LMFDB - Embedded newform 440.2.l.a.309.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [440,2,Mod(309,440)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("440.309"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(440, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 440 = 2^{3} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	440.l (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$3.51341768894$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			309.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	440.309
Dual form			440.2.l.a.309.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.00000 - 1.00000i) q^{2} -2.00000 q^{3} +2.00000i q^{4} +(-1.00000 + 2.00000i) q^{5} +(2.00000 + 2.00000i) q^{6} +4.00000i q^{7} +(2.00000 - 2.00000i) q^{8} +1.00000 q^{9} +(3.00000 - 1.00000i) q^{10} -1.00000i q^{11} -4.00000i q^{12} -2.00000 q^{13} +(4.00000 - 4.00000i) q^{14} +(2.00000 - 4.00000i) q^{15} -4.00000 q^{16} -6.00000i q^{17} +(-1.00000 - 1.00000i) q^{18} +(-4.00000 - 2.00000i) q^{20} -8.00000i q^{21} +(-1.00000 + 1.00000i) q^{22} +8.00000i q^{23} +(-4.00000 + 4.00000i) q^{24} +(-3.00000 - 4.00000i) q^{25} +(2.00000 + 2.00000i) q^{26} +4.00000 q^{27} -8.00000 q^{28} -6.00000i q^{29} +(-6.00000 + 2.00000i) q^{30} -10.0000 q^{31} +(4.00000 + 4.00000i) q^{32} +2.00000i q^{33} +(-6.00000 + 6.00000i) q^{34} +(-8.00000 - 4.00000i) q^{35} +2.00000i q^{36} +6.00000 q^{37} +4.00000 q^{39} +(2.00000 + 6.00000i) q^{40} +6.00000 q^{41} +(-8.00000 + 8.00000i) q^{42} -8.00000 q^{43} +2.00000 q^{44} +(-1.00000 + 2.00000i) q^{45} +(8.00000 - 8.00000i) q^{46} -8.00000i q^{47} +8.00000 q^{48} -9.00000 q^{49} +(-1.00000 + 7.00000i) q^{50} +12.0000i q^{51} -4.00000i q^{52} -10.0000 q^{53} +(-4.00000 - 4.00000i) q^{54} +(2.00000 + 1.00000i) q^{55} +(8.00000 + 8.00000i) q^{56} +(-6.00000 + 6.00000i) q^{58} -4.00000i q^{59} +(8.00000 + 4.00000i) q^{60} -6.00000i q^{61} +(10.0000 + 10.0000i) q^{62} +4.00000i q^{63} -8.00000i q^{64} +(2.00000 - 4.00000i) q^{65} +(2.00000 - 2.00000i) q^{66} -2.00000 q^{67} +12.0000 q^{68} -16.0000i q^{69} +(4.00000 + 12.0000i) q^{70} +6.00000 q^{71} +(2.00000 - 2.00000i) q^{72} -2.00000i q^{73} +(-6.00000 - 6.00000i) q^{74} +(6.00000 + 8.00000i) q^{75} +4.00000 q^{77} +(-4.00000 - 4.00000i) q^{78} +4.00000 q^{79} +(4.00000 - 8.00000i) q^{80} -11.0000 q^{81} +(-6.00000 - 6.00000i) q^{82} -4.00000 q^{83} +16.0000 q^{84} +(12.0000 + 6.00000i) q^{85} +(8.00000 + 8.00000i) q^{86} +12.0000i q^{87} +(-2.00000 - 2.00000i) q^{88} -10.0000 q^{89} +(3.00000 - 1.00000i) q^{90} -8.00000i q^{91} -16.0000 q^{92} +20.0000 q^{93} +(-8.00000 + 8.00000i) q^{94} +(-8.00000 - 8.00000i) q^{96} -4.00000i q^{97} +(9.00000 + 9.00000i) q^{98} -1.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 4 q^{3} - 2 q^{5} + 4 q^{6} + 4 q^{8} + 2 q^{9} + 6 q^{10} - 4 q^{13} + 8 q^{14} + 4 q^{15} - 8 q^{16} - 2 q^{18} - 8 q^{20} - 2 q^{22} - 8 q^{24} - 6 q^{25} + 4 q^{26} + 8 q^{27} - 16 q^{28}+ \cdots + 18 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 - 4 * q^3 - 2 * q^5 + 4 * q^6 + 4 * q^8 + 2 * q^9 + 6 * q^10 - 4 * q^13 + 8 * q^14 + 4 * q^15 - 8 * q^16 - 2 * q^18 - 8 * q^20 - 2 * q^22 - 8 * q^24 - 6 * q^25 + 4 * q^26 + 8 * q^27 - 16 * q^28 - 12 * q^30 - 20 * q^31 + 8 * q^32 - 12 * q^34 - 16 * q^35 + 12 * q^37 + 8 * q^39 + 4 * q^40 + 12 * q^41 - 16 * q^42 - 16 * q^43 + 4 * q^44 - 2 * q^45 + 16 * q^46 + 16 * q^48 - 18 * q^49 - 2 * q^50 - 20 * q^53 - 8 * q^54 + 4 * q^55 + 16 * q^56 - 12 * q^58 + 16 * q^60 + 20 * q^62 + 4 * q^65 + 4 * q^66 - 4 * q^67 + 24 * q^68 + 8 * q^70 + 12 * q^71 + 4 * q^72 - 12 * q^74 + 12 * q^75 + 8 * q^77 - 8 * q^78 + 8 * q^79 + 8 * q^80 - 22 * q^81 - 12 * q^82 - 8 * q^83 + 32 * q^84 + 24 * q^85 + 16 * q^86 - 4 * q^88 - 20 * q^89 + 6 * q^90 - 32 * q^92 + 40 * q^93 - 16 * q^94 - 16 * q^96 + 18 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/440\mathbb{Z}\right)^\times$.

$n$	$111$	$177$	$221$	$321$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$3$	−2.00000			−1.15470			−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000			−1.15470			−0.577350	+	0.816497i	$0.695913\pi$
$4$			2.00000i			1.00000i
$5$	−1.00000	+	2.00000i	−0.447214	+	0.894427i
$5$	−1.00000	+	2.00000i	−0.447214	+	0.894427i
$6$	2.00000	+	2.00000i	0.816497	+	0.816497i
$7$			4.00000i			1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$			4.00000i			1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$	2.00000	−	2.00000i	0.707107	−	0.707107i
$9$	1.00000			0.333333
$10$	3.00000	−	1.00000i	0.948683	−	0.316228i
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$		−	4.00000i		−	1.15470i
$13$	−2.00000			−0.554700			−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000			−0.554700			−0.277350	+	0.960769i	$0.589456\pi$
$14$	4.00000	−	4.00000i	1.06904	−	1.06904i
$15$	2.00000	−	4.00000i	0.516398	−	1.03280i
$16$	−4.00000			−1.00000
$17$		−	6.00000i		−	1.45521i	−0.685994	−	0.727607i	$-0.740633\pi$
$17$		−	6.00000i		−	1.45521i	0.685994	−	0.727607i	$-0.259367\pi$
$18$	−1.00000	−	1.00000i	−0.235702	−	0.235702i
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$	−4.00000	−	2.00000i	−0.894427	−	0.447214i
$21$		−	8.00000i		−	1.74574i
$22$	−1.00000	+	1.00000i	−0.213201	+	0.213201i
$23$			8.00000i			1.66812i	0.551677	+	0.834058i	$0.313988\pi$
$23$			8.00000i			1.66812i	−0.551677	+	0.834058i	$0.686012\pi$
$24$	−4.00000	+	4.00000i	−0.816497	+	0.816497i
$25$	−3.00000	−	4.00000i	−0.600000	−	0.800000i
$26$	2.00000	+	2.00000i	0.392232	+	0.392232i
$27$	4.00000			0.769800
$28$	−8.00000			−1.51186
$29$		−	6.00000i		−	1.11417i	−0.830455	−	0.557086i	$-0.811919\pi$
$29$		−	6.00000i		−	1.11417i	0.830455	−	0.557086i	$-0.188081\pi$
$30$	−6.00000	+	2.00000i	−1.09545	+	0.365148i
$31$	−10.0000			−1.79605			−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000			−1.79605			−0.898027	+	0.439941i	$0.854999\pi$
$32$	4.00000	+	4.00000i	0.707107	+	0.707107i
$33$			2.00000i			0.348155i
$34$	−6.00000	+	6.00000i	−1.02899	+	1.02899i
$35$	−8.00000	−	4.00000i	−1.35225	−	0.676123i
$36$			2.00000i			0.333333i
$37$	6.00000			0.986394			0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000			0.986394			0.493197	+	0.869918i	$0.335828\pi$
$38$		0			0
$39$	4.00000			0.640513
$40$	2.00000	+	6.00000i	0.316228	+	0.948683i
$41$	6.00000			0.937043			0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000			0.937043			0.468521	+	0.883452i	$0.344787\pi$
$42$	−8.00000	+	8.00000i	−1.23443	+	1.23443i
$43$	−8.00000			−1.21999			−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000			−1.21999			−0.609994	+	0.792406i	$0.708828\pi$
$44$	2.00000			0.301511
$45$	−1.00000	+	2.00000i	−0.149071	+	0.298142i
$46$	8.00000	−	8.00000i	1.17954	−	1.17954i
$47$		−	8.00000i		−	1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$		−	8.00000i		−	1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$	8.00000			1.15470
$49$	−9.00000			−1.28571
$50$	−1.00000	+	7.00000i	−0.141421	+	0.989949i
$51$			12.0000i			1.68034i
$52$		−	4.00000i		−	0.554700i
$53$	−10.0000			−1.37361			−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000			−1.37361			−0.686803	+	0.726844i	$0.740986\pi$
$54$	−4.00000	−	4.00000i	−0.544331	−	0.544331i
$55$	2.00000	+	1.00000i	0.269680	+	0.134840i
$56$	8.00000	+	8.00000i	1.06904	+	1.06904i
$57$		0			0
$58$	−6.00000	+	6.00000i	−0.787839	+	0.787839i
$59$		−	4.00000i		−	0.520756i	−0.965507	−	0.260378i	$-0.916153\pi$
$59$		−	4.00000i		−	0.520756i	0.965507	−	0.260378i	$-0.0838471\pi$
$60$	8.00000	+	4.00000i	1.03280	+	0.516398i
$61$		−	6.00000i		−	0.768221i	−0.923287	−	0.384111i	$-0.874508\pi$
