LMFDB - Embedded newform 81.10.a.d.1.6

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			1.6
Root			$-15.7185$ 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)

$f(q)$	$=$	\(q+17.7185 q^{2} -198.054 q^{4} +739.877 q^{5} -6027.34 q^{7} -12581.1 q^{8} +13109.5 q^{10} -18431.0 q^{11} +117586. q^{13} -106795. q^{14} -121515. q^{16} +468308. q^{17} +834842. q^{19} -146536. q^{20} -326571. q^{22} -1.32052e6 q^{23} -1.40571e6 q^{25} +2.08345e6 q^{26} +1.19374e6 q^{28} -64246.8 q^{29} +7.41751e6 q^{31} +4.28847e6 q^{32} +8.29773e6 q^{34} -4.45949e6 q^{35} +3.68044e6 q^{37} +1.47922e7 q^{38} -9.30848e6 q^{40} +2.53611e7 q^{41} +2.42839e7 q^{43} +3.65035e6 q^{44} -2.33977e7 q^{46} -7.03371e6 q^{47} -4.02479e6 q^{49} -2.49070e7 q^{50} -2.32884e7 q^{52} +3.10597e7 q^{53} -1.36367e7 q^{55} +7.58306e7 q^{56} -1.13836e6 q^{58} +1.53654e8 q^{59} -1.55161e8 q^{61} +1.31427e8 q^{62} +1.38201e8 q^{64} +8.69992e7 q^{65} -1.39728e7 q^{67} -9.27506e7 q^{68} -7.90155e7 q^{70} +3.45182e8 q^{71} -3.01511e8 q^{73} +6.52119e7 q^{74} -1.65344e8 q^{76} +1.11090e8 q^{77} +3.59669e8 q^{79} -8.99059e7 q^{80} +4.49361e8 q^{82} +1.05014e8 q^{83} +3.46491e8 q^{85} +4.30274e8 q^{86} +2.31883e8 q^{88} -8.60555e8 q^{89} -7.08730e8 q^{91} +2.61536e8 q^{92} -1.24627e8 q^{94} +6.17681e8 q^{95} +9.48522e7 q^{97} -7.13133e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	17.7185	0.783055	0.391527	−	0.920166i	$-0.371947\pi$
$2$	17.7185	0.783055	0.391527	+	0.920166i	$0.371947\pi$
$3$	0	0
$3$	0	0
$4$	−198.054	−0.386825
$5$	739.877	0.529413	0.264707	−	0.964329i	$-0.414725\pi$
$5$	739.877	0.529413	0.264707	+	0.964329i	$0.414725\pi$
$6$	0	0
$7$	−6027.34	−0.948821	−0.474411	−	0.880304i	$-0.657339\pi$
$7$	−6027.34	−0.948821	−0.474411	+	0.880304i	$0.657339\pi$
$8$	−12581.1	−1.08596
$9$	0	0
$10$	13109.5	0.414560
$11$	−18431.0	−0.379562	−0.189781	−	0.981826i	$-0.560778\pi$
$11$	−18431.0	−0.379562	−0.189781	+	0.981826i	$0.560778\pi$
$12$	0	0
$13$	117586.	1.14185	0.570927	−	0.821001i	$-0.306584\pi$
$13$	117586.	1.14185	0.570927	+	0.821001i	$0.306584\pi$
$14$	−106795.	−0.742979
$15$	0	0
$16$	−121515.	−0.463541
$17$	468308.	1.35992	0.679958	−	0.733251i	$-0.261998\pi$
$17$	468308.	1.35992	0.679958	+	0.733251i	$0.261998\pi$
$18$	0	0
$19$	834842.	1.46965	0.734824	−	0.678258i	$-0.237265\pi$
$19$	834842.	1.46965	0.734824	+	0.678258i	$0.237265\pi$
$20$	−146536.	−0.204790
$21$	0	0
$22$	−326571.	−0.297218
$23$	−1.32052e6	−0.983946	−0.491973	−	0.870610i	$-0.663724\pi$
$23$	−1.32052e6	−0.983946	−0.491973	+	0.870610i	$0.663724\pi$
$24$	0	0
$25$	−1.40571e6	−0.719722
$26$	2.08345e6	0.894134
$27$	0	0
$28$	1.19374e6	0.367028
$29$	−64246.8	−0.0168679	−0.00843394	−	0.999964i	$-0.502685\pi$
$29$	−64246.8	−0.0168679	−0.00843394	+	0.999964i	$0.502685\pi$
$30$	0	0
$31$	7.41751e6	1.44255	0.721274	−	0.692650i	$-0.243557\pi$
$31$	7.41751e6	1.44255	0.721274	+	0.692650i	$0.243557\pi$
$32$	4.28847e6	0.722982
$33$	0	0
$34$	8.29773e6	1.06489
$35$	−4.45949e6	−0.502318
$36$	0	0
$37$	3.68044e6	0.322843	0.161422	−	0.986886i	$-0.448392\pi$
$37$	3.68044e6	0.322843	0.161422	+	0.986886i	$0.448392\pi$
$38$	1.47922e7	1.15081
$39$	0	0
$40$	−9.30848e6	−0.574922
$41$	2.53611e7	1.40165	0.700827	−	0.713331i	$-0.252814\pi$
$41$	2.53611e7	1.40165	0.700827	+	0.713331i	$0.252814\pi$
$42$	0	0
$43$	2.42839e7	1.08320	0.541601	−	0.840636i	$-0.317818\pi$
$43$	2.42839e7	1.08320	0.541601	+	0.840636i	$0.317818\pi$
$44$	3.65035e6	0.146824
$45$	0	0
$46$	−2.33977e7	−0.770484
$47$	−7.03371e6	−0.210254	−0.105127	−	0.994459i	$-0.533525\pi$
$47$	−7.03371e6	−0.210254	−0.105127	+	0.994459i	$0.533525\pi$
$48$	0	0
$49$	−4.02479e6	−0.0997380
$50$	−2.49070e7	−0.563582
$51$	0	0
$52$	−2.32884e7	−0.441698
$53$	3.10597e7	0.540700	0.270350	−	0.962762i	$-0.412861\pi$
$53$	3.10597e7	0.540700	0.270350	+	0.962762i	$0.412861\pi$
$54$	0	0
$55$	−1.36367e7	−0.200945
$56$	7.58306e7	1.03038
$57$	0	0
$58$	−1.13836e6	−0.0132085
$59$	1.53654e8	1.65086	0.825429	−	0.564506i	$-0.190933\pi$
$59$	1.53654e8	1.65086	0.825429	+	0.564506i	$0.190933\pi$
$60$	0	0
$61$	−1.55161e8	−1.43482	−0.717410	−	0.696651i	$-0.754673\pi$
$61$	−1.55161e8	−1.43482	−0.717410	+	0.696651i	$0.754673\pi$
$62$	1.31427e8	1.12959
$63$	0	0
$64$	1.38201e8	1.02968
$65$	8.69992e7	0.604512
$66$	0	0
$67$	−1.39728e7	−0.0847124	−0.0423562	−	0.999103i	$-0.513486\pi$
$67$	−1.39728e7	−0.0847124	−0.0423562	+	0.999103i	$0.513486\pi$
$68$	−9.27506e7	−0.526049
$69$	0	0
$70$	−7.90155e7	−0.393343
$71$	3.45182e8	1.61207	0.806037	−	0.591865i	$-0.201608\pi$
$71$	3.45182e8	1.61207	0.806037	+	0.591865i	$0.201608\pi$
$72$	0	0
$73$	−3.01511e8	−1.24265	−0.621327	−	0.783551i	$-0.713406\pi$
$73$	−3.01511e8	−1.24265	−0.621327	+	0.783551i	$0.713406\pi$
$74$	6.52119e7	0.252804
$75$	0	0
$76$	−1.65344e8	−0.568497
$77$	1.11090e8	0.360137
$78$	0	0
$79$	3.59669e8	1.03892	0.519458	−	0.854496i	$-0.326134\pi$
$79$	3.59669e8	1.03892	0.519458	+	0.854496i	$0.326134\pi$
$80$	−8.99059e7	−0.245405
$81$	0	0
$82$	4.49361e8	1.09757
$83$	1.05014e8	0.242882	0.121441	−	0.992599i	$-0.461249\pi$
$83$	1.05014e8	0.242882	0.121441	+	0.992599i	$0.461249\pi$
$84$	0	0
$85$	3.46491e8	0.719957
$86$	4.30274e8	0.848207
$87$	0	0
