LMFDB - Embedded newform 168.5.z.b.145.2

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} + 130 x^{14} + 6137 x^{12} + 133906 x^{10} + 1360384 x^{8} + 5425142 x^{6} + 5784425 x^{4} + \cdots + 117649 $ x^16 + 130x^14 + 6137x^12 + 133906x^10 + 1360384x^8 + 5425142x^6 + 5784425x^4 + 1717862*x^2 + 117649
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{36}\cdot 7^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			145.2
Root			$4.49703i$ of defining polynomial
Character	$\chi$	$=$	168.145
Dual form			168.5.z.b.73.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.50000 - 2.59808i) q^{3} +(-23.6709 - 13.6664i) q^{5} +(3.37250 - 48.8838i) q^{7} +(13.5000 - 23.3827i) q^{9} +O(q^{10})$	\(q+(4.50000 - 2.59808i) q^{3} +(-23.6709 - 13.6664i) q^{5} +(3.37250 - 48.8838i) q^{7} +(13.5000 - 23.3827i) q^{9} +(12.3579 + 21.4045i) q^{11} +149.394i q^{13} -142.025 q^{15} +(-436.760 + 252.164i) q^{17} +(-226.256 - 130.629i) q^{19} +(-111.828 - 228.739i) q^{21} +(-77.9002 + 134.927i) q^{23} +(61.0396 + 105.724i) q^{25} -140.296i q^{27} -694.247 q^{29} +(-335.838 + 193.896i) q^{31} +(111.221 + 64.2136i) q^{33} +(-747.894 + 1111.03i) q^{35} +(890.004 - 1541.53i) q^{37} +(388.138 + 672.275i) q^{39} +158.644i q^{41} -2145.23 q^{43} +(-639.113 + 368.992i) q^{45} +(1744.60 + 1007.24i) q^{47} +(-2378.25 - 329.721i) q^{49} +(-1310.28 + 2269.47i) q^{51} +(-2280.70 - 3950.29i) q^{53} -675.551i q^{55} -1357.54 q^{57} +(-4882.23 + 2818.75i) q^{59} +(2480.53 + 1432.14i) q^{61} +(-1097.51 - 738.789i) q^{63} +(2041.68 - 3536.29i) q^{65} +(1533.64 + 2656.34i) q^{67} +809.562i q^{69} -1639.00 q^{71} +(4377.19 - 2527.17i) q^{73} +(549.357 + 317.171i) q^{75} +(1088.01 - 531.915i) q^{77} +(3462.00 - 5996.37i) q^{79} +(-364.500 - 631.333i) q^{81} +3214.17i q^{83} +13784.7 q^{85} +(-3124.11 + 1803.71i) q^{87} +(5088.95 + 2938.11i) q^{89} +(7302.97 + 503.832i) q^{91} +(-1007.51 + 1745.07i) q^{93} +(3570.45 + 6184.19i) q^{95} -17613.4i q^{97} +667.327 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 72 q^{3} - 12 q^{5} + 16 q^{7} + 216 q^{9}+O(q^{10}) $ 16 * q + 72 * q^3 - 12 * q^5 + 16 * q^7 + 216 * q^9	$ 16 q + 72 q^{3} - 12 q^{5} + 16 q^{7} + 216 q^{9} + 252 q^{11} - 72 q^{15} - 696 q^{17} + 156 q^{19} + 108 q^{21} - 672 q^{23} + 84 q^{25} + 1992 q^{29} + 2040 q^{31} + 2268 q^{33} + 2712 q^{35} + 2548 q^{37} - 396 q^{39} + 1304 q^{43} - 324 q^{45} - 744 q^{47} - 5608 q^{49} - 2088 q^{51} - 1164 q^{53} + 936 q^{57} + 8988 q^{59} + 816 q^{61} + 540 q^{63} + 8760 q^{65} + 3044 q^{67} - 4464 q^{71} - 15828 q^{73} + 756 q^{75} + 996 q^{77} - 11144 q^{79} - 5832 q^{81} - 15344 q^{85} + 8964 q^{87} + 22248 q^{89} + 1596 q^{91} + 6120 q^{93} + 3840 q^{95} + 13608 q^{99}+O(q^{100}) $ 16 * q + 72 * q^3 - 12 * q^5 + 16 * q^7 + 216 * q^9 + 252 * q^11 - 72 * q^15 - 696 * q^17 + 156 * q^19 + 108 * q^21 - 672 * q^23 + 84 * q^25 + 1992 * q^29 + 2040 * q^31 + 2268 * q^33 + 2712 * q^35 + 2548 * q^37 - 396 * q^39 + 1304 * q^43 - 324 * q^45 - 744 * q^47 - 5608 * q^49 - 2088 * q^51 - 1164 * q^53 + 936 * q^57 + 8988 * q^59 + 816 * q^61 + 540 * q^63 + 8760 * q^65 + 3044 * q^67 - 4464 * q^71 - 15828 * q^73 + 756 * q^75 + 996 * q^77 - 11144 * q^79 - 5832 * q^81 - 15344 * q^85 + 8964 * q^87 + 22248 * q^89 + 1596 * q^91 + 6120 * q^93 + 3840 * q^95 + 13608 * 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
$5$	−23.6709	−	13.6664i	−0.946834	−	0.546655i	−0.0547382	−	0.998501i	$-0.517432\pi$
$5$	−23.6709	−	13.6664i	−0.946834	−	0.546655i	−0.892096	+	0.451846i	$0.850766\pi$
$6$		0			0
$7$	3.37250	−	48.8838i	0.0688265	−	0.997629i
$7$	3.37250	−	48.8838i	0.0688265	−	0.997629i
$8$		0			0
$9$	13.5000	−	23.3827i	0.166667	−	0.288675i
$10$		0			0
$11$	12.3579	+	21.4045i	0.102131	+	0.176897i	0.912563	−	0.408937i	$-0.134100\pi$
$11$	12.3579	+	21.4045i	0.102131	+	0.176897i	−0.810431	+	0.585834i	$0.800767\pi$
$12$		0			0
$13$			149.394i			0.883990i	0.897017	+	0.441995i	$0.145729\pi$
$13$			149.394i			0.883990i	−0.897017	+	0.441995i	$0.854271\pi$
$14$		0			0
$15$	−142.025			−0.631223
$16$		0			0
$17$	−436.760	+	252.164i	−1.51128	+	0.872539i	−0.511368	+	0.859362i	$0.670861\pi$
$17$	−436.760	+	252.164i	−1.51128	+	0.872539i	−0.999913	+	0.0131773i	$0.995805\pi$
$18$		0			0
$19$	−226.256	−	130.629i	−0.626748	−	0.361853i	0.152744	−	0.988266i	$-0.451189\pi$
$19$	−226.256	−	130.629i	−0.626748	−	0.361853i	−0.779491	+	0.626413i	$0.784522\pi$
$20$		0			0
$21$	−111.828	−	228.739i	−0.253577	−	0.518683i
$22$		0			0
$23$	−77.9002	+	134.927i	−0.147259	+	0.255061i	−0.930214	−	0.367019i	$-0.880379\pi$
$23$	−77.9002	+	134.927i	−0.147259	+	0.255061i	0.782954	+	0.622079i	$0.213712\pi$
$24$		0			0
$25$	61.0396	+	105.724i	0.0976634	+	0.169158i
$26$		0			0
$27$		−	140.296i		−	0.192450i
$28$		0			0
$29$	−694.247			−0.825502			−0.412751	−	0.910844i	$-0.635432\pi$
$29$	−694.247			−0.825502			−0.412751	+	0.910844i	$0.635432\pi$
$30$		0			0
$31$	−335.838	+	193.896i	−0.349467	+	0.201765i	−0.664451	−	0.747332i	$-0.731334\pi$
$31$	−335.838	+	193.896i	−0.349467	+	0.201765i	0.314983	+	0.949097i	$0.398001\pi$
$32$		0			0
$33$	111.221	+	64.2136i	0.102131	+	0.0589656i
$34$		0			0
$35$	−747.894	+	1111.03i	−0.610526	+	0.906965i
$36$		0			0
$37$	890.004	−	1541.53i	0.650113	−	1.12603i	−0.332982	−	0.942933i	$-0.608055\pi$
$37$	890.004	−	1541.53i	0.650113	−	1.12603i	0.983095	−	0.183095i	$-0.0586117\pi$
$38$		0			0
$39$	388.138	+	672.275i	0.255186	+	0.441995i
$40$		0			0
$41$			158.644i			0.0943748i	0.998886	+	0.0471874i	$0.0150258\pi$
$41$			158.644i			0.0943748i	−0.998886	+	0.0471874i	$0.984974\pi$
$42$		0			0
$43$	−2145.23			−1.16021			−0.580105	−	0.814542i	$-0.696989\pi$
$43$	−2145.23			−1.16021			−0.580105	+	0.814542i	$0.696989\pi$
$44$		0			0
$45$	−639.113	+	368.992i	−0.315611	+	0.182218i
$46$		0			0
$47$	1744.60	+	1007.24i	0.789769	+	0.455973i	0.839881	−	0.542770i	$-0.182625\pi$
$47$	1744.60	+	1007.24i	0.789769	+	0.455973i	−0.0501124	+	0.998744i	$0.515958\pi$
$48$		0			0
$49$	−2378.25	−	329.721i	−0.990526	−	0.137327i
$50$		0			0
$51$	−1310.28	+	2269.47i	−0.503761	+	0.872539i
$52$		0			0
$53$	−2280.70	−	3950.29i	−0.811926	−	1.40630i	−0.911514	−	0.411268i	$-0.865086\pi$
$53$	−2280.70	−	3950.29i	−0.811926	−	1.40630i	0.0995886	−	0.995029i	$-0.468247\pi$
$54$		0			0
$55$		−	675.551i		−	0.223323i
$56$		0			0
$57$	−1357.54			−0.417832
$58$		0			0
$59$	−4882.23	+	2818.75i	−1.40254	+	0.809754i	−0.994652	−	0.103280i	$-0.967066\pi$
$59$	−4882.23	+	2818.75i	−1.40254	+	0.809754i	−0.407883	+	0.913034i	$0.633733\pi$
$60$		0			0
$61$	2480.53	+	1432.14i	0.666631	+	0.384880i	0.794799	−	0.606873i	$-0.207576\pi$
$61$	2480.53	+	1432.14i	0.666631	+	0.384880i	−0.128168	+	0.991752i	$0.540910\pi$
