LMFDB - Embedded newform 81.10.a.d.1.3

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:	$0$
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			1.3
Root			$23.4832$ 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)

$f(q)$	$=$	\(q-21.4832 q^{2} -50.4713 q^{4} +44.8401 q^{5} +9345.55 q^{7} +12083.7 q^{8} -963.310 q^{10} +55156.9 q^{11} -79968.8 q^{13} -200773. q^{14} -233755. q^{16} -8591.55 q^{17} +643299. q^{19} -2263.14 q^{20} -1.18495e6 q^{22} -533467. q^{23} -1.95111e6 q^{25} +1.71799e6 q^{26} -471682. q^{28} +5.28052e6 q^{29} -3.34323e6 q^{31} -1.16503e6 q^{32} +184574. q^{34} +419056. q^{35} -2.03491e7 q^{37} -1.38201e7 q^{38} +541834. q^{40} +3.18148e7 q^{41} -4.55525e6 q^{43} -2.78384e6 q^{44} +1.14606e7 q^{46} +1.80002e7 q^{47} +4.69858e7 q^{49} +4.19162e7 q^{50} +4.03613e6 q^{52} +6.95534e7 q^{53} +2.47324e6 q^{55} +1.12929e8 q^{56} -1.13443e8 q^{58} -6.48229e7 q^{59} +8.75528e7 q^{61} +7.18233e7 q^{62} +1.44711e8 q^{64} -3.58581e6 q^{65} +1.67405e8 q^{67} +433627. q^{68} -9.00266e6 q^{70} +1.55996e8 q^{71} +3.05377e8 q^{73} +4.37165e8 q^{74} -3.24681e7 q^{76} +5.15472e8 q^{77} -3.59911e8 q^{79} -1.04816e7 q^{80} -6.83484e8 q^{82} -4.63967e8 q^{83} -385246. q^{85} +9.78614e7 q^{86} +6.66500e8 q^{88} +2.70335e8 q^{89} -7.47352e8 q^{91} +2.69248e7 q^{92} -3.86702e8 q^{94} +2.88456e7 q^{95} -1.50985e8 q^{97} -1.00941e9 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 15 q^{2} + 1793 q^{4} + 453 q^{5} + 343 q^{7} + 7239 q^{8} + 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}+ \cdots - 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

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 $

$2$	−21.4832	−0.949433	−0.474717	−	0.880139i	$-0.657449\pi$
$2$	−21.4832	−0.949433	−0.474717	+	0.880139i	$0.657449\pi$
$3$	0	0
$3$	0	0
$4$	−50.4713	−0.0985768
$5$	44.8401	0.0320850	0.0160425	−	0.999871i	$-0.494893\pi$
$5$	44.8401	0.0320850	0.0160425	+	0.999871i	$0.494893\pi$
$6$	0	0
$7$	9345.55	1.47117	0.735587	−	0.677431i	$-0.236907\pi$
$7$	9345.55	1.47117	0.735587	+	0.677431i	$0.236907\pi$
$8$	12083.7	1.04303
$9$	0	0
$10$	−963.310	−0.0304625
$11$	55156.9	1.13588	0.567941	−	0.823069i	$-0.307740\pi$
$11$	55156.9	1.13588	0.567941	+	0.823069i	$0.307740\pi$
$12$	0	0
$13$	−79968.8	−0.776561	−0.388280	−	0.921541i	$-0.626931\pi$
$13$	−79968.8	−0.776561	−0.388280	+	0.921541i	$0.626931\pi$
$14$	−200773.	−1.39678
$15$	0	0
$16$	−233755.	−0.891706
$17$	−8591.55	−0.0249489	−0.0124745	−	0.999922i	$-0.503971\pi$
$17$	−8591.55	−0.0249489	−0.0124745	+	0.999922i	$0.503971\pi$
$18$	0	0
$19$	643299.	1.13246	0.566229	−	0.824248i	$-0.308402\pi$
$19$	643299.	1.13246	0.566229	+	0.824248i	$0.308402\pi$
$20$	−2263.14	−0.00316283
$21$	0	0
$22$	−1.18495e6	−1.07844
$23$	−533467.	−0.397495	−0.198748	−	0.980051i	$-0.563687\pi$
$23$	−533467.	−0.397495	−0.198748	+	0.980051i	$0.563687\pi$
$24$	0	0
$25$	−1.95111e6	−0.998971
$26$	1.71799e6	0.737292
$27$	0	0
$28$	−471682.	−0.145024
$29$	5.28052e6	1.38639	0.693195	−	0.720750i	$-0.256202\pi$
$29$	5.28052e6	1.38639	0.693195	+	0.720750i	$0.256202\pi$
$30$	0	0
$31$	−3.34323e6	−0.650187	−0.325094	−	0.945682i	$-0.605396\pi$
$31$	−3.34323e6	−0.650187	−0.325094	+	0.945682i	$0.605396\pi$
$32$	−1.16503e6	−0.196410
$33$	0	0
$34$	184574.	0.0236873
$35$	419056.	0.0472026
$36$	0	0
$37$	−2.03491e7	−1.78500	−0.892500	−	0.451048i	$-0.851050\pi$
$37$	−2.03491e7	−1.78500	−0.892500	+	0.451048i	$0.851050\pi$
$38$	−1.38201e7	−1.07519
$39$	0	0
$40$	541834.	0.0334654
$41$	3.18148e7	1.75834	0.879168	−	0.476512i	$-0.158099\pi$
$41$	3.18148e7	1.75834	0.879168	+	0.476512i	$0.158099\pi$
$42$	0	0
$43$	−4.55525e6	−0.203191	−0.101595	−	0.994826i	$-0.532395\pi$
$43$	−4.55525e6	−0.203191	−0.101595	+	0.994826i	$0.532395\pi$
$44$	−2.78384e6	−0.111972
$45$	0	0
$46$	1.14606e7	0.377395
$47$	1.80002e7	0.538068	0.269034	−	0.963131i	$-0.413296\pi$
$47$	1.80002e7	0.538068	0.269034	+	0.963131i	$0.413296\pi$
$48$	0	0
$49$	4.69858e7	1.16435
$50$	4.19162e7	0.948456
$51$	0	0
$52$	4.03613e6	0.0765508
$53$	6.95534e7	1.21081	0.605406	−	0.795917i	$-0.293011\pi$
$53$	6.95534e7	1.21081	0.605406	+	0.795917i	$0.293011\pi$
$54$	0	0
$55$	2.47324e6	0.0364447
$56$	1.12929e8	1.53447
$57$	0	0
$58$	−1.13443e8	−1.31629
$59$	−6.48229e7	−0.696457	−0.348229	−	0.937410i	$-0.613217\pi$
$59$	−6.48229e7	−0.696457	−0.348229	+	0.937410i	$0.613217\pi$
$60$	0	0
$61$	8.75528e7	0.809628	0.404814	−	0.914399i	$-0.367336\pi$
$61$	8.75528e7	0.809628	0.404814	+	0.914399i	$0.367336\pi$
$62$	7.18233e7	0.617309
$63$	0	0
$64$	1.44711e8	1.07818
$65$	−3.58581e6	−0.0249159
$66$	0	0
$67$	1.67405e8	1.01492	0.507461	−	0.861675i	$-0.330584\pi$
$67$	1.67405e8	1.01492	0.507461	+	0.861675i	$0.330584\pi$
$68$	433627.	0.00245938
$69$	0	0
$70$	−9.00266e6	−0.0448157
$71$	1.55996e8	0.728537	0.364268	−	0.931294i	$-0.381319\pi$
$71$	1.55996e8	0.728537	0.364268	+	0.931294i	$0.381319\pi$
$72$	0	0
$73$	3.05377e8	1.25859	0.629294	−	0.777168i	$-0.283344\pi$
$73$	3.05377e8	1.25859	0.629294	+	0.777168i	$0.283344\pi$
$74$	4.37165e8	1.69474
$75$	0	0
$76$	−3.24681e7	−0.111634
$77$	5.15472e8	1.67108
$78$	0	0
$79$	−3.59911e8	−1.03962	−0.519808	−	0.854283i	$-0.673996\pi$
$79$	−3.59911e8	−1.03962	−0.519808	+	0.854283i	$0.673996\pi$
