LMFDB - Embedded newform 81.8.c.k.28.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [81,8,Mod(28,81)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(81, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("81.28");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 81 = 3^{4} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	81.c (of order $3$, degree $2$, not 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:	$25.3031870642$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 316x^{10} + 37872x^{8} + 2079550x^{6} + 47948824x^{4} + 251235828x^{2} + 43520409 $ x^12 + 316x^10 + 37872x^8 + 2079550x^6 + 47948824x^4 + 251235828*x^2 + 43520409
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{21} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			28.6
Root			$0.423455i$ of defining polynomial
Character	$\chi$	$=$	81.28
Dual form			81.8.c.k.55.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(9.88711 + 17.1250i) q^{2} +(-131.510 + 227.782i) q^{4} +(-236.803 + 410.154i) q^{5} +(513.424 + 889.277i) q^{7} -2669.91 q^{8} -9365.17 q^{10} +(1113.43 + 1928.51i) q^{11} +(1661.16 - 2877.22i) q^{13} +(-10152.6 + 17584.7i) q^{14} +(-9564.39 - 16566.0i) q^{16} +36094.5 q^{17} -37646.7 q^{19} +(-62283.7 - 107879. i) q^{20} +(-22017.1 + 38134.8i) q^{22} +(35430.3 - 61367.1i) q^{23} +(-73088.5 - 126593. i) q^{25} +65696.4 q^{26} -270081. q^{28} +(14429.5 + 24992.7i) q^{29} +(-7815.15 + 13536.2i) q^{31} +(18254.4 - 31617.5i) q^{32} +(356870. + 618117. i) q^{34} -486321. q^{35} +41084.0 q^{37} +(-372217. - 644698. i) q^{38} +(632241. - 1.09507e6i) q^{40} +(31731.8 - 54961.1i) q^{41} +(75830.5 + 131342. i) q^{43} -585706. q^{44} +1.40121e6 q^{46} +(357366. + 618976. i) q^{47} +(-115437. + 199943. i) q^{49} +(1.44527e6 - 2.50328e6i) q^{50} +(436918. + 756765. i) q^{52} +490289. q^{53} -1.05465e6 q^{55} +(-1.37079e6 - 2.37428e6i) q^{56} +(-285333. + 494210. i) q^{58} +(668114. - 1.15721e6i) q^{59} +(898069. + 1.55550e6i) q^{61} -309077. q^{62} -1.72655e6 q^{64} +(786736. + 1.36267e6i) q^{65} +(-1.27284e6 + 2.20462e6i) q^{67} +(-4.74677e6 + 8.22165e6i) q^{68} +(-4.80831e6 - 8.32823e6i) q^{70} -700407. q^{71} -4.69075e6 q^{73} +(406202. + 703562. i) q^{74} +(4.95090e6 - 8.57522e6i) q^{76} +(-1.14332e6 + 1.98029e6i) q^{77} +(2.49005e6 + 4.31290e6i) q^{79} +9.05950e6 q^{80} +1.25494e6 q^{82} +(-1.69513e6 - 2.93606e6i) q^{83} +(-8.54726e6 + 1.48043e7i) q^{85} +(-1.49949e6 + 2.59719e6i) q^{86} +(-2.97274e6 - 5.14894e6i) q^{88} +1.27747e6 q^{89} +3.41152e6 q^{91} +(9.31886e6 + 1.61407e7i) q^{92} +(-7.06663e6 + 1.22398e7i) q^{94} +(8.91483e6 - 1.54409e7i) q^{95} +(-3.82854e6 - 6.63122e6i) q^{97} -4.56536e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 438 q^{4} + 1932 q^{7} - 29844 q^{10} + 11886 q^{13} - 7314 q^{16} - 134328 q^{19} + 23364 q^{22} - 42324 q^{25} - 1386696 q^{28} + 470832 q^{31} + 1834866 q^{34} - 2052516 q^{37} + 3091374 q^{40}+ \cdots - 26307948 q^{97}+O(q^{100}) $ 12 * q - 438 * q^4 + 1932 * q^7 - 29844 * q^10 + 11886 * q^13 - 7314 * q^16 - 134328 * q^19 + 23364 * q^22 - 42324 * q^25 - 1386696 * q^28 + 470832 * q^31 + 1834866 * q^34 - 2052516 * q^37 + 3091374 * q^40 + 1502268 * q^43 + 209160 * q^46 + 1769490 * q^49 + 2993334 * q^52 - 7334856 * q^55 - 659898 * q^58 + 4564770 * q^61 - 2095188 * q^64 - 2850060 * q^67 - 19286604 * q^70 - 3693684 * q^73 + 5520516 * q^76 + 2878188 * q^79 + 30137904 * q^82 - 32410098 * q^85 - 26598276 * q^88 + 32484216 * q^91 - 51428376 * q^94 - 26307948 * q^97

Character values

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

$n$	$2$
$\chi(n)$	$e\left(\frac{1}{3}\right)$

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 $

$2$	9.88711	+	17.1250i	0.873905	+	1.51365i	0.857925	+	0.513776i	$0.171754\pi$
$2$	9.88711	+	17.1250i	0.873905	+	1.51365i	0.0159805	+	0.999872i	$0.494913\pi$
$3$		0			0
$3$		0			0
$4$	−131.510	+	227.782i	−1.02742	+	1.77954i
$5$	−236.803	+	410.154i	−0.847211	+	1.46741i	0.0364760	+	0.999335i	$0.488387\pi$
$5$	−236.803	+	410.154i	−0.847211	+	1.46741i	−0.883687	+	0.468078i	$0.844947\pi$
$6$		0			0
$7$	513.424	+	889.277i	0.565761	+	0.979927i	0.996978	+	0.0776790i	$0.0247509\pi$
$7$	513.424	+	889.277i	0.565761	+	0.979927i	−0.431217	+	0.902248i	$0.641916\pi$
$8$	−2669.91			−1.84366
$9$		0			0
$10$	−9365.17			−2.96153
$11$	1113.43	+	1928.51i	0.252224	+	0.436865i	0.964138	−	0.265402i	$-0.0855046\pi$
$11$	1113.43	+	1928.51i	0.252224	+	0.436865i	−0.711914	+	0.702267i	$0.752171\pi$
$12$		0			0
$13$	1661.16	−	2877.22i	0.209706	−	0.363221i	−0.741916	−	0.670493i	$-0.766083\pi$
$13$	1661.16	−	2877.22i	0.209706	−	0.363221i	0.951622	+	0.307272i	$0.0994160\pi$
$14$	−10152.6	+	17584.7i	−0.988843	+	1.71273i
$15$		0			0
$16$	−9564.39	−	16566.0i	−0.583764	−	1.01111i
$17$	36094.5			1.78184			0.890921	−	0.454158i	$-0.150060\pi$
$17$	36094.5			1.78184			0.890921	+	0.454158i	$0.150060\pi$
$18$		0			0
$19$	−37646.7			−1.25918			−0.629592	−	0.776926i	$-0.716778\pi$
$19$	−37646.7			−1.25918			−0.629592	+	0.776926i	$0.716778\pi$
$20$	−62283.7	−	107879.i	−1.74088	−	3.01530i
$21$		0			0
$22$	−22017.1	+	38134.8i	−0.440840	+	0.763558i
$23$	35430.3	−	61367.1i	0.607194	−	1.05169i	−0.384507	−	0.923122i	$-0.625628\pi$
$23$	35430.3	−	61367.1i	0.607194	−	1.05169i	0.991701	−	0.128568i	$-0.0410382\pi$
$24$		0			0
$25$	−73088.5	−	126593.i	−0.935533	−	1.62039i
$26$	65696.4			0.733052
$27$		0			0
$28$	−270081.			−2.32510
$29$	14429.5	+	24992.7i	0.109865	+	0.190292i	0.915715	−	0.401827i	$-0.131625\pi$
$29$	14429.5	+	24992.7i	0.109865	+	0.190292i	−0.805850	+	0.592119i	$0.798292\pi$
$30$		0			0
$31$	−7815.15	+	13536.2i	−0.0471163	+	0.0816078i	−0.888622	−	0.458641i	$-0.848336\pi$
$31$	−7815.15	+	13536.2i	−0.0471163	+	0.0816078i	0.841505	+	0.540249i	$0.181670\pi$
$32$	18254.4	−	31617.5i	0.0984786	−	0.170570i
$33$		0			0
$34$	356870.	+	618117.i	1.55716	+	2.69708i
$35$	−486321.			−1.91728
$36$		0			0
$37$	41084.0			0.133342			0.0666709	−	0.997775i	$-0.478762\pi$
$37$	41084.0			0.133342			0.0666709	+	0.997775i	$0.478762\pi$
$38$	−372217.	−	644698.i	−1.10041	−	1.90596i
$39$		0			0
$40$	632241.	−	1.09507e6i	1.56197	−	2.70541i
$41$	31731.8	−	54961.1i	0.0719037	−	0.124541i	−0.827832	−	0.560976i	$-0.810426\pi$
$41$	31731.8	−	54961.1i	0.0719037	−	0.124541i	0.899736	+	0.436435i	$0.143759\pi$
$42$		0			0
$43$	75830.5	+	131342.i	0.145447	+	0.251921i	0.929540	−	0.368722i	$-0.120205\pi$
$43$	75830.5	+	131342.i	0.145447	+	0.251921i	−0.784093	+	0.620644i	$0.786871\pi$
$44$	−585706.			−1.03656
$45$		0			0
$46$	1.40121e6			2.12252
$47$	357366.	+	618976.i	0.502077	+	0.869624i	0.999997	+	0.00240047i	$0.000764093\pi$
$47$	357366.	+	618976.i	0.502077	+	0.869624i	−0.497920	+	0.867223i	$0.665903\pi$
$48$		0			0
$49$	−115437.	+	199943.i	−0.140171	+	0.242784i
$50$	1.44527e6	−	2.50328e6i	1.63513	−	2.83214i
$51$		0			0
$52$	436918.	+	756765.i	0.430912	+	0.746361i
$53$	490289.			0.452363			0.226181	−	0.974085i	$-0.427376\pi$
$53$	490289.			0.452363			0.226181	+	0.974085i	$0.427376\pi$
$54$		0			0
$55$	−1.05465e6			−0.854749
$56$	−1.37079e6	−	2.37428e6i	−1.04307	−	1.80665i
$57$		0			0
$58$	−285333.	+	494210.i	−0.192023	+	0.332594i
$59$	668114.	−	1.15721e6i	0.423515	−	0.733549i	−0.572766	−	0.819719i	$-0.694129\pi$
$59$	668114.	−	1.15721e6i	0.423515	−	0.733549i	0.996280	+	0.0861700i	$0.0274628\pi$
$60$		0			0
$61$	898069.	+	1.55550e6i	0.506588	+	0.877437i	0.999971	+	0.00762444i	$0.00242696\pi$
$61$	898069.	+	1.55550e6i	0.506588	+	0.877437i	−0.493383	+	0.869812i	$0.664240\pi$
