LMFDB - Embedded newform 112.10.p.c.31.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [112,10,Mod(31,112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 1])) N = Newforms(chi, 10, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("112.31"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	112.p (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [24,0,162] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$57.6840136504$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			31.3
Character	$\chi$	$=$	112.31
Dual form			112.10.p.c.47.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-85.3530 + 147.836i) q^{3} +(-1415.83 + 817.429i) q^{5} +(796.914 + 6302.26i) q^{7} +(-4728.77 - 8190.47i) q^{9} +(-76024.2 - 43892.6i) q^{11} +24846.5i q^{13} -279080. i q^{15} +(-98783.3 - 57032.6i) q^{17} +(-120315. - 208392. i) q^{19} +(-999719. - 420105. i) q^{21} +(-1.37851e6 + 795885. i) q^{23} +(359816. - 623220. i) q^{25} -1.74555e6 q^{27} -117163. q^{29} +(-1.04347e6 + 1.80734e6i) q^{31} +(1.29778e7 - 7.49272e6i) q^{33} +(-6.27994e6 - 8.27150e6i) q^{35} +(1.68888e6 + 2.92523e6i) q^{37} +(-3.67320e6 - 2.12072e6i) q^{39} -2.72870e7i q^{41} +1.43490e7i q^{43} +(1.33902e7 + 7.73086e6i) q^{45} +(1.34766e7 + 2.33421e7i) q^{47} +(-3.90835e7 + 1.00447e7i) q^{49} +(1.68629e7 - 9.73580e6i) q^{51} +(-3.45262e7 + 5.98012e7i) q^{53} +1.43516e8 q^{55} +4.10771e7 q^{57} +(-1.41959e7 + 2.45880e7i) q^{59} +(1.15548e8 - 6.67116e7i) q^{61} +(4.78501e7 - 3.63291e7i) q^{63} +(-2.03102e7 - 3.51783e7i) q^{65} +(4.49199e7 + 2.59345e7i) q^{67} -2.71725e8i q^{69} +3.87928e8i q^{71} +(4.09969e8 + 2.36696e8i) q^{73} +(6.14228e7 + 1.06387e8i) q^{75} +(2.16038e8 - 5.14103e8i) q^{77} +(3.35591e8 - 1.93753e8i) q^{79} +(2.42064e8 - 4.19267e8i) q^{81} -5.59811e8 q^{83} +1.86480e8 q^{85} +(1.00002e7 - 1.73208e7i) q^{87} +(4.49942e8 - 2.59774e8i) q^{89} +(-1.56589e8 + 1.98005e7i) q^{91} +(-1.78126e8 - 3.08524e8i) q^{93} +(3.40691e8 + 1.96698e8i) q^{95} -2.35445e8i q^{97} +8.30231e8i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 162 q^{3} - 852 q^{5} + 6744 q^{7} - 77884 q^{9} + 57534 q^{11} + 789336 q^{17} - 469098 q^{19} - 2104376 q^{21} - 1553682 q^{23} + 3602544 q^{25} - 6389244 q^{27} - 2462040 q^{29} + 10306686 q^{31}+ \cdots - 1433917218 q^{95}+O(q^{100}) $ 24 * q + 162 * q^3 - 852 * q^5 + 6744 * q^7 - 77884 * q^9 + 57534 * q^11 + 789336 * q^17 - 469098 * q^19 - 2104376 * q^21 - 1553682 * q^23 + 3602544 * q^25 - 6389244 * q^27 - 2462040 * q^29 + 10306686 * q^31 + 8789748 * q^33 - 21149802 * q^35 + 10422720 * q^37 - 54458664 * q^39 + 2549604 * q^45 - 12751230 * q^47 - 16137504 * q^49 + 107322414 * q^51 + 62342520 * q^53 + 422625348 * q^55 - 317200696 * q^57 + 101234394 * q^59 + 48881304 * q^61 - 726717996 * q^63 + 272864028 * q^65 + 48335310 * q^67 + 672073812 * q^73 - 950672484 * q^75 + 10755684 * q^77 - 144370122 * q^79 - 305389432 * q^81 + 568641984 * q^83 - 734871504 * q^85 - 33665484 * q^87 + 1630891500 * q^89 - 1012044192 * q^91 - 457741376 * q^93 - 1433917218 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/112\mathbb{Z}\right)^\times$.

$n$	$15$	$17$	$85$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$	$1$

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
$3$	−85.3530	+	147.836i	−0.608378	+	1.05374i	0.383130	+	0.923694i	$0.374846\pi$
$3$	−85.3530	+	147.836i	−0.608378	+	1.05374i	−0.991508	+	0.130046i	$0.958487\pi$
$4$		0			0
$5$	−1415.83	+	817.429i	−1.01308	+	0.584904i	−0.912093	−	0.409983i	$-0.865535\pi$
$5$	−1415.83	+	817.429i	−1.01308	+	0.584904i	−0.100991	+	0.994887i	$0.532201\pi$
$6$		0			0
$7$	796.914	+	6302.26i	0.125450	+	0.992100i
$7$	796.914	+	6302.26i	0.125450	+	0.992100i
$8$		0			0
$9$	−4728.77	−	8190.47i	−0.240246	−	0.416119i
$10$		0			0
$11$	−76024.2	−	43892.6i	−1.56561	−	0.903907i	−0.996671	−	0.0815299i	$-0.974019\pi$
$11$	−76024.2	−	43892.6i	−1.56561	−	0.903907i	−0.568942	−	0.822377i	$-0.692647\pi$
$12$		0			0
$13$			24846.5i			0.241279i	0.992696	+	0.120640i	$0.0384945\pi$
$13$			24846.5i			0.241279i	−0.992696	+	0.120640i	$0.961505\pi$
$14$		0			0
$15$		−	279080.i		−	1.42337i
$16$		0			0
$17$	−98783.3	−	57032.6i	−0.286856	−	0.165616i	0.349667	−	0.936874i	$-0.386295\pi$
$17$	−98783.3	−	57032.6i	−0.286856	−	0.165616i	−0.636523	+	0.771258i	$0.719628\pi$
$18$		0			0
$19$	−120315.	−	208392.i	−0.211802	−	0.366851i	0.740477	−	0.672082i	$-0.234600\pi$
$19$	−120315.	−	208392.i	−0.211802	−	0.366851i	−0.952278	+	0.305231i	$0.901266\pi$
$20$		0			0
$21$	−999719.	−	420105.i	−1.12174	−	0.471380i
$22$		0			0
$23$	−1.37851e6	+	795885.i	−1.02715	+	0.593028i	−0.916168	−	0.400794i	$-0.868734\pi$
$23$	−1.37851e6	+	795885.i	−1.02715	+	0.593028i	−0.110986	+	0.993822i	$0.535401\pi$
$24$		0			0
$25$	359816.	−	623220.i	0.184226	−	0.319089i
$26$		0			0
$27$	−1.74555e6			−0.632113
$28$		0			0
$29$	−117163.			−0.0307609			−0.0153804	−	0.999882i	$-0.504896\pi$
$29$	−117163.			−0.0307609			−0.0153804	+	0.999882i	$0.504896\pi$
$30$		0			0
$31$	−1.04347e6	+	1.80734e6i	−0.202932	+	0.351489i	−0.949472	−	0.313852i	$-0.898380\pi$
$31$	−1.04347e6	+	1.80734e6i	−0.202932	+	0.351489i	0.746540	+	0.665341i	$0.231714\pi$
$32$		0			0
$33$	1.29778e7	−	7.49272e6i	1.90497	−	1.09983i
$34$		0			0
$35$	−6.27994e6	−	8.27150e6i	−0.707375	−	0.931704i
$36$		0			0
$37$	1.68888e6	+	2.92523e6i	0.148147	+	0.256598i	0.930543	−	0.366184i	$-0.119336\pi$
$37$	1.68888e6	+	2.92523e6i	0.148147	+	0.256598i	−0.782396	+	0.622782i	$0.786003\pi$
$38$		0			0
$39$	−3.67320e6	−	2.12072e6i	−0.254246	−	0.146789i
$40$		0			0
$41$		−	2.72870e7i		−	1.50810i	−0.656819	−	0.754048i	$-0.728098\pi$
$41$		−	2.72870e7i		−	1.50810i	0.656819	−	0.754048i	$-0.271902\pi$
$42$		0			0
$43$			1.43490e7i			0.640051i	0.947409	+	0.320025i	$0.103691\pi$
$43$			1.43490e7i			0.640051i	−0.947409	+	0.320025i	$0.896309\pi$
$44$		0			0
$45$	1.33902e7	+	7.73086e6i	0.486780	+	0.281042i
$46$		0			0
$47$	1.34766e7	+	2.33421e7i	0.402846	+	0.697749i	0.994068	−	0.108759i	$-0.0346878\pi$
$47$	1.34766e7	+	2.33421e7i	0.402846	+	0.697749i	−0.591222	+	0.806509i	$0.701354\pi$
$48$		0			0
$49$	−3.90835e7	+	1.00447e7i	−0.968525	+	0.248918i
$50$		0			0
$51$	1.68629e7	−	9.73580e6i	0.349033	−	0.201514i
$52$		0			0
$53$	−3.45262e7	+	5.98012e7i	−0.601046	+	1.04104i	0.391617	+	0.920128i	$0.371916\pi$
$53$	−3.45262e7	+	5.98012e7i	−0.601046	+	1.04104i	−0.992663	+	0.120914i	$0.961417\pi$
$54$		0			0
$55$	1.43516e8			2.11480
$56$		0			0
$57$	4.10771e7			0.515421
$58$		0			0
$59$	−1.41959e7	+	2.45880e7i	−0.152521	+	0.264174i	−0.932153	−	0.362063i	$-0.882072\pi$
$59$	−1.41959e7	+	2.45880e7i	−0.152521	+	0.264174i	0.779633	+	0.626237i	$0.215406\pi$
$60$		0			0
$61$	1.15548e8	−	6.67116e7i	1.06851	−	0.616903i	0.140734	−	0.990047i	$-0.455054\pi$
$61$	1.15548e8	−	6.67116e7i	1.06851	−	0.616903i	0.927773	+	0.373144i	$0.121720\pi$
$62$		0			0
$63$	4.78501e7	−	3.63291e7i	0.382693	−	0.290551i
$64$		0			0
$65$	−2.03102e7	−	3.51783e7i	−0.141125	−	0.244436i
$66$		0			0
