LMFDB - Embedded newform 81.10.a.c.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [81,10,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, 10, names="a")

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

chi := DirichletCharacter("81.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 81 = 3^{4} $
Weight:	$ k $	$=$	$ 10 $
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:	$41.7179027293$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 2930 x^{6} - 1276 x^{5} + 2487472 x^{4} + 3423248 x^{3} - 586568096 x^{2} + \cdots + 965565184 $ x^8 - x^7 - 2930x^6 - 1276x^5 + 2487472x^4 + 3423248x^3 - 586568096x^2 - 2825363008x + 965565184
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{7}\cdot 3^{18}\cdot 17 $
Twist minimal:	no (minimal twist has level 9)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-5.66432$ 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)

$f(q)$	$=$	$q-7.66432 q^{2} -453.258 q^{4} +2210.95 q^{5} -6233.15 q^{7} +7398.05 q^{8} +O(q^{10})$	\(q-7.66432 q^{2} -453.258 q^{4} +2210.95 q^{5} -6233.15 q^{7} +7398.05 q^{8} -16945.4 q^{10} +18624.2 q^{11} -169399. q^{13} +47772.9 q^{14} +175367. q^{16} +433113. q^{17} +51200.2 q^{19} -1.00213e6 q^{20} -142742. q^{22} +304563. q^{23} +2.93518e6 q^{25} +1.29833e6 q^{26} +2.82523e6 q^{28} -1.13799e6 q^{29} -778586. q^{31} -5.13187e6 q^{32} -3.31952e6 q^{34} -1.37812e7 q^{35} -6.32050e6 q^{37} -392414. q^{38} +1.63567e7 q^{40} -2.47982e6 q^{41} -3.12600e7 q^{43} -8.44159e6 q^{44} -2.33427e6 q^{46} +1.01408e7 q^{47} -1.50142e6 q^{49} -2.24961e7 q^{50} +7.67816e7 q^{52} -3.32278e7 q^{53} +4.11773e7 q^{55} -4.61132e7 q^{56} +8.72192e6 q^{58} -4.67205e7 q^{59} -6.13804e7 q^{61} +5.96733e6 q^{62} -5.04557e7 q^{64} -3.74533e8 q^{65} -2.74817e8 q^{67} -1.96312e8 q^{68} +1.05623e8 q^{70} -2.93935e8 q^{71} -1.55084e8 q^{73} +4.84423e7 q^{74} -2.32069e7 q^{76} -1.16088e8 q^{77} -2.35767e8 q^{79} +3.87728e8 q^{80} +1.90061e7 q^{82} +3.36140e8 q^{83} +9.57592e8 q^{85} +2.39586e8 q^{86} +1.37783e8 q^{88} -1.03812e9 q^{89} +1.05589e9 q^{91} -1.38046e8 q^{92} -7.77223e7 q^{94} +1.13201e8 q^{95} -2.69573e8 q^{97} +1.15074e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 15 q^{2} + 1793 q^{4} - 453 q^{5} + 343 q^{7} - 7239 q^{8}+O(q^{10}) $ 8 * q - 15 * q^2 + 1793 * q^4 - 453 * q^5 + 343 * q^7 - 7239 * q^8	\( 8 q - 15 q^{2} + 1793 q^{4} - 453 q^{5} + 343 q^{7} - 7239 q^{8} + 510 q^{10} - 99150 q^{11} - 32435 q^{13} - 394824 q^{14} + 328193 q^{16} - 415539 q^{17} - 85277 q^{19} - 1855164 q^{20} - 529359 q^{22} - 1064559 q^{23} + 2293229 q^{25} + 1218156 q^{26} + 612862 q^{28} + 1309053 q^{29} + 2359819 q^{31} - 5760063 q^{32} - 981801 q^{34} - 15533277 q^{35} + 8195758 q^{37} - 39490203 q^{38} + 16760496 q^{40} - 54747318 q^{41} - 15249608 q^{43} - 166254963 q^{44} + 1195260 q^{46} - 156295545 q^{47} - 15239583 q^{49} - 315590163 q^{50} + 19773358 q^{52} - 262758114 q^{53} - 3789885 q^{55} - 470339790 q^{56} - 55408560 q^{58} - 307774074 q^{59} - 69192125 q^{61} - 457218462 q^{62} - 201794239 q^{64} - 482470359 q^{65} - 14328044 q^{67} - 915409575 q^{68} + 229271934 q^{70} - 619800696 q^{71} + 299306599 q^{73} - 1022736000 q^{74} - 119954093 q^{76} - 717995541 q^{77} - 30257531 q^{79} - 1463913264 q^{80} - 101188011 q^{82} - 1176168291 q^{83} - 4818366 q^{85} - 1426944009 q^{86} - 911312427 q^{88} - 1658520648 q^{89} - 369615061 q^{91} + 76813998 q^{92} + 1954316784 q^{94} + 391400652 q^{95} + 267311278 q^{97} + 2413650159 q^{98}+O(q^{100}) \) 8 * q - 15 * q^2 + 1793 * q^4 - 453 * q^5 + 343 * q^7 - 7239 * q^8 + 510 * q^10 - 99150 * q^11 - 32435 * q^13 - 394824 * q^14 + 328193 * q^16 - 415539 * q^17 - 85277 * q^19 - 1855164 * q^20 - 529359 * q^22 - 1064559 * q^23 + 2293229 * q^25 + 1218156 * q^26 + 612862 * q^28 + 1309053 * q^29 + 2359819 * q^31 - 5760063 * q^32 - 981801 * q^34 - 15533277 * q^35 + 8195758 * q^37 - 39490203 * q^38 + 16760496 * q^40 - 54747318 * q^41 - 15249608 * q^43 - 166254963 * q^44 + 1195260 * q^46 - 156295545 * q^47 - 15239583 * q^49 - 315590163 * q^50 + 19773358 * q^52 - 262758114 * q^53 - 3789885 * q^55 - 470339790 * q^56 - 55408560 * q^58 - 307774074 * q^59 - 69192125 * q^61 - 457218462 * q^62 - 201794239 * q^64 - 482470359 * q^65 - 14328044 * q^67 - 915409575 * q^68 + 229271934 * q^70 - 619800696 * q^71 + 299306599 * q^73 - 1022736000 * q^74 - 119954093 * q^76 - 717995541 * q^77 - 30257531 * q^79 - 1463913264 * q^80 - 101188011 * q^82 - 1176168291 * q^83 - 4818366 * q^85 - 1426944009 * q^86 - 911312427 * q^88 - 1658520648 * q^89 - 369615061 * q^91 + 76813998 * q^92 + 1954316784 * q^94 + 391400652 * q^95 + 267311278 * q^97 + 2413650159 * 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 $

$2$	−7.66432	−0.338718	−0.169359	−	0.985554i	$-0.554170\pi$
$2$	−7.66432	−0.338718	−0.169359	+	0.985554i	$0.554170\pi$
$3$	0	0
$3$	0	0
$4$	−453.258	−0.885270
$5$	2210.95	1.58203	0.791014	−	0.611798i	$-0.209554\pi$
$5$	2210.95	1.58203	0.791014	+	0.611798i	$0.209554\pi$
$6$	0	0
$7$	−6233.15	−0.981220	−0.490610	−	0.871379i	$-0.663226\pi$
$7$	−6233.15	−0.981220	−0.490610	+	0.871379i	$0.663226\pi$
$8$	7398.05	0.638575
$9$	0	0
$10$	−16945.4	−0.535862
$11$	18624.2	0.383541	0.191770	−	0.981440i	$-0.438577\pi$
$11$	18624.2	0.383541	0.191770	+	0.981440i	$0.438577\pi$
$12$	0	0
$13$	−169399.	−1.64500	−0.822501	−	0.568764i	$-0.807422\pi$
$13$	−169399.	−1.64500	−0.822501	+	0.568764i	$0.807422\pi$
$14$	47772.9	0.332357
$15$	0	0
$16$	175367.	0.668973
$17$	433113.	1.25771	0.628856	−	0.777522i	$-0.283524\pi$
$17$	433113.	1.25771	0.628856	+	0.777522i	$0.283524\pi$
$18$	0	0
$19$	51200.2	0.0901322	0.0450661	−	0.998984i	$-0.485650\pi$
$19$	51200.2	0.0901322	0.0450661	+	0.998984i	$0.485650\pi$
$20$	−1.00213e6	−1.40052
$21$	0	0
$22$	−142742.	−0.129912
$23$	304563.	0.226935	0.113468	−	0.993542i	$-0.463804\pi$
