LMFDB - Embedded newform 475.2.p.c.293.1

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			293.1
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	475.293
Dual form			475.2.p.c.107.1

$q$-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 $\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$.

$n$	$77$	$401$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$e\left(\frac{1}{6}\right)$

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	0.258819	−	0.965926i	$-0.416667\pi$
$2$	1.36603	−	0.366025i	0.965926	−	0.258819i	0.707107	+	0.707107i	$0.250000\pi$
$3$	0.732051	+	2.73205i	0.422650	+	1.57735i	0.769002	+	0.639246i	$0.220753\pi$
$3$	0.732051	+	2.73205i	0.422650	+	1.57735i	−0.346353	+	0.938104i	$0.612580\pi$
$4$		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
$13$	4.09808	+	1.09808i	1.13660	+	0.304552i	0.777584	−	0.628779i	$-0.216445\pi$
$13$	4.09808	+	1.09808i	1.13660	+	0.304552i	0.359018	+	0.933331i	$0.383112\pi$
$14$		0			0
$15$		0			0
$16$	−2.00000	+	3.46410i	−0.500000	+	0.866025i
$17$	0.633975	+	2.36603i	0.153761	+	0.573845i	0.999208	+	0.0397858i	$0.0126676\pi$
$17$	0.633975	+	2.36603i	0.153761	+	0.573845i	−0.845447	+	0.534060i	$0.820666\pi$
$18$	−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	−0.422496	−	0.906365i	$-0.638846\pi$
$23$	1.90192	−	7.09808i	0.396579	−	1.48005i	0.819075	−	0.573687i	$-0.194487\pi$
$24$	−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	$-0.326920\pi$
$29$	−2.59808	−	4.50000i	−0.482451	−	0.835629i	−0.999797	+	0.0201471i	$0.993587\pi$
$30$		0			0
$31$			8.66025i			1.55543i	0.628619	+	0.777714i	$0.283621\pi$
$31$			8.66025i			1.55543i	−0.628619	+	0.777714i	$0.716379\pi$
$32$		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	0.416115	−	0.909312i	$-0.363391\pi$
$37$	−3.00000	−	3.00000i	−0.493197	−	0.493197i	−0.909312	+	0.416115i	$0.863391\pi$
$38$	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	0.994956	−	0.100309i	$-0.0319833\pi$
$41$	9.00000	+	5.19615i	1.40556	+	0.811503i	0.410608	+	0.911812i	$0.365317\pi$
$42$		0			0
$43$	7.09808	−	1.90192i	1.08245	−	0.290041i	0.326849	−	0.945077i	$-0.394013\pi$
$43$	7.09808	−	1.90192i	1.08245	−	0.290041i	0.755598	+	0.655036i	$0.227347\pi$
$44$		0			0
$45$		0			0
$46$		−	10.3923i		−	1.53226i
$47$	2.36603	+	0.633975i	0.345120	+	0.0924747i	0.427216	−	0.904150i	$-0.359495\pi$
$47$	2.36603	+	0.633975i	0.345120	+	0.0924747i	−0.0820953	+	0.996624i	$0.526161\pi$
$48$	−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	0.0662507	−	0.997803i	$-0.478896\pi$
$53$	−2.73205	−	0.732051i	−0.375276	−	0.100555i	−0.441527	+	0.897248i	$0.645563\pi$
$54$	−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	−0.645861	−	0.763455i	$-0.723502\pi$
$59$	2.59808	−	4.50000i	0.338241	−	0.585850i	0.984102	+	0.177605i	$0.0568349\pi$
$60$		0			0
$61$	0.500000	+	0.866025i	0.0640184	+	0.110883i	0.896258	−	0.443533i	$-0.146275\pi$
$61$	0.500000	+	0.866025i	0.0640184	+	0.110883i	−0.832240	+	0.554416i	$0.812942\pi$
$62$	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	−0.406249	−	0.913762i	$-0.633164\pi$
$67$	3.29423	−	12.2942i	0.402454	−	1.50198i	0.808703	−	0.588217i	$-0.200170\pi$
$68$		0			0
$69$	20.7846			2.50217
$70$		0			0
$71$	−13.5000	−	7.79423i	−1.60216	−	0.925005i	−0.991055	−	0.133451i	$-0.957394\pi$
$71$	−13.5000	−	7.79423i	−1.60216	−	0.925005i	−0.611100	−	0.791554i	$-0.709273\pi$
