LMFDB - Embedded newform 117.9.j.a.73.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [117,9,Mod(73,117)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(117, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 9, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("117.73"); S:= CuspForms(chi, 9); N := Newforms(S);

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	117.j (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [18,2,0,0,-166] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$47.6632973772$
Analytic rank:	$0$
Dimension:	$18$
Relative dimension:	$9$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 2 x^{17} + 13 x^{16} + 10976 x^{15} + 1201625 x^{14} + 122002 x^{13} + 46813351 x^{12} + \cdots + 12\!\cdots\!50 $ x^18 - 2x^17 + 13x^16 + 10976x^15 + 1201625x^14 + 122002x^13 + 46813351x^12 + 7862652956x^11 + 334844194775x^10 + 1307282975658x^9 + 11682756415941x^8 + 633414417845224x^7 + 18032980689164179x^6 + 45002614591373022x^5 + 390994490142779965x^4 + 14232520046144862044x^3 + 162776004109684584700x^2 + 701394949089602778920*x + 1264129294984265466250
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{16}\cdot 3^{8}\cdot 13^{4} $
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			73.8
Root			$18.0289 + 19.0289i$ of defining polynomial
Character	$\chi$	$=$	117.73
Dual form			117.9.j.a.109.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(18.0289 + 18.0289i) q^{2} +394.083i q^{4} +(-706.041 - 706.041i) q^{5} +(1701.26 - 1701.26i) q^{7} +(-2489.49 + 2489.49i) q^{8} -25458.3i q^{10} +(-5801.81 + 5801.81i) q^{11} +(-16499.4 + 23313.1i) q^{13} +61343.7 q^{14} +11119.7 q^{16} +134321. i q^{17} +(84022.1 + 84022.1i) q^{19} +(278239. - 278239. i) q^{20} -209201. q^{22} +136494. i q^{23} +606362. i q^{25} +(-717776. + 122845. i) q^{26} +(670437. + 670437. i) q^{28} +958731. q^{29} +(484682. + 484682. i) q^{31} +(837786. + 837786. i) q^{32} +(-2.42165e6 + 2.42165e6i) q^{34} -2.40232e6 q^{35} +(-1.25356e6 + 1.25356e6i) q^{37} +3.02965e6i q^{38} +3.51536e6 q^{40} +(-687708. - 687708. i) q^{41} -3.17073e6i q^{43} +(-2.28640e6 - 2.28640e6i) q^{44} +(-2.46084e6 + 2.46084e6i) q^{46} +(507919. - 507919. i) q^{47} -23757.9i q^{49} +(-1.09320e7 + 1.09320e7i) q^{50} +(-9.18731e6 - 6.50212e6i) q^{52} +8.75734e6 q^{53} +8.19263e6 q^{55} +8.47053e6i q^{56} +(1.72849e7 + 1.72849e7i) q^{58} +(3.35500e6 - 3.35500e6i) q^{59} +3.53900e6 q^{61} +1.74766e7i q^{62} +2.73621e7i q^{64} +(2.81092e7 - 4.81079e6i) q^{65} +(-4.45919e6 - 4.45919e6i) q^{67} -5.29335e7 q^{68} +(-4.33111e7 - 4.33111e7i) q^{70} +(7.29718e6 + 7.29718e6i) q^{71} +(8.69469e6 - 8.69469e6i) q^{73} -4.52008e7 q^{74} +(-3.31117e7 + 3.31117e7i) q^{76} +1.97408e7i q^{77} -5.57116e7 q^{79} +(-7.85098e6 - 7.85098e6i) q^{80} -2.47972e7i q^{82} +(3.21832e6 + 3.21832e6i) q^{83} +(9.48358e7 - 9.48358e7i) q^{85} +(5.71648e7 - 5.71648e7i) q^{86} -2.88871e7i q^{88} +(-4.69400e7 + 4.69400e7i) q^{89} +(1.15920e7 + 6.77313e7i) q^{91} -5.37901e7 q^{92} +1.83144e7 q^{94} -1.18646e8i q^{95} +(-5.15911e7 - 5.15911e7i) q^{97} +(428329. - 428329. i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 2 q^{2} - 166 q^{5} + 5308 q^{7} - 10464 q^{8} + 31556 q^{11} + 71300 q^{13} + 110260 q^{14} - 522860 q^{16} + 100288 q^{19} - 736268 q^{20} - 977312 q^{22} - 2952238 q^{26} + 4497084 q^{28} + 2479024 q^{29}+ \cdots - 588677614 q^{98}+O(q^{100}) $ 18 * q + 2 * q^2 - 166 * q^5 + 5308 * q^7 - 10464 * q^8 + 31556 * q^11 + 71300 * q^13 + 110260 * q^14 - 522860 * q^16 + 100288 * q^19 - 736268 * q^20 - 977312 * q^22 - 2952238 * q^26 + 4497084 * q^28 + 2479024 * q^29 - 1892664 * q^31 - 947212 * q^32 - 531576 * q^34 + 2918284 * q^35 - 8343978 * q^37 - 12691908 * q^40 - 1140178 * q^41 + 3867188 * q^44 + 2006148 * q^46 + 13368572 * q^47 - 37369598 * q^50 - 14821220 * q^52 - 50561348 * q^53 + 76994128 * q^55 + 22505716 * q^58 - 2127976 * q^59 - 52016516 * q^61 - 10413082 * q^65 + 960292 * q^67 - 283187508 * q^68 - 166635032 * q^70 - 67412140 * q^71 - 145213226 * q^73 + 233620024 * q^74 - 150533640 * q^76 - 76829120 * q^79 - 524889520 * q^80 + 241951556 * q^83 + 260737764 * q^85 + 579480384 * q^86 + 89187110 * q^89 + 232660948 * q^91 + 122690376 * q^92 + 1069637380 * q^94 + 331183146 * q^97 - 588677614 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$.

$n$	$28$	$92$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$

Coefficient data

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

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

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

$2$	18.0289	+	18.0289i	1.12681	+	1.12681i	0.990693	+	0.136114i	$0.0434612\pi$
$2$	18.0289	+	18.0289i	1.12681	+	1.12681i	0.136114	+	0.990693i	$0.456539\pi$
$3$		0			0
$3$		0			0
$4$			394.083i			1.53939i
$5$	−706.041	−	706.041i	−1.12967	−	1.12967i	−0.990231	−	0.139434i	$-0.955472\pi$
$5$	−706.041	−	706.041i	−1.12967	−	1.12967i	−0.139434	−	0.990231i	$-0.544528\pi$
$6$		0			0
$7$	1701.26	−	1701.26i	0.708562	−	0.708562i	−0.257670	−	0.966233i	$-0.582955\pi$
$7$	1701.26	−	1701.26i	0.708562	−	0.708562i	0.966233	+	0.257670i	$0.0829549\pi$
$8$	−2489.49	+	2489.49i	−0.607786	+	0.607786i
$9$		0			0
$10$		−	25458.3i		−	2.54583i
$11$	−5801.81	+	5801.81i	−0.396271	+	0.396271i	−0.876916	−	0.480644i	$-0.840403\pi$
$11$	−5801.81	+	5801.81i	−0.396271	+	0.396271i	0.480644	+	0.876916i	$0.340403\pi$
$12$		0			0
$13$	−16499.4	+	23313.1i	−0.577689	+	0.816257i
$13$	−16499.4	+	23313.1i	−0.577689	+	0.816257i
$14$	61343.7			1.59683
$15$		0			0
$16$	11119.7			0.169674
$17$			134321.i			1.60823i	0.594477	+	0.804113i	$0.297359\pi$
$17$			134321.i			1.60823i	−0.594477	+	0.804113i	$0.702641\pi$
$18$		0			0
$19$	84022.1	+	84022.1i	0.644732	+	0.644732i	0.951715	−	0.306983i	$-0.0993196\pi$
$19$	84022.1	+	84022.1i	0.644732	+	0.644732i	−0.306983	+	0.951715i	$0.599320\pi$
$20$	278239.	−	278239.i	1.73899	−	1.73899i
$21$		0			0
$22$	−209201.			−0.893043
$23$			136494.i			0.487757i	0.969806	+	0.243878i	$0.0784198\pi$
$23$			136494.i			0.487757i	−0.969806	+	0.243878i	$0.921580\pi$
$24$		0			0
$25$			606362.i			1.55229i
$26$	−717776.	+	122845.i	−1.57071	+	0.268821i
$27$		0			0
$28$	670437.	+	670437.i	1.09075	+	1.09075i
$29$	958731.			1.35552			0.677758	−	0.735285i	$-0.262952\pi$
$29$	958731.			1.35552			0.677758	+	0.735285i	$0.262952\pi$
$30$		0			0
$31$	484682.	+	484682.i	0.524819	+	0.524819i	0.919023	−	0.394204i	$-0.128980\pi$
$31$	484682.	+	484682.i	0.524819	+	0.524819i	−0.394204	+	0.919023i	$0.628980\pi$
$32$	837786.	+	837786.i	0.798975	+	0.798975i
$33$		0			0
$34$	−2.42165e6	+	2.42165e6i	−1.81216	+	1.81216i
$35$	−2.40232e6			−1.60088
$36$		0			0
$37$	−1.25356e6	+	1.25356e6i	−0.668867	+	0.668867i	−0.957454	−	0.288587i	$-0.906815\pi$
$37$	−1.25356e6	+	1.25356e6i	−0.668867	+	0.668867i	0.288587	+	0.957454i	$0.406815\pi$
$38$			3.02965e6i			1.45298i
$39$		0			0
$40$	3.51536e6			1.37319
$41$	−687708.	−	687708.i	−0.243371	−	0.243371i	0.574872	−	0.818243i	$-0.305052\pi$
$41$	−687708.	−	687708.i	−0.243371	−	0.243371i	−0.818243	+	0.574872i	$0.805052\pi$
$42$		0			0
$43$		−	3.17073e6i		−	0.927439i	−0.885982	−	0.463720i	$-0.846514\pi$
$43$		−	3.17073e6i		−	0.927439i	0.885982	−	0.463720i	$-0.153486\pi$
$44$	−2.28640e6	−	2.28640e6i	−0.610015	−	0.610015i
$45$		0			0
$46$	−2.46084e6	+	2.46084e6i	−0.549608	+	0.549608i
$47$	507919.	−	507919.i	0.104089	−	0.104089i	−0.653145	−	0.757233i	$-0.726551\pi$
$47$	507919.	−	507919.i	0.104089	−	0.104089i	0.757233	+	0.653145i	$0.226551\pi$
