LMFDB - Embedded newform 81.12.c.i.55.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [81,12,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, 12); 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, 12, names="a")

Level:	$ N $	$=$	$ 81 = 3^{4} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	81.c (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,-1636] 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:	$62.2357976253$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 182x^{6} + 32932x^{4} - 34944x^{2} + 36864 $ x^8 - 182x^6 + 32932x^4 - 34944*x^2 + 36864
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{19} $
Twist minimal:	no (minimal twist has level 27)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			55.2
Root			$-0.892104 + 0.515056i$ of defining polynomial
Character	$\chi$	$=$	81.55
Dual form			81.12.c.i.28.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-2.67631 + 4.63551i) q^{2} +(1009.67 + 1748.81i) q^{4} +(6419.60 + 11119.1i) q^{5} +(15869.3 - 27486.4i) q^{7} -21771.0 q^{8} -68723.4 q^{10} +(-342308. + 592895. i) q^{11} +(-493175. - 854204. i) q^{13} +(84942.3 + 147124. i) q^{14} +(-2.00955e6 + 3.48064e6i) q^{16} -8.95856e6 q^{17} -3.35373e6 q^{19} +(-1.29634e7 + 2.24533e7i) q^{20} +(-1.83225e6 - 3.17355e6i) q^{22} +(5.36262e6 + 9.28832e6i) q^{23} +(-5.80084e7 + 1.00474e8i) q^{25} +5.27956e6 q^{26} +6.40913e7 q^{28} +(7.77565e7 - 1.34678e8i) q^{29} +(-1.12081e8 - 1.94129e8i) q^{31} +(-3.30498e7 - 5.72440e7i) q^{32} +(2.39759e7 - 4.15275e7i) q^{34} +4.07498e8 q^{35} +2.97380e8 q^{37} +(8.97562e6 - 1.55462e7i) q^{38} +(-1.39761e8 - 2.42073e8i) q^{40} +(1.83506e8 + 3.17842e8i) q^{41} +(1.18982e8 - 2.06082e8i) q^{43} -1.38248e9 q^{44} -5.74081e7 q^{46} +(-9.07505e8 + 1.57185e9i) q^{47} +(4.84994e8 + 8.40035e8i) q^{49} +(-3.10497e8 - 5.37797e8i) q^{50} +(9.95892e8 - 1.72494e9i) q^{52} +2.49351e8 q^{53} -8.78993e9 q^{55} +(-3.45490e8 + 5.98407e8i) q^{56} +(4.16201e8 + 7.20881e8i) q^{58} +(2.62950e9 + 4.55442e9i) q^{59} +(-1.95491e9 + 3.38600e9i) q^{61} +1.19985e9 q^{62} -7.87730e9 q^{64} +(6.33197e9 - 1.09673e10i) q^{65} +(-7.99354e9 - 1.38452e10i) q^{67} +(-9.04523e9 - 1.56668e10i) q^{68} +(-1.09059e9 + 1.88896e9i) q^{70} +1.95308e10 q^{71} -9.74734e9 q^{73} +(-7.95881e8 + 1.37851e9i) q^{74} +(-3.38617e9 - 5.86503e9i) q^{76} +(1.08644e10 + 1.88177e10i) q^{77} +(-1.17184e10 + 2.02968e10i) q^{79} -5.16020e10 q^{80} -1.96448e9 q^{82} +(-1.81967e10 + 3.15175e10i) q^{83} +(-5.75104e10 - 9.96109e10i) q^{85} +(6.36864e8 + 1.10308e9i) q^{86} +(7.45239e9 - 1.29079e10i) q^{88} +1.64629e10 q^{89} -3.13053e10 q^{91} +(-1.08290e10 + 1.87564e10i) q^{92} +(-4.85753e9 - 8.41349e9i) q^{94} +(-2.15296e10 - 3.72903e10i) q^{95} +(-1.37713e10 + 2.38526e10i) q^{97} -5.19198e9 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 1636 q^{4} - 28460 q^{7} + 304992 q^{10} - 370868 q^{13} - 4128904 q^{16} + 66418792 q^{19} - 94424400 q^{22} - 142981516 q^{25} + 778081048 q^{28} - 647361104 q^{31} - 1085116176 q^{34} + 1104329752 q^{37}+ \cdots - 137541602900 q^{97}+O(q^{100}) $ 8 * q - 1636 * q^4 - 28460 * q^7 + 304992 * q^10 - 370868 * q^13 - 4128904 * q^16 + 66418792 * q^19 - 94424400 * q^22 - 142981516 * q^25 + 778081048 * q^28 - 647361104 * q^31 - 1085116176 * q^34 + 1104329752 * q^37 - 787910112 * q^40 + 131575840 * q^43 - 11530284768 * q^46 + 1668407928 * q^49 + 8528523820 * q^52 - 29990656512 * q^55 + 4235362560 * q^58 + 6848424772 * q^61 - 85283619040 * q^64 - 43603974212 * q^67 - 17690625936 * q^70 - 26354434232 * q^73 - 126802758164 * q^76 - 140571916436 * q^79 - 284034359808 * q^82 - 265087842048 * q^85 - 38939155488 * q^88 - 272489230984 * q^91 - 105138158544 * q^94 - 137541602900 * 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 $

$2$	−2.67631	+	4.63551i	−0.0591387	+	0.102431i	−0.894079	−	0.447909i	$-0.852169\pi$
$2$	−2.67631	+	4.63551i	−0.0591387	+	0.102431i	0.834940	+	0.550340i	$0.185502\pi$
$3$		0			0
$3$		0			0
$4$	1009.67	+	1748.81i	0.493005	+	0.853910i
$5$	6419.60	+	11119.1i	0.918698	+	1.59123i	0.801395	+	0.598136i	$0.204092\pi$
$5$	6419.60	+	11119.1i	0.918698	+	1.59123i	0.117304	+	0.993096i	$0.462575\pi$
$6$		0			0
$7$	15869.3	−	27486.4i	0.356877	−	0.618129i	−0.630560	−	0.776140i	$-0.717175\pi$
$7$	15869.3	−	27486.4i	0.356877	−	0.618129i	0.987437	+	0.158011i	$0.0505082\pi$
$8$	−21771.0			−0.234900
$9$		0			0
$10$	−68723.4			−0.217322
$11$	−342308.	+	592895.i	−0.640852	+	1.10999i	0.344391	+	0.938826i	$0.388085\pi$
$11$	−342308.	+	592895.i	−0.640852	+	1.10999i	−0.985243	+	0.171162i	$0.945248\pi$
$12$		0			0
$13$	−493175.	−	854204.i	−0.368394	−	0.638077i	0.620921	−	0.783873i	$-0.286759\pi$
$13$	−493175.	−	854204.i	−0.368394	−	0.638077i	−0.989315	+	0.145796i	$0.953426\pi$
$14$	84942.3	+	147124.i	0.0422105	+	0.0731106i
$15$		0			0
$16$	−2.00955e6	+	3.48064e6i	−0.479114	+	0.829849i
$17$	−8.95856e6			−1.53027			−0.765137	−	0.643868i	$-0.777329\pi$
$17$	−8.95856e6			−1.53027			−0.765137	+	0.643868i	$0.777329\pi$
$18$		0			0
$19$	−3.35373e6			−0.310730			−0.155365	−	0.987857i	$-0.549655\pi$
$19$	−3.35373e6			−0.310730			−0.155365	+	0.987857i	$0.549655\pi$
$20$	−1.29634e7	+	2.24533e7i	−0.905846	+	1.56897i
$21$		0			0
$22$	−1.83225e6	−	3.17355e6i	−0.0757983	−	0.131286i
$23$	5.36262e6	+	9.28832e6i	0.173729	+	0.300908i	0.939721	−	0.341943i	$-0.111085\pi$
$23$	5.36262e6	+	9.28832e6i	0.173729	+	0.300908i	−0.765991	+	0.642851i	$0.777751\pi$
$24$		0			0
$25$	−5.80084e7	+	1.00474e8i	−1.18801	+	2.05770i
$26$	5.27956e6			0.0871453
$27$		0			0
$28$	6.40913e7			0.703769
$29$	7.77565e7	−	1.34678e8i	0.703959	−	1.21929i	−0.263107	−	0.964767i	$-0.584747\pi$
$29$	7.77565e7	−	1.34678e8i	0.703959	−	1.21929i	0.967066	−	0.254526i	$-0.0819196\pi$
$30$		0			0
$31$	−1.12081e8	−	1.94129e8i	−0.703140	−	1.21787i	−0.967359	−	0.253411i	$-0.918448\pi$
$31$	−1.12081e8	−	1.94129e8i	−0.703140	−	1.21787i	0.264219	−	0.964463i	$-0.414886\pi$
$32$	−3.30498e7	−	5.72440e7i	−0.174118	−	0.301582i
$33$		0			0
$34$	2.39759e7	−	4.15275e7i	0.0904983	−	0.156748i
$35$	4.07498e8			1.31145
$36$		0			0
$37$	2.97380e8			0.705021			0.352510	−	0.935808i	$-0.385328\pi$
$37$	2.97380e8			0.705021			0.352510	+	0.935808i	$0.385328\pi$
$38$	8.97562e6	−	1.55462e7i	0.0183762	−	0.0318284i
$39$		0			0
$40$	−1.39761e8	−	2.42073e8i	−0.215802	−	0.373781i
$41$	1.83506e8	+	3.17842e8i	0.247365	+	0.428450i	0.962794	−	0.270236i	$-0.0871018\pi$
$41$	1.83506e8	+	3.17842e8i	0.247365	+	0.428450i	−0.715429	+	0.698686i	$0.753769\pi$
$42$		0			0
$43$	1.18982e8	−	2.06082e8i	0.123425	−	0.213778i	−0.797691	−	0.603066i	$-0.793945\pi$
$43$	1.18982e8	−	2.06082e8i	0.123425	−	0.213778i	0.921116	+	0.389288i	$0.127279\pi$
$44$	−1.38248e9			−1.26377
$45$		0			0
$46$	−5.74081e7			−0.0410965
$47$	−9.07505e8	+	1.57185e9i	−0.577180	+	0.999704i	0.418621	+	0.908161i	$0.362513\pi$
$47$	−9.07505e8	+	1.57185e9i	−0.577180	+	0.999704i	−0.995801	+	0.0915436i	$0.970820\pi$
$48$		0			0
$49$	4.84994e8	+	8.40035e8i	0.245278	+	0.424834i
$50$	−3.10497e8	−	5.37797e8i	−0.140515	−	0.243379i
$51$		0			0
$52$	9.95892e8	−	1.72494e9i	0.363240	−	0.629151i
$53$	2.49351e8			0.0819019			0.0409510	−	0.999161i	$-0.486961\pi$
$53$	2.49351e8			0.0819019			0.0409510	+	0.999161i	$0.486961\pi$
$54$		0			0
$55$	−8.78993e9			−2.35500
$56$	−3.45490e8	+	5.98407e8i	−0.0838304	+	0.145199i
$57$		0			0
$58$	4.16201e8	+	7.20881e8i	0.0832624	+	0.144215i
$59$	2.62950e9	+	4.55442e9i	0.478836	+	0.829368i	0.999705	−	0.0242680i	$-0.00772550\pi$
$59$	2.62950e9	+	4.55442e9i	0.478836	+	0.829368i	−0.520869	+	0.853636i	$0.674392\pi$
$60$		0			0
$61$	−1.95491e9	+	3.38600e9i	−0.296355	+	0.513302i	−0.975299	−	0.220888i	$-0.929104\pi$
$61$	−1.95491e9	+	3.38600e9i	−0.296355	+	0.513302i	0.678944	+	0.734190i	$0.262438\pi$
