LMFDB - Embedded newform 20.7.f.a.17.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [20,7,Mod(13,20)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

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			17.1
Root			$2.61494 - 2.61494i$ of defining polynomial
Character	$\chi$	$=$	20.17
Dual form			20.7.f.a.13.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-16.7249 + 16.7249i) q^{3} +(-22.1946 - 123.014i) q^{5} +(-422.022 - 422.022i) q^{7} +169.553i q^{9} +O(q^{10})$	\(q+(-16.7249 + 16.7249i) q^{3} +(-22.1946 - 123.014i) q^{5} +(-422.022 - 422.022i) q^{7} +169.553i q^{9} -1359.94 q^{11} +(1248.94 - 1248.94i) q^{13} +(2428.60 + 1686.20i) q^{15} +(-648.372 - 648.372i) q^{17} +9632.63i q^{19} +14116.6 q^{21} +(886.911 - 886.911i) q^{23} +(-14639.8 + 5460.48i) q^{25} +(-15028.2 - 15028.2i) q^{27} -26789.3i q^{29} -6880.94 q^{31} +(22744.9 - 22744.9i) q^{33} +(-42547.9 + 61281.1i) q^{35} +(-42367.5 - 42367.5i) q^{37} +41777.0i q^{39} +132500. q^{41} +(20888.6 - 20888.6i) q^{43} +(20857.4 - 3763.16i) q^{45} +(-31606.0 - 31606.0i) q^{47} +238556. i q^{49} +21688.0 q^{51} +(80106.6 - 80106.6i) q^{53} +(30183.3 + 167291. i) q^{55} +(-161105. - 161105. i) q^{57} -290136. i q^{59} -253971. q^{61} +(71555.1 - 71555.1i) q^{63} +(-181357. - 125918. i) q^{65} +(11368.7 + 11368.7i) q^{67} +29667.0i q^{69} +179758. q^{71} +(-414484. + 414484. i) q^{73} +(153524. - 336176. i) q^{75} +(573924. + 573924. i) q^{77} -145313. i q^{79} +379089. q^{81} +(-63129.1 + 63129.1i) q^{83} +(-65368.3 + 94149.0i) q^{85} +(448049. + 448049. i) q^{87} -333388. i q^{89} -1.05416e6 q^{91} +(115083. - 115083. i) q^{93} +(1.18495e6 - 213792. i) q^{95} +(519696. + 519696. i) q^{97} -230582. i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 32 q^{3} - 156 q^{5} - 264 q^{7}+O(q^{10}) $ 6 * q + 32 * q^3 - 156 * q^5 - 264 * q^7	$ 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}+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

Display 100 coefficients 10 coefficients

Character values

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

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

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
$3$	−16.7249	+	16.7249i	−0.619442	+	0.619442i	−0.945388	−	0.325946i	$-0.894317\pi$
$3$	−16.7249	+	16.7249i	−0.619442	+	0.619442i	0.325946	+	0.945388i	$0.394317\pi$
$4$		0			0
$5$	−22.1946	−	123.014i	−0.177557	−	0.984111i
$5$	−22.1946	−	123.014i	−0.177557	−	0.984111i
$6$		0			0
$7$	−422.022	−	422.022i	−1.23038	−	1.23038i	−0.963816	−	0.266568i	$-0.914110\pi$
$7$	−422.022	−	422.022i	−1.23038	−	1.23038i	−0.266568	−	0.963816i	$-0.585890\pi$
$8$		0			0
$9$			169.553i			0.232583i
$10$		0			0
$11$	−1359.94			−1.02174			−0.510872	−	0.859657i	$-0.670677\pi$
$11$	−1359.94			−1.02174			−0.510872	+	0.859657i	$0.670677\pi$
$12$		0			0
$13$	1248.94	−	1248.94i	0.568477	−	0.568477i	−0.363224	−	0.931702i	$-0.618324\pi$
$13$	1248.94	−	1248.94i	0.568477	−	0.568477i	0.931702	+	0.363224i	$0.118324\pi$
$14$		0			0
$15$	2428.60	+	1686.20i	0.719586	+	0.499614i
$16$		0			0
$17$	−648.372	−	648.372i	−0.131971	−	0.131971i	0.638036	−	0.770007i	$-0.279747\pi$
$17$	−648.372	−	648.372i	−0.131971	−	0.131971i	−0.770007	+	0.638036i	$0.779747\pi$
$18$		0			0
$19$			9632.63i			1.40438i	0.711990	+	0.702189i	$0.247794\pi$
$19$			9632.63i			1.40438i	−0.711990	+	0.702189i	$0.752206\pi$
$20$		0			0
$21$	14116.6			1.52430
$22$		0			0
$23$	886.911	−	886.911i	0.0728948	−	0.0728948i	−0.669719	−	0.742614i	$-0.733586\pi$
$23$	886.911	−	886.911i	0.0728948	−	0.0728948i	0.742614	+	0.669719i	$0.233586\pi$
$24$		0			0
$25$	−14639.8	+	5460.48i	−0.936947	+	0.349471i
$26$		0			0
$27$	−15028.2	−	15028.2i	−0.763514	−	0.763514i
$28$		0			0
$29$		−	26789.3i		−	1.09842i	−0.835685	−	0.549209i	$-0.814929\pi$
$29$		−	26789.3i		−	1.09842i	0.835685	−	0.549209i	$-0.185071\pi$
$30$		0			0
$31$	−6880.94			−0.230974			−0.115487	−	0.993309i	$-0.536843\pi$
$31$	−6880.94			−0.230974			−0.115487	+	0.993309i	$0.536843\pi$
$32$		0			0
$33$	22744.9	−	22744.9i	0.632911	−	0.632911i
$34$		0			0
$35$	−42547.9	+	61281.1i	−0.992371	+	1.42930i
$36$		0			0
$37$	−42367.5	−	42367.5i	−0.836426	−	0.836426i	0.151960	−	0.988387i	$-0.451441\pi$
$37$	−42367.5	−	42367.5i	−0.836426	−	0.836426i	−0.988387	+	0.151960i	$0.951441\pi$
$38$		0			0
$39$			41777.0i			0.704278i
$40$		0			0
$41$	132500.			1.92250			0.961248	−	0.275684i	$-0.0889042\pi$
$41$	132500.			1.92250			0.961248	+	0.275684i	$0.0889042\pi$
$42$		0			0
$43$	20888.6	−	20888.6i	0.262727	−	0.262727i	−0.563434	−	0.826161i	$-0.690520\pi$
$43$	20888.6	−	20888.6i	0.262727	−	0.262727i	0.826161	+	0.563434i	$0.190520\pi$
$44$		0			0
$45$	20857.4	−	3763.16i	0.228887	−	0.0412966i
$46$		0			0
$47$	−31606.0	−	31606.0i	−0.304422	−	0.304422i	0.538319	−	0.842741i	$-0.319059\pi$
$47$	−31606.0	−	31606.0i	−0.304422	−	0.304422i	−0.842741	+	0.538319i	$0.819059\pi$
$48$		0			0
$49$			238556.i			2.02769i
$50$		0			0
$51$	21688.0			0.163496
$52$		0			0
$53$	80106.6	−	80106.6i	0.538073	−	0.538073i	−0.384890	−	0.922962i	$-0.625761\pi$
$53$	80106.6	−	80106.6i	0.538073	−	0.538073i	0.922962	+	0.384890i	$0.125761\pi$
$54$		0			0
$55$	30183.3	+	167291.i	0.181417	+	1.00551i
$56$		0			0
$57$	−161105.	−	161105.i	−0.869931	−	0.869931i
$58$		0			0
$59$		−	290136.i		−	1.41268i	−0.707871	−	0.706342i	$-0.750344\pi$
$59$		−	290136.i		−	1.41268i	0.707871	−	0.706342i	$-0.249656\pi$
$60$		0			0
$61$	−253971.			−1.11891			−0.559454	−	0.828862i	$-0.688989\pi$
$61$	−253971.			−1.11891			−0.559454	+	0.828862i	$0.688989\pi$
$62$		0			0
$63$	71555.1	−	71555.1i	0.286166	−	0.286166i
$64$		0			0
$65$	−181357.	−	125918.i	−0.660381	−	0.458508i
$66$		0			0
$67$	11368.7	+	11368.7i	0.0377997	+	0.0377997i	0.725754	−	0.687954i	$-0.241491\pi$
$67$	11368.7	+	11368.7i	0.0377997	+	0.0377997i	−0.687954	+	0.725754i	$0.741491\pi$
$68$		0			0
$69$			29667.0i			0.0903082i
$70$		0			0
$71$	179758.			0.502243			0.251122	−	0.967956i	$-0.419201\pi$
$71$	179758.			0.502243			0.251122	+	0.967956i	$0.419201\pi$
$72$		0			0
$73$	−414484.	+	414484.i	−1.06547	+	1.06547i	−0.0677642	+	0.997701i	$0.521587\pi$
$73$	−414484.	+	414484.i	−1.06547	+	1.06547i	−0.997701	+	0.0677642i	$0.978413\pi$
$74$		0			0
$75$	153524.	−	336176.i	0.363908	−	0.796861i
$76$		0			0
$77$	573924.	+	573924.i	1.25714	+	1.25714i
$78$		0			0
$79$		−	145313.i		−	0.294729i	−0.989082	−	0.147364i	$-0.952921\pi$
$79$		−	145313.i		−	0.294729i	0.989082	−	0.147364i	$-0.0470790\pi$
$80$		0			0
$81$	379089.			0.713322
