LMFDB - Embedded newform 420.2.w.a.83.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [420,2,Mod(83,420)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(420, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("420.83");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	420.w (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:	$3.35371688489$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.12745506816.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 23x^{4} + 1 $ x^8 + 23*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			83.4
Root			$-1.54779 - 1.54779i$ of defining polynomial
Character	$\chi$	$=$	420.83
Dual form			420.2.w.a.167.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 + 1.00000i) q^{2} +(1.22474 + 1.22474i) q^{3} -2.00000i q^{4} +(-1.22474 + 1.87083i) q^{5} -2.44949 q^{6} +(1.87083 - 1.87083i) q^{7} +(2.00000 + 2.00000i) q^{8} +3.00000i q^{9} +O(q^{10})$	\(q+(-1.00000 + 1.00000i) q^{2} +(1.22474 + 1.22474i) q^{3} -2.00000i q^{4} +(-1.22474 + 1.87083i) q^{5} -2.44949 q^{6} +(1.87083 - 1.87083i) q^{7} +(2.00000 + 2.00000i) q^{8} +3.00000i q^{9} +(-0.646084 - 3.09557i) q^{10} +5.58258 q^{11} +(2.44949 - 2.44949i) q^{12} +3.74166i q^{14} +(-3.79129 + 0.791288i) q^{15} -4.00000 q^{16} +(-1.80341 + 1.80341i) q^{17} +(-3.00000 - 3.00000i) q^{18} +8.64064i q^{19} +(3.74166 + 2.44949i) q^{20} +4.58258 q^{21} +(-5.58258 + 5.58258i) q^{22} +(-4.58258 - 4.58258i) q^{23} +4.89898i q^{24} +(-2.00000 - 4.58258i) q^{25} +(-3.67423 + 3.67423i) q^{27} +(-3.74166 - 3.74166i) q^{28} +(3.00000 - 4.58258i) q^{30} +5.03383 q^{31} +(4.00000 - 4.00000i) q^{32} +(6.83723 + 6.83723i) q^{33} -3.60681i q^{34} +(1.20871 + 5.79129i) q^{35} +6.00000 q^{36} +(-0.582576 + 0.582576i) q^{37} +(-8.64064 - 8.64064i) q^{38} +(-6.19115 + 1.29217i) q^{40} -3.74166 q^{41} +(-4.58258 + 4.58258i) q^{42} -11.1652i q^{44} +(-5.61249 - 3.67423i) q^{45} +9.16515 q^{46} +(-4.89898 - 4.89898i) q^{48} -7.00000i q^{49} +(6.58258 + 2.58258i) q^{50} -4.41742 q^{51} -7.34847i q^{54} +(-6.83723 + 10.4440i) q^{55} +7.48331 q^{56} +(-10.5826 + 10.5826i) q^{57} +(1.58258 + 7.58258i) q^{60} +(-5.03383 + 5.03383i) q^{62} +(5.61249 + 5.61249i) q^{63} +8.00000i q^{64} -13.6745 q^{66} +(3.60681 + 3.60681i) q^{68} -11.2250i q^{69} +(-7.00000 - 4.58258i) q^{70} +15.5826 q^{71} +(-6.00000 + 6.00000i) q^{72} -1.16515i q^{74} +(3.16300 - 8.06198i) q^{75} +17.2813 q^{76} +(10.4440 - 10.4440i) q^{77} +(4.89898 - 7.48331i) q^{80} -9.00000 q^{81} +(3.74166 - 3.74166i) q^{82} -9.16515i q^{84} +(-1.16515 - 5.58258i) q^{85} +(11.1652 + 11.1652i) q^{88} -2.44949i q^{89} +(9.28672 - 1.93825i) q^{90} +(-9.16515 + 9.16515i) q^{92} +(6.16515 + 6.16515i) q^{93} +(-16.1652 - 10.5826i) q^{95} +9.79796 q^{96} +(7.00000 + 7.00000i) q^{98} +16.7477i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + 16 q^{8}+O(q^{10}) $ 8 * q - 8 * q^2 + 16 * q^8	$ 8 q - 8 q^{2} + 16 q^{8} + 8 q^{11} - 12 q^{15} - 32 q^{16} - 24 q^{18} - 8 q^{22} - 16 q^{25} + 24 q^{30} + 32 q^{32} + 28 q^{35} + 48 q^{36} + 32 q^{37} + 16 q^{50} - 72 q^{51} - 48 q^{57} - 24 q^{60} - 56 q^{70} + 88 q^{71} - 48 q^{72} - 72 q^{81} + 64 q^{85} + 16 q^{88} - 24 q^{93} - 56 q^{95} + 56 q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 + 16 * q^8 + 8 * q^11 - 12 * q^15 - 32 * q^16 - 24 * q^18 - 8 * q^22 - 16 * q^25 + 24 * q^30 + 32 * q^32 + 28 * q^35 + 48 * q^36 + 32 * q^37 + 16 * q^50 - 72 * q^51 - 48 * q^57 - 24 * q^60 - 56 * q^70 + 88 * q^71 - 48 * q^72 - 72 * q^81 + 64 * q^85 + 16 * q^88 - 24 * q^93 - 56 * q^95 + 56 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/420\mathbb{Z}\right)^\times$.

$n$	$211$	$241$	$281$	$337$
$\chi(n)$	$-1$	$-1$	$-1$	$e\left(\frac{3}{4}\right)$

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 $

$2$	−1.00000	+	1.00000i	−0.707107	+	0.707107i
$2$	−1.00000	+	1.00000i	−0.707107	+	0.707107i
$3$	1.22474	+	1.22474i	0.707107	+	0.707107i
$3$	1.22474	+	1.22474i	0.707107	+	0.707107i
$4$		−	2.00000i		−	1.00000i
$5$	−1.22474	+	1.87083i	−0.547723	+	0.836660i
$5$	−1.22474	+	1.87083i	−0.547723	+	0.836660i
$6$	−2.44949			−1.00000
$7$	1.87083	−	1.87083i	0.707107	−	0.707107i
$7$	1.87083	−	1.87083i	0.707107	−	0.707107i
$8$	2.00000	+	2.00000i	0.707107	+	0.707107i
$9$			3.00000i			1.00000i
$10$	−0.646084	−	3.09557i	−0.204310	−	0.978906i
$11$	5.58258			1.68321			0.841605	−	0.540094i	$-0.181611\pi$
$11$	5.58258			1.68321			0.841605	+	0.540094i	$0.181611\pi$
$12$	2.44949	−	2.44949i	0.707107	−	0.707107i
$13$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$13$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$14$			3.74166i			1.00000i
$15$	−3.79129	+	0.791288i	−0.978906	+	0.204310i
$16$	−4.00000			−1.00000
$17$	−1.80341	+	1.80341i	−0.437390	+	0.437390i	−0.891133	−	0.453743i	$-0.850089\pi$
$17$	−1.80341	+	1.80341i	−0.437390	+	0.437390i	0.453743	+	0.891133i	$0.350089\pi$
$18$	−3.00000	−	3.00000i	−0.707107	−	0.707107i
$19$			8.64064i			1.98230i	0.132754	+	0.991149i	$0.457618\pi$
$19$			8.64064i			1.98230i	−0.132754	+	0.991149i	$0.542382\pi$
$20$	3.74166	+	2.44949i	0.836660	+	0.547723i
$21$	4.58258			1.00000
$22$	−5.58258	+	5.58258i	−1.19021	+	1.19021i
$23$	−4.58258	−	4.58258i	−0.955533	−	0.955533i	0.0435195	−	0.999053i	$-0.486143\pi$
$23$	−4.58258	−	4.58258i	−0.955533	−	0.955533i	−0.999053	+	0.0435195i	$0.986143\pi$
$24$			4.89898i			1.00000i
$25$	−2.00000	−	4.58258i	−0.400000	−	0.916515i
$26$		0			0
$27$	−3.67423	+	3.67423i	−0.707107	+	0.707107i
$28$	−3.74166	−	3.74166i	−0.707107	−	0.707107i
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$	3.00000	−	4.58258i	0.547723	−	0.836660i
$31$	5.03383			0.904102			0.452051	−	0.891992i	$-0.350693\pi$
$31$	5.03383			0.904102			0.452051	+	0.891992i	$0.350693\pi$
$32$	4.00000	−	4.00000i	0.707107	−	0.707107i
$33$	6.83723	+	6.83723i	1.19021	+	1.19021i
$34$		−	3.60681i		−	0.618563i
$35$	1.20871	+	5.79129i	0.204310	+	0.978906i
$36$	6.00000			1.00000
$37$	−0.582576	+	0.582576i	−0.0957749	+	0.0957749i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$	−0.582576	+	0.582576i	−0.0957749	+	0.0957749i	0.657596	+	0.753371i	$0.271573\pi$
$38$	−8.64064	−	8.64064i	−1.40170	−	1.40170i
$39$		0			0
$40$	−6.19115	+	1.29217i	−0.978906	+	0.204310i
$41$	−3.74166			−0.584349			−0.292174	−	0.956365i	$-0.594379\pi$
