LMFDB - Embedded newform 84.9.m.b.61.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [84,9,Mod(61,84)]

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, 0, 5]))

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

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

chi := DirichletCharacter("84.61");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 84 = 2^{2} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	84.m (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:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} + 148097 x^{10} + 46071824 x^{9} + 21578502553 x^{8} + 3561445462121 x^{7} + \cdots + 45\!\cdots\!96 $ x^12 - 3x^11 + 148097x^10 + 46071824x^9 + 21578502553x^8 + 3561445462121x^7 + 576413321817541x^6 + 47217566733462528x^5 + 5214056955297543333x^4 + 358752845334081085965x^3 + 30962072851910211245661x^2 + 1221542968331193193318500*x + 45396580558961892385326096
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{20}\cdot 3^{10}\cdot 7^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			61.1
Root			$44.6586 + 77.3509i$ of defining polynomial
Character	$\chi$	$=$	84.61
Dual form			84.9.m.b.73.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(40.5000 - 23.3827i) q^{3} +(-939.615 - 542.487i) q^{5} +(1451.57 + 1912.52i) q^{7} +(1093.50 - 1894.00i) q^{9} +O(q^{10})$	\(q+(40.5000 - 23.3827i) q^{3} +(-939.615 - 542.487i) q^{5} +(1451.57 + 1912.52i) q^{7} +(1093.50 - 1894.00i) q^{9} +(-8435.48 - 14610.7i) q^{11} -9958.41i q^{13} -50739.2 q^{15} +(-91736.7 + 52964.2i) q^{17} +(147650. + 85245.9i) q^{19} +(103509. + 43515.5i) q^{21} +(-105694. + 183067. i) q^{23} +(393272. + 681166. i) q^{25} -102276. i q^{27} -567529. q^{29} +(549248. - 317108. i) q^{31} +(-683274. - 394488. i) q^{33} +(-326400. - 2.58449e6i) q^{35} +(-1.07740e6 + 1.86611e6i) q^{37} +(-232854. - 403316. i) q^{39} +5.45428e6i q^{41} +5.56726e6 q^{43} +(-2.05494e6 + 1.18642e6i) q^{45} +(2.76739e6 + 1.59775e6i) q^{47} +(-1.55068e6 + 5.55233e6i) q^{49} +(-2.47689e6 + 4.29010e6i) q^{51} +(4.38272e6 + 7.59109e6i) q^{53} +1.83046e7i q^{55} +7.97311e6 q^{57} +(-8.08566e6 + 4.66826e6i) q^{59} +(-2.13748e7 - 1.23407e7i) q^{61} +(5.20961e6 - 657931. i) q^{63} +(-5.40231e6 + 9.35707e6i) q^{65} +(-6.91779e6 - 1.19820e7i) q^{67} +9.88562e6i q^{69} +1.19189e7 q^{71} +(-1.83300e7 + 1.05828e7i) q^{73} +(3.18550e7 + 1.83915e7i) q^{75} +(1.56985e7 - 3.73415e7i) q^{77} +(-8.61190e6 + 1.49162e7i) q^{79} +(-2.39148e6 - 4.14217e6i) q^{81} +6.82663e7i q^{83} +1.14930e8 q^{85} +(-2.29849e7 + 1.32703e7i) q^{87} +(-7.87155e7 - 4.54464e7i) q^{89} +(1.90457e7 - 1.44553e7i) q^{91} +(1.48297e7 - 2.56858e7i) q^{93} +(-9.24895e7 - 1.60197e8i) q^{95} +3.95376e7i q^{97} -3.68968e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 486 q^{3} + 285 q^{5} + 198 q^{7} + 13122 q^{9}+O(q^{10}) $ 12 * q + 486 * q^3 + 285 * q^5 + 198 * q^7 + 13122 * q^9	$ 12 q + 486 q^{3} + 285 q^{5} + 198 q^{7} + 13122 q^{9} - 17919 q^{11} + 15390 q^{15} - 205782 q^{17} + 74313 q^{19} - 39609 q^{21} - 62832 q^{23} + 878679 q^{25} - 575454 q^{29} + 1442952 q^{31} - 1451439 q^{33} - 3989514 q^{35} - 2058621 q^{37} - 930933 q^{39} + 7721322 q^{43} + 623295 q^{45} + 12088194 q^{47} - 16964694 q^{49} - 5556114 q^{51} - 5506743 q^{53} + 4012902 q^{57} + 7511901 q^{59} - 37215576 q^{61} - 3641355 q^{63} + 5047122 q^{65} - 36824553 q^{67} - 30011556 q^{71} + 95080185 q^{73} + 71172999 q^{75} - 38333727 q^{77} + 8514456 q^{79} - 28697814 q^{81} + 20121540 q^{85} - 23305887 q^{87} + 83038554 q^{89} - 198538635 q^{91} + 38959704 q^{93} - 221605224 q^{95} - 78377706 q^{99}+O(q^{100}) $ 12 * q + 486 * q^3 + 285 * q^5 + 198 * q^7 + 13122 * q^9 - 17919 * q^11 + 15390 * q^15 - 205782 * q^17 + 74313 * q^19 - 39609 * q^21 - 62832 * q^23 + 878679 * q^25 - 575454 * q^29 + 1442952 * q^31 - 1451439 * q^33 - 3989514 * q^35 - 2058621 * q^37 - 930933 * q^39 + 7721322 * q^43 + 623295 * q^45 + 12088194 * q^47 - 16964694 * q^49 - 5556114 * q^51 - 5506743 * q^53 + 4012902 * q^57 + 7511901 * q^59 - 37215576 * q^61 - 3641355 * q^63 + 5047122 * q^65 - 36824553 * q^67 - 30011556 * q^71 + 95080185 * q^73 + 71172999 * q^75 - 38333727 * q^77 + 8514456 * q^79 - 28697814 * q^81 + 20121540 * q^85 - 23305887 * q^87 + 83038554 * q^89 - 198538635 * q^91 + 38959704 * q^93 - 221605224 * q^95 - 78377706 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$29$	$43$	$73$
$\chi(n)$	$1$	$1$	$e\left(\frac{5}{6}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	40.5000	−	23.3827i	0.500000	−	0.288675i
$3$	40.5000	−	23.3827i	0.500000	−	0.288675i
$4$		0			0
$5$	−939.615	−	542.487i	−1.50338	−	0.867979i	−0.999992	−	0.00392070i	$-0.998752\pi$
$5$	−939.615	−	542.487i	−1.50338	−	0.867979i	−0.503392	−	0.864058i	$-0.667915\pi$
$6$		0			0
$7$	1451.57	+	1912.52i	0.604570	+	0.796552i
$7$	1451.57	+	1912.52i	0.604570	+	0.796552i
$8$		0			0
$9$	1093.50	−	1894.00i	0.166667	−	0.288675i
$10$		0			0
$11$	−8435.48	−	14610.7i	−0.576155	−	0.997929i	−0.995915	−	0.0902942i	$-0.971219\pi$
$11$	−8435.48	−	14610.7i	−0.576155	−	0.997929i	0.419760	−	0.907635i	$-0.362114\pi$
$12$		0			0
$13$		−	9958.41i		−	0.348672i	−0.984686	−	0.174336i	$-0.944222\pi$
$13$		−	9958.41i		−	0.348672i	0.984686	−	0.174336i	$-0.0557778\pi$
$14$		0			0
$15$	−50739.2			−1.00226
$16$		0			0
$17$	−91736.7	+	52964.2i	−1.09837	+	0.634142i	−0.935792	−	0.352554i	$-0.885313\pi$
$17$	−91736.7	+	52964.2i	−1.09837	+	0.634142i	−0.162575	+	0.986696i	$0.551980\pi$
$18$		0			0
$19$	147650.	+	85245.9i	1.13297	+	0.654122i	0.944681	−	0.327991i	$-0.106372\pi$
$19$	147650.	+	85245.9i	1.13297	+	0.654122i	0.188292	+	0.982113i	$0.439705\pi$
$20$		0			0
$21$	103509.	+	43515.5i	0.532230	+	0.223752i
$22$		0			0
$23$	−105694.	+	183067.i	−0.377692	+	0.654183i	−0.990726	−	0.135874i	$-0.956616\pi$
$23$	−105694.	+	183067.i	−0.377692	+	0.654183i	0.613034	+	0.790057i	$0.289949\pi$
$24$		0			0
$25$	393272.	+	681166.i	1.00678	+	1.74379i
$26$		0			0
$27$		−	102276.i		−	0.192450i
$28$		0			0
$29$	−567529.			−0.802409			−0.401205	−	0.915989i	$-0.631408\pi$
$29$	−567529.			−0.802409			−0.401205	+	0.915989i	$0.631408\pi$
$30$		0			0
$31$	549248.	−	317108.i	0.594732	−	0.343369i	−0.172234	−	0.985056i	$-0.555099\pi$
$31$	549248.	−	317108.i	0.594732	−	0.343369i	0.766967	+	0.641687i	$0.221765\pi$
$32$		0			0
$33$	−683274.	−	394488.i	−0.576155	−	0.332643i
$34$		0			0
$35$	−326400.	−	2.58449e6i	−0.217510	−	1.72228i
$36$		0			0
$37$	−1.07740e6	+	1.86611e6i	−0.574870	+	0.995703i	0.421186	+	0.906974i	$0.361614\pi$
$37$	−1.07740e6	+	1.86611e6i	−0.574870	+	0.995703i	−0.996056	+	0.0887291i	$0.971719\pi$
$38$		0			0
$39$	−232854.	−	403316.i	−0.100653	−	0.174336i
$40$		0			0
$41$			5.45428e6i			1.93020i	0.261884	+	0.965099i	$0.415656\pi$
$41$			5.45428e6i			1.93020i	−0.261884	+	0.965099i	$0.584344\pi$
$42$		0			0
$43$	5.56726e6			1.62842			0.814212	−	0.580568i	$-0.197169\pi$
$43$	5.56726e6			1.62842			0.814212	+	0.580568i	$0.197169\pi$
$44$		0			0
$45$	−2.05494e6	+	1.18642e6i	−0.501128	+	0.289326i
$46$		0			0
$47$	2.76739e6	+	1.59775e6i	0.567125	+	0.327430i	0.756000	−	0.654572i	$-0.227151\pi$
$47$	2.76739e6	+	1.59775e6i	0.567125	+	0.327430i	−0.188876	+	0.982001i	$0.560484\pi$
