LMFDB - Embedded newform 475.2.p.c.468.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,2,Mod(107,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([3, 10]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("475.107");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	475.p (of order $12$, degree $4$, 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:	$3.79289409601$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			468.1
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	475.468
Dual form			475.2.p.c.407.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.366025 + 1.36603i) q^{2} +(-2.73205 - 0.732051i) q^{3} +(2.00000 - 3.46410i) q^{6} +(-2.00000 + 2.00000i) q^{8} +(4.33013 + 2.50000i) q^{9} +O(q^{10})$	\(q+(-0.366025 + 1.36603i) q^{2} +(-2.73205 - 0.732051i) q^{3} +(2.00000 - 3.46410i) q^{6} +(-2.00000 + 2.00000i) q^{8} +(4.33013 + 2.50000i) q^{9} -3.00000 q^{11} +(-1.09808 - 4.09808i) q^{13} +(-2.00000 - 3.46410i) q^{16} +(2.36603 + 0.633975i) q^{17} +(-5.00000 + 5.00000i) q^{18} +(-4.33013 - 0.500000i) q^{19} +(1.09808 - 4.09808i) q^{22} +(7.09808 - 1.90192i) q^{23} +(6.92820 - 4.00000i) q^{24} +6.00000 q^{26} +(-4.00000 - 4.00000i) q^{27} +(2.59808 - 4.50000i) q^{29} -8.66025i q^{31} +(8.19615 + 2.19615i) q^{33} +(-1.73205 + 3.00000i) q^{34} +(-3.00000 - 3.00000i) q^{37} +(2.26795 - 5.73205i) q^{38} +12.0000i q^{39} +(9.00000 - 5.19615i) q^{41} +(1.90192 - 7.09808i) q^{43} +10.3923i q^{46} +(0.633975 + 2.36603i) q^{47} +(2.92820 + 10.9282i) q^{48} +7.00000i q^{49} +(-6.00000 - 3.46410i) q^{51} +(0.732051 + 2.73205i) q^{53} +(6.92820 - 4.00000i) q^{54} +(11.4641 + 4.53590i) q^{57} +(5.19615 + 5.19615i) q^{58} +(-2.59808 - 4.50000i) q^{59} +(0.500000 - 0.866025i) q^{61} +(11.8301 + 3.16987i) q^{62} -8.00000i q^{64} +(-6.00000 + 10.3923i) q^{66} +(-12.2942 + 3.29423i) q^{67} -20.7846 q^{69} +(-13.5000 + 7.79423i) q^{71} +(-13.6603 + 3.66025i) q^{72} +(1.90192 - 7.09808i) q^{73} +(5.19615 - 3.00000i) q^{74} +(-16.3923 - 4.39230i) q^{78} +(-2.59808 - 4.50000i) q^{79} +(0.500000 + 0.866025i) q^{81} +(3.80385 + 14.1962i) q^{82} +(-8.66025 - 8.66025i) q^{83} +(9.00000 + 5.19615i) q^{86} +(-10.3923 + 10.3923i) q^{87} +(6.00000 - 6.00000i) q^{88} +(2.59808 - 4.50000i) q^{89} +(-6.33975 + 23.6603i) q^{93} -3.46410 q^{94} +(3.29423 - 12.2942i) q^{97} +(-9.56218 - 2.56218i) q^{98} +(-12.9904 - 7.50000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} - 4 q^{3} + 8 q^{6} - 8 q^{8}+O(q^{10}) $ 4 * q + 2 * q^2 - 4 * q^3 + 8 * q^6 - 8 * q^8	$ 4 q + 2 q^{2} - 4 q^{3} + 8 q^{6} - 8 q^{8} - 12 q^{11} + 6 q^{13} - 8 q^{16} + 6 q^{17} - 20 q^{18} - 6 q^{22} + 18 q^{23} + 24 q^{26} - 16 q^{27} + 12 q^{33} - 12 q^{37} + 16 q^{38} + 36 q^{41} + 18 q^{43} + 6 q^{47} - 16 q^{48} - 24 q^{51} - 4 q^{53} + 32 q^{57} + 2 q^{61} + 30 q^{62} - 24 q^{66} - 18 q^{67} - 54 q^{71} - 20 q^{72} + 18 q^{73} - 24 q^{78} + 2 q^{81} + 36 q^{82} + 36 q^{86} + 24 q^{88} - 60 q^{93} - 18 q^{97} - 14 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 - 4 * q^3 + 8 * q^6 - 8 * q^8 - 12 * q^11 + 6 * q^13 - 8 * q^16 + 6 * q^17 - 20 * q^18 - 6 * q^22 + 18 * q^23 + 24 * q^26 - 16 * q^27 + 12 * q^33 - 12 * q^37 + 16 * q^38 + 36 * q^41 + 18 * q^43 + 6 * q^47 - 16 * q^48 - 24 * q^51 - 4 * q^53 + 32 * q^57 + 2 * q^61 + 30 * q^62 - 24 * q^66 - 18 * q^67 - 54 * q^71 - 20 * q^72 + 18 * q^73 - 24 * q^78 + 2 * q^81 + 36 * q^82 + 36 * q^86 + 24 * q^88 - 60 * q^93 - 18 * q^97 - 14 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$.

$n$	$77$	$401$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$e\left(\frac{5}{6}\right)$

Coefficient data

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

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

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

$2$	−0.366025	+	1.36603i	−0.258819	+	0.965926i	0.707107	+	0.707107i	$0.250000\pi$
$2$	−0.366025	+	1.36603i	−0.258819	+	0.965926i	−0.965926	+	0.258819i	$0.916667\pi$
$3$	−2.73205	−	0.732051i	−1.57735	−	0.422650i	−0.639246	−	0.769002i	$-0.720753\pi$
$3$	−2.73205	−	0.732051i	−1.57735	−	0.422650i	−0.938104	+	0.346353i	$0.887420\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$	2.00000	−	3.46410i	0.816497	−	1.41421i
$7$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$7$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$8$	−2.00000	+	2.00000i	−0.707107	+	0.707107i
$9$	4.33013	+	2.50000i	1.44338	+	0.833333i
$10$		0			0
$11$	−3.00000			−0.904534			−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000			−0.904534			−0.452267	+	0.891883i	$0.649385\pi$
$12$		0			0
$13$	−1.09808	−	4.09808i	−0.304552	−	1.13660i	−0.933331	−	0.359018i	$-0.883112\pi$
$13$	−1.09808	−	4.09808i	−0.304552	−	1.13660i	0.628779	−	0.777584i	$-0.283555\pi$
$14$		0			0
$15$		0			0
$16$	−2.00000	−	3.46410i	−0.500000	−	0.866025i
$17$	2.36603	+	0.633975i	0.573845	+	0.153761i	0.534060	−	0.845447i	$-0.320666\pi$
$17$	2.36603	+	0.633975i	0.573845	+	0.153761i	0.0397858	+	0.999208i	$0.487332\pi$
$18$	−5.00000	+	5.00000i	−1.17851	+	1.17851i
$19$	−4.33013	−	0.500000i	−0.993399	−	0.114708i
$19$	−4.33013	−	0.500000i	−0.993399	−	0.114708i
$20$		0			0
$21$		0			0
$22$	1.09808	−	4.09808i	0.234111	−	0.873713i
$23$	7.09808	−	1.90192i	1.48005	−	0.396579i	0.573687	−	0.819075i	$-0.305513\pi$
$23$	7.09808	−	1.90192i	1.48005	−	0.396579i	0.906365	+	0.422496i	$0.138846\pi$
$24$	6.92820	−	4.00000i	1.41421	−	0.816497i
$25$		0			0
$26$	6.00000			1.17670
$27$	−4.00000	−	4.00000i	−0.769800	−	0.769800i
$28$		0			0
$29$	2.59808	−	4.50000i	0.482451	−	0.835629i	−0.517346	−	0.855776i	$-0.673080\pi$
$29$	2.59808	−	4.50000i	0.482451	−	0.835629i	0.999797	+	0.0201471i	$0.00641344\pi$
$30$		0			0
$31$		−	8.66025i		−	1.55543i	−0.628619	−	0.777714i	$-0.716379\pi$
$31$		−	8.66025i		−	1.55543i	0.628619	−	0.777714i	$-0.283621\pi$
$32$		0			0
$33$	8.19615	+	2.19615i	1.42677	+	0.382301i
$34$	−1.73205	+	3.00000i	−0.297044	+	0.514496i
$35$		0			0
$36$		0			0
$37$	−3.00000	−	3.00000i	−0.493197	−	0.493197i	0.416115	−	0.909312i	$-0.363391\pi$
$37$	−3.00000	−	3.00000i	−0.493197	−	0.493197i	−0.909312	+	0.416115i	$0.863391\pi$
$38$	2.26795	−	5.73205i	0.367910	−	0.929861i
$39$			12.0000i			1.92154i
$40$		0			0
$41$	9.00000	−	5.19615i	1.40556	−	0.811503i	0.410608	−	0.911812i	$-0.365317\pi$
$41$	9.00000	−	5.19615i	1.40556	−	0.811503i	0.994956	+	0.100309i	$0.0319833\pi$
$42$		0			0
$43$	1.90192	−	7.09808i	0.290041	−	1.08245i	−0.655036	−	0.755598i	$-0.727347\pi$
$43$	1.90192	−	7.09808i	0.290041	−	1.08245i	0.945077	−	0.326849i	$-0.105987\pi$
$44$		0			0
$45$		0			0
$46$			10.3923i			1.53226i
$47$	0.633975	+	2.36603i	0.0924747	+	0.345120i	0.996624	−	0.0820953i	$-0.0261612\pi$
$47$	0.633975	+	2.36603i	0.0924747	+	0.345120i	−0.904150	+	0.427216i	$0.859495\pi$
$48$	2.92820	+	10.9282i	0.422650	+	1.57735i
$49$			7.00000i			1.00000i
$50$		0			0
$51$	−6.00000	−	3.46410i	−0.840168	−	0.485071i
$52$		0			0
$53$	0.732051	+	2.73205i	0.100555	+	0.375276i	0.997803	−	0.0662507i	$-0.0211037\pi$
$53$	0.732051	+	2.73205i	0.100555	+	0.375276i	−0.897248	+	0.441527i	$0.854437\pi$
$54$	6.92820	−	4.00000i	0.942809	−	0.544331i
$55$		0			0
$56$		0			0
$57$	11.4641	+	4.53590i	1.51846	+	0.600794i
$58$	5.19615	+	5.19615i	0.682288	+	0.682288i
