LMFDB - Embedded newform 84.9.m.a.73.3

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			73.3
Root			$139.521 + 80.5522i$ of defining polynomial
Character	$\chi$	$=$	84.73
Dual form			84.9.m.a.61.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-40.5000 - 23.3827i) q^{3} +(111.592 - 64.4274i) q^{5} +(2392.99 - 195.963i) q^{7} +(1093.50 + 1894.00i) q^{9} +O(q^{10})$	\(q+(-40.5000 - 23.3827i) q^{3} +(111.592 - 64.4274i) q^{5} +(2392.99 - 195.963i) q^{7} +(1093.50 + 1894.00i) q^{9} +(-14456.4 + 25039.2i) q^{11} -39655.0i q^{13} -6025.94 q^{15} +(-68706.8 - 39667.9i) q^{17} +(61517.2 - 35517.0i) q^{19} +(-101498. - 48018.0i) q^{21} +(182396. + 315919. i) q^{23} +(-187011. + 323912. i) q^{25} -102276. i q^{27} +765205. q^{29} +(-499130. - 288173. i) q^{31} +(1.17097e6 - 676060. i) q^{33} +(254412. - 176042. i) q^{35} +(1.57572e6 + 2.72923e6i) q^{37} +(-927241. + 1.60603e6i) q^{39} +2.40539e6i q^{41} +4.03645e6 q^{43} +(244051. + 140903. i) q^{45} +(2.04429e6 - 1.18027e6i) q^{47} +(5.68800e6 - 937877. i) q^{49} +(1.85508e6 + 3.21310e6i) q^{51} +(-1.86610e6 + 3.23217e6i) q^{53} +3.72556e6i q^{55} -3.32193e6 q^{57} +(1.79677e7 + 1.03737e7i) q^{59} +(-1.72341e7 + 9.95010e6i) q^{61} +(2.98789e6 + 4.31803e6i) q^{63} +(-2.55487e6 - 4.42517e6i) q^{65} +(4.98086e6 - 8.62710e6i) q^{67} -1.70596e7i q^{69} +2.20838e7 q^{71} +(4.52062e7 + 2.60998e7i) q^{73} +(1.51479e7 - 8.74563e6i) q^{75} +(-2.96873e7 + 6.27516e7i) q^{77} +(-1.40366e7 - 2.43122e7i) q^{79} +(-2.39148e6 + 4.14217e6i) q^{81} +3.12932e7i q^{83} -1.02228e7 q^{85} +(-3.09908e7 - 1.78925e7i) q^{87} +(5.95320e7 - 3.43708e7i) q^{89} +(-7.77093e6 - 9.48941e7i) q^{91} +(1.34765e7 + 2.33420e7i) q^{93} +(4.57653e6 - 7.92679e6i) q^{95} -9.71832e7i q^{97} -6.32324e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/84\mathbb{Z}\right)^\times$.

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

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
$5$	111.592	−	64.4274i	0.178546	−	0.103084i	−0.408063	−	0.912954i	$-0.633796\pi$
$5$	111.592	−	64.4274i	0.178546	−	0.103084i	0.586610	+	0.809870i	$0.300462\pi$
$6$		0			0
$7$	2392.99	−	195.963i	0.996664	−	0.0816174i
$7$	2392.99	−	195.963i	0.996664	−	0.0816174i
$8$		0			0
$9$	1093.50	+	1894.00i	0.166667	+	0.288675i
$10$		0			0
$11$	−14456.4	+	25039.2i	−0.987393	+	1.71021i	−0.356612	+	0.934252i	$0.616068\pi$
$11$	−14456.4	+	25039.2i	−0.987393	+	1.71021i	−0.630780	+	0.775962i	$0.717265\pi$
$12$		0			0
$13$		−	39655.0i		−	1.38843i	−0.719766	−	0.694216i	$-0.755751\pi$
$13$		−	39655.0i		−	1.38843i	0.719766	−	0.694216i	$-0.244249\pi$
$14$		0			0
$15$	−6025.94			−0.119031
$16$		0			0
$17$	−68706.8	−	39667.9i	−0.822629	−	0.474945i	0.0286935	−	0.999588i	$-0.490865\pi$
$17$	−68706.8	−	39667.9i	−0.822629	−	0.474945i	−0.851322	+	0.524643i	$0.824199\pi$
$18$		0			0
$19$	61517.2	−	35517.0i	0.472044	−	0.272535i	−0.245051	−	0.969510i	$-0.578805\pi$
$19$	61517.2	−	35517.0i	0.472044	−	0.272535i	0.717095	+	0.696976i	$0.245471\pi$
$20$		0			0
$21$	−101498.	−	48018.0i	−0.521893	−	0.246903i
$22$		0			0
$23$	182396.	+	315919.i	0.651784	+	1.12892i	0.982690	+	0.185259i	$0.0593125\pi$
$23$	182396.	+	315919.i	0.651784	+	1.12892i	−0.330905	+	0.943664i	$0.607354\pi$
$24$		0			0
$25$	−187011.	+	323912.i	−0.478747	+	0.829215i
$26$		0			0
$27$		−	102276.i		−	0.192450i
$28$		0			0
$29$	765205.			1.08190			0.540948	−	0.841056i	$-0.318066\pi$
$29$	765205.			1.08190			0.540948	+	0.841056i	$0.318066\pi$
$30$		0			0
$31$	−499130.	−	288173.i	−0.540464	−	0.312037i	0.204803	−	0.978803i	$-0.434345\pi$
$31$	−499130.	−	288173.i	−0.540464	−	0.312037i	−0.745267	+	0.666766i	$0.767678\pi$
$32$		0			0
$33$	1.17097e6	−	676060.i	0.987393	−	0.570071i
$34$		0			0
$35$	254412.	−	176042.i	0.169537	−	0.117312i
$36$		0			0
$37$	1.57572e6	+	2.72923e6i	0.840762	+	1.45624i	0.889251	+	0.457419i	$0.151226\pi$
$37$	1.57572e6	+	2.72923e6i	0.840762	+	1.45624i	−0.0484895	+	0.998824i	$0.515441\pi$
$38$		0			0
$39$	−927241.	+	1.60603e6i	−0.400806	+	0.694216i
$40$		0			0
$41$			2.40539e6i			0.851235i	0.904903	+	0.425617i	$0.139943\pi$
$41$			2.40539e6i			0.851235i	−0.904903	+	0.425617i	$0.860057\pi$
$42$		0			0
$43$	4.03645e6			1.18066			0.590332	−	0.807161i	$-0.298997\pi$
$43$	4.03645e6			1.18066			0.590332	+	0.807161i	$0.298997\pi$
$44$		0			0
$45$	244051.	+	140903.i	0.0595155	+	0.0343613i
$46$		0			0
$47$	2.04429e6	−	1.18027e6i	0.418939	−	0.241874i	−0.275684	−	0.961248i	$-0.588904\pi$
$47$	2.04429e6	−	1.18027e6i	0.418939	−	0.241874i	0.694623	+	0.719374i	$0.255571\pi$
$48$		0			0
$49$	5.68800e6	−	937877.i	0.986677	−	0.162690i
$50$		0			0
$51$	1.85508e6	+	3.21310e6i	0.274210	+	0.474945i
$52$		0			0
$53$	−1.86610e6	+	3.23217e6i	−0.236500	+	0.409629i	−0.959707	−	0.281001i	$-0.909334\pi$
$53$	−1.86610e6	+	3.23217e6i	−0.236500	+	0.409629i	0.723208	+	0.690630i	$0.242667\pi$
$54$		0			0
$55$			3.72556e6i			0.407137i
$56$		0			0
$57$	−3.32193e6			−0.314696
$58$		0			0
$59$	1.79677e7	+	1.03737e7i	1.48281	+	0.856100i	0.999809	−	0.0195213i	$-0.00621422\pi$
$59$	1.79677e7	+	1.03737e7i	1.48281	+	0.856100i	0.482999	+	0.875621i	$0.339548\pi$
$60$		0			0
$61$	−1.72341e7	+	9.95010e6i	−1.24471	+	0.718634i	−0.970050	−	0.242906i	$-0.921899\pi$
$61$	−1.72341e7	+	9.95010e6i	−1.24471	+	0.718634i	−0.274662	+	0.961541i	$0.588566\pi$
$62$		0			0
$63$	2.98789e6	+	4.31803e6i	0.189672	+	0.274109i
$64$		0			0
$65$	−2.55487e6	−	4.42517e6i	−0.143125	−	0.247900i
$66$		0			0
$67$	4.98086e6	−	8.62710e6i	0.247175	−	0.428120i	−0.715566	−	0.698546i	$-0.753831\pi$
$67$	4.98086e6	−	8.62710e6i	0.247175	−	0.428120i	0.962741	+	0.270425i	$0.0871643\pi$
$68$		0			0
$69$		−	1.70596e7i		−	0.752616i
$70$		0			0
$71$	2.20838e7			0.869041			0.434520	−	0.900662i	$-0.356918\pi$
$71$	2.20838e7			0.869041			0.434520	+	0.900662i	$0.356918\pi$
$72$		0			0
$73$	4.52062e7	+	2.60998e7i	1.59187	+	0.919065i	0.992987	+	0.118226i	$0.0377207\pi$
$73$	4.52062e7	+	2.60998e7i	1.59187	+	0.919065i	0.598880	+	0.800839i	$0.295613\pi$
$74$		0			0
$75$	1.51479e7	−	8.74563e6i	0.478747	−	0.276405i
$76$		0			0
$77$	−2.96873e7	+	6.27516e7i	−0.844515	+	1.78510i
$78$		0			0
$79$	−1.40366e7	−	2.43122e7i	−0.360375	−	0.624188i	0.627647	−	0.778498i	$-0.284018\pi$
$79$	−1.40366e7	−	2.43122e7i	−0.360375	−	0.624188i	−0.988023	+	0.154310i	$0.950685\pi$
$80$		0			0
$81$	−2.39148e6	+	4.14217e6i	−0.0555556	+	0.0962250i
$82$		0			0
$83$			3.12932e7i			0.659382i	0.944089	+	0.329691i	$0.106945\pi$
$83$			3.12932e7i			0.659382i	−0.944089	+	0.329691i	$0.893055\pi$
$84$		0			0
$85$	−1.02228e7			−0.195837
$86$		0			0
$87$	−3.09908e7	−	1.78925e7i	−0.540948	−	0.312317i
$88$		0			0
$89$	5.95320e7	−	3.43708e7i	0.948834	−	0.547810i	0.0561157	−	0.998424i	$-0.482128\pi$
