LMFDB - Embedded newform 84.9.p.b.65.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [84,9,Mod(53,84)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(84, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 4]))

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

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

chi := DirichletCharacter("84.53");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 84 = 2^{2} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	84.p (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$34.2198032451$
Analytic rank:	$0$
Dimension:	$40$
Relative dimension:	$20$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			65.2
Character	$\chi$	$=$	84.65
Dual form			84.9.p.b.53.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-77.6406 + 23.0855i) q^{3} +(726.008 + 419.161i) q^{5} +(-1188.37 - 2086.29i) q^{7} +(5495.12 - 3584.74i) q^{9} +O(q^{10})$	\(q+(-77.6406 + 23.0855i) q^{3} +(726.008 + 419.161i) q^{5} +(-1188.37 - 2086.29i) q^{7} +(5495.12 - 3584.74i) q^{9} +(-11852.6 + 6843.09i) q^{11} -4288.33 q^{13} +(-66044.2 - 15783.7i) q^{15} +(20828.2 - 12025.2i) q^{17} +(-66484.9 + 115155. i) q^{19} +(140428. + 134546. i) q^{21} +(31316.3 + 18080.5i) q^{23} +(156079. + 270338. i) q^{25} +(-343889. + 405179. i) q^{27} -1.12585e6i q^{29} +(-148437. - 257101. i) q^{31} +(762265. - 804924. i) q^{33} +(11726.0 - 2.01278e6i) q^{35} +(1.66179e6 - 2.87830e6i) q^{37} +(332949. - 98998.2i) q^{39} -3.70409e6i q^{41} -1.75904e6 q^{43} +(5.49209e6 - 299212. i) q^{45} +(2.29987e6 + 1.32783e6i) q^{47} +(-2.94037e6 + 4.95854e6i) q^{49} +(-1.33951e6 + 1.41447e6i) q^{51} +(4.36123e6 - 2.51796e6i) q^{53} -1.14734e7 q^{55} +(2.50351e6 - 1.04756e7i) q^{57} +(1.72091e7 - 9.93569e6i) q^{59} +(6.29128e6 - 1.08968e7i) q^{61} +(-1.40090e7 - 7.20440e6i) q^{63} +(-3.11336e6 - 1.79750e6i) q^{65} +(-1.30656e7 - 2.26303e7i) q^{67} +(-2.84881e6 - 680826. i) q^{69} -1.53174e6i q^{71} +(2.43157e7 + 4.21161e7i) q^{73} +(-1.83590e7 - 1.73860e7i) q^{75} +(2.83619e7 + 1.65958e7i) q^{77} +(2.10421e7 - 3.64460e7i) q^{79} +(1.73460e7 - 3.93972e7i) q^{81} -1.42941e7i q^{83} +2.01619e7 q^{85} +(2.59908e7 + 8.74116e7i) q^{87} +(-3.35488e7 - 1.93694e7i) q^{89} +(5.09611e6 + 8.94668e6i) q^{91} +(1.74601e7 + 1.65347e7i) q^{93} +(-9.65372e7 + 5.57358e7i) q^{95} -9.15468e7 q^{97} +(-4.06007e7 + 8.00921e7i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 81 q^{3} - 34 q^{7} + 4771 q^{9}+O(q^{10}) $ 40 * q + 81 * q^3 - 34 * q^7 + 4771 * q^9	$ 40 q + 81 q^{3} - 34 q^{7} + 4771 q^{9} - 55464 q^{13} + 68482 q^{15} + 311690 q^{19} - 172343 q^{21} + 1766792 q^{25} - 3451932 q^{27} + 31596 q^{31} + 1874885 q^{33} - 1853482 q^{37} + 11217526 q^{39} - 13372600 q^{43} - 527785 q^{45} - 12653462 q^{49} - 1103461 q^{51} + 71577224 q^{55} - 17195214 q^{57} - 21761970 q^{61} + 21945045 q^{63} - 26337350 q^{67} - 5588722 q^{69} + 41115682 q^{73} - 17971730 q^{75} - 120916932 q^{79} - 24550133 q^{81} + 139250060 q^{85} - 16321046 q^{87} + 345074940 q^{91} + 25774675 q^{93} - 707216948 q^{97} - 94510994 q^{99}+O(q^{100}) $ 40 * q + 81 * q^3 - 34 * q^7 + 4771 * q^9 - 55464 * q^13 + 68482 * q^15 + 311690 * q^19 - 172343 * q^21 + 1766792 * q^25 - 3451932 * q^27 + 31596 * q^31 + 1874885 * q^33 - 1853482 * q^37 + 11217526 * q^39 - 13372600 * q^43 - 527785 * q^45 - 12653462 * q^49 - 1103461 * q^51 + 71577224 * q^55 - 17195214 * q^57 - 21761970 * q^61 + 21945045 * q^63 - 26337350 * q^67 - 5588722 * q^69 + 41115682 * q^73 - 17971730 * q^75 - 120916932 * q^79 - 24550133 * q^81 + 139250060 * q^85 - 16321046 * q^87 + 345074940 * q^91 + 25774675 * q^93 - 707216948 * q^97 - 94510994 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/84\mathbb{Z}\right)^\times$.

$n$	$29$	$43$	$73$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{3}\right)$

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 $

$2$		0			0
$2$		0			0
$3$	−77.6406	+	23.0855i	−0.958526	+	0.285006i
$3$	−77.6406	+	23.0855i	−0.958526	+	0.285006i
$4$		0			0
$5$	726.008	+	419.161i	1.16161	+	0.670658i	0.951690	−	0.307060i	$-0.0993453\pi$
$5$	726.008	+	419.161i	1.16161	+	0.670658i	0.209923	+	0.977718i	$0.432679\pi$
$6$		0			0
$7$	−1188.37	−	2086.29i	−0.494946	−	0.868924i
$7$	−1188.37	−	2086.29i	−0.494946	−	0.868924i
$8$		0			0
$9$	5495.12	−	3584.74i	0.837543	−	0.546371i
$10$		0			0
$11$	−11852.6	+	6843.09i	−0.809547	+	0.467392i	−0.846799	−	0.531914i	$-0.821473\pi$
$11$	−11852.6	+	6843.09i	−0.809547	+	0.467392i	0.0372513	+	0.999306i	$0.488140\pi$
$12$		0			0
$13$	−4288.33			−0.150146			−0.0750732	−	0.997178i	$-0.523919\pi$
$13$	−4288.33			−0.150146			−0.0750732	+	0.997178i	$0.523919\pi$
$14$		0			0
$15$	−66044.2	−	15783.7i	−1.30458	−	0.311776i
$16$		0			0
$17$	20828.2	−	12025.2i	0.249377	−	0.143978i	−0.370102	−	0.928991i	$-0.620677\pi$
$17$	20828.2	−	12025.2i	0.249377	−	0.143978i	0.619479	+	0.785013i	$0.287344\pi$
$18$		0			0
$19$	−66484.9	+	115155.i	−0.510163	+	0.883628i	0.489768	+	0.871853i	$0.337081\pi$
$19$	−66484.9	+	115155.i	−0.510163	+	0.883628i	−0.999931	+	0.0117749i	$0.996252\pi$
$20$		0			0
$21$	140428.	+	134546.i	0.722067	+	0.691823i
$22$		0			0
$23$	31316.3	+	18080.5i	0.111907	+	0.0646097i	0.554909	−	0.831911i	$-0.312753\pi$
$23$	31316.3	+	18080.5i	0.111907	+	0.0646097i	−0.443002	+	0.896521i	$0.646086\pi$
$24$		0			0
$25$	156079.	+	270338.i	0.399563	+	0.692064i
$26$		0			0
$27$	−343889.	+	405179.i	−0.647087	+	0.762416i
$28$		0			0
$29$		−	1.12585e6i		−	1.59180i	−0.605428	−	0.795900i	$-0.706998\pi$
$29$		−	1.12585e6i		−	1.59180i	0.605428	−	0.795900i	$-0.293002\pi$
$30$		0			0
$31$	−148437.	−	257101.i	−0.160730	−	0.278392i	0.774401	−	0.632695i	$-0.218051\pi$
$31$	−148437.	−	257101.i	−0.160730	−	0.278392i	−0.935131	+	0.354303i	$0.884718\pi$
$32$		0			0
$33$	762265.	−	804924.i	0.642762	−	0.678734i
$34$		0			0
$35$	11726.0	−	2.01278e6i	0.00781406	−	1.34129i
$36$		0			0
$37$	1.66179e6	−	2.87830e6i	0.886684	−	1.53578i	0.0429137	−	0.999079i	$-0.486336\pi$
$37$	1.66179e6	−	2.87830e6i	0.886684	−	1.53578i	0.843771	−	0.536704i	$-0.180331\pi$
$38$		0			0
$39$	332949.	−	98998.2i	0.143919	−	0.0427926i
$40$		0			0
$41$		−	3.70409e6i		−	1.31083i	−0.755270	−	0.655414i	$-0.772494\pi$
$41$		−	3.70409e6i		−	1.31083i	0.755270	−	0.655414i	$-0.227506\pi$
$42$		0			0
$43$	−1.75904e6			−0.514518			−0.257259	−	0.966342i	$-0.582819\pi$
$43$	−1.75904e6			−0.514518			−0.257259	+	0.966342i	$0.582819\pi$
$44$		0			0
$45$	5.49209e6	−	299212.i	1.33933	−	0.0729674i
$46$		0			0
$47$	2.29987e6	+	1.32783e6i	0.471316	+	0.272114i	0.716791	−	0.697289i	$-0.245610\pi$
$47$	2.29987e6	+	1.32783e6i	0.471316	+	0.272114i	−0.245474	+	0.969403i	$0.578944\pi$
$48$		0			0
$49$	−2.94037e6	+	4.95854e6i	−0.510056	+	0.860141i
$50$		0			0
$51$	−1.33951e6	+	1.41447e6i	−0.197999	+	0.209080i
$52$		0			0
$53$	4.36123e6	−	2.51796e6i	0.552721	−	0.319113i	−0.197498	−	0.980303i	$-0.563282\pi$
$53$	4.36123e6	−	2.51796e6i	0.552721	−	0.319113i	0.750219	+	0.661190i	$0.229948\pi$
$54$		0			0
$55$	−1.14734e7			−1.25384
$56$		0			0
$57$	2.50351e6	−	1.04756e7i	0.237165	−	0.992379i
$58$		0			0
$59$	1.72091e7	−	9.93569e6i	1.42020	−	0.819955i	0.423887	−	0.905715i	$-0.360665\pi$
$59$	1.72091e7	−	9.93569e6i	1.42020	−	0.819955i	0.996316	+	0.0857601i	$0.0273319\pi$