$62$	1.19985e9			0.166331
$63$		0			0
$64$	−7.87730e9			−0.917039
$65$	6.33197e9	−	1.09673e10i	0.676886	−	1.17240i
$66$		0			0
$67$	−7.99354e9	−	1.38452e10i	−0.723316	−	1.25282i	−0.959664	−	0.281151i	$-0.909284\pi$
$67$	−7.99354e9	−	1.38452e10i	−0.723316	−	1.25282i	0.236348	−	0.971668i	$-0.424049\pi$
$68$	−9.04523e9	−	1.56668e10i	−0.754433	−	1.30672i
$69$		0			0
$70$	−1.09059e9	+	1.88896e9i	−0.0775573	+	0.134333i
$71$	1.95308e10			1.28469			0.642347	−	0.766414i	$-0.277961\pi$
$71$	1.95308e10			1.28469			0.642347	+	0.766414i	$0.277961\pi$
$72$		0			0
$73$	−9.74734e9			−0.550314			−0.275157	−	0.961399i	$-0.588730\pi$
$73$	−9.74734e9			−0.550314			−0.275157	+	0.961399i	$0.588730\pi$
$74$	−7.95881e8	+	1.37851e9i	−0.0416940	+	0.0722161i
$75$		0			0
$76$	−3.38617e9	−	5.86503e9i	−0.153191	−	0.265335i
$77$	1.08644e10	+	1.88177e10i	0.457411	+	0.792258i
$78$		0			0
$79$	−1.17184e10	+	2.02968e10i	−0.428468	+	0.742128i	−0.996737	−	0.0807145i	$-0.974280\pi$
$79$	−1.17184e10	+	2.02968e10i	−0.428468	+	0.742128i	0.568269	+	0.822843i	$0.307613\pi$
$80$	−5.16020e10			−1.76064
$81$		0			0
$82$	−1.96448e9			−0.0585155
$83$	−1.81967e10	+	3.15175e10i	−0.507063	+	0.878259i	0.492903	+	0.870084i	$0.335936\pi$
$83$	−1.81967e10	+	3.15175e10i	−0.507063	+	0.878259i	−0.999967	+	0.00817506i	$0.997398\pi$
$84$		0			0
$85$	−5.75104e10	−	9.96109e10i	−1.40586	−	2.43502i
$86$	6.36864e8	+	1.10308e9i	0.0145984	+	0.0252852i
$87$		0			0
$88$	7.45239e9	−	1.29079e10i	0.150536	−	0.260736i
$89$	1.64629e10			0.312509			0.156255	−	0.987717i	$-0.450058\pi$
$89$	1.64629e10			0.312509			0.156255	+	0.987717i	$0.450058\pi$
$90$		0			0
$91$	−3.13053e10			−0.525885
$92$	−1.08290e10	+	1.87564e10i	−0.171299	+	0.296699i
$93$		0			0
$94$	−4.85753e9	−	8.41349e9i	−0.0682673	−	0.118242i
$95$	−2.15296e10	−	3.72903e10i	−0.285467	−	0.494443i
$96$		0			0
$97$	−1.37713e10	+	2.38526e10i	−0.162829	+	0.282028i	−0.935882	−	0.352313i	$-0.885395\pi$
$97$	−1.37713e10	+	2.38526e10i	−0.162829	+	0.282028i	0.773053	+	0.634341i	$0.218729\pi$
$98$	−5.19198e9			−0.0580216
$99$		0			0
$100$	−2.34279e11			−2.34279
$101$	7.23245e10	−	1.25270e11i	0.684727	−	1.18598i	−0.288795	−	0.957391i	$-0.593255\pi$
$101$	7.23245e10	−	1.25270e11i	0.684727	−	1.18598i	0.973522	−	0.228592i	$-0.0734121\pi$
$102$		0			0
$103$	−1.96471e10	−	3.40298e10i	−0.166991	−	0.289238i	0.770369	−	0.637598i	$-0.220072\pi$
$103$	−1.96471e10	−	3.40298e10i	−0.166991	−	0.289238i	−0.937361	+	0.348360i	$0.886739\pi$
$104$	1.07369e10	+	1.85969e10i	0.0865357	+	0.149884i
$105$		0			0
$106$	−6.67341e8	+	1.15587e9i	−0.00484357	+	0.00838931i
$107$	7.67258e10			0.528848			0.264424	−	0.964407i	$-0.414818\pi$
$107$	7.67258e10			0.528848			0.264424	+	0.964407i	$0.414818\pi$
$108$		0			0
$109$	−2.43424e11			−1.51537			−0.757683	−	0.652623i	$-0.773669\pi$
$109$	−2.43424e11			−1.51537			−0.757683	+	0.652623i	$0.773669\pi$
$110$	2.35246e10	−	4.07458e10i	0.139271	−	0.241225i
$111$		0			0
$112$	6.37802e10	+	1.10471e11i	0.341969	+	0.592308i
$113$	2.33726e10	+	4.04824e10i	0.119337	+	0.206698i	0.919505	−	0.393078i	$-0.128590\pi$
$113$	2.33726e10	+	4.04824e10i	0.119337	+	0.206698i	−0.800168	+	0.599776i	$0.795256\pi$
$114$		0			0
$115$	−6.88517e10	+	1.19255e11i	−0.319210	+	0.552888i
$116$	3.14035e11			1.38822
$117$		0			0
$118$	−2.81494e10			−0.113271
$119$	−1.42166e11	+	2.46239e11i	−0.546119	+	0.945906i
$120$		0			0
$121$	−9.16942e10	−	1.58819e11i	−0.321383	−	0.556651i
$122$	−1.04639e10	−	1.81240e10i	−0.0350521	−	0.0607120i
$123$		0			0
$124$	2.26330e11	−	3.92015e11i	0.693303	−	1.20084i
$125$	−8.62650e11			−2.52831
$126$		0			0
$127$	2.60792e11			0.700446			0.350223	−	0.936666i	$-0.386106\pi$
$127$	2.60792e11			0.700446			0.350223	+	0.936666i	$0.386106\pi$
$128$	8.87682e10	−	1.53751e11i	0.228351	−	0.395515i
$129$		0			0
$130$	3.38926e10	+	5.87038e10i	0.0800602	+	0.138668i
$131$	−5.51292e10	−	9.54866e10i	−0.124850	−	0.216247i	0.796824	−	0.604211i	$-0.206512\pi$
$131$	−5.51292e10	−	9.54866e10i	−0.124850	−	0.216247i	−0.921674	+	0.387964i	$0.873178\pi$
$132$		0			0
$133$	−5.32213e10	+	9.21820e10i	−0.110892	+	0.192071i
$134$	8.55728e10			0.171104
$135$		0			0
$136$	1.95037e11			0.359461
$137$	−3.72246e11	+	6.44750e11i	−0.658973	+	1.14137i	0.321909	+	0.946771i	$0.395675\pi$
$137$	−3.72246e11	+	6.44750e11i	−0.658973	+	1.14137i	−0.980882	+	0.194604i	$0.937658\pi$
$138$		0			0
$139$	−2.79728e11	−	4.84503e11i	−0.457251	−	0.791982i	0.541563	−	0.840660i	$-0.317833\pi$
$139$	−2.79728e11	−	4.84503e11i	−0.457251	−	0.791982i	−0.998815	+	0.0486776i	$0.984499\pi$
$140$	4.11441e11	+	7.12636e11i	0.646551	+	1.11986i
$141$		0			0
$142$	−5.22706e10	+	9.05353e10i	−0.0759751	+	0.131593i
$143$	6.75271e11			0.944344
$144$		0			0
$145$	1.99666e12			2.58690
$146$	2.60869e10	−	4.51839e10i	0.0325448	−	0.0563693i
$147$		0			0
$148$	3.00257e11	+	5.20060e11i	0.347579	+	0.602024i
$149$	−6.32647e11	−	1.09578e12i	−0.705727	−	1.22236i	−0.966428	−	0.256936i	$-0.917287\pi$
$149$	−6.32647e11	−	1.09578e12i	−0.705727	−	1.22236i	0.260701	−	0.965420i	$-0.416046\pi$
$150$		0			0
$151$	−6.02725e11	+	1.04395e12i	−0.624807	+	1.08220i	0.363771	+	0.931488i	$0.381489\pi$
$151$	−6.02725e11	+	1.04395e12i	−0.624807	+	1.08220i	−0.988578	+	0.150709i	$0.951844\pi$
$152$	7.30140e10			0.0729905
$153$		0			0
$154$	−1.16306e11			−0.108203
$155$	1.43903e12	−	2.49247e12i	1.29195	−	2.23772i
$156$		0			0
$157$	1.01675e12	+	1.76106e12i	0.850679	+	1.47342i	0.880596	+	0.473868i	$0.157143\pi$
$157$	1.01675e12	+	1.76106e12i	0.850679	+	1.47342i	−0.0299166	+	0.999552i	$0.509524\pi$
$158$	−6.27240e10	−	1.08641e11i	−0.0506780	−	0.0877769i
$159$		0			0
$160$	4.24333e11	−	7.34967e11i	0.319924	−	0.554125i
$161$	3.40404e11			0.248000
$162$		0			0
$163$	−3.15622e11			−0.214850			−0.107425	−	0.994213i	$-0.534261\pi$
$163$	−3.15622e11			−0.214850			−0.107425	+	0.994213i	$0.534261\pi$
$164$	−3.70563e11	+	6.41833e11i	−0.243905	+	0.422456i
$165$		0			0
$166$	−9.73998e10	−	1.68701e11i	−0.0599741	−	0.103878i
$167$	7.33624e11	+	1.27067e12i	0.437052	+	0.756995i	0.997461	−	0.0712205i	$-0.0226894\pi$
$167$	7.33624e11	+	1.27067e12i	0.437052	+	0.756995i	−0.560409	+	0.828216i	$0.689356\pi$
$168$		0			0
$169$	4.09637e11	−	7.09513e11i	0.228572	−	0.395898i
$170$	6.15662e11			0.332563
$171$		0			0
$172$	4.80531e11			0.243397
$173$	−1.53808e12	+	2.66404e12i	−0.754616	+	1.30703i	0.190949	+	0.981600i	$0.438844\pi$
$173$	−1.53808e12	+	2.66404e12i	−0.754616	+	1.30703i	−0.945565	+	0.325434i	$0.894490\pi$
$174$		0			0
$175$	1.84111e12	+	3.18889e12i	0.847949	+	1.46869i
$176$	−1.37577e12	−	2.38290e12i	−0.614082	−	1.06362i
$177$		0			0
$178$	−4.40600e10	+	7.63141e10i	−0.0184814	+	0.0320107i
$179$	−2.98789e12			−1.21527			−0.607635	−	0.794216i	$-0.707882\pi$
$179$	−2.98789e12			−1.21527			−0.607635	+	0.794216i	$0.707882\pi$
$180$		0			0
$181$	2.38164e12			0.911265			0.455633	−	0.890168i	$-0.349413\pi$
$181$	2.38164e12			0.911265			0.455633	+	0.890168i	$0.349413\pi$
$182$	8.37828e10	−	1.45116e11i	0.0311001	−	0.0538670i
$183$		0			0
$184$	−1.16749e11	−	2.02216e11i	−0.0408091	−	0.0706834i
$185$	1.90906e12	+	3.30659e12i	0.647701	+	1.12185i
$186$		0			0
$187$	3.06659e12	−	5.31149e12i	0.980679	−	1.69859i
$188$	−3.66514e12			−1.13821
$189$		0			0
$190$	2.30480e11			0.0675286
$191$	−4.48897e11	+	7.77512e11i	−0.127780	+	0.221322i	−0.922816	−	0.385240i	$-0.874119\pi$
$191$	−4.48897e11	+	7.77512e11i	−0.127780	+	0.221322i	0.795036	+	0.606562i	$0.207452\pi$
$192$		0			0
$193$	8.94978e11	+	1.55015e12i	0.240573	+	0.416685i	0.960878	−	0.276973i	$-0.0893313\pi$
$193$	8.94978e11	+	1.55015e12i	0.240573	+	0.416685i	−0.720305	+	0.693658i	$0.755998\pi$
$194$	−7.37127e10	−	1.27674e11i	−0.0192590	−	0.0333575i
$195$		0			0
