LMFDB - Embedded newform 21.5.b.a.8.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [21,5,Mod(8,21)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(21, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("21.8");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 21 = 3 \cdot 7 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	21.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$2.17076922476$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 82x^{6} + 2017x^{4} + 13020x^{2} + 756 $ x^8 + 82x^6 + 2017x^4 + 13020*x^2 + 756
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2\cdot 3^{3}\cdot 7^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			8.2
Root			$-5.97075i$ of defining polynomial
Character	$\chi$	$=$	21.8
Dual form			21.5.b.a.8.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.97075i q^{2} +(5.29264 - 7.27928i) q^{3} -19.6499 q^{4} +22.9681i q^{5} +(-43.4628 - 31.6011i) q^{6} +18.5203 q^{7} +21.7927i q^{8} +(-24.9759 - 77.0533i) q^{9} +O(q^{10})$	\(q-5.97075i q^{2} +(5.29264 - 7.27928i) q^{3} -19.6499 q^{4} +22.9681i q^{5} +(-43.4628 - 31.6011i) q^{6} +18.5203 q^{7} +21.7927i q^{8} +(-24.9759 - 77.0533i) q^{9} +137.137 q^{10} +105.705i q^{11} +(-104.000 + 143.037i) q^{12} +281.865 q^{13} -110.580i q^{14} +(167.191 + 121.562i) q^{15} -184.280 q^{16} -81.9779i q^{17} +(-460.066 + 149.125i) q^{18} -467.830 q^{19} -451.320i q^{20} +(98.0211 - 134.814i) q^{21} +631.137 q^{22} +625.433i q^{23} +(158.635 + 115.341i) q^{24} +97.4678 q^{25} -1682.94i q^{26} +(-693.081 - 226.009i) q^{27} -363.921 q^{28} +806.557i q^{29} +(725.815 - 998.257i) q^{30} +198.508 q^{31} +1448.97i q^{32} +(769.455 + 559.458i) q^{33} -489.470 q^{34} +425.375i q^{35} +(490.774 + 1514.09i) q^{36} -1824.40 q^{37} +2793.30i q^{38} +(1491.81 - 2051.77i) q^{39} -500.536 q^{40} -580.216i q^{41} +(-804.942 - 585.260i) q^{42} -287.843 q^{43} -2077.09i q^{44} +(1769.76 - 573.648i) q^{45} +3734.31 q^{46} -3078.16i q^{47} +(-975.327 + 1341.42i) q^{48} +343.000 q^{49} -581.956i q^{50} +(-596.740 - 433.880i) q^{51} -5538.61 q^{52} -2036.86i q^{53} +(-1349.44 + 4138.22i) q^{54} -2427.84 q^{55} +403.606i q^{56} +(-2476.06 + 3405.47i) q^{57} +4815.76 q^{58} -439.582i q^{59} +(-3285.29 - 2388.68i) q^{60} +1838.00 q^{61} -1185.24i q^{62} +(-462.560 - 1427.05i) q^{63} +5702.98 q^{64} +6473.89i q^{65} +(3340.38 - 4594.23i) q^{66} -4277.43 q^{67} +1610.86i q^{68} +(4552.70 + 3310.19i) q^{69} +2539.81 q^{70} -5140.35i q^{71} +(1679.20 - 544.291i) q^{72} +3471.16 q^{73} +10893.0i q^{74} +(515.862 - 709.496i) q^{75} +9192.82 q^{76} +1957.68i q^{77} +(-12250.6 - 8907.22i) q^{78} -8637.24 q^{79} -4232.55i q^{80} +(-5313.41 + 3848.95i) q^{81} -3464.33 q^{82} +2129.75i q^{83} +(-1926.10 + 2649.09i) q^{84} +1882.87 q^{85} +1718.64i q^{86} +(5871.16 + 4268.82i) q^{87} -2303.59 q^{88} +1460.07i q^{89} +(-3425.11 - 10566.8i) q^{90} +5220.21 q^{91} -12289.7i q^{92} +(1050.63 - 1445.00i) q^{93} -18378.9 q^{94} -10745.2i q^{95} +(10547.5 + 7668.89i) q^{96} +4820.68 q^{97} -2047.97i q^{98} +(8144.90 - 2640.07i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{3} - 36 q^{4} - 34 q^{6} + 64 q^{9}+O(q^{10}) $ 8 * q - 2 * q^3 - 36 * q^4 - 34 * q^6 + 64 * q^9	$ 8 q - 2 q^{3} - 36 q^{4} - 34 q^{6} + 64 q^{9} - 4 q^{10} + 98 q^{12} + 420 q^{13} + 76 q^{15} - 444 q^{16} - 712 q^{18} - 372 q^{19} + 98 q^{21} - 16 q^{22} + 1146 q^{24} + 1056 q^{25} - 1862 q^{27} + 392 q^{28} + 2348 q^{30} - 2776 q^{31} + 1396 q^{33} + 2928 q^{34} - 3268 q^{36} - 2560 q^{37} - 2540 q^{39} - 1980 q^{40} - 2450 q^{42} + 4720 q^{43} + 9700 q^{45} + 7536 q^{46} - 2962 q^{48} + 2744 q^{49} + 4764 q^{51} - 20252 q^{52} + 4886 q^{54} + 184 q^{55} - 14144 q^{57} - 7504 q^{58} - 13828 q^{60} + 972 q^{61} - 6076 q^{63} + 22772 q^{64} + 36020 q^{66} + 10200 q^{67} - 5760 q^{69} + 10780 q^{70} + 14304 q^{72} - 32008 q^{73} + 2114 q^{75} + 17332 q^{76} - 29668 q^{78} - 23168 q^{79} - 17216 q^{81} - 31976 q^{82} - 14798 q^{84} + 32016 q^{85} + 50764 q^{87} + 29208 q^{88} - 24352 q^{90} + 11956 q^{91} + 31848 q^{93} - 64992 q^{94} + 28630 q^{96} + 28112 q^{97} - 32432 q^{99}+O(q^{100}) $ 8 * q - 2 * q^3 - 36 * q^4 - 34 * q^6 + 64 * q^9 - 4 * q^10 + 98 * q^12 + 420 * q^13 + 76 * q^15 - 444 * q^16 - 712 * q^18 - 372 * q^19 + 98 * q^21 - 16 * q^22 + 1146 * q^24 + 1056 * q^25 - 1862 * q^27 + 392 * q^28 + 2348 * q^30 - 2776 * q^31 + 1396 * q^33 + 2928 * q^34 - 3268 * q^36 - 2560 * q^37 - 2540 * q^39 - 1980 * q^40 - 2450 * q^42 + 4720 * q^43 + 9700 * q^45 + 7536 * q^46 - 2962 * q^48 + 2744 * q^49 + 4764 * q^51 - 20252 * q^52 + 4886 * q^54 + 184 * q^55 - 14144 * q^57 - 7504 * q^58 - 13828 * q^60 + 972 * q^61 - 6076 * q^63 + 22772 * q^64 + 36020 * q^66 + 10200 * q^67 - 5760 * q^69 + 10780 * q^70 + 14304 * q^72 - 32008 * q^73 + 2114 * q^75 + 17332 * q^76 - 29668 * q^78 - 23168 * q^79 - 17216 * q^81 - 31976 * q^82 - 14798 * q^84 + 32016 * q^85 + 50764 * q^87 + 29208 * q^88 - 24352 * q^90 + 11956 * q^91 + 31848 * q^93 - 64992 * q^94 + 28630 * q^96 + 28112 * q^97 - 32432 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/21\mathbb{Z}\right)^\times$.

$n$	$8$	$10$
$\chi(n)$	$-1$	$1$

Coefficient data

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

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

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

$2$		−	5.97075i		−	1.49269i	−0.665560	−	0.746344i	$-0.731807\pi$
$2$		−	5.97075i		−	1.49269i	0.665560	−	0.746344i	$-0.268193\pi$
$3$	5.29264	−	7.27928i	0.588071	−	0.808809i
$3$	5.29264	−	7.27928i	0.588071	−	0.808809i
$4$	−19.6499			−1.22812
$5$			22.9681i			0.918723i	0.888249	+	0.459361i	$0.151922\pi$
$5$			22.9681i			0.918723i	−0.888249	+	0.459361i	$0.848078\pi$
$6$	−43.4628	−	31.6011i	−1.20730	−	0.877807i
$7$	18.5203			0.377964
$7$	18.5203			0.377964
$8$			21.7927i			0.340510i
$9$	−24.9759	−	77.0533i	−0.308344	−	0.951275i
$10$	137.137			1.37137
$11$			105.705i			0.873594i	0.899560	+	0.436797i	$0.143887\pi$
$11$			105.705i			0.873594i	−0.899560	+	0.436797i	$0.856113\pi$
$12$	−104.000	+	143.037i	−0.722221	+	0.993314i
$13$	281.865			1.66784			0.833919	−	0.551886i	$-0.186092\pi$
$13$	281.865			1.66784			0.833919	+	0.551886i	$0.186092\pi$
$14$		−	110.580i		−	0.564183i
$15$	167.191	+	121.562i	0.743071	+	0.540274i
$16$	−184.280			−0.719843
$17$		−	81.9779i		−	0.283661i	−0.989891	−	0.141830i	$-0.954701\pi$
$17$		−	81.9779i		−	0.283661i	0.989891	−	0.141830i	$-0.0452987\pi$
$18$	−460.066	+	149.125i	−1.41996	+	0.460262i
$19$	−467.830			−1.29593			−0.647965	−	0.761670i	$-0.724380\pi$
$19$	−467.830			−1.29593			−0.647965	+	0.761670i	$0.724380\pi$
$20$		−	451.320i		−	1.12830i
$21$	98.0211	−	134.814i	0.222270	−	0.305701i
$22$	631.137			1.30400
$23$			625.433i			1.18229i	0.806564	+	0.591146i	$0.201324\pi$
$23$			625.433i			1.18229i	−0.806564	+	0.591146i	$0.798676\pi$
$24$	158.635	+	115.341i	0.275408	+	0.200244i
$25$	97.4678			0.155948
$26$		−	1682.94i		−	2.48956i
$27$	−693.081	−	226.009i	−0.950728	−	0.310026i
$28$	−363.921			−0.464185
$29$			806.557i			0.959046i	0.877529	+	0.479523i	$0.159190\pi$
$29$			806.557i			0.959046i	−0.877529	+	0.479523i	$0.840810\pi$
$30$	725.815	−	998.257i	0.806461	−	1.10917i
$31$	198.508			0.206564			0.103282	−	0.994652i	$-0.467066\pi$
$31$	198.508			0.206564			0.103282	+	0.994652i	$0.467066\pi$
$32$			1448.97i			1.41501i
$33$	769.455	+	559.458i	0.706570	+	0.513735i
$34$	−489.470			−0.423417
$35$			425.375i			0.347245i
$36$	490.774	+	1514.09i	0.378684	+	1.16828i
$37$	−1824.40			−1.33265			−0.666324	−	0.745662i	$-0.732133\pi$
$37$	−1824.40			−1.33265			−0.666324	+	0.745662i	$0.732133\pi$
$38$			2793.30i			1.93442i
$39$	1491.81	−	2051.77i	0.980808	−	1.34896i
$40$	−500.536			−0.312835
$41$		−	580.216i		−	0.345161i	−0.984995	−	0.172581i	$-0.944789\pi$