$82$		0			0
$83$	−63129.1	+	63129.1i	−0.110407	+	0.110407i	−0.760152	−	0.649745i	$-0.774876\pi$
$83$	−63129.1	+	63129.1i	−0.110407	+	0.110407i	0.649745	+	0.760152i	$0.274876\pi$
$84$		0			0
$85$	−65368.3	+	94149.0i	−0.106441	+	0.153306i
$86$		0			0
$87$	448049.	+	448049.i	0.680406	+	0.680406i
$88$		0			0
$89$		−	333388.i		−	0.472912i	−0.971642	−	0.236456i	$-0.924014\pi$
$89$		−	333388.i		−	0.472912i	0.971642	−	0.236456i	$-0.0759859\pi$
$90$		0			0
$91$	−1.05416e6			−1.39889
$92$		0			0
$93$	115083.	−	115083.i	0.143075	−	0.143075i
$94$		0			0
$95$	1.18495e6	−	213792.i	1.38206	−	0.249357i
$96$		0			0
$97$	519696.	+	519696.i	0.569422	+	0.569422i	0.931966	−	0.362545i	$-0.118092\pi$
$97$	519696.	+	519696.i	0.569422	+	0.569422i	−0.362545	+	0.931966i	$0.618092\pi$
$98$		0			0
$99$		−	230582.i		−	0.237640i
$100$		0			0
$101$	−1.97076e6			−1.91280			−0.956399	−	0.292065i	$-0.905658\pi$
$101$	−1.97076e6			−1.91280			−0.956399	+	0.292065i	$0.905658\pi$
$102$		0			0
$103$	−862587.	+	862587.i	−0.789389	+	0.789389i	−0.981394	−	0.192005i	$-0.938501\pi$
$103$	−862587.	+	862587.i	−0.789389	+	0.789389i	0.192005	+	0.981394i	$0.438501\pi$
$104$		0			0
$105$	−313311.	−	1.73653e6i	−0.270650	−	1.50008i
$106$		0			0
$107$	−1.28226e6	−	1.28226e6i	−1.04671	−	1.04671i	−0.998854	−	0.0478536i	$-0.984762\pi$
$107$	−1.28226e6	−	1.28226e6i	−1.04671	−	1.04671i	−0.0478536	−	0.998854i	$-0.515238\pi$
$108$		0			0
$109$			186075.i			0.143684i	0.997416	+	0.0718419i	$0.0228877\pi$
$109$			186075.i			0.143684i	−0.997416	+	0.0718419i	$0.977112\pi$
$110$		0			0
$111$	1.41719e6			1.03624
$112$		0			0
$113$	1.01659e6	−	1.01659e6i	0.704545	−	0.704545i	−0.260838	−	0.965383i	$-0.583999\pi$
$113$	1.01659e6	−	1.01659e6i	0.704545	−	0.704545i	0.965383	+	0.260838i	$0.0839987\pi$
$114$		0			0
$115$	−128787.	−	89417.7i	−0.0846795	−	0.0587936i
$116$		0			0
$117$	211762.	+	211762.i	0.132218	+	0.132218i
$118$		0			0
$119$			547254.i			0.324749i
$120$		0			0
$121$	77877.5			0.0439598
$122$		0			0
$123$	−2.21606e6	+	2.21606e6i	−1.19088	+	1.19088i
$124$		0			0
$125$	996638.	+	1.67971e6i	0.510279	+	0.860009i
$126$		0			0
$127$	1.52424e6	+	1.52424e6i	0.744120	+	0.744120i	0.973368	−	0.229248i	$-0.0736266\pi$
$127$	1.52424e6	+	1.52424e6i	0.744120	+	0.744120i	−0.229248	+	0.973368i	$0.573627\pi$
$128$		0			0
$129$			698721.i			0.325488i
$130$		0			0
$131$	1.10323e6			0.490739			0.245370	−	0.969430i	$-0.421091\pi$
$131$	1.10323e6			0.490739			0.245370	+	0.969430i	$0.421091\pi$
$132$		0			0
$133$	4.06518e6	−	4.06518e6i	1.72792	−	1.72792i
$134$		0			0
$135$	−1.51514e6	+	2.18223e6i	−0.615815	+	0.886949i
$136$		0			0
$137$	720618.	+	720618.i	0.280248	+	0.280248i	0.833208	−	0.552960i	$-0.186502\pi$
$137$	720618.	+	720618.i	0.280248	+	0.280248i	−0.552960	+	0.833208i	$0.686502\pi$
$138$		0			0
$139$		−	2.89720e6i		−	1.07878i	−0.842055	−	0.539391i	$-0.818654\pi$
$139$		−	2.89720e6i		−	1.07878i	0.842055	−	0.539391i	$-0.181346\pi$
$140$		0			0
$141$	1.05722e6			0.377143
$142$		0			0
$143$	−1.69849e6	+	1.69849e6i	−0.580838	+	0.580838i
$144$		0			0
$145$	−3.29545e6	+	594577.i	−1.08096	+	0.195031i
$146$		0			0
$147$	−3.98983e6	−	3.98983e6i	−1.25604	−	1.25604i
$148$		0			0
$149$			1.98382e6i			0.599714i	0.953984	+	0.299857i	$0.0969389\pi$
$149$			1.98382e6i			0.599714i	−0.953984	+	0.299857i	$0.903061\pi$
$150$		0			0
$151$	4.02159e6			1.16807			0.584033	−	0.811730i	$-0.301474\pi$
$151$	4.02159e6			1.16807			0.584033	+	0.811730i	$0.301474\pi$
$152$		0			0
$153$	109933.	−	109933.i	0.0306941	−	0.0306941i
$154$		0			0
$155$	152719.	+	846450.i	0.0410109	+	0.227304i
$156$		0			0
$157$	−3.96237e6	−	3.96237e6i	−1.02390	−	1.02390i	−0.999707	−	0.0241894i	$-0.992300\pi$
$157$	−3.96237e6	−	3.96237e6i	−1.02390	−	1.02390i	−0.0241894	−	0.999707i	$-0.507700\pi$
$158$		0			0
$159$			2.67956e6i			0.666610i
$160$		0			0
$161$	−748591.			−0.179377
$162$		0			0
$163$	2.80620e6	−	2.80620e6i	0.647971	−	0.647971i	−0.304531	−	0.952502i	$-0.598500\pi$
$163$	2.80620e6	−	2.80620e6i	0.647971	−	0.647971i	0.952502	+	0.304531i	$0.0984998\pi$
$164$		0			0
$165$	−3.30275e6	−	2.29313e6i	−0.735232	−	0.510477i
$166$		0			0
$167$	−4.05536e6	−	4.05536e6i	−0.870724	−	0.870724i	0.121827	−	0.992551i	$-0.461125\pi$
$167$	−4.05536e6	−	4.05536e6i	−0.870724	−	0.870724i	−0.992551	+	0.121827i	$0.961125\pi$
$168$		0			0
$169$			1.70708e6i			0.353667i
$170$		0			0
$171$	−1.63324e6			−0.326635
$172$		0			0
$173$	−593924.	+	593924.i	−0.114708	+	0.114708i	−0.762131	−	0.647423i	$-0.775847\pi$
$173$	−593924.	+	593924.i	−0.114708	+	0.114708i	0.647423	+	0.762131i	$0.275847\pi$
$174$		0			0
$175$	8.48275e6	+	3.87387e6i	1.58279	+	0.722822i
$176$		0			0
$177$	4.85250e6	+	4.85250e6i	0.875076	+	0.875076i
$178$		0			0
$179$		−	3.48674e6i		−	0.607939i	−0.952682	−	0.303970i	$-0.901688\pi$
$179$		−	3.48674e6i		−	0.607939i	0.952682	−	0.303970i	$-0.0983122\pi$
$180$		0			0
$181$	−1.41422e6			−0.238496			−0.119248	−	0.992864i	$-0.538048\pi$
$181$	−1.41422e6			−0.238496			−0.119248	+	0.992864i	$0.538048\pi$
$182$		0			0
$183$	4.24764e6	−	4.24764e6i	0.693098	−	0.693098i
$184$		0			0
$185$	−4.27146e6	+	6.15212e6i	−0.674623	+	0.971649i
$186$		0			0
$187$	881747.	+	881747.i	0.134840	+	0.134840i
$188$		0			0
$189$			1.26845e7i			1.87883i
$190$		0			0
$191$	4.51204e6			0.647549			0.323775	−	0.946134i	$-0.395048\pi$
$191$	4.51204e6			0.647549			0.323775	+	0.946134i	$0.395048\pi$
$192$		0			0
$193$	1.70789e6	−	1.70789e6i	0.237569	−	0.237569i	−0.578274	−	0.815843i	$-0.696274\pi$
$193$	1.70789e6	−	1.70789e6i	0.237569	−	0.237569i	0.815843	+	0.578274i	$0.196274\pi$
$194$		0			0
$195$	5.13915e6	−	927223.i	0.693087	−	0.125049i
$196$		0			0
$197$	2.88696e6	+	2.88696e6i	0.377608	+	0.377608i	0.870239	−	0.492630i	$-0.163964\pi$
$197$	2.88696e6	+	2.88696e6i	0.377608	+	0.377608i	−0.492630	+	0.870239i	$0.663964\pi$
$198$		0			0
$199$		−	4.21379e6i		−	0.534704i	−0.963599	−	0.267352i	$-0.913851\pi$
$199$		−	4.21379e6i		−	0.534704i	0.963599	−	0.267352i	$-0.0861487\pi$
$200$		0			0
$201$	−380283.			−0.0468294
$202$		0			0
$203$	−1.13057e7	+	1.13057e7i	−1.35148	+	1.35148i
$204$		0			0
$205$	−2.94079e6	−	1.62994e7i	−0.341352	−	1.89195i
$206$		0			0
$207$	150378.	+	150378.i	0.0169541	+	0.0169541i
$208$		0			0
$209$		−	1.30998e7i		−	1.43491i
$210$		0			0
$211$	−3.89683e6			−0.414825			−0.207412	−	0.978254i	$-0.566504\pi$
$211$	−3.89683e6			−0.414825			−0.207412	+	0.978254i	$0.566504\pi$
$212$		0			0
$213$	−3.00645e6	+	3.00645e6i	−0.311111	+	0.311111i
