LMFDB - Embedded newform 1197.2.j.b.856.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1197,2,Mod(172,1197)] 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("1197.172"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1197, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1197 = 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1197.j (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,2,0,-2,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$9.55809312195$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 399)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			856.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1197.856
Dual form			1197.2.j.b.172.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.00000 - 1.73205i) q^{2} +(-1.00000 - 1.73205i) q^{4} +(0.500000 - 0.866025i) q^{5} +(2.00000 - 1.73205i) q^{7} +(-1.00000 - 1.73205i) q^{10} +(2.00000 + 3.46410i) q^{11} +4.00000 q^{13} +(-1.00000 - 5.19615i) q^{14} +(2.00000 - 3.46410i) q^{16} +(1.50000 + 2.59808i) q^{17} +(0.500000 - 0.866025i) q^{19} -2.00000 q^{20} +8.00000 q^{22} +(-1.50000 + 2.59808i) q^{23} +(2.00000 + 3.46410i) q^{25} +(4.00000 - 6.92820i) q^{26} +(-5.00000 - 1.73205i) q^{28} -10.0000 q^{29} +(-4.00000 - 6.92820i) q^{32} +6.00000 q^{34} +(-0.500000 - 2.59808i) q^{35} +(3.00000 - 5.19615i) q^{37} +(-1.00000 - 1.73205i) q^{38} -2.00000 q^{41} -7.00000 q^{43} +(4.00000 - 6.92820i) q^{44} +(3.00000 + 5.19615i) q^{46} +(1.00000 - 6.92820i) q^{49} +8.00000 q^{50} +(-4.00000 - 6.92820i) q^{52} +(-6.00000 - 10.3923i) q^{53} +4.00000 q^{55} +(-10.0000 + 17.3205i) q^{58} +(6.00000 + 10.3923i) q^{59} +(-5.00000 + 8.66025i) q^{61} -8.00000 q^{64} +(2.00000 - 3.46410i) q^{65} +(-5.00000 - 8.66025i) q^{67} +(3.00000 - 5.19615i) q^{68} +(-5.00000 - 1.73205i) q^{70} -6.00000 q^{71} +(-3.00000 - 5.19615i) q^{73} +(-6.00000 - 10.3923i) q^{74} -2.00000 q^{76} +(10.0000 + 3.46410i) q^{77} +(5.00000 - 8.66025i) q^{79} +(-2.00000 - 3.46410i) q^{80} +(-2.00000 + 3.46410i) q^{82} -3.00000 q^{83} +3.00000 q^{85} +(-7.00000 + 12.1244i) q^{86} +(-7.00000 + 12.1244i) q^{89} +(8.00000 - 6.92820i) q^{91} +6.00000 q^{92} +(-0.500000 - 0.866025i) q^{95} -12.0000 q^{97} +(-11.0000 - 8.66025i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{4} + q^{5} + 4 q^{7} - 2 q^{10} + 4 q^{11} + 8 q^{13} - 2 q^{14} + 4 q^{16} + 3 q^{17} + q^{19} - 4 q^{20} + 16 q^{22} - 3 q^{23} + 4 q^{25} + 8 q^{26} - 10 q^{28} - 20 q^{29} - 8 q^{32}+ \cdots - 22 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 2 * q^4 + q^5 + 4 * q^7 - 2 * q^10 + 4 * q^11 + 8 * q^13 - 2 * q^14 + 4 * q^16 + 3 * q^17 + q^19 - 4 * q^20 + 16 * q^22 - 3 * q^23 + 4 * q^25 + 8 * q^26 - 10 * q^28 - 20 * q^29 - 8 * q^32 + 12 * q^34 - q^35 + 6 * q^37 - 2 * q^38 - 4 * q^41 - 14 * q^43 + 8 * q^44 + 6 * q^46 + 2 * q^49 + 16 * q^50 - 8 * q^52 - 12 * q^53 + 8 * q^55 - 20 * q^58 + 12 * q^59 - 10 * q^61 - 16 * q^64 + 4 * q^65 - 10 * q^67 + 6 * q^68 - 10 * q^70 - 12 * q^71 - 6 * q^73 - 12 * q^74 - 4 * q^76 + 20 * q^77 + 10 * q^79 - 4 * q^80 - 4 * q^82 - 6 * q^83 + 6 * q^85 - 14 * q^86 - 14 * q^89 + 16 * q^91 + 12 * q^92 - q^95 - 24 * q^97 - 22 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1197\mathbb{Z}\right)^\times$.