$41$		−	580.216i		−	0.345161i	0.984995	−	0.172581i	$-0.0552106\pi$
$42$	−804.942	−	585.260i	−0.456317	−	0.331780i
$43$	−287.843			−0.155675			−0.0778374	−	0.996966i	$-0.524802\pi$
$43$	−287.843			−0.155675			−0.0778374	+	0.996966i	$0.524802\pi$
$44$		−	2077.09i		−	1.07288i
$45$	1769.76	−	573.648i	0.873958	−	0.283283i
$46$	3734.31			1.76479
$47$		−	3078.16i		−	1.39346i	−0.717332	−	0.696732i	$-0.754637\pi$
$47$		−	3078.16i		−	1.39346i	0.717332	−	0.696732i	$-0.245363\pi$
$48$	−975.327	+	1341.42i	−0.423319	+	0.582215i
$49$	343.000			0.142857
$50$		−	581.956i		−	0.232783i
$51$	−596.740	−	433.880i	−0.229427	−	0.166813i
$52$	−5538.61			−2.04830
$53$		−	2036.86i		−	0.725121i	−0.931960	−	0.362560i	$-0.881903\pi$
$53$		−	2036.86i		−	0.725121i	0.931960	−	0.362560i	$-0.118097\pi$
$54$	−1349.44	+	4138.22i	−0.462772	+	1.41914i
$55$	−2427.84			−0.802590
$56$			403.606i			0.128701i
$57$	−2476.06	+	3405.47i	−0.762099	+	1.04816i
$58$	4815.76			1.43156
$59$		−	439.582i		−	0.126280i	−0.998005	−	0.0631402i	$-0.979888\pi$
$59$		−	439.582i		−	0.126280i	0.998005	−	0.0631402i	$-0.0201115\pi$
$60$	−3285.29	−	2388.68i	−0.912580	−	0.663521i
$61$	1838.00			0.493952			0.246976	−	0.969022i	$-0.420563\pi$
$61$	1838.00			0.493952			0.246976	+	0.969022i	$0.420563\pi$
$62$		−	1185.24i		−	0.308336i
$63$	−462.560	−	1427.05i	−0.116543	−	0.359548i
$64$	5702.98			1.39233
$65$			6473.89i			1.53228i
$66$	3340.38	−	4594.23i	0.766847	−	1.05469i
$67$	−4277.43			−0.952869			−0.476435	−	0.879210i	$-0.658071\pi$
$67$	−4277.43			−0.952869			−0.476435	+	0.879210i	$0.658071\pi$
$68$			1610.86i			0.348369i
$69$	4552.70	+	3310.19i	0.956249	+	0.695272i
$70$	2539.81			0.518328
$71$		−	5140.35i		−	1.01971i	−0.860261	−	0.509854i	$-0.829700\pi$
$71$		−	5140.35i		−	1.01971i	0.860261	−	0.509854i	$-0.170300\pi$
$72$	1679.20	−	544.291i	0.323919	−	0.104994i
$73$	3471.16			0.651371			0.325686	−	0.945478i	$-0.394405\pi$
$73$	3471.16			0.651371			0.325686	+	0.945478i	$0.394405\pi$
$74$			10893.0i			1.98923i
$75$	515.862	−	709.496i	0.0917088	−	0.126133i
$76$	9192.82			1.59156
$77$			1957.68i			0.330187i
$78$	−12250.6	−	8907.22i	−2.01358	−	1.46404i
$79$	−8637.24			−1.38395			−0.691975	−	0.721921i	$-0.743259\pi$
$79$	−8637.24			−1.38395			−0.691975	+	0.721921i	$0.743259\pi$
$80$		−	4232.55i		−	0.661336i
$81$	−5313.41	+	3848.95i	−0.809847	+	0.586641i
$82$	−3464.33			−0.515219
$83$			2129.75i			0.309152i	0.987981	+	0.154576i	$0.0494011\pi$
$83$			2129.75i			0.309152i	−0.987981	+	0.154576i	$0.950599\pi$
$84$	−1926.10	+	2649.09i	−0.272974	+	0.375437i
$85$	1882.87			0.260605
$86$			1718.64i			0.232374i
$87$	5871.16	+	4268.82i	0.775685	+	0.563987i
$88$	−2303.59			−0.297468
$89$			1460.07i			0.184330i	0.995744	+	0.0921648i	$0.0293786\pi$
$89$			1460.07i			0.184330i	−0.995744	+	0.0921648i	$0.970621\pi$
$90$	−3425.11	−	10566.8i	−0.422853	−	1.30455i
$91$	5220.21			0.630384
$92$		−	12289.7i		−	1.45200i
$93$	1050.63	−	1445.00i	0.121474	−	0.167071i
$94$	−18378.9			−2.08001
$95$		−	10745.2i		−	1.19060i
$96$	10547.5	+	7668.89i	1.14447	+	0.832128i
$97$	4820.68			0.512347			0.256174	−	0.966631i	$-0.417538\pi$
$97$	4820.68			0.512347			0.256174	+	0.966631i	$0.417538\pi$
$98$		−	2047.97i		−	0.213241i
$99$	8144.90	−	2640.07i	0.831027	−	0.269368i
$100$	−1915.23			−0.191523
$101$		−	4805.28i		−	0.471060i	−0.971867	−	0.235530i	$-0.924317\pi$
$101$		−	4805.28i		−	0.471060i	0.971867	−	0.235530i	$-0.0756826\pi$
$102$	−2590.59	+	3562.99i	−0.248999	+	0.342463i
$103$	3139.96			0.295971			0.147986	−	0.988990i	$-0.452721\pi$
$103$	3139.96			0.295971			0.147986	+	0.988990i	$0.452721\pi$
$104$			6142.58i			0.567916i
$105$	3096.42	+	2251.36i	0.280855	+	0.204205i
$106$	−12161.6			−1.08238
$107$		−	3371.76i		−	0.294503i	−0.989099	−	0.147251i	$-0.952957\pi$
$107$		−	3371.76i		−	0.294503i	0.989099	−	0.147251i	$-0.0470426\pi$
$108$	13619.0	+	4441.05i	1.16761	+	0.380748i
$109$	12039.0			1.01330			0.506648	−	0.862153i	$-0.330884\pi$
$109$	12039.0			1.01330			0.506648	+	0.862153i	$0.330884\pi$
$110$			14496.0i			1.19802i
$111$	−9655.87	+	13280.3i	−0.783692	+	1.07786i
$112$	−3412.91			−0.272075
$113$			17357.5i			1.35935i	0.733515	+	0.679673i	$0.237878\pi$
$113$			17357.5i			1.35935i	−0.733515	+	0.679673i	$0.762122\pi$
$114$	20333.2	+	14783.9i	1.56458	+	1.13758i
$115$	−14365.0			−1.08620
$116$		−	15848.8i		−	1.17782i
$117$	−7039.82	−	21718.6i	−0.514269	−	1.58657i
$118$	−2624.64			−0.188497
$119$		−	1518.25i		−	0.107214i
$120$	−2649.15	+	3643.54i	−0.183969	+	0.253024i
$121$	3467.49			0.236834
$122$		−	10974.2i		−	0.737317i
$123$	−4223.56	−	3070.88i	−0.279170	−	0.202980i
$124$	−3900.67			−0.253685
$125$			16593.7i			1.06200i
$126$	−8520.54	+	2761.83i	−0.536693	+	0.173963i
$127$	349.983			0.0216990			0.0108495	−	0.999941i	$-0.496546\pi$
$127$	349.983			0.0216990			0.0108495	+	0.999941i	$0.496546\pi$
$128$		−	10867.5i		−	0.663301i
$129$	−1523.45	+	2095.29i	−0.0915479	+	0.125911i
$130$	38654.0			2.28722
$131$			14276.5i			0.831916i	0.909384	+	0.415958i	$0.136554\pi$
$131$			14276.5i			0.831916i	−0.909384	+	0.415958i	$0.863446\pi$
$132$	−15119.7	−	10993.3i	−0.867752	−	0.630928i
$133$	−8664.34			−0.489815
$134$			25539.5i			1.42234i
$135$	5190.98	−	15918.7i	0.284828	−	0.873456i
$136$	1786.52			0.0965894
$137$		−	24648.4i		−	1.31325i	−0.754217	−	0.656626i	$-0.771983\pi$
$137$		−	24648.4i		−	1.31325i	0.754217	−	0.656626i	$-0.228017\pi$
$138$	19764.3	−	27183.1i	1.03782	−	1.42738i
$139$	15148.0			0.784016			0.392008	−	0.919962i	$-0.371781\pi$
$139$	15148.0			0.784016			0.392008	+	0.919962i	$0.371781\pi$
$140$		−	8358.57i		−	0.426458i
$141$	−22406.8	−	16291.6i	−1.12705	−	0.819456i
$142$	−30691.7			−1.52211
$143$			29794.5i			1.45701i
$144$	4602.55	+	14199.4i	0.221960	+	0.684768i
$145$	−18525.1			−0.881097
$146$		−	20725.4i		−	0.972294i
$147$	1815.38	−	2496.79i	0.0840102	−	0.115544i
$148$	35849.2			1.63665
$149$		−	26719.6i		−	1.20353i	−0.798672	−	0.601766i	$-0.794464\pi$
$149$		−	26719.6i		−	1.20353i	0.798672	−	0.601766i	$-0.205536\pi$
$150$	−4236.22	−	3080.09i	−0.188277	−	0.136893i
$151$	−31206.4			−1.36864			−0.684321	−	0.729181i	$-0.739901\pi$
$151$	−31206.4			−1.36864			−0.684321	+	0.729181i	$0.739901\pi$
$152$		−	10195.3i		−	0.441277i
$153$	−6316.66	+	2047.47i	−0.269839	+	0.0874651i
$154$	11688.8			0.492867
$155$			4559.35i			0.189775i
$156$	−29313.9	+	40317.1i	−1.20455	+	1.65669i
$157$	21210.9			0.860517			0.430259	−	0.902706i	$-0.358422\pi$
$157$	21210.9			0.860517			0.430259	+	0.902706i	$0.358422\pi$
$158$			51570.8i			2.06581i
$159$	−14826.9	−	10780.4i	−0.586484	−	0.426423i
$160$	−33280.1			−1.30000
$161$			11583.2i			0.446865i
$162$	22981.1	+	31725.1i	0.875672	+	1.20885i
$163$	26992.3			1.01593			0.507965	−	0.861378i	$-0.330398\pi$
$163$	26992.3			1.01593			0.507965	+	0.861378i	$0.330398\pi$
$164$			11401.2i			0.423899i
$165$	−12849.7	+	17672.9i	−0.471980	+	0.649142i
$166$	12716.2			0.461468
$167$		−	50310.5i		−	1.80396i	−0.431783	−	0.901978i	$-0.642115\pi$
$167$		−	50310.5i		−	1.80396i	0.431783	−	0.901978i	$-0.357885\pi$
$168$	2937.96	+	2136.14i	0.104094	+	0.0756853i
$169$	50886.7			1.78169
$170$		−	11242.2i		−	0.389003i
$171$	11684.5	+	36047.9i	0.399593	+	1.23278i
$172$	5656.08			0.191187
$173$			1107.17i			0.0369931i	0.999829	+	0.0184966i	$0.00588797\pi$
$173$			1107.17i			0.0369931i	−0.999829	+	0.0184966i	$0.994112\pi$
$174$	25488.1	−	35055.2i	0.841857	−	1.15786i
$175$	1805.13			0.0589430
$176$		−	19479.3i		−	0.628850i
$177$	−3199.84	−	2326.55i	−0.102137	−	0.0742618i
$178$	8717.74			0.275147
$179$			7449.95i			0.232513i	0.993219	+	0.116256i	$0.0370895\pi$
$179$			7449.95i			0.232513i	−0.993219	+	0.116256i	$0.962911\pi$
$180$	−34775.7	+	11272.1i	−1.07332	+	0.347905i
$181$	19250.5			0.587605			0.293803	−	0.955866i	$-0.405079\pi$
$181$	19250.5			0.587605			0.293803	+	0.955866i	$0.405079\pi$