$48$		0			0
$49$		−	23757.9i		−	0.00412120i
$50$	−1.09320e7	+	1.09320e7i	−1.74913	+	1.74913i
$51$		0			0
$52$	−9.18731e6	−	6.50212e6i	−1.25654	−	0.889287i
$53$	8.75734e6			1.10986			0.554931	−	0.831896i	$-0.312745\pi$
$53$	8.75734e6			1.10986			0.554931	+	0.831896i	$0.312745\pi$
$54$		0			0
$55$	8.19263e6			0.895308
$56$			8.47053e6i			0.861308i
$57$		0			0
$58$	1.72849e7	+	1.72849e7i	1.52741	+	1.52741i
$59$	3.35500e6	−	3.35500e6i	0.276875	−	0.276875i	−0.554985	−	0.831860i	$-0.687276\pi$
$59$	3.35500e6	−	3.35500e6i	0.276875	−	0.276875i	0.831860	+	0.554985i	$0.187276\pi$
$60$		0			0
$61$	3.53900e6			0.255600			0.127800	−	0.991800i	$-0.459208\pi$
$61$	3.53900e6			0.255600			0.127800	+	0.991800i	$0.459208\pi$
$62$			1.74766e7i			1.18274i
$63$		0			0
$64$			2.73621e7i			1.63091i
$65$	2.81092e7	−	4.81079e6i	1.57469	−	0.269503i
$66$		0			0
$67$	−4.45919e6	−	4.45919e6i	−0.221287	−	0.221287i	0.587753	−	0.809040i	$-0.300013\pi$
$67$	−4.45919e6	−	4.45919e6i	−0.221287	−	0.221287i	−0.809040	+	0.587753i	$0.800013\pi$
$68$	−5.29335e7			−2.47568
$69$		0			0
$70$	−4.33111e7	−	4.33111e7i	−1.80388	−	1.80388i
$71$	7.29718e6	+	7.29718e6i	0.287158	+	0.287158i	0.835956	−	0.548797i	$-0.184914\pi$
$71$	7.29718e6	+	7.29718e6i	0.287158	+	0.287158i	−0.548797	+	0.835956i	$0.684914\pi$
$72$		0			0
$73$	8.69469e6	−	8.69469e6i	0.306170	−	0.306170i	−0.537252	−	0.843422i	$-0.680538\pi$
$73$	8.69469e6	−	8.69469e6i	0.306170	−	0.306170i	0.843422	+	0.537252i	$0.180538\pi$
$74$	−4.52008e7			−1.50737
$75$		0			0
$76$	−3.31117e7	+	3.31117e7i	−0.992492	+	0.992492i
$77$			1.97408e7i			0.561566i
$78$		0			0
$79$	−5.57116e7			−1.43033			−0.715167	−	0.698954i	$-0.753649\pi$
$79$	−5.57116e7			−1.43033			−0.715167	+	0.698954i	$0.753649\pi$
$80$	−7.85098e6	−	7.85098e6i	−0.191674	−	0.191674i
$81$		0			0
$82$		−	2.47972e7i		−	0.548464i
$83$	3.21832e6	+	3.21832e6i	0.0678136	+	0.0678136i	0.740200	−	0.672387i	$-0.234731\pi$
$83$	3.21832e6	+	3.21832e6i	0.0678136	+	0.0678136i	−0.672387	+	0.740200i	$0.734731\pi$
$84$		0			0
$85$	9.48358e7	−	9.48358e7i	1.81676	−	1.81676i
$86$	5.71648e7	−	5.71648e7i	1.04505	−	1.04505i
$87$		0			0
$88$		−	2.88871e7i		−	0.481696i
$89$	−4.69400e7	+	4.69400e7i	−0.748141	+	0.748141i	−0.974130	−	0.225989i	$-0.927439\pi$
$89$	−4.69400e7	+	4.69400e7i	−0.748141	+	0.748141i	0.225989	+	0.974130i	$0.427439\pi$
$90$		0			0
$91$	1.15920e7	+	6.77313e7i	0.169041	+	0.987698i
$92$	−5.37901e7			−0.750847
$93$		0			0
$94$	1.83144e7			0.234575
$95$		−	1.18646e8i		−	1.45666i
$96$		0			0
$97$	−5.15911e7	−	5.15911e7i	−0.582757	−	0.582757i	0.352903	−	0.935660i	$-0.385195\pi$
$97$	−5.15911e7	−	5.15911e7i	−0.582757	−	0.582757i	−0.935660	+	0.352903i	$0.885195\pi$
$98$	428329.	−	428329.i	0.00464379	−	0.00464379i
$99$		0			0
$100$	−2.38957e8			−2.38957
$101$			1.34398e8i			1.29154i	0.763531	+	0.645771i	$0.223464\pi$
$101$			1.34398e8i			1.29154i	−0.763531	+	0.645771i	$0.776536\pi$
$102$		0			0
$103$			1.40184e8i			1.24552i	0.782415	+	0.622758i	$0.213988\pi$
$103$			1.40184e8i			1.24552i	−0.782415	+	0.622758i	$0.786012\pi$
$104$	−1.69628e7	−	9.91128e7i	−0.144999	−	0.847220i
$105$		0			0
$106$	1.57885e8	+	1.57885e8i	1.25060	+	1.25060i
$107$	−2.20841e8			−1.68479			−0.842394	−	0.538863i	$-0.818854\pi$
$107$	−2.20841e8			−1.68479			−0.842394	+	0.538863i	$0.818854\pi$
$108$		0			0
$109$	3.84233e7	+	3.84233e7i	0.272200	+	0.272200i	0.829985	−	0.557785i	$-0.188349\pi$
$109$	3.84233e7	+	3.84233e7i	0.272200	+	0.272200i	−0.557785	+	0.829985i	$0.688349\pi$
$110$	1.47704e8	+	1.47704e8i	1.00884	+	1.00884i
$111$		0			0
$112$	1.89175e7	−	1.89175e7i	0.120224	−	0.120224i
$113$	−1.23741e7			−0.0758924			−0.0379462	−	0.999280i	$-0.512082\pi$
$113$	−1.23741e7			−0.0758924			−0.0379462	+	0.999280i	$0.512082\pi$
$114$		0			0
$115$	9.63706e7	−	9.63706e7i	0.551002	−	0.551002i
$116$			3.77820e8i			2.08667i
$117$		0			0
$118$	1.20974e8			0.623970
$119$	2.28514e8	+	2.28514e8i	1.13953	+	1.13953i
$120$		0			0
$121$			1.47037e8i			0.685938i
$122$	6.38043e7	+	6.38043e7i	0.288012	+	0.288012i
$123$		0			0
$124$	−1.91005e8	+	1.91005e8i	−0.807900	+	0.807900i
$125$	1.52319e8	−	1.52319e8i	0.623900	−	0.623900i
$126$		0			0
$127$		−	5.22219e7i		−	0.200742i	−0.994950	−	0.100371i	$-0.967997\pi$
$127$		−	5.22219e7i		−	0.200742i	0.994950	−	0.100371i	$-0.0320029\pi$
$128$	−2.78835e8	+	2.78835e8i	−1.03874	+	1.03874i
$129$		0			0
$130$	5.93512e8	+	4.20046e8i	2.07805	+	1.47070i
$131$	3.48669e8			1.18394			0.591968	−	0.805961i	$-0.298351\pi$
$131$	3.48669e8			1.18394			0.591968	+	0.805961i	$0.298351\pi$
$132$		0			0
$133$	2.85886e8			0.913665
$134$		−	1.60789e8i		−	0.498696i
$135$		0			0
$136$	−3.34390e8	−	3.34390e8i	−0.977456	−	0.977456i
$137$	1.37365e8	−	1.37365e8i	0.389937	−	0.389937i	−0.484728	−	0.874665i	$-0.661081\pi$
$137$	1.37365e8	−	1.37365e8i	0.389937	−	0.389937i	0.874665	+	0.484728i	$0.161081\pi$
$138$		0			0
$139$	1.43378e7			0.0384081			0.0192040	−	0.999816i	$-0.493887\pi$
$139$	1.43378e7			0.0384081			0.0192040	+	0.999816i	$0.493887\pi$
$140$		−	9.46712e8i		−	2.46437i
$141$		0			0
$142$			2.63120e8i			0.647144i
$143$	−3.95322e7	−	2.30984e8i	−0.0945380	−	0.552381i
$144$		0			0
$145$	−6.76903e8	−	6.76903e8i	−1.53128	−	1.53128i
$146$	3.13511e8			0.689989
$147$		0			0
$148$	−4.94009e8	−	4.94009e8i	−1.02965	−	1.02965i
$149$	−3.31091e8	−	3.31091e8i	−0.671742	−	0.671742i	0.286376	−	0.958117i	$-0.407550\pi$
$149$	−3.31091e8	−	3.31091e8i	−0.671742	−	0.671742i	−0.958117	+	0.286376i	$0.907550\pi$
$150$		0			0
$151$	−5.37699e7	+	5.37699e7i	−0.103426	+	0.103426i	−0.756926	−	0.653500i	$-0.773300\pi$
$151$	−5.37699e7	+	5.37699e7i	−0.103426	+	0.103426i	0.653500	+	0.756926i	$0.273300\pi$
$152$	−4.18344e8			−0.783717
$153$		0			0
$154$	−3.55904e8	+	3.55904e8i	−0.632776	+	0.632776i
$155$		−	6.84410e8i		−	1.18574i
$156$		0			0
$157$	1.02886e9			1.69339			0.846695	−	0.532079i	$-0.178589\pi$
$157$	1.02886e9			1.69339			0.846695	+	0.532079i	$0.178589\pi$
$158$	−1.00442e9	−	1.00442e9i	−1.61171	−	1.61171i
$159$		0			0
$160$		−	1.18302e9i		−	1.80515i
$161$	2.32212e8	+	2.32212e8i	0.345606	+	0.345606i
$162$		0			0
$163$	−1.25745e7	+	1.25745e7i	−0.0178131	+	0.0178131i	−0.715957	−	0.698144i	$-0.754009\pi$
$163$	−1.25745e7	+	1.25745e7i	−0.0178131	+	0.0178131i	0.698144	+	0.715957i	$0.254009\pi$
$164$	2.71014e8	−	2.71014e8i	0.374642	−	0.374642i
$165$		0			0
$166$			1.16046e8i			0.152826i
$167$	−3.77091e8	+	3.77091e8i	−0.484820	+	0.484820i	−0.906667	−	0.421847i	$-0.861382\pi$
$167$	−3.77091e8	+	3.77091e8i	−0.484820	+	0.484820i	0.421847	+	0.906667i	$0.361382\pi$
$168$		0			0
$169$	−2.71273e8	−	7.69303e8i	−0.332552	−	0.943085i
$170$	3.41957e9			4.09427
$171$		0			0
$172$	1.24953e9			1.42769
$173$		−	1.73349e8i		−	0.193525i	−0.995307	−	0.0967626i	$-0.969151\pi$
$173$		−	1.73349e8i		−	0.193525i	0.995307	−	0.0967626i	$-0.0308488\pi$
$174$		0			0
$175$	1.03158e9	+	1.03158e9i	1.09989	+	1.09989i
$176$	−6.45146e7	+	6.45146e7i	−0.0672368	+	0.0672368i
$177$		0			0
$178$	−1.69256e9			−1.68602
$179$			9.54514e8i			0.929759i	0.885374	+	0.464879i	$0.153902\pi$
$179$			9.54514e8i			0.929759i	−0.885374	+	0.464879i	$0.846098\pi$
$180$		0			0
$181$		−	1.88245e8i		−	0.175391i	−0.996147	−	0.0876957i	$-0.972050\pi$
$181$		−	1.88245e8i		−	0.175391i	0.996147	−	0.0876957i	$-0.0279503\pi$
$182$	−1.01213e9	+	1.43011e9i	−0.922468	+	1.30342i
$183$		0			0
$184$	−3.39801e8	−	3.39801e8i	−0.296452	−	0.296452i
$185$	1.77014e9			1.51119
$186$		0			0