$60$		0			0
$61$	6.29128e6	−	1.08968e7i	0.454381	−	0.787010i	−0.544272	−	0.838909i	$-0.683194\pi$
$61$	6.29128e6	−	1.08968e7i	0.454381	−	0.787010i	0.998652	+	0.0518987i	$0.0165273\pi$
$62$		0			0
$63$	−1.40090e7	−	7.20440e6i	−0.889294	−	0.457336i
$64$		0			0
$65$	−3.11336e6	−	1.79750e6i	−0.174412	−	0.100697i
$66$		0			0
$67$	−1.30656e7	−	2.26303e7i	−0.648383	−	1.12303i	−0.983509	−	0.180858i	$-0.942112\pi$
$67$	−1.30656e7	−	2.26303e7i	−0.648383	−	1.12303i	0.335127	−	0.942173i	$-0.391221\pi$
$68$		0			0
$69$	−2.84881e6	−	680826.i	−0.125680	−	0.0300358i
$70$		0			0
$71$		−	1.53174e6i		−	0.0602770i	−0.999546	−	0.0301385i	$-0.990405\pi$
$71$		−	1.53174e6i		−	0.0602770i	0.999546	−	0.0301385i	$-0.00959483\pi$
$72$		0			0
$73$	2.43157e7	+	4.21161e7i	0.856240	+	1.48305i	0.875490	+	0.483237i	$0.160539\pi$
$73$	2.43157e7	+	4.21161e7i	0.856240	+	1.48305i	−0.0192491	+	0.999815i	$0.506128\pi$
$74$		0			0
$75$	−1.83590e7	−	1.73860e7i	−0.580234	−	0.549483i
$76$		0			0
$77$	2.83619e7	+	1.65958e7i	0.806811	+	0.472101i
$78$		0			0
$79$	2.10421e7	−	3.64460e7i	0.540233	−	0.935711i	−0.458658	−	0.888613i	$-0.651670\pi$
$79$	2.10421e7	−	3.64460e7i	0.540233	−	0.935711i	0.998890	−	0.0470974i	$-0.0149971\pi$
$80$		0			0
$81$	1.73460e7	−	3.93972e7i	0.402957	−	0.915219i
$82$		0			0
$83$		−	1.42941e7i		−	0.301193i	−0.988595	−	0.150597i	$-0.951881\pi$
$83$		−	1.42941e7i		−	0.301193i	0.988595	−	0.150597i	$-0.0481195\pi$
$84$		0			0
$85$	2.01619e7			0.386239
$86$		0			0
$87$	2.59908e7	+	8.74116e7i	0.453673	+	1.52578i
$88$		0			0
$89$	−3.35488e7	−	1.93694e7i	−0.534709	−	0.308714i	0.208223	−	0.978081i	$-0.433232\pi$
$89$	−3.35488e7	−	1.93694e7i	−0.534709	−	0.308714i	−0.742932	+	0.669367i	$0.766565\pi$
$90$		0			0
$91$	5.09611e6	+	8.94668e6i	0.0743144	+	0.130466i
$92$		0			0
$93$	1.74601e7	+	1.65347e7i	0.233407	+	0.221037i
$94$		0			0
$95$	−9.65372e7	+	5.57358e7i	−1.18522	+	0.684289i
$96$		0			0
$97$	−9.15468e7			−1.03409			−0.517043	−	0.855960i	$-0.672967\pi$
$97$	−9.15468e7			−1.03409			−0.517043	+	0.855960i	$0.672967\pi$
$98$		0			0
$99$	−4.06007e7	+	8.00921e7i	−0.422661	+	0.833775i
$100$		0			0
$101$	−1.56504e7	+	9.03574e6i	−0.150397	+	0.0868317i	−0.573310	−	0.819339i	$-0.694341\pi$
$101$	−1.56504e7	+	9.03574e6i	−0.150397	+	0.0868317i	0.422913	+	0.906170i	$0.361008\pi$
$102$		0			0
$103$	8.21411e7	−	1.42273e8i	0.729813	−	1.26407i	−0.227149	−	0.973860i	$-0.572940\pi$
$103$	8.21411e7	−	1.42273e8i	0.729813	−	1.26407i	0.956962	−	0.290213i	$-0.0937263\pi$
$104$		0			0
$105$	4.55555e7	+	1.56544e8i	0.374787	+	1.28789i
$106$		0			0
$107$	−4.09176e7	−	2.36238e7i	−0.312158	−	0.180225i	0.335734	−	0.941957i	$-0.391016\pi$
$107$	−4.09176e7	−	2.36238e7i	−0.312158	−	0.180225i	−0.647892	+	0.761732i	$0.724349\pi$
$108$		0			0
$109$	−6.56157e7	−	1.13650e8i	−0.464838	−	0.805123i	0.534356	−	0.845259i	$-0.320554\pi$
$109$	−6.56157e7	−	1.13650e8i	−0.464838	−	0.805123i	−0.999194	+	0.0401362i	$0.987221\pi$
$110$		0			0
$111$	−6.25752e7	+	2.61836e8i	−0.412203	+	1.72480i
$112$		0			0
$113$			2.75240e8i			1.68810i	0.536265	+	0.844049i	$0.319835\pi$
$113$			2.75240e8i			1.68810i	−0.536265	+	0.844049i	$0.680165\pi$
$114$		0			0
$115$	1.51572e7	+	2.62531e7i	0.0866620	+	0.150103i
$116$		0			0
$117$	−2.35649e7	+	1.53726e7i	−0.125754	+	0.0820357i
$118$		0			0
$119$	−4.98394e7	−	2.91633e7i	−0.248534	−	0.145428i
$120$		0			0
$121$	−1.35236e7	+	2.34236e7i	−0.0630887	+	0.109273i
$122$		0			0
$123$	8.55107e7	+	2.87587e8i	0.373594	+	1.25646i
$124$		0			0
$125$		−	6.57799e7i		−	0.269434i
$126$		0			0
$127$	−3.06495e8			−1.17817			−0.589085	−	0.808071i	$-0.700512\pi$
$127$	−3.06495e8			−1.17817			−0.589085	+	0.808071i	$0.700512\pi$
$128$		0			0
$129$	1.36573e8	−	4.06082e7i	0.493179	−	0.146641i
$130$		0			0
$131$	−2.53627e7	−	1.46432e7i	−0.0861212	−	0.0497221i	0.456321	−	0.889815i	$-0.349167\pi$
$131$	−2.53627e7	−	1.46432e7i	−0.0861212	−	0.0497221i	−0.542442	+	0.840093i	$0.682500\pi$
$132$		0			0
$133$	3.19255e8	+	1.85991e6i	1.02031	+	0.00594408i
$134$		0			0
$135$	−4.19501e8	+	1.50018e8i	−1.26299	+	0.451658i
$136$		0			0
$137$	−3.15124e8	+	1.81937e8i	−0.894539	+	0.516462i	−0.875424	−	0.483355i	$-0.839418\pi$
$137$	−3.15124e8	+	1.81937e8i	−0.894539	+	0.516462i	−0.0191146	+	0.999817i	$0.506085\pi$
$138$		0			0
$139$	5.44570e8			1.45880			0.729398	−	0.684089i	$-0.239800\pi$
$139$	5.44570e8			1.45880			0.729398	+	0.684089i	$0.239800\pi$
$140$		0			0
$141$	−2.09217e8	−	5.00000e7i	−0.529323	−	0.126501i
$142$		0			0
$143$	5.08278e7	−	2.93454e7i	0.121551	−	0.0701773i
$144$		0			0
$145$	4.71912e8	−	8.17376e8i	1.06755	−	1.84906i
$146$		0			0
$147$	1.13822e8	−	4.52864e8i	0.243757	−	0.969836i
$148$		0			0
$149$	8.04115e8	+	4.64256e8i	1.63145	+	0.941916i	0.983647	+	0.180106i	$0.0576440\pi$
$149$	8.04115e8	+	4.64256e8i	1.63145	+	0.941916i	0.647800	+	0.761811i	$0.275689\pi$
$150$		0			0
$151$	−2.34121e8	−	4.05510e8i	−0.450332	−	0.779998i	0.548074	−	0.836430i	$-0.315361\pi$
$151$	−2.34121e8	−	4.05510e8i	−0.450332	−	0.779998i	−0.998406	+	0.0564316i	$0.982028\pi$
$152$		0			0
$153$	7.13463e7	−	1.40743e8i	0.130198	−	0.256840i
$154$		0			0
$155$		−	2.48876e8i		−	0.431178i
$156$		0			0
$157$	−4.57530e8	−	7.92466e8i	−0.753046	−	1.30431i	−0.946340	−	0.323173i	$-0.895250\pi$
$157$	−4.57530e8	−	7.92466e8i	−0.753046	−	1.30431i	0.193294	−	0.981141i	$-0.438083\pi$
$158$		0			0
$159$	−2.80480e8	+	2.96177e8i	−0.438848	+	0.463407i
$160$		0			0
$161$	505798.	−	8.68209e7i	0.000752790	−	0.129217i
$162$		0			0
$163$	−2.22372e8	+	3.85159e8i	−0.315014	+	0.545620i	−0.979440	−	0.201734i	$-0.935342\pi$
$163$	−2.22372e8	+	3.85159e8i	−0.315014	+	0.545620i	0.664427	+	0.747353i	$0.268676\pi$
$164$		0			0
$165$	8.90804e8	−	2.64870e8i	1.20184	−	0.357352i
$166$		0			0
$167$		−	8.98625e8i		−	1.15535i	−0.816268	−	0.577674i	$-0.803961\pi$
$167$		−	8.98625e8i		−	1.15535i	0.816268	−	0.577674i	$-0.196039\pi$
$168$		0			0
$169$	−7.97341e8			−0.977456
$170$		0			0
$171$	4.74593e7	+	8.71123e8i	0.0555056	+	1.01881i
$172$		0			0
$173$	−2.38451e8	−	1.37670e8i	−0.266204	−	0.153693i	0.360957	−	0.932582i	$-0.382450\pi$
$173$	−2.38451e8	−	1.37670e8i	−0.266204	−	0.153693i	−0.627161	+	0.778889i	$0.715783\pi$
$174$		0			0
$175$	3.78522e8	−	6.46886e8i	0.403588	−	0.689725i
$176$		0			0
$177$	−1.10676e9	+	1.16869e9i	−1.12761	+	1.19071i
$178$		0			0
$179$	6.90561e8	−	3.98695e8i	0.672651	−	0.388355i	−0.124429	−	0.992228i	$-0.539710\pi$
$179$	6.90561e8	−	3.98695e8i	0.672651	−	0.388355i	0.797080	+	0.603873i	$0.206377\pi$
$180$		0			0
$181$	−6.51285e8			−0.606816			−0.303408	−	0.952861i	$-0.598124\pi$
$181$	−6.51285e8			−0.606816			−0.303408	+	0.952861i	$0.598124\pi$
$182$		0			0
$183$	−2.36900e8	+	9.91273e8i	−0.211233	+	0.883871i
$184$		0			0
$185$	2.41295e9	−	1.39311e9i	2.05997	−	1.18932i
$186$		0			0
$187$	−1.64579e8	+	2.85058e8i	−0.134588	+	0.233114i
$188$		0			0
$189$	1.25398e9	+	2.35949e8i	0.982755	+	0.184915i
$190$		0			0
$191$	−9.93412e8	−	5.73547e8i	−0.746442	−	0.430959i	0.0779648	−	0.996956i	$-0.475158\pi$
$191$	−9.93412e8	−	5.73547e8i	−0.746442	−	0.430959i	−0.824407	+	0.565998i	$0.808491\pi$
$192$		0			0
$193$	−3.83109e8	−	6.63564e8i	−0.276117	−	0.478248i	0.694299	−	0.719686i	$-0.255714\pi$
