LMFDB - Embedded newform 78.2.g.a.47.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [78,2,Mod(5,78)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(78, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3]))

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

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

chi := DirichletCharacter("78.5");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 78 = 2 \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	78.g (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.622833135766$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	12.0.58498535041007616.52
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 12x^{9} + 72x^{6} - 324x^{3} + 729 $ x^12 - 12x^9 + 72x^6 - 324*x^3 + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			47.2
Root			$-1.44857 - 0.949550i$ of defining polynomial
Character	$\chi$	$=$	78.47
Dual form			78.2.g.a.5.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 + 0.707107i) q^{2} +(0.352860 - 1.69573i) q^{3} -1.00000i q^{4} +(0.499019 - 0.499019i) q^{5} +(0.949550 + 1.44857i) q^{6} +(1.39812 - 1.39812i) q^{7} +(0.707107 + 0.707107i) q^{8} +(-2.75098 - 1.19671i) q^{9} +O(q^{10})$	\(q+(-0.707107 + 0.707107i) q^{2} +(0.352860 - 1.69573i) q^{3} -1.00000i q^{4} +(0.499019 - 0.499019i) q^{5} +(0.949550 + 1.44857i) q^{6} +(1.39812 - 1.39812i) q^{7} +(0.707107 + 0.707107i) q^{8} +(-2.75098 - 1.19671i) q^{9} +0.705720i q^{10} +(3.39145 + 3.39145i) q^{11} +(-1.69573 - 0.352860i) q^{12} +(-2.39812 + 2.69240i) q^{13} +1.97724i q^{14} +(-0.670116 - 1.02228i) q^{15} -1.00000 q^{16} -4.38949 q^{17} +(2.79144 - 1.09904i) q^{18} +(-1.70572 - 1.70572i) q^{19} +(-0.499019 - 0.499019i) q^{20} +(-1.87749 - 2.86417i) q^{21} -4.79624 q^{22} +0.998038 q^{23} +(1.44857 - 0.949550i) q^{24} +4.50196i q^{25} +(-0.208088 - 3.59954i) q^{26} +(-3.00000 + 4.24264i) q^{27} +(-1.39812 - 1.39812i) q^{28} -0.998038i q^{29} +(1.19671 + 0.249020i) q^{30} +(6.50196 + 6.50196i) q^{31} +(0.707107 - 0.707107i) q^{32} +(6.94769 - 4.55427i) q^{33} +(3.10384 - 3.10384i) q^{34} -1.39538i q^{35} +(-1.19671 + 2.75098i) q^{36} +(4.10384 - 4.10384i) q^{37} +2.41225 q^{38} +(3.71938 + 5.01660i) q^{39} +0.705720 q^{40} +(-5.24068 + 5.24068i) q^{41} +(3.35286 + 0.697689i) q^{42} -8.88676i q^{43} +(3.39145 - 3.39145i) q^{44} +(-1.96997 + 0.775612i) q^{45} +(-0.705720 + 0.705720i) q^{46} +(0.352168 + 0.352168i) q^{47} +(-0.352860 + 1.69573i) q^{48} +3.09052i q^{49} +(-3.18337 - 3.18337i) q^{50} +(-1.54888 + 7.44338i) q^{51} +(2.69240 + 2.39812i) q^{52} -14.2702i q^{53} +(-0.878680 - 5.12132i) q^{54} +3.38480 q^{55} +1.97724 q^{56} +(-3.49431 + 2.29055i) q^{57} +(0.705720 + 0.705720i) q^{58} +(-0.998038 - 0.998038i) q^{59} +(-1.02228 + 0.670116i) q^{60} -9.59248 q^{61} -9.19516 q^{62} +(-5.51934 + 2.17306i) q^{63} +1.00000i q^{64} +(0.146852 + 2.54027i) q^{65} +(-1.69240 + 8.13311i) q^{66} +(-5.79624 - 5.79624i) q^{67} +4.38949i q^{68} +(0.352168 - 1.69240i) q^{69} +(0.986681 + 0.986681i) q^{70} +(7.13508 - 7.13508i) q^{71} +(-1.09904 - 2.79144i) q^{72} +(-1.70572 + 1.70572i) q^{73} +5.80371i q^{74} +(7.63409 + 1.58856i) q^{75} +(-1.70572 + 1.70572i) q^{76} +9.48332 q^{77} +(-6.17727 - 0.917274i) q^{78} -0.207679 q^{79} +(-0.499019 + 0.499019i) q^{80} +(6.13578 + 6.58424i) q^{81} -7.41144i q^{82} +(-9.17632 + 9.17632i) q^{83} +(-2.86417 + 1.87749i) q^{84} +(-2.19044 + 2.19044i) q^{85} +(6.28389 + 6.28389i) q^{86} +(-1.69240 - 0.352168i) q^{87} +4.79624i q^{88} +(-2.54027 - 2.54027i) q^{89} +(0.844540 - 1.94142i) q^{90} +(0.411439 + 7.11716i) q^{91} -0.998038i q^{92} +(13.3198 - 8.73127i) q^{93} -0.498040 q^{94} -1.70237 q^{95} +(-0.949550 - 1.44857i) q^{96} +(-3.58856 - 3.58856i) q^{97} +(-2.18533 - 2.18533i) q^{98} +(-5.27124 - 13.3884i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{7}+O(q^{10}) $ 12 * q - 12 * q^7	$ 12 q - 12 q^{7} - 12 q^{16} - 12 q^{19} - 36 q^{27} + 12 q^{28} + 12 q^{31} + 36 q^{33} + 12 q^{37} + 36 q^{42} + 36 q^{45} + 12 q^{52} - 36 q^{54} - 36 q^{57} - 36 q^{63} - 12 q^{67} - 12 q^{73} - 12 q^{76} - 36 q^{78} + 72 q^{79} - 72 q^{85} - 12 q^{91} + 36 q^{93} - 72 q^{94} - 60 q^{97} + 36 q^{99}+O(q^{100}) $ 12 * q - 12 * q^7 - 12 * q^16 - 12 * q^19 - 36 * q^27 + 12 * q^28 + 12 * q^31 + 36 * q^33 + 12 * q^37 + 36 * q^42 + 36 * q^45 + 12 * q^52 - 36 * q^54 - 36 * q^57 - 36 * q^63 - 12 * q^67 - 12 * q^73 - 12 * q^76 - 36 * q^78 + 72 * q^79 - 72 * q^85 - 12 * q^91 + 36 * q^93 - 72 * q^94 - 60 * q^97 + 36 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/78\mathbb{Z}\right)^\times$.

$n$	$53$	$67$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$

Coefficient data

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

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

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

$2$	−0.707107	+	0.707107i	−0.500000	+	0.500000i
$2$	−0.707107	+	0.707107i	−0.500000	+	0.500000i
$3$	0.352860	−	1.69573i	0.203724	−	0.979028i
$3$	0.352860	−	1.69573i	0.203724	−	0.979028i
$4$		−	1.00000i		−	0.500000i
$5$	0.499019	−	0.499019i	0.223168	−	0.223168i	−0.586663	−	0.809831i	$-0.699559\pi$
$5$	0.499019	−	0.499019i	0.223168	−	0.223168i	0.809831	+	0.586663i	$0.199559\pi$
$6$	0.949550	+	1.44857i	0.387652	+	0.591376i
$7$	1.39812	−	1.39812i	0.528440	−	0.528440i	−0.391667	−	0.920107i	$-0.628102\pi$
$7$	1.39812	−	1.39812i	0.528440	−	0.528440i	0.920107	+	0.391667i	$0.128102\pi$
$8$	0.707107	+	0.707107i	0.250000	+	0.250000i
$9$	−2.75098	−	1.19671i	−0.916993	−	0.398903i
$10$			0.705720i			0.223168i
$11$	3.39145	+	3.39145i	1.02256	+	1.02256i	0.999740	+	0.0228223i	$0.00726519\pi$
$11$	3.39145	+	3.39145i	1.02256	+	1.02256i	0.0228223	+	0.999740i	$0.492735\pi$
$12$	−1.69573	−	0.352860i	−0.489514	−	0.101862i
$13$	−2.39812	+	2.69240i	−0.665119	+	0.746738i
$13$	−2.39812	+	2.69240i	−0.665119	+	0.746738i
$14$			1.97724i			0.528440i
$15$	−0.670116	−	1.02228i	−0.173023	−	0.263953i
$16$	−1.00000			−0.250000
$17$	−4.38949			−1.06461			−0.532304	−	0.846553i	$-0.678674\pi$
$17$	−4.38949			−1.06461			−0.532304	+	0.846553i	$0.678674\pi$
$18$	2.79144	−	1.09904i	0.657948	−	0.259045i
$19$	−1.70572	−	1.70572i	−0.391319	−	0.391319i	0.483838	−	0.875157i	$-0.339242\pi$
$19$	−1.70572	−	1.70572i	−0.391319	−	0.391319i	−0.875157	+	0.483838i	$0.839242\pi$
$20$	−0.499019	−	0.499019i	−0.111584	−	0.111584i
$21$	−1.87749	−	2.86417i	−0.409702	−	0.625013i
$22$	−4.79624			−1.02256
$23$	0.998038			0.208105			0.104053	−	0.994572i	$-0.466819\pi$
$23$	0.998038			0.208105			0.104053	+	0.994572i	$0.466819\pi$
$24$	1.44857	−	0.949550i	0.295688	−	0.193826i
$25$			4.50196i			0.900392i
$26$	−0.208088	−	3.59954i	−0.0408093	−	0.705928i
$27$	−3.00000	+	4.24264i	−0.577350	+	0.816497i
$28$	−1.39812	−	1.39812i	−0.264220	−	0.264220i
$29$		−	0.998038i		−	0.185331i	−0.995697	−	0.0926655i	$-0.970461\pi$
$29$		−	0.998038i		−	0.185331i	0.995697	−	0.0926655i	$-0.0295387\pi$
$30$	1.19671	+	0.249020i	0.218488	+	0.0454646i
$31$	6.50196	+	6.50196i	1.16779	+	1.16779i	0.982727	+	0.185059i	$0.0592476\pi$
