LMFDB - Embedded newform 144.8.i.b.49.4

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:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1336x^{6} + 633664x^{4} + 125389995x^{2} + 8783438400 $ x^8 + 1336x^6 + 633664x^4 + 125389995*x^2 + 8783438400
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{10} $
Twist minimal:	no (minimal twist has level 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			49.4
Root			$13.4379i$ of defining polynomial
Character	$\chi$	$=$	144.49
Dual form			144.8.i.b.97.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(46.0138 + 8.35049i) q^{3} +(103.252 - 178.837i) q^{5} +(808.604 + 1400.54i) q^{7} +(2047.54 + 768.476i) q^{9} +(2338.01 + 4049.55i) q^{11} +(-3336.65 + 5779.25i) q^{13} +(6244.39 - 7366.78i) q^{15} -25731.9 q^{17} -22384.5 q^{19} +(25511.7 + 71196.6i) q^{21} +(11710.7 - 20283.5i) q^{23} +(17740.6 + 30727.7i) q^{25} +(87797.9 + 52458.4i) q^{27} +(-81488.4 - 141142. i) q^{29} +(-116733. + 202187. i) q^{31} +(73764.9 + 205859. i) q^{33} +333959. q^{35} +308867. q^{37} +(-201792. + 238063. i) q^{39} +(157555. - 272893. i) q^{41} +(-61173.2 - 105955. i) q^{43} +(348844. - 286830. i) q^{45} +(-97801.6 - 169397. i) q^{47} +(-895910. + 1.55176e6i) q^{49} +(-1.18402e6 - 214874. i) q^{51} -369367. q^{53} +965615. q^{55} +(-1.02999e6 - 186921. i) q^{57} +(-560669. + 971107. i) q^{59} +(-45155.7 - 78212.0i) q^{61} +(579364. + 3.48906e6i) q^{63} +(689031. + 1.19344e6i) q^{65} +(1.76399e6 - 3.05533e6i) q^{67} +(708232. - 835532. i) q^{69} +703989. q^{71} +4.06113e6 q^{73} +(559723. + 1.56204e6i) q^{75} +(-3.78105e6 + 6.54897e6i) q^{77} +(2.33651e6 + 4.04696e6i) q^{79} +(3.60186e6 + 3.14697e6i) q^{81} +(4.38570e6 + 7.59625e6i) q^{83} +(-2.65686e6 + 4.60182e6i) q^{85} +(-2.57099e6 - 7.17495e6i) q^{87} +1.07368e7 q^{89} -1.07921e7 q^{91} +(-7.05968e6 + 8.32862e6i) q^{93} +(-2.31123e6 + 4.00317e6i) q^{95} +(-3.72029e6 - 6.44373e6i) q^{97} +(1.67518e6 + 1.00883e7i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 12 q^{3} + 54 q^{5} + 44 q^{7} - 2238 q^{9} - 2172 q^{11} - 6398 q^{13} - 17604 q^{15} - 51972 q^{17} - 90712 q^{19} + 192390 q^{21} + 2028 q^{23} - 173446 q^{25} + 288360 q^{27} + 283098 q^{29} + 812 q^{31}+ \cdots - 26558784 q^{99}+O(q^{100}) $ 8 * q + 12 * q^3 + 54 * q^5 + 44 * q^7 - 2238 * q^9 - 2172 * q^11 - 6398 * q^13 - 17604 * q^15 - 51972 * q^17 - 90712 * q^19 + 192390 * q^21 + 2028 * q^23 - 173446 * q^25 + 288360 * q^27 + 283098 * q^29 + 812 * q^31 + 342108 * q^33 + 1492992 * q^35 + 441688 * q^37 - 67428 * q^39 + 610704 * q^41 + 84380 * q^43 - 652698 * q^45 - 855708 * q^47 - 1273038 * q^49 - 1748556 * q^51 - 1842600 * q^53 - 7016760 * q^55 - 4893306 * q^57 - 380796 * q^59 - 3151130 * q^61 + 3149076 * q^63 - 326538 * q^65 + 9961580 * q^67 + 15322554 * q^69 - 8970384 * q^71 + 29234932 * q^73 + 26321244 * q^75 - 5954826 * q^77 + 17309396 * q^79 + 6907266 * q^81 + 11520192 * q^83 - 14675148 * q^85 - 37274652 * q^87 + 26066064 * q^89 - 26231936 * q^91 - 48684366 * q^93 + 31016952 * q^95 - 22003112 * q^97 - 26558784 * 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
$3$	46.0138	+	8.35049i	0.983929	+	0.178561i
$3$	46.0138	+	8.35049i	0.983929	+	0.178561i
$4$		0			0
$5$	103.252	−	178.837i	0.369405	−	0.639828i	−0.620068	−	0.784548i	$-0.712895\pi$
$5$	103.252	−	178.837i	0.369405	−	0.639828i	0.989473	+	0.144720i	$0.0462283\pi$
$6$		0			0
$7$	808.604	+	1400.54i	0.891031	+	1.54331i	0.838641	+	0.544684i	$0.183350\pi$
$7$	808.604	+	1400.54i	0.891031	+	1.54331i	0.0523898	+	0.998627i	$0.483316\pi$
$8$		0			0
$9$	2047.54	+	768.476i	0.936232	+	0.351384i
$10$		0			0
$11$	2338.01	+	4049.55i	0.529629	+	0.917345i	0.999403	+	0.0345577i	$0.0110022\pi$
$11$	2338.01	+	4049.55i	0.529629	+	0.917345i	−0.469774	+	0.882787i	$0.655664\pi$
$12$		0			0
$13$	−3336.65	+	5779.25i	−0.421220	+	0.729575i	−0.996059	−	0.0886915i	$-0.971731\pi$
$13$	−3336.65	+	5779.25i	−0.421220	+	0.729575i	0.574839	+	0.818267i	$0.305065\pi$
$14$		0			0
$15$	6244.39	−	7366.78i	0.477717	−	0.563584i
$16$		0			0
$17$	−25731.9			−1.27028			−0.635141	−	0.772396i	$-0.719058\pi$
$17$	−25731.9			−1.27028			−0.635141	+	0.772396i	$0.719058\pi$
$18$		0			0
$19$	−22384.5			−0.748702			−0.374351	−	0.927287i	$-0.622134\pi$
$19$	−22384.5			−0.748702			−0.374351	+	0.927287i	$0.622134\pi$
$20$		0			0
$21$	25511.7	+	71196.6i	0.601135	+	1.67761i
$22$		0			0
$23$	11710.7	−	20283.5i	0.200695	−	0.347613i	−0.748058	−	0.663634i	$-0.769013\pi$
$23$	11710.7	−	20283.5i	0.200695	−	0.347613i	0.948752	+	0.316020i	$0.102347\pi$
$24$		0			0
$25$	17740.6	+	30727.7i	0.227080	+	0.393315i
$26$		0			0
$27$	87797.9	+	52458.4i	0.858442	+	0.512911i
$28$		0			0
$29$	−81488.4	−	141142.i	−0.620444	−	1.07464i	−0.989403	−	0.145195i	$-0.953619\pi$
$29$	−81488.4	−	141142.i	−0.620444	−	1.07464i	0.368959	−	0.929446i	$-0.379714\pi$
$30$		0			0
$31$	−116733.	+	202187.i	−0.703763	+	1.21895i	0.263373	+	0.964694i	$0.415165\pi$
$31$	−116733.	+	202187.i	−0.703763	+	1.21895i	−0.967136	+	0.254260i	$0.918168\pi$
$32$		0			0
$33$	73764.9	+	205859.i	0.357315	+	0.997173i
$34$		0			0
$35$	333959.			1.31660
$36$		0			0
$37$	308867.			1.00246			0.501229	−	0.865315i	$-0.332882\pi$
$37$	308867.			1.00246			0.501229	+	0.865315i	$0.332882\pi$
$38$		0			0
$39$	−201792.	+	238063.i	−0.544725	+	0.642636i
$40$		0			0
$41$	157555.	−	272893.i	0.357017	−	0.618372i	−0.630444	−	0.776235i	$-0.717127\pi$
$41$	157555.	−	272893.i	0.357017	−	0.618372i	0.987461	+	0.157863i	$0.0504605\pi$
$42$		0			0
$43$	−61173.2	−	105955.i	−0.117333	−	0.203228i	0.801377	−	0.598160i	$-0.204101\pi$
$43$	−61173.2	−	105955.i	−0.117333	−	0.203228i	−0.918710	+	0.394933i	$0.870768\pi$
$44$		0			0
$45$	348844.	−	286830.i	0.570673	−	0.469224i
$46$		0			0
$47$	−97801.6	−	169397.i	−0.137405	−	0.237993i	0.789108	−	0.614254i	$-0.210543\pi$
$47$	−97801.6	−	169397.i	−0.137405	−	0.237993i	−0.926514	+	0.376261i	$0.877210\pi$
$48$		0			0
$49$	−895910.	+	1.55176e6i	−1.08787	+	1.88425i
$50$		0			0
$51$	−1.18402e6	−	214874.i	−1.24987	−	0.226823i
$52$		0			0
$53$	−369367.			−0.340794			−0.170397	−	0.985375i	$-0.554505\pi$
$53$	−369367.			−0.340794			−0.170397	+	0.985375i	$0.554505\pi$
$54$		0			0
$55$	965615.			0.782590
$56$		0			0
$57$	−1.02999e6	−	186921.i	−0.736669	−	0.133689i
$58$		0			0
$59$	−560669.	+	971107.i	−0.355406	+	0.615581i	−0.987187	−	0.159565i	$-0.948991\pi$
$59$	−560669.	+	971107.i	−0.355406	+	0.615581i	0.631781	+	0.775147i	$0.282324\pi$
$60$		0			0
$61$	−45155.7	−	78212.0i	−0.0254717	−	0.0441183i	0.853009	−	0.521897i	$-0.174775\pi$
$61$	−45155.7	−	78212.0i	−0.0254717	−	0.0441183i	−0.878480	+	0.477779i	$0.841442\pi$
$62$		0			0
$63$	579364.	+	3.48906e6i	0.291917	+	1.75799i
$64$		0			0
$65$	689031.	+	1.19344e6i	0.311202	+	0.539017i
$66$		0			0
$67$	1.76399e6	−	3.05533e6i	0.716532	−	1.24107i	−0.245834	−	0.969312i	$-0.579062\pi$
$67$	1.76399e6	−	3.05533e6i	0.716532	−	1.24107i	0.962366	−	0.271758i	$-0.0876049\pi$
$68$		0			0