$23$	304563.	0.226935	0.113468	+	0.993542i	$0.463804\pi$
$24$	0	0
$25$	2.93518e6	1.50281
$26$	1.29833e6	0.557192
$27$	0	0
$28$	2.82523e6	0.868645
$29$	−1.13799e6	−0.298777	−0.149389	−	0.988779i	$-0.547731\pi$
$29$	−1.13799e6	−0.298777	−0.149389	+	0.988779i	$0.547731\pi$
$30$	0	0
$31$	−778586.	−0.151418	−0.0757092	−	0.997130i	$-0.524122\pi$
$31$	−778586.	−0.151418	−0.0757092	+	0.997130i	$0.524122\pi$
$32$	−5.13187e6	−0.865169
$33$	0	0
$34$	−3.31952e6	−0.426010
$35$	−1.37812e7	−1.55232
$36$	0	0
$37$	−6.32050e6	−0.554426	−0.277213	−	0.960808i	$-0.589411\pi$
$37$	−6.32050e6	−0.554426	−0.277213	+	0.960808i	$0.589411\pi$
$38$	−392414.	−0.0305294
$39$	0	0
$40$	1.63567e7	1.01024
$41$	−2.47982e6	−0.137054	−0.0685271	−	0.997649i	$-0.521830\pi$
$41$	−2.47982e6	−0.137054	−0.0685271	+	0.997649i	$0.521830\pi$
$42$	0	0
$43$	−3.12600e7	−1.39438	−0.697189	−	0.716888i	$-0.745566\pi$
$43$	−3.12600e7	−1.39438	−0.697189	+	0.716888i	$0.745566\pi$
$44$	−8.44159e6	−0.339537
$45$	0	0
$46$	−2.33427e6	−0.0768671
$47$	1.01408e7	0.303132	0.151566	−	0.988447i	$-0.451568\pi$
$47$	1.01408e7	0.303132	0.151566	+	0.988447i	$0.451568\pi$
$48$	0	0
$49$	−1.50142e6	−0.0372066
$50$	−2.24961e7	−0.509030
$51$	0	0
$52$	7.67816e7	1.45627
$53$	−3.32278e7	−0.578442	−0.289221	−	0.957262i	$-0.593396\pi$
$53$	−3.32278e7	−0.578442	−0.289221	+	0.957262i	$0.593396\pi$
$54$	0	0
$55$	4.11773e7	0.606772
$56$	−4.61132e7	−0.626583
$57$	0	0
$58$	8.72192e6	0.101201
$59$	−4.67205e7	−0.501965	−0.250983	−	0.967992i	$-0.580754\pi$
$59$	−4.67205e7	−0.501965	−0.250983	+	0.967992i	$0.580754\pi$
$60$	0	0
$61$	−6.13804e7	−0.567604	−0.283802	−	0.958883i	$-0.591596\pi$
$61$	−6.13804e7	−0.567604	−0.283802	+	0.958883i	$0.591596\pi$
$62$	5.96733e6	0.0512882
$63$	0	0
$64$	−5.04557e7	−0.375924
$65$	−3.74533e8	−2.60244
$66$	0	0
$67$	−2.74817e8	−1.66612	−0.833062	−	0.553180i	$-0.813414\pi$
$67$	−2.74817e8	−1.66612	−0.833062	+	0.553180i	$0.813414\pi$
$68$	−1.96312e8	−1.11341
$69$	0	0
$70$	1.05623e8	0.525798
$71$	−2.93935e8	−1.37274	−0.686372	−	0.727251i	$-0.740798\pi$
$71$	−2.93935e8	−1.37274	−0.686372	+	0.727251i	$0.740798\pi$
$72$	0	0
$73$	−1.55084e8	−0.639168	−0.319584	−	0.947558i	$-0.603543\pi$
$73$	−1.55084e8	−0.639168	−0.319584	+	0.947558i	$0.603543\pi$
$74$	4.84423e7	0.187794
$75$	0	0
$76$	−2.32069e7	−0.0797913
$77$	−1.16088e8	−0.376338
$78$	0	0
$79$	−2.35767e8	−0.681021	−0.340511	−	0.940241i	$-0.610600\pi$
$79$	−2.35767e8	−0.681021	−0.340511	+	0.940241i	$0.610600\pi$
$80$	3.87728e8	1.05833
$81$	0	0
$82$	1.90061e7	0.0464228
$83$	3.36140e8	0.777444	0.388722	−	0.921355i	$-0.372917\pi$
$83$	3.36140e8	0.777444	0.388722	+	0.921355i	$0.372917\pi$
$84$	0	0
$85$	9.57592e8	1.98973
$86$	2.39586e8	0.472301
$87$	0	0
$88$	1.37783e8	0.244920
$89$	−1.03812e9	−1.75384	−0.876922	−	0.480633i	$-0.840407\pi$
$89$	−1.03812e9	−1.75384	−0.876922	+	0.480633i	$0.840407\pi$
$90$	0	0
$91$	1.05589e9	1.61411
$92$	−1.38046e8	−0.200899
$93$	0	0
$94$	−7.77223e7	−0.102676
$95$	1.13201e8	0.142592
$96$	0	0
$97$	−2.69573e8	−0.309174	−0.154587	−	0.987979i	$-0.549405\pi$
$97$	−2.69573e8	−0.309174	−0.154587	+	0.987979i	$0.549405\pi$
$98$	1.15074e7	0.0126026
$99$	0	0
$100$	−1.33039e9	−1.33039
$101$	9.80093e8	0.937176	0.468588	−	0.883417i	$-0.344763\pi$
$101$	9.80093e8	0.937176	0.468588	+	0.883417i	$0.344763\pi$
$102$	0	0
$103$	−1.67368e8	−0.146522	−0.0732611	−	0.997313i	$-0.523341\pi$
$103$	−1.67368e8	−0.146522	−0.0732611	+	0.997313i	$0.523341\pi$
$104$	−1.25322e9	−1.05046
$105$	0	0
$106$	2.54668e8	0.195929
$107$	1.91944e9	1.41563	0.707813	−	0.706400i	$-0.249682\pi$
$107$	1.91944e9	1.41563	0.707813	+	0.706400i	$0.249682\pi$
$108$	0	0
$109$	2.60871e8	0.177013	0.0885066	−	0.996076i	$-0.471791\pi$
$109$	2.60871e8	0.177013	0.0885066	+	0.996076i	$0.471791\pi$
$110$	−3.15596e8	−0.205525
$111$	0	0
$112$	−1.09309e9	−0.656410
$113$	3.26607e9	1.88439	0.942197	−	0.335059i	$-0.108756\pi$
$113$	3.26607e9	1.88439	0.942197	+	0.335059i	$0.108756\pi$
$114$	0	0
$115$	6.73373e8	0.359018
$116$	5.15803e8	0.264499
$117$	0	0
$118$	3.58081e8	0.170025
$119$	−2.69966e9	−1.23409
$120$	0	0
$121$	−2.01109e9	−0.852896
$122$	4.70439e8	0.192258
$123$	0	0
$124$	3.52900e8	0.134046
$125$	2.17127e9	0.795461
$126$	0	0
$127$	−1.25753e9	−0.428944	−0.214472	−	0.976730i	$-0.568803\pi$
$127$	−1.25753e9	−0.428944	−0.214472	+	0.976730i	$0.568803\pi$
$128$	3.01423e9	0.992501
$129$	0	0
$130$	2.87054e9	0.881493
$131$	−1.81352e9	−0.538022	−0.269011	−	0.963137i	$-0.586697\pi$
$131$	−1.81352e9	−0.538022	−0.269011	+	0.963137i	$0.586697\pi$
$132$	0	0
$133$	−3.19138e8	−0.0884396
$134$	2.10629e9	0.564347
$135$	0	0
$136$	3.20419e9	0.803144
$137$	6.51242e9	1.57943	0.789714	−	0.613475i	$-0.210229\pi$
$137$	6.51242e9	1.57943	0.789714	+	0.613475i	$0.210229\pi$
$138$	0	0
$139$	−1.83332e9	−0.416554	−0.208277	−	0.978070i	$-0.566786\pi$
$139$	−1.83332e9	−0.416554	−0.208277	+	0.978070i	$0.566786\pi$
$140$	6.24644e9	1.37422
$141$	0	0
$142$	2.25281e9	0.464973
$143$	−3.15493e9	−0.630925
$144$	0	0
$145$	−2.51604e9	−0.472674
$146$	1.18862e9	0.216498
$147$	0	0
$148$	2.86482e9	0.490817
$149$	−1.06860e9	−0.177615	−0.0888074	−	0.996049i	$-0.528306\pi$
$149$	−1.06860e9	−0.177615	−0.0888074	+	0.996049i	$0.528306\pi$
$150$	0	0
$151$	−8.07959e9	−1.26472	−0.632358	−	0.774676i	$-0.717913\pi$
$151$	−8.07959e9	−1.26472	−0.632358	+	0.774676i	$0.717913\pi$
$152$	3.78781e8	0.0575562
$153$	0	0
$154$	8.89733e8	0.127473
$155$	−1.72141e9	−0.239548
$156$	0	0
$157$	7.24892e9	0.952193	0.476096	−	0.879393i	$-0.342051\pi$
$157$	7.24892e9	0.952193	0.476096	+	0.879393i	$0.342051\pi$
$158$	1.80699e9	0.230674
$159$	0	0
$160$	−1.13463e10	−1.36872
$161$	−1.89839e9	−0.222673
$162$	0	0
$163$	−1.25726e10	−1.39502	−0.697511	−	0.716574i	$-0.745709\pi$
$163$	−1.25726e10	−1.39502	−0.697511	+	0.716574i	$0.745709\pi$
$164$	1.12400e9	0.121330
$165$	0	0
$166$	−2.57629e9	−0.263335
$167$	−1.41495e10	−1.40772	−0.703860	−	0.710338i	$-0.748542\pi$