$196$	−9.79373e11	+	1.69632e12i	−0.241846	+	0.418890i
$197$	7.38729e12			1.77387			0.886933	−	0.461899i	$-0.152832\pi$
$197$	7.38729e12			1.77387			0.886933	+	0.461899i	$0.152832\pi$
$198$		0			0
$199$	1.19720e12			0.271942			0.135971	−	0.990713i	$-0.456585\pi$
$199$	1.19720e12			0.271942			0.135971	+	0.990713i	$0.456585\pi$
$200$	1.26290e12	−	2.18741e12i	0.279064	−	0.483354i
$201$		0			0
$202$	3.87126e11	+	6.70521e11i	0.0809877	+	0.140275i
$203$	−2.46788e12	−	4.27449e12i	−0.502454	−	0.870275i
$204$		0			0
$205$	−2.35607e12	+	4.08083e12i	−0.454508	+	0.787232i
$206$	2.10327e11			0.0395026
$207$		0			0
$208$	3.96423e12			0.706010
$209$	1.14801e12	−	1.98841e12i	0.199132	−	0.344907i
$210$		0			0
$211$	−3.04783e12	−	5.27899e12i	−0.501692	−	0.868955i	−0.999998	−	0.00195445i	$-0.999378\pi$
$211$	−3.04783e12	−	5.27899e12i	−0.501692	−	0.868955i	0.498306	−	0.867001i	$-0.333955\pi$
$212$	2.51764e11	+	4.36067e11i	0.0403781	+	0.0699369i
$213$		0			0
$214$	−2.05342e11	+	3.55663e11i	−0.0312754	+	0.0541705i
$215$	3.05526e12			0.453562
$216$		0			0
$217$	−7.11457e12			−1.00374
$218$	6.51478e11	−	1.12839e12i	0.0896167	−	0.155221i
$219$		0			0
$220$	−8.87497e12	−	1.53719e13i	−1.16103	−	2.01096i
$221$	4.41814e12	+	7.65243e12i	0.563743	+	0.976432i
$222$		0			0
$223$	−4.51085e12	+	7.81302e12i	−0.547749	+	0.948729i	0.450680	+	0.892686i	$0.351182\pi$
$223$	−4.51085e12	+	7.81302e12i	−0.547749	+	0.948729i	−0.998428	+	0.0560428i	$0.982152\pi$
$224$	−2.09791e12			−0.248555
$225$		0			0
$226$	−2.50209e11			−0.0282297
$227$	1.74181e12	−	3.01691e12i	0.191805	−	0.332216i	−0.754044	−	0.656824i	$-0.771899\pi$
$227$	1.74181e12	−	3.01691e12i	0.191805	−	0.332216i	0.945848	+	0.324609i	$0.105233\pi$
$228$		0			0
$229$	2.05234e12	+	3.55476e12i	0.215355	+	0.373006i	0.953382	−	0.301765i	$-0.0975758\pi$
$229$	2.05234e12	+	3.55476e12i	0.215355	+	0.373006i	−0.738027	+	0.674771i	$0.764243\pi$
$230$	−3.68537e11	−	6.38325e11i	−0.0377553	−	0.0653941i
$231$		0			0
$232$	−1.69283e12	+	2.93208e12i	−0.165360	+	0.286412i
$233$	−5.04001e12			−0.480810			−0.240405	−	0.970673i	$-0.577280\pi$
$233$	−5.04001e12			−0.480810			−0.240405	+	0.970673i	$0.577280\pi$
$234$		0			0
$235$	−2.33033e13			−2.12102
$236$	−5.30988e12	+	9.19698e12i	−0.472137	+	0.817766i
$237$		0			0
$238$	−7.60961e11	−	1.31802e12i	−0.0645935	−	0.111879i
$239$	−7.69620e12	−	1.33302e13i	−0.638393	−	1.10573i	−0.985785	−	0.168009i	$-0.946266\pi$
$239$	−7.69620e12	−	1.33302e13i	−0.638393	−	1.10573i	0.347393	−	0.937720i	$-0.387067\pi$
$240$		0			0
$241$	5.64909e11	−	9.78451e11i	0.0447594	−	0.0775256i	−0.842778	−	0.538262i	$-0.819081\pi$
$241$	5.64909e11	−	9.78451e11i	0.0447594	−	0.0775256i	0.887537	+	0.460736i	$0.152415\pi$
$242$	9.81609e11			0.0760246
$243$		0			0
$244$	−7.89528e12			−0.584418
$245$	−6.22694e12	+	1.07854e13i	−0.450673	+	0.780588i
$246$		0			0
$247$	1.65397e12	+	2.86477e12i	0.114471	+	0.198270i
$248$	2.44011e12	+	4.22639e12i	0.165168	+	0.286079i
$249$		0			0
$250$	2.30872e12	−	3.99882e12i	0.149521	−	0.258977i
$251$	−2.41866e13			−1.53239			−0.766197	−	0.642606i	$-0.777853\pi$
$251$	−2.41866e13			−1.53239			−0.766197	+	0.642606i	$0.777853\pi$
$252$		0			0
$253$	−7.34267e12			−0.445340
$254$	−6.97962e11	+	1.20890e12i	−0.0414234	+	0.0717475i
$255$		0			0
$256$	−7.59121e12	−	1.31484e13i	−0.431511	−	0.747398i
$257$	1.33942e13	+	2.31994e13i	0.745219	+	1.29076i	0.950092	+	0.311969i	$0.100988\pi$
$257$	1.33942e13	+	2.31994e13i	0.745219	+	1.29076i	−0.204873	+	0.978788i	$0.565678\pi$
$258$		0			0
$259$	4.71921e12	−	8.17391e12i	0.251606	−	0.435794i
$260$	2.55729e13			1.33483
$261$		0			0
$262$	5.90172e11			0.0295339
$263$	5.22437e12	−	9.04887e12i	0.256022	−	0.443443i	−0.709151	−	0.705057i	$-0.750921\pi$
$263$	5.22437e12	−	9.04887e12i	0.256022	−	0.443443i	0.965173	+	0.261614i	$0.0842548\pi$
$264$		0			0
$265$	1.60073e12	+	2.77255e12i	0.0752432	+	0.130325i
$266$	−2.84874e11	−	4.93415e11i	−0.0131161	−	0.0227177i
$267$		0			0
$268$	1.61418e13	−	2.79583e13i	0.713197	−	1.23529i
$269$	1.01222e13			0.438167			0.219083	−	0.975706i	$-0.429693\pi$
$269$	1.01222e13			0.438167			0.219083	+	0.975706i	$0.429693\pi$
$270$		0			0
$271$	−2.07098e13			−0.860688			−0.430344	−	0.902665i	$-0.641608\pi$
$271$	−2.07098e13			−0.860688			−0.430344	+	0.902665i	$0.641608\pi$
$272$	1.80027e13	−	3.11815e13i	0.733175	−	1.26990i
$273$		0			0
$274$	−1.99249e12	−	3.45110e12i	−0.0779415	−	0.134999i
$275$	−3.97136e13	−	6.87859e13i	−1.52268	−	2.63736i
$276$		0			0
$277$	−4.41209e12	+	7.64197e12i	−0.162557	+	0.281557i	−0.935785	−	0.352571i	$-0.885308\pi$
$277$	−4.41209e12	+	7.64197e12i	−0.162557	+	0.281557i	0.773228	+	0.634128i	$0.218641\pi$
$278$	2.99456e12			0.108165
$279$		0			0
$280$	−8.87163e12			−0.308059
$281$	3.00188e11	−	5.19941e11i	0.0102214	−	0.0177039i	−0.860869	−	0.508826i	$-0.830080\pi$
$281$	3.00188e11	−	5.19941e11i	0.0102214	−	0.0177039i	0.871091	+	0.491122i	$0.163413\pi$
$282$		0			0
$283$	5.66563e12	+	9.81316e12i	0.185534	+	0.321354i	0.943756	−	0.330642i	$-0.107265\pi$
$283$	5.66563e12	+	9.81316e12i	0.185534	+	0.321354i	−0.758223	+	0.651996i	$0.773932\pi$
$284$	1.97198e13	+	3.41557e13i	0.633361	+	1.09701i
$285$		0			0
$286$	−1.80724e12	+	3.13023e12i	−0.0558472	+	0.0967303i
$287$	1.16484e13			0.353116
$288$		0			0
$289$	4.59839e13			1.34174
$290$	−5.34369e12	+	9.25554e12i	−0.152986	+	0.264980i
$291$		0			0
$292$	−9.84164e12	−	1.70462e13i	−0.271308	−	0.469919i
$293$	2.01021e13	+	3.48179e13i	0.543839	+	0.941957i	0.998679	+	0.0513836i	$0.0163631\pi$
$293$	2.01021e13	+	3.48179e13i	0.543839	+	0.941957i	−0.454840	+	0.890573i	$0.650304\pi$
$294$		0			0
$295$	−3.37607e13	+	5.84752e13i	−0.879812	+	1.52388i
$296$	−6.47425e12			−0.165609
$297$		0			0
$298$	6.77264e12			0.166943
$299$	5.28941e12	−	9.16153e12i	0.128002	−	0.221706i
$300$		0			0
$301$	−3.77631e12	−	6.54076e12i	−0.0880951	−	0.152585i
$302$	−3.22616e12	−	5.58787e12i	−0.0739005	−	0.127999i
$303$		0			0
$304$	6.73948e12	−	1.16731e13i	0.148875	−	0.257859i
$305$	−5.01989e13			−1.08904
$306$		0			0
$307$	7.48437e13			1.56637			0.783185	−	0.621789i	$-0.213594\pi$
$307$	7.48437e13			1.56637			0.783185	+	0.621789i	$0.213594\pi$
$308$	−2.19390e13	+	3.79994e13i	−0.451012	+	0.781175i
$309$		0			0
$310$	7.70257e12	+	1.33412e13i	0.152808	+	0.264671i
$311$	1.61320e13	+	2.79414e13i	0.314416	+	0.544585i	0.979313	−	0.202350i	$-0.0648579\pi$
$311$	1.61320e13	+	2.79414e13i	0.314416	+	0.544585i	−0.664897	+	0.746935i	$0.731525\pi$
$312$		0			0
$313$	−3.12405e13	+	5.41101e13i	−0.587792	+	1.01809i	0.406729	+	0.913549i	$0.366669\pi$
$313$	−3.12405e13	+	5.41101e13i	−0.587792	+	1.01809i	−0.994521	+	0.104537i	$0.966664\pi$
$314$	−1.08846e13			−0.201232
$315$		0			0
$316$	−4.73270e13			−0.844948
$317$	−5.22056e13	+	9.04227e13i	−0.915990	+	1.58654i	−0.110545	+	0.993871i	$0.535260\pi$
$317$	−5.22056e13	+	9.04227e13i	−0.915990	+	1.58654i	−0.805445	+	0.592671i	$0.798074\pi$
$318$		0			0
$319$	5.32334e13	+	9.22029e13i	0.902267	+	1.56277i
$320$	−5.05691e13	−	8.75883e13i	−0.842482	−	1.45922i
$321$		0			0
$322$	−9.11026e11	+	1.57794e12i	−0.0146664	+	0.0254030i
$323$	3.00446e13			0.475502
$324$		0			0
$325$	1.14433e14			1.75063
$326$	8.44703e11	−	1.46307e12i	0.0127060	−	0.0220074i
$327$		0			0
$328$	−3.99511e12	−	6.91973e12i	−0.0581062	−	0.100643i
$329$	2.88029e13	+	4.98882e13i	0.411964	+	0.713543i
$330$		0			0
$331$	3.90501e13	−	6.76368e13i	0.540217	−	0.935683i	−0.458674	−	0.888605i	$-0.651675\pi$
$331$	3.90501e13	−	6.76368e13i	0.540217	−	0.935683i	0.998891	−	0.0470788i	$-0.0149912\pi$
$332$	−7.34908e13			−0.999939
$333$		0			0
$334$	−7.85362e12			−0.103387
$335$	1.02631e14	−	1.77762e14i	1.32902	−	2.30193i
$336$		0			0
$337$	7.05365e12	+	1.22173e13i	0.0883995	+	0.153112i	0.906835	−	0.421486i	$-0.138491\pi$