$182$		−	31168.6i		−	0.940967i
$183$	9727.86	−	13379.3i	0.290479	−	0.399513i
$184$	−13629.8			−0.402583
$185$		−	41902.9i		−	1.22433i
$186$	−8627.72	−	6273.07i	−0.249385	−	0.181324i
$187$	8665.46			0.247804
$188$			60485.6i			1.71134i
$189$	−12836.0	−	4185.74i	−0.359341	−	0.117179i
$190$	−64156.7			−1.77719
$191$			36532.7i			1.00142i	0.865616	+	0.500708i	$0.166927\pi$
$191$			36532.7i			1.00142i	−0.865616	+	0.500708i	$0.833073\pi$
$192$	30183.8	−	41513.6i	0.818788	−	1.12613i
$193$	−65177.2			−1.74977			−0.874886	−	0.484329i	$-0.839064\pi$
$193$	−65177.2			−1.74977			−0.874886	+	0.484329i	$0.839064\pi$
$194$		−	28783.1i		−	0.764775i
$195$	47125.3	+	34264.0i	1.23932	+	0.901091i
$196$	−6739.92			−0.175446
$197$			55856.2i			1.43926i	0.694358	+	0.719629i	$0.255688\pi$
$197$			55856.2i			1.43926i	−0.694358	+	0.719629i	$0.744312\pi$
$198$	−15763.2	−	48631.2i	−0.402082	−	1.24047i
$199$	−10263.9			−0.259184			−0.129592	−	0.991567i	$-0.541367\pi$
$199$	−10263.9			−0.259184			−0.129592	+	0.991567i	$0.541367\pi$
$200$			2124.08i			0.0531021i
$201$	−22638.9	+	31136.6i	−0.560355	+	0.770689i
$202$	−28691.2			−0.703146
$203$			14937.7i			0.362485i
$204$	11725.9	+	8525.69i	0.281764	+	0.204866i
$205$	13326.5			0.317108
$206$		−	18747.9i		−	0.441793i
$207$	48191.6	−	15620.7i	1.12469	−	0.364553i
$208$	−51942.0			−1.20058
$209$		−	49451.9i		−	1.13212i
$210$	13442.3	−	18488.0i	0.304814	−	0.419228i
$211$	−17961.9			−0.403449			−0.201724	−	0.979442i	$-0.564655\pi$
$211$	−17961.9			−0.403449			−0.201724	+	0.979442i	$0.564655\pi$
$212$			40024.2i			0.890534i
$213$	−37418.0	−	27206.0i	−0.824749	−	0.599661i
$214$	−20132.0			−0.439601
$215$		−	6611.19i		−	0.143022i
$216$	4925.33	−	15104.1i	0.105567	−	0.323733i
$217$	3676.42			0.0780739
$218$		−	71881.8i		−	1.51254i
$219$	18371.6	−	25267.5i	0.383053	−	0.526835i
$220$	47706.7			0.985676
$221$		−	23106.7i		−	0.473100i
$222$	79293.4	+	57652.8i	1.60891	+	1.16981i
$223$	−16599.9			−0.333807			−0.166903	−	0.985973i	$-0.553377\pi$
$223$	−16599.9			−0.333807			−0.166903	+	0.985973i	$0.553377\pi$
$224$			26835.3i			0.534824i
$225$	−2434.35	−	7510.21i	−0.0480858	−	0.148350i
$226$	103637.			2.02908
$227$			70168.4i			1.36173i	0.732411	+	0.680863i	$0.238395\pi$
$227$			70168.4i			1.36173i	−0.732411	+	0.680863i	$0.761605\pi$
$228$	48654.3	−	66917.2i	0.935948	−	1.28726i
$229$	54775.6			1.04452			0.522259	−	0.852787i	$-0.325089\pi$
$229$	54775.6			1.04452			0.522259	+	0.852787i	$0.325089\pi$
$230$			85769.8i			1.62136i
$231$	14250.5	+	10361.3i	0.267059	+	0.194174i
$232$	−17577.0			−0.326565
$233$			27616.1i			0.508688i	0.967114	+	0.254344i	$0.0818595\pi$
$233$			27616.1i			0.508688i	−0.967114	+	0.254344i	$0.918141\pi$
$234$	−129676.	+	42033.1i	−2.36826	+	0.767643i
$235$	70699.4			1.28021
$236$			8637.74i			0.155087i
$237$	−45713.8	+	62872.9i	−0.813862	+	1.11935i
$238$	−9065.11			−0.160037
$239$		−	28110.3i		−	0.492118i	−0.969255	−	0.246059i	$-0.920864\pi$
$239$		−	28110.3i		−	0.492118i	0.969255	−	0.246059i	$-0.0791357\pi$
$240$	−30809.9	−	22401.4i	−0.534895	−	0.388913i
$241$	−39152.7			−0.674106			−0.337053	−	0.941486i	$-0.609430\pi$
$241$	−39152.7			−0.674106			−0.337053	+	0.941486i	$0.609430\pi$
$242$		−	20703.5i		−	0.353520i
$243$	−104.384	+	59048.9i	−0.00176775	+	0.999998i
$244$	−36116.5			−0.606632
$245$			7878.05i			0.131246i
$246$	−18335.5	+	25217.8i	−0.302985	+	0.416713i
$247$	−131865.			−2.16140
$248$			4326.02i			0.0703373i
$249$	15503.0	+	11272.0i	0.250045	+	0.181803i
$250$	99076.8			1.58523
$251$		−	48577.1i		−	0.771054i	−0.922697	−	0.385527i	$-0.874020\pi$
$251$		−	48577.1i		−	0.771054i	0.922697	−	0.385527i	$-0.125980\pi$
$252$	9089.26	+	28041.3i	0.143129	+	0.441568i
$253$	−66111.3			−1.03284
$254$		−	2089.66i		−	0.0323898i
$255$	9965.38	−	13706.0i	0.153255	−	0.210780i
$256$	26360.3			0.402226
$257$		−	93467.5i		−	1.41512i	−0.706652	−	0.707562i	$-0.749795\pi$
$257$		−	93467.5i		−	1.41512i	0.706652	−	0.707562i	$-0.250205\pi$
$258$	12510.5	+	9096.14i	0.187946	+	0.136652i
$259$	−33788.3			−0.503694
$260$		−	127211.i		−	1.88182i
$261$	62147.9	−	20144.5i	0.912316	−	0.295716i
$262$	85241.6			1.24179
$263$			16798.9i			0.242868i	0.992600	+	0.121434i	$0.0387492\pi$
$263$			16798.9i			0.242868i	−0.992600	+	0.121434i	$0.961251\pi$
$264$	−12192.1	+	16768.5i	−0.174932	+	0.240595i
$265$	46782.8			0.666185
$266$			51732.7i			0.731142i
$267$	10628.3	+	7727.65i	0.149087	+	0.108399i
$268$	84051.1			1.17024
$269$		−	50281.2i		−	0.694865i	−0.937705	−	0.347433i	$-0.887054\pi$
$269$		−	50281.2i		−	0.694865i	0.937705	−	0.347433i	$-0.112946\pi$
$270$	−95046.8	−	30994.1i	−1.30380	−	0.425159i
$271$	11077.4			0.150834			0.0754171	−	0.997152i	$-0.475971\pi$
$271$	11077.4			0.150834			0.0754171	+	0.997152i	$0.475971\pi$
$272$			15106.9i			0.204191i
$273$	27628.7	−	37999.4i	0.370711	−	0.509860i
$274$	−147170.			−1.96027
$275$			10302.8i			0.136236i
$276$	−89460.1	−	65044.9i	−1.17439	−	0.853877i
$277$	2401.45			0.0312978			0.0156489	−	0.999878i	$-0.495019\pi$
$277$	2401.45			0.0312978			0.0156489	+	0.999878i	$0.495019\pi$
$278$		−	90444.8i		−	1.17029i
$279$	−4957.92	−	15295.7i	−0.0636929	−	0.196499i
$280$	−9270.05			−0.118240
$281$		−	53673.8i		−	0.679751i	−0.940470	−	0.339876i	$-0.889615\pi$
$281$		−	53673.8i		−	0.679751i	0.940470	−	0.339876i	$-0.110385\pi$
$282$	−97273.2	+	133786.i	−1.22319	+	1.68233i
$283$	26258.5			0.327867			0.163933	−	0.986471i	$-0.447582\pi$
$283$	26258.5			0.327867			0.163933	+	0.986471i	$0.447582\pi$
$284$			101007.i			1.25232i
$285$	−78217.1	−	56870.3i	−0.962968	−	0.700157i
$286$	177895.			2.17487
$287$		−	10745.8i		−	0.130459i
$288$	111648.	−	36189.4i	1.34606	−	0.436311i
$289$	76800.6			0.919537
$290$			110609.i			1.31520i
$291$	25514.1	−	35091.1i	0.301297	−	0.414391i
$292$	−68207.9			−0.799961
$293$			72059.1i			0.839370i	0.907670	+	0.419685i	$0.137860\pi$
$293$			72059.1i			0.839370i	−0.907670	+	0.419685i	$0.862140\pi$
$294$	−14907.7	−	10839.2i	−0.172471	−	0.125401i
$295$	10096.3			0.116017
$296$		−	39758.5i		−	0.453781i
$297$	23890.2	−	73262.0i	0.270836	−	0.830550i
$298$	−159536.			−1.79650
$299$			176287.i			1.97187i
$300$	−10136.6	+	13941.5i	−0.112629	+	0.154906i
$301$	−5330.92			−0.0588396
$302$			186326.i			2.04296i
$303$	−34979.0	−	25432.6i	−0.380998	−	0.277017i
$304$	86211.7			0.932865
$305$			42215.2i			0.453805i
$306$	12224.9	+	37715.2i	0.130558	+	0.402786i
$307$	−143206.			−1.51945			−0.759723	−	0.650247i	$-0.774665\pi$
$307$	−143206.			−1.51945			−0.759723	+	0.650247i	$0.774665\pi$
$308$		−	38468.2i		−	0.405509i
$309$	16618.7	−	22856.7i	0.174052	−	0.239384i
$310$	27222.8			0.283275
$311$		−	159322.i		−	1.64723i	−0.567150	−	0.823614i	$-0.691954\pi$
$311$		−	159322.i		−	1.64723i	0.567150	−	0.823614i	$-0.308046\pi$
$312$	44713.6	+	32510.5i	0.459336	+	0.333975i
$313$	65492.5			0.668503			0.334251	−	0.942484i	$-0.391517\pi$
$313$	65492.5			0.668503			0.334251	+	0.942484i	$0.391517\pi$
$314$		−	126645.i		−	1.28448i
$315$	32776.5	−	10624.1i	0.330325	−	0.107071i
$316$	169721.			1.69966
$317$			167671.i			1.66855i	0.551345	+	0.834277i	$0.314115\pi$
$317$			167671.i			1.66855i	−0.551345	+	0.834277i	$0.685885\pi$
$318$	−64367.1	+	88527.8i	−0.636516	+	0.875438i
$319$	−85257.0			−0.837816
$320$			130986.i			1.27916i
$321$	−24544.0	−	17845.5i	−0.238197	−	0.173189i
$322$	69160.3			0.667030
$323$			38351.8i			0.367604i
$324$	104408.	−	75631.5i	0.994589	−	0.720464i
$325$	27472.7			0.260097
$326$		−	161164.i		−	1.51647i
$327$	63718.0	−	87635.1i	0.595891	−	0.819564i
$328$	12644.5			0.117531
$329$		−	57008.3i		−	0.526680i
$330$	105521.	+	76722.2i	0.968967	+	0.704519i
$331$	−126144.			−1.15136			−0.575679	−	0.817676i	$-0.695262\pi$
$331$	−126144.			−1.15136			−0.575679	+	0.817676i	$0.695262\pi$
$332$		−	41849.3i		−	0.379675i