$48$		0			0
$49$	−1.55068e6	+	5.55233e6i	−0.268991	+	0.963143i
$50$		0			0
$51$	−2.47689e6	+	4.29010e6i	−0.366122	+	0.634142i
$52$		0			0
$53$	4.38272e6	+	7.59109e6i	0.555443	+	0.962056i	0.997869	+	0.0652510i	$0.0207848\pi$
$53$	4.38272e6	+	7.59109e6i	0.555443	+	0.962056i	−0.442425	+	0.896805i	$0.645882\pi$
$54$		0			0
$55$			1.83046e7i			2.00036i
$56$		0			0
$57$	7.97311e6			0.755315
$58$		0			0
$59$	−8.08566e6	+	4.66826e6i	−0.667279	+	0.385254i	−0.795045	−	0.606551i	$-0.792553\pi$
$59$	−8.08566e6	+	4.66826e6i	−0.667279	+	0.385254i	0.127766	+	0.991804i	$0.459219\pi$
$60$		0			0
$61$	−2.13748e7	−	1.23407e7i	−1.54377	−	0.891296i	−0.998596	−	0.0529726i	$-0.983130\pi$
$61$	−2.13748e7	−	1.23407e7i	−1.54377	−	0.891296i	−0.545174	−	0.838323i	$-0.683536\pi$
$62$		0			0
$63$	5.20961e6	−	657931.i	0.330706	−	0.0417655i
$64$		0			0
$65$	−5.40231e6	+	9.35707e6i	−0.302640	+	0.524187i
$66$		0			0
$67$	−6.91779e6	−	1.19820e7i	−0.343295	−	0.594605i	0.641747	−	0.766916i	$-0.278210\pi$
$67$	−6.91779e6	−	1.19820e7i	−0.343295	−	0.594605i	−0.985043	+	0.172311i	$0.944877\pi$
$68$		0			0
$69$			9.88562e6i			0.436122i
$70$		0			0
$71$	1.19189e7			0.469031			0.234515	−	0.972112i	$-0.424650\pi$
$71$	1.19189e7			0.469031			0.234515	+	0.972112i	$0.424650\pi$
$72$		0			0
$73$	−1.83300e7	+	1.05828e7i	−0.645463	+	0.372658i	−0.786716	−	0.617315i	$-0.788220\pi$
$73$	−1.83300e7	+	1.05828e7i	−0.645463	+	0.372658i	0.141253	+	0.989974i	$0.454887\pi$
$74$		0			0
$75$	3.18550e7	+	1.83915e7i	1.00678	+	0.581262i
$76$		0			0
$77$	1.56985e7	−	3.73415e7i	0.446577	−	1.06226i
$78$		0			0
$79$	−8.61190e6	+	1.49162e7i	−0.221101	+	0.382958i	−0.955143	−	0.296147i	$-0.904298\pi$
$79$	−8.61190e6	+	1.49162e7i	−0.221101	+	0.382958i	0.734042	+	0.679104i	$0.237632\pi$
$80$		0			0
$81$	−2.39148e6	−	4.14217e6i	−0.0555556	−	0.0962250i
$82$		0			0
$83$			6.82663e7i			1.43845i	0.694779	+	0.719223i	$0.255502\pi$
$83$			6.82663e7i			1.43845i	−0.694779	+	0.719223i	$0.744498\pi$
$84$		0			0
$85$	1.14930e8			2.20169
$86$		0			0
$87$	−2.29849e7	+	1.32703e7i	−0.401205	+	0.231636i
$88$		0			0
$89$	−7.87155e7	−	4.54464e7i	−1.25459	−	0.724335i	−0.282569	−	0.959247i	$-0.591186\pi$
$89$	−7.87155e7	−	4.54464e7i	−1.25459	−	0.724335i	−0.972017	+	0.234912i	$0.924520\pi$
$90$		0			0
$91$	1.90457e7	−	1.44553e7i	0.277735	−	0.210796i
$92$		0			0
$93$	1.48297e7	−	2.56858e7i	0.198244	−	0.343369i
$94$		0			0
$95$	−9.24895e7	−	1.60197e8i	−1.13553	−	1.96679i
$96$		0			0
$97$			3.95376e7i			0.446605i	0.974749	+	0.223302i	$0.0716837\pi$
$97$			3.95376e7i			0.446605i	−0.974749	+	0.223302i	$0.928316\pi$
$98$		0			0
$99$	−3.68968e7			−0.384103
$100$		0			0
$101$	1.01453e8	−	5.85738e7i	0.974941	−	0.562882i	0.0742019	−	0.997243i	$-0.476359\pi$
$101$	1.01453e8	−	5.85738e7i	0.974941	−	0.562882i	0.900739	+	0.434361i	$0.143026\pi$
$102$		0			0
$103$	−4.22305e7	−	2.43818e7i	−0.375212	−	0.216629i	0.300521	−	0.953775i	$-0.402839\pi$
$103$	−4.22305e7	−	2.43818e7i	−0.375212	−	0.216629i	−0.675733	+	0.737146i	$0.736173\pi$
$104$		0			0
$105$	−7.36516e7	−	9.70398e7i	−0.605934	−	0.798349i
$106$		0			0
$107$	5.90508e7	−	1.02279e8i	0.450496	−	0.780282i	−0.547921	−	0.836530i	$-0.684580\pi$
$107$	5.90508e7	−	1.02279e8i	0.450496	−	0.780282i	0.998417	+	0.0562482i	$0.0179138\pi$
$108$		0			0
$109$	−7.54688e6	−	1.30716e7i	−0.0534640	−	0.0926024i	0.838055	−	0.545586i	$-0.183693\pi$
$109$	−7.54688e6	−	1.30716e7i	−0.0534640	−	0.0926024i	−0.891519	+	0.452984i	$0.850360\pi$
$110$		0			0
$111$			1.00770e8i			0.663802i
$112$		0			0
$113$	−6.94293e7			−0.425823			−0.212911	−	0.977072i	$-0.568295\pi$
$113$	−6.94293e7			−0.425823			−0.212911	+	0.977072i	$0.568295\pi$
$114$		0			0
$115$	1.98623e8	−	1.14675e8i	1.13563	−	0.655658i
$116$		0			0
$117$	−1.88612e7	−	1.08895e7i	−0.100653	−	0.0581119i
$118$		0			0
$119$	−2.34458e8	−	9.85671e7i	−1.16917	−	0.491523i
$120$		0			0
$121$	−3.51352e7	+	6.08560e7i	−0.163908	+	0.283898i
$122$		0			0
$123$	1.27536e8	+	2.20898e8i	0.557200	+	0.965099i
$124$		0			0
$125$		−	4.29561e8i		−	1.75948i
$126$		0			0
$127$	−1.00139e8			−0.384934			−0.192467	−	0.981303i	$-0.561649\pi$
$127$	−1.00139e8			−0.384934			−0.192467	+	0.981303i	$0.561649\pi$
$128$		0			0
$129$	2.25474e8	−	1.30177e8i	0.814212	−	0.470085i
$130$		0			0
$131$	−6.73342e6	−	3.88754e6i	−0.0228639	−	0.0132005i	0.488524	−	0.872550i	$-0.337535\pi$
$131$	−6.73342e6	−	3.88754e6i	−0.0228639	−	0.0132005i	−0.511388	+	0.859350i	$0.670869\pi$
$132$		0			0
$133$	5.12902e7	+	4.06125e8i	0.163919	+	1.29793i
$134$		0			0
$135$	−5.54833e7	+	9.60999e7i	−0.167043	+	0.289326i
$136$		0			0
$137$	−5.53803e7	−	9.59215e7i	−0.157207	−	0.272291i	0.776653	−	0.629928i	$-0.216916\pi$
$137$	−5.53803e7	−	9.59215e7i	−0.157207	−	0.272291i	−0.933861	+	0.357637i	$0.883582\pi$
$138$		0			0
$139$		−	3.31664e8i		−	0.888464i	−0.895912	−	0.444232i	$-0.853477\pi$
$139$		−	3.31664e8i		−	0.888464i	0.895912	−	0.444232i	$-0.146523\pi$
$140$		0			0
$141$	1.49439e8			0.378083
$142$		0			0
$143$	−1.45499e8	+	8.40040e7i	−0.347949	+	0.200889i
$144$		0			0
$145$	5.33258e8	+	3.07877e8i	1.20633	+	0.696474i
$146$		0			0
$147$	6.70258e7	+	2.61128e8i	0.143540	+	0.559222i
$148$		0			0
$149$	3.47265e8	−	6.01481e8i	0.704557	−	1.22033i	−0.262294	−	0.964988i	$-0.584479\pi$
$149$	3.47265e8	−	6.01481e8i	0.704557	−	1.22033i	0.966851	−	0.255341i	$-0.0821877\pi$
$150$		0			0
$151$	−4.10896e8	−	7.11694e8i	−0.790359	−	1.36894i	−0.925745	−	0.378150i	$-0.876561\pi$
$151$	−4.10896e8	−	7.11694e8i	−0.790359	−	1.36894i	0.135385	−	0.990793i	$-0.456773\pi$
$152$		0			0
$153$			2.31665e8i			0.422762i
$154$		0			0
$155$	−6.88109e8			−1.19215
$156$		0			0
$157$	−4.16332e8	+	2.40369e8i	−0.685238	+	0.395622i	−0.801825	−	0.597558i	$-0.796138\pi$
$157$	−4.16332e8	+	2.40369e8i	−0.685238	+	0.395622i	0.116588	+	0.993180i	$0.462804\pi$
$158$		0			0
$159$	3.55000e8	+	2.04959e8i	0.555443	+	0.320685i
$160$		0			0
$161$	−5.03542e8	+	6.35933e7i	−0.749432	+	0.0946472i
$162$		0			0
$163$	−3.04664e8	+	5.27694e8i	−0.431590	+	0.747536i	−0.997010	−	0.0772671i	$-0.975381\pi$
$163$	−3.04664e8	+	5.27694e8i	−0.431590	+	0.747536i	0.565420	+	0.824803i	$0.308714\pi$
$164$		0			0
$165$	4.28010e8	+	7.41334e8i	0.577454	+	1.00018i
$166$		0			0
$167$			1.84576e8i			0.237307i	0.992936	+	0.118653i	$0.0378577\pi$
$167$			1.84576e8i			0.237307i	−0.992936	+	0.118653i	$0.962142\pi$
$168$		0			0
$169$	7.16561e8			0.878428
$170$		0			0
$171$	3.22911e8	−	1.86433e8i	0.377658	−	0.218041i
$172$		0			0
$173$	−2.92420e8	−	1.68829e8i	−0.326454	−	0.188478i	0.327812	−	0.944743i	$-0.393689\pi$
$173$	−2.92420e8	−	1.68829e8i	−0.326454	−	0.188478i	−0.654266	+	0.756265i	$0.727022\pi$
$174$		0			0
$175$	−7.31884e8	+	1.74090e9i	−0.780351	+	1.85619i
$176$		0			0
$177$	−2.18313e8	+	3.78129e8i	−0.222426	+	0.385254i
$178$		0			0
$179$	2.92703e8	+	5.06977e8i	0.285112	+	0.493828i	0.972636	−	0.232333i	$-0.0746359\pi$
$179$	2.92703e8	+	5.06977e8i	0.285112	+	0.493828i	−0.687524	+	0.726161i	$0.741303\pi$
$180$		0			0
$181$			1.29407e9i			1.20571i	0.797850	+	0.602856i	$0.205971\pi$
$181$			1.29407e9i			1.20571i	−0.797850	+	0.602856i	$0.794029\pi$
$182$		0			0
$183$	−1.15424e9			−1.02918
$184$		0			0
$185$	2.02468e9	−	1.16895e9i	1.72850	−	0.997950i