$n$	$514$	$533$	$1009$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$1$

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 $

$2$	1.00000	−	1.73205i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$2$	1.00000	−	1.73205i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$3$		0			0
$3$		0			0
$4$	−1.00000	−	1.73205i	−0.500000	−	0.866025i
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i	−0.732294	−	0.680989i	$-0.761550\pi$
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i	0.955901	+	0.293691i	$0.0948835\pi$
$6$		0			0
$7$	2.00000	−	1.73205i	0.755929	−	0.654654i
$7$	2.00000	−	1.73205i	0.755929	−	0.654654i
$8$		0			0
$9$		0			0
$10$	−1.00000	−	1.73205i	−0.316228	−	0.547723i
$11$	2.00000	+	3.46410i	0.603023	+	1.04447i	0.992361	+	0.123371i	$0.0393705\pi$
$11$	2.00000	+	3.46410i	0.603023	+	1.04447i	−0.389338	+	0.921095i	$0.627296\pi$
$12$		0			0
$13$	4.00000			1.10940			0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000			1.10940			0.554700	+	0.832050i	$0.312833\pi$
$14$	−1.00000	−	5.19615i	−0.267261	−	1.38873i
$15$		0			0
$16$	2.00000	−	3.46410i	0.500000	−	0.866025i
$17$	1.50000	+	2.59808i	0.363803	+	0.630126i	0.988583	−	0.150675i	$-0.0481447\pi$
$17$	1.50000	+	2.59808i	0.363803	+	0.630126i	−0.624780	+	0.780801i	$0.714811\pi$
$18$		0			0
$19$	0.500000	−	0.866025i	0.114708	−	0.198680i
$19$	0.500000	−	0.866025i	0.114708	−	0.198680i
$20$	−2.00000			−0.447214
$21$		0			0
$22$	8.00000			1.70561
$23$	−1.50000	+	2.59808i	−0.312772	+	0.541736i	−0.978961	−	0.204046i	$-0.934591\pi$
$23$	−1.50000	+	2.59808i	−0.312772	+	0.541736i	0.666190	+	0.745782i	$0.267924\pi$
$24$		0			0
$25$	2.00000	+	3.46410i	0.400000	+	0.692820i
$26$	4.00000	−	6.92820i	0.784465	−	1.35873i
$27$		0			0
$28$	−5.00000	−	1.73205i	−0.944911	−	0.327327i
$29$	−10.0000			−1.85695			−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−10.0000			−1.85695			−0.928477	+	0.371391i	$0.878881\pi$
$30$		0			0
$31$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$31$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$32$	−4.00000	−	6.92820i	−0.707107	−	1.22474i
$33$		0			0
$34$	6.00000			1.02899
$35$	−0.500000	−	2.59808i	−0.0845154	−	0.439155i
$36$		0			0
$37$	3.00000	−	5.19615i	0.493197	−	0.854242i	−0.506772	−	0.862080i	$-0.669162\pi$
$37$	3.00000	−	5.19615i	0.493197	−	0.854242i	0.999969	+	0.00783774i	$0.00249486\pi$
$38$	−1.00000	−	1.73205i	−0.162221	−	0.280976i
$39$		0			0
$40$		0			0
$41$	−2.00000			−0.312348			−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000			−0.312348			−0.156174	+	0.987730i	$0.549916\pi$
$42$		0			0
$43$	−7.00000			−1.06749			−0.533745	−	0.845645i	$-0.679216\pi$
$43$	−7.00000			−1.06749			−0.533745	+	0.845645i	$0.679216\pi$
$44$	4.00000	−	6.92820i	0.603023	−	1.04447i
$45$		0			0
$46$	3.00000	+	5.19615i	0.442326	+	0.766131i