$193$	−3.83109e8	−	6.63564e8i	−0.276117	−	0.478248i	−0.970416	+	0.241438i	$0.922381\pi$
$194$		0			0
$195$	2.83220e8	+	6.76855e7i	0.195878	+	0.0468120i
$196$		0			0
$197$			3.70739e8i			0.246152i	0.992397	+	0.123076i	$0.0392759\pi$
$197$			3.70739e8i			0.246152i	−0.992397	+	0.123076i	$0.960724\pi$
$198$		0			0
$199$	−8.89835e8	−	1.54124e9i	−0.567410	−	0.982784i	−0.996821	−	0.0796742i	$-0.974612\pi$
$199$	−8.89835e8	−	1.54124e9i	−0.567410	−	0.982784i	0.429411	−	0.903109i	$-0.358721\pi$
$200$		0			0
$201$	1.53686e9	+	1.45541e9i	0.941562	+	0.891662i
$202$		0			0
$203$	−2.34884e9	+	1.33792e9i	−1.38315	+	0.787855i
$204$		0			0
$205$	1.55261e9	−	2.68920e9i	0.879117	−	1.52267i
$206$		0			0
$207$	2.36900e8	−	1.29065e7i	0.129028	−	0.00702952i
$208$		0			0
$209$		−	1.81985e9i		−	0.953785i
$210$		0			0
$211$	3.53555e9			1.78372			0.891862	−	0.452307i	$-0.149399\pi$
$211$	3.53555e9			1.78372			0.891862	+	0.452307i	$0.149399\pi$
$212$		0			0
$213$	3.53610e7	+	1.18925e8i	0.0171793	+	0.0577770i
$214$		0			0
$215$	−1.27707e9	−	7.37319e8i	−0.597671	−	0.345066i
$216$		0			0
$217$	−3.59988e8	+	6.15213e8i	−0.162349	+	0.277451i
$218$		0			0
$219$	−2.86016e9	−	2.70858e9i	−1.24341	−	1.17751i
$220$		0			0
$221$	−8.93182e7	+	5.15679e7i	−0.0374430	+	0.0216177i
$222$		0			0
$223$	−1.83868e9			−0.743510			−0.371755	−	0.928331i	$-0.621244\pi$
$223$	−1.83868e9			−0.743510			−0.371755	+	0.928331i	$0.621244\pi$
$224$		0			0
$225$	1.82677e9	+	9.26033e8i	0.712775	+	0.361323i
$226$		0			0
$227$	−1.39182e9	+	8.03568e8i	−0.524179	+	0.302635i	−0.738643	−	0.674097i	$-0.764533\pi$
$227$	−1.39182e9	+	8.03568e8i	−0.524179	+	0.302635i	0.214464	+	0.976732i	$0.431200\pi$
$228$		0			0
$229$	4.11805e8	−	7.13267e8i	0.149744	−	0.259364i	−0.781389	−	0.624045i	$-0.785488\pi$
$229$	4.11805e8	−	7.13267e8i	0.149744	−	0.259364i	0.931133	+	0.364680i	$0.118822\pi$
$230$		0			0
$231$	−2.58515e9	−	6.33758e8i	−0.907900	−	0.222575i
$232$		0			0
$233$	3.56617e9	+	2.05893e9i	1.20998	+	0.698582i	0.962755	−	0.270377i	$-0.0871482\pi$
$233$	3.56617e9	+	2.05893e9i	1.20998	+	0.698582i	0.247224	+	0.968958i	$0.420482\pi$
$234$		0			0
$235$	1.11315e9	+	1.92803e9i	0.364991	+	0.632184i
$236$		0			0
$237$	−7.92348e8	+	3.31546e9i	−0.251144	+	1.05087i
$238$		0			0
$239$			2.95475e9i			0.905584i	0.891616	+	0.452792i	$0.149572\pi$
$239$			2.95475e9i			0.905584i	−0.891616	+	0.452792i	$0.850428\pi$
$240$		0			0
$241$	2.43372e9	+	4.21532e9i	0.721442	+	1.24958i	0.960422	+	0.278550i	$0.0898539\pi$
$241$	2.43372e9	+	4.21532e9i	0.721442	+	1.24958i	−0.238979	+	0.971025i	$0.576813\pi$
$242$		0			0
$243$	−4.37248e8	+	3.45926e9i	−0.125402	+	0.992106i
$244$		0			0
$245$	−4.21316e9	+	2.36745e9i	−1.16935	+	0.657078i
$246$		0			0
$247$	2.85109e8	−	4.93824e8i	0.0765991	−	0.132674i
$248$		0			0
$249$	3.29987e8	+	1.10980e9i	0.0858419	+	0.288702i
$250$		0			0
$251$			2.63362e9i			0.663526i	0.943363	+	0.331763i	$0.107643\pi$
$251$			2.63362e9i			0.663526i	−0.943363	+	0.331763i	$0.892357\pi$
$252$		0			0
$253$	−4.94905e8			−0.120792
$254$		0			0
$255$	−1.56538e9	+	4.65448e8i	−0.370220	+	0.110080i
$256$		0			0
$257$	6.46260e9	+	3.73119e9i	1.48141	+	0.855292i	0.999778	−	0.0210818i	$-0.00671105\pi$
$257$	6.46260e9	+	3.73119e9i	1.48141	+	0.855292i	0.481631	+	0.876374i	$0.340044\pi$
$258$		0			0
$259$	−7.97978e9	−	4.64883e7i	−1.77334	−	0.0103311i
$260$		0			0
$261$	−4.03588e9	−	6.18668e9i	−0.869714	−	1.33320i
$262$		0			0
$263$	−7.11969e9	+	4.11056e9i	−1.48812	+	0.859167i	−0.999908	−	0.0135574i	$-0.995684\pi$
$263$	−7.11969e9	+	4.11056e9i	−1.48812	+	0.859167i	−0.488213	+	0.872725i	$0.662351\pi$
$264$		0			0
$265$	4.22172e9			0.856063
$266$		0			0
$267$	3.05190e9	+	7.29362e8i	0.600517	+	0.143515i
$268$		0			0
$269$	5.18973e9	−	2.99629e9i	0.991141	−	0.572236i	0.0855260	−	0.996336i	$-0.472743\pi$
$269$	5.18973e9	−	2.99629e9i	0.991141	−	0.572236i	0.905615	+	0.424100i	$0.139410\pi$
$270$		0			0
$271$	−2.24850e9	+	3.89452e9i	−0.416885	+	0.722065i	−0.995624	−	0.0934470i	$-0.970211\pi$
$271$	−2.24850e9	+	3.89452e9i	−0.416885	+	0.722065i	0.578740	+	0.815512i	$0.303545\pi$
$272$		0			0
$273$	−6.02203e8	−	5.76980e8i	−0.108416	−	0.103875i
$274$		0			0
$275$	−3.69989e9	−	2.13613e9i	−0.646931	−	0.373506i
$276$		0			0
$277$	−2.19189e9	−	3.79646e9i	−0.372305	−	0.644852i	0.617614	−	0.786481i	$-0.288099\pi$
$277$	−2.19189e9	−	3.79646e9i	−0.372305	−	0.644852i	−0.989920	+	0.141629i	$0.954766\pi$
$278$		0			0
$279$	−1.73732e9	−	8.80691e8i	−0.286724	−	0.145347i
$280$		0			0
$281$			4.89493e9i			0.785093i	0.919732	+	0.392547i	$0.128406\pi$
$281$			4.89493e9i			0.785093i	−0.919732	+	0.392547i	$0.871594\pi$
$282$		0			0
$283$	−2.71265e9	−	4.69845e9i	−0.422910	−	0.732502i	0.573312	−	0.819337i	$-0.305658\pi$
$283$	−2.71265e9	−	4.69845e9i	−0.422910	−	0.732502i	−0.996223	+	0.0868347i	$0.972325\pi$
$284$		0			0
$285$	6.20852e9	−	6.55597e9i	0.941041	−	0.993704i
$286$		0			0
$287$	−7.72778e9	+	4.40181e9i	−1.13901	+	0.648789i
$288$		0			0
$289$	−3.19867e9	+	5.54026e9i	−0.458541	+	0.794216i
$290$		0			0
$291$	7.10775e9	−	2.11340e9i	0.991197	−	0.294721i
$292$		0			0
$293$		−	1.26502e10i		−	1.71643i	−0.513287	−	0.858217i	$-0.671572\pi$
$293$		−	1.26502e10i		−	1.71643i	0.513287	−	0.858217i	$-0.328428\pi$
$294$		0			0
$295$	1.66586e10			2.19964
$296$		0			0
$297$	1.30329e9	−	7.15568e9i	0.167501	−	0.919655i
$298$		0			0
$299$	−1.34295e8	−	7.75350e7i	−0.0168025	−	0.00970092i
$300$		0			0
$301$	2.09038e9	+	3.66985e9i	0.254659	+	0.447077i
$302$		0			0
$303$	1.00651e9	−	1.06284e9i	0.119412	−	0.126094i
$304$		0			0
$305$	9.13504e9	−	5.27412e9i	1.05563	−	0.609468i
$306$		0			0
$307$	−2.92066e9			−0.328797			−0.164399	−	0.986394i	$-0.552568\pi$
$307$	−2.92066e9			−0.328797			−0.164399	+	0.986394i	$0.552568\pi$
$308$		0			0
$309$	−3.09305e9	+	1.29424e10i	−0.339276	+	1.41965i
$310$		0			0
$311$	1.47356e10	−	8.50759e9i	1.57516	−	0.909421i	0.579643	−	0.814871i	$-0.303192\pi$
$311$	1.47356e10	−	8.50759e9i	1.57516	−	0.909421i	0.995520	−	0.0945502i	$-0.0301413\pi$
$312$		0			0
$313$	−5.43531e9	+	9.41424e9i	−0.566301	+	0.980862i	0.430627	+	0.902530i	$0.358293\pi$
$313$	−5.43531e9	+	9.41424e9i	−0.566301	+	0.980862i	−0.996927	+	0.0783315i	$0.975041\pi$
$314$		0			0
$315$	−7.15085e9	−	1.11025e10i	−0.726299	−	1.12766i
$316$		0			0
$317$	−1.19740e10	−	6.91320e9i	−1.18578	−	0.684608i	−0.228432	−	0.973560i	$-0.573360\pi$
$317$	−1.19740e10	−	6.91320e9i	−1.18578	−	0.684608i	−0.957344	+	0.288952i	$0.906693\pi$
$318$		0			0
$319$	7.70429e9	+	1.33442e10i	0.743995	+	1.28864i
$320$		0			0
$321$	3.72223e9	+	8.89561e8i	0.350577	+	0.0837830i
$322$		0			0
$323$			3.19797e9i			0.293808i
$324$		0			0
$325$	−6.69320e8	−	1.15930e9i	−0.0599930	−	0.103911i
$326$		0			0
$327$	7.71810e9	+	7.30906e9i	0.675024	+	0.639250i
$328$		0			0
$329$	3.71459e7	−	6.37614e9i	0.00317050	−	0.544220i
$330$		0			0
$331$	−3.57896e9	+	6.19894e9i	−0.298157	+	0.516423i	−0.975714	−	0.219047i	$-0.929705\pi$
$331$	−3.57896e9	+	6.19894e9i	−0.298157	+	0.516423i	0.677558	+	0.735470i	$0.263038\pi$
$332$		0			0
$333$	−1.18624e9	−	2.17737e10i	−0.0964710	−	1.77074i
$334$		0			0
$335$		−	2.19064e10i		−	1.73937i
$336$		0			0
$337$	−2.08762e10			−1.61857			−0.809287	−	0.587414i	$-0.800146\pi$
