LMFDB - Embedded newform 20.7.f.a.13.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [20,7,Mod(13,20)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("20.13"); S:= CuspForms(chi, 7); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(20, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3])) N = Newforms(chi, 7, names="a")

Level:	$ N $	$=$	$ 20 = 2^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	20.f (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$4.60108167240$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} - 450x^{3} + 23409x^{2} - 115668x + 285768 $ x^6 - 2x^5 + 2x^4 - 450x^3 + 23409x^2 - 115668*x + 285768
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{7}\cdot 5 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			13.2
Root			$-9.34732 - 9.34732i$ of defining polynomial
Character	$\chi$	$=$	20.13
Dual form			20.7.f.a.17.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-2.90948 - 2.90948i) q^{3} +(59.6880 - 109.829i) q^{5} +(236.070 - 236.070i) q^{7} -712.070i q^{9} +564.122 q^{11} +(134.943 + 134.943i) q^{13} +(-493.205 + 145.883i) q^{15} +(-4797.76 + 4797.76i) q^{17} +3933.56i q^{19} -1373.68 q^{21} +(7894.77 + 7894.77i) q^{23} +(-8499.68 - 13110.9i) q^{25} +(-4192.76 + 4192.76i) q^{27} -33149.9i q^{29} +54733.0 q^{31} +(-1641.30 - 1641.30i) q^{33} +(-11836.7 - 40017.7i) q^{35} +(-42310.9 + 42310.9i) q^{37} -785.228i q^{39} +2806.21 q^{41} +(84084.8 + 84084.8i) q^{43} +(-78205.7 - 42502.1i) q^{45} +(-123469. + 123469. i) q^{47} +6191.32i q^{49} +27918.0 q^{51} +(25238.0 + 25238.0i) q^{53} +(33671.3 - 61956.8i) q^{55} +(11444.6 - 11444.6i) q^{57} -184977. i q^{59} +269076. q^{61} +(-168098. - 168098. i) q^{63} +(22875.1 - 6766.14i) q^{65} +(174862. - 174862. i) q^{67} -45939.3i q^{69} -367159. q^{71} +(296627. + 296627. i) q^{73} +(-13416.3 + 62875.5i) q^{75} +(133172. - 133172. i) q^{77} +765438. i q^{79} -494701. q^{81} +(-599469. - 599469. i) q^{83} +(240563. + 813301. i) q^{85} +(-96448.9 + 96448.9i) q^{87} +210115. i q^{89} +63712.0 q^{91} +(-159245. - 159245. i) q^{93} +(432017. + 234786. i) q^{95} +(984239. - 984239. i) q^{97} -401694. i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 32 q^{3} - 156 q^{5} - 264 q^{7} + 2200 q^{11} + 858 q^{13} - 7768 q^{15} - 3278 q^{17} + 33176 q^{21} + 19984 q^{23} - 24174 q^{25} - 115528 q^{27} + 104976 q^{31} + 177320 q^{33} - 116072 q^{35}+ \cdots + 3338406 q^{97}+O(q^{100}) $ 6 * q + 32 * q^3 - 156 * q^5 - 264 * q^7 + 2200 * q^11 + 858 * q^13 - 7768 * q^15 - 3278 * q^17 + 33176 * q^21 + 19984 * q^23 - 24174 * q^25 - 115528 * q^27 + 104976 * q^31 + 177320 * q^33 - 116072 * q^35 - 241554 * q^37 + 351736 * q^41 + 60720 * q^43 - 287846 * q^45 - 355248 * q^47 + 641872 * q^51 + 346526 * q^53 - 310200 * q^55 - 112816 * q^57 - 492888 * q^61 + 2288 * q^63 - 5082 * q^65 - 230304 * q^67 + 174128 * q^71 - 332442 * q^73 + 1855048 * q^75 + 1618760 * q^77 - 3085166 * q^81 - 2190936 * q^83 - 164934 * q^85 + 2614304 * q^87 - 2186976 * q^91 + 242072 * q^93 + 3484184 * q^95 + 3338406 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/20\mathbb{Z}\right)^\times$.

$n$	$11$	$17$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−2.90948	−	2.90948i	−0.107758	−	0.107758i	0.651172	−	0.758930i	$-0.274278\pi$
$3$	−2.90948	−	2.90948i	−0.107758	−	0.107758i	−0.758930	+	0.651172i	$0.774278\pi$
$4$		0			0
$5$	59.6880	−	109.829i	0.477504	−	0.878629i
$5$	59.6880	−	109.829i	0.477504	−	0.878629i
$6$		0			0
$7$	236.070	−	236.070i	0.688249	−	0.688249i	−0.273595	−	0.961845i	$-0.588213\pi$
$7$	236.070	−	236.070i	0.688249	−	0.688249i	0.961845	+	0.273595i	$0.0882129\pi$
$8$		0			0
$9$		−	712.070i		−	0.976776i
$10$		0			0
$11$	564.122			0.423833			0.211917	−	0.977288i	$-0.432029\pi$
$11$	564.122			0.423833			0.211917	+	0.977288i	$0.432029\pi$
$12$		0			0
$13$	134.943	+	134.943i	0.0614216	+	0.0614216i	0.737150	−	0.675729i	$-0.236171\pi$
$13$	134.943	+	134.943i	0.0614216	+	0.0614216i	−0.675729	+	0.737150i	$0.736171\pi$
$14$		0			0
$15$	−493.205	+	145.883i	−0.146135	+	0.0432246i
$16$		0			0
$17$	−4797.76	+	4797.76i	−0.976545	+	0.976545i	−0.999731	−	0.0231865i	$-0.992619\pi$
$17$	−4797.76	+	4797.76i	−0.976545	+	0.976545i	0.0231865	+	0.999731i	$0.492619\pi$
$18$		0			0
$19$			3933.56i			0.573488i	0.958007	+	0.286744i	$0.0925730\pi$
$19$			3933.56i			0.573488i	−0.958007	+	0.286744i	$0.907427\pi$
$20$		0			0
$21$	−1373.68			−0.148329
$22$		0			0
$23$	7894.77	+	7894.77i	0.648867	+	0.648867i	0.952719	−	0.303852i	$-0.0982729\pi$
$23$	7894.77	+	7894.77i	0.648867	+	0.648867i	−0.303852	+	0.952719i	$0.598273\pi$
$24$		0			0
$25$	−8499.68	−	13110.9i	−0.543979	−	0.839099i
$26$		0			0
$27$	−4192.76	+	4192.76i	−0.213014	+	0.213014i
$28$		0			0
$29$		−	33149.9i		−	1.35921i	−0.733576	−	0.679607i	$-0.762150\pi$
$29$		−	33149.9i		−	1.35921i	0.733576	−	0.679607i	$-0.237850\pi$
$30$		0			0
$31$	54733.0			1.83723			0.918617	−	0.395149i	$-0.129307\pi$
$31$	54733.0			1.83723			0.918617	+	0.395149i	$0.129307\pi$
$32$		0			0
$33$	−1641.30	−	1641.30i	−0.0456716	−	0.0456716i
$34$		0			0
$35$	−11836.7	−	40017.7i	−0.276074	−	0.933358i
$36$		0			0
$37$	−42310.9	+	42310.9i	−0.835309	+	0.835309i	−0.988237	−	0.152928i	$-0.951130\pi$
$37$	−42310.9	+	42310.9i	−0.835309	+	0.835309i	0.152928	+	0.988237i	$0.451130\pi$
$38$		0			0
$39$		−	785.228i		−	0.0132374i
$40$		0			0
$41$	2806.21			0.0407163			0.0203581	−	0.999793i	$-0.493519\pi$
$41$	2806.21			0.0407163			0.0203581	+	0.999793i	$0.493519\pi$
$42$		0			0
$43$	84084.8	+	84084.8i	1.05758	+	1.05758i	0.998238	+	0.0593400i	$0.0188996\pi$
$43$	84084.8	+	84084.8i	1.05758	+	1.05758i	0.0593400	+	0.998238i	$0.481100\pi$
$44$		0			0
$45$	−78205.7	−	42502.1i	−0.858224	−	0.466415i
$46$		0			0
$47$	−123469.	+	123469.i	−1.18923	+	1.18923i	−0.211944	+	0.977282i	$0.567979\pi$
$47$	−123469.	+	123469.i	−1.18923	+	1.18923i	−0.977282	+	0.211944i	$0.932021\pi$
$48$		0			0
$49$			6191.32i			0.0526253i
$50$		0			0
$51$	27918.0			0.210462
$52$		0			0
$53$	25238.0	+	25238.0i	0.169522	+	0.169522i	0.786769	−	0.617247i	$-0.211752\pi$
$53$	25238.0	+	25238.0i	0.169522	+	0.169522i	−0.617247	+	0.786769i	$0.711752\pi$
$54$		0			0
$55$	33671.3	−	61956.8i	0.202382	−	0.372392i
$56$		0			0
$57$	11444.6	−	11444.6i	0.0617982	−	0.0617982i
$58$		0			0
$59$		−	184977.i		−	0.900664i	−0.892861	−	0.450332i	$-0.851306\pi$
$59$		−	184977.i		−	0.900664i	0.892861	−	0.450332i	$-0.148694\pi$
$60$		0			0
$61$	269076.			1.18546			0.592728	−	0.805402i	$-0.298051\pi$
$61$	269076.			1.18546			0.592728	+	0.805402i	$0.298051\pi$
$62$		0			0
$63$	−168098.	−	168098.i	−0.672266	−	0.672266i
$64$		0			0
