LMFDB - Embedded newform 420.2.w.a.83.3

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			83.3
Root			$0.323042 + 0.323042i$ of defining polynomial
Character	$\chi$	$=$	420.83
Dual form			420.2.w.a.167.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 + 1.00000i) q^{2} +(1.22474 + 1.22474i) q^{3} -2.00000i q^{4} +(-1.22474 - 1.87083i) q^{5} -2.44949 q^{6} +(-1.87083 + 1.87083i) q^{7} +(2.00000 + 2.00000i) q^{8} +3.00000i q^{9} +O(q^{10})$	\(q+(-1.00000 + 1.00000i) q^{2} +(1.22474 + 1.22474i) q^{3} -2.00000i q^{4} +(-1.22474 - 1.87083i) q^{5} -2.44949 q^{6} +(-1.87083 + 1.87083i) q^{7} +(2.00000 + 2.00000i) q^{8} +3.00000i q^{9} +(3.09557 + 0.646084i) q^{10} -3.58258 q^{11} +(2.44949 - 2.44949i) q^{12} -3.74166i q^{14} +(0.791288 - 3.79129i) q^{15} -4.00000 q^{16} +(-5.54506 + 5.54506i) q^{17} +(-3.00000 - 3.00000i) q^{18} +1.15732i q^{19} +(-3.74166 + 2.44949i) q^{20} -4.58258 q^{21} +(3.58258 - 3.58258i) q^{22} +(4.58258 + 4.58258i) q^{23} +4.89898i q^{24} +(-2.00000 + 4.58258i) q^{25} +(-3.67423 + 3.67423i) q^{27} +(3.74166 + 3.74166i) q^{28} +(3.00000 + 4.58258i) q^{30} -9.93280 q^{31} +(4.00000 - 4.00000i) q^{32} +(-4.38774 - 4.38774i) q^{33} -11.0901i q^{34} +(5.79129 + 1.20871i) q^{35} +6.00000 q^{36} +(8.58258 - 8.58258i) q^{37} +(-1.15732 - 1.15732i) q^{38} +(1.29217 - 6.19115i) q^{40} +3.74166 q^{41} +(4.58258 - 4.58258i) q^{42} +7.16515i q^{44} +(5.61249 - 3.67423i) q^{45} -9.16515 q^{46} +(-4.89898 - 4.89898i) q^{48} -7.00000i q^{49} +(-2.58258 - 6.58258i) q^{50} -13.5826 q^{51} -7.34847i q^{54} +(4.38774 + 6.70239i) q^{55} -7.48331 q^{56} +(-1.41742 + 1.41742i) q^{57} +(-7.58258 - 1.58258i) q^{60} +(9.93280 - 9.93280i) q^{62} +(-5.61249 - 5.61249i) q^{63} +8.00000i q^{64} +8.77548 q^{66} +(11.0901 + 11.0901i) q^{68} +11.2250i q^{69} +(-7.00000 + 4.58258i) q^{70} +6.41742 q^{71} +(-6.00000 + 6.00000i) q^{72} +17.1652i q^{74} +(-8.06198 + 3.16300i) q^{75} +2.31464 q^{76} +(6.70239 - 6.70239i) q^{77} +(4.89898 + 7.48331i) q^{80} -9.00000 q^{81} +(-3.74166 + 3.74166i) q^{82} +9.16515i q^{84} +(17.1652 + 3.58258i) q^{85} +(-7.16515 - 7.16515i) q^{88} -2.44949i q^{89} +(-1.93825 + 9.28672i) q^{90} +(9.16515 - 9.16515i) q^{92} +(-12.1652 - 12.1652i) q^{93} +(2.16515 - 1.41742i) q^{95} +9.79796 q^{96} +(7.00000 + 7.00000i) q^{98} -10.7477i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$10$	3.09557	+	0.646084i	0.978906	+	0.204310i
$11$	−3.58258			−1.08019			−0.540094	−	0.841605i	$-0.681611\pi$
$11$	−3.58258			−1.08019			−0.540094	+	0.841605i	$0.681611\pi$
$12$	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
$15$	0.791288	−	3.79129i	0.204310	−	0.978906i
$16$	−4.00000			−1.00000
$17$	−5.54506	+	5.54506i	−1.34488	+	1.34488i	−0.453743	+	0.891133i	$0.649911\pi$
$17$	−5.54506	+	5.54506i	−1.34488	+	1.34488i	−0.891133	+	0.453743i	$0.850089\pi$
$18$	−3.00000	−	3.00000i	−0.707107	−	0.707107i
$19$			1.15732i			0.265508i	0.991149	+	0.132754i	$0.0423820\pi$
$19$			1.15732i			0.265508i	−0.991149	+	0.132754i	$0.957618\pi$
$20$	−3.74166	+	2.44949i	−0.836660	+	0.547723i
$21$	−4.58258			−1.00000
$22$	3.58258	−	3.58258i	0.763808	−	0.763808i
$23$	4.58258	+	4.58258i	0.955533	+	0.955533i	0.999053	−	0.0435195i	$-0.0138571\pi$
$23$	4.58258	+	4.58258i	0.955533	+	0.955533i	−0.0435195	+	0.999053i	$0.513857\pi$
$24$			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
$31$	−9.93280			−1.78398			−0.891992	−	0.452051i	$-0.850693\pi$
$31$	−9.93280			−1.78398			−0.891992	+	0.452051i	$0.850693\pi$
$32$	4.00000	−	4.00000i	0.707107	−	0.707107i
$33$	−4.38774	−	4.38774i	−0.763808	−	0.763808i
$34$		−	11.0901i		−	1.90194i
$35$	5.79129	+	1.20871i	0.978906	+	0.204310i
$36$	6.00000			1.00000
$37$	8.58258	−	8.58258i	1.41097	−	1.41097i	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.58258	−	8.58258i	1.41097	−	1.41097i	0.753371	−	0.657596i	$-0.228427\pi$
$38$	−1.15732	−	1.15732i	−0.187742	−	0.187742i
$39$		0			0
$40$	1.29217	−	6.19115i	0.204310	−	0.978906i
$41$	3.74166			0.584349			0.292174	−	0.956365i	$-0.405621\pi$
$41$	3.74166			0.584349			0.292174	+	0.956365i	$0.405621\pi$
$42$	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	$0.250000\pi$
$44$			7.16515i			1.08019i
$45$	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
$50$	−2.58258	−	6.58258i	−0.365231	−	0.930917i
$51$	−13.5826			−1.90194
$52$		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
$55$	4.38774	+	6.70239i	0.591643	+	0.903749i
$56$	−7.48331			−1.00000
$57$	−1.41742	+	1.41742i	−0.187742	+	0.187742i
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$	−7.58258	−	1.58258i	−0.978906	−	0.204310i
