LMFDB - Embedded newform 81.9.b.b.80.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [81,9,Mod(80,81)]

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([1]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("81.80");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 81 = 3^{4} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	81.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$32.9976674150$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 1024 x^{14} + 419701 x^{12} + 88203292 x^{10} + 10121979748 x^{8} + 629108384896 x^{6} + \cdots + 16\!\cdots\!24 $ x^16 + 1024x^14 + 419701x^12 + 88203292x^10 + 10121979748x^8 + 629108384896x^6 + 20221245756928x^4 + 305638522445824*x^2 + 1623834669236224
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{84} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			80.8
Root			$3.33872i$ of defining polynomial
Character	$\chi$	$=$	81.80
Dual form			81.9.b.b.80.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-5.78284i q^{2} +222.559 q^{4} +626.068i q^{5} -2231.44 q^{7} -2767.43i q^{8} +3620.45 q^{10} -18072.2i q^{11} +37655.6 q^{13} +12904.0i q^{14} +40971.5 q^{16} +31224.6i q^{17} +2335.59 q^{19} +139337. i q^{20} -104509. q^{22} +163911. i q^{23} -1336.18 q^{25} -217756. i q^{26} -496626. q^{28} +937335. i q^{29} +1.25199e6 q^{31} -945393. i q^{32} +180567. q^{34} -1.39703e6i q^{35} +2.83025e6 q^{37} -13506.3i q^{38} +1.73260e6 q^{40} +3.60966e6i q^{41} +3.71621e6 q^{43} -4.02214e6i q^{44} +947871. q^{46} +7.41731e6i q^{47} -785489. q^{49} +7726.90i q^{50} +8.38059e6 q^{52} -1.27382e7i q^{53} +1.13145e7 q^{55} +6.17534e6i q^{56} +5.42046e6 q^{58} -1.41849e7i q^{59} -4.92805e6 q^{61} -7.24006e6i q^{62} +5.02164e6 q^{64} +2.35750e7i q^{65} +1.41779e7 q^{67} +6.94930e6i q^{68} -8.07881e6 q^{70} -2.58699e7i q^{71} -1.02185e7 q^{73} -1.63669e7i q^{74} +519806. q^{76} +4.03271e7i q^{77} +6.41821e7 q^{79} +2.56509e7i q^{80} +2.08741e7 q^{82} -2.35963e7i q^{83} -1.95487e7 q^{85} -2.14902e7i q^{86} -5.00136e7 q^{88} +1.04381e8i q^{89} -8.40262e7 q^{91} +3.64798e7i q^{92} +4.28931e7 q^{94} +1.46224e6i q^{95} -1.00259e8 q^{97} +4.54236e6i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2048 q^{4} + 3692 q^{7} + 10752 q^{10} + 63860 q^{13} + 95116 q^{16} + 185108 q^{19} - 691980 q^{22} - 541712 q^{25} - 997132 q^{28} + 571136 q^{31} + 1027656 q^{34} + 4354268 q^{37} - 2973768 q^{40}+ \cdots - 133878688 q^{97}+O(q^{100}) $ 16 * q - 2048 * q^4 + 3692 * q^7 + 10752 * q^10 + 63860 * q^13 + 95116 * q^16 + 185108 * q^19 - 691980 * q^22 - 541712 * q^25 - 997132 * q^28 + 571136 * q^31 + 1027656 * q^34 + 4354268 * q^37 - 2973768 * q^40 + 5453084 * q^43 - 4234044 * q^46 + 14602560 * q^49 - 25384960 * q^52 - 14452572 * q^55 + 1213068 * q^58 + 31037996 * q^61 + 24253000 * q^64 - 62356828 * q^67 + 11075124 * q^70 - 21273664 * q^73 + 69593876 * q^76 - 151045036 * q^79 - 40201140 * q^82 + 160995564 * q^85 + 366976068 * q^88 + 373680796 * q^91 - 544741584 * q^94 - 133878688 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/81\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 5.78284i	− 0.361427i	−0.983536	−	0.180714i	$-0.942159\pi$
$2$	− 5.78284i	− 0.361427i	0.983536	−	0.180714i	$-0.0578407\pi$
$3$	0	0
$3$	0	0
$4$	222.559	0.869370
$5$	626.068i	1.00171i	0.865532	+	0.500854i	$0.166981\pi$
$5$	626.068i	1.00171i	−0.865532	+	0.500854i	$0.833019\pi$
$6$	0	0
$7$	−2231.44	−0.929378	−0.464689	−	0.885474i	$-0.653834\pi$
$7$	−2231.44	−0.929378	−0.464689	+	0.885474i	$0.653834\pi$
$8$	− 2767.43i	− 0.675642i
$9$	0	0
$10$	3620.45	0.362045
$11$	− 18072.2i	− 1.23436i	−0.786823	−	0.617179i	$-0.788275\pi$
$11$	− 18072.2i	− 1.23436i	0.786823	−	0.617179i	$-0.211725\pi$
$12$	0	0
$13$	37655.6	1.31843	0.659214	−	0.751955i	$-0.270889\pi$
$13$	37655.6	1.31843	0.659214	+	0.751955i	$0.270889\pi$
$14$	12904.0i	0.335903i
$15$	0	0
$16$	40971.5	0.625175
$17$	31224.6i	0.373853i	0.982374	+	0.186926i	$0.0598526\pi$
$17$	31224.6i	0.373853i	−0.982374	+	0.186926i	$0.940147\pi$
$18$	0	0
$19$	2335.59	0.0179218	0.00896091	−	0.999960i	$-0.497148\pi$
$19$	2335.59	0.0179218	0.00896091	+	0.999960i	$0.497148\pi$
$20$	139337.i	0.870856i
$21$	0	0
$22$	−104509.	−0.446131
$23$	163911.i	0.585729i	0.956154	+	0.292865i	$0.0946085\pi$
$23$	163911.i	0.585729i	−0.956154	+	0.292865i	$0.905391\pi$
$24$	0	0
$25$	−1336.18	−0.00342062
$26$	− 217756.i	− 0.476516i
$27$	0	0
$28$	−496626.	−0.807974
$29$	937335.i	1.32527i	0.748944	+	0.662633i	$0.230561\pi$
$29$	937335.i	1.32527i	−0.748944	+	0.662633i	$0.769439\pi$
$30$	0	0
$31$	1.25199e6	1.35567	0.677836	−	0.735213i	$-0.262918\pi$
$31$	1.25199e6	1.35567	0.677836	+	0.735213i	$0.262918\pi$
$32$	− 945393.i	− 0.901597i
$33$	0	0
$34$	180567.	0.135121
$35$	− 1.39703e6i	− 0.930966i
$36$	0	0
$37$	2.83025e6	1.51014	0.755070	−	0.655644i	$-0.227603\pi$
$37$	2.83025e6	1.51014	0.755070	+	0.655644i	$0.227603\pi$
$38$	− 13506.3i	− 0.00647743i
$39$	0	0
$40$	1.73260e6	0.676796
$41$	3.60966e6i	1.27741i	0.769451	+	0.638706i	$0.220530\pi$
$41$	3.60966e6i	1.27741i	−0.769451	+	0.638706i	$0.779470\pi$
$42$	0	0
$43$	3.71621e6	1.08699	0.543496	−	0.839412i	$-0.317100\pi$
$43$	3.71621e6	1.08699	0.543496	+	0.839412i	$0.317100\pi$
$44$	− 4.02214e6i	− 1.07311i
$45$	0	0
$46$	947871.	0.211699
$47$	7.41731e6i	1.52004i	0.649899	+	0.760020i	$0.274811\pi$
$47$	7.41731e6i	1.52004i	−0.649899	+	0.760020i	$0.725189\pi$
$48$	0	0
$49$	−785489.	−0.136256
$50$	7726.90i	0.00123630i
$51$	0	0
$52$	8.38059e6	1.14620
$53$	− 1.27382e7i	− 1.61438i	−0.590291	−	0.807190i	$-0.700987\pi$
$53$	− 1.27382e7i	− 1.61438i	0.590291	−	0.807190i	$-0.299013\pi$