$214$		0			0
$215$	−3.03320e6	−	2.10597e6i	−0.305201	−	0.211903i
$216$		0			0
$217$	2.90391e6	+	2.90391e6i	0.284186	+	0.284186i
$218$		0			0
$219$		−	1.38644e7i		−	1.31999i
$220$		0			0
$221$	−1.61956e6			−0.150045
$222$		0			0
$223$	−3.41658e6	+	3.41658e6i	−0.308090	+	0.308090i	−0.844168	−	0.536078i	$-0.819905\pi$
$223$	−3.41658e6	+	3.41658e6i	−0.308090	+	0.308090i	0.536078	+	0.844168i	$0.319905\pi$
$224$		0			0
$225$	−925841.	−	2.48222e6i	−0.0812809	−	0.217918i
$226$		0			0
$227$	7.67846e6	+	7.67846e6i	0.656442	+	0.656442i	0.954537	−	0.298094i	$-0.0963509\pi$
$227$	7.67846e6	+	7.67846e6i	0.656442	+	0.656442i	−0.298094	+	0.954537i	$0.596351\pi$
$228$		0			0
$229$			7.64574e6i			0.636668i	0.947979	+	0.318334i	$0.103123\pi$
$229$			7.64574e6i			0.636668i	−0.947979	+	0.318334i	$0.896877\pi$
$230$		0			0
$231$	−1.91977e7			−1.55745
$232$		0			0
$233$	1.32288e6	−	1.32288e6i	0.104581	−	0.104581i	−0.652880	−	0.757461i	$-0.726440\pi$
$233$	1.32288e6	−	1.32288e6i	0.104581	−	0.104581i	0.757461	+	0.652880i	$0.226440\pi$
$234$		0			0
$235$	−3.18649e6	+	4.58946e6i	−0.245533	+	0.353637i
$236$		0			0
$237$	2.43035e6	+	2.43035e6i	0.182568	+	0.182568i
$238$		0			0
$239$		−	1.04557e7i		−	0.765875i	−0.923774	−	0.382937i	$-0.874913\pi$
$239$		−	1.04557e7i		−	0.765875i	0.923774	−	0.382937i	$-0.125087\pi$
$240$		0			0
$241$	−7.42256e6			−0.530277			−0.265138	−	0.964210i	$-0.585418\pi$
$241$	−7.42256e6			−0.530277			−0.265138	+	0.964210i	$0.585418\pi$
$242$		0			0
$243$	4.61536e6	−	4.61536e6i	0.321652	−	0.321652i
$244$		0			0
$245$	2.93456e7	−	5.29464e6i	1.99547	−	0.360030i
$246$		0			0
$247$	1.20306e7	+	1.20306e7i	0.798357	+	0.798357i
$248$		0			0
$249$		−	2.11166e6i		−	0.136781i
$250$		0			0
$251$	1.25795e7			0.795505			0.397752	−	0.917493i	$-0.369790\pi$
$251$	1.25795e7			0.795505			0.397752	+	0.917493i	$0.369790\pi$
$252$		0			0
$253$	−1.20615e6	+	1.20615e6i	−0.0744798	+	0.0744798i
$254$		0			0
$255$	−481355.	−	2.66792e6i	−0.0290298	−	0.160898i
$256$		0			0
$257$	−3.35530e6	−	3.35530e6i	−0.197666	−	0.197666i	0.601333	−	0.798999i	$-0.294637\pi$
$257$	−3.35530e6	−	3.35530e6i	−0.197666	−	0.197666i	−0.798999	+	0.601333i	$0.794637\pi$
$258$		0			0
$259$			3.57600e7i			2.05825i
$260$		0			0
$261$	4.54221e6			0.255473
$262$		0			0
$263$	−1.97354e7	+	1.97354e7i	−1.08487	+	1.08487i	−0.0888280	+	0.996047i	$0.528312\pi$
$263$	−1.97354e7	+	1.97354e7i	−1.08487	+	1.08487i	−0.996047	+	0.0888280i	$0.971688\pi$
$264$		0			0
$265$	−1.16322e7	−	8.07629e6i	−0.625061	−	0.433985i
$266$		0			0
$267$	5.57590e6	+	5.57590e6i	0.292942	+	0.292942i
$268$		0			0
$269$			3.82612e6i			0.196563i	0.995159	+	0.0982816i	$0.0313346\pi$
$269$			3.82612e6i			0.196563i	−0.995159	+	0.0982816i	$0.968665\pi$
$270$		0			0
$271$	238579.			0.0119874			0.00599368	−	0.999982i	$-0.498092\pi$
$271$	238579.			0.0119874			0.00599368	+	0.999982i	$0.498092\pi$
$272$		0			0
$273$	1.76308e7	−	1.76308e7i	0.866532	−	0.866532i
$274$		0			0
$275$	1.99093e7	−	7.42593e6i	0.957320	−	0.357069i
$276$		0			0
$277$	−8.83629e6	−	8.83629e6i	−0.415749	−	0.415749i	0.467987	−	0.883735i	$-0.344979\pi$
$277$	−8.83629e6	−	8.83629e6i	−0.415749	−	0.415749i	−0.883735	+	0.467987i	$0.844979\pi$
$278$		0			0
$279$		−	1.16668e6i		−	0.0537206i
$280$		0			0
$281$	9.44266e6			0.425574			0.212787	−	0.977099i	$-0.431746\pi$
$281$	9.44266e6			0.425574			0.212787	+	0.977099i	$0.431746\pi$
$282$		0			0
$283$	3.88228e6	−	3.88228e6i	0.171288	−	0.171288i	−0.616257	−	0.787545i	$-0.711352\pi$
$283$	3.88228e6	−	3.88228e6i	0.171288	−	0.171288i	0.787545	+	0.616257i	$0.211352\pi$
$284$		0			0
$285$	−1.62425e7	+	2.33938e7i	−0.701646	+	1.01057i
$286$		0			0
$287$	−5.59180e7	−	5.59180e7i	−2.36541	−	2.36541i
$288$		0			0
$289$		−	2.32968e7i		−	0.965168i
$290$		0			0
$291$	−1.73838e7			−0.705448
$292$		0			0
$293$	4.41027e6	−	4.41027e6i	0.175332	−	0.175332i	−0.613985	−	0.789318i	$-0.710435\pi$
$293$	4.41027e6	−	4.41027e6i	0.175332	−	0.175332i	0.789318	+	0.613985i	$0.210435\pi$
$294$		0			0
$295$	−3.56907e7	+	6.43944e6i	−1.39024	+	0.250831i
$296$		0			0
$297$	2.04375e7	+	2.04375e7i	0.780115	+	0.780115i
$298$		0			0
$299$		−	2.21540e6i		−	0.0828780i
$300$		0			0
$301$	−1.76309e7			−0.646509
$302$		0			0
$303$	3.29608e7	−	3.29608e7i	1.18487	−	1.18487i
$304$		0			0
$305$	5.63677e6	+	3.12419e7i	0.198669	+	1.10113i
$306$		0			0
$307$	2.02549e7	+	2.02549e7i	0.700026	+	0.700026i	0.964416	−	0.264390i	$-0.0851704\pi$
$307$	2.02549e7	+	2.02549e7i	0.700026	+	0.700026i	−0.264390	+	0.964416i	$0.585170\pi$
$308$		0			0
$309$		−	2.88534e7i		−	0.977962i
$310$		0			0
$311$	4.11216e7			1.36707			0.683533	−	0.729920i	$-0.260443\pi$
$311$	4.11216e7			1.36707			0.683533	+	0.729920i	$0.260443\pi$
$312$		0			0
$313$	−3.75640e7	+	3.75640e7i	−1.22501	+	1.22501i	−0.259178	+	0.965830i	$0.583452\pi$
$313$	−3.75640e7	+	3.75640e7i	−1.22501	+	1.22501i	−0.965830	+	0.259178i	$0.916548\pi$
$314$		0			0
$315$	−1.03904e7	−	7.21413e6i	−0.332430	−	0.230809i
$316$		0			0
$317$	−2.01898e7	−	2.01898e7i	−0.633803	−	0.633803i	0.315217	−	0.949020i	$-0.397923\pi$
$317$	−2.01898e7	−	2.01898e7i	−0.633803	−	0.633803i	−0.949020	+	0.315217i	$0.897923\pi$
$318$		0			0
$319$			3.64319e7i			1.12230i
$320$		0			0
$321$	4.28915e7			1.29675
$322$		0			0
$323$	6.24553e6	−	6.24553e6i	0.185337	−	0.185337i
$324$		0			0
$325$	−1.14645e7	+	2.51041e7i	−0.333967	+	0.731299i
$326$		0			0
$327$	−3.11209e6	−	3.11209e6i	−0.0890038	−	0.0890038i
$328$		0			0
$329$			2.66768e7i			0.749112i
$330$		0			0
$331$	5.35648e7			1.47705			0.738526	−	0.674225i	$-0.235522\pi$
$331$	5.35648e7			1.47705			0.738526	+	0.674225i	$0.235522\pi$
$332$		0			0
$333$	7.18354e6	−	7.18354e6i	0.194539	−	0.194539i
$334$		0			0
$335$	1.14619e6	−	1.65084e6i	0.0304875	−	0.0439107i
$336$		0			0
$337$	−3.35730e7	−	3.35730e7i	−0.877203	−	0.877203i	0.116041	−	0.993244i	$-0.462980\pi$
$337$	−3.35730e7	−	3.35730e7i	−0.877203	−	0.877203i	−0.993244	+	0.116041i	$0.962980\pi$
$338$		0			0
$339$			3.40047e7i			0.872850i
$340$		0			0
$341$	9.35767e6			0.235996
$342$		0			0
$343$	5.10252e7	−	5.10252e7i	1.26445	−	1.26445i
$344$		0			0
$345$	3.64946e6	−	658447.i	0.0888732	−	0.0160348i
$346$		0			0
$347$	3.02313e7	+	3.02313e7i	0.723550	+	0.723550i	0.969327	−	0.245776i	$-0.0790428\pi$
$347$	3.02313e7	+	3.02313e7i	0.723550	+	0.723550i	−0.245776	+	0.969327i	$0.579043\pi$
$348$		0			0
$349$			2.12050e7i			0.498842i	0.968395	+	0.249421i	$0.0802402\pi$
$349$			2.12050e7i			0.498842i	−0.968395	+	0.249421i	$0.919760\pi$
$350$		0			0