$62$		0			0
$63$	−1097.51	−	738.789i	−0.276520	−	0.186140i
$64$		0			0
$65$	2041.68	−	3536.29i	0.483238	−	0.836992i
$66$		0			0
$67$	1533.64	+	2656.34i	0.341643	+	0.591744i	0.984738	−	0.174043i	$-0.0556832\pi$
$67$	1533.64	+	2656.34i	0.341643	+	0.591744i	−0.643095	+	0.765787i	$0.722350\pi$
$68$		0			0
$69$			809.562i			0.170040i
$70$		0			0
$71$	−1639.00			−0.325134			−0.162567	−	0.986698i	$-0.551977\pi$
$71$	−1639.00			−0.325134			−0.162567	+	0.986698i	$0.551977\pi$
$72$		0			0
$73$	4377.19	−	2527.17i	0.821390	−	0.474230i	−0.0295057	−	0.999565i	$-0.509393\pi$
$73$	4377.19	−	2527.17i	0.821390	−	0.474230i	0.850896	+	0.525335i	$0.176060\pi$
$74$		0			0
$75$	549.357	+	317.171i	0.0976634	+	0.0563860i
$76$		0			0
$77$	1088.01	−	531.915i	0.183507	−	0.0897141i
$78$		0			0
$79$	3462.00	−	5996.37i	0.554719	−	0.960802i	−0.443206	−	0.896420i	$-0.646159\pi$
$79$	3462.00	−	5996.37i	0.554719	−	0.960802i	0.997925	−	0.0643822i	$-0.0205077\pi$
$80$		0			0
$81$	−364.500	−	631.333i	−0.0555556	−	0.0962250i
$82$		0			0
$83$			3214.17i			0.466565i	0.972409	+	0.233282i	$0.0749467\pi$
$83$			3214.17i			0.466565i	−0.972409	+	0.233282i	$0.925053\pi$
$84$		0			0
$85$	13784.7			1.90791
$86$		0			0
$87$	−3124.11	+	1803.71i	−0.412751	+	0.238302i
$88$		0			0
$89$	5088.95	+	2938.11i	0.642464	+	0.370927i	0.785563	−	0.618782i	$-0.212373\pi$
$89$	5088.95	+	2938.11i	0.642464	+	0.370927i	−0.143099	+	0.989708i	$0.545707\pi$
$90$		0			0
$91$	7302.97	+	503.832i	0.881894	+	0.0608420i
$92$		0			0
$93$	−1007.51	+	1745.07i	−0.116489	+	0.201765i
$94$		0			0
$95$	3570.45	+	6184.19i	0.395617	+	0.685229i
$96$		0			0
$97$		−	17613.4i		−	1.87197i	−0.352040	−	0.935985i	$-0.614512\pi$
$97$		−	17613.4i		−	1.87197i	0.352040	−	0.935985i	$-0.385488\pi$
$98$		0			0
$99$	667.327			0.0680876
$100$		0			0
$101$	10676.2	−	6163.88i	1.04658	−	0.604243i	0.124890	−	0.992171i	$-0.460142\pi$
$101$	10676.2	−	6163.88i	1.04658	−	0.604243i	0.921690	+	0.387927i	$0.126809\pi$
$102$		0			0
$103$	5936.28	+	3427.31i	0.559551	+	0.323057i	0.752965	−	0.658060i	$-0.228623\pi$
$103$	5936.28	+	3427.31i	0.559551	+	0.323057i	−0.193414	+	0.981117i	$0.561956\pi$
$104$		0			0
$105$	−478.980	+	6942.73i	−0.0434449	+	0.629726i
$106$		0			0
$107$	−2812.20	+	4870.88i	−0.245629	+	0.425441i	−0.962308	−	0.271961i	$-0.912328\pi$
$107$	−2812.20	+	4870.88i	−0.245629	+	0.425441i	0.716680	+	0.697403i	$0.245661\pi$
$108$		0			0
$109$	−3690.30	−	6391.79i	−0.310605	−	0.537984i	0.667888	−	0.744262i	$-0.267198\pi$
$109$	−3690.30	−	6391.79i	−0.310605	−	0.537984i	−0.978494	+	0.206277i	$0.933865\pi$
$110$		0			0
$111$		−	9249.20i		−	0.750686i
$112$		0			0
$113$	−19492.6			−1.52656			−0.763279	−	0.646069i	$-0.776412\pi$
$113$	−19492.6			−1.52656			−0.763279	+	0.646069i	$0.776412\pi$
$114$		0			0
$115$	3687.93	−	2129.23i	0.278860	−	0.161000i
$116$		0			0
$117$	3493.24	+	2016.82i	0.255186	+	0.147332i
$118$		0			0
$119$	10853.7	+	22200.9i	0.766454	+	1.56775i
$120$		0			0
$121$	7015.06	−	12150.4i	0.479138	−	0.829892i
$122$		0			0
$123$	412.169	+	713.898i	0.0272436	+	0.0471874i
$124$		0			0
$125$			13746.2i			0.879757i
$126$		0			0
$127$	24198.7			1.50032			0.750162	−	0.661254i	$-0.229975\pi$
$127$	24198.7			1.50032			0.750162	+	0.661254i	$0.229975\pi$
$128$		0			0
$129$	−9653.53	+	5573.47i	−0.580105	+	0.334924i
$130$		0			0
$131$	−16483.1	−	9516.53i	−0.960499	−	0.554544i	−0.0641721	−	0.997939i	$-0.520441\pi$
$131$	−16483.1	−	9516.53i	−0.960499	−	0.554544i	−0.896326	+	0.443395i	$0.853774\pi$
$132$		0			0
$133$	−7148.68	+	10619.7i	−0.404132	+	0.600356i
$134$		0			0
$135$	−1917.34	+	3320.93i	−0.105204	+	0.182218i
$136$		0			0
$137$	−8595.53	−	14887.9i	−0.457964	−	0.793217i	0.540889	−	0.841094i	$-0.318088\pi$
$137$	−8595.53	−	14887.9i	−0.457964	−	0.793217i	−0.998853	+	0.0478770i	$0.984754\pi$
$138$		0			0
$139$		−	29742.1i		−	1.53936i	−0.638428	−	0.769682i	$-0.720415\pi$
$139$		−	29742.1i		−	1.53936i	0.638428	−	0.769682i	$-0.279585\pi$
$140$		0			0
$141$	10467.6			0.526512
$142$		0			0
$143$	−3197.72	+	1846.20i	−0.156375	+	0.0902832i
$144$		0			0
$145$	16433.4	+	9487.85i	0.781614	+	0.451265i
$146$		0			0
$147$	−11558.8	+	4695.14i	−0.534906	+	0.217277i
$148$		0			0
$149$	4132.09	−	7157.00i	0.186122	−	0.322373i	−0.757832	−	0.652450i	$-0.773741\pi$
$149$	4132.09	−	7157.00i	0.186122	−	0.322373i	0.943954	+	0.330077i	$0.107075\pi$
$150$		0			0
$151$	−17444.8	−	30215.2i	−0.765087	−	1.32517i	−0.940200	−	0.340622i	$-0.889362\pi$
$151$	−17444.8	−	30215.2i	−0.765087	−	1.32517i	0.175113	−	0.984548i	$-0.443971\pi$
$152$		0			0
$153$			13616.8i			0.581693i
$154$		0			0
$155$	10599.4			0.441184
$156$		0			0
$157$	−17304.5	+	9990.77i	−0.702037	+	0.405321i	−0.808106	−	0.589037i	$-0.799507\pi$
$157$	−17304.5	+	9990.77i	−0.702037	+	0.405321i	0.106068	+	0.994359i	$0.466174\pi$
$158$		0			0
$159$	−20526.3	−	11850.9i	−0.811926	−	0.468766i
$160$		0			0
$161$	6333.03	+	4263.10i	0.244320	+	0.164465i
$162$		0			0
$163$	−12834.0	+	22229.1i	−0.483044	+	0.836656i	−0.999810	−	0.0194701i	$-0.993802\pi$
$163$	−12834.0	+	22229.1i	−0.483044	+	0.836656i	0.516767	+	0.856126i	$0.327135\pi$
$164$		0			0
$165$	−1755.13	−	3039.98i	−0.0644677	−	0.111661i
$166$		0			0
$167$			6342.22i			0.227409i	0.993515	+	0.113705i	$0.0362718\pi$
$167$			6342.22i			0.227409i	−0.993515	+	0.113705i	$0.963728\pi$
$168$		0			0
$169$	6242.32			0.218561
$170$		0			0
$171$	−6108.91	+	3526.98i	−0.208916	+	0.120618i
$172$		0			0
$173$	−29998.9	−	17319.9i	−1.00234	−	0.578698i	−0.0933968	−	0.995629i	$-0.529773\pi$
$173$	−29998.9	−	17319.9i	−1.00234	−	0.578698i	−0.908938	+	0.416930i	$0.863106\pi$
$174$		0			0
$175$	5374.03	−	2627.30i	0.175479	−	0.0857893i
$176$		0			0
$177$	−14646.7	+	25368.8i	−0.467512	+	0.809754i
$178$		0			0
$179$	−12607.8	−	21837.4i	−0.393491	−	0.681546i	0.599417	−	0.800437i	$-0.295399\pi$
$179$	−12607.8	−	21837.4i	−0.393491	−	0.681546i	−0.992907	+	0.118891i	$0.962066\pi$
$180$		0			0
$181$			53927.8i			1.64610i	0.567971	+	0.823049i	$0.307729\pi$
$181$			53927.8i			1.64610i	−0.567971	+	0.823049i	$0.692271\pi$
$182$		0			0
$183$	14883.2			0.444421
$184$		0			0
$185$	−42134.3	+	24326.3i	−1.23110	+	0.710775i
$186$		0			0
$187$	−10794.9	−	6232.43i	−0.308699	−	0.178227i
$188$		0			0
$189$	−6858.21	−	473.148i	−0.191994	−	0.0132457i
$190$		0			0
$191$	6083.33	−	10536.6i	0.166753	−	0.288826i	−0.770523	−	0.637412i	$-0.780005\pi$
$191$	6083.33	−	10536.6i	0.166753	−	0.288826i	0.937277	+	0.348587i	$0.113338\pi$
$192$		0			0
$193$	3402.39	+	5893.11i	0.0913418	+	0.158209i	0.908076	−	0.418806i	$-0.137551\pi$
$193$	3402.39	+	5893.11i	0.0913418	+	0.158209i	−0.816734	+	0.577014i	$0.804218\pi$
$194$		0			0
$195$		−	21217.8i		−	0.557995i
$196$		0			0