$80$	−1.04816e7	−0.0286104
$81$	0	0
$82$	−6.83484e8	−1.66942
$83$	−4.63967e8	−1.07309	−0.536545	−	0.843872i	$-0.680271\pi$
$83$	−4.63967e8	−1.07309	−0.536545	+	0.843872i	$0.680271\pi$
$84$	0	0
$85$	−385246.	−0.000800485 0
$86$	9.78614e7	0.192916
$87$	0	0
$88$	6.66500e8	1.18475
$89$	2.70335e8	0.456717	0.228358	−	0.973577i	$-0.426664\pi$
$89$	2.70335e8	0.456717	0.228358	+	0.973577i	$0.426664\pi$
$90$	0	0
$91$	−7.47352e8	−1.14246
$92$	2.69248e7	0.0391838
$93$	0	0
$94$	−3.86702e8	−0.510859
$95$	2.88456e7	0.0363349
$96$	0	0
$97$	−1.50985e8	−0.173166	−0.0865828	−	0.996245i	$-0.527595\pi$
$97$	−1.50985e8	−0.173166	−0.0865828	+	0.996245i	$0.527595\pi$
$98$	−1.00941e9	−1.10547
$99$	0	0
$100$	9.84753e7	0.0984753
$101$	−3.12205e8	−0.298534	−0.149267	−	0.988797i	$-0.547691\pi$
$101$	−3.12205e8	−0.298534	−0.149267	+	0.988797i	$0.547691\pi$
$102$	0	0
$103$	8.58637e8	0.751695	0.375848	−	0.926681i	$-0.377352\pi$
$103$	8.58637e8	0.751695	0.375848	+	0.926681i	$0.377352\pi$
$104$	−9.66318e8	−0.809972
$105$	0	0
$106$	−1.49423e9	−1.14959
$107$	2.37971e8	0.175508	0.0877542	−	0.996142i	$-0.472031\pi$
$107$	2.37971e8	0.175508	0.0877542	+	0.996142i	$0.472031\pi$
$108$	0	0
$109$	7.89834e8	0.535940	0.267970	−	0.963427i	$-0.413647\pi$
$109$	7.89834e8	0.535940	0.267970	+	0.963427i	$0.413647\pi$
$110$	−5.31332e7	−0.0346018
$111$	0	0
$112$	−2.18457e9	−1.31185
$113$	5.26286e7	0.0303647	0.0151823	−	0.999885i	$-0.495167\pi$
$113$	5.26286e7	0.0303647	0.0151823	+	0.999885i	$0.495167\pi$
$114$	0	0
$115$	−2.39207e7	−0.0127536
$116$	−2.66515e8	−0.136666
$117$	0	0
$118$	1.39260e9	0.661240
$119$	−8.02928e7	−0.0367042
$120$	0	0
$121$	6.84341e8	0.290227
$122$	−1.88092e9	−0.768688
$123$	0	0
$124$	1.68737e8	0.0640934
$125$	−1.75067e8	−0.0641369
$126$	0	0
$127$	1.15652e9	0.394489	0.197244	−	0.980354i	$-0.436801\pi$
$127$	1.15652e9	0.394489	0.197244	+	0.980354i	$0.436801\pi$
$128$	−2.51237e9	−0.827254
$129$	0	0
$130$	7.70347e7	0.0236560
$131$	−5.94843e9	−1.76474	−0.882371	−	0.470554i	$-0.844054\pi$
$131$	−5.94843e9	−1.76474	−0.882371	+	0.470554i	$0.844054\pi$
$132$	0	0
$133$	6.01199e9	1.66604
$134$	−3.59640e9	−0.963600
$135$	0	0
$136$	−1.03818e8	−0.0260223
$137$	1.49470e9	0.362503	0.181252	−	0.983437i	$-0.441985\pi$
$137$	1.49470e9	0.362503	0.181252	+	0.983437i	$0.441985\pi$
$138$	0	0
$139$	5.12874e9	1.16532	0.582658	−	0.812717i	$-0.302013\pi$
$139$	5.12874e9	1.16532	0.582658	+	0.812717i	$0.302013\pi$
$140$	−2.11503e7	−0.00465307
$141$	0	0
$142$	−3.35130e9	−0.691697
$143$	−4.41083e9	−0.882081
$144$	0	0
$145$	2.36779e8	0.0444823
$146$	−6.56048e9	−1.19494
$147$	0	0
$148$	1.02705e9	0.175959
$149$	1.04170e10	1.73143	0.865715	−	0.500537i	$-0.166864\pi$
$149$	1.04170e10	1.73143	0.865715	+	0.500537i	$0.166864\pi$
$150$	0	0
$151$	9.19920e8	0.143997	0.0719985	−	0.997405i	$-0.477062\pi$
$151$	9.19920e8	0.143997	0.0719985	+	0.997405i	$0.477062\pi$
$152$	7.77343e9	1.18118
$153$	0	0
$154$	−1.10740e10	−1.58658
$155$	−1.49911e8	−0.0208612
$156$	0	0
$157$	−5.34576e9	−0.702200	−0.351100	−	0.936338i	$-0.614192\pi$
$157$	−5.34576e9	−0.702200	−0.351100	+	0.936338i	$0.614192\pi$
$158$	7.73204e9	0.987045
$159$	0	0
$160$	−5.22403e7	−0.00630181
$161$	−4.98554e9	−0.584785
$162$	0	0
$163$	6.82475e9	0.757256	0.378628	−	0.925549i	$-0.376396\pi$
$163$	6.82475e9	0.757256	0.378628	+	0.925549i	$0.376396\pi$
$164$	−1.60573e9	−0.173331
$165$	0	0
$166$	9.96751e9	1.01883
$167$	−5.81556e9	−0.578585	−0.289293	−	0.957241i	$-0.593420\pi$
$167$	−5.81556e9	−0.578585	−0.289293	+	0.957241i	$0.593420\pi$
$168$	0	0
$169$	−4.20949e9	−0.396954
$170$	8.27633e6	0.000760007 0
$171$	0	0
$172$	2.29909e8	0.0200299
$173$	1.23174e10	1.04547	0.522736	−	0.852495i	$-0.324911\pi$
$173$	1.23174e10	1.04547	0.522736	+	0.852495i	$0.324911\pi$
$174$	0	0
$175$	−1.82342e10	−1.46966
$176$	−1.28932e10	−1.01287
$177$	0	0
$178$	−5.80766e9	−0.433622
$179$	1.65764e10	1.20684	0.603422	−	0.797422i	$-0.293804\pi$
$179$	1.65764e10	1.20684	0.603422	+	0.797422i	$0.293804\pi$
$180$	0	0
$181$	−1.06374e9	−0.0736685	−0.0368343	−	0.999321i	$-0.511727\pi$
$181$	−1.06374e9	−0.0736685	−0.0368343	+	0.999321i	$0.511727\pi$
$182$	1.60555e10	1.08468
$183$	0	0
$184$	−6.44625e9	−0.414598
$185$	−9.12457e8	−0.0572716
$186$	0	0
$187$	−4.73884e8	−0.0283390
$188$	−9.08493e8	−0.0530410
$189$	0	0
$190$	−6.19696e8	−0.0344975
$191$	1.29917e10	0.706344	0.353172	−	0.935558i	$-0.385103\pi$
$191$	1.29917e10	0.706344	0.353172	+	0.935558i	$0.385103\pi$
$192$	0	0
$193$	−1.11916e10	−0.580608	−0.290304	−	0.956934i	$-0.593756\pi$
$193$	−1.11916e10	−0.580608	−0.290304	+	0.956934i	$0.593756\pi$
$194$	3.24365e9	0.164409
$195$	0	0
$196$	−2.37143e9	−0.114778
$197$	3.23317e10	1.52943	0.764716	−	0.644368i	$-0.222879\pi$
$197$	3.23317e10	1.52943	0.764716	+	0.644368i	$0.222879\pi$
$198$	0	0
$199$	7.46055e9	0.337234	0.168617	−	0.985682i	$-0.446070\pi$
$199$	7.46055e9	0.337234	0.168617	+	0.985682i	$0.446070\pi$
$200$	−2.35767e10	−1.04195
$201$	0	0
$202$	6.70716e9	0.283438
$203$	4.93494e10	2.03962
$204$	0	0
$205$	1.42658e9	0.0564161
$206$	−1.84463e10	−0.713684
$207$	0	0
$208$	1.86931e10	0.692464
$209$	3.54824e10	1.28634
$210$	0	0
$211$	3.69635e10	1.28381	0.641907	−	0.766783i	$-0.278144\pi$
$211$	3.69635e10	1.28381	0.641907	+	0.766783i	$0.278144\pi$
