LMFDB - Embedded newform 84.9.m.a.61.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(84, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 5]))

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

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

chi := DirichletCharacter("84.61");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 84 = 2^{2} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	84.m (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$34.2198032451$
Analytic rank:	$0$
Dimension:	$10$
Relative dimension:	$5$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 38255 x^{8} + 1483053595 x^{6} - 139470625170 x^{5} + 5194605060018 x^{4} + \cdots + 15\!\cdots\!00 $ x^10 - 38255x^8 + 1483053595x^6 - 139470625170x^5 + 5194605060018x^4 - 71600654137860x^3 + 119846615988780x^2 + 4263507454127400*x + 15758720495290800
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{8}\cdot 7^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			61.5
Root			$16.3207 - 9.42278i$ of defining polynomial
Character	$\chi$	$=$	84.61
Dual form			84.9.m.a.73.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-40.5000 + 23.3827i) q^{3} +(790.151 + 456.194i) q^{5} +(-2318.52 + 623.902i) q^{7} +(1093.50 - 1894.00i) q^{9} +O(q^{10})$	\(q+(-40.5000 + 23.3827i) q^{3} +(790.151 + 456.194i) q^{5} +(-2318.52 + 623.902i) q^{7} +(1093.50 - 1894.00i) q^{9} +(4311.81 + 7468.27i) q^{11} +37108.2i q^{13} -42668.2 q^{15} +(43729.9 - 25247.5i) q^{17} +(-12884.6 - 7438.94i) q^{19} +(79311.7 - 79481.3i) q^{21} +(218851. - 379062. i) q^{23} +(220914. + 382634. i) q^{25} +102276. i q^{27} -155208. q^{29} +(-1.46250e6 + 844374. i) q^{31} +(-349257. - 201643. i) q^{33} +(-2.11660e6 - 564720. i) q^{35} +(-1.15538e6 + 2.00117e6i) q^{37} +(-867690. - 1.50288e6i) q^{39} +4.20233e6i q^{41} -3.43168e6 q^{43} +(1.72806e6 - 997696. i) q^{45} +(-4.11520e6 - 2.37591e6i) q^{47} +(4.98629e6 - 2.89306e6i) q^{49} +(-1.18071e6 + 2.04505e6i) q^{51} +(-1.47580e6 - 2.55615e6i) q^{53} +7.86809e6i q^{55} +695770. q^{57} +(-1.62462e7 + 9.37973e6i) q^{59} +(-1.45547e7 - 8.40319e6i) q^{61} +(-1.35364e6 + 5.07351e6i) q^{63} +(-1.69286e7 + 2.93211e7i) q^{65} +(-9.81510e6 - 1.70003e7i) q^{67} +2.04693e7i q^{69} +3.56826e7 q^{71} +(-1.35470e7 + 7.82137e6i) q^{73} +(-1.78940e7 - 1.03311e7i) q^{75} +(-1.46565e7 - 1.46252e7i) q^{77} +(9.07568e6 - 1.57195e7i) q^{79} +(-2.39148e6 - 4.14217e6i) q^{81} +1.15204e7i q^{83} +4.60710e7 q^{85} +(6.28592e6 - 3.62918e6i) q^{87} +(-2.46432e7 - 1.42278e7i) q^{89} +(-2.31519e7 - 8.60363e7i) q^{91} +(3.94875e7 - 6.83943e7i) q^{93} +(-6.78720e6 - 1.17558e7i) q^{95} +1.52449e8i q^{97} +1.88599e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 405 q^{3} + 1389 q^{5} + 1217 q^{7} + 10935 q^{9}+O(q^{10}) $ 10 * q - 405 * q^3 + 1389 * q^5 + 1217 * q^7 + 10935 * q^9	$ 10 q - 405 q^{3} + 1389 q^{5} + 1217 q^{7} + 10935 q^{9} - 879 q^{11} - 75006 q^{15} - 13674 q^{17} - 29268 q^{19} - 42363 q^{21} + 312732 q^{23} - 22052 q^{25} - 289794 q^{29} + 242787 q^{31} + 71199 q^{33} + 1209372 q^{35} + 1913308 q^{37} - 1232334 q^{39} - 861848 q^{43} + 3037743 q^{45} - 305448 q^{47} + 9821659 q^{49} + 369198 q^{51} - 10663233 q^{53} + 1580472 q^{57} + 18410871 q^{59} - 13937808 q^{61} + 769824 q^{63} - 14966808 q^{65} - 20722822 q^{67} + 113032584 q^{71} + 43436322 q^{73} + 1786212 q^{75} - 98823405 q^{77} - 42189637 q^{79} - 23914845 q^{81} + 142602108 q^{85} + 11736657 q^{87} + 67171914 q^{89} - 246091266 q^{91} - 6555249 q^{93} - 140649894 q^{95} - 3844746 q^{99}+O(q^{100}) $ 10 * q - 405 * q^3 + 1389 * q^5 + 1217 * q^7 + 10935 * q^9 - 879 * q^11 - 75006 * q^15 - 13674 * q^17 - 29268 * q^19 - 42363 * q^21 + 312732 * q^23 - 22052 * q^25 - 289794 * q^29 + 242787 * q^31 + 71199 * q^33 + 1209372 * q^35 + 1913308 * q^37 - 1232334 * q^39 - 861848 * q^43 + 3037743 * q^45 - 305448 * q^47 + 9821659 * q^49 + 369198 * q^51 - 10663233 * q^53 + 1580472 * q^57 + 18410871 * q^59 - 13937808 * q^61 + 769824 * q^63 - 14966808 * q^65 - 20722822 * q^67 + 113032584 * q^71 + 43436322 * q^73 + 1786212 * q^75 - 98823405 * q^77 - 42189637 * q^79 - 23914845 * q^81 + 142602108 * q^85 + 11736657 * q^87 + 67171914 * q^89 - 246091266 * q^91 - 6555249 * q^93 - 140649894 * q^95 - 3844746 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−40.5000	+	23.3827i	−0.500000	+	0.288675i
$3$	−40.5000	+	23.3827i	−0.500000	+	0.288675i
$4$		0			0
$5$	790.151	+	456.194i	1.26424	+	0.729911i	0.973892	−	0.227010i	$-0.0728950\pi$
$5$	790.151	+	456.194i	1.26424	+	0.729911i	0.290350	+	0.956921i	$0.406228\pi$
$6$		0			0
$7$	−2318.52	+	623.902i	−0.965649	+	0.259851i
$7$	−2318.52	+	623.902i	−0.965649	+	0.259851i
$8$		0			0
$9$	1093.50	−	1894.00i	0.166667	−	0.288675i
$10$		0			0
$11$	4311.81	+	7468.27i	0.294502	+	0.510093i	0.974869	−	0.222779i	$-0.0715127\pi$
$11$	4311.81	+	7468.27i	0.294502	+	0.510093i	−0.680367	+	0.732872i	$0.738179\pi$
$12$		0			0
$13$			37108.2i			1.29926i	0.760249	+	0.649631i	$0.225077\pi$
$13$			37108.2i			1.29926i	−0.760249	+	0.649631i	$0.774923\pi$
$14$		0			0
$15$	−42668.2			−0.842828
$16$		0			0
$17$	43729.9	−	25247.5i	0.523580	−	0.302289i	−0.214818	−	0.976654i	$-0.568916\pi$
$17$	43729.9	−	25247.5i	0.523580	−	0.302289i	0.738398	+	0.674365i	$0.235583\pi$
$18$		0			0
$19$	−12884.6	−	7438.94i	−0.0988684	−	0.0570817i	0.449751	−	0.893154i	$-0.351513\pi$
$19$	−12884.6	−	7438.94i	−0.0988684	−	0.0570817i	−0.548619	+	0.836073i	$0.684846\pi$
$20$		0			0
$21$	79311.7	−	79481.3i	0.407812	−	0.408684i
$22$		0			0
$23$	218851.	−	379062.i	0.782056	−	1.35456i	−0.148686	−	0.988885i	$-0.547504\pi$
$23$	218851.	−	379062.i	0.782056	−	1.35456i	0.930742	−	0.365677i	$-0.119162\pi$
$24$		0			0
$25$	220914.	+	382634.i	0.565539	+	0.979542i
$26$		0			0
$27$			102276.i			0.192450i
$28$		0			0
$29$	−155208.			−0.219443			−0.109721	−	0.993962i	$-0.534996\pi$
$29$	−155208.			−0.219443			−0.109721	+	0.993962i	$0.534996\pi$
$30$		0			0
$31$	−1.46250e6	+	844374.i	−1.58361	+	0.914299i	−0.589286	+	0.807924i	$0.700591\pi$
$31$	−1.46250e6	+	844374.i	−1.58361	+	0.914299i	−0.994326	+	0.106375i	$0.966076\pi$
$32$		0			0
$33$	−349257.	−	201643.i	−0.294502	−	0.170031i
$34$		0			0
$35$	−2.11660e6	−	564720.i	−1.41048	−	0.376323i
$36$		0			0
$37$	−1.15538e6	+	2.00117e6i	−0.616476	+	1.06777i	0.373647	+	0.927571i	$0.378107\pi$
$37$	−1.15538e6	+	2.00117e6i	−0.616476	+	1.06777i	−0.990124	+	0.140197i	$0.955226\pi$
$38$		0			0
$39$	−867690.	−	1.50288e6i	−0.375065	−	0.649631i
$40$		0			0
$41$			4.20233e6i			1.48715i	0.668652	+	0.743575i	$0.266872\pi$
$41$			4.20233e6i			1.48715i	−0.668652	+	0.743575i	$0.733128\pi$
$42$		0			0
$43$	−3.43168e6			−1.00377			−0.501884	−	0.864935i	$-0.667360\pi$
$43$	−3.43168e6			−1.00377			−0.501884	+	0.864935i	$0.667360\pi$
$44$		0			0
$45$	1.72806e6	−	997696.i	0.421414	−	0.243304i
$46$		0			0
$47$	−4.11520e6	−	2.37591e6i	−0.843334	−	0.486899i	0.0150621	−	0.999887i	$-0.495205\pi$
$47$	−4.11520e6	−	2.37591e6i	−0.843334	−	0.486899i	−0.858396	+	0.512987i	$0.828539\pi$
