LMFDB - Embedded newform 84.9.m.a.73.4

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			73.4
Root			$-190.225 - 109.826i$ of defining polynomial
Character	$\chi$	$=$	84.73
Dual form			84.9.m.a.61.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-40.5000 - 23.3827i) q^{3} +(656.878 - 379.249i) q^{5} +(1906.96 + 1458.87i) q^{7} +(1093.50 + 1894.00i) q^{9} +O(q^{10})$	\(q+(-40.5000 - 23.3827i) q^{3} +(656.878 - 379.249i) q^{5} +(1906.96 + 1458.87i) q^{7} +(1093.50 + 1894.00i) q^{9} +(6909.25 - 11967.2i) q^{11} +39984.1i q^{13} -35471.4 q^{15} +(33086.2 + 19102.3i) q^{17} +(-119061. + 68739.9i) q^{19} +(-43119.7 - 103674. i) q^{21} +(-74519.7 - 129072. i) q^{23} +(92347.0 - 159950. i) q^{25} -102276. i q^{27} +277619. q^{29} +(1.48847e6 + 859370. i) q^{31} +(-559649. + 323114. i) q^{33} +(1.80592e6 + 235086. i) q^{35} +(235213. + 407400. i) q^{37} +(934936. - 1.61936e6i) q^{39} -2.24961e6i q^{41} +4.59615e6 q^{43} +(1.43659e6 + 829417. i) q^{45} +(3.14538e6 - 1.81598e6i) q^{47} +(1.50821e6 + 5.56401e6i) q^{49} +(-893327. - 1.54729e6i) q^{51} +(-2.54165e6 + 4.40227e6i) q^{53} -1.04813e7i q^{55} +6.42929e6 q^{57} +(2.58932e6 + 1.49495e6i) q^{59} +(1.11403e7 - 6.43183e6i) q^{61} +(-677831. + 5.20705e6i) q^{63} +(1.51639e7 + 2.62647e7i) q^{65} +(-127634. + 221068. i) q^{67} +6.96989e6i q^{69} +4.77334e7 q^{71} +(-1.52724e7 - 8.81754e6i) q^{73} +(-7.48010e6 + 4.31864e6i) q^{75} +(3.06342e7 - 1.27412e7i) q^{77} +(-1.56844e7 - 2.71662e7i) q^{79} +(-2.39148e6 + 4.14217e6i) q^{81} -7.76636e7i q^{83} +2.89781e7 q^{85} +(-1.12436e7 - 6.49147e6i) q^{87} +(-7.90230e7 + 4.56240e7i) q^{89} +(-5.83315e7 + 7.62481e7i) q^{91} +(-4.01887e7 - 6.96090e7i) q^{93} +(-5.21391e7 + 9.03075e7i) q^{95} -2.05936e7i q^{97} +3.02210e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	656.878	−	379.249i	1.05101	−	0.606798i	0.128075	−	0.991765i	$-0.459120\pi$
$5$	656.878	−	379.249i	1.05101	−	0.606798i	0.922931	+	0.384966i	$0.125787\pi$
$6$		0			0
$7$	1906.96	+	1458.87i	0.794236	+	0.607609i
$7$	1906.96	+	1458.87i	0.794236	+	0.607609i
$8$		0			0
$9$	1093.50	+	1894.00i	0.166667	+	0.288675i
$10$		0			0
$11$	6909.25	−	11967.2i	0.471911	−	0.817374i	−0.527573	−	0.849510i	$-0.676898\pi$
$11$	6909.25	−	11967.2i	0.471911	−	0.817374i	0.999483	+	0.0321363i	$0.0102311\pi$
$12$		0			0
$13$			39984.1i			1.39995i	0.714165	+	0.699977i	$0.246806\pi$
$13$			39984.1i			1.39995i	−0.714165	+	0.699977i	$0.753194\pi$
$14$		0			0
$15$	−35471.4			−0.700670
$16$		0			0
$17$	33086.2	+	19102.3i	0.396142	+	0.228713i	0.684818	−	0.728714i	$-0.259882\pi$
$17$	33086.2	+	19102.3i	0.396142	+	0.228713i	−0.288676	+	0.957427i	$0.593215\pi$
$18$		0			0
$19$	−119061.	+	68739.9i	−0.913598	+	0.527466i	−0.881587	−	0.472021i	$-0.843524\pi$
$19$	−119061.	+	68739.9i	−0.913598	+	0.527466i	−0.0320110	+	0.999488i	$0.510191\pi$
$20$		0			0
$21$	−43119.7	−	103674.i	−0.221717	−	0.533081i
$22$		0			0
$23$	−74519.7	−	129072.i	−0.266293	−	0.461233i	0.701608	−	0.712563i	$-0.252466\pi$
$23$	−74519.7	−	129072.i	−0.266293	−	0.461233i	−0.967902	+	0.251329i	$0.919132\pi$
$24$		0			0
$25$	92347.0	−	159950.i	0.236408	−	0.409471i
$26$		0			0
$27$		−	102276.i		−	0.192450i
$28$		0			0
$29$	277619.			0.392515			0.196258	−	0.980552i	$-0.437121\pi$
$29$	277619.			0.392515			0.196258	+	0.980552i	$0.437121\pi$
$30$		0			0
$31$	1.48847e6	+	859370.i	1.61174	+	0.930536i	0.988968	+	0.148130i	$0.0473254\pi$
$31$	1.48847e6	+	859370.i	1.61174	+	0.930536i	0.622768	+	0.782406i	$0.286008\pi$
$32$		0			0
$33$	−559649.	+	323114.i	−0.471911	+	0.272458i
$34$		0			0
$35$	1.80592e6	+	235086.i	1.20344	+	0.156659i
$36$		0			0
$37$	235213.	+	407400.i	0.125503	+	0.217377i	0.921929	−	0.387358i	$-0.126612\pi$
$37$	235213.	+	407400.i	0.125503	+	0.217377i	−0.796426	+	0.604735i	$0.793279\pi$
$38$		0			0
$39$	934936.	−	1.61936e6i	0.404132	−	0.699977i
$40$		0			0
$41$		−	2.24961e6i		−	0.796108i	−0.917362	−	0.398054i	$-0.869686\pi$
$41$		−	2.24961e6i		−	0.796108i	0.917362	−	0.398054i	$-0.130314\pi$
$42$		0			0
$43$	4.59615e6			1.34437			0.672187	−	0.740381i	$-0.265355\pi$
$43$	4.59615e6			1.34437			0.672187	+	0.740381i	$0.265355\pi$
$44$		0			0
$45$	1.43659e6	+	829417.i	0.350335	+	0.202266i
$46$		0			0
$47$	3.14538e6	−	1.81598e6i	0.644586	−	0.372152i	−0.141793	−	0.989896i	$-0.545287\pi$
$47$	3.14538e6	−	1.81598e6i	0.644586	−	0.372152i	0.786379	+	0.617744i	$0.211953\pi$
$48$		0			0
$49$	1.50821e6	+	5.56401e6i	0.261623	+	0.965170i
$50$		0			0
$51$	−893327.	−	1.54729e6i	−0.132047	−	0.228713i
$52$		0			0
$53$	−2.54165e6	+	4.40227e6i	−0.322116	+	0.557921i	−0.980924	−	0.194390i	$-0.937727\pi$
$53$	−2.54165e6	+	4.40227e6i	−0.322116	+	0.557921i	0.658809	+	0.752311i	$0.271061\pi$
$54$		0			0
$55$		−	1.04813e7i		−	1.14542i
$56$		0			0
$57$	6.42929e6			0.609065
$58$		0			0
$59$	2.58932e6	+	1.49495e6i	0.213687	+	0.123372i	0.603024	−	0.797723i	$-0.293962\pi$
$59$	2.58932e6	+	1.49495e6i	0.213687	+	0.123372i	−0.389337	+	0.921095i	$0.627296\pi$
$60$		0			0
$61$	1.11403e7	−	6.43183e6i	0.804592	−	0.464531i	−0.0404824	−	0.999180i	$-0.512889\pi$
$61$	1.11403e7	−	6.43183e6i	0.804592	−	0.464531i	0.845074	+	0.534649i	$0.179556\pi$
$62$		0			0
$63$	−677831.	+	5.20705e6i	−0.0430288	+	0.330544i
$64$		0			0
$65$	1.51639e7	+	2.62647e7i	0.849490	+	1.47136i
$66$		0			0
$67$	−127634.	+	221068.i	−0.00633382	+	0.0109705i	−0.869175	−	0.494505i	$-0.835349\pi$
$67$	−127634.	+	221068.i	−0.00633382	+	0.0109705i	0.862841	+	0.505475i	$0.168683\pi$
$68$		0			0
$69$			6.96989e6i			0.307489i
$70$		0			0
$71$	4.77334e7			1.87840			0.939201	−	0.343367i	$-0.111567\pi$
$71$	4.77334e7			1.87840			0.939201	+	0.343367i	$0.111567\pi$
$72$		0			0
$73$	−1.52724e7	−	8.81754e6i	−0.537795	−	0.310496i	0.206390	−	0.978470i	$-0.433829\pi$
$73$	−1.52724e7	−	8.81754e6i	−0.537795	−	0.310496i	−0.744185	+	0.667974i	$0.767162\pi$
$74$		0			0
$75$	−7.48010e6	+	4.31864e6i	−0.236408	+	0.136490i
$76$		0			0
$77$	3.06342e7	−	1.27412e7i	0.871452	−	0.362451i
$78$		0			0
$79$	−1.56844e7	−	2.71662e7i	−0.402679	−	0.697461i	0.591369	−	0.806401i	$-0.298588\pi$
$79$	−1.56844e7	−	2.71662e7i	−0.402679	−	0.697461i	−0.994048	+	0.108940i	$0.965254\pi$
$80$		0			0
$81$	−2.39148e6	+	4.14217e6i	−0.0555556	+	0.0962250i
$82$		0			0
$83$		−	7.76636e7i		−	1.63646i	−0.574892	−	0.818229i	$-0.694956\pi$
$83$		−	7.76636e7i		−	1.63646i	0.574892	−	0.818229i	$-0.305044\pi$
$84$		0			0
$85$	2.89781e7			0.555130
$86$		0			0
$87$	−1.12436e7	−	6.49147e6i	−0.196258	−	0.113309i
$88$		0			0
$89$	−7.90230e7	+	4.56240e7i	−1.25949	+	0.727165i	−0.972975	−	0.230912i	$-0.925829\pi$
$89$	−7.90230e7	+	4.56240e7i	−1.25949	+	0.727165i	−0.286512	+	0.958077i	$0.592496\pi$
