LMFDB - Embedded newform 144.8.i.a.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [144,8,Mod(49,144)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(144, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("144.49");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 144 = 2^{4} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	144.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$44.9834436697$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.14601465675.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} + 52x^{4} - 99x^{3} + 709x^{2} - 660x + 1872 $ x^6 - 3x^5 + 52x^4 - 99x^3 + 709x^2 - 660*x + 1872
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{6} $
Twist minimal:	no (minimal twist has level 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			49.2
Root			$0.500000 - 5.21243i$ of defining polynomial
Character	$\chi$	$=$	144.49
Dual form			144.8.i.a.97.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(11.7819 - 45.2569i) q^{3} +(187.789 - 325.261i) q^{5} +(528.867 + 916.024i) q^{7} +(-1909.38 - 1066.42i) q^{9} +(849.965 + 1472.18i) q^{11} +(6256.75 - 10837.0i) q^{13} +(-12507.8 - 12330.9i) q^{15} +16879.9 q^{17} +35281.9 q^{19} +(47687.4 - 13142.4i) q^{21} +(35482.7 - 61457.8i) q^{23} +(-31467.1 - 54502.6i) q^{25} +(-70758.9 + 73848.0i) q^{27} +(38827.4 + 67251.0i) q^{29} +(-63093.8 + 109282. i) q^{31} +(76640.6 - 21121.7i) q^{33} +397262. q^{35} -528664. q^{37} +(-416733. - 410841. i) q^{39} +(258960. - 448533. i) q^{41} +(-135830. - 235264. i) q^{43} +(-705425. + 420782. i) q^{45} +(-211814. - 366873. i) q^{47} +(-147628. + 255699. i) q^{49} +(198876. - 763930. i) q^{51} +1.24314e6 q^{53} +638457. q^{55} +(415687. - 1.59675e6i) q^{57} +(-1.34086e6 + 2.32244e6i) q^{59} +(377040. + 653052. i) q^{61} +(-32937.4 - 2.31303e6i) q^{63} +(-2.34990e6 - 4.07014e6i) q^{65} +(-131285. + 227392. i) q^{67} +(-2.36334e6 - 2.32993e6i) q^{69} -990913. q^{71} -1.04389e6 q^{73} +(-2.83736e6 + 781961. i) q^{75} +(-899036. + 1.55718e6i) q^{77} +(-3.66860e6 - 6.35419e6i) q^{79} +(2.50846e6 + 4.07240e6i) q^{81} +(-1.94611e6 - 3.37075e6i) q^{83} +(3.16986e6 - 5.49035e6i) q^{85} +(3.50103e6 - 964865. i) q^{87} +675681. q^{89} +1.32359e7 q^{91} +(4.20239e6 + 4.14298e6i) q^{93} +(6.62556e6 - 1.14758e7i) q^{95} +(-8.12785e6 - 1.40779e7i) q^{97} +(-52935.2 - 3.71737e6i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 27 q^{3} + 54 q^{5} - 210 q^{7} + 2295 q^{9} - 6579 q^{11} + 10092 q^{13} - 756 q^{15} - 29790 q^{17} + 137490 q^{19} + 85464 q^{21} + 39654 q^{23} - 7923 q^{25} - 87480 q^{27} + 239832 q^{29} - 145704 q^{31}+ \cdots + 11453238 q^{99}+O(q^{100}) $ 6 * q + 27 * q^3 + 54 * q^5 - 210 * q^7 + 2295 * q^9 - 6579 * q^11 + 10092 * q^13 - 756 * q^15 - 29790 * q^17 + 137490 * q^19 + 85464 * q^21 + 39654 * q^23 - 7923 * q^25 - 87480 * q^27 + 239832 * q^29 - 145704 * q^31 + 175365 * q^33 + 1087776 * q^35 - 720384 * q^37 - 2386548 * q^39 + 1086993 * q^41 + 299967 * q^43 - 2798874 * q^45 - 131634 * q^47 + 482175 * q^49 + 3719223 * q^51 + 1909152 * q^53 + 5494824 * q^55 + 6076107 * q^57 - 2504853 * q^59 + 7309038 * q^61 + 1254960 * q^63 - 1786698 * q^65 - 3433035 * q^67 - 5815206 * q^69 - 2269368 * q^71 - 15901746 * q^73 - 4230945 * q^75 - 6147432 * q^77 - 7076928 * q^79 + 4749435 * q^81 + 10914444 * q^83 + 17613396 * q^85 + 11533266 * q^87 + 23800356 * q^89 + 37170552 * q^91 + 30872466 * q^93 + 17342424 * q^95 + 519357 * q^97 + 11453238 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$

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
$3$	11.7819	−	45.2569i	0.251936	−	0.967744i
$3$	11.7819	−	45.2569i	0.251936	−	0.967744i
$4$		0			0
$5$	187.789	−	325.261i	0.671855	−	1.16369i	−0.305522	−	0.952185i	$-0.598831\pi$
$5$	187.789	−	325.261i	0.671855	−	1.16369i	0.977377	−	0.211502i	$-0.0678356\pi$
$6$		0			0
$7$	528.867	+	916.024i	0.582778	+	1.00940i	0.995148	+	0.0983845i	$0.0313675\pi$
$7$	528.867	+	916.024i	0.582778	+	1.00940i	−0.412371	+	0.911016i	$0.635299\pi$
$8$		0			0
$9$	−1909.38	−	1066.42i	−0.873057	−	0.487618i
$10$		0			0
$11$	849.965	+	1472.18i	0.192542	+	0.333493i	0.946092	−	0.323898i	$-0.104993\pi$
$11$	849.965	+	1472.18i	0.192542	+	0.333493i	−0.753550	+	0.657391i	$0.771660\pi$
$12$		0			0
$13$	6256.75	−	10837.0i	0.789854	−	1.36807i	−0.136202	−	0.990681i	$-0.543490\pi$
$13$	6256.75	−	10837.0i	0.789854	−	1.36807i	0.926056	−	0.377386i	$-0.123177\pi$
$14$		0			0
$15$	−12507.8	−	12330.9i	−0.956887	−	0.943358i
$16$		0			0
$17$	16879.9			0.833293			0.416646	−	0.909069i	$-0.363205\pi$
$17$	16879.9			0.833293			0.416646	+	0.909069i	$0.363205\pi$
$18$		0			0
$19$	35281.9			1.18009			0.590044	−	0.807371i	$-0.299110\pi$
$19$	35281.9			1.18009			0.590044	+	0.807371i	$0.299110\pi$
$20$		0			0
$21$	47687.4	−	13142.4i	1.12366	−	0.309676i
$22$		0			0
$23$	35482.7	−	61457.8i	0.608092	−	1.05325i	−0.383463	−	0.923556i	$-0.625269\pi$
$23$	35482.7	−	61457.8i	0.608092	−	1.05325i	0.991555	−	0.129690i	$-0.0413981\pi$
$24$		0			0
$25$	−31467.1	−	54502.6i	−0.402779	−	0.697634i
$26$		0			0
$27$	−70758.9	+	73848.0i	−0.691844	+	0.722047i
$28$		0			0
$29$	38827.4	+	67251.0i	0.295628	+	0.512042i	0.975131	−	0.221631i	$-0.0711379\pi$
$29$	38827.4	+	67251.0i	0.295628	+	0.512042i	−0.679503	+	0.733673i	$0.737805\pi$
$30$		0			0
$31$	−63093.8	+	109282.i	−0.380383	+	0.658843i	−0.991117	−	0.132993i	$-0.957541\pi$
$31$	−63093.8	+	109282.i	−0.380383	+	0.658843i	0.610734	+	0.791836i	$0.290874\pi$
$32$		0			0
$33$	76640.6	−	21121.7i	0.371244	−	0.102313i
$34$		0			0
$35$	397262.			1.56617
$36$		0			0
$37$	−528664.			−1.71583			−0.857914	−	0.513794i	$-0.828240\pi$
$37$	−528664.			−1.71583			−0.857914	+	0.513794i	$0.828240\pi$
$38$		0			0
$39$	−416733.	−	410841.i	−1.12495	−	1.10904i
$40$		0			0
$41$	258960.	−	448533.i	0.586800	−	1.01637i	−0.407849	−	0.913050i	$-0.633721\pi$
$41$	258960.	−	448533.i	0.586800	−	1.01637i	0.994648	−	0.103318i	$-0.0329458\pi$
$42$		0			0
$43$	−135830.	−	235264.i	−0.260529	−	0.451249i	0.705854	−	0.708358i	$-0.250564\pi$
$43$	−135830.	−	235264.i	−0.260529	−	0.451249i	−0.966383	+	0.257109i	$0.917230\pi$
$44$		0			0
$45$	−705425.	+	420782.i	−1.15400	+	0.688356i
$46$		0			0
$47$	−211814.	−	366873.i	−0.297586	−	0.515435i	0.677997	−	0.735065i	$-0.262848\pi$
$47$	−211814.	−	366873.i	−0.297586	−	0.515435i	−0.975583	+	0.219630i	$0.929515\pi$
$48$		0			0
$49$	−147628.	+	255699.i	−0.179260	+	0.310487i
$50$		0			0
$51$	198876.	−	763930.i	0.209936	−	0.806414i
$52$		0			0
$53$	1.24314e6			1.14698			0.573490	−	0.819213i	$-0.305589\pi$
$53$	1.24314e6			1.14698			0.573490	+	0.819213i	$0.305589\pi$
$54$		0			0
$55$	638457.			0.517443
$56$		0			0
$57$	415687.	−	1.59675e6i	0.297306	−	1.14202i
$58$		0			0
$59$	−1.34086e6	+	2.32244e6i	−0.849968	+	1.47219i	0.0312678	+	0.999511i	$0.490046\pi$
$59$	−1.34086e6	+	2.32244e6i	−0.849968	+	1.47219i	−0.881236	+	0.472677i	$0.843288\pi$
$60$		0			0
$61$	377040.	+	653052.i	0.212683	+	0.368378i	0.952553	−	0.304372i	$-0.0984465\pi$
