−0.105251	+	0.994446i	$0.533565\pi$
$774$	3.60020e110	0.218675
$775$	2.04363e110	0.118110
$776$	4.56737e110	0.251183
$777$	−1.27404e111	−0.666768
$778$	−6.38410e111	−3.17969
$779$	1.53307e111	0.726726
$780$	1.05187e111	0.474590
$781$	−1.82416e111	−0.783426
$782$	−5.15421e111	−2.10718
$783$	−1.26895e111	−0.493874
$784$	2.37965e111	0.881747
$785$	−2.35555e111	−0.831020
$786$	2.47792e111	0.832374
$787$	3.57848e111	1.14465	0.572324	−	0.820028i	$-0.306042\pi$
$787$	3.57848e111	1.14465	0.572324	+	0.820028i	$0.306042\pi$
$788$	3.04141e111	0.926436
$789$	9.86900e110	0.286291
$790$	8.50471e110	0.234972
$791$	5.80652e111	1.52799
$792$	−9.82634e111	−2.46304
$793$	6.14386e110	0.146698
$794$	−8.77860e111	−1.99681
$795$	5.79778e110	0.125640
$796$	1.30375e112	2.69178
$797$	−6.82080e111	−1.34181	−0.670903	−	0.741545i	$-0.734093\pi$
$797$	−6.82080e111	−1.34181	−0.670903	+	0.741545i	$0.734093\pi$
$798$	−6.66362e111	−1.24911
$799$	2.16536e111	0.386793
$800$	4.63509e110	0.0789031
$801$	6.85446e111	1.11204
$802$	4.29989e110	0.0664882
$803$	1.25218e112	1.84552
$804$	4.53337e111	0.636888
$805$	−1.57547e112	−2.10993
$806$	4.83609e111	0.617440
$807$	−3.35127e111	−0.407923
$808$	−1.02854e112	−1.19366
$809$	1.00115e112	1.10785	0.553926	−	0.832566i	$-0.313129\pi$
$809$	1.00115e112	1.10785	0.553926	+	0.832566i	$0.313129\pi$
$810$	−3.36007e111	−0.354548
$811$	1.35467e112	1.36311	0.681554	−	0.731768i	$-0.261305\pi$
$811$	1.35467e112	1.36311	0.681554	+	0.731768i	$0.261305\pi$
$812$	1.52098e112	1.45954
$813$	−4.02800e111	−0.368641
$814$	−3.40511e112	−2.97229
$815$	−2.14133e111	−0.178286
$816$	−6.38599e111	−0.507174
$817$	2.36512e111	0.179186
$818$	9.87013e111	0.713379
$819$	−5.88803e111	−0.406013
$820$	2.10543e112	1.38519
$821$	−3.04484e112	−1.91141	−0.955703	−	0.294334i	$-0.904902\pi$
$821$	−3.04484e112	−1.91141	−0.955703	+	0.294334i	$0.904902\pi$
$822$	6.78176e111	0.406235
$823$	−2.70754e112	−1.54768	−0.773839	−	0.633382i	$-0.781666\pi$
$823$	−2.70754e112	−1.54768	−0.773839	+	0.633382i	$0.781666\pi$
$824$	−2.05628e112	−1.12172
$825$	2.69833e111	0.140481
$826$	3.33390e111	0.165661
$827$	−7.10593e111	−0.337022	−0.168511	−	0.985700i	$-0.553896\pi$
$827$	−7.10593e111	−0.337022	−0.168511	+	0.985700i	$0.553896\pi$
$828$	6.05151e112	2.73965
$829$	2.05742e112	0.889148	0.444574	−	0.895742i	$-0.353355\pi$
$829$	2.05742e112	0.889148	0.444574	+	0.895742i	$0.353355\pi$
$830$	7.65489e112	3.15815
$831$	−1.15715e112	−0.455774
$832$	−6.10098e111	−0.229431
$833$	−1.13371e112	−0.407069
$834$	3.91972e112	1.34389
$835$	−1.19872e112	−0.392452
$836$	−1.21325e113	−3.79323
$837$	2.39131e112	0.714013
$838$	4.58580e112	1.30774
$839$	3.68294e112	1.00314	0.501568	−	0.865118i	$-0.332757\pi$
$839$	3.68294e112	1.00314	0.501568	+	0.865118i	$0.332757\pi$
$840$	−4.86918e112	−1.26679
$841$	−2.86287e112	−0.711470
$842$	−6.73566e112	−1.59906
$843$	6.27912e111	0.142409
$844$	−5.45317e112	−1.18159
$845$	−3.55163e112	−0.735266
$846$	−3.73196e112	−0.738206