$67$	4.49199e7	+	2.59345e7i	0.272334	+	0.157232i	0.629948	−	0.776637i	$-0.283076\pi$
$67$	4.49199e7	+	2.59345e7i	0.272334	+	0.157232i	−0.357614	+	0.933870i	$0.616410\pi$
$68$		0			0
$69$		−	2.71725e8i		−	1.44314i
$70$		0			0
$71$			3.87928e8i			1.81171i	0.423586	+	0.905856i	$0.360771\pi$
$71$			3.87928e8i			1.81171i	−0.423586	+	0.905856i	$0.639229\pi$
$72$		0			0
$73$	4.09969e8	+	2.36696e8i	1.68966	+	0.975524i	0.954775	+	0.297329i	$0.0960957\pi$
$73$	4.09969e8	+	2.36696e8i	1.68966	+	0.975524i	0.734882	+	0.678195i	$0.237238\pi$
$74$		0			0
$75$	6.14228e7	+	1.06387e8i	0.224158	+	0.388253i
$76$		0			0
$77$	2.16038e8	−	5.14103e8i	0.700360	−	1.66664i
$78$		0			0
$79$	3.35591e8	−	1.93753e8i	0.969367	−	0.559664i	0.0703235	−	0.997524i	$-0.477597\pi$
$79$	3.35591e8	−	1.93753e8i	0.969367	−	0.559664i	0.899043	+	0.437860i	$0.144264\pi$
$80$		0			0
$81$	2.42064e8	−	4.19267e8i	0.624810	−	1.08220i
$82$		0			0
$83$	−5.59811e8			−1.29476			−0.647381	−	0.762166i	$-0.724136\pi$
$83$	−5.59811e8			−1.29476			−0.647381	+	0.762166i	$0.724136\pi$
$84$		0			0
$85$	1.86480e8			0.387478
$86$		0			0
$87$	1.00002e7	−	1.73208e7i	0.0187142	−	0.0324140i
$88$		0			0
$89$	4.49942e8	−	2.59774e8i	0.760154	−	0.438875i	−0.0691969	−	0.997603i	$-0.522044\pi$
$89$	4.49942e8	−	2.59774e8i	0.760154	−	0.438875i	0.829351	+	0.558728i	$0.188710\pi$
$90$		0			0
$91$	−1.56589e8	+	1.98005e7i	−0.239373	+	0.0302684i
$92$		0			0
$93$	−1.78126e8	−	3.08524e8i	−0.246919	−	0.427676i
$94$		0			0
$95$	3.40691e8	+	1.96698e8i	0.429146	+	0.247767i
$96$		0			0
$97$		−	2.35445e8i		−	0.270032i	−0.990843	−	0.135016i	$-0.956891\pi$
$97$		−	2.35445e8i		−	0.270032i	0.990843	−	0.135016i	$-0.0431087\pi$
$98$		0			0
$99$			8.30231e8i			0.868642i
$100$		0			0
$101$	−1.51111e9	−	8.72439e8i	−1.44494	−	0.834236i	−0.446766	−	0.894651i	$-0.647424\pi$
$101$	−1.51111e9	−	8.72439e8i	−1.44494	−	0.834236i	−0.998173	+	0.0604145i	$0.980758\pi$
$102$		0			0
$103$	1.80854e8	+	3.13248e8i	0.158329	+	0.274234i	0.934266	−	0.356576i	$-0.116056\pi$
$103$	1.80854e8	+	3.13248e8i	0.158329	+	0.274234i	−0.775937	+	0.630810i	$0.782723\pi$
$104$		0			0
$105$	1.75884e9	−	2.22403e8i	1.41213	−	0.178562i
$106$		0			0
$107$	3.49756e8	−	2.01932e8i	0.257951	−	0.148928i	−0.365448	−	0.930832i	$-0.619084\pi$
$107$	3.49756e8	−	2.01932e8i	0.257951	−	0.148928i	0.623400	+	0.781903i	$0.285751\pi$
$108$		0			0
$109$	6.92622e8	−	1.19966e9i	0.469977	−	0.814025i	−0.529433	−	0.848352i	$-0.677595\pi$
$109$	6.92622e8	−	1.19966e9i	0.469977	−	0.814025i	0.999411	+	0.0343269i	$0.0109287\pi$
$110$		0			0
$111$	−5.76605e8			−0.360517
$112$		0			0
$113$	1.10316e9			0.636482			0.318241	−	0.948010i	$-0.396908\pi$
$113$	1.10316e9			0.636482			0.318241	+	0.948010i	$0.396908\pi$
$114$		0			0
$115$	1.30116e9	−	2.25367e9i	0.693729	−	1.20157i
$116$		0			0
$117$	2.03504e8	−	1.17493e8i	0.100401	−	0.0579664i
$118$		0			0
$119$	2.80713e8	−	6.68009e8i	0.128322	−	0.305366i
$120$		0			0
$121$	2.67414e9	+	4.63175e9i	1.13410	+	1.96431i
$122$		0			0
$123$	4.03400e9	+	2.32903e9i	1.58914	+	0.917492i
$124$		0			0
$125$		−	2.01658e9i		−	0.738790i
$126$		0			0
$127$			4.53002e9i			1.54520i	0.634895	+	0.772598i	$0.281043\pi$
$127$			4.53002e9i			1.54520i	−0.634895	+	0.772598i	$0.718957\pi$
$128$		0			0
$129$	−2.12130e9	−	1.22473e9i	−0.674448	−	0.389393i
$130$		0			0
$131$	2.34526e9	+	4.06211e9i	0.695777	+	1.20512i	0.969918	+	0.243431i	$0.0782730\pi$
$131$	2.34526e9	+	4.06211e9i	0.695777	+	1.20512i	−0.274142	+	0.961689i	$0.588394\pi$
$132$		0			0
$133$	1.21746e9	−	9.24329e8i	0.337383	−	0.256150i
$134$		0			0
$135$	2.47139e9	−	1.42686e9i	0.640383	−	0.369726i
$136$		0			0
$137$	−3.29257e9	+	5.70290e9i	−0.798533	+	1.38310i	0.122039	+	0.992525i	$0.461057\pi$
$137$	−3.29257e9	+	5.70290e9i	−0.798533	+	1.38310i	−0.920572	+	0.390574i	$0.872277\pi$
$138$		0			0
$139$	−7.24554e9			−1.64628			−0.823140	−	0.567838i	$-0.807780\pi$
$139$	−7.24554e9			−1.64628			−0.823140	+	0.567838i	$0.807780\pi$
$140$		0			0
$141$	−4.60106e9			−0.980329
$142$		0			0
$143$	1.09058e9	−	1.88893e9i	0.218094	−	0.377750i
$144$		0			0
$145$	1.65882e8	−	9.57721e7i	0.0311633	−	0.0179922i
$146$		0			0
$147$	1.85092e9	−	6.63528e9i	0.326934	−	1.17201i
$148$		0			0
$149$	−2.08581e8	−	3.61274e8i	−0.0346687	−	0.0600479i	0.848170	−	0.529723i	$-0.177704\pi$
$149$	−2.08581e8	−	3.61274e8i	−0.0346687	−	0.0600479i	−0.882839	+	0.469675i	$0.844371\pi$
$150$		0			0
$151$	−1.07162e10	−	6.18700e9i	−1.67743	−	0.968465i	−0.963290	−	0.268464i	$-0.913484\pi$
$151$	−1.07162e10	−	6.18700e9i	−1.67743	−	0.968465i	−0.714142	−	0.700001i	$-0.753183\pi$
$152$		0			0
$153$			1.07878e9i			0.159155i
$154$		0			0
$155$		−	3.41184e9i		−	0.474784i
$156$		0			0
$157$	−1.41780e9	−	8.18567e8i	−0.186237	−	0.107524i	0.403983	−	0.914767i	$-0.367626\pi$
$157$	−1.41780e9	−	8.18567e8i	−0.186237	−	0.107524i	−0.590220	+	0.807243i	$0.700959\pi$
$158$		0			0
$159$	−5.89384e9	−	1.02084e10i	−0.731326	−	1.26669i
$160$		0			0
$161$	−6.11443e9	−	8.05350e9i	−0.717199	−	0.944644i
$162$		0			0
$163$	−7.88282e9	+	4.55115e9i	−0.874656	+	0.504983i	−0.868893	−	0.495000i	$-0.835168\pi$
$163$	−7.88282e9	+	4.55115e9i	−0.874656	+	0.504983i	−0.00576346	+	0.999983i	$0.501835\pi$
$164$		0			0
$165$	−1.22495e10	+	2.12168e10i	−1.28659	+	2.22845i
$166$		0			0
$167$	−5.22224e9			−0.519557			−0.259778	−	0.965668i	$-0.583650\pi$
$167$	−5.22224e9			−0.519557			−0.259778	+	0.965668i	$0.583650\pi$
$168$		0			0
$169$	9.98715e9			0.941784
$170$		0			0
$171$	−1.13789e9	+	1.97088e9i	−0.101769	+	0.176269i
$172$		0			0
$173$	−5.66969e9	+	3.27340e9i	−0.481229	+	0.277838i	−0.720929	−	0.693009i	$-0.756284\pi$
$173$	−5.66969e9	+	3.27340e9i	−0.481229	+	0.277838i	0.239699	+	0.970847i	$0.422951\pi$
$174$		0			0
$175$	4.21444e9	+	1.77101e9i	0.339679	+	0.142741i
$176$		0			0
$177$	−2.42332e9	−	4.19732e9i	−0.185580	−	0.321435i
$178$		0			0
$179$	−2.12192e10	−	1.22509e10i	−1.54487	−	0.891930i	−0.998521	−	0.0543705i	$-0.982685\pi$
$179$	−2.12192e10	−	1.22509e10i	−1.54487	−	0.891930i	−0.546347	−	0.837559i	$-0.683982\pi$
$180$		0			0
$181$			5.81750e9i			0.402886i	0.979500	+	0.201443i	$0.0645631\pi$
$181$			5.81750e9i			0.402886i	−0.979500	+	0.201443i	$0.935437\pi$
$182$		0			0
$183$			2.27761e10i			1.50124i
$184$		0			0
$185$	−4.78234e9	−	2.76108e9i	−0.300170	−	0.173303i
$186$		0			0
$187$	5.00661e9	+	8.67171e9i	0.299403	+	0.518582i
$188$		0			0
$189$	−1.39105e9	−	1.10009e10i	−0.0792985	−	0.627119i
$190$		0			0
$191$	2.45560e10	−	1.41774e10i	1.33508	−	0.770809i	0.349006	−	0.937120i	$-0.386519\pi$
$191$	2.45560e10	−	1.41774e10i	1.33508	−	0.770809i	0.986073	+	0.166312i	$0.0531858\pi$
$192$		0			0
$193$	−1.59065e10	+	2.75508e10i	−0.825214	+	1.42931i	0.0765424	+	0.997066i	$0.475612\pi$
$193$	−1.59065e10	+	2.75508e10i	−0.825214	+	1.42931i	−0.901756	+	0.432245i	$0.857721\pi$
$194$		0			0
$195$	6.93415e9			0.343429
$196$		0			0
$197$	1.92054e10			0.908499			0.454250	−	0.890874i	$-0.349907\pi$
$197$	1.92054e10			0.908499			0.454250	+	0.890874i	$0.349907\pi$
$198$		0			0