$186$		0			0
$187$	1.54769e9	+	8.93557e8i	1.26566	+	0.730728i
$188$		0			0
$189$	1.95605e8	−	1.48461e8i	0.153297	−	0.116349i
$190$		0			0
$191$	−7.80043e8	+	1.35107e9i	−0.586118	+	1.01519i	0.408617	+	0.912706i	$0.366011\pi$
$191$	−7.80043e8	+	1.35107e9i	−0.586118	+	1.01519i	−0.994735	+	0.102480i	$0.967322\pi$
$192$		0			0
$193$	8.76806e8	+	1.51867e9i	0.631937	+	1.09455i	0.987155	+	0.159764i	$0.0510734\pi$
$193$	8.76806e8	+	1.51867e9i	0.631937	+	1.09455i	−0.355218	+	0.934784i	$0.615593\pi$
$194$		0			0
$195$			5.05282e8i			0.349458i
$196$		0			0
$197$	−1.78624e9			−1.18598			−0.592988	−	0.805212i	$-0.702052\pi$
$197$	−1.78624e9			−1.18598			−0.592988	+	0.805212i	$0.702052\pi$
$198$		0			0
$199$	−2.60521e9	+	1.50412e9i	−1.66123	+	0.959114i	−0.689107	+	0.724660i	$0.741997\pi$
$199$	−2.60521e9	+	1.50412e9i	−1.66123	+	0.959114i	−0.972127	+	0.234454i	$0.924670\pi$
$200$		0			0
$201$	−5.60341e8	−	3.23513e8i	−0.343295	−	0.198202i
$202$		0			0
$203$	−8.23809e8	−	1.08541e9i	−0.485112	−	0.639161i
$204$		0			0
$205$	2.95888e9	−	5.12492e9i	1.67537	−	2.90183i
$206$		0			0
$207$	2.31152e8	+	4.00368e8i	0.125897	+	0.218061i
$208$		0			0
$209$		−	2.87636e9i		−	1.50750i
$210$		0			0
$211$	3.33988e9			1.68501			0.842503	−	0.538692i	$-0.181081\pi$
$211$	3.33988e9			1.68501			0.842503	+	0.538692i	$0.181081\pi$
$212$		0			0
$213$	4.82714e8	−	2.78695e8i	0.234515	−	0.135398i
$214$		0			0
$215$	−5.23108e9	−	3.02016e9i	−2.44815	−	1.41344i
$216$		0			0
$217$	1.40375e9	+	5.90143e8i	0.633069	+	0.266145i
$218$		0			0
$219$	−4.94911e8	+	8.57210e8i	−0.215154	+	0.372658i
$220$		0			0
$221$	5.27439e8	+	9.13552e8i	0.221107	+	0.382969i
$222$		0			0
$223$			1.94513e9i			0.786556i	0.919420	+	0.393278i	$0.128659\pi$
$223$			1.94513e9i			0.786556i	−0.919420	+	0.393278i	$0.871341\pi$
$224$		0			0
$225$	1.72017e9			0.671184
$226$		0			0
$227$	1.17894e9	−	6.80660e8i	0.444004	−	0.256346i	−0.261290	−	0.965260i	$-0.584148\pi$
$227$	1.17894e9	−	6.80660e8i	0.444004	−	0.256346i	0.705295	+	0.708914i	$0.250815\pi$
$228$		0			0
$229$	−3.23205e9	−	1.86603e9i	−1.17527	−	0.678541i	−0.220352	−	0.975420i	$-0.570721\pi$
$229$	−3.23205e9	−	1.86603e9i	−1.17527	−	0.678541i	−0.954915	+	0.296880i	$0.904054\pi$
$230$		0			0
$231$	−2.37354e8	−	1.87940e9i	−0.0833581	−	0.660043i
$232$		0			0
$233$	−1.72263e8	+	2.98368e8i	−0.0584477	+	0.101234i	−0.893769	−	0.448528i	$-0.851948\pi$
$233$	−1.72263e8	+	2.98368e8i	−0.0584477	+	0.101234i	0.835321	+	0.549762i	$0.185282\pi$
$234$		0			0
$235$	−1.73352e9	−	3.00254e9i	−0.568404	−	0.984505i
$236$		0			0
$237$			8.05477e8i			0.255305i
$238$		0			0
$239$	3.17982e9			0.974565			0.487283	−	0.873244i	$-0.337988\pi$
$239$	3.17982e9			0.974565			0.487283	+	0.873244i	$0.337988\pi$
$240$		0			0
$241$	−4.09977e9	+	2.36701e9i	−1.21532	+	0.701667i	−0.963914	−	0.266214i	$-0.914227\pi$
$241$	−4.09977e9	+	2.36701e9i	−1.21532	+	0.701667i	−0.251409	+	0.967881i	$0.580894\pi$
$242$		0			0
$243$	−1.93710e8	−	1.11839e8i	−0.0555556	−	0.0320750i
$244$		0			0
$245$	4.46911e9	−	4.37583e9i	1.24038	−	1.21449i
$246$		0			0
$247$	8.48913e8	−	1.47036e9i	0.228074	−	0.395035i
$248$		0			0
$249$	1.59625e9	+	2.76478e9i	0.415244	+	0.719223i
$250$		0			0
$251$			7.03152e9i			1.77155i	0.464112	+	0.885776i	$0.346373\pi$
$251$			7.03152e9i			1.77155i	−0.464112	+	0.885776i	$0.653627\pi$
$252$		0			0
$253$	3.56631e9			0.870437
$254$		0			0
$255$	4.65465e9	−	2.68736e9i	1.10084	−	0.635573i
$256$		0			0
$257$	−1.10931e9	−	6.40462e8i	−0.254286	−	0.146812i	0.367439	−	0.930047i	$-0.380235\pi$
$257$	−1.10931e9	−	6.40462e8i	−0.254286	−	0.146812i	−0.621725	+	0.783236i	$0.713568\pi$
$258$		0			0
$259$	−5.13289e9	+	6.48243e8i	−1.14068	+	0.144058i
$260$		0			0
$261$	−6.20593e8	+	1.07490e9i	−0.133735	+	0.231636i
$262$		0			0
$263$	2.37972e8	+	4.12179e8i	0.0497396	+	0.0861516i	0.889823	−	0.456305i	$-0.150828\pi$
$263$	2.37972e8	+	4.12179e8i	0.0497396	+	0.0861516i	−0.840084	+	0.542457i	$0.817494\pi$
$264$		0			0
$265$		−	9.51027e9i		−	1.92845i
$266$		0			0
$267$	−4.25064e9			−0.836390
$268$		0			0
$269$	−4.95417e9	+	2.86029e9i	−0.946154	+	0.546262i	−0.891884	−	0.452264i	$-0.850616\pi$
$269$	−4.95417e9	+	2.86029e9i	−0.946154	+	0.546262i	−0.0542696	+	0.998526i	$0.517283\pi$
$270$		0			0
$271$	−5.53293e9	−	3.19444e9i	−1.02584	−	0.592267i	−0.110048	−	0.993926i	$-0.535100\pi$
$271$	−5.53293e9	−	3.19444e9i	−1.02584	−	0.592267i	−0.915789	+	0.401659i	$0.868434\pi$
$272$		0			0
$273$	4.33345e8	−	1.03078e9i	0.0780159	−	0.185573i
$274$		0			0
$275$	6.63487e9	−	1.14919e10i	1.16012	−	2.00938i
$276$		0			0
$277$	−2.63656e9	−	4.56666e9i	−0.447836	−	0.775675i	0.550409	−	0.834895i	$-0.314472\pi$
$277$	−2.63656e9	−	4.56666e9i	−0.447836	−	0.775675i	−0.998245	+	0.0592204i	$0.981139\pi$
$278$		0			0
$279$		−	1.38703e9i		−	0.228913i
$280$		0			0
$281$	7.48748e9			1.20091			0.600455	−	0.799658i	$-0.294986\pi$
$281$	7.48748e9			1.20091			0.600455	+	0.799658i	$0.294986\pi$
$282$		0			0
$283$	−3.62475e9	+	2.09275e9i	−0.565109	+	0.326266i	−0.755193	−	0.655502i	$-0.772457\pi$
$283$	−3.62475e9	+	2.09275e9i	−0.565109	+	0.326266i	0.190085	+	0.981768i	$0.439124\pi$
$284$		0			0
$285$	−7.49165e9	−	4.32531e9i	−1.13553	−	0.655598i
$286$		0			0
$287$	−1.04314e10	+	7.91728e9i	−1.53750	+	1.16694i
$288$		0			0
$289$	2.12254e9	−	3.67634e9i	0.304273	−	0.527017i
$290$		0			0
$291$	9.24495e8	+	1.60127e9i	0.128924	+	0.223302i
$292$		0			0
$293$			1.12022e9i			0.151997i	0.997108	+	0.0759983i	$0.0242143\pi$
$293$			1.12022e9i			0.151997i	−0.997108	+	0.0759983i	$0.975786\pi$
$294$		0			0
$295$	1.01299e10			1.33757
$296$		0			0
$297$	−1.49432e9	+	8.62746e8i	−0.192052	+	0.110881i
$298$		0			0
$299$	1.82306e9	+	1.05254e9i	0.228095	+	0.131691i
$300$		0			0
$301$	8.08127e9	+	1.06475e10i	0.984496	+	1.29712i
$302$		0			0
$303$	2.73922e9	−	4.74447e9i	0.324980	−	0.562882i
$304$		0			0
$305$	1.33894e10	+	2.31911e10i	1.54725	+	2.67992i
$306$		0			0
$307$			1.02340e10i			1.15210i	0.817413	+	0.576051i	$0.195407\pi$
$307$			1.02340e10i			1.15210i	−0.817413	+	0.576051i	$0.804593\pi$
$308$		0			0
$309$	−2.28044e9			−0.250141
$310$		0			0
$311$	5.99552e9	−	3.46151e9i	0.640892	−	0.370019i	−0.144066	−	0.989568i	$-0.546018\pi$
$311$	5.99552e9	−	3.46151e9i	0.640892	−	0.370019i	0.784958	+	0.619549i	$0.212684\pi$
$312$		0			0
$313$	4.79188e8	+	2.76659e8i	0.0499262	+	0.0288249i	0.524755	−	0.851253i	$-0.324157\pi$
$313$	4.79188e8	+	2.76659e8i	0.0499262	+	0.0288249i	−0.474829	+	0.880078i	$0.657490\pi$
$314$		0			0
$315$	−5.25194e9	−	2.20794e9i	−0.533430	−	0.224257i
$316$		0			0
$317$	−6.72093e8	+	1.16410e9i	−0.0665568	+	0.115280i	−0.897384	−	0.441251i	$-0.854535\pi$
$317$	−6.72093e8	+	1.16410e9i	−0.0665568	+	0.115280i	0.830827	+	0.556531i	$0.187868\pi$
$318$		0			0
$319$	4.78738e9	+	8.29198e9i	0.462312	+	0.800747i
$320$		0			0
$321$		−	5.52307e9i		−	0.520188i
$322$		0			0
$323$	−1.80599e10			−1.65923
$324$		0			0
$325$	6.78333e9	−	3.91636e9i	0.608009	−	0.351034i
$326$		0			0
$327$	−6.11298e8	−	3.52933e8i	−0.0534640	−	0.0308675i
$328$		0			0
$329$	9.61326e8	+	7.61194e9i	0.0820516	+	0.649698i
$330$		0			0
$331$	3.38567e8	−	5.86414e8i	0.0282054	−	0.0488532i	−0.851578	−	0.524228i	$-0.824354\pi$
