LMFDB - Embedded newform 84.9.m.a.61.2

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:	$10$
Relative dimension:	$5$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 38255 x^{8} + 1483053595 x^{6} - 139470625170 x^{5} + 5194605060018 x^{4} + \cdots + 15\!\cdots\!00 $ x^10 - 38255x^8 + 1483053595x^6 - 139470625170x^5 + 5194605060018x^4 - 71600654137860x^3 + 119846615988780x^2 + 4263507454127400*x + 15758720495290800
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{8}\cdot 7^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			61.2
Root			$38.0902 - 21.9914i$ of defining polynomial
Character	$\chi$	$=$	84.61
Dual form			84.9.m.a.73.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-40.5000 + 23.3827i) q^{3} +(-374.307 - 216.106i) q^{5} +(83.7630 + 2399.54i) q^{7} +(1093.50 - 1894.00i) q^{9} +O(q^{10})$	\(q+(-40.5000 + 23.3827i) q^{3} +(-374.307 - 216.106i) q^{5} +(83.7630 + 2399.54i) q^{7} +(1093.50 - 1894.00i) q^{9} +(6902.31 + 11955.2i) q^{11} +3223.79i q^{13} +20212.6 q^{15} +(-21339.9 + 12320.6i) q^{17} +(29141.7 + 16825.0i) q^{19} +(-59500.1 - 95222.7i) q^{21} +(-112905. + 195558. i) q^{23} +(-101909. - 176511. i) q^{25} +102276. i q^{27} -908389. q^{29} +(925773. - 534495. i) q^{31} +(-559087. - 322789. i) q^{33} +(487202. - 916265. i) q^{35} +(979412. - 1.69639e6i) q^{37} +(-75380.8 - 130563. i) q^{39} -2.90040e6i q^{41} -4.01000e6 q^{43} +(-818609. + 472624. i) q^{45} +(-4.29428e6 - 2.47930e6i) q^{47} +(-5.75077e6 + 401985. i) q^{49} +(576177. - 997968. i) q^{51} +(-3.98900e6 - 6.90915e6i) q^{53} -5.96653e6i q^{55} -1.57365e6 q^{57} +(4.69735e6 - 2.71201e6i) q^{59} +(1.84448e6 + 1.06491e6i) q^{61} +(4.63631e6 + 2.46525e6i) q^{63} +(696680. - 1.20669e6i) q^{65} +(-9.53767e6 - 1.65197e7i) q^{67} -1.05601e7i q^{69} -2.08687e7 q^{71} +(-9.66235e6 + 5.57856e6i) q^{73} +(8.25461e6 + 4.76580e6i) q^{75} +(-2.81087e7 + 1.75638e7i) q^{77} +(1.21508e7 - 2.10458e7i) q^{79} +(-2.39148e6 - 4.14217e6i) q^{81} -3.92213e7i q^{83} +1.06502e7 q^{85} +(3.67898e7 - 2.12406e7i) q^{87} +(1.02177e7 + 5.89920e6i) q^{89} +(-7.73560e6 + 270034. i) q^{91} +(-2.49959e7 + 4.32941e7i) q^{93} +(-7.27196e6 - 1.25954e7i) q^{95} +7.80671e7i q^{97} +3.01907e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 405 q^{3} + 1389 q^{5} + 1217 q^{7} + 10935 q^{9}+O(q^{10}) $ 10 * q - 405 * q^3 + 1389 * q^5 + 1217 * q^7 + 10935 * q^9	$ 10 q - 405 q^{3} + 1389 q^{5} + 1217 q^{7} + 10935 q^{9} - 879 q^{11} - 75006 q^{15} - 13674 q^{17} - 29268 q^{19} - 42363 q^{21} + 312732 q^{23} - 22052 q^{25} - 289794 q^{29} + 242787 q^{31} + 71199 q^{33} + 1209372 q^{35} + 1913308 q^{37} - 1232334 q^{39} - 861848 q^{43} + 3037743 q^{45} - 305448 q^{47} + 9821659 q^{49} + 369198 q^{51} - 10663233 q^{53} + 1580472 q^{57} + 18410871 q^{59} - 13937808 q^{61} + 769824 q^{63} - 14966808 q^{65} - 20722822 q^{67} + 113032584 q^{71} + 43436322 q^{73} + 1786212 q^{75} - 98823405 q^{77} - 42189637 q^{79} - 23914845 q^{81} + 142602108 q^{85} + 11736657 q^{87} + 67171914 q^{89} - 246091266 q^{91} - 6555249 q^{93} - 140649894 q^{95} - 3844746 q^{99}+O(q^{100}) $ 10 * q - 405 * q^3 + 1389 * q^5 + 1217 * q^7 + 10935 * q^9 - 879 * q^11 - 75006 * q^15 - 13674 * q^17 - 29268 * q^19 - 42363 * q^21 + 312732 * q^23 - 22052 * q^25 - 289794 * q^29 + 242787 * q^31 + 71199 * q^33 + 1209372 * q^35 + 1913308 * q^37 - 1232334 * q^39 - 861848 * q^43 + 3037743 * q^45 - 305448 * q^47 + 9821659 * q^49 + 369198 * q^51 - 10663233 * q^53 + 1580472 * q^57 + 18410871 * q^59 - 13937808 * q^61 + 769824 * q^63 - 14966808 * q^65 - 20722822 * q^67 + 113032584 * q^71 + 43436322 * q^73 + 1786212 * q^75 - 98823405 * q^77 - 42189637 * q^79 - 23914845 * q^81 + 142602108 * q^85 + 11736657 * q^87 + 67171914 * q^89 - 246091266 * q^91 - 6555249 * q^93 - 140649894 * q^95 - 3844746 * 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
$5$	−374.307	−	216.106i	−0.598891	−	0.345770i	0.169714	−	0.985493i	$-0.445716\pi$
$5$	−374.307	−	216.106i	−0.598891	−	0.345770i	−0.768605	+	0.639724i	$0.779049\pi$
$6$		0			0
$7$	83.7630	+	2399.54i	0.0348867	+	0.999391i
$7$	83.7630	+	2399.54i	0.0348867	+	0.999391i
$8$		0			0
$9$	1093.50	−	1894.00i	0.166667	−	0.288675i
$10$		0			0
$11$	6902.31	+	11955.2i	0.471437	+	0.816553i	0.999466	−	0.0326733i	$-0.0104021\pi$
$11$	6902.31	+	11955.2i	0.471437	+	0.816553i	−0.528029	+	0.849226i	$0.677069\pi$
$12$		0			0
$13$			3223.79i			0.112874i	0.998406	+	0.0564369i	$0.0179740\pi$
$13$			3223.79i			0.112874i	−0.998406	+	0.0564369i	$0.982026\pi$
$14$		0			0
$15$	20212.6			0.399261
$16$		0			0
$17$	−21339.9	+	12320.6i	−0.255503	+	0.147515i	−0.622282	−	0.782794i	$-0.713794\pi$
$17$	−21339.9	+	12320.6i	−0.255503	+	0.147515i	0.366778	+	0.930308i	$0.380461\pi$
$18$		0			0
$19$	29141.7	+	16825.0i	0.223615	+	0.129104i	0.607623	−	0.794226i	$-0.292123\pi$
$19$	29141.7	+	16825.0i	0.223615	+	0.129104i	−0.384008	+	0.923330i	$0.625457\pi$
$20$		0			0
$21$	−59500.1	−	95222.7i	−0.305943	−	0.489625i
$22$		0			0
$23$	−112905.	+	195558.i	−0.403462	+	0.698817i	−0.994141	−	0.108090i	$-0.965527\pi$
$23$	−112905.	+	195558.i	−0.403462	+	0.698817i	0.590679	+	0.806907i	$0.298860\pi$
$24$		0			0
$25$	−101909.	−	176511.i	−0.260887	−	0.451869i
$26$		0			0
$27$			102276.i			0.192450i
$28$		0			0
$29$	−908389.			−1.28434			−0.642170	−	0.766562i	$-0.721966\pi$
$29$	−908389.			−1.28434			−0.642170	+	0.766562i	$0.721966\pi$
$30$		0			0
$31$	925773.	−	534495.i	1.00244	−	0.578758i	0.0934693	−	0.995622i	$-0.470204\pi$
$31$	925773.	−	534495.i	1.00244	−	0.578758i	0.908969	+	0.416864i	$0.136871\pi$
$32$		0			0
$33$	−559087.	−	322789.i	−0.471437	−	0.272184i
$34$		0			0
$35$	487202.	−	916265.i	0.324666	−	0.610589i
$36$		0			0
$37$	979412.	−	1.69639e6i	0.522587	−	0.905147i	−0.477067	−	0.878867i	$-0.658300\pi$
$37$	979412.	−	1.69639e6i	0.522587	−	0.905147i	0.999655	−	0.0262808i	$-0.00836640\pi$
$38$		0			0
$39$	−75380.8	−	130563.i	−0.0325838	−	0.0564369i
$40$		0			0
$41$		−	2.90040e6i		−	1.02641i	−0.858265	−	0.513207i	$-0.828457\pi$
$41$		−	2.90040e6i		−	1.02641i	0.858265	−	0.513207i	$-0.171543\pi$
$42$		0			0
$43$	−4.01000e6			−1.17293			−0.586463	−	0.809976i	$-0.699480\pi$
$43$	−4.01000e6			−1.17293			−0.586463	+	0.809976i	$0.699480\pi$
$44$		0			0
$45$	−818609.	+	472624.i	−0.199630	+	0.115257i
$46$		0			0
$47$	−4.29428e6	−	2.47930e6i	−0.880033	−	0.508087i	−0.00936356	−	0.999956i	$-0.502981\pi$
$47$	−4.29428e6	−	2.47930e6i	−0.880033	−	0.508087i	−0.870669	+	0.491869i	$0.836314\pi$
$48$		0			0
$49$	−5.75077e6	+	401985.i	−0.997566	+	0.0697309i
$50$		0			0
$51$	576177.	−	997968.i	0.0851677	−	0.147515i
$52$		0			0
$53$	−3.98900e6	−	6.90915e6i	−0.505546	−	0.875631i	−0.999979	−	0.00641571i	$-0.997958\pi$
$53$	−3.98900e6	−	6.90915e6i	−0.505546	−	0.875631i	0.494434	−	0.869215i	$-0.335376\pi$
$54$		0			0
$55$		−	5.96653e6i		−	0.652035i
$56$		0			0
$57$	−1.57365e6			−0.149077
$58$		0			0
$59$	4.69735e6	−	2.71201e6i	0.387654	−	0.223812i	−0.293489	−	0.955962i	$-0.594816\pi$
$59$	4.69735e6	−	2.71201e6i	0.387654	−	0.223812i	0.681143	+	0.732150i	$0.261483\pi$
$60$		0			0