$61$		−	6.00000i		−	0.768221i	0.923287	−	0.384111i	$-0.125492\pi$
$62$	10.0000	+	10.0000i	1.27000	+	1.27000i
$63$			4.00000i			0.503953i
$64$		−	8.00000i		−	1.00000i
$65$	2.00000	−	4.00000i	0.248069	−	0.496139i
$66$	2.00000	−	2.00000i	0.246183	−	0.246183i
$67$	−2.00000			−0.244339			−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000			−0.244339			−0.122169	+	0.992509i	$0.538985\pi$
$68$	12.0000			1.45521
$69$		−	16.0000i		−	1.92617i
$70$	4.00000	+	12.0000i	0.478091	+	1.43427i
$71$	6.00000			0.712069			0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000			0.712069			0.356034	+	0.934473i	$0.384129\pi$
$72$	2.00000	−	2.00000i	0.235702	−	0.235702i
$73$		−	2.00000i		−	0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$		−	2.00000i		−	0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	−6.00000	−	6.00000i	−0.697486	−	0.697486i
$75$	6.00000	+	8.00000i	0.692820	+	0.923760i
$76$		0			0
$77$	4.00000			0.455842
$78$	−4.00000	−	4.00000i	−0.452911	−	0.452911i
$79$	4.00000			0.450035			0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000			0.450035			0.225018	+	0.974355i	$0.427756\pi$
$80$	4.00000	−	8.00000i	0.447214	−	0.894427i
$81$	−11.0000			−1.22222
$82$	−6.00000	−	6.00000i	−0.662589	−	0.662589i
$83$	−4.00000			−0.439057			−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000			−0.439057			−0.219529	+	0.975606i	$0.570452\pi$
$84$	16.0000			1.74574
$85$	12.0000	+	6.00000i	1.30158	+	0.650791i
$86$	8.00000	+	8.00000i	0.862662	+	0.862662i
$87$			12.0000i			1.28654i
$88$	−2.00000	−	2.00000i	−0.213201	−	0.213201i
$89$	−10.0000			−1.06000			−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000			−1.06000			−0.529999	+	0.847998i	$0.677808\pi$
$90$	3.00000	−	1.00000i	0.316228	−	0.105409i
$91$		−	8.00000i		−	0.838628i
$92$	−16.0000			−1.66812
$93$	20.0000			2.07390
$94$	−8.00000	+	8.00000i	−0.825137	+	0.825137i
$95$		0			0
$96$	−8.00000	−	8.00000i	−0.816497	−	0.816497i
$97$		−	4.00000i		−	0.406138i	−0.979164	−	0.203069i	$-0.934908\pi$
$97$		−	4.00000i		−	0.406138i	0.979164	−	0.203069i	$-0.0650917\pi$
$98$	9.00000	+	9.00000i	0.909137	+	0.909137i
$99$		−	1.00000i		−	0.100504i
$100$	8.00000	−	6.00000i	0.800000	−	0.600000i
$101$			6.00000i			0.597022i	0.954406	+	0.298511i	$0.0964900\pi$
$101$			6.00000i			0.597022i	−0.954406	+	0.298511i	$0.903510\pi$
$102$	12.0000	−	12.0000i	1.18818	−	1.18818i
$103$		−	8.00000i		−	0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$		−	8.00000i		−	0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$	−4.00000	+	4.00000i	−0.392232	+	0.392232i
$105$	16.0000	+	8.00000i	1.56144	+	0.780720i
$106$	10.0000	+	10.0000i	0.971286	+	0.971286i
$107$	8.00000			0.773389			0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000			0.773389			0.386695	+	0.922208i	$0.373617\pi$
$108$			8.00000i			0.769800i
$109$			10.0000i			0.957826i	0.877862	+	0.478913i	$0.158969\pi$
$109$			10.0000i			0.957826i	−0.877862	+	0.478913i	$0.841031\pi$
$110$	−1.00000	−	3.00000i	−0.0953463	−	0.286039i
$111$	−12.0000			−1.13899
$112$		−	16.0000i		−	1.51186i
$113$			4.00000i			0.376288i	0.982141	+	0.188144i	$0.0602472\pi$
$113$			4.00000i			0.376288i	−0.982141	+	0.188144i	$0.939753\pi$
$114$		0			0
$115$	−16.0000	−	8.00000i	−1.49201	−	0.746004i
$116$	12.0000			1.11417
$117$	−2.00000			−0.184900
$118$	−4.00000	+	4.00000i	−0.368230	+	0.368230i
$119$	24.0000			2.20008
$120$	−4.00000	−	12.0000i	−0.365148	−	1.09545i
$121$	−1.00000			−0.0909091
$122$	−6.00000	+	6.00000i	−0.543214	+	0.543214i
$123$	−12.0000			−1.08200
$124$		−	20.0000i		−	1.79605i
$125$	11.0000	−	2.00000i	0.983870	−	0.178885i
$126$	4.00000	−	4.00000i	0.356348	−	0.356348i
$127$			16.0000i			1.41977i	0.704317	+	0.709885i	$0.251253\pi$
$127$			16.0000i			1.41977i	−0.704317	+	0.709885i	$0.748747\pi$
$128$	−8.00000	+	8.00000i	−0.707107	+	0.707107i
$129$	16.0000			1.40872
$130$	−6.00000	+	2.00000i	−0.526235	+	0.175412i
$131$		−	8.00000i		−	0.698963i	−0.936943	−	0.349482i	$-0.886358\pi$
$131$		−	8.00000i		−	0.698963i	0.936943	−	0.349482i	$-0.113642\pi$
$132$	−4.00000			−0.348155
$133$		0			0
$134$	2.00000	+	2.00000i	0.172774	+	0.172774i
$135$	−4.00000	+	8.00000i	−0.344265	+	0.688530i
$136$	−12.0000	−	12.0000i	−1.02899	−	1.02899i
$137$		−	12.0000i		−	1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$		−	12.0000i		−	1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$	−16.0000	+	16.0000i	−1.36201	+	1.36201i
$139$			16.0000i			1.35710i	0.734553	+	0.678551i	$0.237392\pi$
$139$			16.0000i			1.35710i	−0.734553	+	0.678551i	$0.762608\pi$
$140$	8.00000	−	16.0000i	0.676123	−	1.35225i
$141$			16.0000i			1.34744i
$142$	−6.00000	−	6.00000i	−0.503509	−	0.503509i
$143$			2.00000i			0.167248i
$144$	−4.00000			−0.333333
$145$	12.0000	+	6.00000i	0.996546	+	0.498273i
$146$	−2.00000	+	2.00000i	−0.165521	+	0.165521i
$147$	18.0000			1.48461
$148$			12.0000i			0.986394i
$149$		−	6.00000i		−	0.491539i	−0.969328	−	0.245770i	$-0.920959\pi$
$149$		−	6.00000i		−	0.491539i	0.969328	−	0.245770i	$-0.0790407\pi$
$150$	2.00000	−	14.0000i	0.163299	−	1.14310i
$151$	−16.0000			−1.30206			−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000			−1.30206			−0.651031	+	0.759051i	$0.725663\pi$
$152$		0			0
$153$		−	6.00000i		−	0.485071i
$154$	−4.00000	−	4.00000i	−0.322329	−	0.322329i
$155$	10.0000	−	20.0000i	0.803219	−	1.60644i
$156$			8.00000i			0.640513i
$157$	2.00000			0.159617			0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000			0.159617			0.0798087	+	0.996810i	$0.474569\pi$
$158$	−4.00000	−	4.00000i	−0.318223	−	0.318223i
$159$	20.0000			1.58610
$160$	−12.0000	+	4.00000i	−0.948683	+	0.316228i
$161$	−32.0000			−2.52195
$162$	11.0000	+	11.0000i	0.864242	+	0.864242i
$163$	−14.0000			−1.09656			−0.548282	−	0.836293i	$-0.684718\pi$
$163$	−14.0000			−1.09656			−0.548282	+	0.836293i	$0.684718\pi$
$164$			12.0000i			0.937043i
$165$	−4.00000	−	2.00000i	−0.311400	−	0.155700i
$166$	4.00000	+	4.00000i	0.310460	+	0.310460i
$167$		−	8.00000i		−	0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$		−	8.00000i		−	0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$	−16.0000	−	16.0000i	−1.23443	−	1.23443i
$169$	−9.00000			−0.692308
$170$	−6.00000	−	18.0000i	−0.460179	−	1.38054i
$171$		0			0
$172$		−	16.0000i		−	1.21999i
$173$	2.00000			0.152057			0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000			0.152057			0.0760286	+	0.997106i	$0.475776\pi$
$174$	12.0000	−	12.0000i	0.909718	−	0.909718i
$175$	16.0000	−	12.0000i	1.20949	−	0.907115i
$176$			4.00000i			0.301511i
$177$			8.00000i			0.601317i
$178$	10.0000	+	10.0000i	0.749532	+	0.749532i
$179$			20.0000i			1.49487i	0.664335	+	0.747435i	$0.268715\pi$
$179$			20.0000i			1.49487i	−0.664335	+	0.747435i	$0.731285\pi$
$180$	−4.00000	−	2.00000i	−0.298142	−	0.149071i
$181$			16.0000i			1.18927i	0.803996	+	0.594635i	$0.202704\pi$
$181$			16.0000i			1.18927i	−0.803996	+	0.594635i	$0.797296\pi$
$182$	−8.00000	+	8.00000i	−0.592999	+	0.592999i
$183$			12.0000i			0.887066i
$184$	16.0000	+	16.0000i	1.17954	+	1.17954i
$185$	−6.00000	+	12.0000i	−0.441129	+	0.882258i
$186$	−20.0000	−	20.0000i	−1.46647	−	1.46647i
$187$	−6.00000			−0.438763
$188$	16.0000			1.16692
$189$			16.0000i			1.16383i
$190$		0			0
$191$	6.00000			0.434145			0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000			0.434145			0.217072	+	0.976156i	$0.430349\pi$
$192$			16.0000i			1.15470i
$193$			6.00000i			0.431889i	0.976406	+	0.215945i	$0.0692831\pi$
$193$			6.00000i			0.431889i	−0.976406	+	0.215945i	$0.930717\pi$
$194$	−4.00000	+	4.00000i	−0.287183	+	0.287183i
