LMFDB - Embedded newform 9.74.a.a.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9,74,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, 74, names="a")

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

chi := DirichletCharacter("9.1");

S:= CuspForms(chi, 74);

N := Newforms(S);

Level:	$ N $	$=$	$ 9 = 3^{2} $
Weight:	$ k $	$=$	$ 74 $
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:	$303.735576363$
Analytic rank:	$0$
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 + 17\!\cdots\!00 $ x^5 - x^4 - 10073499617947743056x^3 + 1429272143092482488433869600x^2 + 7661214288514935343595600445215756800*x + 1722510836040319301450745177697157900206688000
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{39}\cdot 3^{22}\cdot 5^{6}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 1)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.60629e8$ 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)

$f(q)$	$=$	$q+5.90767e9 q^{2} -9.40983e21 q^{4} -1.69056e25 q^{5} -2.69951e30 q^{7} -1.11387e32 q^{8} +O(q^{10})$	\(q+5.90767e9 q^{2} -9.40983e21 q^{4} -1.69056e25 q^{5} -2.69951e30 q^{7} -1.11387e32 q^{8} -9.98727e34 q^{10} +1.46803e38 q^{11} -3.34896e40 q^{13} -1.59478e40 q^{14} +8.82153e43 q^{16} +1.35791e45 q^{17} +7.35113e46 q^{19} +1.59079e47 q^{20} +8.67264e47 q^{22} -2.07741e49 q^{23} -7.72992e50 q^{25} -1.97845e50 q^{26} +2.54020e52 q^{28} +3.30017e53 q^{29} +1.78429e54 q^{31} +1.57316e54 q^{32} +8.02211e54 q^{34} +4.56369e55 q^{35} -3.22112e57 q^{37} +4.34280e56 q^{38} +1.88306e57 q^{40} +6.12341e58 q^{41} +2.07892e59 q^{43} -1.38139e60 q^{44} -1.22727e59 q^{46} -5.09761e58 q^{47} -4.19344e61 q^{49} -4.56658e60 q^{50} +3.15131e62 q^{52} +1.65493e62 q^{53} -2.48179e63 q^{55} +3.00690e62 q^{56} +1.94963e63 q^{58} -1.57646e64 q^{59} +3.70457e64 q^{61} +1.05410e64 q^{62} -8.23876e65 q^{64} +5.66162e65 q^{65} -2.21412e66 q^{67} -1.27777e67 q^{68} +2.69608e65 q^{70} +1.08186e67 q^{71} +2.15731e67 q^{73} -1.90293e67 q^{74} -6.91729e68 q^{76} -3.96297e68 q^{77} +1.93510e69 q^{79} -1.49133e69 q^{80} +3.61751e68 q^{82} -1.17002e70 q^{83} -2.29564e70 q^{85} +1.22816e69 q^{86} -1.63519e70 q^{88} -1.65587e71 q^{89} +9.04056e70 q^{91} +1.95481e71 q^{92} -3.01150e68 q^{94} -1.24275e72 q^{95} -9.74539e71 q^{97} -2.47734e71 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 92089333488 q^{2} + 89\!\cdots\!60 q^{4}+ \cdots + 38\!\cdots\!80 q^{8}+O(q^{10}) $ 5 * q + 92089333488 * q^2 + 891110479259563516160 * q^4 - 23099529720469471562826750 * q^5 - 4356178487147579439806268090008 * q^7 + 382321348565178801062509850849280 * q^8	$ 5 q + 92089333488 q^{2} + 89\!\cdots\!60 q^{4}+ \cdots + 76\!\cdots\!16 q^{98}+O(q^{100}) $ 5 * q + 92089333488 * q^2 + 891110479259563516160 * q^4 - 23099529720469471562826750 * q^5 - 4356178487147579439806268090008 * q^7 + 382321348565178801062509850849280 * q^8 - 1082726120403847340426026649690244000 * q^10 - 50083756421782317855880849016778496860 * q^11 + 4753748397471720291047553799357192278886 * q^13 - 263363595637460329420141722180437171694720 * q^14 + 57509595349357883169785087518099233197588480 * q^16 - 663279725175909884624200257401822351394256602 * q^17 + 31387402430958534058855312636541530201995765700 * q^19 - 68535205735011202400715856751699460073169856000 * q^20 - 9492394168845783423791120540922405517833204177216 * q^22 + 41173446223173693726617088414497261153026772718024 * q^23 + 321010940201155914585034686800814006311220324546875 * q^25 - 4496438035379975943261998606627087983483848085006560 * q^26 - 37814301174111882879202704910382387341459683517577216 * q^28 + 217397685843681573854388291537651452827057882148450250 * q^29 - 3959918978904396097569386510196715694840286709585401440 * q^31 + 9479144378484593489541015426974249650207814007382867968 * q^32 - 324638192385754362369074018224584035205208569018875155680 * q^34 + 1051811671165262926972759981500117019527196204743409914000 * q^35 - 6710426612911460903226679780416929578822322356442579950178 * q^37 + 20075776649866586500887218371135359952534984707306414947520 * q^38 - 87443314425210154156114784714128373503740284779633935360000 * q^40 + 89999556000464227434562342749528590851939245788450434103790 * q^41 + 1171421579861687300375905115748875459141635816468653452284156 * q^43 - 3620387401482413711753562750789250834104238410455247119723520 * q^44 + 13122337792181100518478081314861771936057558300377044573644160 * q^46 - 26407117391117866585989236437476333636637017745173062895096432 * q^47 - 47859785088079990485612756297590896342589260563105328803969315 * q^49 + 198341627273982740927873511630518057650356858808871192258250000 * q^50 - 1819820853930882517149424007699825989256997486995920851867734528 * q^52 + 2253672757277210109476755074185818948452790752453662372858468754 * q^53 + 5272301841223840031246441919708126997046467528986443289345081000 * q^55 - 17369101267271748189623122493455314696373137673860233497989120000 * q^56 + 63080485469034924979732075223792805715626817023343777138220605920 * q^58 - 49827032054541030737892137385762950528020446432538215007741437900 * q^59 - 200207946356755779993110124671040230418699916707275462827300034090 * q^61 + 456512716932753516995161459627410738703514050956609499700325060096 * q^62 - 2671988611100492125931167910213768824231319631401339943695669002240 * q^64 + 2221078323698894670419341126197735507928898751783246157619800369500 * q^65 + 1726934463658866670044856129309631728944059099751189118466533017652 * q^67 - 25844159932565717701710366021885253281928511625890180327915132834304 * q^68 + 60019715162941233136874683443010604187189046779871920798680662752000 * q^70 - 29528228383509762365399718501357560736875772117852859701209862331560 * q^71 - 237248424072934328617284197336037660641948064767277176027665176375374 * q^73 + 385668396032745357868624366253781435784895053179379214963132226802080 * q^74 - 402469771116557587325960702574461860644965601604695023105081798528000 * q^76 + 1160921420345569675442649510235573702285949327395662717704239852301856 * q^77 + 1210019475236668608258558597507004925943189831192425236151129201347600 * q^79 - 7695510793835037324754830224890315360926193938062866122926593916928000 * q^80 + 19869905655996015892436375229312420723635630923691072274009372051144864 * q^82 - 10633758535693793129462935403233565634399957806616398467042339222483716 * q^83 - 28641392301460491393930553180957463692521548910270676165772874742366500 * q^85 + 130748824105220136322401247193488038601024385677355691179590030869522240 * q^86 - 221808067516055560593893382793377773024661552295403957235825845052456960 * q^88 + 44230399340933247556508628049428956972940692248797881370004235332026750 * q^89 + 503963419362898329804217905595235878617892804641236823955454638331741360 * q^91 - 1056276182251025158869277711594662138415929570173882076378175681365399552 * q^92 + 1211599009021421323842808603980852406277919665987296133682388929757489920 * q^94 - 1134809355692277446299742482002998127744674957590385906379174039024255000 * q^95 - 4705643796144250410051227741367571092615369303801441364082016351450032918 * q^97 + 7645702282951838206479981702299956043322685777549671658293752892775157616 * 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 $

