LMFDB - Embedded newform 81.8.c.g.55.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [81,8,Mod(28,81)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("81.28"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(81, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 81 = 3^{4} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	81.c (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	no
Analytic conductor:	$25.3031870642$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 27)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			55.1
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	81.55
Dual form			81.8.c.g.28.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-5.19615 + 9.00000i) q^{2} +(10.0000 + 17.3205i) q^{4} +(176.669 + 306.000i) q^{5} +(279.500 - 484.108i) q^{7} -1538.06 q^{8} -3672.00 q^{10} +(-2359.05 + 4086.00i) q^{11} +(4335.50 + 7509.31i) q^{13} +(2904.65 + 5031.00i) q^{14} +(6712.00 - 11625.5i) q^{16} +25128.6 q^{17} -32461.0 q^{19} +(-3533.38 + 6120.00i) q^{20} +(-24516.0 - 42463.0i) q^{22} +(41205.5 + 71370.0i) q^{23} +(-23361.5 + 40463.3i) q^{25} -90111.7 q^{26} +11180.0 q^{28} +(-78898.4 + 136656. i) q^{29} +(-114946. - 199092. i) q^{31} +(-28682.8 - 49680.0i) q^{32} +(-130572. + 226157. i) q^{34} +197516. q^{35} -541177. q^{37} +(168672. - 292149. i) q^{38} +(-271728. - 470647. i) q^{40} +(-176752. - 306144. i) q^{41} +(232556. - 402799. i) q^{43} -94362.1 q^{44} -856440. q^{46} +(415287. - 719298. i) q^{47} +(255531. + 442593. i) q^{49} +(-242780. - 420507. i) q^{50} +(-86710.0 + 150186. i) q^{52} -1.02622e6 q^{53} -1.66709e6 q^{55} +(-429888. + 744588. i) q^{56} +(-819936. - 1.42017e6i) q^{58} +(392632. + 680058. i) q^{59} +(68886.5 - 119315. i) q^{61} +2.38911e6 q^{62} +2.31443e6 q^{64} +(-1.53190e6 + 2.65333e6i) q^{65} +(157020. + 271967. i) q^{67} +(251286. + 435240. i) q^{68} +(-1.02632e6 + 1.77765e6i) q^{70} -2.80979e6 q^{71} +2.66954e6 q^{73} +(2.81204e6 - 4.87059e6i) q^{74} +(-324610. - 562241. i) q^{76} +(1.31871e6 + 2.28407e6i) q^{77} +(-550907. + 954200. i) q^{79} +4.74321e6 q^{80} +3.67373e6 q^{82} +(-3.03952e6 + 5.26460e6i) q^{83} +(4.43945e6 + 7.68935e6i) q^{85} +(2.41679e6 + 4.18601e6i) q^{86} +(3.62837e6 - 6.28452e6i) q^{88} -3.28636e6 q^{89} +4.84709e6 q^{91} +(-824110. + 1.42740e6i) q^{92} +(4.31579e6 + 7.47516e6i) q^{94} +(-5.73486e6 - 9.93307e6i) q^{95} +(1.48969e6 - 2.58022e6i) q^{97} -5.31111e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 40 q^{4} + 1118 q^{7} - 14688 q^{10} + 17342 q^{13} + 26848 q^{16} - 129844 q^{19} - 98064 q^{22} - 93446 q^{25} + 44720 q^{28} - 459784 q^{31} - 522288 q^{34} - 2164708 q^{37} - 1086912 q^{40} + 930224 q^{43}+ \cdots + 5958758 q^{97}+O(q^{100}) $ 4 * q + 40 * q^4 + 1118 * q^7 - 14688 * q^10 + 17342 * q^13 + 26848 * q^16 - 129844 * q^19 - 98064 * q^22 - 93446 * q^25 + 44720 * q^28 - 459784 * q^31 - 522288 * q^34 - 2164708 * q^37 - 1086912 * q^40 + 930224 * q^43 - 3425760 * q^46 + 1022124 * q^49 - 346840 * q^52 - 6668352 * q^55 - 3279744 * q^58 + 275546 * q^61 + 9257728 * q^64 + 628082 * q^67 - 4105296 * q^70 + 10678148 * q^73 - 1298440 * q^76 - 2203630 * q^79 + 14694912 * q^82 + 17757792 * q^85 + 14513472 * q^88 + 19388356 * q^91 + 17263152 * q^94 + 5958758 * q^97

Character values

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

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

Coefficient data

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

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

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

$2$	−5.19615	+	9.00000i	−0.459279	+	0.795495i	−0.998923	−	0.0463984i	$-0.985226\pi$
$2$	−5.19615	+	9.00000i	−0.459279	+	0.795495i	0.539644	+	0.841894i	$0.318559\pi$
$3$		0			0
$3$		0			0
$4$	10.0000	+	17.3205i	0.0781250	+	0.135316i
$5$	176.669	+	306.000i	0.632071	+	1.09478i	0.987128	+	0.159934i	$0.0511282\pi$
$5$	176.669	+	306.000i	0.632071	+	1.09478i	−0.355057	+	0.934845i	$0.615538\pi$
$6$		0			0
$7$	279.500	−	484.108i	0.307991	−	0.533457i	−0.669931	−	0.742423i	$-0.733677\pi$
$7$	279.500	−	484.108i	0.307991	−	0.533457i	0.977923	+	0.208966i	$0.0670098\pi$
$8$	−1538.06			−1.06208
$9$		0			0
$10$	−3672.00			−1.16119
$11$	−2359.05	+	4086.00i	−0.534396	+	0.925601i	0.464796	+	0.885418i	$0.346128\pi$
$11$	−2359.05	+	4086.00i	−0.534396	+	0.925601i	−0.999192	+	0.0401836i	$0.987206\pi$
$12$		0			0
$13$	4335.50	+	7509.31i	0.547315	+	0.947978i	0.998457	+	0.0555255i	$0.0176834\pi$
$13$	4335.50	+	7509.31i	0.547315	+	0.947978i	−0.451142	+	0.892452i	$0.648983\pi$
$14$	2904.65	+	5031.00i	0.282908	+	0.490011i
$15$		0			0
$16$	6712.00	−	11625.5i	0.409668	−	0.709566i
$17$	25128.6			1.24050			0.620250	−	0.784404i	$-0.287031\pi$
$17$	25128.6			1.24050			0.620250	+	0.784404i	$0.287031\pi$
$18$		0			0
$19$	−32461.0			−1.08574			−0.542868	−	0.839818i	$-0.682662\pi$
$19$	−32461.0			−1.08574			−0.542868	+	0.839818i	$0.682662\pi$
$20$	−3533.38	+	6120.00i	−0.0987611	+	0.171059i
$21$		0			0
$22$	−24516.0	−	42463.0i	−0.490874	−	0.850219i
$23$	41205.5	+	71370.0i	0.706167	+	1.22312i	0.966269	+	0.257536i	$0.0829107\pi$
$23$	41205.5	+	71370.0i	0.706167	+	1.22312i	−0.260101	+	0.965581i	$0.583756\pi$
$24$		0			0
$25$	−23361.5	+	40463.3i	−0.299027	+	0.517930i
$26$	−90111.7			−1.00548
$27$		0			0
$28$	11180.0			0.0962473
$29$	−78898.4	+	136656.i	−0.600724	+	1.04048i	0.391987	+	0.919971i	$0.371788\pi$
$29$	−78898.4	+	136656.i	−0.600724	+	1.04048i	−0.992712	+	0.120514i	$0.961546\pi$
$30$		0			0
$31$	−114946.	−	199092.i	−0.692992	−	1.20030i	−0.970853	−	0.239675i	$-0.922959\pi$
$31$	−114946.	−	199092.i	−0.692992	−	1.20030i	0.277862	−	0.960621i	$-0.410374\pi$
$32$	−28682.8	−	49680.0i	−0.154738	−	0.268013i
$33$		0			0
$34$	−130572.	+	226157.i	−0.569736	+	0.986812i
$35$	197516.			0.778690
$36$		0			0
$37$	−541177.			−1.75644			−0.878220	−	0.478257i	$-0.841269\pi$
$37$	−541177.			−1.75644			−0.878220	+	0.478257i	$0.841269\pi$
$38$	168672.	−	292149.i	0.498656	−	0.863698i
$39$		0			0
$40$	−271728.	−	470647.i	−0.671312	−	1.16275i
$41$	−176752.	−	306144.i	−0.400518	−	0.693717i	0.593271	−	0.805003i	$-0.297836\pi$
$41$	−176752.	−	306144.i	−0.400518	−	0.693717i	−0.993788	+	0.111286i	$0.964503\pi$
$42$		0			0
$43$	232556.	−	402799.i	0.446055	−	0.772589i	−0.552070	−	0.833797i	$-0.686162\pi$
$43$	232556.	−	402799.i	0.446055	−	0.772589i	0.998125	+	0.0612083i	$0.0194954\pi$
$44$	−94362.1			−0.166999
$45$		0			0
$46$	−856440.			−1.29731
$47$	415287.	−	719298.i	0.583453	−	1.01057i	−0.411614	−	0.911358i	$-0.635035\pi$
$47$	415287.	−	719298.i	0.583453	−	1.01057i	0.995066	−	0.0992114i	$-0.0316320\pi$
$48$		0			0
$49$	255531.	+	442593.i	0.310283	+	0.537425i
$50$	−242780.	−	420507.i	−0.274674	−	0.475749i
$51$		0			0
$52$	−86710.0	+	150186.i	−0.0855180	+	0.148122i
$53$	−1.02622e6			−0.946836			−0.473418	−	0.880838i	$-0.656980\pi$
$53$	−1.02622e6			−0.946836			−0.473418	+	0.880838i	$0.656980\pi$
$54$		0			0
$55$	−1.66709e6			−1.35111
$56$	−429888.	+	744588.i	−0.327113	+	0.566576i
$57$		0			0
$58$	−819936.	−	1.42017e6i	−0.551800	−	0.955746i
$59$	392632.	+	680058.i	0.248888	+	0.431086i	0.963217	−	0.268723i	$-0.0866017\pi$
$59$	392632.	+	680058.i	0.248888	+	0.431086i	−0.714330	+	0.699809i	$0.753268\pi$
$60$		0			0
$61$	68886.5	−	119315.i	0.0388579	−	0.0673039i	−0.845942	−	0.533274i	$-0.820961\pi$
$61$	68886.5	−	119315.i	0.0388579	−	0.0673039i	0.884800	+	0.465970i	$0.154295\pi$
$62$	2.38911e6			1.27311
$63$		0			0
$64$	2.31443e6			1.10361
$65$	−1.53190e6	+	2.65333e6i	−0.691884	+	1.19838i