$54$	0	0
$55$	1.13145e7	1.23647
$56$	6.17534e6i	0.627927i
$57$	0	0
$58$	5.42046e6	0.478987
$59$	− 1.41849e7i	− 1.17062i	−0.810808	−	0.585312i	$-0.800972\pi$
$59$	− 1.41849e7i	− 1.17062i	0.810808	−	0.585312i	$-0.199028\pi$
$60$	0	0
$61$	−4.92805e6	−0.355923	−0.177961	−	0.984037i	$-0.556950\pi$
$61$	−4.92805e6	−0.355923	−0.177961	+	0.984037i	$0.556950\pi$
$62$	− 7.24006e6i	− 0.489977i
$63$	0	0
$64$	5.02164e6	0.299313
$65$	2.35750e7i	1.32068i
$66$	0	0
$67$	1.41779e7	0.703580	0.351790	−	0.936079i	$-0.385573\pi$
$67$	1.41779e7	0.703580	0.351790	+	0.936079i	$0.385573\pi$
$68$	6.94930e6i	0.325017i
$69$	0	0
$70$	−8.07881e6	−0.336477
$71$	− 2.58699e7i	− 1.01803i	−0.860757	−	0.509017i	$-0.830009\pi$
$71$	− 2.58699e7i	− 1.01803i	0.860757	−	0.509017i	$-0.169991\pi$
$72$	0	0
$73$	−1.02185e7	−0.359827	−0.179914	−	0.983682i	$-0.557582\pi$
$73$	−1.02185e7	−0.359827	−0.179914	+	0.983682i	$0.557582\pi$
$74$	− 1.63669e7i	− 0.545806i
$75$	0	0
$76$	519806.	0.0155807
$77$	4.03271e7i	1.14719i
$78$	0	0
$79$	6.41821e7	1.64780	0.823901	−	0.566733i	$-0.191793\pi$
$79$	6.41821e7	1.64780	0.823901	+	0.566733i	$0.191793\pi$
$80$	2.56509e7i	0.626243i
$81$	0	0
$82$	2.08741e7	0.461692
$83$	− 2.35963e7i	− 0.497200i	−0.968606	−	0.248600i	$-0.920030\pi$
$83$	− 2.35963e7i	− 0.497200i	0.968606	−	0.248600i	$-0.0799704\pi$
$84$	0	0
$85$	−1.95487e7	−0.374492
$86$	− 2.14902e7i	− 0.392868i
$87$	0	0
$88$	−5.00136e7	−0.833984
$89$	1.04381e8i	1.66364i	0.555045	+	0.831820i	$0.312701\pi$
$89$	1.04381e8i	1.66364i	−0.555045	+	0.831820i	$0.687299\pi$
$90$	0	0
$91$	−8.40262e7	−1.22532
$92$	3.64798e7i	0.509216i
$93$	0	0
$94$	4.28931e7	0.549384
$95$	1.46224e6i	0.0179524i
$96$	0	0
$97$	−1.00259e8	−1.13249	−0.566247	−	0.824236i	$-0.691605\pi$
$97$	−1.00259e8	−1.13249	−0.566247	+	0.824236i	$0.691605\pi$
$98$	4.54236e6i	0.0492467i
$99$	0	0
$100$	−297378.	−0.00297378
$101$	− 1.69407e8i	− 1.62796i	−0.580890	−	0.813982i	$-0.697295\pi$
$101$	− 1.69407e8i	− 1.62796i	0.580890	−	0.813982i	$-0.302705\pi$
$102$	0	0
$103$	−1.44761e8	−1.28618	−0.643091	−	0.765790i	$-0.722348\pi$
$103$	−1.44761e8	−1.28618	−0.643091	+	0.765790i	$0.722348\pi$
$104$	− 1.04209e8i	− 0.890785i
$105$	0	0
$106$	−7.36632e7	−0.583481
$107$	4.13111e6i	0.0315161i	0.999876	+	0.0157580i	$0.00501615\pi$
$107$	4.13111e6i	0.0315161i	−0.999876	+	0.0157580i	$0.994984\pi$
$108$	0	0
$109$	−1.19559e8	−0.846987	−0.423493	−	0.905899i	$-0.639196\pi$
$109$	−1.19559e8	−0.846987	−0.423493	+	0.905899i	$0.639196\pi$
$110$	− 6.54296e7i	− 0.446893i
$111$	0	0
$112$	−9.14252e7	−0.581024
$113$	5.39019e6i	0.0330590i	0.999863	+	0.0165295i	$0.00526175\pi$
$113$	5.39019e6i	0.0330590i	−0.999863	+	0.0165295i	$0.994738\pi$
$114$	0	0
$115$	−1.02619e8	−0.586730
$116$	2.08612e8i	1.15215i
$117$	0	0
$118$	−8.20288e7	−0.423096
$119$	− 6.96757e7i	− 0.347451i
$120$	0	0
$121$	−1.12247e8	−0.523641
$122$	2.84981e7i	0.128640i
$123$	0	0
$124$	2.78642e8	1.17858
$125$	2.43721e8i	0.998282i
$126$	0	0
$127$	3.33087e8	1.28039	0.640195	−	0.768212i	$-0.278853\pi$
$127$	3.33087e8	1.28039	0.640195	+	0.768212i	$0.278853\pi$
$128$	− 2.71060e8i	− 1.00978i
$129$	0	0
$130$	1.36330e8	0.477330
$131$	2.49017e8i	0.845558i	0.906233	+	0.422779i	$0.138945\pi$
$131$	2.49017e8i	0.845558i	−0.906233	+	0.422779i	$0.861055\pi$
$132$	0	0
$133$	−5.21172e6	−0.0166561
$134$	− 8.19886e7i	− 0.254293i
$135$	0	0
$136$	8.64118e7	0.252591
$137$	1.95512e8i	0.554997i	0.960726	+	0.277498i	$0.0895053\pi$
$137$	1.95512e8i	0.554997i	−0.960726	+	0.277498i	$0.910495\pi$
$138$	0	0
$139$	−2.90213e8	−0.777424	−0.388712	−	0.921359i	$-0.627080\pi$
$139$	−2.90213e8	−0.777424	−0.388712	+	0.921359i	$0.627080\pi$
$140$	− 3.10922e8i	− 0.809355i
$141$	0	0
$142$	−1.49602e8	−0.367945
$143$	− 6.80522e8i	− 1.62741i
$144$	0	0
$145$	−5.86836e8	−1.32753
$146$	5.90917e7i	0.130051i
$147$	0	0
$148$	6.29896e8	1.31287
$149$	1.37888e8i	0.279757i	0.990169	+	0.139879i	$0.0446713\pi$
$149$	1.37888e8i	0.279757i	−0.990169	+	0.139879i	$0.955329\pi$
$150$	0	0
$151$	−3.15659e8	−0.607169	−0.303585	−	0.952804i	$-0.598184\pi$
$151$	−3.15659e8	−0.607169	−0.303585	+	0.952804i	$0.598184\pi$
$152$	− 6.46357e6i	− 0.0121087i
$153$	0	0
$154$	2.33205e8	0.414624
$155$	7.83832e8i	1.35799i
$156$	0	0
$157$	5.46029e8	0.898704	0.449352	−	0.893355i	$-0.351655\pi$
$157$	5.46029e8	0.898704	0.449352	+	0.893355i	$0.351655\pi$
$158$	− 3.71154e8i	− 0.595561i
$159$	0	0
$160$	5.91880e8	0.903138
$161$	− 3.65757e8i	− 0.544364i
$162$	0	0
$163$	4.38696e8	0.621460	0.310730	−	0.950498i	$-0.399427\pi$
$163$	4.38696e8	0.621460	0.310730	+	0.950498i	$0.399427\pi$
$164$	8.03362e8i	1.11054i
$165$	0	0
$166$	−1.36453e8	−0.179702
$167$	6.86222e8i	0.882264i	0.897442	+	0.441132i	$0.145423\pi$
$167$	6.86222e8i	0.882264i	−0.897442	+	0.441132i	$0.854577\pi$
$168$	0	0
$169$	6.02215e8	0.738253
$170$	1.13047e8i	0.135352i
$171$	0	0
$172$	8.27074e8	0.944998
$173$	4.57941e8i	0.511240i	0.966777	+	0.255620i	$0.0822796\pi$
$173$	4.57941e8i	0.511240i	−0.966777	+	0.255620i	$0.917720\pi$
$174$	0	0
$175$	2.98160e6	0.00317905
$176$	− 7.40446e8i	− 0.771690i
$177$	0	0
$178$	6.03616e8	0.601285
$179$	− 1.82904e9i	− 1.78161i	−0.454390	−	0.890803i	$-0.650143\pi$
$179$	− 1.82904e9i	− 1.78161i	0.454390	−	0.890803i	$-0.349857\pi$
$180$	0	0
$181$	−4.41011e8	−0.410899	−0.205450	−	0.978668i	$-0.565866\pi$
$181$	−4.41011e8	−0.410899	−0.205450	+	0.978668i	$0.565866\pi$
$182$	4.85910e8i	0.442864i
$183$	0	0
$184$	4.53612e8	0.395743
$185$	1.77193e9i	1.51272i
$186$	0	0
$187$	5.64298e8	0.461469
$188$	1.65079e9i	1.32148i
$189$	0	0
$190$	8.45588e6	0.00648850
$191$	− 1.61804e9i	− 1.21578i	−0.794022	−	0.607889i	$-0.792016\pi$