$333$	45565.9	+	140576.i	0.410915	+	1.26772i
$334$	−300392.			−2.69274
$335$		−	98244.3i		−	0.875423i
$336$	−18063.3	+	24843.5i	−0.160000	+	0.220057i
$337$	7098.53			0.0625041			0.0312521	−	0.999512i	$-0.490051\pi$
$337$	7098.53			0.0625041			0.0312521	+	0.999512i	$0.490051\pi$
$338$		−	303832.i		−	2.65950i
$339$	126350.	+	91867.0i	1.09945	+	0.799392i
$340$	−36998.3			−0.320054
$341$			20983.3i			0.180453i
$342$	215233.	−	69765.2i	1.84016	−	0.596467i
$343$	6352.45			0.0539949
$344$		−	6272.86i		−	0.0530089i
$345$	−76028.7	+	104567.i	−0.638762	+	0.878528i
$346$	6610.62			0.0552192
$347$			110747.i			0.919761i	0.887981	+	0.459880i	$0.152108\pi$
$347$			110747.i			0.919761i	−0.887981	+	0.459880i	$0.847892\pi$
$348$	−115368.	−	83881.9i	−0.952633	−	0.692643i
$349$	−102156.			−0.838712			−0.419356	−	0.907822i	$-0.637744\pi$
$349$	−102156.			−0.838712			−0.419356	+	0.907822i	$0.637744\pi$
$350$		−	10778.0i		−	0.0879835i
$351$	−195355.	−	63703.9i	−1.58566	−	0.517073i
$352$	−153163.			−1.23614
$353$			197273.i			1.58313i	0.611084	+	0.791566i	$0.290734\pi$
$353$			197273.i			1.58313i	−0.611084	+	0.791566i	$0.709266\pi$
$354$	−13891.3	+	19105.5i	−0.110850	+	0.152458i
$355$	118064.			0.936828
$356$		−	28690.3i		−	0.226379i
$357$	−11051.8	−	8035.56i	−0.0867153	−	0.0630492i
$358$	44481.8			0.347069
$359$		−	206030.i		−	1.59860i	−0.600930	−	0.799302i	$-0.705203\pi$
$359$		−	206030.i		−	1.59860i	0.600930	−	0.799302i	$-0.294797\pi$
$360$	12501.3	+	38567.9i	0.0964608	+	0.297592i
$361$	88544.3			0.679433
$362$		−	114940.i		−	0.877111i
$363$	18352.2	−	25240.9i	0.139275	−	0.191554i
$364$	−102577.			−0.774186
$365$			79725.8i			0.598429i
$366$	−79884.5	−	58082.6i	−0.596349	−	0.433595i
$367$	25705.6			0.190851			0.0954256	−	0.995437i	$-0.469579\pi$
$367$	25705.6			0.190851			0.0954256	+	0.995437i	$0.469579\pi$
$368$		−	115255.i		−	0.851065i
$369$	−44707.6	+	14491.4i	−0.328343	+	0.106429i
$370$	−250192.			−1.82755
$371$		−	37723.2i		−	0.274070i
$372$	−20644.8	+	28394.0i	−0.149185	+	0.205183i
$373$	74512.2			0.535562			0.267781	−	0.963480i	$-0.413710\pi$
$373$	74512.2			0.535562			0.267781	+	0.963480i	$0.413710\pi$
$374$		−	51739.3i		−	0.369894i
$375$	120790.	+	87824.4i	0.858952	+	0.624529i
$376$	67081.3			0.474489
$377$			227340.i			1.59953i
$378$	−24992.0	+	76640.8i	−0.174911	+	0.536385i
$379$	95609.2			0.665612			0.332806	−	0.942995i	$-0.392005\pi$
$379$	95609.2			0.665612			0.332806	+	0.942995i	$0.392005\pi$
$380$			211141.i			1.46220i
$381$	1852.34	−	2547.63i	0.0127606	−	0.0175503i
$382$	218128.			1.49480
$383$		−	82271.6i		−	0.560857i	−0.959875	−	0.280429i	$-0.909523\pi$
$383$		−	82271.6i		−	0.560857i	0.959875	−	0.280429i	$-0.0904766\pi$
$384$	−79107.8	−	57517.9i	−0.536484	−	0.390068i
$385$	−44964.1			−0.303351
$386$			389157.i			2.61186i
$387$	7189.13	+	22179.2i	0.0480015	+	0.148090i
$388$	−94725.8			−0.629224
$389$		−	160240.i		−	1.05894i	−0.848329	−	0.529470i	$-0.822391\pi$
$389$		−	160240.i		−	1.05894i	0.848329	−	0.529470i	$-0.177609\pi$
$390$	204582.	−	281373.i	1.34505	−	1.84992i
$391$	51271.7			0.335370
$392$			7474.89i			0.0486443i
$393$	103923.	+	75560.5i	0.672862	+	0.489226i
$394$	333504.			2.14836
$395$		−	198381.i		−	1.27147i
$396$	−160046.	+	51877.2i	−1.02060	+	0.330816i
$397$	246394.			1.56332			0.781662	−	0.623702i	$-0.214372\pi$
$397$	246394.			1.56332			0.781662	+	0.623702i	$0.214372\pi$
$398$			61283.4i			0.386880i
$399$	−45857.3	+	63070.2i	−0.288046	+	0.396167i
$400$	−17961.3			−0.112258
$401$			102379.i			0.636681i	0.947976	+	0.318341i	$0.103126\pi$
$401$			102379.i			0.636681i	−0.947976	+	0.318341i	$0.896874\pi$
$402$	185909.	+	135171.i	1.15040	+	0.836436i
$403$	55952.5			0.344516
$404$			94423.4i			0.578518i
$405$	−88402.9	−	122039.i	−0.538960	−	0.744025i
$406$	89189.0			0.541077
$407$		−	192847.i		−	1.16419i
$408$	9455.39	−	13004.6i	0.0568014	−	0.0781224i
$409$	−333297.			−1.99244			−0.996218	−	0.0868875i	$-0.972308\pi$
$409$	−333297.			−1.99244			−0.996218	+	0.0868875i	$0.972308\pi$
$410$		−	79569.0i		−	0.473343i
$411$	−179423.	−	130455.i	−1.06217	−	0.772285i
$412$	−61699.9			−0.363488
$413$		−	8141.17i		−	0.0477295i
$414$	−93267.6	−	287740.i	−0.544165	−	1.67880i
$415$	−48916.2			−0.284025
$416$			408414.i			2.36001i
$417$	80172.8	−	110266.i	0.461057	−	0.634119i
$418$	−295265.			−1.68990
$419$			27105.5i			0.154394i	0.997016	+	0.0771969i	$0.0245970\pi$
$419$			27105.5i			0.154394i	−0.997016	+	0.0771969i	$0.975403\pi$
$420$	−60844.4	−	44238.9i	−0.344923	−	0.250787i
$421$	137329.			0.774817			0.387408	−	0.921908i	$-0.373370\pi$
$421$	137329.			0.774817			0.387408	+	0.921908i	$0.373370\pi$
$422$			107246.i			0.602223i
$423$	−237182.	+	76879.8i	−1.32557	+	0.429667i
$424$	44388.7			0.246911
$425$		−	7990.20i		−	0.0442364i
$426$	−162440.	+	223414.i	−0.895107	+	1.23109i
$427$	34040.2			0.186696
$428$			66254.8i			0.361684i
$429$	216882.	+	157691.i	1.17845	+	0.856827i
$430$	−39473.8			−0.213487
$431$			233453.i			1.25674i	0.777916	+	0.628369i	$0.216277\pi$
$431$			233453.i			1.25674i	−0.777916	+	0.628369i	$0.783723\pi$
$432$	127721.	+	41648.8i	0.684375	+	0.223170i
$433$	321506.			1.71480			0.857400	−	0.514650i	$-0.172078\pi$
$433$	321506.			1.71480			0.857400	+	0.514650i	$0.172078\pi$
$434$		−	21951.0i		−	0.116540i
$435$	−98046.5	+	134849.i	−0.518148	+	0.712639i
$436$	−236565.			−1.24445
$437$		−	292596.i		−	1.53217i
$438$	−150866.	−	109692.i	−0.786400	−	0.571778i
$439$	32218.3			0.167176			0.0835879	−	0.996500i	$-0.473362\pi$
$439$	32218.3			0.167176			0.0835879	+	0.996500i	$0.473362\pi$
$440$		−	52909.0i		−	0.273290i
$441$	−8566.73	−	26429.3i	−0.0440492	−	0.135896i
$442$	−137964.			−0.706191
$443$		−	312740.i		−	1.59359i	−0.604252	−	0.796793i	$-0.706528\pi$
$443$		−	312740.i		−	1.59359i	0.604252	−	0.796793i	$-0.293472\pi$
$444$	189737.	−	260956.i	0.962467	−	1.32374i
$445$	−33535.1			−0.169348
$446$			99113.7i			0.498269i
$447$	−194500.	−	141417.i	−0.973428	−	0.707763i
$448$	105621.			0.526251
$449$		−	167941.i		−	0.833038i	−0.909127	−	0.416519i	$-0.863250\pi$
$449$		−	167941.i		−	0.833038i	0.909127	−	0.416519i	$-0.136750\pi$
$450$	−44841.6	+	14534.9i	−0.221440	+	0.0717772i
$451$	61331.7			0.301531
$452$		−	341073.i		−	1.66944i
$453$	−165164.	+	227160.i	−0.804859	+	1.10697i
$454$	418958.			2.03263
$455$			119898.i			0.579148i
$456$	−74214.3	−	53959.9i	−0.356909	−	0.259503i
$457$	−196337.			−0.940091			−0.470046	−	0.882642i	$-0.655763\pi$
$457$	−196337.			−0.940091			−0.470046	+	0.882642i	$0.655763\pi$
$458$		−	327052.i		−	1.55914i
$459$	−18527.7	+	56817.3i	−0.0879420	+	0.269684i
$460$	282271.			1.33398
$461$		−	237723.i		−	1.11859i	−0.828970	−	0.559293i	$-0.811073\pi$
$461$		−	237723.i		−	1.11859i	0.828970	−	0.559293i	$-0.188927\pi$
$462$	61864.8	−	85086.3i	0.289841	−	0.398635i
$463$	−77983.8			−0.363783			−0.181891	−	0.983319i	$-0.558222\pi$
$463$	−77983.8			−0.363783			−0.181891	+	0.983319i	$0.558222\pi$
$464$		−	148632.i		−	0.690362i
$465$	33188.8	+	24131.0i	0.153492	+	0.111601i
$466$	164889.			0.759312
$467$			352250.i			1.61516i	0.589755	+	0.807582i	$0.299224\pi$
$467$			352250.i			1.61516i	−0.589755	+	0.807582i	$0.700776\pi$
$468$	138332.	+	426768.i	0.631583	+	1.94850i
$469$	−79219.1			−0.360151
$470$		−	422129.i		−	1.91095i
$471$	112262.	−	154400.i	0.506046	−	0.695994i
$472$	9579.66			0.0429998
$473$		−	30426.4i		−	0.135997i
$474$	375398.	+	272946.i	1.67084	+	1.21484i
$475$	−45598.4			−0.202098
$476$			29833.5i			0.131671i
$477$	−156947.	+	50872.5i	−0.689789	+	0.223587i
$478$	−167839.			−0.734579
$479$			422742.i			1.84249i	0.388985	+	0.921244i	$0.372826\pi$
$479$			422742.i			1.84249i	−0.388985	+	0.921244i	$0.627174\pi$
$480$	−176140.	+	242255.i	−0.764495	+	1.05145i
$481$	−514233.			−2.22264
$482$			233771.i			1.00623i
$483$	84317.2	+	61305.6i	0.361428	+	0.262788i