$90$		0			0
$91$	−5.83315e7	+	7.62481e7i	−0.850624	+	1.11189i
$92$		0			0
$93$	−4.01887e7	−	6.96090e7i	−0.537245	−	0.930536i
$94$		0			0
$95$	−5.21391e7	+	9.03075e7i	−0.640131	+	1.10874i
$96$		0			0
$97$		−	2.05936e7i		−	0.232620i	−0.993213	−	0.116310i	$-0.962893\pi$
$97$		−	2.05936e7i		−	0.232620i	0.993213	−	0.116310i	$-0.0371065\pi$
$98$		0			0
$99$	3.02210e7			0.314607
$100$		0			0
$101$	1.11330e8	+	6.42761e7i	1.06985	+	0.617681i	0.928142	−	0.372227i	$-0.121406\pi$
$101$	1.11330e8	+	6.42761e7i	1.06985	+	0.617681i	0.141713	+	0.989908i	$0.454739\pi$
$102$		0			0
$103$	−3.52077e7	+	2.03272e7i	−0.312816	+	0.180604i	−0.648186	−	0.761482i	$-0.724472\pi$
$103$	−3.52077e7	+	2.03272e7i	−0.312816	+	0.180604i	0.335370	+	0.942086i	$0.391139\pi$
$104$		0			0
$105$	−6.76427e7	−	5.17482e7i	−0.556498	−	0.425733i
$106$		0			0
$107$	−5.32367e7	−	9.22087e7i	−0.406140	−	0.703456i	0.588313	−	0.808633i	$-0.299792\pi$
$107$	−5.32367e7	−	9.22087e7i	−0.406140	−	0.703456i	−0.994453	+	0.105178i	$0.966459\pi$
$108$		0			0
$109$	−1.80362e7	+	3.12396e7i	−0.127773	+	0.221309i	−0.922814	−	0.385247i	$-0.874116\pi$
$109$	−1.80362e7	+	3.12396e7i	−0.127773	+	0.221309i	0.795040	+	0.606556i	$0.207450\pi$
$110$		0			0
$111$		−	2.19996e7i		−	0.144918i
$112$		0			0
$113$	−2.06746e8			−1.26801			−0.634006	−	0.773328i	$-0.718591\pi$
$113$	−2.06746e8			−1.26801			−0.634006	+	0.773328i	$0.718591\pi$
$114$		0			0
$115$	−9.79008e7	−	5.65231e7i	−0.559751	−	0.323172i
$116$		0			0
$117$	−7.57298e7	+	4.37226e7i	−0.404132	+	0.233326i
$118$		0			0
$119$	3.52263e7	+	8.46958e7i	0.175663	+	0.422351i
$120$		0			0
$121$	1.17041e7	+	2.02720e7i	0.0546003	+	0.0945704i
$122$		0			0
$123$	−5.26019e7	+	9.11092e7i	−0.229817	+	0.398054i
$124$		0			0
$125$			1.56198e8i			0.639788i
$126$		0			0
$127$	−4.22341e8			−1.62348			−0.811742	−	0.584017i	$-0.801480\pi$
$127$	−4.22341e8			−1.62348			−0.811742	+	0.584017i	$0.801480\pi$
$128$		0			0
$129$	−1.86144e8	−	1.07470e8i	−0.672187	−	0.388088i
$130$		0			0
$131$	4.55181e8	−	2.62799e8i	1.54561	−	0.892356i	0.547137	−	0.837043i	$-0.315718\pi$
$131$	4.55181e8	−	2.62799e8i	1.54561	−	0.892356i	0.998469	−	0.0553133i	$-0.0176158\pi$
$132$		0			0
$133$	−3.27327e8	−	4.26100e7i	−1.04611	−	0.136177i
$134$		0			0
$135$	−3.87880e7	−	6.71828e7i	−0.116778	−	0.202266i
$136$		0			0
$137$	−6.33946e7	+	1.09803e8i	−0.179958	+	0.311696i	−0.941866	−	0.335989i	$-0.890929\pi$
$137$	−6.33946e7	+	1.09803e8i	−0.179958	+	0.311696i	0.761908	+	0.647685i	$0.224263\pi$
$138$		0			0
$139$		−	3.98228e8i		−	1.06678i	−0.845871	−	0.533388i	$-0.820919\pi$
$139$		−	3.98228e8i		−	1.06678i	0.845871	−	0.533388i	$-0.179081\pi$
$140$		0			0
$141$	−1.69850e8			−0.429724
$142$		0			0
$143$	4.78496e8	+	2.76260e8i	1.14429	+	0.660654i
$144$		0			0
$145$	1.82362e8	−	1.05287e8i	0.412536	−	0.238178i
$146$		0			0
$147$	6.90192e7	−	2.60608e8i	0.147809	−	0.558109i
$148$		0			0
$149$	4.11857e8	+	7.13356e8i	0.835605	+	1.44731i	0.893537	+	0.448989i	$0.148216\pi$
$149$	4.11857e8	+	7.13356e8i	0.835605	+	1.44731i	−0.0579324	+	0.998321i	$0.518451\pi$
$150$		0			0
$151$	1.05855e8	−	1.83346e8i	0.203612	−	0.352666i	−0.746078	−	0.665859i	$-0.768065\pi$
$151$	1.05855e8	−	1.83346e8i	0.203612	−	0.352666i	0.949690	+	0.313193i	$0.101399\pi$
$152$		0			0
$153$			8.35535e7i			0.152475i
$154$		0			0
$155$	1.30366e9			2.25859
$156$		0			0
$157$	4.91092e8	+	2.83532e8i	0.808284	+	0.466663i	0.846360	−	0.532612i	$-0.178789\pi$
$157$	4.91092e8	+	2.83532e8i	0.808284	+	0.466663i	−0.0380754	+	0.999275i	$0.512123\pi$
$158$		0			0
$159$	2.05874e8	−	1.18861e8i	0.322116	−	0.185974i
$160$		0			0
$161$	4.61928e7	−	3.54850e8i	0.0687496	−	0.528130i
$162$		0			0
$163$	−4.07796e8	−	7.06324e8i	−0.577688	−	1.00058i	−0.995744	−	0.0921631i	$-0.970622\pi$
$163$	−4.07796e8	−	7.06324e8i	−0.577688	−	1.00058i	0.418056	−	0.908421i	$-0.362711\pi$
$164$		0			0
$165$	−2.45081e8	+	4.24493e8i	−0.330654	+	0.572709i
$166$		0			0
$167$			1.06089e9i			1.36397i	0.731366	+	0.681985i	$0.238883\pi$
$167$			1.06089e9i			1.36397i	−0.731366	+	0.681985i	$0.761117\pi$
$168$		0			0
$169$	−7.82997e8			−0.959872
$170$		0			0
$171$	−2.60386e8	−	1.50334e8i	−0.304533	−	0.175822i
$172$		0			0
$173$	7.06239e8	−	4.07747e8i	0.788437	−	0.455204i	−0.0509750	−	0.998700i	$-0.516233\pi$
$173$	7.06239e8	−	4.07747e8i	0.788437	−	0.455204i	0.839412	+	0.543496i	$0.182900\pi$
$174$		0			0
$175$	4.09448e8	−	1.70296e8i	0.436562	−	0.181573i
$176$		0			0
$177$	−6.99117e7	−	1.21091e8i	−0.0712290	−	0.123372i
$178$		0			0
$179$	−1.06736e8	+	1.84873e8i	−0.103968	+	0.180078i	−0.913316	−	0.407251i	$-0.866487\pi$
$179$	−1.06736e8	+	1.84873e8i	−0.103968	+	0.180078i	0.809348	+	0.587329i	$0.199821\pi$
$180$		0			0
$181$			7.38533e8i			0.688107i	0.938950	+	0.344053i	$0.111800\pi$
$181$			7.38533e8i			0.688107i	−0.938950	+	0.344053i	$0.888200\pi$
$182$		0			0
$183$	−6.01574e8			−0.536395
$184$		0			0
$185$	3.09012e8	+	1.78408e8i	0.263809	+	0.152310i
$186$		0			0
$187$	4.57201e8	−	2.63965e8i	0.373888	−	0.215864i
$188$		0			0
$189$	1.49207e8	−	1.95036e8i	0.116934	−	0.152851i
$190$		0			0
$191$	−1.52734e8	−	2.64542e8i	−0.114763	−	0.198775i	0.802922	−	0.596084i	$-0.203277\pi$
$191$	−1.52734e8	−	2.64542e8i	−0.114763	−	0.198775i	−0.917685	+	0.397309i	$0.869944\pi$
$192$		0			0
$193$	2.32710e8	−	4.03065e8i	0.167720	−	0.290500i	−0.769898	−	0.638167i	$-0.779693\pi$
$193$	2.32710e8	−	4.03065e8i	0.167720	−	0.290500i	0.937618	+	0.347667i	$0.113026\pi$
$194$		0			0
$195$		−	1.41829e9i		−	0.980906i
$196$		0			0
$197$	1.82661e9			1.21278			0.606389	−	0.795168i	$-0.292617\pi$
$197$	1.82661e9			1.21278			0.606389	+	0.795168i	$0.292617\pi$
$198$		0			0
$199$	2.53528e9	+	1.46374e9i	1.61664	+	0.933367i	0.987781	+	0.155846i	$0.0498104\pi$
$199$	2.53528e9	+	1.46374e9i	1.61664	+	0.933367i	0.628857	+	0.777521i	$0.283523\pi$
$200$		0			0
$201$	1.03383e7	−	5.96883e6i	0.00633382	−	0.00365683i
$202$		0			0
$203$	5.29408e8	+	4.05009e8i	0.311750	+	0.238496i
$204$		0			0
$205$	−8.53162e8	−	1.47772e9i	−0.483077	−	0.836714i
$206$		0			0
$207$	1.62975e8	−	2.82280e8i	0.0887644	−	0.153744i
$208$		0			0
$209$			1.89976e9i			0.995668i
$210$		0			0
$211$	−2.28912e9			−1.15489			−0.577443	−	0.816431i	$-0.695949\pi$
$211$	−2.28912e9			−1.15489			−0.577443	+	0.816431i	$0.695949\pi$
$212$		0			0
$213$	−1.93320e9	−	1.11613e9i	−0.939201	−	0.542248i
$214$		0			0
$215$	3.01911e9	−	1.74308e9i	1.41294	−	0.815764i
$216$		0			0
$217$	1.58475e9	+	3.81027e9i	0.714698	+	1.71837i
$218$		0			0
$219$	4.12356e8	+	7.14221e8i	0.179265	+	0.310496i
$220$		0			0
$221$	−7.63789e8	+	1.32292e9i	−0.320187	+	0.554581i
$222$		0			0
$223$		−	2.31777e9i		−	0.937241i	−0.883400	−	0.468620i	$-0.844751\pi$