$65$	22875.1	−	6766.14i	0.0832959	−	0.0246377i
$66$		0			0
$67$	174862.	−	174862.i	0.581396	−	0.581396i	−0.353891	−	0.935287i	$-0.615142\pi$
$67$	174862.	−	174862.i	0.581396	−	0.581396i	0.935287	+	0.353891i	$0.115142\pi$
$68$		0			0
$69$		−	45939.3i		−	0.139842i
$70$		0			0
$71$	−367159.			−1.02584			−0.512920	−	0.858437i	$-0.671436\pi$
$71$	−367159.			−1.02584			−0.512920	+	0.858437i	$0.671436\pi$
$72$		0			0
$73$	296627.	+	296627.i	0.762503	+	0.762503i	0.976774	−	0.214271i	$-0.0687376\pi$
$73$	296627.	+	296627.i	0.762503	+	0.762503i	−0.214271	+	0.976774i	$0.568738\pi$
$74$		0			0
$75$	−13416.3	+	62875.5i	−0.0318016	+	0.149038i
$76$		0			0
$77$	133172.	−	133172.i	0.291703	−	0.291703i
$78$		0			0
$79$			765438.i			1.55249i	0.630432	+	0.776244i	$0.282878\pi$
$79$			765438.i			1.55249i	−0.630432	+	0.776244i	$0.717122\pi$
$80$		0			0
$81$	−494701.			−0.930868
$82$		0			0
$83$	−599469.	−	599469.i	−1.04841	−	1.04841i	−0.998767	−	0.0496472i	$-0.984190\pi$
$83$	−599469.	−	599469.i	−1.04841	−	1.04841i	−0.0496472	−	0.998767i	$-0.515810\pi$
$84$		0			0
$85$	240563.	+	813301.i	0.391717	+	1.32433i
$86$		0			0
$87$	−96448.9	+	96448.9i	−0.146467	+	0.146467i
$88$		0			0
$89$			210115.i			0.298048i	0.988834	+	0.149024i	$0.0476132\pi$
$89$			210115.i			0.298048i	−0.988834	+	0.149024i	$0.952387\pi$
$90$		0			0
$91$	63712.0			0.0845467
$92$		0			0
$93$	−159245.	−	159245.i	−0.197977	−	0.197977i
$94$		0			0
$95$	432017.	+	234786.i	0.503884	+	0.273843i
$96$		0			0
$97$	984239.	−	984239.i	1.07841	−	1.07841i	0.0817614	−	0.996652i	$-0.473945\pi$
$97$	984239.	−	984239.i	1.07841	−	1.07841i	0.996652	−	0.0817614i	$-0.0260546\pi$
$98$		0			0
$99$		−	401694.i		−	0.413990i
$100$		0			0
$101$	−692583.			−0.672214			−0.336107	−	0.941824i	$-0.609110\pi$
$101$	−692583.			−0.672214			−0.336107	+	0.941824i	$0.609110\pi$
$102$		0			0
$103$	−176643.	−	176643.i	−0.161654	−	0.161654i	0.621645	−	0.783299i	$-0.286465\pi$
$103$	−176643.	−	176643.i	−0.161654	−	0.161654i	−0.783299	+	0.621645i	$0.786465\pi$
$104$		0			0
$105$	−81992.2	+	150869.i	−0.0708279	+	0.130327i
$106$		0			0
$107$	860103.	−	860103.i	0.702101	−	0.702101i	−0.262760	−	0.964861i	$-0.584633\pi$
$107$	860103.	−	860103.i	0.702101	−	0.702101i	0.964861	+	0.262760i	$0.0846329\pi$
$108$		0			0
$109$		−	915022.i		−	0.706565i	−0.935517	−	0.353282i	$-0.885065\pi$
$109$		−	915022.i		−	0.706565i	0.935517	−	0.353282i	$-0.114935\pi$
$110$		0			0
$111$	246205.			0.180023
$112$		0			0
$113$	−87133.6	−	87133.6i	−0.0603880	−	0.0603880i	0.676268	−	0.736656i	$-0.263596\pi$
$113$	−87133.6	−	87133.6i	−0.0603880	−	0.0603880i	−0.736656	+	0.676268i	$0.763596\pi$
$114$		0			0
$115$	1.33829e6	−	395849.i	0.879951	−	0.260277i
$116$		0			0
$117$	96089.0	−	96089.0i	0.0599951	−	0.0599951i
$118$		0			0
$119$			2.26521e6i			1.34421i
$120$		0			0
$121$	−1.45333e6			−0.820365
$122$		0			0
$123$	−8164.60	−	8164.60i	−0.00438752	−	0.00438752i
$124$		0			0
$125$	−1.94728e6	+	150944.i	−0.997009	+	0.0772831i
$126$		0			0
$127$	−2.56097e6	+	2.56097e6i	−1.25024	+	1.25024i	−0.294625	+	0.955613i	$0.595195\pi$
$127$	−2.56097e6	+	2.56097e6i	−1.25024	+	1.25024i	−0.955613	+	0.294625i	$0.904805\pi$
$128$		0			0
$129$		−	489286.i		−	0.227926i
$130$		0			0
$131$	2.68025e6			1.19223			0.596116	−	0.802898i	$-0.296710\pi$
$131$	2.68025e6			1.19223			0.596116	+	0.802898i	$0.296710\pi$
$132$		0			0
$133$	928593.	+	928593.i	0.394703	+	0.394703i
$134$		0			0
$135$	210228.	+	710743.i	0.0854454	+	0.288876i
$136$		0			0
$137$	−1.04445e6	+	1.04445e6i	−0.406189	+	0.406189i	−0.880407	−	0.474219i	$-0.842731\pi$
$137$	−1.04445e6	+	1.04445e6i	−0.406189	+	0.406189i	0.474219	+	0.880407i	$0.342731\pi$
$138$		0			0
$139$		−	804949.i		−	0.299726i	−0.988707	−	0.149863i	$-0.952117\pi$
$139$		−	804949.i		−	0.299726i	0.988707	−	0.149863i	$-0.0478832\pi$
$140$		0			0
$141$	718461.			0.256298
$142$		0			0
$143$	76124.4	+	76124.4i	0.0260325	+	0.0260325i
$144$		0			0
$145$	−3.64081e6	−	1.97865e6i	−1.19425	−	0.649031i
$146$		0			0
$147$	18013.5	−	18013.5i	0.00567082	−	0.00567082i
$148$		0			0
$149$		−	2.44790e6i		−	0.740005i	−0.929031	−	0.370002i	$-0.879357\pi$
$149$		−	2.44790e6i		−	0.740005i	0.929031	−	0.370002i	$-0.120643\pi$
$150$		0			0
$151$	623439.			0.181077			0.0905385	−	0.995893i	$-0.471141\pi$
$151$	623439.			0.181077			0.0905385	+	0.995893i	$0.471141\pi$
$152$		0			0
$153$	3.41634e6	+	3.41634e6i	0.953866	+	0.953866i
$154$		0			0
$155$	3.26691e6	−	6.01126e6i	0.877287	−	1.61425i
$156$		0			0
$157$	910203.	−	910203.i	0.235201	−	0.235201i	−0.579658	−	0.814860i	$-0.696814\pi$
$157$	910203.	−	910203.i	0.235201	−	0.235201i	0.814860	+	0.579658i	$0.196814\pi$
$158$		0			0
$159$		−	146859.i		−	0.0365350i
$160$		0			0
$161$	3.72743e6			0.893165
$162$		0			0
$163$	−3.48662e6	−	3.48662e6i	−0.805085	−	0.805085i	0.178800	−	0.983885i	$-0.442778\pi$
$163$	−3.48662e6	−	3.48662e6i	−0.805085	−	0.805085i	−0.983885	+	0.178800i	$0.942778\pi$
$164$		0			0
$165$	−278228.	+	82295.9i	−0.0619368	+	0.0183200i
$166$		0			0
$167$	−4.45760e6	+	4.45760e6i	−0.957087	+	0.957087i	−0.999116	−	0.0420291i	$-0.986618\pi$
$167$	−4.45760e6	+	4.45760e6i	−0.957087	+	0.957087i	0.0420291	+	0.999116i	$0.486618\pi$
$168$		0			0
$169$		−	4.79039e6i		−	0.992455i
$170$		0			0
$171$	2.80097e6			0.560170
$172$		0			0
$173$	4.27506e6	+	4.27506e6i	0.825666	+	0.825666i	0.986914	−	0.161248i	$-0.0515520\pi$
$173$	4.27506e6	+	4.27506e6i	0.825666	+	0.825666i	−0.161248	+	0.986914i	$0.551552\pi$
$174$		0			0
$175$	−5.10160e6	−	1.08857e6i	−0.951903	−	0.203116i
$176$		0			0
$177$	−538188.	+	538188.i	−0.0970541	+	0.0970541i
$178$		0			0
$179$			3.50096e6i			0.610420i	0.952285	+	0.305210i	$0.0987266\pi$
$179$			3.50096e6i			0.610420i	−0.952285	+	0.305210i	$0.901273\pi$
$180$		0			0
$181$	−4.10050e6			−0.691514			−0.345757	−	0.938324i	$-0.612378\pi$
$181$	−4.10050e6			−0.691514			−0.345757	+	0.938324i	$0.612378\pi$
$182$		0			0
$183$	−782871.	−	782871.i	−0.127743	−	0.127743i
$184$		0			0
$185$	2.12150e6	+	7.17240e6i	0.335063	+	1.13279i
$186$		0			0
$187$	−2.70652e6	+	2.70652e6i	−0.413892	+	0.413892i
$188$		0			0
$189$			1.97957e6i			0.293214i
$190$		0			0
$191$	−1.71918e6			−0.246730			−0.123365	−	0.992361i	$-0.539369\pi$
$191$	−1.71918e6			−0.246730			−0.123365	+	0.992361i	$0.539369\pi$
$192$		0			0
$193$	2.18547e6	+	2.18547e6i	0.303999	+	0.303999i	0.842576	−	0.538577i	$-0.181038\pi$
$193$	2.18547e6	+	2.18547e6i	0.303999	+	0.303999i	−0.538577	+	0.842576i	$0.681038\pi$
$194$		0			0
$195$	−86240.6	−	46868.7i	−0.0116308	−	0.00632091i
$196$		0			0
$197$	940651.	−	940651.i	0.123035	−	0.123035i	−0.642908	−	0.765943i	$-0.722272\pi$
