LMFDB - Embedded newform 9.68.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9,68,Mod(1,9)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 68, names="a")

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

chi := DirichletCharacter("9.1");

S:= CuspForms(chi, 68);

N := Newforms(S);

Level:	$ N $	$=$	$ 9 = 3^{2} $
Weight:	$ k $	$=$	$ 68 $
Character orbit:	$[\chi]$	$=$	9.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:	$255.861316737$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} + \cdots + 20\!\cdots\!00 $ x^5 - x^4 - 939384011925257456x^3 + 31046449413968483513911200x^2 + 156793504704482691874379743265203200*x + 20916736226052669578405116700517591609696000
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{40}\cdot 3^{20}\cdot 5^{4}\cdot 7^{2}\cdot 11\cdot 13\cdot 17 $
Twist minimal:	no (minimal twist has level 1)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.05046e8$ of defining polynomial
Character	$\chi$	$=$	9.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.43209e9 q^{2} -7.64738e19 q^{4} -3.83370e23 q^{5} -1.75533e27 q^{7} +1.88919e30 q^{8} +O(q^{10})$	\(q-8.43209e9 q^{2} -7.64738e19 q^{4} -3.83370e23 q^{5} -1.75533e27 q^{7} +1.88919e30 q^{8} +3.23261e33 q^{10} -1.38749e35 q^{11} -6.45579e36 q^{13} +1.48011e37 q^{14} -4.64430e39 q^{16} +8.51154e40 q^{17} -5.19929e42 q^{19} +2.93178e43 q^{20} +1.16995e45 q^{22} -2.49803e45 q^{23} +7.92100e46 q^{25} +5.44359e46 q^{26} +1.34237e47 q^{28} -1.20701e49 q^{29} -6.23103e49 q^{31} -2.39634e50 q^{32} -7.17701e50 q^{34} +6.72941e50 q^{35} +3.56231e51 q^{37} +4.38409e52 q^{38} -7.24259e53 q^{40} -4.51579e53 q^{41} +6.41012e54 q^{43} +1.06107e55 q^{44} +2.10636e55 q^{46} +1.03805e56 q^{47} -4.15297e56 q^{49} -6.67906e56 q^{50} +4.93699e56 q^{52} +9.61057e57 q^{53} +5.31924e58 q^{55} -3.31615e57 q^{56} +1.01776e59 q^{58} +4.77326e58 q^{59} -2.98642e58 q^{61} +5.25406e59 q^{62} +2.70600e60 q^{64} +2.47496e60 q^{65} -2.03266e61 q^{67} -6.50909e60 q^{68} -5.67430e60 q^{70} +6.57721e61 q^{71} -1.20026e62 q^{73} -3.00377e61 q^{74} +3.97609e62 q^{76} +2.43551e62 q^{77} +2.11502e63 q^{79} +1.78049e63 q^{80} +3.80776e63 q^{82} +3.25858e64 q^{83} -3.26307e64 q^{85} -5.40507e64 q^{86} -2.62124e65 q^{88} -6.19321e64 q^{89} +1.13320e64 q^{91} +1.91034e65 q^{92} -8.75297e65 q^{94} +1.99325e66 q^{95} -9.58970e65 q^{97} +3.50182e66 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5554901256 q^{2} + 35\!\cdots\!40 q^{4}+ \cdots - 32\!\cdots\!80 q^{8}+O(q^{10}) $ 5 * q - 5554901256 * q^2 + 350472004382296600640 * q^4 - 330744836907160597528950 * q^5 + 33632726520317603069152899256 * q^7 - 3261706912683890613885787783680 * q^8	$ 5 q - 5554901256 q^{2} + 35\!\cdots\!40 q^{4}+ \cdots - 12\!\cdots\!08 q^{98}+O(q^{100}) $ 5 * q - 5554901256 * q^2 + 350472004382296600640 * q^4 - 330744836907160597528950 * q^5 + 33632726520317603069152899256 * q^7 - 3261706912683890613885787783680 * q^8 - 5287067006725590176865792208438800 * q^10 - 203006981551281631527218766884109060 * q^11 + 177822884499813229570219072356766702 * q^13 - 290753425875732621001638220329334092480 * q^14 + 18266362102387359184851236859301091348480 * q^16 - 75241497376497768433807587544150344102906 * q^17 + 3998884102997801557241177083441346393295100 * q^19 + 15074831378020267075078660994426474705846400 * q^20 + 1841247880573095918698534631505063585473750432 * q^22 + 419660874260151197180020017656265233426170968 * q^23 + 175279270003539476977813620578789764300129011875 * q^25 - 21416337629300654169124097666599354769371062960 * q^26 + 14982728306233364463143494158089725946900023694848 * q^28 - 18733547497571939101380180097730717676662894233950 * q^29 + 365849822471060800690376948081970588059264217288160 * q^31 - 1591165751988992347021834813546084975107380979204096 * q^32 + 7817592154883765712044327776251337623941567500782480 * q^34 - 4524513410554564733118320802153390591800073874285200 * q^35 - 56389392588768330454829645685931746748224606527380394 * q^37 + 311451303114725973135630000941921262878276604287141280 * q^38 - 1886768345048751997029226502271137564956948018338432000 * q^40 - 113555410342193387626817165052765483601006465327103010 * q^41 + 6508304671085340439704487085091235272293711787089861092 * q^43 - 49369449025916528958230735645862721892607029455332199680 * q^44 + 79552980744915855625545425031873854934595203714399466560 * q^46 + 128023594814179612806658270243785955024442659789917751344 * q^47 - 771329931628098140389185999842408514636676231229111640435 * q^49 + 3919481096957516748709549462396523762064894577356951885000 * q^50 + 672997930141532412373441872733517069231734104886640838016 * q^52 - 1046765637663599376128164919371991945530174947374966490822 * q^53 + 66942753107037791618627925698973654692865254681839842449400 * q^55 - 164746697572073652525116376573879338548073684495562965299200 * q^56 - 61137956327686914952144218440883774026433838412764071303120 * q^58 + 305790738450157663862297559072875409310668010528073986627500 * q^59 - 1137073938560830615075915586814815057732339483683661907261890 * q^61 + 2197846158587850633685459639420082896328263471188214086352128 * q^62 + 6319522016046144185339524760669817745425888834305772325437440 * q^64 - 17054811715744261908524663200762497384422620874171150779332900 * q^65 - 11903044989154510038048691316368615691989662189985228212145844 * q^67 - 81678078346863753465296539273887923556029130514802676363013248 * q^68 - 262230520857255225386866306600227822969034558447869634844476800 * q^70 + 112169920751076079402369696382938772997794660792753351047772040 * q^71 - 304061800126793465330511223999919410949872771539142074085150718 * q^73 - 921008183492866283853128598587548033201480125306857491530532080 * q^74 + 2076567511595431334156222539483464646991367936386801535927366400 * q^76 - 5018131462902473316385236580394702266689060843204393245422638432 * q^77 + 2994504004186984867153587379277397704940492543518650040519815600 * q^79 + 15198313539268031787508043655897930060910106707913153957798092800 * q^80 - 17699192859543411700126655967680257958811789647196784631772687408 * q^82 + 47352447490969920981134910324833485175129837165394254819464203588 * q^83 - 39334683419450918052255554891934302881190270223971152032299949300 * q^85 - 49699285824187070765304255878405188987409992188766856866277888160 * q^86 + 430257938373980762031490456670953174964896424231159472833711400960 * q^88 + 11988583288945008097106276879882407358382120120051430459779195150 * q^89 + 232376792470844928685229521265495056030669651812230900159714321360 * q^91 + 986292139147147043544160128863480288571347084605459783959415954944 * q^92 - 4927754861315037106505826725122767884474273389521981763161929704320 * q^94 + 2022377488008256909110920831467942087293430657450319294506545963000 * q^95 + 2897556354128426205551279700626881901316694084191208635967319112106 * q^97 - 12426885478574475523720488645206508880013087100212747539199689691208 * q^98

