LMFDB - Embedded newform 435.2.f.b.289.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [435,2,Mod(289,435)] 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("435.289"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(435, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 435 = 3 \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	435.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,-2,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$3.47349248793$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	435.289
Dual form			435.2.f.b.289.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +(-1.00000 + 2.00000i) q^{5} +1.00000 q^{6} +2.00000i q^{7} +3.00000 q^{8} +1.00000 q^{9} +(1.00000 - 2.00000i) q^{10} -2.00000i q^{11} +1.00000 q^{12} +4.00000i q^{13} -2.00000i q^{14} +(1.00000 - 2.00000i) q^{15} -1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} -2.00000i q^{19} +(1.00000 - 2.00000i) q^{20} -2.00000i q^{21} +2.00000i q^{22} -6.00000i q^{23} -3.00000 q^{24} +(-3.00000 - 4.00000i) q^{25} -4.00000i q^{26} -1.00000 q^{27} -2.00000i q^{28} +(-5.00000 - 2.00000i) q^{29} +(-1.00000 + 2.00000i) q^{30} +2.00000i q^{31} -5.00000 q^{32} +2.00000i q^{33} +6.00000 q^{34} +(-4.00000 - 2.00000i) q^{35} -1.00000 q^{36} -2.00000 q^{37} +2.00000i q^{38} -4.00000i q^{39} +(-3.00000 + 6.00000i) q^{40} +2.00000i q^{42} +4.00000 q^{43} +2.00000i q^{44} +(-1.00000 + 2.00000i) q^{45} +6.00000i q^{46} -8.00000 q^{47} +1.00000 q^{48} +3.00000 q^{49} +(3.00000 + 4.00000i) q^{50} +6.00000 q^{51} -4.00000i q^{52} -12.0000i q^{53} +1.00000 q^{54} +(4.00000 + 2.00000i) q^{55} +6.00000i q^{56} +2.00000i q^{57} +(5.00000 + 2.00000i) q^{58} -4.00000 q^{59} +(-1.00000 + 2.00000i) q^{60} -12.0000i q^{61} -2.00000i q^{62} +2.00000i q^{63} +7.00000 q^{64} +(-8.00000 - 4.00000i) q^{65} -2.00000i q^{66} +6.00000i q^{67} +6.00000 q^{68} +6.00000i q^{69} +(4.00000 + 2.00000i) q^{70} +8.00000 q^{71} +3.00000 q^{72} -6.00000 q^{73} +2.00000 q^{74} +(3.00000 + 4.00000i) q^{75} +2.00000i q^{76} +4.00000 q^{77} +4.00000i q^{78} +2.00000i q^{79} +(1.00000 - 2.00000i) q^{80} +1.00000 q^{81} -2.00000i q^{83} +2.00000i q^{84} +(6.00000 - 12.0000i) q^{85} -4.00000 q^{86} +(5.00000 + 2.00000i) q^{87} -6.00000i q^{88} +16.0000i q^{89} +(1.00000 - 2.00000i) q^{90} -8.00000 q^{91} +6.00000i q^{92} -2.00000i q^{93} +8.00000 q^{94} +(4.00000 + 2.00000i) q^{95} +5.00000 q^{96} -14.0000 q^{97} -3.00000 q^{98} -2.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} - 2 q^{4} - 2 q^{5} + 2 q^{6} + 6 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{12} + 2 q^{15} - 2 q^{16} - 12 q^{17} - 2 q^{18} + 2 q^{20} - 6 q^{24} - 6 q^{25} - 2 q^{27} - 10 q^{29} - 2 q^{30}+ \cdots - 6 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^3 - 2 * q^4 - 2 * q^5 + 2 * q^6 + 6 * q^8 + 2 * q^9 + 2 * q^10 + 2 * q^12 + 2 * q^15 - 2 * q^16 - 12 * q^17 - 2 * q^18 + 2 * q^20 - 6 * q^24 - 6 * q^25 - 2 * q^27 - 10 * q^29 - 2 * q^30 - 10 * q^32 + 12 * q^34 - 8 * q^35 - 2 * q^36 - 4 * q^37 - 6 * q^40 + 8 * q^43 - 2 * q^45 - 16 * q^47 + 2 * q^48 + 6 * q^49 + 6 * q^50 + 12 * q^51 + 2 * q^54 + 8 * q^55 + 10 * q^58 - 8 * q^59 - 2 * q^60 + 14 * q^64 - 16 * q^65 + 12 * q^68 + 8 * q^70 + 16 * q^71 + 6 * q^72 - 12 * q^73 + 4 * q^74 + 6 * q^75 + 8 * q^77 + 2 * q^80 + 2 * q^81 + 12 * q^85 - 8 * q^86 + 10 * q^87 + 2 * q^90 - 16 * q^91 + 16 * q^94 + 8 * q^95 + 10 * q^96 - 28 * q^97 - 6 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/435\mathbb{Z}\right)^\times$.

$n$	$31$	$146$	$262$
$\chi(n)$	$-1$	$1$	$-1$

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			−0.707107			−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000			−0.707107			−0.353553	+	0.935414i	$0.615027\pi$
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$	−1.00000			−0.500000
$5$	−1.00000	+	2.00000i	−0.447214	+	0.894427i
$5$	−1.00000	+	2.00000i	−0.447214	+	0.894427i
$6$	1.00000			0.408248
$7$			2.00000i			0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$			2.00000i			0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	3.00000			1.06066
$9$	1.00000			0.333333
$10$	1.00000	−	2.00000i	0.316228	−	0.632456i
$11$		−	2.00000i		−	0.603023i	−0.953463	−	0.301511i	$-0.902509\pi$
$11$		−	2.00000i		−	0.603023i	0.953463	−	0.301511i	$-0.0974911\pi$
$12$	1.00000			0.288675
$13$			4.00000i			1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$			4.00000i			1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$		−	2.00000i		−	0.534522i
$15$	1.00000	−	2.00000i	0.258199	−	0.516398i
$16$	−1.00000			−0.250000
$17$	−6.00000			−1.45521			−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000			−1.45521			−0.727607	+	0.685994i	$0.759367\pi$
$18$	−1.00000			−0.235702
$19$		−	2.00000i		−	0.458831i	−0.973329	−	0.229416i	$-0.926318\pi$
$19$		−	2.00000i		−	0.458831i	0.973329	−	0.229416i	$-0.0736815\pi$
$20$	1.00000	−	2.00000i	0.223607	−	0.447214i
$21$		−	2.00000i		−	0.436436i
$22$			2.00000i			0.426401i
$23$		−	6.00000i		−	1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$		−	6.00000i		−	1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$	−3.00000			−0.612372
$25$	−3.00000	−	4.00000i	−0.600000	−	0.800000i
$26$		−	4.00000i		−	0.784465i
$27$	−1.00000			−0.192450
$28$		−	2.00000i		−	0.377964i
$29$	−5.00000	−	2.00000i	−0.928477	−	0.371391i
$29$	−5.00000	−	2.00000i	−0.928477	−	0.371391i
$30$	−1.00000	+	2.00000i	−0.182574	+	0.365148i
$31$			2.00000i			0.359211i	0.983739	+	0.179605i	$0.0574821\pi$
$31$			2.00000i			0.359211i	−0.983739	+	0.179605i	$0.942518\pi$
$32$	−5.00000			−0.883883
$33$			2.00000i			0.348155i
$34$	6.00000			1.02899
$35$	−4.00000	−	2.00000i	−0.676123	−	0.338062i
$36$	−1.00000			−0.166667
$37$	−2.00000			−0.328798			−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000			−0.328798			−0.164399	+	0.986394i	$0.552568\pi$
$38$			2.00000i			0.324443i
$39$		−	4.00000i		−	0.640513i
$40$	−3.00000	+	6.00000i	−0.474342	+	0.948683i
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$			2.00000i			0.308607i
$43$	4.00000			0.609994			0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000			0.609994			0.304997	+	0.952353i	$0.401344\pi$
$44$			2.00000i			0.301511i
$45$	−1.00000	+	2.00000i	−0.149071	+	0.298142i
$46$			6.00000i			0.884652i
$47$	−8.00000			−1.16692			−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000			−1.16692			−0.583460	+	0.812142i	$0.698301\pi$
$48$	1.00000			0.144338
$49$	3.00000			0.428571
$50$	3.00000	+	4.00000i	0.424264	+	0.565685i
$51$	6.00000			0.840168
$52$		−	4.00000i		−	0.554700i
$53$		−	12.0000i		−	1.64833i	−0.566352	−	0.824163i	$-0.691646\pi$
$53$		−	12.0000i		−	1.64833i	0.566352	−	0.824163i	$-0.308354\pi$
$54$	1.00000			0.136083
$55$	4.00000	+	2.00000i	0.539360	+	0.269680i
$56$			6.00000i			0.801784i
$57$			2.00000i			0.264906i
$58$	5.00000	+	2.00000i	0.656532	+	0.262613i
$59$	−4.00000			−0.520756			−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000			−0.520756			−0.260378	+	0.965507i	$0.583847\pi$
$60$	−1.00000	+	2.00000i	−0.129099	+	0.258199i
$61$		−	12.0000i		−	1.53644i	−0.640184	−	0.768221i	$-0.721142\pi$