$62$	−309077.			−0.164701
$63$		0			0
$64$	−1.72655e6			−0.823284
$65$	786736.	+	1.36267e6i	0.355330	+	0.615450i
$66$		0			0
$67$	−1.27284e6	+	2.20462e6i	−0.517026	+	0.895515i	0.482779	+	0.875742i	$0.339627\pi$
$67$	−1.27284e6	+	2.20462e6i	−0.517026	+	0.895515i	−0.999805	+	0.0197724i	$0.993706\pi$
$68$	−4.74677e6	+	8.22165e6i	−1.83070	+	3.17087i
$69$		0			0
$70$	−4.80831e6	−	8.32823e6i	−1.67552	−	2.90208i
$71$	−700407.			−0.232245			−0.116122	−	0.993235i	$-0.537047\pi$
$71$	−700407.			−0.232245			−0.116122	+	0.993235i	$0.537047\pi$
$72$		0			0
$73$	−4.69075e6			−1.41128			−0.705638	−	0.708572i	$-0.749340\pi$
$73$	−4.69075e6			−1.41128			−0.705638	+	0.708572i	$0.749340\pi$
$74$	406202.	+	703562.i	0.116528	+	0.201833i
$75$		0			0
$76$	4.95090e6	−	8.57522e6i	1.29371	−	2.24077i
$77$	−1.14332e6	+	1.98029e6i	−0.285397	+	0.494323i
$78$		0			0
$79$	2.49005e6	+	4.31290e6i	0.568217	+	0.984180i	0.996742	+	0.0806508i	$0.0256999\pi$
$79$	2.49005e6	+	4.31290e6i	0.568217	+	0.984180i	−0.428526	+	0.903530i	$0.640967\pi$
$80$	9.05950e6			1.97829
$81$		0			0
$82$	1.25494e6			0.251348
$83$	−1.69513e6	−	2.93606e6i	−0.325410	−	0.563626i	0.656185	−	0.754600i	$-0.272169\pi$
$83$	−1.69513e6	−	2.93606e6i	−0.325410	−	0.563626i	−0.981595	+	0.190973i	$0.938836\pi$
$84$		0			0
$85$	−8.54726e6	+	1.48043e7i	−1.50960	+	2.61470i
$86$	−1.49949e6	+	2.59719e6i	−0.254213	+	0.440311i
$87$		0			0
$88$	−2.97274e6	−	5.14894e6i	−0.465016	−	0.805432i
$89$	1.27747e6			0.192081			0.0960405	−	0.995377i	$-0.469382\pi$
$89$	1.27747e6			0.192081			0.0960405	+	0.995377i	$0.469382\pi$
$90$		0			0
$91$	3.41152e6			0.474574
$92$	9.31886e6	+	1.61407e7i	1.24769	+	2.16106i
$93$		0			0
$94$	−7.06663e6	+	1.22398e7i	−0.877536	+	1.51994i
$95$	8.91483e6	−	1.54409e7i	1.06679	−	1.84774i
$96$		0			0
$97$	−3.82854e6	−	6.63122e6i	−0.425924	−	0.737721i	0.570583	−	0.821240i	$-0.306717\pi$
$97$	−3.82854e6	−	6.63122e6i	−0.425924	−	0.737721i	−0.996506	+	0.0835189i	$0.973384\pi$
$98$	−4.56536e6			−0.489986
$99$		0			0
$100$	3.84474e7			3.84474
$101$	8.52074e6	+	1.47584e7i	0.822911	+	1.42532i	0.903506	+	0.428576i	$0.140985\pi$
$101$	8.52074e6	+	1.47584e7i	0.822911	+	1.42532i	−0.0805948	+	0.996747i	$0.525682\pi$
$102$		0			0
$103$	8.78417e6	−	1.52146e7i	0.792082	−	1.37193i	−0.132593	−	0.991171i	$-0.542330\pi$
$103$	8.78417e6	−	1.52146e7i	0.792082	−	1.37193i	0.924675	−	0.380756i	$-0.124336\pi$
$104$	−4.43515e6	+	7.68190e6i	−0.386626	+	0.669656i
$105$		0			0
$106$	4.84754e6	+	8.39619e6i	0.395322	+	0.684718i
$107$	−1.16418e7			−0.918710			−0.459355	−	0.888253i	$-0.651919\pi$
$107$	−1.16418e7			−0.918710			−0.459355	+	0.888253i	$0.651919\pi$
$108$		0			0
$109$	−2.29184e7			−1.69509			−0.847544	−	0.530725i	$-0.821920\pi$
$109$	−2.29184e7			−1.69509			−0.847544	+	0.530725i	$0.821920\pi$
$110$	−1.04274e7	−	1.80608e7i	−0.746970	−	1.29379i
$111$		0			0
$112$	9.82118e6	−	1.70108e7i	0.660542	−	1.14409i
$113$	−171204.	+	296534.i	−0.0111619	+	0.0193330i	−0.871552	−	0.490302i	$-0.836886\pi$
$113$	−171204.	+	296534.i	−0.0111619	+	0.0193330i	0.860390	+	0.509635i	$0.170220\pi$
$114$		0			0
$115$	1.67800e7	+	2.90638e7i	1.02884	+	1.78201i
$116$	−7.59049e6			−0.451510
$117$		0			0
$118$	2.64229e7			1.48045
$119$	1.85318e7	+	3.20980e7i	1.00810	+	1.74608i
$120$		0			0
$121$	7.26415e6	−	1.25819e7i	0.372766	−	0.645649i
$122$	−1.77586e7	+	3.07588e7i	−0.885420	+	1.53359i
$123$		0			0
$124$	−2.05554e6	−	3.56029e6i	−0.0968165	−	0.167691i
$125$	3.22298e7			1.47595
$126$		0			0
$127$	−1.04923e7			−0.454524			−0.227262	−	0.973834i	$-0.572977\pi$
$127$	−1.04923e7			−0.454524			−0.227262	+	0.973834i	$0.572977\pi$
$128$	−1.94072e7	−	3.36142e7i	−0.817951	−	1.41673i
$129$		0			0
$130$	−1.55571e7	+	2.69457e7i	−0.621050	+	1.07569i
$131$	3.32745e6	−	5.76331e6i	0.129319	−	0.223987i	−0.794094	−	0.607795i	$-0.792054\pi$
$131$	3.32745e6	−	5.76331e6i	0.129319	−	0.223987i	0.923413	+	0.383808i	$0.125388\pi$
$132$		0			0
$133$	−1.93287e7	−	3.34783e7i	−0.712397	−	1.23391i
$134$	−5.03388e7			−1.80733
$135$		0			0
$136$	−9.63688e7			−3.28511
$137$	−2.00068e7	−	3.46527e7i	−0.664744	−	1.15137i	−0.979355	−	0.202150i	$-0.935207\pi$
$137$	−2.00068e7	−	3.46527e7i	−0.664744	−	1.15137i	0.314610	−	0.949221i	$-0.398126\pi$
$138$		0			0
$139$	−9.41311e6	+	1.63040e7i	−0.297291	+	0.514922i	−0.975515	−	0.219932i	$-0.929416\pi$
$139$	−9.41311e6	+	1.63040e7i	−0.297291	+	0.514922i	0.678225	+	0.734855i	$0.262750\pi$
$140$	6.39560e7	−	1.10775e8i	1.96985	−	3.41188i
$141$		0			0
$142$	−6.92500e6	−	1.19945e7i	−0.202960	−	0.351537i
$143$	7.39833e6			0.211572
$144$		0			0
$145$	−1.36678e7			−0.372315
$146$	−4.63779e7	−	8.03290e7i	−1.23332	−	2.13618i
$147$		0			0
$148$	−5.40294e6	+	9.35817e6i	−0.136998	+	0.237288i
$149$	−1.29056e7	+	2.23532e7i	−0.319614	+	0.553589i	−0.980408	−	0.196980i	$-0.936887\pi$
$149$	−1.29056e7	+	2.23532e7i	−0.319614	+	0.553589i	0.660793	+	0.750568i	$0.270220\pi$
$150$		0			0
$151$	4.56751e6	+	7.91116e6i	0.107959	+	0.186991i	0.914943	−	0.403582i	$-0.132235\pi$
$151$	4.56751e6	+	7.91116e6i	0.107959	+	0.186991i	−0.806984	+	0.590573i	$0.798902\pi$
$152$	1.00513e8			2.32151
$153$		0			0
$154$	−4.52165e7			−0.997641
$155$	−3.70130e6	−	6.41083e6i	−0.0798349	−	0.138278i
$156$		0			0
$157$	1.65123e7	−	2.86002e7i	0.340533	−	0.589821i	−0.643999	−	0.765027i	$-0.722726\pi$
$157$	1.65123e7	−	2.86002e7i	0.340533	−	0.589821i	0.984532	+	0.175206i	$0.0560591\pi$
$158$	−4.92389e7	+	8.52842e7i	−0.993135	+	1.72016i
$159$		0			0
$160$	8.64537e6	+	1.49742e7i	0.166864	+	0.289018i
$161$	7.27631e7			1.37411
$162$		0			0
$163$	9.98787e7			1.80641			0.903205	−	0.429210i	$-0.141208\pi$
$163$	9.98787e7			1.80641			0.903205	+	0.429210i	$0.141208\pi$
$164$	8.34608e6	+	1.44558e7i	0.147751	+	0.255912i
$165$		0			0
$166$	3.35199e7	−	5.80582e7i	0.568754	−	0.985112i
$167$	6.13074e6	−	1.06188e7i	0.101860	−	0.176427i	−0.810591	−	0.585613i	$-0.800854\pi$
$167$	6.13074e6	−	1.06188e7i	0.101860	−	0.176427i	0.912451	+	0.409186i	$0.134187\pi$
$168$		0			0
$169$	2.58553e7	+	4.47828e7i	0.412047	+	0.713686i
$170$	−3.38031e8			−5.27698
$171$		0			0
$172$	−3.98898e7			−0.597740
$173$	1.20023e7	+	2.07886e7i	0.176240	+	0.305256i	0.940590	−	0.339546i	$-0.110273\pi$
$173$	1.20023e7	+	2.07886e7i	0.176240	+	0.305256i	−0.764350	+	0.644802i	$0.776940\pi$
$174$		0			0
$175$	7.50508e7	−	1.29992e8i	1.05858	−	1.83351i
$176$	2.12985e7	−	3.68901e7i	0.294479	−	0.510053i
$177$		0			0
$178$	1.26304e7	+	2.18766e7i	0.167861	+	0.290743i
$179$	1.35827e8			1.77011			0.885054	−	0.465488i	$-0.154121\pi$
$179$	1.35827e8			1.77011			0.885054	+	0.465488i	$0.154121\pi$
$180$		0			0
$181$	−7.74602e7			−0.970965			−0.485482	−	0.874246i	$-0.661356\pi$
$181$	−7.74602e7			−0.970965			−0.485482	+	0.874246i	$0.661356\pi$
$182$	3.37301e7	+	5.84223e7i	0.414732	+	0.718337i
$183$		0			0
$184$	−9.45956e7	+	1.63844e8i	−1.11946	+	1.93896i
$185$	−9.72879e6	+	1.68508e7i	−0.112969	+	0.195667i
$186$		0			0
$187$	4.01885e7	+	6.96085e7i	0.449424	+	0.778425i
$188$	−1.87988e8			−2.06338
$189$		0			0
$190$	3.52568e8			3.72911
$191$	1.89245e7	+	3.27782e7i	0.196520	+	0.340383i	0.947398	−	0.320058i	$-0.103702\pi$
$191$	1.89245e7	+	3.27782e7i	0.196520	+	0.340383i	−0.750877	+	0.660442i	$0.770369\pi$
$192$		0			0
$193$	2.15559e7	−	3.73358e7i	0.215831	−	0.373831i	−0.737698	−	0.675131i	$-0.764087\pi$
$193$	2.15559e7	−	3.73358e7i	0.215831	−	0.373831i	0.953529	+	0.301300i	$0.0974205\pi$
$194$	7.57063e7	−	1.31127e8i	0.744434	−	1.28940i
$195$		0			0
