LMFDB - Embedded newform 416.2.bu.a.223.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [416,2,Mod(63,416)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(416, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 0, 7])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("416.63"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 416 = 2^{5} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	416.bu (of order $12$, degree $4$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,-6,0,-2,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$3.32177672409$
Analytic rank:	$1$
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			223.1
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	416.223
Dual form			416.2.bu.a.319.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.633975 + 0.366025i) q^{3} +(-1.36603 + 1.36603i) q^{5} +(-1.00000 - 3.73205i) q^{7} +(-1.23205 + 2.13397i) q^{9} +(2.73205 + 0.732051i) q^{11} +(-3.50000 + 0.866025i) q^{13} +(0.366025 - 1.36603i) q^{15} +(-5.59808 - 3.23205i) q^{17} +(-6.09808 + 1.63397i) q^{19} +(2.00000 + 2.00000i) q^{21} +(-2.63397 - 4.56218i) q^{23} +1.26795i q^{25} -4.00000i q^{27} +(1.23205 + 2.13397i) q^{29} +(-4.46410 - 4.46410i) q^{31} +(-2.00000 + 0.535898i) q^{33} +(6.46410 + 3.73205i) q^{35} +(1.59808 - 5.96410i) q^{37} +(1.90192 - 1.83013i) q^{39} +(9.06218 + 2.42820i) q^{41} +(-2.26795 + 3.92820i) q^{43} +(-1.23205 - 4.59808i) q^{45} +(-4.46410 + 4.46410i) q^{47} +(-6.86603 + 3.96410i) q^{49} +4.73205 q^{51} -1.73205 q^{53} +(-4.73205 + 2.73205i) q^{55} +(3.26795 - 3.26795i) q^{57} +(2.26795 + 8.46410i) q^{59} +(-5.33013 + 9.23205i) q^{61} +(9.19615 + 2.46410i) q^{63} +(3.59808 - 5.96410i) q^{65} +(-0.169873 + 0.633975i) q^{67} +(3.33975 + 1.92820i) q^{69} +(9.56218 - 2.56218i) q^{71} +(6.29423 + 6.29423i) q^{73} +(-0.464102 - 0.803848i) q^{75} -10.9282i q^{77} -2.53590i q^{79} +(-2.23205 - 3.86603i) q^{81} +(-4.19615 - 4.19615i) q^{83} +(12.0622 - 3.23205i) q^{85} +(-1.56218 - 0.901924i) q^{87} +(2.43782 - 9.09808i) q^{89} +(6.73205 + 12.1962i) q^{91} +(4.46410 + 1.19615i) q^{93} +(6.09808 - 10.5622i) q^{95} +(2.83013 + 10.5622i) q^{97} +(-4.92820 + 4.92820i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{3} - 2 q^{5} - 4 q^{7} + 2 q^{9} + 4 q^{11} - 14 q^{13} - 2 q^{15} - 12 q^{17} - 14 q^{19} + 8 q^{21} - 14 q^{23} - 2 q^{29} - 4 q^{31} - 8 q^{33} + 12 q^{35} - 4 q^{37} + 18 q^{39} + 12 q^{41}+ \cdots + 8 q^{99}+O(q^{100}) $ 4 * q - 6 * q^3 - 2 * q^5 - 4 * q^7 + 2 * q^9 + 4 * q^11 - 14 * q^13 - 2 * q^15 - 12 * q^17 - 14 * q^19 + 8 * q^21 - 14 * q^23 - 2 * q^29 - 4 * q^31 - 8 * q^33 + 12 * q^35 - 4 * q^37 + 18 * q^39 + 12 * q^41 - 16 * q^43 + 2 * q^45 - 4 * q^47 - 24 * q^49 + 12 * q^51 - 12 * q^55 + 20 * q^57 + 16 * q^59 - 4 * q^61 + 16 * q^63 + 4 * q^65 - 18 * q^67 + 48 * q^69 + 14 * q^71 - 6 * q^73 + 12 * q^75 - 2 * q^81 + 4 * q^83 + 24 * q^85 + 18 * q^87 + 34 * q^89 + 20 * q^91 + 4 * q^93 + 14 * q^95 - 6 * q^97 + 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/416\mathbb{Z}\right)^\times$.

$n$	$261$	$287$	$353$
$\chi(n)$	$1$	$-1$	$e\left(\frac{1}{12}\right)$

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			0
$2$		0			0
$3$	−0.633975	+	0.366025i	−0.366025	+	0.211325i	−0.671721	−	0.740805i	$-0.734444\pi$
$3$	−0.633975	+	0.366025i	−0.366025	+	0.211325i	0.305695	+	0.952129i	$0.401111\pi$
$4$		0			0
$5$	−1.36603	+	1.36603i	−0.610905	+	0.610905i	−0.943182	−	0.332277i	$-0.892183\pi$
$5$	−1.36603	+	1.36603i	−0.610905	+	0.610905i	0.332277	+	0.943182i	$0.392183\pi$
$6$		0			0
$7$	−1.00000	−	3.73205i	−0.377964	−	1.41058i	−0.848965	−	0.528450i	$-0.822774\pi$
$7$	−1.00000	−	3.73205i	−0.377964	−	1.41058i	0.471000	−	0.882133i	$-0.343893\pi$
$8$		0			0
$9$	−1.23205	+	2.13397i	−0.410684	+	0.711325i
$10$		0			0
$11$	2.73205	+	0.732051i	0.823744	+	0.220722i	0.645983	−	0.763352i	$-0.276448\pi$
$11$	2.73205	+	0.732051i	0.823744	+	0.220722i	0.177762	+	0.984074i	$0.443114\pi$
$12$		0			0
$13$	−3.50000	+	0.866025i	−0.970725	+	0.240192i
$13$	−3.50000	+	0.866025i	−0.970725	+	0.240192i
$14$		0			0
$15$	0.366025	−	1.36603i	0.0945074	−	0.352706i
$16$		0			0
$17$	−5.59808	−	3.23205i	−1.35773	−	0.783887i	−0.368415	−	0.929661i	$-0.620099\pi$
$17$	−5.59808	−	3.23205i	−1.35773	−	0.783887i	−0.989318	+	0.145774i	$0.953433\pi$
$18$		0			0
$19$	−6.09808	+	1.63397i	−1.39899	+	0.374859i	−0.877984	−	0.478691i	$-0.841112\pi$
$19$	−6.09808	+	1.63397i	−1.39899	+	0.374859i	−0.521011	+	0.853550i	$0.674445\pi$
$20$		0			0
$21$	2.00000	+	2.00000i	0.436436	+	0.436436i
$22$		0			0
$23$	−2.63397	−	4.56218i	−0.549222	−	0.951280i	−0.998328	−	0.0578016i	$-0.981591\pi$
$23$	−2.63397	−	4.56218i	−0.549222	−	0.951280i	0.449106	−	0.893478i	$-0.351742\pi$
$24$		0			0
$25$			1.26795i			0.253590i
$26$		0			0
$27$		−	4.00000i		−	0.769800i
$28$		0			0
$29$	1.23205	+	2.13397i	0.228786	+	0.396269i	0.957449	−	0.288604i	$-0.0931910\pi$
$29$	1.23205	+	2.13397i	0.228786	+	0.396269i	−0.728663	+	0.684873i	$0.759858\pi$
$30$		0			0
$31$	−4.46410	−	4.46410i	−0.801776	−	0.801776i	0.181597	−	0.983373i	$-0.441873\pi$
$31$	−4.46410	−	4.46410i	−0.801776	−	0.801776i	−0.983373	+	0.181597i	$0.941873\pi$
$32$		0			0
$33$	−2.00000	+	0.535898i	−0.348155	+	0.0932879i
$34$		0			0
$35$	6.46410	+	3.73205i	1.09263	+	0.630832i
$36$		0			0
$37$	1.59808	−	5.96410i	0.262722	−	0.980492i	−0.700908	−	0.713252i	$-0.747222\pi$
$37$	1.59808	−	5.96410i	0.262722	−	0.980492i	0.963630	−	0.267240i	$-0.0861118\pi$
$38$		0			0
$39$	1.90192	−	1.83013i	0.304552	−	0.293055i
$40$		0			0
$41$	9.06218	+	2.42820i	1.41527	+	0.379222i	0.883805	−	0.467856i	$-0.154973\pi$
$41$	9.06218	+	2.42820i	1.41527	+	0.379222i	0.531470	+	0.847077i	$0.321640\pi$
$42$		0			0
$43$	−2.26795	+	3.92820i	−0.345859	+	0.599045i	−0.985509	−	0.169621i	$-0.945746\pi$
$43$	−2.26795	+	3.92820i	−0.345859	+	0.599045i	0.639650	+	0.768666i	$0.279079\pi$
$44$		0			0
$45$	−1.23205	−	4.59808i	−0.183663	−	0.685441i
$46$		0			0
$47$	−4.46410	+	4.46410i	−0.651156	+	0.651156i	−0.953271	−	0.302115i	$-0.902307\pi$
$47$	−4.46410	+	4.46410i	−0.651156	+	0.651156i	0.302115	+	0.953271i	$0.402307\pi$
$48$		0			0
$49$	−6.86603	+	3.96410i	−0.980861	+	0.566300i
$50$		0			0
$51$	4.73205			0.662620
$52$		0			0
$53$	−1.73205			−0.237915			−0.118958	−	0.992899i	$-0.537955\pi$
$53$	−1.73205			−0.237915			−0.118958	+	0.992899i	$0.537955\pi$
$54$		0			0
$55$	−4.73205	+	2.73205i	−0.638070	+	0.368390i
$56$		0			0
$57$	3.26795	−	3.26795i	0.432850	−	0.432850i
$58$		0			0
$59$	2.26795	+	8.46410i	0.295262	+	1.10193i	0.941009	+	0.338381i	$0.109879\pi$
$59$	2.26795	+	8.46410i	0.295262	+	1.10193i	−0.645747	+	0.763551i	$0.723454\pi$
$60$		0			0
$61$	−5.33013	+	9.23205i	−0.682453	+	1.18204i	0.291777	+	0.956486i	$0.405753\pi$
$61$	−5.33013	+	9.23205i	−0.682453	+	1.18204i	−0.974230	+	0.225557i	$0.927580\pi$
$62$		0			0