$31$	6.50196	+	6.50196i	1.16779	+	1.16779i	0.185059	+	0.982727i	$0.440752\pi$
$32$	0.707107	−	0.707107i	0.125000	−	0.125000i
$33$	6.94769	−	4.55427i	1.20944	−	0.792797i
$34$	3.10384	−	3.10384i	0.532304	−	0.532304i
$35$		−	1.39538i		−	0.235862i
$36$	−1.19671	+	2.75098i	−0.199451	+	0.458497i
$37$	4.10384	−	4.10384i	0.674667	−	0.674667i	−0.284121	−	0.958788i	$-0.591702\pi$
$37$	4.10384	−	4.10384i	0.674667	−	0.674667i	0.958788	+	0.284121i	$0.0917018\pi$
$38$	2.41225			0.391319
$39$	3.71938	+	5.01660i	0.595577	+	0.803298i
$40$	0.705720			0.111584
$41$	−5.24068	+	5.24068i	−0.818457	+	0.818457i	−0.985884	−	0.167428i	$-0.946454\pi$
$41$	−5.24068	+	5.24068i	−0.818457	+	0.818457i	0.167428	+	0.985884i	$0.446454\pi$
$42$	3.35286	+	0.697689i	0.517358	+	0.107656i
$43$		−	8.88676i		−	1.35522i	−0.735422	−	0.677609i	$-0.763016\pi$
$43$		−	8.88676i		−	1.35522i	0.735422	−	0.677609i	$-0.236984\pi$
$44$	3.39145	−	3.39145i	0.511281	−	0.511281i
$45$	−1.96997	+	0.775612i	−0.293666	+	0.115621i
$46$	−0.705720	+	0.705720i	−0.104053	+	0.104053i
$47$	0.352168	+	0.352168i	0.0513689	+	0.0513689i	0.732325	−	0.680956i	$-0.238435\pi$
$47$	0.352168	+	0.352168i	0.0513689	+	0.0513689i	−0.680956	+	0.732325i	$0.738435\pi$
$48$	−0.352860	+	1.69573i	−0.0509309	+	0.244757i
$49$			3.09052i			0.441503i
$50$	−3.18337	−	3.18337i	−0.450196	−	0.450196i
$51$	−1.54888	+	7.44338i	−0.216886	+	1.04228i
$52$	2.69240	+	2.39812i	0.373369	+	0.332559i
$53$		−	14.2702i		−	1.96016i	−0.198613	−	0.980078i	$-0.563644\pi$
$53$		−	14.2702i		−	1.96016i	0.198613	−	0.980078i	$-0.436356\pi$
$54$	−0.878680	−	5.12132i	−0.119573	−	0.696923i
$55$	3.38480			0.456406
$56$	1.97724			0.264220
$57$	−3.49431	+	2.29055i	−0.462833	+	0.303391i
$58$	0.705720	+	0.705720i	0.0926655	+	0.0926655i
$59$	−0.998038	−	0.998038i	−0.129934	−	0.129934i	0.639149	−	0.769083i	$-0.279287\pi$
$59$	−0.998038	−	0.998038i	−0.129934	−	0.129934i	−0.769083	+	0.639149i	$0.779287\pi$
$60$	−1.02228	+	0.670116i	−0.131976	+	0.0865117i
$61$	−9.59248			−1.22819			−0.614096	−	0.789232i	$-0.710479\pi$
$61$	−9.59248			−1.22819			−0.614096	+	0.789232i	$0.710479\pi$
$62$	−9.19516			−1.16779
$63$	−5.51934	+	2.17306i	−0.695372	+	0.273780i
$64$			1.00000i			0.125000i
$65$	0.146852	+	2.54027i	0.0182147	+	0.315081i
$66$	−1.69240	+	8.13311i	−0.208320	+	1.00112i
$67$	−5.79624	−	5.79624i	−0.708123	−	0.708123i	0.258017	−	0.966140i	$-0.416931\pi$
$67$	−5.79624	−	5.79624i	−0.708123	−	0.708123i	−0.966140	+	0.258017i	$0.916931\pi$
$68$			4.38949i			0.532304i
$69$	0.352168	−	1.69240i	0.0423960	−	0.203741i
$70$	0.986681	+	0.986681i	0.117931	+	0.117931i
$71$	7.13508	−	7.13508i	0.846778	−	0.846778i	−0.142952	−	0.989730i	$-0.545659\pi$
$71$	7.13508	−	7.13508i	0.846778	−	0.846778i	0.989730	+	0.142952i	$0.0456594\pi$
$72$	−1.09904	−	2.79144i	−0.129523	−	0.328974i
$73$	−1.70572	+	1.70572i	−0.199639	+	0.199639i	−0.799845	−	0.600206i	$-0.795085\pi$
$73$	−1.70572	+	1.70572i	−0.199639	+	0.199639i	0.600206	+	0.799845i	$0.295085\pi$
$74$			5.80371i			0.674667i
$75$	7.63409	+	1.58856i	0.881509	+	0.183431i
$76$	−1.70572	+	1.70572i	−0.195659	+	0.195659i
$77$	9.48332			1.08072
$78$	−6.17727	−	0.917274i	−0.699438	−	0.103861i
$79$	−0.207679			−0.0233658			−0.0116829	−	0.999932i	$-0.503719\pi$
$79$	−0.207679			−0.0233658			−0.0116829	+	0.999932i	$0.503719\pi$
$80$	−0.499019	+	0.499019i	−0.0557920	+	0.0557920i
$81$	6.13578	+	6.58424i	0.681753	+	0.731582i
$82$		−	7.41144i		−	0.818457i
$83$	−9.17632	+	9.17632i	−1.00723	+	1.00723i	−0.00725871	+	0.999974i	$0.502311\pi$
$83$	−9.17632	+	9.17632i	−1.00723	+	1.00723i	−0.999974	+	0.00725871i	$0.997689\pi$
$84$	−2.86417	+	1.87749i	−0.312507	+	0.204851i
$85$	−2.19044	+	2.19044i	−0.237587	+	0.237587i
$86$	6.28389	+	6.28389i	0.677609	+	0.677609i
$87$	−1.69240	−	0.352168i	−0.181444	−	0.0377563i
$88$			4.79624i			0.511281i
$89$	−2.54027	−	2.54027i	−0.269268	−	0.269268i	0.559537	−	0.828805i	$-0.310979\pi$
$89$	−2.54027	−	2.54027i	−0.269268	−	0.269268i	−0.828805	+	0.559537i	$0.810979\pi$
$90$	0.844540	−	1.94142i	0.0890224	−	0.204644i
$91$	0.411439	+	7.11716i	0.0431306	+	0.746081i
$92$		−	0.998038i		−	0.104053i
$93$	13.3198	−	8.73127i	1.38120	−	0.905390i
$94$	−0.498040			−0.0513689
$95$	−1.70237			−0.174660
$96$	−0.949550	−	1.44857i	−0.0969131	−	0.147844i
$97$	−3.58856	−	3.58856i	−0.364363	−	0.364363i	0.501053	−	0.865416i	$-0.332946\pi$
$97$	−3.58856	−	3.58856i	−0.364363	−	0.364363i	−0.865416	+	0.501053i	$0.832946\pi$
$98$	−2.18533	−	2.18533i	−0.220751	−	0.220751i
$99$	−5.27124	−	13.3884i	−0.529780	−	1.34558i
$100$	4.50196			0.450196
$101$	5.08053			0.505532			0.252766	−	0.967527i	$-0.418660\pi$
$101$	5.08053			0.505532			0.252766	+	0.967527i	$0.418660\pi$
$102$	−4.16804	−	6.35849i	−0.412698	−	0.629584i
$103$		−	12.2077i		−	1.20286i	−0.798926	−	0.601429i	$-0.794598\pi$
$103$		−	12.2077i		−	1.20286i	0.798926	−	0.601429i	$-0.205402\pi$
$104$	−3.59954	+	0.208088i	−0.352964	+	0.0204047i
$105$	−2.36618	−	0.492373i	−0.230915	−	0.0480506i
$106$	10.0905	+	10.0905i	0.980078	+	0.980078i
$107$		−	6.78291i		−	0.655728i	−0.944725	−	0.327864i	$-0.893671\pi$
$107$		−	6.78291i		−	0.655728i	0.944725	−	0.327864i	$-0.106329\pi$
$108$	4.24264	+	3.00000i	0.408248	+	0.288675i
$109$	3.30760	+	3.30760i	0.316811	+	0.316811i	0.847541	−	0.530730i	$-0.178082\pi$
$109$	3.30760	+	3.30760i	0.316811	+	0.316811i	−0.530730	+	0.847541i	$0.678082\pi$
$110$	−2.39342	+	2.39342i	−0.228203	+	0.228203i
$111$	−5.51091	−	8.40707i	−0.523073	−	0.797964i
$112$	−1.39812	+	1.39812i	−0.132110	+	0.132110i
$113$			13.2721i			1.24854i	0.781211	+	0.624268i	$0.214603\pi$
$113$			13.2721i			1.24854i	−0.781211	+	0.624268i	$0.785397\pi$
$114$	0.851187	−	4.09052i	0.0797209	−	0.383112i
$115$	0.498040	−	0.498040i	0.0464425	−	0.0464425i
$116$	−0.998038			−0.0926655
$117$	9.81920	−	4.53689i	0.907785	−	0.419436i
$118$	1.41144			0.129934
$119$	−6.13704	+	6.13704i	−0.562581	+	0.562581i
$120$	0.249020	−	1.19671i	0.0227323	−	0.109244i
$121$			12.0039i			1.09127i
$122$	6.78291	−	6.78291i	0.614096	−	0.614096i
$123$	7.03754	+	10.7360i	0.634553	+	0.968031i
$124$	6.50196	−	6.50196i	0.583893	−	0.583893i
$125$	4.74166	+	4.74166i	0.424107	+	0.424107i
$126$	2.36618	−	5.43935i	0.210796	−	0.484576i
$127$		−	11.6191i		−	1.03103i	−0.856881	−	0.515515i	$-0.827601\pi$
$127$		−	11.6191i		−	1.03103i	0.856881	−	0.515515i	$-0.172399\pi$
$128$	−0.707107	−	0.707107i	−0.0625000	−	0.0625000i
$129$	−15.0695	−	3.13578i	−1.32680	−	0.276090i
$130$	−1.90008	−	1.69240i	−0.166648	−	0.148433i
$131$		−	7.08990i		−	0.619448i	−0.950827	−	0.309724i	$-0.899763\pi$
$131$		−	7.08990i		−	0.619448i	0.950827	−	0.309724i	$-0.100237\pi$
$132$	−4.55427	−	6.94769i	−0.396399	−	0.604719i
$133$	−4.76960			−0.413577
$134$	8.19712			0.708123
$135$	0.620101	+	3.61422i	0.0533698	+	0.311062i
$136$	−3.10384	−	3.10384i	−0.266152	−	0.266152i
$137$	15.4150	+	15.4150i	1.31700	+	1.31700i	0.916142	+	0.400854i	$0.131286\pi$
$137$	15.4150	+	15.4150i	1.31700	+	1.31700i	0.400854	+	0.916142i	$0.368714\pi$
