LMFDB - Embedded newform 2730.2.l.k.1639.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2730,2,Mod(1639,2730)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2730, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))

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

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

chi := DirichletCharacter("2730.1639");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2730 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2730.l (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$21.7991597518$
Analytic rank:	$0$
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			1639.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	2730.1639
Dual form			2730.2.l.k.1639.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +(1.00000 - 2.00000i) q^{5} +1.00000 q^{6} -1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +(1.00000 - 2.00000i) q^{5} +1.00000 q^{6} -1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +(2.00000 + 1.00000i) q^{10} +2.00000 q^{11} +1.00000i q^{12} +1.00000i q^{13} +1.00000 q^{14} +(-2.00000 - 1.00000i) q^{15} +1.00000 q^{16} -6.00000i q^{17} -1.00000i q^{18} -4.00000 q^{19} +(-1.00000 + 2.00000i) q^{20} -1.00000 q^{21} +2.00000i q^{22} -2.00000i q^{23} -1.00000 q^{24} +(-3.00000 - 4.00000i) q^{25} -1.00000 q^{26} +1.00000i q^{27} +1.00000i q^{28} -6.00000 q^{29} +(1.00000 - 2.00000i) q^{30} +8.00000 q^{31} +1.00000i q^{32} -2.00000i q^{33} +6.00000 q^{34} +(-2.00000 - 1.00000i) q^{35} +1.00000 q^{36} -6.00000i q^{37} -4.00000i q^{38} +1.00000 q^{39} +(-2.00000 - 1.00000i) q^{40} +2.00000 q^{41} -1.00000i q^{42} +6.00000i q^{43} -2.00000 q^{44} +(-1.00000 + 2.00000i) q^{45} +2.00000 q^{46} -2.00000i q^{47} -1.00000i q^{48} -1.00000 q^{49} +(4.00000 - 3.00000i) q^{50} -6.00000 q^{51} -1.00000i q^{52} +4.00000i q^{53} -1.00000 q^{54} +(2.00000 - 4.00000i) q^{55} -1.00000 q^{56} +4.00000i q^{57} -6.00000i q^{58} -12.0000 q^{59} +(2.00000 + 1.00000i) q^{60} +8.00000i q^{62} +1.00000i q^{63} -1.00000 q^{64} +(2.00000 + 1.00000i) q^{65} +2.00000 q^{66} +8.00000i q^{67} +6.00000i q^{68} -2.00000 q^{69} +(1.00000 - 2.00000i) q^{70} +4.00000 q^{71} +1.00000i q^{72} +4.00000i q^{73} +6.00000 q^{74} +(-4.00000 + 3.00000i) q^{75} +4.00000 q^{76} -2.00000i q^{77} +1.00000i q^{78} -8.00000 q^{79} +(1.00000 - 2.00000i) q^{80} +1.00000 q^{81} +2.00000i q^{82} -12.0000i q^{83} +1.00000 q^{84} +(-12.0000 - 6.00000i) q^{85} -6.00000 q^{86} +6.00000i q^{87} -2.00000i q^{88} +10.0000 q^{89} +(-2.00000 - 1.00000i) q^{90} +1.00000 q^{91} +2.00000i q^{92} -8.00000i q^{93} +2.00000 q^{94} +(-4.00000 + 8.00000i) q^{95} +1.00000 q^{96} -1.00000i q^{98} -2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 + 2 * q^5 + 2 * q^6 - 2 * q^9	$ 2 q - 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{9} + 4 q^{10} + 4 q^{11} + 2 q^{14} - 4 q^{15} + 2 q^{16} - 8 q^{19} - 2 q^{20} - 2 q^{21} - 2 q^{24} - 6 q^{25} - 2 q^{26} - 12 q^{29} + 2 q^{30} + 16 q^{31} + 12 q^{34} - 4 q^{35} + 2 q^{36} + 2 q^{39} - 4 q^{40} + 4 q^{41} - 4 q^{44} - 2 q^{45} + 4 q^{46} - 2 q^{49} + 8 q^{50} - 12 q^{51} - 2 q^{54} + 4 q^{55} - 2 q^{56} - 24 q^{59} + 4 q^{60} - 2 q^{64} + 4 q^{65} + 4 q^{66} - 4 q^{69} + 2 q^{70} + 8 q^{71} + 12 q^{74} - 8 q^{75} + 8 q^{76} - 16 q^{79} + 2 q^{80} + 2 q^{81} + 2 q^{84} - 24 q^{85} - 12 q^{86} + 20 q^{89} - 4 q^{90} + 2 q^{91} + 4 q^{94} - 8 q^{95} + 2 q^{96} - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^5 + 2 * q^6 - 2 * q^9 + 4 * q^10 + 4 * q^11 + 2 * q^14 - 4 * q^15 + 2 * q^16 - 8 * q^19 - 2 * q^20 - 2 * q^21 - 2 * q^24 - 6 * q^25 - 2 * q^26 - 12 * q^29 + 2 * q^30 + 16 * q^31 + 12 * q^34 - 4 * q^35 + 2 * q^36 + 2 * q^39 - 4 * q^40 + 4 * q^41 - 4 * q^44 - 2 * q^45 + 4 * q^46 - 2 * q^49 + 8 * q^50 - 12 * q^51 - 2 * q^54 + 4 * q^55 - 2 * q^56 - 24 * q^59 + 4 * q^60 - 2 * q^64 + 4 * q^65 + 4 * q^66 - 4 * q^69 + 2 * q^70 + 8 * q^71 + 12 * q^74 - 8 * q^75 + 8 * q^76 - 16 * q^79 + 2 * q^80 + 2 * q^81 + 2 * q^84 - 24 * q^85 - 12 * q^86 + 20 * q^89 - 4 * q^90 + 2 * q^91 + 4 * q^94 - 8 * q^95 + 2 * q^96 - 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2730\mathbb{Z}\right)^\times$.

$n$	$547$	$911$	$1471$	$2341$
$\chi(n)$	$-1$	$1$	$1$	$1$

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 $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$	−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
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		−	1.00000i		−	0.353553i
$9$	−1.00000			−0.333333
$10$	2.00000	+	1.00000i	0.632456	+	0.316228i
$11$	2.00000			0.603023			0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000			0.603023			0.301511	+	0.953463i	$0.402509\pi$
$12$			1.00000i			0.288675i
$13$			1.00000i			0.277350i
$13$			1.00000i			0.277350i
$14$	1.00000			0.267261
$15$	−2.00000	−	1.00000i	−0.516398	−	0.258199i
$16$	1.00000			0.250000
$17$		−	6.00000i		−	1.45521i	−0.685994	−	0.727607i	$-0.740633\pi$
$17$		−	6.00000i		−	1.45521i	0.685994	−	0.727607i	$-0.259367\pi$
$18$		−	1.00000i		−	0.235702i
$19$	−4.00000			−0.917663			−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000			−0.917663			−0.458831	+	0.888523i	$0.651732\pi$
$20$	−1.00000	+	2.00000i	−0.223607	+	0.447214i
$21$	−1.00000			−0.218218
$22$			2.00000i			0.426401i
$23$		−	2.00000i		−	0.417029i	−0.978019	−	0.208514i	$-0.933137\pi$
$23$		−	2.00000i		−	0.417029i	0.978019	−	0.208514i	$-0.0668628\pi$
$24$	−1.00000			−0.204124
$25$	−3.00000	−	4.00000i	−0.600000	−	0.800000i
$26$	−1.00000			−0.196116
$27$			1.00000i			0.192450i
$28$			1.00000i			0.188982i
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$	1.00000	−	2.00000i	0.182574	−	0.365148i
$31$	8.00000			1.43684			0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000			1.43684			0.718421	+	0.695608i	$0.244865\pi$
$32$			1.00000i			0.176777i
$33$		−	2.00000i		−	0.348155i
$34$	6.00000			1.02899
$35$	−2.00000	−	1.00000i	−0.338062	−	0.169031i
$36$	1.00000			0.166667
$37$		−	6.00000i		−	0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$		−	6.00000i		−	0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$		−	4.00000i		−	0.648886i
$39$	1.00000			0.160128
$40$	−2.00000	−	1.00000i	−0.316228	−	0.158114i
$41$	2.00000			0.312348			0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000			0.312348			0.156174	+	0.987730i	$0.450084\pi$
$42$		−	1.00000i		−	0.154303i
$43$			6.00000i			0.914991i	0.889212	+	0.457496i	$0.151253\pi$
$43$			6.00000i			0.914991i	−0.889212	+	0.457496i	$0.848747\pi$
$44$	−2.00000			−0.301511
$45$	−1.00000	+	2.00000i	−0.149071	+	0.298142i
$46$	2.00000			0.294884
$47$		−	2.00000i		−	0.291730i	−0.989305	−	0.145865i	$-0.953403\pi$
$47$		−	2.00000i		−	0.291730i	0.989305	−	0.145865i	$-0.0465965\pi$
$48$		−	1.00000i		−	0.144338i
$49$	−1.00000			−0.142857
$50$	4.00000	−	3.00000i	0.565685	−	0.424264i
$51$	−6.00000			−0.840168
$52$		−	1.00000i		−	0.138675i
$53$			4.00000i			0.549442i	0.961524	+	0.274721i	$0.0885855\pi$
$53$			4.00000i			0.549442i	−0.961524	+	0.274721i	$0.911414\pi$
$54$	−1.00000			−0.136083