$2$	5.90767e9	0.0607885	0.0303943	−	0.999538i	$-0.490324\pi$
$2$	5.90767e9	0.0607885	0.0303943	+	0.999538i	$0.490324\pi$
$3$	0	0
$3$	0	0
$4$	−9.40983e21	−0.996305
$5$	−1.69056e25	−0.519548	−0.259774	−	0.965669i	$-0.583648\pi$
$5$	−1.69056e25	−0.519548	−0.259774	+	0.965669i	$0.583648\pi$
$6$	0	0
$7$	−2.69951e30	−0.384775	−0.192388	−	0.981319i	$-0.561623\pi$
$7$	−2.69951e30	−0.384775	−0.192388	+	0.981319i	$0.561623\pi$
$8$	−1.11387e32	−0.121352
$9$	0	0
$10$	−9.98727e34	−0.0315825
$11$	1.46803e38	1.43187	0.715933	−	0.698169i	$-0.246002\pi$
$11$	1.46803e38	1.43187	0.715933	+	0.698169i	$0.246002\pi$
$12$	0	0
$13$	−3.34896e40	−0.734476	−0.367238	−	0.930127i	$-0.619696\pi$
$13$	−3.34896e40	−0.734476	−0.367238	+	0.930127i	$0.619696\pi$
$14$	−1.59478e40	−0.0233899
$15$	0	0
$16$	8.82153e43	0.988928
$17$	1.35791e45	1.66528	0.832639	−	0.553817i	$-0.186829\pi$
$17$	1.35791e45	1.66528	0.832639	+	0.553817i	$0.186829\pi$
$18$	0	0
$19$	7.35113e46	1.55542	0.777710	−	0.628624i	$-0.216381\pi$
$19$	7.35113e46	1.55542	0.777710	+	0.628624i	$0.216381\pi$
$20$	1.59079e47	0.517628
$21$	0	0
$22$	8.67264e47	0.0870409
$23$	−2.07741e49	−0.411583	−0.205791	−	0.978596i	$-0.565977\pi$
$23$	−2.07741e49	−0.411583	−0.205791	+	0.978596i	$0.565977\pi$
$24$	0	0
$25$	−7.72992e50	−0.730070
$26$	−1.97845e50	−0.0446477
$27$	0	0
$28$	2.54020e52	0.383353
$29$	3.30017e53	1.38360	0.691799	−	0.722091i	$-0.256819\pi$
$29$	3.30017e53	1.38360	0.691799	+	0.722091i	$0.256819\pi$
$30$	0	0
$31$	1.78429e54	0.655790	0.327895	−	0.944714i	$-0.393661\pi$
$31$	1.78429e54	0.655790	0.327895	+	0.944714i	$0.393661\pi$
$32$	1.57316e54	0.181468
$33$	0	0
$34$	8.02211e54	0.101230
$35$	4.56369e55	0.199909
$36$	0	0
$37$	−3.22112e57	−1.85628	−0.928140	−	0.372230i	$-0.878593\pi$
$37$	−3.22112e57	−1.85628	−0.928140	+	0.372230i	$0.878593\pi$
$38$	4.34280e56	0.0945516
$39$	0	0
$40$	1.88306e57	0.0630484
$41$	6.12341e58	0.832497	0.416249	−	0.909251i	$-0.363345\pi$
$41$	6.12341e58	0.832497	0.416249	+	0.909251i	$0.363345\pi$
$42$	0	0
$43$	2.07892e59	0.496867	0.248434	−	0.968649i	$-0.420084\pi$
$43$	2.07892e59	0.496867	0.248434	+	0.968649i	$0.420084\pi$
$44$	−1.38139e60	−1.42657
$45$	0	0
$46$	−1.22727e59	−0.0250195
$47$	−5.09761e58	−0.00474018	−0.00237009	−	0.999997i	$-0.500754\pi$
$47$	−5.09761e58	−0.00474018	−0.00237009	+	0.999997i	$0.500754\pi$
$48$	0	0
$49$	−4.19344e61	−0.851948
$50$	−4.56658e60	−0.0443799
$51$	0	0
$52$	3.15131e62	0.731762
$53$	1.65493e62	0.191739	0.0958697	−	0.995394i	$-0.469437\pi$
$53$	1.65493e62	0.191739	0.0958697	+	0.995394i	$0.469437\pi$
$54$	0	0
$55$	−2.48179e63	−0.743923
$56$	3.00690e62	0.0466934
$57$	0	0
$58$	1.94963e63	0.0841068
$59$	−1.57646e64	−0.364404	−0.182202	−	0.983261i	$-0.558323\pi$
$59$	−1.57646e64	−0.364404	−0.182202	+	0.983261i	$0.558323\pi$
$60$	0	0
$61$	3.70457e64	0.253629	0.126814	−	0.991926i	$-0.459525\pi$
$61$	3.70457e64	0.253629	0.126814	+	0.991926i	$0.459525\pi$
$62$	1.05410e64	0.0398645
$63$	0	0
$64$	−8.23876e65	−0.977897
$65$	5.66162e65	0.381595
$66$	0	0
$67$	−2.21412e66	−0.493709	−0.246855	−	0.969053i	$-0.579397\pi$
$67$	−2.21412e66	−0.493709	−0.246855	+	0.969053i	$0.579397\pi$
$68$	−1.27777e67	−1.65912
$69$	0	0
$70$	2.69608e65	0.0121522
$71$	1.08186e67	0.290564	0.145282	−	0.989390i	$-0.453591\pi$
$71$	1.08186e67	0.290564	0.145282	+	0.989390i	$0.453591\pi$
$72$	0	0
$73$	2.15731e67	0.210197	0.105099	−	0.994462i	$-0.466484\pi$
$73$	2.15731e67	0.210197	0.105099	+	0.994462i	$0.466484\pi$
$74$	−1.90293e67	−0.112841
$75$	0	0
$76$	−6.91729e68	−1.54967
$77$	−3.96297e68	−0.550946
$78$	0	0
$79$	1.93510e69	1.05515	0.527575	−	0.849508i	$-0.323101\pi$
$79$	1.93510e69	1.05515	0.527575	+	0.849508i	$0.323101\pi$
$80$	−1.49133e69	−0.513795
$81$	0	0
$82$	3.61751e68	0.0506063
$83$	−1.17002e70	−1.05158	−0.525791	−	0.850614i	$-0.676231\pi$
$83$	−1.17002e70	−1.05158	−0.525791	+	0.850614i	$0.676231\pi$
$84$	0	0
$85$	−2.29564e70	−0.865191
$86$	1.22816e69	0.0302038
$87$	0	0
$88$	−1.63519e70	−0.173760
$89$	−1.65587e71	−1.16490	−0.582452	−	0.812865i	$-0.697907\pi$
$89$	−1.65587e71	−1.16490	−0.582452	+	0.812865i	$0.697907\pi$
$90$	0	0
$91$	9.04056e70	0.282608
$92$	1.95481e71	0.410062
$93$	0	0
$94$	−3.01150e68	−0.000288149 0
$95$	−1.24275e72	−0.808115
$96$	0	0
$97$	−9.74539e71	−0.296231	−0.148116	−	0.988970i	$-0.547321\pi$
$97$	−9.74539e71	−0.296231	−0.148116	+	0.988970i	$0.547321\pi$
$98$	−2.47734e71	−0.0517886
$99$	0	0
$100$	7.27372e72	0.727372
$101$	1.40279e73	0.975582	0.487791	−	0.872961i	$-0.337803\pi$
$101$	1.40279e73	0.975582	0.487791	+	0.872961i	$0.337803\pi$
$102$	0	0
$103$	4.70385e73	1.59917	0.799586	−	0.600552i	$-0.205052\pi$
$103$	4.70385e73	1.59917	0.799586	+	0.600552i	$0.205052\pi$
$104$	3.73029e72	0.0891304
$105$	0	0
$106$	9.77680e71	0.0116555
$107$	−1.24306e74	−1.05192	−0.525962	−	0.850508i	$-0.676295\pi$
$107$	−1.24306e74	−1.05192	−0.525962	+	0.850508i	$0.676295\pi$
$108$	0	0
$109$	2.39112e74	1.02928	0.514641	−	0.857406i	$-0.327925\pi$
$109$	2.39112e74	1.02928	0.514641	+	0.857406i	$0.327925\pi$
$110$	−1.46616e73	−0.0452219
$111$	0	0
$112$	−2.38139e74	−0.380515
$113$	−7.57224e74	−0.874704	−0.437352	−	0.899290i	$-0.644084\pi$
$113$	−7.57224e74	−0.874704	−0.437352	+	0.899290i	$0.644084\pi$
$114$	0	0
$115$	3.51199e74	0.213837
$116$	−3.10541e75	−1.37848
$117$	0	0
$118$	−9.31319e73	−0.0221516
$119$	−3.66571e75	−0.640758
$120$	0	0
$121$	1.10396e76	1.05024
$122$	2.18854e74	0.0154177
$123$	0	0
$124$	−1.67899e76	−0.653367
$125$	3.09674e76	0.898854
$126$	0	0
$127$	3.10832e76	0.505463	0.252732	−	0.967536i	$-0.418671\pi$
$127$	3.10832e76	0.505463	0.252732	+	0.967536i	$0.418671\pi$
$128$	−1.97253e76	−0.240913
$129$	0	0
$130$	3.34470e75	0.0231966
$131$	2.13454e77	1.11919	0.559593	−	0.828768i	$-0.310958\pi$
$131$	2.13454e77	1.11919	0.559593	+	0.828768i	$0.310958\pi$
$132$	0	0
$133$	−1.98445e77	−0.598487
$134$	−1.30803e76	−0.0300118