$196$	−3.03622e7	−	5.25889e7i	−0.288030	−	0.498882i
$197$	1.35255e8			1.26044			0.630221	−	0.776416i	$-0.282964\pi$
$197$	1.35255e8			1.26044			0.630221	+	0.776416i	$0.282964\pi$
$198$		0			0
$199$	9.83246e7			0.884456			0.442228	−	0.896903i	$-0.354188\pi$
$199$	9.83246e7			0.884456			0.442228	+	0.896903i	$0.354188\pi$
$200$	1.95139e8	+	3.37992e8i	1.72481	+	2.98745i
$201$		0			0
$202$	−1.68491e8	+	2.91835e8i	−1.43829	+	2.49119i
$203$	−1.48169e7	+	2.56637e7i	−0.124315	+	0.215319i
$204$		0			0
$205$	1.50283e7	+	2.60299e7i	0.121835	+	0.211025i
$206$	3.47400e8			2.76882
$207$		0			0
$208$	−6.35520e7			−0.489675
$209$	−4.19168e7	−	7.26020e7i	−0.317597	−	0.550094i
$210$		0			0
$211$	3.41390e7	−	5.91305e7i	0.250186	−	0.433335i	−0.713391	−	0.700766i	$-0.752842\pi$
$211$	3.41390e7	−	5.91305e7i	0.250186	−	0.433335i	0.963577	+	0.267432i	$0.0861750\pi$
$212$	−6.44778e7	+	1.11679e8i	−0.464767	+	0.804999i
$213$		0			0
$214$	−1.15104e8	−	1.99366e8i	−0.802866	−	1.39060i
$215$	−7.18275e7			−0.492897
$216$		0			0
$217$	−1.60499e7			−0.106626
$218$	−2.26597e8	−	3.92478e8i	−1.48135	−	2.56577i
$219$		0			0
$220$	1.38697e8	−	2.40230e8i	0.878186	−	1.52106i
$221$	5.99588e7	−	1.03852e8i	0.373663	−	0.647203i
$222$		0			0
$223$	−1.45930e8	−	2.52758e8i	−0.881207	−	1.52630i	−0.850000	−	0.526782i	$-0.823398\pi$
$223$	−1.45930e8	−	2.52758e8i	−0.881207	−	1.52630i	−0.0312065	−	0.999513i	$-0.509935\pi$
$224$	3.74889e7			0.222862
$225$		0			0
$226$	−6.77085e6			−0.0390179
$227$	1.19293e8	+	2.06622e8i	0.676901	+	1.17243i	0.975909	+	0.218176i	$0.0700108\pi$
$227$	1.19293e8	+	2.06622e8i	0.676901	+	1.17243i	−0.299009	+	0.954250i	$0.596656\pi$
$228$		0			0
$229$	7.65152e7	−	1.32528e8i	0.421040	−	0.729263i	−0.575001	−	0.818152i	$-0.694999\pi$
$229$	7.65152e7	−	1.32528e8i	0.421040	−	0.729263i	0.996041	+	0.0888897i	$0.0283319\pi$
$230$	−3.31811e8	+	5.74713e8i	−1.79822	+	3.11461i
$231$		0			0
$232$	−3.85255e7	−	6.67281e7i	−0.202554	−	0.350833i
$233$	−2.54013e8			−1.31556			−0.657779	−	0.753211i	$-0.728504\pi$
$233$	−2.54013e8			−1.31556			−0.657779	+	0.753211i	$0.728504\pi$
$234$		0			0
$235$	−3.38501e8			−1.70146
$236$	1.75727e8	+	3.04368e8i	0.870255	+	1.50733i
$237$		0			0
$238$	−3.66451e8	+	6.34712e8i	−1.76196	+	3.05181i
$239$	−1.58069e8	+	2.73784e8i	−0.748953	+	1.29722i	0.199372	+	0.979924i	$0.436110\pi$
$239$	−1.58069e8	+	2.73784e8i	−0.748953	+	1.29722i	−0.948325	+	0.317300i	$0.897224\pi$
$240$		0			0
$241$	−1.07767e7	−	1.86659e7i	−0.0495939	−	0.0858991i	0.840163	−	0.542334i	$-0.182459\pi$
$241$	−1.07767e7	−	1.86659e7i	−0.0495939	−	0.0858991i	−0.889757	+	0.456435i	$0.849126\pi$
$242$	2.87286e8			1.30305
$243$		0			0
$244$	−4.72419e8			−2.08192
$245$	−5.46717e7	−	9.46941e7i	−0.237510	−	0.411379i
$246$		0			0
$247$	−6.25372e7	+	1.08318e8i	−0.264058	+	0.457362i
$248$	2.08657e7	−	3.61405e7i	0.0868665	−	0.150457i
$249$		0			0
$250$	3.18660e8	+	5.51935e8i	1.28984	+	2.23408i
$251$	1.38647e8			0.553419			0.276709	−	0.960954i	$-0.410756\pi$
$251$	1.38647e8			0.553419			0.276709	+	0.960954i	$0.410756\pi$
$252$		0			0
$253$	1.57796e8			0.612596
$254$	−1.03738e8	−	1.79680e8i	−0.397211	−	0.687989i
$255$		0			0
$256$	2.73262e8	−	4.73304e8i	1.01798	−	1.76319i
$257$	2.27138e8	−	3.93415e8i	0.834688	−	1.44572i	−0.0595957	−	0.998223i	$-0.518981\pi$
$257$	2.27138e8	−	3.93415e8i	0.834688	−	1.44572i	0.894284	−	0.447500i	$-0.147686\pi$
$258$		0			0
$259$	2.10935e7	+	3.65350e7i	0.0754396	+	0.130665i
$260$	−4.13854e8			−1.46029
$261$		0			0
$262$	1.31595e8			0.452050
$263$	−7.40745e7	−	1.28301e8i	−0.251087	−	0.434895i	0.712739	−	0.701430i	$-0.247455\pi$
$263$	−7.40745e7	−	1.28301e8i	−0.251087	−	0.434895i	−0.963825	+	0.266535i	$0.914121\pi$
$264$		0			0
$265$	−1.16102e8	+	2.01094e8i	−0.383247	+	0.663803i
$266$	3.82210e8	−	6.62007e8i	1.24513	−	2.15664i
$267$		0			0
$268$	−3.34782e8	−	5.79859e8i	−1.06241	−	1.84014i
$269$	2.17316e8			0.680703			0.340352	−	0.940298i	$-0.389454\pi$
$269$	2.17316e8			0.680703			0.340352	+	0.940298i	$0.389454\pi$
$270$		0			0
$271$	−3.83874e8			−1.17165			−0.585823	−	0.810439i	$-0.699229\pi$
$271$	−3.83874e8			−1.17165			−0.585823	+	0.810439i	$0.699229\pi$
$272$	−3.45222e8	−	5.97941e8i	−1.04018	−	1.80164i
$273$		0			0
$274$	3.95618e8	−	6.85230e8i	1.16185	−	2.01238i
$275$	1.62757e8	−	2.81904e8i	0.471929	−	0.817404i
$276$		0			0
$277$	2.08325e8	+	3.60830e8i	0.588928	+	1.02005i	0.994373	+	0.105934i	$0.0337833\pi$
$277$	2.08325e8	+	3.60830e8i	0.588928	+	1.02005i	−0.405445	+	0.914120i	$0.632883\pi$
$278$	−3.72274e8			−1.03921
$279$		0			0
$280$	1.29843e9			3.53481
$281$	3.17153e8	+	5.49325e8i	0.852700	+	1.47692i	0.878762	+	0.477259i	$0.158370\pi$
$281$	3.17153e8	+	5.49325e8i	0.852700	+	1.47692i	−0.0260625	+	0.999660i	$0.508297\pi$
$282$		0			0
$283$	−3.54403e8	+	6.13844e8i	−0.929490	+	1.60992i	−0.145314	+	0.989386i	$0.546419\pi$
$283$	−3.54403e8	+	6.13844e8i	−0.929490	+	1.60992i	−0.784176	+	0.620538i	$0.786914\pi$
$284$	9.21104e7	−	1.59540e8i	0.238613	−	0.413290i
$285$		0			0
$286$	7.31481e7	+	1.26696e8i	0.184894	+	0.320245i
$287$	6.51675e7			0.162721
$288$		0			0
$289$	8.92471e8			2.17496
$290$	−1.35135e8	−	2.34061e8i	−0.325368	−	0.563554i
$291$		0			0
$292$	6.16879e8	−	1.06847e9i	1.44997	−	2.51143i
$293$	−1.82298e7	+	3.15750e7i	−0.0423395	+	0.0733342i	−0.886419	−	0.462885i	$-0.846814\pi$
$293$	−1.82298e7	+	3.15750e7i	−0.0423395	+	0.0733342i	0.844079	+	0.536219i	$0.180148\pi$
$294$		0			0
$295$	3.16422e8	+	5.48060e8i	0.717613	+	1.24294i
$296$	−1.09690e8			−0.245837
$297$		0			0
$298$	−5.10396e8			−1.11725
$299$	−1.17711e8	−	2.03881e8i	−0.254664	−	0.441091i
$300$		0			0
$301$	−7.78664e7	+	1.34869e8i	−0.164576	+	0.285055i
$302$	−9.03189e7	+	1.56437e8i	−0.188692	+	0.326825i
$303$		0			0
$304$	3.60067e8	+	6.23655e8i	0.735066	+	1.27317i
$305$	−8.50661e8			−1.71675
$306$		0			0
$307$	−7.16485e8			−1.41326			−0.706631	−	0.707583i	$-0.749786\pi$
$307$	−7.16485e8			−1.41326			−0.706631	+	0.707583i	$0.749786\pi$
$308$	−3.00715e8	−	5.20854e8i	−0.586446	−	1.01575i
$309$		0			0
$310$	7.31902e7	−	1.26769e8i	0.139536	−	0.241684i
$311$	−1.31464e8	+	2.27703e8i	−0.247826	+	0.429247i	−0.962922	−	0.269779i	$-0.913049\pi$
$311$	−1.31464e8	+	2.27703e8i	−0.247826	+	0.429247i	0.715096	+	0.699026i	$0.246383\pi$
$312$		0			0
$313$	−1.49740e8	−	2.59357e8i	−0.276015	−	0.478071i	0.694376	−	0.719612i	$-0.255681\pi$
$313$	−1.49740e8	−	2.59357e8i	−0.276015	−	0.478071i	−0.970391	+	0.241541i	$0.922347\pi$
$314$	6.53037e8			1.19038
$315$		0			0
$316$	−1.30987e9			−2.33519
$317$	8.87589e7	+	1.53735e8i	0.156497	+	0.271060i	0.933603	−	0.358309i	$-0.116647\pi$
$317$	8.87589e7	+	1.53735e8i	0.156497	+	0.271060i	−0.777106	+	0.629369i	$0.783313\pi$
$318$		0			0
$319$	−3.21324e7	+	5.56550e7i	−0.0554212	+	0.0959924i
$320$	4.08852e8	−	7.08153e8i	0.697496	−	1.20810i
$321$		0			0
$322$	7.19416e8	+	1.24607e9i	1.20084	+	2.07991i
$323$	−1.35884e9			−2.24367
$324$		0			0
$325$	−4.85648e8			−0.784747
$326$	9.87511e8	+	1.71042e9i	1.57863	+	2.73427i
$327$		0			0
$328$	−8.47209e7	+	1.46741e8i	−0.132566	+	0.229611i
$329$	−3.66961e8	+	6.35594e8i	−0.568112	+	0.983999i
$330$		0			0
$331$	−4.09460e8	−	7.09205e8i	−0.620602	−	1.07491i	−0.989374	−	0.145394i	$-0.953555\pi$
$331$	−4.09460e8	−	7.09205e8i	−0.620602	−	1.07491i	0.368772	−	0.929520i	$-0.379778\pi$
$332$	8.91707e8			1.33733
$333$		0			0
$334$	2.42461e8			0.356065
$335$	−6.02824e8	−	1.04412e9i	−0.876060	−	1.51738i
$336$		0			0
$337$	−5.67529e8	+	9.82990e8i	−0.807763	+	1.39909i	0.106647	+	0.994297i	$0.465988\pi$