$69$	708232.	−	835532.i	0.259540	−	0.306190i
$70$		0			0
$71$	703989.			0.233433			0.116716	−	0.993165i	$-0.462763\pi$
$71$	703989.			0.233433			0.116716	+	0.993165i	$0.462763\pi$
$72$		0			0
$73$	4.06113e6			1.22185			0.610923	−	0.791690i	$-0.290799\pi$
$73$	4.06113e6			1.22185			0.610923	+	0.791690i	$0.290799\pi$
$74$		0			0
$75$	559723.	+	1.56204e6i	0.153200	+	0.427541i
$76$		0			0
$77$	−3.78105e6	+	6.54897e6i	−0.943832	+	1.63477i
$78$		0			0
$79$	2.33651e6	+	4.04696e6i	0.533180	+	0.923494i	0.999249	+	0.0387462i	$0.0123364\pi$
$79$	2.33651e6	+	4.04696e6i	0.533180	+	0.923494i	−0.466069	+	0.884748i	$0.654330\pi$
$80$		0			0
$81$	3.60186e6	+	3.14697e6i	0.753059	+	0.657953i
$82$		0			0
$83$	4.38570e6	+	7.59625e6i	0.841910	+	1.45823i	0.888278	+	0.459306i	$0.151902\pi$
$83$	4.38570e6	+	7.59625e6i	0.841910	+	1.45823i	−0.0463686	+	0.998924i	$0.514765\pi$
$84$		0			0
$85$	−2.65686e6	+	4.60182e6i	−0.469248	+	0.812762i
$86$		0			0
$87$	−2.57099e6	−	7.17495e6i	−0.418584	−	1.16816i
$88$		0			0
$89$	1.07368e7			1.61439			0.807194	−	0.590286i	$-0.200985\pi$
$89$	1.07368e7			1.61439			0.807194	+	0.590286i	$0.200985\pi$
$90$		0			0
$91$	−1.07921e7			−1.50128
$92$		0			0
$93$	−7.05968e6	+	8.32862e6i	−0.910111	+	1.07370i
$94$		0			0
$95$	−2.31123e6	+	4.00317e6i	−0.276574	+	0.479040i
$96$		0			0
$97$	−3.72029e6	−	6.44373e6i	−0.413881	−	0.716863i	0.581429	−	0.813597i	$-0.302494\pi$
$97$	−3.72029e6	−	6.44373e6i	−0.413881	−	0.716863i	−0.995310	+	0.0967340i	$0.969160\pi$
$98$		0			0
$99$	1.67518e6	+	1.00883e7i	0.173516	+	1.04495i
$100$		0			0
$101$	−7.40397e6	−	1.28241e7i	−0.715056	−	1.23851i	−0.962938	−	0.269723i	$-0.913068\pi$
$101$	−7.40397e6	−	1.28241e7i	−0.715056	−	1.23851i	0.247882	−	0.968790i	$-0.420265\pi$
$102$		0			0
$103$	−8.62804e6	+	1.49442e7i	−0.778004	+	1.34754i	0.155086	+	0.987901i	$0.450435\pi$
$103$	−8.62804e6	+	1.49442e7i	−0.778004	+	1.34754i	−0.933090	+	0.359642i	$0.882899\pi$
$104$		0			0
$105$	1.53667e7	+	2.78872e6i	1.29544	+	0.235095i
$106$		0			0
$107$	1.31928e7			1.04110			0.520552	−	0.853830i	$-0.325726\pi$
$107$	1.31928e7			1.04110			0.520552	+	0.853830i	$0.325726\pi$
$108$		0			0
$109$	−2.46882e6			−0.182598			−0.0912990	−	0.995824i	$-0.529102\pi$
$109$	−2.46882e6			−0.182598			−0.0912990	+	0.995824i	$0.529102\pi$
$110$		0			0
$111$	1.42122e7	+	2.57919e6i	0.986347	+	0.179000i
$112$		0			0
$113$	−1.72194e6	+	2.98249e6i	−0.112265	+	0.194448i	−0.916683	−	0.399615i	$-0.869144\pi$
$113$	−1.72194e6	+	2.98249e6i	−0.112265	+	0.194448i	0.804418	+	0.594063i	$0.202477\pi$
$114$		0			0
$115$	−2.41830e6	−	4.18862e6i	−0.148275	−	0.256820i
$116$		0			0
$117$	−1.12731e7	+	9.26911e6i	−0.650721	+	0.535041i
$118$		0			0
$119$	−2.08069e7	−	3.60386e7i	−1.13186	−	1.96044i
$120$		0			0
$121$	−1.18899e6	+	2.05940e6i	−0.0610141	+	0.105680i
$122$		0			0
$123$	9.52850e6	−	1.12412e7i	0.461697	−	0.544684i
$124$		0			0
$125$	2.34601e7			1.07435
$126$		0			0
$127$	119100.			0.00515940			0.00257970	−	0.999997i	$-0.499179\pi$
$127$	119100.			0.00515940			0.00257970	+	0.999997i	$0.499179\pi$
$128$		0			0
$129$	−1.93004e6	−	5.38623e6i	−0.0791592	−	0.220913i
$130$		0			0
$131$	1.10622e7	−	1.91602e7i	0.429923	−	0.744648i	−0.566943	−	0.823757i	$-0.691874\pi$
$131$	1.10622e7	−	1.91602e7i	0.429923	−	0.744648i	0.996866	+	0.0791089i	$0.0252075\pi$
$132$		0			0
$133$	−1.81002e7	−	3.13504e7i	−0.667116	−	1.15548i
$134$		0			0
$135$	1.84468e7	−	1.02851e7i	0.645287	−	0.359783i
$136$		0			0
$137$	2.94797e6	+	5.10604e6i	0.0979493	+	0.169653i	0.910836	−	0.412769i	$-0.135438\pi$
$137$	2.94797e6	+	5.10604e6i	0.0979493	+	0.169653i	−0.812886	+	0.582422i	$0.802105\pi$
$138$		0			0
$139$	1.77894e7	−	3.08121e7i	0.561834	−	0.973126i	−0.435502	−	0.900188i	$-0.643429\pi$
$139$	1.77894e7	−	3.08121e7i	0.561834	−	0.973126i	0.997336	−	0.0729381i	$-0.0232376\pi$
$140$		0			0
$141$	−3.08567e6	−	8.61131e6i	−0.0927007	−	0.258703i
$142$		0			0
$143$	−3.12045e7			−0.892363
$144$		0			0
$145$	−3.36553e7			−0.916780
$146$		0			0
$147$	−5.41822e7	+	6.39211e7i	−1.40684	+	1.65972i
$148$		0			0
$149$	1.67110e7	−	2.89443e7i	0.413857	−	0.716821i	−0.581451	−	0.813582i	$-0.697515\pi$
$149$	1.67110e7	−	2.89443e7i	0.413857	−	0.716821i	0.995308	+	0.0967603i	$0.0308480\pi$
$150$		0			0
$151$	−2.49541e7	−	4.32217e7i	−0.589823	−	1.02160i	−0.994255	−	0.107036i	$-0.965864\pi$
$151$	−2.49541e7	−	4.32217e7i	−0.589823	−	1.02160i	0.404432	−	0.914568i	$-0.367469\pi$
$152$		0			0
$153$	−5.26870e7	−	1.97743e7i	−1.18928	−	0.446356i
$154$		0			0
$155$	2.41057e7	+	4.17523e7i	0.519947	+	0.900575i
$156$		0			0
$157$	2.13156e7	−	3.69197e7i	0.439590	−	0.761393i	−0.558067	−	0.829796i	$-0.688457\pi$
$157$	2.13156e7	−	3.69197e7i	0.439590	−	0.761393i	0.997658	+	0.0684026i	$0.0217902\pi$
$158$		0			0
$159$	−1.69960e7	−	3.08440e6i	−0.335317	−	0.0608527i
$160$		0			0
$161$	3.78773e7			0.715300
$162$		0			0
$163$	4.06125e7			0.734519			0.367259	−	0.930119i	$-0.380296\pi$
$163$	4.06125e7			0.734519			0.367259	+	0.930119i	$0.380296\pi$
$164$		0			0
$165$	4.44316e7	+	8.06336e6i	0.770013	+	0.139740i
$166$		0			0
$167$	3.62955e7	−	6.28656e7i	0.603038	−	1.04449i	−0.389320	−	0.921102i	$-0.627290\pi$
$167$	3.62955e7	−	6.28656e7i	0.603038	−	1.04449i	0.992358	−	0.123390i	$-0.0393766\pi$
$168$		0			0
$169$	9.10774e6	+	1.57751e7i	0.145147	+	0.251401i
$170$		0			0
$171$	−4.58330e7	−	1.72019e7i	−0.700958	−	0.263081i
$172$		0			0
$173$	−1.61254e7	−	2.79301e7i	−0.236783	−	0.410120i	0.723006	−	0.690841i	$-0.242760\pi$
$173$	−1.61254e7	−	2.79301e7i	−0.236783	−	0.410120i	−0.959789	+	0.280721i	$0.909426\pi$
$174$		0			0
$175$	−2.86903e7	+	4.96931e7i	−0.404671	+	0.700911i
$176$		0			0
$177$	−3.39077e7	+	4.00025e7i	−0.459613	+	0.542226i
$178$		0			0
$179$	5.74576e7			0.748793			0.374396	−	0.927269i	$-0.377850\pi$
$179$	5.74576e7			0.748793			0.374396	+	0.927269i	$0.377850\pi$
$180$		0			0
$181$	2.89626e7			0.363047			0.181524	−	0.983387i	$-0.441897\pi$
$181$	2.89626e7			0.363047			0.181524	+	0.983387i	$0.441897\pi$
$182$		0			0
$183$	−1.42468e6	−	3.97590e6i	−0.0171845	−	0.0479576i
$184$		0			0
$185$	3.18911e7	−	5.52370e7i	0.370313	−	0.641400i
$186$		0			0
$187$	−6.01614e7	−	1.04203e8i	−0.672779	−	1.16529i
$188$		0			0
$189$	−2.47661e6	+	1.65383e8i	−0.0266834	+	1.78186i
$190$		0			0
$191$	4.94718e7	+	8.56877e7i	0.513737	+	0.889819i	0.999873	+	0.0159358i	$0.00507275\pi$
$191$	4.94718e7	+	8.56877e7i	0.513737	+	0.889819i	−0.486136	+	0.873883i	$0.661594\pi$
$192$		0			0
$193$	7.72389e6	−	1.33782e7i	0.0773367	−	0.133951i	−0.824763	−	0.565478i	$-0.808692\pi$
$193$	7.72389e6	−	1.33782e7i	0.0773367	−	0.133951i	0.902100	+	0.431527i	$0.142025\pi$
$194$		0			0
$195$	2.17391e7	+	6.06683e7i	0.209953	+	0.585923i
$196$		0			0
$197$	−9.85117e7			−0.918028			−0.459014	−	0.888429i	$-0.651797\pi$
$197$	−9.85117e7			−0.918028			−0.459014	+	0.888429i	$0.651797\pi$
$198$		0			0
$199$	−1.40825e8			−1.26676			−0.633380	−	0.773841i	$-0.718333\pi$
$199$	−1.40825e8			−1.26676			−0.633380	+	0.773841i	$0.718333\pi$