$138$	0.947688	+	1.44573i	0.0806725	+	0.123069i
$139$	10.8868			0.923403			0.461701	−	0.887035i	$-0.347239\pi$
$139$	10.8868			0.923403			0.461701	+	0.887035i	$0.347239\pi$
$140$	−1.39538			−0.117931
$141$	0.721446	−	0.472914i	0.0607567	−	0.0398266i
$142$			10.0905i			0.846778i
$143$	−17.2643	+	0.998038i	−1.44371	+	0.0834602i
$144$	2.75098	+	1.19671i	0.229248	+	0.0997257i
$145$	−0.498040	−	0.498040i	−0.0413600	−	0.0413600i
$146$		−	2.41225i		−	0.199639i
$147$	5.24068	+	1.09052i	0.432244	+	0.0899446i
$148$	−4.10384	−	4.10384i	−0.337334	−	0.337334i
$149$	−2.54027	+	2.54027i	−0.208107	+	0.208107i	−0.803462	−	0.595356i	$-0.797011\pi$
$149$	−2.54027	+	2.54027i	−0.208107	+	0.208107i	0.595356	+	0.803462i	$0.297011\pi$
$150$	−6.52140	+	4.27484i	−0.532470	+	0.349039i
$151$	−4.01332	+	4.01332i	−0.326599	+	0.326599i	−0.851292	−	0.524693i	$-0.824180\pi$
$151$	−4.01332	+	4.01332i	−0.326599	+	0.326599i	0.524693	+	0.851292i	$0.324180\pi$
$152$		−	2.41225i		−	0.195659i
$153$	12.0754	+	5.25294i	0.976239	+	0.424675i
$154$	−6.70572	+	6.70572i	−0.540362	+	0.540362i
$155$	6.48920			0.521225
$156$	5.01660	−	3.71938i	0.401649	−	0.297788i
$157$	4.76960			0.380656			0.190328	−	0.981721i	$-0.439045\pi$
$157$	4.76960			0.380656			0.190328	+	0.981721i	$0.439045\pi$
$158$	0.146852	−	0.146852i	0.0116829	−	0.0116829i
$159$	−24.1983	−	5.03536i	−1.91905	−	0.399330i
$160$		−	0.705720i		−	0.0557920i
$161$	1.39538	−	1.39538i	0.109971	−	0.109971i
$162$	−8.99441	−	0.317107i	−0.706668	−	0.0249143i
$163$	−11.4154	+	11.4154i	−0.894120	+	0.894120i	−0.994908	−	0.100788i	$-0.967864\pi$
$163$	−11.4154	+	11.4154i	−0.894120	+	0.894120i	0.100788	+	0.994908i	$0.467864\pi$
$164$	5.24068	+	5.24068i	0.409228	+	0.409228i
$165$	1.19436	−	5.73970i	0.0929808	−	0.446835i
$166$		−	12.9773i		−	1.00723i
$167$	7.78095	+	7.78095i	0.602108	+	0.602108i	0.940871	−	0.338764i	$-0.110009\pi$
$167$	7.78095	+	7.78095i	0.602108	+	0.602108i	−0.338764	+	0.940871i	$0.610009\pi$
$168$	0.697689	−	3.35286i	0.0538279	−	0.258679i
$169$	−1.49804	−	12.9134i	−0.115234	−	0.993338i
$170$		−	3.09775i		−	0.237587i
$171$	2.65115	+	6.73365i	0.202739	+	0.514935i
$172$	−8.88676			−0.677609
$173$	−19.3507			−1.47121			−0.735603	−	0.677413i	$-0.763101\pi$
$173$	−19.3507			−1.47121			−0.735603	+	0.677413i	$0.763101\pi$
$174$	1.44573	−	0.947688i	0.109600	−	0.0718440i
$175$	6.29428	+	6.29428i	0.475803	+	0.475803i
$176$	−3.39145	−	3.39145i	−0.255640	−	0.255640i
$177$	−2.04457	+	1.34023i	−0.153679	+	0.100738i
$178$	3.59248			0.269268
$179$	−7.08990			−0.529924			−0.264962	−	0.964259i	$-0.585359\pi$
$179$	−7.08990			−0.529924			−0.264962	+	0.964259i	$0.585359\pi$
$180$	0.775612	+	1.96997i	0.0578107	+	0.146833i
$181$		−	6.20768i		−	0.461413i	−0.973023	−	0.230707i	$-0.925896\pi$
$181$		−	6.20768i		−	0.461413i	0.973023	−	0.230707i	$-0.0741038\pi$
$182$	−5.32352	−	4.74166i	−0.394606	−	0.351475i
$183$	−3.38480	+	16.2662i	−0.250212	+	1.20243i
$184$	0.705720	+	0.705720i	0.0520263	+	0.0520263i
$185$		−	4.09579i		−	0.301128i
$186$	−3.24460	+	15.5925i	−0.237906	+	1.14330i
$187$	−14.8868	−	14.8868i	−1.08863	−	1.08863i
$188$	0.352168	−	0.352168i	0.0256845	−	0.0256845i
$189$	1.73736	+	10.1261i	0.126374	+	0.736564i
$190$	1.20376	−	1.20376i	0.0873299	−	0.0873299i
$191$		−	9.77702i		−	0.707441i	−0.935351	−	0.353720i	$-0.884916\pi$
$191$		−	9.77702i		−	0.707441i	0.935351	−	0.353720i	$-0.115084\pi$
$192$	1.69573	+	0.352860i	0.122379	+	0.0254655i
$193$	3.79624	−	3.79624i	0.273259	−	0.273259i	−0.557152	−	0.830411i	$-0.688106\pi$
$193$	3.79624	−	3.79624i	0.273259	−	0.273259i	0.830411	+	0.557152i	$0.188106\pi$
$194$	5.07499			0.364363
$195$	4.35942	+	0.647338i	0.312184	+	0.0463568i
$196$	3.09052			0.220751
$197$	7.28193	−	7.28193i	0.518816	−	0.518816i	−0.398397	−	0.917213i	$-0.630433\pi$
$197$	7.28193	−	7.28193i	0.518816	−	0.518816i	0.917213	+	0.398397i	$0.130433\pi$
$198$	13.1944	+	5.73970i	0.937682	+	0.407903i
$199$			18.1544i			1.28693i	0.765475	+	0.643466i	$0.222504\pi$
$199$			18.1544i			1.28693i	−0.765475	+	0.643466i	$0.777496\pi$
$200$	−3.18337	+	3.18337i	−0.225098	+	0.225098i
$201$	−11.8741	+	7.78358i	−0.837535	+	0.549011i
$202$	−3.59248	+	3.59248i	−0.252766	+	0.252766i
$203$	−1.39538	−	1.39538i	−0.0979363	−	0.0979363i
$204$	7.44338	+	1.54888i	0.521141	+	0.108443i
$205$			5.23040i			0.365307i
$206$	8.63213	+	8.63213i	0.601429	+	0.601429i
$207$	−2.74558	−	1.19436i	−0.190831	−	0.0830138i
$208$	2.39812	−	2.69240i	0.166280	−	0.186684i
$209$		−	11.5697i		−	0.800296i
$210$	2.02130	−	1.32498i	0.139483	−	0.0914324i
$211$	11.2943			0.777530			0.388765	−	0.921337i	$-0.372902\pi$
$211$	11.2943			0.777530			0.388765	+	0.921337i	$0.372902\pi$
$212$	−14.2702			−0.980078
$213$	−9.58146	−	14.6168i	−0.656511	−	1.00153i
$214$	4.79624	+	4.79624i	0.327864	+	0.327864i
$215$	−4.43466	−	4.43466i	−0.302442	−	0.302442i
$216$	−5.12132	+	0.878680i	−0.348462	+	0.0597866i
$217$	18.1810			1.23421
$218$	−4.67765			−0.316811
$219$	2.29055	+	3.49431i	0.154781	+	0.236124i
$220$		−	3.38480i		−	0.228203i
$221$	10.5265	−	11.8183i	0.708091	−	0.794983i
$222$	9.84150	+	2.04789i	0.660518	+	0.137446i
$223$	6.19436	+	6.19436i	0.414805	+	0.414805i	0.883409	−	0.468604i	$-0.155243\pi$
$223$	6.19436	+	6.19436i	0.414805	+	0.414805i	−0.468604	+	0.883409i	$0.655243\pi$
$224$		−	1.97724i		−	0.132110i
$225$	5.38753	−	12.3848i	0.359169	−	0.825653i
$226$	−9.38480	−	9.38480i	−0.624268	−	0.624268i
$227$	13.5658	−	13.5658i	0.900395	−	0.900395i	−0.0950753	−	0.995470i	$-0.530309\pi$
$227$	13.5658	−	13.5658i	0.900395	−	0.900395i	0.995470	+	0.0950753i	$0.0303092\pi$
$228$	2.29055	+	3.49431i	0.151696	+	0.231417i
$229$	3.30760	−	3.30760i	0.218572	−	0.218572i	−0.589324	−	0.807897i	$-0.700606\pi$
$229$	3.30760	−	3.30760i	0.218572	−	0.218572i	0.807897	+	0.589324i	$0.200606\pi$
$230$			0.704335i			0.0464425i
$231$	3.34628	−	16.0811i	0.220169	−	1.05806i
$232$	0.705720	−	0.705720i	0.0463328	−	0.0463328i
$233$	−10.8787			−0.712687			−0.356344	−	0.934355i	$-0.615977\pi$
$233$	−10.8787			−0.712687			−0.356344	+	0.934355i	$0.615977\pi$
$234$	−3.73515	+	10.1513i	−0.244175	+	0.663610i
$235$	0.351477			0.0229278
$236$	−0.998038	+	0.998038i	−0.0649668	+	0.0649668i
$237$	−0.0732817	+	0.352168i	−0.00476016	+	0.0228757i
$238$		−	8.67908i		−	0.562581i
$239$	5.13900	−	5.13900i	0.332414	−	0.332414i	−0.521088	−	0.853503i	$-0.674474\pi$
$239$	5.13900	−	5.13900i	0.332414	−	0.332414i	0.853503	+	0.521088i	$0.174474\pi$
$240$	0.670116	+	1.02228i	0.0432558	+	0.0659881i
$241$	−5.06388	+	5.06388i	−0.326193	+	0.326193i	−0.851137	−	0.524944i	$-0.824086\pi$
$241$	−5.06388	+	5.06388i	−0.326193	+	0.326193i	0.524944	+	0.851137i	$0.324086\pi$
$242$	−8.48805	−	8.48805i	−0.545633	−	0.545633i
$243$	13.3301	−	8.08130i	0.855129	−	0.518415i
$244$			9.59248i			0.614096i
$245$	1.54223	+	1.54223i	0.0985294	+	0.0985294i
$246$	−12.5678	−	2.61520i	−0.801292	−	0.166739i
$247$	8.68300	−	0.501960i	0.552486	−	0.0319389i
$248$			9.19516i			0.583893i
$249$	12.3226	+	18.7985i	0.780912	+	1.19131i
$250$	−6.70572			−0.424107