$88$	2.31883e8	0.412189
$89$	−8.60555e8	−1.45386	−0.726932	−	0.686710i	$-0.759054\pi$
$89$	−8.60555e8	−1.45386	−0.726932	+	0.686710i	$0.759054\pi$
$90$	0	0
$91$	−7.08730e8	−1.08342
$92$	2.61536e8	0.380615
$93$	0	0
$94$	−1.24627e8	−0.164640
$95$	6.17681e8	0.778051
$96$	0	0
$97$	9.48522e7	0.108786	0.0543932	−	0.998520i	$-0.482678\pi$
$97$	9.48522e7	0.108786	0.0543932	+	0.998520i	$0.482678\pi$
$98$	−7.13133e7	−0.0781004
$99$	0	0
$100$	2.78406e8	0.278406
$101$	−4.77159e8	−0.456264	−0.228132	−	0.973630i	$-0.573262\pi$
$101$	−4.77159e8	−0.456264	−0.228132	+	0.973630i	$0.573262\pi$
$102$	0	0
$103$	3.37112e8	0.295125	0.147563	−	0.989053i	$-0.452857\pi$
$103$	3.37112e8	0.295125	0.147563	+	0.989053i	$0.452857\pi$
$104$	−1.47936e9	−1.24001
$105$	0	0
$106$	5.50332e8	0.423398
$107$	1.05652e9	0.779203	0.389602	−	0.920983i	$-0.372613\pi$
$107$	1.05652e9	0.779203	0.389602	+	0.920983i	$0.372613\pi$
$108$	0	0
$109$	5.44814e8	0.369683	0.184841	−	0.982768i	$-0.440823\pi$
$109$	5.44814e8	0.369683	0.184841	+	0.982768i	$0.440823\pi$
$110$	−2.41622e8	−0.157351
$111$	0	0
$112$	7.32409e8	0.439818
$113$	−1.09053e9	−0.629193	−0.314597	−	0.949225i	$-0.601869\pi$
$113$	−1.09053e9	−0.629193	−0.314597	+	0.949225i	$0.601869\pi$
$114$	0	0
$115$	−9.77026e8	−0.520914
$116$	1.27244e7	0.00652492
$117$	0	0
$118$	2.72252e9	1.29271
$119$	−2.82265e9	−1.29032
$120$	0	0
$121$	−2.01824e9	−0.855933
$122$	−2.74922e9	−1.12354
$123$	0	0
$124$	−1.46907e9	−0.558014
$125$	−2.48512e9	−0.910443
$126$	0	0
$127$	−1.90290e9	−0.649082	−0.324541	−	0.945872i	$-0.605210\pi$
$127$	−1.90290e9	−0.649082	−0.324541	+	0.945872i	$0.605210\pi$
$128$	2.53015e8	0.0833107
$129$	0	0
$130$	1.54150e9	0.473366
$131$	7.70914e8	0.228710	0.114355	−	0.993440i	$-0.463520\pi$
$131$	7.70914e8	0.228710	0.114355	+	0.993440i	$0.463520\pi$
$132$	0	0
$133$	−5.03188e9	−1.39443
$134$	−2.47577e8	−0.0663345
$135$	0	0
$136$	−5.89184e9	−1.47681
$137$	4.23967e8	0.102823	0.0514114	−	0.998678i	$-0.483628\pi$
$137$	4.23967e8	0.102823	0.0514114	+	0.998678i	$0.483628\pi$
$138$	0	0
$139$	−2.49710e9	−0.567373	−0.283687	−	0.958917i	$-0.591558\pi$
$139$	−2.49710e9	−0.567373	−0.283687	+	0.958917i	$0.591558\pi$
$140$	8.83222e8	0.194309
$141$	0	0
$142$	6.11610e9	1.26234
$143$	−2.16723e9	−0.433404
$144$	0	0
$145$	−4.75347e7	−0.00893007
$146$	−5.34232e9	−0.973066
$147$	0	0
$148$	−7.28927e8	−0.124884
$149$	−3.54791e9	−0.589705	−0.294853	−	0.955543i	$-0.595271\pi$
$149$	−3.54791e9	−0.589705	−0.294853	+	0.955543i	$0.595271\pi$
$150$	0	0
$151$	4.24790e9	0.664933	0.332466	−	0.943115i	$-0.392119\pi$
$151$	4.24790e9	0.664933	0.332466	+	0.943115i	$0.392119\pi$
$152$	−1.05032e10	−1.59598
$153$	0	0
$154$	1.96835e9	0.282007
$155$	5.48804e9	0.763704
$156$	0	0
$157$	−1.83176e9	−0.240613	−0.120307	−	0.992737i	$-0.538388\pi$
$157$	−1.83176e9	−0.240613	−0.120307	+	0.992737i	$0.538388\pi$
$158$	6.37279e9	0.813528
$159$	0	0
$160$	3.17294e9	0.382756
$161$	7.95925e9	0.933589
$162$	0	0
$163$	4.81775e9	0.534564	0.267282	−	0.963618i	$-0.413874\pi$
$163$	4.81775e9	0.534564	0.267282	+	0.963618i	$0.413874\pi$
$164$	−5.02288e9	−0.542195
$165$	0	0
$166$	1.86069e9	0.190190
$167$	1.50711e9	0.149941	0.0749704	−	0.997186i	$-0.476114\pi$
$167$	1.50711e9	0.149941	0.0749704	+	0.997186i	$0.476114\pi$
$168$	0	0
$169$	3.22196e9	0.303830
$170$	6.13930e9	0.563766
$171$	0	0
$172$	−4.80953e9	−0.419010
$173$	2.16043e10	1.83372	0.916859	−	0.399210i	$-0.130716\pi$
$173$	2.16043e10	1.83372	0.916859	+	0.399210i	$0.130716\pi$
$174$	0	0
$175$	8.47267e9	0.682887
$176$	2.23964e9	0.175943
$177$	0	0
$178$	−1.52478e10	−1.13845
$179$	1.94882e10	1.41884	0.709418	−	0.704788i	$-0.248958\pi$
$179$	1.94882e10	1.41884	0.709418	+	0.704788i	$0.248958\pi$
$180$	0	0
$181$	7.45903e9	0.516569	0.258285	−	0.966069i	$-0.416843\pi$
$181$	7.45903e9	0.516569	0.258285	+	0.966069i	$0.416843\pi$
$182$	−1.25576e10	−0.848373
$183$	0	0
$184$	1.66137e10	1.06853
$185$	2.72307e9	0.170917
$186$	0	0
$187$	−8.63142e9	−0.516172
$188$	1.39306e9	0.0813315
$189$	0	0
$190$	1.09444e10	0.609256
$191$	6.56601e9	0.356986	0.178493	−	0.983941i	$-0.442878\pi$
$191$	6.56601e9	0.356986	0.178493	+	0.983941i	$0.442878\pi$
$192$	0	0
$193$	−1.88011e10	−0.975386	−0.487693	−	0.873015i	$-0.662161\pi$
$193$	−1.88011e10	−0.975386	−0.487693	+	0.873015i	$0.662161\pi$
$194$	1.68064e9	0.0851857
$195$	0	0
$196$	7.97127e8	0.0385812
$197$	1.66201e9	0.0786206	0.0393103	−	0.999227i	$-0.487484\pi$
$197$	1.66201e9	0.0786206	0.0393103	+	0.999227i	$0.487484\pi$
$198$	0	0
$199$	1.45345e10	0.656996	0.328498	−	0.944505i	$-0.393458\pi$
$199$	1.45345e10	0.656996	0.328498	+	0.944505i	$0.393458\pi$
$200$	1.76853e10	0.781589
$201$	0	0
$202$	−8.45454e9	−0.357280
$203$	3.87237e8	0.0160046
$204$	0	0
$205$	1.87641e10	0.742054
$206$	5.97312e9	0.231099
$207$	0	0
$208$	−1.42884e10	−0.529296
$209$	−1.53870e10	−0.557823
$210$	0	0
$211$	1.39321e9	0.0483890	0.0241945	−	0.999707i	$-0.492298\pi$
$211$	1.39321e9	0.0483890	0.0241945	+	0.999707i	$0.492298\pi$
$212$	−6.15152e9	−0.209156
$213$	0	0
$214$	1.87200e10	0.610159
$215$	1.79671e10	0.573462
$216$	0	0
$217$	−4.47078e10	−1.36872
$218$	9.65330e9	0.289482
$219$	0	0
$220$	2.70081e9	0.0777307