$61$	377040.	+	653052.i	0.212683	+	0.368378i	−0.739870	+	0.672750i	$0.765113\pi$
$62$		0			0
$63$	−32937.4	−	2.31303e6i	−0.0165958	−	1.16544i
$64$		0			0
$65$	−2.34990e6	−	4.07014e6i	−1.06133	−	1.83829i
$66$		0			0
$67$	−131285.	+	227392.i	−0.0533278	+	0.0923664i	−0.891457	−	0.453105i	$-0.850316\pi$
$67$	−131285.	+	227392.i	−0.0533278	+	0.0923664i	0.838129	+	0.545472i	$0.183649\pi$
$68$		0			0
$69$	−2.36334e6	−	2.32993e6i	−0.866073	−	0.853828i
$70$		0			0
$71$	−990913.			−0.328573			−0.164286	−	0.986413i	$-0.552532\pi$
$71$	−990913.			−0.328573			−0.164286	+	0.986413i	$0.552532\pi$
$72$		0			0
$73$	−1.04389e6			−0.314068			−0.157034	−	0.987593i	$-0.550193\pi$
$73$	−1.04389e6			−0.314068			−0.157034	+	0.987593i	$0.550193\pi$
$74$		0			0
$75$	−2.83736e6	+	781961.i	−0.776605	+	0.214028i
$76$		0			0
$77$	−899036.	+	1.55718e6i	−0.224419	+	0.388705i
$78$		0			0
$79$	−3.66860e6	−	6.35419e6i	−0.837153	−	1.44999i	−0.892265	−	0.451512i	$-0.850885\pi$
$79$	−3.66860e6	−	6.35419e6i	−0.837153	−	1.44999i	0.0551117	−	0.998480i	$-0.482449\pi$
$80$		0			0
$81$	2.50846e6	+	4.07240e6i	0.524457	+	0.851437i
$82$		0			0
$83$	−1.94611e6	−	3.37075e6i	−0.373588	−	0.647074i	0.616526	−	0.787334i	$-0.288539\pi$
$83$	−1.94611e6	−	3.37075e6i	−0.373588	−	0.647074i	−0.990115	+	0.140260i	$0.955206\pi$
$84$		0			0
$85$	3.16986e6	−	5.49035e6i	0.559852	−	0.969692i
$86$		0			0
$87$	3.50103e6	−	964865.i	0.570005	−	0.157090i
$88$		0			0
$89$	675681.			0.101596			0.0507980	−	0.998709i	$-0.483824\pi$
$89$	675681.			0.101596			0.0507980	+	0.998709i	$0.483824\pi$
$90$		0			0
$91$	1.32359e7			1.84124
$92$		0			0
$93$	4.20239e6	+	4.14298e6i	0.541759	+	0.534099i
$94$		0			0
$95$	6.62556e6	−	1.14758e7i	0.792849	−	1.37325i
$96$		0			0
$97$	−8.12785e6	−	1.40779e7i	−0.904221	−	1.56616i	−0.821959	−	0.569546i	$-0.807119\pi$
$97$	−8.12785e6	−	1.40779e7i	−0.904221	−	1.56616i	−0.0822619	−	0.996611i	$-0.526214\pi$
$98$		0			0
$99$	−52935.2	−	3.71737e6i	−0.00548304	−	0.385046i
$100$		0			0
$101$	212270.	+	367662.i	0.0205005	+	0.0355079i	0.876094	−	0.482141i	$-0.160141\pi$
$101$	212270.	+	367662.i	0.0205005	+	0.0355079i	−0.855593	+	0.517649i	$0.826807\pi$
$102$		0			0
$103$	−2.43167e6	+	4.21178e6i	−0.219268	+	0.379783i	−0.954584	−	0.297941i	$-0.903700\pi$
$103$	−2.43167e6	+	4.21178e6i	−0.219268	+	0.379783i	0.735317	+	0.677724i	$0.237033\pi$
$104$		0			0
$105$	4.68048e6	−	1.79788e7i	0.394574	−	1.51565i
$106$		0			0
$107$	−2.83648e6			−0.223839			−0.111920	−	0.993717i	$-0.535700\pi$
$107$	−2.83648e6			−0.223839			−0.111920	+	0.993717i	$0.535700\pi$
$108$		0			0
$109$	6.31389e6			0.466987			0.233493	−	0.972358i	$-0.424984\pi$
$109$	6.31389e6			0.466987			0.233493	+	0.972358i	$0.424984\pi$
$110$		0			0
$111$	−6.22865e6	+	2.39257e7i	−0.432278	+	1.66048i
$112$		0			0
$113$	−1.07218e7	+	1.85708e7i	−0.699029	+	1.21075i	0.269775	+	0.962923i	$0.413051\pi$
$113$	−1.07218e7	+	1.85708e7i	−0.699029	+	1.21075i	−0.968804	+	0.247830i	$0.920283\pi$
$114$		0			0
$115$	−1.33265e7	−	2.30822e7i	−0.817100	−	1.41526i
$116$		0			0
$117$	−2.35033e7	+	1.40196e7i	−1.35668	+	0.809253i
$118$		0			0
$119$	8.92719e6	+	1.54624e7i	0.485624	+	0.841126i
$120$		0			0
$121$	8.29871e6	−	1.43738e7i	0.425855	−	0.737602i
$122$		0			0
$123$	−1.72482e7	−	1.70043e7i	−0.835747	−	0.823931i
$124$		0			0
$125$	5.70534e6			0.261274
$126$		0			0
$127$	−1.35681e7			−0.587768			−0.293884	−	0.955841i	$-0.594948\pi$
$127$	−1.35681e7			−0.587768			−0.293884	+	0.955841i	$0.594948\pi$
$128$		0			0
$129$	−1.22477e7	+	3.37539e6i	−0.502330	+	0.138439i
$130$		0			0
$131$	−1.17657e7	+	2.03788e7i	−0.457265	+	0.792005i	−0.998815	−	0.0486627i	$-0.984504\pi$
$131$	−1.17657e7	+	2.03788e7i	−0.457265	+	0.792005i	0.541551	+	0.840668i	$0.317837\pi$
$132$		0			0
$133$	1.86594e7	+	3.23191e7i	0.687729	+	1.19118i
$134$		0			0
$135$	1.07321e7	+	3.68829e7i	0.375418	+	1.29020i
$136$		0			0
$137$	1.39952e7	+	2.42405e7i	0.465006	+	0.805414i	0.999202	−	0.0399467i	$-0.0127188\pi$
$137$	1.39952e7	+	2.42405e7i	0.465006	+	0.805414i	−0.534196	+	0.845361i	$0.679385\pi$
$138$		0			0
$139$	−4.00094e6	+	6.92982e6i	−0.126360	+	0.218862i	−0.922264	−	0.386561i	$-0.873663\pi$
$139$	−4.00094e6	+	6.92982e6i	−0.126360	+	0.218862i	0.795904	+	0.605423i	$0.206996\pi$
$140$		0			0
$141$	−1.90991e7	+	5.26361e6i	−0.573781	+	0.158131i
$142$		0			0
$143$	2.12721e7			0.608322
$144$		0			0
$145$	2.91655e7			0.794476
$146$		0			0
$147$	9.83283e6	+	9.69381e6i	0.255310	+	0.251700i
$148$		0			0
$149$	1.35829e7	−	2.35262e7i	0.336387	−	0.582639i	−0.647363	−	0.762182i	$-0.724128\pi$
$149$	1.35829e7	−	2.35262e7i	0.336387	−	0.582639i	0.983750	+	0.179542i	$0.0574617\pi$
$150$		0			0
$151$	2.09932e7	+	3.63612e7i	0.496202	+	0.859447i	0.999990	−	0.00438023i	$-0.00139427\pi$
$151$	2.09932e7	+	3.63612e7i	0.496202	+	0.859447i	−0.503789	+	0.863827i	$0.668061\pi$
$152$		0			0
$153$	−3.22300e7	−	1.80010e7i	−0.727512	−	0.406329i
$154$		0			0
$155$	2.36967e7	+	4.10439e7i	0.511125	+	0.885294i
$156$		0			0
$157$	−4.10239e7	+	7.10556e7i	−0.846036	+	1.46538i	0.0386831	+	0.999252i	$0.487684\pi$
$157$	−4.10239e7	+	7.10556e7i	−0.846036	+	1.46538i	−0.884719	+	0.466125i	$0.845650\pi$
$158$		0			0
$159$	1.46465e7	−	5.62608e7i	0.288965	−	1.10998i
$160$		0			0
$161$	7.50625e7			1.41753
$162$		0			0
$163$	2.89286e7			0.523203			0.261602	−	0.965176i	$-0.415749\pi$
$163$	2.89286e7			0.523203			0.261602	+	0.965176i	$0.415749\pi$
$164$		0			0
$165$	7.52221e6	−	2.88946e7i	0.130362	−	0.500752i
$166$		0			0
$167$	1.33585e7	−	2.31376e7i	0.221947	−	0.384424i	−0.733452	−	0.679741i	$-0.762092\pi$
$167$	1.33585e7	−	2.31376e7i	0.221947	−	0.384424i	0.955399	+	0.295318i	$0.0954254\pi$
$168$		0			0
$169$	−4.69195e7	−	8.12669e7i	−0.747738	−	1.29512i
$170$		0			0
$171$	−6.73664e7	−	3.76254e7i	−1.03028	−	0.575433i
$172$		0			0
$173$	1.31561e7	+	2.27871e7i	0.193182	+	0.334601i	0.946303	−	0.323281i	$-0.104786\pi$
$173$	1.31561e7	+	2.27871e7i	0.193182	+	0.334601i	−0.753121	+	0.657882i	$0.771453\pi$
$174$		0			0
$175$	3.32838e7	−	5.76492e7i	0.469461	−	0.813131i
$176$		0			0
$177$	8.93088e7	+	8.80461e7i	1.21056	+	1.19345i
$178$		0			0
$179$	2.19724e7			0.286347			0.143174	−	0.989698i	$-0.454269\pi$
$179$	2.19724e7			0.286347			0.143174	+	0.989698i	$0.454269\pi$
$180$		0			0
$181$	1.26202e8			1.58195			0.790975	−	0.611849i	$-0.209574\pi$
$181$	1.26202e8			1.58195			0.790975	+	0.611849i	$0.209574\pi$
$182$		0			0
$183$	3.39974e7	−	9.36949e6i	0.410078	−	0.113015i
$184$		0			0
$185$	−9.92774e7	+	1.71954e8i	−1.15279	+	1.99669i
$186$		0			0
$187$	1.43473e7	+	2.48502e7i	0.160444	+	0.277898i
$188$		0			0
$189$	−1.05069e8	−	2.57611e7i	−1.13203	−	0.277555i
$190$		0			0
$191$	8.97257e7	+	1.55409e8i	0.931752	+	1.61384i	0.780326	+	0.625372i	$0.215053\pi$
$191$	8.97257e7	+	1.55409e8i	0.931752	+	1.61384i	0.151425	+	0.988469i	$0.451614\pi$
$192$		0			0
$193$	1.49157e7	−	2.58348e7i	0.149346	−	0.258675i	−0.781640	−	0.623730i	$-0.785617\pi$
$193$	1.49157e7	−	2.58348e7i	0.149346	−	0.258675i	0.930986	+	0.365055i	$0.118950\pi$
$194$		0			0
$195$	−2.11888e8	+	5.83953e7i	−2.04638	+	0.563971i