$200$		0			0
$201$	1.06682e8	−	1.25857e8i	0.926623	−	1.09318i
$202$		0			0
$203$	1.31784e8	−	2.28256e8i	1.10567	−	1.91508i
$204$		0			0
$205$	−3.25357e7	−	5.63534e7i	−0.263768	−	0.456859i
$206$		0			0
$207$	3.95655e7	−	3.25319e7i	0.310042	−	0.254926i
$208$		0			0
$209$	−5.23351e7	−	9.06470e7i	−0.396534	−	0.686818i
$210$		0			0
$211$	−3.02133e7	+	5.23309e7i	−0.221416	+	0.383504i	−0.955238	−	0.295838i	$-0.904401\pi$
$211$	−3.02133e7	+	5.23309e7i	−0.221416	+	0.383504i	0.733822	+	0.679342i	$0.237735\pi$
$212$		0			0
$213$	3.23932e7	+	5.87866e6i	0.229681	+	0.0416821i
$214$		0			0
$215$	−2.52650e7			−0.173374
$216$		0			0
$217$	−3.77562e8			−2.50830
$218$		0			0
$219$	1.86868e8	+	3.39124e7i	1.20221	+	0.218175i
$220$		0			0
$221$	8.58584e7	−	1.48711e8i	0.535069	−	0.926767i
$222$		0			0
$223$	−1.42446e8	−	2.46723e8i	−0.860165	−	1.48985i	−0.871769	−	0.489918i	$-0.837027\pi$
$223$	−1.42446e8	−	2.46723e8i	−0.860165	−	1.48985i	0.0116033	−	0.999933i	$-0.496306\pi$
$224$		0			0
$225$	1.27112e7	+	7.65494e7i	0.0743955	+	0.448026i
$226$		0			0
$227$	7.13578e7	+	1.23595e8i	0.404903	+	0.701312i	0.994310	−	0.106524i	$-0.0339720\pi$
$227$	7.13578e7	+	1.23595e8i	0.404903	+	0.701312i	−0.589407	+	0.807836i	$0.700639\pi$
$228$		0			0
$229$	−6.39437e7	+	1.10754e8i	−0.351863	+	0.609445i	−0.986576	−	0.163303i	$-0.947785\pi$
$229$	−6.39437e7	+	1.10754e8i	−0.351863	+	0.609445i	0.634713	+	0.772748i	$0.281118\pi$
$230$		0			0
$231$	−2.28667e8	+	2.69769e8i	−1.22057	+	1.43996i
$232$		0			0
$233$	2.85503e8			1.47865			0.739324	−	0.673350i	$-0.235145\pi$
$233$	2.85503e8			1.47865			0.739324	+	0.673350i	$0.235145\pi$
$234$		0			0
$235$	−4.03928e7			−0.203033
$236$		0			0
$237$	7.37178e7	+	2.05727e8i	0.359710	+	1.00386i
$238$		0			0
$239$	5.90868e6	−	1.02341e7i	0.0279961	−	0.0484907i	−0.851688	−	0.524049i	$-0.824421\pi$
$239$	5.90868e6	−	1.02341e7i	0.0279961	−	0.0484907i	0.879684	+	0.475559i	$0.157754\pi$
$240$		0			0
$241$	−7.41195e7	−	1.28379e8i	−0.341093	−	0.590791i	0.643543	−	0.765410i	$-0.277464\pi$
$241$	−7.41195e7	−	1.28379e8i	−0.341093	−	0.590791i	−0.984636	+	0.174619i	$0.944130\pi$
$242$		0			0
$243$	1.39456e8	+	1.74881e8i	0.623472	+	0.781846i
$244$		0			0
$245$	1.85009e8	+	3.20444e8i	0.803731	+	1.39210i
$246$		0			0
$247$	7.46892e7	−	1.29365e8i	0.315368	−	0.546234i
$248$		0			0
$249$	1.38370e8	+	3.86155e8i	0.567995	+	1.58513i
$250$		0			0
$251$	−6.92645e7			−0.276473			−0.138237	−	0.990399i	$-0.544143\pi$
$251$	−6.92645e7			−0.276473			−0.138237	+	0.990399i	$0.544143\pi$
$252$		0			0
$253$	1.09519e8			0.425175
$254$		0			0
$255$	−1.60680e8	+	1.89561e8i	−0.606835	+	0.715910i
$256$		0			0
$257$	−8.27289e7	+	1.43291e8i	−0.304013	+	0.526565i	−0.977041	−	0.213051i	$-0.931660\pi$
$257$	−8.27289e7	+	1.43291e8i	−0.304013	+	0.526565i	0.673028	+	0.739617i	$0.264993\pi$
$258$		0			0
$259$	2.49751e8	+	4.32582e8i	0.893221	+	1.54710i
$260$		0			0
$261$	−5.83864e7	−	3.51616e8i	−0.203268	−	1.22413i
$262$		0			0
$263$	−8.54750e7	−	1.48047e8i	−0.289730	−	0.501828i	0.684015	−	0.729468i	$-0.260232\pi$
$263$	−8.54750e7	−	1.48047e8i	−0.289730	−	0.501828i	−0.973745	+	0.227640i	$0.926899\pi$
$264$		0			0
$265$	−3.81378e7	+	6.60566e7i	−0.125891	+	0.218050i
$266$		0			0
$267$	4.94039e8	+	8.96572e7i	1.58844	+	0.288267i
$268$		0			0
$269$	2.40056e8			0.751932			0.375966	−	0.926633i	$-0.377311\pi$
$269$	2.40056e8			0.751932			0.375966	+	0.926633i	$0.377311\pi$
$270$		0			0
$271$	−2.92612e8			−0.893098			−0.446549	−	0.894759i	$-0.647347\pi$
$271$	−2.92612e8			−0.893098			−0.446549	+	0.894759i	$0.647347\pi$
$272$		0			0
$273$	−4.96587e8	−	9.01196e7i	−1.47715	−	0.268071i
$274$		0			0
$275$	−8.29556e7	+	1.43683e8i	−0.240537	+	0.416622i
$276$		0			0
$277$	−4.35557e7	−	7.54407e7i	−0.123131	−	0.213269i	0.797870	−	0.602829i	$-0.205960\pi$
$277$	−4.35557e7	−	7.54407e7i	−0.123131	−	0.213269i	−0.921001	+	0.389561i	$0.872627\pi$
$278$		0			0
$279$	−3.94391e8	+	3.24279e8i	−1.08721	+	0.893932i
$280$		0			0
$281$	7.68148e7	+	1.33047e8i	0.206525	+	0.357712i	0.950618	−	0.310365i	$-0.100451\pi$
$281$	7.68148e7	+	1.33047e8i	0.206525	+	0.357712i	−0.744092	+	0.668077i	$0.767118\pi$
$282$		0			0
$283$	−1.09494e8	+	1.89650e8i	−0.287170	+	0.497393i	−0.973133	−	0.230243i	$-0.926048\pi$
$283$	−1.09494e8	+	1.89650e8i	−0.287170	+	0.497393i	0.685963	+	0.727636i	$0.259381\pi$
$284$		0			0
$285$	−1.39777e8	+	1.64901e8i	−0.357667	+	0.421956i
$286$		0			0
$287$	5.09599e8			1.27245
$288$		0			0
$289$	2.51791e8			0.613617
$290$		0			0
$291$	−1.17376e8	−	3.27567e8i	−0.279225	−	0.779245i
$292$		0			0
$293$	1.45004e8	−	2.51154e8i	0.336777	−	0.583315i	−0.647048	−	0.762450i	$-0.723996\pi$
$293$	1.45004e8	−	2.51154e8i	0.336777	−	0.583315i	0.983825	+	0.179135i	$0.0573298\pi$
$294$		0			0
$295$	1.15780e8	+	2.00537e8i	0.262577	+	0.454797i
$296$		0			0
$297$	−7.16090e6	+	4.78190e8i	−0.0158606	+	1.05914i
$298$		0			0
$299$	7.81492e7	+	1.35358e8i	0.169073	+	0.292844i
$300$		0			0
$301$	9.89299e7	−	1.71352e8i	0.209095	−	0.362164i
$302$		0			0
$303$	−2.33598e8	−	6.51910e8i	−0.482413	−	1.34629i
$304$		0			0
$305$	−1.86496e7			−0.0376375
$306$		0			0
$307$	−2.99914e7			−0.0591579			−0.0295789	−	0.999562i	$-0.509417\pi$
$307$	−2.99914e7			−0.0591579			−0.0295789	+	0.999562i	$0.509417\pi$
$308$		0			0
$309$	−5.21801e8	+	6.15591e8i	−1.00612	+	1.18696i
$310$		0			0
$311$	−7.31419e6	+	1.26686e7i	−0.0137881	+	0.0238817i	−0.872837	−	0.488012i	$-0.837722\pi$
$311$	−7.31419e6	+	1.26686e7i	−0.0137881	+	0.0238817i	0.859049	+	0.511893i	$0.171056\pi$
$312$		0			0
$313$	2.52793e7	+	4.37851e7i	0.0465973	+	0.0807089i	0.888383	−	0.459102i	$-0.151829\pi$
$313$	2.52793e7	+	4.37851e7i	0.0465973	+	0.0807089i	−0.841786	+	0.539811i	$0.818496\pi$
$314$		0			0
$315$	6.83794e8	+	2.56640e8i	1.23265	+	0.462633i
$316$		0			0
$317$	3.24798e7	+	5.62566e7i	0.0572671	+	0.0991896i	0.893238	−	0.449585i	$-0.148428\pi$
$317$	3.24798e7	+	5.62566e7i	0.0572671	+	0.0991896i	−0.835971	+	0.548774i	$0.815095\pi$
$318$		0			0
$319$	3.81041e8	−	6.59983e8i	0.657211	−	1.13832i
$320$		0			0
$321$	6.07052e8	+	1.10167e8i	1.02437	+	0.185901i
$322$		0			0
$323$	5.75994e8			0.951063
$324$		0			0
$325$	−2.36778e8			−0.382603
$326$		0			0
$327$	−1.13600e8	−	2.06158e7i	−0.179663	−	0.0326050i
$328$		0			0
$329$	1.58166e8	−	2.73951e8i	0.244865	−	0.424118i
$330$		0			0
$331$	4.42586e7	+	7.66582e7i	0.0670811	+	0.116188i	0.897615	−	0.440780i	$-0.145298\pi$
$331$	4.42586e7	+	7.66582e7i	0.0670811	+	0.116188i	−0.830534	+	0.556968i	$0.811965\pi$
$332$		0			0
$333$	6.32418e8	+	2.37357e8i	0.938532	+	0.352247i
$334$		0			0
$335$	−3.64271e8	−	6.30936e8i	−0.529381	−	0.916914i
$336$		0			0
$337$	2.27637e8	−	3.94279e8i	0.323995	−	0.561176i	−0.657313	−	0.753618i	$-0.728307\pi$
$337$	2.27637e8	−	3.94279e8i	0.323995	−	0.561176i	0.981309	+	0.192441i	$0.0616404\pi$
$338$		0			0
$339$	−1.04138e8	+	1.22857e8i	−0.145181	+	0.171277i
$340$		0			0
$341$	−1.09169e9			−1.49093
$342$		0			0
$343$	−1.56590e9			−2.09525
$344$		0			0