$59$	−2.59808	−	4.50000i	−0.338241	−	0.585850i	0.645861	−	0.763455i	$-0.276498\pi$
$59$	−2.59808	−	4.50000i	−0.338241	−	0.585850i	−0.984102	+	0.177605i	$0.943165\pi$
$60$		0			0
$61$	0.500000	−	0.866025i	0.0640184	−	0.110883i	−0.832240	−	0.554416i	$-0.812942\pi$
$61$	0.500000	−	0.866025i	0.0640184	−	0.110883i	0.896258	+	0.443533i	$0.146275\pi$
$62$	11.8301	+	3.16987i	1.50243	+	0.402574i
$63$		0			0
$64$		−	8.00000i		−	1.00000i
$65$		0			0
$66$	−6.00000	+	10.3923i	−0.738549	+	1.27920i
$67$	−12.2942	+	3.29423i	−1.50198	+	0.402454i	−0.913762	−	0.406249i	$-0.866836\pi$
$67$	−12.2942	+	3.29423i	−1.50198	+	0.402454i	−0.588217	+	0.808703i	$0.700170\pi$
$68$		0			0
$69$	−20.7846			−2.50217
$70$		0			0
$71$	−13.5000	+	7.79423i	−1.60216	+	0.925005i	−0.611100	+	0.791554i	$0.709273\pi$
$71$	−13.5000	+	7.79423i	−1.60216	+	0.925005i	−0.991055	+	0.133451i	$0.957394\pi$
$72$	−13.6603	+	3.66025i	−1.60988	+	0.431365i
$73$	1.90192	−	7.09808i	0.222603	−	0.830767i	−0.760747	−	0.649048i	$-0.775167\pi$
$73$	1.90192	−	7.09808i	0.222603	−	0.830767i	0.983351	−	0.181719i	$-0.0581660\pi$
$74$	5.19615	−	3.00000i	0.604040	−	0.348743i
$75$		0			0
$76$		0			0
$77$		0			0
$78$	−16.3923	−	4.39230i	−1.85606	−	0.497331i
$79$	−2.59808	−	4.50000i	−0.292306	−	0.506290i	0.682048	−	0.731307i	$-0.261089\pi$
$79$	−2.59808	−	4.50000i	−0.292306	−	0.506290i	−0.974355	+	0.225018i	$0.927756\pi$
$80$		0			0
$81$	0.500000	+	0.866025i	0.0555556	+	0.0962250i
$82$	3.80385	+	14.1962i	0.420065	+	1.56770i
$83$	−8.66025	−	8.66025i	−0.950586	−	0.950586i	0.0482490	−	0.998835i	$-0.484636\pi$
$83$	−8.66025	−	8.66025i	−0.950586	−	0.950586i	−0.998835	+	0.0482490i	$0.984636\pi$
$84$		0			0
$85$		0			0
$86$	9.00000	+	5.19615i	0.970495	+	0.560316i
$87$	−10.3923	+	10.3923i	−1.11417	+	1.11417i
$88$	6.00000	−	6.00000i	0.639602	−	0.639602i
$89$	2.59808	−	4.50000i	0.275396	−	0.476999i	−0.694839	−	0.719165i	$-0.744525\pi$
$89$	2.59808	−	4.50000i	0.275396	−	0.476999i	0.970235	+	0.242166i	$0.0778579\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−6.33975	+	23.6603i	−0.657401	+	2.45345i
$94$	−3.46410			−0.357295
$95$		0			0
$96$		0			0
$97$	3.29423	−	12.2942i	0.334478	−	1.24829i	−0.569955	−	0.821676i	$-0.693040\pi$
$97$	3.29423	−	12.2942i	0.334478	−	1.24829i	0.904434	−	0.426614i	$-0.140294\pi$
$98$	−9.56218	−	2.56218i	−0.965926	−	0.258819i
$99$	−12.9904	−	7.50000i	−1.30558	−	0.753778i
$100$		0			0
$101$	−4.50000	+	7.79423i	−0.447767	+	0.775555i	−0.998240	−	0.0592978i	$-0.981114\pi$
$101$	−4.50000	+	7.79423i	−0.447767	+	0.775555i	0.550474	+	0.834853i	$0.314447\pi$
$102$	6.92820	−	6.92820i	0.685994	−	0.685994i
$103$	6.00000	−	6.00000i	0.591198	−	0.591198i	−0.346757	−	0.937955i	$-0.612717\pi$
$103$	6.00000	−	6.00000i	0.591198	−	0.591198i	0.937955	+	0.346757i	$0.112717\pi$
$104$	10.3923	+	6.00000i	1.01905	+	0.588348i
$105$		0			0
$106$	−4.00000			−0.388514
$107$	7.00000	+	7.00000i	0.676716	+	0.676716i	0.959256	−	0.282540i	$-0.0911770\pi$
$107$	7.00000	+	7.00000i	0.676716	+	0.676716i	−0.282540	+	0.959256i	$0.591177\pi$
$108$		0			0
$109$	6.06218	+	10.5000i	0.580651	+	1.00572i	0.995402	+	0.0957826i	$0.0305354\pi$
$109$	6.06218	+	10.5000i	0.580651	+	1.00572i	−0.414751	+	0.909935i	$0.636131\pi$
$110$		0			0
$111$	6.00000	+	10.3923i	0.569495	+	0.986394i
$112$		0			0
$113$	−4.00000	+	4.00000i	−0.376288	+	0.376288i	−0.869761	−	0.493473i	$-0.835727\pi$
$113$	−4.00000	+	4.00000i	−0.376288	+	0.376288i	0.493473	+	0.869761i	$0.335727\pi$
$114$	−10.3923	+	14.0000i	−0.973329	+	1.31122i
$115$		0			0
$116$		0			0
$117$	5.49038	−	20.4904i	0.507586	−	1.89434i
$118$	7.09808	−	1.90192i	0.653431	−	0.175086i
$119$		0			0
$120$		0			0
$121$	−2.00000			−0.181818
$122$	1.00000	+	1.00000i	0.0905357	+	0.0905357i
$123$	−28.3923	+	7.60770i	−2.56005	+	0.685963i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−12.2942	+	3.29423i	−1.09094	+	0.292316i	−0.759069	−	0.651010i	$-0.774345\pi$
$127$	−12.2942	+	3.29423i	−1.09094	+	0.292316i	−0.331868	+	0.943326i	$0.607679\pi$
$128$	10.9282	+	2.92820i	0.965926	+	0.258819i
$129$	−10.3923	+	18.0000i	−0.914991	+	1.58481i
$130$		0			0
$131$	3.00000	+	5.19615i	0.262111	+	0.453990i	0.966803	−	0.255524i	$-0.0822479\pi$
$131$	3.00000	+	5.19615i	0.262111	+	0.453990i	−0.704692	+	0.709514i	$0.748915\pi$
$132$		0			0
$133$		0			0
$134$		−	18.0000i		−	1.55496i
$135$		0			0
$136$	−6.00000	+	3.46410i	−0.514496	+	0.297044i
$137$	−2.53590	−	9.46410i	−0.216656	−	0.808573i	−0.985577	−	0.169228i	$-0.945872\pi$
$137$	−2.53590	−	9.46410i	−0.216656	−	0.808573i	0.768920	−	0.639344i	$-0.220794\pi$
$138$	7.60770	−	28.3923i	0.647610	−	2.41691i
$139$	1.73205	+	1.00000i	0.146911	+	0.0848189i	0.571654	−	0.820495i	$-0.306302\pi$
$139$	1.73205	+	1.00000i	0.146911	+	0.0848189i	−0.424743	+	0.905314i	$0.639635\pi$
$140$		0			0
$141$		−	6.92820i		−	0.583460i
$142$	−5.70577	−	21.2942i	−0.478818	−	1.78697i
$143$	3.29423	+	12.2942i	0.275477	+	1.02810i
$144$		−	20.0000i		−	1.66667i
$145$		0			0
$146$	9.00000	+	5.19615i	0.744845	+	0.430037i
$147$	5.12436	−	19.1244i	0.422650	−	1.57735i
$148$		0			0
$149$	2.59808	−	1.50000i	0.212843	−	0.122885i	−0.389789	−	0.920904i	$-0.627452\pi$
$149$	2.59808	−	1.50000i	0.212843	−	0.122885i	0.602632	+	0.798019i	$0.294119\pi$
$150$		0			0
$151$			8.66025i			0.704761i	0.935857	+	0.352381i	$0.114628\pi$
$151$			8.66025i			0.704761i	−0.935857	+	0.352381i	$0.885372\pi$
$152$	9.66025	−	7.66025i	0.783550	−	0.621329i
$153$	8.66025	+	8.66025i	0.700140	+	0.700140i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	14.1962	+	3.80385i	1.13298	+	0.303580i	0.776124	−	0.630581i	$-0.217183\pi$
$157$	14.1962	+	3.80385i	1.13298	+	0.303580i	0.356853	+	0.934161i	$0.383850\pi$
$158$	7.09808	−	1.90192i	0.564693	−	0.151309i
$159$		−	8.00000i		−	0.634441i
$160$		0			0
$161$		0			0
$162$	−1.36603	+	0.366025i	−0.107325	+	0.0287577i
$163$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$163$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$164$		0			0
$165$		0			0
$166$	15.0000	−	8.66025i	1.16423	−	0.672166i
$167$	−19.1244	+	5.12436i	−1.47989	+	0.396535i	−0.906309	−	0.422616i	$-0.861112\pi$
$167$	−19.1244	+	5.12436i	−1.47989	+	0.396535i	−0.573578	+	0.819151i	$0.694445\pi$
$168$		0			0
$169$	−4.33013	+	2.50000i	−0.333087	+	0.192308i
$170$		0			0
$171$	−17.5000	−	12.9904i	−1.33826	−	0.993399i
$172$		0			0
$173$	−2.73205	−	0.732051i	−0.207714	−	0.0556568i	0.153462	−	0.988155i	$-0.450958\pi$
$173$	−2.73205	−	0.732051i	−0.207714	−	0.0556568i	−0.361176	+	0.932498i	$0.617625\pi$
$174$	−10.3923	−	18.0000i	−0.787839	−	1.36458i
$175$		0			0
$176$	6.00000	+	10.3923i	0.452267	+	0.783349i
$177$	3.80385	+	14.1962i	0.285915	+	1.06705i
$178$	5.19615	+	5.19615i	0.389468	+	0.389468i
$179$	25.9808			1.94189			0.970947	−	0.239296i	$-0.0769166\pi$
$179$	25.9808			1.94189			0.970947	+	0.239296i	$0.0769166\pi$
$180$		0			0
$181$	−21.0000	−	12.1244i	−1.56092	−	0.901196i	−0.997164	−	0.0752530i	$-0.976024\pi$
$181$	−21.0000	−	12.1244i	−1.56092	−	0.901196i	−0.563753	−	0.825943i	$-0.690643\pi$
$182$		0			0
$183$	−2.00000	+	2.00000i	−0.147844	+	0.147844i
$184$	−10.3923	+	18.0000i	−0.766131	+	1.32698i
$185$		0			0
$186$	−30.0000	−	17.3205i	−2.19971	−	1.27000i
$187$	−7.09808	−	1.90192i	−0.519063	−	0.139082i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−3.00000			−0.217072			−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000			−0.217072			−0.108536	+	0.994092i	$0.534616\pi$