$55$	2.00000	−	4.00000i	0.269680	−	0.539360i
$56$	−1.00000			−0.133631
$57$			4.00000i			0.529813i
$58$		−	6.00000i		−	0.787839i
$59$	−12.0000			−1.56227			−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000			−1.56227			−0.781133	+	0.624364i	$0.785358\pi$
$60$	2.00000	+	1.00000i	0.258199	+	0.129099i
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$			8.00000i			1.01600i
$63$			1.00000i			0.125988i
$64$	−1.00000			−0.125000
$65$	2.00000	+	1.00000i	0.248069	+	0.124035i
$66$	2.00000			0.246183
$67$			8.00000i			0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$			8.00000i			0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$			6.00000i			0.727607i
$69$	−2.00000			−0.240772
$70$	1.00000	−	2.00000i	0.119523	−	0.239046i
$71$	4.00000			0.474713			0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000			0.474713			0.237356	+	0.971423i	$0.423719\pi$
$72$			1.00000i			0.117851i
$73$			4.00000i			0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$			4.00000i			0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$	6.00000			0.697486
$75$	−4.00000	+	3.00000i	−0.461880	+	0.346410i
$76$	4.00000			0.458831
$77$		−	2.00000i		−	0.227921i
$78$			1.00000i			0.113228i
$79$	−8.00000			−0.900070			−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000			−0.900070			−0.450035	+	0.893011i	$0.648589\pi$
$80$	1.00000	−	2.00000i	0.111803	−	0.223607i
$81$	1.00000			0.111111
$82$			2.00000i			0.220863i
$83$		−	12.0000i		−	1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$		−	12.0000i		−	1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	1.00000			0.109109
$85$	−12.0000	−	6.00000i	−1.30158	−	0.650791i
$86$	−6.00000			−0.646997
$87$			6.00000i			0.643268i
$88$		−	2.00000i		−	0.213201i
$89$	10.0000			1.06000			0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000			1.06000			0.529999	+	0.847998i	$0.322192\pi$
$90$	−2.00000	−	1.00000i	−0.210819	−	0.105409i
$91$	1.00000			0.104828
$92$			2.00000i			0.208514i
$93$		−	8.00000i		−	0.829561i
$94$	2.00000			0.206284
$95$	−4.00000	+	8.00000i	−0.410391	+	0.820783i
$96$	1.00000			0.102062
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		−	1.00000i		−	0.101015i
$99$	−2.00000			−0.201008
$100$	3.00000	+	4.00000i	0.300000	+	0.400000i
$101$	−10.0000			−0.995037			−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000			−0.995037			−0.497519	+	0.867453i	$0.665755\pi$
$102$		−	6.00000i		−	0.594089i
$103$		−	8.00000i		−	0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$		−	8.00000i		−	0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$	1.00000			0.0980581
$105$	−1.00000	+	2.00000i	−0.0975900	+	0.195180i
$106$	−4.00000			−0.388514
$107$		−	8.00000i		−	0.773389i	−0.922208	−	0.386695i	$-0.873617\pi$
$107$		−	8.00000i		−	0.773389i	0.922208	−	0.386695i	$-0.126383\pi$
$108$		−	1.00000i		−	0.0962250i
$109$	−16.0000			−1.53252			−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000			−1.53252			−0.766261	+	0.642529i	$0.777885\pi$
$110$	4.00000	+	2.00000i	0.381385	+	0.190693i
$111$	−6.00000			−0.569495
$112$		−	1.00000i		−	0.0944911i
$113$		−	6.00000i		−	0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$		−	6.00000i		−	0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	−4.00000			−0.374634
$115$	−4.00000	−	2.00000i	−0.373002	−	0.186501i
$116$	6.00000			0.557086
$117$		−	1.00000i		−	0.0924500i
$118$		−	12.0000i		−	1.10469i
$119$	−6.00000			−0.550019
$120$	−1.00000	+	2.00000i	−0.0912871	+	0.182574i
$121$	−7.00000			−0.636364
$122$		0			0
$123$		−	2.00000i		−	0.180334i
$124$	−8.00000			−0.718421
$125$	−11.0000	+	2.00000i	−0.983870	+	0.178885i
$126$	−1.00000			−0.0890871
$127$		−	16.0000i		−	1.41977i	−0.704317	−	0.709885i	$-0.748747\pi$
$127$		−	16.0000i		−	1.41977i	0.704317	−	0.709885i	$-0.251253\pi$
$128$		−	1.00000i		−	0.0883883i
$129$	6.00000			0.528271
$130$	−1.00000	+	2.00000i	−0.0877058	+	0.175412i
$131$	18.0000			1.57267			0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000			1.57267			0.786334	+	0.617802i	$0.211977\pi$
$132$			2.00000i			0.174078i
$133$			4.00000i			0.346844i
$134$	−8.00000			−0.691095
$135$	2.00000	+	1.00000i	0.172133	+	0.0860663i
$136$	−6.00000			−0.514496
$137$		−	6.00000i		−	0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$		−	6.00000i		−	0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$		−	2.00000i		−	0.170251i
$139$	−16.0000			−1.35710			−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000			−1.35710			−0.678551	+	0.734553i	$0.737392\pi$
$140$	2.00000	+	1.00000i	0.169031	+	0.0845154i
$141$	−2.00000			−0.168430
$142$			4.00000i			0.335673i
$143$			2.00000i			0.167248i
$144$	−1.00000			−0.0833333
$145$	−6.00000	+	12.0000i	−0.498273	+	0.996546i
$146$	−4.00000			−0.331042
$147$			1.00000i			0.0824786i
$148$			6.00000i			0.493197i
$149$	−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
$151$	2.00000			0.162758			0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000			0.162758			0.0813788	+	0.996683i	$0.474068\pi$
$152$			4.00000i			0.324443i
$153$			6.00000i			0.485071i
$154$	2.00000			0.161165
$155$	8.00000	−	16.0000i	0.642575	−	1.28515i
$156$	−1.00000			−0.0800641
$157$		−	2.00000i		−	0.159617i	−0.996810	−	0.0798087i	$-0.974569\pi$
$157$		−	2.00000i		−	0.159617i	0.996810	−	0.0798087i	$-0.0254309\pi$
$158$		−	8.00000i		−	0.636446i
$159$	4.00000			0.317221
$160$	2.00000	+	1.00000i	0.158114	+	0.0790569i
$161$	−2.00000			−0.157622
$162$			1.00000i			0.0785674i
$163$		−	16.0000i		−	1.25322i	−0.779334	−	0.626608i	$-0.784443\pi$
$163$		−	16.0000i		−	1.25322i	0.779334	−	0.626608i	$-0.215557\pi$
$164$	−2.00000			−0.156174
$165$	−4.00000	−	2.00000i	−0.311400	−	0.155700i
$166$	12.0000			0.931381
$167$			18.0000i			1.39288i	0.717614	+	0.696441i	$0.245234\pi$
$167$			18.0000i			1.39288i	−0.717614	+	0.696441i	$0.754766\pi$
$168$			1.00000i			0.0771517i
$169$	−1.00000			−0.0769231
$170$	6.00000	−	12.0000i	0.460179	−	0.920358i
$171$	4.00000			0.305888
$172$		−	6.00000i		−	0.457496i
$173$			2.00000i			0.152057i	0.997106	+	0.0760286i	$0.0242240\pi$
$173$			2.00000i			0.152057i	−0.997106	+	0.0760286i	$0.975776\pi$
$174$	−6.00000			−0.454859
$175$	−4.00000	+	3.00000i	−0.302372	+	0.226779i
$176$	2.00000			0.150756
$177$			12.0000i			0.901975i
$178$			10.0000i			0.749532i
$179$	20.0000			1.49487			0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000			1.49487			0.747435	+	0.664335i	$0.231285\pi$
$180$	1.00000	−	2.00000i	0.0745356	−	0.149071i
$181$	−8.00000			−0.594635			−0.297318	−	0.954779i	$-0.596092\pi$
$181$	−8.00000			−0.594635			−0.297318	+	0.954779i	$0.596092\pi$
$182$			1.00000i			0.0741249i
$183$		0			0
$184$	−2.00000			−0.147442
$185$	−12.0000	−	6.00000i	−0.882258	−	0.441129i
$186$	8.00000			0.586588
$187$		−	12.0000i		−	0.877527i
$188$			2.00000i			0.145865i
$189$	1.00000			0.0727393
$190$	−8.00000	−	4.00000i	−0.580381	−	0.290191i
$191$	−8.00000			−0.578860			−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000			−0.578860			−0.289430	+	0.957199i	$0.593466\pi$
$192$			1.00000i			0.0721688i