$72$	3.66025	−	13.6603i	0.431365	−	1.60988i
$73$	7.09808	−	1.90192i	0.830767	−	0.222603i	0.181719	−	0.983351i	$-0.441834\pi$
$73$	7.09808	−	1.90192i	0.830767	−	0.222603i	0.649048	+	0.760747i	$0.275167\pi$
$74$	−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	−0.682048	−	0.731307i	$-0.738911\pi$
$79$	2.59808	−	4.50000i	0.292306	−	0.506290i	0.974355	+	0.225018i	$0.0722440\pi$
$80$		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	0.998835	−	0.0482490i	$-0.0153641\pi$
$83$	8.66025	+	8.66025i	0.950586	+	0.950586i	−0.0482490	+	0.998835i	$0.515364\pi$
$84$		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	$-0.255475\pi$
$89$	−2.59808	−	4.50000i	−0.275396	−	0.476999i	−0.970235	+	0.242166i	$0.922142\pi$
$90$		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	−0.821676	−	0.569955i	$-0.806960\pi$
$97$	−12.2942	+	3.29423i	−1.24829	+	0.334478i	−0.426614	+	0.904434i	$0.640294\pi$
$98$	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	0.550474	−	0.834853i	$-0.314447\pi$
$101$	−4.50000	−	7.79423i	−0.447767	−	0.775555i	−0.998240	+	0.0592978i	$0.981114\pi$
$102$	−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	0.414751	+	0.909935i	$0.363869\pi$
$109$	−6.06218	+	10.5000i	−0.580651	+	1.00572i	−0.995402	+	0.0957826i	$0.969465\pi$
$110$		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
$117$	−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	−0.651010	−	0.759069i	$-0.725655\pi$
$127$	3.29423	−	12.2942i	0.292316	−	1.09094i	0.943326	−	0.331868i	$-0.107679\pi$
$128$	−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	−0.704692	−	0.709514i	$-0.748915\pi$
$131$	3.00000	−	5.19615i	0.262111	−	0.453990i	0.966803	+	0.255524i	$0.0822479\pi$
$132$		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	−0.169228	−	0.985577i	$-0.554128\pi$
$137$	−9.46410	−	2.53590i	−0.808573	−	0.216656i	−0.639344	+	0.768920i	$0.720794\pi$
$138$	28.3923	−	7.60770i	2.41691	−	0.647610i
$139$	−1.73205	+	1.00000i	−0.146911	+	0.0848189i	−0.571654	−	0.820495i	$-0.693698\pi$
$139$	−1.73205	+	1.00000i	−0.146911	+	0.0848189i	0.424743	+	0.905314i	$0.360365\pi$
$140$		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	0.389789	−	0.920904i	$-0.372548\pi$
$149$	−2.59808	−	1.50000i	−0.212843	−	0.122885i	−0.602632	+	0.798019i	$0.705881\pi$
$150$		0			0
$151$		−	8.66025i		−	0.704761i	−0.935857	−	0.352381i	$-0.885372\pi$
$151$		−	8.66025i		−	0.704761i	0.935857	−	0.352381i	$-0.114628\pi$
$152$	−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	0.934161	+	0.356853i	$0.116150\pi$
$157$	3.80385	+	14.1962i	0.303580	+	1.13298i	−0.630581	+	0.776124i	$0.717183\pi$
$158$	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	$-0.250000\pi$
$163$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$164$		0			0
$165$		0			0
$166$	15.0000	+	8.66025i	1.16423	+	0.672166i
$167$	5.12436	−	19.1244i	0.396535	−	1.47989i	−0.422616	−	0.906309i	$-0.638888\pi$
$167$	5.12436	−	19.1244i	0.396535	−	1.47989i	0.819151	−	0.573578i	$-0.194445\pi$
$168$		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	0.988155	−	0.153462i	$-0.0490422\pi$
$173$	0.732051	+	2.73205i	0.0556568	+	0.207714i	−0.932498	+	0.361176i	$0.882375\pi$
$174$	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	−0.563753	+	0.825943i	$0.690643\pi$
$181$	−21.0000	+	12.1244i	−1.56092	+	0.901196i	−0.997164	+	0.0752530i	$0.976024\pi$
$182$		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	0.922900	+	0.385040i	$0.125812\pi$