$199$	9.72401e9	−	1.68425e10i	0.439548	−	0.761320i	−0.558106	−	0.829769i	$-0.688472\pi$
$199$	9.72401e9	−	1.68425e10i	0.439548	−	0.761320i	0.997655	+	0.0684497i	$0.0218053\pi$
$200$		0			0
$201$	−7.66810e9	+	4.42718e9i	−0.331364	+	0.191313i
$202$		0			0
$203$	−9.33686e7	−	7.38390e8i	−0.00385894	−	0.0305178i
$204$		0			0
$205$	2.23052e10	+	3.86338e10i	0.882092	+	1.52783i
$206$		0			0
$207$	1.30373e10	+	7.52711e9i	0.493540	+	0.284946i
$208$		0			0
$209$			2.11238e10i			0.765796i
$210$		0			0
$211$		−	1.76978e10i		−	0.614679i	−0.951600	−	0.307339i	$-0.900561\pi$
$211$		−	1.76978e10i		−	0.614679i	0.951600	−	0.307339i	$-0.0994387\pi$
$212$		0			0
$213$	−5.73497e10	−	3.31109e10i	−1.90907	−	1.10220i
$214$		0			0
$215$	−1.17293e10	−	2.03158e10i	−0.374368	−	0.648425i
$216$		0			0
$217$	−1.22219e10	−	5.13591e9i	−0.374170	−	0.157235i
$218$		0			0
$219$	−6.99842e10	+	4.04054e10i	−2.05590	+	1.18697i
$220$		0			0
$221$	1.41706e9	−	2.45442e9i	0.0399597	−	0.0692123i
$222$		0			0
$223$	4.26327e10			1.15444			0.577220	−	0.816589i	$-0.304138\pi$
$223$	4.26327e10			1.15444			0.577220	+	0.816589i	$0.304138\pi$
$224$		0			0
$225$	−6.80596e9			−0.177039
$226$		0			0
$227$	1.99382e10	−	3.45339e10i	0.498389	−	0.863236i	−0.501609	−	0.865094i	$-0.667258\pi$
$227$	1.99382e10	−	3.45339e10i	0.498389	−	0.863236i	0.999998	+	0.00185884i	$0.000591688\pi$
$228$		0			0
$229$	5.65491e10	−	3.26487e10i	1.35883	−	0.784523i	0.369367	−	0.929284i	$-0.379575\pi$
$229$	5.65491e10	−	3.26487e10i	1.35883	−	0.784523i	0.989467	+	0.144761i	$0.0462413\pi$
$230$		0			0
$231$	5.75633e10	+	7.58183e10i	1.33012	+	1.75194i
$232$		0			0
$233$	−3.39245e9	−	5.87589e9i	−0.0754070	−	0.130609i	0.825856	−	0.563881i	$-0.190692\pi$
$233$	−3.39245e9	−	5.87589e9i	−0.0754070	−	0.130609i	−0.901263	+	0.433272i	$0.857359\pi$
$234$		0			0
$235$	−3.81610e10	−	2.20323e10i	−0.816233	−	0.471252i
$236$		0			0
$237$			6.61497e10i			1.36195i
$238$		0			0
$239$		−	3.37396e10i		−	0.668881i	−0.942417	−	0.334441i	$-0.891453\pi$
$239$		−	3.37396e10i		−	0.668881i	0.942417	−	0.334441i	$-0.108547\pi$
$240$		0			0
$241$	−7.04664e10	−	4.06838e10i	−1.34557	−	0.776863i	−0.357949	−	0.933741i	$-0.616524\pi$
$241$	−7.04664e10	−	4.06838e10i	−1.34557	−	0.776863i	−0.987618	+	0.156878i	$0.949857\pi$
$242$		0			0
$243$	2.41430e10	+	4.18169e10i	0.444184	+	0.769349i
$244$		0			0
$245$	4.71246e10	−	4.61695e10i	0.835604	−	0.818669i
$246$		0			0
$247$	5.17781e9	−	2.98941e9i	0.0885135	−	0.0511033i
$248$		0			0
$249$	4.77816e10	−	8.27601e10i	0.787704	−	1.36434i
$250$		0			0
$251$	−1.95106e10			−0.310269			−0.155134	−	0.987893i	$-0.549581\pi$
$251$	−1.95106e10			−0.310269			−0.155134	+	0.987893i	$0.549581\pi$
$252$		0			0
$253$	1.39734e11			2.14417
$254$		0			0
$255$	−1.59166e10	+	2.75684e10i	−0.235733	+	0.408302i
$256$		0			0
$257$	7.94705e10	−	4.58823e10i	1.13634	−	0.656064i	0.190816	−	0.981626i	$-0.438887\pi$
$257$	7.94705e10	−	4.58823e10i	1.13634	−	0.656064i	0.945521	+	0.325562i	$0.105553\pi$
$258$		0			0
$259$	−1.70897e10	+	1.29750e10i	−0.235986	+	0.179167i
$260$		0			0
$261$	5.54035e8	+	9.59618e8i	0.00739018	+	0.0128002i
$262$		0			0
$263$	−1.19700e10	−	6.91090e9i	−0.154275	−	0.0890705i	0.420875	−	0.907118i	$-0.361723\pi$
$263$	−1.19700e10	−	6.91090e9i	−0.154275	−	0.0890705i	−0.575150	+	0.818048i	$0.695056\pi$
$264$		0			0
$265$		−	1.12891e11i		−	1.40622i
$266$		0			0
$267$			8.86900e10i			1.06801i
$268$		0			0
$269$	−9.34884e9	−	5.39755e9i	−0.108861	−	0.0628509i	0.444581	−	0.895739i	$-0.353353\pi$
$269$	−9.34884e9	−	5.39755e9i	−0.108861	−	0.0628509i	−0.553442	+	0.832888i	$0.686686\pi$
$270$		0			0
$271$	−2.42736e10	−	4.20430e10i	−0.273383	−	0.473513i	0.696343	−	0.717709i	$-0.254809\pi$
$271$	−2.42736e10	−	4.20430e10i	−0.273383	−	0.473513i	−0.969726	+	0.244196i	$0.921476\pi$
$272$		0			0
$273$	1.04381e10	−	2.48395e10i	0.113734	−	0.270652i
$274$		0			0
$275$	−5.47095e10	+	3.15865e10i	−0.576853	+	0.333046i
$276$		0			0
$277$	−6.65015e9	+	1.15184e10i	−0.0678691	+	0.117553i	−0.897963	−	0.440071i	$-0.854953\pi$
$277$	−6.65015e9	+	1.15184e10i	−0.0678691	+	0.117553i	0.830094	+	0.557624i	$0.188287\pi$
$278$		0			0
$279$	1.97373e10			0.195015
$280$		0			0
$281$	1.02607e11			0.981740			0.490870	−	0.871233i	$-0.336679\pi$
$281$	1.02607e11			0.981740			0.490870	+	0.871233i	$0.336679\pi$
$282$		0			0
$283$	−2.00678e10	+	3.47584e10i	−0.185977	+	0.322122i	−0.943905	−	0.330216i	$-0.892878\pi$
$283$	−2.00678e10	+	3.47584e10i	−0.185977	+	0.322122i	0.757928	+	0.652338i	$0.226212\pi$
$284$		0			0
$285$	−5.81580e10	+	3.35776e10i	−0.522165	+	0.301472i
$286$		0			0
$287$	1.71970e11	−	2.17454e10i	1.49618	−	0.189190i
$288$		0			0
$289$	−5.27885e10	−	9.14324e10i	−0.445143	−	0.771010i
$290$		0			0
$291$	3.48071e10	+	2.00959e10i	0.284544	+	0.164282i
$292$		0			0
$293$		−	4.52666e10i		−	0.358818i	−0.983775	−	0.179409i	$-0.942582\pi$
$293$		−	4.52666e10i		−	0.358818i	0.983775	−	0.179409i	$-0.0574185\pi$
$294$		0			0
$295$		−	4.64165e10i		−	0.356840i
$296$		0			0
$297$	1.32704e11	+	7.66166e10i	0.989645	+	0.571372i
$298$		0			0
$299$	−1.97749e10	−	3.42512e10i	−0.143085	−	0.247831i
$300$		0			0
$301$	−9.04314e10	+	1.14349e10i	−0.634994	+	0.0802943i
$302$		0			0
$303$	2.57955e11	−	1.48931e11i	1.75814	−	1.01506i
$304$		0			0
$305$	−1.09064e11	+	1.88904e11i	−0.721658	+	1.24995i
$306$		0			0
$307$	1.04134e11			0.669068			0.334534	−	0.942384i	$-0.391421\pi$
$307$	1.04134e11			0.669068			0.334534	+	0.942384i	$0.391421\pi$
$308$		0			0
$309$	−6.17457e10			−0.385295
$310$		0			0
$311$	−7.01012e10	+	1.21419e11i	−0.424917	+	0.735977i	−0.996413	−	0.0846275i	$-0.973030\pi$
$311$	−7.01012e10	+	1.21419e11i	−0.424917	+	0.735977i	0.571496	+	0.820605i	$0.306363\pi$
$312$		0			0
$313$	2.45677e10	−	1.41842e10i	0.144682	−	0.0835323i	−0.425911	−	0.904765i	$-0.640046\pi$
$313$	2.45677e10	−	1.41842e10i	0.144682	−	0.0835323i	0.570594	+	0.821233i	$0.306713\pi$
$314$		0			0
$315$	−3.80511e10	+	9.05497e10i	−0.217756	+	0.518191i
$316$		0			0
$317$	−2.23399e9	−	3.86938e9i	−0.0124255	−	0.0215216i	0.859746	−	0.510722i	$-0.170622\pi$
$317$	−2.23399e9	−	3.86938e9i	−0.0124255	−	0.0215216i	−0.872171	+	0.489201i	$0.837289\pi$
$318$		0			0
$319$	8.90720e9	+	5.14257e9i	0.0481596	+	0.0278050i
$320$		0			0
$321$			6.89418e10i			0.362418i
$322$		0			0
$323$			2.74475e10i			0.140311i
$324$		0			0
$325$	1.54848e10	+	8.94017e9i	0.0769894	+	0.0444499i
$326$		0			0
$327$	1.18235e11	+	2.04789e11i	0.571847	+	0.990469i
$328$		0			0
$329$	−1.36368e11	+	1.03535e11i	−0.641700	+	0.487196i
$330$		0			0
$331$	−6.98188e9	+	4.03099e9i	−0.0319703	+	0.0184580i	−0.515900	−	0.856649i	$-0.672542\pi$
$331$	−6.98188e9	+	4.03099e9i	−0.0319703	+	0.0184580i	0.483930	+	0.875107i	$0.339209\pi$
$332$		0			0
$333$	1.59727e10	−	2.76655e10i	0.0711835	−	0.123293i
$334$		0			0
$335$	−8.47985e10			−0.367863
$336$		0			0
$337$	−2.25624e11			−0.952909			−0.476454	−	0.879199i	$-0.658078\pi$
$337$	−2.25624e11			−0.952909			−0.476454	+	0.879199i	$0.658078\pi$
$338$		0			0
$339$	−9.41582e10	+	1.63087e11i	−0.387221	+	0.670687i
