LMFDB - Embedded newform 81.9.d.e.26.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [81,9,Mod(26,81)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([1]))

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

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

chi := DirichletCharacter("81.26");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 81 = 3^{4} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	81.d (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$32.9976674150$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 27)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			26.2
Root			$0.707107 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	81.26
Dual form			81.9.d.e.53.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(25.4558 + 14.6969i) q^{2} +(304.000 + 526.543i) q^{4} +(712.764 - 411.514i) q^{5} +(-983.500 + 1703.47i) q^{7} +10346.6i q^{8} +O(q^{10})$	\(q+(25.4558 + 14.6969i) q^{2} +(304.000 + 526.543i) q^{4} +(712.764 - 411.514i) q^{5} +(-983.500 + 1703.47i) q^{7} +10346.6i q^{8} +24192.0 q^{10} +(-10895.1 - 6290.29i) q^{11} +(22752.5 + 39408.5i) q^{13} +(-50071.6 + 28908.9i) q^{14} +(-74240.0 + 128587. i) q^{16} +59610.8i q^{17} +152399. q^{19} +(433360. + 250201. i) q^{20} +(-184896. - 320249. i) q^{22} +(113737. - 65665.9i) q^{23} +(143376. - 248334. i) q^{25} +1.33757e6i q^{26} -1.19594e6 q^{28} +(-509728. - 294291. i) q^{29} +(82175.0 + 142331. i) q^{31} +(-1.48581e6 + 857831. i) q^{32} +(-876096. + 1.51744e6i) q^{34} +1.61890e6i q^{35} -663937. q^{37} +(3.87945e6 + 2.23980e6i) q^{38} +(4.25779e6 + 7.37471e6i) q^{40} +(812347. - 469009. i) q^{41} +(-287665. + 498250. i) q^{43} -7.64899e6i q^{44} +3.86035e6 q^{46} +(-7.99711e6 - 4.61713e6i) q^{47} +(947856. + 1.64173e6i) q^{49} +(7.29949e6 - 4.21436e6i) q^{50} +(-1.38335e7 + 2.39604e7i) q^{52} -1.03765e7i q^{53} -1.03542e7 q^{55} +(-1.76252e7 - 1.01759e7i) q^{56} +(-8.65037e6 - 1.49829e7i) q^{58} +(4.36466e6 - 2.51994e6i) q^{59} +(9.60648e6 - 1.66389e7i) q^{61} +4.83088e6i q^{62} -1.24191e7 q^{64} +(3.24343e7 + 1.87260e7i) q^{65} +(299016. + 517912. i) q^{67} +(-3.13877e7 + 1.81217e7i) q^{68} +(-2.37928e7 + 4.12104e7i) q^{70} -2.92721e7i q^{71} +1.28502e7 q^{73} +(-1.69011e7 - 9.75784e6i) q^{74} +(4.63293e7 + 8.02447e7i) q^{76} +(2.14307e7 - 1.23730e7i) q^{77} +(1.17923e7 - 2.04249e7i) q^{79} +1.22203e8i q^{80} +2.75720e7 q^{82} +(2.89694e7 + 1.67255e7i) q^{83} +(2.45307e7 + 4.24884e7i) q^{85} +(-1.46455e7 + 8.45559e6i) q^{86} +(6.50834e7 - 1.12728e8i) q^{88} +2.82848e7i q^{89} -8.95083e7 q^{91} +(6.91519e7 + 3.99249e7i) q^{92} +(-1.35715e8 - 2.35066e8i) q^{94} +(1.08624e8 - 6.27144e7i) q^{95} +(-6.82448e7 + 1.18203e8i) q^{97} +5.57223e7i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 1216 q^{4} - 3934 q^{7}+O(q^{10}) $ 4 * q + 1216 * q^4 - 3934 * q^7	$ 4 q + 1216 q^{4} - 3934 q^{7} + 96768 q^{10} + 91010 q^{13} - 296960 q^{16} + 609596 q^{19} - 739584 q^{22} + 573502 q^{25} - 4783744 q^{28} + 328700 q^{31} - 3504384 q^{34} - 2655748 q^{37} + 17031168 q^{40} - 1150660 q^{43} + 15441408 q^{46} + 3791424 q^{49} - 55334080 q^{52} - 41416704 q^{55} - 34601472 q^{58} + 38425922 q^{61} - 49676288 q^{64} + 1196066 q^{67} - 95171328 q^{70} + 51400700 q^{73} + 185317184 q^{76} + 47169314 q^{79} + 110287872 q^{82} + 98122752 q^{85} + 260333568 q^{88} - 358033340 q^{91} - 542861568 q^{94} - 272979262 q^{97}+O(q^{100}) $ 4 * q + 1216 * q^4 - 3934 * q^7 + 96768 * q^10 + 91010 * q^13 - 296960 * q^16 + 609596 * q^19 - 739584 * q^22 + 573502 * q^25 - 4783744 * q^28 + 328700 * q^31 - 3504384 * q^34 - 2655748 * q^37 + 17031168 * q^40 - 1150660 * q^43 + 15441408 * q^46 + 3791424 * q^49 - 55334080 * q^52 - 41416704 * q^55 - 34601472 * q^58 + 38425922 * q^61 - 49676288 * q^64 + 1196066 * q^67 - 95171328 * q^70 + 51400700 * q^73 + 185317184 * q^76 + 47169314 * q^79 + 110287872 * q^82 + 98122752 * q^85 + 260333568 * q^88 - 358033340 * q^91 - 542861568 * q^94 - 272979262 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$2$
$\chi(n)$	$e\left(\frac{1}{6}\right)$

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 $

$2$	25.4558	+	14.6969i	1.59099	+	0.918559i	0.993137	+	0.116953i	$0.0373126\pi$
$2$	25.4558	+	14.6969i	1.59099	+	0.918559i	0.597853	+	0.801606i	$0.296021\pi$
$3$		0			0
$3$		0			0
$4$	304.000	+	526.543i	1.18750	+	2.05681i
$5$	712.764	−	411.514i	1.14042	−	0.658423i	0.193887	−	0.981024i	$-0.437891\pi$
$5$	712.764	−	411.514i	1.14042	−	0.658423i	0.946535	+	0.322601i	$0.104557\pi$
$6$		0			0
$7$	−983.500	+	1703.47i	−0.409621	+	0.709484i	−0.994847	−	0.101386i	$-0.967672\pi$
$7$	−983.500	+	1703.47i	−0.409621	+	0.709484i	0.585226	+	0.810870i	$0.301006\pi$
$8$			10346.6i			2.52604i
$9$		0			0
$10$	24192.0			2.41920
$11$	−10895.1	−	6290.29i	−0.744150	−	0.429635i	0.0794262	−	0.996841i	$-0.474691\pi$
$11$	−10895.1	−	6290.29i	−0.744150	−	0.429635i	−0.823576	+	0.567205i	$0.808025\pi$
$12$		0			0
$13$	22752.5	+	39408.5i	0.796628	+	1.37980i	0.921800	+	0.387666i	$0.126718\pi$
$13$	22752.5	+	39408.5i	0.796628	+	1.37980i	−0.125172	+	0.992135i	$0.539948\pi$
$14$	−50071.6	+	28908.9i	−1.30341	+	0.752522i
$15$		0			0
$16$	−74240.0	+	128587.i	−1.13281	+	1.96209i
$17$			59610.8i			0.713722i	0.934157	+	0.356861i	$0.116153\pi$
$17$			59610.8i			0.713722i	−0.934157	+	0.356861i	$0.883847\pi$
$18$		0			0
$19$	152399.			1.16941			0.584706	−	0.811245i	$-0.301210\pi$
$19$	152399.			1.16941			0.584706	+	0.811245i	$0.301210\pi$
$20$	433360.	+	250201.i	2.70850	+	1.56375i
$21$		0			0
$22$	−184896.	−	320249.i	−0.789290	−	1.36709i
$23$	113737.	−	65665.9i	0.406433	−	0.234654i	−0.282823	−	0.959172i	$-0.591271\pi$
$23$	113737.	−	65665.9i	0.406433	−	0.234654i	0.689256	+	0.724518i	$0.257937\pi$
$24$		0			0
$25$	143376.	−	248334.i	0.367041	−	0.635734i
$26$			1.33757e6i			2.92700i
$27$		0			0
$28$	−1.19594e6			−1.94570
$29$	−509728.	−	294291.i	−0.720686	−	0.416089i	0.0943188	−	0.995542i	$-0.469933\pi$
$29$	−509728.	−	294291.i	−0.720686	−	0.416089i	−0.815005	+	0.579454i	$0.803266\pi$
$30$		0			0
$31$	82175.0	+	142331.i	0.0889801	+	0.154118i	0.907080	−	0.420958i	$-0.138306\pi$
$31$	82175.0	+	142331.i	0.0889801	+	0.154118i	−0.818100	+	0.575076i	$0.804973\pi$
