LMFDB - Embedded newform 475.2.p.c.107.1

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			107.1
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	475.107
Dual form			475.2.p.c.293.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.36603 + 0.366025i) q^{2} +(0.732051 - 2.73205i) q^{3} +(2.00000 - 3.46410i) q^{6} +(-2.00000 - 2.00000i) q^{8} +(-4.33013 - 2.50000i) q^{9} +O(q^{10})$	\(q+(1.36603 + 0.366025i) q^{2} +(0.732051 - 2.73205i) q^{3} +(2.00000 - 3.46410i) q^{6} +(-2.00000 - 2.00000i) q^{8} +(-4.33013 - 2.50000i) q^{9} -3.00000 q^{11} +(4.09808 - 1.09808i) q^{13} +(-2.00000 - 3.46410i) q^{16} +(0.633975 - 2.36603i) q^{17} +(-5.00000 - 5.00000i) q^{18} +(4.33013 + 0.500000i) q^{19} +(-4.09808 - 1.09808i) q^{22} +(1.90192 + 7.09808i) q^{23} +(-6.92820 + 4.00000i) q^{24} +6.00000 q^{26} +(-4.00000 + 4.00000i) q^{27} +(-2.59808 + 4.50000i) q^{29} -8.66025i q^{31} +(-2.19615 + 8.19615i) q^{33} +(1.73205 - 3.00000i) q^{34} +(-3.00000 + 3.00000i) q^{37} +(5.73205 + 2.26795i) q^{38} -12.0000i q^{39} +(9.00000 - 5.19615i) q^{41} +(7.09808 + 1.90192i) q^{43} +10.3923i q^{46} +(2.36603 - 0.633975i) q^{47} +(-10.9282 + 2.92820i) q^{48} -7.00000i q^{49} +(-6.00000 - 3.46410i) q^{51} +(-2.73205 + 0.732051i) q^{53} +(-6.92820 + 4.00000i) q^{54} +(4.53590 - 11.4641i) q^{57} +(-5.19615 + 5.19615i) q^{58} +(2.59808 + 4.50000i) q^{59} +(0.500000 - 0.866025i) q^{61} +(3.16987 - 11.8301i) q^{62} +8.00000i q^{64} +(-6.00000 + 10.3923i) q^{66} +(3.29423 + 12.2942i) q^{67} +20.7846 q^{69} +(-13.5000 + 7.79423i) q^{71} +(3.66025 + 13.6603i) q^{72} +(7.09808 + 1.90192i) q^{73} +(-5.19615 + 3.00000i) q^{74} +(4.39230 - 16.3923i) q^{78} +(2.59808 + 4.50000i) q^{79} +(0.500000 + 0.866025i) q^{81} +(14.1962 - 3.80385i) q^{82} +(8.66025 - 8.66025i) q^{83} +(9.00000 + 5.19615i) q^{86} +(10.3923 + 10.3923i) q^{87} +(6.00000 + 6.00000i) q^{88} +(-2.59808 + 4.50000i) q^{89} +(-23.6603 - 6.33975i) q^{93} +3.46410 q^{94} +(-12.2942 - 3.29423i) q^{97} +(2.56218 - 9.56218i) q^{98} +(12.9904 + 7.50000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$.

$n$	$77$	$401$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$e\left(\frac{5}{6}\right)$

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 $

$2$	1.36603	+	0.366025i	0.965926	+	0.258819i	0.707107	−	0.707107i	$-0.250000\pi$
$2$	1.36603	+	0.366025i	0.965926	+	0.258819i	0.258819	+	0.965926i	$0.416667\pi$
$3$	0.732051	−	2.73205i	0.422650	−	1.57735i	−0.346353	−	0.938104i	$-0.612580\pi$
$3$	0.732051	−	2.73205i	0.422650	−	1.57735i	0.769002	−	0.639246i	$-0.220753\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$	2.00000	−	3.46410i	0.816497	−	1.41421i
$7$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$7$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$8$	−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
$13$	4.09808	−	1.09808i	1.13660	−	0.304552i	0.359018	−	0.933331i	$-0.383112\pi$
$13$	4.09808	−	1.09808i	1.13660	−	0.304552i	0.777584	+	0.628779i	$0.216445\pi$
$14$		0			0
$15$		0			0
$16$	−2.00000	−	3.46410i	−0.500000	−	0.866025i
$17$	0.633975	−	2.36603i	0.153761	−	0.573845i	−0.845447	−	0.534060i	$-0.820666\pi$
$17$	0.633975	−	2.36603i	0.153761	−	0.573845i	0.999208	−	0.0397858i	$-0.0126676\pi$
$18$	−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
$22$	−4.09808	−	1.09808i	−0.873713	−	0.234111i
$23$	1.90192	+	7.09808i	0.396579	+	1.48005i	0.819075	+	0.573687i	$0.194487\pi$
$23$	1.90192	+	7.09808i	0.396579	+	1.48005i	−0.422496	+	0.906365i	$0.638846\pi$
$24$	−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	−0.999797	−	0.0201471i	$-0.993587\pi$
$29$	−2.59808	+	4.50000i	−0.482451	+	0.835629i	0.517346	+	0.855776i	$0.326920\pi$
$30$		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
$33$	−2.19615	+	8.19615i	−0.382301	+	1.42677i
$34$	1.73205	−	3.00000i	0.297044	−	0.514496i
$35$		0			0
$36$		0			0
$37$	−3.00000	+	3.00000i	−0.493197	+	0.493197i	−0.909312	−	0.416115i	$-0.863391\pi$
$37$	−3.00000	+	3.00000i	−0.493197	+	0.493197i	0.416115	+	0.909312i	$0.363391\pi$
$38$	5.73205	+	2.26795i	0.929861	+	0.367910i
$39$		−	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
$43$	7.09808	+	1.90192i	1.08245	+	0.290041i	0.755598	−	0.655036i	$-0.227347\pi$
$43$	7.09808	+	1.90192i	1.08245	+	0.290041i	0.326849	+	0.945077i	$0.394013\pi$
$44$		0			0
$45$		0			0
$46$			10.3923i			1.53226i
$47$	2.36603	−	0.633975i	0.345120	−	0.0924747i	−0.0820953	−	0.996624i	$-0.526161\pi$
$47$	2.36603	−	0.633975i	0.345120	−	0.0924747i	0.427216	+	0.904150i	$0.359495\pi$
$48$	−10.9282	+	2.92820i	−1.57735	+	0.422650i
$49$		−	7.00000i		−	1.00000i
$50$		0			0
$51$	−6.00000	−	3.46410i	−0.840168	−	0.485071i
$52$		0			0
$53$	−2.73205	+	0.732051i	−0.375276	+	0.100555i	−0.441527	−	0.897248i	$-0.645563\pi$
$53$	−2.73205	+	0.732051i	−0.375276	+	0.100555i	0.0662507	+	0.997803i	$0.478896\pi$
$54$	−6.92820	+	4.00000i	−0.942809	+	0.544331i
$55$		0			0
$56$		0			0
$57$	4.53590	−	11.4641i	0.600794	−	1.51846i
$58$	−5.19615	+	5.19615i	−0.682288	+	0.682288i
$59$	2.59808	+	4.50000i	0.338241	+	0.585850i	0.984102	−	0.177605i	$-0.0568349\pi$
$59$	2.59808	+	4.50000i	0.338241	+	0.585850i	−0.645861	+	0.763455i	$0.723502\pi$
$60$		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	$0.146275\pi$