$66$		0			0
$67$	157020.	+	271967.i	0.0637815	+	0.110473i	0.896153	−	0.443746i	$-0.146351\pi$
$67$	157020.	+	271967.i	0.0637815	+	0.110473i	−0.832371	+	0.554218i	$0.813017\pi$
$68$	251286.	+	435240.i	0.0969141	+	0.167860i
$69$		0			0
$70$	−1.02632e6	+	1.77765e6i	−0.357636	+	0.619444i
$71$	−2.80979e6			−0.931686			−0.465843	−	0.884868i	$-0.654249\pi$
$71$	−2.80979e6			−0.931686			−0.465843	+	0.884868i	$0.654249\pi$
$72$		0			0
$73$	2.66954e6			0.803167			0.401584	−	0.915822i	$-0.368460\pi$
$73$	2.66954e6			0.803167			0.401584	+	0.915822i	$0.368460\pi$
$74$	2.81204e6	−	4.87059e6i	0.806697	−	1.39724i
$75$		0			0
$76$	−324610.	−	562241.i	−0.0848231	−	0.146918i
$77$	1.31871e6	+	2.28407e6i	0.329179	+	0.570155i
$78$		0			0
$79$	−550907.	+	954200.i	−0.125714	+	0.217743i	−0.922012	−	0.387162i	$-0.873456\pi$
$79$	−550907.	+	954200.i	−0.125714	+	0.217743i	0.796298	+	0.604905i	$0.206789\pi$
$80$	4.74321e6			1.03576
$81$		0			0
$82$	3.67373e6			0.735798
$83$	−3.03952e6	+	5.26460e6i	−0.583488	+	1.01063i	0.411574	+	0.911376i	$0.364979\pi$
$83$	−3.03952e6	+	5.26460e6i	−0.583488	+	1.01063i	−0.995062	+	0.0992543i	$0.968354\pi$
$84$		0			0
$85$	4.43945e6	+	7.68935e6i	0.784084	+	1.35807i
$86$	2.41679e6	+	4.18601e6i	0.409727	+	0.709668i
$87$		0			0
$88$	3.62837e6	−	6.28452e6i	0.567573	−	0.983066i
$89$	−3.28636e6			−0.494140			−0.247070	−	0.968998i	$-0.579468\pi$
$89$	−3.28636e6			−0.494140			−0.247070	+	0.968998i	$0.579468\pi$
$90$		0			0
$91$	4.84709e6			0.674274
$92$	−824110.	+	1.42740e6i	−0.110339	+	0.191112i
$93$		0			0
$94$	4.31579e6	+	7.47516e6i	0.535936	+	0.928268i
$95$	−5.73486e6	−	9.93307e6i	−0.686262	−	1.18864i
$96$		0			0
$97$	1.48969e6	−	2.58022e6i	0.165728	−	0.287049i	−0.771186	−	0.636610i	$-0.780336\pi$
$97$	1.48969e6	−	2.58022e6i	0.165728	−	0.287049i	0.936913	+	0.349562i	$0.113669\pi$
$98$	−5.31111e6			−0.570025
$99$		0			0
$100$	−934460.			−0.0934460
$101$	1.37370e6	−	2.37931e6i	0.132668	−	0.229788i	−0.792036	−	0.610474i	$-0.790979\pi$
$101$	1.37370e6	−	2.37931e6i	0.132668	−	0.229788i	0.924704	+	0.380686i	$0.124312\pi$
$102$		0			0
$103$	2.93555e6	+	5.08451e6i	0.264703	+	0.458479i	0.967486	−	0.252926i	$-0.0813928\pi$
$103$	2.93555e6	+	5.08451e6i	0.264703	+	0.458479i	−0.702783	+	0.711404i	$0.748059\pi$
$104$	−6.66826e6	−	1.15498e7i	−0.581294	−	1.00683i
$105$		0			0
$106$	5.33239e6	−	9.23597e6i	0.434862	−	0.753203i
$107$	5.84848e6			0.461530			0.230765	−	0.973010i	$-0.425877\pi$
$107$	5.84848e6			0.461530			0.230765	+	0.973010i	$0.425877\pi$
$108$		0			0
$109$	8.94694e6			0.661731			0.330866	−	0.943678i	$-0.392659\pi$
$109$	8.94694e6			0.661731			0.330866	+	0.943678i	$0.392659\pi$
$110$	8.66244e6	−	1.50038e7i	0.620535	−	1.07480i
$111$		0			0
$112$	−3.75201e6	−	6.49867e6i	−0.252348	−	0.437080i
$113$	6.53330e6	+	1.13160e7i	0.425949	+	0.737766i	0.996509	−	0.0834905i	$-0.0266068\pi$
$113$	6.53330e6	+	1.13160e7i	0.425949	+	0.737766i	−0.570559	+	0.821256i	$0.693273\pi$
$114$		0			0
$115$	−1.45595e7	+	2.52178e7i	−0.892696	+	1.54619i
$116$	−3.15594e6			−0.187726
$117$		0			0
$118$	−8.16070e6			−0.457236
$119$	7.02344e6	−	1.21650e7i	0.382064	−	0.661754i
$120$		0			0
$121$	−1.38668e6	−	2.40180e6i	−0.0711585	−	0.123250i
$122$	715890.	+	1.23996e6i	0.0356933	+	0.0618226i
$123$		0			0
$124$	2.29892e6	−	3.98185e6i	0.108280	−	0.187546i
$125$	1.10955e7			0.508116
$126$		0			0
$127$	1.96731e7			0.852235			0.426118	−	0.904668i	$-0.359881\pi$
$127$	1.96731e7			0.852235			0.426118	+	0.904668i	$0.359881\pi$
$128$	−8.35475e6	+	1.44708e7i	−0.352126	+	0.609901i
$129$		0			0
$130$	−1.59200e7	−	2.75742e7i	−0.635536	−	1.10078i
$131$	−4.28375e6	−	7.41967e6i	−0.166485	−	0.288360i	0.770697	−	0.637202i	$-0.219908\pi$
$131$	−4.28375e6	−	7.41967e6i	−0.166485	−	0.288360i	−0.937182	+	0.348842i	$0.886575\pi$
$132$		0			0
$133$	−9.07285e6	+	1.57146e7i	−0.334397	+	0.579193i
$134$	−3.26361e6			−0.117174
$135$		0			0
$136$	−3.86493e7			−1.31752
$137$	−2.78069e7	+	4.81630e7i	−0.923913	+	1.60026i	−0.130613	+	0.991433i	$0.541694\pi$
$137$	−2.78069e7	+	4.81630e7i	−0.923913	+	1.60026i	−0.793300	+	0.608831i	$0.791639\pi$
$138$		0			0
$139$	−1.38995e7	−	2.40746e7i	−0.438981	−	0.760338i	0.558630	−	0.829417i	$-0.311327\pi$
$139$	−1.38995e7	−	2.40746e7i	−0.438981	−	0.760338i	−0.997611	+	0.0690791i	$0.977994\pi$
$140$	1.97516e6	+	3.42108e6i	0.0608351	+	0.105370i
$141$		0			0
$142$	1.46001e7	−	2.52881e7i	0.427904	−	0.741151i
$143$	−4.09107e7			−1.16993
$144$		0			0
$145$	−5.57556e7			−1.51880
$146$	−1.38713e7	+	2.40258e7i	−0.368878	+	0.638916i
$147$		0			0
$148$	−5.41177e6	−	9.37346e6i	−0.137222	−	0.237675i
$149$	3.89185e7	+	6.74089e7i	0.963839	+	1.66942i	0.712702	+	0.701467i	$0.247471\pi$
$149$	3.89185e7	+	6.74089e7i	0.963839	+	1.66942i	0.251137	+	0.967952i	$0.419195\pi$
$150$		0			0
$151$	−5.23161e6	+	9.06141e6i	−0.123656	+	0.214179i	−0.921207	−	0.389073i	$-0.872795\pi$
$151$	−5.23161e6	+	9.06141e6i	−0.123656	+	0.214179i	0.797551	+	0.603252i	$0.206129\pi$
$152$	4.99270e7			1.15314
$153$		0			0
$154$	−2.74089e7			−0.604740
$155$	4.06148e7	−	7.03470e7i	0.876040	−	1.51735i
$156$		0			0
$157$	3.28907e7	+	5.69684e7i	0.678304	+	1.17486i	0.975491	+	0.220038i	$0.0706181\pi$
$157$	3.28907e7	+	5.69684e7i	0.678304	+	1.17486i	−0.297187	+	0.954819i	$0.596049\pi$
$158$	−5.72520e6	−	9.91634e6i	−0.115476	−	0.200010i
$159$		0			0
$160$	1.01347e7	−	1.75538e7i	0.195610	−	0.338807i
$161$	4.60677e7			0.869974
$162$		0			0
$163$	−3.00965e7			−0.544327			−0.272164	−	0.962251i	$-0.587739\pi$
$163$	−3.00965e7			−0.544327			−0.272164	+	0.962251i	$0.587739\pi$
$164$	3.53505e6	−	6.12288e6i	0.0625809	−	0.108393i
$165$		0			0
$166$	−3.15876e7	−	5.47114e7i	−0.535968	−	0.928323i
$167$	−3.95499e7	−	6.85024e7i	−0.657109	−	1.13815i	−0.981361	−	0.192175i	$-0.938446\pi$
$167$	−3.95499e7	−	6.85024e7i	−0.657109	−	1.13815i	0.324252	−	0.945971i	$-0.394887\pi$
$168$		0			0
$169$	−6.21886e6	+	1.07714e7i	−0.0991077	+	0.171660i
$170$	−9.22722e7			−1.44045
$171$		0			0
$172$	9.30224e6			0.139392
$173$	1.40612e7	−	2.43547e7i	0.206472	−	0.357620i	−0.744129	−	0.668036i	$-0.767135\pi$
$173$	1.40612e7	−	2.43547e7i	0.206472	−	0.357620i	0.950601	+	0.310416i	$0.100468\pi$
$174$		0			0
$175$	1.30591e7	+	2.26190e7i	0.184196	+	0.319036i
$176$	3.16679e7	+	5.48505e7i	0.437850	+	0.758378i
$177$		0			0
$178$	1.70764e7	−	2.95772e7i	0.226948	−	0.393086i
$179$	3.04834e7			0.397263			0.198631	−	0.980074i	$-0.436350\pi$
$179$	3.04834e7			0.397263			0.198631	+	0.980074i	$0.436350\pi$
$180$		0			0
$181$	2.98803e6			0.0374550			0.0187275	−	0.999825i	$-0.494039\pi$
$181$	2.98803e6			0.0374550			0.0187275	+	0.999825i	$0.494039\pi$
$182$	−2.51862e7	+	4.36238e7i	−0.309680	+	0.536381i
$183$		0			0
$184$	−6.33766e7	−	1.09771e8i	−0.750009	−	1.29905i
$185$	−9.56093e7	−	1.65600e8i	−1.11019	−	1.92291i
$186$		0			0
$187$	−5.92797e7	+	1.02675e8i	−0.662919	+	1.14821i
$188$	1.66115e7			0.182329
$189$		0			0
$190$	1.19197e8			1.26074
$191$	−1.69714e7	+	2.93953e7i	−0.176238	+	0.305254i	−0.940589	−	0.339547i	$-0.889726\pi$
$191$	−1.69714e7	+	2.93953e7i	−0.176238	+	0.305254i	0.764351	+	0.644801i	$0.223060\pi$
$192$		0			0
$193$	7.86219e7	+	1.36177e8i	0.787214	+	1.36350i	0.927667	+	0.373408i	$0.121811\pi$
$193$	7.86219e7	+	1.36177e8i	0.787214	+	1.36350i	−0.140453	+	0.990087i	$0.544856\pi$
$194$	1.54813e7	+	2.68144e7i	0.152230	+	0.263671i
$195$		0			0
$196$	−5.11062e6	+	8.85185e6i	−0.0484816	+	0.0839727i
$197$	−6.30072e7			−0.587163			−0.293581	−	0.955934i	$-0.594847\pi$