$135$	0	0
$136$	−1.51253e77	−0.202085
$137$	1.04031e78	1.06381	0.531905	−	0.846804i	$-0.321476\pi$
$137$	1.04031e78	1.06381	0.531905	+	0.846804i	$0.321476\pi$
$138$	0	0
$139$	−1.48817e78	−0.896626	−0.448313	−	0.893877i	$-0.647975\pi$
$139$	−1.48817e78	−0.896626	−0.448313	+	0.893877i	$0.647975\pi$
$140$	−4.29436e77	−0.199170
$141$	0	0
$142$	6.39128e76	0.0176630
$143$	−4.91637e78	−1.05167
$144$	0	0
$145$	−5.57914e78	−0.718845
$146$	1.27447e77	0.0127776
$147$	0	0
$148$	3.03102e79	1.84942
$149$	8.64312e78	0.412450	0.206225	−	0.978505i	$-0.433882\pi$
$149$	8.64312e78	0.412450	0.206225	+	0.978505i	$0.433882\pi$
$150$	0	0
$151$	−2.28898e79	−0.671403	−0.335701	−	0.941968i	$-0.608973\pi$
$151$	−2.28898e79	−0.671403	−0.335701	+	0.941968i	$0.608973\pi$
$152$	−8.18817e78	−0.188754
$153$	0	0
$154$	−2.34119e78	−0.0334912
$155$	−3.01646e79	−0.340714
$156$	0	0
$157$	1.93279e80	1.36725	0.683623	−	0.729835i	$-0.260403\pi$
$157$	1.93279e80	1.36725	0.683623	+	0.729835i	$0.260403\pi$
$158$	1.14319e79	0.0641410
$159$	0	0
$160$	−2.65953e79	−0.0942812
$161$	5.60801e79	0.158367
$162$	0	0
$163$	−9.29339e78	−0.0167235	−0.00836174	−	0.999965i	$-0.502662\pi$
$163$	−9.29339e78	−0.0167235	−0.00836174	+	0.999965i	$0.502662\pi$
$164$	−5.76203e80	−0.829421
$165$	0	0
$166$	−6.91210e79	−0.0639241
$167$	1.36355e81	1.01279	0.506397	−	0.862300i	$-0.330977\pi$
$167$	1.36355e81	1.01279	0.506397	+	0.862300i	$0.330977\pi$
$168$	0	0
$169$	−9.57497e80	−0.460546
$170$	−1.35619e80	−0.0525937
$171$	0	0
$172$	−1.95623e81	−0.495031
$173$	−4.09283e81	−0.838188	−0.419094	−	0.907943i	$-0.637652\pi$
$173$	−4.09283e81	−0.838188	−0.419094	+	0.907943i	$0.637652\pi$
$174$	0	0
$175$	2.08670e81	0.280913
$176$	1.29503e82	1.41601
$177$	0	0
$178$	−9.78231e80	−0.0708127
$179$	−6.62628e81	−0.390962	−0.195481	−	0.980708i	$-0.562627\pi$
$179$	−6.62628e81	−0.390962	−0.195481	+	0.980708i	$0.562627\pi$
$180$	0	0
$181$	−2.47756e82	−0.974442	−0.487221	−	0.873279i	$-0.661989\pi$
$181$	−2.47756e82	−0.974442	−0.487221	+	0.873279i	$0.661989\pi$
$182$	5.34087e80	0.0171793
$183$	0	0
$184$	2.31396e81	0.0499465
$185$	5.44549e82	0.964427
$186$	0	0
$187$	1.99346e83	2.38445
$188$	4.79677e80	0.00472267
$189$	0	0
$190$	−7.34177e81	−0.0491241
$191$	−3.23309e83	−1.78608	−0.893038	−	0.449981i	$-0.851431\pi$
$191$	−3.23309e83	−1.78608	−0.893038	+	0.449981i	$0.851431\pi$
$192$	0	0
$193$	−3.80588e83	−1.43752	−0.718759	−	0.695259i	$-0.755290\pi$
$193$	−3.80588e83	−1.43752	−0.718759	+	0.695259i	$0.755290\pi$
$194$	−5.75725e81	−0.0180075
$195$	0	0
$196$	3.94595e83	0.848800
$197$	5.12328e80	0.000915232 0	0.000457616	−	1.00000i	$-0.499854\pi$
$197$	5.12328e80	0.000915232 0	0.000457616		1.00000i	$0.499854\pi$
$198$	0	0
$199$	−2.77399e83	−0.342742	−0.171371	−	0.985207i	$-0.554820\pi$
$199$	−2.77399e83	−0.342742	−0.171371	+	0.985207i	$0.554820\pi$
$200$	8.61009e82	0.0885957
$201$	0	0
$202$	8.28724e82	0.0593042
$203$	−8.90886e83	−0.532374
$204$	0	0
$205$	−1.03520e84	−0.432522
$206$	2.77888e83	0.0972113
$207$	0	0
$208$	−2.95429e84	−0.726343
$209$	1.07917e85	2.22715
$210$	0	0
$211$	4.96555e84	0.723865	0.361932	−	0.932204i	$-0.382117\pi$
$211$	4.96555e84	0.723865	0.361932	+	0.932204i	$0.382117\pi$
$212$	−1.55726e84	−0.191031
$213$	0	0
$214$	−7.34359e83	−0.0639449
$215$	−3.51455e84	−0.258146
$216$	0	0
$217$	−4.81673e84	−0.252332
$218$	1.41259e84	0.0625685
$219$	0	0
$220$	2.33533e85	0.741174
$221$	−4.54760e85	−1.22311
$222$	0	0
$223$	−4.40493e85	−0.852730	−0.426365	−	0.904551i	$-0.640206\pi$
$223$	−4.40493e85	−0.852730	−0.426365	+	0.904551i	$0.640206\pi$
$224$	−4.24678e84	−0.0698243
$225$	0	0
$226$	−4.47343e84	−0.0531720
$227$	−2.14705e85	−0.217220	−0.108610	−	0.994084i	$-0.534640\pi$
$227$	−2.14705e85	−0.217220	−0.108610	+	0.994084i	$0.534640\pi$
$228$	0	0
$229$	1.57271e86	1.15519	0.577595	−	0.816323i	$-0.303991\pi$
$229$	1.57271e86	1.15519	0.577595	+	0.816323i	$0.303991\pi$
$230$	2.07477e84	0.0129988
$231$	0	0
$232$	−3.67595e85	−0.167903
$233$	−7.07062e85	−0.276036	−0.138018	−	0.990430i	$-0.544073\pi$
$233$	−7.07062e85	−0.276036	−0.138018	+	0.990430i	$0.544073\pi$
$234$	0	0
$235$	8.61782e83	0.00246275
$236$	1.48342e86	0.363058
$237$	0	0
$238$	−2.16558e85	−0.0389507
$239$	7.62711e86	1.17716	0.588582	−	0.808438i	$-0.299687\pi$
$239$	7.62711e86	1.17716	0.588582	+	0.808438i	$0.299687\pi$
$240$	0	0
$241$	3.73671e86	0.425468	0.212734	−	0.977110i	$-0.431763\pi$
$241$	3.73671e86	0.425468	0.212734	+	0.977110i	$0.431763\pi$
$242$	6.52184e85	0.0638424
$243$	0	0
$244$	−3.48594e86	−0.252692
$245$	7.08926e86	0.442628
$246$	0	0
$247$	−2.46186e87	−1.14242
$248$	−1.98746e86	−0.0795817
$249$	0	0
$250$	1.82945e86	0.0546400
$251$	−4.47993e87	−1.15659	−0.578296	−	0.815827i	$-0.696282\pi$
$251$	−4.47993e87	−1.15659	−0.578296	+	0.815827i	$0.696282\pi$
$252$	0	0
$253$	−3.04971e87	−0.589331
$254$	1.83629e86	0.0307264
$255$	0	0
$256$	7.66476e87	0.963252
$257$	6.05754e86	0.0660294	0.0330147	−	0.999455i	$-0.489489\pi$
$257$	6.05754e86	0.0660294	0.0330147	+	0.999455i	$0.489489\pi$
$258$	0	0
$259$	8.69545e87	0.714251
$260$	−5.32749e87	−0.380185
$261$	0	0
$262$	1.26102e87	0.0680336
$263$	2.40877e88	1.13086	0.565432	−	0.824795i	$-0.308710\pi$
$263$	2.40877e88	1.13086	0.565432	+	0.824795i	$0.308710\pi$
$264$	0	0
$265$	−2.79776e87	−0.0996178
$266$	−1.17235e87	−0.0363811
$267$	0	0
$268$	2.08345e88	0.491885
$269$	−4.13585e88	−0.852328	−0.426164	−	0.904646i	$-0.640135\pi$
$269$	−4.13585e88	−0.852328	−0.426164	+	0.904646i	$0.640135\pi$
$270$	0	0
$271$	3.37612e87	0.0530932	0.0265466	−	0.999648i	$-0.491549\pi$
$271$	3.37612e87	0.0530932	0.0265466	+	0.999648i	$0.491549\pi$
$272$	1.19789e89	1.64684
$273$	0	0
$274$	6.14583e87	0.0646674
$275$	−1.13478e89	−1.04536
$276$	0	0
$277$	−1.55810e89	−1.10175	−0.550875	−	0.834588i	$-0.685706\pi$
$277$	−1.55810e89	−1.10175	−0.550875	+	0.834588i	$0.685706\pi$
$278$	−8.79160e87	−0.0545046
$279$	0	0
$280$	−5.08334e87	−0.0242595
$281$	4.48983e89	1.88126	0.940630	−	0.339432i	$-0.110235\pi$
$281$	4.48983e89	1.88126	0.940630	+	0.339432i	$0.110235\pi$
$282$	0	0
$283$	−3.25365e88	−0.105236	−0.0526182	−	0.998615i	$-0.516757\pi$
$283$	−3.25365e88	−0.105236	−0.0526182	+	0.998615i	$0.516757\pi$
$284$	−1.01801e89	−0.289490
$285$	0	0
$286$	−2.90443e88	−0.0639295
$287$	−1.65302e89	−0.320324
$288$	0	0
$289$	1.17901e90	1.77315
$290$	−3.29597e88	−0.0436975
$291$	0	0
$292$	−2.02999e89	−0.209421
$293$	1.51494e90	1.37952	0.689760	−	0.724038i	$-0.257716\pi$
$293$	1.51494e90	1.37952	0.689760	+	0.724038i	$0.257716\pi$
$294$	0	0
$295$	2.66509e89	0.189325
$296$	3.58789e89	0.225264
$297$	0	0