$197$	940651.	−	940651.i	0.123035	−	0.123035i	0.765943	+	0.642908i	$0.222272\pi$
$198$		0			0
$199$			3.24582e6i			0.411875i	0.978565	+	0.205938i	$0.0660244\pi$
$199$			3.24582e6i			0.411875i	−0.978565	+	0.205938i	$0.933976\pi$
$200$		0			0
$201$	−1.01752e6			−0.125301
$202$		0			0
$203$	−7.82568e6	−	7.82568e6i	−0.935479	−	0.935479i
$204$		0			0
$205$	167497.	−	308202.i	0.0194422	−	0.0357745i
$206$		0			0
$207$	5.62163e6	−	5.62163e6i	0.633798	−	0.633798i
$208$		0			0
$209$			2.21901e6i			0.243063i
$210$		0			0
$211$	−4.75714e6			−0.506405			−0.253203	−	0.967413i	$-0.581484\pi$
$211$	−4.75714e6			−0.506405			−0.253203	+	0.967413i	$0.581484\pi$
$212$		0			0
$213$	1.06824e6	+	1.06824e6i	0.110543	+	0.110543i
$214$		0			0
$215$	1.42538e7	−	4.21607e6i	1.43422	−	0.424221i
$216$		0			0
$217$	1.29208e7	−	1.29208e7i	1.26448	−	1.26448i
$218$		0			0
$219$		−	1.72606e6i		−	0.164332i
$220$		0			0
$221$	−1.29485e6			−0.119962
$222$		0			0
$223$	−1.04868e6	−	1.04868e6i	−0.0945642	−	0.0945642i	0.658242	−	0.752806i	$-0.271300\pi$
$223$	−1.04868e6	−	1.04868e6i	−0.0945642	−	0.0945642i	−0.752806	+	0.658242i	$0.771300\pi$
$224$		0			0
$225$	−9.33589e6	+	6.05236e6i	−0.819612	+	0.531346i
$226$		0			0
$227$	5.05461e6	−	5.05461e6i	0.432126	−	0.432126i	−0.457225	−	0.889351i	$-0.651157\pi$
$227$	5.05461e6	−	5.05461e6i	0.432126	−	0.432126i	0.889351	+	0.457225i	$0.151157\pi$
$228$		0			0
$229$		−	6.14753e6i		−	0.511910i	−0.966689	−	0.255955i	$-0.917610\pi$
$229$		−	6.14753e6i		−	0.511910i	0.966689	−	0.255955i	$-0.0823900\pi$
$230$		0			0
$231$	−774922.			−0.0628669
$232$		0			0
$233$	−1.32078e7	−	1.32078e7i	−1.04415	−	1.04415i	−0.998979	−	0.0451662i	$-0.985618\pi$
$233$	−1.32078e7	−	1.32078e7i	−1.04415	−	1.04415i	−0.0451662	−	0.998979i	$-0.514382\pi$
$234$		0			0
$235$	6.19082e6	+	2.09301e7i	0.477028	+	1.61275i
$236$		0			0
$237$	2.22702e6	−	2.22702e6i	0.167294	−	0.167294i
$238$		0			0
$239$			8.93927e6i			0.654799i	0.944886	+	0.327400i	$0.106172\pi$
$239$			8.93927e6i			0.654799i	−0.944886	+	0.327400i	$0.893828\pi$
$240$		0			0
$241$	−5.12501e6			−0.366137			−0.183068	−	0.983100i	$-0.558603\pi$
$241$	−5.12501e6			−0.366137			−0.183068	+	0.983100i	$0.558603\pi$
$242$		0			0
$243$	4.49585e6	+	4.49585e6i	0.313323	+	0.313323i
$244$		0			0
$245$	679984.	+	369547.i	0.0462381	+	0.0251288i
$246$		0			0
$247$	−530807.	+	530807.i	−0.0352245	+	0.0352245i
$248$		0			0
$249$			3.48829e6i			0.225951i
$250$		0			0
$251$	−5.48265e6			−0.346712			−0.173356	−	0.984859i	$-0.555461\pi$
$251$	−5.48265e6			−0.346712			−0.173356	+	0.984859i	$0.555461\pi$
$252$		0			0
$253$	4.45361e6	+	4.45361e6i	0.275011	+	0.275011i
$254$		0			0
$255$	1.66637e6	−	3.06619e6i	0.100496	−	0.184918i
$256$		0			0
$257$	−1.34016e7	+	1.34016e7i	−0.789508	+	0.789508i	−0.981413	−	0.191906i	$-0.938533\pi$
$257$	−1.34016e7	+	1.34016e7i	−0.789508	+	0.789508i	0.191906	+	0.981413i	$0.438533\pi$
$258$		0			0
$259$			1.99766e7i			1.14980i
$260$		0			0
$261$	−2.36050e7			−1.32765
$262$		0			0
$263$	−6.02759e6	−	6.02759e6i	−0.331342	−	0.331342i	0.521754	−	0.853096i	$-0.325278\pi$
$263$	−6.02759e6	−	6.02759e6i	−0.331342	−	0.331342i	−0.853096	+	0.521754i	$0.825278\pi$
$264$		0			0
$265$	4.27826e6	−	1.26545e6i	0.229895	−	0.0679997i
$266$		0			0
$267$	611324.	−	611324.i	0.0321172	−	0.0321172i
$268$		0			0
$269$			8.58232e6i			0.440908i	0.975397	+	0.220454i	$0.0707539\pi$
$269$			8.58232e6i			0.440908i	−0.975397	+	0.220454i	$0.929246\pi$
$270$		0			0
$271$	2.09018e7			1.05021			0.525104	−	0.851038i	$-0.324027\pi$
$271$	2.09018e7			1.05021			0.525104	+	0.851038i	$0.324027\pi$
$272$		0			0
$273$	−185369.	−	185369.i	−0.00911062	−	0.00911062i
$274$		0			0
$275$	−4.79486e6	−	7.39616e6i	−0.230557	−	0.355638i
$276$		0			0
$277$	8.53568e6	−	8.53568e6i	0.401605	−	0.401605i	−0.477193	−	0.878798i	$-0.658346\pi$
$277$	8.53568e6	−	8.53568e6i	0.401605	−	0.401605i	0.878798	+	0.477193i	$0.158346\pi$
$278$		0			0
$279$		−	3.89738e7i		−	1.79457i
$280$		0			0
$281$	1.15160e7			0.519017			0.259509	−	0.965741i	$-0.416439\pi$
$281$	1.15160e7			0.519017			0.259509	+	0.965741i	$0.416439\pi$
$282$		0			0
$283$	2.60512e7	+	2.60512e7i	1.14939	+	1.14939i	0.986673	+	0.162719i	$0.0520263\pi$
$283$	2.60512e7	+	2.60512e7i	1.14939	+	1.14939i	0.162719	+	0.986673i	$0.447974\pi$
$284$		0			0
$285$	−573839.	−	1.94005e6i	−0.0247888	−	0.0838066i
$286$		0			0
$287$	662460.	−	662460.i	0.0280230	−	0.0280230i
$288$		0			0
$289$		−	2.18995e7i		−	0.907279i
$290$		0			0
$291$	−5.72724e6			−0.232416
$292$		0			0
$293$	−2.18701e7	−	2.18701e7i	−0.869457	−	0.869457i	0.122956	−	0.992412i	$-0.460763\pi$
$293$	−2.18701e7	−	2.18701e7i	−0.869457	−	0.869457i	−0.992412	+	0.122956i	$0.960763\pi$
$294$		0			0
$295$	−2.03158e7	−	1.10409e7i	−0.791349	−	0.430071i
$296$		0			0
$297$	−2.36523e6	+	2.36523e6i	−0.0902826	+	0.0902826i
$298$		0			0
$299$			2.13069e6i			0.0797089i
$300$		0			0
$301$	3.96997e7			1.45575
$302$		0			0
$303$	2.01505e6	+	2.01505e6i	0.0724367	+	0.0724367i
$304$		0			0
$305$	1.60606e7	−	2.95523e7i	0.566061	−	1.04158i
$306$		0			0
$307$	−1.47333e7	+	1.47333e7i	−0.509194	+	0.509194i	−0.914279	−	0.405085i	$-0.867242\pi$
$307$	−1.47333e7	+	1.47333e7i	−0.509194	+	0.509194i	0.405085	+	0.914279i	$0.367242\pi$
$308$		0			0
$309$			1.02788e6i			0.0348391i
$310$		0			0
$311$	3.35262e7			1.11456			0.557280	−	0.830324i	$-0.311845\pi$
$311$	3.35262e7			1.11456			0.557280	+	0.830324i	$0.311845\pi$
$312$		0			0
$313$	2.80702e7	+	2.80702e7i	0.915403	+	0.915403i	0.996691	−	0.0812875i	$-0.0259032\pi$
$313$	2.80702e7	+	2.80702e7i	0.915403	+	0.915403i	−0.0812875	+	0.996691i	$0.525903\pi$
$314$		0			0
$315$	−2.84954e7	+	8.42854e6i	−0.911682	+	0.269663i
$316$		0			0
$317$	−4.24196e7	+	4.24196e7i	−1.33165	+	1.33165i	−0.427746	+	0.903899i	$0.640692\pi$
$317$	−4.24196e7	+	4.24196e7i	−1.33165	+	1.33165i	−0.903899	+	0.427746i	$0.859308\pi$
$318$		0			0
$319$		−	1.87006e7i		−	0.576081i
$320$		0			0
$321$	−5.00490e6			−0.151315
$322$		0			0
$323$	−1.88723e7	−	1.88723e7i	−0.560037	−	0.560037i
$324$		0			0
$325$	622255.	−	2.91620e6i	0.0181267	−	0.0849508i
$326$		0			0
$327$	−2.66224e6	+	2.66224e6i	−0.0761383	+	0.0761383i
$328$		0			0
$329$			5.82945e7i			1.63697i
$330$		0			0
$331$	−6.25816e7			−1.72569			−0.862844	−	0.505470i	$-0.831319\pi$
$331$	−6.25816e7			−1.72569			−0.862844	+	0.505470i	$0.831319\pi$
$332$		0			0
$333$	3.01283e7	+	3.01283e7i	0.815910	+	0.815910i
$334$		0			0
$335$	−8.76771e6	−	2.96421e7i	−0.233212	−	0.788450i
$336$		0			0
$337$	−8.06605e6	+	8.06605e6i	−0.210752	+	0.210752i	−0.804587	−	0.593835i	$-0.797613\pi$
$337$	−8.06605e6	+	8.06605e6i	−0.210752	+	0.210752i	0.593835	+	0.804587i	$0.297613\pi$