$197$	−6.30072e7			−0.587163			−0.293581	+	0.955934i	$0.594847\pi$
$198$		0			0
$199$	1.92815e8			1.73443			0.867213	−	0.497937i	$-0.165909\pi$
$199$	1.92815e8			1.73443			0.867213	+	0.497937i	$0.165909\pi$
$200$	3.59314e7	−	6.22350e7i	0.317592	−	0.550085i
$201$		0			0
$202$	1.42759e7	+	2.47265e7i	0.121863	+	0.211073i
$203$	4.41042e7	+	7.63907e7i	0.370036	+	0.640921i
$204$		0			0
$205$	6.24534e7	−	1.08172e8i	0.506311	−	0.876956i
$206$	−6.10142e7			−0.486290
$207$		0			0
$208$	1.16400e8			0.896870
$209$	7.65772e7	−	1.32636e8i	0.580213	−	1.00496i
$210$		0			0
$211$	6.39025e7	+	1.10682e8i	0.468305	+	0.811128i	0.999344	−	0.0362192i	$-0.0115314\pi$
$211$	6.39025e7	+	1.10682e8i	0.468305	+	0.811128i	−0.531039	+	0.847348i	$0.678198\pi$
$212$	−1.02622e7	−	1.77746e7i	−0.0739715	−	0.128122i
$213$		0			0
$214$	−3.03896e7	+	5.26363e7i	−0.211971	+	0.367145i
$215$	1.64342e8			1.12775
$216$		0			0
$217$	−1.28510e8			−0.853742
$218$	−4.64897e7	+	8.05224e7i	−0.303920	+	0.526404i
$219$		0			0
$220$	−1.66709e7	−	2.88748e7i	−0.105555	−	0.182827i
$221$	1.08945e8	+	1.88698e8i	0.678945	+	1.17597i
$222$		0			0
$223$	3.32826e7	−	5.76472e7i	0.200979	−	0.348106i	−0.747865	−	0.663851i	$-0.768921\pi$
$223$	3.32826e7	−	5.76472e7i	0.200979	−	0.348106i	0.948844	+	0.315745i	$0.102254\pi$
$224$	−3.20673e7			−0.190632
$225$		0			0
$226$	−1.35792e8			−0.782519
$227$	−7.32073e7	+	1.26799e8i	−0.415397	+	0.719489i	−0.995470	−	0.0950753i	$-0.969691\pi$
$227$	−7.32073e7	+	1.26799e8i	−0.415397	+	0.719489i	0.580073	+	0.814565i	$0.303024\pi$
$228$		0			0
$229$	1.27864e8	+	2.21467e8i	0.703597	+	1.21867i	0.967195	+	0.254033i	$0.0817573\pi$
$229$	1.27864e8	+	2.21467e8i	0.703597	+	1.21867i	−0.263598	+	0.964633i	$0.584909\pi$
$230$	−1.51307e8	−	2.62071e8i	−0.819993	−	1.42027i
$231$		0			0
$232$	1.21351e8	−	2.10185e8i	0.638019	−	1.10508i
$233$	5.51954e7			0.285862			0.142931	−	0.989733i	$-0.454347\pi$
$233$	5.51954e7			0.285862			0.142931	+	0.989733i	$0.454347\pi$
$234$		0			0
$235$	2.93474e8			1.47513
$236$	−7.85263e6	+	1.36012e7i	−0.0388887	+	0.0673572i
$237$		0			0
$238$	7.29897e7	+	1.26422e8i	0.350948	+	0.607859i
$239$	9.12948e7	+	1.58127e8i	0.432567	+	0.749228i	0.997094	−	0.0761871i	$-0.0242746\pi$
$239$	9.12948e7	+	1.58127e8i	0.432567	+	0.749228i	−0.564527	+	0.825415i	$0.690941\pi$
$240$		0			0
$241$	−8.62215e6	+	1.49340e7i	−0.0396786	+	0.0687253i	−0.885183	−	0.465244i	$-0.845967\pi$
$241$	−8.62215e6	+	1.49340e7i	−0.0396786	+	0.0687253i	0.845504	+	0.533969i	$0.179300\pi$
$242$	2.88216e7			0.130727
$243$		0			0
$244$	2.75546e6			0.0121431
$245$	−9.02889e7	+	1.56385e8i	−0.392241	+	0.679382i
$246$		0			0
$247$	−1.40735e8	−	2.43760e8i	−0.594240	−	1.02925i
$248$	1.76794e8	+	3.06216e8i	0.736015	+	1.27482i
$249$		0			0
$250$	−5.76541e7	+	9.98598e7i	−0.233367	+	0.404204i
$251$	−4.88338e7			−0.194923			−0.0974613	−	0.995239i	$-0.531072\pi$
$251$	−4.88338e7			−0.194923			−0.0974613	+	0.995239i	$0.531072\pi$
$252$		0			0
$253$	−3.88824e8			−1.50949
$254$	−1.02224e8	+	1.77058e8i	−0.391414	+	0.677949i
$255$		0			0
$256$	6.12986e7	+	1.06172e8i	0.228355	+	0.395522i
$257$	−4.10712e7	−	7.11374e7i	−0.150929	−	0.261416i	0.780641	−	0.624980i	$-0.214893\pi$
$257$	−4.10712e7	−	7.11374e7i	−0.150929	−	0.261416i	−0.931569	+	0.363564i	$0.881560\pi$
$258$		0			0
$259$	−1.51259e8	+	2.61988e8i	−0.540968	+	0.936985i
$260$	−6.12759e7			−0.216214
$261$		0			0
$262$	8.90361e7			0.305852
$263$	1.03808e8	−	1.79801e8i	0.351874	−	0.609464i	−0.634704	−	0.772756i	$-0.718878\pi$
$263$	1.03808e8	−	1.79801e8i	0.351874	−	0.609464i	0.986578	+	0.163292i	$0.0522112\pi$
$264$		0			0
$265$	−1.81301e8	−	3.14023e8i	−0.598467	−	1.03658i
$266$	−9.42878e7	−	1.63311e8i	−0.307164	−	0.532023i
$267$		0			0
$268$	−3.14041e6	+	5.43935e6i	−0.00996586	+	0.0172614i
$269$	−3.42054e8			−1.07143			−0.535713	−	0.844400i	$-0.679957\pi$
$269$	−3.42054e8			−1.07143			−0.535713	+	0.844400i	$0.679957\pi$
$270$		0			0
$271$	6.74035e7			0.205726			0.102863	−	0.994696i	$-0.467200\pi$
$271$	6.74035e7			0.205726			0.102863	+	0.994696i	$0.467200\pi$
$272$	1.68663e8	−	2.92133e8i	0.508193	−	0.880217i
$273$		0			0
$274$	−2.88978e8	−	5.00525e8i	−0.848668	−	1.46994i
$275$	−1.10222e8	−	1.90910e8i	−0.319598	−	0.553560i
$276$		0			0
$277$	−1.04046e8	+	1.80213e8i	−0.294135	+	0.509457i	−0.974783	−	0.223154i	$-0.928365\pi$
$277$	−1.04046e8	+	1.80213e8i	−0.294135	+	0.509457i	0.680648	+	0.732610i	$0.261698\pi$
$278$	2.88895e8			0.806460
$279$		0			0
$280$	−3.03792e8			−0.827033
$281$	1.27355e8	−	2.20585e8i	0.342407	−	0.593066i	−0.642472	−	0.766309i	$-0.722091\pi$
$281$	1.27355e8	−	2.20585e8i	0.342407	−	0.593066i	0.984879	+	0.173243i	$0.0554245\pi$
$282$		0			0
$283$	−9.10379e7	−	1.57682e8i	−0.238764	−	0.413552i	0.721596	−	0.692315i	$-0.243409\pi$
$283$	−9.10379e7	−	1.57682e8i	−0.238764	−	0.413552i	−0.960360	+	0.278763i	$0.910076\pi$
$284$	−2.80979e7	−	4.86670e7i	−0.0727879	−	0.126072i
$285$		0			0
$286$	2.12578e8	−	3.68196e8i	0.537326	−	0.930676i
$287$	−1.97609e8			−0.493424
$288$		0			0
$289$	2.21108e8			0.538842
$290$	2.89715e8	−	5.01801e8i	0.697554	−	1.20820i
$291$		0			0
$292$	2.66954e7	+	4.62377e7i	0.0627474	+	0.108682i
$293$	−2.72160e8	−	4.71396e8i	−0.632104	−	1.09484i	−0.987121	−	0.159976i	$-0.948858\pi$
$293$	−2.72160e8	−	4.71396e8i	−0.632104	−	1.09484i	0.355017	−	0.934860i	$-0.384475\pi$
$294$		0			0
$295$	−1.38732e8	+	2.40291e8i	−0.314629	+	0.544954i
$296$	8.32363e8			1.86549
$297$		0			0
$298$	−8.08907e8			−1.77069
$299$	−3.57293e8	+	6.18849e8i	−0.772992	+	1.33886i
$300$		0			0
$301$	−1.29999e8	−	2.25165e8i	−0.274762	−	0.475902i
$302$	−5.43685e7	−	9.41690e7i	−0.113586	−	0.196736i
$303$		0			0
$304$	−2.17878e8	+	3.77376e8i	−0.444791	+	0.770401i
$305$	4.86805e7			0.0982439
$306$		0			0
$307$	−5.34564e8			−1.05442			−0.527212	−	0.849734i	$-0.676763\pi$
$307$	−5.34564e8			−1.05442			−0.527212	+	0.849734i	$0.676763\pi$
$308$	−2.63742e7	+	4.56815e7i	−0.0514342	+	0.0890867i
$309$		0			0
$310$	4.22082e8	+	7.31067e8i	0.804694	+	1.39377i
$311$	1.99844e8	+	3.46139e8i	0.376729	+	0.652513i	0.990584	−	0.136905i	$-0.0437156\pi$
$311$	1.99844e8	+	3.46139e8i	0.376729	+	0.652513i	−0.613855	+	0.789418i	$0.710382\pi$
$312$		0			0
$313$	4.56629e8	−	7.90905e8i	0.841702	−	1.45787i	−0.0467530	−	0.998906i	$-0.514887\pi$
$313$	4.56629e8	−	7.90905e8i	0.841702	−	1.45787i	0.888455	−	0.458964i	$-0.151779\pi$
$314$	−6.83621e8			−1.24612
$315$		0			0
$316$	−2.20363e7			−0.0392856
$317$	−2.05328e8	+	3.55639e8i	−0.362027	+	0.627050i	−0.988294	−	0.152559i	$-0.951249\pi$
$317$	−2.05328e8	+	3.55639e8i	−0.362027	+	0.627050i	0.626267	+	0.779609i	$0.284582\pi$
$318$		0			0
$319$	−3.72251e8	−	6.44758e8i	−0.642049	−	1.11206i
$320$	4.08889e8	+	7.08216e8i	0.697558	+	1.20821i
$321$		0			0
$322$	−2.39375e8	+	4.14610e8i	−0.399561	+	0.692060i
$323$	−8.15699e8			−1.34686
$324$		0			0
$325$	−4.05135e8			−0.654648
$326$	1.56386e8	−	2.70869e8i	0.249998	−	0.433010i
$327$		0			0
$328$	2.71856e8	+	4.70868e8i	0.425383	+	0.736785i
$329$	−2.32145e8	−	4.02088e8i	−0.359397	−	0.622494i
$330$		0			0
$331$	4.64872e8	−	8.05183e8i	0.704589	−	1.22038i	−0.262251	−	0.965000i	$-0.584465\pi$
$331$	4.64872e8	−	8.05183e8i	0.704589	−	1.22038i	0.966840	−	0.255384i	$-0.0822019\pi$
$332$	−1.21581e8			−0.182340
$333$		0			0
$334$	8.22028e8			1.20719
$335$	−5.54814e7	+	9.60965e7i	−0.0806288	+	0.139653i
$336$		0			0
$337$	2.86963e8	+	4.97034e8i	0.408433	+	0.707426i	0.994714	−	0.102681i	$-0.0327421\pi$
$337$	2.86963e8	+	4.97034e8i	0.408433	+	0.707426i	−0.586282	+	0.810107i	$0.699409\pi$