$196$		0			0
$197$	6.70863e7			0.625176			0.312588	−	0.949889i	$-0.398804\pi$
$197$	6.70863e7			0.625176			0.312588	+	0.949889i	$0.398804\pi$
$198$		0			0
$199$	1.18177e8			1.06304			0.531519	−	0.847046i	$-0.321621\pi$
$199$	1.18177e8			1.06304			0.531519	+	0.847046i	$0.321621\pi$
$200$		0			0
$201$	8.74429e6	+	8.62066e6i	0.0759519	+	0.0748780i
$202$		0			0
$203$	−4.10690e7	+	7.11336e7i	−0.344570	+	0.596813i
$204$		0			0
$205$	−9.72600e7	−	1.68459e8i	−0.788489	−	1.36570i
$206$		0			0
$207$	−1.33290e8	+	7.95066e7i	−1.04448	+	0.623027i
$208$		0			0
$209$	2.99884e7	+	5.19414e7i	0.227217	+	0.393552i
$210$		0			0
$211$	5.19927e7	−	9.00540e7i	0.381025	−	0.659955i	−0.610184	−	0.792260i	$-0.708904\pi$
$211$	5.19927e7	−	9.00540e7i	0.381025	−	0.659955i	0.991209	+	0.132305i	$0.0422378\pi$
$212$		0			0
$213$	−1.16748e7	+	4.48457e7i	−0.0827791	+	0.317974i
$214$		0			0
$215$	−1.02030e8			−0.700150
$216$		0			0
$217$	−1.33473e8			−0.886715
$218$		0			0
$219$	−1.22989e7	+	4.72431e7i	−0.0791249	+	0.303937i
$220$		0			0
$221$	1.05613e8	−	1.82927e8i	0.658180	−	1.14000i
$222$		0			0
$223$	−8.07648e7	−	1.39889e8i	−0.487703	−	0.844726i	0.512197	−	0.858868i	$-0.328832\pi$
$223$	−8.07648e7	−	1.39889e8i	−0.487703	−	0.844726i	−0.999900	+	0.0141421i	$0.995498\pi$
$224$		0			0
$225$	1.95975e6	+	1.37623e8i	0.0114700	+	0.805476i
$226$		0			0
$227$	1.90267e7	+	3.29553e7i	0.107963	+	0.186997i	0.914945	−	0.403579i	$-0.132234\pi$
$227$	1.90267e7	+	3.29553e7i	0.107963	+	0.186997i	−0.806982	+	0.590576i	$0.798901\pi$
$228$		0			0
$229$	1.44858e8	−	2.50901e8i	0.797110	−	1.38064i	−0.124381	−	0.992235i	$-0.539694\pi$
$229$	1.44858e8	−	2.50901e8i	0.797110	−	1.38064i	0.921491	−	0.388401i	$-0.126972\pi$
$230$		0			0
$231$	5.98806e7	+	5.90340e7i	0.319628	+	0.315109i
$232$		0			0
$233$	−7.45094e7			−0.385892			−0.192946	−	0.981209i	$-0.561804\pi$
$233$	−7.45094e7			−0.385892			−0.192946	+	0.981209i	$0.561804\pi$
$234$		0			0
$235$	−1.59106e8			−0.799740
$236$		0			0
$237$	−3.30794e8	+	9.11650e7i	−1.61413	+	0.444845i
$238$		0			0
$239$	8.98328e7	−	1.55595e8i	0.425640	−	0.737230i	−0.570840	−	0.821061i	$-0.693382\pi$
$239$	8.98328e7	−	1.55595e8i	0.425640	−	0.737230i	0.996480	+	0.0838313i	$0.0267157\pi$
$240$		0			0
$241$	3.04341e7	+	5.27134e7i	0.140056	+	0.242583i	0.927517	−	0.373780i	$-0.121939\pi$
$241$	3.04341e7	+	5.27134e7i	0.140056	+	0.242583i	−0.787462	+	0.616364i	$0.788605\pi$
$242$		0			0
$243$	2.13858e8	−	6.55447e7i	0.956103	−	0.293032i
$244$		0			0
$245$	5.54459e7	+	9.60352e7i	0.240873	+	0.417205i
$246$		0			0
$247$	2.20750e8	−	3.82350e8i	0.932098	−	1.61444i
$248$		0			0
$249$	−1.75479e8	+	4.83610e7i	−0.720322	+	0.198517i
$250$		0			0
$251$	−2.14309e8			−0.855426			−0.427713	−	0.903915i	$-0.640681\pi$
$251$	−2.14309e8			−0.855426			−0.427713	+	0.903915i	$0.640681\pi$
$252$		0			0
$253$	1.20636e8			0.468334
$254$		0			0
$255$	−2.11130e8	−	2.08144e8i	−0.797367	−	0.786094i
$256$		0			0
$257$	2.63538e7	−	4.56461e7i	0.0968450	−	0.167740i	−0.813532	−	0.581520i	$-0.802458\pi$
$257$	2.63538e7	−	4.56461e7i	0.0968450	−	0.167740i	0.910377	+	0.413779i	$0.135792\pi$
$258$		0			0
$259$	−2.79593e8	−	4.84269e8i	−0.999946	−	1.73196i
$260$		0			0
$261$	−2.41814e6	−	1.69814e8i	−0.00841860	−	0.591195i
$262$		0			0
$263$	2.44289e8	+	4.23121e8i	0.828055	+	1.43423i	0.899562	+	0.436793i	$0.143886\pi$
$263$	2.44289e8	+	4.23121e8i	0.828055	+	1.43423i	−0.0715077	+	0.997440i	$0.522781\pi$
$264$		0			0
$265$	2.33449e8	−	4.04346e8i	0.770604	−	1.33473i
$266$		0			0
$267$	7.96078e6	−	3.05792e7i	0.0255956	−	0.0983189i
$268$		0			0
$269$	−1.69083e8			−0.529622			−0.264811	−	0.964300i	$-0.585310\pi$
$269$	−1.69083e8			−0.529622			−0.264811	+	0.964300i	$0.585310\pi$
$270$		0			0
$271$	1.99531e8			0.609001			0.304501	−	0.952512i	$-0.401510\pi$
$271$	1.99531e8			0.609001			0.304501	+	0.952512i	$0.401510\pi$
$272$		0			0
$273$	1.55944e8	−	5.99017e8i	0.463873	−	1.78185i
$274$		0			0
$275$	5.34919e7	−	9.26506e7i	0.155104	−	0.268648i
$276$		0			0
$277$	1.77438e8	+	3.07331e8i	0.501611	+	0.868816i	0.999998	+	0.00186126i	$0.000592457\pi$
$277$	1.77438e8	+	3.07331e8i	0.501611	+	0.868816i	−0.498387	+	0.866955i	$0.666074\pi$
$278$		0			0
$279$	2.37010e8	−	1.41375e8i	0.653360	−	0.389725i
$280$		0			0
$281$	1.40245e8	+	2.42912e8i	0.377065	+	0.653096i	0.990634	−	0.136545i	$-0.0436000\pi$
$281$	1.40245e8	+	2.42912e8i	0.377065	+	0.653096i	−0.613569	+	0.789641i	$0.710267\pi$
$282$		0			0
$283$	−7.95244e7	+	1.37740e8i	−0.208568	+	0.361251i	−0.951264	−	0.308379i	$-0.900214\pi$
$283$	−7.95244e7	+	1.37740e8i	−0.208568	+	0.361251i	0.742696	+	0.669629i	$0.233547\pi$
$284$		0			0
$285$	−4.41298e8	−	4.35059e8i	−1.12921	−	1.11325i
$286$		0			0
$287$	5.47822e8			1.36790
$288$		0			0
$289$	−1.25409e8			−0.305623
$290$		0			0
$291$	−7.32881e8	+	2.01978e8i	−1.74344	+	0.480484i
$292$		0			0
$293$	−1.72190e8	+	2.98242e8i	−0.399918	+	0.692679i	−0.993715	−	0.111936i	$-0.964295\pi$
$293$	−1.72190e8	+	2.98242e8i	−0.399918	+	0.692679i	0.593797	+	0.804615i	$0.297628\pi$
$294$		0			0
$295$	5.03600e8	+	8.72260e8i	1.14211	+	1.97819i
$296$		0			0
$297$	−1.68860e8	−	4.14019e7i	−0.374007	−	0.0917006i
$298$		0			0
$299$	−4.44013e8	−	7.69052e8i	−0.960608	−	1.66382i
$300$		0			0
$301$	1.43672e8	−	2.48847e8i	0.303661	−	0.525956i
$302$		0			0
$303$	1.91402e7	−	5.27494e6i	0.0395273	−	0.0108935i
$304$		0			0
$305$	2.83216e8			0.571569
$306$		0			0
$307$	−5.28744e8			−1.04294			−0.521472	−	0.853269i	$-0.674617\pi$
$307$	−5.28744e8			−1.04294			−0.521472	+	0.853269i	$0.674617\pi$
$308$		0			0
$309$	1.61962e8	+	1.59673e8i	0.312291	+	0.307876i
$310$		0			0
$311$	−1.43382e8	+	2.48345e8i	−0.270292	+	0.468159i	−0.968937	−	0.247310i	$-0.920454\pi$
$311$	−1.43382e8	+	2.48345e8i	−0.270292	+	0.468159i	0.698645	+	0.715469i	$0.253787\pi$
$312$		0			0
$313$	−1.25124e7	−	2.16722e7i	−0.0230641	−	0.0399482i	0.854263	−	0.519841i	$-0.174009\pi$
$313$	−1.25124e7	−	2.16722e7i	−0.0230641	−	0.0399482i	−0.877327	+	0.479893i	$0.840676\pi$
$314$		0			0
$315$	−7.58522e8	−	4.23648e8i	−1.36735	−	0.763693i
$316$		0			0
$317$	2.60137e8	+	4.50570e8i	0.458663	+	0.794428i	0.998891	−	0.0470910i	$-0.0149951\pi$
$317$	2.60137e8	+	4.50570e8i	0.458663	+	0.794428i	−0.540227	+	0.841519i	$0.681662\pi$
$318$		0			0
$319$	−6.60038e7	+	1.14322e8i	−0.113842	+	0.197180i
$320$		0			0
$321$	−3.34190e7	+	1.28370e8i	−0.0563931	+	0.216619i
$322$		0			0
$323$	5.95554e8			0.983359
$324$		0			0
$325$	−7.87527e8			−1.27255
$326$		0			0
$327$	7.43894e7	−	2.85747e8i	0.117651	−	0.451924i
$328$		0			0
$329$	2.24043e8	−	3.88054e8i	0.346853	−	0.600768i
$330$		0			0
$331$	−2.52520e8	−	4.37377e8i	−0.382734	−	0.662915i	0.608718	−	0.793387i	$-0.291684\pi$
$331$	−2.52520e8	−	4.37377e8i	−0.382734	−	0.662915i	−0.991452	+	0.130472i	$0.958351\pi$
$332$		0			0
$333$	1.00942e9	+	5.63779e8i	1.49802	+	0.836669i
$334$		0			0
$335$	4.93078e7	+	8.54037e7i	0.0716571	+	0.124114i
$336$		0			0
$337$	9.18750e7	−	1.59132e8i	0.130765	−	0.226492i	−0.793206	−	0.608953i	$-0.791590\pi$