$193$	4.39230	+	16.3923i	0.316165	+	1.17994i	−0.606735	+	0.794904i	$0.707521\pi$
$194$	−15.5885	+	9.00000i	−1.11919	+	0.646162i
$195$		0			0
$196$		0			0
$197$	8.66025	−	8.66025i	0.617018	−	0.617018i	−0.327748	−	0.944765i	$-0.606290\pi$
$197$	8.66025	−	8.66025i	0.617018	−	0.617018i	0.944765	+	0.327748i	$0.106290\pi$
$198$	15.0000	−	15.0000i	1.06600	−	1.06600i
$199$	−6.06218	+	3.50000i	−0.429736	+	0.248108i	−0.699234	−	0.714893i	$-0.746476\pi$
$199$	−6.06218	+	3.50000i	−0.429736	+	0.248108i	0.269498	+	0.963001i	$0.413142\pi$
$200$		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	0.550743	−	0.834675i	$-0.314345\pi$
$211$	1.50000	+	0.866025i	0.103264	+	0.0596196i	−0.447478	+	0.894295i	$0.647678\pi$
$212$		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	0.919364	−	0.393408i	$-0.128704\pi$
$223$	−1.09808	−	4.09808i	−0.0735326	−	0.274427i	−0.992897	+	0.118981i	$0.962037\pi$
$224$		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
$232$	14.1962	+	3.80385i	0.932023	+	0.249735i
$233$	−4.73205	+	1.26795i	−0.310007	+	0.0830661i	−0.410468	−	0.911875i	$-0.634635\pi$
$233$	−4.73205	+	1.26795i	−0.310007	+	0.0830661i	0.100461	+	0.994941i	$0.467968\pi$
$234$	−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	$-0.762332\pi$
$239$		−	21.0000i		−	1.35838i	0.733964	−	0.679189i	$-0.237668\pi$
$240$		0			0
$241$	16.5000	−	9.52628i	1.06286	−	0.613642i	0.136637	−	0.990621i	$-0.456371\pi$
$241$	16.5000	−	9.52628i	1.06286	−	0.613642i	0.926222	+	0.376980i	$0.123037\pi$
$242$	−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	0.979889	+	0.199543i	$0.0639459\pi$
$251$	10.5000	+	18.1865i	0.662754	+	1.14792i	−0.317135	+	0.948380i	$0.602721\pi$
$252$		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	−0.976402	−	0.215962i	$-0.930711\pi$
$257$	−0.366025	+	1.36603i	−0.0228320	+	0.0852103i	0.953570	+	0.301172i	$0.0973779\pi$
$258$	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	0.377340	−	0.926075i	$-0.376839\pi$
$263$	18.9282	−	5.07180i	1.16716	−	0.312740i	0.789823	+	0.613335i	$0.210172\pi$
$264$	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	$-0.782697\pi$
$269$	2.59808	−	4.50000i	0.158408	−	0.274370i	0.934295	+	0.356502i	$0.116031\pi$
$270$		0			0
$271$	−9.50000	+	16.4545i	−0.577084	+	0.999539i	0.418728	+	0.908112i	$0.362476\pi$
$271$	−9.50000	+	16.4545i	−0.577084	+	0.999539i	−0.995812	+	0.0914269i	$0.970857\pi$
$272$	−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	$-0.750000\pi$
$277$		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	−0.206925	−	0.978357i	$-0.566345\pi$
$281$	9.00000	−	5.19615i	0.536895	−	0.309976i	0.743820	+	0.668380i	$0.233012\pi$
$282$	2.53590	+	9.46410i	0.151011	+	0.563579i
$283$	7.09808	−	1.90192i	0.421937	−	0.113058i	−0.0416020	−	0.999134i	$-0.513246\pi$
$283$	7.09808	−	1.90192i	0.421937	−	0.113058i	0.463539	+	0.886077i	$0.346579\pi$
$284$		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
$298$	−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	−0.871170	+	0.490981i	$0.836639\pi$
$307$	−32.7846	+	8.78461i	−1.87112	+	0.501364i	−0.999946	+	0.0103834i	$0.996695\pi$
$308$		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	−0.891114	−	0.453779i	$-0.850075\pi$
$313$	1.90192	−	7.09808i	0.107503	−	0.401207i	0.998617	+	0.0525725i	$0.0167421\pi$
$314$	10.3923	+	18.0000i	0.586472	+	1.01580i