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$		0			0
$49$	1.00000	−	6.92820i	0.142857	−	0.989743i
$50$	8.00000			1.13137
$51$		0			0
$52$	−4.00000	−	6.92820i	−0.554700	−	0.960769i
$53$	−6.00000	−	10.3923i	−0.824163	−	1.42749i	−0.902557	−	0.430570i	$-0.858312\pi$
$53$	−6.00000	−	10.3923i	−0.824163	−	1.42749i	0.0783936	−	0.996922i	$-0.475021\pi$
$54$		0			0
$55$	4.00000			0.539360
$56$		0			0
$57$		0			0
$58$	−10.0000	+	17.3205i	−1.31306	+	2.27429i
$59$	6.00000	+	10.3923i	0.781133	+	1.35296i	0.931282	+	0.364299i	$0.118692\pi$
$59$	6.00000	+	10.3923i	0.781133	+	1.35296i	−0.150148	+	0.988663i	$0.547975\pi$
$60$		0			0
$61$	−5.00000	+	8.66025i	−0.640184	+	1.10883i	0.345207	+	0.938527i	$0.387809\pi$
$61$	−5.00000	+	8.66025i	−0.640184	+	1.10883i	−0.985391	+	0.170305i	$0.945525\pi$
$62$		0			0
$63$		0			0
$64$	−8.00000			−1.00000
$65$	2.00000	−	3.46410i	0.248069	−	0.429669i
$66$		0			0
$67$	−5.00000	−	8.66025i	−0.610847	−	1.05802i	−0.991098	−	0.133135i	$-0.957496\pi$
$67$	−5.00000	−	8.66025i	−0.610847	−	1.05802i	0.380251	−	0.924883i	$-0.375838\pi$
$68$	3.00000	−	5.19615i	0.363803	−	0.630126i
$69$		0			0
$70$	−5.00000	−	1.73205i	−0.597614	−	0.207020i
$71$	−6.00000			−0.712069			−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000			−0.712069			−0.356034	+	0.934473i	$0.615871\pi$
$72$		0			0
$73$	−3.00000	−	5.19615i	−0.351123	−	0.608164i	0.635323	−	0.772246i	$-0.280867\pi$
$73$	−3.00000	−	5.19615i	−0.351123	−	0.608164i	−0.986447	+	0.164083i	$0.947534\pi$
$74$	−6.00000	−	10.3923i	−0.697486	−	1.20808i
$75$		0			0
$76$	−2.00000			−0.229416
$77$	10.0000	+	3.46410i	1.13961	+	0.394771i
$78$		0			0
$79$	5.00000	−	8.66025i	0.562544	−	0.974355i	−0.434730	−	0.900561i	$-0.643156\pi$
$79$	5.00000	−	8.66025i	0.562544	−	0.974355i	0.997274	−	0.0737937i	$-0.0235106\pi$
$80$	−2.00000	−	3.46410i	−0.223607	−	0.387298i
$81$		0			0
$82$	−2.00000	+	3.46410i	−0.220863	+	0.382546i
$83$	−3.00000			−0.329293			−0.164646	−	0.986353i	$-0.552648\pi$
$83$	−3.00000			−0.329293			−0.164646	+	0.986353i	$0.552648\pi$
$84$		0			0
$85$	3.00000			0.325396
$86$	−7.00000	+	12.1244i	−0.754829	+	1.30740i
$87$		0			0
$88$		0			0
$89$	−7.00000	+	12.1244i	−0.741999	+	1.28518i	0.209585	+	0.977790i	$0.432789\pi$
$89$	−7.00000	+	12.1244i	−0.741999	+	1.28518i	−0.951584	+	0.307389i	$0.900545\pi$
$90$		0			0
$91$	8.00000	−	6.92820i	0.838628	−	0.726273i
$92$	6.00000			0.625543
$93$		0			0
$94$		0			0
$95$	−0.500000	−	0.866025i	−0.0512989	−	0.0888523i
$96$		0			0
$97$	−12.0000			−1.21842			−0.609208	−	0.793011i	$-0.708512\pi$
$97$	−12.0000			−1.21842			−0.609208	+	0.793011i	$0.708512\pi$
$98$	−11.0000	−	8.66025i	−1.11117	−	0.874818i
$99$		0			0
$100$	4.00000	−	6.92820i	0.400000	−	0.692820i
$101$	5.50000	+	9.52628i	0.547270	+	0.947900i	0.998460	+	0.0554722i	$0.0176664\pi$