$298$	5.10607e88	0.0250722
$299$	6.95717e89	0.302297
$300$	0	0
$301$	−5.61209e89	−0.191182
$302$	−1.35225e89	−0.0408136
$303$	0	0
$304$	6.48482e90	1.53820
$305$	−6.26280e89	−0.131772
$306$	0	0
$307$	1.83701e90	0.304482	0.152241	−	0.988343i	$-0.451351\pi$
$307$	1.83701e90	0.304482	0.152241	+	0.988343i	$0.451351\pi$
$308$	3.72909e90	0.548910
$309$	0	0
$310$	−1.78202e89	−0.0207115
$311$	1.36709e91	1.41268	0.706339	−	0.707874i	$-0.250345\pi$
$311$	1.36709e91	1.41268	0.706339	+	0.707874i	$0.250345\pi$
$312$	0	0
$313$	−1.01164e91	−0.827292	−0.413646	−	0.910438i	$-0.635745\pi$
$313$	−1.01164e91	−0.827292	−0.413646	+	0.910438i	$0.635745\pi$
$314$	1.14183e90	0.0831128
$315$	0	0
$316$	−1.82090e91	−1.05125
$317$	8.49508e90	0.437023	0.218512	−	0.975834i	$-0.429880\pi$
$317$	8.49508e90	0.437023	0.218512	+	0.975834i	$0.429880\pi$
$318$	0	0
$319$	4.84475e91	1.98112
$320$	1.39281e91	0.508064
$321$	0	0
$322$	3.31303e89	0.00962688
$323$	9.98220e91	2.59020
$324$	0	0
$325$	2.58872e91	0.536219
$326$	−5.49023e88	−0.00101659
$327$	0	0
$328$	−6.82066e90	−0.101026
$329$	1.37611e89	0.00182390
$330$	0	0
$331$	1.05949e92	1.12558	0.562790	−	0.826600i	$-0.309728\pi$
$331$	1.05949e92	1.12558	0.562790	+	0.826600i	$0.309728\pi$
$332$	1.10097e92	1.04770
$333$	0	0
$334$	8.05543e90	0.0615662
$335$	3.74311e91	0.256506
$336$	0	0
$337$	7.23797e91	0.399140	0.199570	−	0.979884i	$-0.436045\pi$
$337$	7.23797e91	0.399140	0.199570	+	0.979884i	$0.436045\pi$
$338$	−5.65658e90	−0.0279959
$339$	0	0
$340$	2.16015e92	0.861994
$341$	2.61940e92	0.939003
$342$	0	0
$343$	2.46077e92	0.712584
$344$	−2.31564e91	−0.0602960
$345$	0	0
$346$	−2.41791e91	−0.0509522
$347$	6.97644e92	1.32315	0.661574	−	0.749880i	$-0.269889\pi$
$347$	6.97644e92	1.32315	0.661574	+	0.749880i	$0.269889\pi$
$348$	0	0
$349$	−3.78993e92	−0.582778	−0.291389	−	0.956605i	$-0.594117\pi$
$349$	−3.78993e92	−0.582778	−0.291389	+	0.956605i	$0.594117\pi$
$350$	1.23275e91	0.0170763
$351$	0	0
$352$	2.30945e92	0.259837
$353$	1.10203e93	1.11794	0.558970	−	0.829188i	$-0.311197\pi$
$353$	1.10203e93	1.11794	0.558970	+	0.829188i	$0.311197\pi$
$354$	0	0
$355$	−1.82895e92	−0.150962
$356$	1.55814e93	1.16060
$357$	0	0
$358$	−3.91459e91	−0.0237660
$359$	1.96885e93	1.07961	0.539804	−	0.841791i	$-0.318498\pi$
$359$	1.96885e93	1.07961	0.539804	+	0.841791i	$0.318498\pi$
$360$	0	0
$361$	3.17027e93	1.41933
$362$	−1.46366e92	−0.0592349
$363$	0	0
$364$	−8.50702e92	−0.281564
$365$	−3.64706e92	−0.109208
$366$	0	0
$367$	−1.57037e93	−0.385205	−0.192603	−	0.981277i	$-0.561693\pi$
$367$	−1.57037e93	−0.385205	−0.192603	+	0.981277i	$0.561693\pi$
$368$	−1.83260e93	−0.407025
$369$	0	0
$370$	3.21702e92	0.0586261
$371$	−4.46752e92	−0.0737766
$372$	0	0
$373$	9.09346e93	1.23412	0.617059	−	0.786917i	$-0.288324\pi$
$373$	9.09346e93	1.23412	0.617059	+	0.786917i	$0.288324\pi$
$374$	1.17767e93	0.144947
$375$	0	0
$376$	5.67805e90	0.000575232 0
$377$	−1.10521e94	−1.01622
$378$	0	0
$379$	−1.61012e94	−1.22047	−0.610235	−	0.792220i	$-0.708925\pi$
$379$	−1.61012e94	−1.22047	−0.610235	+	0.792220i	$0.708925\pi$
$380$	1.16941e94	0.805129
$381$	0	0
$382$	−1.91000e93	−0.108573
$383$	−3.65366e94	−1.88787	−0.943937	−	0.330125i	$-0.892909\pi$
$383$	−3.65366e94	−1.88787	−0.943937	+	0.330125i	$0.892909\pi$
$384$	0	0
$385$	6.69964e93	0.286243
$386$	−2.24839e93	−0.0873846
$387$	0	0
$388$	9.17025e93	0.295137
$389$	5.06109e94	1.48280	0.741402	−	0.671061i	$-0.234161\pi$
$389$	5.06109e94	1.48280	0.741402	+	0.671061i	$0.234161\pi$
$390$	0	0
$391$	−2.82095e94	−0.685399
$392$	4.67092e93	0.103386
$393$	0	0
$394$	3.02667e90	5.56356e−5 0
$395$	−3.27140e94	−0.548201
$396$	0	0
$397$	−2.77522e94	−0.386762	−0.193381	−	0.981124i	$-0.561945\pi$
$397$	−2.77522e94	−0.386762	−0.193381	+	0.981124i	$0.561945\pi$
$398$	−1.63878e93	−0.0208348
$399$	0	0
$400$	−6.81897e94	−0.721987
$401$	−6.35711e94	−0.614455	−0.307228	−	0.951636i	$-0.599401\pi$
$401$	−6.35711e94	−0.614455	−0.307228	+	0.951636i	$0.599401\pi$
$402$	0	0
$403$	−5.97553e94	−0.481662
$404$	−1.32001e95	−0.971977
$405$	0	0
$406$	−5.26306e93	−0.0323622
$407$	−4.72870e95	−2.65794
$408$	0	0
$409$	1.42308e95	0.668851	0.334426	−	0.942422i	$-0.391458\pi$
$409$	1.42308e95	0.668851	0.334426	+	0.942422i	$0.391458\pi$
$410$	−6.11562e93	−0.0262924
$411$	0	0
$412$	−4.42625e95	−1.59326
$413$	4.25567e94	0.140214
$414$	0	0
$415$	1.97799e95	0.546347
$416$	−5.26846e94	−0.133284
$417$	0	0
$418$	6.37537e94	0.135385
$419$	−3.45923e95	−0.673237	−0.336618	−	0.941641i	$-0.609283\pi$
$419$	−3.45923e95	−0.673237	−0.336618	+	0.941641i	$0.609283\pi$
$420$	0	0
$421$	−3.52446e94	−0.0576495	−0.0288248	−	0.999584i	$-0.509176\pi$
$421$	−3.52446e94	−0.0576495	−0.0288248	+	0.999584i	$0.509176\pi$
$422$	2.93348e94	0.0440026
$423$	0	0
$424$	−1.84337e94	−0.0232680
$425$	−1.04966e96	−1.21577
$426$	0	0
$427$	−1.00005e95	−0.0975901
$428$	1.16970e96	1.04804
$429$	0	0
$430$	−2.07628e94	−0.0156923
$431$	7.25552e95	0.503788	0.251894	−	0.967755i	$-0.418947\pi$
$431$	7.25552e95	0.503788	0.251894	+	0.967755i	$0.418947\pi$
$432$	0	0
$433$	1.48595e95	0.0871354	0.0435677	−	0.999050i	$-0.486128\pi$
$433$	1.48595e95	0.0871354	0.0435677	+	0.999050i	$0.486128\pi$
$434$	−2.84556e94	−0.0153389
$435$	0	0
$436$	−2.25000e96	−1.02548
$437$	−1.52713e96	−0.640183
$438$	0	0
$439$	−1.75652e96	−0.623302	−0.311651	−	0.950197i	$-0.600882\pi$
$439$	−1.75652e96	−0.623302	−0.311651	+	0.950197i	$0.600882\pi$
$440$	2.76439e95	0.0902768
$441$	0	0
$442$	−2.68657e95	−0.0743508
$443$	7.67761e95	0.195654	0.0978272	−	0.995203i	$-0.468811\pi$
$443$	7.67761e95	0.195654	0.0978272	+	0.995203i	$0.468811\pi$
$444$	0	0
$445$	2.79934e96	0.605223
$446$	−2.60229e95	−0.0518362
$447$	0	0
$448$	2.22407e96	0.376270
$449$	2.38413e96	0.371826	0.185913	−	0.982566i	$-0.440476\pi$
$449$	2.38413e96	0.371826	0.185913	+	0.982566i	$0.440476\pi$
$450$	0	0
$451$	8.98936e96	1.19202
$452$	7.12535e96	0.871472
$453$	0	0
$454$	−1.26841e95	−0.0132045
$455$	−1.52836e96	−0.146828
$456$	0	0
$457$	1.08572e97	0.888744	0.444372	−	0.895842i	$-0.353427\pi$
$457$	1.08572e97	0.888744	0.444372	+	0.895842i	$0.353427\pi$
$458$	9.29105e95	0.0702223
$459$	0	0
$460$	−3.30473e96	−0.213047
$461$	2.25885e97	1.34525	0.672625	−	0.739984i	$-0.265167\pi$
$461$	2.25885e97	1.34525	0.672625	+	0.739984i	$0.265167\pi$
$462$	0	0
$463$	1.05828e97	0.538137	0.269068	−	0.963121i	$-0.413284\pi$
$463$	1.05828e97	0.538137	0.269068	+	0.963121i	$0.413284\pi$
$464$	2.91126e97	1.36828
$465$	0	0
$466$	−4.17709e95	−0.0167798
$467$	−2.18220e97	−0.810642	−0.405321	−	0.914174i	$-0.632840\pi$
$467$	−2.18220e97	−0.810642	−0.405321	+	0.914174i	$0.632840\pi$