$62$	3.16987	−	11.8301i	0.402574	−	1.50243i
$63$		0			0
$64$			8.00000i			1.00000i
$65$		0			0
$66$	−6.00000	+	10.3923i	−0.738549	+	1.27920i
$67$	3.29423	+	12.2942i	0.402454	+	1.50198i	0.808703	+	0.588217i	$0.200170\pi$
$67$	3.29423	+	12.2942i	0.402454	+	1.50198i	−0.406249	+	0.913762i	$0.633164\pi$
$68$		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	$0.957394\pi$
$72$	3.66025	+	13.6603i	0.431365	+	1.60988i
$73$	7.09808	+	1.90192i	0.830767	+	0.222603i	0.649048	−	0.760747i	$-0.275167\pi$
$73$	7.09808	+	1.90192i	0.830767	+	0.222603i	0.181719	+	0.983351i	$0.441834\pi$
$74$	−5.19615	+	3.00000i	−0.604040	+	0.348743i
$75$		0			0
$76$		0			0
$77$		0			0
$78$	4.39230	−	16.3923i	0.497331	−	1.85606i
$79$	2.59808	+	4.50000i	0.292306	+	0.506290i	0.974355	−	0.225018i	$-0.0722440\pi$
$79$	2.59808	+	4.50000i	0.292306	+	0.506290i	−0.682048	+	0.731307i	$0.738911\pi$
$80$		0			0
$81$	0.500000	+	0.866025i	0.0555556	+	0.0962250i
$82$	14.1962	−	3.80385i	1.56770	−	0.420065i
$83$	8.66025	−	8.66025i	0.950586	−	0.950586i	−0.0482490	−	0.998835i	$-0.515364\pi$
$83$	8.66025	−	8.66025i	0.950586	−	0.950586i	0.998835	+	0.0482490i	$0.0153641\pi$
$84$		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	−0.970235	−	0.242166i	$-0.922142\pi$
$89$	−2.59808	+	4.50000i	−0.275396	+	0.476999i	0.694839	+	0.719165i	$0.255475\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−23.6603	−	6.33975i	−2.45345	−	0.657401i
$94$	3.46410			0.357295
$95$		0			0
$96$		0			0
$97$	−12.2942	−	3.29423i	−1.24829	−	0.334478i	−0.426614	−	0.904434i	$-0.640294\pi$
$97$	−12.2942	−	3.29423i	−1.24829	−	0.334478i	−0.821676	+	0.569955i	$0.806960\pi$
$98$	2.56218	−	9.56218i	0.258819	−	0.965926i
$99$	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	0.937955	−	0.346757i	$-0.112717\pi$
$103$	6.00000	+	6.00000i	0.591198	+	0.591198i	−0.346757	+	0.937955i	$0.612717\pi$
$104$	−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	−0.282540	−	0.959256i	$-0.591177\pi$
$107$	7.00000	−	7.00000i	0.676716	−	0.676716i	0.959256	+	0.282540i	$0.0911770\pi$
$108$		0			0
$109$	−6.06218	−	10.5000i	−0.580651	−	1.00572i	−0.995402	−	0.0957826i	$-0.969465\pi$
$109$	−6.06218	−	10.5000i	−0.580651	−	1.00572i	0.414751	−	0.909935i	$-0.363869\pi$
$110$		0			0
$111$	6.00000	+	10.3923i	0.569495	+	0.986394i
$112$		0			0
$113$	−4.00000	−	4.00000i	−0.376288	−	0.376288i	0.493473	−	0.869761i	$-0.335727\pi$
$113$	−4.00000	−	4.00000i	−0.376288	−	0.376288i	−0.869761	+	0.493473i	$0.835727\pi$
$114$	10.3923	−	14.0000i	0.973329	−	1.31122i
$115$		0			0
$116$		0			0
$117$	−20.4904	−	5.49038i	−1.89434	−	0.507586i
$118$	1.90192	+	7.09808i	0.175086	+	0.653431i
$119$		0			0
$120$		0			0
$121$	−2.00000			−0.181818
$122$	1.00000	−	1.00000i	0.0905357	−	0.0905357i
$123$	−7.60770	−	28.3923i	−0.685963	−	2.56005i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	3.29423	+	12.2942i	0.292316	+	1.09094i	0.943326	+	0.331868i	$0.107679\pi$
$127$	3.29423	+	12.2942i	0.292316	+	1.09094i	−0.651010	+	0.759069i	$0.725655\pi$
$128$	−2.92820	+	10.9282i	−0.258819	+	0.965926i
$129$	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
$137$	−9.46410	+	2.53590i	−0.808573	+	0.216656i	−0.639344	−	0.768920i	$-0.720794\pi$
$137$	−9.46410	+	2.53590i	−0.808573	+	0.216656i	−0.169228	+	0.985577i	$0.554128\pi$
$138$	28.3923	+	7.60770i	2.41691	+	0.647610i
$139$	−1.73205	−	1.00000i	−0.146911	−	0.0848189i	0.424743	−	0.905314i	$-0.360365\pi$
$139$	−1.73205	−	1.00000i	−0.146911	−	0.0848189i	−0.571654	+	0.820495i	$0.693698\pi$
$140$		0			0
$141$		−	6.92820i		−	0.583460i
$142$	−21.2942	+	5.70577i	−1.78697	+	0.478818i
$143$	−12.2942	+	3.29423i	−1.02810	+	0.275477i
$144$			20.0000i			1.66667i
$145$		0			0
$146$	9.00000	+	5.19615i	0.744845	+	0.430037i
$147$	−19.1244	−	5.12436i	−1.57735	−	0.422650i
$148$		0			0
$149$	−2.59808	+	1.50000i	−0.212843	+	0.122885i	−0.602632	−	0.798019i	$-0.705881\pi$
$149$	−2.59808	+	1.50000i	−0.212843	+	0.122885i	0.389789	+	0.920904i	$0.372548\pi$
$150$		0			0
$151$			8.66025i			0.704761i	0.935857	+	0.352381i	$0.114628\pi$
$151$			8.66025i			0.704761i	−0.935857	+	0.352381i	$0.885372\pi$
$152$	−7.66025	−	9.66025i	−0.621329	−	0.783550i
$153$	−8.66025	+	8.66025i	−0.700140	+	0.700140i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	3.80385	−	14.1962i	0.303580	−	1.13298i	−0.630581	−	0.776124i	$-0.717183\pi$
$157$	3.80385	−	14.1962i	0.303580	−	1.13298i	0.934161	−	0.356853i	$-0.116150\pi$
$158$	1.90192	+	7.09808i	0.151309	+	0.564693i
$159$			8.00000i			0.634441i
$160$		0			0
$161$		0			0
$162$	0.366025	+	1.36603i	0.0287577	+	0.107325i
$163$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$163$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$164$		0			0
$165$		0			0
$166$	15.0000	−	8.66025i	1.16423	−	0.672166i
$167$	5.12436	+	19.1244i	0.396535	+	1.47989i	0.819151	+	0.573578i	$0.194445\pi$
$167$	5.12436	+	19.1244i	0.396535	+	1.47989i	−0.422616	+	0.906309i	$0.638888\pi$
$168$		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
$173$	0.732051	−	2.73205i	0.0556568	−	0.207714i	−0.932498	−	0.361176i	$-0.882375\pi$
$173$	0.732051	−	2.73205i	0.0556568	−	0.207714i	0.988155	+	0.153462i	$0.0490422\pi$
$174$	10.3923	+	18.0000i	0.787839	+	1.36458i
