LMFDB - Embedded newform 84.9.m.a.73.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [84,9,Mod(61,84)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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			73.5
Root			$16.3207 + 9.42278i$ of defining polynomial
Character	$\chi$	$=$	84.73
Dual form			84.9.m.a.61.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

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

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
$5$	790.151	−	456.194i	1.26424	−	0.729911i	0.290350	−	0.956921i	$-0.406228\pi$
$5$	790.151	−	456.194i	1.26424	−	0.729911i	0.973892	+	0.227010i	$0.0728950\pi$
$6$		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	−0.680367	−	0.732872i	$-0.738179\pi$
$11$	4311.81	−	7468.27i	0.294502	−	0.510093i	0.974869	+	0.222779i	$0.0715127\pi$
$12$		0			0
$13$		−	37108.2i		−	1.29926i	−0.760249	−	0.649631i	$-0.774923\pi$
$13$		−	37108.2i		−	1.29926i	0.760249	−	0.649631i	$-0.225077\pi$
$14$		0			0
$15$	−42668.2			−0.842828
$16$		0			0
$17$	43729.9	+	25247.5i	0.523580	+	0.302289i	0.738398	−	0.674365i	$-0.235583\pi$
$17$	43729.9	+	25247.5i	0.523580	+	0.302289i	−0.214818	+	0.976654i	$0.568916\pi$
$18$		0			0
$19$	−12884.6	+	7438.94i	−0.0988684	+	0.0570817i	−0.548619	−	0.836073i	$-0.684846\pi$
$19$	−12884.6	+	7438.94i	−0.0988684	+	0.0570817i	0.449751	+	0.893154i	$0.351513\pi$
$20$		0			0
$21$	79311.7	+	79481.3i	0.407812	+	0.408684i
$22$		0			0
$23$	218851.	+	379062.i	0.782056	+	1.35456i	0.930742	+	0.365677i	$0.119162\pi$
$23$	218851.	+	379062.i	0.782056	+	1.35456i	−0.148686	+	0.988885i	$0.547504\pi$
$24$		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	−0.994326	−	0.106375i	$-0.966076\pi$
$31$	−1.46250e6	−	844374.i	−1.58361	−	0.914299i	−0.589286	−	0.807924i	$-0.700591\pi$
$32$		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	−0.990124	−	0.140197i	$-0.955226\pi$
$37$	−1.15538e6	−	2.00117e6i	−0.616476	−	1.06777i	0.373647	−	0.927571i	$-0.378107\pi$
$38$		0			0
$39$	−867690.	+	1.50288e6i	−0.375065	+	0.649631i
$40$		0			0
$41$		−	4.20233e6i		−	1.48715i	−0.668652	−	0.743575i	$-0.733128\pi$
$41$		−	4.20233e6i		−	1.48715i	0.668652	−	0.743575i	$-0.266872\pi$
$42$		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	−0.858396	−	0.512987i	$-0.828539\pi$
$47$	−4.11520e6	+	2.37591e6i	−0.843334	+	0.486899i	0.0150621	+	0.999887i	$0.495205\pi$
$48$		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	−0.944260	−	0.329200i	$-0.893221\pi$
$53$	−1.47580e6	+	2.55615e6i	−0.187035	+	0.323954i	0.757225	+	0.653154i	$0.226554\pi$
$54$		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	−0.353820	−	0.935313i	$-0.615118\pi$
$59$	−1.62462e7	−	9.37973e6i	−1.34074	−	0.774074i	−0.986915	+	0.161239i	$0.948451\pi$
$60$		0			0
$61$	−1.45547e7	+	8.40319e6i	−1.05120	+	0.606910i	−0.922985	−	0.384836i	$-0.874258\pi$
$61$	−1.45547e7	+	8.40319e6i	−1.05120	+	0.606910i	−0.128215	+	0.991746i	$0.540925\pi$
$62$		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	−0.999890	−	0.0148609i	$-0.995269\pi$
$67$	−9.81510e6	+	1.70003e7i	−0.487075	+	0.843638i	0.512815	+	0.858499i	$0.328603\pi$
$68$		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	0.242144	−	0.970240i	$-0.422149\pi$
$73$	−1.35470e7	−	7.82137e6i	−0.477037	−	0.275417i	−0.719181	+	0.694823i	$0.755483\pi$
$74$		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	0.958692	−	0.284447i	$-0.0918098\pi$