$61$	1.84448e6	+	1.06491e6i	0.133216	+	0.0769120i	0.565127	−	0.825004i	$-0.308827\pi$
$61$	1.84448e6	+	1.06491e6i	0.133216	+	0.0769120i	−0.431911	+	0.901916i	$0.642161\pi$
$62$		0			0
$63$	4.63631e6	+	2.46525e6i	0.294314	+	0.156494i
$64$		0			0
$65$	696680.	−	1.20669e6i	0.0390283	−	0.0675991i
$66$		0			0
$67$	−9.53767e6	−	1.65197e7i	−0.473307	−	0.819792i	0.526226	−	0.850345i	$-0.323607\pi$
$67$	−9.53767e6	−	1.65197e7i	−0.473307	−	0.819792i	−0.999533	+	0.0305527i	$0.990273\pi$
$68$		0			0
$69$		−	1.05601e7i		−	0.465878i
$70$		0			0
$71$	−2.08687e7			−0.821223			−0.410612	−	0.911810i	$-0.634685\pi$
$71$	−2.08687e7			−0.821223			−0.410612	+	0.911810i	$0.634685\pi$
$72$		0			0
$73$	−9.66235e6	+	5.57856e6i	−0.340245	+	0.196440i	−0.660380	−	0.750931i	$-0.729605\pi$
$73$	−9.66235e6	+	5.57856e6i	−0.340245	+	0.196440i	0.320136	+	0.947372i	$0.396272\pi$
$74$		0			0
$75$	8.25461e6	+	4.76580e6i	0.260887	+	0.150623i
$76$		0			0
$77$	−2.81087e7	+	1.75638e7i	−0.799609	+	0.499637i
$78$		0			0
$79$	1.21508e7	−	2.10458e7i	0.311958	−	0.540328i	−0.666828	−	0.745212i	$-0.732348\pi$
$79$	1.21508e7	−	2.10458e7i	0.311958	−	0.540328i	0.978786	+	0.204884i	$0.0656817\pi$
$80$		0			0
$81$	−2.39148e6	−	4.14217e6i	−0.0555556	−	0.0962250i
$82$		0			0
$83$		−	3.92213e7i		−	0.826436i	−0.910632	−	0.413218i	$-0.864405\pi$
$83$		−	3.92213e7i		−	0.826436i	0.910632	−	0.413218i	$-0.135595\pi$
$84$		0			0
$85$	1.06502e7			0.204025
$86$		0			0
$87$	3.67898e7	−	2.12406e7i	0.642170	−	0.370757i
$88$		0			0
$89$	1.02177e7	+	5.89920e6i	0.162852	+	0.0940228i	0.579211	−	0.815178i	$-0.303361\pi$
$89$	1.02177e7	+	5.89920e6i	0.162852	+	0.0940228i	−0.416359	+	0.909200i	$0.636694\pi$
$90$		0			0
$91$	−7.73560e6	+	270034.i	−0.112805	+	0.00393779i
$92$		0			0
$93$	−2.49959e7	+	4.32941e7i	−0.334146	+	0.578758i
$94$		0			0
$95$	−7.27196e6	−	1.25954e7i	−0.0892806	−	0.154638i
$96$		0			0
$97$			7.80671e7i			0.881822i	0.897551	+	0.440911i	$0.145345\pi$
$97$			7.80671e7i			0.881822i	−0.897551	+	0.440911i	$0.854655\pi$
$98$		0			0
$99$	3.01907e7			0.314291
$100$		0			0
$101$	3.07926e7	−	1.77781e7i	0.295911	−	0.170844i	−0.344694	−	0.938715i	$-0.612017\pi$
$101$	3.07926e7	−	1.77781e7i	0.295911	−	0.170844i	0.640604	+	0.767871i	$0.278684\pi$
$102$		0			0
$103$	1.83451e8	+	1.05915e8i	1.62994	+	0.941044i	0.984110	+	0.177560i	$0.0568202\pi$
$103$	1.83451e8	+	1.05915e8i	1.62994	+	0.941044i	0.645826	+	0.763485i	$0.276513\pi$
$104$		0			0
$105$	1.69306e6	+	4.85008e7i	0.0139289	+	0.399017i
$106$		0			0
$107$	−6.59050e7	+	1.14151e8i	−0.502786	+	0.870851i	0.497209	+	0.867631i	$0.334358\pi$
$107$	−6.59050e7	+	1.14151e8i	−0.502786	+	0.870851i	−0.999995	+	0.00322029i	$0.998975\pi$
$108$		0			0
$109$	9.29656e7	+	1.61021e8i	0.658591	+	1.14071i	0.980980	+	0.194107i	$0.0621808\pi$
$109$	9.29656e7	+	1.61021e8i	0.658591	+	1.14071i	−0.322389	+	0.946607i	$0.604486\pi$
$110$		0			0
$111$			9.16052e7i			0.603432i
$112$		0			0
$113$	−4.85733e7			−0.297909			−0.148955	−	0.988844i	$-0.547591\pi$
$113$	−4.85733e7			−0.297909			−0.148955	+	0.988844i	$0.547591\pi$
$114$		0			0
$115$	8.45224e7	−	4.87990e7i	0.483260	−	0.279010i
$116$		0			0
$117$	6.10585e6	+	3.52521e6i	0.0325838	+	0.0188123i
$118$		0			0
$119$	−3.13512e7	−	5.01739e7i	−0.156339	−	0.250201i
$120$		0			0
$121$	1.18956e7	−	2.06038e7i	0.0554940	−	0.0961184i
$122$		0			0
$123$	6.78191e7	+	1.17466e8i	0.296300	+	0.513207i
$124$		0			0
$125$			2.56925e8i			1.05237i
$126$		0			0
$127$	1.32804e8			0.510502			0.255251	−	0.966875i	$-0.417842\pi$
$127$	1.32804e8			0.510502			0.255251	+	0.966875i	$0.417842\pi$
$128$		0			0
$129$	1.62405e8	−	9.37646e7i	0.586463	−	0.338595i
$130$		0			0
$131$	−2.33780e8	−	1.34973e8i	−0.793821	−	0.458313i	0.0474851	−	0.998872i	$-0.484879\pi$
$131$	−2.33780e8	−	1.34973e8i	−0.793821	−	0.458313i	−0.841306	+	0.540559i	$0.818213\pi$
$132$		0			0
$133$	−3.79312e7	+	7.13360e7i	−0.121224	+	0.227983i
$134$		0			0
$135$	2.21024e7	−	3.82825e7i	0.0665434	−	0.115257i
$136$		0			0
$137$	−2.88322e8	−	4.99388e8i	−0.818456	−	1.41761i	−0.906820	−	0.421519i	$-0.861497\pi$
$137$	−2.88322e8	−	4.99388e8i	−0.818456	−	1.41761i	0.0883640	−	0.996088i	$-0.471836\pi$
$138$		0			0
$139$			1.98197e8i			0.530930i	0.964120	+	0.265465i	$0.0855255\pi$
$139$			1.98197e8i			0.530930i	−0.964120	+	0.265465i	$0.914475\pi$
$140$		0			0
$141$	2.31891e8			0.586689
$142$		0			0
$143$	−3.85409e7	+	2.22516e7i	−0.0921674	+	0.0532129i
$144$		0			0
$145$	3.40016e8	+	1.96308e8i	0.769179	+	0.444086i
$146$		0			0
$147$	2.23507e8	−	1.50749e8i	0.478653	−	0.322838i
$148$		0			0
$149$	−1.00595e8	+	1.74236e8i	−0.204095	+	0.353504i	−0.949844	−	0.312724i	$-0.898759\pi$
$149$	−1.00595e8	+	1.74236e8i	−0.204095	+	0.353504i	0.745749	+	0.666227i	$0.232092\pi$
$150$		0			0
$151$	−4.83143e8	−	8.36828e8i	−0.929326	−	1.60964i	−0.784452	−	0.620190i	$-0.787056\pi$
$151$	−4.83143e8	−	8.36828e8i	−0.929326	−	1.60964i	−0.144874	−	0.989450i	$-0.546278\pi$
$152$		0			0
$153$			5.38903e7i			0.0983432i
$154$		0			0
$155$	−4.62031e8			−0.800468
$156$		0			0
$157$	−3.13710e8	+	1.81120e8i	−0.516332	+	0.298105i	−0.735433	−	0.677598i	$-0.763021\pi$
$157$	−3.13710e8	+	1.81120e8i	−0.516332	+	0.298105i	0.219100	+	0.975702i	$0.429688\pi$
$158$		0			0
$159$	3.23109e8	+	1.86547e8i	0.505546	+	0.291877i
$160$		0			0
$161$	−4.78705e8	−	2.54540e8i	−0.712467	−	0.378837i
$162$		0			0
$163$	−3.14898e7	+	5.45419e7i	−0.0446087	+	0.0772645i	−0.887468	−	0.460870i	$-0.847537\pi$
$163$	−3.14898e7	+	5.45419e7i	−0.0446087	+	0.0772645i	0.842859	+	0.538135i	$0.180871\pi$
$164$		0			0
$165$	1.39513e8	+	2.41644e8i	0.188226	+	0.326017i
$166$		0			0
$167$		−	9.92784e8i		−	1.27641i	−0.769868	−	0.638203i	$-0.779678\pi$
$167$		−	9.92784e8i		−	1.27641i	0.769868	−	0.638203i	$-0.220322\pi$
$168$		0			0
$169$	8.05338e8			0.987260
$170$		0			0
$171$	6.37329e7	−	3.67962e7i	0.0745383	−	0.0430347i
$172$		0			0
$173$	−1.20331e9	−	6.94731e8i	−1.34336	−	0.775590i	−0.356062	−	0.934462i	$-0.615881\pi$
$173$	−1.20331e9	−	6.94731e8i	−1.34336	−	0.775590i	−0.987299	+	0.158872i	$0.949214\pi$
$174$		0			0
$175$	4.15009e8	−	2.59319e8i	0.442492	−	0.276492i
$176$		0			0
$177$	−1.26828e8	+	2.19673e8i	−0.129218	+	0.223812i
$178$		0			0
$179$	−4.36947e8	−	7.56814e8i	−0.425614	−	0.737186i	0.570863	−	0.821045i	$-0.306609\pi$
$179$	−4.36947e8	−	7.56814e8i	−0.425614	−	0.737186i	−0.996478	+	0.0838594i	$0.973275\pi$
$180$		0			0
$181$			3.49705e8i			0.325827i	0.986640	+	0.162913i	$0.0520891\pi$
$181$			3.49705e8i			0.325827i	−0.986640	+	0.162913i	$0.947911\pi$
$182$		0			0
$183$	−9.96020e7			−0.0888104
$184$		0			0
$185$	−7.33201e8	+	4.23314e8i	−0.625945	+	0.361390i
$186$		0			0
$187$	−2.94589e8	−	1.70081e8i	−0.240907	−	0.139088i
$188$		0			0
$189$	−2.45415e8	+	8.56693e6i	−0.192333	+	0.00671395i
$190$		0			0
$191$	2.50191e8	−	4.33344e8i	0.187992	−	0.325611i	−0.756589	−	0.653891i	$-0.773136\pi$
$191$	2.50191e8	−	4.33344e8i	0.187992	−	0.325611i	0.944580	+	0.328280i	$0.106469\pi$
$192$		0			0
$193$	−2.34010e8	−	4.05317e8i	−0.168657	−	0.292123i	0.769291	−	0.638899i	$-0.220610\pi$
$193$	−2.34010e8	−	4.05317e8i	−0.168657	−	0.292123i	−0.937948	+	0.346776i	$0.887276\pi$