$484$	−68135.9			−0.290861
$485$			110722.i			0.470705i
$486$	352566.	+	623.249i	1.49269	+	0.00263870i
$487$	−248961.			−1.04972			−0.524859	−	0.851189i	$-0.675882\pi$
$487$	−248961.			−1.04972			−0.524859	+	0.851189i	$0.675882\pi$
$488$			40054.9i			0.168196i
$489$	142860.	−	196484.i	0.597439	−	0.821694i
$490$	47037.9			0.195910
$491$			89109.1i			0.369623i	0.982774	+	0.184812i	$0.0591675\pi$
$491$			89109.1i			0.369623i	−0.982774	+	0.184812i	$0.940833\pi$
$492$	82992.5	+	60342.4i	0.342854	+	0.249283i
$493$	66119.9			0.272043
$494$			787333.i			3.22630i
$495$	60637.4	+	187073.i	0.247474	+	0.763484i
$496$	−36581.0			−0.148694
$497$		−	95200.5i		−	0.385413i
$498$	67302.3	−	92564.8i	0.271376	−	0.373239i
$499$	136136.			0.546727			0.273364	−	0.961911i	$-0.411864\pi$
$499$	136136.			0.546727			0.273364	+	0.961911i	$0.411864\pi$
$500$		−	326064.i		−	1.30426i
$501$	−366224.	−	266275.i	−1.45906	−	1.06085i
$502$	−290042.			−1.15094
$503$			19776.0i			0.0781633i	0.999236	+	0.0390817i	$0.0124432\pi$
$503$			19776.0i			0.0781633i	−0.999236	+	0.0390817i	$0.987557\pi$
$504$	31099.1	−	10080.4i	0.122430	−	0.0396842i
$505$	110368.			0.432774
$506$			394734.i			1.54171i
$507$	269325.	−	370419.i	1.04776	−	1.44104i
$508$	−6877.13			−0.0266489
$509$		−	200323.i		−	0.773208i	−0.922246	−	0.386604i	$-0.873648\pi$
$509$		−	200323.i		−	0.773208i	0.922246	−	0.386604i	$-0.126352\pi$
$510$	−81835.0	−	59500.8i	−0.314629	−	0.228761i
$511$	64286.7			0.246195
$512$		−	331271.i		−	1.26370i
$513$	324244.	+	105734.i	1.23208	+	0.401771i
$514$	−558071.			−2.11234
$515$			72118.8i			0.271916i
$516$	29935.6	−	41172.2i	0.112432	−	0.154634i
$517$	325376.			1.21732
$518$			201742.i			0.751858i
$519$	8059.38	+	5859.84i	0.0299204	+	0.0217546i
$520$	−141083.			−0.521758
$521$			224915.i			0.828596i	0.910141	+	0.414298i	$0.135973\pi$
$521$			224915.i			0.828596i	−0.910141	+	0.414298i	$0.864027\pi$
$522$	−120278.	−	371070.i	−0.441412	−	1.36180i
$523$	489710.			1.79034			0.895170	−	0.445725i	$-0.147054\pi$
$523$	489710.			1.79034			0.895170	+	0.445725i	$0.147054\pi$
$524$		−	280532.i		−	1.02169i
$525$	9553.90	−	13140.0i	0.0346627	−	0.0476736i
$526$	100302.			0.362526
$527$		−	16273.3i		−	0.0585941i
$528$	−141795.	−	103097.i	−0.508620	−	0.369809i
$529$	−111325.			−0.397816
$530$		−	279329.i		−	0.994406i
$531$	−33871.2	+	10979.0i	−0.120127	+	0.0389378i
$532$	170253.			0.601551
$533$		−	163543.i		−	0.575674i
$534$	46139.9	−	63458.9i	0.161806	−	0.222541i
$535$	77442.9			0.270566
$536$		−	93216.6i		−	0.324462i
$537$	54230.3	+	39429.9i	0.188059	+	0.136734i
$538$	−300216.			−1.03722
$539$			36256.8i			0.124799i
$540$	−102002.	+	312801.i	−0.349802	+	1.07271i
$541$	227973.			0.778913			0.389457	−	0.921045i	$-0.372663\pi$
$541$	227973.			0.778913			0.389457	+	0.921045i	$0.372663\pi$
$542$		−	66140.5i		−	0.225148i
$543$	101886.	−	140130.i	0.345554	−	0.475260i
$544$	118784.			0.401383
$545$			276512.i			0.930939i
$546$	−226885.	−	164964.i	−0.761062	−	0.553355i
$547$	−61831.3			−0.206649			−0.103325	−	0.994648i	$-0.532948\pi$
$547$	−61831.3			−0.206649			−0.103325	+	0.994648i	$0.532948\pi$
$548$			484339.i			1.61283i
$549$	−45905.6	−	141624.i	−0.152307	−	0.469884i
$550$	61515.6			0.203357
$551$		−	377332.i		−	1.24286i
$552$	−72137.9	+	99215.5i	−0.236747	+	0.325613i
$553$	−159964.			−0.523084
$554$		−	14338.5i		−	0.0467179i
$555$	−305023.	−	221777.i	−0.990253	−	0.719996i
$556$	−297656.			−0.962865
$557$		−	60242.9i		−	0.194176i	−0.995276	−	0.0970880i	$-0.969047\pi$
$557$		−	60242.9i		−	0.194176i	0.995276	−	0.0970880i	$-0.0309528\pi$
$558$	−91326.9	+	29602.5i	−0.293312	+	0.0950737i
$559$	−81132.7			−0.259641
$560$		−	78387.9i		−	0.249962i
$561$	45863.2	−	63078.3i	0.145726	−	0.200426i
$562$	−320473.			−1.01466
$563$			358187.i			1.13004i	0.825077	+	0.565020i	$0.191131\pi$
$563$			358187.i			1.13004i	−0.825077	+	0.565020i	$0.808869\pi$
$564$	440292.	+	320128.i	1.38415	+	1.00639i
$565$	−398668.			−1.24886
$566$		−	156783.i		−	0.489403i
$567$	−98405.7	+	71283.5i	−0.306094	+	0.221729i
$568$	112022.			0.347221
$569$			174591.i			0.539257i	0.962964	+	0.269629i	$0.0869010\pi$
$569$			174591.i			0.539257i	−0.962964	+	0.269629i	$0.913099\pi$
$570$	−339559.	+	467015.i	−1.04512	+	1.43741i
$571$	372462.			1.14238			0.571189	−	0.820818i	$-0.306482\pi$
$571$	372462.			1.14238			0.571189	+	0.820818i	$0.306482\pi$
$572$		−	585458.i		−	1.78939i
$573$	265932.	+	193354.i	0.809955	+	0.588904i
$574$	−64160.3			−0.194734
$575$			60959.6i			0.184377i
$576$	−142437.	−	439433.i	−0.429317	−	1.32449i
$577$	−411312.			−1.23543			−0.617717	−	0.786401i	$-0.711942\pi$
$577$	−411312.			−1.23543			−0.617717	+	0.786401i	$0.711942\pi$
$578$		−	458558.i		−	1.37258i
$579$	−344960.	+	474444.i	−1.02899	+	1.41523i
$580$	364016.			1.08209
$581$			39443.5i			0.116848i
$582$	−209520.	−	152339.i	−0.618557	−	0.449742i
$583$	215306.			0.633461
$584$			75645.8i			0.221799i
$585$	498834.	−	161691.i	1.45762	−	0.472470i
$586$	430247.			1.25292
$587$			400842.i			1.16331i	0.813434	+	0.581657i	$0.197595\pi$
$587$			400842.i			1.16331i	−0.813434	+	0.581657i	$0.802405\pi$
$588$	−35672.0	+	49061.8i	−0.103174	+	0.141902i
$589$	−92868.2			−0.267693
$590$		−	60282.8i		−	0.173177i
$591$	406593.	+	295627.i	1.16409	+	0.846387i
$592$	336199.			0.959298
$593$		−	407416.i		−	1.15859i	−0.815119	−	0.579294i	$-0.803328\pi$
$593$		−	407416.i		−	1.15859i	0.815119	−	0.579294i	$-0.196672\pi$
$594$	−437429.	−	142643.i	−1.23975	−	0.404274i
$595$	34871.3			0.0984996
$596$			525038.i			1.47808i
$597$	−54323.3	+	74714.1i	−0.152418	+	0.209630i
$598$	1.05257e6			2.94339
$599$			351468.i			0.979562i	0.871846	+	0.489781i	$0.162923\pi$
$599$			351468.i			0.979562i	−0.871846	+	0.489781i	$0.837077\pi$
$600$	15461.8	+	11242.0i	0.0429495	+	0.0312278i
$601$	−455907.			−1.26220			−0.631099	−	0.775702i	$-0.717396\pi$
$601$	−455907.			−1.26220			−0.631099	+	0.775702i	$0.717396\pi$
$602$			31829.6i			0.0878291i
$603$	106833.	+	329590.i	0.293812	+	0.906441i
$604$	613203.			1.68085
$605$			79641.6i			0.217585i
$606$	−151852.	+	208851.i	−0.413500	+	0.568711i
$607$	439664.			1.19328			0.596642	−	0.802508i	$-0.296501\pi$
$607$	439664.			1.19328			0.596642	+	0.802508i	$0.296501\pi$
$608$		−	677873.i		−	1.83375i
$609$	108735.	+	79059.6i	0.293181	+	0.213167i
$610$	252057.			0.677390
$611$		−	867625.i		−	2.32407i
$612$	124122.	−	40232.6i	0.331394	−	0.107418i
$613$	−559629.			−1.48929			−0.744645	−	0.667460i	$-0.767381\pi$
$613$	−559629.			−1.48929			−0.744645	+	0.667460i	$0.767381\pi$
$614$			855049.i			2.26806i
$615$	70532.1	−	97007.0i	0.186482	−	0.256480i
$616$	−42663.1			−0.112432
$617$			383951.i			1.00857i	0.863538	+	0.504284i	$0.168244\pi$
$617$			383951.i			1.00857i	−0.863538	+	0.504284i	$0.831756\pi$
$618$	−136471.	−	99226.0i	−0.357326	−	0.259806i
$619$	−315454.			−0.823293			−0.411646	−	0.911344i	$-0.635046\pi$
$619$	−315454.			−0.823293			−0.411646	+	0.911344i	$0.635046\pi$
$620$		−	89590.8i		−	0.233067i
$621$	141353.	−	433475.i	0.366541	−	1.12404i
$622$	−951270.			−2.45880
$623$			27040.9i			0.0696700i
$624$	−274910.	+	378100.i	−0.706028	+	0.971042i
$625$	−320208.			−0.819732
$626$		−	391040.i		−	0.997866i
$627$	−359975.	−	261731.i	−0.915665	−	0.665765i
$628$	−416792.			−1.05682
$629$			149560.i			0.378020i
$630$	−63434.0	−	195700.i	−0.159824	−	0.493072i
$631$	758425.			1.90482			0.952410	−	0.304821i	$-0.0985965\pi$
$631$	758425.			1.90482			0.952410	+	0.304821i	$0.0985965\pi$
$632$		−	188228.i		−	0.471250i
$633$	−95066.1	+	130750.i	−0.237257	+	0.326313i
$634$	1.00112e6			2.49063
$635$			8038.44i			0.0199354i
$636$	291347.	+	211834.i	0.720272	+	0.523698i
$637$	96679.6			0.238263
$638$			509049.i			1.25060i
$639$	−396080.	+	128385.i	−0.970022	+	0.314421i
$640$	249606.			0.609390