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 $

$2$	−8.43209e9	−0.694114	−0.347057	−	0.937844i	$-0.612819\pi$
$2$	−8.43209e9	−0.694114	−0.347057	+	0.937844i	$0.612819\pi$
$3$	0	0
$3$	0	0
$4$	−7.64738e19	−0.518206
$5$	−3.83370e23	−1.47273	−0.736365	−	0.676584i	$-0.763459\pi$
$5$	−3.83370e23	−1.47273	−0.736365	+	0.676584i	$0.763459\pi$
$6$	0	0
$7$	−1.75533e27	−0.0858172	−0.0429086	−	0.999079i	$-0.513662\pi$
$7$	−1.75533e27	−0.0858172	−0.0429086	+	0.999079i	$0.513662\pi$
$8$	1.88919e30	1.05381
$9$	0	0
$10$	3.23261e33	1.02224
$11$	−1.38749e35	−1.80126	−0.900630	−	0.434587i	$-0.856894\pi$
$11$	−1.38749e35	−1.80126	−0.900630	+	0.434587i	$0.856894\pi$
$12$	0	0
$13$	−6.45579e36	−0.311062	−0.155531	−	0.987831i	$-0.549709\pi$
$13$	−6.45579e36	−0.311062	−0.155531	+	0.987831i	$0.549709\pi$
$14$	1.48011e37	0.0595669
$15$	0	0
$16$	−4.64430e39	−0.213256
$17$	8.51154e40	0.512825	0.256412	−	0.966567i	$-0.417459\pi$
$17$	8.51154e40	0.512825	0.256412	+	0.966567i	$0.417459\pi$
$18$	0	0
$19$	−5.19929e42	−0.754568	−0.377284	−	0.926098i	$-0.623142\pi$
$19$	−5.19929e42	−0.754568	−0.377284	+	0.926098i	$0.623142\pi$
$20$	2.93178e43	0.763178
$21$	0	0
$22$	1.16995e45	1.25028
$23$	−2.49803e45	−0.602164	−0.301082	−	0.953598i	$-0.597348\pi$
$23$	−2.49803e45	−0.602164	−0.301082	+	0.953598i	$0.597348\pi$
$24$	0	0
$25$	7.92100e46	1.16893
$26$	5.44359e46	0.215913
$27$	0	0
$28$	1.34237e47	0.0444710
$29$	−1.20701e49	−1.23417	−0.617087	−	0.786895i	$-0.711687\pi$
$29$	−1.20701e49	−1.23417	−0.617087	+	0.786895i	$0.711687\pi$
$30$	0	0
$31$	−6.23103e49	−0.682249	−0.341124	−	0.940018i	$-0.610808\pi$
$31$	−6.23103e49	−0.682249	−0.341124	+	0.940018i	$0.610808\pi$
$32$	−2.39634e50	−0.905784
$33$	0	0
$34$	−7.17701e50	−0.355959
$35$	6.72941e50	0.126386
$36$	0	0
$37$	3.56231e51	0.103986	0.0519929	−	0.998647i	$-0.483443\pi$
$37$	3.56231e51	0.103986	0.0519929	+	0.998647i	$0.483443\pi$
$38$	4.38409e52	0.523756
$39$	0	0
$40$	−7.24259e53	−1.55197
$41$	−4.51579e53	−0.423131	−0.211565	−	0.977364i	$-0.567856\pi$
$41$	−4.51579e53	−0.423131	−0.211565	+	0.977364i	$0.567856\pi$
$42$	0	0
$43$	6.41012e54	1.21807	0.609036	−	0.793143i	$-0.291557\pi$
$43$	6.41012e54	1.21807	0.609036	+	0.793143i	$0.291557\pi$
$44$	1.06107e55	0.933424
$45$	0	0
$46$	2.10636e55	0.417970
$47$	1.03805e56	1.00217	0.501085	−	0.865398i	$-0.332934\pi$
$47$	1.03805e56	1.00217	0.501085	+	0.865398i	$0.332934\pi$
$48$	0	0
$49$	−4.15297e56	−0.992635
$50$	−6.67906e56	−0.811373
$51$	0	0
$52$	4.93699e56	0.161194
$53$	9.61057e57	1.65771	0.828853	−	0.559466i	$-0.188994\pi$
$53$	9.61057e57	1.65771	0.828853	+	0.559466i	$0.188994\pi$
$54$	0	0
$55$	5.31924e58	2.65277
$56$	−3.31615e57	−0.0904348
$57$	0	0
$58$	1.01776e59	0.856657
$59$	4.77326e58	0.226607	0.113303	−	0.993560i	$-0.463857\pi$
$59$	4.77326e58	0.226607	0.113303	+	0.993560i	$0.463857\pi$
$60$	0	0
$61$	−2.98642e58	−0.0464088	−0.0232044	−	0.999731i	$-0.507387\pi$
$61$	−2.98642e58	−0.0464088	−0.0232044	+	0.999731i	$0.507387\pi$
$62$	5.25406e59	0.473558
$63$	0	0
$64$	2.70600e60	0.841973
$65$	2.47496e60	0.458111
$66$	0	0
$67$	−2.03266e61	−1.36320	−0.681599	−	0.731726i	$-0.738715\pi$
$67$	−2.03266e61	−1.36320	−0.681599	+	0.731726i	$0.738715\pi$
$68$	−6.50909e60	−0.265749
$69$	0	0
$70$	−5.67430e60	−0.0877259
$71$	6.57721e61	0.632247	0.316123	−	0.948718i	$-0.397619\pi$
$71$	6.57721e61	0.632247	0.316123	+	0.948718i	$0.397619\pi$
$72$	0	0
$73$	−1.20026e62	−0.454944	−0.227472	−	0.973785i	$-0.573046\pi$
$73$	−1.20026e62	−0.454944	−0.227472	+	0.973785i	$0.573046\pi$
$74$	−3.00377e61	−0.0721780
$75$	0	0
$76$	3.97609e62	0.391022
$77$	2.43551e62	0.154579
$78$	0	0
$79$	2.11502e63	0.568599	0.284299	−	0.958736i	$-0.408239\pi$
$79$	2.11502e63	0.568599	0.284299	+	0.958736i	$0.408239\pi$
$80$	1.78049e63	0.314068
$81$	0	0
$82$	3.80776e63	0.293701
$83$	3.25858e64	1.67461	0.837303	−	0.546740i	$-0.184131\pi$
$83$	3.25858e64	1.67461	0.837303	+	0.546740i	$0.184131\pi$
$84$	0	0
$85$	−3.26307e64	−0.755253
$86$	−5.40507e64	−0.845480
$87$	0	0
$88$	−2.62124e65	−1.89818
$89$	−6.19321e64	−0.307150	−0.153575	−	0.988137i	$-0.549079\pi$
$89$	−6.19321e64	−0.307150	−0.153575	+	0.988137i	$0.549079\pi$
$90$	0	0
$91$	1.13320e64	0.0266945
$92$	1.91034e65	0.312045
$93$	0	0
$94$	−8.75297e65	−0.695620
$95$	1.99325e66	1.11127
$96$	0	0
$97$	−9.58970e65	−0.266043	−0.133022	−	0.991113i	$-0.542468\pi$
$97$	−9.58970e65	−0.266043	−0.133022	+	0.991113i	$0.542468\pi$
$98$	3.50182e66	0.689002
$99$	0	0
$100$	−6.05749e66	−0.605749
$101$	1.60525e67	1.15021	0.575104	−	0.818080i	$-0.304962\pi$
$101$	1.60525e67	1.15021	0.575104	+	0.818080i	$0.304962\pi$
$102$	0	0
$103$	4.02418e67	1.49496	0.747481	−	0.664284i	$-0.231263\pi$
$103$	4.02418e67	1.49496	0.747481	+	0.664284i	$0.231263\pi$
$104$	−1.21962e67	−0.327800
$105$	0	0
$106$	−8.10373e67	−1.15064
$107$	7.96494e67	0.825707	0.412853	−	0.910798i	$-0.364532\pi$
$107$	7.96494e67	0.825707	0.412853	+	0.910798i	$0.364532\pi$
$108$	0	0
$109$	−1.26408e67	−0.0704673	−0.0352336	−	0.999379i	$-0.511218\pi$
$109$	−1.26408e67	−0.0704673	−0.0352336	+	0.999379i	$0.511218\pi$
$110$	−4.48523e68	−1.84132
$111$	0	0
$112$	8.15228e66	0.0183010
$113$	−4.43655e68	−0.739465	−0.369732	−	0.929138i	$-0.620551\pi$
$113$	−4.43655e68	−0.739465	−0.369732	+	0.929138i	$0.620551\pi$
$114$	0	0
$115$	9.57669e68	0.886825
$116$	9.23042e68	0.639557
$117$	0	0
$118$	−4.02486e68	−0.157291
$119$	−1.49406e68	−0.0440092
$120$	0	0
$121$	1.33179e70	2.24454
$122$	2.51817e68	0.0322130
$123$	0	0
$124$	4.76510e69	0.353546
$125$	−4.38859e69	−0.248794
$126$	0	0
$127$	−3.09560e70	−1.03115	−0.515574	−	0.856845i	$-0.672421\pi$
$127$	−3.09560e70	−1.03115	−0.515574	+	0.856845i	$0.672421\pi$
$128$	1.25466e70	0.321359
$129$	0	0
$130$	−2.08691e70	−0.317981
$131$	6.24988e70	0.736687	0.368343	−	0.929690i	$-0.379925\pi$
$131$	6.24988e70	0.736687	0.368343	+	0.929690i	$0.379925\pi$
$132$	0	0
$133$	9.12646e69	0.0647549
$134$	1.71396e71	0.946215