$41$	−3.74166			−0.584349			−0.292174	+	0.956365i	$0.594379\pi$
$42$	−4.58258	+	4.58258i	−0.707107	+	0.707107i
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$		−	11.1652i		−	1.68321i
$45$	−5.61249	−	3.67423i	−0.836660	−	0.547723i
$46$	9.16515			1.35133
$47$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$47$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$48$	−4.89898	−	4.89898i	−0.707107	−	0.707107i
$49$		−	7.00000i		−	1.00000i
$50$	6.58258	+	2.58258i	0.930917	+	0.365231i
$51$	−4.41742			−0.618563
$52$		0			0
$53$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$53$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$54$		−	7.34847i		−	1.00000i
$55$	−6.83723	+	10.4440i	−0.921932	+	1.40827i
$56$	7.48331			1.00000
$57$	−10.5826	+	10.5826i	−1.40170	+	1.40170i
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$	1.58258	+	7.58258i	0.204310	+	0.978906i
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$	−5.03383	+	5.03383i	−0.639296	+	0.639296i
$63$	5.61249	+	5.61249i	0.707107	+	0.707107i
$64$			8.00000i			1.00000i
$65$		0			0
$66$	−13.6745			−1.68321
$67$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$67$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$68$	3.60681	+	3.60681i	0.437390	+	0.437390i
$69$		−	11.2250i		−	1.35133i
$70$	−7.00000	−	4.58258i	−0.836660	−	0.547723i
$71$	15.5826			1.84931			0.924656	−	0.380804i	$-0.124353\pi$
$71$	15.5826			1.84931			0.924656	+	0.380804i	$0.124353\pi$
$72$	−6.00000	+	6.00000i	−0.707107	+	0.707107i
$73$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$73$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$74$		−	1.16515i		−	0.135446i
$75$	3.16300	−	8.06198i	0.365231	−	0.930917i
$76$	17.2813			1.98230
$77$	10.4440	−	10.4440i	1.19021	−	1.19021i
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	4.89898	−	7.48331i	0.547723	−	0.836660i
$81$	−9.00000			−1.00000
$82$	3.74166	−	3.74166i	0.413197	−	0.413197i
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		−	9.16515i		−	1.00000i
$85$	−1.16515	−	5.58258i	−0.126378	−	0.605515i
$86$		0			0
$87$		0			0
$88$	11.1652	+	11.1652i	1.19021	+	1.19021i
$89$		−	2.44949i		−	0.259645i	−0.991537	−	0.129823i	$-0.958559\pi$
$89$		−	2.44949i		−	0.259645i	0.991537	−	0.129823i	$-0.0414408\pi$
$90$	9.28672	−	1.93825i	0.978906	−	0.204310i
$91$		0			0
$92$	−9.16515	+	9.16515i	−0.955533	+	0.955533i
$93$	6.16515	+	6.16515i	0.639296	+	0.639296i
$94$		0			0
$95$	−16.1652	−	10.5826i	−1.65851	−	1.08575i
$96$	9.79796			1.00000
$97$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$97$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$98$	7.00000	+	7.00000i	0.707107	+	0.707107i
$99$			16.7477i			1.68321i
$100$	−9.16515	+	4.00000i	−0.916515	+	0.400000i
$101$	7.34847			0.731200			0.365600	−	0.930772i	$-0.380864\pi$
$101$	7.34847			0.731200			0.365600	+	0.930772i	$0.380864\pi$
$102$	4.41742	−	4.41742i	0.437390	−	0.437390i
$103$	−7.48331	−	7.48331i	−0.737353	−	0.737353i	0.234712	−	0.972065i	$-0.424585\pi$
$103$	−7.48331	−	7.48331i	−0.737353	−	0.737353i	−0.972065	+	0.234712i	$0.924585\pi$
$104$		0			0
$105$	−5.61249	+	8.57321i	−0.547723	+	0.836660i
$106$		0			0
$107$	13.7477	−	13.7477i	1.32904	−	1.32904i	0.422837	−	0.906206i	$-0.361034\pi$
$107$	13.7477	−	13.7477i	1.32904	−	1.32904i	0.906206	−	0.422837i	$-0.138966\pi$
$108$	7.34847	+	7.34847i	0.707107	+	0.707107i
$109$		−	18.3303i		−	1.75572i	−0.478913	−	0.877862i	$-0.658969\pi$
$109$		−	18.3303i		−	1.75572i	0.478913	−	0.877862i	$-0.341031\pi$
$110$	−3.60681	−	17.2813i	−0.343896	−	1.64770i
$111$	−1.42701			−0.135446
$112$	−7.48331	+	7.48331i	−0.707107	+	0.707107i
$113$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$113$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$114$		−	21.1652i		−	1.98230i
$115$	14.1857	−	2.96073i	1.32282	−	0.276089i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			6.74773i			0.618563i
$120$	−9.16515	−	6.00000i	−0.836660	−	0.547723i
$121$	20.1652			1.83320
$122$		0			0
$123$	−4.58258	−	4.58258i	−0.413197	−	0.413197i
$124$		−	10.0677i		−	0.904102i
$125$	11.0227	+	1.87083i	0.985901	+	0.167332i
$126$	−11.2250			−1.00000
$127$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$127$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$128$	−8.00000	−	8.00000i	−0.707107	−	0.707107i
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$	13.6745	−	13.6745i	1.19021	−	1.19021i
$133$	16.1652	+	16.1652i	1.40170	+	1.40170i
$134$		0			0
$135$	−2.37386	−	11.3739i	−0.204310	−	0.978906i
$136$	−7.21362			−0.618563
$137$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$137$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$138$	11.2250	+	11.2250i	0.955533	+	0.955533i
$139$		−	22.3151i		−	1.89274i	−0.323082	−	0.946371i	$-0.604719\pi$
$139$		−	22.3151i		−	1.89274i	0.323082	−	0.946371i	$-0.395281\pi$
$140$	11.5826	−	2.41742i	0.978906	−	0.204310i
$141$		0			0
$142$	−15.5826	+	15.5826i	−1.30766	+	1.30766i
$143$		0			0
$144$		−	12.0000i		−	1.00000i
$145$		0			0
$146$		0			0
$147$	8.57321	−	8.57321i	0.707107	−	0.707107i
$148$	1.16515	+	1.16515i	0.0957749	+	0.0957749i
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$	4.89898	+	11.2250i	0.400000	+	0.916515i
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$	−17.2813	+	17.2813i	−1.40170	+	1.40170i
$153$	−5.41022	−	5.41022i	−0.437390	−	0.437390i
$154$			20.8881i			1.68321i
$155$	−6.16515	+	9.41742i	−0.495197	+	0.756426i
$156$		0			0
$157$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$157$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$158$		0			0
$159$		0			0
$160$	2.58434	+	12.3823i	0.204310	+	0.978906i
$161$	−17.1464			−1.35133
$162$	9.00000	−	9.00000i	0.707107	−	0.707107i
$163$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$163$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$164$			7.48331i			0.584349i
$165$	−21.1652	+	4.41742i	−1.64770	+	0.343896i
$166$		0			0
$167$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$167$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$168$	9.16515	+	9.16515i	0.707107	+	0.707107i
$169$			13.0000i			1.00000i
$170$	6.74773	+	4.41742i	0.517527	+	0.338801i
$171$	−25.9219			−1.98230
$172$		0			0
$173$	−11.8711	−	11.8711i	−0.902540	−	0.902540i	0.0931156	−	0.995655i	$-0.470317\pi$
