LMFDB - Embedded newform 9.76.a.c.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9,76,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, 76, names="a")

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

chi := DirichletCharacter("9.1");

S:= CuspForms(chi, 76);

N := Newforms(S);

Level:	$ N $	$=$	$ 9 = 3^{2} $
Weight:	$ k $	$=$	$ 76 $
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:	$320.605553540$
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} - 3 x^{5} + \cdots - 67\!\cdots\!50 $ x^6 - 3x^5 - 38457853073924058692x^4 - 10276556354621685339901678086x^3 + 371187556674475060057870954681799784505x^2 + 52686123927652036687598761277591247931691204025*x - 675344021115865838575279495800656435684060652010336995750
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	multiple of $ 2^{58}\cdot 3^{36}\cdot 5^{7}\cdot 7^{3}\cdot 11\cdot 13\cdot 19 $
Twist minimal:	no (minimal twist has level 1)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.60397e9$ 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)

$f(q)$	$=$	$q+1.25000e11 q^{2} -2.21540e22 q^{4} -1.80927e26 q^{5} +2.76229e31 q^{7} -7.49160e33 q^{8} +O(q^{10})$	\(q+1.25000e11 q^{2} -2.21540e22 q^{4} -1.80927e26 q^{5} +2.76229e31 q^{7} -7.49160e33 q^{8} -2.26158e37 q^{10} -7.60456e38 q^{11} +2.57727e41 q^{13} +3.45284e42 q^{14} -9.94899e43 q^{16} +1.39330e46 q^{17} -1.16447e48 q^{19} +4.00827e48 q^{20} -9.50567e49 q^{22} -9.81970e50 q^{23} +6.26484e51 q^{25} +3.22158e52 q^{26} -6.11958e53 q^{28} -3.47815e54 q^{29} +1.19579e56 q^{31} +2.70588e56 q^{32} +1.74162e57 q^{34} -4.99772e57 q^{35} +1.03750e59 q^{37} -1.45558e59 q^{38} +1.35543e60 q^{40} -4.07748e60 q^{41} +2.20097e61 q^{43} +1.68472e61 q^{44} -1.22746e62 q^{46} +4.37785e62 q^{47} -1.64884e63 q^{49} +7.83102e62 q^{50} -5.70970e63 q^{52} -7.68796e64 q^{53} +1.37587e65 q^{55} -2.06939e65 q^{56} -4.34767e65 q^{58} +3.79588e66 q^{59} +5.02447e66 q^{61} +1.49473e67 q^{62} +3.75820e67 q^{64} -4.66298e67 q^{65} +3.39062e68 q^{67} -3.08672e68 q^{68} -6.24713e68 q^{70} +3.17915e69 q^{71} +7.11742e69 q^{73} +1.29687e70 q^{74} +2.57978e70 q^{76} -2.10060e70 q^{77} +9.47303e70 q^{79} +1.80004e70 q^{80} -5.09684e71 q^{82} +1.43323e71 q^{83} -2.52086e72 q^{85} +2.75120e72 q^{86} +5.69703e72 q^{88} -9.94475e72 q^{89} +7.11916e72 q^{91} +2.17546e73 q^{92} +5.47229e73 q^{94} +2.10685e74 q^{95} -7.42743e73 q^{97} -2.06105e74 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 57080822040 q^{2} + 17\!\cdots\!28 q^{4}+ \cdots - 44\!\cdots\!20 q^{8}+O(q^{10}) $ 6 * q + 57080822040 * q^2 + 172600466200162028593728 * q^4 + 38982947396479621874420940 * q^5 + 1924474634802918779239478643600 * q^7 - 4434903997968330342413437320645120 * q^8	$ 6 q + 57080822040 q^{2} + 17\!\cdots\!28 q^{4}+ \cdots + 16\!\cdots\!20 q^{98}+O(q^{100}) $ 6 * q + 57080822040 * q^2 + 172600466200162028593728 * q^4 + 38982947396479621874420940 * q^5 + 1924474634802918779239478643600 * q^7 - 4434903997968330342413437320645120 * q^8 + 133267935677460241037537052811576902960 * q^10 + 945659951949983071650740113735111214088 * q^11 + 533185179266852519941169625089098069764420 * q^13 - 8213946953162263647781694565284746353072576 * q^14 + 2681978005634517266622433394712415602648354816 * q^16 - 18261354095776132566172834671579513215483393580 * q^17 + 1061534856238354155312794852768417904375328867080 * q^19 - 9296557595621768320156983882258818695571369489280 * q^20 + 153315952874260268106194796405555863940172791589920 * q^22 - 1514929449877410771078348320690686508954384916853680 * q^23 + 19310380328261093150952553553121815533899364092239850 * q^25 - 117730941166816926498584842564210483616872200198362352 * q^26 + 1465885017268620651444113586751990931877154821040657920 * q^28 - 14718767377678125816030788545177391174717388984245534820 * q^29 - 41742874787360498453628839393549579933009794702042980288 * q^31 - 1174660148719991756036751189163540495635749096221862952960 * q^32 + 3034358950124956876257603978888653679838045546922892395856 * q^34 - 27299672493656883988453292135086250689495956464846953371360 * q^35 + 98517772547646219221591850851651469983797181732879949140340 * q^37 - 1257660981102762853948269222064665311191946222029530979162080 * q^38 + 8807901623729426290853411185831313455271917763097593104051200 * q^40 - 5039211762207387369185984529408567568647239506257650214613212 * q^41 + 27938749238581541451173935836639198120068956697568041153196600 * q^43 + 86388814623354110240559749283671522466589238091463431115801344 * q^44 - 824145803870667307795487213022331547039596870954611528203613888 * q^46 + 1386854970430427381646742231466788117324292253074354560479038880 * q^47 - 5707868790800975784856935495088511133239544554833878775910738442 * q^49 - 3144970672685230615695776810799603542575044196614572896663392600 * q^50 + 41677832028141179840113326797899392222702339479233001780216150400 * q^52 - 64283727174659970247345252518647455744441825702701764187054106260 * q^53 + 436876078674971654403803167180324983329224614451635548418098895120 * q^55 - 289950797286325881623798664040853221823087783432246887785270865920 * q^56 - 1763005756244374137396902189067511762184157619699352574614137403280 * q^58 + 2478360370933764187935912640189307752170553390931716617591182395560 * q^59 - 25641115321477658815889269525514418267074652297034046321782023152988 * q^61 + 29014701474354643819936101890527484086880802356768235703173807694080 * q^62 + 47683232974651133862719671893097862706511055838864188064811716968448 * q^64 - 122110621788131890149667214965152432737153070969379698276880591977720 * q^65 + 955579073553339181030887342220313909838725364297810737998207107813480 * q^67 - 1237210897217068882936524414359446185548686871817993659403939290386560 * q^68 - 3476902593928340346610001926714053112321275314411181267396530677050240 * q^70 + 255682319306096297182489045146623979225804710540351927309367569173488 * q^71 - 30605188898291905719532895918028188508004452865621142257485594115042020 * q^73 + 24065024027894599201472448362529876891121294631891709256708009194181584 * q^74 + 100020936372377387464010564774983213070641689274186088752725827576144640 * q^76 - 158930415774536768880543239288689183540801263714134710336228496440507200 * q^77 + 116627678166642324332643069323234092884951450625988452824960767925148320 * q^79 - 1239310052714502333188756625709169306278438274940582996524184215206748160 * q^80 - 2557649813228840791149140948716215474705151452500921688788291587877236080 * q^82 + 798780284119889426515570523310162377392459257634002033911814028088739960 * q^83 - 3615500806340871685633297735534398118961343923835923286123685237305018840 * q^85 - 7245194180836090971883743617494948296636529538939897523257528311698092832 * q^86 - 4827001398586167572057494186308233100206033194553368013345656010632181760 * q^88 - 53575610216125238704584826863934604186542249469498220298044079954014547260 * q^89 + 34502419684367451829908407139028781594042460695428817715772307436403470432 * q^91 - 187445224353587479072026786339601818260697439565538078689222489106585838080 * q^92 - 296189325151987924691405038954735773676385503565643130818482899021002578304 * q^94 - 194934959385332686251927514630723902507701573195047654345319995220933314800 * q^95 - 747928220344118582344222376208028579242743608466383778453296378427532508980 * q^97 + 1663265696927109513221859402309517858944410070626232817304447619248634722520 * 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 $