$79$	9.07568e6	+	1.57195e7i	0.233008	+	0.403582i	−0.725684	+	0.688028i	$0.758476\pi$
$80$		0			0
$81$	−2.39148e6	+	4.14217e6i	−0.0555556	+	0.0962250i
$82$		0			0
$83$		−	1.15204e7i		−	0.242747i	−0.992607	−	0.121374i	$-0.961270\pi$
$83$		−	1.15204e7i		−	0.242747i	0.992607	−	0.121374i	$-0.0387299\pi$
$84$		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	−0.683360	−	0.730082i	$-0.739482\pi$
$89$	−2.46432e7	+	1.42278e7i	−0.392770	+	0.226766i	0.290590	+	0.956848i	$0.406148\pi$
$90$		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	$-0.669834\pi$
$97$		−	1.52449e8i		−	1.72201i	0.508593	−	0.861007i	$-0.330166\pi$
$98$		0			0
$99$	1.88599e7			0.196335
$100$		0			0
$101$	1.63696e7	+	9.45100e6i	0.157309	+	0.0908222i	0.576588	−	0.817035i	$-0.304384\pi$
$101$	1.63696e7	+	9.45100e6i	0.157309	+	0.0908222i	−0.419279	+	0.907857i	$0.637717\pi$
$102$		0			0
$103$	1.36241e8	−	7.86589e7i	1.21049	−	0.698874i	0.247621	−	0.968857i	$-0.420351\pi$
$103$	1.36241e8	−	7.86589e7i	1.21049	−	0.698874i	0.962865	+	0.269983i	$0.0870179\pi$
$104$		0			0
$105$	9.89271e7	+	2.66207e7i	0.813876	+	0.219010i
$106$		0			0
$107$	−6.14765e7	−	1.06480e8i	−0.469001	−	0.812334i	0.530371	−	0.847766i	$-0.322053\pi$
$107$	−6.14765e7	−	1.06480e8i	−0.469001	−	0.812334i	−0.999372	+	0.0354319i	$0.988719\pi$
$108$		0			0
$109$	−2.12086e7	+	3.67344e7i	−0.150247	+	0.260236i	−0.931318	−	0.364206i	$-0.881340\pi$
$109$	−2.12086e7	+	3.67344e7i	−0.150247	+	0.260236i	0.781071	+	0.624442i	$0.214674\pi$
$110$		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	−0.254097	−	0.967179i	$-0.581778\pi$
$131$	1.34426e8	−	7.76109e7i	0.456455	−	0.263535i	0.710553	+	0.703644i	$0.248445\pi$
$132$		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	−0.842678	−	0.538418i	$-0.819022\pi$
$137$	1.58330e7	−	2.74236e7i	0.0449450	−	0.0778471i	0.887623	+	0.460571i	$0.152355\pi$
$138$		0			0
$139$			4.96025e8i			1.32875i	0.747398	+	0.664377i	$0.231303\pi$
$139$			4.96025e8i			1.32875i	−0.747398	+	0.664377i	$0.768697\pi$
$140$		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	0.606829	−	0.794833i	$-0.292441\pi$
$149$	−1.89726e8	−	3.28616e8i	−0.384931	−	0.666720i	−0.991760	+	0.128113i	$0.959108\pi$
$150$		0			0
$151$	9.12604e7	−	1.58068e8i	0.175539	−	0.304043i	−0.764808	−	0.644258i	$-0.777166\pi$
$151$	9.12604e7	−	1.58068e8i	0.175539	−	0.304043i	0.940348	+	0.340215i	$0.110500\pi$
$152$		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	0.799727	−	0.600363i	$-0.204977\pi$
$157$	4.12944e8	+	2.38413e8i	0.679661	+	0.392402i	−0.120066	+	0.992766i	$0.538311\pi$
$158$		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	0.867586	−	0.497288i	$-0.165671\pi$
$163$	2.20882e6	+	3.82578e6i	0.00312903	+	0.00541963i	−0.864457	+	0.502707i	$0.832337\pi$
$164$		0			0
$165$	−1.83977e8	+	3.18658e8i	−0.248215	+	0.429921i
$166$		0			0
$167$		−	1.17558e9i		−	1.51142i	−0.654908	−	0.755709i	$-0.727293\pi$
$167$		−	1.17558e9i		−	1.51142i	0.654908	−	0.755709i	$-0.272707\pi$
$168$		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	0.293611	−	0.955925i	$-0.405143\pi$
$173$	1.13605e9	−	6.55898e8i	1.26827	−	0.732237i	0.974661	+	0.223688i	$0.0718097\pi$
$174$		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	0.0135496	+	0.999908i	$0.495687\pi$
$179$	−8.82047e8	+	1.52775e9i	−0.859171	+	1.48813i	−0.872721	+	0.488220i	$0.837646\pi$