$351$	−3.75389e7			−0.868080
$352$		0			0
$353$	−5.10632e7	+	5.10632e7i	−1.16087	+	1.16087i	−0.176586	+	0.984285i	$0.556505\pi$
$353$	−5.10632e7	+	5.10632e7i	−1.16087	+	1.16087i	−0.984285	+	0.176586i	$0.943495\pi$
$354$		0			0
$355$	−3.98966e6	−	2.21128e7i	−0.0891766	−	0.494263i
$356$		0			0
$357$	−9.15279e6	−	9.15279e6i	−0.201163	−	0.201163i
$358$		0			0
$359$		−	6.68160e7i		−	1.44410i	−0.691841	−	0.722050i	$-0.743200\pi$
$359$		−	6.68160e7i		−	1.44410i	0.691841	−	0.722050i	$-0.256800\pi$
$360$		0			0
$361$	−4.57417e7			−0.972279
$362$		0			0
$363$	−1.30250e6	+	1.30250e6i	−0.0272306	+	0.0272306i
$364$		0			0
$365$	6.01866e7	+	4.17880e7i	1.23772	+	0.859356i
$366$		0			0
$367$	−5.04382e6	−	5.04382e6i	−0.102038	−	0.102038i	0.654245	−	0.756283i	$-0.272987\pi$
$367$	−5.04382e6	−	5.04382e6i	−0.102038	−	0.102038i	−0.756283	+	0.654245i	$0.772987\pi$
$368$		0			0
$369$			2.24658e7i			0.447140i
$370$		0			0
$371$	−6.76135e7			−1.32407
$372$		0			0
$373$	5.06926e7	−	5.06926e7i	0.976828	−	0.976828i	−0.0229094	−	0.999738i	$-0.507293\pi$
$373$	5.06926e7	−	5.06926e7i	0.976828	−	0.976828i	0.999738	+	0.0229094i	$0.00729292\pi$
$374$		0			0
$375$	−4.47617e7	−	1.14242e7i	−0.848814	−	0.216638i
$376$		0			0
$377$	−3.34584e7	−	3.34584e7i	−0.624425	−	0.624425i
$378$		0			0
$379$		−	2.97376e7i		−	0.546245i	−0.961979	−	0.273123i	$-0.911944\pi$
$379$		−	2.97376e7i		−	0.546245i	0.961979	−	0.273123i	$-0.0880565\pi$
$380$		0			0
$381$	−5.09857e7			−0.921879
$382$		0			0
$383$	−2.82121e7	+	2.82121e7i	−0.502157	+	0.502157i	−0.912108	−	0.409951i	$-0.865546\pi$
$383$	−2.82121e7	+	2.82121e7i	−0.502157	+	0.502157i	0.409951	+	0.912108i	$0.365546\pi$
$384$		0			0
$385$	5.78626e7	−	8.33387e7i	1.01395	−	1.46037i
$386$		0			0
$387$	3.54173e6	+	3.54173e6i	0.0611058	+	0.0611058i
$388$		0			0
$389$		−	8.32404e7i		−	1.41412i	−0.707155	−	0.707059i	$-0.750022\pi$
$389$		−	8.32404e7i		−	1.41412i	0.707155	−	0.707059i	$-0.249978\pi$
$390$		0			0
$391$	−1.15010e6			−0.0192399
$392$		0			0
$393$	−1.84514e7	+	1.84514e7i	−0.303984	+	0.303984i
$394$		0			0
$395$	−1.78755e7	+	3.22516e6i	−0.290046	+	0.0523311i
$396$		0			0
$397$	4.98898e7	+	4.98898e7i	0.797333	+	0.797333i	0.982674	−	0.185341i	$-0.0593390\pi$
$397$	4.98898e7	+	4.98898e7i	0.797333	+	0.797333i	−0.185341	+	0.982674i	$0.559339\pi$
$398$		0			0
$399$			1.35980e8i			2.14070i
$400$		0			0
$401$	1.12175e8			1.73966			0.869829	−	0.493353i	$-0.164229\pi$
$401$	1.12175e8			1.73966			0.869829	+	0.493353i	$0.164229\pi$
$402$		0			0
$403$	−8.59391e6	+	8.59391e6i	−0.131303	+	0.131303i
$404$		0			0
$405$	−8.41371e6	−	4.66331e7i	−0.126655	−	0.701988i
$406$		0			0
$407$	5.76173e7	+	5.76173e7i	0.854613	+	0.854613i
$408$		0			0
$409$		−	1.66273e7i		−	0.243025i	−0.992590	−	0.121513i	$-0.961226\pi$
$409$		−	1.66273e7i		−	0.243025i	0.992590	−	0.121513i	$-0.0387745\pi$
$410$		0			0
$411$	−2.41046e7			−0.347195
$412$		0			0
$413$	−1.22444e8	+	1.22444e8i	−1.73814	+	1.73814i
$414$		0			0
$415$	9.16688e6	+	6.36463e6i	0.128256	+	0.0890490i
$416$		0			0
$417$	4.84555e7	+	4.84555e7i	0.668244	+	0.668244i
$418$		0			0
$419$		−	7.77132e7i		−	1.05646i	−0.849101	−	0.528230i	$-0.822856\pi$
$419$		−	7.77132e7i		−	1.05646i	0.849101	−	0.528230i	$-0.177144\pi$
$420$		0			0
$421$	4.97347e7			0.666520			0.333260	−	0.942835i	$-0.391851\pi$
$421$	4.97347e7			0.666520			0.333260	+	0.942835i	$0.391851\pi$
$422$		0			0
$423$	5.35889e6	−	5.35889e6i	0.0708034	−	0.0708034i
$424$		0			0
$425$	1.30325e7	+	5.95161e6i	0.169769	+	0.0775297i
$426$		0			0
$427$	1.07181e8	+	1.07181e8i	1.37669	+	1.37669i
$428$		0			0
$429$		−	5.68143e7i		−	0.719591i
$430$		0			0
$431$	−6.70972e7			−0.838056			−0.419028	−	0.907973i	$-0.637629\pi$
$431$	−6.70972e7			−0.838056			−0.419028	+	0.907973i	$0.637629\pi$
$432$		0			0
$433$	9.05301e7	−	9.05301e7i	1.11514	−	1.11514i	0.122696	−	0.992444i	$-0.460846\pi$
$433$	9.05301e7	−	9.05301e7i	1.11514	−	1.11514i	0.992444	−	0.122696i	$-0.0391539\pi$
$434$		0			0
$435$	4.51720e7	−	6.50605e7i	0.548784	−	0.790405i
$436$		0			0
$437$	8.54328e6	+	8.54328e6i	0.102372	+	0.102372i
$438$		0			0
$439$			231381.i			0.00273485i	0.999999	+	0.00136743i	$0.000435265\pi$
$439$			231381.i			0.00273485i	−0.999999	+	0.00136743i	$0.999565\pi$
$440$		0			0
$441$	−4.04478e7			−0.471606
$442$		0			0
$443$	−2.08251e7	+	2.08251e7i	−0.239538	+	0.239538i	−0.816659	−	0.577121i	$-0.804176\pi$
$443$	−2.08251e7	+	2.08251e7i	−0.239538	+	0.239538i	0.577121	+	0.816659i	$0.304176\pi$
$444$		0			0
$445$	−4.10114e7	+	7.39941e6i	−0.465398	+	0.0839686i
$446$		0			0
$447$	−3.31793e7	−	3.31793e7i	−0.371488	−	0.371488i
$448$		0			0
$449$		−	6.49546e7i		−	0.717581i	−0.933418	−	0.358790i	$-0.883189\pi$
$449$		−	6.49546e7i		−	0.717581i	0.933418	−	0.358790i	$-0.116811\pi$
$450$		0			0
$451$	−1.80193e8			−1.96430
$452$		0			0
$453$	−6.72609e7	+	6.72609e7i	−0.723549	+	0.723549i
$454$		0			0
$455$	2.33967e7	+	1.29677e8i	0.248382	+	1.37666i
$456$		0			0
$457$	−6.96439e7	−	6.96439e7i	−0.729684	−	0.729684i	0.240873	−	0.970557i	$-0.422566\pi$
$457$	−6.96439e7	−	6.96439e7i	−0.729684	−	0.729684i	−0.970557	+	0.240873i	$0.922566\pi$
$458$		0			0
$459$			1.94878e7i			0.201523i
$460$		0			0
$461$	8.07689e7			0.824407			0.412203	−	0.911092i	$-0.364759\pi$
$461$	8.07689e7			0.824407			0.412203	+	0.911092i	$0.364759\pi$
$462$		0			0
$463$	8.35152e7	−	8.35152e7i	0.841439	−	0.841439i	−0.147607	−	0.989046i	$-0.547157\pi$
$463$	8.35152e7	−	8.35152e7i	0.841439	−	0.841439i	0.989046	+	0.147607i	$0.0471572\pi$
$464$		0			0
$465$	−1.67111e7	−	1.16026e7i	−0.166205	−	0.115398i
$466$		0			0
$467$	−8.17085e7	−	8.17085e7i	−0.802262	−	0.802262i	0.181186	−	0.983449i	$-0.442006\pi$
$467$	−8.17085e7	−	8.17085e7i	−0.802262	−	0.802262i	−0.983449	+	0.181186i	$0.942006\pi$
$468$		0			0
$469$		−	9.59572e6i		−	0.0930163i
$470$		0			0
$471$	1.32541e8			1.26849
$472$		0			0
$473$	−2.84073e7	+	2.84073e7i	−0.268439	+	0.268439i
$474$		0			0
$475$	−5.25988e7	−	1.41020e8i	−0.490789	−	1.31583i
$476$		0			0
$477$	1.35823e7	+	1.35823e7i	0.125147	+	0.125147i
$478$		0			0
$479$			1.07532e8i			0.978433i	0.872162	+	0.489216i	$0.162717\pi$
$479$			1.07532e8i			0.978433i	−0.872162	+	0.489216i	$0.837283\pi$
$480$		0			0
$481$	−1.05829e8			−0.950979
$482$		0			0
$483$	1.25201e7	−	1.25201e7i	0.111114	−	0.111114i
$484$		0			0
$485$	5.23953e7	−	7.54642e7i	0.459269	−	0.661478i
$486$		0			0
$487$	−4.33036e7	−	4.33036e7i	−0.374919	−	0.374919i	0.494346	−	0.869265i	$-0.335408\pi$