$173$	−11.8711	−	11.8711i	−0.902540	−	0.902540i	−0.995655	+	0.0931156i	$0.970317\pi$
$174$		0			0
$175$	−12.3149	−	4.83156i	−0.930917	−	0.365231i
$176$	−22.3303			−1.68321
$177$		0			0
$178$	2.44949	+	2.44949i	0.183597	+	0.183597i
$179$			26.7477i			1.99922i	0.0279439	+	0.999609i	$0.491104\pi$
$179$			26.7477i			1.99922i	−0.0279439	+	0.999609i	$0.508896\pi$
$180$	−7.34847	+	11.2250i	−0.547723	+	0.836660i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		−	18.3303i		−	1.35133i
$185$	−0.376393	−	1.80341i	−0.0276729	−	0.132589i
$186$	−12.3303			−0.904102
$187$	−10.0677	+	10.0677i	−0.736219	+	0.736219i
$188$		0			0
$189$			13.7477i			1.00000i
$190$	26.7477	−	5.58258i	1.94048	−	0.405003i
$191$	−14.4174			−1.04321			−0.521604	−	0.853188i	$-0.674666\pi$
$191$	−14.4174			−1.04321			−0.521604	+	0.853188i	$0.674666\pi$
$192$	−9.79796	+	9.79796i	−0.707107	+	0.707107i
$193$	−11.7477	−	11.7477i	−0.845620	−	0.845620i	0.143963	−	0.989583i	$-0.454015\pi$
$193$	−11.7477	−	11.7477i	−0.845620	−	0.845620i	−0.989583	+	0.143963i	$0.954015\pi$
$194$		0			0
$195$		0			0
$196$	−14.0000			−1.00000
$197$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$197$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$198$	−16.7477	−	16.7477i	−1.19021	−	1.19021i
$199$		−	15.8543i		−	1.12388i	−0.827178	−	0.561939i	$-0.810055\pi$
$199$		−	15.8543i		−	1.12388i	0.827178	−	0.561939i	$-0.189945\pi$
$200$	5.16515	−	13.1652i	0.365231	−	0.930917i
$201$		0			0
$202$	−7.34847	+	7.34847i	−0.517036	+	0.517036i
$203$		0			0
$204$			8.83485i			0.618563i
$205$	4.58258	−	7.00000i	0.320061	−	0.488901i
$206$	14.9666			1.04277
$207$	13.7477	−	13.7477i	0.955533	−	0.955533i
$208$		0			0
$209$			48.2370i			3.33662i
$210$	−2.96073	−	14.1857i	−0.204310	−	0.978906i
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$	19.0847	+	19.0847i	1.30766	+	1.30766i
$214$			27.4955i			1.87955i
$215$		0			0
$216$	−14.6969			−1.00000
$217$	9.41742	−	9.41742i	0.639296	−	0.639296i
$218$	18.3303	+	18.3303i	1.24148	+	1.24148i
$219$		0			0
$220$	20.8881	+	13.6745i	1.40827	+	0.921932i
$221$		0			0
$222$	1.42701	−	1.42701i	0.0957749	−	0.0957749i
$223$	−9.79796	−	9.79796i	−0.656120	−	0.656120i	0.298340	−	0.954460i	$-0.403567\pi$
$223$	−9.79796	−	9.79796i	−0.656120	−	0.656120i	−0.954460	+	0.298340i	$0.903567\pi$
$224$		−	14.9666i		−	1.00000i
$225$	13.7477	−	6.00000i	0.916515	−	0.400000i
$226$		0			0
$227$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$227$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$228$	21.1652	+	21.1652i	1.40170	+	1.40170i
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$	−11.2250	+	17.1464i	−0.740153	+	1.13060i
$231$	25.5826			1.68321
$232$		0			0
$233$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$233$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$	−6.74773	−	6.74773i	−0.437390	−	0.437390i
$239$		−	3.25227i		−	0.210372i	−0.994453	−	0.105186i	$-0.966456\pi$
$239$		−	3.25227i		−	0.210372i	0.994453	−	0.105186i	$-0.0335438\pi$
$240$	15.1652	−	3.16515i	0.978906	−	0.204310i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	−20.1652	+	20.1652i	−1.29627	+	1.29627i
$243$	−11.0227	−	11.0227i	−0.707107	−	0.707107i
$244$		0			0
$245$	13.0958	+	8.57321i	0.836660	+	0.547723i
$246$	9.16515			0.584349
$247$		0			0
$248$	10.0677	+	10.0677i	0.639296	+	0.639296i
$249$		0			0
$250$	−12.8935	+	9.15188i	−0.815459	+	0.578815i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$	11.2250	−	11.2250i	0.707107	−	0.707107i
$253$	−25.5826	−	25.5826i	−1.60836	−	1.60836i
$254$		0			0
$255$	5.41022	−	8.26424i	0.338801	−	0.517527i
$256$	16.0000			1.00000
$257$	−14.0509	+	14.0509i	−0.876468	+	0.876468i	−0.993167	−	0.116699i	$-0.962769\pi$
$257$	−14.0509	+	14.0509i	−0.876468	+	0.876468i	0.116699	+	0.993167i	$0.462769\pi$
$258$		0			0
$259$			2.17980i			0.135446i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−1.00000	−	1.00000i	−0.0616626	−	0.0616626i	0.675603	−	0.737266i	$-0.263883\pi$
$263$	−1.00000	−	1.00000i	−0.0616626	−	0.0616626i	−0.737266	+	0.675603i	$0.763883\pi$
$264$			27.3489i			1.68321i
$265$		0			0
$266$	−32.3303			−1.98230
$267$	3.00000	−	3.00000i	0.183597	−	0.183597i
$268$		0			0
$269$		−	26.9444i		−	1.64283i	−0.570332	−	0.821414i	$-0.693186\pi$
$269$		−	26.9444i		−	1.64283i	0.570332	−	0.821414i	$-0.306814\pi$
$270$	13.7477	+	9.00000i	0.836660	+	0.547723i
$271$	29.5287			1.79374			0.896871	−	0.442292i	$-0.145835\pi$
$271$	29.5287			1.79374			0.896871	+	0.442292i	$0.145835\pi$
$272$	7.21362	−	7.21362i	0.437390	−	0.437390i
$273$		0			0
$274$		0			0
$275$	−11.1652	−	25.5826i	−0.673284	−	1.54269i
$276$	−22.4499			−1.35133
$277$	−20.5826	+	20.5826i	−1.23669	+	1.23669i	−0.275340	+	0.961347i	$0.588791\pi$
$277$	−20.5826	+	20.5826i	−1.23669	+	1.23669i	−0.961347	+	0.275340i	$0.911209\pi$
$278$	22.3151	+	22.3151i	1.33837	+	1.33837i
$279$			15.1015i			0.904102i
$280$	−9.16515	+	14.0000i	−0.547723	+	0.836660i
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	14.6969	+	14.6969i	0.873642	+	0.873642i	0.992867	−	0.119225i	$-0.0380410\pi$
$283$	14.6969	+	14.6969i	0.873642	+	0.873642i	−0.119225	+	0.992867i	$0.538041\pi$
$284$		−	31.1652i		−	1.84931i
$285$	−6.83723	−	32.7591i	−0.405003	−	1.94048i
$286$		0			0
$287$	−7.00000	+	7.00000i	−0.413197	+	0.413197i
$288$	12.0000	+	12.0000i	0.707107	+	0.707107i
$289$			10.4955i			0.617380i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−24.1185	−	24.1185i	−1.40902	−	1.40902i	−0.765065	−	0.643953i	$-0.777293\pi$
$293$	−24.1185	−	24.1185i	−1.40902	−	1.40902i	−0.643953	−	0.765065i	$-0.722707\pi$
$294$			17.1464i			1.00000i
$295$		0			0
$296$	−2.33030			−0.135446
$297$	−20.5117	+	20.5117i	−1.19021	+	1.19021i
$298$		0			0
$299$		0			0
$300$	−16.1240	−	6.32599i	−0.930917	−	0.365231i
$301$		0			0
$302$		0			0
$303$	9.00000	+	9.00000i	0.517036	+	0.517036i
$304$		−	34.5625i		−	1.98230i
$305$		0			0
$306$	10.8204			0.618563
$307$	3.74166	−	3.74166i	0.213548	−	0.213548i	−0.592225	−	0.805773i	$-0.701750\pi$
$307$	3.74166	−	3.74166i	0.213548	−	0.213548i	0.805773	+	0.592225i	$0.201750\pi$
$308$	−20.8881	−	20.8881i	−1.19021	−	1.19021i
$309$		−	18.3303i		−	1.04277i
$310$	−3.25227	−	15.5826i	−0.184717	−	0.885031i
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$313$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$314$		0			0