$180$		0			0
$181$			7.89659e8i			0.735742i	0.929877	+	0.367871i	$0.119913\pi$
$181$			7.89659e8i			0.735742i	−0.929877	+	0.367871i	$0.880087\pi$
$182$		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	−0.998210	−	0.0598062i	$-0.980952\pi$
$191$	−7.33171e8	−	1.26989e9i	−0.550899	−	0.954185i	0.447311	−	0.894378i	$-0.352382\pi$
$192$		0			0
$193$	1.05021e9	−	1.81901e9i	0.756911	−	1.31101i	−0.187507	−	0.982263i	$-0.560041\pi$
$193$	1.05021e9	−	1.81901e9i	0.756911	−	1.31101i	0.944418	−	0.328746i	$-0.106626\pi$
$194$		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	0.800557	−	0.599256i	$-0.204537\pi$
$199$	1.06933e9	+	6.17377e8i	0.681865	+	0.393675i	−0.118692	+	0.992931i	$0.537870\pi$
$200$		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	$-0.870266\pi$
$223$		−	1.96048e9i		−	0.792761i	0.918086	−	0.396380i	$-0.129734\pi$
$224$		0			0
$225$	9.66276e8			0.377026
$226$		0			0
$227$	−3.85172e8	−	2.22379e8i	−0.145061	−	0.0837510i	0.425713	−	0.904858i	$-0.360023\pi$
$227$	−3.85172e8	−	2.22379e8i	−0.145061	−	0.0837510i	−0.570774	+	0.821107i	$0.693357\pi$
$228$		0			0
$229$	−7.22193e7	+	4.16958e7i	−0.0262610	+	0.0151618i	−0.513073	−	0.858345i	$-0.671493\pi$
$229$	−7.22193e7	+	4.16958e7i	−0.0262610	+	0.0151618i	0.486812	+	0.873507i	$0.338160\pi$
$230$		0			0
$231$	9.35565e8	−	2.49613e8i	0.328569	−	0.0876636i
$232$		0			0
$233$	9.57414e8	+	1.65829e9i	0.324845	+	0.562648i	0.981481	−	0.191560i	$-0.0613546\pi$
$233$	9.57414e8	+	1.65829e9i	0.324845	+	0.562648i	−0.656636	+	0.754208i	$0.728021\pi$
$234$		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	0.967279	+	0.253715i	$0.0816526\pi$
$241$	5.63575e9	+	3.25380e9i	1.67064	+	0.964546i	0.703363	+	0.710831i	$0.251681\pi$
$242$		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	$0.0332167\pi$
$251$			8.26879e8i			0.208328i	−0.994560	+	0.104164i	$0.966783\pi$
$252$		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	0.241576	−	0.970382i	$-0.422336\pi$
$257$	5.24692e9	−	3.02931e9i	1.20274	−	0.694402i	0.961163	+	0.275980i	$0.0890023\pi$
$258$		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	−0.999971	−	0.00763120i	$-0.997571\pi$
$263$	−2.36049e9	+	4.08848e9i	−0.493377	+	0.854553i	0.506594	+	0.862185i	$0.330904\pi$
$264$		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	−0.998899	−	0.0469109i	$-0.985062\pi$
$269$	−8.05825e9	−	4.65243e9i	−1.53897	−	0.888527i	−0.540076	−	0.841617i	$-0.681604\pi$
$270$		0			0
$271$	−7.22249e8	+	4.16991e8i	−0.133909	+	0.0773124i	−0.565458	−	0.824777i	$-0.691301\pi$
$271$	−7.22249e8	+	4.16991e8i	−0.133909	+	0.0773124i	0.431549	+	0.902090i	$0.357967\pi$
$272$		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	0.200904	−	0.979611i	$-0.435612\pi$
$277$	5.58602e9	−	9.67528e9i	0.948820	−	1.64340i	0.747916	−	0.663793i	$-0.231055\pi$
$278$		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	0.0767763	−	0.997048i	$-0.475537\pi$
$283$	−4.79981e9	−	2.77117e9i	−0.748305	−	0.432034i	−0.825081	+	0.565014i	$0.808871\pi$
$284$		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	$0.219748\pi$
$293$			9.38671e9i			1.27363i	−0.771017	+	0.636814i	$0.780252\pi$
$294$		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	$-0.903033\pi$
$307$		−	5.32865e9i		−	0.599879i	0.953958	−	0.299940i	$-0.0969665\pi$
$308$		0			0
$309$	−7.35703e9			−0.806991
$310$		0			0