$337$	−2.08762e10			−1.61857			−0.809287	+	0.587414i	$0.800146\pi$
$338$		0			0
$339$	−6.35405e9	−	2.13698e10i	−0.481118	−	1.61809i
$340$		0			0
$341$	3.51873e9	+	2.03154e9i	0.260237	+	0.150248i
$342$		0			0
$343$	1.38392e10	+	2.41893e8i	0.999847	+	0.0174762i
$344$		0			0
$345$	−1.78288e9	−	1.68839e9i	−0.125848	−	0.119178i
$346$		0			0
$347$	1.19483e10	−	6.89837e9i	0.824118	−	0.475805i	−0.0277165	−	0.999616i	$-0.508824\pi$
$347$	1.19483e10	−	6.89837e9i	0.824118	−	0.475805i	0.851834	+	0.523811i	$0.175490\pi$
$348$		0			0
$349$	−6.40544e9			−0.431765			−0.215882	−	0.976419i	$-0.569263\pi$
$349$	−6.40544e9			−0.431765			−0.215882	+	0.976419i	$0.569263\pi$
$350$		0			0
$351$	1.47471e9	−	1.73754e9i	0.0971579	−	0.114474i
$352$		0			0
$353$	1.02481e9	−	5.91677e8i	0.0660003	−	0.0381053i	−0.466637	−	0.884449i	$-0.654534\pi$
$353$	1.02481e9	−	5.91677e8i	0.0660003	−	0.0381053i	0.532637	+	0.846344i	$0.321201\pi$
$354$		0			0
$355$	6.42046e8	−	1.11206e9i	0.0404252	−	0.0700185i
$356$		0			0
$357$	4.54281e9	+	1.11368e9i	0.279674	+	0.0685629i
$358$		0			0
$359$	2.06909e9	+	1.19459e9i	0.124567	+	0.0719186i	0.560989	−	0.827824i	$-0.310421\pi$
$359$	2.06909e9	+	1.19459e9i	0.124567	+	0.0719186i	−0.436422	+	0.899742i	$0.643754\pi$
$360$		0			0
$361$	−3.48706e8	−	6.03976e8i	−0.0205320	−	0.0355624i
$362$		0			0
$363$	5.09236e8	−	2.13082e9i	0.0293287	−	0.122721i
$364$		0			0
$365$			4.07688e10i			2.29698i
$366$		0			0
$367$	−8.76690e8	−	1.51847e9i	−0.0483261	−	0.0837033i	0.840850	−	0.541267i	$-0.182055\pi$
$367$	−8.76690e8	−	1.51847e9i	−0.0483261	−	0.0837033i	−0.889177	+	0.457564i	$0.848722\pi$
$368$		0			0
$369$	−1.32782e10	−	2.03544e10i	−0.716199	−	1.09787i
$370$		0			0
$371$	−1.04359e10	−	6.10652e9i	−0.550852	−	0.322328i
$372$		0			0
$373$	−1.46590e10	+	2.53901e10i	−0.757302	+	1.31169i	0.186920	+	0.982375i	$0.440150\pi$
$373$	−1.46590e10	+	2.53901e10i	−0.757302	+	1.31169i	−0.944222	+	0.329310i	$0.893184\pi$
$374$		0			0
$375$	1.51856e9	+	5.10719e9i	0.0767904	+	0.258260i
$376$		0			0
$377$			4.82802e9i			0.239003i
$378$		0			0
$379$	−6.47809e9			−0.313972			−0.156986	−	0.987601i	$-0.550178\pi$
$379$	−6.47809e9			−0.313972			−0.156986	+	0.987601i	$0.550178\pi$
$380$		0			0
$381$	2.37964e10	−	7.07558e9i	1.12931	−	0.335786i
$382$		0			0
$383$	2.27121e10	+	1.31128e10i	1.05551	+	0.609398i	0.924186	−	0.381942i	$-0.124745\pi$
$383$	2.27121e10	+	1.31128e10i	1.05551	+	0.609398i	0.131322	+	0.991340i	$0.458078\pi$
$384$		0			0
$385$	1.36346e10	+	2.39369e10i	0.620584	+	1.08949i
$386$		0			0
$387$	−9.66611e9	+	6.30569e9i	−0.430931	+	0.281118i
$388$		0			0
$389$	−2.16168e10	+	1.24804e10i	−0.944044	+	0.545044i	−0.891226	−	0.453560i	$-0.850154\pi$
$389$	−2.16168e10	+	1.24804e10i	−0.944044	+	0.545044i	−0.0528181	+	0.998604i	$0.516820\pi$
$390$		0			0
$391$	8.69682e8			0.0372094
$392$		0			0
$393$	2.30722e9	+	5.51393e8i	0.0967205	+	0.0231148i
$394$		0			0
$395$	3.05535e10	−	1.76401e10i	1.25508	−	0.724622i
$396$		0			0
$397$	1.61597e10	−	2.79894e10i	0.650535	−	1.12676i	−0.332459	−	0.943118i	$-0.607878\pi$
$397$	1.61597e10	−	2.79894e10i	0.650535	−	1.12676i	0.982993	−	0.183641i	$-0.0587884\pi$
$398$		0			0
$399$	−2.48301e10	+	7.22576e9i	−0.979686	+	0.285096i
$400$		0			0
$401$	−3.13160e10	−	1.80803e10i	−1.21112	−	0.699242i	−0.248119	−	0.968730i	$-0.579812\pi$
$401$	−3.13160e10	−	1.80803e10i	−1.21112	−	0.699242i	−0.963004	+	0.269487i	$0.913146\pi$
$402$		0			0
$403$	6.36548e8	+	1.10253e9i	0.0241330	+	0.0417996i
$404$		0			0
$405$	2.91071e10	−	2.13319e10i	1.08188	−	0.792884i
$406$		0			0
$407$			4.54871e10i			1.65772i
$408$		0			0
$409$	6.85527e9	+	1.18737e10i	0.244980	+	0.424319i	0.962126	−	0.272605i	$-0.0878851\pi$
$409$	6.85527e9	+	1.18737e10i	0.244980	+	0.424319i	−0.717146	+	0.696923i	$0.754552\pi$
$410$		0			0
$411$	2.02663e10	−	2.14005e10i	0.710244	−	0.749992i
$412$		0			0
$413$	−4.11794e10	−	2.40959e10i	−1.41540	−	0.828214i
$414$		0			0
$415$	5.99154e9	−	1.03777e10i	0.201998	−	0.349870i
$416$		0			0
$417$	−4.22808e10	+	1.25717e10i	−1.39829	+	0.415766i
$418$		0			0
$419$			4.32974e10i			1.40477i	0.711797	+	0.702386i	$0.247882\pi$
$419$			4.32974e10i			1.40477i	−0.711797	+	0.702386i	$0.752118\pi$
$420$		0			0
$421$	1.86814e10			0.594677			0.297338	−	0.954772i	$-0.403901\pi$
$421$	1.86814e10			0.594677			0.297338	+	0.954772i	$0.403901\pi$
$422$		0			0
$423$	1.73980e10	−	9.47853e8i	0.543423	−	0.0296060i
$424$		0			0
$425$	6.50170e9	+	3.75376e9i	0.199284	+	0.115056i
$426$		0			0
$427$	−3.02102e10	−	1.75998e8i	−0.908746	−	0.00529414i
$428$		0			0
$429$	−3.26885e9	+	3.45178e9i	−0.0965085	+	0.101909i
$430$		0			0
$431$	1.77405e10	−	1.02425e10i	0.514112	−	0.296822i	−0.220411	−	0.975407i	$-0.570740\pi$
$431$	1.77405e10	−	1.02425e10i	0.514112	−	0.296822i	0.734522	+	0.678585i	$0.237406\pi$
$432$		0			0
$433$	1.27553e9			0.0362859			0.0181430	−	0.999835i	$-0.494225\pi$
$433$	1.27553e9			0.0362859			0.0181430	+	0.999835i	$0.494225\pi$
$434$		0			0
$435$	−1.77700e10	+	7.43559e10i	−0.496285	+	2.07663i
$436$		0			0
$437$	−4.16412e9	+	2.40416e9i	−0.114182	+	0.0659230i
$438$		0			0
$439$	1.41474e10	−	2.45041e10i	0.380908	−	0.659751i	−0.610285	−	0.792182i	$-0.708945\pi$
$439$	1.41474e10	−	2.45041e10i	0.380908	−	0.659751i	0.991192	+	0.132431i	$0.0422782\pi$
$440$		0			0
$441$	1.61739e9	+	3.77883e10i	0.0427623	+	0.999085i
$442$		0			0
$443$	−4.36607e10	−	2.52075e10i	−1.13364	−	0.654508i	−0.188793	−	0.982017i	$-0.560457\pi$
$443$	−4.36607e10	−	2.52075e10i	−1.13364	−	0.654508i	−0.944848	+	0.327509i	$0.893791\pi$
$444$		0			0
$445$	−1.62378e10	−	2.81247e10i	−0.414083	−	0.717213i
$446$		0			0
$447$	−7.31495e10	−	1.74817e10i	−1.83224	−	0.437879i
$448$		0			0
$449$			6.43076e10i			1.58226i	0.611650	+	0.791128i	$0.290506\pi$
$449$			6.43076e10i			1.58226i	−0.611650	+	0.791128i	$0.709494\pi$
$450$		0			0
$451$	2.53474e10	+	4.39030e10i	0.612671	+	1.06118i
$452$		0			0
$453$	2.75387e10	+	2.60792e10i	0.653959	+	0.619301i
$454$		0			0
$455$	−5.02848e7	+	8.63146e9i	−0.00117325	+	0.201390i
$456$		0			0
$457$	1.16839e10	−	2.02372e10i	0.267870	−	0.463965i	−0.700441	−	0.713710i	$-0.747013\pi$
$457$	1.16839e10	−	2.02372e10i	0.267870	−	0.463965i	0.968312	+	0.249745i	$0.0803468\pi$
$458$		0			0
$459$	−2.29024e9	+	1.25745e10i	−0.0515977	+	0.283295i
$460$		0			0
$461$		−	7.53854e9i		−	0.166911i	−0.996512	−	0.0834553i	$-0.973404\pi$
$461$		−	7.53854e9i		−	0.166911i	0.996512	−	0.0834553i	$-0.0265956\pi$
$462$		0			0
$463$	3.19399e10			0.695041			0.347520	−	0.937672i	$-0.387024\pi$
$463$	3.19399e10			0.695041			0.347520	+	0.937672i	$0.387024\pi$
$464$		0			0
$465$	5.74544e9	+	1.93229e10i	0.122888	+	0.413296i
$466$		0			0
$467$	−1.57779e10	−	9.10936e9i	−0.331727	−	0.191523i	0.324881	−	0.945755i	$-0.394676\pi$
$467$	−1.57779e10	−	9.10936e9i	−0.331727	−	0.191523i	−0.656608	+	0.754232i	$0.728009\pi$
$468$		0			0
$469$	−3.16866e10	+	5.41518e10i	−0.654914	+	1.11924i
$470$		0			0
$471$	5.38174e10	+	5.09652e10i	1.09355	+	1.03560i
$472$		0			0
$473$	2.08491e10	−	1.20372e10i	0.416527	−	0.240482i
$474$		0			0
$475$	−4.15077e10			−0.815369
$476$		0			0
$477$	1.49393e10	−	2.94704e10i	0.288573	−	0.569262i
$478$		0			0
$479$	4.99587e10	−	2.88436e10i	0.949005	−	0.547909i	0.0562336	−	0.998418i	$-0.482091\pi$