$223$		−	2.31777e9i		−	0.937241i	0.883400	−	0.468620i	$-0.155249\pi$
$224$		0			0
$225$	4.03926e8			0.157605
$226$		0			0
$227$	−3.32328e9	−	1.91869e9i	−1.25159	−	0.722607i	−0.280167	−	0.959951i	$-0.590390\pi$
$227$	−3.32328e9	−	1.91869e9i	−1.25159	−	0.722607i	−0.971426	+	0.237344i	$0.923723\pi$
$228$		0			0
$229$	−4.30131e9	+	2.48336e9i	−1.56408	+	0.903022i	−0.567243	+	0.823550i	$0.691990\pi$
$229$	−4.30131e9	+	2.48336e9i	−1.56408	+	0.903022i	−0.996837	+	0.0794717i	$0.974677\pi$
$230$		0			0
$231$	−1.53861e9	−	2.00289e8i	−0.540357	−	0.0703412i
$232$		0			0
$233$	6.64889e7	+	1.15162e8i	0.0225593	+	0.0390738i	0.877085	−	0.480336i	$-0.159485\pi$
$233$	6.64889e7	+	1.15162e8i	0.0225593	+	0.0390738i	−0.854525	+	0.519410i	$0.826152\pi$
$234$		0			0
$235$	1.37742e9	−	2.38576e9i	0.451643	−	0.782268i
$236$		0			0
$237$			1.46697e9i			0.464974i
$238$		0			0
$239$	−8.69161e8			−0.266384			−0.133192	−	0.991090i	$-0.542523\pi$
$239$	−8.69161e8			−0.266384			−0.133192	+	0.991090i	$0.542523\pi$
$240$		0			0
$241$	−5.15545e9	−	2.97650e9i	−1.52826	−	0.882344i	−0.999435	−	0.0336153i	$-0.989298\pi$
$241$	−5.15545e9	−	2.97650e9i	−1.52826	−	0.882344i	−0.528829	−	0.848728i	$-0.677369\pi$
$242$		0			0
$243$	1.93710e8	−	1.11839e8i	0.0555556	−	0.0320750i
$244$		0			0
$245$	3.10085e9	+	3.08289e9i	0.860631	+	0.855646i
$246$		0			0
$247$	−2.74850e9	−	4.76055e9i	−0.738428	−	1.27900i
$248$		0			0
$249$	−1.81598e9	+	3.14537e9i	−0.472405	+	0.818229i
$250$		0			0
$251$		−	5.85148e9i		−	1.47425i	−0.675757	−	0.737124i	$-0.736183\pi$
$251$		−	5.85148e9i		−	1.47425i	0.675757	−	0.737124i	$-0.263817\pi$
$252$		0			0
$253$	−2.05950e9			−0.502666
$254$		0			0
$255$	−1.17361e9	−	6.77587e8i	−0.277565	−	0.160252i
$256$		0			0
$257$	−5.62716e9	+	3.24884e9i	−1.28990	+	0.744726i	−0.978637	−	0.205597i	$-0.934086\pi$
$257$	−5.62716e9	+	3.24884e9i	−1.28990	+	0.744726i	−0.311266	+	0.950323i	$0.600753\pi$
$258$		0			0
$259$	−1.45802e8	+	1.12004e9i	−0.0324014	+	0.248906i
$260$		0			0
$261$	3.03576e8	+	5.25809e8i	0.0654192	+	0.113309i
$262$		0			0
$263$	−3.97436e9	+	6.88379e9i	−0.830699	+	1.43881i	0.0667853	+	0.997767i	$0.478726\pi$
$263$	−3.97436e9	+	6.88379e9i	−0.830699	+	1.43881i	−0.897485	+	0.441046i	$0.854608\pi$
$264$		0			0
$265$			3.85567e9i			0.781837i
$266$		0			0
$267$	4.26724e9			0.839658
$268$		0			0
$269$	−3.45233e9	−	1.99320e9i	−0.659331	−	0.380665i	0.132691	−	0.991157i	$-0.457638\pi$
$269$	−3.45233e9	−	1.99320e9i	−0.659331	−	0.380665i	−0.792022	+	0.610493i	$0.790972\pi$
$270$		0			0
$271$	4.89106e9	−	2.82386e9i	0.906831	−	0.523559i	0.0274206	−	0.999624i	$-0.491271\pi$
$271$	4.89106e9	−	2.82386e9i	0.906831	−	0.523559i	0.879410	+	0.476065i	$0.157937\pi$
$272$		0			0
$273$	4.14531e9	−	1.72410e9i	0.746289	−	0.310393i
$274$		0			0
$275$	−1.27610e9	−	2.21026e9i	−0.223127	−	0.386468i
$276$		0			0
$277$	−2.20612e9	+	3.82111e9i	−0.374723	+	0.649039i	−0.990285	−	0.139049i	$-0.955595\pi$
$277$	−2.20612e9	+	3.82111e9i	−0.374723	+	0.649039i	0.615563	+	0.788088i	$0.288929\pi$
$278$		0			0
$279$			3.75888e9i			0.620358i
$280$		0			0
$281$	−8.88442e9			−1.42496			−0.712482	−	0.701691i	$-0.752429\pi$
$281$	−8.88442e9			−1.42496			−0.712482	+	0.701691i	$0.752429\pi$
$282$		0			0
$283$	6.15981e9	+	3.55637e9i	0.960333	+	0.554449i	0.896276	−	0.443498i	$-0.146263\pi$
$283$	6.15981e9	+	3.55637e9i	0.960333	+	0.554449i	0.0640575	+	0.997946i	$0.479596\pi$
$284$		0			0
$285$	4.22326e9	−	2.43830e9i	0.640131	−	0.369580i
$286$		0			0
$287$	3.28189e9	−	4.28992e9i	0.483722	−	0.632298i
$288$		0			0
$289$	−2.75808e9	−	4.77714e9i	−0.395381	−	0.684820i
$290$		0			0
$291$	−4.81535e8	+	8.34043e8i	−0.0671515	+	0.116310i
$292$		0			0
$293$			3.43041e9i			0.465453i	0.972542	+	0.232726i	$0.0747647\pi$
$293$			3.43041e9i			0.465453i	−0.972542	+	0.232726i	$0.925235\pi$
$294$		0			0
$295$	2.26783e9			0.299448
$296$		0			0
$297$	−1.22395e9	−	7.06649e8i	−0.157304	−	0.0908193i
$298$		0			0
$299$	5.16083e9	−	2.97960e9i	0.645705	−	0.372798i
$300$		0			0
$301$	8.76468e9	+	6.70518e9i	1.06775	+	0.816854i
$302$		0			0
$303$	−3.00590e9	−	5.20637e9i	−0.356618	−	0.617681i
$304$		0			0
$305$	4.87853e9	−	8.44986e9i	0.563754	−	0.976450i
$306$		0			0
$307$			6.38128e9i			0.718380i	0.933264	+	0.359190i	$0.116947\pi$
$307$			6.38128e9i			0.718380i	−0.933264	+	0.359190i	$0.883053\pi$
$308$		0			0
$309$	1.90122e9			0.208544
$310$		0			0
$311$	−5.32242e9	−	3.07290e9i	−0.568941	−	0.328478i	0.187785	−	0.982210i	$-0.439869\pi$
$311$	−5.32242e9	−	3.07290e9i	−0.568941	−	0.328478i	−0.756726	+	0.653732i	$0.773202\pi$
$312$		0			0
$313$	1.14188e10	−	6.59265e9i	1.18972	−	0.686883i	0.231474	−	0.972841i	$-0.425645\pi$
$313$	1.14188e10	−	6.59265e9i	1.18972	−	0.686883i	0.958242	+	0.285958i	$0.0923119\pi$
$314$		0			0
$315$	1.52952e9	+	3.67747e9i	0.155350	+	0.373514i
$316$		0			0
$317$	−7.40497e9	−	1.28258e10i	−0.733308	−	1.27013i	−0.955462	−	0.295114i	$-0.904642\pi$
$317$	−7.40497e9	−	1.28258e10i	−0.733308	−	1.27013i	0.222154	−	0.975012i	$-0.428691\pi$
$318$		0			0
$319$	1.91814e9	−	3.32231e9i	0.185232	−	0.320832i
$320$		0			0
$321$			4.97927e9i			0.468970i
$322$		0			0
$323$	−5.25237e9			−0.482553
$324$		0			0
$325$	6.39544e9	+	3.69241e9i	0.573241	+	0.330961i
$326$		0			0
$327$	1.46093e9	−	8.43470e8i	0.127773	−	0.0737698i
$328$		0			0
$329$	8.64739e9	+	1.12568e9i	0.738077	+	0.0960795i
$330$		0			0
$331$	6.65412e9	+	1.15253e10i	0.554343	+	0.960151i	0.997954	+	0.0639314i	$0.0203639\pi$
$331$	6.65412e9	+	1.15253e10i	0.554343	+	0.960151i	−0.443611	+	0.896219i	$0.646303\pi$
$332$		0			0
$333$	−5.14410e8	+	8.90985e8i	−0.0418343	+	0.0724592i
$334$		0			0
$335$			1.93620e8i			0.0153734i
$336$		0			0
$337$	−5.05204e9			−0.391695			−0.195847	−	0.980634i	$-0.562746\pi$
$337$	−5.05204e9			−0.391695			−0.195847	+	0.980634i	$0.562746\pi$
$338$		0			0
$339$	8.37321e9	+	4.83428e9i	0.634006	+	0.366043i
$340$		0			0
$341$	2.05684e10	−	1.18752e10i	1.52119	−	0.878260i
$342$		0			0
$343$	−5.24107e9	+	1.28106e10i	−0.378655	+	0.925538i
$344$		0			0
$345$	2.64332e9	+	4.57837e9i	0.186584	+	0.323172i
$346$		0			0
$347$	−1.44081e10	+	2.49556e10i	−0.993777	+	1.72127i	−0.400427	+	0.916329i	$0.631138\pi$
$347$	−1.44081e10	+	2.49556e10i	−0.993777	+	1.72127i	−0.593351	+	0.804944i	$0.702195\pi$
$348$		0			0
$349$			6.85689e9i			0.462195i	0.972931	+	0.231098i	$0.0742316\pi$
$349$			6.85689e9i			0.462195i	−0.972931	+	0.231098i	$0.925768\pi$
$350$		0			0
$351$	4.08941e9			0.269421
$352$		0			0
$353$	1.77313e9	+	1.02372e9i	0.114194	+	0.0659299i	0.556009	−	0.831176i	$-0.312332\pi$
$353$	1.77313e9	+	1.02372e9i	0.114194	+	0.0659299i	−0.441815	+	0.897106i	$0.645665\pi$
$354$		0			0