$337$	7.05365e12	+	1.22173e13i	0.0883995	+	0.153112i	−0.818435	+	0.574599i	$0.805158\pi$
$338$	2.19263e12	+	3.79775e12i	0.0270349	+	0.0468258i
$339$		0			0
$340$	1.16133e14	−	2.01149e14i	1.38619	−	2.40096i
$341$	1.53465e14			1.80243
$342$		0			0
$343$	9.35436e13			1.06389
$344$	−2.59035e12	+	4.48661e12i	−0.0289926	+	0.0502166i
$345$		0			0
$346$	−8.23278e12	−	1.42596e13i	−0.0892540	−	0.154592i
$347$	9.19204e12	+	1.59211e13i	0.0980844	+	0.169887i	0.910892	−	0.412645i	$-0.135395\pi$
$347$	9.19204e12	+	1.59211e13i	0.0980844	+	0.169887i	−0.812807	+	0.582533i	$0.802062\pi$
$348$		0			0
$349$	−4.67968e13	+	8.10545e13i	−0.483812	+	0.837987i	−0.999827	−	0.0185927i	$-0.994081\pi$
$349$	−4.67968e13	+	8.10545e13i	−0.483812	+	0.837987i	0.516015	+	0.856579i	$0.327415\pi$
$350$	−1.97095e13			−0.200586
$351$		0			0
$352$	4.52529e13			0.446336
$353$	−2.21207e13	+	3.83142e13i	−0.214802	+	0.372048i	−0.953211	−	0.302305i	$-0.902244\pi$
$353$	−2.21207e13	+	3.83142e13i	−0.214802	+	0.372048i	0.738409	+	0.674353i	$0.235577\pi$
$354$		0			0
$355$	1.25380e14	+	2.17165e14i	1.18025	+	2.04425i
$356$	1.66222e13	+	2.87905e13i	0.154069	+	0.266855i
$357$		0			0
$358$	7.99652e12	−	1.38504e13i	0.0718695	−	0.124482i
$359$	8.94627e13			0.791813			0.395907	−	0.918291i	$-0.370430\pi$
$359$	8.94627e13			0.791813			0.395907	+	0.918291i	$0.370430\pi$
$360$		0			0
$361$	−1.05243e14			−0.903447
$362$	−6.37402e12	+	1.10401e13i	−0.0538910	+	0.0933420i
$363$		0			0
$364$	−3.16082e13	−	5.47470e13i	−0.259264	−	0.449059i
$365$	−6.25740e13	−	1.08381e14i	−0.505572	−	0.875677i
$366$		0			0
$367$	1.13009e13	−	1.95738e13i	0.0886034	−	0.153466i	−0.818318	−	0.574766i	$-0.805093\pi$
$367$	1.13009e13	−	1.95738e13i	0.0886034	−	0.153466i	0.906921	+	0.421301i	$0.138426\pi$
$368$	−4.31057e13			−0.332945
$369$		0			0
$370$	−2.04369e13			−0.153217
$371$	3.95703e12	−	6.85377e12i	0.0292289	−	0.0506260i
$372$		0			0
$373$	1.64227e13	+	2.84449e13i	0.117773	+	0.203989i	0.918885	−	0.394526i	$-0.129091\pi$
$373$	1.64227e13	+	2.84449e13i	0.117773	+	0.203989i	−0.801112	+	0.598515i	$0.795758\pi$
$374$	1.64143e13	+	2.84304e13i	0.115992	+	0.200904i
$375$		0			0
$376$	1.97573e13	−	3.42206e13i	0.135580	−	0.234831i
$377$	−1.53390e14			−1.03734
$378$		0			0
$379$	−8.18834e13			−0.537874			−0.268937	−	0.963158i	$-0.586672\pi$
$379$	−8.18834e13			−0.537874			−0.268937	+	0.963158i	$0.586672\pi$
$380$	4.34758e13	−	7.53022e13i	0.281473	−	0.487526i
$381$		0			0
$382$	−2.40278e12	−	4.16173e12i	−0.0151135	−	0.0261773i
$383$	1.93612e13	+	3.35345e13i	0.120043	+	0.207921i	0.919785	−	0.392424i	$-0.128363\pi$
$383$	1.93612e13	+	3.35345e13i	0.120043	+	0.207921i	−0.799741	+	0.600345i	$0.795030\pi$
$384$		0			0
$385$	−1.39490e14	+	2.41604e14i	−0.840445	+	1.45569i
$386$	−9.58096e12			−0.0569087
$387$		0			0
$388$	−5.56183e13			−0.321102
$389$	−1.25714e14	+	2.17743e14i	−0.715585	+	1.23943i	0.247149	+	0.968978i	$0.420506\pi$
$389$	−1.25714e14	+	2.17743e14i	−0.715585	+	1.23943i	−0.962734	+	0.270452i	$0.912827\pi$
$390$		0			0
$391$	−4.80413e13	−	8.32100e13i	−0.265854	−	0.460472i
$392$	−1.05588e13	−	1.82884e13i	−0.0576158	−	0.0997934i
$393$		0			0
$394$	−1.97707e13	+	3.42438e13i	−0.104904	+	0.181699i
$395$	−3.00909e14			−1.57453
$396$		0			0
$397$	9.79611e13			0.498547			0.249273	−	0.968433i	$-0.419808\pi$
$397$	9.79611e13			0.498547			0.249273	+	0.968433i	$0.419808\pi$
$398$	−3.20409e12	+	5.54964e12i	−0.0160823	+	0.0278553i
$399$		0			0
$400$	−2.33142e14	−	4.03813e14i	−1.13839	−	1.97174i
$401$	2.28135e13	+	3.95141e13i	0.109875	+	0.190308i	0.915719	−	0.401818i	$-0.131622\pi$
$401$	2.28135e13	+	3.95141e13i	0.109875	+	0.190308i	−0.805845	+	0.592127i	$0.798288\pi$
$402$		0			0
$403$	−1.10551e14	+	1.91480e14i	−0.518065	+	0.897314i
$404$	2.92097e14			1.35030
$405$		0			0
$406$	2.64193e13			0.118858
$407$	−1.01796e14	+	1.76315e14i	−0.451814	+	0.782565i
$408$		0			0
$409$	−9.70854e13	−	1.68157e14i	−0.419446	−	0.726501i	0.576438	−	0.817141i	$-0.304442\pi$
$409$	−9.70854e13	−	1.68157e14i	−0.419446	−	0.726501i	−0.995884	+	0.0906398i	$0.971109\pi$
$410$	−1.26112e13	−	2.18432e13i	−0.0537581	−	0.0931117i
$411$		0			0
$412$	3.96744e13	−	6.87181e13i	0.164655	−	0.285191i
$413$	1.66913e14			0.683542
$414$		0			0
$415$	−4.67261e14			−1.86335
$416$	−3.25987e13	+	5.64626e13i	−0.128288	+	0.222202i
$417$		0			0
$418$	6.14486e12	+	1.06432e13i	0.0235528	+	0.0407946i
$419$	−5.38261e13	−	9.32295e13i	−0.203618	−	0.352676i	0.746074	−	0.665863i	$-0.231937\pi$
$419$	−5.38261e13	−	9.32295e13i	−0.203618	−	0.352676i	−0.949691	+	0.313187i	$0.898603\pi$
$420$		0			0
$421$	−5.06974e11	+	8.78105e11i	−0.00186825	+	0.00323590i	−0.866958	−	0.498381i	$-0.833928\pi$
$421$	−5.06974e11	+	8.78105e11i	−0.00186825	+	0.00323590i	0.865090	+	0.501617i	$0.167261\pi$
$422$	3.26277e13			0.118678
$423$		0			0
$424$	−5.42862e12			−0.0192388
$425$	5.19672e14	−	9.00098e14i	1.81798	−	3.14884i
$426$		0			0
$427$	6.20460e13	+	1.07467e14i	0.211524	+	0.366371i
$428$	7.74681e13	+	1.34179e14i	0.260725	+	0.451589i
$429$		0			0
$430$	−8.17682e12	+	1.41627e13i	−0.0268230	+	0.0464588i
$431$	1.63749e13			0.0530338			0.0265169	−	0.999648i	$-0.491558\pi$
$431$	1.63749e13			0.0530338			0.0265169	+	0.999648i	$0.491558\pi$
$432$		0			0
$433$	2.85219e13			0.0900524			0.0450262	−	0.998986i	$-0.485663\pi$
$433$	2.85219e13			0.0900524			0.0450262	+	0.998986i	$0.485663\pi$
$434$	1.90408e13	−	3.29796e13i	0.0593597	−	0.102814i
$435$		0			0
$436$	−2.45779e14	−	4.25702e14i	−0.747083	−	1.29399i
$437$	−1.79848e13	−	3.11505e13i	−0.0539830	−	0.0935012i
$438$		0			0
$439$	2.84649e14	−	4.93027e14i	0.833212	−	1.44317i	−0.0622659	−	0.998060i	$-0.519833\pi$
$439$	2.84649e14	−	4.93027e14i	0.833212	−	1.44317i	0.895478	−	0.445106i	$-0.146834\pi$
$440$	1.91365e14			0.553189
$441$		0			0
$442$	−4.72972e13			−0.133356
$443$	−1.96977e14	+	3.41174e14i	−0.548523	+	0.950070i	0.449853	+	0.893103i	$0.351476\pi$
$443$	−1.96977e14	+	3.41174e14i	−0.548523	+	0.950070i	−0.998376	+	0.0569675i	$0.981857\pi$
$444$		0			0
$445$	1.05686e14	+	1.83053e14i	0.287102	+	0.497274i
$446$	−2.41449e13	−	4.18201e13i	−0.0647863	−	0.112213i
$447$		0			0
$448$	−1.25007e14	+	2.16519e14i	−0.327270	+	0.566848i
$449$	−2.91972e14			−0.755068			−0.377534	−	0.925996i	$-0.623228\pi$
$449$	−2.91972e14			−0.755068			−0.377534	+	0.925996i	$0.623228\pi$
$450$		0			0
$451$	−2.51263e14			−0.634099
$452$	−4.71973e13	+	8.17482e13i	−0.117667	+	0.203806i
$453$		0			0
$454$	9.32327e12	+	1.61484e13i	0.0226862	+	0.0392936i
$455$	−2.00968e14	−	3.48086e14i	−0.483130	−	0.836805i
$456$		0			0
$457$	−1.20156e13	+	2.08116e13i	−0.0281971	+	0.0488389i	−0.879780	−	0.475382i	$-0.842310\pi$
$457$	−1.20156e13	+	2.08116e13i	−0.0281971	+	0.0488389i	0.851582	+	0.524221i	$0.175643\pi$
$458$	−2.19708e13			−0.0509432
$459$		0			0
$460$	−2.78071e14			−0.629489
$461$	−3.36056e14	+	5.82065e14i	−0.751720	+	1.30202i	0.195269	+	0.980750i	$0.437442\pi$
$461$	−3.36056e14	+	5.82065e14i	−0.751720	+	1.30202i	−0.946989	+	0.321267i	$0.895891\pi$
$462$		0			0
$463$	2.28194e14	+	3.95244e14i	0.498435	+	0.863315i	0.999998	−	0.00180573i	$-0.000574782\pi$
$463$	2.28194e14	+	3.95244e14i	0.498435	+	0.863315i	−0.501563	+	0.865121i	$0.667241\pi$
$464$	3.12511e14	+	5.41284e14i	0.674553	+	1.16836i
$465$		0			0
$466$	1.34886e13	−	2.33630e13i	0.0284345	−	0.0492500i
$467$	1.85626e14			0.386720			0.193360	−	0.981128i	$-0.438061\pi$
$467$	1.85626e14			0.386720			0.193360	+	0.981128i	$0.438061\pi$
$468$		0			0
$469$	−5.07408e14			−1.03254
$470$	6.23668e13	−	1.08023e14i	0.125434	−	0.217258i
$471$		0			0
$472$	−5.72468e13	−	9.91543e13i	−0.112479	−	0.194819i
$473$	8.14568e13	+	1.41087e14i	0.158194	+	0.274001i
$474$		0			0
$475$	1.94545e14	−	3.36961e14i	0.369151	−	0.639389i