$337$	−5.67529e8	+	9.82990e8i	−0.807763	+	1.39909i	−0.914410	+	0.404789i	$0.867345\pi$
$338$	−5.11269e8	+	8.85544e8i	−0.720180	+	1.24739i
$339$		0			0
$340$	−2.24810e9	−	3.89382e9i	−3.10198	−	5.37279i
$341$	−3.48064e7			−0.0475355
$342$		0			0
$343$	6.08581e8			0.814308
$344$	−2.02460e8	−	3.50671e8i	−0.268155	−	0.464457i
$345$		0			0
$346$	−2.37336e8	+	4.11078e8i	−0.308033	+	0.533529i
$347$	−2.61567e8	+	4.53047e8i	−0.336070	+	0.582090i	−0.983690	−	0.179874i	$-0.942431\pi$
$347$	−2.61567e8	+	4.53047e8i	−0.336070	+	0.582090i	0.647620	+	0.761964i	$0.275764\pi$
$348$		0			0
$349$	−1.11792e8	−	1.93630e8i	−0.140774	−	0.243828i	0.787014	−	0.616935i	$-0.211626\pi$
$349$	−1.11792e8	−	1.93630e8i	−0.140774	−	0.243828i	−0.927788	+	0.373107i	$0.878292\pi$
$350$	2.96814e9			3.70038
$351$		0			0
$352$	8.12996e7			0.0993548
$353$	−1.92737e8	−	3.33830e8i	−0.233214	−	0.403938i	0.725538	−	0.688182i	$-0.241591\pi$
$353$	−1.92737e8	−	3.33830e8i	−0.233214	−	0.403938i	−0.958752	+	0.284244i	$0.908257\pi$
$354$		0			0
$355$	1.65858e8	−	2.87275e8i	0.196761	−	0.340799i
$356$	−1.67999e8	+	2.90983e8i	−0.197348	+	0.341816i
$357$		0			0
$358$	1.34293e9	+	2.32603e9i	1.54691	+	2.67932i
$359$	1.05112e9			1.19901			0.599504	−	0.800372i	$-0.295365\pi$
$359$	1.05112e9			1.19901			0.599504	+	0.800372i	$0.295365\pi$
$360$		0			0
$361$	5.23400e8			0.585542
$362$	−7.65857e8	−	1.32650e9i	−0.848531	−	1.46970i
$363$		0			0
$364$	−4.48649e8	+	7.77083e8i	−0.487587	+	0.844525i
$365$	1.11078e9	−	1.92393e9i	1.19565	−	2.07093i
$366$		0			0
$367$	−9.67958e7	−	1.67655e8i	−0.102218	−	0.177046i	0.810380	−	0.585904i	$-0.199260\pi$
$367$	−9.67958e7	−	1.67655e8i	−0.102218	−	0.177046i	−0.912598	+	0.408858i	$0.865927\pi$
$368$	−1.35548e9			−1.41783
$369$		0			0
$370$	−3.84758e8			−0.394895
$371$	2.51726e8	+	4.36003e8i	0.255929	+	0.443282i
$372$		0			0
$373$	1.76946e8	−	3.06480e8i	0.176547	−	0.305788i	−0.764149	−	0.645040i	$-0.776841\pi$
$373$	1.76946e8	−	3.06480e8i	0.176547	−	0.305788i	0.940696	+	0.339252i	$0.110174\pi$
$374$	−7.94696e8	+	1.37645e9i	−0.785508	+	1.36054i
$375$		0			0
$376$	−9.54133e8	−	1.65261e9i	−0.925660	−	1.60329i
$377$	9.58792e7			0.0921572
$378$		0			0
$379$	−4.69956e8			−0.443425			−0.221712	−	0.975112i	$-0.571165\pi$
$379$	−4.69956e8			−0.443425			−0.221712	+	0.975112i	$0.571165\pi$
$380$	2.34478e9	+	4.06127e9i	2.19209	+	3.79681i
$381$		0			0
$382$	−3.74217e8	+	6.48163e8i	−0.343480	+	0.594925i
$383$	−7.65447e8	+	1.32579e9i	−0.696177	+	1.20581i	0.273606	+	0.961842i	$0.411784\pi$
$383$	−7.65447e8	+	1.32579e9i	−0.696177	+	1.20581i	−0.969782	+	0.243971i	$0.921550\pi$
$384$		0			0
$385$	−5.41482e8	−	9.37875e8i	−0.483584	−	0.837592i
$386$	8.52500e8			0.754465
$387$		0			0
$388$	2.01396e9			1.75041
$389$	4.16725e8	+	7.21789e8i	0.358943	+	0.621708i	0.987785	−	0.155826i	$-0.0498038\pi$
$389$	4.16725e8	+	7.21789e8i	0.358943	+	0.621708i	−0.628841	+	0.777534i	$0.716470\pi$
$390$		0			0
$391$	1.27884e9	−	2.21501e9i	1.08192	−	1.87395i
$392$	3.08206e8	−	5.33829e8i	0.258429	−	0.447611i
$393$		0			0
$394$	1.33728e9	+	2.31624e9i	1.10151	+	1.90786i
$395$	−2.35861e9			−1.92560
$396$		0			0
$397$	8.58150e8			0.688330			0.344165	−	0.938909i	$-0.388162\pi$
$397$	8.58150e8			0.688330			0.344165	+	0.938909i	$0.388162\pi$
$398$	9.72146e8	+	1.68381e9i	0.772931	+	1.33876i
$399$		0			0
$400$	−1.39809e9	+	2.42157e9i	−1.09226	+	1.89185i
$401$	−1.03584e8	+	1.79412e8i	−0.0802205	+	0.138946i	−0.903345	−	0.428916i	$-0.858896\pi$
$401$	−1.03584e8	+	1.79412e8i	−0.0802205	+	0.138946i	0.823124	+	0.567862i	$0.192229\pi$
$402$		0			0
$403$	2.59645e7	+	4.49718e7i	0.0197611	+	0.0342273i
$404$	−4.48225e9			−3.38190
$405$		0			0
$406$	−5.85986e8			−0.434557
$407$	4.57440e7	+	7.92309e7i	0.0336320	+	0.0582524i
$408$		0			0
$409$	2.23582e8	−	3.87255e8i	0.161586	−	0.279876i	−0.773851	−	0.633367i	$-0.781672\pi$
$409$	2.23582e8	−	3.87255e8i	0.161586	−	0.279876i	0.935438	+	0.353491i	$0.115006\pi$
$410$	−2.97174e8	+	5.14720e8i	−0.212945	+	0.368831i
$411$		0			0
$412$	2.31041e9	+	4.00174e9i	1.62760	+	2.81909i
$413$	1.37210e9			0.958433
$414$		0			0
$415$	1.60565e9			1.10276
$416$	−6.06470e7	−	1.05044e8i	−0.0413031	−	0.0715390i
$417$		0			0
$418$	8.28871e8	−	1.43565e9i	0.555099	−	0.961459i
$419$	4.02245e8	−	6.96708e8i	0.267141	−	0.462702i	−0.700981	−	0.713180i	$-0.747254\pi$
$419$	4.02245e8	−	6.96708e8i	0.267141	−	0.462702i	0.968122	+	0.250477i	$0.0805876\pi$
$420$		0			0
$421$	−5.98127e8	−	1.03599e9i	−0.390666	−	0.676654i	0.601871	−	0.798593i	$-0.294422\pi$
$421$	−5.98127e8	−	1.03599e9i	−0.390666	−	0.676654i	−0.992538	+	0.121939i	$0.961089\pi$
$422$	1.35015e9			0.874555
$423$		0			0
$424$	−1.30903e9			−0.834003
$425$	−2.63809e9	−	4.56931e9i	−1.66697	−	2.88728i
$426$		0			0
$427$	−9.22181e8	+	1.59726e9i	−0.573216	+	0.992839i
$428$	1.53102e9	−	2.65180e9i	0.943901	−	1.63489i
$429$		0			0
$430$	−7.10166e8	−	1.23004e9i	−0.430745	−	0.746072i
$431$	−8.02879e8			−0.483036			−0.241518	−	0.970396i	$-0.577645\pi$
$431$	−8.02879e8			−0.483036			−0.241518	+	0.970396i	$0.577645\pi$
$432$		0			0
$433$	2.54405e9			1.50597			0.752987	−	0.658035i	$-0.228612\pi$
$433$	2.54405e9			1.50597			0.752987	+	0.658035i	$0.228612\pi$
$434$	−1.58687e8	−	2.74855e8i	−0.0931813	−	0.161395i
$435$		0			0
$436$	3.01400e9	−	5.22040e9i	1.74157	−	3.01648i
$437$	−1.33383e9	+	2.31027e9i	−0.764568	+	1.32427i
$438$		0			0
$439$	4.00339e8	+	6.93408e8i	0.225841	+	0.391168i	0.956571	−	0.291498i	$-0.0941537\pi$
$439$	4.00339e8	+	6.93408e8i	0.225841	+	0.391168i	−0.730731	+	0.682666i	$0.760820\pi$
$440$	2.81581e9			1.57587
$441$		0			0
$442$	2.37127e9			1.30618
$443$	5.39360e8	+	9.34200e8i	0.294758	+	0.510536i	0.974929	−	0.222518i	$-0.0714275\pi$
$443$	5.39360e8	+	9.34200e8i	0.294758	+	0.510536i	−0.680170	+	0.733054i	$0.738094\pi$
$444$		0			0
$445$	−3.02507e8	+	5.23958e8i	−0.162733	+	0.281862i
$446$	2.88565e9	−	4.99810e9i	1.54018	−	2.66767i
$447$		0			0
$448$	−8.86454e8	−	1.53538e9i	−0.465782	−	0.806759i
$449$	1.67071e9			0.871042			0.435521	−	0.900179i	$-0.356564\pi$
$449$	1.67071e9			0.871042			0.435521	+	0.900179i	$0.356564\pi$
$450$		0			0
$451$	1.41324e8			0.0725434
$452$	−4.50300e7	−	7.79942e7i	−0.0229360	−	0.0397263i
$453$		0			0
$454$	−2.35893e9	+	4.08578e9i	−1.18309	+	2.04918i
$455$	−8.07858e8	+	1.39925e9i	−0.402064	+	0.696395i
$456$		0			0
$457$	−9.05761e8	−	1.56882e9i	−0.443922	−	0.768896i	0.554054	−	0.832481i	$-0.313080\pi$
$457$	−9.05761e8	−	1.56882e9i	−0.443922	−	0.768896i	−0.997976	+	0.0635848i	$0.979747\pi$
$458$	3.02605e9			1.47180
$459$		0			0
$460$	−8.82693e9			−4.22822
$461$	−4.54057e8	−	7.86449e8i	−0.215852	−	0.373867i	0.737684	−	0.675147i	$-0.235920\pi$
$461$	−4.54057e8	−	7.86449e8i	−0.215852	−	0.373867i	−0.953536	+	0.301279i	$0.902586\pi$
$462$		0			0
$463$	1.48226e9	−	2.56735e9i	0.694051	−	1.20213i	−0.276449	−	0.961029i	$-0.589158\pi$
$463$	1.48226e9	−	2.56735e9i	0.694051	−	1.20213i	0.970500	−	0.241102i	$-0.0775089\pi$
$464$	2.76019e8	−	4.78079e8i	0.128270	−	0.222171i
$465$		0			0
$466$	−2.51145e9	−	4.34997e9i	−1.14967	−	1.99129i
$467$	−1.15258e9			−0.523675			−0.261837	−	0.965112i	$-0.584328\pi$
$467$	−1.15258e9			−0.523675			−0.261837	+	0.965112i	$0.584328\pi$
$468$		0			0
$469$	−2.61403e9			−1.17005
$470$	−3.34679e9	−	5.79682e9i	−1.48692	−	2.57541i
$471$		0			0
$472$	−1.78380e9	+	3.08963e9i	−0.780818	+	1.35242i
$473$	−1.68863e8	+	2.92480e8i	−0.0733705	+	0.127081i
$474$		0			0
$475$	2.75154e9	+	4.76581e9i	1.17801	+	2.04037i
$476$	−9.74843e9			−4.14296
$477$		0			0
$478$	−6.25139e9			−2.61805