$32$	−1.48581e6	+	857831.i	−1.41698	+	0.818091i
$33$		0			0
$34$	−876096.	+	1.51744e6i	−0.655596	+	1.13552i
$35$			1.61890e6i			1.07882i
$36$		0			0
$37$	−663937.			−0.354258			−0.177129	−	0.984188i	$-0.556681\pi$
$37$	−663937.			−0.354258			−0.177129	+	0.984188i	$0.556681\pi$
$38$	3.87945e6	+	2.23980e6i	1.86052	+	1.07417i
$39$		0			0
$40$	4.25779e6	+	7.37471e6i	1.66320	+	2.88075i
$41$	812347.	−	469009.i	0.287479	−	0.165976i	−0.349325	−	0.937001i	$-0.613589\pi$
$41$	812347.	−	469009.i	0.287479	−	0.165976i	0.636804	+	0.771025i	$0.280256\pi$
$42$		0			0
$43$	−287665.	+	498250.i	−0.0841421	+	0.145738i	−0.905025	−	0.425358i	$-0.860148\pi$
$43$	−287665.	+	498250.i	−0.0841421	+	0.145738i	0.820883	+	0.571096i	$0.193482\pi$
$44$		−	7.64899e6i		−	2.04077i
$45$		0			0
$46$	3.86035e6			0.862175
$47$	−7.99711e6	−	4.61713e6i	−1.63886	−	0.946195i	−0.981227	−	0.192855i	$-0.938225\pi$
$47$	−7.99711e6	−	4.61713e6i	−1.63886	−	0.946195i	−0.657631	−	0.753340i	$-0.728441\pi$
$48$		0			0
$49$	947856.	+	1.64173e6i	0.164421	+	0.284786i
$50$	7.29949e6	−	4.21436e6i	1.16792	−	0.674298i
$51$		0			0
$52$	−1.38335e7	+	2.39604e7i	−1.89199	+	3.27703i
$53$		−	1.03765e7i		−	1.31507i	−0.753425	−	0.657533i	$-0.771600\pi$
$53$		−	1.03765e7i		−	1.31507i	0.753425	−	0.657533i	$-0.228400\pi$
$54$		0			0
$55$	−1.03542e7			−1.13153
$56$	−1.76252e7	−	1.01759e7i	−1.79218	−	1.03472i
$57$		0			0
$58$	−8.65037e6	−	1.49829e7i	−0.764403	−	1.32399i
$59$	4.36466e6	−	2.51994e6i	0.360199	−	0.207961i	−0.308969	−	0.951072i	$-0.599984\pi$
$59$	4.36466e6	−	2.51994e6i	0.360199	−	0.207961i	0.669168	+	0.743111i	$0.266651\pi$
$60$		0			0
$61$	9.60648e6	−	1.66389e7i	0.693817	−	1.20173i	−0.276761	−	0.960939i	$-0.589261\pi$
$61$	9.60648e6	−	1.66389e7i	0.693817	−	1.20173i	0.970578	−	0.240788i	$-0.0774057\pi$
$62$			4.83088e6i			0.326934i
$63$		0			0
$64$	−1.24191e7			−0.740234
$65$	3.24343e7	+	1.87260e7i	1.81698	+	1.04904i
$66$		0			0
$67$	299016.	+	517912.i	0.0148387	+	0.0257014i	0.873349	−	0.487094i	$-0.161943\pi$
$67$	299016.	+	517912.i	0.0148387	+	0.0257014i	−0.858511	+	0.512796i	$0.828610\pi$
$68$	−3.13877e7	+	1.81217e7i	−1.46799	+	0.847545i
$69$		0			0
$70$	−2.37928e7	+	4.12104e7i	−0.990955	+	1.71638i
$71$		−	2.92721e7i		−	1.15191i	−0.817480	−	0.575957i	$-0.804630\pi$
$71$		−	2.92721e7i		−	1.15191i	0.817480	−	0.575957i	$-0.195370\pi$
$72$		0			0
$73$	1.28502e7			0.452499			0.226249	−	0.974069i	$-0.427354\pi$
$73$	1.28502e7			0.452499			0.226249	+	0.974069i	$0.427354\pi$
$74$	−1.69011e7	−	9.75784e6i	−0.563621	−	0.325407i
$75$		0			0
$76$	4.63293e7	+	8.02447e7i	1.38868	+	2.40526i
$77$	2.14307e7	−	1.23730e7i	0.609639	−	0.351975i
$78$		0			0
$79$	1.17923e7	−	2.04249e7i	0.302755	−	0.524387i	−0.674004	−	0.738728i	$-0.735427\pi$
$79$	1.17923e7	−	2.04249e7i	0.302755	−	0.524387i	0.976759	+	0.214341i	$0.0687603\pi$
$80$			1.22203e8i			2.98348i
$81$		0			0
$82$	2.75720e7			0.609835
$83$	2.89694e7	+	1.67255e7i	0.610417	+	0.352424i	0.773129	−	0.634249i	$-0.218691\pi$
$83$	2.89694e7	+	1.67255e7i	0.610417	+	0.352424i	−0.162712	+	0.986674i	$0.552024\pi$
$84$		0			0
$85$	2.45307e7	+	4.24884e7i	0.469931	+	0.813944i
$86$	−1.46455e7	+	8.45559e6i	−0.267738	+	0.154579i
$87$		0			0
$88$	6.50834e7	−	1.12728e8i	1.08527	−	1.87975i
$89$			2.82848e7i			0.450809i	0.974265	+	0.225405i	$0.0723704\pi$
$89$			2.82848e7i			0.450809i	−0.974265	+	0.225405i	$0.927630\pi$
$90$		0			0
$91$	−8.95083e7			−1.30526
$92$	6.91519e7	+	3.99249e7i	0.965279	+	0.557304i
$93$		0			0
$94$	−1.35715e8	−	2.35066e8i	−1.73827	−	3.01078i
$95$	1.08624e8	−	6.27144e7i	1.33362	−	0.769968i
$96$		0			0
$97$	−6.82448e7	+	1.18203e8i	−0.770873	+	1.33519i	0.166212	+	0.986090i	$0.446846\pi$
$97$	−6.82448e7	+	1.18203e8i	−0.770873	+	1.33519i	−0.937085	+	0.349101i	$0.886487\pi$
$98$			5.57223e7i			0.604122i
$99$		0			0
$100$	1.74345e8			1.74345
$101$	3.48570e7	+	2.01247e7i	0.334969	+	0.193394i	0.658045	−	0.752979i	$-0.271384\pi$
$101$	3.48570e7	+	2.01247e7i	0.334969	+	0.193394i	−0.323076	+	0.946373i	$0.604717\pi$
$102$		0			0
$103$	−1.67954e7	−	2.90904e7i	−0.149225	−	0.258465i	0.781716	−	0.623634i	$-0.214344\pi$
$103$	−1.67954e7	−	2.90904e7i	−0.149225	−	0.258465i	−0.930941	+	0.365169i	$0.881011\pi$
$104$	−4.07746e8	+	2.35412e8i	−3.48543	+	2.01231i
$105$		0			0
$106$	1.52503e8	−	2.64143e8i	1.20797	−	2.09226i
$107$		−	1.95483e8i		−	1.49133i	−0.666322	−	0.745664i	$-0.732132\pi$
$107$		−	1.95483e8i		−	1.49133i	0.666322	−	0.745664i	$-0.267868\pi$
$108$		0			0
$109$	−1.01182e8			−0.716800			−0.358400	−	0.933568i	$-0.616678\pi$
$109$	−1.01182e8			−0.716800			−0.358400	+	0.933568i	$0.616678\pi$
$110$	−2.63574e8	−	1.52175e8i	−1.80025	−	1.03937i
$111$		0			0
$112$	−1.46030e8	−	2.52932e8i	−0.928048	−	1.60743i
$113$	2.72369e8	−	1.57252e8i	1.67049	−	0.964457i	0.703126	−	0.711066i	$-0.251787\pi$
$113$	2.72369e8	−	1.57252e8i	1.67049	−	0.964457i	0.967364	−	0.253392i	$-0.0815462\pi$
$114$		0			0
$115$	5.40449e7	−	9.36086e7i	0.309004	−	0.535210i
$116$		−	3.57858e8i		−	1.97642i
$117$		0			0
$118$	1.48141e8			0.764097
$119$	−1.01545e8	−	5.86272e7i	−0.506375	−	0.292356i
$120$		0			0
$121$	−2.80440e7	−	4.85736e7i	−0.130827	−	0.226599i
$122$	4.89082e8	−	2.82372e8i	2.20771	−	1.27462i
$123$		0			0
$124$	−4.99624e7	+	8.65374e7i	−0.211328	+	0.366030i
$125$			8.54913e7i			0.350172i
$126$		0			0
$127$	4.30869e8			1.65627			0.828133	−	0.560532i	$-0.189403\pi$
$127$	4.30869e8			1.65627			0.828133	+	0.560532i	$0.189403\pi$
$128$	6.42286e7	+	3.70824e7i	0.239270	+	0.138143i
$129$		0			0
$130$	5.50428e8	+	9.53370e8i	1.92720	+	3.33801i
$131$	1.63621e8	−	9.44664e7i	0.555588	−	0.320769i	−0.195785	−	0.980647i	$-0.562725\pi$
$131$	1.63621e8	−	9.44664e7i	0.555588	−	0.320769i	0.751373	+	0.659878i	$0.229392\pi$
$132$		0			0
$133$	−1.49884e8	+	2.59607e8i	−0.479016	+	0.829680i
$134$			1.75785e7i			0.0545209i
$135$		0			0
$136$	−6.16772e8			−1.80289
$137$	−1.06519e8	−	6.14989e7i	−0.302375	−	0.174576i	0.341134	−	0.940015i	$-0.389189\pi$
$137$	−1.06519e8	−	6.14989e7i	−0.302375	−	0.174576i	−0.643509	+	0.765438i	$0.722522\pi$
$138$		0			0
$139$	−8.60360e7	−	1.49019e8i	−0.230474	−	0.399192i	0.727474	−	0.686135i	$-0.240694\pi$
$139$	−8.60360e7	−	1.49019e8i	−0.230474	−	0.399192i	−0.957948	+	0.286943i	$0.907361\pi$