$847$	1.31565e113	2.48674
$848$	2.01067e112	0.363162
$849$	−2.84407e112	−0.490901
$850$	−1.07806e112	−0.177833
$851$	1.11576e113	1.75906
$852$	3.47201e112	0.523186
$853$	−7.64268e111	−0.110080	−0.0550399	−	0.998484i	$-0.517529\pi$
$853$	−7.64268e111	−0.110080	−0.0550399	+	0.998484i	$0.517529\pi$
$854$	−5.34527e112	−0.735938
$855$	−5.13823e112	−0.676267
$856$	9.76981e112	1.22927
$857$	−3.36893e112	−0.405257	−0.202629	−	0.979256i	$-0.564948\pi$
$857$	−3.36893e112	−0.405257	−0.202629	+	0.979256i	$0.564948\pi$
$858$	6.38539e112	0.734390
$859$	−1.22161e113	−1.34337	−0.671686	−	0.740836i	$-0.734430\pi$
$859$	−1.22161e113	−1.34337	−0.671686	+	0.740836i	$0.734430\pi$
$860$	3.24811e112	0.341540
$861$	4.78213e112	0.480839
$862$	1.20330e113	1.15703
$863$	−6.32419e112	−0.581553	−0.290776	−	0.956791i	$-0.593914\pi$
$863$	−6.32419e112	−0.581553	−0.290776	+	0.956791i	$0.593914\pi$
$864$	5.42367e112	0.476995
$865$	1.66460e113	1.40020
$866$	−2.88724e113	−2.32297
$867$	−3.93855e112	−0.303111
$868$	−2.86626e113	−2.11012
$869$	3.51706e112	0.247696
$870$	−6.98558e112	−0.470664
$871$	3.86290e112	0.249009
$872$	4.15829e113	2.56466
$873$	1.50342e112	0.0887223
$874$	5.83575e113	3.29539
$875$	−2.49618e113	−1.34885
$876$	−2.38333e113	−1.23247
$877$	9.28246e112	0.459387	0.229694	−	0.973263i	$-0.426228\pi$
$877$	9.28246e112	0.459387	0.229694	+	0.973263i	$0.426228\pi$
$878$	−2.89880e113	−1.37303
$879$	−1.67759e113	−0.760526
$880$	−5.21702e113	−2.26381
$881$	−3.60693e112	−0.149818	−0.0749091	−	0.997190i	$-0.523867\pi$
$881$	−3.60693e112	−0.149818	−0.0749091	+	0.997190i	$0.523867\pi$
$882$	1.95392e113	0.776903
$883$	4.15957e112	0.158329	0.0791646	−	0.996862i	$-0.474775\pi$
$883$	4.15957e112	0.158329	0.0791646	+	0.996862i	$0.474775\pi$
$884$	−1.73791e113	−0.633309
$885$	−1.04310e112	−0.0363923
$886$	−4.88273e113	−1.63104
$887$	3.62921e113	1.16079	0.580394	−	0.814336i	$-0.302899\pi$
$887$	3.62921e113	1.16079	0.580394	+	0.814336i	$0.302899\pi$
$888$	3.44838e113	1.05613
$889$	6.21189e113	1.82182
$890$	9.07788e113	2.54958
$891$	−1.38953e113	−0.373746
$892$	1.74417e113	0.449305
$893$	−2.45168e113	−0.604898
$894$	2.43477e113	0.575391
$895$	−4.48146e113	−1.01446
$896$	8.34994e113	1.81061
$897$	−2.09231e113	−0.434628
$898$	−1.60424e114	−3.19249
$899$	−2.18791e113	−0.417140
$900$	1.26573e113	0.231210
$901$	−9.57918e112	−0.167658
$902$	1.27811e114	2.14346
$903$	7.37752e112	0.118558
$904$	−1.57162e114	−2.42026
$905$	2.76366e113	0.407863
$906$	−3.97709e113	−0.562509
$907$	−1.08819e114	−1.47511	−0.737555	−	0.675287i	$-0.764020\pi$
$907$	−1.08819e114	−1.47511	−0.737555	+	0.675287i	$0.764020\pi$
$908$	1.21794e114	1.58241
$909$	−3.38559e113	−0.421624
$910$	−7.79797e113	−0.930868
$911$	−1.62881e114	−1.86386	−0.931928	−	0.362643i	$-0.881875\pi$
$911$	−1.62881e114	−1.86386	−0.931928	+	0.362643i	$0.881875\pi$
$912$	7.23039e113	0.793160
$913$	3.16562e114	3.32915
$914$	−1.31625e114	−1.32712
$915$	1.67241e113	0.161670
$916$	1.97204e114	1.82784