$338$		0			0
$339$			507027.i			0.0130146i
$340$		0			0
$341$	3.08761e7			0.778681
$342$		0			0
$343$	2.92349e7	+	2.92349e7i	0.724469	+	0.724469i
$344$		0			0
$345$	−5.04545e6	−	2.74203e6i	−0.122869	−	0.0667751i
$346$		0			0
$347$	−4.75806e6	+	4.75806e6i	−0.113878	+	0.113878i	−0.761750	−	0.647871i	$-0.775659\pi$
$347$	−4.75806e6	+	4.75806e6i	−0.113878	+	0.113878i	0.647871	+	0.761750i	$0.275659\pi$
$348$		0			0
$349$		−	2.69269e7i		−	0.633446i	−0.948518	−	0.316723i	$-0.897417\pi$
$349$		−	2.69269e7i		−	0.633446i	0.948518	−	0.316723i	$-0.102583\pi$
$350$		0			0
$351$	−1.13157e6			−0.0261673
$352$		0			0
$353$	−2.55717e7	−	2.55717e7i	−0.581347	−	0.581347i	0.353926	−	0.935273i	$-0.384846\pi$
$353$	−2.55717e7	−	2.55717e7i	−0.581347	−	0.581347i	−0.935273	+	0.353926i	$0.884846\pi$
$354$		0			0
$355$	−2.19150e7	+	4.03246e7i	−0.489843	+	0.901333i
$356$		0			0
$357$	6.59058e6	−	6.59058e6i	0.144850	−	0.144850i
$358$		0			0
$359$		−	4.26875e7i		−	0.922608i	−0.887242	−	0.461304i	$-0.847382\pi$
$359$		−	4.26875e7i		−	0.922608i	0.887242	−	0.461304i	$-0.152618\pi$
$360$		0			0
$361$	3.15730e7			0.671111
$362$		0			0
$363$	4.22842e6	+	4.22842e6i	0.0884013	+	0.0884013i
$364$		0			0
$365$	5.02832e7	−	1.48731e7i	1.03406	−	0.305859i
$366$		0			0
$367$	2.08436e7	−	2.08436e7i	0.421671	−	0.421671i	−0.464107	−	0.885779i	$-0.653625\pi$
$367$	2.08436e7	−	2.08436e7i	0.421671	−	0.421671i	0.885779	+	0.464107i	$0.153625\pi$
$368$		0			0
$369$		−	1.99822e6i		−	0.0397707i
$370$		0			0
$371$	1.19158e7			0.233348
$372$		0			0
$373$	−6.49554e7	−	6.49554e7i	−1.25167	−	1.25167i	−0.954973	−	0.296694i	$-0.904116\pi$
$373$	−6.49554e7	−	6.49554e7i	−1.25167	−	1.25167i	−0.296694	−	0.954973i	$-0.595884\pi$
$374$		0			0
$375$	6.10474e6	+	5.22641e6i	0.115764	+	0.0991082i
$376$		0			0
$377$	4.47335e6	−	4.47335e6i	0.0834851	−	0.0834851i
$378$		0			0
$379$		−	2.86518e7i		−	0.526302i	−0.964755	−	0.263151i	$-0.915238\pi$
$379$		−	2.86518e7i		−	0.526302i	0.964755	−	0.263151i	$-0.0847617\pi$
$380$		0			0
$381$	1.49021e7			0.269447
$382$		0			0
$383$	1.64598e7	+	1.64598e7i	0.292974	+	0.292974i	0.838254	−	0.545280i	$-0.183577\pi$
$383$	1.64598e7	+	1.64598e7i	0.292974	+	0.292974i	−0.545280	+	0.838254i	$0.683577\pi$
$384$		0			0
$385$	−6.67733e6	−	2.25749e7i	−0.117009	−	0.395588i
$386$		0			0
$387$	5.98743e7	−	5.98743e7i	1.03302	−	1.03302i
$388$		0			0
$389$			7.62446e7i			1.29527i	0.761951	+	0.647635i	$0.224242\pi$
$389$			7.62446e7i			1.29527i	−0.761951	+	0.647635i	$0.775758\pi$
$390$		0			0
$391$	−7.57544e7			−1.26730
$392$		0			0
$393$	−7.79812e6	−	7.79812e6i	−0.128473	−	0.128473i
$394$		0			0
$395$	8.40670e7	+	4.56875e7i	1.36406	+	0.741320i
$396$		0			0
$397$	2.81064e7	−	2.81064e7i	0.449194	−	0.449194i	−0.445892	−	0.895087i	$-0.647114\pi$
$397$	2.81064e7	−	2.81064e7i	0.449194	−	0.449194i	0.895087	+	0.445892i	$0.147114\pi$
$398$		0			0
$399$		−	5.40344e6i		−	0.0850651i
$400$		0			0
$401$	−6.91863e7			−1.07297			−0.536484	−	0.843910i	$-0.680248\pi$
$401$	−6.91863e7			−1.07297			−0.536484	+	0.843910i	$0.680248\pi$
$402$		0			0
$403$	7.38585e6	+	7.38585e6i	0.112846	+	0.112846i
$404$		0			0
$405$	−2.95278e7	+	5.43324e7i	−0.444493	+	0.817888i
$406$		0			0
$407$	−2.38685e7	+	2.38685e7i	−0.354032	+	0.354032i
$408$		0			0
$409$			3.71467e7i			0.542938i	0.962447	+	0.271469i	$0.0875095\pi$
$409$			3.71467e7i			0.542938i	−0.962447	+	0.271469i	$0.912491\pi$
$410$		0			0
$411$	6.07763e6			0.0875405
$412$		0			0
$413$	−4.36675e7	−	4.36675e7i	−0.619881	−	0.619881i
$414$		0			0
$415$	−1.01620e8	+	3.00578e7i	−1.42179	+	0.420545i
$416$		0			0
$417$	−2.34198e6	+	2.34198e6i	−0.0322980	+	0.0322980i
$418$		0			0
$419$			8.94839e7i			1.21647i	0.793756	+	0.608237i	$0.208123\pi$
$419$			8.94839e7i			1.21647i	−0.793756	+	0.608237i	$0.791877\pi$
$420$		0			0
$421$	−1.06220e8			−1.42351			−0.711755	−	0.702428i	$-0.752099\pi$
$421$	−1.06220e8			−1.42351			−0.711755	+	0.702428i	$0.752099\pi$
$422$		0			0
$423$	8.79186e7	+	8.79186e7i	1.16161	+	1.16161i
$424$		0			0
$425$	1.03683e8	+	2.21236e7i	1.35064	+	0.288197i
$426$		0			0
$427$	6.35207e7	−	6.35207e7i	0.815890	−	0.815890i
$428$		0			0
$429$		−	442965.i		−	0.00561044i
$430$		0			0
$431$	5.03292e7			0.628620			0.314310	−	0.949320i	$-0.398227\pi$
$431$	5.03292e7			0.628620			0.314310	+	0.949320i	$0.398227\pi$
$432$		0			0
$433$	−7.69336e7	−	7.69336e7i	−0.947659	−	0.947659i	0.0510375	−	0.998697i	$-0.483747\pi$
$433$	−7.69336e7	−	7.69336e7i	−0.947659	−	0.947659i	−0.998697	+	0.0510375i	$0.983747\pi$
$434$		0			0
$435$	4.83601e6	+	1.63497e7i	0.0587515	+	0.198629i
$436$		0			0
$437$	−3.10545e7	+	3.10545e7i	−0.372118	+	0.372118i
$438$		0			0
$439$			3.27093e7i			0.386615i	0.981138	+	0.193307i	$0.0619214\pi$
$439$			3.27093e7i			0.386615i	−0.981138	+	0.193307i	$0.938079\pi$
$440$		0			0
$441$	4.40865e6			0.0514032
$442$		0			0
$443$	−8.00511e7	−	8.00511e7i	−0.920781	−	0.920781i	0.0763037	−	0.997085i	$-0.475688\pi$
$443$	−8.00511e7	−	8.00511e7i	−0.920781	−	0.920781i	−0.997085	+	0.0763037i	$0.975688\pi$
$444$		0			0
$445$	2.30766e7	+	1.25413e7i	0.261874	+	0.142319i
$446$		0			0
$447$	−7.12210e6	+	7.12210e6i	−0.0797417	+	0.0797417i
$448$		0			0
$449$		−	6.87676e7i		−	0.759704i	−0.925047	−	0.379852i	$-0.875975\pi$
$449$		−	6.87676e7i		−	0.759704i	0.925047	−	0.379852i	$-0.124025\pi$
$450$		0			0
$451$	1.58304e6			0.0172569
$452$		0			0
$453$	−1.81388e6	−	1.81388e6i	−0.0195126	−	0.0195126i
$454$		0			0
$455$	3.80284e6	−	6.99740e6i	0.0403714	−	0.0742852i
$456$		0			0
$457$	2.64086e7	−	2.64086e7i	0.276692	−	0.276692i	−0.555095	−	0.831787i	$-0.687318\pi$
$457$	2.64086e7	−	2.64086e7i	0.276692	−	0.276692i	0.831787	+	0.555095i	$0.187318\pi$
$458$		0			0
$459$		−	4.02317e7i		−	0.416036i
$460$		0			0
$461$	9.30861e7			0.950128			0.475064	−	0.879951i	$-0.342425\pi$
$461$	9.30861e7			0.950128			0.475064	+	0.879951i	$0.342425\pi$
$462$		0			0
$463$	2.13010e7	+	2.13010e7i	0.214613	+	0.214613i	0.806224	−	0.591610i	$-0.201508\pi$
$463$	2.13010e7	+	2.13010e7i	0.214613	+	0.214613i	−0.591610	+	0.806224i	$0.701508\pi$
$464$		0			0
$465$	−2.69946e7	+	7.98463e6i	−0.268484	+	0.0794137i
$466$		0			0
$467$	1.05134e8	−	1.05134e8i	1.03226	−	1.03226i	0.0328025	−	0.999462i	$-0.489557\pi$
$467$	1.05134e8	−	1.05134e8i	1.03226	−	1.03226i	0.999462	−	0.0328025i	$-0.0104432\pi$
$468$		0			0
$469$		−	8.25593e7i		−	0.800290i
$470$		0			0
$471$	−5.29643e6			−0.0506898
$472$		0			0
$473$	4.74341e7	+	4.74341e7i	0.448237	+	0.448237i
$474$		0			0
$475$	5.15725e7	−	3.34340e7i	0.481213	−	0.311966i
$476$		0			0
$477$	1.79712e7	−	1.79712e7i	0.165586	−	0.165586i
$478$		0			0