$175$		0			0
$176$	6.00000	+	10.3923i	0.452267	+	0.783349i
$177$	14.1962	−	3.80385i	1.06705	−	0.285915i
$178$	−5.19615	+	5.19615i	−0.389468	+	0.389468i
$179$	−25.9808			−1.94189			−0.970947	−	0.239296i	$-0.923083\pi$
$179$	−25.9808			−1.94189			−0.970947	+	0.239296i	$0.923083\pi$
$180$		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
$187$	−1.90192	+	7.09808i	−0.139082	+	0.519063i
$188$		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	$0.534616\pi$
$192$	21.8564	+	5.85641i	1.57735	+	0.422650i
$193$	4.39230	−	16.3923i	0.316165	−	1.17994i	−0.606735	−	0.794904i	$-0.707521\pi$
$193$	4.39230	−	16.3923i	0.316165	−	1.17994i	0.922900	−	0.385040i	$-0.125812\pi$
$194$	−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	$-0.106290\pi$
$197$	8.66025	+	8.66025i	0.617018	+	0.617018i	−0.327748	+	0.944765i	$0.606290\pi$
$198$	15.0000	+	15.0000i	1.06600	+	1.06600i
$199$	−6.06218	−	3.50000i	−0.429736	−	0.248108i	0.269498	−	0.963001i	$-0.413142\pi$
$199$	−6.06218	−	3.50000i	−0.429736	−	0.248108i	−0.699234	+	0.714893i	$0.746476\pi$
$200$		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
$207$	9.50962	−	35.4904i	0.660964	−	2.46675i
$208$	−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
$213$	11.4115	+	42.5885i	0.781906	+	2.91811i
$214$	12.1244	−	7.00000i	0.828804	−	0.478510i
$215$		0			0
$216$	16.0000			1.08866
$217$		0			0
$218$	−4.43782	−	16.5622i	−0.300567	−	1.12173i
$219$	10.3923	−	18.0000i	0.702247	−	1.21633i
$220$		0			0
$221$		−	10.3923i		−	0.699062i
$222$	4.39230	+	16.3923i	0.294792	+	1.10018i
$223$	−1.09808	+	4.09808i	−0.0735326	+	0.274427i	−0.992897	−	0.118981i	$-0.962037\pi$
$223$	−1.09808	+	4.09808i	−0.0735326	+	0.274427i	0.919364	+	0.393408i	$0.128704\pi$
$224$		0			0
$225$		0			0
$226$	−4.00000	−	6.92820i	−0.266076	−	0.460857i
$227$	7.00000	−	7.00000i	0.464606	−	0.464606i	−0.435556	−	0.900162i	$-0.643448\pi$
$227$	7.00000	−	7.00000i	0.464606	−	0.464606i	0.900162	+	0.435556i	$0.143448\pi$
$228$		0			0
$229$			11.0000i			0.726900i	0.931614	+	0.363450i	$0.118401\pi$
$229$			11.0000i			0.726900i	−0.931614	+	0.363450i	$0.881599\pi$
$230$		0			0
$231$		0			0
$232$	14.1962	−	3.80385i	0.932023	−	0.249735i
$233$	−4.73205	−	1.26795i	−0.310007	−	0.0830661i	0.100461	−	0.994941i	$-0.467968\pi$
$233$	−4.73205	−	1.26795i	−0.310007	−	0.0830661i	−0.410468	+	0.911875i	$0.634635\pi$
$234$	−25.9808	−	15.0000i	−1.69842	−	0.980581i
$235$		0			0
$236$		0			0
$237$	14.1962	−	3.80385i	0.922139	−	0.247086i
$238$		0			0
$239$			21.0000i			1.35838i	0.733964	+	0.679189i	$0.237668\pi$
$239$			21.0000i			1.35838i	−0.733964	+	0.679189i	$0.762332\pi$
$240$		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	$0.456371\pi$
$242$	−2.73205	−	0.732051i	−0.175623	−	0.0470580i
$243$	−13.6603	+	3.66025i	−0.876306	+	0.234805i
$244$		0			0
$245$		0			0
$246$		−	41.5692i		−	2.65036i
$247$	18.2942	−	2.70577i	1.16403	−	0.172164i
$248$	−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
$253$	−5.70577	−	21.2942i	−0.358719	−	1.33876i
$254$			18.0000i			1.12942i
$255$		0			0
$256$		0			0
$257$	−0.366025	−	1.36603i	−0.0228320	−	0.0852103i	0.953570	−	0.301172i	$-0.0973779\pi$
$257$	−0.366025	−	1.36603i	−0.0228320	−	0.0852103i	−0.976402	+	0.215962i	$0.930711\pi$
$258$	20.7846	−	20.7846i	1.29399	−	1.29399i
$259$		0			0
$260$		0			0
$261$	22.5000	−	12.9904i	1.39272	−	0.804084i
$262$	2.19615	+	8.19615i	0.135679	+	0.506360i
$263$	18.9282	+	5.07180i	1.16716	+	0.312740i	0.789823	−	0.613335i	$-0.210172\pi$
$263$	18.9282	+	5.07180i	1.16716	+	0.312740i	0.377340	+	0.926075i	$0.376839\pi$
$264$	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	0.934295	−	0.356502i	$-0.116031\pi$
$269$	2.59808	+	4.50000i	0.158408	+	0.274370i	−0.775887	+	0.630872i	$0.782697\pi$
$270$		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	$-0.362476\pi$
$272$	−9.46410	+	2.53590i	−0.573845	+	0.153761i
$273$		0			0
$274$	−13.8564			−0.837096
$275$		0			0
$276$		0			0
$277$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$277$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$278$	−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	$0.566345\pi$
$282$	2.53590	−	9.46410i	0.151011	−	0.563579i
$283$	7.09808	+	1.90192i	0.421937	+	0.113058i	0.463539	−	0.886077i	$-0.346579\pi$
$283$	7.09808	+	1.90192i	0.421937	+	0.113058i	−0.0416020	+	0.999134i	$0.513246\pi$
$284$		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	0.167913	−	0.985802i	$-0.446297\pi$
$293$	−14.0000	−	14.0000i	−0.817889	−	0.817889i	−0.985802	+	0.167913i	$0.946297\pi$
$294$	−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
$298$	−4.09808	+	1.09808i	−0.237395	+	0.0636098i
$299$	15.5885	+	27.0000i	0.901504	+	1.56145i
$300$		0			0
$301$		0			0
$302$	−3.16987	+	11.8301i	−0.182406	+	0.680747i
$303$	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
$307$	−32.7846	−	8.78461i	−1.87112	−	0.501364i	−0.999946	−	0.0103834i	$-0.996695\pi$
$307$	−32.7846	−	8.78461i	−1.87112	−	0.501364i	−0.871170	−	0.490981i	$-0.836639\pi$
$308$		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
$313$	1.90192	+	7.09808i	0.107503	+	0.401207i	0.998617	−	0.0525725i	$-0.0167421\pi$
$313$	1.90192	+	7.09808i	0.107503	+	0.401207i	−0.891114	+	0.453779i	$0.850075\pi$