$917$	−1.25141e114	−1.11220
$918$	−1.26147e114	−1.07506
$919$	1.03952e114	0.849544	0.424772	−	0.905300i	$-0.360354\pi$
$919$	1.03952e114	0.849544	0.424772	+	0.905300i	$0.360354\pi$
$920$	4.26424e114	3.34203
$921$	1.09875e114	0.825855
$922$	−2.93350e113	−0.211468
$923$	2.95851e113	0.204554
$924$	−3.78451e114	−2.50980
$925$	2.33372e113	0.148454
$926$	−1.89044e114	−1.15356
$927$	−6.76858e113	−0.396212
$928$	−4.96234e113	−0.278669
$929$	−1.28539e114	−0.692515	−0.346258	−	0.938139i	$-0.612548\pi$
$929$	−1.28539e114	−0.692515	−0.346258	+	0.938139i	$0.612548\pi$
$930$	1.31642e114	0.680458
$931$	1.28361e114	0.636607
$932$	−5.66672e114	−2.69662
$933$	1.85948e114	0.849077
$934$	2.94198e114	1.28910
$935$	2.48548e114	1.04511
$936$	1.59368e114	0.643105
$937$	2.31573e114	0.896838	0.448419	−	0.893823i	$-0.351987\pi$
$937$	2.31573e114	0.896838	0.448419	+	0.893823i	$0.351987\pi$
$938$	−3.36079e114	−1.24920
$939$	−2.13978e114	−0.763387
$940$	−3.36699e114	−1.15297
$941$	2.91746e114	0.958970	0.479485	−	0.877550i	$-0.340824\pi$
$941$	2.91746e114	0.958970	0.479485	+	0.877550i	$0.340824\pi$
$942$	2.72168e114	0.858771
$943$	−4.18800e114	−1.26855
$944$	−3.61747e113	−0.105192
$945$	−3.85588e114	−1.07646
$946$	1.97178e114	0.528505
$947$	3.56216e113	0.0916725	0.0458363	−	0.998949i	$-0.485405\pi$
$947$	3.56216e113	0.0916725	0.0458363	+	0.998949i	$0.485405\pi$
$948$	−6.69418e113	−0.165416
$949$	−2.03085e114	−0.481868
$950$	1.22060e114	0.278110
$951$	1.14265e114	0.250013
$952$	8.04493e114	1.69044
$953$	−2.59650e114	−0.523976	−0.261988	−	0.965071i	$-0.584378\pi$
$953$	−2.59650e114	−0.523976	−0.261988	+	0.965071i	$0.584378\pi$
$954$	1.65096e114	0.319980
$955$	−2.66745e114	−0.496554
$956$	8.85383e114	1.58308
$957$	−2.88883e114	−0.496150
$958$	1.04499e115	1.72402
$959$	−3.42497e114	−0.542801
$960$	−1.66074e114	−0.252847
$961$	−2.71368e114	−0.396925
$962$	5.52257e114	0.776070
$963$	3.21589e114	0.434200
$964$	−2.69601e115	−3.49748
$965$	8.19676e114	1.02174
$966$	1.82035e115	2.18039
$967$	1.74914e114	0.201328	0.100664	−	0.994920i	$-0.467903\pi$
$967$	1.74914e114	0.201328	0.100664	+	0.994920i	$0.467903\pi$
$968$	−3.56101e115	−3.93887
$969$	−3.44469e114	−0.366171
$970$	1.99110e114	0.203414
$971$	−6.71765e114	−0.659594	−0.329797	−	0.944052i	$-0.606980\pi$
$971$	−6.71765e114	−0.659594	−0.329797	+	0.944052i	$0.606980\pi$
$972$	2.34651e115	2.21448
$973$	−1.97957e115	−1.79567
$974$	−2.56667e115	−2.23795
$975$	−4.37628e113	−0.0366798
$976$	5.79991e114	0.467308
$977$	1.05958e115	0.820717	0.410358	−	0.911924i	$-0.365404\pi$
$977$	1.05958e115	0.820717	0.410358	+	0.911924i	$0.365404\pi$
$978$	2.47416e114	0.184239
$979$	3.75409e115	2.68764
$980$	1.76284e115	1.21341
$981$	1.36877e115	0.905885
$982$	−1.75128e114	−0.111446
$983$	3.91715e113	0.0239696	0.0119848	−	0.999928i	$-0.496185\pi$
$983$	3.91715e113	0.0239696	0.0119848	+	0.999928i	$0.496185\pi$
$984$	−1.29435e115	−0.761625
$985$	7.05453e114	0.399184
$986$	1.15417e115	0.628070