$221$	5.50665e10	1.55282
$222$	0	0
$223$	−2.87367e10	−0.778153	−0.389077	−	0.921205i	$-0.627206\pi$
$223$	−2.87367e10	−0.778153	−0.389077	+	0.921205i	$0.627206\pi$
$224$	−2.58481e10	−0.685981
$225$	0	0
$226$	−1.93225e10	−0.492693
$227$	3.84175e10	0.960313	0.480157	−	0.877183i	$-0.340580\pi$
$227$	3.84175e10	0.960313	0.480157	+	0.877183i	$0.340580\pi$
$228$	0	0
$229$	−1.33906e10	−0.321765	−0.160883	−	0.986974i	$-0.551434\pi$
$229$	−1.33906e10	−0.321765	−0.160883	+	0.986974i	$0.551434\pi$
$230$	−1.73115e10	−0.407904
$231$	0	0
$232$	8.08296e8	0.0183178
$233$	−4.50134e10	−1.00055	−0.500276	−	0.865866i	$-0.666768\pi$
$233$	−4.50134e10	−1.00055	−0.500276	+	0.865866i	$0.666768\pi$
$234$	0	0
$235$	−5.20409e9	−0.111311
$236$	−3.04319e10	−0.638593
$237$	0	0
$238$	−5.00132e10	−1.01039
$239$	6.55944e10	1.30040	0.650199	−	0.759764i	$-0.274686\pi$
$239$	6.55944e10	1.30040	0.650199	+	0.759764i	$0.274686\pi$
$240$	0	0
$241$	2.68133e10	0.512005	0.256002	−	0.966676i	$-0.417595\pi$
$241$	2.68133e10	0.512005	0.256002	+	0.966676i	$0.417595\pi$
$242$	−3.57603e10	−0.670242
$243$	0	0
$244$	3.07303e10	0.555025
$245$	−2.97785e9	−0.0528026
$246$	0	0
$247$	9.81657e10	1.67812
$248$	−9.33204e10	−1.56655
$249$	0	0
$250$	−4.40327e10	−0.712927
$251$	4.29625e10	0.683216	0.341608	−	0.939842i	$-0.389028\pi$
$251$	4.29625e10	0.683216	0.341608	+	0.939842i	$0.389028\pi$
$252$	0	0
$253$	2.43387e10	0.373469
$254$	−3.37166e10	−0.508267
$255$	0	0
$256$	−6.62757e10	−0.964439
$257$	−9.45505e10	−1.35196	−0.675981	−	0.736919i	$-0.736280\pi$
$257$	−9.45505e10	−1.35196	−0.675981	+	0.736919i	$0.736280\pi$
$258$	0	0
$259$	−2.21832e10	−0.306321
$260$	−1.72306e10	−0.233841
$261$	0	0
$262$	1.36594e10	0.179092
$263$	−1.12469e11	−1.44954	−0.724771	−	0.688990i	$-0.758055\pi$
$263$	−1.12469e11	−1.44954	−0.724771	+	0.688990i	$0.758055\pi$
$264$	0	0
$265$	2.29804e10	0.286254
$266$	−8.91574e10	−1.09192
$267$	0	0
$268$	2.76738e9	0.0327689
$269$	1.11208e11	1.29494	0.647472	−	0.762089i	$-0.275826\pi$
$269$	1.11208e11	1.29494	0.647472	+	0.762089i	$0.275826\pi$
$270$	0	0
$271$	1.30075e10	0.146498	0.0732491	−	0.997314i	$-0.476663\pi$
$271$	1.30075e10	0.146498	0.0732491	+	0.997314i	$0.476663\pi$
$272$	−5.69063e10	−0.630377
$273$	0	0
$274$	7.51207e9	0.0805159
$275$	2.59086e10	0.273179
$276$	0	0
$277$	4.25430e10	0.434180	0.217090	−	0.976152i	$-0.430344\pi$
$277$	4.25430e10	0.434180	0.217090	+	0.976152i	$0.430344\pi$
$278$	−4.42448e10	−0.444284
$279$	0	0
$280$	5.61053e10	0.545498
$281$	1.38366e11	1.32388	0.661941	−	0.749556i	$-0.269733\pi$
$281$	1.38366e11	1.32388	0.661941	+	0.749556i	$0.269733\pi$
$282$	0	0
$283$	8.80789e10	0.816268	0.408134	−	0.912922i	$-0.366180\pi$
$283$	8.80789e10	0.816268	0.408134	+	0.912922i	$0.366180\pi$
$284$	−6.83648e10	−0.623591
$285$	0	0
$286$	−3.84001e10	−0.339379
$287$	−1.52860e11	−1.32992
$288$	0	0
$289$	1.00725e11	0.849370
$290$	−8.42245e8	−0.00699274
$291$	0	0
$292$	5.97156e10	0.480690
$293$	1.51744e11	1.20284	0.601419	−	0.798934i	$-0.294602\pi$
$293$	1.51744e11	1.20284	0.601419	+	0.798934i	$0.294602\pi$
$294$	0	0
$295$	1.13685e11	0.873986
$296$	−4.63040e10	−0.350595
$297$	0	0
$298$	−6.28638e10	−0.461772
$299$	−1.55275e11	−1.12352
$300$	0	0
$301$	−1.46367e11	−1.02777
$302$	7.52664e10	0.520679
$303$	0	0
$304$	−1.01445e11	−0.681242
$305$	−1.14800e11	−0.759613
$306$	0	0
$307$	−1.76213e11	−1.13218	−0.566089	−	0.824344i	$-0.691544\pi$
$307$	−1.76213e11	−1.13218	−0.566089	+	0.824344i	$0.691544\pi$
$308$	−2.20019e10	−0.139310
$309$	0	0
$310$	9.72400e10	0.598022
$311$	−9.98964e10	−0.605519	−0.302760	−	0.953067i	$-0.597908\pi$
$311$	−9.98964e10	−0.605519	−0.302760	+	0.953067i	$0.597908\pi$
$312$	0	0
$313$	1.83425e11	1.08021	0.540106	−	0.841597i	$-0.318384\pi$
$313$	1.83425e11	1.08021	0.540106	+	0.841597i	$0.318384\pi$
$314$	−3.24560e10	−0.188413
$315$	0	0
$316$	−7.12340e10	−0.401879
$317$	2.58351e11	1.43696	0.718479	−	0.695549i	$-0.244839\pi$
$317$	2.58351e11	1.43696	0.718479	+	0.695549i	$0.244839\pi$
$318$	0	0
$319$	1.18414e9	0.00640241
$320$	1.02252e11	0.545124
$321$	0	0
$322$	1.41026e11	0.731051
$323$	3.90964e11	1.99860
$324$	0	0
$325$	−1.65291e11	−0.821817
$326$	8.53633e10	0.418593
$327$	0	0
$328$	−3.19071e11	−1.52214
$329$	4.23946e10	0.199494
$330$	0	0
$331$	−2.83249e11	−1.29701	−0.648503	−	0.761212i	$-0.724605\pi$
$331$	−2.83249e11	−1.29701	−0.648503	+	0.761212i	$0.724605\pi$
$332$	−2.07984e10	−0.0939528
$333$	0	0
$334$	2.67037e10	0.117412
$335$	−1.03382e10	−0.0448479
$336$	0	0
$337$	−2.99587e11	−1.26528	−0.632642	−	0.774444i	$-0.718030\pi$
$337$	−2.99587e11	−1.26528	−0.632642	+	0.774444i	$0.718030\pi$
$338$	5.70883e10	0.237915
$339$	0	0
$340$	−6.86240e10	−0.278497
$341$	−1.36712e11	−0.547537
$342$	0	0
$343$	2.67484e11	1.04345
$344$	−3.05518e11	−1.17631
$345$	0	0
$346$	3.82796e11	1.43590
$347$	5.47340e10	0.202663	0.101332	−	0.994853i	$-0.467690\pi$
$347$	5.47340e10	0.202663	0.101332	+	0.994853i	$0.467690\pi$
$348$	0	0
$349$	−3.04215e11	−1.09766	−0.548828	−	0.835935i	$-0.684926\pi$
$349$	−3.04215e11	−1.09766	−0.548828	+	0.835935i	$0.684926\pi$
$350$	1.50123e11	0.534738
$351$	0	0
$352$	−7.90410e10	−0.274417
$353$	5.30034e11	1.81684	0.908421	−	0.418057i	$-0.137289\pi$