$331$	3.38567e8	−	5.86414e8i	0.0282054	−	0.0488532i	0.879784	+	0.475374i	$0.157687\pi$
$332$		0			0
$333$	2.35627e9	+	4.08118e9i	0.191623	+	0.331901i
$334$		0			0
$335$			1.50112e10i			1.19189i
$336$		0			0
$337$	−1.73692e10			−1.34667			−0.673333	−	0.739340i	$-0.735138\pi$
$337$	−1.73692e10			−1.34667			−0.673333	+	0.739340i	$0.735138\pi$
$338$		0			0
$339$	−2.81189e9	+	1.62344e9i	−0.212911	+	0.122924i
$340$		0			0
$341$	−9.26634e9	−	5.34992e9i	−0.685316	−	0.395667i
$342$		0			0
$343$	−1.28699e10	+	5.09389e9i	−0.929817	+	0.368022i
$344$		0			0
$345$	5.36282e9	−	9.28868e9i	0.378545	−	0.655658i
$346$		0			0
$347$	8.82405e9	+	1.52837e10i	0.608626	+	1.05417i	0.991467	+	0.130356i	$0.0416122\pi$
$347$	8.82405e9	+	1.52837e10i	0.608626	+	1.05417i	−0.382842	+	0.923814i	$0.625054\pi$
$348$		0			0
$349$		−	9.49559e9i		−	0.640059i	−0.947408	−	0.320030i	$-0.896307\pi$
$349$		−	9.49559e9i		−	0.640059i	0.947408	−	0.320030i	$-0.103693\pi$
$350$		0			0
$351$	−1.01850e9			−0.0671019
$352$		0			0
$353$	6.12231e9	−	3.53472e9i	0.394290	−	0.227644i	−0.289727	−	0.957109i	$-0.593565\pi$
$353$	6.12231e9	−	3.53472e9i	0.394290	−	0.227644i	0.684017	+	0.729466i	$0.260231\pi$
$354$		0			0
$355$	−1.11991e10	−	6.46583e9i	−0.705134	−	0.407109i
$356$		0			0
$357$	−1.18003e10	+	1.49028e9i	−0.726474	+	0.0917478i
$358$		0			0
$359$	−6.55145e9	+	1.13474e10i	−0.394421	+	0.683157i	−0.993027	−	0.117887i	$-0.962388\pi$
$359$	−6.55145e9	+	1.13474e10i	−0.394421	+	0.683157i	0.598606	+	0.801043i	$0.295721\pi$
$360$		0			0
$361$	6.04193e9	+	1.04649e10i	0.355752	+	0.616180i
$362$		0			0
$363$			3.28622e9i			0.189265i
$364$		0			0
$365$	2.29642e10			1.29384
$366$		0			0
$367$	2.12630e10	−	1.22762e10i	1.17209	−	0.676707i	0.217919	−	0.975967i	$-0.430073\pi$
$367$	2.12630e10	−	1.22762e10i	1.17209	−	0.676707i	0.954172	+	0.299260i	$0.0967399\pi$
$368$		0			0
$369$	1.03304e10	+	5.96426e9i	0.557200	+	0.321700i
$370$		0			0
$371$	−8.15629e9	+	1.94011e10i	−0.430524	+	1.02407i
$372$		0			0
$373$	5.64618e9	−	9.77947e9i	0.291689	−	0.505219i	−0.682521	−	0.730866i	$-0.739116\pi$
$373$	5.64618e9	−	9.77947e9i	0.291689	−	0.505219i	0.974209	+	0.225647i	$0.0724496\pi$
$374$		0			0
$375$	−1.00443e10	−	1.73972e10i	−0.507919	−	0.879741i
$376$		0			0
$377$			5.65168e9i			0.279777i
$378$		0			0
$379$	−3.52545e9			−0.170867			−0.0854335	−	0.996344i	$-0.527228\pi$
$379$	−3.52545e9			−0.170867			−0.0854335	+	0.996344i	$0.527228\pi$
$380$		0			0
$381$	−4.05561e9	+	2.34151e9i	−0.192467	+	0.111121i
$382$		0			0
$383$	2.74394e10	+	1.58422e10i	1.27521	+	0.736240i	0.975963	−	0.217937i	$-0.0699326\pi$
$383$	2.74394e10	+	1.58422e10i	1.27521	+	0.736240i	0.299243	+	0.954177i	$0.403266\pi$
$384$		0			0
$385$	−3.50079e10	+	2.65704e10i	−1.59339	+	1.20936i
$386$		0			0
$387$	6.08779e9	−	1.05444e10i	0.271404	−	0.470085i
$388$		0			0
$389$	2.01146e9	+	3.48396e9i	0.0878443	+	0.152151i	0.906600	−	0.421992i	$-0.138669\pi$
$389$	2.01146e9	+	3.48396e9i	0.0878443	+	0.152151i	−0.818755	+	0.574143i	$0.805336\pi$
$390$		0			0
$391$		−	2.23920e10i		−	0.958043i
$392$		0			0
$393$	−3.63605e8			−0.0152426
$394$		0			0
$395$	1.61837e10	−	9.34368e9i	0.664799	−	0.383822i
$396$		0			0
$397$	−3.37145e10	−	1.94651e10i	−1.35724	−	0.783600i	−0.367985	−	0.929832i	$-0.619952\pi$
$397$	−3.37145e10	−	1.94651e10i	−1.35724	−	0.783600i	−0.989250	+	0.146232i	$0.953286\pi$
$398$		0			0
$399$	1.15735e10	+	1.52487e10i	0.456641	+	0.601648i
$400$		0			0
$401$	2.03692e10	−	3.52804e10i	0.787764	−	1.36445i	−0.139571	−	0.990212i	$-0.544572\pi$
$401$	2.03692e10	−	3.52804e10i	0.787764	−	1.36445i	0.927334	−	0.374234i	$-0.122094\pi$
$402$		0			0
$403$	−3.15790e9	−	5.46963e9i	−0.119723	−	0.207366i
$404$		0			0
$405$			5.18940e9i			0.192884i
$406$		0			0
$407$	3.63535e10			1.32486
$408$		0			0
$409$	−1.08712e10	+	6.27648e9i	−0.388493	+	0.224297i	−0.681507	−	0.731812i	$-0.738675\pi$
$409$	−1.08712e10	+	6.27648e9i	−0.388493	+	0.224297i	0.293014	+	0.956108i	$0.405342\pi$
$410$		0			0
$411$	−4.48580e9	−	2.58988e9i	−0.157207	−	0.0907637i
$412$		0			0
$413$	−2.06651e10	−	8.68769e9i	−0.710291	−	0.298610i
$414$		0			0
$415$	3.70336e10	−	6.41440e10i	1.24854	−	2.16254i
$416$		0			0
$417$	−7.75521e9	−	1.34324e10i	−0.256477	−	0.444232i
$418$		0			0
$419$			1.94158e9i			0.0629941i	0.999504	+	0.0314970i	$0.0100275\pi$
$419$			1.94158e9i			0.0629941i	−0.999504	+	0.0314970i	$0.989973\pi$
$420$		0			0
$421$	1.38803e9			0.0441846			0.0220923	−	0.999756i	$-0.492967\pi$
$421$	1.38803e9			0.0441846			0.0220923	+	0.999756i	$0.492967\pi$
$422$		0			0
$423$	6.05228e9	−	3.49428e9i	0.189042	−	0.109143i
$424$		0			0
$425$	−7.21549e10	−	4.16586e10i	−2.21162	−	1.27688i
$426$		0			0
$427$	−7.42511e9	−	5.87932e10i	−0.223353	−	1.76854i
$428$		0			0
$429$	−3.92848e9	+	6.80432e9i	−0.115983	+	0.200889i
$430$		0			0
$431$	−1.62135e10	−	2.80826e10i	−0.469860	−	0.813821i	0.529546	−	0.848281i	$-0.322362\pi$
$431$	−1.62135e10	−	2.80826e10i	−0.469860	−	0.813821i	−0.999406	+	0.0344602i	$0.989029\pi$
$432$		0			0
$433$			4.65333e10i			1.32377i	0.749606	+	0.661884i	$0.230243\pi$
$433$			4.65333e10i			1.32377i	−0.749606	+	0.661884i	$0.769757\pi$
$434$		0			0
$435$	2.87960e10			0.804219
$436$		0			0
$437$	−3.12114e10	+	1.80199e10i	−0.855831	+	0.494114i
$438$		0			0
$439$	−4.44106e10	−	2.56404e10i	−1.19572	−	0.690347i	−0.236119	−	0.971724i	$-0.575876\pi$
$439$	−4.44106e10	−	2.56404e10i	−1.19572	−	0.690347i	−0.959597	+	0.281377i	$0.909209\pi$
$440$		0			0
$441$	8.82042e9	+	9.00845e9i	0.233204	+	0.238175i
$442$		0			0
$443$	2.09215e10	−	3.62371e10i	0.543223	−	0.940891i	−0.455493	−	0.890239i	$-0.650537\pi$
$443$	2.09215e10	−	3.62371e10i	0.543223	−	0.940891i	0.998716	−	0.0506512i	$-0.0161297\pi$
$444$		0			0
$445$	4.93082e10	+	8.54042e10i	1.25742	+	2.17791i
$446$		0			0
$447$		−	3.24800e10i		−	0.813553i
$448$		0			0
$449$	2.89337e10			0.711900			0.355950	−	0.934505i	$-0.384157\pi$
$449$	2.89337e10			0.711900			0.355950	+	0.934505i	$0.384157\pi$
$450$		0			0
$451$	7.96907e10	−	4.60095e10i	1.92620	−	1.11209i
$452$		0			0
$453$	−3.32826e10	−	1.92157e10i	−0.790359	−	0.456314i
$454$		0			0
$455$	−2.57374e10	+	3.25043e9i	−0.600509	+	0.0758394i
$456$		0			0
$457$	−3.38299e10	+	5.85951e10i	−0.775597	+	1.34337i	0.158862	+	0.987301i	$0.449218\pi$
$457$	−3.38299e10	+	5.85951e10i	−0.775597	+	1.34337i	−0.934458	+	0.356072i	$0.884116\pi$
$458$		0			0
$459$	5.41696e9	+	9.38245e9i	0.122041	+	0.211381i
$460$		0			0
$461$			2.13195e10i			0.472035i	0.971749	+	0.236017i	$0.0758422\pi$
$461$			2.13195e10i			0.472035i	−0.971749	+	0.236017i	$0.924158\pi$
$462$		0			0
$463$	5.73662e10			1.24834			0.624168	−	0.781290i	$-0.285438\pi$
$463$	5.73662e10			1.24834			0.624168	+	0.781290i	$0.285438\pi$
$464$		0			0
$465$	−2.78684e10	+	1.60898e10i	−0.596074	+	0.344144i
$466$		0			0
$467$	−1.40331e10	−	8.10202e9i	−0.295044	−	0.170343i	0.345171	−	0.938540i	$-0.387821\pi$
$467$	−1.40331e10	−	8.10202e9i	−0.295044	−	0.170343i	−0.640214	+	0.768196i	$0.721154\pi$
$468$		0			0
$469$	1.28741e10	−	3.06231e10i	0.266088	−	0.632933i
$470$		0			0
$471$	−1.12410e10	+	1.94699e10i	−0.228413	+	0.395622i
$472$		0			0
$473$	−4.69625e10	−	8.13414e10i	−0.938224	−	1.62505i
$474$		0			0
$475$			1.34099e11i			2.63422i