$48$		0			0
$49$	4.98629e6	−	2.89306e6i	0.864955	−	0.501849i
$50$		0			0
$51$	−1.18071e6	+	2.04505e6i	−0.174527	+	0.302289i
$52$		0			0
$53$	−1.47580e6	−	2.55615e6i	−0.187035	−	0.323954i	0.757225	−	0.653154i	$-0.226554\pi$
$53$	−1.47580e6	−	2.55615e6i	−0.187035	−	0.323954i	−0.944260	+	0.329200i	$0.893221\pi$
$54$		0			0
$55$			7.86809e6i			0.859842i
$56$		0			0
$57$	695770.			0.0659123
$58$		0			0
$59$	−1.62462e7	+	9.37973e6i	−1.34074	+	0.774074i	−0.986915	−	0.161239i	$-0.948451\pi$
$59$	−1.62462e7	+	9.37973e6i	−1.34074	+	0.774074i	−0.353820	+	0.935313i	$0.615118\pi$
$60$		0			0
$61$	−1.45547e7	−	8.40319e6i	−1.05120	−	0.606910i	−0.128215	−	0.991746i	$-0.540925\pi$
$61$	−1.45547e7	−	8.40319e6i	−1.05120	−	0.606910i	−0.922985	+	0.384836i	$0.874258\pi$
$62$		0			0
$63$	−1.35364e6	+	5.07351e6i	−0.0859290	+	0.322067i
$64$		0			0
$65$	−1.69286e7	+	2.93211e7i	−0.948345	+	1.64258i
$66$		0			0
$67$	−9.81510e6	−	1.70003e7i	−0.487075	−	0.843638i	0.512815	−	0.858499i	$-0.328603\pi$
$67$	−9.81510e6	−	1.70003e7i	−0.487075	−	0.843638i	−0.999890	+	0.0148609i	$0.995269\pi$
$68$		0			0
$69$			2.04693e7i			0.903041i
$70$		0			0
$71$	3.56826e7			1.40418			0.702091	−	0.712087i	$-0.252250\pi$
$71$	3.56826e7			1.40418			0.702091	+	0.712087i	$0.252250\pi$
$72$		0			0
$73$	−1.35470e7	+	7.82137e6i	−0.477037	+	0.275417i	−0.719181	−	0.694823i	$-0.755483\pi$
$73$	−1.35470e7	+	7.82137e6i	−0.477037	+	0.275417i	0.242144	+	0.970240i	$0.422149\pi$
$74$		0			0
$75$	−1.78940e7	−	1.03311e7i	−0.565539	−	0.326514i
$76$		0			0
$77$	−1.46565e7	−	1.46252e7i	−0.416934	−	0.416044i
$78$		0			0
$79$	9.07568e6	−	1.57195e7i	0.233008	−	0.403582i	−0.725684	−	0.688028i	$-0.758476\pi$
$79$	9.07568e6	−	1.57195e7i	0.233008	−	0.403582i	0.958692	+	0.284447i	$0.0918098\pi$
$80$		0			0
$81$	−2.39148e6	−	4.14217e6i	−0.0555556	−	0.0962250i
$82$		0			0
$83$			1.15204e7i			0.242747i	0.992607	+	0.121374i	$0.0387299\pi$
$83$			1.15204e7i			0.242747i	−0.992607	+	0.121374i	$0.961270\pi$
$84$		0			0
$85$	4.60710e7			0.882576
$86$		0			0
$87$	6.28592e6	−	3.62918e6i	0.109721	−	0.0633477i
$88$		0			0
$89$	−2.46432e7	−	1.42278e7i	−0.392770	−	0.226766i	0.290590	−	0.956848i	$-0.406148\pi$
$89$	−2.46432e7	−	1.42278e7i	−0.392770	−	0.226766i	−0.683360	+	0.730082i	$0.739482\pi$
$90$		0			0
$91$	−2.31519e7	−	8.60363e7i	−0.337614	−	1.25463i
$92$		0			0
$93$	3.94875e7	−	6.83943e7i	0.527871	−	0.914299i
$94$		0			0
$95$	−6.78720e6	−	1.17558e7i	−0.0833291	−	0.144330i
$96$		0			0
$97$			1.52449e8i			1.72201i	0.508593	+	0.861007i	$0.330166\pi$
$97$			1.52449e8i			1.72201i	−0.508593	+	0.861007i	$0.669834\pi$
$98$		0			0
$99$	1.88599e7			0.196335
$100$		0			0
$101$	1.63696e7	−	9.45100e6i	0.157309	−	0.0908222i	−0.419279	−	0.907857i	$-0.637717\pi$
$101$	1.63696e7	−	9.45100e6i	0.157309	−	0.0908222i	0.576588	+	0.817035i	$0.304384\pi$
$102$		0			0
$103$	1.36241e8	+	7.86589e7i	1.21049	+	0.698874i	0.962865	−	0.269983i	$-0.0870179\pi$
$103$	1.36241e8	+	7.86589e7i	1.21049	+	0.698874i	0.247621	+	0.968857i	$0.420351\pi$
$104$		0			0
$105$	9.89271e7	−	2.66207e7i	0.813876	−	0.219010i
$106$		0			0
$107$	−6.14765e7	+	1.06480e8i	−0.469001	+	0.812334i	−0.999372	−	0.0354319i	$-0.988719\pi$
$107$	−6.14765e7	+	1.06480e8i	−0.469001	+	0.812334i	0.530371	+	0.847766i	$0.322053\pi$
$108$		0			0
$109$	−2.12086e7	−	3.67344e7i	−0.150247	−	0.260236i	0.781071	−	0.624442i	$-0.214674\pi$
$109$	−2.12086e7	−	3.67344e7i	−0.150247	−	0.260236i	−0.931318	+	0.364206i	$0.881340\pi$
$110$		0			0
$111$		−	1.08063e8i		−	0.711845i
$112$		0			0
$113$	1.70573e8			1.04615			0.523077	−	0.852285i	$-0.324784\pi$
$113$	1.70573e8			1.04615			0.523077	+	0.852285i	$0.324784\pi$
$114$		0			0
$115$	3.45851e8	−	1.99677e8i	1.97742	−	1.14166i
$116$		0			0
$117$	7.02829e7	+	4.05779e7i	0.375065	+	0.216544i
$118$		0			0
$119$	−8.56369e7	+	8.58200e7i	−0.427044	+	0.427958i
$120$		0			0
$121$	6.99960e7	−	1.21237e8i	0.326537	−	0.565578i
$122$		0			0
$123$	−9.82618e7	−	1.70194e8i	−0.429303	−	0.743575i
$124$		0			0
$125$			4.67162e7i			0.191350i
$126$		0			0
$127$	−1.68997e8			−0.649628			−0.324814	−	0.945778i	$-0.605302\pi$
$127$	−1.68997e8			−0.649628			−0.324814	+	0.945778i	$0.605302\pi$
$128$		0			0
$129$	1.38983e8	−	8.02420e7i	0.501884	−	0.289763i
$130$		0			0
$131$	1.34426e8	+	7.76109e7i	0.456455	+	0.263535i	0.710553	−	0.703644i	$-0.248445\pi$
$131$	1.34426e8	+	7.76109e7i	0.456455	+	0.263535i	−0.254097	+	0.967179i	$0.581778\pi$
$132$		0			0
$133$	3.45145e7	+	9.20862e6i	0.110305	+	0.0294298i
$134$		0			0
$135$	−4.66576e7	+	8.08134e7i	−0.140471	+	0.243304i
$136$		0			0
$137$	1.58330e7	+	2.74236e7i	0.0449450	+	0.0778471i	0.887623	−	0.460571i	$-0.152355\pi$
$137$	1.58330e7	+	2.74236e7i	0.0449450	+	0.0778471i	−0.842678	+	0.538418i	$0.819022\pi$
$138$		0			0
$139$		−	4.96025e8i		−	1.32875i	−0.747398	−	0.664377i	$-0.768697\pi$
$139$		−	4.96025e8i		−	1.32875i	0.747398	−	0.664377i	$-0.231303\pi$
$140$		0			0
$141$	2.22221e8			0.562223
$142$		0			0
$143$	−2.77135e8	+	1.60004e8i	−0.662745	+	0.382636i
$144$		0			0
$145$	−1.22638e8	−	7.08049e7i	−0.277429	−	0.160174i
$146$		0			0
$147$	−1.34297e8	+	2.33762e8i	−0.287606	+	0.500616i
$148$		0			0
$149$	−1.89726e8	+	3.28616e8i	−0.384931	+	0.666720i	−0.991760	−	0.128113i	$-0.959108\pi$
$149$	−1.89726e8	+	3.28616e8i	−0.384931	+	0.666720i	0.606829	+	0.794833i	$0.292441\pi$
$150$		0			0
$151$	9.12604e7	+	1.58068e8i	0.175539	+	0.304043i	0.940348	−	0.340215i	$-0.110500\pi$
$151$	9.12604e7	+	1.58068e8i	0.175539	+	0.304043i	−0.764808	+	0.644258i	$0.777166\pi$
$152$		0			0
$153$		−	1.10433e8i		−	0.201526i
$154$		0			0
$155$	−1.54079e9			−2.66943
$156$		0			0
$157$	4.12944e8	−	2.38413e8i	0.679661	−	0.392402i	−0.120066	−	0.992766i	$-0.538311\pi$
$157$	4.12944e8	−	2.38413e8i	0.679661	−	0.392402i	0.799727	+	0.600363i	$0.204977\pi$
$158$		0			0
$159$	1.19539e8	+	6.90161e7i	0.187035	+	0.107985i
$160$		0			0
$161$	−2.70915e8	+	1.01541e9i	−0.403208	+	1.51125i
$162$		0			0
$163$	2.20882e6	−	3.82578e6i	0.00312903	−	0.00541963i	−0.864457	−	0.502707i	$-0.832337\pi$
$163$	2.20882e6	−	3.82578e6i	0.00312903	−	0.00541963i	0.867586	+	0.497288i	$0.165671\pi$
$164$		0			0
$165$	−1.83977e8	−	3.18658e8i	−0.248215	−	0.429921i
$166$		0			0
$167$			1.17558e9i			1.51142i	0.654908	+	0.755709i	$0.272707\pi$
$167$			1.17558e9i			1.51142i	−0.654908	+	0.755709i	$0.727293\pi$
$168$		0			0
$169$	−5.61290e8			−0.688083
$170$		0			0
$171$	−2.81787e7	+	1.62690e7i	−0.0329561	+	0.0190272i
$172$		0			0
$173$	1.13605e9	+	6.55898e8i	1.26827	+	0.732237i	0.974661	−	0.223688i	$-0.0718097\pi$
$173$	1.13605e9	+	6.55898e8i	1.26827	+	0.732237i	0.293611	+	0.955925i	$0.405143\pi$
$174$		0			0
$175$	−7.50919e8	−	7.49316e8i	−0.800647	−	0.798938i
$176$		0			0
$177$	4.38647e8	−	7.59758e8i	0.446912	−	0.774074i
$178$		0			0
$179$	−8.82047e8	−	1.52775e9i	−0.859171	−	1.48813i	−0.872721	−	0.488220i	$-0.837646\pi$
$179$	−8.82047e8	−	1.52775e9i	−0.859171	−	1.48813i	0.0135496	−	0.999908i	$-0.495687\pi$
$180$		0			0