$353$	5.30034e11	1.81684	0.908421	+	0.418057i	$0.137289\pi$
$354$	0	0
$355$	2.55392e11	0.853453
$356$	1.70437e11	0.562391
$357$	0	0
$358$	3.45301e11	1.11103
$359$	4.60506e11	1.46322	0.731611	−	0.681722i	$-0.238769\pi$
$359$	4.60506e11	1.46322	0.731611	+	0.681722i	$0.238769\pi$
$360$	0	0
$361$	3.74274e11	1.15986
$362$	1.32163e11	0.404502
$363$	0	0
$364$	1.40367e11	0.419092
$365$	−2.23081e11	−0.657877
$366$	0	0
$367$	−4.52772e11	−1.30281	−0.651407	−	0.758728i	$-0.725821\pi$
$367$	−4.52772e11	−1.30281	−0.651407	+	0.758728i	$0.725821\pi$
$368$	1.60463e11	0.456100
$369$	0	0
$370$	4.82488e10	0.133838
$371$	−1.87208e11	−0.513028
$372$	0	0
$373$	−1.01130e11	−0.270515	−0.135257	−	0.990810i	$-0.543186\pi$
$373$	−1.01130e11	−0.270515	−0.135257	+	0.990810i	$0.543186\pi$
$374$	−1.52936e11	−0.404191
$375$	0	0
$376$	8.84919e10	0.228327
$377$	−7.55452e9	−0.0192606
$378$	0	0
$379$	−5.06871e11	−1.26189	−0.630944	−	0.775829i	$-0.717332\pi$
$379$	−5.06871e11	−1.26189	−0.630944	+	0.775829i	$0.717332\pi$
$380$	−1.22334e11	−0.300970
$381$	0	0
$382$	1.16340e11	0.279540
$383$	2.96940e11	0.705137	0.352569	−	0.935786i	$-0.385308\pi$
$383$	2.96940e11	0.705137	0.352569	+	0.935786i	$0.385308\pi$
$384$	0	0
$385$	8.21931e10	0.190661
$386$	−3.33128e11	−0.763781
$387$	0	0
$388$	−1.87859e10	−0.0420813
$389$	5.28340e10	0.116988	0.0584939	−	0.998288i	$-0.481370\pi$
$389$	5.28340e10	0.116988	0.0584939	+	0.998288i	$0.481370\pi$
$390$	0	0
$391$	−6.18413e11	−1.33808
$392$	5.06363e10	0.108312
$393$	0	0
$394$	2.94484e10	0.0615642
$395$	2.66111e11	0.550016
$396$	0	0
$397$	4.35046e11	0.878978	0.439489	−	0.898248i	$-0.355160\pi$
$397$	4.35046e11	0.878978	0.439489	+	0.898248i	$0.355160\pi$
$398$	2.57531e11	0.514464
$399$	0	0
$400$	1.70814e11	0.333621
$401$	−4.04110e10	−0.0780459	−0.0390230	−	0.999238i	$-0.512425\pi$
$401$	−4.04110e10	−0.0780459	−0.0390230	+	0.999238i	$0.512425\pi$
$402$	0	0
$403$	8.72195e11	1.64718
$404$	9.45034e10	0.176495
$405$	0	0
$406$	6.86126e9	0.0125325
$407$	−6.78343e10	−0.122539
$408$	0	0
$409$	2.41358e11	0.426487	0.213244	−	0.976999i	$-0.431597\pi$
$409$	2.41358e11	0.426487	0.213244	+	0.976999i	$0.431597\pi$
$410$	3.32472e11	0.581069
$411$	0	0
$412$	−6.67665e10	−0.114162
$413$	−9.26125e11	−1.56637
$414$	0	0
$415$	7.76973e10	0.128585
$416$	5.04264e11	0.825539
$417$	0	0
$418$	−2.72635e11	−0.436806
$419$	3.75371e11	0.594973	0.297486	−	0.954726i	$-0.403852\pi$
$419$	3.75371e11	0.594973	0.297486	+	0.954726i	$0.403852\pi$
$420$	0	0
$421$	7.26393e11	1.12694	0.563472	−	0.826135i	$-0.309465\pi$
$421$	7.26393e11	1.12694	0.563472	+	0.826135i	$0.309465\pi$
$422$	2.46857e10	0.0378912
$423$	0	0
$424$	−3.90766e11	−0.587179
$425$	−6.58304e11	−0.978761
$426$	0	0
$427$	9.35206e11	1.36139
$428$	−2.09249e11	−0.301415
$429$	0	0
$430$	3.18350e11	0.449052
$431$	−8.99614e11	−1.25576	−0.627882	−	0.778308i	$-0.716078\pi$
$431$	−8.99614e11	−1.25576	−0.627882	+	0.778308i	$0.716078\pi$
$432$	0	0
$433$	−7.77024e11	−1.06228	−0.531140	−	0.847284i	$-0.678236\pi$
$433$	−7.77024e11	−1.06228	−0.531140	+	0.847284i	$0.678236\pi$
$434$	−7.92156e11	−1.07178
$435$	0	0
$436$	−1.07903e11	−0.143003
$437$	−1.10243e12	−1.44605
$438$	0	0
$439$	−1.52699e12	−1.96221	−0.981104	−	0.193480i	$-0.938023\pi$
$439$	−1.52699e12	−1.96221	−0.981104	+	0.193480i	$0.938023\pi$
$440$	1.71565e11	0.218218
$441$	0	0
$442$	9.75696e11	1.21595
$443$	−2.78099e11	−0.343070	−0.171535	−	0.985178i	$-0.554873\pi$
$443$	−2.78099e11	−0.343070	−0.171535	+	0.985178i	$0.554873\pi$
$444$	0	0
$445$	−6.36705e11	−0.769694
$446$	−5.09172e11	−0.609337
$447$	0	0
$448$	−8.32983e11	−0.976978
$449$	−3.39438e11	−0.394141	−0.197071	−	0.980389i	$-0.563143\pi$
$449$	−3.39438e11	−0.394141	−0.197071	+	0.980389i	$0.563143\pi$
$450$	0	0
$451$	−4.67432e11	−0.532015
$452$	2.15984e11	0.243388
$453$	0	0
$454$	6.80701e11	0.751978
$455$	−5.24374e11	−0.573574
$456$	0	0
$457$	8.20378e11	0.879814	0.439907	−	0.898043i	$-0.355011\pi$
$457$	8.20378e11	0.879814	0.439907	+	0.898043i	$0.355011\pi$
$458$	−2.37261e11	−0.251960
$459$	0	0
$460$	1.93504e11	0.201503
$461$	−9.91441e11	−1.02238	−0.511190	−	0.859467i	$-0.670795\pi$
$461$	−9.91441e11	−1.02238	−0.511190	+	0.859467i	$0.670795\pi$
$462$	0	0
$463$	−4.92923e11	−0.498499	−0.249250	−	0.968439i	$-0.580184\pi$
$463$	−4.92923e11	−0.498499	−0.249250	+	0.968439i	$0.580184\pi$
$464$	7.80692e9	0.00781896
$465$	0	0
$466$	−7.97570e11	−0.783487
$467$	9.59285e11	0.933301	0.466650	−	0.884442i	$-0.345461\pi$
$467$	9.59285e11	0.933301	0.466650	+	0.884442i	$0.345461\pi$
$468$	0	0
$469$	8.42188e10	0.0803769
$470$	−9.22086e10	−0.0871628
$471$	0	0
$472$	−1.93314e12	−1.79277
$473$	−4.47577e11	−0.411143
$474$	0	0
$475$	−1.17354e12	−1.05774
$476$	5.59039e11	0.499127
$477$	0	0
$478$	1.16223e12	1.01828
$479$	1.05996e11	0.0919979	0.0459989	−	0.998941i	$-0.485353\pi$
$479$	1.05996e11	0.0919979	0.0459989	+	0.998941i	$0.485353\pi$
$480$	0	0
$481$	4.32768e11	0.368640
$482$	4.75092e11	0.400928
$483$	0	0
$484$	3.99722e11	0.331096
$485$	7.01790e10	0.0575929
$486$	0	0
$487$	−4.79462e11	−0.386255	−0.193127	−	0.981174i	$-0.561863\pi$
$487$	−4.79462e11	−0.386255	−0.193127	+	0.981174i	$0.561863\pi$