$311$	1.06252e10	+	6.13448e9i	1.13579	+	0.655747i	0.945384	−	0.325959i	$-0.105687\pi$
$311$	1.06252e10	+	6.13448e9i	1.13579	+	0.655747i	0.190403	+	0.981706i	$0.439020\pi$
$312$		0			0
$313$	3.82607e9	−	2.20898e9i	0.398635	−	0.230152i	−0.287260	−	0.957853i	$-0.592744\pi$
$313$	3.82607e9	−	2.20898e9i	0.398635	−	0.230152i	0.685895	+	0.727701i	$0.259411\pi$
$314$		0			0
$315$	−3.38408e9	−	3.39132e9i	−0.343715	−	0.344451i
$316$		0			0
$317$	−3.64028e9	−	6.30515e9i	−0.360494	−	0.624394i	0.627548	−	0.778578i	$-0.284059\pi$
$317$	−3.64028e9	−	6.30515e9i	−0.360494	−	0.624394i	−0.988042	+	0.154184i	$0.950725\pi$
$318$		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	−0.836648	−	0.547741i	$-0.815488\pi$
$331$	−1.07154e10	−	1.85596e10i	−0.892681	−	1.54617i	−0.0560332	−	0.998429i	$-0.517845\pi$
$332$		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	−0.783678	−	0.621167i	$-0.786659\pi$
$347$	2.11832e9	−	3.66903e9i	0.146108	−	0.253066i	0.929786	+	0.368101i	$0.119992\pi$
$348$		0			0
$349$			5.09547e9i			0.343465i	0.985144	+	0.171732i	$0.0549365\pi$
$349$			5.09547e9i			0.343465i	−0.985144	+	0.171732i	$0.945064\pi$
$350$		0			0
$351$	−3.79528e9			−0.250043
$352$		0			0
$353$	6.67443e9	+	3.85348e9i	0.429848	+	0.248173i	0.699282	−	0.714846i	$-0.253503\pi$
$353$	6.67443e9	+	3.85348e9i	0.429848	+	0.248173i	−0.269434	+	0.963019i	$0.586837\pi$
$354$		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	0.850781	−	0.525521i	$-0.176130\pi$
$359$	−4.93726e8	−	8.55158e8i	−0.0297241	−	0.0514836i	−0.880505	+	0.474037i	$0.842796\pi$
$360$		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	0.845968	−	0.533234i	$-0.179023\pi$
$367$	1.46428e10	+	8.45400e9i	0.807158	+	0.466013i	−0.0388101	+	0.999247i	$0.512357\pi$
$368$		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	0.730877	−	0.682509i	$-0.239111\pi$
$373$	−4.36752e9	−	7.56476e9i	−0.225631	−	0.390805i	−0.956509	+	0.291704i	$0.905778\pi$
$374$		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	0.448359	−	0.893854i	$-0.352009\pi$
$383$	3.11283e10	−	1.79719e10i	1.44664	−	0.835217i	0.998279	+	0.0586370i	$0.0186755\pi$
$384$		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	0.472178	+	0.881503i	$0.343468\pi$
$389$	−1.20745e10	+	2.09136e10i	−0.527315	+	0.913337i	−0.999493	+	0.0318334i	$0.989865\pi$
$390$		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	−0.00573411	−	0.999984i	$-0.501825\pi$
$397$	2.12986e10	−	1.22967e10i	0.857410	−	0.495026i	0.863144	+	0.504958i	$0.168492\pi$
$398$		0			0
$399$	−1.61316e9	−	4.34092e8i	−0.0636481	−	0.0171274i
$400$		0			0
$401$	−2.09789e9	−	3.63366e9i	−0.0811345	−	0.140529i	0.822603	−	0.568616i	$-0.192521\pi$
$401$	−2.09789e9	−	3.63366e9i	−0.0811345	−	0.140529i	−0.903738	+	0.428087i	$0.859188\pi$
$402$		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	0.233123	−	0.972447i	$-0.425106\pi$
$409$	−1.37810e10	−	7.95648e9i	−0.492480	−	0.284333i	−0.725603	+	0.688114i	$0.758439\pi$
$410$		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	$-0.605974\pi$
$419$		−	5.82585e10i		−	1.89018i	0.326809	−	0.945090i	$-0.394026\pi$
$420$		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	−0.991052	−	0.133475i	$-0.957387\pi$
$431$	−1.31104e10	+	2.27079e10i	−0.379934	+	0.658065i	0.611118	+	0.791539i	$0.290720\pi$
$432$		0			0
$433$		−	1.04010e10i		−	0.295884i	−0.988996	−	0.147942i	$-0.952735\pi$