$195$	−4.00000	+	8.00000i	−0.286446	+	0.572892i
$196$		−	18.0000i		−	1.28571i
$197$	−18.0000			−1.28245			−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000			−1.28245			−0.641223	+	0.767354i	$0.721573\pi$
$198$	−1.00000	+	1.00000i	−0.0710669	+	0.0710669i
$199$	−18.0000			−1.27599			−0.637993	−	0.770042i	$-0.720235\pi$
$199$	−18.0000			−1.27599			−0.637993	+	0.770042i	$0.720235\pi$
$200$	−14.0000	−	2.00000i	−0.989949	−	0.141421i
$201$	4.00000			0.282138
$202$	6.00000	−	6.00000i	0.422159	−	0.422159i
$203$	24.0000			1.68447
$204$	−24.0000			−1.68034
$205$	−6.00000	+	12.0000i	−0.419058	+	0.838116i
$206$	−8.00000	+	8.00000i	−0.557386	+	0.557386i
$207$			8.00000i			0.556038i
$208$	8.00000			0.554700
$209$		0			0
$210$	−8.00000	−	24.0000i	−0.552052	−	1.65616i
$211$			12.0000i			0.826114i	0.910705	+	0.413057i	$0.135539\pi$
$211$			12.0000i			0.826114i	−0.910705	+	0.413057i	$0.864461\pi$
$212$		−	20.0000i		−	1.37361i
$213$	−12.0000			−0.822226
$214$	−8.00000	−	8.00000i	−0.546869	−	0.546869i
$215$	8.00000	−	16.0000i	0.545595	−	1.09119i
$216$	8.00000	−	8.00000i	0.544331	−	0.544331i
$217$		−	40.0000i		−	2.71538i
$218$	10.0000	−	10.0000i	0.677285	−	0.677285i
$219$			4.00000i			0.270295i
$220$	−2.00000	+	4.00000i	−0.134840	+	0.269680i
$221$			12.0000i			0.807207i
$222$	12.0000	+	12.0000i	0.805387	+	0.805387i
$223$		−	16.0000i		−	1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$		−	16.0000i		−	1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$	−16.0000	+	16.0000i	−1.06904	+	1.06904i
$225$	−3.00000	−	4.00000i	−0.200000	−	0.266667i
$226$	4.00000	−	4.00000i	0.266076	−	0.266076i
$227$	−16.0000			−1.06196			−0.530979	−	0.847385i	$-0.678176\pi$
$227$	−16.0000			−1.06196			−0.530979	+	0.847385i	$0.678176\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$	8.00000	+	24.0000i	0.527504	+	1.58251i
$231$	−8.00000			−0.526361
$232$	−12.0000	−	12.0000i	−0.787839	−	0.787839i
$233$			18.0000i			1.17922i	0.807688	+	0.589610i	$0.200718\pi$
$233$			18.0000i			1.17922i	−0.807688	+	0.589610i	$0.799282\pi$
$234$	2.00000	+	2.00000i	0.130744	+	0.130744i
$235$	16.0000	+	8.00000i	1.04372	+	0.521862i
$236$	8.00000			0.520756
$237$	−8.00000			−0.519656
$238$	−24.0000	−	24.0000i	−1.55569	−	1.55569i
$239$	−8.00000			−0.517477			−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000			−0.517477			−0.258738	+	0.965947i	$0.583307\pi$
$240$	−8.00000	+	16.0000i	−0.516398	+	1.03280i
$241$	−14.0000			−0.901819			−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000			−0.901819			−0.450910	+	0.892570i	$0.648900\pi$
$242$	1.00000	+	1.00000i	0.0642824	+	0.0642824i
$243$	10.0000			0.641500
$244$	12.0000			0.768221
$245$	9.00000	−	18.0000i	0.574989	−	1.14998i
$246$	12.0000	+	12.0000i	0.765092	+	0.765092i
$247$		0			0
$248$	−20.0000	+	20.0000i	−1.27000	+	1.27000i
$249$	8.00000			0.506979
$250$	−13.0000	−	9.00000i	−0.822192	−	0.569210i
$251$		−	12.0000i		−	0.757433i	−0.925513	−	0.378717i	$-0.876365\pi$
$251$		−	12.0000i		−	0.757433i	0.925513	−	0.378717i	$-0.123635\pi$
$252$	−8.00000			−0.503953
$253$	8.00000			0.502956
$254$	16.0000	−	16.0000i	1.00393	−	1.00393i
$255$	−24.0000	−	12.0000i	−1.50294	−	0.751469i
$256$	16.0000			1.00000
$257$			8.00000i			0.499026i	0.968371	+	0.249513i	$0.0802706\pi$
$257$			8.00000i			0.499026i	−0.968371	+	0.249513i	$0.919729\pi$
$258$	−16.0000	−	16.0000i	−0.996116	−	0.996116i
$259$			24.0000i			1.49129i
$260$	8.00000	+	4.00000i	0.496139	+	0.248069i
$261$		−	6.00000i		−	0.371391i
$262$	−8.00000	+	8.00000i	−0.494242	+	0.494242i
$263$		−	12.0000i		−	0.739952i	−0.929041	−	0.369976i	$-0.879366\pi$
$263$		−	12.0000i		−	0.739952i	0.929041	−	0.369976i	$-0.120634\pi$
$264$	4.00000	+	4.00000i	0.246183	+	0.246183i
$265$	10.0000	−	20.0000i	0.614295	−	1.22859i
$266$		0			0
$267$	20.0000			1.22398
$268$		−	4.00000i		−	0.244339i
$269$		−	24.0000i		−	1.46331i	−0.681677	−	0.731653i	$-0.738749\pi$
$269$		−	24.0000i		−	1.46331i	0.681677	−	0.731653i	$-0.261251\pi$
$270$	12.0000	−	4.00000i	0.730297	−	0.243432i
$271$	8.00000			0.485965			0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000			0.485965			0.242983	+	0.970031i	$0.421874\pi$
$272$			24.0000i			1.45521i
$273$			16.0000i			0.968364i
$274$	−12.0000	+	12.0000i	−0.724947	+	0.724947i
$275$	−4.00000	+	3.00000i	−0.241209	+	0.180907i
$276$	32.0000			1.92617
$277$	10.0000			0.600842			0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000			0.600842			0.300421	+	0.953807i	$0.402873\pi$
$278$	16.0000	−	16.0000i	0.959616	−	0.959616i
$279$	−10.0000			−0.598684
$280$	−24.0000	+	8.00000i	−1.43427	+	0.478091i
$281$	18.0000			1.07379			0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000			1.07379			0.536895	+	0.843649i	$0.319597\pi$
$282$	16.0000	−	16.0000i	0.952786	−	0.952786i
$283$	−4.00000			−0.237775			−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000			−0.237775			−0.118888	+	0.992908i	$0.537933\pi$
$284$			12.0000i			0.712069i
$285$		0			0
$286$	2.00000	−	2.00000i	0.118262	−	0.118262i
$287$			24.0000i			1.41668i
$288$	4.00000	+	4.00000i	0.235702	+	0.235702i
$289$	−19.0000			−1.11765
$290$	−6.00000	−	18.0000i	−0.352332	−	1.05700i
$291$			8.00000i			0.468968i
$292$	4.00000			0.234082
$293$	−6.00000			−0.350524			−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000			−0.350524			−0.175262	+	0.984522i	$0.556077\pi$
$294$	−18.0000	−	18.0000i	−1.04978	−	1.04978i
$295$	8.00000	+	4.00000i	0.465778	+	0.232889i
$296$	12.0000	−	12.0000i	0.697486	−	0.697486i
$297$		−	4.00000i		−	0.232104i
$298$	−6.00000	+	6.00000i	−0.347571	+	0.347571i
$299$		−	16.0000i		−	0.925304i
$300$	−16.0000	+	12.0000i	−0.923760	+	0.692820i
$301$		−	32.0000i		−	1.84445i
$302$	16.0000	+	16.0000i	0.920697	+	0.920697i
$303$		−	12.0000i		−	0.689382i
$304$		0			0
$305$	12.0000	+	6.00000i	0.687118	+	0.343559i
$306$	−6.00000	+	6.00000i	−0.342997	+	0.342997i
$307$	4.00000			0.228292			0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000			0.228292			0.114146	+	0.993464i	$0.463587\pi$
$308$			8.00000i			0.455842i
$309$			16.0000i			0.910208i
$310$	−30.0000	+	10.0000i	−1.70389	+	0.567962i
$311$	−30.0000			−1.70114			−0.850572	−	0.525859i	$-0.823744\pi$
$311$	−30.0000			−1.70114			−0.850572	+	0.525859i	$0.823744\pi$
$312$	8.00000	−	8.00000i	0.452911	−	0.452911i
$313$		−	16.0000i		−	0.904373i	−0.891923	−	0.452187i	$-0.850644\pi$
$313$		−	16.0000i		−	0.904373i	0.891923	−	0.452187i	$-0.149356\pi$
$314$	−2.00000	−	2.00000i	−0.112867	−	0.112867i
$315$	−8.00000	−	4.00000i	−0.450749	−	0.225374i
$316$			8.00000i			0.450035i
$317$	−34.0000			−1.90963			−0.954815	−	0.297200i	$-0.903947\pi$
$317$	−34.0000			−1.90963			−0.954815	+	0.297200i	$0.903947\pi$
$318$	−20.0000	−	20.0000i	−1.12154	−	1.12154i
$319$	−6.00000			−0.335936
$320$	16.0000	+	8.00000i	0.894427	+	0.447214i
$321$	−16.0000			−0.893033
$322$	32.0000	+	32.0000i	1.78329	+	1.78329i
$323$		0			0
$324$		−	22.0000i		−	1.22222i
$325$	6.00000	+	8.00000i	0.332820	+	0.443760i
$326$	14.0000	+	14.0000i	0.775388	+	0.775388i
$327$		−	20.0000i		−	1.10600i
$328$	12.0000	−	12.0000i	0.662589	−	0.662589i
$329$	32.0000			1.76422
$330$	2.00000	+	6.00000i	0.110096	+	0.330289i
$331$		−	4.00000i		−	0.219860i	−0.993939	−	0.109930i	$-0.964937\pi$
$331$		−	4.00000i		−	0.219860i	0.993939	−	0.109930i	$-0.0350627\pi$
$332$		−	8.00000i		−	0.439057i
$333$	6.00000			0.328798
$334$	−8.00000	+	8.00000i	−0.437741	+	0.437741i
$335$	2.00000	−	4.00000i	0.109272	−	0.218543i
$336$			32.0000i			1.74574i
$337$			22.0000i			1.19842i	0.800593	+	0.599208i	$0.204518\pi$
$337$			22.0000i			1.19842i	−0.800593	+	0.599208i	$0.795482\pi$