$187$	−7.79303e8	−	7.79303e8i	−0.637294	−	0.637294i
$188$	2.00162e8	+	2.00162e8i	0.160233	+	0.160233i
$189$		0			0
$190$	2.13906e9	−	2.13906e9i	1.64138	−	1.64138i
$191$	−2.55263e8			−0.191802			−0.0959011	−	0.995391i	$-0.530573\pi$
$191$	−2.55263e8			−0.191802			−0.0959011	+	0.995391i	$0.530573\pi$
$192$		0			0
$193$	1.53214e9	−	1.53214e9i	1.10425	−	1.10425i	0.110360	−	0.993892i	$-0.464800\pi$
$193$	1.53214e9	−	1.53214e9i	1.10425	−	1.10425i	0.993892	−	0.110360i	$-0.0352005\pi$
$194$		−	1.86026e9i		−	1.31331i
$195$		0			0
$196$	9.36259e6			0.00634412
$197$	−8.98261e8	−	8.98261e8i	−0.596400	−	0.596400i	0.342953	−	0.939353i	$-0.388573\pi$
$197$	−8.98261e8	−	8.98261e8i	−0.596400	−	0.596400i	−0.939353	+	0.342953i	$0.888573\pi$
$198$		0			0
$199$			2.75599e9i			1.75738i	0.477393	+	0.878690i	$0.341582\pi$
$199$			2.75599e9i			1.75738i	−0.477393	+	0.878690i	$0.658418\pi$
$200$	−1.50953e9	−	1.50953e9i	−0.943458	−	0.943458i
$201$		0			0
$202$	−2.42306e9	+	2.42306e9i	−1.45532	+	1.45532i
$203$	1.63105e9	−	1.63105e9i	0.960468	−	0.960468i
$204$		0			0
$205$			9.71099e8i			0.549855i
$206$	−2.52736e9	+	2.52736e9i	−1.40346	+	1.40346i
$207$		0			0
$208$	−1.83468e8	+	2.59236e8i	−0.0980185	+	0.138497i
$209$	−9.74960e8			−0.510977
$210$		0			0
$211$	1.56783e9			0.790986			0.395493	−	0.918469i	$-0.370574\pi$
$211$	1.56783e9			0.790986			0.395493	+	0.918469i	$0.370574\pi$
$212$			3.45112e9i			1.70851i
$213$		0			0
$214$	−3.98153e9	−	3.98153e9i	−1.89843	−	1.89843i
$215$	−2.23867e9	+	2.23867e9i	−1.04770	+	1.04770i
$216$		0			0
$217$	1.64914e9			0.743734
$218$			1.38546e9i			0.613435i
$219$		0			0
$220$			3.22858e9i			1.37823i
$221$	−3.13143e9	−	2.21620e9i	−1.31273	−	0.929054i
$222$		0			0
$223$	1.60070e8	+	1.60070e8i	0.0647278	+	0.0647278i	0.738730	−	0.674002i	$-0.235426\pi$
$223$	1.60070e8	+	1.60070e8i	0.0647278	+	0.0647278i	−0.674002	+	0.738730i	$0.735426\pi$
$224$	2.85058e9			1.13225
$225$		0			0
$226$	−2.23091e8	−	2.23091e8i	−0.0855161	−	0.0855161i
$227$	9.32258e8	+	9.32258e8i	0.351102	+	0.351102i	0.860519	−	0.509418i	$-0.170139\pi$
$227$	9.32258e8	+	9.32258e8i	0.351102	+	0.351102i	−0.509418	+	0.860519i	$0.670139\pi$
$228$		0			0
$229$	2.70819e9	−	2.70819e9i	0.984774	−	0.984774i	−0.0151117	−	0.999886i	$-0.504810\pi$
$229$	2.70819e9	−	2.70819e9i	0.984774	−	0.984774i	0.999886	+	0.0151117i	$0.00481040\pi$
$230$	3.47491e9			1.24175
$231$		0			0
$232$	−2.38675e9	+	2.38675e9i	−0.823863	+	0.823863i
$233$		−	7.96832e8i		−	0.270360i	−0.990821	−	0.135180i	$-0.956839\pi$
$233$		−	7.96832e8i		−	0.270360i	0.990821	−	0.135180i	$-0.0431613\pi$
$234$		0			0
$235$	−7.17223e8			−0.235170
$236$	1.32215e9	+	1.32215e9i	0.426218	+	0.426218i
$237$		0			0
$238$			8.23972e9i			2.56806i
$239$	−3.81412e9	−	3.81412e9i	−1.16897	−	1.16897i	−0.982452	−	0.186518i	$-0.940280\pi$
$239$	−3.81412e9	−	3.81412e9i	−1.16897	−	1.16897i	−0.186518	−	0.982452i	$-0.559720\pi$
$240$		0			0
$241$	−2.16281e9	+	2.16281e9i	−0.641135	+	0.641135i	−0.950835	−	0.309699i	$-0.899772\pi$
$241$	−2.16281e9	+	2.16281e9i	−0.641135	+	0.641135i	0.309699	+	0.950835i	$0.399772\pi$
$242$	−2.65091e9	+	2.65091e9i	−0.772920	+	0.772920i
$243$		0			0
$244$			1.39466e9i			0.393468i
$245$	−1.67740e7	+	1.67740e7i	−0.00465557	+	0.00465557i
$246$		0			0
$247$	−3.34513e9	+	5.72506e8i	−0.898721	+	0.153813i
$248$	−2.41322e9			−0.637955
$249$		0			0
$250$	5.49230e9			1.40603
$251$			2.83257e9i			0.713651i	0.934171	+	0.356825i	$0.116141\pi$
$251$			2.83257e9i			0.713651i	−0.934171	+	0.356825i	$0.883859\pi$
$252$		0			0
$253$	−7.91914e8	−	7.91914e8i	−0.193284	−	0.193284i
$254$	9.41504e8	−	9.41504e8i	0.226197	−	0.226197i
$255$		0			0
$256$	−3.04950e9			−0.710017
$257$			2.21483e9i			0.507700i	0.967244	+	0.253850i	$0.0816970\pi$
$257$			2.21483e9i			0.507700i	−0.967244	+	0.253850i	$0.918303\pi$
$258$		0			0
$259$			4.26528e9i			0.947868i
$260$	1.89585e9	+	1.10774e10i	0.414869	+	2.42406i
$261$		0			0
$262$	6.28613e9	+	6.28613e9i	1.33407	+	1.33407i
$263$	−3.78189e8			−0.0790471			−0.0395235	−	0.999219i	$-0.512584\pi$
$263$	−3.78189e8			−0.0790471			−0.0395235	+	0.999219i	$0.512584\pi$
$264$		0			0
$265$	−6.18304e9	−	6.18304e9i	−1.25377	−	1.25377i
$266$	5.15422e9	+	5.15422e9i	1.02952	+	1.02952i
$267$		0			0
$268$	1.75729e9	−	1.75729e9i	0.340647	−	0.340647i
$269$	−9.08136e8			−0.173437			−0.0867185	−	0.996233i	$-0.527638\pi$
$269$	−9.08136e8			−0.173437			−0.0867185	+	0.996233i	$0.527638\pi$
$270$		0			0
$271$	−4.72790e9	+	4.72790e9i	−0.876579	+	0.876579i	−0.993179	−	0.116600i	$-0.962801\pi$
$271$	−4.72790e9	+	4.72790e9i	−0.876579	+	0.876579i	0.116600	+	0.993179i	$0.462801\pi$
$272$			1.49361e9i			0.272874i
$273$		0			0
$274$	4.95309e9			0.878767
$275$	−3.51800e9	−	3.51800e9i	−0.615127	−	0.615127i
$276$		0			0
$277$			1.59344e8i			0.0270655i	0.999908	+	0.0135328i	$0.00430775\pi$
$277$			1.59344e8i			0.0270655i	−0.999908	+	0.0135328i	$0.995692\pi$
$278$	2.58494e8	+	2.58494e8i	0.0432785	+	0.0432785i
$279$		0			0
$280$	5.98054e9	−	5.98054e9i	0.972990	−	0.972990i
$281$	8.53572e9	−	8.53572e9i	1.36904	−	1.36904i	0.507217	−	0.861818i	$-0.330674\pi$
$281$	8.53572e9	−	8.53572e9i	1.36904	−	1.36904i	0.861818	−	0.507217i	$-0.169326\pi$
$282$		0			0
$283$		−	1.14991e10i		−	1.79275i	−0.443298	−	0.896374i	$-0.646192\pi$
$283$		−	1.14991e10i		−	1.79275i	0.443298	−	0.896374i	$-0.353808\pi$
$284$	−2.87569e9	+	2.87569e9i	−0.442048	+	0.442048i
$285$		0			0
$286$	3.45168e9	−	4.87712e9i	0.515901	−	0.728953i
$287$	−2.33994e9			−0.344887
$288$		0			0
$289$	−1.10663e10			−1.58639
$290$		−	2.44077e10i		−	3.45091i
$291$		0			0
$292$	3.42643e9	+	3.42643e9i	0.471314	+	0.471314i
$293$	−9.30877e9	+	9.30877e9i	−1.26305	+	1.26305i	−0.313448	+	0.949605i	$0.601484\pi$
$293$	−9.30877e9	+	9.30877e9i	−1.26305	+	1.26305i	−0.949605	+	0.313448i	$0.898516\pi$
$294$		0			0
$295$	−4.73753e9			−0.625552
$296$		−	6.24147e9i		−	0.813056i
$297$		0			0
$298$		−	1.19384e10i		−	1.51385i
$299$	−3.18211e9	−	2.25207e9i	−0.398135	−	0.281772i
$300$		0			0
$301$	−5.39423e9	−	5.39423e9i	−0.657149	−	0.657149i
$302$	−1.93882e9			−0.233083
$303$		0			0
$304$	9.34303e8	+	9.34303e8i	0.109394	+	0.109394i
$305$	−2.49868e9	−	2.49868e9i	−0.288743	−	0.288743i
$306$		0			0
$307$	−3.72592e9	+	3.72592e9i	−0.419450	+	0.419450i	−0.885014	−	0.465564i	$-0.845851\pi$
$307$	−3.72592e9	+	3.72592e9i	−0.419450	+	0.419450i	0.465564	+	0.885014i	$0.345851\pi$
$308$	−7.77950e9			−0.864468
$309$		0			0
$310$	1.23392e10	−	1.23392e10i	1.33610	−	1.33610i
$311$		−	5.30331e9i		−	0.566898i	−0.958987	−	0.283449i	$-0.908521\pi$
$311$		−	5.30331e9i		−	0.566898i	0.958987	−	0.283449i	$-0.0914787\pi$
$312$		0			0
$313$	1.14891e10			1.19704			0.598521	−	0.801107i	$-0.295755\pi$
$313$	1.14891e10			1.19704			0.598521	+	0.801107i	$0.295755\pi$
$314$	1.85492e10	+	1.85492e10i	1.90812	+	1.90812i
$315$		0			0
$316$		−	2.19550e10i		−	2.20184i
$317$	−3.70571e9	−	3.70571e9i	−0.366973	−	0.366973i	0.499399	−	0.866372i	$-0.333554\pi$
$317$	−3.70571e9	−	3.70571e9i	−0.366973	−	0.366973i	−0.866372	+	0.499399i	$0.833554\pi$
$318$		0			0
$319$	−5.56238e9	+	5.56238e9i	−0.537152	+	0.537152i
$320$	1.93187e10	−	1.93187e10i	1.84238	−	1.84238i
$321$		0			0
$322$			8.37306e9i			0.778863i
$323$	−1.12859e10	+	1.12859e10i	−1.03687	+	1.03687i
$324$		0			0
$325$	−1.41362e10	−	1.00046e10i	−1.26707	−	0.896739i
$326$	−4.53408e8			−0.0401438
$327$		0			0
$328$	3.42408e9			0.295835
$329$		−	1.72820e9i		−	0.147506i
$330$		0			0
$331$	8.91506e8	+	8.91506e8i	0.0742698	+	0.0742698i	0.743266	−	0.668996i	$-0.233276\pi$
