LMFDB - Embedded newform 399.2.j.b.172.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [399,2,Mod(58,399)] 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("399.58"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(399, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 399 = 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	399.j (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,-1,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	no
Analytic conductor:	$3.18603104065$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			172.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	399.172
Dual form			399.2.j.b.58.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.500000 + 0.866025i) q^{3} +(1.00000 - 1.73205i) q^{4} +(-1.00000 - 1.73205i) q^{5} +(0.500000 + 2.59808i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(1.50000 - 2.59808i) q^{11} +(1.00000 + 1.73205i) q^{12} +2.00000 q^{13} +2.00000 q^{15} +(-2.00000 - 3.46410i) q^{16} +(3.50000 - 6.06218i) q^{17} +(0.500000 + 0.866025i) q^{19} -4.00000 q^{20} +(-2.50000 - 0.866025i) q^{21} +(-2.50000 - 4.33013i) q^{23} +(0.500000 - 0.866025i) q^{25} +1.00000 q^{27} +(5.00000 + 1.73205i) q^{28} +2.00000 q^{29} +(-5.00000 + 8.66025i) q^{31} +(1.50000 + 2.59808i) q^{33} +(4.00000 - 3.46410i) q^{35} -2.00000 q^{36} +(-4.00000 - 6.92820i) q^{37} +(-1.00000 + 1.73205i) q^{39} +6.00000 q^{41} +12.0000 q^{43} +(-3.00000 - 5.19615i) q^{44} +(-1.00000 + 1.73205i) q^{45} +(2.50000 + 4.33013i) q^{47} +4.00000 q^{48} +(-6.50000 + 2.59808i) q^{49} +(3.50000 + 6.06218i) q^{51} +(2.00000 - 3.46410i) q^{52} +(-2.00000 + 3.46410i) q^{53} -6.00000 q^{55} -1.00000 q^{57} +(-7.00000 + 12.1244i) q^{59} +(2.00000 - 3.46410i) q^{60} +(6.50000 + 11.2583i) q^{61} +(2.00000 - 1.73205i) q^{63} -8.00000 q^{64} +(-2.00000 - 3.46410i) q^{65} +(1.00000 - 1.73205i) q^{67} +(-7.00000 - 12.1244i) q^{68} +5.00000 q^{69} -10.0000 q^{71} +(-0.500000 + 0.866025i) q^{73} +(0.500000 + 0.866025i) q^{75} +2.00000 q^{76} +(7.50000 + 2.59808i) q^{77} +(2.00000 + 3.46410i) q^{79} +(-4.00000 + 6.92820i) q^{80} +(-0.500000 + 0.866025i) q^{81} -9.00000 q^{83} +(-4.00000 + 3.46410i) q^{84} -14.0000 q^{85} +(-1.00000 + 1.73205i) q^{87} +(-9.00000 - 15.5885i) q^{89} +(1.00000 + 5.19615i) q^{91} -10.0000 q^{92} +(-5.00000 - 8.66025i) q^{93} +(1.00000 - 1.73205i) q^{95} +6.00000 q^{97} -3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} + 2 q^{4} - 2 q^{5} + q^{7} - q^{9} + 3 q^{11} + 2 q^{12} + 4 q^{13} + 4 q^{15} - 4 q^{16} + 7 q^{17} + q^{19} - 8 q^{20} - 5 q^{21} - 5 q^{23} + q^{25} + 2 q^{27} + 10 q^{28} + 4 q^{29}+ \cdots - 6 q^{99}+O(q^{100}) $ 2 * q - q^3 + 2 * q^4 - 2 * q^5 + q^7 - q^9 + 3 * q^11 + 2 * q^12 + 4 * q^13 + 4 * q^15 - 4 * q^16 + 7 * q^17 + q^19 - 8 * q^20 - 5 * q^21 - 5 * q^23 + q^25 + 2 * q^27 + 10 * q^28 + 4 * q^29 - 10 * q^31 + 3 * q^33 + 8 * q^35 - 4 * q^36 - 8 * q^37 - 2 * q^39 + 12 * q^41 + 24 * q^43 - 6 * q^44 - 2 * q^45 + 5 * q^47 + 8 * q^48 - 13 * q^49 + 7 * q^51 + 4 * q^52 - 4 * q^53 - 12 * q^55 - 2 * q^57 - 14 * q^59 + 4 * q^60 + 13 * q^61 + 4 * q^63 - 16 * q^64 - 4 * q^65 + 2 * q^67 - 14 * q^68 + 10 * q^69 - 20 * q^71 - q^73 + q^75 + 4 * q^76 + 15 * q^77 + 4 * q^79 - 8 * q^80 - q^81 - 18 * q^83 - 8 * q^84 - 28 * q^85 - 2 * q^87 - 18 * q^89 + 2 * q^91 - 20 * q^92 - 10 * q^93 + 2 * q^95 + 12 * q^97 - 6 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/399\mathbb{Z}\right)^\times$.

$n$	$115$	$134$	$211$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$1$

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 $

$2$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$2$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$3$	−0.500000	+	0.866025i	−0.288675	+	0.500000i
$3$	−0.500000	+	0.866025i	−0.288675	+	0.500000i
$4$	1.00000	−	1.73205i	0.500000	−	0.866025i
$5$	−1.00000	−	1.73205i	−0.447214	−	0.774597i	0.550990	−	0.834512i	$-0.314250\pi$
$5$	−1.00000	−	1.73205i	−0.447214	−	0.774597i	−0.998203	+	0.0599153i	$0.980917\pi$
$6$		0			0
$7$	0.500000	+	2.59808i	0.188982	+	0.981981i
$7$	0.500000	+	2.59808i	0.188982	+	0.981981i
$8$		0			0
$9$	−0.500000	−	0.866025i	−0.166667	−	0.288675i
$10$		0			0
$11$	1.50000	−	2.59808i	0.452267	−	0.783349i	−0.546259	−	0.837616i	$-0.683949\pi$
$11$	1.50000	−	2.59808i	0.452267	−	0.783349i	0.998526	+	0.0542666i	$0.0172821\pi$
$12$	1.00000	+	1.73205i	0.288675	+	0.500000i
$13$	2.00000			0.554700			0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000			0.554700			0.277350	+	0.960769i	$0.410544\pi$
$14$		0			0
$15$	2.00000			0.516398
$16$	−2.00000	−	3.46410i	−0.500000	−	0.866025i
$17$	3.50000	−	6.06218i	0.848875	−	1.47029i	−0.0333386	−	0.999444i	$-0.510614\pi$
$17$	3.50000	−	6.06218i	0.848875	−	1.47029i	0.882213	−	0.470850i	$-0.156053\pi$
$18$		0			0
$19$	0.500000	+	0.866025i	0.114708	+	0.198680i
$19$	0.500000	+	0.866025i	0.114708	+	0.198680i
$20$	−4.00000			−0.894427
$21$	−2.50000	−	0.866025i	−0.545545	−	0.188982i
$22$		0			0
$23$	−2.50000	−	4.33013i	−0.521286	−	0.902894i	−0.999694	−	0.0247559i	$-0.992119\pi$
$23$	−2.50000	−	4.33013i	−0.521286	−	0.902894i	0.478407	−	0.878138i	$-0.341214\pi$
$24$		0			0
$25$	0.500000	−	0.866025i	0.100000	−	0.173205i
$26$		0			0
$27$	1.00000			0.192450
$28$	5.00000	+	1.73205i	0.944911	+	0.327327i
$29$	2.00000			0.371391			0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000			0.371391			0.185695	+	0.982607i	$0.440546\pi$
$30$		0			0
$31$	−5.00000	+	8.66025i	−0.898027	+	1.55543i	−0.0680129	+	0.997684i	$0.521666\pi$
$31$	−5.00000	+	8.66025i	−0.898027	+	1.55543i	−0.830014	+	0.557743i	$0.811667\pi$
$32$		0			0
$33$	1.50000	+	2.59808i	0.261116	+	0.452267i
$34$		0			0
$35$	4.00000	−	3.46410i	0.676123	−	0.585540i
$36$	−2.00000			−0.333333
$37$	−4.00000	−	6.92820i	−0.657596	−	1.13899i	−0.981236	−	0.192809i	$-0.938240\pi$
$37$	−4.00000	−	6.92820i	−0.657596	−	1.13899i	0.323640	−	0.946180i	$-0.395093\pi$
$38$		0			0
$39$	−1.00000	+	1.73205i	−0.160128	+	0.277350i
$40$		0			0
$41$	6.00000			0.937043			0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000			0.937043			0.468521	+	0.883452i	$0.344787\pi$
$42$		0			0
$43$	12.0000			1.82998			0.914991	−	0.403473i	$-0.132197\pi$
$43$	12.0000			1.82998			0.914991	+	0.403473i	$0.132197\pi$
$44$	−3.00000	−	5.19615i	−0.452267	−	0.783349i
$45$	−1.00000	+	1.73205i	−0.149071	+	0.258199i
$46$		0			0
$47$	2.50000	+	4.33013i	0.364662	+	0.631614i	0.988722	−	0.149763i	$-0.0478510\pi$
$47$	2.50000	+	4.33013i	0.364662	+	0.631614i	−0.624059	+	0.781377i	$0.714518\pi$
$48$	4.00000			0.577350
$49$	−6.50000	+	2.59808i	−0.928571	+	0.371154i
$50$		0			0
$51$	3.50000	+	6.06218i	0.490098	+	0.848875i
$52$	2.00000	−	3.46410i	0.277350	−	0.480384i
$53$	−2.00000	+	3.46410i	−0.274721	+	0.475831i	−0.970065	−	0.242846i	$-0.921919\pi$
$53$	−2.00000	+	3.46410i	−0.274721	+	0.475831i	0.695344	+	0.718677i	$0.255252\pi$
$54$		0			0
$55$	−6.00000			−0.809040
$56$		0			0
$57$	−1.00000			−0.132453
$58$		0			0
$59$	−7.00000	+	12.1244i	−0.911322	+	1.57846i	−0.0991242	+	0.995075i	$0.531604\pi$
$59$	−7.00000	+	12.1244i	−0.911322	+	1.57846i	−0.812198	+	0.583382i	$0.801729\pi$
$60$	2.00000	−	3.46410i	0.258199	−	0.447214i
$61$	6.50000	+	11.2583i	0.832240	+	1.44148i	0.896258	+	0.443533i	$0.146275\pi$