$476$		0			0
$477$	1.91700e10			0.370296
$478$		0			0
$479$	−8.47162e10	+	4.89109e10i	−1.60925	+	0.929103i	−0.619716	+	0.784826i	$0.712752\pi$
$479$	−8.47162e10	+	4.89109e10i	−1.60925	+	0.929103i	−0.989537	+	0.144276i	$0.953915\pi$
$480$		0			0
$481$	1.85835e10	+	1.07292e10i	0.347173	+	0.200441i
$482$		0			0
$483$	−1.89065e10	+	1.43497e10i	−0.347394	+	0.263666i
$484$		0			0
$485$	2.14486e10	−	3.71501e10i	0.387644	−	0.671418i
$486$		0			0
$487$	2.47120e10	+	4.28025e10i	0.439332	+	0.760945i	0.997638	−	0.0686896i	$-0.0218818\pi$
$487$	2.47120e10	+	4.28025e10i	0.439332	+	0.760945i	−0.558306	+	0.829635i	$0.688548\pi$
$488$		0			0
$489$			2.84955e10i			0.498357i
$490$		0			0
$491$	4.36304e10			0.750695			0.375348	−	0.926884i	$-0.377523\pi$
$491$	4.36304e10			0.750695			0.375348	+	0.926884i	$0.377523\pi$
$492$		0			0
$493$	5.20632e10	−	3.00587e10i	0.881340	−	0.508842i
$494$		0			0
$495$	3.46688e10	+	2.00160e10i	0.577454	+	0.333393i
$496$		0			0
$497$	1.73011e10	+	2.27951e10i	0.283562	+	0.373608i
$498$		0			0
$499$	1.70928e10	−	2.96057e10i	0.275684	−	0.477499i	−0.694623	−	0.719374i	$-0.744429\pi$
$499$	1.70928e10	−	2.96057e10i	0.275684	−	0.477499i	0.970308	+	0.241874i	$0.0777622\pi$
$500$		0			0
$501$	4.31589e9	+	7.47534e9i	0.0685045	+	0.118653i
$502$		0			0
$503$		−	1.10624e10i		−	0.172813i	−0.996260	−	0.0864063i	$-0.972462\pi$
$503$		−	1.10624e10i		−	0.172813i	0.996260	−	0.0864063i	$-0.0275383\pi$
$504$		0			0
$505$	−1.27102e11			−1.95428
$506$		0			0
$507$	2.90207e10	−	1.67551e10i	0.439214	−	0.253580i
$508$		0			0
$509$	−9.58080e10	−	5.53148e10i	−1.42735	−	0.824081i	−0.430440	−	0.902619i	$-0.641642\pi$
$509$	−9.58080e10	−	5.53148e10i	−1.42735	−	0.824081i	−0.996911	+	0.0785379i	$0.974975\pi$
$510$		0			0
$511$	−4.68473e10	−	1.96948e10i	−0.687069	−	0.288847i
$512$		0			0
$513$	8.71859e9	−	1.51010e10i	0.125886	−	0.218041i
$514$		0			0
$515$	2.64536e10	+	4.58189e10i	0.376059	+	0.651353i
$516$		0			0
$517$		−	5.39112e10i		−	0.754600i
$518$		0			0
$519$	−1.57907e10			−0.217636
$520$		0			0
$521$	1.11862e11	−	6.45834e10i	1.51820	−	0.876536i	0.518434	−	0.855118i	$-0.326515\pi$
$521$	1.11862e11	−	6.45834e10i	1.51820	−	0.876536i	0.999771	−	0.0214183i	$-0.00681818\pi$
$522$		0			0
$523$	4.41818e10	+	2.55084e10i	0.590523	+	0.340939i	0.765304	−	0.643669i	$-0.222588\pi$
$523$	4.41818e10	+	2.55084e10i	0.590523	+	0.340939i	−0.174781	+	0.984607i	$0.555922\pi$
$524$		0			0
$525$	1.10657e10	+	8.76200e10i	0.145660	+	1.15336i
$526$		0			0
$527$	−3.35908e10	+	5.81810e10i	−0.435490	+	0.754290i
$528$		0			0
$529$	1.68131e10	+	2.91212e10i	0.214697	+	0.371866i
$530$		0			0
$531$			2.04190e10i			0.256836i
$532$		0			0
$533$	5.43159e10			0.673005
$534$		0			0
$535$	−1.10970e11	+	6.40686e10i	−1.35454	+	0.782042i
$536$		0			0
$537$	2.37090e10	+	1.36884e10i	0.285112	+	0.164609i
$538$		0			0
$539$	9.42040e10	−	2.41801e10i	1.11613	−	0.286485i
$540$		0			0
$541$	−5.42593e10	+	9.39799e10i	−0.633410	+	1.09710i	0.353439	+	0.935457i	$0.385012\pi$
$541$	−5.42593e10	+	9.39799e10i	−0.633410	+	1.09710i	−0.986850	+	0.161641i	$0.948321\pi$
$542$		0			0
$543$	3.02588e10	+	5.24098e10i	0.348059	+	0.602856i
$544$		0			0
$545$			1.63763e10i			0.185623i
$546$		0			0
$547$	−6.18018e10			−0.690322			−0.345161	−	0.938544i	$-0.612176\pi$
$547$	−6.18018e10			−0.690322			−0.345161	+	0.938544i	$0.612176\pi$
$548$		0			0
$549$	−4.67467e10	+	2.69892e10i	−0.514590	+	0.297099i
$550$		0			0
$551$	−8.37957e10	−	4.83795e10i	−0.909108	−	0.524874i
$552$		0			0
$553$	−4.10284e10	+	5.18156e9i	−0.438717	+	0.0554064i
$554$		0			0
$555$	5.46663e10	−	9.46849e10i	0.576166	−	0.997950i
$556$		0			0
$557$	2.68485e10	+	4.65029e10i	0.278932	+	0.483125i	0.971120	−	0.238593i	$-0.0766862\pi$
$557$	2.68485e10	+	4.65029e10i	0.278932	+	0.483125i	−0.692188	+	0.721718i	$0.743353\pi$
$558$		0			0
$559$		−	5.54410e10i		−	0.567785i
$560$		0			0
$561$	8.35751e10			0.843772
$562$		0			0
$563$	−1.24343e11	+	7.17894e10i	−1.23762	+	0.714540i	−0.968607	−	0.248596i	$-0.920031\pi$
$563$	−1.24343e11	+	7.17894e10i	−1.23762	+	0.714540i	−0.269013	+	0.963137i	$0.586697\pi$
$564$		0			0
$565$	6.52368e10	+	3.76645e10i	0.640175	+	0.369605i
$566$		0			0
$567$	4.45058e9	−	1.05864e10i	0.0430611	−	0.102428i
$568$		0			0
$569$	4.80492e10	−	8.32237e10i	0.458392	−	0.793958i	−0.540484	−	0.841354i	$-0.681759\pi$
$569$	4.80492e10	−	8.32237e10i	0.458392	−	0.793958i	0.998876	+	0.0473959i	$0.0150922\pi$
$570$		0			0
$571$	1.02075e11	+	1.76799e11i	0.960231	+	1.66317i	0.721915	+	0.691981i	$0.243262\pi$
$571$	1.02075e11	+	1.76799e11i	0.960231	+	1.66317i	0.238316	+	0.971188i	$0.423405\pi$
$572$		0			0
$573$			7.29580e10i			0.676790i
$574$		0			0
$575$	−1.66266e11			−1.52101
$576$		0			0
$577$	−1.23039e10	+	7.10368e9i	−0.111005	+	0.0640885i	−0.554474	−	0.832201i	$-0.687081\pi$
$577$	−1.23039e10	+	7.10368e9i	−0.111005	+	0.0640885i	0.443470	+	0.896289i	$0.353747\pi$
$578$		0			0
$579$	7.10213e10	+	4.10041e10i	0.631937	+	0.364849i
$580$		0			0
$581$	−1.30561e11	+	9.90934e10i	−1.14580	+	0.869641i
$582$		0			0
$583$	7.39406e10	−	1.28069e11i	0.640043	−	1.10859i
$584$		0			0
$585$	1.18148e10	+	2.04639e10i	0.100880	+	0.174729i
$586$		0			0
$587$		−	6.63318e10i		−	0.558688i	−0.960191	−	0.279344i	$-0.909883\pi$
$587$		−	6.63318e10i		−	0.558688i	0.960191	−	0.279344i	$-0.0901170\pi$
$588$		0			0
$589$	1.08129e11			0.898421
$590$		0			0
$591$	−7.23428e10	+	4.17672e10i	−0.592988	+	0.342362i
$592$		0			0
$593$	9.13541e10	+	5.27433e10i	0.738770	+	0.426529i	0.821622	−	0.570033i	$-0.193070\pi$
$593$	9.13541e10	+	5.27433e10i	0.738770	+	0.426529i	−0.0828520	+	0.996562i	$0.526403\pi$
$594$		0			0
$595$	1.66829e11	+	2.19805e11i	1.33107	+	1.75376i
$596$		0			0
$597$	−7.03407e10	+	1.21834e11i	−0.553745	+	0.959114i
$598$		0			0
$599$	−7.35115e10	−	1.27326e11i	−0.571016	−	0.989028i	−0.996462	−	0.0840447i	$-0.973216\pi$
$599$	−7.35115e10	−	1.27326e11i	−0.571016	−	0.989028i	0.425446	−	0.904984i	$-0.360117\pi$
$600$		0			0
$601$		−	1.86033e11i		−	1.42591i	−0.701210	−	0.712954i	$-0.747357\pi$
$601$		−	1.86033e11i		−	1.42591i	0.701210	−	0.712954i	$-0.252643\pi$
$602$		0			0
$603$	−3.02584e10			−0.228864
$604$		0			0
$605$	6.60272e10	−	3.81208e10i	0.492834	−	0.284538i
$606$		0			0
$607$	−5.85799e10	−	3.38211e10i	−0.431513	−	0.249134i	0.268478	−	0.963286i	$-0.413479\pi$
$607$	−5.85799e10	−	3.38211e10i	−0.431513	−	0.249134i	−0.699991	+	0.714152i	$0.746813\pi$
$608$		0			0
$609$	−5.87441e10	−	2.46963e10i	−0.427066	−	0.179541i
$610$		0			0
$611$	1.59111e10	−	2.75588e10i	0.114165	−	0.197740i
$612$		0			0
$613$	−1.19635e11	−	2.07214e11i	−0.847261	−	1.46750i	−0.883643	−	0.468161i	$-0.844917\pi$
$613$	−1.19635e11	−	2.07214e11i	−0.847261	−	1.46750i	0.0363824	−	0.999338i	$-0.488417\pi$
$614$		0			0
$615$		−	2.76746e11i		−	1.93455i
$616$		0			0
$617$	−3.40881e10			−0.235214			−0.117607	−	0.993060i	$-0.537522\pi$
$617$	−3.40881e10			−0.235214			−0.117607	+	0.993060i	$0.537522\pi$
$618$		0			0
$619$	8.42286e10	−	4.86294e10i	0.573716	−	0.331235i	−0.184916	−	0.982754i	$-0.559201\pi$
$619$	8.42286e10	−	4.86294e10i	0.573716	−	0.331235i	0.758632	+	0.651519i	$0.225868\pi$
$620$		0			0
$621$	1.87233e10	+	1.08099e10i	0.125897	+	0.0726870i
