LMFDB - Embedded newform 168.5.z.a.145.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [168,5,Mod(73,168)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(168, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("168.73");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 168 = 2^{3} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	168.z (of order $6$, 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:	$17.3661537981$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6 x^{15} + 99 x^{14} - 1810 x^{13} + 14212 x^{12} - 199882 x^{11} + 1800935 x^{10} + \cdots + 41390114348800 $ x^16 - 6x^15 + 99x^14 - 1810x^13 + 14212x^12 - 199882x^11 + 1800935x^10 - 9073270x^9 + 62016647x^8 - 13497916x^7 - 872612906x^6 + 8208467240x^5 + 11144877252x^4 - 185824445600x^3 + 3750935164160x^2 - 13970712492800*x + 41390114348800
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{36}\cdot 7^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			145.3
Root			$2.19101 - 3.06642i$ of defining polynomial
Character	$\chi$	$=$	168.145
Dual form			168.5.z.a.73.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.50000 + 2.59808i) q^{3} +(-6.72797 - 3.88439i) q^{5} +(-35.4710 + 33.8055i) q^{7} +(13.5000 - 23.3827i) q^{9} +O(q^{10})$	\(q+(-4.50000 + 2.59808i) q^{3} +(-6.72797 - 3.88439i) q^{5} +(-35.4710 + 33.8055i) q^{7} +(13.5000 - 23.3827i) q^{9} +(6.76101 + 11.7104i) q^{11} -75.5898i q^{13} +40.3678 q^{15} +(117.962 - 68.1055i) q^{17} +(36.5610 + 21.1085i) q^{19} +(71.7902 - 244.281i) q^{21} +(185.061 - 320.535i) q^{23} +(-282.323 - 488.998i) q^{25} +140.296i q^{27} +500.906 q^{29} +(771.688 - 445.534i) q^{31} +(-60.8491 - 35.1313i) q^{33} +(369.961 - 89.6589i) q^{35} +(1143.66 - 1980.87i) q^{37} +(196.388 + 340.154i) q^{39} -1038.61i q^{41} +1936.96 q^{43} +(-181.655 + 104.879i) q^{45} +(668.768 + 386.113i) q^{47} +(115.380 - 2398.23i) q^{49} +(-353.886 + 612.949i) q^{51} +(-370.015 - 640.885i) q^{53} -105.050i q^{55} -219.366 q^{57} +(-2540.86 + 1466.96i) q^{59} +(-3016.80 - 1741.75i) q^{61} +(311.605 + 1285.78i) q^{63} +(-293.621 + 508.566i) q^{65} +(-844.320 - 1462.41i) q^{67} +1923.21i q^{69} +2285.68 q^{71} +(-1222.01 + 705.526i) q^{73} +(2540.91 + 1466.99i) q^{75} +(-635.696 - 186.821i) q^{77} +(365.793 - 633.573i) q^{79} +(-364.500 - 631.333i) q^{81} +10031.8i q^{83} -1058.19 q^{85} +(-2254.08 + 1301.39i) q^{87} +(1956.82 + 1129.77i) q^{89} +(2555.35 + 2681.24i) q^{91} +(-2315.07 + 4009.81i) q^{93} +(-163.988 - 284.035i) q^{95} -7671.75i q^{97} +365.095 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 72 q^{3} + 12 q^{5} + 40 q^{7} + 216 q^{9}+O(q^{10}) $ 16 * q - 72 * q^3 + 12 * q^5 + 40 * q^7 + 216 * q^9	$ 16 q - 72 q^{3} + 12 q^{5} + 40 q^{7} + 216 q^{9} - 108 q^{11} - 72 q^{15} - 168 q^{17} + 948 q^{19} - 252 q^{21} - 768 q^{23} + 1908 q^{25} - 1608 q^{29} - 3216 q^{31} + 972 q^{33} + 696 q^{35} - 1820 q^{37} - 1188 q^{39} + 2888 q^{43} + 324 q^{45} + 744 q^{47} - 3784 q^{49} + 504 q^{51} + 4476 q^{53} - 5688 q^{57} - 4668 q^{59} + 17760 q^{61} + 1188 q^{63} + 8760 q^{65} + 1580 q^{67} + 48 q^{71} + 588 q^{73} - 17172 q^{75} - 17508 q^{77} - 3824 q^{79} - 5832 q^{81} + 11440 q^{85} + 7236 q^{87} - 360 q^{89} + 25860 q^{91} + 9648 q^{93} - 21792 q^{95} - 5832 q^{99}+O(q^{100}) $ 16 * q - 72 * q^3 + 12 * q^5 + 40 * q^7 + 216 * q^9 - 108 * q^11 - 72 * q^15 - 168 * q^17 + 948 * q^19 - 252 * q^21 - 768 * q^23 + 1908 * q^25 - 1608 * q^29 - 3216 * q^31 + 972 * q^33 + 696 * q^35 - 1820 * q^37 - 1188 * q^39 + 2888 * q^43 + 324 * q^45 + 744 * q^47 - 3784 * q^49 + 504 * q^51 + 4476 * q^53 - 5688 * q^57 - 4668 * q^59 + 17760 * q^61 + 1188 * q^63 + 8760 * q^65 + 1580 * q^67 + 48 * q^71 + 588 * q^73 - 17172 * q^75 - 17508 * q^77 - 3824 * q^79 - 5832 * q^81 + 11440 * q^85 + 7236 * q^87 - 360 * q^89 + 25860 * q^91 + 9648 * q^93 - 21792 * q^95 - 5832 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/168\mathbb{Z}\right)^\times$.

$n$	$73$	$85$	$113$	$127$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−4.50000	+	2.59808i	−0.500000	+	0.288675i
$3$	−4.50000	+	2.59808i	−0.500000	+	0.288675i
$4$		0			0
$5$	−6.72797	−	3.88439i	−0.269119	−	0.155376i	0.359368	−	0.933196i	$-0.382992\pi$
$5$	−6.72797	−	3.88439i	−0.269119	−	0.155376i	−0.628487	+	0.777820i	$0.716326\pi$
$6$		0			0
$7$	−35.4710	+	33.8055i	−0.723897	+	0.689908i
$7$	−35.4710	+	33.8055i	−0.723897	+	0.689908i
$8$		0			0
$9$	13.5000	−	23.3827i	0.166667	−	0.288675i
$10$		0			0
$11$	6.76101	+	11.7104i	0.0558762	+	0.0967803i	0.892610	−	0.450829i	$-0.148871\pi$
$11$	6.76101	+	11.7104i	0.0558762	+	0.0967803i	−0.836734	+	0.547609i	$0.815538\pi$
$12$		0			0
$13$		−	75.5898i		−	0.447277i	−0.974672	−	0.223638i	$-0.928207\pi$
$13$		−	75.5898i		−	0.447277i	0.974672	−	0.223638i	$-0.0717935\pi$
$14$		0			0
$15$	40.3678			0.179412
$16$		0			0
$17$	117.962	−	68.1055i	0.408174	−	0.235659i	−0.281831	−	0.959464i	$-0.590942\pi$
$17$	117.962	−	68.1055i	0.408174	−	0.235659i	0.690005	+	0.723805i	$0.257608\pi$
$18$		0			0
$19$	36.5610	+	21.1085i	0.101277	+	0.0584723i	0.549783	−	0.835307i	$-0.314710\pi$
$19$	36.5610	+	21.1085i	0.101277	+	0.0584723i	−0.448506	+	0.893780i	$0.648044\pi$
$20$		0			0
$21$	71.7902	−	244.281i	0.162790	−	0.553925i
$22$		0			0
$23$	185.061	−	320.535i	0.349831	−	0.605926i	−0.636388	−	0.771369i	$-0.719572\pi$
$23$	185.061	−	320.535i	0.349831	−	0.605926i	0.986219	+	0.165443i	$0.0529055\pi$
$24$		0			0
$25$	−282.323	−	488.998i	−0.451717	−	0.782396i
$26$		0			0
$27$			140.296i			0.192450i
$28$		0			0
$29$	500.906			0.595607			0.297804	−	0.954627i	$-0.403746\pi$
$29$	500.906			0.595607			0.297804	+	0.954627i	$0.403746\pi$
$30$		0			0
$31$	771.688	−	445.534i	0.803006	−	0.463616i	−0.0415154	−	0.999138i	$-0.513219\pi$
$31$	771.688	−	445.534i	0.803006	−	0.463616i	0.844521	+	0.535522i	$0.179885\pi$
$32$		0			0
$33$	−60.8491	−	35.1313i	−0.0558762	−	0.0322601i
$34$		0			0
$35$	369.961	−	89.6589i	0.302009	−	0.0731909i
$36$		0			0
$37$	1143.66	−	1980.87i	0.835395	−	1.44695i	−0.0583137	−	0.998298i	$-0.518572\pi$
$37$	1143.66	−	1980.87i	0.835395	−	1.44695i	0.893709	−	0.448648i	$-0.148094\pi$
$38$		0			0
$39$	196.388	+	340.154i	0.129118	+	0.223638i
$40$		0			0
$41$		−	1038.61i		−	0.617853i	−0.951086	−	0.308927i	$-0.900030\pi$
$41$		−	1038.61i		−	0.617853i	0.951086	−	0.308927i	$-0.0999698\pi$
$42$		0			0
$43$	1936.96			1.04757			0.523787	−	0.851849i	$-0.324519\pi$
$43$	1936.96			1.04757			0.523787	+	0.851849i	$0.324519\pi$
$44$		0			0
$45$	−181.655	+	104.879i	−0.0897062	+	0.0517919i
$46$		0			0
$47$	668.768	+	386.113i	0.302747	+	0.174791i	0.643676	−	0.765298i	$-0.277408\pi$
$47$	668.768	+	386.113i	0.302747	+	0.174791i	−0.340929	+	0.940089i	$0.610742\pi$
$48$		0			0
$49$	115.380	−	2398.23i	0.0480551	−	0.998845i
$50$		0			0