$101$	5.50000	+	9.52628i	0.547270	+	0.947900i	−0.451190	+	0.892428i	$0.649000\pi$
$102$		0			0
$103$	−4.00000	+	6.92820i	−0.394132	+	0.682656i	−0.992990	−	0.118199i	$-0.962288\pi$
$103$	−4.00000	+	6.92820i	−0.394132	+	0.682656i	0.598858	+	0.800855i	$0.295621\pi$
$104$		0			0
$105$		0			0
$106$	−24.0000			−2.33109
$107$	5.00000	−	8.66025i	0.483368	−	0.837218i	−0.516449	−	0.856318i	$-0.672747\pi$
$107$	5.00000	−	8.66025i	0.483368	−	0.837218i	0.999818	+	0.0190994i	$0.00607989\pi$
$108$		0			0
$109$	3.00000	+	5.19615i	0.287348	+	0.497701i	0.973176	−	0.230063i	$-0.0738931\pi$
$109$	3.00000	+	5.19615i	0.287348	+	0.497701i	−0.685828	+	0.727764i	$0.740560\pi$
$110$	4.00000	−	6.92820i	0.381385	−	0.660578i
$111$		0			0
$112$	−2.00000	−	10.3923i	−0.188982	−	0.981981i
$113$	8.00000			0.752577			0.376288	−	0.926503i	$-0.377200\pi$
$113$	8.00000			0.752577			0.376288	+	0.926503i	$0.377200\pi$
$114$		0			0
$115$	1.50000	+	2.59808i	0.139876	+	0.242272i
$116$	10.0000	+	17.3205i	0.928477	+	1.60817i
$117$		0			0
$118$	24.0000			2.20938
$119$	7.50000	+	2.59808i	0.687524	+	0.238165i
$120$		0			0
$121$	−2.50000	+	4.33013i	−0.227273	+	0.393648i
$122$	10.0000	+	17.3205i	0.905357	+	1.56813i
$123$		0			0
$124$		0			0
$125$	9.00000			0.804984
$126$		0			0
$127$	12.0000			1.06483			0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000			1.06483			0.532414	+	0.846484i	$0.321285\pi$
$128$		0			0
$129$		0			0
$130$	−4.00000	−	6.92820i	−0.350823	−	0.607644i
$131$	−2.50000	+	4.33013i	−0.218426	+	0.378325i	−0.954327	−	0.298764i	$-0.903426\pi$
$131$	−2.50000	+	4.33013i	−0.218426	+	0.378325i	0.735901	+	0.677089i	$0.236759\pi$
$132$		0			0
$133$	−0.500000	−	2.59808i	−0.0433555	−	0.225282i
$134$	−20.0000			−1.72774
$135$		0			0
$136$		0			0
$137$	−9.00000	−	15.5885i	−0.768922	−	1.33181i	−0.938148	−	0.346235i	$-0.887460\pi$
$137$	−9.00000	−	15.5885i	−0.768922	−	1.33181i	0.169226	−	0.985577i	$-0.445873\pi$
$138$		0			0
$139$	−12.0000			−1.01783			−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000			−1.01783			−0.508913	+	0.860818i	$0.669953\pi$
$140$	−4.00000	+	3.46410i	−0.338062	+	0.292770i
$141$		0			0
$142$	−6.00000	+	10.3923i	−0.503509	+	0.872103i
$143$	8.00000	+	13.8564i	0.668994	+	1.15873i
$144$		0			0
$145$	−5.00000	+	8.66025i	−0.415227	+	0.719195i
$146$	−12.0000			−0.993127
$147$		0			0
$148$	−12.0000			−0.986394
$149$	−4.50000	+	7.79423i	−0.368654	+	0.638528i	−0.989355	−	0.145519i	$-0.953515\pi$
$149$	−4.50000	+	7.79423i	−0.368654	+	0.638528i	0.620701	+	0.784047i	$0.286848\pi$
$150$		0			0
$151$	5.00000	+	8.66025i	0.406894	+	0.704761i	0.994540	−	0.104357i	$-0.0332784\pi$
$151$	5.00000	+	8.66025i	0.406894	+	0.704761i	−0.587646	+	0.809118i	$0.699945\pi$
$152$		0			0
$153$		0			0
$154$	16.0000	−	13.8564i	1.28932	−	1.11658i
$155$		0			0
$156$		0			0
$157$	−0.500000	−	0.866025i	−0.0399043	−	0.0691164i	0.845383	−	0.534160i	$-0.179372\pi$