$89$	5.95320e7	−	3.43708e7i	0.948834	−	0.547810i	0.892719	+	0.450615i	$0.148795\pi$
$90$		0			0
$91$	−7.77093e6	−	9.48941e7i	−0.113320	−	1.38380i
$92$		0			0
$93$	1.34765e7	+	2.33420e7i	0.180155	+	0.312037i
$94$		0			0
$95$	4.57653e6	−	7.92679e6i	0.0561878	−	0.0973202i
$96$		0			0
$97$		−	9.71832e7i		−	1.09775i	−0.835904	−	0.548876i	$-0.815056\pi$
$97$		−	9.71832e7i		−	1.09775i	0.835904	−	0.548876i	$-0.184944\pi$
$98$		0			0
$99$	−6.32324e7			−0.658262
$100$		0			0
$101$	−1.77035e8	−	1.02211e8i	−1.70127	−	0.982229i	−0.944479	−	0.328572i	$-0.893433\pi$
$101$	−1.77035e8	−	1.02211e8i	−1.70127	−	0.982229i	−0.756791	−	0.653657i	$-0.773234\pi$
$102$		0			0
$103$	−8.33010e7	+	4.80939e7i	−0.740119	+	0.427308i	−0.822112	−	0.569325i	$-0.807205\pi$
$103$	−8.33010e7	+	4.80939e7i	−0.740119	+	0.427308i	0.0819937	+	0.996633i	$0.473871\pi$
$104$		0			0
$105$	−1.44200e7	+	1.18086e6i	−0.118634	+	0.00971500i
$106$		0			0
$107$	5.45930e7	+	9.45579e7i	0.416487	+	0.721377i	0.995583	−	0.0938820i	$-0.0299276\pi$
$107$	5.45930e7	+	9.45579e7i	0.416487	+	0.721377i	−0.579096	+	0.815259i	$0.696594\pi$
$108$		0			0
$109$	−1.19057e8	+	2.06214e8i	−0.843433	+	1.46087i	0.0435416	+	0.999052i	$0.486136\pi$
$109$	−1.19057e8	+	2.06214e8i	−0.843433	+	1.46087i	−0.886975	+	0.461818i	$0.847197\pi$
$110$		0			0
$111$		−	1.47379e8i		−	0.970828i
$112$		0			0
$113$	6.73456e7			0.413043			0.206521	−	0.978442i	$-0.433786\pi$
$113$	6.73456e7			0.413043			0.206521	+	0.978442i	$0.433786\pi$
$114$		0			0
$115$	4.07077e7	+	2.35026e7i	0.232748	+	0.134377i
$116$		0			0
$117$	7.51065e7	−	4.33628e7i	0.400806	−	0.231405i
$118$		0			0
$119$	−1.72188e8	−	8.14608e7i	−0.858648	−	0.406219i
$120$		0			0
$121$	−3.10796e8	−	5.38315e8i	−1.44989	−	2.51128i
$122$		0			0
$123$	5.62444e7	−	9.74181e7i	0.245730	−	0.425617i
$124$		0			0
$125$			9.85284e7i			0.403572i
$126$		0			0
$127$	1.24587e7			0.0478916			0.0239458	−	0.999713i	$-0.492377\pi$
$127$	1.24587e7			0.0478916			0.0239458	+	0.999713i	$0.492377\pi$
$128$		0			0
$129$	−1.63476e8	−	9.43831e7i	−0.590332	−	0.340828i
$130$		0			0
$131$	1.18661e6	−	685092.i	0.00402925	−	0.00232629i	−0.497984	−	0.867186i	$-0.665926\pi$
$131$	1.18661e6	−	685092.i	0.00402925	−	0.00232629i	0.502013	+	0.864860i	$0.332593\pi$
$132$		0			0
$133$	1.40250e8	−	9.70469e7i	0.448225	−	0.310152i
$134$		0			0
$135$	−6.58937e6	−	1.14131e7i	−0.0198385	−	0.0343613i
$136$		0			0
$137$	−3.28408e8	+	5.68819e8i	−0.932247	+	1.61470i	−0.152777	+	0.988261i	$0.548822\pi$
$137$	−3.28408e8	+	5.68819e8i	−0.932247	+	1.61470i	−0.779470	+	0.626439i	$0.784512\pi$
$138$		0			0
$139$		−	1.36252e8i		−	0.364991i	−0.983207	−	0.182496i	$-0.941582\pi$
$139$		−	1.36252e8i		−	0.364991i	0.983207	−	0.182496i	$-0.0584175\pi$
$140$		0			0
$141$	−1.10392e8			−0.279293
$142$		0			0
$143$	9.92932e8	+	5.73269e8i	2.37452	+	1.37093i
$144$		0			0
$145$	8.53904e7	−	4.93001e7i	0.193169	−	0.111526i
$146$		0			0
$147$	−2.52294e8	−	9.50167e7i	−0.540303	−	0.203484i
$148$		0			0
$149$	2.27005e8	+	3.93184e8i	0.460565	+	0.797721i	0.998989	−	0.0449524i	$-0.0143136\pi$
$149$	2.27005e8	+	3.93184e8i	0.460565	+	0.797721i	−0.538424	+	0.842674i	$0.680980\pi$
$150$		0			0
$151$	3.13822e8	−	5.43555e8i	0.603636	−	1.04553i	−0.388630	−	0.921394i	$-0.627051\pi$
$151$	3.13822e8	−	5.43555e8i	0.603636	−	1.04553i	0.992265	−	0.124134i	$-0.0396152\pi$
$152$		0			0
$153$		−	1.73507e8i		−	0.316630i
$154$		0			0
$155$	−7.42649e7			−0.128664
$156$		0			0
$157$	1.74432e8	+	1.00708e8i	0.287095	+	0.165755i	0.636631	−	0.771168i	$-0.280327\pi$
$157$	1.74432e8	+	1.00708e8i	0.287095	+	0.165755i	−0.349536	+	0.936923i	$0.613661\pi$
$158$		0			0
$159$	1.51154e8	−	8.72686e7i	0.236500	−	0.136543i
$160$		0			0
$161$	4.98380e8	+	7.20248e8i	0.741749	+	1.07196i
$162$		0			0
$163$	1.33704e8	+	2.31582e8i	0.189406	+	0.328061i	0.945052	−	0.326919i	$-0.106010\pi$
$163$	1.33704e8	+	2.31582e8i	0.189406	+	0.328061i	−0.755646	+	0.654980i	$0.772677\pi$
$164$		0			0
$165$	8.71135e7	−	1.50885e8i	0.117530	−	0.203568i
$166$		0			0
$167$			3.34333e8i			0.429846i	0.976631	+	0.214923i	$0.0689501\pi$
$167$			3.34333e8i			0.429846i	−0.976631	+	0.214923i	$0.931050\pi$
$168$		0			0
$169$	−7.56791e8			−0.927746
$170$		0			0
$171$	1.34538e8	+	7.76756e7i	0.157348	+	0.0908449i
$172$		0			0
$173$	−1.44134e7	+	8.32156e6i	−0.0160909	+	0.00929010i	−0.508024	−	0.861343i	$-0.669624\pi$
$173$	−1.44134e7	+	8.32156e6i	−0.0160909	+	0.00929010i	0.491933	+	0.870633i	$0.336291\pi$
$174$		0			0
$175$	−3.84040e8	+	8.11765e8i	−0.409472	+	0.865523i
$176$		0			0
$177$	−4.85129e8	−	8.40267e8i	−0.494269	−	0.856100i
$178$		0			0
$179$	−2.74376e8	+	4.75233e8i	−0.267260	+	0.462908i	−0.968153	−	0.250358i	$-0.919452\pi$
$179$	−2.74376e8	+	4.75233e8i	−0.267260	+	0.462908i	0.700893	+	0.713266i	$0.252785\pi$
$180$		0			0
$181$			6.47149e8i			0.602962i	0.953472	+	0.301481i	$0.0974810\pi$
$181$			6.47149e8i			0.602962i	−0.953472	+	0.301481i	$0.902519\pi$
$182$		0			0
$183$	9.30640e8			0.829808
$184$		0			0
$185$	3.51675e8	+	2.03040e8i	0.300230	+	0.173338i
$186$		0			0
$187$	1.98651e9	−	1.14691e9i	1.62451	−	0.937914i
$188$		0			0
$189$	−2.00423e7	−	2.44745e8i	−0.0157073	−	0.191808i
$190$		0			0
$191$	7.30628e8	+	1.26549e9i	0.548988	+	0.950875i	0.998344	+	0.0575225i	$0.0183201\pi$
$191$	7.30628e8	+	1.26549e9i	0.548988	+	0.950875i	−0.449356	+	0.893353i	$0.648347\pi$
$192$		0			0
$193$	−2.13052e8	+	3.69017e8i	−0.153552	+	0.265961i	−0.932531	−	0.361090i	$-0.882405\pi$
$193$	−2.13052e8	+	3.69017e8i	−0.153552	+	0.265961i	0.778979	+	0.627051i	$0.215738\pi$
$194$		0			0
$195$			2.38959e8i			0.165267i
$196$		0			0
$197$	1.63708e8			0.108694			0.0543469	−	0.998522i	$-0.482692\pi$
$197$	1.63708e8			0.108694			0.0543469	+	0.998522i	$0.482692\pi$
$198$		0			0
$199$	−1.59492e9	−	9.20826e8i	−1.01701	−	0.587172i	−0.103775	−	0.994601i	$-0.533092\pi$
$199$	−1.59492e9	−	9.20826e8i	−1.01701	−	0.587172i	−0.913237	+	0.407429i	$0.866425\pi$
$200$		0			0
$201$	−4.03450e8	+	2.32932e8i	−0.247175	+	0.142707i
$202$		0			0
$203$	1.83113e9	−	1.49952e8i	1.07829	−	0.0883015i
$204$		0			0
$205$	1.54973e8	+	2.68421e8i	0.0877486	+	0.151985i
$206$		0			0
$207$	−3.98900e8	+	6.90915e8i	−0.217261	+	0.376308i
$208$		0			0
$209$			2.05379e9i			1.07639i
$210$		0			0
$211$	2.46982e9			1.24605			0.623024	−	0.782203i	$-0.285904\pi$
$211$	2.46982e9			1.24605			0.623024	+	0.782203i	$0.285904\pi$
$212$		0			0
$213$	−8.94393e8	−	5.16378e8i	−0.434520	−	0.250870i
$214$		0			0
$215$	4.50434e8	−	2.60058e8i	0.210803	−	0.121707i
$216$		0			0
$217$	−1.25088e9	−	5.91784e8i	−0.564129	−	0.266885i
$218$		0			0
$219$	−1.22057e9	−	2.11409e9i	−0.530622	−	0.919065i
$220$		0			0