$51$	−353.886	+	612.949i	−0.136058	+	0.235659i
$52$		0			0
$53$	−370.015	−	640.885i	−0.131725	−	0.228154i	0.792617	−	0.609720i	$-0.208718\pi$
$53$	−370.015	−	640.885i	−0.131725	−	0.228154i	−0.924342	+	0.381566i	$0.875385\pi$
$54$		0			0
$55$		−	105.050i		−	0.0347272i
$56$		0			0
$57$	−219.366			−0.0675180
$58$		0			0
$59$	−2540.86	+	1466.96i	−0.729922	+	0.421420i	−0.818393	−	0.574658i	$-0.805135\pi$
$59$	−2540.86	+	1466.96i	−0.729922	+	0.421420i	0.0884719	+	0.996079i	$0.471802\pi$
$60$		0			0
$61$	−3016.80	−	1741.75i	−0.810750	−	0.468087i	0.0364661	−	0.999335i	$-0.488390\pi$
$61$	−3016.80	−	1741.75i	−0.810750	−	0.468087i	−0.847216	+	0.531248i	$0.821723\pi$
$62$		0			0
$63$	311.605	+	1285.78i	0.0785096	+	0.323956i
$64$		0			0
$65$	−293.621	+	508.566i	−0.0694960	+	0.120371i
$66$		0			0
$67$	−844.320	−	1462.41i	−0.188086	−	0.325775i	0.756526	−	0.653964i	$-0.226895\pi$
$67$	−844.320	−	1462.41i	−0.188086	−	0.325775i	−0.944612	+	0.328189i	$0.893562\pi$
$68$		0			0
$69$			1923.21i			0.403951i
$70$		0			0
$71$	2285.68			0.453417			0.226709	−	0.973963i	$-0.427203\pi$
$71$	2285.68			0.453417			0.226709	+	0.973963i	$0.427203\pi$
$72$		0			0
$73$	−1222.01	+	705.526i	−0.229313	+	0.132394i	−0.610255	−	0.792205i	$-0.708933\pi$
$73$	−1222.01	+	705.526i	−0.229313	+	0.132394i	0.380942	+	0.924599i	$0.375600\pi$
$74$		0			0
$75$	2540.91	+	1466.99i	0.451717	+	0.260799i
$76$		0			0
$77$	−635.696	−	186.821i	−0.107218	−	0.0315097i
$78$		0			0
$79$	365.793	−	633.573i	0.0586113	−	0.101518i	−0.835231	−	0.549899i	$-0.814666\pi$
$79$	365.793	−	633.573i	0.0586113	−	0.101518i	0.893842	+	0.448382i	$0.147999\pi$
$80$		0			0
$81$	−364.500	−	631.333i	−0.0555556	−	0.0962250i
$82$		0			0
$83$			10031.8i			1.45620i	0.685468	+	0.728102i	$0.259597\pi$
$83$			10031.8i			1.45620i	−0.685468	+	0.728102i	$0.740403\pi$
$84$		0			0
$85$	−1058.19			−0.146463
$86$		0			0
$87$	−2254.08	+	1301.39i	−0.297804	+	0.171937i
$88$		0			0
$89$	1956.82	+	1129.77i	0.247042	+	0.142630i	0.618409	−	0.785856i	$-0.287777\pi$
$89$	1956.82	+	1129.77i	0.247042	+	0.142630i	−0.371367	+	0.928486i	$0.621111\pi$
$90$		0			0
$91$	2555.35	+	2681.24i	0.308580	+	0.323783i
$92$		0			0
$93$	−2315.07	+	4009.81i	−0.267669	+	0.463616i
$94$		0			0
$95$	−163.988	−	284.035i	−0.0181704	−	0.0314720i
$96$		0			0
$97$		−	7671.75i		−	0.815363i	−0.913124	−	0.407681i	$-0.866337\pi$
$97$		−	7671.75i		−	0.815363i	0.913124	−	0.407681i	$-0.133663\pi$
$98$		0			0
$99$	365.095			0.0372508
$100$		0			0
$101$	−12570.9	+	7257.81i	−1.23232	+	0.711480i	−0.967513	−	0.252821i	$-0.918641\pi$
$101$	−12570.9	+	7257.81i	−1.23232	+	0.711480i	−0.264807	+	0.964302i	$0.585308\pi$
$102$		0			0
$103$	−2354.59	−	1359.43i	−0.221943	−	0.128139i	0.384907	−	0.922956i	$-0.374234\pi$
$103$	−2354.59	−	1359.43i	−0.221943	−	0.128139i	−0.606850	+	0.794817i	$0.707567\pi$
$104$		0			0
$105$	−1431.89	+	1364.65i	−0.129876	+	0.123778i
$106$		0			0
$107$	−7840.05	+	13579.4i	−0.684780	+	1.18607i	0.288726	+	0.957412i	$0.406768\pi$
$107$	−7840.05	+	13579.4i	−0.684780	+	1.18607i	−0.973506	+	0.228662i	$0.926565\pi$
$108$		0			0
$109$	−7883.54	−	13654.7i	−0.663541	−	1.14929i	−0.979679	−	0.200574i	$-0.935719\pi$
$109$	−7883.54	−	13654.7i	−0.663541	−	1.14929i	0.316137	−	0.948714i	$-0.397614\pi$
$110$		0			0
$111$			11885.2i			0.964631i
$112$		0			0
$113$	11151.6			0.873337			0.436669	−	0.899622i	$-0.356158\pi$
$113$	11151.6			0.873337			0.436669	+	0.899622i	$0.356158\pi$
$114$		0			0
$115$	−2490.17	+	1437.70i	−0.188292	+	0.108711i
$116$		0			0
$117$	−1767.49	−	1020.46i	−0.129118	−	0.0745462i
$118$		0			0
$119$	−1881.89	+	6403.53i	−0.132893	+	0.452195i
$120$		0			0
$121$	7229.08	−	12521.1i	0.493756	−	0.855210i
$122$		0			0
$123$	2698.39	+	4673.75i	0.178359	+	0.308927i
$124$		0			0
$125$			9242.11i			0.591495i
$126$		0			0
$127$	14759.9			0.915114			0.457557	−	0.889180i	$-0.348725\pi$
$127$	14759.9			0.915114			0.457557	+	0.889180i	$0.348725\pi$
$128$		0			0
$129$	−8716.33	+	5032.38i	−0.523787	+	0.302408i
$130$		0			0
$131$	−13084.4	−	7554.29i	−0.762451	−	0.440201i	0.0677241	−	0.997704i	$-0.478426\pi$
$131$	−13084.4	−	7554.29i	−0.762451	−	0.440201i	−0.830175	+	0.557503i	$0.811760\pi$
$132$		0			0
$133$	−2010.44	+	487.223i	−0.113655	+	0.0275438i
$134$		0			0
$135$	544.965	−	943.908i	0.0299021	−	0.0517919i
$136$		0			0
$137$	6215.33	+	10765.3i	0.331149	+	0.573567i	0.982737	−	0.185006i	$-0.0592304\pi$
$137$	6215.33	+	10765.3i	0.331149	+	0.573567i	−0.651588	+	0.758573i	$0.725897\pi$
$138$		0			0
$139$		−	4634.36i		−	0.239861i	−0.992782	−	0.119931i	$-0.961733\pi$
$139$		−	4634.36i		−	0.239861i	0.992782	−	0.119931i	$-0.0382672\pi$
$140$		0			0
$141$	−4012.61			−0.201831
$142$		0			0
$143$	885.188	−	511.064i	0.0432876	−	0.0249921i
$144$		0			0
$145$	−3370.08	−	1945.72i	−0.160289	−	0.0925430i
$146$		0			0
$147$	5711.56	+	11091.8i	0.264314	+	0.513295i
$148$		0			0
$149$	2515.53	−	4357.03i	0.113307	−	0.196254i	−0.803795	−	0.594907i	$-0.797189\pi$
$149$	2515.53	−	4357.03i	0.113307	−	0.196254i	0.917102	+	0.398653i	$0.130522\pi$
$150$		0			0
$151$	−17177.3	−	29752.0i	−0.753359	−	1.30486i	−0.946186	−	0.323623i	$-0.895099\pi$
$151$	−17177.3	−	29752.0i	−0.753359	−	1.30486i	0.192827	−	0.981233i	$-0.438234\pi$
$152$		0			0
$153$		−	3677.70i		−	0.157106i
$154$		0			0
$155$	−6922.53			−0.288138
$156$		0			0
$157$	−4995.16	+	2883.96i	−0.202652	+	0.117001i	−0.597892	−	0.801577i	$-0.703995\pi$
$157$	−4995.16	+	2883.96i	−0.202652	+	0.117001i	0.395240	+	0.918578i	$0.370661\pi$
$158$		0			0
$159$	3330.13	+	1922.65i	0.131725	+	0.0760514i
$160$		0			0
$161$	4271.54	+	17625.7i	0.164791	+	0.679979i
$162$		0			0
$163$	6124.36	−	10607.7i	0.230508	−	0.399251i	−0.727450	−	0.686161i	$-0.759295\pi$
$163$	6124.36	−	10607.7i	0.230508	−	0.399251i	0.957958	+	0.286910i	$0.0926279\pi$
$164$		0			0
$165$	272.927	+	472.724i	0.0100249	+	0.0173636i
$166$		0			0
$167$		−	51247.9i		−	1.83757i	−0.394762	−	0.918783i	$-0.629173\pi$
$167$		−	51247.9i		−	1.83757i	0.394762	−	0.918783i	$-0.370827\pi$
$168$		0			0
$169$	22847.2			0.799943
$170$		0			0
$171$	987.147	−	569.930i	0.0337590	−	0.0194908i
$172$		0			0
$173$	45699.5	+	26384.6i	1.52693	+	0.881574i	0.999488	+	0.0319829i	$0.0101822\pi$
$173$	45699.5	+	26384.6i	1.52693	+	0.881574i	0.527442	+	0.849591i	$0.323151\pi$
$174$		0			0
$175$	26545.1	+	7801.17i	0.866778	+	0.254732i
$176$		0			0
$177$	7622.57	−	13202.7i	0.243307	−	0.421420i
$178$		0			0
$179$	−24843.8	−	43030.8i	−0.775376	−	1.34299i	−0.934583	−	0.355746i	$-0.884227\pi$
$179$	−24843.8	−	43030.8i	−0.775376	−	1.34299i	0.159207	−	0.987245i	$-0.449106\pi$
$180$		0			0
$181$		−	21949.2i		−	0.669980i	−0.942222	−	0.334990i	$-0.891267\pi$
$181$		−	21949.2i		−	0.669980i	0.942222	−	0.334990i	$-0.108733\pi$
$182$		0			0
$183$	18100.8			0.540500
$184$		0			0