$193$		−	2.00000i		−	0.143963i	−0.997406	−	0.0719816i	$-0.977068\pi$
$193$		−	2.00000i		−	0.143963i	0.997406	−	0.0719816i	$-0.0229323\pi$
$194$		0			0
$195$	1.00000	−	2.00000i	0.0716115	−	0.143223i
$196$	1.00000			0.0714286
$197$			22.0000i			1.56744i	0.621117	+	0.783718i	$0.286679\pi$
$197$			22.0000i			1.56744i	−0.621117	+	0.783718i	$0.713321\pi$
$198$		−	2.00000i		−	0.142134i
$199$	−6.00000			−0.425329			−0.212664	−	0.977125i	$-0.568214\pi$
$199$	−6.00000			−0.425329			−0.212664	+	0.977125i	$0.568214\pi$
$200$	−4.00000	+	3.00000i	−0.282843	+	0.212132i
$201$	8.00000			0.564276
$202$		−	10.0000i		−	0.703598i
$203$			6.00000i			0.421117i
$204$	6.00000			0.420084
$205$	2.00000	−	4.00000i	0.139686	−	0.279372i
$206$	8.00000			0.557386
$207$			2.00000i			0.139010i
$208$			1.00000i			0.0693375i
$209$	−8.00000			−0.553372
$210$	−2.00000	−	1.00000i	−0.138013	−	0.0690066i
$211$	−20.0000			−1.37686			−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000			−1.37686			−0.688428	+	0.725304i	$0.741699\pi$
$212$		−	4.00000i		−	0.274721i
$213$		−	4.00000i		−	0.274075i
$214$	8.00000			0.546869
$215$	12.0000	+	6.00000i	0.818393	+	0.409197i
$216$	1.00000			0.0680414
$217$		−	8.00000i		−	0.543075i
$218$		−	16.0000i		−	1.08366i
$219$	4.00000			0.270295
$220$	−2.00000	+	4.00000i	−0.134840	+	0.269680i
$221$	6.00000			0.403604
$222$		−	6.00000i		−	0.402694i
$223$		−	12.0000i		−	0.803579i	−0.915732	−	0.401790i	$-0.868388\pi$
$223$		−	12.0000i		−	0.803579i	0.915732	−	0.401790i	$-0.131612\pi$
$224$	1.00000			0.0668153
$225$	3.00000	+	4.00000i	0.200000	+	0.266667i
$226$	6.00000			0.399114
$227$		−	8.00000i		−	0.530979i	−0.964114	−	0.265489i	$-0.914466\pi$
$227$		−	8.00000i		−	0.530979i	0.964114	−	0.265489i	$-0.0855335\pi$
$228$		−	4.00000i		−	0.264906i
$229$	26.0000			1.71813			0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000			1.71813			0.859064	+	0.511868i	$0.171046\pi$
$230$	2.00000	−	4.00000i	0.131876	−	0.263752i
$231$	−2.00000			−0.131590
$232$			6.00000i			0.393919i
$233$		−	10.0000i		−	0.655122i	−0.944830	−	0.327561i	$-0.893773\pi$
$233$		−	10.0000i		−	0.655122i	0.944830	−	0.327561i	$-0.106227\pi$
$234$	1.00000			0.0653720
$235$	−4.00000	−	2.00000i	−0.260931	−	0.130466i
$236$	12.0000			0.781133
$237$			8.00000i			0.519656i
$238$		−	6.00000i		−	0.388922i
$239$	4.00000			0.258738			0.129369	−	0.991596i	$-0.458705\pi$
$239$	4.00000			0.258738			0.129369	+	0.991596i	$0.458705\pi$
$240$	−2.00000	−	1.00000i	−0.129099	−	0.0645497i
$241$	14.0000			0.901819			0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000			0.901819			0.450910	+	0.892570i	$0.351100\pi$
$242$		−	7.00000i		−	0.449977i
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	−1.00000	+	2.00000i	−0.0638877	+	0.127775i
$246$	2.00000			0.127515
$247$		−	4.00000i		−	0.254514i
$248$		−	8.00000i		−	0.508001i
$249$	−12.0000			−0.760469
$250$	−2.00000	−	11.0000i	−0.126491	−	0.695701i
$251$	30.0000			1.89358			0.946792	−	0.321847i	$-0.104304\pi$
$251$	30.0000			1.89358			0.946792	+	0.321847i	$0.104304\pi$
$252$		−	1.00000i		−	0.0629941i
$253$		−	4.00000i		−	0.251478i
$254$	16.0000			1.00393
$255$	−6.00000	+	12.0000i	−0.375735	+	0.751469i
$256$	1.00000			0.0625000
$257$			30.0000i			1.87135i	0.352865	+	0.935674i	$0.385208\pi$
$257$			30.0000i			1.87135i	−0.352865	+	0.935674i	$0.614792\pi$
$258$			6.00000i			0.373544i
$259$	−6.00000			−0.372822
$260$	−2.00000	−	1.00000i	−0.124035	−	0.0620174i
$261$	6.00000			0.371391
$262$			18.0000i			1.11204i
$263$		−	26.0000i		−	1.60323i	−0.597841	−	0.801614i	$-0.703975\pi$
$263$		−	26.0000i		−	1.60323i	0.597841	−	0.801614i	$-0.296025\pi$
$264$	−2.00000			−0.123091
$265$	8.00000	+	4.00000i	0.491436	+	0.245718i
$266$	−4.00000			−0.245256
$267$		−	10.0000i		−	0.611990i
$268$		−	8.00000i		−	0.488678i
$269$	2.00000			0.121942			0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000			0.121942			0.0609711	+	0.998140i	$0.480580\pi$
$270$	−1.00000	+	2.00000i	−0.0608581	+	0.121716i
$271$	20.0000			1.21491			0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000			1.21491			0.607457	+	0.794353i	$0.292190\pi$
$272$		−	6.00000i		−	0.363803i
$273$		−	1.00000i		−	0.0605228i
$274$	6.00000			0.362473
$275$	−6.00000	−	8.00000i	−0.361814	−	0.482418i
$276$	2.00000			0.120386
$277$		−	2.00000i		−	0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$		−	2.00000i		−	0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$		−	16.0000i		−	0.959616i
$279$	−8.00000			−0.478947
$280$	−1.00000	+	2.00000i	−0.0597614	+	0.119523i
$281$	−20.0000			−1.19310			−0.596550	−	0.802576i	$-0.703462\pi$
$281$	−20.0000			−1.19310			−0.596550	+	0.802576i	$0.703462\pi$
$282$		−	2.00000i		−	0.119098i
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$	−4.00000			−0.237356
$285$	8.00000	+	4.00000i	0.473879	+	0.236940i
$286$	−2.00000			−0.118262
$287$		−	2.00000i		−	0.118056i
$288$		−	1.00000i		−	0.0589256i
$289$	−19.0000			−1.11765
$290$	−12.0000	−	6.00000i	−0.704664	−	0.352332i
$291$		0			0
$292$		−	4.00000i		−	0.234082i
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$	−1.00000			−0.0583212
$295$	−12.0000	+	24.0000i	−0.698667	+	1.39733i
$296$	−6.00000			−0.348743
$297$			2.00000i			0.116052i
$298$		−	10.0000i		−	0.579284i
$299$	2.00000			0.115663
$300$	4.00000	−	3.00000i	0.230940	−	0.173205i
$301$	6.00000			0.345834
$302$			2.00000i			0.115087i
$303$			10.0000i			0.574485i
$304$	−4.00000			−0.229416
$305$		0			0
$306$	−6.00000			−0.342997
$307$		−	2.00000i		−	0.114146i	−0.998370	−	0.0570730i	$-0.981823\pi$
$307$		−	2.00000i		−	0.114146i	0.998370	−	0.0570730i	$-0.0181768\pi$
$308$			2.00000i			0.113961i
$309$	−8.00000			−0.455104
$310$	16.0000	+	8.00000i	0.908739	+	0.454369i
$311$	24.0000			1.36092			0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000			1.36092			0.680458	+	0.732787i	$0.261781\pi$
$312$		−	1.00000i		−	0.0566139i
$313$			2.00000i			0.113047i	0.998401	+	0.0565233i	$0.0180015\pi$
$313$			2.00000i			0.113047i	−0.998401	+	0.0565233i	$0.981998\pi$
$314$	2.00000			0.112867
$315$	2.00000	+	1.00000i	0.112687	+	0.0563436i
$316$	8.00000			0.450035
$317$		−	22.0000i		−	1.23564i	−0.786318	−	0.617822i	$-0.788015\pi$
$317$		−	22.0000i		−	1.23564i	0.786318	−	0.617822i	$-0.211985\pi$
$318$			4.00000i			0.224309i
$319$	−12.0000			−0.671871
$320$	−1.00000	+	2.00000i	−0.0559017	+	0.111803i
$321$	−8.00000			−0.446516
$322$		−	2.00000i		−	0.111456i
$323$			24.0000i			1.33540i
$324$	−1.00000			−0.0555556
$325$	4.00000	−	3.00000i	0.221880	−	0.166410i
$326$	16.0000			0.886158
$327$			16.0000i			0.884802i
$328$		−	2.00000i		−	0.110432i
$329$	−2.00000			−0.110264
$330$	2.00000	−	4.00000i	0.110096	−	0.220193i
$331$	−8.00000			−0.439720			−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000			−0.439720			−0.219860	+	0.975531i	$0.570560\pi$
$332$			12.0000i			0.658586i
$333$			6.00000i			0.328798i
$334$	−18.0000			−0.984916
$335$	16.0000	+	8.00000i	0.874173	+	0.437087i
$336$	−1.00000			−0.0545545
$337$		−	8.00000i		−	0.435788i	−0.975972	−	0.217894i	$-0.930081\pi$