$314$	10.3923	−	18.0000i	0.586472	−	1.01580i
$315$		0			0
$316$		0			0
$317$	−0.366025	−	1.36603i	−0.0205580	−	0.0767236i	0.954885	−	0.296977i	$-0.0959784\pi$
$317$	−0.366025	−	1.36603i	−0.0205580	−	0.0767236i	−0.975443	+	0.220253i	$0.929312\pi$
$318$	−2.92820	+	10.9282i	−0.164205	+	0.612823i
$319$	7.79423	−	13.5000i	0.436393	−	0.755855i
$320$		0			0
$321$	−14.0000	−	24.2487i	−0.781404	−	1.35343i
$322$		0			0
$323$	3.92820	−	9.92820i	0.218571	−	0.552420i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−33.1244	+	8.87564i	−1.83178	+	0.490824i
$328$	−28.3923	−	7.60770i	−1.56770	−	0.420065i
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$	20.4904	−	5.49038i	1.12287	−	0.300871i
$334$			28.0000i			1.53209i
$335$		0			0
$336$		0			0
$337$	−12.2942	−	3.29423i	−0.669709	−	0.179448i	−0.0920854	−	0.995751i	$-0.529353\pi$
$337$	−12.2942	−	3.29423i	−0.669709	−	0.179448i	−0.577624	+	0.816303i	$0.696020\pi$
$338$	6.83013	−	1.83013i	0.371510	−	0.0995458i
$339$	−13.8564	+	8.00000i	−0.752577	+	0.434500i
$340$		0			0
$341$			25.9808i			1.40694i
$342$	−19.1506	−	24.1506i	−1.03555	−	1.30592i
$343$		0			0
$344$	−10.3923	−	18.0000i	−0.560316	−	0.970495i
$345$		0			0
$346$	2.00000	−	3.46410i	0.107521	−	0.186231i
$347$	−2.53590	+	9.46410i	−0.136134	+	0.508060i	0.863857	+	0.503738i	$0.168042\pi$
$347$	−2.53590	+	9.46410i	−0.136134	+	0.508060i	−0.999991	+	0.00432163i	$0.998624\pi$
$348$		0			0
$349$		−	4.00000i		−	0.214115i	−0.994253	−	0.107058i	$-0.965857\pi$
$349$		−	4.00000i		−	0.214115i	0.994253	−	0.107058i	$-0.0341429\pi$
$350$		0			0
$351$	−12.0000	+	20.7846i	−0.640513	+	1.10940i
$352$		0			0
$353$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$353$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$354$	20.7846			1.10469
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−35.4904	−	9.50962i	−1.87572	−	0.502599i
$359$	10.3923	−	6.00000i	0.548485	−	0.316668i	−0.200026	−	0.979791i	$-0.564103\pi$
$359$	10.3923	−	6.00000i	0.548485	−	0.316668i	0.748511	+	0.663123i	$0.230769\pi$
$360$		0			0
$361$	18.5000	+	4.33013i	0.973684	+	0.227901i
$362$	−24.2487	−	24.2487i	−1.27448	−	1.27448i
$363$	−1.46410	+	5.46410i	−0.0768454	+	0.286791i
$364$		0			0
$365$		0			0
$366$	−2.00000	−	3.46410i	−0.104542	−	0.181071i
$367$	14.1962	−	3.80385i	0.741033	−	0.198559i	0.131496	−	0.991317i	$-0.458022\pi$
$367$	14.1962	−	3.80385i	0.741033	−	0.198559i	0.609537	+	0.792757i	$0.291355\pi$
$368$	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	0.845168	−	0.534500i	$-0.179500\pi$
$373$	6.00000	+	6.00000i	0.310668	+	0.310668i	−0.534500	+	0.845168i	$0.679500\pi$
$374$	−5.19615	+	9.00000i	−0.268687	+	0.465379i
$375$		0			0
$376$	−6.00000	−	3.46410i	−0.309426	−	0.178647i
$377$	−5.70577	+	21.2942i	−0.293862	+	1.09671i
$378$		0			0
$379$	25.9808			1.33454			0.667271	−	0.744815i	$-0.267462\pi$
$379$	25.9808			1.33454			0.667271	+	0.744815i	$0.267462\pi$
$380$		0			0
$381$	36.0000			1.84434
$382$	−4.09808	−	1.09808i	−0.209676	−	0.0561825i
$383$	−8.41858	+	31.4186i	−0.430170	+	1.60541i	0.322199	+	0.946672i	$0.395578\pi$
$383$	−8.41858	+	31.4186i	−0.430170	+	1.60541i	−0.752368	+	0.658743i	$0.771089\pi$
$384$	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	0.564416	−	0.825491i	$-0.309102\pi$
$389$	2.59808	+	1.50000i	0.131728	+	0.0760530i	−0.432688	+	0.901544i	$0.642435\pi$
$390$		0			0
$391$	18.0000			0.910299
$392$	−14.0000	+	14.0000i	−0.707107	+	0.707107i
$393$	16.3923	−	4.39230i	0.826882	−	0.221562i
$394$	8.66025	+	15.0000i	0.436297	+	0.755689i
$395$		0			0
$396$		0			0
$397$	3.80385	−	14.1962i	0.190910	−	0.712484i	−0.802378	−	0.596816i	$-0.796432\pi$
$397$	3.80385	−	14.1962i	0.190910	−	0.712484i	0.993288	−	0.115669i	$-0.0369011\pi$
$398$	−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	$0.460591\pi$
$402$	49.1769	+	13.1769i	2.45272	+	0.657205i
$403$	−9.50962	−	35.4904i	−0.473708	−	1.76790i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	9.00000	−	9.00000i	0.446113	−	0.446113i
$408$	5.07180	+	18.9282i	0.251091	+	0.937086i
$409$	−2.59808	+	4.50000i	−0.128467	+	0.222511i	−0.923083	−	0.384602i	$-0.874339\pi$
$409$	−2.59808	+	4.50000i	−0.128467	+	0.222511i	0.794616	+	0.607112i	$0.207672\pi$
$410$		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
$418$	−17.1962	−	6.80385i	−0.841091	−	0.332787i
$419$		−	9.00000i		−	0.439679i	−0.975536	−	0.219839i	$-0.929447\pi$
$419$		−	9.00000i		−	0.439679i	0.975536	−	0.219839i	$-0.0705533\pi$
$420$		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	$-0.0134391\pi$
$422$	2.36603	−	0.633975i	0.115176	−	0.0308614i
$423$	−11.8301	−	3.16987i	−0.575200	−	0.154124i
$424$	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	$0.789173\pi$
$432$	21.8564	+	5.85641i	1.05157	+	0.281766i
$433$	24.5885	−	6.58846i	1.18165	−	0.316621i	0.386067	−	0.922471i	$-0.373833\pi$
$433$	24.5885	−	6.58846i	1.18165	−	0.316621i	0.795579	+	0.605850i	$0.207167\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	4.68653	+	31.6865i	0.224187	+	1.51577i
$438$	20.7846	−	20.7846i	0.993127	−	0.993127i
$439$	−6.06218	−	10.5000i	−0.289332	−	0.501138i	0.684318	−	0.729183i	$-0.260100\pi$
$439$	−6.06218	−	10.5000i	−0.289332	−	0.501138i	−0.973650	+	0.228046i	$0.926766\pi$
$440$		0			0