$61$	6.50000	+	11.2583i	0.832240	+	1.44148i	−0.0640184	+	0.997949i	$0.520392\pi$
$62$		0			0
$63$	2.00000	−	1.73205i	0.251976	−	0.218218i
$64$	−8.00000			−1.00000
$65$	−2.00000	−	3.46410i	−0.248069	−	0.429669i
$66$		0			0
$67$	1.00000	−	1.73205i	0.122169	−	0.211604i	−0.798454	−	0.602056i	$-0.794348\pi$
$67$	1.00000	−	1.73205i	0.122169	−	0.211604i	0.920623	+	0.390453i	$0.127682\pi$
$68$	−7.00000	−	12.1244i	−0.848875	−	1.47029i
$69$	5.00000			0.601929
$70$		0			0
$71$	−10.0000			−1.18678			−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000			−1.18678			−0.593391	+	0.804914i	$0.702211\pi$
$72$		0			0
$73$	−0.500000	+	0.866025i	−0.0585206	+	0.101361i	−0.893801	−	0.448463i	$-0.851972\pi$
$73$	−0.500000	+	0.866025i	−0.0585206	+	0.101361i	0.835281	+	0.549823i	$0.185305\pi$
$74$		0			0
$75$	0.500000	+	0.866025i	0.0577350	+	0.100000i
$76$	2.00000			0.229416
$77$	7.50000	+	2.59808i	0.854704	+	0.296078i
$78$		0			0
$79$	2.00000	+	3.46410i	0.225018	+	0.389742i	0.956325	−	0.292306i	$-0.0944227\pi$
$79$	2.00000	+	3.46410i	0.225018	+	0.389742i	−0.731307	+	0.682048i	$0.761089\pi$
$80$	−4.00000	+	6.92820i	−0.447214	+	0.774597i
$81$	−0.500000	+	0.866025i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	−9.00000			−0.987878			−0.493939	−	0.869496i	$-0.664443\pi$
$83$	−9.00000			−0.987878			−0.493939	+	0.869496i	$0.664443\pi$
$84$	−4.00000	+	3.46410i	−0.436436	+	0.377964i
$85$	−14.0000			−1.51851
$86$		0			0
$87$	−1.00000	+	1.73205i	−0.107211	+	0.185695i
$88$		0			0
$89$	−9.00000	−	15.5885i	−0.953998	−	1.65237i	−0.736644	−	0.676280i	$-0.763591\pi$
$89$	−9.00000	−	15.5885i	−0.953998	−	1.65237i	−0.217354	−	0.976093i	$-0.569742\pi$
$90$		0			0
$91$	1.00000	+	5.19615i	0.104828	+	0.544705i
$92$	−10.0000			−1.04257
$93$	−5.00000	−	8.66025i	−0.518476	−	0.898027i
$94$		0			0
$95$	1.00000	−	1.73205i	0.102598	−	0.177705i
$96$		0			0
$97$	6.00000			0.609208			0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000			0.609208			0.304604	+	0.952479i	$0.401476\pi$
$98$		0			0
$99$	−3.00000			−0.301511
$100$	−1.00000	−	1.73205i	−0.100000	−	0.173205i
$101$	−6.50000	+	11.2583i	−0.646774	+	1.12025i	0.337115	+	0.941464i	$0.390549\pi$
$101$	−6.50000	+	11.2583i	−0.646774	+	1.12025i	−0.983889	+	0.178782i	$0.942784\pi$
$102$		0			0
$103$	−4.00000	−	6.92820i	−0.394132	−	0.682656i	0.598858	−	0.800855i	$-0.295621\pi$
$103$	−4.00000	−	6.92820i	−0.394132	−	0.682656i	−0.992990	+	0.118199i	$0.962288\pi$
$104$		0			0
$105$	1.00000	+	5.19615i	0.0975900	+	0.507093i
$106$		0			0
$107$	9.00000	+	15.5885i	0.870063	+	1.50699i	0.861931	+	0.507026i	$0.169255\pi$
$107$	9.00000	+	15.5885i	0.870063	+	1.50699i	0.00813215	+	0.999967i	$0.497411\pi$
$108$	1.00000	−	1.73205i	0.0962250	−	0.166667i
$109$	−3.00000	+	5.19615i	−0.287348	+	0.497701i	−0.973176	−	0.230063i	$-0.926107\pi$
$109$	−3.00000	+	5.19615i	−0.287348	+	0.497701i	0.685828	+	0.727764i	$0.259440\pi$
$110$		0			0
$111$	8.00000			0.759326
$112$	8.00000	−	6.92820i	0.755929	−	0.654654i
$113$	12.0000			1.12887			0.564433	−	0.825479i	$-0.309095\pi$
$113$	12.0000			1.12887			0.564433	+	0.825479i	$0.309095\pi$
$114$		0			0
$115$	−5.00000	+	8.66025i	−0.466252	+	0.807573i
$116$	2.00000	−	3.46410i	0.185695	−	0.321634i
$117$	−1.00000	−	1.73205i	−0.0924500	−	0.160128i
$118$		0			0
$119$	17.5000	+	6.06218i	1.60422	+	0.555719i
$120$		0			0
$121$	1.00000	+	1.73205i	0.0909091	+	0.157459i
$122$		0			0
$123$	−3.00000	+	5.19615i	−0.270501	+	0.468521i
$124$	10.0000	+	17.3205i	0.898027	+	1.55543i
$125$	−12.0000			−1.07331
$126$		0			0
$127$	16.0000			1.41977			0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000			1.41977			0.709885	+	0.704317i	$0.248747\pi$
$128$		0			0
$129$	−6.00000	+	10.3923i	−0.528271	+	0.914991i
$130$		0			0
$131$	−0.500000	−	0.866025i	−0.0436852	−	0.0756650i	0.843356	−	0.537355i	$-0.180577\pi$
$131$	−0.500000	−	0.866025i	−0.0436852	−	0.0756650i	−0.887041	+	0.461690i	$0.847243\pi$
$132$	6.00000			0.522233
$133$	−2.00000	+	1.73205i	−0.173422	+	0.150188i
$134$		0			0
$135$	−1.00000	−	1.73205i	−0.0860663	−	0.149071i
$136$		0			0
$137$	1.50000	−	2.59808i	0.128154	−	0.221969i	−0.794808	−	0.606861i	$-0.792428\pi$
$137$	1.50000	−	2.59808i	0.128154	−	0.221969i	0.922961	+	0.384893i	$0.125762\pi$
$138$		0			0
$139$	5.00000			0.424094			0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000			0.424094			0.212047	+	0.977259i	$0.431987\pi$
$140$	−2.00000	−	10.3923i	−0.169031	−	0.878310i
$141$	−5.00000			−0.421076
$142$		0			0
$143$	3.00000	−	5.19615i	0.250873	−	0.434524i
$144$	−2.00000	+	3.46410i	−0.166667	+	0.288675i
$145$	−2.00000	−	3.46410i	−0.166091	−	0.287678i
$146$		0			0
$147$	1.00000	−	6.92820i	0.0824786	−	0.571429i
$148$	−16.0000			−1.31519
$149$	−3.00000	−	5.19615i	−0.245770	−	0.425685i	0.716578	−	0.697507i	$-0.245707\pi$
$149$	−3.00000	−	5.19615i	−0.245770	−	0.425685i	−0.962348	+	0.271821i	$0.912374\pi$
$150$		0			0
$151$	4.00000	−	6.92820i	0.325515	−	0.563809i	−0.656101	−	0.754673i	$-0.727796\pi$
$151$	4.00000	−	6.92820i	0.325515	−	0.563809i	0.981617	+	0.190864i	$0.0611289\pi$
$152$		0			0
$153$	−7.00000			−0.565916
$154$		0			0
$155$	20.0000			1.60644
$156$	2.00000	+	3.46410i	0.160128	+	0.277350i
$157$	−2.50000	+	4.33013i	−0.199522	+	0.345582i	−0.948373	−	0.317156i	$-0.897272\pi$
$157$	−2.50000	+	4.33013i	−0.199522	+	0.345582i	0.748852	+	0.662738i	$0.230606\pi$
$158$		0			0
$159$	−2.00000	−	3.46410i	−0.158610	−	0.274721i
$160$		0			0
$161$	10.0000	−	8.66025i	0.788110	−	0.682524i
$162$		0			0
$163$	−4.50000	−	7.79423i	−0.352467	−	0.610491i	0.634214	−	0.773158i	$-0.281324\pi$
$163$	−4.50000	−	7.79423i	−0.352467	−	0.610491i	−0.986681	+	0.162667i	$0.947991\pi$
$164$	6.00000	−	10.3923i	0.468521	−	0.811503i
$165$	3.00000	−	5.19615i	0.233550	−	0.404520i
$166$		0			0
$167$	14.0000			1.08335			0.541676	−	0.840587i	$-0.317790\pi$
$167$	14.0000			1.08335			0.541676	+	0.840587i	$0.317790\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$		0			0
$171$	0.500000	−	0.866025i	0.0382360	−	0.0662266i
$172$	12.0000	−	20.7846i	0.914991	−	1.58481i
$173$	−2.00000	−	3.46410i	−0.152057	−	0.263371i	0.779926	−	0.625871i	$-0.215256\pi$
$173$	−2.00000	−	3.46410i	−0.152057	−	0.263371i	−0.931984	+	0.362500i	$0.881923\pi$
$174$		0			0
$175$	2.50000	+	0.866025i	0.188982	+	0.0654654i
$176$	−12.0000			−0.904534
$177$	−7.00000	−	12.1244i	−0.526152	−	0.911322i
$178$		0			0
$179$	−5.00000	+	8.66025i	−0.373718	+	0.647298i	−0.990134	−	0.140122i	$-0.955250\pi$
$179$	−5.00000	+	8.66025i	−0.373718	+	0.647298i	0.616417	+	0.787420i	$0.288584\pi$
$180$	2.00000	+	3.46410i	0.149071	+	0.258199i
$181$	16.0000			1.18927			0.594635	−	0.803996i	$-0.297296\pi$
$181$	16.0000			1.18927			0.594635	+	0.803996i	$0.297296\pi$
$182$		0			0
$183$	−13.0000			−0.960988
$184$		0			0
$185$	−8.00000	+	13.8564i	−0.588172	+	1.01874i
$186$		0			0
$187$	−10.5000	−	18.1865i	−0.767836	−	1.32993i
$188$	10.0000			0.729325
$189$	0.500000	+	2.59808i	0.0363696	+	0.188982i
$190$		0			0
$191$	4.00000	+	6.92820i	0.289430	+	0.501307i	0.973674	−	0.227946i	$-0.0732010\pi$
$191$	4.00000	+	6.92820i	0.289430	+	0.501307i	−0.684244	+	0.729253i	$0.739868\pi$
$192$	4.00000	−	6.92820i	0.288675	−	0.500000i
$193$	−4.00000	+	6.92820i	−0.287926	+	0.498703i	−0.973315	−	0.229475i	$-0.926299\pi$
$193$	−4.00000	+	6.92820i	−0.287926	+	0.498703i	0.685388	+	0.728178i	$0.259632\pi$