$194$		0			0
$195$			6.51610e7i			0.0450660i
$196$		0			0
$197$	−2.45218e9			−1.62812			−0.814062	−	0.580779i	$-0.802748\pi$
$197$	−2.45218e9			−1.62812			−0.814062	+	0.580779i	$0.802748\pi$
$198$		0			0
$199$	8.11486e8	−	4.68511e8i	0.517450	−	0.298750i	−0.218441	−	0.975850i	$-0.570097\pi$
$199$	8.11486e8	−	4.68511e8i	0.517450	−	0.298750i	0.735891	+	0.677100i	$0.236764\pi$
$200$		0			0
$201$	7.72551e8	+	4.46033e8i	0.473307	+	0.273264i
$202$		0			0
$203$	−7.60894e7	−	2.17971e9i	−0.0448064	−	1.28356i
$204$		0			0
$205$	−6.26794e8	+	1.08564e9i	−0.354903	+	0.614709i
$206$		0			0
$207$	2.46924e8	+	4.27685e8i	0.134487	+	0.232939i
$208$		0			0
$209$			4.64525e8i			0.243458i
$210$		0			0
$211$	1.15290e9			0.581648			0.290824	−	0.956777i	$-0.406071\pi$
$211$	1.15290e9			0.581648			0.290824	+	0.956777i	$0.406071\pi$
$212$		0			0
$213$	8.45181e8	−	4.87966e8i	0.410612	−	0.237067i
$214$		0			0
$215$	1.50097e9	+	8.66586e8i	0.702455	+	0.405563i
$216$		0			0
$217$	1.36009e9	+	2.17666e9i	0.613377	+	0.981637i
$218$		0			0
$219$	2.60883e8	−	4.51863e8i	0.113415	−	0.196440i
$220$		0			0
$221$	−3.97190e7	−	6.87953e7i	−0.0166506	−	0.0288396i
$222$		0			0
$223$		−	4.36603e9i		−	1.76550i	−0.469844	−	0.882749i	$-0.655690\pi$
$223$		−	4.36603e9i		−	1.76550i	0.469844	−	0.882749i	$-0.344310\pi$
$224$		0			0
$225$	−4.45749e8			−0.173924
$226$		0			0
$227$	1.97909e7	−	1.14263e7i	0.00745352	−	0.00430329i	−0.496269	−	0.868169i	$-0.665297\pi$
$227$	1.97909e7	−	1.14263e7i	0.00745352	−	0.00430329i	0.503722	+	0.863866i	$0.331964\pi$
$228$		0			0
$229$	2.21311e9	+	1.27774e9i	0.804749	+	0.464622i	0.845129	−	0.534563i	$-0.179524\pi$
$229$	2.21311e9	+	1.27774e9i	0.804749	+	0.464622i	−0.0403802	+	0.999184i	$0.512857\pi$
$230$		0			0
$231$	7.27714e8	−	1.36859e9i	0.255572	−	0.480646i
$232$		0			0
$233$	−2.38427e9	+	4.12968e9i	−0.808970	+	1.40118i	0.104608	+	0.994513i	$0.466641\pi$
$233$	−2.38427e9	+	4.12968e9i	−0.808970	+	1.40118i	−0.913578	+	0.406663i	$0.866692\pi$
$234$		0			0
$235$	1.07159e9	+	1.85604e9i	0.351362	+	0.608577i
$236$		0			0
$237$			1.13647e9i			0.360219i
$238$		0			0
$239$	−1.03549e9			−0.317360			−0.158680	−	0.987330i	$-0.550724\pi$
$239$	−1.03549e9			−0.317360			−0.158680	+	0.987330i	$0.550724\pi$
$240$		0			0
$241$	−3.22795e9	+	1.86366e9i	−0.956884	+	0.552457i	−0.895213	−	0.445639i	$-0.852976\pi$
$241$	−3.22795e9	+	1.86366e9i	−0.956884	+	0.552457i	−0.0616712	+	0.998097i	$0.519643\pi$
$242$		0			0
$243$	1.93710e8	+	1.11839e8i	0.0555556	+	0.0320750i
$244$		0			0
$245$	2.23942e9	+	1.09231e9i	0.621544	+	0.303167i
$246$		0			0
$247$	−5.42401e7	+	9.39467e7i	−0.0145725	+	0.0252402i
$248$		0			0
$249$	9.17099e8	+	1.58846e9i	0.238572	+	0.413218i
$250$		0			0
$251$		−	6.14337e8i		−	0.154779i	−0.997001	−	0.0773894i	$-0.975342\pi$
$251$		−	6.14337e8i		−	0.154779i	0.997001	−	0.0773894i	$-0.0246585\pi$
$252$		0			0
$253$	−3.11723e9			−0.760828
$254$		0			0
$255$	−4.31334e8	+	2.49031e8i	−0.102012	+	0.0588969i
$256$		0			0
$257$	5.99301e9	+	3.46007e9i	1.37377	+	0.793144i	0.991400	−	0.130868i	$-0.0417763\pi$
$257$	5.99301e9	+	3.46007e9i	1.37377	+	0.793144i	0.382365	+	0.924011i	$0.375110\pi$
$258$		0			0
$259$	4.15260e9	+	2.20804e9i	0.922828	+	0.490691i
$260$		0			0
$261$	−9.93324e8	+	1.72049e9i	−0.214057	+	0.370757i
$262$		0			0
$263$	1.64705e9	+	2.85277e9i	0.344257	+	0.596271i	0.985219	−	0.171302i	$-0.0547974\pi$
$263$	1.64705e9	+	2.85277e9i	0.344257	+	0.596271i	−0.640961	+	0.767573i	$0.721464\pi$
$264$		0			0
$265$			3.44819e9i			0.699210i
$266$		0			0
$267$	−5.51757e8			−0.108568
$268$		0			0
$269$	7.32809e8	−	4.23087e8i	0.139953	−	0.0808018i	−0.428389	−	0.903595i	$-0.640919\pi$
$269$	7.32809e8	−	4.23087e8i	0.139953	−	0.0808018i	0.568341	+	0.822793i	$0.307585\pi$
$270$		0			0
$271$	5.00561e9	+	2.88999e9i	0.928067	+	0.535820i	0.886200	−	0.463303i	$-0.153336\pi$
$271$	5.00561e9	+	2.88999e9i	0.928067	+	0.535820i	0.0418675	+	0.999123i	$0.486669\pi$
$272$		0			0
$273$	3.06978e8	−	1.91816e8i	0.0552658	−	0.0345329i
$274$		0			0
$275$	1.40681e9	−	2.43667e9i	0.245983	−	0.426055i
$276$		0			0
$277$	−4.52504e9	−	7.83759e9i	−0.768604	−	1.33126i	−0.938320	−	0.345769i	$-0.887618\pi$
$277$	−4.52504e9	−	7.83759e9i	−0.768604	−	1.33126i	0.169715	−	0.985493i	$-0.445715\pi$
$278$		0			0
$279$		−	2.33788e9i		−	0.385839i
$280$		0			0
$281$	3.48332e9			0.558687			0.279343	−	0.960191i	$-0.409883\pi$
$281$	3.48332e9			0.558687			0.279343	+	0.960191i	$0.409883\pi$
$282$		0			0
$283$	−5.09155e9	+	2.93961e9i	−0.793788	+	0.458294i	−0.841294	−	0.540577i	$-0.818206\pi$
$283$	−5.09155e9	+	2.93961e9i	−0.793788	+	0.458294i	0.0475064	+	0.998871i	$0.484873\pi$
$284$		0			0
$285$	5.89029e8	+	3.40076e8i	0.0892806	+	0.0515462i
$286$		0			0
$287$	6.95962e9	−	2.42946e8i	1.02579	−	0.0358082i
$288$		0			0
$289$	−3.18428e9	+	5.51534e9i	−0.456479	+	0.790644i
$290$		0			0
$291$	−1.82542e9	−	3.16172e9i	−0.254560	−	0.440911i
$292$		0			0
$293$		−	5.84504e9i		−	0.793080i	−0.918017	−	0.396540i	$-0.870211\pi$
$293$		−	5.84504e9i		−	0.793080i	0.918017	−	0.396540i	$-0.129789\pi$
$294$		0			0
$295$	−2.34433e9			−0.309550
$296$		0			0
$297$	−1.22272e9	+	7.05940e8i	−0.157146	+	0.0907281i
$298$		0			0
$299$	−6.30436e8	−	3.63983e8i	−0.0788781	−	0.0455403i
$300$		0			0
$301$	−3.35890e8	−	9.62215e9i	−0.0409195	−	1.17221i
$302$		0			0
$303$	−8.31400e8	+	1.44003e9i	−0.0986370	+	0.170844i
$304$		0			0
$305$	−4.60268e8	−	7.97207e8i	−0.0531877	−	0.0921238i
$306$		0			0
$307$		−	9.48408e9i		−	1.06768i	−0.845585	−	0.533841i	$-0.820748\pi$
$307$		−	9.48408e9i		−	1.06768i	0.845585	−	0.533841i	$-0.179252\pi$
$308$		0			0
$309$	−9.90634e9			−1.08662
$310$		0			0
$311$	−3.11264e9	+	1.79708e9i	−0.332726	+	0.192099i	−0.657051	−	0.753846i	$-0.728196\pi$
$311$	−3.11264e9	+	1.79708e9i	−0.332726	+	0.192099i	0.324325	+	0.945946i	$0.394863\pi$
$312$		0			0
$313$	7.72906e8	+	4.46237e8i	0.0805284	+	0.0464931i	0.539724	−	0.841842i	$-0.318529\pi$
$313$	7.72906e8	+	4.46237e8i	0.0805284	+	0.0464931i	−0.459195	+	0.888335i	$0.651862\pi$
$314$		0			0
$315$	−1.20265e9	−	1.92469e9i	−0.122151	−	0.195488i
$316$		0			0
$317$	−6.15360e9	+	1.06583e10i	−0.609386	+	1.05549i	0.381956	+	0.924180i	$0.375250\pi$
$317$	−6.15360e9	+	1.06583e10i	−0.609386	+	1.05549i	−0.991342	+	0.131306i	$0.958083\pi$
$318$		0			0
$319$	−6.26998e9	−	1.08599e10i	−0.605486	−	1.04873i
$320$		0			0
$321$		−	6.16415e9i		−	0.580568i
$322$		0			0
$323$	−8.29174e8			−0.0761791
$324$		0			0
$325$	5.69035e8	−	3.28532e8i	0.0510041	−	0.0294472i
$326$		0			0
$327$	−7.53021e9	−	4.34757e9i	−0.658591	−	0.380238i
$328$		0			0
$329$	5.58948e9	−	1.05120e10i	0.477076	−	0.897223i
$330$		0			0
$331$	7.20962e8	−	1.24874e9i	0.0600621	−	0.104031i	−0.834431	−	0.551113i	$-0.814203\pi$
$331$	7.20962e8	−	1.24874e9i	0.0600621	−	0.104031i	0.894493	+	0.447082i	$0.147537\pi$
$332$		0			0
$333$	−2.14198e9	−	3.71001e9i	−0.174196	−	0.301716i
$334$		0			0
$335$			8.24460e9i			0.654621i
$336$		0			0
$337$	−2.10452e9			−0.163168			−0.0815838	−	0.996666i	$-0.525998\pi$
$337$	−2.10452e9			−0.163168			−0.0815838	+	0.996666i	$0.525998\pi$
$338$		0			0
$339$	1.96722e9	−	1.13577e9i	0.148955	−	0.0859990i