$61$		−	12.0000i		−	1.53644i	0.640184	−	0.768221i	$-0.278858\pi$
$62$		−	2.00000i		−	0.254000i
$63$			2.00000i			0.251976i
$64$	7.00000			0.875000
$65$	−8.00000	−	4.00000i	−0.992278	−	0.496139i
$66$		−	2.00000i		−	0.246183i
$67$			6.00000i			0.733017i	0.930415	+	0.366508i	$0.119447\pi$
$67$			6.00000i			0.733017i	−0.930415	+	0.366508i	$0.880553\pi$
$68$	6.00000			0.727607
$69$			6.00000i			0.722315i
$70$	4.00000	+	2.00000i	0.478091	+	0.239046i
$71$	8.00000			0.949425			0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000			0.949425			0.474713	+	0.880141i	$0.342552\pi$
$72$	3.00000			0.353553
$73$	−6.00000			−0.702247			−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000			−0.702247			−0.351123	+	0.936329i	$0.614200\pi$
$74$	2.00000			0.232495
$75$	3.00000	+	4.00000i	0.346410	+	0.461880i
$76$			2.00000i			0.229416i
$77$	4.00000			0.455842
$78$			4.00000i			0.452911i
$79$			2.00000i			0.225018i	0.993651	+	0.112509i	$0.0358886\pi$
$79$			2.00000i			0.225018i	−0.993651	+	0.112509i	$0.964111\pi$
$80$	1.00000	−	2.00000i	0.111803	−	0.223607i
$81$	1.00000			0.111111
$82$		0			0
$83$		−	2.00000i		−	0.219529i	−0.993958	−	0.109764i	$-0.964990\pi$
$83$		−	2.00000i		−	0.219529i	0.993958	−	0.109764i	$-0.0350096\pi$
$84$			2.00000i			0.218218i
$85$	6.00000	−	12.0000i	0.650791	−	1.30158i
$86$	−4.00000			−0.431331
$87$	5.00000	+	2.00000i	0.536056	+	0.214423i
$88$		−	6.00000i		−	0.639602i
$89$			16.0000i			1.69600i	0.529999	+	0.847998i	$0.322192\pi$
$89$			16.0000i			1.69600i	−0.529999	+	0.847998i	$0.677808\pi$
$90$	1.00000	−	2.00000i	0.105409	−	0.210819i
$91$	−8.00000			−0.838628
$92$			6.00000i			0.625543i
$93$		−	2.00000i		−	0.207390i
$94$	8.00000			0.825137
$95$	4.00000	+	2.00000i	0.410391	+	0.205196i
$96$	5.00000			0.510310
$97$	−14.0000			−1.42148			−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000			−1.42148			−0.710742	+	0.703452i	$0.751641\pi$
$98$	−3.00000			−0.303046
$99$		−	2.00000i		−	0.201008i
$100$	3.00000	+	4.00000i	0.300000	+	0.400000i
$101$		−	12.0000i		−	1.19404i	−0.802225	−	0.597022i	$-0.796350\pi$
$101$		−	12.0000i		−	1.19404i	0.802225	−	0.597022i	$-0.203650\pi$
$102$	−6.00000			−0.594089
$103$			18.0000i			1.77359i	0.462160	+	0.886796i	$0.347074\pi$
$103$			18.0000i			1.77359i	−0.462160	+	0.886796i	$0.652926\pi$
$104$			12.0000i			1.17670i
$105$	4.00000	+	2.00000i	0.390360	+	0.195180i
$106$			12.0000i			1.16554i
$107$		−	10.0000i		−	0.966736i	−0.875417	−	0.483368i	$-0.839413\pi$
$107$		−	10.0000i		−	0.966736i	0.875417	−	0.483368i	$-0.160587\pi$
$108$	1.00000			0.0962250
$109$	−10.0000			−0.957826			−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000			−0.957826			−0.478913	+	0.877862i	$0.658969\pi$
$110$	−4.00000	−	2.00000i	−0.381385	−	0.190693i
$111$	2.00000			0.189832
$112$		−	2.00000i		−	0.188982i
$113$	2.00000			0.188144			0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000			0.188144			0.0940721	+	0.995565i	$0.470012\pi$
$114$		−	2.00000i		−	0.187317i
$115$	12.0000	+	6.00000i	1.11901	+	0.559503i
$116$	5.00000	+	2.00000i	0.464238	+	0.185695i
$117$			4.00000i			0.369800i
$118$	4.00000			0.368230
$119$		−	12.0000i		−	1.10004i
$120$	3.00000	−	6.00000i	0.273861	−	0.547723i
$121$	7.00000			0.636364
$122$			12.0000i			1.08643i
$123$		0			0
$124$		−	2.00000i		−	0.179605i
$125$	11.0000	−	2.00000i	0.983870	−	0.178885i
$126$		−	2.00000i		−	0.178174i
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$	3.00000			0.265165
$129$	−4.00000			−0.352180
$130$	8.00000	+	4.00000i	0.701646	+	0.350823i
$131$			22.0000i			1.92215i	0.276289	+	0.961074i	$0.410895\pi$
$131$			22.0000i			1.92215i	−0.276289	+	0.961074i	$0.589105\pi$
$132$		−	2.00000i		−	0.174078i
$133$	4.00000			0.346844
$134$		−	6.00000i		−	0.518321i
$135$	1.00000	−	2.00000i	0.0860663	−	0.172133i
$136$	−18.0000			−1.54349
$137$	−14.0000			−1.19610			−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000			−1.19610			−0.598050	+	0.801459i	$0.704058\pi$
$138$		−	6.00000i		−	0.510754i
$139$	−4.00000			−0.339276			−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000			−0.339276			−0.169638	+	0.985506i	$0.554260\pi$
$140$	4.00000	+	2.00000i	0.338062	+	0.169031i
$141$	8.00000			0.673722
$142$	−8.00000			−0.671345
$143$	8.00000			0.668994
$144$	−1.00000			−0.0833333
$145$	9.00000	−	8.00000i	0.747409	−	0.664364i
$146$	6.00000			0.496564
$147$	−3.00000			−0.247436
$148$	2.00000			0.164399
$149$	−10.0000			−0.819232			−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000			−0.819232			−0.409616	+	0.912258i	$0.634337\pi$
$150$	−3.00000	−	4.00000i	−0.244949	−	0.326599i
$151$	−8.00000			−0.651031			−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000			−0.651031			−0.325515	+	0.945537i	$0.605538\pi$
$152$		−	6.00000i		−	0.486664i
$153$	−6.00000			−0.485071
$154$	−4.00000			−0.322329
$155$	−4.00000	−	2.00000i	−0.321288	−	0.160644i
$156$			4.00000i			0.320256i
$157$	−2.00000			−0.159617			−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000			−0.159617			−0.0798087	+	0.996810i	$0.525431\pi$
$158$		−	2.00000i		−	0.159111i
$159$			12.0000i			0.951662i
$160$	5.00000	−	10.0000i	0.395285	−	0.790569i
$161$	12.0000			0.945732
$162$	−1.00000			−0.0785674
$163$	−20.0000			−1.56652			−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000			−1.56652			−0.783260	+	0.621694i	$0.786445\pi$
$164$		0			0
$165$	−4.00000	−	2.00000i	−0.311400	−	0.155700i
$166$			2.00000i			0.155230i
$167$		−	6.00000i		−	0.464294i	−0.972681	−	0.232147i	$-0.925425\pi$
$167$		−	6.00000i		−	0.464294i	0.972681	−	0.232147i	$-0.0745750\pi$
$168$		−	6.00000i		−	0.462910i
$169$	−3.00000			−0.230769
$170$	−6.00000	+	12.0000i	−0.460179	+	0.920358i
$171$		−	2.00000i		−	0.152944i
$172$	−4.00000			−0.304997
$173$			12.0000i			0.912343i	0.889892	+	0.456172i	$0.150780\pi$
$173$			12.0000i			0.912343i	−0.889892	+	0.456172i	$0.849220\pi$
$174$	−5.00000	−	2.00000i	−0.379049	−	0.151620i
$175$	8.00000	−	6.00000i	0.604743	−	0.453557i
$176$			2.00000i			0.150756i
$177$	4.00000			0.300658
$178$		−	16.0000i		−	1.19925i
$179$	−12.0000			−0.896922			−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$	1.00000	−	2.00000i	0.0745356	−	0.149071i
$181$	−10.0000			−0.743294			−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000			−0.743294			−0.371647	+	0.928374i	$0.621207\pi$
$182$	8.00000			0.592999
$183$			12.0000i			0.887066i
$184$		−	18.0000i		−	1.32698i
$185$	2.00000	−	4.00000i	0.147043	−	0.294086i
$186$			2.00000i			0.146647i
$187$			12.0000i			0.877527i
$188$	8.00000			0.583460
$189$		−	2.00000i		−	0.145479i
$190$	−4.00000	−	2.00000i	−0.290191	−	0.145095i
$191$		−	6.00000i		−	0.434145i	−0.976156	−	0.217072i	$-0.930349\pi$
$191$		−	6.00000i		−	0.434145i	0.976156	−	0.217072i	$-0.0696508\pi$
$192$	−7.00000			−0.505181
$193$	−6.00000			−0.431889			−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000			−0.431889			−0.215945	+	0.976406i	$0.569283\pi$
$194$	14.0000			1.00514
$195$	8.00000	+	4.00000i	0.572892	+	0.286446i
$196$	−3.00000			−0.214286
$197$			4.00000i			0.284988i	0.989796	+	0.142494i	$0.0455122\pi$
$197$			4.00000i			0.284988i	−0.989796	+	0.142494i	$0.954488\pi$
