LMFDB - Embedded newform 81.10.c.j.55.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [81,10,Mod(28,81)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("81.28"); S:= CuspForms(chi, 10); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(81, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 10, names="a")

Level:	$ N $	$=$	$ 81 = 3^{4} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	81.c (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$41.7179027293$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 83x^{6} + 5449x^{4} + 119520x^{2} + 2073600 $ x^8 + 83x^6 + 5449x^4 + 119520*x^2 + 2073600
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{16} $
Twist minimal:	no (minimal twist has level 27)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			55.4
Root			$-3.81773 + 6.61251i$ of defining polynomial
Character	$\chi$	$=$	81.55
Dual form			81.10.c.j.28.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(22.2386 - 38.5183i) q^{2} +(-733.108 - 1269.78i) q^{4} +(-525.311 - 909.865i) q^{5} +(-3295.93 + 5708.72i) q^{7} -42440.8 q^{8} -46728.6 q^{10} +(7232.96 - 12527.8i) q^{11} +(-2501.75 - 4333.15i) q^{13} +(146594. + 253908. i) q^{14} +(-568471. + 984621. i) q^{16} +444500. q^{17} -487431. q^{19} +(-770219. + 1.33406e6i) q^{20} +(-321701. - 557203. i) q^{22} +(-33652.3 - 58287.4i) q^{23} +(424660. - 735532. i) q^{25} -222541. q^{26} +9.66510e6 q^{28} +(-1.70907e6 + 2.96019e6i) q^{29} +(-1.72962e6 - 2.99578e6i) q^{31} +(1.44191e7 + 2.49747e7i) q^{32} +(9.88504e6 - 1.71214e7i) q^{34} +6.92555e6 q^{35} -3.94510e6 q^{37} +(-1.08398e7 + 1.87750e7i) q^{38} +(2.22946e7 + 3.86154e7i) q^{40} +(-8.80305e6 - 1.52473e7i) q^{41} +(-1.65115e7 + 2.85988e7i) q^{43} -2.12102e7 q^{44} -2.99351e6 q^{46} +(-1.85636e7 + 3.21530e7i) q^{47} +(-1.54953e6 - 2.68387e6i) q^{49} +(-1.88876e7 - 3.27144e7i) q^{50} +(-3.66810e6 + 6.35334e6i) q^{52} +7.86818e7 q^{53} -1.51982e7 q^{55} +(1.39882e8 - 2.42283e8i) q^{56} +(7.60144e7 + 1.31661e8i) q^{58} +(2.56527e7 + 4.44318e7i) q^{59} +(9.63615e6 - 1.66903e7i) q^{61} -1.53857e8 q^{62} +7.00529e8 q^{64} +(-2.62839e6 + 4.55250e6i) q^{65} +(-1.07419e8 - 1.86055e8i) q^{67} +(-3.25866e8 - 5.64417e8i) q^{68} +(1.54014e8 - 2.66761e8i) q^{70} -3.25802e8 q^{71} +1.57548e8 q^{73} +(-8.77334e7 + 1.51959e8i) q^{74} +(3.57340e8 + 6.18931e8i) q^{76} +(4.76787e7 + 8.25819e7i) q^{77} +(-1.79317e7 + 3.10586e7i) q^{79} +1.19450e9 q^{80} -7.83069e8 q^{82} +(4.03992e6 - 6.99735e6i) q^{83} +(-2.33501e8 - 4.04435e8i) q^{85} +(7.34386e8 + 1.27199e9i) q^{86} +(-3.06972e8 + 5.31692e8i) q^{88} -9.43654e8 q^{89} +3.29823e7 q^{91} +(-4.93415e7 + 8.54620e7i) q^{92} +(8.25654e8 + 1.43008e9i) q^{94} +(2.56053e8 + 4.43497e8i) q^{95} +(-4.83526e8 + 8.37491e8i) q^{97} -1.37838e8 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2236 q^{4} - 11852 q^{7} - 83520 q^{10} - 179684 q^{13} - 2348680 q^{16} + 2022856 q^{19} - 3473568 q^{22} - 2554060 q^{25} + 39587848 q^{28} - 889136 q^{31} + 43111008 q^{34} - 7610312 q^{37} + 133649280 q^{40}+ \cdots - 1935734516 q^{97}+O(q^{100}) $ 8 * q - 2236 * q^4 - 11852 * q^7 - 83520 * q^10 - 179684 * q^13 - 2348680 * q^16 + 2022856 * q^19 - 3473568 * q^22 - 2554060 * q^25 + 39587848 * q^28 - 889136 * q^31 + 43111008 * q^34 - 7610312 * q^37 + 133649280 * q^40 - 11585024 * q^43 + 26856000 * q^46 + 73622328 * q^49 + 44411860 * q^52 + 629444160 * q^55 + 407685888 * q^58 - 219882572 * q^61 + 3038570720 * q^64 - 390561236 * q^67 + 650481120 * q^70 + 2395899016 * q^73 + 2121017572 * q^76 - 49814324 * q^79 - 3534253056 * q^82 - 370444320 * q^85 + 653198976 * q^88 - 94034344 * q^91 + 4146926688 * q^94 - 1935734516 * q^97

Character values

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

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

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 $

$2$	22.2386	−	38.5183i	0.982815	−	1.70229i	0.331546	−	0.943439i	$-0.392430\pi$
$2$	22.2386	−	38.5183i	0.982815	−	1.70229i	0.651269	−	0.758847i	$-0.274237\pi$
$3$		0			0
$3$		0			0
$4$	−733.108	−	1269.78i	−1.43185	−	2.48004i
$5$	−525.311	−	909.865i	−0.375882	−	0.651046i	0.614577	−	0.788857i	$-0.289327\pi$
$5$	−525.311	−	909.865i	−0.375882	−	0.651046i	−0.990459	+	0.137811i	$0.955993\pi$
$6$		0			0
$7$	−3295.93	+	5708.72i	−0.518844	+	0.898665i	0.480916	+	0.876767i	$0.340304\pi$
$7$	−3295.93	+	5708.72i	−0.518844	+	0.898665i	−0.999760	+	0.0218980i	$0.993029\pi$
$8$	−42440.8			−3.66335
$9$		0			0
$10$	−46728.6			−1.47769
$11$	7232.96	−	12527.8i	0.148953	−	0.257994i	−0.781888	−	0.623419i	$-0.785743\pi$
$11$	7232.96	−	12527.8i	0.148953	−	0.257994i	0.930841	+	0.365425i	$0.119076\pi$
$12$		0			0
$13$	−2501.75	−	4333.15i	−0.0242940	−	0.0420784i	0.853623	−	0.520892i	$-0.174400\pi$
$13$	−2501.75	−	4333.15i	−0.0242940	−	0.0420784i	−0.877917	+	0.478813i	$0.841067\pi$
$14$	146594.	+	253908.i	1.01986	+	1.76644i
$15$		0			0
$16$	−568471.	+	984621.i	−2.16855	+	3.75603i
$17$	444500.			1.29078			0.645389	−	0.763854i	$-0.276695\pi$
$17$	444500.			1.29078			0.645389	+	0.763854i	$0.276695\pi$
$18$		0			0
$19$	−487431.			−0.858069			−0.429035	−	0.903288i	$-0.641146\pi$
$19$	−487431.			−0.858069			−0.429035	+	0.903288i	$0.641146\pi$
$20$	−770219.	+	1.33406e6i	−1.07641	+	1.86440i
$21$		0			0
$22$	−321701.	−	557203.i	−0.292786	−	0.507121i
$23$	−33652.3	−	58287.4i	−0.0250749	−	0.0434310i	0.853216	−	0.521558i	$-0.174649\pi$
$23$	−33652.3	−	58287.4i	−0.0250749	−	0.0434310i	−0.878291	+	0.478127i	$0.841316\pi$
$24$		0			0
$25$	424660.	−	735532.i	0.217426	−	0.376592i
$26$	−222541.			−0.0955059
$27$		0			0
$28$	9.66510e6			2.97163
$29$	−1.70907e6	+	2.96019e6i	−0.448712	+	0.777193i	−0.998303	−	0.0582417i	$-0.981451\pi$
$29$	−1.70907e6	+	2.96019e6i	−0.448712	+	0.777193i	0.549590	+	0.835434i	$0.314784\pi$
$30$		0			0
$31$	−1.72962e6	−	2.99578e6i	−0.336374	−	0.582616i	0.647374	−	0.762172i	$-0.275867\pi$
$31$	−1.72962e6	−	2.99578e6i	−0.336374	−	0.582616i	−0.983748	+	0.179556i	$0.942534\pi$
$32$	1.44191e7	+	2.49747e7i	2.43089	+	4.21042i
$33$		0			0
$34$	9.88504e6	−	1.71214e7i	1.26860	−	2.19727i
$35$	6.92555e6			0.780097
$36$		0			0
$37$	−3.94510e6			−0.346059			−0.173030	−	0.984917i	$-0.555356\pi$
$37$	−3.94510e6			−0.346059			−0.173030	+	0.984917i	$0.555356\pi$
$38$	−1.08398e7	+	1.87750e7i	−0.843323	+	1.46068i
$39$		0			0
$40$	2.22946e7	+	3.86154e7i	1.37699	+	2.38501i
$41$	−8.80305e6	−	1.52473e7i	−0.486526	−	0.842687i	0.513354	−	0.858177i	$-0.328403\pi$
$41$	−8.80305e6	−	1.52473e7i	−0.486526	−	0.842687i	−0.999880	+	0.0154894i	$0.995069\pi$
$42$		0			0
$43$	−1.65115e7	+	2.85988e7i	−0.736512	+	1.27568i	0.217545	+	0.976050i	$0.430195\pi$
$43$	−1.65115e7	+	2.85988e7i	−0.736512	+	1.27568i	−0.954057	+	0.299625i	$0.903138\pi$
$44$	−2.12102e7			−0.853113
$45$		0			0
$46$	−2.99351e6			−0.0985759
$47$	−1.85636e7	+	3.21530e7i	−0.554908	+	0.961129i	0.443003	+	0.896520i	$0.353913\pi$
$47$	−1.85636e7	+	3.21530e7i	−0.554908	+	0.961129i	−0.997911	+	0.0646087i	$0.979420\pi$
$48$		0			0
$49$	−1.54953e6	−	2.68387e6i	−0.0383989	−	0.0665088i
$50$	−1.88876e7	−	3.27144e7i	−0.427379	−	0.740242i
$51$		0			0
$52$	−3.66810e6	+	6.35334e6i	−0.0695707	+	0.120500i
$53$	7.86818e7			1.36972			0.684861	−	0.728673i	$-0.259863\pi$
$53$	7.86818e7			1.36972			0.684861	+	0.728673i	$0.259863\pi$
$54$		0			0
$55$	−1.51982e7			−0.223955
$56$	1.39882e8	−	2.42283e8i	1.90071	−	3.29213i
$57$		0			0
$58$	7.60144e7	+	1.31661e8i	0.882003	+	1.52767i
$59$	2.56527e7	+	4.44318e7i	0.275613	+	0.477375i	0.970290	−	0.241947i	$-0.0777859\pi$
$59$	2.56527e7	+	4.44318e7i	0.275613	+	0.477375i	−0.694677	+	0.719322i	$0.744453\pi$
$60$		0			0
$61$	9.63615e6	−	1.66903e7i	0.0891085	−	0.154340i	−0.818026	−	0.575181i	$-0.804932\pi$