$340$		0			0
$341$	1.27799e10	+	7.37850e9i	0.945173	+	0.545696i
$342$		0			0
$343$	−1.44628e9	−	1.37655e10i	−0.104490	−	0.994526i
$344$		0			0
$345$	−2.28210e9	+	3.95272e9i	−0.161087	+	0.279010i
$346$		0			0
$347$	−3.19525e9	−	5.53434e9i	−0.220388	−	0.381723i	0.734538	−	0.678567i	$-0.237399\pi$
$347$	−3.19525e9	−	5.53434e9i	−0.220388	−	0.381723i	−0.954926	+	0.296845i	$0.904066\pi$
$348$		0			0
$349$			1.17252e9i			0.0790346i	0.999219	+	0.0395173i	$0.0125820\pi$
$349$			1.17252e9i			0.0790346i	−0.999219	+	0.0395173i	$0.987418\pi$
$350$		0			0
$351$	−3.29716e8			−0.0217226
$352$		0			0
$353$	1.60024e10	−	9.23900e9i	1.03059	−	0.595013i	0.113438	−	0.993545i	$-0.463814\pi$
$353$	1.60024e10	−	9.23900e9i	1.03059	−	0.595013i	0.917154	+	0.398532i	$0.130480\pi$
$354$		0			0
$355$	7.81128e9	+	4.50985e9i	0.491823	+	0.283954i
$356$		0			0
$357$	2.44292e9	+	1.29897e9i	0.150396	+	0.0799696i
$358$		0			0
$359$	−1.23847e10	+	2.14508e10i	−0.745600	+	1.29142i	0.204314	+	0.978905i	$0.434504\pi$
$359$	−1.23847e10	+	2.14508e10i	−0.745600	+	1.29142i	−0.949914	+	0.312512i	$0.898830\pi$
$360$		0			0
$361$	−7.92562e9	−	1.37276e10i	−0.466664	−	0.808286i
$362$		0			0
$363$			1.11261e9i			0.0640789i
$364$		0			0
$365$	4.82224e9			0.271693
$366$		0			0
$367$	−2.07383e10	+	1.19733e10i	−1.14316	+	0.660006i	−0.947212	−	0.320607i	$-0.896113\pi$
$367$	−2.07383e10	+	1.19733e10i	−1.14316	+	0.660006i	−0.195952	+	0.980613i	$0.562780\pi$
$368$		0			0
$369$	−5.49335e9	−	3.17159e9i	−0.296300	−	0.171069i
$370$		0			0
$371$	1.62446e10	−	1.01505e10i	0.857461	−	0.535786i
$372$		0			0
$373$	−1.13190e10	+	1.96050e10i	−0.584751	+	1.01282i	0.410155	+	0.912016i	$0.365475\pi$
$373$	−1.13190e10	+	1.96050e10i	−0.584751	+	1.01282i	−0.994906	+	0.100803i	$0.967859\pi$
$374$		0			0
$375$	−6.00760e9	−	1.04055e10i	−0.303792	−	0.526183i
$376$		0			0
$377$		−	2.92845e9i		−	0.144968i
$378$		0			0
$379$	−3.53911e10			−1.71529			−0.857644	−	0.514245i	$-0.828072\pi$
$379$	−3.53911e10			−1.71529			−0.857644	+	0.514245i	$0.828072\pi$
$380$		0			0
$381$	−5.37857e9	+	3.10532e9i	−0.255251	+	0.147369i
$382$		0			0
$383$	3.67174e10	+	2.11988e10i	1.70639	+	0.985183i	0.938947	+	0.344061i	$0.111803\pi$
$383$	3.67174e10	+	2.11988e10i	1.70639	+	0.985183i	0.767439	+	0.641122i	$0.221531\pi$
$384$		0			0
$385$	1.43169e10	−	4.99774e8i	0.651638	−	0.0227473i
$386$		0			0
$387$	−4.38494e9	+	7.59493e9i	−0.195488	+	0.338595i
$388$		0			0
$389$	−1.82067e10	−	3.15350e10i	−0.795121	−	1.37719i	−0.922762	−	0.385370i	$-0.874074\pi$
$389$	−1.82067e10	−	3.15350e10i	−0.795121	−	1.37719i	0.127641	−	0.991820i	$-0.459259\pi$
$390$		0			0
$391$		−	5.56424e9i		−	0.238067i
$392$		0			0
$393$	1.26241e10			0.529214
$394$		0			0
$395$	−9.09626e9	+	5.25173e9i	−0.373658	+	0.215732i
$396$		0			0
$397$	9.66830e9	+	5.58199e9i	0.389214	+	0.224713i	0.681819	−	0.731521i	$-0.261189\pi$
$397$	9.66830e9	+	5.58199e9i	0.389214	+	0.224713i	−0.292606	+	0.956233i	$0.594522\pi$
$398$		0			0
$399$	−1.31814e8	−	3.77604e9i	−0.00520079	−	0.148986i
$400$		0			0
$401$	6.55936e9	−	1.13611e10i	0.253679	−	0.439384i	−0.710857	−	0.703336i	$-0.751693\pi$
$401$	6.55936e9	−	1.13611e10i	0.253679	−	0.439384i	0.964536	+	0.263952i	$0.0850261\pi$
$402$		0			0
$403$	1.72310e9	+	2.98449e9i	0.0653266	+	0.113149i
$404$		0			0
$405$			2.06726e9i			0.0768377i
$406$		0			0
$407$	2.70408e10			0.985468
$408$		0			0
$409$	2.95703e10	−	1.70724e10i	1.05673	−	0.610102i	0.132202	−	0.991223i	$-0.457795\pi$
$409$	2.95703e10	−	1.70724e10i	1.05673	−	0.610102i	0.924525	+	0.381121i	$0.124462\pi$
$410$		0			0
$411$	2.33541e10	+	1.34835e10i	0.818456	+	0.472536i
$412$		0			0
$413$	6.90104e9	+	1.10443e10i	0.237200	+	0.379610i
$414$		0			0
$415$	−8.47596e9	+	1.46808e10i	−0.285757	+	0.494945i
$416$		0			0
$417$	−4.63437e9	−	8.02697e9i	−0.153266	−	0.265465i
$418$		0			0
$419$			3.49852e10i			1.13509i	0.823344	+	0.567543i	$0.192106\pi$
$419$			3.49852e10i			1.13509i	−0.823344	+	0.567543i	$0.807894\pi$
$420$		0			0
$421$	1.41410e10			0.450146			0.225073	−	0.974342i	$-0.427738\pi$
$421$	1.41410e10			0.450146			0.225073	+	0.974342i	$0.427738\pi$
$422$		0			0
$423$	−9.39159e9	+	5.42224e9i	−0.293344	+	0.169362i
$424$		0			0
$425$	4.34944e9	+	2.51115e9i	0.133315	+	0.0769693i
$426$		0			0
$427$	−2.40080e9	+	4.51510e9i	−0.0722178	+	0.135818i
$428$		0			0
$429$	1.04060e9	−	1.80238e9i	0.0307225	−	0.0532129i
$430$		0			0
$431$	7.04983e9	+	1.22107e10i	0.204301	+	0.353859i	0.949910	−	0.312525i	$-0.101175\pi$
$431$	7.04983e9	+	1.22107e10i	0.204301	+	0.353859i	−0.745609	+	0.666384i	$0.767841\pi$
$432$		0			0
$433$			1.05233e10i			0.299365i	0.988734	+	0.149683i	$0.0478252\pi$
$433$			1.05233e10i			0.299365i	−0.988734	+	0.149683i	$0.952175\pi$
$434$		0			0
$435$	−1.83609e10			−0.512786
$436$		0			0
$437$	−6.58050e9	+	3.79926e9i	−0.180440	+	0.104177i
$438$		0			0
$439$	5.71624e10	+	3.30027e10i	1.53905	+	0.888571i	0.998895	+	0.0470049i	$0.0149676\pi$
$439$	5.71624e10	+	3.30027e10i	1.53905	+	0.888571i	0.540155	+	0.841566i	$0.318366\pi$
$440$		0			0
$441$	−5.52711e9	+	1.13315e10i	−0.146131	+	0.299594i
$442$		0			0
$443$	−3.33675e10	+	5.77942e10i	−0.866380	+	1.50061i	−0.000709897		1.00000i	$0.500226\pi$
$443$	−3.33675e10	+	5.77942e10i	−0.866380	+	1.50061i	−0.865670	+	0.500615i	$0.833107\pi$
$444$		0			0
$445$	−2.54971e9	−	4.41622e9i	−0.0650205	−	0.112619i
$446$		0			0
$447$		−	9.40877e9i		−	0.235669i
$448$		0			0
$449$	−4.05035e10			−0.996569			−0.498285	−	0.867014i	$-0.666037\pi$
$449$	−4.05035e10			−0.996569			−0.498285	+	0.867014i	$0.666037\pi$
$450$		0			0
$451$	3.46747e10	−	2.00195e10i	0.838121	−	0.483889i
$452$		0			0
$453$	3.91346e10	+	2.25944e10i	0.929326	+	0.536547i
$454$		0			0
$455$	2.95384e9	+	1.57064e9i	0.0689195	+	0.0366463i
$456$		0			0
$457$	−3.16521e10	+	5.48230e10i	−0.725667	+	1.25689i	0.233032	+	0.972469i	$0.425135\pi$
$457$	−3.16521e10	+	5.48230e10i	−0.725667	+	1.25689i	−0.958699	+	0.284423i	$0.908198\pi$
$458$		0			0
$459$	−1.26010e9	−	2.18256e9i	−0.0283892	−	0.0491716i
$460$		0			0
$461$		−	4.09845e10i		−	0.907436i	−0.891145	−	0.453718i	$-0.850097\pi$
$461$		−	4.09845e10i		−	0.907436i	0.891145	−	0.453718i	$-0.149903\pi$
$462$		0			0
$463$	−1.54289e10			−0.335745			−0.167873	−	0.985809i	$-0.553690\pi$
$463$	−1.54289e10			−0.335745			−0.167873	+	0.985809i	$0.553690\pi$
$464$		0			0
$465$	1.87122e10	−	1.08035e10i	0.400234	−	0.231075i
$466$		0			0
$467$	2.98997e10	+	1.72626e10i	0.628636	+	0.362943i	0.780224	−	0.625501i	$-0.215105\pi$
$467$	2.98997e10	+	1.72626e10i	0.628636	+	0.362943i	−0.151588	+	0.988444i	$0.548439\pi$
$468$		0			0
$469$	3.88408e10	−	2.42697e10i	0.802781	−	0.501619i
$470$		0			0
$471$	8.47016e9	−	1.46707e10i	0.172111	−	0.298105i
$472$		0			0
$473$	−2.76783e10	−	4.79402e10i	−0.552961	−	0.957757i
$474$		0			0
$475$		−	6.85845e9i		−	0.134726i
$476$		0			0
$477$	−1.74479e10			−0.337031
$478$		0			0
$479$	−5.19585e10	+	2.99983e10i	−0.986994	+	0.569842i	−0.904375	−	0.426740i	$-0.859662\pi$
$479$	−5.19585e10	+	2.99983e10i	−0.986994	+	0.569842i	−0.0826199	+	0.996581i	$0.526329\pi$
$480$		0			0
$481$	5.46881e9	+	3.15742e9i	0.102167	+	0.0589864i