$167$	−1.41495e10	−1.40772	−0.703860	+	0.710338i	$0.748542\pi$
$168$	0	0
$169$	1.80916e10	1.70603
$170$	−7.33929e9	−0.673959
$171$	0	0
$172$	1.41688e10	1.23440
$173$	−1.22815e10	−1.04242	−0.521211	−	0.853428i	$-0.674520\pi$
$173$	−1.22815e10	−1.04242	−0.521211	+	0.853428i	$0.674520\pi$
$174$	0	0
$175$	−1.82954e10	−1.47459
$176$	3.26608e9	0.256578
$177$	0	0
$178$	7.95645e9	0.594059
$179$	−5.47007e8	−0.0398249	−0.0199124	−	0.999802i	$-0.506339\pi$
$179$	−5.47007e8	−0.0398249	−0.0199124	+	0.999802i	$0.506339\pi$
$180$	0	0
$181$	8.46578e9	0.586291	0.293146	−	0.956068i	$-0.405298\pi$
$181$	8.46578e9	0.586291	0.293146	+	0.956068i	$0.405298\pi$
$182$	−8.09269e9	−0.546728
$183$	0	0
$184$	2.25317e9	0.144915
$185$	−1.39743e10	−0.877117
$186$	0	0
$187$	8.06640e9	0.482384
$188$	−4.59640e9	−0.268354
$189$	0	0
$190$	−8.67609e8	−0.0482984
$191$	−2.91811e10	−1.58654	−0.793271	−	0.608869i	$-0.791623\pi$
$191$	−2.91811e10	−1.58654	−0.793271	+	0.608869i	$0.791623\pi$
$192$	0	0
$193$	2.12520e10	1.10254	0.551268	−	0.834328i	$-0.314144\pi$
$193$	2.12520e10	1.10254	0.551268	+	0.834328i	$0.314144\pi$
$194$	2.06609e9	0.104723
$195$	0	0
$196$	6.80531e8	0.0329379
$197$	1.91891e10	0.907728	0.453864	−	0.891071i	$-0.350045\pi$
$197$	1.91891e10	0.907728	0.453864	+	0.891071i	$0.350045\pi$
$198$	0	0
$199$	6.01109e9	0.271716	0.135858	−	0.990728i	$-0.456621\pi$
$199$	6.01109e9	0.271716	0.135858	+	0.990728i	$0.456621\pi$
$200$	2.17146e10	0.959658
$201$	0	0
$202$	−7.51175e9	−0.317439
$203$	7.09327e9	0.293166
$204$	0	0
$205$	−5.48276e9	−0.216824
$206$	1.28276e9	0.0496298
$207$	0	0
$208$	−2.97071e10	−1.10046
$209$	9.53564e8	0.0345694
$210$	0	0
$211$	−1.21005e10	−0.420273	−0.210137	−	0.977672i	$-0.567391\pi$
$211$	−1.21005e10	−0.420273	−0.210137	+	0.977672i	$0.567391\pi$
$212$	1.50608e10	0.512077
$213$	0	0
$214$	−1.47112e10	−0.479498
$215$	−6.91142e10	−2.20594
$216$	0	0
$217$	4.85304e9	0.148575
$218$	−1.99939e9	−0.0599576
$219$	0	0
$220$	−1.86639e10	−0.537157
$221$	−7.33690e10	−2.06894
$222$	0	0
$223$	−5.26378e10	−1.42536	−0.712682	−	0.701487i	$-0.752520\pi$
$223$	−5.26378e10	−1.42536	−0.712682	+	0.701487i	$0.752520\pi$
$224$	3.19877e10	0.848921
$225$	0	0
$226$	−2.50322e10	−0.638279
$227$	1.23959e10	0.309856	0.154928	−	0.987926i	$-0.450485\pi$
$227$	1.23959e10	0.309856	0.154928	+	0.987926i	$0.450485\pi$
$228$	0	0
$229$	4.41352e10	1.06054	0.530268	−	0.847830i	$-0.322091\pi$
$229$	4.41352e10	1.06054	0.530268	+	0.847830i	$0.322091\pi$
$230$	−5.16095e9	−0.121606
$231$	0	0
$232$	−8.41890e9	−0.190792
$233$	−5.44229e10	−1.20971	−0.604853	−	0.796337i	$-0.706768\pi$
$233$	−5.44229e10	−1.20971	−0.604853	+	0.796337i	$0.706768\pi$
$234$	0	0
$235$	2.24208e10	0.479563
$236$	2.11765e10	0.444375
$237$	0	0
$238$	2.06911e10	0.418010
$239$	8.43865e10	1.67295	0.836474	−	0.548006i	$-0.184613\pi$
$239$	8.43865e10	1.67295	0.836474	+	0.548006i	$0.184613\pi$
$240$	0	0
$241$	6.49291e10	1.23983	0.619916	−	0.784668i	$-0.287167\pi$
$241$	6.49291e10	1.23983	0.619916	+	0.784668i	$0.287167\pi$
$242$	1.54136e10	0.288892
$243$	0	0
$244$	2.78212e10	0.502483
$245$	−3.31957e9	−0.0588619
$246$	0	0
$247$	−8.67327e9	−0.148268
$248$	−5.76001e9	−0.0966921
$249$	0	0
$250$	−1.66413e10	−0.269437
$251$	4.84990e10	0.771260	0.385630	−	0.922654i	$-0.373984\pi$
$251$	4.84990e10	0.771260	0.385630	+	0.922654i	$0.373984\pi$
$252$	0	0
$253$	5.67225e9	0.0870389
$254$	9.63808e9	0.145291
$255$	0	0
$256$	2.73134e9	0.0397463
$257$	3.64062e10	0.520567	0.260284	−	0.965532i	$-0.416184\pi$
$257$	3.64062e10	0.520567	0.260284	+	0.965532i	$0.416184\pi$
$258$	0	0
$259$	3.93966e10	0.544014
$260$	1.69760e11	2.30386
$261$	0	0
$262$	1.38994e10	0.182238
$263$	−2.73384e10	−0.352349	−0.176174	−	0.984359i	$-0.556372\pi$
$263$	−2.73384e10	−0.352349	−0.176174	+	0.984359i	$0.556372\pi$
$264$	0	0
$265$	−7.34650e10	−0.915111
$266$	2.44598e9	0.0299561
$267$	0	0
$268$	1.24563e11	1.47497
$269$	1.34342e9	0.0156432	0.00782159	−	0.999969i	$-0.497510\pi$
$269$	1.34342e9	0.0156432	0.00782159	+	0.999969i	$0.497510\pi$
$270$	0	0
$271$	−1.98471e10	−0.223530	−0.111765	−	0.993735i	$-0.535650\pi$
$271$	−1.98471e10	−0.223530	−0.111765	+	0.993735i	$0.535650\pi$
$272$	7.59538e10	0.841375
$273$	0	0
$274$	−4.99133e10	−0.534981
$275$	5.46655e10	0.576389
$276$	0	0
$277$	−1.33376e11	−1.36119	−0.680594	−	0.732661i	$-0.738278\pi$
$277$	−1.33376e11	−1.36119	−0.680594	+	0.732661i	$0.738278\pi$
$278$	1.40511e10	0.141094
$279$	0	0
$280$	−1.01954e11	−0.991272
$281$	6.43335e9	0.0615543	0.0307772	−	0.999526i	$-0.490202\pi$
$281$	6.43335e9	0.0615543	0.0307772	+	0.999526i	$0.490202\pi$
$282$	0	0
$283$	−3.84797e10	−0.356609	−0.178305	−	0.983975i	$-0.557061\pi$
$283$	−3.84797e10	−0.356609	−0.178305	+	0.983975i	$0.557061\pi$
$284$	1.33229e11	1.21525
$285$	0	0
$286$	2.41804e10	0.213706
$287$	1.54571e10	0.134480
$288$	0	0
$289$	6.89991e10	0.581839
$290$	1.92837e10	0.160103
$291$	0	0
$292$	7.02932e10	0.565836
$293$	−1.85929e11	−1.47381	−0.736906	−	0.675995i	$-0.763714\pi$
$293$	−1.85929e11	−1.47381	−0.736906	+	0.675995i	$0.763714\pi$
$294$	0	0
$295$	−1.03297e11	−0.794123
$296$	−4.67594e10	−0.354043
$297$	0	0
$298$	8.19013e9	0.0601613
$299$	−5.15927e10	−0.373309
$300$	0	0
$301$	1.94848e11	1.36819
$302$	6.19246e10	0.428383
$303$	0	0
$304$	8.97883e9	0.0602960
$305$	−1.35709e11	−0.897966
$306$	0	0
$307$	1.44816e11	0.930450	0.465225	−	0.885192i	$-0.345973\pi$
$307$	1.44816e11	0.930450	0.465225	+	0.885192i	$0.345973\pi$
$308$	5.26177e10	0.333161
$309$	0	0
$310$	1.31935e10	0.0811393
$311$	1.72871e10	0.104785	0.0523925	−	0.998627i	$-0.483315\pi$
$311$	1.72871e10	0.104785	0.0523925	+	0.998627i	$0.483315\pi$
$312$	0	0
$313$	6.06058e10	0.356915	0.178457	−	0.983948i	$-0.442889\pi$
$313$	6.06058e10	0.356915	0.178457	+	0.983948i	$0.442889\pi$
$314$	−5.55580e10	−0.322525
$315$	0	0
$316$	1.06863e11	0.602887
$317$	2.47858e11	1.37860	0.689298	−	0.724478i	$-0.257919\pi$