$192$	−5.85641	+	21.8564i	−0.422650	+	1.57735i
$193$	−16.3923	−	4.39230i	−1.17994	−	0.316165i	−0.385040	−	0.922900i	$-0.625812\pi$
$193$	−16.3923	−	4.39230i	−1.17994	−	0.316165i	−0.794904	+	0.606735i	$0.792479\pi$
$194$	15.5885	+	9.00000i	1.11919	+	0.646162i
$195$		0			0
$196$		0			0
$197$	−8.66025	+	8.66025i	−0.617018	+	0.617018i	−0.944765	−	0.327748i	$-0.893710\pi$
$197$	−8.66025	+	8.66025i	−0.617018	+	0.617018i	0.327748	+	0.944765i	$0.393710\pi$
$198$	15.0000	−	15.0000i	1.06600	−	1.06600i
$199$	6.06218	+	3.50000i	0.429736	+	0.248108i	0.699234	−	0.714893i	$-0.253524\pi$
$199$	6.06218	+	3.50000i	0.429736	+	0.248108i	−0.269498	+	0.963001i	$0.586858\pi$
$200$		0			0
$201$	36.0000			2.53924
$202$	−9.00000	−	9.00000i	−0.633238	−	0.633238i
$203$		0			0
$204$		0			0
$205$		0			0
$206$	6.00000	+	10.3923i	0.418040	+	0.724066i
$207$	35.4904	+	9.50962i	2.46675	+	0.660964i
$208$	−12.0000	+	12.0000i	−0.832050	+	0.832050i
$209$	12.9904	+	1.50000i	0.898563	+	0.103757i
$210$		0			0
$211$	1.50000	−	0.866025i	0.103264	−	0.0596196i	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	1.50000	−	0.866025i	0.103264	−	0.0596196i	0.550743	+	0.834675i	$0.314345\pi$
$212$		0			0
$213$	42.5885	−	11.4115i	2.91811	−	0.781906i
$214$	−12.1244	+	7.00000i	−0.828804	+	0.478510i
$215$		0			0
$216$	16.0000			1.08866
$217$		0			0
$218$	−16.5622	+	4.43782i	−1.12173	+	0.300567i
$219$	−10.3923	+	18.0000i	−0.702247	+	1.21633i
$220$		0			0
$221$		−	10.3923i		−	0.699062i
$222$	−16.3923	+	4.39230i	−1.10018	+	0.294792i
$223$	4.09808	+	1.09808i	0.274427	+	0.0735326i	0.393408	−	0.919364i	$-0.371296\pi$
$223$	4.09808	+	1.09808i	0.274427	+	0.0735326i	−0.118981	+	0.992897i	$0.537963\pi$
$224$		0			0
$225$		0			0
$226$	−4.00000	−	6.92820i	−0.266076	−	0.460857i
$227$	7.00000	+	7.00000i	0.464606	+	0.464606i	0.900162	−	0.435556i	$-0.143448\pi$
$227$	7.00000	+	7.00000i	0.464606	+	0.464606i	−0.435556	+	0.900162i	$0.643448\pi$
$228$		0			0
$229$		−	11.0000i		−	0.726900i	−0.931614	−	0.363450i	$-0.881599\pi$
$229$		−	11.0000i		−	0.726900i	0.931614	−	0.363450i	$-0.118401\pi$
$230$		0			0
$231$		0			0
$232$	3.80385	+	14.1962i	0.249735	+	0.932023i
$233$	−1.26795	+	4.73205i	−0.0830661	+	0.310007i	−0.994941	−	0.100461i	$-0.967968\pi$
$233$	−1.26795	+	4.73205i	−0.0830661	+	0.310007i	0.911875	+	0.410468i	$0.134635\pi$
$234$	25.9808	+	15.0000i	1.69842	+	0.980581i
$235$		0			0
$236$		0			0
$237$	3.80385	+	14.1962i	0.247086	+	0.922139i
$238$		0			0
$239$		−	21.0000i		−	1.35838i	−0.733964	−	0.679189i	$-0.762332\pi$
$239$		−	21.0000i		−	1.35838i	0.733964	−	0.679189i	$-0.237668\pi$
$240$		0			0
$241$	16.5000	+	9.52628i	1.06286	+	0.613642i	0.926222	−	0.376980i	$-0.123037\pi$
$241$	16.5000	+	9.52628i	1.06286	+	0.613642i	0.136637	+	0.990621i	$0.456371\pi$
$242$	0.732051	−	2.73205i	0.0470580	−	0.175623i
$243$	3.66025	+	13.6603i	0.234805	+	0.876306i
$244$		0			0
$245$		0			0
$246$		−	41.5692i		−	2.65036i
$247$	2.70577	+	18.2942i	0.172164	+	1.16403i
$248$	17.3205	+	17.3205i	1.09985	+	1.09985i
$249$	17.3205	+	30.0000i	1.09764	+	1.90117i
$250$		0			0
$251$	10.5000	−	18.1865i	0.662754	−	1.14792i	−0.317135	−	0.948380i	$-0.602721\pi$
$251$	10.5000	−	18.1865i	0.662754	−	1.14792i	0.979889	−	0.199543i	$-0.0639459\pi$
$252$		0			0
$253$	−21.2942	+	5.70577i	−1.33876	+	0.358719i
$254$		−	18.0000i		−	1.12942i
$255$		0			0
$256$		0			0
$257$	1.36603	−	0.366025i	0.0852103	−	0.0228320i	−0.215962	−	0.976402i	$-0.569289\pi$
$257$	1.36603	−	0.366025i	0.0852103	−	0.0228320i	0.301172	+	0.953570i	$0.402622\pi$
$258$	−20.7846	−	20.7846i	−1.29399	−	1.29399i
$259$		0			0
$260$		0			0
$261$	22.5000	−	12.9904i	1.39272	−	0.804084i
$262$	−8.19615	+	2.19615i	−0.506360	+	0.135679i
$263$	5.07180	−	18.9282i	0.312740	−	1.16716i	−0.613335	−	0.789823i	$-0.710172\pi$
$263$	5.07180	−	18.9282i	0.312740	−	1.16716i	0.926075	−	0.377340i	$-0.123161\pi$
$264$	−20.7846	+	12.0000i	−1.27920	+	0.738549i
$265$		0			0
$266$		0			0
$267$	−10.3923	+	10.3923i	−0.635999	+	0.635999i
$268$		0			0
$269$	−2.59808	−	4.50000i	−0.158408	−	0.274370i	0.775887	−	0.630872i	$-0.217303\pi$
$269$	−2.59808	−	4.50000i	−0.158408	−	0.274370i	−0.934295	+	0.356502i	$0.883969\pi$
$270$		0			0
$271$	−9.50000	−	16.4545i	−0.577084	−	0.999539i	−0.995812	−	0.0914269i	$-0.970857\pi$
$271$	−9.50000	−	16.4545i	−0.577084	−	0.999539i	0.418728	−	0.908112i	$-0.362476\pi$
$272$	−2.53590	−	9.46410i	−0.153761	−	0.573845i
$273$		0			0
$274$	13.8564			0.837096
$275$		0			0
$276$		0			0
$277$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$277$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$278$	−2.00000	+	2.00000i	−0.119952	+	0.119952i
$279$	21.6506	−	37.5000i	1.29619	−	2.24507i
$280$		0			0
$281$	9.00000	+	5.19615i	0.536895	+	0.309976i	0.743820	−	0.668380i	$-0.233012\pi$
$281$	9.00000	+	5.19615i	0.536895	+	0.309976i	−0.206925	+	0.978357i	$0.566345\pi$
$282$	9.46410	+	2.53590i	0.563579	+	0.151011i
$283$	1.90192	−	7.09808i	0.113058	−	0.421937i	−0.886077	−	0.463539i	$-0.846579\pi$
$283$	1.90192	−	7.09808i	0.113058	−	0.421937i	0.999134	+	0.0416020i	$0.0132461\pi$
$284$		0			0
$285$		0			0
$286$	−18.0000			−1.06436
$287$		0			0
$288$		0			0
$289$	−9.52628	−	5.50000i	−0.560369	−	0.323529i
$290$		0			0
$291$	−18.0000	+	31.1769i	−1.05518	+	1.82762i
$292$		0			0
$293$	−14.0000	+	14.0000i	−0.817889	+	0.817889i	−0.985802	−	0.167913i	$-0.946297\pi$
$293$	−14.0000	+	14.0000i	−0.817889	+	0.817889i	0.167913	+	0.985802i	$0.446297\pi$
$294$	24.2487	+	14.0000i	1.41421	+	0.816497i
$295$		0			0
$296$	12.0000			0.697486
$297$	12.0000	+	12.0000i	0.696311	+	0.696311i
$298$	1.09808	+	4.09808i	0.0636098	+	0.237395i
$299$	−15.5885	−	27.0000i	−0.901504	−	1.56145i
$300$		0			0
$301$		0			0
$302$	−11.8301	−	3.16987i	−0.680747	−	0.182406i
$303$	18.0000	−	18.0000i	1.03407	−	1.03407i
$304$	6.92820	+	16.0000i	0.397360	+	0.917663i
$305$		0			0
$306$	−15.0000	+	8.66025i	−0.857493	+	0.495074i
$307$	8.78461	−	32.7846i	0.501364	−	1.87112i	0.0103834	−	0.999946i	$-0.496695\pi$
$307$	8.78461	−	32.7846i	0.501364	−	1.87112i	0.490981	−	0.871170i	$-0.336639\pi$
$308$		0			0
$309$	−20.7846	+	12.0000i	−1.18240	+	0.682656i
$310$		0			0
$311$	−18.0000			−1.02069			−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000			−1.02069			−0.510343	+	0.859971i	$0.670482\pi$
$312$	−24.0000	−	24.0000i	−1.35873	−	1.35873i
$313$	7.09808	−	1.90192i	0.401207	−	0.107503i	−0.0525725	−	0.998617i	$-0.516742\pi$
$313$	7.09808	−	1.90192i	0.401207	−	0.107503i	0.453779	+	0.891114i	$0.350075\pi$
$314$	−10.3923	+	18.0000i	−0.586472	+	1.01580i
$315$		0			0
$316$		0			0
$317$	1.36603	−	0.366025i	0.0767236	−	0.0205580i	−0.220253	−	0.975443i	$-0.570688\pi$
$317$	1.36603	−	0.366025i	0.0767236	−	0.0205580i	0.296977	+	0.954885i	$0.404022\pi$
$318$	10.9282	+	2.92820i	0.612823	+	0.164205i
$319$	−7.79423	+	13.5000i	−0.436393	+	0.755855i
$320$		0			0
$321$	−14.0000	−	24.2487i	−0.781404	−	1.35343i
$322$		0			0
$323$	−9.92820	−	3.92820i	−0.552420	−	0.218571i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−8.87564	−	33.1244i	−0.490824	−	1.83178i
$328$	−7.60770	+	28.3923i	−0.420065	+	1.56770i
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$	−5.49038	−	20.4904i	−0.300871	−	1.12287i