$622$		0			0
$623$	−2.73439e10	−	2.16514e11i	−0.181513	−	1.43725i
$624$		0			0
$625$	−7.94095e10	+	1.37541e11i	−0.520418	+	0.901391i
$626$		0			0
$627$	−6.72570e10	−	1.16493e11i	−0.435178	−	0.753751i
$628$		0			0
$629$		−	2.28254e11i		−	1.45820i
$630$		0			0
$631$	−1.63168e11			−1.02924			−0.514621	−	0.857418i	$-0.672067\pi$
$631$	−1.63168e11			−1.02924			−0.514621	+	0.857418i	$0.672067\pi$
$632$		0			0
$633$	1.35265e11	−	7.80954e10i	0.842503	−	0.486419i
$634$		0			0
$635$	9.40916e10	+	5.43238e10i	0.578704	+	0.334115i
$636$		0			0
$637$	5.52923e10	+	1.54423e10i	0.335820	+	0.0937895i
$638$		0			0
$639$	1.30333e10	−	2.25743e10i	0.0781718	−	0.135398i
$640$		0			0
$641$	−2.04255e10	−	3.53779e10i	−0.120987	−	0.209556i	0.799170	−	0.601105i	$-0.205273\pi$
$641$	−2.04255e10	−	3.53779e10i	−0.120987	−	0.209556i	−0.920157	+	0.391549i	$0.871939\pi$
$642$		0			0
$643$		−	1.66499e11i		−	0.974022i	−0.873396	−	0.487011i	$-0.838087\pi$
$643$		−	1.66499e11i		−	0.974022i	0.873396	−	0.487011i	$-0.161913\pi$
$644$		0			0
$645$	−2.82478e11			−1.63210
$646$		0			0
$647$	−4.33764e10	+	2.50434e10i	−0.247535	+	0.142914i	−0.618635	−	0.785678i	$-0.712314\pi$
$647$	−4.33764e10	+	2.50434e10i	−0.247535	+	0.142914i	0.371100	+	0.928593i	$0.378981\pi$
$648$		0			0
$649$	1.36413e11	+	7.87580e10i	0.768912	+	0.443931i
$650$		0			0
$651$	7.06510e10	−	8.92265e9i	0.393364	−	0.0496786i
$652$		0			0
$653$	−2.99393e10	+	5.18564e10i	−0.164660	+	0.285200i	−0.936535	−	0.350575i	$-0.885986\pi$
$653$	−2.99393e10	+	5.18564e10i	−0.164660	+	0.285200i	0.771874	+	0.635775i	$0.219319\pi$
$654$		0			0
$655$	4.21788e9	+	7.30558e9i	0.0229155	+	0.0396908i
$656$		0			0
$657$			4.62893e10i			0.248439i
$658$		0			0
$659$	−5.62292e10			−0.298140			−0.149070	−	0.988827i	$-0.547628\pi$
$659$	−5.62292e10			−0.298140			−0.149070	+	0.988827i	$0.547628\pi$
$660$		0			0
$661$	1.99084e11	−	1.14941e11i	1.04287	−	0.602102i	0.122226	−	0.992502i	$-0.460997\pi$
$661$	1.99084e11	−	1.14941e11i	1.04287	−	0.602102i	0.920645	+	0.390400i	$0.127663\pi$
$662$		0			0
$663$	4.27226e10	+	2.46659e10i	0.221107	+	0.127656i
$664$		0			0
$665$	1.72124e11	−	4.09425e11i	0.880147	−	2.09357i
$666$		0			0
$667$	5.99843e10	−	1.03896e11i	0.303064	−	0.524922i
$668$		0			0
$669$	4.54824e10	+	7.87779e10i	0.227059	+	0.393278i
$670$		0			0
$671$			4.16400e11i			2.05410i
$672$		0			0
$673$	9.84906e10			0.480103			0.240052	−	0.970760i	$-0.422836\pi$
$673$	9.84906e10			0.480103			0.240052	+	0.970760i	$0.422836\pi$
$674$		0			0
$675$	6.96669e10	−	4.02222e10i	0.335592	−	0.193754i
$676$		0			0
$677$	1.28763e11	+	7.43413e10i	0.612965	+	0.353896i	0.774125	−	0.633033i	$-0.218190\pi$
$677$	1.28763e11	+	7.43413e10i	0.612965	+	0.353896i	−0.161160	+	0.986928i	$0.551523\pi$
$678$		0			0
$679$	−7.56165e10	+	5.73917e10i	−0.355744	+	0.270004i
$680$		0			0
$681$	3.18313e10	−	5.51334e10i	0.148001	−	0.256346i
$682$		0			0
$683$	1.76448e11	+	3.05616e11i	0.810836	+	1.40441i	0.912280	+	0.409568i	$0.134320\pi$
$683$	1.76448e11	+	3.05616e11i	0.810836	+	1.40441i	−0.101444	+	0.994841i	$0.532346\pi$
$684$		0			0
$685$			1.20172e11i			0.545811i
$686$		0			0
$687$	−1.74531e11			−0.783511
$688$		0			0
$689$	7.55951e10	−	4.36449e10i	0.335442	−	0.193667i
$690$		0			0
$691$	−2.91147e11	−	1.68094e11i	−1.27703	−	0.737291i	−0.300725	−	0.953711i	$-0.597229\pi$
$691$	−2.91147e11	−	1.68094e11i	−1.27703	−	0.737291i	−0.976300	+	0.216420i	$0.930562\pi$
$692$		0			0
$693$	−5.35583e10	−	7.05659e10i	−0.232217	−	0.305958i
$694$		0			0
$695$	−1.79924e11	+	3.11637e11i	−0.771168	+	1.33570i
$696$		0			0
$697$	−2.88882e11	−	5.00358e11i	−1.22402	−	2.12007i
$698$		0			0
$699$			1.61119e10i			0.0674896i
$700$		0			0
$701$	4.11369e11			1.70357			0.851784	−	0.523893i	$-0.175521\pi$
$701$	4.11369e11			1.70357			0.851784	+	0.523893i	$0.175521\pi$
$702$		0			0
$703$	−3.18156e11	+	1.83687e11i	−1.30262	+	0.752070i
$704$		0			0
$705$	−1.40415e11	−	8.10687e10i	−0.568404	−	0.328168i
$706$		0			0
$707$	2.59290e11	+	1.09007e11i	1.03778	+	0.436290i
$708$		0			0
$709$	−6.95122e10	+	1.20399e11i	−0.275091	+	0.476472i	−0.970158	−	0.242473i	$-0.922041\pi$
$709$	−6.95122e10	+	1.20399e11i	−0.275091	+	0.476472i	0.695067	+	0.718945i	$0.255375\pi$
$710$		0			0
$711$	1.88342e10	+	3.26218e10i	0.0737003	+	0.127653i
$712$		0			0
$713$			1.34066e11i			0.518751i
$714$		0			0
$715$	1.82284e11			0.697469
$716$		0			0
$717$	1.28783e11	−	7.43527e10i	0.487283	−	0.281333i
$718$		0			0
$719$	−1.01094e11	−	5.83667e10i	−0.378277	−	0.218398i	0.298791	−	0.954318i	$-0.403417\pi$
$719$	−1.01094e11	−	5.83667e10i	−0.378277	−	0.218398i	−0.677068	+	0.735920i	$0.736750\pi$
$720$		0			0
$721$	−1.46699e10	−	1.16159e11i	−0.0542857	−	0.429843i
$722$		0			0
$723$	−1.10694e11	+	1.91727e11i	−0.405108	+	0.701667i
$724$		0			0
$725$	−2.23193e11	−	3.86581e11i	−0.807846	−	1.39923i
$726$		0			0
$727$		−	2.43338e11i		−	0.871109i	−0.900162	−	0.435555i	$-0.856552\pi$
$727$		−	2.43338e11i		−	0.871109i	0.900162	−	0.435555i	$-0.143448\pi$
$728$		0			0
$729$	−1.04604e10			−0.0370370
$730$		0			0
$731$	−5.10722e11	+	2.94865e11i	−1.78861	+	1.03265i
$732$		0			0
$733$	4.20029e11	+	2.42504e11i	1.45500	+	0.840046i	0.998759	−	0.0498097i	$-0.0158615\pi$
$733$	4.20029e11	+	2.42504e11i	1.45500	+	0.840046i	0.456243	+	0.889855i	$0.349195\pi$
$734$		0			0
$735$	7.86802e10	−	2.81721e11i	0.269598	−	0.965315i
$736$		0			0
$737$	−1.16710e11	+	2.02147e11i	−0.395582	+	0.685169i
$738$		0			0
$739$	−1.98657e11	−	3.44083e11i	−0.666078	−	1.15368i	−0.978992	−	0.203900i	$-0.934638\pi$
$739$	−1.98657e11	−	3.44083e11i	−0.666078	−	1.15368i	0.312913	−	0.949782i	$-0.398695\pi$
$740$		0			0
$741$		−	7.93995e10i		−	0.263357i
$742$		0			0
$743$	2.20639e11			0.723980			0.361990	−	0.932182i	$-0.382097\pi$
$743$	2.20639e11			0.723980			0.361990	+	0.932182i	$0.382097\pi$
$744$		0			0
$745$	−6.52591e11	+	3.76774e11i	−2.11844	+	1.22308i
$746$		0			0
$747$	1.29296e11	+	7.46492e10i	0.415244	+	0.239741i
$748$		0			0
$749$	2.81327e11	−	3.55294e10i	0.893892	−	0.112891i
$750$		0			0
$751$	2.44854e8	−	4.24099e8i	0.000769745	−	0.00133324i	−0.865640	−	0.500666i	$-0.833088\pi$
$751$	2.44854e8	−	4.24099e8i	0.000769745	−	0.00133324i	0.866410	+	0.499333i	$0.166422\pi$
$752$		0			0
$753$	1.64416e11	+	2.84776e11i	0.511403	+	0.885776i
$754$		0			0
$755$			8.91624e11i			2.74406i
$756$		0			0
$757$	−6.35724e10			−0.193591			−0.0967955	−	0.995304i	$-0.530859\pi$
$757$	−6.35724e10			−0.193591			−0.0967955	+	0.995304i	$0.530859\pi$
$758$		0			0
$759$	1.44436e11	−	8.33900e10i	0.435219	−	0.251274i
$760$		0			0
$761$	5.15892e10	+	2.97850e10i	0.153823	+	0.0888095i	0.574936	−	0.818199i	$-0.305027\pi$
$761$	5.15892e10	+	2.97850e10i	0.153823	+	0.0888095i	−0.421113	+	0.907008i	$0.638360\pi$
$762$		0			0
$763$	1.40449e10	−	3.34079e10i	0.0414399	−	0.0985715i
$764$		0			0
$765$	1.25675e11	−	2.17676e11i	0.366948	−	0.635573i
$766$		0			0
$767$	4.64884e10	+	8.05203e10i	0.134327	+	0.232661i
$768$		0			0
$769$			3.28466e11i			0.939259i	0.882864	+	0.469629i	$0.155612\pi$
$769$			3.28466e11i			0.939259i	−0.882864	+	0.469629i	$0.844388\pi$
$770$		0			0
$771$	−5.99029e10			−0.169524
$772$		0			0
$773$	4.19975e11	−	2.42473e11i	1.17627	−	0.679117i	0.221118	−	0.975247i	$-0.429030\pi$
