LMFDB - Embedded newform 81.22.a.c.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [81,22,Mod(1,81)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(81, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 22, names="a")

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

chi := DirichletCharacter("81.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 81 = 3^{4} $
Weight:	$ k $	$=$	$ 22 $
Character orbit:	$[\chi]$	$=$	81.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:	$226.376648873$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 3 x^{19} - 30906825 x^{18} + 1599806295 x^{17} + 397632537600480 x^{16} + \cdots + 18\!\cdots\!00 $ x^20 - 3x^19 - 30906825x^18 + 1599806295x^17 + 397632537600480x^16 - 39002822306934732x^15 - 2767501910687405017140x^14 + 391326176564192242210572x^13 + 11337389828314088384663968014x^12 - 2062014444300295623078414274842x^11 - 27853618399037107366577090329328670x^10 + 6152815999004060192586598873467670626x^9 + 39977296093180250047956449416351511766204x^8 - 10437440555436107068148091511291912219002012x^7 - 30979178049576561502756083124979860960535622468x^6 + 9412443148812104482026646046738359353628722883452x^5 + 10839091544581780684520015865967041645826136352388761x^4 - 3593653765955001665256902255875226615482416031331126139x^3 - 1035173030260593239029556168369420911667390769758792727969x^2 + 245833151533526228904784003201706226905806706020142118380223*x + 18500665156024978040775113037246170660034380741167330211264700
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	multiple of $ 2^{56}\cdot 3^{135}\cdot 5^{4}\cdot 7^{6} $
Twist minimal:	no (minimal twist has level 9)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1801.20$ of defining polynomial
Character	$\chi$	$=$	81.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1852.20 q^{2} +1.33350e6 q^{4} -2.91973e7 q^{5} +2.00050e8 q^{7} +1.41443e9 q^{8} +O(q^{10})$	\(q-1852.20 q^{2} +1.33350e6 q^{4} -2.91973e7 q^{5} +2.00050e8 q^{7} +1.41443e9 q^{8} +5.40794e10 q^{10} -1.09054e11 q^{11} +8.00012e11 q^{13} -3.70533e11 q^{14} -5.41637e12 q^{16} +8.78259e12 q^{17} -1.48462e13 q^{19} -3.89347e13 q^{20} +2.01989e14 q^{22} -3.24250e14 q^{23} +3.75647e14 q^{25} -1.48179e15 q^{26} +2.66768e14 q^{28} -2.05722e15 q^{29} +1.38566e15 q^{31} +7.06595e15 q^{32} -1.62671e16 q^{34} -5.84093e15 q^{35} +9.14684e15 q^{37} +2.74982e16 q^{38} -4.12976e16 q^{40} -5.49761e16 q^{41} +3.84581e16 q^{43} -1.45423e17 q^{44} +6.00576e17 q^{46} +5.87517e17 q^{47} -5.18526e17 q^{49} -6.95774e17 q^{50} +1.06682e18 q^{52} -5.47460e17 q^{53} +3.18408e18 q^{55} +2.82957e17 q^{56} +3.81040e18 q^{58} +5.55778e16 q^{59} +1.71645e18 q^{61} -2.56652e18 q^{62} -1.72860e18 q^{64} -2.33582e19 q^{65} +2.59192e16 q^{67} +1.17116e19 q^{68} +1.08186e19 q^{70} +1.92819e19 q^{71} -4.51545e19 q^{73} -1.69418e19 q^{74} -1.97975e19 q^{76} -2.18162e19 q^{77} +1.06501e20 q^{79} +1.58144e20 q^{80} +1.01827e20 q^{82} -2.32315e20 q^{83} -2.56428e20 q^{85} -7.12321e19 q^{86} -1.54249e20 q^{88} -4.71457e20 q^{89} +1.60043e20 q^{91} -4.32388e20 q^{92} -1.08820e21 q^{94} +4.33469e20 q^{95} +6.71658e20 q^{97} +9.60415e20 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 1023 q^{2} + 19922945 q^{4} - 32234853 q^{5} + 189623959 q^{7} - 648135831 q^{8}+O(q^{10}) $ 20 * q - 1023 * q^2 + 19922945 * q^4 - 32234853 * q^5 + 189623959 * q^7 - 648135831 * q^8	$ 20 q - 1023 q^{2} + 19922945 q^{4} - 32234853 q^{5} + 189623959 q^{7} - 648135831 q^{8} + 2097150 q^{10} - 146068576386 q^{11} + 177565977277 q^{13} - 1549677244440 q^{14} + 18691699769345 q^{16} - 9307801874799 q^{17} - 4884366861977 q^{19} - 76202257650204 q^{20} - 86758343554047 q^{22} - 356460494884095 q^{23} + 13\!\cdots\!29 q^{25}+ \cdots + 26\!\cdots\!43 q^{98}+O(q^{100}) $ 20 * q - 1023 * q^2 + 19922945 * q^4 - 32234853 * q^5 + 189623959 * q^7 - 648135831 * q^8 + 2097150 * q^10 - 146068576386 * q^11 + 177565977277 * q^13 - 1549677244440 * q^14 + 18691699769345 * q^16 - 9307801874799 * q^17 - 4884366861977 * q^19 - 76202257650204 * q^20 - 86758343554047 * q^22 - 356460494884095 * q^23 + 1379841857082329 * q^25 + 2254540119546732 * q^26 + 799738574143486 * q^28 + 2772828270755157 * q^29 - 3064842508546901 * q^31 - 9763241440878495 * q^32 - 5532390311666553 * q^34 - 28944789317551797 * q^35 + 15041349243939694 * q^37 - 38173338977409867 * q^38 + 43546396391926176 * q^40 - 4985514195953106 * q^41 + 41986108659289660 * q^43 + 154909775468547021 * q^44 - 370501619585816100 * q^46 - 971751895603333665 * q^47 + 1393736132530446381 * q^49 - 1504393427506051923 * q^50 + 892509168437820382 * q^52 - 3200468347464244650 * q^53 + 1763329427612189115 * q^55 - 9313401575284840590 * q^56 + 985520139125952432 * q^58 - 11335149505478767926 * q^59 + 333769931007960067 * q^61 - 30822195887206546926 * q^62 + 12804600562751272961 * q^64 - 24815104795645584999 * q^65 - 14325054629523996728 * q^67 - 98193137232148637751 * q^68 + 23664205955002266174 * q^70 - 108719113548124731960 * q^71 - 2415544298256170813 * q^73 - 284989967381657148240 * q^74 - 45502930505046158765 * q^76 - 209785238565183158133 * q^77 - 45386959469213650259 * q^79 - 499015504255192380624 * q^80 + 114258090814276483077 * q^82 - 224083700347690062915 * q^83 + 135764416394391341394 * q^85 + 210501162447733500423 * q^86 - 409557505702705259691 * q^88 + 140942904695660466360 * q^89 - 193584504929412672709 * q^91 - 160981359902419031922 * q^92 - 1246412695298803268064 * q^94 - 583111881346045719228 * q^95 + 658984791485125488346 * q^97 + 2671316103447781667343 * 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 $

$2$	−1852.20	−1.27901	−0.639505	−	0.768787i	$-0.720860\pi$
$2$	−1852.20	−1.27901	−0.639505	+	0.768787i	$0.720860\pi$
$3$	0	0
$3$	0	0
$4$	1.33350e6	0.635864
$5$	−2.91973e7	−1.33708	−0.668541	−	0.743675i	$-0.733081\pi$
$5$	−2.91973e7	−1.33708	−0.668541	+	0.743675i	$0.733081\pi$
$6$	0	0
$7$	2.00050e8	0.267676	0.133838	−	0.991003i	$-0.457270\pi$