$251$	19.2603			1.21570			0.607851	−	0.794051i	$-0.292032\pi$
$251$	19.2603			1.21570			0.607851	+	0.794051i	$0.292032\pi$
$252$	2.17306	+	5.51934i	0.136890	+	0.347686i
$253$	3.38480	+	3.38480i	0.212801	+	0.212801i
$254$	8.21596	+	8.21596i	0.515515	+	0.515515i
$255$	2.94147	+	4.48731i	0.184202	+	0.281006i
$256$	1.00000			0.0625000
$257$	−18.2490			−1.13834			−0.569171	−	0.822219i	$-0.692736\pi$
$257$	−18.2490			−1.13834			−0.569171	+	0.822219i	$0.692736\pi$
$258$	12.8731	−	8.43843i	0.801444	−	0.525354i
$259$		−	11.4753i		−	0.713042i
$260$	2.54027	−	0.146852i	0.157541	−	0.00910735i
$261$	−1.19436	+	2.74558i	−0.0739290	+	0.169947i
$262$	5.01332	+	5.01332i	0.309724	+	0.309724i
$263$			3.99215i			0.246167i	0.992396	+	0.123083i	$0.0392783\pi$
$263$			3.99215i			0.246167i	−0.992396	+	0.123083i	$0.960722\pi$
$264$	8.13311	+	1.69240i	0.500559	+	0.104160i
$265$	−7.12108	−	7.12108i	−0.437444	−	0.437444i
$266$	3.37262	−	3.37262i	0.206788	−	0.206788i
$267$	−5.20396	+	3.41124i	−0.318477	+	0.208765i
$268$	−5.79624	+	5.79624i	−0.354062	+	0.354062i
$269$		−	10.4814i		−	0.639060i	−0.947576	−	0.319530i	$-0.896475\pi$
$269$		−	10.4814i		−	0.639060i	0.947576	−	0.319530i	$-0.103525\pi$
$270$	−2.99411	−	2.11716i	−0.182216	−	0.128846i
$271$	−3.37148	+	3.37148i	−0.204803	+	0.204803i	−0.802054	−	0.597251i	$-0.796260\pi$
$271$	−3.37148	+	3.37148i	−0.204803	+	0.204803i	0.597251	+	0.802054i	$0.296260\pi$
$272$	4.38949			0.266152
$273$	12.2139	+	1.81367i	0.739221	+	0.109768i
$274$	−21.8002			−1.31700
$275$	−15.2682	+	15.2682i	−0.920706	+	0.920706i
$276$	−1.69240	−	0.352168i	−0.101871	−	0.0211980i
$277$			27.2382i			1.63659i	0.574801	+	0.818294i	$0.305080\pi$
$277$			27.2382i			1.63659i	−0.574801	+	0.818294i	$0.694920\pi$
$278$	−7.69810	+	7.69810i	−0.461701	+	0.461701i
$279$	−10.1058	−	25.6677i	−0.605019	−	1.53669i
$280$	0.986681	−	0.986681i	0.0589655	−	0.0589655i
$281$	2.24656	+	2.24656i	0.134019	+	0.134019i	0.770934	−	0.636915i	$-0.219790\pi$
$281$	2.24656	+	2.24656i	0.134019	+	0.134019i	−0.636915	+	0.770934i	$0.719790\pi$
$282$	−0.175738	+	0.844540i	−0.0104651	+	0.0502916i
$283$			3.00392i			0.178564i	0.996006	+	0.0892822i	$0.0284573\pi$
$283$			3.00392i			0.178564i	−0.996006	+	0.0892822i	$0.971543\pi$
$284$	−7.13508	−	7.13508i	−0.423389	−	0.423389i
$285$	−0.600699	+	2.88676i	−0.0355824	+	0.170997i
$286$	11.5020	−	12.9134i	0.680125	−	0.763585i
$287$			14.6542i			0.865010i
$288$	−2.79144	+	1.09904i	−0.164487	+	0.0647613i
$289$	2.26764			0.133391
$290$	0.704335			0.0413600
$291$	−7.35148	+	4.81896i	−0.430951	+	0.282492i
$292$	1.70572	+	1.70572i	0.0998197	+	0.0998197i
$293$	−6.98822	−	6.98822i	−0.408256	−	0.408256i	0.472874	−	0.881130i	$-0.343217\pi$
$293$	−6.98822	−	6.98822i	−0.408256	−	0.408256i	−0.881130	+	0.472874i	$0.843217\pi$
$294$	−4.47683	+	2.93461i	−0.261094	+	0.171150i
$295$	−0.996080			−0.0579940
$296$	5.80371			0.337334
$297$	−24.5631	+	4.21436i	−1.42529	+	0.244542i
$298$		−	3.59248i		−	0.208107i
$299$	−2.39342	+	2.68712i	−0.138415	+	0.155400i
$300$	1.58856	−	7.63409i	0.0917156	−	0.440755i
$301$	−12.4248	−	12.4248i	−0.716151	−	0.716151i
$302$		−	5.67569i		−	0.326599i
$303$	1.79272	−	8.61520i	0.102989	−	0.494930i
$304$	1.70572	+	1.70572i	0.0978297	+	0.0978297i
$305$	−4.78683	+	4.78683i	−0.274093	+	0.274093i
$306$	−12.2530	+	4.82421i	−0.700457	+	0.275782i
$307$	−8.88284	+	8.88284i	−0.506971	+	0.506971i	−0.913595	−	0.406625i	$-0.866706\pi$
$307$	−8.88284	+	8.88284i	−0.506971	+	0.506971i	0.406625	+	0.913595i	$0.366706\pi$
$308$		−	9.48332i		−	0.540362i
$309$	−20.7009	−	4.30760i	−1.17763	−	0.245051i
$310$	−4.58856	+	4.58856i	−0.260613	+	0.260613i
$311$	13.2721			0.752592			0.376296	−	0.926499i	$-0.377197\pi$
$311$	13.2721			0.752592			0.376296	+	0.926499i	$0.377197\pi$
$312$	−0.917274	+	6.17727i	−0.0519304	+	0.349719i
$313$	−0.913399			−0.0516284			−0.0258142	−	0.999667i	$-0.508218\pi$
$313$	−0.913399			−0.0516284			−0.0258142	+	0.999667i	$0.508218\pi$
$314$	−3.37262	+	3.37262i	−0.190328	+	0.190328i
$315$	−1.66986	+	3.83866i	−0.0940859	+	0.216284i
$316$			0.207679i			0.0116829i
$317$	0.544191	−	0.544191i	0.0305648	−	0.0305648i	−0.691659	−	0.722224i	$-0.743120\pi$
$317$	0.544191	−	0.544191i	0.0305648	−	0.0305648i	0.722224	+	0.691659i	$0.243120\pi$
$318$	20.6713	−	13.5502i	1.15919	−	0.759859i
$319$	3.38480	−	3.38480i	0.189512	−	0.189512i
$320$	0.499019	+	0.499019i	0.0278960	+	0.0278960i
$321$	−11.5020	−	2.39342i	−0.641977	−	0.133587i
$322$			1.97336i			0.109971i
$323$	7.48724	+	7.48724i	0.416601	+	0.416601i
$324$	6.58424	−	6.13578i	0.365791	−	0.340877i
$325$	−12.1211	−	10.7962i	−0.672356	−	0.598868i
$326$		−	16.1438i		−	0.894120i
$327$	6.77590	−	4.44167i	0.374708	−	0.245625i
$328$	−7.41144			−0.409228
$329$	0.984745			0.0542908
$330$	3.21404	+	4.90312i	0.176927	+	0.269908i
$331$	−10.5925	−	10.5925i	−0.582215	−	0.582215i	0.353296	−	0.935512i	$-0.385061\pi$
$331$	−10.5925	−	10.5925i	−0.582215	−	0.582215i	−0.935512	+	0.353296i	$0.885061\pi$
$332$	9.17632	+	9.17632i	0.503616	+	0.503616i
$333$	−16.2007	+	6.37848i	−0.887792	+	0.349539i
$334$	−11.0039			−0.602108
$335$	−5.78487			−0.316061
$336$	1.87749	+	2.86417i	0.102425	+	0.156253i
$337$			13.7363i			0.748263i	0.927376	+	0.374131i	$0.122059\pi$
$337$			13.7363i			0.748263i	−0.927376	+	0.374131i	$0.877941\pi$
$338$	10.1904	+	8.07188i	0.554286	+	0.439052i
$339$	22.5059	+	4.68320i	1.22235	+	0.254356i
$340$	2.19044	+	2.19044i	0.118793	+	0.118793i
$341$			44.1022i			2.38827i
$342$	−6.63606	−	2.88676i	−0.358837	−	0.156098i
$343$	14.1078	+	14.1078i	0.761747	+	0.761747i
$344$	6.28389	−	6.28389i	0.338805	−	0.338805i
$345$	−0.668802	−	1.02028i	−0.0360071	−	0.0549300i
$346$	13.6830	−	13.6830i	0.735603	−	0.735603i
$347$			24.3542i			1.30740i	0.756754	+	0.653700i	$0.226784\pi$
$347$			24.3542i			1.30740i	−0.756754	+	0.653700i	$0.773216\pi$
$348$	−0.352168	+	1.69240i	−0.0188782	+	0.0907222i
$349$	6.48472	−	6.48472i	0.347119	−	0.347119i	−0.511916	−	0.859035i	$-0.671064\pi$
$349$	6.48472	−	6.48472i	0.347119	−	0.347119i	0.859035	+	0.511916i	$0.171064\pi$
$350$	−8.90146			−0.475803
$351$	−4.22853	−	18.2516i	−0.225702	−	0.974196i
$352$	4.79624			0.255640
$353$	8.63213	−	8.63213i	0.459442	−	0.459442i	−0.439030	−	0.898472i	$-0.644678\pi$
$353$	8.63213	−	8.63213i	0.459442	−	0.459442i	0.898472	+	0.439030i	$0.144678\pi$
$354$	0.498040	−	2.39342i	0.0264705	−	0.127209i
$355$		−	7.12108i		−	0.377948i
$356$	−2.54027	+	2.54027i	−0.134634	+	0.134634i
$357$	8.24123	+	12.5723i	0.436172	+	0.665394i
$358$	5.01332	−	5.01332i	0.264962	−	0.264962i
$359$	12.5678	+	12.5678i	0.663302	+	0.663302i	0.956157	−	0.292855i	$-0.0946053\pi$
$359$	12.5678	+	12.5678i	0.663302	+	0.663302i	−0.292855	+	0.956157i	$0.594605\pi$
$360$	−1.94142	−	0.844540i	−0.102322	−	0.0445112i
$361$		−	13.1810i		−	0.693739i
$362$	4.38949	+	4.38949i	0.230707	+	0.230707i
$363$	20.3554	+	4.23570i	1.06838	+	0.222317i
$364$	7.11716	−	0.411439i	0.373040	−	0.0215653i
$365$			1.70237i			0.0891063i
$366$	−9.10854	−	13.8954i	−0.476111	−	0.726323i