$181$		−	7.89659e8i		−	0.735742i	−0.929877	−	0.367871i	$-0.880087\pi$
$181$		−	7.89659e8i		−	0.735742i	0.929877	−	0.367871i	$-0.119913\pi$
$182$		0			0
$183$	7.85956e8			0.700800
$184$		0			0
$185$	−1.82584e9	+	1.05415e9i	−1.55875	+	0.899945i
$186$		0			0
$187$	3.77110e8	+	2.17725e8i	0.308391	+	0.178050i
$188$		0			0
$189$	−6.38101e7	−	2.37129e8i	−0.0500083	−	0.185839i
$190$		0			0
$191$	−7.33171e8	+	1.26989e9i	−0.550899	+	0.954185i	0.447311	+	0.894378i	$0.352382\pi$
$191$	−7.33171e8	+	1.26989e9i	−0.550899	+	0.954185i	−0.998210	+	0.0598062i	$0.980952\pi$
$192$		0			0
$193$	1.05021e9	+	1.81901e9i	0.756911	+	1.31101i	0.944418	+	0.328746i	$0.106626\pi$
$193$	1.05021e9	+	1.81901e9i	0.756911	+	1.31101i	−0.187507	+	0.982263i	$0.560041\pi$
$194$		0			0
$195$		−	1.58334e9i		−	1.09505i
$196$		0			0
$197$	−3.68170e8			−0.244446			−0.122223	−	0.992503i	$-0.539002\pi$
$197$	−3.68170e8			−0.244446			−0.122223	+	0.992503i	$0.539002\pi$
$198$		0			0
$199$	1.06933e9	−	6.17377e8i	0.681865	−	0.393675i	−0.118692	−	0.992931i	$-0.537870\pi$
$199$	1.06933e9	−	6.17377e8i	0.681865	−	0.393675i	0.800557	+	0.599256i	$0.204537\pi$
$200$		0			0
$201$	7.95023e8	+	4.59007e8i	0.487075	+	0.281213i
$202$		0			0
$203$	3.59853e8	−	9.68344e7i	0.211905	−	0.0570224i
$204$		0			0
$205$	−1.91708e9	+	3.32048e9i	−1.08549	+	1.88012i
$206$		0			0
$207$	−4.78628e8	−	8.29008e8i	−0.260685	−	0.451520i
$208$		0			0
$209$		−	1.28301e8i		−	0.0672428i
$210$		0			0
$211$	−2.16657e9			−1.09306			−0.546528	−	0.837441i	$-0.684051\pi$
$211$	−2.16657e9			−1.09306			−0.546528	+	0.837441i	$0.684051\pi$
$212$		0			0
$213$	−1.44515e9	+	8.34355e8i	−0.702091	+	0.405352i
$214$		0			0
$215$	−2.71155e9	−	1.56551e9i	−1.26901	−	0.732661i
$216$		0			0
$217$	2.86403e9	−	2.87016e9i	1.29163	−	1.29439i
$218$		0			0
$219$	3.65769e8	−	6.33531e8i	0.159012	−	0.275417i
$220$		0			0
$221$	9.36890e8	+	1.62274e9i	0.392753	+	0.680268i
$222$		0			0
$223$			1.96048e9i			0.792761i	0.918086	+	0.396380i	$0.129734\pi$
$223$			1.96048e9i			0.792761i	−0.918086	+	0.396380i	$0.870266\pi$
$224$		0			0
$225$	9.66276e8			0.377026
$226$		0			0
$227$	−3.85172e8	+	2.22379e8i	−0.145061	+	0.0837510i	−0.570774	−	0.821107i	$-0.693357\pi$
$227$	−3.85172e8	+	2.22379e8i	−0.145061	+	0.0837510i	0.425713	+	0.904858i	$0.360023\pi$
$228$		0			0
$229$	−7.22193e7	−	4.16958e7i	−0.0262610	−	0.0151618i	0.486812	−	0.873507i	$-0.338160\pi$
$229$	−7.22193e7	−	4.16958e7i	−0.0262610	−	0.0151618i	−0.513073	+	0.858345i	$0.671493\pi$
$230$		0			0
$231$	9.35565e8	+	2.49613e8i	0.328569	+	0.0876636i
$232$		0			0
$233$	9.57414e8	−	1.65829e9i	0.324845	−	0.562648i	−0.656636	−	0.754208i	$-0.728021\pi$
$233$	9.57414e8	−	1.65829e9i	0.324845	−	0.562648i	0.981481	+	0.191560i	$0.0613546\pi$
$234$		0			0
$235$	−2.16775e9	−	3.75466e9i	−0.710786	−	1.23112i
$236$		0			0
$237$			8.48855e8i			0.269054i
$238$		0			0
$239$	5.74965e9			1.76218			0.881089	−	0.472951i	$-0.156811\pi$
$239$	5.74965e9			1.76218			0.881089	+	0.472951i	$0.156811\pi$
$240$		0			0
$241$	5.63575e9	−	3.25380e9i	1.67064	−	0.964546i	0.703363	−	0.710831i	$-0.251681\pi$
$241$	5.63575e9	−	3.25380e9i	1.67064	−	0.964546i	0.967279	−	0.253715i	$-0.0816526\pi$
$242$		0			0
$243$	1.93710e8	+	1.11839e8i	0.0555556	+	0.0320750i
$244$		0			0
$245$	5.25972e9	−	1.12378e7i	1.45982	−	0.00311900i
$246$		0			0
$247$	2.76046e8	−	4.78126e8i	0.0741641	−	0.128456i
$248$		0			0
$249$	−2.69377e8	−	4.66575e8i	−0.0700751	−	0.121374i
$250$		0			0
$251$		−	8.26879e8i		−	0.208328i	−0.994560	−	0.104164i	$-0.966783\pi$
$251$		−	8.26879e8i		−	0.208328i	0.994560	−	0.104164i	$-0.0332167\pi$
$252$		0			0
$253$	3.77458e9			0.921270
$254$		0			0
$255$	−1.86588e9	+	1.07726e9i	−0.441288	+	0.254778i
$256$		0			0
$257$	5.24692e9	+	3.02931e9i	1.20274	+	0.694402i	0.961163	−	0.275980i	$-0.0890023\pi$
$257$	5.24692e9	+	3.02931e9i	1.20274	+	0.694402i	0.241576	+	0.970382i	$0.422336\pi$
$258$		0			0
$259$	1.43023e9	−	5.36060e9i	0.317839	−	1.19128i
$260$		0			0
$261$	−1.69720e8	+	2.93963e8i	−0.0365738	+	0.0633477i
$262$		0			0
$263$	−2.36049e9	−	4.08848e9i	−0.493377	−	0.854553i	0.506594	−	0.862185i	$-0.330904\pi$
$263$	−2.36049e9	−	4.08848e9i	−0.493377	−	0.854553i	−0.999971	+	0.00763120i	$0.997571\pi$
$264$		0			0
$265$		−	2.69300e9i		−	0.546075i
$266$		0			0
$267$	1.33074e9			0.261846
$268$		0			0
$269$	−8.05825e9	+	4.65243e9i	−1.53897	+	0.888527i	−0.540076	+	0.841617i	$0.681604\pi$
$269$	−8.05825e9	+	4.65243e9i	−1.53897	+	0.888527i	−0.998899	+	0.0469109i	$0.985062\pi$
$270$		0			0
$271$	−7.22249e8	−	4.16991e8i	−0.133909	−	0.0773124i	0.431549	−	0.902090i	$-0.357967\pi$
$271$	−7.22249e8	−	4.16991e8i	−0.133909	−	0.0773124i	−0.565458	+	0.824777i	$0.691301\pi$
$272$		0			0
$273$	2.94941e9	+	2.94312e9i	0.530988	+	0.529855i
$274$		0			0
$275$	−1.90507e9	+	3.29969e9i	−0.333105	+	0.576955i
$276$		0			0
$277$	5.58602e9	+	9.67528e9i	0.948820	+	1.64340i	0.747916	+	0.663793i	$0.231055\pi$
$277$	5.58602e9	+	9.67528e9i	0.948820	+	1.64340i	0.200904	+	0.979611i	$0.435612\pi$
$278$		0			0
$279$			3.69329e9i			0.609533i
$280$		0			0
$281$	5.88765e8			0.0944314			0.0472157	−	0.998885i	$-0.484965\pi$
$281$	5.88765e8			0.0944314			0.0472157	+	0.998885i	$0.484965\pi$
$282$		0			0
$283$	−4.79981e9	+	2.77117e9i	−0.748305	+	0.432034i	−0.825081	−	0.565014i	$-0.808871\pi$
$283$	−4.79981e9	+	2.77117e9i	−0.748305	+	0.432034i	0.0767763	+	0.997048i	$0.475537\pi$
$284$		0			0
$285$	5.49763e8	+	3.17406e8i	0.0833291	+	0.0481100i
$286$		0			0
$287$	−2.62184e9	−	9.74320e9i	−0.386437	−	1.43607i
$288$		0			0
$289$	−2.21301e9	+	3.83304e9i	−0.317243	+	0.549480i
$290$		0			0
$291$	−3.56466e9	−	6.17417e9i	−0.497103	−	0.861007i
$292$		0			0
$293$		−	9.38671e9i		−	1.27363i	−0.771017	−	0.636814i	$-0.780252\pi$
$293$		−	9.38671e9i		−	1.27363i	0.771017	−	0.636814i	$-0.219748\pi$
$294$		0			0
$295$	−1.71159e10			−2.26002
$296$		0			0
$297$	−7.63824e8	+	4.40994e8i	−0.0981675	+	0.0566770i
$298$		0			0
$299$	1.40663e10	+	8.12119e9i	1.75993	+	1.01610i
$300$		0			0
$301$	7.95643e9	−	2.14103e9i	0.969287	−	0.260830i
$302$		0			0
$303$	−4.41979e8	+	7.65531e8i	−0.0524362	+	0.0908222i
$304$		0			0
$305$	−7.66697e9	−	1.32796e10i	−0.885981	−	1.53456i
$306$		0			0
$307$			5.32865e9i			0.599879i	0.953958	+	0.299940i	$0.0969665\pi$
$307$			5.32865e9i			0.599879i	−0.953958	+	0.299940i	$0.903033\pi$
$308$		0			0
$309$	−7.35703e9			−0.806991
$310$		0			0
$311$	1.06252e10	−	6.13448e9i	1.13579	−	0.655747i	0.190403	−	0.981706i	$-0.439020\pi$
$311$	1.06252e10	−	6.13448e9i	1.13579	−	0.655747i	0.945384	+	0.325959i	$0.105687\pi$
$312$		0			0
$313$	3.82607e9	+	2.20898e9i	0.398635	+	0.230152i	0.685895	−	0.727701i	$-0.259411\pi$
$313$	3.82607e9	+	2.20898e9i	0.398635	+	0.230152i	−0.287260	+	0.957853i	$0.592744\pi$
$314$		0			0
$315$	−3.38408e9	+	3.39132e9i	−0.343715	+	0.344451i
$316$		0			0
$317$	−3.64028e9	+	6.30515e9i	−0.360494	+	0.624394i	−0.988042	−	0.154184i	$-0.950725\pi$
$317$	−3.64028e9	+	6.30515e9i	−0.360494	+	0.624394i	0.627548	+	0.778578i	$0.284059\pi$
$318$		0			0