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$	9.93280	−	9.93280i	1.26147	−	1.26147i
$63$	−5.61249	−	5.61249i	−0.707107	−	0.707107i
$64$			8.00000i			1.00000i
$65$		0			0
$66$	8.77548			1.08019
$67$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$67$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$68$	11.0901	+	11.0901i	1.34488	+	1.34488i
$69$			11.2250i			1.35133i
$70$	−7.00000	+	4.58258i	−0.836660	+	0.547723i
$71$	6.41742			0.761608			0.380804	−	0.924656i	$-0.375647\pi$
$71$	6.41742			0.761608			0.380804	+	0.924656i	$0.375647\pi$
$72$	−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	$0.250000\pi$
$74$			17.1652i			1.99541i
$75$	−8.06198	+	3.16300i	−0.930917	+	0.365231i
$76$	2.31464			0.265508
$77$	6.70239	−	6.70239i	0.763808	−	0.763808i
$78$		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
$85$	17.1652	+	3.58258i	1.86182	+	0.388585i
$86$		0			0
$87$		0			0
$88$	−7.16515	−	7.16515i	−0.763808	−	0.763808i
$89$		−	2.44949i		−	0.259645i	−0.991537	−	0.129823i	$-0.958559\pi$
$89$		−	2.44949i		−	0.259645i	0.991537	−	0.129823i	$-0.0414408\pi$
$90$	−1.93825	+	9.28672i	−0.204310	+	0.978906i
$91$		0			0
$92$	9.16515	−	9.16515i	0.955533	−	0.955533i
$93$	−12.1652	−	12.1652i	−1.26147	−	1.26147i
$94$		0			0
$95$	2.16515	−	1.41742i	0.222140	−	0.145425i
$96$	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
$99$		−	10.7477i		−	1.08019i
$100$	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	$0.380864\pi$
$102$	13.5826	−	13.5826i	1.34488	−	1.34488i
$103$	7.48331	+	7.48331i	0.737353	+	0.737353i	0.972065	−	0.234712i	$-0.0754147\pi$
$103$	7.48331	+	7.48331i	0.737353	+	0.737353i	−0.234712	+	0.972065i	$0.575415\pi$
$104$		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	$0.638966\pi$
$107$	−13.7477	+	13.7477i	−1.32904	+	1.32904i	−0.906206	+	0.422837i	$0.861034\pi$
$108$	7.34847	+	7.34847i	0.707107	+	0.707107i
$109$			18.3303i			1.75572i	0.478913	+	0.877862i	$0.341031\pi$
$109$			18.3303i			1.75572i	−0.478913	+	0.877862i	$0.658969\pi$
$110$	−11.0901	−	2.31464i	−1.05740	−	0.220693i
$111$	21.0229			1.99541
$112$	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	$0.250000\pi$
$114$		−	2.83485i		−	0.265508i
$115$	2.96073	−	14.1857i	0.276089	−	1.32282i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	20.7477i		−	1.90194i
$120$	9.16515	−	6.00000i	0.836660	−	0.547723i
$121$	1.83485			0.166804
$122$		0			0
$123$	4.58258	+	4.58258i	0.413197	+	0.413197i
$124$			19.8656i			1.78398i
$125$	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			$\pi$
$132$	−8.77548	+	8.77548i	−0.763808	+	0.763808i
$133$	−2.16515	−	2.16515i	−0.187742	−	0.187742i
$134$		0			0
$135$	11.3739	+	2.37386i	0.978906	+	0.204310i
$136$	−22.1803			−1.90194
$137$		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
$139$			7.61816i			0.646164i	0.946371	+	0.323082i	$0.104719\pi$
$139$			7.61816i			0.646164i	−0.946371	+	0.323082i	$0.895281\pi$
$140$	2.41742	−	11.5826i	0.204310	−	0.978906i
$141$		0			0
$142$	−6.41742	+	6.41742i	−0.538538	+	0.538538i
$143$		0			0
$144$		−	12.0000i		−	1.00000i
$145$		0			0
$146$		0			0
$147$	8.57321	−	8.57321i	0.707107	−	0.707107i
$148$	−17.1652	−	17.1652i	−1.41097	−	1.41097i
$149$		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			$\pi$
$152$	−2.31464	+	2.31464i	−0.187742	+	0.187742i
$153$	−16.6352	−	16.6352i	−1.34488	−	1.34488i
$154$			13.4048i			1.08019i
$155$	12.1652	+	18.5826i	0.977128	+	1.49259i
$156$		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
$160$	−12.3823	−	2.58434i	−0.978906	−	0.204310i
$161$	−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
$165$	−2.83485	+	13.5826i	−0.220693	+	1.05740i
$166$		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
$170$	−20.7477	+	13.5826i	−1.59128	+	1.04174i
$171$	−3.47197			−0.265508
$172$		0			0
$173$	14.3205	+	14.3205i	1.08877	+	1.08877i	0.995655	+	0.0931156i	$0.0296826\pi$
$173$	14.3205	+	14.3205i	1.08877	+	1.08877i	0.0931156	+	0.995655i	$0.470317\pi$
$174$		0			0
$175$	−4.83156	−	12.3149i	−0.365231	−	0.930917i
$176$	14.3303			1.08019
$177$		0			0
$178$	2.44949	+	2.44949i	0.183597	+	0.183597i
$179$		−	0.747727i		−	0.0558877i	−0.999609	−	0.0279439i	$-0.991104\pi$
$179$		−	0.747727i		−	0.0558877i	0.999609	−	0.0279439i	$-0.00889597\pi$
$180$	−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
$185$	−26.5680	−	5.54506i	−1.95332	−	0.407681i
$186$	24.3303			1.78398
$187$	19.8656	−	19.8656i	1.45272	−	1.45272i
$188$		0			0
$189$		−	13.7477i		−	1.00000i
$190$	−0.747727	+	3.58258i	−0.0542458	+	0.259907i
$191$	−23.5826			−1.70638			−0.853188	−	0.521604i	$-0.825334\pi$
$191$	−23.5826			−1.70638			−0.853188	+	0.521604i	$0.825334\pi$