$140$	−8.52420e8	+	4.92145e8i	−2.21892	+	1.28109i
$141$		0			0
$142$	4.30210e8	−	7.45145e8i	1.05810	−	1.83268i
$143$		−	5.72479e8i		−	1.36904i
$144$		0			0
$145$	−4.84421e8			−1.09585
$146$	3.27112e8	+	1.88858e8i	0.719921	+	0.415647i
$147$		0			0
$148$	−2.01837e8	−	3.49592e8i	−0.420682	−	0.728642i
$149$	−6.33642e8	+	3.65834e8i	−1.28558	+	0.742230i	−0.977863	−	0.209248i	$-0.932898\pi$
$149$	−6.33642e8	+	3.65834e8i	−1.28558	+	0.742230i	−0.307717	+	0.951478i	$0.599565\pi$
$150$		0			0
$151$	−9.29762e7	+	1.61039e8i	−0.178840	+	0.309759i	−0.941483	−	0.337059i	$-0.890568\pi$
$151$	−9.29762e7	+	1.61039e8i	−0.178840	+	0.309759i	0.762644	+	0.646819i	$0.223901\pi$
$152$			1.57682e9i			2.95398i
$153$		0			0
$154$	7.27381e8			1.29324
$155$	1.17143e8	+	6.76324e7i	0.202950	+	0.117173i
$156$		0			0
$157$	−4.87004e8	−	8.43515e8i	−0.801556	−	1.38834i	−0.918592	−	0.395207i	$-0.870673\pi$
$157$	−4.87004e8	−	8.43515e8i	−0.801556	−	1.38834i	0.117036	−	0.993128i	$-0.462661\pi$
$158$	6.00367e8	−	3.46622e8i	0.963360	−	0.556196i
$159$		0			0
$160$	−7.06019e8	+	1.22286e9i	−1.07730	+	1.86594i
$161$			2.58330e8i			0.384477i
$162$		0			0
$163$	−1.15499e9			−1.63616			−0.818082	−	0.575102i	$-0.804962\pi$
$163$	−1.15499e9			−1.63616			−0.818082	+	0.575102i	$0.804962\pi$
$164$	4.93907e8	+	2.85157e8i	0.682763	+	0.394193i
$165$		0			0
$166$	4.91626e8	+	8.51522e8i	0.647445	+	1.12141i
$167$	−9.89881e7	+	5.71508e7i	−0.127267	+	0.0734779i	−0.562282	−	0.826945i	$-0.690076\pi$
$167$	−9.89881e7	+	5.71508e7i	−0.127267	+	0.0734779i	0.435015	+	0.900423i	$0.356743\pi$
$168$		0			0
$169$	−6.27487e8	+	1.08684e9i	−0.769233	+	1.33235i
$170$			1.44210e9i			1.72664i
$171$		0			0
$172$	−3.49801e8			−0.399675
$173$	1.43173e9	+	8.26610e8i	1.59837	+	0.922819i	0.991802	+	0.127786i	$0.0407872\pi$
$173$	1.43173e9	+	8.26610e8i	1.59837	+	0.922819i	0.606567	+	0.795032i	$0.292546\pi$
$174$		0			0
$175$	2.82020e8	+	4.88472e8i	0.300696	+	0.520820i
$176$	1.61770e9	−	9.33982e8i	1.68597	−	0.973392i
$177$		0			0
$178$	−4.15700e8	+	7.20013e8i	−0.414095	+	0.717233i
$179$			4.17229e8i			0.406408i	0.979136	+	0.203204i	$0.0651355\pi$
$179$			4.17229e8i			0.406408i	−0.979136	+	0.203204i	$0.934865\pi$
$180$		0			0
$181$	1.16424e9			1.08475			0.542374	−	0.840137i	$-0.317526\pi$
$181$	1.16424e9			1.08475			0.542374	+	0.840137i	$0.317526\pi$
$182$	−2.27851e9	−	1.31550e9i	−2.07666	−	1.19896i
$183$		0			0
$184$	6.79422e8	+	1.17679e9i	0.592746	+	1.02667i
$185$	−4.73230e8	+	2.73220e8i	−0.404004	+	0.233252i
$186$		0			0
$187$	3.74969e8	−	6.49466e8i	0.306640	−	0.531116i
$188$		−	5.61443e9i		−	4.49443i
$189$		0			0
$190$	3.68684e9			2.82904
$191$	−1.39478e9	−	8.05276e8i	−1.04803	−	0.605078i	−0.125930	−	0.992039i	$-0.540191\pi$
$191$	−1.39478e9	−	8.05276e8i	−1.04803	−	0.605078i	−0.922096	+	0.386961i	$0.873525\pi$
$192$		0			0
$193$	3.55139e8	+	6.15120e8i	0.255959	+	0.443333i	0.965155	−	0.261677i	$-0.0842756\pi$
$193$	3.55139e8	+	6.15120e8i	0.255959	+	0.443333i	−0.709197	+	0.705010i	$0.750942\pi$
$194$	−3.47446e9	+	2.00598e9i	−2.45290	+	1.41618i
$195$		0			0
$196$	−5.76296e8	+	9.98175e8i	−0.390501	+	0.676367i
$197$			1.41623e9i			0.940303i	0.882586	+	0.470152i	$0.155801\pi$
$197$			1.41623e9i			0.940303i	−0.882586	+	0.470152i	$0.844199\pi$
$198$		0			0
$199$	−2.34324e9			−1.49419			−0.747093	−	0.664720i	$-0.768551\pi$
$199$	−2.34324e9			−1.49419			−0.747093	+	0.664720i	$0.768551\pi$
$200$	2.56942e9	+	1.48346e9i	1.60589	+	0.927160i
$201$		0			0
$202$	5.91543e8	+	1.02458e9i	0.355288	+	0.615377i
$203$	1.00263e9	−	5.78871e8i	0.590417	−	0.340877i
$204$		0			0
$205$	3.86008e8	−	6.68585e8i	0.218565	−	0.378565i
$206$		−	9.87362e8i		−	0.548286i
$207$		0			0
$208$	−6.75658e9			−3.60972
$209$	−1.66040e9	−	9.58634e8i	−0.870218	−	0.502421i
$210$		0			0
$211$	−5.32585e8	−	9.22464e8i	−0.268695	−	0.465393i	0.699830	−	0.714309i	$-0.253259\pi$
$211$	−5.32585e8	−	9.22464e8i	−0.268695	−	0.465393i	−0.968525	+	0.248916i	$0.919926\pi$
$212$	5.46368e9	−	3.15446e9i	2.70484	−	1.56164i
$213$		0			0
$214$	2.87300e9	−	4.97618e9i	1.36987	−	2.37269i
$215$			4.73513e8i			0.221604i
$216$		0			0
$217$	−3.23276e8			−0.145792
$218$	−2.57568e9	−	1.48707e9i	−1.14042	−	0.658423i
$219$		0			0
$220$	−3.14767e9	−	5.45192e9i	−1.34369	−	2.32734i
$221$	−2.34917e9	+	1.35629e9i	−0.984794	+	0.568571i
$222$		0			0
$223$	−9.97928e8	+	1.72846e9i	−0.403534	+	0.698941i	−0.994150	−	0.108012i	$-0.965552\pi$
$223$	−9.97928e8	+	1.72846e9i	−0.403534	+	0.698941i	0.590616	+	0.806953i	$0.298885\pi$
$224$		−	3.37471e9i		−	1.34043i
$225$		0			0
$226$	9.24451e9			3.54364
$227$	−1.55818e9	−	8.99614e8i	−0.586831	−	0.338807i	0.177012	−	0.984209i	$-0.443357\pi$
$227$	−1.55818e9	−	8.99614e8i	−0.586831	−	0.338807i	−0.763844	+	0.645401i	$0.776690\pi$
$228$		0			0
$229$	−7.98936e8	−	1.38380e9i	−0.290516	−	0.503188i	0.683416	−	0.730029i	$-0.260494\pi$
$229$	−7.98936e8	−	1.38380e9i	−0.290516	−	0.503188i	−0.973932	+	0.226841i	$0.927160\pi$
$230$	2.75152e9	−	1.58859e9i	0.983244	−	0.567676i
$231$		0			0
$232$	3.04493e9	−	5.27397e9i	1.05105	−	1.82048i
$233$			4.98411e9i			1.69108i	0.533912	+	0.845540i	$0.320721\pi$
$233$			4.98411e9i			1.69108i	−0.533912	+	0.845540i	$0.679279\pi$
$234$		0			0
$235$	−7.60006e9			−2.49199
$236$	2.65371e9	+	1.53212e9i	0.855472	+	0.493907i
$237$		0			0
$238$	−1.72328e9	−	2.98481e9i	−0.537091	−	0.930270i
$239$	−5.10742e9	+	2.94877e9i	−1.56535	+	0.903753i	−0.568645	+	0.822583i	$0.692532\pi$
$239$	−5.10742e9	+	2.94877e9i	−1.56535	+	0.903753i	−0.996700	+	0.0811700i	$0.974134\pi$
$240$		0			0
$241$	−1.13087e9	+	1.95873e9i	−0.335231	+	0.580638i	−0.983529	−	0.180749i	$-0.942148\pi$
$241$	−1.13087e9	+	1.95873e9i	−0.335231	+	0.580638i	0.648298	+	0.761387i	$0.275481\pi$
$242$		−	1.64864e9i		−	0.480689i
$243$		0			0
$244$	1.16815e10			3.29563
$245$	1.35119e9	+	7.80113e8i	0.375019	+	0.216517i
$246$		0			0
$247$	3.46746e9	+	6.00581e9i	0.931587	+	1.61356i
$248$	−1.47265e9	+	8.50236e8i	−0.389308	+	0.224767i
$249$		0			0
$250$	−1.25646e9	+	2.17625e9i	−0.321654	+	0.557121i
$251$		−	3.31815e9i		−	0.835989i	−0.908450	−	0.417994i	$-0.862733\pi$
$251$		−	3.31815e9i		−	0.835989i	0.908450	−	0.417994i	$-0.137267\pi$
$252$		0			0
$253$	−1.65223e9			−0.403263
$254$	1.09681e10	+	6.33245e9i	2.63510	+	1.52138i