$194$		0			0
$195$	4.00000			0.286446
$196$	−2.00000	+	13.8564i	−0.142857	+	0.989743i
$197$	−5.00000			−0.356235			−0.178118	−	0.984009i	$-0.557001\pi$
$197$	−5.00000			−0.356235			−0.178118	+	0.984009i	$0.557001\pi$
$198$		0			0
$199$	8.00000	−	13.8564i	0.567105	−	0.982255i	−0.429745	−	0.902950i	$-0.641397\pi$
$199$	8.00000	−	13.8564i	0.567105	−	0.982255i	0.996850	−	0.0793045i	$-0.0252700\pi$
$200$		0			0
$201$	1.00000	+	1.73205i	0.0705346	+	0.122169i
$202$		0			0
$203$	1.00000	+	5.19615i	0.0701862	+	0.364698i
$204$	14.0000			0.980196
$205$	−6.00000	−	10.3923i	−0.419058	−	0.725830i
$206$		0			0
$207$	−2.50000	+	4.33013i	−0.173762	+	0.300965i
$208$	−4.00000	−	6.92820i	−0.277350	−	0.480384i
$209$	3.00000			0.207514
$210$		0			0
$211$	−8.00000			−0.550743			−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000			−0.550743			−0.275371	+	0.961338i	$0.588801\pi$
$212$	4.00000	+	6.92820i	0.274721	+	0.475831i
$213$	5.00000	−	8.66025i	0.342594	−	0.593391i
$214$		0			0
$215$	−12.0000	−	20.7846i	−0.818393	−	1.41750i
$216$		0			0
$217$	−25.0000	−	8.66025i	−1.69711	−	0.587896i
$218$		0			0
$219$	−0.500000	−	0.866025i	−0.0337869	−	0.0585206i
$220$	−6.00000	+	10.3923i	−0.404520	+	0.700649i
$221$	7.00000	−	12.1244i	0.470871	−	0.815572i
$222$		0			0
$223$	2.00000			0.133930			0.0669650	−	0.997755i	$-0.478668\pi$
$223$	2.00000			0.133930			0.0669650	+	0.997755i	$0.478668\pi$
$224$		0			0
$225$	−1.00000			−0.0666667
$226$		0			0
$227$	−5.00000	+	8.66025i	−0.331862	+	0.574801i	−0.982877	−	0.184263i	$-0.941010\pi$
$227$	−5.00000	+	8.66025i	−0.331862	+	0.574801i	0.651015	+	0.759065i	$0.274343\pi$
$228$	−1.00000	+	1.73205i	−0.0662266	+	0.114708i
$229$	−0.500000	−	0.866025i	−0.0330409	−	0.0572286i	0.849032	−	0.528341i	$-0.177186\pi$
$229$	−0.500000	−	0.866025i	−0.0330409	−	0.0572286i	−0.882073	+	0.471113i	$0.843853\pi$
$230$		0			0
$231$	−6.00000	+	5.19615i	−0.394771	+	0.341882i
$232$		0			0
$233$	10.5000	+	18.1865i	0.687878	+	1.19144i	0.972523	+	0.232806i	$0.0747909\pi$
$233$	10.5000	+	18.1865i	0.687878	+	1.19144i	−0.284645	+	0.958633i	$0.591876\pi$
$234$		0			0
$235$	5.00000	−	8.66025i	0.326164	−	0.564933i
$236$	14.0000	+	24.2487i	0.911322	+	1.57846i
$237$	−4.00000			−0.259828
$238$		0			0
$239$	−1.00000			−0.0646846			−0.0323423	−	0.999477i	$-0.510297\pi$
$239$	−1.00000			−0.0646846			−0.0323423	+	0.999477i	$0.510297\pi$
$240$	−4.00000	−	6.92820i	−0.258199	−	0.447214i
$241$	4.00000	−	6.92820i	0.257663	−	0.446285i	−0.707953	−	0.706260i	$-0.750381\pi$
$241$	4.00000	−	6.92820i	0.257663	−	0.446285i	0.965615	+	0.259975i	$0.0837143\pi$
$242$		0			0
$243$	−0.500000	−	0.866025i	−0.0320750	−	0.0555556i
$244$	26.0000			1.66448
$245$	11.0000	+	8.66025i	0.702764	+	0.553283i
$246$		0			0
$247$	1.00000	+	1.73205i	0.0636285	+	0.110208i
$248$		0			0
$249$	4.50000	−	7.79423i	0.285176	−	0.493939i
$250$		0			0
$251$	16.0000			1.00991			0.504956	−	0.863145i	$-0.331509\pi$
$251$	16.0000			1.00991			0.504956	+	0.863145i	$0.331509\pi$
$252$	−1.00000	−	5.19615i	−0.0629941	−	0.327327i
$253$	−15.0000			−0.943042
$254$		0			0
$255$	7.00000	−	12.1244i	0.438357	−	0.759257i
$256$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$257$	6.00000	+	10.3923i	0.374270	+	0.648254i	0.990217	−	0.139533i	$-0.0445601\pi$
$257$	6.00000	+	10.3923i	0.374270	+	0.648254i	−0.615948	+	0.787787i	$0.711227\pi$
$258$		0			0
$259$	16.0000	−	13.8564i	0.994192	−	0.860995i
$260$	−8.00000			−0.496139
$261$	−1.00000	−	1.73205i	−0.0618984	−	0.107211i
$262$		0			0
$263$	5.50000	−	9.52628i	0.339145	−	0.587416i	−0.645128	−	0.764075i	$-0.723196\pi$
$263$	5.50000	−	9.52628i	0.339145	−	0.587416i	0.984272	+	0.176659i	$0.0565291\pi$
$264$		0			0
$265$	8.00000			0.491436
$266$		0			0
$267$	18.0000			1.10158
$268$	−2.00000	−	3.46410i	−0.122169	−	0.211604i
$269$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$269$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$270$		0			0
$271$	0.500000	+	0.866025i	0.0303728	+	0.0526073i	0.880812	−	0.473466i	$-0.156997\pi$
$271$	0.500000	+	0.866025i	0.0303728	+	0.0526073i	−0.850439	+	0.526073i	$0.823664\pi$
$272$	−28.0000			−1.69775
$273$	−5.00000	−	1.73205i	−0.302614	−	0.104828i
$274$		0			0
$275$	−1.50000	−	2.59808i	−0.0904534	−	0.156670i
$276$	5.00000	−	8.66025i	0.300965	−	0.521286i
$277$	−8.50000	+	14.7224i	−0.510716	+	0.884585i	0.489207	+	0.872167i	$0.337286\pi$
$277$	−8.50000	+	14.7224i	−0.510716	+	0.884585i	−0.999923	+	0.0124177i	$0.996047\pi$
$278$		0			0
$279$	10.0000			0.598684
$280$		0			0
$281$	−20.0000			−1.19310			−0.596550	−	0.802576i	$-0.703462\pi$
$281$	−20.0000			−1.19310			−0.596550	+	0.802576i	$0.703462\pi$
$282$		0			0
$283$	2.00000	−	3.46410i	0.118888	−	0.205919i	−0.800439	−	0.599414i	$-0.795400\pi$
$283$	2.00000	−	3.46410i	0.118888	−	0.205919i	0.919327	+	0.393494i	$0.128734\pi$
$284$	−10.0000	+	17.3205i	−0.593391	+	1.02778i
$285$	1.00000	+	1.73205i	0.0592349	+	0.102598i
$286$		0			0
$287$	3.00000	+	15.5885i	0.177084	+	0.920158i
$288$		0			0
$289$	−16.0000	−	27.7128i	−0.941176	−	1.63017i
$290$		0			0
$291$	−3.00000	+	5.19615i	−0.175863	+	0.304604i
$292$	1.00000	+	1.73205i	0.0585206	+	0.101361i
$293$	−18.0000			−1.05157			−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000			−1.05157			−0.525786	+	0.850617i	$0.676229\pi$
$294$		0			0
$295$	28.0000			1.63022
$296$		0			0
$297$	1.50000	−	2.59808i	0.0870388	−	0.150756i
$298$		0			0
$299$	−5.00000	−	8.66025i	−0.289157	−	0.500835i
$300$	2.00000			0.115470
$301$	6.00000	+	31.1769i	0.345834	+	1.79701i
$302$		0			0
$303$	−6.50000	−	11.2583i	−0.373415	−	0.646774i
$304$	2.00000	−	3.46410i	0.114708	−	0.198680i
$305$	13.0000	−	22.5167i	0.744378	−	1.28930i
$306$		0			0
$307$	16.0000			0.913168			0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000			0.913168			0.456584	+	0.889680i	$0.349073\pi$
$308$	12.0000	−	10.3923i	0.683763	−	0.592157i
$309$	8.00000			0.455104
$310$		0			0
$311$	−2.50000	+	4.33013i	−0.141762	+	0.245539i	−0.928160	−	0.372181i	$-0.878610\pi$
$311$	−2.50000	+	4.33013i	−0.141762	+	0.245539i	0.786398	+	0.617720i	$0.211943\pi$
$312$		0			0
$313$	−9.50000	−	16.4545i	−0.536972	−	0.930062i	−0.999065	−	0.0432311i	$-0.986235\pi$
$313$	−9.50000	−	16.4545i	−0.536972	−	0.930062i	0.462093	−	0.886831i	$-0.347098\pi$
$314$		0			0
$315$	−5.00000	−	1.73205i	−0.281718	−	0.0975900i
$316$	8.00000			0.450035
$317$	15.0000	+	25.9808i	0.842484	+	1.45922i	0.887788	+	0.460252i	$0.152241\pi$
$317$	15.0000	+	25.9808i	0.842484	+	1.45922i	−0.0453045	+	0.998973i	$0.514426\pi$
$318$		0			0
$319$	3.00000	−	5.19615i	0.167968	−	0.290929i
$320$	8.00000	+	13.8564i	0.447214	+	0.774597i
$321$	−18.0000			−1.00466
$322$		0			0
$323$	7.00000			0.389490
$324$	1.00000	+	1.73205i	0.0555556	+	0.0962250i
$325$	1.00000	−	1.73205i	0.0554700	−	0.0960769i
$326$		0			0
$327$	−3.00000	−	5.19615i	−0.165900	−	0.287348i
$328$		0			0
$329$	−10.0000	+	8.66025i	−0.551318	+	0.477455i
$330$		0			0
$331$	−13.0000	−	22.5167i	−0.714545	−	1.23763i	−0.963135	−	0.269019i	$-0.913301\pi$
$331$	−13.0000	−	22.5167i	−0.714545	−	1.23763i	0.248590	−	0.968609i	$-0.420033\pi$
$332$	−9.00000	+	15.5885i	−0.493939	+	0.855528i
$333$	−4.00000	+	6.92820i	−0.219199	+	0.379663i
$334$		0			0
$335$	−4.00000			−0.218543