$337$	9.18750e7	−	1.59132e8i	0.130765	−	0.226492i	0.923972	+	0.382461i	$0.124923\pi$
$338$		0			0
$339$	7.14133e8	+	7.04036e8i	0.995589	+	0.981513i
$340$		0			0
$341$	−2.14510e8			−0.292960
$342$		0			0
$343$	5.58786e8			0.747681
$344$		0			0
$345$	−1.20164e9	+	3.31166e8i	−1.57546	+	0.434189i
$346$		0			0
$347$	−1.43066e8	+	2.47797e8i	−0.183816	+	0.318378i	−0.943177	−	0.332291i	$-0.892178\pi$
$347$	−1.43066e8	+	2.47797e8i	−0.183816	+	0.318378i	0.759361	+	0.650670i	$0.225512\pi$
$348$		0			0
$349$	1.83182e7	+	3.17281e7i	0.0230672	+	0.0399535i	0.877329	−	0.479890i	$-0.159324\pi$
$349$	1.83182e7	+	3.17281e7i	0.0230672	+	0.0399535i	−0.854261	+	0.519844i	$0.825990\pi$
$350$		0			0
$351$	3.57570e8	+	1.22886e9i	0.441353	+	1.51680i
$352$		0			0
$353$	−2.38469e8	−	4.13040e8i	−0.288549	−	0.499782i	0.684914	−	0.728624i	$-0.259840\pi$
$353$	−2.38469e8	−	4.13040e8i	−0.288549	−	0.499782i	−0.973464	+	0.228841i	$0.926506\pi$
$354$		0			0
$355$	−1.86083e8	+	3.22305e8i	−0.220753	+	0.382356i
$356$		0			0
$357$	8.04957e8	−	2.21842e8i	0.936341	−	0.258050i
$358$		0			0
$359$	3.73983e8			0.426600			0.213300	−	0.976987i	$-0.431579\pi$
$359$	3.73983e8			0.426600			0.213300	+	0.976987i	$0.431579\pi$
$360$		0			0
$361$	3.50942e8			0.392609
$362$		0			0
$363$	−5.52739e8	−	5.44924e8i	−0.606522	−	0.597947i
$364$		0			0
$365$	−1.96031e8	+	3.39535e8i	−0.211008	+	0.365477i
$366$		0			0
$367$	−2.91624e8	−	5.05107e8i	−0.307958	−	0.533399i	0.669957	−	0.742400i	$-0.266312\pi$
$367$	−2.91624e8	−	5.05107e8i	−0.307958	−	0.533399i	−0.977916	+	0.209000i	$0.932979\pi$
$368$		0			0
$369$	−9.72778e8	+	5.80256e8i	−1.00791	+	0.601212i
$370$		0			0
$371$	6.57457e8	+	1.13875e9i	0.668434	+	1.15776i
$372$		0			0
$373$	−7.38605e8	+	1.27930e9i	−0.736939	+	1.27642i	0.216928	+	0.976188i	$0.430396\pi$
$373$	−7.38605e8	+	1.27930e9i	−0.736939	+	1.27642i	−0.953867	+	0.300228i	$0.902937\pi$
$374$		0			0
$375$	6.72195e7	−	2.58206e8i	0.0658242	−	0.252846i
$376$		0			0
$377$	9.71732e8			0.934011
$378$		0			0
$379$	6.08958e8			0.574580			0.287290	−	0.957844i	$-0.407246\pi$
$379$	6.08958e8			0.574580			0.287290	+	0.957844i	$0.407246\pi$
$380$		0			0
$381$	−1.59857e8	+	6.14050e8i	−0.148080	+	0.568809i
$382$		0			0
$383$	−4.24357e8	+	7.35008e8i	−0.385954	+	0.668492i	−0.991901	−	0.127013i	$-0.959461\pi$
$383$	−4.24357e8	+	7.35008e8i	−0.385954	+	0.668492i	0.605947	+	0.795505i	$0.292794\pi$
$384$		0			0
$385$	3.37659e8	+	5.84842e8i	0.301554	+	0.522307i
$386$		0			0
$387$	8.45939e6	+	5.94060e8i	0.00741909	+	0.521005i
$388$		0			0
$389$	1.26651e8	+	2.19367e8i	0.109090	+	0.188950i	0.915402	−	0.402541i	$-0.131873\pi$
$389$	1.26651e8	+	2.19367e8i	0.109090	+	0.188950i	−0.806312	+	0.591491i	$0.798540\pi$
$390$		0			0
$391$	5.98943e8	−	1.03740e9i	0.506719	−	0.877662i
$392$		0			0
$393$	7.83658e8	+	7.72578e8i	0.651257	+	0.642049i
$394$		0			0
$395$	−2.75569e9			−2.24978
$396$		0			0
$397$	9.92215e8			0.795865			0.397932	−	0.917415i	$-0.369728\pi$
$397$	9.92215e8			0.795865			0.397932	+	0.917415i	$0.369728\pi$
$398$		0			0
$399$	1.68250e9	−	4.63689e8i	1.32602	−	0.365445i
$400$		0			0
$401$	4.74376e8	−	8.21644e8i	0.367382	−	0.636324i	−0.621774	−	0.783197i	$-0.713588\pi$
$401$	4.74376e8	−	8.21644e8i	0.367382	−	0.636324i	0.989155	+	0.146873i	$0.0469209\pi$
$402$		0			0
$403$	7.89524e8	+	1.36750e9i	0.600894	+	1.04078i
$404$		0			0
$405$	1.79565e9	−	5.11504e7i	1.34317	−	0.0382610i
$406$		0			0
$407$	−4.49346e8	−	7.78290e8i	−0.330370	−	0.572217i
$408$		0			0
$409$	−3.88383e8	+	6.72699e8i	−0.280691	+	0.486171i	−0.971555	−	0.236813i	$-0.923897\pi$
$409$	−3.88383e8	+	6.72699e8i	−0.280691	+	0.486171i	0.690864	+	0.722985i	$0.257230\pi$
$410$		0			0
$411$	1.26194e9	−	3.47784e8i	0.896586	−	0.247094i
$412$		0			0
$413$	−2.83655e9			−1.98137
$414$		0			0
$415$	−1.46183e9			−1.00399
$416$		0			0
$417$	2.66484e8	+	2.62716e8i	0.179968	+	0.177423i
$418$		0			0
$419$	−3.14943e8	+	5.45498e8i	−0.209162	+	0.362280i	−0.951451	−	0.307801i	$-0.900407\pi$
$419$	−3.14943e8	+	5.45498e8i	−0.209162	+	0.362280i	0.742289	+	0.670080i	$0.233740\pi$
$420$		0			0
$421$	−5.52425e8	−	9.56829e8i	−0.360816	−	0.624952i	0.627279	−	0.778795i	$-0.284169\pi$
$421$	−5.52425e8	−	9.56829e8i	−0.360816	−	0.624952i	−0.988095	+	0.153842i	$0.950835\pi$
$422$		0			0
$423$	1.31917e7	+	9.26382e8i	0.00847438	+	0.595112i
$424$		0			0
$425$	−5.31160e8	−	9.19997e8i	−0.335633	−	0.581333i
$426$		0			0
$427$	−3.98808e8	+	6.90755e8i	−0.247894	+	0.429365i
$428$		0			0
$429$	2.50624e8	−	9.62707e8i	0.153258	−	0.588700i
$430$		0			0
$431$	2.85661e8			0.171863			0.0859313	−	0.996301i	$-0.472613\pi$
$431$	2.85661e8			0.171863			0.0859313	+	0.996301i	$0.472613\pi$
$432$		0			0
$433$	4.76788e8			0.282239			0.141120	−	0.989993i	$-0.454930\pi$
$433$	4.76788e8			0.282239			0.141120	+	0.989993i	$0.454930\pi$
$434$		0			0
$435$	3.43623e8	−	1.31994e9i	0.200157	−	0.768849i
$436$		0			0
$437$	1.25190e9	−	2.16835e9i	0.717602	−	1.24292i
$438$		0			0
$439$	1.25879e9	+	2.18029e9i	0.710112	+	1.22995i	0.964815	+	0.262931i	$0.0846890\pi$
$439$	1.25879e9	+	2.18029e9i	0.710112	+	1.22995i	−0.254703	+	0.967019i	$0.581978\pi$
$440$		0			0
$441$	5.54561e8	−	3.30792e8i	0.307903	−	0.183662i
$442$		0			0
$443$	−2.29324e8	−	3.97201e8i	−0.125325	−	0.217069i	0.796535	−	0.604592i	$-0.206664\pi$
$443$	−2.29324e8	−	3.97201e8i	−0.125325	−	0.217069i	−0.921860	+	0.387523i	$0.873331\pi$
$444$		0			0
$445$	1.26886e8	−	2.19772e8i	0.0682578	−	0.118226i
$446$		0			0
$447$	−9.04691e8	−	8.91900e8i	−0.479098	−	0.472324i
$448$		0			0
$449$	−1.46433e9			−0.763444			−0.381722	−	0.924277i	$-0.624669\pi$
$449$	−1.46433e9			−0.763444			−0.381722	+	0.924277i	$0.624669\pi$
$450$		0			0
$451$	8.80429e8			0.451936
$452$		0			0
$453$	1.89293e9	−	5.21682e8i	0.956735	−	0.263671i
$454$		0			0
$455$	2.48557e9	−	4.30513e9i	1.23704	−	2.14262i
$456$		0			0
$457$	−6.98299e8	−	1.20949e9i	−0.342243	−	0.592783i	0.642606	−	0.766197i	$-0.277853\pi$
$457$	−6.98299e8	−	1.20949e9i	−0.342243	−	0.592783i	−0.984849	+	0.173414i	$0.944520\pi$
$458$		0			0
$459$	−1.19440e9	+	1.24654e9i	−0.576509	+	0.601677i
$460$		0			0
$461$	1.98110e9	+	3.43137e9i	0.941789	+	1.63123i	0.762056	+	0.647511i	$0.224190\pi$
$461$	1.98110e9	+	3.43137e9i	0.941789	+	1.63123i	0.179733	+	0.983715i	$0.442476\pi$
$462$		0			0
$463$	−9.40813e8	+	1.62954e9i	−0.440524	+	0.763010i	−0.997728	−	0.0673653i	$-0.978541\pi$
$463$	−9.40813e8	+	1.62954e9i	−0.440524	+	0.763010i	0.557204	+	0.830375i	$0.311874\pi$
$464$		0			0
$465$	2.13671e9	−	5.88866e8i	0.985508	−	0.271601i
$466$		0			0
$467$	−9.69518e8			−0.440501			−0.220250	−	0.975443i	$-0.570687\pi$
$467$	−9.69518e8			−0.440501			−0.220250	+	0.975443i	$0.570687\pi$
$468$		0			0
$469$	−2.77729e8			−0.124313
$470$		0			0
$471$	2.73242e9	+	2.69378e9i	1.20496	+	1.18793i
$472$		0			0
$473$	2.30901e8	−	3.99933e8i	0.100326	−	0.173769i
$474$		0			0
$475$	−1.11022e9	−	1.92296e9i	−0.475315	−	0.823269i
$476$		0			0
$477$	−2.37363e9	−	1.32571e9i	−1.00138	−	0.559288i
$478$		0			0