$135$	0	0
$136$	1.60799e71	0.540419
$137$	−9.38159e69	−0.0246682	−0.0123341	−	0.999924i	$-0.503926\pi$
$137$	−9.38159e69	−0.0246682	−0.0123341	+	0.999924i	$0.503926\pi$
$138$	0	0
$139$	6.21722e71	1.00601	0.503003	−	0.864285i	$-0.332228\pi$
$139$	6.21722e71	1.00601	0.503003	+	0.864285i	$0.332228\pi$
$140$	−5.14623e70	−0.0654938
$141$	0	0
$142$	−5.54596e71	−0.438851
$143$	8.95738e71	0.560304
$144$	0	0
$145$	4.62730e72	1.81760
$146$	1.01207e72	0.315783
$147$	0	0
$148$	−2.72423e71	−0.0538861
$149$	−2.36862e72	−0.373900	−0.186950	−	0.982369i	$-0.559860\pi$
$149$	−2.36862e72	−0.373900	−0.186950	+	0.982369i	$0.559860\pi$
$150$	0	0
$151$	4.21606e72	0.425774	0.212887	−	0.977077i	$-0.431713\pi$
$151$	4.21606e72	0.425774	0.212887	+	0.977077i	$0.431713\pi$
$152$	−9.82245e72	−0.795169
$153$	0	0
$154$	−2.05365e72	−0.107295
$155$	2.38879e73	1.00477
$156$	0	0
$157$	2.28213e73	0.624744	0.312372	−	0.949960i	$-0.398876\pi$
$157$	2.28213e73	0.624744	0.312372	+	0.949960i	$0.398876\pi$
$158$	−1.78340e73	−0.394672
$159$	0	0
$160$	9.18686e73	1.33398
$161$	4.38486e72	0.0516760
$162$	0	0
$163$	−1.50864e74	−1.17571	−0.587854	−	0.808967i	$-0.700027\pi$
$163$	−1.50864e74	−1.17571	−0.587854	+	0.808967i	$0.700027\pi$
$164$	3.45340e73	0.219269
$165$	0	0
$166$	−2.74767e74	−1.16237
$167$	−2.98109e73	−0.103127	−0.0515637	−	0.998670i	$-0.516421\pi$
$167$	−2.98109e73	−0.103127	−0.0515637	+	0.998670i	$0.516421\pi$
$168$	0	0
$169$	−3.89052e74	−0.903240
$170$	2.75145e74	0.524231
$171$	0	0
$172$	−4.90206e74	−0.631212
$173$	−5.87795e74	−0.623277	−0.311639	−	0.950201i	$-0.600878\pi$
$173$	−5.87795e74	−0.623277	−0.311639	+	0.950201i	$0.600878\pi$
$174$	0	0
$175$	−1.39040e74	−0.100315
$176$	6.44395e74	0.384129
$177$	0	0
$178$	5.22217e74	0.213197
$179$	−2.89879e75	−0.980935	−0.490467	−	0.871460i	$-0.663174\pi$
$179$	−2.89879e75	−0.980935	−0.490467	+	0.871460i	$0.663174\pi$
$180$	0	0
$181$	7.66482e75	1.78760	0.893798	−	0.448469i	$-0.148031\pi$
$181$	7.66482e75	1.78760	0.893798	+	0.448469i	$0.148031\pi$
$182$	−9.55529e73	−0.0185290
$183$	0	0
$184$	−4.71925e75	−0.634565
$185$	−1.36568e75	−0.153143
$186$	0	0
$187$	−1.18097e76	−0.923731
$188$	−7.93839e75	−0.519331
$189$	0	0
$190$	−1.68073e76	−0.771351
$191$	−4.40959e75	−0.169739	−0.0848693	−	0.996392i	$-0.527047\pi$
$191$	−4.40959e75	−0.169739	−0.0848693	+	0.996392i	$0.527047\pi$
$192$	0	0
$193$	3.04412e76	0.826592	0.413296	−	0.910597i	$-0.364377\pi$
$193$	3.04412e76	0.826592	0.413296	+	0.910597i	$0.364377\pi$
$194$	8.08613e75	0.184664
$195$	0	0
$196$	3.17593e76	0.514390
$197$	6.11000e76	0.834492	0.417246	−	0.908793i	$-0.362995\pi$
$197$	6.11000e76	0.834492	0.417246	+	0.908793i	$0.362995\pi$
$198$	0	0
$199$	6.76401e76	0.658607	0.329303	−	0.944224i	$-0.393186\pi$
$199$	6.76401e76	0.658607	0.329303	+	0.944224i	$0.393186\pi$
$200$	1.49643e77	1.23183
$201$	0	0
$202$	−1.35356e77	−0.798375
$203$	2.11869e76	0.105913
$204$	0	0
$205$	1.73122e77	0.623158
$206$	−3.39322e77	−1.03767
$207$	0	0
$208$	2.99827e76	0.0663359
$209$	7.21399e77	1.35917
$210$	0	0
$211$	4.61607e77	0.632136	0.316068	−	0.948737i	$-0.397637\pi$
$211$	4.61607e77	0.632136	0.316068	+	0.948737i	$0.397637\pi$
$212$	−7.34957e77	−0.859034
$213$	0	0
$214$	−6.71611e77	−0.573134
$215$	−2.45745e78	−1.79389
$216$	0	0
$217$	1.09375e77	0.0585487
$218$	1.06589e77	0.0489123
$219$	0	0
$220$	−4.06782e78	−1.37468
$221$	−5.49487e77	−0.159521
$222$	0	0
$223$	5.73226e78	1.23059	0.615295	−	0.788297i	$-0.289037\pi$
$223$	5.73226e78	1.23059	0.615295	+	0.788297i	$0.289037\pi$
$224$	4.20637e77	0.0777318
$225$	0	0
$226$	3.74094e78	0.513272
$227$	1.30879e79	1.54883	0.774414	−	0.632680i	$-0.218045\pi$
$227$	1.30879e79	1.54883	0.774414	+	0.632680i	$0.218045\pi$
$228$	0	0
$229$	−1.09438e79	−0.965342	−0.482671	−	0.875802i	$-0.660333\pi$
$229$	−1.09438e79	−0.965342	−0.482671	+	0.875802i	$0.660333\pi$
$230$	−8.07515e78	−0.615557
$231$	0	0
$232$	−2.28026e79	−1.30058
$233$	−2.12939e79	−1.05155	−0.525777	−	0.850622i	$-0.676225\pi$
$233$	−2.12939e79	−1.05155	−0.525777	+	0.850622i	$0.676225\pi$
$234$	0	0
$235$	−3.97959e79	−1.47593
$236$	−3.65029e78	−0.117429
$237$	0	0
$238$	1.25980e78	0.0305474
$239$	−3.58283e77	−0.00754912	−0.00377456	−	0.999993i	$-0.501201\pi$
$239$	−3.58283e77	−0.00754912	−0.00377456	+	0.999993i	$0.501201\pi$
$240$	0	0
$241$	9.84113e79	1.56846	0.784231	−	0.620469i	$-0.213058\pi$
$241$	9.84113e79	1.56846	0.784231	+	0.620469i	$0.213058\pi$
$242$	−1.12298e80	−1.55796
$243$	0	0
$244$	2.28382e78	0.0240494
$245$	1.59212e80	1.46188
$246$	0	0
$247$	3.35655e79	0.234718
$248$	−1.17716e80	−0.718959
$249$	0	0
$250$	3.70050e79	0.172691
$251$	−2.75351e80	−1.12413	−0.562065	−	0.827093i	$-0.689993\pi$
$251$	−2.75351e80	−1.12413	−0.562065	+	0.827093i	$0.689993\pi$
$252$	0	0
$253$	3.46600e80	1.08465
$254$	2.61024e80	0.715733
$255$	0	0
$256$	−5.05128e80	−1.06503
$257$	−4.62111e80	−0.855041	−0.427520	−	0.904006i	$-0.640613\pi$
$257$	−4.62111e80	−0.855041	−0.427520	+	0.904006i	$0.640613\pi$
$258$	0	0
$259$	−6.25303e78	−0.00892377
$260$	−1.89269e80	−0.237396
$261$	0	0
$262$	−5.26995e80	−0.511344
$263$	−2.15218e81	−1.83806	−0.919031	−	0.394184i	$-0.871027\pi$
$263$	−2.15218e81	−1.83806	−0.919031	+	0.394184i	$0.871027\pi$
$264$	0	0
$265$	−3.68441e81	−2.44135
$266$	−7.69552e79	−0.0449472
$267$	0	0
$268$	1.55445e81	0.706418
$269$	1.18612e81	0.475803	0.237901	−	0.971289i	$-0.423540\pi$
$269$	1.18612e81	0.475803	0.237901	+	0.971289i	$0.423540\pi$
$270$	0	0
$271$	−2.43922e81	−0.763448	−0.381724	−	0.924276i	$-0.624669\pi$
$271$	−2.43922e81	−0.763448	−0.381724	+	0.924276i	$0.624669\pi$
$272$	−3.95302e80	−0.109363
$273$	0	0
$274$	7.91064e79	0.0171225
$275$	−1.09904e82	−2.10555
$276$	0	0
$277$	−4.94623e81	−0.743365	−0.371682	−	0.928360i	$-0.621219\pi$
$277$	−4.94623e81	−0.743365	−0.371682	+	0.928360i	$0.621219\pi$
$278$	−5.24242e81	−0.698283
$279$	0	0
$280$	1.27131e81	0.133186
$281$	−1.83232e82	−1.70349	−0.851746	−	0.523955i	$-0.824456\pi$
$281$	−1.83232e82	−1.70349	−0.851746	+	0.523955i	$0.824456\pi$
$282$	0	0
$283$	−5.08970e81	−0.373119	−0.186559	−	0.982444i	$-0.559734\pi$
$283$	−5.08970e81	−0.373119	−0.186559	+	0.982444i	$0.559734\pi$
$284$	−5.02984e81	−0.327634
$285$	0	0
$286$	−7.55295e81	−0.388915
$287$	7.92670e80	0.0363119
$288$	0	0
$289$	−2.03026e82	−0.737011
$290$	−3.90178e82	−1.26162
$291$	0	0
$292$	9.17881e81	0.235755
$293$	−4.16071e82	−0.953020	−0.476510	−	0.879169i	$-0.658098\pi$
$293$	−4.16071e82	−0.953020	−0.476510	+	0.879169i	$0.658098\pi$