$192$	−9.79796	+	9.79796i	−0.707107	+	0.707107i
$193$	15.7477	+	15.7477i	1.13355	+	1.13355i	0.989583	+	0.143963i	$0.0459847\pi$
$193$	15.7477	+	15.7477i	1.13355	+	1.13355i	0.143963	+	0.989583i	$0.454015\pi$
$194$		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	$0.750000\pi$
$198$	10.7477	+	10.7477i	0.763808	+	0.763808i
$199$		−	23.3376i		−	1.65436i	−0.561939	−	0.827178i	$-0.689945\pi$
$199$		−	23.3376i		−	1.65436i	0.561939	−	0.827178i	$-0.310055\pi$
$200$	−13.1652	+	5.16515i	−0.930917	+	0.365231i
$201$		0			0
$202$	−7.34847	+	7.34847i	−0.517036	+	0.517036i
$203$		0			0
$204$			27.1652i			1.90194i
$205$	−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
$209$		−	4.14619i		−	0.286798i
$210$	−14.1857	−	2.96073i	−0.978906	−	0.204310i
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$	7.85971	+	7.85971i	0.538538	+	0.538538i
$214$		−	27.4955i		−	1.87955i
$215$		0			0
$216$	−14.6969			−1.00000
$217$	18.5826	−	18.5826i	1.26147	−	1.26147i
$218$	−18.3303	−	18.3303i	−1.24148	−	1.24148i
$219$		0			0
$220$	13.4048	−	8.77548i	0.903749	−	0.591643i
$221$		0			0
$222$	−21.0229	+	21.0229i	−1.41097	+	1.41097i
$223$	−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	$0.750000\pi$
$228$	2.83485	+	2.83485i	0.187742	+	0.187742i
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$	11.2250	+	17.1464i	0.740153	+	1.13060i
$231$	16.4174			1.08019
$232$		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
$238$	20.7477	+	20.7477i	1.34488	+	1.34488i
$239$		−	30.7477i		−	1.98891i	−0.105186	−	0.994453i	$-0.533544\pi$
$239$		−	30.7477i		−	1.98891i	0.105186	−	0.994453i	$-0.466456\pi$
$240$	−3.16515	+	15.1652i	−0.204310	+	0.978906i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	−1.83485	+	1.83485i	−0.117949	+	0.117949i
$243$	−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
$248$	−19.8656	−	19.8656i	−1.26147	−	1.26147i
$249$		0			0
$250$	−9.15188	+	12.8935i	−0.578815	+	0.815459i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$	−11.2250	+	11.2250i	−0.707107	+	0.707107i
$253$	−16.4174	−	16.4174i	−1.03215	−	1.03215i
$254$		0			0
$255$	16.6352	+	25.4107i	1.04174	+	1.59128i
$256$	16.0000			1.00000
$257$	−17.7925	+	17.7925i	−1.10987	+	1.10987i	−0.116699	+	0.993167i	$0.537231\pi$
$257$	−17.7925	+	17.7925i	−1.10987	+	1.10987i	−0.993167	+	0.116699i	$0.962769\pi$
$258$		0			0
$259$			32.1131i			1.99541i
$260$		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	$0.763883\pi$
$264$		−	17.5510i		−	1.08019i
$265$		0			0
$266$	4.33030			0.265508
$267$	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
$271$	14.5621			0.884584			0.442292	−	0.896871i	$-0.354165\pi$
$271$	14.5621			0.884584			0.442292	+	0.896871i	$0.354165\pi$
$272$	22.1803	−	22.1803i	1.34488	−	1.34488i
$273$		0			0
$274$		0			0
$275$	7.16515	−	16.4174i	0.432075	−	0.990008i
$276$	22.4499			1.35133
$277$	−11.4174	+	11.4174i	−0.686007	+	0.686007i	−0.961347	−	0.275340i	$-0.911209\pi$
$277$	−11.4174	+	11.4174i	−0.686007	+	0.686007i	0.275340	+	0.961347i	$0.411209\pi$
$278$	−7.61816	−	7.61816i	−0.456907	−	0.456907i
$279$		−	29.7984i		−	1.78398i
$280$	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	$0.538041\pi$
$284$		−	12.8348i		−	0.761608i
$285$	4.38774	+	0.915775i	0.259907	+	0.0542458i
$286$		0			0
$287$	−7.00000	+	7.00000i	−0.413197	+	0.413197i
$288$	12.0000	+	12.0000i	0.707107	+	0.707107i
$289$		−	44.4955i		−	2.61738i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2.07310	+	2.07310i	0.121112	+	0.121112i	0.765065	−	0.643953i	$-0.222707\pi$
$293$	2.07310	+	2.07310i	0.121112	+	0.121112i	−0.643953	+	0.765065i	$0.722707\pi$
$294$			17.1464i			1.00000i
$295$		0			0
$296$	34.3303			1.99541
$297$	13.1632	−	13.1632i	0.763808	−	0.763808i
$298$		0			0
$299$		0			0
$300$	6.32599	+	16.1240i	0.365231	+	0.930917i
$301$		0			0
$302$		0			0
$303$	9.00000	+	9.00000i	0.517036	+	0.517036i
$304$		−	4.62929i		−	0.265508i
$305$		0			0
$306$	33.2704			1.90194
$307$	−3.74166	+	3.74166i	−0.213548	+	0.213548i	−0.805773	−	0.592225i	$-0.798250\pi$
$307$	−3.74166	+	3.74166i	−0.213548	+	0.213548i	0.592225	+	0.805773i	$0.298250\pi$
$308$	−13.4048	−	13.4048i	−0.763808	−	0.763808i
$309$			18.3303i			1.04277i
$310$	−30.7477	−	6.41742i	−1.74635	−	0.364485i
$311$		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
$315$	−3.62614	+	17.3739i	−0.204310	+	0.978906i
$316$		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
$323$	−6.41742	−	6.41742i	−0.357075	−	0.357075i
$324$			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
$330$	−10.7477	−	16.4174i	−0.591643	−	0.903749i
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$	25.7477	+	25.7477i	1.41097	+	1.41097i
$334$		0			0