$221$	−1.57303e9	+	2.72457e9i	−0.659429	+	1.14216i
$222$		0			0
$223$		−	3.01259e9i		−	1.21821i	−0.793091	−	0.609103i	$-0.791530\pi$
$223$		−	3.01259e9i		−	1.21821i	0.793091	−	0.609103i	$-0.208470\pi$
$224$		0			0
$225$	−8.17985e8			−0.319165
$226$		0			0
$227$	−5.04253e7	−	2.91131e7i	−0.0189909	−	0.0109644i	0.490474	−	0.871456i	$-0.336823\pi$
$227$	−5.04253e7	−	2.91131e7i	−0.0189909	−	0.0109644i	−0.509465	+	0.860491i	$0.670157\pi$
$228$		0			0
$229$	−5.41882e8	+	3.12856e8i	−0.197044	+	0.113763i	−0.595276	−	0.803521i	$-0.702957\pi$
$229$	−5.41882e8	+	3.12856e8i	−0.197044	+	0.113763i	0.398232	+	0.917285i	$0.369624\pi$
$230$		0			0
$231$	2.66964e9	−	1.84727e9i	0.937571	−	0.648758i
$232$		0			0
$233$	−2.06028e9	−	3.56850e9i	−0.699040	−	1.21077i	−0.968800	−	0.247845i	$-0.920278\pi$
$233$	−2.06028e9	−	3.56850e9i	−0.699040	−	1.21077i	0.269760	−	0.962928i	$-0.413056\pi$
$234$		0			0
$235$	1.52084e8	−	2.63416e8i	0.0498667	−	0.0863717i
$236$		0			0
$237$			1.31286e9i			0.416125i
$238$		0			0
$239$	−1.56676e9			−0.480186			−0.240093	−	0.970750i	$-0.577178\pi$
$239$	−1.56676e9			−0.480186			−0.240093	+	0.970750i	$0.577178\pi$
$240$		0			0
$241$	−2.00073e9	−	1.15512e9i	−0.593091	−	0.342421i	0.173228	−	0.984882i	$-0.444580\pi$
$241$	−2.00073e9	−	1.15512e9i	−0.593091	−	0.342421i	−0.766319	+	0.642461i	$0.777914\pi$
$242$		0			0
$243$	1.93710e8	−	1.11839e8i	0.0555556	−	0.0320750i
$244$		0			0
$245$	5.74307e8	−	4.71122e8i	0.159397	−	0.130758i
$246$		0			0
$247$	−1.40843e9	−	2.43947e9i	−0.378396	−	0.655401i
$248$		0			0
$249$	7.31718e8	−	1.26737e9i	0.190347	−	0.329691i
$250$		0			0
$251$		−	6.63498e7i		−	0.0167165i	−0.999965	−	0.00835823i	$-0.997339\pi$
$251$		−	6.63498e7i		−	0.0167165i	0.999965	−	0.00835823i	$-0.00266054\pi$
$252$		0			0
$253$	−1.05472e10			−2.57427
$254$		0			0
$255$	4.14023e8	+	2.39036e8i	0.0979183	+	0.0565331i
$256$		0			0
$257$	−1.28769e9	+	7.43447e8i	−0.295174	+	0.170419i	−0.640273	−	0.768148i	$-0.721179\pi$
$257$	−1.28769e9	+	7.43447e8i	−0.295174	+	0.170419i	0.345099	+	0.938566i	$0.387845\pi$
$258$		0			0
$259$	4.30552e9	+	6.22224e9i	0.956812	+	1.38276i
$260$		0			0
$261$	8.36751e8	+	1.44930e9i	0.180316	+	0.312317i
$262$		0			0
$263$	2.54599e9	−	4.40979e9i	0.532151	−	0.921712i	−0.467145	−	0.884181i	$-0.654717\pi$
$263$	2.54599e9	−	4.40979e9i	0.532151	−	0.921712i	0.999295	−	0.0375311i	$-0.0119493\pi$
$264$		0			0
$265$			4.80911e8i			0.0975171i
$266$		0			0
$267$	−3.21473e9			−0.632556
$268$		0			0
$269$	−7.13647e9	−	4.12024e9i	−1.36293	−	0.786890i	−0.372920	−	0.927864i	$-0.621643\pi$
$269$	−7.13647e9	−	4.12024e9i	−1.36293	−	0.786890i	−0.990013	+	0.140974i	$0.954977\pi$
$270$		0			0
$271$	4.02331e9	−	2.32286e9i	0.745944	−	0.430671i	−0.0782825	−	0.996931i	$-0.524944\pi$
$271$	4.02331e9	−	2.32286e9i	0.745944	−	0.430671i	0.824227	+	0.566260i	$0.191610\pi$
$272$		0			0
$273$	−1.90416e9	+	4.02492e9i	−0.342809	+	0.724613i
$274$		0			0
$275$	−5.40701e9	−	9.36521e9i	−0.945423	−	1.63752i
$276$		0			0
$277$	1.25437e9	−	2.17264e9i	0.213063	−	0.369036i	−0.739609	−	0.673037i	$-0.764989\pi$
$277$	1.25437e9	−	2.17264e9i	0.213063	−	0.369036i	0.952672	+	0.304001i	$0.0983228\pi$
$278$		0			0
$279$		−	1.26047e9i		−	0.208025i
$280$		0			0
$281$	2.18908e9			0.351104			0.175552	−	0.984470i	$-0.443829\pi$
$281$	2.18908e9			0.351104			0.175552	+	0.984470i	$0.443829\pi$
$282$		0			0
$283$	5.54119e8	+	3.19921e8i	0.0863887	+	0.0498766i	0.542572	−	0.840009i	$-0.317451\pi$
$283$	5.54119e8	+	3.19921e8i	0.0863887	+	0.0498766i	−0.456183	+	0.889886i	$0.650784\pi$
$284$		0			0
$285$	−3.70699e8	+	2.14023e8i	−0.0561878	+	0.0324401i
$286$		0			0
$287$	4.71368e8	+	5.75606e9i	0.0694756	+	0.848395i
$288$		0			0
$289$	−3.40799e8	−	5.90282e8i	−0.0488548	−	0.0846191i
$290$		0			0
$291$	−2.27240e9	+	3.93592e9i	−0.316894	+	0.548876i
$292$		0			0
$293$			1.48526e9i			0.201527i	0.994910	+	0.100763i	$0.0321285\pi$
$293$			1.48526e9i			0.201527i	−0.994910	+	0.100763i	$0.967871\pi$
$294$		0			0
$295$	2.67339e9			0.353000
$296$		0			0
$297$	2.56091e9	+	1.47854e9i	0.329131	+	0.190024i
$298$		0			0
$299$	1.25278e10	−	7.23292e9i	1.56743	−	0.904958i
$300$		0			0
$301$	9.65919e9	−	7.90997e8i	1.17672	−	0.0963627i
$302$		0			0
$303$	4.77994e9	+	8.27910e9i	0.567090	+	0.982229i
$304$		0			0
$305$	−1.28212e9	+	2.22069e9i	−0.148159	+	0.256619i
$306$		0			0
$307$			1.33187e10i			1.49937i	0.661793	+	0.749687i	$0.269796\pi$
$307$			1.33187e10i			1.49937i	−0.661793	+	0.749687i	$0.730204\pi$
$308$		0			0
$309$	4.49825e9			0.493412
$310$		0			0
$311$	−3.31513e9	−	1.91399e9i	−0.354372	−	0.204597i	0.312237	−	0.950004i	$-0.398922\pi$
$311$	−3.31513e9	−	1.91399e9i	−0.354372	−	0.204597i	−0.666609	+	0.745407i	$0.732255\pi$
$312$		0			0
$313$	1.91930e9	−	1.10811e9i	0.199971	−	0.115453i	−0.396671	−	0.917961i	$-0.629835\pi$
$313$	1.91930e9	−	1.10811e9i	0.199971	−	0.115453i	0.596642	+	0.802508i	$0.296501\pi$
$314$		0			0
$315$	6.11623e8	+	2.89354e8i	0.0621214	+	0.0293891i
$316$		0			0
$317$	−2.52992e8	−	4.38195e8i	−0.0250536	−	0.0433940i	0.853227	−	0.521540i	$-0.174642\pi$
$317$	−2.52992e8	−	4.38195e8i	−0.0250536	−	0.0433940i	−0.878280	+	0.478146i	$0.841309\pi$
$318$		0			0
$319$	−1.10621e10	+	1.91601e10i	−1.06826	+	1.85027i
$320$		0			0
$321$		−	5.10612e9i		−	0.480918i
$322$		0			0
$323$	−5.63553e9			−0.517756
$324$		0			0
$325$	1.28447e10	+	7.41592e9i	1.15131	+	0.664709i
$326$		0			0
$327$	9.64366e9	−	5.56777e9i	0.843433	−	0.486956i
$328$		0			0
$329$	4.66067e9	−	3.22498e9i	0.397800	−	0.275260i
$330$		0			0
$331$	−7.85009e9	−	1.35968e10i	−0.653978	−	1.13272i	−0.982149	−	0.188104i	$-0.939766\pi$
$331$	−7.85009e9	−	1.35968e10i	−0.653978	−	1.13272i	0.328171	−	0.944618i	$-0.393568\pi$
$332$		0			0
$333$	−3.44611e9	+	5.96883e9i	−0.280254	+	0.485414i
$334$		0			0
$335$		−	1.28362e9i		−	0.101919i
$336$		0			0
$337$	1.56196e9			0.121101			0.0605507	−	0.998165i	$-0.480714\pi$
$337$	1.56196e9			0.121101			0.0605507	+	0.998165i	$0.480714\pi$
$338$		0			0
$339$	−2.72749e9	−	1.57472e9i	−0.206521	−	0.119235i
$340$		0			0
$341$	1.44313e10	−	8.33189e9i	1.06730	−	0.616206i
$342$		0			0
$343$	1.34275e10	−	3.35897e9i	0.970107	−	0.242677i
$344$		0			0
$345$	−1.09911e9	−	1.90371e9i	−0.0775825	−	0.134377i
$346$		0			0
$347$	−5.09009e9	+	8.81629e9i	−0.351081	+	0.608090i	−0.986439	−	0.164127i	$-0.947519\pi$
$347$	−5.09009e9	+	8.81629e9i	−0.351081	+	0.608090i	0.635358	+	0.772218i	$0.280853\pi$
$348$		0			0
$349$			2.88299e10i			1.94331i	0.236403	+	0.971655i	$0.424032\pi$
$349$			2.88299e10i			1.94331i	−0.236403	+	0.971655i	$0.575968\pi$
$350$		0			0
$351$	−4.05575e9			−0.267204
$352$		0			0
$353$	−7.38733e9	−	4.26508e9i	−0.475761	−	0.274681i	0.242887	−	0.970055i	$-0.421905\pi$