$336$	2.00000	+	10.3923i	0.109109	+	0.566947i
$337$	12.0000			0.653682			0.326841	−	0.945079i	$-0.394016\pi$
$337$	12.0000			0.653682			0.326841	+	0.945079i	$0.394016\pi$
$338$		0			0
$339$	−6.00000	+	10.3923i	−0.325875	+	0.564433i
$340$	−14.0000	+	24.2487i	−0.759257	+	1.31507i
$341$	15.0000	+	25.9808i	0.812296	+	1.40694i
$342$		0			0
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$		0			0
$345$	−5.00000	−	8.66025i	−0.269191	−	0.466252i
$346$		0			0
$347$	6.00000	−	10.3923i	0.322097	−	0.557888i	−0.658824	−	0.752297i	$-0.728946\pi$
$347$	6.00000	−	10.3923i	0.322097	−	0.557888i	0.980921	+	0.194409i	$0.0622790\pi$
$348$	2.00000	+	3.46410i	0.107211	+	0.185695i
$349$	−27.0000			−1.44528			−0.722638	−	0.691226i	$-0.757071\pi$
$349$	−27.0000			−1.44528			−0.722638	+	0.691226i	$0.757071\pi$
$350$		0			0
$351$	2.00000			0.106752
$352$		0			0
$353$	7.00000	−	12.1244i	0.372572	−	0.645314i	−0.617388	−	0.786659i	$-0.711809\pi$
$353$	7.00000	−	12.1244i	0.372572	−	0.645314i	0.989960	+	0.141344i	$0.0451425\pi$
$354$		0			0
$355$	10.0000	+	17.3205i	0.530745	+	0.919277i
$356$	−36.0000			−1.90800
$357$	−14.0000	+	12.1244i	−0.740959	+	0.641689i
$358$		0			0
$359$	15.5000	+	26.8468i	0.818059	+	1.41692i	0.907111	+	0.420892i	$0.138283\pi$
$359$	15.5000	+	26.8468i	0.818059	+	1.41692i	−0.0890519	+	0.996027i	$0.528384\pi$
$360$		0			0
$361$	−0.500000	+	0.866025i	−0.0263158	+	0.0455803i
$362$		0			0
$363$	−2.00000			−0.104973
$364$	10.0000	+	3.46410i	0.524142	+	0.181568i
$365$	2.00000			0.104685
$366$		0			0
$367$	−8.00000	+	13.8564i	−0.417597	+	0.723299i	−0.995697	−	0.0926670i	$-0.970461\pi$
$367$	−8.00000	+	13.8564i	−0.417597	+	0.723299i	0.578101	+	0.815966i	$0.303794\pi$
$368$	−10.0000	+	17.3205i	−0.521286	+	0.902894i
$369$	−3.00000	−	5.19615i	−0.156174	−	0.270501i
$370$		0			0
$371$	−10.0000	−	3.46410i	−0.519174	−	0.179847i
$372$	−20.0000			−1.03695
$373$	5.00000	+	8.66025i	0.258890	+	0.448411i	0.965945	−	0.258748i	$-0.0833099\pi$
$373$	5.00000	+	8.66025i	0.258890	+	0.448411i	−0.707055	+	0.707159i	$0.749977\pi$
$374$		0			0
$375$	6.00000	−	10.3923i	0.309839	−	0.536656i
$376$		0			0
$377$	4.00000			0.206010
$378$		0			0
$379$	8.00000			0.410932			0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000			0.410932			0.205466	+	0.978664i	$0.434129\pi$
$380$	−2.00000	−	3.46410i	−0.102598	−	0.177705i
$381$	−8.00000	+	13.8564i	−0.409852	+	0.709885i
$382$		0			0
$383$	−3.00000	−	5.19615i	−0.153293	−	0.265511i	0.779143	−	0.626846i	$-0.215654\pi$
$383$	−3.00000	−	5.19615i	−0.153293	−	0.265511i	−0.932436	+	0.361335i	$0.882321\pi$
$384$		0			0
$385$	−3.00000	−	15.5885i	−0.152894	−	0.794461i
$386$		0			0
$387$	−6.00000	−	10.3923i	−0.304997	−	0.528271i
$388$	6.00000	−	10.3923i	0.304604	−	0.527589i
$389$	14.5000	−	25.1147i	0.735179	−	1.27337i	−0.219465	−	0.975620i	$-0.570431\pi$
$389$	14.5000	−	25.1147i	0.735179	−	1.27337i	0.954645	−	0.297747i	$-0.0962353\pi$
$390$		0			0
$391$	−35.0000			−1.77003
$392$		0			0
$393$	1.00000			0.0504433
$394$		0			0
$395$	4.00000	−	6.92820i	0.201262	−	0.348596i
$396$	−3.00000	+	5.19615i	−0.150756	+	0.261116i
$397$	11.5000	+	19.9186i	0.577168	+	0.999685i	0.995802	+	0.0915300i	$0.0291757\pi$
$397$	11.5000	+	19.9186i	0.577168	+	0.999685i	−0.418634	+	0.908155i	$0.637491\pi$
$398$		0			0
$399$	−0.500000	−	2.59808i	−0.0250313	−	0.130066i
$400$	−4.00000			−0.200000
$401$	3.00000	+	5.19615i	0.149813	+	0.259483i	0.931158	−	0.364615i	$-0.118800\pi$
$401$	3.00000	+	5.19615i	0.149813	+	0.259483i	−0.781345	+	0.624099i	$0.785466\pi$
$402$		0			0
$403$	−10.0000	+	17.3205i	−0.498135	+	0.862796i
$404$	13.0000	+	22.5167i	0.646774	+	1.12025i
$405$	2.00000			0.0993808
$406$		0			0
$407$	−24.0000			−1.18964
$408$		0			0
$409$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$409$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$410$		0			0
$411$	1.50000	+	2.59808i	0.0739895	+	0.128154i
$412$	−16.0000			−0.788263
$413$	−35.0000	−	12.1244i	−1.72224	−	0.596601i
$414$		0			0
$415$	9.00000	+	15.5885i	0.441793	+	0.765207i
$416$		0			0
$417$	−2.50000	+	4.33013i	−0.122426	+	0.212047i
$418$		0			0
$419$	−9.00000			−0.439679			−0.219839	−	0.975536i	$-0.570553\pi$
$419$	−9.00000			−0.439679			−0.219839	+	0.975536i	$0.570553\pi$
$420$	10.0000	+	3.46410i	0.487950	+	0.169031i
$421$	4.00000			0.194948			0.0974740	−	0.995238i	$-0.468924\pi$
$421$	4.00000			0.194948			0.0974740	+	0.995238i	$0.468924\pi$
$422$		0			0
$423$	2.50000	−	4.33013i	0.121554	−	0.210538i
$424$		0			0
$425$	−3.50000	−	6.06218i	−0.169775	−	0.294059i
$426$		0			0
$427$	−26.0000	+	22.5167i	−1.25823	+	1.08966i
$428$	36.0000			1.74013
$429$	3.00000	+	5.19615i	0.144841	+	0.250873i
$430$		0			0
$431$	3.00000	−	5.19615i	0.144505	−	0.250290i	−0.784683	−	0.619897i	$-0.787174\pi$
$431$	3.00000	−	5.19615i	0.144505	−	0.250290i	0.929188	+	0.369607i	$0.120508\pi$
$432$	−2.00000	−	3.46410i	−0.0962250	−	0.166667i
$433$	−18.0000			−0.865025			−0.432512	−	0.901628i	$-0.642373\pi$
$433$	−18.0000			−0.865025			−0.432512	+	0.901628i	$0.642373\pi$
$434$		0			0
$435$	4.00000			0.191785
$436$	6.00000	+	10.3923i	0.287348	+	0.497701i
$437$	2.50000	−	4.33013i	0.119591	−	0.207138i
$438$		0			0
$439$	−7.00000	−	12.1244i	−0.334092	−	0.578664i	0.649218	−	0.760602i	$-0.275096\pi$
$439$	−7.00000	−	12.1244i	−0.334092	−	0.578664i	−0.983310	+	0.181938i	$0.941763\pi$
$440$		0			0
$441$	5.50000	+	4.33013i	0.261905	+	0.206197i
$442$		0			0
$443$	−10.5000	−	18.1865i	−0.498870	−	0.864068i	0.501129	−	0.865373i	$-0.332918\pi$
$443$	−10.5000	−	18.1865i	−0.498870	−	0.864068i	−0.999999	+	0.00130426i	$0.999585\pi$
$444$	8.00000	−	13.8564i	0.379663	−	0.657596i
$445$	−18.0000	+	31.1769i	−0.853282	+	1.47793i
$446$		0			0
$447$	6.00000			0.283790
$448$	−4.00000	−	20.7846i	−0.188982	−	0.981981i
$449$	−18.0000			−0.849473			−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000			−0.849473			−0.424736	+	0.905317i	$0.639633\pi$
$450$		0			0
$451$	9.00000	−	15.5885i	0.423793	−	0.734032i
$452$	12.0000	−	20.7846i	0.564433	−	0.977626i
$453$	4.00000	+	6.92820i	0.187936	+	0.325515i
$454$		0			0
$455$	8.00000	−	6.92820i	0.375046	−	0.324799i
$456$		0			0
$457$	12.5000	+	21.6506i	0.584725	+	1.01277i	0.994910	+	0.100771i	$0.0321310\pi$
$457$	12.5000	+	21.6506i	0.584725	+	1.01277i	−0.410184	+	0.912003i	$0.634536\pi$
$458$		0			0
$459$	3.50000	−	6.06218i	0.163366	−	0.282958i
$460$	10.0000	+	17.3205i	0.466252	+	0.807573i
$461$	−15.0000			−0.698620			−0.349310	−	0.937007i	$-0.613584\pi$
$461$	−15.0000			−0.698620			−0.349310	+	0.937007i	$0.613584\pi$
$462$		0			0
$463$	−9.00000			−0.418265			−0.209133	−	0.977887i	$-0.567064\pi$
$463$	−9.00000			−0.418265			−0.209133	+	0.977887i	$0.567064\pi$
$464$	−4.00000	−	6.92820i	−0.185695	−	0.321634i
$465$	−10.0000	+	17.3205i	−0.463739	+	0.803219i
$466$		0			0
$467$	2.50000	+	4.33013i	0.115686	+	0.200374i	0.918054	−	0.396456i	$-0.129760\pi$
$467$	2.50000	+	4.33013i	0.115686	+	0.200374i	−0.802368	+	0.596830i	$0.796427\pi$
$468$	−4.00000			−0.184900
$469$	5.00000	+	1.73205i	0.230879	+	0.0799787i
$470$		0			0
$471$	−2.50000	−	4.33013i	−0.115194	−	0.199522i
$472$		0			0
$473$	18.0000	−	31.1769i	0.827641	−	1.43352i
$474$		0			0
$475$	1.00000			0.0458831
$476$	28.0000	−	24.2487i	1.28338	−	1.11144i
$477$	4.00000			0.183147
$478$		0			0