$338$	−6.46283e7	−	1.11940e8i	−0.0910362	−	0.157679i
$339$		0			0
$340$	−8.87890e7	+	1.53787e8i	−0.122513	+	0.212199i
$341$	1.08465e9			1.48133
$342$		0			0
$343$	7.46044e8			0.998240
$344$	−3.57685e8	+	6.19529e8i	−0.473747	+	0.820554i
$345$		0			0
$346$	1.46128e8	+	2.53102e8i	0.189657	+	0.328495i
$347$	5.46477e8	+	9.46527e8i	0.702133	+	1.21613i	0.967716	+	0.252041i	$0.0811019\pi$
$347$	5.46477e8	+	9.46527e8i	0.702133	+	1.21613i	−0.265584	+	0.964088i	$0.585565\pi$
$348$		0			0
$349$	−3.64264e8	+	6.30924e8i	−0.458699	+	0.794489i	−0.998892	−	0.0470513i	$-0.985018\pi$
$349$	−3.64264e8	+	6.30924e8i	−0.458699	+	0.794489i	0.540194	+	0.841541i	$0.318351\pi$
$350$	−2.71428e8			−0.338389
$351$		0			0
$352$	2.70657e8			0.330765
$353$	2.93456e8	−	5.08281e8i	0.355084	−	0.615024i	−0.632048	−	0.774929i	$-0.717786\pi$
$353$	2.93456e8	−	5.08281e8i	0.355084	−	0.615024i	0.987132	+	0.159905i	$0.0511188\pi$
$354$		0			0
$355$	−4.96403e8	−	8.59795e8i	−0.588891	−	1.01999i
$356$	−3.28636e7	−	5.69214e7i	−0.0386047	−	0.0668653i
$357$		0			0
$358$	−1.58396e8	+	2.74351e8i	−0.182455	+	0.316021i
$359$	1.19985e9			1.36866			0.684330	−	0.729173i	$-0.260095\pi$
$359$	1.19985e9			1.36866			0.684330	+	0.729173i	$0.260095\pi$
$360$		0			0
$361$	1.59845e8			0.178823
$362$	−1.55263e7	+	2.68923e7i	−0.0172023	+	0.0297953i
$363$		0			0
$364$	4.84709e7	+	8.39540e7i	0.0526776	+	0.0912403i
$365$	4.71625e8	+	8.16878e8i	0.507659	+	0.879290i
$366$		0			0
$367$	6.57753e8	−	1.13926e9i	0.694595	−	1.20307i	−0.275722	−	0.961237i	$-0.588917\pi$
$367$	6.57753e8	−	1.13926e9i	0.694595	−	1.20307i	0.970317	−	0.241836i	$-0.0777497\pi$
$368$	1.10628e9			1.15718
$369$		0			0
$370$	1.98720e9			2.03956
$371$	−2.86828e8	+	4.96801e8i	−0.291617	+	0.505096i
$372$		0			0
$373$	5.38626e8	+	9.32928e8i	0.537411	+	0.930823i	0.999042	+	0.0437512i	$0.0139309\pi$
$373$	5.38626e8	+	9.32928e8i	0.537411	+	0.930823i	−0.461632	+	0.887072i	$0.652736\pi$
$374$	−6.16053e8	−	1.06703e9i	−0.608930	−	1.05470i
$375$		0			0
$376$	−6.38737e8	+	1.10632e9i	−0.619676	+	1.07331i
$377$	−1.36826e9			−1.31514
$378$		0			0
$379$	7.73954e8			0.730261			0.365130	−	0.930956i	$-0.381024\pi$
$379$	7.73954e8			0.730261			0.365130	+	0.930956i	$0.381024\pi$
$380$	1.14697e8	−	1.98661e8i	0.107228	−	0.185725i
$381$		0			0
$382$	−1.76372e8	−	3.05485e8i	−0.161885	−	0.280394i
$383$	−6.35193e8	−	1.10019e9i	−0.577710	−	1.00062i	−0.995741	−	0.0921911i	$-0.970613\pi$
$383$	−6.35193e8	−	1.10019e9i	−0.577710	−	1.00062i	0.418031	−	0.908433i	$-0.362720\pi$
$384$		0			0
$385$	−4.65951e8	+	8.07051e8i	−0.416129	+	0.720756i
$386$	−1.63413e9			−1.44620
$387$		0			0
$388$	5.95876e7			0.0517899
$389$	5.43188e8	−	9.40829e8i	0.467871	−	0.810377i	−0.531455	−	0.847087i	$-0.678354\pi$
$389$	5.43188e8	−	9.40829e8i	0.467871	−	0.810377i	0.999326	+	0.0367098i	$0.0116877\pi$
$390$		0			0
$391$	1.03544e9	+	1.79343e9i	0.876001	+	1.51728i
$392$	−3.93022e8	−	6.80735e8i	−0.329546	−	0.570790i
$393$		0			0
$394$	3.27395e8	−	5.67065e8i	0.269672	−	0.467085i
$395$	−3.89314e8			−0.317841
$396$		0			0
$397$	9.52195e7			0.0763764			0.0381882	−	0.999271i	$-0.487841\pi$
$397$	9.52195e7			0.0763764			0.0381882	+	0.999271i	$0.487841\pi$
$398$	−1.00190e9	+	1.73534e9i	−0.796586	+	1.37973i
$399$		0			0
$400$	3.13605e8	+	5.43179e8i	0.245004	+	0.424359i
$401$	7.21578e8	+	1.24981e9i	0.558828	+	0.967918i	0.997595	+	0.0693170i	$0.0220820\pi$
$401$	7.21578e8	+	1.24981e9i	0.558828	+	0.967918i	−0.438767	+	0.898601i	$0.644585\pi$
$402$		0			0
$403$	9.96697e8	−	1.72633e9i	0.758570	−	1.31388i
$404$	5.49479e7			0.0414587
$405$		0			0
$406$	−9.16688e8			−0.679799
$407$	1.27667e9	−	2.21125e9i	0.938635	−	1.62576i
$408$		0			0
$409$	−1.19362e9	−	2.06741e9i	−0.862650	−	1.49415i	−0.869361	−	0.494177i	$-0.835470\pi$
$409$	−1.19362e9	−	2.06741e9i	−0.862650	−	1.49415i	0.00671115	−	0.999977i	$-0.497864\pi$
$410$	6.49035e8	+	1.12416e9i	0.465076	+	0.805536i
$411$		0			0
$412$	−5.87109e7	+	1.01690e8i	−0.0413598	+	0.0716373i
$413$	4.38962e8			0.306621
$414$		0			0
$415$	−2.14796e9			−1.47522
$416$	2.48708e8	−	4.30775e8i	0.169381	−	0.293376i
$417$		0			0
$418$	7.95814e8	+	1.37839e9i	0.532960	+	0.923114i
$419$	−1.15268e9	−	1.99650e9i	−0.765527	−	1.32593i	−0.939968	−	0.341264i	$-0.889145\pi$
$419$	−1.15268e9	−	1.99650e9i	−0.765527	−	1.32593i	0.174441	−	0.984668i	$-0.444188\pi$
$420$		0			0
$421$	6.04688e8	−	1.04735e9i	0.394952	−	0.684077i	−0.598143	−	0.801389i	$-0.704095\pi$
$421$	6.04688e8	−	1.04735e9i	0.394952	−	0.684077i	0.993095	+	0.117313i	$0.0374280\pi$
$422$	−1.32819e9			−0.860332
$423$		0			0
$424$	1.57839e9			1.00562
$425$	−5.87042e8	+	1.01679e9i	−0.370943	+	0.642493i
$426$		0			0
$427$	−3.85076e7	−	6.66970e7i	−0.0239358	−	0.0414581i
$428$	5.84848e7	+	1.01299e8i	0.0360570	+	0.0624526i
$429$		0			0
$430$	−8.53946e8	+	1.47908e9i	−0.517953	+	0.897121i
$431$	2.87283e9			1.72838			0.864189	−	0.503167i	$-0.167832\pi$
$431$	2.87283e9			1.72838			0.864189	+	0.503167i	$0.167832\pi$
$432$		0			0
$433$	−1.79302e9			−1.06140			−0.530699	−	0.847560i	$-0.678070\pi$
$433$	−1.79302e9			−1.06140			−0.530699	+	0.847560i	$0.678070\pi$
$434$	6.67756e8	−	1.15659e9i	0.392106	−	0.679148i
$435$		0			0
$436$	8.94694e7	+	1.54966e8i	0.0516978	+	0.0895431i
$437$	−1.33757e9	−	2.31674e9i	−0.766711	−	1.32798i
$438$		0			0
$439$	1.55122e8	−	2.68679e8i	0.0875081	−	0.151568i	−0.818949	−	0.573866i	$-0.805443\pi$
$439$	1.55122e8	−	2.68679e8i	0.0875081	−	0.151568i	0.906457	+	0.422298i	$0.138776\pi$
$440$	2.56408e9			1.43499
$441$		0			0
$442$	−2.26438e9			−1.24730
$443$	3.06327e8	−	5.30574e8i	0.167406	−	0.289957i	−0.770101	−	0.637922i	$-0.779794\pi$
$443$	3.06327e8	−	5.30574e8i	0.167406	−	0.289957i	0.937507	+	0.347966i	$0.113127\pi$
$444$		0			0
$445$	−5.80598e8	−	1.00563e9i	−0.312331	−	0.540974i
$446$	3.45883e8	+	5.99087e8i	0.184611	+	0.319756i
$447$		0			0
$448$	6.46884e8	−	1.12044e9i	0.339902	−	0.588727i
$449$	−8.72232e8			−0.454747			−0.227374	−	0.973808i	$-0.573014\pi$
$449$	−8.72232e8			−0.454747			−0.227374	+	0.973808i	$0.573014\pi$
$450$		0			0
$451$	1.66787e9			0.856140
$452$	−1.30666e8	+	2.26320e8i	−0.0665546	+	0.115276i
$453$		0			0
$454$	−7.60792e8	−	1.31773e9i	−0.381567	−	0.660893i
$455$	8.56331e8	+	1.48321e9i	0.426189	+	0.738180i
$456$		0			0
$457$	−1.95028e9	+	3.37798e9i	−0.955851	+	1.65558i	−0.223443	+	0.974717i	$0.571730\pi$
$457$	−1.95028e9	+	3.37798e9i	−0.955851	+	1.65558i	−0.732408	+	0.680866i	$0.761604\pi$
$458$	−2.65760e9			−1.29259
$459$		0			0
$460$	−5.82379e8			−0.278967
$461$	−1.05420e9	+	1.82594e9i	−0.501155	+	0.868025i	0.498844	+	0.866692i	$0.333758\pi$
$461$	−1.05420e9	+	1.82594e9i	−0.501155	+	0.868025i	−0.999999	+	0.00133382i	$0.999575\pi$
$462$		0			0
$463$	−7.21743e8	−	1.25010e9i	−0.337948	−	0.585342i	0.646099	−	0.763254i	$-0.276399\pi$
$463$	−7.21743e8	−	1.25010e9i	−0.337948	−	0.585342i	−0.984047	+	0.177911i	$0.943066\pi$
$464$	1.05913e9	+	1.83447e9i	0.492195	+	0.852507i
$465$		0			0
$466$	−2.86804e8	+	4.96758e8i	−0.131291	+	0.227402i
$467$	−2.03696e9			−0.925493			−0.462746	−	0.886491i	$-0.653136\pi$
$467$	−2.03696e9			−0.925493			−0.462746	+	0.886491i	$0.653136\pi$
$468$		0			0
$469$	1.75549e8			0.0785766
$470$	−1.52493e9	+	2.64126e9i	−0.677499	+	1.17346i
$471$		0			0
$472$	−6.03892e8	−	1.04597e9i	−0.264339	−	0.457849i
$473$	1.09722e9	+	1.90045e9i	0.476740	+	0.825737i
$474$		0			0
$475$	7.58338e8	−	1.31348e9i	0.324665	−	0.562336i
$476$	2.80938e8			0.119395
$477$		0			0
$478$	−1.89753e9			−0.794676
$479$	3.50416e8	−	6.06938e8i	0.145683	−	0.252331i	−0.783944	−	0.620831i	$-0.786795\pi$