$2$	1.25000e11	0.643108	0.321554	−	0.946891i	$-0.395795\pi$
$2$	1.25000e11	0.643108	0.321554	+	0.946891i	$0.395795\pi$
$3$	0	0
$3$	0	0
$4$	−2.21540e22	−0.586413
$5$	−1.80927e26	−1.11206	−0.556030	−	0.831162i	$-0.687676\pi$
$5$	−1.80927e26	−1.11206	−0.556030	+	0.831162i	$0.687676\pi$
$6$	0	0
$7$	2.76229e31	0.562461	0.281230	−	0.959640i	$-0.409258\pi$
$7$	2.76229e31	0.562461	0.281230	+	0.959640i	$0.409258\pi$
$8$	−7.49160e33	−1.02023
$9$	0	0
$10$	−2.26158e37	−0.715175
$11$	−7.60456e38	−0.674293	−0.337146	−	0.941452i	$-0.609462\pi$
$11$	−7.60456e38	−0.674293	−0.337146	+	0.941452i	$0.609462\pi$
$12$	0	0
$13$	2.57727e41	0.434795	0.217397	−	0.976083i	$-0.430243\pi$
$13$	2.57727e41	0.434795	0.217397	+	0.976083i	$0.430243\pi$
$14$	3.45284e42	0.361723
$15$	0	0
$16$	−9.94899e43	−0.0697076
$17$	1.39330e46	1.00510	0.502551	−	0.864548i	$-0.332395\pi$
$17$	1.39330e46	1.00510	0.502551	+	0.864548i	$0.332395\pi$
$18$	0	0
$19$	−1.16447e48	−1.29679	−0.648394	−	0.761305i	$-0.724559\pi$
$19$	−1.16447e48	−1.29679	−0.648394	+	0.761305i	$0.724559\pi$
$20$	4.00827e48	0.652126
$21$	0	0
$22$	−9.50567e49	−0.433643
$23$	−9.81970e50	−0.845872	−0.422936	−	0.906160i	$-0.639000\pi$
$23$	−9.81970e50	−0.845872	−0.422936	+	0.906160i	$0.639000\pi$
$24$	0	0
$25$	6.26484e51	0.236679
$26$	3.22158e52	0.279620
$27$	0	0
$28$	−6.11958e53	−0.329834
$29$	−3.47815e54	−0.502832	−0.251416	−	0.967879i	$-0.580896\pi$
$29$	−3.47815e54	−0.502832	−0.251416	+	0.967879i	$0.580896\pi$
$30$	0	0
$31$	1.19579e56	1.41773	0.708863	−	0.705346i	$-0.249209\pi$
$31$	1.19579e56	1.41773	0.708863	+	0.705346i	$0.249209\pi$
$32$	2.70588e56	0.975405
$33$	0	0
$34$	1.74162e57	0.646388
$35$	−4.99772e57	−0.625490
$36$	0	0
$37$	1.03750e59	1.61593	0.807963	−	0.589233i	$-0.200570\pi$
$37$	1.03750e59	1.61593	0.807963	+	0.589233i	$0.200570\pi$
$38$	−1.45558e59	−0.833974
$39$	0	0
$40$	1.35543e60	1.13456
$41$	−4.07748e60	−1.35207	−0.676033	−	0.736872i	$-0.736302\pi$
$41$	−4.07748e60	−1.35207	−0.676033	+	0.736872i	$0.736302\pi$
$42$	0	0
$43$	2.20097e61	1.22334	0.611670	−	0.791113i	$-0.290498\pi$
$43$	2.20097e61	1.22334	0.611670	+	0.791113i	$0.290498\pi$
$44$	1.68472e61	0.395414
$45$	0	0
$46$	−1.22746e62	−0.543987
$47$	4.37785e62	0.866146	0.433073	−	0.901359i	$-0.357429\pi$
$47$	4.37785e62	0.866146	0.433073	+	0.901359i	$0.357429\pi$
$48$	0	0
$49$	−1.64884e63	−0.683638
$50$	7.83102e62	0.152210
$51$	0	0
$52$	−5.70970e63	−0.254969
$53$	−7.68796e64	−1.68061	−0.840303	−	0.542117i	$-0.817623\pi$
$53$	−7.68796e64	−1.68061	−0.840303	+	0.542117i	$0.817623\pi$
$54$	0	0
$55$	1.37587e65	0.749855
$56$	−2.06939e65	−0.573841
$57$	0	0
$58$	−4.34767e65	−0.323375
$59$	3.79588e66	1.48717	0.743587	−	0.668639i	$-0.233123\pi$
$59$	3.79588e66	1.48717	0.743587	+	0.668639i	$0.233123\pi$
$60$	0	0
$61$	5.02447e66	0.563925	0.281962	−	0.959425i	$-0.409015\pi$
$61$	5.02447e66	0.563925	0.281962	+	0.959425i	$0.409015\pi$
$62$	1.49473e67	0.911750
$63$	0	0
$64$	3.75820e67	0.696998
$65$	−4.66298e67	−0.483518
$66$	0	0
$67$	3.39062e68	1.12843	0.564214	−	0.825629i	$-0.309179\pi$
$67$	3.39062e68	1.12843	0.564214	+	0.825629i	$0.309179\pi$
$68$	−3.08672e68	−0.589404
$69$	0	0
$70$	−6.24713e68	−0.402258
$71$	3.17915e69	1.20261	0.601303	−	0.799021i	$-0.294648\pi$
$71$	3.17915e69	1.20261	0.601303	+	0.799021i	$0.294648\pi$
$72$	0	0
$73$	7.11742e69	0.949982	0.474991	−	0.879991i	$-0.342451\pi$
$73$	7.11742e69	0.949982	0.474991	+	0.879991i	$0.342451\pi$
$74$	1.29687e70	1.03921
$75$	0	0
$76$	2.57978e70	0.760453
$77$	−2.10060e70	−0.379263
$78$	0	0
$79$	9.47303e70	0.653842	0.326921	−	0.945052i	$-0.393989\pi$
$79$	9.47303e70	0.653842	0.326921	+	0.945052i	$0.393989\pi$
$80$	1.80004e70	0.0775190
$81$	0	0
$82$	−5.09684e71	−0.869523
$83$	1.43323e71	0.155198	0.0775992	−	0.996985i	$-0.475275\pi$
$83$	1.43323e71	0.155198	0.0775992	+	0.996985i	$0.475275\pi$
$84$	0	0
$85$	−2.52086e72	−1.11773
$86$	2.75120e72	0.786739
$87$	0	0
$88$	5.69703e72	0.687937
$89$	−9.94475e72	−0.786083	−0.393042	−	0.919521i	$-0.628577\pi$
$89$	−9.94475e72	−0.786083	−0.393042	+	0.919521i	$0.628577\pi$
$90$	0	0
$91$	7.11916e72	0.244555
$92$	2.17546e73	0.496030
$93$	0	0
$94$	5.47229e73	0.557025
$95$	2.10685e74	1.44211
$96$	0	0
$97$	−7.42743e73	−0.232755	−0.116377	−	0.993205i	$-0.537128\pi$
$97$	−7.42743e73	−0.232755	−0.116377	+	0.993205i	$0.537128\pi$
$98$	−2.06105e74	−0.439653
$99$	0	0
$100$	−1.38791e74	−0.138791
$101$	−1.42243e74	−0.0979445	−0.0489722	−	0.998800i	$-0.515595\pi$
$101$	−1.42243e74	−0.0979445	−0.0489722	+	0.998800i	$0.515595\pi$
$102$	0	0
$103$	4.64143e75	1.53199	0.765996	−	0.642846i	$-0.222246\pi$
$103$	4.64143e75	1.53199	0.765996	+	0.642846i	$0.222246\pi$
$104$	−1.93079e75	−0.443592
$105$	0	0
$106$	−9.60991e75	−1.08081
$107$	−5.79865e75	−0.458601	−0.229301	−	0.973356i	$-0.573644\pi$
$107$	−5.79865e75	−0.458601	−0.229301	+	0.973356i	$0.573644\pi$
$108$	0	0
$109$	−4.52515e76	−1.78706	−0.893531	−	0.449001i	$-0.851780\pi$
$109$	−4.52515e76	−1.78706	−0.893531	+	0.449001i	$0.851780\pi$
$110$	1.71983e76	0.482237
$111$	0	0
$112$	−2.74820e75	−0.0392078
$113$	−7.93579e76	−0.811239	−0.405619	−	0.914042i	$-0.632944\pi$
$113$	−7.93579e76	−0.811239	−0.405619	+	0.914042i	$0.632944\pi$
$114$	0	0
$115$	1.77665e77	0.940661
$116$	7.70550e76	0.294867
$117$	0	0
$118$	4.74483e77	0.956413
$119$	3.84869e77	0.565330
$120$	0	0
$121$	−6.93601e77	−0.545329
$122$	6.28056e77	0.362664
$123$	0	0
$124$	−2.64916e78	−0.831372
$125$	3.65562e78	0.848859
$126$	0	0
$127$	9.37430e78	1.20033	0.600163	−	0.799878i	$-0.295102\pi$
$127$	9.37430e78	1.20033	0.600163	+	0.799878i	$0.295102\pi$
$128$	−5.52480e78	−0.527160
$129$	0	0
$130$	−5.82871e78	−0.310954
$131$	−2.91749e79	−1.16771	−0.583857	−	0.811857i	$-0.698457\pi$