$315$		0			0
$316$		0			0
$317$	−0.366025	+	1.36603i	−0.0205580	+	0.0767236i	−0.975443	−	0.220253i	$-0.929312\pi$
$317$	−0.366025	+	1.36603i	−0.0205580	+	0.0767236i	0.954885	+	0.296977i	$0.0959784\pi$
$318$	−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	−0.577624	−	0.816303i	$-0.696020\pi$
$337$	−12.2942	+	3.29423i	−0.669709	+	0.179448i	−0.0920854	+	0.995751i	$0.529353\pi$
$338$	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	−0.999991	−	0.00432163i	$-0.998624\pi$
$347$	−2.53590	−	9.46410i	−0.136134	−	0.508060i	0.863857	−	0.503738i	$-0.168042\pi$
$348$		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
$358$	−35.4904	+	9.50962i	−1.87572	+	0.502599i
$359$	10.3923	+	6.00000i	0.548485	+	0.316668i	0.748511	−	0.663123i	$-0.230769\pi$
$359$	10.3923	+	6.00000i	0.548485	+	0.316668i	−0.200026	+	0.979791i	$0.564103\pi$
$360$		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	0.609537	−	0.792757i	$-0.291355\pi$
$367$	14.1962	+	3.80385i	0.741033	+	0.198559i	0.131496	+	0.991317i	$0.458022\pi$
$368$	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
$377$	−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	−0.752368	−	0.658743i	$-0.771089\pi$
$383$	−8.41858	−	31.4186i	−0.430170	−	1.60541i	0.322199	−	0.946672i	$-0.395578\pi$
$384$	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	$-0.642435\pi$
$389$	2.59808	−	1.50000i	0.131728	−	0.0760530i	0.564416	+	0.825491i	$0.309102\pi$
$390$		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	0.993288	+	0.115669i	$0.0369011\pi$
$397$	3.80385	+	14.1962i	0.190910	+	0.712484i	−0.802378	+	0.596816i	$0.796432\pi$
$398$	−7.00000	+	7.00000i	−0.350878	+	0.350878i
$399$		0			0
$400$		0			0
$401$	−13.5000	−	7.79423i	−0.674158	−	0.389225i	0.123492	−	0.992346i	$-0.460591\pi$
$401$	−13.5000	−	7.79423i	−0.674158	−	0.389225i	−0.797650	+	0.603120i	$0.793924\pi$
$402$	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	0.794616	−	0.607112i	$-0.207672\pi$
$409$	−2.59808	−	4.50000i	−0.128467	−	0.222511i	−0.923083	+	0.384602i	$0.874339\pi$
$410$		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	$0.0705533\pi$
$419$			9.00000i			0.439679i	−0.975536	+	0.219839i	$0.929447\pi$
$420$		0			0
$421$	31.5000	+	18.1865i	1.53522	+	0.886357i	0.999109	+	0.0422075i	$0.0134391\pi$
$421$	31.5000	+	18.1865i	1.53522	+	0.886357i	0.536107	+	0.844150i	$0.319894\pi$
$422$	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	−0.788560	−	0.614957i	$-0.789173\pi$
$431$	−13.5000	+	7.79423i	−0.650272	+	0.375435i	0.138288	+	0.990392i	$0.455840\pi$
$432$	21.8564	−	5.85641i	1.05157	−	0.281766i
$433$	24.5885	+	6.58846i	1.18165	+	0.316621i	0.795579	−	0.605850i	$-0.207167\pi$
$433$	24.5885	+	6.58846i	1.18165	+	0.316621i	0.386067	+	0.922471i	$0.373833\pi$
$434$		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	−0.973650	−	0.228046i	$-0.926766\pi$
$439$	−6.06218	+	10.5000i	−0.289332	+	0.501138i	0.684318	+	0.729183i	$0.260100\pi$
$440$		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	−0.989483	−	0.144647i	$-0.953795\pi$
$443$	−1.26795	+	4.73205i	−0.0602421	+	0.224827i	0.929241	+	0.369474i	$0.120462\pi$
$444$		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	0.246079	−	0.969250i	$-0.420858\pi$
$457$	25.9808	−	25.9808i	1.21533	−	1.21533i	0.969250	−	0.246079i	$-0.0791423\pi$
$458$	−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	−0.951584	−	0.307388i	$-0.900545\pi$