$63$	9.19615	+	2.46410i	1.15861	+	0.310448i
$64$		0			0
$65$	3.59808	−	5.96410i	0.446286	−	0.739756i
$66$		0			0
$67$	−0.169873	+	0.633975i	−0.0207533	+	0.0774523i	−0.975526	−	0.219885i	$-0.929432\pi$
$67$	−0.169873	+	0.633975i	−0.0207533	+	0.0774523i	0.954773	+	0.297337i	$0.0960985\pi$
$68$		0			0
$69$	3.33975	+	1.92820i	0.402058	+	0.232128i
$70$		0			0
$71$	9.56218	−	2.56218i	1.13482	−	0.304075i	0.357955	−	0.933739i	$-0.383474\pi$
$71$	9.56218	−	2.56218i	1.13482	−	0.304075i	0.776867	+	0.629664i	$0.216808\pi$
$72$		0			0
$73$	6.29423	+	6.29423i	0.736684	+	0.736684i	0.971935	−	0.235251i	$-0.0755912\pi$
$73$	6.29423	+	6.29423i	0.736684	+	0.736684i	−0.235251	+	0.971935i	$0.575591\pi$
$74$		0			0
$75$	−0.464102	−	0.803848i	−0.0535898	−	0.0928203i
$76$		0			0
$77$		−	10.9282i		−	1.24538i
$78$		0			0
$79$		−	2.53590i		−	0.285311i	−0.989772	−	0.142655i	$-0.954436\pi$
$79$		−	2.53590i		−	0.285311i	0.989772	−	0.142655i	$-0.0455641\pi$
$80$		0			0
$81$	−2.23205	−	3.86603i	−0.248006	−	0.429558i
$82$		0			0
$83$	−4.19615	−	4.19615i	−0.460588	−	0.460588i	0.438260	−	0.898848i	$-0.355595\pi$
$83$	−4.19615	−	4.19615i	−0.460588	−	0.460588i	−0.898848	+	0.438260i	$0.855595\pi$
$84$		0			0
$85$	12.0622	−	3.23205i	1.30833	−	0.350565i
$86$		0			0
$87$	−1.56218	−	0.901924i	−0.167483	−	0.0966964i
$88$		0			0
$89$	2.43782	−	9.09808i	0.258409	−	0.964394i	−0.707754	−	0.706459i	$-0.750291\pi$
$89$	2.43782	−	9.09808i	0.258409	−	0.964394i	0.966162	−	0.257935i	$-0.0830420\pi$
$90$		0			0
$91$	6.73205	+	12.1962i	0.705711	+	1.27850i
$92$		0			0
$93$	4.46410	+	1.19615i	0.462906	+	0.124035i
$94$		0			0
$95$	6.09808	−	10.5622i	0.625649	−	1.08366i
$96$		0			0
$97$	2.83013	+	10.5622i	0.287356	+	1.07243i	0.947101	+	0.320937i	$0.103998\pi$
$97$	2.83013	+	10.5622i	0.287356	+	1.07243i	−0.659745	+	0.751490i	$0.729336\pi$
$98$		0			0
$99$	−4.92820	+	4.92820i	−0.495303	+	0.495303i
$100$		0			0
$101$	1.79423	−	1.03590i	0.178532	−	0.103076i	−0.408071	−	0.912950i	$-0.633798\pi$
$101$	1.79423	−	1.03590i	0.178532	−	0.103076i	0.586603	+	0.809875i	$0.300465\pi$
$102$		0			0
$103$	−3.26795			−0.322001			−0.161000	−	0.986954i	$-0.551472\pi$
$103$	−3.26795			−0.322001			−0.161000	+	0.986954i	$0.551472\pi$
$104$		0			0
$105$	−5.46410			−0.533242
$106$		0			0
$107$	2.36603	−	1.36603i	0.228732	−	0.132059i	−0.381255	−	0.924470i	$-0.624508\pi$
$107$	2.36603	−	1.36603i	0.228732	−	0.132059i	0.609987	+	0.792411i	$0.291175\pi$
$108$		0			0
$109$	−0.803848	+	0.803848i	−0.0769946	+	0.0769946i	−0.744555	−	0.667561i	$-0.767338\pi$
$109$	−0.803848	+	0.803848i	−0.0769946	+	0.0769946i	0.667561	+	0.744555i	$0.267338\pi$
$110$		0			0
$111$	1.16987	+	4.36603i	0.111039	+	0.414405i
$112$		0			0
$113$	3.59808	−	6.23205i	0.338479	−	0.586262i	−0.645668	−	0.763618i	$-0.723421\pi$
$113$	3.59808	−	6.23205i	0.338479	−	0.586262i	0.984147	+	0.177356i	$0.0567544\pi$
$114$		0			0
$115$	9.83013	+	2.63397i	0.916664	+	0.245619i
$116$		0			0
$117$	2.46410	−	8.53590i	0.227806	−	0.789144i
$118$		0			0
$119$	−6.46410	+	24.1244i	−0.592563	+	2.21148i
$120$		0			0
$121$	−2.59808	−	1.50000i	−0.236189	−	0.136364i
$122$		0			0
$123$	−6.63397	+	1.77757i	−0.598165	+	0.160278i
$124$		0			0
$125$	−8.56218	−	8.56218i	−0.765824	−	0.765824i
$126$		0			0
$127$	8.09808	+	14.0263i	0.718588	+	1.24463i	0.961559	+	0.274598i	$0.0885446\pi$
$127$	8.09808	+	14.0263i	0.718588	+	1.24463i	−0.242971	+	0.970034i	$0.578122\pi$
$128$		0			0
$129$		−	3.32051i		−	0.292354i
$130$		0			0
$131$		−	14.1962i		−	1.24032i	−0.784474	−	0.620162i	$-0.787067\pi$
$131$		−	14.1962i		−	1.24032i	0.784474	−	0.620162i	$-0.212933\pi$
$132$		0			0
$133$	12.1962	+	21.1244i	1.05754	+	1.83171i
$134$		0			0
$135$	5.46410	+	5.46410i	0.470275	+	0.470275i
$136$		0			0
$137$	−1.50000	+	0.401924i	−0.128154	+	0.0343387i	−0.322326	−	0.946629i	$-0.604465\pi$
$137$	−1.50000	+	0.401924i	−0.128154	+	0.0343387i	0.194172	+	0.980967i	$0.437798\pi$
$138$		0			0
$139$	−18.1244	−	10.4641i	−1.53729	−	0.887554i	−0.998996	−	0.0447948i	$-0.985737\pi$
$139$	−18.1244	−	10.4641i	−1.53729	−	0.887554i	−0.538292	−	0.842759i	$-0.680930\pi$
$140$		0			0
$141$	1.19615	−	4.46410i	0.100734	−	0.375945i
$142$		0			0
$143$	−10.1962	−	0.196152i	−0.852645	−	0.0164031i
$144$		0			0
$145$	−4.59808	−	1.23205i	−0.381849	−	0.102316i
$146$		0			0
$147$	2.90192	−	5.02628i	0.239347	−	0.414561i
$148$		0			0
$149$	4.13397	+	15.4282i	0.338668	+	1.26393i	0.899837	+	0.436226i	$0.143685\pi$
$149$	4.13397	+	15.4282i	0.338668	+	1.26393i	−0.561169	+	0.827701i	$0.689648\pi$
$150$		0			0
$151$	−6.46410	+	6.46410i	−0.526041	+	0.526041i	−0.919390	−	0.393348i	$-0.871317\pi$
$151$	−6.46410	+	6.46410i	−0.526041	+	0.526041i	0.393348	+	0.919390i	$0.371317\pi$
$152$		0			0
$153$	13.7942	−	7.96410i	1.11520	−	0.643859i
$154$		0			0
$155$	12.1962			0.979619
$156$		0			0
$157$	−15.5359			−1.23990			−0.619950	−	0.784641i	$-0.712847\pi$
$157$	−15.5359			−1.23990			−0.619950	+	0.784641i	$0.712847\pi$
$158$		0			0
$159$	1.09808	−	0.633975i	0.0870831	−	0.0502775i
$160$		0			0
$161$	−14.3923	+	14.3923i	−1.13427	+	1.13427i
$162$		0			0
$163$	−4.26795	−	15.9282i	−0.334292	−	1.24759i	−0.904635	−	0.426187i	$-0.859857\pi$
$163$	−4.26795	−	15.9282i	−0.334292	−	1.24759i	0.570344	−	0.821406i	$-0.306810\pi$
$164$		0			0
$165$	2.00000	−	3.46410i	0.155700	−	0.269680i
$166$		0			0
$167$	6.19615	+	1.66025i	0.479473	+	0.128474i	0.490457	−	0.871465i	$-0.336830\pi$
$167$	6.19615	+	1.66025i	0.479473	+	0.128474i	−0.0109844	+	0.999940i	$0.503497\pi$
$168$		0			0
$169$	11.5000	−	6.06218i	0.884615	−	0.466321i
$170$		0			0
$171$	4.02628	−	15.0263i	0.307897	−	1.14909i
$172$		0			0
$173$	−12.5885	−	7.26795i	−0.957083	−	0.552572i	−0.0618087	−	0.998088i	$-0.519687\pi$
$173$	−12.5885	−	7.26795i	−0.957083	−	0.552572i	−0.895274	+	0.445516i	$0.853020\pi$
$174$		0			0
$175$	4.73205	−	1.26795i	0.357709	−	0.0958479i
$176$		0			0
$177$	−4.53590	−	4.53590i	−0.340939	−	0.340939i
$178$		0			0
$179$	−0.732051	−	1.26795i	−0.0547160	−	0.0947710i	0.837370	−	0.546636i	$-0.184092\pi$
$179$	−0.732051	−	1.26795i	−0.0547160	−	0.0947710i	−0.892086	+	0.451865i	$0.850759\pi$
$180$		0			0
$181$		−	3.19615i		−	0.237568i	−0.992920	−	0.118784i	$-0.962100\pi$
$181$		−	3.19615i		−	0.237568i	0.992920	−	0.118784i	$-0.0378996\pi$
$182$		0			0
$183$		−	7.80385i		−	0.576877i
$184$		0			0
$185$	5.96410	+	10.3301i	0.438489	+	0.759486i
$186$		0			0
$187$	−12.9282	−	12.9282i	−0.945404	−	0.945404i
$188$		0			0
$189$	−14.9282	+	4.00000i	−1.08587	+	0.290957i
$190$		0			0
$191$	−19.0981	−	11.0263i	−1.38189	−	0.797834i	−0.389505	−	0.921024i	$-0.627354\pi$
$191$	−19.0981	−	11.0263i	−1.38189	−	0.797834i	−0.992383	+	0.123191i	$0.960687\pi$
$192$		0			0
$193$	1.33013	−	4.96410i	0.0957446	−	0.357324i	−0.901387	−	0.433015i	$-0.857450\pi$
$193$	1.33013	−	4.96410i	0.0957446	−	0.357324i	0.997131	+	0.0756913i	$0.0241163\pi$