$487$	−4.33036e7	−	4.33036e7i	−0.374919	−	0.374919i	−0.869265	+	0.494346i	$0.835408\pi$
$488$		0			0
$489$			9.38670e7i			0.802761i
$490$		0			0
$491$	−6.23191e7			−0.526474			−0.263237	−	0.964731i	$-0.584790\pi$
$491$	−6.23191e7			−0.526474			−0.263237	+	0.964731i	$0.584790\pi$
$492$		0			0
$493$	−1.73694e7	+	1.73694e7i	−0.144959	+	0.144959i
$494$		0			0
$495$	−2.83648e7	+	5.11767e6i	−0.233864	+	0.0421946i
$496$		0			0
$497$	−7.58619e7	−	7.58619e7i	−0.617952	−	0.617952i
$498$		0			0
$499$		−	1.67847e6i		−	0.0135086i	−0.999977	−	0.00675432i	$-0.997850\pi$
$499$		−	1.67847e6i		−	0.0135086i	0.999977	−	0.00675432i	$-0.00214998\pi$
$500$		0			0
$501$	1.35651e8			1.07873
$502$		0			0
$503$	5.94281e7	−	5.94281e7i	0.466969	−	0.466969i	−0.433962	−	0.900931i	$-0.642885\pi$
$503$	5.94281e7	−	5.94281e7i	0.466969	−	0.466969i	0.900931	+	0.433962i	$0.142885\pi$
$504$		0			0
$505$	4.37401e7	+	2.42430e8i	0.339630	+	1.88240i
$506$		0			0
$507$	−2.85509e7	−	2.85509e7i	−0.219076	−	0.219076i
$508$		0			0
$509$			1.54356e8i			1.17049i	0.810855	+	0.585247i	$0.199003\pi$
$509$			1.54356e8i			1.17049i	−0.810855	+	0.585247i	$0.800997\pi$
$510$		0			0
$511$	3.49843e8			2.62186
$512$		0			0
$513$	1.44762e8	−	1.44762e8i	1.07226	−	1.07226i
$514$		0			0
$515$	1.25255e8	+	8.69654e7i	0.917008	+	0.636685i
$516$		0			0
$517$	4.29823e7	+	4.29823e7i	0.311041	+	0.311041i
$518$		0			0
$519$		−	1.98667e7i		−	0.142110i
$520$		0			0
$521$	−1.71124e8			−1.21004			−0.605019	−	0.796211i	$-0.706834\pi$
$521$	−1.71124e8			−1.21004			−0.605019	+	0.796211i	$0.706834\pi$
$522$		0			0
$523$	−1.28297e8	+	1.28297e8i	−0.896833	+	0.896833i	−0.995155	−	0.0983219i	$-0.968653\pi$
$523$	−1.28297e8	+	1.28297e8i	−0.896833	+	0.896833i	0.0983219	+	0.995155i	$0.468653\pi$
$524$		0			0
$525$	−2.06664e8	+	7.70832e7i	−1.42819	+	0.532699i
$526$		0			0
$527$	4.46141e6	+	4.46141e6i	0.0304817	+	0.0304817i
$528$		0			0
$529$			1.46463e8i			0.989373i
$530$		0			0
$531$	4.91934e7			0.328566
$532$		0			0
$533$	1.65486e8	−	1.65486e8i	1.09290	−	1.09290i
$534$		0			0
$535$	−1.29277e8	+	1.86195e8i	−0.844226	+	1.21593i
$536$		0			0
$537$	5.83155e7	+	5.83155e7i	0.376583	+	0.376583i
$538$		0			0
$539$		−	3.24422e8i		−	2.07178i
$540$		0			0
$541$	−1.12555e8			−0.710841			−0.355420	−	0.934707i	$-0.615662\pi$
$541$	−1.12555e8			−0.710841			−0.355420	+	0.934707i	$0.615662\pi$
$542$		0			0
$543$	2.36528e7	−	2.36528e7i	0.147735	−	0.147735i
$544$		0			0
$545$	2.28898e7	−	4.12985e6i	0.141401	−	0.0255120i
$546$		0			0
$547$	−3.86772e7	−	3.86772e7i	−0.236316	−	0.236316i	0.579007	−	0.815323i	$-0.303440\pi$
$547$	−3.86772e7	−	3.86772e7i	−0.236316	−	0.236316i	−0.815323	+	0.579007i	$0.803440\pi$
$548$		0			0
$549$		−	4.30615e7i		−	0.260239i
$550$		0			0
$551$	2.58051e8			1.54259
$552$		0			0
$553$	−6.13252e7	+	6.13252e7i	−0.362630	+	0.362630i
$554$		0			0
$555$	−3.14539e7	−	1.74334e8i	−0.183990	−	1.01977i
$556$		0			0
$557$	−1.77413e8	−	1.77413e8i	−1.02664	−	1.02664i	−0.999635	−	0.0270085i	$-0.991402\pi$
$557$	−1.77413e8	−	1.77413e8i	−1.02664	−	1.02664i	−0.0270085	−	0.999635i	$-0.508598\pi$
$558$		0			0
$559$		−	5.21774e7i		−	0.298708i
$560$		0			0
$561$	−2.94943e7			−0.167051
$562$		0			0
$563$	−2.61909e7	+	2.61909e7i	−0.146766	+	0.146766i	−0.776672	−	0.629906i	$-0.783094\pi$
$563$	−2.61909e7	+	2.61909e7i	−0.146766	+	0.146766i	0.629906	+	0.776672i	$0.283094\pi$
$564$		0			0
$565$	−1.47617e8	−	1.02491e8i	−0.818447	−	0.568254i
$566$		0			0
$567$	−1.59984e8	−	1.59984e8i	−0.877660	−	0.877660i
$568$		0			0
$569$			9.04667e7i			0.491080i	0.969387	+	0.245540i	$0.0789652\pi$
$569$			9.04667e7i			0.491080i	−0.969387	+	0.245540i	$0.921035\pi$
$570$		0			0
$571$	−3.72767e7			−0.200230			−0.100115	−	0.994976i	$-0.531921\pi$
$571$	−3.72767e7			−0.200230			−0.100115	+	0.994976i	$0.531921\pi$
$572$		0			0
$573$	−7.54636e7	+	7.54636e7i	−0.401119	+	0.401119i
$574$		0			0
$575$	−8.14124e6	+	1.78272e7i	−0.0428240	+	0.0937731i
$576$		0			0
$577$	6.48027e7	+	6.48027e7i	0.337339	+	0.337339i	0.855365	−	0.518026i	$-0.173333\pi$
$577$	6.48027e7	+	6.48027e7i	0.337339	+	0.337339i	−0.518026	+	0.855365i	$0.673333\pi$
$578$		0			0
$579$			5.71288e7i			0.294320i
$580$		0			0
$581$	5.32837e7			0.271685
$582$		0			0
$583$	−1.08940e8	+	1.08940e8i	−0.549772	+	0.549772i
$584$		0			0
$585$	2.13497e7	−	3.07497e7i	0.106641	−	0.153593i
$586$		0			0
$587$	1.02171e8	+	1.02171e8i	0.505144	+	0.505144i	0.913032	−	0.407888i	$-0.133735\pi$
$587$	1.02171e8	+	1.02171e8i	0.505144	+	0.505144i	−0.407888	+	0.913032i	$0.633735\pi$
$588$		0			0
$589$		−	6.62815e7i		−	0.324375i
$590$		0			0
$591$	−9.65684e7			−0.467813
$592$		0			0
$593$	1.09929e8	−	1.09929e8i	0.527166	−	0.527166i	−0.392560	−	0.919726i	$-0.628410\pi$
$593$	1.09929e8	−	1.09929e8i	0.527166	−	0.527166i	0.919726	+	0.392560i	$0.128410\pi$
$594$		0			0
$595$	6.73198e7	−	1.21461e7i	0.319589	−	0.0576613i
$596$		0			0
$597$	7.04754e7	+	7.04754e7i	0.331218	+	0.331218i
$598$		0			0
$599$			2.43420e8i			1.13260i	0.824200	+	0.566298i	$0.191625\pi$
$599$			2.43420e8i			1.13260i	−0.824200	+	0.566298i	$0.808375\pi$
$600$		0			0
$601$	−1.12016e7			−0.0516007			−0.0258004	−	0.999667i	$-0.508213\pi$
$601$	−1.12016e7			−0.0516007			−0.0258004	+	0.999667i	$0.508213\pi$
$602$		0			0
$603$	−1.92761e6	+	1.92761e6i	−0.00879156	+	0.00879156i
$604$		0			0
$605$	−1.72846e6	−	9.58001e6i	−0.00780535	−	0.0432613i
$606$		0			0
$607$	−1.50111e8	−	1.50111e8i	−0.671192	−	0.671192i	0.286799	−	0.957991i	$-0.407409\pi$
$607$	−1.50111e8	−	1.50111e8i	−0.671192	−	0.671192i	−0.957991	+	0.286799i	$0.907409\pi$
$608$		0			0
$609$		−	3.78173e8i		−	1.67432i
$610$		0			0
$611$	−7.89483e7			−0.346114
$612$		0			0
$613$	1.74572e8	−	1.74572e8i	0.757868	−	0.757868i	−0.218066	−	0.975934i	$-0.569975\pi$
$613$	1.74572e8	−	1.74572e8i	0.757868	−	0.757868i	0.975934	+	0.218066i	$0.0699749\pi$
$614$		0			0
$615$	3.21791e8	+	2.23422e8i	1.38340	+	0.960505i
$616$		0			0
$617$	2.43703e8	+	2.43703e8i	1.03754	+	1.03754i	0.999267	+	0.0382750i	$0.0121863\pi$
$617$	2.43703e8	+	2.43703e8i	1.03754	+	1.03754i	0.0382750	+	0.999267i	$0.487814\pi$
$618$		0			0
$619$			1.67973e8i			0.708220i	0.935204	+	0.354110i	$0.115216\pi$
$619$			1.67973e8i			0.708220i	−0.935204	+	0.354110i	$0.884784\pi$
$620$		0			0
$621$	−2.66574e7			−0.111312
$622$		0			0
$623$	−1.40697e8	+	1.40697e8i	−0.581863	+	0.581863i
$624$		0			0
$625$	1.84507e8	−	1.59881e8i	0.755741	−	0.654871i
$626$		0			0
$627$	2.19093e8	+	2.19093e8i	0.888846	+	0.888846i
$628$		0			0
$629$			5.49398e7i			0.220767i
$630$		0			0
