LMFDB - Embedded newform 84.9.p.b.53.16

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			53.16
Character	$\chi$	$=$	84.53
Dual form			84.9.p.b.65.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(58.8129 + 55.6960i) q^{3} +(-726.008 + 419.161i) q^{5} +(-1188.37 + 2086.29i) q^{7} +(356.917 + 6551.28i) q^{9} +O(q^{10})$	\(q+(58.8129 + 55.6960i) q^{3} +(-726.008 + 419.161i) q^{5} +(-1188.37 + 2086.29i) q^{7} +(356.917 + 6551.28i) q^{9} +(11852.6 + 6843.09i) q^{11} -4288.33 q^{13} +(-66044.2 - 15783.7i) q^{15} +(-20828.2 - 12025.2i) q^{17} +(-66484.9 - 115155. i) q^{19} +(-186089. + 56513.3i) q^{21} +(-31316.3 + 18080.5i) q^{23} +(156079. - 270338. i) q^{25} +(-343889. + 405179. i) q^{27} -1.12585e6i q^{29} +(-148437. + 257101. i) q^{31} +(315952. + 1.06260e6i) q^{33} +(-11726.0 - 2.01278e6i) q^{35} +(1.66179e6 + 2.87830e6i) q^{37} +(-252209. - 238843. i) q^{39} -3.70409e6i q^{41} -1.75904e6 q^{43} +(-3.00517e6 - 4.60668e6i) q^{45} +(-2.29987e6 + 1.32783e6i) q^{47} +(-2.94037e6 - 4.95854e6i) q^{49} +(-555213. - 1.86728e6i) q^{51} +(-4.36123e6 - 2.51796e6i) q^{53} -1.14734e7 q^{55} +(2.50351e6 - 1.04756e7i) q^{57} +(-1.72091e7 - 9.93569e6i) q^{59} +(6.29128e6 + 1.08968e7i) q^{61} +(-1.40920e7 - 7.04069e6i) q^{63} +(3.11336e6 - 1.79750e6i) q^{65} +(-1.30656e7 + 2.26303e7i) q^{67} +(-2.84881e6 - 680826. i) q^{69} -1.53174e6i q^{71} +(2.43157e7 - 4.21161e7i) q^{73} +(2.42362e7 - 7.20634e6i) q^{75} +(-2.83619e7 + 1.65958e7i) q^{77} +(2.10421e7 + 3.64460e7i) q^{79} +(-4.27919e7 + 4.67654e6i) q^{81} -1.42941e7i q^{83} +2.01619e7 q^{85} +(6.27053e7 - 6.62145e7i) q^{87} +(3.35488e7 - 1.93694e7i) q^{89} +(5.09611e6 - 8.94668e6i) q^{91} +(-2.30495e7 + 6.85350e6i) q^{93} +(9.65372e7 + 5.57358e7i) q^{95} -9.15468e7 q^{97} +(-4.06007e7 + 8.00921e7i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/84\mathbb{Z}\right)^\times$.

$n$	$29$	$43$	$73$
$\chi(n)$	$-1$	$1$	$e\left(\frac{2}{3}\right)$

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 $

$2$		0			0
$2$		0			0
$3$	58.8129	+	55.6960i	0.726085	+	0.687605i
$3$	58.8129	+	55.6960i	0.726085	+	0.687605i
$4$		0			0
$5$	−726.008	+	419.161i	−1.16161	+	0.670658i	−0.951690	−	0.307060i	$-0.900655\pi$
$5$	−726.008	+	419.161i	−1.16161	+	0.670658i	−0.209923	+	0.977718i	$0.567321\pi$
$6$		0			0
$7$	−1188.37	+	2086.29i	−0.494946	+	0.868924i
$7$	−1188.37	+	2086.29i	−0.494946	+	0.868924i
$8$		0			0
$9$	356.917	+	6551.28i	0.0543999	+	0.998519i
$10$		0			0
$11$	11852.6	+	6843.09i	0.809547	+	0.467392i	0.846799	−	0.531914i	$-0.178527\pi$
$11$	11852.6	+	6843.09i	0.809547	+	0.467392i	−0.0372513	+	0.999306i	$0.511860\pi$
$12$		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	$-0.379323\pi$
$17$	−20828.2	−	12025.2i	−0.249377	−	0.143978i	−0.619479	+	0.785013i	$0.712656\pi$
$18$		0			0
$19$	−66484.9	−	115155.i	−0.510163	−	0.883628i	−0.999931	−	0.0117749i	$-0.996252\pi$
$19$	−66484.9	−	115155.i	−0.510163	−	0.883628i	0.489768	−	0.871853i	$-0.337081\pi$
$20$		0			0
$21$	−186089.	+	56513.3i	−0.956849	+	0.290585i
$22$		0			0
$23$	−31316.3	+	18080.5i	−0.111907	+	0.0646097i	−0.554909	−	0.831911i	$-0.687247\pi$
$23$	−31316.3	+	18080.5i	−0.111907	+	0.0646097i	0.443002	+	0.896521i	$0.353914\pi$
$24$		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	−0.935131	−	0.354303i	$-0.884718\pi$
$31$	−148437.	+	257101.i	−0.160730	+	0.278392i	0.774401	+	0.632695i	$0.218051\pi$
$32$		0			0
$33$	315952.	+	1.06260e6i	0.266419	+	0.896015i
$34$		0			0
$35$	−11726.0	−	2.01278e6i	−0.00781406	−	1.34129i
$36$		0			0
$37$	1.66179e6	+	2.87830e6i	0.886684	+	1.53578i	0.843771	+	0.536704i	$0.180331\pi$
$37$	1.66179e6	+	2.87830e6i	0.886684	+	1.53578i	0.0429137	+	0.999079i	$0.486336\pi$
$38$		0			0
$39$	−252209.	−	238843.i	−0.109019	−	0.103241i
$40$		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
$45$	−3.00517e6	−	4.60668e6i	−0.732856	−	1.12341i
$46$		0			0
$47$	−2.29987e6	+	1.32783e6i	−0.471316	+	0.272114i	−0.716791	−	0.697289i	$-0.754390\pi$
$47$	−2.29987e6	+	1.32783e6i	−0.471316	+	0.272114i	0.245474	+	0.969403i	$0.421056\pi$
$48$		0			0
$49$	−2.94037e6	−	4.95854e6i	−0.510056	−	0.860141i
$50$		0			0
$51$	−555213.	−	1.86728e6i	−0.0820690	−	0.276013i
$52$		0			0
$53$	−4.36123e6	−	2.51796e6i	−0.552721	−	0.319113i	0.197498	−	0.980303i	$-0.436718\pi$
$53$	−4.36123e6	−	2.51796e6i	−0.552721	−	0.319113i	−0.750219	+	0.661190i	$0.770052\pi$
$54$		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	$-0.639335\pi$
$59$	−1.72091e7	−	9.93569e6i	−1.42020	−	0.819955i	−0.996316	+	0.0857601i	$0.972668\pi$
$60$		0			0
$61$	6.29128e6	+	1.08968e7i	0.454381	+	0.787010i	0.998652	−	0.0518987i	$-0.0165273\pi$
$61$	6.29128e6	+	1.08968e7i	0.454381	+	0.787010i	−0.544272	+	0.838909i	$0.683194\pi$
$62$		0			0
$63$	−1.40920e7	−	7.04069e6i	−0.894562	−	0.446944i