$479$			1.75946e7i			0.160093i	0.996791	+	0.0800466i	$0.0255069\pi$
$479$			1.75946e7i			0.160093i	−0.996791	+	0.0800466i	$0.974493\pi$
$480$		0			0
$481$	−1.14191e7			−0.102612
$482$		0			0
$483$	−1.08449e7	−	1.08449e7i	−0.0962460	−	0.0962460i
$484$		0			0
$485$	−4.93504e7	−	1.66845e8i	−0.432579	−	1.46247i
$486$		0			0
$487$	−1.32038e8	+	1.32038e8i	−1.14317	+	1.14317i	−0.155307	+	0.987866i	$0.549637\pi$
$487$	−1.32038e8	+	1.32038e8i	−1.14317	+	1.14317i	−0.987866	+	0.155307i	$0.950363\pi$
$488$		0			0
$489$			2.02885e7i			0.173509i
$490$		0			0
$491$	−9.92255e7			−0.838260			−0.419130	−	0.907926i	$-0.637665\pi$
$491$	−9.92255e7			−0.838260			−0.419130	+	0.907926i	$0.637665\pi$
$492$		0			0
$493$	1.59045e8	+	1.59045e8i	1.32733	+	1.32733i
$494$		0			0
$495$	−4.41176e7	−	2.39763e7i	−0.363744	−	0.197682i
$496$		0			0
$497$	−8.66751e7	+	8.66751e7i	−0.706033	+	0.706033i
$498$		0			0
$499$		−	1.16816e8i		−	0.940155i	−0.882625	−	0.470077i	$-0.844226\pi$
$499$		−	1.16816e8i		−	0.940155i	0.882625	−	0.470077i	$-0.155774\pi$
$500$		0			0
$501$	2.59386e7			0.206268
$502$		0			0
$503$	−4.24523e7	−	4.24523e7i	−0.333578	−	0.333578i	0.520366	−	0.853944i	$-0.325796\pi$
$503$	−4.24523e7	−	4.24523e7i	−0.333578	−	0.333578i	−0.853944	+	0.520366i	$0.825796\pi$
$504$		0			0
$505$	−4.13389e7	+	7.60654e7i	−0.320985	+	0.590627i
$506$		0			0
$507$	−1.39375e7	+	1.39375e7i	−0.106945	+	0.106945i
$508$		0			0
$509$			1.86649e8i			1.41538i	0.706523	+	0.707690i	$0.250263\pi$
$509$			1.86649e8i			1.41538i	−0.706523	+	0.707690i	$0.749737\pi$
$510$		0			0
$511$	1.40049e8			1.04958
$512$		0			0
$513$	−1.64925e7	−	1.64925e7i	−0.122161	−	0.122161i
$514$		0			0
$515$	−2.99440e7	+	8.85702e6i	−0.219224	+	0.0648434i
$516$		0			0
$517$	−6.96516e7	+	6.96516e7i	−0.504034	+	0.504034i
$518$		0			0
$519$		−	2.48764e7i		−	0.177945i
$520$		0			0
$521$	1.92823e8			1.36347			0.681735	−	0.731600i	$-0.261226\pi$
$521$	1.92823e8			1.36347			0.681735	+	0.731600i	$0.261226\pi$
$522$		0			0
$523$	1.45933e7	+	1.45933e7i	0.102011	+	0.102011i	0.756270	−	0.654259i	$-0.227019\pi$
$523$	1.45933e7	+	1.45933e7i	0.102011	+	0.102011i	−0.654259	+	0.756270i	$0.727019\pi$
$524$		0			0
$525$	1.16758e7	+	1.80102e7i	0.0806881	+	0.124463i
$526$		0			0
$527$	−2.62596e8	+	2.62596e8i	−1.79414	+	1.79414i
$528$		0			0
$529$		−	2.33812e7i		−	0.157943i
$530$		0			0
$531$	−1.31717e8			−0.879747
$532$		0			0
$533$	378679.	+	378679.i	0.00250086	+	0.00250086i
$534$		0			0
$535$	−4.31261e7	−	1.45802e8i	−0.281630	−	0.952142i
$536$		0			0
$537$	1.01860e7	−	1.01860e7i	0.0657779	−	0.0657779i
$538$		0			0
$539$			3.49266e6i			0.0223044i
$540$		0			0
$541$	2.21291e7			0.139757			0.0698783	−	0.997556i	$-0.477739\pi$
$541$	2.21291e7			0.139757			0.0698783	+	0.997556i	$0.477739\pi$
$542$		0			0
$543$	1.19303e7	+	1.19303e7i	0.0745165	+	0.0745165i
$544$		0			0
$545$	−1.00496e8	−	5.46159e7i	−0.620809	−	0.337388i
$546$		0			0
$547$	1.18791e8	−	1.18791e8i	0.725810	−	0.725810i	−0.243972	−	0.969782i	$-0.578451\pi$
$547$	1.18791e8	−	1.18791e8i	0.725810	−	0.725810i	0.969782	+	0.243972i	$0.0784506\pi$
$548$		0			0
$549$		−	1.91601e8i		−	1.15793i
$550$		0			0
$551$	1.30397e8			0.779494
$552$		0			0
$553$	1.80697e8	+	1.80697e8i	1.06850	+	1.06850i
$554$		0			0
$555$	1.46955e7	−	2.70404e7i	0.0859618	−	0.158174i
$556$		0			0
$557$	1.56212e8	−	1.56212e8i	0.903959	−	0.903959i	−0.0918165	−	0.995776i	$-0.529267\pi$
$557$	1.56212e8	−	1.56212e8i	0.903959	−	0.903959i	0.995776	+	0.0918165i	$0.0292673\pi$
$558$		0			0
$559$			2.26934e7i			0.129916i
$560$		0			0
$561$	1.57491e7			0.0892007
$562$		0			0
$563$	−9.92959e7	−	9.92959e7i	−0.556425	−	0.556425i	0.371863	−	0.928288i	$-0.378719\pi$
$563$	−9.92959e7	−	9.92959e7i	−0.556425	−	0.556425i	−0.928288	+	0.371863i	$0.878719\pi$
$564$		0			0
$565$	−1.47706e7	+	4.36894e6i	−0.0818941	+	0.0242231i
$566$		0			0
$567$	−1.16784e8	+	1.16784e8i	−0.640669	+	0.640669i
$568$		0			0
$569$		−	1.84249e8i		−	1.00016i	−0.865980	−	0.500079i	$-0.833304\pi$
$569$		−	1.84249e8i		−	1.00016i	0.865980	−	0.500079i	$-0.166696\pi$
$570$		0			0
$571$	1.86560e8			1.00210			0.501049	−	0.865419i	$-0.332948\pi$
$571$	1.86560e8			1.00210			0.501049	+	0.865419i	$0.332948\pi$
$572$		0			0
$573$	5.00192e6	+	5.00192e6i	0.0265872	+	0.0265872i
$574$		0			0
$575$	3.64046e7	−	1.70611e8i	0.191493	−	0.897434i
$576$		0			0
$577$	5.66241e7	−	5.66241e7i	0.294764	−	0.294764i	−0.544195	−	0.838959i	$-0.683165\pi$
$577$	5.66241e7	−	5.66241e7i	0.294764	−	0.294764i	0.838959	+	0.544195i	$0.183165\pi$
$578$		0			0
$579$		−	1.27171e7i		−	0.0655170i
$580$		0			0
$581$	−2.83033e8			−1.44314
$582$		0			0
$583$	1.42373e7	+	1.42373e7i	0.0718493	+	0.0718493i
$584$		0			0
$585$	−4.81796e6	−	1.62887e7i	−0.0240656	−	0.0813614i
$586$		0			0
$587$	8.92386e7	−	8.92386e7i	0.441203	−	0.441203i	−0.451213	−	0.892416i	$-0.649009\pi$
$587$	8.92386e7	−	8.92386e7i	0.441203	−	0.441203i	0.892416	+	0.451213i	$0.149009\pi$
$588$		0			0
$589$			2.15296e8i			1.05363i
$590$		0			0
$591$	−5.47360e6			−0.0265162
$592$		0			0
$593$	−4.45215e7	−	4.45215e7i	−0.213504	−	0.213504i	0.592250	−	0.805754i	$-0.298240\pi$
$593$	−4.45215e7	−	4.45215e7i	−0.213504	−	0.213504i	−0.805754	+	0.592250i	$0.798240\pi$
$594$		0			0
$595$	2.48785e8	+	1.35206e8i	1.18106	+	0.641867i
$596$		0			0
$597$	9.44365e6	−	9.44365e6i	0.0443830	−	0.0443830i
$598$		0			0
$599$			7.04179e7i			0.327644i	0.986490	+	0.163822i	$0.0523823\pi$
$599$			7.04179e7i			0.327644i	−0.986490	+	0.163822i	$0.947618\pi$
$600$		0			0
$601$	−4.19019e7			−0.193024			−0.0965118	−	0.995332i	$-0.530769\pi$
$601$	−4.19019e7			−0.193024			−0.0965118	+	0.995332i	$0.530769\pi$
$602$		0			0
$603$	−1.24514e8	−	1.24514e8i	−0.567893	−	0.567893i
$604$		0			0
$605$	−8.67462e7	+	1.59617e8i	−0.391728	+	0.720797i
$606$		0			0
$607$	−6.07259e7	+	6.07259e7i	−0.271524	+	0.271524i	−0.829713	−	0.558190i	$-0.811496\pi$
$607$	−6.07259e7	+	6.07259e7i	−0.271524	+	0.271524i	0.558190	+	0.829713i	$0.311496\pi$
$608$		0			0
$609$			4.55373e7i			0.201611i
$610$		0			0
$611$	−3.33226e7			−0.146088
$612$		0			0
$613$	−4.74161e7	−	4.74161e7i	−0.205847	−	0.205847i	0.596653	−	0.802500i	$-0.296497\pi$
$613$	−4.74161e7	−	4.74161e7i	−0.205847	−	0.205847i	−0.802500	+	0.596653i	$0.796497\pi$
$614$		0			0
$615$	−1.38404e6	+	409378.i	−0.00595007	+	0.00175995i
$616$		0			0
$617$	−4.80708e7	+	4.80708e7i	−0.204657	+	0.204657i	−0.801992	−	0.597335i	$-0.796226\pi$
$617$	−4.80708e7	+	4.80708e7i	−0.204657	+	0.204657i	0.597335	+	0.801992i	$0.296226\pi$
$618$		0			0
$619$			3.05975e8i			1.29007i	0.764152	+	0.645037i	$0.223158\pi$
$619$			3.05975e8i			1.29007i	−0.764152	+	0.645037i	$0.776842\pi$