$479$	2.83332e8	+	4.90745e8i	0.117793	+	0.204024i	0.918893	−	0.394507i	$-0.129085\pi$
$479$	2.83332e8	+	4.90745e8i	0.117793	+	0.204024i	−0.801100	+	0.598531i	$0.795751\pi$
$480$		0			0
$481$	6.82471e7	−	1.18208e8i	0.0279625	−	0.0484326i
$482$	2.13102e8	−	3.69103e8i	0.0866807	−	0.150135i
$483$		0			0
$484$	1.91061e9	+	3.30928e9i	0.765974	+	1.32671i
$485$	3.62643e9			1.44339
$486$		0			0
$487$	3.19934e9			1.25519			0.627595	−	0.778540i	$-0.284040\pi$
$487$	3.19934e9			1.25519			0.627595	+	0.778540i	$0.284040\pi$
$488$	−2.39776e9	−	4.15304e9i	−0.933977	−	1.61770i
$489$		0			0
$490$	1.08109e9	−	1.87250e9i	0.415122	−	0.719012i
$491$	−1.06617e9	+	1.84667e9i	−0.406483	+	0.704049i	−0.994493	−	0.104804i	$-0.966578\pi$
$491$	−1.06617e9	+	1.84667e9i	−0.406483	+	0.704049i	0.588010	+	0.808854i	$0.299912\pi$
$492$		0			0
$493$	5.20826e8	+	9.02097e8i	0.195762	+	0.339070i
$494$	−2.47325e9			−0.923047
$495$		0			0
$496$	2.98989e8			0.110019
$497$	−3.59606e8	−	6.22856e8i	−0.131395	−	0.227583i
$498$		0			0
$499$	−5.21456e8	+	9.03188e8i	−0.187874	+	0.325407i	−0.944541	−	0.328393i	$-0.893493\pi$
$499$	−5.21456e8	+	9.03188e8i	−0.187874	+	0.325407i	0.756667	+	0.653800i	$0.226826\pi$
$500$	−4.23854e9	+	7.34136e9i	−1.51643	+	2.62653i
$501$		0			0
$502$	1.37082e9	+	2.37433e9i	0.483635	+	0.837681i
$503$	2.59463e9			0.909051			0.454526	−	0.890734i	$-0.349809\pi$
$503$	2.59463e9			0.909051			0.454526	+	0.890734i	$0.349809\pi$
$504$		0			0
$505$	−8.07094e9			−2.78872
$506$	1.56015e9	+	2.70225e9i	0.535351	+	0.927255i
$507$		0			0
$508$	1.37984e9	−	2.38995e9i	0.466987	−	0.808845i
$509$	2.52142e9	−	4.36723e9i	0.847488	−	1.46789i	−0.0359549	−	0.999353i	$-0.511447\pi$
$509$	2.52142e9	−	4.36723e9i	0.847488	−	1.46789i	0.883443	−	0.468539i	$-0.155219\pi$
$510$		0			0
$511$	−2.40834e9	−	4.17137e9i	−0.798446	−	1.38295i
$512$	5.83885e9			1.92257
$513$		0			0
$514$	8.98296e9			2.91775
$515$	4.16023e9	+	7.20573e9i	1.34212	+	2.32462i
$516$		0			0
$517$	−7.95801e8	+	1.37837e9i	−0.253272	+	0.438681i
$518$	−4.17107e8	+	7.22451e8i	−0.131854	+	0.228378i
$519$		0			0
$520$	−2.10051e9	−	3.63819e9i	−0.655108	−	1.13468i
$521$	−3.81880e9			−1.18303			−0.591514	−	0.806295i	$-0.701469\pi$
$521$	−3.81880e9			−1.18303			−0.591514	+	0.806295i	$0.701469\pi$
$522$		0			0
$523$	−1.81806e9			−0.555715			−0.277857	−	0.960622i	$-0.589624\pi$
$523$	−1.81806e9			−0.555715			−0.277857	+	0.960622i	$0.589624\pi$
$524$	8.75185e8	+	1.51586e9i	0.265730	+	0.460257i
$525$		0			0
$526$	1.46477e9	−	2.53705e9i	0.438852	−	0.760114i
$527$	−2.82083e8	+	4.88583e8i	−0.0839538	+	0.145412i
$528$		0			0
$529$	−8.08199e8	−	1.39984e9i	−0.237369	−	0.411135i
$530$	−4.59164e9			−1.33968
$531$		0			0
$532$	1.01677e10			2.92772
$533$	−1.05423e8	−	1.82599e8i	−0.0301572	−	0.0522339i
$534$		0			0
$535$	2.75682e9	−	4.77495e9i	0.778342	−	1.34813i
$536$	3.39836e9	−	5.88614e9i	0.953220	−	1.65103i
$537$		0			0
$538$	2.14862e9	+	3.72152e9i	0.594870	+	1.03034i
$539$	−5.14123e8			−0.141419
$540$		0			0
$541$	3.76234e9			1.02157			0.510784	−	0.859709i	$-0.329355\pi$
$541$	3.76234e9			1.02157			0.510784	+	0.859709i	$0.329355\pi$
$542$	−3.79541e9	−	6.57384e9i	−1.02391	−	1.77346i
$543$		0			0
$544$	6.58882e8	−	1.14122e9i	0.175473	−	0.303929i
$545$	5.42715e9	−	9.40010e9i	1.43610	−	2.48739i
$546$		0			0
$547$	2.99728e9	+	5.19144e9i	0.783018	+	1.35623i	0.930176	+	0.367114i	$0.119654\pi$
$547$	2.99728e9	+	5.19144e9i	0.783018	+	1.35623i	−0.147158	+	0.989113i	$0.547013\pi$
$548$	1.05243e10			2.73189
$549$		0			0
$550$	6.43680e9			1.64968
$551$	−5.43223e8	−	9.40891e8i	−0.138340	−	0.239612i
$552$		0			0
$553$	−2.55691e9	+	4.42870e9i	−0.642950	+	1.11362i
$554$	−4.11947e9	+	7.13512e9i	−1.02933	+	1.78286i
$555$		0			0
$556$	−2.47583e9	−	4.28826e9i	−0.610885	−	1.05808i
$557$	4.74153e9			1.16259			0.581294	−	0.813694i	$-0.302547\pi$
$557$	4.74153e9			1.16259			0.581294	+	0.813694i	$0.302547\pi$
$558$		0			0
$559$	5.03867e8			0.122004
$560$	4.65136e9	+	8.05640e9i	1.11924	+	1.93858i
$561$		0			0
$562$	−6.27144e9	+	1.08625e10i	−1.49036	+	2.58138i
$563$	1.66572e7	−	2.88510e7i	0.00393388	−	0.00681368i	−0.864052	−	0.503403i	$-0.832081\pi$
$563$	1.66572e7	−	2.88510e7i	0.00393388	−	0.00681368i	0.867986	+	0.496589i	$0.165414\pi$
$564$		0			0
$565$	−8.10831e7	−	1.40440e8i	−0.0189130	−	0.0327583i
$566$	−1.40161e10			−3.24914
$567$		0			0
$568$	1.87002e9			0.428181
$569$	1.85350e9	+	3.21036e9i	0.421794	+	0.730569i	0.996115	−	0.0880612i	$-0.0280671\pi$
$569$	1.85350e9	+	3.21036e9i	0.421794	+	0.730569i	−0.574321	+	0.818630i	$0.694734\pi$
$570$		0			0
$571$	−3.41860e9	+	5.92119e9i	−0.768462	+	1.33102i	0.169935	+	0.985455i	$0.445644\pi$
$571$	−3.41860e9	+	5.92119e9i	−0.768462	+	1.33102i	−0.938397	+	0.345560i	$0.887689\pi$
$572$	−9.72953e8	+	1.68520e9i	−0.217373	+	0.376501i
$573$		0			0
$574$	6.44318e8	+	1.11599e9i	0.142203	+	0.246303i
$575$	−1.03582e10			−2.27220
$576$		0			0
$577$	4.39006e9			0.951382			0.475691	−	0.879612i	$-0.342198\pi$
$577$	4.39006e9			0.951382			0.475691	+	0.879612i	$0.342198\pi$
$578$	8.82396e9	+	1.52835e10i	1.90071	+	3.29213i
$579$		0			0
$580$	1.79745e9	−	3.11327e9i	0.382524	−	0.662551i
$581$	1.74064e9	−	3.01489e9i	0.368208	−	0.637756i
$582$		0			0
$583$	5.45901e8	+	9.45528e8i	0.114097	+	0.197622i
$584$	1.25239e10			2.60192
$585$		0			0
$586$	−7.20961e8			−0.148003
$587$	−4.09956e8	−	7.10064e8i	−0.0836573	−	0.144899i	0.821161	−	0.570697i	$-0.193327\pi$
$587$	−4.09956e8	−	7.10064e8i	−0.0836573	−	0.144899i	−0.904818	+	0.425798i	$0.859993\pi$
$588$		0			0
$589$	2.94214e8	−	5.09594e8i	0.0593281	−	0.102759i
$590$	−6.25701e9	+	1.08375e10i	−1.25425	+	2.17243i
$591$		0			0
$592$	−3.92943e8	−	6.80597e8i	−0.0778402	−	0.134823i
$593$	4.33599e9			0.853880			0.426940	−	0.904280i	$-0.359592\pi$
$593$	4.33599e9			0.853880			0.426940	+	0.904280i	$0.359592\pi$
$594$		0			0
$595$	−1.75535e10			−3.41628
$596$	−3.39443e9	−	5.87932e9i	−0.656757	−	1.13754i
$597$		0			0
$598$	2.32764e9	−	4.03159e9i	0.445105	−	0.770944i
$599$	−4.96206e9	+	8.59454e9i	−0.943339	+	1.63391i	−0.184297	+	0.982871i	$0.559001\pi$
$599$	−4.96206e9	+	8.59454e9i	−0.943339	+	1.63391i	−0.759042	+	0.651041i	$0.774332\pi$
$600$		0			0
$601$	−9.72471e8	−	1.68437e9i	−0.182733	−	0.316502i	0.760078	−	0.649832i	$-0.225161\pi$
$601$	−9.72471e8	−	1.68437e9i	−0.182733	−	0.316502i	−0.942810	+	0.333330i	$0.891828\pi$
$602$	−3.07949e9			−0.575296
$603$		0			0
$604$	−2.40269e9			−0.443678
$605$	3.44034e9	+	5.95884e9i	0.631623	+	1.09400i
$606$		0			0
$607$	3.99796e9	−	6.92467e9i	0.725568	−	1.25672i	−0.233172	−	0.972435i	$-0.574911\pi$
$607$	3.99796e9	−	6.92467e9i	0.725568	−	1.25672i	0.958740	−	0.284285i	$-0.0917561\pi$
$608$	−6.87216e8	+	1.19029e9i	−0.124003	+	0.214779i
$609$		0			0
$610$	−8.41057e9	−	1.45675e10i	−1.50028	−	2.59855i
$611$	2.37457e9			0.421154
$612$		0			0
$613$	5.19028e8			0.0910079			0.0455040	−	0.998964i	$-0.485511\pi$
$613$	5.19028e8			0.0910079			0.0455040	+	0.998964i	$0.485511\pi$
$614$	−7.08396e9	−	1.22698e10i	−1.23506	−	2.13918i
$615$		0			0
$616$	3.05256e9	−	5.28718e9i	0.526176	−	0.911364i
$617$	−1.16882e9	+	2.02445e9i	−0.200331	+	0.346984i	−0.948635	−	0.316372i	$-0.897535\pi$
$617$	−1.16882e9	+	2.02445e9i	−0.200331	+	0.346984i	0.748304	+	0.663356i	$0.230868\pi$
$618$		0			0
$619$	3.37653e9	+	5.84833e9i	0.572208	+	0.991093i	0.996339	+	0.0854921i	$0.0272462\pi$
$619$	3.37653e9	+	5.84833e9i	0.572208	+	0.991093i	−0.424131	+	0.905601i	$0.639420\pi$
$620$	1.94703e9			0.328096
$621$		0			0
$622$	−5.19921e9			−0.866306