$331$	8.91506e8	+	8.91506e8i	0.0742698	+	0.0742698i	−0.668996	+	0.743266i	$0.733276\pi$
$332$	−1.26829e9	+	1.26829e9i	−0.104391	+	0.104391i
$333$		0			0
$334$	−1.35971e10			−1.09260
$335$			6.29674e9i			0.499961i
$336$		0			0
$337$			2.86060e8i			0.0221788i	0.999939	+	0.0110894i	$0.00352993\pi$
$337$			2.86060e8i			0.0221788i	−0.999939	+	0.0110894i	$0.996470\pi$
$338$	8.97895e9	−	1.87605e10i	0.687953	−	1.43740i
$339$		0			0
$340$	3.73732e10	+	3.73732e10i	2.79669	+	2.79669i
$341$	−5.62406e9			−0.415942
$342$		0			0
$343$	9.76700e9	+	9.76700e9i	0.705642	+	0.705642i
$344$	7.89350e9	+	7.89350e9i	0.563684	+	0.563684i
$345$		0			0
$346$	3.12530e9	−	3.12530e9i	0.218065	−	0.218065i
$347$	2.29430e10			1.58246			0.791230	−	0.611519i	$-0.209441\pi$
$347$	2.29430e10			1.58246			0.791230	+	0.611519i	$0.209441\pi$
$348$		0			0
$349$	−8.38687e9	+	8.38687e9i	−0.565325	+	0.565325i	−0.930815	−	0.365490i	$-0.880901\pi$
$349$	−8.38687e9	+	8.38687e9i	−0.565325	+	0.565325i	0.365490	+	0.930815i	$0.380901\pi$
$350$			3.71965e10i			2.47873i
$351$		0			0
$352$	−9.72135e9			−0.633222
$353$	−1.24878e10	−	1.24878e10i	−0.804245	−	0.804245i	0.179511	−	0.983756i	$-0.442548\pi$
$353$	−1.24878e10	−	1.24878e10i	−0.804245	−	0.804245i	−0.983756	+	0.179511i	$0.942548\pi$
$354$		0			0
$355$		−	1.03042e10i		−	0.648786i
$356$	−1.84983e10	−	1.84983e10i	−1.15168	−	1.15168i
$357$		0			0
$358$	−1.72088e10	+	1.72088e10i	−1.04766	+	1.04766i
$359$	1.89252e10	−	1.89252e10i	1.13936	−	1.13936i	0.150799	−	0.988564i	$-0.451815\pi$
$359$	1.89252e10	−	1.89252e10i	1.13936	−	1.13936i	0.988564	−	0.150799i	$-0.0481846\pi$
$360$		0			0
$361$		−	2.86415e9i		−	0.168642i
$362$	3.39385e9	−	3.39385e9i	0.197632	−	0.197632i
$363$		0			0
$364$	−2.66918e10	+	4.56820e9i	−1.52045	+	0.260219i
$365$	−1.22776e10			−0.691739
$366$		0			0
$367$	6.23990e9			0.343964			0.171982	−	0.985100i	$-0.444983\pi$
$367$	6.23990e9			0.343964			0.171982	+	0.985100i	$0.444983\pi$
$368$			1.51778e9i			0.0827595i
$369$		0			0
$370$	3.19136e10	+	3.19136e10i	1.70282	+	1.70282i
$371$	1.48985e10	−	1.48985e10i	0.786406	−	0.786406i
$372$		0			0
$373$	−2.96446e9			−0.153148			−0.0765739	−	0.997064i	$-0.524398\pi$
$373$	−2.96446e9			−0.153148			−0.0765739	+	0.997064i	$0.524398\pi$
$374$		−	2.81000e10i		−	1.43621i
$375$		0			0
$376$			2.52892e9i			0.126527i
$377$	−1.58185e10	+	2.23510e10i	−0.783066	+	1.10645i
$378$		0			0
$379$	1.71245e10	+	1.71245e10i	0.829970	+	0.829970i	0.987512	−	0.157542i	$-0.0503571\pi$
$379$	1.71245e10	+	1.71245e10i	0.829970	+	0.829970i	−0.157542	+	0.987512i	$0.550357\pi$
$380$	4.67564e10			2.24237
$381$		0			0
$382$	−4.60211e9	−	4.60211e9i	−0.216124	−	0.216124i
$383$	−1.46047e10	−	1.46047e10i	−0.678732	−	0.678732i	0.280982	−	0.959713i	$-0.409340\pi$
$383$	−1.46047e10	−	1.46047e10i	−0.678732	−	0.678732i	−0.959713	+	0.280982i	$0.909340\pi$
$384$		0			0
$385$	1.39378e10	−	1.39378e10i	0.634382	−	0.634382i
$386$	5.52455e10			2.48856
$387$		0			0
$388$	2.03312e10	−	2.03312e10i	0.897089	−	0.897089i
$389$			2.88195e10i			1.25860i	0.777162	+	0.629301i	$0.216659\pi$
$389$			2.88195e10i			1.25860i	−0.777162	+	0.629301i	$0.783341\pi$
$390$		0			0
$391$	−1.83340e10			−0.784423
$392$	5.91450e7	+	5.91450e7i	0.00250480	+	0.00250480i
$393$		0			0
$394$		−	3.23893e10i		−	1.34406i
$395$	3.93347e10	+	3.93347e10i	1.61580	+	1.61580i
$396$		0			0
$397$	1.55229e10	−	1.55229e10i	0.624902	−	0.624902i	−0.321879	−	0.946781i	$-0.604314\pi$
$397$	1.55229e10	−	1.55229e10i	0.624902	−	0.624902i	0.946781	+	0.321879i	$0.104314\pi$
$398$	−4.96875e10	+	4.96875e10i	−1.98023	+	1.98023i
$399$		0			0
$400$			6.74258e9i			0.263382i
$401$	2.81886e10	−	2.81886e10i	1.09017	−	1.09017i	0.0946646	−	0.995509i	$-0.469822\pi$
$401$	2.81886e10	−	2.81886e10i	1.09017	−	1.09017i	0.995509	−	0.0946646i	$-0.0301779\pi$
$402$		0			0
$403$	−1.92964e10	+	3.30251e9i	−0.731570	+	0.125205i
$404$	−5.29642e10			−1.98818
$405$		0			0
$406$	5.88121e10			2.16452
$407$		−	1.45459e10i		−	0.530106i
$408$		0			0
$409$	−1.87784e10	−	1.87784e10i	−0.671066	−	0.671066i	0.286896	−	0.957962i	$-0.407377\pi$
$409$	−1.87784e10	−	1.87784e10i	−0.671066	−	0.671066i	−0.957962	+	0.286896i	$0.907377\pi$
$410$	−1.75079e10	+	1.75079e10i	−0.619581	+	0.619581i
$411$		0			0
$412$	−5.52441e10			−1.91733
$413$		−	1.14154e10i		−	0.392367i
$414$		0			0
$415$		−	4.54453e9i		−	0.153213i
$416$	−3.33543e10	+	5.70847e9i	−1.11373	+	0.190610i
$417$		0			0
$418$	−1.75775e10	−	1.75775e10i	−0.575773	−	0.575773i
$419$	−5.34767e10			−1.73503			−0.867517	−	0.497407i	$-0.834286\pi$
$419$	−5.34767e10			−1.73503			−0.867517	+	0.497407i	$0.834286\pi$
$420$		0			0
$421$	2.38177e10	+	2.38177e10i	0.758180	+	0.758180i	0.975991	−	0.217811i	$-0.0698916\pi$
$421$	2.38177e10	+	2.38177e10i	0.758180	+	0.758180i	−0.217811	+	0.975991i	$0.569892\pi$
$422$	2.82663e10	+	2.82663e10i	0.891289	+	0.891289i
$423$		0			0
$424$	−2.18013e10	+	2.18013e10i	−0.674558	+	0.674558i
$425$	−8.14470e10			−2.49643
$426$		0			0
$427$	6.02075e9	−	6.02075e9i	0.181109	−	0.181109i
$428$		−	8.70298e10i		−	2.59354i
$429$		0			0
$430$	−8.07214e10			−2.36110
$431$	−3.39333e10	−	3.39333e10i	−0.983370	−	0.983370i	0.0164943	−	0.999864i	$-0.494749\pi$
$431$	−3.39333e10	−	3.39333e10i	−0.983370	−	0.983370i	−0.999864	+	0.0164943i	$0.994749\pi$
$432$		0			0
$433$			6.21384e10i			1.76770i	0.467771	+	0.883849i	$0.345057\pi$
$433$			6.21384e10i			1.76770i	−0.467771	+	0.883849i	$0.654943\pi$
$434$	2.97321e10	+	2.97321e10i	0.838045	+	0.838045i
$435$		0			0
$436$	−1.51420e10	+	1.51420e10i	−0.419022	+	0.419022i
$437$	−1.14685e10	+	1.14685e10i	−0.314472	+	0.314472i
$438$		0			0
$439$		−	7.19465e10i		−	1.93710i	−0.248822	−	0.968549i	$-0.580043\pi$
$439$		−	7.19465e10i		−	1.93710i	0.248822	−	0.968549i	$-0.419957\pi$
$440$	−2.03955e10	+	2.03955e10i	−0.544155	+	0.544155i
$441$		0			0
$442$	−1.65006e10	−	9.64121e10i	−0.432325	−	2.52605i
$443$	2.02774e10			0.526499			0.263249	−	0.964728i	$-0.415206\pi$
$443$	2.02774e10			0.526499			0.263249	+	0.964728i	$0.415206\pi$
$444$		0			0
$445$	6.62832e10			1.69030
$446$			5.77178e9i			0.145871i
$447$		0			0
$448$	4.65500e10	+	4.65500e10i	1.15560	+	1.15560i
$449$	−7.54656e9	+	7.54656e9i	−0.185679	+	0.185679i	−0.793825	−	0.608146i	$-0.791914\pi$
$449$	−7.54656e9	+	7.54656e9i	−0.185679	+	0.185679i	0.608146	+	0.793825i	$0.291914\pi$
$450$		0			0
$451$	7.97990e9			0.192882
$452$		−	4.87641e9i		−	0.116828i
$453$		0			0
$454$			3.36152e10i			0.791247i
$455$	3.96367e10	−	5.60055e10i	0.924808	−	1.30673i
$456$		0			0
$457$	−7.97853e9	−	7.97853e9i	−0.182919	−	0.182919i	0.609708	−	0.792626i	$-0.291287\pi$
$457$	−7.97853e9	−	7.97853e9i	−0.182919	−	0.182919i	−0.792626	+	0.609708i	$0.791287\pi$
$458$	9.76513e10			2.21930
$459$		0			0
$460$	3.79780e10	+	3.79780e10i	0.848206	+	0.848206i
$461$	3.37578e10	+	3.37578e10i	0.747430	+	0.747430i	0.973996	−	0.226565i	$-0.0727497\pi$
$461$	3.37578e10	+	3.37578e10i	0.747430	+	0.747430i	−0.226565	+	0.973996i	$0.572750\pi$
$462$		0			0
$463$	2.56807e9	−	2.56807e9i	0.0558835	−	0.0558835i	−0.678613	−	0.734496i	$-0.737419\pi$
$463$	2.56807e9	−	2.56807e9i	0.0558835	−	0.0558835i	0.734496	+	0.678613i	$0.237419\pi$
$464$	1.06608e10			0.229995
$465$		0			0
$466$	1.43660e10	−	1.43660e10i	0.304644	−	0.304644i
$467$			6.24287e10i			1.31255i	0.754521	+	0.656276i	$0.227869\pi$
$467$			6.24287e10i			1.31255i	−0.754521	+	0.656276i	$0.772131\pi$
$468$		0			0
$469$	−1.51725e10			−0.313592