$319$	−6.69227e8	−	1.15913e9i	−0.0646265	−	0.111936i
$320$		0			0
$321$		−	5.74994e9i		−	0.541556i
$322$		0			0
$323$	−7.51258e8			−0.0690207
$324$		0			0
$325$	−1.41989e10	+	8.19771e9i	−1.27268	+	0.734783i
$326$		0			0
$327$	1.71790e9	+	9.91828e8i	0.150247	+	0.0867452i
$328$		0			0
$329$	1.10235e10	+	2.94113e9i	0.940886	+	0.251033i
$330$		0			0
$331$	−1.07154e10	+	1.85596e10i	−0.892681	+	1.54617i	−0.0560332	+	0.998429i	$0.517845\pi$
$331$	−1.07154e10	+	1.85596e10i	−0.892681	+	1.54617i	−0.836648	+	0.547741i	$0.815488\pi$
$332$		0			0
$333$	2.52681e9	+	4.37656e9i	0.205492	+	0.355923i
$334$		0			0
$335$		−	1.79104e10i		−	1.42208i
$336$		0			0
$337$	−2.19922e10			−1.70510			−0.852550	−	0.522645i	$-0.824945\pi$
$337$	−2.19922e10			−1.70510			−0.852550	+	0.522645i	$0.824945\pi$
$338$		0			0
$339$	−6.90819e9	+	3.98845e9i	−0.523077	+	0.301999i
$340$		0			0
$341$	−1.26120e10	−	7.28157e9i	−0.932756	−	0.538527i
$342$		0			0
$343$	−9.75585e9	+	9.81858e9i	−0.704837	+	0.709369i
$344$		0			0
$345$	−9.33799e9	+	1.61739e10i	−0.659139	+	1.14166i
$346$		0			0
$347$	2.11832e9	+	3.66903e9i	0.146108	+	0.253066i	0.929786	−	0.368101i	$-0.119992\pi$
$347$	2.11832e9	+	3.66903e9i	0.146108	+	0.253066i	−0.783678	+	0.621167i	$0.786659\pi$
$348$		0			0
$349$		−	5.09547e9i		−	0.343465i	−0.985144	−	0.171732i	$-0.945064\pi$
$349$		−	5.09547e9i		−	0.343465i	0.985144	−	0.171732i	$-0.0549365\pi$
$350$		0			0
$351$	−3.79528e9			−0.250043
$352$		0			0
$353$	6.67443e9	−	3.85348e9i	0.429848	−	0.248173i	−0.269434	−	0.963019i	$-0.586837\pi$
$353$	6.67443e9	−	3.85348e9i	0.429848	−	0.248173i	0.699282	+	0.714846i	$0.253503\pi$
$354$		0			0
$355$	2.81947e10	+	1.62782e10i	1.77523	+	1.02493i
$356$		0			0
$357$	1.46159e9	−	5.47813e9i	0.0899814	−	0.337256i
$358$		0			0
$359$	−4.93726e8	+	8.55158e8i	−0.0297241	+	0.0514836i	−0.880505	−	0.474037i	$-0.842796\pi$
$359$	−4.93726e8	+	8.55158e8i	−0.0297241	+	0.0514836i	0.850781	+	0.525521i	$0.176130\pi$
$360$		0			0
$361$	−8.38111e9	−	1.45165e10i	−0.493483	−	0.854738i
$362$		0			0
$363$			6.54678e9i			0.377052i
$364$		0			0
$365$	−1.42723e10			−0.804120
$366$		0			0
$367$	1.46428e10	−	8.45400e9i	0.807158	−	0.466013i	−0.0388101	−	0.999247i	$-0.512357\pi$
$367$	1.46428e10	−	8.45400e9i	0.807158	−	0.466013i	0.845968	+	0.533234i	$0.179023\pi$
$368$		0			0
$369$	7.95921e9	+	4.59525e9i	0.429303	+	0.247858i
$370$		0			0
$371$	5.01645e9	+	5.00575e9i	0.264790	+	0.264225i
$372$		0			0
$373$	−4.36752e9	+	7.56476e9i	−0.225631	+	0.390805i	−0.956509	−	0.291704i	$-0.905778\pi$
$373$	−4.36752e9	+	7.56476e9i	−0.225631	+	0.390805i	0.730877	+	0.682509i	$0.239111\pi$
$374$		0			0
$375$	−1.09235e9	−	1.89201e9i	−0.0552379	−	0.0956748i
$376$		0			0
$377$		−	5.75949e9i		−	0.285114i
$378$		0			0
$379$	1.34632e10			0.652516			0.326258	−	0.945281i	$-0.394212\pi$
$379$	1.34632e10			0.652516			0.326258	+	0.945281i	$0.394212\pi$
$380$		0			0
$381$	6.84439e9	−	3.95161e9i	0.324814	−	0.187531i
$382$		0			0
$383$	3.11283e10	+	1.79719e10i	1.44664	+	0.835217i	0.998279	−	0.0586370i	$-0.0186755\pi$
$383$	3.11283e10	+	1.79719e10i	1.44664	+	0.835217i	0.448359	+	0.893854i	$0.352009\pi$
$384$		0			0
$385$	−4.90891e9	−	1.82423e10i	−0.223431	−	0.830305i
$386$		0			0
$387$	−3.75255e9	+	6.49960e9i	−0.167295	+	0.289763i
$388$		0			0
$389$	−1.20745e10	−	2.09136e10i	−0.527315	−	0.913337i	−0.999493	−	0.0318334i	$-0.989865\pi$
$389$	−1.20745e10	−	2.09136e10i	−0.527315	−	0.913337i	0.472178	−	0.881503i	$-0.343468\pi$
$390$		0			0
$391$		−	2.21018e10i		−	0.945628i
$392$		0			0
$393$	−7.25900e9			−0.304303
$394$		0			0
$395$	1.43423e10	−	8.28054e9i	0.589157	−	0.340150i
$396$		0			0
$397$	2.12986e10	+	1.22967e10i	0.857410	+	0.495026i	0.863144	−	0.504958i	$-0.168492\pi$
$397$	2.12986e10	+	1.22967e10i	0.857410	+	0.495026i	−0.00573411	+	0.999984i	$0.501825\pi$
$398$		0			0
$399$	−1.61316e9	+	4.34092e8i	−0.0636481	+	0.0171274i
$400$		0			0
$401$	−2.09789e9	+	3.63366e9i	−0.0811345	+	0.140529i	−0.903738	−	0.428087i	$-0.859188\pi$
$401$	−2.09789e9	+	3.63366e9i	−0.0811345	+	0.140529i	0.822603	+	0.568616i	$0.192521\pi$
$402$		0			0
$403$	−3.13332e10	−	5.42708e10i	−1.18791	−	2.05753i
$404$		0			0
$405$		−	4.36392e9i		−	0.162202i
$406$		0			0
$407$	−1.99270e10			−0.726215
$408$		0			0
$409$	−1.37810e10	+	7.95648e9i	−0.492480	+	0.284333i	−0.725603	−	0.688114i	$-0.758439\pi$
$409$	−1.37810e10	+	7.95648e9i	−0.492480	+	0.284333i	0.233123	+	0.972447i	$0.425106\pi$
$410$		0			0
$411$	−1.28248e9	−	7.40438e8i	−0.0449450	−	0.0259490i
$412$		0			0
$413$	3.18151e10	−	3.18831e10i	1.09354	−	1.09587i
$414$		0			0
$415$	−5.25553e9	+	9.10284e9i	−0.177184	+	0.306891i
$416$		0			0
$417$	1.15984e10	+	2.00890e10i	0.383578	+	0.664377i
$418$		0			0
$419$			5.82585e10i			1.89018i	0.326809	+	0.945090i	$0.394026\pi$
$419$			5.82585e10i			1.89018i	−0.326809	+	0.945090i	$0.605974\pi$
$420$		0			0
$421$	2.37307e10			0.755410			0.377705	−	0.925926i	$-0.376713\pi$
$421$	2.37307e10			0.755410			0.377705	+	0.925926i	$0.376713\pi$
$422$		0			0
$423$	−8.99994e9	+	5.19612e9i	−0.281111	+	0.162300i
$424$		0			0
$425$	1.93211e10	+	1.11550e10i	0.592210	+	0.341912i
$426$		0			0
$427$	3.89883e10	+	1.04022e10i	1.17280	+	0.312907i
$428$		0			0
$429$	7.48263e9	−	1.29603e10i	0.220915	−	0.382636i
$430$		0			0
$431$	−1.31104e10	−	2.27079e10i	−0.379934	−	0.658065i	0.611118	−	0.791539i	$-0.290720\pi$
$431$	−1.31104e10	−	2.27079e10i	−0.379934	−	0.658065i	−0.991052	+	0.133475i	$0.957387\pi$
$432$		0			0
$433$			1.04010e10i			0.295884i	0.988996	+	0.147942i	$0.0472649\pi$
$433$			1.04010e10i			0.295884i	−0.988996	+	0.147942i	$0.952735\pi$
$434$		0			0
$435$	6.62244e9			0.184953
$436$		0			0
$437$	−5.63964e9	+	3.25605e9i	−0.154641	+	0.0892822i
$438$		0			0
$439$	4.94931e10	+	2.85749e10i	1.33256	+	0.769354i	0.985691	−	0.168560i	$-0.0539117\pi$
$439$	4.94931e10	+	2.85749e10i	1.33256	+	0.769354i	0.346868	+	0.937914i	$0.387245\pi$
$440$		0			0
$441$	−2.69370e7	−	1.26076e10i	−0.000712187	−	0.333333i
$442$		0			0
$443$	−1.46702e9	+	2.54096e9i	−0.0380909	+	0.0659754i	−0.884442	−	0.466649i	$-0.845461\pi$
$443$	−1.46702e9	+	2.54096e9i	−0.0380909	+	0.0659754i	0.846351	+	0.532625i	$0.178794\pi$
$444$		0			0
$445$	−1.29813e10	−	2.24842e10i	−0.331037	−	0.573373i
$446$		0			0
$447$		−	1.77453e10i		−	0.444480i
$448$		0			0
$449$	2.90570e10			0.714932			0.357466	−	0.933926i	$-0.383641\pi$
$449$	2.90570e10			0.714932			0.357466	+	0.933926i	$0.383641\pi$
$450$		0			0
$451$	−3.13842e10	+	1.81197e10i	−0.758585	+	0.437969i
$452$		0			0
$453$	−7.39209e9	−	4.26783e9i	−0.175539	−	0.101348i
$454$		0			0
$455$	2.09557e10	−	7.85434e10i	0.488942	−	1.83259i
$456$		0			0
$457$	−1.10752e10	+	1.91828e10i	−0.253914	+	0.439792i	−0.964600	−	0.263717i	$-0.915051\pi$
$457$	−1.10752e10	+	1.91828e10i	−0.253914	+	0.439792i	0.710686	+	0.703509i	$0.248385\pi$
$458$		0			0
$459$	2.58221e9	+	4.47252e9i	0.0581756	+	0.100763i
$460$		0			0
$461$		−	1.24751e10i		−	0.276210i	−0.990418	−	0.138105i	$-0.955899\pi$
$461$		−	1.24751e10i		−	0.276210i	0.990418	−	0.138105i	$-0.0441011\pi$