$345$	−7.62982e7	−	2.12928e8i	−0.100034	−	0.279169i
$346$		0			0
$347$	1.48016e8	−	2.56372e8i	0.190176	−	0.329395i	−0.755132	−	0.655572i	$-0.772427\pi$
$347$	1.48016e8	−	2.56372e8i	0.190176	−	0.329395i	0.945309	+	0.326178i	$0.105761\pi$
$348$		0			0
$349$	−3.80328e8	−	6.58748e8i	−0.478927	−	0.829526i	0.520781	−	0.853691i	$-0.325641\pi$
$349$	−3.80328e8	−	6.58748e8i	−0.478927	−	0.829526i	−0.999708	+	0.0241640i	$0.992308\pi$
$350$		0			0
$351$	−5.96122e8	+	3.32371e8i	−0.735800	+	0.410249i
$352$		0			0
$353$	−7.00254e8	−	1.21288e9i	−0.847313	−	1.46759i	−0.883597	−	0.468248i	$-0.844885\pi$
$353$	−7.00254e8	−	1.21288e9i	−0.847313	−	1.46759i	0.0362837	−	0.999342i	$-0.488448\pi$
$354$		0			0
$355$	7.26882e7	−	1.25900e8i	0.0862312	−	0.149357i
$356$		0			0
$357$	−6.56464e8	−	1.83202e9i	−0.763611	−	2.13104i
$358$		0			0
$359$	−1.32132e9			−1.50722			−0.753612	−	0.657320i	$-0.771690\pi$
$359$	−1.32132e9			−1.50722			−0.753612	+	0.657320i	$0.771690\pi$
$360$		0			0
$361$	−3.92808e8			−0.439446
$362$		0			0
$363$	−7.19071e7	+	8.48320e7i	−0.0789039	+	0.0930864i
$364$		0			0
$365$	4.19318e8	−	7.26281e8i	0.451356	−	0.781771i
$366$		0			0
$367$	−3.70961e8	−	6.42523e8i	−0.391739	−	0.678511i	0.600940	−	0.799294i	$-0.294793\pi$
$367$	−3.70961e8	−	6.42523e8i	−0.391739	−	0.678511i	−0.992679	+	0.120783i	$0.961460\pi$
$368$		0			0
$369$	5.32312e8	−	4.37682e8i	0.551536	−	0.453489i
$370$		0			0
$371$	−2.98672e8	−	5.17314e8i	−0.303658	−	0.525952i
$372$		0			0
$373$	−4.39730e8	+	7.61635e8i	−0.438738	+	0.759916i	−0.997592	−	0.0693494i	$-0.977908\pi$
$373$	−4.39730e8	+	7.61635e8i	−0.438738	+	0.759916i	0.558855	+	0.829266i	$0.311241\pi$
$374$		0			0
$375$	1.07949e9	+	1.95903e8i	1.05708	+	0.191837i
$376$		0			0
$377$	1.08759e9			1.04538
$378$		0			0
$379$	1.99736e9			1.88460			0.942301	−	0.334767i	$-0.108658\pi$
$379$	1.99736e9			1.88460			0.942301	+	0.334767i	$0.108658\pi$
$380$		0			0
$381$	5.48024e6	+	994544.i	0.00507648	+	0.000921269i
$382$		0			0
$383$	1.54585e8	−	2.67750e8i	0.140596	−	0.243519i	−0.787125	−	0.616793i	$-0.788432\pi$
$383$	1.54585e8	−	2.67750e8i	0.140596	−	0.243519i	0.927721	+	0.373274i	$0.121765\pi$
$384$		0			0
$385$	7.80800e8	+	1.35239e9i	0.697312	+	1.20778i
$386$		0			0
$387$	−4.38306e7	−	2.63957e8i	−0.0384405	−	0.231497i
$388$		0			0
$389$	9.64068e8	+	1.66982e9i	0.830394	+	1.43828i	0.897726	+	0.440554i	$0.145218\pi$
$389$	9.64068e8	+	1.66982e9i	0.830394	+	1.43828i	−0.0673322	+	0.997731i	$0.521449\pi$
$390$		0			0
$391$	−3.01338e8	+	5.21934e8i	−0.254939	+	0.441567i
$392$		0			0
$393$	6.69009e8	−	7.89260e8i	0.555979	−	0.655913i
$394$		0			0
$395$	9.64997e8			0.787837
$396$		0			0
$397$	−1.08397e9			−0.869460			−0.434730	−	0.900561i	$-0.643156\pi$
$397$	−1.08397e9			−0.869460			−0.434730	+	0.900561i	$0.643156\pi$
$398$		0			0
$399$	−5.71066e8	−	1.59370e9i	−0.450071	−	1.25603i
$400$		0			0
$401$	−4.81857e8	+	8.34600e8i	−0.373175	+	0.646358i	−0.990052	−	0.140701i	$-0.955064\pi$
$401$	−4.81857e8	+	8.34600e8i	−0.373175	+	0.646358i	0.616877	+	0.787060i	$0.288398\pi$
$402$		0			0
$403$	−7.78993e8	−	1.34926e9i	−0.592879	−	1.02690i
$404$		0			0
$405$	9.34694e8	−	3.19217e8i	0.699160	−	0.238777i
$406$		0			0
$407$	7.22135e8	+	1.25077e9i	0.530931	+	0.919599i
$408$		0			0
$409$	6.07460e7	−	1.05215e8i	0.0439022	−	0.0760408i	−0.843239	−	0.537538i	$-0.819354\pi$
$409$	6.07460e7	−	1.05215e8i	0.0439022	−	0.0760408i	0.887141	+	0.461498i	$0.152688\pi$
$410$		0			0
$411$	9.30095e7	+	2.59565e8i	0.0660816	+	0.184417i
$412$		0			0
$413$	−1.81344e9			−1.26671
$414$		0			0
$415$	1.81132e9			1.24402
$416$		0			0
$417$	1.07585e9	−	1.26923e9i	0.726568	−	0.857165i
$418$		0			0
$419$	−4.52373e8	+	7.83532e8i	−0.300433	+	0.520365i	−0.976234	−	0.216719i	$-0.930464\pi$
$419$	−4.52373e8	+	7.83532e8i	−0.300433	+	0.520365i	0.675801	+	0.737084i	$0.263798\pi$
$420$		0			0
$421$	5.94491e8	+	1.02969e9i	0.388292	+	0.672541i	0.992220	−	0.124498i	$-0.0397320\pi$
$421$	5.94491e8	+	1.02969e9i	0.388292	+	0.672541i	−0.603928	+	0.797039i	$0.706399\pi$
$422$		0			0
$423$	−7.00748e7	−	4.22006e8i	−0.0450164	−	0.271098i
$424$		0			0
$425$	−4.56500e8	−	7.90681e8i	−0.288456	−	0.499621i
$426$		0			0
$427$	7.30262e7	−	1.26485e8i	0.0453922	−	0.0786216i
$428$		0			0
$429$	−1.43584e9	−	2.60573e8i	−0.878021	−	0.159342i
$430$		0			0
$431$	−1.77750e9			−1.06940			−0.534700	−	0.845042i	$-0.679575\pi$
$431$	−1.77750e9			−1.06940			−0.534700	+	0.845042i	$0.679575\pi$
$432$		0			0
$433$	1.98626e9			1.17579			0.587894	−	0.808938i	$-0.299957\pi$
$433$	1.98626e9			1.17579			0.587894	+	0.808938i	$0.299957\pi$
$434$		0			0
$435$	−1.54861e9	−	2.81038e8i	−0.902047	−	0.163702i
$436$		0			0
$437$	−2.62138e8	+	4.54036e8i	−0.150260	+	0.260259i
$438$		0			0
$439$	6.54719e8	+	1.13401e9i	0.369342	+	0.639719i	0.989463	−	0.144787i	$-0.0462497\pi$
$439$	6.54719e8	+	1.13401e9i	0.369342	+	0.639719i	−0.620121	+	0.784506i	$0.712916\pi$
$440$		0			0
$441$	−3.02690e9	+	2.48881e9i	−1.68060	+	1.38183i
$442$		0			0
$443$	9.12222e8	+	1.58002e9i	0.498526	+	0.863472i	0.999999	−	0.00170136i	$-0.000541559\pi$
$443$	9.12222e8	+	1.58002e9i	0.498526	+	0.863472i	−0.501473	+	0.865173i	$0.667208\pi$
$444$		0			0
$445$	1.10859e9	−	1.92013e9i	0.596363	−	1.03293i
$446$		0			0
$447$	1.01064e9	−	1.19229e9i	0.535202	−	0.631402i
$448$		0			0
$449$	−3.58855e9			−1.87093			−0.935464	−	0.353423i	$-0.885018\pi$
$449$	−3.58855e9			−1.87093			−0.935464	+	0.353423i	$0.885018\pi$
$450$		0			0
$451$	1.47346e9			0.756346
$452$		0			0
$453$	−7.87308e8	−	2.19717e9i	−0.397925	−	1.11051i
$454$		0			0
$455$	−1.11431e9	+	1.93004e9i	−0.554581	+	0.960562i
$456$		0			0
$457$	−1.59845e9	−	2.76860e9i	−0.783417	−	1.35692i	−0.929940	−	0.367711i	$-0.880141\pi$
$457$	−1.59845e9	−	2.76860e9i	−0.783417	−	1.35692i	0.146522	−	0.989207i	$-0.453192\pi$
$458$		0			0
$459$	−2.25920e9	−	1.34985e9i	−1.09046	−	0.651542i
$460$		0			0
$461$	1.87364e9	+	3.24524e9i	0.890704	+	1.54275i	0.839033	+	0.544081i	$0.183122\pi$
$461$	1.87364e9	+	3.24524e9i	0.890704	+	1.54275i	0.0516717	+	0.998664i	$0.483545\pi$
$462$		0			0
$463$	−1.09312e9	+	1.89333e9i	−0.511838	+	0.886530i	0.488068	+	0.872806i	$0.337702\pi$
$463$	−1.09312e9	+	1.89333e9i	−0.511838	+	0.886530i	−0.999906	+	0.0137240i	$0.995631\pi$
$464$		0			0
$465$	7.60543e8	+	2.12248e9i	0.350783	+	0.978944i
$466$		0			0
$467$	−9.63843e8			−0.437922			−0.218961	−	0.975734i	$-0.570267\pi$
$467$	−9.63843e8			−0.437922			−0.218961	+	0.975734i	$0.570267\pi$
$468$		0			0
$469$	5.70549e9			2.55381
$470$		0			0
$471$	1.28911e9	−	1.52082e9i	0.568481	−	0.670663i
$472$		0			0
$473$	2.86047e8	−	4.95448e8i	0.124286	−	0.215270i
$474$		0			0
$475$	−3.97115e8	−	6.87823e8i	−0.170015	−	0.294475i
$476$		0			0
$477$	−7.56293e8	−	2.83850e8i	−0.319063	−	0.119750i
$478$		0			0
$479$	−8.61804e8	−	1.49269e9i	−0.358290	−	0.620576i	0.629386	−	0.777093i	$-0.283307\pi$
$479$	−8.61804e8	−	1.49269e9i	−0.358290	−	0.620576i	−0.987675	+	0.156517i	$0.949973\pi$
$480$		0			0