$335$		0			0
$336$	18.3303			1.00000
$337$	−19.3303	+	19.3303i	−1.05299	+	1.05299i	−0.0544735	+	0.998515i	$0.517348\pi$
$337$	−19.3303	+	19.3303i	−1.05299	+	1.05299i	−0.998515	+	0.0544735i	$0.982652\pi$
$338$	−13.0000	−	13.0000i	−0.707107	−	0.707107i
$339$		0			0
$340$	7.16515	−	34.3303i	0.388585	−	1.86182i
$341$	35.5850			1.92704
$342$	3.47197	−	3.47197i	0.187742	−	0.187742i
$343$	13.0958	+	13.0958i	0.707107	+	0.707107i
$344$		0			0
$345$	21.0000	−	13.7477i	1.13060	−	0.740153i
$346$	−28.6411			−1.53975
$347$	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
$352$	−14.3303	+	14.3303i	−0.763808	+	0.763808i
$353$	26.5680	+	26.5680i	1.41407	+	1.41407i	0.717053	+	0.697019i	$0.245491\pi$
$353$	26.5680	+	26.5680i	1.41407	+	1.41407i	0.697019	+	0.717053i	$0.254509\pi$
$354$		0			0
$355$	−7.85971	−	12.0059i	−0.417150	−	0.637207i
$356$	−4.89898			−0.259645
$357$	25.4107	−	25.4107i	1.34488	−	1.34488i
$358$	0.747727	+	0.747727i	0.0395186	+	0.0395186i
$359$			9.25227i			0.488316i	0.969735	+	0.244158i	$0.0785116\pi$
$359$			9.25227i			0.488316i	−0.969735	+	0.244158i	$0.921488\pi$
$360$	18.5734	+	3.87650i	0.978906	+	0.204310i
$361$	17.6606			0.929506
$362$		0			0
$363$	2.24722	+	2.24722i	0.117949	+	0.117949i
$364$		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
$370$	32.1131	−	21.0229i	1.66948	−	1.09293i
$371$		0			0
$372$	−24.3303	+	24.3303i	−1.26147	+	1.26147i
$373$	7.83485	+	7.83485i	0.405673	+	0.405673i	0.880227	−	0.474554i	$-0.157390\pi$
$373$	7.83485	+	7.83485i	0.405673	+	0.405673i	−0.474554	+	0.880227i	$0.657390\pi$
$374$			39.7312i			2.05445i
$375$	15.7913	+	11.2087i	0.815459	+	0.578815i
$376$		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	$0.5\pi$
$380$	−2.83485	−	4.33030i	−0.145425	−	0.222140i
$381$		0			0
$382$	23.5826	−	23.5826i	1.20659	−	1.20659i
$383$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$383$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$384$		−	19.5959i		−	1.00000i
$385$	−20.7477	−	4.33030i	−1.05740	−	0.220693i
$386$	−31.4955			−1.60308
$387$		0			0
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$	−50.8213			−2.57015
$392$	14.0000	−	14.0000i	0.707107	−	0.707107i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	−21.4955			−1.08019
$397$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$397$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$398$	23.3376	+	23.3376i	1.16981	+	1.16981i
$399$		−	5.30352i		−	0.265508i
$400$	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
$407$	−30.7477	+	30.7477i	−1.52411	+	1.52411i
$408$	−27.1652	−	27.1652i	−1.34488	−	1.34488i
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$	11.5826	+	2.41742i	0.572023	+	0.119388i
$411$		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
$417$	−9.33030	+	9.33030i	−0.456907	+	0.456907i
$418$	4.14619	+	4.14619i	0.202797	+	0.202797i
$419$		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	$-0.571696\pi$
$421$	−9.16515			−0.446682			−0.223341	+	0.974740i	$0.571696\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−14.3205	−	36.5008i	−0.694649	−	1.77055i
$426$	−15.7194			−0.761608
$427$		0			0
$428$	27.4955	+	27.4955i	1.32904	+	1.32904i
$429$		0			0
$430$		0			0
$431$	−33.5826			−1.61762			−0.808808	−	0.588073i	$-0.799887\pi$
$431$	−33.5826			−1.61762			−0.808808	+	0.588073i	$0.799887\pi$
$432$	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	$0.250000\pi$
$434$			37.1652i			1.78398i
$435$		0			0
$436$	36.6606			1.75572
$437$	−5.30352	+	5.30352i	−0.253702	+	0.253702i
$438$		0			0
$439$			25.6522i			1.22431i	0.790736	+	0.612157i	$0.209698\pi$
$439$			25.6522i			1.22431i	−0.790736	+	0.612157i	$0.790302\pi$
$440$	−4.62929	+	22.1803i	−0.220693	+	1.05740i
$441$	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	$0.529631\pi$
$444$		−	42.0459i		−	1.99541i
$445$	−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	$0.5\pi$
$450$	19.7477	−	7.74773i	0.930917	−	0.365231i
$451$	−13.4048			−0.631206
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$	−5.66970			−0.265508
$457$	−29.3303	+	29.3303i	−1.37201	+	1.37201i	−0.514558	+	0.857455i	$0.672044\pi$
$457$	−29.3303	+	29.3303i	−1.37201	+	1.37201i	−0.857455	+	0.514558i	$0.827956\pi$
$458$		0			0
$459$		−	40.7477i		−	1.90194i
$460$	−28.3714	−	5.92146i	−1.32282	−	0.276089i
$461$	41.1582			1.91693			0.958465	−	0.285210i	$-0.0920634\pi$
$461$	41.1582			1.91693			0.958465	+	0.285210i	$0.0920634\pi$
$462$	−16.4174	+	16.4174i	−0.763808	+	0.763808i
$463$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$463$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$464$		0			0
$465$	−7.85971	+	37.6581i	−0.364485	+	1.74635i