$334$		−	28.0000i		−	1.53209i
$335$		0			0
$336$		0			0
$337$	3.29423	−	12.2942i	0.179448	−	0.669709i	−0.816303	−	0.577624i	$-0.803980\pi$
$337$	3.29423	−	12.2942i	0.179448	−	0.669709i	0.995751	−	0.0920854i	$-0.0293533\pi$
$338$	−1.83013	−	6.83013i	−0.0995458	−	0.371510i
$339$	13.8564	−	8.00000i	0.752577	−	0.434500i
$340$		0			0
$341$			25.9808i			1.40694i
$342$	24.1506	−	19.1506i	1.30592	−	1.03555i
$343$		0			0
$344$	10.3923	+	18.0000i	0.560316	+	0.970495i
$345$		0			0
$346$	2.00000	−	3.46410i	0.107521	−	0.186231i
$347$	−9.46410	−	2.53590i	−0.508060	−	0.136134i	−0.00432163	−	0.999991i	$-0.501376\pi$
$347$	−9.46410	−	2.53590i	−0.508060	−	0.136134i	−0.503738	+	0.863857i	$0.668042\pi$
$348$		0			0
$349$			4.00000i			0.214115i	0.994253	+	0.107058i	$0.0341429\pi$
$349$			4.00000i			0.214115i	−0.994253	+	0.107058i	$0.965857\pi$
$350$		0			0
$351$	−12.0000	+	20.7846i	−0.640513	+	1.10940i
$352$		0			0
$353$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$353$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$354$	−20.7846			−1.10469
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−9.50962	+	35.4904i	−0.502599	+	1.87572i
$359$	−10.3923	+	6.00000i	−0.548485	+	0.316668i	−0.748511	−	0.663123i	$-0.769231\pi$
$359$	−10.3923	+	6.00000i	−0.548485	+	0.316668i	0.200026	+	0.979791i	$0.435897\pi$
$360$		0			0
$361$	18.5000	+	4.33013i	0.973684	+	0.227901i
$362$	24.2487	−	24.2487i	1.27448	−	1.27448i
$363$	5.46410	+	1.46410i	0.286791	+	0.0768454i
$364$		0			0
$365$		0			0
$366$	−2.00000	−	3.46410i	−0.104542	−	0.181071i
$367$	3.80385	+	14.1962i	0.198559	+	0.741033i	0.991317	+	0.131496i	$0.0419780\pi$
$367$	3.80385	+	14.1962i	0.198559	+	0.741033i	−0.792757	+	0.609537i	$0.791355\pi$
$368$	−20.7846	−	20.7846i	−1.08347	−	1.08347i
$369$	51.9615			2.70501
$370$		0			0
$371$		0			0
$372$		0			0
$373$	6.00000	−	6.00000i	0.310668	−	0.310668i	−0.534500	−	0.845168i	$-0.679500\pi$
$373$	6.00000	−	6.00000i	0.310668	−	0.310668i	0.845168	+	0.534500i	$0.179500\pi$
$374$	5.19615	−	9.00000i	0.268687	−	0.465379i
$375$		0			0
$376$	−6.00000	−	3.46410i	−0.309426	−	0.178647i
$377$	−21.2942	−	5.70577i	−1.09671	−	0.293862i
$378$		0			0
$379$	−25.9808			−1.33454			−0.667271	−	0.744815i	$-0.732538\pi$
$379$	−25.9808			−1.33454			−0.667271	+	0.744815i	$0.732538\pi$
$380$		0			0
$381$	36.0000			1.84434
$382$	1.09808	−	4.09808i	0.0561825	−	0.209676i
$383$	31.4186	+	8.41858i	1.60541	+	0.430170i	0.946672	−	0.322199i	$-0.104422\pi$
$383$	31.4186	+	8.41858i	1.60541	+	0.430170i	0.658743	+	0.752368i	$0.271089\pi$
$384$	−27.7128	−	16.0000i	−1.41421	−	0.816497i
$385$		0			0
$386$	12.0000	−	20.7846i	0.610784	−	1.05791i
$387$	25.9808	−	25.9808i	1.32068	−	1.32068i
$388$		0			0
$389$	−2.59808	−	1.50000i	−0.131728	−	0.0760530i	0.432688	−	0.901544i	$-0.357565\pi$
$389$	−2.59808	−	1.50000i	−0.131728	−	0.0760530i	−0.564416	+	0.825491i	$0.690898\pi$
$390$		0			0
$391$	18.0000			0.910299
$392$	−14.0000	−	14.0000i	−0.707107	−	0.707107i
$393$	−4.39230	−	16.3923i	−0.221562	−	0.826882i
$394$	−8.66025	−	15.0000i	−0.436297	−	0.755689i
$395$		0			0
$396$		0			0
$397$	14.1962	+	3.80385i	0.712484	+	0.190910i	0.596816	−	0.802378i	$-0.296432\pi$
$397$	14.1962	+	3.80385i	0.712484	+	0.190910i	0.115669	+	0.993288i	$0.463099\pi$
$398$	−7.00000	+	7.00000i	−0.350878	+	0.350878i
$399$		0			0
$400$		0			0
$401$	−13.5000	+	7.79423i	−0.674158	+	0.389225i	−0.797650	−	0.603120i	$-0.793924\pi$
$401$	−13.5000	+	7.79423i	−0.674158	+	0.389225i	0.123492	+	0.992346i	$0.460591\pi$
$402$	−13.1769	+	49.1769i	−0.657205	+	2.45272i
$403$	−35.4904	+	9.50962i	−1.76790	+	0.473708i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	9.00000	+	9.00000i	0.446113	+	0.446113i
$408$	18.9282	−	5.07180i	0.937086	−	0.251091i
$409$	2.59808	−	4.50000i	0.128467	−	0.222511i	−0.794616	−	0.607112i	$-0.792328\pi$
$409$	2.59808	−	4.50000i	0.128467	−	0.222511i	0.923083	+	0.384602i	$0.125661\pi$
$410$		0			0
$411$			27.7128i			1.36697i
$412$		0			0
$413$		0			0
$414$	−25.9808	+	45.0000i	−1.27688	+	2.21163i
$415$		0			0
$416$		0			0
$417$	−4.00000	−	4.00000i	−0.195881	−	0.195881i
$418$	−6.80385	+	17.1962i	−0.332787	+	0.841091i
$419$			9.00000i			0.439679i	0.975536	+	0.219839i	$0.0705533\pi$
$419$			9.00000i			0.439679i	−0.975536	+	0.219839i	$0.929447\pi$
$420$		0			0
$421$	31.5000	−	18.1865i	1.53522	−	0.886357i	0.536107	−	0.844150i	$-0.319894\pi$
$421$	31.5000	−	18.1865i	1.53522	−	0.886357i	0.999109	−	0.0422075i	$-0.0134391\pi$
$422$	0.633975	+	2.36603i	0.0308614	+	0.115176i
$423$	−3.16987	+	11.8301i	−0.154124	+	0.575200i
$424$	−6.92820	−	4.00000i	−0.336463	−	0.194257i
$425$		0			0
$426$			62.3538i			3.02105i
$427$		0			0
$428$		0			0
$429$		−	36.0000i		−	1.73810i
$430$		0			0
$431$	−13.5000	−	7.79423i	−0.650272	−	0.375435i	0.138288	−	0.990392i	$-0.455840\pi$
$431$	−13.5000	−	7.79423i	−0.650272	−	0.375435i	−0.788560	+	0.614957i	$0.789173\pi$
$432$	−5.85641	+	21.8564i	−0.281766	+	1.05157i
$433$	−6.58846	−	24.5885i	−0.316621	−	1.18165i	−0.922471	−	0.386067i	$-0.873833\pi$
$433$	−6.58846	−	24.5885i	−0.316621	−	1.18165i	0.605850	−	0.795579i	$-0.292833\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−31.6865	+	4.68653i	−1.51577	+	0.224187i
$438$	−20.7846	−	20.7846i	−0.993127	−	0.993127i
$439$	6.06218	+	10.5000i	0.289332	+	0.501138i	0.973650	−	0.228046i	$-0.0732335\pi$
$439$	6.06218	+	10.5000i	0.289332	+	0.501138i	−0.684318	+	0.729183i	$0.739900\pi$
$440$		0			0
$441$	−17.5000	+	30.3109i	−0.833333	+	1.44338i
$442$	14.1962	+	3.80385i	0.675242	+	0.180931i
$443$	−4.73205	+	1.26795i	−0.224827	+	0.0602421i	−0.369474	−	0.929241i	$-0.620462\pi$
$443$	−4.73205	+	1.26795i	−0.224827	+	0.0602421i	0.144647	+	0.989483i	$0.453795\pi$
$444$		0			0
$445$		0			0
$446$	−3.00000	+	5.19615i	−0.142054	+	0.246045i
$447$	−8.19615	+	2.19615i	−0.387665	+	0.103874i
$448$		0			0
$449$	25.9808			1.22611			0.613054	−	0.790041i	$-0.289941\pi$
$449$	25.9808			1.22611			0.613054	+	0.790041i	$0.289941\pi$
$450$		0			0
$451$	−27.0000	+	15.5885i	−1.27138	+	0.734032i
$452$		0			0
$453$	6.33975	−	23.6603i	0.297867	−	1.11166i
$454$	−12.1244	+	7.00000i	−0.569024	+	0.328526i
$455$		0			0
$456$	−32.0000	+	13.8564i	−1.49854	+	0.648886i
$457$	−25.9808	+	25.9808i	−1.21533	+	1.21533i	−0.246079	+	0.969250i	$0.579142\pi$
$457$	−25.9808	+	25.9808i	−1.21533	+	1.21533i	−0.969250	+	0.246079i	$0.920858\pi$
$458$	15.0263	+	4.02628i	0.702132	+	0.188136i
$459$	−6.92820	−	12.0000i	−0.323381	−	0.560112i
$460$		0			0
$461$	−4.50000	−	7.79423i	−0.209586	−	0.363013i	0.741998	−	0.670402i	$-0.233878\pi$
$461$	−4.50000	−	7.79423i	−0.209586	−	0.363013i	−0.951584	+	0.307388i	$0.900545\pi$
$462$		0			0
$463$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$	−20.7846			−0.964901
$465$		0			0
$466$	−6.00000	−	3.46410i	−0.277945	−	0.160471i
$467$	−8.66025	+	8.66025i	−0.400749	+	0.400749i	−0.878497	−	0.477748i	$-0.841453\pi$
$467$	−8.66025	+	8.66025i	−0.400749	+	0.400749i	0.477748	+	0.878497i	$0.341453\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−36.0000	−	20.7846i	−1.65879	−	0.957704i
$472$	14.1962	+	3.80385i	0.653431	+	0.175086i
$473$	−5.70577	+	21.2942i	−0.262352	+	0.979110i
$474$	−20.7846			−0.954669
$475$		0			0
$476$		0			0
$477$	−3.66025	+	13.6603i	−0.167592	+	0.625460i
$478$	28.6865	+	7.68653i	1.31209	+	0.351574i