$212$	−3.51045e9	−0.119358
$213$	0	0
$214$	−5.11239e9	−0.166633
$215$	−2.04258e8	−0.00651937
$216$	0	0
$217$	−3.12443e10	−0.956538
$218$	−1.69682e10	−0.508839
$219$	0	0
$220$	−1.24828e8	−0.00359260
$221$	6.87056e8	0.0193743
$222$	0	0
$223$	1.75526e10	0.475302	0.237651	−	0.971351i	$-0.423622\pi$
$223$	1.75526e10	0.475302	0.237651	+	0.971351i	$0.423622\pi$
$224$	−1.08879e10	−0.288953
$225$	0	0
$226$	−1.13063e9	−0.0288292
$227$	−3.52219e10	−0.880434	−0.440217	−	0.897891i	$-0.645099\pi$
$227$	−3.52219e10	−0.880434	−0.440217	+	0.897891i	$0.645099\pi$
$228$	0	0
$229$	3.59990e10	0.865028	0.432514	−	0.901627i	$-0.357627\pi$
$229$	3.59990e10	0.865028	0.432514	+	0.901627i	$0.357627\pi$
$230$	5.13894e8	0.0121087
$231$	0	0
$232$	6.38082e10	1.44604
$233$	−1.22534e10	−0.272367	−0.136184	−	0.990684i	$-0.543484\pi$
$233$	−1.22534e10	−0.272367	−0.136184	+	0.990684i	$0.543484\pi$
$234$	0	0
$235$	8.07131e8	0.0172639
$236$	3.27170e9	0.0686545
$237$	0	0
$238$	1.72495e9	0.0348482
$239$	−1.25661e10	−0.249121	−0.124561	−	0.992212i	$-0.539752\pi$
$239$	−1.25661e10	−0.249121	−0.124561	+	0.992212i	$0.539752\pi$
$240$	0	0
$241$	1.14905e10	0.219413	0.109707	−	0.993964i	$-0.465009\pi$
$241$	1.14905e10	0.219413	0.109707	+	0.993964i	$0.465009\pi$
$242$	−1.47018e10	−0.275551
$243$	0	0
$244$	−4.41890e9	−0.0798105
$245$	2.10685e9	0.0373582
$246$	0	0
$247$	−5.14438e10	−0.879422
$248$	−4.03986e10	−0.678162
$249$	0	0
$250$	3.76099e9	0.0608937
$251$	4.00421e10	0.636774	0.318387	−	0.947961i	$-0.396859\pi$
$251$	4.00421e10	0.636774	0.318387	+	0.947961i	$0.396859\pi$
$252$	0	0
$253$	−2.94244e10	−0.451508
$254$	−2.48457e10	−0.374541
$255$	0	0
$256$	−2.01185e10	−0.292762
$257$	4.41455e10	0.631229	0.315615	−	0.948887i	$-0.397789\pi$
$257$	4.41455e10	0.631229	0.315615	+	0.948887i	$0.397789\pi$
$258$	0	0
$259$	−1.90174e11	−2.62604
$260$	1.80980e8	0.00245613
$261$	0	0
$262$	1.27791e11	1.67551
$263$	2.72553e10	0.351278	0.175639	−	0.984455i	$-0.443801\pi$
$263$	2.72553e10	0.351278	0.175639	+	0.984455i	$0.443801\pi$
$264$	0	0
$265$	3.11878e9	0.0388489
$266$	−1.29157e11	−1.58179
$267$	0	0
$268$	−8.44916e9	−0.100048
$269$	−1.37243e11	−1.59810	−0.799051	−	0.601263i	$-0.794664\pi$
$269$	−1.37243e11	−1.59810	−0.799051	+	0.601263i	$0.794664\pi$
$270$	0	0
$271$	9.04089e9	0.101824	0.0509119	−	0.998703i	$-0.483787\pi$
$271$	9.04089e9	0.101824	0.0509119	+	0.998703i	$0.483787\pi$
$272$	2.00832e9	0.0222471
$273$	0	0
$274$	−3.21110e10	−0.344173
$275$	−1.07618e11	−1.13471
$276$	0	0
$277$	4.12144e10	0.420620	0.210310	−	0.977635i	$-0.432553\pi$
$277$	4.12144e10	0.420620	0.210310	+	0.977635i	$0.432553\pi$
$278$	−1.10182e11	−1.10639
$279$	0	0
$280$	5.06374e9	0.0492335
$281$	−1.44729e11	−1.38477	−0.692383	−	0.721531i	$-0.743439\pi$
$281$	−1.44729e11	−1.38477	−0.692383	+	0.721531i	$0.743439\pi$
$282$	0	0
$283$	−3.60385e10	−0.333986	−0.166993	−	0.985958i	$-0.553406\pi$
$283$	−3.60385e10	−0.333986	−0.166993	+	0.985958i	$0.553406\pi$
$284$	−7.87333e9	−0.0718168
$285$	0	0
$286$	9.47589e10	0.837477
$287$	2.97327e11	2.58682
$288$	0	0
$289$	−1.18514e11	−0.999378
$290$	−5.08678e9	−0.0422330
$291$	0	0
$292$	−1.54128e10	−0.124067
$293$	3.92901e10	0.311443	0.155722	−	0.987801i	$-0.450230\pi$
$293$	3.92901e10	0.311443	0.155722	+	0.987801i	$0.450230\pi$
$294$	0	0
$295$	−2.90667e9	−0.0223458
$296$	−2.45893e11	−1.86180
$297$	0	0
$298$	−2.23791e11	−1.64388
$299$	4.26607e10	0.308679
$300$	0	0
$301$	−4.25713e10	−0.298929
$302$	−1.97628e10	−0.136716
$303$	0	0
$304$	−1.50375e11	−1.00982
$305$	3.92588e9	0.0259769
$306$	0	0
$307$	6.02560e10	0.387148	0.193574	−	0.981086i	$-0.437992\pi$
$307$	6.02560e10	0.387148	0.193574	+	0.981086i	$0.437992\pi$
$308$	−2.60166e10	−0.164730
$309$	0	0
$310$	3.22057e9	0.0198064
$311$	2.81885e11	1.70864	0.854319	−	0.519750i	$-0.173975\pi$
$311$	2.81885e11	1.70864	0.854319	+	0.519750i	$0.173975\pi$
$312$	0	0
$313$	1.48600e11	0.875122	0.437561	−	0.899189i	$-0.355842\pi$
$313$	1.48600e11	0.875122	0.437561	+	0.899189i	$0.355842\pi$
$314$	1.14844e11	0.666692
$315$	0	0
$316$	1.81652e10	0.102482
$317$	1.40319e11	0.780458	0.390229	−	0.920718i	$-0.372396\pi$
$317$	1.40319e11	0.780458	0.390229	+	0.920718i	$0.372396\pi$
$318$	0	0
$319$	2.91257e11	1.57478
$320$	6.48888e9	0.0345935
$321$	0	0
$322$	1.07106e11	0.555214
$323$	−5.52694e9	−0.0282536
$324$	0	0
$325$	1.56028e11	0.775761
$326$	−1.46618e11	−0.718963
$327$	0	0
$328$	3.84440e11	1.83399
$329$	1.68222e11	0.791591
$330$	0	0
$331$	−1.19210e11	−0.545867	−0.272934	−	0.962033i	$-0.587994\pi$
$331$	−1.19210e11	−0.545867	−0.272934	+	0.962033i	$0.587994\pi$
$332$	2.34170e10	0.105782
$333$	0	0
$334$	1.24937e11	0.549328
$335$	7.50647e9	0.0325637
$336$	0	0
$337$	−3.51241e11	−1.48344	−0.741720	−	0.670709i	$-0.765990\pi$
$337$	−3.51241e11	−1.48344	−0.741720	+	0.670709i	$0.765990\pi$
$338$	9.04335e10	0.376881
$339$	0	0
$340$	1.94439e7	7.89092e−5 0
$341$	−1.84402e11	−0.738536
$342$	0	0
$343$	6.19811e10	0.241789
$344$	−5.50443e10	−0.211933
$345$	0	0
$346$	−2.64618e11	−0.992606
$347$	−3.57306e11	−1.32299	−0.661496	−	0.749949i	$-0.730078\pi$
$347$	−3.57306e11	−1.32299	−0.661496	+	0.749949i	$0.730078\pi$
$348$	0	0
$349$	4.30212e11	1.55227	0.776136	−	0.630565i	$-0.217177\pi$