$476$	−5.74166e14			−1.07696
$477$		0			0
$478$	8.23897e13			0.151015
$479$	−1.83959e14	+	3.18627e14i	−0.333332	+	0.577347i	−0.983163	−	0.182731i	$-0.941506\pi$
$479$	−1.83959e14	+	3.18627e14i	−0.333332	+	0.577347i	0.649831	+	0.760078i	$0.274840\pi$
$480$		0			0
$481$	−1.46660e14	−	2.54023e14i	−0.259725	−	0.449858i
$482$	3.02374e12	+	5.23728e12i	0.00529403	+	0.00916953i
$483$		0			0
$484$	1.85163e14	−	3.20711e14i	0.316887	−	0.548864i
$485$	−3.53626e14			−0.598362
$486$		0			0
$487$	−6.47703e14			−1.07144			−0.535718	−	0.844397i	$-0.679959\pi$
$487$	−6.47703e14			−1.07144			−0.535718	+	0.844397i	$0.679959\pi$
$488$	4.25602e13	−	7.37165e13i	0.0696138	−	0.120575i
$489$		0			0
$490$	−3.33304e13	−	5.77300e13i	−0.0533044	−	0.0923258i
$491$	2.73516e14	+	4.73744e14i	0.432548	+	0.749195i	0.997092	−	0.0762077i	$-0.0242812\pi$
$491$	2.73516e14	+	4.73744e14i	0.432548	+	0.749195i	−0.564544	+	0.825403i	$0.690948\pi$
$492$		0			0
$493$	−6.96586e14	+	1.20652e15i	−1.07725	+	1.86585i
$494$	−1.77062e13			−0.0270787
$495$		0			0
$496$	9.00926e14			1.34753
$497$	3.09940e14	−	5.36833e14i	0.458478	−	0.794106i
$498$		0			0
$499$	5.18322e14	+	8.97760e14i	0.749975	+	1.29899i	0.947834	+	0.318764i	$0.103268\pi$
$499$	5.18322e14	+	8.97760e14i	0.749975	+	1.29899i	−0.197860	+	0.980230i	$0.563399\pi$
$500$	−8.70996e14	−	1.50861e15i	−1.24647	−	2.15895i
$501$		0			0
$502$	6.47310e13	−	1.12117e14i	0.0906237	−	0.156965i
$503$	8.18026e14			1.13277			0.566387	−	0.824139i	$-0.308341\pi$
$503$	8.18026e14			1.13277			0.566387	+	0.824139i	$0.308341\pi$
$504$		0			0
$505$	1.85718e15			2.51623
$506$	1.96513e13	−	3.40370e13i	0.0263368	−	0.0456167i
$507$		0			0
$508$	2.63315e14	+	4.56076e14i	0.345323	+	0.598118i
$509$	1.52241e13	+	2.63689e13i	0.0197508	+	0.0342093i	0.875732	−	0.482798i	$-0.160379\pi$
$509$	1.52241e13	+	2.63689e13i	0.0197508	+	0.0342093i	−0.855981	+	0.517007i	$0.827046\pi$
$510$		0			0
$511$	−1.54683e14	+	2.67920e14i	−0.196394	+	0.340165i
$512$	4.44860e14			0.558777
$513$		0			0
$514$	−1.43388e14			−0.176285
$515$	2.52253e14	−	4.36915e14i	0.306829	−	0.531444i
$516$		0			0
$517$	−6.21293e14	−	1.07611e15i	−0.739773	−	1.28133i
$518$	2.52601e13	+	4.37518e13i	0.0297593	+	0.0515445i
$519$		0			0
$520$	−1.37853e14	+	2.38769e14i	−0.159000	+	0.275397i
$521$	5.08860e14			0.580752			0.290376	−	0.956913i	$-0.406220\pi$
$521$	5.08860e14			0.580752			0.290376	+	0.956913i	$0.406220\pi$
$522$		0			0
$523$	1.56904e15			1.75337			0.876686	−	0.481062i	$-0.159749\pi$
$523$	1.56904e15			1.75337			0.876686	+	0.481062i	$0.159749\pi$
$524$	1.11325e14	−	1.92821e14i	0.123104	−	0.213222i
$525$		0			0
$526$	2.79641e13	+	4.84352e13i	0.0302816	+	0.0524492i
$527$	1.00408e15	+	1.73912e15i	1.07600	+	1.86368i
$528$		0			0
$529$	4.18890e14	−	7.25538e14i	0.439636	−	0.761472i
$530$	−1.71363e13			−0.0177991
$531$		0			0
$532$	−2.14945e14			−0.218682
$533$	1.81001e14	−	3.13503e14i	0.182256	−	0.315676i
$534$		0			0
$535$	4.92549e14	+	8.53120e14i	0.485852	+	0.841520i
$536$	1.74027e14	+	3.01424e14i	0.169907	+	0.294287i
$537$		0			0
$538$	−2.70903e13	+	4.69217e13i	−0.0259126	+	0.0448819i
$539$	−6.64070e14			−0.628747
$540$		0			0
$541$	1.48428e15			1.37699			0.688496	−	0.725240i	$-0.258271\pi$
$541$	1.48428e15			1.37699			0.688496	+	0.725240i	$0.258271\pi$
$542$	5.54260e13	−	9.60006e13i	0.0508999	−	0.0881613i
$543$		0			0
$544$	2.96079e14	+	5.12824e14i	0.266449	+	0.461503i
$545$	−1.56268e15	−	2.70665e15i	−1.39216	−	2.41130i
$546$		0			0
$547$	5.46153e14	−	9.45965e14i	0.476853	−	0.825933i	−0.522796	−	0.852458i	$-0.675111\pi$
$547$	5.46153e14	−	9.45965e14i	0.476853	−	0.825933i	0.999648	+	0.0265251i	$0.00844420\pi$
$548$	−1.50339e15			−1.29951
$549$		0			0
$550$	4.25143e14			0.360197
$551$	−2.60774e14	+	4.51674e14i	−0.218741	+	0.378871i
$552$		0			0
$553$	3.71925e14	+	6.44192e14i	0.305821	+	0.529697i
$554$	−2.36163e13	−	4.09046e13i	−0.0192268	−	0.0333018i
$555$		0			0
$556$	5.64869e14	−	9.78382e14i	0.450854	−	0.780903i
$557$	1.37505e15			1.08671			0.543356	−	0.839503i	$-0.317153\pi$
$557$	1.37505e15			1.08671			0.543356	+	0.839503i	$0.317153\pi$
$558$		0			0
$559$	−2.34715e14			−0.181876
$560$	−8.18887e14	+	1.41835e15i	−0.628333	+	1.08830i
$561$		0			0
$562$	1.60679e12	+	2.78305e12i	0.00120895	+	0.00209397i
$563$	3.14519e14	+	5.44763e14i	0.234342	+	0.405893i	0.959081	−	0.283131i	$-0.0913730\pi$
$563$	3.14519e14	+	5.44763e14i	0.234342	+	0.405893i	−0.724739	+	0.689023i	$0.758040\pi$
$564$		0			0
$565$	−3.00085e14	+	5.19762e14i	−0.219269	+	0.379785i
$566$	−6.06520e13			−0.0438889
$567$		0			0
$568$	−4.25205e14			−0.301775
$569$	−1.74567e14	+	3.02358e14i	−0.122700	+	0.212522i	−0.920831	−	0.389961i	$-0.872489\pi$
$569$	−1.74567e14	+	3.02358e14i	−0.122700	+	0.212522i	0.798132	+	0.602483i	$0.205822\pi$
$570$		0			0
$571$	−1.11876e15	−	1.93774e15i	−0.771324	−	1.33597i	−0.936837	−	0.349765i	$-0.886261\pi$
$571$	−1.11876e15	−	1.93774e15i	−0.771324	−	1.33597i	0.165513	−	0.986208i	$-0.447072\pi$
$572$	6.81804e14	+	1.18092e15i	0.465566	+	0.806385i
$573$		0			0
$574$	−3.11749e13	+	5.39964e13i	−0.0208828	+	0.0361701i
$575$	−1.24431e15			−0.825572
$576$		0			0
$577$	−6.58500e14			−0.428636			−0.214318	−	0.976764i	$-0.568753\pi$
$577$	−6.58500e14			−0.428636			−0.214318	+	0.976764i	$0.568753\pi$
$578$	−1.23067e14	+	2.13159e14i	−0.0793485	+	0.137436i
$579$		0			0
$580$	2.01598e15	+	3.49178e15i	1.27536	+	2.20898i
$581$	5.77536e14	+	1.00032e15i	0.361918	+	0.626861i
$582$		0			0
$583$	−8.53550e13	+	1.47839e14i	−0.0524870	+	0.0909102i
$584$	2.12209e14			0.129269
$585$		0			0
$586$	−2.15198e14			−0.128648
$587$	−7.70528e14	+	1.33459e15i	−0.456330	+	0.790386i	−0.998764	−	0.0497124i	$-0.984170\pi$
$587$	−7.70528e14	+	1.33459e15i	−0.456330	+	0.790386i	0.542434	+	0.840098i	$0.317503\pi$
$588$		0			0
$589$	3.75888e14	+	6.51058e14i	0.218487	+	0.378430i
$590$	−1.80708e14	−	3.12995e14i	−0.104062	−	0.180240i
$591$		0			0
$592$	−5.97599e14	+	1.03507e15i	−0.337785	+	0.585061i
$593$	−1.02033e15			−0.571398			−0.285699	−	0.958319i	$-0.592226\pi$
$593$	−1.02033e15			−0.571398			−0.285699	+	0.958319i	$0.592226\pi$
$594$		0			0
$595$	−3.65060e15			−2.00688
$596$	1.27754e15	−	2.21276e15i	0.695854	−	1.20526i
$597$		0			0
$598$	2.83122e13	+	4.90382e13i	0.0151397	+	0.0262227i
$599$	−4.15264e14	−	7.19258e14i	−0.220027	−	0.381098i	0.734789	−	0.678296i	$-0.237281\pi$
$599$	−4.15264e14	−	7.19258e14i	−0.220027	−	0.381098i	−0.954816	+	0.297198i	$0.903948\pi$
$600$		0			0
$601$	−7.36989e12	+	1.27650e13i	−0.00383399	+	0.00664067i	−0.867936	−	0.496676i	$-0.834554\pi$
$601$	−7.36989e12	+	1.27650e13i	−0.00383399	+	0.00664067i	0.864102	+	0.503317i	$0.167887\pi$
$602$	4.04263e13			0.0208393
$603$		0			0
$604$	−2.43422e15			−1.23213
$605$	1.17728e15	−	2.03911e15i	0.590507	−	1.02279i
$606$		0			0
$607$	−4.25308e14	−	7.36655e14i	−0.209491	−	0.362850i	0.742063	−	0.670330i	$-0.233847\pi$
$607$	−4.25308e14	−	7.36655e14i	−0.209491	−	0.362850i	−0.951554	+	0.307481i	$0.900514\pi$
$608$	1.10840e14	+	1.91981e14i	0.0541038	+	0.0937105i
$609$		0			0
$610$	1.34348e14	−	2.32697e14i	0.0644045	−	0.111552i
$611$	1.79024e15			0.850518
$612$		0			0
$613$	2.88157e15			1.34461			0.672306	−	0.740274i	$-0.265304\pi$
$613$	2.88157e15			1.34461			0.672306	+	0.740274i	$0.265304\pi$
$614$	−2.00305e14	+	3.46938e14i	−0.0926330	+	0.160445i
$615$		0			0
$616$	−2.36528e14	−	4.09679e14i	−0.107446	−	0.186102i
$617$	−1.31694e15	−	2.28100e15i	−0.592920	−	1.02697i	−0.993837	−	0.110853i	$-0.964642\pi$
$617$	−1.31694e15	−	2.28100e15i	−0.592920	−	1.02697i	0.400916	−	0.916115i	$-0.368692\pi$
$618$		0			0
$619$	8.69568e14	−	1.50614e15i	0.384596	−	0.666140i	−0.607117	−	0.794612i	$-0.707674\pi$
$619$	8.69568e14	−	1.50614e15i	0.384596	−	0.666140i	0.991713	+	0.128472i	$0.0410074\pi$