$7$	2.00050e8	0.267676	0.133838	+	0.991003i	$0.457270\pi$
$8$	1.41443e9	0.465733
$9$	0	0
$10$	5.40794e10	1.71014
$11$	−1.09054e11	−1.26770	−0.633851	−	0.773455i	$-0.718527\pi$
$11$	−1.09054e11	−1.26770	−0.633851	+	0.773455i	$0.718527\pi$
$12$	0	0
$13$	8.00012e11	1.60950	0.804751	−	0.593612i	$-0.202299\pi$
$13$	8.00012e11	1.60950	0.804751	+	0.593612i	$0.202299\pi$
$14$	−3.70533e11	−0.342360
$15$	0	0
$16$	−5.41637e12	−1.23154
$17$	8.78259e12	1.05660	0.528298	−	0.849059i	$-0.322830\pi$
$17$	8.78259e12	1.05660	0.528298	+	0.849059i	$0.322830\pi$
$18$	0	0
$19$	−1.48462e13	−0.555524	−0.277762	−	0.960650i	$-0.589593\pi$
$19$	−1.48462e13	−0.555524	−0.277762	+	0.960650i	$0.589593\pi$
$20$	−3.89347e13	−0.850202
$21$	0	0
$22$	2.01989e14	1.62140
$23$	−3.24250e14	−1.63207	−0.816033	−	0.578006i	$-0.803831\pi$
$23$	−3.24250e14	−1.63207	−0.816033	+	0.578006i	$0.803831\pi$
$24$	0	0
$25$	3.75647e14	0.787788
$26$	−1.48179e15	−2.05857
$27$	0	0
$28$	2.66768e14	0.170206
$29$	−2.05722e15	−0.908035	−0.454017	−	0.890993i	$-0.650010\pi$
$29$	−2.05722e15	−0.908035	−0.454017	+	0.890993i	$0.650010\pi$
$30$	0	0
$31$	1.38566e15	0.303639	0.151820	−	0.988408i	$-0.451487\pi$
$31$	1.38566e15	0.303639	0.151820	+	0.988408i	$0.451487\pi$
$32$	7.06595e15	1.10942
$33$	0	0
$34$	−1.62671e16	−1.35140
$35$	−5.84093e15	−0.357905
$36$	0	0
$37$	9.14684e15	0.312718	0.156359	−	0.987700i	$-0.450024\pi$
$37$	9.14684e15	0.312718	0.156359	+	0.987700i	$0.450024\pi$
$38$	2.74982e16	0.710520
$39$	0	0
$40$	−4.12976e16	−0.622723
$41$	−5.49761e16	−0.639652	−0.319826	−	0.947476i	$-0.603624\pi$
$41$	−5.49761e16	−0.639652	−0.319826	+	0.947476i	$0.603624\pi$
$42$	0	0
$43$	3.84581e16	0.271374	0.135687	−	0.990752i	$-0.456676\pi$
$43$	3.84581e16	0.271374	0.135687	+	0.990752i	$0.456676\pi$
$44$	−1.45423e17	−0.806086
$45$	0	0
$46$	6.00576e17	2.08743
$47$	5.87517e17	1.62927	0.814635	−	0.579973i	$-0.196937\pi$
$47$	5.87517e17	1.62927	0.814635	+	0.579973i	$0.196937\pi$
$48$	0	0
$49$	−5.18526e17	−0.928350
$50$	−6.95774e17	−1.00759
$51$	0	0
$52$	1.06682e18	1.02342
$53$	−5.47460e17	−0.429988	−0.214994	−	0.976615i	$-0.568973\pi$
$53$	−5.47460e17	−0.429988	−0.214994	+	0.976615i	$0.568973\pi$
$54$	0	0
$55$	3.18408e18	1.69502
$56$	2.82957e17	0.124666
$57$	0	0
$58$	3.81040e18	1.16138
$59$	5.55778e16	0.0141565	0.00707825	−	0.999975i	$-0.497747\pi$
$59$	5.55778e16	0.0141565	0.00707825	+	0.999975i	$0.497747\pi$
$60$	0	0
$61$	1.71645e18	0.308083	0.154041	−	0.988064i	$-0.450771\pi$
$61$	1.71645e18	0.308083	0.154041	+	0.988064i	$0.450771\pi$
$62$	−2.56652e18	−0.388358
$63$	0	0
$64$	−1.72860e18	−0.187416
$65$	−2.33582e19	−2.15204
$66$	0	0
$67$	2.59192e16	0.00173714	0.000868572	−	1.00000i	$-0.499724\pi$
$67$	2.59192e16	0.00173714	0.000868572		1.00000i	$0.499724\pi$
$68$	1.17116e19	0.671851
$69$	0	0
$70$	1.08186e19	0.457763
$71$	1.92819e19	0.702971	0.351485	−	0.936193i	$-0.385677\pi$
$71$	1.92819e19	0.702971	0.351485	+	0.936193i	$0.385677\pi$
$72$	0	0
$73$	−4.51545e19	−1.22973	−0.614867	−	0.788631i	$-0.710790\pi$
$73$	−4.51545e19	−1.22973	−0.614867	+	0.788631i	$0.710790\pi$
$74$	−1.69418e19	−0.399969
$75$	0	0
$76$	−1.97975e19	−0.353238
$77$	−2.18162e19	−0.339333
$78$	0	0
$79$	1.06501e20	1.26552	0.632760	−	0.774348i	$-0.281922\pi$
$79$	1.06501e20	1.26552	0.632760	+	0.774348i	$0.281922\pi$
$80$	1.58144e20	1.64667
$81$	0	0
$82$	1.01827e20	0.818121
$83$	−2.32315e20	−1.64345	−0.821727	−	0.569882i	$-0.806989\pi$
$83$	−2.32315e20	−1.64345	−0.821727	+	0.569882i	$0.806989\pi$
$84$	0	0
$85$	−2.56428e20	−1.41276
$86$	−7.12321e19	−0.347090
$87$	0	0
$88$	−1.54249e20	−0.590411
$89$	−4.71457e20	−1.60268	−0.801340	−	0.598210i	$-0.795879\pi$
$89$	−4.71457e20	−1.60268	−0.801340	+	0.598210i	$0.795879\pi$
$90$	0	0
$91$	1.60043e20	0.430825
$92$	−4.32388e20	−1.03777
$93$	0	0
$94$	−1.08820e21	−2.08385
$95$	4.33469e20	0.742781
$96$	0	0
$97$	6.71658e20	0.924794	0.462397	−	0.886673i	$-0.346989\pi$
$97$	6.71658e20	0.924794	0.462397	+	0.886673i	$0.346989\pi$
$98$	9.60415e20	1.18737
$99$	0	0
$100$	5.00926e20	0.500926
$101$	1.75309e21	1.57917	0.789585	−	0.613641i	$-0.210296\pi$
$101$	1.75309e21	1.57917	0.789585	+	0.613641i	$0.210296\pi$
$102$	0	0
$103$	1.12370e21	0.823868	0.411934	−	0.911214i	$-0.364853\pi$
$103$	1.12370e21	0.823868	0.411934	+	0.911214i	$0.364853\pi$
$104$	1.13156e21	0.749599
$105$	0	0
$106$	1.01401e21	0.549958
$107$	3.43408e21	1.68764	0.843822	−	0.536623i	$-0.180300\pi$
$107$	3.43408e21	1.68764	0.843822	+	0.536623i	$0.180300\pi$
$108$	0	0
$109$	3.77657e21	1.52799	0.763994	−	0.645224i	$-0.223236\pi$
$109$	3.77657e21	1.52799	0.763994	+	0.645224i	$0.223236\pi$
$110$	−5.89755e21	−2.16795
$111$	0	0
$112$	−1.08355e21	−0.329654
$113$	2.43915e20	0.0675951	0.0337976	−	0.999429i	$-0.489240\pi$
$113$	2.43915e20	0.0675951	0.0337976	+	0.999429i	$0.489240\pi$
$114$	0	0
$115$	9.46723e21	2.18221
$116$	−2.74332e21	−0.577387
$117$	0	0
$118$	−1.02941e20	−0.0181063
$119$	1.75696e21	0.282825
$120$	0	0
$121$	4.49245e21	0.607067
$122$	−3.17921e21	−0.394041
$123$	0	0
$124$	1.84778e21	0.193073
$125$	2.95449e21	0.283744
$126$	0	0
$127$	−1.66455e21	−0.135318	−0.0676592	−	0.997708i	$-0.521553\pi$
$127$	−1.66455e21	−0.135318	−0.0676592	+	0.997708i	$0.521553\pi$
$128$	−1.16166e22	−0.869712
$129$	0	0
$130$	4.32642e22	2.75247
$131$	4.88409e21	0.286705	0.143353	−	0.989672i	$-0.454212\pi$
$131$	4.88409e21	0.286705	0.143353	+	0.989672i	$0.454212\pi$
$132$	0	0
$133$	−2.96998e21	−0.148700
$134$	−4.80076e19	−0.00222182
$135$	0	0
$136$	1.24224e22	0.492092
$137$	4.39375e22	1.61164	0.805822	−	0.592158i	$-0.201724\pi$
$137$	4.39375e22	1.61164	0.805822	+	0.592158i	$0.201724\pi$
$138$	0	0
$139$	−2.99698e21	−0.0944123	−0.0472062	−	0.998885i	$-0.515032\pi$
$139$	−2.99698e21	−0.0944123	−0.0472062	+	0.998885i	$0.515032\pi$
$140$	−7.78890e21	−0.227579
$141$	0	0
$142$	−3.57140e22	−0.899106
$143$	−8.72443e22	−2.04037
$144$	0	0
$145$	6.00654e22	1.21412
$146$	8.36354e22	1.57284
$147$	0	0
$148$	1.21973e22	0.198846
$149$	−3.88496e22	−0.590108	−0.295054	−	0.955481i	$-0.595338\pi$