$620$		0			0
$621$	−6.62017e7			−0.276436
$622$		0			0
$623$	4.96017e7	+	4.96017e7i	0.205131	+	0.205131i
$624$		0			0
$625$	−9.96516e7	+	2.22877e8i	−0.408173	+	0.912905i
$626$		0			0
$627$	6.45615e6	−	6.45615e6i	0.0261921	−	0.0261921i
$628$		0			0
$629$		−	4.05995e8i		−	1.63143i
$630$		0			0
$631$	5.69063e7			0.226502			0.113251	−	0.993566i	$-0.463874\pi$
$631$	5.69063e7			0.226502			0.113251	+	0.993566i	$0.463874\pi$
$632$		0			0
$633$	1.38408e7	+	1.38408e7i	0.0545695	+	0.0545695i
$634$		0			0
$635$	1.28408e8	+	4.34126e8i	0.501502	+	1.69549i
$636$		0			0
$637$	−835476.	+	835476.i	−0.00323233	+	0.00323233i
$638$		0			0
$639$			2.61443e8i			1.00202i
$640$		0			0
$641$	−4.60271e8			−1.74759			−0.873796	−	0.486293i	$-0.838349\pi$
$641$	−4.60271e8			−1.74759			−0.873796	+	0.486293i	$0.838349\pi$
$642$		0			0
$643$	−1.75048e8	−	1.75048e8i	−0.658452	−	0.658452i	0.296562	−	0.955014i	$-0.404160\pi$
$643$	−1.75048e8	−	1.75048e8i	−0.658452	−	0.658452i	−0.955014	+	0.296562i	$0.904160\pi$
$644$		0			0
$645$	−5.37376e7	−	2.92045e7i	−0.200262	−	0.108836i
$646$		0			0
$647$	1.80134e8	−	1.80134e8i	0.665093	−	0.665093i	−0.291483	−	0.956576i	$-0.594149\pi$
$647$	1.80134e8	−	1.80134e8i	0.665093	−	0.665093i	0.956576	+	0.291483i	$0.0941487\pi$
$648$		0			0
$649$		−	1.04350e8i		−	0.381731i
$650$		0			0
$651$	−7.51856e7			−0.272516
$652$		0			0
$653$	1.29029e8	+	1.29029e8i	0.463391	+	0.463391i	0.899765	−	0.436374i	$-0.143738\pi$
$653$	1.29029e8	+	1.29029e8i	0.463391	+	0.463391i	−0.436374	+	0.899765i	$0.643738\pi$
$654$		0			0
$655$	1.59979e8	−	2.94368e8i	0.569296	−	1.04753i
$656$		0			0
$657$	2.11219e8	−	2.11219e8i	0.744795	−	0.744795i
$658$		0			0
$659$			4.30915e8i			1.50569i	0.658198	+	0.752845i	$0.271319\pi$
$659$			4.30915e8i			1.50569i	−0.658198	+	0.752845i	$0.728681\pi$
$660$		0			0
$661$	−3.78322e7			−0.130996			−0.0654978	−	0.997853i	$-0.520864\pi$
$661$	−3.78322e7			−0.130996			−0.0654978	+	0.997853i	$0.520864\pi$
$662$		0			0
$663$	3.76734e6	+	3.76734e6i	0.0129269	+	0.0129269i
$664$		0			0
$665$	1.57412e8	−	4.65602e7i	0.535270	−	0.158325i
$666$		0			0
$667$	2.61711e8	−	2.61711e8i	0.881950	−	0.881950i
$668$		0			0
$669$			6.10220e6i			0.0203802i
$670$		0			0
$671$	1.51792e8			0.502436
$672$		0			0
$673$	−2.07010e8	−	2.07010e8i	−0.679118	−	0.679118i	0.280683	−	0.959801i	$-0.409439\pi$
$673$	−2.07010e8	−	2.07010e8i	−0.679118	−	0.679118i	−0.959801	+	0.280683i	$0.909439\pi$
$674$		0			0
$675$	9.06080e7	+	1.93338e7i	0.294615	+	0.0628646i
$676$		0			0
$677$	1.70158e8	−	1.70158e8i	0.548385	−	0.548385i	−0.377588	−	0.925974i	$-0.623247\pi$
$677$	1.70158e8	−	1.70158e8i	0.548385	−	0.548385i	0.925974	+	0.377588i	$0.123247\pi$
$678$		0			0
$679$		−	4.64698e8i		−	1.48443i
$680$		0			0
$681$	−2.94126e7			−0.0931304
$682$		0			0
$683$	−1.12935e8	−	1.12935e8i	−0.354461	−	0.354461i	0.507306	−	0.861766i	$-0.330642\pi$
$683$	−1.12935e8	−	1.12935e8i	−0.354461	−	0.354461i	−0.861766	+	0.507306i	$0.830642\pi$
$684$		0			0
$685$	5.23696e7	+	1.77052e8i	0.162932	+	0.550846i
$686$		0			0
$687$	−1.78861e7	+	1.78861e7i	−0.0551627	+	0.0551627i
$688$		0			0
$689$			6.81139e6i			0.0208247i
$690$		0			0
$691$	−3.93382e8			−1.19228			−0.596142	−	0.802879i	$-0.703301\pi$
$691$	−3.93382e8			−1.19228			−0.596142	+	0.802879i	$0.703301\pi$
$692$		0			0
$693$	−9.48278e7	−	9.48278e7i	−0.284929	−	0.284929i
$694$		0			0
$695$	−8.84065e7	−	4.80458e7i	−0.263348	−	0.143120i
$696$		0			0
$697$	−1.34635e7	+	1.34635e7i	−0.0397613	+	0.0397613i
$698$		0			0
$699$			7.68553e7i			0.225031i
$700$		0			0
$701$	4.73676e8			1.37508			0.687539	−	0.726148i	$-0.258691\pi$
$701$	4.73676e8			1.37508			0.687539	+	0.726148i	$0.258691\pi$
$702$		0			0
$703$	−1.66432e8	−	1.66432e8i	−0.479040	−	0.479040i
$704$		0			0
$705$	4.28835e7	−	7.89076e7i	0.122383	−	0.225191i
$706$		0			0
$707$	−1.63498e8	+	1.63498e8i	−0.462651	+	0.462651i
$708$		0			0
$709$			2.86349e8i			0.803447i	0.915761	+	0.401724i	$0.131589\pi$
$709$			2.86349e8i			0.803447i	−0.915761	+	0.401724i	$0.868411\pi$
$710$		0			0
$711$	5.45045e8			1.51643
$712$		0			0
$713$	4.32105e8	+	4.32105e8i	1.19212	+	1.19212i
$714$		0			0
$715$	1.29044e7	−	3.81693e6i	0.0353036	−	0.0104423i
$716$		0			0
$717$	2.60086e7	−	2.60086e7i	0.0705602	−	0.0705602i
$718$		0			0
$719$		−	5.54458e8i		−	1.49170i	−0.666113	−	0.745851i	$-0.732043\pi$
$719$		−	5.54458e8i		−	1.49170i	0.666113	−	0.745851i	$-0.267957\pi$
$720$		0			0
$721$	−8.34003e7			−0.222516
$722$		0			0
$723$	1.49111e7	+	1.49111e7i	0.0394543	+	0.0394543i
$724$		0			0
$725$	−4.34625e8	+	2.81763e8i	−1.14052	+	0.739385i
$726$		0			0
$727$	−2.97383e8	+	2.97383e8i	−0.773950	+	0.773950i	−0.978794	−	0.204845i	$-0.934331\pi$
$727$	−2.97383e8	+	2.97383e8i	−0.773950	+	0.773950i	0.204845	+	0.978794i	$0.434331\pi$
$728$		0			0
$729$			3.34476e8i			0.863342i
$730$		0			0
$731$	−8.06838e8			−2.06554
$732$		0			0
$733$	−6.13761e7	−	6.13761e7i	−0.155843	−	0.155843i	0.624879	−	0.780722i	$-0.285148\pi$
$733$	−6.13761e7	−	6.13761e7i	−0.155843	−	0.155843i	−0.780722	+	0.624879i	$0.785148\pi$
$734$		0			0
$735$	−903208.	−	3.05359e6i	−0.00227471	−	0.00769039i
$736$		0			0
$737$	9.86437e7	−	9.86437e7i	0.246415	−	0.246415i
$738$		0			0
$739$			3.80209e8i			0.942084i	0.882111	+	0.471042i	$0.156122\pi$
$739$			3.80209e8i			0.942084i	−0.882111	+	0.471042i	$0.843878\pi$
$740$		0			0
$741$	3.08874e6			0.00759148
$742$		0			0
$743$	3.43942e8	+	3.43942e8i	0.838531	+	0.838531i	0.988666	−	0.150135i	$-0.0479708\pi$
$743$	3.43942e8	+	3.43942e8i	0.838531	+	0.838531i	−0.150135	+	0.988666i	$0.547971\pi$
$744$		0			0
$745$	−2.68849e8	−	1.46110e8i	−0.650190	−	0.353355i
$746$		0			0
$747$	−4.26864e8	+	4.26864e8i	−1.02407	+	1.02407i
$748$		0			0
$749$		−	4.06089e8i		−	0.966441i
$750$		0			0
$751$	2.34620e8			0.553918			0.276959	−	0.960882i	$-0.410673\pi$
$751$	2.34620e8			0.553918			0.276959	+	0.960882i	$0.410673\pi$
$752$		0			0
$753$	1.59517e7	+	1.59517e7i	0.0373612	+	0.0373612i
$754$		0			0
$755$	3.72119e7	−	6.84715e7i	0.0864650	−	0.159100i
$756$		0			0
$757$	−3.86385e8	+	3.86385e8i	−0.890701	+	0.890701i	−0.994589	−	0.103888i	$-0.966872\pi$
$757$	−3.86385e8	+	3.86385e8i	−0.890701	+	0.890701i	0.103888	+	0.994589i	$0.466872\pi$
$758$		0			0
$759$		−	2.59154e7i		−	0.0592696i
$760$		0			0
$761$	6.44521e8			1.46246			0.731228	−	0.682133i	$-0.238947\pi$
$761$	6.44521e8			1.46246			0.731228	+	0.682133i	$0.238947\pi$
$762$		0			0
$763$	−2.16009e8	−	2.16009e8i	−0.486293	−	0.486293i
$764$		0			0
$765$	5.79127e8	−	1.71298e8i	1.29357	−	0.382619i
$766$		0			0
$767$	2.49614e7	−	2.49614e7i	0.0553202	−	0.0553202i
$768$		0			0