$488$	1.95209e12	1.55816
$489$	0	0
$490$	−5.27631e10	−0.0413474
$491$	−1.21845e12	−0.946108	−0.473054	−	0.881033i	$-0.656848\pi$
$491$	−1.21845e12	−0.946108	−0.473054	+	0.881033i	$0.656848\pi$
$492$	0	0
$493$	−3.00873e10	−0.0229389
$494$	1.73935e12	1.31406
$495$	0	0
$496$	−9.01335e11	−0.668680
$497$	−2.08053e12	−1.52957
$498$	0	0
$499$	1.87731e12	1.35545	0.677724	−	0.735316i	$-0.262966\pi$
$499$	1.87731e12	1.35545	0.677724	+	0.735316i	$0.262966\pi$
$500$	4.92190e11	0.352182
$501$	0	0
$502$	7.61232e11	0.534996
$503$	−1.91484e12	−1.33375	−0.666877	−	0.745168i	$-0.732370\pi$
$503$	−1.91484e12	−1.33375	−0.666877	+	0.745168i	$0.732370\pi$
$504$	0	0
$505$	−3.53039e11	−0.241552
$506$	4.31245e11	0.292447
$507$	0	0
$508$	3.76878e11	0.251081
$509$	1.34641e12	0.889095	0.444548	−	0.895755i	$-0.353364\pi$
$509$	1.34641e12	0.889095	0.444548	+	0.895755i	$0.353364\pi$
$510$	0	0
$511$	1.81731e12	1.17906
$512$	−1.30385e12	−0.838519
$513$	0	0
$514$	−1.67529e12	−1.05866
$515$	2.49421e11	0.156243
$516$	0	0
$517$	1.29639e11	0.0798045
$518$	−3.93054e11	−0.239866
$519$	0	0
$520$	−1.09455e12	−0.656476
$521$	−1.22696e12	−0.729557	−0.364778	−	0.931094i	$-0.618855\pi$
$521$	−1.22696e12	−0.729557	−0.364778	+	0.931094i	$0.618855\pi$
$522$	0	0
$523$	2.16646e12	1.26617	0.633087	−	0.774080i	$-0.281787\pi$
$523$	2.16646e12	1.26617	0.633087	+	0.774080i	$0.281787\pi$
$524$	−1.52683e11	−0.0884707
$525$	0	0
$526$	−1.99278e12	−1.13507
$527$	3.47368e12	1.96174
$528$	0	0
$529$	−5.73668e10	−0.0318500
$530$	4.07178e11	0.224152
$531$	0	0
$532$	9.96586e11	0.539402
$533$	2.98211e12	1.60048
$534$	0	0
$535$	7.81695e11	0.412520
$536$	1.75793e11	0.0919943
$537$	0	0
$538$	1.97044e12	1.01401
$539$	7.41811e10	0.0378568
$540$	0	0
$541$	−1.66976e12	−0.838041	−0.419021	−	0.907977i	$-0.637626\pi$
$541$	−1.66976e12	−0.838041	−0.419021	+	0.907977i	$0.637626\pi$
$542$	2.30473e11	0.114716
$543$	0	0
$544$	2.00833e12	0.983194
$545$	4.03096e11	0.195715
$546$	0	0
$547$	1.74317e12	0.832526	0.416263	−	0.909244i	$-0.363340\pi$
$547$	1.74317e12	0.832526	0.416263	+	0.909244i	$0.363340\pi$
$548$	−8.39686e10	−0.0397745
$549$	0	0
$550$	4.59063e11	0.213914
$551$	−5.36359e10	−0.0247898
$552$	0	0
$553$	−2.16784e12	−0.985746
$554$	7.53799e11	0.339987
$555$	0	0
$556$	4.94561e11	0.219474
$557$	3.88681e12	1.71098	0.855490	−	0.517820i	$-0.173256\pi$
$557$	3.88681e12	1.71098	0.855490	+	0.517820i	$0.173256\pi$
$558$	0	0
$559$	2.85544e12	1.23686
$560$	5.41893e11	0.232845
$561$	0	0
$562$	2.45163e12	1.03667
$563$	−2.81208e12	−1.17961	−0.589806	−	0.807545i	$-0.700796\pi$
$563$	−2.81208e12	−1.17961	−0.589806	+	0.807545i	$0.700796\pi$
$564$	0	0
$565$	−8.06858e11	−0.333103
$566$	1.56063e12	0.639183
$567$	0	0
$568$	−4.34277e12	−1.75065
$569$	−1.82196e12	−0.728675	−0.364338	−	0.931267i	$-0.618705\pi$
$569$	−1.82196e12	−0.728675	−0.364338	+	0.931267i	$0.618705\pi$
$570$	0	0
$571$	−3.36495e12	−1.32470	−0.662348	−	0.749196i	$-0.730440\pi$
$571$	−3.36495e12	−1.32470	−0.662348	+	0.749196i	$0.730440\pi$
$572$	4.29230e11	0.167652
$573$	0	0
$574$	−2.70845e12	−1.04140
$575$	1.85627e12	0.708167
$576$	0	0
$577$	2.47809e12	0.930737	0.465368	−	0.885117i	$-0.345922\pi$
$577$	2.47809e12	0.930737	0.465368	+	0.885117i	$0.345922\pi$
$578$	1.78470e12	0.665103
$579$	0	0
$580$	9.41447e9	0.00345438
$581$	−6.32954e11	−0.230451
$582$	0	0
$583$	−5.72464e11	−0.205229
$584$	3.79334e12	1.34947
$585$	0	0
$586$	2.68868e12	0.941888
$587$	1.14701e12	0.398746	0.199373	−	0.979924i	$-0.436110\pi$
$587$	1.14701e12	0.398746	0.199373	+	0.979924i	$0.436110\pi$
$588$	0	0
$589$	6.19245e12	2.12004
$590$	2.01433e12	0.684379
$591$	0	0
$592$	−4.47227e11	−0.149651
$593$	−3.07354e12	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$593$	−3.07354e12	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$594$	0	0
$595$	−2.08842e12	−0.683111
$596$	7.02680e11	0.228113
$597$	0	0
$598$	−2.75124e12	−0.879780
$599$	−1.30326e12	−0.413630	−0.206815	−	0.978380i	$-0.566310\pi$
$599$	−1.30326e12	−0.413630	−0.206815	+	0.978380i	$0.566310\pi$
$600$	0	0
$601$	−6.05451e12	−1.89297	−0.946486	−	0.322745i	$-0.895394\pi$
$601$	−6.05451e12	−1.89297	−0.946486	+	0.322745i	$0.895394\pi$
$602$	−2.59341e12	−0.804797
$603$	0	0
$604$	−8.41315e11	−0.257213
$605$	−1.49325e12	−0.453142
$606$	0	0
$607$	−4.07835e12	−1.21937	−0.609685	−	0.792643i	$-0.708704\pi$
$607$	−4.07835e12	−1.21937	−0.609685	+	0.792643i	$0.708704\pi$
$608$	3.58020e12	1.06253
$609$	0	0
$610$	−2.03408e12	−0.594819
$611$	−8.27066e11	−0.240079
$612$	0	0
$613$	2.84192e12	0.812904	0.406452	−	0.913672i	$-0.366766\pi$
$613$	2.84192e12	0.812904	0.406452	+	0.913672i	$0.366766\pi$
$614$	−3.12223e12	−0.886557
$615$	0	0
$616$	−1.39764e12	−0.391094
$617$	−5.82246e12	−1.61742	−0.808711	−	0.588206i	$-0.799835\pi$
$617$	−5.82246e12	−1.61742	−0.808711	+	0.588206i	$0.799835\pi$
$618$	0	0
$619$	5.50829e12	1.50803	0.754014	−	0.656859i	$-0.228115\pi$
$619$	5.50829e12	1.50803	0.754014	+	0.656859i	$0.228115\pi$
$620$	−1.08693e12	−0.295420
$621$	0	0
$622$	−1.77001e12	−0.474155
$623$	5.18686e12	1.37946
$624$	0	0
$625$	9.06834e11	0.237721
$626$	3.25002e12	0.845865
$627$	0	0
$628$	3.62788e11	0.0930752
$629$	1.72358e12	0.439039