$338$	9.00000	+	9.00000i	0.489535	+	0.489535i
$339$		−	8.00000i		−	0.434500i
$340$	−12.0000	+	24.0000i	−0.650791	+	1.30158i
$341$			10.0000i			0.541530i
$342$		0			0
$343$		−	8.00000i		−	0.431959i
$344$	−16.0000	+	16.0000i	−0.862662	+	0.862662i
$345$	32.0000	+	16.0000i	1.72282	+	0.861411i
$346$	−2.00000	−	2.00000i	−0.107521	−	0.107521i
$347$	16.0000			0.858925			0.429463	−	0.903085i	$-0.358703\pi$
$347$	16.0000			0.858925			0.429463	+	0.903085i	$0.358703\pi$
$348$	−24.0000			−1.28654
$349$			10.0000i			0.535288i	0.963518	+	0.267644i	$0.0862451\pi$
$349$			10.0000i			0.535288i	−0.963518	+	0.267644i	$0.913755\pi$
$350$	−28.0000	−	4.00000i	−1.49666	−	0.213809i
$351$	−8.00000			−0.427008
$352$	4.00000	−	4.00000i	0.213201	−	0.213201i
$353$		−	8.00000i		−	0.425797i	−0.977074	−	0.212899i	$-0.931710\pi$
$353$		−	8.00000i		−	0.425797i	0.977074	−	0.212899i	$-0.0682904\pi$
$354$	8.00000	−	8.00000i	0.425195	−	0.425195i
$355$	−6.00000	+	12.0000i	−0.318447	+	0.636894i
$356$		−	20.0000i		−	1.06000i
$357$	−48.0000			−2.54043
$358$	20.0000	−	20.0000i	1.05703	−	1.05703i
$359$	−12.0000			−0.633336			−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000			−0.633336			−0.316668	+	0.948536i	$0.602564\pi$
$360$	2.00000	+	6.00000i	0.105409	+	0.316228i
$361$	19.0000			1.00000
$362$	16.0000	−	16.0000i	0.840941	−	0.840941i
$363$	2.00000			0.104973
$364$	16.0000			0.838628
$365$	4.00000	+	2.00000i	0.209370	+	0.104685i
$366$	12.0000	−	12.0000i	0.627250	−	0.627250i
$367$			8.00000i			0.417597i	0.977959	+	0.208798i	$0.0669552\pi$
$367$			8.00000i			0.417597i	−0.977959	+	0.208798i	$0.933045\pi$
$368$		−	32.0000i		−	1.66812i
$369$	6.00000			0.312348
$370$	18.0000	−	6.00000i	0.935775	−	0.311925i
$371$		−	40.0000i		−	2.07670i
$372$			40.0000i			2.07390i
$373$	10.0000			0.517780			0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000			0.517780			0.258890	+	0.965907i	$0.416643\pi$
$374$	6.00000	+	6.00000i	0.310253	+	0.310253i
$375$	−22.0000	+	4.00000i	−1.13608	+	0.206559i
$376$	−16.0000	−	16.0000i	−0.825137	−	0.825137i
$377$			12.0000i			0.618031i
$378$	16.0000	−	16.0000i	0.822951	−	0.822951i
$379$		−	4.00000i		−	0.205466i	−0.994709	−	0.102733i	$-0.967241\pi$
$379$		−	4.00000i		−	0.205466i	0.994709	−	0.102733i	$-0.0327588\pi$
$380$		0			0
$381$		−	32.0000i		−	1.63941i
$382$	−6.00000	−	6.00000i	−0.306987	−	0.306987i
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$	16.0000	−	16.0000i	0.816497	−	0.816497i
$385$	−4.00000	+	8.00000i	−0.203859	+	0.407718i
$386$	6.00000	−	6.00000i	0.305392	−	0.305392i
$387$	−8.00000			−0.406663
$388$	8.00000			0.406138
$389$			12.0000i			0.608424i	0.952604	+	0.304212i	$0.0983931\pi$
$389$			12.0000i			0.608424i	−0.952604	+	0.304212i	$0.901607\pi$
$390$	12.0000	−	4.00000i	0.607644	−	0.202548i
$391$	48.0000			2.42746
$392$	−18.0000	+	18.0000i	−0.909137	+	0.909137i
$393$			16.0000i			0.807093i
$394$	18.0000	+	18.0000i	0.906827	+	0.906827i
$395$	−4.00000	+	8.00000i	−0.201262	+	0.402524i
$396$	2.00000			0.100504
$397$	−22.0000			−1.10415			−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000			−1.10415			−0.552074	+	0.833795i	$0.686163\pi$
$398$	18.0000	+	18.0000i	0.902258	+	0.902258i
$399$		0			0
$400$	12.0000	+	16.0000i	0.600000	+	0.800000i
$401$	18.0000			0.898877			0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000			0.898877			0.449439	+	0.893311i	$0.351624\pi$
$402$	−4.00000	−	4.00000i	−0.199502	−	0.199502i
$403$	20.0000			0.996271
$404$	−12.0000			−0.597022
$405$	11.0000	−	22.0000i	0.546594	−	1.09319i
$406$	−24.0000	−	24.0000i	−1.19110	−	1.19110i
$407$		−	6.00000i		−	0.297409i
$408$	24.0000	+	24.0000i	1.18818	+	1.18818i
$409$	30.0000			1.48340			0.741702	−	0.670729i	$-0.234019\pi$
$409$	30.0000			1.48340			0.741702	+	0.670729i	$0.234019\pi$
$410$	18.0000	−	6.00000i	0.888957	−	0.296319i
$411$			24.0000i			1.18383i
$412$	16.0000			0.788263
$413$	16.0000			0.787309
$414$	8.00000	−	8.00000i	0.393179	−	0.393179i
$415$	4.00000	−	8.00000i	0.196352	−	0.392705i
$416$	−8.00000	−	8.00000i	−0.392232	−	0.392232i
$417$		−	32.0000i		−	1.56705i
$418$		0			0
$419$		−	36.0000i		−	1.75872i	−0.476162	−	0.879358i	$-0.657972\pi$
$419$		−	36.0000i		−	1.75872i	0.476162	−	0.879358i	$-0.342028\pi$
$420$	−16.0000	+	32.0000i	−0.780720	+	1.56144i
$421$			20.0000i			0.974740i	0.873195	+	0.487370i	$0.162044\pi$
$421$			20.0000i			0.974740i	−0.873195	+	0.487370i	$0.837956\pi$
$422$	12.0000	−	12.0000i	0.584151	−	0.584151i
$423$		−	8.00000i		−	0.388973i
$424$	−20.0000	+	20.0000i	−0.971286	+	0.971286i
$425$	−24.0000	+	18.0000i	−1.16417	+	0.873128i
$426$	12.0000	+	12.0000i	0.581402	+	0.581402i
$427$	24.0000			1.16144
$428$			16.0000i			0.773389i
$429$		−	4.00000i		−	0.193122i
$430$	−24.0000	+	8.00000i	−1.15738	+	0.385794i
$431$	−16.0000			−0.770693			−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000			−0.770693			−0.385346	+	0.922772i	$0.625918\pi$
$432$	−16.0000			−0.769800
$433$		−	4.00000i		−	0.192228i	−0.995370	−	0.0961139i	$-0.969359\pi$
$433$		−	4.00000i		−	0.192228i	0.995370	−	0.0961139i	$-0.0306413\pi$
$434$	−40.0000	+	40.0000i	−1.92006	+	1.92006i
$435$	−24.0000	−	12.0000i	−1.15071	−	0.575356i
$436$	−20.0000			−0.957826
$437$		0			0
$438$	4.00000	−	4.00000i	0.191127	−	0.191127i
$439$	28.0000			1.33637			0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000			1.33637			0.668184	+	0.743996i	$0.267072\pi$
$440$	6.00000	−	2.00000i	0.286039	−	0.0953463i
$441$	−9.00000			−0.428571
$442$	12.0000	−	12.0000i	0.570782	−	0.570782i
$443$	−22.0000			−1.04525			−0.522626	−	0.852562i	$-0.675047\pi$
$443$	−22.0000			−1.04525			−0.522626	+	0.852562i	$0.675047\pi$
$444$		−	24.0000i		−	1.13899i
$445$	10.0000	−	20.0000i	0.474045	−	0.948091i
$446$	−16.0000	+	16.0000i	−0.757622	+	0.757622i
$447$			12.0000i			0.567581i
$448$	32.0000			1.51186
$449$	−10.0000			−0.471929			−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000			−0.471929			−0.235965	+	0.971762i	$0.575825\pi$
$450$	−1.00000	+	7.00000i	−0.0471405	+	0.329983i
$451$		−	6.00000i		−	0.282529i
$452$	−8.00000			−0.376288
$453$	32.0000			1.50349
$454$	16.0000	+	16.0000i	0.750917	+	0.750917i
$455$	16.0000	+	8.00000i	0.750092	+	0.375046i
$456$		0			0
$457$			26.0000i			1.21623i	0.793849	+	0.608114i	$0.208074\pi$
$457$			26.0000i			1.21623i	−0.793849	+	0.608114i	$0.791926\pi$
$458$		0			0
$459$		−	24.0000i		−	1.12022i
$460$	16.0000	−	32.0000i	0.746004	−	1.49201i
$461$			18.0000i			0.838344i	0.907907	+	0.419172i	$0.137680\pi$
$461$			18.0000i			0.838344i	−0.907907	+	0.419172i	$0.862320\pi$
$462$	8.00000	+	8.00000i	0.372194	+	0.372194i
$463$			16.0000i			0.743583i	0.928316	+	0.371792i	$0.121256\pi$
$463$			16.0000i			0.743583i	−0.928316	+	0.371792i	$0.878744\pi$
$464$			24.0000i			1.11417i
$465$	−20.0000	+	40.0000i	−0.927478	+	1.85496i
$466$	18.0000	−	18.0000i	0.833834	−	0.833834i
$467$	6.00000			0.277647			0.138823	−	0.990317i	$-0.455668\pi$
$467$	6.00000			0.277647			0.138823	+	0.990317i	$0.455668\pi$
$468$		−	4.00000i		−	0.184900i
$469$		−	8.00000i		−	0.369406i
$470$	−8.00000	−	24.0000i	−0.369012	−	1.10704i
$471$	−4.00000			−0.184310
$472$	−8.00000	−	8.00000i	−0.368230	−	0.368230i
$473$			8.00000i			0.367840i
$474$	8.00000	+	8.00000i	0.367452	+	0.367452i
$475$		0			0
$476$			48.0000i			2.20008i
$477$	−10.0000			−0.457869
$478$	8.00000	+	8.00000i	0.365911	+	0.365911i
$479$	−24.0000			−1.09659			−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000			−1.09659			−0.548294	+	0.836286i	$0.684723\pi$
$480$	24.0000	−	8.00000i	1.09545	−	0.365148i
$481$	−12.0000			−0.547153