$441$	−17.5000	+	30.3109i	−0.833333	+	1.44338i
$442$	3.80385	−	14.1962i	0.180931	−	0.675242i
$443$	−1.26795	−	4.73205i	−0.0602421	−	0.224827i	0.929241	−	0.369474i	$-0.120462\pi$
$443$	−1.26795	−	4.73205i	−0.0602421	−	0.224827i	−0.989483	+	0.144647i	$0.953795\pi$
$444$		0			0
$445$		0			0
$446$	−3.00000	+	5.19615i	−0.142054	+	0.246045i
$447$	2.19615	+	8.19615i	0.103874	+	0.387665i
$448$		0			0
$449$	−25.9808			−1.22611			−0.613054	−	0.790041i	$-0.710059\pi$
$449$	−25.9808			−1.22611			−0.613054	+	0.790041i	$0.710059\pi$
$450$		0			0
$451$	−27.0000	+	15.5885i	−1.27138	+	0.734032i
$452$		0			0
$453$	23.6603	+	6.33975i	1.11166	+	0.297867i
$454$	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	0.969250	+	0.246079i	$0.0791423\pi$
$457$	25.9808	+	25.9808i	1.21533	+	1.21533i	0.246079	+	0.969250i	$0.420858\pi$
$458$	−4.02628	+	15.0263i	−0.188136	+	0.702132i
$459$	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	$-0.750000\pi$
$463$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$464$	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	$-0.158547\pi$
$467$	8.66025	+	8.66025i	0.400749	+	0.400749i	−0.477748	+	0.878497i	$0.658547\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−36.0000	−	20.7846i	−1.65879	−	0.957704i
$472$	3.80385	−	14.1962i	0.175086	−	0.653431i
$473$	−21.2942	−	5.70577i	−0.979110	−	0.262352i
$474$	20.7846			0.954669
$475$		0			0
$476$		0			0
$477$	13.6603	+	3.66025i	0.625460	+	0.167592i
$478$	−7.68653	+	28.6865i	−0.351574	+	1.31209i
$479$	2.59808	+	1.50000i	0.118709	+	0.0685367i	0.558179	−	0.829721i	$-0.311500\pi$
$479$	2.59808	+	1.50000i	0.118709	+	0.0685367i	−0.439470	+	0.898257i	$0.644834\pi$
$480$		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	0.257103	−	0.966384i	$-0.417232\pi$
$487$	27.0000	−	27.0000i	1.22349	−	1.22349i	0.966384	−	0.257103i	$-0.0827679\pi$
$488$	−2.73205	+	0.732051i	−0.123674	+	0.0331384i
$489$		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
$498$	−12.6795	−	47.3205i	−0.568182	−	2.12048i
$499$	−32.9090	+	19.0000i	−1.47321	+	0.850557i	−0.999545	−	0.0301498i	$-0.990402\pi$
$499$	−32.9090	+	19.0000i	−1.47321	+	0.850557i	−0.473662	+	0.880707i	$0.657068\pi$
$500$		0			0
$501$	56.0000			2.50190
$502$	21.0000	−	21.0000i	0.937276	−	0.937276i
$503$	8.24167	+	30.7583i	0.367478	+	1.37145i	0.864030	+	0.503440i	$0.167932\pi$
$503$	8.24167	+	30.7583i	0.367478	+	1.37145i	−0.496553	+	0.868007i	$0.665401\pi$
$504$		0			0
$505$		0			0
$506$		−	31.1769i		−	1.38598i
$507$	−3.66025	−	13.6603i	−0.162558	−	0.606673i
$508$		0			0
$509$	10.3923	−	18.0000i	0.460631	−	0.797836i	−0.538362	−	0.842714i	$-0.680957\pi$
$509$	10.3923	−	18.0000i	0.460631	−	0.797836i	0.998992	+	0.0448779i	$0.0142899\pi$
$510$		0			0
$511$		0			0
$512$	16.0000	−	16.0000i	0.707107	−	0.707107i
$513$	−19.3205	+	15.3205i	−0.853021	+	0.676417i
$514$		−	2.00000i		−	0.0882162i
$515$		0			0
$516$		0			0
$517$	−7.09808	+	1.90192i	−0.312173	+	0.0836465i
$518$		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	$0.807284\pi$
$522$	35.4904	−	9.50962i	1.55337	−	0.416225i
$523$	−16.3923	+	4.39230i	−0.716785	+	0.192062i	−0.598737	−	0.800946i	$-0.704331\pi$
$523$	−16.3923	+	4.39230i	−0.716785	+	0.192062i	−0.118049	+	0.993008i	$0.537664\pi$
$524$		0			0
$525$		0			0
$526$	24.0000	+	13.8564i	1.04645	+	0.604168i
$527$	−20.4904	−	5.49038i	−0.892575	−	0.239165i
$528$	32.7846	−	8.78461i	1.42677	−	0.382301i
$529$	−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
$537$	−19.0192	+	70.9808i	−0.820741	+	3.06305i
$538$	1.90192	+	7.09808i	0.0819978	+	0.306020i
$539$			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	$0.300593\pi$
$542$	−6.95448	−	25.9545i	−0.298721	−	1.11484i
$543$	−48.4974	+	48.4974i	−2.08122	+	2.08122i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−7.68653	−	28.6865i	−0.328652	−	1.22655i	−0.910589	−	0.413313i	$-0.864371\pi$
$547$	−7.68653	−	28.6865i	−0.328652	−	1.22655i	0.581936	−	0.813234i	$-0.302295\pi$
$548$		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
$557$	−21.2942	+	5.70577i	−0.902265	+	0.241761i	−0.679989	−	0.733222i	$-0.738015\pi$
$557$	−21.2942	+	5.70577i	−0.902265	+	0.241761i	−0.222276	+	0.974984i	$0.571349\pi$
$558$	−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	0.347610	−	0.937639i	$-0.386993\pi$
$563$	−14.0000	−	14.0000i	−0.590030	−	0.590030i	−0.937639	+	0.347610i	$0.886993\pi$
$564$		0			0
$565$		0			0
$566$	9.00000	+	5.19615i	0.378298	+	0.218411i
$567$		0			0
$568$	42.5885	+	11.4115i	1.78697	+	0.478818i
$569$	25.9808			1.08917			0.544585	−	0.838706i	$-0.316687\pi$
$569$	25.9808			1.08917			0.544585	+	0.838706i	$0.316687\pi$
$570$		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
$573$	−2.19615	+	8.19615i	−0.0917456	+	0.342399i
$574$		0			0
$575$		0			0
$576$	20.0000	−	34.6410i	0.833333	−	1.44338i
$577$	25.9808	+	25.9808i	1.08159	+	1.08159i	0.996361	+	0.0852322i	$0.0271632\pi$
$577$	25.9808	+	25.9808i	1.08159	+	1.08159i	0.0852322	+	0.996361i	$0.472837\pi$
$578$	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
$583$	8.19615	−	2.19615i	0.339450	−	0.0909553i
$584$	−10.3923	−	18.0000i	−0.430037	−	0.744845i
$585$		0			0
$586$	−14.0000	−	24.2487i	−0.578335	−	1.00171i
$587$	−2.53590	+	9.46410i	−0.104668	+	0.390625i	−0.998307	−	0.0581602i	$-0.981477\pi$