$353$	−7.38733e9	−	4.26508e9i	−0.475761	−	0.274681i	−0.718648	+	0.695374i	$0.755239\pi$
$354$		0			0
$355$	2.46436e9	−	1.42280e9i	0.155164	−	0.0895840i
$356$		0			0
$357$	5.06884e9	+	7.32538e9i	0.312058	+	0.450980i
$358$		0			0
$359$	−5.99690e9	−	1.03869e10i	−0.361035	−	0.625330i	0.627097	−	0.778941i	$-0.284243\pi$
$359$	−5.99690e9	−	1.03869e10i	−0.361035	−	0.625330i	−0.988131	+	0.153611i	$0.950910\pi$
$360$		0			0
$361$	−5.96887e9	+	1.03384e10i	−0.351450	+	0.608729i
$362$		0			0
$363$			2.90690e10i			1.67419i
$364$		0			0
$365$	6.72618e9			0.378963
$366$		0			0
$367$	1.66442e9	+	9.60953e8i	0.0917484	+	0.0529710i	0.545172	−	0.838324i	$-0.316464\pi$
$367$	1.66442e9	+	9.60953e8i	0.0917484	+	0.0529710i	−0.453424	+	0.891295i	$0.649798\pi$
$368$		0			0
$369$	−4.55580e9	+	2.63029e9i	−0.245730	+	0.141872i
$370$		0			0
$371$	−3.83216e9	+	8.10024e9i	−0.202278	+	0.427565i
$372$		0			0
$373$	−1.22841e10	−	2.12767e10i	−0.634611	−	1.09918i	−0.986597	−	0.163173i	$-0.947827\pi$
$373$	−1.22841e10	−	2.12767e10i	−0.634611	−	1.09918i	0.351987	−	0.936005i	$-0.385506\pi$
$374$		0			0
$375$	2.30386e9	−	3.99040e9i	0.116501	−	0.201786i
$376$		0			0
$377$		−	3.03442e10i		−	1.50214i
$378$		0			0
$379$	1.64148e10			0.795569			0.397785	−	0.917479i	$-0.369779\pi$
$379$	1.64148e10			0.795569			0.397785	+	0.917479i	$0.369779\pi$
$380$		0			0
$381$	−5.04579e8	−	2.91319e8i	−0.0239458	−	0.0138251i
$382$		0			0
$383$	1.04919e10	−	6.05749e9i	0.487594	−	0.281512i	−0.235982	−	0.971757i	$-0.575831\pi$
$383$	1.04919e10	−	6.05749e9i	0.487594	−	0.281512i	0.723576	+	0.690245i	$0.242497\pi$
$384$		0			0
$385$	7.30073e8	+	8.91522e9i	0.0332295	+	0.405779i
$386$		0			0
$387$	4.41386e9	+	7.64503e9i	0.196777	+	0.340828i
$388$		0			0
$389$	−2.60896e9	+	4.51886e9i	−0.113938	+	0.197347i	−0.917355	−	0.398070i	$-0.869680\pi$
$389$	−2.60896e9	+	4.51886e9i	−0.113938	+	0.197347i	0.803417	+	0.595417i	$0.203013\pi$
$390$		0			0
$391$		−	2.89410e10i		−	1.23825i
$392$		0			0
$393$	−6.40771e7			−0.00268617
$394$		0			0
$395$	−3.13274e9	−	1.80869e9i	−0.128687	−	0.0742977i
$396$		0			0
$397$	2.21404e10	−	1.27828e10i	0.891298	−	0.514591i	0.0169314	−	0.999857i	$-0.494610\pi$
$397$	2.21404e10	−	1.27828e10i	0.891298	−	0.514591i	0.874367	+	0.485265i	$0.161277\pi$
$398$		0			0
$399$	−7.94934e9	+	6.50977e8i	−0.313646	+	0.0256847i
$400$		0			0
$401$	−1.85899e10	−	3.21987e10i	−0.718953	−	1.24526i	−0.961415	−	0.275102i	$-0.911288\pi$
$401$	−1.85899e10	−	3.21987e10i	−0.718953	−	1.24526i	0.242462	−	0.970161i	$-0.422045\pi$
$402$		0			0
$403$	−1.14275e10	+	1.97930e10i	−0.433243	+	0.750398i
$404$		0			0
$405$			6.16309e8i			0.0229075i
$406$		0			0
$407$	−9.11172e10			−3.32065
$408$		0			0
$409$	1.06294e10	+	6.13691e9i	0.379854	+	0.219309i	0.677755	−	0.735288i	$-0.262953\pi$
$409$	1.06294e10	+	6.13691e9i	0.379854	+	0.219309i	−0.297901	+	0.954597i	$0.596286\pi$
$410$		0			0
$411$	2.66010e10	−	1.53581e10i	0.932247	−	0.538233i
$412$		0			0
$413$	4.50294e10	+	2.13031e10i	1.54773	+	0.732221i
$414$		0			0
$415$	2.01614e9	+	3.49205e9i	0.0679717	+	0.117730i
$416$		0			0
$417$	−3.18593e9	+	5.51819e9i	−0.105364	+	0.182496i
$418$		0			0
$419$		−	1.76170e10i		−	0.571577i	−0.958293	−	0.285788i	$-0.907745\pi$
$419$		−	1.76170e10i		−	0.571577i	0.958293	−	0.285788i	$-0.0922555\pi$
$420$		0			0
$421$	−5.40629e10			−1.72096			−0.860480	−	0.509485i	$-0.829836\pi$
$421$	−5.40629e10			−1.72096			−0.860480	+	0.509485i	$0.829836\pi$
$422$		0			0
$423$	4.47086e9	+	2.58125e9i	0.139646	+	0.0806248i
$424$		0			0
$425$	2.56978e10	−	1.48366e10i	0.787663	−	0.454757i
$426$		0			0
$427$	−3.92911e10	+	2.71877e10i	−1.18191	+	0.817827i
$428$		0			0
$429$	−2.68092e10	−	4.64348e10i	−0.791506	−	1.37093i
$430$		0			0
$431$	−8.23020e9	+	1.42551e10i	−0.238507	+	0.413106i	−0.960286	−	0.279017i	$-0.909991\pi$
$431$	−8.23020e9	+	1.42551e10i	−0.238507	+	0.413106i	0.721779	+	0.692124i	$0.243325\pi$
$432$		0			0
$433$		−	2.03523e10i		−	0.578978i	−0.957181	−	0.289489i	$-0.906515\pi$
$433$		−	2.03523e10i		−	0.578978i	0.957181	−	0.289489i	$-0.0934853\pi$
$434$		0			0
$435$	−4.61108e9			−0.128779
$436$		0			0
$437$	2.24410e10	+	1.29563e10i	0.615341	+	0.355268i
$438$		0			0
$439$	1.98300e10	−	1.14488e10i	0.533905	−	0.308250i	−0.208700	−	0.977980i	$-0.566923\pi$
$439$	1.98300e10	−	1.14488e10i	0.533905	−	0.308250i	0.742605	+	0.669730i	$0.233590\pi$
$440$		0			0
$441$	7.99616e9	+	9.74749e9i	0.211411	+	0.257714i
$442$		0			0
$443$	−1.69660e10	−	2.93860e10i	−0.440520	−	0.763003i	0.557208	−	0.830373i	$-0.311873\pi$
$443$	−1.69660e10	−	2.93860e10i	−0.440520	−	0.763003i	−0.997728	+	0.0673700i	$0.978539\pi$
$444$		0			0
$445$	4.42884e9	−	7.67098e9i	0.112941	−	0.195619i
$446$		0			0
$447$		−	2.12320e10i		−	0.531814i
$448$		0			0
$449$	5.99826e10			1.47584			0.737921	−	0.674887i	$-0.235808\pi$
$449$	5.99826e10			1.47584			0.737921	+	0.674887i	$0.235808\pi$
$450$		0			0
$451$	−6.02291e10	−	3.47733e10i	−1.45579	−	0.840503i
$452$		0			0
$453$	−2.54195e10	+	1.46760e10i	−0.603636	+	0.348509i
$454$		0			0
$455$	−6.98095e9	−	1.00887e10i	−0.162880	−	0.235391i
$456$		0			0
$457$	−1.57618e10	−	2.73002e10i	−0.361361	−	0.625895i	0.626824	−	0.779161i	$-0.284354\pi$
$457$	−1.57618e10	−	2.73002e10i	−0.361361	−	0.625895i	−0.988185	+	0.153266i	$0.951021\pi$
$458$		0			0
$459$	−4.05707e9	+	7.02704e9i	−0.0914032	+	0.158315i
$460$		0			0
$461$			5.38684e10i			1.19270i	0.802725	+	0.596349i	$0.203383\pi$
$461$			5.38684e10i			1.19270i	−0.802725	+	0.596349i	$0.796617\pi$
$462$		0			0
$463$	6.39076e9			0.139068			0.0695342	−	0.997580i	$-0.477849\pi$
$463$	6.39076e9			0.139068			0.0695342	+	0.997580i	$0.477849\pi$
$464$		0			0
$465$	3.00773e9	+	1.73651e9i	0.0643320	+	0.0371421i
$466$		0			0
$467$	−5.01832e10	+	2.89733e10i	−1.05509	+	0.609158i	−0.924071	−	0.382222i	$-0.875159\pi$
$467$	−5.01832e10	+	2.89733e10i	−1.05509	+	0.609158i	−0.131022	+	0.991380i	$0.541826\pi$
$468$		0			0
$469$	1.02286e10	−	2.16206e10i	0.211409	−	0.446866i
$470$		0			0
$471$	−4.70965e9	−	8.15736e9i	−0.0956985	−	0.165755i
$472$		0			0
$473$	−5.83527e10	+	1.01070e11i	−1.16578	+	2.01919i
$474$		0			0
$475$			2.65682e10i			0.521901i
$476$		0			0
$477$	−8.16230e9			−0.157666
$478$		0			0
$479$	−2.54905e10	−	1.47170e10i	−0.484214	−	0.279561i	0.237957	−	0.971276i	$-0.423522\pi$
$479$	−2.54905e10	−	1.47170e10i	−0.484214	−	0.279561i	−0.722171	+	0.691715i	$0.756856\pi$
$480$		0			0
$481$	1.08228e11	−	6.24853e10i	2.02189	−	1.16734i
$482$		0			0
$483$	−3.34306e9	−	4.08235e10i	−0.0614265	−	0.750105i
$484$		0			0
$485$	−6.26126e9	−	1.08448e10i	−0.113161	−	0.196000i
$486$		0			0
$487$	−5.23988e8	+	9.07574e8i	−0.00931549	+	0.0161349i	−0.870646	−	0.491911i	$-0.836299\pi$