$197$	−2995.35			−0.0771819			−0.0385910	−	0.999255i	$-0.512287\pi$
$197$	−2995.35			−0.0771819			−0.0385910	+	0.999255i	$0.512287\pi$
$198$		0			0
$199$	−27169.6	+	15686.4i	−0.686083	+	0.396110i	−0.802143	−	0.597132i	$-0.796307\pi$
$199$	−27169.6	+	15686.4i	−0.686083	+	0.396110i	0.116060	+	0.993242i	$0.462973\pi$
$200$		0			0
$201$	13802.7	+	7969.01i	0.341643	+	0.197248i
$202$		0			0
$203$	−2341.35	+	33937.5i	−0.0568164	+	0.823545i
$204$		0			0
$205$	2168.09	−	3755.24i	0.0515904	−	0.0893573i
$206$		0			0
$207$	2103.30	+	3643.03i	0.0490864	+	0.0850202i
$208$		0			0
$209$		−	6457.20i		−	0.147826i
$210$		0			0
$211$	4762.43			0.106970			0.0534852	−	0.998569i	$-0.482967\pi$
$211$	4762.43			0.106970			0.0534852	+	0.998569i	$0.482967\pi$
$212$		0			0
$213$	−7375.49	+	4258.24i	−0.162567	+	0.0938580i
$214$		0			0
$215$	50779.4	+	29317.5i	1.09853	+	0.634235i
$216$		0			0
$217$	8345.77	+	17071.0i	0.177234	+	0.362525i
$218$		0			0
$219$	13131.6	−	22744.5i	0.273797	−	0.474230i
$220$		0			0
$221$	−37671.8	−	65249.5i	−0.771316	−	1.33596i
$222$		0			0
$223$		−	66913.9i		−	1.34557i	−0.739838	−	0.672786i	$-0.765098\pi$
$223$		−	66913.9i		−	1.34557i	0.739838	−	0.672786i	$-0.234902\pi$
$224$		0			0
$225$	3296.14			0.0651089
$226$		0			0
$227$	−61644.1	+	35590.2i	−1.19630	+	0.690684i	−0.959728	−	0.280931i	$-0.909357\pi$
$227$	−61644.1	+	35590.2i	−1.19630	+	0.690684i	−0.236571	+	0.971614i	$0.576024\pi$
$228$		0			0
$229$	−4917.28	−	2838.99i	−0.0937679	−	0.0541369i	0.452383	−	0.891824i	$-0.350574\pi$
$229$	−4917.28	−	2838.99i	−0.0937679	−	0.0541369i	−0.546151	+	0.837687i	$0.683907\pi$
$230$		0			0
$231$	3514.10	−	5220.35i	0.0658551	−	0.0978309i
$232$		0			0
$233$	39621.4	−	68626.2i	0.729823	−	1.26409i	−0.227135	−	0.973863i	$-0.572936\pi$
$233$	39621.4	−	68626.2i	0.729823	−	1.26409i	0.956958	−	0.290227i	$-0.0937309\pi$
$234$		0			0
$235$	−27530.8	−	47684.7i	−0.498520	−	0.863462i
$236$		0			0
$237$		−	35978.2i		−	0.640535i
$238$		0			0
$239$	71123.0			1.24513			0.622564	−	0.782569i	$-0.286091\pi$
$239$	71123.0			1.24513			0.622564	+	0.782569i	$0.286091\pi$
$240$		0			0
$241$	−94065.5	+	54308.7i	−1.61956	+	0.935052i	−0.632523	+	0.774542i	$0.717981\pi$
$241$	−94065.5	+	54308.7i	−1.61956	+	0.935052i	−0.987034	+	0.160510i	$0.948686\pi$
$242$		0			0
$243$	−3280.50	−	1894.00i	−0.0555556	−	0.0320750i
$244$		0			0
$245$	51789.2	+	40306.9i	0.862794	+	0.671501i
$246$		0			0
$247$	19515.2	−	33801.4i	0.319874	−	0.554039i
$248$		0			0
$249$	8350.65	+	14463.7i	0.134686	+	0.233282i
$250$		0			0
$251$			59487.9i			0.944238i	0.881535	+	0.472119i	$0.156511\pi$
$251$			59487.9i			0.944238i	−0.881535	+	0.472119i	$0.843489\pi$
$252$		0			0
$253$	−3850.73			−0.0601592
$254$		0			0
$255$	62031.0	−	35813.6i	0.953955	−	0.550766i
$256$		0			0
$257$	39432.1	+	22766.1i	0.597012	+	0.344685i	0.767865	−	0.640611i	$-0.221319\pi$
$257$	39432.1	+	22766.1i	0.597012	+	0.344685i	−0.170853	+	0.985297i	$0.554652\pi$
$258$		0			0
$259$	−72354.4	−	48705.6i	−1.07861	−	0.726072i
$260$		0			0
$261$	−9372.34	+	16233.4i	−0.137584	+	0.238302i
$262$		0			0
$263$	−14537.1	−	25179.0i	−0.210168	−	0.364022i	0.741599	−	0.670844i	$-0.234068\pi$
$263$	−14537.1	−	25179.0i	−0.210168	−	0.364022i	−0.951767	+	0.306822i	$0.900734\pi$
$264$		0			0
$265$			124676.i			1.77537i
$266$		0			0
$267$	30533.7			0.428309
$268$		0			0
$269$	−45684.8	+	26376.1i	−0.631345	+	0.364507i	−0.781273	−	0.624190i	$-0.785429\pi$
$269$	−45684.8	+	26376.1i	−0.631345	+	0.364507i	0.149928	+	0.988697i	$0.452096\pi$
$270$		0			0
$271$	94981.3	+	54837.5i	1.29330	+	0.746687i	0.979238	−	0.202715i	$-0.0649764\pi$
$271$	94981.3	+	54837.5i	1.29330	+	0.746687i	0.314063	+	0.949402i	$0.398310\pi$
$272$		0			0
$273$	34172.3	−	16706.4i	0.458511	−	0.224160i
$274$		0			0
$275$	−1508.64	+	2613.05i	−0.0199490	+	0.0345527i
$276$		0			0
$277$	15144.9	+	26231.7i	0.197382	+	0.341875i	0.947679	−	0.319226i	$-0.103423\pi$
$277$	15144.9	+	26231.7i	0.197382	+	0.341875i	−0.750297	+	0.661101i	$0.770090\pi$
$278$		0			0
$279$			10470.4i			0.134510i
$280$		0			0
$281$	103643.			1.31258			0.656290	−	0.754509i	$-0.272125\pi$
$281$	103643.			1.31258			0.656290	+	0.754509i	$0.272125\pi$
$282$		0			0
$283$	−100067.	+	57773.8i	−1.24945	+	0.721370i	−0.971000	−	0.239081i	$-0.923154\pi$
$283$	−100067.	+	57773.8i	−1.24945	+	0.721370i	−0.278449	+	0.960451i	$0.589820\pi$
$284$		0			0
$285$	32134.0	+	18552.6i	0.395617	+	0.228410i
$286$		0			0
$287$	7755.12	+	535.027i	0.0941510	+	0.00649548i
$288$		0			0
$289$	85412.6	−	147939.i	1.02265	−	1.77128i
$290$		0			0
$291$	−45760.9	−	79260.1i	−0.540391	−	0.935985i
$292$		0			0
$293$		−	91168.1i		−	1.06196i	−0.847385	−	0.530980i	$-0.821824\pi$
$293$		−	91168.1i		−	1.06196i	0.847385	−	0.530980i	$-0.178176\pi$
$294$		0			0
$295$	154089.			1.77063
$296$		0			0
$297$	3002.97	−	1733.77i	0.0340438	−	0.0196552i
$298$		0			0
$299$	−20157.3	−	11637.8i	−0.225471	−	0.130176i
$300$		0			0
$301$	−7234.78	+	104867.i	−0.0798532	+	1.15746i
$302$		0			0
$303$	32028.5	−	55475.0i	0.348860	−	0.604243i
$304$		0			0
$305$	−39144.2	−	67799.8i	−0.420793	−	0.728834i
$306$		0			0
$307$			79064.7i			0.838891i	0.907780	+	0.419446i	$0.137776\pi$
$307$			79064.7i			0.838891i	−0.907780	+	0.419446i	$0.862224\pi$
$308$		0			0
$309$	35617.7			0.373034
$310$		0			0
$311$	154471.	−	89183.9i	1.59708	−	0.922074i	0.605033	−	0.796200i	$-0.293160\pi$
$311$	154471.	−	89183.9i	1.59708	−	0.922074i	0.992046	−	0.125874i	$-0.0401734\pi$
$312$		0			0
$313$	−4546.16	−	2624.73i	−0.0464041	−	0.0267914i	0.476618	−	0.879110i	$-0.341862\pi$
$313$	−4546.16	−	2624.73i	−0.0464041	−	0.0267914i	−0.523023	+	0.852319i	$0.675196\pi$
$314$		0			0
$315$	15882.3	+	32486.7i	0.160064	+	0.327404i
$316$		0			0
$317$	32566.2	−	56406.3i	0.324077	−	0.561318i	−0.657248	−	0.753674i	$-0.728280\pi$
$317$	32566.2	−	56406.3i	0.324077	−	0.561318i	0.981325	+	0.192356i	$0.0616129\pi$
$318$		0			0
$319$	−8579.44	−	14860.0i	−0.0843098	−	0.146029i
$320$		0			0
$321$			29225.3i			0.283627i
$322$		0			0
$323$	131759.			1.26292
$324$		0			0
$325$	−15794.5	+	9118.98i	−0.149534	+	0.0863335i
$326$		0			0
$327$	−33212.7	−	19175.4i	−0.310605	−	0.179328i
$328$		0			0
$329$	55121.6	−	81885.7i	0.509249	−	0.756513i
$330$		0			0
$331$	−86682.9	+	150139.i	−0.791184	+	1.37037i	0.134051	+	0.990974i	$0.457201\pi$
$331$	−86682.9	+	150139.i	−0.791184	+	1.37037i	−0.925234	+	0.379396i	$0.876132\pi$
$332$		0			0
$333$	−24030.1	−	41621.4i	−0.216704	−	0.375343i
$334$		0			0
$335$		−	83837.0i		−	0.747044i
$336$		0			0
$337$	−94176.4			−0.829244			−0.414622	−	0.909994i	$-0.636086\pi$
$337$	−94176.4			−0.829244			−0.414622	+	0.909994i	$0.636086\pi$
$338$		0			0
$339$	−87716.8	+	50643.3i	−0.763279	+	0.440679i