$623$	6.55882e8	+	1.13602e9i	0.108672	+	0.188225i
$624$		0			0
$625$	−1.92207e9	+	3.32912e9i	−0.314912	+	0.545443i
$626$	2.96099e9	−	5.12858e9i	0.482421	−	0.835578i
$627$		0			0
$628$	4.34307e9	+	7.52241e9i	0.699742	+	1.21199i
$629$	1.48290e9			0.237594
$630$		0			0
$631$	−9.84840e9			−1.56050			−0.780249	−	0.625469i	$-0.784908\pi$
$631$	−9.84840e9			−1.56050			−0.780249	+	0.625469i	$0.784908\pi$
$632$	−6.64821e9	−	1.15150e10i	−1.04760	−	1.81449i
$633$		0			0
$634$	−1.75514e9	+	3.03999e9i	−0.273526	+	0.473761i
$635$	2.48460e9	−	4.30345e9i	0.385077	−	0.666974i
$636$		0			0
$637$	3.83520e8	+	6.64276e8i	0.0587895	+	0.101826i
$638$	−1.27079e9			−0.193732
$639$		0			0
$640$	1.83827e10			2.77191
$641$	9.64590e8	+	1.67072e9i	0.144657	+	0.250553i	0.929245	−	0.369464i	$-0.120459\pi$
$641$	9.64590e8	+	1.67072e9i	0.144657	+	0.250553i	−0.784588	+	0.620018i	$0.787125\pi$
$642$		0			0
$643$	−1.84715e9	+	3.19936e9i	−0.274009	+	0.474597i	−0.969885	−	0.243565i	$-0.921683\pi$
$643$	−1.84715e9	+	3.19936e9i	−0.274009	+	0.474597i	0.695876	+	0.718162i	$0.255016\pi$
$644$	−9.56906e9	+	1.65741e10i	−1.41179	+	2.44528i
$645$		0			0
$646$	−1.34350e10	−	2.32700e10i	−1.96075	−	3.39612i
$647$	1.04870e10			1.52225			0.761127	−	0.648602i	$-0.224646\pi$
$647$	1.04870e10			1.52225			0.761127	+	0.648602i	$0.224646\pi$
$648$		0			0
$649$	2.97558e9			0.427283
$650$	−4.80165e9	−	8.31671e9i	−0.685794	−	1.18783i
$651$		0			0
$652$	−1.31350e10	+	2.27505e10i	−1.85594	+	3.21459i
$653$	3.38105e9	−	5.85615e9i	0.475177	−	0.823031i	−0.524419	−	0.851461i	$-0.675717\pi$
$653$	3.38105e9	−	5.85615e9i	0.475177	−	0.823031i	0.999596	+	0.0284296i	$0.00905063\pi$
$654$		0			0
$655$	1.57590e9	+	2.72954e9i	0.219121	+	0.379529i
$656$	−1.21398e9			−0.167899
$657$		0			0
$658$	−1.45127e10			−1.98590
$659$	3.11470e8	+	5.39481e8i	0.0423952	+	0.0734306i	0.886444	−	0.462835i	$-0.153168\pi$
$659$	3.11470e8	+	5.39481e8i	0.0423952	+	0.0734306i	−0.844049	+	0.536266i	$0.819834\pi$
$660$		0			0
$661$	−7.64249e8	+	1.32372e9i	−0.102927	+	0.178275i	−0.912889	−	0.408207i	$-0.866154\pi$
$661$	−7.64249e8	+	1.32372e9i	−0.102927	+	0.178275i	0.809962	+	0.586482i	$0.199488\pi$
$662$	8.09674e9	−	1.40240e10i	1.08469	−	1.87875i
$663$		0			0
$664$	4.52585e9	+	7.83900e9i	0.599945	+	1.03914i
$665$	1.83084e10			2.41420
$666$		0			0
$667$	2.04497e9			0.266837
$668$	1.61251e9	+	2.79294e9i	0.209307	+	0.362530i
$669$		0			0
$670$	1.19204e10	−	2.06467e10i	1.53119	−	2.65209i
$671$	−1.99987e9	+	3.46387e9i	−0.255548	+	0.442622i
$672$		0			0
$673$	5.64752e9	+	9.78180e9i	0.714176	+	1.23699i	0.963276	+	0.268511i	$0.0865317\pi$
$673$	5.64752e9	+	9.78180e9i	0.714176	+	1.23699i	−0.249101	+	0.968478i	$0.580135\pi$
$674$	−2.24449e10			−2.82363
$675$		0			0
$676$	−1.36009e10			−1.69338
$677$	−4.02738e9	−	6.97562e9i	−0.498841	−	0.864017i	0.501159	−	0.865355i	$-0.332907\pi$
$677$	−4.02738e9	−	6.97562e9i	−0.498841	−	0.864017i	−0.999999	+	0.00133832i	$0.999574\pi$
$678$		0			0
$679$	3.93133e9	−	6.80926e9i	0.481942	−	0.834748i
$680$	2.28204e10	−	3.95261e10i	2.78318	−	4.82062i
$681$		0			0
$682$	−3.44134e8	−	5.96058e8i	−0.0415415	−	0.0719520i
$683$	−1.53230e10			−1.84023			−0.920113	−	0.391652i	$-0.871904\pi$
$683$	−1.53230e10			−1.84023			−0.920113	+	0.391652i	$0.871904\pi$
$684$		0			0
$685$	1.89506e10			2.25272
$686$	6.01710e9	+	1.04219e10i	0.711628	+	1.23258i
$687$		0			0
$688$	1.45054e9	−	2.51242e9i	0.169813	−	0.294125i
$689$	8.14450e8	−	1.41067e9i	0.0948631	−	0.164308i
$690$		0			0
$691$	−1.82587e9	−	3.16250e9i	−0.210521	−	0.364634i	0.741356	−	0.671112i	$-0.234183\pi$
$691$	−1.82587e9	−	3.16250e9i	−0.210521	−	0.364634i	−0.951878	+	0.306478i	$0.900849\pi$
$692$	−6.31369e9			−0.724289
$693$		0			0
$694$	−1.03446e10			−1.17477
$695$	−4.45810e9	−	7.72165e9i	−0.503736	−	0.872496i
$696$		0			0
$697$	1.14534e9	−	1.98379e9i	0.128121	−	0.221912i
$698$	2.21060e9	−	3.82888e9i	0.246046	−	0.426165i
$699$		0			0
$700$	1.97398e10	+	3.41904e10i	2.17521	+	3.76757i
$701$	−1.42036e10			−1.55735			−0.778675	−	0.627427i	$-0.784108\pi$
$701$	−1.42036e10			−1.55735			−0.778675	+	0.627427i	$0.784108\pi$
$702$		0			0
$703$	−1.54667e9			−0.167902
$704$	−1.92239e9	−	3.32967e9i	−0.207652	−	0.359664i
$705$		0			0
$706$	3.81122e9	−	6.60124e9i	0.407613	−	0.706007i
$707$	−8.74951e9	+	1.51546e10i	−0.931142	+	1.61279i
$708$		0			0
$709$	−5.84215e9	−	1.01189e10i	−0.615617	−	1.06628i	−0.990276	−	0.139118i	$-0.955573\pi$
$709$	−5.84215e9	−	1.01189e10i	−0.615617	−	1.06628i	0.374659	−	0.927163i	$-0.377760\pi$
$710$	6.55943e9			0.687800
$711$		0			0
$712$	−3.41071e9			−0.354132
$713$	5.53786e8	+	9.59185e8i	0.0572175	+	0.0991035i
$714$		0			0
$715$	−1.75194e9	+	3.03446e9i	−0.179246	+	0.310463i
$716$	−1.78626e10	+	3.09389e10i	−1.81865	+	3.14999i
$717$		0			0
$718$	1.03925e10	+	1.80004e10i	1.04782	+	1.81488i
$719$	8.68481e9			0.871383			0.435691	−	0.900096i	$-0.356504\pi$
$719$	8.68481e9			0.871383			0.435691	+	0.900096i	$0.356504\pi$
$720$		0			0
$721$	1.80400e10			1.79252
$722$	5.17491e9	+	8.96320e9i	0.511708	+	0.886305i
$723$		0			0
$724$	1.01868e10	−	1.76440e10i	0.997589	−	1.72787i
$725$	2.10927e9	−	3.65336e9i	0.205565	−	0.356048i
$726$		0			0
$727$	−1.74815e9	−	3.02788e9i	−0.168736	−	0.292259i	0.769240	−	0.638960i	$-0.220635\pi$
$727$	−1.74815e9	−	3.02788e9i	−0.168736	−	0.292259i	−0.937976	+	0.346701i	$0.887302\pi$
$728$	−9.10845e9			−0.874953
$729$		0			0
$730$	4.39297e10			4.17954
$731$	2.73706e9	+	4.74073e9i	0.259163	+	0.448884i
$732$		0			0
$733$	4.65551e9	−	8.06359e9i	0.436620	−	0.756248i	−0.560806	−	0.827947i	$-0.689509\pi$
$733$	4.65551e9	−	8.06359e9i	0.436620	−	0.756248i	0.997426	+	0.0716989i	$0.0228421\pi$
$734$	1.91406e9	−	3.31525e9i	0.178657	−	0.309443i
$735$		0			0
$736$	−1.29352e9	−	2.24044e9i	−0.119591	−	0.207138i
$737$	−5.66885e9			−0.521626
$738$		0			0
$739$	−7.46338e9			−0.680268			−0.340134	−	0.940377i	$-0.610472\pi$
$739$	−7.46338e9			−0.680268			−0.340134	+	0.940377i	$0.610472\pi$
$740$	−2.55886e9	−	4.43208e9i	−0.232133	−	0.402065i
$741$		0			0
$742$	−4.97769e9	+	8.62161e9i	−0.447316	+	0.774774i
$743$	−3.84486e9	+	6.65950e9i	−0.343890	+	0.595636i	−0.985152	−	0.171687i	$-0.945078\pi$
$743$	−3.84486e9	+	6.65950e9i	−0.343890	+	0.595636i	0.641261	+	0.767323i	$0.278412\pi$
$744$		0			0
$745$	−6.11216e9	−	1.05866e10i	−0.541562	−	0.938013i
$746$	6.99795e9			0.617141
$747$		0			0
$748$	−2.11407e10			−1.84699
$749$	−5.97720e9	−	1.03528e10i	−0.519771	−	0.900269i
$750$		0			0
$751$	9.74148e9	−	1.68727e10i	0.839238	−	1.45360i	−0.0512949	−	0.998684i	$-0.516335\pi$
$751$	9.74148e9	−	1.68727e10i	0.839238	−	1.45360i	0.890533	−	0.454919i	$-0.150332\pi$
$752$	6.83598e9	−	1.18403e10i	0.586190	−	1.01531i
$753$		0			0
$754$	9.47968e8	+	1.64193e9i	0.0805367	+	0.139494i
$755$	−4.32639e9			−0.365857
$756$		0			0
$757$	−4.45370e9			−0.373152			−0.186576	−	0.982441i	$-0.559739\pi$
$757$	−4.45370e9			−0.373152			−0.186576	+	0.982441i	$0.559739\pi$
$758$	−4.64650e9	−	8.04798e9i	−0.387511	−	0.671189i
$759$		0			0
$760$	−2.38018e10	+	4.12259e10i	−1.96681	+	3.40661i
$761$	9.24329e9	−	1.60099e10i	0.760292	−	1.31686i	−0.182408	−	0.983223i	$-0.558389\pi$
$761$	9.24329e9	−	1.60099e10i	0.760292	−	1.31686i	0.942700	−	0.333641i	$-0.108277\pi$
$762$		0			0
$763$	−1.17669e10	−	2.03808e10i	−0.959015	−	1.66106i
$764$	−9.95503e9			−0.807636
$765$		0			0
$766$	−3.02722e10			−2.43357
$767$	−2.21969e9	−	3.84462e9i	−0.177627	−	0.307659i
$768$		0			0
$769$	5.53663e8	−	9.58973e8i	0.0439039	−	0.0760438i	−0.843238	−	0.537540i	$-0.819354\pi$