$631$	−4.57816e8			−1.82223			−0.911114	−	0.412154i	$-0.864777\pi$
$631$	−4.57816e8			−1.82223			−0.911114	+	0.412154i	$0.864777\pi$
$632$		0			0
$633$	6.51743e7	−	6.51743e7i	0.256960	−	0.256960i
$634$		0			0
$635$	1.53673e8	−	2.21333e8i	0.600173	−	0.864420i
$636$		0			0
$637$	2.97943e8	+	2.97943e8i	1.15270	+	1.15270i
$638$		0			0
$639$			3.04786e7i			0.116813i
$640$		0			0
$641$	−4.41603e7			−0.167671			−0.0838356	−	0.996480i	$-0.526717\pi$
$641$	−4.41603e7			−0.167671			−0.0838356	+	0.996480i	$0.526717\pi$
$642$		0			0
$643$	−2.97415e8	+	2.97415e8i	−1.11874	+	1.11874i	−0.126816	+	0.991926i	$0.540476\pi$
$643$	−2.97415e8	+	2.97415e8i	−1.11874	+	1.11874i	−0.991926	+	0.126816i	$0.959524\pi$
$644$		0			0
$645$	8.59524e7	−	1.55078e7i	0.320316	−	0.0577925i
$646$		0			0
$647$	−8.69716e7	−	8.69716e7i	−0.321118	−	0.321118i	0.528078	−	0.849196i	$-0.322913\pi$
$647$	−8.69716e7	−	8.69716e7i	−0.321118	−	0.321118i	−0.849196	+	0.528078i	$0.822913\pi$
$648$		0			0
$649$			3.94567e8i			1.44340i
$650$		0			0
$651$	−9.71353e7			−0.352074
$652$		0			0
$653$	2.33636e7	−	2.33636e7i	0.0839074	−	0.0839074i	−0.663907	−	0.747815i	$-0.731103\pi$
$653$	2.33636e7	−	2.33636e7i	0.0839074	−	0.0839074i	0.747815	+	0.663907i	$0.231103\pi$
$654$		0			0
$655$	−2.44856e7	−	1.35712e8i	−0.0871339	−	0.482942i
$656$		0			0
$657$	−7.02770e7	−	7.02770e7i	−0.247809	−	0.247809i
$658$		0			0
$659$		−	4.40867e8i		−	1.54046i	−0.637764	−	0.770232i	$-0.720141\pi$
$659$		−	4.40867e8i		−	1.54046i	0.637764	−	0.770232i	$-0.279859\pi$
$660$		0			0
$661$	2.14414e8			0.742419			0.371209	−	0.928549i	$-0.378943\pi$
$661$	2.14414e8			0.742419			0.371209	+	0.928549i	$0.378943\pi$
$662$		0			0
$663$	2.70870e7	−	2.70870e7i	0.0929439	−	0.0929439i
$664$		0			0
$665$	−5.90298e8	−	4.09848e8i	−2.00727	−	1.39366i
$666$		0			0
$667$	−2.37597e7	−	2.37597e7i	−0.0800689	−	0.0800689i
$668$		0			0
$669$		−	1.14284e8i		−	0.381688i
$670$		0			0
$671$	3.45385e8			1.14324
$672$		0			0
$673$	−7.41496e7	+	7.41496e7i	−0.243256	+	0.243256i	−0.818196	−	0.574940i	$-0.805026\pi$
$673$	−7.41496e7	+	7.41496e7i	−0.243256	+	0.243256i	0.574940	+	0.818196i	$0.305026\pi$
$674$		0			0
$675$	3.02072e8	+	1.37949e8i	0.982198	+	0.448547i
$676$		0			0
$677$	−1.19427e8	−	1.19427e8i	−0.384890	−	0.384890i	0.487970	−	0.872860i	$-0.337737\pi$
$677$	−1.19427e8	−	1.19427e8i	−0.384890	−	0.384890i	−0.872860	+	0.487970i	$0.837737\pi$
$678$		0			0
$679$		−	4.38646e8i		−	1.40121i
$680$		0			0
$681$	−2.56844e8			−0.813256
$682$		0			0
$683$	−3.67224e8	+	3.67224e8i	−1.15257	+	1.15257i	−0.166540	+	0.986035i	$0.553259\pi$
$683$	−3.67224e8	+	3.67224e8i	−1.15257	+	1.15257i	−0.986035	+	0.166540i	$0.946741\pi$
$684$		0			0
$685$	7.26521e7	−	1.04640e8i	0.226036	−	0.325555i
$686$		0			0
$687$	−1.27875e8	−	1.27875e8i	−0.394379	−	0.394379i
$688$		0			0
$689$		−	2.00097e8i		−	0.611764i
$690$		0			0
$691$	−1.40286e8			−0.425187			−0.212594	−	0.977141i	$-0.568191\pi$
$691$	−1.40286e8			−0.425187			−0.212594	+	0.977141i	$0.568191\pi$
$692$		0			0
$693$	−9.73106e7	+	9.73106e7i	−0.292389	+	0.292389i
$694$		0			0
$695$	−3.56396e8	+	6.43021e7i	−1.06164	+	0.191545i
$696$		0			0
$697$	−8.59095e7	−	8.59095e7i	−0.253713	−	0.253713i
$698$		0			0
$699$			4.42502e7i			0.129564i
$700$		0			0
$701$	−5.48014e8			−1.59088			−0.795440	−	0.606032i	$-0.792760\pi$
$701$	−5.48014e8			−1.59088			−0.795440	+	0.606032i	$0.792760\pi$
$702$		0			0
$703$	4.08111e8	−	4.08111e8i	1.17466	−	1.17466i
$704$		0			0
$705$	−2.34645e7	−	1.30052e8i	−0.0669643	−	0.371151i
$706$		0			0
$707$	8.31702e8	+	8.31702e8i	2.35348	+	2.35348i
$708$		0			0
$709$		−	6.10430e6i		−	0.0171276i	−0.999963	−	0.00856382i	$-0.997274\pi$
$709$		−	6.10430e6i		−	0.0171276i	0.999963	−	0.00856382i	$-0.00272598\pi$
$710$		0			0
$711$	2.46382e7			0.0685489
$712$		0			0
$713$	−6.10278e6	+	6.10278e6i	−0.0168368	+	0.0168368i
$714$		0			0
$715$	2.46635e8	+	1.71241e8i	0.674740	+	0.468477i
$716$		0			0
$717$	1.74870e8	+	1.74870e8i	0.474415	+	0.474415i
$718$		0			0
$719$			2.94556e7i			0.0792467i	0.999215	+	0.0396233i	$0.0126158\pi$
$719$			2.94556e7i			0.0792467i	−0.999215	+	0.0396233i	$0.987384\pi$
$720$		0			0
$721$	7.28061e8			1.94250
$722$		0			0
$723$	1.24142e8	−	1.24142e8i	0.328476	−	0.328476i
$724$		0			0
$725$	1.46282e8	+	3.92190e8i	0.383865	+	1.02916i
$726$		0			0
$727$	−5.16355e8	−	5.16355e8i	−1.34383	−	1.34383i	−0.892207	−	0.451627i	$-0.850844\pi$
$727$	−5.16355e8	−	5.16355e8i	−1.34383	−	1.34383i	−0.451627	−	0.892207i	$-0.649156\pi$
$728$		0			0
$729$			4.30739e8i			1.11181i
$730$		0			0
$731$	−2.70872e7			−0.0693444
$732$		0			0
$733$	−1.26280e8	+	1.26280e8i	−0.320643	+	0.320643i	−0.849014	−	0.528371i	$-0.822803\pi$
$733$	−1.26280e8	+	1.26280e8i	−0.320643	+	0.320643i	0.528371	+	0.849014i	$0.322803\pi$
$734$		0			0
$735$	−4.02251e8	+	5.79357e8i	−1.01306	+	1.45910i
$736$		0			0
$737$	−1.54608e7	−	1.54608e7i	−0.0386216	−	0.0386216i
$738$		0			0
$739$		−	6.71482e8i		−	1.66380i	−0.554927	−	0.831899i	$-0.687254\pi$
$739$		−	6.71482e8i		−	1.66380i	0.554927	−	0.831899i	$-0.312746\pi$
$740$		0			0
$741$	−4.02423e8			−0.989072
$742$		0			0
$743$	3.38189e8	−	3.38189e8i	0.824505	−	0.824505i	−0.162246	−	0.986750i	$-0.551874\pi$
$743$	3.38189e8	−	3.38189e8i	0.824505	−	0.824505i	0.986750	+	0.162246i	$0.0518737\pi$
$744$		0			0
$745$	2.44038e8	−	4.40301e7i	0.590185	−	0.106483i
$746$		0			0
$747$	−1.07037e7	−	1.07037e7i	−0.0256787	−	0.0256787i
$748$		0			0
$749$			1.08228e9i			2.57571i
$750$		0			0
$751$	7.72625e8			1.82410			0.912050	−	0.410078i	$-0.134499\pi$
$751$	7.72625e8			1.82410			0.912050	+	0.410078i	$0.134499\pi$
$752$		0			0
$753$	−2.10392e8	+	2.10392e8i	−0.492769	+	0.492769i
$754$		0			0
$755$	−8.92575e7	−	4.94711e8i	−0.207398	−	1.14951i
$756$		0			0
$757$	−6.84105e7	−	6.84105e7i	−0.157701	−	0.157701i	0.623846	−	0.781547i	$-0.285569\pi$
$757$	−6.84105e7	−	6.84105e7i	−0.157701	−	0.157701i	−0.781547	+	0.623846i	$0.785569\pi$
$758$		0			0
$759$		−	4.03454e7i		−	0.0922718i
$760$		0			0
$761$	7.07244e8			1.60478			0.802389	−	0.596801i	$-0.203562\pi$
$761$	7.07244e8			1.60478			0.802389	+	0.596801i	$0.203562\pi$
$762$		0			0
$763$	7.85276e7	−	7.85276e7i	0.176786	−	0.176786i
$764$		0			0
$765$	−1.59632e7	−	1.10834e7i	−0.0356564	−	0.0247565i
$766$		0			0
$767$	−3.62363e8	−	3.62363e8i	−0.803079	−	0.803079i
$768$		0			0
$769$			1.14644e8i			0.252100i	0.992024	+	0.126050i	$0.0402300\pi$
$769$			1.14644e8i			0.252100i	−0.992024	+	0.126050i	$0.959770\pi$
$770$		0			0
$771$	1.12234e8			0.244885