$255$		0			0
$256$	2.67964e9	+	4.64127e9i	0.623901	+	1.08063i
$257$	2.64153e9	−	1.52509e9i	0.605512	−	0.349593i	−0.165695	−	0.986177i	$-0.552987\pi$
$257$	2.64153e9	−	1.52509e9i	0.605512	−	0.349593i	0.771207	+	0.636584i	$0.219653\pi$
$258$		0			0
$259$	6.52982e8	−	1.13100e9i	0.145112	−	0.251341i
$260$			2.27708e10i			4.98292i
$261$		0			0
$262$	5.55347e9			1.17858
$263$	1.70972e9	+	9.87108e8i	0.357357	+	0.206320i	0.667921	−	0.744232i	$-0.267184\pi$
$263$	1.70972e9	+	9.87108e8i	0.357357	+	0.206320i	−0.310564	+	0.950553i	$0.600518\pi$
$264$		0			0
$265$	−4.27008e9	−	7.39600e9i	−0.865870	−	1.49973i
$266$	−7.63087e9	+	4.40568e9i	−1.52422	+	0.880008i
$267$		0			0
$268$	−1.81802e8	+	3.14890e8i	−0.0352419	+	0.0610408i
$269$		−	3.49334e9i		−	0.667163i	−0.942721	−	0.333581i	$-0.891743\pi$
$269$		−	3.49334e9i		−	0.667163i	0.942721	−	0.333581i	$-0.108257\pi$
$270$		0			0
$271$	1.43226e9			0.265549			0.132774	−	0.991146i	$-0.457611\pi$
$271$	1.43226e9			0.265549			0.132774	+	0.991146i	$0.457611\pi$
$272$	−7.66520e9	−	4.42550e9i	−1.40039	−	0.808513i
$273$		0			0
$274$	−1.80769e9	−	3.13101e9i	−0.320717	−	0.555498i
$275$	−3.12418e9	+	1.80375e9i	−0.546268	+	0.315388i
$276$		0			0
$277$	−1.67547e9	+	2.90201e9i	−0.284589	+	0.492923i	−0.972510	−	0.232863i	$-0.925191\pi$
$277$	−1.67547e9	+	2.90201e9i	−0.284589	+	0.492923i	0.687920	+	0.725786i	$0.258524\pi$
$278$		−	5.05787e9i		−	0.846814i
$279$		0			0
$280$	−1.67502e10			−2.72513
$281$	−3.71560e9	−	2.14520e9i	−0.595941	−	0.344067i	0.171502	−	0.985184i	$-0.445138\pi$
$281$	−3.71560e9	−	2.14520e9i	−0.595941	−	0.344067i	−0.767443	+	0.641117i	$0.778471\pi$
$282$		0			0
$283$	4.95573e7	+	8.58357e7i	0.00772612	+	0.0133820i	0.869863	−	0.493294i	$-0.164207\pi$
$283$	4.95573e7	+	8.58357e7i	0.00772612	+	0.0133820i	−0.862136	+	0.506676i	$0.830874\pi$
$284$	1.54130e10	−	8.89871e9i	2.36927	−	1.36790i
$285$		0			0
$286$	8.41369e9	−	1.45729e10i	1.25754	−	2.17813i
$287$			1.84508e9i			0.271949i
$288$		0			0
$289$	3.42231e9			0.490601
$290$	−1.23313e10	−	7.11950e9i	−1.74348	−	1.00660i
$291$		0			0
$292$	3.90645e9	+	6.76618e9i	0.537343	+	0.930705i
$293$	−7.81997e8	+	4.51486e8i	−0.106105	+	0.0612596i	−0.552113	−	0.833769i	$-0.686178\pi$
$293$	−7.81997e8	+	4.51486e8i	−0.106105	+	0.0612596i	0.446009	+	0.895029i	$0.352845\pi$
$294$		0			0
$295$	2.07398e9	−	3.59224e9i	0.273852	−	0.474326i
$296$		−	6.86952e9i		−	0.894869i
$297$		0			0
$298$	−2.15065e10			−2.72713
$299$	5.17559e9	+	2.98813e9i	0.647553	+	0.373865i
$300$		0			0
$301$	−5.65837e8	−	9.80059e8i	−0.0689327	−	0.119395i
$302$	−4.73357e9	+	2.73293e9i	−0.569064	+	0.328549i
$303$		0			0
$304$	−1.13141e10	+	1.95966e10i	−1.32473	+	2.29449i
$305$		−	1.58128e10i		−	1.82730i
$306$		0			0
$307$	3.36397e9			0.378703			0.189352	−	0.981909i	$-0.439361\pi$
$307$	3.36397e9			0.378703			0.189352	+	0.981909i	$0.439361\pi$
$308$	1.30298e10	+	7.52278e9i	1.44789	+	0.835941i
$309$		0			0
$310$	1.98798e9	+	3.44328e9i	0.215261	+	0.372842i
$311$	1.15202e10	−	6.65116e9i	1.23145	−	0.710978i	0.264117	−	0.964491i	$-0.414919\pi$
$311$	1.15202e10	−	6.65116e9i	1.23145	−	0.710978i	0.967332	+	0.253513i	$0.0815860\pi$
$312$		0			0
$313$	−6.56247e9	+	1.13665e10i	−0.683738	+	1.18427i	0.290094	+	0.956998i	$0.406314\pi$
$313$	−6.56247e9	+	1.13665e10i	−0.683738	+	1.18427i	−0.973832	+	0.227271i	$0.927020\pi$
$314$		−	2.86299e10i		−	2.94510i
$315$		0			0
$316$	1.43395e10			1.43809
$317$	6.04936e9	+	3.49260e9i	0.599063	+	0.345869i	0.768673	−	0.639642i	$-0.220917\pi$
$317$	6.04936e9	+	3.49260e9i	0.599063	+	0.345869i	−0.169610	+	0.985511i	$0.554251\pi$
$318$		0			0
$319$	3.70236e9	+	6.41267e9i	0.357533	+	0.619265i
$320$	−8.85186e9	+	5.11063e9i	−0.844179	+	0.487387i
$321$		0			0
$322$	−3.79666e9	+	6.57600e9i	−0.353165	+	0.611700i
$323$			9.08462e9i			0.834635i
$324$		0			0
$325$	1.30486e10			1.16958
$326$	−2.94012e10	−	1.69748e10i	−2.60312	−	1.50291i
$327$		0			0
$328$	4.85267e9	+	8.40506e9i	0.419262	+	0.726182i
$329$	1.57303e10	−	9.08190e9i	1.34262	−	0.775163i
$330$		0			0
$331$	5.93663e9	−	1.02825e10i	0.494570	−	0.856620i	−0.505410	−	0.862879i	$-0.668659\pi$
$331$	5.93663e9	−	1.02825e10i	0.494570	−	0.856620i	0.999980	+	0.00625875i	$0.00199223\pi$
$332$			2.03382e10i			1.67402i
$333$		0			0
$334$	−3.35977e9			−0.269975
$335$	4.26256e8	+	2.46099e8i	0.0338448	+	0.0195403i
$336$		0			0
$337$	−4.57709e9	−	7.92776e9i	−0.354871	−	0.614654i	0.632225	−	0.774785i	$-0.282142\pi$
$337$	−4.57709e9	−	7.92776e9i	−0.354871	−	0.614654i	−0.987096	+	0.160131i	$0.948808\pi$
$338$	−3.19464e10	+	1.84443e10i	−2.44769	+	1.41317i
$339$		0			0
$340$	−1.49147e10	+	2.58329e10i	−1.11609	+	1.93312i
$341$		−	2.06762e9i		−	0.152916i
$342$		0			0
$343$	−1.50682e10			−1.08864
$344$	−5.15522e9	−	2.97637e9i	−0.368140	−	0.212546i
$345$		0			0
$346$	2.42973e10	+	4.20841e10i	1.69533	+	2.93639i
$347$	1.77605e10	−	1.02540e10i	1.22500	−	0.707257i	0.259024	−	0.965871i	$-0.416599\pi$
$347$	1.77605e10	−	1.02540e10i	1.22500	−	0.707257i	0.965981	+	0.258614i	$0.0832659\pi$
$348$		0			0
$349$	−6.73994e9	+	1.16739e10i	−0.454312	+	0.786892i	−0.998648	−	0.0519753i	$-0.983448\pi$
$349$	−6.73994e9	+	1.16739e10i	−0.454312	+	0.786892i	0.544336	+	0.838867i	$0.316782\pi$
$350$			1.65793e10i			1.10483i
$351$		0			0
$352$	2.15840e10			1.40592
$353$	−8.47197e9	−	4.89130e9i	−0.545614	−	0.315011i	0.201737	−	0.979440i	$-0.435341\pi$
$353$	−8.47197e9	−	4.89130e9i	−0.545614	−	0.315011i	−0.747351	+	0.664429i	$0.768675\pi$
$354$		0			0
$355$	−1.20459e10	−	2.08641e10i	−0.758446	−	1.31367i
$356$	−1.48932e10	+	8.59858e9i	−0.927229	+	0.535336i
$357$		0			0
$358$	−6.13199e9	+	1.06209e10i	−0.373310	+	0.646591i
$359$			1.67338e10i			1.00743i	0.863868	+	0.503717i	$0.168035\pi$
$359$			1.67338e10i			1.00743i	−0.863868	+	0.503717i	$0.831965\pi$
$360$		0			0
$361$	6.24189e9			0.367525
$362$	2.96368e10	+	1.71108e10i	1.72582	+	0.996405i
$363$		0			0
$364$	−2.72105e10	−	4.71300e10i	−1.55000	−	2.68468i
$365$	9.15914e9	−	5.28803e9i	0.516040	−	0.297936i
$366$		0			0
$367$	7.49396e9	−	1.29799e10i	0.413092	−	0.715497i	−0.582134	−	0.813093i	$-0.697782\pi$
$367$	7.49396e9	−	1.29799e10i	0.413092	−	0.715497i	0.995226	+	0.0975963i	$0.0311154\pi$
$368$			1.95002e10i			1.06328i