$641$		−	333749.i		−	0.812276i	−0.913812	−	0.406138i	$-0.866875\pi$
$641$		−	333749.i		−	0.812276i	0.913812	−	0.406138i	$-0.133125\pi$
$642$	−106551.	+	146546.i	−0.258517	+	0.355553i
$643$	−222406.			−0.537928			−0.268964	−	0.963150i	$-0.586681\pi$
$643$	−222406.			−0.537928			−0.268964	+	0.963150i	$0.586681\pi$
$644$		−	227608.i		−	0.548803i
$645$	−48124.7	−	34990.7i	−0.115678	−	0.0841071i
$646$	228989.			0.548718
$647$		−	75495.6i		−	0.180349i	−0.995926	−	0.0901743i	$-0.971258\pi$
$647$		−	75495.6i		−	0.180349i	0.995926	−	0.0901743i	$-0.0287424\pi$
$648$	−83878.9	−	115793.i	−0.199757	−	0.275762i
$649$	46465.9			0.110318
$650$		−	164033.i		−	0.388244i
$651$	19458.0	−	26761.7i	0.0459130	−	0.0631469i
$652$	−530395.			−1.24768
$653$			134162.i			0.314632i	0.987548	+	0.157316i	$0.0502841\pi$
$653$			134162.i			0.314632i	−0.987548	+	0.157316i	$0.949716\pi$
$654$	−523248.	−	380444.i	−1.22335	−	0.889479i
$655$	−327904.			−0.764300
$656$			106922.i			0.248462i
$657$	−86695.2	−	267464.i	−0.200847	−	0.619633i
$658$	−340383.			−0.786169
$659$		−	323350.i		−	0.744563i	−0.928120	−	0.372282i	$-0.878576\pi$
$659$		−	323350.i		−	0.744563i	0.928120	−	0.372282i	$-0.121424\pi$
$660$	252495.	−	347271.i	0.579648	−	0.797224i
$661$	250103.			0.572422			0.286211	−	0.958167i	$-0.407604\pi$
$661$	250103.			0.572422			0.286211	+	0.958167i	$0.407604\pi$
$662$			753174.i			1.71862i
$663$	−168200.	−	122295.i	−0.382648	−	0.278216i
$664$	−46412.9			−0.105269
$665$		−	199003.i		−	0.450004i
$666$	839343.	−	272063.i	1.89230	−	0.613368i
$667$	−504447.			−1.13387
$668$			988597.i			2.21547i
$669$	−87857.1	+	120835.i	−0.196302	+	0.269986i
$670$	−586593.			−1.30673
$671$			194285.i			0.431514i
$672$	195342.	+	142030.i	0.432571	+	0.314515i
$673$	−267526.			−0.590657			−0.295328	−	0.955396i	$-0.595429\pi$
$673$	−267526.			−0.590657			−0.295328	+	0.955396i	$0.595429\pi$
$674$		−	42383.6i		−	0.0932992i
$675$	−67553.1	−	22028.6i	−0.148265	−	0.0483480i
$676$	−999919.			−2.18812
$677$		−	149336.i		−	0.325828i	−0.986640	−	0.162914i	$-0.947911\pi$
$677$		−	149336.i		−	0.325828i	0.986640	−	0.162914i	$-0.0520893\pi$
$678$	548515.	−	754405.i	1.19324	−	1.64114i
$679$	89280.2			0.193649
$680$			41032.8i			0.0887389i
$681$	510775.	+	371376.i	1.10138	+	0.800792i
$682$	125286.			0.269360
$683$		−	669204.i		−	1.43455i	−0.696788	−	0.717277i	$-0.745388\pi$
$683$		−	669204.i		−	1.43455i	0.696788	−	0.717277i	$-0.254612\pi$
$684$	−229599.	−	708337.i	−0.490747	−	1.51401i
$685$	566126.			1.20651
$686$		−	37928.9i		−	0.0805976i
$687$	289908.	−	398727.i	0.614251	−	0.844816i
$688$	53043.6			0.112061
$689$		−	574120.i		−	1.20938i
$690$	624342.	+	453949.i	1.31137	+	0.953473i
$691$	−404843.			−0.847873			−0.423936	−	0.905692i	$-0.639352\pi$
$691$	−404843.			−0.847873			−0.423936	+	0.905692i	$0.639352\pi$
$692$		−	21755.7i		−	0.0454319i
$693$	150846.	−	48894.8i	0.314099	−	0.101811i
$694$	661246.			1.37292
$695$			347920.i			0.720293i
$696$	−93029.0	+	127948.i	−0.192044	+	0.264129i
$697$	−47564.9			−0.0979087
$698$			609948.i			1.25194i
$699$	201026.	+	146162.i	0.411431	+	0.299145i
$700$	−35470.6			−0.0723890
$701$			144738.i			0.294542i	0.989096	+	0.147271i	$0.0470489\pi$
$701$			144738.i			0.294542i	−0.989096	+	0.147271i	$0.952951\pi$
$702$	−380360.	+	1.16642e6i	−0.771828	+	2.36690i
$703$	853508.			1.72702
$704$			602832.i			1.21633i
$705$	374187.	−	514641.i	0.752853	−	1.03544i
$706$	1.17787e6			2.36312
$707$		−	88995.1i		−	0.178044i
$708$	62876.6	+	45716.5i	0.125436	+	0.0912024i
$709$	−331379.			−0.659223			−0.329611	−	0.944117i	$-0.606918\pi$
$709$	−331379.			−0.659223			−0.329611	+	0.944117i	$0.606918\pi$
$710$		−	704930.i		−	1.39839i
$711$	215723.	+	665527.i	0.426733	+	1.31652i
$712$	−31818.9			−0.0627661
$713$			124154.i			0.244219i
$714$	−47978.4	+	65987.5i	−0.0941129	+	0.129439i
$715$	−684321.			−1.33859
$716$		−	146391.i		−	0.285554i
$717$	−204623.	−	148778.i	−0.398029	−	0.289400i
$718$	−1.23015e6			−2.38622
$719$		−	33141.0i		−	0.0641073i	−0.999486	−	0.0320537i	$-0.989795\pi$
$719$		−	33141.0i		−	0.0641073i	0.999486	−	0.0320537i	$-0.0102047\pi$
$720$	−326132.	+	105712.i	−0.629112	+	0.203919i
$721$	58152.9			0.111867
$722$		−	528677.i		−	1.01418i
$723$	−207221.	+	285004.i	−0.396422	+	0.545223i
$724$	−378271.			−0.721649
$725$			78613.4i			0.149562i
$726$	−150707.	−	109576.i	−0.285930	−	0.207895i
$727$	270597.			0.511982			0.255991	−	0.966679i	$-0.417598\pi$
$727$	270597.			0.511982			0.255991	+	0.966679i	$0.417598\pi$
$728$			113762.i			0.214652i
$729$	429281.	+	313285.i	0.807768	+	0.589500i
$730$	476023.			0.893269
$731$			23596.7i			0.0441588i
$732$	−191151.	+	262902.i	−0.356743	+	0.490650i
$733$	58711.2			0.109273			0.0546365	−	0.998506i	$-0.482600\pi$
$733$	58711.2			0.109273			0.0546365	+	0.998506i	$0.482600\pi$
$734$		−	153482.i		−	0.284881i
$735$	57346.5	+	41695.7i	0.106153	+	0.0771821i
$736$	−906234.			−1.67296
$737$		−	452145.i		−	0.832421i
$738$	86524.7	+	266938.i	0.158865	+	0.490114i
$739$	174370.			0.319288			0.159644	−	0.987175i	$-0.448965\pi$
$739$	174370.			0.319288			0.159644	+	0.987175i	$0.448965\pi$
$740$			823387.i			1.50363i
$741$	−697914.	+	959882.i	−1.27106	+	1.74816i
$742$	−225236.			−0.409101
$743$			190117.i			0.344384i	0.985063	+	0.172192i	$0.0550850\pi$
$743$			190117.i			0.344384i	−0.985063	+	0.172192i	$0.944915\pi$
$744$	31490.3	+	22896.1i	0.0568894	+	0.0413633i
$745$	613698.			1.10571
$746$		−	444894.i		−	0.799427i
$747$	164104.	−	53192.4i	0.294089	−	0.0953253i
$748$	−170275.			−0.304333
$749$		−	62445.9i		−	0.111312i
$750$	524378.	−	721208.i	0.932228	−	1.28215i
$751$	−622793.			−1.10424			−0.552121	−	0.833764i	$-0.686181\pi$
$751$	−622793.			−1.10424			−0.552121	+	0.833764i	$0.686181\pi$
$752$			567243.i			1.00307i
$753$	−353607.	−	257101.i	−0.623635	−	0.453434i
$754$	1.35739e6			2.38761
$755$		−	716751.i		−	1.25740i
$756$	252227.	+	82249.4i	0.441314	+	0.143909i
$757$	−15450.7			−0.0269623			−0.0134811	−	0.999909i	$-0.504291\pi$
$757$	−15450.7			−0.0269623			−0.0134811	+	0.999909i	$0.504291\pi$
$758$		−	570859.i		−	0.993551i
$759$	−349903.	+	481242.i	−0.607385	+	0.835373i
$760$	234166.			0.405412
$761$			548629.i			0.947347i	0.880700	+	0.473674i	$0.157072\pi$
$761$			548629.i			0.947347i	−0.880700	+	0.473674i	$0.842928\pi$
$762$	−15211.2	−	11059.8i	−0.0261972	−	0.0190475i
$763$	222965.			0.382990
$764$		−	717864.i		−	1.22986i
$765$	−47026.5	−	145082.i	−0.0803562	−	0.247907i
$766$	−491224.			−0.837185
$767$		−	123903.i		−	0.210615i
$768$	139516.	−	191884.i	0.236538	−	0.325324i
$769$	−381937.			−0.645861			−0.322931	−	0.946423i	$-0.604668\pi$
$769$	−381937.			−0.645861			−0.322931	+	0.946423i	$0.604668\pi$
$770$			268470.i			0.452808i
$771$	−680376.	−	494690.i	−1.14456	−	0.832193i
$772$	1.28073e6			2.14893
$773$		−	314565.i		−	0.526443i	−0.964735	−	0.263222i	$-0.915215\pi$
$773$		−	314565.i		−	0.526443i	0.964735	−	0.263222i	$-0.0847850\pi$
$774$	132427.	−	42924.5i	0.221052	−	0.0716512i
$775$	19348.2			0.0322134
$776$			105055.i			0.174460i
$777$	−178829.	+	245954.i	−0.296208	+	0.407392i
$778$	−956752.			−1.58067
$779$			271443.i			0.447305i
$780$	−926007.	−	673284.i	−1.52204	−	1.10665i
$781$	543359.			0.890810
$782$		−	306130.i		−	0.500603i
$783$	182289.	−	559009.i	0.297329	−	0.911792i
$784$	−63208.0			−0.102835
$785$			487173.i			0.790577i
$786$	451153.	−	620497.i	0.730262	−	1.00437i
$787$	644572.			1.04069			0.520346	−	0.853956i	$-0.325803\pi$
$787$	644572.			1.04069			0.520346	+	0.853956i	$0.325803\pi$
$788$		−	1.09757e6i		−	1.76758i
$789$	122284.	+	88910.6i	0.196434	+	0.142824i
$790$	−1.18448e6			−1.89790
$791$			321465.i			0.513785i
$792$	57534.2	+	177499.i	0.0917225	+	0.282974i
$793$	518066.			0.823833
$794$		−	1.47116e6i		−	2.33356i
$795$	247605.	−	340545.i	0.391764	−	0.538816i
$796$	201685.			0.318308
