LMFDB - Embedded newform 72.8.d.c.37.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,8,Mod(37,72)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("72.37");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 72 = 2^{3} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	72.d (of order $2$, degree $1$, 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:	$22.4917218349$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 206x^{10} + 24336x^{8} - 1510912x^{6} + 398721024x^{4} - 55297703936x^{2} + 4398046511104 $ x^12 - 206x^10 + 24336x^8 - 1510912x^6 + 398721024x^4 - 55297703936*x^2 + 4398046511104
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{30}\cdot 3^{6}\cdot 5^{2}\cdot 13^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			37.6
Root			$4.93370 + 10.1813i$ of defining polynomial
Character	$\chi$	$=$	72.37
Dual form			72.8.d.c.37.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.93370 + 10.1813i) q^{2} +(-79.3172 - 100.463i) q^{4} -181.720i q^{5} +142.301 q^{7} +(1414.17 - 311.897i) q^{8} +O(q^{10})$	\(q+(-4.93370 + 10.1813i) q^{2} +(-79.3172 - 100.463i) q^{4} -181.720i q^{5} +142.301 q^{7} +(1414.17 - 311.897i) q^{8} +(1850.14 + 896.551i) q^{10} -2456.40i q^{11} +5781.09i q^{13} +(-702.069 + 1448.80i) q^{14} +(-3801.58 + 15936.9i) q^{16} -15704.0 q^{17} +29121.8i q^{19} +(-18256.1 + 14413.5i) q^{20} +(25009.3 + 12119.1i) q^{22} -58467.2 q^{23} +45103.0 q^{25} +(-58858.9 - 28522.2i) q^{26} +(-11286.9 - 14295.9i) q^{28} -26818.8i q^{29} -170990. q^{31} +(-143502. - 117333. i) q^{32} +(77479.0 - 159887. i) q^{34} -25858.8i q^{35} -21445.4i q^{37} +(-296497. - 143678. i) q^{38} +(-56677.7 - 256982. i) q^{40} -586675. q^{41} +197379. i q^{43} +(-246777. + 194835. i) q^{44} +(288460. - 595271. i) q^{46} -597097. q^{47} -803294. q^{49} +(-222525. + 459206. i) q^{50} +(580785. - 458540. i) q^{52} -675222. i q^{53} -446376. q^{55} +(201237. - 44383.1i) q^{56} +(273050. + 132316. i) q^{58} +587777. i q^{59} +3.33803e6i q^{61} +(843613. - 1.74090e6i) q^{62} +(1.90259e6 - 882149. i) q^{64} +1.05054e6 q^{65} +3.90475e6i q^{67} +(1.24560e6 + 1.57767e6i) q^{68} +(263276. + 127580. i) q^{70} +4.33729e6 q^{71} -3.72507e6 q^{73} +(218342. + 105805. i) q^{74} +(2.92566e6 - 2.30986e6i) q^{76} -349547. i q^{77} -3.21723e6 q^{79} +(2.89604e6 + 690821. i) q^{80} +(2.89448e6 - 5.97310e6i) q^{82} +5.92166e6i q^{83} +2.85373e6i q^{85} +(-2.00957e6 - 973810. i) q^{86} +(-766142. - 3.47376e6i) q^{88} +1.73709e6 q^{89} +822653. i q^{91} +(4.63745e6 + 5.87378e6i) q^{92} +(2.94590e6 - 6.07921e6i) q^{94} +5.29200e6 q^{95} -1.95917e6 q^{97} +(3.96321e6 - 8.17856e6i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 412 q^{4} + 136 q^{7}+O(q^{10}) $ 12 * q + 412 * q^4 + 136 * q^7	$ 12 q + 412 q^{4} + 136 q^{7} + 4680 q^{10} - 12472 q^{16} + 32624 q^{22} + 229820 q^{25} - 157288 q^{28} - 37224 q^{31} - 74432 q^{34} - 937520 q^{40} - 1264256 q^{46} + 2668188 q^{49} - 1539680 q^{52} + 6928960 q^{55} - 4035448 q^{58} - 3530192 q^{64} - 10228720 q^{70} + 13619048 q^{73} - 2441920 q^{76} + 20470552 q^{79} - 2507200 q^{82} - 26170912 q^{88} - 22132608 q^{94} + 27442456 q^{97}+O(q^{100}) $ 12 * q + 412 * q^4 + 136 * q^7 + 4680 * q^10 - 12472 * q^16 + 32624 * q^22 + 229820 * q^25 - 157288 * q^28 - 37224 * q^31 - 74432 * q^34 - 937520 * q^40 - 1264256 * q^46 + 2668188 * q^49 - 1539680 * q^52 + 6928960 * q^55 - 4035448 * q^58 - 3530192 * q^64 - 10228720 * q^70 + 13619048 * q^73 - 2441920 * q^76 + 20470552 * q^79 - 2507200 * q^82 - 26170912 * q^88 - 22132608 * q^94 + 27442456 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/72\mathbb{Z}\right)^\times$.

$n$	$37$	$55$	$65$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	−4.93370	+	10.1813i	−0.436082	+	0.899907i
$2$	−4.93370	+	10.1813i	−0.436082	+	0.899907i
$3$		0			0
$3$		0			0
$4$	−79.3172	−	100.463i	−0.619665	−	0.784866i
$5$		−	181.720i		−	0.650140i	−0.945690	−	0.325070i	$-0.894612\pi$
$5$		−	181.720i		−	0.650140i	0.945690	−	0.325070i	$-0.105388\pi$
$6$		0			0
$7$	142.301			0.156806			0.0784032	−	0.996922i	$-0.475018\pi$
$7$	142.301			0.156806			0.0784032	+	0.996922i	$0.475018\pi$
$8$	1414.17	−	311.897i	0.976531	−	0.215375i
$9$		0			0
$10$	1850.14	+	896.551i	0.585065	+	0.283514i
$11$		−	2456.40i		−	0.556448i	−0.960516	−	0.278224i	$-0.910254\pi$
$11$		−	2456.40i		−	0.556448i	0.960516	−	0.278224i	$-0.0897457\pi$
$12$		0			0
$13$			5781.09i			0.729807i	0.931045	+	0.364904i	$0.118898\pi$
$13$			5781.09i			0.729807i	−0.931045	+	0.364904i	$0.881102\pi$
$14$	−702.069	+	1448.80i	−0.0683804	+	0.141111i
$15$		0			0
$16$	−3801.58	+	15936.9i	−0.232030	+	0.972709i
$17$	−15704.0			−0.775246			−0.387623	−	0.921818i	$-0.626704\pi$
$17$	−15704.0			−0.775246			−0.387623	+	0.921818i	$0.626704\pi$
$18$		0			0
$19$			29121.8i			0.974049i	0.873388	+	0.487024i	$0.161918\pi$
$19$			29121.8i			0.974049i	−0.873388	+	0.487024i	$0.838082\pi$
$20$	−18256.1	+	14413.5i	−0.510273	+	0.402869i
$21$		0			0
$22$	25009.3	+	12119.1i	0.500751	+	0.242657i
$23$	−58467.2			−1.00199			−0.500997	−	0.865449i	$-0.667033\pi$
$23$	−58467.2			−1.00199			−0.500997	+	0.865449i	$0.667033\pi$
$24$		0			0
$25$	45103.0			0.577318
$26$	−58858.9	−	28522.2i	−0.656759	−	0.318256i
$27$		0			0
$28$	−11286.9	−	14295.9i	−0.0971675	−	0.123072i
$29$		−	26818.8i		−	0.204195i	−0.994774	−	0.102098i	$-0.967445\pi$
$29$		−	26818.8i		−	0.204195i	0.994774	−	0.102098i	$-0.0325554\pi$
$30$		0			0
$31$	−170990.			−1.03087			−0.515435	−	0.856928i	$-0.672370\pi$
$31$	−170990.			−1.03087			−0.515435	+	0.856928i	$0.672370\pi$
$32$	−143502.	−	117333.i	−0.774163	−	0.632986i
$33$		0			0
$34$	77479.0	−	159887.i	0.338071	−	0.697650i
$35$		−	25858.8i		−	0.101946i
$36$		0			0
$37$		−	21445.4i		−	0.0696030i	−0.999394	−	0.0348015i	$-0.988920\pi$
$37$		−	21445.4i		−	0.0696030i	0.999394	−	0.0348015i	$-0.0110799\pi$
$38$	−296497.	−	143678.i	−0.876553	−	0.424765i
$39$		0			0
$40$	−56677.7	−	256982.i	−0.140024	−	0.634882i
$41$	−586675.			−1.32939			−0.664697	−	0.747113i	$-0.731439\pi$
$41$	−586675.			−1.32939			−0.664697	+	0.747113i	$0.731439\pi$
$42$		0			0
$43$			197379.i			0.378584i	0.981921	+	0.189292i	$0.0606192\pi$
$43$			197379.i			0.378584i	−0.981921	+	0.189292i	$0.939381\pi$
$44$	−246777.	+	194835.i	−0.436737	+	0.344811i
$45$		0			0
$46$	288460.	−	595271.i	0.436951	−	0.901701i
$47$	−597097.			−0.838885			−0.419442	−	0.907782i	$-0.637774\pi$
$47$	−597097.			−0.838885			−0.419442	+	0.907782i	$0.637774\pi$
$48$		0			0
$49$	−803294.			−0.975412
$50$	−222525.	+	459206.i	−0.251758	+	0.519533i
$51$		0			0
$52$	580785.	−	458540.i	0.572801	−	0.452236i
$53$		−	675222.i		−	0.622990i	−0.950248	−	0.311495i	$-0.899170\pi$
$53$		−	675222.i		−	0.622990i	0.950248	−	0.311495i	$-0.100830\pi$
$54$		0			0
$55$	−446376.			−0.361769
$56$	201237.	−	44383.1i	0.153126	−	0.0337722i
$57$		0			0
$58$	273050.	+	132316.i	0.183757	+	0.0890459i
$59$			587777.i			0.372590i	0.982494	+	0.186295i	$0.0596479\pi$
$59$			587777.i			0.372590i	−0.982494	+	0.186295i	$0.940352\pi$
$60$		0			0