$482$	14.0000	+	14.0000i	0.637683	+	0.637683i
$483$	64.0000			2.91210
$484$		−	2.00000i		−	0.0909091i
$485$	8.00000	+	4.00000i	0.363261	+	0.181631i
$486$	−10.0000	−	10.0000i	−0.453609	−	0.453609i
$487$		−	8.00000i		−	0.362515i	−0.983436	−	0.181257i	$-0.941983\pi$
$487$		−	8.00000i		−	0.362515i	0.983436	−	0.181257i	$-0.0580167\pi$
$488$	−12.0000	−	12.0000i	−0.543214	−	0.543214i
$489$	28.0000			1.26620
$490$	−27.0000	+	9.00000i	−1.21974	+	0.406579i
$491$		−	24.0000i		−	1.08310i	−0.840667	−	0.541552i	$-0.817837\pi$
$491$		−	24.0000i		−	1.08310i	0.840667	−	0.541552i	$-0.182163\pi$
$492$		−	24.0000i		−	1.08200i
$493$	−36.0000			−1.62136
$494$		0			0
$495$	2.00000	+	1.00000i	0.0898933	+	0.0449467i
$496$	40.0000			1.79605
$497$			24.0000i			1.07655i
$498$	−8.00000	−	8.00000i	−0.358489	−	0.358489i
$499$		−	20.0000i		−	0.895323i	−0.894203	−	0.447661i	$-0.852257\pi$
$499$		−	20.0000i		−	0.895323i	0.894203	−	0.447661i	$-0.147743\pi$
$500$	4.00000	+	22.0000i	0.178885	+	0.983870i
$501$			16.0000i			0.714827i
$502$	−12.0000	+	12.0000i	−0.535586	+	0.535586i
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$	8.00000	+	8.00000i	0.356348	+	0.356348i
$505$	−12.0000	−	6.00000i	−0.533993	−	0.266996i
$506$	−8.00000	−	8.00000i	−0.355643	−	0.355643i
$507$	18.0000			0.799408
$508$	−32.0000			−1.41977
$509$			8.00000i			0.354594i	0.984157	+	0.177297i	$0.0567353\pi$
$509$			8.00000i			0.354594i	−0.984157	+	0.177297i	$0.943265\pi$
$510$	12.0000	+	36.0000i	0.531369	+	1.59411i
$511$	8.00000			0.353899
$512$	−16.0000	−	16.0000i	−0.707107	−	0.707107i
$513$		0			0
$514$	8.00000	−	8.00000i	0.352865	−	0.352865i
$515$	16.0000	+	8.00000i	0.705044	+	0.352522i
$516$			32.0000i			1.40872i
$517$	−8.00000			−0.351840
$518$	24.0000	−	24.0000i	1.05450	−	1.05450i
$519$	−4.00000			−0.175581
$520$	−4.00000	−	12.0000i	−0.175412	−	0.526235i
$521$	−26.0000			−1.13908			−0.569540	−	0.821963i	$-0.692879\pi$
$521$	−26.0000			−1.13908			−0.569540	+	0.821963i	$0.692879\pi$
$522$	−6.00000	+	6.00000i	−0.262613	+	0.262613i
$523$	−36.0000			−1.57417			−0.787085	−	0.616844i	$-0.788411\pi$
$523$	−36.0000			−1.57417			−0.787085	+	0.616844i	$0.788411\pi$
$524$	16.0000			0.698963
$525$	−32.0000	+	24.0000i	−1.39659	+	1.04745i
$526$	−12.0000	+	12.0000i	−0.523225	+	0.523225i
$527$			60.0000i			2.61364i
$528$		−	8.00000i		−	0.348155i
$529$	−41.0000			−1.78261
$530$	−30.0000	+	10.0000i	−1.30312	+	0.434372i
$531$		−	4.00000i		−	0.173585i
$532$		0			0
$533$	−12.0000			−0.519778
$534$	−20.0000	−	20.0000i	−0.865485	−	0.865485i
$535$	−8.00000	+	16.0000i	−0.345870	+	0.691740i
$536$	−4.00000	+	4.00000i	−0.172774	+	0.172774i
$537$		−	40.0000i		−	1.72613i
$538$	−24.0000	+	24.0000i	−1.03471	+	1.03471i
$539$			9.00000i			0.387657i
$540$	−16.0000	−	8.00000i	−0.688530	−	0.344265i
$541$		−	34.0000i		−	1.46177i	−0.682498	−	0.730887i	$-0.739107\pi$
$541$		−	34.0000i		−	1.46177i	0.682498	−	0.730887i	$-0.260893\pi$
$542$	−8.00000	−	8.00000i	−0.343629	−	0.343629i
$543$		−	32.0000i		−	1.37325i
$544$	24.0000	−	24.0000i	1.02899	−	1.02899i
$545$	−20.0000	−	10.0000i	−0.856706	−	0.428353i
$546$	16.0000	−	16.0000i	0.684737	−	0.684737i
$547$	−20.0000			−0.855138			−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000			−0.855138			−0.427569	+	0.903983i	$0.640630\pi$
$548$	24.0000			1.02523
$549$		−	6.00000i		−	0.256074i
$550$	7.00000	+	1.00000i	0.298481	+	0.0426401i
$551$		0			0
$552$	−32.0000	−	32.0000i	−1.36201	−	1.36201i
$553$			16.0000i			0.680389i
$554$	−10.0000	−	10.0000i	−0.424859	−	0.424859i
$555$	12.0000	−	24.0000i	0.509372	−	1.01874i
$556$	−32.0000			−1.35710
$557$	−14.0000			−0.593199			−0.296600	−	0.955002i	$-0.595853\pi$
$557$	−14.0000			−0.593199			−0.296600	+	0.955002i	$0.595853\pi$
$558$	10.0000	+	10.0000i	0.423334	+	0.423334i
$559$	16.0000			0.676728
$560$	32.0000	+	16.0000i	1.35225	+	0.676123i
$561$	12.0000			0.506640
$562$	−18.0000	−	18.0000i	−0.759284	−	0.759284i
$563$	−12.0000			−0.505740			−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000			−0.505740			−0.252870	+	0.967500i	$0.581374\pi$
$564$	−32.0000			−1.34744
$565$	−8.00000	−	4.00000i	−0.336563	−	0.168281i
$566$	4.00000	+	4.00000i	0.168133	+	0.168133i
$567$		−	44.0000i		−	1.84783i
$568$	12.0000	−	12.0000i	0.503509	−	0.503509i
$569$	22.0000			0.922288			0.461144	−	0.887325i	$-0.347439\pi$
$569$	22.0000			0.922288			0.461144	+	0.887325i	$0.347439\pi$
$570$		0			0
$571$		−	28.0000i		−	1.17176i	−0.810397	−	0.585882i	$-0.800748\pi$
$571$		−	28.0000i		−	1.17176i	0.810397	−	0.585882i	$-0.199252\pi$
$572$	−4.00000			−0.167248
$573$	−12.0000			−0.501307
$574$	24.0000	−	24.0000i	1.00174	−	1.00174i
$575$	32.0000	−	24.0000i	1.33449	−	1.00087i
$576$		−	8.00000i		−	0.333333i
$577$		−	20.0000i		−	0.832611i	−0.909225	−	0.416305i	$-0.863325\pi$
$577$		−	20.0000i		−	0.832611i	0.909225	−	0.416305i	$-0.136675\pi$
$578$	19.0000	+	19.0000i	0.790296	+	0.790296i
$579$		−	12.0000i		−	0.498703i
$580$	−12.0000	+	24.0000i	−0.498273	+	0.996546i
$581$		−	16.0000i		−	0.663792i
$582$	8.00000	−	8.00000i	0.331611	−	0.331611i
$583$			10.0000i			0.414158i
$584$	−4.00000	−	4.00000i	−0.165521	−	0.165521i
$585$	2.00000	−	4.00000i	0.0826898	−	0.165380i
$586$	6.00000	+	6.00000i	0.247858	+	0.247858i
$587$	−30.0000			−1.23823			−0.619116	−	0.785299i	$-0.712509\pi$
$587$	−30.0000			−1.23823			−0.619116	+	0.785299i	$0.712509\pi$
$588$			36.0000i			1.48461i
$589$		0			0
$590$	−4.00000	−	12.0000i	−0.164677	−	0.494032i
$591$	36.0000			1.48084
$592$	−24.0000			−0.986394
$593$		−	6.00000i		−	0.246390i	−0.992382	−	0.123195i	$-0.960686\pi$
$593$		−	6.00000i		−	0.246390i	0.992382	−	0.123195i	$-0.0393141\pi$
$594$	−4.00000	+	4.00000i	−0.164122	+	0.164122i
$595$	−24.0000	+	48.0000i	−0.983904	+	1.96781i
$596$	12.0000			0.491539
$597$	36.0000			1.47338
$598$	−16.0000	+	16.0000i	−0.654289	+	0.654289i
$599$	−46.0000			−1.87951			−0.939755	−	0.341850i	$-0.888947\pi$
$599$	−46.0000			−1.87951			−0.939755	+	0.341850i	$0.888947\pi$
$600$	28.0000	+	4.00000i	1.14310	+	0.163299i
$601$	−22.0000			−0.897399			−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000			−0.897399			−0.448699	+	0.893683i	$0.648113\pi$
$602$	−32.0000	+	32.0000i	−1.30422	+	1.30422i
$603$	−2.00000			−0.0814463
$604$		−	32.0000i		−	1.30206i
$605$	1.00000	−	2.00000i	0.0406558	−	0.0813116i
$606$	−12.0000	+	12.0000i	−0.487467	+	0.487467i
$607$			16.0000i			0.649420i	0.945814	+	0.324710i	$0.105267\pi$
$607$			16.0000i			0.649420i	−0.945814	+	0.324710i	$0.894733\pi$
$608$		0			0
$609$	−48.0000			−1.94506
$610$	−6.00000	−	18.0000i	−0.242933	−	0.728799i
$611$			16.0000i			0.647291i
$612$	12.0000			0.485071
$613$	38.0000			1.53481			0.767403	−	0.641165i	$-0.221549\pi$
$613$	38.0000			1.53481			0.767403	+	0.641165i	$0.221549\pi$
$614$	−4.00000	−	4.00000i	−0.161427	−	0.161427i
$615$	12.0000	−	24.0000i	0.483887	−	0.967773i
$616$	8.00000	−	8.00000i	0.322329	−	0.322329i
$617$			12.0000i			0.483102i	0.970388	+	0.241551i	$0.0776561\pi$
$617$			12.0000i			0.483102i	−0.970388	+	0.241551i	$0.922344\pi$
$618$	16.0000	−	16.0000i	0.643614	−	0.643614i
$619$			4.00000i			0.160774i	0.996764	+	0.0803868i	$0.0256155\pi$
$619$			4.00000i			0.160774i	−0.996764	+	0.0803868i	$0.974384\pi$
$620$	40.0000	+	20.0000i	1.60644	+	0.803219i
$621$			32.0000i			1.28412i
$622$	30.0000	+	30.0000i	1.20289	+	1.20289i
$623$		−	40.0000i		−	1.60257i
$624$	−16.0000			−0.640513
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$	−16.0000	+	16.0000i	−0.639489	+	0.639489i
$627$		0			0
$628$			4.00000i			0.159617i
$629$		−	36.0000i		−	1.43541i
$630$	4.00000	+	12.0000i	0.159364	+	0.478091i