$185$	−15389.0	+	8884.82i	−0.449641	+	0.259600i
$186$		0			0
$187$	1595.09	+	920.924i	0.0456143	+	0.0263354i
$188$		0			0
$189$	−4742.78	−	4976.44i	−0.132773	−	0.139314i
$190$		0			0
$191$	30573.5	−	52954.9i	0.838067	−	1.45157i	−0.0534418	−	0.998571i	$-0.517019\pi$
$191$	30573.5	−	52954.9i	0.838067	−	1.45157i	0.891509	−	0.453003i	$-0.149647\pi$
$192$		0			0
$193$	−7496.00	−	12983.5i	−0.201240	−	0.348559i	0.747688	−	0.664050i	$-0.231164\pi$
$193$	−7496.00	−	12983.5i	−0.201240	−	0.348559i	−0.948928	+	0.315492i	$0.897831\pi$
$194$		0			0
$195$		−	3051.39i		−	0.0802471i
$196$		0			0
$197$	69960.1			1.80268			0.901339	−	0.433114i	$-0.142585\pi$
$197$	69960.1			1.80268			0.901339	+	0.433114i	$0.142585\pi$
$198$		0			0
$199$	20926.9	−	12082.2i	0.528444	−	0.305098i	−0.211938	−	0.977283i	$-0.567978\pi$
$199$	20926.9	−	12082.2i	0.528444	−	0.305098i	0.740383	+	0.672186i	$0.234644\pi$
$200$		0			0
$201$	7598.88	+	4387.22i	0.188086	+	0.108592i
$202$		0			0
$203$	−17767.6	+	16933.4i	−0.431159	+	0.410914i
$204$		0			0
$205$	−4034.38	+	6987.75i	−0.0959995	+	0.166276i
$206$		0			0
$207$	−4996.64	−	8654.44i	−0.116610	−	0.201975i
$208$		0			0
$209$			570.860i			0.0130688i
$210$		0			0
$211$	−10962.5			−0.246232			−0.123116	−	0.992392i	$-0.539289\pi$
$211$	−10962.5			−0.246232			−0.123116	+	0.992392i	$0.539289\pi$
$212$		0			0
$213$	−10285.5	+	5938.36i	−0.226709	+	0.130890i
$214$		0			0
$215$	−13031.8	−	7523.93i	−0.281922	−	0.162768i
$216$		0			0
$217$	−12311.0	+	41890.8i	−0.261442	+	0.889610i
$218$		0			0
$219$	3666.02	−	6349.74i	0.0764376	−	0.132394i
$220$		0			0
$221$	−5148.08	−	8916.74i	−0.105405	−	0.182567i
$222$		0			0
$223$		−	21364.5i		−	0.429618i	−0.976656	−	0.214809i	$-0.931087\pi$
$223$		−	21364.5i		−	0.429618i	0.976656	−	0.214809i	$-0.0689129\pi$
$224$		0			0
$225$	−15245.4			−0.301144
$226$		0			0
$227$	−49907.4	+	28814.1i	−0.968531	+	0.559182i	−0.898788	−	0.438383i	$-0.855551\pi$
$227$	−49907.4	+	28814.1i	−0.968531	+	0.559182i	−0.0697428	+	0.997565i	$0.522218\pi$
$228$		0			0
$229$	30360.2	+	17528.5i	0.578940	+	0.334251i	0.760712	−	0.649090i	$-0.224850\pi$
$229$	30360.2	+	17528.5i	0.578940	+	0.334251i	−0.181772	+	0.983341i	$0.558183\pi$
$230$		0			0
$231$	3346.01	−	810.893i	0.0627051	−	0.0151964i
$232$		0			0
$233$	−4261.59	+	7381.29i	−0.0784982	+	0.135963i	−0.902602	−	0.430476i	$-0.858346\pi$
$233$	−4261.59	+	7381.29i	−0.0784982	+	0.135963i	0.824104	+	0.566439i	$0.191679\pi$
$234$		0			0
$235$	−2999.63	−	5195.52i	−0.0543166	−	0.0940790i
$236$		0			0
$237$			3801.44i			0.0676785i
$238$		0			0
$239$	−26435.0			−0.462789			−0.231395	−	0.972860i	$-0.574329\pi$
$239$	−26435.0			−0.462789			−0.231395	+	0.972860i	$0.574329\pi$
$240$		0			0
$241$	−66270.4	+	38261.2i	−1.14100	+	0.658756i	−0.946678	−	0.322181i	$-0.895584\pi$
$241$	−66270.4	+	38261.2i	−1.14100	+	0.658756i	−0.194322	+	0.980938i	$0.562251\pi$
$242$		0			0
$243$	3280.50	+	1894.00i	0.0555556	+	0.0320750i
$244$		0			0
$245$	−10091.9	+	15687.0i	−0.168129	+	0.261341i
$246$		0			0
$247$	1595.59	−	2763.64i	0.0261533	−	0.0452989i
$248$		0			0
$249$	−26063.4	−	45143.1i	−0.420370	−	0.728102i
$250$		0			0
$251$			32807.4i			0.520743i	0.965508	+	0.260372i	$0.0838451\pi$
$251$			32807.4i			0.520743i	−0.965508	+	0.260372i	$0.916155\pi$
$252$		0			0
$253$	5004.80			0.0781889
$254$		0			0
$255$	4761.87	−	2749.27i	0.0732314	−	0.0422802i
$256$		0			0
$257$	62322.5	+	35981.9i	0.943579	+	0.544776i	0.891080	−	0.453845i	$-0.149948\pi$
$257$	62322.5	+	35981.9i	0.943579	+	0.544776i	0.0524986	+	0.998621i	$0.483281\pi$
$258$		0			0
$259$	26397.6	+	108925.i	0.393519	+	1.62379i
$260$		0			0
$261$	6762.23	−	11712.5i	0.0992679	−	0.171937i
$262$		0			0
$263$	−50349.3	−	87207.6i	−0.727917	−	1.26079i	−0.957762	−	0.287562i	$-0.907155\pi$
$263$	−50349.3	−	87207.6i	−0.727917	−	1.26079i	0.229845	−	0.973227i	$-0.426178\pi$
$264$		0			0
$265$			5749.14i			0.0818674i
$266$		0			0
$267$	−11740.9			−0.164695
$268$		0			0
$269$	−38269.8	+	22095.1i	−0.528874	+	0.305345i	−0.740558	−	0.671993i	$-0.765439\pi$
$269$	−38269.8	+	22095.1i	−0.528874	+	0.305345i	0.211684	+	0.977338i	$0.432105\pi$
$270$		0			0
$271$	32492.4	+	18759.5i	0.442428	+	0.255436i	0.704627	−	0.709578i	$-0.251114\pi$
$271$	32492.4	+	18759.5i	0.442428	+	0.255436i	−0.262199	+	0.965014i	$0.584448\pi$
$272$		0			0
$273$	−18465.1	−	5426.61i	−0.247758	−	0.0728120i
$274$		0			0
$275$	3817.58	−	6612.24i	0.0504804	−	0.0874346i
$276$		0			0
$277$	12551.1	+	21739.2i	0.163577	+	0.283324i	0.936149	−	0.351603i	$-0.114363\pi$
$277$	12551.1	+	21739.2i	0.163577	+	0.283324i	−0.772572	+	0.634927i	$0.781030\pi$
$278$		0			0
$279$		−	24058.9i		−	0.309077i
$280$		0			0
$281$	22667.1			0.287067			0.143533	−	0.989645i	$-0.454154\pi$
$281$	22667.1			0.287067			0.143533	+	0.989645i	$0.454154\pi$
$282$		0			0
$283$	−119088.	+	68755.7i	−1.48695	+	0.858491i	−0.999889	−	0.0148762i	$-0.995265\pi$
$283$	−119088.	+	68755.7i	−1.48695	+	0.858491i	−0.487061	+	0.873368i	$0.661931\pi$
$284$		0			0
$285$	1475.89	+	852.104i	0.0181704	+	0.0104907i
$286$		0			0
$287$	35110.8	+	36840.6i	0.426262	+	0.447263i
$288$		0			0
$289$	−32483.8	+	56263.6i	−0.388930	+	0.673646i
$290$		0			0
$291$	19931.8	+	34522.9i	0.235375	+	0.407681i
$292$		0			0
$293$		−	88726.2i		−	1.03351i	−0.856132	−	0.516757i	$-0.827139\pi$
$293$		−	88726.2i		−	1.03351i	0.856132	−	0.516757i	$-0.172861\pi$
$294$		0			0
$295$	22793.1			0.261914
$296$		0			0
$297$	−1642.93	+	948.544i	−0.0186254	+	0.0107534i
$298$		0			0
$299$	−24229.2	−	13988.7i	−0.271017	−	0.156472i
$300$		0			0
$301$	−68706.0	+	65479.9i	−0.758336	+	0.722729i
$302$		0			0
$303$	37712.7	−	65320.3i	0.410773	−	0.711480i
$304$		0			0
$305$	13531.3	+	23436.9i	0.145459	+	0.251942i
$306$		0			0
$307$			168464.i			1.78744i	0.448629	+	0.893718i	$0.351912\pi$
$307$			168464.i			1.78744i	−0.448629	+	0.893718i	$0.648088\pi$
$308$		0			0
$309$	14127.6			0.147962
$310$		0			0
$311$	−43728.1	+	25246.5i	−0.452106	+	0.261024i	−0.708719	−	0.705491i	$-0.750727\pi$
$311$	−43728.1	+	25246.5i	−0.452106	+	0.261024i	0.256613	+	0.966514i	$0.417393\pi$
$312$		0			0
$313$	−73627.9	−	42509.1i	−0.751543	−	0.433903i	0.0747084	−	0.997205i	$-0.476197\pi$
$313$	−73627.9	−	42509.1i	−0.751543	−	0.433903i	−0.826251	+	0.563302i	$0.809531\pi$
$314$		0			0
$315$	2898.01	−	9861.09i	0.0292065	−	0.0993811i
$316$		0			0
$317$	30813.9	−	53371.3i	0.306640	−	0.531116i	−0.670985	−	0.741471i	$-0.734129\pi$
$317$	30813.9	−	53371.3i	0.306640	−	0.531116i	0.977625	+	0.210355i	$0.0674619\pi$
$318$		0			0
$319$	3386.63	+	5865.82i	0.0332803	+	0.0576431i
$320$		0			0
$321$		−	81476.2i		−	0.790716i
$322$		0			0
$323$	5750.42			0.0551181
$324$		0			0
$325$	−36963.2	+	21340.7i	−0.349948	+	0.202042i
$326$		0			0
$327$	70951.8	+	40964.1i	0.663541	+	0.383096i
$328$		0			0
$329$	−36774.6	+	8912.19i	−0.339747	+	0.0823366i