$479$	−1.00435e9	−	1.73959e9i	−0.417552	−	0.723222i	0.578140	−	0.815937i	$-0.303779\pi$
$479$	−1.00435e9	−	1.73959e9i	−0.417552	−	0.723222i	−0.995693	+	0.0927156i	$0.970445\pi$
$480$		0			0
$481$	−3.30772e9	+	5.72913e9i	−1.35525	+	2.34737i
$482$		0			0
$483$	8.84376e8	−	3.39709e9i	0.357126	−	1.37181i
$484$		0			0
$485$	−6.10529e9			−2.43002
$486$		0			0
$487$	−3.02884e9			−1.18830			−0.594148	−	0.804356i	$-0.702510\pi$
$487$	−3.02884e9			−1.18830			−0.594148	+	0.804356i	$0.702510\pi$
$488$		0			0
$489$	3.40833e8	−	1.30922e9i	0.131814	−	0.506327i
$490$		0			0
$491$	2.25862e9	−	3.91204e9i	0.861107	−	1.49148i	−0.00975413	−	0.999952i	$-0.503105\pi$
$491$	2.25862e9	−	3.91204e9i	0.861107	−	1.49148i	0.870861	−	0.491529i	$-0.163562\pi$
$492$		0			0
$493$	6.55401e8	+	1.13519e9i	0.246344	+	0.426681i
$494$		0			0
$495$	−1.21905e9	−	6.80864e8i	−0.451757	−	0.252315i
$496$		0			0
$497$	−5.24061e8	−	9.07700e8i	−0.191485	−	0.331661i
$498$		0			0
$499$	−2.32832e9	+	4.03277e9i	−0.838862	+	1.45295i	0.0519839	+	0.998648i	$0.483446\pi$
$499$	−2.32832e9	+	4.03277e9i	−0.838862	+	1.45295i	−0.890846	+	0.454305i	$0.849888\pi$
$500$		0			0
$501$	−8.89747e8	−	8.77167e8i	−0.316107	−	0.311638i
$502$		0			0
$503$	−2.88413e9			−1.01048			−0.505239	−	0.862980i	$-0.668596\pi$
$503$	−2.88413e9			−1.01048			−0.505239	+	0.862980i	$0.668596\pi$
$504$		0			0
$505$	1.59448e8			0.0550934
$506$		0			0
$507$	−4.23069e9	+	1.16595e9i	−1.44173	+	0.397332i
$508$		0			0
$509$	6.60362e8	−	1.14378e9i	0.221957	−	0.384442i	−0.733445	−	0.679749i	$-0.762089\pi$
$509$	6.60362e8	−	1.14378e9i	0.221957	−	0.384442i	0.955402	+	0.295307i	$0.0954221\pi$
$510$		0			0
$511$	−5.52077e8	−	9.56225e8i	−0.183032	−	0.317020i
$512$		0			0
$513$	−2.49651e9	+	2.60550e9i	−0.816437	+	0.852079i
$514$		0			0
$515$	9.13283e8	+	1.58185e9i	0.294632	+	0.510318i
$516$		0			0
$517$	3.60070e8	−	6.23659e8i	0.114596	−	0.198486i
$518$		0			0
$519$	1.18628e9	−	3.26931e8i	0.372477	−	0.102653i
$520$		0			0
$521$	5.14654e9			1.59435			0.797175	−	0.603749i	$-0.206327\pi$
$521$	5.14654e9			1.59435			0.797175	+	0.603749i	$0.206327\pi$
$522$		0			0
$523$	2.17050e9			0.663444			0.331722	−	0.943377i	$-0.392370\pi$
$523$	2.17050e9			0.663444			0.331722	+	0.943377i	$0.392370\pi$
$524$		0			0
$525$	−2.21688e9	−	2.18554e9i	−0.668628	−	0.659175i
$526$		0			0
$527$	−1.06502e9	+	1.84466e9i	−0.316970	+	0.549009i
$528$		0			0
$529$	−8.15632e8	−	1.41272e9i	−0.239552	−	0.414916i
$530$		0			0
$531$	5.03692e9	−	3.00449e9i	1.45994	−	0.870844i
$532$		0			0
$533$	−3.24050e9	−	5.61271e9i	−0.926972	−	1.60556i
$534$		0			0
$535$	−5.32661e8	+	9.22595e8i	−0.150388	+	0.260479i
$536$		0			0
$537$	2.58876e8	−	9.94405e8i	0.0721410	−	0.277111i
$538$		0			0
$539$	−5.01915e8			−0.138060
$540$		0			0
$541$	−9.27955e7			−0.0251963			−0.0125981	−	0.999921i	$-0.504010\pi$
$541$	−9.27955e7			−0.0251963			−0.0125981	+	0.999921i	$0.504010\pi$
$542$		0			0
$543$	1.48690e9	−	5.71153e9i	0.398549	−	1.53092i
$544$		0			0
$545$	1.18568e9	−	2.05366e9i	0.313747	−	0.543427i
$546$		0			0
$547$	−2.73518e9	−	4.73747e9i	−0.714547	−	1.23763i	−0.963134	−	0.269022i	$-0.913300\pi$
$547$	−2.73518e9	−	4.73747e9i	−0.714547	−	1.23763i	0.248588	−	0.968609i	$-0.420034\pi$
$548$		0			0
$549$	−2.34818e7	−	1.64901e9i	−0.00605659	−	0.425323i
$550$		0			0
$551$	1.36990e9	+	2.37274e9i	0.348867	+	0.604255i
$552$		0			0
$553$	3.88039e9	−	6.72104e9i	0.975749	−	1.69005i
$554$		0			0
$555$	6.61241e9	+	6.51892e9i	1.64185	+	1.61864i
$556$		0			0
$557$	4.50060e9			1.10351			0.551756	−	0.834006i	$-0.313958\pi$
$557$	4.50060e9			1.10351			0.551756	+	0.834006i	$0.313958\pi$
$558$		0			0
$559$	−3.39941e9			−0.823119
$560$		0			0
$561$	1.29368e9	−	3.56532e8i	0.309355	−	0.0852567i
$562$		0			0
$563$	3.06529e9	−	5.30923e9i	0.723922	−	1.25387i	−0.235494	−	0.971876i	$-0.575671\pi$
$563$	3.06529e9	−	5.30923e9i	0.723922	−	1.25387i	0.959416	−	0.281994i	$-0.0909957\pi$
$564$		0			0
$565$	4.02690e9	+	6.97479e9i	0.939292	+	1.62690i
$566$		0			0
$567$	−2.40377e9	+	4.45156e9i	−0.553800	+	1.02559i
$568$		0			0
$569$	1.71059e9	+	2.96282e9i	0.389271	+	0.674238i	0.992352	−	0.123443i	$-0.0393935\pi$
$569$	1.71059e9	+	2.96282e9i	0.389271	+	0.674238i	−0.603080	+	0.797680i	$0.706060\pi$
$570$		0			0
$571$	1.14292e9	−	1.97960e9i	0.256916	−	0.444991i	−0.708498	−	0.705712i	$-0.750627\pi$
$571$	1.14292e9	−	1.97960e9i	0.256916	−	0.444991i	0.965414	+	0.260721i	$0.0839604\pi$
$572$		0			0
$573$	8.09049e9	−	2.22969e9i	1.79653	−	0.495113i
$574$		0			0
$575$	−4.46615e9			−0.979706
$576$		0			0
$577$	−7.03121e9			−1.52375			−0.761877	−	0.647722i	$-0.775722\pi$
$577$	−7.03121e9			−1.52375			−0.761877	+	0.647722i	$0.775722\pi$
$578$		0			0
$579$	−9.93467e8	−	9.79420e8i	−0.212705	−	0.209698i
$580$		0			0
$581$	2.05846e9	−	3.56536e9i	0.435438	−	0.754200i
$582$		0			0
$583$	1.05663e9	+	1.83013e9i	0.220842	+	0.382510i
$584$		0			0
$585$	1.46350e8	+	1.02774e10i	0.0302237	+	2.12245i
$586$		0			0
$587$	−4.47158e8	−	7.74500e8i	−0.0912488	−	0.158048i	0.816788	−	0.576938i	$-0.195753\pi$
$587$	−4.47158e8	−	7.74500e8i	−0.0912488	−	0.158048i	−0.908037	+	0.418890i	$0.862419\pi$
$588$		0			0
$589$	−2.22607e9	+	3.85567e9i	−0.448886	+	0.777493i
$590$		0			0
$591$	7.90402e8	−	3.03612e9i	0.157504	−	0.605010i
$592$		0			0
$593$	−8.80907e9			−1.73476			−0.867379	−	0.497649i	$-0.834197\pi$
$593$	−8.80907e9			−1.73476			−0.867379	+	0.497649i	$0.834197\pi$
$594$		0			0
$595$	6.70572e9			1.30508
$596$		0			0
$597$	1.39235e9	−	5.34835e9i	0.267817	−	1.02875i
$598$		0			0
$599$	4.91595e8	−	8.51468e8i	0.0934574	−	0.161873i	−0.815506	−	0.578748i	$-0.803541\pi$
$599$	4.91595e8	−	8.51468e8i	0.0934574	−	0.161873i	0.908964	+	0.416875i	$0.136875\pi$
$600$		0			0
$601$	−1.02236e9	−	1.77078e9i	−0.192107	−	0.332739i	0.753841	−	0.657056i	$-0.228199\pi$
$601$	−1.02236e9	−	1.77078e9i	−0.192107	−	0.332739i	−0.945948	+	0.324317i	$0.894865\pi$
$602$		0			0
$603$	4.93169e8	−	2.94172e8i	0.0915978	−	0.0546375i
$604$		0			0
$605$	−3.11682e9	−	5.39848e9i	−0.572226	−	0.991124i
$606$		0			0
$607$	1.78806e9	−	3.09701e9i	0.324505	−	0.562059i	−0.656907	−	0.753971i	$-0.728136\pi$
$607$	1.78806e9	−	3.09701e9i	0.324505	−	0.562059i	0.981412	+	0.191913i	$0.0614690\pi$
$608$		0			0
$609$	2.73542e9	+	2.69674e9i	0.490753	+	0.483815i
$610$		0			0
$611$	−5.30108e9			−0.940199
$612$		0			0
$613$	1.22567e9			0.214913			0.107457	−	0.994210i	$-0.465729\pi$
$613$	1.22567e9			0.214913			0.107457	+	0.994210i	$0.465729\pi$
$614$		0			0
$615$	−8.76985e9	+	2.41692e9i	−1.52030	+	0.418986i
$616$		0			0
$617$	−3.80166e9	+	6.58468e9i	−0.651592	+	1.12859i	0.331145	+	0.943580i	$0.392565\pi$
$617$	−3.80166e9	+	6.58468e9i	−0.651592	+	1.12859i	−0.982737	+	0.185010i	$0.940768\pi$
$618$		0			0
$619$	2.87850e9	+	4.98571e9i	0.487808	+	0.844908i	0.999902	−	0.0140213i	$-0.00446327\pi$
$619$	2.87850e9	+	4.98571e9i	0.487808	+	0.844908i	−0.512094	+	0.858930i	$0.671130\pi$
$620$		0			0
$621$	2.02782e9	+	6.96902e9i	0.339789	+	1.16775i
$622$		0			0
$623$	3.57345e8	+	6.18940e8i	0.0592079	+	0.102551i
$624$		0			0