$340$		0			0
$341$	1.58657e11	−	9.16009e10i	0.635427	−	0.366864i
$342$		0			0
$343$	−9.44506e10	−	2.38310e11i	−0.368452	−	0.929647i
$344$		0			0
$345$	2.22115e11	+	3.84715e11i	0.844098	+	1.46202i
$346$		0			0
$347$	−1.52483e11	−	8.80360e10i	−0.564597	−	0.325970i	0.190392	−	0.981708i	$-0.439024\pi$
$347$	−1.52483e11	−	8.80360e10i	−0.564597	−	0.325970i	−0.754988	+	0.655738i	$0.772358\pi$
$348$		0			0
$349$		−	3.27852e11i		−	1.18294i	−0.806327	−	0.591470i	$-0.798548\pi$
$349$		−	3.27852e11i		−	1.18294i	0.806327	−	0.591470i	$-0.201452\pi$
$350$		0			0
$351$		−	4.33707e10i		−	0.152516i
$352$		0			0
$353$	2.30212e11	+	1.32913e11i	0.789117	+	0.455597i	0.839652	−	0.543125i	$-0.182759\pi$
$353$	2.30212e11	+	1.32913e11i	0.789117	+	0.455597i	−0.0505347	+	0.998722i	$0.516093\pi$
$354$		0			0
$355$	−3.17104e11	−	5.49240e11i	−1.05968	−	1.83542i
$356$		0			0
$357$	7.47959e10	+	9.85159e10i	0.243708	+	0.320996i
$358$		0			0
$359$	2.70703e11	−	1.56291e11i	0.860139	−	0.496602i	−0.00391966	−	0.999992i	$-0.501248\pi$
$359$	2.70703e11	−	1.56291e11i	0.860139	−	0.496602i	0.864059	+	0.503391i	$0.167914\pi$
$360$		0			0
$361$	1.32392e11	−	2.29310e11i	0.410280	−	0.710626i
$362$		0			0
$363$	−9.12984e11			−2.75984
$364$		0			0
$365$	−7.73928e11			−2.28235
$366$		0			0
$367$	1.67183e11	−	2.89569e11i	0.481055	−	0.833212i	−0.518708	−	0.854951i	$-0.673587\pi$
$367$	1.67183e11	−	2.89569e11i	0.481055	−	0.833212i	0.999764	+	0.0217392i	$0.00692033\pi$
$368$		0			0
$369$	−2.23494e11	+	1.29034e11i	−0.627548	+	0.362315i
$370$		0			0
$371$	−4.04398e11	−	1.69937e11i	−1.10822	−	0.465699i
$372$		0			0
$373$	7.56342e10	+	1.31002e11i	0.202315	+	0.350420i	0.949274	−	0.314450i	$-0.101820\pi$
$373$	7.56342e10	+	1.31002e11i	0.202315	+	0.350420i	−0.746959	+	0.664870i	$0.768487\pi$
$374$		0			0
$375$	2.98123e11	+	1.72121e11i	0.778493	+	0.449463i
$376$		0			0
$377$		−	2.91108e9i		−	0.00742195i
$378$		0			0
$379$			4.74193e11i			1.18053i	0.807208	+	0.590267i	$0.200978\pi$
$379$			4.74193e11i			1.18053i	−0.807208	+	0.590267i	$0.799022\pi$
$380$		0			0
$381$	−6.69699e11	−	3.86651e11i	−1.62824	−	0.940063i
$382$		0			0
$383$	−1.36095e11	−	2.35724e11i	−0.323184	−	0.559770i	0.657960	−	0.753053i	$-0.271420\pi$
$383$	−1.36095e11	−	2.35724e11i	−0.323184	−	0.559770i	−0.981143	+	0.193283i	$0.938086\pi$
$384$		0			0
$385$	1.14370e11	+	9.04477e11i	0.265301	+	2.09809i
$386$		0			0
$387$	1.17525e11	−	6.78533e10i	0.266337	−	0.153770i
$388$		0			0
$389$	5.63162e10	−	9.75424e10i	0.124698	−	0.215983i	−0.796917	−	0.604089i	$-0.793537\pi$
$389$	5.63162e10	−	9.75424e10i	0.124698	−	0.215983i	0.921615	+	0.388106i	$0.126870\pi$
$390$		0			0
$391$	1.81565e11			0.392860
$392$		0			0
$393$	−8.00699e11			−1.69318
$394$		0			0
$395$	−3.16759e11	+	5.48643e11i	−0.654700	+	1.13397i
$396$		0			0
$397$	−6.39536e11	+	3.69236e11i	−1.29213	+	0.746014i	−0.979032	−	0.203706i	$-0.934701\pi$
$397$	−6.39536e11	+	3.69236e11i	−1.29213	+	0.746014i	−0.313102	+	0.949720i	$0.601368\pi$
$398$		0			0
$399$	3.27349e10	+	2.58878e11i	0.0646595	+	0.511350i
$400$		0			0
$401$	−9.80108e10	−	1.69760e11i	−0.189288	−	0.327857i	0.755725	−	0.654889i	$-0.227285\pi$
$401$	−9.80108e10	−	1.69760e11i	−0.189288	−	0.327857i	−0.945013	+	0.327032i	$0.893951\pi$
$402$		0			0
$403$	−4.49060e10	−	2.59265e10i	−0.0848070	−	0.0489633i
$404$		0			0
$405$			7.91480e11i			1.46182i
$406$		0			0
$407$		−	2.96518e11i		−	0.535644i
$408$		0			0
$409$	6.86903e11	+	3.96584e11i	1.21378	+	0.700777i	0.963581	−	0.267417i	$-0.0861701\pi$
$409$	6.86903e11	+	3.96584e11i	1.21378	+	0.700777i	0.250201	+	0.968194i	$0.419503\pi$
$410$		0			0
$411$	−5.62062e11	−	9.73519e11i	−0.971619	−	1.68289i
$412$		0			0
$413$	−1.66273e11	−	6.98718e10i	−0.281220	−	0.118175i
$414$		0			0
$415$	7.92596e11	−	4.57606e11i	1.31170	−	0.757312i
$416$		0			0
$417$	6.18428e11	−	1.07115e12i	1.00156	−	1.73475i
$418$		0			0
$419$	−2.73830e11			−0.434029			−0.217014	−	0.976168i	$-0.569632\pi$
$419$	−2.73830e11			−0.434029			−0.217014	+	0.976168i	$0.569632\pi$
$420$		0			0
$421$	9.35590e11			1.45150			0.725749	−	0.687960i	$-0.241494\pi$
$421$	9.35590e11			1.45150			0.725749	+	0.687960i	$0.241494\pi$
$422$		0			0
$423$	1.27455e11	−	2.20759e11i	0.193565	−	0.335264i
$424$		0			0
$425$	−7.10877e10	+	4.10425e10i	−0.105693	+	0.0610216i
$426$		0			0
$427$	5.12516e11	+	6.75050e11i	0.746074	+	0.982676i
$428$		0			0
$429$	1.86168e11	+	3.22452e11i	0.265367	+	0.459629i
$430$		0			0
$431$	9.42075e11	+	5.43907e11i	1.31504	+	0.759237i	0.982926	−	0.184004i	$-0.0589058\pi$
$431$	9.42075e11	+	5.43907e11i	1.31504	+	0.759237i	0.332111	+	0.943240i	$0.392239\pi$
$432$		0			0
$433$		−	6.43180e11i		−	0.879300i	−0.898169	−	0.439650i	$-0.855102\pi$
$433$		−	6.43180e11i		−	0.879300i	0.898169	−	0.439650i	$-0.144898\pi$
$434$		0			0
$435$			3.26978e10i			0.0437841i
$436$		0			0
$437$	3.31712e11	+	1.91514e11i	0.435106	+	0.251208i
$438$		0			0
$439$	2.79648e11	+	4.84365e11i	0.359353	+	0.622418i	0.987853	−	0.155392i	$-0.0496640\pi$
$439$	2.79648e11	+	4.84365e11i	0.359353	+	0.622418i	−0.628500	+	0.777810i	$0.716331\pi$
$440$		0			0
$441$	2.67088e11	+	2.72613e11i	0.336264	+	0.343220i
$442$		0			0
$443$	1.01336e12	−	5.85066e11i	1.25011	−	0.721752i	0.278980	−	0.960297i	$-0.410004\pi$
$443$	1.01336e12	−	5.85066e11i	1.25011	−	0.721752i	0.971131	+	0.238545i	$0.0766704\pi$
$444$		0			0
$445$	−4.24694e11	+	7.35591e11i	−0.513400	+	0.889235i
$446$		0			0
$447$	7.12122e10			0.0843666
$448$		0			0
$449$	−1.60346e12			−1.86187			−0.930936	−	0.365181i	$-0.881007\pi$
$449$	−1.60346e12			−1.86187			−0.930936	+	0.365181i	$0.881007\pi$
$450$		0			0
$451$	−1.19770e12	+	2.07447e12i	−1.36318	+	2.36110i
$452$		0			0
$453$	1.82932e12	−	1.05616e12i	2.04102	−	1.17839i
$454$		0			0
$455$	2.05518e11	−	1.56034e11i	0.224801	−	0.170675i
$456$		0			0
$457$	−3.53869e11	−	6.12920e11i	−0.379507	−	0.657326i	0.611483	−	0.791257i	$-0.290573\pi$
$457$	−3.53869e11	−	6.12920e11i	−0.379507	−	0.657326i	−0.990991	+	0.133931i	$0.957240\pi$
$458$		0			0
$459$	1.72431e11	+	9.95531e10i	0.181325	+	0.104688i
$460$		0			0
$461$		−	6.31509e11i		−	0.651217i	−0.945505	−	0.325608i	$-0.894431\pi$
$461$		−	6.31509e11i		−	0.651217i	0.945505	−	0.325608i	$-0.105569\pi$
$462$		0			0
$463$			5.41413e11i			0.547538i	0.961795	+	0.273769i	$0.0882704\pi$
$463$			5.41413e11i			0.547538i	−0.961795	+	0.273769i	$0.911730\pi$
$464$		0			0
$465$	5.04392e11	+	2.91211e11i	0.500299	+	0.288848i
$466$		0			0
$467$	−1.88032e11	−	3.25681e11i	−0.182939	−	0.316859i	0.759941	−	0.649992i	$-0.225228\pi$
$467$	−1.88032e11	−	3.25681e11i	−0.182939	−	0.316859i	−0.942880	+	0.333133i	$0.891894\pi$
$468$		0			0
$469$	−1.27649e11	+	3.03765e11i	−0.121826	+	0.289907i
$470$		0			0
$471$	2.42027e11	−	1.39734e11i	0.226605	−	0.130830i
$472$		0			0
$473$	6.29816e11	−	1.09087e12i	0.578547	−	1.00207i
$474$		0			0
$475$	−1.73166e11			−0.156077
$476$		0			0
$477$	6.53067e11			0.577597
$478$		0			0
$479$	−3.31624e11	+	5.74389e11i	−0.287830	+	0.498536i	−0.973291	−	0.229573i	$-0.926267\pi$