$61$			3.33803e6i			1.88293i	0.337104	+	0.941467i	$0.390552\pi$
$61$			3.33803e6i			1.88293i	−0.337104	+	0.941467i	$0.609448\pi$
$62$	843613.	−	1.74090e6i	0.449544	−	0.927688i
$63$		0			0
$64$	1.90259e6	−	882149.i	0.907227	−	0.420641i
$65$	1.05054e6			0.474477
$66$		0			0
$67$			3.90475e6i			1.58610i	0.609156	+	0.793051i	$0.291509\pi$
$67$			3.90475e6i			1.58610i	−0.609156	+	0.793051i	$0.708491\pi$
$68$	1.24560e6	+	1.57767e6i	0.480393	+	0.608465i
$69$		0			0
$70$	263276.	+	127580.i	0.0917420	+	0.0444568i
$71$	4.33729e6			1.43818			0.719092	−	0.694915i	$-0.244558\pi$
$71$	4.33729e6			1.43818			0.719092	+	0.694915i	$0.244558\pi$
$72$		0			0
$73$	−3.72507e6			−1.12074			−0.560370	−	0.828243i	$-0.689341\pi$
$73$	−3.72507e6			−1.12074			−0.560370	+	0.828243i	$0.689341\pi$
$74$	218342.	+	105805.i	0.0626362	+	0.0303526i
$75$		0			0
$76$	2.92566e6	−	2.30986e6i	0.764498	−	0.603584i
$77$		−	349547.i		−	0.0872546i
$78$		0			0
$79$	−3.21723e6			−0.734154			−0.367077	−	0.930190i	$-0.619642\pi$
$79$	−3.21723e6			−0.734154			−0.367077	+	0.930190i	$0.619642\pi$
$80$	2.89604e6	+	690821.i	0.632397	+	0.150852i
$81$		0			0
$82$	2.89448e6	−	5.97310e6i	0.579725	−	1.19633i
$83$			5.92166e6i			1.13676i	0.822765	+	0.568382i	$0.192430\pi$
$83$			5.92166e6i			1.13676i	−0.822765	+	0.568382i	$0.807570\pi$
$84$		0			0
$85$			2.85373e6i			0.504019i
$86$	−2.00957e6	−	973810.i	−0.340690	−	0.165093i
$87$		0			0
$88$	−766142.	−	3.47376e6i	−0.119845	−	0.543389i
$89$	1.73709e6			0.261191			0.130595	−	0.991436i	$-0.458311\pi$
$89$	1.73709e6			0.261191			0.130595	+	0.991436i	$0.458311\pi$
$90$		0			0
$91$			822653.i			0.114438i
$92$	4.63745e6	+	5.87378e6i	0.620901	+	0.786431i
$93$		0			0
$94$	2.94590e6	−	6.07921e6i	0.365822	−	0.754918i
$95$	5.29200e6			0.633268
$96$		0			0
$97$	−1.95917e6			−0.217957			−0.108979	−	0.994044i	$-0.534758\pi$
$97$	−1.95917e6			−0.217957			−0.108979	+	0.994044i	$0.534758\pi$
$98$	3.96321e6	−	8.17856e6i	0.425359	−	0.877780i
$99$		0			0
$100$	−3.57744e6	−	4.53117e6i	−0.357744	−	0.453117i
$101$		−	1.71734e7i		−	1.65856i	−0.558831	−	0.829281i	$-0.688750\pi$
$101$		−	1.71734e7i		−	1.65856i	0.558831	−	0.829281i	$-0.311250\pi$
$102$		0			0
$103$	−9.43145e6			−0.850449			−0.425225	−	0.905088i	$-0.639805\pi$
$103$	−9.43145e6			−0.850449			−0.425225	+	0.905088i	$0.639805\pi$
$104$	1.80310e6	+	8.17544e6i	0.157182	+	0.712680i
$105$		0			0
$106$	6.87462e6	+	3.33134e6i	0.560633	+	0.271674i
$107$			1.14291e7i			0.901919i	0.892544	+	0.450959i	$0.148918\pi$
$107$			1.14291e7i			0.901919i	−0.892544	+	0.450959i	$0.851082\pi$
$108$		0			0
$109$		−	2.40670e6i		−	0.178004i	−0.996031	−	0.0890018i	$-0.971632\pi$
$109$		−	2.40670e6i		−	0.178004i	0.996031	−	0.0890018i	$-0.0283677\pi$
$110$	2.20229e6	−	4.54468e6i	0.157761	−	0.325558i
$111$		0			0
$112$	−540967.	+	2.26783e6i	−0.0363838	+	0.152527i
$113$	−2.18386e7			−1.42380			−0.711902	−	0.702279i	$-0.752166\pi$
$113$	−2.18386e7			−1.42380			−0.711902	+	0.702279i	$0.752166\pi$
$114$		0			0
$115$			1.06246e7i			0.651436i
$116$	−2.69429e6	+	2.12719e6i	−0.160266	+	0.126533i
$117$		0			0
$118$	−5.98433e6	−	2.89992e6i	−0.335296	−	0.162480i
$119$	−2.23469e6			−0.121564
$120$		0			0
$121$	1.34533e7			0.690366
$122$	−3.39854e7	−	1.64688e7i	−1.69447	−	0.821114i
$123$		0			0
$124$	1.35624e7	+	1.71781e7i	0.638795	+	0.809096i
$125$		−	2.23929e7i		−	1.02548i
$126$		0			0
$127$	1.27930e7			0.554191			0.277095	−	0.960842i	$-0.410628\pi$
$127$	1.27930e7			0.554191			0.277095	+	0.960842i	$0.410628\pi$
$128$	−405421.	+	2.37231e7i	−0.0170872	+	0.999854i
$129$		0			0
$130$	−5.18304e6	+	1.06958e7i	−0.206911	+	0.426985i
$131$			4.02317e7i			1.56357i	0.623545	+	0.781787i	$0.285692\pi$
$131$			4.02317e7i			1.56357i	−0.623545	+	0.781787i	$0.714308\pi$
$132$		0			0
$133$			4.14405e6i			0.152737i
$134$	−3.97553e7	−	1.92649e7i	−1.42734	−	0.691670i
$135$		0			0
$136$	−2.22081e7	+	4.89803e6i	−0.757053	+	0.166969i
$137$	−3.46309e7			−1.15065			−0.575323	−	0.817926i	$-0.695124\pi$
$137$	−3.46309e7			−1.15065			−0.575323	+	0.817926i	$0.695124\pi$
$138$		0			0
$139$		−	2.76664e7i		−	0.873779i	−0.899515	−	0.436889i	$-0.856080\pi$
$139$		−	2.76664e7i		−	0.873779i	0.899515	−	0.436889i	$-0.143920\pi$
$140$	−2.59785e6	+	2.05105e6i	−0.0800140	+	0.0631725i
$141$		0			0
$142$	−2.13989e7	+	4.41592e7i	−0.627166	+	1.29423i
$143$	1.42007e7			0.406100
$144$		0			0
$145$	−4.87350e6			−0.132756
$146$	1.83784e7	−	3.79260e7i	0.488734	−	1.00856i
$147$		0			0
$148$	−2.15447e6	+	1.70099e6i	−0.0546290	+	0.0431305i
$149$		−	6.35611e7i		−	1.57413i	−0.616872	−	0.787063i	$-0.711600\pi$
$149$		−	6.35611e7i		−	1.57413i	0.616872	−	0.787063i	$-0.288400\pi$
$150$		0			0
$151$	5.40032e7			1.27644			0.638219	−	0.769855i	$-0.279671\pi$
$151$	5.40032e7			1.27644			0.638219	+	0.769855i	$0.279671\pi$
$152$	9.08299e6	+	4.11831e7i	0.209786	+	0.951189i
$153$		0			0
$154$	3.55884e6	+	1.72456e6i	0.0785210	+	0.0380501i
$155$			3.10722e7i			0.670210i
$156$		0			0
$157$		−	6.17904e7i		−	1.27430i	−0.770739	−	0.637151i	$-0.780113\pi$
$157$		−	6.17904e7i		−	1.27430i	0.770739	−	0.637151i	$-0.219887\pi$
$158$	1.58729e7	−	3.27555e7i	0.320151	−	0.660671i
$159$		0			0
$160$	−2.13217e7	+	2.60771e7i	−0.411529	+	0.503314i
$161$	−8.31992e6			−0.157119
$162$		0			0
$163$		−	9.03941e7i		−	1.63487i	−0.576021	−	0.817435i	$-0.695395\pi$
$163$		−	9.03941e7i		−	1.63487i	0.576021	−	0.817435i	$-0.304605\pi$
$164$	4.65334e7	+	5.89390e7i	0.823780	+	1.04340i
$165$		0			0
$166$	−6.02901e7	−	2.92157e7i	−1.02298	−	0.495722i
$167$	3.52259e7			0.585268			0.292634	−	0.956225i	$-0.405468\pi$
$167$	3.52259e7			0.585268			0.292634	+	0.956225i	$0.405468\pi$
$168$		0			0
$169$	2.93275e7			0.467382
$170$	−2.90546e7	−	1.40795e7i	−0.453570	−	0.219793i
$171$		0			0
$172$	1.98293e7	−	1.56556e7i	0.297137	−	0.234595i
$173$		−	6.72578e7i		−	0.987601i	−0.869575	−	0.493800i	$-0.835607\pi$
$173$		−	6.72578e7i		−	0.987601i	0.869575	−	0.493800i	$-0.164393\pi$
$174$		0			0
$175$	6.41818e6			0.0905272
$176$	3.91473e7	+	9.33819e6i	0.541262	+	0.129113i
$177$		0			0
$178$	−8.57030e6	+	1.76858e7i	−0.113901	+	0.235048i
$179$			1.33915e8i			1.74520i	0.488437	+	0.872599i	$0.337567\pi$
$179$			1.33915e8i			1.74520i	−0.488437	+	0.872599i	$0.662433\pi$
$180$		0			0
$181$		−	6.55914e7i		−	0.822189i	−0.911593	−	0.411095i	$-0.865147\pi$
$181$		−	6.55914e7i		−	0.822189i	0.911593	−	0.411095i	$-0.134853\pi$
$182$	−8.37567e6	−	4.05873e6i	−0.102984	−	0.0499045i
$183$		0			0
$184$	−8.26825e7	+	1.82357e7i	−0.978478	+	0.215805i
$185$	−3.89705e6			−0.0452517
$186$		0			0
$187$			3.85754e7i			0.431384i
$188$	4.73600e7	+	5.99861e7i	0.519828	+	0.658412i
$189$		0			0
$190$	−2.61092e7	+	5.38794e7i	−0.276157	+	0.569882i
$191$	1.48094e8			1.53787			0.768937	−	0.639325i	$-0.220786\pi$
$191$	1.48094e8			1.53787			0.768937	+	0.639325i	$0.220786\pi$
$192$		0			0
$193$	8.31622e7			0.832674			0.416337	−	0.909210i	$-0.363314\pi$
$193$	8.31622e7			0.832674			0.416337	+	0.909210i	$0.363314\pi$
$194$	9.66596e6	−	1.99469e7i	0.0950471	−	0.196141i
$195$		0			0