$481$	−1.03058e9	+	1.78502e9i	−0.422256	+	0.731368i
$482$		0			0
$483$	1.74288e9	+	3.16294e8i	0.703805	+	0.127725i
$484$		0			0
$485$	−1.53651e9			−0.611558
$486$		0			0
$487$	6.57115e7			0.0257804			0.0128902	−	0.999917i	$-0.495897\pi$
$487$	6.57115e7			0.0257804			0.0128902	+	0.999917i	$0.495897\pi$
$488$		0			0
$489$	1.86873e9	+	3.39134e8i	0.722714	+	0.131157i
$490$		0			0
$491$	9.79411e8	−	1.69639e9i	0.373405	−	0.646756i	−0.616682	−	0.787212i	$-0.711524\pi$
$491$	9.79411e8	−	1.69639e9i	0.373405	−	0.646756i	0.990087	+	0.140456i	$0.0448570\pi$
$492$		0			0
$493$	2.09685e9	+	3.63185e9i	0.788140	+	1.36510i
$494$		0			0
$495$	1.97713e9	+	7.42051e8i	0.732686	+	0.274989i
$496$		0			0
$497$	5.69249e8	+	9.85968e8i	0.207996	+	0.360259i
$498$		0			0
$499$	−1.54210e9	+	2.67100e9i	−0.555599	+	0.962325i	0.442258	+	0.896888i	$0.354178\pi$
$499$	−1.54210e9	+	2.67100e9i	−0.555599	+	0.962325i	−0.997857	+	0.0654372i	$0.979156\pi$
$500$		0			0
$501$	2.19505e9	−	2.58960e9i	0.779852	−	0.920027i
$502$		0			0
$503$	−3.95457e9			−1.38552			−0.692758	−	0.721170i	$-0.743605\pi$
$503$	−3.95457e9			−1.38552			−0.692758	+	0.721170i	$0.743605\pi$
$504$		0			0
$505$	−3.05789e9			−1.05658
$506$		0			0
$507$	2.87352e8	+	8.01925e8i	0.0979234	+	0.273279i
$508$		0			0
$509$	1.92422e9	−	3.33285e9i	0.646760	−	1.12022i	−0.337132	−	0.941457i	$-0.609457\pi$
$509$	1.92422e9	−	3.33285e9i	0.646760	−	1.12022i	0.983892	−	0.178764i	$-0.0572099\pi$
$510$		0			0
$511$	3.28384e9	+	5.68778e9i	1.08870	+	1.88569i
$512$		0			0
$513$	−1.96531e9	−	1.17425e9i	−0.642717	−	0.384018i
$514$		0			0
$515$	1.78172e9	+	3.08603e9i	0.574797	+	0.995578i
$516$		0			0
$517$	4.57322e8	−	7.92105e8i	0.145548	−	0.252096i
$518$		0			0
$519$	−5.08763e8	−	1.41982e9i	−0.159746	−	0.445809i
$520$		0			0
$521$	−1.90239e8			−0.0589344			−0.0294672	−	0.999566i	$-0.509381\pi$
$521$	−1.90239e8			−0.0589344			−0.0294672	+	0.999566i	$0.509381\pi$
$522$		0			0
$523$	9.89767e8			0.302536			0.151268	−	0.988493i	$-0.451664\pi$
$523$	9.89767e8			0.302536			0.151268	+	0.988493i	$0.451664\pi$
$524$		0			0
$525$	−1.73511e9	+	2.04699e9i	−0.523323	+	0.617388i
$526$		0			0
$527$	3.00375e9	−	5.20265e9i	0.893978	−	1.54842i
$528$		0			0
$529$	1.42813e9	+	2.47360e9i	0.419443	+	0.726497i
$530$		0			0
$531$	−1.89426e9	+	1.55752e9i	−0.549047	+	0.451443i
$532$		0			0
$533$	1.05141e9	+	1.82110e9i	0.300766	+	0.520941i
$534$		0			0
$535$	1.36218e9	−	2.35937e9i	0.384589	−	0.666128i
$536$		0			0
$537$	2.64384e9	+	4.79799e8i	0.736759	+	0.133706i
$538$		0			0
$539$	−8.37858e9			−2.30468
$540$		0			0
$541$	−3.06000e9			−0.830865			−0.415433	−	0.909624i	$-0.636370\pi$
$541$	−3.06000e9			−0.830865			−0.415433	+	0.909624i	$0.636370\pi$
$542$		0			0
$543$	1.33268e9	+	2.41852e8i	0.357213	+	0.0648263i
$544$		0			0
$545$	−2.54910e8	+	4.41517e8i	−0.0674526	+	0.116831i
$546$		0			0
$547$	2.69716e9	+	4.67161e9i	0.704612	+	1.22042i	0.966831	+	0.255416i	$0.0822124\pi$
$547$	2.69716e9	+	4.67161e9i	0.704612	+	1.22042i	−0.262219	+	0.965008i	$0.584454\pi$
$548$		0			0
$549$	−3.23541e7	−	1.94843e8i	−0.00834499	−	0.0502553i
$550$		0			0
$551$	1.82407e9	+	3.15939e9i	0.464528	+	0.804586i
$552$		0			0
$553$	−3.77863e9	+	6.54478e9i	−0.950159	+	1.64572i
$554$		0			0
$555$	1.92869e9	−	2.27536e9i	0.478891	−	0.564969i
$556$		0			0
$557$	−704537.			−0.000172747			−8.63736e−5	−	1.00000i	$-0.500027\pi$
$557$	−704537.			−0.000172747			−8.63736e−5		1.00000i	$0.500027\pi$
$558$		0			0
$559$	8.16456e8			0.197693
$560$		0			0
$561$	−1.89811e9	−	5.29713e9i	−0.453891	−	1.26669i
$562$		0			0
$563$	−1.58230e9	+	2.74063e9i	−0.373688	+	0.647247i	−0.990130	−	0.140153i	$-0.955240\pi$
$563$	−1.58230e9	+	2.74063e9i	−0.373688	+	0.647247i	0.616441	+	0.787401i	$0.288574\pi$
$564$		0			0
$565$	3.55587e8	+	6.15894e8i	0.0829423	+	0.143660i
$566$		0			0
$567$	−1.49499e9	+	7.58921e9i	−0.344426	+	1.74846i
$568$		0			0
$569$	−6.94504e8	−	1.20292e9i	−0.158045	−	0.273742i	0.776118	−	0.630587i	$-0.217186\pi$
$569$	−6.94504e8	−	1.20292e9i	−0.158045	−	0.273742i	−0.934164	+	0.356845i	$0.883853\pi$
$570$		0			0
$571$	2.83085e9	−	4.90317e9i	0.636341	−	1.10217i	−0.349889	−	0.936791i	$-0.613781\pi$
$571$	2.83085e9	−	4.90317e9i	0.636341	−	1.10217i	0.986229	−	0.165383i	$-0.0528861\pi$
$572$		0			0
$573$	1.56085e9	+	4.35593e9i	0.346594	+	0.967252i
$574$		0			0
$575$	8.31022e8			0.182295
$576$		0			0
$577$	3.72547e9			0.807358			0.403679	−	0.914901i	$-0.367731\pi$
$577$	3.72547e9			0.807358			0.403679	+	0.914901i	$0.367731\pi$
$578$		0			0
$579$	4.67120e8	−	5.51082e8i	0.100012	−	0.117989i
$580$		0			0
$581$	−7.09259e9	+	1.22847e10i	−1.50034	+	2.59866i
$582$		0			0
$583$	−8.63583e8	−	1.49577e9i	−0.180495	−	0.312626i
$584$		0			0
$585$	4.93690e8	+	2.97311e9i	0.101955	+	0.613996i
$586$		0			0
$587$	−6.23155e8	−	1.07934e9i	−0.127164	−	0.220254i	0.795413	−	0.606068i	$-0.207254\pi$
$587$	−6.23155e8	−	1.07934e9i	−0.127164	−	0.220254i	−0.922577	+	0.385814i	$0.873921\pi$
$588$		0			0
$589$	2.61300e9	−	4.52584e9i	0.526909	−	0.912633i
$590$		0			0
$591$	−4.53290e9	−	8.22621e8i	−0.903274	−	0.163924i
$592$		0			0
$593$	2.28414e9			0.449813			0.224906	−	0.974380i	$-0.427792\pi$
$593$	2.28414e9			0.449813			0.224906	+	0.974380i	$0.427792\pi$
$594$		0			0
$595$	−8.59340e9			−1.67246
$596$		0			0
$597$	−6.47990e9	−	1.17596e9i	−1.24640	−	0.226194i
$598$		0			0
$599$	9.17038e8	−	1.58836e9i	0.174339	−	0.301963i	−0.765594	−	0.643325i	$-0.777555\pi$
$599$	9.17038e8	−	1.58836e9i	0.174339	−	0.301963i	0.939932	+	0.341361i	$0.110888\pi$
$600$		0			0
$601$	−1.04820e9	−	1.81554e9i	−0.196962	−	0.341149i	0.750580	−	0.660780i	$-0.229774\pi$
$601$	−1.04820e9	−	1.81554e9i	−0.196962	−	0.341149i	−0.947542	+	0.319631i	$0.896441\pi$
$602$		0			0
$603$	5.95979e9	−	4.90032e9i	1.10693	−	0.910151i
$604$		0			0
$605$	2.45531e8	+	4.25273e8i	0.0450778	+	0.0780771i
$606$		0			0
$607$	1.96750e9	−	3.40781e9i	0.357070	−	0.618464i	−0.630400	−	0.776271i	$-0.717109\pi$
$607$	1.96750e9	−	3.40781e9i	0.357070	−	0.618464i	0.987470	+	0.157807i	$0.0504423\pi$
$608$		0			0
$609$	7.96992e9	−	9.40247e9i	1.42986	−	1.68687i
$610$		0			0
$611$	1.30532e9			0.231512
$612$		0			0
$613$	8.35535e9			1.46505			0.732526	−	0.680739i	$-0.238341\pi$
$613$	8.35535e9			1.46505			0.732526	+	0.680739i	$0.238341\pi$
$614$		0			0
$615$	−1.02651e9	−	2.86472e9i	−0.177951	−	0.496615i
$616$		0			0
$617$	5.33262e9	−	9.23638e9i	0.913993	−	1.58308i	0.105624	−	0.994406i	$-0.466316\pi$
$617$	5.33262e9	−	9.23638e9i	0.913993	−	1.58308i	0.808369	−	0.588676i	$-0.200351\pi$
$618$		0			0
$619$	−3.59538e9	−	6.22739e9i	−0.609295	−	1.05533i	−0.991357	−	0.131194i	$-0.958119\pi$
$619$	−3.59538e9	−	6.22739e9i	−0.609295	−	1.05533i	0.382061	−	0.924137i	$-0.375214\pi$
$620$		0			0
$621$	2.09222e9	−	1.16653e9i	0.350579	−	0.195467i
$622$		0			0
$623$	8.68178e9	+	1.50373e10i	1.43847	+	2.49150i
$624$		0			0
$625$	1.03631e9	−	1.79494e9i	0.169789	−	0.294083i
$626$		0			0
$627$	−1.65119e9	−	4.60804e9i	−0.267522	−	0.746585i
$628$		0			0