$479$	−3.31624e11	+	5.74389e11i	−0.287830	+	0.498536i	0.685462	+	0.728109i	$0.259600\pi$
$480$		0			0
$481$	−7.26817e10	+	4.19628e10i	−0.0619117	+	0.0357447i
$482$		0			0
$483$	1.71248e12	−	2.16541e11i	1.43174	−	0.181042i
$484$		0			0
$485$	1.92459e11	+	3.33349e11i	0.157943	+	0.273566i
$486$		0			0
$487$	−1.75884e12	−	1.01546e12i	−1.41692	−	0.818059i	−0.420893	−	0.907110i	$-0.638283\pi$
$487$	−1.75884e12	−	1.01546e12i	−1.41692	−	0.818059i	−0.996027	+	0.0890514i	$0.971616\pi$
$488$		0			0
$489$		−	1.55382e12i		−	1.22888i
$490$		0			0
$491$		−	1.25834e12i		−	0.977086i	−0.872540	−	0.488543i	$-0.837529\pi$
$491$		−	1.25834e12i		−	0.977086i	0.872540	−	0.488543i	$-0.162471\pi$
$492$		0			0
$493$	1.15737e10	+	6.68209e9i	0.00882392	+	0.00509450i
$494$		0			0
$495$	−6.78655e11	−	1.17546e12i	−0.508072	−	0.880007i
$496$		0			0
$497$	−2.44483e12	+	3.09146e11i	−1.79740	+	0.227279i
$498$		0			0
$499$	−4.40134e10	+	2.54111e10i	−0.0317784	+	0.0183473i	−0.515805	−	0.856706i	$-0.672507\pi$
$499$	−4.40134e10	+	2.54111e10i	−0.0317784	+	0.0183473i	0.484027	+	0.875053i	$0.339174\pi$
$500$		0			0
$501$	4.45734e11	−	7.72034e11i	0.316087	−	0.547478i
$502$		0			0
$503$	1.48741e12			1.03604			0.518019	−	0.855369i	$-0.326670\pi$
$503$	1.48741e12			1.03604			0.518019	+	0.855369i	$0.326670\pi$
$504$		0			0
$505$	2.85263e12			1.95179
$506$		0			0
$507$	−8.52433e11	+	1.47646e12i	−0.572960	+	0.992397i
$508$		0			0
$509$	1.80197e12	−	1.04037e12i	1.18992	−	0.687000i	0.231630	−	0.972804i	$-0.425594\pi$
$509$	1.80197e12	−	1.04037e12i	1.18992	−	0.687000i	0.958288	+	0.285804i	$0.0922607\pi$
$510$		0			0
$511$	−1.16501e12	+	2.77236e12i	−0.755850	+	1.79869i
$512$		0			0
$513$	2.10016e11	+	3.63758e11i	0.133883	+	0.231891i
$514$		0			0
$515$	−5.12116e11	−	2.95670e11i	−0.320801	−	0.185215i
$516$		0			0
$517$		−	2.36608e12i		−	1.45654i
$518$		0			0
$519$		−	1.11758e12i		−	0.676121i
$520$		0			0
$521$	−2.19762e12	−	1.26880e12i	−1.30672	−	0.754437i	−0.325175	−	0.945654i	$-0.605423\pi$
$521$	−2.19762e12	−	1.26880e12i	−1.30672	−	0.754437i	−0.981548	+	0.191217i	$0.938757\pi$
$522$		0			0
$523$	−3.49782e11	−	6.05841e11i	−0.204428	−	0.354079i	0.745522	−	0.666481i	$-0.232200\pi$
$523$	−3.49782e11	−	6.05841e11i	−0.204428	−	0.354079i	−0.949950	+	0.312401i	$0.898867\pi$
$524$		0			0
$525$	−6.21533e11	+	4.71884e11i	−0.357065	+	0.271093i
$526$		0			0
$527$	2.06154e11	−	1.19023e11i	0.116425	−	0.0672178i
$528$		0			0
$529$	3.66289e11	−	6.34431e11i	0.203364	−	0.352236i
$530$		0			0
$531$	2.68517e11			0.146570
$532$		0			0
$533$	6.77987e11			0.363872
$534$		0			0
$535$	−3.30129e11	+	5.71800e11i	−0.174218	+	0.301754i
$536$		0			0
$537$	3.62225e12	−	2.09131e12i	1.87973	−	1.08526i
$538$		0			0
$539$	3.41218e12	+	9.51832e11i	1.74133	+	0.485748i
$540$		0			0
$541$	9.98576e11	+	1.72958e12i	0.501180	+	0.868069i	0.999999	+	0.00136264i	$0.000433743\pi$
$541$	9.98576e11	+	1.72958e12i	0.501180	+	0.868069i	−0.498819	+	0.866706i	$0.666233\pi$
$542$		0			0
$543$	−8.60034e11	−	4.96541e11i	−0.424538	−	0.245107i
$544$		0			0
$545$			2.26468e12i			1.09957i
$546$		0			0
$547$			3.30491e12i			1.57840i	0.614136	+	0.789200i	$0.289505\pi$
$547$			3.30491e12i			1.57840i	−0.614136	+	0.789200i	$0.710495\pi$
$548$		0			0
$549$	−1.09280e12	−	6.30927e11i	−0.513410	−	0.296417i
$550$		0			0
$551$	1.40965e10	+	2.44158e10i	0.00651520	+	0.0112847i
$552$		0			0
$553$	1.48852e12	+	1.96058e12i	0.676850	+	0.891499i
$554$		0			0
$555$	8.16374e11	−	4.71334e11i	0.365234	−	0.210868i
$556$		0			0
$557$	−1.34411e12	+	2.32807e12i	−0.591679	+	1.02482i	0.402327	+	0.915496i	$0.368202\pi$
$557$	−1.34411e12	+	2.32807e12i	−0.591679	+	1.02482i	−0.994006	+	0.109323i	$0.965132\pi$
$558$		0			0
$559$	−3.56523e11			−0.154431
$560$		0			0
$561$	−1.70932e12			−0.728601
$562$		0			0
$563$	1.00707e12	−	1.74430e12i	0.422448	−	0.731701i	−0.573731	−	0.819044i	$-0.694504\pi$
$563$	1.00707e12	−	1.74430e12i	0.422448	−	0.731701i	0.996178	+	0.0873431i	$0.0278377\pi$
$564$		0			0
$565$	−1.56189e12	+	9.01756e11i	−0.644810	+	0.372281i
$566$		0			0
$567$	2.83524e12	+	1.19143e12i	1.15204	+	0.484112i
$568$		0			0
$569$	3.83942e11	+	6.65007e11i	0.153554	+	0.265963i	0.932531	−	0.361089i	$-0.117595\pi$
$569$	3.83942e11	+	6.65007e11i	0.153554	+	0.265963i	−0.778978	+	0.627052i	$0.784262\pi$
$570$		0			0
$571$	2.33966e12	+	1.35081e12i	0.921066	+	0.531778i	0.883975	−	0.467534i	$-0.154857\pi$
$571$	2.33966e12	+	1.35081e12i	0.921066	+	0.531778i	0.0370913	+	0.999312i	$0.488191\pi$
$572$		0			0
$573$			4.84034e12i			1.87577i
$574$		0			0
$575$			1.14549e12i			0.437004i
$576$		0			0
$577$	−2.53310e12	−	1.46249e12i	−0.951396	−	0.549289i	−0.0578819	−	0.998323i	$-0.518435\pi$
$577$	−2.53310e12	−	1.46249e12i	−0.951396	−	0.549289i	−0.893514	+	0.449035i	$0.851768\pi$
$578$		0			0
$579$	−2.71533e12	−	4.70309e12i	−1.00408	−	1.73912i
$580$		0			0
$581$	−4.46121e11	−	3.52808e12i	−0.162428	−	1.28453i
$582$		0			0
$583$	5.24966e12	−	3.03089e12i	1.88201	−	1.08658i
$584$		0			0
$585$	−1.92085e11	+	3.32700e11i	−0.0678096	+	0.117450i
$586$		0			0
$587$	−2.60113e12			−0.904253			−0.452127	−	0.891954i	$-0.649334\pi$
$587$	−2.60113e12			−0.904253			−0.452127	+	0.891954i	$0.649334\pi$
$588$		0			0
$589$	5.02180e11			0.171926
$590$		0			0
$591$	−1.63924e12	+	2.83924e12i	−0.552711	+	0.957323i
$592$		0			0
$593$	−3.45745e12	+	1.99616e12i	−1.14818	+	0.662901i	−0.948443	−	0.316949i	$-0.897342\pi$
$593$	−3.45745e12	+	1.99616e12i	−1.14818	+	0.662901i	−0.199736	+	0.979850i	$0.564008\pi$
$594$		0			0
$595$	1.48609e11	+	1.17525e12i	0.0486091	+	0.384417i
$596$		0			0
$597$	1.65995e12	+	2.87511e12i	0.534822	+	0.926340i
$598$		0			0
$599$	−3.50447e12	−	2.02331e12i	−1.11225	−	0.642156i	−0.172837	−	0.984950i	$-0.555293\pi$
$599$	−3.50447e12	−	2.02331e12i	−1.11225	−	0.642156i	−0.939411	+	0.342794i	$0.888627\pi$
$600$		0			0
$601$			1.98218e12i			0.619737i	0.950779	+	0.309869i	$0.100285\pi$
$601$			1.98218e12i			0.619737i	−0.950779	+	0.309869i	$0.899715\pi$
$602$		0			0
$603$		−	4.90553e11i		−	0.151098i
$604$		0			0
$605$	−7.57225e12	−	4.37184e12i	−2.29787	−	1.32668i
$606$		0			0
$607$	−2.18180e12	−	3.77898e12i	−0.652326	−	1.12986i	−0.982557	−	0.185962i	$-0.940460\pi$
$607$	−2.18180e12	−	3.77898e12i	−0.652326	−	1.12986i	0.330230	−	0.943900i	$-0.392874\pi$
$608$		0			0
$609$	1.17130e11	+	4.92206e10i	0.0345056	+	0.0145000i
$610$		0			0
$611$	−5.79969e11	+	3.34845e11i	−0.168352	+	0.0971983i
$612$		0			0
$613$	−1.60668e12	+	2.78285e12i	−0.459576	+	0.796010i	−0.998938	−	0.0460643i	$-0.985332\pi$
$613$	−1.60668e12	+	2.78285e12i	−0.459576	+	0.796010i	0.539362	+	0.842074i	$0.318665\pi$
$614$		0			0
$615$	−7.61527e12			−2.14658
$616$		0			0
$617$	−5.12033e12			−1.42238			−0.711189	−	0.703001i	$-0.751843\pi$
$617$	−5.12033e12			−1.42238			−0.711189	+	0.703001i	$0.751843\pi$
$618$		0			0
$619$	1.30204e12	−	2.25520e12i	0.356464	−	0.617414i	−0.630903	−	0.775862i	$-0.717315\pi$
$619$	1.30204e12	−	2.25520e12i	0.356464	−	0.617414i	0.987367	+	0.158447i	$0.0506488\pi$
$620$		0			0