$355$	3.13550e10	−	1.81028e10i	1.97421	−	1.13981i
$356$		0			0
$357$	5.53749e8	−	4.25387e9i	0.0340910	−	0.261885i
$358$		0			0
$359$	−1.16378e10	−	2.01573e10i	−0.700640	−	1.21354i	−0.968242	−	0.250014i	$-0.919565\pi$
$359$	−1.16378e10	−	2.01573e10i	−0.700640	−	1.21354i	0.267603	−	0.963529i	$-0.413769\pi$
$360$		0			0
$361$	9.58567e8	−	1.66029e9i	0.0564409	−	0.0977585i
$362$		0			0
$363$		−	1.09469e9i		−	0.0630470i
$364$		0			0
$365$	−1.33762e10			−0.753634
$366$		0			0
$367$	2.90688e9	+	1.67829e9i	0.160237	+	0.0925128i	0.577974	−	0.816055i	$-0.303843\pi$
$367$	2.90688e9	+	1.67829e9i	0.160237	+	0.0925128i	−0.417737	+	0.908568i	$0.637177\pi$
$368$		0			0
$369$	4.26076e9	−	2.45995e9i	0.229817	−	0.132685i
$370$		0			0
$371$	−1.12692e10	+	4.68702e9i	−0.594834	+	0.247401i
$372$		0			0
$373$	−2.69315e9	−	4.66467e9i	−0.139131	−	0.240982i	0.788037	−	0.615628i	$-0.211098\pi$
$373$	−2.69315e9	−	4.66467e9i	−0.139131	−	0.240982i	−0.927168	+	0.374646i	$0.877764\pi$
$374$		0			0
$375$	3.65234e9	−	6.32603e9i	0.184691	−	0.319894i
$376$		0			0
$377$			1.11003e10i			0.549504i
$378$		0			0
$379$	−2.41287e10			−1.16944			−0.584718	−	0.811237i	$-0.698795\pi$
$379$	−2.41287e10			−1.16944			−0.584718	+	0.811237i	$0.698795\pi$
$380$		0			0
$381$	1.71048e10	+	9.87546e9i	0.811742	+	0.468659i
$382$		0			0
$383$	3.42280e8	−	1.97616e8i	0.0159069	−	0.00918388i	−0.492025	−	0.870581i	$-0.663743\pi$
$383$	3.42280e8	−	1.97616e8i	0.0159069	−	0.00918388i	0.507932	+	0.861397i	$0.330410\pi$
$384$		0			0
$385$	1.52908e10	−	1.99874e10i	0.695966	−	0.909733i
$386$		0			0
$387$	5.02589e9	+	8.70510e9i	0.224062	+	0.388088i
$388$		0			0
$389$	−2.08469e9	+	3.61078e9i	−0.0910421	+	0.157690i	−0.907950	−	0.419079i	$-0.862353\pi$
$389$	−2.08469e9	+	3.61078e9i	−0.0910421	+	0.157690i	0.816908	+	0.576768i	$0.195686\pi$
$390$		0			0
$391$		−	5.69400e9i		−	0.243619i
$392$		0			0
$393$	−2.45798e10			−1.03040
$394$		0			0
$395$	−2.06055e10	−	1.18966e10i	−0.846437	−	0.488690i
$396$		0			0
$397$	4.84338e9	−	2.79632e9i	0.194978	−	0.112571i	−0.399333	−	0.916806i	$-0.630758\pi$
$397$	4.84338e9	−	2.79632e9i	0.194978	−	0.112571i	0.594311	+	0.804235i	$0.297425\pi$
$398$		0			0
$399$	1.22604e10	+	9.37950e9i	0.483742	+	0.370073i
$400$		0			0
$401$	−2.33080e10	−	4.03706e10i	−0.901420	−	1.56130i	−0.825653	−	0.564179i	$-0.809193\pi$
$401$	−2.33080e10	−	4.03706e10i	−0.901420	−	1.56130i	−0.0757670	−	0.997126i	$-0.524141\pi$
$402$		0			0
$403$	−3.43611e10	+	5.95152e10i	−1.30271	+	2.25636i
$404$		0			0
$405$			3.62787e9i			0.134844i
$406$		0			0
$407$	6.50057e9			0.236905
$408$		0			0
$409$	−2.88978e10	−	1.66841e10i	−1.03269	−	0.596225i	−0.114937	−	0.993373i	$-0.536667\pi$
$409$	−2.88978e10	−	1.66841e10i	−1.03269	−	0.596225i	−0.917755	+	0.397148i	$0.870000\pi$
$410$		0			0
$411$	5.13497e9	−	2.96467e9i	0.179958	−	0.103899i
$412$		0			0
$413$	2.75681e9	+	6.62828e9i	0.0947559	+	0.227825i
$414$		0			0
$415$	−2.94538e10	−	5.10155e10i	−0.993000	−	1.71993i
$416$		0			0
$417$	−9.31165e9	+	1.61283e10i	−0.307952	+	0.533388i
$418$		0			0
$419$			1.29982e10i			0.421724i	0.977516	+	0.210862i	$0.0676270\pi$
$419$			1.29982e10i			0.421724i	−0.977516	+	0.210862i	$0.932373\pi$
$420$		0			0
$421$	2.85590e10			0.909105			0.454552	−	0.890720i	$-0.349799\pi$
$421$	2.85590e10			0.909105			0.454552	+	0.890720i	$0.349799\pi$
$422$		0			0
$423$	6.87894e9	+	3.97156e9i	0.214862	+	0.124051i
$424$		0			0
$425$	6.11082e9	−	3.52808e9i	0.187303	−	0.108139i
$426$		0			0
$427$	3.06272e10	+	3.98691e9i	0.921290	+	0.119929i
$428$		0			0
$429$	−1.29194e10	−	2.23771e10i	−0.381429	−	0.660654i
$430$		0			0
$431$	−1.78709e10	+	3.09533e10i	−0.517890	+	0.897012i	0.481894	+	0.876229i	$0.339949\pi$
$431$	−1.78709e10	+	3.09533e10i	−0.517890	+	0.897012i	−0.999784	+	0.0207821i	$0.993384\pi$
$432$		0			0
$433$			3.07905e9i			0.0875920i	0.999040	+	0.0437960i	$0.0139452\pi$
$433$			3.07905e9i			0.0875920i	−0.999040	+	0.0437960i	$0.986055\pi$
$434$		0			0
$435$	−9.84753e9			−0.275024
$436$		0			0
$437$	1.77448e10	+	1.02450e10i	0.486570	+	0.280921i
$438$		0			0
$439$	7.67947e9	−	4.43374e9i	0.206763	−	0.119375i	−0.393043	−	0.919520i	$-0.628578\pi$
$439$	7.67947e9	−	4.43374e9i	0.206763	−	0.119375i	0.599806	+	0.800145i	$0.295244\pi$
$440$		0			0
$441$	−8.88901e9	+	8.94079e9i	−0.235017	+	0.236386i
$442$		0			0
$443$	3.80719e9	+	6.59425e9i	0.0988529	+	0.171218i	0.911210	−	0.411942i	$-0.135149\pi$
$443$	3.80719e9	+	6.59425e9i	0.0988529	+	0.171218i	−0.812357	+	0.583160i	$0.801816\pi$
$444$		0			0
$445$	−3.46057e10	+	5.99388e10i	−0.882485	+	1.52851i
$446$		0			0
$447$		−	3.85212e10i		−	0.964873i
$448$		0			0
$449$	−6.30579e8			−0.0155151			−0.00775754	−	0.999970i	$-0.502469\pi$
$449$	−6.30579e8			−0.0155151			−0.00775754	+	0.999970i	$0.502469\pi$
$450$		0			0
$451$	−2.69215e10	−	1.55431e10i	−0.650718	−	0.375692i
$452$		0			0
$453$	−8.57425e9	+	4.95034e9i	−0.203612	+	0.117555i
$454$		0			0
$455$	−9.39970e9	+	7.22079e10i	−0.219315	+	1.68476i
$456$		0			0
$457$	−3.72887e9	−	6.45859e9i	−0.0854894	−	0.148072i	0.820110	−	0.572206i	$-0.193912\pi$
$457$	−3.72887e9	−	6.45859e9i	−0.0854894	−	0.148072i	−0.905600	+	0.424134i	$0.860579\pi$
$458$		0			0
$459$	1.95371e9	−	3.38392e9i	0.0440158	−	0.0762376i
$460$		0			0
$461$		−	7.83758e10i		−	1.73532i	−0.497162	−	0.867658i	$-0.665625\pi$
$461$		−	7.83758e10i		−	1.73532i	0.497162	−	0.867658i	$-0.334375\pi$
$462$		0			0
$463$	−3.00729e10			−0.654413			−0.327206	−	0.944953i	$-0.606107\pi$
$463$	−3.00729e10			−0.654413			−0.327206	+	0.944953i	$0.606107\pi$
$464$		0			0
$465$	−5.27982e10	−	3.04831e10i	−1.12930	−	0.651999i
$466$		0			0
$467$	−1.07043e10	+	6.18012e9i	−0.225056	+	0.129936i	−0.608289	−	0.793716i	$-0.708144\pi$
$467$	−1.07043e10	+	6.18012e9i	−0.225056	+	0.129936i	0.383233	+	0.923652i	$0.374811\pi$
$468$		0			0
$469$	−5.65902e8	+	2.35367e8i	−0.0116963	+	0.00486469i
$470$		0			0
$471$	−1.32595e10	−	2.29661e10i	−0.269428	−	0.466663i
$472$		0			0
$473$	3.17559e10	−	5.50029e10i	0.634425	−	1.09886i
$474$		0			0
$475$			2.53917e10i			0.498789i
$476$		0			0
$477$	−1.11172e10			−0.214744
$478$		0			0
$479$	1.02421e10	+	5.91328e9i	0.194557	+	0.112328i	0.594114	−	0.804381i	$-0.297503\pi$
$479$	1.02421e10	+	5.91328e9i	0.194557	+	0.112328i	−0.399557	+	0.916708i	$0.630836\pi$
$480$		0			0
$481$	−1.62895e10	+	9.40477e9i	−0.304319	+	0.175698i
$482$		0			0
$483$	−1.01681e10	+	1.32913e10i	−0.186833	+	0.244219i
$484$		0			0
$485$	−7.81012e9	−	1.35275e10i	−0.141153	−	0.244484i
$486$		0			0
$487$	−1.29037e10	+	2.23499e10i	−0.229403	+	0.397337i	−0.957631	−	0.287997i	$-0.907011\pi$
$487$	−1.29037e10	+	2.23499e10i	−0.229403	+	0.397337i	0.728229	+	0.685334i	$0.240344\pi$