$773$	4.19975e11	−	2.42473e11i	1.17627	−	0.679117i	0.955148	+	0.296130i	$0.0956962\pi$
$774$		0			0
$775$	4.32007e11	+	2.49420e11i	1.19752	+	0.691391i
$776$		0			0
$777$	−1.92725e11	+	1.46275e11i	−0.528753	+	0.401315i
$778$		0			0
$779$	−4.64955e11	+	8.05325e11i	−1.26259	+	2.18686i
$780$		0			0
$781$	−1.00541e11	−	1.74143e11i	−0.270234	−	0.468060i
$782$		0			0
$783$			5.80445e10i			0.154424i
$784$		0			0
$785$	5.21589e11			1.37357
$786$		0			0
$787$	2.48776e11	−	1.43631e11i	0.648500	−	0.374412i	−0.139381	−	0.990239i	$-0.544511\pi$
$787$	2.48776e11	−	1.43631e11i	0.648500	−	0.374412i	0.787881	+	0.615827i	$0.211178\pi$
$788$		0			0
$789$	1.92757e10	+	1.11288e10i	0.0497396	+	0.0287172i
$790$		0			0
$791$	−1.00782e11	−	1.32785e11i	−0.257440	−	0.339190i
$792$		0			0
$793$	−1.22894e11	+	2.12859e11i	−0.310769	+	0.538269i
$794$		0			0
$795$	−2.22376e11	−	3.85166e11i	−0.556697	−	0.964227i
$796$		0			0
$797$			2.39071e11i			0.592506i	0.955109	+	0.296253i	$0.0957372\pi$
$797$			2.39071e11i			0.592506i	−0.955109	+	0.296253i	$0.904263\pi$
$798$		0			0
$799$	−3.38495e11			−0.830548
$800$		0			0
$801$	−1.72151e11	+	9.93913e10i	−0.418195	+	0.241445i
$802$		0			0
$803$	3.09245e11	+	1.78543e11i	0.743773	+	0.429418i
$804$		0			0
$805$	5.07634e11	+	2.13412e11i	1.20884	+	0.508200i
$806$		0			0
$807$	−1.33763e11	+	2.31684e11i	−0.315385	+	0.546262i
$808$		0			0
$809$	5.36214e10	+	9.28750e10i	0.125183	+	0.216823i	0.921804	−	0.387656i	$-0.126715\pi$
$809$	5.36214e10	+	9.28750e10i	0.125183	+	0.216823i	−0.796622	+	0.604478i	$0.793382\pi$
$810$		0			0
$811$		−	1.36934e11i		−	0.316540i	−0.987396	−	0.158270i	$-0.949408\pi$
$811$		−	1.36934e11i		−	0.316540i	0.987396	−	0.158270i	$-0.0505916\pi$
$812$		0			0
$813$	−2.98778e11			−0.683891
$814$		0			0
$815$	5.72535e11	−	3.30553e11i	1.29769	−	0.749222i
$816$		0			0
$817$	8.22006e11	+	4.74586e11i	1.84496	+	1.06519i
$818$		0			0
$819$	−6.55194e9	−	5.18794e10i	−0.0145625	−	0.115308i
$820$		0			0
$821$	−1.88762e11	+	3.26945e11i	−0.415472	+	0.719619i	−0.995478	−	0.0949936i	$-0.969717\pi$
$821$	−1.88762e11	+	3.26945e11i	−0.415472	+	0.719619i	0.580006	+	0.814612i	$0.303050\pi$
$822$		0			0
$823$	−3.58340e11	−	6.20662e11i	−0.781080	−	1.35287i	−0.931313	−	0.364220i	$-0.881336\pi$
$823$	−3.58340e11	−	6.20662e11i	−0.781080	−	1.35287i	0.150233	−	0.988651i	$-0.451998\pi$
$824$		0			0
$825$		−	6.20564e11i		−	1.33959i
$826$		0			0
$827$	−4.45077e11			−0.951510			−0.475755	−	0.879578i	$-0.657825\pi$
$827$	−4.45077e11			−0.951510			−0.475755	+	0.879578i	$0.657825\pi$
$828$		0			0
$829$	−3.52361e11	+	2.03436e11i	−0.746053	+	0.430734i	−0.824266	−	0.566203i	$-0.808412\pi$
$829$	−3.52361e11	+	2.03436e11i	−0.746053	+	0.430734i	0.0782131	+	0.996937i	$0.475079\pi$
$830$		0			0
$831$	−2.13562e11	−	1.23300e11i	−0.447836	−	0.258558i
$832$		0			0
$833$	−1.51820e11	−	5.91483e11i	−0.315319	−	1.22846i
$834$		0			0
$835$	1.00130e11	−	1.73431e11i	0.205977	−	0.356763i
$836$		0			0
$837$	−3.24325e10	−	5.61748e10i	−0.0660814	−	0.114456i
$838$		0			0
$839$		−	2.89090e11i		−	0.583424i	−0.956506	−	0.291712i	$-0.905775\pi$
$839$		−	2.89090e11i		−	0.583424i	0.956506	−	0.291712i	$-0.0942249\pi$
$840$		0			0
$841$	−1.78158e11			−0.356140
$842$		0			0
$843$	3.03243e11	−	1.75077e11i	0.600455	−	0.346673i
$844$		0			0
$845$	−6.73291e11	−	3.88725e11i	−1.32061	−	0.762457i
$846$		0			0
$847$	−1.67390e11	+	2.11400e10i	−0.325233	+	0.0410743i
$848$		0			0
$849$	−9.78682e10	+	1.69513e11i	−0.188370	+	0.326266i
$850$		0			0
$851$	−2.27749e11	−	3.94472e11i	−0.434248	−	0.752139i
$852$		0			0
$853$		−	8.33755e11i		−	1.57486i	−0.616404	−	0.787430i	$-0.711411\pi$
$853$		−	8.33755e11i		−	1.57486i	0.616404	−	0.787430i	$-0.288589\pi$
$854$		0			0
$855$	−4.04549e11			−0.757019
$856$		0			0
$857$	−5.63733e11	+	3.25471e11i	−1.04508	+	0.603378i	−0.921268	−	0.388927i	$-0.872846\pi$
$857$	−5.63733e11	+	3.25471e11i	−1.04508	+	0.603378i	−0.123813	+	0.992306i	$0.539512\pi$
$858$		0			0
$859$	7.76570e11	+	4.48353e11i	1.42629	+	0.823469i	0.996826	−	0.0796151i	$-0.0253691\pi$
$859$	7.76570e11	+	4.48353e11i	1.42629	+	0.823469i	0.429464	+	0.903084i	$0.358702\pi$
$860$		0			0
$861$	−2.37346e11	+	5.64565e11i	−0.431886	+	1.02731i
$862$		0			0
$863$	4.04187e9	−	7.00073e9i	0.00728684	−	0.0126212i	−0.862359	−	0.506297i	$-0.831014\pi$
$863$	4.04187e9	−	7.00073e9i	0.00728684	−	0.0126212i	0.869646	+	0.493676i	$0.164347\pi$
$864$		0			0
$865$	1.83175e11	+	3.17268e11i	0.327191	+	0.566711i
$866$		0			0
$867$		−	1.98522e11i		−	0.351345i
$868$		0			0
$869$	2.90582e11			0.509553
$870$		0			0
$871$	−1.19321e11	+	6.88901e10i	−0.207322	+	0.119697i
$872$		0			0
$873$	7.48841e10	+	4.32344e10i	0.128924	+	0.0744341i
$874$		0			0
$875$	8.21545e11	−	6.23539e11i	1.40152	−	1.06373i
$876$		0			0
$877$	−2.91022e11	+	5.04065e11i	−0.491957	+	0.852095i	−0.999957	−	0.00926195i	$-0.997052\pi$
$877$	−2.91022e11	+	5.04065e11i	−0.491957	+	0.852095i	0.508000	+	0.861357i	$0.330385\pi$
$878$		0			0
$879$	2.61938e10	+	4.53690e10i	0.0438776	+	0.0759983i
$880$		0			0
$881$			3.79168e11i			0.629403i	0.949191	+	0.314701i	$0.101904\pi$
$881$			3.79168e11i			0.629403i	−0.949191	+	0.314701i	$0.898096\pi$
$882$		0			0
$883$	2.21424e11			0.364235			0.182118	−	0.983277i	$-0.441705\pi$
$883$	2.21424e11			0.364235			0.182118	+	0.983277i	$0.441705\pi$
$884$		0			0
$885$	4.10260e11	−	2.36864e11i	0.668784	−	0.386123i
$886$		0			0
$887$	2.45661e11	+	1.41832e11i	0.396863	+	0.229129i	0.685130	−	0.728421i	$-0.259745\pi$
$887$	2.45661e11	+	1.41832e11i	0.396863	+	0.229129i	−0.288266	+	0.957550i	$0.593079\pi$
$888$		0			0
$889$	−1.45358e11	−	1.91517e11i	−0.232719	−	0.306620i
$890$		0			0
$891$	−4.03466e10	+	6.98824e10i	−0.0640172	+	0.110881i
$892$		0			0
$893$	2.72403e11	+	4.71817e11i	0.428358	+	0.741938i
$894$		0			0
$895$		−	6.35151e11i		−	0.989885i
$896$		0			0
$897$	9.84451e10			0.152063
$898$		0			0
$899$	−3.11714e11	+	1.79968e11i	−0.477219	+	0.275522i
$900$		0			0
$901$	−8.04112e11	−	4.64254e11i	−1.22016	−	0.704461i
$902$		0			0
$903$	5.76259e11	+	2.42262e11i	0.866695	+	0.364363i
$904$		0			0
$905$	7.02016e11	−	1.21593e12i	1.04653	−	1.81265i
$906$		0			0
$907$	3.30665e11	+	5.72728e11i	0.488606	+	0.846290i	0.999914	−	0.0131074i	$-0.00417235\pi$
$907$	3.30665e11	+	5.72728e11i	0.488606	+	0.846290i	−0.511308	+	0.859397i	$0.670839\pi$
$908$		0			0
$909$		−	2.56202e11i		−	0.375255i
$910$		0			0
$911$	−9.94922e11			−1.44449			−0.722247	−	0.691635i	$-0.756890\pi$
$911$	−9.94922e11			−1.44449			−0.722247	+	0.691635i	$0.756890\pi$
$912$		0			0
$913$	9.97417e11	−	5.75859e11i	1.43547	−	0.828768i
$914$		0			0
$915$	1.08454e12	+	6.26159e11i	1.54725	+	0.893306i
$916$		0			0
$917$	−2.33903e9	−	1.85209e10i	−0.00330795	−	0.0261929i
$918$		0			0
$919$	3.13835e11	−	5.43578e11i	0.439986	−	0.762079i	−0.557702	−	0.830042i	$-0.688317\pi$
$919$	3.13835e11	−	5.43578e11i	0.439986	−	0.762079i	0.997688	+	0.0679630i	$0.0216500\pi$
$920$		0			0
$921$	2.39298e11	+	4.14476e11i	0.332583	+	0.576051i
$922$		0			0
$923$		−	1.18693e11i		−	0.163538i
$924$		0			0
$925$	−1.69484e12			−2.31506
$926$		0			0
$927$	−9.23580e10	+	5.33229e10i	−0.125071	+	0.0722096i
$928$		0			0