$194$		0			0
$195$	−0.0980762	+	5.09808i	−0.00702338	+	0.365081i
$196$		0			0
$197$	−3.83013	−	1.02628i	−0.272885	−	0.0731194i	0.119781	−	0.992800i	$-0.461781\pi$
$197$	−3.83013	−	1.02628i	−0.272885	−	0.0731194i	−0.392667	+	0.919681i	$0.628447\pi$
$198$		0			0
$199$	7.29423	−	12.6340i	0.517074	−	0.895599i	−0.482729	−	0.875770i	$-0.660354\pi$
$199$	7.29423	−	12.6340i	0.517074	−	0.895599i	0.999803	−	0.0198291i	$-0.00631220\pi$
$200$		0			0
$201$	−0.124356	−	0.464102i	−0.00877137	−	0.0327352i
$202$		0			0
$203$	6.73205	−	6.73205i	0.472497	−	0.472497i
$204$		0			0
$205$	−15.6962	+	9.06218i	−1.09627	+	0.632930i
$206$		0			0
$207$	12.9808			0.902225
$208$		0			0
$209$	−17.8564			−1.23515
$210$		0			0
$211$	−21.6340	+	12.4904i	−1.48934	+	0.859873i	−0.999926	−	0.0121758i	$-0.996124\pi$
$211$	−21.6340	+	12.4904i	−1.48934	+	0.859873i	−0.489418	+	0.872049i	$0.662791\pi$
$212$		0			0
$213$	−5.12436	+	5.12436i	−0.351115	+	0.351115i
$214$		0			0
$215$	−2.26795	−	8.46410i	−0.154673	−	0.577247i
$216$		0			0
$217$	−12.1962	+	21.1244i	−0.827929	+	1.43401i
$218$		0			0
$219$	−6.29423	−	1.68653i	−0.425325	−	0.113965i
$220$		0			0
$221$	22.3923	+	6.46410i	1.50627	+	0.434823i
$222$		0			0
$223$	−7.56218	+	28.2224i	−0.506401	+	1.88991i	−0.0530277	+	0.998593i	$0.516887\pi$
$223$	−7.56218	+	28.2224i	−0.506401	+	1.88991i	−0.453373	+	0.891321i	$0.649780\pi$
$224$		0			0
$225$	−2.70577	−	1.56218i	−0.180385	−	0.104145i
$226$		0			0
$227$	4.36603	−	1.16987i	0.289783	−	0.0776472i	−0.110999	−	0.993821i	$-0.535405\pi$
$227$	4.36603	−	1.16987i	0.289783	−	0.0776472i	0.400782	+	0.916173i	$0.368738\pi$
$228$		0			0
$229$	12.2679	+	12.2679i	0.810689	+	0.810689i	0.984737	−	0.174048i	$-0.0556849\pi$
$229$	12.2679	+	12.2679i	0.810689	+	0.810689i	−0.174048	+	0.984737i	$0.555685\pi$
$230$		0			0
$231$	4.00000	+	6.92820i	0.263181	+	0.455842i
$232$		0			0
$233$			15.4641i			1.01309i	0.862214	+	0.506543i	$0.169077\pi$
$233$			15.4641i			1.01309i	−0.862214	+	0.506543i	$0.830923\pi$
$234$		0			0
$235$		−	12.1962i		−	0.795589i
$236$		0			0
$237$	0.928203	+	1.60770i	0.0602933	+	0.104431i
$238$		0			0
$239$	7.26795	+	7.26795i	0.470125	+	0.470125i	0.901955	−	0.431830i	$-0.142132\pi$
$239$	7.26795	+	7.26795i	0.470125	+	0.470125i	−0.431830	+	0.901955i	$0.642132\pi$
$240$		0			0
$241$	−1.50000	+	0.401924i	−0.0966235	+	0.0258902i	−0.306807	−	0.951772i	$-0.599261\pi$
$241$	−1.50000	+	0.401924i	−0.0966235	+	0.0258902i	0.210183	+	0.977662i	$0.432594\pi$
$242$		0			0
$243$	13.2224	+	7.63397i	0.848219	+	0.489720i
$244$		0			0
$245$	3.96410	−	14.7942i	0.253257	−	0.945169i
$246$		0			0
$247$	19.9282	−	11.0000i	1.26800	−	0.699913i
$248$		0			0
$249$	4.19615	+	1.12436i	0.265920	+	0.0712531i
$250$		0			0
$251$	4.63397	−	8.02628i	0.292494	−	0.506614i	−0.681905	−	0.731441i	$-0.738848\pi$
$251$	4.63397	−	8.02628i	0.292494	−	0.506614i	0.974399	+	0.224827i	$0.0721816\pi$
$252$		0			0
$253$	−3.85641	−	14.3923i	−0.242450	−	0.904836i
$254$		0			0
$255$	−6.46410	+	6.46410i	−0.404798	+	0.404798i
$256$		0			0
$257$	−3.23205	+	1.86603i	−0.201610	+	0.116399i	−0.597406	−	0.801939i	$-0.703802\pi$
$257$	−3.23205	+	1.86603i	−0.201610	+	0.116399i	0.395796	+	0.918338i	$0.370469\pi$
$258$		0			0
$259$	−23.8564			−1.48236
$260$		0			0
$261$	−6.07180			−0.375835
$262$		0			0
$263$	−21.5885	+	12.4641i	−1.33120	+	0.768569i	−0.985484	−	0.169769i	$-0.945698\pi$
$263$	−21.5885	+	12.4641i	−1.33120	+	0.768569i	−0.345717	+	0.938339i	$0.612364\pi$
$264$		0			0
$265$	2.36603	−	2.36603i	0.145344	−	0.145344i
$266$		0			0
$267$	1.78461	+	6.66025i	0.109216	+	0.407601i
$268$		0			0
$269$	4.00000	−	6.92820i	0.243884	−	0.422420i	−0.717933	−	0.696112i	$-0.754912\pi$
$269$	4.00000	−	6.92820i	0.243884	−	0.422420i	0.961817	+	0.273692i	$0.0882449\pi$
$270$		0			0
$271$	−17.1962	−	4.60770i	−1.04459	−	0.279898i	−0.304577	−	0.952488i	$-0.598515\pi$
$271$	−17.1962	−	4.60770i	−1.04459	−	0.279898i	−0.740015	+	0.672590i	$0.765182\pi$
$272$		0			0
$273$	−8.73205	−	5.26795i	−0.528488	−	0.318831i
$274$		0			0
$275$	−0.928203	+	3.46410i	−0.0559728	+	0.208893i
$276$		0			0
$277$	−7.66987	−	4.42820i	−0.460838	−	0.266065i	0.251559	−	0.967842i	$-0.419057\pi$
$277$	−7.66987	−	4.42820i	−0.460838	−	0.266065i	−0.712397	+	0.701777i	$0.752390\pi$
$278$		0			0
$279$	15.0263	−	4.02628i	0.899600	−	0.241047i
$280$		0			0
$281$	−11.6340	−	11.6340i	−0.694025	−	0.694025i	0.269090	−	0.963115i	$-0.413277\pi$
$281$	−11.6340	−	11.6340i	−0.694025	−	0.694025i	−0.963115	+	0.269090i	$0.913277\pi$
$282$		0			0
$283$	−7.00000	−	12.1244i	−0.416107	−	0.720718i	0.579437	−	0.815017i	$-0.303272\pi$
$283$	−7.00000	−	12.1244i	−0.416107	−	0.720718i	−0.995544	+	0.0942988i	$0.969939\pi$
$284$		0			0
$285$			8.92820i			0.528861i
$286$		0			0
$287$		−	36.2487i		−	2.13969i
$288$		0			0
$289$	12.3923	+	21.4641i	0.728959	+	1.26259i
$290$		0			0
$291$	−5.66025	−	5.66025i	−0.331810	−	0.331810i
$292$		0			0
$293$	−0.232051	+	0.0621778i	−0.0135566	+	0.00363247i	−0.265591	−	0.964086i	$-0.585567\pi$
$293$	−0.232051	+	0.0621778i	−0.0135566	+	0.00363247i	0.252034	+	0.967718i	$0.418900\pi$
$294$		0			0
$295$	−14.6603	−	8.46410i	−0.853553	−	0.492799i
$296$		0			0
$297$	2.92820	−	10.9282i	0.169912	−	0.634119i
$298$		0			0
$299$	13.1699	+	13.6865i	0.761633	+	0.791513i
$300$		0			0
$301$	16.9282	+	4.53590i	0.975725	+	0.261445i
$302$		0			0
$303$	−0.758330	+	1.31347i	−0.0435649	+	0.0754567i
$304$		0			0
$305$	−5.33013	−	19.8923i	−0.305202	−	1.13903i
$306$		0			0
$307$	15.7321	−	15.7321i	0.897876	−	0.897876i	−0.0973724	−	0.995248i	$-0.531044\pi$
$307$	15.7321	−	15.7321i	0.897876	−	0.897876i	0.995248	+	0.0973724i	$0.0310438\pi$
$308$		0			0
$309$	2.07180	−	1.19615i	0.117860	−	0.0680467i
$310$		0			0
$311$	6.58846			0.373597			0.186799	−	0.982398i	$-0.440189\pi$
$311$	6.58846			0.373597			0.186799	+	0.982398i	$0.440189\pi$
$312$		0			0
$313$	−4.53590			−0.256384			−0.128192	−	0.991749i	$-0.540917\pi$
$313$	−4.53590			−0.256384			−0.128192	+	0.991749i	$0.540917\pi$
$314$		0			0
$315$	−15.9282	+	9.19615i	−0.897453	+	0.518144i
$316$		0			0
$317$	3.09808	−	3.09808i	0.174005	−	0.174005i	−0.614731	−	0.788737i	$-0.710736\pi$
$317$	3.09808	−	3.09808i	0.174005	−	0.174005i	0.788737	+	0.614731i	$0.210736\pi$
$318$		0			0
$319$	1.80385	+	6.73205i	0.100996	+	0.376922i
$320$		0			0
$321$	−1.00000	+	1.73205i	−0.0558146	+	0.0966736i
$322$		0			0
$323$	39.4186	+	10.5622i	2.19331	+	0.587695i
$324$		0			0
$325$	−1.09808	−	4.43782i	−0.0609103	−	0.246166i
$326$		0			0
$327$	0.215390	−	0.803848i	0.0119111	−	0.0444529i
$328$		0			0
$329$	21.1244	+	12.1962i	1.16462	+	0.672396i
$330$		0			0
$331$	−29.8564	+	8.00000i	−1.64106	+	0.439720i	−0.957089	−	0.289793i	$-0.906414\pi$
$331$	−29.8564	+	8.00000i	−1.64106	+	0.439720i	−0.683967	+	0.729513i	$0.739747\pi$
$332$		0			0
$333$	10.7583	+	10.7583i	0.589553	+	0.589553i
$334$		0			0