$487$	−5.23988e8	+	9.07574e8i	−0.00931549	+	0.0161349i	0.861330	+	0.508046i	$0.169632\pi$
$488$		0			0
$489$		−	1.25054e10i		−	0.218707i
$490$		0			0
$491$	8.26283e10			1.42168			0.710842	−	0.703352i	$-0.248314\pi$
$491$	8.26283e10			1.42168			0.710842	+	0.703352i	$0.248314\pi$
$492$		0			0
$493$	−5.25747e10	−	3.03540e10i	−0.889999	−	0.513841i
$494$		0			0
$495$	−7.05620e9	+	4.07390e9i	−0.117530	+	0.0678562i
$496$		0			0
$497$	5.28463e10	−	4.32761e9i	0.866141	−	0.0709288i
$498$		0			0
$499$	1.74791e10	+	3.02747e10i	0.281914	+	0.488290i	0.971856	−	0.235575i	$-0.0756972\pi$
$499$	1.74791e10	+	3.02747e10i	0.281914	+	0.488290i	−0.689942	+	0.723865i	$0.742364\pi$
$500$		0			0
$501$	7.81760e9	−	1.35405e10i	0.124086	−	0.214923i
$502$		0			0
$503$			4.71506e10i			0.736573i	0.929712	+	0.368286i	$0.120055\pi$
$503$			4.71506e10i			0.736573i	−0.929712	+	0.368286i	$0.879945\pi$
$504$		0			0
$505$	−2.63408e10			−0.405008
$506$		0			0
$507$	3.06500e10	+	1.76958e10i	0.463873	+	0.267817i
$508$		0			0
$509$	1.53536e10	−	8.86441e9i	0.228739	−	0.132062i	−0.381251	−	0.924471i	$-0.624507\pi$
$509$	1.53536e10	−	8.86441e9i	0.228739	−	0.132062i	0.609990	+	0.792409i	$0.291173\pi$
$510$		0			0
$511$	1.13293e11	+	5.35978e10i	1.66157	+	0.786074i
$512$		0			0
$513$	−3.63253e9	−	6.29173e9i	−0.0524493	−	0.0908449i
$514$		0			0
$515$	−6.19713e9	+	1.07337e10i	−0.0880971	+	0.152589i
$516$		0			0
$517$			6.82499e10i			0.955300i
$518$		0			0
$519$	7.78322e8			0.0107273
$520$		0			0
$521$	9.72635e10	+	5.61551e10i	1.32008	+	0.762147i	0.983741	−	0.179595i	$-0.0574788\pi$
$521$	9.72635e10	+	5.61551e10i	1.32008	+	0.762147i	0.336336	+	0.941742i	$0.390812\pi$
$522$		0			0
$523$	−2.21210e10	+	1.27716e10i	−0.295664	+	0.170702i	−0.640494	−	0.767964i	$-0.721270\pi$
$523$	−2.21210e10	+	1.27716e10i	−0.295664	+	0.170702i	0.344829	+	0.938665i	$0.387937\pi$
$524$		0			0
$525$	3.45349e10	−	2.38966e10i	0.454591	−	0.314557i
$526$		0			0
$527$	2.28624e10	+	3.95989e10i	0.296401	+	0.513381i
$528$		0			0
$529$	−2.73811e10	+	4.74254e10i	−0.349645	+	0.605603i
$530$		0			0
$531$			4.53744e10i			0.570733i
$532$		0			0
$533$	9.53857e10			1.18188
$534$		0			0
$535$	1.21842e10	+	7.03457e9i	0.148725	+	0.0858663i
$536$		0			0
$537$	2.22244e10	−	1.28313e10i	0.267260	−	0.154303i
$538$		0			0
$539$	−5.87443e10	+	1.55981e11i	−0.696003	+	1.84807i
$540$		0			0
$541$	8.31041e9	+	1.43941e10i	0.0970138	+	0.168033i	0.910447	−	0.413625i	$-0.135738\pi$
$541$	8.31041e9	+	1.43941e10i	0.0970138	+	0.168033i	−0.813433	+	0.581658i	$0.802404\pi$
$542$		0			0
$543$	1.51321e10	−	2.62095e10i	0.174060	−	0.301481i
$544$		0			0
$545$			3.06823e10i			0.347777i
$546$		0			0
$547$	−2.80622e10			−0.313453			−0.156727	−	0.987642i	$-0.550094\pi$
$547$	−2.80622e10			−0.313453			−0.156727	+	0.987642i	$0.550094\pi$
$548$		0			0
$549$	−3.76909e10	−	2.17609e10i	−0.414904	−	0.239545i
$550$		0			0
$551$	4.70733e10	−	2.71778e10i	0.510702	−	0.294854i
$552$		0			0
$553$	−3.83538e10	−	5.54281e10i	−0.410117	−	0.592693i
$554$		0			0
$555$	−9.49522e9	−	1.64462e10i	−0.100077	−	0.173338i
$556$		0			0
$557$	7.66577e10	−	1.32775e11i	0.796407	−	1.37942i	−0.125535	−	0.992089i	$-0.540065\pi$
$557$	7.66577e10	−	1.32775e11i	0.796407	−	1.37942i	0.921942	−	0.387328i	$-0.126602\pi$
$558$		0			0
$559$		−	1.60066e11i		−	1.63927i
$560$		0			0
$561$	−1.07271e11			−1.08301
$562$		0			0
$563$	−3.23010e10	−	1.86490e10i	−0.321501	−	0.185619i	0.330560	−	0.943785i	$-0.392762\pi$
$563$	−3.23010e10	−	1.86490e10i	−0.321501	−	0.185619i	−0.652062	+	0.758166i	$0.726096\pi$
$564$		0			0
$565$	7.51519e9	−	4.33890e9i	0.0737473	−	0.0425781i
$566$		0			0
$567$	−4.91108e9	+	1.03808e10i	−0.0475166	+	0.100438i
$568$		0			0
$569$	7.60551e10	+	1.31731e11i	0.725570	+	1.25672i	0.958739	+	0.284288i	$0.0917571\pi$
$569$	7.60551e10	+	1.31731e11i	0.725570	+	1.25672i	−0.233169	+	0.972436i	$0.574910\pi$
$570$		0			0
$571$	3.27147e10	−	5.66635e10i	0.307750	−	0.533039i	−0.670120	−	0.742253i	$-0.733757\pi$
$571$	3.27147e10	−	5.66635e10i	0.307750	−	0.533039i	0.977870	+	0.209214i	$0.0670906\pi$
$572$		0			0
$573$		−	6.83362e10i		−	0.633917i
$574$		0			0
$575$	−1.36440e11			−1.24816
$576$		0			0
$577$	9.48040e10	+	5.47351e10i	0.855310	+	0.493813i	0.862439	−	0.506161i	$-0.168936\pi$
$577$	9.48040e10	+	5.47351e10i	0.855310	+	0.493813i	−0.00712906	+	0.999975i	$0.502269\pi$
$578$		0			0
$579$	1.72572e10	−	9.96347e9i	0.153552	−	0.0886536i
$580$		0			0
$581$	6.13232e9	+	7.48842e10i	0.0538171	+	0.657182i
$582$		0			0
$583$	−5.39541e10	−	9.34512e10i	−0.467036	−	0.808930i
$584$		0			0
$585$	5.58750e9	−	9.67784e9i	0.0477083	−	0.0826333i
$586$		0			0
$587$			1.75131e11i			1.47507i	0.675311	+	0.737533i	$0.264009\pi$
$587$			1.75131e11i			1.47507i	−0.675311	+	0.737533i	$0.735991\pi$
$588$		0			0
$589$	−4.09401e10			−0.340164
$590$		0			0
$591$	−6.63017e9	−	3.82793e9i	−0.0543469	−	0.0313772i
$592$		0			0
$593$	−1.28936e11	+	7.44413e10i	−1.04269	+	0.601998i	−0.920595	−	0.390520i	$-0.872295\pi$
$593$	−1.28936e11	+	7.44413e10i	−1.04269	+	0.601998i	−0.122097	+	0.992518i	$0.538962\pi$
$594$		0			0
$595$	−2.44630e10	+	2.00329e9i	−0.195183	+	0.0159837i
$596$		0			0
$597$	4.30628e10	+	7.45869e10i	0.339004	+	0.587172i
$598$		0			0
$599$	−2.71938e10	+	4.71010e10i	−0.211233	+	0.365867i	−0.952101	−	0.305785i	$-0.901081\pi$
$599$	−2.71938e10	+	4.71010e10i	−0.211233	+	0.365867i	0.740868	+	0.671651i	$0.234415\pi$
$600$		0			0
$601$			6.23469e10i			0.477878i	0.971035	+	0.238939i	$0.0767996\pi$
$601$			6.23469e10i			0.477878i	−0.971035	+	0.238939i	$0.923200\pi$
$602$		0			0
$603$	2.17863e10			0.164784
$604$		0			0
$605$	−6.93645e10	−	4.00476e10i	−0.517745	−	0.298920i
$606$		0			0
$607$	8.11628e9	−	4.68593e9i	0.0597864	−	0.0345177i	−0.469809	−	0.882768i	$-0.655677\pi$
$607$	8.11628e9	−	4.68593e9i	0.0597864	−	0.0345177i	0.529595	+	0.848250i	$0.322344\pi$
$608$		0			0
$609$	−7.76669e10	−	3.67436e10i	−0.564634	−	0.267124i
$610$		0			0
$611$	−4.68037e10	−	8.10663e10i	−0.335826	−	0.581668i
$612$		0			0
$613$	4.19383e10	−	7.26393e10i	0.297009	−	0.514434i	−0.678441	−	0.734654i	$-0.737344\pi$
$613$	4.19383e10	−	7.26393e10i	0.297009	−	0.514434i	0.975450	+	0.220220i	$0.0706776\pi$
$614$		0			0
$615$		−	1.44947e10i		−	0.101323i
$616$		0			0
$617$	2.87511e10			0.198387			0.0991935	−	0.995068i	$-0.468374\pi$
$617$	2.87511e10			0.198387			0.0991935	+	0.995068i	$0.468374\pi$
$618$		0			0
$619$	3.02776e10	+	1.74808e10i	0.206233	+	0.119069i	0.599560	−	0.800330i	$-0.295342\pi$
$619$	3.02776e10	+	1.74808e10i	0.206233	+	0.119069i	−0.393327	+	0.919399i	$0.628676\pi$
$620$		0			0
$621$	3.23109e10	−	1.86547e10i	0.217261	−	0.125436i
$622$		0			0