$488$		0			0
$489$			3.81415e10i			0.667056i
$490$		0			0
$491$	−6.27542e9			−0.107973			−0.0539867	−	0.998542i	$-0.517193\pi$
$491$	−6.27542e9			−0.107973			−0.0539867	+	0.998542i	$0.517193\pi$
$492$		0			0
$493$	9.18534e9	+	5.30316e9i	0.155492	+	0.0897733i
$494$		0			0
$495$	1.98516e10	−	1.14613e10i	0.330654	−	0.190903i
$496$		0			0
$497$	9.10257e10	+	6.96367e10i	1.49190	+	1.14133i
$498$		0			0
$499$	6.03302e9	+	1.04495e10i	0.0973044	+	0.168536i	0.910568	−	0.413359i	$-0.135645\pi$
$499$	6.03302e9	+	1.04495e10i	0.0973044	+	0.168536i	−0.813264	+	0.581895i	$0.802311\pi$
$500$		0			0
$501$	2.48065e10	−	4.29661e10i	0.393744	−	0.681985i
$502$		0			0
$503$			3.03272e10i			0.473763i	0.971539	+	0.236881i	$0.0761253\pi$
$503$			3.03272e10i			0.473763i	−0.971539	+	0.236881i	$0.923875\pi$
$504$		0			0
$505$	9.75066e10			1.49923
$506$		0			0
$507$	3.17114e10	+	1.83086e10i	0.479936	+	0.277091i
$508$		0			0
$509$	9.94432e10	−	5.74136e10i	1.48151	−	0.855349i	0.481728	−	0.876321i	$-0.340009\pi$
$509$	9.94432e10	−	5.74136e10i	1.48151	−	0.855349i	0.999780	+	0.0209721i	$0.00667611\pi$
$510$		0			0
$511$	−1.62603e10	−	3.90952e10i	−0.238476	−	0.573376i
$512$		0			0
$513$	7.03043e9	+	1.21771e10i	0.101511	+	0.175822i
$514$		0			0
$515$	−1.54181e10	+	2.67050e10i	−0.219181	+	0.379632i
$516$		0			0
$517$		−	5.01883e10i		−	0.702491i
$518$		0			0
$519$	−3.81369e10			−0.525625
$520$		0			0
$521$	1.01705e11	+	5.87193e10i	1.38035	+	0.796948i	0.992201	−	0.124647i	$-0.0397797\pi$
$521$	1.01705e11	+	5.87193e10i	1.38035	+	0.796948i	0.388153	+	0.921595i	$0.373113\pi$
$522$		0			0
$523$	1.14302e11	−	6.59924e10i	1.52773	−	0.882037i	0.528277	−	0.849072i	$-0.322838\pi$
$523$	1.14302e11	−	6.59924e10i	1.52773	−	0.882037i	0.999457	−	0.0329648i	$-0.0104949\pi$
$524$		0			0
$525$	−2.05646e10	−	2.67701e9i	−0.270697	−	0.0352381i
$526$		0			0
$527$	3.28319e10	+	5.68665e10i	0.425651	+	0.737249i
$528$		0			0
$529$	2.80491e10	−	4.85825e10i	0.358176	−	0.620379i
$530$		0			0
$531$			6.53889e9i			0.0822481i
$532$		0			0
$533$	8.99487e10			1.11451
$534$		0			0
$535$	−6.99401e10	−	4.03799e10i	−0.853711	−	0.492890i
$536$		0			0
$537$	8.64563e9	−	4.99156e9i	0.103968	−	0.0600260i
$538$		0			0
$539$	7.70060e10	+	2.03942e10i	0.912367	+	0.241630i
$540$		0			0
$541$	−5.07062e10	−	8.78258e10i	−0.591933	−	1.02526i	−0.993972	−	0.109635i	$-0.965032\pi$
$541$	−5.07062e10	−	8.78258e10i	−0.591933	−	1.02526i	0.402039	−	0.915622i	$-0.368302\pi$
$542$		0			0
$543$	1.72689e10	−	2.99106e10i	0.198639	−	0.344053i
$544$		0			0
$545$			2.73609e10i			0.310130i
$546$		0			0
$547$	1.33269e10			0.148861			0.0744305	−	0.997226i	$-0.476286\pi$
$547$	1.33269e10			0.148861			0.0744305	+	0.997226i	$0.476286\pi$
$548$		0			0
$549$	2.43637e10	+	1.40664e10i	0.268197	+	0.154844i
$550$		0			0
$551$	−3.30536e10	+	1.90835e10i	−0.358601	+	0.207039i
$552$		0			0
$553$	9.72233e9	−	7.46863e10i	0.103961	−	0.798621i
$554$		0			0
$555$	−8.34333e9	−	1.44511e10i	−0.0879362	−	0.152310i
$556$		0			0
$557$	−6.36147e10	+	1.10184e11i	−0.660901	+	1.14471i	0.319478	+	0.947594i	$0.396492\pi$
$557$	−6.36147e10	+	1.10184e11i	−0.660901	+	1.14471i	−0.980379	+	0.197121i	$0.936841\pi$
$558$		0			0
$559$			1.83773e11i			1.88206i
$560$		0			0
$561$	−2.46889e10			−0.249258
$562$		0			0
$563$	−1.34889e10	−	7.78784e9i	−0.134259	−	0.0775146i	0.431366	−	0.902177i	$-0.358032\pi$
$563$	−1.34889e10	−	7.78784e9i	−0.134259	−	0.0775146i	−0.565625	+	0.824662i	$0.691365\pi$
$564$		0			0
$565$	−1.35807e11	+	7.84082e10i	−1.33269	+	0.769427i
$566$		0			0
$567$	−1.06034e10	+	4.41010e9i	−0.102591	+	0.0426694i
$568$		0			0
$569$	1.62497e10	+	2.81452e10i	0.155023	+	0.268507i	0.933067	−	0.359702i	$-0.117122\pi$
$569$	1.62497e10	+	2.81452e10i	0.155023	+	0.268507i	−0.778045	+	0.628209i	$0.783788\pi$
$570$		0			0
$571$	5.39206e10	−	9.33932e10i	0.507236	−	0.878559i	−0.492729	−	0.870183i	$-0.664001\pi$
$571$	5.39206e10	−	9.33932e10i	0.507236	−	0.878559i	0.999965	−	0.00837569i	$-0.00266610\pi$
$572$		0			0
$573$			1.42853e10i			0.132517i
$574$		0			0
$575$	−2.75267e10			−0.251816
$576$		0			0
$577$	−7.22142e10	−	4.16929e10i	−0.651507	−	0.376148i	0.137526	−	0.990498i	$-0.456085\pi$
$577$	−7.22142e10	−	4.16929e10i	−0.651507	−	0.376148i	−0.789033	+	0.614350i	$0.789418\pi$
$578$		0			0
$579$	−1.88495e10	+	1.08828e10i	−0.167720	+	0.0968332i
$580$		0			0
$581$	1.13301e11	−	1.48101e11i	0.994327	−	1.29974i
$582$		0			0
$583$	3.51218e10	+	6.08327e10i	0.304020	+	0.526578i
$584$		0			0
$585$	−3.31635e10	+	5.74409e10i	−0.283163	+	0.490453i
$586$		0			0
$587$			7.46672e9i			0.0628894i	0.999505	+	0.0314447i	$0.0100108\pi$
$587$			7.46672e9i			0.0628894i	−0.999505	+	0.0314447i	$0.989989\pi$
$588$		0			0
$589$	−2.36292e11			−1.96331
$590$		0			0
$591$	−7.39778e10	−	4.27111e10i	−0.606389	−	0.350099i
$592$		0			0
$593$	1.56081e10	−	9.01133e9i	0.126221	−	0.0728736i	−0.435560	−	0.900160i	$-0.643450\pi$
$593$	1.56081e10	−	9.01133e9i	0.126221	−	0.0728736i	0.561781	+	0.827286i	$0.310116\pi$
$594$		0			0
$595$	5.52602e10	+	4.22753e10i	0.440905	+	0.337302i
$596$		0			0
$597$	−6.84525e10	−	1.18563e11i	−0.538880	−	0.933367i
$598$		0			0
$599$	1.04457e11	−	1.80925e11i	0.811390	−	1.40537i	−0.100502	−	0.994937i	$-0.532045\pi$
$599$	1.04457e11	−	1.80925e11i	0.811390	−	1.40537i	0.911891	−	0.410432i	$-0.134622\pi$
$600$		0			0
$601$		−	2.08426e10i		−	0.159755i	−0.996805	−	0.0798775i	$-0.974547\pi$
$601$		−	2.08426e10i		−	0.159755i	0.996805	−	0.0798775i	$-0.0254529\pi$
$602$		0			0
$603$	−5.58270e8			−0.00422255
$604$		0			0
$605$	1.53763e10	+	8.87750e9i	0.114770	+	0.0662627i
$606$		0			0
$607$	1.34325e11	−	7.75528e10i	0.989472	−	0.571272i	0.0843557	−	0.996436i	$-0.473117\pi$
$607$	1.34325e11	−	7.75528e10i	0.989472	−	0.571272i	0.905116	+	0.425164i	$0.139783\pi$
$608$		0			0
$609$	−1.19708e10	−	2.87819e10i	−0.0870272	−	0.209242i
$610$		0			0
$611$	7.26105e10	+	1.25765e11i	0.520996	+	0.902391i
$612$		0			0
$613$	−8.32583e10	+	1.44208e11i	−0.589639	+	1.02128i	0.404641	+	0.914476i	$0.367396\pi$
$613$	−8.32583e10	+	1.44208e11i	−0.589639	+	1.02128i	−0.994280	+	0.106808i	$0.965937\pi$
$614$		0			0
$615$			7.97969e10i			0.557809i
$616$		0			0
$617$	8.79412e10			0.606808			0.303404	−	0.952862i	$-0.401877\pi$
$617$	8.79412e10			0.606808			0.303404	+	0.952862i	$0.401877\pi$
$618$		0			0
$619$	5.96984e10	+	3.44669e10i	0.406630	+	0.234768i	0.689341	−	0.724437i	$-0.257900\pi$
$619$	5.96984e10	+	3.44669e10i	0.406630	+	0.234768i	−0.282710	+	0.959205i	$0.591234\pi$
$620$		0			0
$621$	−1.32009e10	+	7.62157e9i	−0.0887644	+	0.0512481i
$622$		0			0
$623$	−2.17253e11	−	2.82811e10i	−1.44216	−	0.187734i
$624$		0			0