$367$	19.3848			1.01188			0.505939	−	0.862569i	$-0.331146\pi$
$367$	19.3848			1.01188			0.505939	+	0.862569i	$0.331146\pi$
$368$	−0.998038			−0.0520263
$369$	20.6886	−	8.14544i	1.07700	−	0.424035i
$370$	2.89616	+	2.89616i	0.150564	+	0.150564i
$371$	−19.9514	−	19.9514i	−1.03582	−	1.03582i
$372$	−8.73127	−	13.3198i	−0.452695	−	0.690601i
$373$	24.4154			1.26418			0.632090	−	0.774895i	$-0.282197\pi$
$373$	24.4154			1.26418			0.632090	+	0.774895i	$0.282197\pi$
$374$	21.0531			1.08863
$375$	9.71370	−	6.36742i	0.501613	−	0.328812i
$376$			0.498040i			0.0256845i
$377$	2.68712	+	2.39342i	0.138394	+	0.123267i
$378$	−8.38872	−	5.93172i	−0.431469	−	0.305095i
$379$	−2.17712	−	2.17712i	−0.111831	−	0.111831i	0.648977	−	0.760808i	$-0.275197\pi$
$379$	−2.17712	−	2.17712i	−0.111831	−	0.111831i	−0.760808	+	0.648977i	$0.775197\pi$
$380$			1.70237i			0.0873299i
$381$	−19.7029	−	4.09992i	−1.00941	−	0.210045i
$382$	6.91340	+	6.91340i	0.353720	+	0.353720i
$383$	−4.14096	+	4.14096i	−0.211593	+	0.211593i	−0.804944	−	0.593351i	$-0.797805\pi$
$383$	−4.14096	+	4.14096i	−0.211593	+	0.211593i	0.593351	+	0.804944i	$0.297805\pi$
$384$	−1.44857	+	0.949550i	−0.0739220	+	0.0484565i
$385$	4.73236	−	4.73236i	0.241183	−	0.241183i
$386$			5.36869i			0.273259i
$387$	−10.6349	+	24.4473i	−0.540600	+	1.24273i
$388$	−3.58856	+	3.58856i	−0.182182	+	0.182182i
$389$	−6.78291			−0.343907			−0.171954	−	0.985105i	$-0.555008\pi$
$389$	−6.78291			−0.343907			−0.171954	+	0.985105i	$0.555008\pi$
$390$	−3.54031	+	2.62484i	−0.179271	+	0.132914i
$391$	−4.38088			−0.221551
$392$	−2.18533	+	2.18533i	−0.110376	+	0.110376i
$393$	−12.0225	−	2.50174i	−0.606457	−	0.126196i
$394$			10.2982i			0.518816i
$395$	−0.103636	+	0.103636i	−0.00521449	+	0.00521449i
$396$	−13.3884	+	5.27124i	−0.672792	+	0.264890i
$397$	17.2716	−	17.2716i	0.866835	−	0.866835i	−0.125286	−	0.992121i	$-0.539985\pi$
$397$	17.2716	−	17.2716i	0.866835	−	0.866835i	0.992121	+	0.125286i	$0.0399848\pi$
$398$	−12.8371	−	12.8371i	−0.643466	−	0.643466i
$399$	−1.68300	+	8.08794i	−0.0842554	+	0.404904i
$400$		−	4.50196i		−	0.225098i
$401$	−22.1979	−	22.1979i	−1.10851	−	1.10851i	−0.993346	−	0.115166i	$-0.963260\pi$
$401$	−22.1979	−	22.1979i	−1.10851	−	1.10851i	−0.115166	−	0.993346i	$-0.536740\pi$
$402$	2.89243	−	13.9001i	0.144262	−	0.693273i
$403$	−33.0984	+	1.91340i	−1.64875	+	0.0953132i
$404$		−	5.08053i		−	0.252766i
$405$	6.34753	+	0.223789i	0.315411	+	0.0111202i
$406$	1.97336			0.0979363
$407$	27.8360			1.37978
$408$	−6.35849	+	4.16804i	−0.314792	+	0.206349i
$409$	3.02664	+	3.02664i	0.149658	+	0.149658i	0.777965	−	0.628307i	$-0.216252\pi$
$409$	3.02664	+	3.02664i	0.149658	+	0.149658i	−0.628307	+	0.777965i	$0.716252\pi$
$410$	−3.69845	−	3.69845i	−0.182653	−	0.182653i
$411$	31.5790	−	20.7004i	1.55768	−	1.02107i
$412$	−12.2077			−0.601429
$413$	−2.79075			−0.137324
$414$	2.78596	−	1.09688i	0.136923	−	0.0539087i
$415$			9.15832i			0.449564i
$416$	0.208088	+	3.59954i	0.0102023	+	0.176482i
$417$	3.84150	−	18.4610i	0.188119	−	0.904038i
$418$	8.18104	+	8.18104i	0.400148	+	0.400148i
$419$		−	34.2215i		−	1.67183i	−0.548858	−	0.835916i	$-0.684937\pi$
$419$		−	34.2215i		−	1.67183i	0.548858	−	0.835916i	$-0.315063\pi$
$420$	−0.492373	+	2.36618i	−0.0240253	+	0.115458i
$421$	2.87344	+	2.87344i	0.140043	+	0.140043i	0.773653	−	0.633610i	$-0.218428\pi$
$421$	2.87344	+	2.87344i	0.140043	+	0.140043i	−0.633610	+	0.773653i	$0.718428\pi$
$422$	−7.98626	+	7.98626i	−0.388765	+	0.388765i
$423$	−0.547364	−	1.39025i	−0.0266138	−	0.0675962i
$424$	10.0905	−	10.0905i	0.490039	−	0.490039i
$425$		−	19.7613i		−	0.958565i
$426$	17.1108	+	3.56054i	0.829019	+	0.172509i
$427$	−13.4114	+	13.4114i	−0.649025	+	0.649025i
$428$	−6.78291			−0.327864
$429$	−4.39947	+	29.6276i	−0.212408	+	1.43044i
$430$	6.27156			0.302442
$431$	6.84137	−	6.84137i	0.329537	−	0.329537i	−0.522873	−	0.852410i	$-0.675140\pi$
$431$	6.84137	−	6.84137i	0.329537	−	0.329537i	0.852410	+	0.522873i	$0.175140\pi$
$432$	3.00000	−	4.24264i	0.144338	−	0.204124i
$433$		−	29.2833i		−	1.40727i	−0.710563	−	0.703633i	$-0.751560\pi$
$433$		−	29.2833i		−	1.40727i	0.710563	−	0.703633i	$-0.248440\pi$
$434$	−12.8559	+	12.8559i	−0.617105	+	0.617105i
$435$	−1.02028	+	0.668802i	−0.0489186	+	0.0320666i
$436$	3.30760	−	3.30760i	0.158405	−	0.158405i
$437$	−1.70237	−	1.70237i	−0.0814356	−	0.0814356i
$438$	−4.09052	−	0.851187i	−0.195453	−	0.0406713i
$439$			18.0000i			0.859093i	0.903045	+	0.429547i	$0.141327\pi$
$439$			18.0000i			0.859093i	−0.903045	+	0.429547i	$0.858673\pi$
$440$	2.39342	+	2.39342i	0.114102	+	0.114102i
$441$	3.69845	−	8.50196i	0.176117	−	0.404855i
$442$	0.913399	+	15.8002i	0.0434460	+	0.751537i
$443$			26.6439i			1.26589i	0.774196	+	0.632946i	$0.218155\pi$
$443$			26.6439i			1.26589i	−0.774196	+	0.632946i	$0.781845\pi$
$444$	−8.40707	+	5.51091i	−0.398982	+	0.261536i
$445$	−2.53528			−0.120184
$446$	−8.76015			−0.414805
$447$	3.41124	+	5.20396i	0.161346	+	0.246139i
$448$	1.39812	+	1.39812i	0.0660550	+	0.0660550i
$449$	25.9867	+	25.9867i	1.22639	+	1.22639i	0.965320	+	0.261070i	$0.0840755\pi$
$449$	25.9867	+	25.9867i	1.22639	+	1.22639i	0.261070	+	0.965320i	$0.415925\pi$
$450$	4.94782	+	12.5669i	0.233242	+	0.592411i
$451$	−35.5470			−1.67384
$452$	13.2721			0.624268
$453$	5.38935	+	8.22163i	0.253214	+	0.386286i
$454$			19.1850i			0.900395i
$455$	3.75691	+	3.34628i	0.176127	+	0.156876i
$456$	−4.09052	−	0.851187i	−0.191556	−	0.0398605i
$457$	−18.8907	−	18.8907i	−0.883669	−	0.883669i	0.110237	−	0.993905i	$-0.464839\pi$
$457$	−18.8907	−	18.8907i	−0.883669	−	0.883669i	−0.993905	+	0.110237i	$0.964839\pi$
$458$			4.67765i			0.218572i
$459$	13.1685	−	18.6230i	0.614652	−	0.869249i
$460$	−0.498040	−	0.498040i	−0.0232212	−	0.0232212i
$461$	−24.8399	+	24.8399i	−1.15691	+	1.15691i	−0.171773	+	0.985137i	$0.554949\pi$
$461$	−24.8399	+	24.8399i	−1.15691	+	1.15691i	−0.985137	+	0.171773i	$0.945051\pi$
$462$	9.00489	+	13.7372i	0.418945	+	0.639115i
$463$	−14.0944	+	14.0944i	−0.655024	+	0.655024i	−0.954198	−	0.299174i	$-0.903289\pi$
$463$	−14.0944	+	14.0944i	−0.655024	+	0.655024i	0.299174	+	0.954198i	$0.403289\pi$
$464$			0.998038i			0.0463328i
$465$	2.28978	−	11.0039i	0.106186	−	0.510295i
$466$	7.69240	−	7.69240i	0.356344	−	0.356344i
$467$	−16.6769			−0.771713			−0.385857	−	0.922559i	$-0.626094\pi$
$467$	−16.6769			−0.771713			−0.385857	+	0.922559i	$0.626094\pi$
$468$	−4.53689	−	9.81920i	−0.209718	−	0.453893i
$469$	−16.2077			−0.748401
$470$	−0.248532	+	0.248532i	−0.0114639	+	0.0114639i
$471$	1.68300	−	8.08794i	0.0775486	−	0.372673i
$472$		−	1.41144i		−	0.0649668i
$473$	30.1390	−	30.1390i	1.38579	−	1.38579i
$474$	−0.197202	−	0.300838i	−0.00905779	−	0.0138180i
$475$	7.67908	−	7.67908i	0.352340	−	0.352340i
$476$	6.13704	+	6.13704i	0.281291	+	0.281291i
$477$	−17.0772	+	39.2569i	−0.781911	+	1.79745i
$478$			7.26764i			0.332414i
$479$	−26.4858	−	26.4858i	−1.21017	−	1.21017i	−0.970972	−	0.239193i	$-0.923117\pi$
$479$	−26.4858	−	26.4858i	−1.21017	−	1.21017i	−0.239193	−	0.970972i	$-0.576883\pi$
$480$	−1.19671	−	0.249020i	−0.0546220	−	0.0113662i