$433$		−	1.04010e10i		−	0.295884i	0.988996	−	0.147942i	$-0.0472649\pi$
$434$		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	0.346868	−	0.937914i	$-0.387245\pi$
$439$	4.94931e10	−	2.85749e10i	1.33256	−	0.769354i	0.985691	+	0.168560i	$0.0539117\pi$
$440$		0			0
$441$	−2.69370e7	+	1.26076e10i	−0.000712187	+	0.333333i
$442$		0			0
$443$	−1.46702e9	−	2.54096e9i	−0.0380909	−	0.0659754i	0.846351	−	0.532625i	$-0.178794\pi$
$443$	−1.46702e9	−	2.54096e9i	−0.0380909	−	0.0659754i	−0.884442	+	0.466649i	$0.845461\pi$
$444$		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	0.710686	−	0.703509i	$-0.248385\pi$
$457$	−1.10752e10	−	1.91828e10i	−0.253914	−	0.439792i	−0.964600	+	0.263717i	$0.915051\pi$
$458$		0			0
$459$	2.58221e9	−	4.47252e9i	0.0581756	−	0.100763i
$460$		0			0
$461$			1.24751e10i			0.276210i	0.990418	+	0.138105i	$0.0441011\pi$
$461$			1.24751e10i			0.276210i	−0.990418	+	0.138105i	$0.955899\pi$
$462$		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	−0.243600	−	0.969876i	$-0.578328\pi$
$467$	2.25703e10	−	1.30310e10i	0.474537	−	0.273974i	0.718137	+	0.695902i	$0.244995\pi$
$468$		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	0.655902	−	0.754846i	$-0.272288\pi$
$479$	1.73795e10	+	1.00341e10i	0.330138	+	0.190605i	−0.325765	+	0.945451i	$0.605622\pi$
$480$		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	−0.906225	−	0.422795i	$-0.861049\pi$
$487$	−4.89148e9	+	8.47229e9i	−0.0869610	+	0.150621i	0.819264	+	0.573416i	$0.194382\pi$
$488$		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	0.811525	−	0.584317i	$-0.198638\pi$
$499$	−6.21696e9	−	1.07681e10i	−0.100271	−	0.173675i	−0.911796	+	0.410643i	$0.865304\pi$
$500$		0			0
$501$	−2.74881e10	+	4.76108e10i	−0.436309	+	0.755709i
$502$		0			0
$503$		−	8.85735e10i		−	1.38367i	−0.722057	−	0.691834i	$-0.756803\pi$
$503$		−	8.85735e10i		−	1.38367i	0.722057	−	0.691834i	$-0.243197\pi$
$504$		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	−0.999978	−	0.00668109i	$-0.997873\pi$
$509$	−1.00294e11	+	5.79047e10i	−1.49418	+	0.862666i	−0.494203	+	0.869347i	$0.664540\pi$
$510$		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	−0.951670	−	0.307122i	$-0.900634\pi$
$521$	−1.24776e11	−	7.20395e10i	−1.69348	−	0.977732i	−0.741811	−	0.670609i	$-0.766033\pi$
$522$		0			0
$523$	2.73214e10	−	1.57740e10i	0.365170	−	0.210831i	−0.306176	−	0.951975i	$-0.599050\pi$
$523$	2.73214e10	−	1.57740e10i	0.365170	−	0.210831i	0.671346	+	0.741144i	$0.265716\pi$
$524$		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	0.986845	+	0.161667i	$0.0516871\pi$
$541$	5.42610e10	+	9.39828e10i	0.633430	+	1.09713i	−0.353415	+	0.935467i	$0.614980\pi$
$542$		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	−0.949155	−	0.314809i	$-0.898059\pi$
$557$	−1.94381e10	+	3.36677e10i	−0.201945	+	0.349778i	0.747210	+	0.664588i	$0.231393\pi$
$558$		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	−0.138171	−	0.990408i	$-0.544122\pi$
$563$	−1.06997e11	−	6.17750e10i	−1.06498	−	0.614864i	−0.926804	+	0.375545i	$0.877456\pi$
$564$		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	0.628991	−	0.777412i	$-0.283468\pi$
$569$	−3.76060e10	−	6.51355e10i	−0.358763	−	0.621396i	−0.987754	+	0.156016i	$0.950135\pi$
$570$		0			0
$571$	6.39802e9	−	1.10817e10i	0.0601868	−	0.104247i	−0.834362	−	0.551217i	$-0.814164\pi$