$317$	2.47858e11	1.37860	0.689298	+	0.724478i	$0.257919\pi$
$318$	0	0
$319$	−2.11942e10	−0.114593
$320$	−1.11555e11	−0.594723
$321$	0	0
$322$	1.45498e10	0.0754235
$323$	2.21755e10	0.113360
$324$	0	0
$325$	−4.97217e11	−2.47213
$326$	9.63605e10	0.472520
$327$	0	0
$328$	−1.83458e10	−0.0875195
$329$	−6.32091e10	−0.297439
$330$	0	0
$331$	3.04696e10	0.139521	0.0697607	−	0.997564i	$-0.477776\pi$
$331$	3.04696e10	0.139521	0.0697607	+	0.997564i	$0.477776\pi$
$332$	−1.52358e11	−0.688248
$333$	0	0
$334$	1.08446e11	0.476821
$335$	−6.07607e11	−2.63585
$336$	0	0
$337$	5.33543e10	0.225338	0.112669	−	0.993633i	$-0.464060\pi$
$337$	5.33543e10	0.225338	0.112669	+	0.993633i	$0.464060\pi$
$338$	−1.38660e11	−0.577864
$339$	0	0
$340$	−4.34036e11	−1.76145
$341$	−1.45006e10	−0.0580751
$342$	0	0
$343$	2.60889e11	1.01773
$344$	−2.31263e11	−0.890415
$345$	0	0
$346$	9.41293e10	0.353088
$347$	−1.25123e11	−0.463291	−0.231646	−	0.972800i	$-0.574411\pi$
$347$	−1.25123e11	−0.463291	−0.231646	+	0.972800i	$0.574411\pi$
$348$	0	0
$349$	2.55017e11	0.920140	0.460070	−	0.887883i	$-0.347824\pi$
$349$	2.55017e11	0.920140	0.460070	+	0.887883i	$0.347824\pi$
$350$	1.40222e11	0.499470
$351$	0	0
$352$	−9.55772e10	−0.331827
$353$	3.33062e10	0.114167	0.0570834	−	0.998369i	$-0.481820\pi$
$353$	3.33062e10	0.114167	0.0570834	+	0.998369i	$0.481820\pi$
$354$	0	0
$355$	−6.49877e11	−2.17172
$356$	4.70535e11	1.55263
$357$	0	0
$358$	4.19244e9	0.0134894
$359$	−8.10854e10	−0.257643	−0.128821	−	0.991668i	$-0.541119\pi$
$359$	−8.10854e10	−0.257643	−0.128821	+	0.991668i	$0.541119\pi$
$360$	0	0
$361$	−3.20066e11	−0.991876
$362$	−6.48845e10	−0.198588
$363$	0	0
$364$	−4.78591e11	−1.42892
$365$	−3.42884e11	−1.01118
$366$	0	0
$367$	4.71294e11	1.35611	0.678054	−	0.735012i	$-0.262824\pi$
$367$	4.71294e11	1.35611	0.678054	+	0.735012i	$0.262824\pi$
$368$	5.34103e10	0.151813
$369$	0	0
$370$	1.07104e11	0.297096
$371$	2.07114e11	0.567579
$372$	0	0
$373$	5.06336e11	1.35441	0.677204	−	0.735795i	$-0.263192\pi$
$373$	5.06336e11	1.35441	0.677204	+	0.735795i	$0.263192\pi$
$374$	−6.18235e10	−0.163392
$375$	0	0
$376$	7.50221e10	0.193573
$377$	1.92775e11	0.491489
$378$	0	0
$379$	3.43255e11	0.854557	0.427278	−	0.904120i	$-0.359472\pi$
$379$	3.43255e11	0.854557	0.427278	+	0.904120i	$0.359472\pi$
$380$	−5.13093e10	−0.126232
$381$	0	0
$382$	2.23653e11	0.537390
$383$	−4.39589e11	−1.04388	−0.521942	−	0.852981i	$-0.674792\pi$
$383$	−4.39589e11	−1.04388	−0.521942	+	0.852981i	$0.674792\pi$
$384$	0	0
$385$	−2.56664e11	−0.595377
$386$	−1.62882e11	−0.373449
$387$	0	0
$388$	1.22186e11	0.273703
$389$	−5.04529e11	−1.11715	−0.558577	−	0.829453i	$-0.688653\pi$
$389$	−5.04529e11	−1.11715	−0.558577	+	0.829453i	$0.688653\pi$
$390$	0	0
$391$	1.31910e11	0.285419
$392$	−1.11076e10	−0.0237592
$393$	0	0
$394$	−1.47071e11	−0.307464
$395$	−5.21269e11	−1.07739
$396$	0	0
$397$	−2.62244e11	−0.529844	−0.264922	−	0.964270i	$-0.585346\pi$
$397$	−2.62244e11	−0.529844	−0.264922	+	0.964270i	$0.585346\pi$
$398$	−4.60709e10	−0.0920350
$399$	0	0
$400$	5.14734e11	1.00534
$401$	2.47525e11	0.478046	0.239023	−	0.971014i	$-0.423173\pi$
$401$	2.47525e11	0.478046	0.239023	+	0.971014i	$0.423173\pi$
$402$	0	0
$403$	1.31892e11	0.249084
$404$	−4.44235e11	−0.829654
$405$	0	0
$406$	−5.43651e10	−0.0993008
$407$	−1.17715e11	−0.212645
$408$	0	0
$409$	1.55744e11	0.275205	0.137603	−	0.990488i	$-0.456060\pi$
$409$	1.55744e11	0.275205	0.137603	+	0.990488i	$0.456060\pi$
$410$	4.20216e10	0.0734421
$411$	0	0
$412$	7.58607e10	0.129712
$413$	2.91216e11	0.492538
$414$	0	0
$415$	7.43190e11	1.22994
$416$	8.69335e11	1.42320
$417$	0	0
$418$	−7.30842e9	−0.0117093
$419$	9.20355e11	1.45879	0.729394	−	0.684094i	$-0.239802\pi$
$419$	9.20355e11	1.45879	0.729394	+	0.684094i	$0.239802\pi$
$420$	0	0
$421$	−8.56838e11	−1.32932	−0.664660	−	0.747146i	$-0.731423\pi$
$421$	−8.56838e11	−1.32932	−0.664660	+	0.747146i	$0.731423\pi$
$422$	9.27419e10	0.142354
$423$	0	0
$424$	−2.45821e11	−0.369379
$425$	1.27126e12	1.89010
$426$	0	0
$427$	3.82594e11	0.556945
$428$	−8.70004e11	−1.25321
$429$	0	0
$430$	5.29713e11	0.747193
$431$	5.94731e11	0.830181	0.415090	−	0.909780i	$-0.363750\pi$
$431$	5.94731e11	0.830181	0.415090	+	0.909780i	$0.363750\pi$
$432$	0	0
$433$	−1.28268e12	−1.75357	−0.876787	−	0.480880i	$-0.840317\pi$
$433$	−1.28268e12	−1.75357	−0.876787	+	0.480880i	$0.840317\pi$
$434$	−3.71953e10	−0.0503250
$435$	0	0
$436$	−1.18242e11	−0.156704
$437$	1.55937e10	0.0204542
$438$	0	0
$439$	−8.01266e11	−1.02964	−0.514821	−	0.857298i	$-0.672142\pi$
$439$	−8.01266e11	−1.02964	−0.514821	+	0.857298i	$0.672142\pi$
$440$	3.04631e11	0.387470
$441$	0	0
$442$	5.62324e11	0.700787
$443$	7.15042e11	0.882094	0.441047	−	0.897484i	$-0.354607\pi$
$443$	7.15042e11	0.882094	0.441047	+	0.897484i	$0.354607\pi$
$444$	0	0
$445$	−2.29522e12	−2.77463
$446$	4.03433e11	0.482797
$447$	0	0
$448$	3.14498e11	0.368865
$449$	8.30808e11	0.964700	0.482350	−	0.875979i	$-0.339783\pi$
$449$	8.30808e11	0.964700	0.482350	+	0.875979i	$0.339783\pi$
$450$	0	0
$451$	−4.61847e10	−0.0525659
$452$	−1.48037e12	−1.66820
$453$	0	0
$454$	−9.50058e10	−0.104954
$455$	2.33452e12	2.55357
$456$	0	0
$457$	5.61439e11	0.602115	0.301058	−	0.953606i	$-0.402660\pi$
$457$	5.61439e11	0.602115	0.301058	+	0.953606i	$0.402660\pi$
$458$	−3.38266e11	−0.359223
$459$	0	0
$460$	−3.05212e11	−0.317827
$461$	5.35589e11	0.552303	0.276151	−	0.961114i	$-0.410941\pi$
$461$	5.35589e11	0.552303	0.276151	+	0.961114i	$0.410941\pi$
$462$	0	0
$463$	1.26735e12	1.28168	0.640841	−	0.767673i	$-0.278586\pi$
$463$	1.26735e12	1.28168	0.640841	+	0.767673i	$0.278586\pi$
$464$	−1.99566e11	−0.199874
$465$	0	0
$466$	4.17114e11	0.409749
$467$	9.53957e11	0.928117	0.464059	−	0.885804i	$-0.346393\pi$
$467$	9.53957e11	0.928117	0.464059	+	0.885804i	$0.346393\pi$
$468$	0	0
$469$	1.71298e12	1.63483
$470$	−1.71840e11	−0.162437
$471$	0	0
$472$	−3.45641e11	−0.320543