$198$			2.00000i			0.142134i
$199$	8.00000			0.567105			0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000			0.567105			0.283552	+	0.958957i	$0.408487\pi$
$200$	−9.00000	−	12.0000i	−0.636396	−	0.848528i
$201$		−	6.00000i		−	0.423207i
$202$			12.0000i			0.844317i
$203$	4.00000	−	10.0000i	0.280745	−	0.701862i
$204$	−6.00000			−0.420084
$205$		0			0
$206$		−	18.0000i		−	1.25412i
$207$		−	6.00000i		−	0.417029i
$208$		−	4.00000i		−	0.277350i
$209$	−4.00000			−0.276686
$210$	−4.00000	−	2.00000i	−0.276026	−	0.138013i
$211$		−	2.00000i		−	0.137686i	−0.997628	−	0.0688428i	$-0.978069\pi$
$211$		−	2.00000i		−	0.137686i	0.997628	−	0.0688428i	$-0.0219307\pi$
$212$			12.0000i			0.824163i
$213$	−8.00000			−0.548151
$214$			10.0000i			0.683586i
$215$	−4.00000	+	8.00000i	−0.272798	+	0.545595i
$216$	−3.00000			−0.204124
$217$	−4.00000			−0.271538
$218$	10.0000			0.677285
$219$	6.00000			0.405442
$220$	−4.00000	−	2.00000i	−0.269680	−	0.134840i
$221$		−	24.0000i		−	1.61441i
$222$	−2.00000			−0.134231
$223$		−	22.0000i		−	1.47323i	−0.676313	−	0.736614i	$-0.736423\pi$
$223$		−	22.0000i		−	1.47323i	0.676313	−	0.736614i	$-0.263577\pi$
$224$		−	10.0000i		−	0.668153i
$225$	−3.00000	−	4.00000i	−0.200000	−	0.266667i
$226$	−2.00000			−0.133038
$227$		−	2.00000i		−	0.132745i	−0.997795	−	0.0663723i	$-0.978857\pi$
$227$		−	2.00000i		−	0.132745i	0.997795	−	0.0663723i	$-0.0211425\pi$
$228$		−	2.00000i		−	0.132453i
$229$		−	20.0000i		−	1.32164i	−0.750546	−	0.660819i	$-0.770209\pi$
$229$		−	20.0000i		−	1.32164i	0.750546	−	0.660819i	$-0.229791\pi$
$230$	−12.0000	−	6.00000i	−0.791257	−	0.395628i
$231$	−4.00000			−0.263181
$232$	−15.0000	−	6.00000i	−0.984798	−	0.393919i
$233$			16.0000i			1.04819i	0.851658	+	0.524097i	$0.175597\pi$
$233$			16.0000i			1.04819i	−0.851658	+	0.524097i	$0.824403\pi$
$234$		−	4.00000i		−	0.261488i
$235$	8.00000	−	16.0000i	0.521862	−	1.04372i
$236$	4.00000			0.260378
$237$		−	2.00000i		−	0.129914i
$238$			12.0000i			0.777844i
$239$	16.0000			1.03495			0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000			1.03495			0.517477	+	0.855697i	$0.326871\pi$
$240$	−1.00000	+	2.00000i	−0.0645497	+	0.129099i
$241$	−22.0000			−1.41714			−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000			−1.41714			−0.708572	+	0.705638i	$0.750660\pi$
$242$	−7.00000			−0.449977
$243$	−1.00000			−0.0641500
$244$			12.0000i			0.768221i
$245$	−3.00000	+	6.00000i	−0.191663	+	0.383326i
$246$		0			0
$247$	8.00000			0.509028
$248$			6.00000i			0.381000i
$249$			2.00000i			0.126745i
$250$	−11.0000	+	2.00000i	−0.695701	+	0.126491i
$251$			30.0000i			1.89358i	0.321847	+	0.946792i	$0.395696\pi$
$251$			30.0000i			1.89358i	−0.321847	+	0.946792i	$0.604304\pi$
$252$		−	2.00000i		−	0.125988i
$253$	−12.0000			−0.754434
$254$	−8.00000			−0.501965
$255$	−6.00000	+	12.0000i	−0.375735	+	0.751469i
$256$	−17.0000			−1.06250
$257$			24.0000i			1.49708i	0.663090	+	0.748539i	$0.269245\pi$
$257$			24.0000i			1.49708i	−0.663090	+	0.748539i	$0.730755\pi$
$258$	4.00000			0.249029
$259$		−	4.00000i		−	0.248548i
$260$	8.00000	+	4.00000i	0.496139	+	0.248069i
$261$	−5.00000	−	2.00000i	−0.309492	−	0.123797i
$262$		−	22.0000i		−	1.35916i
$263$	16.0000			0.986602			0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000			0.986602			0.493301	+	0.869859i	$0.335790\pi$
$264$			6.00000i			0.369274i
$265$	24.0000	+	12.0000i	1.47431	+	0.737154i
$266$	−4.00000			−0.245256
$267$		−	16.0000i		−	0.979184i
$268$		−	6.00000i		−	0.366508i
$269$		−	4.00000i		−	0.243884i	−0.992537	−	0.121942i	$-0.961088\pi$
$269$		−	4.00000i		−	0.243884i	0.992537	−	0.121942i	$-0.0389122\pi$
$270$	−1.00000	+	2.00000i	−0.0608581	+	0.121716i
$271$			2.00000i			0.121491i	0.998153	+	0.0607457i	$0.0193479\pi$
$271$			2.00000i			0.121491i	−0.998153	+	0.0607457i	$0.980652\pi$
$272$	6.00000			0.363803
$273$	8.00000			0.484182
$274$	14.0000			0.845771
$275$	−8.00000	+	6.00000i	−0.482418	+	0.361814i
$276$		−	6.00000i		−	0.361158i
$277$		−	4.00000i		−	0.240337i	−0.992754	−	0.120168i	$-0.961657\pi$
$277$		−	4.00000i		−	0.240337i	0.992754	−	0.120168i	$-0.0383434\pi$
$278$	4.00000			0.239904
$279$			2.00000i			0.119737i
$280$	−12.0000	−	6.00000i	−0.717137	−	0.358569i
$281$	−30.0000			−1.78965			−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000			−1.78965			−0.894825	+	0.446417i	$0.852700\pi$
$282$	−8.00000			−0.476393
$283$		−	2.00000i		−	0.118888i	−0.998232	−	0.0594438i	$-0.981067\pi$
$283$		−	2.00000i		−	0.118888i	0.998232	−	0.0594438i	$-0.0189327\pi$
$284$	−8.00000			−0.474713
$285$	−4.00000	−	2.00000i	−0.236940	−	0.118470i
$286$	−8.00000			−0.473050
$287$		0			0
$288$	−5.00000			−0.294628
$289$	19.0000			1.11765
$290$	−9.00000	+	8.00000i	−0.528498	+	0.469776i
$291$	14.0000			0.820695
$292$	6.00000			0.351123
$293$	−26.0000			−1.51894			−0.759468	−	0.650545i	$-0.774541\pi$
$293$	−26.0000			−1.51894			−0.759468	+	0.650545i	$0.774541\pi$
$294$	3.00000			0.174964
$295$	4.00000	−	8.00000i	0.232889	−	0.465778i
$296$	−6.00000			−0.348743
$297$			2.00000i			0.116052i
$298$	10.0000			0.579284
$299$	24.0000			1.38796
$300$	−3.00000	−	4.00000i	−0.173205	−	0.230940i
$301$			8.00000i			0.461112i
$302$	8.00000			0.460348
$303$			12.0000i			0.689382i
$304$			2.00000i			0.114708i
$305$	24.0000	+	12.0000i	1.37424	+	0.687118i
$306$	6.00000			0.342997
$307$	12.0000			0.684876			0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000			0.684876			0.342438	+	0.939540i	$0.388747\pi$
$308$	−4.00000			−0.227921
$309$		−	18.0000i		−	1.02398i
$310$	4.00000	+	2.00000i	0.227185	+	0.113592i
$311$		−	14.0000i		−	0.793867i	−0.917847	−	0.396934i	$-0.870074\pi$
$311$		−	14.0000i		−	0.793867i	0.917847	−	0.396934i	$-0.129926\pi$
$312$		−	12.0000i		−	0.679366i
$313$		−	8.00000i		−	0.452187i	−0.974106	−	0.226093i	$-0.927405\pi$
$313$		−	8.00000i		−	0.452187i	0.974106	−	0.226093i	$-0.0725954\pi$
$314$	2.00000			0.112867
$315$	−4.00000	−	2.00000i	−0.225374	−	0.112687i
$316$		−	2.00000i		−	0.112509i
$317$	6.00000			0.336994			0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000			0.336994			0.168497	+	0.985702i	$0.446109\pi$
$318$		−	12.0000i		−	0.672927i
$319$	−4.00000	+	10.0000i	−0.223957	+	0.559893i
$320$	−7.00000	+	14.0000i	−0.391312	+	0.782624i
$321$			10.0000i			0.558146i
$322$	−12.0000			−0.668734
$323$			12.0000i			0.667698i
$324$	−1.00000			−0.0555556
$325$	16.0000	−	12.0000i	0.887520	−	0.665640i
$326$	20.0000			1.10770
$327$	10.0000			0.553001
$328$		0			0
$329$		−	16.0000i		−	0.882109i
$330$	4.00000	+	2.00000i	0.220193	+	0.110096i
$331$			22.0000i			1.20923i	0.796518	+	0.604615i	$0.206673\pi$
$331$			22.0000i			1.20923i	−0.796518	+	0.604615i	$0.793327\pi$
$332$			2.00000i			0.109764i
$333$	−2.00000			−0.109599
$334$			6.00000i			0.328305i
$335$	−12.0000	−	6.00000i	−0.655630	−	0.327815i
$336$			2.00000i			0.109109i
$337$	34.0000			1.85210			0.926049	−	0.377403i	$-0.123183\pi$
$337$	34.0000			1.85210			0.926049	+	0.377403i	$0.123183\pi$
$338$	3.00000			0.163178
$339$	−2.00000			−0.108625
$340$	−6.00000	+	12.0000i	−0.325396	+	0.650791i
$341$	4.00000			0.216612