$470$	−1.29307e10	−	1.29307e10i	−0.264992	−	0.264992i
$471$		0			0
$472$			1.67045e10i			0.336561i
$473$	1.83960e10	+	1.83960e10i	0.367518	+	0.367518i
$474$		0			0
$475$	−5.09478e10	+	5.09478e10i	−1.00081	+	1.00081i
$476$	−9.00536e10	+	9.00536e10i	−1.75418	+	1.75418i
$477$		0			0
$478$		−	1.37529e11i		−	2.63441i
$479$	5.07009e10	−	5.07009e10i	0.963104	−	0.963104i	−0.0362388	−	0.999343i	$-0.511538\pi$
$479$	5.07009e10	−	5.07009e10i	0.963104	−	0.963104i	0.999343	+	0.0362388i	$0.0115377\pi$
$480$		0			0
$481$	−8.54149e9	−	4.99075e10i	−0.159571	−	0.932365i
$482$	−7.79861e10			−1.44487
$483$		0			0
$484$	−5.79448e10			−1.05592
$485$			7.28508e10i			1.31664i
$486$		0			0
$487$	−1.81769e10	−	1.81769e10i	−0.323150	−	0.323150i	0.526824	−	0.849974i	$-0.323383\pi$
$487$	−1.81769e10	−	1.81769e10i	−0.323150	−	0.323150i	−0.849974	+	0.526824i	$0.823383\pi$
$488$	−8.81030e9	+	8.81030e9i	−0.155350	+	0.155350i
$489$		0			0
$490$	−6.04835e8			−0.0104919
$491$		−	1.13568e11i		−	1.95402i	−0.213184	−	0.977012i	$-0.568383\pi$
$491$		−	1.13568e11i		−	1.95402i	0.213184	−	0.977012i	$-0.431617\pi$
$492$		0			0
$493$			1.28777e11i			2.17998i
$494$	−7.06307e10	−	4.99873e10i	−1.18600	−	0.839367i
$495$		0			0
$496$	5.38953e9	+	5.38953e9i	0.0890480	+	0.0890480i
$497$	2.48288e10			0.406939
$498$		0			0
$499$	5.85848e9	+	5.85848e9i	0.0944894	+	0.0944894i	0.752771	−	0.658282i	$-0.228717\pi$
$499$	5.85848e9	+	5.85848e9i	0.0944894	+	0.0944894i	−0.658282	+	0.752771i	$0.728717\pi$
$500$	6.00265e10	+	6.00265e10i	0.960423	+	0.960423i
$501$		0			0
$502$	−5.10682e10	+	5.10682e10i	−0.804147	+	0.804147i
$503$	−8.18123e10			−1.27805			−0.639023	−	0.769188i	$-0.720661\pi$
$503$	−8.18123e10			−1.27805			−0.639023	+	0.769188i	$0.720661\pi$
$504$		0			0
$505$	9.48908e10	−	9.48908e10i	1.45901	−	1.45901i
$506$		−	2.85547e10i		−	0.435588i
$507$		0			0
$508$	2.05798e10			0.309019
$509$	6.63020e10	+	6.63020e10i	0.987770	+	0.987770i	0.999926	−	0.0121563i	$-0.00386957\pi$
$509$	6.63020e10	+	6.63020e10i	0.987770	+	0.987770i	−0.0121563	+	0.999926i	$0.503870\pi$
$510$		0			0
$511$		−	2.95838e10i		−	0.433881i
$512$	1.64027e10	+	1.64027e10i	0.238690	+	0.238690i
$513$		0			0
$514$	−3.99309e10	+	3.99309e10i	−0.572080	+	0.572080i
$515$	9.89755e10	−	9.89755e10i	1.40702	−	1.40702i
$516$		0			0
$517$			5.89370e9i			0.0824946i
$518$	−7.68983e10	+	7.68983e10i	−1.06806	+	1.06806i
$519$		0			0
$520$	−5.80012e10	+	8.19541e10i	−0.793275	+	1.12088i
$521$	8.77167e9			0.119051			0.0595253	−	0.998227i	$-0.481041\pi$
$521$	8.77167e9			0.119051			0.0595253	+	0.998227i	$0.481041\pi$
$522$		0			0
$523$	−8.20061e10			−1.09607			−0.548037	−	0.836454i	$-0.684625\pi$
$523$	−8.20061e10			−1.09607			−0.548037	+	0.836454i	$0.684625\pi$
$524$			1.37405e11i			1.82254i
$525$		0			0
$526$	−6.81834e9	−	6.81834e9i	−0.0890708	−	0.0890708i
$527$	−6.51027e10	+	6.51027e10i	−0.844028	+	0.844028i
$528$		0			0
$529$	5.96803e10			0.762093
$530$		−	2.22947e11i		−	2.82552i
$531$		0			0
$532$			1.12663e11i			1.40648i
$533$	2.73794e10	−	4.68588e9i	0.339246	−	0.0580607i
$534$		0			0
$535$	1.55923e11	+	1.55923e11i	1.90325	+	1.90325i
$536$	2.22022e10			0.268990
$537$		0			0
$538$	−1.63727e10	−	1.63727e10i	−0.195430	−	0.195430i
$539$	1.37839e8	+	1.37839e8i	0.00163311	+	0.00163311i
$540$		0			0
$541$	6.06619e10	−	6.06619e10i	0.708153	−	0.708153i	−0.257994	−	0.966147i	$-0.583061\pi$
$541$	6.06619e10	−	6.06619e10i	0.708153	−	0.708153i	0.966147	+	0.257994i	$0.0830614\pi$
$542$	−1.70478e11			−1.97547
$543$		0			0
$544$	−1.12532e11	+	1.12532e11i	−1.28493	+	1.28493i
$545$		−	5.42569e10i		−	0.614991i
$546$		0			0
$547$	1.54001e10			0.172018			0.0860090	−	0.996294i	$-0.472589\pi$
$547$	1.54001e10			0.172018			0.0860090	+	0.996294i	$0.472589\pi$
$548$	5.41333e10	+	5.41333e10i	0.600264	+	0.600264i
$549$		0			0
$550$		−	1.26851e11i		−	1.38626i
$551$	8.05546e10	+	8.05546e10i	0.873944	+	0.873944i
$552$		0			0
$553$	−9.47798e10	+	9.47798e10i	−1.01348	+	1.01348i
$554$	−2.87280e9	+	2.87280e9i	−0.0304976	+	0.0304976i
$555$		0			0
$556$			5.65027e9i			0.0591249i
$557$	−4.93168e10	+	4.93168e10i	−0.512359	+	0.512359i	−0.915248	−	0.402890i	$-0.868006\pi$
$557$	−4.93168e10	+	4.93168e10i	−0.512359	+	0.512359i	0.402890	+	0.915248i	$0.368006\pi$
$558$		0			0
$559$	7.39196e10	+	5.23150e10i	0.757029	+	0.535771i
$560$	−2.67131e10			−0.271627
$561$		0			0
$562$	3.07779e11			3.08528
$563$			5.82242e10i			0.579522i	0.957099	+	0.289761i	$0.0935758\pi$
$563$			5.82242e10i			0.579522i	−0.957099	+	0.289761i	$0.906424\pi$
$564$		0			0
$565$	8.73659e9	+	8.73659e9i	0.0857330	+	0.0857330i
$566$	2.07317e11	−	2.07317e11i	2.02008	−	2.02008i
$567$		0			0
$568$	−3.63325e10			−0.349061
$569$			1.06030e11i			1.01154i	0.862669	+	0.505768i	$0.168791\pi$
$569$			1.06030e11i			1.01154i	−0.862669	+	0.505768i	$0.831209\pi$
$570$		0			0
$571$		−	9.54463e10i		−	0.897872i	−0.893564	−	0.448936i	$-0.851803\pi$
$571$		−	9.54463e10i		−	0.897872i	0.893564	−	0.448936i	$-0.148197\pi$
$572$	9.10271e10	−	1.55790e10i	0.850328	−	0.145531i
$573$		0			0
$574$	−4.21865e10	−	4.21865e10i	−0.388621	−	0.388621i
$575$	−8.27650e10			−0.757139
$576$		0			0
$577$	−8.36851e9	−	8.36851e9i	−0.0754997	−	0.0754997i	0.668349	−	0.743848i	$-0.267001\pi$
$577$	−8.36851e9	−	8.36851e9i	−0.0754997	−	0.0754997i	−0.743848	+	0.668349i	$0.767001\pi$
$578$	−1.99513e11	−	1.99513e11i	−1.78756	−	1.78756i
$579$		0			0
$580$	2.66756e11	−	2.66756e11i	2.35723	−	2.35723i
$581$	1.09504e10			0.0961003
$582$		0			0
$583$	−5.08084e10	+	5.08084e10i	−0.439807	+	0.439807i
$584$			4.32907e10i			0.372171i
$585$		0			0
$586$	−3.35654e11			−2.84643
$587$	1.12403e11	+	1.12403e11i	0.946731	+	0.946731i	0.998651	−	0.0519204i	$-0.0165342\pi$
$587$	1.12403e11	+	1.12403e11i	0.946731	+	0.946731i	−0.0519204	+	0.998651i	$0.516534\pi$
$588$		0			0
$589$			8.14479e10i			0.676735i
$590$	−8.54125e10	−	8.54125e10i	−0.704877	−	0.704877i
$591$		0			0
$592$	−1.39393e10	+	1.39393e10i	−0.113489	+	0.113489i
$593$	8.34367e10	−	8.34367e10i	0.674743	−	0.674743i	−0.284063	−	0.958806i	$-0.591682\pi$
$593$	8.34367e10	−	8.34367e10i	0.674743	−	0.674743i	0.958806	+	0.284063i	$0.0916825\pi$
$594$		0			0
$595$		−	3.22681e11i		−	2.57457i
$596$	1.30477e11	−	1.30477e11i	1.03407	−	1.03407i
$597$		0			0
$598$	−1.67676e10	−	9.79723e10i	−0.131119	−	0.766123i
$599$	−2.58135e10			−0.200512			−0.100256	−	0.994962i	$-0.531966\pi$
$599$	−2.58135e10			−0.200512			−0.100256	+	0.994962i	$0.531966\pi$
$600$		0			0
$601$	−1.14049e11			−0.874162			−0.437081	−	0.899422i	$-0.643988\pi$
$601$	−1.14049e11			−0.874162			−0.437081	+	0.899422i	$0.643988\pi$
$602$		−	1.94504e11i		−	1.48096i
$603$		0			0
$604$	−2.11898e10	−	2.11898e10i	−0.159213	−	0.159213i
$605$	1.03814e11	−	1.03814e11i	0.774880	−	0.774880i
$606$		0			0
$607$	−1.31365e9			−0.00967668			−0.00483834	−	0.999988i	$-0.501540\pi$
$607$	−1.31365e9			−0.00967668			−0.00483834	+	0.999988i	$0.501540\pi$
$608$			1.40785e11i			1.03025i
$609$		0			0
$610$		−	9.00969e10i		−	0.650715i
$611$	3.46084e9	+	2.02215e10i	0.0248323	+	0.145094i
$612$		0			0
$613$	6.67181e10	+	6.67181e10i	0.472500	+	0.472500i	0.902723	−	0.430223i	$-0.141565\pi$
$613$	6.67181e10	+	6.67181e10i	0.472500	+	0.472500i	−0.430223	+	0.902723i	$0.641565\pi$
$614$	−1.34349e11			−0.945278
$615$		0			0
$616$	−4.91444e10	−	4.91444e10i	−0.341312	−	0.341312i
$617$	1.24710e11	+	1.24710e11i	0.860519	+	0.860519i	0.991398	−	0.130880i	$-0.0417801\pi$
$617$	1.24710e11	+	1.24710e11i	0.860519	+	0.860519i	−0.130880	+	0.991398i	$0.541780\pi$