$157$	−0.500000	−	0.866025i	−0.0399043	−	0.0691164i	−0.885288	+	0.465044i	$0.846039\pi$
$158$	−10.0000	−	17.3205i	−0.795557	−	1.37795i
$159$		0			0
$160$	−8.00000			−0.632456
$161$	1.50000	+	7.79423i	0.118217	+	0.614271i
$162$		0			0
$163$	−9.50000	+	16.4545i	−0.744097	+	1.28881i	0.206518	+	0.978443i	$0.433787\pi$
$163$	−9.50000	+	16.4545i	−0.744097	+	1.28881i	−0.950615	+	0.310372i	$0.899546\pi$
$164$	2.00000	+	3.46410i	0.156174	+	0.270501i
$165$		0			0
$166$	−3.00000	+	5.19615i	−0.232845	+	0.403300i
$167$	2.00000			0.154765			0.0773823	−	0.997001i	$-0.475344\pi$
$167$	2.00000			0.154765			0.0773823	+	0.997001i	$0.475344\pi$
$168$		0			0
$169$	3.00000			0.230769
$170$	3.00000	−	5.19615i	0.230089	−	0.398527i
$171$		0			0
$172$	7.00000	+	12.1244i	0.533745	+	0.924473i
$173$	7.00000	−	12.1244i	0.532200	−	0.921798i	−0.467093	−	0.884208i	$-0.654699\pi$
$173$	7.00000	−	12.1244i	0.532200	−	0.921798i	0.999293	−	0.0375896i	$-0.0119679\pi$
$174$		0			0
$175$	10.0000	+	3.46410i	0.755929	+	0.261861i
$176$	16.0000			1.20605
$177$		0			0
$178$	14.0000	+	24.2487i	1.04934	+	1.81752i
$179$	−8.00000	−	13.8564i	−0.597948	−	1.03568i	−0.993124	−	0.117071i	$-0.962650\pi$
$179$	−8.00000	−	13.8564i	−0.597948	−	1.03568i	0.395175	−	0.918606i	$-0.370684\pi$
$180$		0			0
$181$	−14.0000			−1.04061			−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000			−1.04061			−0.520306	+	0.853980i	$0.674182\pi$
$182$	−4.00000	−	20.7846i	−0.296500	−	1.54066i
$183$		0			0
$184$		0			0
$185$	−3.00000	−	5.19615i	−0.220564	−	0.382029i
$186$		0			0
$187$	−6.00000	+	10.3923i	−0.438763	+	0.759961i
$188$		0			0
$189$		0			0
$190$	−2.00000			−0.145095
$191$	−8.50000	+	14.7224i	−0.615038	+	1.06528i	0.375339	+	0.926887i	$0.377526\pi$
$191$	−8.50000	+	14.7224i	−0.615038	+	1.06528i	−0.990378	+	0.138390i	$0.955807\pi$
$192$		0			0
$193$	8.00000	+	13.8564i	0.575853	+	0.997406i	0.995948	+	0.0899262i	$0.0286631\pi$
$193$	8.00000	+	13.8564i	0.575853	+	0.997406i	−0.420096	+	0.907480i	$0.638004\pi$
$194$	−12.0000	+	20.7846i	−0.861550	+	1.49225i
$195$		0			0
$196$	−13.0000	+	5.19615i	−0.928571	+	0.371154i
$197$	11.0000			0.783718			0.391859	−	0.920025i	$-0.371832\pi$
$197$	11.0000			0.783718			0.391859	+	0.920025i	$0.371832\pi$
$198$		0			0
$199$	13.5000	+	23.3827i	0.956990	+	1.65755i	0.729748	+	0.683716i	$0.239637\pi$
$199$	13.5000	+	23.3827i	0.956990	+	1.65755i	0.227242	+	0.973838i	$0.427029\pi$
$200$		0			0
$201$		0			0
$202$	22.0000			1.54791
$203$	−20.0000	+	17.3205i	−1.40372	+	1.21566i
$204$		0			0
$205$	−1.00000	+	1.73205i	−0.0698430	+	0.120972i
$206$	8.00000	+	13.8564i	0.557386	+	0.965422i
$207$		0			0
$208$	8.00000	−	13.8564i	0.554700	−	0.960769i
$209$	4.00000			0.276686
$210$		0			0
$211$	4.00000			0.275371			0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000			0.275371			0.137686	+	0.990476i	$0.456034\pi$