$342$			2.00000i			0.108148i
$343$			20.0000i			1.07990i
$344$	12.0000			0.646997
$345$	−12.0000	−	6.00000i	−0.646058	−	0.323029i
$346$		−	12.0000i		−	0.645124i
$347$		−	10.0000i		−	0.536828i	−0.963304	−	0.268414i	$-0.913500\pi$
$347$		−	10.0000i		−	0.536828i	0.963304	−	0.268414i	$-0.0864995\pi$
$348$	−5.00000	−	2.00000i	−0.268028	−	0.107211i
$349$	−26.0000			−1.39175			−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000			−1.39175			−0.695874	+	0.718164i	$0.744983\pi$
$350$	−8.00000	+	6.00000i	−0.427618	+	0.320713i
$351$		−	4.00000i		−	0.213504i
$352$			10.0000i			0.533002i
$353$			24.0000i			1.27739i	0.769460	+	0.638696i	$0.220526\pi$
$353$			24.0000i			1.27739i	−0.769460	+	0.638696i	$0.779474\pi$
$354$	−4.00000			−0.212598
$355$	−8.00000	+	16.0000i	−0.424596	+	0.849192i
$356$		−	16.0000i		−	0.847998i
$357$			12.0000i			0.635107i
$358$	12.0000			0.634220
$359$			18.0000i			0.950004i	0.879985	+	0.475002i	$0.157553\pi$
$359$			18.0000i			0.950004i	−0.879985	+	0.475002i	$0.842447\pi$
$360$	−3.00000	+	6.00000i	−0.158114	+	0.316228i
$361$	15.0000			0.789474
$362$	10.0000			0.525588
$363$	−7.00000			−0.367405
$364$	8.00000			0.419314
$365$	6.00000	−	12.0000i	0.314054	−	0.628109i
$366$		−	12.0000i		−	0.627250i
$367$	24.0000			1.25279			0.626395	−	0.779506i	$-0.284530\pi$
$367$	24.0000			1.25279			0.626395	+	0.779506i	$0.284530\pi$
$368$			6.00000i			0.312772i
$369$		0			0
$370$	−2.00000	+	4.00000i	−0.103975	+	0.207950i
$371$	24.0000			1.24602
$372$			2.00000i			0.103695i
$373$		−	4.00000i		−	0.207112i	−0.994624	−	0.103556i	$-0.966978\pi$
$373$		−	4.00000i		−	0.207112i	0.994624	−	0.103556i	$-0.0330221\pi$
$374$		−	12.0000i		−	0.620505i
$375$	−11.0000	+	2.00000i	−0.568038	+	0.103280i
$376$	−24.0000			−1.23771
$377$	8.00000	−	20.0000i	0.412021	−	1.03005i
$378$			2.00000i			0.102869i
$379$			38.0000i			1.95193i	0.217930	+	0.975964i	$0.430070\pi$
$379$			38.0000i			1.95193i	−0.217930	+	0.975964i	$0.569930\pi$
$380$	−4.00000	−	2.00000i	−0.205196	−	0.102598i
$381$	−8.00000			−0.409852
$382$			6.00000i			0.306987i
$383$		−	30.0000i		−	1.53293i	−0.642287	−	0.766464i	$-0.722014\pi$
$383$		−	30.0000i		−	1.53293i	0.642287	−	0.766464i	$-0.277986\pi$
$384$	−3.00000			−0.153093
$385$	−4.00000	+	8.00000i	−0.203859	+	0.407718i
$386$	6.00000			0.305392
$387$	4.00000			0.203331
$388$	14.0000			0.710742
$389$		−	12.0000i		−	0.608424i	−0.952604	−	0.304212i	$-0.901607\pi$
$389$		−	12.0000i		−	0.608424i	0.952604	−	0.304212i	$-0.0983931\pi$
$390$	−8.00000	−	4.00000i	−0.405096	−	0.202548i
$391$			36.0000i			1.82060i
$392$	9.00000			0.454569
$393$		−	22.0000i		−	1.10975i
$394$		−	4.00000i		−	0.201517i
$395$	−4.00000	−	2.00000i	−0.201262	−	0.100631i
$396$			2.00000i			0.100504i
$397$		−	28.0000i		−	1.40528i	−0.711546	−	0.702640i	$-0.752005\pi$
$397$		−	28.0000i		−	1.40528i	0.711546	−	0.702640i	$-0.247995\pi$
$398$	−8.00000			−0.401004
$399$	−4.00000			−0.200250
$400$	3.00000	+	4.00000i	0.150000	+	0.200000i
$401$	18.0000			0.898877			0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000			0.898877			0.449439	+	0.893311i	$0.351624\pi$
$402$			6.00000i			0.299253i
$403$	−8.00000			−0.398508
$404$			12.0000i			0.597022i
$405$	−1.00000	+	2.00000i	−0.0496904	+	0.0993808i
$406$	−4.00000	+	10.0000i	−0.198517	+	0.496292i
$407$			4.00000i			0.198273i
$408$	18.0000			0.891133
$409$			8.00000i			0.395575i	0.980245	+	0.197787i	$0.0633755\pi$
$409$			8.00000i			0.395575i	−0.980245	+	0.197787i	$0.936624\pi$
$410$		0			0
$411$	14.0000			0.690569
$412$		−	18.0000i		−	0.886796i
$413$		−	8.00000i		−	0.393654i
$414$			6.00000i			0.294884i
$415$	4.00000	+	2.00000i	0.196352	+	0.0981761i
$416$		−	20.0000i		−	0.980581i
$417$	4.00000			0.195881
$418$	4.00000			0.195646
$419$	−12.0000			−0.586238			−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000			−0.586238			−0.293119	+	0.956076i	$0.594693\pi$
$420$	−4.00000	−	2.00000i	−0.195180	−	0.0975900i
$421$			28.0000i			1.36464i	0.731055	+	0.682318i	$0.239028\pi$
$421$			28.0000i			1.36464i	−0.731055	+	0.682318i	$0.760972\pi$
$422$			2.00000i			0.0973585i
$423$	−8.00000			−0.388973
$424$		−	36.0000i		−	1.74831i
$425$	18.0000	+	24.0000i	0.873128	+	1.16417i
$426$	8.00000			0.387601
$427$	24.0000			1.16144
$428$			10.0000i			0.483368i
$429$	−8.00000			−0.386244
$430$	4.00000	−	8.00000i	0.192897	−	0.385794i
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	1.00000			0.0481125
$433$	−22.0000			−1.05725			−0.528626	−	0.848855i	$-0.677293\pi$
$433$	−22.0000			−1.05725			−0.528626	+	0.848855i	$0.677293\pi$
$434$	4.00000			0.192006
$435$	−9.00000	+	8.00000i	−0.431517	+	0.383571i
$436$	10.0000			0.478913
$437$	−12.0000			−0.574038
$438$	−6.00000			−0.286691
$439$	24.0000			1.14546			0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000			1.14546			0.572729	+	0.819745i	$0.305885\pi$
$440$	12.0000	+	6.00000i	0.572078	+	0.286039i
$441$	3.00000			0.142857
$442$			24.0000i			1.14156i
$443$	−28.0000			−1.33032			−0.665160	−	0.746701i	$-0.731637\pi$
$443$	−28.0000			−1.33032			−0.665160	+	0.746701i	$0.731637\pi$
$444$	−2.00000			−0.0949158
$445$	−32.0000	−	16.0000i	−1.51695	−	0.758473i
$446$			22.0000i			1.04173i
$447$	10.0000			0.472984
$448$			14.0000i			0.661438i
$449$			24.0000i			1.13263i	0.824189	+	0.566315i	$0.191631\pi$
$449$			24.0000i			1.13263i	−0.824189	+	0.566315i	$0.808369\pi$
$450$	3.00000	+	4.00000i	0.141421	+	0.188562i
$451$		0			0
$452$	−2.00000			−0.0940721
$453$	8.00000			0.375873
$454$			2.00000i			0.0938647i
$455$	8.00000	−	16.0000i	0.375046	−	0.750092i
$456$			6.00000i			0.280976i
$457$		−	8.00000i		−	0.374224i	−0.982339	−	0.187112i	$-0.940087\pi$
$457$		−	8.00000i		−	0.374224i	0.982339	−	0.187112i	$-0.0599128\pi$
$458$			20.0000i			0.934539i
$459$	6.00000			0.280056
$460$	−12.0000	−	6.00000i	−0.559503	−	0.279751i
$461$			12.0000i			0.558896i	0.960161	+	0.279448i	$0.0901514\pi$
$461$			12.0000i			0.558896i	−0.960161	+	0.279448i	$0.909849\pi$
$462$	4.00000			0.186097
$463$			26.0000i			1.20832i	0.796862	+	0.604161i	$0.206492\pi$
$463$			26.0000i			1.20832i	−0.796862	+	0.604161i	$0.793508\pi$
$464$	5.00000	+	2.00000i	0.232119	+	0.0928477i
$465$	4.00000	+	2.00000i	0.185496	+	0.0927478i
$466$		−	16.0000i		−	0.741186i
$467$	12.0000			0.555294			0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000			0.555294			0.277647	+	0.960683i	$0.410445\pi$
$468$		−	4.00000i		−	0.184900i
$469$	−12.0000			−0.554109
$470$	−8.00000	+	16.0000i	−0.369012	+	0.738025i
$471$	2.00000			0.0921551
$472$	−12.0000			−0.552345
$473$		−	8.00000i		−	0.367840i
$474$			2.00000i			0.0918630i
$475$	−8.00000	+	6.00000i	−0.367065	+	0.275299i
$476$			12.0000i			0.550019i
$477$		−	12.0000i		−	0.549442i
$478$	−16.0000			−0.731823
$479$			42.0000i			1.91903i	0.281659	+	0.959514i	$0.409115\pi$
$479$			42.0000i			1.91903i	−0.281659	+	0.959514i	$0.590885\pi$
$480$	−5.00000	+	10.0000i	−0.228218	+	0.456435i
$481$		−	8.00000i		−	0.364769i
$482$	22.0000			1.00207
$483$	−12.0000			−0.546019
$484$	−7.00000			−0.318182
$485$	14.0000	−	28.0000i	0.635707	−	1.27141i
$486$	1.00000			0.0453609