$625$	9.53111e10	+	1.65084e11i	0.624631	+	1.08189i
$626$		0			0
$627$	4.44216e10	−	7.69404e10i	0.287425	−	0.497834i
$628$		0			0
$629$			1.79724e10i			0.114817i
$630$		0			0
$631$	−5.94652e10			−0.375099			−0.187549	−	0.982255i	$-0.560054\pi$
$631$	−5.94652e10			−0.375099			−0.187549	+	0.982255i	$0.560054\pi$
$632$		0			0
$633$	9.27094e10	+	5.35258e10i	0.577443	+	0.333387i
$634$		0			0
$635$	−2.77426e11	+	1.60172e11i	−1.70629	+	0.985127i
$636$		0			0
$637$	−2.22472e11	+	6.03042e10i	−1.35119	+	0.366261i
$638$		0			0
$639$	5.21964e10	+	9.04069e10i	0.313067	+	0.542248i
$640$		0			0
$641$	−1.97978e10	+	3.42908e10i	−0.117269	+	0.203117i	−0.918685	−	0.394992i	$-0.870747\pi$
$641$	−1.97978e10	+	3.42908e10i	−0.117269	+	0.203117i	0.801415	+	0.598108i	$0.204081\pi$
$642$		0			0
$643$		−	5.06967e10i		−	0.296576i	−0.988944	−	0.148288i	$-0.952624\pi$
$643$		−	5.06967e10i		−	0.296576i	0.988944	−	0.148288i	$-0.0473763\pi$
$644$		0			0
$645$	−1.63032e11			−0.941963
$646$		0			0
$647$	−2.68870e11	−	1.55232e11i	−1.53435	−	0.885860i	−0.999154	−	0.0411356i	$-0.986902\pi$
$647$	−2.68870e11	−	1.55232e11i	−1.53435	−	0.885860i	−0.535201	−	0.844725i	$-0.679764\pi$
$648$		0			0
$649$	3.57805e10	−	2.06579e10i	0.201682	−	0.116441i
$650$		0			0
$651$	2.49119e10	−	1.91372e11i	0.138702	−	1.06550i
$652$		0			0
$653$	−5.56400e10	−	9.63713e10i	−0.306009	−	0.530023i	0.671476	−	0.741026i	$-0.265660\pi$
$653$	−5.56400e10	−	9.63713e10i	−0.306009	−	0.530023i	−0.977485	+	0.211003i	$0.932327\pi$
$654$		0			0
$655$	1.99332e11	−	3.45254e11i	1.08296	−	1.87574i
$656$		0			0
$657$		−	3.85679e10i		−	0.206997i
$658$		0			0
$659$	−3.65142e10			−0.193607			−0.0968033	−	0.995304i	$-0.530862\pi$
$659$	−3.65142e10			−0.193607			−0.0968033	+	0.995304i	$0.530862\pi$
$660$		0			0
$661$	1.90939e11	+	1.10239e11i	1.00020	+	0.577468i	0.908309	−	0.418301i	$-0.137374\pi$
$661$	1.90939e11	+	1.10239e11i	1.00020	+	0.577468i	0.0918951	+	0.995769i	$0.470708\pi$
$662$		0			0
$663$	6.18669e10	−	3.57189e10i	0.320187	−	0.184860i
$664$		0			0
$665$	−2.31174e11	+	9.61489e10i	−1.18210	+	0.491652i
$666$		0			0
$667$	−2.06881e10	−	3.58328e10i	−0.104524	−	0.181041i
$668$		0			0
$669$	−5.41957e10	+	9.38698e10i	−0.270558	+	0.468620i
$670$		0			0
$671$		−	1.77756e11i		−	0.876870i
$672$		0			0
$673$	−3.08399e11			−1.50333			−0.751663	−	0.659548i	$-0.770748\pi$
$673$	−3.08399e11			−1.50333			−0.751663	+	0.659548i	$0.770748\pi$
$674$		0			0
$675$	−1.63590e10	−	9.44487e9i	−0.0788027	−	0.0454968i
$676$		0			0
$677$	−1.10257e11	+	6.36567e10i	−0.524868	+	0.303033i	−0.738924	−	0.673789i	$-0.764666\pi$
$677$	−1.10257e11	+	6.36567e10i	−0.524868	+	0.303033i	0.214056	+	0.976821i	$0.431332\pi$
$678$		0			0
$679$	3.00434e10	−	3.92713e10i	0.141342	−	0.184755i
$680$		0			0
$681$	8.97285e10	+	1.55414e11i	0.417198	+	0.722607i
$682$		0			0
$683$	−6.36226e9	+	1.10198e10i	−0.0292367	+	0.0506395i	−0.880274	−	0.474467i	$-0.842641\pi$
$683$	−6.36226e9	+	1.10198e10i	−0.0292367	+	0.0506395i	0.851037	+	0.525106i	$0.175974\pi$
$684$		0			0
$685$			9.61694e10i			0.436792i
$686$		0			0
$687$	2.32271e11			1.04272
$688$		0			0
$689$	−1.76021e11	−	1.01626e11i	−0.781064	−	0.450948i
$690$		0			0
$691$	−2.25426e11	+	1.30150e11i	−0.988764	+	0.570863i	−0.904905	−	0.425614i	$-0.860058\pi$
$691$	−2.25426e11	+	1.30150e11i	−0.988764	+	0.570863i	−0.0838595	+	0.996478i	$0.526725\pi$
$692$		0			0
$693$	5.76304e10	+	4.40885e10i	0.249873	+	0.191158i
$694$		0			0
$695$	−1.51028e11	−	2.61588e11i	−0.647318	−	1.12119i
$696$		0			0
$697$	4.29728e10	−	7.44311e10i	0.182080	−	0.315372i
$698$		0			0
$699$		−	6.21875e9i		−	0.0260492i
$700$		0			0
$701$	4.00585e11			1.65891			0.829454	−	0.558575i	$-0.188652\pi$
$701$	4.00585e11			1.65891			0.829454	+	0.558575i	$0.188652\pi$
$702$		0			0
$703$	−5.60093e10	−	3.23370e10i	−0.229319	−	0.132397i
$704$		0			0
$705$	−1.11571e11	+	6.44155e10i	−0.451643	+	0.260756i
$706$		0			0
$707$	1.18531e11	+	2.84987e11i	0.474409	+	1.14064i
$708$		0			0
$709$	1.90688e11	+	3.30281e11i	0.754637	+	1.30707i	0.945555	+	0.325463i	$0.105520\pi$
$709$	1.90688e11	+	3.30281e11i	0.754637	+	1.30707i	−0.190918	+	0.981606i	$0.561146\pi$
$710$		0			0
$711$	3.43018e10	−	5.94124e10i	0.134226	−	0.232487i
$712$		0			0
$713$		−	2.56160e11i		−	0.991182i
$714$		0			0
$715$	4.19085e11			1.60353
$716$		0			0
$717$	3.52010e10	+	2.03233e10i	0.133192	+	0.0768985i
$718$		0			0
$719$	−2.96410e11	+	1.71132e11i	−1.10912	+	0.640349i	−0.938600	−	0.345006i	$-0.887877\pi$
$719$	−2.96410e11	+	1.71132e11i	−1.10912	+	0.640349i	−0.170516	+	0.985355i	$0.554544\pi$
$720$		0			0
$721$	−9.67944e10	−	1.26003e10i	−0.358186	−	0.0466271i
$722$		0			0
$723$	1.39197e11	+	2.41097e11i	0.509421	+	0.882344i
$724$		0			0
$725$	2.56372e10	−	4.44050e10i	0.0927939	−	0.160724i
$726$		0			0
$727$		−	3.71217e11i		−	1.32889i	−0.747335	−	0.664447i	$-0.768667\pi$
$727$		−	3.71217e11i		−	1.32889i	0.747335	−	0.664447i	$-0.231333\pi$
$728$		0			0
$729$	−1.04604e10			−0.0370370
$730$		0			0
$731$	1.52069e11	+	8.77971e10i	0.532563	+	0.307476i
$732$		0			0
$733$	−2.92685e11	+	1.68982e11i	−1.01387	+	0.585361i	−0.912324	−	0.409470i	$-0.865714\pi$
$733$	−2.92685e11	+	1.68982e11i	−1.01387	+	0.585361i	−0.101551	+	0.994830i	$0.532380\pi$
$734$		0			0
$735$	−5.34982e10	−	1.97364e11i	−0.183312	−	0.676266i
$736$		0			0
$737$	1.76370e9	+	3.05483e9i	0.00597800	+	0.0103542i
$738$		0			0
$739$	−4.94965e10	+	8.57304e10i	−0.165957	+	0.287447i	−0.936995	−	0.349343i	$-0.886405\pi$
$739$	−4.94965e10	+	8.57304e10i	−0.165957	+	0.287447i	0.771038	+	0.636790i	$0.219738\pi$
$740$		0			0
$741$			2.57070e11i			0.852664i
$742$		0			0
$743$	2.43942e11			0.800445			0.400222	−	0.916418i	$-0.368933\pi$
$743$	2.43942e11			0.800445			0.400222	+	0.916418i	$0.368933\pi$
$744$		0			0
$745$	5.41079e11	+	3.12392e11i	1.75645	+	1.01409i
$746$		0			0
$747$	1.47095e11	−	8.49251e10i	0.472405	−	0.272743i
$748$		0			0
$749$	3.30000e10	−	2.53504e11i	0.104854	−	0.805484i
$750$		0			0
$751$	−1.90079e11	−	3.29227e11i	−0.597552	−	1.03499i	−0.993181	−	0.116580i	$-0.962807\pi$
$751$	−1.90079e11	−	3.29227e11i	−0.597552	−	1.03499i	0.395630	−	0.918410i	$-0.370526\pi$
$752$		0			0
$753$	−1.36823e11	+	2.36985e11i	−0.425579	+	0.737124i
$754$		0			0
$755$		−	1.60581e11i		−	0.494205i
$756$		0			0
$757$	2.45316e11			0.747038			0.373519	−	0.927623i	$-0.378151\pi$
$757$	2.45316e11			0.747038			0.373519	+	0.927623i	$0.378151\pi$
$758$		0			0
$759$	8.34098e10	+	4.81567e10i	0.251333	+	0.145107i
$760$		0			0
$761$	−1.86360e11	+	1.07595e11i	−0.555666	+	0.320814i	−0.751404	−	0.659842i	$-0.770623\pi$
$761$	−1.86360e11	+	1.07595e11i	−0.555666	+	0.320814i	0.195738	+	0.980656i	$0.437290\pi$
$762$		0			0