$330$		0			0
$331$	−58966.7	+	102133.i	−0.538208	+	0.932204i	0.460792	+	0.887508i	$0.347565\pi$
$331$	−58966.7	+	102133.i	−0.538208	+	0.932204i	−0.999001	+	0.0446963i	$0.985768\pi$
$332$		0			0
$333$	−30878.7	−	53483.5i	−0.278465	−	0.482315i
$334$		0			0
$335$			13118.7i			0.116896i
$336$		0			0
$337$	168112.			1.48027			0.740134	−	0.672460i	$-0.234762\pi$
$337$	168112.			1.48027			0.740134	+	0.672460i	$0.234762\pi$
$338$		0			0
$339$	−50182.4	+	28972.8i	−0.436669	+	0.252111i
$340$		0			0
$341$	10434.8	+	6024.53i	0.0897377	+	0.0518101i
$342$		0			0
$343$	76980.5	+	88967.9i	0.654324	+	0.756215i
$344$		0			0
$345$	7470.50	−	12939.3i	0.0627641	−	0.108711i
$346$		0			0
$347$	−34494.3	−	59745.9i	−0.286476	−	0.496191i	0.686490	−	0.727139i	$-0.259151\pi$
$347$	−34494.3	−	59745.9i	−0.286476	−	0.496191i	−0.972966	+	0.230948i	$0.925817\pi$
$348$		0			0
$349$		−	55130.4i		−	0.452626i	−0.974055	−	0.226313i	$-0.927333\pi$
$349$		−	55130.4i		−	0.452626i	0.974055	−	0.226313i	$-0.0726673\pi$
$350$		0			0
$351$	10605.0			0.0860785
$352$		0			0
$353$	−41940.6	+	24214.4i	−0.336578	+	0.194323i	−0.658758	−	0.752355i	$-0.728918\pi$
$353$	−41940.6	+	24214.4i	−0.336578	+	0.194323i	0.322180	+	0.946678i	$0.395584\pi$
$354$		0			0
$355$	−15378.0	−	8878.47i	−0.122023	−	0.0704501i
$356$		0			0
$357$	−8168.34	−	33705.2i	−0.0640911	−	0.264460i
$358$		0			0
$359$	95322.9	−	165104.i	0.739619	−	1.28106i	−0.213048	−	0.977042i	$-0.568339\pi$
$359$	95322.9	−	165104.i	0.739619	−	1.28106i	0.952667	−	0.304016i	$-0.0983277\pi$
$360$		0			0
$361$	−64269.4	−	111318.i	−0.493162	−	0.854182i
$362$		0			0
$363$			75126.8i			0.570140i
$364$		0			0
$365$	10962.2			0.0822831
$366$		0			0
$367$	124732.	−	72014.1i	0.926075	−	0.534669i	0.0405067	−	0.999179i	$-0.487103\pi$
$367$	124732.	−	72014.1i	0.926075	−	0.534669i	0.885568	+	0.464510i	$0.153769\pi$
$368$		0			0
$369$	−24285.5	−	14021.3i	−0.178359	−	0.102976i
$370$		0			0
$371$	34790.2	+	10224.3i	0.252760	+	0.0742822i
$372$		0			0
$373$	−43720.7	+	75726.5i	−0.314246	+	0.544290i	−0.979277	−	0.202526i	$-0.935085\pi$
$373$	−43720.7	+	75726.5i	−0.314246	+	0.544290i	0.665031	+	0.746816i	$0.268418\pi$
$374$		0			0
$375$	−24011.7	−	41589.5i	−0.170750	−	0.295747i
$376$		0			0
$377$		−	37863.4i		−	0.266401i
$378$		0			0
$379$	20646.4			0.143736			0.0718682	−	0.997414i	$-0.477104\pi$
$379$	20646.4			0.143736			0.0718682	+	0.997414i	$0.477104\pi$
$380$		0			0
$381$	−66419.4	+	38347.3i	−0.457557	+	0.264171i
$382$		0			0
$383$	38723.4	+	22357.0i	0.263983	+	0.152411i	0.626150	−	0.779702i	$-0.284630\pi$
$383$	38723.4	+	22357.0i	0.263983	+	0.152411i	−0.362167	+	0.932113i	$0.617963\pi$
$384$		0			0
$385$	3551.26	+	3726.22i	0.0239586	+	0.0251389i
$386$		0			0
$387$	26149.0	−	45291.4i	0.174596	−	0.302408i
$388$		0			0
$389$	−51436.1	−	89090.0i	−0.339914	−	0.588749i	0.644502	−	0.764603i	$-0.277065\pi$
$389$	−51436.1	−	89090.0i	−0.339914	−	0.588749i	−0.984416	+	0.175854i	$0.943731\pi$
$390$		0			0
$391$		−	50414.6i		−	0.329764i
$392$		0			0
$393$	78506.5			0.508301
$394$		0			0
$395$	−4922.09	+	2841.77i	−0.0315468	+	0.0182136i
$396$		0			0
$397$	−57688.7	−	33306.6i	−0.366024	−	0.211324i	0.305696	−	0.952129i	$-0.401111\pi$
$397$	−57688.7	−	33306.6i	−0.366024	−	0.211324i	−0.671720	+	0.740805i	$0.734444\pi$
$398$		0			0
$399$	7781.13	−	7415.77i	0.0488761	−	0.0465812i
$400$		0			0
$401$	−15514.3	+	26871.6i	−0.0964816	+	0.167111i	−0.910226	−	0.414112i	$-0.864092\pi$
$401$	−15514.3	+	26871.6i	−0.0964816	+	0.167111i	0.813744	+	0.581223i	$0.197426\pi$
$402$		0			0
$403$	−33677.9	−	58331.8i	−0.207365	−	0.359166i
$404$		0			0
$405$			5663.45i			0.0345279i
$406$		0			0
$407$	30929.1			0.186715
$408$		0			0
$409$	112001.	−	64663.7i	0.669537	−	0.386557i	−0.126364	−	0.991984i	$-0.540331\pi$
$409$	112001.	−	64663.7i	0.669537	−	0.386557i	0.795901	+	0.605427i	$0.206998\pi$
$410$		0			0
$411$	−55938.0	−	32295.8i	−0.331149	−	0.191189i
$412$		0			0
$413$	40535.3	−	137930.i	0.237647	−	0.808644i
$414$		0			0
$415$	38967.4	−	67493.6i	0.226259	−	0.391892i
$416$		0			0
$417$	12040.4	+	20854.6i	0.0692420	+	0.119931i
$418$		0			0
$419$			226622.i			1.29085i	0.763826	+	0.645423i	$0.223319\pi$
$419$			226622.i			1.29085i	−0.763826	+	0.645423i	$0.776681\pi$
$420$		0			0
$421$	174587.			0.985024			0.492512	−	0.870306i	$-0.336079\pi$
$421$	174587.			0.985024			0.492512	+	0.870306i	$0.336079\pi$
$422$		0			0
$423$	18056.7	−	10425.1i	0.100916	−	0.0582637i
$424$		0			0
$425$	−66606.8	−	38455.5i	−0.368758	−	0.212902i
$426$		0			0
$427$	165890.	−	40202.8i	0.909837	−	0.220496i
$428$		0			0
$429$	−2655.57	+	4599.57i	−0.0144292	+	0.0249921i
$430$		0			0
$431$	22934.0	+	39722.8i	0.123459	+	0.213838i	0.921130	−	0.389256i	$-0.127268\pi$
$431$	22934.0	+	39722.8i	0.123459	+	0.213838i	−0.797670	+	0.603094i	$0.793934\pi$
$432$		0			0
$433$		−	278272.i		−	1.48420i	−0.670287	−	0.742102i	$-0.733829\pi$
$433$		−	278272.i		−	1.48420i	0.670287	−	0.742102i	$-0.266171\pi$
$434$		0			0
$435$	20220.5			0.106859
$436$		0			0
$437$	13532.0	−	7812.72i	0.0708598	−	0.0409109i
$438$		0			0
$439$	−66629.0	−	38468.3i	−0.345728	−	0.199606i	0.317074	−	0.948401i	$-0.397300\pi$
$439$	−66629.0	−	38468.3i	−0.345728	−	0.199606i	−0.662802	+	0.748795i	$0.730633\pi$
$440$		0			0
$441$	−54519.3	−	35074.0i	−0.280332	−	0.180346i
$442$		0			0
$443$	43415.8	−	75198.4i	0.221228	−	0.383178i	−0.733953	−	0.679200i	$-0.762327\pi$
$443$	43415.8	−	75198.4i	0.221228	−	0.383178i	0.955181	+	0.296022i	$0.0956601\pi$
$444$		0			0
$445$	−8776.95	−	15202.1i	−0.0443225	−	0.0767687i
$446$		0			0
$447$			26142.2i			0.130836i
$448$		0			0
$449$	238675.			1.18390			0.591949	−	0.805976i	$-0.298359\pi$
$449$	238675.			1.18390			0.591949	+	0.805976i	$0.298359\pi$
$450$		0			0
$451$	12162.6	−	7022.07i	0.0597961	−	0.0345233i
$452$		0			0
$453$	154596.	+	89256.1i	0.753359	+	0.434952i
$454$		0			0
$455$	−6777.30	−	27965.3i	−0.0327366	−	0.135082i
$456$		0			0
$457$	−72057.2	+	124807.i	−0.345021	+	0.597593i	−0.985358	−	0.170501i	$-0.945461\pi$
$457$	−72057.2	+	124807.i	−0.345021	+	0.597593i	0.640337	+	0.768094i	$0.278795\pi$
$458$		0			0
$459$	9554.93	+	16549.6i	0.0453526	+	0.0785530i
$460$		0			0
$461$		−	195405.i		−	0.919462i	−0.888058	−	0.459731i	$-0.847946\pi$
$461$		−	195405.i		−	0.919462i	0.888058	−	0.459731i	$-0.152054\pi$
$462$		0			0
$463$	−155163.			−0.723813			−0.361906	−	0.932214i	$-0.617874\pi$
$463$	−155163.			−0.723813			−0.361906	+	0.932214i	$0.617874\pi$
$464$		0			0
$465$	31151.4	−	17985.3i	0.144069	−	0.0831784i
$466$		0			0
$467$	266635.	+	153942.i	1.22260	+	0.705866i	0.965471	−	0.260512i	$-0.0838914\pi$
$467$	266635.	+	153942.i	1.22260	+	0.705866i	0.257125	+	0.966378i	$0.417225\pi$
$468$		0			0
$469$	79386.2	+	23330.3i	0.360910	+	0.106066i
$470$		0			0