$369$		0			0
$370$	−1.60620e10			−0.857022
$371$	1.76761e10	+	1.02053e10i	0.933019	+	0.538679i
$372$		0			0
$373$	9.89819e9	+	1.71442e10i	0.511353	+	0.885689i	0.999913	+	0.0131590i	$0.00418876\pi$
$373$	9.89819e9	+	1.71442e10i	0.511353	+	0.885689i	−0.488561	+	0.872530i	$0.662478\pi$
$374$	1.90903e10	−	1.10218e10i	0.975723	−	0.563334i
$375$		0			0
$376$	4.77718e10	−	8.27432e10i	2.39012	−	4.13982i
$377$		−	2.67835e10i		−	1.32587i
$378$		0			0
$379$	−5.34899e9			−0.259248			−0.129624	−	0.991563i	$-0.541377\pi$
$379$	−5.34899e9			−0.259248			−0.129624	+	0.991563i	$0.541377\pi$
$380$	6.60437e10	+	3.81303e10i	3.16736	+	1.82867i
$381$		0			0
$382$	−2.36702e10	−	4.09980e10i	−1.11160	−	1.92535i
$383$	−1.25520e10	+	7.24690e9i	−0.583335	+	0.336788i	−0.762457	−	0.647038i	$-0.776007\pi$
$383$	−1.25520e10	+	7.24690e9i	−0.583335	+	0.336788i	0.179123	+	0.983827i	$0.442674\pi$
$384$		0			0
$385$	1.01833e10	−	1.76380e10i	0.463497	−	0.802800i
$386$			2.08778e10i			0.940452i
$387$		0			0
$388$	−8.29857e10			−3.66165
$389$	−1.56324e10	−	9.02536e9i	−0.682695	−	0.394154i	0.118175	−	0.992993i	$-0.462296\pi$
$389$	−1.56324e10	−	9.02536e9i	−0.682695	−	0.394154i	−0.800870	+	0.598839i	$0.795629\pi$
$390$		0			0
$391$	3.91440e9	+	6.77993e9i	0.167478	+	0.290080i
$392$	−1.69864e10	+	9.80713e9i	−0.719380	+	0.415334i
$393$		0			0
$394$	−2.08142e10	+	3.60513e10i	−0.863724	+	1.49601i
$395$		−	1.94108e10i		−	0.797363i
$396$		0			0
$397$	3.82640e10			1.54038			0.770191	−	0.637814i	$-0.220161\pi$
$397$	3.82640e10			1.54038			0.770191	+	0.637814i	$0.220161\pi$
$398$	−5.96491e10	−	3.44384e10i	−2.37723	−	1.37250i
$399$		0			0
$400$	2.12884e10	+	3.68726e10i	0.831578	+	1.44034i
$401$	−3.69448e10	+	2.13301e10i	−1.42881	+	0.824926i	−0.997027	−	0.0770497i	$-0.975450\pi$
$401$	−3.69448e10	+	2.13301e10i	−1.42881	+	0.824926i	−0.431787	+	0.901976i	$0.642117\pi$
$402$		0			0
$403$	−3.73937e9	+	6.47678e9i	−0.141768	+	0.245550i
$404$			2.44716e10i			0.918623i
$405$		0			0
$406$	3.40305e10			1.25246
$407$	7.23366e9	+	4.17636e9i	0.263621	+	0.152202i
$408$		0			0
$409$	1.76976e10	+	3.06532e10i	0.632444	+	1.09542i	0.987051	+	0.160409i	$0.0512813\pi$
$409$	1.76976e10	+	3.06532e10i	0.632444	+	1.09542i	−0.354607	+	0.935015i	$0.615385\pi$
$410$	1.96523e10	−	1.13463e10i	0.695469	−	0.401529i
$411$		0			0
$412$	1.02116e10	−	1.76870e10i	0.354409	−	0.613854i
$413$			9.91343e9i			0.340741i
$414$		0			0
$415$	2.75311e10			0.928177
$416$	−6.76116e10	−	3.90356e10i	−2.25761	−	1.30343i
$417$		0			0
$418$	−2.81780e10	−	4.88057e10i	−0.923006	−	1.59869i
$419$	−1.26376e10	+	7.29630e9i	−0.410022	+	0.236726i	−0.690799	−	0.723047i	$-0.742741\pi$
$419$	−1.26376e10	+	7.29630e9i	−0.410022	+	0.236726i	0.280777	+	0.959773i	$0.409408\pi$
$420$		0			0
$421$	1.32550e10	−	2.29583e10i	0.421939	−	0.730820i	−0.574190	−	0.818722i	$-0.694683\pi$
$421$	1.32550e10	−	2.29583e10i	0.421939	−	0.730820i	0.996129	+	0.0879018i	$0.0280162\pi$
$422$		−	3.13095e10i		−	0.987247i
$423$		0			0
$424$	1.07362e11			3.32191
$425$	1.48034e10	+	8.54673e9i	0.453738	+	0.261965i
$426$		0			0
$427$	1.88959e10	+	3.27287e10i	0.568404	+	0.984505i
$428$	1.02930e11	−	5.94268e10i	3.06738	−	1.77095i
$429$		0			0
$430$	−6.95919e9	+	1.20537e10i	−0.203557	+	0.352570i
$431$		−	4.61676e9i		−	0.133791i	−0.997760	−	0.0668957i	$-0.978691\pi$
$431$		−	4.61676e9i		−	0.133791i	0.997760	−	0.0668957i	$-0.0213095\pi$
$432$		0			0
$433$	−1.64494e10			−0.467948			−0.233974	−	0.972243i	$-0.575173\pi$
$433$	−1.64494e10			−0.467948			−0.233974	+	0.972243i	$0.575173\pi$
$434$	−8.22927e9	−	4.75117e9i	−0.231954	−	0.133919i
$435$		0			0
$436$	−3.07594e10	−	5.32768e10i	−0.851200	−	1.47432i
$437$	1.73334e10	−	1.00074e10i	0.475288	−	0.274408i
$438$		0			0
$439$	1.30737e10	−	2.26444e10i	0.351999	−	0.609681i	−0.634600	−	0.772840i	$-0.718835\pi$
$439$	1.30737e10	−	2.26444e10i	0.351999	−	0.609681i	0.986600	+	0.163160i	$0.0521686\pi$
$440$		−	1.07131e11i		−	2.85828i
$441$		0			0
$442$	−7.97335e10			−2.08906
$443$	−3.73651e10	−	2.15728e10i	−0.970179	−	0.560133i	−0.0708880	−	0.997484i	$-0.522583\pi$
$443$	−3.73651e10	−	2.15728e10i	−0.970179	−	0.560133i	−0.899291	+	0.437351i	$0.855917\pi$
$444$		0			0
$445$	1.16396e10	+	2.01604e10i	0.296823	+	0.514113i
$446$	−5.08062e10	+	2.93330e10i	−1.28404	+	0.741339i
$447$		0			0
$448$	1.22142e10	−	2.11555e10i	0.303216	−	0.525185i
$449$			2.18287e10i			0.537084i	0.963268	+	0.268542i	$0.0865418\pi$
$449$			2.18287e10i			0.537084i	−0.963268	+	0.268542i	$0.913458\pi$
$450$		0			0
$451$	−1.18008e10			−0.285237
$452$	1.65600e11	+	9.56094e10i	3.96741	+	2.29059i
$453$		0			0
$454$	−2.64431e10	−	4.58009e10i	−0.622429	−	1.07808i
$455$	−6.37983e10	+	3.68340e10i	−1.48855	+	0.859415i
$456$		0			0
$457$	−1.23453e10	+	2.13826e10i	−0.283032	+	0.490226i	−0.972130	−	0.234442i	$-0.924674\pi$
$457$	−1.23453e10	+	2.13826e10i	−0.283032	+	0.490226i	0.689098	+	0.724668i	$0.258007\pi$
$458$		−	4.69676e10i		−	1.06742i
$459$		0			0
$460$	6.57186e10			1.46777
$461$	−2.26667e10	−	1.30866e10i	−0.501862	−	0.289750i	0.227620	−	0.973750i	$-0.426906\pi$
$461$	−2.26667e10	−	1.30866e10i	−0.501862	−	0.289750i	−0.729482	+	0.684000i	$0.760239\pi$
$462$		0			0
$463$	−3.52051e10	−	6.09770e10i	−0.766093	−	1.32691i	−0.939667	−	0.342092i	$-0.888865\pi$
$463$	−3.52051e10	−	6.09770e10i	−0.766093	−	1.32691i	0.173573	−	0.984821i	$-0.444469\pi$
$464$	7.56844e10	−	4.36964e10i	1.63281	−	0.942701i
$465$		0			0
$466$	−7.32512e10	+	1.26875e11i	−1.55336	+	2.69049i
$467$		−	1.07220e10i		−	0.225428i	−0.993627	−	0.112714i	$-0.964046\pi$
$467$		−	1.07220e10i		−	0.225428i	0.993627	−	0.112714i	$-0.0359543\pi$
$468$		0			0
$469$	−1.17633e9			−0.0243130
$470$	−1.93466e11	−	1.11698e11i	−3.96473	−	2.28904i
$471$		0			0
$472$	2.60729e10	+	4.51596e10i	0.525317	+	0.909875i
$473$	6.26828e9	−	3.61899e9i	0.125229	−	0.0723008i
$474$		0			0
$475$	2.18503e10	−	3.78458e10i	0.429223	−	0.743435i
$476$		−	7.12907e10i		−	1.38869i
$477$		0			0
$478$	−1.73352e11			−3.32060
$479$	−1.21273e10	−	7.00168e9i	−0.230367	−	0.133003i	0.380374	−	0.924833i	$-0.375795\pi$
$479$	−1.21273e10	−	7.00168e9i	−0.230367	−	0.133003i	−0.610742	+	0.791830i	$0.709129\pi$
$480$		0			0
$481$	−1.51062e10	−	2.61648e10i	−0.282212	−	0.488806i