$461$	−4.50000	+	7.79423i	−0.209586	+	0.363013i	0.741998	+	0.670402i	$0.233878\pi$
$462$		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	−0.477748	−	0.878497i	$-0.658547\pi$
$467$	8.66025	−	8.66025i	0.400749	−	0.400749i	0.878497	+	0.477748i	$0.158547\pi$
$468$		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	−0.439470	−	0.898257i	$-0.644834\pi$
$479$	2.59808	−	1.50000i	0.118709	−	0.0685367i	0.558179	+	0.829721i	$0.311500\pi$
$480$		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	$0.417232\pi$
$488$	−2.73205	−	0.732051i	−0.123674	−	0.0331384i
$489$		0			0
$490$		0			0
$491$	10.5000	−	18.1865i	0.473858	−	0.820747i	−0.525694	−	0.850674i	$-0.676194\pi$
$491$	10.5000	−	18.1865i	0.473858	−	0.820747i	0.999552	+	0.0299272i	$0.00952753\pi$
$492$		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	−0.473662	−	0.880707i	$-0.657068\pi$
$499$	−32.9090	−	19.0000i	−1.47321	−	0.850557i	−0.999545	+	0.0301498i	$0.990402\pi$
$500$		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	−0.496553	−	0.868007i	$-0.665401\pi$
$503$	8.24167	−	30.7583i	0.367478	−	1.37145i	0.864030	−	0.503440i	$-0.167932\pi$
$504$		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	0.998992	−	0.0448779i	$-0.0142899\pi$
$509$	10.3923	+	18.0000i	0.460631	+	0.797836i	−0.538362	+	0.842714i	$0.680957\pi$
$510$		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	$-0.807284\pi$
$521$		−	25.9808i		−	1.13824i	0.822255	−	0.569119i	$-0.192716\pi$
$522$	35.4904	+	9.50962i	1.55337	+	0.416225i
$523$	−16.3923	−	4.39230i	−0.716785	−	0.192062i	−0.118049	−	0.993008i	$-0.537664\pi$
$523$	−16.3923	−	4.39230i	−0.716785	−	0.192062i	−0.598737	+	0.800946i	$0.704331\pi$
$524$		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	0.586278	−	0.810110i	$-0.300593\pi$
$541$	−9.50000	−	16.4545i	−0.408437	−	0.707433i	−0.994715	+	0.102677i	$0.967259\pi$
$542$	−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	0.581936	+	0.813234i	$0.302295\pi$
$547$	−7.68653	+	28.6865i	−0.328652	+	1.22655i	−0.910589	+	0.413313i	$0.864371\pi$
$548$		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	−0.222276	−	0.974984i	$-0.571349\pi$
$557$	−21.2942	−	5.70577i	−0.902265	−	0.241761i	−0.679989	+	0.733222i	$0.738015\pi$
$558$	−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
$568$	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	0.0852322	−	0.996361i	$-0.472837\pi$
$577$	25.9808	−	25.9808i	1.08159	−	1.08159i	0.996361	−	0.0852322i	$-0.0271632\pi$
$578$	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	0.893640	−	0.448785i	$-0.148143\pi$
$587$	−2.53590	−	9.46410i	−0.104668	−	0.390625i	−0.998307	+	0.0581602i	$0.981477\pi$
$588$		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	−0.780153	−	0.625589i	$-0.784859\pi$
$593$	5.07180	−	18.9282i	0.208274	−	0.777288i	0.988427	−	0.151699i	$-0.0484745\pi$
$594$	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	−0.986103	−	0.166133i	$-0.946872\pi$
$599$	−15.5885	−	27.0000i	−0.636927	−	1.10319i	0.349176	−	0.937057i	$-0.386461\pi$
$600$		0			0
$601$			25.9808i			1.05978i	0.848067	+	0.529889i	$0.177766\pi$
$601$			25.9808i			1.05978i	−0.848067	+	0.529889i	$0.822234\pi$
$602$		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	0.643598	−	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
$613$	−28.3923	+	7.60770i	−1.14675	+	0.307272i	−0.781664	−	0.623700i	$-0.785629\pi$