$479$	−15.5000	+	26.8468i	−0.708213	+	1.22666i	0.257306	+	0.966330i	$0.417165\pi$
$479$	−15.5000	+	26.8468i	−0.708213	+	1.22666i	−0.965519	+	0.260331i	$0.916168\pi$
$480$		0			0
$481$	−8.00000	−	13.8564i	−0.364769	−	0.631798i
$482$		0			0
$483$	2.50000	+	12.9904i	0.113754	+	0.591083i
$484$	4.00000			0.181818
$485$	−6.00000	−	10.3923i	−0.272446	−	0.471890i
$486$		0			0
$487$	−1.00000	+	1.73205i	−0.0453143	+	0.0784867i	−0.887793	−	0.460243i	$-0.847762\pi$
$487$	−1.00000	+	1.73205i	−0.0453143	+	0.0784867i	0.842479	+	0.538730i	$0.181096\pi$
$488$		0			0
$489$	9.00000			0.406994
$490$		0			0
$491$	27.0000			1.21849			0.609246	−	0.792981i	$-0.291472\pi$
$491$	27.0000			1.21849			0.609246	+	0.792981i	$0.291472\pi$
$492$	6.00000	+	10.3923i	0.270501	+	0.468521i
$493$	7.00000	−	12.1244i	0.315264	−	0.546054i
$494$		0			0
$495$	3.00000	+	5.19615i	0.134840	+	0.233550i
$496$	40.0000			1.79605
$497$	−5.00000	−	25.9808i	−0.224281	−	1.16540i
$498$		0			0
$499$	6.00000	+	10.3923i	0.268597	+	0.465223i	0.968500	−	0.249015i	$-0.0801067\pi$
$499$	6.00000	+	10.3923i	0.268597	+	0.465223i	−0.699903	+	0.714238i	$0.746773\pi$
$500$	−12.0000	+	20.7846i	−0.536656	+	0.929516i
$501$	−7.00000	+	12.1244i	−0.312737	+	0.541676i
$502$		0			0
$503$	−11.0000			−0.490466			−0.245233	−	0.969464i	$-0.578864\pi$
$503$	−11.0000			−0.490466			−0.245233	+	0.969464i	$0.578864\pi$
$504$		0			0
$505$	26.0000			1.15698
$506$		0			0
$507$	4.50000	−	7.79423i	0.199852	−	0.346154i
$508$	16.0000	−	27.7128i	0.709885	−	1.22956i
$509$	−15.0000	−	25.9808i	−0.664863	−	1.15158i	−0.979322	−	0.202306i	$-0.935156\pi$
$509$	−15.0000	−	25.9808i	−0.664863	−	1.15158i	0.314459	−	0.949271i	$-0.398177\pi$
$510$		0			0
$511$	−2.50000	−	0.866025i	−0.110593	−	0.0383107i
$512$		0			0
$513$	0.500000	+	0.866025i	0.0220755	+	0.0382360i
$514$		0			0
$515$	−8.00000	+	13.8564i	−0.352522	+	0.610586i
$516$	12.0000	+	20.7846i	0.528271	+	0.914991i
$517$	15.0000			0.659699
$518$		0			0
$519$	4.00000			0.175581
$520$		0			0
$521$	14.0000	−	24.2487i	0.613351	−	1.06236i	−0.377320	−	0.926083i	$-0.623154\pi$
$521$	14.0000	−	24.2487i	0.613351	−	1.06236i	0.990671	−	0.136272i	$-0.0435123\pi$
$522$		0			0
$523$	−17.0000	−	29.4449i	−0.743358	−	1.28753i	−0.950958	−	0.309320i	$-0.899899\pi$
$523$	−17.0000	−	29.4449i	−0.743358	−	1.28753i	0.207600	−	0.978214i	$-0.433435\pi$
$524$	−2.00000			−0.0873704
$525$	−2.00000	+	1.73205i	−0.0872872	+	0.0755929i
$526$		0			0
$527$	35.0000	+	60.6218i	1.52462	+	2.64073i
$528$	6.00000	−	10.3923i	0.261116	−	0.452267i
$529$	−1.00000	+	1.73205i	−0.0434783	+	0.0753066i
$530$		0			0
$531$	14.0000			0.607548
$532$	1.00000	+	5.19615i	0.0433555	+	0.225282i
$533$	12.0000			0.519778
$534$		0			0
$535$	18.0000	−	31.1769i	0.778208	−	1.34790i
$536$		0			0
$537$	−5.00000	−	8.66025i	−0.215766	−	0.373718i
$538$		0			0
$539$	−3.00000	+	20.7846i	−0.129219	+	0.895257i
$540$	−4.00000			−0.172133
$541$	7.00000	+	12.1244i	0.300954	+	0.521267i	0.976352	−	0.216186i	$-0.0693618\pi$
$541$	7.00000	+	12.1244i	0.300954	+	0.521267i	−0.675399	+	0.737453i	$0.736028\pi$
$542$		0			0
$543$	−8.00000	+	13.8564i	−0.343313	+	0.594635i
$544$		0			0
$545$	12.0000			0.514024
$546$		0			0
$547$	−14.0000			−0.598597			−0.299298	−	0.954160i	$-0.596753\pi$
$547$	−14.0000			−0.598597			−0.299298	+	0.954160i	$0.596753\pi$
$548$	−3.00000	−	5.19615i	−0.128154	−	0.221969i
$549$	6.50000	−	11.2583i	0.277413	−	0.480494i
$550$		0			0
$551$	1.00000	+	1.73205i	0.0426014	+	0.0737878i
$552$		0			0
$553$	−8.00000	+	6.92820i	−0.340195	+	0.294617i
$554$		0			0
$555$	−8.00000	−	13.8564i	−0.339581	−	0.588172i
$556$	5.00000	−	8.66025i	0.212047	−	0.367277i
$557$	5.00000	−	8.66025i	0.211857	−	0.366947i	−0.740439	−	0.672124i	$-0.765382\pi$
$557$	5.00000	−	8.66025i	0.211857	−	0.366947i	0.952296	+	0.305177i	$0.0987156\pi$
$558$		0			0
$559$	24.0000			1.01509
$560$	−20.0000	−	6.92820i	−0.845154	−	0.292770i
$561$	21.0000			0.886621
$562$		0			0
$563$	5.00000	−	8.66025i	0.210725	−	0.364986i	−0.741217	−	0.671266i	$-0.765751\pi$
$563$	5.00000	−	8.66025i	0.210725	−	0.364986i	0.951942	+	0.306280i	$0.0990842\pi$
$564$	−5.00000	+	8.66025i	−0.210538	+	0.364662i
$565$	−12.0000	−	20.7846i	−0.504844	−	0.874415i
$566$		0			0
$567$	−2.50000	−	0.866025i	−0.104990	−	0.0363696i
$568$		0			0
$569$	−10.0000	−	17.3205i	−0.419222	−	0.726113i	0.576640	−	0.816999i	$-0.304364\pi$
$569$	−10.0000	−	17.3205i	−0.419222	−	0.726113i	−0.995861	+	0.0908852i	$0.971030\pi$
$570$		0			0
$571$	−23.5000	+	40.7032i	−0.983444	+	1.70338i	−0.334790	+	0.942293i	$0.608665\pi$
$571$	−23.5000	+	40.7032i	−0.983444	+	1.70338i	−0.648655	+	0.761083i	$0.724668\pi$
$572$	−6.00000	−	10.3923i	−0.250873	−	0.434524i
$573$	−8.00000			−0.334205
$574$		0			0
$575$	−5.00000			−0.208514
$576$	4.00000	+	6.92820i	0.166667	+	0.288675i
$577$	14.5000	−	25.1147i	0.603643	−	1.04554i	−0.388621	−	0.921397i	$-0.627049\pi$
$577$	14.5000	−	25.1147i	0.603643	−	1.04554i	0.992264	−	0.124143i	$-0.0396180\pi$
$578$		0			0
$579$	−4.00000	−	6.92820i	−0.166234	−	0.287926i
$580$	−8.00000			−0.332182
$581$	−4.50000	−	23.3827i	−0.186691	−	0.970077i
$582$		0			0
$583$	6.00000	+	10.3923i	0.248495	+	0.430405i
$584$		0			0
$585$	−2.00000	+	3.46410i	−0.0826898	+	0.143223i
$586$		0			0
$587$	−15.0000			−0.619116			−0.309558	−	0.950881i	$-0.600181\pi$
$587$	−15.0000			−0.619116			−0.309558	+	0.950881i	$0.600181\pi$
$588$	−11.0000	−	8.66025i	−0.453632	−	0.357143i
$589$	−10.0000			−0.412043
$590$		0			0
$591$	2.50000	−	4.33013i	0.102836	−	0.178118i
$592$	−16.0000	+	27.7128i	−0.657596	+	1.13899i
$593$	−3.50000	−	6.06218i	−0.143728	−	0.248944i	0.785170	−	0.619281i	$-0.212576\pi$
$593$	−3.50000	−	6.06218i	−0.143728	−	0.248944i	−0.928898	+	0.370337i	$0.879242\pi$
$594$		0			0
$595$	−7.00000	−	36.3731i	−0.286972	−	1.49115i
$596$	−12.0000			−0.491539
$597$	8.00000	+	13.8564i	0.327418	+	0.567105i
$598$		0			0
$599$	8.00000	−	13.8564i	0.326871	−	0.566157i	−0.655018	−	0.755613i	$-0.727339\pi$
$599$	8.00000	−	13.8564i	0.326871	−	0.566157i	0.981889	+	0.189456i	$0.0606724\pi$
$600$		0			0
$601$	−20.0000			−0.815817			−0.407909	−	0.913023i	$-0.633742\pi$
$601$	−20.0000			−0.815817			−0.407909	+	0.913023i	$0.633742\pi$
$602$		0			0
$603$	−2.00000			−0.0814463
$604$	−8.00000	−	13.8564i	−0.325515	−	0.563809i
$605$	2.00000	−	3.46410i	0.0813116	−	0.140836i
$606$		0			0
$607$	9.00000	+	15.5885i	0.365299	+	0.632716i	0.988824	−	0.149087i	$-0.0476335\pi$
$607$	9.00000	+	15.5885i	0.365299	+	0.632716i	−0.623525	+	0.781803i	$0.714300\pi$
$608$		0			0
$609$	−5.00000	−	1.73205i	−0.202610	−	0.0701862i
$610$		0			0
$611$	5.00000	+	8.66025i	0.202278	+	0.350356i
$612$	−7.00000	+	12.1244i	−0.282958	+	0.490098i
$613$	21.5000	−	37.2391i	0.868377	−	1.50407i	0.00472215	−	0.999989i	$-0.498497\pi$
$613$	21.5000	−	37.2391i	0.868377	−	1.50407i	0.863655	−	0.504084i	$-0.168170\pi$
$614$		0			0
$615$	12.0000			0.483887
$616$		0			0
$617$	10.0000			0.402585			0.201292	−	0.979531i	$-0.435486\pi$
$617$	10.0000			0.402585			0.201292	+	0.979531i	$0.435486\pi$
$618$		0			0
$619$	−19.5000	+	33.7750i	−0.783771	+	1.35753i	0.145959	+	0.989291i	$0.453373\pi$
$619$	−19.5000	+	33.7750i	−0.783771	+	1.35753i	−0.929730	+	0.368241i	$0.879960\pi$
$620$	20.0000	−	34.6410i	0.803219	−	1.39122i
$621$	−2.50000	−	4.33013i	−0.100322	−	0.173762i