$349$	4.30212e11	1.55227	0.776136	+	0.630565i	$0.217177\pi$
$350$	3.91730e11	1.39534
$351$	0	0
$352$	−6.42597e10	−0.223099
$353$	5.18997e11	1.77901	0.889505	−	0.456925i	$-0.151049\pi$
$353$	5.18997e11	1.77901	0.889505	+	0.456925i	$0.151049\pi$
$354$	0	0
$355$	6.99489e9	0.0233751
$356$	−1.36441e10	−0.0450217
$357$	0	0
$358$	−3.56114e11	−1.14582
$359$	4.22565e11	1.34267	0.671334	−	0.741155i	$-0.265721\pi$
$359$	4.22565e11	1.34267	0.671334	+	0.741155i	$0.265721\pi$
$360$	0	0
$361$	9.11460e10	0.282459
$362$	2.28526e10	0.0699433
$363$	0	0
$364$	3.77199e10	0.112620
$365$	1.36931e10	0.0403817
$366$	0	0
$367$	−1.87479e11	−0.539456	−0.269728	−	0.962937i	$-0.586934\pi$
$367$	−1.87479e11	−0.539456	−0.269728	+	0.962937i	$0.586934\pi$
$368$	1.24701e11	0.354449
$369$	0	0
$370$	1.96025e10	0.0543756
$371$	6.50015e11	1.78131
$372$	0	0
$373$	3.54499e10	0.0948254	0.0474127	−	0.998875i	$-0.484902\pi$
$373$	3.54499e10	0.0948254	0.0474127	+	0.998875i	$0.484902\pi$
$374$	1.01805e10	0.0269060
$375$	0	0
$376$	2.17509e11	0.561218
$377$	−4.22277e11	−1.07662
$378$	0	0
$379$	2.58921e11	0.644602	0.322301	−	0.946637i	$-0.395544\pi$
$379$	2.58921e11	0.644602	0.322301	+	0.946637i	$0.395544\pi$
$380$	−1.45588e9	−0.00358177
$381$	0	0
$382$	−2.79104e11	−0.670626
$383$	−1.17011e11	−0.277864	−0.138932	−	0.990302i	$-0.544367\pi$
$383$	−1.17011e11	−0.277864	−0.138932	+	0.990302i	$0.544367\pi$
$384$	0	0
$385$	2.31138e10	0.0536165
$386$	2.40431e11	0.551249
$387$	0	0
$388$	7.62042e9	0.0170701
$389$	−4.16368e11	−0.921942	−0.460971	−	0.887415i	$-0.652499\pi$
$389$	−4.16368e11	−0.921942	−0.460971	+	0.887415i	$0.652499\pi$
$390$	0	0
$391$	4.58331e9	0.00991708
$392$	5.67762e11	1.21445
$393$	0	0
$394$	−6.94588e11	−1.45209
$395$	−1.61384e10	−0.0333560
$396$	0	0
$397$	−4.63043e11	−0.935544	−0.467772	−	0.883849i	$-0.654943\pi$
$397$	−4.63043e11	−0.935544	−0.467772	+	0.883849i	$0.654943\pi$
$398$	−1.60277e11	−0.320181
$399$	0	0
$400$	4.56083e11	0.890788
$401$	6.35971e11	1.22825	0.614127	−	0.789208i	$-0.289508\pi$
$401$	6.35971e11	1.22825	0.614127	+	0.789208i	$0.289508\pi$
$402$	0	0
$403$	2.67354e11	0.504910
$404$	1.57574e10	0.0294285
$405$	0	0
$406$	−1.06018e12	−1.93648
$407$	−1.12240e12	−2.02755
$408$	0	0
$409$	−1.22939e11	−0.217237	−0.108619	−	0.994084i	$-0.534643\pi$
$409$	−1.22939e11	−0.217237	−0.108619	+	0.994084i	$0.534643\pi$
$410$	−3.06475e10	−0.0535634
$411$	0	0
$412$	−4.33365e10	−0.0740997
$413$	−6.05806e11	−1.02461
$414$	0	0
$415$	−2.08043e10	−0.0344300
$416$	9.31664e10	0.152524
$417$	0	0
$418$	−7.62276e11	−1.22129
$419$	−8.58084e11	−1.36009	−0.680044	−	0.733171i	$-0.738039\pi$
$419$	−8.58084e11	−1.36009	−0.680044	+	0.733171i	$0.738039\pi$
$420$	0	0
$421$	2.65092e11	0.411270	0.205635	−	0.978629i	$-0.434074\pi$
$421$	2.65092e11	0.411270	0.205635	+	0.978629i	$0.434074\pi$
$422$	−7.94095e11	−1.21890
$423$	0	0
$424$	8.40462e11	1.26291
$425$	1.67631e10	0.0249232
$426$	0	0
$427$	8.18229e11	1.19110
$428$	−1.20107e10	−0.0173011
$429$	0	0
$430$	4.38812e9	0.00618971
$431$	2.69116e11	0.375658	0.187829	−	0.982202i	$-0.439855\pi$
$431$	2.69116e11	0.375658	0.187829	+	0.982202i	$0.439855\pi$
$432$	0	0
$433$	−1.07088e12	−1.46401	−0.732006	−	0.681298i	$-0.761416\pi$
$433$	−1.07088e12	−1.46401	−0.732006	+	0.681298i	$0.761416\pi$
$434$	6.71229e11	0.908169
$435$	0	0
$436$	−3.98639e10	−0.0528313
$437$	−3.43179e11	−0.450146
$438$	0	0
$439$	−5.12835e11	−0.659003	−0.329502	−	0.944155i	$-0.606881\pi$
$439$	−5.12835e11	−0.659003	−0.329502	+	0.944155i	$0.606881\pi$
$440$	2.98859e10	0.0380128
$441$	0	0
$442$	−1.47602e10	−0.0183946
$443$	−2.30225e11	−0.284011	−0.142006	−	0.989866i	$-0.545355\pi$
$443$	−2.30225e11	−0.284011	−0.142006	+	0.989866i	$0.545355\pi$
$444$	0	0
$445$	1.21218e10	0.0146537
$446$	−3.77087e11	−0.451268
$447$	0	0
$448$	1.35241e12	1.58620
$449$	1.09130e12	1.26717	0.633586	−	0.773672i	$-0.281582\pi$
$449$	1.09130e12	1.26717	0.633586	+	0.773672i	$0.281582\pi$
$450$	0	0
$451$	1.75481e12	1.99726
$452$	−2.65623e9	−0.00299325
$453$	0	0
$454$	7.56681e11	0.835914
$455$	−3.35114e10	−0.0366556
$456$	0	0
$457$	−9.15864e10	−0.0982219	−0.0491109	−	0.998793i	$-0.515639\pi$
$457$	−9.15864e10	−0.0982219	−0.0491109	+	0.998793i	$0.515639\pi$
$458$	−7.73373e11	−0.821286
$459$	0	0
$460$	1.20731e9	0.00125721
$461$	−4.65818e11	−0.480355	−0.240177	−	0.970729i	$-0.577206\pi$
$461$	−4.65818e11	−0.480355	−0.240177	+	0.970729i	$0.577206\pi$
$462$	0	0
$463$	−1.06973e12	−1.08183	−0.540916	−	0.841077i	$-0.681922\pi$
$463$	−1.06973e12	−1.08183	−0.540916	+	0.841077i	$0.681922\pi$
$464$	−1.23435e12	−1.23625
$465$	0	0
$466$	2.63243e11	0.258595
$467$	−1.30619e12	−1.27081	−0.635406	−	0.772178i	$-0.719167\pi$
$467$	−1.30619e12	−1.27081	−0.635406	+	0.772178i	$0.719167\pi$
$468$	0	0
$469$	1.56449e12	1.49313
$470$	−1.73398e10	−0.0163909
$471$	0	0
$472$	−7.83300e11	−0.726422
$473$	−2.51254e11	−0.230801
$474$	0	0
$475$	−1.25515e12	−1.13129
$476$	4.05248e9	0.00361818
$477$	0	0
$478$	2.69961e11	0.236524
$479$	−2.13641e12	−1.85428	−0.927139	−	0.374717i	$-0.877740\pi$
$479$	−2.13641e12	−1.85428	−0.927139	+	0.374717i	$0.877740\pi$
$480$	0	0
$481$	1.62729e12	1.38616
$482$	−2.46853e11	−0.208318
$483$	0	0
$484$	−3.45396e10	−0.0286097
$485$	−6.77019e9	−0.00555601
$486$	0	0