$294$	0	0
$295$	−1.82992e82	−0.333730
$296$	6.72989e81	0.109581
$297$	0	0
$298$	1.99724e82	0.259529
$299$	1.61268e82	0.187310
$300$	0	0
$301$	−1.12519e82	−0.104531
$302$	−3.55502e82	−0.295535
$303$	0	0
$304$	2.41471e82	0.160916
$305$	1.14490e82	0.0683477
$306$	0	0
$307$	−7.74779e82	−0.371572	−0.185786	−	0.982590i	$-0.559483\pi$
$307$	−7.74779e82	−0.371572	−0.185786	+	0.982590i	$0.559483\pi$
$308$	−1.86253e82	−0.0801038
$309$	0	0
$310$	−2.01425e83	−0.697423
$311$	−5.78665e83	−1.79868	−0.899341	−	0.437248i	$-0.855953\pi$
$311$	−5.78665e83	−1.79868	−0.899341	+	0.437248i	$0.855953\pi$
$312$	0	0
$313$	−4.31694e83	−1.08253	−0.541266	−	0.840851i	$-0.682055\pi$
$313$	−4.31694e83	−1.08253	−0.541266	+	0.840851i	$0.682055\pi$
$314$	−1.92432e83	−0.433644
$315$	0	0
$316$	−1.61743e83	−0.294651
$317$	9.02240e83	1.47855	0.739276	−	0.673402i	$-0.235168\pi$
$317$	9.02240e83	1.47855	0.739276	+	0.673402i	$0.235168\pi$
$318$	0	0
$319$	1.67471e84	2.22307
$320$	−1.03740e84	−1.24000
$321$	0	0
$322$	−3.69735e82	−0.0358690
$323$	−4.42539e83	−0.386961
$324$	0	0
$325$	−5.11364e83	−0.363611
$326$	1.27210e84	0.816075
$327$	0	0
$328$	−8.53120e83	−0.445899
$329$	−1.82213e83	−0.0860035
$330$	0	0
$331$	−1.45028e84	−0.558744	−0.279372	−	0.960183i	$-0.590126\pi$
$331$	−1.45028e84	−0.558744	−0.279372	+	0.960183i	$0.590126\pi$
$332$	−2.49196e84	−0.867791
$333$	0	0
$334$	2.51369e83	0.0715821
$335$	7.79262e84	2.00762
$336$	0	0
$337$	1.36629e83	0.0288365	0.0144183	−	0.999896i	$-0.495410\pi$
$337$	1.36629e83	0.0288365	0.0144183	+	0.999896i	$0.495410\pi$
$338$	3.28052e84	0.626951
$339$	0	0
$340$	2.49539e84	0.391377
$341$	8.64552e84	1.22891
$342$	0	0
$343$	1.46337e84	0.171002
$344$	1.21099e85	1.28361
$345$	0	0
$346$	4.95634e84	0.432625
$347$	−3.52969e84	−0.279704	−0.139852	−	0.990172i	$-0.544663\pi$
$347$	−3.52969e84	−0.279704	−0.139852	+	0.990172i	$0.544663\pi$
$348$	0	0
$349$	2.68026e85	1.75196	0.875980	−	0.482347i	$-0.160216\pi$
$349$	2.68026e85	1.75196	0.875980	+	0.482347i	$0.160216\pi$
$350$	1.17240e84	0.0696297
$351$	0	0
$352$	3.32491e85	1.63155
$353$	1.77646e82	0.000792688 0	0.000396344	−	1.00000i	$-0.499874\pi$
$353$	1.77646e82	0.000792688 0	0.000396344		1.00000i	$0.499874\pi$
$354$	0	0
$355$	−2.52150e85	−0.931129
$356$	4.73618e84	0.159167
$357$	0	0
$358$	2.44429e85	0.680880
$359$	7.24629e85	1.83845	0.919225	−	0.393732i	$-0.128816\pi$
$359$	7.24629e85	1.83845	0.919225	+	0.393732i	$0.128816\pi$
$360$	0	0
$361$	−2.04453e85	−0.430627
$362$	−6.46304e85	−1.24079
$363$	0	0
$364$	−8.66604e83	−0.0138333
$365$	4.60143e85	0.670010
$366$	0	0
$367$	−9.89279e85	−1.19952	−0.599758	−	0.800182i	$-0.704736\pi$
$367$	−9.89279e85	−1.19952	−0.599758	+	0.800182i	$0.704736\pi$
$368$	1.16016e85	0.128415
$369$	0	0
$370$	1.15156e85	0.106299
$371$	−1.68697e85	−0.142260
$372$	0	0
$373$	−2.64810e84	−0.0186504	−0.00932521	−	0.999957i	$-0.502968\pi$
$373$	−2.64810e84	−0.0186504	−0.00932521	+	0.999957i	$0.502968\pi$
$374$	9.95806e85	0.641174
$375$	0	0
$376$	1.96108e86	1.05610
$377$	7.79218e85	0.383905
$378$	0	0
$379$	−3.32517e86	−1.37215	−0.686073	−	0.727533i	$-0.740667\pi$
$379$	−3.32517e86	−1.37215	−0.686073	+	0.727533i	$0.740667\pi$
$380$	−1.52431e86	−0.575870
$381$	0	0
$382$	3.71821e85	0.117818
$383$	6.25868e85	0.181687	0.0908435	−	0.995865i	$-0.471044\pi$
$383$	6.25868e85	0.181687	0.0908435	+	0.995865i	$0.471044\pi$
$384$	0	0
$385$	−9.33702e85	−0.227653
$386$	−2.56683e86	−0.573749
$387$	0	0
$388$	7.33361e85	0.137865
$389$	−6.09926e86	−1.05188	−0.525939	−	0.850522i	$-0.676286\pi$
$389$	−6.09926e86	−1.05188	−0.525939	+	0.850522i	$0.676286\pi$
$390$	0	0
$391$	−2.12621e86	−0.308805
$392$	−7.84575e86	−1.04605
$393$	0	0
$394$	−5.15201e86	−0.579233
$395$	−8.10834e86	−0.837393
$396$	0	0
$397$	1.14304e87	0.996736	0.498368	−	0.866966i	$-0.333933\pi$
$397$	1.14304e87	0.996736	0.498368	+	0.866966i	$0.333933\pi$
$398$	−5.70347e86	−0.457148
$399$	0	0
$400$	−3.67875e86	−0.249282
$401$	1.93243e87	1.20439	0.602196	−	0.798348i	$-0.294293\pi$
$401$	1.93243e87	1.20439	0.602196	+	0.798348i	$0.294293\pi$
$402$	0	0
$403$	4.02262e86	0.212222
$404$	−1.22759e87	−0.596045
$405$	0	0
$406$	−1.78650e86	−0.0735159
$407$	−4.94269e86	−0.187306
$408$	0	0
$409$	4.40444e87	1.41632	0.708158	−	0.706054i	$-0.249526\pi$
$409$	4.40444e87	1.41632	0.708158	+	0.706054i	$0.249526\pi$
$410$	−1.45978e87	−0.432542
$411$	0	0
$412$	−3.07744e87	−0.774698
$413$	−8.37864e85	−0.0194467
$414$	0	0
$415$	−1.24924e88	−2.46624
$416$	1.54703e87	0.281755
$417$	0	0
$418$	−6.08290e87	−0.943420
$419$	−9.02411e87	−1.29192	−0.645958	−	0.763373i	$-0.723542\pi$
$419$	−9.02411e87	−1.29192	−0.645958	+	0.763373i	$0.723542\pi$
$420$	0	0
$421$	−1.07202e88	−1.30843	−0.654215	−	0.756308i	$-0.727001\pi$
$421$	−1.07202e88	−1.30843	−0.654215	+	0.756308i	$0.727001\pi$
$422$	−3.89231e87	−0.438774
$423$	0	0
$424$	1.81562e88	1.74690
$425$	6.74199e87	0.599458
$426$	0	0
$427$	5.24214e85	0.00398268
$428$	−6.09109e87	−0.427886
$429$	0	0
$430$	2.07214e88	1.24516
$431$	−7.24337e87	−0.402673	−0.201336	−	0.979522i	$-0.564528\pi$
$431$	−7.24337e87	−0.402673	−0.201336	+	0.979522i	$0.564528\pi$
$432$	0	0
$433$	−7.28174e86	−0.0346649	−0.0173325	−	0.999850i	$-0.505517\pi$
$433$	−7.28174e86	−0.0346649	−0.0173325	+	0.999850i	$0.505517\pi$
$434$	−9.22261e86	−0.0406394
$435$	0	0
$436$	9.66691e86	0.0365166
$437$	1.29880e88	0.454373
$438$	0	0
$439$	1.22243e88	0.366998	0.183499	−	0.983020i	$-0.441258\pi$
$439$	1.22243e88	0.366998	0.183499	+	0.983020i	$0.441258\pi$
$440$	1.00491e89	2.79551
$441$	0	0
$442$	4.63333e87	0.110725
$443$	−2.48797e88	−0.551215	−0.275608	−	0.961270i	$-0.588879\pi$
$443$	−2.48797e88	−0.551215	−0.275608	+	0.961270i	$0.588879\pi$
$444$	0	0
$445$	2.37429e88	0.452349
$446$	−4.83350e88	−0.854169
$447$	0	0
$448$	−4.74991e87	−0.0722557
$449$	1.29903e89	1.83386	0.916931	−	0.399046i	$-0.130659\pi$
$449$	1.29903e89	1.83386	0.916931	+	0.399046i	$0.130659\pi$
$450$	0	0
$451$	6.26564e88	0.762169
$452$	3.39280e88	0.383195
$453$	0	0
$454$	−1.10358e89	−1.07506
$455$	−4.34437e87	−0.0393138
$456$	0	0
$457$	1.20530e89	0.941681	0.470841	−	0.882218i	$-0.343951\pi$
$457$	1.20530e89	0.941681	0.470841	+	0.882218i	$0.343951\pi$
$458$	9.22796e88	0.670057
$459$	0	0
$460$	−7.32365e88	−0.459558
$461$	2.20679e88	0.128760	0.0643800	−	0.997925i	$-0.479493\pi$
$461$	2.20679e88	0.128760	0.0643800	+	0.997925i	$0.479493\pi$
$462$	0	0
$463$	1.20440e89	0.607867	0.303934	−	0.952693i	$-0.401700\pi$