$482$		0			0
$483$	2.53394e10	−	8.84546e8i	0.465594	−	0.0162529i
$484$		0			0
$485$	1.68708e10	−	2.92210e10i	0.304907	−	0.528115i
$486$		0			0
$487$	7.27140e8	+	1.25944e9i	0.0129271	+	0.0223904i	0.872417	−	0.488763i	$-0.162552\pi$
$487$	7.27140e8	+	1.25944e9i	0.0129271	+	0.0223904i	−0.859489	+	0.511153i	$0.829218\pi$
$488$		0			0
$489$		−	2.94526e9i		−	0.0515097i
$490$		0			0
$491$	−6.42504e10			−1.10548			−0.552739	−	0.833355i	$-0.686417\pi$
$491$	−6.42504e10			−1.10548			−0.552739	+	0.833355i	$0.686417\pi$
$492$		0			0
$493$	1.93849e10	−	1.11919e10i	0.328153	−	0.189459i
$494$		0			0
$495$	−1.13006e10	−	6.52440e9i	−0.188226	−	0.108672i
$496$		0			0
$497$	−1.74802e9	−	5.00752e10i	−0.0286498	−	0.820724i
$498$		0			0
$499$	−1.47773e10	+	2.55951e10i	−0.238338	+	0.412813i	−0.960238	−	0.279184i	$-0.909936\pi$
$499$	−1.47773e10	+	2.55951e10i	−0.238338	+	0.412813i	0.721900	+	0.691998i	$0.243269\pi$
$500$		0			0
$501$	2.32140e10	+	4.02077e10i	0.368467	+	0.638203i
$502$		0			0
$503$		−	3.65527e10i		−	0.571015i	−0.958376	−	0.285508i	$-0.907838\pi$
$503$		−	3.65527e10i		−	0.571015i	0.958376	−	0.285508i	$-0.0921622\pi$
$504$		0			0
$505$	−1.53678e10			−0.236291
$506$		0			0
$507$	−3.26162e10	+	1.88310e10i	−0.493630	+	0.284997i
$508$		0			0
$509$	−6.42278e10	−	3.70819e10i	−0.956867	−	0.552447i	−0.0616596	−	0.998097i	$-0.519639\pi$
$509$	−6.42278e10	−	3.70819e10i	−0.956867	−	0.552447i	−0.895207	+	0.445650i	$0.852973\pi$
$510$		0			0
$511$	−1.41953e10	−	2.27179e10i	−0.208191	−	0.333184i
$512$		0			0
$513$	−1.72079e9	+	2.98049e9i	−0.0248461	+	0.0430347i
$514$		0			0
$515$	−4.57779e10	−	7.92897e10i	−0.650769	−	1.12717i
$516$		0			0
$517$		−	6.84517e10i		−	0.958125i
$518$		0			0
$519$	6.49787e10			0.895574
$520$		0			0
$521$	1.33656e10	−	7.71665e9i	0.181401	−	0.104732i	−0.406550	−	0.913629i	$-0.633268\pi$
$521$	1.33656e10	−	7.71665e9i	0.181401	−	0.104732i	0.587951	+	0.808897i	$0.299935\pi$
$522$		0			0
$523$	−3.44919e10	−	1.99139e10i	−0.461010	−	0.266164i	0.251459	−	0.967868i	$-0.419090\pi$
$523$	−3.44919e10	−	1.99139e10i	−0.461010	−	0.266164i	−0.712469	+	0.701704i	$0.752423\pi$
$524$		0			0
$525$	−1.07443e10	+	2.02065e10i	−0.141430	+	0.265982i
$526$		0			0
$527$	−1.31706e10	+	2.28121e10i	−0.170751	+	0.295749i
$528$		0			0
$529$	1.36603e10	+	2.36603e10i	0.174437	+	0.302133i
$530$		0			0
$531$		−	1.18623e10i		−	0.149208i
$532$		0			0
$533$	9.35027e9			0.115855
$534$		0			0
$535$	4.93374e10	−	2.84850e10i	0.602228	−	0.347697i
$536$		0			0
$537$	3.53927e10	+	2.04340e10i	0.425614	+	0.245729i
$538$		0			0
$539$	−4.44994e10	−	6.59767e10i	−0.527229	−	0.781692i
$540$		0			0
$541$	3.80963e9	−	6.59847e9i	0.0444727	−	0.0770290i	−0.842932	−	0.538020i	$-0.819173\pi$
$541$	3.80963e9	−	6.59847e9i	0.0444727	−	0.0770290i	0.887405	+	0.460991i	$0.152506\pi$
$542$		0			0
$543$	−8.17703e9	−	1.41630e10i	−0.0940582	−	0.162913i
$544$		0			0
$545$		−	8.03617e10i		−	0.910884i
$546$		0			0
$547$	−9.03887e10			−1.00964			−0.504818	−	0.863226i	$-0.668441\pi$
$547$	−9.03887e10			−1.00964			−0.504818	+	0.863226i	$0.668441\pi$
$548$		0			0
$549$	4.03388e9	−	2.32896e9i	0.0444052	−	0.0256373i
$550$		0			0
$551$	−2.64720e10	−	1.52836e10i	−0.287197	−	0.165814i
$552$		0			0
$553$	5.15180e10	+	2.73935e10i	0.550882	+	0.292918i
$554$		0			0
$555$	1.97964e10	−	3.42884e10i	0.208648	−	0.361390i
$556$		0			0
$557$	−4.86872e10	−	8.43287e10i	−0.505817	−	0.876101i	−0.999977	−	0.00673046i	$-0.997858\pi$
$557$	−4.86872e10	−	8.43287e10i	−0.505817	−	0.876101i	0.494160	−	0.869371i	$-0.335476\pi$
$558$		0			0
$559$		−	1.29274e10i		−	0.132393i
$560$		0			0
$561$	1.59078e10			0.160605
$562$		0			0
$563$	−1.63982e11	+	9.46751e10i	−1.63216	+	0.942329i	−0.648736	+	0.761014i	$0.724702\pi$
$563$	−1.63982e11	+	9.46751e10i	−1.63216	+	0.942329i	−0.983425	+	0.181315i	$0.941965\pi$
$564$		0			0
$565$	1.81813e10	+	1.04970e10i	0.178415	+	0.103008i
$566$		0			0
$567$	9.73898e9	−	6.08542e9i	0.0942283	−	0.0588787i
$568$		0			0
$569$	5.52884e9	−	9.57622e9i	0.0527454	−	0.0913577i	−0.838447	−	0.544983i	$-0.816536\pi$
$569$	5.52884e9	−	9.57622e9i	0.0527454	−	0.0913577i	0.891193	+	0.453625i	$0.149869\pi$
$570$		0			0
$571$	2.80453e9	+	4.85758e9i	0.0263824	+	0.0456957i	0.878915	−	0.476978i	$-0.158268\pi$
$571$	2.80453e9	+	4.85758e9i	0.0263824	+	0.0456957i	−0.852533	+	0.522674i	$0.824935\pi$
$572$		0			0
$573$			2.34006e10i			0.217074i
$574$		0			0
$575$	4.60242e10			0.421031
$576$		0			0
$577$	−1.30662e11	+	7.54375e10i	−1.17881	+	0.680587i	−0.955739	−	0.294215i	$-0.904942\pi$
$577$	−1.30662e11	+	7.54375e10i	−1.17881	+	0.680587i	−0.223072	+	0.974802i	$0.571609\pi$
$578$		0			0
$579$	1.89548e10	+	1.09435e10i	0.168657	+	0.0973742i
$580$		0			0
$581$	9.41130e10	−	3.28529e9i	0.825933	−	0.0288316i
$582$		0			0
$583$	5.50666e10	−	9.53782e10i	0.476666	−	0.825610i
$584$		0			0
$585$	−1.52364e9	−	2.63902e9i	−0.0130094	−	0.0225330i
$586$		0			0
$587$			2.18697e11i			1.84201i	0.389556	+	0.921003i	$0.372629\pi$
$587$			2.18697e11i			1.84201i	−0.389556	+	0.921003i	$0.627371\pi$
$588$		0			0
$589$	3.59715e10			0.298880
$590$		0			0
$591$	9.93132e10	−	5.73385e10i	0.814062	−	0.469999i
$592$		0			0
$593$	−1.98077e11	−	1.14360e11i	−1.60183	−	0.924816i	−0.991121	−	0.132960i	$-0.957552\pi$
$593$	−1.98077e11	−	1.14360e11i	−1.60183	−	0.924816i	−0.610708	−	0.791856i	$-0.709115\pi$
$594$		0			0
$595$	8.92094e8	+	2.55556e10i	0.00711775	+	0.203901i
$596$		0			0
$597$	−2.19101e10	+	3.79494e10i	−0.172483	+	0.298750i
$598$		0			0
$599$	8.69028e10	+	1.50520e11i	0.675036	+	1.16920i	0.976458	+	0.215705i	$0.0692051\pi$
$599$	8.69028e10	+	1.50520e11i	0.675036	+	1.16920i	−0.301423	+	0.953491i	$0.597462\pi$
$600$		0			0
$601$			3.36846e10i			0.258187i	0.991632	+	0.129093i	$0.0412067\pi$
$601$			3.36846e10i			0.258187i	−0.991632	+	0.129093i	$0.958793\pi$
$602$		0			0
$603$	−4.17178e10			−0.315538
$604$		0			0
$605$	−8.90523e9	+	5.14144e9i	−0.0664697	+	0.0383763i
$606$		0			0
$607$	9.20478e10	+	5.31438e10i	0.678045	+	0.391470i	0.799118	−	0.601174i	$-0.205300\pi$
$607$	9.20478e10	+	5.31438e10i	0.678045	+	0.391470i	−0.121073	+	0.992644i	$0.538633\pi$
$608$		0			0
$609$	5.40492e10	+	8.64993e10i	0.392934	+	0.628845i
$610$		0			0
$611$	7.99275e9	−	1.38438e10i	0.0573497	−	0.0993326i
$612$		0			0
$613$	5.34031e10	+	9.24969e10i	0.378203	+	0.655066i	0.990801	−	0.135328i	$-0.0432089\pi$
$613$	5.34031e10	+	9.24969e10i	0.378203	+	0.655066i	−0.612598	+	0.790395i	$0.709876\pi$
$614$		0			0
$615$		−	5.86245e10i		−	0.409806i
$616$		0			0
$617$	1.04431e11			0.720593			0.360297	−	0.932838i	$-0.382675\pi$
$617$	1.04431e11			0.720593			0.360297	+	0.932838i	$0.382675\pi$
$618$		0			0
$619$	−1.91873e10	+	1.10778e10i	−0.130693	+	0.0754554i	−0.563921	−	0.825829i	$-0.690708\pi$
$619$	−1.91873e10	+	1.10778e10i	−0.130693	+	0.0754554i	0.433228	+	0.901284i	$0.357374\pi$
$620$		0			0
$621$	−2.00008e10	−	1.15475e10i	−0.134487	−	0.0776463i
$622$		0			0
$623$	−1.32995e10	+	2.50119e10i	−0.0882842	+	0.166033i
$624$		0			0
$625$	1.57150e10	−	2.72192e10i	0.102990	−	0.178384i
$626$		0			0
$627$	−1.08618e10	−	1.88133e10i	−0.0702802	−	0.121729i