$61$	9.63615e6	−	1.66903e7i	0.0891085	−	0.154340i	0.907135	+	0.420841i	$0.138265\pi$
$62$	−1.53857e8			−1.32237
$63$		0			0
$64$	7.00529e8			5.21935
$65$	−2.62839e6	+	4.55250e6i	−0.0182633	+	0.0316330i
$66$		0			0
$67$	−1.07419e8	−	1.86055e8i	−0.651245	−	1.12799i	−0.982821	−	0.184562i	$-0.940913\pi$
$67$	−1.07419e8	−	1.86055e8i	−0.651245	−	1.12799i	0.331575	−	0.943429i	$-0.392420\pi$
$68$	−3.25866e8	−	5.64417e8i	−1.84820	−	3.20118i
$69$		0			0
$70$	1.54014e8	−	2.66761e8i	0.766691	−	1.32795i
$71$	−3.25802e8			−1.52157			−0.760784	−	0.649006i	$-0.775185\pi$
$71$	−3.25802e8			−1.52157			−0.760784	+	0.649006i	$0.775185\pi$
$72$		0			0
$73$	1.57548e8			0.649322			0.324661	−	0.945830i	$-0.394750\pi$
$73$	1.57548e8			0.649322			0.324661	+	0.945830i	$0.394750\pi$
$74$	−8.77334e7	+	1.51959e8i	−0.340112	+	0.589092i
$75$		0			0
$76$	3.57340e8	+	6.18931e8i	1.22863	+	2.12805i
$77$	4.76787e7	+	8.25819e7i	0.154567	+	0.267717i
$78$		0			0
$79$	−1.79317e7	+	3.10586e7i	−0.0517964	+	0.0897140i	−0.890761	−	0.454472i	$-0.849828\pi$
$79$	−1.79317e7	+	3.10586e7i	−0.0517964	+	0.0897140i	0.838965	+	0.544186i	$0.183161\pi$
$80$	1.19450e9			3.26047
$81$		0			0
$82$	−7.83069e8			−1.91266
$83$	4.03992e6	−	6.99735e6i	0.00934375	−	0.0161839i	−0.861316	−	0.508070i	$-0.830359\pi$
$83$	4.03992e6	−	6.99735e6i	0.00934375	−	0.0161839i	0.870659	+	0.491886i	$0.163692\pi$
$84$		0			0
$85$	−2.33501e8	−	4.04435e8i	−0.485180	−	0.840356i
$86$	7.34386e8	+	1.27199e9i	1.44771	+	2.50751i
$87$		0			0
$88$	−3.06972e8	+	5.31692e8i	−0.545667	+	0.945122i
$89$	−9.43654e8			−1.59425			−0.797127	−	0.603812i	$-0.793648\pi$
$89$	−9.43654e8			−1.59425			−0.797127	+	0.603812i	$0.793648\pi$
$90$		0			0
$91$	3.29823e7			0.0504191
$92$	−4.93415e7	+	8.54620e7i	−0.0718070	+	0.124373i
$93$		0			0
$94$	8.25654e8	+	1.43008e9i	1.09074	+	1.88922i
$95$	2.56053e8	+	4.43497e8i	0.322533	+	0.558643i
$96$		0			0
$97$	−4.83526e8	+	8.37491e8i	−0.554558	+	0.960522i	0.443380	+	0.896334i	$0.353779\pi$
$97$	−4.83526e8	+	8.37491e8i	−0.554558	+	0.960522i	−0.997938	+	0.0641886i	$0.979554\pi$
$98$	−1.37838e8			−0.150956
$99$		0			0
$100$	−1.24529e9			−1.24529
$101$	−8.41991e8	+	1.45837e9i	−0.805121	+	1.39451i	0.111088	+	0.993811i	$0.464566\pi$
$101$	−8.41991e8	+	1.45837e9i	−0.805121	+	1.39451i	−0.916209	+	0.400700i	$0.868767\pi$
$102$		0			0
$103$	−5.03057e7	−	8.71321e7i	−0.0440403	−	0.0762800i	0.843165	−	0.537655i	$-0.180690\pi$
$103$	−5.03057e7	−	8.71321e7i	−0.0440403	−	0.0762800i	−0.887205	+	0.461375i	$0.847356\pi$
$104$	1.06176e8	+	1.83902e8i	0.0889973	+	0.154148i
$105$		0			0
$106$	1.74977e9	−	3.03069e9i	1.34618	−	2.33166i
$107$	−5.50494e8			−0.406000			−0.203000	−	0.979179i	$-0.565069\pi$
$107$	−5.50494e8			−0.406000			−0.203000	+	0.979179i	$0.565069\pi$
$108$		0			0
$109$	1.31023e9			0.889055			0.444527	−	0.895765i	$-0.353372\pi$
$109$	1.31023e9			0.889055			0.444527	+	0.895765i	$0.353372\pi$
$110$	−3.37986e8	+	5.85409e8i	−0.220106	+	0.381235i
$111$		0			0
$112$	−3.74729e9	−	6.49049e9i	−2.25028	−	3.89759i
$113$	−3.87701e7	−	6.71518e7i	−0.0223689	−	0.0387440i	0.854624	−	0.519247i	$-0.173787\pi$
$113$	−3.87701e7	−	6.71518e7i	−0.0223689	−	0.0387440i	−0.876993	+	0.480503i	$0.840454\pi$
$114$		0			0
$115$	−3.53558e7	+	6.12380e7i	−0.0188504	+	0.0326498i
$116$	5.01172e9			2.56996
$117$		0			0
$118$	2.28192e9			1.08351
$119$	−1.46504e9	+	2.53753e9i	−0.669713	+	1.15998i
$120$		0			0
$121$	1.07434e9	+	1.86082e9i	0.455626	+	0.789168i
$122$	−4.28588e8	−	7.42337e8i	−0.175154	−	0.303376i
$123$		0			0
$124$	−2.53599e9	+	4.39246e9i	−0.963274	+	1.66844i
$125$	−2.94431e9			−1.07867
$126$		0			0
$127$	−1.89287e9			−0.645661			−0.322830	−	0.946457i	$-0.604634\pi$
$127$	−1.89287e9			−0.645661			−0.322830	+	0.946457i	$0.604634\pi$
$128$	8.19617e9	−	1.41962e10i	2.69877	−	4.67441i
$129$		0			0
$130$	1.16903e8	+	2.02482e8i	0.0358989	+	0.0621788i
$131$	−2.31093e9	−	4.00265e9i	−0.685593	−	1.18748i	−0.973250	−	0.229749i	$-0.926210\pi$
$131$	−2.31093e9	−	4.00265e9i	−0.685593	−	1.18748i	0.287657	−	0.957734i	$-0.407124\pi$
$132$		0			0
$133$	1.60654e9	−	2.78261e9i	0.445204	−	0.771116i
$134$	−9.55538e9			−2.56022
$135$		0			0
$136$	−1.88649e10			−4.72857
$137$	−3.41347e9	+	5.91231e9i	−0.827854	+	1.43389i	0.0718644	+	0.997414i	$0.477105\pi$
$137$	−3.41347e9	+	5.91231e9i	−0.827854	+	1.43389i	−0.899718	+	0.436471i	$0.856228\pi$
$138$		0			0
$139$	−2.43834e9	−	4.22333e9i	−0.554023	−	0.959596i	−0.997979	−	0.0635471i	$-0.979759\pi$
$139$	−2.43834e9	−	4.22333e9i	−0.554023	−	0.959596i	0.443956	−	0.896049i	$-0.353575\pi$
$140$	−5.07718e9	−	8.79393e9i	−1.11698	−	1.93467i
$141$		0			0
$142$	−7.24537e9	+	1.25493e10i	−1.49542	+	2.59014i
$143$	−7.23801e7			−0.0144746
$144$		0			0
$145$	3.59117e9			0.674651
$146$	3.50364e9	−	6.06849e9i	0.638163	−	1.10533i
$147$		0			0
$148$	2.89219e9	+	5.00941e9i	0.495506	+	0.858241i
$149$	3.82468e9	+	6.62455e9i	0.635708	+	1.10108i	0.986365	+	0.164574i	$0.0526250\pi$
$149$	3.82468e9	+	6.62455e9i	0.635708	+	1.10108i	−0.350657	+	0.936504i	$0.614042\pi$
$150$		0			0
$151$	2.09794e9	−	3.63374e9i	0.328396	−	0.568798i	−0.653798	−	0.756669i	$-0.726825\pi$
$151$	2.09794e9	−	3.63374e9i	0.328396	−	0.568798i	0.982194	+	0.187871i	$0.0601587\pi$
$152$	2.06870e10			3.14341
$153$		0			0
$154$	4.24122e9			0.607642
$155$	−1.81717e9	+	3.14743e9i	−0.252873	+	0.437990i
$156$		0			0
$157$	3.80239e8	+	6.58594e8i	0.0499469	+	0.0865106i	0.889918	−	0.456121i	$-0.150761\pi$
$157$	3.80239e8	+	6.58594e8i	0.0499469	+	0.0865106i	−0.839971	+	0.542631i	$0.817428\pi$
$158$	7.97550e8	+	1.38140e9i	0.101813	+	0.176344i
$159$		0			0
$160$	1.51491e10	−	2.62389e10i	1.82745	−	3.16524i
$161$	4.43662e8			0.0520398
$162$		0			0
$163$	−7.55113e9			−0.837853			−0.418926	−	0.908020i	$-0.637593\pi$
$163$	−7.55113e9			−0.837853			−0.418926	+	0.908020i	$0.637593\pi$
$164$	−1.29072e10	+	2.23559e10i	−1.39327	+	2.41321i
$165$		0			0
$166$	−1.79684e8	−	3.11222e8i	−0.0183664	−	0.0318115i
$167$	4.02882e9	+	6.97812e9i	0.400824	+	0.694248i	0.993826	−	0.110954i	$-0.0353905\pi$
$167$	4.02882e9	+	6.97812e9i	0.400824	+	0.694248i	−0.593002	+	0.805201i	$0.702057\pi$
$168$		0			0
$169$	5.28973e9	−	9.16208e9i	0.498820	−	0.863981i
$170$	−2.07709e10			−1.90737
$171$		0			0
$172$	4.84190e10			4.21830
$173$	2.06159e9	−	3.57078e9i	0.174982	−	0.303079i	−0.765173	−	0.643825i	$-0.777346\pi$
$173$	2.06159e9	−	3.57078e9i	0.174982	−	0.303079i	0.940155	+	0.340747i	$0.110680\pi$
$174$		0			0
$175$	2.79930e9	+	4.84853e9i	0.225620	+	0.390786i
$176$	8.22346e9	+	1.42434e10i	0.646022	+	1.11894i
$177$		0			0
$178$	−2.09855e10	+	3.63480e10i	−1.56686	+	2.71388i
$179$	−2.46170e10			−1.79224			−0.896120	−	0.443811i	$-0.853626\pi$
$179$	−2.46170e10			−1.79224			−0.896120	+	0.443811i	$0.853626\pi$
$180$		0			0
$181$	−4.17661e9			−0.289248			−0.144624	−	0.989487i	$-0.546197\pi$
$181$	−4.17661e9			−0.289248			−0.144624	+	0.989487i	$0.546197\pi$
$182$	7.33480e8	−	1.27043e9i	0.0495527	−	0.0858278i
$183$		0			0
$184$	1.42823e9	+	2.47376e9i	0.0918581	+	0.159103i
$185$	2.07240e9	+	3.58951e9i	0.130077	+	0.225301i
$186$		0			0
$187$	3.21505e9	−	5.56862e9i	0.192265	−	0.333013i
$188$	5.44364e10			3.17818
$189$		0			0
$190$	2.27770e10			1.26796
$191$	7.32150e9	−	1.26812e10i	0.398061	−	0.689462i	−0.595426	−	0.803411i	$-0.703017\pi$
$191$	7.32150e9	−	1.26812e10i	0.398061	−	0.689462i	0.993487	+	0.113948i	$0.0363499\pi$
$192$		0			0
$193$	−7.93330e9	−	1.37409e10i	−0.411572	−	0.712864i	0.583490	−	0.812120i	$-0.301687\pi$
$193$	−7.93330e9	−	1.37409e10i	−0.411572	−	0.712864i	−0.995062	+	0.0992567i	$0.968354\pi$
$194$	2.15058e10	+	3.72492e10i	1.09006	+	1.88803i
$195$		0			0