$131$	−2.91749e79	−1.16771	−0.583857	+	0.811857i	$0.698457\pi$
$132$	0	0
$133$	−3.21660e79	−0.729392
$134$	4.23826e79	0.725700
$135$	0	0
$136$	−1.04380e80	−1.02544
$137$	−9.29317e79	−0.693653	−0.346827	−	0.937929i	$-0.612741\pi$
$137$	−9.29317e79	−0.693653	−0.346827	+	0.937929i	$0.612741\pi$
$138$	0	0
$139$	2.02447e80	0.877518	0.438759	−	0.898605i	$-0.355418\pi$
$139$	2.02447e80	0.877518	0.438759	+	0.898605i	$0.355418\pi$
$140$	1.10720e80	0.366795
$141$	0	0
$142$	3.97393e80	0.773405
$143$	−1.95990e80	−0.293179
$144$	0	0
$145$	6.29291e80	0.559180
$146$	8.89674e80	0.610940
$147$	0	0
$148$	−2.29847e81	−0.947600
$149$	−2.75595e81	−0.882646	−0.441323	−	0.897348i	$-0.645491\pi$
$149$	−2.75595e81	−0.882646	−0.441323	+	0.897348i	$0.645491\pi$
$150$	0	0
$151$	−5.55612e81	−1.07928	−0.539642	−	0.841894i	$-0.681441\pi$
$151$	−5.55612e81	−1.07928	−0.539642	+	0.841894i	$0.681441\pi$
$152$	8.72375e81	1.32303
$153$	0	0
$154$	−2.62574e81	−0.243907
$155$	−2.16351e82	−1.57660
$156$	0	0
$157$	2.68318e82	1.20896	0.604481	−	0.796619i	$-0.293380\pi$
$157$	2.68318e82	1.20896	0.604481	+	0.796619i	$0.293380\pi$
$158$	1.18412e82	0.420491
$159$	0	0
$160$	−4.89568e82	−1.08471
$161$	−2.71248e82	−0.475770
$162$	0	0
$163$	−4.12916e82	−0.455855	−0.227927	−	0.973678i	$-0.573195\pi$
$163$	−4.12916e82	−0.455855	−0.227927	+	0.973678i	$0.573195\pi$
$164$	9.03328e82	0.792868
$165$	0	0
$166$	1.79153e82	0.0998093
$167$	−1.65503e83	−0.736103	−0.368052	−	0.929805i	$-0.619975\pi$
$167$	−1.65503e83	−0.736103	−0.368052	+	0.929805i	$0.619975\pi$
$168$	0	0
$169$	−2.84936e83	−0.810954
$170$	−3.15106e83	−0.718823
$171$	0	0
$172$	−4.87604e83	−0.717382
$173$	−2.24737e83	−0.266039	−0.133019	−	0.991113i	$-0.542467\pi$
$173$	−2.24737e83	−0.266039	−0.133019	+	0.991113i	$0.542467\pi$
$174$	0	0
$175$	1.73053e83	0.133123
$176$	7.56578e82	0.0470033
$177$	0	0
$178$	−1.24309e84	−0.505536
$179$	2.15324e84	0.709748	0.354874	−	0.934914i	$-0.384524\pi$
$179$	2.15324e84	0.709748	0.354874	+	0.934914i	$0.384524\pi$
$180$	0	0
$181$	−5.53508e84	−1.20275	−0.601377	−	0.798965i	$-0.705381\pi$
$181$	−5.53508e84	−1.20275	−0.601377	+	0.798965i	$0.705381\pi$
$182$	8.89892e83	0.157275
$183$	0	0
$184$	7.35652e84	0.862987
$185$	−1.87711e85	−1.79701
$186$	0	0
$187$	−1.05954e85	−0.677733
$188$	−9.69871e84	−0.507919
$189$	0	0
$190$	2.63355e85	0.927430
$191$	4.53880e85	1.31277	0.656387	−	0.754424i	$-0.272084\pi$
$191$	4.53880e85	1.31277	0.656387	+	0.754424i	$0.272084\pi$
$192$	0	0
$193$	−9.26819e85	−1.81383	−0.906914	−	0.421316i	$-0.861568\pi$
$193$	−9.26819e85	−1.81383	−0.906914	+	0.421316i	$0.861568\pi$
$194$	−9.28425e84	−0.149686
$195$	0	0
$196$	3.65285e85	0.400894
$197$	−4.97463e85	−0.451105	−0.225552	−	0.974231i	$-0.572419\pi$
$197$	−4.97463e85	−0.451105	−0.225552	+	0.974231i	$0.572419\pi$
$198$	0	0
$199$	−7.14033e85	−0.443331	−0.221666	−	0.975123i	$-0.571149\pi$
$199$	−7.14033e85	−0.443331	−0.221666	+	0.975123i	$0.571149\pi$
$200$	−4.69336e85	−0.241468
$201$	0	0
$202$	−1.77803e85	−0.0629888
$203$	−9.60764e85	−0.282823
$204$	0	0
$205$	7.37727e86	1.50358
$206$	5.80177e86	0.985235
$207$	0	0
$208$	−2.56413e85	−0.0303085
$209$	8.85530e86	0.874415
$210$	0	0
$211$	1.15343e87	0.796892	0.398446	−	0.917192i	$-0.369550\pi$
$211$	1.15343e87	0.796892	0.398446	+	0.917192i	$0.369550\pi$
$212$	1.70319e87	0.985529
$213$	0	0
$214$	−7.24828e86	−0.294930
$215$	−3.98215e87	−1.36043
$216$	0	0
$217$	3.30312e87	0.797415
$218$	−5.65642e87	−1.14927
$219$	0	0
$220$	−3.04811e87	−0.439724
$221$	3.59091e87	0.437013
$222$	0	0
$223$	−2.95594e87	−0.256604	−0.128302	−	0.991735i	$-0.540953\pi$
$223$	−2.95594e87	−0.256604	−0.128302	+	0.991735i	$0.540953\pi$
$224$	7.47442e87	0.548627
$225$	0	0
$226$	−9.91971e87	−0.521714
$227$	9.67879e87	0.431371	0.215686	−	0.976463i	$-0.430801\pi$
$227$	9.67879e87	0.431371	0.215686	+	0.976463i	$0.430801\pi$
$228$	0	0
$229$	4.56764e88	1.46508	0.732540	−	0.680723i	$-0.238334\pi$
$229$	4.56764e88	1.46508	0.732540	+	0.680723i	$0.238334\pi$
$230$	2.22081e88	0.604946
$231$	0	0
$232$	2.60569e88	0.513007
$233$	−1.95700e88	−0.327902	−0.163951	−	0.986468i	$-0.552424\pi$
$233$	−1.95700e88	−0.327902	−0.163951	+	0.986468i	$0.552424\pi$
$234$	0	0
$235$	−7.92072e88	−0.963207
$236$	−8.40941e88	−0.872098
$237$	0	0
$238$	4.81084e88	0.363568
$239$	5.75419e88	0.371589	0.185795	−	0.982589i	$-0.440514\pi$
$239$	5.75419e88	0.371589	0.185795	+	0.982589i	$0.440514\pi$
$240$	0	0
$241$	−3.37052e89	−1.59242	−0.796210	−	0.605020i	$-0.793165\pi$
$241$	−3.37052e89	−1.59242	−0.796210	+	0.605020i	$0.793165\pi$
$242$	−8.66999e88	−0.350705
$243$	0	0
$244$	−1.11312e89	−0.330693
$245$	2.98320e89	0.760247
$246$	0	0
$247$	−3.00116e89	−0.563837
$248$	−8.95839e89	−1.44641
$249$	0	0
$250$	4.56951e89	0.545908
$251$	−8.76805e89	−0.901858	−0.450929	−	0.892560i	$-0.648907\pi$
$251$	−8.76805e89	−0.901858	−0.450929	+	0.892560i	$0.648907\pi$
$252$	0	0
$253$	7.46746e89	0.570365
$254$	1.17178e90	0.771939
$255$	0	0
$256$	−2.11041e90	−1.03602
$257$	1.45805e90	0.618416	0.309208	−	0.950995i	$-0.399936\pi$
$257$	1.45805e90	0.618416	0.309208	+	0.950995i	$0.399936\pi$
$258$	0	0
$259$	2.86586e90	0.908895
$260$	1.03304e90	0.283541
$261$	0	0
$262$	−3.64685e90	−0.750965
$263$	1.04913e90	0.187280	0.0936398	−	0.995606i	$-0.470150\pi$
$263$	1.04913e90	0.187280	0.0936398	+	0.995606i	$0.470150\pi$
$264$	0	0
$265$	1.39096e91	1.86894
$266$	−4.02074e90	−0.469078
$267$	0	0
$268$	−7.51159e90	−0.661724
$269$	−1.36688e90	−0.104717	−0.0523587	−	0.998628i	$-0.516674\pi$
$269$	−1.36688e90	−0.104717	−0.0523587	+	0.998628i	$0.516674\pi$
$270$	0	0
$271$	−3.50139e90	−0.203185	−0.101593	−	0.994826i	$-0.532394\pi$
$271$	−3.50139e90	−0.203185	−0.101593	+	0.994826i	$0.532394\pi$
$272$	−1.38619e90	−0.0700631
$273$	0	0
$274$	−1.16164e91	−0.446094
$275$	−4.76414e90	−0.159591
$276$	0	0
$277$	3.00814e90	0.0767903	0.0383951	−	0.999263i	$-0.487775\pi$
$277$	3.00814e90	0.0767903	0.0383951	+	0.999263i	$0.487775\pi$
$278$	2.53058e91	0.564338
$279$	0	0
$280$	3.74409e91	0.638146
$281$	1.95826e91	0.292000	0.146000	−	0.989285i	$-0.453360\pi$
$281$	1.95826e91	0.292000	0.146000	+	0.989285i	$0.453360\pi$
$282$	0	0
$283$	7.34668e91	0.839653	0.419826	−	0.907604i	$-0.362091\pi$
$283$	7.34668e91	0.839653	0.419826	+	0.907604i	$0.362091\pi$
$284$	−7.04311e91	−0.705224
$285$	0	0
$286$	−2.44987e91	−0.188546
$287$	−1.12632e92	−0.760483
$288$	0	0
$289$	1.96546e90	0.0102281
$290$	7.86611e91	0.359613
$291$	0	0
$292$	−1.57680e92	−0.557081
$293$	3.78620e92	1.17670	0.588352	−	0.808605i	$-0.299777\pi$
$293$	3.78620e92	1.17670	0.588352	+	0.808605i	$0.299777\pi$
$294$	0	0
$295$	−6.86777e92	−1.65383