$479$	−2.59808	−	1.50000i	−0.118709	−	0.0685367i	0.439470	−	0.898257i	$-0.355166\pi$
$479$	−2.59808	−	1.50000i	−0.118709	−	0.0685367i	−0.558179	+	0.829721i	$0.688500\pi$
$480$		0			0
$481$	−9.00000	+	15.5885i	−0.410365	+	0.710772i
$482$	−19.0526	+	19.0526i	−0.867820	+	0.867820i
$483$		0			0
$484$		0			0
$485$		0			0
$486$	−20.0000			−0.907218
$487$	27.0000	+	27.0000i	1.22349	+	1.22349i	0.966384	+	0.257103i	$0.0827679\pi$
$487$	27.0000	+	27.0000i	1.22349	+	1.22349i	0.257103	+	0.966384i	$0.417232\pi$
$488$	0.732051	+	2.73205i	0.0331384	+	0.123674i
$489$		0			0
$490$		0			0
$491$	10.5000	+	18.1865i	0.473858	+	0.820747i	0.999552	−	0.0299272i	$-0.00952753\pi$
$491$	10.5000	+	18.1865i	0.473858	+	0.820747i	−0.525694	+	0.850674i	$0.676194\pi$
$492$		0			0
$493$	9.00000	−	9.00000i	0.405340	−	0.405340i
$494$	−25.9808	−	3.00000i	−1.16893	−	0.134976i
$495$		0			0
$496$	−30.0000	+	17.3205i	−1.34704	+	0.777714i
$497$		0			0
$498$	−47.3205	+	12.6795i	−2.12048	+	0.568182i
$499$	32.9090	−	19.0000i	1.47321	−	0.850557i	0.473662	−	0.880707i	$-0.342932\pi$
$499$	32.9090	−	19.0000i	1.47321	−	0.850557i	0.999545	+	0.0301498i	$0.00959843\pi$
$500$		0			0
$501$	56.0000			2.50190
$502$	21.0000	+	21.0000i	0.937276	+	0.937276i
$503$	30.7583	−	8.24167i	1.37145	−	0.367478i	0.503440	−	0.864030i	$-0.332068\pi$
$503$	30.7583	−	8.24167i	1.37145	−	0.367478i	0.868007	+	0.496553i	$0.165401\pi$
$504$		0			0
$505$		0			0
$506$		−	31.1769i		−	1.38598i
$507$	13.6603	−	3.66025i	0.606673	−	0.162558i
$508$		0			0
$509$	−10.3923	+	18.0000i	−0.460631	+	0.797836i	−0.998992	−	0.0448779i	$-0.985710\pi$
$509$	−10.3923	+	18.0000i	−0.460631	+	0.797836i	0.538362	+	0.842714i	$0.319043\pi$
$510$		0			0
$511$		0			0
$512$	16.0000	+	16.0000i	0.707107	+	0.707107i
$513$	15.3205	+	19.3205i	0.676417	+	0.853021i
$514$			2.00000i			0.0882162i
$515$		0			0
$516$		0			0
$517$	−1.90192	−	7.09808i	−0.0836465	−	0.312173i
$518$		0			0
$519$	6.92820	+	4.00000i	0.304114	+	0.175581i
$520$		0			0
$521$			25.9808i			1.13824i	0.822255	+	0.569119i	$0.192716\pi$
$521$			25.9808i			1.13824i	−0.822255	+	0.569119i	$0.807284\pi$
$522$	9.50962	+	35.4904i	0.416225	+	1.55337i
$523$	4.39230	+	16.3923i	0.192062	+	0.716785i	0.993008	+	0.118049i	$0.0376638\pi$
$523$	4.39230	+	16.3923i	0.192062	+	0.716785i	−0.800946	+	0.598737i	$0.795669\pi$
$524$		0			0
$525$		0			0
$526$	24.0000	+	13.8564i	1.04645	+	0.604168i
$527$	5.49038	−	20.4904i	0.239165	−	0.892575i
$528$	−8.78461	−	32.7846i	−0.382301	−	1.42677i
$529$	26.8468	−	15.5000i	1.16725	−	0.673913i
$530$		0			0
$531$		−	25.9808i		−	1.12747i
$532$		0			0
$533$	−31.1769	−	31.1769i	−1.35042	−	1.35042i
$534$	−10.3923	−	18.0000i	−0.449719	−	0.778936i
$535$		0			0
$536$	18.0000	−	31.1769i	0.777482	−	1.34664i
$537$	−70.9808	−	19.0192i	−3.06305	−	0.820741i
$538$	7.09808	−	1.90192i	0.306020	−	0.0819978i
$539$		−	21.0000i		−	0.904534i
$540$		0			0
$541$	−9.50000	+	16.4545i	−0.408437	+	0.707433i	−0.994715	−	0.102677i	$-0.967259\pi$
$541$	−9.50000	+	16.4545i	−0.408437	+	0.707433i	0.586278	+	0.810110i	$0.300593\pi$
$542$	25.9545	−	6.95448i	1.11484	−	0.298721i
$543$	48.4974	+	48.4974i	2.08122	+	2.08122i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	28.6865	−	7.68653i	1.22655	−	0.328652i	0.413313	−	0.910589i	$-0.364371\pi$
$547$	28.6865	−	7.68653i	1.22655	−	0.328652i	0.813234	+	0.581936i	$0.197705\pi$
$548$		0			0
$549$	4.33013	−	2.50000i	0.184805	−	0.106697i
$550$		0			0
$551$	−13.5000	+	18.1865i	−0.575119	+	0.774772i
$552$	41.5692	−	41.5692i	1.76930	−	1.76930i
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−5.70577	−	21.2942i	−0.241761	−	0.902265i	−0.974984	−	0.222276i	$-0.928651\pi$
$557$	−5.70577	−	21.2942i	−0.241761	−	0.902265i	0.733222	−	0.679989i	$-0.238015\pi$
$558$	43.3013	+	43.3013i	1.83309	+	1.83309i
$559$	−31.1769			−1.31864
$560$		0			0
$561$	18.0000	+	10.3923i	0.759961	+	0.438763i
$562$	−10.3923	+	10.3923i	−0.438373	+	0.438373i
$563$	−14.0000	+	14.0000i	−0.590030	+	0.590030i	−0.937639	−	0.347610i	$-0.886993\pi$
$563$	−14.0000	+	14.0000i	−0.590030	+	0.590030i	0.347610	+	0.937639i	$0.386993\pi$
$564$		0			0
$565$		0			0
$566$	9.00000	+	5.19615i	0.378298	+	0.218411i
$567$		0			0
$568$	11.4115	−	42.5885i	0.478818	−	1.78697i
$569$	−25.9808			−1.08917			−0.544585	−	0.838706i	$-0.683313\pi$
$569$	−25.9808			−1.08917			−0.544585	+	0.838706i	$0.683313\pi$
$570$		0			0
$571$	7.00000			0.292941			0.146470	−	0.989215i	$-0.453209\pi$
$571$	7.00000			0.292941			0.146470	+	0.989215i	$0.453209\pi$
$572$		0			0
$573$	8.19615	+	2.19615i	0.342399	+	0.0917456i
$574$		0			0
$575$		0			0
$576$	20.0000	−	34.6410i	0.833333	−	1.44338i
$577$	−25.9808	+	25.9808i	−1.08159	+	1.08159i	−0.0852322	+	0.996361i	$0.527163\pi$
$577$	−25.9808	+	25.9808i	−1.08159	+	1.08159i	−0.996361	+	0.0852322i	$0.972837\pi$
$578$	11.0000	−	11.0000i	0.457540	−	0.457540i
$579$	41.5692	+	24.0000i	1.72756	+	0.997406i
$580$		0			0
$581$		0			0
$582$	−36.0000	−	36.0000i	−1.49225	−	1.49225i
$583$	−2.19615	−	8.19615i	−0.0909553	−	0.339450i
$584$	10.3923	+	18.0000i	0.430037	+	0.744845i
$585$		0			0
$586$	−14.0000	−	24.2487i	−0.578335	−	1.00171i
$587$	−9.46410	−	2.53590i	−0.390625	−	0.104668i	0.0581602	−	0.998307i	$-0.481477\pi$
$587$	−9.46410	−	2.53590i	−0.390625	−	0.104668i	−0.448785	+	0.893640i	$0.648143\pi$
$588$		0			0
$589$	−4.33013	+	37.5000i	−0.178420	+	1.54516i
$590$		0			0
$591$	30.0000	−	17.3205i	1.23404	−	0.712470i
$592$	−4.39230	+	16.3923i	−0.180523	+	0.673720i
$593$	18.9282	−	5.07180i	0.777288	−	0.208274i	0.151699	−	0.988427i	$-0.451525\pi$
$593$	18.9282	−	5.07180i	0.777288	−	0.208274i	0.625589	+	0.780153i	$0.284859\pi$
$594$	−20.7846	+	12.0000i	−0.852803	+	0.492366i
$595$		0			0
$596$		0			0
$597$	−14.0000	−	14.0000i	−0.572982	−	0.572982i
$598$	42.5885	−	11.4115i	1.74157	−	0.466653i
$599$	15.5885	−	27.0000i	0.636927	−	1.10319i	−0.349176	−	0.937057i	$-0.613539\pi$
$599$	15.5885	−	27.0000i	0.636927	−	1.10319i	0.986103	−	0.166133i	$-0.0531281\pi$
$600$		0			0
$601$		−	25.9808i		−	1.05978i	−0.848067	−	0.529889i	$-0.822234\pi$
$601$		−	25.9808i		−	1.05978i	0.848067	−	0.529889i	$-0.177766\pi$
$602$		0			0
$603$	−61.4711	−	16.4711i	−2.50330	−	0.670757i
$604$		0			0
$605$		0			0
$606$	18.0000	+	31.1769i	0.731200	+	1.26648i
$607$	−3.00000	−	3.00000i	−0.121766	−	0.121766i	0.643598	−	0.765364i	$-0.277441\pi$
$607$	−3.00000	−	3.00000i	−0.121766	−	0.121766i	−0.765364	+	0.643598i	$0.777441\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	9.00000	−	5.19615i	0.364101	−	0.210214i
$612$		0			0
$613$	−7.60770	+	28.3923i	−0.307272	+	1.14675i	0.623700	+	0.781664i	$0.285629\pi$
$613$	−7.60770	+	28.3923i	−0.307272	+	1.14675i	−0.930972	+	0.365091i	$0.881038\pi$
$614$	41.5692	+	24.0000i	1.67760	+	0.968561i
$615$		0			0
$616$		0			0
$617$	0.633975	+	2.36603i	0.0255229	+	0.0952526i	0.977512	−	0.210878i	$-0.0676323\pi$
$617$	0.633975	+	2.36603i	0.0255229	+	0.0952526i	−0.951990	+	0.306131i	$0.900966\pi$
$618$	−8.78461	−	32.7846i	−0.353369	−	1.31879i
$619$			14.0000i			0.562708i	0.959604	+	0.281354i	$0.0907834\pi$
$619$			14.0000i			0.562708i	−0.959604	+	0.281354i	$0.909217\pi$
$620$		0			0
$621$	−36.0000	−	20.7846i	−1.44463	−	0.834058i
$622$	6.58846	−	24.5885i	0.264173	−	0.985907i
$623$		0			0
$624$	41.5692	−	24.0000i	1.66410	−	0.960769i
$625$		0			0