$479$	3.50416e8	−	6.06938e8i	0.145683	−	0.252331i	0.929628	+	0.368500i	$0.120129\pi$
$480$		0			0
$481$	−2.34627e9	−	4.06386e9i	−0.961326	−	1.66507i
$482$	−8.96040e7	−	1.55199e8i	−0.0364471	−	0.0631282i
$483$		0			0
$484$	2.77336e7	−	4.80360e7i	0.0111185	−	0.0192578i
$485$	1.05273e9			0.419006
$486$		0			0
$487$	−2.15479e8			−0.0845384			−0.0422692	−	0.999106i	$-0.513459\pi$
$487$	−2.15479e8			−0.0845384			−0.0422692	+	0.999106i	$0.513459\pi$
$488$	−1.05952e8	+	1.83514e8i	−0.0412704	+	0.0714824i
$489$		0			0
$490$	−9.38310e8	−	1.62520e9i	−0.360296	−	0.624052i
$491$	5.78546e8	+	1.00207e9i	0.220573	+	0.382044i	0.954982	−	0.296663i	$-0.0958739\pi$
$491$	5.78546e8	+	1.00207e9i	0.220573	+	0.382044i	−0.734409	+	0.678707i	$0.762541\pi$
$492$		0			0
$493$	−1.98261e9	+	3.43397e9i	−0.745199	+	1.29072i
$494$	2.92512e9			1.09169
$495$		0			0
$496$	−3.08607e9			−1.13559
$497$	−7.85336e8	+	1.36024e9i	−0.286951	+	0.497014i
$498$		0			0
$499$	1.62240e9	+	2.81008e9i	0.584529	+	1.01243i	0.994934	+	0.100530i	$0.0320539\pi$
$499$	1.62240e9	+	2.81008e9i	0.584529	+	1.01243i	−0.410405	+	0.911903i	$0.634613\pi$
$500$	1.10955e8	+	1.92180e8i	0.0396966	+	0.0687565i
$501$		0			0
$502$	2.53748e8	−	4.39504e8i	0.0895239	−	0.155060i
$503$	−1.71289e9			−0.600125			−0.300062	−	0.953920i	$-0.597008\pi$
$503$	−1.71289e9			−0.600125			−0.300062	+	0.953920i	$0.597008\pi$
$504$		0			0
$505$	9.70759e8			0.335422
$506$	2.02039e9	−	3.49941e9i	0.693279	−	1.20079i
$507$		0			0
$508$	1.96731e8	+	3.40748e8i	0.0665809	+	0.115321i
$509$	1.13724e8	+	1.96975e8i	0.0382242	+	0.0662063i	0.884505	−	0.466531i	$-0.154497\pi$
$509$	1.13724e8	+	1.96975e8i	0.0382242	+	0.0662063i	−0.846280	+	0.532738i	$0.821163\pi$
$510$		0			0
$511$	7.46136e8	−	1.29234e9i	0.247369	−	0.428455i
$512$	−3.41288e9			−1.12377
$513$		0			0
$514$	8.53649e8			0.277273
$515$	−1.03724e9	+	1.79655e9i	−0.334622	+	0.579582i
$516$		0			0
$517$	1.95937e9	+	3.39372e9i	0.623590	+	1.08009i
$518$	−1.57193e9	−	2.72266e9i	−0.496911	−	0.860676i
$519$		0			0
$520$	2.35615e9	−	4.08098e9i	0.734838	−	1.27278i
$521$	4.19551e9			1.29973			0.649864	−	0.760050i	$-0.274826\pi$
$521$	4.19551e9			1.29973			0.649864	+	0.760050i	$0.274826\pi$
$522$		0			0
$523$	1.77394e9			0.542228			0.271114	−	0.962547i	$-0.412608\pi$
$523$	1.77394e9			0.542228			0.271114	+	0.962547i	$0.412608\pi$
$524$	8.56750e7	−	1.48393e8i	0.0260132	−	0.0450563i
$525$		0			0
$526$	1.07881e9	+	1.86855e9i	0.323217	+	0.559828i
$527$	−2.88843e9	−	5.00291e9i	−0.859656	−	1.48897i
$528$		0			0
$529$	−1.69337e9	+	2.93301e9i	−0.497345	+	0.861426i
$530$	3.76828e9			1.09945
$531$		0			0
$532$	−3.62914e8			−0.104499
$533$	1.53262e9	−	2.65457e9i	0.438419	−	0.759363i
$534$		0			0
$535$	1.03325e9	+	1.78963e9i	0.291719	+	0.505273i
$536$	−2.41507e8	−	4.18303e8i	−0.0677412	−	0.117331i
$537$		0			0
$538$	1.77737e9	−	3.07849e9i	0.492084	−	0.852314i
$539$	−2.41124e9			−0.663255
$540$		0			0
$541$	−3.74560e9			−1.01702			−0.508511	−	0.861055i	$-0.669804\pi$
$541$	−3.74560e9			−1.01702			−0.508511	+	0.861055i	$0.669804\pi$
$542$	−3.50239e8	+	6.06632e8i	−0.0944859	+	0.163654i
$543$		0			0
$544$	−7.20757e8	−	1.24839e9i	−0.191952	−	0.332471i
$545$	1.58065e9	+	2.73776e9i	0.418261	+	0.724449i
$546$		0			0
$547$	1.07176e9	−	1.85635e9i	0.279990	−	0.484957i	−0.691392	−	0.722480i	$-0.743002\pi$
$547$	1.07176e9	−	1.85635e9i	0.279990	−	0.484957i	0.971382	+	0.237523i	$0.0763354\pi$
$548$	−1.11228e9			−0.288723
$549$		0			0
$550$	2.29092e9			0.587139
$551$	2.56112e9	−	4.43599e9i	0.652228	−	1.12969i
$552$		0			0
$553$	3.07957e8	+	5.33398e8i	0.0774377	+	0.134126i
$554$	−1.08128e9	−	1.87283e9i	−0.270180	−	0.467966i
$555$		0			0
$556$	2.77989e8	−	4.81491e8i	0.0685908	−	0.118803i
$557$	−3.54423e9			−0.869019			−0.434509	−	0.900667i	$-0.643078\pi$
$557$	−3.54423e9			−0.869019			−0.434509	+	0.900667i	$0.643078\pi$
$558$		0			0
$559$	4.03299e9			0.976530
$560$	1.32573e9	−	2.29623e9i	0.319004	−	0.552532i
$561$		0			0
$562$	1.32351e9	+	2.29238e9i	0.314521	+	0.544766i
$563$	6.76819e8	+	1.17229e9i	0.159843	+	0.276856i	0.934812	−	0.355143i	$-0.115568\pi$
$563$	6.76819e8	+	1.17229e9i	0.159843	+	0.276856i	−0.774969	+	0.631999i	$0.782235\pi$
$564$		0			0
$565$	−2.30847e9	+	3.99838e9i	−0.538460	+	0.932641i
$566$	1.89219e9			0.438638
$567$		0			0
$568$	4.32163e9			0.989528
$569$	−5.55537e8	+	9.62219e8i	−0.126421	+	0.218968i	−0.922288	−	0.386504i	$-0.873682\pi$
$569$	−5.55537e8	+	9.62219e8i	−0.126421	+	0.218968i	0.795866	+	0.605472i	$0.207016\pi$
$570$		0			0
$571$	9.28825e8	+	1.60877e9i	0.208789	+	0.361633i	0.951333	−	0.308164i	$-0.0997145\pi$
$571$	9.28825e8	+	1.60877e9i	0.208789	+	0.361633i	−0.742544	+	0.669797i	$0.766381\pi$
$572$	−4.09107e8	−	7.08594e8i	−0.0914010	−	0.158311i
$573$		0			0
$574$	1.02681e9	−	1.77848e9i	0.226619	−	0.392516i
$575$	−3.85049e9			−0.844653
$576$		0			0
$577$	3.26056e9			0.706606			0.353303	−	0.935509i	$-0.385059\pi$
$577$	3.26056e9			0.706606			0.353303	+	0.935509i	$0.385059\pi$
$578$	−1.14891e9	+	1.98997e9i	−0.247479	+	0.428646i
$579$		0			0
$580$	−5.57556e8	−	9.65716e8i	−0.118656	−	0.205519i
$581$	1.69909e9	+	2.94291e9i	0.359418	+	0.622531i
$582$		0			0
$583$	2.42091e9	−	4.19313e9i	0.505985	−	0.876392i
$584$	−4.10591e9			−0.853031
$585$		0			0
$586$	5.65675e9			1.16125
$587$	−4.24886e9	+	7.35925e9i	−0.867041	+	1.50176i	−0.00203426	+	0.999998i	$0.500648\pi$
$587$	−4.24886e9	+	7.35925e9i	−0.867041	+	1.50176i	−0.865006	+	0.501761i	$0.832686\pi$
$588$		0			0
$589$	3.73126e9	+	6.46274e9i	0.752406	+	1.30321i
$590$	−1.44174e9	−	2.49717e9i	−0.289005	−	0.500572i
$591$		0			0
$592$	−3.63238e9	+	6.29147e9i	−0.719557	+	1.24631i
$593$	4.65379e9			0.916464			0.458232	−	0.888832i	$-0.348483\pi$
$593$	4.65379e9			0.916464			0.458232	+	0.888832i	$0.348483\pi$
$594$		0			0
$595$	4.96330e9			0.965965
$596$	−7.78371e8	+	1.34818e9i	−0.150600	+	0.260847i
$597$		0			0
$598$	−3.71310e9	−	6.43127e9i	−0.710039	−	1.22982i
$599$	5.97413e8	+	1.03475e9i	0.113575	+	0.196717i	0.917209	−	0.398406i	$-0.130437\pi$
$599$	5.97413e8	+	1.03475e9i	0.113575	+	0.196717i	−0.803635	+	0.595123i	$0.797103\pi$
$600$		0			0
$601$	−3.59466e9	+	6.22614e9i	−0.675457	+	1.16993i	0.300879	+	0.953662i	$0.402720\pi$
$601$	−3.59466e9	+	6.22614e9i	−0.675457	+	1.16993i	−0.976335	+	0.216263i	$0.930613\pi$
$602$	2.70197e9			0.504770
$603$		0			0
$604$	−2.09264e8			−0.0386426
$605$	4.89967e8	−	8.48647e8i	0.0899545	−	0.155806i
$606$		0			0
$607$	1.86704e8	+	3.23380e8i	0.0338838	+	0.0586885i	0.882470	−	0.470369i	$-0.155879\pi$
$607$	1.86704e8	+	3.23380e8i	0.0338838	+	0.0586885i	−0.848586	+	0.529057i	$0.822546\pi$
$608$	9.31071e8	+	1.61266e9i	0.168004	+	0.290992i
$609$		0			0
$610$	−2.52951e8	+	4.38124e8i	−0.0451214	+	0.0781525i
$611$	7.20191e9			1.27733
$612$		0			0
$613$	−2.57761e9			−0.451965			−0.225982	−	0.974131i	$-0.572559\pi$
$613$	−2.57761e9			−0.451965			−0.225982	+	0.974131i	$0.572559\pi$
$614$	2.77768e9	−	4.81108e9i	0.484275	−	0.838790i
$615$		0			0
$616$	−2.02826e9	−	3.51305e9i	−0.349615	−	0.605552i
$617$	−4.29290e9	−	7.43552e9i	−0.735788	−	1.27442i	−0.954377	−	0.298605i	$-0.903479\pi$
$617$	−4.29290e9	−	7.43552e9i	−0.735788	−	1.27442i	0.218589	−	0.975817i	$-0.429855\pi$
$618$		0			0
$619$	3.95632e9	−	6.85255e9i	0.670462	−	1.16127i	−0.307311	−	0.951609i	$-0.599429\pi$
$619$	3.95632e9	−	6.85255e9i	0.670462	−	1.16127i	0.977773	−	0.209666i	$-0.0672375\pi$
$620$	1.62459e9			0.273762
$621$		0			0
$622$	−4.15367e9			−0.692095