$622$		0			0
$623$	36.0000	−	31.1769i	1.44231	−	1.24908i
$624$	8.00000			0.320256
$625$	9.50000	+	16.4545i	0.380000	+	0.658179i
$626$		0			0
$627$	−1.50000	+	2.59808i	−0.0599042	+	0.103757i
$628$	5.00000	+	8.66025i	0.199522	+	0.345582i
$629$	−56.0000			−2.23287
$630$		0			0
$631$	48.0000			1.91085			0.955425	−	0.295234i	$-0.0953977\pi$
$631$	48.0000			1.91085			0.955425	+	0.295234i	$0.0953977\pi$
$632$		0			0
$633$	4.00000	−	6.92820i	0.158986	−	0.275371i
$634$		0			0
$635$	−16.0000	−	27.7128i	−0.634941	−	1.09975i
$636$	−8.00000			−0.317221
$637$	−13.0000	+	5.19615i	−0.515079	+	0.205879i
$638$		0			0
$639$	5.00000	+	8.66025i	0.197797	+	0.342594i
$640$		0			0
$641$	−13.0000	+	22.5167i	−0.513469	+	0.889355i	0.486409	+	0.873731i	$0.338307\pi$
$641$	−13.0000	+	22.5167i	−0.513469	+	0.889355i	−0.999878	+	0.0156233i	$0.995027\pi$
$642$		0			0
$643$	−9.00000			−0.354925			−0.177463	−	0.984128i	$-0.556789\pi$
$643$	−9.00000			−0.354925			−0.177463	+	0.984128i	$0.556789\pi$
$644$	−5.00000	−	25.9808i	−0.197028	−	1.02379i
$645$	24.0000			0.944999
$646$		0			0
$647$	−12.0000	+	20.7846i	−0.471769	+	0.817127i	−0.999478	−	0.0322975i	$-0.989718\pi$
$647$	−12.0000	+	20.7846i	−0.471769	+	0.817127i	0.527710	+	0.849425i	$0.323051\pi$
$648$		0			0
$649$	21.0000	+	36.3731i	0.824322	+	1.42777i
$650$		0			0
$651$	20.0000	−	17.3205i	0.783862	−	0.678844i
$652$	−18.0000			−0.704934
$653$	11.5000	+	19.9186i	0.450030	+	0.779474i	0.998387	−	0.0567696i	$-0.0180800\pi$
$653$	11.5000	+	19.9186i	0.450030	+	0.779474i	−0.548358	+	0.836244i	$0.684747\pi$
$654$		0			0
$655$	−1.00000	+	1.73205i	−0.0390732	+	0.0676768i
$656$	−12.0000	−	20.7846i	−0.468521	−	0.811503i
$657$	1.00000			0.0390137
$658$		0			0
$659$	−18.0000			−0.701180			−0.350590	−	0.936529i	$-0.614019\pi$
$659$	−18.0000			−0.701180			−0.350590	+	0.936529i	$0.614019\pi$
$660$	−6.00000	−	10.3923i	−0.233550	−	0.404520i
$661$	−6.00000	+	10.3923i	−0.233373	+	0.404214i	−0.958799	−	0.284087i	$-0.908310\pi$
$661$	−6.00000	+	10.3923i	−0.233373	+	0.404214i	0.725426	+	0.688301i	$0.241643\pi$
$662$		0			0
$663$	7.00000	+	12.1244i	0.271857	+	0.470871i
$664$		0			0
$665$	5.00000	+	1.73205i	0.193892	+	0.0671660i
$666$		0			0
$667$	−5.00000	−	8.66025i	−0.193601	−	0.335326i
$668$	14.0000	−	24.2487i	0.541676	−	0.938211i
$669$	−1.00000	+	1.73205i	−0.0386622	+	0.0669650i
$670$		0			0
$671$	39.0000			1.50558
$672$		0			0
$673$	34.0000			1.31060			0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000			1.31060			0.655302	+	0.755367i	$0.272541\pi$
$674$		0			0
$675$	0.500000	−	0.866025i	0.0192450	−	0.0333333i
$676$	−9.00000	+	15.5885i	−0.346154	+	0.599556i
$677$	6.00000	+	10.3923i	0.230599	+	0.399409i	0.957984	−	0.286820i	$-0.0925982\pi$
$677$	6.00000	+	10.3923i	0.230599	+	0.399409i	−0.727386	+	0.686229i	$0.759265\pi$
$678$		0			0
$679$	3.00000	+	15.5885i	0.115129	+	0.598230i
$680$		0			0
$681$	−5.00000	−	8.66025i	−0.191600	−	0.331862i
$682$		0			0
$683$	−22.0000	+	38.1051i	−0.841807	+	1.45805i	0.0465592	+	0.998916i	$0.485174\pi$
$683$	−22.0000	+	38.1051i	−0.841807	+	1.45805i	−0.888366	+	0.459136i	$0.848159\pi$
$684$	−1.00000	−	1.73205i	−0.0382360	−	0.0662266i
$685$	−6.00000			−0.229248
$686$		0			0
$687$	1.00000			0.0381524
$688$	−24.0000	−	41.5692i	−0.914991	−	1.58481i
$689$	−4.00000	+	6.92820i	−0.152388	+	0.263944i
$690$		0			0
$691$	9.50000	+	16.4545i	0.361397	+	0.625958i	0.988191	−	0.153227i	$-0.0489666\pi$
$691$	9.50000	+	16.4545i	0.361397	+	0.625958i	−0.626794	+	0.779185i	$0.715633\pi$
$692$	−8.00000			−0.304114
$693$	−1.50000	−	7.79423i	−0.0569803	−	0.296078i
$694$		0			0
$695$	−5.00000	−	8.66025i	−0.189661	−	0.328502i
$696$		0			0
$697$	21.0000	−	36.3731i	0.795432	−	1.37773i
$698$		0			0
$699$	−21.0000			−0.794293
$700$	4.00000	−	3.46410i	0.151186	−	0.130931i
$701$	−15.0000			−0.566542			−0.283271	−	0.959040i	$-0.591420\pi$
$701$	−15.0000			−0.566542			−0.283271	+	0.959040i	$0.591420\pi$
$702$		0			0
$703$	4.00000	−	6.92820i	0.150863	−	0.261302i
$704$	−12.0000	+	20.7846i	−0.452267	+	0.783349i
$705$	5.00000	+	8.66025i	0.188311	+	0.326164i
$706$		0			0
$707$	−32.5000	−	11.2583i	−1.22229	−	0.423413i
$708$	−28.0000			−1.05230
$709$	−0.500000	−	0.866025i	−0.0187779	−	0.0325243i	0.856484	−	0.516174i	$-0.172644\pi$
$709$	−0.500000	−	0.866025i	−0.0187779	−	0.0325243i	−0.875262	+	0.483650i	$0.839311\pi$
$710$		0			0
$711$	2.00000	−	3.46410i	0.0750059	−	0.129914i
$712$		0			0
$713$	50.0000			1.87251
$714$		0			0
$715$	−12.0000			−0.448775
$716$	10.0000	+	17.3205i	0.373718	+	0.647298i
$717$	0.500000	−	0.866025i	0.0186728	−	0.0323423i
$718$		0			0
$719$	−13.5000	−	23.3827i	−0.503465	−	0.872027i	−0.999992	−	0.00400572i	$-0.998725\pi$
$719$	−13.5000	−	23.3827i	−0.503465	−	0.872027i	0.496527	−	0.868021i	$-0.334608\pi$
$720$	8.00000			0.298142
$721$	16.0000	−	13.8564i	0.595871	−	0.516040i
$722$		0			0
$723$	4.00000	+	6.92820i	0.148762	+	0.257663i
$724$	16.0000	−	27.7128i	0.594635	−	1.02994i
$725$	1.00000	−	1.73205i	0.0371391	−	0.0643268i
$726$		0			0
$727$	28.0000			1.03846			0.519231	−	0.854634i	$-0.326218\pi$
$727$	28.0000			1.03846			0.519231	+	0.854634i	$0.326218\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	42.0000	−	72.7461i	1.55343	−	2.69061i
$732$	−13.0000	+	22.5167i	−0.480494	+	0.832240i
$733$	9.00000	+	15.5885i	0.332423	+	0.575773i	0.982986	−	0.183679i	$-0.0588007\pi$
$733$	9.00000	+	15.5885i	0.332423	+	0.575773i	−0.650564	+	0.759452i	$0.725467\pi$
$734$		0			0
$735$	−13.0000	+	5.19615i	−0.479512	+	0.191663i
$736$		0			0
$737$	−3.00000	−	5.19615i	−0.110506	−	0.191403i
$738$		0			0
$739$	20.5000	−	35.5070i	0.754105	−	1.30615i	−0.191714	−	0.981451i	$-0.561404\pi$
$739$	20.5000	−	35.5070i	0.754105	−	1.30615i	0.945818	−	0.324697i	$-0.105262\pi$
$740$	16.0000	+	27.7128i	0.588172	+	1.01874i
$741$	−2.00000			−0.0734718
$742$		0			0
$743$	−44.0000			−1.61420			−0.807102	−	0.590412i	$-0.798965\pi$
$743$	−44.0000			−1.61420			−0.807102	+	0.590412i	$0.798965\pi$
$744$		0			0
$745$	−6.00000	+	10.3923i	−0.219823	+	0.380745i
$746$		0			0
$747$	4.50000	+	7.79423i	0.164646	+	0.285176i
$748$	−42.0000			−1.53567
$749$	−36.0000	+	31.1769i	−1.31541	+	1.13918i
$750$		0			0
$751$	19.0000	+	32.9090i	0.693320	+	1.20087i	0.970744	+	0.240118i	$0.0771860\pi$
$751$	19.0000	+	32.9090i	0.693320	+	1.20087i	−0.277424	+	0.960748i	$0.589481\pi$
$752$	10.0000	−	17.3205i	0.364662	−	0.631614i
$753$	−8.00000	+	13.8564i	−0.291536	+	0.504956i
$754$		0			0
$755$	−16.0000			−0.582300
$756$	5.00000	+	1.73205i	0.181848	+	0.0629941i
$757$	−17.0000			−0.617876			−0.308938	−	0.951082i	$-0.599973\pi$
$757$	−17.0000			−0.617876			−0.308938	+	0.951082i	$0.599973\pi$
$758$		0			0
$759$	7.50000	−	12.9904i	0.272233	−	0.471521i
$760$		0			0
$761$	−5.00000	−	8.66025i	−0.181250	−	0.313934i	0.761057	−	0.648686i	$-0.224681\pi$
$761$	−5.00000	−	8.66025i	−0.181250	−	0.313934i	−0.942306	+	0.334752i	$0.891348\pi$
$762$		0			0
$763$	−15.0000	−	5.19615i	−0.543036	−	0.188113i
$764$	16.0000			0.578860
$765$	7.00000	+	12.1244i	0.253086	+	0.438357i
$766$		0			0
$767$	−14.0000	+	24.2487i	−0.505511	+	0.875570i
$768$	−8.00000	−	13.8564i	−0.288675	−	0.500000i
$769$	30.0000			1.08183			0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000			1.08183			0.540914	+	0.841078i	$0.318079\pi$