$487$	−1.44314e12	−1.16259	−0.581297	−	0.813692i	$-0.697454\pi$
$487$	−1.44314e12	−1.16259	−0.581297	+	0.813692i	$0.697454\pi$
$488$	1.05796e12	0.844463
$489$	0	0
$490$	−4.52618e10	−0.0354691
$491$	7.73531e11	0.600636	0.300318	−	0.953839i	$-0.402907\pi$
$491$	7.73531e11	0.600636	0.300318	+	0.953839i	$0.402907\pi$
$492$	0	0
$493$	−4.53679e10	−0.0345889
$494$	1.10518e12	0.834952
$495$	0	0
$496$	7.81498e11	0.579776
$497$	1.45787e12	1.07180
$498$	0	0
$499$	−5.19884e11	−0.375366	−0.187683	−	0.982230i	$-0.560098\pi$
$499$	−5.19884e11	−0.375366	−0.187683	+	0.982230i	$0.560098\pi$
$500$	8.83584e9	0.00632241
$501$	0	0
$502$	−8.60234e11	−0.604575
$503$	−1.32324e12	−0.921683	−0.460842	−	0.887482i	$-0.652452\pi$
$503$	−1.32324e12	−0.921683	−0.460842	+	0.887482i	$0.652452\pi$
$504$	0	0
$505$	−1.39993e10	−0.00957844
$506$	6.32131e11	0.428676
$507$	0	0
$508$	−5.83708e10	−0.0388874
$509$	−1.65550e12	−1.09320	−0.546600	−	0.837394i	$-0.684078\pi$
$509$	−1.65550e12	−1.09320	−0.546600	+	0.837394i	$0.684078\pi$
$510$	0	0
$511$	2.85392e12	1.85160
$512$	1.71854e12	1.10521
$513$	0	0
$514$	−9.48387e11	−0.599310
$515$	3.85014e10	0.0241181
$516$	0	0
$517$	9.92836e11	0.611181
$518$	4.08555e12	2.49325
$519$	0	0
$520$	−4.33298e10	−0.0259879
$521$	−3.36913e11	−0.200331	−0.100165	−	0.994971i	$-0.531937\pi$
$521$	−3.36913e11	−0.200331	−0.100165	+	0.994971i	$0.531937\pi$
$522$	0	0
$523$	−2.63804e12	−1.54179	−0.770893	−	0.636964i	$-0.780190\pi$
$523$	−2.63804e12	−1.54179	−0.770893	+	0.636964i	$0.780190\pi$
$524$	3.00225e11	0.173963
$525$	0	0
$526$	−5.85532e11	−0.333515
$527$	2.87235e10	0.0162215
$528$	0	0
$529$	−1.51657e12	−0.841997
$530$	−6.70014e10	−0.0368844
$531$	0	0
$532$	−3.03433e11	−0.164233
$533$	−2.54419e12	−1.36545
$534$	0	0
$535$	1.06707e10	0.00563118
$536$	2.02287e12	1.05859
$537$	0	0
$538$	2.94842e12	1.51729
$539$	2.59159e12	1.32256
$540$	0	0
$541$	−2.84689e12	−1.42884	−0.714419	−	0.699718i	$-0.753309\pi$
$541$	−2.84689e12	−1.42884	−0.714419	+	0.699718i	$0.753309\pi$
$542$	−1.94227e11	−0.0966749
$543$	0	0
$544$	1.00095e10	0.00490022
$545$	3.54162e10	0.0171956
$546$	0	0
$547$	−3.97570e12	−1.89876	−0.949381	−	0.314127i	$-0.898288\pi$
$547$	−3.97570e12	−1.89876	−0.949381	+	0.314127i	$0.898288\pi$
$548$	−7.54396e10	−0.0357344
$549$	0	0
$550$	2.31197e12	1.07733
$551$	3.39695e12	1.57003
$552$	0	0
$553$	−3.36356e12	−1.52945
$554$	−8.85417e11	−0.399350
$555$	0	0
$556$	−2.58854e11	−0.114873
$557$	−3.04474e12	−1.34030	−0.670151	−	0.742225i	$-0.733771\pi$
$557$	−3.04474e12	−1.34030	−0.670151	+	0.742225i	$0.733771\pi$
$558$	0	0
$559$	3.64278e11	0.157790
$560$	−9.79565e10	−0.0420908
$561$	0	0
$562$	3.10924e12	1.31474
$563$	2.41196e12	1.01177	0.505886	−	0.862600i	$-0.331166\pi$
$563$	2.41196e12	1.01177	0.505886	+	0.862600i	$0.331166\pi$
$564$	0	0
$565$	2.35987e9	0.000974250 0
$566$	7.74224e11	0.317097
$567$	0	0
$568$	1.88501e12	0.759882
$569$	−1.43197e12	−0.572702	−0.286351	−	0.958125i	$-0.592442\pi$
$569$	−1.43197e12	−0.572702	−0.286351	+	0.958125i	$0.592442\pi$
$570$	0	0
$571$	2.17567e10	0.00856508	0.00428254	−	0.999991i	$-0.498637\pi$
$571$	2.17567e10	0.00856508	0.00428254	+	0.999991i	$0.498637\pi$
$572$	2.22620e11	0.0869527
$573$	0	0
$574$	−6.38754e12	−2.45601
$575$	1.04085e12	0.397086
$576$	0	0
$577$	2.44386e12	0.917879	0.458940	−	0.888467i	$-0.348229\pi$
$577$	2.44386e12	0.917879	0.458940	+	0.888467i	$0.348229\pi$
$578$	2.54606e12	0.948842
$579$	0	0
$580$	−1.19505e10	−0.00438492
$581$	−4.33603e12	−1.57870
$582$	0	0
$583$	3.83635e12	1.37534
$584$	3.69008e12	1.31274
$585$	0	0
$586$	−8.44078e11	−0.295694
$587$	4.14752e12	1.44184	0.720919	−	0.693019i	$-0.243720\pi$
$587$	4.14752e12	1.44184	0.720919	+	0.693019i	$0.243720\pi$
$588$	0	0
$589$	−2.15070e12	−0.736309
$590$	6.24446e10	0.0212159
$591$	0	0
$592$	4.75672e12	1.59169
$593$	1.12036e12	0.372059	0.186029	−	0.982544i	$-0.440438\pi$
$593$	1.12036e12	0.372059	0.186029	+	0.982544i	$0.440438\pi$
$594$	0	0
$595$	−3.60034e9	−0.00117765
$596$	−5.25760e11	−0.170679
$597$	0	0
$598$	−9.16489e11	−0.293070
$599$	−9.68704e10	−0.0307447	−0.0153724	−	0.999882i	$-0.504893\pi$
$599$	−9.68704e10	−0.0307447	−0.0153724	+	0.999882i	$0.504893\pi$
$600$	0	0
$601$	8.43997e11	0.263880	0.131940	−	0.991258i	$-0.457879\pi$
$601$	8.43997e11	0.263880	0.131940	+	0.991258i	$0.457879\pi$
$602$	9.14569e11	0.283813
$603$	0	0
$604$	−4.64295e10	−0.0141948
$605$	3.06859e10	0.00931193
$606$	0	0
$607$	−3.12241e12	−0.933558	−0.466779	−	0.884374i	$-0.654586\pi$
$607$	−3.12241e12	−0.933558	−0.466779	+	0.884374i	$0.654586\pi$
$608$	−7.49465e11	−0.222426
$609$	0	0
$610$	−8.43405e10	−0.0246633
$611$	−1.43945e12	−0.417842
$612$	0	0
$613$	5.54993e12	1.58751	0.793753	−	0.608240i	$-0.208124\pi$
$613$	5.54993e12	1.58751	0.793753	+	0.608240i	$0.208124\pi$
$614$	−1.29449e12	−0.367572
$615$	0	0
$616$	6.22881e12	1.74298
$617$	2.74184e12	0.761655	0.380828	−	0.924646i	$-0.375639\pi$
$617$	2.74184e12	0.761655	0.380828	+	0.924646i	$0.375639\pi$
$618$	0	0
$619$	2.83694e12	0.776679	0.388339	−	0.921516i	$-0.373049\pi$
$619$	2.83694e12	0.776679	0.388339	+	0.921516i	$0.373049\pi$
$620$	7.56619e9	0.00205643
$621$	0	0
$622$	−6.05579e12	−1.62224
$623$	2.52643e12	0.671909
$624$	0	0
$625$	3.80292e12	0.996913