$337$		−	8.00000i		−	0.435788i	0.975972	−	0.217894i	$-0.0699187\pi$
$338$		−	1.00000i		−	0.0543928i
$339$	−6.00000			−0.325875
$340$	12.0000	+	6.00000i	0.650791	+	0.325396i
$341$	16.0000			0.866449
$342$			4.00000i			0.216295i
$343$			1.00000i			0.0539949i
$344$	6.00000			0.323498
$345$	−2.00000	+	4.00000i	−0.107676	+	0.215353i
$346$	−2.00000			−0.107521
$347$			16.0000i			0.858925i	0.903085	+	0.429463i	$0.141297\pi$
$347$			16.0000i			0.858925i	−0.903085	+	0.429463i	$0.858703\pi$
$348$		−	6.00000i		−	0.321634i
$349$	6.00000			0.321173			0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000			0.321173			0.160586	+	0.987022i	$0.448662\pi$
$350$	−3.00000	−	4.00000i	−0.160357	−	0.213809i
$351$	−1.00000			−0.0533761
$352$			2.00000i			0.106600i
$353$		−	10.0000i		−	0.532246i	−0.963939	−	0.266123i	$-0.914257\pi$
$353$		−	10.0000i		−	0.532246i	0.963939	−	0.266123i	$-0.0857428\pi$
$354$	−12.0000			−0.637793
$355$	4.00000	−	8.00000i	0.212298	−	0.424596i
$356$	−10.0000			−0.529999
$357$			6.00000i			0.317554i
$358$			20.0000i			1.05703i
$359$	24.0000			1.26667			0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000			1.26667			0.633336	+	0.773877i	$0.281685\pi$
$360$	2.00000	+	1.00000i	0.105409	+	0.0527046i
$361$	−3.00000			−0.157895
$362$		−	8.00000i		−	0.420471i
$363$			7.00000i			0.367405i
$364$	−1.00000			−0.0524142
$365$	8.00000	+	4.00000i	0.418739	+	0.209370i
$366$		0			0
$367$			8.00000i			0.417597i	0.977959	+	0.208798i	$0.0669552\pi$
$367$			8.00000i			0.417597i	−0.977959	+	0.208798i	$0.933045\pi$
$368$		−	2.00000i		−	0.104257i
$369$	−2.00000			−0.104116
$370$	6.00000	−	12.0000i	0.311925	−	0.623850i
$371$	4.00000			0.207670
$372$			8.00000i			0.414781i
$373$		−	6.00000i		−	0.310668i	−0.987862	−	0.155334i	$-0.950355\pi$
$373$		−	6.00000i		−	0.310668i	0.987862	−	0.155334i	$-0.0496454\pi$
$374$	12.0000			0.620505
$375$	2.00000	+	11.0000i	0.103280	+	0.568038i
$376$	−2.00000			−0.103142
$377$		−	6.00000i		−	0.309016i
$378$			1.00000i			0.0514344i
$379$	−20.0000			−1.02733			−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000			−1.02733			−0.513665	+	0.857991i	$0.671713\pi$
$380$	4.00000	−	8.00000i	0.205196	−	0.410391i
$381$	−16.0000			−0.819705
$382$		−	8.00000i		−	0.409316i
$383$		−	30.0000i		−	1.53293i	−0.642287	−	0.766464i	$-0.722014\pi$
$383$		−	30.0000i		−	1.53293i	0.642287	−	0.766464i	$-0.277986\pi$
$384$	−1.00000			−0.0510310
$385$	−4.00000	−	2.00000i	−0.203859	−	0.101929i
$386$	2.00000			0.101797
$387$		−	6.00000i		−	0.304997i
$388$		0			0
$389$	38.0000			1.92668			0.963338	−	0.268290i	$-0.0864585\pi$
$389$	38.0000			1.92668			0.963338	+	0.268290i	$0.0864585\pi$
$390$	2.00000	+	1.00000i	0.101274	+	0.0506370i
$391$	−12.0000			−0.606866
$392$			1.00000i			0.0505076i
$393$		−	18.0000i		−	0.907980i
$394$	−22.0000			−1.10834
$395$	−8.00000	+	16.0000i	−0.402524	+	0.805047i
$396$	2.00000			0.100504
$397$			18.0000i			0.903394i	0.892171	+	0.451697i	$0.149181\pi$
$397$			18.0000i			0.903394i	−0.892171	+	0.451697i	$0.850819\pi$
$398$		−	6.00000i		−	0.300753i
$399$	4.00000			0.200250
$400$	−3.00000	−	4.00000i	−0.150000	−	0.200000i
$401$	36.0000			1.79775			0.898877	−	0.438201i	$-0.144384\pi$
$401$	36.0000			1.79775			0.898877	+	0.438201i	$0.144384\pi$
$402$			8.00000i			0.399004i
$403$			8.00000i			0.398508i
$404$	10.0000			0.497519
$405$	1.00000	−	2.00000i	0.0496904	−	0.0993808i
$406$	−6.00000			−0.297775
$407$		−	12.0000i		−	0.594818i
$408$			6.00000i			0.297044i
$409$	−26.0000			−1.28562			−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000			−1.28562			−0.642809	+	0.766027i	$0.722231\pi$
$410$	4.00000	+	2.00000i	0.197546	+	0.0987730i
$411$	−6.00000			−0.295958
$412$			8.00000i			0.394132i
$413$			12.0000i			0.590481i
$414$	−2.00000			−0.0982946
$415$	−24.0000	−	12.0000i	−1.17811	−	0.589057i
$416$	−1.00000			−0.0490290
$417$			16.0000i			0.783523i
$418$		−	8.00000i		−	0.391293i
$419$	10.0000			0.488532			0.244266	−	0.969708i	$-0.421453\pi$
$419$	10.0000			0.488532			0.244266	+	0.969708i	$0.421453\pi$
$420$	1.00000	−	2.00000i	0.0487950	−	0.0975900i
$421$	−36.0000			−1.75453			−0.877266	−	0.480004i	$-0.840635\pi$
$421$	−36.0000			−1.75453			−0.877266	+	0.480004i	$0.840635\pi$
$422$		−	20.0000i		−	0.973585i
$423$			2.00000i			0.0972433i
$424$	4.00000			0.194257
$425$	−24.0000	+	18.0000i	−1.16417	+	0.873128i
$426$	4.00000			0.193801
$427$		0			0
$428$			8.00000i			0.386695i
$429$	2.00000			0.0965609
$430$	−6.00000	+	12.0000i	−0.289346	+	0.578691i
$431$	12.0000			0.578020			0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000			0.578020			0.289010	+	0.957326i	$0.406674\pi$
$432$			1.00000i			0.0481125i
$433$			26.0000i			1.24948i	0.780833	+	0.624740i	$0.214795\pi$
$433$			26.0000i			1.24948i	−0.780833	+	0.624740i	$0.785205\pi$
$434$	8.00000			0.384012
$435$	12.0000	+	6.00000i	0.575356	+	0.287678i
$436$	16.0000			0.766261
$437$			8.00000i			0.382692i
$438$			4.00000i			0.191127i
$439$	−2.00000			−0.0954548			−0.0477274	−	0.998860i	$-0.515198\pi$
$439$	−2.00000			−0.0954548			−0.0477274	+	0.998860i	$0.515198\pi$
$440$	−4.00000	−	2.00000i	−0.190693	−	0.0953463i
$441$	1.00000			0.0476190
$442$			6.00000i			0.285391i
$443$			12.0000i			0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$			12.0000i			0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$	6.00000			0.284747
$445$	10.0000	−	20.0000i	0.474045	−	0.948091i
$446$	12.0000			0.568216
$447$			10.0000i			0.472984i
$448$			1.00000i			0.0472456i
$449$	16.0000			0.755087			0.377543	−	0.925992i	$-0.376769\pi$
$449$	16.0000			0.755087			0.377543	+	0.925992i	$0.376769\pi$
$450$	−4.00000	+	3.00000i	−0.188562	+	0.141421i
$451$	4.00000			0.188353
$452$			6.00000i			0.282216i
$453$		−	2.00000i		−	0.0939682i
$454$	8.00000			0.375459
$455$	1.00000	−	2.00000i	0.0468807	−	0.0937614i
$456$	4.00000			0.187317
$457$		−	22.0000i		−	1.02912i	−0.857455	−	0.514558i	$-0.827956\pi$
$457$		−	22.0000i		−	1.02912i	0.857455	−	0.514558i	$-0.172044\pi$
$458$			26.0000i			1.21490i
$459$	6.00000			0.280056
$460$	4.00000	+	2.00000i	0.186501	+	0.0932505i
$461$	−18.0000			−0.838344			−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000			−0.838344			−0.419172	+	0.907907i	$0.637680\pi$
$462$		−	2.00000i		−	0.0930484i
$463$		−	16.0000i		−	0.743583i	−0.928316	−	0.371792i	$-0.878744\pi$
$463$		−	16.0000i		−	0.743583i	0.928316	−	0.371792i	$-0.121256\pi$
$464$	−6.00000			−0.278543
$465$	−16.0000	−	8.00000i	−0.741982	−	0.370991i
$466$	10.0000			0.463241
$467$		−	4.00000i		−	0.185098i	−0.995708	−	0.0925490i	$-0.970499\pi$
$467$		−	4.00000i		−	0.185098i	0.995708	−	0.0925490i	$-0.0295015\pi$
$468$			1.00000i			0.0462250i
$469$	8.00000			0.369406
$470$	2.00000	−	4.00000i	0.0922531	−	0.184506i
$471$	−2.00000			−0.0921551
$472$			12.0000i			0.552345i
$473$			12.0000i			0.551761i
$474$	−8.00000			−0.367452
$475$	12.0000	+	16.0000i	0.550598	+	0.734130i
$476$	6.00000			0.275010
$477$		−	4.00000i		−	0.183147i
$478$			4.00000i			0.182956i
$479$	32.0000			1.46212			0.731059	−	0.682315i	$-0.239027\pi$