$571$	6.39802e9	−	1.10817e10i	0.0601868	−	0.104247i	0.894549	+	0.446970i	$0.147497\pi$
$572$		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	0.999589	−	0.0286728i	$-0.00912810\pi$
$577$	1.63442e11	+	9.43632e10i	1.47455	+	0.851333i	0.474963	+	0.880006i	$0.342461\pi$
$578$		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	$0.0965393\pi$
$587$			7.09182e10i			0.597318i	−0.954360	+	0.298659i	$0.903461\pi$
$588$		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	0.00906300	−	0.999959i	$-0.497115\pi$
$593$	1.08767e11	−	6.27965e10i	0.879584	−	0.507828i	0.870521	+	0.492131i	$0.163782\pi$
$594$		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	−0.683141	−	0.730286i	$-0.739387\pi$
$599$	3.74468e10	−	6.48598e10i	0.290876	−	0.503812i	0.974017	+	0.226475i	$0.0727200\pi$
$600$		0			0
$601$		−	7.91457e10i		−	0.606638i	−0.952889	−	0.303319i	$-0.901905\pi$
$601$		−	7.91457e10i		−	0.606638i	0.952889	−	0.303319i	$-0.0980948\pi$
$602$		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	−0.517464	−	0.855705i	$-0.673124\pi$
$607$	−4.76961e9	+	2.75374e9i	−0.0351340	+	0.0202846i	0.482330	+	0.875990i	$0.339791\pi$
$608$		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	−0.137550	−	0.990495i	$-0.543923\pi$
$613$	1.11411e11	−	1.92970e11i	0.789018	−	1.36662i	0.926569	−	0.376125i	$-0.122744\pi$
$614$		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	0.119453	−	0.992840i	$-0.461886\pi$
$619$	−9.99270e10	−	5.76929e10i	−0.680644	−	0.392970i	−0.800098	+	0.599870i	$0.795219\pi$
$620$		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	−0.994918	−	0.100685i	$-0.967897\pi$
$641$	−6.92620e10	+	1.19965e11i	−0.410264	+	0.710597i	0.584655	+	0.811282i	$0.301230\pi$
$642$		0			0
$643$			6.27928e10i			0.367338i	0.982988	+	0.183669i	$0.0587975\pi$
$643$			6.27928e10i			0.367338i	−0.982988	+	0.183669i	$0.941203\pi$
$644$		0			0
$645$	1.46424e11			0.846004
$646$		0			0
$647$	3.40232e10	+	1.96433e10i	0.194159	+	0.112098i	0.593928	−	0.804518i	$-0.297576\pi$
$647$	3.40232e10	+	1.96433e10i	0.194159	+	0.112098i	−0.399769	+	0.916616i	$0.630910\pi$
$648$		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	0.987065	−	0.160322i	$-0.0512534\pi$
$653$	6.44912e10	+	1.11702e11i	0.354689	+	0.614340i	−0.632376	+	0.774662i	$0.717920\pi$
$654$		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	0.983968	+	0.178345i	$0.0570744\pi$
$661$	3.11244e11	+	1.79697e11i	1.63040	+	0.941314i	0.646435	+	0.762969i	$0.276259\pi$
$662$		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	−0.917268	−	0.398269i	$-0.869611\pi$
$677$	−2.16576e11	+	1.25040e11i	−1.03099	+	0.595243i	−0.113723	+	0.993513i	$0.536278\pi$
$678$		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	−0.707513	−	0.706701i	$-0.750183\pi$
$683$	5.62014e10	−	9.73438e10i	0.258264	−	0.447327i	0.965777	+	0.259374i	$0.0835161\pi$
$684$		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	−0.307376	−	0.951588i	$-0.599451\pi$
$691$	8.27679e10	−	4.77861e10i	0.363036	−	0.209599i	0.670412	+	0.741989i	$0.266118\pi$
$692$		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	−0.999266	−	0.0382955i	$-0.987807\pi$
$709$	−1.34632e11	−	2.33189e11i	−0.532798	−	0.922833i	0.466468	−	0.884538i	$-0.345526\pi$
$710$		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	−0.833712	−	0.552199i	$-0.813789\pi$
$719$	−2.06410e11	+	1.19171e11i	−0.772350	+	0.445917i	0.0613622	+	0.998116i	$0.480456\pi$