$623$	1.35724e11	−	9.39151e10i	0.900958	−	0.623423i
$624$		0			0
$625$	−6.67031e10	−	1.15533e11i	−0.437146	−	0.757158i
$626$		0			0
$627$	4.80232e10	−	8.31786e10i	0.310728	−	0.538197i
$628$		0			0
$629$		−	2.50022e11i		−	1.59726i
$630$		0			0
$631$	3.82364e10			0.241190			0.120595	−	0.992702i	$-0.461520\pi$
$631$	3.82364e10			0.241190			0.120595	+	0.992702i	$0.461520\pi$
$632$		0			0
$633$	−1.00028e11	−	5.77509e10i	−0.623024	−	0.359703i
$634$		0			0
$635$	1.39029e9	−	8.02684e8i	0.00855088	−	0.00493685i
$636$		0			0
$637$	−3.71915e10	−	2.25558e11i	−0.225884	−	1.36993i
$638$		0			0
$639$	2.41486e10	+	4.18266e10i	0.144840	+	0.250870i
$640$		0			0
$641$	4.73930e10	−	8.20871e10i	0.280726	−	0.486231i	−0.690838	−	0.723010i	$-0.742758\pi$
$641$	4.73930e10	−	8.20871e10i	0.280726	−	0.486231i	0.971564	+	0.236779i	$0.0760916\pi$
$642$		0			0
$643$			2.11036e11i			1.23456i	0.786743	+	0.617281i	$0.211766\pi$
$643$			2.11036e11i			1.23456i	−0.786743	+	0.617281i	$0.788234\pi$
$644$		0			0
$645$	−2.43234e10			−0.140536
$646$		0			0
$647$	−2.30606e11	−	1.33140e11i	−1.31599	−	0.759788i	−0.332910	−	0.942959i	$-0.608031\pi$
$647$	−2.30606e11	−	1.33140e11i	−1.31599	−	0.759788i	−0.983081	+	0.183171i	$0.941364\pi$
$648$		0			0
$649$	−5.19498e11	+	2.99932e11i	−2.92823	+	1.69061i
$650$		0			0
$651$	3.68233e10	+	5.32163e10i	0.205021	+	0.296292i
$652$		0			0
$653$	3.84667e9	+	6.66262e9i	0.0211559	+	0.0366431i	0.876410	−	0.481567i	$-0.159932\pi$
$653$	3.84667e9	+	6.66262e9i	0.0211559	+	0.0366431i	−0.855254	+	0.518210i	$0.826599\pi$
$654$		0			0
$655$	8.82774e7	−	1.52901e8i	0.000479605	−	0.000830701i
$656$		0			0
$657$			1.14161e11i			0.612710i
$658$		0			0
$659$	1.44191e11			0.764533			0.382267	−	0.924052i	$-0.375144\pi$
$659$	1.44191e11			0.764533			0.382267	+	0.924052i	$0.375144\pi$
$660$		0			0
$661$	−8.57512e10	−	4.95085e10i	−0.449195	−	0.259343i	0.258295	−	0.966066i	$-0.416839\pi$
$661$	−8.57512e10	−	4.95085e10i	−0.449195	−	0.259343i	−0.707490	+	0.706723i	$0.750173\pi$
$662$		0			0
$663$	1.27415e11	−	7.35633e10i	0.659429	−	0.380721i
$664$		0			0
$665$	9.39824e9	−	1.98656e10i	0.0480574	−	0.101581i
$666$		0			0
$667$	1.39570e11	+	2.41743e11i	0.705163	+	1.22138i
$668$		0			0
$669$	−7.04425e10	+	1.22010e11i	−0.351666	+	0.609103i
$670$		0			0
$671$		−	5.75371e11i		−	2.83830i
$672$		0			0
$673$	1.97102e10			0.0960796			0.0480398	−	0.998845i	$-0.484703\pi$
$673$	1.97102e10			0.0960796			0.0480398	+	0.998845i	$0.484703\pi$
$674$		0			0
$675$	3.31284e10	+	1.91267e10i	0.159582	+	0.0921350i
$676$		0			0
$677$	−2.24608e11	+	1.29677e11i	−1.06923	+	0.617319i	−0.927970	−	0.372654i	$-0.878448\pi$
$677$	−2.24608e11	+	1.29677e11i	−1.06923	+	0.617319i	−0.141257	+	0.989973i	$0.545114\pi$
$678$		0			0
$679$	−1.90444e10	−	2.32558e11i	−0.0895957	−	1.09409i
$680$		0			0
$681$	1.36148e9	+	2.35816e9i	0.00633030	+	0.0109644i
$682$		0			0
$683$	−4.61231e10	+	7.98876e10i	−0.211951	+	0.367110i	−0.952325	−	0.305085i	$-0.901315\pi$
$683$	−4.61231e10	+	7.98876e10i	−0.211951	+	0.367110i	0.740374	+	0.672195i	$0.234648\pi$
$684$		0			0
$685$			8.46339e10i			0.384399i
$686$		0			0
$687$	2.92616e10			0.131362
$688$		0			0
$689$	1.28172e11	+	7.40001e10i	0.568743	+	0.328364i
$690$		0			0
$691$	3.27987e11	−	1.89364e11i	1.43862	−	0.830585i	0.440862	−	0.897575i	$-0.354673\pi$
$691$	3.27987e11	−	1.89364e11i	1.43862	−	0.830585i	0.997754	+	0.0669894i	$0.0213394\pi$
$692$		0			0
$693$	−1.51314e11	+	1.23912e10i	−0.656066	+	0.0537256i
$694$		0			0
$695$	−8.77834e9	−	1.52045e10i	−0.0376247	−	0.0651679i
$696$		0			0
$697$	9.54165e10	−	1.65266e11i	0.404290	−	0.700250i
$698$		0			0
$699$			1.92699e11i			0.807182i
$700$		0			0
$701$	4.60282e10			0.190613			0.0953064	−	0.995448i	$-0.469617\pi$
$701$	4.60282e10			0.190613			0.0953064	+	0.995448i	$0.469617\pi$
$702$		0			0
$703$	1.93868e11	+	1.11930e11i	0.793753	+	0.458273i
$704$		0			0
$705$	−1.23188e10	+	7.11224e9i	−0.0498667	+	0.0287906i
$706$		0			0
$707$	−4.43672e11	−	2.09898e11i	−1.77576	−	0.840099i
$708$		0			0
$709$	1.64597e10	+	2.85091e10i	0.0651385	+	0.112823i	0.896755	−	0.442527i	$-0.145918\pi$
$709$	1.64597e10	+	2.85091e10i	0.0651385	+	0.112823i	−0.831617	+	0.555350i	$0.812584\pi$
$710$		0			0
$711$	3.06981e10	−	5.31707e10i	0.120125	−	0.208063i
$712$		0			0
$713$		−	2.10246e11i		−	0.813524i
$714$		0			0
$715$	1.47737e11			0.565282
$716$		0			0
$717$	6.34536e10	+	3.66350e10i	0.240093	+	0.138618i
$718$		0			0
$719$	−1.28200e11	+	7.40164e10i	−0.479704	+	0.276957i	−0.720293	−	0.693670i	$-0.755993\pi$
$719$	−1.28200e11	+	7.40164e10i	−0.479704	+	0.276957i	0.240589	+	0.970627i	$0.422659\pi$
$720$		0			0
$721$	−1.89914e11	+	1.31412e11i	−0.702774	+	0.486289i
$722$		0			0
$723$	5.40198e10	+	9.35650e10i	0.197697	+	0.342421i
$724$		0			0
$725$	−1.43101e11	+	2.47859e11i	−0.517955	+	0.897124i
$726$		0			0
$727$			1.87221e11i			0.670218i	0.942179	+	0.335109i	$0.108773\pi$
$727$			1.87221e11i			0.670218i	−0.942179	+	0.335109i	$0.891227\pi$
$728$		0			0
$729$	−1.04604e10			−0.0370370
$730$		0			0
$731$	−2.77332e11	−	1.60118e11i	−0.971248	−	0.560750i
$732$		0			0
$733$	−2.72527e11	+	1.57344e11i	−0.944047	+	0.545046i	−0.891227	−	0.453558i	$-0.850154\pi$
$733$	−2.72527e11	+	1.57344e11i	−0.944047	+	0.545046i	−0.0528204	+	0.998604i	$0.516821\pi$
$734$		0			0
$735$	−3.42756e10	+	5.65159e9i	−0.117445	+	0.0193652i
$736$		0			0
$737$	1.44011e11	+	2.49434e11i	0.488118	+	0.845446i
$738$		0			0
$739$	1.91103e11	−	3.31001e11i	0.640753	−	1.10982i	−0.344512	−	0.938782i	$-0.611956\pi$
$739$	1.91103e11	−	3.31001e11i	0.640753	−	1.10982i	0.985265	−	0.171035i	$-0.0547111\pi$
$740$		0			0
$741$			1.31731e11i			0.436934i
$742$		0			0
$743$	3.66837e11			1.20370			0.601850	−	0.798609i	$-0.294431\pi$
$743$	3.66837e11			1.20370			0.601850	+	0.798609i	$0.294431\pi$
$744$		0			0
$745$	5.06637e10	+	2.92507e10i	0.164464	+	0.0949536i
$746$		0			0
$747$	−5.92692e10	+	3.42191e10i	−0.190347	+	0.109897i
$748$		0			0
$749$	1.49170e11	+	2.15578e11i	0.473975	+	0.684978i
$750$		0			0
$751$	1.58136e11	+	2.73899e11i	0.497130	+	0.861054i	0.999995	−	0.00331119i	$-0.00105399\pi$
$751$	1.58136e11	+	2.73899e11i	0.497130	+	0.861054i	−0.502865	+	0.864365i	$0.667721\pi$
$752$		0			0
$753$	−1.55144e9	+	2.68717e9i	−0.00482563	+	0.00835823i
$754$		0			0
$755$		−	8.08748e10i		−	0.248900i
$756$		0			0
$757$	−3.84918e10			−0.117215			−0.0586077	−	0.998281i	$-0.518666\pi$
$757$	−3.84918e10			−0.117215			−0.0586077	+	0.998281i	$0.518666\pi$
$758$		0			0
$759$	4.27160e11	+	2.46621e11i	1.28713	+	0.743127i
$760$		0			0
$761$	2.29914e11	−	1.32741e11i	0.685532	−	0.395792i	−0.116404	−	0.993202i	$-0.537137\pi$
$761$	2.29914e11	−	1.32741e11i	0.685532	−	0.395792i	0.801936	+	0.597410i	$0.203804\pi$