$629$	−7.94774e9			−1.27340
$630$		0			0
$631$	6.01287e9			0.952750			0.476375	−	0.879242i	$-0.341951\pi$
$631$	6.01287e9			0.952750			0.476375	+	0.879242i	$0.341951\pi$
$632$		0			0
$633$	−1.82722e9	+	2.15565e9i	−0.286337	+	0.337804i
$634$		0			0
$635$	1.22973e7	−	2.12995e7i	0.00190591	−	0.00330113i
$636$		0			0
$637$	−5.97868e9	−	1.03554e10i	−0.916468	−	1.58737i
$638$		0			0
$639$	1.44145e9	+	5.40999e8i	0.218547	+	0.0820245i
$640$		0			0
$641$	−1.26243e9	−	2.18659e9i	−0.189323	−	0.327917i	0.755702	−	0.654916i	$-0.227296\pi$
$641$	−1.26243e9	−	2.18659e9i	−0.189323	−	0.327917i	−0.945025	+	0.326999i	$0.893963\pi$
$642$		0			0
$643$	−4.05815e9	+	7.02892e9i	−0.601990	+	1.04268i	0.390529	+	0.920590i	$0.372292\pi$
$643$	−4.05815e9	+	7.02892e9i	−0.601990	+	1.04268i	−0.992519	+	0.122087i	$0.961041\pi$
$644$		0			0
$645$	−1.16254e9	−	2.10975e8i	−0.170588	−	0.0309579i
$646$		0			0
$647$	6.70712e9			0.973578			0.486789	−	0.873519i	$-0.338168\pi$
$647$	6.70712e9			0.973578			0.486789	+	0.873519i	$0.338168\pi$
$648$		0			0
$649$	−5.24340e9			−0.752933
$650$		0			0
$651$	−1.73731e10	−	3.15283e9i	−2.46799	−	0.447886i
$652$		0			0
$653$	3.33252e9	−	5.77210e9i	0.468357	−	0.811218i	−0.530989	−	0.847379i	$-0.678179\pi$
$653$	3.33252e9	−	5.77210e9i	0.468357	−	0.811218i	0.999346	+	0.0361606i	$0.0115128\pi$
$654$		0			0
$655$	−2.28438e9	−	3.95665e9i	−0.317631	−	0.550153i
$656$		0			0
$657$	8.31531e9	+	3.12088e9i	1.14393	+	0.429337i
$658$		0			0
$659$	−6.56243e9	−	1.13665e10i	−0.893235	−	1.54713i	−0.835973	−	0.548770i	$-0.815096\pi$
$659$	−6.56243e9	−	1.13665e10i	−0.893235	−	1.54713i	−0.0572622	−	0.998359i	$-0.518237\pi$
$660$		0			0
$661$	8.40716e8	−	1.45616e9i	0.113226	−	0.196112i	−0.803843	−	0.594841i	$-0.797215\pi$
$661$	8.40716e8	−	1.45616e9i	0.113226	−	0.196112i	0.917069	+	0.398728i	$0.130548\pi$
$662$		0			0
$663$	5.19248e9	−	6.12580e9i	0.691954	−	0.816330i
$664$		0			0
$665$	−7.47549e9			−0.985744
$666$		0			0
$667$	−3.81715e9			−0.498079
$668$		0			0
$669$	−4.49420e9	−	1.25422e10i	−0.580312	−	1.61950i
$670$		0			0
$671$	2.11149e8	−	3.65721e8i	0.0269811	−	0.0467327i
$672$		0			0
$673$	1.90889e9	+	3.30630e9i	0.241395	+	0.418109i	0.961112	−	0.276159i	$-0.0890616\pi$
$673$	1.90889e9	+	3.30630e9i	0.241395	+	0.418109i	−0.719717	+	0.694268i	$0.755728\pi$
$674$		0			0
$675$	−5.43364e7	+	3.62847e9i	−0.00680029	+	0.454110i
$676$		0			0
$677$	−4.08674e9	−	7.07845e9i	−0.506194	−	0.876754i	−0.999974	−	0.00716716i	$-0.997719\pi$
$677$	−4.08674e9	−	7.07845e9i	−0.506194	−	0.876754i	0.493780	−	0.869587i	$-0.335615\pi$
$678$		0			0
$679$	6.01648e9	−	1.04209e10i	0.737562	−	1.27749i
$680$		0			0
$681$	2.25136e9	+	6.28296e9i	0.273168	+	0.762341i
$682$		0			0
$683$	−1.11162e10			−1.33501			−0.667507	−	0.744604i	$-0.732638\pi$
$683$	−1.11162e10			−1.33501			−0.667507	+	0.744604i	$0.732638\pi$
$684$		0			0
$685$	1.21753e9			0.144732
$686$		0			0
$687$	−3.86714e9	+	4.56224e9i	−0.455032	+	0.536821i
$688$		0			0
$689$	1.23245e9	−	2.13467e9i	0.143550	−	0.248635i
$690$		0			0
$691$	−1.06038e9	−	1.83664e9i	−0.122262	−	0.211763i	0.798398	−	0.602131i	$-0.205681\pi$
$691$	−1.06038e9	−	1.83664e9i	−0.122262	−	0.211763i	−0.920659	+	0.390367i	$0.872348\pi$
$692$		0			0
$693$	−1.27746e10	+	1.05036e10i	−1.45807	+	1.19887i
$694$		0			0
$695$	−3.67356e9	−	6.36280e9i	−0.415089	−	0.718955i
$696$		0			0
$697$	−4.05419e9	+	7.02206e9i	−0.453512	+	0.785506i
$698$		0			0
$699$	1.31371e10	+	2.38409e9i	1.45488	+	0.264029i
$700$		0			0
$701$	1.27398e10			1.39684			0.698422	−	0.715686i	$-0.253886\pi$
$701$	1.27398e10			1.39684			0.698422	+	0.715686i	$0.253886\pi$
$702$		0			0
$703$	−6.91383e9			−0.750542
$704$		0			0
$705$	−1.85862e9	−	3.37299e8i	−0.199770	−	0.0362538i
$706$		0			0
$707$	1.19738e10	−	2.07392e10i	1.27427	−	2.20711i
$708$		0			0
$709$	3.84065e9	+	6.65221e9i	0.404710	+	0.700977i	0.994288	−	0.106734i	$-0.0340393\pi$
$709$	3.84065e9	+	6.65221e9i	0.404710	+	0.700977i	−0.589578	+	0.807711i	$0.700706\pi$
$710$		0			0
$711$	1.67411e9	+	1.00819e10i	0.174679	+	1.05196i
$712$		0			0
$713$	2.73405e9	+	4.73551e9i	0.282483	+	0.489275i
$714$		0			0
$715$	−3.22192e9	+	5.58053e9i	−0.329643	+	0.570958i
$716$		0			0
$717$	3.57341e8	−	4.21571e8i	0.0362047	−	0.0427123i
$718$		0			0
$719$	1.29899e10			1.30333			0.651666	−	0.758506i	$-0.274071\pi$
$719$	1.29899e10			1.30333			0.651666	+	0.758506i	$0.274071\pi$
$720$		0			0
$721$	−2.79067e10			−2.77290
$722$		0			0
$723$	−2.33849e9	−	6.52613e9i	−0.230119	−	0.642202i
$724$		0			0
$725$	2.89131e9	−	5.00790e9i	0.281781	−	0.488060i
$726$		0			0
$727$	−4.67747e8	−	8.10161e8i	−0.0451482	−	0.0781990i	0.842568	−	0.538590i	$-0.181043\pi$
$727$	−4.67747e8	−	8.10161e8i	−0.0451482	−	0.0781990i	−0.887716	+	0.460391i	$0.847709\pi$
$728$		0			0
$729$	4.95658e9	+	9.21148e9i	0.473844	+	0.880609i
$730$		0			0
$731$	1.57410e9	+	2.72643e9i	0.149047	+	0.258156i
$732$		0			0
$733$	5.82375e9	−	1.00870e10i	0.546184	−	0.946019i	−0.452347	−	0.891842i	$-0.649413\pi$
$733$	5.82375e9	−	1.00870e10i	0.546184	−	0.946019i	0.998531	−	0.0541767i	$-0.0172534\pi$
$734$		0			0
$735$	5.83708e9	+	1.62898e10i	0.542238	+	1.51324i
$736$		0			0
$737$	1.64969e10			1.51798
$738$		0			0
$739$	1.36168e10			1.24114			0.620570	−	0.784151i	$-0.286901\pi$
$739$	1.36168e10			1.24114			0.620570	+	0.784151i	$0.286901\pi$
$740$		0			0
$741$	4.51700e9	−	5.32890e9i	0.407836	−	0.481143i
$742$		0			0
$743$	7.55834e8	−	1.30914e9i	0.0676029	−	0.117092i	−0.830243	−	0.557402i	$-0.811798\pi$
$743$	7.55834e8	−	1.30914e9i	0.0676029	−	0.117092i	0.897846	+	0.440310i	$0.145132\pi$
$744$		0			0
$745$	−3.45088e9	−	5.97710e9i	−0.305762	−	0.529594i
$746$		0			0
$747$	3.14235e9	+	1.89239e10i	0.275824	+	1.66107i
$748$		0			0
$749$	1.06678e10	+	1.84771e10i	0.927657	+	1.60675i
$750$		0			0
$751$	−9.52029e9	+	1.64896e10i	−0.820183	+	1.42060i	0.0853635	+	0.996350i	$0.472795\pi$
$751$	−9.52029e9	+	1.64896e10i	−0.820183	+	1.42060i	−0.905546	+	0.424248i	$0.860538\pi$
$752$		0			0
$753$	−3.18712e9	−	5.78393e8i	−0.272030	−	0.0493674i
$754$		0			0
$755$	−1.03062e10			−0.871534
$756$		0			0
$757$	−6.00984e9			−0.503532			−0.251766	−	0.967788i	$-0.581011\pi$
$757$	−6.00984e9			−0.503532			−0.251766	+	0.967788i	$0.581011\pi$
$758$		0			0
$759$	5.03938e9	+	9.14537e8i	0.418342	+	0.0759198i
$760$		0			0
$761$	1.12107e10	−	1.94175e10i	0.922116	−	1.59715i	0.125981	−	0.992033i	$-0.459792\pi$
$761$	1.12107e10	−	1.94175e10i	0.922116	−	1.59715i	0.796135	−	0.605119i	$-0.206874\pi$
$762$		0			0
$763$	−1.99630e9	−	3.45769e9i	−0.162701	−	0.281806i
$764$		0			0
$765$	−8.97641e9	+	7.38067e9i	−0.724916	+	0.596047i
$766$		0			0
$767$	−3.74152e9	−	6.48050e9i	−0.299408	−	0.518591i
$768$		0			0
$769$	3.36877e9	−	5.83488e9i	0.267134	−	0.462690i	−0.700987	−	0.713175i	$-0.747257\pi$
$769$	3.36877e9	−	5.83488e9i	0.267134	−	0.462690i	0.968121	+	0.250485i	$0.0805900\pi$
$770$		0			0
$771$	−5.00322e9	+	5.90252e9i	−0.393151	+	0.463818i
$772$		0			0
$773$	−9.36137e9			−0.728973			−0.364486	−	0.931209i	$-0.618755\pi$