$625$	3.52977e9	−	6.11374e9i	0.578317	−	1.00167i
$626$		0			0
$627$	2.70403e9	−	7.45215e8i	0.438101	−	0.120738i
$628$		0			0
$629$	−8.92377e9			−1.42979
$630$		0			0
$631$	2.41912e9			0.383313			0.191657	−	0.981462i	$-0.438614\pi$
$631$	2.41912e9			0.383313			0.191657	+	0.981462i	$0.438614\pi$
$632$		0			0
$633$	−3.46299e9	−	3.41403e9i	−0.542674	−	0.535001i
$634$		0			0
$635$	−2.54794e9	+	4.41316e9i	−0.394895	+	0.683978i
$636$		0			0
$637$	1.84734e9	+	3.19969e9i	0.283178	+	0.490479i
$638$		0			0
$639$	1.89202e9	+	1.05673e9i	0.286863	+	0.160218i
$640$		0			0
$641$	3.95891e9	+	6.85703e9i	0.593707	+	1.02833i	0.993728	+	0.111825i	$0.0356696\pi$
$641$	3.95891e9	+	6.85703e9i	0.593707	+	1.02833i	−0.400021	+	0.916506i	$0.630997\pi$
$642$		0			0
$643$	−5.35854e9	+	9.28126e9i	−0.794891	+	1.37679i	0.128017	+	0.991772i	$0.459139\pi$
$643$	−5.35854e9	+	9.28126e9i	−0.794891	+	1.37679i	−0.922908	+	0.385020i	$0.874195\pi$
$644$		0			0
$645$	−1.20210e9	+	4.61754e9i	−0.176393	+	0.677566i
$646$		0			0
$647$	−8.06643e9			−1.17089			−0.585446	−	0.810712i	$-0.699080\pi$
$647$	−8.06643e9			−1.17089			−0.585446	+	0.810712i	$0.699080\pi$
$648$		0			0
$649$	−4.55875e9			−0.654620
$650$		0			0
$651$	−1.57256e9	+	6.04057e9i	−0.223395	+	0.858113i
$652$		0			0
$653$	−5.00680e9	+	8.67202e9i	−0.703661	+	1.21878i	0.263511	+	0.964656i	$0.415119\pi$
$653$	−5.00680e9	+	8.67202e9i	−0.703661	+	1.21878i	−0.967172	+	0.254121i	$0.918214\pi$
$654$		0			0
$655$	4.41894e9	+	7.65382e9i	0.614431	+	1.06423i
$656$		0			0
$657$	1.99317e9	+	1.11322e9i	0.274199	+	0.153145i
$658$		0			0
$659$	7.05570e9	+	1.22208e10i	0.960376	+	1.66342i	0.721555	+	0.692357i	$0.243428\pi$
$659$	7.05570e9	+	1.22208e10i	0.960376	+	1.66342i	0.238821	+	0.971064i	$0.423239\pi$
$660$		0			0
$661$	−4.81075e9	+	8.33246e9i	−0.647899	+	1.12219i	0.335725	+	0.941960i	$0.391019\pi$
$661$	−4.81075e9	+	8.33246e9i	−0.647899	+	1.12219i	−0.983624	+	0.180234i	$0.942315\pi$
$662$		0			0
$663$	−7.03440e9	−	6.93494e9i	−0.937410	−	0.924156i
$664$		0			0
$665$	1.40162e10			1.84822
$666$		0			0
$667$	5.51080e9			0.719075
$668$		0			0
$669$	−7.28249e9	+	2.00701e9i	−0.940348	+	0.259155i
$670$		0			0
$671$	−6.40941e8	+	1.11014e9i	−0.0819011	+	0.141857i
$672$		0			0
$673$	−5.84290e9	−	1.01202e10i	−0.738884	−	1.27978i	−0.952998	−	0.302975i	$-0.902020\pi$
$673$	−5.84290e9	−	1.01202e10i	−0.738884	−	1.27978i	0.214115	−	0.976808i	$-0.431313\pi$
$674$		0			0
$675$	6.25149e9	+	1.53276e9i	0.782384	+	0.191828i
$676$		0			0
$677$	5.55719e9	+	9.62534e9i	0.688327	+	1.19222i	0.972379	+	0.233409i	$0.0749881\pi$
$677$	5.55719e9	+	9.62534e9i	0.688327	+	1.19222i	−0.284051	+	0.958809i	$0.591679\pi$
$678$		0			0
$679$	8.59710e9	−	1.48906e10i	1.05392	−	1.82544i
$680$		0			0
$681$	1.71562e9	−	4.72817e8i	0.208165	−	0.0573691i
$682$		0			0
$683$	1.61450e10			1.93894			0.969472	−	0.245201i	$-0.0788539\pi$
$683$	1.61450e10			1.93894			0.969472	+	0.245201i	$0.0788539\pi$
$684$		0			0
$685$	1.05126e10			1.24967
$686$		0			0
$687$	−9.64833e9	−	9.51191e9i	−1.13528	−	1.11923i
$688$		0			0
$689$	7.77803e9	−	1.34719e10i	0.905946	−	1.56915i
$690$		0			0
$691$	−6.98902e8	−	1.21053e9i	−0.0805829	−	0.139574i	0.822918	−	0.568161i	$-0.192345\pi$
$691$	−6.98902e8	−	1.21053e9i	−0.0805829	−	0.139574i	−0.903500	+	0.428587i	$0.859012\pi$
$692$		0			0
$693$	3.37720e9	−	2.01448e9i	0.385470	−	0.229931i
$694$		0			0
$695$	1.50267e9	+	2.60269e9i	0.169791	+	0.294087i
$696$		0			0
$697$	4.37122e9	−	7.57117e9i	0.488976	−	0.846931i
$698$		0			0
$699$	−8.77860e8	+	3.37207e9i	−0.0972199	+	0.373445i
$700$		0			0
$701$	−1.34244e9			−0.147191			−0.0735956	−	0.997288i	$-0.523447\pi$
$701$	−1.34244e9			−0.147191			−0.0735956	+	0.997288i	$0.523447\pi$
$702$		0			0
$703$	−1.86523e10			−2.02483
$704$		0			0
$705$	−1.87456e9	+	7.20064e9i	−0.201483	+	0.773943i
$706$		0			0
$707$	−2.24525e8	+	3.88889e8i	−0.0238944	+	0.0413864i
$708$		0			0
$709$	7.93153e9	+	1.37378e10i	0.835786	+	1.44762i	0.893389	+	0.449285i	$0.148321\pi$
$709$	7.93153e9	+	1.37378e10i	0.835786	+	1.44762i	−0.0576025	+	0.998340i	$0.518346\pi$
$710$		0			0
$711$	2.28478e8	+	1.60448e10i	0.0238397	+	1.67414i
$712$		0			0
$713$	4.47748e9	+	7.75522e9i	0.462616	+	0.801274i
$714$		0			0
$715$	3.99466e9	−	6.91896e9i	0.408704	−	0.707896i
$716$		0			0
$717$	−5.98335e9	−	5.89875e9i	−0.606216	−	0.597645i
$718$		0			0
$719$	−2.14766e9			−0.215483			−0.107742	−	0.994179i	$-0.534362\pi$
$719$	−2.14766e9			−0.215483			−0.107742	+	0.994179i	$0.534362\pi$
$720$		0			0
$721$	−5.14412e9			−0.511137
$722$		0			0
$723$	2.74421e9	−	7.56291e8i	0.270044	−	0.0744226i
$724$		0			0
$725$	2.44357e9	−	4.23239e9i	0.238145	−	0.412480i
$726$		0			0
$727$	−5.14336e9	−	8.90856e9i	−0.496451	−	0.859879i	0.503540	−	0.863972i	$-0.332031\pi$
$727$	−5.14336e9	−	8.90856e9i	−0.496451	−	0.859879i	−0.999992	+	0.00409288i	$0.998697\pi$
$728$		0			0
$729$	−4.46699e8	−	1.04508e10i	−0.0427040	−	0.999088i
$730$		0			0
$731$	−2.29279e9	−	3.97123e9i	−0.217097	−	0.376023i
$732$		0			0
$733$	1.60210e9	−	2.77493e9i	0.150254	−	0.260248i	−0.781067	−	0.624448i	$-0.785324\pi$
$733$	1.60210e9	−	2.77493e9i	0.150254	−	0.260248i	0.931321	+	0.364200i	$0.118657\pi$
$734$		0			0
$735$	4.99951e9	−	1.37784e9i	0.464432	−	0.127995i
$736$		0			0
$737$	−4.46351e8			−0.0410715
$738$		0			0
$739$	−1.29766e10			−1.18279			−0.591394	−	0.806383i	$-0.701422\pi$
$739$	−1.29766e10			−1.18279			−0.591394	+	0.806383i	$0.701422\pi$
$740$		0			0
$741$	−1.47031e10	−	1.44953e10i	−1.32754	−	1.30877i
$742$		0			0
$743$	3.62353e9	−	6.27614e9i	0.324094	−	0.561348i	−0.657234	−	0.753686i	$-0.728274\pi$
$743$	3.62353e9	−	6.27614e9i	0.324094	−	0.561348i	0.981329	+	0.192339i	$0.0616072\pi$
$744$		0			0
$745$	−5.10143e9	−	8.83593e9i	−0.452007	−	0.782898i
$746$		0			0
$747$	1.21202e8	+	8.51141e9i	0.0106387	+	0.747101i
$748$		0			0
$749$	−1.50012e9	−	2.59828e9i	−0.130449	−	0.225944i
$750$		0			0
$751$	−4.47123e9	+	7.74440e9i	−0.385201	+	0.667188i	−0.991797	−	0.127822i	$-0.959201\pi$
$751$	−4.47123e9	+	7.74440e9i	−0.385201	+	0.667188i	0.606596	+	0.795010i	$0.292535\pi$
$752$		0			0
$753$	−2.52496e9	+	9.69897e9i	−0.215512	+	0.827834i
$754$		0			0
$755$	1.57692e10			1.33350
$756$		0			0
$757$	1.36108e10			1.14037			0.570187	−	0.821515i	$-0.306871\pi$
$757$	1.36108e10			1.14037			0.570187	+	0.821515i	$0.306871\pi$
$758$		0			0
$759$	1.42132e9	−	5.45962e9i	0.117990	−	0.453228i
$760$		0			0
$761$	1.10122e10	−	1.90738e10i	0.905794	−	1.56888i	0.0859464	−	0.996300i	$-0.472609\pi$
$761$	1.10122e10	−	1.90738e10i	0.905794	−	1.56888i	0.819848	−	0.572582i	$-0.194058\pi$
$762$		0			0
$763$	3.33921e9	+	5.78368e9i	0.272149	+	0.471377i
$764$		0			0
$765$	−1.19075e10	+	7.10274e9i	−0.961623	+	0.573602i
$766$		0			0
$767$	1.67789e10	+	2.90619e10i	1.34270	+	2.32563i
$768$		0			0
$769$	−5.17112e9	+	8.95663e9i	−0.410055	+	0.710236i	−0.994895	−	0.100912i	$-0.967824\pi$
$769$	−5.17112e9	+	8.95663e9i	−0.410055	+	0.710236i	0.584840	+	0.811148i	$0.301157\pi$
$770$		0			0
$771$	−1.75531e9	−	1.73049e9i	−0.137931	−	0.135981i