$471$	14985.5	−	25955.6i	0.0675505	−	0.117001i
$472$		0			0
$473$	13095.8	+	22682.7i	0.0585344	+	0.101385i
$474$		0			0
$475$		−	23837.7i		−	0.105652i
$476$		0			0
$477$	−19980.8			−0.0878165
$478$		0			0
$479$	82401.4	−	47574.5i	0.359140	−	0.207349i	−0.309564	−	0.950879i	$-0.600183\pi$
$479$	82401.4	−	47574.5i	0.359140	−	0.207349i	0.668703	+	0.743529i	$0.266850\pi$
$480$		0			0
$481$	−149734.	−	86448.7i	−0.647186	−	0.373653i
$482$		0			0
$483$	−65015.0	−	68218.1i	−0.278689	−	0.292419i
$484$		0			0
$485$	−29800.1	+	51615.3i	−0.126688	+	0.219429i
$486$		0			0
$487$	−54230.5	−	93929.9i	−0.228657	−	0.396046i	0.728753	−	0.684777i	$-0.240100\pi$
$487$	−54230.5	−	93929.9i	−0.228657	−	0.396046i	−0.957410	+	0.288730i	$0.906767\pi$
$488$		0			0
$489$			63646.2i			0.266167i
$490$		0			0
$491$	−328876.			−1.36417			−0.682087	−	0.731271i	$-0.738927\pi$
$491$	−328876.			−1.36417			−0.682087	+	0.731271i	$0.738927\pi$
$492$		0			0
$493$	59087.9	−	34114.4i	0.243111	−	0.140360i
$494$		0			0
$495$	−2456.35	−	1418.17i	−0.0100249	−	0.00578787i
$496$		0			0
$497$	−81075.2	+	77268.4i	−0.328228	+	0.312816i
$498$		0			0
$499$	22002.2	−	38108.9i	0.0883618	−	0.153047i	−0.818457	−	0.574568i	$-0.805170\pi$
$499$	22002.2	−	38108.9i	0.0883618	−	0.153047i	0.906819	+	0.421521i	$0.138503\pi$
$500$		0			0
$501$	133146.	+	230616.i	0.530460	+	0.918783i
$502$		0			0
$503$			481813.i			1.90433i	0.305580	+	0.952166i	$0.401150\pi$
$503$			481813.i			1.90433i	−0.305580	+	0.952166i	$0.598850\pi$
$504$		0			0
$505$	112769.			0.442187
$506$		0			0
$507$	−102812.	+	59358.7i	−0.399972	+	0.230924i
$508$		0			0
$509$	−232170.	−	134043.i	−0.896128	−	0.517380i	−0.0201862	−	0.999796i	$-0.506426\pi$
$509$	−232170.	−	134043.i	−0.896128	−	0.517380i	−0.875942	+	0.482416i	$0.839759\pi$
$510$		0			0
$511$	19495.1	−	66336.2i	0.0746594	−	0.254044i
$512$		0			0
$513$	−2961.44	+	5129.37i	−0.0112530	+	0.0194908i
$514$		0			0
$515$	10561.1	+	18292.3i	0.0398194	+	0.0689692i
$516$		0			0
$517$			10442.1i			0.0390666i
$518$		0			0
$519$	−274197.			−1.01795
$520$		0			0
$521$	−396125.	+	228703.i	−1.45934	+	0.842552i	−0.998979	−	0.0451784i	$-0.985614\pi$
$521$	−396125.	+	228703.i	−1.45934	+	0.842552i	−0.460364	+	0.887730i	$0.652281\pi$
$522$		0			0
$523$	−382306.	−	220724.i	−1.39768	−	0.806951i	−0.403531	−	0.914966i	$-0.632217\pi$
$523$	−382306.	−	220724.i	−1.39768	−	0.806951i	−0.994149	+	0.108015i	$0.965551\pi$
$524$		0			0
$525$	−139721.	+	33860.9i	−0.506924	+	0.122851i
$526$		0			0
$527$	60686.7	−	105112.i	0.218510	−	0.378471i
$528$		0			0
$529$	71425.5	+	123713.i	0.255236	+	0.442082i
$530$		0			0
$531$			79216.1i			0.280947i
$532$		0			0
$533$	−78508.5			−0.276352
$534$		0			0
$535$	105495.	−	60907.7i	0.368574	−	0.212797i
$536$		0			0
$537$	223594.	+	129092.i	0.775376	+	0.447664i
$538$		0			0
$539$	28864.3	−	14863.3i	0.0993537	−	0.0511608i
$540$		0			0
$541$	−234260.	+	405750.i	−0.800394	+	1.38632i	0.118964	+	0.992899i	$0.462043\pi$
$541$	−234260.	+	405750.i	−0.800394	+	1.38632i	−0.919357	+	0.393424i	$0.871290\pi$
$542$		0			0
$543$	57025.7	+	98771.4i	0.193406	+	0.334990i
$544$		0			0
$545$			122491.i			0.412393i
$546$		0			0
$547$	−30223.3			−0.101011			−0.0505053	−	0.998724i	$-0.516083\pi$
$547$	−30223.3			−0.101011			−0.0505053	+	0.998724i	$0.516083\pi$
$548$		0			0
$549$	−81453.6	+	47027.3i	−0.270250	+	0.156029i
$550$		0			0
$551$	18313.6	+	10573.4i	0.0603214	+	0.0348265i
$552$		0			0
$553$	8443.17	+	34839.3i	0.0276093	+	0.113925i
$554$		0			0
$555$	46166.9	−	79963.4i	0.149880	−	0.259600i
$556$		0			0
$557$	269643.	+	467035.i	0.869117	+	1.50535i	0.862901	+	0.505373i	$0.168645\pi$
$557$	269643.	+	467035.i	0.869117	+	1.50535i	0.00621580	+	0.999981i	$0.498021\pi$
$558$		0			0
$559$		−	146415.i		−	0.468555i
$560$		0			0
$561$	−9570.53			−0.0304096
$562$		0			0
$563$	69013.6	−	39845.0i	0.217730	−	0.125706i	−0.387169	−	0.922009i	$-0.626547\pi$
$563$	69013.6	−	39845.0i	0.217730	−	0.125706i	0.604899	+	0.796302i	$0.293214\pi$
$564$		0			0
$565$	−75027.9	−	43317.4i	−0.235031	−	0.135695i
$566$		0			0
$567$	34271.7	+	10071.9i	0.106603	+	0.0313289i
$568$		0			0
$569$	−205774.	+	356410.i	−0.635572	+	1.10084i	0.350821	+	0.936442i	$0.385902\pi$
$569$	−205774.	+	356410.i	−0.635572	+	1.10084i	−0.986394	+	0.164401i	$0.947431\pi$
$570$		0			0
$571$	−164913.	−	285637.i	−0.505803	−	0.876077i	−0.999977	−	0.00671401i	$-0.997863\pi$
$571$	−164913.	−	285637.i	−0.505803	−	0.876077i	0.494174	−	0.869363i	$-0.335470\pi$
$572$		0			0
$573$			317729.i			0.967716i
$574$		0			0
$575$	−208988.			−0.632099
$576$		0			0
$577$	492831.	−	284536.i	1.48029	−	0.854645i	0.480539	−	0.876974i	$-0.340441\pi$
$577$	492831.	−	284536.i	1.48029	−	0.854645i	0.999751	+	0.0223281i	$0.00710786\pi$
$578$		0			0
$579$	67464.0	+	38950.4i	0.201240	+	0.116186i
$580$		0			0
$581$	−339130.	−	355838.i	−1.00465	−	1.05414i
$582$		0			0
$583$	5003.35	−	8666.06i	0.0147206	−	0.0254967i
$584$		0			0
$585$	7927.76	+	13731.3i	0.0231653	+	0.0401235i
$586$		0			0
$587$			261714.i			0.759540i	0.925081	+	0.379770i	$0.123997\pi$
$587$			261714.i			0.759540i	−0.925081	+	0.379770i	$0.876003\pi$
$588$		0			0
$589$	37618.3			0.108435
$590$		0			0
$591$	−314821.	+	181762.i	−0.901339	+	0.520388i
$592$		0			0
$593$	472133.	+	272586.i	1.34263	+	0.775165i	0.987192	−	0.159537i	$-0.0510000\pi$
$593$	472133.	+	272586.i	1.34263	+	0.775165i	0.355433	+	0.934702i	$0.384333\pi$
$594$		0			0
$595$	37535.2	−	35772.7i	0.106024	−	0.101046i
$596$		0			0
$597$	−62780.8	+	108739.i	−0.176148	+	0.305098i
$598$		0			0
$599$	24712.7	+	42803.7i	0.0688759	+	0.119296i	0.898407	−	0.439164i	$-0.144725\pi$
$599$	24712.7	+	42803.7i	0.0688759	+	0.119296i	−0.829531	+	0.558461i	$0.811392\pi$
$600$		0			0
$601$		−	352114.i		−	0.974843i	−0.873167	−	0.487422i	$-0.837937\pi$
$601$		−	352114.i		−	0.974843i	0.873167	−	0.487422i	$-0.162063\pi$
$602$		0			0
$603$	−45593.3			−0.125391
$604$		0			0
$605$	−97274.0	+	56161.2i	−0.265758	+	0.153435i
$606$		0			0
$607$	−190770.	−	110141.i	−0.517765	−	0.298932i	0.218255	−	0.975892i	$-0.429964\pi$
$607$	−190770.	−	110141.i	−0.517765	−	0.298932i	−0.736020	+	0.676960i	$0.763297\pi$
$608$		0			0
$609$	35960.1	−	122362.i	0.0969587	−	0.329922i
$610$		0			0
$611$	29186.2	−	50552.0i	0.0781800	−	0.135412i
$612$		0			0
$613$	252311.	+	437016.i	0.671453	+	1.16299i	0.977492	+	0.210972i	$0.0676628\pi$
$613$	252311.	+	437016.i	0.671453	+	1.16299i	−0.306039	+	0.952019i	$0.599004\pi$
$614$		0			0
$615$		−	41926.5i		−	0.110851i
$616$		0			0
$617$	−402406.			−1.05705			−0.528524	−	0.848919i	$-0.677254\pi$
$617$	−402406.			−1.05705			−0.528524	+	0.848919i	$0.677254\pi$
$618$		0			0
$619$	578914.	−	334236.i	1.51089	−	0.872313i	0.510971	−	0.859598i	$-0.329286\pi$
$619$	578914.	−	334236.i	1.51089	−	0.872313i	0.999919	−	0.0127147i	$-0.00404734\pi$