$763$	−7.99689e10	+	3.32603e10i	−0.235952	+	0.0981360i
$764$		0			0
$765$	3.16876e10	+	5.48845e10i	0.0925217	+	0.160252i
$766$		0			0
$767$	−5.97740e10	+	1.03532e11i	−0.172715	+	0.299152i
$768$		0			0
$769$		−	1.78234e11i		−	0.509664i	−0.966985	−	0.254832i	$-0.917980\pi$
$769$		−	1.78234e11i		−	0.509664i	0.966985	−	0.254832i	$-0.0820202\pi$
$770$		0			0
$771$	3.03867e11			0.859935
$772$		0			0
$773$	−1.49619e11	−	8.63826e10i	−0.419053	−	0.241940i	0.275619	−	0.961267i	$-0.411117\pi$
$773$	−1.49619e11	−	8.63826e10i	−0.419053	−	0.241940i	−0.694672	+	0.719327i	$0.744451\pi$
$774$		0			0
$775$	2.74912e11	−	1.58720e11i	0.762055	−	0.439973i
$776$		0			0
$777$	3.20946e10	−	4.19524e10i	0.0880536	−	0.115099i
$778$		0			0
$779$	1.54638e11	+	2.67841e11i	0.419920	+	0.727323i
$780$		0			0
$781$	3.29802e11	−	5.71233e11i	0.886439	−	1.53536i
$782$		0			0
$783$		−	2.83937e10i		−	0.0755396i
$784$		0			0
$785$	4.30117e11			1.13268
$786$		0			0
$787$	3.97454e11	+	2.29470e11i	1.03607	+	0.598173i	0.918717	−	0.394918i	$-0.129227\pi$
$787$	3.97454e11	+	2.29470e11i	1.03607	+	0.598173i	0.117350	+	0.993091i	$0.462560\pi$
$788$		0			0
$789$	3.21923e11	−	1.85862e11i	0.830699	−	0.479604i
$790$		0			0
$791$	−3.94257e11	−	3.01615e11i	−1.00710	−	0.770455i
$792$		0			0
$793$	2.57171e11	+	4.45433e11i	0.650323	+	1.12639i
$794$		0			0
$795$	9.01559e10	−	1.56155e11i	0.225697	−	0.390919i
$796$		0			0
$797$		−	7.17227e11i		−	1.77756i	−0.458337	−	0.888778i	$-0.651555\pi$
$797$		−	7.17227e11i		−	1.77756i	0.458337	−	0.888778i	$-0.348445\pi$
$798$		0			0
$799$	1.38758e11			0.340464
$800$		0			0
$801$	−1.72823e11	−	9.97796e10i	−0.419829	−	0.242388i
$802$		0			0
$803$	−2.11042e11	+	1.21845e11i	−0.507583	+	0.293053i
$804$		0			0
$805$	−1.04233e11	−	2.50612e11i	−0.248212	−	0.596785i
$806$		0			0
$807$	9.32130e10	+	1.61450e11i	0.219777	+	0.380665i
$808$		0			0
$809$	−1.77589e10	+	3.07592e10i	−0.0414592	+	0.0718094i	−0.886010	−	0.463665i	$-0.846534\pi$
$809$	−1.77589e10	+	3.07592e10i	−0.0414592	+	0.0718094i	0.844551	+	0.535475i	$0.179867\pi$
$810$		0			0
$811$			3.23237e11i			0.747202i	0.927590	+	0.373601i	$0.121877\pi$
$811$			3.23237e11i			0.747202i	−0.927590	+	0.373601i	$0.878123\pi$
$812$		0			0
$813$	−2.64117e11			−0.604554
$814$		0			0
$815$	−5.35745e11	−	3.09313e11i	−1.21431	−	0.701080i
$816$		0			0
$817$	−5.47222e11	+	3.15939e11i	−1.22822	+	0.709112i
$818$		0			0
$819$	−2.08199e11	−	2.71025e10i	−0.462747	−	0.0602383i
$820$		0			0
$821$	3.30060e11	+	5.71681e11i	0.726475	+	1.25829i	0.958364	+	0.285550i	$0.0921762\pi$
$821$	3.30060e11	+	5.71681e11i	0.726475	+	1.25829i	−0.231889	+	0.972742i	$0.574490\pi$
$822$		0			0
$823$	3.85109e10	−	6.67029e10i	0.0839430	−	0.145394i	−0.820997	−	0.570932i	$-0.806582\pi$
$823$	3.85109e10	−	6.67029e10i	0.0839430	−	0.145394i	0.904940	+	0.425538i	$0.139915\pi$
$824$		0			0
$825$			1.19354e11i			0.257645i
$826$		0			0
$827$	3.29730e11			0.704914			0.352457	−	0.935828i	$-0.385346\pi$
$827$	3.29730e11			0.704914			0.352457	+	0.935828i	$0.385346\pi$
$828$		0			0
$829$	4.47097e11	+	2.58132e11i	0.946637	+	0.546541i	0.892035	−	0.451967i	$-0.149278\pi$
$829$	4.47097e11	+	2.58132e11i	0.946637	+	0.546541i	0.0546025	+	0.998508i	$0.482611\pi$
$830$		0			0
$831$	1.78696e11	−	1.03170e11i	0.374723	−	0.216346i
$832$		0			0
$833$	−5.63848e10	+	2.12902e11i	−0.117107	+	0.442181i
$834$		0			0
$835$	4.02342e11	+	6.96876e11i	0.827654	+	1.43354i
$836$		0			0
$837$	8.78928e10	−	1.52235e11i	0.179082	−	0.310179i
$838$		0			0
$839$		−	7.12614e11i		−	1.43816i	−0.694928	−	0.719079i	$-0.744564\pi$
$839$		−	7.12614e11i		−	1.43816i	0.694928	−	0.719079i	$-0.255436\pi$
$840$		0			0
$841$	−4.23174e11			−0.845932
$842$		0			0
$843$	3.59819e11	+	2.07742e11i	0.712482	+	0.411352i
$844$		0			0
$845$	−5.14334e11	+	2.96951e11i	−1.00883	+	0.582449i
$846$		0			0
$847$	−7.25502e9	+	5.57326e10i	−0.0140963	+	0.108287i
$848$		0			0
$849$	−1.66315e11	−	2.88066e11i	−0.320111	−	0.554449i
$850$		0			0
$851$	3.50560e10	−	6.07187e10i	0.0668411	−	0.115772i
$852$		0			0
$853$			8.51195e11i			1.60780i	0.594762	+	0.803901i	$0.297246\pi$
$853$			8.51195e11i			1.60780i	−0.594762	+	0.803901i	$0.702754\pi$
$854$		0			0
$855$	−2.28056e11			−0.426754
$856$		0			0
$857$	−8.97102e10	−	5.17942e10i	−0.166310	−	0.0960192i	0.414535	−	0.910034i	$-0.363944\pi$
$857$	−8.97102e10	−	5.17942e10i	−0.166310	−	0.0960192i	−0.580845	+	0.814014i	$0.697278\pi$
$858$		0			0
$859$	2.62089e11	−	1.51317e11i	0.481366	−	0.277917i	−0.239620	−	0.970867i	$-0.577023\pi$
$859$	2.62089e11	−	1.51317e11i	0.481366	−	0.277917i	0.720986	+	0.692950i	$0.243689\pi$
$860$		0			0
$861$	−2.33226e11	+	9.70025e10i	−0.424390	+	0.176510i
$862$		0			0
$863$	−1.58567e10	−	2.74647e10i	−0.0285871	−	0.0495144i	0.851378	−	0.524553i	$-0.175767\pi$
$863$	−1.58567e10	−	2.74647e10i	−0.0285871	−	0.0495144i	−0.879965	+	0.475038i	$0.842434\pi$
$864$		0			0
$865$	3.09275e11	−	5.35680e11i	0.552434	−	0.956844i
$866$		0			0
$867$			2.57965e11i			0.456547i
$868$		0			0
$869$	−4.33470e11			−0.760115
$870$		0			0
$871$	−8.83920e9	−	5.10332e9i	−0.0153582	−	0.00886706i
$872$		0			0
$873$	3.90043e10	−	2.25191e10i	0.0671515	−	0.0387699i
$874$		0			0
$875$	−2.27873e11	+	2.97864e11i	−0.388741	+	0.508143i
$876$		0			0
$877$	−3.80322e11	−	6.58737e11i	−0.642914	−	1.11356i	−0.984779	−	0.173811i	$-0.944392\pi$
$877$	−3.80322e11	−	6.58737e11i	−0.642914	−	1.11356i	0.341865	−	0.939749i	$-0.388941\pi$
$878$		0			0
$879$	8.02122e10	−	1.38932e11i	0.134365	−	0.232726i
$880$		0			0
$881$		−	1.08310e12i		−	1.79790i	−0.438047	−	0.898952i	$-0.644330\pi$
$881$		−	1.08310e12i		−	1.79790i	0.438047	−	0.898952i	$-0.355670\pi$
$882$		0			0
$883$	−8.89279e11			−1.46284			−0.731418	−	0.681930i	$-0.761141\pi$
$883$	−8.89279e11			−1.46284			−0.731418	+	0.681930i	$0.761141\pi$
$884$		0			0
$885$	−9.18469e10	−	5.30279e10i	−0.149724	−	0.0864432i
$886$		0			0
$887$	−5.90203e11	+	3.40754e11i	−0.953469	+	0.550486i	−0.894157	−	0.447754i	$-0.852224\pi$
$887$	−5.90203e11	+	3.40754e11i	−0.953469	+	0.550486i	−0.0593124	+	0.998239i	$0.518891\pi$
$888$		0			0
$889$	−8.05387e11	−	6.16139e11i	−1.28943	−	0.986443i
$890$		0			0
$891$	3.30467e10	+	5.72386e10i	0.0524345	+	0.0908193i
$892$		0			0
$893$	−2.49661e11	+	4.32426e11i	−0.392595	+	0.679995i
$894$		0			0
$895$			1.61918e11i			0.252350i
$896$		0			0
$897$	−2.78685e11			−0.430470
$898$		0			0
$899$	4.13228e11	+	2.38577e11i	0.632631	+	0.365250i
$900$		0			0
$901$	−1.68187e11	+	9.71028e10i	−0.255207	+	0.147344i
$902$		0			0
$903$	−1.98185e11	−	4.76501e11i	−0.298070	−	0.716660i
$904$		0			0
$905$	2.80088e11	+	4.85127e11i	0.417542	+	0.723204i
$906$		0			0