$468$	0	0
$469$	5.97705e96	0.189967
$470$	5.09112e93	0.000149707 0
$471$	0	0
$472$	1.75596e96	0.0442213
$473$	3.05193e97	0.711447
$474$	0	0
$475$	−5.68236e97	−1.13557
$476$	3.44937e97	0.638390
$477$	0	0
$478$	4.50585e96	0.0715580
$479$	1.96022e97	0.288442	0.144221	−	0.989546i	$-0.453932\pi$
$479$	1.96022e97	0.288442	0.144221	+	0.989546i	$0.453932\pi$
$480$	0	0
$481$	1.07874e98	1.36339
$482$	2.20753e96	0.0258636
$483$	0	0
$484$	−1.03881e98	−1.04636
$485$	1.64752e97	0.153906
$486$	0	0
$487$	−4.70183e97	−0.377972	−0.188986	−	0.981980i	$-0.560520\pi$
$487$	−4.70183e97	−0.377972	−0.188986	+	0.981980i	$0.560520\pi$
$488$	−4.12639e96	−0.0307785
$489$	0	0
$490$	4.18810e96	0.0269067
$491$	8.71377e97	0.519675	0.259838	−	0.965652i	$-0.416331\pi$
$491$	8.71377e97	0.519675	0.259838	+	0.965652i	$0.416331\pi$
$492$	0	0
$493$	4.48135e98	2.30407
$494$	−1.45439e97	−0.0694459
$495$	0	0
$496$	1.57402e98	0.648529
$497$	−2.92050e97	−0.111802
$498$	0	0
$499$	−4.82312e98	−1.59462	−0.797312	−	0.603567i	$-0.793745\pi$
$499$	−4.82312e98	−1.59462	−0.797312	+	0.603567i	$0.793745\pi$
$500$	−2.91398e98	−0.895533
$501$	0	0
$502$	−2.64659e97	−0.0703075
$503$	−2.59870e98	−0.641985	−0.320992	−	0.947082i	$-0.604016\pi$
$503$	−2.59870e98	−0.641985	−0.320992	+	0.947082i	$0.604016\pi$
$504$	0	0
$505$	−2.37151e98	−0.506861
$506$	−1.80167e97	−0.0358245
$507$	0	0
$508$	−2.92488e98	−0.503596
$509$	−7.47042e98	−1.19714	−0.598572	−	0.801069i	$-0.704265\pi$
$509$	−7.47042e98	−1.19714	−0.598572	+	0.801069i	$0.704265\pi$
$510$	0	0
$511$	−5.82368e97	−0.0808788
$512$	2.31581e98	0.299467
$513$	0	0
$514$	3.57859e96	0.00401383
$515$	−7.95215e98	−0.830846
$516$	0	0
$517$	−7.48345e96	−0.00678730
$518$	5.13699e97	0.0434182
$519$	0	0
$520$	−6.30628e97	−0.0463075
$521$	−2.02966e98	−0.138946	−0.0694729	−	0.997584i	$-0.522132\pi$
$521$	−2.02966e98	−0.138946	−0.0694729	+	0.997584i	$0.522132\pi$
$522$	0	0
$523$	−2.22396e99	−1.32378	−0.661890	−	0.749601i	$-0.730245\pi$
$523$	−2.22396e99	−1.32378	−0.661890	+	0.749601i	$0.730245\pi$
$524$	−2.00857e99	−1.11505
$525$	0	0
$526$	1.42302e98	0.0687435
$527$	2.42292e99	1.09207
$528$	0	0
$529$	−2.11604e99	−0.830600
$530$	−1.65283e97	−0.00605561
$531$	0	0
$532$	1.86733e99	0.596275
$533$	−2.05071e99	−0.611449
$534$	0	0
$535$	2.10147e99	0.546525
$536$	2.46623e98	0.0599128
$537$	0	0
$538$	−2.44333e98	−0.0518117
$539$	−6.15609e99	−1.21987
$540$	0	0
$541$	−1.61237e99	−0.279102	−0.139551	−	0.990215i	$-0.544566\pi$
$541$	−1.61237e99	−0.279102	−0.139551	+	0.990215i	$0.544566\pi$
$542$	1.99450e97	0.00322746
$543$	0	0
$544$	2.13622e99	0.302194
$545$	−4.04233e99	−0.534761
$546$	0	0
$547$	1.79750e99	0.208033	0.104016	−	0.994576i	$-0.466831\pi$
$547$	1.79750e99	0.208033	0.104016	+	0.994576i	$0.466831\pi$
$548$	−9.78918e99	−1.05988
$549$	0	0
$550$	−6.70388e98	−0.0635460
$551$	2.42600e100	2.15207
$552$	0	0
$553$	−5.22383e99	−0.405996
$554$	−9.20473e98	−0.0669738
$555$	0	0
$556$	1.40034e100	0.893313
$557$	8.70004e99	0.519764	0.259882	−	0.965640i	$-0.416316\pi$
$557$	8.70004e99	0.519764	0.259882	+	0.965640i	$0.416316\pi$
$558$	0	0
$559$	−6.96223e99	−0.364937
$560$	4.02588e99	0.197696
$561$	0	0
$562$	2.65244e99	0.114359
$563$	1.34453e100	0.543267	0.271634	−	0.962401i	$-0.412436\pi$
$563$	1.34453e100	0.543267	0.271634	+	0.962401i	$0.412436\pi$
$564$	0	0
$565$	1.28013e100	0.454451
$566$	−1.92215e98	−0.00639717
$567$	0	0
$568$	−1.20505e99	−0.0352606
$569$	4.71804e100	1.29468	0.647342	−	0.762200i	$-0.275881\pi$
$569$	4.71804e100	1.29468	0.647342	+	0.762200i	$0.275881\pi$
$570$	0	0
$571$	−2.57174e100	−0.620882	−0.310441	−	0.950593i	$-0.600477\pi$
$571$	−2.57174e100	−0.620882	−0.310441	+	0.950593i	$0.600477\pi$
$572$	4.62623e100	1.04778
$573$	0	0
$574$	−9.76552e98	−0.0194720
$575$	1.60582e100	0.300484
$576$	0	0
$577$	5.80535e100	0.956998	0.478499	−	0.878088i	$-0.341181\pi$
$577$	5.80535e100	0.956998	0.478499	+	0.878088i	$0.341181\pi$
$578$	6.96519e99	0.107787
$579$	0	0
$580$	5.24988e100	0.716189
$581$	3.15849e100	0.404623
$582$	0	0
$583$	2.42949e100	0.274545
$584$	−2.40295e99	−0.0255080
$585$	0	0
$586$	8.94978e99	0.0838589
$587$	−1.38113e101	−1.21603	−0.608013	−	0.793927i	$-0.708033\pi$
$587$	−1.38113e101	−1.21603	−0.608013	+	0.793927i	$0.708033\pi$
$588$	0	0
$589$	1.31166e101	1.02003
$590$	1.57445e99	0.0115088
$591$	0	0
$592$	−2.84152e101	−1.83573
$593$	−9.03719e100	−0.548956	−0.274478	−	0.961593i	$-0.588505\pi$
$593$	−9.03719e100	−0.548956	−0.274478	+	0.961593i	$0.588505\pi$
$594$	0	0
$595$	6.19710e100	0.332904
$596$	−8.13303e100	−0.410926
$597$	0	0
$598$	4.11007e99	0.0183762
$599$	4.41983e101	1.85920	0.929601	−	0.368568i	$-0.120152\pi$
$599$	4.41983e101	1.85920	0.929601	+	0.368568i	$0.120152\pi$
$600$	0	0
$601$	5.61076e100	0.208980	0.104490	−	0.994526i	$-0.466679\pi$
$601$	5.61076e100	0.208980	0.104490	+	0.994526i	$0.466679\pi$
$602$	−3.31544e99	−0.0116217
$603$	0	0
$604$	2.15389e101	0.668922
$605$	−1.86631e101	−0.545649
$606$	0	0
$607$	−3.42040e101	−0.886524	−0.443262	−	0.896392i	$-0.646179\pi$
$607$	−3.42040e101	−0.886524	−0.443262	+	0.896392i	$0.646179\pi$
$608$	1.15645e101	0.282259
$609$	0	0
$610$	−3.69985e99	−0.00801024
$611$	1.70717e99	0.00348155
$612$	0	0
$613$	−1.01689e102	−1.84062	−0.920312	−	0.391185i	$-0.872065\pi$
$613$	−1.01689e102	−1.84062	−0.920312	+	0.391185i	$0.872065\pi$
$614$	1.08525e100	0.0185090
$615$	0	0
$616$	4.41422e100	0.0668586
$617$	−1.26801e102	−1.81015	−0.905077	−	0.425247i	$-0.860187\pi$
$617$	−1.26801e102	−1.81015	−0.905077	+	0.425247i	$0.860187\pi$
$618$	0	0
$619$	8.97168e101	1.13806	0.569030	−	0.822317i	$-0.307319\pi$
$619$	8.97168e101	1.13806	0.569030	+	0.822317i	$0.307319\pi$
$620$	2.83844e101	0.339455
$621$	0	0
$622$	8.07632e100	0.0858746
$623$	4.47003e101	0.448226
$624$	0	0
$625$	2.94914e101	0.263072
$626$	−5.97645e100	−0.0502898
$627$	0	0
$628$	−1.81872e102	−1.36219
$629$	−4.37400e102	−3.09122
$630$	0	0
$631$	−1.05597e102	−0.664629	−0.332314	−	0.943169i	$-0.607829\pi$
$631$	−1.05597e102	−0.664629	−0.332314	+	0.943169i	$0.607829\pi$
$632$	−2.15544e101	−0.128045
$633$	0	0
$634$	5.01862e100	0.0265660
$635$	−5.25480e101	−0.262612
$636$	0	0
$637$	1.40436e102	0.625735
$638$	2.86212e101	0.120430
$639$	0	0
$640$	3.33468e101	0.125166
$641$	1.64809e102	0.584336	0.292168	−	0.956367i	$-0.405623\pi$
$641$	1.64809e102	0.584336	0.292168	+	0.956367i	$0.405623\pi$
$642$	0	0
$643$	1.19000e102	0.376570	0.188285	−	0.982114i	$-0.439707\pi$
$643$	1.19000e102	0.376570	0.188285	+	0.982114i	$0.439707\pi$
$644$	−5.27704e101	−0.157782
$645$	0	0
$646$	5.89715e101	0.157455