$196$	−2.27195e9	+	3.93514e9i	−0.109963	+	0.190462i
$197$	4.01582e9			0.189966			0.0949832	−	0.995479i	$-0.469720\pi$
$197$	4.01582e9			0.189966			0.0949832	+	0.995479i	$0.469720\pi$
$198$		0			0
$199$	−3.90309e10			−1.76429			−0.882144	−	0.470980i	$-0.843901\pi$
$199$	−3.90309e10			−1.76429			−0.882144	+	0.470980i	$0.843901\pi$
$200$	−1.80229e10	+	3.12166e10i	−0.796507	+	1.37959i
$201$		0			0
$202$	3.74493e10	+	6.48642e10i	1.58257	+	2.74109i
$203$	−1.12659e10	−	1.95132e10i	−0.465624	−	0.806484i
$204$		0			0
$205$	−9.24867e9	+	1.60192e10i	−0.365752	+	0.633502i
$206$	−4.47491e9			−0.173134
$207$		0			0
$208$	5.68869e9			0.210730
$209$	−3.52557e9	+	6.10647e9i	−0.127812	+	0.221377i
$210$		0			0
$211$	−1.35050e10	−	2.33913e10i	−0.469054	−	0.812425i	0.530320	−	0.847797i	$-0.322072\pi$
$211$	−1.35050e10	−	2.33913e10i	−0.469054	−	0.812425i	−0.999374	+	0.0353723i	$0.988738\pi$
$212$	−5.76822e10	−	9.99086e10i	−1.96124	−	3.39697i
$213$		0			0
$214$	−1.22422e10	+	2.12041e10i	−0.399023	+	0.691128i
$215$	3.46948e10			1.10737
$216$		0			0
$217$	2.28028e10			0.698102
$218$	2.91377e10	−	5.04679e10i	0.873776	−	1.51343i
$219$		0			0
$220$	1.11419e10	+	1.92984e10i	0.320670	+	0.555416i
$221$	−1.11203e9	−	1.92608e9i	−0.0313581	−	0.0543138i
$222$		0			0
$223$	2.86518e10	−	4.96264e10i	0.775854	−	1.34382i	−0.158459	−	0.987366i	$-0.550653\pi$
$223$	2.86518e10	−	4.96264e10i	0.775854	−	1.34382i	0.934313	−	0.356453i	$-0.116014\pi$
$224$	−1.90098e11			−5.04500
$225$		0			0
$226$	−3.44877e9			−0.0879379
$227$	1.75028e10	−	3.03158e10i	0.437514	−	0.757797i	−0.559983	−	0.828504i	$-0.689192\pi$
$227$	1.75028e10	−	3.03158e10i	0.437514	−	0.757797i	0.997497	+	0.0707073i	$0.0225256\pi$
$228$		0			0
$229$	−3.23765e10	−	5.60778e10i	−0.777984	−	1.34751i	−0.933102	−	0.359611i	$-0.882909\pi$
$229$	−3.23765e10	−	5.60778e10i	−0.777984	−	1.34751i	0.155118	−	0.987896i	$-0.450424\pi$
$230$	1.57252e9	+	2.72369e9i	0.0370529	+	0.0641775i
$231$		0			0
$232$	7.25342e10	−	1.25633e11i	1.64379	−	2.84713i
$233$	5.91061e10			1.31380			0.656902	−	0.753976i	$-0.271866\pi$
$233$	5.91061e10			1.31380			0.656902	+	0.753976i	$0.271866\pi$
$234$		0			0
$235$	3.90066e10			0.834319
$236$	3.76124e10	−	6.51466e10i	0.789273	−	1.36706i
$237$		0			0
$238$	6.51608e10	+	1.12862e11i	1.31641	+	2.28008i
$239$	−2.02351e10	−	3.50482e10i	−0.401158	−	0.694825i	0.592708	−	0.805417i	$-0.298059\pi$
$239$	−2.02351e10	−	3.50482e10i	−0.401158	−	0.694825i	−0.993866	+	0.110592i	$0.964725\pi$
$240$		0			0
$241$	−1.68545e9	+	2.91928e9i	−0.0321839	+	0.0557441i	−0.881669	−	0.471869i	$-0.843580\pi$
$241$	−1.68545e9	+	2.91928e9i	−0.0321839	+	0.0557441i	0.849485	+	0.527613i	$0.176913\pi$
$242$	9.55674e10			1.79119
$243$		0			0
$244$	−2.82573e10			−0.510361
$245$	−1.62797e9	+	2.81973e9i	−0.0288669	+	0.0499989i
$246$		0			0
$247$	1.21943e9	+	2.11211e9i	0.0208459	+	0.0361061i
$248$	7.34063e10	+	1.27143e11i	1.23225	+	2.13433i
$249$		0			0
$250$	−6.54772e10	+	1.13410e11i	−1.06013	+	1.83620i
$251$	4.69295e10			0.746301			0.373151	−	0.927771i	$-0.378277\pi$
$251$	4.69295e10			0.746301			0.373151	+	0.927771i	$0.378277\pi$
$252$		0			0
$253$	−9.73621e8			−0.0149399
$254$	−4.20947e10	+	7.29102e10i	−0.634565	+	1.09910i
$255$		0			0
$256$	−1.85207e11	−	3.20788e11i	−2.69511	−	4.66808i
$257$	−8.74728e9	−	1.51507e10i	−0.125076	−	0.216638i	0.796687	−	0.604393i	$-0.206584\pi$
$257$	−8.74728e9	−	1.51507e10i	−0.125076	−	0.216638i	−0.921763	+	0.387755i	$0.873251\pi$
$258$		0			0
$259$	1.30028e10	−	2.25215e10i	0.179551	−	0.310991i
$260$	7.70757e9			0.104601
$261$		0			0
$262$	−2.05567e11			−2.69525
$263$	5.03147e10	−	8.71475e10i	0.648476	−	1.12319i	−0.335011	−	0.942214i	$-0.608740\pi$
$263$	5.03147e10	−	8.71475e10i	0.648476	−	1.12319i	0.983487	−	0.180979i	$-0.0579265\pi$
$264$		0			0
$265$	−4.13324e10	−	7.15898e10i	−0.514854	−	0.891753i
$266$	−7.14543e10	−	1.23763e11i	−0.875107	−	1.51573i
$267$		0			0
$268$	−1.57500e11	+	2.72797e11i	−1.86497	+	3.23023i
$269$	−2.18445e10			−0.254365			−0.127182	−	0.991879i	$-0.540593\pi$
$269$	−2.18445e10			−0.254365			−0.127182	+	0.991879i	$0.540593\pi$
$270$		0			0
$271$	5.19462e10			0.585049			0.292524	−	0.956258i	$-0.405505\pi$
$271$	5.19462e10			0.585049			0.292524	+	0.956258i	$0.405505\pi$
$272$	−2.52685e11	+	4.37664e11i	−2.79911	+	4.84820i
$273$		0			0
$274$	1.51821e11	+	2.62962e11i	1.62726	+	2.81849i
$275$	−6.14309e9	−	1.06401e10i	−0.0647724	−	0.112189i
$276$		0			0
$277$	4.18852e9	−	7.25473e9i	0.0427466	−	0.0740393i	−0.843861	−	0.536563i	$-0.819723\pi$
$277$	4.18852e9	−	7.25473e9i	0.0427466	−	0.0740393i	0.886607	+	0.462523i	$0.153056\pi$
$278$	−2.16901e11			−2.17801
$279$		0			0
$280$	−2.93926e11			−2.85777
$281$	4.08163e10	−	7.06959e10i	0.390530	−	0.676419i	−0.601989	−	0.798504i	$-0.705625\pi$
$281$	4.08163e10	−	7.06959e10i	0.390530	−	0.676419i	0.992520	+	0.122086i	$0.0389583\pi$
$282$		0			0
$283$	3.05738e10	+	5.29553e10i	0.283341	+	0.490762i	0.972206	−	0.234129i	$-0.0752237\pi$
$283$	3.05738e10	+	5.29553e10i	0.283341	+	0.490762i	−0.688864	+	0.724890i	$0.741890\pi$
$284$	2.38848e11	+	4.13697e11i	2.17866	+	3.77355i
$285$		0			0
$286$	−1.60963e9	+	2.78796e9i	−0.0142259	+	0.0246399i
$287$	1.16057e11			1.00972
$288$		0			0
$289$	7.89921e10			0.666106
$290$	7.98624e10	−	1.38326e11i	0.663058	−	1.14845i
$291$		0			0
$292$	−1.15500e11	−	2.00051e11i	−0.929732	−	1.61034i
$293$	8.67297e10	+	1.50220e11i	0.687486	+	1.19076i	0.972649	+	0.232281i	$0.0746189\pi$
$293$	8.67297e10	+	1.50220e11i	0.687486	+	1.19076i	−0.285163	+	0.958479i	$0.592048\pi$
$294$		0			0
$295$	2.69513e10	−	4.66810e10i	0.207196	−	0.358873i
$296$	1.67433e11			1.26774
$297$		0			0
$298$	3.40222e11			2.49913
$299$	−1.68379e8	+	2.91641e8i	−0.00121834	+	0.00211022i
$300$		0			0
$301$	−1.08842e11	−	1.88520e11i	−0.764270	−	1.32375i
$302$	−9.33105e10	−	1.61619e11i	−0.645505	−	1.11805i
$303$		0			0
$304$	2.77091e11	−	4.79935e11i	1.86076	−	3.22294i
$305$	−2.02479e10			−0.133977
$306$		0			0
$307$	−1.89397e11			−1.21689			−0.608443	−	0.793598i	$-0.708205\pi$
$307$	−1.89397e11			−1.21689			−0.608443	+	0.793598i	$0.708205\pi$
$308$	6.99072e10	−	1.21083e11i	0.442633	−	0.766663i
$309$		0			0
$310$	8.08226e10	+	1.39989e11i	0.497056	+	0.860926i
$311$	7.38382e10	+	1.27892e11i	0.447569	+	0.775212i	0.998227	−	0.0595190i	$-0.0189567\pi$
$311$	7.38382e10	+	1.27892e11i	0.447569	+	0.775212i	−0.550659	+	0.834731i	$0.685623\pi$
$312$		0			0
$313$	−2.32124e9	+	4.02051e9i	−0.0136701	+	0.0236773i	−0.872780	−	0.488115i	$-0.837685\pi$
$313$	−2.32124e9	+	4.02051e9i	−0.0136701	+	0.0236773i	0.859109	+	0.511792i	$0.171018\pi$
$314$	3.38239e10			0.196354
$315$		0			0
$316$	5.25835e10			0.296659
$317$	8.81821e10	−	1.52736e11i	0.490471	−	0.849521i	−0.509468	−	0.860489i	$-0.670158\pi$
$317$	8.81821e10	−	1.52736e11i	0.490471	−	0.849521i	0.999940	+	0.0109679i	$0.00349126\pi$
$318$		0			0
$319$	2.47232e10	+	4.28219e10i	0.133674	+	0.231530i
$320$	−3.67996e11	−	6.37387e11i	−1.96186	−	3.39804i
$321$		0			0
$322$	9.86641e9	−	1.70891e10i	0.0511456	−	0.0885867i
$323$	−2.16663e11			−1.10758
$324$		0			0
$325$	−4.24956e9			−0.0211285
$326$	−1.67926e11	+	2.90857e11i	−0.823454	+	1.42626i
$327$		0			0
$328$	3.73609e11	+	6.47109e11i	1.78232	+	3.08706i
$329$	−1.22369e11	−	2.11948e11i	−0.575822	−	0.997353i
$330$		0			0
$331$	1.54928e11	−	2.68344e11i	0.709423	−	1.22876i	−0.255649	−	0.966770i	$-0.582289\pi$
$331$	1.54928e11	−	2.68344e11i	0.709423	−	1.22876i	0.965072	−	0.261987i	$-0.0843776\pi$
$332$	−1.18468e10			−0.0535155
$333$		0			0
$334$	3.58381e11			1.57574
$335$	−1.12857e11	+	1.95474e11i	−0.489583	+	0.847982i
$336$		0			0
$337$	−1.56829e11	−	2.71636e11i	−0.662356	−	1.14723i	−0.979995	−	0.199023i	$-0.936223\pi$