$196$	6.37150e7	+	8.07012e7i	0.604429	+	0.765568i
$197$		−	9.69265e6i		−	0.0903255i	−0.998980	−	0.0451628i	$-0.985619\pi$
$197$		−	9.69265e6i		−	0.0903255i	0.998980	−	0.0451628i	$-0.0143806\pi$
$198$		0			0
$199$	−4.43216e7			−0.398685			−0.199343	−	0.979930i	$-0.563881\pi$
$199$	−4.43216e7			−0.398685			−0.199343	+	0.979930i	$0.563881\pi$
$200$	6.37832e7	−	1.40675e7i	0.563769	−	0.124340i
$201$		0			0
$202$	1.74847e8	+	8.47285e7i	1.49255	+	0.723269i
$203$		−	3.81633e6i		−	0.0320191i
$204$		0			0
$205$			1.06610e8i			0.864292i
$206$	4.65320e7	−	9.60243e7i	0.370865	−	0.765325i
$207$		0			0
$208$	−9.21324e7	−	2.19773e7i	−0.709890	−	0.169337i
$209$	7.15348e7			0.542007
$210$		0			0
$211$			6.50991e7i			0.477075i	0.971133	+	0.238537i	$0.0766679\pi$
$211$			6.50991e7i			0.477075i	−0.971133	+	0.238537i	$0.923332\pi$
$212$	−6.78347e7	+	5.35567e7i	−0.488963	+	0.386045i
$213$		0			0
$214$	−1.16363e8	−	5.63876e7i	−0.811643	−	0.393310i
$215$	3.58677e7			0.246132
$216$		0			0
$217$	−2.43320e7			−0.161647
$218$	2.45033e7	+	1.18739e7i	0.160187	+	0.0776241i
$219$		0			0
$220$	3.54053e7	+	4.48442e7i	0.224176	+	0.283940i
$221$		−	9.07864e7i		−	0.565780i
$222$		0			0
$223$	641903.			0.00387616			0.00193808	−	0.999998i	$-0.499383\pi$
$223$	641903.			0.00387616			0.00193808	+	0.999998i	$0.499383\pi$
$224$	−2.04204e7	−	1.66965e7i	−0.121394	−	0.0992562i
$225$		0			0
$226$	1.07745e8	−	2.22345e8i	0.620895	−	1.28129i
$227$			9.67017e7i			0.548711i	0.961628	+	0.274355i	$0.0884645\pi$
$227$			9.67017e7i			0.548711i	−0.961628	+	0.274355i	$0.911536\pi$
$228$		0			0
$229$			2.38606e7i			0.131298i	0.997843	+	0.0656489i	$0.0209117\pi$
$229$			2.38606e7i			0.131298i	−0.997843	+	0.0656489i	$0.979088\pi$
$230$	−1.08173e8	−	5.24188e7i	−0.586232	−	0.284079i
$231$		0			0
$232$	−8.36468e6	−	3.79263e7i	−0.0439786	−	0.199403i
$233$	−8.69266e7			−0.450202			−0.225101	−	0.974335i	$-0.572271\pi$
$233$	−8.69266e7			−0.450202			−0.225101	+	0.974335i	$0.572271\pi$
$234$		0			0
$235$			1.08504e8i			0.545392i
$236$	5.90498e7	−	4.66208e7i	0.292433	−	0.230881i
$237$		0			0
$238$	1.10253e7	−	2.27521e7i	0.0530117	−	0.109396i
$239$	−3.83176e8			−1.81554			−0.907770	−	0.419468i	$-0.862217\pi$
$239$	−3.83176e8			−1.81554			−0.907770	+	0.419468i	$0.862217\pi$
$240$		0			0
$241$	−2.54040e8			−1.16907			−0.584537	−	0.811367i	$-0.698724\pi$
$241$	−2.54040e8			−1.16907			−0.584537	+	0.811367i	$0.698724\pi$
$242$	−6.63745e7	+	1.36972e8i	−0.301056	+	0.621265i
$243$		0			0
$244$	3.35348e8	−	2.64763e8i	1.47785	−	1.16679i
$245$			1.45974e8i			0.634154i
$246$		0			0
$247$	−1.68356e8			−0.710868
$248$	−2.41808e8	+	5.33311e7i	−1.00668	+	0.222024i
$249$		0			0
$250$	2.27989e8	+	1.10480e8i	0.922834	+	0.447192i
$251$		−	3.97000e8i		−	1.58465i	−0.610101	−	0.792324i	$-0.708871\pi$
$251$		−	3.97000e8i		−	1.58465i	0.610101	−	0.792324i	$-0.291129\pi$
$252$		0			0
$253$			1.43619e8i			0.557557i
$254$	−6.31168e7	+	1.30249e8i	−0.241673	+	0.498720i
$255$		0			0
$256$	−2.39531e8	−	1.21170e8i	−0.892324	−	0.451395i
$257$	−3.45658e8			−1.27023			−0.635113	−	0.772419i	$-0.719047\pi$
$257$	−3.45658e8			−1.27023			−0.635113	+	0.772419i	$0.719047\pi$
$258$		0			0
$259$		−	3.05169e6i		−	0.0109142i
$260$	−8.33257e7	−	1.05540e8i	−0.294017	−	0.372401i
$261$		0			0
$262$	−4.09610e8	−	1.98491e8i	−1.40707	−	0.681846i
$263$	−5.36439e6			−0.0181834			−0.00909170	−	0.999959i	$-0.502894\pi$
$263$	−5.36439e6			−0.0181834			−0.00909170	+	0.999959i	$0.502894\pi$
$264$		0			0
$265$	−1.22701e8			−0.405030
$266$	−4.21918e7	−	2.04455e7i	−0.137449	−	0.0666059i
$267$		0			0
$268$	3.92282e8	−	3.09713e8i	1.24488	−	0.982852i
$269$			1.51836e8i			0.475601i	0.971314	+	0.237800i	$0.0764264\pi$
$269$			1.51836e8i			0.475601i	−0.971314	+	0.237800i	$0.923574\pi$
$270$		0			0
$271$	−9.10422e7			−0.277875			−0.138938	−	0.990301i	$-0.544369\pi$
$271$	−9.10422e7			−0.277875			−0.138938	+	0.990301i	$0.544369\pi$
$272$	5.97001e7	−	2.50273e8i	0.179880	−	0.754089i
$273$		0			0
$274$	1.70859e8	−	3.52587e8i	0.501776	−	1.03547i
$275$		−	1.10791e8i		−	0.321247i
$276$		0			0
$277$			2.62715e8i			0.742686i	0.928496	+	0.371343i	$0.121103\pi$
$277$			2.62715e8i			0.742686i	−0.928496	+	0.371343i	$0.878897\pi$
$278$	2.81680e8	+	1.36498e8i	0.786320	+	0.381039i
$279$		0			0
$280$	−8.06528e6	−	3.65687e7i	−0.0219567	−	0.0995536i
$281$	4.72035e8			1.26912			0.634559	−	0.772874i	$-0.281182\pi$
$281$	4.72035e8			1.26912			0.634559	+	0.772874i	$0.281182\pi$
$282$		0			0
$283$			1.09703e8i			0.287717i	0.989598	+	0.143858i	$0.0459510\pi$
$283$			1.09703e8i			0.287717i	−0.989598	+	0.143858i	$0.954049\pi$
$284$	−3.44022e8	−	4.35737e8i	−0.891193	−	1.12878i
$285$		0			0
$286$	−7.00619e7	+	1.44581e8i	−0.177093	+	0.365452i
$287$	−8.34842e7			−0.208458
$288$		0			0
$289$	−1.63722e8			−0.398993
$290$	2.40444e7	−	4.96185e7i	0.0578923	−	0.119468i
$291$		0			0
$292$	2.95462e8	+	3.74231e8i	0.694483	+	0.879630i
$293$			3.98243e8i			0.924936i	0.886636	+	0.462468i	$0.153036\pi$
$293$			3.98243e8i			0.924936i	−0.886636	+	0.462468i	$0.846964\pi$
$294$		0			0
$295$	1.06811e8			0.242235
$296$	−6.68874e6	−	3.03274e7i	−0.0149908	−	0.0679695i
$297$		0			0
$298$	6.47134e8	+	3.13592e8i	1.41657	+	0.686448i
$299$		−	3.38004e8i		−	0.731262i
$300$		0			0
$301$			2.80872e7i			0.0593643i
$302$	−2.66436e8	+	5.49822e8i	−0.556632	+	1.14868i
$303$		0			0
$304$	−4.64110e8	−	1.10709e8i	−0.947466	−	0.226008i
$305$	6.06585e8			1.22417
$306$		0			0
$307$			6.56108e8i			1.29417i	0.762419	+	0.647084i	$0.224012\pi$
$307$			6.56108e8i			1.29417i	−0.762419	+	0.647084i	$0.775988\pi$
$308$	−3.51165e7	+	2.77251e7i	−0.0684832	+	0.0540686i
$309$		0			0
$310$	−3.16355e8	−	1.53301e8i	−0.603127	−	0.292267i
$311$	−4.05434e8			−0.764290			−0.382145	−	0.924102i	$-0.624815\pi$
$311$	−4.05434e8			−0.764290			−0.382145	+	0.924102i	$0.624815\pi$
$312$		0			0
$313$	2.41061e7			0.0444346			0.0222173	−	0.999753i	$-0.492927\pi$
$313$	2.41061e7			0.0444346			0.0222173	+	0.999753i	$0.492927\pi$
$314$	6.29105e8	+	3.04855e8i	1.14675	+	0.555700i
$315$		0			0
$316$	2.55182e8	+	3.23212e8i	0.454930	+	0.576213i
$317$			7.64888e8i			1.34862i	0.738447	+	0.674311i	$0.235559\pi$
$317$			7.64888e8i			1.34862i	−0.738447	+	0.674311i	$0.764441\pi$
$318$		0			0
$319$	−6.58776e7			−0.113624
$320$	−1.60304e8	−	3.45739e8i	−0.273476	−	0.589825i
$321$		0			0
$322$	4.10480e7	−	8.47075e7i	0.0685168	−	0.141393i
$323$		−	4.57330e8i		−	0.755128i
$324$		0			0
$325$			2.60744e8i			0.421331i
$326$	9.20328e8	+	4.45977e8i	1.47123	+	0.712937i
$327$		0			0
$328$	−8.29657e8	+	1.82982e8i	−1.29820	+	0.286319i
$329$	−8.49673e7			−0.131542
$330$		0			0
$331$		−	7.22579e8i		−	1.09518i	−0.836745	−	0.547592i	$-0.815545\pi$
$331$		−	7.22579e8i		−	1.09518i	0.836745	−	0.547592i	$-0.184455\pi$
$332$	5.94907e8	−	4.69689e8i	0.892207	−	0.704413i
$333$		0			0
$334$	−1.73794e8	+	3.58645e8i	−0.255225	+	0.526687i
$335$	7.09569e8			1.03119
$336$		0			0
$337$	1.02011e9			1.45191			0.725957	−	0.687740i	$-0.241397\pi$
$337$	1.02011e9			1.45191			0.725957	+	0.687740i	$0.241397\pi$