$473$	−5.82193e11	−0.534800
$474$	0	0
$475$	1.50282e11	0.135452
$476$	1.22364e12	1.09251
$477$	0	0
$478$	−6.46765e11	−0.566658
$479$	−1.28688e12	−1.11694	−0.558469	−	0.829525i	$-0.688611\pi$
$479$	−1.28688e12	−1.11694	−0.558469	+	0.829525i	$0.688611\pi$
$480$	0	0
$481$	1.07069e12	0.912032
$482$	−4.97637e11	−0.419953
$483$	0	0
$484$	9.11541e11	0.755044
$485$	−5.96012e11	−0.489122
$486$	0	0
$487$	−1.22356e12	−0.985704	−0.492852	−	0.870113i	$-0.664046\pi$
$487$	−1.22356e12	−0.985704	−0.492852	+	0.870113i	$0.664046\pi$
$488$	−4.54095e11	−0.362458
$489$	0	0
$490$	2.54422e10	0.0199376
$491$	−1.17058e12	−0.908940	−0.454470	−	0.890762i	$-0.650171\pi$
$491$	−1.17058e12	−0.908940	−0.454470	+	0.890762i	$0.650171\pi$
$492$	0	0
$493$	−4.92878e11	−0.375776
$494$	6.64747e10	0.0502210
$495$	0	0
$496$	−1.36538e11	−0.101295
$497$	1.83214e12	1.34696
$498$	0	0
$499$	−2.21768e12	−1.60120	−0.800602	−	0.599197i	$-0.795487\pi$
$499$	−2.21768e12	−1.60120	−0.800602	+	0.599197i	$0.795487\pi$
$500$	−9.84146e11	−0.704198
$501$	0	0
$502$	−3.71711e11	−0.261240
$503$	−2.31868e12	−1.61505	−0.807524	−	0.589834i	$-0.799193\pi$
$503$	−2.31868e12	−1.61505	−0.807524	+	0.589834i	$0.799193\pi$
$504$	0	0
$505$	2.16694e12	1.48264
$506$	−4.34739e10	−0.0294816
$507$	0	0
$508$	5.69984e11	0.379731
$509$	2.53321e12	1.67279	0.836396	−	0.548126i	$-0.184658\pi$
$509$	2.53321e12	1.67279	0.836396	+	0.548126i	$0.184658\pi$
$510$	0	0
$511$	9.66664e11	0.627164
$512$	−1.56422e12	−1.00596
$513$	0	0
$514$	−2.79029e11	−0.176326
$515$	−3.70041e11	−0.231802
$516$	0	0
$517$	1.88865e11	0.116263
$518$	−3.01948e11	−0.184268
$519$	0	0
$520$	−2.77082e12	−1.66185
$521$	9.44125e11	0.561384	0.280692	−	0.959798i	$-0.409436\pi$
$521$	9.44125e11	0.561384	0.280692	+	0.959798i	$0.409436\pi$
$522$	0	0
$523$	−1.67874e12	−0.981128	−0.490564	−	0.871405i	$-0.663209\pi$
$523$	−1.67874e12	−0.981128	−0.490564	+	0.871405i	$0.663209\pi$
$524$	8.21991e11	0.476295
$525$	0	0
$526$	2.09531e11	0.119347
$527$	−3.37216e11	−0.190441
$528$	0	0
$529$	−1.70839e12	−0.948500
$530$	5.63059e11	0.309965
$531$	0	0
$532$	1.44652e11	0.0782929
$533$	4.20079e11	0.225454
$534$	0	0
$535$	4.24380e12	2.23956
$536$	−2.03311e12	−1.06395
$537$	0	0
$538$	−1.02964e10	−0.00529863
$539$	−2.79628e10	−0.0142703
$540$	0	0
$541$	−1.34365e12	−0.674372	−0.337186	−	0.941438i	$-0.609475\pi$
$541$	−1.34365e12	−0.674372	−0.337186	+	0.941438i	$0.609475\pi$
$542$	1.52115e11	0.0757137
$543$	0	0
$544$	−2.22268e12	−1.08813
$545$	5.76772e11	0.280040
$546$	0	0
$547$	−6.08783e11	−0.290750	−0.145375	−	0.989377i	$-0.546439\pi$
$547$	−6.08783e11	−0.290750	−0.145375	+	0.989377i	$0.546439\pi$
$548$	−2.95181e12	−1.39822
$549$	0	0
$550$	−4.18974e11	−0.195234
$551$	−5.82653e10	−0.0269295
$552$	0	0
$553$	1.46957e12	0.668232
$554$	1.02223e12	0.461059
$555$	0	0
$556$	8.30967e11	0.368763
$557$	−6.12226e11	−0.269503	−0.134751	−	0.990879i	$-0.543024\pi$
$557$	−6.12226e11	−0.269503	−0.134751	+	0.990879i	$0.543024\pi$
$558$	0	0
$559$	5.29541e12	2.29375
$560$	−2.41677e12	−1.03846
$561$	0	0
$562$	−4.93072e10	−0.0208496
$563$	−4.24388e12	−1.78023	−0.890114	−	0.455737i	$-0.849376\pi$
$563$	−4.24388e12	−1.78023	−0.890114	+	0.455737i	$0.849376\pi$
$564$	0	0
$565$	7.22111e12	2.98116
$566$	2.94921e11	0.120790
$567$	0	0
$568$	−2.17455e12	−0.876600
$569$	−4.10789e12	−1.64291	−0.821456	−	0.570273i	$-0.806838\pi$
$569$	−4.10789e12	−1.64291	−0.821456	+	0.570273i	$0.806838\pi$
$570$	0	0
$571$	−3.81741e12	−1.50282	−0.751409	−	0.659837i	$-0.770625\pi$
$571$	−3.81741e12	−1.50282	−0.751409	+	0.659837i	$0.770625\pi$
$572$	1.43000e12	0.558539
$573$	0	0
$574$	−1.18468e11	−0.0455510
$575$	8.93946e11	0.341041
$576$	0	0
$577$	1.58536e12	0.595440	0.297720	−	0.954653i	$-0.403774\pi$
$577$	1.58536e12	0.595440	0.297720	+	0.954653i	$0.403774\pi$
$578$	−5.28831e11	−0.197080
$579$	0	0
$580$	1.14042e12	0.418444
$581$	−2.09521e12	−0.762844
$582$	0	0
$583$	−6.18842e11	−0.221856
$584$	−1.14732e12	−0.408157
$585$	0	0
$586$	1.42502e12	0.499207
$587$	1.43205e12	0.497836	0.248918	−	0.968525i	$-0.419925\pi$
$587$	1.43205e12	0.497836	0.248918	+	0.968525i	$0.419925\pi$
$588$	0	0
$589$	−3.98637e10	−0.0136477
$590$	7.91699e11	0.268984
$591$	0	0
$592$	−1.10841e12	−0.370896
$593$	4.74549e12	1.57592	0.787961	−	0.615725i	$-0.211137\pi$
$593$	4.74549e12	1.57592	0.787961	+	0.615725i	$0.211137\pi$
$594$	0	0
$595$	−5.96881e12	−1.95237
$596$	4.84354e11	0.157237
$597$	0	0
$598$	3.95423e11	0.126446
$599$	1.57237e11	0.0499038	0.0249519	−	0.999689i	$-0.492057\pi$
$599$	1.57237e11	0.0499038	0.0249519	+	0.999689i	$0.492057\pi$
$600$	0	0
$601$	−3.30017e12	−1.03181	−0.515906	−	0.856645i	$-0.672545\pi$
$601$	−3.30017e12	−1.03181	−0.515906	+	0.856645i	$0.672545\pi$
$602$	−1.49338e12	−0.463431
$603$	0	0
$604$	3.66214e12	1.11962
$605$	−4.44641e12	−1.34931
$606$	0	0
$607$	−3.90342e12	−1.16707	−0.583534	−	0.812088i	$-0.698331\pi$
$607$	−3.90342e12	−1.16707	−0.583534	+	0.812088i	$0.698331\pi$
$608$	−2.62753e11	−0.0779796
$609$	0	0
$610$	1.04012e12	0.304157
$611$	−1.71784e12	−0.498653
$612$	0	0
$613$	−2.67659e12	−0.765615	−0.382807	−	0.923828i	$-0.625043\pi$
$613$	−2.67659e12	−0.765615	−0.382807	+	0.923828i	$0.625043\pi$
$614$	−1.10991e12	−0.315160
$615$	0	0
$616$	−8.58822e11	−0.240320
$617$	3.64857e12	1.01354	0.506769	−	0.862082i	$-0.330840\pi$
$617$	3.64857e12	1.01354	0.506769	+	0.862082i	$0.330840\pi$
$618$	0	0
$619$	9.66513e11	0.264606	0.132303	−	0.991209i	$-0.457763\pi$
$619$	9.66513e11	0.264606	0.132303	+	0.991209i	$0.457763\pi$
$620$	7.80245e11	0.212065
$621$	0	0
$622$	−1.32493e11	−0.0354926
$623$	6.47074e12	1.72091
$624$	0	0
$625$	−9.32197e11	−0.244370
$626$	−4.64502e11	−0.120894
$627$	0	0
$628$	−3.28563e12	−0.842948
$629$	−2.73749e12	−0.697308
$630$	0	0
$631$	−4.92249e12	−1.23610	−0.618049	−	0.786139i	$-0.712077\pi$
$631$	−4.92249e12	−1.23610	−0.618049	+	0.786139i	$0.712077\pi$
$632$	−1.74421e12	−0.434883