$212$	−12.0000	+	20.7846i	−0.824163	+	1.42749i
$213$		0			0
$214$	−10.0000	−	17.3205i	−0.683586	−	1.18401i
$215$	−3.50000	+	6.06218i	−0.238698	+	0.413437i
$216$		0			0
$217$		0			0
$218$	12.0000			0.812743
$219$		0			0
$220$	−4.00000	−	6.92820i	−0.269680	−	0.467099i
$221$	6.00000	+	10.3923i	0.403604	+	0.699062i
$222$		0			0
$223$	10.0000			0.669650			0.334825	−	0.942280i	$-0.391323\pi$
$223$	10.0000			0.669650			0.334825	+	0.942280i	$0.391323\pi$
$224$	−20.0000	−	6.92820i	−1.33631	−	0.462910i
$225$		0			0
$226$	8.00000	−	13.8564i	0.532152	−	0.921714i
$227$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$227$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$228$		0			0
$229$	3.50000	−	6.06218i	0.231287	−	0.400600i	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	3.50000	−	6.06218i	0.231287	−	0.400600i	0.958187	+	0.286143i	$0.0923732\pi$
$230$	6.00000			0.395628
$231$		0			0
$232$		0			0
$233$	4.50000	−	7.79423i	0.294805	−	0.510617i	−0.680135	−	0.733087i	$-0.738079\pi$
$233$	4.50000	−	7.79423i	0.294805	−	0.510617i	0.974939	+	0.222470i	$0.0714120\pi$
$234$		0			0
$235$		0			0
$236$	12.0000	−	20.7846i	0.781133	−	1.35296i
$237$		0			0
$238$	12.0000	−	10.3923i	0.777844	−	0.673633i
$239$	−9.00000			−0.582162			−0.291081	−	0.956698i	$-0.594015\pi$
$239$	−9.00000			−0.582162			−0.291081	+	0.956698i	$0.594015\pi$
$240$		0			0
$241$	14.0000	+	24.2487i	0.901819	+	1.56200i	0.825131	+	0.564942i	$0.191101\pi$
$241$	14.0000	+	24.2487i	0.901819	+	1.56200i	0.0766885	+	0.997055i	$0.475565\pi$
$242$	5.00000	+	8.66025i	0.321412	+	0.556702i
$243$		0			0
$244$	20.0000			1.28037
$245$	−5.50000	−	4.33013i	−0.351382	−	0.276642i
$246$		0			0
$247$	2.00000	−	3.46410i	0.127257	−	0.220416i
$248$		0			0
$249$		0			0
$250$	9.00000	−	15.5885i	0.569210	−	0.985901i
$251$	23.0000			1.45175			0.725874	−	0.687828i	$-0.241436\pi$
$251$	23.0000			1.45175			0.725874	+	0.687828i	$0.241436\pi$
$252$		0			0
$253$	−12.0000			−0.754434
$254$	12.0000	−	20.7846i	0.752947	−	1.30414i
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$	1.00000	−	1.73205i	0.0623783	−	0.108042i	−0.833150	−	0.553047i	$-0.813465\pi$
$257$	1.00000	−	1.73205i	0.0623783	−	0.108042i	0.895528	+	0.445005i	$0.146798\pi$
$258$		0			0
$259$	−3.00000	−	15.5885i	−0.186411	−	0.968620i
$260$	−8.00000			−0.496139
$261$		0			0
$262$	5.00000	+	8.66025i	0.308901	+	0.535032i
$263$	−8.50000	−	14.7224i	−0.524132	−	0.907824i	−0.999605	−	0.0280936i	$-0.991056\pi$
$263$	−8.50000	−	14.7224i	−0.524132	−	0.907824i	0.475473	−	0.879730i	$-0.342277\pi$
$264$		0			0
$265$	−12.0000			−0.737154
$266$	−5.00000	−	1.73205i	−0.306570	−	0.106199i
$267$		0			0
$268$	−10.0000	+	17.3205i	−0.610847	+	1.05802i
$269$	−9.00000	−	15.5885i	−0.548740	−	0.950445i	−0.998361	−	0.0572259i	$-0.981774\pi$