$618$		0			0
$619$	5.75160e10	−	5.75160e10i	0.391766	−	0.391766i	−0.483551	−	0.875316i	$-0.660653\pi$
$619$	5.75160e10	−	5.75160e10i	0.391766	−	0.391766i	0.875316	+	0.483551i	$0.160653\pi$
$620$	2.69714e11			1.82531
$621$		0			0
$622$	9.56128e10	−	9.56128e10i	0.638785	−	0.638785i
$623$			1.59714e11i			1.06021i
$624$		0			0
$625$	2.17730e10			0.142692
$626$	2.07136e11	+	2.07136e11i	1.34884	+	1.34884i
$627$		0			0
$628$			4.05456e11i			2.60678i
$629$	−1.68380e11	−	1.68380e11i	−1.07569	−	1.07569i
$630$		0			0
$631$	1.45885e11	−	1.45885e11i	0.920223	−	0.920223i	−0.0768214	−	0.997045i	$-0.524477\pi$
$631$	1.45885e11	−	1.45885e11i	0.920223	−	0.920223i	0.997045	+	0.0768214i	$0.0244771\pi$
$632$	1.38693e11	−	1.38693e11i	0.869336	−	0.869336i
$633$		0			0
$634$		−	1.33620e11i		−	0.827015i
$635$	−3.68708e10	+	3.68708e10i	−0.226771	+	0.226771i
$636$		0			0
$637$	5.53871e8	+	3.91990e8i	0.00336396	+	0.00238077i
$638$	−2.00567e11			−1.21053
$639$		0			0
$640$	3.93738e11			2.34686
$641$		−	5.42742e10i		−	0.321486i	−0.986996	−	0.160743i	$-0.948611\pi$
$641$		−	5.42742e10i		−	0.321486i	0.986996	−	0.160743i	$-0.0513890\pi$
$642$		0			0
$643$	1.17827e11	+	1.17827e11i	0.689287	+	0.689287i	0.962074	−	0.272787i	$-0.0879456\pi$
$643$	1.17827e11	+	1.17827e11i	0.689287	+	0.689287i	−0.272787	+	0.962074i	$0.587946\pi$
$644$	−9.15109e10	+	9.15109e10i	−0.532022	+	0.532022i
$645$		0			0
$646$	−4.06945e11			−2.33671
$647$		−	2.59864e11i		−	1.48296i	−0.670977	−	0.741479i	$-0.734125\pi$
$647$		−	2.59864e11i		−	1.48296i	0.670977	−	0.741479i	$-0.265875\pi$
$648$		0			0
$649$			3.89301e10i			0.219435i
$650$	−7.44884e10	−	4.35232e11i	−0.417287	−	2.43819i
$651$		0			0
$652$	−4.95539e9	−	4.95539e9i	−0.0274212	−	0.0274212i
$653$	2.92048e11			1.60621			0.803105	−	0.595838i	$-0.203180\pi$
$653$	2.92048e11			1.60621			0.803105	+	0.595838i	$0.203180\pi$
$654$		0			0
$655$	−2.46175e11	−	2.46175e11i	−1.33745	−	1.33745i
$656$	−7.64713e9	−	7.64713e9i	−0.0412936	−	0.0412936i
$657$		0			0
$658$	3.11576e10	−	3.11576e10i	0.166211	−	0.166211i
$659$	−1.68819e11			−0.895115			−0.447557	−	0.894255i	$-0.647706\pi$
$659$	−1.68819e11			−0.895115			−0.447557	+	0.894255i	$0.647706\pi$
$660$		0			0
$661$	−1.99672e11	+	1.99672e11i	−1.04595	+	1.04595i	−0.0470593	+	0.998892i	$0.514985\pi$
$661$	−1.99672e11	+	1.99672e11i	−1.04595	+	1.04595i	−0.998892	+	0.0470593i	$0.985015\pi$
$662$			3.21458e10i			0.167375i
$663$		0			0
$664$	−1.60239e10			−0.0824323
$665$	−2.01847e11	−	2.01847e11i	−1.03214	−	1.03214i
$666$		0			0
$667$			1.30861e11i			0.661162i
$668$	−1.48605e11	−	1.48605e11i	−0.746325	−	0.746325i
$669$		0			0
$670$	−1.13523e11	+	1.13523e11i	−0.563360	+	0.563360i
$671$	−2.05326e10	+	2.05326e10i	−0.101287	+	0.101287i
$672$		0			0
$673$		−	1.11564e11i		−	0.543828i	−0.962321	−	0.271914i	$-0.912343\pi$
$673$		−	1.11564e11i		−	0.543828i	0.962321	−	0.271914i	$-0.0876567\pi$
$674$	−5.15735e9	+	5.15735e9i	−0.0249912	+	0.0249912i
$675$		0			0
$676$	3.03170e11	−	1.06904e11i	1.45177	−	0.511926i
$677$	2.36899e10			0.112774			0.0563869	−	0.998409i	$-0.482042\pi$
$677$	2.36899e10			0.112774			0.0563869	+	0.998409i	$0.482042\pi$
$678$		0			0
$679$	−1.75539e11			−0.825840
$680$			4.72186e11i			2.20840i
$681$		0			0
$682$	−1.01396e11	−	1.01396e11i	−0.468686	−	0.468686i
$683$	1.10040e11	−	1.10040e11i	0.505673	−	0.505673i	−0.407522	−	0.913195i	$-0.633607\pi$
$683$	1.10040e11	−	1.10040e11i	0.505673	−	0.505673i	0.913195	+	0.407522i	$0.133607\pi$
$684$		0			0
$685$	−1.93971e11			−0.880996
$686$			3.52177e11i			1.59025i
$687$		0			0
$688$		−	3.52577e10i		−	0.157362i
$689$	−1.44491e11	+	2.04161e11i	−0.641155	+	0.905933i
$690$		0			0
$691$	9.26218e10	+	9.26218e10i	0.406257	+	0.406257i	0.880431	−	0.474174i	$-0.157253\pi$
$691$	9.26218e10	+	9.26218e10i	0.406257	+	0.406257i	−0.474174	+	0.880431i	$0.657253\pi$
$692$	6.83140e10			0.297910
$693$		0			0
$694$	4.13638e11	+	4.13638e11i	1.78313	+	1.78313i
$695$	−1.01231e10	−	1.01231e10i	−0.0433883	−	0.0433883i
$696$		0			0
$697$	9.23733e10	−	9.23733e10i	0.391395	−	0.391395i
$698$	−3.02412e11			−1.27402
$699$		0			0
$700$	−4.06528e11	+	4.06528e11i	−1.69316	+	1.69316i
$701$			9.17227e10i			0.379844i	0.981799	+	0.189922i	$0.0608234\pi$
$701$			9.17227e10i			0.379844i	−0.981799	+	0.189922i	$0.939177\pi$
$702$		0			0
$703$	−2.10654e11			−0.862480
$704$	−1.58750e11	−	1.58750e11i	−0.646282	−	0.646282i
$705$		0			0
$706$		−	4.50284e11i		−	1.81246i
$707$	2.28646e11	+	2.28646e11i	0.915138	+	0.915138i
$708$		0			0
$709$	−7.57204e10	+	7.57204e10i	−0.299659	+	0.299659i	−0.840880	−	0.541221i	$-0.817962\pi$
$709$	−7.57204e10	+	7.57204e10i	−0.299659	+	0.299659i	0.541221	+	0.840880i	$0.317962\pi$
$710$	1.85774e11	−	1.85774e11i	0.731056	−	0.731056i
$711$		0			0
$712$		−	2.33713e11i		−	0.909419i
$713$	−6.61563e10	+	6.61563e10i	−0.255984	+	0.255984i
$714$		0			0
$715$	−1.35173e11	+	1.90996e11i	−0.517209	+	0.730802i
$716$	−3.76158e11			−1.43126
$717$		0			0
$718$	6.82401e11			2.56769
$719$			1.51493e11i			0.566861i	0.958993	+	0.283430i	$0.0914725\pi$
$719$			1.51493e11i			0.566861i	−0.958993	+	0.283430i	$0.908528\pi$
$720$		0			0
$721$	2.38489e11	+	2.38489e11i	0.882526	+	0.882526i
$722$	5.16375e10	−	5.16375e10i	0.190028	−	0.190028i
$723$		0			0
$724$	7.41841e10			0.269995
$725$			5.81338e11i			2.10415i
$726$		0			0
$727$			3.09159e11i			1.10674i	0.832937	+	0.553368i	$0.186658\pi$
$727$			3.09159e11i			1.10674i	−0.832937	+	0.553368i	$0.813342\pi$
$728$	−1.97475e11	−	1.39758e11i	−0.703049	−	0.497568i
$729$		0			0
$730$	−2.21352e11	−	2.21352e11i	−0.779456	−	0.779456i
$731$	4.25895e11			1.49153
$732$		0			0
$733$	−3.53287e11	−	3.53287e11i	−1.22380	−	1.22380i	−0.966269	−	0.257533i	$-0.917090\pi$
$733$	−3.53287e11	−	3.53287e11i	−1.22380	−	1.22380i	−0.257533	−	0.966269i	$-0.582910\pi$
$734$	1.12499e11	+	1.12499e11i	0.387581	+	0.387581i
$735$		0			0
$736$	−1.14353e11	+	1.14353e11i	−0.389705	+	0.389705i
$737$	5.17427e10			0.175380
$738$		0			0
$739$	1.74930e11	−	1.74930e11i	0.586524	−	0.586524i	−0.350164	−	0.936688i	$-0.613874\pi$
$739$	1.74930e11	−	1.74930e11i	0.586524	−	0.586524i	0.936688	+	0.350164i	$0.113874\pi$
$740$			6.97581e11i			2.32631i
$741$		0			0
$742$	5.37208e11			1.77226
$743$	−1.82265e11	−	1.82265e11i	−0.598066	−	0.598066i	0.341732	−	0.939798i	$-0.388987\pi$
$743$	−1.82265e11	−	1.82265e11i	−0.598066	−	0.598066i	−0.939798	+	0.341732i	$0.888987\pi$
$744$		0			0
$745$			4.67528e11i			1.51769i
$746$	−5.34461e10	−	5.34461e10i	−0.172568	−	0.172568i
$747$		0			0
$748$	3.07110e11	−	3.07110e11i	0.981042	−	0.981042i
$749$	−3.75708e11	+	3.75708e11i	−1.19378	+	1.19378i
$750$		0			0
$751$			4.21734e11i			1.32580i	0.748706	+	0.662902i	$0.230675\pi$
$751$			4.21734e11i			1.32580i	−0.748706	+	0.662902i	$0.769325\pi$
$752$	5.64792e9	−	5.64792e9i	0.0176611	−	0.0176611i
$753$		0			0
$754$	−6.88154e11	+	1.17775e11i	−2.12912	+	0.364391i
$755$	7.59274e10			0.233674
$756$		0			0
$757$	−1.34616e11			−0.409932			−0.204966	−	0.978769i	$-0.565708\pi$
$757$	−1.34616e11			−0.409932			−0.204966	+	0.978769i	$0.565708\pi$
$758$			6.17474e11i			1.87043i
$759$		0			0
$760$	2.95368e11	+	2.95368e11i	0.885338	+	0.885338i
$761$	−2.98532e9	+	2.98532e9i	−0.00890126	+	0.00890126i	−0.711543	−	0.702642i	$-0.752004\pi$
$761$	−2.98532e9	+	2.98532e9i	−0.00890126	+	0.00890126i	0.702642	+	0.711543i	$0.252004\pi$
$762$		0			0
$763$	1.30736e11			0.385742
$764$		−	1.00595e11i		−	0.295258i
$765$		0			0
$766$		−	5.26614e11i		−	1.52960i