$337$	−1.56829e11	−	2.71636e11i	−0.662356	−	1.14723i	0.317639	−	0.948212i	$-0.397110\pi$
$338$	−2.35272e11	−	4.07503e11i	−0.980495	−	1.69827i
$339$		0			0
$340$	−3.42362e11	+	5.92989e11i	−1.38941	+	2.40653i
$341$	−5.00409e10			−0.200415
$342$		0			0
$343$	−2.45577e11			−0.957996
$344$	7.00763e11	−	1.21376e12i	2.69810	−	4.67325i
$345$		0			0
$346$	−9.16936e10	−	1.58818e11i	−0.343951	−	0.595740i
$347$	−4.05420e10	−	7.02209e10i	−0.150115	−	0.260006i	0.781155	−	0.624337i	$-0.214631\pi$
$347$	−4.05420e10	−	7.02209e10i	−0.150115	−	0.260006i	−0.931269	+	0.364331i	$0.881298\pi$
$348$		0			0
$349$	−1.34389e11	+	2.32768e11i	−0.484896	+	0.839864i	−0.999849	−	0.0173540i	$-0.994476\pi$
$349$	−1.34389e11	+	2.32768e11i	−0.484896	+	0.839864i	0.514954	+	0.857218i	$0.327809\pi$
$350$	2.49010e11			0.886972
$351$		0			0
$352$	4.17172e11			1.44835
$353$	−1.96480e11	+	3.40313e11i	−0.673491	+	1.16652i	0.303416	+	0.952858i	$0.401873\pi$
$353$	−1.96480e11	+	3.40313e11i	−0.673491	+	1.16652i	−0.976907	+	0.213663i	$0.931461\pi$
$354$		0			0
$355$	1.71147e11	+	2.96436e11i	0.571929	+	0.990611i
$356$	6.91800e11	+	1.19823e12i	2.28274	+	3.95381i
$357$		0			0
$358$	−5.47447e11	+	9.48206e11i	−1.76144	+	3.05091i
$359$	−2.48781e11			−0.790484			−0.395242	−	0.918577i	$-0.629339\pi$
$359$	−2.48781e11			−0.790484			−0.395242	+	0.918577i	$0.629339\pi$
$360$		0			0
$361$	−8.50984e10			−0.263718
$362$	−9.28819e10	+	1.60876e11i	−0.284277	+	0.492383i
$363$		0			0
$364$	−2.41796e10	−	4.18803e10i	−0.0721927	−	0.125041i
$365$	−8.27617e10	−	1.43347e11i	−0.244068	−	0.422739i
$366$		0			0
$367$	1.62624e11	−	2.81673e11i	0.467937	−	0.810491i	−0.531391	−	0.847126i	$-0.678331\pi$
$367$	1.62624e11	−	2.81673e11i	0.467937	−	0.810491i	0.999329	+	0.0366351i	$0.0116639\pi$
$368$	7.65214e10			0.217504
$369$		0			0
$370$	1.84349e11			0.511368
$371$	−2.59330e11	+	4.49172e11i	−0.710673	+	1.23092i
$372$		0			0
$373$	1.77616e11	+	3.07640e11i	0.475108	+	0.822912i	0.999594	−	0.0285077i	$-0.00907553\pi$
$373$	1.77616e11	+	3.07640e11i	0.475108	+	0.822912i	−0.524485	+	0.851420i	$0.675742\pi$
$374$	−1.42996e11	−	2.47677e11i	−0.377922	−	0.654580i
$375$		0			0
$376$	7.87853e11	−	1.36460e12i	2.03282	−	3.52095i
$377$	1.71026e10			0.0436040
$378$		0			0
$379$	−4.17590e11			−1.03962			−0.519809	−	0.854283i	$-0.673997\pi$
$379$	−4.17590e11			−1.03962			−0.519809	+	0.854283i	$0.673997\pi$
$380$	3.75429e11	−	6.50262e11i	0.923637	−	1.59979i
$381$		0			0
$382$	−3.25639e11	−	5.64024e11i	−0.782441	−	1.35523i
$383$	−4.19861e9	−	7.27221e9i	−0.00997038	−	0.0172692i	0.860997	−	0.508610i	$-0.169840\pi$
$383$	−4.19861e9	−	7.27221e9i	−0.00997038	−	0.0172692i	−0.870968	+	0.491341i	$0.836507\pi$
$384$		0			0
$385$	5.00922e10	−	8.67623e10i	0.116198	−	0.201260i
$386$	−7.05701e11			−1.61800
$387$		0			0
$388$	1.41791e12			3.17618
$389$	−2.23902e11	+	3.87809e11i	−0.495774	+	0.858706i	−0.999988	−	0.00487245i	$-0.998449\pi$
$389$	−2.23902e11	+	3.87809e11i	−0.495774	+	0.858706i	0.504214	+	0.863579i	$0.331782\pi$
$390$		0			0
$391$	−1.49584e10	−	2.59087e10i	−0.0323661	−	0.0560597i
$392$	6.57635e10	+	1.13906e11i	0.140669	+	0.243645i
$393$		0			0
$394$	8.93062e10	−	1.54683e11i	0.186702	−	0.323377i
$395$	3.76788e10			0.0778773
$396$		0			0
$397$	8.93449e11			1.80515			0.902573	−	0.430537i	$-0.141676\pi$
$397$	8.93449e11			1.80515			0.902573	+	0.430537i	$0.141676\pi$
$398$	−8.67991e11	+	1.50340e12i	−1.73397	+	3.00332i
$399$		0			0
$400$	4.82814e11	+	8.36258e11i	0.942996	+	1.63332i
$401$	1.10853e10	+	1.92004e10i	0.0214092	+	0.0370817i	0.876531	−	0.481345i	$-0.159851\pi$
$401$	1.10853e10	+	1.92004e10i	0.0214092	+	0.0370817i	−0.855122	+	0.518426i	$0.826518\pi$
$402$		0			0
$403$	−8.65412e9	+	1.49894e10i	−0.0163437	+	0.0283081i
$404$	2.46908e12			4.61126
$405$		0			0
$406$	−1.00215e12			−1.83049
$407$	−2.85347e10	+	4.94236e10i	−0.0515465	+	0.0892812i
$408$		0			0
$409$	−2.39862e11	−	4.15453e11i	−0.423844	−	0.734120i	0.572468	−	0.819927i	$-0.305986\pi$
$409$	−2.39862e11	−	4.15453e11i	−0.423844	−	0.734120i	−0.996312	+	0.0858078i	$0.972653\pi$
$410$	4.11355e11	+	7.12487e11i	0.718934	+	1.24523i
$411$		0			0
$412$	−7.37590e10	+	1.27754e11i	−0.126118	+	0.218443i
$413$	−3.38198e11			−0.572000
$414$		0			0
$415$	−8.48885e9			−0.0140486
$416$	7.21461e10	−	1.24961e11i	0.118112	−	0.204575i
$417$		0			0
$418$	1.56807e11	+	2.71598e11i	0.251231	+	0.435144i
$419$	4.17661e11	+	7.23411e11i	0.662005	+	1.14663i	0.980088	+	0.198563i	$0.0636275\pi$
$419$	4.17661e11	+	7.23411e11i	0.662005	+	1.14663i	−0.318083	+	0.948063i	$0.603039\pi$
$420$		0			0
$421$	−1.03140e11	+	1.78644e11i	−0.160014	+	0.277153i	−0.934874	−	0.354981i	$-0.884487\pi$
$421$	−1.03140e11	+	1.78644e11i	−0.160014	+	0.277153i	0.774859	+	0.632134i	$0.217821\pi$
$422$	−1.20133e12			−1.84397
$423$		0			0
$424$	−3.33932e12			−5.01778
$425$	1.88761e11	−	3.26944e11i	0.280648	−	0.486097i
$426$		0			0
$427$	6.35202e10	+	1.10020e11i	0.0924669	+	0.160157i
$428$	4.03572e11	+	6.99007e11i	0.581332	+	1.00690i
$429$		0			0
$430$	7.71562e11	−	1.33638e12i	1.08834	−	1.88505i
$431$	−3.46311e11			−0.483414			−0.241707	−	0.970349i	$-0.577707\pi$
$431$	−3.46311e11			−0.483414			−0.241707	+	0.970349i	$0.577707\pi$
$432$		0			0
$433$	−7.41282e9			−0.0101342			−0.00506708	−	0.999987i	$-0.501613\pi$
$433$	−7.41282e9			−0.0101342			−0.00506708	+	0.999987i	$0.501613\pi$
$434$	5.07101e11	−	8.78325e11i	0.686105	−	1.18837i
$435$		0			0
$436$	−9.60541e11	−	1.66370e12i	−1.27299	−	2.20489i
$437$	1.64032e10	+	2.84111e10i	0.0215160	+	0.0372668i
$438$		0			0
$439$	−4.79907e11	+	8.31224e11i	−0.616690	+	1.06814i	0.373396	+	0.927672i	$0.378193\pi$
$439$	−4.79907e11	+	8.31224e11i	−0.616690	+	1.06814i	−0.990085	+	0.140466i	$0.955140\pi$
$440$	6.45024e11			0.820425
$441$		0			0
$442$	−9.89194e10			−0.123277
$443$	1.44256e11	−	2.49858e11i	0.177958	−	0.308232i	−0.763223	−	0.646135i	$-0.776384\pi$
$443$	1.44256e11	−	2.49858e11i	0.177958	−	0.308232i	0.941181	+	0.337903i	$0.109718\pi$
$444$		0			0
$445$	4.95711e11	+	8.58597e11i	0.599251	+	1.03793i
$446$	−1.27435e12	−	2.20724e12i	−1.52504	−	2.64145i
$447$		0			0
$448$	−2.30890e12	+	3.99913e12i	−2.70803	+	4.69045i
$449$	−1.00029e12			−1.16150			−0.580749	−	0.814083i	$-0.697240\pi$
$449$	−1.00029e12			−1.16150			−0.580749	+	0.814083i	$0.697240\pi$
$450$		0			0
$451$	−2.54688e11			−0.289878
$452$	−5.68454e10	+	9.84591e10i	−0.0640578	+	0.110951i
$453$		0			0
$454$	−7.78476e11	−	1.34836e12i	−0.859991	−	1.48955i
$455$	−1.73260e10	−	3.00095e10i	−0.0189516	−	0.0328252i
$456$		0			0
$457$	−1.41927e11	+	2.45826e11i	−0.152210	+	0.263636i	−0.932040	−	0.362356i	$-0.881972\pi$
$457$	−1.41927e11	+	2.45826e11i	−0.152210	+	0.263636i	0.779830	+	0.625992i	$0.215306\pi$
$458$	−2.88003e12			−3.05846
$459$		0			0
$460$	1.03678e11			0.107964
$461$	−3.22910e8	+	5.59296e8i	−0.000332987	+	0.000576750i	−0.866192	−	0.499712i	$-0.833439\pi$
$461$	−3.22910e8	+	5.59296e8i	−0.000332987	+	0.000576750i	0.865859	+	0.500288i	$0.166773\pi$
$462$		0			0
$463$	2.41535e11	+	4.18352e11i	0.244268	+	0.423084i	0.961926	−	0.273312i	$-0.0881190\pi$
$463$	2.41535e11	+	4.18352e11i	0.244268	+	0.423084i	−0.717658	+	0.696396i	$0.754786\pi$
$464$	−1.94311e12	−	3.36557e12i	−1.94611	−	3.37076i
$465$		0			0
$466$	1.31444e12	−	2.27667e12i	1.29123	−	2.23647i
$467$	7.10128e11			0.690893			0.345447	−	0.938438i	$-0.387727\pi$
$467$	7.10128e11			0.690893			0.345447	+	0.938438i	$0.387727\pi$
$468$		0			0
$469$	1.41618e12			1.35158
$470$	8.67450e11	−	1.50247e12i	0.819982	−	1.42025i
$471$		0			0
$472$	−1.08872e12	−	1.88572e12i	−1.00967	−	1.74879i
$473$	2.38855e11	+	4.13708e11i	0.219411	+	0.380031i
$474$		0			0
$475$	−2.06992e11	+	3.58521e11i	−0.186566	+	0.323142i