$628$		0			0
$629$			4.82677e10i			0.308357i
$630$		0			0
$631$	2.53563e11			1.59944			0.799722	−	0.600371i	$-0.204980\pi$
$631$	2.53563e11			1.59944			0.799722	+	0.600371i	$0.204980\pi$
$632$		0			0
$633$	−4.66923e10	+	2.69578e10i	−0.290824	+	0.167907i
$634$		0			0
$635$	−4.97095e10	−	2.86998e10i	−0.305735	−	0.176516i
$636$		0			0
$637$	−1.29591e9	−	1.85393e10i	−0.00787079	−	0.112599i
$638$		0			0
$639$	−2.28199e10	+	3.95252e10i	−0.136871	+	0.237067i
$640$		0			0
$641$	4.44352e10	+	7.69641e10i	0.263206	+	0.455886i	0.967092	−	0.254427i	$-0.0818869\pi$
$641$	4.44352e10	+	7.69641e10i	0.263206	+	0.455886i	−0.703886	+	0.710313i	$0.748554\pi$
$642$		0			0
$643$			1.06138e10i			0.0620909i	0.999518	+	0.0310455i	$0.00988367\pi$
$643$			1.06138e10i			0.0620909i	−0.999518	+	0.0310455i	$0.990116\pi$
$644$		0			0
$645$	−8.10524e10			−0.468303
$646$		0			0
$647$	2.13261e11	−	1.23126e11i	1.21701	−	0.702642i	0.252734	−	0.967536i	$-0.418670\pi$
$647$	2.13261e11	−	1.23126e11i	1.21701	−	0.702642i	0.964278	+	0.264894i	$0.0853369\pi$
$648$		0			0
$649$	6.48451e10	+	3.74383e10i	0.365509	+	0.211027i
$650$		0			0
$651$	−1.05980e11	−	5.63521e10i	−0.590063	−	0.313752i
$652$		0			0
$653$	−3.72691e10	+	6.45521e10i	−0.204973	+	0.355024i	−0.950124	−	0.311872i	$-0.899044\pi$
$653$	−3.72691e10	+	6.45521e10i	−0.204973	+	0.355024i	0.745151	+	0.666896i	$0.232377\pi$
$654$		0			0
$655$	5.83370e10	+	1.01043e11i	0.316941	+	0.548958i
$656$		0			0
$657$			2.44006e10i			0.130960i
$658$		0			0
$659$	1.29564e11			0.686975			0.343488	−	0.939157i	$-0.388392\pi$
$659$	1.29564e11			0.686975			0.343488	+	0.939157i	$0.388392\pi$
$660$		0			0
$661$	−2.62586e10	+	1.51604e10i	−0.137552	+	0.0794155i	−0.567197	−	0.823582i	$-0.691972\pi$
$661$	−2.62586e10	+	1.51604e10i	−0.137552	+	0.0794155i	0.429645	+	0.902998i	$0.358639\pi$
$662$		0			0
$663$	3.21724e9	+	1.85747e9i	0.0166506	+	0.00961320i
$664$		0			0
$665$	2.96140e10	−	1.85044e10i	0.151430	−	0.0946211i
$666$		0			0
$667$	1.02562e11	−	1.77642e11i	0.518183	−	0.897519i
$668$		0			0
$669$	1.02090e11	+	1.76824e11i	0.509655	+	0.882749i
$670$		0			0
$671$			2.94014e10i			0.145037i
$672$		0			0
$673$	−2.10696e11			−1.02706			−0.513530	−	0.858071i	$-0.671663\pi$
$673$	−2.10696e11			−1.02706			−0.513530	+	0.858071i	$0.671663\pi$
$674$		0			0
$675$	1.80528e10	−	1.04228e10i	0.0869622	−	0.0502076i
$676$		0			0
$677$	3.82781e10	+	2.20999e10i	0.182220	+	0.105205i	0.588335	−	0.808617i	$-0.299784\pi$
$677$	3.82781e10	+	2.20999e10i	0.182220	+	0.105205i	−0.406115	+	0.913822i	$0.633117\pi$
$678$		0			0
$679$	−1.87325e11	+	6.53913e9i	−0.881285	+	0.0307639i
$680$		0			0
$681$	−5.34354e8	+	9.25528e8i	−0.00248451	+	0.00430329i
$682$		0			0
$683$	6.69506e10	+	1.15962e11i	0.307660	+	0.532883i	0.977850	−	0.209307i	$-0.0671206\pi$
$683$	6.69506e10	+	1.15962e11i	0.307660	+	0.532883i	−0.670190	+	0.742190i	$0.733787\pi$
$684$		0			0
$685$			2.49232e11i			1.13199i
$686$		0			0
$687$	−1.19508e11			−0.536499
$688$		0			0
$689$	2.22736e10	−	1.28597e10i	0.0988358	−	0.0570629i
$690$		0			0
$691$	−3.36307e11	−	1.94167e11i	−1.47511	−	0.851653i	−0.475501	−	0.879715i	$-0.657733\pi$
$691$	−3.36307e11	−	1.94167e11i	−1.47511	−	0.851653i	−0.999606	+	0.0280622i	$0.991066\pi$
$692$		0			0
$693$	2.52886e9	+	7.24438e10i	0.0109646	+	0.314100i
$694$		0			0
$695$	4.28315e10	−	7.41864e10i	0.183580	−	0.317969i
$696$		0			0
$697$	3.57346e10	+	6.18942e10i	0.151411	+	0.262252i
$698$		0			0
$699$		−	2.23003e11i		−	0.934118i
$700$		0			0
$701$	1.53431e11			0.635392			0.317696	−	0.948193i	$-0.397091\pi$
$701$	1.53431e11			0.635392			0.317696	+	0.948193i	$0.397091\pi$
$702$		0			0
$703$	5.70835e10	−	3.29572e10i	0.233716	−	0.134936i
$704$		0			0
$705$	−8.67984e10	−	5.01131e10i	−0.351362	−	0.202859i
$706$		0			0
$707$	4.52386e10	+	7.23989e10i	0.181064	+	0.289771i
$708$		0			0
$709$	−5.25372e10	+	9.09971e10i	−0.207913	+	0.360116i	−0.951057	−	0.309016i	$-0.900000\pi$
$709$	−5.25372e10	+	9.09971e10i	−0.207913	+	0.360116i	0.743144	+	0.669132i	$0.233334\pi$
$710$		0			0
$711$	−2.65738e10	−	4.60272e10i	−0.103986	−	0.180109i
$712$		0			0
$713$			2.41389e11i			0.934028i
$714$		0			0
$715$	1.92348e10			0.0735976
$716$		0			0
$717$	4.19372e10	−	2.42124e10i	0.158680	−	0.0916140i
$718$		0			0
$719$	1.52234e11	+	8.78923e10i	0.569634	+	0.328878i	0.757003	−	0.653411i	$-0.226663\pi$
$719$	1.52234e11	+	8.78923e10i	0.569634	+	0.328878i	−0.187369	+	0.982290i	$0.559996\pi$
$720$		0			0
$721$	−2.38782e11	+	4.49069e11i	−0.883608	+	1.66177i
$722$		0			0
$723$	8.71548e10	−	1.50956e11i	0.318961	−	0.552457i
$724$		0			0
$725$	9.25729e10	+	1.60341e11i	0.335067	+	0.580353i
$726$		0			0
$727$		−	3.71190e11i		−	1.32880i	−0.747379	−	0.664398i	$-0.768688\pi$
$727$		−	3.71190e11i		−	1.32880i	0.747379	−	0.664398i	$-0.231312\pi$
$728$		0			0
$729$	−1.04604e10			−0.0370370
$730$		0			0
$731$	8.55730e10	−	4.94056e10i	0.299686	−	0.173024i
$732$		0			0
$733$	3.30816e10	+	1.90996e10i	0.114596	+	0.0661621i	0.556203	−	0.831047i	$-0.312258\pi$
$733$	3.30816e10	+	1.90996e10i	0.114596	+	0.0661621i	−0.441606	+	0.897209i	$0.645591\pi$
$734$		0			0
$735$	−1.16238e11	+	8.12515e9i	−0.398289	+	0.0278408i
$736$		0			0
$737$	1.31664e11	−	2.28049e11i	0.446269	−	0.772961i
$738$		0			0
$739$	−1.92789e11	−	3.33920e11i	−0.646404	−	1.11960i	−0.983975	−	0.178305i	$-0.942939\pi$
$739$	−1.92789e11	−	3.33920e11i	−0.646404	−	1.11960i	0.337571	−	0.941300i	$-0.390395\pi$
$740$		0			0
$741$		−	5.07312e9i		−	0.0168268i
$742$		0			0
$743$	5.07236e11			1.66439			0.832194	−	0.554485i	$-0.187085\pi$
$743$	5.07236e11			1.66439			0.832194	+	0.554485i	$0.187085\pi$
$744$		0			0
$745$	7.53071e10	−	4.34786e10i	0.244462	−	0.141140i
$746$		0			0
$747$	−7.42850e10	−	4.28885e10i	−0.238572	−	0.137739i
$748$		0			0
$749$	−2.79430e11	−	1.48580e11i	−0.887862	−	0.472099i
$750$		0			0
$751$	3.14299e11	−	5.44382e11i	0.988060	−	1.71137i	0.360600	−	0.932721i	$-0.382572\pi$
$751$	3.14299e11	−	5.44382e11i	0.988060	−	1.71137i	0.627460	−	0.778649i	$-0.284095\pi$
$752$		0			0
$753$	1.43648e10	+	2.48806e10i	0.0446808	+	0.0773894i
$754$		0			0
$755$			4.17641e11i			1.28533i
$756$		0			0
$757$	−3.39756e11			−1.03463			−0.517314	−	0.855796i	$-0.673068\pi$
$757$	−3.39756e11			−1.03463			−0.517314	+	0.855796i	$0.673068\pi$
$758$		0			0
$759$	1.26248e11	−	7.28892e10i	0.380414	−	0.219632i
$760$		0			0
$761$	−5.74796e11	−	3.31859e11i	−1.71386	−	0.989497i	−0.929205	−	0.369564i	$-0.879507\pi$
$761$	−5.74796e11	−	3.31859e11i	−1.71386	−	0.989497i	−0.784655	−	0.619933i	$-0.787160\pi$
$762$		0			0
$763$	−3.78589e11	+	2.36562e11i	−1.11704	+	0.697986i
$764$		0			0
$765$	1.16460e10	−	2.01715e10i	0.0340041	−	0.0588969i
$766$		0			0
$767$	8.74296e9	+	1.51432e10i	0.0252625	+	0.0437560i
$768$		0			0
$769$		−	3.99685e11i		−	1.14291i	−0.820633	−	0.571456i	$-0.806379\pi$
$769$		−	3.99685e11i		−	1.14291i	0.820633	−	0.571456i	$-0.193621\pi$
$770$		0			0
$771$	−3.23623e11			−0.915843
$772$		0			0
$773$	−8.18869e10	+	4.72774e10i	−0.229349	+	0.132415i	−0.610272	−	0.792192i	$-0.708940\pi$
$773$	−8.18869e10	+	4.72774e10i	−0.229349	+	0.132415i	0.380923	+	0.924607i	$0.375606\pi$