$620$	5.81180e15			2.54775
$621$		0			0
$622$	−1.72697e14			−0.0743767
$623$	2.61255e14	−	4.52508e14i	0.111527	−	0.193171i
$624$		0			0
$625$	−2.70542e15	−	4.68593e15i	−1.13474	−	1.96542i
$626$	−1.67218e14	−	2.89631e14i	−0.0695225	−	0.120416i
$627$		0			0
$628$	−2.05317e15	+	3.55620e15i	−0.838779	+	1.45281i
$629$	−2.66409e15			−1.07887
$630$		0			0
$631$	−8.33659e14			−0.331762			−0.165881	−	0.986146i	$-0.553047\pi$
$631$	−8.33659e14			−0.331762			−0.165881	+	0.986146i	$0.553047\pi$
$632$	2.55120e14	−	4.41882e14i	0.100647	−	0.174326i
$633$		0			0
$634$	−2.79437e14	−	4.83998e14i	−0.108341	−	0.187652i
$635$	1.67418e15	+	2.89977e15i	0.643498	+	1.11457i
$636$		0			0
$637$	4.78374e14	−	8.28568e14i	0.180718	−	0.313012i
$638$	−5.69876e14			−0.213436
$639$		0			0
$640$	2.27942e15			0.839142
$641$	−2.19992e14	+	3.81037e14i	−0.0802948	+	0.139075i	−0.903377	−	0.428848i	$-0.858920\pi$
$641$	−2.19992e14	+	3.81037e14i	−0.0802948	+	0.139075i	0.823082	+	0.567923i	$0.192253\pi$
$642$		0			0
$643$	−2.57010e15	−	4.45155e15i	−0.922126	−	1.59717i	−0.796119	−	0.605139i	$-0.793117\pi$
$643$	−2.57010e15	−	4.45155e15i	−0.922126	−	1.59717i	−0.126006	−	0.992029i	$-0.540216\pi$
$644$	3.43697e14	+	5.95301e14i	0.122265	+	0.211770i
$645$		0			0
$646$	−8.04086e13	+	1.39272e14i	−0.0281205	+	0.0487062i
$647$	−3.06586e15			−1.06311			−0.531555	−	0.847024i	$-0.678392\pi$
$647$	−3.06586e15			−1.06311			−0.531555	+	0.847024i	$0.678392\pi$
$648$		0			0
$649$	−3.60040e15			−1.22745
$650$	−3.06259e14	+	5.30456e14i	−0.103530	+	0.179319i
$651$		0			0
$652$	−3.18676e14	−	5.51963e14i	−0.105922	−	0.183463i
$653$	−3.77335e14	−	6.53564e14i	−0.124367	−	0.215410i	0.797118	−	0.603823i	$-0.206357\pi$
$653$	−3.77335e14	−	6.53564e14i	−0.124367	−	0.215410i	−0.921485	+	0.388413i	$0.873023\pi$
$654$		0			0
$655$	7.07815e14	−	1.22597e15i	0.229400	−	0.397332i
$656$	−1.47506e15			−0.474065
$657$		0			0
$658$	−3.08343e14			−0.0974521
$659$	−1.24077e15	+	2.14907e15i	−0.388884	+	0.673567i	−0.992300	−	0.123860i	$-0.960473\pi$
$659$	−1.24077e15	+	2.14907e15i	−0.388884	+	0.673567i	0.603416	+	0.797427i	$0.293806\pi$
$660$		0			0
$661$	−3.53113e14	−	6.11609e14i	−0.108844	−	0.188524i	0.806458	−	0.591291i	$-0.201382\pi$
$661$	−3.53113e14	−	6.11609e14i	−0.108844	−	0.188524i	−0.915302	+	0.402768i	$0.868048\pi$
$662$	2.09020e14	+	3.62034e14i	0.0638954	+	0.110670i
$663$		0			0
$664$	3.96159e14	−	6.86168e14i	0.119109	−	0.206303i
$665$	−1.36664e15			−0.407506
$666$		0			0
$667$	1.66791e15			0.489194
$668$	−1.48144e15	+	2.56593e15i	−0.430937	+	0.746405i
$669$		0			0
$670$	5.49343e14	+	9.51490e14i	0.157193	+	0.272266i
$671$	−1.33836e15	−	2.31811e15i	−0.379839	−	0.657901i
$672$		0			0
$673$	−2.58308e15	+	4.47403e15i	−0.721200	+	1.24915i	0.239319	+	0.970941i	$0.423076\pi$
$673$	−2.58308e15	+	4.47403e15i	−0.721200	+	1.24915i	−0.960519	+	0.278214i	$0.910258\pi$
$674$	−7.55111e13			−0.0209113
$675$		0			0
$676$	1.65440e15			0.450748
$677$	−1.61865e15	+	2.80359e15i	−0.437438	+	0.757665i	−0.997491	−	0.0707921i	$-0.977447\pi$
$677$	−1.61865e15	+	2.80359e15i	−0.437438	+	0.757665i	0.560053	+	0.828457i	$0.310781\pi$
$678$		0			0
$679$	4.37083e14	+	7.57049e14i	0.116220	+	0.201298i
$680$	1.25206e15	+	2.16863e15i	0.330236	+	0.571986i
$681$		0			0
$682$	−4.10719e14	+	7.11386e14i	−0.106594	+	0.184625i
$683$	6.24600e15			1.60801			0.804003	−	0.594625i	$-0.202699\pi$
$683$	6.24600e15			1.60801			0.804003	+	0.594625i	$0.202699\pi$
$684$		0			0
$685$	−9.55869e15			−2.42159
$686$	−2.50352e14	+	4.33622e14i	−0.0629170	+	0.108975i
$687$		0			0
$688$	4.78198e14	+	8.28264e14i	0.118269	+	0.204848i
$689$	−1.22974e14	−	2.12997e14i	−0.0301722	−	0.0522597i
$690$		0			0
$691$	3.24923e15	−	5.62783e15i	0.784605	−	1.35898i	−0.144630	−	0.989486i	$-0.546199\pi$
$691$	3.24923e15	−	5.62783e15i	0.784605	−	1.35898i	0.929235	−	0.369490i	$-0.120468\pi$
$692$	−6.21185e15			−1.48812
$693$		0			0
$694$	−9.84031e13			−0.0232023
$695$	3.59149e15	−	6.22064e15i	0.840152	−	1.45519i
$696$		0			0
$697$	−1.64395e15	−	2.84740e15i	−0.378537	−	0.655645i
$698$	−2.50486e14	−	4.33854e14i	−0.0572240	−	0.0991148i
$699$		0			0
$700$	−3.71784e15	+	6.43948e15i	−0.836086	+	1.44814i
$701$	−1.70842e14			−0.0381194			−0.0190597	−	0.999818i	$-0.506067\pi$
$701$	−1.70842e14			−0.0381194			−0.0190597	+	0.999818i	$0.506067\pi$
$702$		0			0
$703$	−9.97331e14			−0.219071
$704$	2.69647e15	−	4.67042e15i	0.587686	−	1.01790i
$705$		0			0
$706$	−1.18404e14	−	2.05081e14i	−0.0254062	−	0.0440049i
$707$	−2.29548e15	−	3.97588e15i	−0.488727	−	0.846500i
$708$		0			0
$709$	2.09960e15	−	3.63661e15i	0.440130	−	0.762328i	−0.557568	−	0.830131i	$-0.688266\pi$
$709$	2.09960e15	−	3.63661e15i	0.440130	−	0.762328i	0.997699	+	0.0678028i	$0.0215989\pi$
$710$	−1.34222e15			−0.279193
$711$		0			0
$712$	−3.58415e14			−0.0734084
$713$	1.20209e15	−	2.08208e15i	0.244312	−	0.423161i
$714$		0			0
$715$	4.33497e15	+	7.50839e15i	0.867567	+	1.50267i
$716$	−3.01680e15	−	5.22524e15i	−0.599135	−	1.03773i
$717$		0			0
$718$	−2.39430e14	+	4.14705e14i	−0.0468268	+	0.0811064i
$719$	7.52148e15			1.45980			0.729901	−	0.683553i	$-0.239566\pi$
$719$	7.52148e15			1.45980			0.729901	+	0.683553i	$0.239566\pi$
$720$		0			0
$721$	−1.24714e15			−0.238382
$722$	2.81662e14	−	4.87854e14i	0.0534287	−	0.0925411i
$723$		0			0
$724$	2.40469e15	+	4.16504e15i	0.449259	+	0.778139i
$725$	9.02106e15	+	1.56249e16i	1.67263	+	2.89707i
$726$		0			0
$727$	−3.62296e14	+	6.27514e14i	−0.0661643	+	0.114600i	−0.897210	−	0.441604i	$-0.854409\pi$
$727$	−3.62296e14	+	6.27514e14i	−0.0661643	+	0.114600i	0.831046	+	0.556204i	$0.187743\pi$
$728$	6.81548e14			0.123530
$729$		0			0
$730$	6.69870e14			0.119596
$731$	−1.06590e15	+	1.84620e15i	−0.188874	+	0.327140i
$732$		0			0
$733$	9.69462e14	+	1.67916e15i	0.169223	+	0.293103i	0.938147	−	0.346238i	$-0.112541\pi$
$733$	9.69462e14	+	1.67916e15i	0.169223	+	0.293103i	−0.768924	+	0.639340i	$0.779208\pi$
$734$	6.04896e13	+	1.04771e14i	0.0104798	+	0.0181515i
$735$		0			0
$736$	3.54467e14	−	6.13955e14i	0.0604990	−	0.104787i
$737$	1.09450e16			1.85415
$738$		0			0
$739$	−3.45034e15			−0.575861			−0.287930	−	0.957651i	$-0.592967\pi$
$739$	−3.45034e15			−0.575861			−0.287930	+	0.957651i	$0.592967\pi$
$740$	−3.85506e15	+	6.67716e15i	−0.638640	+	1.10616i
$741$		0			0
$742$	2.11805e13	+	3.66857e13i	0.00345712	+	0.00598790i
$743$	−4.87568e15	−	8.44493e15i	−0.789945	−	1.36822i	−0.926000	−	0.377524i	$-0.876776\pi$
$743$	−4.87568e15	−	8.44493e15i	−0.789945	−	1.36822i	0.136055	−	0.990701i	$-0.456558\pi$
$744$		0			0
$745$	8.12268e15	−	1.40689e16i	1.29670	−	2.24595i
$746$	−1.75809e14			−0.0278597
$747$		0			0
$748$	1.23850e16			1.93392
$749$	1.21759e15	−	2.10892e15i	0.188734	−	0.326896i
$750$		0			0
$751$	−2.61949e15	−	4.53708e15i	−0.400126	−	0.693038i	0.593615	−	0.804749i	$-0.297700\pi$
$751$	−2.61949e15	−	4.53708e15i	−0.400126	−	0.693038i	−0.993741	+	0.111711i	$0.964367\pi$
$752$	−3.64735e15	−	6.31740e15i	−0.553069	−	0.957944i
$753$		0			0
$754$	4.10520e14	−	7.11041e14i	0.0613467	−	0.106256i
$755$	−1.54770e16			−2.29604
$756$		0			0
$757$	7.83764e15			1.14593			0.572965	−	0.819580i	$-0.305793\pi$
$757$	7.83764e15			1.14593			0.572965	+	0.819580i	$0.305793\pi$
$758$	2.19145e14	−	3.79571e14i	0.0318091	−	0.0550950i
$759$		0			0
$760$	4.68720e14	+	8.11848e14i	0.0670562	+	0.116145i
$761$	4.03075e15	+	6.98147e15i	0.572494	+	0.991588i	0.996309	+	0.0858395i	$0.0273572\pi$
$761$	4.03075e15	+	6.98147e15i	0.572494	+	0.991588i	−0.423815	+	0.905749i	$0.639309\pi$
$762$		0			0
$763$	−3.86296e15	+	6.69085e15i	−0.540799	+	0.936691i
$764$	−1.81296e15			−0.251985
$765$		0			0
$766$	−2.07266e14			−0.0283968
$767$	2.59360e15	−	4.49226e15i	0.352801	−	0.611069i
$768$		0			0