$769$			1.48521e8i			0.326595i	0.986577	+	0.163298i	$0.0522131\pi$
$769$			1.48521e8i			0.326595i	−0.986577	+	0.163298i	$0.947787\pi$
$770$		0			0
$771$	7.79831e7			0.170152
$772$		0			0
$773$	2.74600e8	+	2.74600e8i	0.594514	+	0.594514i	0.938847	−	0.344333i	$-0.111895\pi$
$773$	2.74600e8	+	2.74600e8i	0.594514	+	0.594514i	−0.344333	+	0.938847i	$0.611895\pi$
$774$		0			0
$775$	−4.65213e8	−	7.17600e8i	−0.999417	−	1.54162i
$776$		0			0
$777$	5.81216e7	−	5.81216e7i	0.123901	−	0.123901i
$778$		0			0
$779$			1.10384e7i			0.0233503i
$780$		0			0
$781$	−2.07123e8			−0.434785
$782$		0			0
$783$	1.38990e8	+	1.38990e8i	0.289532	+	0.289532i
$784$		0			0
$785$	−4.56382e7	−	1.54295e8i	−0.0943451	−	0.318964i
$786$		0			0
$787$	7.67750e7	−	7.67750e7i	0.157506	−	0.157506i	−0.623955	−	0.781460i	$-0.714475\pi$
$787$	7.67750e7	−	7.67750e7i	0.157506	−	0.157506i	0.781460	+	0.623955i	$0.214475\pi$
$788$		0			0
$789$			3.50743e7i			0.0714098i
$790$		0			0
$791$	−4.11392e7			−0.0831240
$792$		0			0
$793$	3.63100e7	+	3.63100e7i	0.0728126	+	0.0728126i
$794$		0			0
$795$	−1.61293e7	−	8.76571e6i	−0.0321007	−	0.0174456i
$796$		0			0
$797$	−1.79997e8	+	1.79997e8i	−0.355541	+	0.355541i	−0.862166	−	0.506626i	$-0.830893\pi$
$797$	−1.79997e8	+	1.79997e8i	−0.355541	+	0.355541i	0.506626	+	0.862166i	$0.330893\pi$
$798$		0			0
$799$		−	1.18475e9i		−	2.32266i
$800$		0			0
$801$	1.49616e8			0.291126
$802$		0			0
$803$	1.67334e8	+	1.67334e8i	0.323174	+	0.323174i
$804$		0			0
$805$	2.22483e8	−	4.09378e8i	0.426490	−	0.784761i
$806$		0			0
$807$	2.49701e7	−	2.49701e7i	0.0475116	−	0.0475116i
$808$		0			0
$809$		−	3.46422e8i		−	0.654275i	−0.944977	−	0.327138i	$-0.893916\pi$
$809$		−	3.46422e8i		−	0.654275i	0.944977	−	0.327138i	$-0.106084\pi$
$810$		0			0
$811$	9.03611e8			1.69402			0.847011	−	0.531575i	$-0.178400\pi$
$811$	9.03611e8			1.69402			0.847011	+	0.531575i	$0.178400\pi$
$812$		0			0
$813$	−6.08132e7	−	6.08132e7i	−0.113169	−	0.113169i
$814$		0			0
$815$	−5.91041e8	+	1.74821e8i	−1.09180	+	0.322940i
$816$		0			0
$817$	−3.30752e8	+	3.30752e8i	−0.606508	+	0.606508i
$818$		0			0
$819$		−	4.53674e7i		−	0.0825832i
$820$		0			0
$821$	5.35905e8			0.968408			0.484204	−	0.874955i	$-0.339109\pi$
$821$	5.35905e8			0.968408			0.484204	+	0.874955i	$0.339109\pi$
$822$		0			0
$823$	−7.68486e7	−	7.68486e7i	−0.137859	−	0.137859i	0.634809	−	0.772669i	$-0.281079\pi$
$823$	−7.68486e7	−	7.68486e7i	−0.137859	−	0.137859i	−0.772669	+	0.634809i	$0.781079\pi$
$824$		0			0
$825$	−7.56843e6	+	3.54695e7i	−0.0134786	+	0.0631674i
$826$		0			0
$827$	6.48830e8	−	6.48830e8i	1.14714	−	1.14714i	0.160022	−	0.987114i	$-0.448844\pi$
$827$	6.48830e8	−	6.48830e8i	1.14714	−	1.14714i	0.987114	−	0.160022i	$-0.0511564\pi$
$828$		0			0
$829$		−	2.57599e8i		−	0.452147i	−0.974110	−	0.226074i	$-0.927411\pi$
$829$		−	2.57599e8i		−	0.452147i	0.974110	−	0.226074i	$-0.0725890\pi$
$830$		0			0
$831$	−4.96688e7			−0.0865526
$832$		0			0
$833$	−2.97045e7	−	2.97045e7i	−0.0513910	−	0.0513910i
$834$		0			0
$835$	2.23507e8	+	7.55637e8i	0.383912	+	1.29794i
$836$		0			0
$837$	−2.29483e8	+	2.29483e8i	−0.391357	+	0.391357i
$838$		0			0
$839$			8.50884e8i			1.44074i	0.693592	+	0.720368i	$0.256027\pi$
$839$			8.50884e8i			1.44074i	−0.693592	+	0.720368i	$0.743973\pi$
$840$		0			0
$841$	−5.04092e8			−0.847465
$842$		0			0
$843$	−3.35055e7	−	3.35055e7i	−0.0559285	−	0.0559285i
$844$		0			0
$845$	−5.26122e8	−	2.85929e8i	−0.872000	−	0.473901i
$846$		0			0
$847$	−3.43086e8	+	3.43086e8i	−0.564616	+	0.564616i
$848$		0			0
$849$		−	1.51591e8i		−	0.247713i
$850$		0			0
$851$	−6.68069e8			−1.08401
$852$		0			0
$853$	−2.03613e8	−	2.03613e8i	−0.328065	−	0.328065i	0.523786	−	0.851850i	$-0.324519\pi$
$853$	−2.03613e8	−	2.03613e8i	−0.328065	−	0.328065i	−0.851850	+	0.523786i	$0.824519\pi$
$854$		0			0
$855$	1.67184e8	−	3.07626e8i	0.267483	−	0.492182i
$856$		0			0
$857$	−2.98255e8	+	2.98255e8i	−0.473856	+	0.473856i	−0.903160	−	0.429304i	$-0.858759\pi$
$857$	−2.98255e8	+	2.98255e8i	−0.473856	+	0.473856i	0.429304	+	0.903160i	$0.358759\pi$
$858$		0			0
$859$			7.72806e8i			1.21924i	0.792692	+	0.609622i	$0.208679\pi$
$859$			7.72806e8i			1.21924i	−0.792692	+	0.609622i	$0.791321\pi$
$860$		0			0
$861$	−3.85483e6			−0.00603942
$862$		0			0
$863$	7.80898e8	+	7.80898e8i	1.21496	+	1.21496i	0.969375	+	0.245585i	$0.0789800\pi$
$863$	7.80898e8	+	7.80898e8i	1.21496	+	1.21496i	0.245585	+	0.969375i	$0.421020\pi$
$864$		0			0
$865$	7.24695e8	−	2.14354e8i	1.11971	−	0.331195i
$866$		0			0
$867$	−6.37161e7	+	6.37161e7i	−0.0977669	+	0.0977669i
$868$		0			0
$869$			4.31800e8i			0.657996i
$870$		0			0
$871$	4.71929e7			0.0714204
$872$		0			0
$873$	−7.00847e8	−	7.00847e8i	−1.05337	−	1.05337i
$874$		0			0
$875$	−4.24061e8	+	4.95328e8i	−0.633001	+	0.739381i
$876$		0			0
$877$	1.28762e8	−	1.28762e8i	0.190892	−	0.190892i	−0.605189	−	0.796082i	$-0.706903\pi$
$877$	1.28762e8	−	1.28762e8i	0.190892	−	0.190892i	0.796082	+	0.605189i	$0.206903\pi$
$878$		0			0
$879$			1.27261e8i			0.187383i
$880$		0			0
$881$	−4.42468e8			−0.647075			−0.323537	−	0.946215i	$-0.604872\pi$
$881$	−4.42468e8			−0.647075			−0.323537	+	0.946215i	$0.604872\pi$
$882$		0			0
$883$	4.54363e8	+	4.54363e8i	0.659965	+	0.659965i	0.955372	−	0.295407i	$-0.0954552\pi$
$883$	4.54363e8	+	4.54363e8i	0.659965	+	0.659965i	−0.295407	+	0.955372i	$0.595455\pi$
$884$		0			0
$885$	2.69851e7	+	9.12318e7i	0.0389308	+	0.131618i
$886$		0			0
$887$	3.87480e8	−	3.87480e8i	0.555236	−	0.555236i	−0.372711	−	0.927947i	$-0.621572\pi$
$887$	3.87480e8	−	3.87480e8i	0.555236	−	0.555236i	0.927947	+	0.372711i	$0.121572\pi$
$888$		0			0
$889$			1.20913e9i			1.72095i
$890$		0			0
$891$	−2.79072e8			−0.394533
$892$		0			0
$893$	−4.85672e8	−	4.85672e8i	−0.682007	−	0.682007i
$894$		0			0
$895$	3.84506e8	+	2.08966e8i	0.536333	+	0.291478i
$896$		0			0
$897$	6.19919e6	−	6.19919e6i	0.00858930	−	0.00858930i
$898$		0			0
$899$		−	1.81439e9i		−	2.49720i
$900$		0			0
$901$	−2.42172e8			−0.331093
$902$		0			0
$903$	−1.15506e8	−	1.15506e8i	−0.156870	−	0.156870i
$904$		0			0
$905$	−2.44751e8	+	4.50352e8i	−0.330201	+	0.607585i
$906$		0			0
$907$	5.76878e8	−	5.76878e8i	0.773147	−	0.773147i	−0.205508	−	0.978655i	$-0.565885\pi$
$907$	5.76878e8	−	5.76878e8i	0.773147	−	0.773147i	0.978655	+	0.205508i	$0.0658847\pi$
$908$		0			0
$909$			4.93167e8i			0.656603i
$910$		0			0
$911$	2.36936e8			0.313383			0.156691	−	0.987648i	$-0.449917\pi$
$911$	2.36936e8			0.313383			0.156691	+	0.987648i	$0.449917\pi$
$912$		0			0
$913$	−3.38174e8	−	3.38174e8i	−0.444353	−	0.444353i
$914$		0			0
$915$	−1.32710e8	+	3.92537e7i	−0.173236	+	0.0512409i
$916$		0			0
$917$	6.32725e8	−	6.32725e8i	0.820554	−	0.820554i