$64$		0			0
$65$	3.11336e6	−	1.79750e6i	0.174412	−	0.100697i
$66$		0			0
$67$	−1.30656e7	+	2.26303e7i	−0.648383	+	1.12303i	0.335127	+	0.942173i	$0.391221\pi$
$67$	−1.30656e7	+	2.26303e7i	−0.648383	+	1.12303i	−0.983509	+	0.180858i	$0.942112\pi$
$68$		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	−0.0192491	−	0.999815i	$-0.506128\pi$
$73$	2.43157e7	−	4.21161e7i	0.856240	−	1.48305i	0.875490	−	0.483237i	$-0.160539\pi$
$74$		0			0
$75$	2.42362e7	−	7.20634e6i	0.765983	−	0.227756i
$76$		0			0
$77$	−2.83619e7	+	1.65958e7i	−0.806811	+	0.472101i
$78$		0			0
$79$	2.10421e7	+	3.64460e7i	0.540233	+	0.935711i	0.998890	+	0.0470974i	$0.0149971\pi$
$79$	2.10421e7	+	3.64460e7i	0.540233	+	0.935711i	−0.458658	+	0.888613i	$0.651670\pi$
$80$		0			0
$81$	−4.27919e7	+	4.67654e6i	−0.994081	+	0.108639i
$82$		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
$87$	6.27053e7	−	6.62145e7i	1.09453	−	1.15578i
$88$		0			0
$89$	3.35488e7	−	1.93694e7i	0.534709	−	0.308714i	−0.208223	−	0.978081i	$-0.566768\pi$
$89$	3.35488e7	−	1.93694e7i	0.534709	−	0.308714i	0.742932	+	0.669367i	$0.233435\pi$
$90$		0			0
$91$	5.09611e6	−	8.94668e6i	0.0743144	−	0.130466i
$92$		0			0
$93$	−2.30495e7	+	6.85350e6i	−0.308127	+	0.0916179i
$94$		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	$-0.305659\pi$
$101$	1.56504e7	+	9.03574e6i	0.150397	+	0.0868317i	−0.422913	+	0.906170i	$0.638992\pi$
$102$		0			0
$103$	8.21411e7	+	1.42273e8i	0.729813	+	1.26407i	0.956962	+	0.290213i	$0.0937263\pi$
$103$	8.21411e7	+	1.42273e8i	0.729813	+	1.26407i	−0.227149	+	0.973860i	$0.572940\pi$
$104$		0			0
$105$	1.11414e8	−	1.19030e8i	0.916605	−	0.979266i
$106$		0			0
$107$	4.09176e7	−	2.36238e7i	0.312158	−	0.180225i	−0.335734	−	0.941957i	$-0.608984\pi$
$107$	4.09176e7	−	2.36238e7i	0.312158	−	0.180225i	0.647892	+	0.761732i	$0.275651\pi$
$108$		0			0
$109$	−6.56157e7	+	1.13650e8i	−0.464838	+	0.805123i	−0.999194	−	0.0401362i	$-0.987221\pi$
$109$	−6.56157e7	+	1.13650e8i	−0.464838	+	0.805123i	0.534356	+	0.845259i	$0.320554\pi$
$110$		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
$117$	−1.53058e6	−	2.80941e7i	−0.00816794	−	0.149924i
$118$		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
$123$	2.06303e8	−	2.17848e8i	0.901331	−	0.951773i
$124$		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
$129$	−1.03454e8	−	9.79712e7i	−0.373584	−	0.353785i
$130$		0			0
$131$	2.53627e7	−	1.46432e7i	0.0861212	−	0.0497221i	−0.456321	−	0.889815i	$-0.650833\pi$
$131$	2.53627e7	−	1.46432e7i	0.0861212	−	0.0497221i	0.542442	+	0.840093i	$0.317500\pi$
$132$		0			0
$133$	3.19255e8	−	1.85991e6i	1.02031	−	0.00594408i
$134$		0			0
$135$	7.98308e7	−	4.38308e8i	0.240345	−	1.31961i
$136$		0			0
$137$	3.15124e8	+	1.81937e8i	0.894539	+	0.516462i	0.875424	−	0.483355i	$-0.160582\pi$
$137$	3.15124e8	+	1.81937e8i	0.894539	+	0.516462i	0.0191146	+	0.999817i	$0.493915\pi$
$138$		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
$147$	1.03239e8	−	4.55393e8i	0.221093	−	0.975253i
$148$		0			0
$149$	−8.04115e8	+	4.64256e8i	−1.63145	+	0.941916i	−0.647800	+	0.761811i	$0.724311\pi$
$149$	−8.04115e8	+	4.64256e8i	−1.63145	+	0.941916i	−0.983647	+	0.180106i	$0.942356\pi$
$150$		0			0
$151$	−2.34121e8	+	4.05510e8i	−0.450332	+	0.779998i	−0.998406	−	0.0564316i	$-0.982028\pi$
$151$	−2.34121e8	+	4.05510e8i	−0.450332	+	0.779998i	0.548074	+	0.836430i	$0.315361\pi$
$152$		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	0.193294	+	0.981141i	$0.438083\pi$
$157$	−4.57530e8	+	7.92466e8i	−0.753046	+	1.30431i	−0.946340	+	0.323173i	$0.895250\pi$
$158$		0			0
$159$	−1.16257e8	−	3.90991e8i	−0.181899	−	0.611757i
$160$		0			0
$161$	−505798.	−	8.68209e7i	−0.000752790	−	0.129217i
$162$		0			0
$163$	−2.22372e8	−	3.85159e8i	−0.315014	−	0.545620i	0.664427	−	0.747353i	$-0.268676\pi$
$163$	−2.22372e8	−	3.85159e8i	−0.315014	−	0.545620i	−0.979440	+	0.201734i	$0.935342\pi$
$164$		0			0
$165$	−6.74786e8	−	6.39024e8i	−0.910396	−	0.862147i
$166$		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
$171$	7.30685e8	−	4.76663e8i	0.854566	−	0.557476i
$172$		0			0
$173$	2.38451e8	−	1.37670e8i	0.266204	−	0.153693i	−0.360957	−	0.932582i	$-0.617550\pi$
$173$	2.38451e8	−	1.37670e8i	0.266204	−	0.153693i	0.627161	+	0.778889i	$0.284217\pi$
$174$		0			0
$175$	3.78522e8	+	6.46886e8i	0.403588	+	0.689725i
$176$		0			0
$177$	−4.58740e8	−	1.54283e9i	−0.467384	−	1.57190i
$178$		0			0
$179$	−6.90561e8	−	3.98695e8i	−0.672651	−	0.388355i	0.124429	−	0.992228i	$-0.460290\pi$
$179$	−6.90561e8	−	3.98695e8i	−0.672651	−	0.388355i	−0.797080	+	0.603873i	$0.793623\pi$
$180$		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
$189$	−4.36653e8	−	1.19895e9i	−0.342207	−	0.939624i
$190$		0			0
$191$	9.93412e8	−	5.73547e8i	0.746442	−	0.430959i	−0.0779648	−	0.996956i	$-0.524842\pi$