$626$			10.3923i			0.415360i
$627$	−34.3923	−	13.6077i	−1.37350	−	0.543439i
$628$		0			0
$629$	−5.19615	−	9.00000i	−0.207184	−	0.358854i
$630$		0			0
$631$	0.500000	−	0.866025i	0.0199047	−	0.0344759i	−0.855901	−	0.517139i	$-0.826997\pi$
$631$	0.500000	−	0.866025i	0.0199047	−	0.0344759i	0.875806	+	0.482663i	$0.160330\pi$
$632$	14.1962	+	3.80385i	0.564693	+	0.151309i
$633$	−4.73205	+	1.26795i	−0.188082	+	0.0503965i
$634$			2.00000i			0.0794301i
$635$		0			0
$636$		0			0
$637$	28.6865	−	7.68653i	1.13660	−	0.304552i
$638$	−15.5885	−	15.5885i	−0.617153	−	0.617153i
$639$	−77.9423			−3.08335
$640$		0			0
$641$	−13.5000	+	7.79423i	−0.533218	+	0.307854i	−0.742326	−	0.670039i	$-0.766277\pi$
$641$	−13.5000	+	7.79423i	−0.533218	+	0.307854i	0.209108	+	0.977893i	$0.432944\pi$
$642$	38.2487	−	10.2487i	1.50956	−	0.404484i
$643$	1.90192	−	7.09808i	0.0750046	−	0.279921i	−0.918230	−	0.396048i	$-0.870381\pi$
$643$	1.90192	−	7.09808i	0.0750046	−	0.279921i	0.993234	+	0.116127i	$0.0370480\pi$
$644$		0			0
$645$		0			0
$646$	9.00000	−	12.1244i	0.354100	−	0.477026i
$647$	8.66025	−	8.66025i	0.340470	−	0.340470i	−0.516074	−	0.856544i	$-0.672607\pi$
$647$	8.66025	−	8.66025i	0.340470	−	0.340470i	0.856544	+	0.516074i	$0.172607\pi$
$648$	−2.73205	−	0.732051i	−0.107325	−	0.0287577i
$649$	7.79423	+	13.5000i	0.305950	+	0.529921i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	25.9808	+	25.9808i	1.01671	+	1.01671i	0.999858	+	0.0168477i	$0.00536305\pi$
$653$	25.9808	+	25.9808i	1.01671	+	1.01671i	0.0168477	+	0.999858i	$0.494637\pi$
$654$	48.4974			1.89640
$655$		0			0
$656$	−36.0000	−	20.7846i	−1.40556	−	0.811503i
$657$	25.9808	−	25.9808i	1.01361	−	1.01361i
$658$		0			0
$659$	−10.3923	+	18.0000i	−0.404827	+	0.701180i	−0.994301	−	0.106606i	$-0.966001\pi$
$659$	−10.3923	+	18.0000i	−0.404827	+	0.701180i	0.589475	+	0.807787i	$0.299335\pi$
$660$		0			0
$661$	16.5000	+	9.52628i	0.641776	+	0.370529i	0.785298	−	0.619118i	$-0.212510\pi$
$661$	16.5000	+	9.52628i	0.641776	+	0.370529i	−0.143523	+	0.989647i	$0.545843\pi$
$662$		0			0
$663$	−7.60770	+	28.3923i	−0.295458	+	1.10267i
$664$	34.6410			1.34433
$665$		0			0
$666$	30.0000			1.16248
$667$	9.88269	−	36.8827i	0.382659	−	1.42810i
$668$		0			0
$669$	−10.3923	−	6.00000i	−0.401790	−	0.231973i
$670$		0			0
$671$	−1.50000	+	2.59808i	−0.0579069	+	0.100298i
$672$		0			0
$673$	−24.0000	+	24.0000i	−0.925132	+	0.925132i	−0.997386	−	0.0722542i	$-0.976981\pi$
$673$	−24.0000	+	24.0000i	−0.925132	+	0.925132i	0.0722542	+	0.997386i	$0.476981\pi$
$674$	15.5885	+	9.00000i	0.600445	+	0.346667i
$675$		0			0
$676$		0			0
$677$	−8.00000	−	8.00000i	−0.307465	−	0.307465i	0.536460	−	0.843925i	$-0.319761\pi$
$677$	−8.00000	−	8.00000i	−0.307465	−	0.307465i	−0.843925	+	0.536460i	$0.819761\pi$
$678$	5.85641	+	21.8564i	0.224914	+	0.839390i
$679$		0			0
$680$		0			0
$681$	−14.0000	−	24.2487i	−0.536481	−	0.929213i
$682$	−35.4904	−	9.50962i	−1.35900	−	0.364142i
$683$	11.0000	−	11.0000i	0.420903	−	0.420903i	−0.464611	−	0.885515i	$-0.653806\pi$
$683$	11.0000	−	11.0000i	0.420903	−	0.420903i	0.885515	+	0.464611i	$0.153806\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−8.05256	+	30.0526i	−0.307224	+	1.14658i
$688$	−28.3923	+	7.60770i	−1.08245	+	0.290041i
$689$	10.3923	−	6.00000i	0.395915	−	0.228582i
$690$		0			0
$691$	−13.0000			−0.494543			−0.247272	−	0.968946i	$-0.579534\pi$
$691$	−13.0000			−0.494543			−0.247272	+	0.968946i	$0.579534\pi$
$692$		0			0
$693$		0			0
$694$	6.92820	−	12.0000i	0.262991	−	0.455514i
$695$		0			0
$696$		−	41.5692i		−	1.57568i
$697$	24.5885	−	6.58846i	0.931354	−	0.249556i
$698$	−5.46410	−	1.46410i	−0.206819	−	0.0554171i
$699$	6.92820	−	12.0000i	0.262049	−	0.453882i
$700$		0			0
$701$	−12.0000	−	20.7846i	−0.453234	−	0.785024i	0.545351	−	0.838208i	$-0.316396\pi$
$701$	−12.0000	−	20.7846i	−0.453234	−	0.785024i	−0.998585	+	0.0531839i	$0.983063\pi$
$702$	−24.0000	−	24.0000i	−0.905822	−	0.905822i
$703$	11.4904	+	14.4904i	0.433368	+	0.546515i
$704$			24.0000i			0.904534i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−11.2583	−	6.50000i	−0.422815	−	0.244113i	0.273466	−	0.961882i	$-0.411830\pi$
$709$	−11.2583	−	6.50000i	−0.422815	−	0.244113i	−0.696281	+	0.717769i	$0.745163\pi$
$710$		0			0
$711$		−	25.9808i		−	0.974355i
$712$	3.80385	+	14.1962i	0.142555	+	0.532023i
$713$	−16.4711	−	61.4711i	−0.616849	−	2.30211i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−15.3731	+	57.3731i	−0.574118	+	2.14264i
$718$	−4.39230	−	16.3923i	−0.163919	−	0.611755i
$719$	−23.3827	+	13.5000i	−0.872027	+	0.503465i	−0.868021	−	0.496527i	$-0.834608\pi$
$719$	−23.3827	+	13.5000i	−0.872027	+	0.503465i	−0.00400572	+	0.999992i	$0.501275\pi$
$720$		0			0
$721$		0			0
$722$	−12.6865	+	23.6865i	−0.472144	+	0.881521i
$723$	−38.1051	−	38.1051i	−1.41714	−	1.41714i
$724$		0			0
$725$		0			0
$726$	−4.00000	+	6.92820i	−0.148454	+	0.257130i
$727$	14.1962	+	3.80385i	0.526506	+	0.141077i	0.512274	−	0.858822i	$-0.328803\pi$
$727$	14.1962	+	3.80385i	0.526506	+	0.141077i	0.0142317	+	0.999899i	$0.495470\pi$
$728$		0			0
$729$		−	43.0000i		−	1.59259i
$730$		0			0
$731$	9.00000	−	15.5885i	0.332877	−	0.576560i
$732$		0			0
$733$	−25.9808	−	25.9808i	−0.959621	−	0.959621i	0.0395945	−	0.999216i	$-0.487393\pi$
$733$	−25.9808	−	25.9808i	−0.959621	−	0.959621i	−0.999216	+	0.0395945i	$0.987393\pi$
$734$	−20.7846			−0.767174
$735$		0			0
$736$		0			0
$737$	36.8827	−	9.88269i	1.35859	−	0.364033i
$738$	−19.0192	+	70.9808i	−0.700108	+	2.61284i
$739$	45.8993	−	26.5000i	1.68843	−	0.974818i	0.732717	−	0.680534i	$-0.238252\pi$
$739$	45.8993	−	26.5000i	1.68843	−	0.974818i	0.955718	−	0.294285i	$-0.0950814\pi$
$740$		0			0
$741$	6.00000	−	51.9615i	0.220416	−	1.90885i
$742$		0			0
$743$	31.4186	+	8.41858i	1.15264	+	0.308848i	0.784021	−	0.620734i	$-0.213165\pi$
$743$	31.4186	+	8.41858i	1.15264	+	0.308848i	0.368615	+	0.929582i	$0.379832\pi$
$744$	−34.6410	−	60.0000i	−1.27000	−	2.19971i
$745$		0			0
$746$	6.00000	+	10.3923i	0.219676	+	0.380489i
$747$	−15.8494	−	59.1506i	−0.579898	−	2.16421i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	46.5000	+	26.8468i	1.69681	+	0.979653i	0.948753	+	0.316017i	$0.102346\pi$
$751$	46.5000	+	26.8468i	1.69681	+	0.979653i	0.748056	+	0.663636i	$0.230988\pi$
$752$	6.92820	−	6.92820i	0.252646	−	0.252646i
$753$	−42.0000	+	42.0000i	−1.53057	+	1.53057i
$754$	15.5885	−	27.0000i	0.567698	−	0.983282i
$755$		0			0
$756$		0			0
$757$	−21.2942	−	5.70577i	−0.773952	−	0.207380i	−0.149835	−	0.988711i	$-0.547874\pi$
$757$	−21.2942	−	5.70577i	−0.773952	−	0.207380i	−0.624117	+	0.781331i	$0.714541\pi$
$758$	9.50962	−	35.4904i	0.345405	−	1.28907i
$759$	62.3538			2.26330
$760$		0			0
$761$	−48.0000			−1.74000			−0.869999	−	0.493053i	$-0.835881\pi$
$761$	−48.0000			−1.74000			−0.869999	+	0.493053i	$0.835881\pi$
$762$	−13.1769	+	49.1769i	−0.477349	+	1.78149i
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−23.0000	+	39.8372i	−0.831024	+	1.43938i
$767$	−15.5885	+	15.5885i	−0.562867	+	0.562867i
$768$		0			0
$769$	6.06218	+	3.50000i	0.218608	+	0.126213i	0.605305	−	0.795993i	$-0.293051\pi$
$769$	6.06218	+	3.50000i	0.218608	+	0.126213i	−0.386698	+	0.922207i	$0.626384\pi$
$770$		0			0
$771$	−4.00000			−0.144056
$772$		0			0
$773$	−8.41858	−	31.4186i	−0.302795	−	1.13005i	−0.934826	−	0.355105i	$-0.884445\pi$