$463$	1.20440e89	0.607867	0.303934	+	0.952693i	$0.401700\pi$
$464$	5.60570e88	0.263195
$465$	0	0
$466$	1.79552e89	0.729898
$467$	2.22415e89	0.841491	0.420745	−	0.907179i	$-0.361769\pi$
$467$	2.22415e89	0.841491	0.420745	+	0.907179i	$0.361769\pi$
$468$	0	0
$469$	3.56799e88	0.116986
$470$	3.35563e89	1.02446
$471$	0	0
$472$	9.01760e88	0.238800
$473$	−8.89401e89	−2.19406
$474$	0	0
$475$	−4.11836e89	−0.882040
$476$	1.14256e88	0.0228058
$477$	0	0
$478$	3.02108e87	0.00523995
$479$	−3.75550e89	−0.607335	−0.303667	−	0.952778i	$-0.598211\pi$
$479$	−3.75550e89	−0.607335	−0.303667	+	0.952778i	$0.598211\pi$
$480$	0	0
$481$	−2.29976e88	−0.0323461
$482$	−8.29813e89	−1.08869
$483$	0	0
$484$	−1.01847e90	−1.16313
$485$	3.67641e89	0.391810
$486$	0	0
$487$	3.60986e89	0.335174	0.167587	−	0.985857i	$-0.446403\pi$
$487$	3.60986e89	0.335174	0.167587	+	0.985857i	$0.446403\pi$
$488$	−5.64191e88	−0.0489060
$489$	0	0
$490$	−1.34249e90	−1.01471
$491$	−9.21555e89	−0.650566	−0.325283	−	0.945617i	$-0.605460\pi$
$491$	−9.21555e89	−0.650566	−0.325283	+	0.945617i	$0.605460\pi$
$492$	0	0
$493$	−1.02735e90	−0.632915
$494$	−2.83028e89	−0.162921
$495$	0	0
$496$	2.89388e89	0.145494
$497$	−1.15452e89	−0.0542577
$498$	0	0
$499$	2.88968e90	1.18708	0.593542	−	0.804803i	$-0.297729\pi$
$499$	2.88968e90	1.18708	0.593542	+	0.804803i	$0.297729\pi$
$500$	3.35612e89	0.128926
$501$	0	0
$502$	2.32178e90	0.780273
$503$	3.03767e90	0.955021	0.477511	−	0.878626i	$-0.341539\pi$
$503$	3.03767e90	0.955021	0.477511	+	0.878626i	$0.341539\pi$
$504$	0	0
$505$	−6.15405e90	−1.69395
$506$	−2.92256e90	−0.752873
$507$	0	0
$508$	2.36732e90	0.534347
$509$	−6.10246e90	−1.28961	−0.644806	−	0.764346i	$-0.723062\pi$
$509$	−6.10246e90	−1.28961	−0.644806	+	0.764346i	$0.723062\pi$
$510$	0	0
$511$	2.10685e89	0.0390420
$512$	2.40774e90	0.417894
$513$	0	0
$514$	3.89656e90	0.593495
$515$	−1.54275e91	−2.20167
$516$	0	0
$517$	−1.44029e91	−1.80517
$518$	5.27261e88	0.00619411
$519$	0	0
$520$	4.67567e90	0.482761
$521$	1.36791e91	1.32433	0.662163	−	0.749360i	$-0.269638\pi$
$521$	1.36791e91	1.32433	0.662163	+	0.749360i	$0.269638\pi$
$522$	0	0
$523$	7.08965e90	0.603695	0.301848	−	0.953356i	$-0.402397\pi$
$523$	7.08965e90	0.603695	0.301848	+	0.953356i	$0.402397\pi$
$524$	−4.77951e90	−0.381756
$525$	0	0
$526$	1.81474e91	1.27582
$527$	−5.30356e90	−0.349874
$528$	0	0
$529$	−1.09692e91	−0.637399
$530$	3.10673e91	1.69458
$531$	0	0
$532$	−6.97935e89	−0.0335564
$533$	2.91530e90	0.131620
$534$	0	0
$535$	−3.05352e91	−1.21604
$536$	−3.84009e91	−1.43655
$537$	0	0
$538$	−1.00015e91	−0.330261
$539$	5.76222e91	1.78799
$540$	0	0
$541$	−6.80713e90	−0.186576	−0.0932879	−	0.995639i	$-0.529738\pi$
$541$	−6.80713e90	−0.186576	−0.0932879	+	0.995639i	$0.529738\pi$
$542$	2.05677e91	0.529919
$543$	0	0
$544$	−2.03966e91	−0.464509
$545$	4.84611e90	0.103779
$546$	0	0
$547$	7.97357e91	1.51035	0.755174	−	0.655524i	$-0.227552\pi$
$547$	7.97357e91	1.51035	0.755174	+	0.655524i	$0.227552\pi$
$548$	7.17445e89	0.0127832
$549$	0	0
$550$	9.26717e91	1.46149
$551$	6.27557e91	0.931268
$552$	0	0
$553$	−3.71255e90	−0.0487955
$554$	4.17071e91	0.515980
$555$	0	0
$556$	−4.75455e91	−0.521319
$557$	6.17265e91	0.637269	0.318634	−	0.947878i	$-0.396776\pi$
$557$	6.17265e91	0.637269	0.318634	+	0.947878i	$0.396776\pi$
$558$	0	0
$559$	−4.13824e91	−0.378896
$560$	−3.12534e90	−0.0269525
$561$	0	0
$562$	1.54503e92	1.18242
$563$	−2.12849e92	−1.53476	−0.767379	−	0.641194i	$-0.778440\pi$
$563$	−2.12849e92	−1.53476	−0.767379	+	0.641194i	$0.778440\pi$
$564$	0	0
$565$	1.70084e92	1.08903
$566$	4.29169e91	0.258987
$567$	0	0
$568$	1.24256e92	0.666267
$569$	2.23483e92	1.12975	0.564876	−	0.825176i	$-0.308924\pi$
$569$	2.23483e92	1.12975	0.564876	+	0.825176i	$0.308924\pi$
$570$	0	0
$571$	−4.99687e91	−0.224589	−0.112294	−	0.993675i	$-0.535820\pi$
$571$	−4.99687e91	−0.224589	−0.112294	+	0.993675i	$0.535820\pi$
$572$	−6.85005e91	−0.290353
$573$	0	0
$574$	−6.68387e90	−0.0252046
$575$	−1.97869e92	−0.703890
$576$	0	0
$577$	−1.98229e92	−0.627736	−0.313868	−	0.949467i	$-0.601625\pi$
$577$	−1.98229e92	−0.627736	−0.313868	+	0.949467i	$0.601625\pi$
$578$	1.71193e92	0.511569
$579$	0	0
$580$	−3.53867e92	−0.941894
$581$	−5.71988e91	−0.143710
$582$	0	0
$583$	−1.33346e93	−2.98596
$584$	−2.26751e92	−0.479424
$585$	0	0
$586$	3.50835e92	0.661504
$587$	−1.77753e92	−0.316548	−0.158274	−	0.987395i	$-0.550593\pi$
$587$	−1.77753e92	−0.316548	−0.158274	+	0.987395i	$0.550593\pi$
$588$	0	0
$589$	3.23969e92	0.514803
$590$	1.54301e92	0.231647
$591$	0	0
$592$	−1.65445e91	−0.0221756
$593$	−1.10546e93	−1.40026	−0.700132	−	0.714014i	$-0.746875\pi$
$593$	−1.10546e93	−1.40026	−0.700132	+	0.714014i	$0.746875\pi$
$594$	0	0
$595$	5.72776e91	0.0648137
$596$	1.81137e92	0.193758
$597$	0	0
$598$	−1.35982e92	−0.130015
$599$	−2.74758e92	−0.248400	−0.124200	−	0.992257i	$-0.539636\pi$
$599$	−2.74758e92	−0.248400	−0.124200	+	0.992257i	$0.539636\pi$
$600$	0	0
$601$	1.24095e93	1.00337	0.501683	−	0.865052i	$-0.332714\pi$
$601$	1.24095e93	1.00337	0.501683	+	0.865052i	$0.332714\pi$
$602$	9.48768e91	0.0725567
$603$	0	0
$604$	−3.22418e92	−0.220639
$605$	−5.10570e93	−3.30560
$606$	0	0
$607$	−1.85465e93	−1.07508	−0.537541	−	0.843238i	$-0.680647\pi$
$607$	−1.85465e93	−1.07508	−0.537541	+	0.843238i	$0.680647\pi$
$608$	1.24593e93	0.683475
$609$	0	0
$610$	−9.65393e91	−0.0474411
$611$	−6.70146e92	−0.311738
$612$	0	0
$613$	−9.71814e92	−0.405189	−0.202594	−	0.979263i	$-0.564937\pi$
$613$	−9.71814e92	−0.405189	−0.202594	+	0.979263i	$0.564937\pi$
$614$	6.53301e92	0.257913
$615$	0	0
$616$	4.60114e92	0.162897
$617$	5.79595e93	1.94344	0.971722	−	0.236129i	$-0.0758789\pi$
$617$	5.79595e93	1.94344	0.971722	+	0.236129i	$0.0758789\pi$
$618$	0	0
$619$	1.80658e93	0.543527	0.271764	−	0.962364i	$-0.412393\pi$
$619$	1.80658e93	0.543527	0.271764	+	0.962364i	$0.412393\pi$
$620$	−1.82680e93	−0.520677
$621$	0	0
$622$	4.87936e93	1.24849
$623$	1.08711e92	0.0263587
$624$	0	0
$625$	−3.68503e93	−0.802528
$626$	3.64008e93	0.751400
$627$	0	0
$628$	−1.74523e93	−0.323747
$629$	3.03208e92	0.0533265
$630$	0	0
$631$	1.24799e94	1.97346	0.986731	−	0.162366i	$-0.0519124\pi$
$631$	1.24799e94	1.97346	0.986731	+	0.162366i	$0.0519124\pi$
$632$	3.99567e93	0.599194
$633$	0	0
$634$	−7.60777e93	−1.02628
$635$	1.18676e94	1.51860
$636$	0	0
$637$	2.68107e93	0.308771
$638$	−1.41213e94	−1.54306
$639$	0	0
$640$	−4.80997e93	−0.473275
$641$	1.46544e94	1.36843	0.684217	−	0.729279i	$-0.260144\pi$
$641$	1.46544e94	1.36843	0.684217	+	0.729279i	$0.260144\pi$