$479$	4.99587e10	−	2.88436e10i	0.949005	−	0.547909i	0.892772	+	0.450509i	$0.148758\pi$
$480$		0			0
$481$	−7.12630e9	+	1.23431e10i	−0.133132	+	0.230592i
$482$		0			0
$483$	1.96503e9	+	6.75250e9i	0.0361061	+	0.124073i
$484$		0			0
$485$	−6.64637e10	−	3.83729e10i	−1.20121	−	0.693517i
$486$		0			0
$487$	−2.63352e10	−	4.56138e10i	−0.468188	−	0.810925i	0.531151	−	0.847277i	$-0.321760\pi$
$487$	−2.63352e10	−	4.56138e10i	−0.468188	−	0.810925i	−0.999339	+	0.0363521i	$0.988426\pi$
$488$		0			0
$489$	8.37349e9	−	3.50376e10i	0.146444	−	0.612771i
$490$		0			0
$491$		−	4.95042e10i		−	0.851758i	−0.904780	−	0.425879i	$-0.859965\pi$
$491$		−	4.95042e10i		−	0.851758i	0.904780	−	0.425879i	$-0.140035\pi$
$492$		0			0
$493$	−1.35385e10	−	2.34494e10i	−0.229184	−	0.396958i
$494$		0			0
$495$	−6.30479e10	+	4.11293e10i	−1.05015	+	0.685063i
$496$		0			0
$497$	−3.19565e9	+	1.82027e9i	−0.0523761	+	0.0298339i
$498$		0			0
$499$	3.65446e10	−	6.32970e10i	0.589414	−	1.02090i	−0.404895	−	0.914363i	$-0.632692\pi$
$499$	3.65446e10	−	6.32970e10i	0.589414	−	1.02090i	0.994309	−	0.106532i	$-0.0339747\pi$
$500$		0			0
$501$	2.07452e10	+	6.97697e10i	0.329281	+	1.10743i
$502$		0			0
$503$			5.81734e10i			0.908766i	0.890806	+	0.454383i	$0.150140\pi$
$503$			5.81734e10i			0.908766i	−0.890806	+	0.454383i	$0.849860\pi$
$504$		0			0
$505$	−1.51497e10			−0.232937
$506$		0			0
$507$	6.19060e10	−	1.84070e10i	0.936917	−	0.278581i
$508$		0			0
$509$	2.67878e9	+	1.54660e9i	0.0399086	+	0.0230412i	0.519822	−	0.854275i	$-0.325998\pi$
$509$	2.67878e9	+	1.54660e9i	0.0399086	+	0.0230412i	−0.479913	+	0.877316i	$0.659332\pi$
$510$		0			0
$511$	5.89702e10	−	1.00779e11i	0.864866	−	1.47804i
$512$		0			0
$513$	−2.37951e10	−	6.65389e10i	−0.343572	−	0.960741i
$514$		0			0
$515$	1.19270e11	−	6.88607e10i	1.69552	−	0.978909i
$516$		0			0
$517$	−3.63459e10			−0.508737
$518$		0			0
$519$	2.16917e10	+	5.18400e9i	0.298967	+	0.0714490i
$520$		0			0
$521$	2.52830e10	−	1.45971e10i	0.343145	−	0.198115i	−0.318517	−	0.947917i	$-0.603185\pi$
$521$	2.52830e10	−	1.45971e10i	0.343145	−	0.198115i	0.661662	+	0.749803i	$0.269852\pi$
$522$		0			0
$523$	6.17406e10	−	1.06938e11i	0.825210	−	1.42931i	−0.0765488	−	0.997066i	$-0.524390\pi$
$523$	6.17406e10	−	1.06938e11i	0.825210	−	1.42931i	0.901759	−	0.432240i	$-0.142277\pi$
$524$		0			0
$525$	−1.44550e10	+	5.89630e10i	−0.190274	+	0.776144i
$526$		0			0
$527$	−6.18336e9	−	3.56996e9i	−0.0801645	−	0.0462830i
$528$		0			0
$529$	−3.85017e10	−	6.66869e10i	−0.491651	−	0.851565i
$530$		0			0
$531$	5.89493e10	−	1.16288e11i	0.741482	−	1.46271i
$532$		0			0
$533$			1.58844e10i			0.196816i
$534$		0			0
$535$	−1.98043e10	−	3.43021e10i	−0.241738	−	0.418703i
$536$		0			0
$537$	−4.44114e10	+	4.68969e10i	−0.534070	+	0.563958i
$538$		0			0
$539$	9.19253e8	−	7.88928e10i	0.0108913	−	0.934721i
$540$		0			0
$541$	1.85709e10	−	3.21657e10i	0.216792	−	0.375495i	−0.737033	−	0.675856i	$-0.763774\pi$
$541$	1.85709e10	−	3.21657e10i	0.216792	−	0.375495i	0.953825	+	0.300362i	$0.0971073\pi$
$542$		0			0
$543$	5.05661e10	−	1.50352e10i	0.581648	−	0.172946i
$544$		0			0
$545$		−	1.10014e11i		−	1.24699i
$546$		0			0
$547$	8.32028e10			0.929370			0.464685	−	0.885476i	$-0.346168\pi$
$547$	8.32028e10			0.929370			0.464685	+	0.885476i	$0.346168\pi$
$548$		0			0
$549$	−4.49094e9	−	8.24320e10i	−0.0494365	−	0.907416i
$550$		0			0
$551$	1.29647e11	+	7.48520e10i	1.40656	+	0.812077i
$552$		0			0
$553$	−1.01042e11	−	5.88650e8i	−1.08045	−	0.00629443i
$554$		0			0
$555$	−1.55182e11	+	1.63866e11i	−1.63557	+	1.72710i
$556$		0			0
$557$	8.57873e9	−	4.95293e9i	0.0891256	−	0.0514567i	−0.454775	−	0.890606i	$-0.650280\pi$
$557$	8.57873e9	−	4.95293e9i	0.0891256	−	0.0514567i	0.543900	+	0.839150i	$0.316947\pi$
$558$		0			0
$559$	7.54333e9			0.0772531
$560$		0			0
$561$	6.19726e9	−	2.59315e10i	0.0625674	−	0.261804i
$562$		0			0
$563$	−8.62703e10	+	4.98082e10i	−0.858672	+	0.495755i	−0.863568	−	0.504233i	$-0.831775\pi$
$563$	−8.62703e10	+	4.98082e10i	−0.858672	+	0.495755i	0.00489502	+	0.999988i	$0.498442\pi$
$564$		0			0
$565$	−1.15370e11	+	1.99827e11i	−1.13214	+	1.96092i
$566$		0			0
$567$	−1.02807e11	+	1.06296e10i	−0.994697	+	0.102846i
$568$		0			0
$569$	6.89683e10	+	3.98189e10i	0.657962	+	0.379874i	0.791500	−	0.611169i	$-0.209301\pi$
$569$	6.89683e10	+	3.98189e10i	0.657962	+	0.379874i	−0.133538	+	0.991044i	$0.542634\pi$
$570$		0			0
$571$	7.14946e10	+	1.23832e11i	0.672556	+	1.16490i	0.977177	+	0.212428i	$0.0681371\pi$
$571$	7.14946e10	+	1.23832e11i	0.672556	+	1.16490i	−0.304620	+	0.952474i	$0.598530\pi$
$572$		0			0
$573$	9.03697e10	+	2.15971e10i	0.838310	+	0.200344i
$574$		0			0
$575$			1.12879e10i			0.103263i
$576$		0			0
$577$	−1.00892e11	−	1.74750e11i	−0.910235	−	1.57657i	−0.813732	−	0.581240i	$-0.802568\pi$
$577$	−1.00892e11	−	1.74750e11i	−0.910235	−	1.57657i	−0.0965021	−	0.995333i	$-0.530765\pi$
$578$		0			0
$579$	4.50635e10	+	4.26752e10i	0.400969	+	0.379718i
$580$		0			0
$581$	−2.98216e10	+	1.69867e10i	−0.261714	+	0.149075i
$582$		0			0
$583$	−3.44612e10	+	5.96886e10i	−0.298302	+	0.516675i
$584$		0			0
$585$	−2.35519e10	+	1.28312e9i	−0.201095	+	0.0109558i
$586$		0			0
$587$			2.93750e10i			0.247415i	0.992319	+	0.123707i	$0.0394784\pi$
$587$			2.93750e10i			0.247415i	−0.992319	+	0.123707i	$0.960522\pi$
$588$		0			0
$589$	3.94754e10			0.327993
$590$		0			0
$591$	−8.55870e9	−	2.87844e10i	−0.0701549	−	0.235943i
$592$		0			0
$593$	4.24481e10	+	2.45074e10i	0.343273	+	0.198189i	0.661718	−	0.749752i	$-0.269827\pi$
$593$	4.24481e10	+	2.45074e10i	0.343273	+	0.198189i	−0.318445	+	0.947941i	$0.603161\pi$
$594$		0			0
$595$	−2.39597e10	−	4.20635e10i	−0.191168	−	0.335612i
$596$		0			0
$597$	1.04668e11	+	9.91205e10i	0.823977	+	0.780308i
$598$		0			0
$599$	−1.62609e11	+	9.38821e10i	−1.26310	+	0.729248i	−0.973672	−	0.227953i	$-0.926797\pi$
$599$	−1.62609e11	+	9.38821e10i	−1.26310	+	0.729248i	−0.289423	+	0.957201i	$0.593463\pi$
$600$		0			0
$601$	1.40135e11			1.07411			0.537053	−	0.843548i	$-0.319537\pi$
$601$	1.40135e11			1.07411			0.537053	+	0.843548i	$0.319537\pi$
$602$		0			0
$603$	−1.52921e11	−	7.75195e10i	−1.15664	−	0.586330i
$604$		0			0
$605$	−1.96365e10	+	1.13371e10i	−0.146569	+	0.0846218i
$606$		0			0
$607$	7.63717e10	−	1.32280e11i	0.562572	−	0.974403i	−0.434699	−	0.900576i	$-0.643145\pi$
$607$	7.63717e10	−	1.32280e11i	0.562572	−	0.974403i	0.997271	−	0.0738270i	$-0.0235213\pi$
$608$		0			0
$609$	1.51479e11	−	1.58101e11i	1.10124	−	1.14939i
$610$		0			0
$611$	−9.86262e9	−	5.69418e9i	−0.0707664	−	0.0408570i
$612$		0			0
$613$	2.80686e10	+	4.86162e10i	0.198783	+	0.344302i	0.948134	−	0.317871i	$-0.102968\pi$
$613$	2.80686e10	+	4.86162e10i	0.198783	+	0.344302i	−0.749351	+	0.662173i	$0.769635\pi$
$614$		0			0
$615$	−5.84640e10	+	2.44634e11i	−0.408684	+	1.71008i
$616$		0			0
$617$		−	4.29400e10i		−	0.296293i	−0.988965	−	0.148147i	$-0.952669\pi$
$617$		−	4.29400e10i		−	0.296293i	0.988965	−	0.148147i	$-0.0473308\pi$
$618$		0			0
$619$	5.43304e9	+	9.41031e9i	0.0370067	+	0.0640975i	0.883936	−	0.467609i	$-0.154884\pi$
$619$	5.43304e9	+	9.41031e9i	0.0370067	+	0.0640975i	−0.846929	+	0.531706i	$0.821551\pi$
$620$		0			0
$621$	−1.80951e10	+	6.47103e9i	−0.121673	+	0.0435118i
$622$		0			0