$631$	−2.00000			−0.0796187			−0.0398094	−	0.999207i	$-0.512675\pi$
$631$	−2.00000			−0.0796187			−0.0398094	+	0.999207i	$0.512675\pi$
$632$	8.00000	−	8.00000i	0.318223	−	0.318223i
$633$		−	24.0000i		−	0.953914i
$634$	34.0000	+	34.0000i	1.35031	+	1.35031i
$635$	−32.0000	−	16.0000i	−1.26988	−	0.634941i
$636$			40.0000i			1.58610i
$637$	18.0000			0.713186
$638$	6.00000	+	6.00000i	0.237542	+	0.237542i
$639$	6.00000			0.237356
$640$	−8.00000	−	24.0000i	−0.316228	−	0.948683i
$641$	18.0000			0.710957			0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000			0.710957			0.355479	+	0.934684i	$0.384318\pi$
$642$	16.0000	+	16.0000i	0.631470	+	0.631470i
$643$	−10.0000			−0.394362			−0.197181	−	0.980367i	$-0.563179\pi$
$643$	−10.0000			−0.394362			−0.197181	+	0.980367i	$0.563179\pi$
$644$		−	64.0000i		−	2.52195i
$645$	−16.0000	+	32.0000i	−0.629999	+	1.26000i
$646$		0			0
$647$		−	32.0000i		−	1.25805i	−0.777385	−	0.629025i	$-0.783454\pi$
$647$		−	32.0000i		−	1.25805i	0.777385	−	0.629025i	$-0.216546\pi$
$648$	−22.0000	+	22.0000i	−0.864242	+	0.864242i
$649$	−4.00000			−0.157014
$650$	2.00000	−	14.0000i	0.0784465	−	0.549125i
$651$			80.0000i			3.13545i
$652$		−	28.0000i		−	1.09656i
$653$	−34.0000			−1.33052			−0.665261	−	0.746611i	$-0.731680\pi$
$653$	−34.0000			−1.33052			−0.665261	+	0.746611i	$0.731680\pi$
$654$	−20.0000	+	20.0000i	−0.782062	+	0.782062i
$655$	16.0000	+	8.00000i	0.625172	+	0.312586i
$656$	−24.0000			−0.937043
$657$		−	2.00000i		−	0.0780274i
$658$	−32.0000	−	32.0000i	−1.24749	−	1.24749i
$659$			12.0000i			0.467454i	0.972302	+	0.233727i	$0.0750921\pi$
$659$			12.0000i			0.467454i	−0.972302	+	0.233727i	$0.924908\pi$
$660$	4.00000	−	8.00000i	0.155700	−	0.311400i
$661$			4.00000i			0.155582i	0.996970	+	0.0777910i	$0.0247867\pi$
$661$			4.00000i			0.155582i	−0.996970	+	0.0777910i	$0.975213\pi$
$662$	−4.00000	+	4.00000i	−0.155464	+	0.155464i
$663$		−	24.0000i		−	0.932083i
$664$	−8.00000	+	8.00000i	−0.310460	+	0.310460i
$665$		0			0
$666$	−6.00000	−	6.00000i	−0.232495	−	0.232495i
$667$	48.0000			1.85857
$668$	16.0000			0.619059
$669$			32.0000i			1.23719i
$670$	−6.00000	+	2.00000i	−0.231800	+	0.0772667i
$671$	−6.00000			−0.231627
$672$	32.0000	−	32.0000i	1.23443	−	1.23443i
$673$		−	10.0000i		−	0.385472i	−0.981251	−	0.192736i	$-0.938264\pi$
$673$		−	10.0000i		−	0.385472i	0.981251	−	0.192736i	$-0.0617360\pi$
$674$	22.0000	−	22.0000i	0.847408	−	0.847408i
$675$	−12.0000	−	16.0000i	−0.461880	−	0.615840i
$676$		−	18.0000i		−	0.692308i
$677$	−18.0000			−0.691796			−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000			−0.691796			−0.345898	+	0.938272i	$0.612426\pi$
$678$	−8.00000	+	8.00000i	−0.307238	+	0.307238i
$679$	16.0000			0.614024
$680$	36.0000	−	12.0000i	1.38054	−	0.460179i
$681$	32.0000			1.22624
$682$	10.0000	−	10.0000i	0.382920	−	0.382920i
$683$	6.00000			0.229584			0.114792	−	0.993390i	$-0.463380\pi$
$683$	6.00000			0.229584			0.114792	+	0.993390i	$0.463380\pi$
$684$		0			0
$685$	24.0000	+	12.0000i	0.916993	+	0.458496i
$686$	−8.00000	+	8.00000i	−0.305441	+	0.305441i
$687$		0			0
$688$	32.0000			1.21999
$689$	20.0000			0.761939
$690$	−16.0000	−	48.0000i	−0.609110	−	1.82733i
$691$			20.0000i			0.760836i	0.924815	+	0.380418i	$0.124220\pi$
$691$			20.0000i			0.760836i	−0.924815	+	0.380418i	$0.875780\pi$
$692$			4.00000i			0.152057i
$693$	4.00000			0.151947
$694$	−16.0000	−	16.0000i	−0.607352	−	0.607352i
$695$	−32.0000	−	16.0000i	−1.21383	−	0.606915i
$696$	24.0000	+	24.0000i	0.909718	+	0.909718i
$697$		−	36.0000i		−	1.36360i
$698$	10.0000	−	10.0000i	0.378506	−	0.378506i
$699$		−	36.0000i		−	1.36165i
$700$	24.0000	+	32.0000i	0.907115	+	1.20949i
$701$		−	30.0000i		−	1.13308i	−0.824033	−	0.566542i	$-0.808281\pi$
$701$		−	30.0000i		−	1.13308i	0.824033	−	0.566542i	$-0.191719\pi$
$702$	8.00000	+	8.00000i	0.301941	+	0.301941i
$703$		0			0
$704$	−8.00000			−0.301511
$705$	−32.0000	−	16.0000i	−1.20519	−	0.602595i
$706$	−8.00000	+	8.00000i	−0.301084	+	0.301084i
$707$	−24.0000			−0.902613
$708$	−16.0000			−0.601317
$709$			8.00000i			0.300446i	0.988652	+	0.150223i	$0.0479992\pi$
$709$			8.00000i			0.300446i	−0.988652	+	0.150223i	$0.952001\pi$
$710$	18.0000	−	6.00000i	0.675528	−	0.225176i
$711$	4.00000			0.150012
$712$	−20.0000	+	20.0000i	−0.749532	+	0.749532i
$713$		−	80.0000i		−	2.99602i
$714$	48.0000	+	48.0000i	1.79635	+	1.79635i
$715$	−4.00000	−	2.00000i	−0.149592	−	0.0747958i
$716$	−40.0000			−1.49487
$717$	16.0000			0.597531
$718$	12.0000	+	12.0000i	0.447836	+	0.447836i
$719$	6.00000			0.223762			0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000			0.223762			0.111881	+	0.993722i	$0.464312\pi$
$720$	4.00000	−	8.00000i	0.149071	−	0.298142i
$721$	32.0000			1.19174
$722$	−19.0000	−	19.0000i	−0.707107	−	0.707107i
$723$	28.0000			1.04133
$724$	−32.0000			−1.18927
$725$	−24.0000	+	18.0000i	−0.891338	+	0.668503i
$726$	−2.00000	−	2.00000i	−0.0742270	−	0.0742270i
$727$			48.0000i			1.78022i	0.455744	+	0.890111i	$0.349373\pi$
$727$			48.0000i			1.78022i	−0.455744	+	0.890111i	$0.650627\pi$
$728$	−16.0000	−	16.0000i	−0.592999	−	0.592999i
$729$	13.0000			0.481481
$730$	−2.00000	−	6.00000i	−0.0740233	−	0.222070i
$731$			48.0000i			1.77534i
$732$	−24.0000			−0.887066
$733$	50.0000			1.84679			0.923396	−	0.383849i	$-0.125402\pi$
$733$	50.0000			1.84679			0.923396	+	0.383849i	$0.125402\pi$
$734$	8.00000	−	8.00000i	0.295285	−	0.295285i
$735$	−18.0000	+	36.0000i	−0.663940	+	1.32788i
$736$	−32.0000	+	32.0000i	−1.17954	+	1.17954i
$737$			2.00000i			0.0736709i
$738$	−6.00000	−	6.00000i	−0.220863	−	0.220863i
$739$		−	20.0000i		−	0.735712i	−0.929883	−	0.367856i	$-0.880092\pi$
$739$		−	20.0000i		−	0.735712i	0.929883	−	0.367856i	$-0.119908\pi$
$740$	−24.0000	−	12.0000i	−0.882258	−	0.441129i
$741$		0			0
$742$	−40.0000	+	40.0000i	−1.46845	+	1.46845i
$743$		−	8.00000i		−	0.293492i	−0.989174	−	0.146746i	$-0.953120\pi$
$743$		−	8.00000i		−	0.293492i	0.989174	−	0.146746i	$-0.0468799\pi$
$744$	40.0000	−	40.0000i	1.46647	−	1.46647i
$745$	12.0000	+	6.00000i	0.439646	+	0.219823i
$746$	−10.0000	−	10.0000i	−0.366126	−	0.366126i
$747$	−4.00000			−0.146352
$748$		−	12.0000i		−	0.438763i
$749$			32.0000i			1.16925i
$750$	26.0000	+	18.0000i	0.949386	+	0.657267i
$751$	−22.0000			−0.802791			−0.401396	−	0.915905i	$-0.631475\pi$
$751$	−22.0000			−0.802791			−0.401396	+	0.915905i	$0.631475\pi$
$752$			32.0000i			1.16692i
$753$			24.0000i			0.874609i
$754$	12.0000	−	12.0000i	0.437014	−	0.437014i
$755$	16.0000	−	32.0000i	0.582300	−	1.16460i
$756$	−32.0000			−1.16383
$757$	46.0000			1.67190			0.835949	−	0.548807i	$-0.184918\pi$
$757$	46.0000			1.67190			0.835949	+	0.548807i	$0.184918\pi$
$758$	−4.00000	+	4.00000i	−0.145287	+	0.145287i
$759$	−16.0000			−0.580763
$760$		0			0
$761$	50.0000			1.81250			0.906249	−	0.422744i	$-0.138933\pi$
$761$	50.0000			1.81250			0.906249	+	0.422744i	$0.138933\pi$
$762$	−32.0000	+	32.0000i	−1.15924	+	1.15924i
$763$	−40.0000			−1.44810
$764$			12.0000i			0.434145i
$765$	12.0000	+	6.00000i	0.433861	+	0.216930i
$766$		0			0
$767$			8.00000i			0.288863i
$768$	−32.0000			−1.15470
$769$	−2.00000			−0.0721218			−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000			−0.0721218			−0.0360609	+	0.999350i	$0.511481\pi$
$770$	12.0000	−	4.00000i	0.432450	−	0.144150i
$771$		−	16.0000i		−	0.576226i
$772$	−12.0000			−0.431889
$773$	−6.00000			−0.215805			−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000			−0.215805			−0.107903	+	0.994161i	$0.534413\pi$
$774$	8.00000	+	8.00000i	0.287554	+	0.287554i
$775$	30.0000	+	40.0000i	1.07763	+	1.43684i