$772$		0			0
$773$	3.65169e9			0.284358			0.142179	−	0.989841i	$-0.454589\pi$
$773$	3.65169e9			0.284358			0.142179	+	0.989841i	$0.454589\pi$
$774$		0			0
$775$	7.94152e9			0.612841
$776$		0			0
$777$	−2.52106e10	+	6.94791e9i	−1.92801	+	0.531350i
$778$		0			0
$779$	9.13662e9	−	1.58251e10i	0.692476	−	1.19940i
$780$		0			0
$781$	−8.42241e8	−	1.45880e9i	−0.0632642	−	0.109577i
$782$		0			0
$783$	−7.71373e9	−	1.89128e9i	−0.574247	−	0.140796i
$784$		0			0
$785$	1.54077e10	+	2.66869e10i	1.13683	+	1.96904i
$786$		0			0
$787$	1.19537e10	−	2.07044e10i	0.874158	−	1.51409i	0.0165008	−	0.999864i	$-0.494747\pi$
$787$	1.19537e10	−	2.07044e10i	0.874158	−	1.51409i	0.857657	−	0.514222i	$-0.171919\pi$
$788$		0			0
$789$	2.20273e10	−	6.07061e9i	1.59659	−	0.440011i
$790$		0			0
$791$	−2.26817e10			−1.62951
$792$		0			0
$793$	9.43617e9			0.671954
$794$		0			0
$795$	−1.55490e10	−	1.53291e10i	−1.09753	−	1.08201i
$796$		0			0
$797$	−1.36082e10	+	2.35700e10i	−0.952128	+	1.64913i	−0.211321	+	0.977417i	$0.567777\pi$
$797$	−1.36082e10	+	2.35700e10i	−0.952128	+	1.64913i	−0.740807	+	0.671718i	$0.765557\pi$
$798$		0			0
$799$	−3.57540e9	−	6.19277e9i	−0.247977	−	0.429508i
$800$		0			0
$801$	−1.29013e9	−	7.20560e8i	−0.0886991	−	0.0495401i
$802$		0			0
$803$	−8.87267e8	−	1.53679e9i	−0.0604714	−	0.104740i
$804$		0			0
$805$	1.40959e10	−	2.44149e10i	0.952375	−	1.64956i
$806$		0			0
$807$	−1.99211e9	+	7.65216e9i	−0.133431	+	0.512538i
$808$		0			0
$809$	−2.09999e10			−1.39443			−0.697216	−	0.716861i	$-0.745578\pi$
$809$	−2.09999e10			−1.39443			−0.697216	+	0.716861i	$0.745578\pi$
$810$		0			0
$811$	1.21414e10			0.799276			0.399638	−	0.916673i	$-0.369136\pi$
$811$	1.21414e10			0.799276			0.399638	+	0.916673i	$0.369136\pi$
$812$		0			0
$813$	2.35085e9	−	9.03016e9i	0.153429	−	0.589357i
$814$		0			0
$815$	5.43248e9	−	9.40933e9i	0.351517	−	0.608845i
$816$		0			0
$817$	−4.79234e9	−	8.30057e9i	−0.307447	−	0.532514i
$818$		0			0
$819$	−2.52724e10	−	1.41151e10i	−1.60750	−	0.897821i
$820$		0			0
$821$	5.66101e9	+	9.80516e9i	0.357020	+	0.618377i	0.987462	−	0.157860i	$-0.0504593\pi$
$821$	5.66101e9	+	9.80516e9i	0.357020	+	0.618377i	−0.630441	+	0.776237i	$0.717126\pi$
$822$		0			0
$823$	−2.91399e9	+	5.04718e9i	−0.182217	+	0.315609i	−0.942635	−	0.333825i	$-0.891661\pi$
$823$	−2.91399e9	+	5.04718e9i	−0.182217	+	0.315609i	0.760418	+	0.649434i	$0.224994\pi$
$824$		0			0
$825$	−3.56285e9	−	3.51247e9i	−0.220906	−	0.217783i
$826$		0			0
$827$	−2.11794e10			−1.30210			−0.651050	−	0.759034i	$-0.725671\pi$
$827$	−2.11794e10			−1.30210			−0.651050	+	0.759034i	$0.725671\pi$
$828$		0			0
$829$	−1.77492e10			−1.08203			−0.541013	−	0.841014i	$-0.681959\pi$
$829$	−1.77492e10			−1.08203			−0.541013	+	0.841014i	$0.681959\pi$
$830$		0			0
$831$	1.59994e10	−	4.40935e9i	0.967165	−	0.266545i
$832$		0			0
$833$	−2.49194e9	+	4.31617e9i	−0.149376	+	0.258727i
$834$		0			0
$835$	−5.01716e9	−	8.68997e9i	−0.298233	−	0.516554i
$836$		0			0
$837$	−3.60578e9	−	1.23920e10i	−0.212550	−	0.730471i
$838$		0			0
$839$	−1.31911e10	−	2.28476e10i	−0.771104	−	1.33559i	−0.936959	−	0.349441i	$-0.886372\pi$
$839$	−1.31911e10	−	2.28476e10i	−0.771104	−	1.33559i	0.165855	−	0.986150i	$-0.446962\pi$
$840$		0			0
$841$	5.60981e9	−	9.71647e9i	0.325209	−	0.563278i
$842$		0			0
$843$	1.26458e10	−	3.48511e9i	0.727026	−	0.200364i
$844$		0			0
$845$	−3.52439e10			−2.00949
$846$		0			0
$847$	1.75556e10			0.992715
$848$		0			0
$849$	5.29676e9	+	5.22187e9i	0.297052	+	0.292853i
$850$		0			0
$851$	−1.87584e10	+	3.24905e10i	−1.04338	+	1.80719i
$852$		0			0
$853$	−1.33903e10	−	2.31927e10i	−0.738701	−	1.27947i	−0.953080	−	0.302717i	$-0.902106\pi$
$853$	−1.33903e10	−	2.31927e10i	−0.738701	−	1.27947i	0.214379	−	0.976750i	$-0.431227\pi$
$854$		0			0
$855$	−2.48887e10	+	1.48460e10i	−1.36183	+	0.812321i
$856$		0			0
$857$	−6.44399e9	−	1.11613e10i	−0.349721	−	0.605735i	0.636479	−	0.771294i	$-0.280390\pi$
$857$	−6.44399e9	−	1.11613e10i	−0.349721	−	0.605735i	−0.986200	+	0.165560i	$0.947057\pi$
$858$		0			0
$859$	2.13794e9	−	3.70302e9i	0.115085	−	0.199333i	−0.802729	−	0.596344i	$-0.796619\pi$
$859$	2.13794e9	−	3.70302e9i	0.115085	−	0.199333i	0.917814	+	0.397011i	$0.129953\pi$
$860$		0			0
$861$	6.45437e9	−	2.47927e10i	0.344622	−	1.32377i
$862$		0			0
$863$	2.63998e10			1.39818			0.699089	−	0.715035i	$-0.253589\pi$
$863$	2.63998e10			1.39818			0.699089	+	0.715035i	$0.253589\pi$
$864$		0			0
$865$	9.88231e9			0.519161
$866$		0			0
$867$	−1.47755e9	+	5.67562e9i	−0.0769973	+	0.295765i
$868$		0			0
$869$	6.23635e9	−	1.08017e10i	0.322375	−	0.558370i
$870$		0			0
$871$	1.64283e9	+	2.84547e9i	0.0842423	+	0.145912i
$872$		0			0
$873$	5.06197e8	+	3.55476e10i	0.0257495	+	1.80826i
$874$		0			0
$875$	3.01736e9	+	5.22622e9i	0.152265	+	0.263730i
$876$		0			0
$877$	−7.19624e9	+	1.24642e10i	−0.360252	+	0.623975i	−0.988002	−	0.154440i	$-0.950643\pi$
$877$	−7.19624e9	+	1.24642e10i	−0.360252	+	0.623975i	0.627750	+	0.778415i	$0.283976\pi$
$878$		0			0
$879$	1.14688e10	+	1.13066e10i	0.569582	+	0.561529i
$880$		0			0
$881$	2.86878e10			1.41345			0.706727	−	0.707487i	$-0.250171\pi$
$881$	2.86878e10			1.41345			0.706727	+	0.707487i	$0.250171\pi$
$882$		0			0
$883$	−1.10176e10			−0.538550			−0.269275	−	0.963063i	$-0.586784\pi$
$883$	−1.10176e10			−0.538550			−0.269275	+	0.963063i	$0.586784\pi$
$884$		0			0
$885$	4.54091e10	−	1.25145e10i	2.20212	−	0.606893i
$886$		0			0
$887$	8.77534e9	−	1.51993e10i	0.422213	−	0.731294i	−0.573943	−	0.818895i	$-0.694587\pi$
$887$	8.77534e9	−	1.51993e10i	0.422213	−	0.731294i	0.996156	+	0.0876014i	$0.0279202\pi$
$888$		0			0
$889$	−7.17571e9	−	1.24287e10i	−0.342538	−	0.593293i
$890$		0			0
$891$	−3.86321e9	+	7.15430e9i	−0.182968	+	0.338841i
$892$		0			0
$893$	−7.47322e9	−	1.29440e10i	−0.351178	−	0.608259i
$894$		0			0
$895$	4.12619e9	−	7.14677e9i	0.192384	−	0.333218i
$896$		0			0
$897$	−4.00362e10	+	1.10338e10i	−1.85216	+	0.510446i
$898$		0			0
$899$	−9.79907e9			−0.449807
$900$		0			0
$901$	2.09841e10			0.955770
$902$		0			0
$903$	−9.56931e9	−	9.43402e9i	−0.432488	−	0.426373i
$904$		0			0
$905$	2.36994e10	−	4.10486e10i	1.06284	−	1.84089i
$906$		0			0
$907$	−8.42225e9	−	1.45878e10i	−0.374803	−	0.649177i	0.615495	−	0.788141i	$-0.288956\pi$
$907$	−8.42225e9	−	1.45878e10i	−0.374803	−	0.649177i	−0.990297	+	0.138964i	$0.955623\pi$
$908$		0			0
$909$	−1.32200e7	−	9.28375e8i	−0.000583793	−	0.0409968i
$910$		0			0
$911$	7.36357e9	+	1.27541e10i	0.322682	+	0.558901i	0.981040	−	0.193803i	$-0.0620823\pi$
$911$	7.36357e9	+	1.27541e10i	0.322682	+	0.558901i	−0.658359	+	0.752704i	$0.728749\pi$
$912$		0			0
$913$	3.30824e9	−	5.73005e9i	0.143863	−	0.249178i
$914$		0			0
$915$	3.33682e9	−	1.28175e10i	0.143999	−	0.553133i
$916$		0			0
$917$	−2.48899e10			−1.06593
$918$		0			0
$919$	−7.11506e9			−0.302395			−0.151197	−	0.988504i	$-0.548313\pi$
$919$	−7.11506e9			−0.302395			−0.151197	+	0.988504i	$0.548313\pi$
$920$		0			0
$921$	−6.22958e9	+	2.39293e10i	−0.262755	+	1.00930i
$922$		0			0
$923$	−6.19989e9	+	1.07385e10i	−0.259524	+	0.449509i