$338$	−1.44693e8	+	2.98592e8i	−0.203817	+	0.420600i
$339$		0			0
$340$	2.86694e8	−	2.26350e8i	0.395587	−	0.312323i
$341$			4.20019e8i			0.573626i
$342$		0			0
$343$	−2.31500e8			−0.309757
$344$	6.15619e7	+	2.79127e8i	0.0815375	+	0.369699i
$345$		0			0
$346$	6.84771e8	+	3.31830e8i	0.888749	+	0.430675i
$347$		−	1.05676e9i		−	1.35776i	−0.734248	−	0.678881i	$-0.762465\pi$
$347$		−	1.05676e9i		−	1.35776i	0.734248	−	0.678881i	$-0.237535\pi$
$348$		0			0
$349$		−	1.41141e9i		−	1.77731i	−0.458575	−	0.888656i	$-0.651640\pi$
$349$		−	1.41141e9i		−	1.77731i	0.458575	−	0.888656i	$-0.348360\pi$
$350$	−3.16654e7	+	6.53454e7i	−0.0394773	+	0.0814660i
$351$		0			0
$352$	−2.88216e8	+	3.52498e8i	−0.352224	+	0.430781i
$353$	1.08394e9			1.31158			0.655789	−	0.754944i	$-0.272336\pi$
$353$	1.08394e9			1.31158			0.655789	+	0.754944i	$0.272336\pi$
$354$		0			0
$355$		−	7.88171e8i		−	0.935021i
$356$	−1.37781e8	−	1.74513e8i	−0.161851	−	0.205000i
$357$		0			0
$358$	−1.36343e9	−	6.60699e8i	−1.57052	−	0.761049i
$359$	−3.53330e8			−0.403041			−0.201521	−	0.979484i	$-0.564588\pi$
$359$	−3.53330e8			−0.403041			−0.201521	+	0.979484i	$0.564588\pi$
$360$		0			0
$361$	4.57922e7			0.0512290
$362$	6.67805e8	+	3.23608e8i	0.739894	+	0.358542i
$363$		0			0
$364$	8.26461e7	−	6.52505e7i	0.0898189	−	0.0709135i
$365$			6.76919e8i			0.728637i
$366$		0			0
$367$	7.10822e8			0.750636			0.375318	−	0.926896i	$-0.377533\pi$
$367$	7.10822e8			0.750636			0.375318	+	0.926896i	$0.377533\pi$
$368$	2.22268e8	−	9.31784e8i	0.232493	−	0.974648i
$369$		0			0
$370$	1.92269e7	−	3.96770e7i	0.0197334	−	0.0407223i
$371$		−	9.60845e7i		−	0.0976888i
$372$		0			0
$373$		−	1.37070e9i		−	1.36761i	−0.729667	−	0.683803i	$-0.760325\pi$
$373$		−	1.37070e9i		−	1.36761i	0.729667	−	0.683803i	$-0.239675\pi$
$374$	−3.92747e8	−	1.90319e8i	−0.388206	−	0.188119i
$375$		0			0
$376$	−8.44395e8	+	1.86232e8i	−0.819197	+	0.180675i
$377$	1.55042e8			0.149023
$378$		0			0
$379$		−	3.29099e8i		−	0.310520i	−0.987874	−	0.155260i	$-0.950379\pi$
$379$		−	3.29099e8i		−	0.310520i	0.987874	−	0.155260i	$-0.0496215\pi$
$380$	−4.19747e8	−	5.31650e8i	−0.392414	−	0.497031i
$381$		0			0
$382$	−7.30652e8	+	1.50779e9i	−0.670639	+	1.38394i
$383$	−1.14897e9			−1.04499			−0.522497	−	0.852641i	$-0.675000\pi$
$383$	−1.14897e9			−1.04499			−0.522497	+	0.852641i	$0.675000\pi$
$384$		0			0
$385$	−6.35196e7			−0.0567277
$386$	−4.10297e8	+	8.46698e8i	−0.363114	+	0.749330i
$387$		0			0
$388$	1.55396e8	+	1.96824e8i	0.135060	+	0.171067i
$389$		−	1.10441e8i		−	0.0951275i	−0.998868	−	0.0475638i	$-0.984854\pi$
$389$		−	1.10441e8i		−	0.0951275i	0.998868	−	0.0475638i	$-0.0151457\pi$
$390$		0			0
$391$	9.18171e8			0.776792
$392$	−1.13599e9	+	2.50545e8i	−0.952520	+	0.210080i
$393$		0			0
$394$	9.86836e7	+	4.78206e7i	0.0812846	+	0.0393893i
$395$			5.84634e8i			0.477303i
$396$		0			0
$397$			1.60005e9i			1.28341i	0.766951	+	0.641705i	$0.221773\pi$
$397$			1.60005e9i			1.28341i	−0.766951	+	0.641705i	$0.778227\pi$
$398$	2.18670e8	−	4.51251e8i	0.173859	−	0.358780i
$399$		0			0
$400$	−1.71463e8	+	7.18800e8i	−0.133955	+	0.561562i
$401$	−1.49732e9			−1.15960			−0.579800	−	0.814759i	$-0.696869\pi$
$401$	−1.49732e9			−1.15960			−0.579800	+	0.814759i	$0.696869\pi$
$402$		0			0
$403$		−	9.88508e8i		−	0.752337i
$404$	−1.72529e9	+	1.36215e9i	−1.30175	+	1.02775i
$405$		0			0
$406$	3.88551e7	+	1.88286e7i	0.0288142	+	0.0139630i
$407$	−5.26784e7			−0.0387304
$408$		0			0
$409$	3.79465e8			0.274246			0.137123	−	0.990554i	$-0.456214\pi$
$409$	3.79465e8			0.274246			0.137123	+	0.990554i	$0.456214\pi$
$410$	−1.08543e9	−	5.25984e8i	−0.777783	−	0.376902i
$411$		0			0
$412$	7.48076e8	+	9.47511e8i	0.526994	+	0.667489i
$413$			8.36411e7i			0.0584244i
$414$		0			0
$415$	1.07608e9			0.739055
$416$	6.78311e8	−	8.29597e8i	0.461958	−	0.564990i
$417$		0			0
$418$	−3.52931e8	+	7.28316e8i	−0.236360	+	0.487756i
$419$		−	8.90387e8i		−	0.591330i	−0.955292	−	0.295665i	$-0.904459\pi$
$419$		−	8.90387e8i		−	0.591330i	0.955292	−	0.295665i	$-0.0955412\pi$
$420$		0			0
$421$		−	3.90442e7i		−	0.0255017i	−0.999919	−	0.0127509i	$-0.995941\pi$
$421$		−	3.90442e7i		−	0.0255017i	0.999919	−	0.0127509i	$-0.00405883\pi$
$422$	−6.62792e8	−	3.21180e8i	−0.429323	−	0.208044i
$423$		0			0
$424$	−2.10599e8	−	9.54877e8i	−0.134177	−	0.608369i
$425$	−7.08298e8			−0.447564
$426$		0			0
$427$			4.75003e8i			0.295256i
$428$	1.14820e9	−	9.06521e8i	0.707886	−	0.558888i
$429$		0			0
$430$	−1.76960e8	+	3.65179e8i	−0.107334	+	0.221496i
$431$	−1.82055e8			−0.109530			−0.0547648	−	0.998499i	$-0.517441\pi$
$431$	−1.82055e8			−0.109530			−0.0547648	+	0.998499i	$0.517441\pi$
$432$		0			0
$433$	−7.90002e8			−0.467650			−0.233825	−	0.972279i	$-0.575124\pi$
$433$	−7.90002e8			−0.467650			−0.233825	+	0.972279i	$0.575124\pi$
$434$	1.20047e8	−	2.47731e8i	0.0704914	−	0.145467i
$435$		0			0
$436$	−2.41784e8	+	1.90892e8i	−0.139709	+	0.110303i
$437$		−	1.70267e9i		−	0.975991i
$438$		0			0
$439$	−1.02051e9			−0.575691			−0.287845	−	0.957677i	$-0.592939\pi$
$439$	−1.02051e9			−0.575691			−0.287845	+	0.957677i	$0.592939\pi$
$440$	−6.31251e8	+	1.39223e8i	−0.353279	+	0.0779161i
$441$		0			0
$442$	9.24322e8	+	4.47913e8i	0.509150	+	0.246727i
$443$			1.20175e9i			0.656750i	0.944547	+	0.328375i	$0.106501\pi$
$443$			1.20175e9i			0.656750i	−0.944547	+	0.328375i	$0.893499\pi$
$444$		0			0
$445$		−	3.15664e8i		−	0.169811i
$446$	−3.16696e6	+	6.53539e6i	−0.00169032	+	0.00348819i
$447$		0			0
$448$	2.70740e8	−	1.25530e8i	0.142259	−	0.0659592i
$449$	−1.07902e9			−0.562556			−0.281278	−	0.959626i	$-0.590758\pi$
$449$	−1.07902e9			−0.562556			−0.281278	+	0.959626i	$0.590758\pi$
$450$		0			0
$451$			1.44111e9i			0.739739i
$452$	1.73217e9	+	2.19397e9i	0.882282	+	1.11750i
$453$		0			0
$454$	−9.84547e8	−	4.77097e8i	−0.493789	−	0.239283i
$455$	1.49492e8			0.0744010
$456$		0			0
$457$	5.48576e8			0.268862			0.134431	−	0.990923i	$-0.457079\pi$
$457$	5.48576e8			0.268862			0.134431	+	0.990923i	$0.457079\pi$
$458$	−2.42932e8	−	1.17721e8i	−0.118156	−	0.0572566i
$459$		0			0
$460$	1.06738e9	−	8.42716e8i	0.511290	−	0.403672i
$461$		−	2.05577e9i		−	0.977287i	−0.872484	−	0.488643i	$-0.837492\pi$
$461$		−	2.05577e9i		−	0.977287i	0.872484	−	0.488643i	$-0.162508\pi$
$462$		0			0
$463$	−2.59836e8			−0.121665			−0.0608324	−	0.998148i	$-0.519376\pi$
$463$	−2.59836e8			−0.121665			−0.0608324	+	0.998148i	$0.519376\pi$
$464$	4.27407e8	+	1.01954e8i	0.198623	+	0.0473794i
$465$		0			0
$466$	4.28870e8	−	8.85025e8i	0.196325	−	0.405140i
$467$			7.98568e8i			0.362830i	0.983407	+	0.181415i	$0.0580677\pi$
$467$			7.98568e8i			0.362830i	−0.983407	+	0.181415i	$0.941932\pi$
$468$		0			0
$469$			5.55648e8i			0.248711i
$470$	−1.10471e9	−	5.35327e8i	−0.490802	−	0.237836i
$471$		0			0
$472$	1.83326e8	+	8.31216e8i	0.0802465	+	0.363845i
$473$	4.84842e8			0.210662
$474$		0			0
$475$			1.31348e9i			0.562336i
$476$	1.77250e8	+	2.24504e8i	0.0753287	+	0.0954112i
$477$		0			0
$478$	1.89048e9	−	3.90123e9i	0.791724	−	1.63382i
$479$	1.34662e9			0.559849			0.279924	−	0.960022i	$-0.409691\pi$