$987$	−7.64753e114	−0.400231
$988$	1.96771e115	0.990419
$989$	−6.46095e114	−0.312780
$990$	−4.28369e115	−1.99463
$991$	−2.31884e115	−1.03857	−0.519284	−	0.854602i	$-0.673801\pi$
$991$	−2.31884e115	−1.03857	−0.519284	+	0.854602i	$0.673801\pi$
$992$	9.35145e114	0.402883
$993$	1.54925e114	0.0642060
$994$	−2.57396e115	−1.02618
$995$	3.02403e115	1.15984
$996$	−6.02527e115	−2.22327
$997$	−2.46494e115	−0.875069	−0.437535	−	0.899202i	$-0.644148\pi$
$997$	−2.46494e115	−0.875069	−0.437535	+	0.899202i	$0.644148\pi$
$998$	2.77300e115	0.947163
$999$	2.73076e115	0.897453

Twists

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

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

\(f(q)\)	\(=\)	\(q-6.88520e11 q^{2} +1.25704e18 q^{3} +3.22944e23 q^{4} +7.49066e26 q^{5} -8.65494e29 q^{6} +4.37098e32 q^{7} -1.18307e35 q^{8} -3.89426e36 q^{9} +O(q^{10})\)	\(q-6.88520e11 q^{2} +1.25704e18 q^{3} +3.22944e23 q^{4} +7.49066e26 q^{5} -8.65494e29 q^{6} +4.37098e32 q^{7} -1.18307e35 q^{8} -3.89426e36 q^{9} -5.15747e38 q^{10} -2.13283e40 q^{11} +4.05952e41 q^{12} +3.45913e42 q^{13} -3.00950e44 q^{14} +9.41603e44 q^{15} +3.26547e46 q^{16} -1.55573e47 q^{17} +2.68128e48 q^{18} +1.76144e49 q^{19} +2.41906e50 q^{20} +5.49447e50 q^{21} +1.46850e52 q^{22} -4.81185e52 q^{23} -1.48716e53 q^{24} -1.00645e53 q^{25} -2.38168e54 q^{26} -1.17767e55 q^{27} +1.41158e56 q^{28} +1.07750e56 q^{29} -6.48312e56 q^{30} -2.03054e57 q^{31} -4.60540e57 q^{32} -2.68105e58 q^{33} +1.07115e59 q^{34} +3.27415e59 q^{35} -1.25763e60 q^{36} -2.31877e60 q^{37} -1.21279e61 q^{38} +4.34825e60 q^{39} -8.86196e61 q^{40} +8.70352e61 q^{41} -3.78305e62 q^{42} +1.34272e62 q^{43} -6.88784e63 q^{44} -2.91706e63 q^{45} +3.31305e64 q^{46} -1.39186e64 q^{47} +4.10482e64 q^{48} +7.28729e64 q^{49} +6.92958e64 q^{50} -1.95561e65 q^{51} +1.11710e66 q^{52} +6.15735e65 q^{53} +8.10852e66 q^{54} -1.59763e67 q^{55} -5.17116e67 q^{56} +2.21419e67 q^{57} -7.41882e67 q^{58} -1.10779e67 q^{59} +3.04085e68 q^{60} +1.77613e68 q^{61} +1.39806e69 q^{62} -1.70217e69 q^{63} -1.76373e69 q^{64} +2.59112e69 q^{65} +1.84595e70 q^{66} +1.11673e70 q^{67} -5.02413e70 q^{68} -6.04867e70 q^{69} -2.25432e71 q^{70} +8.55276e70 q^{71} +4.60718e71 q^{72} -5.87097e71 q^{73} +1.59652e72 q^{74} -1.26514e71 q^{75} +5.68846e72 q^{76} -9.32255e72 q^{77} -2.99386e72 q^{78} -1.64901e72 q^{79} +2.44606e73 q^{80} +6.51496e72 q^{81} -5.99255e73 q^{82} -1.48423e74 q^{83} +1.77440e74 q^{84} -1.16534e74 q^{85} -9.24487e73 q^{86} +1.35446e74 q^{87} +2.52329e75 q^{88} -1.76014e75 q^{89} +2.00845e75 q^{90} +1.51198e75 q^{91} -1.55396e76 q^{92} -2.55246e75 q^{93} +9.58322e75 q^{94} +1.31944e76 q^{95} -5.78916e75 q^{96} -3.86061e75 q^{97} -5.01744e76 q^{98} +8.30580e76 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 264721893120 q^{2} + 14\!\cdots\!80 q^{3}+ \cdots - 48\!\cdots\!42 q^{9}+O(q^{10}) \) 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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