$770$		0			0
$771$	−12.0000			−0.432169
$772$	8.00000	+	13.8564i	0.287926	+	0.498703i
$773$	18.0000	−	31.1769i	0.647415	−	1.12136i	−0.336323	−	0.941747i	$-0.609183\pi$
$773$	18.0000	−	31.1769i	0.647415	−	1.12136i	0.983738	−	0.179609i	$-0.0574833\pi$
$774$		0			0
$775$	5.00000	+	8.66025i	0.179605	+	0.311086i
$776$		0			0
$777$	4.00000	+	20.7846i	0.143499	+	0.745644i
$778$		0			0
$779$	3.00000	+	5.19615i	0.107486	+	0.186171i
$780$	4.00000	−	6.92820i	0.143223	−	0.248069i
$781$	−15.0000	+	25.9808i	−0.536742	+	0.929665i
$782$		0			0
$783$	2.00000			0.0714742
$784$	22.0000	+	17.3205i	0.785714	+	0.618590i
$785$	10.0000			0.356915
$786$		0			0
$787$	−10.0000	+	17.3205i	−0.356462	+	0.617409i	−0.987367	−	0.158450i	$-0.949350\pi$
$787$	−10.0000	+	17.3205i	−0.356462	+	0.617409i	0.630905	+	0.775860i	$0.282684\pi$
$788$	−5.00000	+	8.66025i	−0.178118	+	0.308509i
$789$	5.50000	+	9.52628i	0.195805	+	0.339145i
$790$		0			0
$791$	6.00000	+	31.1769i	0.213335	+	1.10852i
$792$		0			0
$793$	13.0000	+	22.5167i	0.461644	+	0.799590i
$794$		0			0
$795$	−4.00000	+	6.92820i	−0.141865	+	0.245718i
$796$	−16.0000	−	27.7128i	−0.567105	−	0.982255i
$797$	−22.0000			−0.779280			−0.389640	−	0.920967i	$-0.627401\pi$
$797$	−22.0000			−0.779280			−0.389640	+	0.920967i	$0.627401\pi$
$798$		0			0
$799$	35.0000			1.23821
$800$		0			0
$801$	−9.00000	+	15.5885i	−0.317999	+	0.550791i
$802$		0			0
$803$	1.50000	+	2.59808i	0.0529339	+	0.0916841i
$804$	4.00000			0.141069
$805$	−25.0000	−	8.66025i	−0.881134	−	0.305234i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	25.0000	−	43.3013i	0.878953	−	1.52239i	0.0264621	−	0.999650i	$-0.491576\pi$
$809$	25.0000	−	43.3013i	0.878953	−	1.52239i	0.852491	−	0.522742i	$-0.175091\pi$
$810$		0			0
$811$	14.0000			0.491606			0.245803	−	0.969320i	$-0.420948\pi$
$811$	14.0000			0.491606			0.245803	+	0.969320i	$0.420948\pi$
$812$	10.0000	+	3.46410i	0.350931	+	0.121566i
$813$	−1.00000			−0.0350715
$814$		0			0
$815$	−9.00000	+	15.5885i	−0.315256	+	0.546040i
$816$	14.0000	−	24.2487i	0.490098	−	0.848875i
$817$	6.00000	+	10.3923i	0.209913	+	0.363581i
$818$		0			0
$819$	4.00000	−	3.46410i	0.139771	−	0.121046i
$820$	−24.0000			−0.838116
$821$	25.5000	+	44.1673i	0.889956	+	1.54145i	0.839926	+	0.542702i	$0.182599\pi$
$821$	25.5000	+	44.1673i	0.889956	+	1.54145i	0.0500305	+	0.998748i	$0.484068\pi$
$822$		0			0
$823$	7.50000	−	12.9904i	0.261434	−	0.452816i	−0.705190	−	0.709019i	$-0.749138\pi$
$823$	7.50000	−	12.9904i	0.261434	−	0.452816i	0.966623	+	0.256203i	$0.0824714\pi$
$824$		0			0
$825$	3.00000			0.104447
$826$		0			0
$827$	2.00000			0.0695468			0.0347734	−	0.999395i	$-0.488929\pi$
$827$	2.00000			0.0695468			0.0347734	+	0.999395i	$0.488929\pi$
$828$	5.00000	+	8.66025i	0.173762	+	0.300965i
$829$	−17.0000	+	29.4449i	−0.590434	+	1.02266i	0.403739	+	0.914874i	$0.367710\pi$
$829$	−17.0000	+	29.4449i	−0.590434	+	1.02266i	−0.994174	+	0.107788i	$0.965623\pi$
$830$		0			0
$831$	−8.50000	−	14.7224i	−0.294862	−	0.510716i
$832$	−16.0000			−0.554700
$833$	−7.00000	+	48.4974i	−0.242536	+	1.68034i
$834$		0			0
$835$	−14.0000	−	24.2487i	−0.484490	−	0.839161i
$836$	3.00000	−	5.19615i	0.103757	−	0.179713i
$837$	−5.00000	+	8.66025i	−0.172825	+	0.299342i
$838$		0			0
$839$	−12.0000			−0.414286			−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000			−0.414286			−0.207143	+	0.978311i	$0.566417\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$	10.0000	−	17.3205i	0.344418	−	0.596550i
$844$	−8.00000	+	13.8564i	−0.275371	+	0.476957i
$845$	9.00000	+	15.5885i	0.309609	+	0.536259i
$846$		0			0
$847$	−4.00000	+	3.46410i	−0.137442	+	0.119028i
$848$	16.0000			0.549442
$849$	2.00000	+	3.46410i	0.0686398	+	0.118888i
$850$		0			0
$851$	−20.0000	+	34.6410i	−0.685591	+	1.18748i
$852$	−10.0000	−	17.3205i	−0.342594	−	0.593391i
$853$	−5.00000			−0.171197			−0.0855984	−	0.996330i	$-0.527280\pi$
$853$	−5.00000			−0.171197			−0.0855984	+	0.996330i	$0.527280\pi$
$854$		0			0
$855$	−2.00000			−0.0683986
$856$		0			0
$857$	3.00000	−	5.19615i	0.102478	−	0.177497i	−0.810227	−	0.586116i	$-0.800656\pi$
$857$	3.00000	−	5.19615i	0.102478	−	0.177497i	0.912705	+	0.408619i	$0.133990\pi$
$858$		0			0
$859$	−2.00000	−	3.46410i	−0.0682391	−	0.118194i	0.829887	−	0.557931i	$-0.188405\pi$
$859$	−2.00000	−	3.46410i	−0.0682391	−	0.118194i	−0.898126	+	0.439738i	$0.855071\pi$
$860$	−48.0000			−1.63679
$861$	−15.0000	−	5.19615i	−0.511199	−	0.177084i
$862$		0			0
$863$	6.00000	+	10.3923i	0.204242	+	0.353758i	0.949891	−	0.312581i	$-0.101194\pi$
$863$	6.00000	+	10.3923i	0.204242	+	0.353758i	−0.745649	+	0.666339i	$0.767860\pi$
$864$		0			0
$865$	−4.00000	+	6.92820i	−0.136004	+	0.235566i
$866$		0			0
$867$	32.0000			1.08678
$868$	−40.0000	+	34.6410i	−1.35769	+	1.17579i
$869$	12.0000			0.407072
$870$		0			0
$871$	2.00000	−	3.46410i	0.0677674	−	0.117377i
$872$		0			0
$873$	−3.00000	−	5.19615i	−0.101535	−	0.175863i
$874$		0			0
$875$	−6.00000	−	31.1769i	−0.202837	−	1.05397i
$876$	−2.00000			−0.0675737
$877$	−18.0000	−	31.1769i	−0.607817	−	1.05277i	−0.991600	−	0.129346i	$-0.958712\pi$
$877$	−18.0000	−	31.1769i	−0.607817	−	1.05277i	0.383783	−	0.923423i	$-0.374621\pi$
$878$		0			0
$879$	9.00000	−	15.5885i	0.303562	−	0.525786i
$880$	12.0000	+	20.7846i	0.404520	+	0.700649i
$881$	33.0000			1.11180			0.555899	−	0.831250i	$-0.312374\pi$
$881$	33.0000			1.11180			0.555899	+	0.831250i	$0.312374\pi$
$882$		0			0
$883$	−9.00000			−0.302874			−0.151437	−	0.988467i	$-0.548390\pi$
$883$	−9.00000			−0.302874			−0.151437	+	0.988467i	$0.548390\pi$
$884$	−14.0000	−	24.2487i	−0.470871	−	0.815572i
$885$	−14.0000	+	24.2487i	−0.470605	+	0.815112i
$886$		0			0
$887$	−21.0000	−	36.3731i	−0.705111	−	1.22129i	−0.966651	−	0.256096i	$-0.917564\pi$
$887$	−21.0000	−	36.3731i	−0.705111	−	1.22129i	0.261540	−	0.965193i	$-0.415770\pi$
$888$		0			0
$889$	8.00000	+	41.5692i	0.268311	+	1.39419i
$890$		0			0
$891$	1.50000	+	2.59808i	0.0502519	+	0.0870388i
$892$	2.00000	−	3.46410i	0.0669650	−	0.115987i
$893$	−2.50000	+	4.33013i	−0.0836593	+	0.144902i
$894$		0			0
$895$	20.0000			0.668526
$896$		0			0
$897$	10.0000			0.333890
$898$		0			0
$899$	−10.0000	+	17.3205i	−0.333519	+	0.577671i
$900$	−1.00000	+	1.73205i	−0.0333333	+	0.0577350i
$901$	14.0000	+	24.2487i	0.466408	+	0.807842i
$902$		0			0
$903$	−30.0000	−	10.3923i	−0.998337	−	0.345834i
$904$		0			0
$905$	−16.0000	−	27.7128i	−0.531858	−	0.921205i
$906$		0			0
$907$	1.00000	−	1.73205i	0.0332045	−	0.0575118i	−0.848946	−	0.528480i	$-0.822762\pi$
$907$	1.00000	−	1.73205i	0.0332045	−	0.0575118i	0.882150	+	0.470968i	$0.156095\pi$
$908$	10.0000	+	17.3205i	0.331862	+	0.574801i
$909$	13.0000			0.431183
$910$		0			0
$911$	−50.0000			−1.65657			−0.828287	−	0.560304i	$-0.810684\pi$
$911$	−50.0000			−1.65657			−0.828287	+	0.560304i	$0.810684\pi$
$912$	2.00000	+	3.46410i	0.0662266	+	0.114708i
$913$	−13.5000	+	23.3827i	−0.446785	+	0.773854i
$914$		0			0
$915$	13.0000	+	22.5167i	0.429767	+	0.744378i
$916$	−2.00000			−0.0660819
$917$	2.00000	−	1.73205i	0.0660458	−	0.0571974i
$918$		0			0
$919$	−1.50000	−	2.59808i	−0.0494804	−	0.0857026i	0.840224	−	0.542239i	$-0.182423\pi$
$919$	−1.50000	−	2.59808i	−0.0494804	−	0.0857026i	−0.889705	+	0.456536i	$0.849090\pi$
$920$		0			0
$921$	−8.00000	+	13.8564i	−0.263609	+	0.456584i
$922$		0			0