$773$	−8.41858	−	31.4186i	−0.302795	−	1.13005i	0.632031	−	0.774943i	$-0.282222\pi$
$774$	25.9808	+	45.0000i	0.933859	+	1.61749i
$775$		0			0
$776$	18.0000	+	31.1769i	0.646162	+	1.11919i
$777$		0			0
$778$	3.00000	−	3.00000i	0.107555	−	0.107555i
$779$	−41.5692	+	18.0000i	−1.48937	+	0.644917i
$780$		0			0
$781$	40.5000	−	23.3827i	1.44920	−	0.836698i
$782$	−6.58846	+	24.5885i	−0.235603	+	0.879281i
$783$	−28.3923	+	7.60770i	−1.01466	+	0.271877i
$784$	24.2487	−	14.0000i	0.866025	−	0.500000i
$785$		0			0
$786$	24.0000			0.856052
$787$	27.0000	+	27.0000i	0.962446	+	0.962446i	0.999320	−	0.0368739i	$-0.0117400\pi$
$787$	27.0000	+	27.0000i	0.962446	+	0.962446i	−0.0368739	+	0.999320i	$0.511740\pi$
$788$		0			0
$789$	−27.7128	+	48.0000i	−0.986602	+	1.70885i
$790$		0			0
$791$		0			0
$792$	40.9808	−	10.9808i	1.45619	−	0.390184i
$793$	−4.09808	−	1.09808i	−0.145527	−	0.0389938i
$794$	−10.3923	+	18.0000i	−0.368809	+	0.638796i
$795$		0			0
$796$		0			0
$797$	32.0000	+	32.0000i	1.13350	+	1.13350i	0.989591	+	0.143907i	$0.0459666\pi$
$797$	32.0000	+	32.0000i	1.13350	+	1.13350i	0.143907	+	0.989591i	$0.454033\pi$
$798$		0			0
$799$			6.00000i			0.212265i
$800$		0			0
$801$	22.5000	−	12.9904i	0.794998	−	0.458993i
$802$	−5.70577	−	21.2942i	−0.201478	−	0.751925i
$803$	−5.70577	+	21.2942i	−0.201352	+	0.751457i
$804$		0			0
$805$		0			0
$806$		−	51.9615i		−	1.83027i
$807$	3.80385	+	14.1962i	0.133902	+	0.499728i
$808$	−6.58846	−	24.5885i	−0.231781	−	0.865019i
$809$			39.0000i			1.37117i	0.727994	+	0.685583i	$0.240453\pi$
$809$			39.0000i			1.37117i	−0.727994	+	0.685583i	$0.759547\pi$
$810$		0			0
$811$	1.50000	+	0.866025i	0.0526721	+	0.0304103i	0.526105	−	0.850420i	$-0.323652\pi$
$811$	1.50000	+	0.866025i	0.0526721	+	0.0304103i	−0.473433	+	0.880830i	$0.656985\pi$
$812$		0			0
$813$	13.9090	+	51.9090i	0.487809	+	1.82053i
$814$	−15.5885	+	9.00000i	−0.546375	+	0.315450i
$815$		0			0
$816$			27.7128i			0.970143i
$817$	−11.7846	+	29.7846i	−0.412291	+	1.04203i
$818$	5.19615	+	5.19615i	0.181679	+	0.181679i
$819$		0			0
$820$		0			0
$821$	10.5000	−	18.1865i	0.366453	−	0.634714i	−0.622556	−	0.782576i	$-0.713906\pi$
$821$	10.5000	−	18.1865i	0.366453	−	0.634714i	0.989008	+	0.147861i	$0.0472389\pi$
$822$	−37.8564	−	10.1436i	−1.32039	−	0.353798i
$823$	7.09808	−	1.90192i	0.247423	−	0.0662969i	−0.132976	−	0.991119i	$-0.542453\pi$
$823$	7.09808	−	1.90192i	0.247423	−	0.0662969i	0.380399	+	0.924822i	$0.375787\pi$
$824$			24.0000i			0.836080i
$825$		0			0
$826$		0			0
$827$	15.0263	−	4.02628i	0.522515	−	0.140007i	0.0120853	−	0.999927i	$-0.496153\pi$
$827$	15.0263	−	4.02628i	0.522515	−	0.140007i	0.510430	+	0.859920i	$0.329486\pi$
$828$		0			0
$829$	−17.3205			−0.601566			−0.300783	−	0.953693i	$-0.597248\pi$
$829$	−17.3205			−0.601566			−0.300783	+	0.953693i	$0.597248\pi$
$830$		0			0
$831$		0			0
$832$	−32.7846	+	8.78461i	−1.13660	+	0.304552i
$833$	−4.43782	+	16.5622i	−0.153761	+	0.573845i
$834$	6.92820	−	4.00000i	0.239904	−	0.138509i
$835$		0			0
$836$		0			0
$837$	−34.6410	+	34.6410i	−1.19737	+	1.19737i
$838$	−12.2942	−	3.29423i	−0.424697	−	0.113797i
$839$	10.3923	+	18.0000i	0.358782	+	0.621429i	0.987758	−	0.155996i	$-0.0498587\pi$
$839$	10.3923	+	18.0000i	0.358782	+	0.621429i	−0.628975	+	0.777425i	$0.716525\pi$
$840$		0			0
$841$	1.00000	+	1.73205i	0.0344828	+	0.0597259i
$842$	13.3135	+	49.6865i	0.458812	+	1.71231i
$843$	−20.7846	−	20.7846i	−0.715860	−	0.715860i
$844$		0			0
$845$		0			0
$846$	−15.0000	−	8.66025i	−0.515711	−	0.297746i
$847$		0			0
$848$	8.00000	−	8.00000i	0.274721	−	0.274721i
$849$	−10.3923	+	18.0000i	−0.356663	+	0.617758i
$850$		0			0
$851$	−27.0000	−	15.5885i	−0.925548	−	0.534365i
$852$		0			0
$853$	11.4115	−	42.5885i	0.390724	−	1.45820i	−0.438219	−	0.898868i	$-0.644391\pi$
$853$	11.4115	−	42.5885i	0.390724	−	1.45820i	0.828943	−	0.559333i	$-0.188943\pi$
$854$		0			0
$855$		0			0
$856$	−28.0000			−0.957020
$857$	6.95448	−	25.9545i	0.237561	−	0.886588i	−0.739417	−	0.673247i	$-0.764899\pi$
$857$	6.95448	−	25.9545i	0.237561	−	0.886588i	0.976978	−	0.213341i	$-0.0684345\pi$
$858$	49.1769	+	13.1769i	1.67887	+	0.449852i
$859$	40.7032	+	23.5000i	1.38878	+	0.801810i	0.993177	−	0.116614i	$-0.0372041\pi$
$859$	40.7032	+	23.5000i	1.38878	+	0.801810i	0.395598	+	0.918424i	$0.370537\pi$
$860$		0			0
$861$		0			0
$862$	15.5885	−	15.5885i	0.530945	−	0.530945i
$863$	−14.0000	+	14.0000i	−0.476566	+	0.476566i	−0.904031	−	0.427466i	$-0.859406\pi$
$863$	−14.0000	+	14.0000i	−0.476566	+	0.476566i	0.427466	+	0.904031i	$0.359406\pi$
$864$		0			0
$865$		0			0
$866$	36.0000			1.22333
$867$	22.0000	+	22.0000i	0.747159	+	0.747159i
$868$		0			0
$869$	7.79423	+	13.5000i	0.264401	+	0.457956i
$870$		0			0
$871$	27.0000	+	46.7654i	0.914860	+	1.58458i
$872$	−33.1244	−	8.87564i	−1.12173	−	0.300567i
$873$	45.0000	−	45.0000i	1.52302	−	1.52302i
$874$	5.19615	−	45.0000i	0.175762	−	1.52215i
$875$		0			0
$876$		0			0
$877$	3.29423	−	12.2942i	0.111238	−	0.415147i	−0.887740	−	0.460346i	$-0.847726\pi$
$877$	3.29423	−	12.2942i	0.111238	−	0.415147i	0.998978	+	0.0451990i	$0.0143922\pi$
$878$	−16.5622	+	4.43782i	−0.558946	+	0.149769i
$879$	48.4974	−	28.0000i	1.63578	−	0.944417i
$880$		0			0
$881$	−33.0000			−1.11180			−0.555899	−	0.831250i	$-0.687626\pi$
$881$	−33.0000			−1.11180			−0.555899	+	0.831250i	$0.687626\pi$
$882$	−35.0000	−	35.0000i	−1.17851	−	1.17851i
$883$	42.5885	−	11.4115i	1.43322	−	0.384029i	0.543063	−	0.839692i	$-0.317264\pi$
$883$	42.5885	−	11.4115i	1.43322	−	0.384029i	0.890152	+	0.455663i	$0.150598\pi$
$884$		0			0
$885$		0			0
$886$		−	6.92820i		−	0.232758i
$887$	−19.1244	+	5.12436i	−0.642133	+	0.172059i	−0.565169	−	0.824975i	$-0.691189\pi$
$887$	−19.1244	+	5.12436i	−0.642133	+	0.172059i	−0.0769636	+	0.997034i	$0.524523\pi$
$888$	−32.7846	−	8.78461i	−1.10018	−	0.294792i
$889$		0			0
$890$		0			0
$891$	−1.50000	−	2.59808i	−0.0502519	−	0.0870388i
$892$		0			0
$893$	−1.56218	−	10.5622i	−0.0522763	−	0.353450i
$894$		−	12.0000i		−	0.401340i
$895$		0			0
$896$		0			0
$897$	22.8231	+	85.1769i	0.762041	+	2.84397i
$898$	−9.50962	+	35.4904i	−0.317340	+	1.18433i
$899$	−38.9711	−	22.5000i	−1.29976	−	0.750417i
$900$		0			0
$901$			6.92820i			0.230812i
$902$	−11.4115	−	42.5885i	−0.379963	−	1.41804i
$903$		0			0
$904$		−	16.0000i		−	0.532152i
$905$		0			0
$906$	30.0000	+	17.3205i	0.996683	+	0.575435i
$907$	−2.19615	+	8.19615i	−0.0729220	+	0.272149i	−0.992754	−	0.120164i	$-0.961658\pi$
$907$	−2.19615	+	8.19615i	−0.0729220	+	0.272149i	0.919832	+	0.392312i	$0.128325\pi$
$908$		0			0
$909$	−38.9711	+	22.5000i	−1.29259	+	0.746278i
$910$		0			0
$911$		−	25.9808i		−	0.860781i	−0.902643	−	0.430391i	$-0.858376\pi$
$911$		−	25.9808i		−	0.860781i	0.902643	−	0.430391i	$-0.141624\pi$
$912$	−7.21539	−	48.7846i	−0.238925	−	1.61542i
$913$	25.9808	+	25.9808i	0.859838	+	0.859838i
$914$	−25.9808	−	45.0000i	−0.859367	−	1.48847i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	18.9282	−	5.07180i	0.624724	−	0.167394i
$919$		−	46.0000i		−	1.51740i	−0.651440	−	0.758700i	$-0.725835\pi$
$919$		−	46.0000i		−	1.51740i	0.651440	−	0.758700i	$-0.274165\pi$
$920$		0			0
$921$	−48.0000	+	83.1384i	−1.58165	+	2.73950i
$922$	12.2942	−	3.29423i	0.404889	−	0.108490i
$923$	46.7654	+	46.7654i	1.53930	+	1.53930i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	40.9808	−	10.9808i	1.34598	−	0.360656i