$642$	0	0
$643$	−1.83843e93	−0.154661	−0.0773305	−	0.997006i	$-0.524640\pi$
$643$	−1.83843e93	−0.154661	−0.0773305	+	0.997006i	$0.524640\pi$
$644$	−3.35327e92	−0.0267788
$645$	0	0
$646$	3.73153e93	0.268595
$647$	−1.37225e94	−0.937867	−0.468933	−	0.883234i	$-0.655362\pi$
$647$	−1.37225e94	−0.937867	−0.468933	+	0.883234i	$0.655362\pi$
$648$	0	0
$649$	−6.62287e93	−0.408177
$650$	4.31187e93	0.252388
$651$	0	0
$652$	1.15371e94	0.609259
$653$	−9.27456e93	−0.465266	−0.232633	−	0.972565i	$-0.574734\pi$
$653$	−9.27456e93	−0.465266	−0.232633	+	0.972565i	$0.574734\pi$
$654$	0	0
$655$	−2.39602e94	−1.08494
$656$	2.09727e93	0.0902352
$657$	0	0
$658$	1.53643e93	0.0596962
$659$	3.82564e94	1.41268	0.706339	−	0.707874i	$-0.250345\pi$
$659$	3.82564e94	1.41268	0.706339	+	0.707874i	$0.250345\pi$
$660$	0	0
$661$	3.63043e94	1.21118	0.605590	−	0.795777i	$-0.292937\pi$
$661$	3.63043e94	1.21118	0.605590	+	0.795777i	$0.292937\pi$
$662$	1.22289e94	0.387832
$663$	0	0
$664$	6.15608e94	1.76471
$665$	−3.49881e93	−0.0953665
$666$	0	0
$667$	3.01513e94	0.743175
$668$	2.27975e93	0.0534412
$669$	0	0
$670$	−6.57081e94	−1.39352
$671$	4.14364e93	0.0835944
$672$	0	0
$673$	−8.84206e94	−1.61454	−0.807270	−	0.590182i	$-0.799056\pi$
$673$	−8.84206e94	−1.61454	−0.807270	+	0.590182i	$0.799056\pi$
$674$	−1.15207e93	−0.0200158
$675$	0	0
$676$	2.97523e94	0.468065
$677$	−2.44389e94	−0.365899	−0.182949	−	0.983122i	$-0.558564\pi$
$677$	−2.44389e94	−0.365899	−0.182949	+	0.983122i	$0.558564\pi$
$678$	0	0
$679$	1.68331e93	0.0228311
$680$	−6.16456e94	−0.795891
$681$	0	0
$682$	−7.28998e94	−0.853002
$683$	4.41098e94	0.491408	0.245704	−	0.969345i	$-0.420981\pi$
$683$	4.41098e94	0.491408	0.245704	+	0.969345i	$0.420981\pi$
$684$	0	0
$685$	3.59662e93	0.0363296
$686$	−1.23393e94	−0.118695
$687$	0	0
$688$	−2.97705e94	−0.259761
$689$	−6.20439e94	−0.515650
$690$	0	0
$691$	7.50986e94	0.566392	0.283196	−	0.959062i	$-0.408605\pi$
$691$	7.50986e94	0.566392	0.283196	+	0.959062i	$0.408605\pi$
$692$	4.49509e94	0.322986
$693$	0	0
$694$	2.97627e94	0.194147
$695$	−2.38350e95	−1.48158
$696$	0	0
$697$	−3.84363e94	−0.216992
$698$	−2.26002e95	−1.21606
$699$	0	0
$700$	1.06329e94	0.0519837
$701$	1.61698e95	0.753617	0.376808	−	0.926291i	$-0.377022\pi$
$701$	1.61698e95	0.753617	0.376808	+	0.926291i	$0.377022\pi$
$702$	0	0
$703$	−1.85215e94	−0.0784644
$704$	−3.75456e95	−1.51661
$705$	0	0
$706$	−1.49793e92	−0.000550215 0
$707$	−2.81774e94	−0.0987076
$708$	0	0
$709$	2.05667e95	0.655413	0.327707	−	0.944780i	$-0.393724\pi$
$709$	2.05667e95	0.655413	0.327707	+	0.944780i	$0.393724\pi$
$710$	2.12616e95	0.646309
$711$	0	0
$712$	−1.17002e95	−0.323677
$713$	1.55653e95	0.410826
$714$	0	0
$715$	−3.43399e95	−0.825177
$716$	2.21681e95	0.508327
$717$	0	0
$718$	−6.11014e95	−1.27609
$719$	−2.35820e95	−0.470070	−0.235035	−	0.971987i	$-0.575521\pi$
$719$	−2.35820e95	−0.470070	−0.235035	+	0.971987i	$0.575521\pi$
$720$	0	0
$721$	−7.06375e94	−0.128293
$722$	1.72396e95	0.298904
$723$	0	0
$724$	−5.86157e95	−0.926344
$725$	−9.56069e95	−1.44267
$726$	0	0
$727$	4.63503e95	0.637750	0.318875	−	0.947797i	$-0.396695\pi$
$727$	4.63503e95	0.637750	0.318875	+	0.947797i	$0.396695\pi$
$728$	2.14084e94	0.0281309
$729$	0	0
$730$	−3.87997e95	−0.465063
$731$	5.45600e95	0.624658
$732$	0	0
$733$	1.41477e96	1.47809	0.739047	−	0.673654i	$-0.235276\pi$
$733$	1.41477e96	1.47809	0.739047	+	0.673654i	$0.235276\pi$
$734$	8.34170e95	0.832600
$735$	0	0
$736$	5.98613e95	0.545430
$737$	2.82031e96	2.45548
$738$	0	0
$739$	2.00156e96	1.59141	0.795703	−	0.605688i	$-0.207102\pi$
$739$	2.00156e96	1.59141	0.795703	+	0.605688i	$0.207102\pi$
$740$	1.04439e95	0.0793597
$741$	0	0
$742$	1.42247e95	0.0987444
$743$	6.29529e95	0.417725	0.208863	−	0.977945i	$-0.433024\pi$
$743$	6.29529e95	0.417725	0.208863	+	0.977945i	$0.433024\pi$
$744$	0	0
$745$	9.08059e95	0.550654
$746$	2.23290e94	0.0129455
$747$	0	0
$748$	9.03133e95	0.478683
$749$	−1.39811e95	−0.0708598
$750$	0	0
$751$	4.41895e95	0.204824	0.102412	−	0.994742i	$-0.467344\pi$
$751$	4.41895e95	0.204824	0.102412	+	0.994742i	$0.467344\pi$
$752$	−4.82104e95	−0.213719
$753$	0	0
$754$	−6.57044e95	−0.266474
$755$	−1.61631e96	−0.627050
$756$	0	0
$757$	−5.26073e96	−1.86782	−0.933908	−	0.357513i	$-0.883625\pi$
$757$	−5.26073e96	−1.86782	−0.933908	+	0.357513i	$0.883625\pi$
$758$	2.80381e96	0.952425
$759$	0	0
$760$	3.76563e96	1.17107
$761$	−2.28223e96	−0.679161	−0.339581	−	0.940577i	$-0.610285\pi$
$761$	−2.28223e96	−0.679161	−0.339581	+	0.940577i	$0.610285\pi$
$762$	0	0
$763$	2.21888e94	0.00604730
$764$	3.37218e95	0.0879596
$765$	0	0
$766$	−5.27738e95	−0.126111
$767$	−3.08152e95	−0.0704888
$768$	0	0
$769$	−3.29099e95	−0.0689913	−0.0344956	−	0.999405i	$-0.510982\pi$
$769$	−3.29099e95	−0.0689913	−0.0344956	+	0.999405i	$0.510982\pi$
$770$	7.87306e95	0.158017
$771$	0	0
$772$	−2.32795e96	−0.428345
$773$	1.15830e96	0.204084	0.102042	−	0.994780i	$-0.467462\pi$
$773$	1.15830e96	0.204084	0.102042	+	0.994780i	$0.467462\pi$
$774$	0	0
$775$	−4.93560e96	−0.797504
$776$	−1.81168e96	−0.280358
$777$	0	0
$778$	5.14295e96	0.730123
$779$	2.34789e96	0.319281
$780$	0	0
$781$	−9.12584e96	−1.13884
$782$	1.79284e96	0.214346
$783$	0	0
$784$	1.92876e96	0.211685
$785$	−8.74901e96	−0.920080
$786$	0	0
$787$	−8.38013e96	−0.809277	−0.404639	−	0.914477i	$-0.632603\pi$
$787$	−8.38013e96	−0.809277	−0.404639	+	0.914477i	$0.632603\pi$
$788$	−4.67254e96	−0.432439
$789$	0	0
$790$	6.83703e96	0.581246
$791$	7.78760e95	0.0634588
$792$	0	0
$793$	1.92797e95	0.0144360
$794$	−9.63823e96	−0.691848
$795$	0	0
$796$	−5.17269e96	−0.341294
$797$	−1.06380e97	−0.672988	−0.336494	−	0.941686i	$-0.609241\pi$
$797$	−1.06380e97	−0.672988	−0.336494	+	0.941686i	$0.609241\pi$
$798$	0	0
$799$	8.83544e96	0.513938
$800$	−1.89814e97	−1.05880
$801$	0	0
$802$	−1.62944e97	−0.835985
$803$	1.66535e97	0.819473
$804$	0	0
$805$	−1.68102e96	−0.0761048
$806$	−3.39191e96	−0.147306
$807$	0	0
$808$	3.03262e97	1.21210
$809$	−4.90920e97	−1.88250	−0.941250	−	0.337711i	$-0.890347\pi$
$809$	−4.90920e97	−1.88250	−0.941250	+	0.337711i	$0.890347\pi$
$810$	0	0
$811$	2.45004e97	0.864917	0.432458	−	0.901654i	$-0.357646\pi$
$811$	2.45004e97	0.864917	0.432458	+	0.901654i	$0.357646\pi$
$812$	−1.62024e96	−0.0548849
$813$	0	0
$814$	4.16772e96	0.130011
$815$	5.78366e97	1.73150
$816$	0	0
$817$	−3.33281e97	−0.919117
$818$	−3.71387e97	−0.983085
$819$	0	0
$820$	−1.32393e97	−0.322924
$821$	2.49187e97	0.583485	0.291742	−	0.956497i	$-0.405765\pi$