$335$	−0.633975	−	1.09808i	−0.0346377	−	0.0599943i
$336$		0			0
$337$		−	30.8564i		−	1.68086i	−0.541924	−	0.840428i	$-0.682304\pi$
$337$		−	30.8564i		−	1.68086i	0.541924	−	0.840428i	$-0.317696\pi$
$338$		0			0
$339$			5.26795i			0.286116i
$340$		0			0
$341$	−8.92820	−	15.4641i	−0.483489	−	0.837428i
$342$		0			0
$343$	2.53590	+	2.53590i	0.136926	+	0.136926i
$344$		0			0
$345$	−7.19615	+	1.92820i	−0.387428	+	0.103811i
$346$		0			0
$347$	30.7583	+	17.7583i	1.65119	+	0.953317i	0.976582	+	0.215144i	$0.0690221\pi$
$347$	30.7583	+	17.7583i	1.65119	+	0.953317i	0.674611	+	0.738173i	$0.264311\pi$
$348$		0			0
$349$	−2.63397	+	9.83013i	−0.140993	+	0.526195i	0.858908	+	0.512130i	$0.171144\pi$
$349$	−2.63397	+	9.83013i	−0.140993	+	0.526195i	−0.999901	+	0.0140642i	$0.995523\pi$
$350$		0			0
$351$	3.46410	+	14.0000i	0.184900	+	0.747265i
$352$		0			0
$353$	34.3564	+	9.20577i	1.82861	+	0.489974i	0.997782	−	0.0665606i	$-0.0212026\pi$
$353$	34.3564	+	9.20577i	1.82861	+	0.489974i	0.830825	+	0.556534i	$0.187869\pi$
$354$		0			0
$355$	−9.56218	+	16.5622i	−0.507508	+	0.879029i
$356$		0			0
$357$	−4.73205	−	17.6603i	−0.250447	−	0.934680i
$358$		0			0
$359$	−9.07180	+	9.07180i	−0.478791	+	0.478791i	−0.904745	−	0.425954i	$-0.859939\pi$
$359$	−9.07180	+	9.07180i	−0.478791	+	0.478791i	0.425954	+	0.904745i	$0.359939\pi$
$360$		0			0
$361$	18.0622	−	10.4282i	0.950641	−	0.548853i
$362$		0			0
$363$	2.19615			0.115268
$364$		0			0
$365$	−17.1962			−0.900088
$366$		0			0
$367$	20.4904	−	11.8301i	1.06959	−	0.617528i	0.141520	−	0.989935i	$-0.454801\pi$
$367$	20.4904	−	11.8301i	1.06959	−	0.617528i	0.928069	+	0.372408i	$0.121468\pi$
$368$		0			0
$369$	−16.3468	+	16.3468i	−0.850980	+	0.850980i
$370$		0			0
$371$	1.73205	+	6.46410i	0.0899236	+	0.335599i
$372$		0			0
$373$	−5.03590	+	8.72243i	−0.260749	+	0.451630i	−0.966441	−	0.256888i	$-0.917303\pi$
$373$	−5.03590	+	8.72243i	−0.260749	+	0.451630i	0.705692	+	0.708519i	$0.250636\pi$
$374$		0			0
$375$	8.56218	+	2.29423i	0.442149	+	0.118473i
$376$		0			0
$377$	−6.16025	−	6.40192i	−0.317269	−	0.329716i
$378$		0			0
$379$	−1.02628	+	3.83013i	−0.0527164	+	0.196740i	−0.987262	−	0.159103i	$-0.949140\pi$
$379$	−1.02628	+	3.83013i	−0.0527164	+	0.196740i	0.934546	+	0.355844i	$0.115806\pi$
$380$		0			0
$381$	−10.2679	−	5.92820i	−0.526043	−	0.303711i
$382$		0			0
$383$	−6.92820	+	1.85641i	−0.354015	+	0.0948579i	−0.431444	−	0.902140i	$-0.641996\pi$
$383$	−6.92820	+	1.85641i	−0.354015	+	0.0948579i	0.0774289	+	0.996998i	$0.475329\pi$
$384$		0			0
$385$	14.9282	+	14.9282i	0.760812	+	0.760812i
$386$		0			0
$387$	−5.58846	−	9.67949i	−0.284077	−	0.492036i
$388$		0			0
$389$			10.6603i			0.540496i	0.962791	+	0.270248i	$0.0871057\pi$
$389$			10.6603i			0.540496i	−0.962791	+	0.270248i	$0.912894\pi$
$390$		0			0
$391$			34.0526i			1.72211i
$392$		0			0
$393$	5.19615	+	9.00000i	0.262111	+	0.453990i
$394$		0			0
$395$	3.46410	+	3.46410i	0.174298	+	0.174298i
$396$		0			0
$397$	−4.56218	+	1.22243i	−0.228969	+	0.0613521i	−0.371479	−	0.928441i	$-0.621149\pi$
$397$	−4.56218	+	1.22243i	−0.228969	+	0.0613521i	0.142510	+	0.989793i	$0.454483\pi$
$398$		0			0
$399$	−15.4641	−	8.92820i	−0.774173	−	0.446969i
$400$		0			0
$401$	3.03590	−	11.3301i	0.151606	−	0.565800i	−0.847767	−	0.530369i	$-0.822053\pi$
$401$	3.03590	−	11.3301i	0.151606	−	0.565800i	0.999372	−	0.0354301i	$-0.0112801\pi$
$402$		0			0
$403$	19.4904	+	11.7583i	0.970885	+	0.585724i
$404$		0			0
$405$	8.33013	+	2.23205i	0.413927	+	0.110911i
$406$		0			0
$407$	8.73205	−	15.1244i	0.432832	−	0.749686i
$408$		0			0
$409$	−1.25833	−	4.69615i	−0.0622204	−	0.232210i	0.927812	−	0.373047i	$-0.121687\pi$
$409$	−1.25833	−	4.69615i	−0.0622204	−	0.232210i	−0.990033	+	0.140838i	$0.955020\pi$
$410$		0			0
$411$	0.803848	−	0.803848i	0.0396509	−	0.0396509i
$412$		0			0
$413$	29.3205	−	16.9282i	1.44277	−	0.832982i
$414$		0			0
$415$	11.4641			0.562751
$416$		0			0
$417$	15.3205			0.750249
$418$		0			0
$419$	−19.9019	+	11.4904i	−0.972273	+	0.561342i	−0.899928	−	0.436038i	$-0.856381\pi$
$419$	−19.9019	+	11.4904i	−0.972273	+	0.561342i	−0.0723443	+	0.997380i	$0.523048\pi$
$420$		0			0
$421$	19.2942	−	19.2942i	0.940343	−	0.940343i	−0.0579749	−	0.998318i	$-0.518464\pi$
$421$	19.2942	−	19.2942i	0.940343	−	0.940343i	0.998318	+	0.0579749i	$0.0184643\pi$
$422$		0			0
$423$	−4.02628	−	15.0263i	−0.195764	−	0.730603i
$424$		0			0
$425$	4.09808	−	7.09808i	0.198786	−	0.344307i
$426$		0			0
$427$	39.7846	+	10.6603i	1.92531	+	0.515886i
$428$		0			0
$429$	6.53590	−	3.60770i	0.315556	−	0.174181i
$430$		0			0
$431$	0.875644	−	3.26795i	0.0421783	−	0.157412i	−0.941625	−	0.336664i	$-0.890701\pi$
$431$	0.875644	−	3.26795i	0.0421783	−	0.157412i	0.983803	+	0.179253i	$0.0573679\pi$
$432$		0			0
$433$	−28.2846	−	16.3301i	−1.35927	−	0.784776i	−0.369746	−	0.929133i	$-0.620555\pi$
$433$	−28.2846	−	16.3301i	−1.35927	−	0.784776i	−0.989526	+	0.144357i	$0.953889\pi$
$434$		0			0
$435$	3.36603	−	0.901924i	0.161389	−	0.0432439i
$436$		0			0
$437$	23.5167	+	23.5167i	1.12495	+	1.12495i
$438$		0			0
$439$	−1.53590	−	2.66025i	−0.0733044	−	0.126967i	0.827043	−	0.562138i	$-0.190021\pi$
$439$	−1.53590	−	2.66025i	−0.0733044	−	0.126967i	−0.900348	+	0.435171i	$0.856688\pi$
$440$		0			0
$441$		−	19.5359i		−	0.930281i
$442$		0			0
$443$			17.8564i			0.848383i	0.905572	+	0.424192i	$0.139442\pi$
$443$			17.8564i			0.848383i	−0.905572	+	0.424192i	$0.860558\pi$
$444$		0			0
$445$	9.09808	+	15.7583i	0.431290	+	0.747016i
$446$		0			0
$447$	−8.26795	−	8.26795i	−0.391061	−	0.391061i
$448$		0			0
$449$	30.9545	−	8.29423i	1.46083	−	0.391429i	0.561055	−	0.827778i	$-0.310396\pi$
$449$	30.9545	−	8.29423i	1.46083	−	0.391429i	0.899777	+	0.436350i	$0.143729\pi$
$450$		0			0
$451$	22.9808	+	13.2679i	1.08212	+	0.624763i
$452$		0			0
$453$	1.73205	−	6.46410i	0.0813788	−	0.303710i
$454$		0			0
$455$	−25.8564	−	7.46410i	−1.21217	−	0.349922i
$456$		0			0
$457$	−14.2583	−	3.82051i	−0.666977	−	0.178716i	−0.0905841	−	0.995889i	$-0.528873\pi$
$457$	−14.2583	−	3.82051i	−0.666977	−	0.178716i	−0.576393	+	0.817173i	$0.695540\pi$
$458$		0			0
$459$	−12.9282	+	22.3923i	−0.603437	+	1.04518i
$460$		0			0
$461$	8.66987	+	32.3564i	0.403796	+	1.50699i	0.806265	+	0.591554i	$0.201486\pi$
$461$	8.66987	+	32.3564i	0.403796	+	1.50699i	−0.402469	+	0.915434i	$0.631848\pi$
$462$		0			0
$463$	10.7321	−	10.7321i	0.498761	−	0.498761i	−0.412291	−	0.911052i	$-0.635271\pi$
$463$	10.7321	−	10.7321i	0.498761	−	0.498761i	0.911052	+	0.412291i	$0.135271\pi$
$464$		0			0
$465$	−7.73205	+	4.46410i	−0.358565	+	0.207018i
$466$		0			0
$467$	−13.8564			−0.641198			−0.320599	−	0.947215i	$-0.603884\pi$
$467$	−13.8564			−0.641198			−0.320599	+	0.947215i	$0.603884\pi$
$468$		0			0
$469$	2.53590			0.117097
$470$		0			0
$471$	9.84936	−	5.68653i	0.453835	−	0.262022i
$472$		0			0
$473$	−9.07180	+	9.07180i	−0.417122	+	0.417122i
$474$		0			0