$923$	−20.0000			−0.658308
$924$	3.00000	+	15.5885i	0.0986928	+	0.512823i
$925$	−8.00000			−0.263038
$926$		0			0
$927$	−4.00000	+	6.92820i	−0.131377	+	0.227552i
$928$		0			0
$929$	−17.5000	−	30.3109i	−0.574156	−	0.994468i	−0.996133	−	0.0878612i	$-0.971997\pi$
$929$	−17.5000	−	30.3109i	−0.574156	−	0.994468i	0.421976	−	0.906607i	$-0.361337\pi$
$930$		0			0
$931$	−5.50000	−	4.33013i	−0.180255	−	0.141914i
$932$	42.0000			1.37576
$933$	−2.50000	−	4.33013i	−0.0818463	−	0.141762i
$934$		0			0
$935$	−21.0000	+	36.3731i	−0.686773	+	1.18953i
$936$		0			0
$937$	−10.0000			−0.326686			−0.163343	−	0.986569i	$-0.552228\pi$
$937$	−10.0000			−0.326686			−0.163343	+	0.986569i	$0.552228\pi$
$938$		0			0
$939$	19.0000			0.620042
$940$	−10.0000	−	17.3205i	−0.326164	−	0.564933i
$941$	−19.0000	+	32.9090i	−0.619382	+	1.07280i	0.370216	+	0.928946i	$0.379284\pi$
$941$	−19.0000	+	32.9090i	−0.619382	+	1.07280i	−0.989599	+	0.143856i	$0.954050\pi$
$942$		0			0
$943$	−15.0000	−	25.9808i	−0.488467	−	0.846050i
$944$	56.0000			1.82264
$945$	4.00000	−	3.46410i	0.130120	−	0.112687i
$946$		0			0
$947$	12.0000	+	20.7846i	0.389948	+	0.675409i	0.992442	−	0.122714i	$-0.0391598\pi$
$947$	12.0000	+	20.7846i	0.389948	+	0.675409i	−0.602494	+	0.798123i	$0.705826\pi$
$948$	−4.00000	+	6.92820i	−0.129914	+	0.225018i
$949$	−1.00000	+	1.73205i	−0.0324614	+	0.0562247i
$950$		0			0
$951$	−30.0000			−0.972817
$952$		0			0
$953$	26.0000			0.842223			0.421111	−	0.907009i	$-0.361640\pi$
$953$	26.0000			0.842223			0.421111	+	0.907009i	$0.361640\pi$
$954$		0			0
$955$	8.00000	−	13.8564i	0.258874	−	0.448383i
$956$	−1.00000	+	1.73205i	−0.0323423	+	0.0560185i
$957$	3.00000	+	5.19615i	0.0969762	+	0.167968i
$958$		0			0
$959$	7.50000	+	2.59808i	0.242188	+	0.0838963i
$960$	−16.0000			−0.516398
$961$	−34.5000	−	59.7558i	−1.11290	−	1.92760i
$962$		0			0
$963$	9.00000	−	15.5885i	0.290021	−	0.502331i
$964$	−8.00000	−	13.8564i	−0.257663	−	0.446285i
$965$	16.0000			0.515058
$966$		0			0
$967$	−32.0000			−1.02905			−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000			−1.02905			−0.514525	+	0.857475i	$0.672032\pi$
$968$		0			0
$969$	−3.50000	+	6.06218i	−0.112436	+	0.194745i
$970$		0			0
$971$	9.00000	+	15.5885i	0.288824	+	0.500257i	0.973529	−	0.228562i	$-0.0734025\pi$
$971$	9.00000	+	15.5885i	0.288824	+	0.500257i	−0.684706	+	0.728820i	$0.740069\pi$
$972$	−2.00000			−0.0641500
$973$	2.50000	+	12.9904i	0.0801463	+	0.416452i
$974$		0			0
$975$	1.00000	+	1.73205i	0.0320256	+	0.0554700i
$976$	26.0000	−	45.0333i	0.832240	−	1.44148i
$977$	12.0000	−	20.7846i	0.383914	−	0.664959i	−0.607704	−	0.794164i	$-0.707909\pi$
$977$	12.0000	−	20.7846i	0.383914	−	0.664959i	0.991618	+	0.129205i	$0.0412426\pi$
$978$		0			0
$979$	−54.0000			−1.72585
$980$	26.0000	−	10.3923i	0.830540	−	0.331970i
$981$	6.00000			0.191565
$982$		0			0
$983$	−20.0000	+	34.6410i	−0.637901	+	1.10488i	0.347992	+	0.937498i	$0.386864\pi$
$983$	−20.0000	+	34.6410i	−0.637901	+	1.10488i	−0.985893	+	0.167379i	$0.946470\pi$
$984$		0			0
$985$	5.00000	+	8.66025i	0.159313	+	0.275939i
$986$		0			0
$987$	−2.50000	−	12.9904i	−0.0795759	−	0.413488i
$988$	4.00000			0.127257
$989$	−30.0000	−	51.9615i	−0.953945	−	1.65228i
$990$		0			0
$991$	21.0000	−	36.3731i	0.667087	−	1.15543i	−0.311628	−	0.950204i	$-0.600874\pi$
$991$	21.0000	−	36.3731i	0.667087	−	1.15543i	0.978715	−	0.205224i	$-0.0657924\pi$
$992$		0			0
$993$	26.0000			0.825085
$994$		0			0
$995$	−32.0000			−1.01447
$996$	−9.00000	−	15.5885i	−0.285176	−	0.493939i
$997$	−0.500000	+	0.866025i	−0.0158352	+	0.0274273i	−0.873834	−	0.486224i	$-0.838374\pi$
$997$	−0.500000	+	0.866025i	−0.0158352	+	0.0274273i	0.857999	+	0.513651i	$0.171707\pi$
$998$		0			0
$999$	−4.00000	−	6.92820i	−0.126554	−	0.219199i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	399.2.j.b.172.1	yes	2
3.2	odd	2		1197.2.j.a.172.1		2
7.2	even	3	inner	399.2.j.b.58.1	✓	2
7.3	odd	6		2793.2.a.g.1.1		1
7.4	even	3		2793.2.a.h.1.1		1
21.2	odd	6		1197.2.j.a.856.1		2
21.11	odd	6		8379.2.a.h.1.1		1
21.17	even	6		8379.2.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
399.2.j.b.58.1	✓	2	7.2	even	3	inner
399.2.j.b.172.1	yes	2	1.1	even	1	trivial
1197.2.j.a.172.1		2	3.2	odd	2
1197.2.j.a.856.1		2	21.2	odd	6
2793.2.a.g.1.1		1	7.3	odd	6
2793.2.a.h.1.1		1	7.4	even	3
8379.2.a.h.1.1		1	21.11	odd	6
8379.2.a.i.1.1		1	21.17	even	6

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:	\( N \)	\(=\)	\( 399 = 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	399.j (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{3} +(1.00000 - 1.73205i) q^{4} +(-1.00000 - 1.73205i) q^{5} +(0.500000 + 2.59808i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(1.50000 - 2.59808i) q^{11} +(1.00000 + 1.73205i) q^{12} +2.00000 q^{13} +2.00000 q^{15} +(-2.00000 - 3.46410i) q^{16} +(3.50000 - 6.06218i) q^{17} +(0.500000 + 0.866025i) q^{19} -4.00000 q^{20} +(-2.50000 - 0.866025i) q^{21} +(-2.50000 - 4.33013i) q^{23} +(0.500000 - 0.866025i) q^{25} +1.00000 q^{27} +(5.00000 + 1.73205i) q^{28} +2.00000 q^{29} +(-5.00000 + 8.66025i) q^{31} +(1.50000 + 2.59808i) q^{33} +(4.00000 - 3.46410i) q^{35} -2.00000 q^{36} +(-4.00000 - 6.92820i) q^{37} +(-1.00000 + 1.73205i) q^{39} +6.00000 q^{41} +12.0000 q^{43} +(-3.00000 - 5.19615i) q^{44} +(-1.00000 + 1.73205i) q^{45} +(2.50000 + 4.33013i) q^{47} +4.00000 q^{48} +(-6.50000 + 2.59808i) q^{49} +(3.50000 + 6.06218i) q^{51} +(2.00000 - 3.46410i) q^{52} +(-2.00000 + 3.46410i) q^{53} -6.00000 q^{55} -1.00000 q^{57} +(-7.00000 + 12.1244i) q^{59} +(2.00000 - 3.46410i) q^{60} +(6.50000 + 11.2583i) q^{61} +(2.00000 - 1.73205i) q^{63} -8.00000 q^{64} +(-2.00000 - 3.46410i) q^{65} +(1.00000 - 1.73205i) q^{67} +(-7.00000 - 12.1244i) q^{68} +5.00000 q^{69} -10.0000 q^{71} +(-0.500000 + 0.866025i) q^{73} +(0.500000 + 0.866025i) q^{75} +2.00000 q^{76} +(7.50000 + 2.59808i) q^{77} +(2.00000 + 3.46410i) q^{79} +(-4.00000 + 6.92820i) q^{80} +(-0.500000 + 0.866025i) q^{81} -9.00000 q^{83} +(-4.00000 + 3.46410i) q^{84} -14.0000 q^{85} +(-1.00000 + 1.73205i) q^{87} +(-9.00000 - 15.5885i) q^{89} +(1.00000 + 5.19615i) q^{91} -10.0000 q^{92} +(-5.00000 - 8.66025i) q^{93} +(1.00000 - 1.73205i) q^{95} +6.00000 q^{97} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + 2 q^{4} - 2 q^{5} + q^{7} - q^{9} + 3 q^{11} + 2 q^{12} + 4 q^{13} + 4 q^{15} - 4 q^{16} + 7 q^{17} + q^{19} - 8 q^{20} - 5 q^{21} - 5 q^{23} + q^{25} + 2 q^{27} + 10 q^{28} + 4 q^{29}+ \cdots - 6 q^{99}+O(q^{100}) \) 2 * q - q^3 + 2 * q^4 - 2 * q^5 + q^7 - q^9 + 3 * q^11 + 2 * q^12 + 4 * q^13 + 4 * q^15 - 4 * q^16 + 7 * q^17 + q^19 - 8 * q^20 - 5 * q^21 - 5 * q^23 + q^25 + 2 * q^27 + 10 * q^28 + 4 * q^29 - 10 * q^31 + 3 * q^33 + 8 * q^35 - 4 * q^36 - 8 * q^37 - 2 * q^39 + 12 * q^41 + 24 * q^43 - 6 * q^44 - 2 * q^45 + 5 * q^47 + 8 * q^48 - 13 * q^49 + 7 * q^51 + 4 * q^52 - 4 * q^53 - 12 * q^55 - 2 * q^57 - 14 * q^59 + 4 * q^60 + 13 * q^61 + 4 * q^63 - 16 * q^64 - 4 * q^65 + 2 * q^67 - 14 * q^68 + 10 * q^69 - 20 * q^71 - q^73 + q^75 + 4 * q^76 + 15 * q^77 + 4 * q^79 - 8 * q^80 - q^81 - 18 * q^83 - 8 * q^84 - 28 * q^85 - 2 * q^87 - 18 * q^89 + 2 * q^91 - 20 * q^92 - 10 * q^93 + 2 * q^95 + 12 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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