$487$		−	14.0000i		−	0.634401i	−0.948359	−	0.317200i	$-0.897257\pi$
$487$		−	14.0000i		−	0.634401i	0.948359	−	0.317200i	$-0.102743\pi$
$488$		−	36.0000i		−	1.62964i
$489$	20.0000			0.904431
$490$	3.00000	−	6.00000i	0.135526	−	0.271052i
$491$		−	2.00000i		−	0.0902587i	−0.998981	−	0.0451294i	$-0.985630\pi$
$491$		−	2.00000i		−	0.0902587i	0.998981	−	0.0451294i	$-0.0143700\pi$
$492$		0			0
$493$	30.0000	+	12.0000i	1.35113	+	0.540453i
$494$	−8.00000			−0.359937
$495$	4.00000	+	2.00000i	0.179787	+	0.0898933i
$496$		−	2.00000i		−	0.0898027i
$497$			16.0000i			0.717698i
$498$		−	2.00000i		−	0.0896221i
$499$	−12.0000			−0.537194			−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000			−0.537194			−0.268597	+	0.963253i	$0.586560\pi$
$500$	−11.0000	+	2.00000i	−0.491935	+	0.0894427i
$501$			6.00000i			0.268060i
$502$		−	30.0000i		−	1.33897i
$503$	−16.0000			−0.713405			−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000			−0.713405			−0.356702	+	0.934218i	$0.616099\pi$
$504$			6.00000i			0.267261i
$505$	24.0000	+	12.0000i	1.06799	+	0.533993i
$506$	12.0000			0.533465
$507$	3.00000			0.133235
$508$	−8.00000			−0.354943
$509$	−10.0000			−0.443242			−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000			−0.443242			−0.221621	+	0.975133i	$0.571135\pi$
$510$	6.00000	−	12.0000i	0.265684	−	0.531369i
$511$		−	12.0000i		−	0.530849i
$512$	11.0000			0.486136
$513$			2.00000i			0.0883022i
$514$		−	24.0000i		−	1.05859i
$515$	−36.0000	−	18.0000i	−1.58635	−	0.793175i
$516$	4.00000			0.176090
$517$			16.0000i			0.703679i
$518$			4.00000i			0.175750i
$519$		−	12.0000i		−	0.526742i
$520$	−24.0000	−	12.0000i	−1.05247	−	0.526235i
$521$	−14.0000			−0.613351			−0.306676	−	0.951814i	$-0.599217\pi$
$521$	−14.0000			−0.613351			−0.306676	+	0.951814i	$0.599217\pi$
$522$	5.00000	+	2.00000i	0.218844	+	0.0875376i
$523$		−	34.0000i		−	1.48672i	−0.668894	−	0.743358i	$-0.733232\pi$
$523$		−	34.0000i		−	1.48672i	0.668894	−	0.743358i	$-0.266768\pi$
$524$		−	22.0000i		−	0.961074i
$525$	−8.00000	+	6.00000i	−0.349149	+	0.261861i
$526$	−16.0000			−0.697633
$527$		−	12.0000i		−	0.522728i
$528$		−	2.00000i		−	0.0870388i
$529$	−13.0000			−0.565217
$530$	−24.0000	−	12.0000i	−1.04249	−	0.521247i
$531$	−4.00000			−0.173585
$532$	−4.00000			−0.173422
$533$		0			0
$534$			16.0000i			0.692388i
$535$	20.0000	+	10.0000i	0.864675	+	0.432338i
$536$			18.0000i			0.777482i
$537$	12.0000			0.517838
$538$			4.00000i			0.172452i
$539$		−	6.00000i		−	0.258438i
$540$	−1.00000	+	2.00000i	−0.0430331	+	0.0860663i
$541$		−	44.0000i		−	1.89171i	−0.324593	−	0.945854i	$-0.605227\pi$
$541$		−	44.0000i		−	1.89171i	0.324593	−	0.945854i	$-0.394773\pi$
$542$		−	2.00000i		−	0.0859074i
$543$	10.0000			0.429141
$544$	30.0000			1.28624
$545$	10.0000	−	20.0000i	0.428353	−	0.856706i
$546$	−8.00000			−0.342368
$547$		−	26.0000i		−	1.11168i	−0.831289	−	0.555840i	$-0.812397\pi$
$547$		−	26.0000i		−	1.11168i	0.831289	−	0.555840i	$-0.187603\pi$
$548$	14.0000			0.598050
$549$		−	12.0000i		−	0.512148i
$550$	8.00000	−	6.00000i	0.341121	−	0.255841i
$551$	−4.00000	+	10.0000i	−0.170406	+	0.426014i
$552$			18.0000i			0.766131i
$553$	−4.00000			−0.170097
$554$			4.00000i			0.169944i
$555$	−2.00000	+	4.00000i	−0.0848953	+	0.169791i
$556$	4.00000			0.169638
$557$			12.0000i			0.508456i	0.967144	+	0.254228i	$0.0818214\pi$
$557$			12.0000i			0.508456i	−0.967144	+	0.254228i	$0.918179\pi$
$558$		−	2.00000i		−	0.0846668i
$559$			16.0000i			0.676728i
$560$	4.00000	+	2.00000i	0.169031	+	0.0845154i
$561$		−	12.0000i		−	0.506640i
$562$	30.0000			1.26547
$563$	12.0000			0.505740			0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000			0.505740			0.252870	+	0.967500i	$0.418626\pi$
$564$	−8.00000			−0.336861
$565$	−2.00000	+	4.00000i	−0.0841406	+	0.168281i
$566$			2.00000i			0.0840663i
$567$			2.00000i			0.0839921i
$568$	24.0000			1.00702
$569$		−	32.0000i		−	1.34151i	−0.741679	−	0.670755i	$-0.765970\pi$
$569$		−	32.0000i		−	1.34151i	0.741679	−	0.670755i	$-0.234030\pi$
$570$	4.00000	+	2.00000i	0.167542	+	0.0837708i
$571$	−20.0000			−0.836974			−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000			−0.836974			−0.418487	+	0.908223i	$0.637439\pi$
$572$	−8.00000			−0.334497
$573$			6.00000i			0.250654i
$574$		0			0
$575$	−24.0000	+	18.0000i	−1.00087	+	0.750652i
$576$	7.00000			0.291667
$577$	−38.0000			−1.58196			−0.790980	−	0.611842i	$-0.790429\pi$
$577$	−38.0000			−1.58196			−0.790980	+	0.611842i	$0.790429\pi$
$578$	−19.0000			−0.790296
$579$	6.00000			0.249351
$580$	−9.00000	+	8.00000i	−0.373705	+	0.332182i
$581$	4.00000			0.165948
$582$	−14.0000			−0.580319
$583$	−24.0000			−0.993978
$584$	−18.0000			−0.744845
$585$	−8.00000	−	4.00000i	−0.330759	−	0.165380i
$586$	26.0000			1.07405
$587$		−	42.0000i		−	1.73353i	−0.498721	−	0.866763i	$-0.666197\pi$
$587$		−	42.0000i		−	1.73353i	0.498721	−	0.866763i	$-0.333803\pi$
$588$	3.00000			0.123718
$589$	4.00000			0.164817
$590$	−4.00000	+	8.00000i	−0.164677	+	0.329355i
$591$		−	4.00000i		−	0.164538i
$592$	2.00000			0.0821995
$593$		−	8.00000i		−	0.328521i	−0.986417	−	0.164260i	$-0.947476\pi$
$593$		−	8.00000i		−	0.328521i	0.986417	−	0.164260i	$-0.0525237\pi$
$594$		−	2.00000i		−	0.0820610i
$595$	24.0000	+	12.0000i	0.983904	+	0.491952i
$596$	10.0000			0.409616
$597$	−8.00000			−0.327418
$598$	−24.0000			−0.981433
$599$			18.0000i			0.735460i	0.929933	+	0.367730i	$0.119865\pi$
$599$			18.0000i			0.735460i	−0.929933	+	0.367730i	$0.880135\pi$
$600$	9.00000	+	12.0000i	0.367423	+	0.489898i
$601$		−	8.00000i		−	0.326327i	−0.986599	−	0.163163i	$-0.947830\pi$
$601$		−	8.00000i		−	0.326327i	0.986599	−	0.163163i	$-0.0521698\pi$
$602$		−	8.00000i		−	0.326056i
$603$			6.00000i			0.244339i
$604$	8.00000			0.325515
$605$	−7.00000	+	14.0000i	−0.284590	+	0.569181i
$606$		−	12.0000i		−	0.487467i
$607$	8.00000			0.324710			0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000			0.324710			0.162355	+	0.986732i	$0.448091\pi$
$608$			10.0000i			0.405554i
$609$	−4.00000	+	10.0000i	−0.162088	+	0.405220i
$610$	−24.0000	−	12.0000i	−0.971732	−	0.485866i
$611$		−	32.0000i		−	1.29458i
$612$	6.00000			0.242536
$613$			12.0000i			0.484675i	0.970192	+	0.242338i	$0.0779142\pi$
$613$			12.0000i			0.484675i	−0.970192	+	0.242338i	$0.922086\pi$
$614$	−12.0000			−0.484281
$615$		0			0
$616$	12.0000			0.483494
$617$	−6.00000			−0.241551			−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000			−0.241551			−0.120775	+	0.992680i	$0.538538\pi$
$618$			18.0000i			0.724066i
$619$		−	26.0000i		−	1.04503i	−0.852631	−	0.522514i	$-0.824994\pi$
$619$		−	26.0000i		−	1.04503i	0.852631	−	0.522514i	$-0.175006\pi$
$620$	4.00000	+	2.00000i	0.160644	+	0.0803219i
$621$			6.00000i			0.240772i
$622$			14.0000i			0.561349i
$623$	−32.0000			−1.28205
$624$			4.00000i			0.160128i
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$			8.00000i			0.319744i
$627$	4.00000			0.159745
$628$	2.00000			0.0798087
$629$	12.0000			0.478471
$630$	4.00000	+	2.00000i	0.159364	+	0.0796819i
$631$	8.00000			0.318475			0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000			0.318475			0.159237	+	0.987240i	$0.449096\pi$