$479$	1.34662e9			0.559849			0.279924	+	0.960022i	$0.409691\pi$
$480$		0			0
$481$	1.23978e8			0.0507968
$482$	1.25336e9	−	2.58645e9i	0.509812	−	1.05206i
$483$		0			0
$484$	−1.06708e9	−	1.35155e9i	−0.427796	−	0.541845i
$485$			3.56020e8i			0.141703i
$486$		0			0
$487$	−4.31480e9			−1.69282			−0.846408	−	0.532535i	$-0.821239\pi$
$487$	−4.31480e9			−1.69282			−0.846408	+	0.532535i	$0.821239\pi$
$488$	1.04112e9	+	4.72053e9i	0.405537	+	1.83874i
$489$		0			0
$490$	−1.48621e9	−	7.20193e8i	−0.570680	−	0.276543i
$491$			4.76477e9i			1.81659i	0.418333	+	0.908294i	$0.362615\pi$
$491$			4.76477e9i			1.81659i	−0.418333	+	0.908294i	$0.637385\pi$
$492$		0			0
$493$			4.21163e8i			0.158302i
$494$	8.30618e8	−	1.71408e9i	0.309997	−	0.639715i
$495$		0			0
$496$	6.50031e8	−	2.72504e9i	0.239193	−	1.00274i
$497$	6.17199e8			0.225516
$498$		0			0
$499$		−	2.16221e9i		−	0.779015i	−0.921023	−	0.389508i	$-0.872645\pi$
$499$		−	2.16221e9i		−	0.779015i	0.921023	−	0.389508i	$-0.127355\pi$
$500$	−2.24966e9	+	1.77614e9i	−0.804863	+	0.635453i
$501$		0			0
$502$	4.04197e9	+	1.95868e9i	1.42604	+	0.691036i
$503$	4.33465e9			1.51868			0.759340	−	0.650694i	$-0.225522\pi$
$503$	4.33465e9			1.51868			0.759340	+	0.650694i	$0.225522\pi$
$504$		0			0
$505$	−3.12075e9			−1.07830
$506$	−1.46222e9	−	7.08572e8i	−0.501750	−	0.243141i
$507$		0			0
$508$	−1.01470e9	−	1.28522e9i	−0.343413	−	0.434966i
$509$		−	2.22997e9i		−	0.749527i	−0.927120	−	0.374764i	$-0.877724\pi$
$509$		−	2.22997e9i		−	0.749527i	0.927120	−	0.374764i	$-0.122276\pi$
$510$		0			0
$511$	−5.30080e8			−0.175739
$512$	2.41545e9	−	1.84092e9i	0.795340	−	0.606164i
$513$		0			0
$514$	1.70538e9	−	3.51925e9i	0.553923	−	1.14309i
$515$			1.71388e9i			0.552911i
$516$		0			0
$517$			1.46671e9i			0.466795i
$518$	3.10702e7	+	1.50561e7i	0.00982176	+	0.00475948i
$519$		0			0
$520$	1.48564e9	−	3.27659e8i	0.463341	−	0.102191i
$521$	1.18275e9			0.366406			0.183203	−	0.983075i	$-0.441354\pi$
$521$	1.18275e9			0.366406			0.183203	+	0.983075i	$0.441354\pi$
$522$		0			0
$523$			4.29293e9i			1.31219i	0.754677	+	0.656097i	$0.227794\pi$
$523$			4.29293e9i			1.31219i	−0.754677	+	0.656097i	$0.772206\pi$
$524$	4.04179e9	−	3.19106e9i	1.22720	−	0.968893i
$525$		0			0
$526$	2.64663e7	−	5.46164e7i	0.00792945	−	0.0163634i
$527$	2.68523e9			0.799179
$528$		0			0
$529$	1.35898e7			0.00399132
$530$	6.05370e8	−	1.24925e9i	0.176626	−	0.364490i
$531$		0			0
$532$	4.16323e8	−	3.28694e8i	0.119878	−	0.0946459i
$533$		−	3.39162e9i		−	0.970202i
$534$		0			0
$535$	2.07689e9			0.586373
$536$	1.21788e9	+	5.52197e9i	0.341607	+	1.54888i
$537$		0			0
$538$	−1.54589e9	−	7.49115e8i	−0.427997	−	0.207401i
$539$			1.97321e9i			0.542766i
$540$		0			0
$541$			2.80766e9i			0.762350i	0.924503	+	0.381175i	$0.124481\pi$
$541$			2.80766e9i			0.762350i	−0.924503	+	0.381175i	$0.875519\pi$
$542$	4.49175e8	−	9.26926e8i	0.121176	−	0.250062i
$543$		0			0
$544$	2.25356e9	+	1.84260e9i	0.600167	+	0.490720i
$545$	−4.37344e8			−0.115727
$546$		0			0
$547$			4.82553e9i			1.26064i	0.776337	+	0.630318i	$0.217075\pi$
$547$			4.82553e9i			1.26064i	−0.776337	+	0.630318i	$0.782925\pi$
$548$	2.74683e9	+	3.47912e9i	0.713016	+	0.903104i
$549$		0			0
$550$	1.12799e9	+	5.46609e8i	0.289093	+	0.140090i
$551$	7.81011e8			0.198896
$552$		0			0
$553$	−4.57814e8			−0.115120
$554$	−2.67477e9	−	1.29616e9i	−0.668348	−	0.323872i
$555$		0			0
$556$	−2.77945e9	+	2.19442e9i	−0.685800	+	0.541450i
$557$		−	3.91335e9i		−	0.959523i	−0.877399	−	0.479761i	$-0.840723\pi$
$557$		−	3.91335e9i		−	0.959523i	0.877399	−	0.479761i	$-0.159277\pi$
$558$		0			0
$559$	−1.14107e9			−0.276293
$560$	4.12108e8	+	9.83044e7i	0.0991639	+	0.0236545i
$561$		0			0
$562$	−2.32888e9	+	4.80593e9i	−0.553440	+	1.14209i
$563$			8.31928e8i			0.196475i	0.995163	+	0.0982373i	$0.0313204\pi$
$563$			8.31928e8i			0.196475i	−0.995163	+	0.0982373i	$0.968680\pi$
$564$		0			0
$565$			3.96850e9i			0.925671i
$566$	−1.11692e9	−	5.41241e8i	−0.258918	−	0.125468i
$567$		0			0
$568$	6.13366e9	−	1.35279e9i	1.40443	−	0.309749i
$569$	−7.99458e9			−1.81929			−0.909647	−	0.415382i	$-0.863648\pi$
$569$	−7.99458e9			−1.81929			−0.909647	+	0.415382i	$0.863648\pi$
$570$		0			0
$571$		−	4.78215e9i		−	1.07497i	−0.843273	−	0.537486i	$-0.819374\pi$
$571$		−	4.78215e9i		−	1.07497i	0.843273	−	0.537486i	$-0.180626\pi$
$572$	−1.12636e9	−	1.42664e9i	−0.251646	−	0.318734i
$573$		0			0
$574$	4.11886e8	−	8.49977e8i	0.0909046	−	0.187592i
$575$	−2.63705e9			−0.578469
$576$		0			0
$577$	−2.64830e9			−0.573921			−0.286960	−	0.957942i	$-0.592645\pi$
$577$	−2.64830e9			−0.573921			−0.286960	+	0.957942i	$0.592645\pi$
$578$	8.07757e8	−	1.66690e9i	0.173994	−	0.359056i
$579$		0			0
$580$	3.86552e8	+	4.89605e8i	0.0822640	+	0.104195i
$581$			8.42656e8i			0.178252i
$582$		0			0
$583$	−1.65861e9			−0.346661
$584$	−5.26788e9	+	1.16184e9i	−1.09444	+	0.241379i
$585$		0			0
$586$	−4.05463e9	−	1.96481e9i	−0.832356	−	0.403348i
$587$			4.26739e9i			0.870821i	0.900232	+	0.435411i	$0.143397\pi$
$587$			4.26739e9i			0.870821i	−0.900232	+	0.435411i	$0.856603\pi$
$588$		0			0
$589$		−	4.97953e9i		−	1.00412i
$590$	−5.26972e8	+	1.08747e9i	−0.105634	+	0.217989i
$591$		0			0
$592$	3.41772e8	+	8.15263e7i	0.0677034	+	0.0161500i
$593$	−9.48384e8			−0.186764			−0.0933819	−	0.995630i	$-0.529768\pi$
$593$	−9.48384e8			−0.186764			−0.0933819	+	0.995630i	$0.529768\pi$
$594$		0			0
$595$			4.06088e8i			0.0790334i
$596$	−6.38553e9	+	5.04149e9i	−1.23548	+	0.975432i
$597$		0			0
$598$	3.44132e9	+	1.66761e9i	0.658068	+	0.318890i
$599$	−4.15802e9			−0.790482			−0.395241	−	0.918577i	$-0.629339\pi$
$599$	−4.15802e9			−0.790482			−0.395241	+	0.918577i	$0.629339\pi$
$600$		0			0
$601$	4.31252e9			0.810346			0.405173	−	0.914240i	$-0.367211\pi$
$601$	4.31252e9			0.810346			0.405173	+	0.914240i	$0.367211\pi$
$602$	−2.85964e8	−	1.38574e8i	−0.0534224	−	0.0258877i
$603$		0			0
$604$	−4.28338e9	−	5.42531e9i	−0.790965	−	1.00183i
$605$		−	2.44472e9i		−	0.448834i
$606$		0			0
$607$	1.06773e10			1.93777			0.968885	−	0.247511i	$-0.0796126\pi$
$607$	1.06773e10			1.93777			0.968885	+	0.247511i	$0.0796126\pi$
$608$	3.41694e9	−	4.17903e9i	0.616559	−	0.754073i
$609$		0			0
$610$	−2.99271e9	+	6.17581e9i	−0.533839	+	1.10164i
$611$		−	3.45187e9i		−	0.612224i
$612$		0			0
$613$		−	1.93607e9i		−	0.339477i	−0.985489	−	0.169738i	$-0.945708\pi$
$613$		−	1.93607e9i		−	0.339477i	0.985489	−	0.169738i	$-0.0542923\pi$
$614$	−6.68002e9	−	3.23704e9i	−1.16463	−	0.564363i
$615$		0			0
$616$	−1.09023e8	−	4.94319e8i	−0.0187925	−	0.0852068i
$617$	2.55319e9			0.437607			0.218804	−	0.975769i	$-0.429785\pi$
$617$	2.55319e9			0.437607			0.218804	+	0.975769i	$0.429785\pi$
$618$		0			0
$619$		−	6.26163e9i		−	1.06113i	−0.847643	−	0.530567i	$-0.821979\pi$
$619$		−	6.26163e9i		−	1.06113i	0.847643	−	0.530567i	$-0.178021\pi$
$620$	3.12160e9	−	2.46456e9i	0.526025	−	0.415306i
$621$		0			0
$622$	2.00029e9	−	4.12783e9i	0.333293	−	0.687790i
$623$	2.47190e8			0.0409564
$624$		0			0
$625$	−5.45567e8			−0.0893857
$626$	−1.18932e8	+	2.45431e8i	−0.0193771	+	0.0399870i
$627$		0			0