$191$	9.93412e8	−	5.73547e8i	0.746442	−	0.430959i	0.824407	+	0.565998i	$0.191509\pi$
$192$		0			0
$193$	−3.83109e8	+	6.63564e8i	−0.276117	+	0.478248i	−0.970416	−	0.241438i	$-0.922381\pi$
$193$	−3.83109e8	+	6.63564e8i	−0.276117	+	0.478248i	0.694299	+	0.719686i	$0.255714\pi$
$194$		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	0.429411	+	0.903109i	$0.358721\pi$
$199$	−8.89835e8	+	1.54124e9i	−0.567410	+	0.982784i	−0.996821	+	0.0796742i	$0.974612\pi$
$200$		0			0
$201$	−2.02885e9	+	6.03253e8i	−1.24298	+	0.369586i
$202$		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
$207$	−1.29628e8	−	1.98709e8i	−0.0706018	−	0.108227i
$208$		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
$213$	8.53117e7	−	9.00861e7i	0.0414467	−	0.0437662i
$214$		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
$219$	3.77577e9	−	1.12268e9i	1.64146	−	0.488067i
$220$		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	$-0.235467\pi$
$227$	1.39182e9	+	8.03568e8i	0.524179	+	0.302635i	−0.214464	+	0.976732i	$0.568800\pi$
$228$		0			0
$229$	4.11805e8	+	7.13267e8i	0.149744	+	0.259364i	0.931133	−	0.364680i	$-0.118822\pi$
$229$	4.11805e8	+	7.13267e8i	0.149744	+	0.259364i	−0.781389	+	0.624045i	$0.785488\pi$
$230$		0			0
$231$	−2.59236e9	−	6.03595e8i	−0.910432	−	0.211981i
$232$		0			0
$233$	−3.56617e9	+	2.05893e9i	−1.20998	+	0.698582i	−0.962755	−	0.270377i	$-0.912852\pi$
$233$	−3.56617e9	+	2.05893e9i	−1.20998	+	0.698582i	−0.247224	+	0.968958i	$0.579518\pi$
$234$		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	−0.238979	−	0.971025i	$-0.576813\pi$
$241$	2.43372e9	−	4.21532e9i	0.721442	−	1.24958i	0.960422	−	0.278550i	$-0.0898539\pi$
$242$		0			0
$243$	−2.77718e9	−	2.10830e9i	−0.796488	−	0.604654i
$244$		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
$249$	7.96125e8	−	8.40679e8i	0.207102	−	0.218692i
$250$		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
$255$	1.18578e9	+	1.12294e9i	0.280442	+	0.265580i
$256$		0			0
$257$	−6.46260e9	+	3.73119e9i	−1.48141	+	0.855292i	−0.999778	−	0.0210818i	$-0.993289\pi$
$257$	−6.46260e9	+	3.73119e9i	−1.48141	+	0.855292i	−0.481631	+	0.876374i	$0.659956\pi$
$258$		0			0
$259$	−7.97978e9	+	4.64883e7i	−1.77334	+	0.0103311i
$260$		0			0
$261$	7.37576e9	−	4.01835e8i	1.58944	−	0.0865937i
$262$		0			0
$263$	7.11969e9	+	4.11056e9i	1.48812	+	0.859167i	0.999908	−	0.0135574i	$-0.00431560\pi$
$263$	7.11969e9	+	4.11056e9i	1.48812	+	0.859167i	0.488213	+	0.872725i	$0.337649\pi$
$264$		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	$-0.527257\pi$
$269$	−5.18973e9	−	2.99629e9i	−0.991141	−	0.572236i	−0.905615	+	0.424100i	$0.860590\pi$
$270$		0			0
$271$	−2.24850e9	−	3.89452e9i	−0.416885	−	0.722065i	0.578740	−	0.815512i	$-0.303545\pi$
$271$	−2.24850e9	−	3.89452e9i	−0.416885	−	0.722065i	−0.995624	+	0.0934470i	$0.970211\pi$
$272$		0			0
$273$	7.98011e8	−	2.42348e8i	0.143667	−	0.0436303i
$274$		0			0
$275$	3.69989e9	−	2.13613e9i	0.646931	−	0.373506i
$276$		0			0
$277$	−2.19189e9	+	3.79646e9i	−0.372305	+	0.644852i	−0.989920	−	0.141629i	$-0.954766\pi$
$277$	−2.19189e9	+	3.79646e9i	−0.372305	+	0.644852i	0.617614	+	0.786481i	$0.288099\pi$
$278$		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	−0.996223	−	0.0868347i	$-0.972325\pi$
$283$	−2.71265e9	+	4.69845e9i	−0.422910	+	0.732502i	0.573312	+	0.819337i	$0.305658\pi$
$284$		0			0
$285$	2.57338e9	+	8.65472e9i	0.390053	+	1.31182i
$286$		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
$291$	−5.38413e9	−	5.09879e9i	−0.750834	−	0.711042i
$292$		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
$297$	−6.84865e9	+	2.44916e9i	−0.880195	+	0.314768i
$298$		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
$303$	4.17189e8	+	1.40308e9i	0.0494951	+	0.166461i
$304$		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	−0.995520	−	0.0945502i	$-0.969859\pi$
$311$	−1.47356e10	−	8.50759e9i	−1.57516	−	0.909421i	−0.579643	−	0.814871i	$-0.696808\pi$
$312$		0			0
$313$	−5.43531e9	−	9.41424e9i	−0.566301	−	0.980862i	−0.996927	−	0.0783315i	$-0.975041\pi$
$313$	−5.43531e9	−	9.41424e9i	−0.566301	−	0.980862i	0.430627	−	0.902530i	$-0.358293\pi$
$314$		0			0
$315$	1.31821e10	−	7.95216e8i	1.33888	−	0.0807686i
$316$		0			0
$317$	1.19740e10	−	6.91320e9i	1.18578	−	0.684608i	0.228432	−	0.973560i	$-0.426640\pi$
$317$	1.19740e10	−	6.91320e9i	1.18578	−	0.684608i	0.957344	+	0.288952i	$0.0933069\pi$
$318$		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
$327$	−1.01889e10	+	3.02954e9i	−0.891119	+	0.264963i
$328$		0			0
$329$	−3.71459e7	−	6.37614e9i	−0.00317050	−	0.544220i
$330$		0			0
$331$	−3.57896e9	−	6.19894e9i	−0.298157	−	0.516423i	0.677558	−	0.735470i	$-0.263038\pi$
$331$	−3.57896e9	−	6.19894e9i	−0.298157	−	0.516423i	−0.975714	+	0.219047i	$0.929705\pi$