$769$	6.20915e15	+	1.07546e16i	0.832602	+	1.44211i	0.895968	+	0.444118i	$0.146483\pi$
$769$	6.20915e15	+	1.07546e16i	0.832602	+	1.44211i	−0.0633665	+	0.997990i	$0.520184\pi$
$770$	−7.46637e14	−	1.29321e15i	−0.0994056	−	0.172175i
$771$		0			0
$772$	−1.80727e15	+	3.13029e15i	−0.237208	+	0.410856i
$773$	−9.96695e15			−1.29890			−0.649449	−	0.760405i	$-0.725000\pi$
$773$	−9.96695e15			−1.29890			−0.649449	+	0.760405i	$0.725000\pi$
$774$		0			0
$775$	2.60065e16			3.34136
$776$	2.99815e14	−	5.19296e14i	0.0382485	−	0.0662484i
$777$		0			0
$778$	−6.72900e14	−	1.16550e15i	−0.0846375	−	0.146596i
$779$	−6.15429e14	−	1.06595e15i	−0.0768638	−	0.133132i
$780$		0			0
$781$	−6.68556e15	+	1.15797e16i	−0.823299	+	1.42600i
$782$	5.14294e14			0.0628889
$783$		0			0
$784$	−3.89848e15			−0.470064
$785$	−1.30543e16	+	2.26106e16i	−1.56304	+	2.70726i
$786$		0			0
$787$	−3.08384e15	−	5.34136e15i	−0.364108	−	0.630654i	0.624524	−	0.781005i	$-0.285293\pi$
$787$	−3.08384e15	−	5.34136e15i	−0.364108	−	0.630654i	−0.988633	+	0.150351i	$0.951959\pi$
$788$	7.45876e15	+	1.29189e16i	0.874525	+	1.51472i
$789$		0			0
$790$	8.05326e14	−	1.39487e15i	0.0931157	−	0.161281i
$791$	1.48362e15			0.170354
$792$		0			0
$793$	3.85644e15			0.436701
$794$	−2.62174e14	+	4.54100e14i	−0.0294834	+	0.0510668i
$795$		0			0
$796$	1.20878e15	+	2.09368e15i	0.134069	+	0.232214i
$797$	−1.94585e15	−	3.37032e15i	−0.214333	−	0.371236i	0.738733	−	0.673998i	$-0.235424\pi$
$797$	−1.94585e15	−	3.37032e15i	−0.214333	−	0.371236i	−0.953066	+	0.302762i	$0.902091\pi$
$798$		0			0
$799$	8.12994e15	−	1.40815e16i	0.883243	−	1.52982i
$800$	7.66868e15			0.827419
$801$		0			0
$802$	−2.44224e14			−0.0259914
$803$	3.33660e15	−	5.77915e15i	0.352670	−	0.610842i
$804$		0			0
$805$	2.18526e15	+	3.78497e15i	0.227837	+	0.394626i
$806$	−5.91736e14	−	1.02492e15i	−0.0612753	−	0.106132i
$807$		0			0
$808$	−1.57457e15	+	2.72724e15i	−0.160842	+	0.278587i
$809$	1.34438e16			1.36397			0.681986	−	0.731366i	$-0.261117\pi$
$809$	1.34438e16			1.36397			0.681986	+	0.731366i	$0.261117\pi$
$810$		0			0
$811$	8.13011e15			0.813733			0.406866	−	0.913488i	$-0.366621\pi$
$811$	8.13011e15			0.813733			0.406866	+	0.913488i	$0.366621\pi$
$812$	4.98351e15	−	8.63170e15i	0.495424	−	0.858100i
$813$		0			0
$814$	−5.44873e14	−	9.43748e14i	−0.0534394	−	0.0925597i
$815$	−2.02617e15	−	3.50943e15i	−0.197382	−	0.341876i
$816$		0			0
$817$	−3.99032e14	+	6.91144e14i	−0.0383519	+	0.0664274i
$818$	1.03932e15			0.0992218
$819$		0			0
$820$	−9.51546e15			−0.896300
$821$	2.17310e15	−	3.76392e15i	0.203326	−	0.352170i	−0.746272	−	0.665641i	$-0.768158\pi$
$821$	2.17310e15	−	3.76392e15i	0.203326	−	0.352170i	0.949598	+	0.313470i	$0.101492\pi$
$822$		0			0
$823$	−1.37996e15	−	2.39016e15i	−0.127399	−	0.220662i	0.795269	−	0.606257i	$-0.207330\pi$
$823$	−1.37996e15	−	2.39016e15i	−0.127399	−	0.220662i	−0.922668	+	0.385595i	$0.873996\pi$
$824$	4.27737e14	+	7.40862e14i	0.0392263	+	0.0679419i
$825$		0			0
$826$	−4.46711e14	+	7.73727e14i	−0.0404238	+	0.0700160i
$827$	−1.63880e16			−1.47314			−0.736571	−	0.676361i	$-0.763556\pi$
$827$	−1.63880e16			−1.47314			−0.736571	+	0.676361i	$0.763556\pi$
$828$		0			0
$829$	−9.26443e15			−0.821805			−0.410902	−	0.911679i	$-0.634786\pi$
$829$	−9.26443e15			−0.821805			−0.410902	+	0.911679i	$0.634786\pi$
$830$	1.25054e15	−	2.16599e15i	0.110196	−	0.190865i
$831$		0			0
$832$	3.88489e15	+	6.72882e15i	0.337831	+	0.585141i
$833$	−4.34485e15	−	7.52550e15i	−0.375342	−	0.650112i
$834$		0			0
$835$	−9.41914e15	+	1.63144e16i	−0.803037	+	1.39090i
$836$	4.63646e15			0.392692
$837$		0			0
$838$	5.76221e14			0.0481667
$839$	9.03463e15	−	1.56484e16i	0.750273	−	1.29951i	−0.197417	−	0.980320i	$-0.563255\pi$
$839$	9.03463e15	−	1.56484e16i	0.750273	−	1.29951i	0.947690	−	0.319191i	$-0.103411\pi$
$840$		0			0
$841$	−5.99188e15	−	1.03782e16i	−0.491117	−	0.850640i
$842$	−2.71364e12	−	4.70016e12i	−0.000220971	−	0.000382733i
$843$		0			0
$844$	6.15463e15	−	1.06601e16i	0.494673	−	0.856799i
$845$	1.05188e16			0.839954
$846$		0			0
$847$	−5.82049e15			−0.458776
$848$	−5.01083e14	+	8.67902e14i	−0.0392403	+	0.0679662i
$849$		0			0
$850$	2.78161e15	+	4.81789e15i	0.215026	+	0.372437i
$851$	1.59473e15	+	2.76216e15i	0.122483	+	0.212147i
$852$		0			0
$853$	9.83630e14	−	1.70370e15i	0.0745783	−	0.129173i	−0.826325	−	0.563194i	$-0.809572\pi$
$853$	9.83630e14	−	1.70370e15i	0.0745783	−	0.129173i	0.900903	+	0.434021i	$0.142906\pi$
$854$	−6.64218e14			−0.0500371
$855$		0			0
$856$	−1.67040e15			−0.124226
$857$	−6.04333e13	+	1.04674e14i	−0.00446562	+	0.00773468i	−0.868250	−	0.496128i	$-0.834755\pi$
$857$	−6.04333e13	+	1.04674e14i	−0.00446562	+	0.00773468i	0.863784	+	0.503862i	$0.168088\pi$
$858$		0			0
$859$	6.77934e15	+	1.17422e16i	0.494567	+	0.856614i	0.999980	−	0.00626268i	$-0.00199349\pi$
$859$	6.77934e15	+	1.17422e16i	0.494567	+	0.856614i	−0.505414	+	0.862877i	$0.668660\pi$
$860$	3.08482e15	+	5.34306e15i	0.223608	+	0.387301i
$861$		0			0
$862$	−4.38242e13	+	7.59058e13i	−0.00313635	+	0.00543231i
$863$	3.86625e15			0.274935			0.137468	−	0.990506i	$-0.456104\pi$
$863$	3.86625e15			0.274935			0.137468	+	0.990506i	$0.456104\pi$
$864$		0			0
$865$	−3.94955e16			−2.77306
$866$	−7.63335e13	+	1.32213e14i	−0.00532558	+	0.00922417i
$867$		0			0
$868$	−7.18340e15	−	1.24420e16i	−0.494848	−	0.857101i
$869$	−8.02259e15	−	1.38955e16i	−0.549169	−	0.951189i
$870$		0			0
$871$	−7.88443e15	+	1.36562e16i	−0.532930	+	0.923062i
$872$	5.29958e15			0.355959
$873$		0			0
$874$	1.92531e14			0.0127699
$875$	−1.36896e16	+	2.37112e16i	−0.902294	+	1.56282i
$876$		0			0
$877$	1.15648e16	+	2.00307e16i	0.752729	+	1.30377i	0.946495	+	0.322717i	$0.104596\pi$
$877$	1.15648e16	+	2.00307e16i	0.752729	+	1.30377i	−0.193766	+	0.981048i	$0.562070\pi$
$878$	1.52362e15	+	2.63899e15i	0.0985501	+	0.170694i
$879$		0			0
$880$	1.76638e16	−	3.05946e16i	1.12831	−	1.95429i
$881$	−1.74728e16			−1.10916			−0.554581	−	0.832130i	$-0.687121\pi$
$881$	−1.74728e16			−1.10916			−0.554581	+	0.832130i	$0.687121\pi$
$882$		0			0
$883$	2.15253e16			1.34948			0.674738	−	0.738057i	$-0.264257\pi$
$883$	2.15253e16			1.34948			0.674738	+	0.738057i	$0.264257\pi$
$884$	−8.92176e15	+	1.54529e16i	−0.555857	+	0.962772i
$885$		0			0
$886$	−1.05434e15	−	1.82618e15i	−0.0648779	−	0.112372i
$887$	4.09458e15	+	7.09202e15i	0.250397	+	0.433701i	0.963635	−	0.267221i	$-0.0861055\pi$
$887$	4.09458e15	+	7.09202e15i	0.250397	+	0.433701i	−0.713238	+	0.700922i	$0.752772\pi$
$888$		0			0
$889$	4.13859e15	−	7.16825e15i	0.249973	−	0.432966i
$890$	−1.13139e15			−0.0679152
$891$		0			0
$892$	−1.82180e16			−1.08017
$893$	3.04353e15	−	5.27154e15i	0.179347	−	0.310638i
$894$		0			0
$895$	−1.91811e16	−	3.32226e16i	−1.11647	−	1.93378i
$896$	−2.81738e15	−	4.87984e15i	−0.162986	−	0.282300i
$897$		0			0
$898$	7.81407e14	−	1.35344e15i	0.0446537	−	0.0773425i
$899$	−3.48600e16			−1.97993
$900$		0			0
$901$	−2.23383e15			−0.125332
$902$	6.72457e14	−	1.16473e15i	0.0374998	−	0.0649515i
$903$		0			0
$904$	−5.08843e14	−	8.81343e14i	−0.0280322	−	0.0485533i
$905$	1.52892e16	+	2.64817e16i	0.837178	+	1.45003i
$906$		0			0
$907$	−9.30777e13	+	1.61215e14i	−0.00503507	+	0.00872100i	−0.868532	−	0.495633i	$-0.834936\pi$
$907$	−9.30777e13	+	1.61215e14i	−0.00503507	+	0.00872100i	0.863497	+	0.504354i	$0.168269\pi$
$908$	7.03466e15			0.378243
$909$		0			0
$910$	2.15141e15			0.114287
$911$	1.13431e15	−	1.96468e15i	0.0598935	−	0.103739i	−0.834524	−	0.550972i	$-0.814257\pi$
$911$	1.13431e15	−	1.96468e15i	0.0598935	−	0.103739i	0.894417	+	0.447233i	$0.147591\pi$
$912$		0			0
$913$	−1.24577e16	−	2.15774e16i	−0.649905	−	1.12567i
$914$	−6.43148e13	−	1.11396e14i	−0.00333508	−	0.00577653i
$915$		0			0
$916$	−4.14440e15	+	7.17831e15i	−0.212342	+	0.367787i