$774$		0			0
$775$	−1.88689e11	−	1.08940e11i	−0.523045	−	0.301980i
$776$		0			0
$777$	−2.19810e11	+	7.67312e9i	−0.603064	+	0.0210517i
$778$		0			0
$779$	4.87991e10	−	8.45226e10i	0.132514	−	0.229521i
$780$		0			0
$781$	−1.44042e11	−	2.49488e11i	−0.387155	−	0.670573i
$782$		0			0
$783$		−	9.29063e10i		−	0.247171i
$784$		0			0
$785$	1.56565e11			0.412302
$786$		0			0
$787$	−2.38365e10	+	1.37620e10i	−0.0621362	+	0.0358743i	−0.530746	−	0.847531i	$-0.678088\pi$
$787$	−2.38365e10	+	1.37620e10i	−0.0621362	+	0.0358743i	0.468610	+	0.883405i	$0.344755\pi$
$788$		0			0
$789$	−1.33411e11	−	7.70248e10i	−0.344257	−	0.198757i
$790$		0			0
$791$	−4.06864e9	−	1.16554e11i	−0.0103931	−	0.297728i
$792$		0			0
$793$	−3.43305e9	+	5.94622e9i	−0.00868135	+	0.0150365i
$794$		0			0
$795$	−8.06279e10	−	1.39652e11i	−0.201845	−	0.349605i
$796$		0			0
$797$			1.49359e11i			0.370167i	0.982723	+	0.185084i	$0.0592556\pi$
$797$			1.49359e11i			0.370167i	−0.982723	+	0.185084i	$0.940744\pi$
$798$		0			0
$799$	1.22186e11			0.299802
$800$		0			0
$801$	2.23462e10	−	1.29016e10i	0.0542841	−	0.0313409i
$802$		0			0
$803$	−1.33385e11	−	7.70099e10i	−0.320808	−	0.185219i
$804$		0			0
$805$	1.24175e11	+	1.98727e11i	0.295700	+	0.473232i
$806$		0			0
$807$	−1.97858e10	+	3.42701e10i	−0.0466509	+	0.0808018i
$808$		0			0
$809$	−3.45374e9	−	5.98206e9i	−0.00806298	−	0.0139655i	0.861966	−	0.506966i	$-0.169233\pi$
$809$	−3.45374e9	−	5.98206e9i	−0.00806298	−	0.0139655i	−0.870029	+	0.493001i	$0.835900\pi$
$810$		0			0
$811$		−	3.74372e11i		−	0.865406i	−0.901537	−	0.432703i	$-0.857560\pi$
$811$		−	3.74372e11i		−	0.865406i	0.901537	−	0.432703i	$-0.142440\pi$
$812$		0			0
$813$	−2.70303e11			−0.618712
$814$		0			0
$815$	2.35737e10	−	1.36103e10i	0.0534315	−	0.0308487i
$816$		0			0
$817$	−1.16858e11	−	6.74682e10i	−0.262284	−	0.151430i
$818$		0			0
$819$	−7.94744e9	+	1.49465e10i	−0.0176641	+	0.0332203i
$820$		0			0
$821$	4.25087e11	−	7.36272e11i	0.935632	−	1.62056i	0.162128	−	0.986770i	$-0.448164\pi$
$821$	4.25087e11	−	7.36272e11i	0.935632	−	1.62056i	0.773504	−	0.633792i	$-0.218502\pi$
$822$		0			0
$823$	1.48472e11	+	2.57161e11i	0.323627	+	0.560539i	0.981234	−	0.192823i	$-0.0617642\pi$
$823$	1.48472e11	+	2.57161e11i	0.323627	+	0.560539i	−0.657606	+	0.753362i	$0.728431\pi$
$824$		0			0
$825$			1.31580e11i			0.284037i
$826$		0			0
$827$	3.43560e11			0.734482			0.367241	−	0.930126i	$-0.380302\pi$
$827$	3.43560e11			0.734482			0.367241	+	0.930126i	$0.380302\pi$
$828$		0			0
$829$	1.65330e11	−	9.54533e10i	0.350053	−	0.202103i	−0.314656	−	0.949206i	$-0.601889\pi$
$829$	1.65330e11	−	9.54533e10i	0.350053	−	0.202103i	0.664708	+	0.747103i	$0.268556\pi$
$830$		0			0
$831$	3.66528e11	+	2.11615e11i	0.768604	+	0.443754i
$832$		0			0
$833$	1.17768e11	−	7.94312e10i	0.244595	−	0.164972i
$834$		0			0
$835$	−2.14547e11	+	3.71606e11i	−0.441343	+	0.764428i
$836$		0			0
$837$	5.46659e10	+	9.46842e10i	0.111382	+	0.192919i
$838$		0			0
$839$			3.92637e11i			0.792399i	0.918164	+	0.396199i	$0.129671\pi$
$839$			3.92637e11i			0.792399i	−0.918164	+	0.396199i	$0.870329\pi$
$840$		0			0
$841$	3.24924e11			0.649529
$842$		0			0
$843$	−1.41075e11	+	8.14495e10i	−0.279343	+	0.161279i
$844$		0			0
$845$	−3.01443e11	−	1.74038e11i	−0.591261	−	0.341364i
$846$		0			0
$847$	5.04361e10	+	2.68182e10i	0.0979959	+	0.0521070i
$848$		0			0
$849$	1.37472e11	−	2.38108e11i	0.264596	−	0.458294i
$850$		0			0
$851$	2.21162e11	+	3.83063e11i	0.421688	+	0.730386i
$852$		0			0
$853$			8.40881e11i			1.58832i	0.607707	+	0.794161i	$0.292089\pi$
$853$			8.40881e11i			1.58832i	−0.607707	+	0.794161i	$0.707911\pi$
$854$		0			0
$855$	−3.18075e10			−0.0595204
$856$		0			0
$857$	6.17115e11	−	3.56292e11i	1.14404	−	0.660514i	0.196615	−	0.980481i	$-0.437005\pi$
$857$	6.17115e11	−	3.56292e11i	1.14404	−	0.660514i	0.947429	+	0.319966i	$0.103672\pi$
$858$		0			0
$859$	−4.48728e11	−	2.59073e11i	−0.824158	−	0.475828i	0.0276900	−	0.999617i	$-0.491185\pi$
$859$	−4.48728e11	−	2.59073e11i	−0.824158	−	0.475828i	−0.851848	+	0.523789i	$0.824518\pi$
$860$		0			0
$861$	−2.76184e11	+	1.72574e11i	−0.502557	+	0.314024i
$862$		0			0
$863$	−3.65673e11	+	6.33365e11i	−0.659250	+	1.14185i	0.321560	+	0.946889i	$0.395793\pi$
$863$	−3.65673e11	+	6.33365e11i	−0.659250	+	1.14185i	−0.980810	+	0.194965i	$0.937541\pi$
$864$		0			0
$865$	3.00271e11	+	5.20085e11i	0.536351	+	0.928987i
$866$		0			0
$867$		−	2.97829e11i		−	0.527096i
$868$		0			0
$869$	3.35475e11			0.588275
$870$		0			0
$871$	5.32561e10	−	3.07474e10i	0.0925330	−	0.0534240i
$872$		0			0
$873$	1.47859e11	+	8.53663e10i	0.254560	+	0.146970i
$874$		0			0
$875$	−6.16502e11	+	2.15208e10i	−1.05173	+	0.0367136i
$876$		0			0
$877$	3.40974e11	−	5.90584e11i	0.576398	−	0.998351i	−0.419490	−	0.907760i	$-0.637791\pi$
$877$	3.40974e11	−	5.90584e11i	0.576398	−	0.998351i	0.995888	−	0.0905912i	$-0.0288757\pi$
$878$		0			0
$879$	1.36673e11	+	2.36724e11i	0.228943	+	0.396540i
$880$		0			0
$881$			1.12611e12i			1.86929i	0.355580	+	0.934646i	$0.384283\pi$
$881$			1.12611e12i			1.86929i	−0.355580	+	0.934646i	$0.615717\pi$
$882$		0			0
$883$	−2.79315e10			−0.0459464			−0.0229732	−	0.999736i	$-0.507313\pi$
$883$	−2.79315e10			−0.0459464			−0.0229732	+	0.999736i	$0.507313\pi$
$884$		0			0
$885$	9.49454e10	−	5.48168e10i	0.154775	−	0.0893594i
$886$		0			0
$887$	−3.70745e11	−	2.14050e11i	−0.598937	−	0.345796i	0.169687	−	0.985498i	$-0.445724\pi$
$887$	−3.70745e11	−	2.14050e11i	−0.598937	−	0.345796i	−0.768623	+	0.639702i	$0.779058\pi$
$888$		0			0
$889$	1.11241e10	+	3.18669e11i	0.0178097	+	0.510191i
$890$		0			0
$891$	3.30135e10	−	5.71811e10i	0.0523819	−	0.0907281i
$892$		0			0
$893$	−8.34284e10	−	1.44502e11i	−0.131192	−	0.227232i
$894$		0			0
$895$			3.77707e11i			0.588658i
$896$		0			0
$897$	3.40436e10			0.0525854
$898$		0			0
$899$	−8.40962e11	+	4.85530e11i	−1.28747	+	0.743322i
$900$		0			0
$901$	1.70250e11	+	9.82937e10i	0.258337	+	0.149151i
$902$		0			0
$903$	2.38595e11	+	3.81843e11i	0.358848	+	0.574294i
$904$		0			0
$905$	7.55733e10	−	1.30897e11i	0.112661	−	0.195135i
$906$		0			0
$907$	3.94184e11	+	6.82746e11i	0.582465	+	1.00886i	0.995186	+	0.0980011i	$0.0312449\pi$
$907$	3.94184e11	+	6.82746e11i	0.582465	+	1.00886i	−0.412722	+	0.910857i	$0.635422\pi$
$908$		0			0
$909$		−	7.77615e10i		−	0.113896i
$910$		0			0
$911$	8.76508e11			1.27257			0.636286	−	0.771453i	$-0.280470\pi$
$911$	8.76508e11			1.27257			0.636286	+	0.771453i	$0.280470\pi$
$912$		0			0
$913$	4.68897e11	−	2.70718e11i	0.674829	−	0.389613i
$914$		0			0
$915$	3.72817e10	+	2.15246e10i	0.0531877	+	0.0307079i
$916$		0			0
$917$	3.04291e11	−	5.72270e11i	0.430340	−	0.809327i
$918$		0			0
$919$	6.03038e10	−	1.04449e11i	0.0845440	−	0.146434i	−0.820653	−	0.571427i	$-0.806390\pi$
$919$	6.03038e10	−	1.04449e11i	0.0845440	−	0.146434i	0.905197	+	0.424993i	$0.139723\pi$
$920$		0			0
$921$	2.21763e11	+	3.84105e11i	0.308213	+	0.533841i
$922$		0			0
$923$		−	6.72761e10i		−	0.0926946i
$924$		0			0
$925$	−3.99243e11			−0.545344
$926$		0			0
$927$	4.01207e11	−	2.31637e11i	0.543312	−	0.313681i
$928$		0			0