$776$	−8.00000	−	8.00000i	−0.287183	−	0.287183i
$777$		−	48.0000i		−	1.72199i
$778$	12.0000	−	12.0000i	0.430221	−	0.430221i
$779$		0			0
$780$	−16.0000	−	8.00000i	−0.572892	−	0.286446i
$781$		−	6.00000i		−	0.214697i
$782$	−48.0000	−	48.0000i	−1.71648	−	1.71648i
$783$		−	24.0000i		−	0.857690i
$784$	36.0000			1.28571
$785$	−2.00000	+	4.00000i	−0.0713831	+	0.142766i
$786$	16.0000	−	16.0000i	0.570701	−	0.570701i
$787$	4.00000			0.142585			0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000			0.142585			0.0712923	+	0.997455i	$0.477288\pi$
$788$		−	36.0000i		−	1.28245i
$789$			24.0000i			0.854423i
$790$	12.0000	−	4.00000i	0.426941	−	0.142314i
$791$	−16.0000			−0.568895
$792$	−2.00000	−	2.00000i	−0.0710669	−	0.0710669i
$793$			12.0000i			0.426132i
$794$	22.0000	+	22.0000i	0.780751	+	0.780751i
$795$	−20.0000	+	40.0000i	−0.709327	+	1.41865i
$796$		−	36.0000i		−	1.27599i
$797$	30.0000			1.06265			0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000			1.06265			0.531327	+	0.847167i	$0.321693\pi$
$798$		0			0
$799$	−48.0000			−1.69812
$800$	4.00000	−	28.0000i	0.141421	−	0.989949i
$801$	−10.0000			−0.353333
$802$	−18.0000	−	18.0000i	−0.635602	−	0.635602i
$803$	−2.00000			−0.0705785
$804$			8.00000i			0.282138i
$805$	32.0000	−	64.0000i	1.12785	−	2.25570i
$806$	−20.0000	−	20.0000i	−0.704470	−	0.704470i
$807$			48.0000i			1.68968i
$808$	12.0000	+	12.0000i	0.422159	+	0.422159i
$809$	18.0000			0.632846			0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000			0.632846			0.316423	+	0.948618i	$0.397518\pi$
$810$	−33.0000	+	11.0000i	−1.15950	+	0.386501i
$811$			28.0000i			0.983213i	0.870817	+	0.491606i	$0.163590\pi$
$811$			28.0000i			0.983213i	−0.870817	+	0.491606i	$0.836410\pi$
$812$			48.0000i			1.68447i
$813$	−16.0000			−0.561144
$814$	−6.00000	+	6.00000i	−0.210300	+	0.210300i
$815$	14.0000	−	28.0000i	0.490399	−	0.980797i
$816$		−	48.0000i		−	1.68034i
$817$		0			0
$818$	−30.0000	−	30.0000i	−1.04893	−	1.04893i
$819$		−	8.00000i		−	0.279543i
$820$	−24.0000	−	12.0000i	−0.838116	−	0.419058i
$821$		−	6.00000i		−	0.209401i	−0.994504	−	0.104701i	$-0.966612\pi$
$821$		−	6.00000i		−	0.209401i	0.994504	−	0.104701i	$-0.0333885\pi$
$822$	24.0000	−	24.0000i	0.837096	−	0.837096i
$823$			48.0000i			1.67317i	0.547833	+	0.836587i	$0.315453\pi$
$823$			48.0000i			1.67317i	−0.547833	+	0.836587i	$0.684547\pi$
$824$	−16.0000	−	16.0000i	−0.557386	−	0.557386i
$825$	8.00000	−	6.00000i	0.278524	−	0.208893i
$826$	−16.0000	−	16.0000i	−0.556711	−	0.556711i
$827$	12.0000			0.417281			0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000			0.417281			0.208640	+	0.977992i	$0.433096\pi$
$828$	−16.0000			−0.556038
$829$			12.0000i			0.416777i	0.978046	+	0.208389i	$0.0668219\pi$
$829$			12.0000i			0.416777i	−0.978046	+	0.208389i	$0.933178\pi$
$830$	−12.0000	+	4.00000i	−0.416526	+	0.138842i
$831$	−20.0000			−0.693792
$832$			16.0000i			0.554700i
$833$			54.0000i			1.87099i
$834$	−32.0000	+	32.0000i	−1.10807	+	1.10807i
$835$	16.0000	+	8.00000i	0.553703	+	0.276851i
$836$		0			0
$837$	−40.0000			−1.38260
$838$	−36.0000	+	36.0000i	−1.24360	+	1.24360i
$839$	−6.00000			−0.207143			−0.103572	−	0.994622i	$-0.533027\pi$
$839$	−6.00000			−0.207143			−0.103572	+	0.994622i	$0.533027\pi$
$840$	48.0000	−	16.0000i	1.65616	−	0.552052i
$841$	−7.00000			−0.241379
$842$	20.0000	−	20.0000i	0.689246	−	0.689246i
$843$	−36.0000			−1.23991
$844$	−24.0000			−0.826114
$845$	9.00000	−	18.0000i	0.309609	−	0.619219i
$846$	−8.00000	+	8.00000i	−0.275046	+	0.275046i
$847$		−	4.00000i		−	0.137442i
$848$	40.0000			1.37361
$849$	8.00000			0.274559
$850$	42.0000	+	6.00000i	1.44059	+	0.205798i
$851$			48.0000i			1.64542i
$852$		−	24.0000i		−	0.822226i
$853$	−2.00000			−0.0684787			−0.0342393	−	0.999414i	$-0.510901\pi$
$853$	−2.00000			−0.0684787			−0.0342393	+	0.999414i	$0.510901\pi$
$854$	−24.0000	−	24.0000i	−0.821263	−	0.821263i
$855$		0			0
$856$	16.0000	−	16.0000i	0.546869	−	0.546869i
$857$			54.0000i			1.84460i	0.386469	+	0.922302i	$0.373695\pi$
$857$			54.0000i			1.84460i	−0.386469	+	0.922302i	$0.626305\pi$
$858$	−4.00000	+	4.00000i	−0.136558	+	0.136558i
$859$			12.0000i			0.409435i	0.978821	+	0.204717i	$0.0656275\pi$
$859$			12.0000i			0.409435i	−0.978821	+	0.204717i	$0.934372\pi$
$860$	32.0000	+	16.0000i	1.09119	+	0.545595i
$861$		−	48.0000i		−	1.63584i
$862$	16.0000	+	16.0000i	0.544962	+	0.544962i
$863$			48.0000i			1.63394i	0.576681	+	0.816970i	$0.304348\pi$
$863$			48.0000i			1.63394i	−0.576681	+	0.816970i	$0.695652\pi$
$864$	16.0000	+	16.0000i	0.544331	+	0.544331i
$865$	−2.00000	+	4.00000i	−0.0680020	+	0.136004i
$866$	−4.00000	+	4.00000i	−0.135926	+	0.135926i
$867$	38.0000			1.29055
$868$	80.0000			2.71538
$869$		−	4.00000i		−	0.135691i
$870$	12.0000	+	36.0000i	0.406838	+	1.22051i
$871$	4.00000			0.135535
$872$	20.0000	+	20.0000i	0.677285	+	0.677285i
$873$		−	4.00000i		−	0.135379i
$874$		0			0
$875$	8.00000	+	44.0000i	0.270449	+	1.48747i
$876$	−8.00000			−0.270295
$877$	18.0000			0.607817			0.303908	−	0.952701i	$-0.401708\pi$
$877$	18.0000			0.607817			0.303908	+	0.952701i	$0.401708\pi$
$878$	−28.0000	−	28.0000i	−0.944954	−	0.944954i
$879$	12.0000			0.404750
$880$	−8.00000	−	4.00000i	−0.269680	−	0.134840i
$881$	−30.0000			−1.01073			−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000			−1.01073			−0.505363	+	0.862907i	$0.668641\pi$
$882$	9.00000	+	9.00000i	0.303046	+	0.303046i
$883$	−10.0000			−0.336527			−0.168263	−	0.985742i	$-0.553816\pi$
$883$	−10.0000			−0.336527			−0.168263	+	0.985742i	$0.553816\pi$
$884$	−24.0000			−0.807207
$885$	−16.0000	−	8.00000i	−0.537834	−	0.268917i
$886$	22.0000	+	22.0000i	0.739104	+	0.739104i
$887$			8.00000i			0.268614i	0.990940	+	0.134307i	$0.0428808\pi$
$887$			8.00000i			0.268614i	−0.990940	+	0.134307i	$0.957119\pi$
$888$	−24.0000	+	24.0000i	−0.805387	+	0.805387i
$889$	−64.0000			−2.14649
$890$	−30.0000	+	10.0000i	−1.00560	+	0.335201i
$891$			11.0000i			0.368514i
$892$	32.0000			1.07144
$893$		0			0
$894$	12.0000	−	12.0000i	0.401340	−	0.401340i
$895$	−40.0000	−	20.0000i	−1.33705	−	0.668526i
$896$	−32.0000	−	32.0000i	−1.06904	−	1.06904i
$897$			32.0000i			1.06845i
$898$	10.0000	+	10.0000i	0.333704	+	0.333704i
$899$			60.0000i			2.00111i
$900$	8.00000	−	6.00000i	0.266667	−	0.200000i
$901$			60.0000i			1.99889i
$902$	−6.00000	+	6.00000i	−0.199778	+	0.199778i
$903$			64.0000i			2.12979i
$904$	8.00000	+	8.00000i	0.266076	+	0.266076i
$905$	−32.0000	−	16.0000i	−1.06372	−	0.531858i
$906$	−32.0000	−	32.0000i	−1.06313	−	1.06313i
$907$	46.0000			1.52740			0.763702	−	0.645568i	$-0.223379\pi$
$907$	46.0000			1.52740			0.763702	+	0.645568i	$0.223379\pi$
$908$		−	32.0000i		−	1.06196i
$909$			6.00000i			0.199007i
$910$	−8.00000	−	24.0000i	−0.265197	−	0.795592i
$911$	18.0000			0.596367			0.298183	−	0.954509i	$-0.403619\pi$
$911$	18.0000			0.596367			0.298183	+	0.954509i	$0.403619\pi$
$912$		0			0
$913$			4.00000i			0.132381i
$914$	26.0000	−	26.0000i	0.860004	−	0.860004i
$915$	−24.0000	−	12.0000i	−0.793416	−	0.396708i
$916$		0			0
$917$	32.0000			1.05673
$918$	−24.0000	+	24.0000i	−0.792118	+	0.792118i
$919$	4.00000			0.131948			0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000			0.131948			0.0659739	+	0.997821i	$0.478985\pi$
$920$	−48.0000	+	16.0000i	−1.58251	+	0.527504i
$921$	−8.00000			−0.263609
$922$	18.0000	−	18.0000i	0.592798	−	0.592798i
$923$	−12.0000			−0.394985
$924$		−	16.0000i		−	0.526361i
$925$	−18.0000	−	24.0000i	−0.591836	−	0.789115i
$926$	16.0000	−	16.0000i	0.525793	−	0.525793i
$927$		−	8.00000i		−	0.262754i
$928$	24.0000	−	24.0000i	0.787839	−	0.787839i