$647$	−3.27438e102	−0.826273	−0.413136	−	0.910669i	$-0.635567\pi$
$647$	−3.27438e102	−0.826273	−0.413136	+	0.910669i	$0.635567\pi$
$648$	0	0
$649$	−2.31429e102	−0.521778
$650$	1.52933e101	0.0325959
$651$	0	0
$652$	8.74492e100	0.0166617
$653$	7.85065e102	1.41441	0.707204	−	0.707010i	$-0.249956\pi$
$653$	7.85065e102	1.41441	0.707204	+	0.707010i	$0.249956\pi$
$654$	0	0
$655$	−3.60857e102	−0.581470
$656$	5.40179e102	0.823280
$657$	0	0
$658$	8.12959e98	0.000110872 0
$659$	2.63756e102	0.340318	0.170159	−	0.985417i	$-0.445572\pi$
$659$	2.63756e102	0.340318	0.170159	+	0.985417i	$0.445572\pi$
$660$	0	0
$661$	4.82101e102	0.556910	0.278455	−	0.960449i	$-0.410178\pi$
$661$	4.82101e102	0.556910	0.278455	+	0.960449i	$0.410178\pi$
$662$	6.25914e101	0.0684224
$663$	0	0
$664$	1.30325e102	0.127612
$665$	3.35483e102	0.310943
$666$	0	0
$667$	−6.85582e102	−0.569464
$668$	−1.28308e103	−1.00905
$669$	0	0
$670$	2.21130e101	0.0155926
$671$	5.43842e102	0.363162
$672$	0	0
$673$	1.51214e103	0.905822	0.452911	−	0.891556i	$-0.350386\pi$
$673$	1.51214e103	0.905822	0.452911	+	0.891556i	$0.350386\pi$
$674$	4.27595e101	0.0242631
$675$	0	0
$676$	9.00988e102	0.458844
$677$	2.33256e103	1.12550	0.562751	−	0.826626i	$-0.309743\pi$
$677$	2.33256e103	1.12550	0.562751	+	0.826626i	$0.309743\pi$
$678$	0	0
$679$	2.63078e102	0.113982
$680$	2.55703e102	0.104993
$681$	0	0
$682$	1.54745e102	0.0570806
$683$	1.30508e103	0.456334	0.228167	−	0.973622i	$-0.426727\pi$
$683$	1.30508e103	0.456334	0.228167	+	0.973622i	$0.426727\pi$
$684$	0	0
$685$	−1.75871e103	−0.552700
$686$	1.45374e102	0.0433169
$687$	0	0
$688$	1.83393e103	0.491366
$689$	−5.54230e102	−0.140828
$690$	0	0
$691$	6.78656e103	1.55132	0.775660	−	0.631151i	$-0.217417\pi$
$691$	6.78656e103	1.55132	0.775660	+	0.631151i	$0.217417\pi$
$692$	3.85129e103	0.835090
$693$	0	0
$694$	4.12145e102	0.0804321
$695$	2.51584e103	0.465840
$696$	0	0
$697$	8.31507e103	1.38634
$698$	−2.23897e102	−0.0354262
$699$	0	0
$700$	−1.96355e103	−0.279875
$701$	7.61065e103	1.02971	0.514854	−	0.857278i	$-0.327846\pi$
$701$	7.61065e103	1.02971	0.514854	+	0.857278i	$0.327846\pi$
$702$	0	0
$703$	−2.36788e104	−2.88730
$704$	−1.20948e104	−1.40022
$705$	0	0
$706$	6.51045e102	0.0679579
$707$	−3.78686e103	−0.375380
$708$	0	0
$709$	−1.25155e104	−1.11908	−0.559538	−	0.828805i	$-0.689021\pi$
$709$	−1.25155e104	−1.11908	−0.559538	+	0.828805i	$0.689021\pi$
$710$	−1.08048e102	−0.00917675
$711$	0	0
$712$	1.84441e103	0.141364
$713$	−3.70672e103	−0.269912
$714$	0	0
$715$	8.31143e103	0.546393
$716$	6.23522e103	0.389517
$717$	0	0
$718$	1.16313e103	0.0656277
$719$	−2.65219e104	−1.42233	−0.711164	−	0.703026i	$-0.751832\pi$
$719$	−2.65219e104	−1.42233	−0.711164	+	0.703026i	$0.751832\pi$
$720$	0	0
$721$	−1.26981e104	−0.615322
$722$	1.87289e103	0.0862789
$723$	0	0
$724$	2.33134e104	0.970842
$725$	−2.55101e104	−1.01012
$726$	0	0
$727$	2.80516e103	0.100451	0.0502253	−	0.998738i	$-0.484006\pi$
$727$	2.80516e103	0.100451	0.0502253	+	0.998738i	$0.484006\pi$
$728$	−1.00700e103	−0.0342952
$729$	0	0
$730$	−2.15456e102	−0.00663857
$731$	2.82300e104	0.827421
$732$	0	0
$733$	−4.71011e104	−1.24950	−0.624750	−	0.780825i	$-0.714799\pi$
$733$	−4.71011e104	−1.24950	−0.624750	+	0.780825i	$0.714799\pi$
$734$	−9.27726e102	−0.0234160
$735$	0	0
$736$	−3.26811e103	−0.0746890
$737$	−3.25040e104	−0.706925
$738$	0	0
$739$	7.23905e104	1.42613	0.713065	−	0.701098i	$-0.247306\pi$
$739$	7.23905e104	1.42613	0.713065	+	0.701098i	$0.247306\pi$
$740$	−5.12412e104	−0.960863
$741$	0	0
$742$	−2.63926e102	−0.00448477
$743$	−7.11987e104	−1.15181	−0.575905	−	0.817517i	$-0.695350\pi$
$743$	−7.11987e104	−1.15181	−0.575905	+	0.817517i	$0.695350\pi$
$744$	0	0
$745$	−1.46117e104	−0.214288
$746$	5.37212e103	0.0750202
$747$	0	0
$748$	−1.87581e105	−2.37564
$749$	3.35566e104	0.404754
$750$	0	0
$751$	1.54302e105	1.68855	0.844277	−	0.535907i	$-0.180030\pi$
$751$	1.54302e105	1.68855	0.844277	+	0.535907i	$0.180030\pi$
$752$	−4.49687e102	−0.00468770
$753$	0	0
$754$	−6.52924e103	−0.0617744
$755$	3.86966e104	0.348826
$756$	0	0
$757$	−1.58740e105	−1.29923	−0.649617	−	0.760262i	$-0.725071\pi$
$757$	−1.58740e105	−1.29923	−0.649617	+	0.760262i	$0.725071\pi$
$758$	−9.51207e103	−0.0741906
$759$	0	0
$760$	1.38426e104	0.0980666
$761$	−7.55382e104	−0.510066	−0.255033	−	0.966932i	$-0.582086\pi$
$761$	−7.55382e104	−0.510066	−0.255033	+	0.966932i	$0.582086\pi$
$762$	0	0
$763$	−6.45486e104	−0.396042
$764$	3.04228e105	1.77948
$765$	0	0
$766$	−2.15846e104	−0.114761
$767$	5.27949e104	0.267646
$768$	0	0
$769$	3.16605e105	1.45951	0.729754	−	0.683710i	$-0.239635\pi$
$769$	3.16605e105	1.45951	0.729754	+	0.683710i	$0.239635\pi$
$770$	3.95793e103	0.0174003
$771$	0	0
$772$	3.58127e105	1.43221
$773$	6.80947e104	0.259754	0.129877	−	0.991530i	$-0.458542\pi$
$773$	6.80947e104	0.259754	0.129877	+	0.991530i	$0.458542\pi$
$774$	0	0
$775$	−1.37925e105	−0.478773
$776$	1.08551e104	0.0359484
$777$	0	0
$778$	2.98993e104	0.0901374
$779$	4.50140e105	1.29488
$780$	0	0
$781$	1.58821e105	0.416049
$782$	−1.66652e104	−0.0416644
$783$	0	0
$784$	−3.69925e105	−0.842515
$785$	−3.26749e105	−0.710350
$786$	0	0
$787$	5.43922e105	1.07760	0.538801	−	0.842433i	$-0.318877\pi$
$787$	5.43922e105	1.07760	0.538801	+	0.842433i	$0.318877\pi$
$788$	−4.82092e102	−0.000911850 0
$789$	0	0
$790$	−1.93264e104	−0.0333243
$791$	2.04414e105	0.336565
$792$	0	0
$793$	−1.24064e105	−0.186284
$794$	−1.63951e104	−0.0235107
$795$	0	0
$796$	2.61028e105	0.341476
$797$	7.11246e105	0.888773	0.444387	−	0.895835i	$-0.353422\pi$
$797$	7.11246e105	0.888773	0.444387	+	0.895835i	$0.353422\pi$
$798$	0	0
$799$	−6.92212e103	−0.00789372
$800$	−1.21604e105	−0.132484
$801$	0	0
$802$	−3.75557e104	−0.0373518
$803$	3.16699e105	0.300974
$804$	0	0
$805$	−9.48068e104	−0.0822791
$806$	−3.53015e104	−0.0292795
$807$	0	0
$808$	−1.56252e105	−0.118389
$809$	−7.46404e105	−0.540571	−0.270286	−	0.962780i	$-0.587118\pi$
$809$	−7.46404e105	−0.540571	−0.270286	+	0.962780i	$0.587118\pi$
$810$	0	0
$811$	−1.03340e106	−0.683920	−0.341960	−	0.939715i	$-0.611091\pi$
$811$	−1.03340e106	−0.683920	−0.341960	+	0.939715i	$0.611091\pi$
$812$	8.38309e105	0.530407
$813$	0	0
$814$	−2.79356e105	−0.161572
$815$	1.57110e104	0.00868864
$816$	0	0
$817$	1.52824e106	0.772837
$818$	8.40711e104	0.0406585
$819$	0	0
$820$	9.74106e105	0.430924
$821$	1.85960e106	0.786855	0.393428	−	0.919356i	$-0.371289\pi$
$821$	1.85960e106	0.786855	0.393428	+	0.919356i	$0.371289\pi$
$822$	0	0
$823$	−1.97114e106	−0.763172	−0.381586	−	0.924333i	$-0.624622\pi$
$823$	−1.97114e106	−0.763172	−0.381586	+	0.924333i	$0.624622\pi$