$620$		0			0
$621$	44969.8	+	25963.3i	0.116610	+	0.0673251i
$622$		0			0
$623$	−107603.	+	26077.2i	−0.277235	+	0.0671869i
$624$		0			0
$625$	−140552.	+	243443.i	−0.359813	+	0.623214i
$626$		0			0
$627$	−1483.14	−	2568.87i	−0.00377265	−	0.00653442i
$628$		0			0
$629$		−	311557.i		−	0.787474i
$630$		0			0
$631$	−703475.			−1.76681			−0.883405	−	0.468610i	$-0.844755\pi$
$631$	−703475.			−1.76681			−0.883405	+	0.468610i	$0.844755\pi$
$632$		0			0
$633$	49331.3	−	28481.4i	0.123116	−	0.0710811i
$634$		0			0
$635$	−99303.9	−	57333.1i	−0.246274	−	0.142186i
$636$		0			0
$637$	−181281.	−	8721.58i	−0.446760	−	0.0214940i
$638$		0			0
$639$	30856.6	−	53445.3i	0.0755696	−	0.130890i
$640$		0			0
$641$	151858.	+	263026.i	0.369592	+	0.640152i	0.989502	−	0.144521i	$-0.0461642\pi$
$641$	151858.	+	263026.i	0.369592	+	0.640152i	−0.619910	+	0.784673i	$0.712831\pi$
$642$		0			0
$643$		−	73377.4i		−	0.177476i	−0.996055	−	0.0887382i	$-0.971717\pi$
$643$		−	73377.4i		−	0.177476i	0.996055	−	0.0887382i	$-0.0282834\pi$
$644$		0			0
$645$	78191.0			0.187948
$646$		0			0
$647$	523716.	−	302368.i	1.25109	−	0.722316i	0.279762	−	0.960069i	$-0.409744\pi$
$647$	523716.	−	302368.i	1.25109	−	0.722316i	0.971325	+	0.237754i	$0.0764111\pi$
$648$		0			0
$649$	−34357.5	−	19836.3i	−0.0815704	−	0.0470947i
$650$		0			0
$651$	−53435.9	−	220494.i	−0.126087	−	0.520277i
$652$		0			0
$653$	213363.	−	369556.i	0.500372	−	0.866669i	−0.499628	−	0.866240i	$-0.666530\pi$
$653$	213363.	−	369556.i	0.500372	−	0.866669i	1.00000		0.000429258i	$-0.000136637\pi$
$654$		0			0
$655$	58687.7	+	101650.i	0.136793	+	0.236933i
$656$		0			0
$657$			38098.4i			0.0882625i
$658$		0			0
$659$	108430.			0.249678			0.124839	−	0.992177i	$-0.460159\pi$
$659$	108430.			0.249678			0.124839	+	0.992177i	$0.460159\pi$
$660$		0			0
$661$	544794.	−	314537.i	1.24689	−	0.719894i	0.276405	−	0.961041i	$-0.410857\pi$
$661$	544794.	−	314537.i	1.24689	−	0.719894i	0.970489	+	0.241147i	$0.0775236\pi$
$662$		0			0
$663$	46332.7	+	26750.2i	0.105405	+	0.0608555i
$664$		0			0
$665$	15418.7	+	4531.31i	0.0348662	+	0.0102466i
$666$		0			0
$667$	92698.0	−	160558.i	0.208362	−	0.360894i
$668$		0			0
$669$	55506.5	+	96140.1i	0.124020	+	0.214809i
$670$		0			0
$671$		−	47104.0i		−	0.104620i
$672$		0			0
$673$	103653.			0.228851			0.114425	−	0.993432i	$-0.463497\pi$
$673$	103653.			0.228851			0.114425	+	0.993432i	$0.463497\pi$
$674$		0			0
$675$	68604.5	−	39608.8i	0.150572	−	0.0869329i
$676$		0			0
$677$	98794.4	+	57039.0i	0.215553	+	0.124450i	0.603890	−	0.797068i	$-0.293617\pi$
$677$	98794.4	+	57039.0i	0.215553	+	0.124450i	−0.388336	+	0.921518i	$0.626950\pi$
$678$		0			0
$679$	259347.	+	272124.i	0.562525	+	0.590239i
$680$		0			0
$681$	149722.	−	259327.i	0.322844	−	0.559182i
$682$		0			0
$683$	220633.	+	382148.i	0.472966	+	0.819201i	0.999521	−	0.0309401i	$-0.00985011\pi$
$683$	220633.	+	382148.i	0.472966	+	0.819201i	−0.526556	+	0.850141i	$0.676517\pi$
$684$		0			0
$685$		−	96571.2i		−	0.205810i
$686$		0			0
$687$	−182161.			−0.385960
$688$		0			0
$689$	−48444.4	+	27969.4i	−0.102048	+	0.0589175i
$690$		0			0
$691$	−557701.	−	321989.i	−1.16801	−	0.674350i	−0.214798	−	0.976659i	$-0.568909\pi$
$691$	−557701.	−	321989.i	−1.16801	−	0.674350i	−0.953210	+	0.302309i	$0.902243\pi$
$692$		0			0
$693$	−12950.3	+	12342.2i	−0.0269657	+	0.0256996i
$694$		0			0
$695$	−18001.7	+	31179.8i	−0.0372686	+	0.0645512i
$696$		0			0
$697$	−70735.1	−	122517.i	−0.145603	−	0.252191i
$698$		0			0
$699$		−	44287.7i		−	0.0906419i
$700$		0			0
$701$	−337689.			−0.687196			−0.343598	−	0.939117i	$-0.611646\pi$
$701$	−337689.			−0.687196			−0.343598	+	0.939117i	$0.611646\pi$
$702$		0			0
$703$	83626.4	−	48281.7i	0.169213	−	0.0976950i
$704$		0			0
$705$	26996.7	+	15586.5i	0.0543166	+	0.0313597i
$706$		0			0
$707$	200548.	−	682406.i	0.401218	−	1.36523i
$708$		0			0
$709$	−214167.	+	370949.i	−0.426050	+	0.737941i	−0.996518	−	0.0833795i	$-0.973429\pi$
$709$	−214167.	+	370949.i	−0.426050	+	0.737941i	0.570468	+	0.821320i	$0.306762\pi$
$710$		0			0
$711$	−9876.42	−	17106.5i	−0.0195371	−	0.0338393i
$712$		0			0
$713$		−	329804.i		−	0.648749i
$714$		0			0
$715$	−7940.69			−0.0155327
$716$		0			0
$717$	118957.	−	68680.1i	0.231395	−	0.133596i
$718$		0			0
$719$	858771.	+	495812.i	1.66119	+	0.959090i	0.972147	+	0.234370i	$0.0753029\pi$
$719$	858771.	+	495812.i	1.66119	+	0.959090i	0.689044	+	0.724719i	$0.258030\pi$
$720$		0			0
$721$	129476.	−	31378.0i	0.249068	−	0.0603608i
$722$		0			0
$723$	198811.	−	344351.i	0.380333	−	0.658756i
$724$		0			0
$725$	−141417.	−	244942.i	−0.269046	−	0.466001i
$726$		0			0
$727$		−	88010.4i		−	0.166519i	−0.996528	−	0.0832597i	$-0.973467\pi$
$727$		−	88010.4i		−	0.166519i	0.996528	−	0.0832597i	$-0.0265331\pi$
$728$		0			0
$729$	−19683.0			−0.0370370
$730$		0			0
$731$	228488.	−	131918.i	0.427592	−	0.246870i
$732$		0			0
$733$	−280457.	−	161922.i	−0.521986	−	0.301369i	0.215761	−	0.976446i	$-0.430777\pi$
$733$	−280457.	−	161922.i	−0.521986	−	0.301369i	−0.737747	+	0.675078i	$0.764110\pi$
$734$		0			0
$735$	4657.65	−	96811.1i	0.00862169	−	0.179205i
$736$		0			0
$737$	11416.9	−	19774.7i	0.0210191	−	0.0364061i
$738$		0			0
$739$	182318.	+	315784.i	0.333841	+	0.578230i	0.983262	−	0.182200i	$-0.0583217\pi$
$739$	182318.	+	315784.i	0.333841	+	0.578230i	−0.649420	+	0.760430i	$0.724988\pi$
$740$		0			0
$741$			16581.8i			0.0301993i
$742$		0			0
$743$	−695021.			−1.25898			−0.629492	−	0.777007i	$-0.716737\pi$
$743$	−695021.			−1.25898			−0.629492	+	0.777007i	$0.716737\pi$
$744$		0			0
$745$	−33848.8	+	19542.6i	−0.0609861	+	0.0352104i
$746$		0			0
$747$	234570.	+	135429.i	0.420370	+	0.242701i
$748$		0			0
$749$	−180963.	−	746710.i	−0.322571	−	1.33103i
$750$		0			0
$751$	−513684.	+	889726.i	−0.910785	+	1.57753i	−0.0978278	+	0.995203i	$0.531189\pi$
$751$	−513684.	+	889726.i	−0.910785	+	1.57753i	−0.812957	+	0.582323i	$0.802144\pi$
$752$		0			0
$753$	−85236.0	−	147633.i	−0.150326	−	0.260372i
$754$		0			0
$755$			266894.i			0.468215i
$756$		0			0
$757$	419375.			0.731832			0.365916	−	0.930648i	$-0.380756\pi$
$757$	419375.			0.731832			0.365916	+	0.930648i	$0.380756\pi$
$758$		0			0
$759$	−22521.6	+	13002.8i	−0.0390945	+	0.0225712i
$760$		0			0
$761$	−766840.	−	442735.i	−1.32415	−	0.764496i	−0.339758	−	0.940513i	$-0.610345\pi$
$761$	−766840.	−	442735.i	−1.32415	−	0.764496i	−0.984387	+	0.176017i	$0.943679\pi$
$762$		0			0
$763$	741240.	+	217838.i	1.27324	+	0.374184i
$764$		0			0
$765$	−14285.6	+	24743.4i	−0.0244105	+	0.0422802i
$766$		0			0
$767$	110888.	+	192063.i	0.188492	+	0.326477i
$768$		0			0
$769$		−	333255.i		−	0.563540i	−0.959482	−	0.281770i	$-0.909079\pi$
$769$		−	333255.i		−	0.563540i	0.959482	−	0.281770i	$-0.0909215\pi$
$770$		0			0
$771$	−373935.			−0.629053
$772$		0			0