$630$	0	0
$631$	−3.74010e12	−0.939184	−0.469592	−	0.882884i	$-0.655599\pi$
$631$	−3.74010e12	−0.939184	−0.469592	+	0.882884i	$0.655599\pi$
$632$	−4.52503e12	−1.12822
$633$	0	0
$634$	4.57760e12	1.12522
$635$	−1.40791e12	−0.343633
$636$	0	0
$637$	−4.73259e11	−0.113886
$638$	2.09811e10	0.00501344
$639$	0	0
$640$	1.87200e11	0.0441058
$641$	1.41848e12	0.331866	0.165933	−	0.986137i	$-0.446936\pi$
$641$	1.41848e12	0.331866	0.165933	+	0.986137i	$0.446936\pi$
$642$	0	0
$643$	−7.10131e11	−0.163828	−0.0819142	−	0.996639i	$-0.526103\pi$
$643$	−7.10131e11	−0.163828	−0.0819142	+	0.996639i	$0.526103\pi$
$644$	−1.57637e12	−0.361136
$645$	0	0
$646$	6.92729e12	1.56501
$647$	−1.71001e12	−0.383645	−0.191823	−	0.981430i	$-0.561440\pi$
$647$	−1.71001e12	−0.383645	−0.191823	+	0.981430i	$0.561440\pi$
$648$	0	0
$649$	−2.83200e12	−0.626603
$650$	−2.92872e12	−0.643528
$651$	0	0
$652$	−9.54176e11	−0.206783
$653$	1.87672e12	0.403916	0.201958	−	0.979394i	$-0.435270\pi$
$653$	1.87672e12	0.403916	0.201958	+	0.979394i	$0.435270\pi$
$654$	0	0
$655$	5.70382e11	0.121082
$656$	−3.08175e12	−0.649725
$657$	0	0
$658$	7.51169e11	0.156214
$659$	−2.10240e12	−0.434242	−0.217121	−	0.976145i	$-0.569667\pi$
$659$	−2.10240e12	−0.434242	−0.217121	+	0.976145i	$0.569667\pi$
$660$	0	0
$661$	−6.14986e12	−1.25302	−0.626511	−	0.779413i	$-0.715518\pi$
$661$	−6.14986e12	−1.25302	−0.626511	+	0.779413i	$0.715518\pi$
$662$	−5.01874e12	−1.01563
$663$	0	0
$664$	−1.32119e12	−0.263760
$665$	−3.72297e12	−0.738231
$666$	0	0
$667$	8.48395e10	0.0165971
$668$	−2.98489e11	−0.0580009
$669$	0	0
$670$	−1.83177e11	−0.0351183
$671$	2.85978e12	0.544604
$672$	0	0
$673$	3.74185e12	0.703102	0.351551	−	0.936169i	$-0.385654\pi$
$673$	3.74185e12	0.703102	0.351551	+	0.936169i	$0.385654\pi$
$674$	−5.30823e12	−0.990787
$675$	0	0
$676$	−6.38124e11	−0.117529
$677$	−3.95982e12	−0.724480	−0.362240	−	0.932085i	$-0.617988\pi$
$677$	−3.95982e12	−0.724480	−0.362240	+	0.932085i	$0.617988\pi$
$678$	0	0
$679$	−5.71706e11	−0.103219
$680$	−4.35924e12	−0.781845
$681$	0	0
$682$	−2.42234e12	−0.428751
$683$	−3.62977e12	−0.638243	−0.319121	−	0.947714i	$-0.603388\pi$
$683$	−3.62977e12	−0.638243	−0.319121	+	0.947714i	$0.603388\pi$
$684$	0	0
$685$	3.13684e11	0.0544358
$686$	4.73941e12	0.817082
$687$	0	0
$688$	−2.95084e12	−0.502109
$689$	3.65219e12	0.617400
$690$	0	0
$691$	−3.26815e12	−0.545319	−0.272660	−	0.962111i	$-0.587903\pi$
$691$	−3.26815e12	−0.545319	−0.272660	+	0.962111i	$0.587903\pi$
$692$	−4.27883e12	−0.709328
$693$	0	0
$694$	9.69805e11	0.158696
$695$	−1.84755e12	−0.300375
$696$	0	0
$697$	1.18768e13	1.90613
$698$	−5.39023e12	−0.859525
$699$	0	0
$700$	−1.67805e12	−0.264158
$701$	5.82077e12	0.910435	0.455218	−	0.890380i	$-0.349561\pi$
$701$	5.82077e12	0.910435	0.455218	+	0.890380i	$0.349561\pi$
$702$	0	0
$703$	3.07258e12	0.474466
$704$	−2.54718e12	−0.390826
$705$	0	0
$706$	9.39140e12	1.42269
$707$	2.87600e12	0.432913
$708$	0	0
$709$	−6.83280e12	−1.01553	−0.507763	−	0.861497i	$-0.669527\pi$
$709$	−6.83280e12	−1.01553	−0.507763	+	0.861497i	$0.669527\pi$
$710$	4.52517e12	0.668301
$711$	0	0
$712$	1.08267e13	1.57884
$713$	−9.79500e12	−1.41939
$714$	0	0
$715$	−1.60349e12	−0.229450
$716$	−3.85972e12	−0.548841
$717$	0	0
$718$	8.15948e12	1.14578
$719$	5.93643e12	0.828411	0.414205	−	0.910183i	$-0.364060\pi$
$719$	5.93643e12	0.828411	0.414205	+	0.910183i	$0.364060\pi$
$720$	0	0
$721$	−2.03189e12	−0.280021
$722$	6.63158e12	0.908237
$723$	0	0
$724$	−1.47729e12	−0.199822
$725$	9.03121e10	0.0121402
$726$	0	0
$727$	5.97848e12	0.793754	0.396877	−	0.917872i	$-0.370094\pi$
$727$	5.97848e12	0.793754	0.396877	+	0.917872i	$0.370094\pi$
$728$	8.91661e12	1.17655
$729$	0	0
$730$	−3.95266e12	−0.515154
$731$	1.13723e13	1.47306
$732$	0	0
$733$	9.47010e11	0.121168	0.0605838	−	0.998163i	$-0.480704\pi$
$733$	9.47010e11	0.121168	0.0605838	+	0.998163i	$0.480704\pi$
$734$	−8.02245e12	−1.02018
$735$	0	0
$736$	−5.66303e12	−0.711375
$737$	2.57533e11	0.0321536
$738$	0	0
$739$	8.03141e12	0.990585	0.495293	−	0.868726i	$-0.335061\pi$
$739$	8.03141e12	0.990585	0.495293	+	0.868726i	$0.335061\pi$
$740$	−5.39317e11	−0.0661152
$741$	0	0
$742$	−3.31704e12	−0.401729
$743$	4.44579e12	0.535179	0.267590	−	0.963533i	$-0.413773\pi$
$743$	4.44579e12	0.535179	0.267590	+	0.963533i	$0.413773\pi$
$744$	0	0
$745$	−2.62502e12	−0.312198
$746$	−1.79188e12	−0.211828
$747$	0	0
$748$	1.70949e12	0.199668
$749$	−6.36800e12	−0.739325
$750$	0	0
$751$	−2.21770e12	−0.254403	−0.127202	−	0.991877i	$-0.540600\pi$
$751$	−2.21770e12	−0.254403	−0.127202	+	0.991877i	$0.540600\pi$
$752$	8.54699e11	0.0974614
$753$	0	0
$754$	−1.33855e11	−0.0150821
$755$	3.14292e12	0.352024
$756$	0	0
$757$	−9.93039e12	−1.09909	−0.549547	−	0.835463i	$-0.685200\pi$
$757$	−9.93039e12	−1.09909	−0.549547	+	0.835463i	$0.685200\pi$
$758$	−8.98099e12	−0.988127
$759$	0	0
$760$	−7.77111e12	−0.844932
$761$	8.52851e11	0.0921811	0.0460906	−	0.998937i	$-0.485324\pi$
$761$	8.52851e11	0.0921811	0.0460906	+	0.998937i	$0.485324\pi$
$762$	0	0
$763$	−3.28378e12	−0.350763
$764$	−1.30043e12	−0.138091
$765$	0	0
$766$	5.26133e12	0.552161
$767$	1.80676e13	1.88504
$768$	0	0
$769$	−6.77247e12	−0.698359	−0.349179	−	0.937056i	$-0.613540\pi$