$315$	−17.3739	+	3.62614i	−0.978906	+	0.204310i
$316$		0			0
$317$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$317$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$318$		0			0
$319$		0			0
$320$	−14.9666	−	9.79796i	−0.836660	−	0.547723i
$321$	33.6749			1.87955
$322$	17.1464	−	17.1464i	0.955533	−	0.955533i
$323$	−15.5826	−	15.5826i	−0.867038	−	0.867038i
$324$			18.0000i			1.00000i
$325$		0			0
$326$		0			0
$327$	22.4499	−	22.4499i	1.24148	−	1.24148i
$328$	−7.48331	−	7.48331i	−0.413197	−	0.413197i
$329$		0			0
$330$	16.7477	−	25.5826i	0.921932	−	1.40827i
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$	−1.74773	−	1.74773i	−0.0957749	−	0.0957749i
$334$		0			0
$335$		0			0
$336$	−18.3303			−1.00000
$337$	17.3303	−	17.3303i	0.944042	−	0.944042i	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	17.3303	−	17.3303i	0.944042	−	0.944042i	0.998515	+	0.0544735i	$0.0173480\pi$
$338$	−13.0000	−	13.0000i	−0.707107	−	0.707107i
$339$		0			0
$340$	−11.1652	+	2.33030i	−0.605515	+	0.126378i
$341$	28.1017			1.52179
$342$	25.9219	−	25.9219i	1.40170	−	1.40170i
$343$	−13.0958	−	13.0958i	−0.707107	−	0.707107i
$344$		0			0
$345$	21.0000	+	13.7477i	1.13060	+	0.740153i
$346$	23.7421			1.27638
$347$	13.0000	−	13.0000i	0.697877	−	0.697877i	−0.266076	−	0.963952i	$-0.585727\pi$
$347$	13.0000	−	13.0000i	0.697877	−	0.697877i	0.963952	+	0.266076i	$0.0857271\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$	17.1464	−	7.48331i	0.916515	−	0.400000i
$351$		0			0
$352$	22.3303	−	22.3303i	1.19021	−	1.19021i
$353$	0.376393	+	0.376393i	0.0200334	+	0.0200334i	0.717053	−	0.697019i	$-0.245491\pi$
$353$	0.376393	+	0.376393i	0.0200334	+	0.0200334i	−0.697019	+	0.717053i	$0.745491\pi$
$354$		0			0
$355$	−19.0847	+	29.1523i	−1.01291	+	1.54725i
$356$	−4.89898			−0.259645
$357$	−8.26424	+	8.26424i	−0.437390	+	0.437390i
$358$	−26.7477	−	26.7477i	−1.41366	−	1.41366i
$359$			36.7477i			1.93947i	0.244158	+	0.969735i	$0.421488\pi$
$359$			36.7477i			1.93947i	−0.244158	+	0.969735i	$0.578512\pi$
$360$	−3.87650	−	18.5734i	−0.204310	−	0.978906i
$361$	−55.6606			−2.92951
$362$		0			0
$363$	24.6972	+	24.6972i	1.29627	+	1.29627i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−19.5959	+	19.5959i	−1.02290	+	1.02290i	−0.0231670	+	0.999732i	$0.507375\pi$
$367$	−19.5959	+	19.5959i	−1.02290	+	1.02290i	−0.999732	+	0.0231670i	$0.992625\pi$
$368$	18.3303	+	18.3303i	0.955533	+	0.955533i
$369$		−	11.2250i		−	0.584349i
$370$	2.17980	+	1.42701i	0.113322	+	0.0741869i
$371$		0			0
$372$	12.3303	−	12.3303i	0.639296	−	0.639296i
$373$	26.1652	+	26.1652i	1.35478	+	1.35478i	0.880227	+	0.474554i	$0.157390\pi$
$373$	26.1652	+	26.1652i	1.35478	+	1.35478i	0.474554	+	0.880227i	$0.342610\pi$
$374$		−	20.1353i		−	1.04117i
$375$	11.2087	+	15.7913i	0.578815	+	0.815459i
$376$		0			0
$377$		0			0
$378$	−13.7477	−	13.7477i	−0.707107	−	0.707107i
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$	−21.1652	+	32.3303i	−1.08575	+	1.65851i
$381$		0			0
$382$	14.4174	−	14.4174i	0.737660	−	0.737660i
$383$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$383$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$384$		−	19.5959i		−	1.00000i
$385$	6.74773	+	32.3303i	0.343896	+	1.64770i
$386$	23.4955			1.19589
$387$		0			0
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$	16.5285			0.835882
$392$	14.0000	−	14.0000i	0.707107	−	0.707107i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	33.4955			1.68321
$397$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$397$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$398$	15.8543	+	15.8543i	0.794702	+	0.794702i
$399$			39.5964i			1.98230i
$400$	8.00000	+	18.3303i	0.400000	+	0.916515i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$		−	14.6969i		−	0.731200i
$405$	11.0227	−	16.8375i	0.547723	−	0.836660i
$406$		0			0
$407$	−3.25227	+	3.25227i	−0.161209	+	0.161209i
$408$	−8.83485	−	8.83485i	−0.437390	−	0.437390i
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$	2.41742	+	11.5826i	0.119388	+	0.572023i
$411$		0			0
$412$	−14.9666	+	14.9666i	−0.737353	+	0.737353i
$413$		0			0
$414$			27.4955i			1.35133i
$415$		0			0
$416$		0			0
$417$	27.3303	−	27.3303i	1.33837	−	1.33837i
$418$	−48.2370	−	48.2370i	−2.35935	−	2.35935i
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$	17.1464	+	11.2250i	0.836660	+	0.547723i
$421$	9.16515			0.446682			0.223341	−	0.974740i	$-0.428304\pi$
$421$	9.16515			0.446682			0.223341	+	0.974740i	$0.428304\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	11.8711	+	4.65743i	0.575831	+	0.225919i
$426$	−38.1694			−1.84931
$427$		0			0
$428$	−27.4955	−	27.4955i	−1.32904	−	1.32904i
$429$		0			0
$430$		0			0
$431$	−24.4174			−1.17615			−0.588073	−	0.808808i	$-0.700113\pi$
$431$	−24.4174			−1.17615			−0.588073	+	0.808808i	$0.700113\pi$
$432$	14.6969	−	14.6969i	0.707107	−	0.707107i
$433$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$433$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$434$			18.8348i			0.904102i
$435$		0			0
$436$	−36.6606			−1.75572
$437$	39.5964	−	39.5964i	1.89415	−	1.89415i
$438$		0			0
$439$			33.1355i			1.58147i	0.612157	+	0.790736i	$0.290302\pi$
$439$			33.1355i			1.58147i	−0.612157	+	0.790736i	$0.709698\pi$
$440$	−34.5625	+	7.21362i	−1.64770	+	0.343896i
$441$	21.0000			1.00000
$442$		0			0
$443$	19.0000	+	19.0000i	0.902717	+	0.902717i	0.995670	−	0.0929532i	$-0.0296307\pi$
$443$	19.0000	+	19.0000i	0.902717	+	0.902717i	−0.0929532	+	0.995670i	$0.529631\pi$
$444$			2.85403i			0.135446i
$445$	4.58258	+	3.00000i	0.217235	+	0.142214i
$446$	19.5959			0.927894
$447$		0			0
$448$	14.9666	+	14.9666i	0.707107	+	0.707107i
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$	−7.74773	+	19.7477i	−0.365231	+	0.930917i
$451$	−20.8881			−0.983582
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$	−42.3303			−1.98230
$457$	7.33030	−	7.33030i	0.342897	−	0.342897i	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	7.33030	−	7.33030i	0.342897	−	0.342897i	0.857455	+	0.514558i	$0.172044\pi$
$458$		0			0
$459$		−	13.2523i		−	0.618563i
$460$	−5.92146	−	28.3714i	−0.276089	−	1.32282i
$461$	−41.1582			−1.91693			−0.958465	−	0.285210i	$-0.907937\pi$
$461$	−41.1582			−1.91693			−0.958465	+	0.285210i	$0.907937\pi$
$462$	−25.5826	+	25.5826i	−1.19021	+	1.19021i