$466$		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
$475$	−5.30352	−	2.31464i	−0.243342	−	0.106203i
$476$	−41.4955			−1.90194
$477$		0			0
$478$	30.7477	+	30.7477i	1.40637	+	1.40637i
$479$		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
$484$		−	3.66970i		−	0.166804i
$485$		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
$490$	4.52259	−	21.6690i	0.204310	−	0.978906i
$491$	26.4174			1.19220			0.596101	−	0.802910i	$-0.296716\pi$
$491$	26.4174			1.19220			0.596101	+	0.802910i	$0.296716\pi$
$492$	9.16515	−	9.16515i	0.413197	−	0.413197i
$493$		0			0
$494$		0			0
$495$	−20.1072	+	13.1632i	−0.903749	+	0.591643i
$496$	39.7312			1.78398
$497$	−12.0059	+	12.0059i	−0.538538	+	0.538538i
$498$		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
$506$	32.8348			1.45969
$507$	−15.9217	+	15.9217i	−0.707107	+	0.707107i
$508$		0			0
$509$		−	26.1916i		−	1.16092i	−0.814288	−	0.580461i	$-0.802872\pi$
$509$		−	26.1916i		−	1.16092i	0.814288	−	0.580461i	$-0.197128\pi$
$510$	−42.0459	−	8.77548i	−1.86182	−	0.388585i
$511$		0			0
$512$	−16.0000	+	16.0000i	−0.707107	+	0.707107i
$513$	−4.25227	−	4.25227i	−0.187742	−	0.187742i
$514$		−	35.5850i		−	1.56959i
$515$	4.83485	−	23.1652i	0.213049	−	1.02078i
$516$		0			0
$517$		0			0
$518$	−32.1131	−	32.1131i	−1.41097	−	1.41097i
$519$			35.0780i			1.53975i
$520$		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	$-0.873624\pi$
$523$	−29.9333	−	29.9333i	−1.30889	−	1.30889i	−0.386673	−	0.922217i	$-0.626376\pi$
$524$		0			0
$525$	9.16515	−	21.0000i	0.400000	−	0.916515i
$526$	2.00000			0.0872041
$527$	55.0780	−	55.0780i	2.39924	−	2.39924i
$528$	17.5510	+	17.5510i	0.763808	+	0.763808i
$529$			19.0000i			0.826087i
$530$		0			0
$531$		0			0
$532$	−4.33030	+	4.33030i	−0.187742	+	0.187742i
$533$		0			0
$534$			6.00000i			0.259645i
$535$	42.5571	+	8.88218i	1.83990	+	0.384010i
$536$		0			0
$537$	0.915775	−	0.915775i	0.0395186	−	0.0395186i
$538$	26.9444	+	26.9444i	1.16166	+	1.16166i
$539$			25.0780i			1.08019i
$540$	4.74773	−	22.7477i	0.204310	−	0.978906i
$541$	−8.00000			−0.343947			−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000			−0.343947			−0.171973	+	0.985102i	$0.555014\pi$
$542$	−14.5621	+	14.5621i	−0.625495	+	0.625495i
$543$		0			0
$544$			44.3605i			1.90194i
$545$	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
$550$	9.25227	+	23.5826i	0.394518	+	1.00556i
$551$		0			0
$552$	−22.4499	+	22.4499i	−0.955533	+	0.955533i
$553$		0			0
$554$		−	22.8348i		−	0.970160i
$555$	−25.7477	−	39.3303i	−1.09293	−	1.66948i
$556$	15.2363			0.646164
$557$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$557$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$558$	29.7984	+	29.7984i	1.26147	+	1.26147i
$559$		0			0
$560$	−23.1652	−	4.83485i	−0.978906	−	0.204310i
$561$	48.6606			2.05445
$562$		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
$568$	12.8348	+	12.8348i	0.538538	+	0.538538i
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$	−5.30352	+	3.47197i	−0.222140	+	0.145425i
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$	−28.8826	−	28.8826i	−1.20659	−	1.20659i
$574$		−	14.0000i		−	0.584349i
$575$	−30.1652	+	11.8348i	−1.25797	+	0.493547i
$576$	−24.0000			−1.00000
$577$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$577$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$578$	44.4955	+	44.4955i	1.85077	+	1.85077i
$579$			38.5739i			1.60308i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−4.14619			−0.171278
$587$		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
$589$		−	11.4955i		−	0.473662i
$590$		0			0
$591$		0			0
$592$	−34.3303	+	34.3303i	−1.41097	+	1.41097i
$593$	−23.0960	−	23.0960i	−0.948440	−	0.948440i	0.0502942	−	0.998734i	$-0.483984\pi$
$593$	−23.0960	−	23.0960i	−0.948440	−	0.948440i	−0.998734	+	0.0502942i	$0.983984\pi$
$594$			26.3264i			1.08019i
$595$	−38.8154	+	25.4107i	−1.59128	+	1.04174i
$596$		0			0
$597$	28.5826	−	28.5826i	1.16981	−	1.16981i
$598$		0			0
$599$			45.0780i			1.84184i	0.389754	+	0.920919i	$0.372560\pi$
$599$			45.0780i			1.84184i	−0.389754	+	0.920919i	$0.627440\pi$
$600$	−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
$605$	−2.24722	−	3.43269i	−0.0913625	−	0.139559i
$606$	−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	$-0.0695807\pi$
$608$	4.62929	+	4.62929i	0.187742	+	0.187742i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−33.2704	+	33.2704i	−1.34488	+	1.34488i
$613$	−32.1652	−	32.1652i	−1.29914	−	1.29914i	−0.928961	−	0.370177i	$-0.879297\pi$
$613$	−32.1652	−	32.1652i	−1.29914	−	1.29914i	−0.370177	−	0.928961i	$-0.620703\pi$
$614$		−	7.48331i		−	0.302002i