$769$	5.53663e8	−	9.58973e8i	0.0439039	−	0.0760438i	0.887142	+	0.461496i	$0.152687\pi$
$770$	1.07074e10	−	1.85457e10i	0.845213	−	1.46395i
$771$		0			0
$772$	5.66961e9	+	9.82005e9i	0.443499	+	0.768163i
$773$	1.36030e10			1.05927			0.529633	−	0.848227i	$-0.322330\pi$
$773$	1.36030e10			1.05927			0.529633	+	0.848227i	$0.322330\pi$
$774$		0			0
$775$	2.28479e9			0.176315
$776$	1.02218e10	+	1.77047e10i	0.785259	+	1.36011i
$777$		0			0
$778$	−8.24041e9	+	1.42728e10i	−0.627365	+	1.08663i
$779$	−1.19460e9	+	2.06910e9i	−0.0905399	+	0.156820i
$780$		0			0
$781$	−7.79852e8	−	1.35074e9i	−0.0585778	−	0.101460i
$782$	5.05760e10			3.78199
$783$		0			0
$784$	4.41635e9			0.327308
$785$	7.82033e9	+	1.35452e10i	0.577007	+	0.999406i
$786$		0			0
$787$	9.97966e9	−	1.72853e10i	0.729800	−	1.26405i	−0.227167	−	0.973856i	$-0.572946\pi$
$787$	9.97966e9	−	1.72853e10i	0.729800	−	1.26405i	0.956967	−	0.290195i	$-0.0937203\pi$
$788$	−1.77874e10	+	3.08087e10i	−1.29500	+	2.24301i
$789$		0			0
$790$	−2.33198e10	−	4.03911e10i	−1.68279	−	2.91468i
$791$	−3.51601e8			−0.0252599
$792$		0			0
$793$	5.96736e9			0.424938
$794$	8.48462e9	+	1.46958e10i	0.601535	+	1.04189i
$795$		0			0
$796$	−1.29306e10	+	2.23965e10i	−0.908708	+	1.57393i
$797$	6.15857e9	−	1.06670e10i	0.430899	−	0.746340i	−0.566052	−	0.824370i	$-0.691530\pi$
$797$	6.15857e9	−	1.06670e10i	0.430899	−	0.746340i	0.996951	+	0.0780302i	$0.0248631\pi$
$798$		0			0
$799$	1.28989e10	+	2.23416e10i	0.894623	+	1.54953i
$800$	−5.33674e9			−0.368520
$801$		0			0
$802$	−4.09657e9			−0.280421
$803$	−5.22280e9	−	9.04616e9i	−0.355958	−	0.616538i
$804$		0			0
$805$	−1.72305e10	+	2.98441e10i	−1.16416	+	2.01638i
$806$	−5.13427e8	+	8.89281e8i	−0.0345387	+	0.0598228i
$807$		0			0
$808$	−2.27496e10	−	3.94034e10i	−1.51717	−	2.62781i
$809$	1.27792e10			0.848559			0.424280	−	0.905531i	$-0.360527\pi$
$809$	1.27792e10			0.848559			0.424280	+	0.905531i	$0.360527\pi$
$810$		0			0
$811$	−2.84223e9			−0.187105			−0.0935526	−	0.995614i	$-0.529822\pi$
$811$	−2.84223e9			−0.187105			−0.0935526	+	0.995614i	$0.529822\pi$
$812$	−3.89714e9	−	6.75005e9i	−0.255447	−	0.442447i
$813$		0			0
$814$	−9.04551e8	+	1.56673e9i	−0.0587824	+	0.101814i
$815$	−2.36515e10	+	4.09657e10i	−1.53041	+	2.65075i
$816$		0			0
$817$	−2.85476e9	−	4.94460e9i	−0.183144	−	0.317215i
$818$	8.84231e9			0.564845
$819$		0			0
$820$	−7.90550e9			−0.500704
$821$	−8.89206e9	−	1.54015e10i	−0.560791	−	0.971319i	−0.997428	−	0.0716808i	$-0.977164\pi$
$821$	−8.89206e9	−	1.54015e10i	−0.560791	−	0.971319i	0.436636	−	0.899638i	$-0.356170\pi$
$822$		0			0
$823$	−2.43059e9	+	4.20991e9i	−0.151989	+	0.263253i	−0.931959	−	0.362565i	$-0.881901\pi$
$823$	−2.43059e9	+	4.20991e9i	−0.151989	+	0.263253i	0.779969	+	0.625818i	$0.215235\pi$
$824$	−2.34529e10	+	4.06216e10i	−1.46033	+	2.52937i
$825$		0			0
$826$	1.35661e10	+	2.34972e10i	0.837579	+	1.45073i
$827$	−2.39461e10			−1.47220			−0.736099	−	0.676874i	$-0.763334\pi$
$827$	−2.39461e10			−1.47220			−0.736099	+	0.676874i	$0.763334\pi$
$828$		0			0
$829$	−1.19577e10			−0.728968			−0.364484	−	0.931210i	$-0.618755\pi$
$829$	−1.19577e10			−0.728968			−0.364484	+	0.931210i	$0.618755\pi$
$830$	1.58752e10	+	2.74967e10i	0.963710	+	1.66920i
$831$		0			0
$832$	−2.86808e9	+	4.96767e9i	−0.172648	+	0.299034i
$833$	−4.16664e9	+	7.21684e9i	−0.249763	+	0.432603i
$834$		0			0
$835$	2.90355e9	+	5.02910e9i	0.172595	+	0.298943i
$836$	2.20499e10			1.30522
$837$		0			0
$838$	1.59081e10			0.933825
$839$	4.93588e9	+	8.54920e9i	0.288535	+	0.499757i	0.973460	−	0.228856i	$-0.0734986\pi$
$839$	4.93588e9	+	8.54920e9i	0.288535	+	0.499757i	−0.684926	+	0.728613i	$0.740165\pi$
$840$		0			0
$841$	8.20852e9	−	1.42176e10i	0.475859	−	0.824213i
$842$	1.18275e10	−	2.04858e10i	0.682811	−	1.18266i
$843$		0			0
$844$	8.97923e9	+	1.55525e10i	0.514092	+	0.890433i
$845$	−2.44905e10			−1.39636
$846$		0			0
$847$	1.49184e10			0.843586
$848$	−4.68932e9	−	8.12214e9i	−0.264073	−	0.457388i
$849$		0			0
$850$	5.21662e10	−	9.03545e10i	2.91355	−	5.04642i
$851$	1.45562e9	−	2.52120e9i	0.0809643	−	0.140234i
$852$		0			0
$853$	−6.19893e9	−	1.07369e10i	−0.341976	−	0.592320i	0.642824	−	0.766014i	$-0.277763\pi$
$853$	−6.19893e9	−	1.07369e10i	−0.341976	−	0.592320i	−0.984800	+	0.173695i	$0.944429\pi$
$854$	−3.64708e10			−2.00375
$855$		0			0
$856$	3.10826e10			1.69379
$857$	−1.76003e10	−	3.04847e10i	−0.955186	−	1.65443i	−0.733942	−	0.679212i	$-0.762322\pi$
$857$	−1.76003e10	−	3.04847e10i	−0.955186	−	1.65443i	−0.221244	−	0.975219i	$-0.571012\pi$
$858$		0			0
$859$	−6.50686e9	+	1.12702e10i	−0.350264	+	0.606675i	−0.986296	−	0.164988i	$-0.947242\pi$
$859$	−6.50686e9	+	1.12702e10i	−0.350264	+	0.606675i	0.636031	+	0.771663i	$0.280575\pi$
$860$	9.44601e9	−	1.63610e10i	0.506412	−	0.877131i
$861$		0			0
$862$	−7.93815e9	−	1.37493e10i	−0.422128	−	0.731147i
$863$	−3.11852e10			−1.65162			−0.825811	−	0.563947i	$-0.809282\pi$
$863$	−3.11852e10			−1.65162			−0.825811	+	0.563947i	$0.809282\pi$
$864$		0			0
$865$	−1.13687e10			−0.597249
$866$	2.51533e10	+	4.35667e10i	1.31608	+	2.27952i
$867$		0			0
$868$	2.11072e9	−	3.65588e9i	0.109550	−	0.189746i
$869$	−5.54498e9	+	9.60419e9i	−0.286636	+	0.496469i
$870$		0			0
$871$	4.22879e9	+	7.32448e9i	0.216847	+	0.375589i
$872$	6.11901e10			3.12517
$873$		0			0
$874$	−5.27510e10			−2.67264
$875$	1.65476e10	+	2.86612e10i	0.835038	+	1.44633i
$876$		0			0
$877$	−1.01903e10	+	1.76501e10i	−0.510140	+	0.883588i	0.489791	+	0.871840i	$0.337073\pi$
$877$	−1.01903e10	+	1.76501e10i	−0.510140	+	0.883588i	−0.999931	+	0.0117482i	$0.996260\pi$
$878$	−7.91640e9	+	1.37116e10i	−0.394727	+	0.683687i
$879$		0			0
$880$	1.00871e10	+	1.74713e10i	0.498972	+	0.864245i
$881$	1.59058e10			0.783681			0.391841	−	0.920033i	$-0.371838\pi$
$881$	1.59058e10			0.783681			0.391841	+	0.920033i	$0.371838\pi$
$882$		0			0
$883$	3.50773e10			1.71460			0.857302	−	0.514814i	$-0.172139\pi$
$883$	3.50773e10			1.71460			0.857302	+	0.514814i	$0.172139\pi$
$884$	1.57703e10	+	2.73150e10i	0.767817	+	1.32990i
$885$		0			0
$886$	−1.06654e10	+	1.84731e10i	−0.515181	+	0.892321i
$887$	1.44830e10	−	2.50853e10i	0.696828	−	1.20694i	−0.272732	−	0.962090i	$-0.587927\pi$
$887$	1.44830e10	−	2.50853e10i	0.696828	−	1.20694i	0.969560	−	0.244852i	$-0.0787394\pi$
$888$		0			0
$889$	−5.38699e9	−	9.33053e9i	−0.257152	−	0.445400i
$890$	−1.19637e10			−0.568853
$891$		0			0
$892$	7.67649e10			3.62148
$893$	−1.34536e10	−	2.33024e10i	−0.632207	−	1.09502i
$894$		0			0
$895$	−3.21642e10	+	5.57100e10i	−1.49966	+	2.59748i
$896$	1.99282e10	−	3.45167e10i	0.925530	−	1.60306i
$897$		0			0
$898$	1.65185e10	+	2.86109e10i	0.761208	+	1.31845i
$899$	−4.51075e8			−0.0207057
$900$		0			0
$901$	1.76967e10			0.806039
$902$	1.39729e9	+	2.42017e9i	0.0633961	+	0.109805i
$903$		0			0
$904$	4.57098e8	−	7.91718e8i	0.0205788	−	0.0356435i
$905$	1.83428e10	−	3.17706e10i	0.822612	−	1.42481i
$906$		0			0
$907$	1.21829e10	+	2.11013e10i	0.542155	+	0.939041i	0.998780	+	0.0493813i	$0.0157250\pi$
$907$	1.21829e10	+	2.11013e10i	0.542155	+	0.939041i	−0.456625	+	0.889660i	$0.650942\pi$
$908$	−6.27529e10			−2.78185
$909$		0			0
$910$	−3.19495e10			−1.40546
$911$	1.38488e9	+	2.39868e9i	0.0606872	+	0.105113i	0.894773	−	0.446522i	$-0.147337\pi$
$911$	1.38488e9	+	2.39868e9i	0.0606872	+	0.105113i	−0.834086	+	0.551635i	$0.814004\pi$
$912$		0			0
$913$	3.77481e9	−	6.53817e9i	0.164153	−	0.284321i
$914$	1.79107e10	−	3.10223e10i	0.775892	−	1.34388i
$915$		0			0
$916$	2.01250e10	+	3.48575e10i	0.865170	+	1.49852i