$824$	−5.23946e105	−0.194063
$825$	0	0
$826$	2.51411e104	0.00852339
$827$	−1.57846e106	−0.512014	−0.256007	−	0.966675i	$-0.582407\pi$
$827$	−1.57846e106	−0.512014	−0.256007	+	0.966675i	$0.582407\pi$
$828$	0	0
$829$	−3.37721e106	−1.00304	−0.501519	−	0.865146i	$-0.667225\pi$
$829$	−3.37721e106	−1.00304	−0.501519	+	0.865146i	$0.667225\pi$
$830$	1.16853e105	0.0332116
$831$	0	0
$832$	2.75913e106	0.718241
$833$	−5.69432e106	−1.41873
$834$	0	0
$835$	−2.30517e106	−0.526195
$836$	−1.01548e107	−2.21892
$837$	0	0
$838$	−2.04360e105	−0.0409251
$839$	−4.15571e106	−0.796771	−0.398385	−	0.917218i	$-0.630429\pi$
$839$	−4.15571e106	−0.796771	−0.398385	+	0.917218i	$0.630429\pi$
$840$	0	0
$841$	5.20190e106	0.914341
$842$	−2.08213e104	−0.00350443
$843$	0	0
$844$	−4.67250e106	−0.721190
$845$	1.61871e106	0.239275
$846$	0	0
$847$	−2.98016e106	−0.404106
$848$	1.45990e106	0.189616
$849$	0	0
$850$	−6.20102e105	−0.0739048
$851$	6.69159e106	0.764013
$852$	0	0
$853$	3.24533e106	0.340110	0.170055	−	0.985435i	$-0.445605\pi$
$853$	3.24533e106	0.340110	0.170055	+	0.985435i	$0.445605\pi$
$854$	−5.90799e104	−0.00593236
$855$	0	0
$856$	1.38460e106	0.127654
$857$	−1.48340e107	−1.31056	−0.655282	−	0.755385i	$-0.727450\pi$
$857$	−1.48340e107	−1.31056	−0.655282	+	0.755385i	$0.727450\pi$
$858$	0	0
$859$	−1.87004e107	−1.51740	−0.758699	−	0.651441i	$-0.774165\pi$
$859$	−1.87004e107	−1.51740	−0.758699	+	0.651441i	$0.774165\pi$
$860$	3.30713e106	0.257192
$861$	0	0
$862$	4.28632e105	0.0306245
$863$	1.06435e107	0.728939	0.364469	−	0.931215i	$-0.381250\pi$
$863$	1.06435e107	0.728939	0.364469	+	0.931215i	$0.381250\pi$
$864$	0	0
$865$	6.91918e106	0.435479
$866$	8.77848e104	0.00529683
$867$	0	0
$868$	4.53246e106	0.251399
$869$	2.84078e107	1.51083
$870$	0	0
$871$	7.41500e106	0.362617
$872$	−2.66339e106	−0.124906
$873$	0	0
$874$	−9.02180e105	−0.0389158
$875$	−8.35969e106	−0.345857
$876$	0	0
$877$	−7.50757e106	−0.285768	−0.142884	−	0.989739i	$-0.545638\pi$
$877$	−7.50757e106	−0.285768	−0.142884	+	0.989739i	$0.545638\pi$
$878$	−1.03769e106	−0.0378896
$879$	0	0
$880$	−2.18932e107	−0.735686
$881$	1.46469e107	0.472199	0.236099	−	0.971729i	$-0.424131\pi$
$881$	1.46469e107	0.472199	0.236099	+	0.971729i	$0.424131\pi$
$882$	0	0
$883$	6.01925e107	1.78639	0.893196	−	0.449667i	$-0.148457\pi$
$883$	6.01925e107	1.78639	0.893196	+	0.449667i	$0.148457\pi$
$884$	4.27921e107	1.21859
$885$	0	0
$886$	4.53568e105	0.0118935
$887$	−5.72253e107	−1.44004	−0.720022	−	0.693951i	$-0.755868\pi$
$887$	−5.72253e107	−1.44004	−0.720022	+	0.693951i	$0.755868\pi$
$888$	0	0
$889$	−8.39095e106	−0.194490
$890$	1.65376e106	0.0367906
$891$	0	0
$892$	4.14497e107	0.849579
$893$	−3.74732e105	−0.00737297
$894$	0	0
$895$	1.12021e107	0.203123
$896$	5.32487e106	0.0926972
$897$	0	0
$898$	1.40847e106	0.0226027
$899$	5.88848e107	0.907349
$900$	0	0
$901$	2.24726e107	0.319299
$902$	5.31062e106	0.0724613
$903$	0	0
$904$	8.43446e106	0.106147
$905$	4.18846e107	0.506269
$906$	0	0
$907$	4.68331e107	0.522261	0.261130	−	0.965304i	$-0.415905\pi$
$907$	4.68331e107	0.522261	0.261130	+	0.965304i	$0.415905\pi$
$908$	2.02034e107	0.216417
$909$	0	0
$910$	−9.02906e105	−0.00892548
$911$	1.35405e108	1.28592	0.642959	−	0.765901i	$-0.277707\pi$
$911$	1.35405e108	1.28592	0.642959	+	0.765901i	$0.277707\pi$
$912$	0	0
$913$	−1.71763e108	−1.50572
$914$	6.41408e106	0.0540254
$915$	0	0
$916$	−1.47989e108	−1.15092
$917$	−5.76222e107	−0.430635
$918$	0	0
$919$	1.02309e108	0.706148	0.353074	−	0.935595i	$-0.385136\pi$
$919$	1.02309e108	0.706148	0.353074	+	0.935595i	$0.385136\pi$
$920$	−3.91189e106	−0.0259496
$921$	0	0
$922$	1.33445e107	0.0817757
$923$	−3.62311e107	−0.213412
$924$	0	0
$925$	2.48990e108	1.35522
$926$	6.25194e106	0.0327125
$927$	0	0
$928$	5.19171e107	0.251078
$929$	−4.67277e107	−0.217271	−0.108635	−	0.994082i	$-0.534648\pi$
$929$	−4.67277e107	−0.217271	−0.108635	+	0.994082i	$0.534648\pi$
$930$	0	0
$931$	−3.08265e108	−1.32514
$932$	6.65333e107	0.275016
$933$	0	0
$934$	−1.28917e107	−0.0492777
$935$	−3.37006e108	−1.23884
$936$	0	0
$937$	−1.30332e108	−0.443156	−0.221578	−	0.975143i	$-0.571121\pi$
$937$	−1.30332e108	−0.443156	−0.221578	+	0.975143i	$0.571121\pi$
$938$	3.53105e106	0.0115478
$939$	0	0
$940$	−8.10923e105	−0.00245365
$941$	−3.96389e108	−1.15372	−0.576860	−	0.816843i	$-0.695722\pi$
$941$	−3.96389e108	−1.15372	−0.576860	+	0.816843i	$0.695722\pi$
$942$	0	0
$943$	−1.27209e108	−0.342641
$944$	−1.39068e108	−0.360370
$945$	0	0
$946$	1.80298e107	0.0432478
$947$	−7.26911e108	−1.67767	−0.838835	−	0.544385i	$-0.816763\pi$
$947$	−7.26911e108	−1.67767	−0.838835	+	0.544385i	$0.816763\pi$
$948$	0	0
$949$	−7.22473e107	−0.154385
$950$	−3.35695e107	−0.0690293
$951$	0	0
$952$	4.08311e107	0.0777575
$953$	9.62132e108	1.76337	0.881685	−	0.471838i	$-0.156409\pi$
$953$	9.62132e108	1.76337	0.881685	+	0.471838i	$0.156409\pi$
$954$	0	0
$955$	5.46573e108	0.927952
$956$	−7.17698e108	−1.17281
$957$	0	0
$958$	1.15803e107	0.0175340
$959$	−2.80834e108	−0.409328
$960$	0	0
$961$	−4.21922e108	−0.569940
$962$	6.37283e107	0.0828786
$963$	0	0
$964$	−3.51618e108	−0.423896
$965$	6.43407e108	0.746859
$966$	0	0
$967$	−1.54885e108	−0.166703	−0.0833513	−	0.996520i	$-0.526562\pi$
$967$	−1.54885e108	−0.166703	−0.0833513	+	0.996520i	$0.526562\pi$
$968$	−1.22966e108	−0.127449
$969$	0	0
$970$	9.73299e106	0.00935573
$971$	5.52113e108	0.511124	0.255562	−	0.966793i	$-0.417740\pi$
$971$	5.52113e108	0.511124	0.255562	+	0.966793i	$0.417740\pi$
$972$	0	0
$973$	4.01733e108	0.345000
$974$	−2.77769e107	−0.0229764
$975$	0	0
$976$	3.26800e108	0.250821
$977$	−2.30000e108	−0.170050	−0.0850250	−	0.996379i	$-0.527097\pi$
$977$	−2.30000e108	−0.170050	−0.0850250	+	0.996379i	$0.527097\pi$
$978$	0	0
$979$	−2.43086e109	−1.66799
$980$	−6.67087e108	−0.440992
$981$	0	0
$982$	5.14781e107	0.0315903
$983$	8.64281e108	0.511036	0.255518	−	0.966804i	$-0.417754\pi$
$983$	8.64281e108	0.511036	0.255518	+	0.966804i	$0.417754\pi$
$984$	0	0
$985$	−8.66122e105	−0.000475507 0
$986$	2.64743e108	0.140061
$987$	0	0
$988$	2.31657e109	1.13820
$989$	−4.31879e108	−0.204502
$990$	0	0
$991$	−1.02125e109	−0.449202	−0.224601	−	0.974451i	$-0.572108\pi$
$991$	−1.02125e109	−0.449202	−0.224601	+	0.974451i	$0.572108\pi$
$992$	2.80699e108	0.119005
$993$	0	0
$994$	−1.72534e107	−0.00679627
$995$	4.68960e108	0.178071
$996$	0	0
$997$	4.58848e109	1.61918	0.809592	−	0.586994i	$-0.199689\pi$
$997$	4.58848e109	1.61918	0.809592	+	0.586994i	$0.199689\pi$
$998$	−2.84934e108	−0.0969348
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9.74.a.a.1.3		5