$928$		0			0
$929$	2.59808	−	1.50000i	0.0852401	−	0.0492134i	−0.456774	−	0.889583i	$-0.650995\pi$
$929$	2.59808	−	1.50000i	0.0852401	−	0.0492134i	0.542014	+	0.840369i	$0.317662\pi$
$930$		0			0
$931$	3.50000	−	30.3109i	0.114708	−	0.993399i
$932$		0			0
$933$	49.1769	+	13.1769i	1.60998	+	0.431393i
$934$	−8.66025	−	15.0000i	−0.283372	−	0.490815i
$935$		0			0
$936$	30.0000	+	51.9615i	0.980581	+	1.69842i
$937$	−15.2154	−	56.7846i	−0.497065	−	1.85507i	−0.518142	−	0.855294i	$-0.673376\pi$
$937$	−15.2154	−	56.7846i	−0.497065	−	1.85507i	0.0210771	−	0.999778i	$-0.493290\pi$
$938$		0			0
$939$	−20.7846			−0.678280
$940$		0			0
$941$	−13.5000	−	7.79423i	−0.440087	−	0.254085i	0.263547	−	0.964646i	$-0.415107\pi$
$941$	−13.5000	−	7.79423i	−0.440087	−	0.254085i	−0.703635	+	0.710562i	$0.748441\pi$
$942$	41.5692	−	41.5692i	1.35440	−	1.35440i
$943$	54.0000	−	54.0000i	1.75848	−	1.75848i
$944$	−10.3923	+	18.0000i	−0.338241	+	0.585850i
$945$		0			0
$946$	−27.0000	−	15.5885i	−0.877846	−	0.506824i
$947$	49.6865	+	13.3135i	1.61460	+	0.432630i	0.949408	−	0.314047i	$-0.101685\pi$
$947$	49.6865	+	13.3135i	1.61460	+	0.432630i	0.665188	+	0.746676i	$0.268351\pi$
$948$		0			0
$949$	−31.1769			−1.01205
$950$		0			0
$951$	−4.00000			−0.129709
$952$		0			0
$953$	−2.73205	−	0.732051i	−0.0884998	−	0.0237135i	0.214297	−	0.976768i	$-0.431254\pi$
$953$	−2.73205	−	0.732051i	−0.0884998	−	0.0237135i	−0.302797	+	0.953055i	$0.597920\pi$
$954$	−17.3205	−	10.0000i	−0.560772	−	0.323762i
$955$		0			0
$956$		0			0
$957$	31.1769	−	31.1769i	1.00781	−	1.00781i
$958$	3.00000	−	3.00000i	0.0969256	−	0.0969256i
$959$		0			0
$960$		0			0
$961$	−44.0000			−1.41935
$962$	−18.0000	−	18.0000i	−0.580343	−	0.580343i
$963$	12.8109	+	47.8109i	0.412825	+	1.54068i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−21.2942	−	5.70577i	−0.684776	−	0.183485i	−0.100374	−	0.994950i	$-0.532004\pi$
$967$	−21.2942	−	5.70577i	−0.684776	−	0.183485i	−0.584402	+	0.811464i	$0.698671\pi$
$968$	4.00000	−	4.00000i	0.128565	−	0.128565i
$969$	24.2487	+	18.0000i	0.778981	+	0.578243i
$970$		0			0
$971$	−36.0000	+	20.7846i	−1.15529	+	0.667010i	−0.950172	−	0.311726i	$-0.899093\pi$
$971$	−36.0000	+	20.7846i	−1.15529	+	0.667010i	−0.205123	+	0.978736i	$0.565759\pi$
$972$		0			0
$973$		0			0
$974$	−46.7654	+	27.0000i	−1.49846	+	0.865136i
$975$		0			0
$976$	−4.00000			−0.128037
$977$	32.0000	+	32.0000i	1.02377	+	1.02377i	0.999711	+	0.0240602i	$0.00765934\pi$
$977$	32.0000	+	32.0000i	1.02377	+	1.02377i	0.0240602	+	0.999711i	$0.492341\pi$
$978$		0			0
$979$	−7.79423	+	13.5000i	−0.249105	+	0.431462i
$980$		0			0
$981$			60.6218i			1.93550i
$982$	−28.6865	+	7.68653i	−0.915424	+	0.245287i
$983$	10.9282	+	2.92820i	0.348556	+	0.0933952i	0.428849	−	0.903376i	$-0.358919\pi$
$983$	10.9282	+	2.92820i	0.348556	+	0.0933952i	−0.0802937	+	0.996771i	$0.525586\pi$
$984$	41.5692	−	72.0000i	1.32518	−	2.29528i
$985$		0			0
$986$	9.00000	+	15.5885i	0.286618	+	0.496438i
$987$		0			0
$988$		0			0
$989$		−	54.0000i		−	1.71710i
$990$		0			0
$991$	−21.0000	+	12.1244i	−0.667087	+	0.385143i	−0.794972	−	0.606646i	$-0.792514\pi$
$991$	−21.0000	+	12.1244i	−0.667087	+	0.385143i	0.127885	+	0.991789i	$0.459181\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−5.70577	−	21.2942i	−0.180704	−	0.674395i	−0.995510	−	0.0946612i	$-0.969823\pi$
$997$	−5.70577	−	21.2942i	−0.180704	−	0.674395i	0.814806	−	0.579734i	$-0.196843\pi$
$998$	13.9090	+	51.9090i	0.440281	+	1.64315i
$999$			24.0000i			0.759326i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.p.c.468.1	yes	4
5.2	odd	4		475.2.p.a.107.1	✓	4
5.3	odd	4	inner	475.2.p.c.107.1	yes	4
5.4	even	2		475.2.p.a.468.1	yes	4
19.8	odd	6		475.2.p.a.293.1	yes	4
95.8	even	12		475.2.p.a.407.1	yes	4
95.27	even	12	inner	475.2.p.c.407.1	yes	4
95.84	odd	6	inner	475.2.p.c.293.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.p.a.107.1	✓	4	5.2	odd	4
475.2.p.a.293.1	yes	4	19.8	odd	6
475.2.p.a.407.1	yes	4	95.8	even	12
475.2.p.a.468.1	yes	4	5.4	even	2
475.2.p.c.107.1	yes	4	5.3	odd	4	inner
475.2.p.c.293.1	yes	4	95.84	odd	6	inner
475.2.p.c.407.1	yes	4	95.27	even	12	inner
475.2.p.c.468.1	yes	4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.p (of order \(12\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.366025 + 1.36603i) q^{2} +(-2.73205 - 0.732051i) q^{3} +(2.00000 - 3.46410i) q^{6} +(-2.00000 + 2.00000i) q^{8} +(4.33013 + 2.50000i) q^{9} +O(q^{10})\)	\(q+(-0.366025 + 1.36603i) q^{2} +(-2.73205 - 0.732051i) q^{3} +(2.00000 - 3.46410i) q^{6} +(-2.00000 + 2.00000i) q^{8} +(4.33013 + 2.50000i) q^{9} -3.00000 q^{11} +(-1.09808 - 4.09808i) q^{13} +(-2.00000 - 3.46410i) q^{16} +(2.36603 + 0.633975i) q^{17} +(-5.00000 + 5.00000i) q^{18} +(-4.33013 - 0.500000i) q^{19} +(1.09808 - 4.09808i) q^{22} +(7.09808 - 1.90192i) q^{23} +(6.92820 - 4.00000i) q^{24} +6.00000 q^{26} +(-4.00000 - 4.00000i) q^{27} +(2.59808 - 4.50000i) q^{29} -8.66025i q^{31} +(8.19615 + 2.19615i) q^{33} +(-1.73205 + 3.00000i) q^{34} +(-3.00000 - 3.00000i) q^{37} +(2.26795 - 5.73205i) q^{38} +12.0000i q^{39} +(9.00000 - 5.19615i) q^{41} +(1.90192 - 7.09808i) q^{43} +10.3923i q^{46} +(0.633975 + 2.36603i) q^{47} +(2.92820 + 10.9282i) q^{48} +7.00000i q^{49} +(-6.00000 - 3.46410i) q^{51} +(0.732051 + 2.73205i) q^{53} +(6.92820 - 4.00000i) q^{54} +(11.4641 + 4.53590i) q^{57} +(5.19615 + 5.19615i) q^{58} +(-2.59808 - 4.50000i) q^{59} +(0.500000 - 0.866025i) q^{61} +(11.8301 + 3.16987i) q^{62} -8.00000i q^{64} +(-6.00000 + 10.3923i) q^{66} +(-12.2942 + 3.29423i) q^{67} -20.7846 q^{69} +(-13.5000 + 7.79423i) q^{71} +(-13.6603 + 3.66025i) q^{72} +(1.90192 - 7.09808i) q^{73} +(5.19615 - 3.00000i) q^{74} +(-16.3923 - 4.39230i) q^{78} +(-2.59808 - 4.50000i) q^{79} +(0.500000 + 0.866025i) q^{81} +(3.80385 + 14.1962i) q^{82} +(-8.66025 - 8.66025i) q^{83} +(9.00000 + 5.19615i) q^{86} +(-10.3923 + 10.3923i) q^{87} +(6.00000 - 6.00000i) q^{88} +(2.59808 - 4.50000i) q^{89} +(-6.33975 + 23.6603i) q^{93} -3.46410 q^{94} +(3.29423 - 12.2942i) q^{97} +(-9.56218 - 2.56218i) q^{98} +(-12.9904 - 7.50000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 4 q^{3} + 8 q^{6} - 8 q^{8}+O(q^{10}) \) 4 * q + 2 * q^2 - 4 * q^3 + 8 * q^6 - 8 * q^8	\( 4 q + 2 q^{2} - 4 q^{3} + 8 q^{6} - 8 q^{8} - 12 q^{11} + 6 q^{13} - 8 q^{16} + 6 q^{17} - 20 q^{18} - 6 q^{22} + 18 q^{23} + 24 q^{26} - 16 q^{27} + 12 q^{33} - 12 q^{37} + 16 q^{38} + 36 q^{41} + 18 q^{43} + 6 q^{47} - 16 q^{48} - 24 q^{51} - 4 q^{53} + 32 q^{57} + 2 q^{61} + 30 q^{62} - 24 q^{66} - 18 q^{67} - 54 q^{71} - 20 q^{72} + 18 q^{73} - 24 q^{78} + 2 q^{81} + 36 q^{82} + 36 q^{86} + 24 q^{88} - 60 q^{93} - 18 q^{97} - 14 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 - 4 * q^3 + 8 * q^6 - 8 * q^8 - 12 * q^11 + 6 * q^13 - 8 * q^16 + 6 * q^17 - 20 * q^18 - 6 * q^22 + 18 * q^23 + 24 * q^26 - 16 * q^27 + 12 * q^33 - 12 * q^37 + 16 * q^38 + 36 * q^41 + 18 * q^43 + 6 * q^47 - 16 * q^48 - 24 * q^51 - 4 * q^53 + 32 * q^57 + 2 * q^61 + 30 * q^62 - 24 * q^66 - 18 * q^67 - 54 * q^71 - 20 * q^72 + 18 * q^73 - 24 * q^78 + 2 * q^81 + 36 * q^82 + 36 * q^86 + 24 * q^88 - 60 * q^93 - 18 * q^97 - 14 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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