$628$	−6.20764e9	+	4.90104e9i	−1.00016	+	0.789640i
$629$			3.36779e8i			0.0539595i
$630$		0			0
$631$	−7.84943e9			−1.24376			−0.621879	−	0.783114i	$-0.713630\pi$
$631$	−7.84943e9			−1.24376			−0.621879	+	0.783114i	$0.713630\pi$
$632$	−4.54971e9	+	1.00344e9i	−0.716925	+	0.158119i
$633$		0			0
$634$	−7.78754e9	−	3.77373e9i	−1.21364	−	0.588110i
$635$		−	2.32474e9i		−	0.360302i
$636$		0			0
$637$		−	4.64391e9i		−	0.711862i
$638$	3.25020e8	−	6.70718e8i	0.0495494	−	0.102251i
$639$		0			0
$640$	4.31095e9	+	7.36729e7i	0.650045	+	0.0111091i
$641$	−1.85142e9			−0.277653			−0.138827	−	0.990317i	$-0.544333\pi$
$641$	−1.85142e9			−0.277653			−0.138827	+	0.990317i	$0.544333\pi$
$642$		0			0
$643$		−	7.03822e9i		−	1.04406i	−0.852928	−	0.522029i	$-0.825175\pi$
$643$		−	7.03822e9i		−	1.04406i	0.852928	−	0.522029i	$-0.174825\pi$
$644$	6.59913e8	+	8.35843e8i	0.0973612	+	0.123317i
$645$		0			0
$646$	4.65620e9	+	2.25633e9i	0.679545	+	0.329298i
$647$	−4.22196e9			−0.612843			−0.306422	−	0.951896i	$-0.599132\pi$
$647$	−4.22196e9			−0.612843			−0.306422	+	0.951896i	$0.599132\pi$
$648$		0			0
$649$	1.44381e9			0.207327
$650$	−2.65471e9	−	1.28644e9i	−0.379159	−	0.183735i
$651$		0			0
$652$	−9.08125e9	+	7.16980e9i	−1.28315	+	1.01307i
$653$			5.61072e9i			0.788538i	0.918995	+	0.394269i	$0.129002\pi$
$653$			5.61072e9i			0.788538i	−0.918995	+	0.394269i	$0.870998\pi$
$654$		0			0
$655$	7.31088e9			1.01654
$656$	2.23029e9	−	9.34975e9i	0.308459	−	1.29311i
$657$		0			0
$658$	4.19203e8	−	8.65076e8i	0.0573633	−	0.118376i
$659$		−	6.24707e8i		−	0.0850310i	−0.999096	−	0.0425155i	$-0.986463\pi$
$659$		−	6.24707e8i		−	0.0850310i	0.999096	−	0.0425155i	$-0.0135372\pi$
$660$		0			0
$661$		−	1.10194e10i		−	1.48407i	−0.670363	−	0.742033i	$-0.733862\pi$
$661$		−	1.10194e10i		−	1.48407i	0.670363	−	0.742033i	$-0.266138\pi$
$662$	7.35678e9	+	3.56499e9i	0.985564	+	0.477590i
$663$		0			0
$664$	1.84694e9	+	8.37422e9i	0.244831	+	1.11008i
$665$	7.53056e8			0.0993005
$666$		0			0
$667$			1.56802e9i			0.204602i
$668$	−2.79402e9	−	3.53890e9i	−0.362670	−	0.459357i
$669$		0			0
$670$	−3.50080e9	+	7.22433e9i	−0.449682	+	0.927973i
$671$	8.19952e9			1.04775
$672$		0			0
$673$	8.24352e9			1.04246			0.521231	−	0.853416i	$-0.325473\pi$
$673$	8.24352e9			1.04246			0.521231	+	0.853416i	$0.325473\pi$
$674$	−5.03290e9	+	1.03860e10i	−0.633153	+	1.30659i
$675$		0			0
$676$	−2.32617e9	−	2.94632e9i	−0.289620	−	0.366832i
$677$			3.47037e9i			0.429849i	0.976631	+	0.214924i	$0.0689505\pi$
$677$			3.47037e9i			0.429849i	−0.976631	+	0.214924i	$0.931049\pi$
$678$		0			0
$679$	−2.78791e8			−0.0341771
$680$	8.90069e8	+	4.03566e9i	0.108553	+	0.492190i
$681$		0			0
$682$	−4.27633e9	−	2.07225e9i	−0.516210	−	0.250148i
$683$		−	9.66831e9i		−	1.16112i	−0.814216	−	0.580562i	$-0.802833\pi$
$683$		−	9.66831e9i		−	1.16112i	0.814216	−	0.580562i	$-0.197167\pi$
$684$		0			0
$685$			6.29312e9i			0.748081i
$686$	1.14215e9	−	2.35697e9i	0.135079	−	0.278753i
$687$		0			0
$688$	−3.14560e9	−	7.50353e8i	−0.368252	−	0.0878427i
$689$	3.90352e9			0.454662
$690$		0			0
$691$			2.19307e9i			0.252859i	0.991976	+	0.126430i	$0.0403518\pi$
$691$			2.19307e9i			0.252859i	−0.991976	+	0.126430i	$0.959648\pi$
$692$	−6.75691e9	+	5.33470e9i	−0.775134	+	0.611982i
$693$		0			0
$694$	1.07592e10	+	5.21374e9i	1.22186	+	0.592095i
$695$	−5.02754e9			−0.568079
$696$		0			0
$697$	9.21316e9			1.03061
$698$	1.43699e10	+	6.96347e9i	1.59942	+	0.775053i
$699$		0			0
$700$	−5.09072e8	−	6.44789e8i	−0.0560965	−	0.0710517i
$701$			1.27990e10i			1.40335i	0.712500	+	0.701673i	$0.247563\pi$
$701$			1.27990e10i			1.40335i	−0.712500	+	0.701673i	$0.752437\pi$
$702$		0			0
$703$	6.24528e8			0.0677967
$704$	−2.16691e9	−	4.67353e9i	−0.234065	−	0.504825i
$705$		0			0
$706$	−5.34784e9	+	1.10359e10i	−0.571955	+	1.18030i
$707$		−	2.44379e9i		−	0.260073i
$708$		0			0
$709$		−	4.33284e9i		−	0.456574i	−0.973594	−	0.228287i	$-0.926688\pi$
$709$		−	4.33284e9i		−	0.456574i	0.973594	−	0.228287i	$-0.0733125\pi$
$710$	8.02459e9	+	3.88860e9i	0.841432	+	0.407746i
$711$		0			0
$712$	2.45654e9	−	5.41793e8i	0.255061	−	0.0562541i
$713$	9.99730e9			1.03293
$714$		0			0
$715$		−	2.58054e9i		−	0.264022i
$716$	1.34535e10	−	1.06218e10i	1.36975	−	1.08144i
$717$		0			0
$718$	1.74322e9	−	3.59735e9i	0.175759	−	0.362700i
$719$	−1.22304e10			−1.22713			−0.613566	−	0.789643i	$-0.710266\pi$
$719$	−1.22304e10			−1.22713			−0.613566	+	0.789643i	$0.710266\pi$
$720$		0			0
$721$	−1.34210e9			−0.133356
$722$	−2.25925e8	+	4.66223e8i	−0.0223400	+	0.0461013i
$723$		0			0
$724$	−6.58950e9	+	5.20252e9i	−0.645309	+	0.509482i
$725$		−	1.20961e9i		−	0.117886i
$726$		0			0
$727$	−6.51792e9			−0.629127			−0.314564	−	0.949236i	$-0.601858\pi$
$727$	−6.51792e9			−0.629127			−0.314564	+	0.949236i	$0.601858\pi$
$728$	2.56583e8	+	1.16337e9i	0.0246472	+	0.111753i
$729$		0			0
$730$	−6.89190e9	−	3.33971e9i	−0.655706	−	0.317745i
$731$		−	3.09965e9i		−	0.293496i
$732$		0			0
$733$			4.48583e9i			0.420706i	0.977626	+	0.210353i	$0.0674613\pi$
$733$			4.48583e9i			0.420706i	−0.977626	+	0.210353i	$0.932539\pi$
$734$	−3.50698e9	+	7.23708e9i	−0.327339	+	0.675503i
$735$		0			0
$736$	8.39015e9	+	6.86012e9i	0.775707	+	0.634248i
$737$	9.59161e9			0.882583
$738$		0			0
$739$			6.10477e9i			0.556434i	0.960518	+	0.278217i	$0.0897435\pi$
$739$			6.10477e9i			0.556434i	−0.960518	+	0.278217i	$0.910256\pi$
$740$	3.09103e8	+	3.91509e8i	0.0280409	+	0.0355165i
$741$		0			0
$742$	9.78264e8	+	4.74052e8i	0.0879108	+	0.0426003i
$743$	1.71868e10			1.53722			0.768608	−	0.639720i	$-0.220950\pi$
$743$	1.71868e10			1.53722			0.768608	+	0.639720i	$0.220950\pi$
$744$		0			0
$745$	−1.15503e10			−1.02340
$746$	1.39555e10	+	6.76262e9i	1.23072	+	0.596388i
$747$		0			0
$748$	3.87539e9	−	3.05969e9i	0.338579	−	0.267314i
$749$			1.62636e9i			0.141427i
$750$		0			0
$751$	1.24259e10			1.07051			0.535254	−	0.844691i	$-0.320216\pi$
$751$	1.24259e10			1.07051			0.535254	+	0.844691i	$0.320216\pi$
$752$	2.26991e9	−	9.51585e9i	0.194646	−	0.815990i
$753$		0			0
$754$	−7.64930e8	+	1.57852e9i	−0.0649863	+	0.134107i
$755$		−	9.81343e9i		−	0.829864i
$756$		0			0
$757$			1.14395e10i			0.958454i	0.877691	+	0.479227i	$0.159083\pi$
$757$			1.14395e10i			0.958454i	−0.877691	+	0.479227i	$0.840917\pi$
$758$	3.35065e9	+	1.62368e9i	0.279439	+	0.135412i
$759$		0			0
$760$	7.48379e9	−	1.65056e9i	0.618406	−	0.136390i
$761$	1.56409e10			1.28651			0.643257	−	0.765650i	$-0.277583\pi$
$761$	1.56409e10			1.28651			0.643257	+	0.765650i	$0.277583\pi$
$762$		0			0
$763$		−	3.42475e8i		−	0.0279121i
$764$	−1.17464e10	−	1.48779e10i	−0.952967	−	1.20702i
$765$		0			0
$766$	5.66869e9	−	1.16980e10i	0.455703	−	0.940398i
$767$	−3.39799e9			−0.271918
$768$		0			0
$769$	−1.12978e9			−0.0895887			−0.0447943	−	0.998996i	$-0.514263\pi$
$769$	−1.12978e9			−0.0895887			−0.0447943	+	0.998996i	$0.514263\pi$
$770$	3.13387e8	−	6.46711e8i	0.0247379	−	0.0510496i
$771$		0			0
$772$	−6.59619e9	−	8.35471e9i	−0.515979	−	0.653538i
$773$			2.40878e10i			1.87572i	0.347011	+	0.937861i	$0.387197\pi$