$632$			6.00000i			0.238667i
$633$			2.00000i			0.0794929i
$634$	−6.00000			−0.238290
$635$	−8.00000	+	16.0000i	−0.317470	+	0.634941i
$636$		−	12.0000i		−	0.475831i
$637$			12.0000i			0.475457i
$638$	4.00000	−	10.0000i	0.158362	−	0.395904i
$639$	8.00000			0.316475
$640$	−3.00000	+	6.00000i	−0.118585	+	0.237171i
$641$		−	24.0000i		−	0.947943i	−0.880540	−	0.473972i	$-0.842820\pi$
$641$		−	24.0000i		−	0.947943i	0.880540	−	0.473972i	$-0.157180\pi$
$642$		−	10.0000i		−	0.394669i
$643$			6.00000i			0.236617i	0.992977	+	0.118308i	$0.0377472\pi$
$643$			6.00000i			0.236617i	−0.992977	+	0.118308i	$0.962253\pi$
$644$	−12.0000			−0.472866
$645$	4.00000	−	8.00000i	0.157500	−	0.315000i
$646$		−	12.0000i		−	0.472134i
$647$			10.0000i			0.393141i	0.980490	+	0.196570i	$0.0629804\pi$
$647$			10.0000i			0.393141i	−0.980490	+	0.196570i	$0.937020\pi$
$648$	3.00000			0.117851
$649$			8.00000i			0.314027i
$650$	−16.0000	+	12.0000i	−0.627572	+	0.470679i
$651$	4.00000			0.156772
$652$	20.0000			0.783260
$653$	−18.0000			−0.704394			−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000			−0.704394			−0.352197	+	0.935926i	$0.614565\pi$
$654$	−10.0000			−0.391031
$655$	−44.0000	−	22.0000i	−1.71922	−	0.859611i
$656$		0			0
$657$	−6.00000			−0.234082
$658$			16.0000i			0.623745i
$659$		−	26.0000i		−	1.01282i	−0.862294	−	0.506408i	$-0.830973\pi$
$659$		−	26.0000i		−	1.01282i	0.862294	−	0.506408i	$-0.169027\pi$
$660$	4.00000	+	2.00000i	0.155700	+	0.0778499i
$661$	22.0000			0.855701			0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000			0.855701			0.427850	+	0.903850i	$0.359271\pi$
$662$		−	22.0000i		−	0.855054i
$663$			24.0000i			0.932083i
$664$		−	6.00000i		−	0.232845i
$665$	−4.00000	+	8.00000i	−0.155113	+	0.310227i
$666$	2.00000			0.0774984
$667$	−12.0000	+	30.0000i	−0.464642	+	1.16160i
$668$			6.00000i			0.232147i
$669$			22.0000i			0.850569i
$670$	12.0000	+	6.00000i	0.463600	+	0.231800i
$671$	−24.0000			−0.926510
$672$			10.0000i			0.385758i
$673$			32.0000i			1.23351i	0.787155	+	0.616755i	$0.211553\pi$
$673$			32.0000i			1.23351i	−0.787155	+	0.616755i	$0.788447\pi$
$674$	−34.0000			−1.30963
$675$	3.00000	+	4.00000i	0.115470	+	0.153960i
$676$	3.00000			0.115385
$677$	6.00000			0.230599			0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000			0.230599			0.115299	+	0.993331i	$0.463217\pi$
$678$	2.00000			0.0768095
$679$		−	28.0000i		−	1.07454i
$680$	18.0000	−	36.0000i	0.690268	−	1.38054i
$681$			2.00000i			0.0766402i
$682$	−4.00000			−0.153168
$683$			6.00000i			0.229584i	0.993390	+	0.114792i	$0.0366201\pi$
$683$			6.00000i			0.229584i	−0.993390	+	0.114792i	$0.963380\pi$
$684$			2.00000i			0.0764719i
$685$	14.0000	−	28.0000i	0.534913	−	1.06983i
$686$		−	20.0000i		−	0.763604i
$687$			20.0000i			0.763048i
$688$	−4.00000			−0.152499
$689$	48.0000			1.82865
$690$	12.0000	+	6.00000i	0.456832	+	0.228416i
$691$	20.0000			0.760836			0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000			0.760836			0.380418	+	0.924815i	$0.375780\pi$
$692$		−	12.0000i		−	0.456172i
$693$	4.00000			0.151947
$694$			10.0000i			0.379595i
$695$	4.00000	−	8.00000i	0.151729	−	0.303457i
$696$	15.0000	+	6.00000i	0.568574	+	0.227429i
$697$		0			0
$698$	26.0000			0.984115
$699$		−	16.0000i		−	0.605176i
$700$	−8.00000	+	6.00000i	−0.302372	+	0.226779i
$701$	30.0000			1.13308			0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000			1.13308			0.566542	+	0.824033i	$0.308281\pi$
$702$			4.00000i			0.150970i
$703$			4.00000i			0.150863i
$704$		−	14.0000i		−	0.527645i
$705$	−8.00000	+	16.0000i	−0.301297	+	0.602595i
$706$		−	24.0000i		−	0.903252i
$707$	24.0000			0.902613
$708$	−4.00000			−0.150329
$709$	−50.0000			−1.87779			−0.938895	−	0.344204i	$-0.888149\pi$
$709$	−50.0000			−1.87779			−0.938895	+	0.344204i	$0.888149\pi$
$710$	8.00000	−	16.0000i	0.300235	−	0.600469i
$711$			2.00000i			0.0750059i
$712$			48.0000i			1.79888i
$713$	12.0000			0.449404
$714$		−	12.0000i		−	0.449089i
$715$	−8.00000	+	16.0000i	−0.299183	+	0.598366i
$716$	12.0000			0.448461
$717$	−16.0000			−0.597531
$718$		−	18.0000i		−	0.671754i
$719$	−48.0000			−1.79010			−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000			−1.79010			−0.895049	+	0.445968i	$0.852860\pi$
$720$	1.00000	−	2.00000i	0.0372678	−	0.0745356i
$721$	−36.0000			−1.34071
$722$	−15.0000			−0.558242
$723$	22.0000			0.818189
$724$	10.0000			0.371647
$725$	7.00000	+	26.0000i	0.259973	+	0.965616i
$726$	7.00000			0.259794
$727$	48.0000			1.78022			0.890111	−	0.455744i	$-0.150627\pi$
$727$	48.0000			1.78022			0.890111	+	0.455744i	$0.150627\pi$
$728$	−24.0000			−0.889499
$729$	1.00000			0.0370370
$730$	−6.00000	+	12.0000i	−0.222070	+	0.444140i
$731$	−24.0000			−0.887672
$732$		−	12.0000i		−	0.443533i
$733$	22.0000			0.812589			0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000			0.812589			0.406294	+	0.913742i	$0.366821\pi$
$734$	−24.0000			−0.885856
$735$	3.00000	−	6.00000i	0.110657	−	0.221313i
$736$			30.0000i			1.10581i
$737$	12.0000			0.442026
$738$		0			0
$739$		−	34.0000i		−	1.25071i	−0.780340	−	0.625355i	$-0.784954\pi$
$739$		−	34.0000i		−	1.25071i	0.780340	−	0.625355i	$-0.215046\pi$
$740$	−2.00000	+	4.00000i	−0.0735215	+	0.147043i
$741$	−8.00000			−0.293887
$742$	−24.0000			−0.881068
$743$	−48.0000			−1.76095			−0.880475	−	0.474093i	$-0.842776\pi$
$743$	−48.0000			−1.76095			−0.880475	+	0.474093i	$0.842776\pi$
$744$		−	6.00000i		−	0.219971i
$745$	10.0000	−	20.0000i	0.366372	−	0.732743i
$746$			4.00000i			0.146450i
$747$		−	2.00000i		−	0.0731762i
$748$		−	12.0000i		−	0.438763i
$749$	20.0000			0.730784
$750$	11.0000	−	2.00000i	0.401663	−	0.0730297i
$751$		−	46.0000i		−	1.67856i	−0.543696	−	0.839282i	$-0.682976\pi$
$751$		−	46.0000i		−	1.67856i	0.543696	−	0.839282i	$-0.317024\pi$
$752$	8.00000			0.291730
$753$		−	30.0000i		−	1.09326i
$754$	−8.00000	+	20.0000i	−0.291343	+	0.728357i
$755$	8.00000	−	16.0000i	0.291150	−	0.582300i
$756$			2.00000i			0.0727393i
$757$	−42.0000			−1.52652			−0.763258	−	0.646094i	$-0.776401\pi$
$757$	−42.0000			−1.52652			−0.763258	+	0.646094i	$0.776401\pi$
$758$		−	38.0000i		−	1.38022i
$759$	12.0000			0.435572
$760$	12.0000	+	6.00000i	0.435286	+	0.217643i
$761$	2.00000			0.0724999			0.0362500	−	0.999343i	$-0.488459\pi$
$761$	2.00000			0.0724999			0.0362500	+	0.999343i	$0.488459\pi$
$762$	8.00000			0.289809
$763$		−	20.0000i		−	0.724049i
$764$			6.00000i			0.217072i
$765$	6.00000	−	12.0000i	0.216930	−	0.433861i
$766$			30.0000i			1.08394i
$767$		−	16.0000i		−	0.577727i
$768$	17.0000			0.613435
$769$			32.0000i			1.15395i	0.816762	+	0.576975i	$0.195767\pi$
$769$			32.0000i			1.15395i	−0.816762	+	0.576975i	$0.804233\pi$
$770$	4.00000	−	8.00000i	0.144150	−	0.288300i
$771$		−	24.0000i		−	0.864339i
$772$	6.00000			0.215945
$773$	−18.0000			−0.647415			−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000			−0.647415			−0.323708	+	0.946157i	$0.604929\pi$
$774$	−4.00000			−0.143777
$775$	8.00000	−	6.00000i	0.287368	−	0.215526i
$776$	−42.0000			−1.50771
$777$			4.00000i			0.143499i
$778$			12.0000i			0.430221i
$779$		0			0
$780$	−8.00000	−	4.00000i	−0.286446	−	0.143223i