$462$		0			0
$463$	5.14297e10			1.11916			0.559578	−	0.828778i	$-0.310963\pi$
$463$	5.14297e10			1.11916			0.559578	+	0.828778i	$0.310963\pi$
$464$		0			0
$465$	6.24022e10	−	3.60279e10i	1.33471	−	0.770597i
$466$		0			0
$467$	2.25703e10	+	1.30310e10i	0.474537	+	0.273974i	0.718137	−	0.695902i	$-0.244995\pi$
$467$	2.25703e10	+	1.30310e10i	0.474537	+	0.273974i	−0.243600	+	0.969876i	$0.578328\pi$
$468$		0			0
$469$	3.33630e10	+	3.32918e10i	0.689563	+	0.688092i
$470$		0			0
$471$	−1.11495e10	+	1.93115e10i	−0.226554	+	0.392402i
$472$		0			0
$473$	−1.47968e10	−	2.56288e10i	−0.295612	−	0.512015i
$474$		0			0
$475$		−	6.57345e9i		−	0.129128i
$476$		0			0
$477$	−6.45513e9			−0.124690
$478$		0			0
$479$	1.73795e10	−	1.00341e10i	0.330138	−	0.190605i	−0.325765	−	0.945451i	$-0.605622\pi$
$479$	1.73795e10	−	1.00341e10i	0.330138	−	0.190605i	0.655902	+	0.754846i	$0.272288\pi$
$480$		0			0
$481$	−7.42599e10	−	4.28740e10i	−1.38731	−	0.800964i
$482$		0			0
$483$	−1.27709e10	−	4.74586e10i	−0.234656	−	0.872020i
$484$		0			0
$485$	−6.95462e10	+	1.20457e11i	−1.25692	+	2.17704i
$486$		0			0
$487$	−4.89148e9	−	8.47229e9i	−0.0869610	−	0.150621i	0.819264	−	0.573416i	$-0.194382\pi$
$487$	−4.89148e9	−	8.47229e9i	−0.0869610	−	0.150621i	−0.906225	+	0.422795i	$0.861049\pi$
$488$		0			0
$489$			2.06592e8i			0.00361309i
$490$		0			0
$491$	−4.29568e8			−0.00739105			−0.00369553	−	0.999993i	$-0.501176\pi$
$491$	−4.29568e8			−0.00739105			−0.00369553	+	0.999993i	$0.501176\pi$
$492$		0			0
$493$	−6.78723e9	+	3.91861e9i	−0.114896	+	0.0663352i
$494$		0			0
$495$	1.49021e10	+	8.60376e9i	0.248215	+	0.143307i
$496$		0			0
$497$	−8.27309e10	+	2.22624e10i	−1.35595	+	0.364878i
$498$		0			0
$499$	−6.21696e9	+	1.07681e10i	−0.100271	+	0.173675i	−0.911796	−	0.410643i	$-0.865304\pi$
$499$	−6.21696e9	+	1.07681e10i	−0.100271	+	0.173675i	0.811525	+	0.584317i	$0.198638\pi$
$500$		0			0
$501$	−2.74881e10	−	4.76108e10i	−0.436309	−	0.755709i
$502$		0			0
$503$			8.85735e10i			1.38367i	0.722057	+	0.691834i	$0.243197\pi$
$503$			8.85735e10i			1.38367i	−0.722057	+	0.691834i	$0.756803\pi$
$504$		0			0
$505$	1.72460e10			0.265168
$506$		0			0
$507$	2.27323e10	−	1.31245e10i	0.344041	−	0.198632i
$508$		0			0
$509$	−1.00294e11	−	5.79047e10i	−1.49418	−	0.862666i	−0.494203	−	0.869347i	$-0.664540\pi$
$509$	−1.00294e11	−	5.79047e10i	−1.49418	−	0.862666i	−0.999978	+	0.00668109i	$0.997873\pi$
$510$		0			0
$511$	2.65293e10	−	2.65860e10i	0.389083	−	0.389915i
$512$		0			0
$513$	7.60824e8	−	1.31779e9i	0.0109854	−	0.0190272i
$514$		0			0
$515$	7.17675e10	+	1.24305e11i	1.02023	+	1.76709i
$516$		0			0
$517$		−	4.09779e10i		−	0.573572i
$518$		0			0
$519$	−6.13466e10			−0.845514
$520$		0			0
$521$	−1.24776e11	+	7.20395e10i	−1.69348	+	0.977732i	−0.741811	+	0.670609i	$0.766033\pi$
$521$	−1.24776e11	+	7.20395e10i	−1.69348	+	0.977732i	−0.951670	+	0.307122i	$0.900634\pi$
$522$		0			0
$523$	2.73214e10	+	1.57740e10i	0.365170	+	0.210831i	0.671346	−	0.741144i	$-0.265716\pi$
$523$	2.73214e10	+	1.57740e10i	0.365170	+	0.210831i	−0.306176	+	0.951975i	$0.599050\pi$
$524$		0			0
$525$	4.79332e10	+	1.27888e10i	0.630957	+	0.168342i
$526$		0			0
$527$	−4.26367e10	+	7.38489e10i	−0.552765	+	0.957418i
$528$		0			0
$529$	−5.66364e10	−	9.80971e10i	−0.723224	−	1.25266i
$530$		0			0
$531$			4.10270e10i			0.516049i
$532$		0			0
$533$	−1.55941e11			−1.93220
$534$		0			0
$535$	−9.71514e10	+	5.60904e10i	−1.18586	+	0.684658i
$536$		0			0
$537$	7.14458e10	+	4.12493e10i	0.859171	+	0.496043i
$538$		0			0
$539$	4.31061e10	+	2.47647e10i	0.510721	+	0.293412i
$540$		0			0
$541$	5.42610e10	−	9.39828e10i	0.633430	−	1.09713i	−0.353415	−	0.935467i	$-0.614980\pi$
$541$	5.42610e10	−	9.39828e10i	0.633430	−	1.09713i	0.986845	−	0.161667i	$-0.0516871\pi$
$542$		0			0
$543$	1.84644e10	+	3.19812e10i	0.212390	+	0.367871i
$544$		0			0
$545$		−	3.87010e10i		−	0.438668i
$546$		0			0
$547$	−1.21262e11			−1.35449			−0.677246	−	0.735757i	$-0.736827\pi$
$547$	−1.21262e11			−1.35449			−0.677246	+	0.735757i	$0.736827\pi$
$548$		0			0
$549$	−3.18312e10	+	1.83778e10i	−0.350400	+	0.202303i
$550$		0			0
$551$	1.99980e9	+	1.15458e9i	0.0216960	+	0.0125262i
$552$		0			0
$553$	−1.12347e10	+	4.21084e10i	−0.120133	+	0.450265i
$554$		0			0
$555$	4.92978e10	−	8.53862e10i	0.519583	−	0.899945i
$556$		0			0
$557$	−1.94381e10	−	3.36677e10i	−0.201945	−	0.349778i	0.747210	−	0.664588i	$-0.231393\pi$
$557$	−1.94381e10	−	3.36677e10i	−0.201945	−	0.349778i	−0.949155	+	0.314809i	$0.898059\pi$
$558$		0			0
$559$		−	1.27344e11i		−	1.30416i
$560$		0			0
$561$	−2.03640e10			−0.205594
$562$		0			0
$563$	−1.06997e11	+	6.17750e10i	−1.06498	+	0.614864i	−0.926804	−	0.375545i	$-0.877456\pi$
$563$	−1.06997e11	+	6.17750e10i	−1.06498	+	0.614864i	−0.138171	+	0.990408i	$0.544122\pi$
$564$		0			0
$565$	1.34778e11	+	7.78143e10i	1.32259	+	0.763599i
$566$		0			0
$567$	8.12902e9	+	8.11167e9i	0.0786513	+	0.0784834i
$568$		0			0
$569$	−3.76060e10	+	6.51355e10i	−0.358763	+	0.621396i	−0.987754	−	0.156016i	$-0.950135\pi$
$569$	−3.76060e10	+	6.51355e10i	−0.358763	+	0.621396i	0.628991	+	0.777412i	$0.283468\pi$
$570$		0			0
$571$	6.39802e9	+	1.10817e10i	0.0601868	+	0.104247i	0.894549	−	0.446970i	$-0.147497\pi$
$571$	6.39802e9	+	1.10817e10i	0.0601868	+	0.104247i	−0.834362	+	0.551217i	$0.814164\pi$
$572$		0			0
$573$		−	6.85740e10i		−	0.636123i
$574$		0			0
$575$	1.93389e11			1.76913
$576$		0			0
$577$	1.63442e11	−	9.43632e10i	1.47455	−	0.851333i	0.474963	−	0.880006i	$-0.342461\pi$
$577$	1.63442e11	−	9.43632e10i	1.47455	−	0.851333i	0.999589	+	0.0286728i	$0.00912810\pi$
$578$		0			0
$579$	−8.50666e10	−	4.91133e10i	−0.756911	−	0.437003i
$580$		0			0
$581$	−7.18758e9	−	2.67103e10i	−0.0630780	−	0.234409i
$582$		0			0
$583$	1.27267e10	−	2.20433e10i	0.110164	−	0.190810i
$584$		0			0
$585$	3.70228e10	+	6.41253e10i	0.316115	+	0.547527i
$586$		0			0
$587$		−	7.09182e10i		−	0.597318i	−0.954360	−	0.298659i	$-0.903461\pi$
$587$		−	7.09182e10i		−	0.597318i	0.954360	−	0.298659i	$-0.0965393\pi$
$588$		0			0
$589$	2.51250e10			0.208759
$590$		0			0
$591$	1.49109e10	−	8.60881e9i	0.122223	−	0.0705656i
$592$		0			0
$593$	1.08767e11	+	6.27965e10i	0.879584	+	0.507828i	0.870521	−	0.492131i	$-0.163782\pi$
$593$	1.08767e11	+	6.27965e10i	0.879584	+	0.507828i	0.00906300	+	0.999959i	$0.497115\pi$
$594$		0			0
$595$	−1.06817e11	+	2.87438e10i	−0.852258	+	0.229338i
$596$		0			0
$597$	−2.88718e10	+	5.00075e10i	−0.227288	+	0.393675i
$598$		0			0
$599$	3.74468e10	+	6.48598e10i	0.290876	+	0.503812i	0.974017	−	0.226475i	$-0.0727200\pi$
$599$	3.74468e10	+	6.48598e10i	0.290876	+	0.503812i	−0.683141	+	0.730286i	$0.739387\pi$
$600$		0			0
$601$			7.91457e10i			0.606638i	0.952889	+	0.303319i	$0.0980948\pi$
$601$			7.91457e10i			0.606638i	−0.952889	+	0.303319i	$0.901905\pi$
$602$		0			0
$603$	−4.29313e10			−0.324717
$604$		0			0
$605$	1.10615e11	−	6.38635e10i	0.825643	−	0.476685i
$606$		0			0
$607$	−4.76961e9	−	2.75374e9i	−0.0351340	−	0.0202846i	0.482330	−	0.875990i	$-0.339791\pi$