$773$	−9.36137e9			−0.728973			−0.364486	+	0.931209i	$0.618755\pi$
$774$		0			0
$775$	−8.28365e9			−0.639243
$776$		0			0
$777$	7.87974e9	+	2.19903e10i	0.602612	+	1.68173i
$778$		0			0
$779$	−3.52678e9	+	6.10857e9i	−0.267299	+	0.462976i
$780$		0			0
$781$	1.64593e9	+	2.85084e9i	0.123633	+	0.214138i
$782$		0			0
$783$	2.49584e8	−	1.66667e10i	0.0185802	−	1.24075i
$784$		0			0
$785$	−4.40174e9	−	7.62404e9i	−0.324774	−	0.562524i
$786$		0			0
$787$	3.25442e9	−	5.63682e9i	0.237992	−	0.412214i	−0.722146	−	0.691740i	$-0.756844\pi$
$787$	3.25442e9	−	5.63682e9i	0.237992	−	0.412214i	0.960138	+	0.279527i	$0.0901776\pi$
$788$		0			0
$789$	−2.69676e9	−	7.52597e9i	−0.195467	−	0.545497i
$790$		0			0
$791$	−5.56947e9			−0.400125
$792$		0			0
$793$	6.02676e8			0.0429169
$794$		0			0
$795$	−2.30647e9	+	2.72105e9i	−0.162803	+	0.192066i
$796$		0			0
$797$	−8.12967e9	+	1.40810e10i	−0.568812	+	0.985211i	0.427872	+	0.903839i	$0.359263\pi$
$797$	−8.12967e9	+	1.40810e10i	−0.568812	+	0.985211i	−0.996684	+	0.0813716i	$0.974070\pi$
$798$		0			0
$799$	2.51662e9	+	4.35891e9i	0.174544	+	0.302318i
$800$		0			0
$801$	2.19839e10	+	8.25093e9i	1.51144	+	0.567269i
$802$		0			0
$803$	9.49495e9	+	1.64457e10i	0.647125	+	1.12085i
$804$		0			0
$805$	3.91090e9	−	6.77387e9i	0.264235	−	0.457669i
$806$		0			0
$807$	1.10459e10	+	2.00458e9i	0.739848	+	0.134266i
$808$		0			0
$809$	−2.11275e10			−1.40291			−0.701454	−	0.712715i	$-0.747465\pi$
$809$	−2.11275e10			−1.40291			−0.701454	+	0.712715i	$0.747465\pi$
$810$		0			0
$811$	−1.05784e10			−0.696383			−0.348191	−	0.937423i	$-0.613204\pi$
$811$	−1.05784e10			−0.696383			−0.348191	+	0.937423i	$0.613204\pi$
$812$		0			0
$813$	−1.34642e10	−	2.44345e9i	−0.878745	−	0.159473i
$814$		0			0
$815$	4.19331e9	−	7.26303e9i	0.271335	−	0.469965i
$816$		0			0
$817$	1.36933e9	+	2.37175e9i	0.0878478	+	0.152157i
$818$		0			0
$819$	−2.20973e10	−	8.29349e9i	−1.40555	−	0.527526i
$820$		0			0
$821$	1.46609e10	+	2.53934e10i	0.924610	+	1.60147i	0.792188	+	0.610277i	$0.208942\pi$
$821$	1.46609e10	+	2.53934e10i	0.924610	+	1.60147i	0.132422	+	0.991193i	$0.457725\pi$
$822$		0			0
$823$	4.13709e9	−	7.16565e9i	0.258700	−	0.448081i	−0.707194	−	0.707019i	$-0.750039\pi$
$823$	4.13709e9	−	7.16565e9i	0.258700	−	0.448081i	0.965894	+	0.258939i	$0.0833727\pi$
$824$		0			0
$825$	−5.01693e9	+	5.91869e9i	−0.311064	+	0.366975i
$826$		0			0
$827$	1.68712e10			1.03723			0.518616	−	0.855007i	$-0.326448\pi$
$827$	1.68712e10			1.03723			0.518616	+	0.855007i	$0.326448\pi$
$828$		0			0
$829$	1.62532e10			0.990830			0.495415	−	0.868657i	$-0.335016\pi$
$829$	1.62532e10			0.990830			0.495415	+	0.868657i	$0.335016\pi$
$830$		0			0
$831$	−1.37420e9	−	3.83503e9i	−0.0830702	−	0.231827i
$832$		0			0
$833$	2.30534e10	−	3.99297e10i	1.38191	−	2.39353i
$834$		0			0
$835$	−7.49514e9	−	1.29820e10i	−0.445530	−	0.771681i
$836$		0			0
$837$	−2.08553e10	+	1.16280e10i	−1.22935	+	0.685433i
$838$		0			0
$839$	1.12134e10	+	1.94222e10i	0.655498	+	1.13536i	0.981769	+	0.190079i	$0.0608745\pi$
$839$	1.12134e10	+	1.94222e10i	0.655498	+	1.13536i	−0.326271	+	0.945276i	$0.605792\pi$
$840$		0			0
$841$	−4.65578e9	+	8.06405e9i	−0.269902	+	0.467485i
$842$		0			0
$843$	2.42353e9	+	6.76345e9i	0.139332	+	0.388840i
$844$		0			0
$845$	3.76156e9			0.214471
$846$		0			0
$847$	−3.84570e9			−0.217462
$848$		0			0
$849$	−6.62192e9	+	7.81217e9i	−0.371370	+	0.438122i
$850$		0			0
$851$	3.61706e9	−	6.26492e9i	0.201188	−	0.348468i
$852$		0			0
$853$	−1.11195e10	−	1.92595e10i	−0.613428	−	1.06249i	−0.990658	−	0.136369i	$-0.956457\pi$
$853$	−1.11195e10	−	1.92595e10i	−0.613428	−	1.06249i	0.377230	−	0.926120i	$-0.376877\pi$
$854$		0			0
$855$	−7.80868e9	+	6.42053e9i	−0.427264	+	0.351309i
$856$		0			0
$857$	8.25516e9	+	1.42984e10i	0.448015	+	0.775985i	0.998257	−	0.0590205i	$-0.0187977\pi$
$857$	8.25516e9	+	1.42984e10i	0.448015	+	0.775985i	−0.550242	+	0.835005i	$0.685464\pi$
$858$		0			0
$859$	−1.01406e10	+	1.75640e10i	−0.545866	+	0.945468i	0.452686	+	0.891670i	$0.350466\pi$
$859$	−1.01406e10	+	1.75640e10i	−0.545866	+	0.945468i	−0.998552	+	0.0537978i	$0.982867\pi$
$860$		0			0
$861$	2.34486e10	+	4.25540e9i	1.25200	+	0.227211i
$862$		0			0
$863$	−2.39028e10			−1.26593			−0.632967	−	0.774179i	$-0.718163\pi$
$863$	−2.39028e10			−1.26593			−0.632967	+	0.774179i	$0.718163\pi$
$864$		0			0
$865$	−6.65992e9			−0.349875
$866$		0			0
$867$	1.15859e10	+	2.10258e9i	0.603756	+	0.109568i
$868$		0			0
$869$	−1.09256e10	+	1.89237e10i	−0.564775	+	0.978219i
$870$		0			0
$871$	1.17717e10	+	2.03891e10i	0.603636	+	1.04553i
$872$		0			0
$873$	−2.66558e9	−	1.60527e10i	−0.135595	−	0.816581i
$874$		0			0
$875$	1.89699e10	+	3.28569e10i	0.957277	+	1.65805i
$876$		0			0
$877$	1.28617e9	−	2.22771e9i	0.0643873	−	0.111522i	−0.832035	−	0.554724i	$-0.812824\pi$
$877$	1.28617e9	−	2.22771e9i	0.0643873	−	0.111522i	0.896422	+	0.443201i	$0.146157\pi$
$878$		0			0
$879$	8.76943e9	−	1.03457e10i	0.435522	−	0.513805i
$880$		0			0
$881$	−3.40624e10			−1.67826			−0.839130	−	0.543931i	$-0.816935\pi$
$881$	−3.40624e10			−1.67826			−0.839130	+	0.543931i	$0.816935\pi$
$882$		0			0
$883$	−5.47993e9			−0.267863			−0.133931	−	0.990991i	$-0.542760\pi$
$883$	−5.47993e9			−0.267863			−0.133931	+	0.990991i	$0.542760\pi$
$884$		0			0
$885$	3.65290e9	+	1.01943e10i	0.177148	+	0.494374i
$886$		0			0
$887$	1.08266e10	−	1.87522e10i	0.520904	−	0.902233i	−0.478800	−	0.877924i	$-0.658928\pi$
$887$	1.08266e10	−	1.87522e10i	0.520904	−	0.902233i	0.999704	−	0.0243090i	$-0.00773856\pi$
$888$		0			0
$889$	9.63048e7	+	1.66805e8i	0.00459718	+	0.00796255i
$890$		0			0
$891$	−4.32263e9	+	2.19436e10i	−0.204727	+	1.03929i
$892$		0			0
$893$	2.18924e9	+	3.79187e9i	0.102876	+	0.178186i
$894$		0			0
$895$	5.93260e9	−	1.02756e10i	0.276608	−	0.479098i
$896$		0			0
$897$	2.46563e9	+	6.88093e9i	0.114066	+	0.318327i
$898$		0			0
$899$	3.80495e10			1.74658
$900$		0			0
$901$	9.50451e9			0.432905
$902$		0			0
$903$	5.98301e9	−	7.05842e9i	0.270404	−	0.319007i
$904$		0			0
$905$	2.99044e9	−	5.17960e9i	0.134111	−	0.232288i
$906$		0			0
$907$	−1.64024e10	−	2.84098e10i	−0.729931	−	1.26428i	−0.956912	−	0.290378i	$-0.906219\pi$
$907$	−1.64024e10	−	2.84098e10i	−0.729931	−	1.26428i	0.226982	−	0.973899i	$-0.427114\pi$
$908$		0			0
$909$	−5.30494e9	−	3.19475e10i	−0.234265	−	1.41079i
$910$		0			0
$911$	−9.25174e9	−	1.60245e10i	−0.405423	−	0.702214i	0.588947	−	0.808172i	$-0.299543\pi$
$911$	−9.25174e9	−	1.60245e10i	−0.405423	−	0.702214i	−0.994371	+	0.105957i	$0.966209\pi$
$912$		0			0
$913$	−2.05076e10	+	3.55202e10i	−0.891800	+	1.54464i
$914$		0			0
$915$	−8.58141e8	−	1.55734e8i	−0.0370326	−	0.00672061i
$916$		0			0
$917$	3.57796e10			1.53230
$918$		0			0
$919$	−1.47226e10			−0.625720			−0.312860	−	0.949799i	$-0.601287\pi$
$919$	−1.47226e10			−0.625720			−0.312860	+	0.949799i	$0.601287\pi$
$920$		0			0
$921$	−1.38002e9	−	2.50443e8i	−0.0582071	−	0.0105633i
$922$		0			0
$923$	−2.34897e9	+	4.06853e9i	−0.0983267	+	0.170307i