$476$	4.29613e12			3.83572
$477$		0			0
$478$	−1.80000e12			−1.57706
$479$	7.73097e9	−	1.33904e10i	0.00671002	−	0.0116221i	−0.862651	−	0.505800i	$-0.831197\pi$
$479$	7.73097e9	−	1.33904e10i	0.00671002	−	0.0116221i	0.869361	+	0.494178i	$0.164531\pi$
$480$		0			0
$481$	9.86964e9	+	1.70947e10i	0.00840715	+	0.0145616i
$482$	7.49638e10	+	1.29841e11i	0.0632616	+	0.109572i
$483$		0			0
$484$	1.57522e12	−	2.72836e12i	1.30478	−	2.25994i
$485$	1.01601e12			0.833793
$486$		0			0
$487$	−5.65405e11			−0.455490			−0.227745	−	0.973721i	$-0.573135\pi$
$487$	−5.65405e11			−0.455490			−0.227745	+	0.973721i	$0.573135\pi$
$488$	−4.08966e11	+	7.08349e11i	−0.326436	+	0.565403i
$489$		0			0
$490$	7.24076e10	+	1.25414e11i	0.0567416	+	0.0982794i
$491$	−1.86366e11	−	3.22796e11i	−0.144711	−	0.250646i	0.784554	−	0.620060i	$-0.212892\pi$
$491$	−1.86366e11	−	3.22796e11i	−0.144711	−	0.250646i	−0.929265	+	0.369414i	$0.879558\pi$
$492$		0			0
$493$	−7.59680e11	+	1.31580e12i	−0.579188	+	1.00318i
$494$	1.08473e11			0.0819506
$495$		0			0
$496$	3.93295e12			2.91777
$497$	1.07382e12	−	1.85991e12i	0.789456	−	1.36738i
$498$		0			0
$499$	1.78807e11	+	3.09703e11i	0.129102	+	0.223611i	0.923329	−	0.384010i	$-0.125457\pi$
$499$	1.78807e11	+	3.09703e11i	0.129102	+	0.223611i	−0.794227	+	0.607621i	$0.792124\pi$
$500$	2.15850e12	+	3.73863e12i	1.54449	+	2.67514i
$501$		0			0
$502$	1.04365e12	−	1.80765e12i	0.733476	−	1.27042i
$503$	1.08145e12			0.753267			0.376633	−	0.926362i	$-0.377082\pi$
$503$	1.08145e12			0.753267			0.376633	+	0.926362i	$0.377082\pi$
$504$		0			0
$505$	1.76923e12			1.21052
$506$	−2.16519e10	+	3.75023e10i	−0.0146832	+	0.0254320i
$507$		0			0
$508$	1.38768e12	+	2.40353e12i	0.924490	+	1.60126i
$509$	−3.48650e11	−	6.03880e11i	−0.230229	−	0.398768i	0.727646	−	0.685952i	$-0.240614\pi$
$509$	−3.48650e11	−	6.03880e11i	−0.230229	−	0.398768i	−0.957875	+	0.287184i	$0.907281\pi$
$510$		0			0
$511$	−5.19267e11	+	8.99398e11i	−0.336897	+	0.583523i
$512$	−8.08206e12			−5.19765
$513$		0			0
$514$	−7.78108e11			−0.491706
$515$	−5.28523e10	+	9.15428e10i	−0.0331079	+	0.0573445i
$516$		0			0
$517$	2.68539e11	+	4.65123e11i	0.165310	+	0.286326i
$518$	−5.78327e11	−	1.00169e12i	−0.352931	−	0.611294i
$519$		0			0
$520$	1.11551e11	−	1.93212e11i	0.0669049	−	0.115883i
$521$	1.96135e12			1.16623			0.583116	−	0.812389i	$-0.301833\pi$
$521$	1.96135e12			1.16623			0.583116	+	0.812389i	$0.301833\pi$
$522$		0			0
$523$	−6.03647e10			−0.0352797			−0.0176399	−	0.999844i	$-0.505615\pi$
$523$	−6.03647e10			−0.0352797			−0.0176399	+	0.999844i	$0.505615\pi$
$524$	−3.38833e12	+	5.86875e12i	−1.96334	+	3.40060i
$525$		0			0
$526$	−2.23785e12	−	3.87607e12i	−1.27466	−	2.20778i
$527$	−7.68814e11	−	1.33162e12i	−0.434183	−	0.752028i
$528$		0			0
$529$	8.98311e11	−	1.55592e12i	0.498743	−	0.863847i
$530$	−3.67669e12			−2.02402
$531$		0			0
$532$	−4.71107e12			−2.54987
$533$	−4.40460e10	+	7.62899e10i	−0.0236393	+	0.0409444i
$534$		0			0
$535$	2.89181e11	+	5.00876e11i	0.152608	+	0.264325i
$536$	4.55895e12	+	7.89633e12i	2.38574	+	4.13223i
$537$		0			0
$538$	−4.85790e11	+	8.41413e11i	−0.249993	+	0.433001i
$539$	−4.48308e10			−0.0228785
$540$		0			0
$541$	−1.58437e12			−0.795185			−0.397592	−	0.917562i	$-0.630154\pi$
$541$	−1.58437e12			−0.795185			−0.397592	+	0.917562i	$0.630154\pi$
$542$	1.15521e12	−	2.00088e12i	0.574995	−	0.995920i
$543$		0			0
$544$	6.40930e12	+	1.11012e13i	3.13773	+	5.43471i
$545$	−6.88278e11	−	1.19213e12i	−0.334179	−	0.578816i
$546$		0			0
$547$	−7.36182e11	+	1.27510e12i	−0.351595	+	0.608980i	−0.986529	−	0.163586i	$-0.947694\pi$
$547$	−7.36182e11	+	1.27510e12i	−0.351595	+	0.608980i	0.634934	+	0.772566i	$0.281027\pi$
$548$	1.00098e13			4.74146
$549$		0			0
$550$	−5.46454e11			−0.254637
$551$	8.33053e11	−	1.44289e12i	0.385026	−	0.666885i
$552$		0			0
$553$	−1.18203e11	−	2.04734e11i	−0.0537485	−	0.0930952i
$554$	−1.86293e11	−	3.22669e11i	−0.0840240	−	0.145534i
$555$		0			0
$556$	−3.57513e12	+	6.19231e12i	−1.58656	+	2.74800i
$557$	1.23459e12			0.543470			0.271735	−	0.962372i	$-0.412403\pi$
$557$	1.23459e12			0.543470			0.271735	+	0.962372i	$0.412403\pi$
$558$		0			0
$559$	1.65231e11			0.0715711
$560$	−3.93698e12	+	6.81905e12i	−1.69168	+	2.93007i
$561$		0			0
$562$	−1.81539e12	−	3.14435e12i	−0.767638	−	1.32959i
$563$	1.61399e11	+	2.79551e11i	0.0677037	+	0.117266i	0.897890	−	0.440220i	$-0.145099\pi$
$563$	1.61399e11	+	2.79551e11i	0.0677037	+	0.117266i	−0.830186	+	0.557486i	$0.811766\pi$
$564$		0			0
$565$	−4.07327e10	+	7.05511e10i	−0.0168161	+	0.0291263i
$566$	2.71967e12			1.11389
$567$		0			0
$568$	1.38273e13			5.57404
$569$	−7.58978e11	+	1.31459e12i	−0.303546	+	0.525756i	−0.976936	−	0.213530i	$-0.931504\pi$
$569$	−7.58978e11	+	1.31459e12i	−0.303546	+	0.525756i	0.673391	+	0.739287i	$0.264837\pi$
$570$		0			0
$571$	−3.02035e11	−	5.23139e11i	−0.118903	−	0.205947i	0.800430	−	0.599426i	$-0.204605\pi$
$571$	−3.02035e11	−	5.23139e11i	−0.118903	−	0.205947i	−0.919333	+	0.393480i	$0.871271\pi$
$572$	5.30624e10	+	9.19068e10i	0.0207255	+	0.0358976i
$573$		0			0
$574$	2.58094e12	−	4.47032e12i	0.992373	−	1.71884i
$575$	−5.71630e10			−0.0218077
$576$		0			0
$577$	−4.16132e12			−1.56293			−0.781466	−	0.623948i	$-0.785528\pi$
$577$	−4.16132e12			−1.56293			−0.781466	+	0.623948i	$0.785528\pi$
$578$	1.75667e12	−	3.04265e12i	0.654660	−	1.13390i
$579$		0			0
$580$	−2.63271e12	−	4.55999e12i	−0.966001	−	1.67316i
$581$	2.66306e10	+	4.61256e10i	0.00969591	+	0.0167938i
$582$		0			0
$583$	5.69102e11	−	9.85713e11i	0.204024	−	0.353380i
$584$	−6.68646e12			−2.37869
$585$		0			0
$586$	7.71498e12			2.70269
$587$	6.48959e11	−	1.12403e12i	0.225604	−	0.390757i	−0.730897	−	0.682488i	$-0.760898\pi$
$587$	6.48959e11	−	1.12403e12i	0.225604	−	0.390757i	0.956500	+	0.291731i	$0.0942312\pi$
$588$		0			0
$589$	8.43069e11	+	1.46024e12i	0.288632	+	0.499925i
$590$	−1.19872e12	−	2.07624e12i	−0.407270	−	0.705412i
$591$		0			0
$592$	2.24268e12	−	3.88443e12i	0.750446	−	1.29981i
$593$	−6.89642e11			−0.229022			−0.114511	−	0.993422i	$-0.536530\pi$
$593$	−6.89642e11			−0.229022			−0.114511	+	0.993422i	$0.536530\pi$
$594$		0			0
$595$	3.07841e12			1.00693
$596$	5.60781e12	−	9.71302e12i	1.82048	−	3.15316i
$597$		0			0
$598$	7.48901e9	+	1.29713e10i	0.00239480	+	0.00414791i
$599$	−3.40832e11	−	5.90339e11i	−0.108173	−	0.187362i	0.806857	−	0.590747i	$-0.201167\pi$
$599$	−3.40832e11	−	5.90339e11i	−0.108173	−	0.187362i	−0.915030	+	0.403385i	$0.867833\pi$
$600$		0			0
$601$	2.36698e12	−	4.09974e12i	0.740049	−	1.28180i	−0.212424	−	0.977178i	$-0.568136\pi$
$601$	2.36698e12	−	4.09974e12i	0.740049	−	1.28180i	0.952473	−	0.304624i	$-0.0985309\pi$
$602$	−9.68195e12			−3.00454
$603$		0			0
$604$	−6.15208e12			−1.88086
$605$	1.12873e12	−	1.95501e12i	0.342523	−	0.593267i
$606$		0			0
$607$	1.69211e12	+	2.93082e12i	0.505917	+	0.876273i	0.999977	+	0.00684543i	$0.00217898\pi$
$607$	1.69211e12	+	2.93082e12i	0.505917	+	0.876273i	−0.494060	+	0.869428i	$0.664488\pi$
$608$	−7.02834e12	−	1.21734e13i	−2.08587	−	3.61283i
$609$		0			0
$610$	−4.50284e11	+	7.79915e11i	−0.131675	+	0.228067i
$611$	1.85765e11			0.0539237
$612$		0			0
$613$	−2.77900e12			−0.794906			−0.397453	−	0.917622i	$-0.630106\pi$
$613$	−2.77900e12			−0.794906			−0.397453	+	0.917622i	$0.630106\pi$
$614$	−4.21191e12	+	7.29525e12i	−1.19597	+	2.07149i
$615$		0			0
$616$	−2.02352e12	−	3.50484e12i	−0.566232	−	0.980743i
$617$	2.85307e12	+	4.94166e12i	0.792554	+	1.37274i	0.924381	+	0.381470i	$0.124582\pi$
$617$	2.85307e12	+	4.94166e12i	0.792554	+	1.37274i	−0.131827	+	0.991273i	$0.542084\pi$
$618$		0			0
$619$	−2.85011e12	+	4.93653e12i	−0.780285	+	1.35149i	0.151490	+	0.988459i	$0.451593\pi$