$773$			2.40878e10i			1.87572i	−0.347011	+	0.937861i	$0.612803\pi$
$774$		0			0
$775$	−7.71215e9			−0.595140
$776$	−2.77060e9	+	6.11058e8i	−0.212842	+	0.0469425i
$777$		0			0
$778$	1.12443e9	+	5.44883e8i	0.0856059	+	0.0414834i
$779$		−	1.70850e10i		−	1.29490i
$780$		0			0
$781$		−	1.06541e10i		−	0.800274i
$782$	−4.52998e9	+	9.34816e9i	−0.338745	+	0.699041i
$783$		0			0
$784$	3.05378e9	−	1.28020e10i	0.226325	−	0.948791i
$785$	−1.12285e10			−0.828474
$786$		0			0
$787$		−	1.09646e10i		−	0.801829i	−0.916115	−	0.400915i	$-0.868692\pi$
$787$		−	1.09646e10i		−	0.801829i	0.916115	−	0.400915i	$-0.131308\pi$
$788$	−9.73751e8	+	7.68793e8i	−0.0708935	+	0.0559716i
$789$		0			0
$790$	−5.95233e9	−	2.88441e9i	−0.429528	−	0.208143i
$791$	−3.10765e9			−0.223261
$792$		0			0
$793$	−1.92974e10			−1.37418
$794$	−1.62905e10	−	7.89415e9i	−1.15495	−	0.559672i
$795$		0			0
$796$	3.51547e9	+	4.45268e9i	0.247051	+	0.312915i
$797$			1.88303e10i			1.31751i	0.752358	+	0.658755i	$0.228916\pi$
$797$			1.88303e10i			1.31751i	−0.752358	+	0.658755i	$0.771084\pi$
$798$		0			0
$799$	9.37682e9			0.650342
$800$	−6.47236e9	−	5.29205e9i	−0.446938	−	0.365434i
$801$		0			0
$802$	7.38732e9	−	1.52446e10i	0.505681	−	1.04353i
$803$			9.15026e9i			0.623633i
$804$		0			0
$805$			1.51189e9i			0.102149i
$806$	1.00643e10	+	4.87700e9i	0.677033	+	0.328081i
$807$		0			0
$808$	−5.35633e9	−	2.42861e10i	−0.357213	−	1.61964i
$809$	−1.44051e10			−0.956526			−0.478263	−	0.878217i	$-0.658733\pi$
$809$	−1.44051e10			−0.956526			−0.478263	+	0.878217i	$0.658733\pi$
$810$		0			0
$811$			1.71616e10i			1.12976i	0.825173	+	0.564879i	$0.191077\pi$
$811$			1.71616e10i			1.12976i	−0.825173	+	0.564879i	$0.808923\pi$
$812$	−3.83399e8	+	3.02700e8i	−0.0251307	+	0.0198411i
$813$		0			0
$814$	2.59900e8	−	5.36334e8i	0.0168896	−	0.0348538i
$815$	−1.64264e10			−1.06289
$816$		0			0
$817$	−5.74804e9			−0.368759
$818$	−1.87217e9	+	3.86344e9i	−0.119594	+	0.246796i
$819$		0			0
$820$	1.07104e10	−	8.45603e9i	0.678354	−	0.535572i
$821$		−	1.99385e10i		−	1.25745i	−0.777627	−	0.628725i	$-0.783577\pi$
$821$		−	1.99385e10i		−	1.25745i	0.777627	−	0.628725i	$-0.216423\pi$
$822$		0			0
$823$	−2.79176e10			−1.74573			−0.872867	−	0.487957i	$-0.837742\pi$
$823$	−2.79176e10			−1.74573			−0.872867	+	0.487957i	$0.837742\pi$
$824$	−1.33377e10	+	2.94164e9i	−0.830490	+	0.183166i
$825$		0			0
$826$	−8.51574e8	−	4.12660e8i	−0.0525765	−	0.0254778i
$827$		−	1.70900e10i		−	1.05068i	−0.850892	−	0.525341i	$-0.823938\pi$
$827$		−	1.70900e10i		−	1.05068i	0.850892	−	0.525341i	$-0.176062\pi$
$828$		0			0
$829$			2.23316e10i			1.36138i	0.732571	+	0.680690i	$0.238320\pi$
$829$			2.23316e10i			1.36138i	−0.732571	+	0.680690i	$0.761680\pi$
$830$	−5.30907e9	+	1.09559e10i	−0.322288	+	0.665081i
$831$		0			0
$832$	5.09978e9	+	1.09991e10i	0.306987	+	0.662101i
$833$	1.26149e10			0.756185
$834$		0			0
$835$		−	6.40124e9i		−	0.380506i
$836$	−5.67393e9	−	7.18659e9i	−0.335863	−	0.425403i
$837$		0			0
$838$	9.06529e9	+	4.39291e9i	0.532142	+	0.257868i
$839$	−2.43266e10			−1.42205			−0.711025	−	0.703166i	$-0.751769\pi$
$839$	−2.43266e10			−1.42205			−0.711025	+	0.703166i	$0.751769\pi$
$840$		0			0
$841$	1.65306e10			0.958304
$842$	3.97520e8	+	1.92633e8i	0.0229492	+	0.0111208i
$843$		0			0
$844$	6.54004e9	−	5.16347e9i	0.374440	−	0.295627i
$845$		−	5.32938e9i		−	0.303863i
$846$		0			0
$847$	1.91441e9			0.108254
$848$	1.07609e10	+	2.56691e9i	0.605987	+	0.144552i
$849$		0			0
$850$	3.49453e9	−	7.21139e9i	0.195174	−	0.402766i
$851$			1.25385e9i			0.0697417i
$852$		0			0
$853$			1.79927e10i			0.992600i	0.868151	+	0.496300i	$0.165308\pi$
$853$			1.79927e10i			0.992600i	−0.868151	+	0.496300i	$0.834692\pi$
$854$	−4.83614e9	−	2.34352e9i	−0.265703	−	0.128756i
$855$		0			0
$856$	3.56469e9	+	1.61626e10i	0.194251	+	0.880752i
$857$	−1.96490e10			−1.06637			−0.533185	−	0.845999i	$-0.679005\pi$
$857$	−1.96490e10			−1.06637			−0.533185	+	0.845999i	$0.679005\pi$
$858$		0			0
$859$			2.66738e10i			1.43585i	0.696120	+	0.717926i	$0.254908\pi$
$859$			2.66738e10i			1.43585i	−0.696120	+	0.717926i	$0.745092\pi$
$860$	−2.84492e9	−	3.60337e9i	−0.152520	−	0.193181i
$861$		0			0
$862$	8.98204e8	−	1.85355e9i	0.0477639	−	0.0985665i
$863$	−4.34618e8			−0.0230181			−0.0115091	−	0.999934i	$-0.503664\pi$
$863$	−4.34618e8			−0.0230181			−0.0115091	+	0.999934i	$0.503664\pi$
$864$		0			0
$865$	−1.22221e10			−0.642079
$866$	3.89764e9	−	8.04324e9i	0.203934	−	0.420841i
$867$		0			0
$868$	1.92994e9	+	2.44446e9i	0.100167	+	0.126871i
$869$			7.90280e9i			0.408519i
$870$		0			0
$871$	−2.25737e10			−1.15755
$872$	−7.50641e8	−	3.40348e9i	−0.0383376	−	0.173826i
$873$		0			0
$874$	1.73354e10	+	8.40047e9i	0.878301	+	0.425612i
$875$		−	3.18653e9i		−	0.160801i
$876$		0			0
$877$		−	1.78067e10i		−	0.891424i	−0.895176	−	0.445712i	$-0.852951\pi$
$877$		−	1.78067e10i		−	0.891424i	0.895176	−	0.445712i	$-0.147049\pi$
$878$	5.03487e9	−	1.03901e10i	0.251048	−	0.518068i
$879$		0			0
$880$	1.69693e9	−	7.11383e9i	0.0839412	−	0.351896i
$881$	1.14351e10			0.563412			0.281706	−	0.959501i	$-0.409100\pi$
$881$	1.14351e10			0.563412			0.281706	+	0.959501i	$0.409100\pi$
$882$		0			0
$883$			3.69847e10i			1.80784i	0.427703	+	0.903919i	$0.359323\pi$
$883$			3.69847e10i			1.80784i	−0.427703	+	0.903919i	$0.640677\pi$
$884$	−9.12066e9	+	7.20092e9i	−0.444062	+	0.350594i
$885$		0			0
$886$	−1.22353e10	−	5.92907e9i	−0.591014	−	0.286397i
$887$	1.79445e9			0.0863373			0.0431686	−	0.999068i	$-0.486255\pi$
$887$	1.79445e9			0.0863373			0.0431686	+	0.999068i	$0.486255\pi$
$888$		0			0
$889$	1.82045e9			0.0869007
$890$	3.21386e9	+	1.55739e9i	0.152814	+	0.0740513i
$891$		0			0
$892$	−5.09139e7	−	6.44874e7i	−0.00240192	−	0.00304227i
$893$		−	1.73885e10i		−	0.817114i
$894$		0			0
$895$	2.43350e10			1.13462
$896$	−5.76916e7	+	3.37581e9i	−0.00267938	+	0.156784i
$897$		0			0
$898$	5.32354e9	−	1.09858e10i	0.245320	−	0.506248i
$899$			4.58574e9i			0.210499i
$900$		0			0
$901$			1.06037e10i			0.482970i
$902$	−1.46723e10	−	7.10999e9i	−0.665696	−	0.322587i
$903$		0			0
$904$	−3.08834e10	+	6.81138e9i	−1.39039	+	0.306652i
$905$	−1.19192e10			−0.534538
$906$		0			0
$907$		−	2.15533e10i		−	0.959156i	−0.877499	−	0.479578i	$-0.840790\pi$
$907$		−	2.15533e10i		−	0.959156i	0.877499	−	0.479578i	$-0.159210\pi$
$908$	9.71493e9	−	7.67010e9i	0.430664	−	0.340017i
$909$		0			0
$910$	−7.37550e8	+	1.52202e9i	−0.0324449	+	0.0669540i
$911$	3.75841e10			1.64699			0.823493	−	0.567326i	$-0.192022\pi$
$911$	3.75841e10			1.64699			0.823493	+	0.567326i	$0.192022\pi$
$912$		0			0
$913$	1.45460e10			0.632549
$914$	−2.70651e9	+	5.58521e9i	−0.117246	+	0.241951i
$915$		0			0
$916$	2.39711e9	−	1.89256e9i	0.103051	−	0.0813607i
$917$			5.72499e9i			0.245178i
$918$		0			0
$919$	−2.35611e10			−1.00136			−0.500681	−	0.865632i	$-0.666917\pi$
$919$	−2.35611e10			−1.00136			−0.500681	+	0.865632i	$0.666917\pi$
$920$	3.31379e9	+	1.50250e10i	0.140303	+	0.636148i
$921$		0			0
$922$	2.09304e10	+	1.01426e10i	0.879467	+	0.426177i
$923$			2.50743e10i			1.04960i
$924$		0			0
$925$		−	9.67251e8i		−	0.0401831i