$587$	−2.53590	+	9.46410i	−0.104668	+	0.390625i	0.893640	+	0.448785i	$0.148143\pi$
$588$		0			0
$589$	4.33013	−	37.5000i	0.178420	−	1.54516i
$590$		0			0
$591$	30.0000	−	17.3205i	1.23404	−	0.712470i
$592$	16.3923	+	4.39230i	0.673720	+	0.180523i
$593$	5.07180	+	18.9282i	0.208274	+	0.777288i	0.988427	+	0.151699i	$0.0484745\pi$
$593$	5.07180	+	18.9282i	0.208274	+	0.777288i	−0.780153	+	0.625589i	$0.784859\pi$
$594$	20.7846	−	12.0000i	0.852803	−	0.492366i
$595$		0			0
$596$		0			0
$597$	−14.0000	+	14.0000i	−0.572982	+	0.572982i
$598$	11.4115	+	42.5885i	0.466653	+	1.74157i
$599$	−15.5885	+	27.0000i	−0.636927	+	1.10319i	0.349176	+	0.937057i	$0.386461\pi$
$599$	−15.5885	+	27.0000i	−0.636927	+	1.10319i	−0.986103	+	0.166133i	$0.946872\pi$
$600$		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
$603$	16.4711	−	61.4711i	0.670757	−	2.50330i
$604$		0			0
$605$		0			0
$606$	18.0000	+	31.1769i	0.731200	+	1.26648i
$607$	−3.00000	+	3.00000i	−0.121766	+	0.121766i	−0.765364	−	0.643598i	$-0.777441\pi$
$607$	−3.00000	+	3.00000i	−0.121766	+	0.121766i	0.643598	+	0.765364i	$0.277441\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	9.00000	−	5.19615i	0.364101	−	0.210214i
$612$		0			0
$613$	−28.3923	−	7.60770i	−1.14675	−	0.307272i	−0.365091	−	0.930972i	$-0.618962\pi$
$613$	−28.3923	−	7.60770i	−1.14675	−	0.307272i	−0.781664	+	0.623700i	$0.785629\pi$
$614$	−41.5692	−	24.0000i	−1.67760	−	0.968561i
$615$		0			0
$616$		0			0
$617$	2.36603	−	0.633975i	0.0952526	−	0.0255229i	−0.210878	−	0.977512i	$-0.567632\pi$
$617$	2.36603	−	0.633975i	0.0952526	−	0.0255229i	0.306131	+	0.951990i	$0.400966\pi$
$618$	32.7846	−	8.78461i	1.31879	−	0.353369i
$619$		−	14.0000i		−	0.562708i	−0.959604	−	0.281354i	$-0.909217\pi$
$619$		−	14.0000i		−	0.562708i	0.959604	−	0.281354i	$-0.0907834\pi$
$620$		0			0
$621$	−36.0000	−	20.7846i	−1.44463	−	0.834058i
$622$	−24.5885	−	6.58846i	−0.985907	−	0.264173i
$623$		0			0
$624$	−41.5692	+	24.0000i	−1.66410	+	0.960769i
$625$		0			0
$626$			10.3923i			0.415360i
$627$	−13.6077	+	34.3923i	−0.543439	+	1.37350i
$628$		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	$0.160330\pi$
$632$	3.80385	−	14.1962i	0.151309	−	0.564693i
$633$	−1.26795	−	4.73205i	−0.0503965	−	0.188082i
$634$		−	2.00000i		−	0.0794301i
$635$		0			0
$636$		0			0
$637$	−7.68653	−	28.6865i	−0.304552	−	1.13660i
$638$	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	$0.432944\pi$
$642$	−10.2487	−	38.2487i	−0.404484	−	1.50956i
$643$	7.09808	+	1.90192i	0.279921	+	0.0750046i	0.396048	−	0.918230i	$-0.370381\pi$
$643$	7.09808	+	1.90192i	0.279921	+	0.0750046i	−0.116127	+	0.993234i	$0.537048\pi$
$644$		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	$-0.327393\pi$
$647$	−8.66025	−	8.66025i	−0.340470	−	0.340470i	−0.856544	+	0.516074i	$0.827393\pi$
$648$	0.732051	−	2.73205i	0.0287577	−	0.107325i
$649$	−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	−0.0168477	+	0.999858i	$0.505363\pi$
$653$	−25.9808	+	25.9808i	−1.01671	+	1.01671i	−0.999858	+	0.0168477i	$0.994637\pi$
$654$	−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	−0.589475	−	0.807787i	$-0.700665\pi$
$659$	10.3923	−	18.0000i	0.404827	−	0.701180i	0.994301	+	0.106606i	$0.0339985\pi$
$660$		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
$663$	−28.3923	−	7.60770i	−1.10267	−	0.295458i
$664$	−34.6410			−1.34433
$665$		0			0
$666$	30.0000			1.16248
$667$	−36.8827	−	9.88269i	−1.42810	−	0.382659i
$668$		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	0.0722542	−	0.997386i	$-0.476981\pi$
$673$	−24.0000	−	24.0000i	−0.925132	−	0.925132i	−0.997386	+	0.0722542i	$0.976981\pi$
$674$	−15.5885	−	9.00000i	−0.600445	−	0.346667i
$675$		0			0
$676$		0			0
$677$	−8.00000	+	8.00000i	−0.307465	+	0.307465i	−0.843925	−	0.536460i	$-0.819761\pi$
$677$	−8.00000	+	8.00000i	−0.307465	+	0.307465i	0.536460	+	0.843925i	$0.319761\pi$
$678$	−21.8564	+	5.85641i	−0.839390	+	0.224914i
$679$		0			0
$680$		0			0
$681$	−14.0000	−	24.2487i	−0.536481	−	0.929213i
$682$	−9.50962	+	35.4904i	−0.364142	+	1.35900i
$683$	11.0000	+	11.0000i	0.420903	+	0.420903i	0.885515	−	0.464611i	$-0.153806\pi$
$683$	11.0000	+	11.0000i	0.420903	+	0.420903i	−0.464611	+	0.885515i	$0.653806\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	30.0526	+	8.05256i	1.14658	+	0.307224i
$688$	−7.60770	−	28.3923i	−0.290041	−	1.08245i
$689$	−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
$697$	−6.58846	−	24.5885i	−0.249556	−	0.931354i
$698$	1.46410	−	5.46410i	0.0554171	−	0.206819i
$699$	−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
$703$	−14.4904	+	11.4904i	−0.546515	+	0.433368i
$704$		−	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	0.696281	−	0.717769i	$-0.254837\pi$
$709$	11.2583	+	6.50000i	0.422815	+	0.244113i	−0.273466	+	0.961882i	$0.588170\pi$
$710$		0			0
$711$		−	25.9808i		−	0.974355i
$712$	14.1962	−	3.80385i	0.532023	−	0.142555i
$713$	61.4711	−	16.4711i	2.30211	−	0.616849i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	57.3731	+	15.3731i	2.14264	+	0.574118i
$718$	16.3923	−	4.39230i	0.611755	−	0.163919i
$719$	23.3827	−	13.5000i	0.872027	−	0.503465i	0.00400572	−	0.999992i	$-0.498725\pi$
$719$	23.3827	−	13.5000i	0.872027	−	0.503465i	0.868021	+	0.496527i	$0.165392\pi$