$481$	1.20768	+	20.8907i	0.0550654	+	0.952533i
$482$		−	7.16141i		−	0.326193i
$483$	−1.87381	−	2.85855i	−0.0852611	−	0.130069i
$484$	12.0039			0.545633
$485$	−3.58152			−0.162628
$486$	−3.71149	+	15.1402i	−0.168357	+	0.686772i
$487$	3.67908	+	3.67908i	0.166715	+	0.166715i	0.785534	−	0.618819i	$-0.212388\pi$
$487$	3.67908	+	3.67908i	0.166715	+	0.166715i	−0.618819	+	0.785534i	$0.712388\pi$
$488$	−6.78291	−	6.78291i	−0.307048	−	0.307048i
$489$	15.3293	+	23.3854i	0.693215	+	1.05752i
$490$	−2.18104			−0.0985294
$491$	40.7107			1.83725			0.918625	−	0.395131i	$-0.129301\pi$
$491$	40.7107			1.83725			0.918625	+	0.395131i	$0.129301\pi$
$492$	10.7360	−	7.03754i	0.484016	−	0.317277i
$493$			4.38088i			0.197305i
$494$	−5.78487	+	6.49475i	−0.260274	+	0.292213i
$495$	−9.31152	−	4.05062i	−0.418522	−	0.182062i
$496$	−6.50196	−	6.50196i	−0.291947	−	0.291947i
$497$		−	19.9514i		−	0.894942i
$498$	−22.0059	−	4.57916i	−0.986109	−	0.205197i
$499$	−10.5020	−	10.5020i	−0.470132	−	0.470132i	0.431825	−	0.901957i	$-0.357870\pi$
$499$	−10.5020	−	10.5020i	−0.470132	−	0.470132i	−0.901957	+	0.431825i	$0.857870\pi$
$500$	4.74166	−	4.74166i	0.212053	−	0.212053i
$501$	15.9399	−	10.4488i	0.712144	−	0.466817i
$502$	−13.6191	+	13.6191i	−0.607851	+	0.607851i
$503$			1.38208i			0.0616241i	0.999525	+	0.0308120i	$0.00980933\pi$
$503$			1.38208i			0.0616241i	−0.999525	+	0.0308120i	$0.990191\pi$
$504$	−5.43935	−	2.36618i	−0.242288	−	0.105398i
$505$	2.53528	−	2.53528i	0.112819	−	0.112819i
$506$	−4.78683			−0.212801
$507$	−22.4262	−	2.01635i	−0.995982	−	0.0895494i
$508$	−11.6191			−0.515515
$509$	7.32710	−	7.32710i	0.324768	−	0.324768i	−0.525825	−	0.850593i	$-0.676243\pi$
$509$	7.32710	−	7.32710i	0.324768	−	0.324768i	0.850593	+	0.525825i	$0.176243\pi$
$510$	−5.25294	−	1.09307i	−0.232604	−	0.0484020i
$511$			4.76960i			0.210995i
$512$	−0.707107	+	0.707107i	−0.0312500	+	0.0312500i
$513$	12.3539	−	2.11960i	0.545439	−	0.0935825i
$514$	12.9040	−	12.9040i	0.569171	−	0.569171i
$515$	−6.09187	−	6.09187i	−0.268440	−	0.268440i
$516$	−3.13578	+	15.0695i	−0.138045	+	0.663399i
$517$			2.38872i			0.105056i
$518$	8.11428	+	8.11428i	0.356521	+	0.356521i
$519$	−6.82808	+	32.8135i	−0.299719	+	1.44035i
$520$	−1.69240	+	1.90008i	−0.0742167	+	0.0833240i
$521$			2.09971i			0.0919901i	0.998942	+	0.0459950i	$0.0146458\pi$
$521$			2.09971i			0.0919901i	−0.998942	+	0.0459950i	$0.985354\pi$
$522$	−1.09688	−	2.78596i	−0.0480091	−	0.121938i
$523$	−2.76960			−0.121106			−0.0605531	−	0.998165i	$-0.519286\pi$
$523$	−2.76960			−0.121106			−0.0605531	+	0.998165i	$0.519286\pi$
$524$	−7.08990			−0.309724
$525$	12.8944	−	8.45238i	0.562757	−	0.368892i
$526$	−2.82288	−	2.82288i	−0.123083	−	0.123083i
$527$	−28.5403	−	28.5403i	−1.24324	−	1.24324i
$528$	−6.94769	+	4.55427i	−0.302359	+	0.198199i
$529$	−22.0039			−0.956692
$530$	10.0707			0.437444
$531$	1.55122	+	3.93994i	0.0673173	+	0.170979i
$532$			4.76960i			0.206788i
$533$	−1.54223	−	26.6778i	−0.0668013	−	1.15554i
$534$	1.26764	−	6.09187i	0.0548562	−	0.263621i
$535$	−3.38480	−	3.38480i	−0.146338	−	0.146338i
$536$		−	8.19712i		−	0.354062i
$537$	−2.50174	+	12.0225i	−0.107958	+	0.518811i
$538$	7.41144	+	7.41144i	0.319530	+	0.319530i
$539$	−10.4814	+	10.4814i	−0.451464	+	0.451464i
$540$	3.61422	−	0.620101i	0.155531	−	0.0266849i
$541$	23.0545	−	23.0545i	0.991190	−	0.991190i	−0.00877191	−	0.999962i	$-0.502792\pi$
$541$	23.0545	−	23.0545i	0.991190	−	0.991190i	0.999962	+	0.00877191i	$0.00279222\pi$
$542$		−	4.76800i		−	0.204803i
$543$	−10.5265	−	2.19044i	−0.451737	−	0.0940008i
$544$	−3.10384	+	3.10384i	−0.133076	+	0.133076i
$545$	3.30111			0.141404
$546$	−9.91902	+	7.35410i	−0.424495	+	0.314726i
$547$	28.1172			1.20220			0.601101	−	0.799173i	$-0.294729\pi$
$547$	28.1172			1.20220			0.601101	+	0.799173i	$0.294729\pi$
$548$	15.4150	−	15.4150i	0.658498	−	0.658498i
$549$	26.3887	+	11.4794i	1.12624	+	0.489929i
$550$		−	21.5925i		−	0.920706i
$551$	−1.70237	+	1.70237i	−0.0725235	+	0.0725235i
$552$	1.44573	−	0.947688i	0.0615343	−	0.0403363i
$553$	−0.290361	+	0.290361i	−0.0123474	+	0.0123474i
$554$	−19.2603	−	19.2603i	−0.818294	−	0.818294i
$555$	−6.94534	−	1.44524i	−0.294813	−	0.0613470i
$556$		−	10.8868i		−	0.461701i
$557$	14.0648	+	14.0648i	0.595946	+	0.595946i	0.939231	−	0.343285i	$-0.111540\pi$
$557$	14.0648	+	14.0648i	0.595946	+	0.595946i	−0.343285	+	0.939231i	$0.611540\pi$
$558$	25.2957	+	11.0039i	1.07085	+	0.465833i
$559$	23.9267	+	21.3115i	1.01199	+	0.901381i
$560$			1.39538i			0.0589655i
$561$	−30.4968	+	19.9909i	−1.28758	+	0.844018i
$562$	−3.17712			−0.134019
$563$	−28.6400			−1.20703			−0.603517	−	0.797350i	$-0.706234\pi$
$563$	−28.6400			−1.20703			−0.603517	+	0.797350i	$0.706234\pi$
$564$	−0.472914	−	0.721446i	−0.0199133	−	0.0303784i
$565$	6.62304	+	6.62304i	0.278633	+	0.278633i
$566$	−2.12409	−	2.12409i	−0.0892822	−	0.0892822i
$567$	17.7841	+	0.626998i	0.746863	+	0.0263314i
$568$	10.0905			0.423389
$569$	−3.48180			−0.145965			−0.0729823	−	0.997333i	$-0.523252\pi$
$569$	−3.48180			−0.145965			−0.0729823	+	0.997333i	$0.523252\pi$
$570$	−1.61649	−	2.46601i	−0.0677073	−	0.103290i
$571$		−	5.34756i		−	0.223788i	−0.993720	−	0.111894i	$-0.964308\pi$
$571$		−	5.34756i		−	0.223788i	0.993720	−	0.111894i	$-0.0356918\pi$
$572$	0.998038	+	17.2643i	0.0417301	+	0.721855i
$573$	−16.5792	−	3.44992i	−0.692604	−	0.144122i
$574$	−10.3621	−	10.3621i	−0.432505	−	0.432505i
$575$			4.49313i			0.187376i
$576$	1.19671	−	2.75098i	0.0498628	−	0.114624i
$577$	26.4753	+	26.4753i	1.10218	+	1.10218i	0.994147	+	0.108035i	$0.0344558\pi$
$577$	26.4753	+	26.4753i	1.10218	+	1.10218i	0.108035	+	0.994147i	$0.465544\pi$
$578$	−1.60347	+	1.60347i	−0.0666954	+	0.0666954i
$579$	−5.09785	−	7.77693i	−0.211859	−	0.323198i
$580$	−0.498040	+	0.498040i	−0.0206800	+	0.0206800i
$581$			25.6592i			1.06452i
$582$	1.79076	−	8.60580i	0.0742294	−	0.356722i
$583$	48.3966	−	48.3966i	2.00438	−	2.00438i
$584$	−2.41225			−0.0998197
$585$	2.63597	−	7.16396i	0.108984	−	0.296193i
$586$	9.88284			0.408256
$587$	19.3640	−	19.3640i	0.799237	−	0.799237i	−0.183738	−	0.982975i	$-0.558820\pi$
$587$	19.3640	−	19.3640i	0.799237	−	0.799237i	0.982975	+	0.183738i	$0.0588199\pi$
$588$	1.09052	−	5.24068i	0.0449723	−	0.216122i
$589$		−	22.1810i		−	0.913954i
$590$	0.704335	−	0.704335i	0.0289970	−	0.0289970i
$591$	−9.77866	−	14.9177i	−0.402240	−	0.613631i
$592$	−4.10384	+	4.10384i	−0.168667	+	0.168667i
$593$	3.14097	+	3.14097i	0.128984	+	0.128984i	0.768652	−	0.639668i	$-0.220928\pi$
$593$	3.14097	+	3.14097i	0.128984	+	0.128984i	−0.639668	+	0.768652i	$0.720928\pi$
$594$	14.3887	−	20.3487i	0.590376	−	0.834918i
$595$			6.12500i			0.251100i
$596$	2.54027	+	2.54027i	0.104053	+	0.104053i
$597$	30.7849	+	6.40596i	1.25994	+	0.262178i
$598$	−0.207679	−	3.59248i	−0.00849264	−	0.146907i
$599$			38.7280i			1.58238i	0.611570	+	0.791191i	$0.290538\pi$
$599$			38.7280i			1.58238i	−0.611570	+	0.791191i	$0.709462\pi$
$600$	4.27484	+	6.52140i	0.174520	+	0.266235i
$601$	16.2794			0.664051			0.332025	−	0.943270i	$-0.392268\pi$