$762$		0			0
$763$	−2.44493e11	+	5.16798e11i	−0.721387	+	1.52483i
$764$		0			0
$765$	−1.11786e10	−	1.93619e10i	−0.0326394	−	0.0565331i
$766$		0			0
$767$	4.11368e11	−	7.12511e11i	1.18864	−	2.05878i
$768$		0			0
$769$		−	5.74390e11i		−	1.64248i	−0.570580	−	0.821242i	$-0.693281\pi$
$769$		−	5.74390e11i		−	1.64248i	0.570580	−	0.821242i	$-0.306719\pi$
$770$		0			0
$771$	6.95351e10			0.196783
$772$		0			0
$773$	3.81782e11	+	2.20422e11i	1.06930	+	0.617358i	0.927989	−	0.372608i	$-0.121536\pi$
$773$	3.81782e11	+	2.20422e11i	1.06930	+	0.617358i	0.141306	+	0.989966i	$0.454870\pi$
$774$		0			0
$775$	1.86685e11	−	1.07783e11i	0.517492	−	0.298774i
$776$		0			0
$777$	−2.88808e10	−	3.52675e11i	−0.0792365	−	0.967589i
$778$		0			0
$779$	8.54321e10	+	1.47973e11i	0.231991	+	0.401820i
$780$		0			0
$781$	−3.19252e11	+	5.52961e11i	−0.858084	+	1.48625i
$782$		0			0
$783$		−	7.82620e10i		−	0.208211i
$784$		0			0
$785$	2.59534e10			0.0683465
$786$		0			0
$787$	5.86998e11	+	3.38904e11i	1.53016	+	0.883441i	0.999354	+	0.0359500i	$0.0114457\pi$
$787$	5.86998e11	+	3.38904e11i	1.53016	+	0.883441i	0.530810	+	0.847491i	$0.321888\pi$
$788$		0			0
$789$	−2.06226e11	+	1.19064e11i	−0.532151	+	0.307237i
$790$		0			0
$791$	1.61157e11	−	1.31973e10i	0.411665	−	0.0337115i
$792$		0			0
$793$	3.94571e11	+	6.83418e11i	0.997776	+	1.72820i
$794$		0			0
$795$	1.12450e10	−	1.94769e10i	0.0281508	−	0.0487586i
$796$		0			0
$797$		−	4.73315e11i		−	1.17305i	−0.809931	−	0.586525i	$-0.800495\pi$
$797$		−	4.73315e11i		−	1.17305i	0.809931	−	0.586525i	$-0.199505\pi$
$798$		0			0
$799$	−1.87275e11			−0.459508
$800$		0			0
$801$	1.30196e11	+	7.51690e10i	0.316278	+	0.182603i
$802$		0			0
$803$	−1.30704e12	+	7.54620e11i	−3.14359	+	1.81496i
$804$		0			0
$805$	1.02019e11	+	4.82643e10i	0.242939	+	0.114932i
$806$		0			0
$807$	1.92685e11	+	3.33740e11i	0.454311	+	0.786890i
$808$		0			0
$809$	2.95659e11	−	5.12096e11i	0.690235	−	1.19552i	−0.281526	−	0.959554i	$-0.590841\pi$
$809$	2.95659e11	−	5.12096e11i	0.690235	−	1.19552i	0.971761	−	0.235968i	$-0.0758261\pi$
$810$		0			0
$811$			4.08161e11i			0.943513i	0.881729	+	0.471756i	$0.156380\pi$
$811$			4.08161e11i			0.943513i	−0.881729	+	0.471756i	$0.843620\pi$
$812$		0			0
$813$	−2.17259e11			−0.497296
$814$		0			0
$815$	2.98404e10	+	1.72284e10i	0.0676355	+	0.0390494i
$816$		0			0
$817$	2.48311e11	−	1.43363e11i	0.557325	−	0.321772i
$818$		0			0
$819$	1.71232e11	−	1.18485e11i	0.380582	−	0.263346i
$820$		0			0
$821$	−2.37680e11	−	4.11675e11i	−0.523144	−	0.906111i	−0.999637	−	0.0269333i	$-0.991426\pi$
$821$	−2.37680e11	−	4.11675e11i	−0.523144	−	0.906111i	0.476494	−	0.879178i	$-0.341908\pi$
$822$		0			0
$823$	2.56842e11	−	4.44863e11i	0.559844	−	0.969678i	−0.437665	−	0.899138i	$-0.644195\pi$
$823$	2.56842e11	−	4.44863e11i	0.559844	−	0.969678i	0.997509	−	0.0705396i	$-0.0224721\pi$
$824$		0			0
$825$			5.05722e11i			1.09168i
$826$		0			0
$827$	7.48614e11			1.60043			0.800214	−	0.599715i	$-0.204719\pi$
$827$	7.48614e11			1.60043			0.800214	+	0.599715i	$0.204719\pi$
$828$		0			0
$829$	−6.14063e10	−	3.54529e10i	−0.130015	−	0.0750644i	0.433582	−	0.901114i	$-0.357250\pi$
$829$	−6.14063e10	−	3.54529e10i	−0.130015	−	0.0750644i	−0.563597	+	0.826050i	$0.690583\pi$
$830$		0			0
$831$	−1.01604e11	+	5.86612e10i	−0.213063	+	0.123012i
$832$		0			0
$833$	−4.28007e11	−	1.61192e11i	−0.888938	−	0.334784i
$834$		0			0
$835$	2.15402e10	+	3.73087e10i	0.0443102	+	0.0767476i
$836$		0			0
$837$	−2.94731e10	+	5.10490e10i	−0.0600516	+	0.104012i
$838$		0			0
$839$		−	3.83142e11i		−	0.773235i	−0.922240	−	0.386618i	$-0.873643\pi$
$839$		−	3.83142e11i		−	0.773235i	0.922240	−	0.386618i	$-0.126357\pi$
$840$		0			0
$841$	8.52917e10			0.170499
$842$		0			0
$843$	−8.86576e10	−	5.11865e10i	−0.175552	−	0.101355i
$844$		0			0
$845$	−8.44514e10	+	4.87581e10i	−0.165646	+	0.0956356i
$846$		0			0
$847$	−8.49223e11	−	1.22728e12i	−1.65001	−	2.38457i
$848$		0			0
$849$	−1.49612e10	−	2.59136e10i	−0.0287962	−	0.0498766i
$850$		0			0
$851$	−5.74811e11	+	9.95602e11i	−1.09599	+	1.89831i
$852$		0			0
$853$		−	2.00261e11i		−	0.378268i	−0.981951	−	0.189134i	$-0.939432\pi$
$853$		−	2.00261e11i		−	0.378268i	0.981951	−	0.189134i	$-0.0605680\pi$
$854$		0			0
$855$	2.00178e10			0.0374586
$856$		0			0
$857$	−5.12447e11	−	2.95862e11i	−0.950005	−	0.548486i	−0.0569227	−	0.998379i	$-0.518129\pi$
$857$	−5.12447e11	−	2.95862e11i	−0.950005	−	0.548486i	−0.893083	+	0.449893i	$0.851462\pi$
$858$		0			0
$859$	−1.34764e10	+	7.78060e9i	−0.0247515	+	0.0142903i	−0.512325	−	0.858792i	$-0.671216\pi$
$859$	−1.34764e10	+	7.78060e9i	−0.0247515	+	0.0142903i	0.487573	+	0.873082i	$0.337882\pi$
$860$		0			0
$861$	1.15502e11	−	2.44142e11i	0.210173	−	0.444253i
$862$		0			0
$863$	−3.22577e11	−	5.58719e11i	−0.581553	−	1.00728i	−0.995295	−	0.0968860i	$-0.969112\pi$
$863$	−3.22577e11	−	5.58719e11i	−0.581553	−	1.00728i	0.413742	−	0.910394i	$-0.364222\pi$
$864$		0			0
$865$	−1.07227e9	+	1.85723e9i	−0.00191532	+	0.00331743i
$866$		0			0
$867$			3.18752e10i			0.0564127i
$868$		0			0
$869$	8.11678e11			1.42333
$870$		0			0
$871$	−3.42108e11	−	1.97516e11i	−0.594416	−	0.343186i
$872$		0			0
$873$	1.84065e11	−	1.06270e11i	0.316894	−	0.182959i
$874$		0			0
$875$	1.93080e10	+	2.35777e11i	0.0329385	+	0.402226i
$876$		0			0
$877$	3.51275e11	+	6.08427e11i	0.593812	+	1.02851i	0.993713	+	0.111955i	$0.0357112\pi$
$877$	3.51275e11	+	6.08427e11i	0.593812	+	1.02851i	−0.399901	+	0.916558i	$0.630955\pi$
$878$		0			0
$879$	3.47294e10	−	6.01531e10i	0.0581758	−	0.100763i
$880$		0			0
$881$			4.60476e11i			0.764369i	0.924086	+	0.382185i	$0.124828\pi$
$881$			4.60476e11i			0.764369i	−0.924086	+	0.382185i	$0.875172\pi$
$882$		0			0
$883$	−5.63870e11			−0.927547			−0.463774	−	0.885954i	$-0.653505\pi$
$883$	−5.63870e11			−0.927547			−0.463774	+	0.885954i	$0.653505\pi$
$884$		0			0
$885$	−1.08272e11	−	6.25111e10i	−0.176500	−	0.101902i
$886$		0			0
$887$	−3.07767e11	+	1.77690e11i	−0.497197	+	0.287057i	−0.727555	−	0.686049i	$-0.759343\pi$
$887$	−3.07767e11	+	1.77690e11i	−0.497197	+	0.287057i	0.230358	+	0.973106i	$0.426010\pi$
$888$		0			0
$889$	2.98136e10	−	2.44146e9i	0.0477318	−	0.00390879i
$890$		0			0
$891$	−6.91446e10	−	1.19762e11i	−0.109710	−	0.190024i
$892$		0			0
$893$	8.38393e10	−	1.45214e11i	0.131838	−	0.228351i
$894$		0			0
$895$			7.07093e10i			0.110201i
$896$		0			0
$897$	−6.76500e11			−1.04496
$898$		0			0
$899$	−3.81937e11	−	2.20511e11i	−0.584726	−	0.337592i
$900$		0			0
$901$	2.56427e11	−	1.48048e11i	0.389103	−	0.224648i
$902$		0			0
$903$	−4.09693e11	−	1.93822e11i	−0.616180	−	0.291510i
$904$		0			0
$905$	4.16941e10	+	7.22164e10i	0.0621557	+	0.107657i
$906$		0			0