$918$		0			0
$919$			1.05932e8i			0.136483i	0.997669	+	0.0682417i	$0.0217389\pi$
$919$			1.05932e8i			0.136483i	−0.997669	+	0.0682417i	$0.978261\pi$
$920$		0			0
$921$	8.57321e7			0.109740
$922$		0			0
$923$	−4.95456e7	−	4.95456e7i	−0.0630087	−	0.0630087i
$924$		0			0
$925$	9.14364e8	+	1.95106e8i	1.15530	+	0.246516i
$926$		0			0
$927$	−1.25782e8	+	1.25782e8i	−0.157900	+	0.157900i
$928$		0			0
$929$		−	9.20051e8i		−	1.14753i	−0.819019	−	0.573766i	$-0.805482\pi$
$929$		−	9.20051e8i		−	1.14753i	0.819019	−	0.573766i	$-0.194518\pi$
$930$		0			0
$931$	−2.43539e7			−0.0301800
$932$		0			0
$933$	−9.75439e7	−	9.75439e7i	−0.120103	−	0.120103i
$934$		0			0
$935$	1.35707e8	+	4.58801e8i	0.166023	+	0.561293i
$936$		0			0
$937$	7.29862e8	−	7.29862e8i	0.887201	−	0.887201i	−0.107053	−	0.994253i	$-0.534141\pi$
$937$	7.29862e8	−	7.29862e8i	0.887201	−	0.887201i	0.994253	+	0.107053i	$0.0341413\pi$
$938$		0			0
$939$		−	1.63339e8i		−	0.197285i
$940$		0			0
$941$	−1.35134e9			−1.62179			−0.810895	−	0.585191i	$-0.801019\pi$
$941$	−1.35134e9			−1.62179			−0.810895	+	0.585191i	$0.801019\pi$
$942$		0			0
$943$	2.21544e7	+	2.21544e7i	0.0264195	+	0.0264195i
$944$		0			0
$945$	2.17413e8	+	1.18156e8i	0.257626	+	0.140011i
$946$		0			0
$947$	9.91390e8	−	9.91390e8i	1.16733	−	1.16733i	0.184500	−	0.982832i	$-0.440933\pi$
$947$	9.91390e8	−	9.91390e8i	1.16733	−	1.16733i	0.982832	−	0.184500i	$-0.0590666\pi$
$948$		0			0
$949$			8.00555e7i			0.0936683i
$950$		0			0
$951$	2.46838e8			0.286992
$952$		0			0
$953$	−2.78775e8	−	2.78775e8i	−0.322088	−	0.322088i	0.527479	−	0.849568i	$-0.323137\pi$
$953$	−2.78775e8	−	2.78775e8i	−0.322088	−	0.322088i	−0.849568	+	0.527479i	$0.823137\pi$
$954$		0			0
$955$	−1.02614e8	+	1.88815e8i	−0.117814	+	0.216784i
$956$		0			0
$957$	−5.44089e7	+	5.44089e7i	−0.0620775	+	0.0620775i
$958$		0			0
$959$			4.93128e8i			0.559118i
$960$		0			0
$961$	2.10820e9			2.37543
$962$		0			0
$963$	−6.12454e8	−	6.12454e8i	−0.685795	−	0.685795i
$964$		0			0
$965$	3.70473e8	−	1.09581e8i	0.412264	−	0.121942i
$966$		0			0
$967$	−6.86367e8	+	6.86367e8i	−0.759061	+	0.759061i	−0.976152	−	0.217090i	$-0.930343\pi$
$967$	−6.86367e8	+	6.86367e8i	−0.759061	+	0.759061i	0.217090	+	0.976152i	$0.430343\pi$
$968$		0			0
$969$			1.09817e8i			0.120697i
$970$		0			0
$971$	−1.10147e9			−1.20313			−0.601566	−	0.798823i	$-0.705456\pi$
$971$	−1.10147e9			−1.20313			−0.601566	+	0.798823i	$0.705456\pi$
$972$		0			0
$973$	−1.90024e8	−	1.90024e8i	−0.206286	−	0.206286i
$974$		0			0
$975$	−1.02951e7	+	6.67419e6i	−0.0111075	+	0.00720086i
$976$		0			0
$977$	−8.32735e8	+	8.32735e8i	−0.892942	+	0.892942i	−0.994799	−	0.101857i	$-0.967522\pi$
$977$	−8.32735e8	+	8.32735e8i	−0.892942	+	0.892942i	0.101857	+	0.994799i	$0.467522\pi$
$978$		0			0
$979$			1.18530e8i			0.126323i
$980$		0			0
$981$	−6.51560e8			−0.690156
$982$		0			0
$983$	4.76445e7	+	4.76445e7i	0.0501594	+	0.0501594i	0.731742	−	0.681582i	$-0.238708\pi$
$983$	4.76445e7	+	4.76445e7i	0.0501594	+	0.0501594i	−0.681582	+	0.731742i	$0.738708\pi$
$984$		0			0
$985$	−4.71648e7	−	1.59456e8i	−0.0493525	−	0.166852i
$986$		0			0
$987$	1.69607e8	−	1.69607e8i	0.176397	−	0.176397i
$988$		0			0
$989$			1.32766e9i			1.37245i
$990$		0			0
$991$	−1.65244e9			−1.69787			−0.848935	−	0.528497i	$-0.822756\pi$
$991$	−1.65244e9			−1.69787			−0.848935	+	0.528497i	$0.822756\pi$
$992$		0			0
$993$	1.82080e8	+	1.82080e8i	0.185958	+	0.185958i
$994$		0			0
$995$	3.56484e8	+	1.93737e8i	0.361886	+	0.196672i
$996$		0			0
$997$	5.19940e8	−	5.19940e8i	0.524648	−	0.524648i	−0.394324	−	0.918972i	$-0.629021\pi$
$997$	5.19940e8	−	5.19940e8i	0.524648	−	0.524648i	0.918972	+	0.394324i	$0.129021\pi$
$998$		0			0
$999$		−	3.54799e8i		−	0.355865i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.7.f.a.13.2	✓	6
3.2	odd	2		180.7.l.a.73.1		6
4.3	odd	2		80.7.p.d.33.2		6
5.2	odd	4	inner	20.7.f.a.17.2	yes	6
5.3	odd	4		100.7.f.b.57.2		6
5.4	even	2		100.7.f.b.93.2		6
15.2	even	4		180.7.l.a.37.1		6
20.7	even	4		80.7.p.d.17.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.7.f.a.13.2	✓	6	1.1	even	1	trivial
20.7.f.a.17.2	yes	6	5.2	odd	4	inner
80.7.p.d.17.2		6	20.7	even	4
80.7.p.d.33.2		6	4.3	odd	2
100.7.f.b.57.2		6	5.3	odd	4
100.7.f.b.93.2		6	5.4	even	2
180.7.l.a.37.1		6	15.2	even	4
180.7.l.a.73.1		6	3.2	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 20 = 2^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	20.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.90948 - 2.90948i) q^{3} +(59.6880 - 109.829i) q^{5} +(236.070 - 236.070i) q^{7} -712.070i q^{9} +564.122 q^{11} +(134.943 + 134.943i) q^{13} +(-493.205 + 145.883i) q^{15} +(-4797.76 + 4797.76i) q^{17} +3933.56i q^{19} -1373.68 q^{21} +(7894.77 + 7894.77i) q^{23} +(-8499.68 - 13110.9i) q^{25} +(-4192.76 + 4192.76i) q^{27} -33149.9i q^{29} +54733.0 q^{31} +(-1641.30 - 1641.30i) q^{33} +(-11836.7 - 40017.7i) q^{35} +(-42310.9 + 42310.9i) q^{37} -785.228i q^{39} +2806.21 q^{41} +(84084.8 + 84084.8i) q^{43} +(-78205.7 - 42502.1i) q^{45} +(-123469. + 123469. i) q^{47} +6191.32i q^{49} +27918.0 q^{51} +(25238.0 + 25238.0i) q^{53} +(33671.3 - 61956.8i) q^{55} +(11444.6 - 11444.6i) q^{57} -184977. i q^{59} +269076. q^{61} +(-168098. - 168098. i) q^{63} +(22875.1 - 6766.14i) q^{65} +(174862. - 174862. i) q^{67} -45939.3i q^{69} -367159. q^{71} +(296627. + 296627. i) q^{73} +(-13416.3 + 62875.5i) q^{75} +(133172. - 133172. i) q^{77} +765438. i q^{79} -494701. q^{81} +(-599469. - 599469. i) q^{83} +(240563. + 813301. i) q^{85} +(-96448.9 + 96448.9i) q^{87} +210115. i q^{89} +63712.0 q^{91} +(-159245. - 159245. i) q^{93} +(432017. + 234786. i) q^{95} +(984239. - 984239. i) q^{97} -401694. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 32 q^{3} - 156 q^{5} - 264 q^{7} + 2200 q^{11} + 858 q^{13} - 7768 q^{15} - 3278 q^{17} + 33176 q^{21} + 19984 q^{23} - 24174 q^{25} - 115528 q^{27} + 104976 q^{31} + 177320 q^{33} - 116072 q^{35}+ \cdots + 3338406 q^{97}+O(q^{100}) \) 6 * q + 32 * q^3 - 156 * q^5 - 264 * q^7 + 2200 * q^11 + 858 * q^13 - 7768 * q^15 - 3278 * q^17 + 33176 * q^21 + 19984 * q^23 - 24174 * q^25 - 115528 * q^27 + 104976 * q^31 + 177320 * q^33 - 116072 * q^35 - 241554 * q^37 + 351736 * q^41 + 60720 * q^43 - 287846 * q^45 - 355248 * q^47 + 641872 * q^51 + 346526 * q^53 - 310200 * q^55 - 112816 * q^57 - 492888 * q^61 + 2288 * q^63 - 5082 * q^65 - 230304 * q^67 + 174128 * q^71 - 332442 * q^73 + 1855048 * q^75 + 1618760 * q^77 - 3085166 * q^81 - 2190936 * q^83 - 164934 * q^85 + 2614304 * q^87 - 2186976 * q^91 + 242072 * q^93 + 3484184 * q^95 + 3338406 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more