$332$		0			0
$333$	−1.82635e10	+	1.19142e10i	−1.48527	+	0.968918i
$334$		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
$339$	−1.53298e10	+	1.61877e10i	−1.16074	+	1.22570i
$340$		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
$345$	2.35363e9	−	6.99825e8i	0.166136	−	0.0493984i
$346$		0			0
$347$	−1.19483e10	−	6.89837e9i	−0.824118	−	0.475805i	0.0277165	−	0.999616i	$-0.491176\pi$
$347$	−1.19483e10	−	6.89837e9i	−0.824118	−	0.475805i	−0.851834	+	0.523811i	$0.824510\pi$
$348$		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	$-0.345466\pi$
$353$	−1.02481e9	−	5.91677e8i	−0.0660003	−	0.0381053i	−0.532637	+	0.846344i	$0.678799\pi$
$354$		0			0
$355$	6.42046e8	+	1.11206e9i	0.0404252	+	0.0700185i
$356$		0			0
$357$	4.55548e9	+	1.06068e9i	0.280454	+	0.0652997i
$358$		0			0
$359$	−2.06909e9	+	1.19459e9i	−0.124567	+	0.0719186i	−0.560989	−	0.827824i	$-0.689579\pi$
$359$	−2.06909e9	+	1.19459e9i	−0.124567	+	0.0719186i	0.436422	+	0.899742i	$0.356246\pi$
$360$		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	−0.889177	−	0.457564i	$-0.848722\pi$
$367$	−8.76690e8	+	1.51847e9i	−0.0483261	+	0.0837033i	0.840850	+	0.541267i	$0.182055\pi$
$368$		0			0
$369$	2.42665e10	−	1.32205e9i	1.30889	−	0.0713089i
$370$		0			0
$371$	1.04359e10	−	6.10652e9i	0.550852	−	0.322328i
$372$		0			0
$373$	−1.46590e10	−	2.53901e10i	−0.757302	−	1.31169i	−0.944222	−	0.329310i	$-0.893184\pi$
$373$	−1.46590e10	−	2.53901e10i	−0.757302	−	1.31169i	0.186920	−	0.982375i	$-0.440150\pi$
$374$		0			0
$375$	3.66368e9	−	3.86871e9i	0.185264	−	0.195632i
$376$		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
$381$	−1.80258e10	−	1.70705e10i	−0.855452	−	0.810115i
$382$		0			0
$383$	−2.27121e10	+	1.31128e10i	−1.05551	+	0.609398i	−0.924186	−	0.381942i	$-0.875255\pi$
$383$	−2.27121e10	+	1.31128e10i	−1.05551	+	0.609398i	−0.131322	+	0.991340i	$0.541922\pi$
$384$		0			0
$385$	1.36346e10	−	2.39369e10i	0.620584	−	1.08949i
$386$		0			0
$387$	−6.27831e8	−	1.15239e10i	−0.0279897	−	0.513756i
$388$		0			0
$389$	2.16168e10	+	1.24804e10i	0.944044	+	0.545044i	0.891226	−	0.453560i	$-0.149846\pi$
$389$	2.16168e10	+	1.24804e10i	0.944044	+	0.545044i	0.0528181	+	0.998604i	$0.483180\pi$
$390$		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	0.982993	+	0.183641i	$0.0587884\pi$
$397$	1.61597e10	+	2.79894e10i	0.650535	+	1.12676i	−0.332459	+	0.943118i	$0.607878\pi$
$398$		0			0
$399$	1.88799e10	+	1.76718e10i	0.744918	+	0.697253i
$400$		0			0
$401$	3.13160e10	−	1.80803e10i	1.21112	−	0.699242i	0.248119	−	0.968730i	$-0.420188\pi$
$401$	3.13160e10	−	1.80803e10i	1.21112	−	0.699242i	0.963004	+	0.269487i	$0.0868542\pi$
$402$		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	−0.717146	−	0.696923i	$-0.754552\pi$
$409$	6.85527e9	−	1.18737e10i	0.244980	−	0.424319i	0.962126	+	0.272605i	$0.0878851\pi$
$410$		0			0
$411$	8.40021e9	+	2.82514e10i	0.294390	+	0.990085i
$412$		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
$417$	3.20278e10	+	3.03304e10i	1.05921	+	1.00308i
$418$		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
$423$	−9.51987e9	−	1.45932e10i	−0.297351	−	0.455815i
$424$		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
$429$	−1.35491e9	−	4.55680e9i	−0.0400019	−	0.134533i
$430$		0			0
$431$	−1.77405e10	−	1.02425e10i	−0.514112	−	0.296822i	0.220411	−	0.975407i	$-0.429260\pi$
$431$	−1.77405e10	−	1.02425e10i	−0.514112	−	0.296822i	−0.734522	+	0.678585i	$0.762594\pi$
$432$		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	0.991192	−	0.132431i	$-0.0422782\pi$
$439$	1.41474e10	+	2.45041e10i	0.380908	+	0.659751i	−0.610285	+	0.792182i	$0.708945\pi$
$440$		0			0
$441$	3.14353e10	−	2.10330e10i	0.831120	−	0.556093i
$442$		0			0
$443$	4.36607e10	−	2.52075e10i	1.13364	−	0.654508i	0.188793	−	0.982017i	$-0.439543\pi$
$443$	4.36607e10	−	2.52075e10i	1.13364	−	0.654508i	0.944848	+	0.327509i	$0.106209\pi$
$444$		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
$453$	−3.63546e10	+	1.08096e10i	−0.863310	+	0.256695i
$454$		0			0
$455$	5.02848e7	+	8.63146e9i	0.00117325	+	0.201390i
$456$		0			0
$457$	1.16839e10	+	2.02372e10i	0.267870	+	0.463965i	0.968312	−	0.249745i	$-0.0803468\pi$
$457$	1.16839e10	+	2.02372e10i	0.267870	+	0.463965i	−0.700441	+	0.713710i	$0.747013\pi$
$458$		0			0
$459$	1.20349e10	−	4.30383e9i	0.271139	−	0.0969626i
$460$		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
$465$	1.38614e10	−	1.46372e10i	0.296480	−	0.313072i
$466$		0			0
$467$	1.57779e10	−	9.10936e9i	0.331727	−	0.191523i	−0.324881	−	0.945755i	$-0.605324\pi$
$467$	1.57779e10	−	9.10936e9i	0.331727	−	0.191523i	0.656608	+	0.754232i	$0.271991\pi$
$468$		0			0
$469$	−3.16866e10	−	5.41518e10i	−0.654914	−	1.11924i
$470$		0			0
$471$	−7.10459e10	+	2.11246e10i	−1.44363	+	0.429245i