$907$	−1.67768e11	+	2.90583e11i	−0.247902	+	0.429379i	−0.962944	−	0.269703i	$-0.913075\pi$
$907$	−1.67768e11	+	2.90583e11i	−0.247902	+	0.429379i	0.715041	+	0.699082i	$0.246408\pi$
$908$		0			0
$909$		−	4.47071e11i		−	0.654819i
$910$		0			0
$911$	8.91134e11			1.29381			0.646904	−	0.762572i	$-0.276064\pi$
$911$	8.91134e11			1.29381			0.646904	+	0.762572i	$0.276064\pi$
$912$		0			0
$913$	−7.83557e11	−	4.52387e11i	−1.12768	−	0.651069i
$914$		0			0
$915$	1.03852e11	−	5.99587e10i	0.148159	−	0.0855398i
$916$		0			0
$917$	2.70530e9	−	1.87195e9i	0.00382594	−	0.00264738i
$918$		0			0
$919$	−1.24743e11	−	2.16061e11i	−0.174885	−	0.302911i	0.765236	−	0.643750i	$-0.222622\pi$
$919$	−1.24743e11	−	2.16061e11i	−0.174885	−	0.302911i	−0.940122	+	0.340839i	$0.889289\pi$
$920$		0			0
$921$	3.11428e11	−	5.39409e11i	0.432832	−	0.749687i
$922$		0			0
$923$		−	8.75733e11i		−	1.20660i
$924$		0			0
$925$	−1.17871e12			−1.61005
$926$		0			0
$927$	−1.82179e11	−	1.05181e11i	−0.246706	−	0.142436i
$928$		0			0
$929$	−4.86525e11	+	2.80895e11i	−0.653194	+	0.377122i	−0.789679	−	0.613521i	$-0.789753\pi$
$929$	−4.86525e11	+	2.80895e11i	−0.653194	+	0.377122i	0.136485	+	0.990642i	$0.456419\pi$
$930$		0			0
$931$	3.16599e11	−	2.59716e11i	0.421416	−	0.345701i
$932$		0			0
$933$	8.95086e10	+	1.55033e11i	0.118124	+	0.204597i
$934$		0			0
$935$	1.47785e11	−	2.55971e11i	0.193368	−	0.334922i
$936$		0			0
$937$		−	6.30931e11i		−	0.818510i	−0.912420	−	0.409255i	$-0.865789\pi$
$937$		−	6.30931e11i		−	0.818510i	0.912420	−	0.409255i	$-0.134211\pi$
$938$		0			0
$939$	−1.03642e11			−0.133314
$940$		0			0
$941$	9.15139e11	+	5.28356e11i	1.16716	+	0.673857i	0.953008	−	0.302944i	$-0.0979695\pi$
$941$	9.15139e11	+	5.28356e11i	1.16716	+	0.673857i	0.214147	+	0.976801i	$0.431303\pi$
$942$		0			0
$943$	−7.59907e11	+	4.38733e11i	−0.960979	+	0.554821i
$944$		0			0
$945$	−1.80048e10	−	2.60202e10i	−0.0225768	−	0.0326275i
$946$		0			0
$947$	7.00683e11	+	1.21362e12i	0.871207	+	1.50898i	0.860749	+	0.509030i	$0.169996\pi$
$947$	7.00683e11	+	1.21362e12i	0.871207	+	1.50898i	0.0104586	+	0.999945i	$0.496671\pi$
$948$		0			0
$949$	1.03499e12	−	1.79265e12i	1.27606	−	2.21020i
$950$		0			0
$951$			2.36625e10i			0.0289294i
$952$		0			0
$953$	−6.29698e11			−0.763415			−0.381707	−	0.924283i	$-0.624664\pi$
$953$	−6.29698e11			−0.763415			−0.381707	+	0.924283i	$0.624664\pi$
$954$		0			0
$955$	1.63064e11	+	9.41450e10i	0.196040	+	0.113184i
$956$		0			0
$957$	8.96031e11	−	5.17324e11i	1.06826	−	0.616758i
$958$		0			0
$959$	−6.74409e11	+	1.42553e12i	−0.797350	+	1.68540i
$960$		0			0
$961$	−2.60358e11	−	4.50954e11i	−0.305266	−	0.528736i
$962$		0			0
$963$	−1.19395e11	+	2.06798e11i	−0.138829	+	0.240459i
$964$		0			0
$965$			5.49056e10i			0.0633151i
$966$		0			0
$967$	8.76815e11			1.00277			0.501386	−	0.865224i	$-0.332824\pi$
$967$	8.76815e11			1.00277			0.501386	+	0.865224i	$0.332824\pi$
$968$		0			0
$969$	2.28239e11	+	1.31774e11i	0.258878	+	0.149463i
$970$		0			0
$971$	−2.84265e11	+	1.64121e11i	−0.319777	+	0.184623i	−0.651293	−	0.758826i	$-0.725773\pi$
$971$	−2.84265e11	+	1.64121e11i	−0.319777	+	0.184623i	0.331516	+	0.943449i	$0.392440\pi$
$972$		0			0
$973$	−2.67003e10	−	3.26049e11i	−0.0297896	−	0.363774i
$974$		0			0
$975$	−3.46808e11	−	6.00689e11i	−0.383770	−	0.664709i
$976$		0			0
$977$	−1.77711e11	+	3.07804e11i	−0.195046	+	0.337829i	−0.946915	−	0.321483i	$-0.895819\pi$
$977$	−1.77711e11	+	3.07804e11i	−0.195046	+	0.337829i	0.751870	+	0.659311i	$0.229152\pi$
$978$		0			0
$979$			1.98751e12i			2.16361i
$980$		0			0
$981$	−5.20757e11			−0.562289
$982$		0			0
$983$	−1.12439e12	−	6.49165e11i	−1.20421	−	0.695250i	−0.242720	−	0.970096i	$-0.578040\pi$
$983$	−1.12439e12	−	6.49165e11i	−1.20421	−	0.695250i	−0.961488	+	0.274846i	$0.911373\pi$
$984$		0			0
$985$	1.82684e10	−	1.05473e10i	0.0194069	−	0.0112046i
$986$		0			0
$987$	−2.64166e11	+	2.16327e10i	−0.278361	+	0.0227951i
$988$		0			0
$989$	7.36233e11	+	1.27519e12i	0.769538	+	1.33288i
$990$		0			0
$991$	5.60644e11	−	9.71063e11i	0.581289	−	1.00682i	−0.414038	−	0.910260i	$-0.635882\pi$
$991$	5.60644e11	−	9.71063e11i	0.581289	−	1.00682i	0.995327	−	0.0965627i	$-0.0307848\pi$
$992$		0			0
$993$			7.34225e11i			0.755148i
$994$		0			0
$995$	−2.37306e11			−0.242112
$996$		0			0
$997$	1.13941e12	+	6.57840e11i	1.15319	+	0.665794i	0.949662	−	0.313275i	$-0.101426\pi$
$997$	1.13941e12	+	6.57840e11i	1.15319	+	0.665794i	0.203527	+	0.979069i	$0.434760\pi$
$998$		0			0
$999$	2.79135e11	−	1.61158e11i	0.280254	−	0.161805i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.9.m.a.73.3	yes	10
3.2	odd	2		252.9.z.b.73.3		10
7.5	odd	6	inner	84.9.m.a.61.3	✓	10
21.5	even	6		252.9.z.b.145.3		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.9.m.a.61.3	✓	10	7.5	odd	6	inner
84.9.m.a.73.3	yes	10	1.1	even	1	trivial
252.9.z.b.73.3		10	3.2	odd	2
252.9.z.b.145.3		10	21.5	even	6

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)

\(f(q)\)	\(=\)	\(q+(-40.5000 - 23.3827i) q^{3} +(111.592 - 64.4274i) q^{5} +(2392.99 - 195.963i) q^{7} +(1093.50 + 1894.00i) q^{9} +O(q^{10})\)	\(q+(-40.5000 - 23.3827i) q^{3} +(111.592 - 64.4274i) q^{5} +(2392.99 - 195.963i) q^{7} +(1093.50 + 1894.00i) q^{9} +(-14456.4 + 25039.2i) q^{11} -39655.0i q^{13} -6025.94 q^{15} +(-68706.8 - 39667.9i) q^{17} +(61517.2 - 35517.0i) q^{19} +(-101498. - 48018.0i) q^{21} +(182396. + 315919. i) q^{23} +(-187011. + 323912. i) q^{25} -102276. i q^{27} +765205. q^{29} +(-499130. - 288173. i) q^{31} +(1.17097e6 - 676060. i) q^{33} +(254412. - 176042. i) q^{35} +(1.57572e6 + 2.72923e6i) q^{37} +(-927241. + 1.60603e6i) q^{39} +2.40539e6i q^{41} +4.03645e6 q^{43} +(244051. + 140903. i) q^{45} +(2.04429e6 - 1.18027e6i) q^{47} +(5.68800e6 - 937877. i) q^{49} +(1.85508e6 + 3.21310e6i) q^{51} +(-1.86610e6 + 3.23217e6i) q^{53} +3.72556e6i q^{55} -3.32193e6 q^{57} +(1.79677e7 + 1.03737e7i) q^{59} +(-1.72341e7 + 9.95010e6i) q^{61} +(2.98789e6 + 4.31803e6i) q^{63} +(-2.55487e6 - 4.42517e6i) q^{65} +(4.98086e6 - 8.62710e6i) q^{67} -1.70596e7i q^{69} +2.20838e7 q^{71} +(4.52062e7 + 2.60998e7i) q^{73} +(1.51479e7 - 8.74563e6i) q^{75} +(-2.96873e7 + 6.27516e7i) q^{77} +(-1.40366e7 - 2.43122e7i) q^{79} +(-2.39148e6 + 4.14217e6i) q^{81} +3.12932e7i q^{83} -1.02228e7 q^{85} +(-3.09908e7 - 1.78925e7i) q^{87} +(5.95320e7 - 3.43708e7i) q^{89} +(-7.77093e6 - 9.48941e7i) q^{91} +(1.34765e7 + 2.33420e7i) q^{93} +(4.57653e6 - 7.92679e6i) q^{95} -9.71832e7i q^{97} -6.32324e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 405 q^{3} + 1389 q^{5} + 1217 q^{7} + 10935 q^{9}+O(q^{10}) \) 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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