$623$	−5.41857e8	+	9.30104e10i	−0.00359693	+	0.617418i
$624$		0			0
$625$	8.85409e10	−	1.53357e11i	0.580262	−	1.00504i
$626$		0			0
$627$	4.20121e10	+	1.41294e11i	0.271834	+	0.914227i
$628$		0			0
$629$		−	7.99331e10i		−	0.510651i
$630$		0			0
$631$	−3.12531e10			−0.197140			−0.0985701	−	0.995130i	$-0.531427\pi$
$631$	−3.12531e10			−0.197140			−0.0985701	+	0.995130i	$0.531427\pi$
$632$		0			0
$633$	−2.74502e11	+	8.16200e10i	−1.70975	+	0.508372i
$634$		0			0
$635$	−2.22518e11	−	1.28471e11i	−1.36858	−	0.790149i
$636$		0			0
$637$	1.26093e10	−	2.12639e10i	0.0765831	−	0.129147i
$638$		0			0
$639$	−5.49089e9	−	8.41709e9i	−0.0329336	−	0.0504846i
$640$		0			0
$641$	1.01252e9	−	5.84579e8i	0.00599752	−	0.00346267i	−0.496998	−	0.867752i	$-0.665564\pi$
$641$	1.01252e9	−	5.84579e8i	0.00599752	−	0.00346267i	0.502996	+	0.864289i	$0.332231\pi$
$642$		0			0
$643$	3.03236e10			0.177393			0.0886966	−	0.996059i	$-0.471730\pi$
$643$	3.03236e10			0.177393			0.0886966	+	0.996059i	$0.471730\pi$
$644$		0			0
$645$	1.16174e11	+	2.77640e10i	0.671229	+	0.160414i
$646$		0			0
$647$	−1.76301e11	+	1.01788e11i	−1.00609	+	0.580868i	−0.910045	−	0.414509i	$-0.863953\pi$
$647$	−1.76301e11	+	1.01788e11i	−1.00609	+	0.580868i	−0.0960476	+	0.995377i	$0.530620\pi$
$648$		0			0
$649$	−1.35982e11	+	2.35527e11i	−0.766481	+	1.32758i
$650$		0			0
$651$	1.37472e10	−	5.60760e10i	0.0765403	−	0.312214i
$652$		0			0
$653$	−6.23592e10	−	3.60031e10i	−0.342964	−	0.198010i	0.318618	−	0.947883i	$-0.396781\pi$
$653$	−6.23592e10	−	3.60031e10i	−0.342964	−	0.198010i	−0.661582	+	0.749873i	$0.730115\pi$
$654$		0			0
$655$	−1.22757e10	−	2.12621e10i	−0.0666930	−	0.115516i
$656$		0			0
$657$	2.84593e11	+	1.44267e11i	1.52744	+	0.774295i
$658$		0			0
$659$		−	2.86928e11i		−	1.52136i	−0.649128	−	0.760679i	$-0.724866\pi$
$659$		−	2.86928e11i		−	1.52136i	0.649128	−	0.760679i	$-0.275134\pi$
$660$		0			0
$661$	1.11955e11	+	1.93912e11i	0.586460	+	1.01578i	0.994692	+	0.102900i	$0.0328123\pi$
$661$	1.11955e11	+	1.93912e11i	0.586460	+	1.01578i	−0.408232	+	0.912878i	$0.633854\pi$
$662$		0			0
$663$	5.74425e9	−	6.06571e9i	0.0297289	−	0.0313926i
$664$		0			0
$665$	2.31002e11	+	1.35170e11i	1.18122	+	0.691182i
$666$		0			0
$667$	2.03559e10	−	3.52574e10i	0.102846	−	0.178134i
$668$		0			0
$669$	1.42756e11	−	4.24468e10i	0.712673	−	0.211905i
$670$		0			0
$671$			1.72207e11i			0.849496i
$672$		0			0
$673$	−5.41750e10			−0.264082			−0.132041	−	0.991244i	$-0.542153\pi$
$673$	−5.41750e10			−0.264082			−0.132041	+	0.991244i	$0.542153\pi$
$674$		0			0
$675$	−1.63209e11	−	2.97259e10i	−0.786193	−	0.143193i
$676$		0			0
$677$	9.84930e10	+	5.68650e10i	0.468868	+	0.270701i	0.715766	−	0.698340i	$-0.246078\pi$
$677$	9.84930e10	+	5.68650e10i	0.468868	+	0.270701i	−0.246898	+	0.969042i	$0.579411\pi$
$678$		0			0
$679$	1.08791e11	+	1.90993e11i	0.511817	+	0.898541i
$680$		0			0
$681$	8.95110e10	−	9.45204e10i	0.416187	−	0.439478i
$682$		0			0
$683$	−3.73633e11	+	2.15717e11i	−1.71697	+	0.991291i	−0.792639	+	0.609691i	$0.791293\pi$
$683$	−3.73633e11	+	2.15717e11i	−1.71697	+	0.991291i	−0.924328	+	0.381600i	$0.875373\pi$
$684$		0			0
$685$	−3.05044e11			−1.38548
$686$		0			0
$687$	−1.55067e10	+	6.48852e10i	−0.0696131	+	0.291285i
$688$		0			0
$689$	−1.87024e10	+	1.07978e10i	−0.0829890	+	0.0479137i
$690$		0			0
$691$	6.06267e10	−	1.05009e11i	0.265921	−	0.460588i	−0.701884	−	0.712291i	$-0.747657\pi$
$691$	6.06267e10	−	1.05009e11i	0.265921	−	0.460588i	0.967804	+	0.251704i	$0.0809908\pi$
$692$		0			0
$693$	2.15343e11	−	1.04742e10i	0.933681	−	0.0454136i
$694$		0			0
$695$	3.95363e11	+	2.28263e11i	1.69456	+	0.978353i
$696$		0			0
$697$	−4.45422e10	−	7.71494e10i	−0.188730	−	0.326890i
$698$		0			0
$699$	−3.24410e11	−	7.75296e10i	−1.35890	−	0.324757i
$700$		0			0
$701$			1.58865e11i			0.657895i	0.944348	+	0.328947i	$0.106694\pi$
$701$			1.58865e11i			0.657895i	−0.944348	+	0.328947i	$0.893306\pi$
$702$		0			0
$703$	2.20968e11	+	3.82728e11i	0.904707	+	1.56700i
$704$		0			0
$705$	−1.30935e11	−	1.23996e11i	−0.530030	−	0.501939i
$706$		0			0
$707$	3.74495e10	+	2.19134e10i	0.149889	+	0.0877064i
$708$		0			0
$709$	−6.06898e10	+	1.05118e11i	−0.240177	+	0.415998i	−0.960764	−	0.277365i	$-0.910539\pi$
$709$	−6.06898e10	+	1.05118e11i	−0.240177	+	0.415998i	0.720588	+	0.693364i	$0.243872\pi$
$710$		0			0
$711$	−1.50206e10	−	2.75706e11i	−0.0587772	−	1.07887i
$712$		0			0
$713$		−	1.07353e10i		−	0.0415388i
$714$		0			0
$715$	4.92019e10			0.188260
$716$		0			0
$717$	−6.82118e10	−	2.29408e11i	−0.258097	−	0.868025i
$718$		0			0
$719$	−3.18390e11	−	1.83822e11i	−1.19136	−	0.687832i	−0.232746	−	0.972538i	$-0.574771\pi$
$719$	−3.18390e11	−	1.83822e11i	−1.19136	−	0.687832i	−0.958615	+	0.284705i	$0.908104\pi$
$720$		0			0
$721$	−3.94435e11	−	2.29789e9i	−1.45960	−	0.00850330i
$722$		0			0
$723$	−2.86268e11	−	2.71096e11i	−1.04766	−	0.992134i
$724$		0			0
$725$	3.04359e11	−	1.75722e11i	1.10163	−	0.636025i
$726$		0			0
$727$	−1.43071e11			−0.512168			−0.256084	−	0.966654i	$-0.582432\pi$
$727$	−1.43071e11			−0.512168			−0.256084	+	0.966654i	$0.582432\pi$
$728$		0			0
$729$	−4.59105e10	−	2.78673e11i	−0.162556	−	0.986699i
$730$		0			0
$731$	−3.66375e10	+	2.11527e10i	−0.128309	+	0.0740791i
$732$		0			0
$733$	5.43153e10	−	9.40769e10i	0.188151	−	0.325887i	−0.756483	−	0.654014i	$-0.773084\pi$
$733$	5.43153e10	−	9.40769e10i	0.188151	−	0.325887i	0.944634	+	0.328127i	$0.106417\pi$
$734$		0			0
$735$	2.72459e11	−	2.81073e11i	0.933579	−	0.963097i
$736$		0			0
$737$	3.09723e11	+	1.78819e11i	1.04979	+	0.606098i
$738$		0			0
$739$	−7.57046e10	−	1.31124e11i	−0.253831	−	0.439648i	0.710746	−	0.703448i	$-0.248357\pi$
$739$	−7.57046e10	−	1.31124e11i	−0.253831	−	0.439648i	−0.964577	+	0.263800i	$0.915024\pi$
$740$		0			0
$741$	−1.07359e10	+	4.49227e10i	−0.0356094	+	0.149002i
$742$		0			0
$743$		−	3.17021e11i		−	1.04024i	−0.854094	−	0.520119i	$-0.825888\pi$
$743$		−	3.17021e11i		−	1.04024i	0.854094	−	0.520119i	$-0.174112\pi$
$744$		0			0
$745$	3.89196e11	+	6.74107e11i	1.26341	+	2.18828i
$746$		0			0
$747$	−5.12408e10	−	7.85480e10i	−0.164563	−	0.252262i
$748$		0			0
$749$	−6.60872e8	+	1.13439e11i	−0.00209986	+	0.360443i
$750$		0			0
$751$	8.57533e10	−	1.48529e11i	0.269582	−	0.466930i	−0.699172	−	0.714954i	$-0.746448\pi$
$751$	8.57533e10	−	1.48529e11i	0.269582	−	0.466930i	0.968754	+	0.248024i	$0.0797811\pi$
$752$		0			0
$753$	−6.07984e10	−	2.04476e11i	−0.189109	−	0.636007i
$754$		0			0
$755$		−	3.92538e11i		−	1.20807i
$756$		0			0
$757$	−3.69046e10			−0.112382			−0.0561911	−	0.998420i	$-0.517896\pi$
$757$	−3.69046e10			−0.112382			−0.0561911	+	0.998420i	$0.517896\pi$
$758$		0			0
$759$	3.84247e10	−	1.14251e10i	0.115783	−	0.0344266i
$760$		0			0
$761$	−4.32044e10	−	2.49441e10i	−0.128822	−	0.0743754i	0.434204	−	0.900814i	$-0.357030\pi$
$761$	−4.32044e10	−	2.49441e10i	−0.128822	−	0.0743754i	−0.563026	+	0.826439i	$0.690363\pi$
$762$		0			0
$763$	−1.59130e11	+	2.71951e11i	−0.469521	+	0.802402i
$764$		0			0
$765$	1.10792e11	−	7.22753e10i	0.323492	−	0.211030i
$766$		0			0
$767$	−7.37984e10	+	4.26075e10i	−0.213238	+	0.123113i
$768$		0			0
$769$	−4.96051e10			−0.141847			−0.0709236	−	0.997482i	$-0.522595\pi$
$769$	−4.96051e10			−0.141847			−0.0709236	+	0.997482i	$0.522595\pi$
$770$		0			0