$472$		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	$-0.517909\pi$
$479$	−4.99587e10	−	2.88436e10i	−0.949005	−	0.547909i	−0.892772	+	0.450509i	$0.851242\pi$
$480$		0			0
$481$	−7.12630e9	−	1.23431e10i	−0.133132	−	0.230592i
$482$		0			0
$483$	4.80583e9	−	5.13436e9i	0.0883038	−	0.0943404i
$484$		0			0
$485$	6.64637e10	−	3.83729e10i	1.20121	−	0.693517i
$486$		0			0
$487$	−2.63352e10	+	4.56138e10i	−0.468188	+	0.810925i	−0.999339	−	0.0363521i	$-0.988426\pi$
$487$	−2.63352e10	+	4.56138e10i	−0.468188	+	0.810925i	0.531151	+	0.847277i	$0.321760\pi$
$488$		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
$495$	−4.09507e9	−	7.51657e10i	−0.0682088	−	1.25198i
$496$		0			0
$497$	3.19565e9	+	1.82027e9i	0.0523761	+	0.0298339i
$498$		0			0
$499$	3.65446e10	+	6.32970e10i	0.589414	+	1.02090i	0.994309	+	0.106532i	$0.0339747\pi$
$499$	3.65446e10	+	6.32970e10i	0.589414	+	1.02090i	−0.404895	+	0.914363i	$0.632692\pi$
$500$		0			0
$501$	5.00498e10	−	5.28507e10i	0.794422	−	0.838881i
$502$		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
$507$	−4.68939e10	−	4.44087e10i	−0.709717	−	0.672103i
$508$		0			0
$509$	−2.67878e9	+	1.54660e9i	−0.0399086	+	0.0230412i	−0.519822	−	0.854275i	$-0.674002\pi$
$509$	−2.67878e9	+	1.54660e9i	−0.0399086	+	0.0230412i	0.479913	+	0.877316i	$0.340668\pi$
$510$		0			0
$511$	5.89702e10	+	1.00779e11i	0.864866	+	1.47804i
$512$		0			0
$513$	6.95219e10	+	1.26623e10i	1.00381	+	0.182828i
$514$		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	$-0.396815\pi$
$521$	−2.52830e10	−	1.45971e10i	−0.343145	−	0.198115i	−0.661662	+	0.749803i	$0.730148\pi$
$522$		0			0
$523$	6.17406e10	+	1.06938e11i	0.825210	+	1.42931i	0.901759	+	0.432240i	$0.142277\pi$
$523$	6.17406e10	+	1.06938e11i	0.825210	+	1.42931i	−0.0765488	+	0.997066i	$0.524390\pi$
$524$		0			0
$525$	−1.37670e10	+	5.91274e10i	−0.181218	+	0.778308i
$526$		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
$537$	−1.84082e10	−	6.19099e10i	−0.221367	−	0.744497i
$538$		0			0
$539$	−9.19253e8	−	7.88928e10i	−0.0108913	−	0.934721i
$540$		0			0
$541$	1.85709e10	+	3.21657e10i	0.216792	+	0.375495i	0.953825	−	0.300362i	$-0.0971073\pi$
$541$	1.85709e10	+	3.21657e10i	0.216792	+	0.375495i	−0.737033	+	0.675856i	$0.763774\pi$
$542$		0			0
$543$	−3.83040e10	−	3.62739e10i	−0.440600	−	0.417249i
$544$		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
$549$	−6.91427e10	+	4.51052e10i	−0.761127	+	0.496521i
$550$		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
$555$	−6.43215e10	−	2.16324e11i	−0.677929	−	2.27999i
$556$		0			0
$557$	−8.57873e9	−	4.95293e9i	−0.0891256	−	0.0514567i	0.454775	−	0.890606i	$-0.349720\pi$
$557$	−8.57873e9	−	4.95293e9i	−0.0891256	−	0.0514567i	−0.543900	+	0.839150i	$0.683053\pi$
$558$		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	$-0.168225\pi$
$563$	8.62703e10	+	4.98082e10i	0.858672	+	0.495755i	−0.00489502	+	0.999988i	$0.501558\pi$
$564$		0			0
$565$	−1.15370e11	−	1.99827e11i	−1.13214	−	1.96092i
$566$		0			0
$567$	4.10959e10	−	9.48336e10i	0.397618	−	0.917551i
$568$		0			0
$569$	−6.89683e10	+	3.98189e10i	−0.657962	+	0.379874i	−0.791500	−	0.611169i	$-0.790699\pi$
$569$	−6.89683e10	+	3.98189e10i	−0.657962	+	0.379874i	0.133538	+	0.991044i	$0.457366\pi$
$570$		0			0
$571$	7.14946e10	−	1.23832e11i	0.672556	−	1.16490i	−0.304620	−	0.952474i	$-0.598530\pi$
$571$	7.14946e10	−	1.23832e11i	0.672556	−	1.16490i	0.977177	−	0.212428i	$-0.0681371\pi$
$572$		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	−0.0965021	+	0.995333i	$0.530765\pi$
$577$	−1.00892e11	+	1.74750e11i	−0.910235	+	1.57657i	−0.813732	+	0.581240i	$0.802568\pi$
$578$		0			0
$579$	−5.94896e10	+	1.76885e10i	−0.529330	+	0.157390i
$580$		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
$585$	1.28872e10	+	1.97550e10i	0.110036	+	0.168676i
$586$		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
$591$	−2.06487e10	+	2.18043e10i	−0.169255	+	0.178728i
$592$		0			0
$593$	−4.24481e10	+	2.45074e10i	−0.343273	+	0.198189i	−0.661718	−	0.749752i	$-0.730173\pi$
$593$	−4.24481e10	+	2.45074e10i	−0.343273	+	0.198189i	0.318445	+	0.947941i	$0.396839\pi$
$594$		0			0
$595$	−2.39597e10	+	4.20635e10i	−0.191168	+	0.335612i
$596$		0			0
$597$	−1.38175e11	+	4.10846e10i	−1.08775	+	0.323431i
$598$		0			0
$599$	1.62609e11	+	9.38821e10i	1.26310	+	0.729248i	0.973672	−	0.227953i	$-0.0732033\pi$
$599$	1.62609e11	+	9.38821e10i	1.26310	+	0.729248i	0.289423	+	0.957201i	$0.406537\pi$
$600$		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	0.997271	+	0.0738270i	$0.0235213\pi$
$607$	7.63717e10	+	1.32280e11i	0.562572	+	0.974403i	−0.434699	+	0.900576i	$0.643145\pi$
$608$		0			0
$609$	6.36255e10	+	2.09508e11i	0.462554	+	1.52311i
$610$		0			0
$611$	9.86262e9	−	5.69418e9i	0.0707664	−	0.0408570i
$612$		0			0