$463$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$463$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$464$		0			0
$465$	−19.0847	+	3.98320i	−0.885031	+	0.184717i
$466$		0			0
$467$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$467$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	39.5964	−	17.2813i	1.81681	−	0.792919i
$476$	13.4955			0.618563
$477$		0			0
$478$	3.25227	+	3.25227i	0.148756	+	0.148756i
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$	−12.0000	+	18.3303i	−0.547723	+	0.836660i
$481$		0			0
$482$		0			0
$483$	−21.0000	−	21.0000i	−0.955533	−	0.955533i
$484$		−	40.3303i		−	1.83320i
$485$		0			0
$486$	22.0454			1.00000
$487$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$487$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$488$		0			0
$489$		0			0
$490$	−21.6690	+	4.52259i	−0.978906	+	0.204310i
$491$	35.5826			1.60582			0.802910	−	0.596101i	$-0.203284\pi$
$491$	35.5826			1.60582			0.802910	+	0.596101i	$0.203284\pi$
$492$	−9.16515	+	9.16515i	−0.413197	+	0.413197i
$493$		0			0
$494$		0			0
$495$	−31.3321	−	20.5117i	−1.40827	−	0.921932i
$496$	−20.1353			−0.904102
$497$	29.1523	−	29.1523i	1.30766	−	1.30766i
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$	3.74166	−	22.0454i	0.167332	−	0.985901i
$501$		0			0
$502$		0			0
$503$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$503$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$504$			22.4499i			1.00000i
$505$	−9.00000	+	13.7477i	−0.400495	+	0.611766i
$506$	51.1652			2.27457
$507$	−15.9217	+	15.9217i	−0.707107	+	0.707107i
$508$		0			0
$509$			26.1916i			1.16092i	0.814288	+	0.580461i	$0.197128\pi$
$509$			26.1916i			1.16092i	−0.814288	+	0.580461i	$0.802872\pi$
$510$	2.85403	+	13.6745i	0.126378	+	0.605515i
$511$		0			0
$512$	−16.0000	+	16.0000i	−0.707107	+	0.707107i
$513$	−31.7477	−	31.7477i	−1.40170	−	1.40170i
$514$		−	28.1017i		−	1.23951i
$515$	23.1652	−	4.83485i	1.02078	−	0.213049i
$516$		0			0
$517$		0			0
$518$	−2.17980	−	2.17980i	−0.0957749	−	0.0957749i
$519$		−	29.0780i		−	1.27638i
$520$		0			0
$521$	−41.6413			−1.82434			−0.912170	−	0.409812i	$-0.865594\pi$
$521$	−41.6413			−1.82434			−0.912170	+	0.409812i	$0.865594\pi$
$522$		0			0
$523$	29.9333	+	29.9333i	1.30889	+	1.30889i	0.922217	+	0.386673i	$0.126376\pi$
$523$	29.9333	+	29.9333i	1.30889	+	1.30889i	0.386673	+	0.922217i	$0.373624\pi$
$524$		0			0
$525$	−9.16515	−	21.0000i	−0.400000	−	0.916515i
$526$	2.00000			0.0872041
$527$	−9.07803	+	9.07803i	−0.395445	+	0.395445i
$528$	−27.3489	−	27.3489i	−1.19021	−	1.19021i
$529$			19.0000i			0.826087i
$530$		0			0
$531$		0			0
$532$	32.3303	−	32.3303i	1.40170	−	1.40170i
$533$		0			0
$534$			6.00000i			0.259645i
$535$	8.88218	+	42.5571i	0.384010	+	1.83990i
$536$		0			0
$537$	−32.7591	+	32.7591i	−1.41366	+	1.41366i
$538$	26.9444	+	26.9444i	1.16166	+	1.16166i
$539$		−	39.0780i		−	1.68321i
$540$	−22.7477	+	4.74773i	−0.978906	+	0.204310i
$541$	−8.00000			−0.343947			−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000			−0.343947			−0.171973	+	0.985102i	$0.555014\pi$
$542$	−29.5287	+	29.5287i	−1.26837	+	1.26837i
$543$		0			0
$544$			14.4272i			0.618563i
$545$	34.2929	+	22.4499i	1.46894	+	0.961650i
$546$		0			0
$547$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$547$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$548$		0			0
$549$		0			0
$550$	36.7477	+	14.4174i	1.56693	+	0.614761i
$551$		0			0
$552$	22.4499	−	22.4499i	0.955533	−	0.955533i
$553$		0			0
$554$		−	41.1652i		−	1.74894i
$555$	1.74773	−	2.66970i	0.0741869	−	0.113322i
$556$	−44.6302			−1.89274
$557$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$557$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$558$	−15.1015	−	15.1015i	−0.639296	−	0.639296i
$559$		0			0
$560$	−4.83485	−	23.1652i	−0.204310	−	0.978906i
$561$	−24.6606			−1.04117
$562$		0			0
$563$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$563$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$564$		0			0
$565$		0			0
$566$	−29.3939			−1.23552
$567$	−16.8375	+	16.8375i	−0.707107	+	0.707107i
$568$	31.1652	+	31.1652i	1.30766	+	1.30766i
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$	39.5964	+	25.9219i	1.65851	+	1.08575i
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$	−17.6577	−	17.6577i	−0.737660	−	0.737660i
$574$		−	14.0000i		−	0.584349i
$575$	−11.8348	+	30.1652i	−0.493547	+	1.25797i
$576$	−24.0000			−1.00000
$577$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$577$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$578$	−10.4955	−	10.4955i	−0.436553	−	0.436553i
$579$		−	28.7759i		−	1.19589i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	48.2370			1.99265
$587$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$587$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$588$	−17.1464	−	17.1464i	−0.707107	−	0.707107i
$589$			43.4955i			1.79220i
$590$		0			0
$591$		0			0
$592$	2.33030	−	2.33030i	0.0957749	−	0.0957749i
$593$	25.5455	+	25.5455i	1.04903	+	1.04903i	0.998734	+	0.0502942i	$0.0160159\pi$
$593$	25.5455	+	25.5455i	1.04903	+	1.04903i	0.0502942	+	0.998734i	$0.483984\pi$
$594$		−	41.0234i		−	1.68321i
$595$	−12.6238	−	8.26424i	−0.517527	−	0.338801i
$596$		0			0
$597$	19.4174	−	19.4174i	0.794702	−	0.794702i
$598$		0			0
$599$		−	19.0780i		−	0.779507i	−0.920919	−	0.389754i	$-0.872560\pi$
$599$		−	19.0780i		−	0.779507i	0.920919	−	0.389754i	$-0.127440\pi$
$600$	22.4499	−	9.79796i	0.916515	−	0.400000i
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−24.6972	+	37.7255i	−1.00408	+	1.53376i
$606$	−18.0000			−0.731200
$607$	29.3939	−	29.3939i	1.19306	−	1.19306i	0.216857	−	0.976203i	$-0.430419\pi$
$607$	29.3939	−	29.3939i	1.19306	−	1.19306i	0.976203	−	0.216857i	$-0.0695807\pi$
$608$	34.5625	+	34.5625i	1.40170	+	1.40170i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−10.8204	+	10.8204i	−0.437390	+	0.437390i
$613$	−13.8348	−	13.8348i	−0.558784	−	0.558784i	0.370177	−	0.928961i	$-0.379297\pi$
$613$	−13.8348	−	13.8348i	−0.558784	−	0.558784i	−0.928961	+	0.370177i	$0.879297\pi$
$614$			7.48331i			0.302002i
$615$	14.1857	−	2.96073i	0.572023	−	0.119388i
$616$	41.7762			1.68321
$617$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$617$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$618$	18.3303	+	18.3303i	0.737353	+	0.737353i