$797$			1.10088e6i			1.73310i	0.499089	+	0.866551i	$0.333668\pi$
$797$			1.10088e6i			1.73310i	−0.499089	+	0.866551i	$0.666332\pi$
$798$	376577.	+	273802.i	0.591354	+	0.429963i
$799$	−252341.			−0.395271
$800$			141228.i			0.220669i
$801$	112503.	−	36466.7i	0.175348	−	0.0568370i
$802$	611280.			0.950367
$803$			366918.i			0.569034i
$804$	444852.	−	611832.i	0.688183	−	0.946498i
$805$	−266043.			−0.410545
$806$		−	334078.i		−	0.514255i
$807$	−366011.	−	266120.i	−0.562013	−	0.408630i
$808$	104720.			0.160401
$809$		−	188233.i		−	0.287606i	−0.989606	−	0.143803i	$-0.954067\pi$
$809$		−	188233.i		−	0.287606i	0.989606	−	0.143803i	$-0.0459332\pi$
$810$	−728663.	+	527832.i	−1.11060	+	0.804499i
$811$	−699153.			−1.06299			−0.531497	−	0.847060i	$-0.678370\pi$
$811$	−699153.			−1.06299			−0.531497	+	0.847060i	$0.678370\pi$
$812$		−	293523.i		−	0.445175i
$813$	58628.8	−	80635.6i	0.0887013	−	0.121996i
$814$	−1.15144e6			−1.73778
$815$			619960.i			0.933358i
$816$	109967.	+	79955.2i	0.165152	+	0.120079i
$817$	134662.			0.201744
$818$			1.99003e6i			2.97409i
$819$	−130379.	−	402234.i	−0.194375	−	0.599668i
$820$	−261863.			−0.389446
$821$			742049.i			1.10090i	0.834870	+	0.550448i	$0.185543\pi$
$821$			742049.i			1.10090i	−0.834870	+	0.550448i	$0.814457\pi$
$822$	−778916.	+	1.07129e6i	−1.15278	+	1.58549i
$823$	352649.			0.520647			0.260323	−	0.965522i	$-0.416171\pi$
$823$	352649.			0.520647			0.260323	+	0.965522i	$0.416171\pi$
$824$			68428.1i			0.100781i
$825$	74997.1	+	54529.1i	0.110189	+	0.0801162i
$826$	−48608.9			−0.0712453
$827$			247464.i			0.361827i	0.983499	+	0.180913i	$0.0579054\pi$
$827$			247464.i			0.361827i	−0.983499	+	0.180913i	$0.942095\pi$
$828$	−946961.	+	306946.i	−1.38125	+	0.447715i
$829$	−130730.			−0.190224			−0.0951120	−	0.995467i	$-0.530321\pi$
$829$	−130730.			−0.190224			−0.0951120	+	0.995467i	$0.530321\pi$
$830$			292067.i			0.423961i
$831$	12710.0	−	17480.8i	0.0184053	−	0.0253140i
$832$	1.60747e6			2.32218
$833$		−	28118.4i		−	0.0405229i
$834$	−658373.	−	478692.i	−0.946542	−	0.688215i
$835$	1.15554e6			1.65733
$836$			971726.i			1.39037i
$837$	−137582.	−	44864.6i	−0.196386	−	0.0640402i
$838$	161841.			0.230462
$839$		−	360914.i		−	0.512720i	−0.966581	−	0.256360i	$-0.917477\pi$
$839$		−	360914.i		−	0.512720i	0.966581	−	0.256360i	$-0.0825233\pi$
$840$	−49063.0	+	67479.3i	−0.0695338	+	0.0956339i
$841$	56746.2			0.0802315
$842$		−	819960.i		−	1.15656i
$843$	−390707.	−	284076.i	−0.549789	−	0.399742i
$844$	352950.			0.495483
$845$			1.16877e6i			1.63688i
$846$	459031.	+	1.41616e6i	0.641359	+	1.97866i
$847$	64218.8			0.0895150
$848$			375353.i			0.521973i
$849$	138977.	−	191143.i	0.192809	−	0.265181i
$850$	−47707.5			−0.0660312
$851$		−	1.14104e6i		−	1.57558i
$852$	735261.	+	534595.i	1.01289	+	0.736455i
$853$	911063.			1.25213			0.626067	−	0.779770i	$-0.284664\pi$
$853$	911063.			1.25213			0.626067	+	0.779770i	$0.284664\pi$
$854$		−	203246.i		−	0.278680i
$855$	−827950.	+	268370.i	−1.13259	+	0.367115i
$856$	73479.7			0.100281
$857$		−	498246.i		−	0.678394i	−0.940715	−	0.339197i	$-0.889845\pi$
$857$		−	498246.i		−	0.678394i	0.940715	−	0.339197i	$-0.110155\pi$
$858$	941536.	−	1.29495e6i	1.27898	−	1.75905i
$859$	−245384.			−0.332553			−0.166276	−	0.986079i	$-0.553174\pi$
$859$	−245384.			−0.332553			−0.166276	+	0.986079i	$0.553174\pi$
$860$			129909.i			0.175648i
$861$	−78221.4	−	56873.4i	−0.105516	−	0.0767191i
$862$	1.39389e6			1.87592
$863$			986142.i			1.32409i	0.749463	+	0.662046i	$0.230312\pi$
$863$			986142.i			1.32409i	−0.749463	+	0.662046i	$0.769688\pi$
$864$	327480.	−	1.00425e6i	0.438690	−	1.34529i
$865$	−25429.5			−0.0339864
$866$		−	1.91963e6i		−	2.55966i
$867$	406478.	−	559053.i	0.540753	−	0.743730i
$868$	−72241.3			−0.0958841
$869$		−	912997.i		−	1.20901i
$870$	805151.	+	585412.i	1.06375	+	0.773433i
$871$	−1.20566e6			−1.58923
$872$			262361.i			0.345038i
$873$	−120401.	−	371449.i	−0.157979	−	0.487383i
$874$	−1.74702e6			−2.28705
$875$			307319.i			0.401397i
$876$	−361000.	+	496504.i	−0.470434	+	0.647016i
$877$	1.21508e6			1.57981			0.789904	−	0.613231i	$-0.210130\pi$
$877$	1.21508e6			1.57981			0.789904	+	0.613231i	$0.210130\pi$
$878$		−	192368.i		−	0.249542i
$879$	524539.	+	381383.i	0.678890	+	0.493610i
$880$	447401.			0.577739
$881$		−	1.14816e6i		−	1.47928i	−0.673004	−	0.739639i	$-0.734996\pi$
$881$		−	1.14816e6i		−	1.47928i	0.673004	−	0.739639i	$-0.265004\pi$
$882$	−157803.	+	51149.9i	−0.202851	+	0.0657517i
$883$	266109.			0.341302			0.170651	−	0.985332i	$-0.445413\pi$
$883$	266109.			0.341302			0.170651	+	0.985332i	$0.445413\pi$
$884$			454044.i			0.581023i
$885$	53436.3	−	73494.2i	0.0682260	−	0.0938353i
$886$	−1.86729e6			−2.37873
$887$		−	692218.i		−	0.879824i	−0.898041	−	0.439912i	$-0.855010\pi$
$887$		−	692218.i		−	0.879824i	0.898041	−	0.439912i	$-0.144990\pi$
$888$	−289413.	−	210427.i	−0.367022	−	0.266855i
$889$	6481.78			0.00820145
$890$			200230.i			0.252783i
$891$	−406852.	−	561653.i	−0.512485	−	0.707477i
$892$	326186.			0.409954
$893$			1.44006e6i			1.80583i
$894$	−844368.	+	1.16131e6i	−1.05647	+	1.45302i
$895$	−171111.			−0.213615
$896$		−	201269.i		−	0.250704i
$897$	1.28325e6	+	933026.i	1.59487	+	1.15960i
$898$	−1.00274e6			−1.24347
$899$			160108.i			0.198104i
$900$	47834.7	+	147575.i	0.0590551	+	0.182191i
$901$	−166978.			−0.205688
$902$		−	366196.i		−	0.450092i
$903$	−28214.7	+	38805.3i	−0.0346019	+	0.0475900i
$904$	−378266.			−0.462872
$905$			442147.i			0.539846i
$906$	1.35632e6	+	986155.i	1.65236	+	1.20140i
$907$	−798565.			−0.970724			−0.485362	−	0.874313i	$-0.661312\pi$
$907$	−798565.			−0.970724			−0.485362	+	0.874313i	$0.661312\pi$
$908$		−	1.37880e6i		−	1.67236i
$909$	−370263.	+	120016.i	−0.448108	+	0.145249i
$910$	715882.			0.864487
$911$		−	620184.i		−	0.747281i	−0.927574	−	0.373640i	$-0.878109\pi$
$911$		−	620184.i		−	0.747281i	0.927574	−	0.373640i	$-0.121891\pi$
$912$	456288.	−	627559.i	0.548591	−	0.754510i
$913$	−225125.			−0.270073
$914$			1.17228e6i			1.40326i
$915$	307297.	+	223430.i	0.367042	+	0.266870i
$916$	−1.07634e6			−1.28279
$917$			264405.i			0.314435i
$918$	339242.	+	110624.i	0.402554	+	0.131270i
$919$	−54791.9			−0.0648762			−0.0324381	−	0.999474i	$-0.510327\pi$
$919$	−54791.9			−0.0648762			−0.0324381	+	0.999474i	$0.510327\pi$
$920$		−	313051.i		−	0.369862i
$921$	−757939.	+	1.04244e6i	−0.893542	+	1.22894i
$922$	−1.41939e6			−1.66970
$923$		−	1.44888e6i		−	1.70071i
$924$	−280021.	−	203599.i	−0.327980	−	0.238468i
$925$	−177820.			−0.207825
$926$			465622.i			0.543014i
$927$	−78423.3	−	241944.i	−0.0912611	−	0.281550i
$928$	−1.16868e6			−1.35706
$929$		−	1.12958e6i		−	1.30884i	−0.756133	−	0.654418i	$-0.772914\pi$
$929$		−	1.12958e6i		−	1.30884i	0.756133	−	0.654418i	$-0.227086\pi$
$930$	144080.	−	198162.i	0.166586	−	0.229116i
$931$	−160466.			−0.185133
$932$		−	542655.i		−	0.624729i
$933$	−1.15975e6	−	843232.i	−1.33229	−	0.968688i
$934$	2.10320e6			2.41094
$935$			199029.i			0.227663i
$936$	473306.	−	153417.i	0.540245	−	0.175114i
$937$	−441974.			−0.503406			−0.251703	−	0.967805i	$-0.580991\pi$
$937$	−441974.			−0.503406			−0.251703	+	0.967805i	$0.580991\pi$
$938$			472998.i			0.537593i
$939$	346628.	−	476739.i	0.393127	−	0.540691i
$940$	−1.38924e6			−1.57225
$941$		−	366621.i		−	0.414036i	−0.978337	−	0.207018i	$-0.933624\pi$
$941$		−	366621.i		−	0.414036i	0.978337	−	0.207018i	$-0.0663759\pi$
$942$	−921885.	−	670287.i	−1.03890	−	0.755368i
$943$	362886.			0.408082
$944$			81006.1i			0.0909020i
$945$	96138.3	−	294819.i	0.107655	−	0.330135i
$946$	−181668.			−0.203000
$947$		−	1.49170e6i		−	1.66334i	−0.555272	−	0.831669i	$-0.687386\pi$
$947$		−	1.49170e6i		−	1.66334i	0.555272	−	0.831669i	$-0.312614\pi$
$948$	898272.	−	1.23545e6i	0.999519	−	1.37470i
$949$	978397.			1.08638
$950$			272257.i			0.301670i