$149$	−3.88496e22	−0.590108	−0.295054	+	0.955481i	$0.595338\pi$
$150$	0	0
$151$	1.14855e23	1.51667	0.758335	−	0.651866i	$-0.226013\pi$
$151$	1.14855e23	1.51667	0.758335	+	0.651866i	$0.226013\pi$
$152$	−2.09989e22	−0.258726
$153$	0	0
$154$	4.04080e22	0.434010
$155$	−4.04575e22	−0.405991
$156$	0	0
$157$	5.88793e22	0.516436	0.258218	−	0.966087i	$-0.416865\pi$
$157$	5.88793e22	0.516436	0.258218	+	0.966087i	$0.416865\pi$
$158$	−1.97261e23	−1.61861
$159$	0	0
$160$	−2.06307e23	−1.48338
$161$	−6.48662e22	−0.436865
$162$	0	0
$163$	−9.94689e22	−0.588460	−0.294230	−	0.955735i	$-0.595063\pi$
$163$	−9.94689e22	−0.588460	−0.294230	+	0.955735i	$0.595063\pi$
$164$	−7.33109e22	−0.406732
$165$	0	0
$166$	4.30295e23	2.10199
$167$	2.36763e23	1.08590	0.542952	−	0.839764i	$-0.317306\pi$
$167$	2.36763e23	1.08590	0.542952	+	0.839764i	$0.317306\pi$
$168$	0	0
$169$	3.92955e23	1.59050
$170$	4.74957e23	1.80693
$171$	0	0
$172$	5.12840e22	0.172557
$173$	−1.24941e23	−0.395567	−0.197783	−	0.980246i	$-0.563374\pi$
$173$	−1.24941e23	−0.395567	−0.197783	+	0.980246i	$0.563374\pi$
$174$	0	0
$175$	7.51482e22	0.210872
$176$	5.90675e23	1.56123
$177$	0	0
$178$	8.73233e23	2.04984
$179$	−4.10576e23	−0.908734	−0.454367	−	0.890815i	$-0.650134\pi$
$179$	−4.10576e23	−0.908734	−0.454367	+	0.890815i	$0.650134\pi$
$180$	0	0
$181$	6.48730e23	1.27773	0.638865	−	0.769319i	$-0.279404\pi$
$181$	6.48730e23	1.27773	0.638865	+	0.769319i	$0.279404\pi$
$182$	−2.96431e23	−0.551029
$183$	0	0
$184$	−4.58629e23	−0.760107
$185$	−2.67063e23	−0.418130
$186$	0	0
$187$	−9.57774e23	−1.33945
$188$	7.83457e23	1.03599
$189$	0	0
$190$	−8.02873e23	−0.950023
$191$	−9.81019e23	−1.09857	−0.549284	−	0.835635i	$-0.685100\pi$
$191$	−9.81019e23	−1.09857	−0.549284	+	0.835635i	$0.685100\pi$
$192$	0	0
$193$	−1.57239e24	−1.57836	−0.789182	−	0.614159i	$-0.789495\pi$
$193$	−1.57239e24	−1.57836	−0.789182	+	0.614159i	$0.789495\pi$
$194$	−1.24405e24	−1.18282
$195$	0	0
$196$	−6.91456e23	−0.590304
$197$	−8.48520e22	−0.0686700	−0.0343350	−	0.999410i	$-0.510931\pi$
$197$	−8.48520e22	−0.0686700	−0.0343350	+	0.999410i	$0.510931\pi$
$198$	0	0
$199$	1.56650e24	1.14018	0.570088	−	0.821584i	$-0.306909\pi$
$199$	1.56650e24	1.14018	0.570088	+	0.821584i	$0.306909\pi$
$200$	5.31327e23	0.366899
$201$	0	0
$202$	−3.24707e24	−2.01977
$203$	−4.11548e23	−0.243059
$204$	0	0
$205$	1.60516e24	0.855267
$206$	−2.08131e24	−1.05373
$207$	0	0
$208$	−4.33317e24	−1.98217
$209$	1.61903e24	0.704238
$210$	0	0
$211$	−1.87409e24	−0.737608	−0.368804	−	0.929507i	$-0.620233\pi$
$211$	−1.87409e24	−0.737608	−0.368804	+	0.929507i	$0.620233\pi$
$212$	−7.30040e23	−0.273414
$213$	0	0
$214$	−6.36061e24	−2.15851
$215$	−1.12287e24	−0.362850
$216$	0	0
$217$	2.77201e23	0.0812770
$218$	−6.99498e24	−1.95431
$219$	0	0
$220$	4.24598e24	1.07780
$221$	7.02618e24	1.70059
$222$	0	0
$223$	9.91574e23	0.218335	0.109168	−	0.994023i	$-0.465181\pi$
$223$	9.91574e23	0.218335	0.109168	+	0.994023i	$0.465181\pi$
$224$	1.41354e24	0.296965
$225$	0	0
$226$	−4.51781e23	−0.0864548
$227$	−2.74891e24	−0.502214	−0.251107	−	0.967959i	$-0.580795\pi$
$227$	−2.74891e24	−0.502214	−0.251107	+	0.967959i	$0.580795\pi$
$228$	0	0
$229$	9.31324e24	1.55177	0.775887	−	0.630872i	$-0.217303\pi$
$229$	9.31324e24	1.55177	0.775887	+	0.630872i	$0.217303\pi$
$230$	−1.75352e25	−2.79106
$231$	0	0
$232$	−2.90980e24	−0.422902
$233$	−8.67981e24	−1.20579	−0.602897	−	0.797819i	$-0.705987\pi$
$233$	−8.67981e24	−1.20579	−0.602897	+	0.797819i	$0.705987\pi$
$234$	0	0
$235$	−1.71539e25	−2.17847
$236$	7.41133e22	0.00900160
$237$	0	0
$238$	−3.25424e24	−0.361736
$239$	−4.50025e24	−0.478695	−0.239347	−	0.970934i	$-0.576933\pi$
$239$	−4.50025e24	−0.478695	−0.239347	+	0.970934i	$0.576933\pi$
$240$	0	0
$241$	−1.17659e23	−0.0114669	−0.00573343	−	0.999984i	$-0.501825\pi$
$241$	−1.17659e23	−0.0114669	−0.00573343	+	0.999984i	$0.501825\pi$
$242$	−8.32093e24	−0.776444
$243$	0	0
$244$	2.28889e24	0.195899
$245$	1.51396e25	1.24128
$246$	0	0
$247$	−1.18771e25	−0.894117
$248$	1.95992e24	0.141415
$249$	0	0
$250$	−5.47231e24	−0.362912
$251$	1.70983e24	0.108738	0.0543688	−	0.998521i	$-0.482685\pi$
$251$	1.70983e24	0.108738	0.0543688	+	0.998521i	$0.482685\pi$
$252$	0	0
$253$	3.53606e25	2.06897
$254$	3.08308e24	0.173073
$255$	0	0
$256$	2.51415e25	1.29979
$257$	−1.41747e25	−0.703424	−0.351712	−	0.936108i	$-0.614400\pi$
$257$	−1.41747e25	−0.703424	−0.351712	+	0.936108i	$0.614400\pi$
$258$	0	0
$259$	1.82983e24	0.0837071
$260$	−3.11483e25	−1.36840
$261$	0	0
$262$	−9.04633e24	−0.366698
$263$	−1.13381e25	−0.441574	−0.220787	−	0.975322i	$-0.570863\pi$
$263$	−1.13381e25	−0.441574	−0.220787	+	0.975322i	$0.570863\pi$
$264$	0	0
$265$	1.59844e25	0.574929
$266$	5.50101e24	0.190189
$267$	0	0
$268$	3.45633e22	0.00110459
$269$	−1.04993e25	−0.322671	−0.161336	−	0.986900i	$-0.551580\pi$
$269$	−1.04993e25	−0.322671	−0.161336	+	0.986900i	$0.551580\pi$
$270$	0	0
$271$	1.34053e25	0.381152	0.190576	−	0.981672i	$-0.438964\pi$
$271$	1.34053e25	0.381152	0.190576	+	0.981672i	$0.438964\pi$
$272$	−4.75698e25	−1.30124
$273$	0	0
$274$	−8.13811e25	−2.06131
$275$	−4.09657e25	−0.998681
$276$	0	0
$277$	−1.66932e25	−0.377140	−0.188570	−	0.982060i	$-0.560385\pi$
$277$	−1.66932e25	−0.377140	−0.188570	+	0.982060i	$0.560385\pi$
$278$	5.55102e24	0.120754
$279$	0	0
$280$	−8.26160e24	−0.166688
$281$	−2.10088e25	−0.408304	−0.204152	−	0.978939i	$-0.565444\pi$
$281$	−2.10088e25	−0.408304	−0.204152	+	0.978939i	$0.565444\pi$
$282$	0	0
$283$	8.09632e25	1.46060	0.730298	−	0.683129i	$-0.239381\pi$
$283$	8.09632e25	1.46060	0.730298	+	0.683129i	$0.239381\pi$
$284$	2.57125e25	0.446994
$285$	0	0
$286$	1.61594e26	2.60965
$287$	−1.09980e25	−0.171219
$288$	0	0
$289$	8.04198e24	0.116395
$290$	−1.11253e26	−1.55287
$291$	0	0
$292$	−6.02137e25	−0.781944
$293$	−6.06612e25	−0.759978	−0.379989	−	0.924991i	$-0.624072\pi$
$293$	−6.06612e25	−0.759978	−0.379989	+	0.924991i	$0.624072\pi$
$294$	0	0
$295$	−1.62272e24	−0.0189284
$296$	1.29376e25	0.145643
$297$	0	0
$298$	7.19574e25	0.754753