$482$	−5.75745e10	+	3.32407e10i	−1.06670	+	0.615860i
$483$		0			0
$484$	1.70507e10	−	2.95327e10i	0.310714	−	0.538173i
$485$			1.12335e11i			2.03024i
$486$		0			0
$487$	1.72271e9			0.0306264			0.0153132	−	0.999883i	$-0.495125\pi$
$487$	1.72271e9			0.0306264			0.0153132	+	0.999883i	$0.495125\pi$
$488$	1.72157e11	+	9.93948e10i	3.03560	+	1.75261i
$489$		0			0
$490$	2.29305e10	+	3.97168e10i	0.397768	+	0.688954i
$491$	3.20792e10	−	1.85209e10i	0.551947	−	0.318667i	−0.197960	−	0.980210i	$-0.563432\pi$
$491$	3.20792e10	−	1.85209e10i	0.551947	−	0.318667i	0.749907	+	0.661543i	$0.230098\pi$
$492$		0			0
$493$	1.75429e10	−	3.03853e10i	0.296972	−	0.514370i
$494$			2.03844e11i			3.42287i
$495$		0			0
$496$	−2.44027e10			−0.403191
$497$	4.98641e10	+	2.87891e10i	0.817265	+	0.471848i
$498$		0			0
$499$	6.67483e9	+	1.15612e10i	0.107656	+	0.186466i	0.914820	−	0.403861i	$-0.132332\pi$
$499$	6.67483e9	+	1.15612e10i	0.107656	+	0.186466i	−0.807164	+	0.590327i	$0.798999\pi$
$500$	−4.50149e10	+	2.59893e10i	−0.720238	+	0.415830i
$501$		0			0
$502$	4.87666e10	−	8.44662e10i	0.767905	−	1.33005i
$503$			5.91175e10i			0.923516i	0.887006	+	0.461758i	$0.152781\pi$
$503$			5.91175e10i			0.923516i	−0.887006	+	0.461758i	$0.847219\pi$
$504$		0			0
$505$	3.31264e10			0.509341
$506$	−4.20589e10	−	2.42827e10i	−0.641588	−	0.370421i
$507$		0			0
$508$	1.30984e11	+	2.26871e11i	1.96682	+	3.40663i
$509$	−2.87565e10	+	1.66026e10i	−0.428415	+	0.247345i	−0.698671	−	0.715443i	$-0.746225\pi$
$509$	−2.87565e10	+	1.66026e10i	−0.428415	+	0.247345i	0.270256	+	0.962788i	$0.412892\pi$
$510$		0			0
$511$	−1.26381e10	+	2.18899e10i	−0.185353	+	0.321041i
$512$			1.38544e11i			2.01607i
$513$		0			0
$514$	8.96565e10			1.28449
$515$	−2.39423e10	−	1.38231e10i	−0.340358	−	0.196506i
$516$		0			0
$517$	5.80862e10	+	1.00608e11i	0.813038	+	1.40822i
$518$	3.32444e10	−	1.91937e10i	0.461742	−	0.266587i
$519$		0			0
$520$	−1.93751e11	+	3.35586e11i	−2.64990	+	4.58977i
$521$			2.53746e10i			0.344388i	0.985063	+	0.172194i	$0.0550856\pi$
$521$			2.53746e10i			0.344388i	−0.985063	+	0.172194i	$0.944914\pi$
$522$		0			0
$523$	1.27775e11			1.70781			0.853903	−	0.520432i	$-0.174229\pi$
$523$	1.27775e11			1.70781			0.853903	+	0.520432i	$0.174229\pi$
$524$	9.94813e10	+	5.74356e10i	1.31952	+	0.761826i
$525$		0			0
$526$	2.90149e10	+	5.02553e10i	0.379034	+	0.656507i
$527$	−8.48448e9	+	4.89852e9i	−0.109997	+	0.0635071i
$528$		0			0
$529$	−3.05315e10	+	5.28821e10i	−0.389875	+	0.675283i
$530$		−	2.51029e11i		−	3.18141i
$531$		0			0
$532$	−1.82259e11			−2.27533
$533$	3.69658e10	+	2.13422e10i	0.458028	+	0.264442i
$534$		0			0
$535$	−8.04440e10	−	1.39333e11i	−0.981925	−	1.70074i
$536$	−5.35865e9	+	3.09382e9i	−0.0649226	+	0.0374831i
$537$		0			0
$538$	5.13414e10	−	8.89259e10i	0.612828	−	1.06145i
$539$		−	2.38492e10i		−	0.282565i
$540$		0			0
$541$	6.16835e9			0.0720078			0.0360039	−	0.999352i	$-0.488537\pi$
$541$	6.16835e9			0.0720078			0.0360039	+	0.999352i	$0.488537\pi$
$542$	3.64594e10	+	2.10498e10i	0.422486	+	0.243922i
$543$		0			0
$544$	−5.11360e10	−	8.85701e10i	−0.583890	−	1.01133i
$545$	−7.21190e10	+	4.16379e10i	−0.817454	+	0.471958i
$546$		0			0
$547$	−5.11547e10	+	8.86025e10i	−0.571394	+	0.989684i	0.425029	+	0.905180i	$0.360264\pi$
$547$	−5.11547e10	+	8.86025e10i	−0.571394	+	0.989684i	−0.996423	+	0.0845041i	$0.973069\pi$
$548$		−	7.47826e10i		−	0.829237i
$549$		0			0
$550$	−1.06038e11			−1.15881
$551$	−7.76820e10	−	4.48497e10i	−0.842780	−	0.486579i
$552$		0			0
$553$	2.31955e10	+	4.01758e10i	0.248030	+	0.429600i
$554$	−8.53012e10	+	4.92487e10i	−0.905558	+	0.522824i
$555$		0			0
$556$	5.23099e10	−	9.06034e10i	0.547375	−	0.948081i
$557$		−	1.00553e11i		−	1.04466i	−0.852743	−	0.522331i	$-0.825063\pi$
$557$		−	1.00553e11i		−	1.04466i	0.852743	−	0.522331i	$-0.174937\pi$
$558$		0			0
$559$	−2.61804e10			−0.268120
$560$	−2.08170e11	−	1.20187e11i	−2.11673	−	1.22210i
$561$		0			0
$562$	−6.30558e10	−	1.09216e11i	−0.632091	−	1.09481i
$563$	1.11721e11	−	6.45022e10i	1.11199	−	0.642009i	0.172648	−	0.984984i	$-0.444768\pi$
$563$	1.11721e11	−	6.45022e10i	1.11199	−	0.642009i	0.939345	+	0.342974i	$0.111434\pi$
$564$		0			0
$565$	1.29423e11	−	2.24167e11i	1.27004	−	2.19978i
$566$			2.91336e9i			0.0283876i
$567$		0			0
$568$	3.02868e11			2.90978
$569$	1.19625e10	+	6.90655e9i	0.114123	+	0.0658889i	0.555975	−	0.831199i	$-0.312345\pi$
$569$	1.19625e10	+	6.90655e9i	0.114123	+	0.0658889i	−0.441852	+	0.897088i	$0.645678\pi$
$570$		0			0
$571$	−3.91580e10	−	6.78237e10i	−0.368363	−	0.638024i	0.620946	−	0.783853i	$-0.286748\pi$
$571$	−3.91580e10	−	6.78237e10i	−0.368363	−	0.638024i	−0.989310	+	0.145829i	$0.953415\pi$
$572$	3.01435e11	−	1.74034e11i	2.81585	−	1.62573i
$573$		0			0
$574$	−2.71170e10	+	4.69681e10i	−0.249801	+	0.432668i
$575$		−	3.76595e10i		−	0.344511i
$576$		0			0
$577$	−6.44834e10			−0.581761			−0.290880	−	0.956759i	$-0.593948\pi$
$577$	−6.44834e10			−0.581761			−0.290880	+	0.956759i	$0.593948\pi$
$578$	8.71178e10	+	5.02975e10i	0.780541	+	0.450646i
$579$		0			0
$580$	−1.47264e11	−	2.55068e11i	−1.30132	−	2.25395i
$581$	−5.69827e10	+	3.28990e10i	−0.500079	+	0.288721i
$582$		0			0
$583$	−6.52712e10	+	1.13053e11i	−0.564999	+	0.978607i
$584$			1.32956e11i			1.14303i
$585$		0			0
$586$	−2.65418e10			−0.225082
$587$	9.52651e10	+	5.50013e10i	0.802383	+	0.463256i	0.844304	−	0.535865i	$-0.180014\pi$
$587$	9.52651e10	+	5.50013e10i	0.802383	+	0.463256i	−0.0419210	+	0.999121i	$0.513348\pi$
$588$		0			0
$589$	1.25234e10	+	2.16911e10i	0.104054	+	0.180228i
$590$	1.05590e11	−	6.09623e10i	0.871393	−	0.503099i
$591$		0			0
$592$	4.92907e10	−	8.53740e10i	0.401308	−	0.695086i
$593$			9.33048e9i			0.0754545i	0.999288	+	0.0377273i	$0.0120118\pi$
$593$			9.33048e9i			0.0754545i	−0.999288	+	0.0377273i	$0.987988\pi$
$594$		0			0
$595$	−9.65037e10			−0.769974
$596$	−3.85255e11	−	2.22427e11i	−3.05325	−	1.76280i
$597$		0			0
$598$	8.78327e10	+	1.52131e11i	0.686833	+	1.18963i
$599$	−7.03813e10	+	4.06347e10i	−0.546701	+	0.315638i	−0.747790	−	0.663935i	$-0.768885\pi$
$599$	−7.03813e10	+	4.06347e10i	−0.546701	+	0.315638i	0.201089	+	0.979573i	$0.435552\pi$
$600$		0			0
$601$	−9.02916e10	+	1.56390e11i	−0.692069	+	1.19870i	0.279090	+	0.960265i	$0.409967\pi$