$626$	−3.19240e12	−0.830870
$627$	0	0
$628$	2.69808e11	0.0692207
$629$	1.74831e11	0.0445338
$630$	0	0
$631$	4.50954e12	1.13240	0.566200	−	0.824268i	$-0.308413\pi$
$631$	4.50954e12	1.13240	0.566200	+	0.824268i	$0.308413\pi$
$632$	−4.34905e12	−1.08434
$633$	0	0
$634$	−3.01450e12	−0.740993
$635$	5.18583e10	0.0126572
$636$	0	0
$637$	−3.75739e12	−0.904189
$638$	−6.25715e12	−1.49514
$639$	0	0
$640$	−1.12655e11	−0.0265424
$641$	−5.63735e12	−1.31891	−0.659453	−	0.751746i	$-0.729212\pi$
$641$	−5.63735e12	−1.31891	−0.659453	+	0.751746i	$0.729212\pi$
$642$	0	0
$643$	4.49329e12	1.03661	0.518304	−	0.855196i	$-0.326563\pi$
$643$	4.49329e12	1.03661	0.518304	+	0.855196i	$0.326563\pi$
$644$	2.51627e11	0.0576462
$645$	0	0
$646$	1.18736e11	0.0268249
$647$	−2.52863e12	−0.567304	−0.283652	−	0.958927i	$-0.591546\pi$
$647$	−2.52863e12	−0.567304	−0.283652	+	0.958927i	$0.591546\pi$
$648$	0	0
$649$	−3.57543e12	−0.791093
$650$	−3.35199e12	−0.736533
$651$	0	0
$652$	−3.44454e11	−0.0746478
$653$	−9.58850e11	−0.206368	−0.103184	−	0.994662i	$-0.532903\pi$
$653$	−9.58850e11	−0.206368	−0.103184	+	0.994662i	$0.532903\pi$
$654$	0	0
$655$	−2.66728e11	−0.0566217
$656$	−7.43688e12	−1.56792
$657$	0	0
$658$	−3.61395e12	−0.751562
$659$	−1.37907e12	−0.284840	−0.142420	−	0.989806i	$-0.545488\pi$
$659$	−1.37907e12	−0.284840	−0.142420	+	0.989806i	$0.545488\pi$
$660$	0	0
$661$	−1.16187e12	−0.236729	−0.118365	−	0.992970i	$-0.537765\pi$
$661$	−1.16187e12	−0.236729	−0.118365	+	0.992970i	$0.537765\pi$
$662$	2.56102e12	0.518265
$663$	0	0
$664$	−5.60644e12	−1.11926
$665$	2.69578e11	0.0534549
$666$	0	0
$667$	−2.81698e12	−0.551084
$668$	2.93519e11	0.0570351
$669$	0	0
$670$	−1.61263e11	−0.0309171
$671$	4.82914e12	0.919642
$672$	0	0
$673$	2.27994e12	0.428406	0.214203	−	0.976789i	$-0.431285\pi$
$673$	2.27994e12	0.428406	0.214203	+	0.976789i	$0.431285\pi$
$674$	7.54578e12	1.40843
$675$	0	0
$676$	2.12459e11	0.0391304
$677$	−2.13897e12	−0.391340	−0.195670	−	0.980670i	$-0.562688\pi$
$677$	−2.13897e12	−0.391340	−0.195670	+	0.980670i	$0.562688\pi$
$678$	0	0
$679$	−1.41104e12	−0.254757
$680$	−4.65520e9	−0.000834926 0
$681$	0	0
$682$	3.96155e12	0.701190
$683$	−4.62393e11	−0.0813051	−0.0406526	−	0.999173i	$-0.512944\pi$
$683$	−4.62393e11	−0.0813051	−0.0406526	+	0.999173i	$0.512944\pi$
$684$	0	0
$685$	6.70226e10	0.0116309
$686$	−1.33155e12	−0.229562
$687$	0	0
$688$	1.06481e12	0.181186
$689$	−5.56210e12	−0.940269
$690$	0	0
$691$	9.21097e12	1.53693	0.768465	−	0.639892i	$-0.221021\pi$
$691$	9.21097e12	1.53693	0.768465	+	0.639892i	$0.221021\pi$
$692$	−6.21676e11	−0.103059
$693$	0	0
$694$	7.67607e12	1.25609
$695$	2.29973e11	0.0373891
$696$	0	0
$697$	−2.73338e11	−0.0438685
$698$	−9.24234e12	−1.47378
$699$	0	0
$700$	9.20306e11	0.144874
$701$	−1.30164e12	−0.203592	−0.101796	−	0.994805i	$-0.532459\pi$
$701$	−1.30164e12	−0.203592	−0.101796	+	0.994805i	$0.532459\pi$
$702$	0	0
$703$	−1.30906e13	−2.02143
$704$	7.98184e12	1.22469
$705$	0	0
$706$	−1.11497e13	−1.68905
$707$	−2.91772e12	−0.439195
$708$	0	0
$709$	−1.23312e13	−1.83272	−0.916361	−	0.400353i	$-0.868888\pi$
$709$	−1.23312e13	−1.83272	−0.916361	+	0.400353i	$0.868888\pi$
$710$	−1.50273e11	−0.0221931
$711$	0	0
$712$	3.26664e12	0.476367
$713$	1.78350e12	0.258446
$714$	0	0
$715$	−1.97782e11	−0.0283015
$716$	−8.36631e11	−0.118967
$717$	0	0
$718$	−9.07807e12	−1.27477
$719$	1.63948e12	0.228784	0.114392	−	0.993436i	$-0.463508\pi$
$719$	1.63948e12	0.228784	0.114392	+	0.993436i	$0.463508\pi$
$720$	0	0
$721$	8.02443e12	1.10587
$722$	−1.95811e12	−0.268176
$723$	0	0
$724$	5.36883e10	0.00726200
$725$	−1.03029e13	−1.38496
$726$	0	0
$727$	−6.03339e12	−0.801045	−0.400522	−	0.916287i	$-0.631171\pi$
$727$	−6.03339e12	−0.801045	−0.400522	+	0.916287i	$0.631171\pi$
$728$	−9.03078e12	−1.19161
$729$	0	0
$730$	−2.94173e11	−0.0383398
$731$	3.91367e10	0.00506939
$732$	0	0
$733$	9.43666e11	0.120740	0.0603699	−	0.998176i	$-0.480772\pi$
$733$	9.43666e11	0.120740	0.0603699	+	0.998176i	$0.480772\pi$
$734$	4.02766e12	0.512177
$735$	0	0
$736$	6.21507e11	0.0780721
$737$	9.23356e12	1.15283
$738$	0	0
$739$	2.35026e12	0.289879	0.144939	−	0.989441i	$-0.453701\pi$
$739$	2.35026e12	0.289879	0.144939	+	0.989441i	$0.453701\pi$
$740$	4.60529e10	0.00564565
$741$	0	0
$742$	−1.39644e13	−1.69124
$743$	1.06015e13	1.27620	0.638098	−	0.769955i	$-0.279722\pi$
$743$	1.06015e13	1.27620	0.638098	+	0.769955i	$0.279722\pi$
$744$	0	0
$745$	4.67100e11	0.0555529
$746$	−7.61577e11	−0.0900304
$747$	0	0
$748$	2.39175e10	0.00279357
$749$	2.22398e12	0.258203
$750$	0	0
$751$	1.42801e13	1.63814	0.819072	−	0.573691i	$-0.194489\pi$
$751$	1.42801e13	1.63814	0.819072	+	0.573691i	$0.194489\pi$
$752$	−4.20764e12	−0.479798
$753$	0	0
$754$	9.07186e12	1.02218
$755$	4.12493e10	0.00462014
$756$	0	0
$757$	2.98242e12	0.330094	0.165047	−	0.986286i	$-0.447222\pi$
$757$	2.98242e12	0.330094	0.165047	+	0.986286i	$0.447222\pi$
$758$	−5.56247e12	−0.612006
$759$	0	0
$760$	3.48561e11	0.0378982
$761$	1.15072e13	1.24377	0.621885	−	0.783109i	$-0.286367\pi$
$761$	1.15072e13	1.24377	0.621885	+	0.783109i	$0.286367\pi$
$762$	0	0
$763$	7.38143e12	0.788461
$764$	−6.55709e11	−0.0696291
$765$	0	0
$766$	2.51377e12	0.263813
$767$	5.18381e12	0.540841
$768$	0	0
$769$	1.53574e13	1.58361	0.791807	−	0.610771i	$-0.209140\pi$