$621$	2.40626e12	−	1.38925e12i	0.649277	−	0.374860i
$622$		0			0
$623$	1.99573e12	+	2.62864e12i	0.530769	+	0.699092i
$624$		0			0
$625$	2.35118e12	+	4.07236e12i	0.616348	+	1.06755i
$626$		0			0
$627$	−3.12285e12	−	1.80298e12i	−0.806951	−	0.465893i
$628$		0			0
$629$		−	3.85286e11i		−	0.0981420i
$630$		0			0
$631$			1.69466e12i			0.425549i	0.977101	+	0.212774i	$0.0682500\pi$
$631$			1.69466e12i			0.425549i	−0.977101	+	0.212774i	$0.931750\pi$
$632$		0			0
$633$	2.61637e12	+	1.51056e12i	0.647712	+	0.373957i
$634$		0			0
$635$	−3.70297e12	−	6.41373e12i	−0.903792	−	1.56541i
$636$		0			0
$637$	−2.49576e11	−	9.71086e11i	−0.0600586	−	0.233685i
$638$		0			0
$639$	3.17732e12	−	1.83442e12i	0.753888	−	0.435257i
$640$		0			0
$641$	−9.15438e11	+	1.58559e12i	−0.214175	+	0.370961i	−0.953017	−	0.302917i	$-0.902039\pi$
$641$	−9.15438e11	+	1.58559e12i	−0.214175	+	0.370961i	0.738842	+	0.673878i	$0.235373\pi$
$642$		0			0
$643$	4.26517e12			0.983982			0.491991	−	0.870600i	$-0.336269\pi$
$643$	4.26517e12			0.983982			0.491991	+	0.870600i	$0.336269\pi$
$644$		0			0
$645$	4.00453e12			0.911029
$646$		0			0
$647$	−2.48457e12	+	4.30340e12i	−0.557419	+	0.965478i	0.440292	+	0.897855i	$0.354875\pi$
$647$	−2.48457e12	+	4.30340e12i	−0.557419	+	0.965478i	−0.997711	+	0.0676232i	$0.978458\pi$
$648$		0			0
$649$	2.15846e12	−	1.24619e12i	0.477577	−	0.275729i
$650$		0			0
$651$	1.80245e12	−	1.36846e12i	0.393322	−	0.298620i
$652$		0			0
$653$	5.01849e11	+	8.69228e11i	0.108010	+	0.187079i	0.914964	−	0.403536i	$-0.132219\pi$
$653$	5.01849e11	+	8.69228e11i	0.108010	+	0.187079i	−0.806954	+	0.590614i	$0.798886\pi$
$654$		0			0
$655$	−6.64096e12	−	3.83416e12i	−1.40976	−	0.813925i
$656$		0			0
$657$		−	4.47712e12i		−	0.937464i
$658$		0			0
$659$			4.12361e12i			0.851714i	0.904791	+	0.425857i	$0.140027\pi$
$659$			4.12361e12i			0.851714i	−0.904791	+	0.425857i	$0.859973\pi$
$660$		0			0
$661$	−6.67049e12	−	3.85121e12i	−1.35910	−	0.784676i	−0.369596	−	0.929193i	$-0.620504\pi$
$661$	−6.67049e12	−	3.85121e12i	−1.35910	−	0.784676i	−0.989502	+	0.144517i	$0.953837\pi$
$662$		0			0
$663$	2.41900e11	+	4.18984e11i	0.0486212	+	0.0842144i
$664$		0			0
$665$	−9.68142e11	+	2.30388e12i	−0.191974	+	0.456838i
$666$		0			0
$667$	1.61510e11	−	9.32480e10i	0.0315961	−	0.0182420i
$668$		0			0
$669$	−3.63883e12	+	6.30264e12i	−0.702335	+	1.21648i
$670$		0			0
$671$	−1.17126e13			−2.23049
$672$		0			0
$673$	5.08885e12			0.956206			0.478103	−	0.878304i	$-0.341325\pi$
$673$	5.08885e12			0.956206			0.478103	+	0.878304i	$0.341325\pi$
$674$		0			0
$675$	−6.28077e11	+	1.08786e12i	−0.116452	+	0.201700i
$676$		0			0
$677$	2.84738e12	−	1.64394e12i	0.520951	−	0.300771i	−0.216373	−	0.976311i	$-0.569423\pi$
$677$	2.84738e12	−	1.64394e12i	0.520951	−	0.300771i	0.737324	+	0.675540i	$0.236089\pi$
$678$		0			0
$679$	1.48383e12	−	1.87629e11i	0.267899	−	0.0338755i
$680$		0			0
$681$	3.40356e12	+	5.89515e12i	0.606418	+	1.05035i
$682$		0			0
$683$	6.16040e12	+	3.55671e12i	1.08322	+	0.625396i	0.931763	−	0.363068i	$-0.118271\pi$
$683$	6.16040e12	+	3.55671e12i	1.08322	+	0.625396i	0.151455	+	0.988464i	$0.451604\pi$
$684$		0			0
$685$		−	1.07658e13i		−	1.86826i
$686$		0			0
$687$			1.11466e13i			1.90914i
$688$		0			0
$689$	−1.48585e12	−	8.57856e11i	−0.251182	−	0.145020i
$690$		0			0
$691$	1.42639e12	+	2.47057e12i	0.238005	+	0.412236i	0.960142	−	0.279514i	$-0.0901733\pi$
$691$	1.42639e12	+	2.47057e12i	0.238005	+	0.412236i	−0.722137	+	0.691750i	$0.756840\pi$
$692$		0			0
$693$	−5.23234e12	+	6.61623e11i	−0.861780	+	0.108971i
$694$		0			0
$695$	1.02584e13	−	5.92271e12i	1.66782	−	0.962917i
$696$		0			0
$697$	−1.55625e12	+	2.69550e12i	−0.249765	+	0.432606i
$698$		0			0
$699$	1.15822e12			0.183504
$700$		0			0
$701$	−1.02186e13			−1.59831			−0.799153	−	0.601127i	$-0.794718\pi$
$701$	−1.02186e13			−1.59831			−0.799153	+	0.601127i	$0.794718\pi$
$702$		0			0
$703$	4.06397e11	−	7.03900e11i	0.0627555	−	0.108696i
$704$		0			0
$705$	6.51431e12	−	3.76104e12i	0.993156	−	0.573399i
$706$		0			0
$707$	4.29412e12	−	1.02187e13i	0.646378	−	1.53818i
$708$		0			0
$709$	4.08353e12	+	7.07288e12i	0.606915	+	1.05121i	0.991746	+	0.128220i	$0.0409264\pi$
$709$	4.08353e12	+	7.07288e12i	0.606915	+	1.05121i	−0.384831	+	0.922987i	$0.625740\pi$
$710$		0			0
$711$	−3.17386e12	−	1.83243e12i	−0.465774	−	0.268915i
$712$		0			0
$713$		−	3.32192e12i		−	0.481378i
$714$		0			0
$715$			3.56587e12i			0.510256i
$716$		0			0
$717$	4.98792e12	+	2.87977e12i	0.704828	+	0.406932i
$718$		0			0
$719$	−1.13356e12	−	1.96338e12i	−0.158185	−	0.273984i	0.776029	−	0.630697i	$-0.217231\pi$
$719$	−1.13356e12	−	1.96338e12i	−0.158185	−	0.273984i	−0.934214	+	0.356713i	$0.883897\pi$
$720$		0			0
$721$	−1.83005e12	+	1.38942e12i	−0.252205	+	0.191481i
$722$		0			0
$723$	1.20290e13	−	6.94496e12i	1.63722	−	0.945252i
$724$		0			0
$725$	−4.21571e10	+	7.30182e10i	−0.00566695	+	0.00981544i
$726$		0			0
$727$	−1.26124e13			−1.67453			−0.837267	−	0.546794i	$-0.815848\pi$
$727$	−1.26124e13			−1.67453			−0.837267	+	0.546794i	$0.815848\pi$
$728$		0			0
$729$	1.28639e12			0.168693
$730$		0			0
$731$	8.18362e11	−	1.41744e12i	0.106003	−	0.183602i
$732$		0			0
$733$	6.36963e12	−	3.67751e12i	0.814978	−	0.470528i	−0.0337033	−	0.999432i	$-0.510730\pi$
$733$	6.36963e12	−	3.67751e12i	0.814978	−	0.470528i	0.848682	+	0.528904i	$0.177397\pi$
$734$		0			0
$735$	2.80328e12	+	1.09074e13i	0.354302	+	1.37857i
$736$		0			0
$737$	−2.27666e12	−	3.94330e12i	−0.284247	−	0.492330i
$738$		0			0
$739$	4.72198e12	+	2.72624e12i	0.582403	+	0.336251i	0.762088	−	0.647473i	$-0.224174\pi$
$739$	4.72198e12	+	2.72624e12i	0.582403	+	0.336251i	−0.179685	+	0.983724i	$0.557508\pi$
$740$		0			0
$741$			1.02062e12i			0.124360i
$742$		0			0
$743$		−	1.39812e13i		−	1.68304i	−0.540229	−	0.841518i	$-0.681662\pi$
$743$		−	1.39812e13i		−	1.68304i	0.540229	−	0.841518i	$-0.318338\pi$
$744$		0			0
$745$	5.90631e11	+	3.41001e11i	0.0702446	+	0.0405557i
$746$		0			0
$747$	2.64722e12	+	4.58512e12i	0.311062	+	0.538775i
$748$		0			0
$749$	1.55135e12	+	2.04333e12i	0.180112	+	0.237230i
$750$		0			0
$751$	2.51802e12	−	1.45378e12i	0.288855	−	0.166771i	−0.348570	−	0.937283i	$-0.613333\pi$
$751$	2.51802e12	−	1.45378e12i	0.288855	−	0.166771i	0.637425	+	0.770512i	$0.279999\pi$
$752$		0			0
$753$	1.66528e12	−	2.88436e12i	0.188760	−	0.326943i
$754$		0			0
$755$	2.02297e13			2.26584
$756$		0			0
$757$	2.43736e12			0.269767			0.134883	−	0.990861i	$-0.456934\pi$
$757$	2.43736e12			0.269767			0.134883	+	0.990861i	$0.456934\pi$
$758$		0			0
$759$	−1.19267e13	+	2.06576e13i	−1.30446	+	2.25940i
$760$		0			0
$761$	2.41393e11	−	1.39369e11i	0.0260912	−	0.0150638i	−0.486898	−	0.873459i	$-0.661872\pi$
$761$	2.41393e11	−	1.39369e11i	0.0260912	−	0.0150638i	0.512989	+	0.858395i	$0.328538\pi$
$762$		0			0
$763$	8.11251e12	+	3.40906e12i	0.866552	+	0.364145i
$764$		0			0
$765$	−8.81822e11	−	1.52736e12i	−0.0930903	−	0.161237i
$766$		0			0
$767$	−6.10925e11	−	3.52718e11i	−0.0637396	−	0.0368001i
$768$		0			0
$769$		−	1.07508e12i		−	0.110859i	−0.998463	−	0.0554296i	$-0.982347\pi$