$475$	−2.07180	−	7.73205i	−0.0950606	−	0.354771i
$476$		0			0
$477$	2.13397	−	3.69615i	0.0977080	−	0.169235i
$478$		0			0
$479$	5.63397	+	1.50962i	0.257423	+	0.0689763i	0.385222	−	0.922824i	$-0.374125\pi$
$479$	5.63397	+	1.50962i	0.257423	+	0.0689763i	−0.127800	+	0.991800i	$0.540791\pi$
$480$		0			0
$481$	−0.428203	+	22.2583i	−0.0195244	+	1.01489i
$482$		0			0
$483$	3.85641	−	14.3923i	0.175473	−	0.654873i
$484$		0			0
$485$	−18.2942	−	10.5622i	−0.830698	−	0.479604i
$486$		0			0
$487$	−5.00000	+	1.33975i	−0.226572	+	0.0607097i	−0.370319	−	0.928905i	$-0.620752\pi$
$487$	−5.00000	+	1.33975i	−0.226572	+	0.0607097i	0.143747	+	0.989614i	$0.454085\pi$
$488$		0			0
$489$	8.53590	+	8.53590i	0.386007	+	0.386007i
$490$		0			0
$491$	5.53590	+	9.58846i	0.249832	+	0.432721i	0.963479	−	0.267784i	$-0.0862914\pi$
$491$	5.53590	+	9.58846i	0.249832	+	0.432721i	−0.713647	+	0.700505i	$0.752958\pi$
$492$		0			0
$493$		−	15.9282i		−	0.717370i
$494$		0			0
$495$		−	13.4641i		−	0.605166i
$496$		0			0
$497$	−19.1244	−	33.1244i	−0.857845	−	1.48583i
$498$		0			0
$499$	−7.39230	−	7.39230i	−0.330925	−	0.330925i	0.522013	−	0.852938i	$-0.325181\pi$
$499$	−7.39230	−	7.39230i	−0.330925	−	0.330925i	−0.852938	+	0.522013i	$0.825181\pi$
$500$		0			0
$501$	−4.53590	+	1.21539i	−0.202649	+	0.0542996i
$502$		0			0
$503$	−28.3923	−	16.3923i	−1.26595	−	0.730897i	−0.291731	−	0.956500i	$-0.594231\pi$
$503$	−28.3923	−	16.3923i	−1.26595	−	0.730897i	−0.974219	+	0.225604i	$0.927565\pi$
$504$		0			0
$505$	−1.03590	+	3.86603i	−0.0460969	+	0.172036i
$506$		0			0
$507$	−5.07180	+	8.05256i	−0.225246	+	0.357627i
$508$		0			0
$509$	3.96410	+	1.06218i	0.175706	+	0.0470802i	0.345599	−	0.938382i	$-0.387676\pi$
$509$	3.96410	+	1.06218i	0.175706	+	0.0470802i	−0.169893	+	0.985462i	$0.554342\pi$
$510$		0			0
$511$	17.1962	−	29.7846i	0.760713	−	1.31759i
$512$		0			0
$513$	6.53590	+	24.3923i	0.288567	+	1.07695i
$514$		0			0
$515$	4.46410	−	4.46410i	0.196712	−	0.196712i
$516$		0			0
$517$	−15.4641	+	8.92820i	−0.680110	+	0.392662i
$518$		0			0
$519$	10.6410			0.467089
$520$		0			0
$521$	−38.6603			−1.69374			−0.846868	−	0.531803i	$-0.821515\pi$
$521$	−38.6603			−1.69374			−0.846868	+	0.531803i	$0.821515\pi$
$522$		0			0
$523$	−8.32051	+	4.80385i	−0.363830	+	0.210058i	−0.670760	−	0.741675i	$-0.734032\pi$
$523$	−8.32051	+	4.80385i	−0.363830	+	0.210058i	0.306929	+	0.951732i	$0.400698\pi$
$524$		0			0
$525$	−2.53590	+	2.53590i	−0.110676	+	0.110676i
$526$		0			0
$527$	10.5622	+	39.4186i	0.460096	+	1.71710i
$528$		0			0
$529$	−2.37564	+	4.11474i	−0.103289	+	0.178902i
$530$		0			0
$531$	−20.8564	−	5.58846i	−0.905091	−	0.242518i
$532$		0			0
$533$	−33.8205	−	0.650635i	−1.46493	−	0.0281821i
$534$		0			0
$535$	−1.36603	+	5.09808i	−0.0590584	+	0.220409i
$536$		0			0
$537$	0.928203	+	0.535898i	0.0400549	+	0.0231257i
$538$		0			0
$539$	−21.6603	+	5.80385i	−0.932973	+	0.249989i
$540$		0			0
$541$	−31.5622	−	31.5622i	−1.35696	−	1.35696i	−0.877636	−	0.479328i	$-0.840880\pi$
$541$	−31.5622	−	31.5622i	−1.35696	−	1.35696i	−0.479328	−	0.877636i	$-0.659120\pi$
$542$		0			0
$543$	1.16987	+	2.02628i	0.0502041	+	0.0869560i
$544$		0			0
$545$		−	2.19615i		−	0.0940728i
$546$		0			0
$547$			7.80385i			0.333668i	0.985985	+	0.166834i	$0.0533545\pi$
$547$			7.80385i			0.333668i	−0.985985	+	0.166834i	$0.946646\pi$
$548$		0			0
$549$	−13.1340	−	22.7487i	−0.560544	−	0.970891i
$550$		0			0
$551$	−11.0000	−	11.0000i	−0.468616	−	0.468616i
$552$		0			0
$553$	−9.46410	+	2.53590i	−0.402455	+	0.107837i
$554$		0			0
$555$	−7.56218	−	4.36603i	−0.320997	−	0.185327i
$556$		0			0
$557$	3.03590	−	11.3301i	0.128635	−	0.480073i	−0.871308	−	0.490737i	$-0.836728\pi$
$557$	3.03590	−	11.3301i	0.128635	−	0.480073i	0.999943	+	0.0106637i	$0.00339442\pi$
$558$		0			0
$559$	4.53590	−	15.7128i	0.191848	−	0.664581i
$560$		0			0
$561$	12.9282	+	3.46410i	0.545829	+	0.146254i
$562$		0			0
$563$	−6.46410	+	11.1962i	−0.272429	+	0.471862i	−0.969483	−	0.245157i	$-0.921160\pi$
$563$	−6.46410	+	11.1962i	−0.272429	+	0.471862i	0.697054	+	0.717019i	$0.254494\pi$
$564$		0			0
$565$	3.59808	+	13.4282i	0.151372	+	0.564929i
$566$		0			0
$567$	−12.1962	+	12.1962i	−0.512190	+	0.512190i
$568$		0			0
$569$	−15.0000	+	8.66025i	−0.628833	+	0.363057i	−0.780300	−	0.625406i	$-0.784934\pi$
$569$	−15.0000	+	8.66025i	−0.628833	+	0.363057i	0.151467	+	0.988462i	$0.451600\pi$
$570$		0			0
$571$	43.3205			1.81291			0.906453	−	0.422306i	$-0.138779\pi$
$571$	43.3205			1.81291			0.906453	+	0.422306i	$0.138779\pi$
$572$		0			0
$573$	16.1436			0.674408
$574$		0			0
$575$	5.78461	−	3.33975i	0.241235	−	0.139277i
$576$		0			0
$577$	19.1506	−	19.1506i	0.797251	−	0.797251i	−0.185410	−	0.982661i	$-0.559361\pi$
$577$	19.1506	−	19.1506i	0.797251	−	0.797251i	0.982661	+	0.185410i	$0.0593613\pi$
$578$		0			0
$579$	0.973721	+	3.63397i	0.0404664	+	0.151023i
$580$		0			0
$581$	−11.4641	+	19.8564i	−0.475611	+	0.823783i
$582$		0			0
$583$	−4.73205	−	1.26795i	−0.195982	−	0.0525131i
$584$		0			0
$585$	8.29423	+	15.0263i	0.342924	+	0.621260i
$586$		0			0
$587$	9.73205	−	36.3205i	0.401685	−	1.49911i	−0.408404	−	0.912801i	$-0.633915\pi$
$587$	9.73205	−	36.3205i	0.401685	−	1.49911i	0.810089	−	0.586307i	$-0.199419\pi$
$588$		0			0
$589$	34.5167	+	19.9282i	1.42223	+	0.821127i
$590$		0			0
$591$	2.80385	−	0.751289i	0.115335	−	0.0309039i
$592$		0			0
$593$	20.8301	+	20.8301i	0.855391	+	0.855391i	0.990791	−	0.135400i	$-0.0432320\pi$
$593$	20.8301	+	20.8301i	0.855391	+	0.855391i	−0.135400	+	0.990791i	$0.543232\pi$
$594$		0			0
$595$	−24.1244	−	41.7846i	−0.989002	−	1.71300i
$596$		0			0
$597$			10.6795i			0.437083i
$598$		0			0
$599$		−	22.0526i		−	0.901043i	−0.892766	−	0.450522i	$-0.851238\pi$
$599$		−	22.0526i		−	0.901043i	0.892766	−	0.450522i	$-0.148762\pi$
$600$		0			0
$601$	−0.330127	−	0.571797i	−0.0134662	−	0.0233241i	0.859214	−	0.511617i	$-0.170953\pi$
$601$	−0.330127	−	0.571797i	−0.0134662	−	0.0233241i	−0.872680	+	0.488293i	$0.837620\pi$
$602$		0			0
$603$	−1.14359	−	1.14359i	−0.0465707	−	0.0465707i
$604$		0			0
$605$	5.59808	−	1.50000i	0.227594	−	0.0609837i
$606$		0			0
$607$	−27.8827	−	16.0981i	−1.13172	−	0.653401i	−0.187356	−	0.982292i	$-0.559992\pi$
$607$	−27.8827	−	16.0981i	−1.13172	−	0.653401i	−0.944368	+	0.328891i	$0.893325\pi$
$608$		0			0
$609$	−1.80385	+	6.73205i	−0.0730956	+	0.272796i
$610$		0			0
$611$	11.7583	−	19.4904i	0.475691	−	0.788496i
$612$		0			0
$613$	−31.9186	−	8.55256i	−1.28918	−	0.345435i	−0.451831	−	0.892104i	$-0.649229\pi$
$613$	−31.9186	−	8.55256i	−1.28918	−	0.345435i	−0.837349	+	0.546669i	$0.815896\pi$
$614$		0			0
$615$	6.63397	−	11.4904i	0.267508	−	0.463337i
$616$		0			0
$617$	−3.37564	−	12.5981i	−0.135898	−	0.507179i	−0.999993	−	0.00384944i	$-0.998775\pi$
$617$	−3.37564	−	12.5981i	−0.135898	−	0.507179i	0.864094	−	0.503330i	$-0.167892\pi$
$618$		0			0