$296$	−7.77250e92	−1.64862
$297$	0	0
$298$	−3.44493e92	−0.567637
$299$	−2.53080e92	−0.367781
$300$	0	0
$301$	6.07971e92	0.688080
$302$	−6.94513e92	−0.694096
$303$	0	0
$304$	1.15853e92	0.0903959
$305$	−9.09062e92	−0.627119
$306$	0	0
$307$	1.04256e92	0.0562878	0.0281439	−	0.999604i	$-0.491040\pi$
$307$	1.04256e92	0.0562878	0.0281439	+	0.999604i	$0.491040\pi$
$308$	4.65367e92	0.222405
$309$	0	0
$310$	−2.70438e93	−1.01392
$311$	−2.13035e93	−0.707843	−0.353921	−	0.935275i	$-0.615152\pi$
$311$	−2.13035e93	−0.707843	−0.353921	+	0.935275i	$0.615152\pi$
$312$	0	0
$313$	−2.58095e91	−0.00674322	−0.00337161	−	0.999994i	$-0.501073\pi$
$313$	−2.58095e91	−0.00674322	−0.00337161	+	0.999994i	$0.501073\pi$
$314$	3.35396e93	0.777493
$315$	0	0
$316$	−2.09866e93	−0.383421
$317$	−1.87491e93	−0.304269	−0.152135	−	0.988360i	$-0.548615\pi$
$317$	−1.87491e93	−0.304269	−0.152135	+	0.988360i	$0.548615\pi$
$318$	0	0
$319$	2.64498e93	0.339056
$320$	−6.79961e93	−0.775104
$321$	0	0
$322$	−3.39059e93	−0.305971
$323$	−1.62246e94	−1.30340
$324$	0	0
$325$	1.61462e93	0.102907
$326$	−5.16143e93	−0.293164
$327$	0	0
$328$	3.05469e94	1.37942
$329$	1.20929e94	0.487173
$330$	0	0
$331$	−5.42826e94	−1.74225	−0.871125	−	0.491061i	$-0.836609\pi$
$331$	−5.42826e94	−1.74225	−0.871125	+	0.491061i	$0.836609\pi$
$332$	−3.17518e93	−0.0910103
$333$	0	0
$334$	−2.06878e94	−0.473394
$335$	−6.13454e94	−1.25488
$336$	0	0
$337$	1.16963e94	0.191393	0.0956965	−	0.995411i	$-0.469492\pi$
$337$	1.16963e94	0.191393	0.0956965	+	0.995411i	$0.469492\pi$
$338$	−3.56169e94	−0.521530
$339$	0	0
$340$	5.58471e94	0.655453
$341$	−9.09348e94	−0.955962
$342$	0	0
$343$	−1.12168e95	−0.946980
$344$	−1.64888e95	−1.24809
$345$	0	0
$346$	−2.80920e94	−0.171092
$347$	−7.78575e94	−0.425545	−0.212772	−	0.977102i	$-0.568249\pi$
$347$	−7.78575e94	−0.425545	−0.212772	+	0.977102i	$0.568249\pi$
$348$	0	0
$349$	−3.88388e95	−1.71125	−0.855623	−	0.517600i	$-0.826826\pi$
$349$	−3.88388e95	−1.71125	−0.855623	+	0.517600i	$0.826826\pi$
$350$	2.16315e94	0.0856121
$351$	0	0
$352$	−2.05771e95	−0.657708
$353$	−6.23918e95	−1.79298	−0.896492	−	0.443060i	$-0.853893\pi$
$353$	−6.23918e95	−1.79298	−0.896492	+	0.443060i	$0.853893\pi$
$354$	0	0
$355$	−5.75195e95	−1.33737
$356$	2.20316e95	0.460969
$357$	0	0
$358$	2.69154e95	0.456444
$359$	−7.02370e95	−1.07281	−0.536407	−	0.843960i	$-0.680219\pi$
$359$	−7.02370e95	−1.07281	−0.536407	+	0.843960i	$0.680219\pi$
$360$	0	0
$361$	5.49651e95	0.681659
$362$	−6.91882e95	−0.773500
$363$	0	0
$364$	−1.57718e95	−0.143410
$365$	−1.28773e96	−1.05644
$366$	0	0
$367$	4.84188e95	0.323621	0.161810	−	0.986822i	$-0.448267\pi$
$367$	4.84188e95	0.323621	0.161810	+	0.986822i	$0.448267\pi$
$368$	9.76962e94	0.0589637
$369$	0	0
$370$	−2.34638e96	−1.15567
$371$	−2.12363e96	−0.945275
$372$	0	0
$373$	−3.19767e96	−1.16346	−0.581732	−	0.813380i	$-0.697625\pi$
$373$	−3.19767e96	−1.16346	−0.581732	+	0.813380i	$0.697625\pi$
$374$	−1.32442e96	−0.435855
$375$	0	0
$376$	−3.27971e96	−0.883672
$377$	−8.96413e95	−0.218629
$378$	0	0
$379$	6.31380e96	1.26276	0.631380	−	0.775474i	$-0.282489\pi$
$379$	6.31380e96	1.26276	0.631380	+	0.775474i	$0.282489\pi$
$380$	−4.66751e96	−0.845670
$381$	0	0
$382$	5.67348e96	0.844255
$383$	1.05836e97	1.42785	0.713923	−	0.700225i	$-0.246917\pi$
$383$	1.05836e97	1.42785	0.713923	+	0.700225i	$0.246917\pi$
$384$	0	0
$385$	3.80055e96	0.421764
$386$	−1.15852e97	−1.16649
$387$	0	0
$388$	1.64548e96	0.136490
$389$	6.81840e96	0.513538	0.256769	−	0.966473i	$-0.417342\pi$
$389$	6.81840e96	0.513538	0.256769	+	0.966473i	$0.417342\pi$
$390$	0	0
$391$	−1.36818e97	−0.850187
$392$	1.23525e97	0.697471
$393$	0	0
$394$	−6.21826e96	−0.290109
$395$	−1.71393e97	−0.727112
$396$	0	0
$397$	−3.56797e96	−0.125250	−0.0626250	−	0.998037i	$-0.519947\pi$
$397$	−3.56797e96	−0.125250	−0.0626250	+	0.998037i	$0.519947\pi$
$398$	−8.92539e96	−0.285110
$399$	0	0
$400$	−6.23288e95	−0.0164983
$401$	1.50986e97	0.363934	0.181967	−	0.983305i	$-0.441754\pi$
$401$	1.50986e97	0.363934	0.181967	+	0.983305i	$0.441754\pi$
$402$	0	0
$403$	3.08188e97	0.616420
$404$	3.15126e96	0.0574359
$405$	0	0
$406$	−1.20095e97	−0.181886
$407$	−7.88970e97	−1.08961
$408$	0	0
$409$	−9.27122e96	−0.106540	−0.0532700	−	0.998580i	$-0.516964\pi$
$409$	−9.27122e96	−0.106540	−0.0532700	+	0.998580i	$0.516964\pi$
$410$	9.22156e97	0.966963
$411$	0	0
$412$	−1.02827e98	−0.898379
$413$	1.04853e98	0.836477
$414$	0	0
$415$	−2.59310e97	−0.172590
$416$	6.97379e97	0.424101
$417$	0	0
$418$	1.10691e98	0.562343
$419$	2.22993e98	1.03577	0.517887	−	0.855449i	$-0.326719\pi$
$419$	2.22993e98	1.03577	0.517887	+	0.855449i	$0.326719\pi$
$420$	0	0
$421$	2.56359e98	0.996023	0.498012	−	0.867170i	$-0.334064\pi$
$421$	2.56359e98	0.996023	0.498012	+	0.867170i	$0.334064\pi$
$422$	1.44178e98	0.512487
$423$	0	0
$424$	5.75951e98	1.71461
$425$	8.72879e97	0.237886
$426$	0	0
$427$	1.38790e98	0.317185
$428$	1.28464e98	0.268929
$429$	0	0
$430$	−4.97767e98	−0.874902
$431$	5.84060e98	0.940935	0.470468	−	0.882417i	$-0.344085\pi$
$431$	5.84060e98	0.940935	0.470468	+	0.882417i	$0.344085\pi$
$432$	0	0
$433$	−2.66887e97	−0.0361436	−0.0180718	−	0.999837i	$-0.505753\pi$
$433$	−2.66887e97	−0.0361436	−0.0180718	+	0.999837i	$0.505753\pi$
$434$	4.12888e98	0.512823
$435$	0	0
$436$	1.00250e99	1.04796
$437$	1.14348e99	1.09692
$438$	0	0
$439$	−1.07199e99	−0.866509	−0.433255	−	0.901272i	$-0.642635\pi$
$439$	−1.07199e99	−0.866509	−0.433255	+	0.901272i	$0.642635\pi$
$440$	−1.03075e99	−0.765027
$441$	0	0
$442$	4.48862e98	0.281046
$443$	2.88329e98	0.165863	0.0829315	−	0.996555i	$-0.473572\pi$
$443$	2.88329e98	0.165863	0.0829315	+	0.996555i	$0.473572\pi$
$444$	0	0
$445$	1.79927e99	0.874172
$446$	−3.69491e98	−0.165024
$447$	0	0
$448$	1.03812e99	0.392034
$449$	3.37230e99	1.17135	0.585676	−	0.810545i	$-0.300829\pi$
$449$	3.37230e99	1.17135	0.585676	+	0.810545i	$0.300829\pi$
$450$	0	0
$451$	3.10075e99	0.911688
$452$	1.75810e99	0.475721
$453$	0	0
$454$	1.20984e99	0.277418
$455$	−1.28805e99	−0.271960
$456$	0	0
$457$	−6.02179e99	−1.07862	−0.539310	−	0.842107i	$-0.681315\pi$
$457$	−6.02179e99	−1.07862	−0.539310	+	0.842107i	$0.681315\pi$
$458$	5.70953e99	0.942205
$459$	0	0
$460$	−3.93600e99	−0.551615
$461$	−8.06561e99	−1.04196	−0.520982	−	0.853568i	$-0.674434\pi$
$461$	−8.06561e99	−1.04196	−0.520982	+	0.853568i	$0.674434\pi$
$462$	0	0
$463$	−8.25998e99	−0.907176	−0.453588	−	0.891211i	$-0.649856\pi$
$463$	−8.25998e99	−0.907176	−0.453588	+	0.891211i	$0.649856\pi$
$464$	3.46041e98	0.0350512
$465$	0	0
$466$	−2.44624e99	−0.210876
$467$	−2.12206e100	−1.68801	−0.844005	−	0.536335i	$-0.819808\pi$