$924$		0			0
$925$	5.47951e9	+	9.49078e9i	0.227638	+	0.394281i
$926$		0			0
$927$	−2.91505e10	+	2.39684e10i	−1.20190	+	0.988235i
$928$		0			0
$929$	2.24590e9	+	3.89000e9i	0.0919040	+	0.159182i	0.908312	−	0.418293i	$-0.137371\pi$
$929$	2.24590e9	+	3.89000e9i	0.0919040	+	0.159182i	−0.816408	+	0.577475i	$0.804038\pi$
$930$		0			0
$931$	2.00544e10	−	3.47353e10i	0.814492	−	1.41074i
$932$		0			0
$933$	−4.42343e8	+	5.21851e8i	−0.0178309	+	0.0210359i
$934$		0			0
$935$	−2.48471e10			−0.994111
$936$		0			0
$937$	−3.45477e10			−1.37193			−0.685963	−	0.727637i	$-0.740619\pi$
$937$	−3.45477e10			−1.37193			−0.685963	+	0.727637i	$0.740619\pi$
$938$		0			0
$939$	7.97571e8	+	2.22581e9i	0.0314369	+	0.0877322i
$940$		0			0
$941$	1.06261e10	−	1.84050e10i	0.415729	−	0.720065i	−0.579775	−	0.814776i	$-0.696860\pi$
$941$	1.06261e10	−	1.84050e10i	0.415729	−	0.720065i	0.995505	+	0.0947119i	$0.0301930\pi$
$942$		0			0
$943$	−3.69016e9	−	6.39155e9i	−0.143303	−	0.248208i
$944$		0			0
$945$	2.93209e10	+	1.75190e10i	1.13023	+	0.675301i
$946$		0			0
$947$	−1.52119e10	−	2.63477e10i	−0.582046	−	1.00813i	−0.995237	−	0.0974893i	$-0.968919\pi$
$947$	−1.52119e10	−	2.63477e10i	−0.582046	−	1.00813i	0.413190	−	0.910645i	$-0.364415\pi$
$948$		0			0
$949$	−1.35506e10	+	2.34703e10i	−0.514666	+	0.891429i
$950$		0			0
$951$	1.02475e9	+	2.85980e9i	0.0386353	+	0.107821i
$952$		0			0
$953$	−3.86478e10			−1.44644			−0.723218	−	0.690620i	$-0.757338\pi$
$953$	−3.86478e10			−1.44644			−0.723218	+	0.690620i	$0.757338\pi$
$954$		0			0
$955$	2.04322e10			0.759108
$956$		0			0
$957$	2.30443e10	−	2.71864e10i	0.849909	−	1.00268i
$958$		0			0
$959$	−4.76749e9	+	8.25753e9i	−0.174552	+	0.302333i
$960$		0			0
$961$	−1.34967e10	−	2.33771e10i	−0.490566	−	0.849685i
$962$		0			0
$963$	2.70128e10	+	1.01384e10i	0.974715	+	0.365827i
$964$		0			0
$965$	−1.59501e9	−	2.76264e9i	−0.0571371	−	0.0989643i
$966$		0			0
$967$	−1.70024e10	+	2.94491e10i	−0.604670	+	1.04732i	0.387434	+	0.921898i	$0.373362\pi$
$967$	−1.70024e10	+	2.94491e10i	−0.604670	+	1.04732i	−0.992104	+	0.125421i	$0.959972\pi$
$968$		0			0
$969$	2.65037e10	+	4.80983e9i	0.935778	+	0.169823i
$970$		0			0
$971$	−2.08691e10			−0.731538			−0.365769	−	0.930706i	$-0.619194\pi$
$971$	−2.08691e10			−0.731538			−0.365769	+	0.930706i	$0.619194\pi$
$972$		0			0
$973$	5.75382e10			2.00245
$974$		0			0
$975$	−1.08950e10	−	1.97721e9i	−0.376454	−	0.0683182i
$976$		0			0
$977$	2.13334e10	−	3.69505e10i	0.731862	−	1.26762i	−0.224225	−	0.974537i	$-0.571985\pi$
$977$	2.13334e10	−	3.69505e10i	0.731862	−	1.26762i	0.956087	−	0.293085i	$-0.0946817\pi$
$978$		0			0
$979$	2.51026e10	+	4.34790e10i	0.855027	+	1.48095i
$980$		0			0
$981$	−5.05500e9	−	1.89723e9i	−0.170954	−	0.0641619i
$982$		0			0
$983$	−9.99083e9	−	1.73046e10i	−0.335478	−	0.581065i	0.648098	−	0.761557i	$-0.275565\pi$
$983$	−9.99083e9	−	1.73046e10i	−0.335478	−	0.581065i	−0.983577	+	0.180491i	$0.942231\pi$
$984$		0			0
$985$	−1.01715e10	+	1.76176e10i	−0.339124	+	0.587380i
$986$		0			0
$987$	9.56542e9	−	1.12848e10i	0.316661	−	0.373579i
$988$		0			0
$989$	−2.86553e9			−0.0941928
$990$		0			0
$991$	3.14380e10			1.02612			0.513059	−	0.858353i	$-0.328512\pi$
$991$	3.14380e10			1.02612			0.513059	+	0.858353i	$0.328512\pi$
$992$		0			0
$993$	1.39637e9	+	3.89692e9i	0.0452563	+	0.126299i
$994$		0			0
$995$	−1.45404e10	+	2.51848e10i	−0.467947	+	0.810508i
$996$		0			0
$997$	−1.75392e10	−	3.03787e10i	−0.560500	−	0.970814i	−0.997453	−	0.0713299i	$-0.977276\pi$
$997$	−1.75392e10	−	3.03787e10i	−0.560500	−	0.970814i	0.436953	−	0.899484i	$-0.356058\pi$
$998$		0			0
$999$	2.71179e10	+	1.62027e10i	0.860551	+	0.514172i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.8.i.b.49.4		8
3.2	odd	2		432.8.i.b.145.2		8
4.3	odd	2		18.8.c.b.13.1	yes	8
9.2	odd	6		432.8.i.b.289.2		8
9.7	even	3	inner	144.8.i.b.97.4		8
12.11	even	2		54.8.c.b.37.2		8
36.7	odd	6		18.8.c.b.7.1	✓	8
36.11	even	6		54.8.c.b.19.2		8
36.23	even	6		162.8.a.i.1.3		4
36.31	odd	6		162.8.a.h.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.8.c.b.7.1	✓	8	36.7	odd	6
18.8.c.b.13.1	yes	8	4.3	odd	2
54.8.c.b.19.2		8	36.11	even	6
54.8.c.b.37.2		8	12.11	even	2
144.8.i.b.49.4		8	1.1	even	1	trivial
144.8.i.b.97.4		8	9.7	even	3	inner
162.8.a.h.1.2		4	36.31	odd	6
162.8.a.i.1.3		4	36.23	even	6
432.8.i.b.145.2		8	3.2	odd	2
432.8.i.b.289.2		8	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)

\(f(q)\)	\(=\)	\(q+(46.0138 + 8.35049i) q^{3} +(103.252 - 178.837i) q^{5} +(808.604 + 1400.54i) q^{7} +(2047.54 + 768.476i) q^{9} +(2338.01 + 4049.55i) q^{11} +(-3336.65 + 5779.25i) q^{13} +(6244.39 - 7366.78i) q^{15} -25731.9 q^{17} -22384.5 q^{19} +(25511.7 + 71196.6i) q^{21} +(11710.7 - 20283.5i) q^{23} +(17740.6 + 30727.7i) q^{25} +(87797.9 + 52458.4i) q^{27} +(-81488.4 - 141142. i) q^{29} +(-116733. + 202187. i) q^{31} +(73764.9 + 205859. i) q^{33} +333959. q^{35} +308867. q^{37} +(-201792. + 238063. i) q^{39} +(157555. - 272893. i) q^{41} +(-61173.2 - 105955. i) q^{43} +(348844. - 286830. i) q^{45} +(-97801.6 - 169397. i) q^{47} +(-895910. + 1.55176e6i) q^{49} +(-1.18402e6 - 214874. i) q^{51} -369367. q^{53} +965615. q^{55} +(-1.02999e6 - 186921. i) q^{57} +(-560669. + 971107. i) q^{59} +(-45155.7 - 78212.0i) q^{61} +(579364. + 3.48906e6i) q^{63} +(689031. + 1.19344e6i) q^{65} +(1.76399e6 - 3.05533e6i) q^{67} +(708232. - 835532. i) q^{69} +703989. q^{71} +4.06113e6 q^{73} +(559723. + 1.56204e6i) q^{75} +(-3.78105e6 + 6.54897e6i) q^{77} +(2.33651e6 + 4.04696e6i) q^{79} +(3.60186e6 + 3.14697e6i) q^{81} +(4.38570e6 + 7.59625e6i) q^{83} +(-2.65686e6 + 4.60182e6i) q^{85} +(-2.57099e6 - 7.17495e6i) q^{87} +1.07368e7 q^{89} -1.07921e7 q^{91} +(-7.05968e6 + 8.32862e6i) q^{93} +(-2.31123e6 + 4.00317e6i) q^{95} +(-3.72029e6 - 6.44373e6i) q^{97} +(1.67518e6 + 1.00883e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 12 q^{3} + 54 q^{5} + 44 q^{7} - 2238 q^{9} - 2172 q^{11} - 6398 q^{13} - 17604 q^{15} - 51972 q^{17} - 90712 q^{19} + 192390 q^{21} + 2028 q^{23} - 173446 q^{25} + 288360 q^{27} + 283098 q^{29} + 812 q^{31}+ \cdots - 26558784 q^{99}+O(q^{100}) \) 8 * q + 12 * q^3 + 54 * q^5 + 44 * q^7 - 2238 * q^9 - 2172 * q^11 - 6398 * q^13 - 17604 * q^15 - 51972 * q^17 - 90712 * q^19 + 192390 * q^21 + 2028 * q^23 - 173446 * q^25 + 288360 * q^27 + 283098 * q^29 + 812 * q^31 + 342108 * q^33 + 1492992 * q^35 + 441688 * q^37 - 67428 * q^39 + 610704 * q^41 + 84380 * q^43 - 652698 * q^45 - 855708 * q^47 - 1273038 * q^49 - 1748556 * q^51 - 1842600 * q^53 - 7016760 * q^55 - 4893306 * q^57 - 380796 * q^59 - 3151130 * q^61 + 3149076 * q^63 - 326538 * q^65 + 9961580 * q^67 + 15322554 * q^69 - 8970384 * q^71 + 29234932 * q^73 + 26321244 * q^75 - 5954826 * q^77 + 17309396 * q^79 + 6907266 * q^81 + 11520192 * q^83 - 14675148 * q^85 - 37274652 * q^87 + 26066064 * q^89 - 26231936 * q^91 - 48684366 * q^93 + 31016952 * q^95 - 22003112 * q^97 - 26558784 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more