$299$	−2.59404e26	−2.62681
$300$	0	0
$301$	7.69354e24	0.0726404
$302$	−2.12734e26	−1.93983
$303$	0	0
$304$	8.04126e25	0.684150
$305$	−5.01157e25	−0.411932
$306$	0	0
$307$	−3.77222e25	−0.289497	−0.144748	−	0.989468i	$-0.546237\pi$
$307$	−3.77222e25	−0.289497	−0.144748	+	0.989468i	$0.546237\pi$
$308$	−2.90920e25	−0.215770
$309$	0	0
$310$	7.49355e25	0.519266
$311$	9.23482e25	0.618649	0.309325	−	0.950956i	$-0.399897\pi$
$311$	9.23482e25	0.618649	0.309325	+	0.950956i	$0.399897\pi$
$312$	0	0
$313$	−1.65526e25	−0.103669	−0.0518346	−	0.998656i	$-0.516507\pi$
$313$	−1.65526e25	−0.103669	−0.0518346	+	0.998656i	$0.516507\pi$
$314$	−1.09056e26	−0.660526
$315$	0	0
$316$	1.42019e26	0.804699
$317$	2.02490e26	1.10990	0.554948	−	0.831885i	$-0.312738\pi$
$317$	2.02490e26	1.10990	0.554948	+	0.831885i	$0.312738\pi$
$318$	0	0
$319$	2.24348e26	1.15112
$320$	5.04706e25	0.250590
$321$	0	0
$322$	1.20145e26	0.558754
$323$	−1.30388e26	−0.586964
$324$	0	0
$325$	3.00522e26	1.26795
$326$	1.84237e26	0.752646
$327$	0	0
$328$	−7.77600e25	−0.297907
$329$	1.17533e26	0.436117
$330$	0	0
$331$	−3.94086e26	−1.37214	−0.686068	−	0.727537i	$-0.740665\pi$
$331$	−3.94086e26	−1.37214	−0.686068	+	0.727537i	$0.740665\pi$
$332$	−3.09793e26	−1.04501
$333$	0	0
$334$	−4.38534e26	−1.38888
$335$	−7.56770e23	−0.00232270
$336$	0	0
$337$	−2.91824e26	−0.841410	−0.420705	−	0.907198i	$-0.638217\pi$
$337$	−2.91824e26	−0.841410	−0.420705	+	0.907198i	$0.638217\pi$
$338$	−7.27833e26	−2.03426
$339$	0	0
$340$	−3.41948e26	−0.898321
$341$	−1.51111e26	−0.384924
$342$	0	0
$343$	−2.15468e26	−0.516173
$344$	5.43963e25	0.126388
$345$	0	0
$346$	2.31415e26	0.505933
$347$	−9.89485e25	−0.209870	−0.104935	−	0.994479i	$-0.533463\pi$
$347$	−9.89485e25	−0.209870	−0.104935	+	0.994479i	$0.533463\pi$
$348$	0	0
$349$	−7.46849e26	−1.49130	−0.745651	−	0.666336i	$-0.767862\pi$
$349$	−7.46849e26	−1.49130	−0.745651	+	0.666336i	$0.767862\pi$
$350$	−1.39190e26	−0.269707
$351$	0	0
$352$	−7.70567e26	−1.40641
$353$	5.91911e26	1.04863	0.524315	−	0.851524i	$-0.324322\pi$
$353$	5.91911e26	1.04863	0.524315	+	0.851524i	$0.324322\pi$
$354$	0	0
$355$	−5.62980e26	−0.939930
$356$	−6.28689e26	−1.01909
$357$	0	0
$358$	7.60470e26	1.16228
$359$	−6.37467e25	−0.0946162	−0.0473081	−	0.998880i	$-0.515064\pi$
$359$	−6.37467e25	−0.0946162	−0.0473081	+	0.998880i	$0.515064\pi$
$360$	0	0
$361$	−4.93800e26	−0.691393
$362$	−1.20158e27	−1.63423
$363$	0	0
$364$	2.13417e26	0.273946
$365$	1.31839e27	1.64426
$366$	0	0
$367$	−1.40377e27	−1.65311	−0.826553	−	0.562858i	$-0.809702\pi$
$367$	−1.40377e27	−1.65311	−0.826553	+	0.562858i	$0.809702\pi$
$368$	1.75626e27	2.00996
$369$	0	0
$370$	4.94655e26	0.534792
$371$	−1.09520e26	−0.115097
$372$	0	0
$373$	−5.10048e26	−0.506604	−0.253302	−	0.967387i	$-0.581517\pi$
$373$	−5.10048e26	−0.506604	−0.253302	+	0.967387i	$0.581517\pi$
$374$	1.77399e27	1.71317
$375$	0	0
$376$	8.31003e26	0.758805
$377$	−1.64580e27	−1.46148
$378$	0	0
$379$	−1.62832e27	−1.36782	−0.683909	−	0.729567i	$-0.739722\pi$
$379$	−1.62832e27	−1.36782	−0.683909	+	0.729567i	$0.739722\pi$
$380$	5.78033e26	0.472308
$381$	0	0
$382$	1.81705e27	1.40508
$383$	−1.24728e27	−0.938374	−0.469187	−	0.883099i	$-0.655453\pi$
$383$	−1.24728e27	−0.938374	−0.469187	+	0.883099i	$0.655453\pi$
$384$	0	0
$385$	6.36975e26	0.453716
$386$	2.91238e27	2.01874
$387$	0	0
$388$	8.95659e26	0.588043
$389$	8.59750e26	0.549416	0.274708	−	0.961528i	$-0.411419\pi$
$389$	8.59750e26	0.549416	0.274708	+	0.961528i	$0.411419\pi$
$390$	0	0
$391$	−2.84775e27	−1.72443
$392$	−7.33419e26	−0.432363
$393$	0	0
$394$	1.57163e26	0.0878295
$395$	−3.10954e27	−1.69210
$396$	0	0
$397$	2.80257e27	1.44629	0.723145	−	0.690696i	$-0.242696\pi$
$397$	2.80257e27	1.44629	0.723145	+	0.690696i	$0.242696\pi$
$398$	−2.90147e27	−1.45829
$399$	0	0
$400$	−2.03464e27	−0.970194
$401$	1.14248e27	0.530678	0.265339	−	0.964155i	$-0.414516\pi$
$401$	1.14248e27	0.530678	0.265339	+	0.964155i	$0.414516\pi$
$402$	0	0
$403$	1.10854e27	0.488708
$404$	2.33775e27	1.00414
$405$	0	0
$406$	7.62270e26	0.310875
$407$	−9.97496e26	−0.396433
$408$	0	0
$409$	2.59685e27	0.980285	0.490142	−	0.871642i	$-0.336945\pi$
$409$	2.59685e27	0.980285	0.490142	+	0.871642i	$0.336945\pi$
$410$	−2.97308e27	−1.09389
$411$	0	0
$412$	1.49845e27	0.523868
$413$	1.11184e25	0.00378935
$414$	0	0
$415$	6.78298e27	2.19743
$416$	5.65284e27	1.78561
$417$	0	0
$418$	−2.99878e27	−0.900727
$419$	5.81060e27	1.70206	0.851028	−	0.525120i	$-0.175979\pi$
$419$	5.81060e27	1.70206	0.851028	+	0.525120i	$0.175979\pi$
$420$	0	0
$421$	3.52401e27	0.981918	0.490959	−	0.871183i	$-0.336647\pi$
$421$	3.52401e27	0.981918	0.490959	+	0.871183i	$0.336647\pi$
$422$	3.47120e27	0.943407
$423$	0	0
$424$	−7.74346e26	−0.200260
$425$	3.29915e27	0.832374
$426$	0	0
$427$	3.43376e26	0.0824663
$428$	4.57936e27	1.07311
$429$	0	0
$430$	2.07979e27	0.464088
$431$	4.48426e27	0.976516	0.488258	−	0.872699i	$-0.337633\pi$
$431$	4.48426e27	0.976516	0.488258	+	0.872699i	$0.337633\pi$
$432$	0	0
$433$	−7.30886e27	−1.51610	−0.758048	−	0.652199i	$-0.773847\pi$
$433$	−7.30886e27	−1.51610	−0.758048	+	0.652199i	$0.773847\pi$
$434$	−5.13432e26	−0.103954
$435$	0	0
$436$	5.03607e27	0.971592
$437$	4.81388e27	0.906651
$438$	0	0
$439$	−8.98749e27	−1.61347	−0.806735	−	0.590914i	$-0.798767\pi$
$439$	−8.98749e27	−1.61347	−0.806735	+	0.590914i	$0.798767\pi$
$440$	4.50366e27	0.789428
$441$	0	0
$442$	−1.30139e28	−2.17507
$443$	4.92817e27	0.804352	0.402176	−	0.915562i	$-0.368254\pi$
$443$	4.92817e27	0.804352	0.402176	+	0.915562i	$0.368254\pi$
$444$	0	0
$445$	1.37653e28	2.14291
$446$	−1.83660e27	−0.279253
$447$	0	0
$448$	−3.45807e26	−0.0501667
$449$	3.59096e27	0.508890	0.254445	−	0.967087i	$-0.418107\pi$
$449$	3.59096e27	0.508890	0.254445	+	0.967087i	$0.418107\pi$
$450$	0	0
$451$	5.99535e27	0.810888
$452$	3.25262e26	0.0429813
$453$	0	0
$454$	5.09154e27	0.642336
$455$	−4.67282e27	−0.576048
$456$	0	0
$457$	−5.32972e27	−0.627457	−0.313729	−	0.949513i	$-0.601578\pi$
$457$	−5.32972e27	−0.627457	−0.313729	+	0.949513i	$0.601578\pi$