$467$	−2.12206e100	−1.68801	−0.844005	+	0.536335i	$0.819808\pi$
$468$	0	0
$469$	9.36585e99	0.634696
$470$	−9.90086e99	−0.619446
$471$	0	0
$472$	−2.84372e100	−1.51727
$473$	−1.67374e100	−0.824889
$474$	0	0
$475$	−7.29523e99	−0.306922
$476$	−8.52640e99	−0.331516
$477$	0	0
$478$	7.19271e99	0.238972
$479$	6.07031e100	1.86479	0.932394	−	0.361445i	$-0.117716\pi$
$479$	6.07031e100	1.86479	0.932394	+	0.361445i	$0.117716\pi$
$480$	0	0
$481$	2.67391e100	0.702597
$482$	−4.21314e100	−1.02410
$483$	0	0
$484$	1.53661e100	0.319788
$485$	1.34382e100	0.258837
$486$	0	0
$487$	−3.82403e100	−0.631226	−0.315613	−	0.948888i	$-0.602210\pi$
$487$	−3.82403e100	−0.631226	−0.315613	+	0.948888i	$0.602210\pi$
$488$	−3.76413e100	−0.575335
$489$	0	0
$490$	3.72899e100	0.488921
$491$	−1.61347e101	−1.95977	−0.979884	−	0.199566i	$-0.936047\pi$
$491$	−1.61347e101	−1.95977	−0.979884	+	0.199566i	$0.936047\pi$
$492$	0	0
$493$	−4.84610e100	−0.505397
$494$	−3.75144e100	−0.362608
$495$	0	0
$496$	−1.18969e100	−0.0988262
$497$	8.78173e100	0.676419
$498$	0	0
$499$	−7.09381e99	−0.0470012	−0.0235006	−	0.999724i	$-0.507481\pi$
$499$	−7.09381e99	−0.0470012	−0.0235006	+	0.999724i	$0.507481\pi$
$500$	−8.09868e100	−0.497782
$501$	0	0
$502$	−1.09600e101	−0.579992
$503$	−2.88098e100	−0.141495	−0.0707475	−	0.997494i	$-0.522538\pi$
$503$	−2.88098e100	−0.141495	−0.0707475	+	0.997494i	$0.522538\pi$
$504$	0	0
$505$	2.57356e100	0.108920
$506$	9.33429e100	0.366806
$507$	0	0
$508$	−2.07679e101	−0.703886
$509$	−3.40536e100	−0.107213	−0.0536063	−	0.998562i	$-0.517072\pi$
$509$	−3.40536e100	−0.107213	−0.0536063	+	0.998562i	$0.517072\pi$
$510$	0	0
$511$	1.96603e101	0.534327
$512$	−5.50789e100	−0.139111
$513$	0	0
$514$	1.82256e101	0.397708
$515$	−8.39761e101	−1.70367
$516$	0	0
$517$	−3.32916e101	−0.584036
$518$	3.58231e101	0.584517
$519$	0	0
$520$	3.49332e101	0.493302
$521$	2.83196e100	0.0372111	0.0186055	−	0.999827i	$-0.494077\pi$
$521$	2.83196e100	0.0372111	0.0186055	+	0.999827i	$0.494077\pi$
$522$	0	0
$523$	−6.78214e101	−0.771887	−0.385943	−	0.922522i	$-0.626124\pi$
$523$	−6.78214e101	−0.771887	−0.385943	+	0.922522i	$0.626124\pi$
$524$	6.46343e101	0.684762
$525$	0	0
$526$	1.31141e101	0.120441
$527$	1.66610e102	1.42496
$528$	0	0
$529$	−3.83417e101	−0.284501
$530$	1.73869e102	1.20193
$531$	0	0
$532$	7.12608e101	0.427725
$533$	−1.05088e102	−0.587871
$534$	0	0
$535$	1.04913e102	0.509992
$536$	−2.54011e102	−1.15126
$537$	0	0
$538$	−1.70859e101	−0.0673445
$539$	1.25387e102	0.460972
$540$	0	0
$541$	−1.95501e101	−0.0625536	−0.0312768	−	0.999511i	$-0.509957\pi$
$541$	−1.95501e101	−0.0625536	−0.0312768	+	0.999511i	$0.509957\pi$
$542$	−4.37673e101	−0.130670
$543$	0	0
$544$	3.77010e102	0.980380
$545$	8.18723e102	1.98732
$546$	0	0
$547$	5.36210e102	1.13451	0.567257	−	0.823541i	$-0.308004\pi$
$547$	5.36210e102	1.13451	0.567257	+	0.823541i	$0.308004\pi$
$548$	2.05881e102	0.406767
$549$	0	0
$550$	−5.95515e101	−0.102634
$551$	4.05021e102	0.652067
$552$	0	0
$553$	2.61672e102	0.367760
$554$	3.76016e101	0.0493844
$555$	0	0
$556$	−4.48502e102	−0.514588
$557$	1.00958e103	1.08286	0.541430	−	0.840746i	$-0.317883\pi$
$557$	1.00958e103	1.08286	0.541430	+	0.840746i	$0.317883\pi$
$558$	0	0
$559$	5.67250e102	0.531902
$560$	4.97223e101	0.0436014
$561$	0	0
$562$	2.44781e102	0.187787
$563$	−1.59566e103	−1.14518	−0.572591	−	0.819841i	$-0.694062\pi$
$563$	−1.59566e103	−1.14518	−0.572591	+	0.819841i	$0.694062\pi$
$564$	0	0
$565$	1.43580e103	0.902147
$566$	9.18332e102	0.539987
$567$	0	0
$568$	−2.38169e103	−1.22694
$569$	−6.65325e102	−0.320866	−0.160433	−	0.987047i	$-0.551289\pi$
$569$	−6.65325e102	−0.320866	−0.160433	+	0.987047i	$0.551289\pi$
$570$	0	0
$571$	−1.13478e103	−0.479798	−0.239899	−	0.970798i	$-0.577114\pi$
$571$	−1.13478e103	−0.479798	−0.239899	+	0.970798i	$0.577114\pi$
$572$	4.34198e102	0.171924
$573$	0	0
$574$	−1.40789e103	−0.489073
$575$	−6.15188e102	−0.200200
$576$	0	0
$577$	−1.02337e103	−0.292374	−0.146187	−	0.989257i	$-0.546700\pi$
$577$	−1.02337e103	−0.292374	−0.146187	+	0.989257i	$0.546700\pi$
$578$	2.45682e101	0.00657778
$579$	0	0
$580$	−1.39413e103	−0.327910
$581$	3.95899e102	0.0872930
$582$	0	0
$583$	5.84636e103	1.13322
$584$	−5.33208e103	−0.969204
$585$	0	0
$586$	4.73273e103	0.756748
$587$	−1.04432e104	−1.56641	−0.783206	−	0.621763i	$-0.786417\pi$
$587$	−1.04432e104	−1.56641	−0.783206	+	0.621763i	$0.786417\pi$
$588$	0	0
$589$	−1.39247e104	−1.83849
$590$	−8.58469e103	−1.06359
$591$	0	0
$592$	−1.03220e103	−0.112642
$593$	9.71824e103	0.995489	0.497745	−	0.867324i	$-0.334162\pi$
$593$	9.71824e103	0.995489	0.497745	+	0.867324i	$0.334162\pi$
$594$	0	0
$595$	−6.96332e103	−0.628681
$596$	6.10555e103	0.517595
$597$	0	0
$598$	−3.16349e103	−0.236523
$599$	−2.58692e104	−1.81667	−0.908337	−	0.418240i	$-0.862647\pi$
$599$	−2.58692e104	−1.81667	−0.908337	+	0.418240i	$0.862647\pi$
$600$	0	0
$601$	−2.18067e104	−1.35144	−0.675720	−	0.737158i	$-0.736167\pi$
$601$	−2.18067e104	−1.35144	−0.675720	+	0.737158i	$0.736167\pi$
$602$	7.59961e103	0.442510
$603$	0	0
$604$	1.23091e104	0.632906
$605$	1.25491e104	0.606439
$606$	0	0
$607$	3.95671e104	1.68950	0.844750	−	0.535161i	$-0.179749\pi$
$607$	3.95671e104	1.68950	0.844750	+	0.535161i	$0.179749\pi$
$608$	−3.15092e104	−1.26489
$609$	0	0
$610$	−1.13632e104	−0.403305
$611$	1.12829e104	0.376596
$612$	0	0
$613$	5.60714e102	0.0165567	0.00827834	−	0.999966i	$-0.497365\pi$
$613$	5.60714e102	0.0165567	0.00827834	+	0.999966i	$0.497365\pi$
$614$	1.30320e103	0.0361991
$615$	0	0
$616$	1.57368e104	0.386937
$617$	−4.75635e104	−1.10047	−0.550237	−	0.835009i	$-0.685463\pi$
$617$	−4.75635e104	−1.10047	−0.550237	+	0.835009i	$0.685463\pi$
$618$	0	0
$619$	3.45849e104	0.708740	0.354370	−	0.935105i	$-0.384695\pi$
$619$	3.45849e104	0.708740	0.354370	+	0.935105i	$0.384695\pi$
$620$	4.79305e104	0.924536
$621$	0	0
$622$	−2.66293e104	−0.455219
$623$	−2.74702e104	−0.442141
$624$	0	0
$625$	−8.27230e104	−1.18066
$626$	−3.22618e102	−0.00433662
$627$	0	0
$628$	−5.94433e104	−0.708951
$629$	1.44554e105	1.62417
$630$	0	0
$631$	1.13799e105	1.13510	0.567552	−	0.823337i	$-0.307890\pi$
$631$	1.13799e105	1.13510	0.567552	+	0.823337i	$0.307890\pi$
$632$	−7.09681e104	−0.667072
$633$	0	0
$634$	−2.34363e104	−0.195678
$635$	−1.69606e105	−1.33484
$636$	0	0
$637$	−4.24951e104	−0.297242
$638$	3.30621e104	0.218050
$639$	0	0
$640$	9.99586e104	0.586234
$641$	1.25933e105	0.696569	0.348285	−	0.937389i	$-0.386764\pi$
$641$	1.25933e105	0.696569	0.348285	+	0.937389i	$0.386764\pi$
$642$	0	0
$643$	2.15243e105	1.05930	0.529648	−	0.848218i	$-0.322324\pi$
$643$	2.15243e105	1.05930	0.529648	+	0.848218i	$0.322324\pi$