$821$	2.49187e97	0.583485	0.291742	+	0.956497i	$0.405765\pi$
$822$	0	0
$823$	2.64932e97	0.571794	0.285897	−	0.958260i	$-0.407708\pi$
$823$	2.64932e97	0.571794	0.285897	+	0.958260i	$0.407708\pi$
$824$	7.60244e97	1.57540
$825$	0	0
$826$	7.06495e95	0.0134982
$827$	−2.19369e97	−0.402477	−0.201238	−	0.979542i	$-0.564497\pi$
$827$	−2.19369e97	−0.402477	−0.201238	+	0.979542i	$0.564497\pi$
$828$	0	0
$829$	−8.91046e97	−1.50773	−0.753864	−	0.657030i	$-0.771812\pi$
$829$	−8.91046e97	−1.50773	−0.753864	+	0.657030i	$0.771812\pi$
$830$	1.05337e98	1.71185
$831$	0	0
$832$	−1.74694e97	−0.261906
$833$	−3.53481e97	−0.509048
$834$	0	0
$835$	1.14286e97	0.151879
$836$	−5.51681e97	−0.704332
$837$	0	0
$838$	7.60921e97	0.896737
$839$	−1.07395e98	−1.21607	−0.608035	−	0.793910i	$-0.708042\pi$
$839$	−1.07395e98	−1.21607	−0.608035	+	0.793910i	$0.708042\pi$
$840$	0	0
$841$	5.00404e97	0.523185
$842$	9.03933e97	0.908199
$843$	0	0
$844$	−3.53008e97	−0.327577
$845$	1.49151e98	1.33023
$846$	0	0
$847$	−2.33774e97	−0.192620
$848$	−4.46344e97	−0.353516
$849$	0	0
$850$	−5.68491e97	−0.416092
$851$	−8.89875e96	−0.0626165
$852$	0	0
$853$	−2.41773e97	−0.157259	−0.0786295	−	0.996904i	$-0.525054\pi$
$853$	−2.41773e97	−0.157259	−0.0786295	+	0.996904i	$0.525054\pi$
$854$	−4.42022e95	−0.00276443
$855$	0	0
$856$	1.50473e98	0.870136
$857$	−1.45082e97	−0.0806780	−0.0403390	−	0.999186i	$-0.512844\pi$
$857$	−1.45082e97	−0.0806780	−0.0403390	+	0.999186i	$0.512844\pi$
$858$	0	0
$859$	2.11191e98	1.08619	0.543093	−	0.839672i	$-0.317253\pi$
$859$	2.11191e98	1.08619	0.543093	+	0.839672i	$0.317253\pi$
$860$	1.87930e98	0.929605
$861$	0	0
$862$	6.10768e97	0.279501
$863$	2.19104e98	0.964470	0.482235	−	0.876042i	$-0.339825\pi$
$863$	2.19104e98	0.964470	0.482235	+	0.876042i	$0.339825\pi$
$864$	0	0
$865$	2.25343e98	0.917919
$866$	6.14003e96	0.0240614
$867$	0	0
$868$	−8.36432e96	−0.0303403
$869$	−2.93457e98	−1.02419
$870$	0	0
$871$	1.31224e98	0.424040
$872$	−2.38809e97	−0.0742590
$873$	0	0
$874$	−1.09516e98	−0.315387
$875$	7.70342e96	0.0213508
$876$	0	0
$877$	−6.51102e98	−1.67172	−0.835858	−	0.548946i	$-0.815029\pi$
$877$	−6.51102e98	−1.67172	−0.835858	+	0.548946i	$0.815029\pi$
$878$	−1.03077e98	−0.254738
$879$	0	0
$880$	−2.47042e98	−0.565719
$881$	6.96595e98	1.53563	0.767817	−	0.640670i	$-0.221343\pi$
$881$	6.96595e98	1.53563	0.767817	+	0.640670i	$0.221343\pi$
$882$	0	0
$883$	−1.65040e98	−0.337215	−0.168607	−	0.985683i	$-0.553927\pi$
$883$	−1.65040e98	−0.337215	−0.168607	+	0.985683i	$0.553927\pi$
$884$	4.20214e97	0.0826645
$885$	0	0
$886$	2.09788e98	0.382606
$887$	−5.98960e98	−1.05186	−0.525929	−	0.850529i	$-0.676282\pi$
$887$	−5.98960e98	−1.05186	−0.525929	+	0.850529i	$0.676282\pi$
$888$	0	0
$889$	5.43380e97	0.0884902
$890$	−2.00202e98	−0.313982
$891$	0	0
$892$	−4.38368e98	−0.637699
$893$	−5.39714e98	−0.756206
$894$	0	0
$895$	1.11131e99	1.44465
$896$	−2.20233e97	−0.0275781
$897$	0	0
$898$	−1.09535e99	−1.27291
$899$	7.52088e98	0.842014
$900$	0	0
$901$	8.18008e98	0.850113
$902$	−5.28325e98	−0.529032
$903$	0	0
$904$	−8.38149e98	−0.779253
$905$	−2.93846e99	−2.63265
$906$	0	0
$907$	1.20263e99	1.00067	0.500333	−	0.865833i	$-0.333211\pi$
$907$	1.20263e99	1.00067	0.500333	+	0.865833i	$0.333211\pi$
$908$	−1.00088e99	−0.802612
$909$	0	0
$910$	3.66321e97	0.0272882
$911$	5.27192e98	0.378532	0.189266	−	0.981926i	$-0.439389\pi$
$911$	5.27192e98	0.378532	0.189266	+	0.981926i	$0.439389\pi$
$912$	0	0
$913$	−4.52126e99	−3.01640
$914$	−1.01632e99	−0.653634
$915$	0	0
$916$	8.36917e98	0.500247
$917$	−1.09706e98	−0.0632204
$918$	0	0
$919$	−1.45226e98	−0.0777993	−0.0388997	−	0.999243i	$-0.512385\pi$
$919$	−1.45226e98	−0.0777993	−0.0388997	+	0.999243i	$0.512385\pi$
$920$	1.80922e99	0.934543
$921$	0	0
$922$	−1.86079e98	−0.0893741
$923$	−4.24611e98	−0.196668
$924$	0	0
$925$	2.82171e98	0.121553
$926$	−1.01556e99	−0.421929
$927$	0	0
$928$	2.89240e99	1.11789
$929$	1.56446e99	0.583227	0.291614	−	0.956536i	$-0.405808\pi$
$929$	1.56446e99	0.583227	0.291614	+	0.956536i	$0.405808\pi$
$930$	0	0
$931$	2.15925e99	0.749011
$932$	1.62842e99	0.544922
$933$	0	0
$934$	−1.87543e99	−0.584090
$935$	4.52749e99	1.36041
$936$	0	0
$937$	−4.36725e99	−1.22161	−0.610804	−	0.791782i	$-0.709154\pi$
$937$	−4.36725e99	−1.22161	−0.610804	+	0.791782i	$0.709154\pi$
$938$	−3.00856e98	−0.0812015
$939$	0	0
$940$	3.04334e99	0.764835
$941$	5.74065e99	1.39222	0.696111	−	0.717934i	$-0.254912\pi$
$941$	5.74065e99	1.39222	0.696111	+	0.717934i	$0.254912\pi$
$942$	0	0
$943$	1.12806e99	0.254794
$944$	−2.21685e98	−0.0483252
$945$	0	0
$946$	7.49951e99	1.52293
$947$	−1.46937e99	−0.288010	−0.144005	−	0.989577i	$-0.545998\pi$
$947$	−1.46937e99	−0.288010	−0.144005	+	0.989577i	$0.545998\pi$
$948$	0	0
$949$	7.74861e98	0.141516
$950$	3.47264e99	0.612236
$951$	0	0
$952$	−2.82256e98	−0.0463772
$953$	−3.58980e99	−0.569453	−0.284727	−	0.958609i	$-0.591903\pi$
$953$	−3.58980e99	−0.569453	−0.284727	+	0.958609i	$0.591903\pi$
$954$	0	0
$955$	1.69051e99	0.249979
$956$	2.73992e97	0.00391200
$957$	0	0
$958$	3.16667e99	0.421559
$959$	1.64678e97	0.00211696
$960$	0	0
$961$	−4.45873e99	−0.534536
$962$	1.93918e98	0.0224519
$963$	0	0
$964$	−7.52588e99	−0.812787
$965$	−1.16702e100	−1.21735
$966$	0	0
$967$	1.01618e100	0.988969	0.494484	−	0.869187i	$-0.335357\pi$
$967$	1.01618e100	0.988969	0.494484	+	0.869187i	$0.335357\pi$
$968$	2.51601e100	2.36531
$969$	0	0
$970$	−3.09998e99	−0.271960
$971$	6.18477e99	0.524179	0.262089	−	0.965044i	$-0.415588\pi$
$971$	6.18477e99	0.524179	0.262089	+	0.965044i	$0.415588\pi$
$972$	0	0
$973$	−1.09133e99	−0.0863326
$974$	−3.04387e99	−0.232649
$975$	0	0
$976$	1.38698e98	0.00989696
$977$	−9.04350e99	−0.623546	−0.311773	−	0.950157i	$-0.600923\pi$
$977$	−9.04350e99	−0.623546	−0.311773	+	0.950157i	$0.600923\pi$
$978$	0	0
$979$	8.59305e99	0.553257
$980$	−1.21756e100	−0.757557
$981$	0	0
$982$	7.77064e99	0.451567
$983$	9.29144e99	0.521844	0.260922	−	0.965360i	$-0.415974\pi$
$983$	9.29144e99	0.521844	0.260922	+	0.965360i	$0.415974\pi$
$984$	0	0
$985$	−2.34239e100	−1.22898
$986$	8.66269e99	0.439315
$987$	0	0
$988$	−2.56688e99	−0.121632
$989$	−1.60126e100	−0.733479
$990$	0	0
$991$	−3.03921e100	−1.30105	−0.650524	−	0.759485i	$-0.725451\pi$
$991$	−3.03921e100	−1.30105	−0.650524	+	0.759485i	$0.725451\pi$
$992$	1.49317e100	0.617970
$993$	0	0
$994$	9.73499e98	0.0376610
$995$	−2.59312e100	−0.969950
$996$	0	0
$997$	2.16012e100	0.755422	0.377711	−	0.925924i	$-0.376711\pi$