$613$	−28.3923	+	7.60770i	−1.14675	+	0.307272i	−0.365091	+	0.930972i	$0.618962\pi$
$614$	−41.5692	+	24.0000i	−1.67760	+	0.968561i
$615$		0			0
$616$		0			0
$617$	2.36603	+	0.633975i	0.0952526	+	0.0255229i	0.306131	−	0.951990i	$-0.400966\pi$
$617$	2.36603	+	0.633975i	0.0952526	+	0.0255229i	−0.210878	+	0.977512i	$0.567632\pi$
$618$	32.7846	+	8.78461i	1.31879	+	0.353369i
$619$			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
$622$	−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	0.875806	−	0.482663i	$-0.160330\pi$
$631$	0.500000	+	0.866025i	0.0199047	+	0.0344759i	−0.855901	+	0.517139i	$0.826997\pi$
$632$	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	0.209108	−	0.977893i	$-0.432944\pi$
$641$	−13.5000	−	7.79423i	−0.533218	−	0.307854i	−0.742326	+	0.670039i	$0.766277\pi$
$642$	−10.2487	+	38.2487i	−0.404484	+	1.50956i
$643$	7.09808	−	1.90192i	0.279921	−	0.0750046i	−0.116127	−	0.993234i	$-0.537048\pi$
$643$	7.09808	−	1.90192i	0.279921	−	0.0750046i	0.396048	+	0.918230i	$0.370381\pi$
$644$		0			0
$645$		0			0
$646$	9.00000	+	12.1244i	0.354100	+	0.477026i
$647$	−8.66025	+	8.66025i	−0.340470	+	0.340470i	−0.856544	−	0.516074i	$-0.827393\pi$
$647$	−8.66025	+	8.66025i	−0.340470	+	0.340470i	0.516074	+	0.856544i	$0.327393\pi$
$648$	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	−0.999858	−	0.0168477i	$-0.994637\pi$
$653$	−25.9808	−	25.9808i	−1.01671	−	1.01671i	−0.0168477	−	0.999858i	$-0.505363\pi$
$654$	−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	$-0.0339985\pi$
$659$	10.3923	+	18.0000i	0.404827	+	0.701180i	−0.589475	+	0.807787i	$0.700665\pi$
$660$		0			0
$661$	16.5000	−	9.52628i	0.641776	−	0.370529i	−0.143523	−	0.989647i	$-0.545843\pi$
$661$	16.5000	−	9.52628i	0.641776	−	0.370529i	0.785298	+	0.619118i	$0.212510\pi$
$662$		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	−0.997386	−	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	$0.819761\pi$
$678$	−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	−0.464611	−	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
$687$	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	−0.998585	−	0.0531839i	$-0.983063\pi$
$701$	−12.0000	+	20.7846i	−0.453234	+	0.785024i	0.545351	+	0.838208i	$0.316396\pi$
$702$	−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	−0.273466	−	0.961882i	$-0.588170\pi$
$709$	11.2583	−	6.50000i	0.422815	−	0.244113i	0.696281	+	0.717769i	$0.254837\pi$
$710$		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	0.868021	−	0.496527i	$-0.165392\pi$
$719$	23.3827	+	13.5000i	0.872027	+	0.503465i	0.00400572	+	0.999992i	$0.498725\pi$
$720$		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	0.999899	+	0.0142317i	$0.00453026\pi$
$727$	3.80385	+	14.1962i	0.141077	+	0.526506i	−0.858822	+	0.512274i	$0.828803\pi$
$728$		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	0.999216	−	0.0395945i	$-0.0126066\pi$
$733$	25.9808	+	25.9808i	0.959621	+	0.959621i	−0.0395945	+	0.999216i	$0.512607\pi$
$734$	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	−0.955718	−	0.294285i	$-0.904919\pi$
$739$	−45.8993	−	26.5000i	−1.68843	−	0.974818i	−0.732717	−	0.680534i	$-0.761748\pi$
$740$		0			0
$741$	6.00000	+	51.9615i	0.220416	+	1.90885i
$742$		0			0
$743$	−8.41858	−	31.4186i	−0.308848	−	1.15264i	−0.929582	−	0.368615i	$-0.879832\pi$
$743$	−8.41858	−	31.4186i	−0.308848	−	1.15264i	0.620734	−	0.784021i	$-0.286835\pi$