$781$		−	16.0000i		−	0.572525i
$782$		−	36.0000i		−	1.28736i
$783$	5.00000	+	2.00000i	0.178685	+	0.0714742i
$784$	−3.00000			−0.107143
$785$	2.00000	−	4.00000i	0.0713831	−	0.142766i
$786$			22.0000i			0.784714i
$787$			6.00000i			0.213877i	0.994266	+	0.106938i	$0.0341048\pi$
$787$			6.00000i			0.213877i	−0.994266	+	0.106938i	$0.965895\pi$
$788$		−	4.00000i		−	0.142494i
$789$	−16.0000			−0.569615
$790$	4.00000	+	2.00000i	0.142314	+	0.0711568i
$791$			4.00000i			0.142224i
$792$		−	6.00000i		−	0.213201i
$793$	48.0000			1.70453
$794$			28.0000i			0.993683i
$795$	−24.0000	−	12.0000i	−0.851192	−	0.425596i
$796$	−8.00000			−0.283552
$797$	−2.00000			−0.0708436			−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000			−0.0708436			−0.0354218	+	0.999372i	$0.511277\pi$
$798$	4.00000			0.141598
$799$	48.0000			1.69812
$800$	15.0000	+	20.0000i	0.530330	+	0.707107i
$801$			16.0000i			0.565332i
$802$	−18.0000			−0.635602
$803$			12.0000i			0.423471i
$804$			6.00000i			0.211604i
$805$	−12.0000	+	24.0000i	−0.422944	+	0.845889i
$806$	8.00000			0.281788
$807$			4.00000i			0.140807i
$808$		−	36.0000i		−	1.26648i
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$	1.00000	−	2.00000i	0.0351364	−	0.0702728i
$811$	−20.0000			−0.702295			−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000			−0.702295			−0.351147	+	0.936320i	$0.614208\pi$
$812$	−4.00000	+	10.0000i	−0.140372	+	0.350931i
$813$		−	2.00000i		−	0.0701431i
$814$		−	4.00000i		−	0.140200i
$815$	20.0000	−	40.0000i	0.700569	−	1.40114i
$816$	−6.00000			−0.210042
$817$		−	8.00000i		−	0.279885i
$818$		−	8.00000i		−	0.279713i
$819$	−8.00000			−0.279543
$820$		0			0
$821$	−10.0000			−0.349002			−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000			−0.349002			−0.174501	+	0.984657i	$0.555831\pi$
$822$	−14.0000			−0.488306
$823$	16.0000			0.557725			0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000			0.557725			0.278862	+	0.960331i	$0.410043\pi$
$824$			54.0000i			1.88118i
$825$	8.00000	−	6.00000i	0.278524	−	0.208893i
$826$			8.00000i			0.278356i
$827$	36.0000			1.25184			0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000			1.25184			0.625921	+	0.779886i	$0.284723\pi$
$828$			6.00000i			0.208514i
$829$			4.00000i			0.138926i	0.997585	+	0.0694629i	$0.0221285\pi$
$829$			4.00000i			0.138926i	−0.997585	+	0.0694629i	$0.977871\pi$
$830$	−4.00000	−	2.00000i	−0.138842	−	0.0694210i
$831$			4.00000i			0.138758i
$832$			28.0000i			0.970725i
$833$	−18.0000			−0.623663
$834$	−4.00000			−0.138509
$835$	12.0000	+	6.00000i	0.415277	+	0.207639i
$836$	4.00000			0.138343
$837$		−	2.00000i		−	0.0691301i
$838$	12.0000			0.414533
$839$		−	14.0000i		−	0.483334i	−0.970359	−	0.241667i	$-0.922306\pi$
$839$		−	14.0000i		−	0.483334i	0.970359	−	0.241667i	$-0.0776941\pi$
$840$	12.0000	+	6.00000i	0.414039	+	0.207020i
$841$	21.0000	+	20.0000i	0.724138	+	0.689655i
$842$		−	28.0000i		−	0.964944i
$843$	30.0000			1.03325
$844$			2.00000i			0.0688428i
$845$	3.00000	−	6.00000i	0.103203	−	0.206406i
$846$	8.00000			0.275046
$847$			14.0000i			0.481046i
$848$			12.0000i			0.412082i
$849$			2.00000i			0.0686398i
$850$	−18.0000	−	24.0000i	−0.617395	−	0.823193i
$851$			12.0000i			0.411355i
$852$	8.00000			0.274075
$853$	−18.0000			−0.616308			−0.308154	−	0.951336i	$-0.599711\pi$
$853$	−18.0000			−0.616308			−0.308154	+	0.951336i	$0.599711\pi$
$854$	−24.0000			−0.821263
$855$	4.00000	+	2.00000i	0.136797	+	0.0683986i
$856$		−	30.0000i		−	1.02538i
$857$		−	16.0000i		−	0.546550i	−0.961936	−	0.273275i	$-0.911893\pi$
$857$		−	16.0000i		−	0.546550i	0.961936	−	0.273275i	$-0.0881068\pi$
$858$	8.00000			0.273115
$859$			38.0000i			1.29654i	0.761409	+	0.648272i	$0.224508\pi$
$859$			38.0000i			1.29654i	−0.761409	+	0.648272i	$0.775492\pi$
$860$	4.00000	−	8.00000i	0.136399	−	0.272798i
$861$		0			0
$862$		0			0
$863$			2.00000i			0.0680808i	0.999420	+	0.0340404i	$0.0108375\pi$
$863$			2.00000i			0.0680808i	−0.999420	+	0.0340404i	$0.989163\pi$
$864$	5.00000			0.170103
$865$	−24.0000	−	12.0000i	−0.816024	−	0.408012i
$866$	22.0000			0.747590
$867$	−19.0000			−0.645274
$868$	4.00000			0.135769
$869$	4.00000			0.135691
$870$	9.00000	−	8.00000i	0.305129	−	0.271225i
$871$	−24.0000			−0.813209
$872$	−30.0000			−1.01593
$873$	−14.0000			−0.473828
$874$	12.0000			0.405906
$875$	4.00000	+	22.0000i	0.135225	+	0.743736i
$876$	−6.00000			−0.202721
$877$			20.0000i			0.675352i	0.941262	+	0.337676i	$0.109641\pi$
$877$			20.0000i			0.675352i	−0.941262	+	0.337676i	$0.890359\pi$
$878$	−24.0000			−0.809961
$879$	26.0000			0.876958
$880$	−4.00000	−	2.00000i	−0.134840	−	0.0674200i
$881$			24.0000i			0.808581i	0.914631	+	0.404290i	$0.132481\pi$
$881$			24.0000i			0.808581i	−0.914631	+	0.404290i	$0.867519\pi$
$882$	−3.00000			−0.101015
$883$		−	26.0000i		−	0.874970i	−0.899226	−	0.437485i	$-0.855869\pi$
$883$		−	26.0000i		−	0.874970i	0.899226	−	0.437485i	$-0.144131\pi$
$884$			24.0000i			0.807207i
$885$	−4.00000	+	8.00000i	−0.134459	+	0.268917i
$886$	28.0000			0.940678
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$	6.00000			0.201347
$889$			16.0000i			0.536623i
$890$	32.0000	+	16.0000i	1.07264	+	0.536321i
$891$		−	2.00000i		−	0.0670025i
$892$			22.0000i			0.736614i
$893$			16.0000i			0.535420i
$894$	−10.0000			−0.334450
$895$	12.0000	−	24.0000i	0.401116	−	0.802232i
$896$			6.00000i			0.200446i
$897$	−24.0000			−0.801337
$898$		−	24.0000i		−	0.800890i
$899$	4.00000	−	10.0000i	0.133407	−	0.333519i
$900$	3.00000	+	4.00000i	0.100000	+	0.133333i
$901$			72.0000i			2.39867i
$902$		0			0
$903$		−	8.00000i		−	0.266223i
$904$	6.00000			0.199557
$905$	10.0000	−	20.0000i	0.332411	−	0.664822i
$906$	−8.00000			−0.265782
$907$	4.00000			0.132818			0.0664089	−	0.997792i	$-0.478846\pi$
$907$	4.00000			0.132818			0.0664089	+	0.997792i	$0.478846\pi$
$908$			2.00000i			0.0663723i
$909$		−	12.0000i		−	0.398015i
$910$	−8.00000	+	16.0000i	−0.265197	+	0.530395i
$911$			42.0000i			1.39152i	0.718273	+	0.695761i	$0.244933\pi$
$911$			42.0000i			1.39152i	−0.718273	+	0.695761i	$0.755067\pi$
$912$		−	2.00000i		−	0.0662266i
$913$	−4.00000			−0.132381
$914$			8.00000i			0.264616i
$915$	−24.0000	−	12.0000i	−0.793416	−	0.396708i
$916$			20.0000i			0.660819i
$917$	−44.0000			−1.45301
$918$	−6.00000			−0.198030
$919$	−24.0000			−0.791687			−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000			−0.791687			−0.395843	+	0.918318i	$0.629548\pi$
$920$	36.0000	+	18.0000i	1.18688	+	0.593442i
$921$	−12.0000			−0.395413
$922$		−	12.0000i		−	0.395199i
$923$			32.0000i			1.05329i
$924$	4.00000			0.131590
$925$	6.00000	+	8.00000i	0.197279	+	0.263038i
$926$		−	26.0000i		−	0.854413i
$927$			18.0000i			0.591198i
$928$	25.0000	+	10.0000i	0.820665	+	0.328266i
$929$	34.0000			1.11550			0.557752	−	0.830008i	$-0.311664\pi$
$929$	34.0000			1.11550			0.557752	+	0.830008i	$0.311664\pi$
$930$	−4.00000	−	2.00000i	−0.131165	−	0.0655826i
$931$		−	6.00000i		−	0.196642i
$932$		−	16.0000i		−	0.524097i
$933$			14.0000i			0.458339i
$934$	−12.0000			−0.392652
$935$	−24.0000	−	12.0000i	−0.784884	−	0.392442i
$936$			12.0000i			0.392232i