$479$	32.0000			1.46212			0.731059	+	0.682315i	$0.239027\pi$
$480$	1.00000	−	2.00000i	0.0456435	−	0.0912871i
$481$	6.00000			0.273576
$482$			14.0000i			0.637683i
$483$			2.00000i			0.0910032i
$484$	7.00000			0.318182
$485$		0			0
$486$	1.00000			0.0453609
$487$		−	8.00000i		−	0.362515i	−0.983436	−	0.181257i	$-0.941983\pi$
$487$		−	8.00000i		−	0.362515i	0.983436	−	0.181257i	$-0.0580167\pi$
$488$		0			0
$489$	−16.0000			−0.723545
$490$	−2.00000	−	1.00000i	−0.0903508	−	0.0451754i
$491$	20.0000			0.902587			0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000			0.902587			0.451294	+	0.892375i	$0.350963\pi$
$492$			2.00000i			0.0901670i
$493$			36.0000i			1.62136i
$494$	4.00000			0.179969
$495$	−2.00000	+	4.00000i	−0.0898933	+	0.179787i
$496$	8.00000			0.359211
$497$		−	4.00000i		−	0.179425i
$498$		−	12.0000i		−	0.537733i
$499$	32.0000			1.43252			0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000			1.43252			0.716258	+	0.697835i	$0.245853\pi$
$500$	11.0000	−	2.00000i	0.491935	−	0.0894427i
$501$	18.0000			0.804181
$502$			30.0000i			1.33897i
$503$			32.0000i			1.42681i	0.700752	+	0.713405i	$0.252848\pi$
$503$			32.0000i			1.42681i	−0.700752	+	0.713405i	$0.747152\pi$
$504$	1.00000			0.0445435
$505$	−10.0000	+	20.0000i	−0.444994	+	0.889988i
$506$	4.00000			0.177822
$507$			1.00000i			0.0444116i
$508$			16.0000i			0.709885i
$509$	30.0000			1.32973			0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000			1.32973			0.664863	+	0.746965i	$0.268490\pi$
$510$	−12.0000	−	6.00000i	−0.531369	−	0.265684i
$511$	4.00000			0.176950
$512$			1.00000i			0.0441942i
$513$		−	4.00000i		−	0.176604i
$514$	−30.0000			−1.32324
$515$	−16.0000	−	8.00000i	−0.705044	−	0.352522i
$516$	−6.00000			−0.264135
$517$		−	4.00000i		−	0.175920i
$518$		−	6.00000i		−	0.263625i
$519$	2.00000			0.0877903
$520$	1.00000	−	2.00000i	0.0438529	−	0.0877058i
$521$	−36.0000			−1.57719			−0.788594	−	0.614914i	$-0.789191\pi$
$521$	−36.0000			−1.57719			−0.788594	+	0.614914i	$0.789191\pi$
$522$			6.00000i			0.262613i
$523$		−	12.0000i		−	0.524723i	−0.964970	−	0.262362i	$-0.915499\pi$
$523$		−	12.0000i		−	0.524723i	0.964970	−	0.262362i	$-0.0845013\pi$
$524$	−18.0000			−0.786334
$525$	3.00000	+	4.00000i	0.130931	+	0.174574i
$526$	26.0000			1.13365
$527$		−	48.0000i		−	2.09091i
$528$		−	2.00000i		−	0.0870388i
$529$	19.0000			0.826087
$530$	−4.00000	+	8.00000i	−0.173749	+	0.347498i
$531$	12.0000			0.520756
$532$		−	4.00000i		−	0.173422i
$533$			2.00000i			0.0866296i
$534$	10.0000			0.432742
$535$	−16.0000	−	8.00000i	−0.691740	−	0.345870i
$536$	8.00000			0.345547
$537$		−	20.0000i		−	0.863064i
$538$			2.00000i			0.0862261i
$539$	−2.00000			−0.0861461
$540$	−2.00000	−	1.00000i	−0.0860663	−	0.0430331i
$541$	32.0000			1.37579			0.687894	−	0.725811i	$-0.258536\pi$
$541$	32.0000			1.37579			0.687894	+	0.725811i	$0.258536\pi$
$542$			20.0000i			0.859074i
$543$			8.00000i			0.343313i
$544$	6.00000			0.257248
$545$	−16.0000	+	32.0000i	−0.685365	+	1.37073i
$546$	1.00000			0.0427960
$547$			30.0000i			1.28271i	0.767245	+	0.641354i	$0.221627\pi$
$547$			30.0000i			1.28271i	−0.767245	+	0.641354i	$0.778373\pi$
$548$			6.00000i			0.256307i
$549$		0			0
$550$	8.00000	−	6.00000i	0.341121	−	0.255841i
$551$	24.0000			1.02243
$552$			2.00000i			0.0851257i
$553$			8.00000i			0.340195i
$554$	2.00000			0.0849719
$555$	−6.00000	+	12.0000i	−0.254686	+	0.509372i
$556$	16.0000			0.678551
$557$		−	2.00000i		−	0.0847427i	−0.999102	−	0.0423714i	$-0.986509\pi$
$557$		−	2.00000i		−	0.0847427i	0.999102	−	0.0423714i	$-0.0134913\pi$
$558$		−	8.00000i		−	0.338667i
$559$	−6.00000			−0.253773
$560$	−2.00000	−	1.00000i	−0.0845154	−	0.0422577i
$561$	−12.0000			−0.506640
$562$		−	20.0000i		−	0.843649i
$563$		−	36.0000i		−	1.51722i	−0.651546	−	0.758610i	$-0.725879\pi$
$563$		−	36.0000i		−	1.51722i	0.651546	−	0.758610i	$-0.274121\pi$
$564$	2.00000			0.0842152
$565$	−12.0000	−	6.00000i	−0.504844	−	0.252422i
$566$		0			0
$567$		−	1.00000i		−	0.0419961i
$568$		−	4.00000i		−	0.167836i
$569$	6.00000			0.251533			0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000			0.251533			0.125767	+	0.992060i	$0.459861\pi$
$570$	−4.00000	+	8.00000i	−0.167542	+	0.335083i
$571$	44.0000			1.84134			0.920671	−	0.390339i	$-0.127642\pi$
$571$	44.0000			1.84134			0.920671	+	0.390339i	$0.127642\pi$
$572$		−	2.00000i		−	0.0836242i
$573$			8.00000i			0.334205i
$574$	2.00000			0.0834784
$575$	−8.00000	+	6.00000i	−0.333623	+	0.250217i
$576$	1.00000			0.0416667
$577$			24.0000i			0.999133i	0.866276	+	0.499567i	$0.166507\pi$
$577$			24.0000i			0.999133i	−0.866276	+	0.499567i	$0.833493\pi$
$578$		−	19.0000i		−	0.790296i
$579$	−2.00000			−0.0831172
$580$	6.00000	−	12.0000i	0.249136	−	0.498273i
$581$	−12.0000			−0.497844
$582$		0			0
$583$			8.00000i			0.331326i
$584$	4.00000			0.165521
$585$	−2.00000	−	1.00000i	−0.0826898	−	0.0413449i
$586$		0			0
$587$			8.00000i			0.330195i	0.986277	+	0.165098i	$0.0527939\pi$
$587$			8.00000i			0.330195i	−0.986277	+	0.165098i	$0.947206\pi$
$588$		−	1.00000i		−	0.0412393i
$589$	−32.0000			−1.31854
$590$	−24.0000	−	12.0000i	−0.988064	−	0.494032i
$591$	22.0000			0.904959
$592$		−	6.00000i		−	0.246598i
$593$			18.0000i			0.739171i	0.929197	+	0.369586i	$0.120500\pi$
$593$			18.0000i			0.739171i	−0.929197	+	0.369586i	$0.879500\pi$
$594$	−2.00000			−0.0820610
$595$	−6.00000	+	12.0000i	−0.245976	+	0.491952i
$596$	10.0000			0.409616
$597$			6.00000i			0.245564i
$598$			2.00000i			0.0817861i
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$	3.00000	+	4.00000i	0.122474	+	0.163299i
$601$	−2.00000			−0.0815817			−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000			−0.0815817			−0.0407909	+	0.999168i	$0.512988\pi$
$602$			6.00000i			0.244542i
$603$		−	8.00000i		−	0.325785i
$604$	−2.00000			−0.0813788
$605$	−7.00000	+	14.0000i	−0.284590	+	0.569181i
$606$	−10.0000			−0.406222
$607$		−	32.0000i		−	1.29884i	−0.760430	−	0.649420i	$-0.775012\pi$
$607$		−	32.0000i		−	1.29884i	0.760430	−	0.649420i	$-0.224988\pi$
$608$		−	4.00000i		−	0.162221i
$609$	6.00000			0.243132
$610$		0			0
$611$	2.00000			0.0809113
$612$		−	6.00000i		−	0.242536i
$613$			42.0000i			1.69636i	0.529705	+	0.848182i	$0.322303\pi$
$613$			42.0000i			1.69636i	−0.529705	+	0.848182i	$0.677697\pi$
$614$	2.00000			0.0807134
$615$	−4.00000	−	2.00000i	−0.161296	−	0.0806478i
$616$	−2.00000			−0.0805823
$617$		−	6.00000i		−	0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$		−	6.00000i		−	0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$		−	8.00000i		−	0.321807i
$619$	−28.0000			−1.12542			−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000			−1.12542			−0.562708	+	0.826656i	$0.690240\pi$
$620$	−8.00000	+	16.0000i	−0.321288	+	0.642575i
$621$	2.00000			0.0802572
$622$			24.0000i			0.962312i
$623$		−	10.0000i		−	0.400642i
$624$	1.00000			0.0400320
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$	−2.00000			−0.0799361
$627$			8.00000i			0.319489i
$628$			2.00000i			0.0798087i
$629$	−36.0000			−1.43541
$630$	−1.00000	+	2.00000i	−0.0398410	+	0.0796819i