$744$	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	0.748056	−	0.663636i	$-0.230988\pi$
$751$	46.5000	−	26.8468i	1.69681	−	0.979653i	0.948753	−	0.316017i	$-0.102346\pi$
$752$	−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	−0.988711	−	0.149835i	$-0.952126\pi$
$757$	−5.70577	−	21.2942i	−0.207380	−	0.773952i	0.781331	−	0.624117i	$-0.214541\pi$
$758$	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	−0.605305	−	0.795993i	$-0.706949\pi$
$769$	−6.06218	+	3.50000i	−0.218608	+	0.126213i	0.386698	+	0.922207i	$0.373616\pi$
$770$		0			0
$771$	−4.00000			−0.144056
$772$		0			0
$773$	31.4186	+	8.41858i	1.13005	+	0.302795i	0.774943	−	0.632031i	$-0.217778\pi$
$773$	31.4186	+	8.41858i	1.13005	+	0.302795i	0.355105	+	0.934826i	$0.384445\pi$
$774$	−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	0.999320	−	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
$792$	−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	0.989591	+	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
$802$	−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	$0.240453\pi$
$809$			39.0000i			1.37117i	−0.727994	+	0.685583i	$0.759547\pi$
$810$		0			0
$811$	1.50000	−	0.866025i	0.0526721	−	0.0304103i	−0.473433	−	0.880830i	$-0.656985\pi$
$811$	1.50000	−	0.866025i	0.0526721	−	0.0304103i	0.526105	+	0.850420i	$0.323652\pi$
$812$		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	0.989008	−	0.147861i	$-0.0472389\pi$
$821$	10.5000	+	18.1865i	0.366453	+	0.634714i	−0.622556	+	0.782576i	$0.713906\pi$
$822$	−10.1436	−	37.8564i	−0.353798	−	1.32039i
$823$	1.90192	−	7.09808i	0.0662969	−	0.247423i	−0.924822	−	0.380399i	$-0.875787\pi$
$823$	1.90192	−	7.09808i	0.0662969	−	0.247423i	0.991119	+	0.132976i	$0.0424533\pi$
$824$			24.0000i			0.836080i
$825$		0			0
$826$		0			0
$827$	−4.02628	+	15.0263i	−0.140007	+	0.522515i	0.859920	+	0.510430i	$0.170514\pi$
$827$	−4.02628	+	15.0263i	−0.140007	+	0.522515i	−0.999927	+	0.0120853i	$0.996153\pi$
$828$		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	−0.987758	−	0.155996i	$-0.950141\pi$
$839$	−10.3923	+	18.0000i	−0.358782	+	0.621429i	0.628975	+	0.777425i	$0.283475\pi$
$840$		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	0.559333	−	0.828943i	$-0.311057\pi$
$853$	42.5885	−	11.4115i	1.45820	−	0.390724i	0.898868	+	0.438219i	$0.144391\pi$
$854$		0			0
$855$		0			0
$856$	−28.0000			−0.957020
$857$	−25.9545	+	6.95448i	−0.886588	+	0.237561i	−0.673247	−	0.739417i	$-0.735101\pi$
$857$	−25.9545	+	6.95448i	−0.886588	+	0.237561i	−0.213341	+	0.976978i	$0.568434\pi$
$858$	−13.1769	−	49.1769i	−0.449852	−	1.67887i
$859$	−40.7032	+	23.5000i	−1.38878	+	0.801810i	−0.993177	−	0.116614i	$-0.962796\pi$
$859$	−40.7032	+	23.5000i	−1.38878	+	0.801810i	−0.395598	+	0.918424i	$0.629463\pi$
$860$		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
$872$	−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	−0.460346	−	0.887740i	$-0.652274\pi$
$877$	−12.2942	+	3.29423i	−0.415147	+	0.111238i	0.0451990	+	0.998978i	$0.485608\pi$
$878$	−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	−0.455663	−	0.890152i	$-0.650598\pi$
$883$	11.4115	−	42.5885i	0.384029	−	1.43322i	0.839692	−	0.543063i	$-0.182736\pi$
$884$		0			0
$885$		0			0
$886$			6.92820i			0.232758i
$887$	5.12436	−	19.1244i	0.172059	−	0.642133i	−0.824975	−	0.565169i	$-0.808811\pi$
$887$	5.12436	−	19.1244i	0.172059	−	0.642133i	0.997034	−	0.0769636i	$-0.0245225\pi$