$772$		0			0
$773$	6.12226e8	−	6.12226e8i	1.32548	−	1.32548i	0.416215	−	0.909266i	$-0.363356\pi$
$773$	6.12226e8	−	6.12226e8i	1.32548	−	1.32548i	0.909266	−	0.416215i	$-0.136644\pi$
$774$		0			0
$775$	1.00736e8	−	3.75732e7i	0.216410	−	0.0807185i
$776$		0			0
$777$	−5.98084e8	−	5.98084e8i	−1.27497	−	1.27497i
$778$		0			0
$779$			1.27633e9i			2.69991i
$780$		0			0
$781$	−2.44461e8			−0.513164
$782$		0			0
$783$	−4.02596e8	+	4.02596e8i	−0.838657	+	0.838657i
$784$		0			0
$785$	−3.99483e8	+	5.75370e8i	−0.825828	+	1.18943i
$786$		0			0
$787$	−7.26664e7	−	7.26664e7i	−0.149077	−	0.149077i	0.628629	−	0.777705i	$-0.283616\pi$
$787$	−7.26664e7	−	7.26664e7i	−0.149077	−	0.149077i	−0.777705	+	0.628629i	$0.783616\pi$
$788$		0			0
$789$		−	6.60148e8i		−	1.34403i
$790$		0			0
$791$	−8.58043e8			−1.73372
$792$		0			0
$793$	−3.17195e8	+	3.17195e8i	−0.636073	+	0.636073i
$794$		0			0
$795$	3.29622e8	−	5.94716e7i	0.656017	−	0.118361i
$796$		0			0
$797$	−4.52017e8	−	4.52017e8i	−0.892853	−	0.892853i	0.101938	−	0.994791i	$-0.467496\pi$
$797$	−4.52017e8	−	4.52017e8i	−0.892853	−	0.892853i	−0.994791	+	0.101938i	$0.967496\pi$
$798$		0			0
$799$			4.09849e7i			0.0803495i
$800$		0			0
$801$	5.65270e7			0.109991
$802$		0			0
$803$	5.63674e8	−	5.63674e8i	1.08863	−	1.08863i
$804$		0			0
$805$	1.66147e7	+	9.20871e7i	0.0318496	+	0.176527i
$806$		0			0
$807$	−6.39917e7	−	6.39917e7i	−0.121759	−	0.121759i
$808$		0			0
$809$		−	7.32339e8i		−	1.38314i	−0.722309	−	0.691570i	$-0.756919\pi$
$809$		−	7.32339e8i		−	1.38314i	0.722309	−	0.691570i	$-0.243081\pi$
$810$		0			0
$811$	3.74076e8			0.701290			0.350645	−	0.936509i	$-0.385962\pi$
$811$	3.74076e8			0.701290			0.350645	+	0.936509i	$0.385962\pi$
$812$		0			0
$813$	−3.99021e6	+	3.99021e6i	−0.00742548	+	0.00742548i
$814$		0			0
$815$	−4.07484e8	−	2.82919e8i	−0.752727	−	0.522624i
$816$		0			0
$817$	2.01212e8	+	2.01212e8i	0.368968	+	0.368968i
$818$		0			0
$819$		−	1.78737e8i		−	0.325358i
$820$		0			0
$821$	−4.05908e8			−0.733496			−0.366748	−	0.930320i	$-0.619529\pi$
$821$	−4.05908e8			−0.733496			−0.366748	+	0.930320i	$0.619529\pi$
$822$		0			0
$823$	5.03821e8	−	5.03821e8i	0.903809	−	0.903809i	−0.0919544	−	0.995763i	$-0.529311\pi$
$823$	5.03821e8	−	5.03821e8i	0.903809	−	0.903809i	0.995763	+	0.0919544i	$0.0293114\pi$
$824$		0			0
$825$	−2.08783e8	+	4.57179e8i	−0.371820	+	0.814188i
$826$		0			0
$827$	−1.75124e8	−	1.75124e8i	−0.309620	−	0.309620i	0.535142	−	0.844762i	$-0.320258\pi$
$827$	−1.75124e8	−	1.75124e8i	−0.309620	−	0.309620i	−0.844762	+	0.535142i	$0.820258\pi$
$828$		0			0
$829$		−	4.90749e8i		−	0.861381i	−0.902500	−	0.430691i	$-0.858270\pi$
$829$		−	4.90749e8i		−	0.861381i	0.902500	−	0.430691i	$-0.141730\pi$
$830$		0			0
$831$	2.95573e8			0.515064
$832$		0			0
$833$	1.54673e8	−	1.54673e8i	0.267595	−	0.267595i
$834$		0			0
$835$	−4.08859e8	+	5.88873e8i	−0.702286	+	1.01149i
$836$		0			0
$837$	1.03408e8	+	1.03408e8i	0.176352	+	0.176352i
$838$		0			0
$839$			6.07365e8i			1.02840i	0.857669	+	0.514202i	$0.171912\pi$
$839$			6.07365e8i			1.02840i	−0.857669	+	0.514202i	$0.828088\pi$
$840$		0			0
$841$	−1.22843e8			−0.206521
$842$		0			0
$843$	−1.57928e8	+	1.57928e8i	−0.263618	+	0.263618i
$844$		0			0
$845$	2.09995e8	−	3.78880e7i	0.348048	−	0.0627959i
$846$		0			0
$847$	−3.28660e7	−	3.28660e7i	−0.0540874	−	0.0540874i
$848$		0			0
$849$			1.29862e8i			0.212206i
$850$		0			0
$851$	−7.51524e7			−0.121942
$852$		0			0
$853$	−3.41127e8	+	3.41127e8i	−0.549628	+	0.549628i	−0.926333	−	0.376705i	$-0.877057\pi$
$853$	−3.41127e8	+	3.41127e8i	−0.549628	+	0.549628i	0.376705	+	0.926333i	$0.377057\pi$
$854$		0			0
$855$	3.62491e7	+	2.00911e8i	0.0579961	+	0.321445i
$856$		0			0
$857$	7.87515e8	+	7.87515e8i	1.25117	+	1.25117i	0.955196	+	0.295974i	$0.0956441\pi$
$857$	7.87515e8	+	7.87515e8i	1.25117	+	1.25117i	0.295974	+	0.955196i	$0.404356\pi$
$858$		0			0
$859$			2.57007e8i			0.405476i	0.979233	+	0.202738i	$0.0649840\pi$
$859$			2.57007e8i			0.405476i	−0.979233	+	0.202738i	$0.935016\pi$
$860$		0			0
$861$	1.87045e9			2.93047
$862$		0			0
$863$	−3.70540e8	+	3.70540e8i	−0.576504	+	0.576504i	−0.933938	−	0.357434i	$-0.883652\pi$
$863$	−3.70540e8	+	3.70540e8i	−0.576504	+	0.576504i	0.357434	+	0.933938i	$0.383652\pi$
$864$		0			0
$865$	8.62428e7	+	5.98790e7i	0.133252	+	0.0925180i
$866$		0			0
$867$	3.89637e8	+	3.89637e8i	0.597865	+	0.597865i
$868$		0			0
$869$			1.97617e8i			0.301137i
$870$		0			0
$871$	2.83979e7			0.0429765
$872$		0			0
$873$	−8.81160e7	+	8.81160e7i	−0.132438	+	0.132438i
$874$		0			0
$875$	2.88269e8	−	1.12948e9i	0.430302	−	1.68598i
$876$		0			0
$877$	−1.21352e8	−	1.21352e8i	−0.179907	−	0.179907i	0.611408	−	0.791315i	$-0.290603\pi$
$877$	−1.21352e8	−	1.21352e8i	−0.179907	−	0.179907i	−0.791315	+	0.611408i	$0.790603\pi$
$878$		0			0
$879$			1.47523e8i			0.217216i
$880$		0			0
$881$	−4.29574e8			−0.628218			−0.314109	−	0.949387i	$-0.601706\pi$
$881$	−4.29574e8			−0.628218			−0.314109	+	0.949387i	$0.601706\pi$
$882$		0			0
$883$	−8.67016e8	+	8.67016e8i	−1.25935	+	1.25935i	−0.307939	+	0.951406i	$0.599639\pi$
$883$	−8.67016e8	+	8.67016e8i	−1.25935	+	1.25935i	−0.951406	+	0.307939i	$0.900361\pi$
$884$		0			0
$885$	4.89226e8	−	7.04624e8i	0.705796	−	1.01655i
$886$		0			0
$887$	5.00413e8	+	5.00413e8i	0.717064	+	0.717064i	0.968003	−	0.250939i	$-0.0807394\pi$
$887$	5.00413e8	+	5.00413e8i	0.717064	+	0.717064i	−0.250939	+	0.968003i	$0.580739\pi$
$888$		0			0
$889$		−	1.28653e9i		−	1.83111i
$890$		0			0
$891$	−5.15538e8			−0.728832
$892$		0			0
$893$	3.04449e8	−	3.04449e8i	0.427524	−	0.427524i
$894$		0			0
$895$	−4.28917e8	+	7.73867e7i	−0.598280	+	0.107944i
$896$		0			0
$897$	3.70525e7	+	3.70525e7i	0.0513381	+	0.0513381i
$898$		0			0
$899$			1.84336e8i			0.253706i
$900$		0			0
$901$	−1.03878e8			−0.142020
$902$		0			0
$903$	2.94876e8	−	2.94876e8i	0.400475	−	0.400475i
$904$		0			0
$905$	3.13881e7	+	1.73969e8i	0.0423466	+	0.234707i
$906$		0			0
$907$	2.77507e8	+	2.77507e8i	0.371922	+	0.371922i	0.868177	−	0.496255i	$-0.165292\pi$
$907$	2.77507e8	+	2.77507e8i	0.371922	+	0.371922i	−0.496255	+	0.868177i	$0.665292\pi$
$908$		0			0
$909$		−	3.34148e8i		−	0.444884i
$910$		0			0
$911$	7.67626e7			0.101530			0.0507650	−	0.998711i	$-0.483834\pi$
$911$	7.67626e7			0.101530			0.0507650	+	0.998711i	$0.483834\pi$
$912$		0			0
$913$	8.58518e7	−	8.58518e7i	0.112807	−	0.112807i
$914$		0			0
$915$	−6.16794e8	−	4.28244e8i	−0.805149	−	0.559021i
$916$		0			0
$917$	−4.65585e8	−	4.65585e8i	−0.603798	−	0.603798i
$918$		0			0