$773$	787057.	−	454408.i	1.31719	−	0.760478i	0.333911	−	0.942605i	$-0.391631\pi$
$773$	787057.	−	454408.i	1.31719	−	0.760478i	0.983275	+	0.182127i	$0.0582981\pi$
$774$		0			0
$775$	−435731.	−	251569.i	−0.725462	−	0.418846i
$776$		0			0
$777$	−401785.	−	421580.i	−0.665506	−	0.698294i
$778$		0			0
$779$	21923.5	−	37972.7i	0.0361273	−	0.0625744i
$780$		0			0
$781$	15453.5	+	26766.2i	0.0253352	+	0.0438819i
$782$		0			0
$783$			70275.1i			0.114625i
$784$		0			0
$785$	44809.7			0.0727165
$786$		0			0
$787$	703181.	−	405982.i	1.13532	−	0.655477i	0.190052	−	0.981774i	$-0.439134\pi$
$787$	703181.	−	405982.i	1.13532	−	0.655477i	0.945267	+	0.326298i	$0.105801\pi$
$788$		0			0
$789$	453144.	+	261623.i	0.727917	+	0.420263i
$790$		0			0
$791$	−395560.	+	376987.i	−0.632207	+	0.602522i
$792$		0			0
$793$	−131659.	+	228039.i	−0.209364	+	0.362630i
$794$		0			0
$795$	−14936.7	−	25871.1i	−0.0236331	−	0.0409337i
$796$		0			0
$797$		−	918678.i		−	1.44626i	−0.690711	−	0.723130i	$-0.742703\pi$
$797$		−	918678.i		−	1.44626i	0.690711	−	0.723130i	$-0.257297\pi$
$798$		0			0
$799$	105186.			0.164764
$800$		0			0
$801$	52834.2	−	30503.8i	0.0823474	−	0.0475433i
$802$		0			0
$803$	−16524.0	−	9540.15i	−0.0256262	−	0.0147953i
$804$		0			0
$805$	39726.6	−	135178.i	0.0613041	−	0.208600i
$806$		0			0
$807$	114810.	−	198856.i	0.176291	−	0.305345i
$808$		0			0
$809$	402027.	+	696332.i	0.614269	+	1.06395i	0.990512	+	0.137424i	$0.0438824\pi$
$809$	402027.	+	696332.i	0.614269	+	1.06395i	−0.376243	+	0.926521i	$0.622784\pi$
$810$		0			0
$811$			308731.i			0.469396i	0.972068	+	0.234698i	$0.0754100\pi$
$811$			308731.i			0.469396i	−0.972068	+	0.234698i	$0.924590\pi$
$812$		0			0
$813$	−194954.			−0.294952
$814$		0			0
$815$	−82409.0	+	47578.9i	−0.124068	+	0.0716307i
$816$		0			0
$817$	70817.3	+	40886.4i	0.106095	+	0.0612540i
$818$		0			0
$819$	97191.9	−	23554.1i	0.144898	−	0.0351155i
$820$		0			0
$821$	334029.	−	578555.i	0.495561	−	0.858337i	−0.504426	−	0.863455i	$-0.668296\pi$
$821$	334029.	−	578555.i	0.495561	−	0.858337i	0.999987	+	0.00511783i	$0.00162906\pi$
$822$		0			0
$823$	457854.	+	793026.i	0.675969	+	1.17081i	0.976185	+	0.216942i	$0.0696081\pi$
$823$	457854.	+	793026.i	0.675969	+	1.17081i	−0.300215	+	0.953871i	$0.597059\pi$
$824$		0			0
$825$			39673.5i			0.0582897i
$826$		0			0
$827$	1.04336e6			1.52554			0.762769	−	0.646672i	$-0.223840\pi$
$827$	1.04336e6			1.52554			0.762769	+	0.646672i	$0.223840\pi$
$828$		0			0
$829$	462255.	−	266883.i	0.672625	−	0.388340i	−0.124446	−	0.992226i	$-0.539715\pi$
$829$	462255.	−	266883.i	0.672625	−	0.388340i	0.797070	+	0.603886i	$0.206382\pi$
$830$		0			0
$831$	−112960.	−	65217.5i	−0.163577	−	0.0944414i
$832$		0			0
$833$	−149722.	−	290758.i	−0.215772	−	0.419027i
$834$		0			0
$835$	−199067.	+	344794.i	−0.285513	+	0.494524i
$836$		0			0
$837$	62506.8	+	108265.i	0.0892228	+	0.154539i
$838$		0			0
$839$			657882.i			0.934597i	0.884100	+	0.467298i	$0.154773\pi$
$839$			657882.i			0.934597i	−0.884100	+	0.467298i	$0.845227\pi$
$840$		0			0
$841$	−456374.			−0.645252
$842$		0			0
$843$	−102002.	+	58890.8i	−0.143533	+	0.0828690i
$844$		0			0
$845$	−153715.	−	88747.5i	−0.215280	−	0.124292i
$846$		0			0
$847$	166860.	+	688519.i	0.232587	+	0.959730i
$848$		0			0
$849$	357265.	−	618801.i	0.495650	−	0.858491i
$850$		0			0
$851$	−423292.	−	733163.i	−0.584495	−	1.01237i
$852$		0			0
$853$			994103.i			1.36626i	0.730297	+	0.683130i	$0.239382\pi$
$853$			994103.i			1.36626i	−0.730297	+	0.683130i	$0.760618\pi$
$854$		0			0
$855$	−8855.33			−0.0121136
$856$		0			0
$857$	−1.08942e6	+	628977.i	−1.48332	+	0.856393i	−0.999820	−	0.0189528i	$-0.993967\pi$
$857$	−1.08942e6	+	628977.i	−1.48332	+	0.856393i	−0.483497	+	0.875346i	$0.660633\pi$
$858$		0			0
$859$	−267518.	−	154452.i	−0.362549	−	0.209318i	0.307649	−	0.951500i	$-0.400458\pi$
$859$	−267518.	−	154452.i	−0.362549	−	0.209318i	−0.670199	+	0.742182i	$0.733791\pi$
$860$		0			0
$861$	−253713.	−	74562.1i	−0.342244	−	0.100580i
$862$		0			0
$863$	−156153.	+	270466.i	−0.209667	+	0.363154i	−0.951610	−	0.307310i	$-0.900571\pi$
$863$	−156153.	+	270466.i	−0.209667	+	0.363154i	0.741943	+	0.670463i	$0.233905\pi$
$864$		0			0
$865$	−204977.	−	355030.i	−0.273950	−	0.474496i
$866$		0			0
$867$		−	337581.i		−	0.449097i
$868$		0			0
$869$	9892.54			0.0130999
$870$		0			0
$871$	−110543.	+	63822.0i	−0.145712	+	0.0841267i
$872$		0			0
$873$	−179386.	−	103569.i	−0.235375	−	0.135894i
$874$		0			0
$875$	−312434.	−	327827.i	−0.408077	−	0.428182i
$876$		0			0
$877$	−726266.	+	1.25793e6i	−0.944270	+	1.63552i	−0.187065	+	0.982348i	$0.559897\pi$
$877$	−726266.	+	1.25793e6i	−0.944270	+	1.63552i	−0.757206	+	0.653177i	$0.773436\pi$
$878$		0			0
$879$	230517.	+	399268.i	0.298350	+	0.516757i
$880$		0			0
$881$			880963.i			1.13503i	0.823364	+	0.567513i	$0.192094\pi$
$881$			880963.i			1.13503i	−0.823364	+	0.567513i	$0.807906\pi$
$882$		0			0
$883$	−324767.			−0.416534			−0.208267	−	0.978072i	$-0.566782\pi$
$883$	−324767.			−0.416534			−0.208267	+	0.978072i	$0.566782\pi$
$884$		0			0
$885$	−102569.	+	59218.1i	−0.130957	+	0.0756081i
$886$		0			0
$887$	−394415.	−	227716.i	−0.501310	−	0.289431i	0.227945	−	0.973674i	$-0.426800\pi$
$887$	−394415.	−	227716.i	−0.501310	−	0.289431i	−0.729254	+	0.684243i	$0.760133\pi$
$888$		0			0
$889$	−523547.	+	498964.i	−0.662448	+	0.631344i
$890$		0			0
$891$	4928.78	−	8536.90i	0.00620846	−	0.0107534i
$892$		0			0
$893$	16300.5	+	28233.4i	0.0204409	+	0.0354046i
$894$		0			0
$895$			386013.i			0.481899i
$896$		0			0
$897$	145375.			0.180678
$898$		0			0
$899$	386543.	−	223171.i	0.478276	−	0.276133i
$900$		0			0
$901$	−87295.5	−	50400.1i	−0.107533	−	0.0620843i
$902$		0			0
$903$	139055.	−	473163.i	0.170534	−	0.580277i
$904$		0			0
$905$	−85259.4	+	147674.i	−0.104099	+	0.180304i
$906$		0			0
$907$	375844.	+	650981.i	0.456871	+	0.791323i	0.998794	−	0.0491051i	$-0.0156369\pi$
$907$	375844.	+	650981.i	0.456871	+	0.791323i	−0.541923	+	0.840428i	$0.682304\pi$
$908$		0			0
$909$			391922.i			0.474320i
$910$		0			0
$911$	−938776.			−1.13116			−0.565581	−	0.824692i	$-0.691348\pi$
$911$	−938776.			−1.13116			−0.565581	+	0.824692i	$0.691348\pi$
$912$		0			0
$913$	−117477.	+	67825.1i	−0.140932	+	0.0813671i
$914$		0			0
$915$	−121782.	−	70310.7i	−0.145459	−	0.0839806i
$916$		0			0
$917$	719494.	−	174367.i	0.855634	−	0.207360i
$918$		0			0
$919$	−703984.	+	1.21934e6i	−0.833551	+	1.44375i	0.0616544	+	0.998098i	$0.480362\pi$
$919$	−703984.	+	1.21934e6i	−0.833551	+	1.44375i	−0.895205	+	0.445655i	$0.852971\pi$
$920$		0			0
$921$	−437682.	−	758088.i	−0.515988	−	0.893718i
$922$		0			0
$923$		−	172774.i		−	0.202803i
$924$		0			0
$925$	−1.29152e6			−1.50945
$926$		0			0
$927$	−63574.1	+	36704.5i	−0.0739810	+	0.0427130i
$928$		0			0
$929$	857278.	+	494950.i	0.993323	+	0.573495i	0.906266	−	0.422708i	$-0.138920\pi$