$613$	2.80686e10	−	4.86162e10i	0.198783	−	0.344302i	−0.749351	−	0.662173i	$-0.769635\pi$
$613$	2.80686e10	−	4.86162e10i	0.198783	−	0.344302i	0.948134	+	0.317871i	$0.102968\pi$
$614$		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	−0.846929	−	0.531706i	$-0.821551\pi$
$619$	5.43304e9	−	9.41031e9i	0.0370067	−	0.0640975i	0.883936	+	0.467609i	$0.154884\pi$
$620$		0			0
$621$	3.44349e9	−	1.89064e10i	0.0231544	−	0.127128i
$622$		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
$627$	1.01358e11	−	1.07031e11i	0.655827	−	0.692529i
$628$		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
$633$	2.07936e11	+	1.96916e11i	1.29514	+	1.22650i
$634$		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
$639$	1.00349e10	−	5.46705e8i	0.0601877	−	0.00327906i
$640$		0			0
$641$	−1.01252e9	−	5.84579e8i	−0.00599752	−	0.00346267i	0.496998	−	0.867752i	$-0.334436\pi$
$641$	−1.01252e9	−	5.84579e8i	−0.00599752	−	0.00346267i	−0.502996	+	0.864289i	$0.667769\pi$
$642$		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	$-0.136047\pi$
$647$	1.76301e11	+	1.01788e11i	1.00609	+	0.580868i	0.0960476	+	0.995377i	$0.469380\pi$
$648$		0			0
$649$	−1.35982e11	−	2.35527e11i	−0.766481	−	1.32758i
$650$		0			0
$651$	1.30929e10	−	5.62323e10i	0.0728975	−	0.313085i
$652$		0			0
$653$	6.23592e10	−	3.60031e10i	0.342964	−	0.198010i	−0.318618	−	0.947883i	$-0.603219\pi$
$653$	6.23592e10	−	3.60031e10i	0.342964	−	0.198010i	0.661582	+	0.749873i	$0.269885\pi$
$654$		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	−0.408232	−	0.912878i	$-0.633854\pi$
$661$	1.11955e11	−	1.93912e11i	0.586460	−	1.01578i	0.994692	−	0.102900i	$-0.0328123\pi$
$662$		0			0
$663$	2.38094e9	+	8.00752e9i	0.0123224	+	0.0414423i
$664$		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
$669$	−1.08138e11	−	1.02407e11i	−0.539852	−	0.511241i
$670$		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
$675$	5.58611e10	+	1.56206e11i	0.269088	+	0.752459i
$676$		0			0
$677$	−9.84930e10	+	5.68650e10i	−0.468868	+	0.270701i	−0.715766	−	0.698340i	$-0.753922\pi$
$677$	−9.84930e10	+	5.68650e10i	−0.468868	+	0.270701i	0.246898	+	0.969042i	$0.420589\pi$
$678$		0			0
$679$	1.08791e11	−	1.90993e11i	0.511817	−	0.898541i
$680$		0			0
$681$	3.71015e10	+	1.24779e11i	0.172506	+	0.580167i
$682$		0			0
$683$	3.73633e11	+	2.15717e11i	1.71697	+	0.991291i	0.924328	+	0.381600i	$0.124627\pi$
$683$	3.73633e11	+	2.15717e11i	1.71697	+	0.991291i	0.792639	+	0.609691i	$0.208707\pi$
$684$		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	0.967804	−	0.251704i	$-0.0809908\pi$
$691$	6.06267e10	+	1.05009e11i	0.265921	+	0.460588i	−0.701884	+	0.712291i	$0.747657\pi$
$692$		0			0
$693$	−1.18846e11	−	1.79883e11i	−0.515292	−	0.779934i
$694$		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
$705$	1.72851e11	−	5.13953e10i	0.699707	−	0.208049i
$706$		0			0
$707$	−3.74495e10	+	2.19134e10i	−0.149889	+	0.0877064i
$708$		0			0
$709$	−6.06898e10	−	1.05118e11i	−0.240177	−	0.415998i	0.720588	−	0.693364i	$-0.243872\pi$
$709$	−6.06898e10	−	1.05118e11i	−0.240177	−	0.415998i	−0.960764	+	0.277365i	$0.910539\pi$
$710$		0			0
$711$	−2.31258e11	+	1.50861e11i	−0.904936	+	0.590335i
$712$		0			0
$713$		−	1.07353e10i		−	0.0415388i
$714$		0			0
$715$	4.92019e10			0.188260
$716$		0			0
$717$	−1.64567e11	+	1.73777e11i	−0.622684	+	0.657531i
$718$		0			0
$719$	3.18390e11	−	1.83822e11i	1.19136	−	0.687832i	0.232746	−	0.972538i	$-0.425229\pi$
$719$	3.18390e11	−	1.83822e11i	1.19136	−	0.687832i	0.958615	+	0.284705i	$0.0918957\pi$
$720$		0			0
$721$	−3.94435e11	+	2.29789e9i	−1.45960	+	0.00850330i
$722$		0			0
$723$	3.77910e11	−	1.12367e11i	1.38304	−	0.411231i
$724$		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	0.944634	−	0.328127i	$-0.106417\pi$
$733$	5.43153e10	+	9.40769e10i	0.188151	+	0.325887i	−0.756483	+	0.654014i	$0.773084\pi$
$734$		0			0
$735$	1.15931e11	+	3.73893e11i	0.397237	+	1.28114i
$736$		0			0
$737$	−3.09723e11	+	1.78819e11i	−1.04979	+	0.606098i
$738$		0			0
$739$	−7.57046e10	+	1.31124e11i	−0.253831	+	0.439648i	−0.964577	−	0.263800i	$-0.915024\pi$
$739$	−7.57046e10	+	1.31124e11i	−0.253831	+	0.439648i	0.710746	+	0.703448i	$0.248357\pi$
$740$		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
$747$	9.36449e10	−	5.10183e9i	0.300747	−	0.0163849i
$748$		0			0
$749$	6.60872e8	+	1.13439e11i	0.00209986	+	0.360443i
$750$		0			0
$751$	8.57533e10	+	1.48529e11i	0.269582	+	0.466930i	0.968754	−	0.248024i	$-0.0797811\pi$
$751$	8.57533e10	+	1.48529e11i	0.269582	+	0.466930i	−0.699172	+	0.714954i	$0.746448\pi$
$752$		0			0
$753$	−1.46682e11	+	1.54891e11i	−0.456244	+	0.481777i
$754$		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
$759$	−2.91068e10	−	2.75642e10i	−0.0877056	−	0.0830574i
$760$		0			0
$761$	4.32044e10	−	2.49441e10i	0.128822	−	0.0743754i	−0.434204	−	0.900814i	$-0.642970\pi$