$607$	−4.76961e9	−	2.75374e9i	−0.0351340	−	0.0202846i	−0.517464	+	0.855705i	$0.673124\pi$
$608$		0			0
$609$	−1.23098e10	+	1.23361e10i	−0.0894915	+	0.0896829i
$610$		0			0
$611$	8.81659e10	−	1.52708e11i	0.632610	−	1.09571i
$612$		0			0
$613$	1.11411e11	+	1.92970e11i	0.789018	+	1.36662i	0.926569	+	0.376125i	$0.122744\pi$
$613$	1.11411e11	+	1.92970e11i	0.789018	+	1.36662i	−0.137550	+	0.990495i	$0.543923\pi$
$614$		0			0
$615$		−	1.79306e11i		−	1.25341i
$616$		0			0
$617$	−1.45466e11			−1.00374			−0.501868	−	0.864944i	$-0.667354\pi$
$617$	−1.45466e11			−1.00374			−0.501868	+	0.864944i	$0.667354\pi$
$618$		0			0
$619$	−9.99270e10	+	5.76929e10i	−0.680644	+	0.392970i	−0.800098	−	0.599870i	$-0.795219\pi$
$619$	−9.99270e10	+	5.76929e10i	−0.680644	+	0.392970i	0.119453	+	0.992840i	$0.461886\pi$
$620$		0			0
$621$	3.87689e10	+	2.23832e10i	0.260685	+	0.150507i
$622$		0			0
$623$	6.60127e10	+	1.76125e10i	0.438203	+	0.116914i
$624$		0			0
$625$	6.49827e10	−	1.12553e11i	0.425871	−	0.737630i
$626$		0			0
$627$	3.00003e9	+	5.19620e9i	0.0194113	+	0.0336214i
$628$		0			0
$629$			1.16681e11i			0.745416i
$630$		0			0
$631$	−1.98489e11			−1.25204			−0.626020	−	0.779807i	$-0.715317\pi$
$631$	−1.98489e11			−1.25204			−0.626020	+	0.779807i	$0.715317\pi$
$632$		0			0
$633$	8.77460e10	−	5.06602e10i	0.546528	−	0.315538i
$634$		0			0
$635$	−1.33533e11	−	7.70956e10i	−0.821287	−	0.474170i
$636$		0			0
$637$	1.07356e11	+	1.85033e11i	0.652034	+	1.12380i
$638$		0			0
$639$	3.90189e10	−	6.75828e10i	0.234030	−	0.405352i
$640$		0			0
$641$	−6.92620e10	−	1.19965e11i	−0.410264	−	0.710597i	0.584655	−	0.811282i	$-0.301230\pi$
$641$	−6.92620e10	−	1.19965e11i	−0.410264	−	0.710597i	−0.994918	+	0.100685i	$0.967897\pi$
$642$		0			0
$643$		−	6.27928e10i		−	0.367338i	−0.982988	−	0.183669i	$-0.941203\pi$
$643$		−	6.27928e10i		−	0.367338i	0.982988	−	0.183669i	$-0.0587975\pi$
$644$		0			0
$645$	1.46424e11			0.846004
$646$		0			0
$647$	3.40232e10	−	1.96433e10i	0.194159	−	0.112098i	−0.399769	−	0.916616i	$-0.630910\pi$
$647$	3.40232e10	−	1.96433e10i	0.194159	−	0.112098i	0.593928	+	0.804518i	$0.297576\pi$
$648$		0			0
$649$	−1.40101e11	−	8.08873e10i	−0.789700	−	0.455933i
$650$		0			0
$651$	−4.88813e10	+	1.83210e11i	−0.272156	+	1.02006i
$652$		0			0
$653$	6.44912e10	−	1.11702e11i	0.354689	−	0.614340i	−0.632376	−	0.774662i	$-0.717920\pi$
$653$	6.44912e10	−	1.11702e11i	0.354689	−	0.614340i	0.987065	+	0.160322i	$0.0512534\pi$
$654$		0			0
$655$	7.08113e10	+	1.22649e11i	0.384713	+	0.666343i
$656$		0			0
$657$			3.42107e10i			0.183612i
$658$		0			0
$659$	2.68793e11			1.42520			0.712600	−	0.701571i	$-0.247518\pi$
$659$	2.68793e11			1.42520			0.712600	+	0.701571i	$0.247518\pi$
$660$		0			0
$661$	3.11244e11	−	1.79697e11i	1.63040	−	0.941314i	0.646435	−	0.762969i	$-0.276259\pi$
$661$	3.11244e11	−	1.79697e11i	1.63040	−	0.941314i	0.983968	−	0.178345i	$-0.0570744\pi$
$662$		0			0
$663$	−7.58881e10	−	4.38140e10i	−0.392753	−	0.226756i
$664$		0			0
$665$	2.30707e10	+	2.30215e10i	0.117971	+	0.117719i
$666$		0			0
$667$	−3.39675e10	+	5.88334e10i	−0.171617	+	0.297249i
$668$		0			0
$669$	−4.58412e10	−	7.93993e10i	−0.228850	−	0.396380i
$670$		0			0
$671$		−	1.44932e11i		−	0.714946i
$672$		0			0
$673$	2.68409e11			1.30839			0.654195	−	0.756326i	$-0.273007\pi$
$673$	2.68409e11			1.30839			0.654195	+	0.756326i	$0.273007\pi$
$674$		0			0
$675$	−3.91342e10	+	2.25941e10i	−0.188513	+	0.108838i
$676$		0			0
$677$	−2.16576e11	−	1.25040e11i	−1.03099	−	0.595243i	−0.113723	−	0.993513i	$-0.536278\pi$
$677$	−2.16576e11	−	1.25040e11i	−1.03099	−	0.595243i	−0.917268	+	0.398269i	$0.869611\pi$
$678$		0			0
$679$	−9.51130e10	−	3.53456e11i	−0.447467	−	1.66286i
$680$		0			0
$681$	1.03996e10	−	1.80127e10i	0.0483537	−	0.0837510i
$682$		0			0
$683$	5.62014e10	+	9.73438e10i	0.258264	+	0.447327i	0.965777	−	0.259374i	$-0.0835161\pi$
$683$	5.62014e10	+	9.73438e10i	0.258264	+	0.447327i	−0.707513	+	0.706701i	$0.750183\pi$
$684$		0			0
$685$			2.88917e10i			0.131223i
$686$		0			0
$687$	3.89984e9			0.0175073
$688$		0			0
$689$	9.48543e10	−	5.47642e10i	0.420901	−	0.243007i
$690$		0			0
$691$	8.27679e10	+	4.77861e10i	0.363036	+	0.209599i	0.670412	−	0.741989i	$-0.266118\pi$
$691$	8.27679e10	+	4.77861e10i	0.363036	+	0.209599i	−0.307376	+	0.951588i	$0.599451\pi$
$692$		0			0
$693$	−4.37270e10	+	1.17667e10i	−0.189591	+	0.0510178i
$694$		0			0
$695$	2.26284e11	−	3.91935e11i	0.969871	−	1.67987i
$696$		0			0
$697$	1.06098e11	+	1.83768e11i	0.449549	+	0.778642i
$698$		0			0
$699$			8.95476e10i			0.375099i
$700$		0			0
$701$	3.06300e11			1.26846			0.634228	−	0.773146i	$-0.281318\pi$
$701$	3.06300e11			1.26846			0.634228	+	0.773146i	$0.281318\pi$
$702$		0			0
$703$	2.97732e10	−	1.71895e10i	0.121900	−	0.0703790i
$704$		0			0
$705$	1.75588e11	+	1.01376e11i	0.710786	+	0.410372i
$706$		0			0
$707$	−3.20568e10	+	3.21254e10i	−0.128305	+	0.128579i
$708$		0			0
$709$	−1.34632e11	+	2.33189e11i	−0.532798	+	0.922833i	0.466468	+	0.884538i	$0.345526\pi$
$709$	−1.34632e11	+	2.33189e11i	−0.532798	+	0.922833i	−0.999266	+	0.0382955i	$0.987807\pi$
$710$		0			0
$711$	−1.98485e10	−	3.43786e10i	−0.0776693	−	0.134527i
$712$		0			0
$713$			7.39170e11i			2.86013i
$714$		0			0
$715$	−2.91971e11			−1.11716
$716$		0			0
$717$	−2.32861e11	+	1.34442e11i	−0.881089	+	0.508697i
$718$		0			0
$719$	−2.06410e11	−	1.19171e11i	−0.772350	−	0.445917i	0.0613622	−	0.998116i	$-0.480456\pi$
$719$	−2.06410e11	−	1.19171e11i	−0.772350	−	0.445917i	−0.833712	+	0.552199i	$0.813789\pi$
$720$		0			0
$721$	−3.64954e11	−	9.73714e10i	−1.35051	−	0.360322i
$722$		0			0
$723$	−1.52165e11	+	2.63558e11i	−0.556881	+	0.964546i
$724$		0			0
$725$	−3.42875e10	−	5.93877e10i	−0.124104	−	0.214954i
$726$		0			0
$727$		−	1.68567e11i		−	0.603442i	−0.953396	−	0.301721i	$-0.902439\pi$
$727$		−	1.68567e11i		−	0.603442i	0.953396	−	0.301721i	$-0.0975611\pi$
$728$		0			0
$729$	−1.04604e10			−0.0370370
$730$		0			0
$731$	−1.50067e11	+	8.66414e10i	−0.525553	+	0.303428i
$732$		0			0
$733$	−2.59349e11	−	1.49735e11i	−0.898399	−	0.518691i	−0.0217184	−	0.999764i	$-0.506914\pi$
$733$	−2.59349e11	−	1.49735e11i	−0.898399	−	0.518691i	−0.876680	+	0.481073i	$0.840247\pi$
$734$		0			0
$735$	−2.12756e11	+	1.23442e11i	−0.729008	+	0.422973i
$736$		0			0
$737$	8.46417e10	−	1.46604e11i	0.286890	−	0.496907i
$738$		0			0
$739$	1.66690e11	+	2.88716e11i	0.558897	+	0.968038i	0.997589	+	0.0694004i	$0.0221086\pi$
$739$	1.66690e11	+	2.88716e11i	0.558897	+	0.968038i	−0.438692	+	0.898638i	$0.644558\pi$
$740$		0			0
$741$			2.58188e10i			0.0856373i
$742$		0			0
$743$	5.61696e11			1.84309			0.921545	−	0.388272i	$-0.126928\pi$
$743$	5.61696e11			1.84309			0.921545	+	0.388272i	$0.126928\pi$
$744$		0			0
$745$	−2.99825e11	+	1.73104e11i	−0.973292	+	0.561930i
$746$		0			0
$747$	2.18196e10	+	1.25975e10i	0.0700751	+	0.0404579i
$748$		0			0
$749$	7.61013e10	−	2.85232e11i	0.241805	−	0.906299i
$750$		0			0
$751$	8.17182e10	−	1.41540e11i	0.256897	−	0.444959i	−0.708512	−	0.705699i	$-0.750633\pi$