$767$	2.28601e10	+	1.33571e11i	0.0660537	+	0.385949i
$768$		0			0
$769$	−4.30116e11	−	4.30116e11i	−1.22993	−	1.22993i	−0.963990	−	0.265939i	$-0.914318\pi$
$769$	−4.30116e11	−	4.30116e11i	−1.22993	−	1.22993i	−0.265939	−	0.963990i	$-0.585682\pi$
$770$	5.02566e11			1.42965
$771$		0			0
$772$	6.03789e11	+	6.03789e11i	1.69987	+	1.69987i
$773$	−8.08122e10	−	8.08122e10i	−0.226339	−	0.226339i	0.584823	−	0.811161i	$-0.301164\pi$
$773$	−8.08122e10	−	8.08122e10i	−0.226339	−	0.226339i	−0.811161	+	0.584823i	$0.801164\pi$
$774$		0			0
$775$	−2.93893e11	+	2.93893e11i	−0.814670	+	0.814670i
$776$	2.56871e11			0.708383
$777$		0			0
$778$	−5.19585e11	+	5.19585e11i	−1.41820	+	1.41820i
$779$		−	1.15565e11i		−	0.313818i
$780$		0			0
$781$	−8.46736e10			−0.227585
$782$	−3.30542e11	−	3.30542e11i	−0.883893	−	0.883893i
$783$		0			0
$784$		−	2.64181e8i		−	0.000699259i
$785$	−7.26416e11	−	7.26416e11i	−1.91296	−	1.91296i
$786$		0			0
$787$	3.45600e11	−	3.45600e11i	0.900896	−	0.900896i	−0.0946180	−	0.995514i	$-0.530163\pi$
$787$	3.45600e11	−	3.45600e11i	0.900896	−	0.900896i	0.995514	+	0.0946180i	$0.0301630\pi$
$788$	3.53990e11	−	3.53990e11i	0.918091	−	0.918091i
$789$		0			0
$790$			1.41832e12i			3.64138i
$791$	−2.10515e10	+	2.10515e10i	−0.0537745	+	0.0537745i
$792$		0			0
$793$	−5.83912e10	+	8.25052e10i	−0.147657	+	0.208636i
$794$	5.59724e11			1.40829
$795$		0			0
$796$	−1.08609e12			−2.70529
$797$		−	3.23312e11i		−	0.801287i	−0.916234	−	0.400644i	$-0.868787\pi$
$797$		−	3.23312e11i		−	0.801287i	0.916234	−	0.400644i	$-0.131213\pi$
$798$		0			0
$799$	6.82240e10	+	6.82240e10i	0.167398	+	0.167398i
$800$	−5.08002e11	+	5.08002e11i	−1.24024	+	1.24024i
$801$		0			0
$802$	1.01642e12			2.45683
$803$			1.00890e11i			0.242653i
$804$		0			0
$805$		−	3.27902e11i		−	0.780838i
$806$	−4.07433e11	−	2.88352e11i	−0.965420	−	0.683255i
$807$		0			0
$808$	−3.34584e11	−	3.34584e11i	−0.784981	−	0.784981i
$809$	−6.43334e10			−0.150190			−0.0750952	−	0.997176i	$-0.523926\pi$
$809$	−6.43334e10			−0.150190			−0.0750952	+	0.997176i	$0.523926\pi$
$810$		0			0
$811$	1.97439e11	+	1.97439e11i	0.456404	+	0.456404i	0.897473	−	0.441069i	$-0.145401\pi$
$811$	1.97439e11	+	1.97439e11i	0.456404	+	0.456404i	−0.441069	+	0.897473i	$0.645401\pi$
$812$	6.42769e11	+	6.42769e11i	1.47853	+	1.47853i
$813$		0			0
$814$	2.62247e11	−	2.62247e11i	0.597327	−	0.597327i
$815$	1.77562e10			0.0402457
$816$		0			0
$817$	2.66411e11	−	2.66411e11i	0.597949	−	0.597949i
$818$		−	6.77108e11i		−	1.51232i
$819$		0			0
$820$	−3.82694e11			−0.846440
$821$	2.21288e11	+	2.21288e11i	0.487063	+	0.487063i	0.907378	−	0.420315i	$-0.138081\pi$
$821$	2.21288e11	+	2.21288e11i	0.487063	+	0.487063i	−0.420315	+	0.907378i	$0.638081\pi$
$822$		0			0
$823$		−	2.35397e11i		−	0.513101i	−0.966531	−	0.256550i	$-0.917414\pi$
$823$		−	2.35397e11i		−	0.513101i	0.966531	−	0.256550i	$-0.0825859\pi$
$824$	−3.48986e11	−	3.48986e11i	−0.757006	−	0.757006i
$825$		0			0
$826$	2.05808e11	−	2.05808e11i	0.442121	−	0.442121i
$827$	5.32581e11	−	5.32581e11i	1.13858	−	1.13858i	0.149874	−	0.988705i	$-0.452113\pi$
$827$	5.32581e11	−	5.32581e11i	1.13858	−	1.13858i	0.988705	−	0.149874i	$-0.0478869\pi$
$828$		0			0
$829$		−	7.96173e11i		−	1.68574i	−0.538121	−	0.842868i	$-0.680866\pi$
$829$		−	7.96173e11i		−	1.68574i	0.538121	−	0.842868i	$-0.319134\pi$
$830$	8.19329e10	−	8.19329e10i	0.172642	−	0.172642i
$831$		0			0
$832$	−6.37896e11	−	4.51457e11i	−1.33124	−	0.942156i
$833$	3.19117e9			0.00662782
$834$		0			0
$835$	5.32483e11			1.09537
$836$		−	3.84215e11i		−	0.786592i
$837$		0			0
$838$	−9.64126e11	−	9.64126e11i	−1.95505	−	1.95505i
$839$	−7.78054e10	+	7.78054e10i	−0.157023	+	0.157023i	−0.781246	−	0.624223i	$-0.785416\pi$
$839$	−7.78054e10	+	7.78054e10i	−0.157023	+	0.157023i	0.624223	+	0.781246i	$0.285416\pi$
$840$		0			0
$841$	4.18919e11			0.837425
$842$			8.58816e11i			1.70864i
$843$		0			0
$844$			6.17855e11i			1.21763i
$845$	−3.51630e11	+	7.34689e11i	−0.689698	+	1.44104i
$846$		0			0
$847$	2.50148e11	+	2.50148e11i	0.486030	+	0.486030i
$848$	9.73793e10			0.188314
$849$		0			0
$850$	−1.46840e12	−	1.46840e12i	−2.81299	−	2.81299i
$851$	−1.71105e11	−	1.71105e11i	−0.326245	−	0.326245i
$852$		0			0
$853$	−4.02094e10	+	4.02094e10i	−0.0759506	+	0.0759506i	−0.744062	−	0.668111i	$-0.767103\pi$
$853$	−4.02094e10	+	4.02094e10i	−0.0759506	+	0.0759506i	0.668111	+	0.744062i	$0.267103\pi$
$854$	2.17095e11			0.408149
$855$		0			0
$856$	5.49782e11	−	5.49782e11i	1.02399	−	1.02399i
$857$		−	8.03555e11i		−	1.48968i	−0.667245	−	0.744839i	$-0.732526\pi$
$857$		−	8.03555e11i		−	1.48968i	0.667245	−	0.744839i	$-0.267474\pi$
$858$		0			0
$859$	6.03941e11			1.10923			0.554615	−	0.832107i	$-0.312866\pi$
$859$	6.03941e11			1.10923			0.554615	+	0.832107i	$0.312866\pi$
$860$	−8.82220e11	−	8.82220e11i	−1.61281	−	1.61281i
$861$		0			0
$862$		−	1.22356e12i		−	2.21614i
$863$	−7.14366e11	−	7.14366e11i	−1.28789	−	1.28789i	−0.936068	−	0.351818i	$-0.885564\pi$
$863$	−7.14366e11	−	7.14366e11i	−1.28789	−	1.28789i	−0.351818	−	0.936068i	$-0.614436\pi$
$864$		0			0
$865$	−1.22392e11	+	1.22392e11i	−0.218619	+	0.218619i
$866$	−1.12029e12	+	1.12029e12i	−1.99186	+	1.99186i
$867$		0			0
$868$			6.49897e11i			1.14490i
$869$	3.23228e11	−	3.23228e11i	0.566800	−	0.566800i
$870$		0			0
$871$	1.77531e11	−	3.03838e10i	0.308462	−	0.0527922i
$872$	−1.91309e11			−0.330879
$873$		0			0
$874$	−4.13530e11			−0.708699
$875$		−	5.18269e11i		−	0.884144i
$876$		0			0
$877$	5.61395e11	+	5.61395e11i	0.949008	+	0.949008i	0.998762	−	0.0497532i	$-0.0158435\pi$
$877$	5.61395e11	+	5.61395e11i	0.949008	+	0.949008i	−0.0497532	+	0.998762i	$0.515843\pi$
$878$	1.29712e12	−	1.29712e12i	2.18274	−	2.18274i
$879$		0			0
$880$	9.10998e10			0.151910
$881$		−	1.38278e11i		−	0.229536i	−0.993392	−	0.114768i	$-0.963388\pi$
$881$		−	1.38278e11i		−	0.229536i	0.993392	−	0.114768i	$-0.0366125\pi$
$882$		0			0
$883$		−	6.30093e11i		−	1.03648i	−0.855235	−	0.518241i	$-0.826587\pi$
$883$		−	6.30093e11i		−	1.03648i	0.855235	−	0.518241i	$-0.173413\pi$
$884$	8.73369e11	−	1.23405e12i	1.43017	−	2.02079i
$885$		0			0
$886$	3.65579e11	+	3.65579e11i	0.593263	+	0.593263i
$887$	7.83211e11			1.26527			0.632637	−	0.774449i	$-0.281973\pi$
$887$	7.83211e11			1.26527			0.632637	+	0.774449i	$0.281973\pi$
$888$		0			0
$889$	−8.88429e10	−	8.88429e10i	−0.142238	−	0.142238i
$890$	1.19501e12	+	1.19501e12i	1.90464	+	1.90464i
$891$		0			0
$892$	−6.30809e10	+	6.30809e10i	−0.0996411	+	0.0996411i
$893$	8.53528e10			0.134218
$894$		0			0
$895$	6.73926e11	−	6.73926e11i	1.05032	−	1.05032i
$896$			9.48742e11i			1.47203i
$897$		0			0
$898$	−2.72112e11			−0.418450
$899$	4.64679e11	+	4.64679e11i	0.711401	+	0.711401i
$900$		0			0
$901$			1.17629e12i			1.78491i
$902$	1.43869e11	+	1.43869e11i	0.217341	+	0.217341i
$903$		0			0
$904$	3.08051e10	−	3.08051e10i	0.0461263	−	0.0461263i
$905$	−1.32908e11	+	1.32908e11i	−0.198134	+	0.198134i
$906$		0			0
$907$			4.86298e10i			0.0718577i	0.999354	+	0.0359288i	$0.0114390\pi$
$907$			4.86298e10i			0.0718577i	−0.999354	+	0.0359288i	$0.988561\pi$
$908$	−3.67387e11	+	3.67387e11i	−0.540481	+	0.540481i
$909$		0			0
$910$	1.72432e12	−	2.95112e11i	2.51451	−	0.430349i
$911$	−5.89747e10			−0.0856234			−0.0428117	−	0.999083i	$-0.513632\pi$
$911$	−5.89747e10			−0.0856234			−0.0428117	+	0.999083i	$0.513632\pi$
$912$		0			0
$913$	−3.73442e10			−0.0537452
$914$		−	2.87688e11i		−	0.412228i
$915$		0			0
$916$	1.06725e12	+	1.06725e12i	1.51595	+	1.51595i