$269$	−9.00000	−	15.5885i	−0.548740	−	0.950445i	0.449622	−	0.893219i	$-0.351559\pi$
$270$		0			0
$271$	11.5000	−	19.9186i	0.698575	−	1.20997i	−0.270385	−	0.962752i	$-0.587151\pi$
$271$	11.5000	−	19.9186i	0.698575	−	1.20997i	0.968960	−	0.247216i	$-0.0795156\pi$
$272$	12.0000			0.727607
$273$		0			0
$274$	−36.0000			−2.17484
$275$	−8.00000	+	13.8564i	−0.482418	+	0.835573i
$276$		0			0
$277$	−11.0000	−	19.0526i	−0.660926	−	1.14476i	−0.980373	−	0.197153i	$-0.936830\pi$
$277$	−11.0000	−	19.0526i	−0.660926	−	1.14476i	0.319447	−	0.947604i	$-0.396503\pi$
$278$	−12.0000	+	20.7846i	−0.719712	+	1.24658i
$279$		0			0
$280$		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
$283$	−0.500000	−	0.866025i	−0.0297219	−	0.0514799i	0.850782	−	0.525519i	$-0.176129\pi$
$283$	−0.500000	−	0.866025i	−0.0297219	−	0.0514799i	−0.880504	+	0.474039i	$0.842796\pi$
$284$	6.00000	+	10.3923i	0.356034	+	0.616670i
$285$		0			0
$286$	32.0000			1.89220
$287$	−4.00000	+	3.46410i	−0.236113	+	0.204479i
$288$		0			0
$289$	4.00000	−	6.92820i	0.235294	−	0.407541i
$290$	10.0000	+	17.3205i	0.587220	+	1.01710i
$291$		0			0
$292$	−6.00000	+	10.3923i	−0.351123	+	0.608164i
$293$	16.0000			0.934730			0.467365	−	0.884064i	$-0.345203\pi$
$293$	16.0000			0.934730			0.467365	+	0.884064i	$0.345203\pi$
$294$		0			0
$295$	12.0000			0.698667
$296$		0			0
$297$		0			0
$298$	9.00000	+	15.5885i	0.521356	+	0.903015i
$299$	−6.00000	+	10.3923i	−0.346989	+	0.601003i
$300$		0			0
$301$	−14.0000	+	12.1244i	−0.806947	+	0.698836i
$302$	20.0000			1.15087
$303$		0			0
$304$	−2.00000	−	3.46410i	−0.114708	−	0.198680i
$305$	5.00000	+	8.66025i	0.286299	+	0.495885i
$306$		0			0
$307$	22.0000			1.25561			0.627803	−	0.778372i	$-0.283954\pi$
$307$	22.0000			1.25561			0.627803	+	0.778372i	$0.283954\pi$
$308$	−4.00000	−	20.7846i	−0.227921	−	1.18431i
$309$		0			0
$310$		0			0
$311$	1.50000	+	2.59808i	0.0850572	+	0.147323i	0.905416	−	0.424526i	$-0.139559\pi$
$311$	1.50000	+	2.59808i	0.0850572	+	0.147323i	−0.820358	+	0.571850i	$0.806226\pi$
$312$		0			0
$313$	−5.50000	+	9.52628i	−0.310878	+	0.538457i	−0.978553	−	0.205996i	$-0.933957\pi$
$313$	−5.50000	+	9.52628i	−0.310878	+	0.538457i	0.667674	+	0.744453i	$0.267290\pi$
$314$	−2.00000			−0.112867
$315$		0			0
$316$	−20.0000			−1.12509
$317$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$317$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$318$		0			0
$319$	−20.0000	−	34.6410i	−1.11979	−	1.93952i
$320$	−4.00000	+	6.92820i	−0.223607	+	0.387298i
$321$		0			0
$322$	15.0000	+	5.19615i	0.835917	+	0.289570i
$323$	3.00000			0.166924
$324$		0			0
$325$	8.00000	+	13.8564i	0.443760	+	0.768615i
$326$	19.0000	+	32.9090i	1.05231	+	1.82266i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	11.0000	−	19.0526i	0.604615	−	1.04722i	−0.387498	−	0.921871i	$-0.626660\pi$