3.2	odd	2		1.74.a.a.1.3	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.74.a.a.1.3	✓	5	3.2	odd	2
9.74.a.a.1.3		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:	\( k \)	\(=\)	\( 74 \)
Character orbit:	\([\chi]\)	\(=\)	9.a (trivial)

\(f(q)\)	\(=\)	\(q+5.90767e9 q^{2} -9.40983e21 q^{4} -1.69056e25 q^{5} -2.69951e30 q^{7} -1.11387e32 q^{8} +O(q^{10})\)	\(q+5.90767e9 q^{2} -9.40983e21 q^{4} -1.69056e25 q^{5} -2.69951e30 q^{7} -1.11387e32 q^{8} -9.98727e34 q^{10} +1.46803e38 q^{11} -3.34896e40 q^{13} -1.59478e40 q^{14} +8.82153e43 q^{16} +1.35791e45 q^{17} +7.35113e46 q^{19} +1.59079e47 q^{20} +8.67264e47 q^{22} -2.07741e49 q^{23} -7.72992e50 q^{25} -1.97845e50 q^{26} +2.54020e52 q^{28} +3.30017e53 q^{29} +1.78429e54 q^{31} +1.57316e54 q^{32} +8.02211e54 q^{34} +4.56369e55 q^{35} -3.22112e57 q^{37} +4.34280e56 q^{38} +1.88306e57 q^{40} +6.12341e58 q^{41} +2.07892e59 q^{43} -1.38139e60 q^{44} -1.22727e59 q^{46} -5.09761e58 q^{47} -4.19344e61 q^{49} -4.56658e60 q^{50} +3.15131e62 q^{52} +1.65493e62 q^{53} -2.48179e63 q^{55} +3.00690e62 q^{56} +1.94963e63 q^{58} -1.57646e64 q^{59} +3.70457e64 q^{61} +1.05410e64 q^{62} -8.23876e65 q^{64} +5.66162e65 q^{65} -2.21412e66 q^{67} -1.27777e67 q^{68} +2.69608e65 q^{70} +1.08186e67 q^{71} +2.15731e67 q^{73} -1.90293e67 q^{74} -6.91729e68 q^{76} -3.96297e68 q^{77} +1.93510e69 q^{79} -1.49133e69 q^{80} +3.61751e68 q^{82} -1.17002e70 q^{83} -2.29564e70 q^{85} +1.22816e69 q^{86} -1.63519e70 q^{88} -1.65587e71 q^{89} +9.04056e70 q^{91} +1.95481e71 q^{92} -3.01150e68 q^{94} -1.24275e72 q^{95} -9.74539e71 q^{97} -2.47734e71 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 92089333488 q^{2} + 89\!\cdots\!60 q^{4}+ \cdots + 38\!\cdots\!80 q^{8}+O(q^{10}) \) 5 * q + 92089333488 * q^2 + 891110479259563516160 * q^4 - 23099529720469471562826750 * q^5 - 4356178487147579439806268090008 * q^7 + 382321348565178801062509850849280 * q^8	\( 5 q + 92089333488 q^{2} + 89\!\cdots\!60 q^{4}+ \cdots + 76\!\cdots\!16 q^{98}+O(q^{100}) \) 5 * q + 92089333488 * q^2 + 891110479259563516160 * q^4 - 23099529720469471562826750 * q^5 - 4356178487147579439806268090008 * q^7 + 382321348565178801062509850849280 * q^8 - 1082726120403847340426026649690244000 * q^10 - 50083756421782317855880849016778496860 * q^11 + 4753748397471720291047553799357192278886 * q^13 - 263363595637460329420141722180437171694720 * q^14 + 57509595349357883169785087518099233197588480 * q^16 - 663279725175909884624200257401822351394256602 * q^17 + 31387402430958534058855312636541530201995765700 * q^19 - 68535205735011202400715856751699460073169856000 * q^20 - 9492394168845783423791120540922405517833204177216 * q^22 + 41173446223173693726617088414497261153026772718024 * q^23 + 321010940201155914585034686800814006311220324546875 * q^25 - 4496438035379975943261998606627087983483848085006560 * q^26 - 37814301174111882879202704910382387341459683517577216 * q^28 + 217397685843681573854388291537651452827057882148450250 * q^29 - 3959918978904396097569386510196715694840286709585401440 * q^31 + 9479144378484593489541015426974249650207814007382867968 * q^32 - 324638192385754362369074018224584035205208569018875155680 * q^34 + 1051811671165262926972759981500117019527196204743409914000 * q^35 - 6710426612911460903226679780416929578822322356442579950178 * q^37 + 20075776649866586500887218371135359952534984707306414947520 * q^38 - 87443314425210154156114784714128373503740284779633935360000 * q^40 + 89999556000464227434562342749528590851939245788450434103790 * q^41 + 1171421579861687300375905115748875459141635816468653452284156 * q^43 - 3620387401482413711753562750789250834104238410455247119723520 * q^44 + 13122337792181100518478081314861771936057558300377044573644160 * q^46 - 26407117391117866585989236437476333636637017745173062895096432 * q^47 - 47859785088079990485612756297590896342589260563105328803969315 * q^49 + 198341627273982740927873511630518057650356858808871192258250000 * q^50 - 1819820853930882517149424007699825989256997486995920851867734528 * q^52 + 2253672757277210109476755074185818948452790752453662372858468754 * q^53 + 5272301841223840031246441919708126997046467528986443289345081000 * q^55 - 17369101267271748189623122493455314696373137673860233497989120000 * q^56 + 63080485469034924979732075223792805715626817023343777138220605920 * q^58 - 49827032054541030737892137385762950528020446432538215007741437900 * q^59 - 200207946356755779993110124671040230418699916707275462827300034090 * q^61 + 456512716932753516995161459627410738703514050956609499700325060096 * q^62 - 2671988611100492125931167910213768824231319631401339943695669002240 * q^64 + 2221078323698894670419341126197735507928898751783246157619800369500 * q^65 + 1726934463658866670044856129309631728944059099751189118466533017652 * q^67 - 25844159932565717701710366021885253281928511625890180327915132834304 * q^68 + 60019715162941233136874683443010604187189046779871920798680662752000 * q^70 - 29528228383509762365399718501357560736875772117852859701209862331560 * q^71 - 237248424072934328617284197336037660641948064767277176027665176375374 * q^73 + 385668396032745357868624366253781435784895053179379214963132226802080 * q^74 - 402469771116557587325960702574461860644965601604695023105081798528000 * q^76 + 1160921420345569675442649510235573702285949327395662717704239852301856 * q^77 + 1210019475236668608258558597507004925943189831192425236151129201347600 * q^79 - 7695510793835037324754830224890315360926193938062866122926593916928000 * q^80 + 19869905655996015892436375229312420723635630923691072274009372051144864 * q^82 - 10633758535693793129462935403233565634399957806616398467042339222483716 * q^83 - 28641392301460491393930553180957463692521548910270676165772874742366500 * q^85 + 130748824105220136322401247193488038601024385677355691179590030869522240 * q^86 - 221808067516055560593893382793377773024661552295403957235825845052456960 * q^88 + 44230399340933247556508628049428956972940692248797881370004235332026750 * q^89 + 503963419362898329804217905595235878617892804641236823955454638331741360 * q^91 - 1056276182251025158869277711594662138415929570173882076378175681365399552 * q^92 + 1211599009021421323842808603980852406277919665987296133682388929757489920 * q^94 - 1134809355692277446299742482002998127744674957590385906379174039024255000 * q^95 - 4705643796144250410051227741367571092615369303801441364082016351450032918 * q^97 + 7645702282951838206479981702299956043322685777549671658293752892775157616 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more