$769$		−	1.07508e12i		−	0.110859i	0.998463	−	0.0554296i	$-0.0176528\pi$
$770$		0			0
$771$			1.56648e13i			1.59654i
$772$		0			0
$773$	1.21102e12	+	6.99185e11i	0.121996	+	0.0704343i	0.559756	−	0.828657i	$-0.310895\pi$
$773$	1.21102e12	+	6.99185e11i	0.121996	+	0.0704343i	−0.437760	+	0.899092i	$0.644228\pi$
$774$		0			0
$775$	7.50913e11	+	1.30062e12i	0.0747708	+	0.129507i
$776$		0			0
$777$	−4.59505e11	−	3.63392e12i	−0.0452268	−	0.357669i
$778$		0			0
$779$	−5.68640e12	+	3.28305e12i	−0.553247	+	0.319417i
$780$		0			0
$781$	1.70272e13	−	2.94919e13i	1.63762	−	2.83644i
$782$		0			0
$783$	2.04513e11			0.0194443
$784$		0			0
$785$	2.67648e12			0.251565
$786$		0			0
$787$	−4.35554e12	+	7.54402e12i	−0.404721	+	0.700997i	−0.994289	−	0.106722i	$-0.965965\pi$
$787$	−4.35554e12	+	7.54402e12i	−0.404721	+	0.700997i	0.589568	+	0.807719i	$0.299298\pi$
$788$		0			0
$789$	2.04336e12	−	1.17973e12i	0.187715	−	0.108377i
$790$		0			0
$791$	8.79125e11	+	6.95242e12i	0.0798466	+	0.631454i
$792$		0			0
$793$	1.65755e12	+	2.87096e12i	0.148846	+	0.257808i
$794$		0			0
$795$	1.66893e13	+	9.63558e12i	1.48179	+	0.855512i
$796$		0			0
$797$		−	2.41480e12i		−	0.211992i	−0.994367	−	0.105996i	$-0.966197\pi$
$797$		−	2.41480e12i		−	0.211992i	0.994367	−	0.105996i	$-0.0338031\pi$
$798$		0			0
$799$		−	3.07441e12i		−	0.266871i
$800$		0			0
$801$	−4.25535e12	−	2.45683e12i	−0.365249	−	0.210876i
$802$		0			0
$803$	−2.07784e13	−	3.59892e13i	−1.76357	−	3.05459i
$804$		0			0
$805$	1.52401e13	+	6.40426e12i	1.27911	+	0.537511i
$806$		0			0
$807$	1.59590e12	−	9.21395e11i	0.132457	−	0.0764742i
$808$		0			0
$809$	3.03123e12	−	5.25024e12i	0.248800	−	0.430934i	−0.714393	−	0.699744i	$-0.753297\pi$
$809$	3.03123e12	−	5.25024e12i	0.248800	−	0.430934i	0.963193	+	0.268810i	$0.0866305\pi$
$810$		0			0
$811$	−2.03991e13			−1.65584			−0.827919	−	0.560848i	$-0.810475\pi$
$811$	−2.03991e13			−1.65584			−0.827919	+	0.560848i	$0.810475\pi$
$812$		0			0
$813$	8.28728e12			0.665280
$814$		0			0
$815$	7.44048e12	−	1.28873e13i	0.590733	−	1.02318i
$816$		0			0
$817$	2.99022e12	−	1.72641e12i	0.234803	−	0.135564i
$818$		0			0
$819$	9.02649e11	+	1.18891e12i	0.0701038	+	0.0923358i
$820$		0			0
$821$	−9.17133e12	−	1.58852e13i	−0.704511	−	1.22025i	−0.966868	−	0.255278i	$-0.917833\pi$
$821$	−9.17133e12	−	1.58852e13i	−0.704511	−	1.22025i	0.262356	−	0.964971i	$-0.415500\pi$
$822$		0			0
$823$	−8.75404e12	−	5.05415e12i	−0.665135	−	0.384016i	0.129096	−	0.991632i	$-0.458792\pi$
$823$	−8.75404e12	−	5.05415e12i	−0.665135	−	0.384016i	−0.794231	+	0.607617i	$0.792126\pi$
$824$		0			0
$825$		−	1.07840e13i		−	0.810472i
$826$		0			0
$827$		−	1.17899e13i		−	0.876465i	−0.898862	−	0.438232i	$-0.855605\pi$
$827$		−	1.17899e13i		−	0.876465i	0.898862	−	0.438232i	$-0.144395\pi$
$828$		0			0
$829$	1.68645e13	+	9.73674e12i	1.24016	+	0.716008i	0.969127	−	0.246561i	$-0.0793005\pi$
$829$	1.68645e13	+	9.73674e12i	1.24016	+	0.716008i	0.271036	+	0.962569i	$0.412634\pi$
$830$		0			0
$831$	−1.13522e12	−	1.96626e12i	−0.0825801	−	0.143033i
$832$		0			0
$833$	4.43367e12	+	1.23678e12i	0.319052	+	0.0890000i
$834$		0			0
$835$	7.39380e12	−	4.26881e12i	0.526355	−	0.303891i
$836$		0			0
$837$	1.82142e12	−	3.15479e12i	0.128276	−	0.222181i
$838$		0			0
$839$	1.86414e12			0.129882			0.0649412	−	0.997889i	$-0.479314\pi$
$839$	1.86414e12			0.129882			0.0649412	+	0.997889i	$0.479314\pi$
$840$		0			0
$841$	−1.44934e13			−0.999054
$842$		0			0
$843$	−8.75778e12	+	1.51689e13i	−0.597269	+	1.03450i
$844$		0			0
$845$	−1.41401e13	+	8.16378e12i	−0.954107	+	0.550854i
$846$		0			0
$847$	−2.70594e13	+	2.05442e13i	−1.80652	+	1.37156i
$848$		0			0
$849$	−3.42569e12	−	5.93346e12i	−0.226289	−	0.391944i
$850$		0			0
$851$	−4.65630e12	−	2.68831e12i	−0.304339	−	0.175710i
$852$		0			0
$853$			2.83795e13i			1.83542i	0.397255	+	0.917708i	$0.369963\pi$
$853$			2.83795e13i			1.83542i	−0.397255	+	0.917708i	$0.630037\pi$
$854$		0			0
$855$		−	3.72056e12i		−	0.238101i
$856$		0			0
$857$	−2.45848e13	−	1.41940e13i	−1.55687	−	0.898860i	−0.997554	−	0.0699060i	$-0.977730\pi$
$857$	−2.45848e13	−	1.41940e13i	−1.55687	−	0.898860i	−0.559317	−	0.828954i	$-0.688937\pi$
$858$		0			0
$859$	2.96769e12	+	5.14019e12i	0.185973	+	0.322114i	0.943904	−	0.330220i	$-0.107123\pi$
$859$	2.96769e12	+	5.14019e12i	0.185973	+	0.322114i	−0.757931	+	0.652335i	$0.773790\pi$
$860$		0			0
$861$	−1.14634e13	+	2.72794e13i	−0.710886	+	1.69169i
$862$		0			0
$863$	1.13888e13	−	6.57533e12i	0.698923	−	0.403523i	−0.108023	−	0.994148i	$-0.534452\pi$
$863$	1.13888e13	−	6.57533e12i	0.698923	−	0.403523i	0.806946	+	0.590625i	$0.201119\pi$
$864$		0			0
$865$	5.35154e12	−	9.26914e12i	0.325017	−	0.562946i
$866$		0			0
$867$	1.80226e13			1.08326
$868$		0			0
$869$	−3.40173e13			−2.02354
$870$		0			0
$871$	−6.44381e11	+	1.11610e12i	−0.0379368	+	0.0657085i
$872$		0			0
$873$	−1.92840e12	+	1.11336e12i	−0.112366	+	0.0648743i
$874$		0			0
$875$	1.27090e13	−	1.60704e12i	0.732954	−	0.0926811i
$876$		0			0
$877$	1.38775e13	+	2.40365e13i	0.792161	+	1.37206i	0.924627	+	0.380875i	$0.124377\pi$
$877$	1.38775e13	+	2.40365e13i	0.792161	+	1.37206i	−0.132466	+	0.991188i	$0.542290\pi$
$878$		0			0
$879$	6.69202e12	+	3.86364e12i	0.378101	+	0.218297i
$880$		0			0
$881$		−	7.81392e11i		−	0.0436996i	−0.999761	−	0.0218498i	$-0.993044\pi$
$881$		−	7.81392e11i		−	0.0436996i	0.999761	−	0.0218498i	$-0.00695556\pi$
$882$		0			0
$883$		−	2.74212e13i		−	1.51797i	−0.651108	−	0.758985i	$-0.725696\pi$
$883$		−	2.74212e13i		−	1.51797i	0.651108	−	0.758985i	$-0.274304\pi$
$884$		0			0
$885$	6.86202e12	+	3.96179e12i	0.376017	+	0.217093i
$886$		0			0
$887$	7.82207e12	+	1.35482e13i	0.424293	+	0.734897i	0.996354	−	0.0853142i	$-0.0271894\pi$
$887$	7.82207e12	+	1.35482e13i	0.424293	+	0.734897i	−0.572061	+	0.820211i	$0.693856\pi$
$888$		0			0
$889$	−2.85494e13	+	3.61004e12i	−1.53299	+	0.193845i
$890$		0			0
$891$	−3.68054e13	+	2.12496e13i	−1.95642	+	1.12954i
$892$		0			0
$893$	3.24287e12	−	5.61682e12i	0.170647	−	0.295569i
$894$		0			0
$895$	4.00571e13			2.08677
$896$		0			0
$897$	6.75140e12			0.348199
$898$		0			0
$899$	1.22255e11	−	2.11753e11i	0.00624237	−	0.0108121i
$900$		0			0
$901$	6.82123e12	−	3.93824e12i	0.344827	−	0.199086i
$902$		0			0
$903$	6.02810e12	−	1.43450e13i	0.301707	−	0.717969i
$904$		0			0
$905$	−4.75539e12	−	8.23657e12i	−0.235650	−	0.408158i
$906$		0			0
$907$	2.35999e12	+	1.36254e12i	0.115792	+	0.0668524i	0.556777	−	0.830662i	$-0.312038\pi$
$907$	2.35999e12	+	1.36254e12i	0.115792	+	0.0668524i	−0.440986	+	0.897514i	$0.645371\pi$
$908$		0			0
$909$			1.65023e13i			0.801689i
$910$		0			0
$911$		−	2.33160e12i		−	0.112156i	−0.998426	−	0.0560779i	$-0.982140\pi$
$911$		−	2.33160e12i		−	0.112156i	0.998426	−	0.0560779i	$-0.0178595\pi$
$912$		0			0
$913$	4.25592e13	+	2.45715e13i	2.02710	+	1.17035i
$914$		0			0
$915$	−1.86179e13	−	3.22471e13i	−0.878081	−	1.52088i
$916$		0			0
$917$	−2.37315e13	+	1.80176e13i	−1.10831	+	0.841462i
$918$		0			0