$340$		0			0
$341$	−8300.51	−	4792.30i	−0.0713832	−	0.0412131i
$342$		0			0
$343$	−24138.7	+	115146.i	−0.205175	+	0.978725i
$344$		0			0
$345$	11063.8	−	19163.0i	0.0929535	−	0.161000i
$346$		0			0
$347$	42968.0	+	74422.8i	0.356850	+	0.618083i	0.987433	−	0.158039i	$-0.0505172\pi$
$347$	42968.0	+	74422.8i	0.356850	+	0.618083i	−0.630582	+	0.776122i	$0.717184\pi$
$348$		0			0
$349$		−	80938.5i		−	0.664514i	−0.943189	−	0.332257i	$-0.892190\pi$
$349$		−	80938.5i		−	0.664514i	0.943189	−	0.332257i	$-0.107810\pi$
$350$		0			0
$351$	20959.5			0.170124
$352$		0			0
$353$	47601.7	−	27482.8i	0.382008	−	0.220553i	−0.296683	−	0.954976i	$-0.595881\pi$
$353$	47601.7	−	27482.8i	0.382008	−	0.220553i	0.678692	+	0.734423i	$0.262547\pi$
$354$		0			0
$355$	38796.5	+	22399.2i	0.307848	+	0.177736i
$356$		0			0
$357$	106522.	+	71705.3i	0.835798	+	0.562620i
$358$		0			0
$359$	−100636.	+	174307.i	−0.780845	+	1.35246i	0.150605	+	0.988594i	$0.451878\pi$
$359$	−100636.	+	174307.i	−0.780845	+	1.35246i	−0.931450	+	0.363870i	$0.881455\pi$
$360$		0			0
$361$	−31032.7	−	53750.2i	−0.238125	−	0.412445i
$362$		0			0
$363$		−	72902.7i		−	0.553261i
$364$		0			0
$365$	−138149.			−1.03696
$366$		0			0
$367$	−197955.	+	114289.i	−1.46972	+	0.848543i	−0.999423	−	0.0339657i	$-0.989186\pi$
$367$	−197955.	+	114289.i	−1.46972	+	0.848543i	−0.470296	+	0.882509i	$0.655853\pi$
$368$		0			0
$369$	3709.52	+	2141.69i	0.0272436	+	0.0157291i
$370$		0			0
$371$	−200797.	+	98166.9i	−1.45884	+	0.713210i
$372$		0			0
$373$	52586.5	−	91082.5i	0.377969	−	0.654662i	−0.612797	−	0.790240i	$-0.709956\pi$
$373$	52586.5	−	91082.5i	0.377969	−	0.654662i	0.990767	+	0.135578i	$0.0432891\pi$
$374$		0			0
$375$	35713.7	+	61857.9i	0.253964	+	0.439879i
$376$		0			0
$377$		−	103717.i		−	0.729736i
$378$		0			0
$379$	−221091.			−1.53919			−0.769595	−	0.638532i	$-0.779542\pi$
$379$	−221091.			−1.53919			−0.769595	+	0.638532i	$0.779542\pi$
$380$		0			0
$381$	108894.	−	62870.1i	0.750162	−	0.433106i
$382$		0			0
$383$	−17227.8	−	9946.46i	−0.117444	−	0.0678064i	0.440127	−	0.897935i	$-0.354933\pi$
$383$	−17227.8	−	9946.46i	−0.117444	−	0.0678064i	−0.557571	+	0.830129i	$0.688267\pi$
$384$		0			0
$385$	−33023.5	−	2278.29i	−0.222793	−	0.0153705i
$386$		0			0
$387$	−28960.6	+	50161.2i	−0.193368	+	0.334924i
$388$		0			0
$389$	−36139.5	−	62595.5i	−0.238827	−	0.413660i	0.721551	−	0.692361i	$-0.243430\pi$
$389$	−36139.5	−	62595.5i	−0.238827	−	0.413660i	−0.960378	+	0.278701i	$0.910096\pi$
$390$		0			0
$391$		−	78574.4i		−	0.513958i
$392$		0			0
$393$	−98898.7			−0.640332
$394$		0			0
$395$	−163897.	+	94626.1i	−1.05045	+	0.606480i
$396$		0			0
$397$	−156524.	−	90369.4i	−0.993119	−	0.573377i	−0.0869137	−	0.996216i	$-0.527700\pi$
$397$	−156524.	−	90369.4i	−0.993119	−	0.573377i	−0.906205	+	0.422838i	$0.861034\pi$
$398$		0			0
$399$	−4578.28	+	66361.5i	−0.0287579	+	0.416841i
$400$		0			0
$401$	−40246.6	+	69709.1i	−0.250288	+	0.433512i	−0.963605	−	0.267330i	$-0.913859\pi$
$401$	−40246.6	+	69709.1i	−0.250288	+	0.433512i	0.713317	+	0.700842i	$0.247192\pi$
$402$		0			0
$403$	−28967.0	−	50172.3i	−0.178358	−	0.308926i
$404$		0			0
$405$			19925.6i			0.121479i
$406$		0			0
$407$	43994.4			0.265588
$408$		0			0
$409$	−28413.3	+	16404.5i	−0.169854	+	0.0980653i	−0.582517	−	0.812819i	$-0.697932\pi$
$409$	−28413.3	+	16404.5i	−0.169854	+	0.0980653i	0.412663	+	0.910884i	$0.364599\pi$
$410$		0			0
$411$	−77359.7	−	44663.7i	−0.457964	−	0.264406i
$412$		0			0
$413$	121326.	+	248168.i	0.711303	+	1.45494i
$414$		0			0
$415$	43926.0	−	76082.0i	0.255050	−	0.441760i
$416$		0			0
$417$	−77272.1	−	133839.i	−0.444376	−	0.769682i
$418$		0			0
$419$		−	219452.i		−	1.25001i	−0.780623	−	0.625003i	$-0.785098\pi$
$419$		−	219452.i		−	1.25001i	0.780623	−	0.625003i	$-0.214902\pi$
$420$		0			0
$421$	66401.6			0.374640			0.187320	−	0.982299i	$-0.440020\pi$
$421$	66401.6			0.374640			0.187320	+	0.982299i	$0.440020\pi$
$422$		0			0
$423$	47104.2	−	27195.6i	0.263256	−	0.151991i
$424$		0			0
$425$	−53319.4	−	30784.0i	−0.295194	−	0.170430i
$426$		0			0
$427$	78373.9	−	116428.i	0.429849	−	0.638560i
$428$		0			0
$429$	−9593.15	+	16615.8i	−0.0521250	+	0.0902832i
$430$		0			0
$431$	76560.1	+	132606.i	0.412143	+	0.713853i	0.995124	−	0.0986333i	$-0.0314471\pi$
$431$	76560.1	+	132606.i	0.412143	+	0.713853i	−0.582981	+	0.812486i	$0.698114\pi$
$432$		0			0
$433$			266246.i			1.42006i	0.704171	+	0.710030i	$0.251319\pi$
$433$			266246.i			1.42006i	−0.704171	+	0.710030i	$0.748681\pi$
$434$		0			0
$435$	98600.6			0.521076
$436$		0			0
$437$	35250.7	−	20352.0i	0.184589	−	0.106572i
$438$		0			0
$439$	285221.	+	164672.i	1.47997	+	0.854460i	0.999743	−	0.0226845i	$-0.00722131\pi$
$439$	285221.	+	164672.i	1.47997	+	0.854460i	0.480226	+	0.877145i	$0.340555\pi$
$440$		0			0
$441$	−39816.2	+	51158.7i	−0.204730	+	0.263052i
$442$		0			0
$443$	−45733.5	+	79212.8i	−0.233038	+	0.403634i	−0.958701	−	0.284417i	$-0.908200\pi$
$443$	−45733.5	+	79212.8i	−0.233038	+	0.403634i	0.725662	+	0.688051i	$0.241533\pi$
$444$		0			0
$445$	−80306.6	−	139095.i	−0.405538	−	0.702412i
$446$		0			0
$447$		−	42942.0i		−	0.214915i
$448$		0			0
$449$	16690.5			0.0827897			0.0413949	−	0.999143i	$-0.486820\pi$
$449$	16690.5			0.0827897			0.0413949	+	0.999143i	$0.486820\pi$
$450$		0			0
$451$	−3395.70	+	1960.51i	−0.0166946	+	0.00963863i
$452$		0			0
$453$	−157003.	−	90645.6i	−0.765087	−	0.441723i
$454$		0			0
$455$	−165982.	−	111731.i	−0.801748	−	0.539699i
$456$		0			0
$457$	23113.8	−	40034.2i	0.110672	−	0.191690i	−0.805369	−	0.592773i	$-0.798033\pi$
$457$	23113.8	−	40034.2i	0.110672	−	0.191690i	0.916041	+	0.401084i	$0.131366\pi$
$458$		0			0
$459$	35377.6	+	61275.8i	0.167920	+	0.290846i
$460$		0			0
$461$			152806.i			0.719017i	0.933142	+	0.359508i	$0.117056\pi$
$461$			152806.i			0.719017i	−0.933142	+	0.359508i	$0.882944\pi$
$462$		0			0
$463$	−97906.5			−0.456719			−0.228360	−	0.973577i	$-0.573336\pi$
$463$	−97906.5			−0.456719			−0.228360	+	0.973577i	$0.573336\pi$
$464$		0			0
$465$	47697.5	−	27538.1i	0.220592	−	0.127359i
$466$		0			0
$467$	−68132.3	−	39336.2i	−0.312406	−	0.180368i	0.335597	−	0.942006i	$-0.391062\pi$
$467$	−68132.3	−	39336.2i	−0.312406	−	0.180368i	−0.648003	+	0.761638i	$0.724395\pi$
$468$		0			0
$469$	135024.	−	66011.5i	0.613854	−	0.300106i
$470$		0			0
$471$	−51913.6	+	89916.9i	−0.234012	+	0.405321i
$472$		0			0
$473$	−26510.5	−	45917.6i	−0.118494	−	0.205238i
$474$		0			0
$475$		−	31894.2i		−	0.141359i
$476$		0			0
$477$	−123158.			−0.541284
$478$		0			0
$479$	−150006.	+	86606.0i	−0.653789	+	0.377465i	−0.789906	−	0.613227i	$-0.789871\pi$
$479$	−150006.	+	86606.0i	−0.653789	+	0.377465i	0.136117	+	0.990693i	$0.456538\pi$
$480$		0			0