$623$	−9.18537e8	+	1.59095e9i	−0.152191	+	0.263602i
$624$		0			0
$625$	3.78536e9	+	6.55643e9i	0.620193	+	1.07421i
$626$	4.74543e9	+	8.21932e9i	0.773153	+	1.33914i
$627$		0			0
$628$	−6.57814e8	+	1.13937e9i	−0.105985	+	0.183571i
$629$	−1.35990e10			−2.17886
$630$		0			0
$631$	1.12326e9			0.177983			0.0889916	−	0.996032i	$-0.471636\pi$
$631$	1.12326e9			0.177983			0.0889916	+	0.996032i	$0.471636\pi$
$632$	8.47329e8	−	1.46762e9i	0.133519	−	0.231261i
$633$		0			0
$634$	−2.13383e9	−	3.69591e9i	−0.332543	−	0.575982i
$635$	3.47563e9	+	6.01996e9i	0.538673	+	0.933009i
$636$		0			0
$637$	−2.21571e9	+	3.83772e9i	−0.339645	+	0.588282i
$638$	7.73709e9			1.17952
$639$		0			0
$640$	−5.90411e9			−0.890275
$641$	−2.30695e9	+	3.99575e9i	−0.345967	+	0.599232i	−0.985529	−	0.169507i	$-0.945782\pi$
$641$	−2.30695e9	+	3.99575e9i	−0.345967	+	0.599232i	0.639562	+	0.768739i	$0.279116\pi$
$642$		0			0
$643$	1.41163e8	+	2.44501e8i	0.0209402	+	0.0362696i	0.876306	−	0.481756i	$-0.160001\pi$
$643$	1.41163e8	+	2.44501e8i	0.0209402	+	0.0362696i	−0.855365	+	0.518025i	$0.826667\pi$
$644$	4.60677e8	+	7.97917e8i	0.0679667	+	0.117722i
$645$		0			0
$646$	4.23850e9	−	7.34129e9i	0.618583	−	1.07142i
$647$	1.09978e10			1.59639			0.798197	−	0.602396i	$-0.205787\pi$
$647$	1.09978e10			1.59639			0.798197	+	0.602396i	$0.205787\pi$
$648$		0			0
$649$	−3.70496e9			−0.532018
$650$	2.10514e9	−	3.64622e9i	0.300666	−	0.520770i
$651$		0			0
$652$	−3.00965e8	−	5.21287e8i	−0.0425256	−	0.0736564i
$653$	−7.76089e8	−	1.34423e9i	−0.109073	−	0.188919i	0.806322	−	0.591476i	$-0.201455\pi$
$653$	−7.76089e8	−	1.34423e9i	−0.109073	−	0.188919i	−0.915395	+	0.402557i	$0.868121\pi$
$654$		0			0
$655$	1.51361e9	−	2.62165e9i	0.210460	−	0.364528i
$656$	−4.74545e9			−0.656317
$657$		0			0
$658$	4.82505e9			0.660254
$659$	−4.10032e9	+	7.10196e9i	−0.558108	+	0.966672i	0.439546	+	0.898220i	$0.355139\pi$
$659$	−4.10032e9	+	7.10196e9i	−0.558108	+	0.966672i	−0.997654	+	0.0684519i	$0.978194\pi$
$660$		0			0
$661$	4.11606e8	+	7.12922e8i	0.0554340	+	0.0960145i	0.892411	−	0.451224i	$-0.149012\pi$
$661$	4.11606e8	+	7.12922e8i	0.0554340	+	0.0960145i	−0.836977	+	0.547238i	$0.815679\pi$
$662$	4.83110e9	+	8.36770e9i	0.647206	+	1.12099i
$663$		0			0
$664$	4.67497e9	−	8.09728e9i	0.619713	−	1.07337i
$665$	−6.41157e9			−0.845452
$666$		0			0
$667$	−1.30042e10			−1.69685
$668$	7.90997e8	−	1.37005e9i	0.102673	−	0.177835i
$669$		0			0
$670$	−5.76579e8	−	9.98665e8i	−0.0740623	−	0.128280i
$671$	3.25014e8	+	5.62940e8i	0.0415311	+	0.0719339i
$672$		0			0
$673$	−7.33607e8	+	1.27064e9i	−0.0927707	+	0.160684i	−0.908676	−	0.417502i	$-0.862906\pi$
$673$	−7.33607e8	+	1.27064e9i	−0.0927707	+	0.160684i	0.815905	+	0.578186i	$0.196239\pi$
$674$	−5.96440e9			−0.750339
$675$		0			0
$676$	−2.48754e8			−0.0309712
$677$	1.74584e9	−	3.02388e9i	0.216244	−	0.374545i	−0.737413	−	0.675442i	$-0.763953\pi$
$677$	1.74584e9	−	3.02388e9i	0.216244	−	0.374545i	0.953657	+	0.300897i	$0.0972861\pi$
$678$		0			0
$679$	−8.32736e8	−	1.44234e9i	−0.102085	−	0.176817i
$680$	−6.82814e9	−	1.18267e10i	−0.832763	−	1.44239i
$681$		0			0
$682$	−5.63603e9	+	9.76189e9i	−0.680343	+	1.17839i
$683$	−1.51044e10			−1.81398			−0.906988	−	0.421156i	$-0.861624\pi$
$683$	−1.51044e10			−1.81398			−0.906988	+	0.421156i	$0.861624\pi$
$684$		0			0
$685$	−1.96505e10			−2.33591
$686$	−3.87656e9	+	6.71440e9i	−0.458471	+	0.794095i
$687$		0			0
$688$	−3.12183e9	−	5.40717e9i	−0.365469	−	0.633010i
$689$	−4.44917e9	−	7.70620e9i	−0.518218	−	0.897579i
$690$		0			0
$691$	−6.31402e9	+	1.09362e10i	−0.728003	+	1.26094i	0.229723	+	0.973256i	$0.426218\pi$
$691$	−6.31402e9	+	1.09362e10i	−0.728003	+	1.26094i	−0.957726	+	0.287682i	$0.907115\pi$
$692$	5.62448e8			0.0645225
$693$		0			0
$694$	−1.13583e10			−1.28990
$695$	4.91121e9	−	8.50647e9i	0.554935	−	0.961175i
$696$		0			0
$697$	−4.44154e9	−	7.69297e9i	−0.496842	−	0.860556i
$698$	−3.78554e9	−	6.55675e9i	−0.421342	−	0.729785i
$699$		0			0
$700$	−2.61182e8	+	4.52380e8i	−0.0287806	+	0.0498494i
$701$	1.52554e10			1.67267			0.836336	−	0.548217i	$-0.184693\pi$
$701$	1.52554e10			1.67267			0.836336	+	0.548217i	$0.184693\pi$
$702$		0			0
$703$	1.75671e10			1.90703
$704$	−5.45987e9	+	9.45677e9i	−0.589763	+	1.02150i
$705$		0			0
$706$	3.04968e9	+	5.28221e9i	0.326166	+	0.564936i
$707$	−7.67896e8	−	1.33004e9i	−0.0817212	−	0.141545i
$708$		0			0
$709$	8.79991e9	−	1.52419e10i	0.927292	−	1.60612i	0.139459	−	0.990228i	$-0.455464\pi$
$709$	8.79991e9	−	1.52419e10i	0.927292	−	1.60612i	0.787833	−	0.615889i	$-0.211203\pi$
$710$	1.03175e10			1.08186
$711$		0			0
$712$	5.05462e9			0.524818
$713$	9.47281e9	−	1.64074e10i	0.978736	−	1.69522i
$714$		0			0
$715$	−7.22766e9	−	1.25187e10i	−0.739480	−	1.28082i
$716$	3.04834e8	+	5.27988e8i	0.0310362	+	0.0537562i
$717$		0			0
$718$	−6.23459e9	+	1.07986e10i	−0.628597	+	1.08876i
$719$	1.01872e10			1.02212			0.511061	−	0.859544i	$-0.329253\pi$
$719$	1.01872e10			1.02212			0.511061	+	0.859544i	$0.329253\pi$
$720$		0			0
$721$	3.28194e9			0.326105
$722$	−8.30578e8	+	1.43860e9i	−0.0821297	+	0.142253i
$723$		0			0
$724$	2.98803e7	+	5.17542e7i	0.00292617	+	0.00506828i
$725$	−3.68637e9	−	6.38498e9i	−0.359266	−	0.622266i
$726$		0			0
$727$	−3.68253e8	+	6.37834e8i	−0.0355448	+	0.0615654i	−0.883251	−	0.468901i	$-0.844650\pi$
$727$	−3.68253e8	+	6.37834e8i	−0.0355448	+	0.0615654i	0.847706	+	0.530467i	$0.177983\pi$
$728$	−7.45512e9			−0.716135
$729$		0			0
$730$	−9.80254e9			−0.932628
$731$	5.84381e9	−	1.01218e10i	0.553331	−	0.958397i
$732$		0			0
$733$	−8.27019e8	−	1.43244e9i	−0.0775625	−	0.134342i	0.824635	−	0.565665i	$-0.191380\pi$
$733$	−8.27019e8	−	1.43244e9i	−0.0775625	−	0.134342i	−0.902198	+	0.431323i	$0.858047\pi$
$734$	6.83557e9	+	1.18396e10i	0.638026	+	1.10509i
$735$		0			0
$736$	2.36377e9	−	4.09418e9i	0.218541	−	0.378525i
$737$	−1.48168e9			−0.136338
$738$		0			0
$739$	1.56333e10			1.42493			0.712467	−	0.701706i	$-0.247578\pi$
$739$	1.56333e10			1.42493			0.712467	+	0.701706i	$0.247578\pi$
$740$	1.91219e9	−	3.31200e9i	0.173468	−	0.300455i
$741$		0			0
$742$	−2.98081e9	−	5.16291e9i	−0.267868	−	0.463960i
$743$	5.75316e9	+	9.96477e9i	0.514571	+	0.891264i	0.999857	+	0.0169082i	$0.00538230\pi$
$743$	5.75316e9	+	9.96477e9i	0.514571	+	0.891264i	−0.485286	+	0.874356i	$0.661284\pi$
$744$		0			0
$745$	−1.37514e10	+	2.38181e10i	−1.21843	+	2.11038i
$746$	−1.11951e10			−0.987287
$747$		0			0
$748$	−2.37119e9			−0.207162
$749$	1.63465e9	−	2.83130e9i	0.142147	−	0.246206i
$750$		0			0
$751$	−5.60814e9	−	9.71359e9i	−0.483147	−	0.836835i	0.516666	−	0.856187i	$-0.327173\pi$
$751$	−5.60814e9	−	9.71359e9i	−0.483147	−	0.836835i	−0.999813	+	0.0193522i	$0.993840\pi$
$752$	−5.57481e9	−	9.65586e9i	−0.478044	−	0.827996i
$753$		0			0
$754$	7.10967e9	−	1.23143e10i	0.604017	−	1.04619i
$755$	−3.69706e9			−0.312638
$756$		0			0
$757$	−1.70722e10			−1.43039			−0.715194	−	0.698926i	$-0.753661\pi$
$757$	−1.70722e10			−1.43039			−0.715194	+	0.698926i	$0.753661\pi$
$758$	−4.02158e9	+	6.96559e9i	−0.335394	+	0.580919i
$759$		0			0
$760$	8.82056e9	+	1.52777e10i	0.728868	+	1.26244i
$761$	1.35383e9	+	2.34490e9i	0.111357	+	0.192876i	0.916318	−	0.400452i	$-0.131147\pi$
$761$	1.35383e9	+	2.34490e9i	0.111357	+	0.192876i	−0.804961	+	0.593328i	$0.797814\pi$
$762$		0			0
$763$	2.50067e9	−	4.33129e9i	0.203808	−	0.353005i
$764$	−6.78856e8			−0.0550745
$765$		0			0
$766$	1.32022e10			1.06132
$767$	−3.40451e9	+	5.89678e9i	−0.272440	+	0.471880i
$768$		0			0
$769$	4.10669e9	+	7.11300e9i	0.325649	+	0.564041i	0.981643	−	0.190725i	$-0.0610839\pi$