$631$	−34.0000			−1.35352			−0.676759	−	0.736204i	$-0.736616\pi$
$631$	−34.0000			−1.35352			−0.676759	+	0.736204i	$0.736616\pi$
$632$			8.00000i			0.318223i
$633$			20.0000i			0.794929i
$634$	22.0000			0.873732
$635$	−32.0000	−	16.0000i	−1.26988	−	0.634941i
$636$	−4.00000			−0.158610
$637$		−	1.00000i		−	0.0396214i
$638$		−	12.0000i		−	0.475085i
$639$	−4.00000			−0.158238
$640$	−2.00000	−	1.00000i	−0.0790569	−	0.0395285i
$641$	−6.00000			−0.236986			−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000			−0.236986			−0.118493	+	0.992955i	$0.537806\pi$
$642$		−	8.00000i		−	0.315735i
$643$			18.0000i			0.709851i	0.934895	+	0.354925i	$0.115494\pi$
$643$			18.0000i			0.709851i	−0.934895	+	0.354925i	$0.884506\pi$
$644$	2.00000			0.0788110
$645$	6.00000	−	12.0000i	0.236250	−	0.472500i
$646$	−24.0000			−0.944267
$647$			36.0000i			1.41531i	0.706560	+	0.707653i	$0.250246\pi$
$647$			36.0000i			1.41531i	−0.706560	+	0.707653i	$0.749754\pi$
$648$		−	1.00000i		−	0.0392837i
$649$	−24.0000			−0.942082
$650$	3.00000	+	4.00000i	0.117670	+	0.156893i
$651$	−8.00000			−0.313545
$652$			16.0000i			0.626608i
$653$		−	24.0000i		−	0.939193i	−0.882881	−	0.469596i	$-0.844399\pi$
$653$		−	24.0000i		−	0.939193i	0.882881	−	0.469596i	$-0.155601\pi$
$654$	−16.0000			−0.625650
$655$	18.0000	−	36.0000i	0.703318	−	1.40664i
$656$	2.00000			0.0780869
$657$		−	4.00000i		−	0.156055i
$658$		−	2.00000i		−	0.0779681i
$659$	36.0000			1.40236			0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000			1.40236			0.701180	+	0.712984i	$0.252657\pi$
$660$	4.00000	+	2.00000i	0.155700	+	0.0778499i
$661$	18.0000			0.700119			0.350059	−	0.936727i	$-0.386161\pi$
$661$	18.0000			0.700119			0.350059	+	0.936727i	$0.386161\pi$
$662$		−	8.00000i		−	0.310929i
$663$		−	6.00000i		−	0.233021i
$664$	−12.0000			−0.465690
$665$	8.00000	+	4.00000i	0.310227	+	0.155113i
$666$	−6.00000			−0.232495
$667$			12.0000i			0.464642i
$668$		−	18.0000i		−	0.696441i
$669$	−12.0000			−0.463947
$670$	−8.00000	+	16.0000i	−0.309067	+	0.618134i
$671$		0			0
$672$		−	1.00000i		−	0.0385758i
$673$		−	32.0000i		−	1.23351i	−0.787155	−	0.616755i	$-0.788447\pi$
$673$		−	32.0000i		−	1.23351i	0.787155	−	0.616755i	$-0.211553\pi$
$674$	8.00000			0.308148
$675$	4.00000	−	3.00000i	0.153960	−	0.115470i
$676$	1.00000			0.0384615
$677$			30.0000i			1.15299i	0.817099	+	0.576497i	$0.195581\pi$
$677$			30.0000i			1.15299i	−0.817099	+	0.576497i	$0.804419\pi$
$678$		−	6.00000i		−	0.230429i
$679$		0			0
$680$	−6.00000	+	12.0000i	−0.230089	+	0.460179i
$681$	−8.00000			−0.306561
$682$			16.0000i			0.612672i
$683$		−	4.00000i		−	0.153056i	−0.997067	−	0.0765279i	$-0.975617\pi$
$683$		−	4.00000i		−	0.153056i	0.997067	−	0.0765279i	$-0.0243834\pi$
$684$	−4.00000			−0.152944
$685$	−12.0000	−	6.00000i	−0.458496	−	0.229248i
$686$	−1.00000			−0.0381802
$687$		−	26.0000i		−	0.991962i
$688$			6.00000i			0.228748i
$689$	−4.00000			−0.152388
$690$	−4.00000	−	2.00000i	−0.152277	−	0.0761387i
$691$	−12.0000			−0.456502			−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000			−0.456502			−0.228251	+	0.973602i	$0.573301\pi$
$692$		−	2.00000i		−	0.0760286i
$693$			2.00000i			0.0759737i
$694$	−16.0000			−0.607352
$695$	−16.0000	+	32.0000i	−0.606915	+	1.21383i
$696$	6.00000			0.227429
$697$		−	12.0000i		−	0.454532i
$698$			6.00000i			0.227103i
$699$	−10.0000			−0.378235
$700$	4.00000	−	3.00000i	0.151186	−	0.113389i
$701$	−14.0000			−0.528773			−0.264386	−	0.964417i	$-0.585169\pi$
$701$	−14.0000			−0.528773			−0.264386	+	0.964417i	$0.585169\pi$
$702$		−	1.00000i		−	0.0377426i
$703$			24.0000i			0.905177i
$704$	−2.00000			−0.0753778
$705$	−2.00000	+	4.00000i	−0.0753244	+	0.150649i
$706$	10.0000			0.376355
$707$			10.0000i			0.376089i
$708$		−	12.0000i		−	0.450988i
$709$	−44.0000			−1.65245			−0.826227	−	0.563337i	$-0.809517\pi$
$709$	−44.0000			−1.65245			−0.826227	+	0.563337i	$0.809517\pi$
$710$	8.00000	+	4.00000i	0.300235	+	0.150117i
$711$	8.00000			0.300023
$712$		−	10.0000i		−	0.374766i
$713$		−	16.0000i		−	0.599205i
$714$	−6.00000			−0.224544
$715$	4.00000	+	2.00000i	0.149592	+	0.0747958i
$716$	−20.0000			−0.747435
$717$		−	4.00000i		−	0.149383i
$718$			24.0000i			0.895672i
$719$	24.0000			0.895049			0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000			0.895049			0.447524	+	0.894272i	$0.352306\pi$
$720$	−1.00000	+	2.00000i	−0.0372678	+	0.0745356i
$721$	−8.00000			−0.297936
$722$		−	3.00000i		−	0.111648i
$723$		−	14.0000i		−	0.520666i
$724$	8.00000			0.297318
$725$	18.0000	+	24.0000i	0.668503	+	0.891338i
$726$	−7.00000			−0.259794
$727$		−	16.0000i		−	0.593407i	−0.954970	−	0.296704i	$-0.904113\pi$
$727$		−	16.0000i		−	0.593407i	0.954970	−	0.296704i	$-0.0958873\pi$
$728$		−	1.00000i		−	0.0370625i
$729$	−1.00000			−0.0370370
$730$	−4.00000	+	8.00000i	−0.148047	+	0.296093i
$731$	36.0000			1.33151
$732$		0			0
$733$			26.0000i			0.960332i	0.877178	+	0.480166i	$0.159424\pi$
$733$			26.0000i			0.960332i	−0.877178	+	0.480166i	$0.840576\pi$
$734$	−8.00000			−0.295285
$735$	2.00000	+	1.00000i	0.0737711	+	0.0368856i
$736$	2.00000			0.0737210
$737$			16.0000i			0.589368i
$738$		−	2.00000i		−	0.0736210i
$739$	44.0000			1.61857			0.809283	−	0.587419i	$-0.199856\pi$
$739$	44.0000			1.61857			0.809283	+	0.587419i	$0.199856\pi$
$740$	12.0000	+	6.00000i	0.441129	+	0.220564i
$741$	−4.00000			−0.146944
$742$			4.00000i			0.146845i
$743$		−	12.0000i		−	0.440237i	−0.975473	−	0.220119i	$-0.929356\pi$
$743$		−	12.0000i		−	0.440237i	0.975473	−	0.220119i	$-0.0706445\pi$
$744$	−8.00000			−0.293294
$745$	−10.0000	+	20.0000i	−0.366372	+	0.732743i
$746$	6.00000			0.219676
$747$			12.0000i			0.439057i
$748$			12.0000i			0.438763i
$749$	−8.00000			−0.292314
$750$	−11.0000	+	2.00000i	−0.401663	+	0.0730297i
$751$	40.0000			1.45962			0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000			1.45962			0.729810	+	0.683650i	$0.239608\pi$
$752$		−	2.00000i		−	0.0729325i
$753$		−	30.0000i		−	1.09326i
$754$	6.00000			0.218507
$755$	2.00000	−	4.00000i	0.0727875	−	0.145575i
$756$	−1.00000			−0.0363696
$757$		−	2.00000i		−	0.0726912i	−0.999339	−	0.0363456i	$-0.988428\pi$
$757$		−	2.00000i		−	0.0726912i	0.999339	−	0.0363456i	$-0.0115717\pi$
$758$		−	20.0000i		−	0.726433i
$759$	−4.00000			−0.145191
$760$	8.00000	+	4.00000i	0.290191	+	0.145095i
$761$	−26.0000			−0.942499			−0.471250	−	0.882000i	$-0.656197\pi$
$761$	−26.0000			−0.942499			−0.471250	+	0.882000i	$0.656197\pi$
$762$		−	16.0000i		−	0.579619i
$763$			16.0000i			0.579239i
$764$	8.00000			0.289430
$765$	12.0000	+	6.00000i	0.433861	+	0.216930i
$766$	30.0000			1.08394
$767$		−	12.0000i		−	0.433295i
$768$		−	1.00000i		−	0.0360844i
$769$	10.0000			0.360609			0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000			0.360609			0.180305	+	0.983611i	$0.442292\pi$
$770$	2.00000	−	4.00000i	0.0720750	−	0.144150i
$771$	30.0000			1.08042
$772$			2.00000i			0.0719816i
$773$		−	40.0000i		−	1.43870i	−0.694648	−	0.719350i	$-0.744440\pi$
$773$		−	40.0000i		−	1.43870i	0.694648	−	0.719350i	$-0.255560\pi$