$644$	6.00925e104	0.278997
$645$	0	0
$646$	−2.02806e105	−0.838228
$647$	4.52853e105	1.76623	0.883115	−	0.469157i	$-0.155442\pi$
$647$	4.52853e105	1.76623	0.883115	+	0.469157i	$0.155442\pi$
$648$	0	0
$649$	−2.88660e105	−1.00279
$650$	2.01827e104	0.0661801
$651$	0	0
$652$	9.14776e104	0.267319
$653$	4.91238e105	1.35534	0.677669	−	0.735367i	$-0.262990\pi$
$653$	4.91238e105	1.35534	0.677669	+	0.735367i	$0.262990\pi$
$654$	0	0
$655$	5.27854e105	1.29857
$656$	4.05669e104	0.0942492
$657$	0	0
$658$	1.51160e105	0.313305
$659$	−1.98171e105	−0.388005	−0.194003	−	0.981001i	$-0.562147\pi$
$659$	−1.98171e105	−0.388005	−0.194003	+	0.981001i	$0.562147\pi$
$660$	0	0
$661$	2.87807e105	0.502975	0.251487	−	0.967861i	$-0.419080\pi$
$661$	2.87807e105	0.502975	0.251487	+	0.967861i	$0.419080\pi$
$662$	−6.78530e105	−1.12045
$663$	0	0
$664$	−1.07372e105	−0.158339
$665$	5.81971e105	0.811128
$666$	0	0
$667$	3.41544e105	0.425332
$668$	3.66656e105	0.431660
$669$	0	0
$670$	−7.66815e105	−0.807023
$671$	−3.82089e105	−0.380250
$672$	0	0
$673$	2.37451e105	0.211353	0.105676	−	0.994401i	$-0.466299\pi$
$673$	2.37451e105	0.211353	0.105676	+	0.994401i	$0.466299\pi$
$674$	1.46203e105	0.123086
$675$	0	0
$676$	6.31249e105	0.475553
$677$	5.88403e105	0.419372	0.209686	−	0.977769i	$-0.432756\pi$
$677$	5.88403e105	0.419372	0.209686	+	0.977769i	$0.432756\pi$
$678$	0	0
$679$	−2.05167e105	−0.130915
$680$	1.88852e106	1.14035
$681$	0	0
$682$	−1.13668e106	−0.614787
$683$	−3.15565e105	−0.161552	−0.0807760	−	0.996732i	$-0.525740\pi$
$683$	−3.15565e105	−0.161552	−0.0807760	+	0.996732i	$0.525740\pi$
$684$	0	0
$685$	1.68139e106	0.771385
$686$	−1.40210e106	−0.609010
$687$	0	0
$688$	−2.18974e105	−0.0852760
$689$	−1.98140e106	−0.730719
$690$	0	0
$691$	4.44799e106	1.47142	0.735710	−	0.677296i	$-0.236849\pi$
$691$	4.44799e106	1.47142	0.735710	+	0.677296i	$0.236849\pi$
$692$	4.97883e105	0.156009
$693$	0	0
$694$	−9.73215e105	−0.273671
$695$	−3.66281e106	−0.975853
$696$	0	0
$697$	−5.68115e106	−1.35896
$698$	−4.85483e106	−1.10052
$699$	0	0
$700$	−3.83382e105	−0.0780647
$701$	2.53368e106	0.489020	0.244510	−	0.969647i	$-0.421373\pi$
$701$	2.53368e106	0.489020	0.244510	+	0.969647i	$0.421373\pi$
$702$	0	0
$703$	−1.20813e107	−2.09551
$704$	−2.85795e106	−0.469981
$705$	0	0
$706$	−7.79895e106	−1.15308
$707$	−3.92916e105	−0.0550899
$708$	0	0
$709$	−1.30039e106	−0.163998	−0.0819990	−	0.996632i	$-0.526130\pi$
$709$	−1.30039e106	−0.163998	−0.0819990	+	0.996632i	$0.526130\pi$
$710$	−7.18991e106	−0.860074
$711$	0	0
$712$	7.45020e106	0.801989
$713$	−1.17423e107	−1.19921
$714$	0	0
$715$	3.54599e106	0.326033
$716$	−4.77030e106	−0.416205
$717$	0	0
$718$	−8.77960e106	−0.689935
$719$	4.99955e106	0.372905	0.186452	−	0.982464i	$-0.440301\pi$
$719$	4.99955e106	0.372905	0.186452	+	0.982464i	$0.440301\pi$
$720$	0	0
$721$	1.28210e107	0.861685
$722$	6.87061e106	0.438380
$723$	0	0
$724$	1.22624e107	0.705310
$725$	−2.17900e106	−0.119010
$726$	0	0
$727$	1.34392e106	0.0661963	0.0330981	−	0.999452i	$-0.489463\pi$
$727$	1.34392e106	0.0661963	0.0330981	+	0.999452i	$0.489463\pi$
$728$	−5.33338e106	−0.249503
$729$	0	0
$730$	−1.60966e107	−0.679403
$731$	3.06661e107	1.22958
$732$	0	0
$733$	2.35888e107	0.853703	0.426851	−	0.904322i	$-0.359623\pi$
$733$	2.35888e107	0.853703	0.426851	+	0.904322i	$0.359623\pi$
$734$	6.05233e106	0.208123
$735$	0	0
$736$	−2.65710e107	−0.825067
$737$	−2.57842e107	−0.760890
$738$	0	0
$739$	2.54158e107	0.677543	0.338771	−	0.940869i	$-0.389989\pi$
$739$	2.54158e107	0.677543	0.338771	+	0.940869i	$0.389989\pi$
$740$	4.15856e107	1.05379
$741$	0	0
$742$	−2.65453e107	−0.607913
$743$	1.66703e107	0.362964	0.181482	−	0.983394i	$-0.441911\pi$
$743$	1.66703e107	0.362964	0.181482	+	0.983394i	$0.441911\pi$
$744$	0	0
$745$	4.98627e107	0.981556
$746$	−3.99708e107	−0.748233
$747$	0	0
$748$	2.34732e107	0.397431
$749$	−1.60175e107	−0.257945
$750$	0	0
$751$	−4.43790e105	−0.00646666	−0.00323333	−	0.999995i	$-0.501029\pi$
$751$	−4.43790e105	−0.00646666	−0.00323333	+	0.999995i	$0.501029\pi$
$752$	−4.35552e106	−0.0603769
$753$	0	0
$754$	−1.12051e107	−0.140602
$755$	1.00525e108	1.20023
$756$	0	0
$757$	1.15903e108	1.25314	0.626571	−	0.779364i	$-0.284458\pi$
$757$	1.15903e108	1.25314	0.626571	+	0.779364i	$0.284458\pi$
$758$	7.89223e107	0.812090
$759$	0	0
$760$	−1.57836e108	−1.47129
$761$	1.37920e108	1.22377	0.611887	−	0.790945i	$-0.290411\pi$
$761$	1.37920e108	1.22377	0.611887	+	0.790945i	$0.290411\pi$
$762$	0	0
$763$	−1.24998e108	−1.00515
$764$	−1.00553e108	−0.769827
$765$	0	0
$766$	1.32295e108	0.918258
$767$	9.78301e107	0.646616
$768$	0	0
$769$	4.00119e107	0.239856	0.119928	−	0.992783i	$-0.461734\pi$
$769$	4.00119e107	0.239856	0.119928	+	0.992783i	$0.461734\pi$
$770$	4.75067e107	0.271239
$771$	0	0
$772$	2.05328e108	1.06365
$773$	−1.87052e108	−0.923064	−0.461532	−	0.887124i	$-0.652700\pi$
$773$	−1.87052e108	−0.923064	−0.461532	+	0.887124i	$0.652700\pi$
$774$	0	0
$775$	7.49144e107	0.335546
$776$	5.56433e107	0.237464
$777$	0	0
$778$	8.52296e107	0.330260
$779$	4.74812e108	1.75334
$780$	0	0
$781$	−2.41761e108	−0.810909
$782$	−1.71022e108	−0.546761
$783$	0	0
$784$	1.64043e107	0.0476547
$785$	−4.85460e108	−1.34444
$786$	0	0
$787$	−6.55380e108	−1.64983	−0.824917	−	0.565253i	$-0.808778\pi$
$787$	−6.55380e108	−1.64983	−0.824917	+	0.565253i	$0.808778\pi$
$788$	1.10208e108	0.264533
$789$	0	0
$790$	−2.14240e108	−0.467611
$791$	−2.19209e108	−0.456290
$792$	0	0
$793$	1.29494e108	0.245192
$794$	−4.45995e107	−0.0805492
$795$	0	0
$796$	1.58187e108	0.259975
$797$	9.83900e106	0.0154264	0.00771319	−	0.999970i	$-0.497545\pi$
$797$	9.83900e106	0.0154264	0.00771319	+	0.999970i	$0.497545\pi$
$798$	0	0
$799$	6.09965e108	0.870564
$800$	1.69519e108	0.230858
$801$	0	0
$802$	1.88732e108	0.234049
$803$	−5.41249e108	−0.640566
$804$	0	0
$805$	4.90762e108	0.529085
$806$	3.85234e108	0.396424
$807$	0	0
$808$	1.06563e108	0.0999263
$809$	−1.09522e109	−0.980463	−0.490231	−	0.871592i	$-0.663088\pi$
$809$	−1.09522e109	−0.980463	−0.490231	+	0.871592i	$0.663088\pi$
$810$	0	0
$811$	1.44228e109	1.17698	0.588488	−	0.808506i	$-0.299724\pi$
$811$	1.44228e109	1.17698	0.588488	+	0.808506i	$0.299724\pi$
$812$	2.12848e108	0.165851
$813$	0	0
$814$	−9.86209e108	−0.700735
$815$	7.47077e108	0.506938
$816$	0	0
$817$	−2.56297e109	−1.58641
$818$	−1.15890e108	−0.0685167
$819$	0	0
$820$	−1.63436e109	−0.881717
$821$	−1.67032e109	−0.860858	−0.430429	−	0.902624i	$-0.641638\pi$
$821$	−1.67032e109	−0.860858	−0.430429	+	0.902624i	$0.641638\pi$
$822$	0	0
$823$	−2.67065e109	−1.25639	−0.628193	−	0.778057i	$-0.716205\pi$
$823$	−2.67065e109	−1.25639	−0.628193	+	0.778057i	$0.716205\pi$