$769$	4.10669e9	+	7.11300e9i	0.325649	+	0.564041i	−0.655995	+	0.754766i	$0.727751\pi$
$770$	−4.84231e9	−	8.38712e9i	−0.382239	−	0.662057i
$771$		0			0
$772$	−1.57244e9	+	2.72354e9i	−0.123002	+	0.213046i
$773$	1.01337e10			0.789115			0.394557	−	0.918871i	$-0.370898\pi$
$773$	1.01337e10			0.789115			0.394557	+	0.918871i	$0.370898\pi$
$774$		0			0
$775$	1.07412e10			0.828893
$776$	−2.29123e9	+	3.96853e9i	−0.176016	+	0.304869i
$777$		0			0
$778$	5.64497e9	+	9.77738e9i	0.429767	+	0.744379i
$779$	5.73756e9	+	9.93774e9i	0.434856	+	0.753193i
$780$		0			0
$781$	6.62844e9	−	1.14808e10i	0.497889	−	0.862369i
$782$	−2.15211e10			−1.60932
$783$		0			0
$784$	6.86050e9			0.508451
$785$	−1.16215e10	+	2.01291e10i	−0.857473	+	1.48519i
$786$		0			0
$787$	3.95403e9	+	6.84858e9i	0.289153	+	0.500828i	0.973608	−	0.228227i	$-0.0732929\pi$
$787$	3.95403e9	+	6.84858e9i	0.289153	+	0.500828i	−0.684455	+	0.729055i	$0.739960\pi$
$788$	−6.30072e8	−	1.09132e9i	−0.0458721	−	0.0794528i
$789$		0			0
$790$	2.02293e9	−	3.50382e9i	0.145978	−	0.252841i
$791$	7.30423e9			0.524755
$792$		0			0
$793$	1.19463e9			0.0850701
$794$	−4.94775e8	+	8.56975e8i	−0.0350781	+	0.0607570i
$795$		0			0
$796$	1.92815e9	+	3.33966e9i	0.135502	+	0.234696i
$797$	6.84380e9	+	1.18538e10i	0.478843	+	0.829381i	0.999706	−	0.0242598i	$-0.00772290\pi$
$797$	6.84380e9	+	1.18538e10i	0.478843	+	0.829381i	−0.520862	+	0.853641i	$0.674390\pi$
$798$		0			0
$799$	1.04356e10	−	1.80749e10i	0.723773	−	1.25361i
$800$	2.68029e9			0.185083
$801$		0			0
$802$	−1.49977e10			−1.02663
$803$	−6.29758e9	+	1.09077e10i	−0.429209	+	0.743413i
$804$		0			0
$805$	8.13875e9	+	1.40967e10i	0.549885	+	0.952429i
$806$	1.03580e10	+	1.79405e10i	0.696791	+	1.20688i
$807$		0			0
$808$	−2.11283e9	+	3.65953e9i	−0.140904	+	0.244054i
$809$	9.65209e9			0.640917			0.320458	−	0.947263i	$-0.396163\pi$
$809$	9.65209e9			0.640917			0.320458	+	0.947263i	$0.396163\pi$
$810$		0			0
$811$	2.93565e10			1.93256			0.966278	−	0.257502i	$-0.0828994\pi$
$811$	2.93565e10			1.93256			0.966278	+	0.257502i	$0.0828994\pi$
$812$	−8.82084e8	+	1.52781e9i	−0.0578181	+	0.100144i
$813$		0			0
$814$	1.32675e10	+	2.29800e10i	0.862191	+	1.49336i
$815$	−5.31713e9	−	9.20954e9i	−0.344053	−	0.595918i
$816$		0			0
$817$	−7.54900e9	+	1.30753e10i	−0.484298	+	0.838828i
$818$	2.48089e10			1.58479
$819$		0			0
$820$	2.49814e9			0.158222
$821$	7.30438e8	−	1.26516e9i	0.0460662	−	0.0797889i	−0.842073	−	0.539364i	$-0.818665\pi$
$821$	7.30438e8	−	1.26516e9i	0.0460662	−	0.0797889i	0.888139	+	0.459575i	$0.151998\pi$
$822$		0			0
$823$	−1.14909e10	−	1.99029e10i	−0.718548	−	1.24456i	−0.961575	−	0.274542i	$-0.911474\pi$
$823$	−1.14909e10	−	1.99029e10i	−0.718548	−	1.24456i	0.243027	−	0.970019i	$-0.421860\pi$
$824$	−4.51505e9	−	7.82029e9i	−0.281136	−	0.486943i
$825$		0			0
$826$	−2.28091e9	+	3.95066e9i	−0.140825	+	0.243916i
$827$	−1.26270e10			−0.776300			−0.388150	−	0.921596i	$-0.626886\pi$
$827$	−1.26270e10			−0.776300			−0.388150	+	0.921596i	$0.626886\pi$
$828$		0			0
$829$	−1.63249e10			−0.995198			−0.497599	−	0.867407i	$-0.665785\pi$
$829$	−1.63249e10			−0.995198			−0.497599	+	0.867407i	$0.665785\pi$
$830$	1.11611e10	−	1.93316e10i	0.677539	−	1.17353i
$831$		0			0
$832$	1.00342e10	+	1.73798e10i	0.604021	+	1.04619i
$833$	6.42113e9	+	1.11217e10i	0.384906	+	0.666676i
$834$		0			0
$835$	1.39745e10	−	2.42045e10i	0.830679	−	1.43878i
$836$	3.06309e9			0.181317
$837$		0			0
$838$	2.39580e10			1.40636
$839$	5.83269e9	−	1.01025e10i	0.340959	−	0.590558i	−0.643652	−	0.765318i	$-0.722582\pi$
$839$	5.83269e9	−	1.01025e10i	0.340959	−	0.590558i	0.984611	+	0.174760i	$0.0559150\pi$
$840$		0			0
$841$	−3.82497e9	−	6.62504e9i	−0.221739	−	0.384063i
$842$	6.28410e9	+	1.08844e10i	0.362786	+	0.628364i
$843$		0			0
$844$	−1.27805e9	+	2.21365e9i	−0.0731727	+	0.126739i
$845$	−4.39473e9			−0.250572
$846$		0			0
$847$	−1.55031e9			−0.0876649
$848$	−6.88798e9	+	1.19303e10i	−0.387888	+	0.671842i
$849$		0			0
$850$	−6.10072e9	−	1.05667e10i	−0.340733	−	0.590167i
$851$	−2.22995e10	−	3.86238e10i	−1.24034	−	2.14833i
$852$		0			0
$853$	6.13508e8	−	1.06263e9i	0.0338453	−	0.0586218i	−0.848607	−	0.529024i	$-0.822558\pi$
$853$	6.13508e8	−	1.06263e9i	0.0338453	−	0.0586218i	0.882452	+	0.470403i	$0.155891\pi$
$854$	8.00364e8			0.0439729
$855$		0			0
$856$	−8.99532e9			−0.490183
$857$	1.11466e10	−	1.93064e10i	0.604935	−	1.04778i	−0.387127	−	0.922027i	$-0.626532\pi$
$857$	1.11466e10	−	1.93064e10i	0.604935	−	1.04778i	0.992062	−	0.125752i	$-0.0401343\pi$
$858$		0			0
$859$	−5.56626e9	−	9.64105e9i	−0.299632	−	0.518977i	0.676420	−	0.736516i	$-0.263530\pi$
$859$	−5.56626e9	−	9.64105e9i	−0.299632	−	0.518977i	−0.976052	+	0.217539i	$0.930197\pi$
$860$	1.64342e9	+	2.84649e9i	0.0881057	+	0.152603i
$861$		0			0
$862$	−1.49276e10	+	2.58554e10i	−0.793809	+	1.37492i
$863$	−1.07748e10			−0.570653			−0.285327	−	0.958430i	$-0.592102\pi$
$863$	−1.07748e10			−0.570653			−0.285327	+	0.958430i	$0.592102\pi$
$864$		0			0
$865$	9.93673e9			0.522020
$866$	9.31682e9	−	1.61372e10i	0.487478	−	0.844337i
$867$		0			0
$868$	−1.28510e9	−	2.22585e9i	−0.0666986	−	0.115525i
$869$	−2.59924e9	−	4.50202e9i	−0.134362	−	0.232722i
$870$		0			0
$871$	−1.36152e9	+	2.35823e9i	−0.0698171	+	0.120927i
$872$	−1.37609e10			−0.702814
$873$		0			0
$874$	2.78009e10			1.40854
$875$	3.10120e9	−	5.37144e9i	0.156495	−	0.271058i
$876$		0			0
$877$	2.15943e9	+	3.74024e9i	0.108104	+	0.187241i	0.915002	−	0.403449i	$-0.132189\pi$
$877$	2.15943e9	+	3.74024e9i	0.108104	+	0.187241i	−0.806898	+	0.590690i	$0.798855\pi$
$878$	1.61208e9	+	2.79220e9i	0.0803813	+	0.139224i
$879$		0			0
$880$	−1.11895e10	+	1.93808e10i	−0.553504	+	0.958698i
$881$	−2.49223e10			−1.22793			−0.613963	−	0.789335i	$-0.710426\pi$
$881$	−2.49223e10			−1.22793			−0.613963	+	0.789335i	$0.710426\pi$
$882$		0			0
$883$	8.75421e9			0.427912			0.213956	−	0.976843i	$-0.431365\pi$
$883$	8.75421e9			0.427912			0.213956	+	0.976843i	$0.431365\pi$
$884$	−2.17890e9	+	3.77397e9i	−0.106085	+	0.183745i
$885$		0			0
$886$	3.18344e9	+	5.51389e9i	0.153773	+	0.266342i
$887$	3.59185e8	+	6.22127e8i	0.0172817	+	0.0299327i	0.874537	−	0.484959i	$-0.161165\pi$
$887$	3.59185e8	+	6.22127e8i	0.0172817	+	0.0299327i	−0.857255	+	0.514892i	$0.827832\pi$
$888$		0			0
$889$	5.49863e9	−	9.52390e9i	0.262481	−	0.454631i
$890$	1.20675e10			0.573789
$891$		0			0
$892$	1.33131e9			0.0628059
$893$	−1.34806e10	+	2.33491e10i	−0.633476	+	1.09721i
$894$		0			0
$895$	5.38548e9	+	9.32793e9i	0.251098	+	0.434915i
$896$	4.67030e9	+	8.08920e9i	0.216904	+	0.375688i
$897$		0			0
$898$	4.53225e9	−	7.85009e9i	0.208856	−	0.361749i
$899$	3.62762e10			1.66519
$900$		0			0
$901$	−2.57874e10			−1.17455
$902$	−8.66652e9	+	1.50109e10i	−0.393208	+	0.681055i
$903$		0			0
$904$	−1.00486e10	−	1.74047e10i	−0.452394	−	0.783569i
$905$	5.27893e8	+	9.14337e8i	0.0236742	+	0.0410049i
$906$		0			0
$907$	−1.01488e10	+	1.75782e10i	−0.451635	+	0.782255i	−0.998488	−	0.0549737i	$-0.982493\pi$
$907$	−1.01488e10	+	1.75782e10i	−0.451635	+	0.782255i	0.546852	+	0.837229i	$0.315826\pi$
$908$	−2.92829e9			−0.129812
$909$		0			0
$910$	−1.77985e10			−0.782959
$911$	−1.45808e10	+	2.52547e10i	−0.638951	+	1.10670i	0.346712	+	0.937972i	$0.387298\pi$
$911$	−1.45808e10	+	2.52547e10i	−0.638951	+	1.10670i	−0.985663	+	0.168725i	$0.946035\pi$
$912$		0			0
$913$	−1.43408e10	−	2.48390e10i	−0.623627	−	1.08015i
$914$	−2.02679e10	−	3.51050e10i	−0.878005	−	1.52075i
$915$		0			0
$916$	−2.55728e9	+	4.42934e9i	−0.109937	+	0.190417i
$917$	−4.78923e9			−0.205104
$918$		0			0