$774$	6.00000			0.215666
$775$	−24.0000	−	32.0000i	−0.862105	−	1.14947i
$776$		0			0
$777$			6.00000i			0.215249i
$778$			38.0000i			1.36237i
$779$	−8.00000			−0.286630
$780$	−1.00000	+	2.00000i	−0.0358057	+	0.0716115i
$781$	8.00000			0.286263
$782$		−	12.0000i		−	0.429119i
$783$		−	6.00000i		−	0.214423i
$784$	−1.00000			−0.0357143
$785$	−4.00000	−	2.00000i	−0.142766	−	0.0713831i
$786$	18.0000			0.642039
$787$		−	22.0000i		−	0.784215i	−0.919919	−	0.392108i	$-0.871746\pi$
$787$		−	22.0000i		−	0.784215i	0.919919	−	0.392108i	$-0.128254\pi$
$788$		−	22.0000i		−	0.783718i
$789$	−26.0000			−0.925625
$790$	−16.0000	−	8.00000i	−0.569254	−	0.284627i
$791$	−6.00000			−0.213335
$792$			2.00000i			0.0710669i
$793$		0			0
$794$	−18.0000			−0.638796
$795$	4.00000	−	8.00000i	0.141865	−	0.283731i
$796$	6.00000			0.212664
$797$			42.0000i			1.48772i	0.668338	+	0.743858i	$0.267006\pi$
$797$			42.0000i			1.48772i	−0.668338	+	0.743858i	$0.732994\pi$
$798$			4.00000i			0.141598i
$799$	−12.0000			−0.424529
$800$	4.00000	−	3.00000i	0.141421	−	0.106066i
$801$	−10.0000			−0.353333
$802$			36.0000i			1.27120i
$803$			8.00000i			0.282314i
$804$	−8.00000			−0.282138
$805$	−2.00000	+	4.00000i	−0.0704907	+	0.140981i
$806$	−8.00000			−0.281788
$807$		−	2.00000i		−	0.0704033i
$808$			10.0000i			0.351799i
$809$	30.0000			1.05474			0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000			1.05474			0.527372	+	0.849635i	$0.323177\pi$
$810$	2.00000	+	1.00000i	0.0702728	+	0.0351364i
$811$	−4.00000			−0.140459			−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000			−0.140459			−0.0702295	+	0.997531i	$0.522373\pi$
$812$		−	6.00000i		−	0.210559i
$813$		−	20.0000i		−	0.701431i
$814$	12.0000			0.420600
$815$	−32.0000	−	16.0000i	−1.12091	−	0.560456i
$816$	−6.00000			−0.210042
$817$		−	24.0000i		−	0.839654i
$818$		−	26.0000i		−	0.909069i
$819$	−1.00000			−0.0349428
$820$	−2.00000	+	4.00000i	−0.0698430	+	0.139686i
$821$	2.00000			0.0698005			0.0349002	−	0.999391i	$-0.488889\pi$
$821$	2.00000			0.0698005			0.0349002	+	0.999391i	$0.488889\pi$
$822$		−	6.00000i		−	0.209274i
$823$			8.00000i			0.278862i	0.990232	+	0.139431i	$0.0445274\pi$
$823$			8.00000i			0.278862i	−0.990232	+	0.139431i	$0.955473\pi$
$824$	−8.00000			−0.278693
$825$	−8.00000	+	6.00000i	−0.278524	+	0.208893i
$826$	−12.0000			−0.417533
$827$		−	20.0000i		−	0.695468i	−0.937593	−	0.347734i	$-0.886951\pi$
$827$		−	20.0000i		−	0.695468i	0.937593	−	0.347734i	$-0.113049\pi$
$828$		−	2.00000i		−	0.0695048i
$829$	32.0000			1.11141			0.555703	−	0.831381i	$-0.312449\pi$
$829$	32.0000			1.11141			0.555703	+	0.831381i	$0.312449\pi$
$830$	12.0000	−	24.0000i	0.416526	−	0.833052i
$831$	−2.00000			−0.0693792
$832$		−	1.00000i		−	0.0346688i
$833$			6.00000i			0.207888i
$834$	−16.0000			−0.554035
$835$	36.0000	+	18.0000i	1.24583	+	0.622916i
$836$	8.00000			0.276686
$837$			8.00000i			0.276520i
$838$			10.0000i			0.345444i
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$	2.00000	+	1.00000i	0.0690066	+	0.0345033i
$841$	7.00000			0.241379
$842$		−	36.0000i		−	1.24064i
$843$			20.0000i			0.688837i
$844$	20.0000			0.688428
$845$	−1.00000	+	2.00000i	−0.0344010	+	0.0688021i
$846$	−2.00000			−0.0687614
$847$			7.00000i			0.240523i
$848$			4.00000i			0.137361i
$849$		0			0
$850$	−18.0000	−	24.0000i	−0.617395	−	0.823193i
$851$	−12.0000			−0.411355
$852$			4.00000i			0.137038i
$853$		−	18.0000i		−	0.616308i	−0.951336	−	0.308154i	$-0.900289\pi$
$853$		−	18.0000i		−	0.616308i	0.951336	−	0.308154i	$-0.0997113\pi$
$854$		0			0
$855$	4.00000	−	8.00000i	0.136797	−	0.273594i
$856$	−8.00000			−0.273434
$857$		−	26.0000i		−	0.888143i	−0.895991	−	0.444072i	$-0.853534\pi$
$857$		−	26.0000i		−	0.888143i	0.895991	−	0.444072i	$-0.146466\pi$
$858$			2.00000i			0.0682789i
$859$	40.0000			1.36478			0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000			1.36478			0.682391	+	0.730987i	$0.260940\pi$
$860$	−12.0000	−	6.00000i	−0.409197	−	0.204598i
$861$	−2.00000			−0.0681598
$862$			12.0000i			0.408722i
$863$			48.0000i			1.63394i	0.576681	+	0.816970i	$0.304348\pi$
$863$			48.0000i			1.63394i	−0.576681	+	0.816970i	$0.695652\pi$
$864$	−1.00000			−0.0340207
$865$	4.00000	+	2.00000i	0.136004	+	0.0680020i
$866$	−26.0000			−0.883516
$867$			19.0000i			0.645274i
$868$			8.00000i			0.271538i
$869$	−16.0000			−0.542763
$870$	−6.00000	+	12.0000i	−0.203419	+	0.406838i
$871$	−8.00000			−0.271070
$872$			16.0000i			0.541828i
$873$		0			0
$874$	−8.00000			−0.270604
$875$	2.00000	+	11.0000i	0.0676123	+	0.371868i
$876$	−4.00000			−0.135147
$877$			2.00000i			0.0675352i	0.999430	+	0.0337676i	$0.0107506\pi$
$877$			2.00000i			0.0675352i	−0.999430	+	0.0337676i	$0.989249\pi$
$878$		−	2.00000i		−	0.0674967i
$879$		0			0
$880$	2.00000	−	4.00000i	0.0674200	−	0.134840i
$881$	40.0000			1.34763			0.673817	−	0.738898i	$-0.264654\pi$
$881$	40.0000			1.34763			0.673817	+	0.738898i	$0.264654\pi$
$882$			1.00000i			0.0336718i
$883$		−	14.0000i		−	0.471138i	−0.971858	−	0.235569i	$-0.924305\pi$
$883$		−	14.0000i		−	0.471138i	0.971858	−	0.235569i	$-0.0756953\pi$
$884$	−6.00000			−0.201802
$885$	24.0000	+	12.0000i	0.806751	+	0.403376i
$886$	−12.0000			−0.403148
$887$		−	20.0000i		−	0.671534i	−0.941945	−	0.335767i	$-0.891004\pi$
$887$		−	20.0000i		−	0.671534i	0.941945	−	0.335767i	$-0.108996\pi$
$888$			6.00000i			0.201347i
$889$	−16.0000			−0.536623
$890$	20.0000	+	10.0000i	0.670402	+	0.335201i
$891$	2.00000			0.0670025
$892$			12.0000i			0.401790i
$893$			8.00000i			0.267710i
$894$	−10.0000			−0.334450
$895$	20.0000	−	40.0000i	0.668526	−	1.33705i
$896$	−1.00000			−0.0334077
$897$		−	2.00000i		−	0.0667781i
$898$			16.0000i			0.533927i
$899$	−48.0000			−1.60089
$900$	−3.00000	−	4.00000i	−0.100000	−	0.133333i
$901$	24.0000			0.799556
$902$			4.00000i			0.133185i
$903$		−	6.00000i		−	0.199667i
$904$	−6.00000			−0.199557
$905$	−8.00000	+	16.0000i	−0.265929	+	0.531858i
$906$	2.00000			0.0664455
$907$		−	6.00000i		−	0.199227i	−0.995026	−	0.0996134i	$-0.968239\pi$
$907$		−	6.00000i		−	0.199227i	0.995026	−	0.0996134i	$-0.0317606\pi$
$908$			8.00000i			0.265489i
$909$	10.0000			0.331679
$910$	2.00000	+	1.00000i	0.0662994	+	0.0331497i
$911$	16.0000			0.530104			0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000			0.530104			0.265052	+	0.964234i	$0.414611\pi$
$912$			4.00000i			0.132453i
$913$		−	24.0000i		−	0.794284i
$914$	22.0000			0.727695
$915$		0			0
$916$	−26.0000			−0.859064
$917$		−	18.0000i		−	0.594412i
$918$			6.00000i			0.198030i
$919$	−40.0000			−1.31948			−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000			−1.31948			−0.659739	+	0.751495i	$0.729333\pi$
$920$	−2.00000	+	4.00000i	−0.0659380	+	0.131876i
$921$	−2.00000			−0.0659022
$922$		−	18.0000i		−	0.592798i
$923$			4.00000i			0.131662i
$924$	2.00000			0.0657952
$925$	−24.0000	+	18.0000i	−0.789115	+	0.591836i
$926$	16.0000			0.525793
$927$			8.00000i			0.262754i
$928$		−	6.00000i		−	0.196960i
$929$	−30.0000			−0.984268			−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000			−0.984268			−0.492134	+	0.870519i	$0.663783\pi$