$619$	−16.7321	+	16.7321i	−0.672518	+	0.672518i	−0.958296	−	0.285778i	$-0.907748\pi$
$619$	−16.7321	+	16.7321i	−0.672518	+	0.672518i	0.285778	+	0.958296i	$0.407748\pi$
$620$		0			0
$621$	−18.2487	+	10.5359i	−0.732296	+	0.422791i
$622$		0			0
$623$	−36.3923			−1.45803
$624$		0			0
$625$	17.0526			0.682102
$626$		0			0
$627$	11.3205	−	6.53590i	0.452098	−	0.261019i
$628$		0			0
$629$	−28.2224	+	28.2224i	−1.12530	+	1.12530i
$630$		0			0
$631$	3.32051	+	12.3923i	0.132187	+	0.493330i	0.999994	−	0.00357280i	$-0.00113726\pi$
$631$	3.32051	+	12.3923i	0.132187	+	0.493330i	−0.867806	+	0.496903i	$0.834471\pi$
$632$		0			0
$633$	9.14359	−	15.8372i	0.363425	−	0.629471i
$634$		0			0
$635$	−30.2224	−	8.09808i	−1.19934	−	0.321362i
$636$		0			0
$637$	20.5981	−	19.8205i	0.816125	−	0.785317i
$638$		0			0
$639$	−6.31347	+	23.5622i	−0.249757	+	0.932105i
$640$		0			0
$641$	25.6699	+	14.8205i	1.01390	+	0.585375i	0.912331	−	0.409454i	$-0.134281\pi$
$641$	25.6699	+	14.8205i	1.01390	+	0.585375i	0.101568	+	0.994829i	$0.467614\pi$
$642$		0			0
$643$	−14.1962	+	3.80385i	−0.559842	+	0.150009i	−0.527634	−	0.849472i	$-0.676921\pi$
$643$	−14.1962	+	3.80385i	−0.559842	+	0.150009i	−0.0322080	+	0.999481i	$0.510254\pi$
$644$		0			0
$645$	4.53590	+	4.53590i	0.178601	+	0.178601i
$646$		0			0
$647$	9.49038	+	16.4378i	0.373105	+	0.646237i	0.990042	−	0.140776i	$-0.0449597\pi$
$647$	9.49038	+	16.4378i	0.373105	+	0.646237i	−0.616936	+	0.787013i	$0.711626\pi$
$648$		0			0
$649$			24.7846i			0.972881i
$650$		0			0
$651$		−	17.8564i		−	0.699848i
$652$		0			0
$653$	8.53590	+	14.7846i	0.334036	+	0.578566i	0.983299	−	0.181997i	$-0.0582561\pi$
$653$	8.53590	+	14.7846i	0.334036	+	0.578566i	−0.649264	+	0.760563i	$0.724923\pi$
$654$		0			0
$655$	19.3923	+	19.3923i	0.757720	+	0.757720i
$656$		0			0
$657$	−21.1865	+	5.67691i	−0.826565	+	0.221478i
$658$		0			0
$659$	16.0526	+	9.26795i	0.625319	+	0.361028i	0.778937	−	0.627102i	$-0.215759\pi$
$659$	16.0526	+	9.26795i	0.625319	+	0.361028i	−0.153618	+	0.988130i	$0.549093\pi$
$660$		0			0
$661$	4.54552	−	16.9641i	0.176800	−	0.659827i	−0.819438	−	0.573168i	$-0.805714\pi$
$661$	4.54552	−	16.9641i	0.176800	−	0.659827i	0.996238	−	0.0866591i	$-0.0276191\pi$
$662$		0			0
$663$	−16.5622	+	4.09808i	−0.643222	+	0.159156i
$664$		0			0
$665$	−45.5167	−	12.1962i	−1.76506	−	0.472947i
$666$		0			0
$667$	6.49038	−	11.2417i	0.251309	−	0.435279i
$668$		0			0
$669$	−5.53590	−	20.6603i	−0.214030	−	0.798772i
$670$		0			0
$671$	−21.3205	+	21.3205i	−0.823069	+	0.823069i
$672$		0			0
$673$	−30.6506	+	17.6962i	−1.18150	+	0.682137i	−0.956360	−	0.292191i	$-0.905616\pi$
$673$	−30.6506	+	17.6962i	−1.18150	+	0.682137i	−0.225135	+	0.974328i	$0.572282\pi$
$674$		0			0
$675$	5.07180			0.195214
$676$		0			0
$677$	−5.07180			−0.194925			−0.0974625	−	0.995239i	$-0.531073\pi$
$677$	−5.07180			−0.194925			−0.0974625	+	0.995239i	$0.531073\pi$
$678$		0			0
$679$	36.5885	−	21.1244i	1.40414	−	0.810678i
$680$		0			0
$681$	−2.33975	+	2.33975i	−0.0896593	+	0.0896593i
$682$		0			0
$683$	−10.6340	−	39.6865i	−0.406898	−	1.51856i	−0.800529	−	0.599294i	$-0.795448\pi$
$683$	−10.6340	−	39.6865i	−0.406898	−	1.51856i	0.393631	−	0.919269i	$-0.371219\pi$
$684$		0			0
$685$	1.50000	−	2.59808i	0.0573121	−	0.0992674i
$686$		0			0
$687$	−12.2679	−	3.28719i	−0.468051	−	0.125414i
$688$		0			0
$689$	6.06218	−	1.50000i	0.230951	−	0.0571454i
$690$		0			0
$691$	−2.67949	+	10.0000i	−0.101933	+	0.380418i	−0.997979	−	0.0635424i	$-0.979760\pi$
$691$	−2.67949	+	10.0000i	−0.101933	+	0.380418i	0.896046	+	0.443960i	$0.146427\pi$
$692$		0			0
$693$	23.3205	+	13.4641i	0.885873	+	0.511459i
$694$		0			0
$695$	39.0526	−	10.4641i	1.48135	−	0.396926i
$696$		0			0
$697$	−42.8827	−	42.8827i	−1.62430	−	1.62430i
$698$		0			0
$699$	−5.66025	−	9.80385i	−0.214090	−	0.370816i
$700$		0			0
$701$			11.0718i			0.418176i	0.977897	+	0.209088i	$0.0670495\pi$
$701$			11.0718i			0.418176i	−0.977897	+	0.209088i	$0.932950\pi$
$702$		0			0
$703$			38.9808i			1.47019i
$704$		0			0
$705$	4.46410	+	7.73205i	0.168128	+	0.291206i
$706$		0			0
$707$	−5.66025	−	5.66025i	−0.212876	−	0.212876i
$708$		0			0
$709$	26.5263	−	7.10770i	0.996215	−	0.266935i	0.276356	−	0.961055i	$-0.410873\pi$
$709$	26.5263	−	7.10770i	0.996215	−	0.266935i	0.719859	+	0.694120i	$0.244206\pi$
$710$		0			0
$711$	5.41154	+	3.12436i	0.202949	+	0.117172i
$712$		0			0
$713$	−8.60770	+	32.1244i	−0.322361	+	1.20307i
$714$		0			0
$715$	14.1962	−	13.6603i	0.530906	−	0.510865i
$716$		0			0
$717$	−7.26795	−	1.94744i	−0.271427	−	0.0727285i
$718$		0			0
$719$	−5.53590	+	9.58846i	−0.206454	+	0.357589i	−0.950595	−	0.310434i	$-0.899526\pi$
$719$	−5.53590	+	9.58846i	−0.206454	+	0.357589i	0.744141	+	0.668023i	$0.232859\pi$
$720$		0			0
$721$	3.26795	+	12.1962i	0.121705	+	0.454208i
$722$		0			0
$723$	0.803848	−	0.803848i	0.0298954	−	0.0298954i
$724$		0			0
$725$	−2.70577	+	1.56218i	−0.100490	+	0.0580178i
$726$		0			0
$727$	3.21539			0.119252			0.0596261	−	0.998221i	$-0.481009\pi$
$727$	3.21539			0.119252			0.0596261	+	0.998221i	$0.481009\pi$
$728$		0			0
$729$	2.21539			0.0820515
$730$		0			0
$731$	25.3923	−	14.6603i	0.939168	−	0.542229i
$732$		0			0
$733$	−23.6147	+	23.6147i	−0.872230	+	0.872230i	−0.992715	−	0.120485i	$-0.961555\pi$
$733$	−23.6147	+	23.6147i	−0.872230	+	0.872230i	0.120485	+	0.992715i	$0.461555\pi$
$734$		0			0
$735$	2.90192	+	10.8301i	0.107039	+	0.399475i
$736$		0			0
$737$	−0.928203	+	1.60770i	−0.0341908	+	0.0592202i
$738$		0			0
$739$	−9.66025	−	2.58846i	−0.355358	−	0.0952179i	0.0767235	−	0.997052i	$-0.475554\pi$
$739$	−9.66025	−	2.58846i	−0.355358	−	0.0952179i	−0.432082	+	0.901834i	$0.642221\pi$
$740$		0			0
$741$	−8.60770	+	14.2679i	−0.316212	+	0.524146i
$742$		0			0
$743$	−5.39230	+	20.1244i	−0.197824	+	0.738291i	0.793693	+	0.608318i	$0.208156\pi$
$743$	−5.39230	+	20.1244i	−0.197824	+	0.738291i	−0.991518	+	0.129973i	$0.958511\pi$
$744$		0			0
$745$	−26.7224	−	15.4282i	−0.979034	−	0.565246i
$746$		0			0
$747$	14.1244	−	3.78461i	0.516783	−	0.138472i
$748$		0			0
$749$	−7.46410	−	7.46410i	−0.272732	−	0.272732i
$750$		0			0
$751$	0.124356	+	0.215390i	0.00453780	+	0.00785970i	0.868285	−	0.496065i	$-0.165222\pi$
$751$	0.124356	+	0.215390i	0.00453780	+	0.00785970i	−0.863748	+	0.503925i	$0.831889\pi$
$752$		0			0
$753$			6.78461i			0.247245i
$754$		0			0
$755$		−	17.6603i		−	0.642722i
$756$		0			0
$757$	−19.3923	−	33.5885i	−0.704825	−	1.22079i	−0.966755	−	0.255706i	$-0.917692\pi$
$757$	−19.3923	−	33.5885i	−0.704825	−	1.22079i	0.261929	−	0.965087i	$-0.415641\pi$
$758$		0			0
$759$	7.71281	+	7.71281i	0.279957	+	0.279957i
$760$		0			0
$761$	3.83013	−	1.02628i	0.138842	−	0.0372026i	−0.188729	−	0.982029i	$-0.560437\pi$
$761$	3.83013	−	1.02628i	0.138842	−	0.0372026i	0.327571	+	0.944827i	$0.393770\pi$
$762$		0			0
$763$	3.80385	+	2.19615i	0.137709	+	0.0795061i
$764$		0			0
$765$	−7.96410	+	29.7224i	−0.287943	+	1.07462i