$997$	2.16012e100	0.755422	0.377711	+	0.925924i	$0.376711\pi$
$998$	−2.43661e100	−0.823971
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9.68.a.a.1.2		5
3.2	odd	2		1.68.a.a.1.4	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.68.a.a.1.4	✓	5	3.2	odd	2
9.68.a.a.1.2		5	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 9 = 3^{2} \)
Weight:	\( k \)	\(=\)	\( 68 \)
Character orbit:	\([\chi]\)	\(=\)	9.a (trivial)

\(f(q)\)	\(=\)	\(q-8.43209e9 q^{2} -7.64738e19 q^{4} -3.83370e23 q^{5} -1.75533e27 q^{7} +1.88919e30 q^{8} +O(q^{10})\)	\(q-8.43209e9 q^{2} -7.64738e19 q^{4} -3.83370e23 q^{5} -1.75533e27 q^{7} +1.88919e30 q^{8} +3.23261e33 q^{10} -1.38749e35 q^{11} -6.45579e36 q^{13} +1.48011e37 q^{14} -4.64430e39 q^{16} +8.51154e40 q^{17} -5.19929e42 q^{19} +2.93178e43 q^{20} +1.16995e45 q^{22} -2.49803e45 q^{23} +7.92100e46 q^{25} +5.44359e46 q^{26} +1.34237e47 q^{28} -1.20701e49 q^{29} -6.23103e49 q^{31} -2.39634e50 q^{32} -7.17701e50 q^{34} +6.72941e50 q^{35} +3.56231e51 q^{37} +4.38409e52 q^{38} -7.24259e53 q^{40} -4.51579e53 q^{41} +6.41012e54 q^{43} +1.06107e55 q^{44} +2.10636e55 q^{46} +1.03805e56 q^{47} -4.15297e56 q^{49} -6.67906e56 q^{50} +4.93699e56 q^{52} +9.61057e57 q^{53} +5.31924e58 q^{55} -3.31615e57 q^{56} +1.01776e59 q^{58} +4.77326e58 q^{59} -2.98642e58 q^{61} +5.25406e59 q^{62} +2.70600e60 q^{64} +2.47496e60 q^{65} -2.03266e61 q^{67} -6.50909e60 q^{68} -5.67430e60 q^{70} +6.57721e61 q^{71} -1.20026e62 q^{73} -3.00377e61 q^{74} +3.97609e62 q^{76} +2.43551e62 q^{77} +2.11502e63 q^{79} +1.78049e63 q^{80} +3.80776e63 q^{82} +3.25858e64 q^{83} -3.26307e64 q^{85} -5.40507e64 q^{86} -2.62124e65 q^{88} -6.19321e64 q^{89} +1.13320e64 q^{91} +1.91034e65 q^{92} -8.75297e65 q^{94} +1.99325e66 q^{95} -9.58970e65 q^{97} +3.50182e66 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5554901256 q^{2} + 35\!\cdots\!40 q^{4}+ \cdots - 32\!\cdots\!80 q^{8}+O(q^{10}) \) 5 * q - 5554901256 * q^2 + 350472004382296600640 * q^4 - 330744836907160597528950 * q^5 + 33632726520317603069152899256 * q^7 - 3261706912683890613885787783680 * q^8	\( 5 q - 5554901256 q^{2} + 35\!\cdots\!40 q^{4}+ \cdots - 12\!\cdots\!08 q^{98}+O(q^{100}) \) 5 * q - 5554901256 * q^2 + 350472004382296600640 * q^4 - 330744836907160597528950 * q^5 + 33632726520317603069152899256 * q^7 - 3261706912683890613885787783680 * q^8 - 5287067006725590176865792208438800 * q^10 - 203006981551281631527218766884109060 * q^11 + 177822884499813229570219072356766702 * q^13 - 290753425875732621001638220329334092480 * q^14 + 18266362102387359184851236859301091348480 * q^16 - 75241497376497768433807587544150344102906 * q^17 + 3998884102997801557241177083441346393295100 * q^19 + 15074831378020267075078660994426474705846400 * q^20 + 1841247880573095918698534631505063585473750432 * q^22 + 419660874260151197180020017656265233426170968 * q^23 + 175279270003539476977813620578789764300129011875 * q^25 - 21416337629300654169124097666599354769371062960 * q^26 + 14982728306233364463143494158089725946900023694848 * q^28 - 18733547497571939101380180097730717676662894233950 * q^29 + 365849822471060800690376948081970588059264217288160 * q^31 - 1591165751988992347021834813546084975107380979204096 * q^32 + 7817592154883765712044327776251337623941567500782480 * q^34 - 4524513410554564733118320802153390591800073874285200 * q^35 - 56389392588768330454829645685931746748224606527380394 * q^37 + 311451303114725973135630000941921262878276604287141280 * q^38 - 1886768345048751997029226502271137564956948018338432000 * q^40 - 113555410342193387626817165052765483601006465327103010 * q^41 + 6508304671085340439704487085091235272293711787089861092 * q^43 - 49369449025916528958230735645862721892607029455332199680 * q^44 + 79552980744915855625545425031873854934595203714399466560 * q^46 + 128023594814179612806658270243785955024442659789917751344 * q^47 - 771329931628098140389185999842408514636676231229111640435 * q^49 + 3919481096957516748709549462396523762064894577356951885000 * q^50 + 672997930141532412373441872733517069231734104886640838016 * q^52 - 1046765637663599376128164919371991945530174947374966490822 * q^53 + 66942753107037791618627925698973654692865254681839842449400 * q^55 - 164746697572073652525116376573879338548073684495562965299200 * q^56 - 61137956327686914952144218440883774026433838412764071303120 * q^58 + 305790738450157663862297559072875409310668010528073986627500 * q^59 - 1137073938560830615075915586814815057732339483683661907261890 * q^61 + 2197846158587850633685459639420082896328263471188214086352128 * q^62 + 6319522016046144185339524760669817745425888834305772325437440 * q^64 - 17054811715744261908524663200762497384422620874171150779332900 * q^65 - 11903044989154510038048691316368615691989662189985228212145844 * q^67 - 81678078346863753465296539273887923556029130514802676363013248 * q^68 - 262230520857255225386866306600227822969034558447869634844476800 * q^70 + 112169920751076079402369696382938772997794660792753351047772040 * q^71 - 304061800126793465330511223999919410949872771539142074085150718 * q^73 - 921008183492866283853128598587548033201480125306857491530532080 * q^74 + 2076567511595431334156222539483464646991367936386801535927366400 * q^76 - 5018131462902473316385236580394702266689060843204393245422638432 * q^77 + 2994504004186984867153587379277397704940492543518650040519815600 * q^79 + 15198313539268031787508043655897930060910106707913153957798092800 * q^80 - 17699192859543411700126655967680257958811789647196784631772687408 * q^82 + 47352447490969920981134910324833485175129837165394254819464203588 * q^83 - 39334683419450918052255554891934302881190270223971152032299949300 * q^85 - 49699285824187070765304255878405188987409992188766856866277888160 * q^86 + 430257938373980762031490456670953174964896424231159472833711400960 * q^88 + 11988583288945008097106276879882407358382120120051430459779195150 * q^89 + 232376792470844928685229521265495056030669651812230900159714321360 * q^91 + 986292139147147043544160128863480288571347084605459783959415954944 * q^92 - 4927754861315037106505826725122767884474273389521981763161929704320 * q^94 + 2022377488008256909110920831467942087293430657450319294506545963000 * q^95 + 2897556354128426205551279700626881901316694084191208635967319112106 * q^97 - 12426885478574475523720488645206508880013087100212747539199689691208 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more