$458$	−1.72500e28	−1.98473
$459$	0	0
$460$	1.26246e28	1.38759
$461$	−1.01133e28	−1.08650	−0.543252	−	0.839569i	$-0.682807\pi$
$461$	−1.01133e28	−1.08650	−0.543252	+	0.839569i	$0.682807\pi$
$462$	0	0
$463$	−6.79427e27	−0.697497	−0.348749	−	0.937216i	$-0.613393\pi$
$463$	−6.79427e27	−0.697497	−0.348749	+	0.937216i	$0.613393\pi$
$464$	1.11427e28	1.11828
$465$	0	0
$466$	1.60768e28	1.54222
$467$	8.03175e27	0.753326	0.376663	−	0.926350i	$-0.377071\pi$
$467$	8.03175e27	0.753326	0.376663	+	0.926350i	$0.377071\pi$
$468$	0	0
$469$	5.18513e24	0.000464991 0
$470$	3.17726e28	2.78628
$471$	0	0
$472$	7.86111e25	0.00659315
$473$	−4.19399e27	−0.344022
$474$	0	0
$475$	−5.57693e27	−0.437635
$476$	2.34291e27	0.179838
$477$	0	0
$478$	8.33537e27	0.612255
$479$	1.40711e28	1.01113	0.505564	−	0.862789i	$-0.331284\pi$
$479$	1.40711e28	1.01113	0.505564	+	0.862789i	$0.331284\pi$
$480$	0	0
$481$	7.31759e27	0.503321
$482$	2.17927e26	0.0146662
$483$	0	0
$484$	5.99070e27	0.386012
$485$	−1.96106e28	−1.23653
$486$	0	0
$487$	1.10422e28	0.666810	0.333405	−	0.942784i	$-0.391802\pi$
$487$	1.10422e28	0.666810	0.333405	+	0.942784i	$0.391802\pi$
$488$	2.42780e27	0.143484
$489$	0	0
$490$	−2.80416e28	−1.58761
$491$	−2.32217e28	−1.28688	−0.643440	−	0.765496i	$-0.722493\pi$
$491$	−2.32217e28	−1.28688	−0.643440	+	0.765496i	$0.722493\pi$
$492$	0	0
$493$	−1.80678e28	−0.959426
$494$	2.19989e28	1.14358
$495$	0	0
$496$	−7.50524e27	−0.373944
$497$	3.85735e27	0.188168
$498$	0	0
$499$	1.00636e28	0.470650	0.235325	−	0.971917i	$-0.424385\pi$
$499$	1.00636e28	0.470650	0.235325	+	0.971917i	$0.424385\pi$
$500$	3.93982e27	0.180423
$501$	0	0
$502$	−3.16696e27	−0.139076
$503$	1.53739e28	0.661180	0.330590	−	0.943774i	$-0.392752\pi$
$503$	1.53739e28	0.661180	0.330590	+	0.943774i	$0.392752\pi$
$504$	0	0
$505$	−5.11854e28	−2.11148
$506$	−6.54951e28	−2.64623
$507$	0	0
$508$	−2.21968e27	−0.0860441
$509$	3.53664e28	1.34293	0.671466	−	0.741036i	$-0.265665\pi$
$509$	3.53664e28	1.34293	0.671466	+	0.741036i	$0.265665\pi$
$510$	0	0
$511$	−9.03317e27	−0.329170
$512$	−2.22053e28	−0.792725
$513$	0	0
$514$	2.62545e28	0.899686
$515$	−3.28089e28	−1.10158
$516$	0	0
$517$	−6.40709e28	−2.06543
$518$	−3.38921e27	−0.107062
$519$	0	0
$520$	−3.30386e28	−1.00227
$521$	9.09473e27	0.270392	0.135196	−	0.990819i	$-0.456834\pi$
$521$	9.09473e27	0.270392	0.135196	+	0.990819i	$0.456834\pi$
$522$	0	0
$523$	−4.58369e28	−1.30902	−0.654511	−	0.756052i	$-0.727126\pi$
$523$	−4.58369e28	−1.30902	−0.654511	+	0.756052i	$0.727126\pi$
$524$	6.51296e27	0.182305
$525$	0	0
$526$	2.10004e28	0.564778
$527$	1.21697e28	0.320824
$528$	0	0
$529$	6.56664e28	1.66364
$530$	−2.96063e28	−0.735339
$531$	0	0
$532$	−3.96048e27	−0.0945532
$533$	−4.39816e28	−1.02952
$534$	0	0
$535$	−1.00266e29	−2.25652
$536$	3.66609e25	0.000809045 0
$537$	0	0
$538$	1.94468e28	0.412700
$539$	5.65471e28	1.17687
$540$	0	0
$541$	−5.06673e28	−1.01428	−0.507139	−	0.861864i	$-0.669297\pi$
$541$	−5.06673e28	−1.01428	−0.507139	+	0.861864i	$0.669297\pi$
$542$	−2.48293e28	−0.487497
$543$	0	0
$544$	6.20573e28	1.17221
$545$	−1.10266e29	−2.04304
$546$	0	0
$547$	−6.17248e28	−1.10051	−0.550253	−	0.834998i	$-0.685469\pi$
$547$	−6.17248e28	−1.10051	−0.550253	+	0.834998i	$0.685469\pi$
$548$	5.85908e28	1.02479
$549$	0	0
$550$	7.58767e28	1.27732
$551$	3.05420e28	0.504435
$552$	0	0
$553$	2.13055e28	0.338749
$554$	3.09192e28	0.482365
$555$	0	0
$556$	−3.99649e27	−0.0600334
$557$	−3.48582e28	−0.513837	−0.256919	−	0.966433i	$-0.582707\pi$
$557$	−3.48582e28	−0.513837	−0.256919	+	0.966433i	$0.582707\pi$
$558$	0	0
$559$	3.07669e28	0.436777
$560$	3.16367e28	0.440774
$561$	0	0
$562$	3.89125e28	0.522225
$563$	3.79156e28	0.499435	0.249718	−	0.968319i	$-0.419662\pi$
$563$	3.79156e28	0.499435	0.249718	+	0.968319i	$0.419662\pi$
$564$	0	0
$565$	−7.12167e27	−0.0903802
$566$	−1.49960e29	−1.86812
$567$	0	0
$568$	2.72729e28	0.327397
$569$	1.04572e29	1.23236	0.616178	−	0.787607i	$-0.288680\pi$
$569$	1.04572e29	1.23236	0.616178	+	0.787607i	$0.288680\pi$
$570$	0	0
$571$	−3.48760e28	−0.396138	−0.198069	−	0.980188i	$-0.563467\pi$
$571$	−3.48760e28	−0.396138	−0.198069	+	0.980188i	$0.563467\pi$
$572$	−1.16341e29	−1.29740
$573$	0	0
$574$	2.03705e28	0.218991
$575$	−1.21803e29	−1.28572
$576$	0	0
$577$	6.73102e27	0.0685070	0.0342535	−	0.999413i	$-0.489095\pi$
$577$	6.73102e27	0.0685070	0.0342535	+	0.999413i	$0.489095\pi$
$578$	−1.48954e28	−0.148871
$579$	0	0
$580$	8.00975e28	0.772013
$581$	−4.64747e28	−0.439913
$582$	0	0
$583$	5.97026e28	0.545096
$584$	−6.38680e28	−0.572728
$585$	0	0
$586$	1.12357e29	0.972018
$587$	−1.03612e29	−0.880465	−0.440232	−	0.897884i	$-0.645104\pi$
$587$	−1.03612e29	−0.880465	−0.440232	+	0.897884i	$0.645104\pi$
$588$	0	0
$589$	−2.05718e28	−0.168679
$590$	3.00562e27	0.0242096
$591$	0	0
$592$	−4.95427e28	−0.385125
$593$	−2.46895e29	−1.88555	−0.942775	−	0.333430i	$-0.891794\pi$
$593$	−2.46895e29	−1.88555	−0.942775	+	0.333430i	$0.891794\pi$
$594$	0	0
$595$	−5.12985e28	−0.378161
$596$	−5.18061e28	−0.375228
$597$	0	0
$598$	4.80469e29	3.35972
$599$	−1.21495e29	−0.834787	−0.417393	−	0.908726i	$-0.637056\pi$
$599$	−1.21495e29	−0.834787	−0.417393	+	0.908726i	$0.637056\pi$
$600$	0	0
$601$	−2.94874e28	−0.195639	−0.0978193	−	0.995204i	$-0.531187\pi$
$601$	−2.94874e28	−0.195639	−0.0978193	+	0.995204i	$0.531187\pi$
$602$	−1.42500e28	−0.0929077
$603$	0	0
$604$	1.53159e29	0.964395
$605$	−1.31167e29	−0.811699
$606$	0	0
$607$	1.03895e29	0.621033	0.310516	−	0.950568i	$-0.399498\pi$
$607$	1.03895e29	0.621033	0.310516	+	0.950568i	$0.399498\pi$
$608$	−1.04902e29	−0.616308
$609$	0	0
$610$	9.28244e28	0.526865
$611$	4.70021e29	2.62231
$612$	0	0
$613$	−7.95397e28	−0.428794	−0.214397	−	0.976747i	$-0.568779\pi$
$613$	−7.95397e28	−0.428794	−0.214397	+	0.976747i	$0.568779\pi$
$614$	6.98691e28	0.370269
$615$	0	0
$616$	−3.08575e28	−0.158039
$617$	5.43588e28	0.273701	0.136850	−	0.990592i	$-0.456302\pi$
$617$	5.43588e28	0.273701	0.136850	+	0.990592i	$0.456302\pi$
$618$	0	0
$619$	2.60180e29	1.26626	0.633128	−	0.774047i	$-0.281771\pi$