$929$	−4.76098e11	−	2.74876e11i	−0.639196	−	0.369040i	0.145109	−	0.989416i	$-0.453647\pi$
$929$	−4.76098e11	−	2.74876e11i	−0.639196	−	0.369040i	−0.784305	+	0.620376i	$0.786980\pi$
$930$		0			0
$931$	−1.74351e11	−	8.50420e10i	−0.232073	−	0.113197i
$932$		0			0
$933$	8.40412e10	−	1.45564e11i	0.110909	−	0.192099i
$934$		0			0
$935$	7.35111e10	+	1.27325e11i	0.0961848	+	0.166597i
$936$		0			0
$937$		−	7.46093e11i		−	0.967909i	−0.875093	−	0.483955i	$-0.839200\pi$
$937$		−	7.46093e11i		−	0.967909i	0.875093	−	0.483955i	$-0.160800\pi$
$938$		0			0
$939$	−4.17369e10			−0.0536856
$940$		0			0
$941$	−7.72272e11	+	4.45871e11i	−0.984945	+	0.568658i	−0.903759	−	0.428041i	$-0.859204\pi$
$941$	−7.72272e11	+	4.45871e11i	−0.984945	+	0.568658i	−0.0811853	+	0.996699i	$0.525871\pi$
$942$		0			0
$943$	5.67195e11	+	3.27470e11i	0.717275	+	0.414119i
$944$		0			0
$945$	9.37118e10	+	4.98290e10i	0.117508	+	0.0624820i
$946$		0			0
$947$	−5.89215e11	+	1.02055e12i	−0.732611	+	1.26892i	0.223152	+	0.974784i	$0.428365\pi$
$947$	−5.89215e11	+	1.02055e12i	−0.732611	+	1.26892i	−0.955764	+	0.294136i	$0.904968\pi$
$948$		0			0
$949$	−1.79841e10	−	3.11494e10i	−0.0221730	−	0.0384047i
$950$		0			0
$951$		−	5.75551e11i		−	0.703658i
$952$		0			0
$953$	6.62724e11			0.803454			0.401727	−	0.915759i	$-0.368410\pi$
$953$	6.62724e11			0.803454			0.401727	+	0.915759i	$0.368410\pi$
$954$		0			0
$955$	−1.87296e11	+	1.08136e11i	−0.225173	+	0.130004i
$956$		0			0
$957$	5.07869e11	+	2.93218e11i	0.605486	+	0.349577i
$958$		0			0
$959$	1.17415e12	−	7.33669e11i	1.38819	−	0.867413i
$960$		0			0
$961$	1.44924e11	−	2.51016e11i	0.169921	−	0.294312i
$962$		0			0
$963$	1.44134e11	+	2.49648e11i	0.167595	+	0.290284i
$964$		0			0
$965$			2.02284e11i			0.233266i
$966$		0			0
$967$	7.71443e11			0.882263			0.441131	−	0.897443i	$-0.354577\pi$
$967$	7.71443e11			0.882263			0.441131	+	0.897443i	$0.354577\pi$
$968$		0			0
$969$	3.35816e10	−	1.93883e10i	0.0380895	−	0.0219910i
$970$		0			0
$971$	−1.42118e12	−	8.20516e11i	−1.59871	−	0.923018i	−0.991735	−	0.128301i	$-0.959048\pi$
$971$	−1.42118e12	−	8.20516e11i	−1.59871	−	0.923018i	−0.606979	−	0.794718i	$-0.707619\pi$
$972$		0			0
$973$	−4.75581e11	+	1.66015e10i	−0.530607	+	0.0185224i
$974$		0			0
$975$	−1.53639e10	+	2.66111e10i	−0.0170014	+	0.0294472i
$976$		0			0
$977$	−5.74522e11	−	9.95102e11i	−0.630563	−	1.09217i	−0.987437	−	0.158015i	$-0.949491\pi$
$977$	−5.74522e11	−	9.95102e11i	−0.630563	−	1.09217i	0.356874	−	0.934153i	$-0.383843\pi$
$978$		0			0
$979$			1.62873e11i			0.177303i
$980$		0			0
$981$	4.06631e11			0.439061
$982$		0			0
$983$	−9.51092e11	+	5.49113e11i	−1.01861	+	0.588096i	−0.913701	−	0.406386i	$-0.866789\pi$
$983$	−9.51092e11	+	5.49113e11i	−1.01861	+	0.588096i	−0.104910	+	0.994482i	$0.533456\pi$
$984$		0			0
$985$	9.17867e11	+	5.29931e11i	0.975068	+	0.562956i
$986$		0			0
$987$	1.94239e10	+	5.56432e11i	0.0204676	+	0.586331i
$988$		0			0
$989$	4.52750e11	−	7.84187e11i	0.473231	−	0.819661i
$990$		0			0
$991$	5.20799e11	+	9.02050e11i	0.539977	+	0.935267i	0.998905	+	0.0467938i	$0.0149004\pi$
$991$	5.20799e11	+	9.02050e11i	0.539977	+	0.935267i	−0.458928	+	0.888474i	$0.651766\pi$
$992$		0			0
$993$			6.74321e10i			0.0693538i
$994$		0			0
$995$	−4.04993e11			−0.413195
$996$		0			0
$997$	−5.42668e11	+	3.13310e11i	−0.549229	+	0.317098i	−0.748811	−	0.662783i	$-0.769375\pi$
$997$	−5.42668e11	+	3.13310e11i	−0.549229	+	0.317098i	0.199582	+	0.979881i	$0.436042\pi$
$998$		0			0
$999$	1.73500e11	+	1.00170e11i	0.174196	+	0.100572i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.9.m.a.61.2	✓	10
3.2	odd	2		252.9.z.b.145.4		10
7.3	odd	6	inner	84.9.m.a.73.2	yes	10
21.17	even	6		252.9.z.b.73.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.9.m.a.61.2	✓	10	1.1	even	1	trivial
84.9.m.a.73.2	yes	10	7.3	odd	6	inner
252.9.z.b.73.4		10	21.17	even	6
252.9.z.b.145.4		10	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)

\(f(q)\)	\(=\)	\(q+(-40.5000 + 23.3827i) q^{3} +(-374.307 - 216.106i) q^{5} +(83.7630 + 2399.54i) q^{7} +(1093.50 - 1894.00i) q^{9} +O(q^{10})\)	\(q+(-40.5000 + 23.3827i) q^{3} +(-374.307 - 216.106i) q^{5} +(83.7630 + 2399.54i) q^{7} +(1093.50 - 1894.00i) q^{9} +(6902.31 + 11955.2i) q^{11} +3223.79i q^{13} +20212.6 q^{15} +(-21339.9 + 12320.6i) q^{17} +(29141.7 + 16825.0i) q^{19} +(-59500.1 - 95222.7i) q^{21} +(-112905. + 195558. i) q^{23} +(-101909. - 176511. i) q^{25} +102276. i q^{27} -908389. q^{29} +(925773. - 534495. i) q^{31} +(-559087. - 322789. i) q^{33} +(487202. - 916265. i) q^{35} +(979412. - 1.69639e6i) q^{37} +(-75380.8 - 130563. i) q^{39} -2.90040e6i q^{41} -4.01000e6 q^{43} +(-818609. + 472624. i) q^{45} +(-4.29428e6 - 2.47930e6i) q^{47} +(-5.75077e6 + 401985. i) q^{49} +(576177. - 997968. i) q^{51} +(-3.98900e6 - 6.90915e6i) q^{53} -5.96653e6i q^{55} -1.57365e6 q^{57} +(4.69735e6 - 2.71201e6i) q^{59} +(1.84448e6 + 1.06491e6i) q^{61} +(4.63631e6 + 2.46525e6i) q^{63} +(696680. - 1.20669e6i) q^{65} +(-9.53767e6 - 1.65197e7i) q^{67} -1.05601e7i q^{69} -2.08687e7 q^{71} +(-9.66235e6 + 5.57856e6i) q^{73} +(8.25461e6 + 4.76580e6i) q^{75} +(-2.81087e7 + 1.75638e7i) q^{77} +(1.21508e7 - 2.10458e7i) q^{79} +(-2.39148e6 - 4.14217e6i) q^{81} -3.92213e7i q^{83} +1.06502e7 q^{85} +(3.67898e7 - 2.12406e7i) q^{87} +(1.02177e7 + 5.89920e6i) q^{89} +(-7.73560e6 + 270034. i) q^{91} +(-2.49959e7 + 4.32941e7i) q^{93} +(-7.27196e6 - 1.25954e7i) q^{95} +7.80671e7i q^{97} +3.01907e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 405 q^{3} + 1389 q^{5} + 1217 q^{7} + 10935 q^{9}+O(q^{10}) \) 10 * q - 405 * q^3 + 1389 * q^5 + 1217 * q^7 + 10935 * q^9	\( 10 q - 405 q^{3} + 1389 q^{5} + 1217 q^{7} + 10935 q^{9} - 879 q^{11} - 75006 q^{15} - 13674 q^{17} - 29268 q^{19} - 42363 q^{21} + 312732 q^{23} - 22052 q^{25} - 289794 q^{29} + 242787 q^{31} + 71199 q^{33} + 1209372 q^{35} + 1913308 q^{37} - 1232334 q^{39} - 861848 q^{43} + 3037743 q^{45} - 305448 q^{47} + 9821659 q^{49} + 369198 q^{51} - 10663233 q^{53} + 1580472 q^{57} + 18410871 q^{59} - 13937808 q^{61} + 769824 q^{63} - 14966808 q^{65} - 20722822 q^{67} + 113032584 q^{71} + 43436322 q^{73} + 1786212 q^{75} - 98823405 q^{77} - 42189637 q^{79} - 23914845 q^{81} + 142602108 q^{85} + 11736657 q^{87} + 67171914 q^{89} - 246091266 q^{91} - 6555249 q^{93} - 140649894 q^{95} - 3844746 q^{99}+O(q^{100}) \) 10 * q - 405 * q^3 + 1389 * q^5 + 1217 * q^7 + 10935 * q^9 - 879 * q^11 - 75006 * q^15 - 13674 * q^17 - 29268 * q^19 - 42363 * q^21 + 312732 * q^23 - 22052 * q^25 - 289794 * q^29 + 242787 * q^31 + 71199 * q^33 + 1209372 * q^35 + 1913308 * q^37 - 1232334 * q^39 - 861848 * q^43 + 3037743 * q^45 - 305448 * q^47 + 9821659 * q^49 + 369198 * q^51 - 10663233 * q^53 + 1580472 * q^57 + 18410871 * q^59 - 13937808 * q^61 + 769824 * q^63 - 14966808 * q^65 - 20722822 * q^67 + 113032584 * q^71 + 43436322 * q^73 + 1786212 * q^75 - 98823405 * q^77 - 42189637 * q^79 - 23914845 * q^81 + 142602108 * q^85 + 11736657 * q^87 + 67171914 * q^89 - 246091266 * q^91 - 6555249 * q^93 - 140649894 * q^95 - 3844746 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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