$766$		0			0
$767$	−15.2679	−	27.6603i	−0.551294	−	0.998754i
$768$		0			0
$769$	−3.09808	−	0.830127i	−0.111719	−	0.0299351i	0.202526	−	0.979277i	$-0.435085\pi$
$769$	−3.09808	−	0.830127i	−0.111719	−	0.0299351i	−0.314246	+	0.949342i	$0.601752\pi$
$770$		0			0
$771$	1.36603	−	2.36603i	0.0491962	−	0.0852103i
$772$		0			0
$773$	2.22243	+	8.29423i	0.0799353	+	0.298323i	0.994307	−	0.106553i	$-0.0339814\pi$
$773$	2.22243	+	8.29423i	0.0799353	+	0.298323i	−0.914372	+	0.404876i	$0.867315\pi$
$774$		0			0
$775$	5.66025	−	5.66025i	0.203322	−	0.203322i
$776$		0			0
$777$	15.1244	−	8.73205i	0.542583	−	0.313261i
$778$		0			0
$779$	−59.2295			−2.12212
$780$		0			0
$781$	28.0000			1.00192
$782$		0			0
$783$	8.53590	−	4.92820i	0.305048	−	0.176120i
$784$		0			0
$785$	21.2224	−	21.2224i	0.757461	−	0.757461i
$786$		0			0
$787$	3.04552	+	11.3660i	0.108561	+	0.405155i	0.998725	−	0.0504856i	$-0.0160769\pi$
$787$	3.04552	+	11.3660i	0.108561	+	0.405155i	−0.890164	+	0.455641i	$0.849410\pi$
$788$		0			0
$789$	9.12436	−	15.8038i	0.324836	−	0.562632i
$790$		0			0
$791$	−26.8564	−	7.19615i	−0.954904	−	0.255866i
$792$		0			0
$793$	10.6603	−	36.9282i	0.378557	−	1.31136i
$794$		0			0
$795$	−0.633975	+	2.36603i	−0.0224848	+	0.0839143i
$796$		0			0
$797$	4.85641	+	2.80385i	0.172023	+	0.0993174i	0.583539	−	0.812085i	$-0.301667\pi$
$797$	4.85641	+	2.80385i	0.172023	+	0.0993174i	−0.411517	+	0.911402i	$0.635001\pi$
$798$		0			0
$799$	39.4186	−	10.5622i	1.39453	−	0.373663i
$800$		0			0
$801$	16.4115	+	16.4115i	0.579873	+	0.579873i
$802$		0			0
$803$	12.5885	+	21.8038i	0.444237	+	0.769441i
$804$		0			0
$805$		−	39.3205i		−	1.38587i
$806$		0			0
$807$			5.85641i			0.206155i
$808$		0			0
$809$	4.79423	+	8.30385i	0.168556	+	0.291948i	0.937912	−	0.346872i	$-0.112756\pi$
$809$	4.79423	+	8.30385i	0.168556	+	0.291948i	−0.769356	+	0.638820i	$0.779423\pi$
$810$		0			0
$811$	10.6077	+	10.6077i	0.372487	+	0.372487i	0.868382	−	0.495896i	$-0.165160\pi$
$811$	10.6077	+	10.6077i	0.372487	+	0.372487i	−0.495896	+	0.868382i	$0.665160\pi$
$812$		0			0
$813$	12.5885	−	3.37307i	0.441496	−	0.118299i
$814$		0			0
$815$	27.5885	+	15.9282i	0.966382	+	0.557941i
$816$		0			0
$817$	7.41154	−	27.6603i	0.259297	−	0.967710i
$818$		0			0
$819$	−34.3205	−	0.660254i	−1.19926	−	0.0230711i
$820$		0			0
$821$	20.9545	+	5.61474i	0.731316	+	0.195956i	0.605215	−	0.796062i	$-0.293087\pi$
$821$	20.9545	+	5.61474i	0.731316	+	0.195956i	0.126101	+	0.992017i	$0.459754\pi$
$822$		0			0
$823$	−19.0000	+	32.9090i	−0.662298	+	1.14713i	0.317712	+	0.948187i	$0.397086\pi$
$823$	−19.0000	+	32.9090i	−0.662298	+	1.14713i	−0.980010	+	0.198947i	$0.936248\pi$
$824$		0			0
$825$	−0.679492	−	2.53590i	−0.0236569	−	0.0882886i
$826$		0			0
$827$	23.3205	−	23.3205i	0.810934	−	0.810934i	−0.173840	−	0.984774i	$-0.555618\pi$
$827$	23.3205	−	23.3205i	0.810934	−	0.810934i	0.984774	+	0.173840i	$0.0556177\pi$
$828$		0			0
$829$	35.0429	−	20.2321i	1.21709	−	0.702688i	0.252797	−	0.967519i	$-0.418650\pi$
$829$	35.0429	−	20.2321i	1.21709	−	0.702688i	0.964295	+	0.264831i	$0.0853162\pi$
$830$		0			0
$831$	6.48334			0.224905
$832$		0			0
$833$	51.2487			1.77566
$834$		0			0
$835$	−10.7321	+	6.19615i	−0.371398	+	0.214427i
$836$		0			0
$837$	−17.8564	+	17.8564i	−0.617208	+	0.617208i
$838$		0			0
$839$	−3.90192	−	14.5622i	−0.134709	−	0.502742i	−0.999999	−	0.00146984i	$-0.999532\pi$
$839$	−3.90192	−	14.5622i	−0.134709	−	0.502742i	0.865290	−	0.501272i	$-0.167135\pi$
$840$		0			0
$841$	11.4641	−	19.8564i	0.395314	−	0.684704i
$842$		0			0
$843$	11.6340	+	3.11731i	0.400695	+	0.107366i
$844$		0			0
$845$	−7.42820	+	23.9904i	−0.255538	+	0.825294i
$846$		0			0
$847$	−3.00000	+	11.1962i	−0.103081	+	0.384704i
$848$		0			0
$849$	8.87564	+	5.12436i	0.304611	+	0.175867i
$850$		0			0
$851$	−31.4186	+	8.41858i	−1.07702	+	0.288585i
$852$		0			0
$853$	−25.2942	−	25.2942i	−0.866058	−	0.866058i	0.125975	−	0.992033i	$-0.459794\pi$
$853$	−25.2942	−	25.2942i	−0.866058	−	0.866058i	−0.992033	+	0.125975i	$0.959794\pi$
$854$		0			0
$855$	15.0263	+	26.0263i	0.513888	+	0.890080i
$856$		0			0
$857$			47.0526i			1.60728i	0.595113	+	0.803642i	$0.297107\pi$
$857$			47.0526i			1.60728i	−0.595113	+	0.803642i	$0.702893\pi$
$858$		0			0
$859$			13.2679i			0.452697i	0.974046	+	0.226348i	$0.0726788\pi$
$859$			13.2679i			0.452697i	−0.974046	+	0.226348i	$0.927321\pi$
$860$		0			0
$861$	13.2679	+	22.9808i	0.452170	+	0.783182i
$862$		0			0
$863$	29.8038	+	29.8038i	1.01453	+	1.01453i	0.999893	+	0.0146420i	$0.00466085\pi$
$863$	29.8038	+	29.8038i	1.01453	+	1.01453i	0.0146420	+	0.999893i	$0.495339\pi$
$864$		0			0
$865$	27.1244	−	7.26795i	0.922256	−	0.247118i
$866$		0			0
$867$	−15.7128	−	9.07180i	−0.533635	−	0.308094i
$868$		0			0
$869$	1.85641	−	6.92820i	0.0629743	−	0.235023i
$870$		0			0
$871$	0.0455173	−	2.36603i	0.00154230	−	0.0801697i
$872$		0			0
$873$	−26.0263	−	6.97372i	−0.880856	−	0.236025i
$874$		0			0
$875$	−23.3923	+	40.5167i	−0.790804	+	1.36971i
$876$		0			0
$877$	−1.83975	−	6.86603i	−0.0621238	−	0.231849i	0.927882	−	0.372873i	$-0.121627\pi$
$877$	−1.83975	−	6.86603i	−0.0621238	−	0.231849i	−0.990006	+	0.141024i	$0.954961\pi$
$878$		0			0
$879$	0.124356	−	0.124356i	0.00419441	−	0.00419441i
$880$		0			0
$881$	−11.8923	+	6.86603i	−0.400662	+	0.231322i	−0.686770	−	0.726875i	$-0.740972\pi$
$881$	−11.8923	+	6.86603i	−0.400662	+	0.231322i	0.286108	+	0.958197i	$0.407639\pi$
$882$		0			0
$883$	24.3397			0.819098			0.409549	−	0.912288i	$-0.365686\pi$
$883$	24.3397			0.819098			0.409549	+	0.912288i	$0.365686\pi$
$884$		0			0
$885$	12.3923			0.416563
$886$		0			0
$887$	−16.3923	+	9.46410i	−0.550400	+	0.317773i	−0.749283	−	0.662250i	$-0.769602\pi$
$887$	−16.3923	+	9.46410i	−0.550400	+	0.317773i	0.198883	+	0.980023i	$0.436268\pi$
$888$		0			0
$889$	44.2487	−	44.2487i	1.48405	−	1.48405i
$890$		0			0
$891$	−3.26795	−	12.1962i	−0.109480	−	0.408586i
$892$		0			0
$893$	19.9282	−	34.5167i	0.666872	−	1.15506i
$894$		0			0
$895$	2.73205	+	0.732051i	0.0913224	+	0.0244698i
$896$		0			0
$897$	−13.3590	−	3.85641i	−0.446043	−	0.128762i
$898$		0			0
$899$	4.02628	−	15.0263i	0.134284	−	0.501154i
$900$		0			0
$901$	9.69615	+	5.59808i	0.323026	+	0.186499i
$902$		0			0
$903$	−12.3923	+	3.32051i	−0.412390	+	0.110500i
$904$		0			0
$905$	4.36603	+	4.36603i	0.145132	+	0.145132i
$906$		0			0
$907$	15.0526	+	26.0718i	0.499812	+	0.865700i	1.00000		0.000217054i	$-6.90903e-5\pi$
$907$	15.0526	+	26.0718i	0.499812	+	0.865700i	−0.500188	+	0.865917i	$0.666736\pi$
$908$		0			0
$909$			5.10512i			0.169326i
$910$		0			0
$911$			43.1769i			1.43051i	0.698861	+	0.715257i	$0.253691\pi$
$911$			43.1769i			1.43051i	−0.698861	+	0.715257i	$0.746309\pi$
$912$		0			0
$913$	−8.39230	−	14.5359i	−0.277745	−	0.481068i
$914$		0			0
$915$	10.6603	+	10.6603i	0.352417	+	0.352417i
$916$		0			0
$917$	−52.9808	+	14.1962i	−1.74958	+	0.468798i
$918$		0			0