$824$	−3.47717e109	−1.56299
$825$	0	0
$826$	1.31066e109	0.537945
$827$	3.63648e108	0.142634	0.0713172	−	0.997454i	$-0.477280\pi$
$827$	3.63648e108	0.142634	0.0713172	+	0.997454i	$0.477280\pi$
$828$	0	0
$829$	−8.52744e108	−0.305509	−0.152754	−	0.988264i	$-0.548814\pi$
$829$	−8.52744e108	−0.305509	−0.152754	+	0.988264i	$0.548814\pi$
$830$	−3.24136e108	−0.110994
$831$	0	0
$832$	9.68591e108	0.303051
$833$	−2.29733e109	−0.687125
$834$	0	0
$835$	2.99440e109	0.818591
$836$	−1.96181e109	−0.512768
$837$	0	0
$838$	2.78740e109	0.666115
$839$	1.65765e109	0.378809	0.189404	−	0.981899i	$-0.439344\pi$
$839$	1.65765e109	0.378809	0.189404	+	0.981899i	$0.439344\pi$
$840$	0	0
$841$	−3.57490e109	−0.747160
$842$	3.20447e109	0.640550
$843$	0	0
$844$	−2.55532e109	−0.467308
$845$	5.15527e109	0.901829
$846$	0	0
$847$	−1.91593e109	−0.306726
$848$	7.64875e108	0.117151
$849$	0	0
$850$	1.09109e109	0.152986
$851$	−1.01879e110	−1.36687
$852$	0	0
$853$	−5.77263e109	−0.709227	−0.354613	−	0.935013i	$-0.615387\pi$
$853$	−5.77263e109	−0.709227	−0.354613	+	0.935013i	$0.615387\pi$
$854$	1.73487e109	0.203984
$855$	0	0
$856$	4.34411e109	0.467880
$857$	1.55041e110	1.59833	0.799164	−	0.601113i	$-0.205276\pi$
$857$	1.55041e110	1.59833	0.799164	+	0.601113i	$0.205276\pi$
$858$	0	0
$859$	1.13730e110	1.07431	0.537156	−	0.843483i	$-0.319499\pi$
$859$	1.13730e110	1.07431	0.537156	+	0.843483i	$0.319499\pi$
$860$	8.82208e109	0.797772
$861$	0	0
$862$	7.30072e109	0.605122
$863$	−2.94758e109	−0.233917	−0.116958	−	0.993137i	$-0.537314\pi$
$863$	−2.94758e109	−0.233917	−0.116958	+	0.993137i	$0.537314\pi$
$864$	0	0
$865$	4.06610e109	0.295851
$866$	−3.33608e108	−0.0232442
$867$	0	0
$868$	−7.31774e109	−0.467614
$869$	−7.20382e109	−0.440881
$870$	0	0
$871$	8.73854e109	0.490634
$872$	3.39006e110	1.82322
$873$	0	0
$874$	1.42934e110	0.705435
$875$	1.00979e110	0.477450
$876$	0	0
$877$	−5.57654e109	−0.242036	−0.121018	−	0.992650i	$-0.538616\pi$
$877$	−5.57654e109	−0.242036	−0.121018	+	0.992650i	$0.538616\pi$
$878$	−1.33999e110	−0.557259
$879$	0	0
$880$	−1.36885e109	−0.0522705
$881$	5.10259e110	1.86721	0.933607	−	0.358299i	$-0.116643\pi$
$881$	5.10259e110	1.86721	0.933607	+	0.358299i	$0.116643\pi$
$882$	0	0
$883$	1.13575e110	0.381730	0.190865	−	0.981616i	$-0.438871\pi$
$883$	1.13575e110	0.381730	0.190865	+	0.981616i	$0.438871\pi$
$884$	−7.95531e109	−0.256270
$885$	0	0
$886$	3.60411e109	0.106668
$887$	−3.22462e110	−0.914835	−0.457417	−	0.889252i	$-0.651225\pi$
$887$	−3.22462e110	−0.914835	−0.457417	+	0.889252i	$0.651225\pi$
$888$	0	0
$889$	2.58945e110	0.675136
$890$	2.24909e110	0.562187
$891$	0	0
$892$	6.54860e109	0.150476
$893$	−5.09788e110	−1.12321
$894$	0	0
$895$	−3.89580e110	−0.789283
$896$	−1.52611e110	−0.296507
$897$	0	0
$898$	4.21535e110	0.753306
$899$	−4.15914e110	−0.712878
$900$	0	0
$901$	−1.07116e111	−1.68918
$902$	3.87592e110	0.586313
$903$	0	0
$904$	5.94518e110	0.827654
$905$	1.00145e111	1.33754
$906$	0	0
$907$	−1.38032e111	−1.69709	−0.848547	−	0.529120i	$-0.822522\pi$
$907$	−1.38032e111	−1.69709	−0.848547	+	0.529120i	$0.822522\pi$
$908$	−2.14424e110	−0.252962
$909$	0	0
$910$	−1.61005e110	−0.174899
$911$	−2.99939e110	−0.312676	−0.156338	−	0.987704i	$-0.549969\pi$
$911$	−2.99939e110	−0.312676	−0.156338	+	0.987704i	$0.549969\pi$
$912$	0	0
$913$	−1.08991e110	−0.104649
$914$	−7.52721e110	−0.693668
$915$	0	0
$916$	−1.01192e111	−0.859142
$917$	−8.05895e110	−0.656793
$918$	0	0
$919$	2.25950e111	1.69700	0.848498	−	0.529199i	$-0.177508\pi$
$919$	2.25950e111	1.69700	0.848498	+	0.529199i	$0.177508\pi$
$920$	−1.33099e111	−0.959694
$921$	0	0
$922$	−1.00820e111	−0.670095
$923$	8.19354e110	0.522887
$924$	0	0
$925$	6.49974e110	0.382456
$926$	−1.03249e111	−0.583412
$927$	0	0
$928$	−9.41146e110	−0.490465
$929$	−3.73615e111	−1.86997	−0.934985	−	0.354688i	$-0.884587\pi$
$929$	−3.73615e111	−1.86997	−0.934985	+	0.354688i	$0.884587\pi$
$930$	0	0
$931$	1.92003e111	0.886534
$932$	4.33555e110	0.192286
$933$	0	0
$934$	−2.65257e111	−1.08557
$935$	1.91700e111	0.753680
$936$	0	0
$937$	−2.04035e111	−0.740405	−0.370203	−	0.928951i	$-0.620712\pi$
$937$	−2.04035e111	−0.740405	−0.370203	+	0.928951i	$0.620712\pi$
$938$	1.17073e111	0.408178
$939$	0	0
$940$	1.75476e111	0.564837
$941$	−1.19436e111	−0.369425	−0.184712	−	0.982793i	$-0.559135\pi$
$941$	−1.19436e111	−0.369425	−0.184712	+	0.982793i	$0.559135\pi$
$942$	0	0
$943$	4.00397e111	1.14367
$944$	−3.77652e110	−0.103667
$945$	0	0
$946$	−2.09217e111	−0.530493
$947$	5.14584e111	1.25410	0.627050	−	0.778979i	$-0.284262\pi$
$947$	5.14584e111	1.25410	0.627050	+	0.778979i	$0.284262\pi$
$948$	0	0
$949$	1.83435e111	0.413047
$950$	−9.11900e110	−0.197384
$951$	0	0
$952$	−2.88328e111	−0.576769
$953$	−8.87914e111	−1.70760	−0.853802	−	0.520597i	$-0.825709\pi$
$953$	−8.87914e111	−1.70760	−0.853802	+	0.520597i	$0.825709\pi$
$954$	0	0
$955$	−8.21192e111	−1.45988
$956$	−1.27479e111	−0.217905
$957$	0	0
$958$	7.58786e111	1.19926
$959$	−2.56704e111	−0.390153
$960$	0	0
$961$	7.18497e111	1.00995
$962$	3.34237e111	0.451845
$963$	0	0
$964$	7.46707e111	0.933816
$965$	1.67687e112	2.01709
$966$	0	0
$967$	−1.49205e112	−1.66070	−0.830351	−	0.557241i	$-0.811860\pi$
$967$	−1.49205e112	−1.66070	−0.830351	+	0.557241i	$0.811860\pi$
$968$	5.19618e111	0.556363
$969$	0	0
$970$	1.67977e111	0.166460
$971$	2.06690e111	0.197060	0.0985299	−	0.995134i	$-0.468586\pi$
$971$	2.06690e111	0.197060	0.0985299	+	0.995134i	$0.468586\pi$
$972$	0	0
$973$	5.59216e111	0.493569
$974$	−4.78002e111	−0.405946
$975$	0	0
$976$	−4.99884e110	−0.0393098
$977$	1.87496e112	1.41888	0.709439	−	0.704767i	$-0.248949\pi$
$977$	1.87496e112	1.41888	0.709439	+	0.704767i	$0.248949\pi$
$978$	0	0
$979$	7.56255e111	0.530050
$980$	−6.60900e111	−0.445818
$981$	0	0
$982$	−2.01683e112	−1.26034
$983$	−7.83556e111	−0.471317	−0.235659	−	0.971836i	$-0.575725\pi$
$983$	−7.83556e111	−0.471317	−0.235659	+	0.971836i	$0.575725\pi$
$984$	0	0
$985$	9.00045e111	0.501656
$986$	−6.05760e111	−0.325025
$987$	0	0
$988$	6.64878e111	0.330641
$989$	−2.16129e112	−1.03479
$990$	0	0
$991$	−1.41350e112	−0.627383	−0.313692	−	0.949525i	$-0.601566\pi$
$991$	−1.41350e112	−0.627383	−0.313692	+	0.949525i	$0.601566\pi$
$992$	3.23567e112	1.38286
$993$	0	0
$994$	1.09771e112	0.435010
$995$	1.29188e112	0.493011
$996$	0	0
$997$	−4.15637e111	−0.147111	−0.0735556	−	0.997291i	$-0.523435\pi$
$997$	−4.15637e111	−0.147111	−0.0735556	+	0.997291i	$0.523435\pi$
$998$	−8.86724e110	−0.0302269
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9.76.a.c.1.4		6
3.2	odd	2		1.76.a.a.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.76.a.a.1.3	✓	6	3.2	odd	2