$191$	− 1.61804e9i	− 1.21578i	0.794022	−	0.607889i	$-0.207984\pi$
$192$	0	0
$193$	1.13241e9	0.816158	0.408079	−	0.912947i	$-0.366199\pi$
$193$	1.13241e9	0.816158	0.408079	+	0.912947i	$0.366199\pi$
$194$	5.79781e8i	0.409314i
$195$	0	0
$196$	−1.74817e8	−0.118457
$197$	− 2.90283e9i	− 1.92733i	−0.267113	−	0.963665i	$-0.586070\pi$
$197$	− 2.90283e9i	− 1.92733i	0.267113	−	0.963665i	$-0.413930\pi$
$198$	0	0
$199$	7.60460e8	0.484914	0.242457	−	0.970162i	$-0.422047\pi$
$199$	7.60460e8	0.484914	0.242457	+	0.970162i	$0.422047\pi$
$200$	3.69778e6i	0.00231111i
$201$	0	0
$202$	−9.79651e8	−0.588391
$203$	− 2.09160e9i	− 1.23167i
$204$	0	0
$205$	−2.25989e9	−1.27959
$206$	8.37129e8i	0.464861i
$207$	0	0
$208$	1.54281e9	0.824248
$209$	− 4.22093e7i	− 0.0221219i
$210$	0	0
$211$	−2.41950e9	−1.22066	−0.610332	−	0.792146i	$-0.708964\pi$
$211$	−2.41950e9	−1.22066	−0.610332	+	0.792146i	$0.708964\pi$
$212$	− 2.83501e9i	− 1.40349i
$213$	0	0
$214$	2.38896e7	0.0113908
$215$	2.32660e9i	1.08885i
$216$	0	0
$217$	−2.79374e9	−1.25993
$218$	6.91391e8i	0.306124i
$219$	0	0
$220$	2.51813e9	1.07495
$221$	1.17578e9i	0.492898i
$222$	0	0
$223$	−1.22791e9	−0.496533	−0.248266	−	0.968692i	$-0.579861\pi$
$223$	−1.22791e9	−0.496533	−0.248266	+	0.968692i	$0.579861\pi$
$224$	2.10958e9i	0.837925i
$225$	0	0
$226$	3.11706e7	0.0119484
$227$	− 1.92651e9i	− 0.725552i	−0.931876	−	0.362776i	$-0.881829\pi$
$227$	− 1.92651e9i	− 0.725552i	0.931876	−	0.362776i	$-0.118171\pi$
$228$	0	0
$229$	−5.33004e9	−1.93816	−0.969078	−	0.246754i	$-0.920636\pi$
$229$	−5.33004e9	−1.93816	−0.969078	+	0.246754i	$0.920636\pi$
$230$	5.93432e8i	0.212060i
$231$	0	0
$232$	2.59401e9	0.895405
$233$	2.11463e9i	0.717481i	0.933437	+	0.358741i	$0.116794\pi$
$233$	2.11463e9i	0.717481i	−0.933437	+	0.358741i	$0.883206\pi$
$234$	0	0
$235$	−4.64374e9	−1.52264
$236$	− 3.15697e9i	− 1.01771i
$237$	0	0
$238$	−4.02923e8	−0.125578
$239$	3.99616e9i	1.22476i	0.790563	+	0.612380i	$0.209788\pi$
$239$	3.99616e9i	1.22476i	−0.790563	+	0.612380i	$0.790212\pi$
$240$	0	0
$241$	−3.53535e9	−1.04801	−0.524003	−	0.851716i	$-0.675562\pi$
$241$	−3.53535e9	−1.04801	−0.524003	+	0.851716i	$0.675562\pi$
$242$	6.49107e8i	0.189258i
$243$	0	0
$244$	−1.09678e9	−0.309429
$245$	− 4.91770e8i	− 0.136489i
$246$	0	0
$247$	8.79480e7	0.0236286
$248$	− 3.46479e9i	− 0.915948i
$249$	0	0
$250$	1.40940e9	0.360807
$251$	1.12042e9i	0.282285i	0.989989	+	0.141142i	$0.0450775\pi$
$251$	1.12042e9i	0.282285i	−0.989989	+	0.141142i	$0.954922\pi$
$252$	0	0
$253$	2.96224e9	0.723000
$254$	− 1.92619e9i	− 0.462768i
$255$	0	0
$256$	−2.81955e8	−0.0656478
$257$	2.16052e9i	0.495251i	0.968856	+	0.247625i	$0.0796503\pi$
$257$	2.16052e9i	0.495251i	−0.968856	+	0.247625i	$0.920350\pi$
$258$	0	0
$259$	−6.31552e9	−1.40349
$260$	5.24682e9i	1.14816i
$261$	0	0
$262$	1.44002e9	0.305608
$263$	− 5.23202e9i	− 1.09357i	−0.837274	−	0.546784i	$-0.815852\pi$
$263$	− 5.23202e9i	− 1.09357i	0.837274	−	0.546784i	$-0.184148\pi$
$264$	0	0
$265$	7.97500e9	1.61714
$266$	3.01385e7i	0.00601999i
$267$	0	0
$268$	3.15542e9	0.611672
$269$	− 1.12846e9i	− 0.215514i	−0.994177	−	0.107757i	$-0.965633\pi$
$269$	− 1.12846e9i	− 0.215514i	0.994177	−	0.107757i	$-0.0343669\pi$
$270$	0	0
$271$	−4.03545e9	−0.748194	−0.374097	−	0.927390i	$-0.622047\pi$
$271$	−4.03545e9	−0.748194	−0.374097	+	0.927390i	$0.622047\pi$
$272$	1.27932e9i	0.233724i
$273$	0	0
$274$	1.13061e9	0.200591
$275$	2.41477e7i	0.00422227i
$276$	0	0
$277$	8.22794e8	0.139756	0.0698782	−	0.997556i	$-0.477739\pi$
$277$	8.22794e8	0.139756	0.0698782	+	0.997556i	$0.477739\pi$
$278$	1.67826e9i	0.280982i
$279$	0	0
$280$	−3.86618e9	−0.629000
$281$	6.04095e9i	0.968902i	0.874818	+	0.484451i	$0.160981\pi$
$281$	6.04095e9i	0.968902i	−0.874818	+	0.484451i	$0.839019\pi$
$282$	0	0
$283$	−6.08283e9	−0.948330	−0.474165	−	0.880436i	$-0.657250\pi$
$283$	−6.08283e9	−0.948330	−0.474165	+	0.880436i	$0.657250\pi$
$284$	− 5.75758e9i	− 0.885048i
$285$	0	0
$286$	−3.93535e9	−0.588192
$287$	− 8.05473e9i	− 1.18720i
$288$	0	0
$289$	6.00078e9	0.860234
$290$	3.39358e9i	0.479806i
$291$	0	0
$292$	−2.27421e9	−0.312823
$293$	− 7.51565e8i	− 0.101976i	−0.998699	−	0.0509878i	$-0.983763\pi$
$293$	− 7.51565e8i	− 0.101976i	0.998699	−	0.0509878i	$-0.0162370\pi$
$294$	0	0
$295$	8.88070e9	1.17262
$296$	− 7.83250e9i	− 1.02031i
$297$	0	0
$298$	7.97385e8	0.101112
$299$	6.17217e9i	0.772242i
$300$	0	0
$301$	−8.29248e9	−1.01023
$302$	1.82540e9i	0.219448i
$303$	0	0
$304$	9.56925e7	0.0112043
$305$	− 3.08529e9i	− 0.356531i
$306$	0	0
$307$	2.09132e9	0.235432	0.117716	−	0.993047i	$-0.462443\pi$
$307$	2.09132e9	0.235432	0.117716	+	0.993047i	$0.462443\pi$
$308$	8.97514e9i	0.997329i
$309$	0	0
$310$	4.53277e9	0.490814
$311$	9.71696e9i	1.03870i	0.854562	+	0.519349i	$0.173825\pi$
$311$	9.71696e9i	1.03870i	−0.854562	+	0.519349i	$0.826175\pi$
$312$	0	0
$313$	9.61959e9	1.00226	0.501129	−	0.865373i	$-0.332918\pi$
$313$	9.61959e9	1.00226	0.501129	+	0.865373i	$0.332918\pi$
$314$	− 3.15760e9i	− 0.324816i
$315$	0	0
$316$	1.42843e10	1.43255
$317$	− 3.06390e9i	− 0.303415i	−0.988425	−	0.151707i	$-0.951523\pi$
$317$	− 3.06390e9i	− 0.303415i	0.988425	−	0.151707i	$-0.0484772\pi$
$318$	0	0
$319$	1.69398e10	1.63585
$320$	3.14389e9i	0.299825i
$321$	0	0
$322$	−2.11511e9	−0.196748
$323$	7.29278e7i	0.00670012i
$324$	0	0
$325$	−5.03146e7	−0.00450984
$326$	− 2.53691e9i	− 0.224613i
$327$	0	0
$328$	9.98947e9	0.863072
$329$	− 1.65513e10i	− 1.41269i
$330$	0	0
$331$	3.64592e9	0.303735	0.151867	−	0.988401i	$-0.451471\pi$
$331$	3.64592e9	0.303735	0.151867	+	0.988401i	$0.451471\pi$
$332$	− 5.25156e9i	− 0.432251i
$333$	0	0