$615$	2.96073	−	14.1857i	0.119388	−	0.572023i
$616$	26.8095			1.08019
$617$		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
$619$		−	16.8767i		−	0.678333i	−0.940726	−	0.339167i	$-0.889855\pi$
$619$		−	16.8767i		−	0.678333i	0.940726	−	0.339167i	$-0.110145\pi$
$620$	37.1652	−	24.3303i	1.49259	−	0.977128i
$621$	−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
$627$	5.07803	−	5.07803i	0.202797	−	0.202797i
$628$		0			0
$629$			95.1819i			3.79515i
$630$	−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
$639$			19.2523i			0.761608i
$640$	−5.16867	+	24.7646i	−0.204310	+	0.978906i
$641$		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	$0.0106513\pi$
$643$	26.1916	+	26.1916i	1.03290	+	1.03290i	0.0334557	+	0.999440i	$0.489349\pi$
$644$			34.2929i			1.35133i
$645$		0			0
$646$	12.8348			0.504980
$647$		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
$651$	45.5178			1.78398
$652$		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
$659$			15.0780i			0.587357i	0.955904	+	0.293678i	$0.0948794\pi$
$659$			15.0780i			0.587357i	−0.955904	+	0.293678i	$0.905121\pi$
$660$	27.1652	+	5.66970i	1.05740	+	0.220693i
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1.39887	+	6.70239i	−0.0542458	+	0.259907i
$666$	−51.4955			−1.99541
$667$		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
$673$	35.7477	+	35.7477i	1.37797	+	1.37797i	0.848038	+	0.529936i	$0.177784\pi$
$673$	35.7477	+	35.7477i	1.37797	+	1.37797i	0.529936	+	0.848038i	$0.322216\pi$
$674$		−	38.6606i		−	1.48915i
$675$	−9.48899	−	24.1859i	−0.365231	−	0.930917i
$676$	26.0000			1.00000
$677$	−36.5008	+	36.5008i	−1.40284	+	1.40284i	−0.611920	+	0.790920i	$0.709603\pi$
$677$	−36.5008	+	36.5008i	−1.40284	+	1.40284i	−0.790920	+	0.611920i	$0.790397\pi$
$678$		0			0
$679$		0			0
$680$	27.1652	+	41.4955i	1.04174	+	1.59128i
$681$		0			0
$682$	−35.5850	+	35.5850i	−1.36262	+	1.36262i
$683$	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	$0.462846\pi$
$684$			6.94393i			0.265508i
$685$		0			0
$686$	−26.1916			−1.00000
$687$		0			0
$688$		0			0
$689$		0			0
$690$	−7.25227	+	34.7477i	−0.276089	+	1.32282i
$691$	51.9787			1.97736			0.988681	−	0.150034i	$-0.0479383\pi$
$691$	51.9787			1.97736			0.988681	+	0.150034i	$0.0479383\pi$
$692$	28.6411	−	28.6411i	1.08877	−	1.08877i
$693$	20.1072	+	20.1072i	0.763808	+	0.763808i
$694$			26.0000i			0.986947i
$695$	14.2523	−	9.33030i	0.540620	−	0.353919i
$696$		0			0
$697$	−20.7477	+	20.7477i	−0.785876	+	0.785876i
$698$		0			0
$699$		0			0
$700$	−24.6297	+	9.66311i	−0.930917	+	0.365231i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	9.93280	+	9.93280i	0.374623	+	0.374623i
$704$		−	28.6606i		−	1.08019i
$705$		0			0
$706$	−53.1360			−1.99980
$707$	−13.7477	+	13.7477i	−0.517036	+	0.517036i
$708$		0			0
$709$			18.3303i			0.688409i	0.938895	+	0.344204i	$0.111851\pi$
$709$			18.3303i			0.688409i	−0.938895	+	0.344204i	$0.888149\pi$
$710$	19.8656	+	4.14619i	0.745543	+	0.155604i
$711$		0			0
$712$	4.89898	−	4.89898i	0.183597	−	0.183597i
$713$	−45.5178	−	45.5178i	−1.70466	−	1.70466i
$714$			50.8213i			1.90194i
$715$		0			0
$716$	−1.49545			−0.0558877
$717$	37.6581	−	37.6581i	1.40637	−	1.40637i
$718$	−9.25227	−	9.25227i	−0.345292	−	0.345292i
$719$		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
$722$	−17.6606	+	17.6606i	−0.657260	+	0.657260i
$723$		0			0
$724$		0			0
$725$		0			0
$726$	−4.49444			−0.166804
$727$	−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
$735$	−26.5390	−	5.53901i	−0.978906	−	0.204310i
$736$	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	$0.5\pi$
$740$	−11.0901	+	53.1360i	−0.407681	+	1.95332i
$741$		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	$-0.547394\pi$
$744$		−	48.6606i		−	1.78398i
$745$		0			0
$746$	−15.6697			−0.573708
$747$		0			0
$748$	−39.7312	−	39.7312i	−1.45272	−	1.45272i
$749$		−	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
$757$	26.4955	−	26.4955i	0.962994	−	0.962994i	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	26.4955	−	26.4955i	0.962994	−	0.962994i	0.999339	+	0.0363456i	$0.0115717\pi$
$758$		0			0
$759$		−	40.2143i		−	1.45969i
$760$	7.16515	+	1.49545i	0.259907	+	0.0542458i
$761$	41.1582			1.49198			0.745992	−	0.665955i	$-0.231976\pi$
$761$	41.1582			1.49198			0.745992	+	0.665955i	$0.231976\pi$
$762$		0			0
$763$	−34.2929	−	34.2929i	−1.24148	−	1.24148i
$764$			47.1652i			1.70638i
$765$	−10.7477	+	51.4955i	−0.388585	+	1.86182i
$766$		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	$0.5\pi$
$770$	25.0780	−	16.4174i	0.903749	−	0.591643i
$771$	−43.5826			−1.56959
$772$	31.4955	−	31.4955i	1.13355	−	1.13355i