9.76.a.c.1.4		6	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:	\( k \)	\(=\)	\( 76 \)
Character orbit:	\([\chi]\)	\(=\)	9.a (trivial)

\(f(q)\)	\(=\)	\(q+1.25000e11 q^{2} -2.21540e22 q^{4} -1.80927e26 q^{5} +2.76229e31 q^{7} -7.49160e33 q^{8} +O(q^{10})\)	\(q+1.25000e11 q^{2} -2.21540e22 q^{4} -1.80927e26 q^{5} +2.76229e31 q^{7} -7.49160e33 q^{8} -2.26158e37 q^{10} -7.60456e38 q^{11} +2.57727e41 q^{13} +3.45284e42 q^{14} -9.94899e43 q^{16} +1.39330e46 q^{17} -1.16447e48 q^{19} +4.00827e48 q^{20} -9.50567e49 q^{22} -9.81970e50 q^{23} +6.26484e51 q^{25} +3.22158e52 q^{26} -6.11958e53 q^{28} -3.47815e54 q^{29} +1.19579e56 q^{31} +2.70588e56 q^{32} +1.74162e57 q^{34} -4.99772e57 q^{35} +1.03750e59 q^{37} -1.45558e59 q^{38} +1.35543e60 q^{40} -4.07748e60 q^{41} +2.20097e61 q^{43} +1.68472e61 q^{44} -1.22746e62 q^{46} +4.37785e62 q^{47} -1.64884e63 q^{49} +7.83102e62 q^{50} -5.70970e63 q^{52} -7.68796e64 q^{53} +1.37587e65 q^{55} -2.06939e65 q^{56} -4.34767e65 q^{58} +3.79588e66 q^{59} +5.02447e66 q^{61} +1.49473e67 q^{62} +3.75820e67 q^{64} -4.66298e67 q^{65} +3.39062e68 q^{67} -3.08672e68 q^{68} -6.24713e68 q^{70} +3.17915e69 q^{71} +7.11742e69 q^{73} +1.29687e70 q^{74} +2.57978e70 q^{76} -2.10060e70 q^{77} +9.47303e70 q^{79} +1.80004e70 q^{80} -5.09684e71 q^{82} +1.43323e71 q^{83} -2.52086e72 q^{85} +2.75120e72 q^{86} +5.69703e72 q^{88} -9.94475e72 q^{89} +7.11916e72 q^{91} +2.17546e73 q^{92} +5.47229e73 q^{94} +2.10685e74 q^{95} -7.42743e73 q^{97} -2.06105e74 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 57080822040 q^{2} + 17\!\cdots\!28 q^{4}+ \cdots - 44\!\cdots\!20 q^{8}+O(q^{10}) \) 6 * q + 57080822040 * q^2 + 172600466200162028593728 * q^4 + 38982947396479621874420940 * q^5 + 1924474634802918779239478643600 * q^7 - 4434903997968330342413437320645120 * q^8	\( 6 q + 57080822040 q^{2} + 17\!\cdots\!28 q^{4}+ \cdots + 16\!\cdots\!20 q^{98}+O(q^{100}) \) 6 * q + 57080822040 * q^2 + 172600466200162028593728 * q^4 + 38982947396479621874420940 * q^5 + 1924474634802918779239478643600 * q^7 - 4434903997968330342413437320645120 * q^8 + 133267935677460241037537052811576902960 * q^10 + 945659951949983071650740113735111214088 * q^11 + 533185179266852519941169625089098069764420 * q^13 - 8213946953162263647781694565284746353072576 * q^14 + 2681978005634517266622433394712415602648354816 * q^16 - 18261354095776132566172834671579513215483393580 * q^17 + 1061534856238354155312794852768417904375328867080 * q^19 - 9296557595621768320156983882258818695571369489280 * q^20 + 153315952874260268106194796405555863940172791589920 * q^22 - 1514929449877410771078348320690686508954384916853680 * q^23 + 19310380328261093150952553553121815533899364092239850 * q^25 - 117730941166816926498584842564210483616872200198362352 * q^26 + 1465885017268620651444113586751990931877154821040657920 * q^28 - 14718767377678125816030788545177391174717388984245534820 * q^29 - 41742874787360498453628839393549579933009794702042980288 * q^31 - 1174660148719991756036751189163540495635749096221862952960 * q^32 + 3034358950124956876257603978888653679838045546922892395856 * q^34 - 27299672493656883988453292135086250689495956464846953371360 * q^35 + 98517772547646219221591850851651469983797181732879949140340 * q^37 - 1257660981102762853948269222064665311191946222029530979162080 * q^38 + 8807901623729426290853411185831313455271917763097593104051200 * q^40 - 5039211762207387369185984529408567568647239506257650214613212 * q^41 + 27938749238581541451173935836639198120068956697568041153196600 * q^43 + 86388814623354110240559749283671522466589238091463431115801344 * q^44 - 824145803870667307795487213022331547039596870954611528203613888 * q^46 + 1386854970430427381646742231466788117324292253074354560479038880 * q^47 - 5707868790800975784856935495088511133239544554833878775910738442 * q^49 - 3144970672685230615695776810799603542575044196614572896663392600 * q^50 + 41677832028141179840113326797899392222702339479233001780216150400 * q^52 - 64283727174659970247345252518647455744441825702701764187054106260 * q^53 + 436876078674971654403803167180324983329224614451635548418098895120 * q^55 - 289950797286325881623798664040853221823087783432246887785270865920 * q^56 - 1763005756244374137396902189067511762184157619699352574614137403280 * q^58 + 2478360370933764187935912640189307752170553390931716617591182395560 * q^59 - 25641115321477658815889269525514418267074652297034046321782023152988 * q^61 + 29014701474354643819936101890527484086880802356768235703173807694080 * q^62 + 47683232974651133862719671893097862706511055838864188064811716968448 * q^64 - 122110621788131890149667214965152432737153070969379698276880591977720 * q^65 + 955579073553339181030887342220313909838725364297810737998207107813480 * q^67 - 1237210897217068882936524414359446185548686871817993659403939290386560 * q^68 - 3476902593928340346610001926714053112321275314411181267396530677050240 * q^70 + 255682319306096297182489045146623979225804710540351927309367569173488 * q^71 - 30605188898291905719532895918028188508004452865621142257485594115042020 * q^73 + 24065024027894599201472448362529876891121294631891709256708009194181584 * q^74 + 100020936372377387464010564774983213070641689274186088752725827576144640 * q^76 - 158930415774536768880543239288689183540801263714134710336228496440507200 * q^77 + 116627678166642324332643069323234092884951450625988452824960767925148320 * q^79 - 1239310052714502333188756625709169306278438274940582996524184215206748160 * q^80 - 2557649813228840791149140948716215474705151452500921688788291587877236080 * q^82 + 798780284119889426515570523310162377392459257634002033911814028088739960 * q^83 - 3615500806340871685633297735534398118961343923835923286123685237305018840 * q^85 - 7245194180836090971883743617494948296636529538939897523257528311698092832 * q^86 - 4827001398586167572057494186308233100206033194553368013345656010632181760 * q^88 - 53575610216125238704584826863934604186542249469498220298044079954014547260 * q^89 + 34502419684367451829908407139028781594042460695428817715772307436403470432 * q^91 - 187445224353587479072026786339601818260697439565538078689222489106585838080 * q^92 - 296189325151987924691405038954735773676385503565643130818482899021002578304 * q^94 - 194934959385332686251927514630723902507701573195047654345319995220933314800 * q^95 - 747928220344118582344222376208028579242743608466383778453296378427532508980 * q^97 + 1663265696927109513221859402309517858944410070626232817304447619248634722520 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more