$334$	3.96831e9	0.318875
$335$	8.87635e9i	0.704782i
$336$	0	0
$337$	−7.40093e9	−0.573808	−0.286904	−	0.957959i	$-0.592626\pi$
$337$	−7.40093e9	−0.573808	−0.286904	+	0.957959i	$0.592626\pi$
$338$	− 3.48251e9i	− 0.266825i
$339$	0	0
$340$	−4.35074e9	−0.325572
$341$	− 2.26263e10i	− 1.67338i
$342$	0	0
$343$	1.46166e10	1.05601
$344$	− 1.02843e10i	− 0.734416i
$345$	0	0
$346$	2.64820e9	0.184776
$347$	6.99169e9i	0.482241i	0.970495	+	0.241120i	$0.0775149\pi$
$347$	6.99169e9i	0.482241i	−0.970495	+	0.241120i	$0.922485\pi$
$348$	0	0
$349$	1.60970e10	1.08503	0.542516	−	0.840045i	$-0.317472\pi$
$349$	1.60970e10	1.08503	0.542516	+	0.840045i	$0.317472\pi$
$350$	− 1.72421e7i	− 0.00114899i
$351$	0	0
$352$	−1.70854e10	−1.11289
$353$	2.96109e9i	0.190701i	0.995444	+	0.0953504i	$0.0303972\pi$
$353$	2.96109e9i	0.190701i	−0.995444	+	0.0953504i	$0.969603\pi$
$354$	0	0
$355$	1.61963e10	1.01977
$356$	2.32308e10i	1.44632i
$357$	0	0
$358$	−1.05771e10	−0.643921
$359$	9.16347e9i	0.551674i	0.961204	+	0.275837i	$0.0889549\pi$
$359$	9.16347e9i	0.551674i	−0.961204	+	0.275837i	$0.911045\pi$
$360$	0	0
$361$	−1.69781e10	−0.999679
$362$	2.55030e9i	0.148510i
$363$	0	0
$364$	−1.87008e10	−1.06526
$365$	− 6.39746e9i	− 0.360442i
$366$	0	0
$367$	−1.62858e10	−0.897727	−0.448864	−	0.893600i	$-0.648171\pi$
$367$	−1.62858e10	−0.897727	−0.448864	+	0.893600i	$0.648171\pi$
$368$	6.71568e9i	0.366183i
$369$	0	0
$370$	1.02468e10	0.546739
$371$	2.84246e10i	1.50037i
$372$	0	0
$373$	−9.04189e9	−0.467115	−0.233558	−	0.972343i	$-0.575037\pi$
$373$	−9.04189e9	−0.467115	−0.233558	+	0.972343i	$0.575037\pi$
$374$	− 3.26324e9i	− 0.166787i
$375$	0	0
$376$	2.05269e10	1.02700
$377$	3.52959e10i	1.74727i
$378$	0	0
$379$	−1.57162e10	−0.761712	−0.380856	−	0.924634i	$-0.624371\pi$
$379$	−1.57162e10	−0.761712	−0.380856	+	0.924634i	$0.624371\pi$
$380$	3.25434e8i	0.0156073i
$381$	0	0
$382$	−9.35683e9	−0.439416
$383$	− 1.98122e10i	− 0.920743i	−0.887726	−	0.460371i	$-0.847716\pi$
$383$	− 1.98122e10i	− 0.920743i	0.887726	−	0.460371i	$-0.152284\pi$
$384$	0	0
$385$	−2.52475e10	−1.14915
$386$	− 6.54854e9i	− 0.294982i
$387$	0	0
$388$	−2.23135e10	−0.984557
$389$	− 3.05719e10i	− 1.33513i	−0.744550	−	0.667566i	$-0.767336\pi$
$389$	− 3.05719e10i	− 1.33513i	0.744550	−	0.667566i	$-0.232664\pi$
$390$	0	0
$391$	−5.11805e9	−0.218977
$392$	2.17378e9i	0.0920603i
$393$	0	0
$394$	−1.67866e10	−0.696590
$395$	4.01823e10i	1.65062i
$396$	0	0
$397$	2.04581e10	0.823576	0.411788	−	0.911280i	$-0.364904\pi$
$397$	2.04581e10	0.823576	0.411788	+	0.911280i	$0.364904\pi$
$398$	− 4.39762e9i	− 0.175261i
$399$	0	0
$400$	−5.47452e7	−0.00213848
$401$	− 1.12170e10i	− 0.433810i	−0.976193	−	0.216905i	$-0.930404\pi$
$401$	− 1.12170e10i	− 0.433810i	0.976193	−	0.216905i	$-0.0695962\pi$
$402$	0	0
$403$	4.71445e10	1.78736
$404$	− 3.77029e10i	− 1.41530i
$405$	0	0
$406$	−1.20954e10	−0.445160
$407$	− 5.11489e10i	− 1.86406i
$408$	0	0
$409$	5.07523e9	0.181369	0.0906844	−	0.995880i	$-0.471095\pi$
$409$	5.07523e9	0.181369	0.0906844	+	0.995880i	$0.471095\pi$
$410$	1.30686e10i	0.462480i
$411$	0	0
$412$	−3.22178e10	−1.11817
$413$	3.16527e10i	1.08795i
$414$	0	0
$415$	1.47729e10	0.498049
$416$	− 3.55994e10i	− 1.18869i
$417$	0	0
$418$	−2.44090e8	−0.00799548
$419$	1.32217e10i	0.428974i	0.976727	+	0.214487i	$0.0688080\pi$
$419$	1.32217e10i	0.428974i	−0.976727	+	0.214487i	$0.931192\pi$
$420$	0	0
$421$	−5.70068e10	−1.81467	−0.907337	−	0.420404i	$-0.861888\pi$
$421$	−5.70068e10	−1.81467	−0.907337	+	0.420404i	$0.861888\pi$
$422$	1.39916e10i	0.441181i
$423$	0	0
$424$	−3.52522e10	−1.09074
$425$	− 4.17216e7i	− 0.00127881i
$426$	0	0
$427$	1.09966e10	0.330787
$428$	9.19416e8i	0.0273991i
$429$	0	0
$430$	1.34543e10	0.393540
$431$	1.25350e10i	0.363258i	0.983367	+	0.181629i	$0.0581370\pi$
$431$	1.25350e10i	0.363258i	−0.983367	+	0.181629i	$0.941863\pi$
$432$	0	0
$433$	−4.07711e9	−0.115985	−0.0579924	−	0.998317i	$-0.518470\pi$
$433$	−4.07711e9	−0.115985	−0.0579924	+	0.998317i	$0.518470\pi$
$434$	1.61557e10i	0.455374i
$435$	0	0
$436$	−2.66089e10	−0.736345
$437$	3.82829e8i	0.0104973i
$438$	0	0
$439$	2.42860e9	0.0653881	0.0326940	−	0.999465i	$-0.489591\pi$
$439$	2.42860e9	0.0653881	0.0326940	+	0.999465i	$0.489591\pi$
$440$	− 3.13119e10i	− 0.835409i
$441$	0	0
$442$	6.79935e9	0.178147
$443$	− 3.25859e10i	− 0.846088i	−0.906109	−	0.423044i	$-0.860962\pi$
$443$	− 3.25859e10i	− 0.846088i	0.906109	−	0.423044i	$-0.139038\pi$
$444$	0	0
$445$	−6.53493e10	−1.66648
$446$	7.10082e9i	0.179461i
$447$	0	0
$448$	−1.12055e10	−0.278175
$449$	2.69566e10i	0.663255i	0.943410	+	0.331627i	$0.107598\pi$
$449$	2.69566e10i	0.663255i	−0.943410	+	0.331627i	$0.892402\pi$
$450$	0	0
$451$	6.52347e10	1.57678
$452$	1.19963e9i	0.0287406i
$453$	0	0
$454$	−1.11407e10	−0.262234
$455$	− 5.26061e10i	− 1.22741i
$456$	0	0
$457$	6.93862e10	1.59077	0.795387	−	0.606102i	$-0.207268\pi$
$457$	6.93862e10	1.59077	0.795387	+	0.606102i	$0.207268\pi$
$458$	3.08228e10i	0.700503i
$459$	0	0
$460$	−2.28389e10	−0.510086
$461$	5.77668e10i	1.27901i	0.768787	+	0.639505i	$0.220861\pi$
$461$	5.77668e10i	1.27901i	−0.768787	+	0.639505i	$0.779139\pi$
$462$	0	0
$463$	5.29309e10	1.15182	0.575911	−	0.817512i	$-0.304648\pi$
$463$	5.29309e10	1.15182	0.575911	+	0.817512i	$0.304648\pi$
$464$	3.84040e10i	0.828523i
$465$	0	0
$466$	1.22286e10	0.259317
$467$	− 3.42083e10i	− 0.719224i	−0.933102	−	0.359612i	$-0.882909\pi$
$467$	− 3.42083e10i	− 0.719224i	0.933102	−	0.359612i	$-0.117091\pi$
$468$	0	0
$469$	−3.16371e10	−0.653892
$470$	2.68540e10i	0.550323i
$471$	0	0
$472$	−3.92556e10	−0.790922
$473$	− 6.71602e10i	− 1.34174i
$474$	0	0
$475$	−3.12076e6	−6.13037e−5 0