$930$	8.00000	−	16.0000i	0.262330	−	0.524661i
$931$	4.00000			0.131095
$932$			10.0000i			0.327561i
$933$		−	24.0000i		−	0.785725i
$934$	4.00000			0.130884
$935$	−24.0000	−	12.0000i	−0.784884	−	0.392442i
$936$	−1.00000			−0.0326860
$937$			6.00000i			0.196011i	0.995186	+	0.0980057i	$0.0312463\pi$
$937$			6.00000i			0.196011i	−0.995186	+	0.0980057i	$0.968754\pi$
$938$			8.00000i			0.261209i
$939$	2.00000			0.0652675
$940$	4.00000	+	2.00000i	0.130466	+	0.0652328i
$941$	−30.0000			−0.977972			−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000			−0.977972			−0.488986	+	0.872292i	$0.662633\pi$
$942$		−	2.00000i		−	0.0651635i
$943$		−	4.00000i		−	0.130258i
$944$	−12.0000			−0.390567
$945$	1.00000	−	2.00000i	0.0325300	−	0.0650600i
$946$	−12.0000			−0.390154
$947$		−	12.0000i		−	0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$		−	12.0000i		−	0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$		−	8.00000i		−	0.259828i
$949$	−4.00000			−0.129845
$950$	−16.0000	+	12.0000i	−0.519109	+	0.389331i
$951$	−22.0000			−0.713399
$952$			6.00000i			0.194461i
$953$			30.0000i			0.971795i	0.874016	+	0.485898i	$0.161507\pi$
$953$			30.0000i			0.971795i	−0.874016	+	0.485898i	$0.838493\pi$
$954$	4.00000			0.129505
$955$	−8.00000	+	16.0000i	−0.258874	+	0.517748i
$956$	−4.00000			−0.129369
$957$			12.0000i			0.387905i
$958$			32.0000i			1.03387i
$959$	−6.00000			−0.193750
$960$	2.00000	+	1.00000i	0.0645497	+	0.0322749i
$961$	33.0000			1.06452
$962$			6.00000i			0.193448i
$963$			8.00000i			0.257796i
$964$	−14.0000			−0.450910
$965$	−4.00000	−	2.00000i	−0.128765	−	0.0643823i
$966$	−2.00000			−0.0643489
$967$		−	24.0000i		−	0.771788i	−0.922543	−	0.385894i	$-0.873893\pi$
$967$		−	24.0000i		−	0.771788i	0.922543	−	0.385894i	$-0.126107\pi$
$968$			7.00000i			0.224989i
$969$	24.0000			0.770991
$970$		0			0
$971$	46.0000			1.47621			0.738105	−	0.674686i	$-0.235721\pi$
$971$	46.0000			1.47621			0.738105	+	0.674686i	$0.235721\pi$
$972$			1.00000i			0.0320750i
$973$			16.0000i			0.512936i
$974$	8.00000			0.256337
$975$	−3.00000	−	4.00000i	−0.0960769	−	0.128103i
$976$		0			0
$977$		−	18.0000i		−	0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$		−	18.0000i		−	0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$		−	16.0000i		−	0.511624i
$979$	20.0000			0.639203
$980$	1.00000	−	2.00000i	0.0319438	−	0.0638877i
$981$	16.0000			0.510841
$982$			20.0000i			0.638226i
$983$			26.0000i			0.829271i	0.909988	+	0.414636i	$0.136091\pi$
$983$			26.0000i			0.829271i	−0.909988	+	0.414636i	$0.863909\pi$
$984$	−2.00000			−0.0637577
$985$	44.0000	+	22.0000i	1.40196	+	0.700978i
$986$	−36.0000			−1.14647
$987$			2.00000i			0.0636607i
$988$			4.00000i			0.127257i
$989$	12.0000			0.381578
$990$	−4.00000	−	2.00000i	−0.127128	−	0.0635642i
$991$	24.0000			0.762385			0.381193	−	0.924496i	$-0.375513\pi$
$991$	24.0000			0.762385			0.381193	+	0.924496i	$0.375513\pi$
$992$			8.00000i			0.254000i
$993$			8.00000i			0.253872i
$994$	4.00000			0.126872
$995$	−6.00000	+	12.0000i	−0.190213	+	0.380426i
$996$	12.0000			0.380235
$997$		−	42.0000i		−	1.33015i	−0.746775	−	0.665077i	$-0.768399\pi$
$997$		−	42.0000i		−	1.33015i	0.746775	−	0.665077i	$-0.231601\pi$
$998$			32.0000i			1.01294i
$999$	6.00000			0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2730.2.l.k.1639.2	yes	2
5.4	even	2	inner	2730.2.l.k.1639.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2730.2.l.k.1639.1	✓	2	5.4	even	2	inner
2730.2.l.k.1639.2	yes	2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2730 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2730.l (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +(1.00000 - 2.00000i) q^{5} +1.00000 q^{6} -1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +(1.00000 - 2.00000i) q^{5} +1.00000 q^{6} -1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +(2.00000 + 1.00000i) q^{10} +2.00000 q^{11} +1.00000i q^{12} +1.00000i q^{13} +1.00000 q^{14} +(-2.00000 - 1.00000i) q^{15} +1.00000 q^{16} -6.00000i q^{17} -1.00000i q^{18} -4.00000 q^{19} +(-1.00000 + 2.00000i) q^{20} -1.00000 q^{21} +2.00000i q^{22} -2.00000i q^{23} -1.00000 q^{24} +(-3.00000 - 4.00000i) q^{25} -1.00000 q^{26} +1.00000i q^{27} +1.00000i q^{28} -6.00000 q^{29} +(1.00000 - 2.00000i) q^{30} +8.00000 q^{31} +1.00000i q^{32} -2.00000i q^{33} +6.00000 q^{34} +(-2.00000 - 1.00000i) q^{35} +1.00000 q^{36} -6.00000i q^{37} -4.00000i q^{38} +1.00000 q^{39} +(-2.00000 - 1.00000i) q^{40} +2.00000 q^{41} -1.00000i q^{42} +6.00000i q^{43} -2.00000 q^{44} +(-1.00000 + 2.00000i) q^{45} +2.00000 q^{46} -2.00000i q^{47} -1.00000i q^{48} -1.00000 q^{49} +(4.00000 - 3.00000i) q^{50} -6.00000 q^{51} -1.00000i q^{52} +4.00000i q^{53} -1.00000 q^{54} +(2.00000 - 4.00000i) q^{55} -1.00000 q^{56} +4.00000i q^{57} -6.00000i q^{58} -12.0000 q^{59} +(2.00000 + 1.00000i) q^{60} +8.00000i q^{62} +1.00000i q^{63} -1.00000 q^{64} +(2.00000 + 1.00000i) q^{65} +2.00000 q^{66} +8.00000i q^{67} +6.00000i q^{68} -2.00000 q^{69} +(1.00000 - 2.00000i) q^{70} +4.00000 q^{71} +1.00000i q^{72} +4.00000i q^{73} +6.00000 q^{74} +(-4.00000 + 3.00000i) q^{75} +4.00000 q^{76} -2.00000i q^{77} +1.00000i q^{78} -8.00000 q^{79} +(1.00000 - 2.00000i) q^{80} +1.00000 q^{81} +2.00000i q^{82} -12.0000i q^{83} +1.00000 q^{84} +(-12.0000 - 6.00000i) q^{85} -6.00000 q^{86} +6.00000i q^{87} -2.00000i q^{88} +10.0000 q^{89} +(-2.00000 - 1.00000i) q^{90} +1.00000 q^{91} +2.00000i q^{92} -8.00000i q^{93} +2.00000 q^{94} +(-4.00000 + 8.00000i) q^{95} +1.00000 q^{96} -1.00000i q^{98} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^5 + 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{9} + 4 q^{10} + 4 q^{11} + 2 q^{14} - 4 q^{15} + 2 q^{16} - 8 q^{19} - 2 q^{20} - 2 q^{21} - 2 q^{24} - 6 q^{25} - 2 q^{26} - 12 q^{29} + 2 q^{30} + 16 q^{31} + 12 q^{34} - 4 q^{35} + 2 q^{36} + 2 q^{39} - 4 q^{40} + 4 q^{41} - 4 q^{44} - 2 q^{45} + 4 q^{46} - 2 q^{49} + 8 q^{50} - 12 q^{51} - 2 q^{54} + 4 q^{55} - 2 q^{56} - 24 q^{59} + 4 q^{60} - 2 q^{64} + 4 q^{65} + 4 q^{66} - 4 q^{69} + 2 q^{70} + 8 q^{71} + 12 q^{74} - 8 q^{75} + 8 q^{76} - 16 q^{79} + 2 q^{80} + 2 q^{81} + 2 q^{84} - 24 q^{85} - 12 q^{86} + 20 q^{89} - 4 q^{90} + 2 q^{91} + 4 q^{94} - 8 q^{95} + 2 q^{96} - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^5 + 2 * q^6 - 2 * q^9 + 4 * q^10 + 4 * q^11 + 2 * q^14 - 4 * q^15 + 2 * q^16 - 8 * q^19 - 2 * q^20 - 2 * q^21 - 2 * q^24 - 6 * q^25 - 2 * q^26 - 12 * q^29 + 2 * q^30 + 16 * q^31 + 12 * q^34 - 4 * q^35 + 2 * q^36 + 2 * q^39 - 4 * q^40 + 4 * q^41 - 4 * q^44 - 2 * q^45 + 4 * q^46 - 2 * q^49 + 8 * q^50 - 12 * q^51 - 2 * q^54 + 4 * q^55 - 2 * q^56 - 24 * q^59 + 4 * q^60 - 2 * q^64 + 4 * q^65 + 4 * q^66 - 4 * q^69 + 2 * q^70 + 8 * q^71 + 12 * q^74 - 8 * q^75 + 8 * q^76 - 16 * q^79 + 2 * q^80 + 2 * q^81 + 2 * q^84 - 24 * q^85 - 12 * q^86 + 20 * q^89 - 4 * q^90 + 2 * q^91 + 4 * q^94 - 8 * q^95 + 2 * q^96 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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