$919$	−2.07384e10			−0.881394			−0.440697	−	0.897656i	$-0.645269\pi$
$919$	−2.07384e10			−0.881394			−0.440697	+	0.897656i	$0.645269\pi$
$920$	2.23934e10	−	3.87865e10i	0.948117	−	1.64219i
$921$		0			0
$922$	−1.09556e10	−	1.89757e10i	−0.460340	−	0.797332i
$923$	−1.21818e10	−	2.10996e10i	−0.509926	−	0.883217i
$924$		0			0
$925$	1.26427e10	−	2.18978e10i	0.525223	−	0.909713i
$926$	1.50012e10			0.620849
$927$		0			0
$928$	9.05209e9			0.371819
$929$	4.13088e9	−	7.15489e9i	0.169039	−	0.292784i	−0.769043	−	0.639197i	$-0.779267\pi$
$929$	4.13088e9	−	7.15489e9i	0.169039	−	0.292784i	0.938082	+	0.346413i	$0.112600\pi$
$930$		0			0
$931$	−8.29479e9	−	1.43670e10i	−0.336885	−	0.583502i
$932$	5.51954e8	+	9.56012e8i	0.0223330	+	0.0386819i
$933$		0			0
$934$	1.05843e10	−	1.83326e10i	0.425060	−	0.736225i
$935$	−4.18916e10			−1.67605
$936$		0			0
$937$	−4.78884e10			−1.90170			−0.950849	−	0.309655i	$-0.899786\pi$
$937$	−4.78884e10			−1.90170			−0.950849	+	0.309655i	$0.899786\pi$
$938$	−9.12179e8	+	1.57994e9i	−0.0360886	+	0.0625073i
$939$		0			0
$940$	2.93474e9	+	5.08311e9i	0.115245	+	0.199610i
$941$	1.27018e10	+	2.20002e10i	0.496939	+	0.860723i	0.999994	−	0.00353107i	$-0.00112398\pi$
$941$	1.27018e10	+	2.20002e10i	0.496939	+	0.860723i	−0.503055	+	0.864254i	$0.667791\pi$
$942$		0			0
$943$	1.45663e10	−	2.52296e10i	0.565665	−	0.979760i
$944$	1.05414e10			0.407845
$945$		0			0
$946$	−2.28054e10			−0.875827
$947$	2.05977e10	−	3.56763e10i	0.788123	−	1.36507i	−0.138992	−	0.990293i	$-0.544386\pi$
$947$	2.05977e10	−	3.56763e10i	0.788123	−	1.36507i	0.927115	−	0.374776i	$-0.122280\pi$
$948$		0			0
$949$	1.15738e10	+	2.00464e10i	0.439586	+	0.761384i
$950$	7.88088e9	+	1.36501e10i	0.298224	+	0.516538i
$951$		0			0
$952$	−1.08025e10	+	1.87104e10i	−0.405783	+	0.702837i
$953$	−1.74074e10			−0.651491			−0.325745	−	0.945458i	$-0.605615\pi$
$953$	−1.74074e10			−0.651491			−0.325745	+	0.945458i	$0.605615\pi$
$954$		0			0
$955$	−1.19933e10			−0.445581
$956$	−1.82590e9	+	3.16254e9i	−0.0675886	+	0.117067i
$957$		0			0
$958$	3.64163e9	+	6.30749e9i	0.133819	+	0.231781i
$959$	1.55441e10	+	2.69231e10i	0.569115	+	0.985735i
$960$		0			0
$961$	−1.26689e10	+	2.19431e10i	−0.460475	+	0.797565i
$962$	4.87664e10			1.76607
$963$		0			0
$964$	−3.44886e8			−0.0123996
$965$	−2.77801e10	+	4.81166e10i	−0.995150	+	1.72365i
$966$		0			0
$967$	4.65290e9	+	8.05906e9i	0.165474	+	0.286610i	0.936824	−	0.349802i	$-0.113751\pi$
$967$	4.65290e9	+	8.05906e9i	0.165474	+	0.286610i	−0.771349	+	0.636412i	$0.780418\pi$
$968$	2.13280e9	+	3.69411e9i	0.0755763	+	0.130902i
$969$		0			0
$970$	−5.47014e9	+	9.47456e9i	−0.192441	+	0.333317i
$971$	−2.04072e10			−0.715347			−0.357673	−	0.933847i	$-0.616430\pi$
$971$	−2.04072e10			−0.715347			−0.357673	+	0.933847i	$0.616430\pi$
$972$		0			0
$973$	−1.55396e10			−0.540810
$974$	1.11966e9	−	1.93931e9i	0.0388268	−	0.0672499i
$975$		0			0
$976$	−9.24732e8	−	1.60168e9i	−0.0318377	−	0.0551445i
$977$	4.18921e9	+	7.25592e9i	0.143715	+	0.248921i	0.928893	−	0.370349i	$-0.120762\pi$
$977$	4.18921e9	+	7.25592e9i	0.143715	+	0.248921i	−0.785178	+	0.619270i	$0.787429\pi$
$978$		0			0
$979$	7.75269e9	−	1.34281e10i	0.264066	−	0.457377i
$980$	−3.61156e9			−0.122575
$981$		0			0
$982$	−1.20249e10			−0.405219
$983$	1.60476e10	−	2.77952e10i	0.538855	−	0.933324i	−0.460111	−	0.887861i	$-0.652190\pi$
$983$	1.60476e10	−	2.77952e10i	0.538855	−	0.933324i	0.998966	−	0.0454630i	$-0.0144763\pi$
$984$		0			0
$985$	−1.11314e10	−	1.92802e10i	−0.371129	−	0.642814i
$986$	−2.06038e10	−	3.56869e10i	−0.684509	−	1.18560i
$987$		0			0
$988$	2.81469e9	−	4.87519e9i	0.0928500	−	0.160821i
$989$	3.83303e10			1.25996
$990$		0			0
$991$	−2.26449e9			−0.0739115			−0.0369558	−	0.999317i	$-0.511766\pi$
$991$	−2.26449e9			−0.0739115			−0.0369558	+	0.999317i	$0.511766\pi$
$992$	−6.59394e9	+	1.14210e10i	−0.214464	+	0.371462i
$993$		0			0
$994$	−8.16145e9	−	1.41360e10i	−0.263582	−	0.456537i
$995$	3.40645e10	+	5.90015e10i	1.09628	+	1.89881i
$996$		0			0
$997$	−4.64663e9	+	8.04819e9i	−0.148493	+	0.257197i	−0.930671	−	0.365858i	$-0.880775\pi$
$997$	−4.64663e9	+	8.04819e9i	−0.148493	+	0.257197i	0.782178	+	0.623055i	$0.214109\pi$
$998$	−3.37209e10			−1.07385
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.8.c.g.55.1		4
3.2	odd	2	inner	81.8.c.g.55.2		4
9.2	odd	6		27.8.a.c.1.1	✓	2
9.4	even	3	inner	81.8.c.g.28.1		4
9.5	odd	6	inner	81.8.c.g.28.2		4
9.7	even	3		27.8.a.c.1.2	yes	2
36.7	odd	6		432.8.a.n.1.1		2
36.11	even	6		432.8.a.n.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.8.a.c.1.1	✓	2	9.2	odd	6
27.8.a.c.1.2	yes	2	9.7	even	3
81.8.c.g.28.1		4	9.4	even	3	inner
81.8.c.g.28.2		4	9.5	odd	6	inner
81.8.c.g.55.1		4	1.1	even	1	trivial
81.8.c.g.55.2		4	3.2	odd	2	inner
432.8.a.n.1.1		2	36.7	odd	6
432.8.a.n.1.2		2	36.11	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	81.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-5.19615 + 9.00000i) q^{2} +(10.0000 + 17.3205i) q^{4} +(176.669 + 306.000i) q^{5} +(279.500 - 484.108i) q^{7} -1538.06 q^{8} -3672.00 q^{10} +(-2359.05 + 4086.00i) q^{11} +(4335.50 + 7509.31i) q^{13} +(2904.65 + 5031.00i) q^{14} +(6712.00 - 11625.5i) q^{16} +25128.6 q^{17} -32461.0 q^{19} +(-3533.38 + 6120.00i) q^{20} +(-24516.0 - 42463.0i) q^{22} +(41205.5 + 71370.0i) q^{23} +(-23361.5 + 40463.3i) q^{25} -90111.7 q^{26} +11180.0 q^{28} +(-78898.4 + 136656. i) q^{29} +(-114946. - 199092. i) q^{31} +(-28682.8 - 49680.0i) q^{32} +(-130572. + 226157. i) q^{34} +197516. q^{35} -541177. q^{37} +(168672. - 292149. i) q^{38} +(-271728. - 470647. i) q^{40} +(-176752. - 306144. i) q^{41} +(232556. - 402799. i) q^{43} -94362.1 q^{44} -856440. q^{46} +(415287. - 719298. i) q^{47} +(255531. + 442593. i) q^{49} +(-242780. - 420507. i) q^{50} +(-86710.0 + 150186. i) q^{52} -1.02622e6 q^{53} -1.66709e6 q^{55} +(-429888. + 744588. i) q^{56} +(-819936. - 1.42017e6i) q^{58} +(392632. + 680058. i) q^{59} +(68886.5 - 119315. i) q^{61} +2.38911e6 q^{62} +2.31443e6 q^{64} +(-1.53190e6 + 2.65333e6i) q^{65} +(157020. + 271967. i) q^{67} +(251286. + 435240. i) q^{68} +(-1.02632e6 + 1.77765e6i) q^{70} -2.80979e6 q^{71} +2.66954e6 q^{73} +(2.81204e6 - 4.87059e6i) q^{74} +(-324610. - 562241. i) q^{76} +(1.31871e6 + 2.28407e6i) q^{77} +(-550907. + 954200. i) q^{79} +4.74321e6 q^{80} +3.67373e6 q^{82} +(-3.03952e6 + 5.26460e6i) q^{83} +(4.43945e6 + 7.68935e6i) q^{85} +(2.41679e6 + 4.18601e6i) q^{86} +(3.62837e6 - 6.28452e6i) q^{88} -3.28636e6 q^{89} +4.84709e6 q^{91} +(-824110. + 1.42740e6i) q^{92} +(4.31579e6 + 7.47516e6i) q^{94} +(-5.73486e6 - 9.93307e6i) q^{95} +(1.48969e6 - 2.58022e6i) q^{97} -5.31111e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 40 q^{4} + 1118 q^{7} - 14688 q^{10} + 17342 q^{13} + 26848 q^{16} - 129844 q^{19} - 98064 q^{22} - 93446 q^{25} + 44720 q^{28} - 459784 q^{31} - 522288 q^{34} - 2164708 q^{37} - 1086912 q^{40} + 930224 q^{43}+ \cdots + 5958758 q^{97}+O(q^{100}) \) 4 * q + 40 * q^4 + 1118 * q^7 - 14688 * q^10 + 17342 * q^13 + 26848 * q^16 - 129844 * q^19 - 98064 * q^22 - 93446 * q^25 + 44720 * q^28 - 459784 * q^31 - 522288 * q^34 - 2164708 * q^37 - 1086912 * q^40 + 930224 * q^43 - 3425760 * q^46 + 1022124 * q^49 - 346840 * q^52 - 6668352 * q^55 - 3279744 * q^58 + 275546 * q^61 + 9257728 * q^64 + 628082 * q^67 - 4105296 * q^70 + 10678148 * q^73 - 1298440 * q^76 - 2203630 * q^79 + 14694912 * q^82 + 17757792 * q^85 + 14513472 * q^88 + 19388356 * q^91 + 17263152 * q^94 + 5958758 * q^97

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