$476$	− 1.55069e10i	− 0.302063i
$477$	0	0
$478$	2.31091e10	0.442662
$479$	3.35282e10i	0.636896i	0.947940	+	0.318448i	$0.103162\pi$
$479$	3.35282e10i	0.636896i	−0.947940	+	0.318448i	$0.896838\pi$
$480$	0	0
$481$	1.06575e11	1.99101
$482$	2.04444e10i	0.378778i
$483$	0	0
$484$	−2.49816e10	−0.455238
$485$	− 6.27689e10i	− 1.13443i
$486$	0	0
$487$	−8.39697e10	−1.49282	−0.746409	−	0.665488i	$-0.768224\pi$
$487$	−8.39697e10	−1.49282	−0.746409	+	0.665488i	$0.768224\pi$
$488$	1.36380e10i	0.240476i
$489$	0	0
$490$	−2.84382e9	−0.0493308
$491$	1.98925e10i	0.342266i	0.985248	+	0.171133i	$0.0547427\pi$
$491$	1.98925e10i	0.342266i	−0.985248	+	0.171133i	$0.945257\pi$
$492$	0	0
$493$	−2.92679e10	−0.495455
$494$	− 5.08589e8i	− 0.00854003i
$495$	0	0
$496$	5.12959e10	0.847532
$497$	5.77272e10i	0.946138i
$498$	0	0
$499$	−3.78469e10	−0.610419	−0.305209	−	0.952285i	$-0.598726\pi$
$499$	−3.78469e10	−0.610419	−0.305209	+	0.952285i	$0.598726\pi$
$500$	5.42423e10i	0.867877i
$501$	0	0
$502$	6.47923e9	0.102025
$503$	− 6.98859e10i	− 1.09174i	−0.837871	−	0.545868i	$-0.816200\pi$
$503$	− 6.98859e10i	− 1.09174i	0.837871	−	0.545868i	$-0.183800\pi$
$504$	0	0
$505$	1.06060e11	1.63075
$506$	− 1.71302e10i	− 0.261312i
$507$	0	0
$508$	7.41314e10	1.11313
$509$	− 2.96345e9i	− 0.0441496i	−0.999756	−	0.0220748i	$-0.992973\pi$
$509$	− 2.96345e9i	− 0.0441496i	0.999756	−	0.0220748i	$-0.00702720\pi$
$510$	0	0
$511$	2.28019e10	0.334416
$512$	− 6.77608e10i	− 0.986050i
$513$	0	0
$514$	1.24939e10	0.178997
$515$	− 9.06302e10i	− 1.28838i
$516$	0	0
$517$	1.34047e11	1.87628
$518$	3.65216e10i	0.507260i
$519$	0	0
$520$	6.52421e10	0.892307
$521$	4.90458e10i	0.665657i	0.942987	+	0.332829i	$0.108003\pi$
$521$	4.90458e10i	0.665657i	−0.942987	+	0.332829i	$0.891997\pi$
$522$	0	0
$523$	5.03289e10	0.672683	0.336342	−	0.941740i	$-0.390810\pi$
$523$	5.03289e10	0.672683	0.336342	+	0.941740i	$0.390810\pi$
$524$	5.54209e10i	0.735103i
$525$	0	0
$526$	−3.02559e10	−0.395246
$527$	3.90929e10i	0.506822i
$528$	0	0
$529$	5.14441e10	0.656921
$530$	− 4.61182e10i	− 0.584478i
$531$	0	0
$532$	−1.15991e9	−0.0144804
$533$	1.35924e11i	1.68418i
$534$	0	0
$535$	−2.58636e9	−0.0315699
$536$	− 3.92364e10i	− 0.475368i
$537$	0	0
$538$	−6.52569e9	−0.0778928
$539$	1.41955e10i	0.168189i
$540$	0	0
$541$	−1.30236e11	−1.52034	−0.760170	−	0.649724i	$-0.774885\pi$
$541$	−1.30236e11	−1.52034	−0.760170	+	0.649724i	$0.774885\pi$
$542$	2.33363e10i	0.270418i
$543$	0	0
$544$	2.95195e10	0.337065
$545$	− 7.48521e10i	− 0.848434i
$546$	0	0
$547$	1.13948e11	1.27279	0.636396	−	0.771362i	$-0.280424\pi$
$547$	1.13948e11	1.27279	0.636396	+	0.771362i	$0.280424\pi$
$548$	4.35128e10i	0.482498i
$549$	0	0
$550$	1.39642e8	0.00152604
$551$	2.18923e9i	0.0237512i
$552$	0	0
$553$	−1.43218e11	−1.53143
$554$	− 4.75808e9i	− 0.0505118i
$555$	0	0
$556$	−6.45895e10	−0.675870
$557$	− 4.23916e10i	− 0.440411i	−0.975453	−	0.220206i	$-0.929327\pi$
$557$	− 4.23916e10i	− 0.440411i	0.975453	−	0.220206i	$-0.0706729\pi$
$558$	0	0
$559$	1.39936e11	1.43312
$560$	− 5.72384e10i	− 0.582017i
$561$	0	0
$562$	3.49338e10	0.350188
$563$	1.20039e11i	1.19478i	0.801949	+	0.597392i	$0.203796\pi$
$563$	1.20039e11i	1.19478i	−0.801949	+	0.597392i	$0.796204\pi$
$564$	0	0
$565$	−3.37463e9	−0.0331155
$566$	3.51760e10i	0.342753i
$567$	0	0
$568$	−7.15932e10	−0.687826
$569$	4.59201e10i	0.438080i	0.975716	+	0.219040i	$0.0702926\pi$
$569$	4.59201e10i	0.438080i	−0.975716	+	0.219040i	$0.929707\pi$
$570$	0	0
$571$	−6.54486e10	−0.615681	−0.307840	−	0.951438i	$-0.599606\pi$
$571$	−6.54486e10	−0.615681	−0.307840	+	0.951438i	$0.599606\pi$
$572$	− 1.51456e11i	− 1.41482i
$573$	0	0
$574$	−4.65792e10	−0.429086
$575$	− 2.19014e8i	− 0.00200356i
$576$	0	0
$577$	−1.82107e10	−0.164295	−0.0821474	−	0.996620i	$-0.526178\pi$
$577$	−1.82107e10	−0.164295	−0.0821474	+	0.996620i	$0.526178\pi$
$578$	− 3.47016e10i	− 0.310912i
$579$	0	0
$580$	−1.30605e11	−1.15412
$581$	5.26536e10i	0.462087i
$582$	0	0
$583$	−2.30209e11	−1.99272
$584$	2.82789e10i	0.243114i
$585$	0	0
$586$	−4.34618e9	−0.0368567
$587$	− 1.80419e11i	− 1.51960i	−0.650158	−	0.759799i	$-0.725297\pi$
$587$	− 1.80419e11i	− 1.51960i	0.650158	−	0.759799i	$-0.274703\pi$
$588$	0	0
$589$	2.92414e9	0.0242961
$590$	− 5.13556e10i	− 0.423819i
$591$	0	0
$592$	1.15959e11	0.944102
$593$	− 1.26101e11i	− 1.01977i	−0.860244	−	0.509883i	$-0.829689\pi$
$593$	− 1.26101e11i	− 1.01977i	0.860244	−	0.509883i	$-0.170311\pi$
$594$	0	0
$595$	4.36217e10	0.348045
$596$	3.06882e10i	0.243213i
$597$	0	0
$598$	3.56927e10	0.279109
$599$	− 5.56354e10i	− 0.432159i	−0.976376	−	0.216080i	$-0.930673\pi$
$599$	− 5.56354e10i	− 0.432159i	0.976376	−	0.216080i	$-0.0693271\pi$
$600$	0	0
$601$	−2.14824e11	−1.64658	−0.823292	−	0.567618i	$-0.807865\pi$
$601$	−2.14824e11	−1.64658	−0.823292	+	0.567618i	$0.807865\pi$
$602$	4.79541e10i	0.365123i
$603$	0	0
$604$	−7.02526e10	−0.527855
$605$	− 7.02743e10i	− 0.524536i
$606$	0	0
$607$	4.35584e10	0.320861	0.160430	−	0.987047i	$-0.448712\pi$
$607$	4.35584e10	0.320861	0.160430	+	0.987047i	$0.448712\pi$
$608$	− 2.20805e9i	− 0.0161583i
$609$	0	0
$610$	−1.78418e10	−0.128860
$611$	2.79304e11i	2.00406i
$612$	0	0
$613$	−2.07092e11	−1.46663	−0.733316	−	0.679888i	$-0.762028\pi$
$613$	−2.07092e11	−1.46663	−0.733316	+	0.679888i	$0.762028\pi$
$614$	− 1.20937e10i	− 0.0850917i
$615$	0	0
$616$	1.11602e11	0.775087
$617$	2.39440e10i	0.165218i	0.996582	+	0.0826089i	$0.0263252\pi$
$617$	2.39440e10i	0.165218i	−0.996582	+	0.0826089i	$0.973675\pi$
$618$	0	0
$619$	1.50995e11	1.02849	0.514244	−	0.857644i	$-0.328073\pi$
$619$	1.50995e11	1.02849	0.514244	+	0.857644i	$0.328073\pi$