$929$	857278.	+	494950.i	0.993323	+	0.573495i	0.0870569	+	0.996203i	$0.472254\pi$
$930$		0			0
$931$	54841.4	−	85246.1i	0.0632716	−	0.0983501i
$932$		0			0
$933$	131184.	−	227218.i	0.150702	−	0.261024i
$934$		0			0
$935$	−7154.47	−	12391.9i	−0.00818378	−	0.0141747i
$936$		0			0
$937$			88009.0i			0.100242i	0.998743	+	0.0501208i	$0.0159606\pi$
$937$			88009.0i			0.100242i	−0.998743	+	0.0501208i	$0.984039\pi$
$938$		0			0
$939$	441767.			0.501028
$940$		0			0
$941$	−933046.	+	538694.i	−1.05372	+	0.608364i	−0.923688	−	0.383147i	$-0.874840\pi$
$941$	−933046.	+	538694.i	−1.05372	+	0.608364i	−0.130029	+	0.991510i	$0.541507\pi$
$942$		0			0
$943$	−332911.	−	192206.i	−0.374373	−	0.216145i
$944$		0			0
$945$	12578.8	+	51904.1i	0.0140856	+	0.0581217i
$946$		0			0
$947$	270050.	−	467740.i	0.301123	−	0.521561i	−0.675267	−	0.737573i	$-0.735972\pi$
$947$	270050.	−	467740.i	0.301123	−	0.521561i	0.976391	+	0.216012i	$0.0693051\pi$
$948$		0			0
$949$	53330.6	+	92371.3i	0.0592167	+	0.102566i
$950$		0			0
$951$			320228.i			0.354077i
$952$		0			0
$953$	675288.			0.743538			0.371769	−	0.928325i	$-0.378751\pi$
$953$	675288.			0.743538			0.371769	+	0.928325i	$0.378751\pi$
$954$		0			0
$955$	−411395.	+	237519.i	−0.451079	+	0.260431i
$956$		0			0
$957$	−30479.7	−	17597.5i	−0.0332803	−	0.0192144i
$958$		0			0
$959$	−584389.	−	171743.i	−0.635426	−	0.186741i
$960$		0			0
$961$	−64758.5	+	112165.i	−0.0701213	+	0.121454i
$962$		0			0
$963$	211681.	+	366643.i	0.228260	+	0.395358i
$964$		0			0
$965$			116470.i			0.125072i
$966$		0			0
$967$	1.01409e6			1.08449			0.542244	−	0.840221i	$-0.317575\pi$
$967$	1.01409e6			1.08449			0.542244	+	0.840221i	$0.317575\pi$
$968$		0			0
$969$	−25876.9	+	14940.0i	−0.0275591	+	0.0159112i
$970$		0			0
$971$	66792.2	+	38562.5i	0.0708414	+	0.0409003i	0.535002	−	0.844851i	$-0.320311\pi$
$971$	66792.2	+	38562.5i	0.0708414	+	0.0409003i	−0.464161	+	0.885751i	$0.653644\pi$
$972$		0			0
$973$	156667.	+	164385.i	0.165482	+	0.173635i
$974$		0			0
$975$	110890.	−	192067.i	0.116649	−	0.202042i
$976$		0			0
$977$	436977.	+	756866.i	0.457793	+	0.792921i	0.998844	−	0.0480689i	$-0.0153067\pi$
$977$	436977.	+	756866.i	0.457793	+	0.792921i	−0.541051	+	0.840990i	$0.681973\pi$
$978$		0			0
$979$			30553.6i			0.0318784i
$980$		0			0
$981$	−425711.			−0.442361
$982$		0			0
$983$	−269678.	+	155698.i	−0.279086	+	0.161130i	−0.633010	−	0.774144i	$-0.718181\pi$
$983$	−269678.	+	155698.i	−0.279086	+	0.161130i	0.353924	+	0.935274i	$0.384847\pi$
$984$		0			0
$985$	−470690.	−	271753.i	−0.485134	−	0.280092i
$986$		0			0
$987$	142331.	−	135648.i	0.146105	−	0.139245i
$988$		0			0
$989$	358456.	−	620864.i	0.366474	−	0.634752i
$990$		0			0
$991$	−137603.	−	238335.i	−0.140113	−	0.242683i	0.787426	−	0.616409i	$-0.211413\pi$
$991$	−137603.	−	238335.i	−0.140113	−	0.242683i	−0.927539	+	0.373726i	$0.878080\pi$
$992$		0			0
$993$		−	612799.i		−	0.621470i
$994$		0			0
$995$	−187728.			−0.189619
$996$		0			0
$997$	389290.	−	224757.i	0.391637	−	0.226112i	−0.291232	−	0.956652i	$-0.594065\pi$
$997$	389290.	−	224757.i	0.391637	−	0.226112i	0.682869	+	0.730541i	$0.260732\pi$
$998$		0			0
$999$	277908.	+	160450.i	0.278465	+	0.160772i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	168.5.z.a.145.3	yes	16
3.2	odd	2		504.5.by.a.145.6		16
4.3	odd	2		336.5.bh.i.145.3		16
7.2	even	3		1176.5.f.b.97.6		16
7.3	odd	6	inner	168.5.z.a.73.3	✓	16
7.5	odd	6		1176.5.f.b.97.11		16
21.17	even	6		504.5.by.a.73.6		16
28.3	even	6		336.5.bh.i.241.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.5.z.a.73.3	✓	16	7.3	odd	6	inner
168.5.z.a.145.3	yes	16	1.1	even	1	trivial
336.5.bh.i.145.3		16	4.3	odd	2
336.5.bh.i.241.3		16	28.3	even	6
504.5.by.a.73.6		16	21.17	even	6
504.5.by.a.145.6		16	3.2	odd	2
1176.5.f.b.97.6		16	7.2	even	3
1176.5.f.b.97.11		16	7.5	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	168.z (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-4.50000 + 2.59808i) q^{3} +(-6.72797 - 3.88439i) q^{5} +(-35.4710 + 33.8055i) q^{7} +(13.5000 - 23.3827i) q^{9} +O(q^{10})\)	\(q+(-4.50000 + 2.59808i) q^{3} +(-6.72797 - 3.88439i) q^{5} +(-35.4710 + 33.8055i) q^{7} +(13.5000 - 23.3827i) q^{9} +(6.76101 + 11.7104i) q^{11} -75.5898i q^{13} +40.3678 q^{15} +(117.962 - 68.1055i) q^{17} +(36.5610 + 21.1085i) q^{19} +(71.7902 - 244.281i) q^{21} +(185.061 - 320.535i) q^{23} +(-282.323 - 488.998i) q^{25} +140.296i q^{27} +500.906 q^{29} +(771.688 - 445.534i) q^{31} +(-60.8491 - 35.1313i) q^{33} +(369.961 - 89.6589i) q^{35} +(1143.66 - 1980.87i) q^{37} +(196.388 + 340.154i) q^{39} -1038.61i q^{41} +1936.96 q^{43} +(-181.655 + 104.879i) q^{45} +(668.768 + 386.113i) q^{47} +(115.380 - 2398.23i) q^{49} +(-353.886 + 612.949i) q^{51} +(-370.015 - 640.885i) q^{53} -105.050i q^{55} -219.366 q^{57} +(-2540.86 + 1466.96i) q^{59} +(-3016.80 - 1741.75i) q^{61} +(311.605 + 1285.78i) q^{63} +(-293.621 + 508.566i) q^{65} +(-844.320 - 1462.41i) q^{67} +1923.21i q^{69} +2285.68 q^{71} +(-1222.01 + 705.526i) q^{73} +(2540.91 + 1466.99i) q^{75} +(-635.696 - 186.821i) q^{77} +(365.793 - 633.573i) q^{79} +(-364.500 - 631.333i) q^{81} +10031.8i q^{83} -1058.19 q^{85} +(-2254.08 + 1301.39i) q^{87} +(1956.82 + 1129.77i) q^{89} +(2555.35 + 2681.24i) q^{91} +(-2315.07 + 4009.81i) q^{93} +(-163.988 - 284.035i) q^{95} -7671.75i q^{97} +365.095 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 72 q^{3} + 12 q^{5} + 40 q^{7} + 216 q^{9}+O(q^{10}) \) 16 * q - 72 * q^3 + 12 * q^5 + 40 * q^7 + 216 * q^9	\( 16 q - 72 q^{3} + 12 q^{5} + 40 q^{7} + 216 q^{9} - 108 q^{11} - 72 q^{15} - 168 q^{17} + 948 q^{19} - 252 q^{21} - 768 q^{23} + 1908 q^{25} - 1608 q^{29} - 3216 q^{31} + 972 q^{33} + 696 q^{35} - 1820 q^{37} - 1188 q^{39} + 2888 q^{43} + 324 q^{45} + 744 q^{47} - 3784 q^{49} + 504 q^{51} + 4476 q^{53} - 5688 q^{57} - 4668 q^{59} + 17760 q^{61} + 1188 q^{63} + 8760 q^{65} + 1580 q^{67} + 48 q^{71} + 588 q^{73} - 17172 q^{75} - 17508 q^{77} - 3824 q^{79} - 5832 q^{81} + 11440 q^{85} + 7236 q^{87} - 360 q^{89} + 25860 q^{91} + 9648 q^{93} - 21792 q^{95} - 5832 q^{99}+O(q^{100}) \) 16 * q - 72 * q^3 + 12 * q^5 + 40 * q^7 + 216 * q^9 - 108 * q^11 - 72 * q^15 - 168 * q^17 + 948 * q^19 - 252 * q^21 - 768 * q^23 + 1908 * q^25 - 1608 * q^29 - 3216 * q^31 + 972 * q^33 + 696 * q^35 - 1820 * q^37 - 1188 * q^39 + 2888 * q^43 + 324 * q^45 + 744 * q^47 - 3784 * q^49 + 504 * q^51 + 4476 * q^53 - 5688 * q^57 - 4668 * q^59 + 17760 * q^61 + 1188 * q^63 + 8760 * q^65 + 1580 * q^67 + 48 * q^71 + 588 * q^73 - 17172 * q^75 - 17508 * q^77 - 3824 * q^79 - 5832 * q^81 + 11440 * q^85 + 7236 * q^87 - 360 * q^89 + 25860 * q^91 + 9648 * q^93 - 21792 * q^95 - 5832 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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