$769$	1.53574e13	1.58361	0.791807	+	0.610771i	$0.209140\pi$
$770$	−4.96559e11	−0.0509053
$771$	0	0
$772$	5.64853e11	0.0572345
$773$	1.55153e13	1.56298	0.781491	−	0.623917i	$-0.214460\pi$
$773$	1.55153e13	1.56298	0.781491	+	0.623917i	$0.214460\pi$
$774$	0	0
$775$	6.52302e12	0.649518
$776$	−1.82446e12	−0.180616
$777$	0	0
$778$	8.94491e12	0.875322
$779$	2.04664e13	1.99124
$780$	0	0
$781$	8.60428e12	0.827532
$782$	−9.84642e10	−0.00941560
$783$	0	0
$784$	−1.09832e13	−1.03826
$785$	−2.39705e11	−0.0225301
$786$	0	0
$787$	−1.49230e13	−1.38666	−0.693329	−	0.720621i	$-0.743857\pi$
$787$	−1.49230e13	−1.38666	−0.693329	+	0.720621i	$0.743857\pi$
$788$	−1.63182e12	−0.150766
$789$	0	0
$790$	3.46705e11	0.0316693
$791$	4.91843e11	0.0446717
$792$	0	0
$793$	−7.00149e12	−0.628725
$794$	9.94766e12	0.888237
$795$	0	0
$796$	−3.76544e11	−0.0332435
$797$	8.06094e11	0.0707657	0.0353829	−	0.999374i	$-0.488735\pi$
$797$	8.06094e11	0.0707657	0.0353829	+	0.999374i	$0.488735\pi$
$798$	0	0
$799$	−1.54650e11	−0.0134242
$800$	2.27312e12	0.196208
$801$	0	0
$802$	−1.36627e13	−1.16614
$803$	1.68437e13	1.42961
$804$	0	0
$805$	−2.23552e11	−0.0187628
$806$	−5.74362e12	−0.479378
$807$	0	0
$808$	−3.77258e12	−0.311378
$809$	−1.37229e13	−1.12636	−0.563179	−	0.826335i	$-0.690422\pi$
$809$	−1.37229e13	−1.12636	−0.563179	+	0.826335i	$0.690422\pi$
$810$	0	0
$811$	6.34290e12	0.514866	0.257433	−	0.966296i	$-0.417123\pi$
$811$	6.34290e12	0.514866	0.257433	+	0.966296i	$0.417123\pi$
$812$	−2.49073e12	−0.201059
$813$	0	0
$814$	2.41127e13	1.92502
$815$	3.06023e11	0.0242965
$816$	0	0
$817$	−2.93039e12	−0.230105
$818$	2.64112e12	0.206252
$819$	0	0
$820$	−7.20013e10	−0.00556132
$821$	−1.23943e13	−0.952093	−0.476046	−	0.879420i	$-0.657931\pi$
$821$	−1.23943e13	−0.952093	−0.476046	+	0.879420i	$0.657931\pi$
$822$	0	0
$823$	−1.14112e13	−0.867023	−0.433511	−	0.901148i	$-0.642726\pi$
$823$	−1.14112e13	−0.867023	−0.433511	+	0.901148i	$0.642726\pi$
$824$	1.03755e13	0.784037
$825$	0	0
$826$	1.30147e13	0.972798
$827$	1.74662e13	1.29844	0.649222	−	0.760599i	$-0.275095\pi$
$827$	1.74662e13	1.29844	0.649222	+	0.760599i	$0.275095\pi$
$828$	0	0
$829$	−4.32053e12	−0.317718	−0.158859	−	0.987301i	$-0.550781\pi$
$829$	−4.32053e12	−0.317718	−0.158859	+	0.987301i	$0.550781\pi$
$830$	4.46944e11	0.0326890
$831$	0	0
$832$	−1.15724e13	−0.837275
$833$	−4.03681e11	−0.0290493
$834$	0	0
$835$	−2.60770e11	−0.0185639
$836$	−1.79084e12	−0.126803
$837$	0	0
$838$	1.84344e13	1.29131
$839$	−3.95249e12	−0.275386	−0.137693	−	0.990475i	$-0.543969\pi$
$839$	−3.95249e12	−0.275386	−0.137693	+	0.990475i	$0.543969\pi$
$840$	0	0
$841$	1.33767e13	0.922080
$842$	−5.69503e12	−0.390473
$843$	0	0
$844$	−1.86560e12	−0.126554
$845$	−1.88754e11	−0.0127362
$846$	0	0
$847$	6.39554e12	0.426975
$848$	−1.62585e13	−1.07969
$849$	0	0
$850$	−3.60125e11	−0.0236629
$851$	1.08556e13	0.709529
$852$	0	0
$853$	−1.49915e13	−0.969560	−0.484780	−	0.874636i	$-0.661100\pi$
$853$	−1.49915e13	−0.969560	−0.484780	+	0.874636i	$0.661100\pi$
$854$	−1.75782e13	−1.13087
$855$	0	0
$856$	2.87557e12	0.183060
$857$	1.41058e13	0.893272	0.446636	−	0.894716i	$-0.352622\pi$
$857$	1.41058e13	0.893272	0.446636	+	0.894716i	$0.352622\pi$
$858$	0	0
$859$	−2.24956e13	−1.40971	−0.704853	−	0.709353i	$-0.748987\pi$
$859$	−2.24956e13	−1.40971	−0.704853	+	0.709353i	$0.748987\pi$
$860$	1.03092e10	0.000642659 0
$861$	0	0
$862$	−5.78149e12	−0.356662
$863$	1.15572e13	0.709259	0.354629	−	0.935007i	$-0.384607\pi$
$863$	1.15572e13	0.709259	0.354629	+	0.935007i	$0.384607\pi$
$864$	0	0
$865$	5.52315e11	0.0335439
$866$	2.30059e13	1.38998
$867$	0	0
$868$	1.57694e12	0.0942924
$869$	−1.98516e13	−1.18088
$870$	0	0
$871$	−1.33872e13	−0.788148
$872$	9.54411e12	0.558999
$873$	0	0
$874$	7.37258e12	0.427384
$875$	−1.63609e12	−0.0943565
$876$	0	0
$877$	−1.94644e13	−1.11107	−0.555536	−	0.831493i	$-0.687487\pi$
$877$	−1.94644e13	−1.11107	−0.555536	+	0.831493i	$0.687487\pi$
$878$	1.10174e13	0.625680
$879$	0	0
$880$	−5.78134e11	−0.0324980
$881$	8.46829e12	0.473592	0.236796	−	0.971559i	$-0.423903\pi$
$881$	8.46829e12	0.473592	0.236796	+	0.971559i	$0.423903\pi$
$882$	0	0
$883$	−1.77577e13	−0.983025	−0.491513	−	0.870870i	$-0.663556\pi$
$883$	−1.77577e13	−0.983025	−0.491513	+	0.870870i	$0.663556\pi$
$884$	−3.46766e10	−0.00190986
$885$	0	0
$886$	4.94597e12	0.269650
$887$	−2.46237e13	−1.33567	−0.667833	−	0.744311i	$-0.732778\pi$
$887$	−2.46237e13	−1.33567	−0.667833	+	0.744311i	$0.732778\pi$
$888$	0	0
$889$	1.08083e13	0.580361
$890$	−2.60416e11	−0.0139127
$891$	0	0
$892$	−8.85903e11	−0.0468538
$893$	1.15795e13	0.609338
$894$	0	0
$895$	7.43286e11	0.0387215
$896$	−2.34795e13	−1.21703
$897$	0	0
$898$	−2.34446e13	−1.20310
$899$	−1.76540e13	−0.901414
$900$	0	0
$901$	−5.97571e11	−0.0302084
$902$	−3.76989e13	−1.89627
$903$	0	0
$904$	6.35948e11	0.0316711
$905$	−4.76982e10	−0.00236365
$906$	0	0
$907$	−2.26386e12	−0.111075	−0.0555376	−	0.998457i	$-0.517687\pi$
$907$	−2.26386e12	−0.111075	−0.0555376	+	0.998457i	$0.517687\pi$
$908$	1.77770e12	0.0867904
$909$	0	0
$910$	7.19932e11	0.0348021
$911$	2.44219e13	1.17475	0.587377	−	0.809313i	$-0.300160\pi$
$911$	2.44219e13	1.17475	0.587377	+	0.809313i	$0.300160\pi$
$912$	0	0