$620$	1.74449e11i	1.18059i
$621$	0	0
$622$	5.61916e10	0.375414
$623$	− 2.32919e11i	− 1.54615i
$624$	0	0
$625$	−1.53108e11	−1.00341
$626$	− 5.56285e10i	− 0.362243i
$627$	0	0
$628$	1.21523e11	0.781307
$629$	8.83733e10i	0.564571i
$630$	0	0
$631$	6.53454e10	0.412190	0.206095	−	0.978532i	$-0.433924\pi$
$631$	6.53454e10	0.412190	0.206095	+	0.978532i	$0.433924\pi$
$632$	− 1.77619e11i	− 1.11332i
$633$	0	0
$634$	−1.77180e10	−0.109662
$635$	2.08535e11i	1.28258i
$636$	0	0
$637$	−2.95781e10	−0.179644
$638$	− 9.79598e10i	− 0.591242i
$639$	0	0
$640$	1.69702e11	1.01150
$641$	− 3.63671e10i	− 0.215415i	−0.994183	−	0.107708i	$-0.965649\pi$
$641$	− 3.63671e10i	− 0.215415i	0.994183	−	0.107708i	$-0.0343510\pi$
$642$	0	0
$643$	−1.92078e11	−1.12366	−0.561828	−	0.827254i	$-0.689902\pi$
$643$	−1.92078e11	−1.12366	−0.561828	+	0.827254i	$0.689902\pi$
$644$	− 8.14025e10i	− 0.473254i
$645$	0	0
$646$	4.21729e8	0.00242161
$647$	4.70466e10i	0.268479i	0.990949	+	0.134240i	$0.0428592\pi$
$647$	4.70466e10i	0.268479i	−0.990949	+	0.134240i	$0.957141\pi$
$648$	0	0
$649$	−2.56353e11	−1.44497
$650$	2.90961e8i	0.00162998i
$651$	0	0
$652$	9.76356e10	0.540279
$653$	1.41222e11i	0.776692i	0.921514	+	0.388346i	$0.126953\pi$
$653$	1.41222e11i	0.776692i	−0.921514	+	0.388346i	$0.873047\pi$
$654$	0	0
$655$	−1.55902e11	−0.847003
$656$	1.47893e11i	0.798606i
$657$	0	0
$658$	−9.57133e10	−0.510586
$659$	2.07124e11i	1.09822i	0.835750	+	0.549109i	$0.185033\pi$
$659$	2.07124e11i	1.09822i	−0.835750	+	0.549109i	$0.814967\pi$
$660$	0	0
$661$	−1.87243e11	−0.980841	−0.490421	−	0.871486i	$-0.663157\pi$
$661$	−1.87243e11	−0.980841	−0.490421	+	0.871486i	$0.663157\pi$
$662$	− 2.10837e10i	− 0.109778i
$663$	0	0
$664$	−6.53010e10	−0.335929
$665$	− 3.26289e9i	− 0.0166846i
$666$	0	0
$667$	−1.53640e11	−0.776247
$668$	1.52725e11i	0.767014i
$669$	0	0
$670$	5.13305e10	0.254728
$671$	8.90609e10i	0.439336i
$672$	0	0
$673$	−4.53506e10	−0.221066	−0.110533	−	0.993872i	$-0.535256\pi$
$673$	−4.53506e10	−0.221066	−0.110533	+	0.993872i	$0.535256\pi$
$674$	4.27984e10i	0.207390i
$675$	0	0
$676$	1.34028e11	0.641815
$677$	2.60088e11i	1.23813i	0.785340	+	0.619064i	$0.212488\pi$
$677$	2.60088e11i	1.23813i	−0.785340	+	0.619064i	$0.787512\pi$
$678$	0	0
$679$	2.23721e11	1.05252
$680$	5.40996e10i	0.253022i
$681$	0	0
$682$	−1.30844e11	−0.604807
$683$	− 3.08050e11i	− 1.41559i	−0.706417	−	0.707796i	$-0.749690\pi$
$683$	− 3.08050e11i	− 1.41559i	0.706417	−	0.707796i	$-0.250310\pi$
$684$	0	0
$685$	−1.22404e11	−0.555945
$686$	− 8.45252e10i	− 0.381672i
$687$	0	0
$688$	1.52258e11	0.679560
$689$	− 4.79666e11i	− 2.12844i
$690$	0	0
$691$	2.67485e11	1.17324	0.586620	−	0.809862i	$-0.300458\pi$
$691$	2.67485e11	1.17324	0.586620	+	0.809862i	$0.300458\pi$
$692$	1.01919e11i	0.444457i
$693$	0	0
$694$	4.04318e10	0.174295
$695$	− 1.81693e11i	− 0.778753i
$696$	0	0
$697$	−1.12710e11	−0.477564
$698$	− 9.30862e10i	− 0.392160i
$699$	0	0
$700$	6.63581e8	0.00276377
$701$	− 1.32742e11i	− 0.549713i	−0.961485	−	0.274857i	$-0.911370\pi$
$701$	− 1.32742e11i	− 0.549713i	0.961485	−	0.274857i	$-0.0886304\pi$
$702$	0	0
$703$	6.61029e9	0.0270645
$704$	− 9.07523e10i	− 0.369460i
$705$	0	0
$706$	1.71235e10	0.0689245
$707$	3.78020e11i	1.51299i
$708$	0	0
$709$	−1.35390e11	−0.535800	−0.267900	−	0.963447i	$-0.586330\pi$
$709$	−1.35390e11	−0.535800	−0.267900	+	0.963447i	$0.586330\pi$
$710$	− 9.36608e10i	− 0.368574i
$711$	0	0
$712$	2.88866e11	1.12402
$713$	2.05215e11i	0.794056i
$714$	0	0
$715$	4.26053e11	1.63019
$716$	− 4.07069e11i	− 1.54887i
$717$	0	0
$718$	5.29909e10	0.199390
$719$	− 4.62350e11i	− 1.73004i	−0.501739	−	0.865019i	$-0.667306\pi$
$719$	− 4.62350e11i	− 1.73004i	0.501739	−	0.865019i	$-0.332694\pi$
$720$	0	0
$721$	3.23025e11	1.19535
$722$	9.81816e10i	0.361311i
$723$	0	0
$724$	−9.81509e10	−0.357224
$725$	− 1.25245e9i	− 0.00453323i
$726$	0	0
$727$	3.60324e11	1.28990	0.644950	−	0.764225i	$-0.276878\pi$
$727$	3.60324e11	1.28990	0.644950	+	0.764225i	$0.276878\pi$
$728$	2.32536e11i	0.827876i
$729$	0	0
$730$	−3.69954e10	−0.130274
$731$	1.16037e11i	0.406375i
$732$	0	0
$733$	−3.39027e11	−1.17441	−0.587204	−	0.809439i	$-0.699771\pi$
$733$	−3.39027e11	−1.17441	−0.587204	+	0.809439i	$0.699771\pi$
$734$	9.41780e10i	0.324463i
$735$	0	0
$736$	1.54960e11	0.528092
$737$	− 2.56227e11i	− 0.868470i
$738$	0	0
$739$	−5.40896e10	−0.181358	−0.0906788	−	0.995880i	$-0.528904\pi$
$739$	−5.40896e10	−0.181358	−0.0906788	+	0.995880i	$0.528904\pi$
$740$	3.94358e11i	1.31511i
$741$	0	0
$742$	1.64375e11	0.542275
$743$	4.20299e11i	1.37912i	0.724226	+	0.689562i	$0.242197\pi$
$743$	4.20299e11i	1.37912i	−0.724226	+	0.689562i	$0.757803\pi$
$744$	0	0
$745$	−8.63273e10	−0.280236
$746$	5.22878e10i	0.168828i
$747$	0	0
$748$	1.25590e11	0.401187
$749$	− 9.21832e9i	− 0.0292904i
$750$	0	0
$751$	2.60430e11	0.818712	0.409356	−	0.912375i	$-0.365753\pi$
$751$	2.60430e11	0.818712	0.409356	+	0.912375i	$0.365753\pi$
$752$	3.03898e11i	0.950291i
$753$	0	0
$754$	2.04111e11	0.631510
$755$	− 1.97624e11i	− 0.608207i
$756$	0	0
$757$	4.52924e11	1.37925	0.689623	−	0.724168i	$-0.257776\pi$
$757$	4.52924e11	1.37925	0.689623	+	0.724168i	$0.257776\pi$
$758$	9.08842e10i	0.275303i
$759$	0	0
$760$	4.04664e9	0.0121294
$761$	5.51539e10i	0.164452i	0.996614	+	0.0822258i	$0.0262029\pi$
$761$	5.51539e10i	0.164452i	−0.996614	+	0.0822258i	$0.973797\pi$
$762$	0	0
$763$	2.66789e11	0.787171
$764$	− 3.60108e11i	− 1.05696i
$765$	0	0
$766$	−1.14571e11	−0.332782
$767$	− 5.34140e11i	− 1.54338i
$768$	0	0
$769$	−1.66570e11	−0.476313	−0.238156	−	0.971227i	$-0.576543\pi$
$769$	−1.66570e11	−0.476313	−0.238156	+	0.971227i	$0.576543\pi$