$929$	2.80765e11	+	1.62100e11i	0.376947	+	0.217631i	0.676489	−	0.736453i	$-0.263501\pi$
$929$	2.80765e11	+	1.62100e11i	0.376947	+	0.217631i	−0.299542	+	0.954083i	$0.596834\pi$
$930$		0			0
$931$	−7.02271e11	+	6.87613e11i	−0.934772	+	0.915262i
$932$		0			0
$933$	1.61879e11	−	2.80383e11i	0.213631	−	0.370019i
$934$		0			0
$935$	−9.69486e11	−	1.67920e12i	−1.26851	−	2.19713i
$936$		0			0
$937$		−	1.21189e12i		−	1.57219i	−0.618108	−	0.786093i	$-0.712101\pi$
$937$		−	1.21189e12i		−	1.57219i	0.618108	−	0.786093i	$-0.287899\pi$
$938$		0			0
$939$	2.58762e10			0.0332841
$940$		0			0
$941$	−1.35199e11	+	7.80573e10i	−0.172431	+	0.0995531i	−0.583732	−	0.811947i	$-0.698408\pi$
$941$	−1.35199e11	+	7.80573e10i	−0.172431	+	0.0995531i	0.411301	+	0.911500i	$0.365075\pi$
$942$		0			0
$943$	−9.98499e11	−	5.76484e11i	−1.26270	−	0.729022i
$944$		0			0
$945$	−2.64331e11	+	3.33829e10i	−0.331452	+	0.0418598i
$946$		0			0
$947$	1.78356e10	−	3.08921e10i	0.0221762	−	0.0384103i	−0.854724	−	0.519082i	$-0.826274\pi$
$947$	1.78356e10	−	3.08921e10i	0.0221762	−	0.0384103i	0.876901	+	0.480672i	$0.159607\pi$
$948$		0			0
$949$	1.05388e11	+	1.82538e11i	0.129935	+	0.225055i
$950$		0			0
$951$			6.28614e10i			0.0768532i
$952$		0			0
$953$	−1.75336e11			−0.212569			−0.106284	−	0.994336i	$-0.533895\pi$
$953$	−1.75336e11			−0.212569			−0.106284	+	0.994336i	$0.533895\pi$
$954$		0			0
$955$	1.46588e12	−	8.46326e11i	1.76232	−	1.01748i
$956$		0			0
$957$	3.87778e11	+	2.23883e11i	0.462312	+	0.266916i
$958$		0			0
$959$	1.03063e11	−	2.45153e11i	0.121851	−	0.289843i
$960$		0			0
$961$	−2.25330e11	+	3.90283e11i	−0.264196	+	0.457600i
$962$		0			0
$963$	−1.29144e11	−	2.23684e11i	−0.150165	−	0.260094i
$964$		0			0
$965$		−	1.90262e12i		−	2.19403i
$966$		0			0
$967$	−6.32296e11			−0.723127			−0.361563	−	0.932348i	$-0.617757\pi$
$967$	−6.32296e11			−0.723127			−0.361563	+	0.932348i	$0.617757\pi$
$968$		0			0
$969$	−7.31427e11	+	4.22289e11i	−0.829613	+	0.478977i
$970$		0			0
$971$	−4.20551e11	−	2.42805e11i	−0.473088	−	0.273137i	0.244444	−	0.969664i	$-0.421395\pi$
$971$	−4.20551e11	−	2.42805e11i	−0.473088	−	0.273137i	−0.717531	+	0.696526i	$0.754728\pi$
$972$		0			0
$973$	6.34316e11	−	4.81435e11i	0.707708	−	0.537138i
$974$		0			0
$975$	1.83150e11	−	3.17225e11i	0.202670	−	0.351034i
$976$		0			0
$977$	−3.84268e11	−	6.65572e11i	−0.421751	−	0.730495i	0.574360	−	0.818603i	$-0.305251\pi$
$977$	−3.84268e11	−	6.65572e11i	−0.421751	−	0.730495i	−0.996111	+	0.0881084i	$0.971918\pi$
$978$		0			0
$979$			1.53345e12i			1.66932i
$980$		0			0
$981$	−3.30101e10			−0.0356427
$982$		0			0
$983$	−9.58883e10	+	5.53611e10i	−0.102695	+	0.0592913i	−0.550468	−	0.834856i	$-0.685551\pi$
$983$	−9.58883e10	+	5.53611e10i	−0.102695	+	0.0592913i	0.447773	+	0.894147i	$0.352217\pi$
$984$		0			0
$985$	1.67838e12	+	9.69014e11i	1.78298	+	1.02940i
$986$		0			0
$987$	2.16921e11	+	2.85805e11i	0.228578	+	0.301163i
$988$		0			0
$989$	−5.88425e11	+	1.01918e12i	−0.615043	+	1.06529i
$990$		0			0
$991$	5.66493e11	+	9.81195e11i	0.587354	+	1.01733i	0.994577	+	0.103999i	$0.0331638\pi$
$991$	5.66493e11	+	9.81195e11i	0.587354	+	1.01733i	−0.407223	+	0.913329i	$0.633503\pi$
$992$		0			0
$993$		−	3.16664e10i		−	0.0325688i
$994$		0			0
$995$	3.26386e12			3.32996
$996$		0			0
$997$	−1.13793e12	+	6.56982e11i	−1.15168	+	0.664925i	−0.949297	−	0.314380i	$-0.898203\pi$
$997$	−1.13793e12	+	6.56982e11i	−1.15168	+	0.664925i	−0.202387	+	0.979306i	$0.564870\pi$
$998$		0			0
$999$	1.90858e11	+	1.10192e11i	0.191623	+	0.110634i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.9.m.b.61.1	✓	12
3.2	odd	2		252.9.z.d.145.6		12
7.3	odd	6	inner	84.9.m.b.73.1	yes	12
21.17	even	6		252.9.z.d.73.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.9.m.b.61.1	✓	12	1.1	even	1	trivial
84.9.m.b.73.1	yes	12	7.3	odd	6	inner
252.9.z.d.73.6		12	21.17	even	6
252.9.z.d.145.6		12	3.2	odd	2

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.m (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(40.5000 - 23.3827i) q^{3} +(-939.615 - 542.487i) q^{5} +(1451.57 + 1912.52i) q^{7} +(1093.50 - 1894.00i) q^{9} +O(q^{10})\)	\(q+(40.5000 - 23.3827i) q^{3} +(-939.615 - 542.487i) q^{5} +(1451.57 + 1912.52i) q^{7} +(1093.50 - 1894.00i) q^{9} +(-8435.48 - 14610.7i) q^{11} -9958.41i q^{13} -50739.2 q^{15} +(-91736.7 + 52964.2i) q^{17} +(147650. + 85245.9i) q^{19} +(103509. + 43515.5i) q^{21} +(-105694. + 183067. i) q^{23} +(393272. + 681166. i) q^{25} -102276. i q^{27} -567529. q^{29} +(549248. - 317108. i) q^{31} +(-683274. - 394488. i) q^{33} +(-326400. - 2.58449e6i) q^{35} +(-1.07740e6 + 1.86611e6i) q^{37} +(-232854. - 403316. i) q^{39} +5.45428e6i q^{41} +5.56726e6 q^{43} +(-2.05494e6 + 1.18642e6i) q^{45} +(2.76739e6 + 1.59775e6i) q^{47} +(-1.55068e6 + 5.55233e6i) q^{49} +(-2.47689e6 + 4.29010e6i) q^{51} +(4.38272e6 + 7.59109e6i) q^{53} +1.83046e7i q^{55} +7.97311e6 q^{57} +(-8.08566e6 + 4.66826e6i) q^{59} +(-2.13748e7 - 1.23407e7i) q^{61} +(5.20961e6 - 657931. i) q^{63} +(-5.40231e6 + 9.35707e6i) q^{65} +(-6.91779e6 - 1.19820e7i) q^{67} +9.88562e6i q^{69} +1.19189e7 q^{71} +(-1.83300e7 + 1.05828e7i) q^{73} +(3.18550e7 + 1.83915e7i) q^{75} +(1.56985e7 - 3.73415e7i) q^{77} +(-8.61190e6 + 1.49162e7i) q^{79} +(-2.39148e6 - 4.14217e6i) q^{81} +6.82663e7i q^{83} +1.14930e8 q^{85} +(-2.29849e7 + 1.32703e7i) q^{87} +(-7.87155e7 - 4.54464e7i) q^{89} +(1.90457e7 - 1.44553e7i) q^{91} +(1.48297e7 - 2.56858e7i) q^{93} +(-9.24895e7 - 1.60197e8i) q^{95} +3.95376e7i q^{97} -3.68968e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 486 q^{3} + 285 q^{5} + 198 q^{7} + 13122 q^{9}+O(q^{10}) \) 12 * q + 486 * q^3 + 285 * q^5 + 198 * q^7 + 13122 * q^9	\( 12 q + 486 q^{3} + 285 q^{5} + 198 q^{7} + 13122 q^{9} - 17919 q^{11} + 15390 q^{15} - 205782 q^{17} + 74313 q^{19} - 39609 q^{21} - 62832 q^{23} + 878679 q^{25} - 575454 q^{29} + 1442952 q^{31} - 1451439 q^{33} - 3989514 q^{35} - 2058621 q^{37} - 930933 q^{39} + 7721322 q^{43} + 623295 q^{45} + 12088194 q^{47} - 16964694 q^{49} - 5556114 q^{51} - 5506743 q^{53} + 4012902 q^{57} + 7511901 q^{59} - 37215576 q^{61} - 3641355 q^{63} + 5047122 q^{65} - 36824553 q^{67} - 30011556 q^{71} + 95080185 q^{73} + 71172999 q^{75} - 38333727 q^{77} + 8514456 q^{79} - 28697814 q^{81} + 20121540 q^{85} - 23305887 q^{87} + 83038554 q^{89} - 198538635 q^{91} + 38959704 q^{93} - 221605224 q^{95} - 78377706 q^{99}+O(q^{100}) \) 12 * q + 486 * q^3 + 285 * q^5 + 198 * q^7 + 13122 * q^9 - 17919 * q^11 + 15390 * q^15 - 205782 * q^17 + 74313 * q^19 - 39609 * q^21 - 62832 * q^23 + 878679 * q^25 - 575454 * q^29 + 1442952 * q^31 - 1451439 * q^33 - 3989514 * q^35 - 2058621 * q^37 - 930933 * q^39 + 7721322 * q^43 + 623295 * q^45 + 12088194 * q^47 - 16964694 * q^49 - 5556114 * q^51 - 5506743 * q^53 + 4012902 * q^57 + 7511901 * q^59 - 37215576 * q^61 - 3641355 * q^63 + 5047122 * q^65 - 36824553 * q^67 - 30011556 * q^71 + 95080185 * q^73 + 71172999 * q^75 - 38333727 * q^77 + 8514456 * q^79 - 28697814 * q^81 + 20121540 * q^85 - 23305887 * q^87 + 83038554 * q^89 - 198538635 * q^91 + 38959704 * q^93 - 221605224 * q^95 - 78377706 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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