$720$		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	$0.0975611\pi$
$727$			1.68567e11i			0.603442i	−0.953396	+	0.301721i	$0.902439\pi$
$728$		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	−0.876680	−	0.481073i	$-0.840247\pi$
$733$	−2.59349e11	+	1.49735e11i	−0.898399	+	0.518691i	−0.0217184	+	0.999764i	$0.506914\pi$
$734$		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	−0.438692	−	0.898638i	$-0.644558\pi$
$739$	1.66690e11	−	2.88716e11i	0.558897	−	0.968038i	0.997589	−	0.0694004i	$-0.0221086\pi$
$740$		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	0.965409	−	0.260740i	$-0.0839665\pi$
$751$	8.17182e10	+	1.41540e11i	0.256897	+	0.444959i	−0.708512	+	0.705699i	$0.750633\pi$
$752$		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	−0.967235	−	0.253881i	$-0.918293\pi$
$761$	−4.12849e11	+	2.38359e11i	−1.23099	+	0.710710i	−0.263750	+	0.964591i	$0.584960\pi$
$762$		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	$-0.687405\pi$
$769$		−	5.81659e11i		−	1.66327i	0.555323	−	0.831635i	$-0.312595\pi$
$770$		0			0
$771$	−2.83333e11			−0.801826
$772$		0			0
$773$	1.79880e10	+	1.03854e10i	0.0503808	+	0.0290874i	0.524979	−	0.851115i	$-0.324073\pi$
$773$	1.79880e10	+	1.03854e10i	0.0503808	+	0.0290874i	−0.474598	+	0.880203i	$0.657407\pi$
$774$		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	0.834723	−	0.550670i	$-0.185628\pi$
$787$	2.97377e11	+	1.71691e11i	0.775191	+	0.447557i	−0.0595323	+	0.998226i	$0.518961\pi$
$788$		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	$0.0229797\pi$
$797$			5.82077e10i			0.144260i	−0.997395	+	0.0721302i	$0.977020\pi$
$798$		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	−0.911611	−	0.411054i	$-0.865161\pi$
$809$	−4.27586e10	+	7.40600e10i	−0.0998226	+	0.172898i	0.811789	+	0.583952i	$0.198494\pi$
$810$		0			0
$811$		−	7.56023e11i		−	1.74764i	−0.486252	−	0.873819i	$-0.661636\pi$
$811$		−	7.56023e11i		−	1.74764i	0.486252	−	0.873819i	$-0.338364\pi$
$812$		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	0.668923	−	0.743332i	$-0.266756\pi$
$821$	−1.40517e11	−	2.43383e11i	−0.309283	−	0.535694i	−0.978206	+	0.207638i	$0.933422\pi$
$822$		0			0
$823$	−2.21259e11	+	3.83231e11i	−0.482282	+	0.835336i	−0.999793	−	0.0203400i	$-0.993525\pi$
$823$	−2.21259e11	+	3.83231e11i	−0.482282	+	0.835336i	0.517512	+	0.855676i	$0.326858\pi$
$824$		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	0.998022	+	0.0628712i	$0.0200257\pi$
$829$	7.32765e11	+	4.23062e11i	1.55148	+	0.895748i	0.553459	+	0.832877i	$0.313308\pi$
$830$		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	$-0.956861\pi$
$839$		−	1.33895e11i		−	0.270220i	0.990831	−	0.135110i	$-0.0431388\pi$
$840$		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	$0.0978486\pi$
$853$			3.20383e11i			0.605164i	−0.953123	+	0.302582i	$0.902151\pi$
$854$		0			0
$855$	−2.96872e10			−0.0555527
$856$		0			0
$857$	−3.52411e11	−	2.03465e11i	−0.653320	−	0.377195i	0.136407	−	0.990653i	$-0.456445\pi$
$857$	−3.52411e11	−	2.03465e11i	−0.653320	−	0.377195i	−0.789727	+	0.613458i	$0.789778\pi$
$858$		0			0
$859$	1.83612e11	−	1.06008e11i	0.337231	−	0.194700i	−0.321816	−	0.946802i	$-0.604293\pi$
$859$	1.83612e11	−	1.06008e11i	0.337231	−	0.194700i	0.659047	+	0.752102i	$0.270960\pi$
$860$		0			0
$861$	3.34007e11	−	3.33294e11i	0.607775	−	0.606478i
$862$		0			0