$751$	8.17182e10	−	1.41540e11i	0.256897	−	0.444959i	0.965409	+	0.260740i	$0.0839665\pi$
$752$		0			0
$753$	1.93347e10	+	3.34886e10i	0.0601391	+	0.104164i
$754$		0			0
$755$			1.66530e11i			0.512512i
$756$		0			0
$757$	−7.42672e10			−0.226159			−0.113079	−	0.993586i	$-0.536071\pi$
$757$	−7.42672e10			−0.226159			−0.113079	+	0.993586i	$0.536071\pi$
$758$		0			0
$759$	−1.52871e11	+	8.82599e10i	−0.460635	+	0.265948i
$760$		0			0
$761$	−4.12849e11	−	2.38359e11i	−1.23099	−	0.710710i	−0.263750	−	0.964591i	$-0.584960\pi$
$761$	−4.12849e11	−	2.38359e11i	−1.23099	−	0.710710i	−0.967235	+	0.253881i	$0.918293\pi$
$762$		0			0
$763$	7.20913e10	+	7.19374e10i	0.212708	+	0.212254i
$764$		0			0
$765$	5.03787e10	−	8.72584e10i	0.147096	−	0.254778i
$766$		0			0
$767$	−3.48065e11	−	6.02867e11i	−1.00573	−	1.74197i
$768$		0			0
$769$			5.81659e11i			1.66327i	0.555323	+	0.831635i	$0.312595\pi$
$769$			5.81659e11i			1.66327i	−0.555323	+	0.831635i	$0.687405\pi$
$770$		0			0
$771$	−2.83333e11			−0.801826
$772$		0			0
$773$	1.79880e10	−	1.03854e10i	0.0503808	−	0.0290874i	−0.474598	−	0.880203i	$-0.657407\pi$
$773$	1.79880e10	−	1.03854e10i	0.0503808	−	0.0290874i	0.524979	+	0.851115i	$0.324073\pi$
$774$		0			0
$775$	−6.46172e11	−	3.73068e11i	−1.79119	−	1.03414i
$776$		0			0
$777$	6.74208e10	+	2.50547e11i	0.184974	+	0.687393i
$778$		0			0
$779$	3.12609e10	−	5.41455e10i	0.0848891	−	0.147032i
$780$		0			0
$781$	1.53857e11	+	2.66488e11i	0.413535	+	0.716263i
$782$		0			0
$783$		−	1.58740e10i		−	0.0422318i
$784$		0			0
$785$	4.35051e11			1.14567
$786$		0			0
$787$	2.97377e11	−	1.71691e11i	0.775191	−	0.447557i	−0.0595323	−	0.998226i	$-0.518961\pi$
$787$	2.97377e11	−	1.71691e11i	0.775191	−	0.447557i	0.834723	+	0.550670i	$0.185628\pi$
$788$		0			0
$789$	1.91199e11	+	1.10389e11i	0.493377	+	0.284851i
$790$		0			0
$791$	−3.95477e11	+	1.06421e11i	−1.01022	+	0.271844i
$792$		0			0
$793$	3.11827e11	−	5.40101e11i	0.788536	−	1.36578i
$794$		0			0
$795$	6.29695e10	+	1.09066e11i	0.157638	+	0.273037i
$796$		0			0
$797$		−	5.82077e10i		−	0.144260i	−0.997395	−	0.0721302i	$-0.977020\pi$
$797$		−	5.82077e10i		−	0.144260i	0.997395	−	0.0721302i	$-0.0229797\pi$
$798$		0			0
$799$	−2.39943e11			−0.588737
$800$		0			0
$801$	−5.38948e10	+	3.11162e10i	−0.130923	+	0.0755886i
$802$		0			0
$803$	−1.16824e11	−	6.74485e10i	−0.280977	−	0.162222i
$804$		0			0
$805$	−6.77285e11	+	6.78734e11i	−1.61283	+	1.61628i
$806$		0			0
$807$	2.17573e11	−	3.76847e11i	0.512992	−	0.888527i
$808$		0			0
$809$	−4.27586e10	−	7.40600e10i	−0.0998226	−	0.172898i	0.811789	−	0.583952i	$-0.198494\pi$
$809$	−4.27586e10	−	7.40600e10i	−0.0998226	−	0.172898i	−0.911611	+	0.411054i	$0.865161\pi$
$810$		0			0
$811$			7.56023e11i			1.74764i	0.486252	+	0.873819i	$0.338364\pi$
$811$			7.56023e11i			1.74764i	−0.486252	+	0.873819i	$0.661636\pi$
$812$		0			0
$813$	3.90015e10			0.0892727
$814$		0			0
$815$	3.49060e9	−	2.01530e9i	0.00791170	−	0.00456782i
$816$		0			0
$817$	4.42160e10	+	2.55281e10i	0.0992409	+	0.0572968i
$818$		0			0
$819$	−1.88269e11	−	5.02311e10i	−0.418450	−	0.111644i
$820$		0			0
$821$	−1.40517e11	+	2.43383e11i	−0.309283	+	0.535694i	−0.978206	−	0.207638i	$-0.933422\pi$
$821$	−1.40517e11	+	2.43383e11i	−0.309283	+	0.535694i	0.668923	+	0.743332i	$0.266756\pi$
$822$		0			0
$823$	−2.21259e11	−	3.83231e11i	−0.482282	−	0.835336i	0.517512	−	0.855676i	$-0.326858\pi$
$823$	−2.21259e11	−	3.83231e11i	−0.482282	−	0.835336i	−0.999793	+	0.0203400i	$0.993525\pi$
$824$		0			0
$825$		−	1.78183e11i		−	0.384637i
$826$		0			0
$827$	3.33766e11			0.713543			0.356772	−	0.934192i	$-0.383877\pi$
$827$	3.33766e11			0.713543			0.356772	+	0.934192i	$0.383877\pi$
$828$		0			0
$829$	7.32765e11	−	4.23062e11i	1.55148	−	0.895748i	0.553459	−	0.832877i	$-0.313308\pi$
$829$	7.32765e11	−	4.23062e11i	1.55148	−	0.895748i	0.998022	−	0.0628712i	$-0.0200257\pi$
$830$		0			0
$831$	−4.52468e11	−	2.61233e11i	−0.948820	−	0.547801i
$832$		0			0
$833$	1.45008e11	−	2.52405e11i	0.301170	−	0.524225i
$834$		0			0
$835$	−5.36290e11	+	9.28882e11i	−1.10320	+	1.91080i
$836$		0			0
$837$	−8.63591e10	−	1.49578e11i	−0.175957	−	0.304766i
$838$		0			0
$839$			1.33895e11i			0.270220i	0.990831	+	0.135110i	$0.0431388\pi$
$839$			1.33895e11i			0.270220i	−0.990831	+	0.135110i	$0.956861\pi$
$840$		0			0
$841$	−4.76157e11			−0.951845
$842$		0			0
$843$	−2.38450e10	+	1.37669e10i	−0.0472157	+	0.0272600i
$844$		0			0
$845$	−4.43504e11	−	2.56057e11i	−0.869903	−	0.502239i
$846$		0			0
$847$	−8.66476e10	+	3.24761e11i	−0.168354	+	0.631000i
$848$		0			0
$849$	1.29595e11	−	2.24465e11i	0.249435	−	0.432034i
$850$		0			0
$851$	5.05711e11	+	8.75918e11i	0.964238	+	1.67011i
$852$		0			0
$853$		−	3.20383e11i		−	0.605164i	−0.953123	−	0.302582i	$-0.902151\pi$
$853$		−	3.20383e11i		−	0.605164i	0.953123	−	0.302582i	$-0.0978486\pi$
$854$		0			0
$855$	−2.96872e10			−0.0555527
$856$		0			0
$857$	−3.52411e11	+	2.03465e11i	−0.653320	+	0.377195i	−0.789727	−	0.613458i	$-0.789778\pi$
$857$	−3.52411e11	+	2.03465e11i	−0.653320	+	0.377195i	0.136407	+	0.990653i	$0.456445\pi$
$858$		0			0
$859$	1.83612e11	+	1.06008e11i	0.337231	+	0.194700i	0.659047	−	0.752102i	$-0.270960\pi$
$859$	1.83612e11	+	1.06008e11i	0.337231	+	0.194700i	−0.321816	+	0.946802i	$0.604293\pi$
$860$		0			0
$861$	3.34007e11	+	3.33294e11i	0.607775	+	0.606478i
$862$		0			0
$863$	2.51111e11	−	4.34936e11i	0.452712	−	0.784120i	−0.545841	−	0.837888i	$-0.683790\pi$
$863$	2.51111e11	−	4.34936e11i	0.452712	−	0.784120i	0.998553	+	0.0537683i	$0.0171232\pi$
$864$		0			0
$865$	5.98433e11	+	1.03652e12i	1.06893	+	1.85145i
$866$		0			0
$867$		−	2.06984e11i		−	0.366320i
$868$		0			0
$869$	1.56530e11			0.274486
$870$		0			0
$871$	6.30850e11	−	3.64221e11i	1.09611	−	0.632838i
$872$		0			0
$873$	2.88737e11	+	1.66703e11i	0.497103	+	0.287002i
$874$		0			0
$875$	−2.91463e10	−	1.08313e11i	−0.0497224	−	0.184777i
$876$		0			0
$877$	5.16027e11	−	8.93786e11i	0.872317	−	1.51090i	0.0127234	−	0.999919i	$-0.495950\pi$
$877$	5.16027e11	−	8.93786e11i	0.872317	−	1.51090i	0.859594	−	0.510978i	$-0.170717\pi$
$878$		0			0
$879$	2.19486e11	+	3.80162e11i	0.367665	+	0.636814i
$880$		0			0
$881$		−	7.05174e11i		−	1.17056i	−0.810832	−	0.585279i	$-0.800985\pi$
$881$		−	7.05174e11i		−	1.17056i	0.810832	−	0.585279i	$-0.199015\pi$
$882$		0			0
$883$	7.95305e11			1.30825			0.654125	−	0.756386i	$-0.273037\pi$
$883$	7.95305e11			1.30825			0.654125	+	0.756386i	$0.273037\pi$
$884$		0			0
$885$	6.93195e11	−	4.00216e11i	1.13001	−	0.652411i
$886$		0			0
$887$	4.17819e11	+	2.41228e11i	0.674985	+	0.389703i	0.797963	−	0.602707i	$-0.205911\pi$
$887$	4.17819e11	+	2.41228e11i	0.674985	+	0.389703i	−0.122978	+	0.992409i	$0.539244\pi$
$888$		0			0
$889$	3.91824e11	−	1.05438e11i	0.627313	−	0.168806i
$890$		0			0
$891$	2.06233e10	−	3.57205e10i	0.0327225	−	0.0566770i
$892$		0			0
$893$	3.53486e10	+	6.12255e10i	0.0555861	+	0.0962779i
$894$		0			0
$895$		−	1.60954e12i		−	2.50847i
$896$		0			0
$897$	−7.59581e11			−1.17329
$898$		0			0
$899$	2.26991e11	−	1.31054e11i	0.347513	−	0.200637i
$900$		0			0