$619$		−	46.8100i		−	1.88145i	−0.339167	−	0.940726i	$-0.610145\pi$
$619$		−	46.8100i		−	1.88145i	0.339167	−	0.940726i	$-0.389855\pi$
$620$	18.8348	+	12.3303i	0.756426	+	0.495197i
$621$	33.6749			1.35133
$622$		0			0
$623$	−4.58258	−	4.58258i	−0.183597	−	0.183597i
$624$		0			0
$625$	−17.0000	+	18.3303i	−0.680000	+	0.733212i
$626$		0			0
$627$	−59.0780	+	59.0780i	−2.35935	+	2.35935i
$628$		0			0
$629$		−	2.10124i		−	0.0837820i
$630$	13.7477	−	21.0000i	0.547723	−	0.836660i
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$			46.7477i			1.84931i
$640$	24.7646	−	5.16867i	0.978906	−	0.204310i
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$	−33.6749	+	33.6749i	−1.32904	+	1.32904i
$643$	−26.1916	−	26.1916i	−1.03290	−	1.03290i	−0.999440	−	0.0334557i	$-0.989349\pi$
$643$	−26.1916	−	26.1916i	−1.03290	−	1.03290i	−0.0334557	−	0.999440i	$-0.510651\pi$
$644$			34.2929i			1.35133i
$645$		0			0
$646$	31.1652			1.22618
$647$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$647$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$648$	−18.0000	−	18.0000i	−0.707107	−	0.707107i
$649$		0			0
$650$		0			0
$651$	23.0679			0.904102
$652$		0			0
$653$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$653$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$654$			44.8999i			1.75572i
$655$		0			0
$656$	14.9666			0.584349
$657$		0			0
$658$		0			0
$659$		−	49.0780i		−	1.91181i	−0.293678	−	0.955904i	$-0.594879\pi$
$659$		−	49.0780i		−	1.91181i	0.293678	−	0.955904i	$-0.405121\pi$
$660$	8.83485	+	42.3303i	0.343896	+	1.64770i
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−50.0404	+	10.4440i	−1.94048	+	0.405003i
$666$	3.49545			0.135446
$667$		0			0
$668$		0			0
$669$		−	24.0000i		−	0.927894i
$670$		0			0
$671$		0			0
$672$	18.3303	−	18.3303i	0.707107	−	0.707107i
$673$	8.25227	+	8.25227i	0.318102	+	0.318102i	0.848038	−	0.529936i	$-0.177784\pi$
$673$	8.25227	+	8.25227i	0.318102	+	0.318102i	−0.529936	+	0.848038i	$0.677784\pi$
$674$			34.6606i			1.33508i
$675$	24.1859	+	9.48899i	0.930917	+	0.365231i
$676$	26.0000			1.00000
$677$	4.65743	−	4.65743i	0.179000	−	0.179000i	−0.611920	−	0.790920i	$-0.709603\pi$
$677$	4.65743	−	4.65743i	0.179000	−	0.179000i	0.790920	+	0.611920i	$0.209603\pi$
$678$		0			0
$679$		0			0
$680$	8.83485	−	13.4955i	0.338801	−	0.517527i
$681$		0			0
$682$	−28.1017	+	28.1017i	−1.07607	+	1.07607i
$683$	29.0000	+	29.0000i	1.10965	+	1.10965i	0.993196	+	0.116459i	$0.0371542\pi$
$683$	29.0000	+	29.0000i	1.10965	+	1.10965i	0.116459	+	0.993196i	$0.462846\pi$
$684$			51.8438i			1.98230i
$685$		0			0
$686$	26.1916			1.00000
$687$		0			0
$688$		0			0
$689$		0			0
$690$	−34.7477	+	7.25227i	−1.32282	+	0.276089i
$691$	−7.88785			−0.300068			−0.150034	−	0.988681i	$-0.547938\pi$
$691$	−7.88785			−0.300068			−0.150034	+	0.988681i	$0.547938\pi$
$692$	−23.7421	+	23.7421i	−0.902540	+	0.902540i
$693$	31.3321	+	31.3321i	1.19021	+	1.19021i
$694$			26.0000i			0.986947i
$695$	41.7477	+	27.3303i	1.58358	+	1.03670i
$696$		0			0
$697$	6.74773	−	6.74773i	0.255588	−	0.255588i
$698$		0			0
$699$		0			0
$700$	−9.66311	+	24.6297i	−0.365231	+	0.930917i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	−5.03383	−	5.03383i	−0.189854	−	0.189854i
$704$			44.6606i			1.68321i
$705$		0			0
$706$	−0.752785			−0.0283315
$707$	13.7477	−	13.7477i	0.517036	−	0.517036i
$708$		0			0
$709$		−	18.3303i		−	0.688409i	−0.938895	−	0.344204i	$-0.888149\pi$
$709$		−	18.3303i		−	0.688409i	0.938895	−	0.344204i	$-0.111851\pi$
$710$	−10.0677	−	48.2370i	−0.377832	−	1.81030i
$711$		0			0
$712$	4.89898	−	4.89898i	0.183597	−	0.183597i
$713$	−23.0679	−	23.0679i	−0.863899	−	0.863899i
$714$		−	16.5285i		−	0.618563i
$715$		0			0
$716$	53.4955			1.99922
$717$	3.98320	−	3.98320i	0.148756	−	0.148756i
$718$	−36.7477	−	36.7477i	−1.37141	−	1.37141i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	22.4499	+	14.6969i	0.836660	+	0.547723i
$721$	−28.0000			−1.04277
$722$	55.6606	−	55.6606i	2.07147	−	2.07147i
$723$		0			0
$724$		0			0
$725$		0			0
$726$	−49.3943			−1.83320
$727$	−7.34847	+	7.34847i	−0.272540	+	0.272540i	−0.830122	−	0.557582i	$-0.811729\pi$
$727$	−7.34847	+	7.34847i	−0.272540	+	0.272540i	0.557582	+	0.830122i	$0.311729\pi$
$728$		0			0
$729$		−	27.0000i		−	1.00000i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$733$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$734$		−	39.1918i		−	1.44660i
$735$	5.53901	+	26.5390i	0.204310	+	0.978906i
$736$	−36.6606			−1.35133
$737$		0			0
$738$	11.2250	+	11.2250i	0.413197	+	0.413197i
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$	−3.60681	+	0.752785i	−0.132589	+	0.0276729i
$741$		0			0
$742$		0			0
$743$	−31.0000	−	31.0000i	−1.13728	−	1.13728i	−0.988936	−	0.148344i	$-0.952606\pi$
$743$	−31.0000	−	31.0000i	−1.13728	−	1.13728i	−0.148344	−	0.988936i	$-0.547394\pi$
$744$			24.6606i			0.904102i
$745$		0			0
$746$	−52.3303			−1.91595
$747$		0			0
$748$	20.1353	+	20.1353i	0.736219	+	0.736219i
$749$		−	51.4393i		−	1.87955i
$750$	−27.0000	−	4.58258i	−0.985901	−	0.167332i
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	27.4955			1.00000
$757$	−28.4955	+	28.4955i	−1.03568	+	1.03568i	−0.0363456	+	0.999339i	$0.511572\pi$
$757$	−28.4955	+	28.4955i	−1.03568	+	1.03568i	−0.999339	+	0.0363456i	$0.988428\pi$
$758$		0			0
$759$		−	62.6643i		−	2.27457i
$760$	−11.1652	−	53.4955i	−0.405003	−	1.94048i
$761$	−41.1582			−1.49198			−0.745992	−	0.665955i	$-0.768024\pi$
$761$	−41.1582			−1.49198			−0.745992	+	0.665955i	$0.768024\pi$
$762$		0			0
$763$	−34.2929	−	34.2929i	−1.24148	−	1.24148i
$764$			28.8348i			1.04321i
$765$	16.7477	−	3.49545i	0.605515	−	0.126378i
$766$		0			0
$767$		0			0
$768$	19.5959	+	19.5959i	0.707107	+	0.707107i
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$	−39.0780	−	25.5826i	−1.40827	−	0.921932i
$771$	−34.4174			−1.23951
$772$	−23.4955	+	23.4955i	−0.845620	+	0.845620i
$773$	37.7930	+	37.7930i	1.35932	+	1.35932i	0.874757	+	0.484561i	$0.161021\pi$
$773$	37.7930	+	37.7930i	1.35932	+	1.35932i	0.484561	+	0.874757i	$0.338979\pi$
$774$		0			0
$775$	−10.0677	−	23.0679i	−0.361641	−	0.828623i
$776$		0			0
$777$	−2.66970	+	2.66970i	−0.0957749	+	0.0957749i
$778$		0			0
$779$		−	32.3303i		−	1.15835i
$780$		0			0
$781$	86.9909			3.11278
$782$	−16.5285	+	16.5285i	−0.591058	+	0.591058i
$783$		0			0