$720$		0			0
$721$		0			0
$722$	23.6865	+	12.6865i	0.881521	+	0.472144i
$723$	38.1051	−	38.1051i	1.41714	−	1.41714i
$724$		0			0
$725$		0			0
$726$	−4.00000	+	6.92820i	−0.148454	+	0.257130i
$727$	3.80385	−	14.1962i	0.141077	−	0.526506i	−0.858822	−	0.512274i	$-0.828803\pi$
$727$	3.80385	−	14.1962i	0.141077	−	0.526506i	0.999899	−	0.0142317i	$-0.00453026\pi$
$728$		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	$-0.512607\pi$
$733$	25.9808	−	25.9808i	0.959621	−	0.959621i	0.999216	+	0.0395945i	$0.0126066\pi$
$734$	20.7846			0.767174
$735$		0			0
$736$		0			0
$737$	−9.88269	−	36.8827i	−0.364033	−	1.35859i
$738$	−70.9808	−	19.0192i	−2.61284	−	0.700108i
$739$	−45.8993	+	26.5000i	−1.68843	+	0.974818i	−0.732717	+	0.680534i	$0.761748\pi$
$739$	−45.8993	+	26.5000i	−1.68843	+	0.974818i	−0.955718	+	0.294285i	$0.904919\pi$
$740$		0			0
$741$	6.00000	−	51.9615i	0.220416	−	1.90885i
$742$		0			0
$743$	−8.41858	+	31.4186i	−0.308848	+	1.15264i	0.620734	+	0.784021i	$0.286835\pi$
$743$	−8.41858	+	31.4186i	−0.308848	+	1.15264i	−0.929582	+	0.368615i	$0.879832\pi$
$744$	34.6410	+	60.0000i	1.27000	+	2.19971i
$745$		0			0
$746$	6.00000	+	10.3923i	0.219676	+	0.380489i
$747$	−59.1506	+	15.8494i	−2.16421	+	0.579898i
$748$		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
$757$	−5.70577	+	21.2942i	−0.207380	+	0.773952i	0.781331	+	0.624117i	$0.214541\pi$
$757$	−5.70577	+	21.2942i	−0.207380	+	0.773952i	−0.988711	+	0.149835i	$0.952126\pi$
$758$	35.4904	+	9.50962i	1.28907	+	0.345405i
$759$	−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	$0.835881\pi$
$762$	49.1769	+	13.1769i	1.78149	+	0.477349i
$763$		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	0.386698	−	0.922207i	$-0.373616\pi$
$769$	−6.06218	−	3.50000i	−0.218608	−	0.126213i	−0.605305	+	0.795993i	$0.706949\pi$
$770$		0			0
$771$	−4.00000			−0.144056
$772$		0			0
$773$	31.4186	−	8.41858i	1.13005	−	0.302795i	0.355105	−	0.934826i	$-0.384445\pi$
$773$	31.4186	−	8.41858i	1.13005	−	0.302795i	0.774943	+	0.632031i	$0.217778\pi$
$774$	−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
$782$	24.5885	+	6.58846i	0.879281	+	0.235603i
$783$	−7.60770	−	28.3923i	−0.271877	−	1.01466i
$784$	−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	−0.0368739	−	0.999320i	$-0.511740\pi$
$787$	27.0000	−	27.0000i	0.962446	−	0.962446i	0.999320	+	0.0368739i	$0.0117400\pi$
$788$		0			0
$789$	27.7128	−	48.0000i	0.986602	−	1.70885i
$790$		0			0
$791$		0			0
$792$	−10.9808	−	40.9808i	−0.390184	−	1.45619i
$793$	1.09808	−	4.09808i	0.0389938	−	0.145527i
$794$	10.3923	−	18.0000i	0.368809	−	0.638796i
$795$		0			0
$796$		0			0
$797$	32.0000	−	32.0000i	1.13350	−	1.13350i	0.143907	−	0.989591i	$-0.454033\pi$
$797$	32.0000	−	32.0000i	1.13350	−	1.13350i	0.989591	−	0.143907i	$-0.0459666\pi$
$798$		0			0
$799$		−	6.00000i		−	0.212265i
$800$		0			0
$801$	22.5000	−	12.9904i	0.794998	−	0.458993i
$802$	−21.2942	+	5.70577i	−0.751925	+	0.201478i
$803$	−21.2942	−	5.70577i	−0.751457	−	0.201352i
$804$		0			0
$805$		0			0
$806$		−	51.9615i		−	1.83027i
$807$	14.1962	−	3.80385i	0.499728	−	0.133902i
$808$	24.5885	−	6.58846i	0.865019	−	0.231781i
$809$		−	39.0000i		−	1.37117i	−0.727994	−	0.685583i	$-0.759547\pi$
$809$		−	39.0000i		−	1.37117i	0.727994	−	0.685583i	$-0.240453\pi$
$810$		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
$813$	−51.9090	+	13.9090i	−1.82053	+	0.487809i
$814$	15.5885	−	9.00000i	0.546375	−	0.315450i
$815$		0			0
$816$			27.7128i			0.970143i
$817$	29.7846	+	11.7846i	1.04203	+	0.412291i
$818$	−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	$0.0472389\pi$
$822$	−10.1436	+	37.8564i	−0.353798	+	1.32039i
$823$	1.90192	+	7.09808i	0.0662969	+	0.247423i	0.991119	−	0.132976i	$-0.0424533\pi$
$823$	1.90192	+	7.09808i	0.0662969	+	0.247423i	−0.924822	+	0.380399i	$0.875787\pi$
$824$		−	24.0000i		−	0.836080i
$825$		0			0
$826$		0			0
$827$	−4.02628	−	15.0263i	−0.140007	−	0.522515i	−0.999927	−	0.0120853i	$-0.996153\pi$
$827$	−4.02628	−	15.0263i	−0.140007	−	0.522515i	0.859920	−	0.510430i	$-0.170514\pi$
$828$		0			0
$829$	17.3205			0.601566			0.300783	−	0.953693i	$-0.402752\pi$
$829$	17.3205			0.601566			0.300783	+	0.953693i	$0.402752\pi$
$830$		0			0
$831$		0			0
$832$	8.78461	+	32.7846i	0.304552	+	1.13660i
$833$	−16.5622	−	4.43782i	−0.573845	−	0.153761i
$834$	−6.92820	+	4.00000i	−0.239904	+	0.138509i
$835$		0			0
$836$		0			0
$837$	34.6410	+	34.6410i	1.19737	+	1.19737i
$838$	3.29423	−	12.2942i	0.113797	−	0.424697i
$839$	−10.3923	−	18.0000i	−0.358782	−	0.621429i	0.628975	−	0.777425i	$-0.283475\pi$
$839$	−10.3923	−	18.0000i	−0.358782	−	0.621429i	−0.987758	+	0.155996i	$0.950141\pi$
$840$		0			0
$841$	1.00000	+	1.73205i	0.0344828	+	0.0597259i
$842$	49.6865	−	13.3135i	1.71231	−	0.458812i
$843$	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
$853$	42.5885	+	11.4115i	1.45820	+	0.390724i	0.898868	−	0.438219i	$-0.144391\pi$
$853$	42.5885	+	11.4115i	1.45820	+	0.390724i	0.559333	+	0.828943i	$0.311057\pi$
$854$		0			0
$855$		0			0
$856$	−28.0000			−0.957020
$857$	−25.9545	−	6.95448i	−0.886588	−	0.237561i	−0.213341	−	0.976978i	$-0.568434\pi$
$857$	−25.9545	−	6.95448i	−0.886588	−	0.237561i	−0.673247	+	0.739417i	$0.735101\pi$