$633$	0	0
$634$	−1.89966e12	−0.466955
$635$	−2.78033e12	−0.678601
$636$	0	0
$637$	2.54340e11	0.0612049
$638$	1.62439e11	0.0388148
$639$	0	0
$640$	6.66431e12	1.57016
$641$	4.81168e12	1.12573	0.562866	−	0.826548i	$-0.309699\pi$
$641$	4.81168e12	1.12573	0.562866	+	0.826548i	$0.309699\pi$
$642$	0	0
$643$	5.33301e12	1.23033	0.615167	−	0.788397i	$-0.289089\pi$
$643$	5.33301e12	1.23033	0.615167	+	0.788397i	$0.289089\pi$
$644$	8.60459e11	0.197126
$645$	0	0
$646$	−1.69960e11	−0.0383972
$647$	6.50961e12	1.46045	0.730223	−	0.683209i	$-0.239416\pi$
$647$	6.50961e12	1.46045	0.730223	+	0.683209i	$0.239416\pi$
$648$	0	0
$649$	−8.70134e11	−0.192524
$650$	3.81083e12	0.837355
$651$	0	0
$652$	5.69864e12	1.23497
$653$	5.35684e12	1.15292	0.576460	−	0.817125i	$-0.304434\pi$
$653$	5.35684e12	1.15292	0.576460	+	0.817125i	$0.304434\pi$
$654$	0	0
$655$	−4.00959e12	−0.851166
$656$	−4.34879e11	−0.0916856
$657$	0	0
$658$	4.84455e11	0.100748
$659$	1.30702e12	0.269959	0.134980	−	0.990848i	$-0.456903\pi$
$659$	1.30702e12	0.269959	0.134980	+	0.990848i	$0.456903\pi$
$660$	0	0
$661$	1.21681e11	0.0247922	0.0123961	−	0.999923i	$-0.496054\pi$
$661$	1.21681e11	0.0247922	0.0123961	+	0.999923i	$0.496054\pi$
$662$	−2.33529e11	−0.0472585
$663$	0	0
$664$	2.48678e12	0.496457
$665$	−7.05599e11	−0.139914
$666$	0	0
$667$	−3.46590e11	−0.0678030
$668$	6.41337e12	1.24621
$669$	0	0
$670$	4.65690e12	0.892812
$671$	−1.14316e12	−0.217699
$672$	0	0
$673$	3.29507e12	0.619151	0.309575	−	0.950875i	$-0.399813\pi$
$673$	3.29507e12	0.619151	0.309575	+	0.950875i	$0.399813\pi$
$674$	−4.08924e11	−0.0763261
$675$	0	0
$676$	−8.20017e12	−1.51030
$677$	1.16832e12	0.213754	0.106877	−	0.994272i	$-0.465915\pi$
$677$	1.16832e12	0.213754	0.106877	+	0.994272i	$0.465915\pi$
$678$	0	0
$679$	1.68029e12	0.303368
$680$	7.08431e12	1.27060
$681$	0	0
$682$	1.11137e11	0.0196711
$683$	7.05520e11	0.124055	0.0620277	−	0.998074i	$-0.480243\pi$
$683$	7.05520e11	0.124055	0.0620277	+	0.998074i	$0.480243\pi$
$684$	0	0
$685$	1.43986e13	2.49870
$686$	−1.99953e12	−0.344723
$687$	0	0
$688$	−5.48197e12	−0.932800
$689$	5.62876e12	0.951538
$690$	0	0
$691$	−5.69125e12	−0.949635	−0.474817	−	0.880084i	$-0.657486\pi$
$691$	−5.69125e12	−0.949635	−0.474817	+	0.880084i	$0.657486\pi$
$692$	5.56669e12	0.922825
$693$	0	0
$694$	9.58982e11	0.156925
$695$	−4.05338e12	−0.659000
$696$	0	0
$697$	−1.07404e12	−0.172375
$698$	−1.95453e12	−0.311668
$699$	0	0
$700$	8.29255e12	1.30541
$701$	−3.75961e12	−0.588046	−0.294023	−	0.955798i	$-0.594994\pi$
$701$	−3.75961e12	−0.588046	−0.294023	+	0.955798i	$0.594994\pi$
$702$	0	0
$703$	−3.23611e11	−0.0499717
$704$	−9.39700e11	−0.144182
$705$	0	0
$706$	−2.55270e11	−0.0386703
$707$	−6.10907e12	−0.919576
$708$	0	0
$709$	−3.70506e12	−0.550665	−0.275333	−	0.961349i	$-0.588788\pi$
$709$	−3.70506e12	−0.550665	−0.275333	+	0.961349i	$0.588788\pi$
$710$	4.98086e12	0.735600
$711$	0	0
$712$	−7.68003e12	−1.11996
$713$	−2.37128e11	−0.0343622
$714$	0	0
$715$	−6.97540e12	−0.998141
$716$	2.47936e11	0.0352558
$717$	0	0
$718$	6.21464e11	0.0872683
$719$	−1.23468e11	−0.0172296	−0.00861478	−	0.999963i	$-0.502742\pi$
$719$	−1.23468e11	−0.0172296	−0.00861478	+	0.999963i	$0.502742\pi$
$720$	0	0
$721$	1.04323e12	0.143771
$722$	2.45309e12	0.335967
$723$	0	0
$724$	−3.83719e12	−0.519026
$725$	−3.34020e12	−0.449006
$726$	0	0
$727$	1.17989e13	1.56653	0.783264	−	0.621689i	$-0.213553\pi$
$727$	1.17989e13	1.56653	0.783264	+	0.621689i	$0.213553\pi$
$728$	7.81153e12	1.03073
$729$	0	0
$730$	2.62797e12	0.342505
$731$	−1.35391e13	−1.75372
$732$	0	0
$733$	−7.06822e12	−0.904361	−0.452181	−	0.891926i	$-0.649354\pi$
$733$	−7.06822e12	−0.904361	−0.452181	+	0.891926i	$0.649354\pi$
$734$	−3.61214e12	−0.459338
$735$	0	0
$736$	−1.56298e12	−0.196337
$737$	−5.11826e12	−0.639026
$738$	0	0
$739$	−4.61726e12	−0.569488	−0.284744	−	0.958604i	$-0.591909\pi$
$739$	−4.61726e12	−0.569488	−0.284744	+	0.958604i	$0.591909\pi$
$740$	6.33397e12	0.776486
$741$	0	0
$742$	−1.58739e12	−0.192249
$743$	−1.51136e12	−0.181935	−0.0909677	−	0.995854i	$-0.528996\pi$
$743$	−1.51136e12	−0.181935	−0.0909677	+	0.995854i	$0.528996\pi$
$744$	0	0
$745$	−2.36263e12	−0.280991
$746$	−3.88072e12	−0.458763
$747$	0	0
$748$	−3.65616e12	−0.427040
$749$	−1.19642e13	−1.38904
$750$	0	0
$751$	1.49806e13	1.71849	0.859247	−	0.511560i	$-0.170932\pi$
$751$	1.49806e13	1.71849	0.859247	+	0.511560i	$0.170932\pi$
$752$	1.77836e12	0.202787
$753$	0	0
$754$	−1.47749e12	−0.166476
$755$	−1.78636e13	−2.00082
$756$	0	0
$757$	3.27715e12	0.362714	0.181357	−	0.983417i	$-0.441951\pi$
$757$	3.27715e12	0.362714	0.181357	+	0.983417i	$0.441951\pi$
$758$	−2.63082e12	−0.289454
$759$	0	0
$760$	8.37466e11	0.0910555
$761$	5.83618e12	0.630809	0.315404	−	0.948957i	$-0.397860\pi$
$761$	5.83618e12	0.630809	0.315404	+	0.948957i	$0.397860\pi$
$762$	0	0
$763$	−1.62605e12	−0.173689
$764$	1.32266e13	1.40452
$765$	0	0
$766$	3.36915e12	0.353583
$767$	7.91442e12	0.825734
$768$	0	0
$769$	−1.57659e12	−0.162574	−0.0812870	−	0.996691i	$-0.525903\pi$
$769$	−1.57659e12	−0.162574	−0.0812870	+	0.996691i	$0.525903\pi$
$770$	1.96716e12	0.201665
$771$	0	0
$772$	−9.63266e12	−0.976042
$773$	1.49007e13	1.50106	0.750529	−	0.660837i	$-0.229799\pi$
$773$	1.49007e13	1.50106	0.750529	+	0.660837i	$0.229799\pi$
$774$	0	0
$775$	−2.28529e12	−0.227553
$776$	−1.99431e12	−0.197431
$777$	0	0
$778$	3.86687e12	0.378401
$779$	−1.26967e11	−0.0123530
$780$	0	0
$781$	−5.47432e12	−0.526503
$782$	−1.01100e12	−0.0966766
$783$	0	0
$784$	−2.63300e11	−0.0248902
$785$	1.60270e13	1.50640
$786$	0	0
$787$	9.89437e12	0.919394	0.459697	−	0.888076i	$-0.347958\pi$
$787$	9.89437e12	0.919394	0.459697	+	0.888076i	$0.347958\pi$
$788$	−8.69760e12	−0.803584
$789$	0	0
$790$	3.99517e12	0.364933
$791$	−2.03579e13	−1.84901
$792$	0	0
$793$	1.03978e13	0.933710
$794$	2.00992e12	0.179468
$795$	0	0
$796$	−2.72458e12	−0.240542
$797$	2.55007e11	0.0223866	0.0111933	−	0.999937i	$-0.496437\pi$