$913$	−2.55910e13	−1.21890
$914$	1.96757e12	0.0932551
$915$	0	0
$916$	−1.81691e12	−0.0852717
$917$	−5.55914e13	−2.59624
$918$	0	0
$919$	−9.43274e12	−0.436233	−0.218116	−	0.975923i	$-0.569991\pi$
$919$	−9.43274e12	−0.436233	−0.218116	+	0.975923i	$0.569991\pi$
$920$	−2.89051e11	−0.0133024
$921$	0	0
$922$	1.00073e13	0.456065
$923$	−1.24748e13	−0.565753
$924$	0	0
$925$	3.97035e13	1.78316
$926$	2.29812e13	1.02713
$927$	0	0
$928$	−6.15199e12	−0.272301
$929$	−2.76401e13	−1.21750	−0.608749	−	0.793363i	$-0.708328\pi$
$929$	−2.76401e13	−1.21750	−0.608749	+	0.793363i	$0.708328\pi$
$930$	0	0
$931$	3.02259e13	1.31858
$932$	6.18445e11	0.0268491
$933$	0	0
$934$	2.80612e13	1.20655
$935$	−2.12490e10	−0.000909256 0
$936$	0	0
$937$	−2.24945e13	−0.953341	−0.476671	−	0.879082i	$-0.658157\pi$
$937$	−2.24945e13	−0.953341	−0.476671	+	0.879082i	$0.658157\pi$
$938$	−3.36104e13	−1.41762
$939$	0	0
$940$	−4.07369e10	−0.00170182
$941$	3.28332e12	0.136508	0.0682542	−	0.997668i	$-0.478257\pi$
$941$	3.28332e12	0.136508	0.0682542	+	0.997668i	$0.478257\pi$
$942$	0	0
$943$	−1.69721e13	−0.698930
$944$	1.51527e13	0.621035
$945$	0	0
$946$	5.39774e12	0.219130
$947$	−4.65124e13	−1.87929	−0.939645	−	0.342151i	$-0.888845\pi$
$947$	−4.65124e13	−1.87929	−0.939645	+	0.342151i	$0.888845\pi$
$948$	0	0
$949$	−2.44206e13	−0.977370
$950$	2.69647e13	1.07409
$951$	0	0
$952$	−9.70234e11	−0.0382834
$953$	−3.34707e13	−1.31446	−0.657229	−	0.753691i	$-0.728271\pi$
$953$	−3.34707e13	−1.31446	−0.657229	+	0.753691i	$0.728271\pi$
$954$	0	0
$955$	5.82550e11	0.0226630
$956$	6.34229e11	0.0245576
$957$	0	0
$958$	4.58970e13	1.76051
$959$	1.39688e13	0.533305
$960$	0	0
$961$	−1.52624e13	−0.577257
$962$	−3.49595e13	−1.31607
$963$	0	0
$964$	−5.79941e11	−0.0216290
$965$	−5.01831e11	−0.0186288
$966$	0	0
$967$	4.98325e13	1.83271	0.916354	−	0.400370i	$-0.131118\pi$
$967$	4.98325e13	1.83271	0.916354	+	0.400370i	$0.131118\pi$
$968$	8.26936e12	0.302714
$969$	0	0
$970$	1.45445e11	0.00527506
$971$	2.07241e13	0.748149	0.374075	−	0.927399i	$-0.377960\pi$
$971$	2.07241e13	0.748149	0.374075	+	0.927399i	$0.377960\pi$
$972$	0	0
$973$	4.79309e13	1.71438
$974$	3.10033e13	1.10380
$975$	0	0
$976$	−2.04659e13	−0.721950
$977$	7.94484e11	0.0278971	0.0139486	−	0.999903i	$-0.495560\pi$
$977$	7.94484e11	0.0278971	0.0139486	+	0.999903i	$0.495560\pi$
$978$	0	0
$979$	1.49108e13	0.518776
$980$	−1.06335e11	−0.00368265
$981$	0	0
$982$	−1.66179e13	−0.570263
$983$	5.33290e13	1.82168	0.910840	−	0.412759i	$-0.135435\pi$
$983$	5.33290e13	1.82168	0.910840	+	0.412759i	$0.135435\pi$
$984$	0	0
$985$	1.44976e12	0.0490718
$986$	9.74648e11	0.0328399
$987$	0	0
$988$	2.59644e12	0.0866905
$989$	2.43007e12	0.0807674
$990$	0	0
$991$	3.10566e13	1.02288	0.511438	−	0.859320i	$-0.329113\pi$
$991$	3.10566e13	1.02288	0.511438	+	0.859320i	$0.329113\pi$
$992$	3.89498e12	0.127703
$993$	0	0
$994$	−3.13198e13	−1.01761
$995$	3.34532e11	0.0108202
$996$	0	0
$997$	−2.49263e13	−0.798969	−0.399484	−	0.916740i	$-0.630811\pi$
$997$	−2.49263e13	−0.798969	−0.399484	+	0.916740i	$0.630811\pi$
$998$	1.11688e13	0.356384
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.10.a.d.1.3		8
3.2	odd	2		81.10.a.c.1.6		8
9.2	odd	6		9.10.c.a.4.3	✓	16
9.4	even	3		27.10.c.a.19.6		16
9.5	odd	6		9.10.c.a.7.3	yes	16
9.7	even	3		27.10.c.a.10.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.10.c.a.4.3	✓	16	9.2	odd	6
9.10.c.a.7.3	yes	16	9.5	odd	6
27.10.c.a.10.6		16	9.7	even	3
27.10.c.a.19.6		16	9.4	even	3
81.10.a.c.1.6		8	3.2	odd	2
81.10.a.d.1.3		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	81.a (trivial)

\(f(q)\)	\(=\)	\(q-21.4832 q^{2} -50.4713 q^{4} +44.8401 q^{5} +9345.55 q^{7} +12083.7 q^{8} -963.310 q^{10} +55156.9 q^{11} -79968.8 q^{13} -200773. q^{14} -233755. q^{16} -8591.55 q^{17} +643299. q^{19} -2263.14 q^{20} -1.18495e6 q^{22} -533467. q^{23} -1.95111e6 q^{25} +1.71799e6 q^{26} -471682. q^{28} +5.28052e6 q^{29} -3.34323e6 q^{31} -1.16503e6 q^{32} +184574. q^{34} +419056. q^{35} -2.03491e7 q^{37} -1.38201e7 q^{38} +541834. q^{40} +3.18148e7 q^{41} -4.55525e6 q^{43} -2.78384e6 q^{44} +1.14606e7 q^{46} +1.80002e7 q^{47} +4.69858e7 q^{49} +4.19162e7 q^{50} +4.03613e6 q^{52} +6.95534e7 q^{53} +2.47324e6 q^{55} +1.12929e8 q^{56} -1.13443e8 q^{58} -6.48229e7 q^{59} +8.75528e7 q^{61} +7.18233e7 q^{62} +1.44711e8 q^{64} -3.58581e6 q^{65} +1.67405e8 q^{67} +433627. q^{68} -9.00266e6 q^{70} +1.55996e8 q^{71} +3.05377e8 q^{73} +4.37165e8 q^{74} -3.24681e7 q^{76} +5.15472e8 q^{77} -3.59911e8 q^{79} -1.04816e7 q^{80} -6.83484e8 q^{82} -4.63967e8 q^{83} -385246. q^{85} +9.78614e7 q^{86} +6.66500e8 q^{88} +2.70335e8 q^{89} -7.47352e8 q^{91} +2.69248e7 q^{92} -3.86702e8 q^{94} +2.88456e7 q^{95} -1.50985e8 q^{97} -1.00941e9 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 15 q^{2} + 1793 q^{4} + 453 q^{5} + 343 q^{7} + 7239 q^{8} + 510 q^{10} + 99150 q^{11} - 32435 q^{13} + 394824 q^{14} + 328193 q^{16} + 415539 q^{17} - 85277 q^{19} + 1855164 q^{20} - 529359 q^{22}+ \cdots - 2413650159 q^{98}+O(q^{100}) \) 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

L-functions

Modular forms

Varieties

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Database

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