$601$	−9.02916e10	+	1.56390e11i	−0.692069	+	1.19870i	−0.971159	+	0.238434i	$0.923366\pi$
$602$		−	3.32643e10i		−	0.253275i
$603$		0			0
$604$	−1.13059e11			−0.849488
$605$	−3.99774e10	−	2.30810e10i	−0.298396	−	0.172279i
$606$		0			0
$607$	−5.78752e10	−	1.00243e11i	−0.426322	−	0.738411i	0.570221	−	0.821491i	$-0.306858\pi$
$607$	−5.78752e10	−	1.00243e11i	−0.426322	−	0.738411i	−0.996543	+	0.0830800i	$0.973524\pi$
$608$	−2.26435e11	+	1.30733e11i	−1.65703	+	0.956686i
$609$		0			0
$610$	2.32400e11	−	4.02529e11i	1.67848	−	2.90722i
$611$		−	4.20205e11i		−	3.01506i
$612$		0			0
$613$	7.01257e10			0.496632			0.248316	−	0.968679i	$-0.420123\pi$
$613$	7.01257e10			0.496632			0.248316	+	0.968679i	$0.420123\pi$
$614$	8.56328e10	+	4.94401e10i	0.602513	+	0.347861i
$615$		0			0
$616$	1.28019e11	+	2.21735e11i	0.889102	+	1.53997i
$617$	1.55050e10	−	8.95183e9i	0.106987	−	0.0617691i	−0.445552	−	0.895256i	$-0.646992\pi$
$617$	1.55050e10	−	8.95183e9i	0.106987	−	0.0617691i	0.552539	+	0.833487i	$0.313659\pi$
$618$		0			0
$619$	−6.79961e10	+	1.17773e11i	−0.463150	+	0.802199i	−0.999116	−	0.0420404i	$-0.986614\pi$
$619$	−6.79961e10	+	1.17773e11i	−0.463150	+	0.802199i	0.535966	+	0.844240i	$0.319948\pi$
$620$			8.22410e10i			0.556572i
$621$		0			0
$622$	3.91007e11			2.61230
$623$	−4.81823e10	−	2.78181e10i	−0.319842	−	0.184661i
$624$		0			0
$625$	9.11869e10	+	1.57940e11i	0.597603	+	1.03508i
$626$	−3.34106e11	+	1.92896e11i	−2.17564	+	1.25611i
$627$		0			0
$628$	2.96098e11	−	5.12857e11i	1.90369	−	3.29730i
$629$		−	3.95778e10i		−	0.252842i
$630$		0			0
$631$	−2.50988e11			−1.58320			−0.791600	−	0.611040i	$-0.790752\pi$
$631$	−2.50988e11			−1.58320			−0.791600	+	0.611040i	$0.790752\pi$
$632$	2.11329e11	+	1.22011e11i	1.32462	+	0.764770i
$633$		0			0
$634$	1.02661e11	+	1.77814e11i	0.635402	+	1.10055i
$635$	3.07108e11	−	1.77309e11i	1.88884	−	1.09052i
$636$		0			0
$637$	−4.31322e10	+	7.47071e10i	−0.261965	+	0.453737i
$638$			2.17653e11i			1.31366i
$639$		0			0
$640$	6.10397e10			0.363825
$641$	2.22817e10	+	1.28643e10i	0.131982	+	0.0761999i	0.564537	−	0.825408i	$-0.309055\pi$
$641$	2.22817e10	+	1.28643e10i	0.131982	+	0.0761999i	−0.432555	+	0.901607i	$0.642388\pi$
$642$		0			0
$643$	−2.14853e10	−	3.72136e10i	−0.125689	−	0.217700i	0.796313	−	0.604885i	$-0.206781\pi$
$643$	−2.14853e10	−	3.72136e10i	−0.125689	−	0.217700i	−0.922002	+	0.387185i	$0.873448\pi$
$644$	−1.36022e11	+	7.85322e10i	−0.790797	+	0.456567i
$645$		0			0
$646$	−1.33516e11	+	2.31257e11i	−0.766662	+	1.32790i
$647$			2.20975e11i			1.26103i	0.776177	+	0.630516i	$0.217157\pi$
$647$			2.20975e11i			1.26103i	−0.776177	+	0.630516i	$0.782843\pi$
$648$		0			0
$649$	−6.34045e10			−0.357389
$650$	3.32163e11	+	1.91775e11i	1.86079	+	1.07433i
$651$		0			0
$652$	−3.51116e11	−	6.08151e11i	−1.94294	−	3.36528i
$653$	2.58951e11	−	1.49505e11i	1.42418	−	0.822251i	0.427527	−	0.904002i	$-0.359385\pi$
$653$	2.58951e11	−	1.49505e11i	1.42418	−	0.822251i	0.996653	+	0.0817518i	$0.0260515\pi$
$654$		0			0
$655$	7.77485e10	−	1.34664e11i	0.422403	−	0.731624i
$656$			1.39277e11i			0.752079i
$657$		0			0
$658$	5.33904e11			2.84813
$659$	2.59640e11	+	1.49903e11i	1.37667	+	0.794820i	0.991757	−	0.128134i	$-0.0408987\pi$
$659$	2.59640e11	+	1.49903e11i	1.37667	+	0.794820i	0.384911	+	0.922954i	$0.374232\pi$
$660$		0			0
$661$	1.05724e11	+	1.83120e11i	0.553820	+	0.959244i	0.997994	+	0.0633036i	$0.0201636\pi$
$661$	1.05724e11	+	1.83120e11i	0.553820	+	0.959244i	−0.444175	+	0.895940i	$0.646503\pi$
$662$	3.02244e11	−	1.74500e11i	1.57371	−	0.908583i
$663$		0			0
$664$	−1.73052e11	+	2.99736e11i	−0.890237	+	1.54194i
$665$			2.46718e11i			1.26158i
$666$		0			0
$667$	−7.72997e10			−0.390548
$668$	−6.01848e10	−	3.47477e10i	−0.302260	−	0.174510i
$669$		0			0
$670$	7.23381e9	+	1.25293e10i	0.0358978	+	0.0621768i
$671$	−2.09327e11	+	1.20855e11i	−1.03261	+	0.596177i
$672$		0			0
$673$	9.50831e10	−	1.64689e11i	0.463493	−	0.802793i	−0.535639	−	0.844447i	$-0.679929\pi$
$673$	9.50831e10	−	1.64689e11i	0.463493	−	0.802793i	0.999132	+	0.0416539i	$0.0132627\pi$
$674$		−	2.69077e11i		−	1.30388i
$675$		0			0
$676$	−7.63024e11			−3.65386
$677$	−9.94331e10	−	5.74077e10i	−0.473343	−	0.273285i	0.244295	−	0.969701i	$-0.421443\pi$
$677$	−9.94331e10	−	5.74077e10i	−0.473343	−	0.273285i	−0.717638	+	0.696416i	$0.754777\pi$
$678$		0			0
$679$	−1.34238e11	−	2.32506e11i	−0.631531	−	1.09384i
$680$	−4.39612e11	+	2.53810e11i	−2.05605	+	1.18706i
$681$		0			0
$682$	3.03877e10	−	5.26330e10i	0.140462	−	0.243288i
$683$			3.07050e11i			1.41100i	0.708710	+	0.705500i	$0.249277\pi$
$683$			3.07050e11i			1.41100i	−0.708710	+	0.705500i	$0.750723\pi$
$684$		0			0
$685$	−1.01231e11			−0.459780
$686$	−3.83574e11	−	2.21457e11i	−1.73202	−	0.999983i
$687$		0			0
$688$	−4.27125e10	−	7.39802e10i	−0.190634	−	0.330188i
$689$	4.08923e11	−	2.36092e11i	1.81453	−	1.04762i
$690$		0			0
$691$	1.10299e11	−	1.91043e11i	0.483791	−	0.837951i	−0.516036	−	0.856567i	$-0.672593\pi$
$691$	1.10299e11	−	1.91043e11i	0.483791	−	0.837951i	0.999827	+	0.0186164i	$0.00592614\pi$
$692$			1.00516e12i			4.38339i
$693$		0			0
$694$	6.02812e11			2.59863
$695$	−1.22647e11	−	7.08101e10i	−0.525674	−	0.303498i
$696$		0			0
$697$	2.79580e10	+	4.84246e10i	0.118461	+	0.205180i
$698$	−3.43142e11	+	1.98113e11i	−1.44561	+	0.834625i
$699$		0			0
$700$	−1.71468e11	+	2.96991e11i	−0.714152	+	1.23695i
$701$		−	4.11156e11i		−	1.70269i	−0.524609	−	0.851343i	$-0.675788\pi$
$701$		−	4.11156e11i		−	1.70269i	0.524609	−	0.851343i	$-0.324212\pi$
$702$		0			0
$703$	−1.01183e11			−0.414274
$704$	1.35307e11	+	7.81196e10i	0.550845	+	0.318031i
$705$		0			0
$706$	−1.43774e11	−	2.49024e11i	−0.578711	−	1.00236i
$707$	−6.85637e10	+	3.95853e10i	−0.274421	+	0.158437i
$708$		0			0
$709$	−7.05213e10	+	1.22146e11i	−0.279084	+	0.483388i	−0.971157	−	0.238439i	$-0.923364\pi$
$709$	−7.05213e10	+	1.22146e11i	−0.279084	+	0.483388i	0.692073	+	0.721827i	$0.256698\pi$
$710$		−	7.08150e11i		−	2.78671i
$711$		0			0
$712$	−2.92653e11			−1.13876
$713$	1.86926e10	+	1.07922e10i	0.0723290	+	0.0417591i
$714$		0			0
$715$	−2.35583e11	−	4.08042e11i	−0.901406	−	1.56128i
$716$	−2.19689e11	+	1.26838e11i	−0.835904	+	0.482610i
$717$		0			0
$718$	−2.45936e11	+	4.25973e11i	−0.925388	+	1.60282i