$917$	6.83358e9			0.292655
$918$		0			0
$919$	6.03812e9			0.256624			0.128312	−	0.991734i	$-0.459044\pi$
$919$	6.03812e9			0.256624			0.128312	+	0.991734i	$0.459044\pi$
$920$	−4.48010e10	−	7.75975e10i	−1.89684	−	3.28542i
$921$		0			0
$922$	8.97862e9	−	1.55514e10i	0.377269	−	0.653449i
$923$	−1.16349e9	+	2.01522e9i	−0.0487031	+	0.0843563i
$924$		0			0
$925$	−3.00277e9	−	5.20094e9i	−0.124746	−	0.216066i
$926$	5.86211e10			2.42614
$927$		0			0
$928$	1.05361e9			0.0432774
$929$	1.52786e9	+	2.64632e9i	0.0625212	+	0.108290i	0.895592	−	0.444877i	$-0.146753\pi$
$929$	1.52786e9	+	2.64632e9i	0.0625212	+	0.108290i	−0.833071	+	0.553167i	$0.813419\pi$
$930$		0			0
$931$	4.34583e9	−	7.52719e9i	0.176502	−	0.305710i
$932$	3.34052e10	−	5.78595e10i	1.35163	−	2.34109i
$933$		0			0
$934$	−1.13957e10	−	1.97379e10i	−0.457642	−	0.792659i
$935$	−3.80670e10			−1.52303
$936$		0			0
$937$	−4.12174e10			−1.63679			−0.818393	−	0.574658i	$-0.805135\pi$
$937$	−4.12174e10			−1.63679			−0.818393	+	0.574658i	$0.805135\pi$
$938$	−2.58452e10	−	4.47651e10i	−1.02251	−	1.77105i
$939$		0			0
$940$	4.45162e10	−	7.71043e10i	1.74812	−	3.02783i
$941$	−1.22425e10	+	2.12046e10i	−0.478967	+	0.829594i	−0.999709	−	0.0241193i	$-0.992322\pi$
$941$	−1.22425e10	+	2.12046e10i	−0.478967	+	0.829594i	0.520742	+	0.853714i	$0.325655\pi$
$942$		0			0
$943$	−2.24853e9	−	3.89457e9i	−0.0873189	−	0.151241i
$944$	−2.55604e10			−0.988931
$945$		0			0
$946$	−6.67828e9			−0.256475
$947$	1.87516e10	+	3.24788e10i	0.717488	+	1.24273i	0.961992	+	0.273077i	$0.0880414\pi$
$947$	1.87516e10	+	3.24788e10i	0.717488	+	1.24273i	−0.244504	+	0.969648i	$0.578625\pi$
$948$		0			0
$949$	−7.79210e9	+	1.34963e10i	−0.295953	+	0.512606i
$950$	−5.44095e10	+	9.42401e10i	−2.05893	+	3.56618i
$951$		0			0
$952$	−4.94781e10	−	8.56985e10i	−1.85859	−	3.21917i
$953$	−1.27579e10			−0.477480			−0.238740	−	0.971084i	$-0.576734\pi$
$953$	−1.27579e10			−0.477480			−0.238740	+	0.971084i	$0.576734\pi$
$954$		0			0
$955$	−1.79255e10			−0.665977
$956$	−4.15753e10	−	7.20105e10i	−1.53898	−	2.66559i
$957$		0			0
$958$	−5.60266e9	+	9.70409e9i	−0.205880	+	0.356595i
$959$	2.05439e10	−	3.55831e10i	0.752173	−	1.30280i
$960$		0			0
$961$	1.36342e10	+	2.36150e10i	0.495560	+	0.858335i
$962$	2.69907e9			0.0977464
$963$		0			0
$964$	5.66899e9			0.203815
$965$	1.02090e10	+	1.76824e10i	0.365709	+	0.633427i
$966$		0			0
$967$	1.08441e10	−	1.87826e10i	0.385658	−	0.667980i	−0.606202	−	0.795311i	$-0.707308\pi$
$967$	1.08441e10	−	1.87826e10i	0.385658	−	0.667980i	0.991860	+	0.127331i	$0.0406411\pi$
$968$	−1.93946e10	+	3.35924e10i	−0.687254	+	1.19036i
$969$		0			0
$970$	3.58549e10	+	6.21025e10i	1.26138	+	2.18478i
$971$	4.12029e10			1.44431			0.722155	−	0.691732i	$-0.243152\pi$
$971$	4.12029e10			1.44431			0.722155	+	0.691732i	$0.243152\pi$
$972$		0			0
$973$	−1.93317e10			−0.672782
$974$	3.16322e10	+	5.47886e10i	1.09692	+	1.89991i
$975$		0			0
$976$	1.71790e10	−	2.97548e10i	0.591456	−	1.02443i
$977$	2.35348e8	−	4.07635e8i	0.00807384	−	0.0139843i	−0.861960	−	0.506976i	$-0.830763\pi$
$977$	2.35348e8	−	4.07635e8i	0.00807384	−	0.0139843i	0.870034	+	0.492992i	$0.164097\pi$
$978$		0			0
$979$	1.42236e9	+	2.46361e9i	0.0484475	+	0.0839135i
$980$	2.87594e10			0.976089
$981$		0			0
$982$	−4.21655e10			−1.42091
$983$	−3.40287e9	−	5.89394e9i	−0.114264	−	0.197910i	0.803222	−	0.595680i	$-0.203117\pi$
$983$	−3.40287e9	−	5.89394e9i	−0.114264	−	0.197910i	−0.917485	+	0.397770i	$0.869784\pi$
$984$		0			0
$985$	−3.20288e10	+	5.54756e10i	−1.06786	+	1.84959i
$986$	−1.02989e10	+	1.78383e10i	−0.342155	+	0.592629i
$987$		0			0
$988$	−1.64485e10	−	2.84897e10i	−0.542597	−	0.939806i
$989$	1.07468e10			0.353258
$990$		0			0
$991$	−5.55681e10			−1.81371			−0.906855	−	0.421442i	$-0.861524\pi$
$991$	−5.55681e10			−1.81371			−0.906855	+	0.421442i	$0.861524\pi$
$992$	2.85321e8	+	4.94191e8i	0.00927990	+	0.0160733i
$993$		0			0
$994$	7.11092e9	−	1.23165e10i	0.229654	−	0.397772i
$995$	−2.32835e10	+	4.03283e10i	−0.749321	+	1.29786i
$996$		0			0
$997$	−1.68657e10	−	2.92123e10i	−0.538979	−	0.933539i	−0.998959	−	0.0456096i	$-0.985477\pi$
$997$	−1.68657e10	−	2.92123e10i	−0.538979	−	0.933539i	0.459981	−	0.887929i	$-0.347856\pi$
$998$	−2.06228e10			−0.656735
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.8.c.k.28.6		12
3.2	odd	2	inner	81.8.c.k.28.1		12
9.2	odd	6	inner	81.8.c.k.55.1		12
9.4	even	3		81.8.a.d.1.1	✓	6
9.5	odd	6		81.8.a.d.1.6	yes	6
9.7	even	3	inner	81.8.c.k.55.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.8.a.d.1.1	✓	6	9.4	even	3
81.8.a.d.1.6	yes	6	9.5	odd	6
81.8.c.k.28.1		12	3.2	odd	2	inner
81.8.c.k.28.6		12	1.1	even	1	trivial
81.8.c.k.55.1		12	9.2	odd	6	inner
81.8.c.k.55.6		12	9.7	even	3	inner

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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	81.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(9.88711 + 17.1250i) q^{2} +(-131.510 + 227.782i) q^{4} +(-236.803 + 410.154i) q^{5} +(513.424 + 889.277i) q^{7} -2669.91 q^{8} -9365.17 q^{10} +(1113.43 + 1928.51i) q^{11} +(1661.16 - 2877.22i) q^{13} +(-10152.6 + 17584.7i) q^{14} +(-9564.39 - 16566.0i) q^{16} +36094.5 q^{17} -37646.7 q^{19} +(-62283.7 - 107879. i) q^{20} +(-22017.1 + 38134.8i) q^{22} +(35430.3 - 61367.1i) q^{23} +(-73088.5 - 126593. i) q^{25} +65696.4 q^{26} -270081. q^{28} +(14429.5 + 24992.7i) q^{29} +(-7815.15 + 13536.2i) q^{31} +(18254.4 - 31617.5i) q^{32} +(356870. + 618117. i) q^{34} -486321. q^{35} +41084.0 q^{37} +(-372217. - 644698. i) q^{38} +(632241. - 1.09507e6i) q^{40} +(31731.8 - 54961.1i) q^{41} +(75830.5 + 131342. i) q^{43} -585706. q^{44} +1.40121e6 q^{46} +(357366. + 618976. i) q^{47} +(-115437. + 199943. i) q^{49} +(1.44527e6 - 2.50328e6i) q^{50} +(436918. + 756765. i) q^{52} +490289. q^{53} -1.05465e6 q^{55} +(-1.37079e6 - 2.37428e6i) q^{56} +(-285333. + 494210. i) q^{58} +(668114. - 1.15721e6i) q^{59} +(898069. + 1.55550e6i) q^{61} -309077. q^{62} -1.72655e6 q^{64} +(786736. + 1.36267e6i) q^{65} +(-1.27284e6 + 2.20462e6i) q^{67} +(-4.74677e6 + 8.22165e6i) q^{68} +(-4.80831e6 - 8.32823e6i) q^{70} -700407. q^{71} -4.69075e6 q^{73} +(406202. + 703562. i) q^{74} +(4.95090e6 - 8.57522e6i) q^{76} +(-1.14332e6 + 1.98029e6i) q^{77} +(2.49005e6 + 4.31290e6i) q^{79} +9.05950e6 q^{80} +1.25494e6 q^{82} +(-1.69513e6 - 2.93606e6i) q^{83} +(-8.54726e6 + 1.48043e7i) q^{85} +(-1.49949e6 + 2.59719e6i) q^{86} +(-2.97274e6 - 5.14894e6i) q^{88} +1.27747e6 q^{89} +3.41152e6 q^{91} +(9.31886e6 + 1.61407e7i) q^{92} +(-7.06663e6 + 1.22398e7i) q^{94} +(8.91483e6 - 1.54409e7i) q^{95} +(-3.82854e6 - 6.63122e6i) q^{97} -4.56536e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 438 q^{4} + 1932 q^{7} - 29844 q^{10} + 11886 q^{13} - 7314 q^{16} - 134328 q^{19} + 23364 q^{22} - 42324 q^{25} - 1386696 q^{28} + 470832 q^{31} + 1834866 q^{34} - 2052516 q^{37} + 3091374 q^{40}+ \cdots - 26307948 q^{97}+O(q^{100}) \) 12 * q - 438 * q^4 + 1932 * q^7 - 29844 * q^10 + 11886 * q^13 - 7314 * q^16 - 134328 * q^19 + 23364 * q^22 - 42324 * q^25 - 1386696 * q^28 + 470832 * q^31 + 1834866 * q^34 - 2052516 * q^37 + 3091374 * q^40 + 1502268 * q^43 + 209160 * q^46 + 1769490 * q^49 + 2993334 * q^52 - 7334856 * q^55 - 659898 * q^58 + 4564770 * q^61 - 2095188 * q^64 - 2850060 * q^67 - 19286604 * q^70 - 3693684 * q^73 + 5520516 * q^76 + 2878188 * q^79 + 30137904 * q^82 - 32410098 * q^85 - 26598276 * q^88 + 32484216 * q^91 - 51428376 * q^94 - 26307948 * q^97

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