$619$	2.60180e29	1.26626	0.633128	+	0.774047i	$0.281771\pi$
$620$	−5.39502e28	−0.258155
$621$	0	0
$622$	−1.71048e29	−0.791258
$623$	−9.43149e28	−0.428999
$624$	0	0
$625$	−2.65386e29	−1.16718
$626$	3.06587e28	0.132594
$627$	0	0
$628$	7.85158e28	0.328383
$629$	8.03330e28	0.330417
$630$	0	0
$631$	−2.95926e29	−1.17726	−0.588632	−	0.808401i	$-0.700333\pi$
$631$	−2.95926e29	−1.17726	−0.588632	+	0.808401i	$0.700333\pi$
$632$	1.50638e29	0.589395
$633$	0	0
$634$	−3.75053e29	−1.41957
$635$	4.86003e28	0.180932
$636$	0	0
$637$	−4.14827e29	−1.49418
$638$	−4.15538e29	−1.47229
$639$	0	0
$640$	3.39175e29	1.16288
$641$	−5.96787e28	−0.201284	−0.100642	−	0.994923i	$-0.532090\pi$
$641$	−5.96787e28	−0.201284	−0.100642	+	0.994923i	$0.532090\pi$
$642$	0	0
$643$	−9.03178e28	−0.294821	−0.147410	−	0.989075i	$-0.547094\pi$
$643$	−9.03178e28	−0.294821	−0.147410	+	0.989075i	$0.547094\pi$
$644$	−8.64993e28	−0.277786
$645$	0	0
$646$	2.41505e29	0.750732
$647$	1.85763e29	0.568150	0.284075	−	0.958802i	$-0.408313\pi$
$647$	1.85763e29	0.568150	0.284075	+	0.958802i	$0.408313\pi$
$648$	0	0
$649$	−6.06097e27	−0.0179462
$650$	−5.56628e29	−1.62172
$651$	0	0
$652$	−1.32642e29	−0.374181
$653$	2.12263e29	0.589231	0.294615	−	0.955616i	$-0.404809\pi$
$653$	2.12263e29	0.589231	0.294615	+	0.955616i	$0.404809\pi$
$654$	0	0
$655$	−1.42602e29	−0.383348
$656$	2.97771e29	0.787758
$657$	0	0
$658$	−2.17695e29	−0.557797
$659$	−4.06420e28	−0.102489	−0.0512447	−	0.998686i	$-0.516319\pi$
$659$	−4.06420e28	−0.102489	−0.0512447	+	0.998686i	$0.516319\pi$
$660$	0	0
$661$	−2.72047e29	−0.664551	−0.332276	−	0.943182i	$-0.607816\pi$
$661$	−2.72047e29	−0.664551	−0.332276	+	0.943182i	$0.607816\pi$
$662$	7.29928e29	1.75498
$663$	0	0
$664$	−3.28594e29	−0.765411
$665$	8.67156e28	0.198825
$666$	0	0
$667$	6.67054e29	1.48197
$668$	3.15725e29	0.690488
$669$	0	0
$670$	1.40169e27	0.00297076
$671$	−1.87185e29	−0.390557
$672$	0	0
$673$	−3.22064e28	−0.0651305	−0.0325652	−	0.999470i	$-0.510368\pi$
$673$	−3.22064e28	−0.0651305	−0.0325652	+	0.999470i	$0.510368\pi$
$674$	5.40518e29	1.07617
$675$	0	0
$676$	5.24007e29	1.01134
$677$	6.03838e29	1.14746	0.573732	−	0.819043i	$-0.305495\pi$
$677$	6.03838e29	1.14746	0.573732	+	0.819043i	$0.305495\pi$
$678$	0	0
$679$	1.34365e29	0.247545
$680$	−3.62700e29	−0.657967
$681$	0	0
$682$	2.79888e29	0.492322
$683$	−4.89575e29	−0.848011	−0.424006	−	0.905660i	$-0.639376\pi$
$683$	−4.89575e29	−0.848011	−0.424006	+	0.905660i	$0.639376\pi$
$684$	0	0
$685$	−1.28286e30	−2.15490
$686$	3.99091e29	0.660190
$687$	0	0
$688$	−2.08303e29	−0.334209
$689$	−4.37975e29	−0.692066
$690$	0	0
$691$	5.98983e29	0.918110	0.459055	−	0.888408i	$-0.348188\pi$
$691$	5.98983e29	0.918110	0.459055	+	0.888408i	$0.348188\pi$
$692$	−1.66609e29	−0.251527
$693$	0	0
$694$	1.83273e29	0.268425
$695$	8.75039e28	0.126237
$696$	0	0
$697$	−4.82833e29	−0.675854
$698$	1.38332e30	1.90739
$699$	0	0
$700$	1.00210e29	0.134086
$701$	9.91472e29	1.30690	0.653448	−	0.756971i	$-0.273322\pi$
$701$	9.91472e29	1.30690	0.653448	+	0.756971i	$0.273322\pi$
$702$	0	0
$703$	−1.35796e29	−0.173722
$704$	1.88511e29	0.237587
$705$	0	0
$706$	−1.09634e30	−1.34121
$707$	3.50705e29	0.422706
$708$	0	0
$709$	−1.26216e30	−1.47682	−0.738412	−	0.674350i	$-0.764424\pi$
$709$	−1.26216e30	−1.47682	−0.738412	+	0.674350i	$0.764424\pi$
$710$	1.04275e30	1.20218
$711$	0	0
$712$	−6.66843e29	−0.746421
$713$	−4.49299e29	−0.495559
$714$	0	0
$715$	2.54730e30	2.72814
$716$	−5.47505e29	−0.577831
$717$	0	0
$718$	1.18072e29	0.121015
$719$	−1.02638e30	−1.03671	−0.518353	−	0.855167i	$-0.673455\pi$
$719$	−1.02638e30	−1.03671	−0.518353	+	0.855167i	$0.673455\pi$
$720$	0	0
$721$	2.24795e29	0.220530
$722$	9.14617e29	0.884298
$723$	0	0
$724$	8.65084e29	0.812462
$725$	−7.72790e29	−0.715339
$726$	0	0
$727$	−1.22533e30	−1.10190	−0.550951	−	0.834538i	$-0.685735\pi$
$727$	−1.22533e30	−1.10190	−0.550951	+	0.834538i	$0.685735\pi$
$728$	2.26369e29	0.200650
$729$	0	0
$730$	−2.44193e30	−2.10302
$731$	3.37761e29	0.286733
$732$	0	0
$733$	−1.47706e29	−0.121845	−0.0609224	−	0.998143i	$-0.519404\pi$
$733$	−1.47706e29	−0.121845	−0.0609224	+	0.998143i	$0.519404\pi$
$734$	2.60006e30	2.11434
$735$	0	0
$736$	−2.29113e30	−1.81064
$737$	−2.82658e27	−0.00220218
$738$	0	0
$739$	9.02599e29	0.683484	0.341742	−	0.939794i	$-0.388983\pi$
$739$	9.02599e29	0.683484	0.341742	+	0.939794i	$0.388983\pi$
$740$	−3.56130e29	−0.265874
$741$	0	0
$742$	2.02852e29	0.147211
$743$	2.18667e29	0.156459	0.0782295	−	0.996935i	$-0.475073\pi$
$743$	2.18667e29	0.156459	0.0782295	+	0.996935i	$0.475073\pi$
$744$	0	0
$745$	1.13431e30	0.789022
$746$	9.44713e29	0.647951
$747$	0	0
$748$	−1.27719e30	−0.851707
$749$	6.86988e29	0.451742
$750$	0	0
$751$	−8.64189e29	−0.552573	−0.276286	−	0.961075i	$-0.589104\pi$
$751$	−8.64189e29	−0.552573	−0.276286	+	0.961075i	$0.589104\pi$
$752$	−3.18221e30	−2.00651
$753$	0	0
$754$	3.04836e30	1.86925
$755$	−3.35345e30	−2.02791
$756$	0	0
$757$	2.79807e30	1.64570	0.822851	−	0.568257i	$-0.192382\pi$
$757$	2.79807e30	1.64570	0.822851	+	0.568257i	$0.192382\pi$
$758$	3.01598e30	1.74945
$759$	0	0
$760$	6.13113e29	0.345938
$761$	2.13977e30	1.19077	0.595387	−	0.803439i	$-0.296999\pi$
$761$	2.13977e30	1.19077	0.595387	+	0.803439i	$0.296999\pi$
$762$	0	0
$763$	7.55504e29	0.409005
$764$	−1.30819e30	−0.698540
$765$	0	0
$766$	2.31021e30	1.20019
$767$	4.44630e28	0.0227849
$768$	0	0
$769$	1.02877e30	0.512967	0.256484	−	0.966549i	$-0.417436\pi$
$769$	1.02877e30	0.512967	0.256484	+	0.966549i	$0.417436\pi$
$770$	−1.17981e30	−0.580307
$771$	0	0
$772$	−2.09678e30	−1.00363
$773$	1.48505e30	0.701225	0.350612	−	0.936521i	$-0.385973\pi$
$773$	1.48505e30	0.701225	0.350612	+	0.936521i	$0.385973\pi$
$774$	0	0
$775$	5.20518e29	0.239204
$776$	9.50015e29	0.430707
$777$	0	0
$778$	−1.59243e30	−0.702708
$779$	8.16187e29	0.355342
$780$	0	0
$781$	−2.10276e30	−0.891157
$782$	5.27462e30	2.20557
$783$	0	0
$784$	2.80853e30	1.14330
$785$	−1.71912e30	−0.690517
$786$	0	0
$787$	−2.38369e30	−0.932212	−0.466106	−	0.884729i	$-0.654343\pi$