$771$	−5.87897e11	−	1.40499e11i	−1.66373	−	0.397609i
$772$		0			0
$773$	2.63590e11	−	1.52184e11i	0.738262	−	0.426236i	−0.0831749	−	0.996535i	$-0.526506\pi$
$773$	2.63590e11	−	1.52184e11i	0.738262	−	0.426236i	0.821437	+	0.570299i	$0.193173\pi$
$774$		0			0
$775$	4.63360e10	−	8.02563e10i	0.128443	−	0.222471i
$776$		0			0
$777$	6.20628e11	−	1.80608e11i	1.70274	−	0.495510i
$778$		0			0
$779$	4.26545e11	+	2.46266e11i	1.15828	+	0.668736i
$780$		0			0
$781$	1.04818e10	+	1.81551e10i	0.0281730	+	0.0487971i
$782$		0			0
$783$	4.56171e11	+	3.87167e11i	1.21361	+	1.03003i
$784$		0			0
$785$		−	7.67116e11i		−	2.02014i
$786$		0			0
$787$	2.54476e11	+	4.40765e11i	0.663358	+	1.14897i	0.979728	+	0.200334i	$0.0642026\pi$
$787$	2.54476e11	+	4.40765e11i	0.663358	+	1.14897i	−0.316370	+	0.948636i	$0.602464\pi$
$788$		0			0
$789$	4.57883e11	−	4.83508e11i	1.18153	−	1.24766i
$790$		0			0
$791$	5.74229e11	−	3.27086e11i	1.46683	−	0.835518i
$792$		0			0
$793$	−2.69791e10	+	4.67292e10i	−0.0682236	+	0.118167i
$794$		0			0
$795$	−3.27777e11	+	9.74605e10i	−0.820559	+	0.243983i
$796$		0			0
$797$			1.31085e11i			0.324876i	0.986719	+	0.162438i	$0.0519358\pi$
$797$			1.31085e11i			0.324876i	−0.986719	+	0.162438i	$0.948064\pi$
$798$		0			0
$799$	6.38696e10			0.156714
$800$		0			0
$801$	−2.53789e11	+	1.38266e10i	−0.616514	+	0.0335880i
$802$		0			0
$803$	−5.76408e11	−	3.32789e11i	−1.38633	−	0.800401i
$804$		0			0
$805$	3.67591e10	−	6.28207e10i	0.0875350	−	0.149596i
$806$		0			0
$807$	−3.33763e11	+	3.52441e11i	−0.786944	+	0.830984i
$808$		0			0
$809$	−3.16264e11	+	1.82595e11i	−0.738339	+	0.426280i	−0.821465	−	0.570259i	$-0.806843\pi$
$809$	−3.16264e11	+	1.82595e11i	−0.738339	+	0.426280i	0.0831258	+	0.996539i	$0.473510\pi$
$810$		0			0
$811$	−2.53712e11			−0.586487			−0.293244	−	0.956038i	$-0.594735\pi$
$811$	−2.53712e11			−0.586487			−0.293244	+	0.956038i	$0.594735\pi$
$812$		0			0
$813$	8.46681e10	−	3.54280e11i	0.193802	−	0.810933i
$814$		0			0
$815$	−3.22888e11	+	1.86419e11i	−0.731848	+	0.422533i
$816$		0			0
$817$	1.16949e11	−	2.02562e11i	0.262488	−	0.454643i
$818$		0			0
$819$	6.00753e10	+	3.08949e10i	0.133524	+	0.0686674i
$820$		0			0
$821$	9.29739e10	+	5.36785e10i	0.204639	+	0.118148i	0.598817	−	0.800886i	$-0.295637\pi$
$821$	9.29739e10	+	5.36785e10i	0.204639	+	0.118148i	−0.394179	+	0.919034i	$0.628971\pi$
$822$		0			0
$823$	9.17659e10	+	1.58943e11i	0.200024	+	0.346452i	0.948536	−	0.316670i	$-0.102565\pi$
$823$	9.17659e10	+	1.58943e11i	0.200024	+	0.346452i	−0.748512	+	0.663121i	$0.769231\pi$
$824$		0			0
$825$	3.36575e11	+	8.04368e10i	0.726551	+	0.173636i
$826$		0			0
$827$		−	8.71283e11i		−	1.86267i	−0.364157	−	0.931337i	$-0.618643\pi$
$827$		−	8.71283e11i		−	1.86267i	0.364157	−	0.931337i	$-0.381357\pi$
$828$		0			0
$829$	−1.08452e11	−	1.87845e11i	−0.229625	−	0.397723i	0.728072	−	0.685501i	$-0.240417\pi$
$829$	−1.08452e11	−	1.87845e11i	−0.229625	−	0.397723i	−0.957697	+	0.287778i	$0.907083\pi$
$830$		0			0
$831$	2.57823e11	+	2.44159e11i	0.540651	+	0.511998i
$832$		0			0
$833$	−1.61538e9	+	1.38636e11i	−0.00335501	+	0.287936i
$834$		0			0
$835$	3.76668e11	−	6.52409e11i	0.774842	−	1.34207i
$836$		0			0
$837$	1.55218e11	+	2.82705e10i	0.316257	+	0.0576011i
$838$		0			0
$839$			4.64214e11i			0.936852i	0.883503	+	0.468426i	$0.155179\pi$
$839$			4.64214e11i			0.936852i	−0.883503	+	0.468426i	$0.844821\pi$
$840$		0			0
$841$	−7.67291e11			−1.53383
$842$		0			0
$843$	−1.13002e11	−	3.80045e11i	−0.223756	−	0.752532i
$844$		0			0
$845$	−5.78876e11	−	3.34214e11i	−1.13543	−	0.655538i
$846$		0			0
$847$	6.49393e10	+	3.78321e8i	0.126175	+	0.000735067i
$848$		0			0
$849$	3.19078e11	+	3.02168e11i	0.614138	+	0.581590i
$850$		0			0
$851$	1.04082e11	−	6.00918e10i	0.198453	−	0.114577i
$852$		0			0
$853$	−4.67940e11			−0.883882			−0.441941	−	0.897044i	$-0.645710\pi$
$853$	−4.67940e11			−0.883882			−0.441941	+	0.897044i	$0.645710\pi$
$854$		0			0
$855$	−3.30685e11	+	6.52336e11i	−0.618800	+	1.22069i
$856$		0			0
$857$	−4.80557e11	+	2.77449e11i	−0.890884	+	0.514352i	−0.874232	−	0.485509i	$-0.838634\pi$
$857$	−4.80557e11	+	2.77449e11i	−0.890884	+	0.514352i	−0.0166525	+	0.999861i	$0.505301\pi$
$858$		0			0
$859$	−2.15341e11	+	3.72981e11i	−0.395506	+	0.685037i	−0.993166	−	0.116713i	$-0.962764\pi$
$859$	−2.15341e11	+	3.72981e11i	−0.395506	+	0.685037i	0.597660	+	0.801750i	$0.296097\pi$
$860$		0			0
$861$	4.98371e11	−	5.20159e11i	0.906861	−	0.946506i
$862$		0			0
$863$	5.39592e11	+	3.11533e11i	0.972797	+	0.561644i	0.900088	−	0.435709i	$-0.143502\pi$
$863$	5.39592e11	+	3.11533e11i	0.972797	+	0.561644i	0.0727089	+	0.997353i	$0.476836\pi$
$864$		0			0
$865$	−1.15412e11	−	1.99899e11i	−0.206151	−	0.357064i
$866$		0			0
$867$	1.20447e11	−	5.03992e11i	0.213167	−	0.891963i
$868$		0			0
$869$			5.75972e11i			1.01000i
$870$		0			0
$871$	5.60298e10	+	9.70464e10i	0.0973523	+	0.168619i
$872$		0			0
$873$	−5.03061e11	+	3.28172e11i	−0.866091	+	0.564994i
$874$		0			0
$875$	−1.37236e11	+	7.81706e10i	−0.234118	+	0.133356i
$876$		0			0
$877$	−1.84222e11	+	3.19082e11i	−0.311418	+	0.539392i	−0.978670	−	0.205441i	$-0.934137\pi$
$877$	−1.84222e11	+	3.19082e11i	−0.311418	+	0.539392i	0.667252	+	0.744832i	$0.267471\pi$
$878$		0			0
$879$	2.92036e11	+	9.82169e11i	0.489194	+	1.64525i
$880$		0			0
$881$			7.03606e11i			1.16795i	0.811770	+	0.583977i	$0.198504\pi$
$881$			7.03606e11i			1.16795i	−0.811770	+	0.583977i	$0.801496\pi$
$882$		0			0
$883$	5.02291e10			0.0826252			0.0413126	−	0.999146i	$-0.486846\pi$
$883$	5.02291e10			0.0826252			0.0413126	+	0.999146i	$0.486846\pi$
$884$		0			0
$885$	−1.29338e12	+	3.84572e11i	−2.10841	+	0.626910i
$886$		0			0
$887$	9.99354e11	+	5.76977e11i	1.61445	+	0.932104i	0.988321	+	0.152385i	$0.0486953\pi$
$887$	9.99354e11	+	5.76977e11i	1.61445	+	0.932104i	0.626130	+	0.779719i	$0.284638\pi$
$888$		0			0
$889$	3.64228e11	+	6.39435e11i	0.583131	+	1.02374i
$890$		0			0
$891$	6.40039e10	+	5.85658e11i	0.101554	+	0.929252i
$892$		0			0
$893$	−3.05814e11	+	1.76562e11i	−0.480896	+	0.277645i
$894$		0			0
$895$	6.68470e11			1.04181
$896$		0			0
$897$	1.22166e10	+	2.91961e9i	0.0188704	+	0.00450977i
$898$		0			0
$899$	−2.89457e11	+	1.67118e11i	−0.443144	+	0.255850i
$900$		0			0
$901$	6.05577e10	−	1.04889e11i	0.0918904	−	0.159159i
$902$		0			0
$903$	−2.47018e11	−	2.36672e11i	−0.371517	−	0.355955i
$904$		0			0
$905$	−4.72838e11	−	2.72993e11i	−0.704885	−	0.406966i
$906$		0			0
$907$	6.66818e10	+	1.15496e11i	0.0985322	+	0.170663i	0.911077	−	0.412236i	$-0.135252\pi$
$907$	6.66818e10	+	1.15496e11i	0.0985322	+	0.170663i	−0.812545	+	0.582898i	$0.801919\pi$
$908$		0			0
$909$	−5.36098e10	+	1.05755e11i	−0.0785216	+	0.154898i
$910$		0			0
$911$			1.47755e11i			0.214520i	0.994231	+	0.107260i	$0.0342078\pi$
$911$			1.47755e11i			0.214520i	−0.994231	+	0.107260i	$0.965792\pi$
$912$		0			0
$913$	9.78160e10	+	1.69422e11i	0.140775	+	0.243830i
$914$		0			0
$915$	−5.87494e11	+	6.20373e11i	−0.838146	+	0.885051i
$916$		0			0
$917$	−4.09640e8	+	7.03152e10i	−0.000579329	+	0.0994425i
$918$		0			0
$919$	3.80288e11	−	6.58679e11i	0.533152	−	0.923446i	−0.466098	−	0.884733i	$-0.654341\pi$
$919$	3.80288e11	−	6.58679e11i	0.533152	−	0.923446i	0.999250	−	0.0387135i	$-0.0123260\pi$
$920$		0			0