$761$	4.32044e10	−	2.49441e10i	0.128822	−	0.0743754i	0.563026	+	0.826439i	$0.309637\pi$
$762$		0			0
$763$	−1.59130e11	−	2.71951e11i	−0.469521	−	0.802402i
$764$		0			0
$765$	7.19614e9	+	1.32086e11i	0.0210113	+	0.385667i
$766$		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	$-0.473494\pi$
$773$	−2.63590e11	−	1.52184e11i	−0.738262	−	0.426236i	−0.821437	+	0.570299i	$0.806827\pi$
$774$		0			0
$775$	4.63360e10	+	8.02563e10i	0.128443	+	0.222471i
$776$		0			0
$777$	−4.71903e11	−	4.41707e11i	−1.29470	−	1.21185i
$778$		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	−0.316370	−	0.948636i	$-0.602464\pi$
$787$	2.54476e11	−	4.40765e11i	0.663358	−	1.14897i	0.979728	−	0.200334i	$-0.0642026\pi$
$788$		0			0
$789$	1.89788e11	+	6.38292e11i	0.489736	+	1.64707i
$790$		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
$795$	2.48292e11	+	2.35133e11i	0.621575	+	0.588633i
$796$		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
$801$	1.38869e11	+	2.12875e11i	0.337345	+	0.517123i
$802$		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
$807$	−1.38342e11	−	4.65268e11i	−0.326181	−	1.09701i
$808$		0			0
$809$	3.16264e11	+	1.82595e11i	0.738339	+	0.426280i	0.821465	−	0.570259i	$-0.193157\pi$
$809$	3.16264e11	+	1.82595e11i	0.738339	+	0.426280i	−0.0831258	+	0.996539i	$0.526490\pi$
$810$		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
$819$	6.04312e10	+	3.01928e10i	0.134315	+	0.0671071i
$820$		0			0
$821$	−9.29739e10	+	5.36785e10i	−0.204639	+	0.118148i	−0.598817	−	0.800886i	$-0.704363\pi$
$821$	−9.29739e10	+	5.36785e10i	−0.204639	+	0.118148i	0.394179	+	0.919034i	$0.371029\pi$
$822$		0			0
$823$	9.17659e10	−	1.58943e11i	0.200024	−	0.346452i	−0.748512	−	0.663121i	$-0.769231\pi$
$823$	9.17659e10	−	1.58943e11i	0.200024	−	0.346452i	0.948536	+	0.316670i	$0.102565\pi$
$824$		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	−0.957697	−	0.287778i	$-0.907083\pi$
$829$	−1.08452e11	+	1.87845e11i	−0.229625	+	0.397723i	0.728072	+	0.685501i	$0.240417\pi$
$830$		0			0
$831$	−3.40359e11	+	1.01202e11i	−0.713728	+	0.212219i
$832$		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
$837$	−5.31260e10	−	1.48558e11i	−0.108244	−	0.302687i
$838$		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
$843$	−2.72628e11	+	2.87885e11i	−0.539834	+	0.570045i
$844$		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
$849$	−4.21224e11	+	1.25246e11i	−0.810741	+	0.241064i
$850$		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	$-0.161366\pi$
$857$	4.80557e11	+	2.77449e11i	0.890884	+	0.514352i	0.0166525	+	0.999861i	$0.494699\pi$
$858$		0			0
$859$	−2.15341e11	−	3.72981e11i	−0.395506	−	0.685037i	0.597660	−	0.801750i	$-0.296097\pi$
$859$	−2.15341e11	−	3.72981e11i	−0.395506	−	0.685037i	−0.993166	+	0.116713i	$0.962764\pi$
$860$		0			0
$861$	2.09330e11	+	6.89290e11i	0.380907	+	1.25426i
$862$		0			0
$863$	−5.39592e11	+	3.11533e11i	−0.972797	+	0.561644i	−0.900088	−	0.435709i	$-0.856498\pi$
$863$	−5.39592e11	+	3.11533e11i	−0.972797	+	0.561644i	−0.0727089	+	0.997353i	$0.523164\pi$
$864$		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
$873$	−3.26747e10	−	5.99749e11i	−0.0562541	−	1.03255i
$874$		0			0
$875$	1.37236e11	+	7.81706e10i	0.234118	+	0.133356i
$876$		0			0
$877$	−1.84222e11	−	3.19082e11i	−0.311418	−	0.539392i	0.667252	−	0.744832i	$-0.267471\pi$
$877$	−1.84222e11	−	3.19082e11i	−0.311418	−	0.539392i	−0.978670	+	0.205441i	$0.934137\pi$
$878$		0			0
$879$	7.04565e11	−	7.43995e11i	1.18023	−	1.24628i
$880$		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
$885$	9.79742e11	+	9.27818e11i	1.59712	+	1.51248i
$886$		0			0
$887$	−9.99354e11	+	5.76977e11i	−1.61445	+	0.932104i	−0.626130	+	0.779719i	$0.715362\pi$
$887$	−9.99354e11	+	5.76977e11i	−1.61445	+	0.932104i	−0.988321	+	0.152385i	$0.951305\pi$
$888$		0			0
$889$	3.64228e11	−	6.39435e11i	0.583131	−	1.02374i
$890$		0			0
$891$	−5.39197e11	−	2.37400e11i	−0.855533	−	0.376678i
$892$		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
$903$	3.27337e11	−	9.94089e10i	0.492316	−	0.149511i
$904$		0			0
$905$	4.72838e11	−	2.72993e11i	0.704885	−	0.406966i
$906$		0			0
$907$	6.66818e10	−	1.15496e11i	0.0985322	−	0.170663i	−0.812545	−	0.582898i	$-0.801919\pi$
$907$	6.66818e10	−	1.15496e11i	0.0985322	−	0.170663i	0.911077	+	0.412236i	$0.135252\pi$
$908$		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
$915$	−2.43511e11	−	8.18972e11i	−0.347404	−	1.16838i
$916$		0			0
$917$	4.09640e8	+	7.03152e10i	0.000579329	+	0.0994425i
$918$		0			0
$919$	3.80288e11	+	6.58679e11i	0.533152	+	0.923446i	0.999250	+	0.0387135i	$0.0123260\pi$
$919$	3.80288e11	+	6.58679e11i	0.533152	+	0.923446i	−0.466098	+	0.884733i	$0.654341\pi$
$920$		0			0
$921$	−1.71773e11	−	1.62669e11i	−0.238735	−	0.226082i