$929$	34.0000			1.11550			0.557752	−	0.830008i	$-0.311664\pi$
$929$	34.0000			1.11550			0.557752	+	0.830008i	$0.311664\pi$
$930$	60.0000	−	20.0000i	1.96748	−	0.655826i
$931$		0			0
$932$	−36.0000			−1.17922
$933$	60.0000			1.96431
$934$	−6.00000	−	6.00000i	−0.196326	−	0.196326i
$935$	6.00000	−	12.0000i	0.196221	−	0.392442i
$936$	−4.00000	+	4.00000i	−0.130744	+	0.130744i
$937$			14.0000i			0.457360i	0.973502	+	0.228680i	$0.0734410\pi$
$937$			14.0000i			0.457360i	−0.973502	+	0.228680i	$0.926559\pi$
$938$	−8.00000	+	8.00000i	−0.261209	+	0.261209i
$939$			32.0000i			1.04428i
$940$	−16.0000	+	32.0000i	−0.521862	+	1.04372i
$941$		−	50.0000i		−	1.62995i	−0.579494	−	0.814977i	$-0.696750\pi$
$941$		−	50.0000i		−	1.62995i	0.579494	−	0.814977i	$-0.303250\pi$
$942$	4.00000	+	4.00000i	0.130327	+	0.130327i
$943$			48.0000i			1.56310i
$944$			16.0000i			0.520756i
$945$	−32.0000	−	16.0000i	−1.04096	−	0.520480i
$946$	8.00000	−	8.00000i	0.260102	−	0.260102i
$947$	50.0000			1.62478			0.812391	−	0.583113i	$-0.198166\pi$
$947$	50.0000			1.62478			0.812391	+	0.583113i	$0.198166\pi$
$948$		−	16.0000i		−	0.519656i
$949$			4.00000i			0.129845i
$950$		0			0
$951$	68.0000			2.20505
$952$	48.0000	−	48.0000i	1.55569	−	1.55569i
$953$			22.0000i			0.712650i	0.934362	+	0.356325i	$0.115970\pi$
$953$			22.0000i			0.712650i	−0.934362	+	0.356325i	$0.884030\pi$
$954$	10.0000	+	10.0000i	0.323762	+	0.323762i
$955$	−6.00000	+	12.0000i	−0.194155	+	0.388311i
$956$		−	16.0000i		−	0.517477i
$957$	12.0000			0.387905
$958$	24.0000	+	24.0000i	0.775405	+	0.775405i
$959$	48.0000			1.55000
$960$	−32.0000	−	16.0000i	−1.03280	−	0.516398i
$961$	69.0000			2.22581
$962$	12.0000	+	12.0000i	0.386896	+	0.386896i
$963$	8.00000			0.257796
$964$		−	28.0000i		−	0.901819i
$965$	−12.0000	−	6.00000i	−0.386294	−	0.193147i
$966$	−64.0000	−	64.0000i	−2.05917	−	2.05917i
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$	−2.00000	+	2.00000i	−0.0642824	+	0.0642824i
$969$		0			0
$970$	−4.00000	−	12.0000i	−0.128432	−	0.385297i
$971$		−	60.0000i		−	1.92549i	−0.270408	−	0.962746i	$-0.587159\pi$
$971$		−	60.0000i		−	1.92549i	0.270408	−	0.962746i	$-0.412841\pi$
$972$			20.0000i			0.641500i
$973$	−64.0000			−2.05175
$974$	−8.00000	+	8.00000i	−0.256337	+	0.256337i
$975$	−12.0000	−	16.0000i	−0.384308	−	0.512410i
$976$			24.0000i			0.768221i
$977$		−	28.0000i		−	0.895799i	−0.894084	−	0.447900i	$-0.852172\pi$
$977$		−	28.0000i		−	0.895799i	0.894084	−	0.447900i	$-0.147828\pi$
$978$	−28.0000	−	28.0000i	−0.895341	−	0.895341i
$979$			10.0000i			0.319601i
$980$	36.0000	+	18.0000i	1.14998	+	0.574989i
$981$			10.0000i			0.319275i
$982$	−24.0000	+	24.0000i	−0.765871	+	0.765871i
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$	−24.0000	+	24.0000i	−0.765092	+	0.765092i
$985$	18.0000	−	36.0000i	0.573528	−	1.14706i
$986$	36.0000	+	36.0000i	1.14647	+	1.14647i
$987$	−64.0000			−2.03714
$988$		0			0
$989$		−	64.0000i		−	2.03508i
$990$	−1.00000	−	3.00000i	−0.0317821	−	0.0953463i
$991$	38.0000			1.20711			0.603555	−	0.797321i	$-0.293750\pi$
$991$	38.0000			1.20711			0.603555	+	0.797321i	$0.293750\pi$
$992$	−40.0000	−	40.0000i	−1.27000	−	1.27000i
$993$			8.00000i			0.253872i
$994$	24.0000	−	24.0000i	0.761234	−	0.761234i
$995$	18.0000	−	36.0000i	0.570638	−	1.14128i
$996$			16.0000i			0.506979i
$997$	−10.0000			−0.316703			−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000			−0.316703			−0.158352	+	0.987383i	$0.550618\pi$
$998$	−20.0000	+	20.0000i	−0.633089	+	0.633089i
$999$	24.0000			0.759326

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	440.2.l.a.309.1	✓	2
4.3	odd	2		1760.2.l.b.529.2		2
5.4	even	2		440.2.l.b.309.2	yes	2
8.3	odd	2		1760.2.l.a.529.1		2
8.5	even	2		440.2.l.b.309.1	yes	2
20.19	odd	2		1760.2.l.a.529.2		2
40.19	odd	2		1760.2.l.b.529.1		2
40.29	even	2	inner	440.2.l.a.309.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.l.a.309.1	✓	2	1.1	even	1	trivial
440.2.l.a.309.2	yes	2	40.29	even	2	inner
440.2.l.b.309.1	yes	2	8.5	even	2
440.2.l.b.309.2	yes	2	5.4	even	2
1760.2.l.a.529.1		2	8.3	odd	2
1760.2.l.a.529.2		2	20.19	odd	2
1760.2.l.b.529.1		2	40.19	odd	2
1760.2.l.b.529.2		2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	440.l (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.00000i) q^{2} -2.00000 q^{3} +2.00000i q^{4} +(-1.00000 + 2.00000i) q^{5} +(2.00000 + 2.00000i) q^{6} +4.00000i q^{7} +(2.00000 - 2.00000i) q^{8} +1.00000 q^{9} +(3.00000 - 1.00000i) q^{10} -1.00000i q^{11} -4.00000i q^{12} -2.00000 q^{13} +(4.00000 - 4.00000i) q^{14} +(2.00000 - 4.00000i) q^{15} -4.00000 q^{16} -6.00000i q^{17} +(-1.00000 - 1.00000i) q^{18} +(-4.00000 - 2.00000i) q^{20} -8.00000i q^{21} +(-1.00000 + 1.00000i) q^{22} +8.00000i q^{23} +(-4.00000 + 4.00000i) q^{24} +(-3.00000 - 4.00000i) q^{25} +(2.00000 + 2.00000i) q^{26} +4.00000 q^{27} -8.00000 q^{28} -6.00000i q^{29} +(-6.00000 + 2.00000i) q^{30} -10.0000 q^{31} +(4.00000 + 4.00000i) q^{32} +2.00000i q^{33} +(-6.00000 + 6.00000i) q^{34} +(-8.00000 - 4.00000i) q^{35} +2.00000i q^{36} +6.00000 q^{37} +4.00000 q^{39} +(2.00000 + 6.00000i) q^{40} +6.00000 q^{41} +(-8.00000 + 8.00000i) q^{42} -8.00000 q^{43} +2.00000 q^{44} +(-1.00000 + 2.00000i) q^{45} +(8.00000 - 8.00000i) q^{46} -8.00000i q^{47} +8.00000 q^{48} -9.00000 q^{49} +(-1.00000 + 7.00000i) q^{50} +12.0000i q^{51} -4.00000i q^{52} -10.0000 q^{53} +(-4.00000 - 4.00000i) q^{54} +(2.00000 + 1.00000i) q^{55} +(8.00000 + 8.00000i) q^{56} +(-6.00000 + 6.00000i) q^{58} -4.00000i q^{59} +(8.00000 + 4.00000i) q^{60} -6.00000i q^{61} +(10.0000 + 10.0000i) q^{62} +4.00000i q^{63} -8.00000i q^{64} +(2.00000 - 4.00000i) q^{65} +(2.00000 - 2.00000i) q^{66} -2.00000 q^{67} +12.0000 q^{68} -16.0000i q^{69} +(4.00000 + 12.0000i) q^{70} +6.00000 q^{71} +(2.00000 - 2.00000i) q^{72} -2.00000i q^{73} +(-6.00000 - 6.00000i) q^{74} +(6.00000 + 8.00000i) q^{75} +4.00000 q^{77} +(-4.00000 - 4.00000i) q^{78} +4.00000 q^{79} +(4.00000 - 8.00000i) q^{80} -11.0000 q^{81} +(-6.00000 - 6.00000i) q^{82} -4.00000 q^{83} +16.0000 q^{84} +(12.0000 + 6.00000i) q^{85} +(8.00000 + 8.00000i) q^{86} +12.0000i q^{87} +(-2.00000 - 2.00000i) q^{88} -10.0000 q^{89} +(3.00000 - 1.00000i) q^{90} -8.00000i q^{91} -16.0000 q^{92} +20.0000 q^{93} +(-8.00000 + 8.00000i) q^{94} +(-8.00000 - 8.00000i) q^{96} -4.00000i q^{97} +(9.00000 + 9.00000i) q^{98} -1.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 4 q^{3} - 2 q^{5} + 4 q^{6} + 4 q^{8} + 2 q^{9} + 6 q^{10} - 4 q^{13} + 8 q^{14} + 4 q^{15} - 8 q^{16} - 2 q^{18} - 8 q^{20} - 2 q^{22} - 8 q^{24} - 6 q^{25} + 4 q^{26} + 8 q^{27} - 16 q^{28}+ \cdots + 18 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 - 4 * q^3 - 2 * q^5 + 4 * q^6 + 4 * q^8 + 2 * q^9 + 6 * q^10 - 4 * q^13 + 8 * q^14 + 4 * q^15 - 8 * q^16 - 2 * q^18 - 8 * q^20 - 2 * q^22 - 8 * q^24 - 6 * q^25 + 4 * q^26 + 8 * q^27 - 16 * q^28 - 12 * q^30 - 20 * q^31 + 8 * q^32 - 12 * q^34 - 16 * q^35 + 12 * q^37 + 8 * q^39 + 4 * q^40 + 12 * q^41 - 16 * q^42 - 16 * q^43 + 4 * q^44 - 2 * q^45 + 16 * q^46 + 16 * q^48 - 18 * q^49 - 2 * q^50 - 20 * q^53 - 8 * q^54 + 4 * q^55 + 16 * q^56 - 12 * q^58 + 16 * q^60 + 20 * q^62 + 4 * q^65 + 4 * q^66 - 4 * q^67 + 24 * q^68 + 8 * q^70 + 12 * q^71 + 4 * q^72 - 12 * q^74 + 12 * q^75 + 8 * q^77 - 8 * q^78 + 8 * q^79 + 8 * q^80 - 22 * q^81 - 12 * q^82 - 8 * q^83 + 32 * q^84 + 24 * q^85 + 16 * q^86 - 4 * q^88 - 20 * q^89 + 6 * q^90 - 32 * q^92 + 40 * q^93 - 16 * q^94 - 16 * q^96 + 18 * q^98

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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