$907$	−2.31902e11	+	4.01666e11i	−0.342669	+	0.593520i	−0.984927	−	0.172968i	$-0.944664\pi$
$907$	−2.31902e11	+	4.01666e11i	−0.342669	+	0.593520i	0.642258	+	0.766488i	$0.277998\pi$
$908$		0			0
$909$			2.81144e11i			0.411787i
$910$		0			0
$911$	−1.51575e11			−0.220066			−0.110033	−	0.993928i	$-0.535096\pi$
$911$	−1.51575e11			−0.220066			−0.110033	+	0.993928i	$0.535096\pi$
$912$		0			0
$913$	−9.29413e11	−	5.36597e11i	−1.33760	−	0.772263i
$914$		0			0
$915$	−3.95161e11	+	2.28146e11i	−0.563754	+	0.325483i
$916$		0			0
$917$	1.25140e12	+	1.62902e11i	1.76978	+	0.230382i
$918$		0			0
$919$	−6.58090e11	−	1.13985e12i	−0.922621	−	1.59803i	−0.795343	−	0.606159i	$-0.792709\pi$
$919$	−6.58090e11	−	1.13985e12i	−0.922621	−	1.59803i	−0.127278	−	0.991867i	$-0.540624\pi$
$920$		0			0
$921$	1.49211e11	−	2.58442e11i	0.207378	−	0.359190i
$922$		0			0
$923$			1.90858e12i			2.62968i
$924$		0			0
$925$	8.68847e10			0.118680
$926$		0			0
$927$	−7.69992e10	−	4.44555e10i	−0.104272	−	0.0602014i
$928$		0			0
$929$	3.55698e11	−	2.05362e11i	0.477549	−	0.275713i	−0.241845	−	0.970315i	$-0.577753\pi$
$929$	3.55698e11	−	2.05362e11i	0.477549	−	0.275713i	0.719395	+	0.694602i	$0.244419\pi$
$930$		0			0
$931$	−5.62038e11	−	5.58783e11i	−0.748113	−	0.743780i
$932$		0			0
$933$	1.43705e11	+	2.48905e11i	0.189647	+	0.328478i
$934$		0			0
$935$	2.00217e11	−	3.46786e11i	0.261972	−	0.453749i
$936$		0			0
$937$		−	3.44178e11i		−	0.446503i	−0.974761	−	0.223252i	$-0.928333\pi$
$937$		−	3.44178e11i		−	0.446503i	0.974761	−	0.223252i	$-0.0716672\pi$
$938$		0			0
$939$	−6.16615e11			−0.793144
$940$		0			0
$941$	−2.62624e10	−	1.51626e10i	−0.0334947	−	0.0193382i	0.483159	−	0.875533i	$-0.339489\pi$
$941$	−2.62624e10	−	1.51626e10i	−0.0334947	−	0.0193382i	−0.516654	+	0.856194i	$0.672823\pi$
$942$		0			0
$943$	−2.90362e11	+	1.67640e11i	−0.367191	+	0.211998i
$944$		0			0
$945$	2.40436e10	−	1.84702e11i	0.0301490	−	0.231603i
$946$		0			0
$947$	−2.03677e11	−	3.52778e11i	−0.253245	−	0.438634i	0.711172	−	0.703018i	$-0.248165\pi$
$947$	−2.03677e11	−	3.52778e11i	−0.253245	−	0.438634i	−0.964417	+	0.264384i	$0.914831\pi$
$948$		0			0
$949$	3.52561e11	−	6.10654e11i	0.434680	−	0.752888i
$950$		0			0
$951$			6.92592e11i			0.846751i
$952$		0			0
$953$	1.16996e12			1.41841			0.709204	−	0.705003i	$-0.249055\pi$
$953$	1.16996e12			1.41841			0.709204	+	0.705003i	$0.249055\pi$
$954$		0			0
$955$	−2.00655e11	−	1.15848e11i	−0.241233	−	0.139276i
$956$		0			0
$957$	−1.55369e11	+	8.97023e10i	−0.185232	+	0.106944i
$958$		0			0
$959$	−2.81079e11	+	1.16905e11i	−0.332318	+	0.138216i
$960$		0			0
$961$	1.05059e12	+	1.81967e12i	1.23180	+	2.13353i
$962$		0			0
$963$	1.16429e11	−	2.01660e11i	0.135380	−	0.234485i
$964$		0			0
$965$		−	3.53019e11i		−	0.407089i
$966$		0			0
$967$	−3.11579e11			−0.356338			−0.178169	−	0.984000i	$-0.557017\pi$
$967$	−3.11579e11			−0.356338			−0.178169	+	0.984000i	$0.557017\pi$
$968$		0			0
$969$	2.12721e11	+	1.22814e11i	0.241276	+	0.139301i
$970$		0			0
$971$	−2.44094e11	+	1.40928e11i	−0.274587	+	0.158533i	−0.630970	−	0.775807i	$-0.717343\pi$
$971$	−2.44094e11	+	1.40928e11i	−0.274587	+	0.158533i	0.356383	+	0.934340i	$0.384010\pi$
$972$		0			0
$973$	5.80963e11	−	7.59407e11i	0.648182	−	0.847272i
$974$		0			0
$975$	−1.72677e11	−	2.99085e11i	−0.191080	−	0.330961i
$976$		0			0
$977$	4.13846e11	−	7.16803e11i	0.454214	−	0.786723i	−0.544428	−	0.838807i	$-0.683253\pi$
$977$	4.13846e11	−	7.16803e11i	0.454214	−	0.786723i	0.998643	+	0.0520849i	$0.0165866\pi$
$978$		0			0
$979$			1.26091e12i			1.37263i
$980$		0			0
$981$	−7.88904e10			−0.0851820
$982$		0			0
$983$	7.85992e11	+	4.53793e11i	0.841791	+	0.486008i	0.857873	−	0.513862i	$-0.171786\pi$
$983$	7.85992e11	+	4.53793e11i	0.841791	+	0.486008i	−0.0160817	+	0.999871i	$0.505119\pi$
$984$		0			0
$985$	1.19986e12	−	6.92740e11i	1.27464	−	0.735912i
$986$		0			0
$987$	−3.23898e11	−	2.47789e11i	−0.341303	−	0.261104i
$988$		0			0
$989$	−3.42504e11	−	5.93234e11i	−0.357998	−	0.620070i
$990$		0			0
$991$	−5.53863e11	+	9.59320e11i	−0.574259	+	0.994646i	0.421862	+	0.906660i	$0.361377\pi$
$991$	−5.53863e11	+	9.59320e11i	−0.574259	+	0.994646i	−0.996122	+	0.0879865i	$0.971957\pi$
$992$		0			0
$993$		−	6.22365e11i		−	0.640101i
$994$		0			0
$995$	2.22049e12			2.26546
$996$		0			0
$997$	−8.14412e11	−	4.70201e11i	−0.824259	−	0.475886i	0.0276241	−	0.999618i	$-0.491206\pi$
$997$	−8.14412e11	−	4.70201e11i	−0.824259	−	0.475886i	−0.851883	+	0.523732i	$0.824539\pi$
$998$		0			0
$999$	4.16672e10	−	2.40566e10i	0.0418343	−	0.0241531i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.9.m.a.73.4	yes	10
3.2	odd	2		252.9.z.b.73.2		10
7.5	odd	6	inner	84.9.m.a.61.4	✓	10
21.5	even	6		252.9.z.b.145.2		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.9.m.a.61.4	✓	10	7.5	odd	6	inner
84.9.m.a.73.4	yes	10	1.1	even	1	trivial
252.9.z.b.73.2		10	3.2	odd	2
252.9.z.b.145.2		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)

\(f(q)\)	\(=\)	\(q+(-40.5000 - 23.3827i) q^{3} +(656.878 - 379.249i) q^{5} +(1906.96 + 1458.87i) q^{7} +(1093.50 + 1894.00i) q^{9} +O(q^{10})\)	\(q+(-40.5000 - 23.3827i) q^{3} +(656.878 - 379.249i) q^{5} +(1906.96 + 1458.87i) q^{7} +(1093.50 + 1894.00i) q^{9} +(6909.25 - 11967.2i) q^{11} +39984.1i q^{13} -35471.4 q^{15} +(33086.2 + 19102.3i) q^{17} +(-119061. + 68739.9i) q^{19} +(-43119.7 - 103674. i) q^{21} +(-74519.7 - 129072. i) q^{23} +(92347.0 - 159950. i) q^{25} -102276. i q^{27} +277619. q^{29} +(1.48847e6 + 859370. i) q^{31} +(-559649. + 323114. i) q^{33} +(1.80592e6 + 235086. i) q^{35} +(235213. + 407400. i) q^{37} +(934936. - 1.61936e6i) q^{39} -2.24961e6i q^{41} +4.59615e6 q^{43} +(1.43659e6 + 829417. i) q^{45} +(3.14538e6 - 1.81598e6i) q^{47} +(1.50821e6 + 5.56401e6i) q^{49} +(-893327. - 1.54729e6i) q^{51} +(-2.54165e6 + 4.40227e6i) q^{53} -1.04813e7i q^{55} +6.42929e6 q^{57} +(2.58932e6 + 1.49495e6i) q^{59} +(1.11403e7 - 6.43183e6i) q^{61} +(-677831. + 5.20705e6i) q^{63} +(1.51639e7 + 2.62647e7i) q^{65} +(-127634. + 221068. i) q^{67} +6.96989e6i q^{69} +4.77334e7 q^{71} +(-1.52724e7 - 8.81754e6i) q^{73} +(-7.48010e6 + 4.31864e6i) q^{75} +(3.06342e7 - 1.27412e7i) q^{77} +(-1.56844e7 - 2.71662e7i) q^{79} +(-2.39148e6 + 4.14217e6i) q^{81} -7.76636e7i q^{83} +2.89781e7 q^{85} +(-1.12436e7 - 6.49147e6i) q^{87} +(-7.90230e7 + 4.56240e7i) q^{89} +(-5.83315e7 + 7.62481e7i) q^{91} +(-4.01887e7 - 6.96090e7i) q^{93} +(-5.21391e7 + 9.03075e7i) q^{95} -2.05936e7i q^{97} +3.02210e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 405 q^{3} + 1389 q^{5} + 1217 q^{7} + 10935 q^{9}+O(q^{10}) \) 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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