$797$	2.55007e11	0.0223866	0.0111933	+	0.999937i	$0.496437\pi$
$798$	0	0
$799$	4.39211e12	0.381253
$800$	−1.50630e13	−1.30018
$801$	0	0
$802$	−1.89711e12	−0.161923
$803$	−2.88833e12	−0.245147
$804$	0	0
$805$	−4.19724e12	−0.352275
$806$	−1.01086e12	−0.0843692
$807$	0	0
$808$	7.25078e12	0.598458
$809$	−1.62301e13	−1.33215	−0.666075	−	0.745885i	$-0.732027\pi$
$809$	−1.62301e13	−1.33215	−0.666075	+	0.745885i	$0.732027\pi$
$810$	0	0
$811$	−4.16753e12	−0.338287	−0.169143	−	0.985591i	$-0.554100\pi$
$811$	−4.16753e12	−0.338287	−0.169143	+	0.985591i	$0.554100\pi$
$812$	−3.21508e12	−0.259531
$813$	0	0
$814$	9.02202e11	0.0720267
$815$	−2.77974e13	−2.20696
$816$	0	0
$817$	−1.60051e12	−0.125678
$818$	−1.19367e12	−0.0932171
$819$	0	0
$820$	2.48510e12	0.191947
$821$	−2.21789e12	−0.170371	−0.0851855	−	0.996365i	$-0.527148\pi$
$821$	−2.21789e12	−0.170371	−0.0851855	+	0.996365i	$0.527148\pi$
$822$	0	0
$823$	−4.73373e11	−0.0359670	−0.0179835	−	0.999838i	$-0.505725\pi$
$823$	−4.73373e11	−0.0359670	−0.0179835	+	0.999838i	$0.505725\pi$
$824$	−1.23819e12	−0.0935655
$825$	0	0
$826$	−2.23197e12	−0.166832
$827$	−3.43656e12	−0.255475	−0.127738	−	0.991808i	$-0.540772\pi$
$827$	−3.43656e12	−0.255475	−0.127738	+	0.991808i	$0.540772\pi$
$828$	0	0
$829$	−1.73227e13	−1.27386	−0.636929	−	0.770922i	$-0.719796\pi$
$829$	−1.73227e13	−1.27386	−0.636929	+	0.770922i	$0.719796\pi$
$830$	−5.69604e12	−0.416603
$831$	0	0
$832$	8.54716e12	0.618397
$833$	−6.50285e11	−0.0467952
$834$	0	0
$835$	−3.12838e13	−2.22705
$836$	−4.32211e11	−0.0306032
$837$	0	0
$838$	−7.05389e12	−0.494118
$839$	−2.44994e13	−1.70697	−0.853486	−	0.521116i	$-0.825516\pi$
$839$	−2.44994e13	−1.70697	−0.853486	+	0.521116i	$0.825516\pi$
$840$	0	0
$841$	−1.32121e13	−0.910732
$842$	6.56708e12	0.450265
$843$	0	0
$844$	5.48464e12	0.372055
$845$	3.99997e13	2.69899
$846$	0	0
$847$	1.25354e13	0.836879
$848$	−5.82706e12	−0.386962
$849$	0	0
$850$	−9.74337e12	−0.640212
$851$	−1.92499e12	−0.125819
$852$	0	0
$853$	3.85406e12	0.249257	0.124629	−	0.992203i	$-0.460226\pi$
$853$	3.85406e12	0.249257	0.124629	+	0.992203i	$0.460226\pi$
$854$	−2.93232e12	−0.188647
$855$	0	0
$856$	1.42001e13	0.903984
$857$	−3.76399e12	−0.238361	−0.119181	−	0.992873i	$-0.538027\pi$
$857$	−3.76399e12	−0.238361	−0.119181	+	0.992873i	$0.538027\pi$
$858$	0	0
$859$	−2.35407e12	−0.147520	−0.0737599	−	0.997276i	$-0.523500\pi$
$859$	−2.35407e12	−0.147520	−0.0737599	+	0.997276i	$0.523500\pi$
$860$	3.13266e13	1.95286
$861$	0	0
$862$	−4.55821e12	−0.281197
$863$	−1.83375e13	−1.12536	−0.562680	−	0.826675i	$-0.690230\pi$
$863$	−1.83375e13	−1.12536	−0.562680	+	0.826675i	$0.690230\pi$
$864$	0	0
$865$	−2.71538e13	−1.64914
$866$	9.83089e12	0.593967
$867$	0	0
$868$	−2.19968e12	−0.131529
$869$	−4.39098e12	−0.261199
$870$	0	0
$871$	4.65538e13	2.74078
$872$	1.92993e12	0.113036
$873$	0	0
$874$	−1.19515e11	−0.00692820
$875$	−1.35339e13	−0.780523
$876$	0	0
$877$	1.15063e12	0.0656805	0.0328402	−	0.999461i	$-0.489545\pi$
$877$	1.15063e12	0.0656805	0.0328402	+	0.999461i	$0.489545\pi$
$878$	6.14116e12	0.348758
$879$	0	0
$880$	7.22115e12	0.405914
$881$	2.37153e13	1.32628	0.663142	−	0.748494i	$-0.269223\pi$
$881$	2.37153e13	1.32628	0.663142	+	0.748494i	$0.269223\pi$
$882$	0	0
$883$	2.59683e13	1.43754	0.718771	−	0.695247i	$-0.244705\pi$
$883$	2.59683e13	1.43754	0.718771	+	0.695247i	$0.244705\pi$
$884$	3.32551e13	1.83157
$885$	0	0
$886$	−5.48031e12	−0.298781
$887$	−1.58210e13	−0.858178	−0.429089	−	0.903262i	$-0.641165\pi$
$887$	−1.58210e13	−0.858178	−0.429089	+	0.903262i	$0.641165\pi$
$888$	0	0
$889$	7.83835e12	0.420888
$890$	1.75913e13	0.939818
$891$	0	0
$892$	2.38585e13	1.26183
$893$	5.19210e11	0.0273220
$894$	0	0
$895$	−1.20941e12	−0.0630041
$896$	−1.87881e13	−0.973862
$897$	0	0
$898$	−6.36758e12	−0.326762
$899$	8.86023e11	0.0452404
$900$	0	0
$901$	−1.43914e13	−0.727513
$902$	3.53975e11	0.0178050
$903$	0	0
$904$	2.41625e13	1.20333
$905$	1.87174e13	0.927529
$906$	0	0
$907$	1.55355e13	0.762243	0.381122	−	0.924525i	$-0.375538\pi$
$907$	1.55355e13	0.762243	0.381122	+	0.924525i	$0.375538\pi$
$908$	−5.61852e12	−0.274306
$909$	0	0
$910$	−1.78925e13	−0.864939
$911$	5.12476e12	0.246514	0.123257	−	0.992375i	$-0.460666\pi$
$911$	5.12476e12	0.246514	0.123257	+	0.992375i	$0.460666\pi$
$912$	0	0
$913$	6.26036e12	0.298182
$914$	−4.30305e12	−0.203947
$915$	0	0
$916$	−2.00046e13	−0.938861
$917$	1.13039e13	0.527919
$918$	0	0
$919$	2.60261e13	1.20362	0.601810	−	0.798639i	$-0.294446\pi$
$919$	2.60261e13	1.20362	0.601810	+	0.798639i	$0.294446\pi$
$920$	4.98165e12	0.229260
$921$	0	0
$922$	−4.10492e12	−0.187075
$923$	4.97924e13	2.25817
$924$	0	0
$925$	−1.85518e13	−0.833198
$926$	−9.71334e12	−0.434129
$927$	0	0
$928$	5.84002e12	0.258493
$929$	4.03671e13	1.77810	0.889052	−	0.457806i	$-0.151365\pi$
$929$	4.03671e13	1.77810	0.889052	+	0.457806i	$0.151365\pi$
$930$	0	0
$931$	−7.68730e10	−0.00335351
$932$	2.46676e13	1.07092
$933$	0	0
$934$	−7.31143e12	−0.314370
$935$	1.78344e13	0.763144
$936$	0	0
$937$	−1.54039e12	−0.0652835	−0.0326418	−	0.999467i	$-0.510392\pi$
$937$	−1.54039e12	−0.0652835	−0.0326418	+	0.999467i	$0.510392\pi$
$938$	−1.31288e13	−0.553748
$939$	0	0
$940$	−1.01624e13	−0.424543
$941$	−5.54554e12	−0.230564	−0.115282	−	0.993333i	$-0.536777\pi$
$941$	−5.54554e12	−0.230564	−0.115282	+	0.993333i	$0.536777\pi$
$942$	0	0
$943$	−7.55260e11	−0.0311024
$944$	−8.19325e12	−0.335801
$945$	0	0
$946$	4.46211e12	0.181147
$947$	1.76140e13	0.711676	0.355838	−	0.934548i	$-0.384195\pi$
$947$	1.76140e13	0.711676	0.355838	+	0.934548i	$0.384195\pi$
$948$	0	0
$949$	2.62712e13	1.05143
$950$	−1.15181e12	−0.0458800
$951$	0	0
$952$	−1.99722e13	−0.788061
$953$	−4.21531e13	−1.65543	−0.827717	−	0.561146i	$-0.810361\pi$
$953$	−4.21531e13	−1.65543	−0.827717	+	0.561146i	$0.810361\pi$
$954$	0	0
$955$	−6.45180e13	−2.50995
$956$	−3.82489e13	−1.48101
$957$	0	0