$901$	−1.29073e11	−	7.45202e10i	−0.195855	−	0.113077i
$902$		0			0
$903$	−2.72173e11	+	2.72755e11i	−0.409349	+	0.410224i
$904$		0			0
$905$	3.60238e11	−	6.23950e11i	0.537026	−	0.930156i
$906$		0			0
$907$	−1.39196e11	−	2.41094e11i	−0.205682	−	0.356252i	0.744668	−	0.667436i	$-0.232608\pi$
$907$	−1.39196e11	−	2.41094e11i	−0.205682	−	0.356252i	−0.950350	+	0.311183i	$0.899275\pi$
$908$		0			0
$909$		−	4.13387e10i		−	0.0605481i
$910$		0			0
$911$	9.45540e11			1.37280			0.686399	−	0.727225i	$-0.259190\pi$
$911$	9.45540e11			1.37280			0.686399	+	0.727225i	$0.259190\pi$
$912$		0			0
$913$	−8.60373e10	+	4.96737e10i	−0.123824	+	0.0714896i
$914$		0			0
$915$	6.21024e11	+	3.58549e11i	0.885981	+	0.511521i
$916$		0			0
$917$	−3.60091e11	−	9.60740e10i	−0.509255	−	0.135872i
$918$		0			0
$919$	2.32980e11	−	4.03533e11i	0.326630	−	0.565740i	−0.655211	−	0.755446i	$-0.727420\pi$
$919$	2.32980e11	−	4.03533e11i	0.326630	−	0.565740i	0.981841	+	0.189706i	$0.0607535\pi$
$920$		0			0
$921$	−1.24598e11	−	2.15810e11i	−0.173170	−	0.299940i
$922$		0			0
$923$			1.32412e12i			1.82440i
$924$		0			0
$925$	−1.02095e12			−1.39456
$926$		0			0
$927$	2.97960e11	−	1.72027e11i	0.403495	−	0.232958i
$928$		0			0
$929$	−3.20399e11	−	1.84983e11i	−0.430159	−	0.248352i	0.269255	−	0.963069i	$-0.413222\pi$
$929$	−3.20399e11	−	1.84983e11i	−0.430159	−	0.248352i	−0.699414	+	0.714716i	$0.746556\pi$
$930$		0			0
$931$	−8.57679e10	+	1.83249e8i	−0.114163	+	0.000243917i
$932$		0			0
$933$	−2.86881e11	+	4.96893e11i	−0.378596	+	0.655747i
$934$		0			0
$935$	1.98649e11	+	3.44071e11i	0.259921	+	0.450196i
$936$		0			0
$937$		−	9.36393e11i		−	1.21479i	−0.794401	−	0.607393i	$-0.792215\pi$
$937$		−	9.36393e11i		−	1.21479i	0.794401	−	0.607393i	$-0.207785\pi$
$938$		0			0
$939$	−2.06608e11			−0.265757
$940$		0			0
$941$	−7.35252e11	+	4.24498e11i	−0.937729	+	0.541398i	−0.889248	−	0.457426i	$-0.848772\pi$
$941$	−7.35252e11	+	4.24498e11i	−0.937729	+	0.541398i	−0.0484815	+	0.998824i	$0.515438\pi$
$942$		0			0
$943$	1.59294e12	+	9.19686e11i	2.01444	+	1.16304i
$944$		0			0
$945$	5.77572e10	−	2.16478e11i	0.0724234	−	0.271447i
$946$		0			0
$947$	−3.10771e11	+	5.38271e11i	−0.386403	+	0.669270i	−0.991963	−	0.126530i	$-0.959616\pi$
$947$	−3.10771e11	+	5.38271e11i	−0.386403	+	0.669270i	0.605560	+	0.795800i	$0.292949\pi$
$948$		0			0
$949$	−2.90237e11	−	5.02706e11i	−0.357839	−	0.619796i
$950$		0			0
$951$		−	3.40478e11i		−	0.416263i
$952$		0			0
$953$	−8.09972e11			−0.981970			−0.490985	−	0.871168i	$-0.663363\pi$
$953$	−8.09972e11			−0.981970			−0.490985	+	0.871168i	$0.663363\pi$
$954$		0			0
$955$	−1.15863e12	+	6.68936e11i	−1.39294	+	0.804214i
$956$		0			0
$957$	5.42074e10	+	3.12966e10i	0.0646265	+	0.0373121i
$958$		0			0
$959$	−5.38189e10	−	5.37040e10i	−0.0636298	−	0.0634939i
$960$		0			0
$961$	9.99491e11	−	1.73117e12i	1.17189	−	2.02977i
$962$		0			0
$963$	1.34449e11	+	2.32873e11i	0.156334	+	0.270778i
$964$		0			0
$965$			1.91639e12i			2.20991i
$966$		0			0
$967$	1.48206e11			0.169496			0.0847480	−	0.996402i	$-0.472991\pi$
$967$	1.48206e11			0.169496			0.0847480	+	0.996402i	$0.472991\pi$
$968$		0			0
$969$	3.04260e10	−	1.75664e10i	0.0345103	−	0.0199246i
$970$		0			0
$971$	−1.32961e12	−	7.67652e11i	−1.49571	−	0.863550i	−0.495725	−	0.868480i	$-0.665097\pi$
$971$	−1.32961e12	−	7.67652e11i	−1.49571	−	0.863550i	−0.999988	+	0.00492975i	$0.998431\pi$
$972$		0			0
$973$	3.09471e11	+	1.15005e12i	0.345278	+	1.28311i
$974$		0			0
$975$	3.83369e11	−	6.64015e11i	0.424227	−	0.734783i
$976$		0			0
$977$	8.81173e10	+	1.52624e11i	0.0967126	+	0.167511i	0.910322	−	0.413901i	$-0.135834\pi$
$977$	8.81173e10	+	1.52624e11i	0.0967126	+	0.167511i	−0.813609	+	0.581412i	$0.802501\pi$
$978$		0			0
$979$		−	2.45390e11i		−	0.267132i
$980$		0			0
$981$	−9.27664e10			−0.100165
$982$		0			0
$983$	7.05620e11	−	4.07390e11i	0.755712	−	0.436311i	−0.0720418	−	0.997402i	$-0.522951\pi$
$983$	7.05620e11	−	4.07390e11i	0.755712	−	0.436311i	0.827754	+	0.561091i	$0.189618\pi$
$984$		0			0
$985$	−2.90910e11	−	1.67957e11i	−0.309039	−	0.178424i
$986$		0			0
$987$	−5.15224e11	+	1.38644e11i	−0.542910	+	0.146094i
$988$		0			0
$989$	−7.51029e11	+	1.30082e12i	−0.785003	+	1.35967i
$990$		0			0
$991$	−2.70351e11	−	4.68262e11i	−0.280307	−	0.485506i	0.691153	−	0.722708i	$-0.257103\pi$
$991$	−2.70351e11	−	4.68262e11i	−0.280307	−	0.485506i	−0.971460	+	0.237202i	$0.923770\pi$
$992$		0			0
$993$		−	1.00222e12i		−	1.03078i
$994$		0			0
$995$	1.12657e12			1.14939
$996$		0			0
$997$	−7.07745e11	+	4.08617e11i	−0.716302	+	0.413557i	−0.813390	−	0.581719i	$-0.802380\pi$
$997$	−7.07745e11	+	4.08617e11i	−0.716302	+	0.413557i	0.0970882	+	0.995276i	$0.469047\pi$
$998$		0			0
$999$	−2.04671e11	−	1.18167e11i	−0.205492	−	0.118641i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.9.m.a.61.5	✓	10
3.2	odd	2		252.9.z.b.145.1		10
7.3	odd	6	inner	84.9.m.a.73.5	yes	10
21.17	even	6		252.9.z.b.73.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.9.m.a.61.5	✓	10	1.1	even	1	trivial
84.9.m.a.73.5	yes	10	7.3	odd	6	inner
252.9.z.b.73.1		10	21.17	even	6
252.9.z.b.145.1		10	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	84.m (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-40.5000 + 23.3827i) q^{3} +(790.151 + 456.194i) q^{5} +(-2318.52 + 623.902i) q^{7} +(1093.50 - 1894.00i) q^{9} +O(q^{10})\)	\(q+(-40.5000 + 23.3827i) q^{3} +(790.151 + 456.194i) q^{5} +(-2318.52 + 623.902i) q^{7} +(1093.50 - 1894.00i) q^{9} +(4311.81 + 7468.27i) q^{11} +37108.2i q^{13} -42668.2 q^{15} +(43729.9 - 25247.5i) q^{17} +(-12884.6 - 7438.94i) q^{19} +(79311.7 - 79481.3i) q^{21} +(218851. - 379062. i) q^{23} +(220914. + 382634. i) q^{25} +102276. i q^{27} -155208. q^{29} +(-1.46250e6 + 844374. i) q^{31} +(-349257. - 201643. i) q^{33} +(-2.11660e6 - 564720. i) q^{35} +(-1.15538e6 + 2.00117e6i) q^{37} +(-867690. - 1.50288e6i) q^{39} +4.20233e6i q^{41} -3.43168e6 q^{43} +(1.72806e6 - 997696. i) q^{45} +(-4.11520e6 - 2.37591e6i) q^{47} +(4.98629e6 - 2.89306e6i) q^{49} +(-1.18071e6 + 2.04505e6i) q^{51} +(-1.47580e6 - 2.55615e6i) q^{53} +7.86809e6i q^{55} +695770. q^{57} +(-1.62462e7 + 9.37973e6i) q^{59} +(-1.45547e7 - 8.40319e6i) q^{61} +(-1.35364e6 + 5.07351e6i) q^{63} +(-1.69286e7 + 2.93211e7i) q^{65} +(-9.81510e6 - 1.70003e7i) q^{67} +2.04693e7i q^{69} +3.56826e7 q^{71} +(-1.35470e7 + 7.82137e6i) q^{73} +(-1.78940e7 - 1.03311e7i) q^{75} +(-1.46565e7 - 1.46252e7i) q^{77} +(9.07568e6 - 1.57195e7i) q^{79} +(-2.39148e6 - 4.14217e6i) q^{81} +1.15204e7i q^{83} +4.60710e7 q^{85} +(6.28592e6 - 3.62918e6i) q^{87} +(-2.46432e7 - 1.42278e7i) q^{89} +(-2.31519e7 - 8.60363e7i) q^{91} +(3.94875e7 - 6.83943e7i) q^{93} +(-6.78720e6 - 1.17558e7i) q^{95} +1.52449e8i q^{97} +1.88599e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 405 q^{3} + 1389 q^{5} + 1217 q^{7} + 10935 q^{9}+O(q^{10}) \) 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

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Groups

Database

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