$481$	230296.	+	132962.i	0.995398	+	0.574693i
$482$		0			0
$483$	39574.5	+	2730.25i	0.169637	+	0.0117033i
$484$		0			0
$485$	−240711.	+	416923.i	−1.02332	+	1.77245i
$486$		0			0
$487$	44804.2	+	77603.2i	0.188913	+	0.327206i	0.944888	−	0.327394i	$-0.106170\pi$
$487$	44804.2	+	77603.2i	0.188913	+	0.327206i	−0.755975	+	0.654600i	$0.772837\pi$
$488$		0			0
$489$			133375.i			0.557771i
$490$		0			0
$491$	325059.			1.34834			0.674169	−	0.738577i	$-0.264502\pi$
$491$	325059.			1.34834			0.674169	+	0.738577i	$0.264502\pi$
$492$		0			0
$493$	303220.	−	175064.i	1.24757	−	0.720283i
$494$		0			0
$495$	−15796.2	−	9119.94i	−0.0644677	−	0.0372204i
$496$		0			0
$497$	−5527.52	+	80120.5i	−0.0223778	+	0.324363i
$498$		0			0
$499$	195906.	−	339320.i	0.786770	−	1.36273i	−0.141166	−	0.989986i	$-0.545085\pi$
$499$	195906.	−	339320.i	0.786770	−	1.36273i	0.927936	−	0.372740i	$-0.121582\pi$
$500$		0			0
$501$	16477.6	+	28540.0i	0.0656474	+	0.113705i
$502$		0			0
$503$		−	259346.i		−	1.02505i	−0.858674	−	0.512523i	$-0.828711\pi$
$503$		−	259346.i		−	1.02505i	0.858674	−	0.512523i	$-0.171289\pi$
$504$		0			0
$505$	−336952.			−1.32125
$506$		0			0
$507$	28090.4	−	16218.0i	0.109281	−	0.0630931i
$508$		0			0
$509$	−196342.	−	113358.i	−0.757841	−	0.437540i	0.0706791	−	0.997499i	$-0.477483\pi$
$509$	−196342.	−	113358.i	−0.757841	−	0.437540i	−0.828520	+	0.559959i	$0.810817\pi$
$510$		0			0
$511$	−108776.	−	222496.i	−0.416572	−	0.852082i
$512$		0			0
$513$	−18326.7	+	31742.8i	−0.0696386	+	0.120618i
$514$		0			0
$515$	−93677.9	−	162255.i	−0.353202	−	0.611763i
$516$		0			0
$517$			49789.7i			0.186277i
$518$		0			0
$519$	−179993.			−0.668223
$520$		0			0
$521$	219919.	−	126970.i	0.810190	−	0.467764i	−0.0368317	−	0.999321i	$-0.511727\pi$
$521$	219919.	−	126970.i	0.810190	−	0.467764i	0.847022	+	0.531558i	$0.178393\pi$
$522$		0			0
$523$	155738.	+	89915.4i	0.569365	+	0.328723i	0.756896	−	0.653536i	$-0.226715\pi$
$523$	155738.	+	89915.4i	0.569365	+	0.328723i	−0.187530	+	0.982259i	$0.560048\pi$
$524$		0			0
$525$	17357.2	−	25785.0i	0.0629741	−	0.0935510i
$526$		0			0
$527$	97787.2	−	169372.i	0.352096	−	0.609848i
$528$		0			0
$529$	127784.	+	221328.i	0.456629	+	0.790905i
$530$		0			0
$531$			152213.i			0.539836i
$532$		0			0
$533$	−23700.5			−0.0834264
$534$		0			0
$535$	133134.	−	76865.2i	0.465139	−	0.268548i
$536$		0			0
$537$	−113470.	−	65512.2i	−0.393491	−	0.227182i
$538$		0			0
$539$	−22332.7	−	54980.0i	−0.0768712	−	0.189246i
$540$		0			0
$541$	239795.	−	415338.i	0.819307	−	1.41908i	−0.0868872	−	0.996218i	$-0.527692\pi$
$541$	239795.	−	415338.i	0.819307	−	1.41908i	0.906194	−	0.422863i	$-0.138975\pi$
$542$		0			0
$543$	140108.	+	242675.i	0.475187	+	0.823049i
$544$		0			0
$545$			201732.i			0.679176i
$546$		0			0
$547$	161964.			0.541308			0.270654	−	0.962677i	$-0.412760\pi$
$547$	161964.			0.541308			0.270654	+	0.962677i	$0.412760\pi$
$548$		0			0
$549$	66974.4	−	38667.7i	0.222210	−	0.128293i
$550$		0			0
$551$	157078.	+	90688.8i	0.517382	+	0.298710i
$552$		0			0
$553$	−281450.	−	189459.i	−0.920344	−	0.619532i
$554$		0			0
$555$	−126403.	+	218936.i	−0.410366	+	0.710775i
$556$		0			0
$557$	225560.	+	390682.i	0.727029	+	1.25925i	0.958133	+	0.286322i	$0.0924329\pi$
$557$	225560.	+	390682.i	0.727029	+	1.25925i	−0.231104	+	0.972929i	$0.574234\pi$
$558$		0			0
$559$		−	320485.i		−	1.02561i
$560$		0			0
$561$	−64769.3			−0.205799
$562$		0			0
$563$	304567.	−	175842.i	0.960873	−	0.554760i	0.0644315	−	0.997922i	$-0.479477\pi$
$563$	304567.	−	175842.i	0.960873	−	0.554760i	0.896442	+	0.443162i	$0.146143\pi$
$564$		0			0
$565$	461407.	+	266394.i	1.44540	+	0.834501i
$566$		0			0
$567$	−32091.2	+	15689.0i	−0.0998206	+	0.0488010i
$568$		0			0
$569$	228936.	−	396528.i	0.707113	−	1.22476i	−0.258810	−	0.965928i	$-0.583330\pi$
$569$	228936.	−	396528.i	0.707113	−	1.22476i	0.965923	−	0.258828i	$-0.0833362\pi$
$570$		0			0
$571$	310705.	+	538157.i	0.952963	+	1.65058i	0.738962	+	0.673747i	$0.235316\pi$
$571$	310705.	+	538157.i	0.952963	+	1.65058i	0.214001	+	0.976833i	$0.431350\pi$
$572$		0			0
$573$		−	63219.9i		−	0.192550i
$574$		0			0
$575$	−19020.0			−0.0575274
$576$		0			0
$577$	−2926.91	+	1689.85i	−0.00879140	+	0.00507572i	−0.504389	−	0.863476i	$-0.668282\pi$
$577$	−2926.91	+	1689.85i	−0.00879140	+	0.00507572i	0.495598	+	0.868552i	$0.334949\pi$
$578$		0			0
$579$	30621.5	+	17679.3i	0.0913418	+	0.0527362i
$580$		0			0
$581$	157121.	+	10839.8i	0.465458	+	0.0321120i
$582$		0			0
$583$	56369.3	−	97634.6i	0.165846	−	0.287254i
$584$		0			0
$585$	−55125.3	−	95479.9i	−0.161079	−	0.278997i
$586$		0			0
$587$		−	353245.i		−	1.02518i	−0.858634	−	0.512590i	$-0.828686\pi$
$587$		−	353245.i		−	1.02518i	0.858634	−	0.512590i	$-0.171314\pi$
$588$		0			0
$589$	101314.			0.292037
$590$		0			0
$591$	−13479.1	+	7782.15i	−0.0385910	+	0.0222805i
$592$		0			0
$593$	253975.	+	146633.i	0.722241	+	0.416986i	0.815577	−	0.578648i	$-0.196420\pi$
$593$	253975.	+	146633.i	0.722241	+	0.416986i	−0.0933357	+	0.995635i	$0.529753\pi$
$594$		0			0
$595$	46488.7	−	673846.i	0.131315	−	1.90339i
$596$		0			0
$597$	−81508.7	+	141177.i	−0.228694	+	0.396110i
$598$		0			0
$599$	−80838.5	−	140016.i	−0.225302	−	0.390234i	0.731108	−	0.682262i	$-0.239003\pi$
$599$	−80838.5	−	140016.i	−0.225302	−	0.390234i	−0.956410	+	0.292027i	$0.905670\pi$
$600$		0			0
$601$			253089.i			0.700687i	0.936621	+	0.350344i	$0.113935\pi$
$601$			253089.i			0.700687i	−0.936621	+	0.350344i	$0.886065\pi$
$602$		0			0
$603$	82816.4			0.227762
$604$		0			0
$605$	−332105.	+	191741.i	−0.907329	+	0.523847i
$606$		0			0
$607$	−114620.	−	66175.7i	−0.311087	−	0.179606i	0.336326	−	0.941746i	$-0.390816\pi$
$607$	−114620.	−	66175.7i	−0.311087	−	0.179606i	−0.647413	+	0.762139i	$0.724149\pi$
$608$		0			0
$609$	77636.0	+	158802.i	0.209329	+	0.428174i
$610$		0			0
$611$	−150477.	+	260633.i	−0.403076	+	0.698148i
$612$		0			0
$613$	−198555.	−	343907.i	−0.528396	−	0.915209i	−0.999452	−	0.0331052i	$-0.989460\pi$
$613$	−198555.	−	343907.i	−0.528396	−	0.915209i	0.471056	−	0.882103i	$-0.343873\pi$
$614$		0			0
$615$		−	22531.4i		−	0.0595715i
$616$		0			0
$617$	−661554.			−1.73778			−0.868891	−	0.495004i	$-0.835167\pi$
$617$	−661554.			−1.73778			−0.868891	+	0.495004i	$0.835167\pi$
$618$		0			0
$619$	−181433.	+	104751.i	−0.473517	+	0.273385i	−0.717711	−	0.696341i	$-0.754810\pi$
$619$	−181433.	+	104751.i	−0.473517	+	0.273385i	0.244194	+	0.969726i	$0.421477\pi$
$620$		0			0
$621$	18929.7	+	10929.1i	0.0490864	+	0.0283401i
$622$		0			0
$623$	160788.	−	238859.i	0.414265	−	0.615411i
$624$		0			0
$625$	226011.	−	391462.i	0.578587	−	1.00214i
$626$		0			0
$627$	−16776.3	−	29057.4i	−0.0426738	−	0.0739131i
$628$		0			0
$629$			897707.i			2.26899i