$601$	16.2794			0.664051			0.332025	+	0.943270i	$0.392268\pi$
$602$	17.5713			0.716151
$603$	9.00893	+	22.8817i	0.366872	+	0.931817i
$604$	4.01332	+	4.01332i	0.163300	+	0.163300i
$605$	5.99019	+	5.99019i	0.243536	+	0.243536i
$606$	4.82422	+	7.35951i	0.195971	+	0.298960i
$607$	−29.0118			−1.17755			−0.588775	−	0.808297i	$-0.700390\pi$
$607$	−29.0118			−1.17755			−0.588775	+	0.808297i	$0.700390\pi$
$608$	−2.41225			−0.0978297
$609$	−2.85855	+	1.87381i	−0.115834	+	0.0759305i
$610$		−	6.76960i		−	0.274093i
$611$	−1.79272	+	0.103636i	−0.0725255	+	0.00419266i
$612$	5.25294	−	12.0754i	0.212338	−	0.488119i
$613$	−9.94004	−	9.94004i	−0.401474	−	0.401474i	0.477278	−	0.878752i	$-0.341623\pi$
$613$	−9.94004	−	9.94004i	−0.401474	−	0.401474i	−0.878752	+	0.477278i	$0.841623\pi$
$614$		−	12.5622i		−	0.506971i
$615$	8.86933	+	1.84560i	0.357646	+	0.0744217i
$616$	6.70572	+	6.70572i	0.270181	+	0.270181i
$617$	−2.15622	+	2.15622i	−0.0868062	+	0.0868062i	−0.749177	−	0.662370i	$-0.769551\pi$
$617$	−2.15622	+	2.15622i	−0.0868062	+	0.0868062i	0.662370	+	0.749177i	$0.269551\pi$
$618$	17.6837	−	11.5918i	0.711342	−	0.466291i
$619$	0.683001	−	0.683001i	0.0274521	−	0.0274521i	−0.693247	−	0.720700i	$-0.743821\pi$
$619$	0.683001	−	0.683001i	0.0274521	−	0.0274521i	0.720700	+	0.693247i	$0.243821\pi$
$620$		−	6.48920i		−	0.260613i
$621$	−2.99411	+	4.23432i	−0.120150	+	0.169917i
$622$	−9.38480	+	9.38480i	−0.376296	+	0.376296i
$623$	−7.10320			−0.284584
$624$	−3.71938	−	5.01660i	−0.148894	−	0.200825i
$625$	−17.7774			−0.711098
$626$	0.645871	−	0.645871i	0.0258142	−	0.0258142i
$627$	−19.6191	−	4.08250i	−0.783512	−	0.163039i
$628$		−	4.76960i		−	0.190328i
$629$	−18.0138	+	18.0138i	−0.718256	+	0.718256i
$630$	−1.53357	−	3.89511i	−0.0610989	−	0.155185i
$631$	−7.98668	+	7.98668i	−0.317945	+	0.317945i	−0.847977	−	0.530033i	$-0.822180\pi$
$631$	−7.98668	+	7.98668i	−0.317945	+	0.317945i	0.530033	+	0.847977i	$0.322180\pi$
$632$	−0.146852	−	0.146852i	−0.00584144	−	0.00584144i
$633$	3.98530	−	19.1520i	0.158401	−	0.761224i
$634$			0.769602i			0.0305648i
$635$	−5.79816	−	5.79816i	−0.230093	−	0.230093i
$636$	−5.03536	+	24.1983i	−0.199665	+	0.959524i
$637$	−8.32092	−	7.41144i	−0.329687	−	0.293652i
$638$			4.78683i			0.189512i
$639$	−28.1670	+	11.0898i	−1.11427	+	0.438708i
$640$	−0.705720			−0.0278960
$641$	29.7417			1.17473			0.587363	−	0.809323i	$-0.300166\pi$
$641$	29.7417			1.17473			0.587363	+	0.809323i	$0.300166\pi$
$642$	9.82551	−	6.44071i	0.387782	−	0.254195i
$643$	13.3887	+	13.3887i	0.528000	+	0.528000i	0.919975	−	0.391976i	$-0.128208\pi$
$643$	13.3887	+	13.3887i	0.528000	+	0.528000i	−0.391976	+	0.919975i	$0.628208\pi$
$644$	−1.39538	−	1.39538i	−0.0549856	−	0.0549856i
$645$	−9.08479	+	5.95516i	−0.357713	+	0.234484i
$646$	−10.5886			−0.416601
$647$	−8.77898			−0.345137			−0.172569	−	0.984997i	$-0.555207\pi$
$647$	−8.77898			−0.345137			−0.172569	+	0.984997i	$0.555207\pi$
$648$	−0.317107	+	8.99441i	−0.0124572	+	0.353334i
$649$		−	6.76960i		−	0.265730i
$650$	16.2050	−	0.936802i	0.635612	−	0.0367444i
$651$	6.41536	−	30.8301i	0.251438	−	1.20833i
$652$	11.4154	+	11.4154i	0.447060	+	0.447060i
$653$			30.2161i			1.18245i	0.806508	+	0.591223i	$0.201355\pi$
$653$			30.2161i			1.18245i	−0.806508	+	0.591223i	$0.798645\pi$
$654$	−1.65056	+	7.93202i	−0.0645418	+	0.310167i
$655$	−3.53800	−	3.53800i	−0.138241	−	0.138241i
$656$	5.24068	−	5.24068i	0.204614	−	0.204614i
$657$	6.73365	−	2.65115i	0.262705	−	0.103431i
$658$	−0.696320	+	0.696320i	−0.0271454	+	0.0271454i
$659$		−	5.10712i		−	0.198945i	−0.995040	−	0.0994726i	$-0.968284\pi$
$659$		−	5.10712i		−	0.198945i	0.995040	−	0.0994726i	$-0.0317156\pi$
$660$	−5.73970	−	1.19436i	−0.223417	−	0.0464904i
$661$	−30.2488	+	30.2488i	−1.17654	+	1.17654i	−0.195925	+	0.980619i	$0.562771\pi$
$661$	−30.2488	+	30.2488i	−1.17654	+	1.17654i	−0.980619	+	0.195925i	$0.937229\pi$
$662$	14.9800			0.582215
$663$	−16.3262	−	22.0203i	−0.634056	−	0.855198i
$664$	−12.9773			−0.503616
$665$	−2.38012	+	2.38012i	−0.0922972	+	0.0922972i
$666$	6.94534	−	15.9659i	0.269126	−	0.618665i
$667$		−	0.996080i		−	0.0385684i
$668$	7.78095	−	7.78095i	0.301054	−	0.301054i
$669$	12.6897	−	8.31820i	0.490612	−	0.321600i
$670$	4.09052	−	4.09052i	0.158031	−	0.158031i
$671$	−32.5325	−	32.5325i	−1.25590	−	1.25590i
$672$	−3.35286	−	0.697689i	−0.129339	−	0.0269139i
$673$			40.8601i			1.57504i	0.616288	+	0.787521i	$0.288636\pi$
$673$			40.8601i			1.57504i	−0.616288	+	0.787521i	$0.711364\pi$
$674$	−9.71302	−	9.71302i	−0.374131	−	0.374131i
$675$	−19.1002	−	13.5059i	−0.735167	−	0.519842i
$676$	−12.9134	+	1.49804i	−0.496669	+	0.0576169i
$677$			25.8399i			0.993108i	0.868006	+	0.496554i	$0.165401\pi$
$677$			25.8399i			0.993108i	−0.868006	+	0.496554i	$0.834599\pi$
$678$	−19.2256	+	12.6025i	−0.738354	+	0.483998i
$679$	−10.0345			−0.385088
$680$	−3.09775			−0.118793
$681$	−18.2171	−	27.7908i	−0.698080	−	1.06494i
$682$	−31.1850	−	31.1850i	−1.19413	−	1.19413i
$683$	2.09971	+	2.09971i	0.0803433	+	0.0803433i	0.746136	−	0.665793i	$-0.231907\pi$
$683$	2.09971	+	2.09971i	0.0803433	+	0.0803433i	−0.665793	+	0.746136i	$0.731907\pi$
$684$	6.73365	−	2.65115i	0.257467	−	0.101369i
$685$	15.3848			0.587823
$686$	−19.9514			−0.761747
$687$	−4.44167	−	6.77590i	−0.169460	−	0.258517i
$688$			8.88676i			0.338805i
$689$	38.4210	+	34.2215i	1.46372	+	1.30374i
$690$	1.19436	+	0.248532i	0.0454685	+	0.00946144i
$691$	24.5020	+	24.5020i	0.932098	+	0.932098i	0.997837	−	0.0657384i	$-0.0209403\pi$
$691$	24.5020	+	24.5020i	0.932098	+	0.932098i	−0.0657384	+	0.997837i	$0.520940\pi$
$692$			19.3507i			0.735603i
$693$	−26.0884	−	11.3488i	−0.991017	−	0.431104i
$694$	−17.2210	−	17.2210i	−0.653700	−	0.653700i
$695$	5.43270	−	5.43270i	0.206074	−	0.206074i
$696$	−0.947688	−	1.44573i	−0.0359220	−	0.0548002i
$697$	23.0039	−	23.0039i	0.871336	−	0.871336i
$698$			9.17078i			0.347119i
$699$	−3.83866	+	18.4473i	−0.145191	+	0.697741i
$700$	6.29428	−	6.29428i	0.237901	−	0.237901i
$701$	−19.6710			−0.742962			−0.371481	−	0.928440i	$-0.621150\pi$
$701$	−19.6710			−0.742962			−0.371481	+	0.928440i	$0.621150\pi$
$702$	15.8958	+	9.91578i	0.599949	+	0.374247i
$703$	−14.0000			−0.528020
$704$	−3.39145	+	3.39145i	−0.127820	+	0.127820i
$705$	0.124022	−	0.596009i	0.00467094	−	0.0224470i
$706$			12.2077i			0.459442i
$707$	7.10320	−	7.10320i	0.267143	−	0.267143i
$708$	1.34023	+	2.04457i	0.0503690	+	0.0768396i
$709$	−26.0944	+	26.0944i	−0.979997	+	0.979997i	−0.999804	−	0.0198066i	$-0.993695\pi$
$709$	−26.0944	+	26.0944i	−0.979997	+	0.979997i	0.0198066	+	0.999804i	$0.493695\pi$
$710$	5.03536	+	5.03536i	0.188974	+	0.188974i
$711$	0.571322	+	0.248532i	0.0214262	+	0.00932066i
$712$		−	3.59248i		−	0.134634i
$713$	6.48920	+	6.48920i	0.243023	+	0.243023i
$714$	−14.7174	−	3.06250i	−0.550783	−	0.114611i
$715$	−8.11716	+	9.11324i	−0.303565	+	0.340816i
$716$			7.08990i			0.264962i
$717$	−6.90099	−	10.5277i	−0.257722	−	0.393164i
$718$	−17.7735			−0.663302
$719$	28.7437			1.07196			0.535979	−	0.844231i	$-0.319943\pi$