$858$	−13.1769	+	49.1769i	−0.449852	+	1.67887i
$859$	−40.7032	−	23.5000i	−1.38878	−	0.801810i	−0.395598	−	0.918424i	$-0.629463\pi$
$859$	−40.7032	−	23.5000i	−1.38878	−	0.801810i	−0.993177	+	0.116614i	$0.962796\pi$
$860$		0			0
$861$		0			0
$862$	−15.5885	−	15.5885i	−0.530945	−	0.530945i
$863$	−14.0000	−	14.0000i	−0.476566	−	0.476566i	0.427466	−	0.904031i	$-0.359406\pi$
$863$	−14.0000	−	14.0000i	−0.476566	−	0.476566i	−0.904031	+	0.427466i	$0.859406\pi$
$864$		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
$872$	−8.87564	+	33.1244i	−0.300567	+	1.12173i
$873$	45.0000	+	45.0000i	1.52302	+	1.52302i
$874$	−5.19615	+	45.0000i	−0.175762	+	1.52215i
$875$		0			0
$876$		0			0
$877$	−12.2942	−	3.29423i	−0.415147	−	0.111238i	0.0451990	−	0.998978i	$-0.485608\pi$
$877$	−12.2942	−	3.29423i	−0.415147	−	0.111238i	−0.460346	+	0.887740i	$0.652274\pi$
$878$	−4.43782	−	16.5622i	−0.149769	−	0.558946i
$879$	−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
$883$	11.4115	+	42.5885i	0.384029	+	1.43322i	0.839692	+	0.543063i	$0.182736\pi$
$883$	11.4115	+	42.5885i	0.384029	+	1.43322i	−0.455663	+	0.890152i	$0.650598\pi$
$884$		0			0
$885$		0			0
$886$		−	6.92820i		−	0.232758i
$887$	5.12436	+	19.1244i	0.172059	+	0.642133i	0.997034	+	0.0769636i	$0.0245225\pi$
$887$	5.12436	+	19.1244i	0.172059	+	0.642133i	−0.824975	+	0.565169i	$0.808811\pi$
$888$	8.78461	−	32.7846i	0.294792	−	1.10018i
$889$		0			0
$890$		0			0
$891$	−1.50000	−	2.59808i	−0.0502519	−	0.0870388i
$892$		0			0
$893$	10.5622	−	1.56218i	0.353450	−	0.0522763i
$894$			12.0000i			0.401340i
$895$		0			0
$896$		0			0
$897$	85.1769	−	22.8231i	2.84397	−	0.762041i
$898$	−35.4904	−	9.50962i	−1.18433	−	0.317340i
$899$	38.9711	+	22.5000i	1.29976	+	0.750417i
$900$		0			0
$901$			6.92820i			0.230812i
$902$	−42.5885	+	11.4115i	−1.41804	+	0.379963i
$903$		0			0
$904$			16.0000i			0.532152i
$905$		0			0
$906$	30.0000	+	17.3205i	0.996683	+	0.575435i
$907$	8.19615	+	2.19615i	0.272149	+	0.0729220i	0.392312	−	0.919832i	$-0.371675\pi$
$907$	8.19615	+	2.19615i	0.272149	+	0.0729220i	−0.120164	+	0.992754i	$0.538342\pi$
$908$		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	$-0.141624\pi$
$912$	−48.7846	+	7.21539i	−1.61542	+	0.238925i
$913$	−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
$918$	5.07180	+	18.9282i	0.167394	+	0.624724i
$919$			46.0000i			1.51740i	0.651440	+	0.758700i	$0.274165\pi$
$919$			46.0000i			1.51740i	−0.651440	+	0.758700i	$0.725835\pi$
$920$		0			0
$921$	−48.0000	+	83.1384i	−1.58165	+	2.73950i
$922$	−3.29423	−	12.2942i	−0.108490	−	0.404889i
$923$	−46.7654	+	46.7654i	−1.53930	+	1.53930i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−10.9808	−	40.9808i	−0.360656	−	1.34598i
$928$		0			0
$929$	−2.59808	+	1.50000i	−0.0852401	+	0.0492134i	−0.542014	−	0.840369i	$-0.682338\pi$
$929$	−2.59808	+	1.50000i	−0.0852401	+	0.0492134i	0.456774	+	0.889583i	$0.349005\pi$
$930$		0			0
$931$	3.50000	−	30.3109i	0.114708	−	0.993399i
$932$		0			0
$933$	−13.1769	+	49.1769i	−0.431393	+	1.60998i
$934$	8.66025	+	15.0000i	0.283372	+	0.490815i
$935$		0			0
$936$	30.0000	+	51.9615i	0.980581	+	1.69842i
$937$	−56.7846	+	15.2154i	−1.85507	+	0.497065i	−0.999778	−	0.0210771i	$-0.993290\pi$
$937$	−56.7846	+	15.2154i	−1.85507	+	0.497065i	−0.855294	+	0.518142i	$0.826624\pi$
$938$		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
$947$	13.3135	−	49.6865i	0.432630	−	1.61460i	−0.314047	−	0.949408i	$-0.601685\pi$
$947$	13.3135	−	49.6865i	0.432630	−	1.61460i	0.746676	−	0.665188i	$-0.231649\pi$
$948$		0			0
$949$	31.1769			1.01205
$950$		0			0
$951$	−4.00000			−0.129709
$952$		0			0
$953$	0.732051	−	2.73205i	0.0237135	−	0.0884998i	−0.953055	−	0.302797i	$-0.902080\pi$
$953$	0.732051	−	2.73205i	0.0237135	−	0.0884998i	0.976768	+	0.214297i	$0.0687462\pi$
$954$	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
$963$	−47.8109	+	12.8109i	−1.54068	+	0.412825i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−5.70577	+	21.2942i	−0.183485	+	0.684776i	0.811464	+	0.584402i	$0.198671\pi$
$967$	−5.70577	+	21.2942i	−0.183485	+	0.684776i	−0.994950	+	0.100374i	$0.967996\pi$
$968$	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	0.0240602	−	0.999711i	$-0.492341\pi$
$977$	32.0000	−	32.0000i	1.02377	−	1.02377i	0.999711	−	0.0240602i	$-0.00765934\pi$
$978$		0			0
$979$	7.79423	−	13.5000i	0.249105	−	0.431462i
$980$		0			0
$981$			60.6218i			1.93550i
$982$	7.68653	+	28.6865i	0.245287	+	0.915424i
$983$	−2.92820	+	10.9282i	−0.0933952	+	0.348556i	−0.996771	−	0.0802937i	$-0.974414\pi$
$983$	−2.92820	+	10.9282i	−0.0933952	+	0.348556i	0.903376	+	0.428849i	$0.141081\pi$
$984$	−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
$997$	−21.2942	+	5.70577i	−0.674395	+	0.180704i	−0.579734	−	0.814806i	$-0.696843\pi$
$997$	−21.2942	+	5.70577i	−0.674395	+	0.180704i	−0.0946612	+	0.995510i	$0.530177\pi$
$998$	−51.9090	+	13.9090i	−1.64315	+	0.440281i
$999$		−	24.0000i		−	0.759326i

Twists

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

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

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)

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

Properties

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