$630$		0			0
$631$	529135.			1.32895			0.664473	−	0.747312i	$-0.268656\pi$
$631$	529135.			1.32895			0.664473	+	0.747312i	$0.268656\pi$
$632$		0			0
$633$	21430.9	−	12373.2i	0.0534852	−	0.0308797i
$634$		0			0
$635$	−572805.	−	330709.i	−1.42056	−	0.820160i
$636$		0			0
$637$	49258.5	−	355298.i	0.121395	−	0.875615i
$638$		0			0
$639$	−22126.5	+	38324.2i	−0.0541889	+	0.0938580i
$640$		0			0
$641$	363986.	+	630442.i	0.885867	+	1.53437i	0.844717	+	0.535214i	$0.179769\pi$
$641$	363986.	+	630442.i	0.885867	+	1.53437i	0.0411505	+	0.999153i	$0.486898\pi$
$642$		0			0
$643$			402810.i			0.974267i	0.873327	+	0.487134i	$0.161958\pi$
$643$			402810.i			0.974267i	−0.873327	+	0.487134i	$0.838042\pi$
$644$		0			0
$645$	304676.			0.732351
$646$		0			0
$647$	241852.	−	139633.i	0.577752	−	0.333565i	−0.182487	−	0.983208i	$-0.558415\pi$
$647$	241852.	−	139633.i	0.577752	−	0.333565i	0.760240	+	0.649643i	$0.225082\pi$
$648$		0			0
$649$	−120668.	−	69667.8i	−0.286486	−	0.165403i
$650$		0			0
$651$	81907.6	+	55136.4i	0.193269	+	0.130100i
$652$		0			0
$653$	−298755.	+	517458.i	−0.700629	+	1.21353i	0.267616	+	0.963526i	$0.413764\pi$
$653$	−298755.	+	517458.i	−0.700629	+	1.21353i	−0.968246	+	0.250000i	$0.919569\pi$
$654$		0			0
$655$	260113.	+	450529.i	0.606289	+	1.05012i
$656$		0			0
$657$		−	136467.i		−	0.316153i
$658$		0			0
$659$	−88942.9			−0.204805			−0.102402	−	0.994743i	$-0.532653\pi$
$659$	−88942.9			−0.204805			−0.102402	+	0.994743i	$0.532653\pi$
$660$		0			0
$661$	22581.2	−	13037.2i	0.0516825	−	0.0298389i	−0.473936	−	0.880559i	$-0.657167\pi$
$661$	22581.2	−	13037.2i	0.0516825	−	0.0298389i	0.525619	+	0.850720i	$0.323834\pi$
$662$		0			0
$663$	−339047.	−	195749.i	−0.771316	−	0.445319i
$664$		0			0
$665$	314348.	−	153681.i	0.710833	−	0.347517i
$666$		0			0
$667$	54082.0	−	93672.8i	0.121563	−	0.210553i
$668$		0			0
$669$	−173847.	−	301113.i	−0.388433	−	0.672786i
$670$		0			0
$671$			70792.9i			0.157233i
$672$		0			0
$673$	−492245.			−1.08680			−0.543402	−	0.839473i	$-0.682864\pi$
$673$	−492245.			−1.08680			−0.543402	+	0.839473i	$0.682864\pi$
$674$		0			0
$675$	14832.6	−	8563.62i	0.0325545	−	0.0187953i
$676$		0			0
$677$	−565901.	−	326723.i	−1.23470	−	0.712857i	−0.266697	−	0.963780i	$-0.585932\pi$
$677$	−565901.	−	326723.i	−1.23470	−	0.712857i	−0.968007	+	0.250923i	$0.919266\pi$
$678$		0			0
$679$	−861008.	−	59401.0i	−1.86753	−	0.128841i
$680$		0			0
$681$	−184932.	+	320312.i	−0.398766	+	0.690684i
$682$		0			0
$683$	−11245.0	−	19476.8i	−0.0241055	−	0.0417520i	0.853721	−	0.520731i	$-0.174340\pi$
$683$	−11245.0	−	19476.8i	−0.0241055	−	0.0417520i	−0.877827	+	0.478979i	$0.841007\pi$
$684$		0			0
$685$			469879.i			1.00139i
$686$		0			0
$687$	−29503.7			−0.0625119
$688$		0			0
$689$	590151.	−	340724.i	1.24315	−	0.717735i
$690$		0			0
$691$	261381.	+	150908.i	0.547416	+	0.316051i	0.748079	−	0.663609i	$-0.230976\pi$
$691$	261381.	+	150908.i	0.547416	+	0.316051i	−0.200663	+	0.979660i	$0.564310\pi$
$692$		0			0
$693$	2250.56	−	32621.5i	0.00468623	−	0.0679262i
$694$		0			0
$695$	−406466.	+	704020.i	−0.841501	+	1.45752i
$696$		0			0
$697$	−40004.3	−	69289.4i	−0.0823457	−	0.142627i
$698$		0			0
$699$		−	411757.i		−	0.842727i
$700$		0			0
$701$	−442335.			−0.900150			−0.450075	−	0.892991i	$-0.648603\pi$
$701$	−442335.			−0.900150			−0.450075	+	0.892991i	$0.648603\pi$
$702$		0			0
$703$	−402737.	+	232521.i	−0.814913	+	0.470490i
$704$		0			0
$705$	−247777.	−	143054.i	−0.498520	−	0.287821i
$706$		0			0
$707$	−265309.	−	542679.i	−0.530778	−	1.08569i
$708$		0			0
$709$	−6587.70	+	11410.2i	−0.0131051	+	0.0226987i	−0.872504	−	0.488608i	$-0.837505\pi$
$709$	−6587.70	+	11410.2i	−0.0131051	+	0.0226987i	0.859398	+	0.511306i	$0.170838\pi$
$710$		0			0
$711$	−93474.1	−	161902.i	−0.184906	−	0.320267i
$712$		0			0
$713$		−	60418.2i		−	0.118847i
$714$		0			0
$715$	100924.			0.197415
$716$		0			0
$717$	320053.	−	184783.i	0.622564	−	0.359438i
$718$		0			0
$719$	−7244.64	−	4182.69i	−0.0140139	−	0.00809093i	0.492977	−	0.870043i	$-0.335909\pi$
$719$	−7244.64	−	4182.69i	−0.0140139	−	0.00809093i	−0.506991	+	0.861952i	$0.669242\pi$
$720$		0			0
$721$	187560.	−	278629.i	0.360803	−	0.535989i
$722$		0			0
$723$	−282196.	+	488779.i	−0.539852	+	0.935052i
$724$		0			0
$725$	−42376.6	−	73398.4i	−0.0806214	−	0.139640i
$726$		0			0
$727$		−	330324.i		−	0.624988i	−0.949920	−	0.312494i	$-0.898836\pi$
$727$		−	330324.i		−	0.624988i	0.949920	−	0.312494i	$-0.101164\pi$
$728$		0			0
$729$	−19683.0			−0.0370370
$730$		0			0
$731$	936951.	−	540949.i	1.75340	−	1.01233i
$732$		0			0
$733$	−243140.	−	140377.i	−0.452531	−	0.261269i	0.256368	−	0.966579i	$-0.417474\pi$
$733$	−243140.	−	140377.i	−0.452531	−	0.261269i	−0.708898	+	0.705311i	$0.750808\pi$
$734$		0			0
$735$	337772.	+	46828.7i	0.625243	+	0.0866837i
$736$		0			0
$737$	−37905.1	+	65653.5i	−0.0697851	+	0.120871i
$738$		0			0
$739$	235776.	+	408376.i	0.431728	+	0.747775i	0.997022	−	0.0771144i	$-0.0245707\pi$
$739$	235776.	+	408376.i	0.431728	+	0.747775i	−0.565294	+	0.824889i	$0.691237\pi$
$740$		0			0
$741$		−	202808.i		−	0.369359i
$742$		0			0
$743$	−1.08961e6			−1.97376			−0.986880	−	0.161453i	$-0.948382\pi$
$743$	−1.08961e6			−1.97376			−0.986880	+	0.161453i	$0.948382\pi$
$744$		0			0
$745$	−195620.	+	112941.i	−0.352453	+	0.203489i
$746$		0			0
$747$	75155.8	+	43391.2i	0.134686	+	0.0777608i
$748$		0			0
$749$	228623.	+	153898.i	0.407526	+	0.274328i
$750$		0			0
$751$	−295585.	+	511968.i	−0.524086	+	0.907743i	0.475521	+	0.879704i	$0.342260\pi$
$751$	−295585.	+	511968.i	−0.524086	+	0.907743i	−0.999607	+	0.0280387i	$0.991074\pi$
$752$		0			0
$753$	154554.	+	267696.i	0.272578	+	0.472119i
$754$		0			0
$755$			953626.i			1.67296i
$756$		0			0
$757$	−1.05386e6			−1.83904			−0.919522	−	0.393038i	$-0.871424\pi$
$757$	−1.05386e6			−1.83904			−0.919522	+	0.393038i	$0.871424\pi$
$758$		0			0
$759$	−17328.3	+	10004.5i	−0.0300796	+	0.0173665i
$760$		0			0
$761$	−706768.	−	408053.i	−1.22041	−	0.704607i	−0.255408	−	0.966833i	$-0.582210\pi$
$761$	−706768.	−	408053.i	−1.22041	−	0.704607i	−0.965006	+	0.262226i	$0.915543\pi$
$762$		0			0
$763$	−324901.	+	158840.i	−0.558086	+	0.272841i
$764$		0			0
$765$	186093.	−	322322.i	0.317985	−	0.550766i
$766$		0			0
$767$	−421106.	−	729377.i	−0.715815	−	1.23983i
$768$		0			0
$769$		−	18599.2i		−	0.0314515i	−0.999876	−	0.0157257i	$-0.994994\pi$
$769$		−	18599.2i		−	0.0314515i	0.999876	−	0.0157257i	$-0.00500586\pi$
$770$		0			0
$771$	236592.			0.398008
$772$		0			0
$773$	33672.0	−	19440.6i	0.0563521	−	0.0325349i	−0.471559	−	0.881834i	$-0.656309\pi$
$773$	33672.0	−	19440.6i	0.0563521	−	0.0325349i	0.527911	+	0.849299i	$0.322975\pi$
$774$		0			0
$775$	−40998.9	−	23670.7i	−0.0682604	−	0.0394101i