$619$	−2.85011e12	+	4.93653e12i	−0.780285	+	1.35149i	−0.931776	+	0.363035i	$0.881741\pi$
$620$	5.32873e12			1.44831
$621$		0			0
$622$	6.56823e12			1.75951
$623$	3.11022e12	−	5.38706e12i	0.827170	−	1.43270i
$624$		0			0
$625$	7.17264e11	+	1.24234e12i	0.188026	+	0.325671i
$626$	1.03242e11	+	1.78821e11i	0.0268703	+	0.0465408i
$627$		0			0
$628$	5.57513e11	−	9.65641e11i	0.143033	−	0.247741i
$629$	−1.75360e12			−0.446685
$630$		0			0
$631$	−7.58053e11			−0.190356			−0.0951782	−	0.995460i	$-0.530342\pi$
$631$	−7.58053e11			−0.190356			−0.0951782	+	0.995460i	$0.530342\pi$
$632$	7.61035e11	−	1.31815e12i	0.189748	−	0.328654i
$633$		0			0
$634$	−3.92209e12	−	6.79325e12i	−0.964086	−	1.66985i
$635$	9.94346e11	+	1.72226e12i	0.242692	+	0.420355i
$636$		0			0
$637$	−7.75308e9	+	1.34287e10i	−0.00186572	+	0.00323153i
$638$	2.19924e12			0.525507
$639$		0			0
$640$	−1.72222e13			−4.05768
$641$	−1.53875e12	+	2.66519e12i	−0.360003	+	0.623544i	−0.987961	−	0.154703i	$-0.950558\pi$
$641$	−1.53875e12	+	2.66519e12i	−0.360003	+	0.623544i	0.627958	+	0.778248i	$0.283891\pi$
$642$		0			0
$643$	−1.05213e12	−	1.82234e12i	−0.242727	−	0.420416i	0.718763	−	0.695255i	$-0.244709\pi$
$643$	−1.05213e12	−	1.82234e12i	−0.242727	−	0.420416i	−0.961490	+	0.274839i	$0.911375\pi$
$644$	−3.25252e11	−	5.63354e11i	−0.0745133	−	0.129061i
$645$		0			0
$646$	−4.81828e12	+	8.34550e12i	−1.08854	+	1.88541i
$647$	1.41884e12			0.318319			0.159160	−	0.987253i	$-0.449122\pi$
$647$	1.41884e12			0.318319			0.159160	+	0.987253i	$0.449122\pi$
$648$		0			0
$649$	7.42180e11			0.164213
$650$	−9.45042e10	+	1.63686e11i	−0.0207654	+	0.0359668i
$651$		0			0
$652$	5.53579e12	+	9.58827e12i	1.19968	+	2.07791i
$653$	1.70144e11	+	2.94698e11i	0.0366191	+	0.0634262i	0.883754	−	0.467952i	$-0.155008\pi$
$653$	1.70144e11	+	2.94698e11i	0.0366191	+	0.0634262i	−0.847135	+	0.531378i	$0.821674\pi$
$654$		0			0
$655$	−2.42792e12	+	4.20527e12i	−0.515404	+	0.892706i
$656$	2.00171e13			4.22022
$657$		0			0
$658$	−1.08852e13			−2.26371
$659$	−2.31012e12	+	4.00125e12i	−0.477145	+	0.826440i	−0.999657	−	0.0261924i	$-0.991662\pi$
$659$	−2.31012e12	+	4.00125e12i	−0.477145	+	0.826440i	0.522512	+	0.852632i	$0.324995\pi$
$660$		0			0
$661$	2.31748e12	+	4.01400e12i	0.472183	+	0.817845i	0.999493	−	0.0318281i	$-0.0101329\pi$
$661$	2.31748e12	+	4.01400e12i	0.472183	+	0.817845i	−0.527311	+	0.849673i	$0.676800\pi$
$662$	−6.89077e12	−	1.19352e13i	−1.39446	−	2.41528i
$663$		0			0
$664$	−1.71457e11	+	2.96973e11i	−0.0342295	+	0.0592872i
$665$	−3.37573e12			−0.669377
$666$		0			0
$667$	2.30056e11			0.0450057
$668$	5.90712e12	−	1.02314e13i	1.14784	−	1.98812i
$669$		0			0
$670$	5.01955e12	+	8.69411e12i	0.962339	+	1.66682i
$671$	−1.39396e11	−	2.41440e11i	−0.0265459	−	0.0459789i
$672$		0			0
$673$	5.02585e12	−	8.70503e12i	0.944370	−	1.63570i	0.187361	−	0.982291i	$-0.440007\pi$
$673$	5.02585e12	−	8.70503e12i	0.944370	−	1.63570i	0.757009	−	0.653405i	$-0.226660\pi$
$674$	−1.39506e13			−2.60390
$675$		0			0
$676$	−1.55118e13			−2.85694
$677$	−4.41589e12	+	7.64855e12i	−0.807922	+	1.39936i	0.106378	+	0.994326i	$0.466075\pi$
$677$	−4.41589e12	+	7.64855e12i	−0.807922	+	1.39936i	−0.914300	+	0.405037i	$0.867259\pi$
$678$		0			0
$679$	−3.18734e12	−	5.52063e12i	−0.575458	−	0.996723i
$680$	9.90995e12	+	1.71645e13i	1.77738	+	3.07852i
$681$		0			0
$682$	−1.11284e12	+	1.92749e12i	−0.196971	+	0.341164i
$683$	−1.10788e13			−1.94805			−0.974027	−	0.226430i	$-0.927294\pi$
$683$	−1.10788e13			−1.94805			−0.974027	+	0.226430i	$0.927294\pi$
$684$		0			0
$685$	7.17253e12			1.24470
$686$	−5.46128e12	+	9.45921e12i	−0.941534	+	1.63078i
$687$		0			0
$688$	−1.87727e13	−	3.25152e13i	−3.19432	−	5.53272i
$689$	−1.96842e11	−	3.40940e11i	−0.0332760	−	0.0576357i
$690$		0			0
$691$	3.16465e11	−	5.48134e11i	0.0528050	−	0.0914609i	−0.838415	−	0.545033i	$-0.816517\pi$
$691$	3.16465e11	−	5.48134e11i	0.0528050	−	0.0914609i	0.891220	+	0.453572i	$0.149850\pi$
$692$	−6.04547e12			−1.00220
$693$		0			0
$694$	−3.60639e12			−0.590140
$695$	−2.56177e12	+	4.43712e12i	−0.416494	+	0.721389i
$696$		0			0
$697$	−3.91295e12	−	6.77743e12i	−0.627996	−	1.08772i
$698$	5.97723e12	+	1.03529e13i	0.953126	+	1.65086i
$699$		0			0
$700$	4.10438e12	−	7.10899e12i	0.646109	−	1.11909i
$701$	3.84382e12			0.601219			0.300609	−	0.953747i	$-0.402810\pi$
$701$	3.84382e12			0.601219			0.300609	+	0.953747i	$0.402810\pi$
$702$		0			0
$703$	1.92297e12			0.296943
$704$	5.06690e12	−	8.77613e12i	0.777437	−	1.34656i
$705$		0			0
$706$	8.73887e12	+	1.51362e13i	1.32384	+	2.29295i
$707$	−5.55029e12	−	9.61338e12i	−0.835465	−	1.44707i
$708$		0			0
$709$	3.07967e12	−	5.33415e12i	0.457716	−	0.792788i	−0.541123	−	0.840943i	$-0.682001\pi$
$709$	3.07967e12	−	5.33415e12i	0.457716	−	0.792788i	0.998840	+	0.0481550i	$0.0153341\pi$
$710$	1.52243e13			2.24840
$711$		0			0
$712$	4.00494e13			5.84031
$713$	−1.16411e11	+	2.01630e11i	−0.0168691	+	0.0292181i
$714$		0			0
$715$	3.80220e10	+	6.58561e10i	0.00544074	+	0.00942364i
$716$	1.80469e13	+	3.12582e13i	2.56622	+	4.44483i
$717$		0			0
$718$	−5.53254e12	+	9.58265e12i	−0.776900	+	1.34563i
$719$	6.51194e12			0.908721			0.454361	−	0.890818i	$-0.349868\pi$
$719$	6.51194e12			0.908721			0.454361	+	0.890818i	$0.349868\pi$
$720$		0			0
$721$	6.63217e11			0.0914001
$722$	−1.89247e12	+	3.27785e12i	−0.259186	+	0.448923i
$723$		0			0
$724$	3.06191e12	+	5.30338e12i	0.414160	+	0.717347i
$725$	1.45154e12	+	2.51415e12i	0.195123	+	0.337963i
$726$		0			0
$727$	−1.02893e12	+	1.78215e12i	−0.136609	+	0.236614i	−0.926211	−	0.377006i	$-0.876954\pi$
$727$	−1.02893e12	+	1.78215e12i	−0.136609	+	0.236614i	0.789602	+	0.613620i	$0.210287\pi$
$728$	−1.39980e12			−0.184703
$729$		0			0
$730$	−7.36200e12			−0.959496
$731$	−7.33938e12	+	1.27122e13i	−0.950673	+	1.64661i
$732$		0			0
$733$	−3.98122e12	−	6.89568e12i	−0.509388	−	0.882286i	−0.999941	−	0.0108743i	$-0.996539\pi$
$733$	−3.98122e12	−	6.89568e12i	−0.509388	−	0.882286i	0.490553	−	0.871411i	$-0.336795\pi$
$734$	−7.23306e12	−	1.25280e13i	−0.919792	−	1.59313i
$735$		0			0
$736$	9.70473e11	−	1.68091e12i	0.121908	−	0.211151i
$737$	−3.10783e12			−0.388019
$738$		0			0
$739$	−4.20258e12			−0.518341			−0.259171	−	0.965832i	$-0.583449\pi$
$739$	−4.20258e12			−0.518341			−0.259171	+	0.965832i	$0.583449\pi$
$740$	3.03859e12	−	5.26300e12i	0.372503	−	0.645194i
$741$		0			0
$742$	1.15342e13	+	1.99779e13i	1.39692	+	2.41954i
$743$	2.73405e12	+	4.73551e12i	0.329122	+	0.570056i	0.982338	−	0.187116i	$-0.0599139\pi$
$743$	2.73405e12	+	4.73551e12i	0.329122	+	0.570056i	−0.653216	+	0.757172i	$0.726581\pi$
$744$		0			0
$745$	4.01830e12	−	6.95989e12i	0.477902	−	0.827750i
$746$	1.57997e13			1.86777
$747$		0			0
$748$	−9.42791e12			−1.10118
$749$	1.81439e12	−	3.14262e12i	0.210651	−	0.364858i
$750$		0			0
$751$	−3.73761e12	−	6.47374e12i	−0.428760	−	0.742635i	0.568003	−	0.823026i	$-0.307716\pi$
$751$	−3.73761e12	−	6.47374e12i	−0.428760	−	0.742635i	−0.996763	+	0.0803918i	$0.974383\pi$
$752$	−2.11057e13	−	3.65562e13i	−2.40669	−	4.16851i
$753$		0			0
$754$	3.80338e11	−	6.58764e11i	0.0428547	−	0.0742265i
$755$	−4.40829e12			−0.493752
$756$		0			0
$757$	1.40395e12			0.155389			0.0776947	−	0.996977i	$-0.475244\pi$
$757$	1.40395e12			0.155389			0.0776947	+	0.996977i	$0.475244\pi$
$758$	−9.28661e12	+	1.60849e13i	−1.02175	+	1.76973i
$759$		0			0
$760$	−1.08671e13	−	1.88224e13i	−1.18155	−	2.04650i
$761$	−3.97524e12	−	6.88531e12i	−0.429667	−	0.744205i	0.567177	−	0.823596i	$-0.308036\pi$
$761$	−3.97524e12	−	6.88531e12i	−0.429667	−	0.744205i	−0.996844	+	0.0793912i	$0.974702\pi$
$762$		0			0
$763$	−4.31843e12	+	7.47974e12i	−0.461281	+	0.798962i
$764$	−2.14698e13			−2.27986
$765$		0			0