$769$	−6.77247e12	−0.698359	−0.349179	+	0.937056i	$0.613540\pi$
$770$	1.45634e12	0.149298
$771$	0	0
$772$	3.72365e12	0.377304
$773$	−3.60630e12	−0.363291	−0.181645	−	0.983364i	$-0.558142\pi$
$773$	−3.60630e12	−0.363291	−0.181645	+	0.983364i	$0.558142\pi$
$774$	0	0
$775$	−1.04268e13	−1.03823
$776$	−1.19335e12	−0.118138
$777$	0	0
$778$	9.36140e11	0.0916079
$779$	2.11725e13	2.05994
$780$	0	0
$781$	−6.36206e12	−0.611883
$782$	−1.09574e13	−1.04779
$783$	0	0
$784$	4.89071e11	0.0462327
$785$	−1.35528e12	−0.127384
$786$	0	0
$787$	3.58864e12	0.333460	0.166730	−	0.986003i	$-0.446679\pi$
$787$	3.58864e12	0.333460	0.166730	+	0.986003i	$0.446679\pi$
$788$	−3.29169e11	−0.0304124
$789$	0	0
$790$	4.71508e12	0.430693
$791$	6.57299e12	0.596992
$792$	0	0
$793$	−1.82447e13	−1.63835
$794$	7.70837e12	0.688288
$795$	0	0
$796$	−2.87863e12	−0.254143
$797$	−5.25346e12	−0.461194	−0.230597	−	0.973049i	$-0.574068\pi$
$797$	−5.25346e12	−0.461194	−0.230597	+	0.973049i	$0.574068\pi$
$798$	0	0
$799$	−3.29395e12	−0.285928
$800$	−6.02833e12	−0.520346
$801$	0	0
$802$	−7.16023e11	−0.0611142
$803$	5.55716e12	0.471664
$804$	0	0
$805$	5.88887e12	0.494254
$806$	1.54540e13	1.28983
$807$	0	0
$808$	6.00318e12	0.495485
$809$	7.22017e12	0.592623	0.296312	−	0.955091i	$-0.404243\pi$
$809$	7.22017e12	0.592623	0.296312	+	0.955091i	$0.404243\pi$
$810$	0	0
$811$	−8.41322e12	−0.682918	−0.341459	−	0.939897i	$-0.610921\pi$
$811$	−8.41322e12	−0.682918	−0.341459	+	0.939897i	$0.610921\pi$
$812$	−7.66940e10	−0.00619098
$813$	0	0
$814$	−1.20192e12	−0.0959548
$815$	3.56454e12	0.283005
$816$	0	0
$817$	2.02732e13	1.59193
$818$	4.27650e12	0.333963
$819$	0	0
$820$	−3.71632e12	−0.287045
$821$	3.47602e12	0.267016	0.133508	−	0.991048i	$-0.457376\pi$
$821$	3.47602e12	0.267016	0.133508	+	0.991048i	$0.457376\pi$
$822$	0	0
$823$	1.32343e12	0.100555	0.0502774	−	0.998735i	$-0.483989\pi$
$823$	1.32343e12	0.100555	0.0502774	+	0.998735i	$0.483989\pi$
$824$	−4.24124e12	−0.320494
$825$	0	0
$826$	−1.64095e13	−1.22655
$827$	−3.65944e11	−0.0272044	−0.0136022	−	0.999907i	$-0.504330\pi$
$827$	−3.65944e11	−0.0272044	−0.0136022	+	0.999907i	$0.504330\pi$
$828$	0	0
$829$	2.59469e13	1.90805	0.954025	−	0.299728i	$-0.0968960\pi$
$829$	2.59469e13	1.90805	0.954025	+	0.299728i	$0.0968960\pi$
$830$	1.37668e12	0.100689
$831$	0	0
$832$	1.62505e13	1.17574
$833$	−1.88484e12	−0.135635
$834$	0	0
$835$	1.11507e12	0.0793806
$836$	3.04747e12	0.215780
$837$	0	0
$838$	6.65101e12	0.465896
$839$	1.42569e13	0.993337	0.496669	−	0.867940i	$-0.334556\pi$
$839$	1.42569e13	0.993337	0.496669	+	0.867940i	$0.334556\pi$
$840$	0	0
$841$	−1.45030e13	−0.999715
$842$	1.28706e13	0.882459
$843$	0	0
$844$	−2.75932e11	−0.0187181
$845$	2.38386e12	0.160851
$846$	0	0
$847$	1.21646e13	0.812127
$848$	−3.77421e12	−0.250637
$849$	0	0
$850$	−1.16642e13	−0.766423
$851$	−4.86011e12	−0.317660
$852$	0	0
$853$	−3.58103e10	−0.00231600	−0.00115800	−	0.999999i	$-0.500369\pi$
$853$	−3.58103e10	−0.00231600	−0.00115800	+	0.999999i	$0.500369\pi$
$854$	1.65705e13	1.06604
$855$	0	0
$856$	−1.32922e13	−0.846184
$857$	−1.39468e13	−0.883207	−0.441603	−	0.897210i	$-0.645590\pi$
$857$	−1.39468e13	−0.883207	−0.441603	+	0.897210i	$0.645590\pi$
$858$	0	0
$859$	−1.41204e13	−0.884864	−0.442432	−	0.896802i	$-0.645884\pi$
$859$	−1.41204e13	−0.884864	−0.442432	+	0.896802i	$0.645884\pi$
$860$	−3.55846e12	−0.221829
$861$	0	0
$862$	−1.59398e13	−0.983333
$863$	2.25669e13	1.38492	0.692458	−	0.721458i	$-0.256528\pi$
$863$	2.25669e13	1.38492	0.692458	+	0.721458i	$0.256528\pi$
$864$	0	0
$865$	1.59845e13	0.970795
$866$	−1.37677e13	−0.831823
$867$	0	0
$868$	8.85458e12	0.529455
$869$	−6.62907e12	−0.394333
$870$	0	0
$871$	−1.64301e12	−0.0967292
$872$	−6.85437e12	−0.401461
$873$	0	0
$874$	−1.95334e13	−1.13234
$875$	1.49787e13	0.863848
$876$	0	0
$877$	3.05269e13	1.74255	0.871273	−	0.490799i	$-0.163295\pi$
$877$	3.05269e13	1.74255	0.871273	+	0.490799i	$0.163295\pi$
$878$	−2.70559e13	−1.53652
$879$	0	0
$880$	1.65706e12	0.0931464
$881$	−2.00261e13	−1.11997	−0.559983	−	0.828504i	$-0.689192\pi$
$881$	−2.00261e13	−1.11997	−0.559983	+	0.828504i	$0.689192\pi$
$882$	0	0
$883$	1.87899e13	1.04016	0.520080	−	0.854117i	$-0.325902\pi$
$883$	1.87899e13	1.04016	0.520080	+	0.854117i	$0.325902\pi$
$884$	−1.09062e13	−0.600671
$885$	0	0
$886$	−4.92749e12	−0.268642
$887$	−2.88989e13	−1.56756	−0.783781	−	0.621037i	$-0.786712\pi$
$887$	−2.88989e13	−1.56756	−0.783781	+	0.621037i	$0.786712\pi$
$888$	0	0
$889$	1.14694e13	0.615863
$890$	−1.12815e13	−0.602713
$891$	0	0
$892$	5.69143e12	0.301009
$893$	−5.87204e12	−0.308999
$894$	0	0
$895$	1.44188e13	0.751150
$896$	−1.52500e12	−0.0790469
$897$	0	0
$898$	−6.01433e12	−0.308634
$899$	−4.76551e11	−0.0243327
$900$	0	0
$901$	1.45455e13	0.735306
$902$	−8.28220e12	−0.416597
$903$	0	0
$904$	1.37201e13	0.683279
$905$	5.51876e12	0.273479
$906$	0	0
$907$	2.77468e12	0.136138	0.0680691	−	0.997681i	$-0.478316\pi$
$907$	2.77468e12	0.136138	0.0680691	+	0.997681i	$0.478316\pi$
$908$	−7.60876e12	−0.371473
$909$	0	0
$910$	−9.29112e12	−0.449140
$911$	−3.58019e13	−1.72216	−0.861079	−	0.508471i	$-0.830211\pi$
$911$	−3.58019e13	−1.72216	−0.861079	+	0.508471i	$0.830211\pi$