$951$	1.22053e6	+	887424.i	1.34954	+	0.981229i
$952$	33086.8			0.0365073
$953$			539755.i			0.594308i	0.954830	+	0.297154i	$0.0960374\pi$
$953$			539755.i			0.594308i	−0.954830	+	0.297154i	$0.903963\pi$
$954$	303747.	+	937092.i	0.333746	+	1.02964i
$955$	−839085.			−0.920024
$956$			552364.i			0.604379i
$957$	−451235.	+	620610.i	−0.492696	+	0.677633i
$958$	2.52409e6			2.75026
$959$		−	456495.i		−	0.496362i
$960$	953487.	+	693264.i	1.03460	+	0.752240i
$961$	−884116.			−0.957331
$962$			3.07036e6i			3.31771i
$963$	−259805.	+	84212.8i	−0.280153	+	0.0908083i
$964$	769347.			0.827882
$965$		−	1.49700e6i		−	1.60756i
$966$	366041.	−	503437.i	0.392261	−	0.539500i
$967$	884115.			0.945488			0.472744	−	0.881200i	$-0.343264\pi$
$967$	884115.			0.945488			0.472744	+	0.881200i	$0.343264\pi$
$968$			75565.9i			0.0806446i
$969$	279173.	+	202982.i	0.297321	+	0.216177i
$970$	661092.			0.702616
$971$			1.49754e6i			1.58833i	0.607703	+	0.794164i	$0.292091\pi$
$971$			1.49754e6i			1.58833i	−0.607703	+	0.794164i	$0.707909\pi$
$972$	2051.13	−	1.16031e6i	0.00217100	−	1.22812i
$973$	280544.			0.296330
$974$			1.48648e6i			1.56690i
$975$	145403.	−	199982.i	0.152956	−	0.210369i
$976$	−338706.			−0.355568
$977$			1.72743e6i			1.80972i	0.425710	+	0.904859i	$0.360024\pi$
$977$			1.72743e6i			1.80972i	−0.425710	+	0.904859i	$0.639976\pi$
$978$	−1.17316e6	−	852984.i	−1.22653	−	0.891791i
$979$	−154337.			−0.161029
$980$		−	154803.i		−	0.161186i
$981$	−300684.	−	927642.i	−0.312444	−	0.963923i
$982$	532049.			0.551732
$983$		−	56276.6i		−	0.0582400i	−0.999576	−	0.0291200i	$-0.990730\pi$
$983$		−	56276.6i		−	0.0582400i	0.999576	−	0.0291200i	$-0.00927049\pi$
$984$	66922.6	−	92042.6i	0.0691166	−	0.0950602i
$985$	−1.28291e6			−1.32228
$986$		−	394785.i		−	0.406076i
$987$	−414980.	−	301725.i	−0.425983	−	0.309725i
$988$	2.59113e6			2.65446
$989$		−	180026.i		−	0.184053i
$990$	1.11696e6	−	362051.i	1.13964	−	0.369402i
$991$	798690.			0.813262			0.406631	−	0.913592i	$-0.366703\pi$
$991$	798690.			0.813262			0.406631	+	0.913592i	$0.366703\pi$
$992$			287633.i			0.292291i
$993$	−667634.	+	918237.i	−0.677080	+	0.931229i
$994$	−568419.			−0.575302
$995$		−	235743.i		−	0.238118i
$996$	−304633.	−	221494.i	−0.307085	−	0.223276i
$997$	−451024.			−0.453743			−0.226871	−	0.973925i	$-0.572850\pi$
$997$	−451024.			−0.453743			−0.226871	+	0.973925i	$0.572850\pi$
$998$		−	812833.i		−	0.816094i
$999$	1.26445e6	+	412329.i	1.26699	+	0.413155i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.5.b.a.8.2	✓	8
3.2	odd	2	inner	21.5.b.a.8.7	yes	8
4.3	odd	2		336.5.d.b.113.4		8
7.2	even	3		147.5.h.e.116.2		16
7.3	odd	6		147.5.h.c.128.7		16
7.4	even	3		147.5.h.e.128.7		16
7.5	odd	6		147.5.h.c.116.2		16
7.6	odd	2		147.5.b.e.50.2		8
12.11	even	2		336.5.d.b.113.3		8
21.2	odd	6		147.5.h.e.116.7		16
21.5	even	6		147.5.h.c.116.7		16
21.11	odd	6		147.5.h.e.128.2		16
21.17	even	6		147.5.h.c.128.2		16
21.20	even	2		147.5.b.e.50.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.5.b.a.8.2	✓	8	1.1	even	1	trivial
21.5.b.a.8.7	yes	8	3.2	odd	2	inner
147.5.b.e.50.2		8	7.6	odd	2
147.5.b.e.50.7		8	21.20	even	2
147.5.h.c.116.2		16	7.5	odd	6
147.5.h.c.116.7		16	21.5	even	6
147.5.h.c.128.2		16	21.17	even	6
147.5.h.c.128.7		16	7.3	odd	6
147.5.h.e.116.2		16	7.2	even	3
147.5.h.e.116.7		16	21.2	odd	6
147.5.h.e.128.2		16	21.11	odd	6
147.5.h.e.128.7		16	7.4	even	3
336.5.d.b.113.3		8	12.11	even	2
336.5.d.b.113.4		8	4.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	21.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.97075i q^{2} +(5.29264 - 7.27928i) q^{3} -19.6499 q^{4} +22.9681i q^{5} +(-43.4628 - 31.6011i) q^{6} +18.5203 q^{7} +21.7927i q^{8} +(-24.9759 - 77.0533i) q^{9} +O(q^{10})\)	\(q-5.97075i q^{2} +(5.29264 - 7.27928i) q^{3} -19.6499 q^{4} +22.9681i q^{5} +(-43.4628 - 31.6011i) q^{6} +18.5203 q^{7} +21.7927i q^{8} +(-24.9759 - 77.0533i) q^{9} +137.137 q^{10} +105.705i q^{11} +(-104.000 + 143.037i) q^{12} +281.865 q^{13} -110.580i q^{14} +(167.191 + 121.562i) q^{15} -184.280 q^{16} -81.9779i q^{17} +(-460.066 + 149.125i) q^{18} -467.830 q^{19} -451.320i q^{20} +(98.0211 - 134.814i) q^{21} +631.137 q^{22} +625.433i q^{23} +(158.635 + 115.341i) q^{24} +97.4678 q^{25} -1682.94i q^{26} +(-693.081 - 226.009i) q^{27} -363.921 q^{28} +806.557i q^{29} +(725.815 - 998.257i) q^{30} +198.508 q^{31} +1448.97i q^{32} +(769.455 + 559.458i) q^{33} -489.470 q^{34} +425.375i q^{35} +(490.774 + 1514.09i) q^{36} -1824.40 q^{37} +2793.30i q^{38} +(1491.81 - 2051.77i) q^{39} -500.536 q^{40} -580.216i q^{41} +(-804.942 - 585.260i) q^{42} -287.843 q^{43} -2077.09i q^{44} +(1769.76 - 573.648i) q^{45} +3734.31 q^{46} -3078.16i q^{47} +(-975.327 + 1341.42i) q^{48} +343.000 q^{49} -581.956i q^{50} +(-596.740 - 433.880i) q^{51} -5538.61 q^{52} -2036.86i q^{53} +(-1349.44 + 4138.22i) q^{54} -2427.84 q^{55} +403.606i q^{56} +(-2476.06 + 3405.47i) q^{57} +4815.76 q^{58} -439.582i q^{59} +(-3285.29 - 2388.68i) q^{60} +1838.00 q^{61} -1185.24i q^{62} +(-462.560 - 1427.05i) q^{63} +5702.98 q^{64} +6473.89i q^{65} +(3340.38 - 4594.23i) q^{66} -4277.43 q^{67} +1610.86i q^{68} +(4552.70 + 3310.19i) q^{69} +2539.81 q^{70} -5140.35i q^{71} +(1679.20 - 544.291i) q^{72} +3471.16 q^{73} +10893.0i q^{74} +(515.862 - 709.496i) q^{75} +9192.82 q^{76} +1957.68i q^{77} +(-12250.6 - 8907.22i) q^{78} -8637.24 q^{79} -4232.55i q^{80} +(-5313.41 + 3848.95i) q^{81} -3464.33 q^{82} +2129.75i q^{83} +(-1926.10 + 2649.09i) q^{84} +1882.87 q^{85} +1718.64i q^{86} +(5871.16 + 4268.82i) q^{87} -2303.59 q^{88} +1460.07i q^{89} +(-3425.11 - 10566.8i) q^{90} +5220.21 q^{91} -12289.7i q^{92} +(1050.63 - 1445.00i) q^{93} -18378.9 q^{94} -10745.2i q^{95} +(10547.5 + 7668.89i) q^{96} +4820.68 q^{97} -2047.97i q^{98} +(8144.90 - 2640.07i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{3} - 36 q^{4} - 34 q^{6} + 64 q^{9}+O(q^{10}) \) 8 * q - 2 * q^3 - 36 * q^4 - 34 * q^6 + 64 * q^9	\( 8 q - 2 q^{3} - 36 q^{4} - 34 q^{6} + 64 q^{9} - 4 q^{10} + 98 q^{12} + 420 q^{13} + 76 q^{15} - 444 q^{16} - 712 q^{18} - 372 q^{19} + 98 q^{21} - 16 q^{22} + 1146 q^{24} + 1056 q^{25} - 1862 q^{27} + 392 q^{28} + 2348 q^{30} - 2776 q^{31} + 1396 q^{33} + 2928 q^{34} - 3268 q^{36} - 2560 q^{37} - 2540 q^{39} - 1980 q^{40} - 2450 q^{42} + 4720 q^{43} + 9700 q^{45} + 7536 q^{46} - 2962 q^{48} + 2744 q^{49} + 4764 q^{51} - 20252 q^{52} + 4886 q^{54} + 184 q^{55} - 14144 q^{57} - 7504 q^{58} - 13828 q^{60} + 972 q^{61} - 6076 q^{63} + 22772 q^{64} + 36020 q^{66} + 10200 q^{67} - 5760 q^{69} + 10780 q^{70} + 14304 q^{72} - 32008 q^{73} + 2114 q^{75} + 17332 q^{76} - 29668 q^{78} - 23168 q^{79} - 17216 q^{81} - 31976 q^{82} - 14798 q^{84} + 32016 q^{85} + 50764 q^{87} + 29208 q^{88} - 24352 q^{90} + 11956 q^{91} + 31848 q^{93} - 64992 q^{94} + 28630 q^{96} + 28112 q^{97} - 32432 q^{99}+O(q^{100}) \) 8 * q - 2 * q^3 - 36 * q^4 - 34 * q^6 + 64 * q^9 - 4 * q^10 + 98 * q^12 + 420 * q^13 + 76 * q^15 - 444 * q^16 - 712 * q^18 - 372 * q^19 + 98 * q^21 - 16 * q^22 + 1146 * q^24 + 1056 * q^25 - 1862 * q^27 + 392 * q^28 + 2348 * q^30 - 2776 * q^31 + 1396 * q^33 + 2928 * q^34 - 3268 * q^36 - 2560 * q^37 - 2540 * q^39 - 1980 * q^40 - 2450 * q^42 + 4720 * q^43 + 9700 * q^45 + 7536 * q^46 - 2962 * q^48 + 2744 * q^49 + 4764 * q^51 - 20252 * q^52 + 4886 * q^54 + 184 * q^55 - 14144 * q^57 - 7504 * q^58 - 13828 * q^60 + 972 * q^61 - 6076 * q^63 + 22772 * q^64 + 36020 * q^66 + 10200 * q^67 - 5760 * q^69 + 10780 * q^70 + 14304 * q^72 - 32008 * q^73 + 2114 * q^75 + 17332 * q^76 - 29668 * q^78 - 23168 * q^79 - 17216 * q^81 - 31976 * q^82 - 14798 * q^84 + 32016 * q^85 + 50764 * q^87 + 29208 * q^88 - 24352 * q^90 + 11956 * q^91 + 31848 * q^93 - 64992 * q^94 + 28630 * q^96 + 28112 * q^97 - 32432 * q^99

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