$766$	−3.73485e11			−0.0391961
$767$	1.28353e11	−	2.22314e11i	0.0133914	−	0.0231947i
$768$		0			0
$769$	3.47949e12	+	6.02666e12i	0.358796	+	0.621453i	0.987760	−	0.155982i	$-0.0498541\pi$
$769$	3.47949e12	+	6.02666e12i	0.358796	+	0.621453i	−0.628964	+	0.777434i	$0.716521\pi$
$770$	−2.22796e12	−	3.85894e12i	−0.228402	−	0.395603i
$771$		0			0
$772$	−1.16319e13	+	2.01471e13i	−1.17862	+	2.04143i
$773$	−1.22597e13			−1.23501			−0.617506	−	0.786566i	$-0.711857\pi$
$773$	−1.22597e13			−1.23501			−0.617506	+	0.786566i	$0.711857\pi$
$774$		0			0
$775$	−2.93799e12			−0.292545
$776$	2.05212e13	−	3.55438e13i	2.03154	−	3.51873i
$777$		0			0
$778$	9.95851e12	+	1.72486e13i	0.974509	+	1.68790i
$779$	4.29088e12	+	7.43203e12i	0.417473	+	0.723084i
$780$		0			0
$781$	−2.35651e12	+	4.08160e12i	−0.226642	+	0.392555i
$782$	−1.33062e12			−0.127240
$783$		0			0
$784$	3.52346e12			0.333079
$785$	3.99488e11	−	6.91933e11i	0.0375483	−	0.0650355i
$786$		0			0
$787$	−5.99070e12	−	1.03762e13i	−0.556662	−	0.964167i	−0.997772	−	0.0667139i	$-0.978749\pi$
$787$	−5.99070e12	−	1.03762e13i	−0.556662	−	0.964167i	0.441110	−	0.897453i	$-0.354585\pi$
$788$	−2.94403e12	−	5.09921e12i	−0.272004	−	0.471124i
$789$		0			0
$790$	8.37924e11	−	1.45133e12i	0.0765390	−	0.132569i
$791$	5.11135e11			0.0464238
$792$		0			0
$793$	−9.64288e10			−0.00865919
$794$	1.98690e13	−	3.44142e13i	1.77413	−	3.07288i
$795$		0			0
$796$	2.86139e13	+	4.95606e13i	2.52620	+	4.37550i
$797$	−1.45080e12	−	2.51285e12i	−0.127363	−	0.220599i	0.795291	−	0.606228i	$-0.207318\pi$
$797$	−1.45080e12	−	2.51285e12i	−0.127363	−	0.220599i	−0.922654	+	0.385628i	$0.873985\pi$
$798$		0			0
$799$	−8.25150e12	+	1.42920e13i	−0.716263	+	1.24060i
$800$	2.44929e13			2.11415
$801$		0			0
$802$	9.86089e11			0.0841650
$803$	1.13954e12	−	1.97374e12i	0.0967183	−	0.167521i
$804$		0			0
$805$	−2.33061e11	−	4.03673e11i	−0.0195608	−	0.0338804i
$806$	3.84910e11	+	6.66684e11i	0.0321257	+	0.0556433i
$807$		0			0
$808$	3.57348e13	−	6.18944e13i	2.94944	−	5.10858i
$809$	1.25435e13			1.02956			0.514779	−	0.857323i	$-0.327874\pi$
$809$	1.25435e13			1.02956			0.514779	+	0.857323i	$0.327874\pi$
$810$		0			0
$811$	−1.31788e13			−1.06975			−0.534874	−	0.844932i	$-0.679641\pi$
$811$	−1.31788e13			−1.06975			−0.534874	+	0.844932i	$0.679641\pi$
$812$	−1.65183e13	+	2.86105e13i	−1.33341	+	2.30953i
$813$		0			0
$814$	1.26914e12	+	2.19822e12i	0.101321	+	0.175494i
$815$	3.96669e12	+	6.87051e12i	0.314934	+	0.545481i
$816$		0			0
$817$	8.04824e12	−	1.39400e13i	0.631978	−	1.09462i
$818$	−2.13367e13			−1.66624
$819$		0			0
$820$	2.71211e13			2.09481
$821$	6.97251e12	−	1.20767e13i	0.535606	−	0.927696i	−0.463528	−	0.886082i	$-0.653417\pi$
$821$	6.97251e12	−	1.20767e13i	0.535606	−	0.927696i	0.999134	−	0.0416139i	$-0.0132500\pi$
$822$		0			0
$823$	−9.05731e12	−	1.56877e13i	−0.688177	−	1.19196i	−0.972427	−	0.233207i	$-0.925078\pi$
$823$	−9.05731e12	−	1.56877e13i	−0.688177	−	1.19196i	0.284250	−	0.958750i	$-0.408255\pi$
$824$	2.13501e12	+	3.69795e12i	0.161335	+	0.279440i
$825$		0			0
$826$	−7.52105e12	+	1.30268e13i	−0.562171	+	0.973708i
$827$	−9.15727e12			−0.680756			−0.340378	−	0.940289i	$-0.610555\pi$
$827$	−9.15727e12			−0.680756			−0.340378	+	0.940289i	$0.610555\pi$
$828$		0			0
$829$	1.41975e13			1.04404			0.522020	−	0.852933i	$-0.325179\pi$
$829$	1.41975e13			1.04404			0.522020	+	0.852933i	$0.325179\pi$
$830$	−1.88780e11	+	3.26976e11i	−0.0138072	+	0.0239147i
$831$		0			0
$832$	−1.75255e12	−	3.03550e12i	−0.126799	−	0.219622i
$833$	−6.88767e11	−	1.19298e12i	−0.0495644	−	0.0858481i
$834$		0			0
$835$	4.23276e12	−	7.33136e12i	0.301325	−	0.521910i
$836$	1.03385e13			0.732030
$837$		0			0
$838$	3.71528e13			2.60251
$839$	1.13179e13	−	1.96031e13i	0.788561	−	1.36583i	−0.138287	−	0.990392i	$-0.544160\pi$
$839$	1.13179e13	−	1.96031e13i	0.788561	−	1.36583i	0.926848	−	0.375436i	$-0.122507\pi$
$840$		0			0
$841$	1.41175e12	+	2.44523e12i	0.0973143	+	0.168553i
$842$	4.58739e12	+	7.94559e12i	0.314529	+	0.544781i
$843$		0			0
$844$	−1.98012e13	+	3.42967e13i	−1.34323	+	2.32654i
$845$	−1.11150e13			−0.749989
$846$		0			0
$847$	−1.41638e13			−0.945596
$848$	−4.47283e13	+	7.74718e13i	−2.97031	+	5.14472i
$849$		0			0
$850$	−8.39555e12	−	1.45415e13i	−0.551651	−	0.955487i
$851$	1.32762e11	+	2.29950e11i	0.00867740	+	0.0150297i
$852$		0			0
$853$	1.37739e13	−	2.38571e13i	0.890813	−	1.54293i	0.0519103	−	0.998652i	$-0.483469\pi$
$853$	1.37739e13	−	2.38571e13i	0.890813	−	1.54293i	0.838903	−	0.544281i	$-0.183198\pi$
$854$	5.65039e12			0.363511
$855$		0			0
$856$	2.33634e13			1.48732
$857$	−5.97538e12	+	1.03497e13i	−0.378401	+	0.655409i	−0.990830	−	0.135116i	$-0.956859\pi$
$857$	−5.97538e12	+	1.03497e13i	−0.378401	+	0.655409i	0.612429	+	0.790526i	$0.290193\pi$
$858$		0			0
$859$	8.43166e12	+	1.46041e13i	0.528377	+	0.915176i	0.999453	+	0.0330829i	$0.0105326\pi$
$859$	8.43166e12	+	1.46041e13i	0.528377	+	0.915176i	−0.471076	+	0.882093i	$0.656134\pi$
$860$	−2.54350e13	−	4.40547e13i	−1.58558	−	2.74631i
$861$		0			0
$862$	−7.70147e12	+	1.33393e13i	−0.475107	+	0.822909i
$863$	2.70257e13			1.65855			0.829275	−	0.558840i	$-0.188753\pi$
$863$	2.70257e13			1.65855			0.829275	+	0.558840i	$0.188753\pi$
$864$		0			0
$865$	−4.33190e12			−0.263091
$866$	−1.64851e11	+	2.85530e11i	−0.00996001	+	0.0172512i
$867$		0			0
$868$	−1.67169e13	−	2.89545e13i	−0.999579	−	1.73132i
$869$	2.59398e11	+	4.49291e11i	0.0154304	+	0.0267263i
$870$		0			0
$871$	−5.37470e11	+	9.30926e11i	−0.0316427	+	0.0548067i
$872$	−5.56072e13			−3.25692
$873$		0			0
$874$	1.45913e12			0.0845849
$875$	9.70424e12	−	1.68082e13i	0.559661	−	0.969362i
$876$		0			0
$877$	−7.36962e12	−	1.27646e13i	−0.420675	−	0.728631i	0.575331	−	0.817921i	$-0.304873\pi$
$877$	−7.36962e12	−	1.27646e13i	−0.420675	−	0.728631i	−0.996006	+	0.0892904i	$0.971540\pi$
$878$	2.13449e13	+	3.69704e13i	1.21218	+	2.09957i
$879$		0			0
$880$	8.63974e12	−	1.49645e13i	0.485656	−	0.841181i
$881$	2.25153e13			1.25918			0.629589	−	0.776928i	$-0.283223\pi$
$881$	2.25153e13			1.25918			0.629589	+	0.776928i	$0.283223\pi$
$882$		0			0
$883$	−2.11103e13			−1.16861			−0.584307	−	0.811533i	$-0.698634\pi$
$883$	−2.11103e13			−1.16861			−0.584307	+	0.811533i	$0.698634\pi$
$884$	−1.63047e12	+	2.82406e12i	−0.0898003	+	0.155539i
$885$		0			0
$886$	−6.41609e12	−	1.11130e13i	−0.349799	−	0.605870i
$887$	2.36351e12	+	4.09372e12i	0.128204	+	0.222056i	0.922981	−	0.384846i	$-0.125745\pi$
$887$	2.36351e12	+	4.09372e12i	0.128204	+	0.222056i	−0.794777	+	0.606902i	$0.792412\pi$
$888$		0			0
$889$	6.23877e12	−	1.08059e13i	0.334997	−	0.580232i
$890$	4.40956e13			2.35581
$891$		0			0
$892$	−8.40195e13			−4.44363
$893$	9.04846e12	−	1.56724e13i	0.476149	−	0.824715i
$894$		0			0
$895$	1.29316e13	+	2.23981e13i	0.673671	+	1.16683i
$896$	5.40281e13	+	9.35794e13i	2.80049	+	4.85058i
$897$		0			0
$898$	−2.22451e13	+	3.85296e13i	−1.14154	+	1.97720i
$899$	1.18241e13			0.603740
$900$		0			0
$901$	3.49740e13			1.76801
$902$	−5.66390e12	+	9.81017e12i	−0.284896	+	0.493454i
$903$		0			0
$904$	1.64543e12	+	2.84998e12i	0.0819450	+	0.141933i
$905$	2.19402e12	+	3.80015e12i	0.108723	+	0.188314i
$906$		0			0
$907$	−6.64325e12	+	1.15064e13i	−0.325948	+	0.564558i	−0.981704	−	0.190415i	$-0.939017\pi$
$907$	−6.64325e12	+	1.15064e13i	−0.325948	+	0.564558i	0.655756	+	0.754973i	$0.272350\pi$
$908$	−5.13259e13			−2.50582
$909$		0			0
$910$	−1.54122e12			−0.0745038
$911$	−1.04554e13	+	1.81093e13i	−0.502930	+	0.871100i	0.497064	+	0.867714i	$0.334411\pi$
$911$	−1.04554e13	+	1.81093e13i	−0.502930	+	0.871100i	−0.999994	+	0.00338636i	$0.998922\pi$
$912$		0			0
$913$	−5.84411e10	−	1.01223e11i	−0.00278356	−	0.00482126i
$914$	6.31253e12	+	1.09336e13i	0.299189	+	0.518210i