$912$	0	0
$913$	−1.93551e12	−0.0921887
$914$	1.45359e13	0.688943
$915$	0	0
$916$	2.65206e12	0.124467
$917$	−4.64656e12	−0.217005
$918$	0	0
$919$	−1.46068e13	−0.675516	−0.337758	−	0.941233i	$-0.609669\pi$
$919$	−1.46068e13	−0.675516	−0.337758	+	0.941233i	$0.609669\pi$
$920$	1.22921e13	0.565692
$921$	0	0
$922$	−1.75669e13	−0.800580
$923$	4.05885e13	1.84075
$924$	0	0
$925$	−5.17361e12	−0.232357
$926$	−8.73386e12	−0.390352
$927$	0	0
$928$	−2.75520e11	−0.0121952
$929$	3.96447e13	1.74628	0.873142	−	0.487467i	$-0.162079\pi$
$929$	3.96447e13	1.74628	0.873142	+	0.487467i	$0.162079\pi$
$930$	0	0
$931$	−3.36006e12	−0.146580
$932$	8.91510e12	0.387039
$933$	0	0
$934$	1.69971e13	0.730826
$935$	−6.38619e12	−0.273268
$936$	0	0
$937$	−1.72240e13	−0.729973	−0.364987	−	0.931013i	$-0.618926\pi$
$937$	−1.72240e13	−0.729973	−0.364987	+	0.931013i	$0.618926\pi$
$938$	1.49223e12	0.0629396
$939$	0	0
$940$	1.03069e12	0.0430580
$941$	−4.12899e13	−1.71668	−0.858341	−	0.513079i	$-0.828505\pi$
$941$	−4.12899e13	−1.71668	−0.858341	+	0.513079i	$0.828505\pi$
$942$	0	0
$943$	−3.34900e13	−1.37915
$944$	−1.86712e13	−0.765241
$945$	0	0
$946$	−7.93040e12	−0.321947
$947$	−1.76185e13	−0.711859	−0.355929	−	0.934513i	$-0.615836\pi$
$947$	−1.76185e13	−0.711859	−0.355929	+	0.934513i	$0.615836\pi$
$948$	0	0
$949$	−3.54535e13	−1.41893
$950$	−2.07934e13	−0.828266
$951$	0	0
$952$	3.55121e13	1.40123
$953$	−4.44553e13	−1.74584	−0.872922	−	0.487859i	$-0.837778\pi$
$953$	−4.44553e13	−1.74584	−0.872922	+	0.487859i	$0.837778\pi$
$954$	0	0
$955$	4.85804e12	0.188993
$956$	−1.29913e13	−0.503026
$957$	0	0
$958$	1.87808e12	0.0720394
$959$	−2.55539e12	−0.0975605
$960$	0	0
$961$	2.85798e13	1.08094
$962$	7.66800e12	0.288665
$963$	0	0
$964$	−5.31050e12	−0.198056
$965$	−1.39105e13	−0.516382
$966$	0	0
$967$	−1.48376e13	−0.545688	−0.272844	−	0.962058i	$-0.587964\pi$
$967$	−1.48376e13	−0.545688	−0.272844	+	0.962058i	$0.587964\pi$
$968$	2.53917e13	0.929509
$969$	0	0
$970$	1.24347e12	0.0450984
$971$	3.34482e13	1.20750	0.603749	−	0.797174i	$-0.293673\pi$
$971$	3.34482e13	1.20750	0.603749	+	0.797174i	$0.293673\pi$
$972$	0	0
$973$	1.50508e13	0.538336
$974$	−8.49535e12	−0.302459
$975$	0	0
$976$	1.88543e13	0.665098
$977$	−2.04042e13	−0.716463	−0.358231	−	0.933633i	$-0.616620\pi$
$977$	−2.04042e13	−0.716463	−0.358231	+	0.933633i	$0.616620\pi$
$978$	0	0
$979$	1.58609e13	0.551832
$980$	5.89777e11	0.0204254
$981$	0	0
$982$	−2.15891e13	−0.740855
$983$	−2.30999e13	−0.789077	−0.394539	−	0.918879i	$-0.629096\pi$
$983$	−2.30999e13	−0.789077	−0.394539	+	0.918879i	$0.629096\pi$
$984$	0	0
$985$	1.22968e12	0.0416228
$986$	−5.33102e11	−0.0179624
$987$	0	0
$988$	−1.94422e13	−0.649140
$989$	−3.20674e13	−1.06581
$990$	0	0
$991$	−4.59514e13	−1.51345	−0.756724	−	0.653734i	$-0.773201\pi$
$991$	−4.59514e13	−1.51345	−0.756724	+	0.653734i	$0.773201\pi$
$992$	3.18097e13	1.04294
$993$	0	0
$994$	−3.68638e13	−1.19774
$995$	1.07538e13	0.347822
$996$	0	0
$997$	−4.39528e13	−1.40883	−0.704414	−	0.709790i	$-0.748790\pi$
$997$	−4.39528e13	−1.40883	−0.704414	+	0.709790i	$0.748790\pi$
$998$	3.32631e13	1.06139
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.10.a.d.1.6		8
3.2	odd	2		81.10.a.c.1.3		8
9.2	odd	6		9.10.c.a.4.6	✓	16
9.4	even	3		27.10.c.a.19.3		16
9.5	odd	6		9.10.c.a.7.6	yes	16
9.7	even	3		27.10.c.a.10.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.10.c.a.4.6	✓	16	9.2	odd	6
9.10.c.a.7.6	yes	16	9.5	odd	6
27.10.c.a.10.3		16	9.7	even	3
27.10.c.a.19.3		16	9.4	even	3
81.10.a.c.1.3		8	3.2	odd	2
81.10.a.d.1.6		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)

\(f(q)\)	\(=\)	\(q+17.7185 q^{2} -198.054 q^{4} +739.877 q^{5} -6027.34 q^{7} -12581.1 q^{8} +13109.5 q^{10} -18431.0 q^{11} +117586. q^{13} -106795. q^{14} -121515. q^{16} +468308. q^{17} +834842. q^{19} -146536. q^{20} -326571. q^{22} -1.32052e6 q^{23} -1.40571e6 q^{25} +2.08345e6 q^{26} +1.19374e6 q^{28} -64246.8 q^{29} +7.41751e6 q^{31} +4.28847e6 q^{32} +8.29773e6 q^{34} -4.45949e6 q^{35} +3.68044e6 q^{37} +1.47922e7 q^{38} -9.30848e6 q^{40} +2.53611e7 q^{41} +2.42839e7 q^{43} +3.65035e6 q^{44} -2.33977e7 q^{46} -7.03371e6 q^{47} -4.02479e6 q^{49} -2.49070e7 q^{50} -2.32884e7 q^{52} +3.10597e7 q^{53} -1.36367e7 q^{55} +7.58306e7 q^{56} -1.13836e6 q^{58} +1.53654e8 q^{59} -1.55161e8 q^{61} +1.31427e8 q^{62} +1.38201e8 q^{64} +8.69992e7 q^{65} -1.39728e7 q^{67} -9.27506e7 q^{68} -7.90155e7 q^{70} +3.45182e8 q^{71} -3.01511e8 q^{73} +6.52119e7 q^{74} -1.65344e8 q^{76} +1.11090e8 q^{77} +3.59669e8 q^{79} -8.99059e7 q^{80} +4.49361e8 q^{82} +1.05014e8 q^{83} +3.46491e8 q^{85} +4.30274e8 q^{86} +2.31883e8 q^{88} -8.60555e8 q^{89} -7.08730e8 q^{91} +2.61536e8 q^{92} -1.24627e8 q^{94} +6.17681e8 q^{95} +9.48522e7 q^{97} -7.13133e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 15 q^{2} + 1793 q^{4} + 453 q^{5} + 343 q^{7} + 7239 q^{8} + 510 q^{10} + 99150 q^{11} - 32435 q^{13} + 394824 q^{14} + 328193 q^{16} + 415539 q^{17} - 85277 q^{19} + 1855164 q^{20} - 529359 q^{22}+ \cdots - 2413650159 q^{98}+O(q^{100}) \) 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

Properties

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