$917$	5.93177e11	−	5.93177e11i	0.838893	−	0.838893i
$918$		0			0
$919$	−1.20228e12			−1.68556			−0.842781	−	0.538257i	$-0.819083\pi$
$919$	−1.20228e12			−1.68556			−0.842781	+	0.538257i	$0.819083\pi$
$920$			4.79827e11i			0.669782i
$921$		0			0
$922$			1.21723e12i			1.68442i
$923$	−2.90519e11	+	4.97212e10i	−0.400283	+	0.0685070i
$924$		0			0
$925$	−7.60114e11	−	7.60114e11i	−1.03827	−	1.03827i
$926$	9.25992e10			0.125940
$927$		0			0
$928$	8.03211e11	+	8.03211e11i	1.08302	+	1.08302i
$929$	5.36726e11	+	5.36726e11i	0.720593	+	0.720593i	0.968726	−	0.248133i	$-0.0798170\pi$
$929$	5.36726e11	+	5.36726e11i	0.720593	+	0.720593i	−0.248133	+	0.968726i	$0.579817\pi$
$930$		0			0
$931$	1.99619e9	−	1.99619e9i	0.00265707	−	0.00265707i
$932$	3.14018e11			0.416189
$933$		0			0
$934$	−1.12552e12	+	1.12552e12i	−1.47899	+	1.47899i
$935$			1.10044e12i			1.43986i
$936$		0			0
$937$	5.61778e11			0.728796			0.364398	−	0.931243i	$-0.381275\pi$
$937$	5.61778e11			0.728796			0.364398	+	0.931243i	$0.381275\pi$
$938$	−2.73543e11	−	2.73543e11i	−0.353357	−	0.353357i
$939$		0			0
$940$		−	2.82646e11i		−	0.362018i
$941$	5.81505e11	+	5.81505e11i	0.741643	+	0.741643i	0.972894	−	0.231251i	$-0.0742820\pi$
$941$	5.81505e11	+	5.81505e11i	0.741643	+	0.741643i	−0.231251	+	0.972894i	$0.574282\pi$
$942$		0			0
$943$	9.38682e10	−	9.38682e10i	0.118706	−	0.118706i
$944$	3.73066e10	−	3.73066e10i	0.0469784	−	0.0469784i
$945$		0			0
$946$			6.63319e11i			0.828243i
$947$	−5.95719e11	+	5.95719e11i	−0.740698	+	0.740698i	−0.972712	−	0.232014i	$-0.925468\pi$
$947$	−5.95719e11	+	5.95719e11i	−0.740698	+	0.740698i	0.232014	+	0.972712i	$0.425468\pi$
$948$		0			0
$949$	5.92435e10	+	3.46157e11i	0.0730426	+	0.426784i
$950$	−1.83707e12			−2.25544
$951$		0			0
$952$	−1.13777e12			−1.38518
$953$			5.10820e11i			0.619293i	0.950852	+	0.309646i	$0.100211\pi$
$953$			5.10820e11i			0.619293i	−0.950852	+	0.309646i	$0.899789\pi$
$954$		0			0
$955$	1.80226e11	+	1.80226e11i	0.216672	+	0.216672i
$956$	1.50308e12	−	1.50308e12i	1.79950	−	1.79950i
$957$		0			0
$958$	1.82816e12			2.17047
$959$		−	4.67387e11i		−	0.552589i
$960$		0			0
$961$		−	3.83058e11i		−	0.449129i
$962$	7.45785e11	−	1.05377e12i	0.870789	−	1.23040i
$963$		0			0
$964$	−8.52326e11	−	8.52326e11i	−0.986956	−	0.986956i
$965$	−2.16350e12			−2.49487
$966$		0			0
$967$	−3.84454e11	−	3.84454e11i	−0.439682	−	0.439682i	0.452223	−	0.891905i	$-0.350631\pi$
$967$	−3.84454e11	−	3.84454e11i	−0.439682	−	0.439682i	−0.891905	+	0.452223i	$0.850631\pi$
$968$	−3.66047e11	−	3.66047e11i	−0.416903	−	0.416903i
$969$		0			0
$970$	−1.31342e12	+	1.31342e12i	−1.48360	+	1.48360i
$971$	−5.37749e11			−0.604926			−0.302463	−	0.953161i	$-0.597809\pi$
$971$	−5.37749e11			−0.604926			−0.302463	+	0.953161i	$0.597809\pi$
$972$		0			0
$973$	2.43922e10	−	2.43922e10i	0.0272145	−	0.0272145i
$974$		−	6.55419e11i		−	0.728255i
$975$		0			0
$976$	3.93527e10			0.0433686
$977$	8.39362e10	+	8.39362e10i	0.0921237	+	0.0921237i	0.751667	−	0.659543i	$-0.229250\pi$
$977$	8.39362e10	+	8.39362e10i	0.0921237	+	0.0921237i	−0.659543	+	0.751667i	$0.729250\pi$
$978$		0			0
$979$		−	5.44674e11i		−	0.592934i
$980$	−6.61037e9	−	6.61037e9i	−0.00716673	−	0.00716673i
$981$		0			0
$982$	2.04751e12	−	2.04751e12i	2.20181	−	2.20181i
$983$	−2.54267e11	+	2.54267e11i	−0.272318	+	0.272318i	−0.830033	−	0.557715i	$-0.811678\pi$
$983$	−2.54267e11	+	2.54267e11i	−0.272318	+	0.272318i	0.557715	+	0.830033i	$0.311678\pi$
$984$		0			0
$985$			1.26842e12i			1.34746i
$986$	−2.32172e12	+	2.32172e12i	−2.45641	+	2.45641i
$987$		0			0
$988$	−2.25615e11	−	1.31826e12i	−0.236778	−	1.38348i
$989$	4.32787e11			0.452365
$990$		0			0
$991$	−1.73726e12			−1.80123			−0.900615	−	0.434617i	$-0.856884\pi$
$991$	−1.73726e12			−1.80123			−0.900615	+	0.434617i	$0.856884\pi$
$992$			8.12119e11i			0.838635i
$993$		0			0
$994$	4.47635e11	+	4.47635e11i	0.458542	+	0.458542i
$995$	1.94584e12	−	1.94584e12i	1.98525	−	1.98525i
$996$		0			0
$997$	1.32018e12			1.33614			0.668072	−	0.744096i	$-0.267120\pi$
$997$	1.32018e12			1.33614			0.668072	+	0.744096i	$0.267120\pi$
$998$			2.11244e11i			0.212943i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.9.j.a.73.8		18
3.2	odd	2		13.9.d.a.8.2	yes	18
13.5	odd	4	inner	117.9.j.a.109.8		18
39.5	even	4		13.9.d.a.5.2	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.9.d.a.5.2	✓	18	39.5	even	4
13.9.d.a.8.2	yes	18	3.2	odd	2
117.9.j.a.73.8		18	1.1	even	1	trivial
117.9.j.a.109.8		18	13.5	odd	4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	117.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(18.0289 + 18.0289i) q^{2} +394.083i q^{4} +(-706.041 - 706.041i) q^{5} +(1701.26 - 1701.26i) q^{7} +(-2489.49 + 2489.49i) q^{8} -25458.3i q^{10} +(-5801.81 + 5801.81i) q^{11} +(-16499.4 + 23313.1i) q^{13} +61343.7 q^{14} +11119.7 q^{16} +134321. i q^{17} +(84022.1 + 84022.1i) q^{19} +(278239. - 278239. i) q^{20} -209201. q^{22} +136494. i q^{23} +606362. i q^{25} +(-717776. + 122845. i) q^{26} +(670437. + 670437. i) q^{28} +958731. q^{29} +(484682. + 484682. i) q^{31} +(837786. + 837786. i) q^{32} +(-2.42165e6 + 2.42165e6i) q^{34} -2.40232e6 q^{35} +(-1.25356e6 + 1.25356e6i) q^{37} +3.02965e6i q^{38} +3.51536e6 q^{40} +(-687708. - 687708. i) q^{41} -3.17073e6i q^{43} +(-2.28640e6 - 2.28640e6i) q^{44} +(-2.46084e6 + 2.46084e6i) q^{46} +(507919. - 507919. i) q^{47} -23757.9i q^{49} +(-1.09320e7 + 1.09320e7i) q^{50} +(-9.18731e6 - 6.50212e6i) q^{52} +8.75734e6 q^{53} +8.19263e6 q^{55} +8.47053e6i q^{56} +(1.72849e7 + 1.72849e7i) q^{58} +(3.35500e6 - 3.35500e6i) q^{59} +3.53900e6 q^{61} +1.74766e7i q^{62} +2.73621e7i q^{64} +(2.81092e7 - 4.81079e6i) q^{65} +(-4.45919e6 - 4.45919e6i) q^{67} -5.29335e7 q^{68} +(-4.33111e7 - 4.33111e7i) q^{70} +(7.29718e6 + 7.29718e6i) q^{71} +(8.69469e6 - 8.69469e6i) q^{73} -4.52008e7 q^{74} +(-3.31117e7 + 3.31117e7i) q^{76} +1.97408e7i q^{77} -5.57116e7 q^{79} +(-7.85098e6 - 7.85098e6i) q^{80} -2.47972e7i q^{82} +(3.21832e6 + 3.21832e6i) q^{83} +(9.48358e7 - 9.48358e7i) q^{85} +(5.71648e7 - 5.71648e7i) q^{86} -2.88871e7i q^{88} +(-4.69400e7 + 4.69400e7i) q^{89} +(1.15920e7 + 6.77313e7i) q^{91} -5.37901e7 q^{92} +1.83144e7 q^{94} -1.18646e8i q^{95} +(-5.15911e7 - 5.15911e7i) q^{97} +(428329. - 428329. i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 2 q^{2} - 166 q^{5} + 5308 q^{7} - 10464 q^{8} + 31556 q^{11} + 71300 q^{13} + 110260 q^{14} - 522860 q^{16} + 100288 q^{19} - 736268 q^{20} - 977312 q^{22} - 2952238 q^{26} + 4497084 q^{28} + 2479024 q^{29}+ \cdots - 588677614 q^{98}+O(q^{100}) \) 18 * q + 2 * q^2 - 166 * q^5 + 5308 * q^7 - 10464 * q^8 + 31556 * q^11 + 71300 * q^13 + 110260 * q^14 - 522860 * q^16 + 100288 * q^19 - 736268 * q^20 - 977312 * q^22 - 2952238 * q^26 + 4497084 * q^28 + 2479024 * q^29 - 1892664 * q^31 - 947212 * q^32 - 531576 * q^34 + 2918284 * q^35 - 8343978 * q^37 - 12691908 * q^40 - 1140178 * q^41 + 3867188 * q^44 + 2006148 * q^46 + 13368572 * q^47 - 37369598 * q^50 - 14821220 * q^52 - 50561348 * q^53 + 76994128 * q^55 + 22505716 * q^58 - 2127976 * q^59 - 52016516 * q^61 - 10413082 * q^65 + 960292 * q^67 - 283187508 * q^68 - 166635032 * q^70 - 67412140 * q^71 - 145213226 * q^73 + 233620024 * q^74 - 150533640 * q^76 - 76829120 * q^79 - 524889520 * q^80 + 241951556 * q^83 + 260737764 * q^85 + 579480384 * q^86 + 89187110 * q^89 + 232660948 * q^91 + 122690376 * q^92 + 1069637380 * q^94 + 331183146 * q^97 - 588677614 * q^98

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