$922$		0			0
$923$			6.56861e9i			0.00905037i
$924$		0			0
$925$	1.03748e12			1.41715
$926$		0			0
$927$	−9.02750e11	+	5.88909e11i	−1.22250	+	0.797498i
$928$		0			0
$929$	2.97498e11	−	1.71761e11i	0.399413	−	0.230601i	−0.286818	−	0.957985i	$-0.592597\pi$
$929$	2.97498e11	−	1.71761e11i	0.399413	−	0.230601i	0.686230	+	0.727384i	$0.259264\pi$
$930$		0			0
$931$	−3.75512e11	+	6.68268e11i	−0.499833	+	0.889512i
$932$		0			0
$933$	−3.92804e11	−	1.32107e12i	−0.518381	−	1.74341i
$934$		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	$-0.297354\pi$
$941$	1.53177e11	+	8.84367e10i	0.195360	+	0.112791i	−0.399130	+	0.916895i	$0.630688\pi$
$942$		0			0
$943$	6.69716e10	+	1.15998e11i	0.0846923	+	0.146691i
$944$		0			0
$945$	8.19567e11	+	6.87420e11i	1.02768	+	0.861976i
$946$		0			0
$947$	5.66637e11	−	3.27148e11i	0.704539	−	0.406766i	−0.104497	−	0.994525i	$-0.533323\pi$
$947$	5.66637e11	−	3.27148e11i	0.704539	−	0.406766i	0.809036	+	0.587759i	$0.199990\pi$
$948$		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
$957$	1.19633e12	−	3.55715e11i	1.42628	−	0.424086i
$958$		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
$963$	1.69370e11	+	2.59631e11i	0.196939	+	0.301892i
$964$		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
$969$	−1.78114e11	+	1.88082e11i	−0.202024	+	0.213330i
$970$		0			0
$971$	−8.01034e11	+	4.62477e11i	−0.901102	+	0.520252i	−0.877558	−	0.479471i	$-0.840829\pi$
$971$	−8.01034e11	+	4.62477e11i	−0.901102	+	0.520252i	−0.0235447	+	0.999723i	$0.507495\pi$
$972$		0			0
$973$	−6.47149e11	+	1.13613e12i	−0.722026	+	1.26758i
$974$		0			0
$975$	−1.03933e11	+	3.09032e10i	−0.115010	+	0.0341967i
$976$		0			0
$977$	−1.22820e12	−	7.09101e11i	−1.34800	−	0.778269i	−0.360036	−	0.932939i	$-0.617235\pi$
$977$	−1.22820e12	−	7.09101e11i	−1.34800	−	0.778269i	−0.987966	+	0.154669i	$0.950569\pi$
$978$		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	0.999569	+	0.0293398i	$0.00934049\pi$
$983$	1.42369e12	+	8.21970e11i	1.52476	+	0.880322i	0.525194	+	0.850983i	$0.323993\pi$
$984$		0			0
$985$	−1.55399e11	−	2.69160e11i	−0.165084	−	0.285934i
$986$		0			0
$987$	3.52941e11	−	3.77068e11i	0.371906	−	0.397330i
$988$		0			0
$989$	5.50864e10	−	3.18042e10i	0.0575784	−	0.0332429i
$990$		0			0
$991$	−7.61463e11	+	1.31889e12i	−0.789504	+	1.36746i	0.136768	+	0.990603i	$0.456329\pi$
$991$	−7.61463e11	+	1.31889e12i	−0.789504	+	1.36746i	−0.926271	+	0.376857i	$0.877005\pi$
$992$		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	−0.173502	−	0.984833i	$-0.555508\pi$
$997$	7.56987e11	−	1.31114e12i	0.766140	−	1.32699i	0.939642	−	0.342159i	$-0.111158\pi$
$998$		0			0
$999$	−1.73770e12	−	3.16494e11i	−1.74467	−	0.317763i

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.9.p.b.53.2	✓	40	3.2	odd	2	inner
84.9.p.b.53.16	yes	40	1.1	even	1	trivial
84.9.p.b.65.2	yes	40	7.2	even	3	inner
84.9.p.b.65.16	yes	40	21.2	odd	6	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)

\(f(q)\)	\(=\)	\(q+(58.8129 + 55.6960i) q^{3} +(-726.008 + 419.161i) q^{5} +(-1188.37 + 2086.29i) q^{7} +(356.917 + 6551.28i) q^{9} +O(q^{10})\)	\(q+(58.8129 + 55.6960i) q^{3} +(-726.008 + 419.161i) q^{5} +(-1188.37 + 2086.29i) q^{7} +(356.917 + 6551.28i) q^{9} +(11852.6 + 6843.09i) q^{11} -4288.33 q^{13} +(-66044.2 - 15783.7i) q^{15} +(-20828.2 - 12025.2i) q^{17} +(-66484.9 - 115155. i) q^{19} +(-186089. + 56513.3i) q^{21} +(-31316.3 + 18080.5i) q^{23} +(156079. - 270338. i) q^{25} +(-343889. + 405179. i) q^{27} -1.12585e6i q^{29} +(-148437. + 257101. i) q^{31} +(315952. + 1.06260e6i) q^{33} +(-11726.0 - 2.01278e6i) q^{35} +(1.66179e6 + 2.87830e6i) q^{37} +(-252209. - 238843. i) q^{39} -3.70409e6i q^{41} -1.75904e6 q^{43} +(-3.00517e6 - 4.60668e6i) q^{45} +(-2.29987e6 + 1.32783e6i) q^{47} +(-2.94037e6 - 4.95854e6i) q^{49} +(-555213. - 1.86728e6i) q^{51} +(-4.36123e6 - 2.51796e6i) q^{53} -1.14734e7 q^{55} +(2.50351e6 - 1.04756e7i) q^{57} +(-1.72091e7 - 9.93569e6i) q^{59} +(6.29128e6 + 1.08968e7i) q^{61} +(-1.40920e7 - 7.04069e6i) q^{63} +(3.11336e6 - 1.79750e6i) q^{65} +(-1.30656e7 + 2.26303e7i) q^{67} +(-2.84881e6 - 680826. i) q^{69} -1.53174e6i q^{71} +(2.43157e7 - 4.21161e7i) q^{73} +(2.42362e7 - 7.20634e6i) q^{75} +(-2.83619e7 + 1.65958e7i) q^{77} +(2.10421e7 + 3.64460e7i) q^{79} +(-4.27919e7 + 4.67654e6i) q^{81} -1.42941e7i q^{83} +2.01619e7 q^{85} +(6.27053e7 - 6.62145e7i) q^{87} +(3.35488e7 - 1.93694e7i) q^{89} +(5.09611e6 - 8.94668e6i) q^{91} +(-2.30495e7 + 6.85350e6i) q^{93} +(9.65372e7 + 5.57358e7i) q^{95} -9.15468e7 q^{97} +(-4.06007e7 + 8.00921e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 81 q^{3} - 34 q^{7} + 4771 q^{9}+O(q^{10}) \) 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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