$787$	−2.38369e30	−0.932212	−0.466106	+	0.884729i	$0.654343\pi$
$788$	−1.13150e29	−0.0436648
$789$	0	0
$790$	5.75950e30	2.16422
$791$	4.87953e28	0.0180936
$792$	0	0
$793$	1.37318e30	0.495860
$794$	−5.19092e30	−1.84982
$795$	0	0
$796$	2.08893e30	0.724997
$797$	3.16177e30	1.08297	0.541486	−	0.840710i	$-0.317862\pi$
$797$	3.16177e30	1.08297	0.541486	+	0.840710i	$0.317862\pi$
$798$	0	0
$799$	5.15993e30	1.72148
$800$	2.65430e30	0.873987
$801$	0	0
$802$	−2.11610e30	−0.678742
$803$	4.92427e30	1.55894
$804$	0	0
$805$	1.89392e30	0.584124
$806$	−2.05325e30	−0.625062
$807$	0	0
$808$	2.47962e30	0.735472
$809$	8.32657e29	0.243785	0.121892	−	0.992543i	$-0.461104\pi$
$809$	8.32657e29	0.243785	0.121892	+	0.992543i	$0.461104\pi$
$810$	0	0
$811$	7.75437e29	0.221222	0.110611	−	0.993864i	$-0.464719\pi$
$811$	7.75437e29	0.221222	0.110611	+	0.993864i	$0.464719\pi$
$812$	−5.48800e29	−0.154553
$813$	0	0
$814$	1.84757e30	0.507042
$815$	2.90423e30	0.786820
$816$	0	0
$817$	−5.70956e29	−0.150755
$818$	−4.80989e30	−1.25379
$819$	0	0
$820$	2.14048e30	0.543834
$821$	−6.41299e30	−1.60864	−0.804318	−	0.594199i	$-0.797469\pi$
$821$	−6.41299e30	−1.60864	−0.804318	+	0.594199i	$0.797469\pi$
$822$	0	0
$823$	1.14028e29	0.0278813	0.0139406	−	0.999903i	$-0.495562\pi$
$823$	1.14028e29	0.0278813	0.0139406	+	0.999903i	$0.495562\pi$
$824$	1.58939e30	0.383703
$825$	0	0
$826$	−2.05934e28	−0.00484662
$827$	−6.08952e29	−0.141506	−0.0707531	−	0.997494i	$-0.522540\pi$
$827$	−6.08952e29	−0.141506	−0.0707531	+	0.997494i	$0.522540\pi$
$828$	0	0
$829$	−3.05382e30	−0.691863	−0.345932	−	0.938260i	$-0.612437\pi$
$829$	−3.05382e30	−0.691863	−0.345932	+	0.938260i	$0.612437\pi$
$830$	−1.25635e31	−2.81054
$831$	0	0
$832$	−1.38291e30	−0.301646
$833$	−4.55400e30	−0.980891
$834$	0	0
$835$	−6.91286e30	−1.45194
$836$	2.15899e30	0.447800
$837$	0	0
$838$	−1.07624e31	−2.17695
$839$	−3.83614e30	−0.766292	−0.383146	−	0.923688i	$-0.625159\pi$
$839$	−3.83614e30	−0.766292	−0.383146	+	0.923688i	$0.625159\pi$
$840$	0	0
$841$	−9.00673e29	−0.175473
$842$	−6.52718e30	−1.25588
$843$	0	0
$844$	−2.49911e30	−0.469018
$845$	−1.14732e31	−2.12663
$846$	0	0
$847$	8.98715e29	0.162497
$848$	2.96525e30	0.529548
$849$	0	0
$850$	−6.11070e30	−1.06461
$851$	−2.96586e30	−0.510376
$852$	0	0
$853$	3.96446e30	0.665609	0.332805	−	0.942996i	$-0.392005\pi$
$853$	3.96446e30	0.665609	0.332805	+	0.942996i	$0.392005\pi$
$854$	−6.36001e29	−0.105475
$855$	0	0
$856$	4.85727e30	0.785992
$857$	6.37093e30	1.01837	0.509183	−	0.860658i	$-0.329948\pi$
$857$	6.37093e30	1.01837	0.509183	+	0.860658i	$0.329948\pi$
$858$	0	0
$859$	2.94165e30	0.458841	0.229420	−	0.973327i	$-0.426317\pi$
$859$	2.94165e30	0.458841	0.229420	+	0.973327i	$0.426317\pi$
$860$	−1.49735e30	−0.230723
$861$	0	0
$862$	−8.30575e30	−1.24897
$863$	−1.79374e30	−0.266469	−0.133234	−	0.991085i	$-0.542536\pi$
$863$	−1.79374e30	−0.266469	−0.133234	+	0.991085i	$0.542536\pi$
$864$	0	0
$865$	3.64793e30	0.528905
$866$	1.35375e31	1.93910
$867$	0	0
$868$	3.69648e29	0.0516811
$869$	−1.16143e31	−1.60430
$870$	0	0
$871$	2.07357e28	0.00279594
$872$	5.34171e30	0.711634
$873$	0	0
$874$	−8.91628e30	−1.15961
$875$	5.91046e29	0.0759515
$876$	0	0
$877$	−1.28521e31	−1.61242	−0.806209	−	0.591631i	$-0.798484\pi$
$877$	−1.28521e31	−1.61242	−0.806209	+	0.591631i	$0.798484\pi$
$878$	1.66466e31	2.06364
$879$	0	0
$880$	−1.72461e31	−2.08749
$881$	−1.19067e30	−0.142412	−0.0712059	−	0.997462i	$-0.522685\pi$
$881$	−1.19067e30	−0.142412	−0.0712059	+	0.997462i	$0.522685\pi$
$882$	0	0
$883$	−1.50906e30	−0.176246	−0.0881229	−	0.996110i	$-0.528087\pi$
$883$	−1.50906e30	−0.176246	−0.0881229	+	0.996110i	$0.528087\pi$
$884$	9.36944e30	1.08135
$885$	0	0
$886$	−9.12797e30	−1.02877
$887$	−1.05464e31	−1.17465	−0.587324	−	0.809352i	$-0.699819\pi$
$887$	−1.05464e31	−1.17465	−0.587324	+	0.809352i	$0.699819\pi$
$888$	0	0
$889$	−3.32993e29	−0.0362215
$890$	−2.54961e31	−2.74081
$891$	0	0
$892$	1.32227e30	0.138832
$893$	−8.72240e30	−0.905098
$894$	0	0
$895$	1.19877e31	1.21505
$896$	−2.32391e30	−0.232801
$897$	0	0
$898$	−6.65118e30	−0.650874
$899$	−2.85061e30	−0.275715
$900$	0	0
$901$	−4.80812e30	−0.454323
$902$	−1.11046e31	−1.03713
$903$	0	0
$904$	3.45002e29	0.0314813
$905$	−1.89412e31	−1.70843
$906$	0	0
$907$	1.18403e31	1.04349	0.521743	−	0.853103i	$-0.325282\pi$
$907$	1.18403e31	1.04349	0.521743	+	0.853103i	$0.325282\pi$
$908$	−3.66568e30	−0.319340
$909$	0	0
$910$	8.65500e30	0.736771
$911$	−1.24297e31	−1.04597	−0.522983	−	0.852343i	$-0.675181\pi$
$911$	−1.24297e31	−1.04597	−0.522983	+	0.852343i	$0.675181\pi$
$912$	0	0
$913$	2.53348e31	2.08341
$914$	9.87172e30	0.802524
$915$	0	0
$916$	1.24192e31	0.986717
$917$	9.77063e29	0.0767440
$918$	0	0
$919$	−4.46865e30	−0.343055	−0.171528	−	0.985179i	$-0.554870\pi$
$919$	−4.46865e30	−0.343055	−0.171528	+	0.985179i	$0.554870\pi$
$920$	1.33908e31	1.01633
$921$	0	0
$922$	1.87318e31	1.38965
$923$	1.54258e31	1.13143
$924$	0	0
$925$	3.43598e30	0.246356
$926$	1.25844e31	0.892105
$927$	0	0
$928$	−1.45362e31	−1.00739
$929$	−1.04801e31	−0.718124	−0.359062	−	0.933314i	$-0.616903\pi$
$929$	−1.04801e31	−0.718124	−0.359062	+	0.933314i	$0.616903\pi$
$930$	0	0
$931$	7.69814e30	0.515720
$932$	−1.15746e31	−0.766721
$933$	0	0
$934$	−1.48764e31	−0.963511
$935$	2.79644e31	1.79095
$936$	0	0
$937$	−9.17367e30	−0.574483	−0.287242	−	0.957858i	$-0.592738\pi$
$937$	−9.17367e30	−0.574483	−0.287242	+	0.957858i	$0.592738\pi$
$938$	−9.60392e27	−0.000594728 0
$939$	0	0
$940$	−2.28748e31	−1.38521
$941$	1.65714e31	0.992358	0.496179	−	0.868220i	$-0.334736\pi$
$941$	1.65714e31	0.992358	0.496179	+	0.868220i	$0.334736\pi$
$942$	0	0
$943$	1.78260e31	1.04395
$944$	−3.01030e29	−0.0174343
$945$	0	0
$946$	7.76813e30	0.440007
$947$	−1.78965e31	−1.00252	−0.501260	−	0.865296i	$-0.667130\pi$
$947$	−1.78965e31	−1.00252	−0.501260	+	0.865296i	$0.667130\pi$
$948$	0	0
$949$	−3.61242e31	−1.97926
$950$	1.03296e31	0.559739
$951$	0	0
$952$	2.48510e30	0.131721
$953$	1.60157e31	0.839599	0.419799	−	0.907617i	$-0.362100\pi$