$926$	1.28195e9	−	2.64546e9i	0.0530558	−	0.109487i
$927$		0			0
$928$	−3.14672e9	+	3.84854e9i	−0.129253	+	0.158081i
$929$	−2.90619e10			−1.18924			−0.594619	−	0.804008i	$-0.702697\pi$
$929$	−2.90619e10			−1.18924			−0.594619	+	0.804008i	$0.702697\pi$
$930$		0			0
$931$		−	2.33934e10i		−	0.950099i
$932$	6.89477e9	+	8.73290e9i	0.278974	+	0.353348i
$933$		0			0
$934$	−8.13045e9	−	3.93990e9i	−0.326513	−	0.158223i
$935$	7.00990e9			0.280460
$936$		0			0
$937$	−4.04747e9			−0.160729			−0.0803646	−	0.996766i	$-0.525608\pi$
$937$	−4.04747e9			−0.160729			−0.0803646	+	0.996766i	$0.525608\pi$
$938$	−5.65721e9	−	2.74140e9i	−0.223817	−	0.108458i
$939$		0			0
$940$	1.09006e10	−	8.60624e9i	0.428060	−	0.337961i
$941$		−	1.55484e10i		−	0.608306i	−0.952623	−	0.304153i	$-0.901627\pi$
$941$		−	1.55484e10i		−	0.608306i	0.952623	−	0.304153i	$-0.0983734\pi$
$942$		0			0
$943$	3.43012e10			1.33204
$944$	−9.36732e9	−	2.23448e9i	−0.362421	−	0.0864519i
$945$		0			0
$946$	−2.39207e9	+	4.93632e9i	−0.0918659	+	0.189576i
$947$			3.68953e10i			1.41171i	0.708355	+	0.705856i	$0.249438\pi$
$947$			3.68953e10i			1.41171i	−0.708355	+	0.705856i	$0.750562\pi$
$948$		0			0
$949$		−	2.15350e10i		−	0.817923i
$950$	−1.33729e10	−	6.48032e9i	−0.506050	−	0.245225i
$951$		0			0
$952$	−3.16023e9	+	6.96993e8i	−0.118711	+	0.0261818i
$953$	4.82887e10			1.80726			0.903629	−	0.428316i	$-0.140893\pi$
$953$	4.82887e10			1.80726			0.903629	+	0.428316i	$0.140893\pi$
$954$		0			0
$955$		−	2.69116e10i		−	0.999833i
$956$	3.03925e10	+	3.84950e10i	1.12503	+	1.42496i
$957$		0			0
$958$	−6.64382e9	+	1.37103e10i	−0.244140	+	0.503812i
$959$	−4.92800e9			−0.180429
$960$		0			0
$961$	1.72490e9			0.0626949
$962$	−6.11669e8	+	1.26225e9i	−0.0221515	+	0.0457124i
$963$		0			0
$964$	2.01497e10	+	2.55216e10i	0.724435	+	0.917567i
$965$		−	1.51122e10i		−	0.541355i
$966$		0			0
$967$	−2.75840e10			−0.980989			−0.490494	−	0.871444i	$-0.663184\pi$
$967$	−2.75840e10			−0.980989			−0.490494	+	0.871444i	$0.663184\pi$
$968$	1.90252e10	−	4.19603e9i	0.674164	−	0.148688i
$969$		0			0
$970$	−3.62474e9	−	1.75649e9i	−0.127519	−	0.0617939i
$971$			7.26027e8i			0.0254499i	0.999919	+	0.0127249i	$0.00405058\pi$
$971$			7.26027e8i			0.0254499i	−0.999919	+	0.0127249i	$0.995949\pi$
$972$		0			0
$973$		−	3.93695e9i		−	0.137014i
$974$	2.12880e10	−	4.39303e10i	0.738206	−	1.52338i
$975$		0			0
$976$	−5.31976e10	−	1.26898e10i	−1.83155	−	0.436897i
$977$	5.61457e9			0.192613			0.0963065	−	0.995352i	$-0.469297\pi$
$977$	5.61457e9			0.192613			0.0963065	+	0.995352i	$0.469297\pi$
$978$		0			0
$979$		−	4.26699e9i		−	0.145339i
$980$	1.46650e10	−	1.15783e10i	0.497726	−	0.392963i
$981$		0			0
$982$	−4.85114e10	−	2.35079e10i	−1.63476	−	0.792181i
$983$	4.31402e10			1.44859			0.724294	−	0.689491i	$-0.242166\pi$
$983$	4.31402e10			1.44859			0.724294	+	0.689491i	$0.242166\pi$
$984$		0			0
$985$	−1.76134e9			−0.0587242
$986$	−4.28798e9	−	2.07789e9i	−0.142457	−	0.0690325i
$987$		0			0
$988$	1.33535e10	+	1.69135e10i	0.440500	+	0.557936i
$989$		−	1.15402e10i		−	0.379338i
$990$		0			0
$991$	5.15407e10			1.68226			0.841129	−	0.540834i	$-0.181891\pi$
$991$	5.15407e10			1.68226			0.841129	+	0.540834i	$0.181891\pi$
$992$	2.45374e10	+	2.00627e10i	0.798062	+	0.652527i
$993$		0			0
$994$	−3.04508e9	+	6.28388e9i	−0.0983436	+	0.202944i
$995$			8.05411e9i			0.259201i
$996$		0			0
$997$		−	3.99678e10i		−	1.27725i	−0.769516	−	0.638627i	$-0.779503\pi$
$997$		−	3.99678e10i		−	1.27725i	0.769516	−	0.638627i	$-0.220497\pi$
$998$	2.20141e10	+	1.06677e10i	0.701041	+	0.339714i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.8.d.c.37.6	yes	12
3.2	odd	2	inner	72.8.d.c.37.7	yes	12
4.3	odd	2		288.8.d.c.145.6		12
8.3	odd	2		288.8.d.c.145.7		12
8.5	even	2	inner	72.8.d.c.37.5	✓	12
12.11	even	2		288.8.d.c.145.8		12
24.5	odd	2	inner	72.8.d.c.37.8	yes	12
24.11	even	2		288.8.d.c.145.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.8.d.c.37.5	✓	12	8.5	even	2	inner
72.8.d.c.37.6	yes	12	1.1	even	1	trivial
72.8.d.c.37.7	yes	12	3.2	odd	2	inner
72.8.d.c.37.8	yes	12	24.5	odd	2	inner
288.8.d.c.145.5		12	24.11	even	2
288.8.d.c.145.6		12	4.3	odd	2
288.8.d.c.145.7		12	8.3	odd	2
288.8.d.c.145.8		12	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	72.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-4.93370 + 10.1813i) q^{2} +(-79.3172 - 100.463i) q^{4} -181.720i q^{5} +142.301 q^{7} +(1414.17 - 311.897i) q^{8} +O(q^{10})\)	\(q+(-4.93370 + 10.1813i) q^{2} +(-79.3172 - 100.463i) q^{4} -181.720i q^{5} +142.301 q^{7} +(1414.17 - 311.897i) q^{8} +(1850.14 + 896.551i) q^{10} -2456.40i q^{11} +5781.09i q^{13} +(-702.069 + 1448.80i) q^{14} +(-3801.58 + 15936.9i) q^{16} -15704.0 q^{17} +29121.8i q^{19} +(-18256.1 + 14413.5i) q^{20} +(25009.3 + 12119.1i) q^{22} -58467.2 q^{23} +45103.0 q^{25} +(-58858.9 - 28522.2i) q^{26} +(-11286.9 - 14295.9i) q^{28} -26818.8i q^{29} -170990. q^{31} +(-143502. - 117333. i) q^{32} +(77479.0 - 159887. i) q^{34} -25858.8i q^{35} -21445.4i q^{37} +(-296497. - 143678. i) q^{38} +(-56677.7 - 256982. i) q^{40} -586675. q^{41} +197379. i q^{43} +(-246777. + 194835. i) q^{44} +(288460. - 595271. i) q^{46} -597097. q^{47} -803294. q^{49} +(-222525. + 459206. i) q^{50} +(580785. - 458540. i) q^{52} -675222. i q^{53} -446376. q^{55} +(201237. - 44383.1i) q^{56} +(273050. + 132316. i) q^{58} +587777. i q^{59} +3.33803e6i q^{61} +(843613. - 1.74090e6i) q^{62} +(1.90259e6 - 882149. i) q^{64} +1.05054e6 q^{65} +3.90475e6i q^{67} +(1.24560e6 + 1.57767e6i) q^{68} +(263276. + 127580. i) q^{70} +4.33729e6 q^{71} -3.72507e6 q^{73} +(218342. + 105805. i) q^{74} +(2.92566e6 - 2.30986e6i) q^{76} -349547. i q^{77} -3.21723e6 q^{79} +(2.89604e6 + 690821. i) q^{80} +(2.89448e6 - 5.97310e6i) q^{82} +5.92166e6i q^{83} +2.85373e6i q^{85} +(-2.00957e6 - 973810. i) q^{86} +(-766142. - 3.47376e6i) q^{88} +1.73709e6 q^{89} +822653. i q^{91} +(4.63745e6 + 5.87378e6i) q^{92} +(2.94590e6 - 6.07921e6i) q^{94} +5.29200e6 q^{95} -1.95917e6 q^{97} +(3.96321e6 - 8.17856e6i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 412 q^{4} + 136 q^{7}+O(q^{10}) \) 12 * q + 412 * q^4 + 136 * q^7	\( 12 q + 412 q^{4} + 136 q^{7} + 4680 q^{10} - 12472 q^{16} + 32624 q^{22} + 229820 q^{25} - 157288 q^{28} - 37224 q^{31} - 74432 q^{34} - 937520 q^{40} - 1264256 q^{46} + 2668188 q^{49} - 1539680 q^{52} + 6928960 q^{55} - 4035448 q^{58} - 3530192 q^{64} - 10228720 q^{70} + 13619048 q^{73} - 2441920 q^{76} + 20470552 q^{79} - 2507200 q^{82} - 26170912 q^{88} - 22132608 q^{94} + 27442456 q^{97}+O(q^{100}) \) 12 * q + 412 * q^4 + 136 * q^7 + 4680 * q^10 - 12472 * q^16 + 32624 * q^22 + 229820 * q^25 - 157288 * q^28 - 37224 * q^31 - 74432 * q^34 - 937520 * q^40 - 1264256 * q^46 + 2668188 * q^49 - 1539680 * q^52 + 6928960 * q^55 - 4035448 * q^58 - 3530192 * q^64 - 10228720 * q^70 + 13619048 * q^73 - 2441920 * q^76 + 20470552 * q^79 - 2507200 * q^82 - 26170912 * q^88 - 22132608 * q^94 + 27442456 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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