$770$	1.46002e11i	0.415333i
$771$	0	0
$772$	2.52028e11	0.709543
$773$	− 7.29931e9i	− 0.0204439i	−0.999948	−	0.0102219i	$-0.996746\pi$
$773$	− 7.29931e9i	− 0.0204439i	0.999948	−	0.0102219i	$-0.00325380\pi$
$774$	0	0
$775$	−1.67288e9	−0.00463723
$776$	2.77459e11i	0.765160i
$777$	0	0
$778$	−1.76792e11	−0.482553
$779$	8.43068e9i	0.0228935i
$780$	0	0
$781$	−4.67528e11	−1.25662
$782$	2.95969e10i	0.0791441i
$783$	0	0
$784$	−3.21826e10	−0.0851839
$785$	3.41851e11i	0.900240i
$786$	0	0
$787$	1.51500e10	0.0394923	0.0197461	−	0.999805i	$-0.493714\pi$
$787$	1.51500e10	0.0394923	0.0197461	+	0.999805i	$0.493714\pi$
$788$	− 6.46050e11i	− 1.67556i
$789$	0	0
$790$	2.32368e11	0.596579
$791$	− 1.20279e10i	− 0.0307244i
$792$	0	0
$793$	−1.85569e11	−0.469259
$794$	− 1.18306e11i	− 0.297663i
$795$	0	0
$796$	1.69247e11	0.421569
$797$	− 6.63280e11i	− 1.64386i	−0.569592	−	0.821928i	$-0.692899\pi$
$797$	− 6.63280e11i	− 1.64386i	0.569592	−	0.821928i	$-0.307101\pi$
$798$	0	0
$799$	−2.31602e11	−0.568272
$800$	1.26321e9i	0.00308402i
$801$	0	0
$802$	−6.48662e10	−0.156791
$803$	1.84671e11i	0.444156i
$804$	0	0
$805$	2.28989e11	0.545294
$806$	− 2.72629e11i	− 0.645999i
$807$	0	0
$808$	−4.68820e11	−1.09992
$809$	1.85171e11i	0.432293i	0.976361	+	0.216146i	$0.0693488\pi$
$809$	1.85171e11i	0.432293i	−0.976361	+	0.216146i	$0.930651\pi$
$810$	0	0
$811$	4.16615e11	0.963057	0.481528	−	0.876430i	$-0.340082\pi$
$811$	4.16615e11	0.963057	0.481528	+	0.876430i	$0.340082\pi$
$812$	− 4.65505e11i	− 1.07078i
$813$	0	0
$814$	−2.95786e11	−0.673721
$815$	2.74653e11i	0.622522i
$816$	0	0
$817$	8.67953e9	0.0194809
$818$	− 2.93492e10i	− 0.0655517i
$819$	0	0
$820$	−5.02959e11	−1.11244
$821$	5.47092e11i	1.20417i	0.798432	+	0.602085i	$0.205663\pi$
$821$	5.47092e11i	1.20417i	−0.798432	+	0.602085i	$0.794337\pi$
$822$	0	0
$823$	4.57975e11	0.998258	0.499129	−	0.866528i	$-0.333653\pi$
$823$	4.57975e11	0.998258	0.499129	+	0.866528i	$0.333653\pi$
$824$	4.00615e11i	0.868998i
$825$	0	0
$826$	1.83042e11	0.393216
$827$	5.62301e11i	1.20212i	0.799205	+	0.601058i	$0.205254\pi$
$827$	5.62301e11i	1.20212i	−0.799205	+	0.601058i	$0.794746\pi$
$828$	0	0
$829$	−7.49184e11	−1.58625	−0.793123	−	0.609062i	$-0.791546\pi$
$829$	−7.49184e11	−1.58625	−0.793123	+	0.609062i	$0.791546\pi$
$830$	− 8.54291e10i	− 0.180009i
$831$	0	0
$832$	1.89093e11	0.394623
$833$	− 2.45266e10i	− 0.0509397i
$834$	0	0
$835$	−4.29622e11	−0.883772
$836$	− 9.39406e9i	− 0.0192322i
$837$	0	0
$838$	7.64590e10	0.155043
$839$	− 1.99264e11i	− 0.402143i	−0.979577	−	0.201072i	$-0.935558\pi$
$839$	− 1.99264e11i	− 0.402143i	0.979577	−	0.201072i	$-0.0644424\pi$
$840$	0	0
$841$	−3.78351e11	−0.756329
$842$	3.29661e11i	0.655873i
$843$	0	0
$844$	−5.38481e11	−1.06121
$845$	3.77028e11i	0.739514i
$846$	0	0
$847$	2.50472e11	0.486660
$848$	− 5.21904e11i	− 1.00927i
$849$	0	0
$850$	−2.41269e8	−0.000462196 0
$851$	4.63909e11i	0.884534i
$852$	0	0
$853$	1.58598e11	0.299572	0.149786	−	0.988718i	$-0.452141\pi$
$853$	1.58598e11	0.299572	0.149786	+	0.988718i	$0.452141\pi$
$854$	− 6.35918e10i	− 0.119555i
$855$	0	0
$856$	1.14326e10	0.0212936
$857$	− 6.91737e11i	− 1.28238i	−0.767381	−	0.641191i	$-0.778441\pi$
$857$	− 6.91737e11i	− 1.28238i	0.767381	−	0.641191i	$-0.221559\pi$
$858$	0	0
$859$	−6.97094e11	−1.28032	−0.640160	−	0.768241i	$-0.721132\pi$
$859$	−6.97094e11	−1.28032	−0.640160	+	0.768241i	$0.721132\pi$
$860$	5.17805e11i	0.946613i
$861$	0	0
$862$	7.24879e10	0.131291
$863$	5.88063e11i	1.06018i	0.847940	+	0.530092i	$0.177842\pi$
$863$	5.88063e11i	1.06018i	−0.847940	+	0.530092i	$0.822158\pi$
$864$	0	0
$865$	−2.86702e11	−0.512113
$866$	2.35773e10i	0.0419201i
$867$	0	0
$868$	−6.21771e11	−1.09535
$869$	− 1.15991e12i	− 2.03398i
$870$	0	0
$871$	5.33879e11	0.927620
$872$	3.30871e11i	0.572259i
$873$	0	0
$874$	2.21384e9	0.00379402
$875$	− 5.43849e11i	− 0.927782i
$876$	0	0
$877$	−1.40313e11	−0.237192	−0.118596	−	0.992943i	$-0.537839\pi$
$877$	−1.40313e11	−0.237192	−0.118596	+	0.992943i	$0.537839\pi$
$878$	− 1.40442e10i	− 0.0236330i
$879$	0	0
$880$	4.63570e11	0.773009
$881$	− 7.87575e10i	− 0.130734i	−0.997861	−	0.0653670i	$-0.979178\pi$
$881$	− 7.87575e10i	− 0.130734i	0.997861	−	0.0653670i	$-0.0208218\pi$
$882$	0	0
$883$	−6.05141e11	−0.995437	−0.497719	−	0.867339i	$-0.665829\pi$
$883$	−6.05141e11	−0.995437	−0.497719	+	0.867339i	$0.665829\pi$
$884$	2.61680e11i	0.428511i
$885$	0	0
$886$	−1.88439e11	−0.305799
$887$	3.28250e11i	0.530286i	0.964209	+	0.265143i	$0.0854191\pi$
$887$	3.28250e11i	0.530286i	−0.964209	+	0.265143i	$0.914581\pi$
$888$	0	0
$889$	−7.43262e11	−1.18997
$890$	3.77905e11i	0.602313i
$891$	0	0
$892$	−2.73283e11	−0.431671
$893$	1.73238e10i	0.0272419i
$894$	0	0
$895$	1.14510e12	1.78465
$896$	6.04853e11i	0.938465i
$897$	0	0
$898$	1.55886e11	0.239718
$899$	1.17354e12i	1.79662i
$900$	0	0
$901$	3.97746e11	0.603541
$902$	− 3.77241e11i	− 0.569893i
$903$	0	0
$904$	1.49170e10	0.0223361
$905$	− 2.76103e11i	− 0.411601i
$906$	0	0
$907$	3.15443e11	0.466114	0.233057	−	0.972463i	$-0.425127\pi$
$907$	3.15443e11	0.466114	0.233057	+	0.972463i	$0.425127\pi$
$908$	− 4.28763e11i	− 0.630774i
$909$	0	0
$910$	−3.04213e11	−0.443620
$911$	− 1.27525e12i	− 1.85149i	−0.378149	−	0.925745i	$-0.623439\pi$
$911$	− 1.27525e12i	− 1.85149i	0.378149	−	0.925745i	$-0.376561\pi$
$912$	0	0
$913$	−4.26437e11	−0.613723
$914$	− 4.01249e11i	− 0.574949i
$915$	0	0
$916$	−1.18625e12	−1.68498
$917$	− 5.55666e11i	− 0.785844i
$918$	0	0
$919$	9.91643e11	1.39025	0.695126	−	0.718888i	$-0.255349\pi$
$919$	9.91643e11	1.39025	0.695126	+	0.718888i	$0.255349\pi$
$920$	2.83992e11i	0.396419i
$921$	0	0
$922$	3.34056e11	0.462270