$863$	2.51111e11	+	4.34936e11i	0.452712	+	0.784120i	0.998553	−	0.0537683i	$-0.0171232\pi$
$863$	2.51111e11	+	4.34936e11i	0.452712	+	0.784120i	−0.545841	+	0.837888i	$0.683790\pi$
$864$		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	0.859594	+	0.510978i	$0.170717\pi$
$877$	5.16027e11	+	8.93786e11i	0.872317	+	1.51090i	0.0127234	+	0.999919i	$0.495950\pi$
$878$		0			0
$879$	2.19486e11	−	3.80162e11i	0.367665	−	0.636814i
$880$		0			0
$881$			7.05174e11i			1.17056i	0.810832	+	0.585279i	$0.199015\pi$
$881$			7.05174e11i			1.17056i	−0.810832	+	0.585279i	$0.800985\pi$
$882$		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	−0.122978	−	0.992409i	$-0.539244\pi$
$887$	4.17819e11	−	2.41228e11i	0.674985	−	0.389703i	0.797963	+	0.602707i	$0.205911\pi$
$888$		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	−0.950350	−	0.311183i	$-0.899275\pi$
$907$	−1.39196e11	+	2.41094e11i	−0.205682	+	0.356252i	0.744668	+	0.667436i	$0.232608\pi$
$908$		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	0.981841	−	0.189706i	$-0.0607535\pi$
$919$	2.32980e11	+	4.03533e11i	0.326630	+	0.565740i	−0.655211	+	0.755446i	$0.727420\pi$
$920$		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	−0.699414	−	0.714716i	$-0.746556\pi$
$929$	−3.20399e11	+	1.84983e11i	−0.430159	+	0.248352i	0.269255	+	0.963069i	$0.413222\pi$
$930$		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	$0.207785\pi$
$937$			9.36393e11i			1.21479i	−0.794401	+	0.607393i	$0.792215\pi$
$938$		0			0
$939$	−2.06608e11			−0.265757
$940$		0			0
$941$	−7.35252e11	−	4.24498e11i	−0.937729	−	0.541398i	−0.0484815	−	0.998824i	$-0.515438\pi$
$941$	−7.35252e11	−	4.24498e11i	−0.937729	−	0.541398i	−0.889248	+	0.457426i	$0.848772\pi$
$942$		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	0.605560	−	0.795800i	$-0.292949\pi$
$947$	−3.10771e11	−	5.38271e11i	−0.386403	−	0.669270i	−0.991963	+	0.126530i	$0.959616\pi$
$948$		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	−0.999988	−	0.00492975i	$-0.998431\pi$
$971$	−1.32961e12	+	7.67652e11i	−1.49571	+	0.863550i	−0.495725	+	0.868480i	$0.665097\pi$
$972$		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	−0.813609	−	0.581412i	$-0.802501\pi$
$977$	8.81173e10	−	1.52624e11i	0.0967126	−	0.167511i	0.910322	+	0.413901i	$0.135834\pi$
$978$		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	0.827754	−	0.561091i	$-0.189618\pi$
$983$	7.05620e11	+	4.07390e11i	0.755712	+	0.436311i	−0.0720418	+	0.997402i	$0.522951\pi$
$984$		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	−0.971460	−	0.237202i	$-0.923770\pi$
$991$	−2.70351e11	+	4.68262e11i	−0.280307	+	0.485506i	0.691153	+	0.722708i	$0.257103\pi$
$992$		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	0.0970882	−	0.995276i	$-0.469047\pi$
$997$	−7.07745e11	−	4.08617e11i	−0.716302	−	0.413557i	−0.813390	+	0.581719i	$0.802380\pi$
$998$		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.73.5	yes	10
3.2	odd	2		252.9.z.b.73.1		10
7.5	odd	6	inner	84.9.m.a.61.5	✓	10
21.5	even	6		252.9.z.b.145.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.9.m.a.61.5	✓	10	7.5	odd	6	inner
84.9.m.a.73.5	yes	10	1.1	even	1	trivial
252.9.z.b.73.1		10	3.2	odd	2
252.9.z.b.145.1		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 $\F_{q}$
Belyi maps

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

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