$773$	−10.8486	−	10.8486i	−0.390196	−	0.390196i	0.484561	−	0.874757i	$-0.338979\pi$
$773$	−10.8486	−	10.8486i	−0.390196	−	0.390196i	−0.874757	+	0.484561i	$0.838979\pi$
$774$		0			0
$775$	19.8656	−	45.5178i	0.713594	−	1.63505i
$776$		0			0
$777$	−39.3303	+	39.3303i	−1.41097	+	1.41097i
$778$		0			0
$779$			4.33030i			0.155149i
$780$		0			0
$781$	−22.9909			−0.822679
$782$	50.8213	−	50.8213i	1.81737	−	1.81737i
$783$		0			0
$784$			28.0000i			1.00000i
$785$		0			0
$786$		0			0
$787$	14.9666	−	14.9666i	0.533503	−	0.533503i	−0.388110	−	0.921613i	$-0.626872\pi$
$787$	14.9666	−	14.9666i	0.533503	−	0.533503i	0.921613	+	0.388110i	$0.126872\pi$
$788$		0			0
$789$		−	2.44949i		−	0.0872041i
$790$		0			0
$791$		0			0
$792$	21.4955	−	21.4955i	0.763808	−	0.763808i
$793$		0			0
$794$		0			0
$795$		0			0
$796$	−46.6752			−1.65436
$797$	−30.0400	+	30.0400i	−1.06407	+	1.06407i	−0.0662682	+	0.997802i	$0.521109\pi$
$797$	−30.0400	+	30.0400i	−1.06407	+	1.06407i	−0.997802	+	0.0662682i	$0.978891\pi$
$798$	5.30352	+	5.30352i	0.187742	+	0.187742i
$799$		0			0
$800$	10.3303	+	26.3303i	0.365231	+	0.930917i
$801$	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	$0.5\pi$
$810$	−27.8602	−	5.81475i	−0.978906	−	0.204310i
$811$	2.98887			0.104953			0.0524767	−	0.998622i	$-0.483288\pi$
$811$	2.98887			0.104953			0.0524767	+	0.998622i	$0.483288\pi$
$812$		0			0
$813$	17.8348	+	17.8348i	0.625495	+	0.625495i
$814$		−	61.4955i		−	2.15541i
$815$		0			0
$816$	54.3303			1.90194
$817$		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
$825$	28.8826	−	11.3317i	1.00556	−	0.394518i
$826$		0			0
$827$	32.0780	−	32.0780i	1.11546	−	1.11546i	0.123064	−	0.992399i	$-0.460728\pi$
$827$	32.0780	−	32.0780i	1.11546	−	1.11546i	0.992399	−	0.123064i	$-0.0392719\pi$
$828$	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
$831$	−27.9669			−0.970160
$832$		0			0
$833$	38.8154	+	38.8154i	1.34488	+	1.34488i
$834$		−	18.6606i		−	0.646164i
$835$		0			0
$836$	−8.29239			−0.286798
$837$	36.4955	−	36.4955i	1.26147	−	1.26147i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$	−5.92146	+	28.3714i	−0.204310	+	0.978906i
$841$	−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
$847$	−3.43269	+	3.43269i	−0.117949	+	0.117949i
$848$		0			0
$849$			36.0000i			1.23552i
$850$	50.8213	+	22.1803i	1.74316	+	0.760776i
$851$	78.6606			2.69645
$852$	15.7194	−	15.7194i	0.538538	−	0.538538i
$853$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$853$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$854$		0			0
$855$	4.25227	+	6.49545i	0.145425	+	0.222140i
$856$	−54.9909			−1.87955
$857$	0.241547	−	0.241547i	0.00825109	−	0.00825109i	−0.702969	−	0.711220i	$-0.748143\pi$
$857$	0.241547	−	0.241547i	0.00825109	−	0.00825109i	0.711220	+	0.702969i	$0.248143\pi$
$858$		0			0
$859$		−	36.2593i		−	1.23715i	−0.785726	−	0.618575i	$-0.787710\pi$
$859$		−	36.2593i		−	1.23715i	0.785726	−	0.618575i	$-0.212290\pi$
$860$		0			0
$861$	−17.1464			−0.584349
$862$	33.5826	−	33.5826i	1.14383	−	1.14383i
$863$	−41.2432	−	41.2432i	−1.40393	−	1.40393i	−0.787068	−	0.616866i	$-0.788402\pi$
$863$	−41.2432	−	41.2432i	−1.40393	−	1.40393i	−0.616866	−	0.787068i	$-0.711598\pi$
$864$			29.3939i			1.00000i
$865$	9.25227	−	44.3303i	0.314587	−	1.50728i
$866$		0			0
$867$	54.4956	−	54.4956i	1.85077	−	1.85077i
$868$	−37.1652	−	37.1652i	−1.26147	−	1.26147i
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−36.6606	+	36.6606i	−1.24148	+	1.24148i
$873$		0			0
$874$		−	10.6070i		−	0.358788i
$875$	−17.1216	+	24.1216i	−0.578815	+	0.815459i
$876$		0			0
$877$	16.4955	−	16.4955i	0.557012	−	0.557012i	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	16.4955	−	16.4955i	0.557012	−	0.557012i	0.928456	+	0.371444i	$0.121137\pi$
$878$	−25.6522	−	25.6522i	−0.865720	−	0.865720i
$879$			5.07803i			0.171278i
$880$	−17.5510	−	26.8095i	−0.591643	−	0.903749i
$881$	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	$0.750000\pi$
$888$	42.0459	+	42.0459i	1.41097	+	1.41097i
$889$		0			0
$890$	1.58258	−	7.58258i	0.0530481	−	0.254169i
$891$	32.2432			1.08019
$892$	−19.5959	+	19.5959i	−0.656120	+	0.656120i
$893$		0			0
$894$		0			0
$895$	−1.39887	+	0.915775i	−0.0467590	+	0.0306110i
$896$	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
$902$	13.4048	−	13.4048i	0.446330	−	0.446330i
$903$		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
$911$	52.2432			1.73089			0.865447	−	0.501001i	$-0.167035\pi$
$911$	52.2432			1.73089			0.865447	+	0.501001i	$0.167035\pi$
$912$	5.66970	−	5.66970i	0.187742	−	0.187742i
$913$		0			0
$914$		−	58.6606i		−	1.94032i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	40.7477	+	40.7477i	1.34488	+	1.34488i