$923$	− 9.74149e11i	− 1.34220i
$924$	0	0
$925$	−3.78171e9	−0.00516561
$926$	− 3.06091e11i	− 0.416300i
$927$	0	0
$928$	8.86150e11	1.19486
$929$	1.41759e12i	1.90322i	0.307308	+	0.951610i	$0.400572\pi$
$929$	1.41759e12i	1.90322i	−0.307308	+	0.951610i	$0.599428\pi$
$930$	0	0
$931$	−1.83458e9	−0.00244196
$932$	4.70629e11i	0.623757i
$933$	0	0
$934$	−1.97821e11	−0.259947
$935$	3.53289e11i	0.462257i
$936$	0	0
$937$	−1.38380e12	−1.79521	−0.897606	−	0.440800i	$-0.854695\pi$
$937$	−1.38380e12	−1.79521	−0.897606	+	0.440800i	$0.854695\pi$
$938$	1.82953e11i	0.236334i
$939$	0	0
$940$	−1.03351e12	−1.32374
$941$	− 6.38123e11i	− 0.813853i	−0.913461	−	0.406926i	$-0.866601\pi$
$941$	− 6.38123e11i	− 0.813853i	0.913461	−	0.406926i	$-0.133399\pi$
$942$	0	0
$943$	−5.91663e11	−0.748217
$944$	− 5.81175e11i	− 0.731845i
$945$	0	0
$946$	−3.88376e11	−0.484940
$947$	− 1.28017e12i	− 1.59173i	−0.605476	−	0.795864i	$-0.707017\pi$
$947$	− 1.28017e12i	− 1.59173i	0.605476	−	0.795864i	$-0.292983\pi$
$948$	0	0
$949$	−3.84783e11	−0.474407
$950$	1.80469e7i	0 2.21568e-5i
$951$	0	0
$952$	−1.92822e11	−0.234752
$953$	− 3.43083e11i	− 0.415936i	−0.978136	−	0.207968i	$-0.933315\pi$
$953$	− 3.43083e11i	− 0.415936i	0.978136	−	0.207968i	$-0.0666851\pi$
$954$	0	0
$955$	1.01300e12	1.21786
$956$	8.89381e11i	1.06477i
$957$	0	0
$958$	1.93888e11	0.230192
$959$	− 4.36272e11i	− 0.515802i
$960$	0	0
$961$	7.14591e11	0.837845
$962$	− 6.16304e11i	− 0.719606i
$963$	0	0
$964$	−7.86823e11	−0.911106
$965$	7.08965e11i	0.817553i
$966$	0	0
$967$	−2.04278e11	−0.233623	−0.116811	−	0.993154i	$-0.537267\pi$
$967$	−2.04278e11	−0.233623	−0.116811	+	0.993154i	$0.537267\pi$
$968$	3.10636e11i	0.353794i
$969$	0	0
$970$	−3.62982e11	−0.410014
$971$	− 9.01348e11i	− 1.01395i	−0.861962	−	0.506974i	$-0.830764\pi$
$971$	− 9.01348e11i	− 1.01395i	0.861962	−	0.506974i	$-0.169236\pi$
$972$	0	0
$973$	6.47593e11	0.722521
$974$	4.85583e11i	0.539545i
$975$	0	0
$976$	−2.01909e11	−0.222514
$977$	1.22590e12i	1.34548i	0.739880	+	0.672739i	$0.234882\pi$
$977$	1.22590e12i	1.34548i	−0.739880	+	0.672739i	$0.765118\pi$
$978$	0	0
$979$	1.88639e12	2.05353
$980$	− 1.09448e11i	− 0.118659i
$981$	0	0
$982$	1.15035e11	0.123704
$983$	4.96952e11i	0.532231i	0.963941	+	0.266116i	$0.0857403\pi$
$983$	4.96952e11i	0.532231i	−0.963941	+	0.266116i	$0.914260\pi$
$984$	0	0
$985$	1.81737e12	1.93062
$986$	1.69252e11i	0.179071i
$987$	0	0
$988$	1.95736e10	0.0205420
$989$	6.09127e11i	0.636682i
$990$	0	0
$991$	6.31886e10	0.0655155	0.0327578	−	0.999463i	$-0.489571\pi$
$991$	6.31886e10	0.0655155	0.0327578	+	0.999463i	$0.489571\pi$
$992$	− 1.18362e12i	− 1.22227i
$993$	0	0
$994$	3.33827e11	0.341960
$995$	4.76100e11i	0.485742i
$996$	0	0
$997$	1.78193e12	1.80347	0.901736	−	0.432287i	$-0.142293\pi$
$997$	1.78193e12	1.80347	0.901736	+	0.432287i	$0.142293\pi$
$998$	2.18862e11i	0.220622i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.9.b.b.80.8	✓	16
3.2	odd	2	inner	81.9.b.b.80.9	yes	16
9.2	odd	6		81.9.d.g.53.9		32
9.4	even	3		81.9.d.g.26.9		32
9.5	odd	6		81.9.d.g.26.8		32
9.7	even	3		81.9.d.g.53.8		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.9.b.b.80.8	✓	16	1.1	even	1	trivial
81.9.b.b.80.9	yes	16	3.2	odd	2	inner
81.9.d.g.26.8		32	9.5	odd	6
81.9.d.g.26.9		32	9.4	even	3
81.9.d.g.53.8		32	9.7	even	3
81.9.d.g.53.9		32	9.2	odd	6

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:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	81.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.78284i q^{2} +222.559 q^{4} +626.068i q^{5} -2231.44 q^{7} -2767.43i q^{8} +3620.45 q^{10} -18072.2i q^{11} +37655.6 q^{13} +12904.0i q^{14} +40971.5 q^{16} +31224.6i q^{17} +2335.59 q^{19} +139337. i q^{20} -104509. q^{22} +163911. i q^{23} -1336.18 q^{25} -217756. i q^{26} -496626. q^{28} +937335. i q^{29} +1.25199e6 q^{31} -945393. i q^{32} +180567. q^{34} -1.39703e6i q^{35} +2.83025e6 q^{37} -13506.3i q^{38} +1.73260e6 q^{40} +3.60966e6i q^{41} +3.71621e6 q^{43} -4.02214e6i q^{44} +947871. q^{46} +7.41731e6i q^{47} -785489. q^{49} +7726.90i q^{50} +8.38059e6 q^{52} -1.27382e7i q^{53} +1.13145e7 q^{55} +6.17534e6i q^{56} +5.42046e6 q^{58} -1.41849e7i q^{59} -4.92805e6 q^{61} -7.24006e6i q^{62} +5.02164e6 q^{64} +2.35750e7i q^{65} +1.41779e7 q^{67} +6.94930e6i q^{68} -8.07881e6 q^{70} -2.58699e7i q^{71} -1.02185e7 q^{73} -1.63669e7i q^{74} +519806. q^{76} +4.03271e7i q^{77} +6.41821e7 q^{79} +2.56509e7i q^{80} +2.08741e7 q^{82} -2.35963e7i q^{83} -1.95487e7 q^{85} -2.14902e7i q^{86} -5.00136e7 q^{88} +1.04381e8i q^{89} -8.40262e7 q^{91} +3.64798e7i q^{92} +4.28931e7 q^{94} +1.46224e6i q^{95} -1.00259e8 q^{97} +4.54236e6i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2048 q^{4} + 3692 q^{7} + 10752 q^{10} + 63860 q^{13} + 95116 q^{16} + 185108 q^{19} - 691980 q^{22} - 541712 q^{25} - 997132 q^{28} + 571136 q^{31} + 1027656 q^{34} + 4354268 q^{37} - 2973768 q^{40}+ \cdots - 133878688 q^{97}+O(q^{100}) \) 16 * q - 2048 * q^4 + 3692 * q^7 + 10752 * q^10 + 63860 * q^13 + 95116 * q^16 + 185108 * q^19 - 691980 * q^22 - 541712 * q^25 - 997132 * q^28 + 571136 * q^31 + 1027656 * q^34 + 4354268 * q^37 - 2973768 * q^40 + 5453084 * q^43 - 4234044 * q^46 + 14602560 * q^49 - 25384960 * q^52 - 14452572 * q^55 + 1213068 * q^58 + 31037996 * q^61 + 24253000 * q^64 - 62356828 * q^67 + 11075124 * q^70 - 21273664 * q^73 + 69593876 * q^76 - 151045036 * q^79 - 40201140 * q^82 + 160995564 * q^85 + 366976068 * q^88 + 373680796 * q^91 - 544741584 * q^94 - 133878688 * q^97

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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