$919$	1.10859e13	−	6.40045e12i	0.512686	−	0.295999i	−0.221251	−	0.975217i	$-0.571014\pi$
$919$	1.10859e13	−	6.40045e12i	0.512686	−	0.295999i	0.733937	+	0.679217i	$0.237681\pi$
$920$		0			0
$921$	−8.88816e12	+	1.53947e13i	−0.407046	+	0.705025i
$922$		0			0
$923$	−9.63865e12			−0.437128
$924$		0			0
$925$	2.43075e12			0.109170
$926$		0			0
$927$	1.71043e12	−	2.96256e12i	0.0760759	−	0.131767i
$928$		0			0
$929$	−6.95688e12	+	4.01655e12i	−0.306439	+	0.176922i	−0.645332	−	0.763902i	$-0.723281\pi$
$929$	−6.95688e12	+	4.01655e12i	−0.306439	+	0.176922i	0.338893	+	0.940825i	$0.389948\pi$
$930$		0			0
$931$	6.79558e12	+	6.93615e12i	0.296451	+	0.302583i
$932$		0			0
$933$	−1.19667e13	−	2.07269e13i	−0.517020	−	0.895504i
$934$		0			0
$935$	−1.41770e13	−	8.18509e12i	−0.606641	−	0.350245i
$936$		0			0
$937$			1.66323e13i			0.704893i	0.935832	+	0.352446i	$0.114650\pi$
$937$			1.66323e13i			0.704893i	−0.935832	+	0.352446i	$0.885350\pi$
$938$		0			0
$939$			4.84265e12i			0.203277i
$940$		0			0
$941$	−2.01805e12	−	1.16512e12i	−0.0839031	−	0.0484415i	0.457461	−	0.889229i	$-0.348759\pi$
$941$	−2.01805e12	−	1.16512e12i	−0.0839031	−	0.0484415i	−0.541365	+	0.840788i	$0.682092\pi$
$942$		0			0
$943$	2.17173e13	+	3.76155e13i	0.894343	+	1.54905i
$944$		0			0
$945$	1.09619e13	+	1.44383e13i	0.447141	+	0.588942i
$946$		0			0
$947$	−2.69273e12	+	1.55465e12i	−0.108797	+	0.0628141i	−0.553411	−	0.832908i	$-0.686674\pi$
$947$	−2.69273e12	+	1.55465e12i	−0.108797	+	0.0628141i	0.444614	+	0.895722i	$0.353341\pi$
$948$		0			0
$949$	−5.88106e12	+	1.01863e13i	−0.235374	+	0.407679i
$950$		0			0
$951$	7.62710e11			0.0302376
$952$		0			0
$953$	−7.44898e12			−0.292535			−0.146268	−	0.989245i	$-0.546726\pi$
$953$	−7.44898e12			−0.292535			−0.146268	+	0.989245i	$0.546726\pi$
$954$		0			0
$955$	−2.31780e13	+	4.01455e13i	−0.901698	+	1.56179i
$956$		0			0
$957$	−1.52051e12	+	8.77868e11i	−0.0585984	+	0.0338318i
$958$		0			0
$959$	−3.85651e13	−	1.62059e13i	−1.47235	−	0.618715i
$960$		0			0
$961$	1.10422e13	+	1.91256e13i	0.417637	+	0.723368i
$962$		0			0
$963$	−3.30783e12	−	1.90978e12i	−0.123944	−	0.0715590i
$964$		0			0
$965$		−	5.20097e13i		−	1.93068i
$966$		0			0
$967$		−	4.67375e13i		−	1.71888i	−0.511233	−	0.859442i	$-0.670811\pi$
$967$		−	4.67375e13i		−	1.71888i	0.511233	−	0.859442i	$-0.329189\pi$
$968$		0			0
$969$	−4.05773e12	−	2.34273e12i	−0.147852	−	0.0853621i
$970$		0			0
$971$	−4.16152e12	−	7.20797e12i	−0.150233	−	0.260212i	0.781080	−	0.624431i	$-0.214669\pi$
$971$	−4.16152e12	−	7.20797e12i	−0.150233	−	0.260212i	−0.931313	+	0.364220i	$0.881336\pi$
$972$		0			0
$973$	−5.77407e12	−	4.56633e13i	−0.206526	−	1.63328i
$974$		0			0
$975$	−2.64335e12	+	1.52614e12i	−0.0936773	+	0.0540846i
$976$		0			0
$977$	−2.18149e13	+	3.77845e13i	−0.765998	+	1.32675i	0.173720	+	0.984795i	$0.444421\pi$
$977$	−2.18149e13	+	3.77845e13i	−0.765998	+	1.32675i	−0.939718	+	0.341952i	$0.888912\pi$
$978$		0			0
$979$	−4.56086e13			−1.58681
$980$		0			0
$981$	−1.31010e13			−0.451641
$982$		0			0
$983$	1.53980e13	−	2.66702e13i	0.525986	−	0.911034i	−0.473556	−	0.880764i	$-0.657030\pi$
$983$	1.53980e13	−	2.66702e13i	0.525986	−	0.911034i	0.999542	−	0.0302705i	$-0.00963688\pi$
$984$		0			0
$985$	−2.71915e13	+	1.56990e13i	−0.920386	+	0.531385i
$986$		0			0
$987$	−3.66665e12	−	2.89971e13i	−0.122982	−	0.972585i
$988$		0			0
$989$	−1.14202e13	−	1.97803e13i	−0.379568	−	0.657431i
$990$		0			0
$991$	−2.79061e13	−	1.61116e13i	−0.919109	−	0.530648i	−0.0357585	−	0.999360i	$-0.511385\pi$
$991$	−2.79061e13	−	1.61116e13i	−0.919109	−	0.530648i	−0.883351	+	0.468712i	$0.844718\pi$
$992$		0			0
$993$		−	1.37623e12i		−	0.0449178i
$994$		0			0
$995$			3.17947e13i			1.02837i
$996$		0			0
$997$	2.01141e13	+	1.16129e13i	0.644723	+	0.372231i	0.786432	−	0.617677i	$-0.211926\pi$
$997$	2.01141e13	+	1.16129e13i	0.644723	+	0.372231i	−0.141708	+	0.989908i	$0.545260\pi$
$998$		0			0
$999$	−2.94803e12	−	5.10613e12i	−0.0936455	−	0.162199i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.10.p.c.31.3	yes	24
4.3	odd	2		112.10.p.a.31.10	✓	24
7.5	odd	6		112.10.p.a.47.10	yes	24
28.19	even	6	inner	112.10.p.c.47.3	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.10.p.a.31.10	✓	24	4.3	odd	2
112.10.p.a.47.10	yes	24	7.5	odd	6
112.10.p.c.31.3	yes	24	1.1	even	1	trivial
112.10.p.c.47.3	yes	24	28.19	even	6	inner

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:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	112.p (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-85.3530 + 147.836i) q^{3} +(-1415.83 + 817.429i) q^{5} +(796.914 + 6302.26i) q^{7} +(-4728.77 - 8190.47i) q^{9} +(-76024.2 - 43892.6i) q^{11} +24846.5i q^{13} -279080. i q^{15} +(-98783.3 - 57032.6i) q^{17} +(-120315. - 208392. i) q^{19} +(-999719. - 420105. i) q^{21} +(-1.37851e6 + 795885. i) q^{23} +(359816. - 623220. i) q^{25} -1.74555e6 q^{27} -117163. q^{29} +(-1.04347e6 + 1.80734e6i) q^{31} +(1.29778e7 - 7.49272e6i) q^{33} +(-6.27994e6 - 8.27150e6i) q^{35} +(1.68888e6 + 2.92523e6i) q^{37} +(-3.67320e6 - 2.12072e6i) q^{39} -2.72870e7i q^{41} +1.43490e7i q^{43} +(1.33902e7 + 7.73086e6i) q^{45} +(1.34766e7 + 2.33421e7i) q^{47} +(-3.90835e7 + 1.00447e7i) q^{49} +(1.68629e7 - 9.73580e6i) q^{51} +(-3.45262e7 + 5.98012e7i) q^{53} +1.43516e8 q^{55} +4.10771e7 q^{57} +(-1.41959e7 + 2.45880e7i) q^{59} +(1.15548e8 - 6.67116e7i) q^{61} +(4.78501e7 - 3.63291e7i) q^{63} +(-2.03102e7 - 3.51783e7i) q^{65} +(4.49199e7 + 2.59345e7i) q^{67} -2.71725e8i q^{69} +3.87928e8i q^{71} +(4.09969e8 + 2.36696e8i) q^{73} +(6.14228e7 + 1.06387e8i) q^{75} +(2.16038e8 - 5.14103e8i) q^{77} +(3.35591e8 - 1.93753e8i) q^{79} +(2.42064e8 - 4.19267e8i) q^{81} -5.59811e8 q^{83} +1.86480e8 q^{85} +(1.00002e7 - 1.73208e7i) q^{87} +(4.49942e8 - 2.59774e8i) q^{89} +(-1.56589e8 + 1.98005e7i) q^{91} +(-1.78126e8 - 3.08524e8i) q^{93} +(3.40691e8 + 1.96698e8i) q^{95} -2.35445e8i q^{97} +8.30231e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 162 q^{3} - 852 q^{5} + 6744 q^{7} - 77884 q^{9} + 57534 q^{11} + 789336 q^{17} - 469098 q^{19} - 2104376 q^{21} - 1553682 q^{23} + 3602544 q^{25} - 6389244 q^{27} - 2462040 q^{29} + 10306686 q^{31}+ \cdots - 1433917218 q^{95}+O(q^{100}) \) 24 * q + 162 * q^3 - 852 * q^5 + 6744 * q^7 - 77884 * q^9 + 57534 * q^11 + 789336 * q^17 - 469098 * q^19 - 2104376 * q^21 - 1553682 * q^23 + 3602544 * q^25 - 6389244 * q^27 - 2462040 * q^29 + 10306686 * q^31 + 8789748 * q^33 - 21149802 * q^35 + 10422720 * q^37 - 54458664 * q^39 + 2549604 * q^45 - 12751230 * q^47 - 16137504 * q^49 + 107322414 * q^51 + 62342520 * q^53 + 422625348 * q^55 - 317200696 * q^57 + 101234394 * q^59 + 48881304 * q^61 - 726717996 * q^63 + 272864028 * q^65 + 48335310 * q^67 + 672073812 * q^73 - 950672484 * q^75 + 10755684 * q^77 - 144370122 * q^79 - 305389432 * q^81 + 568641984 * q^83 - 734871504 * q^85 - 33665484 * q^87 + 1630891500 * q^89 - 1012044192 * q^91 - 457741376 * q^93 - 1433917218 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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