$917$	−3.49945e15			−0.178225
$918$		0			0
$919$	−8.13214e15			−0.409232			−0.204616	−	0.978842i	$-0.565595\pi$
$919$	−8.13214e15			−0.409232			−0.204616	+	0.978842i	$0.565595\pi$
$920$	1.49897e15	−	2.59629e15i	0.0749824	−	0.129873i
$921$		0			0
$922$	−1.79878e15	−	3.11558e15i	−0.0889114	−	0.153999i
$923$	−9.63211e15	−	1.66833e16i	−0.473273	−	0.819734i
$924$		0			0
$925$	−1.72505e16	+	2.98788e16i	−0.837574	+	1.45072i
$926$	−2.44287e15			−0.117907
$927$		0			0
$928$	−1.02794e16			−0.490289
$929$	9.14760e15	−	1.58441e16i	0.433732	−	0.751245i	−0.563460	−	0.826144i	$-0.690530\pi$
$929$	9.14760e15	−	1.58441e16i	0.433732	−	0.751245i	0.997191	+	0.0748986i	$0.0238633\pi$
$930$		0			0
$931$	−1.62654e15	−	2.81725e15i	−0.0762151	−	0.132008i
$932$	−5.08877e15	−	8.81401e15i	−0.237042	−	0.410569i
$933$		0			0
$934$	−4.96794e14	+	8.60473e14i	−0.0228701	+	0.0396122i
$935$	7.87451e16			3.60379
$936$		0			0
$937$	1.24402e16			0.562677			0.281338	−	0.959609i	$-0.409222\pi$
$937$	1.24402e16			0.562677			0.281338	+	0.959609i	$0.409222\pi$
$938$	1.35798e15	−	2.35209e15i	0.0610630	−	0.105764i
$939$		0			0
$940$	−2.35287e16	−	4.07530e16i	−1.04567	−	1.81116i
$941$	−1.20714e16	−	2.09083e16i	−0.533355	−	0.923797i	−0.999241	−	0.0389527i	$-0.987598\pi$
$941$	−1.20714e16	−	2.09083e16i	−0.533355	−	0.923797i	0.465886	−	0.884845i	$-0.345736\pi$
$942$		0			0
$943$	−1.96814e15	+	3.40893e15i	−0.0859494	+	0.148869i
$944$	−2.11364e16			−0.917667
$945$		0			0
$946$	−8.72015e14			−0.0374216
$947$	−4.24023e15	+	7.34429e15i	−0.180911	+	0.313347i	−0.942191	−	0.335076i	$-0.891238\pi$
$947$	−4.24023e15	+	7.34429e15i	−0.180911	+	0.313347i	0.761280	+	0.648423i	$0.224571\pi$
$948$		0			0
$949$	4.80714e15	+	8.32622e15i	0.202732	+	0.351143i
$950$	1.04132e15	+	1.80363e15i	0.0436622	+	0.0756252i
$951$		0			0
$952$	3.09509e15	−	5.36086e15i	0.128283	−	0.222193i
$953$	−1.81730e16			−0.748884			−0.374442	−	0.927250i	$-0.622166\pi$
$953$	−1.81730e16			−0.748884			−0.374442	+	0.927250i	$0.622166\pi$
$954$		0			0
$955$	−1.15270e16			−0.469565
$956$	1.55413e16	−	2.69184e16i	0.629462	−	1.09026i
$957$		0			0
$958$	−9.84665e14	−	1.70549e15i	−0.0394256	−	0.0682871i
$959$	1.18146e16	+	2.04634e16i	0.470344	+	0.814660i
$960$		0			0
$961$	−1.24199e16	+	2.15120e16i	−0.488811	+	0.846645i
$962$	1.57003e15			0.0614393
$963$		0			0
$964$	2.28150e15			0.0882666
$965$	−1.14908e16	+	1.99027e16i	−0.442028	+	0.765616i
$966$		0			0
$967$	5.24249e15	+	9.08026e15i	0.199385	+	0.345345i	0.948329	−	0.317288i	$-0.102772\pi$
$967$	5.24249e15	+	9.08026e15i	0.199385	+	0.345345i	−0.748944	+	0.662633i	$0.769439\pi$
$968$	1.99627e15	+	3.45765e15i	0.0754928	+	0.130757i
$969$		0			0
$970$	9.46412e14	−	1.63923e15i	0.0353864	−	0.0612910i
$971$	3.04177e16			1.13089			0.565445	−	0.824786i	$-0.308704\pi$
$971$	3.04177e16			1.13089			0.565445	+	0.824786i	$0.308704\pi$
$972$		0			0
$973$	−1.77564e16			−0.652730
$974$	1.73345e15	−	3.00243e15i	0.0633634	−	0.109749i
$975$		0			0
$976$	−7.85696e15	−	1.36087e16i	−0.283975	−	0.491860i
$977$	1.46347e16	+	2.53480e16i	0.525972	+	0.911011i	0.999542	+	0.0302547i	$0.00963183\pi$
$977$	1.46347e16	+	2.53480e16i	0.525972	+	0.911011i	−0.473570	+	0.880756i	$0.657035\pi$
$978$		0			0
$979$	−5.63541e15	+	9.76081e15i	−0.200272	+	0.346881i
$980$	−2.51487e16			−0.888736
$981$		0			0
$982$	−2.92806e15			−0.102321
$983$	6.21882e15	−	1.07713e16i	0.216104	−	0.374304i	−0.737509	−	0.675337i	$-0.763998\pi$
$983$	6.21882e15	−	1.07713e16i	0.216104	−	0.374304i	0.953614	+	0.301033i	$0.0973315\pi$
$984$		0			0
$985$	4.74234e16	+	8.21398e16i	1.62965	+	2.82263i
$986$	−3.72856e15	−	6.45806e15i	−0.127414	−	0.220688i
$987$		0			0
$988$	−3.33995e15	+	5.78497e15i	−0.112870	+	0.195496i
$989$	2.55221e15			0.0857703
$990$		0			0
$991$	1.42821e16			0.474666			0.237333	−	0.971428i	$-0.423727\pi$
$991$	1.42821e16			0.474666			0.237333	+	0.971428i	$0.423727\pi$
$992$	−7.40850e15	+	1.28319e16i	−0.244859	+	0.424108i
$993$		0			0
$994$	1.65899e15	+	2.87346e15i	0.0542275	+	0.0939248i
$995$	7.68556e15	+	1.33118e16i	0.249832	+	0.432722i
$996$		0			0
$997$	−4.55514e15	+	7.88974e15i	−0.146446	+	0.253653i	−0.929912	−	0.367783i	$-0.880117\pi$
$997$	−4.55514e15	+	7.88974e15i	−0.146446	+	0.253653i	0.783465	+	0.621436i	$0.213450\pi$
$998$	−5.54876e15			−0.177410
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.12.c.i.55.2		8
3.2	odd	2	inner	81.12.c.i.55.3		8
9.2	odd	6		27.12.a.d.1.2	✓	4
9.4	even	3	inner	81.12.c.i.28.2		8
9.5	odd	6	inner	81.12.c.i.28.3		8
9.7	even	3		27.12.a.d.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.12.a.d.1.2	✓	4	9.2	odd	6
27.12.a.d.1.3	yes	4	9.7	even	3
81.12.c.i.28.2		8	9.4	even	3	inner
81.12.c.i.28.3		8	9.5	odd	6	inner
81.12.c.i.55.2		8	1.1	even	1	trivial
81.12.c.i.55.3		8	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	81.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.67631 + 4.63551i) q^{2} +(1009.67 + 1748.81i) q^{4} +(6419.60 + 11119.1i) q^{5} +(15869.3 - 27486.4i) q^{7} -21771.0 q^{8} -68723.4 q^{10} +(-342308. + 592895. i) q^{11} +(-493175. - 854204. i) q^{13} +(84942.3 + 147124. i) q^{14} +(-2.00955e6 + 3.48064e6i) q^{16} -8.95856e6 q^{17} -3.35373e6 q^{19} +(-1.29634e7 + 2.24533e7i) q^{20} +(-1.83225e6 - 3.17355e6i) q^{22} +(5.36262e6 + 9.28832e6i) q^{23} +(-5.80084e7 + 1.00474e8i) q^{25} +5.27956e6 q^{26} +6.40913e7 q^{28} +(7.77565e7 - 1.34678e8i) q^{29} +(-1.12081e8 - 1.94129e8i) q^{31} +(-3.30498e7 - 5.72440e7i) q^{32} +(2.39759e7 - 4.15275e7i) q^{34} +4.07498e8 q^{35} +2.97380e8 q^{37} +(8.97562e6 - 1.55462e7i) q^{38} +(-1.39761e8 - 2.42073e8i) q^{40} +(1.83506e8 + 3.17842e8i) q^{41} +(1.18982e8 - 2.06082e8i) q^{43} -1.38248e9 q^{44} -5.74081e7 q^{46} +(-9.07505e8 + 1.57185e9i) q^{47} +(4.84994e8 + 8.40035e8i) q^{49} +(-3.10497e8 - 5.37797e8i) q^{50} +(9.95892e8 - 1.72494e9i) q^{52} +2.49351e8 q^{53} -8.78993e9 q^{55} +(-3.45490e8 + 5.98407e8i) q^{56} +(4.16201e8 + 7.20881e8i) q^{58} +(2.62950e9 + 4.55442e9i) q^{59} +(-1.95491e9 + 3.38600e9i) q^{61} +1.19985e9 q^{62} -7.87730e9 q^{64} +(6.33197e9 - 1.09673e10i) q^{65} +(-7.99354e9 - 1.38452e10i) q^{67} +(-9.04523e9 - 1.56668e10i) q^{68} +(-1.09059e9 + 1.88896e9i) q^{70} +1.95308e10 q^{71} -9.74734e9 q^{73} +(-7.95881e8 + 1.37851e9i) q^{74} +(-3.38617e9 - 5.86503e9i) q^{76} +(1.08644e10 + 1.88177e10i) q^{77} +(-1.17184e10 + 2.02968e10i) q^{79} -5.16020e10 q^{80} -1.96448e9 q^{82} +(-1.81967e10 + 3.15175e10i) q^{83} +(-5.75104e10 - 9.96109e10i) q^{85} +(6.36864e8 + 1.10308e9i) q^{86} +(7.45239e9 - 1.29079e10i) q^{88} +1.64629e10 q^{89} -3.13053e10 q^{91} +(-1.08290e10 + 1.87564e10i) q^{92} +(-4.85753e9 - 8.41349e9i) q^{94} +(-2.15296e10 - 3.72903e10i) q^{95} +(-1.37713e10 + 2.38526e10i) q^{97} -5.19198e9 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 1636 q^{4} - 28460 q^{7} + 304992 q^{10} - 370868 q^{13} - 4128904 q^{16} + 66418792 q^{19} - 94424400 q^{22} - 142981516 q^{25} + 778081048 q^{28} - 647361104 q^{31} - 1085116176 q^{34} + 1104329752 q^{37}+ \cdots - 137541602900 q^{97}+O(q^{100}) \) 8 * q - 1636 * q^4 - 28460 * q^7 + 304992 * q^10 - 370868 * q^13 - 4128904 * q^16 + 66418792 * q^19 - 94424400 * q^22 - 142981516 * q^25 + 778081048 * q^28 - 647361104 * q^31 - 1085116176 * q^34 + 1104329752 * q^37 - 787910112 * q^40 + 131575840 * q^43 - 11530284768 * q^46 + 1668407928 * q^49 + 8528523820 * q^52 - 29990656512 * q^55 + 4235362560 * q^58 + 6848424772 * q^61 - 85283619040 * q^64 - 43603974212 * q^67 - 17690625936 * q^70 - 26354434232 * q^73 - 126802758164 * q^76 - 140571916436 * q^79 - 284034359808 * q^82 - 265087842048 * q^85 - 38939155488 * q^88 - 272489230984 * q^91 - 105138158544 * q^94 - 137541602900 * q^97

Introduction

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