$919$	7.56218	+	4.36603i	0.249453	+	0.144022i	0.619514	−	0.784986i	$-0.287330\pi$
$919$	7.56218	+	4.36603i	0.249453	+	0.144022i	−0.370061	+	0.929008i	$0.620663\pi$
$920$		0			0
$921$	−4.21539	+	15.7321i	−0.138902	+	0.518389i
$922$		0			0
$923$	−31.2487	+	17.2487i	−1.02856	+	0.567748i
$924$		0			0
$925$	7.56218	+	2.02628i	0.248643	+	0.0666237i
$926$		0			0
$927$	4.02628	−	6.97372i	0.132240	−	0.229047i
$928$		0			0
$929$	1.65064	+	6.16025i	0.0541556	+	0.202111i	0.987703	−	0.156343i	$-0.0499705\pi$
$929$	1.65064	+	6.16025i	0.0541556	+	0.202111i	−0.933547	+	0.358454i	$0.883304\pi$
$930$		0			0
$931$	35.3923	−	35.3923i	1.15994	−	1.15994i
$932$		0			0
$933$	−4.17691	+	2.41154i	−0.136746	+	0.0789504i
$934$		0			0
$935$	35.3205			1.15510
$936$		0			0
$937$	31.0000			1.01273			0.506363	−	0.862320i	$-0.330990\pi$
$937$	31.0000			1.01273			0.506363	+	0.862320i	$0.330990\pi$
$938$		0			0
$939$	2.87564	−	1.66025i	0.0938431	−	0.0541803i
$940$		0			0
$941$	39.7846	−	39.7846i	1.29694	−	1.29694i	0.366538	−	0.930403i	$-0.380543\pi$
$941$	39.7846	−	39.7846i	1.29694	−	1.29694i	0.930403	−	0.366538i	$-0.119457\pi$
$942$		0			0
$943$	−12.7917	−	47.7391i	−0.416553	−	1.55460i
$944$		0			0
$945$	14.9282	−	25.8564i	0.485614	−	0.841109i
$946$		0			0
$947$	−2.16987	−	0.581416i	−0.0705114	−	0.0188935i	0.223391	−	0.974729i	$-0.428287\pi$
$947$	−2.16987	−	0.581416i	−0.0705114	−	0.0188935i	−0.293902	+	0.955835i	$0.594954\pi$
$948$		0			0
$949$	−27.4808	−	16.5788i	−0.892063	−	0.538172i
$950$		0			0
$951$	−0.830127	+	3.09808i	−0.0269187	+	0.100462i
$952$		0			0
$953$	−28.6410	−	16.5359i	−0.927774	−	0.535650i	−0.0416668	−	0.999132i	$-0.513267\pi$
$953$	−28.6410	−	16.5359i	−0.927774	−	0.535650i	−0.886107	+	0.463481i	$0.846600\pi$
$954$		0			0
$955$	41.1506	−	11.0263i	1.33160	−	0.356802i
$956$		0			0
$957$	−3.60770	−	3.60770i	−0.116620	−	0.116620i
$958$		0			0
$959$	3.00000	+	5.19615i	0.0968751	+	0.167793i
$960$		0			0
$961$			8.85641i			0.285691i
$962$		0			0
$963$			6.73205i			0.216937i
$964$		0			0
$965$	4.96410	+	8.59808i	0.159800	+	0.276782i
$966$		0			0
$967$	20.6603	+	20.6603i	0.664389	+	0.664389i	0.956411	−	0.292022i	$-0.0943282\pi$
$967$	20.6603	+	20.6603i	0.664389	+	0.664389i	−0.292022	+	0.956411i	$0.594328\pi$
$968$		0			0
$969$	−28.8564	+	7.73205i	−0.927001	+	0.248389i
$970$		0			0
$971$	49.0070	+	28.2942i	1.57271	+	0.908005i	0.995835	+	0.0911731i	$0.0290617\pi$
$971$	49.0070	+	28.2942i	1.57271	+	0.908005i	0.576876	+	0.816832i	$0.304272\pi$
$972$		0			0
$973$	−20.9282	+	78.1051i	−0.670927	+	2.50394i
$974$		0			0
$975$	2.32051	+	2.41154i	0.0743157	+	0.0772312i
$976$		0			0
$977$	−26.5263	−	7.10770i	−0.848651	−	0.227395i	−0.191817	−	0.981431i	$-0.561438\pi$
$977$	−26.5263	−	7.10770i	−0.848651	−	0.227395i	−0.656834	+	0.754035i	$0.728105\pi$
$978$		0			0
$979$	13.3205	−	23.0718i	0.425725	−	0.737378i
$980$		0			0
$981$	−0.725009	−	2.70577i	−0.0231478	−	0.0863886i
$982$		0			0
$983$	−17.7846	+	17.7846i	−0.567241	+	0.567241i	−0.931355	−	0.364114i	$-0.881372\pi$
$983$	−17.7846	+	17.7846i	−0.567241	+	0.567241i	0.364114	+	0.931355i	$0.381372\pi$
$984$		0			0
$985$	6.63397	−	3.83013i	0.211376	−	0.122038i
$986$		0			0
$987$	−17.8564			−0.568376
$988$		0			0
$989$	23.8949			0.759813
$990$		0			0
$991$	−26.8301	+	15.4904i	−0.852287	+	0.492068i	−0.861422	−	0.507890i	$-0.830426\pi$
$991$	−26.8301	+	15.4904i	−0.852287	+	0.492068i	0.00913480	+	0.999958i	$0.497092\pi$
$992$		0			0
$993$	16.0000	−	16.0000i	0.507745	−	0.507745i
$994$		0			0
$995$	7.29423	+	27.2224i	0.231243	+	0.863009i
$996$		0			0
$997$	−5.52628	+	9.57180i	−0.175019	+	0.303142i	−0.940168	−	0.340712i	$-0.889332\pi$
$997$	−5.52628	+	9.57180i	−0.175019	+	0.303142i	0.765149	+	0.643853i	$0.222665\pi$
$998$		0			0
$999$	−23.8564	−	6.39230i	−0.754783	−	0.202244i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	416.2.bu.a.223.1	✓	4
4.3	odd	2		416.2.bu.b.223.1	yes	4
8.3	odd	2		832.2.bu.b.639.1		4
8.5	even	2		832.2.bu.g.639.1		4
13.7	odd	12		416.2.bu.b.319.1	yes	4
52.7	even	12	inner	416.2.bu.a.319.1	yes	4
104.59	even	12		832.2.bu.g.319.1		4
104.85	odd	12		832.2.bu.b.319.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
416.2.bu.a.223.1	✓	4	1.1	even	1	trivial
416.2.bu.a.319.1	yes	4	52.7	even	12	inner
416.2.bu.b.223.1	yes	4	4.3	odd	2
416.2.bu.b.319.1	yes	4	13.7	odd	12
832.2.bu.b.319.1		4	104.85	odd	12
832.2.bu.b.639.1		4	8.3	odd	2
832.2.bu.g.319.1		4	104.59	even	12
832.2.bu.g.639.1		4	8.5	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.bu (of order \(12\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.633975 + 0.366025i) q^{3} +(-1.36603 + 1.36603i) q^{5} +(-1.00000 - 3.73205i) q^{7} +(-1.23205 + 2.13397i) q^{9} +(2.73205 + 0.732051i) q^{11} +(-3.50000 + 0.866025i) q^{13} +(0.366025 - 1.36603i) q^{15} +(-5.59808 - 3.23205i) q^{17} +(-6.09808 + 1.63397i) q^{19} +(2.00000 + 2.00000i) q^{21} +(-2.63397 - 4.56218i) q^{23} +1.26795i q^{25} -4.00000i q^{27} +(1.23205 + 2.13397i) q^{29} +(-4.46410 - 4.46410i) q^{31} +(-2.00000 + 0.535898i) q^{33} +(6.46410 + 3.73205i) q^{35} +(1.59808 - 5.96410i) q^{37} +(1.90192 - 1.83013i) q^{39} +(9.06218 + 2.42820i) q^{41} +(-2.26795 + 3.92820i) q^{43} +(-1.23205 - 4.59808i) q^{45} +(-4.46410 + 4.46410i) q^{47} +(-6.86603 + 3.96410i) q^{49} +4.73205 q^{51} -1.73205 q^{53} +(-4.73205 + 2.73205i) q^{55} +(3.26795 - 3.26795i) q^{57} +(2.26795 + 8.46410i) q^{59} +(-5.33013 + 9.23205i) q^{61} +(9.19615 + 2.46410i) q^{63} +(3.59808 - 5.96410i) q^{65} +(-0.169873 + 0.633975i) q^{67} +(3.33975 + 1.92820i) q^{69} +(9.56218 - 2.56218i) q^{71} +(6.29423 + 6.29423i) q^{73} +(-0.464102 - 0.803848i) q^{75} -10.9282i q^{77} -2.53590i q^{79} +(-2.23205 - 3.86603i) q^{81} +(-4.19615 - 4.19615i) q^{83} +(12.0622 - 3.23205i) q^{85} +(-1.56218 - 0.901924i) q^{87} +(2.43782 - 9.09808i) q^{89} +(6.73205 + 12.1962i) q^{91} +(4.46410 + 1.19615i) q^{93} +(6.09808 - 10.5622i) q^{95} +(2.83013 + 10.5622i) q^{97} +(-4.92820 + 4.92820i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{3} - 2 q^{5} - 4 q^{7} + 2 q^{9} + 4 q^{11} - 14 q^{13} - 2 q^{15} - 12 q^{17} - 14 q^{19} + 8 q^{21} - 14 q^{23} - 2 q^{29} - 4 q^{31} - 8 q^{33} + 12 q^{35} - 4 q^{37} + 18 q^{39} + 12 q^{41}+ \cdots + 8 q^{99}+O(q^{100}) \) 4 * q - 6 * q^3 - 2 * q^5 - 4 * q^7 + 2 * q^9 + 4 * q^11 - 14 * q^13 - 2 * q^15 - 12 * q^17 - 14 * q^19 + 8 * q^21 - 14 * q^23 - 2 * q^29 - 4 * q^31 - 8 * q^33 + 12 * q^35 - 4 * q^37 + 18 * q^39 + 12 * q^41 - 16 * q^43 + 2 * q^45 - 4 * q^47 - 24 * q^49 + 12 * q^51 - 12 * q^55 + 20 * q^57 + 16 * q^59 - 4 * q^61 + 16 * q^63 + 4 * q^65 - 18 * q^67 + 48 * q^69 + 14 * q^71 - 6 * q^73 + 12 * q^75 - 2 * q^81 + 4 * q^83 + 24 * q^85 + 18 * q^87 + 34 * q^89 + 20 * q^91 + 4 * q^93 + 14 * q^95 - 6 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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