$958$	9.86308e12	0.378327
$959$	−4.05929e13	−1.54977
$960$	0	0
$961$	−2.58334e13	−0.977072
$962$	−8.20609e12	−0.308922
$963$	0	0
$964$	−2.94296e13	−1.09759
$965$	4.69872e13	1.74424
$966$	0	0
$967$	−3.25155e13	−1.19584	−0.597918	−	0.801558i	$-0.704005\pi$
$967$	−3.25155e13	−1.19584	−0.597918	+	0.801558i	$0.704005\pi$
$968$	−1.48781e13	−0.544639
$969$	0	0
$970$	4.56803e12	0.165675
$971$	4.38405e13	1.58266	0.791332	−	0.611386i	$-0.209388\pi$
$971$	4.38405e13	1.58266	0.791332	+	0.611386i	$0.209388\pi$
$972$	0	0
$973$	1.14274e13	0.408731
$974$	9.37778e12	0.333876
$975$	0	0
$976$	−1.07641e13	−0.379712
$977$	1.31296e13	0.461027	0.230514	−	0.973069i	$-0.425959\pi$
$977$	1.31296e13	0.461027	0.230514	+	0.973069i	$0.425959\pi$
$978$	0	0
$979$	−1.93341e13	−0.672671
$980$	1.50462e12	0.0521087
$981$	0	0
$982$	8.97171e12	0.307874
$983$	−1.85255e13	−0.632817	−0.316408	−	0.948623i	$-0.602477\pi$
$983$	−1.85255e13	−0.632817	−0.316408	+	0.948623i	$0.602477\pi$
$984$	0	0
$985$	4.24261e13	1.43605
$986$	3.77758e12	0.127282
$987$	0	0
$988$	3.93123e12	0.131257
$989$	−9.52062e12	−0.316433
$990$	0	0
$991$	3.95986e13	1.30421	0.652106	−	0.758127i	$-0.273885\pi$
$991$	3.95986e13	1.30421	0.652106	+	0.758127i	$0.273885\pi$
$992$	3.99560e12	0.131002
$993$	0	0
$994$	−1.40421e13	−0.456241
$995$	1.32902e13	0.429862
$996$	0	0
$997$	4.79505e13	1.53697	0.768485	−	0.639868i	$-0.221011\pi$
$997$	4.79505e13	1.53697	0.768485	+	0.639868i	$0.221011\pi$
$998$	1.69970e13	0.542357
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.10.a.c.1.4		8
3.2	odd	2		81.10.a.d.1.5		8
9.2	odd	6		27.10.c.a.10.4		16
9.4	even	3		9.10.c.a.7.5	yes	16
9.5	odd	6		27.10.c.a.19.4		16
9.7	even	3		9.10.c.a.4.5	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.10.c.a.4.5	✓	16	9.7	even	3
9.10.c.a.7.5	yes	16	9.4	even	3
27.10.c.a.10.4		16	9.2	odd	6
27.10.c.a.19.4		16	9.5	odd	6
81.10.a.c.1.4		8	1.1	even	1	trivial
81.10.a.d.1.5		8	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:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	81.a (trivial)

\(f(q)\)	\(=\)	\(q-7.66432 q^{2} -453.258 q^{4} +2210.95 q^{5} -6233.15 q^{7} +7398.05 q^{8} +O(q^{10})\)	\(q-7.66432 q^{2} -453.258 q^{4} +2210.95 q^{5} -6233.15 q^{7} +7398.05 q^{8} -16945.4 q^{10} +18624.2 q^{11} -169399. q^{13} +47772.9 q^{14} +175367. q^{16} +433113. q^{17} +51200.2 q^{19} -1.00213e6 q^{20} -142742. q^{22} +304563. q^{23} +2.93518e6 q^{25} +1.29833e6 q^{26} +2.82523e6 q^{28} -1.13799e6 q^{29} -778586. q^{31} -5.13187e6 q^{32} -3.31952e6 q^{34} -1.37812e7 q^{35} -6.32050e6 q^{37} -392414. q^{38} +1.63567e7 q^{40} -2.47982e6 q^{41} -3.12600e7 q^{43} -8.44159e6 q^{44} -2.33427e6 q^{46} +1.01408e7 q^{47} -1.50142e6 q^{49} -2.24961e7 q^{50} +7.67816e7 q^{52} -3.32278e7 q^{53} +4.11773e7 q^{55} -4.61132e7 q^{56} +8.72192e6 q^{58} -4.67205e7 q^{59} -6.13804e7 q^{61} +5.96733e6 q^{62} -5.04557e7 q^{64} -3.74533e8 q^{65} -2.74817e8 q^{67} -1.96312e8 q^{68} +1.05623e8 q^{70} -2.93935e8 q^{71} -1.55084e8 q^{73} +4.84423e7 q^{74} -2.32069e7 q^{76} -1.16088e8 q^{77} -2.35767e8 q^{79} +3.87728e8 q^{80} +1.90061e7 q^{82} +3.36140e8 q^{83} +9.57592e8 q^{85} +2.39586e8 q^{86} +1.37783e8 q^{88} -1.03812e9 q^{89} +1.05589e9 q^{91} -1.38046e8 q^{92} -7.77223e7 q^{94} +1.13201e8 q^{95} -2.69573e8 q^{97} +1.15074e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 15 q^{2} + 1793 q^{4} - 453 q^{5} + 343 q^{7} - 7239 q^{8}+O(q^{10}) \) 8 * q - 15 * q^2 + 1793 * q^4 - 453 * q^5 + 343 * q^7 - 7239 * q^8	\( 8 q - 15 q^{2} + 1793 q^{4} - 453 q^{5} + 343 q^{7} - 7239 q^{8} + 510 q^{10} - 99150 q^{11} - 32435 q^{13} - 394824 q^{14} + 328193 q^{16} - 415539 q^{17} - 85277 q^{19} - 1855164 q^{20} - 529359 q^{22} - 1064559 q^{23} + 2293229 q^{25} + 1218156 q^{26} + 612862 q^{28} + 1309053 q^{29} + 2359819 q^{31} - 5760063 q^{32} - 981801 q^{34} - 15533277 q^{35} + 8195758 q^{37} - 39490203 q^{38} + 16760496 q^{40} - 54747318 q^{41} - 15249608 q^{43} - 166254963 q^{44} + 1195260 q^{46} - 156295545 q^{47} - 15239583 q^{49} - 315590163 q^{50} + 19773358 q^{52} - 262758114 q^{53} - 3789885 q^{55} - 470339790 q^{56} - 55408560 q^{58} - 307774074 q^{59} - 69192125 q^{61} - 457218462 q^{62} - 201794239 q^{64} - 482470359 q^{65} - 14328044 q^{67} - 915409575 q^{68} + 229271934 q^{70} - 619800696 q^{71} + 299306599 q^{73} - 1022736000 q^{74} - 119954093 q^{76} - 717995541 q^{77} - 30257531 q^{79} - 1463913264 q^{80} - 101188011 q^{82} - 1176168291 q^{83} - 4818366 q^{85} - 1426944009 q^{86} - 911312427 q^{88} - 1658520648 q^{89} - 369615061 q^{91} + 76813998 q^{92} + 1954316784 q^{94} + 391400652 q^{95} + 267311278 q^{97} + 2413650159 q^{98}+O(q^{100}) \) 8 * q - 15 * q^2 + 1793 * q^4 - 453 * q^5 + 343 * q^7 - 7239 * q^8 + 510 * q^10 - 99150 * q^11 - 32435 * q^13 - 394824 * q^14 + 328193 * q^16 - 415539 * q^17 - 85277 * q^19 - 1855164 * q^20 - 529359 * q^22 - 1064559 * q^23 + 2293229 * q^25 + 1218156 * q^26 + 612862 * q^28 + 1309053 * q^29 + 2359819 * q^31 - 5760063 * q^32 - 981801 * q^34 - 15533277 * q^35 + 8195758 * q^37 - 39490203 * q^38 + 16760496 * q^40 - 54747318 * q^41 - 15249608 * q^43 - 166254963 * q^44 + 1195260 * q^46 - 156295545 * q^47 - 15239583 * q^49 - 315590163 * q^50 + 19773358 * q^52 - 262758114 * q^53 - 3789885 * q^55 - 470339790 * q^56 - 55408560 * q^58 - 307774074 * q^59 - 69192125 * q^61 - 457218462 * q^62 - 201794239 * q^64 - 482470359 * q^65 - 14328044 * q^67 - 915409575 * q^68 + 229271934 * q^70 - 619800696 * q^71 + 299306599 * q^73 - 1022736000 * q^74 - 119954093 * q^76 - 717995541 * q^77 - 30257531 * q^79 - 1463913264 * q^80 - 101188011 * q^82 - 1176168291 * q^83 - 4818366 * q^85 - 1426944009 * q^86 - 911312427 * q^88 - 1658520648 * q^89 - 369615061 * q^91 + 76813998 * q^92 + 1954316784 * q^94 + 391400652 * q^95 + 267311278 * q^97 + 2413650159 * q^98

Introduction

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