$919$			7.93161e8i			1.02192i	0.859606	+	0.510958i	$0.170709\pi$
$919$			7.93161e8i			1.02192i	−0.859606	+	0.510958i	$0.829291\pi$
$920$		0			0
$921$	−6.77523e8			−0.867252
$922$		0			0
$923$	2.24508e8	−	2.24508e8i	0.285514	−	0.285514i
$924$		0			0
$925$	8.51599e8	+	3.88905e8i	1.07599	+	0.491381i
$926$		0			0
$927$	−1.46254e8	−	1.46254e8i	−0.183599	−	0.183599i
$928$		0			0
$929$		−	5.48777e8i		−	0.684461i	−0.939616	−	0.342231i	$-0.888818\pi$
$929$		−	5.48777e8i		−	0.684461i	0.939616	−	0.342231i	$-0.111182\pi$
$930$		0			0
$931$	−2.29792e9			−2.84764
$932$		0			0
$933$	−6.87757e8	+	6.87757e8i	−0.846818	+	0.846818i
$934$		0			0
$935$	8.88971e7	−	1.28037e8i	0.108756	−	0.156639i
$936$		0			0
$937$	−5.56840e7	−	5.56840e7i	−0.0676880	−	0.0676880i	0.672452	−	0.740140i	$-0.265241\pi$
$937$	−5.56840e7	−	5.56840e7i	−0.0676880	−	0.0676880i	−0.740140	+	0.672452i	$0.765241\pi$
$938$		0			0
$939$		−	1.25651e9i		−	1.51764i
$940$		0			0
$941$	−8.88360e8			−1.06616			−0.533078	−	0.846066i	$-0.678965\pi$
$941$	−8.88360e8			−1.06616			−0.533078	+	0.846066i	$0.678965\pi$
$942$		0			0
$943$	1.17516e8	−	1.17516e8i	0.140140	−	0.140140i
$944$		0			0
$945$	1.56037e9	−	2.81527e8i	1.84898	−	0.333599i
$946$		0			0
$947$	6.61360e8	+	6.61360e8i	0.778732	+	0.778732i	0.979615	−	0.200883i	$-0.0643811\pi$
$947$	6.61360e8	+	6.61360e8i	0.778732	+	0.778732i	−0.200883	+	0.979615i	$0.564381\pi$
$948$		0			0
$949$			1.03534e9i			1.21139i
$950$		0			0
$951$	6.75346e8			0.785208
$952$		0			0
$953$	2.59406e8	−	2.59406e8i	0.299710	−	0.299710i	−0.541190	−	0.840900i	$-0.682026\pi$
$953$	2.59406e8	−	2.59406e8i	0.299710	−	0.299710i	0.840900	+	0.541190i	$0.182026\pi$
$954$		0			0
$955$	−1.00143e8	−	5.55043e8i	−0.114977	−	0.637260i
$956$		0			0
$957$	−6.09321e8	−	6.09321e8i	−0.695200	−	0.695200i
$958$		0			0
$959$		−	6.08233e8i		−	0.689626i
$960$		0			0
$961$	−8.40156e8			−0.946651
$962$		0			0
$963$	2.17411e8	−	2.17411e8i	0.243446	−	0.243446i
$964$		0			0
$965$	−2.48001e8	−	1.72189e8i	−0.275976	−	0.191612i
$966$		0			0
$967$	5.21710e8	+	5.21710e8i	0.576965	+	0.576965i	0.934066	−	0.357101i	$-0.116235\pi$
$967$	5.21710e8	+	5.21710e8i	0.576965	+	0.576965i	−0.357101	+	0.934066i	$0.616235\pi$
$968$		0			0
$969$			2.08912e8i			0.229611i
$970$		0			0
$971$	4.50595e8			0.492186			0.246093	−	0.969246i	$-0.420853\pi$
$971$	4.50595e8			0.492186			0.246093	+	0.969246i	$0.420853\pi$
$972$		0			0
$973$	−1.22268e9	+	1.22268e9i	−1.32732	+	1.32732i
$974$		0			0
$975$	−2.28123e8	−	6.11608e8i	−0.246124	−	0.659871i
$976$		0			0
$977$	5.75242e8	+	5.75242e8i	0.616832	+	0.616832i	0.944717	−	0.327886i	$-0.106336\pi$
$977$	5.75242e8	+	5.75242e8i	0.616832	+	0.616832i	−0.327886	+	0.944717i	$0.606336\pi$
$978$		0			0
$979$			4.53388e8i			0.483195i
$980$		0			0
$981$	−3.15495e7			−0.0334184
$982$		0			0
$983$	9.06660e8	−	9.06660e8i	0.954518	−	0.954518i	−0.0444920	−	0.999010i	$-0.514167\pi$
$983$	9.06660e8	−	9.06660e8i	0.954518	−	0.954518i	0.999010	+	0.0444920i	$0.0141669\pi$
$984$		0			0
$985$	2.91061e8	−	4.19210e8i	0.304562	−	0.438655i
$986$		0			0
$987$	−4.46168e8	−	4.46168e8i	−0.464031	−	0.464031i
$988$		0			0
$989$		−	3.70527e7i		−	0.0383028i
$990$		0			0
$991$	1.01456e9			1.04245			0.521227	−	0.853418i	$-0.325474\pi$
$991$	1.01456e9			1.04245			0.521227	+	0.853418i	$0.325474\pi$
$992$		0			0
$993$	−8.95868e8	+	8.95868e8i	−0.914948	+	0.914948i
$994$		0			0
$995$	−5.18354e8	+	9.35233e7i	−0.526208	+	0.0949402i
$996$		0			0
$997$	2.49097e8	+	2.49097e8i	0.251353	+	0.251353i	0.821525	−	0.570172i	$-0.193124\pi$
$997$	2.49097e8	+	2.49097e8i	0.251353	+	0.251353i	−0.570172	+	0.821525i	$0.693124\pi$
$998$		0			0
$999$			1.27342e9i			1.27725i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.7.f.a.17.1	yes	6
3.2	odd	2		180.7.l.a.37.2		6
4.3	odd	2		80.7.p.d.17.3		6
5.2	odd	4		100.7.f.b.93.3		6
5.3	odd	4	inner	20.7.f.a.13.1	✓	6
5.4	even	2		100.7.f.b.57.3		6
15.8	even	4		180.7.l.a.73.2		6
20.3	even	4		80.7.p.d.33.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.7.f.a.13.1	✓	6	5.3	odd	4	inner
20.7.f.a.17.1	yes	6	1.1	even	1	trivial
80.7.p.d.17.3		6	4.3	odd	2
80.7.p.d.33.3		6	20.3	even	4
100.7.f.b.57.3		6	5.4	even	2
100.7.f.b.93.3		6	5.2	odd	4
180.7.l.a.37.2		6	3.2	odd	2
180.7.l.a.73.2		6	15.8	even	4

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

\(f(q)\)	\(=\)	\(q+(-16.7249 + 16.7249i) q^{3} +(-22.1946 - 123.014i) q^{5} +(-422.022 - 422.022i) q^{7} +169.553i q^{9} +O(q^{10})\)	\(q+(-16.7249 + 16.7249i) q^{3} +(-22.1946 - 123.014i) q^{5} +(-422.022 - 422.022i) q^{7} +169.553i q^{9} -1359.94 q^{11} +(1248.94 - 1248.94i) q^{13} +(2428.60 + 1686.20i) q^{15} +(-648.372 - 648.372i) q^{17} +9632.63i q^{19} +14116.6 q^{21} +(886.911 - 886.911i) q^{23} +(-14639.8 + 5460.48i) q^{25} +(-15028.2 - 15028.2i) q^{27} -26789.3i q^{29} -6880.94 q^{31} +(22744.9 - 22744.9i) q^{33} +(-42547.9 + 61281.1i) q^{35} +(-42367.5 - 42367.5i) q^{37} +41777.0i q^{39} +132500. q^{41} +(20888.6 - 20888.6i) q^{43} +(20857.4 - 3763.16i) q^{45} +(-31606.0 - 31606.0i) q^{47} +238556. i q^{49} +21688.0 q^{51} +(80106.6 - 80106.6i) q^{53} +(30183.3 + 167291. i) q^{55} +(-161105. - 161105. i) q^{57} -290136. i q^{59} -253971. q^{61} +(71555.1 - 71555.1i) q^{63} +(-181357. - 125918. i) q^{65} +(11368.7 + 11368.7i) q^{67} +29667.0i q^{69} +179758. q^{71} +(-414484. + 414484. i) q^{73} +(153524. - 336176. i) q^{75} +(573924. + 573924. i) q^{77} -145313. i q^{79} +379089. q^{81} +(-63129.1 + 63129.1i) q^{83} +(-65368.3 + 94149.0i) q^{85} +(448049. + 448049. i) q^{87} -333388. i q^{89} -1.05416e6 q^{91} +(115083. - 115083. i) q^{93} +(1.18495e6 - 213792. i) q^{95} +(519696. + 519696. i) q^{97} -230582. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 32 q^{3} - 156 q^{5} - 264 q^{7}+O(q^{10}) \) 6 * q + 32 * q^3 - 156 * q^5 - 264 * q^7	\( 6 q + 32 q^{3} - 156 q^{5} - 264 q^{7} + 2200 q^{11} + 858 q^{13} - 7768 q^{15} - 3278 q^{17} + 33176 q^{21} + 19984 q^{23} - 24174 q^{25} - 115528 q^{27} + 104976 q^{31} + 177320 q^{33} - 116072 q^{35} - 241554 q^{37} + 351736 q^{41} + 60720 q^{43} - 287846 q^{45} - 355248 q^{47} + 641872 q^{51} + 346526 q^{53} - 310200 q^{55} - 112816 q^{57} - 492888 q^{61} + 2288 q^{63} - 5082 q^{65} - 230304 q^{67} + 174128 q^{71} - 332442 q^{73} + 1855048 q^{75} + 1618760 q^{77} - 3085166 q^{81} - 2190936 q^{83} - 164934 q^{85} + 2614304 q^{87} - 2186976 q^{91} + 242072 q^{93} + 3484184 q^{95} + 3338406 q^{97}+O(q^{100}) \) 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