$776$		0			0
$777$	−452136.	−	31192.9i	−0.748905	−	0.0516671i
$778$		0			0
$779$	20723.5	−	35894.1i	0.0341498	−	0.0591491i
$780$		0			0
$781$	−20254.6	−	35082.0i	−0.0332064	−	0.0575151i
$782$		0			0
$783$			97400.2i			0.158868i
$784$		0			0
$785$	546150.			0.886284
$786$		0			0
$787$	82864.9	−	47842.1i	0.133789	−	0.0772433i	−0.431612	−	0.902060i	$-0.642055\pi$
$787$	82864.9	−	47842.1i	0.133789	−	0.0772433i	0.565401	+	0.824816i	$0.308722\pi$
$788$		0			0
$789$	−130834.	−	75537.1i	−0.210168	−	0.121341i
$790$		0			0
$791$	−65738.8	+	952874.i	−0.105068	+	1.52294i
$792$		0			0
$793$	−213953.	+	370578.i	−0.340230	+	0.589295i
$794$		0			0
$795$	323917.	+	561040.i	0.512506	+	0.887687i
$796$		0			0
$797$		−	1.13357e6i		−	1.78457i	−0.451474	−	0.892284i	$-0.649102\pi$
$797$		−	1.13357e6i		−	1.78457i	0.451474	−	0.892284i	$-0.350898\pi$
$798$		0			0
$799$	−1.01596e6			−1.59142
$800$		0			0
$801$	137402.	−	79328.9i	0.214155	−	0.123642i
$802$		0			0
$803$	108186.	+	62461.1i	0.167779	+	0.0968675i
$804$		0			0
$805$	−91647.1	−	187461.i	−0.141425	−	0.289280i
$806$		0			0
$807$	−137054.	+	237385.i	−0.210448	+	0.364507i
$808$		0			0
$809$	335524.	+	581144.i	0.512656	+	0.887947i	0.999892	+	0.0146766i	$0.00467187\pi$
$809$	335524.	+	581144.i	0.512656	+	0.887947i	−0.487236	+	0.873270i	$0.661995\pi$
$810$		0			0
$811$			385128.i			0.585550i	0.956181	+	0.292775i	$0.0945787\pi$
$811$			385128.i			0.585550i	−0.956181	+	0.292775i	$0.905421\pi$
$812$		0			0
$813$	569888.			0.862200
$814$		0			0
$815$	607583.	−	350788.i	0.914725	−	0.528116i
$816$		0			0
$817$	485371.	+	280229.i	0.727159	+	0.419825i
$818$		0			0
$819$	110371.	−	163961.i	0.164546	−	0.244441i
$820$		0			0
$821$	305035.	−	528336.i	0.452547	−	0.783834i	−0.545997	−	0.837787i	$-0.683849\pi$
$821$	305035.	−	528336.i	0.452547	−	0.783834i	0.998543	+	0.0539532i	$0.0171822\pi$
$822$		0			0
$823$	408343.	+	707271.i	0.602872	+	1.04421i	0.992384	+	0.123184i	$0.0393105\pi$
$823$	408343.	+	707271.i	0.602872	+	1.04421i	−0.389512	+	0.921022i	$0.627356\pi$
$824$		0			0
$825$			15678.3i			0.0230351i
$826$		0			0
$827$	900643.			1.31687			0.658433	−	0.752639i	$-0.271219\pi$
$827$	900643.			1.31687			0.658433	+	0.752639i	$0.271219\pi$
$828$		0			0
$829$	−422528.	+	243947.i	−0.614818	+	0.354965i	−0.774849	−	0.632147i	$-0.782174\pi$
$829$	−422528.	+	243947.i	−0.614818	+	0.354965i	0.160031	+	0.987112i	$0.448841\pi$
$830$		0			0
$831$	136304.	+	78695.2i	0.197382	+	0.113958i
$832$		0			0
$833$	1.12187e6	−	455700.i	1.61679	−	0.656733i
$834$		0			0
$835$	86675.1	−	150126.i	0.124314	−	0.215319i
$836$		0			0
$837$	27202.9	+	47116.8i	0.0388297	+	0.0672550i
$838$		0			0
$839$		−	171949.i		−	0.244272i	−0.992513	−	0.122136i	$-0.961026\pi$
$839$		−	171949.i		−	0.244272i	0.992513	−	0.122136i	$-0.0389745\pi$
$840$		0			0
$841$	−225301.			−0.318546
$842$		0			0
$843$	466392.	−	269271.i	0.656290	−	0.378909i
$844$		0			0
$845$	−147761.	−	85309.9i	−0.206941	−	0.119477i
$846$		0			0
$847$	−570302.	−	383900.i	−0.794947	−	0.535121i
$848$		0			0
$849$	−300201.	+	519964.i	−0.416483	+	0.721370i
$850$		0			0
$851$	138663.	+	240171.i	0.191470	+	0.331636i
$852$		0			0
$853$			265601.i			0.365033i	0.983203	+	0.182517i	$0.0584243\pi$
$853$			265601.i			0.365033i	−0.983203	+	0.182517i	$0.941576\pi$
$854$		0			0
$855$	192804.			0.263745
$856$		0			0
$857$	563065.	−	325086.i	0.766650	−	0.442626i	−0.0650281	−	0.997883i	$-0.520714\pi$
$857$	563065.	−	325086.i	0.766650	−	0.442626i	0.831678	+	0.555258i	$0.187380\pi$
$858$		0			0
$859$	−240129.	−	138639.i	−0.325431	−	0.187888i	0.328380	−	0.944546i	$-0.393497\pi$
$859$	−240129.	−	138639.i	−0.325431	−	0.187888i	−0.653811	+	0.756658i	$0.726831\pi$
$860$		0			0
$861$	36288.1	−	17740.8i	0.0489506	−	0.0239313i
$862$		0			0
$863$	−198026.	+	342991.i	−0.265889	+	0.460533i	−0.967796	−	0.251735i	$-0.918999\pi$
$863$	−198026.	+	342991.i	−0.265889	+	0.460533i	0.701907	+	0.712268i	$0.252332\pi$
$864$		0			0
$865$	473400.	+	819952.i	0.632697	+	1.09586i
$866$		0			0
$867$		−	887634.i		−	1.18085i
$868$		0			0
$869$	171132.			0.226617
$870$		0			0
$871$	−396842.	+	229117.i	−0.523096	+	0.302009i
$872$		0			0
$873$	−411848.	−	237780.i	−0.540391	−	0.311995i
$874$		0			0
$875$	671967.	+	46359.1i	0.877671	+	0.0605506i
$876$		0			0
$877$	430511.	−	745667.i	0.559738	−	0.969495i	−0.437780	−	0.899082i	$-0.644235\pi$
$877$	430511.	−	745667.i	0.559738	−	0.969495i	0.997518	−	0.0704126i	$-0.0224316\pi$
$878$		0			0
$879$	−236862.	−	410257.i	−0.306561	−	0.530980i
$880$		0			0
$881$		−	47080.1i		−	0.0606577i	−0.999540	−	0.0303289i	$-0.990345\pi$
$881$		−	47080.1i		−	0.0606577i	0.999540	−	0.0303289i	$-0.00965545\pi$
$882$		0			0
$883$	656966.			0.842599			0.421300	−	0.906921i	$-0.361574\pi$
$883$	656966.			0.842599			0.421300	+	0.906921i	$0.361574\pi$
$884$		0			0
$885$	693399.	−	400334.i	0.885313	−	0.511135i
$886$		0			0
$887$	−464028.	−	267907.i	−0.589790	−	0.340515i	0.175225	−	0.984529i	$-0.443935\pi$
$887$	−464028.	−	267907.i	−0.589790	−	0.340515i	−0.765014	+	0.644013i	$0.777268\pi$
$888$		0			0
$889$	81610.2	−	1.18293e6i	0.103262	−	1.49677i
$890$		0			0
$891$	9008.91	−	15603.9i	0.0113479	−	0.0196552i
$892$		0			0
$893$	−263150.	−	455790.i	−0.329990	−	0.571560i
$894$		0			0
$895$			689213.i			0.860414i
$896$		0			0
$897$	−120944.			−0.150314
$898$		0			0
$899$	233155.	−	134612.i	0.288486	−	0.166558i
$900$		0			0
$901$	1.99224e6	+	1.15022e6i	2.45410	+	1.41687i
$902$		0			0
$903$	239896.	+	490698.i	0.294203	+	0.601781i
$904$		0			0
$905$	736997.	−	1.27652e6i	0.899847	−	1.55858i
$906$		0			0
$907$	−668602.	−	1.15805e6i	−0.812743	−	1.40771i	−0.910937	−	0.412545i	$-0.864640\pi$
$907$	−668602.	−	1.15805e6i	−0.812743	−	1.40771i	0.0981945	−	0.995167i	$-0.468693\pi$
$908$		0			0
$909$		−	332850.i		−	0.402829i
$910$		0			0
$911$	74248.9			0.0894650			0.0447325	−	0.998999i	$-0.485756\pi$
$911$	74248.9			0.0894650			0.0447325	+	0.998999i	$0.485756\pi$
$912$		0			0
$913$	−68797.7	+	39720.4i	−0.0825339	+	0.0476509i
$914$		0			0
$915$	−352298.	−	203399.i	−0.420793	−	0.242945i
$916$		0			0
$917$	−520794.	+	773663.i	−0.619337	+	0.920054i
$918$		0			0
$919$	−179674.	+	311205.i	−0.212743	+	0.368482i	−0.952572	−	0.304313i	$-0.901573\pi$
$919$	−179674.	+	311205.i	−0.212743	+	0.368482i	0.739829	+	0.672795i	$0.234906\pi$
$920$		0			0
$921$	205416.	+	355791.i	0.242167	+	0.419446i
$922$		0			0
$923$		−	244857.i		−	0.287415i
$924$		0			0
$925$	217302.			0.253969
$926$		0			0
$927$	160280.	−	92537.4i	0.186517	−	0.107686i
$928$		0			0
$929$	−160839.	−	92860.6i	−0.186363	−	0.107597i	0.403916	−	0.914796i	$-0.367649\pi$
$929$	−160839.	−	92860.6i	−0.186363	−	0.107597i	−0.590279	+	0.807199i	$0.700982\pi$