$921$	2.26762e11	−	6.74249e10i	0.315160	−	0.0937092i
$922$		0			0
$923$			6.56861e9i			0.00905037i
$924$		0			0
$925$	1.03748e12			1.41715
$926$		0			0
$927$	−5.86352e10	−	1.07626e12i	−0.0794035	−	1.45746i
$928$		0			0
$929$	−2.97498e11	−	1.71761e11i	−0.399413	−	0.230601i	0.286818	−	0.957985i	$-0.407403\pi$
$929$	−2.97498e11	−	1.71761e11i	−0.399413	−	0.230601i	−0.686230	+	0.727384i	$0.740736\pi$
$930$		0			0
$931$	−3.75512e11	−	6.68268e11i	−0.499833	−	0.889512i
$932$		0			0
$933$	−9.47677e11	+	1.00071e12i	−1.25064	+	1.32063i
$934$		0			0
$935$	−2.38971e11	+	1.37970e11i	−0.312679	+	0.180525i
$936$		0			0
$937$	−2.22618e11			−0.288803			−0.144401	−	0.989519i	$-0.546126\pi$
$937$	−2.22618e11			−0.288803			−0.144401	+	0.989519i	$0.546126\pi$
$938$		0			0
$939$	2.04669e11	−	8.56404e11i	0.263262	−	1.10158i
$940$		0			0
$941$	−1.53177e11	+	8.84367e10i	−0.195360	+	0.112791i	−0.594489	−	0.804104i	$-0.702646\pi$
$941$	−1.53177e11	+	8.84367e10i	−0.195360	+	0.112791i	0.399130	+	0.916895i	$0.369312\pi$
$942$		0			0
$943$	6.69716e10	−	1.15998e11i	0.0846923	−	0.146691i
$944$		0			0
$945$	8.11503e11	+	6.96923e11i	1.01757	+	0.873891i
$946$		0			0
$947$	−5.66637e11	−	3.27148e11i	−0.704539	−	0.406766i	0.104497	−	0.994525i	$-0.466677\pi$
$947$	−5.66637e11	−	3.27148e11i	−0.704539	−	0.406766i	−0.809036	+	0.587759i	$0.800010\pi$
$948$		0			0
$949$	−1.04274e11	−	1.80608e11i	−0.128561	−	0.222675i
$950$		0			0
$951$	1.08926e12	+	2.60319e11i	1.33171	+	0.318261i
$952$		0			0
$953$			1.08868e12i			1.31987i	0.751324	+	0.659934i	$0.229416\pi$
$953$			1.08868e12i			1.31987i	−0.751324	+	0.659934i	$0.770584\pi$
$954$		0			0
$955$	−4.80817e11	−	8.32800e11i	−0.578051	−	1.00121i
$956$		0			0
$957$	−9.06224e11	−	8.58196e11i	−1.08041	−	1.02315i
$958$		0			0
$959$	7.54055e11	+	4.41231e11i	0.891515	+	0.521665i
$960$		0			0
$961$	3.82378e11	−	6.62299e11i	0.448332	−	0.776534i
$962$		0			0
$963$	−3.09532e11	+	1.68635e10i	−0.359916	+	0.0196084i
$964$		0			0
$965$		−	6.42337e11i		−	0.740720i
$966$		0			0
$967$	1.11340e12			1.27334			0.636669	−	0.771137i	$-0.280312\pi$
$967$	1.11340e12			1.27334			0.636669	+	0.771137i	$0.280312\pi$
$968$		0			0
$969$	−7.38266e10	−	2.48292e11i	−0.0837371	−	0.281623i
$970$		0			0
$971$	8.01034e11	+	4.62477e11i	0.901102	+	0.520252i	0.877558	−	0.479471i	$-0.159171\pi$
$971$	8.01034e11	+	4.62477e11i	0.901102	+	0.520252i	0.0235447	+	0.999723i	$0.492505\pi$
$972$		0			0
$973$	−6.47149e11	−	1.13613e12i	−0.722026	−	1.26758i
$974$		0			0
$975$	7.87294e10	+	7.45569e10i	0.0871201	+	0.0825029i
$976$		0			0
$977$	1.22820e12	−	7.09101e11i	1.34800	−	0.778269i	0.360036	−	0.932939i	$-0.382765\pi$
$977$	1.22820e12	−	7.09101e11i	1.34800	−	0.778269i	0.987966	+	0.154669i	$0.0494312\pi$
$978$		0			0
$979$	5.30187e11			0.577163
$980$		0			0
$981$	−7.67971e11	−	3.89304e11i	−0.829218	−	0.420351i
$982$		0			0
$983$	−1.42369e12	+	8.21970e11i	−1.52476	+	0.880322i	−0.525194	+	0.850983i	$0.676007\pi$
$983$	−1.42369e12	+	8.21970e11i	−1.52476	+	0.880322i	−0.999569	+	0.0293398i	$0.990660\pi$
$984$		0			0
$985$	−1.55399e11	+	2.69160e11i	−0.165084	+	0.285934i
$986$		0			0
$987$	1.44312e11	+	4.95905e11i	0.152067	+	0.522552i
$988$		0			0
$989$	−5.50864e10	−	3.18042e10i	−0.0575784	−	0.0332429i
$990$		0			0
$991$	−7.61463e11	−	1.31889e12i	−0.789504	−	1.36746i	−0.926271	−	0.376857i	$-0.877005\pi$
$991$	−7.61463e11	−	1.31889e12i	−0.789504	−	1.36746i	0.136768	−	0.990603i	$-0.456329\pi$
$992$		0			0
$993$	1.34767e11	−	5.63911e11i	0.138607	−	0.579981i
$994$		0			0
$995$		−	1.49194e12i		−	1.52215i
$996$		0			0
$997$	7.56987e11	+	1.31114e12i	0.766140	+	1.32699i	0.939642	+	0.342159i	$0.111158\pi$
$997$	7.56987e11	+	1.31114e12i	0.766140	+	1.32699i	−0.173502	+	0.984833i	$0.555508\pi$
$998$		0			0
$999$	5.94757e11	+	1.66314e12i	0.597142	+	1.66981i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.9.p.b.65.2	yes	40
3.2	odd	2	inner	84.9.p.b.65.16	yes	40
7.4	even	3	inner	84.9.p.b.53.16	yes	40
21.11	odd	6	inner	84.9.p.b.53.2	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.9.p.b.53.2	✓	40	21.11	odd	6	inner
84.9.p.b.53.16	yes	40	7.4	even	3	inner
84.9.p.b.65.2	yes	40	1.1	even	1	trivial
84.9.p.b.65.16	yes	40	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	84.p (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-77.6406 + 23.0855i) q^{3} +(726.008 + 419.161i) q^{5} +(-1188.37 - 2086.29i) q^{7} +(5495.12 - 3584.74i) q^{9} +O(q^{10})\)	\(q+(-77.6406 + 23.0855i) q^{3} +(726.008 + 419.161i) q^{5} +(-1188.37 - 2086.29i) q^{7} +(5495.12 - 3584.74i) q^{9} +(-11852.6 + 6843.09i) q^{11} -4288.33 q^{13} +(-66044.2 - 15783.7i) q^{15} +(20828.2 - 12025.2i) q^{17} +(-66484.9 + 115155. i) q^{19} +(140428. + 134546. i) q^{21} +(31316.3 + 18080.5i) q^{23} +(156079. + 270338. i) q^{25} +(-343889. + 405179. i) q^{27} -1.12585e6i q^{29} +(-148437. - 257101. i) q^{31} +(762265. - 804924. i) q^{33} +(11726.0 - 2.01278e6i) q^{35} +(1.66179e6 - 2.87830e6i) q^{37} +(332949. - 98998.2i) q^{39} -3.70409e6i q^{41} -1.75904e6 q^{43} +(5.49209e6 - 299212. i) q^{45} +(2.29987e6 + 1.32783e6i) q^{47} +(-2.94037e6 + 4.95854e6i) q^{49} +(-1.33951e6 + 1.41447e6i) q^{51} +(4.36123e6 - 2.51796e6i) q^{53} -1.14734e7 q^{55} +(2.50351e6 - 1.04756e7i) q^{57} +(1.72091e7 - 9.93569e6i) q^{59} +(6.29128e6 - 1.08968e7i) q^{61} +(-1.40090e7 - 7.20440e6i) q^{63} +(-3.11336e6 - 1.79750e6i) q^{65} +(-1.30656e7 - 2.26303e7i) q^{67} +(-2.84881e6 - 680826. i) q^{69} -1.53174e6i q^{71} +(2.43157e7 + 4.21161e7i) q^{73} +(-1.83590e7 - 1.73860e7i) q^{75} +(2.83619e7 + 1.65958e7i) q^{77} +(2.10421e7 - 3.64460e7i) q^{79} +(1.73460e7 - 3.93972e7i) q^{81} -1.42941e7i q^{83} +2.01619e7 q^{85} +(2.59908e7 + 8.74116e7i) q^{87} +(-3.35488e7 - 1.93694e7i) q^{89} +(5.09611e6 + 8.94668e6i) q^{91} +(1.74601e7 + 1.65347e7i) q^{93} +(-9.65372e7 + 5.57358e7i) q^{95} -9.15468e7 q^{97} +(-4.06007e7 + 8.00921e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 81 q^{3} - 34 q^{7} + 4771 q^{9}+O(q^{10}) \) 40 * q + 81 * q^3 - 34 * q^7 + 4771 * q^9	\( 40 q + 81 q^{3} - 34 q^{7} + 4771 q^{9} - 55464 q^{13} + 68482 q^{15} + 311690 q^{19} - 172343 q^{21} + 1766792 q^{25} - 3451932 q^{27} + 31596 q^{31} + 1874885 q^{33} - 1853482 q^{37} + 11217526 q^{39} - 13372600 q^{43} - 527785 q^{45} - 12653462 q^{49} - 1103461 q^{51} + 71577224 q^{55} - 17195214 q^{57} - 21761970 q^{61} + 21945045 q^{63} - 26337350 q^{67} - 5588722 q^{69} + 41115682 q^{73} - 17971730 q^{75} - 120916932 q^{79} - 24550133 q^{81} + 139250060 q^{85} - 16321046 q^{87} + 345074940 q^{91} + 25774675 q^{93} - 707216948 q^{97} - 94510994 q^{99}+O(q^{100}) \) 40 * q + 81 * q^3 - 34 * q^7 + 4771 * q^9 - 55464 * q^13 + 68482 * q^15 + 311690 * q^19 - 172343 * q^21 + 1766792 * q^25 - 3451932 * q^27 + 31596 * q^31 + 1874885 * q^33 - 1853482 * q^37 + 11217526 * q^39 - 13372600 * q^43 - 527785 * q^45 - 12653462 * q^49 - 1103461 * q^51 + 71577224 * q^55 - 17195214 * q^57 - 21761970 * q^61 + 21945045 * q^63 - 26337350 * q^67 - 5588722 * q^69 + 41115682 * q^73 - 17971730 * q^75 - 120916932 * q^79 - 24550133 * q^81 + 139250060 * q^85 - 16321046 * q^87 + 345074940 * q^91 + 25774675 * q^93 - 707216948 * q^97 - 94510994 * q^99

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