$888$	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	−0.120164	−	0.992754i	$-0.538342\pi$
$907$	8.19615	−	2.19615i	0.272149	−	0.0729220i	0.392312	+	0.919832i	$0.371675\pi$
$908$		0			0
$909$	38.9711	+	22.5000i	1.29259	+	0.746278i
$910$		0			0
$911$			25.9808i			0.860781i	0.902643	+	0.430391i	$0.141624\pi$
$911$			25.9808i			0.860781i	−0.902643	+	0.430391i	$0.858376\pi$
$912$	−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	$-0.725835\pi$
$919$		−	46.0000i		−	1.51740i	0.651440	−	0.758700i	$-0.274165\pi$
$920$		0			0
$921$	−48.0000	−	83.1384i	−1.58165	−	2.73950i
$922$	−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	0.456774	−	0.889583i	$-0.349005\pi$
$929$	−2.59808	−	1.50000i	−0.0852401	−	0.0492134i	−0.542014	+	0.840369i	$0.682338\pi$
$930$		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	−0.855294	−	0.518142i	$-0.826624\pi$
$937$	−56.7846	−	15.2154i	−1.85507	−	0.497065i	−0.999778	+	0.0210771i	$0.993290\pi$
$938$		0			0
$939$	20.7846			0.678280
$940$		0			0
$941$	−13.5000	+	7.79423i	−0.440087	+	0.254085i	−0.703635	−	0.710562i	$-0.748441\pi$
$941$	−13.5000	+	7.79423i	−0.440087	+	0.254085i	0.263547	+	0.964646i	$0.415107\pi$
$942$	−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	0.746676	+	0.665188i	$0.231649\pi$
$947$	13.3135	+	49.6865i	0.432630	+	1.61460i	−0.314047	+	0.949408i	$0.601685\pi$
$948$		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	0.976768	−	0.214297i	$-0.0687462\pi$
$953$	0.732051	+	2.73205i	0.0237135	+	0.0884998i	−0.953055	+	0.302797i	$0.902080\pi$
$954$	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	−0.994950	−	0.100374i	$-0.967996\pi$
$967$	−5.70577	−	21.2942i	−0.183485	−	0.684776i	0.811464	−	0.584402i	$-0.198671\pi$
$968$	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	−0.205123	−	0.978736i	$-0.565759\pi$
$971$	−36.0000	−	20.7846i	−1.15529	−	0.667010i	−0.950172	+	0.311726i	$0.899093\pi$
$972$		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
$982$	7.68653	−	28.6865i	0.245287	−	0.915424i
$983$	−2.92820	−	10.9282i	−0.0933952	−	0.348556i	0.903376	−	0.428849i	$-0.141081\pi$
$983$	−2.92820	−	10.9282i	−0.0933952	−	0.348556i	−0.996771	+	0.0802937i	$0.974414\pi$
$984$	−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	0.127885	−	0.991789i	$-0.459181\pi$
$991$	−21.0000	−	12.1244i	−0.667087	−	0.385143i	−0.794972	+	0.606646i	$0.792514\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−21.2942	−	5.70577i	−0.674395	−	0.180704i	−0.0946612	−	0.995510i	$-0.530177\pi$
$997$	−21.2942	−	5.70577i	−0.674395	−	0.180704i	−0.579734	+	0.814806i	$0.696843\pi$
$998$	−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.293.1	yes	4
5.2	odd	4		475.2.p.a.407.1	yes	4
5.3	odd	4	inner	475.2.p.c.407.1	yes	4
5.4	even	2		475.2.p.a.293.1	yes	4
19.12	odd	6		475.2.p.a.468.1	yes	4
95.12	even	12	inner	475.2.p.c.107.1	yes	4
95.69	odd	6	inner	475.2.p.c.468.1	yes	4
95.88	even	12		475.2.p.a.107.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.p.a.107.1	✓	4	95.88	even	12
475.2.p.a.293.1	yes	4	5.4	even	2
475.2.p.a.407.1	yes	4	5.2	odd	4
475.2.p.a.468.1	yes	4	19.12	odd	6
475.2.p.c.107.1	yes	4	95.12	even	12	inner
475.2.p.c.293.1	yes	4	1.1	even	1	trivial
475.2.p.c.407.1	yes	4	5.3	odd	4	inner
475.2.p.c.468.1	yes	4	95.69	odd	6	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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