$930$		0			0
$931$	495022.	+	385270.i	0.571118	+	0.444494i
$932$		0			0
$933$	463413.	−	802655.i	0.532360	−	0.922074i
$934$		0			0
$935$	170349.	+	295054.i	0.194858	+	0.337503i
$936$		0			0
$937$		−	99162.5i		−	0.112945i	−0.998404	−	0.0564727i	$-0.982015\pi$
$937$		−	99162.5i		−	0.112945i	0.998404	−	0.0564727i	$-0.0179854\pi$
$938$		0			0
$939$	−27277.0			−0.0309361
$940$		0			0
$941$	−1.06307e6	+	613762.i	−1.20055	+	0.693139i	−0.960678	−	0.277665i	$-0.910440\pi$
$941$	−1.06307e6	+	613762.i	−1.20055	+	0.693139i	−0.239874	+	0.970804i	$0.577106\pi$
$942$		0			0
$943$	−21405.4	−	12358.4i	−0.0240713	−	0.0138976i
$944$		0			0
$945$	155873.	+	104927.i	0.174545	+	0.117496i
$946$		0			0
$947$	−477796.	+	827567.i	−0.532773	+	0.922791i	0.466494	+	0.884524i	$0.345517\pi$
$947$	−477796.	+	827567.i	−0.532773	+	0.922791i	−0.999268	+	0.0382664i	$0.987816\pi$
$948$		0			0
$949$	377545.	+	653927.i	0.419214	+	0.726101i
$950$		0			0
$951$		−	338438.i		−	0.374212i
$952$		0			0
$953$	−273680.			−0.301340			−0.150670	−	0.988584i	$-0.548143\pi$
$953$	−273680.			−0.301340			−0.150670	+	0.988584i	$0.548143\pi$
$954$		0			0
$955$	−287995.	+	166274.i	−0.315776	+	0.182313i
$956$		0			0
$957$	−77215.0	−	44580.1i	−0.0843098	−	0.0486763i
$958$		0			0
$959$	−756765.	+	369973.i	−0.822856	+	0.402284i
$960$		0			0
$961$	−386569.	+	669557.i	−0.418582	+	0.725005i
$962$		0			0
$963$	75929.4	+	131514.i	0.0818762	+	0.141814i
$964$		0			0
$965$		−	185993.i		−	0.199730i
$966$		0			0
$967$	−1.01033e6			−1.08047			−0.540233	−	0.841515i	$-0.681664\pi$
$967$	−1.01033e6			−1.08047			−0.540233	+	0.841515i	$0.681664\pi$
$968$		0			0
$969$	592918.	−	342321.i	0.631461	−	0.364574i
$970$		0			0
$971$	1.41613e6	+	817602.i	1.50198	+	0.867169i	0.999997	+	0.00229077i	$0.000729175\pi$
$971$	1.41613e6	+	817602.i	1.50198	+	0.867169i	0.501983	+	0.864878i	$0.332604\pi$
$972$		0			0
$973$	−1.45390e6	−	100305.i	−1.53571	−	0.105949i
$974$		0			0
$975$	−47383.6	+	82070.8i	−0.0498447	+	0.0863335i
$976$		0			0
$977$	−440094.	−	762265.i	−0.461059	−	0.798577i	0.537955	−	0.842973i	$-0.319197\pi$
$977$	−440094.	−	762265.i	−0.461059	−	0.798577i	−0.999014	+	0.0443965i	$0.985864\pi$
$978$		0			0
$979$			145236.i			0.151533i
$980$		0			0
$981$	−199276.			−0.207070
$982$		0			0
$983$	491660.	−	283860.i	0.508812	−	0.293763i	−0.223533	−	0.974696i	$-0.571759\pi$
$983$	491660.	−	283860.i	0.508812	−	0.293763i	0.732345	+	0.680933i	$0.238426\pi$
$984$		0			0
$985$	70902.6	+	40935.6i	0.0730785	+	0.0421919i
$986$		0			0
$987$	35301.9	−	511696.i	0.0362380	−	0.525264i
$988$		0			0
$989$	167114.	−	289449.i	0.170852	−	0.295924i
$990$		0			0
$991$	611764.	+	1.05961e6i	0.622927	+	1.07894i	0.988938	+	0.148330i	$0.0473898\pi$
$991$	611764.	+	1.05961e6i	0.622927	+	1.07894i	−0.366011	+	0.930610i	$0.619277\pi$
$992$		0			0
$993$			900835.i			0.913580i
$994$		0			0
$995$	857502.			0.866142
$996$		0			0
$997$	−255157.	+	147315.i	−0.256695	+	0.148203i	−0.622826	−	0.782360i	$-0.714016\pi$
$997$	−255157.	+	147315.i	−0.256695	+	0.148203i	0.366131	+	0.930563i	$0.380682\pi$
$998$		0			0
$999$	−216271.	−	124864.i	−0.216704	−	0.125114i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	168.5.z.b.145.2	yes	16
3.2	odd	2		504.5.by.c.145.7		16
4.3	odd	2		336.5.bh.h.145.2		16
7.2	even	3		1176.5.f.a.97.15		16
7.3	odd	6	inner	168.5.z.b.73.2	✓	16
7.5	odd	6		1176.5.f.a.97.2		16
21.17	even	6		504.5.by.c.73.7		16
28.3	even	6		336.5.bh.h.241.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.5.z.b.73.2	✓	16	7.3	odd	6	inner
168.5.z.b.145.2	yes	16	1.1	even	1	trivial
336.5.bh.h.145.2		16	4.3	odd	2
336.5.bh.h.241.2		16	28.3	even	6
504.5.by.c.73.7		16	21.17	even	6
504.5.by.c.145.7		16	3.2	odd	2
1176.5.f.a.97.2		16	7.5	odd	6
1176.5.f.a.97.15		16	7.2	even	3

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)

\(f(q)\)	\(=\)	\(q+(4.50000 - 2.59808i) q^{3} +(-23.6709 - 13.6664i) q^{5} +(3.37250 - 48.8838i) q^{7} +(13.5000 - 23.3827i) q^{9} +O(q^{10})\)	\(q+(4.50000 - 2.59808i) q^{3} +(-23.6709 - 13.6664i) q^{5} +(3.37250 - 48.8838i) q^{7} +(13.5000 - 23.3827i) q^{9} +(12.3579 + 21.4045i) q^{11} +149.394i q^{13} -142.025 q^{15} +(-436.760 + 252.164i) q^{17} +(-226.256 - 130.629i) q^{19} +(-111.828 - 228.739i) q^{21} +(-77.9002 + 134.927i) q^{23} +(61.0396 + 105.724i) q^{25} -140.296i q^{27} -694.247 q^{29} +(-335.838 + 193.896i) q^{31} +(111.221 + 64.2136i) q^{33} +(-747.894 + 1111.03i) q^{35} +(890.004 - 1541.53i) q^{37} +(388.138 + 672.275i) q^{39} +158.644i q^{41} -2145.23 q^{43} +(-639.113 + 368.992i) q^{45} +(1744.60 + 1007.24i) q^{47} +(-2378.25 - 329.721i) q^{49} +(-1310.28 + 2269.47i) q^{51} +(-2280.70 - 3950.29i) q^{53} -675.551i q^{55} -1357.54 q^{57} +(-4882.23 + 2818.75i) q^{59} +(2480.53 + 1432.14i) q^{61} +(-1097.51 - 738.789i) q^{63} +(2041.68 - 3536.29i) q^{65} +(1533.64 + 2656.34i) q^{67} +809.562i q^{69} -1639.00 q^{71} +(4377.19 - 2527.17i) q^{73} +(549.357 + 317.171i) q^{75} +(1088.01 - 531.915i) q^{77} +(3462.00 - 5996.37i) q^{79} +(-364.500 - 631.333i) q^{81} +3214.17i q^{83} +13784.7 q^{85} +(-3124.11 + 1803.71i) q^{87} +(5088.95 + 2938.11i) q^{89} +(7302.97 + 503.832i) q^{91} +(-1007.51 + 1745.07i) q^{93} +(3570.45 + 6184.19i) q^{95} -17613.4i q^{97} +667.327 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 72 q^{3} - 12 q^{5} + 16 q^{7} + 216 q^{9}+O(q^{10}) \) 16 * q + 72 * q^3 - 12 * q^5 + 16 * q^7 + 216 * q^9	\( 16 q + 72 q^{3} - 12 q^{5} + 16 q^{7} + 216 q^{9} + 252 q^{11} - 72 q^{15} - 696 q^{17} + 156 q^{19} + 108 q^{21} - 672 q^{23} + 84 q^{25} + 1992 q^{29} + 2040 q^{31} + 2268 q^{33} + 2712 q^{35} + 2548 q^{37} - 396 q^{39} + 1304 q^{43} - 324 q^{45} - 744 q^{47} - 5608 q^{49} - 2088 q^{51} - 1164 q^{53} + 936 q^{57} + 8988 q^{59} + 816 q^{61} + 540 q^{63} + 8760 q^{65} + 3044 q^{67} - 4464 q^{71} - 15828 q^{73} + 756 q^{75} + 996 q^{77} - 11144 q^{79} - 5832 q^{81} - 15344 q^{85} + 8964 q^{87} + 22248 q^{89} + 1596 q^{91} + 6120 q^{93} + 3840 q^{95} + 13608 q^{99}+O(q^{100}) \) 16 * q + 72 * q^3 - 12 * q^5 + 16 * q^7 + 216 * q^9 + 252 * q^11 - 72 * q^15 - 696 * q^17 + 156 * q^19 + 108 * q^21 - 672 * q^23 + 84 * q^25 + 1992 * q^29 + 2040 * q^31 + 2268 * q^33 + 2712 * q^35 + 2548 * q^37 - 396 * q^39 + 1304 * q^43 - 324 * q^45 - 744 * q^47 - 5608 * q^49 - 2088 * q^51 - 1164 * q^53 + 936 * q^57 + 8988 * q^59 + 816 * q^61 + 540 * q^63 + 8760 * q^65 + 3044 * q^67 - 4464 * q^71 - 15828 * q^73 + 756 * q^75 + 996 * q^77 - 11144 * q^79 - 5832 * q^81 - 15344 * q^85 + 8964 * q^87 + 22248 * q^89 + 1596 * q^91 + 6120 * q^93 + 3840 * q^95 + 13608 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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