$719$		−	1.87514e11i		−	0.701645i	−0.936442	−	0.350823i	$-0.885902\pi$
$719$		−	1.87514e11i		−	0.701645i	0.936442	−	0.350823i	$-0.114098\pi$
$720$		0			0
$721$	6.60730e10			0.244502
$722$	1.58893e11	+	9.17367e10i	0.584729	+	0.337594i
$723$		0			0
$724$	3.53929e11	+	6.13024e11i	1.28814	+	2.23112i
$725$	−1.46165e11	+	8.43884e10i	−0.529043	+	0.305443i
$726$		0			0
$727$	−3.37310e10	+	5.84239e10i	−0.120751	+	0.209148i	−0.920064	−	0.391768i	$-0.871864\pi$
$727$	−3.37310e10	+	5.84239e10i	−0.120751	+	0.209148i	0.799313	+	0.600915i	$0.205197\pi$
$728$		−	9.26111e11i		−	3.29714i
$729$		0			0
$730$	3.10871e11			1.09469
$731$	−2.97011e10	−	1.71479e10i	−0.104017	−	0.0600541i
$732$		0			0
$733$	2.75674e11	+	4.77481e11i	0.954948	+	1.65402i	0.734487	+	0.678623i	$0.237423\pi$
$733$	2.75674e11	+	4.77481e11i	0.954948	+	1.65402i	0.220461	+	0.975396i	$0.429244\pi$
$734$	3.81530e11	−	2.20276e11i	1.31445	−	0.758899i
$735$		0			0
$736$	−1.12661e11	+	1.95134e11i	−0.383937	+	0.664999i
$737$		−	7.52360e9i		−	0.0255009i
$738$		0			0
$739$	−3.53894e11			−1.18658			−0.593288	−	0.804990i	$-0.702171\pi$
$739$	−3.53894e11			−1.18658			−0.593288	+	0.804990i	$0.702171\pi$
$740$	−2.87724e11	−	1.66117e11i	−0.959509	−	0.553973i
$741$		0			0
$742$	2.99973e11	+	5.19569e11i	0.989616	+	1.71407i
$743$	−3.99633e10	+	2.30728e10i	−0.131131	+	0.0757086i	−0.564131	−	0.825686i	$-0.690789\pi$
$743$	−3.99633e10	+	2.30728e10i	−0.131131	+	0.0757086i	0.432999	+	0.901394i	$0.357455\pi$
$744$		0			0
$745$	−3.01092e11	+	5.21506e11i	−0.977402	+	1.69291i
$746$			5.81893e11i			1.87883i
$747$		0			0
$748$	4.55962e11			1.45654
$749$	3.32999e11	+	1.92257e11i	1.05807	+	0.610880i
$750$		0			0
$751$	1.39277e11	+	2.41235e11i	0.437845	+	0.758370i	0.997523	−	0.0703401i	$-0.0224085\pi$
$751$	1.39277e11	+	2.41235e11i	0.437845	+	0.758370i	−0.559678	+	0.828710i	$0.689075\pi$
$752$	1.18741e12	−	6.85552e11i	3.71304	−	2.14372i
$753$		0			0
$754$	3.93635e11	−	6.81796e11i	1.21789	−	2.10945i
$755$			1.53044e11i			0.471008i
$756$		0			0
$757$	−2.54100e11			−0.773786			−0.386893	−	0.922125i	$-0.626452\pi$
$757$	−2.54100e11			−0.773786			−0.386893	+	0.922125i	$0.626452\pi$
$758$	−1.36163e11	−	7.86138e10i	−0.412461	−	0.238134i
$759$		0			0
$760$	6.48883e11	+	1.12390e12i	1.94497	+	3.36878i
$761$	−4.36794e11	+	2.52183e11i	−1.30238	+	0.751930i	−0.980812	−	0.194956i	$-0.937543\pi$
$761$	−4.36794e11	+	2.52183e11i	−1.30238	+	0.751930i	−0.321569	+	0.946886i	$0.604210\pi$
$762$		0			0
$763$	9.95127e10	−	1.72361e11i	0.293616	−	0.508558i
$764$		−	9.79216e11i		−	2.87412i
$765$		0			0
$766$	−4.26029e11			−1.23744
$767$	1.98614e11	+	1.14670e11i	0.573889	+	0.331335i
$768$		0			0
$769$	−1.08224e11	−	1.87449e11i	−0.309468	−	0.536015i	0.668778	−	0.743462i	$-0.266818\pi$
$769$	−1.08224e11	−	1.87449e11i	−0.309468	−	0.536015i	−0.978246	+	0.207447i	$0.933484\pi$

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	81.d (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(25.4558 + 14.6969i) q^{2} +(304.000 + 526.543i) q^{4} +(712.764 - 411.514i) q^{5} +(-983.500 + 1703.47i) q^{7} +10346.6i q^{8} +O(q^{10})\)	\(q+(25.4558 + 14.6969i) q^{2} +(304.000 + 526.543i) q^{4} +(712.764 - 411.514i) q^{5} +(-983.500 + 1703.47i) q^{7} +10346.6i q^{8} +24192.0 q^{10} +(-10895.1 - 6290.29i) q^{11} +(22752.5 + 39408.5i) q^{13} +(-50071.6 + 28908.9i) q^{14} +(-74240.0 + 128587. i) q^{16} +59610.8i q^{17} +152399. q^{19} +(433360. + 250201. i) q^{20} +(-184896. - 320249. i) q^{22} +(113737. - 65665.9i) q^{23} +(143376. - 248334. i) q^{25} +1.33757e6i q^{26} -1.19594e6 q^{28} +(-509728. - 294291. i) q^{29} +(82175.0 + 142331. i) q^{31} +(-1.48581e6 + 857831. i) q^{32} +(-876096. + 1.51744e6i) q^{34} +1.61890e6i q^{35} -663937. q^{37} +(3.87945e6 + 2.23980e6i) q^{38} +(4.25779e6 + 7.37471e6i) q^{40} +(812347. - 469009. i) q^{41} +(-287665. + 498250. i) q^{43} -7.64899e6i q^{44} +3.86035e6 q^{46} +(-7.99711e6 - 4.61713e6i) q^{47} +(947856. + 1.64173e6i) q^{49} +(7.29949e6 - 4.21436e6i) q^{50} +(-1.38335e7 + 2.39604e7i) q^{52} -1.03765e7i q^{53} -1.03542e7 q^{55} +(-1.76252e7 - 1.01759e7i) q^{56} +(-8.65037e6 - 1.49829e7i) q^{58} +(4.36466e6 - 2.51994e6i) q^{59} +(9.60648e6 - 1.66389e7i) q^{61} +4.83088e6i q^{62} -1.24191e7 q^{64} +(3.24343e7 + 1.87260e7i) q^{65} +(299016. + 517912. i) q^{67} +(-3.13877e7 + 1.81217e7i) q^{68} +(-2.37928e7 + 4.12104e7i) q^{70} -2.92721e7i q^{71} +1.28502e7 q^{73} +(-1.69011e7 - 9.75784e6i) q^{74} +(4.63293e7 + 8.02447e7i) q^{76} +(2.14307e7 - 1.23730e7i) q^{77} +(1.17923e7 - 2.04249e7i) q^{79} +1.22203e8i q^{80} +2.75720e7 q^{82} +(2.89694e7 + 1.67255e7i) q^{83} +(2.45307e7 + 4.24884e7i) q^{85} +(-1.46455e7 + 8.45559e6i) q^{86} +(6.50834e7 - 1.12728e8i) q^{88} +2.82848e7i q^{89} -8.95083e7 q^{91} +(6.91519e7 + 3.99249e7i) q^{92} +(-1.35715e8 - 2.35066e8i) q^{94} +(1.08624e8 - 6.27144e7i) q^{95} +(-6.82448e7 + 1.18203e8i) q^{97} +5.57223e7i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 1216 q^{4} - 3934 q^{7}+O(q^{10}) \) 4 * q + 1216 * q^4 - 3934 * q^7	\( 4 q + 1216 q^{4} - 3934 q^{7} + 96768 q^{10} + 91010 q^{13} - 296960 q^{16} + 609596 q^{19} - 739584 q^{22} + 573502 q^{25} - 4783744 q^{28} + 328700 q^{31} - 3504384 q^{34} - 2655748 q^{37} + 17031168 q^{40} - 1150660 q^{43} + 15441408 q^{46} + 3791424 q^{49} - 55334080 q^{52} - 41416704 q^{55} - 34601472 q^{58} + 38425922 q^{61} - 49676288 q^{64} + 1196066 q^{67} - 95171328 q^{70} + 51400700 q^{73} + 185317184 q^{76} + 47169314 q^{79} + 110287872 q^{82} + 98122752 q^{85} + 260333568 q^{88} - 358033340 q^{91} - 542861568 q^{94} - 272979262 q^{97}+O(q^{100}) \) 4 * q + 1216 * q^4 - 3934 * q^7 + 96768 * q^10 + 91010 * q^13 - 296960 * q^16 + 609596 * q^19 - 739584 * q^22 + 573502 * q^25 - 4783744 * q^28 + 328700 * q^31 - 3504384 * q^34 - 2655748 * q^37 + 17031168 * q^40 - 1150660 * q^43 + 15441408 * q^46 + 3791424 * q^49 - 55334080 * q^52 - 41416704 * q^55 - 34601472 * q^58 + 38425922 * q^61 - 49676288 * q^64 + 1196066 * q^67 - 95171328 * q^70 + 51400700 * q^73 + 185317184 * q^76 + 47169314 * q^79 + 110287872 * q^82 + 98122752 * q^85 + 260333568 * q^88 - 358033340 * q^91 - 542861568 * q^94 - 272979262 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more