$719$	28.7437			1.07196			0.535979	+	0.844231i	$0.319943\pi$
$720$	1.96997	−	0.775612i	0.0734165	−	0.0289053i
$721$	−17.0678	−	17.0678i	−0.635638	−	0.635638i
$722$	9.32040	+	9.32040i	0.346870	+	0.346870i
$723$	6.80012	+	10.3738i	0.252899	+	0.385806i
$724$	−6.20768			−0.230707
$725$	4.49313			0.166871
$726$	−17.3885	+	11.3983i	−0.645348	+	0.423032i
$727$		−	11.3848i		−	0.422239i	−0.977460	−	0.211119i	$-0.932289\pi$
$727$		−	11.3848i		−	0.422239i	0.977460	−	0.211119i	$-0.0677109\pi$
$728$	−4.74166	+	5.32352i	−0.175738	+	0.197303i
$729$	−9.00000	−	25.4558i	−0.333333	−	0.942809i
$730$	−1.20376	−	1.20376i	−0.0445532	−	0.0445532i
$731$			39.0084i			1.44278i
$732$	16.2662	+	3.38480i	0.601217	+	0.125106i
$733$	27.3154	+	27.3154i	1.00892	+	1.00892i	0.999960	+	0.00895888i	$0.00285174\pi$
$733$	27.3154	+	27.3154i	1.00892	+	1.00892i	0.00895888	+	0.999960i	$0.497148\pi$
$734$	−13.7071	+	13.7071i	−0.505939	+	0.505939i
$735$	3.15939	−	2.07101i	0.116536	−	0.0763903i
$736$	0.705720	−	0.705720i	0.0260132	−	0.0260132i
$737$		−	39.3154i		−	1.44820i
$738$	−8.86933	+	20.3887i	−0.326484	+	0.750519i
$739$	−2.26764	+	2.26764i	−0.0834166	+	0.0834166i	−0.747584	−	0.664167i	$-0.768786\pi$
$739$	−2.26764	+	2.26764i	−0.0834166	+	0.0834166i	0.664167	+	0.747584i	$0.268786\pi$
$740$	−4.09579			−0.150564
$741$	2.21270	−	14.9011i	0.0812854	−	0.547406i
$742$	28.2155			1.03582
$743$	0.645871	−	0.645871i	0.0236947	−	0.0236947i	−0.695160	−	0.718855i	$-0.744667\pi$
$743$	0.645871	−	0.645871i	0.0236947	−	0.0236947i	0.718855	+	0.695160i	$0.244667\pi$
$744$	15.5925	+	3.24460i	0.571648	+	0.118953i
$745$			2.53528i			0.0928856i
$746$	−17.2643	+	17.2643i	−0.632090	+	0.632090i
$747$	36.2253	−	14.2625i	1.32541	−	0.521838i
$748$	−14.8868	+	14.8868i	−0.544314	+	0.544314i
$749$	−9.48332	−	9.48332i	−0.346513	−	0.346513i
$750$	−2.36618	+	11.3711i	−0.0864006	+	0.415213i
$751$		−	26.5964i		−	0.970516i	−0.874371	−	0.485258i	$-0.838726\pi$
$751$		−	26.5964i		−	0.970516i	0.874371	−	0.485258i	$-0.161274\pi$
$752$	−0.352168	−	0.352168i	−0.0128422	−	0.0128422i
$753$	6.79620	−	32.6603i	0.247667	−	1.19021i
$754$	−3.59248	+	0.207679i	−0.130830	+	0.00756324i
$755$			4.00545i			0.145773i
$756$	10.1261	−	1.73736i	0.368282	−	0.0631872i
$757$	−8.19984			−0.298028			−0.149014	−	0.988835i	$-0.547610\pi$
$757$	−8.19984			−0.298028			−0.149014	+	0.988835i	$0.547610\pi$
$758$	3.07891			0.111831
$759$	6.93406	−	4.54534i	0.251690	−	0.164985i
$760$	−1.20376	−	1.20376i	−0.0436650	−	0.0436650i
$761$	−7.64739	−	7.64739i	−0.277218	−	0.277218i	0.554780	−	0.831997i	$-0.312803\pi$
$761$	−7.64739	−	7.64739i	−0.277218	−	0.277218i	−0.831997	+	0.554780i	$0.812803\pi$
$762$	16.8311	−	11.0329i	0.609727	−	0.399681i
$763$	9.24884			0.334831
$764$	−9.77702			−0.353720
$765$	8.64718	−	3.40454i	0.312639	−	0.123091i
$766$		−	5.85620i		−	0.211593i
$767$	5.08053	−	0.293703i	0.183447	−	0.0106050i
$768$	0.352860	−	1.69573i	0.0127327	−	0.0611893i
$769$	6.14772	+	6.14772i	0.221692	+	0.221692i	0.809211	−	0.587518i	$-0.199895\pi$
$769$	6.14772	+	6.14772i	0.221692	+	0.221692i	−0.587518	+	0.809211i	$0.699895\pi$
$770$			6.69256i			0.241183i
$771$	−6.43934	+	30.9453i	−0.231907	+	1.11447i
$772$	−3.79624	−	3.79624i	−0.136630

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:	\( N \)	\(=\)	\( 78 = 2 \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	78.g (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 + 0.707107i) q^{2} +(0.352860 - 1.69573i) q^{3} -1.00000i q^{4} +(0.499019 - 0.499019i) q^{5} +(0.949550 + 1.44857i) q^{6} +(1.39812 - 1.39812i) q^{7} +(0.707107 + 0.707107i) q^{8} +(-2.75098 - 1.19671i) q^{9} +O(q^{10})\)	\(q+(-0.707107 + 0.707107i) q^{2} +(0.352860 - 1.69573i) q^{3} -1.00000i q^{4} +(0.499019 - 0.499019i) q^{5} +(0.949550 + 1.44857i) q^{6} +(1.39812 - 1.39812i) q^{7} +(0.707107 + 0.707107i) q^{8} +(-2.75098 - 1.19671i) q^{9} +0.705720i q^{10} +(3.39145 + 3.39145i) q^{11} +(-1.69573 - 0.352860i) q^{12} +(-2.39812 + 2.69240i) q^{13} +1.97724i q^{14} +(-0.670116 - 1.02228i) q^{15} -1.00000 q^{16} -4.38949 q^{17} +(2.79144 - 1.09904i) q^{18} +(-1.70572 - 1.70572i) q^{19} +(-0.499019 - 0.499019i) q^{20} +(-1.87749 - 2.86417i) q^{21} -4.79624 q^{22} +0.998038 q^{23} +(1.44857 - 0.949550i) q^{24} +4.50196i q^{25} +(-0.208088 - 3.59954i) q^{26} +(-3.00000 + 4.24264i) q^{27} +(-1.39812 - 1.39812i) q^{28} -0.998038i q^{29} +(1.19671 + 0.249020i) q^{30} +(6.50196 + 6.50196i) q^{31} +(0.707107 - 0.707107i) q^{32} +(6.94769 - 4.55427i) q^{33} +(3.10384 - 3.10384i) q^{34} -1.39538i q^{35} +(-1.19671 + 2.75098i) q^{36} +(4.10384 - 4.10384i) q^{37} +2.41225 q^{38} +(3.71938 + 5.01660i) q^{39} +0.705720 q^{40} +(-5.24068 + 5.24068i) q^{41} +(3.35286 + 0.697689i) q^{42} -8.88676i q^{43} +(3.39145 - 3.39145i) q^{44} +(-1.96997 + 0.775612i) q^{45} +(-0.705720 + 0.705720i) q^{46} +(0.352168 + 0.352168i) q^{47} +(-0.352860 + 1.69573i) q^{48} +3.09052i q^{49} +(-3.18337 - 3.18337i) q^{50} +(-1.54888 + 7.44338i) q^{51} +(2.69240 + 2.39812i) q^{52} -14.2702i q^{53} +(-0.878680 - 5.12132i) q^{54} +3.38480 q^{55} +1.97724 q^{56} +(-3.49431 + 2.29055i) q^{57} +(0.705720 + 0.705720i) q^{58} +(-0.998038 - 0.998038i) q^{59} +(-1.02228 + 0.670116i) q^{60} -9.59248 q^{61} -9.19516 q^{62} +(-5.51934 + 2.17306i) q^{63} +1.00000i q^{64} +(0.146852 + 2.54027i) q^{65} +(-1.69240 + 8.13311i) q^{66} +(-5.79624 - 5.79624i) q^{67} +4.38949i q^{68} +(0.352168 - 1.69240i) q^{69} +(0.986681 + 0.986681i) q^{70} +(7.13508 - 7.13508i) q^{71} +(-1.09904 - 2.79144i) q^{72} +(-1.70572 + 1.70572i) q^{73} +5.80371i q^{74} +(7.63409 + 1.58856i) q^{75} +(-1.70572 + 1.70572i) q^{76} +9.48332 q^{77} +(-6.17727 - 0.917274i) q^{78} -0.207679 q^{79} +(-0.499019 + 0.499019i) q^{80} +(6.13578 + 6.58424i) q^{81} -7.41144i q^{82} +(-9.17632 + 9.17632i) q^{83} +(-2.86417 + 1.87749i) q^{84} +(-2.19044 + 2.19044i) q^{85} +(6.28389 + 6.28389i) q^{86} +(-1.69240 - 0.352168i) q^{87} +4.79624i q^{88} +(-2.54027 - 2.54027i) q^{89} +(0.844540 - 1.94142i) q^{90} +(0.411439 + 7.11716i) q^{91} -0.998038i q^{92} +(13.3198 - 8.73127i) q^{93} -0.498040 q^{94} -1.70237 q^{95} +(-0.949550 - 1.44857i) q^{96} +(-3.58856 - 3.58856i) q^{97} +(-2.18533 - 2.18533i) q^{98} +(-5.27124 - 13.3884i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{7}+O(q^{10}) \) 12 * q - 12 * q^7	\( 12 q - 12 q^{7} - 12 q^{16} - 12 q^{19} - 36 q^{27} + 12 q^{28} + 12 q^{31} + 36 q^{33} + 12 q^{37} + 36 q^{42} + 36 q^{45} + 12 q^{52} - 36 q^{54} - 36 q^{57} - 36 q^{63} - 12 q^{67} - 12 q^{73} - 12 q^{76} - 36 q^{78} + 72 q^{79} - 72 q^{85} - 12 q^{91} + 36 q^{93} - 72 q^{94} - 60 q^{97} + 36 q^{99}+O(q^{100}) \) 12 * q - 12 * q^7 - 12 * q^16 - 12 * q^19 - 36 * q^27 + 12 * q^28 + 12 * q^31 + 36 * q^33 + 12 * q^37 + 36 * q^42 + 36 * q^45 + 12 * q^52 - 36 * q^54 - 36 * q^57 - 36 * q^63 - 12 * q^67 - 12 * q^73 - 12 * q^76 - 36 * q^78 + 72 * q^79 - 72 * q^85 - 12 * q^91 + 36 * q^93 - 72 * q^94 - 60 * q^97 + 36 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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