$953$	1.60157e31	0.839599	0.419799	+	0.907617i	$0.362100\pi$
$954$	0	0
$955$	2.86431e31	1.46888
$956$	−6.00110e30	−0.304385
$957$	0	0
$958$	−2.60626e31	−1.29324
$959$	8.78970e30	0.431398
$960$	0	0
$961$	−1.89055e31	−0.907803
$962$	−1.35537e31	−0.643751
$963$	0	0
$964$	−1.56898e29	−0.00729136
$965$	4.59094e31	2.11040
$966$	0	0
$967$	2.99659e31	1.34787	0.673936	−	0.738789i	$-0.264602\pi$
$967$	2.99659e31	1.34787	0.673936	+	0.738789i	$0.264602\pi$
$968$	6.35426e30	0.282731
$969$	0	0
$970$	3.63229e31	1.58153
$971$	4.76501e29	0.0205240	0.0102620	−	0.999947i	$-0.496733\pi$
$971$	4.76501e29	0.0205240	0.0102620	+	0.999947i	$0.496733\pi$
$972$	0	0
$973$	−5.99547e29	−0.0252719
$974$	−2.04524e31	−0.852855
$975$	0	0
$976$	−9.29692e30	−0.379416
$977$	−3.80897e31	−1.53785	−0.768926	−	0.639338i	$-0.779209\pi$
$977$	−3.80897e31	−1.53785	−0.768926	+	0.639338i	$0.779209\pi$
$978$	0	0
$979$	5.14141e31	2.03172
$980$	2.01887e31	0.789285
$981$	0	0
$982$	4.30113e31	1.64593
$983$	−1.27281e31	−0.481894	−0.240947	−	0.970538i	$-0.577458\pi$
$983$	−1.27281e31	−0.481894	−0.240947	+	0.970538i	$0.577458\pi$
$984$	0	0
$985$	2.47745e30	0.0918174
$986$	3.34652e31	1.22711
$987$	0	0
$988$	−1.58382e31	−0.568537
$989$	−1.24700e31	−0.442900
$990$	0	0
$991$	2.45208e31	0.852630	0.426315	−	0.904575i	$-0.359812\pi$
$991$	2.45208e31	0.852630	0.426315	+	0.904575i	$0.359812\pi$
$992$	9.79098e30	0.336863
$993$	0	0
$994$	−7.14459e30	−0.240669
$995$	−4.57375e31	−1.52451
$996$	0	0
$997$	−2.11344e31	−0.689749	−0.344875	−	0.938649i	$-0.612079\pi$
$997$	−2.11344e31	−0.689749	−0.344875	+	0.938649i	$0.612079\pi$
$998$	−1.86399e31	−0.601966
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.22.a.c.1.5		20
3.2	odd	2		81.22.a.d.1.16		20
9.2	odd	6		27.22.c.a.10.5		40
9.4	even	3		9.22.c.a.7.16	yes	40
9.5	odd	6		27.22.c.a.19.5		40
9.7	even	3		9.22.c.a.4.16	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.22.c.a.4.16	✓	40	9.7	even	3
9.22.c.a.7.16	yes	40	9.4	even	3
27.22.c.a.10.5		40	9.2	odd	6
27.22.c.a.19.5		40	9.5	odd	6
81.22.a.c.1.5		20	1.1	even	1	trivial
81.22.a.d.1.16		20	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	81.a (trivial)

\(f(q)\)	\(=\)	\(q-1852.20 q^{2} +1.33350e6 q^{4} -2.91973e7 q^{5} +2.00050e8 q^{7} +1.41443e9 q^{8} +O(q^{10})\)	\(q-1852.20 q^{2} +1.33350e6 q^{4} -2.91973e7 q^{5} +2.00050e8 q^{7} +1.41443e9 q^{8} +5.40794e10 q^{10} -1.09054e11 q^{11} +8.00012e11 q^{13} -3.70533e11 q^{14} -5.41637e12 q^{16} +8.78259e12 q^{17} -1.48462e13 q^{19} -3.89347e13 q^{20} +2.01989e14 q^{22} -3.24250e14 q^{23} +3.75647e14 q^{25} -1.48179e15 q^{26} +2.66768e14 q^{28} -2.05722e15 q^{29} +1.38566e15 q^{31} +7.06595e15 q^{32} -1.62671e16 q^{34} -5.84093e15 q^{35} +9.14684e15 q^{37} +2.74982e16 q^{38} -4.12976e16 q^{40} -5.49761e16 q^{41} +3.84581e16 q^{43} -1.45423e17 q^{44} +6.00576e17 q^{46} +5.87517e17 q^{47} -5.18526e17 q^{49} -6.95774e17 q^{50} +1.06682e18 q^{52} -5.47460e17 q^{53} +3.18408e18 q^{55} +2.82957e17 q^{56} +3.81040e18 q^{58} +5.55778e16 q^{59} +1.71645e18 q^{61} -2.56652e18 q^{62} -1.72860e18 q^{64} -2.33582e19 q^{65} +2.59192e16 q^{67} +1.17116e19 q^{68} +1.08186e19 q^{70} +1.92819e19 q^{71} -4.51545e19 q^{73} -1.69418e19 q^{74} -1.97975e19 q^{76} -2.18162e19 q^{77} +1.06501e20 q^{79} +1.58144e20 q^{80} +1.01827e20 q^{82} -2.32315e20 q^{83} -2.56428e20 q^{85} -7.12321e19 q^{86} -1.54249e20 q^{88} -4.71457e20 q^{89} +1.60043e20 q^{91} -4.32388e20 q^{92} -1.08820e21 q^{94} +4.33469e20 q^{95} +6.71658e20 q^{97} +9.60415e20 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 1023 q^{2} + 19922945 q^{4} - 32234853 q^{5} + 189623959 q^{7} - 648135831 q^{8}+O(q^{10}) \) 20 * q - 1023 * q^2 + 19922945 * q^4 - 32234853 * q^5 + 189623959 * q^7 - 648135831 * q^8	\( 20 q - 1023 q^{2} + 19922945 q^{4} - 32234853 q^{5} + 189623959 q^{7} - 648135831 q^{8} + 2097150 q^{10} - 146068576386 q^{11} + 177565977277 q^{13} - 1549677244440 q^{14} + 18691699769345 q^{16} - 9307801874799 q^{17} - 4884366861977 q^{19} - 76202257650204 q^{20} - 86758343554047 q^{22} - 356460494884095 q^{23} + 13\!\cdots\!29 q^{25}+ \cdots + 26\!\cdots\!43 q^{98}+O(q^{100}) \) 20 * q - 1023 * q^2 + 19922945 * q^4 - 32234853 * q^5 + 189623959 * q^7 - 648135831 * q^8 + 2097150 * q^10 - 146068576386 * q^11 + 177565977277 * q^13 - 1549677244440 * q^14 + 18691699769345 * q^16 - 9307801874799 * q^17 - 4884366861977 * q^19 - 76202257650204 * q^20 - 86758343554047 * q^22 - 356460494884095 * q^23 + 1379841857082329 * q^25 + 2254540119546732 * q^26 + 799738574143486 * q^28 + 2772828270755157 * q^29 - 3064842508546901 * q^31 - 9763241440878495 * q^32 - 5532390311666553 * q^34 - 28944789317551797 * q^35 + 15041349243939694 * q^37 - 38173338977409867 * q^38 + 43546396391926176 * q^40 - 4985514195953106 * q^41 + 41986108659289660 * q^43 + 154909775468547021 * q^44 - 370501619585816100 * q^46 - 971751895603333665 * q^47 + 1393736132530446381 * q^49 - 1504393427506051923 * q^50 + 892509168437820382 * q^52 - 3200468347464244650 * q^53 + 1763329427612189115 * q^55 - 9313401575284840590 * q^56 + 985520139125952432 * q^58 - 11335149505478767926 * q^59 + 333769931007960067 * q^61 - 30822195887206546926 * q^62 + 12804600562751272961 * q^64 - 24815104795645584999 * q^65 - 14325054629523996728 * q^67 - 98193137232148637751 * q^68 + 23664205955002266174 * q^70 - 108719113548124731960 * q^71 - 2415544298256170813 * q^73 - 284989967381657148240 * q^74 - 45502930505046158765 * q^76 - 209785238565183158133 * q^77 - 45386959469213650259 * q^79 - 499015504255192380624 * q^80 + 114258090814276483077 * q^82 - 224083700347690062915 * q^83 + 135764416394391341394 * q^85 + 210501162447733500423 * q^86 - 409557505702705259691 * q^88 + 140942904695660466360 * q^89 - 193584504929412672709 * q^91 - 160981359902419031922 * q^92 - 1246412695298803268064 * q^94 - 583111881346045719228 * q^95 + 658984791485125488346 * q^97 + 2671316103447781667343 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more