$784$			28.0000i			1.00000i
$785$		0			0
$786$		0			0
$787$	−14.9666	+	14.9666i	−0.533503	+	0.533503i	−0.921613	−	0.388110i	$-0.873128\pi$
$787$	−14.9666	+	14.9666i	−0.533503	+	0.533503i	0.388110	+	0.921613i	$0.373128\pi$
$788$		0			0
$789$		−	2.44949i		−	0.0872041i
$790$		0			0
$791$		0			0
$792$	−33.4955	+	33.4955i	−1.19021	+	1.19021i
$793$		0			0
$794$		0			0
$795$		0			0
$796$	−31.7085			−1.12388
$797$	−26.2983	+	26.2983i	−0.931534	+	0.931534i	−0.997802	−	0.0662682i	$-0.978891\pi$
$797$	−26.2983	+	26.2983i	−0.931534	+	0.931534i	0.0662682	+	0.997802i	$0.478891\pi$
$798$	−39.5964	−	39.5964i	−1.40170	−	1.40170i
$799$		0			0
$800$	−26.3303	−	10.3303i	−0.930917	−	0.365231i
$801$	7.34847			0.259645
$802$		0			0
$803$		0			0
$804$		0			0
$805$	21.0000	−	32.0780i	0.740153	−	1.13060i
$806$		0			0
$807$	33.0000	−	33.0000i	1.16166	−	1.16166i
$808$	14.6969	+	14.6969i	0.517036	+	0.517036i
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$	5.81475	+	27.8602i	0.204310	+	0.978906i
$811$	−56.8776			−1.99724			−0.998622	−	0.0524767i	$-0.983288\pi$
$811$	−56.8776			−1.99724			−0.998622	+	0.0524767i	$0.983288\pi$
$812$		0			0
$813$	36.1652	+	36.1652i	1.26837	+	1.26837i
$814$		−	6.50455i		−	0.227984i
$815$		0			0
$816$	17.6697			0.618563
$817$		0			0
$818$		0			0
$819$		0			0
$820$	−14.0000	−	9.16515i	−0.488901	−	0.320061i
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$823$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$824$		−	29.9333i		−	1.04277i
$825$	17.6577	−	45.0066i	0.614761	−	1.56693i
$826$		0			0
$827$	−32.0780	+	32.0780i	−1.11546	+	1.11546i	−0.123064	+	0.992399i	$0.539272\pi$
$827$	−32.0780	+	32.0780i	−1.11546	+	1.11546i	−0.992399	+	0.123064i	$0.960728\pi$
$828$	−27.4955	−	27.4955i	−0.955533	−	0.955533i
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		0			0
$831$	−50.4168			−1.74894
$832$		0			0
$833$	12.6238	+	12.6238i	0.437390	+	0.437390i
$834$			54.6606i			1.89274i
$835$		0			0
$836$	96.4740			3.33662
$837$	−18.4955	+	18.4955i	−0.639296	+	0.639296i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$	−28.3714	+	5.92146i	−0.978906	+	0.204310i
$841$	−29.0000			−1.00000
$842$	−9.16515	+	9.16515i	−0.315852	+	0.315852i
$843$		0			0
$844$		0			0
$845$	−24.3208	−	15.9217i	−0.836660	−	0.547723i
$846$		0			0
$847$	37.7255	−	37.7255i	1.29627	−	1.29627i
$848$		0			0
$849$			36.0000i			1.23552i
$850$	−16.5285	+	7.21362i	−0.566922	+	0.247425i
$851$	5.33939			0.183032
$852$	38.1694	−	38.1694i	1.30766	−	1.30766i
$853$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$853$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$854$		0			0
$855$	31.7477	−	48.4955i	1.08575	−	1.65851i
$856$	54.9909			1.87955
$857$	41.3998	−	41.3998i	1.41419	−	1.41419i	0.702969	−	0.711220i	$-0.251857\pi$
$857$	41.3998	−	41.3998i	1.41419	−	1.41419i	0.711220	−	0.702969i	$-0.248143\pi$
$858$		0			0
$859$			46.0572i			1.57145i	0.618575	+	0.785726i	$0.287710\pi$
$859$			46.0572i			1.57145i	−0.618575	+	0.785726i	$0.712290\pi$
$860$		0			0
$861$	−17.1464			−0.584349
$862$	24.4174	−	24.4174i	0.831661	−	0.831661i
$863$	41.2432	+	41.2432i	1.40393	+	1.40393i	0.787068	+	0.616866i	$0.211598\pi$
$863$	41.2432	+	41.2432i	1.40393	+	1.40393i	0.616866	+	0.787068i	$0.288402\pi$
$864$			29.3939i			1.00000i
$865$	36.7477	−	7.66970i	1.24946	−	0.260778i
$866$		0			0
$867$	−12.8543	+	12.8543i	−0.436553	+	0.436553i
$868$	−18.8348	−	18.8348i	−0.639296	−	0.639296i
$869$		0			0
$870$		0			0
$871$		0			0
$872$	36.6606	−	36.6606i	1.24148	−	1.24148i
$873$		0			0
$874$			79.1927i			2.67873i
$875$	24.1216	−	17.1216i	0.815459	−	0.578815i
$876$		0			0
$877$	−38.4955	+	38.4955i	−1.29990	+	1.29990i	−0.371444	+	0.928456i	$0.621137\pi$
$877$	−38.4955	+	38.4955i	−1.29990	+	1.29990i	−0.928456	+	0.371444i	$0.878863\pi$
$878$	−33.1355	−	33.1355i	−1.11827	−	1.11827i
$879$		−	59.0780i		−	1.99265i
$880$	27.3489	−	41.7762i	0.921932	−	1.40827i
$881$	56.3383			1.89808			0.949042	−	0.315149i	$-0.102055\pi$
$881$	56.3383			1.89808			0.949042	+	0.315149i	$0.102055\pi$
$882$	−21.0000	+	21.0000i	−0.707107	+	0.707107i
$883$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$883$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$884$		0			0
$885$		0			0
$886$	−38.0000			−1.27663
$887$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$887$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$888$	−2.85403	−	2.85403i	−0.0957749	−	0.0957749i
$889$		0			0
$890$	−7.58258	+	1.58258i	−0.254169	+	0.0530481i
$891$	−50.2432			−1.68321
$892$	−19.5959	+	19.5959i	−0.656120	+	0.656120i
$893$		0			0
$894$		0			0
$895$	−50.0404	−	32.7591i	−1.67267	−	1.09502i
$896$	−29.9333			−1.00000
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−12.0000	−	27.4955i	−0.400000	−	0.916515i
$901$		0			0
$902$	20.8881	−	20.8881i	0.695497	−	0.695497i
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$907$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$908$		0			0
$909$			22.0454i			0.731200i
$910$		0			0
$911$	−30.2432			−1.00200			−0.501001	−	0.865447i	$-0.667035\pi$
$911$	−30.2432			−1.00200			−0.501001	+	0.865447i	$0.667035\pi$
$912$	42.3303	−	42.3303i	1.40170	−	1.40170i
$913$		0			0
$914$			14.6606i			0.484930i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	13.2523	+	13.2523i	0.437390	+	0.437390i
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$	34.2929	+	22.4499i	1.13060	+	0.740153i
$921$	9.16515			0.302002
$922$	41.1582	−	41.1582i	1.35547	−	1.35547i
$923$		0			0
$924$		−	51.1652i		−	1.68321i
$925$	3.83485	+	1.50455i	0.126089	+	0.0494692i
$926$		0			0
$927$	22.4499	−	22.4499i	0.737353	−	0.737353i
$928$		0			0
$929$		−	48.6415i		−	1.59588i	−0.602739	−	0.797939i	$-0.705924\pi$
$929$		−	48.6415i		−	1.59588i	0.602739	−	0.797939i	$-0.294076\pi$
$930$	15.1015	−	23.0679i	0.495197	−	0.756426i
$931$	60.4845			1.98230
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−6.50455	−	31.1652i	−0.212721	−	1.01921i
$936$		0			0
$937$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$937$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−3.74166			−0.121975			−0.0609873	−	0.998139i	$-0.519425\pi$
$941$	−3.74166			−0.121975			−0.0609873	+	0.998139i	$0.519425\pi$
$942$		0			0
$943$	17.1464	+	17.1464i	0.558365	+	0.558365i
$944$		0			0
$945$	−25.7196	−	16.8375i	−0.836660	−	0.547723i
$946$		0			0