$915$		0			0
$916$	−4.74710e13	+	8.22221e13i	−2.22791	+	3.85886i
$917$	3.04667e13			1.42286
$918$		0			0
$919$	−1.12670e13			−0.521060			−0.260530	−	0.965466i	$-0.583897\pi$
$919$	−1.12670e13			−0.521060			−0.260530	+	0.965466i	$0.583897\pi$
$920$	1.50053e12	−	2.59899e12i	0.0690556	−	0.119608i
$921$		0			0
$922$	1.43621e10	+	2.48759e10i	0.000654529	+	0.00113368i
$923$	8.15074e11	+	1.41175e12i	0.0369649	+	0.0640250i
$924$		0			0
$925$	−1.67533e12	+	2.90175e12i	−0.0752422	+	0.130323i
$926$	2.14856e13			0.960281
$927$		0			0
$928$	−9.85731e13			−4.36307
$929$	1.24126e13	−	2.14992e13i	0.546752	−	0.947003i	−0.451742	−	0.892149i	$-0.649197\pi$
$929$	1.24126e13	−	2.14992e13i	0.546752	−	0.947003i	0.998494	−	0.0548543i	$-0.0174694\pi$
$930$		0			0
$931$	7.55291e11	+	1.30820e12i	0.0329489	+	0.0570692i
$932$	−4.33312e13	−	7.50518e13i	−1.88117	−	3.25829i
$933$		0			0
$934$	1.57922e13	−	2.73530e13i	0.679020	−	1.17610i
$935$	−6.75560e12			−0.289076
$936$		0			0
$937$	2.05299e13			0.870081			0.435040	−	0.900411i	$-0.356734\pi$
$937$	2.05299e13			0.870081			0.435040	+	0.900411i	$0.356734\pi$
$938$	3.14939e13	−	5.45490e13i	1.32835	−	2.30078i
$939$		0			0
$940$	−2.85960e13	−	4.95298e13i	−1.19462	−	2.06915i
$941$	−1.44919e13	−	2.51006e13i	−0.602519	−	1.04359i	−0.992438	−	0.122745i	$-0.960830\pi$
$941$	−1.44919e13	−	2.51006e13i	−0.602519	−	1.04359i	0.389919	−	0.920849i	$-0.372503\pi$
$942$		0			0
$943$	−5.92485e11	+	1.02621e12i	−0.0243992	+	0.0422606i
$944$	−5.83313e13			−2.39072
$945$		0			0
$946$	2.12471e13			0.862562
$947$	−1.39104e13	+	2.40935e13i	−0.562036	+	0.973475i	0.435283	+	0.900294i	$0.356648\pi$
$947$	−1.39104e13	+	2.40935e13i	−0.562036	+	0.973475i	−0.997319	+	0.0731809i	$0.976685\pi$
$948$		0			0
$949$	−3.94145e11	−	6.82679e11i	−0.0157746	−	0.0273224i
$950$	9.20643e12	+	1.59460e13i	0.366720	+	0.635178i
$951$		0			0
$952$	6.21775e13	−	1.07695e14i	2.45339	−	4.24940i
$953$	−2.85882e13			−1.12271			−0.561357	−	0.827573i	$-0.689721\pi$
$953$	−2.85882e13			−1.12271			−0.561357	+	0.827573i	$0.689721\pi$
$954$		0			0
$955$	−1.53842e13			−0.598496
$956$	−2.96690e13	+	5.13883e13i	−1.14880	+	1.98977i
$957$		0			0
$958$	−3.43851e11	−	5.95568e11i	−0.0131894	−	0.0228448i
$959$	−2.25011e13	−	3.89731e13i	−0.859055	−	1.48793i
$960$		0			0
$961$	7.23667e12	−	1.25343e13i	0.273706	−	0.474072i
$962$	8.77947e11			0.0330507
$963$		0			0
$964$	4.94246e12			0.184330
$965$	−8.33489e12	+	1.44365e13i	−0.309405	+	0.535905i
$966$		0			0
$967$	2.57467e13	+	4.45947e13i	0.946898	+	1.64008i	0.751906	+	0.659271i	$0.229135\pi$
$967$	2.57467e13	+	4.45947e13i	0.946898	+	1.64008i	0.194993	+	0.980805i	$0.437532\pi$
$968$	−4.55960e13	−	7.89745e13i	−1.66912	−	2.89100i
$969$		0			0
$970$	2.25945e13	−	3.91348e13i	0.819464	−	1.41935i
$971$	3.75474e13			1.35548			0.677740	−	0.735301i	$-0.262959\pi$
$971$	3.75474e13			1.35548			0.677740	+	0.735301i	$0.262959\pi$
$972$		0			0
$973$	3.21464e13			1.14981
$974$	−1.25738e13	+	2.17784e13i	−0.447663	+	0.775374i
$975$		0			0
$976$	1.09557e13	+	1.89759e13i	0.386472	+	0.669389i
$977$	−1.62693e13	−	2.81792e13i	−0.571271	−	0.989470i	−0.996436	−	0.0843544i	$-0.973117\pi$
$977$	−1.62693e13	−	2.81792e13i	−0.571271	−	0.989470i	0.425165	−	0.905116i	$-0.360216\pi$
$978$		0			0
$979$	−6.82540e12	+	1.18219e13i	−0.237469	+	0.411308i
$980$	4.77392e12			0.165332
$981$		0			0
$982$	−1.65781e13			−0.568895
$983$	2.81683e13	−	4.87889e13i	0.962209	−	1.66660i	0.245277	−	0.969453i	$-0.421121\pi$
$983$	2.81683e13	−	4.87889e13i	0.962209	−	1.66660i	0.716932	−	0.697143i	$-0.245546\pi$
$984$		0			0
$985$	−2.10956e12	−	3.65386e12i	−0.0714049	−	0.123677i
$986$	3.37884e13	+	5.85232e13i	1.13847	+	1.97189i
$987$		0			0
$988$	1.78795e12	−	3.09682e12i	0.0596964	−	0.103397i
$989$	2.22260e12			0.0738718
$990$		0			0
$991$	−2.09674e13			−0.690578			−0.345289	−	0.938496i	$-0.612219\pi$
$991$	−2.09674e13			−0.690578			−0.345289	+	0.938496i	$0.612219\pi$
$992$	4.98791e13	−	8.63932e13i	1.63537	−	2.83255i
$993$		0			0
$994$	−4.77605e13	−	8.27236e13i	−1.55178	−	2.68776i
$995$	2.05033e13	+	3.55128e13i	0.663164	+	1.14863i
$996$		0			0
$997$	−2.42890e13	+	4.20698e13i	−0.778541	+	1.34847i	0.154241	+	0.988033i	$0.450707\pi$
$997$	−2.42890e13	+	4.20698e13i	−0.778541	+	1.34847i	−0.932782	+	0.360440i	$0.882627\pi$
$998$	1.59057e13			0.507533
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.10.c.j.55.4		8
3.2	odd	2	inner	81.10.c.j.55.1		8
9.2	odd	6		27.10.a.d.1.4	yes	4
9.4	even	3	inner	81.10.c.j.28.4		8
9.5	odd	6	inner	81.10.c.j.28.1		8
9.7	even	3		27.10.a.d.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.10.a.d.1.1	✓	4	9.7	even	3
27.10.a.d.1.4	yes	4	9.2	odd	6
81.10.c.j.28.1		8	9.5	odd	6	inner
81.10.c.j.28.4		8	9.4	even	3	inner
81.10.c.j.55.1		8	3.2	odd	2	inner
81.10.c.j.55.4		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	81.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(22.2386 - 38.5183i) q^{2} +(-733.108 - 1269.78i) q^{4} +(-525.311 - 909.865i) q^{5} +(-3295.93 + 5708.72i) q^{7} -42440.8 q^{8} -46728.6 q^{10} +(7232.96 - 12527.8i) q^{11} +(-2501.75 - 4333.15i) q^{13} +(146594. + 253908. i) q^{14} +(-568471. + 984621. i) q^{16} +444500. q^{17} -487431. q^{19} +(-770219. + 1.33406e6i) q^{20} +(-321701. - 557203. i) q^{22} +(-33652.3 - 58287.4i) q^{23} +(424660. - 735532. i) q^{25} -222541. q^{26} +9.66510e6 q^{28} +(-1.70907e6 + 2.96019e6i) q^{29} +(-1.72962e6 - 2.99578e6i) q^{31} +(1.44191e7 + 2.49747e7i) q^{32} +(9.88504e6 - 1.71214e7i) q^{34} +6.92555e6 q^{35} -3.94510e6 q^{37} +(-1.08398e7 + 1.87750e7i) q^{38} +(2.22946e7 + 3.86154e7i) q^{40} +(-8.80305e6 - 1.52473e7i) q^{41} +(-1.65115e7 + 2.85988e7i) q^{43} -2.12102e7 q^{44} -2.99351e6 q^{46} +(-1.85636e7 + 3.21530e7i) q^{47} +(-1.54953e6 - 2.68387e6i) q^{49} +(-1.88876e7 - 3.27144e7i) q^{50} +(-3.66810e6 + 6.35334e6i) q^{52} +7.86818e7 q^{53} -1.51982e7 q^{55} +(1.39882e8 - 2.42283e8i) q^{56} +(7.60144e7 + 1.31661e8i) q^{58} +(2.56527e7 + 4.44318e7i) q^{59} +(9.63615e6 - 1.66903e7i) q^{61} -1.53857e8 q^{62} +7.00529e8 q^{64} +(-2.62839e6 + 4.55250e6i) q^{65} +(-1.07419e8 - 1.86055e8i) q^{67} +(-3.25866e8 - 5.64417e8i) q^{68} +(1.54014e8 - 2.66761e8i) q^{70} -3.25802e8 q^{71} +1.57548e8 q^{73} +(-8.77334e7 + 1.51959e8i) q^{74} +(3.57340e8 + 6.18931e8i) q^{76} +(4.76787e7 + 8.25819e7i) q^{77} +(-1.79317e7 + 3.10586e7i) q^{79} +1.19450e9 q^{80} -7.83069e8 q^{82} +(4.03992e6 - 6.99735e6i) q^{83} +(-2.33501e8 - 4.04435e8i) q^{85} +(7.34386e8 + 1.27199e9i) q^{86} +(-3.06972e8 + 5.31692e8i) q^{88} -9.43654e8 q^{89} +3.29823e7 q^{91} +(-4.93415e7 + 8.54620e7i) q^{92} +(8.25654e8 + 1.43008e9i) q^{94} +(2.56053e8 + 4.43497e8i) q^{95} +(-4.83526e8 + 8.37491e8i) q^{97} -1.37838e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2236 q^{4} - 11852 q^{7} - 83520 q^{10} - 179684 q^{13} - 2348680 q^{16} + 2022856 q^{19} - 3473568 q^{22} - 2554060 q^{25} + 39587848 q^{28} - 889136 q^{31} + 43111008 q^{34} - 7610312 q^{37} + 133649280 q^{40}+ \cdots - 1935734516 q^{97}+O(q^{100}) \) 8 * q - 2236 * q^4 - 11852 * q^7 - 83520 * q^10 - 179684 * q^13 - 2348680 * q^16 + 2022856 * q^19 - 3473568 * q^22 - 2554060 * q^25 + 39587848 * q^28 - 889136 * q^31 + 43111008 * q^34 - 7610312 * q^37 + 133649280 * q^40 - 11585024 * q^43 + 26856000 * q^46 + 73622328 * q^49 + 44411860 * q^52 + 629444160 * q^55 + 407685888 * q^58 - 219882572 * q^61 + 3038570720 * q^64 - 390561236 * q^67 + 650481120 * q^70 + 2395899016 * q^73 + 2121017572 * q^76 - 49814324 * q^79 - 3534253056 * q^82 - 370444320 * q^85 + 653198976 * q^88 - 94034344 * q^91 + 4146926688 * q^94 - 1935734516 * q^97

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