$924$		0			0
$925$	1.66355e10	+	2.88136e10i	0.691099	+	1.19702i
$926$		0			0
$927$	9.13450e9	−	5.44868e9i	0.376622	−	0.224653i
$928$		0			0
$929$	1.69702e10	+	2.93933e10i	0.694437	+	1.20280i	0.970370	+	0.241624i	$0.0776799\pi$
$929$	1.69702e10	+	2.93933e10i	0.694437	+	1.20280i	−0.275933	+	0.961177i	$0.588987\pi$
$930$		0			0
$931$	−5.20860e9	+	9.02156e9i	−0.211542	+	0.366402i
$932$		0			0
$933$	9.55001e9	+	9.41499e9i	0.384962	+	0.379519i
$934$		0			0
$935$	1.07771e10			0.431181
$936$		0			0
$937$	−3.78364e10			−1.50252			−0.751261	−	0.660005i	$-0.770554\pi$
$937$	−3.78364e10			−1.50252			−0.751261	+	0.660005i	$0.770554\pi$
$938$		0			0
$939$	−1.12823e9	+	3.10935e8i	−0.0444703	+	0.0122558i
$940$		0			0
$941$	−8.44461e9	+	1.46265e10i	−0.330382	+	0.572238i	−0.982587	−	0.185805i	$-0.940511\pi$
$941$	−8.44461e9	+	1.46265e10i	−0.330382	+	0.572238i	0.652205	+	0.758043i	$0.273844\pi$
$942$		0			0
$943$	−1.83772e10	−	3.18303e10i	−0.713657	−	1.23609i
$944$		0			0
$945$	−2.81098e10	+	2.93370e10i	−1.08354	+	1.13085i
$946$		0			0
$947$	5.84561e9	+	1.01249e10i	0.223668	+	0.387405i	0.955919	−	0.293630i	$-0.0948634\pi$
$947$	5.84561e9	+	1.01249e10i	0.223668	+	0.387405i	−0.732251	+	0.681035i	$0.761530\pi$
$948$		0			0
$949$	−6.53134e9	+	1.13126e10i	−0.248068	+	0.429666i
$950$		0			0
$951$	2.34563e10	−	6.46443e9i	0.884357	−	0.243724i
$952$		0			0
$953$	5.10551e10			1.91079			0.955397	−	0.295323i	$-0.0954274\pi$
$953$	5.10551e10			1.91079			0.955397	+	0.295323i	$0.0954274\pi$
$954$		0			0
$955$	6.73981e10			2.50401
$956$		0			0
$957$	4.39621e9	+	4.33405e9i	0.162139	+	0.159846i
$958$		0			0
$959$	−1.48032e10	+	2.56400e10i	−0.541990	+	0.938755i
$960$		0			0
$961$	5.79464e9	+	1.00366e10i	0.210618	+	0.364800i
$962$		0			0
$963$	5.41591e9	+	3.02488e9i	0.195425	+	0.109148i
$964$		0			0
$965$	−5.60202e9	−	9.70298e9i	−0.200678	−	0.347584i
$966$		0			0
$967$	−4.34365e9	+	7.52343e9i	−0.154476	+	0.267561i	−0.932868	−	0.360218i	$-0.882702\pi$
$967$	−4.34365e9	+	7.52343e9i	−0.154476	+	0.267561i	0.778392	+	0.627779i	$0.216036\pi$
$968$		0			0
$969$	7.01673e9	−	2.69529e10i	0.247743	−	0.951640i
$970$		0			0
$971$	−3.91347e10			−1.37181			−0.685906	−	0.727690i	$-0.740594\pi$
$971$	−3.91347e10			−1.37181			−0.685906	+	0.727690i	$0.740594\pi$
$972$		0			0
$973$	−8.46384e9			−0.294559
$974$		0			0
$975$	−9.27853e9	+	3.56410e10i	−0.320600	+	1.23150i
$976$		0			0
$977$	1.36255e10	−	2.36001e10i	0.467437	−	0.809624i	−0.531871	−	0.846825i	$-0.678511\pi$
$977$	1.36255e10	−	2.36001e10i	0.467437	−	0.809624i	0.999308	+	0.0372010i	$0.0118442\pi$
$978$		0			0
$979$	5.74305e8	+	9.94725e8i	0.0195615	+	0.0338816i
$980$		0			0
$981$	−1.20556e10	−	6.73327e9i	−0.407706	−	0.227711i
$982$		0			0
$983$	3.56677e9	+	6.17782e9i	0.119767	+	0.207443i	0.919675	−	0.392680i	$-0.128452\pi$
$983$	3.56677e9	+	6.17782e9i	0.119767	+	0.207443i	−0.799908	+	0.600122i	$0.795119\pi$
$984$		0			0
$985$	1.25981e10	−	2.18205e10i	0.420028	−	0.727509i
$986$		0			0
$987$	−1.49225e10	−	1.47115e10i	−0.494005	−	0.487020i
$988$		0			0
$989$	−1.92784e10			−0.633702
$990$		0			0
$991$	1.99378e10			0.650760			0.325380	−	0.945583i	$-0.394508\pi$
$991$	1.99378e10			0.650760			0.325380	+	0.945583i	$0.394508\pi$
$992$		0			0
$993$	−2.27695e10	+	6.27514e9i	−0.737956	+	0.203377i
$994$		0			0
$995$	2.21925e10	−	3.84385e10i	0.714208	−	1.23704i
$996$		0			0
$997$	6.37020e9	+	1.10335e10i	0.203573	+	0.352598i	0.949677	−	0.313231i	$-0.101411\pi$
$997$	6.37020e9	+	1.10335e10i	0.203573	+	0.352598i	−0.746104	+	0.665829i	$0.768078\pi$
$998$		0			0
$999$	3.74077e10	−	3.90408e10i	1.18708	−	1.23891i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.8.i.a.49.2		6
3.2	odd	2		432.8.i.a.145.1		6
4.3	odd	2		18.8.c.a.13.2	yes	6
9.2	odd	6		432.8.i.a.289.1		6
9.7	even	3	inner	144.8.i.a.97.2		6
12.11	even	2		54.8.c.a.37.1		6
36.7	odd	6		18.8.c.a.7.2	✓	6
36.11	even	6		54.8.c.a.19.1		6
36.23	even	6		162.8.a.e.1.3		3
36.31	odd	6		162.8.a.f.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.8.c.a.7.2	✓	6	36.7	odd	6
18.8.c.a.13.2	yes	6	4.3	odd	2
54.8.c.a.19.1		6	36.11	even	6
54.8.c.a.37.1		6	12.11	even	2
144.8.i.a.49.2		6	1.1	even	1	trivial
144.8.i.a.97.2		6	9.7	even	3	inner
162.8.a.e.1.3		3	36.23	even	6
162.8.a.f.1.1		3	36.31	odd	6
432.8.i.a.145.1		6	3.2	odd	2
432.8.i.a.289.1		6	9.2	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	144.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(11.7819 - 45.2569i) q^{3} +(187.789 - 325.261i) q^{5} +(528.867 + 916.024i) q^{7} +(-1909.38 - 1066.42i) q^{9} +(849.965 + 1472.18i) q^{11} +(6256.75 - 10837.0i) q^{13} +(-12507.8 - 12330.9i) q^{15} +16879.9 q^{17} +35281.9 q^{19} +(47687.4 - 13142.4i) q^{21} +(35482.7 - 61457.8i) q^{23} +(-31467.1 - 54502.6i) q^{25} +(-70758.9 + 73848.0i) q^{27} +(38827.4 + 67251.0i) q^{29} +(-63093.8 + 109282. i) q^{31} +(76640.6 - 21121.7i) q^{33} +397262. q^{35} -528664. q^{37} +(-416733. - 410841. i) q^{39} +(258960. - 448533. i) q^{41} +(-135830. - 235264. i) q^{43} +(-705425. + 420782. i) q^{45} +(-211814. - 366873. i) q^{47} +(-147628. + 255699. i) q^{49} +(198876. - 763930. i) q^{51} +1.24314e6 q^{53} +638457. q^{55} +(415687. - 1.59675e6i) q^{57} +(-1.34086e6 + 2.32244e6i) q^{59} +(377040. + 653052. i) q^{61} +(-32937.4 - 2.31303e6i) q^{63} +(-2.34990e6 - 4.07014e6i) q^{65} +(-131285. + 227392. i) q^{67} +(-2.36334e6 - 2.32993e6i) q^{69} -990913. q^{71} -1.04389e6 q^{73} +(-2.83736e6 + 781961. i) q^{75} +(-899036. + 1.55718e6i) q^{77} +(-3.66860e6 - 6.35419e6i) q^{79} +(2.50846e6 + 4.07240e6i) q^{81} +(-1.94611e6 - 3.37075e6i) q^{83} +(3.16986e6 - 5.49035e6i) q^{85} +(3.50103e6 - 964865. i) q^{87} +675681. q^{89} +1.32359e7 q^{91} +(4.20239e6 + 4.14298e6i) q^{93} +(6.62556e6 - 1.14758e7i) q^{95} +(-8.12785e6 - 1.40779e7i) q^{97} +(-52935.2 - 3.71737e6i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 27 q^{3} + 54 q^{5} - 210 q^{7} + 2295 q^{9} - 6579 q^{11} + 10092 q^{13} - 756 q^{15} - 29790 q^{17} + 137490 q^{19} + 85464 q^{21} + 39654 q^{23} - 7923 q^{25} - 87480 q^{27} + 239832 q^{29} - 145704 q^{31}+ \cdots + 11453238 q^{99}+O(q^{100}) \) 6 * q + 27 * q^3 + 54 * q^5 - 210 * q^7 + 2295 * q^9 - 6579 * q^11 + 10092 * q^13 - 756 * q^15 - 29790 * q^17 + 137490 * q^19 + 85464 * q^21 + 39654 * q^23 - 7923 * q^25 - 87480 * q^27 + 239832 * q^29 - 145704 * q^31 + 175365 * q^33 + 1087776 * q^35 - 720384 * q^37 - 2386548 * q^39 + 1086993 * q^41 + 299967 * q^43 - 2798874 * q^45 - 131634 * q^47 + 482175 * q^49 + 3719223 * q^51 + 1909152 * q^53 + 5494824 * q^55 + 6076107 * q^57 - 2504853 * q^59 + 7309038 * q^61 + 1254960 * q^63 - 1786698 * q^65 - 3433035 * q^67 - 5815206 * q^69 - 2269368 * q^71 - 15901746 * q^73 - 4230945 * q^75 - 6147432 * q^77 - 7076928 * q^79 + 4749435 * q^81 + 10914444 * q^83 + 17613396 * q^85 + 11533266 * q^87 + 23800356 * q^89 + 37170552 * q^91 + 30872466 * q^93 + 17342424 * q^95 + 519357 * q^97 + 11453238 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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