$947$	−37.0000	+	37.0000i	−1.20234	+	1.20234i	−0.228885	+	0.973453i	$0.573508\pi$
$947$	−37.0000	+	37.0000i	−1.20234	+	1.20234i	−0.973453	+	0.228885i	$0.926492\pi$
$948$		0			0
$949$		0			0
$950$	−22.3151	+	56.8776i	−0.723997	+	1.84535i
$951$		0			0
$952$	−13.4955	+	13.4955i	−0.437390	+	0.437390i
$953$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$953$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$954$		0			0
$955$	17.6577	−	26.9725i	0.571389	−	0.872811i
$956$	−6.50455			−0.210372
$957$		0			0
$958$		0			0
$959$		0			0
$960$	−6.33030	−	30.3303i	−0.204310	−	0.978906i
$961$	−5.66061			−0.182600
$962$		0			0
$963$	41.2432	+	41.2432i	1.32904	+	1.32904i
$964$		0			0
$965$	36.3660	−	7.59002i	1.17066	−	0.244331i
$966$	42.0000			1.35133
$967$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$967$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$968$	40.3303	+	40.3303i	1.29627	+	1.29627i
$969$		−	38.1694i		−	1.22618i
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$	−22.0454	+	22.0454i	−0.707107	+	0.707107i
$973$	−41.7477	−	41.7477i	−1.33837	−	1.33837i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$977$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$978$		0			0
$979$		−	13.6745i		−	0.437038i
$980$	17.1464	−	26.1916i	0.547723	−	0.836660i
$981$	54.9909			1.75572
$982$	−35.5826	+	35.5826i	−1.13549	+	1.13549i
$983$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$983$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$984$		−	18.3303i		−	0.584349i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$	51.8438	−	10.8204i	1.64770	−	0.343896i
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$	20.1353	−	20.1353i	0.639296	−	0.639296i
$993$		0			0
$994$			58.3047i			1.84931i
$995$	29.6606	+	19.4174i	0.940304	+	0.615574i
$996$		0			0
$997$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$997$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$998$		0			0
$999$		−	4.28104i		−	0.135446i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	420.2.w.a.83.4	yes	8
3.2	odd	2		420.2.w.b.83.3	yes	8
4.3	odd	2		420.2.w.b.83.2	yes	8
5.2	odd	4	inner	420.2.w.a.167.3	yes	8
7.6	odd	2	inner	420.2.w.a.83.1	✓	8
12.11	even	2	inner	420.2.w.a.83.1	✓	8
15.2	even	4		420.2.w.b.167.4	yes	8
20.7	even	4		420.2.w.b.167.1	yes	8
21.20	even	2		420.2.w.b.83.2	yes	8
28.27	even	2		420.2.w.b.83.3	yes	8
35.27	even	4	inner	420.2.w.a.167.2	yes	8
60.47	odd	4	inner	420.2.w.a.167.2	yes	8
84.83	odd	2	CM	420.2.w.a.83.4	yes	8
105.62	odd	4		420.2.w.b.167.1	yes	8
140.27	odd	4		420.2.w.b.167.4	yes	8
420.167	even	4	inner	420.2.w.a.167.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.w.a.83.1	✓	8	7.6	odd	2	inner
420.2.w.a.83.1	✓	8	12.11	even	2	inner
420.2.w.a.83.4	yes	8	1.1	even	1	trivial
420.2.w.a.83.4	yes	8	84.83	odd	2	CM
420.2.w.a.167.2	yes	8	35.27	even	4	inner
420.2.w.a.167.2	yes	8	60.47	odd	4	inner
420.2.w.a.167.3	yes	8	5.2	odd	4	inner
420.2.w.a.167.3	yes	8	420.167	even	4	inner
420.2.w.b.83.2	yes	8	4.3	odd	2
420.2.w.b.83.2	yes	8	21.20	even	2
420.2.w.b.83.3	yes	8	3.2	odd	2
420.2.w.b.83.3	yes	8	28.27	even	2
420.2.w.b.167.1	yes	8	20.7	even	4
420.2.w.b.167.1	yes	8	105.62	odd	4
420.2.w.b.167.4	yes	8	15.2	even	4
420.2.w.b.167.4	yes	8	140.27	odd	4

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:	\( N \)	\(=\)	\( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	420.w (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.00000i) q^{2} +(1.22474 + 1.22474i) q^{3} -2.00000i q^{4} +(-1.22474 + 1.87083i) q^{5} -2.44949 q^{6} +(1.87083 - 1.87083i) q^{7} +(2.00000 + 2.00000i) q^{8} +3.00000i q^{9} +O(q^{10})\)	\(q+(-1.00000 + 1.00000i) q^{2} +(1.22474 + 1.22474i) q^{3} -2.00000i q^{4} +(-1.22474 + 1.87083i) q^{5} -2.44949 q^{6} +(1.87083 - 1.87083i) q^{7} +(2.00000 + 2.00000i) q^{8} +3.00000i q^{9} +(-0.646084 - 3.09557i) q^{10} +5.58258 q^{11} +(2.44949 - 2.44949i) q^{12} +3.74166i q^{14} +(-3.79129 + 0.791288i) q^{15} -4.00000 q^{16} +(-1.80341 + 1.80341i) q^{17} +(-3.00000 - 3.00000i) q^{18} +8.64064i q^{19} +(3.74166 + 2.44949i) q^{20} +4.58258 q^{21} +(-5.58258 + 5.58258i) q^{22} +(-4.58258 - 4.58258i) q^{23} +4.89898i q^{24} +(-2.00000 - 4.58258i) q^{25} +(-3.67423 + 3.67423i) q^{27} +(-3.74166 - 3.74166i) q^{28} +(3.00000 - 4.58258i) q^{30} +5.03383 q^{31} +(4.00000 - 4.00000i) q^{32} +(6.83723 + 6.83723i) q^{33} -3.60681i q^{34} +(1.20871 + 5.79129i) q^{35} +6.00000 q^{36} +(-0.582576 + 0.582576i) q^{37} +(-8.64064 - 8.64064i) q^{38} +(-6.19115 + 1.29217i) q^{40} -3.74166 q^{41} +(-4.58258 + 4.58258i) q^{42} -11.1652i q^{44} +(-5.61249 - 3.67423i) q^{45} +9.16515 q^{46} +(-4.89898 - 4.89898i) q^{48} -7.00000i q^{49} +(6.58258 + 2.58258i) q^{50} -4.41742 q^{51} -7.34847i q^{54} +(-6.83723 + 10.4440i) q^{55} +7.48331 q^{56} +(-10.5826 + 10.5826i) q^{57} +(1.58258 + 7.58258i) q^{60} +(-5.03383 + 5.03383i) q^{62} +(5.61249 + 5.61249i) q^{63} +8.00000i q^{64} -13.6745 q^{66} +(3.60681 + 3.60681i) q^{68} -11.2250i q^{69} +(-7.00000 - 4.58258i) q^{70} +15.5826 q^{71} +(-6.00000 + 6.00000i) q^{72} -1.16515i q^{74} +(3.16300 - 8.06198i) q^{75} +17.2813 q^{76} +(10.4440 - 10.4440i) q^{77} +(4.89898 - 7.48331i) q^{80} -9.00000 q^{81} +(3.74166 - 3.74166i) q^{82} -9.16515i q^{84} +(-1.16515 - 5.58258i) q^{85} +(11.1652 + 11.1652i) q^{88} -2.44949i q^{89} +(9.28672 - 1.93825i) q^{90} +(-9.16515 + 9.16515i) q^{92} +(6.16515 + 6.16515i) q^{93} +(-16.1652 - 10.5826i) q^{95} +9.79796 q^{96} +(7.00000 + 7.00000i) q^{98} +16.7477i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 16 q^{8}+O(q^{10}) \) 8 * q - 8 * q^2 + 16 * q^8	\( 8 q - 8 q^{2} + 16 q^{8} + 8 q^{11} - 12 q^{15} - 32 q^{16} - 24 q^{18} - 8 q^{22} - 16 q^{25} + 24 q^{30} + 32 q^{32} + 28 q^{35} + 48 q^{36} + 32 q^{37} + 16 q^{50} - 72 q^{51} - 48 q^{57} - 24 q^{60} - 56 q^{70} + 88 q^{71} - 48 q^{72} - 72 q^{81} + 64 q^{85} + 16 q^{88} - 24 q^{93} - 56 q^{95} + 56 q^{98}+O(q^{100}) \) 8 * q - 8 * q^2 + 16 * q^8 + 8 * q^11 - 12 * q^15 - 32 * q^16 - 24 * q^18 - 8 * q^22 - 16 * q^25 + 24 * q^30 + 32 * q^32 + 28 * q^35 + 48 * q^36 + 32 * q^37 + 16 * q^50 - 72 * q^51 - 48 * q^57 - 24 * q^60 - 56 * q^70 + 88 * q^71 - 48 * q^72 - 72 * q^81 + 64 * q^85 + 16 * q^88 - 24 * q^93 - 56 * q^95 + 56 * q^98

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