$937$		−	56.0000i		−	1.82944i	−0.404088	−	0.914720i	$-0.632411\pi$
$937$		−	56.0000i		−	1.82944i	0.404088	−	0.914720i	$-0.367589\pi$
$938$	12.0000			0.391814
$939$			8.00000i			0.261070i
$940$	−8.00000	+	16.0000i	−0.260931	+	0.521862i
$941$	−18.0000			−0.586783			−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000			−0.586783			−0.293392	+	0.955992i	$0.594784\pi$
$942$	−2.00000			−0.0651635
$943$		0			0
$944$	4.00000			0.130189
$945$	4.00000	+	2.00000i	0.130120	+	0.0650600i
$946$			8.00000i			0.260102i
$947$	12.0000			0.389948			0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000			0.389948			0.194974	+	0.980808i	$0.437538\pi$
$948$			2.00000i			0.0649570i
$949$		−	24.0000i		−	0.779073i
$950$	8.00000	−	6.00000i	0.259554	−	0.194666i
$951$	−6.00000			−0.194563
$952$		−	36.0000i		−	1.16677i
$953$			16.0000i			0.518291i	0.965838	+	0.259145i	$0.0834409\pi$
$953$			16.0000i			0.518291i	−0.965838	+	0.259145i	$0.916559\pi$
$954$			12.0000i			0.388514i
$955$	12.0000	+	6.00000i	0.388311	+	0.194155i
$956$	−16.0000			−0.517477
$957$	4.00000	−	10.0000i	0.129302	−	0.323254i
$958$		−	42.0000i		−	1.35696i
$959$		−	28.0000i		−	0.904167i
$960$	7.00000	−	14.0000i	0.225924	−	0.451848i
$961$	27.0000			0.870968
$962$			8.00000i			0.257930i
$963$		−	10.0000i		−	0.322245i
$964$	22.0000			0.708572
$965$	6.00000	−	12.0000i	0.193147	−	0.386294i
$966$	12.0000			0.386094
$967$	−32.0000			−1.02905			−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000			−1.02905			−0.514525	+	0.857475i	$0.672032\pi$
$968$	21.0000			0.674966
$969$		−	12.0000i		−	0.385496i
$970$	−14.0000	+	28.0000i	−0.449513	+	0.899026i
$971$		−	18.0000i		−	0.577647i	−0.957382	−	0.288824i	$-0.906736\pi$
$971$		−	18.0000i		−	0.577647i	0.957382	−	0.288824i	$-0.0932642\pi$
$972$	1.00000			0.0320750
$973$		−	8.00000i		−	0.256468i
$974$			14.0000i			0.448589i
$975$	−16.0000	+	12.0000i	−0.512410	+	0.384308i
$976$			12.0000i			0.384111i
$977$			56.0000i			1.79160i	0.444459	+	0.895799i	$0.353396\pi$
$977$			56.0000i			1.79160i	−0.444459	+	0.895799i	$0.646604\pi$
$978$	−20.0000			−0.639529
$979$	32.0000			1.02272
$980$	3.00000	−	6.00000i	0.0958315	−	0.191663i
$981$	−10.0000			−0.319275
$982$			2.00000i			0.0638226i
$983$	48.0000			1.53096			0.765481	−	0.643458i	$-0.222501\pi$
$983$	48.0000			1.53096			0.765481	+	0.643458i	$0.222501\pi$
$984$		0			0
$985$	−8.00000	−	4.00000i	−0.254901	−	0.127451i
$986$	−30.0000	−	12.0000i	−0.955395	−	0.382158i
$987$			16.0000i			0.509286i
$988$	−8.00000			−0.254514
$989$		−	24.0000i		−	0.763156i
$990$	−4.00000	−	2.00000i	−0.127128	−	0.0635642i
$991$	16.0000			0.508257			0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000			0.508257			0.254128	+	0.967170i	$0.418211\pi$
$992$		−	10.0000i		−	0.317500i
$993$		−	22.0000i		−	0.698149i
$994$		−	16.0000i		−	0.507489i
$995$	−8.00000	+	16.0000i	−0.253617	+	0.507234i
$996$		−	2.00000i		−	0.0633724i
$997$	−26.0000			−0.823428			−0.411714	−	0.911313i	$-0.635070\pi$
$997$	−26.0000			−0.823428			−0.411714	+	0.911313i	$0.635070\pi$
$998$	12.0000			0.379853
$999$	2.00000			0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	435.2.f.b.289.2	yes	2
3.2	odd	2		1305.2.f.c.289.1		2
5.2	odd	4		2175.2.d.c.376.1		2
5.3	odd	4		2175.2.d.d.376.2		2
5.4	even	2		435.2.f.c.289.1	yes	2
15.14	odd	2		1305.2.f.b.289.2		2
29.28	even	2		435.2.f.c.289.2	yes	2
87.86	odd	2		1305.2.f.b.289.1		2
145.28	odd	4		2175.2.d.d.376.1		2
145.57	odd	4		2175.2.d.c.376.2		2
145.144	even	2	inner	435.2.f.b.289.1	✓	2
435.434	odd	2		1305.2.f.c.289.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.2.f.b.289.1	✓	2	145.144	even	2	inner
435.2.f.b.289.2	yes	2	1.1	even	1	trivial
435.2.f.c.289.1	yes	2	5.4	even	2
435.2.f.c.289.2	yes	2	29.28	even	2
1305.2.f.b.289.1		2	87.86	odd	2
1305.2.f.b.289.2		2	15.14	odd	2
1305.2.f.c.289.1		2	3.2	odd	2
1305.2.f.c.289.2		2	435.434	odd	2
2175.2.d.c.376.1		2	5.2	odd	4
2175.2.d.c.376.2		2	145.57	odd	4
2175.2.d.d.376.1		2	145.28	odd	4
2175.2.d.d.376.2		2	5.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 435 = 3 \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	435.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +(-1.00000 + 2.00000i) q^{5} +1.00000 q^{6} +2.00000i q^{7} +3.00000 q^{8} +1.00000 q^{9} +(1.00000 - 2.00000i) q^{10} -2.00000i q^{11} +1.00000 q^{12} +4.00000i q^{13} -2.00000i q^{14} +(1.00000 - 2.00000i) q^{15} -1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} -2.00000i q^{19} +(1.00000 - 2.00000i) q^{20} -2.00000i q^{21} +2.00000i q^{22} -6.00000i q^{23} -3.00000 q^{24} +(-3.00000 - 4.00000i) q^{25} -4.00000i q^{26} -1.00000 q^{27} -2.00000i q^{28} +(-5.00000 - 2.00000i) q^{29} +(-1.00000 + 2.00000i) q^{30} +2.00000i q^{31} -5.00000 q^{32} +2.00000i q^{33} +6.00000 q^{34} +(-4.00000 - 2.00000i) q^{35} -1.00000 q^{36} -2.00000 q^{37} +2.00000i q^{38} -4.00000i q^{39} +(-3.00000 + 6.00000i) q^{40} +2.00000i q^{42} +4.00000 q^{43} +2.00000i q^{44} +(-1.00000 + 2.00000i) q^{45} +6.00000i q^{46} -8.00000 q^{47} +1.00000 q^{48} +3.00000 q^{49} +(3.00000 + 4.00000i) q^{50} +6.00000 q^{51} -4.00000i q^{52} -12.0000i q^{53} +1.00000 q^{54} +(4.00000 + 2.00000i) q^{55} +6.00000i q^{56} +2.00000i q^{57} +(5.00000 + 2.00000i) q^{58} -4.00000 q^{59} +(-1.00000 + 2.00000i) q^{60} -12.0000i q^{61} -2.00000i q^{62} +2.00000i q^{63} +7.00000 q^{64} +(-8.00000 - 4.00000i) q^{65} -2.00000i q^{66} +6.00000i q^{67} +6.00000 q^{68} +6.00000i q^{69} +(4.00000 + 2.00000i) q^{70} +8.00000 q^{71} +3.00000 q^{72} -6.00000 q^{73} +2.00000 q^{74} +(3.00000 + 4.00000i) q^{75} +2.00000i q^{76} +4.00000 q^{77} +4.00000i q^{78} +2.00000i q^{79} +(1.00000 - 2.00000i) q^{80} +1.00000 q^{81} -2.00000i q^{83} +2.00000i q^{84} +(6.00000 - 12.0000i) q^{85} -4.00000 q^{86} +(5.00000 + 2.00000i) q^{87} -6.00000i q^{88} +16.0000i q^{89} +(1.00000 - 2.00000i) q^{90} -8.00000 q^{91} +6.00000i q^{92} -2.00000i q^{93} +8.00000 q^{94} +(4.00000 + 2.00000i) q^{95} +5.00000 q^{96} -14.0000 q^{97} -3.00000 q^{98} -2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} - 2 q^{4} - 2 q^{5} + 2 q^{6} + 6 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{12} + 2 q^{15} - 2 q^{16} - 12 q^{17} - 2 q^{18} + 2 q^{20} - 6 q^{24} - 6 q^{25} - 2 q^{27} - 10 q^{29} - 2 q^{30}+ \cdots - 6 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 - 2 * q^4 - 2 * q^5 + 2 * q^6 + 6 * q^8 + 2 * q^9 + 2 * q^10 + 2 * q^12 + 2 * q^15 - 2 * q^16 - 12 * q^17 - 2 * q^18 + 2 * q^20 - 6 * q^24 - 6 * q^25 - 2 * q^27 - 10 * q^29 - 2 * q^30 - 10 * q^32 + 12 * q^34 - 8 * q^35 - 2 * q^36 - 4 * q^37 - 6 * q^40 + 8 * q^43 - 2 * q^45 - 16 * q^47 + 2 * q^48 + 6 * q^49 + 6 * q^50 + 12 * q^51 + 2 * q^54 + 8 * q^55 + 10 * q^58 - 8 * q^59 - 2 * q^60 + 14 * q^64 - 16 * q^65 + 12 * q^68 + 8 * q^70 + 16 * q^71 + 6 * q^72 - 12 * q^73 + 4 * q^74 + 6 * q^75 + 8 * q^77 + 2 * q^80 + 2 * q^81 + 12 * q^85 - 8 * q^86 + 10 * q^87 + 2 * q^90 - 16 * q^91 + 16 * q^94 + 8 * q^95 + 10 * q^96 - 28 * q^97 - 6 * q^98

Introduction

L-functions

Modular forms

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Database

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