$919$		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
$924$		−	32.8348i		−	1.08019i
$925$	22.1652	+	56.4955i	0.728786	+	1.85756i
$926$		0			0
$927$	−22.4499	+	22.4499i	−0.737353	+	0.737353i
$928$		0			0
$929$			48.6415i			1.59588i	0.602739	+	0.797939i	$0.294076\pi$
$929$			48.6415i			1.59588i	−0.602739	+	0.797939i	$0.705924\pi$
$930$	−29.7984	−	45.5178i	−0.977128	−	1.49259i
$931$	8.10125			0.265508
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−61.4955	−	12.8348i	−2.01112	−	0.419744i
$936$		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	$-0.480575\pi$
$941$	3.74166			0.121975			0.0609873	+	0.998139i	$0.480575\pi$
$942$		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
$950$	7.61816	−	2.98887i	0.247166	−	0.0969718i
$951$		0			0
$952$	41.4955	−	41.4955i	1.34488	−	1.34488i
$953$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$953$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$954$		0			0
$955$	28.8826	+	44.1190i	0.934620	+	1.42766i
$956$	−61.4955			−1.98891
$957$		0			0
$958$		0			0
$959$		0			0
$960$	30.3303	+	6.33030i	0.978906	+	0.204310i
$961$	67.6606			2.18260
$962$		0			0
$963$	−41.2432	−	41.2432i	−1.32904	−	1.32904i
$964$		0			0
$965$	10.1744	−	48.7482i	0.327524	−	1.56926i
$966$	42.0000			1.35133
$967$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$967$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$968$	3.66970	+	3.66970i	0.117949	+	0.117949i
$969$		−	15.7194i		−	0.504980i
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$	−22.0454	+	22.0454i	−0.707107	+	0.707107i
$973$	−14.2523	−	14.2523i	−0.456907	−	0.456907i
$974$		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
$979$			8.77548i			0.280466i
$980$	17.1464	+	26.1916i	0.547723	+	0.836660i
$981$	−54.9909			−1.75572
$982$	−26.4174	+	26.4174i	−0.843014	+	0.843014i
$983$		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
$990$	6.94393	−	33.2704i	0.220693	−	1.05740i
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$	−39.7312	+	39.7312i	−1.26147	+	1.26147i
$993$		0			0
$994$		−	24.0118i		−	0.761608i
$995$	−43.6606	+	28.5826i	−1.38413	+	0.906129i
$996$		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
$999$			63.0688i			1.99541i

Twists

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

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

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.00000i) q^{2} +(1.22474 + 1.22474i) q^{3} -2.00000i q^{4} +(-1.22474 - 1.87083i) q^{5} -2.44949 q^{6} +(-1.87083 + 1.87083i) q^{7} +(2.00000 + 2.00000i) q^{8} +3.00000i q^{9} +O(q^{10})\)	\(q+(-1.00000 + 1.00000i) q^{2} +(1.22474 + 1.22474i) q^{3} -2.00000i q^{4} +(-1.22474 - 1.87083i) q^{5} -2.44949 q^{6} +(-1.87083 + 1.87083i) q^{7} +(2.00000 + 2.00000i) q^{8} +3.00000i q^{9} +(3.09557 + 0.646084i) q^{10} -3.58258 q^{11} +(2.44949 - 2.44949i) q^{12} -3.74166i q^{14} +(0.791288 - 3.79129i) q^{15} -4.00000 q^{16} +(-5.54506 + 5.54506i) q^{17} +(-3.00000 - 3.00000i) q^{18} +1.15732i q^{19} +(-3.74166 + 2.44949i) q^{20} -4.58258 q^{21} +(3.58258 - 3.58258i) q^{22} +(4.58258 + 4.58258i) q^{23} +4.89898i q^{24} +(-2.00000 + 4.58258i) q^{25} +(-3.67423 + 3.67423i) q^{27} +(3.74166 + 3.74166i) q^{28} +(3.00000 + 4.58258i) q^{30} -9.93280 q^{31} +(4.00000 - 4.00000i) q^{32} +(-4.38774 - 4.38774i) q^{33} -11.0901i q^{34} +(5.79129 + 1.20871i) q^{35} +6.00000 q^{36} +(8.58258 - 8.58258i) q^{37} +(-1.15732 - 1.15732i) q^{38} +(1.29217 - 6.19115i) q^{40} +3.74166 q^{41} +(4.58258 - 4.58258i) q^{42} +7.16515i q^{44} +(5.61249 - 3.67423i) q^{45} -9.16515 q^{46} +(-4.89898 - 4.89898i) q^{48} -7.00000i q^{49} +(-2.58258 - 6.58258i) q^{50} -13.5826 q^{51} -7.34847i q^{54} +(4.38774 + 6.70239i) q^{55} -7.48331 q^{56} +(-1.41742 + 1.41742i) q^{57} +(-7.58258 - 1.58258i) q^{60} +(9.93280 - 9.93280i) q^{62} +(-5.61249 - 5.61249i) q^{63} +8.00000i q^{64} +8.77548 q^{66} +(11.0901 + 11.0901i) q^{68} +11.2250i q^{69} +(-7.00000 + 4.58258i) q^{70} +6.41742 q^{71} +(-6.00000 + 6.00000i) q^{72} +17.1652i q^{74} +(-8.06198 + 3.16300i) q^{75} +2.31464 q^{76} +(6.70239 - 6.70239i) q^{77} +(4.89898 + 7.48331i) q^{80} -9.00000 q^{81} +(-3.74166 + 3.74166i) q^{82} +9.16515i q^{84} +(17.1652 + 3.58258i) q^{85} +(-7.16515 - 7.16515i) q^{88} -2.44949i q^{89} +(-1.93825 + 9.28672i) q^{90} +(9.16515 - 9.16515i) q^{92} +(-12.1652 - 12.1652i) q^{93} +(2.16515 - 1.41742i) q^{95} +9.79796 q^{96} +(7.00000 + 7.00000i) q^{98} -10.7477i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 16 q^{8}+O(q^{10}) \) 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

Introduction

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