$331$	11.0000	−	19.0526i	0.604615	−	1.04722i	0.992112	−	0.125353i	$-0.0400062\pi$
$332$	3.00000	+	5.19615i	0.164646	+	0.285176i
$333$		0			0
$334$	2.00000	−	3.46410i	0.109435	−	0.189547i
$335$	−10.0000			−0.546358
$336$		0			0
$337$	−8.00000			−0.435788			−0.217894	−	0.975972i	$-0.569919\pi$
$337$	−8.00000			−0.435788			−0.217894	+	0.975972i	$0.569919\pi$
$338$	3.00000	−	5.19615i	0.163178	−	0.282633i
$339$		0			0
$340$	−3.00000	−	5.19615i	−0.162698	−	0.281801i
$341$		0			0
$342$		0			0
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$		0			0
$345$		0			0
$346$	−14.0000	−	24.2487i	−0.752645	−	1.30362i
$347$	14.5000	+	25.1147i	0.778401	+	1.34823i	0.932863	+	0.360231i	$0.117302\pi$
$347$	14.5000	+	25.1147i	0.778401	+	1.34823i	−0.154462	+	0.987999i	$0.549365\pi$
$348$		0			0
$349$	13.0000			0.695874			0.347937	−	0.937518i	$-0.386882\pi$
$349$	13.0000			0.695874			0.347937	+	0.937518i	$0.386882\pi$
$350$	16.0000	−	13.8564i	0.855236	−	0.740656i
$351$		0			0
$352$	16.0000	−	27.7128i	0.852803	−	1.47710i
$353$	−13.0000	−	22.5167i	−0.691920	−	1.19844i	−0.971208	−	0.238233i	$-0.923432\pi$
$353$	−13.0000	−	22.5167i	−0.691920	−	1.19844i	0.279288	−	0.960207i	$-0.409902\pi$
$354$		0			0
$355$	−3.00000	+	5.19615i	−0.159223	+	0.275783i
$356$	28.0000			1.48400
$357$		0			0
$358$	−32.0000			−1.69125
$359$	10.0000	−	17.3205i	0.527780	−	0.914141i	−0.471696	−	0.881761i	$-0.656358\pi$
$359$	10.0000	−	17.3205i	0.527780	−	0.914141i	0.999476	−	0.0323801i	$-0.0103087\pi$
$360$		0			0
$361$	−0.500000	−	0.866025i	−0.0263158	−	0.0455803i
$362$	−14.0000	+	24.2487i	−0.735824	+	1.27448i
$363$		0			0
$364$	−20.0000	−	6.92820i	−1.04828	−	0.363137i
$365$	−6.00000			−0.314054
$366$		0			0
$367$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$367$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$368$	6.00000	+	10.3923i	0.312772	+	0.541736i
$369$		0			0
$370$	−12.0000			−0.623850
$371$	−30.0000	−	10.3923i	−1.55752	−	0.539542i
$372$		0			0
$373$	−16.0000	+	27.7128i	−0.828449	+	1.43492i	0.0708063	+	0.997490i	$0.477443\pi$
$373$	−16.0000	+	27.7128i	−0.828449	+	1.43492i	−0.899255	+	0.437425i	$0.855891\pi$
$374$	12.0000	+	20.7846i	0.620505	+	1.07475i
$375$		0			0
$376$		0			0
$377$	−40.0000			−2.06010
$378$		0			0
$379$	12.0000			0.616399			0.308199	−	0.951322i	$-0.400274\pi$
$379$	12.0000			0.616399			0.308199	+	0.951322i	$0.400274\pi$
$380$	−1.00000	+	1.73205i	−0.0512989	+	0.0888523i
$381$		0			0
$382$	17.0000	+	29.4449i	0.869796	+	1.50653i
$383$	6.00000	−	10.3923i	0.306586	−	0.531022i	−0.671027	−	0.741433i	$-0.734147\pi$
$383$	6.00000	−	10.3923i	0.306586	−	0.531022i	0.977613	+	0.210411i	$0.0674801\pi$
$384$		0			0
$385$	8.00000	−	6.92820i	0.407718	−	0.353094i
$386$	32.0000			1.62876
$387$		0			0
$388$	12.0000	+	20.7846i	0.609208	+	1.05518i
$389$	−1.50000	−	2.59808i	−0.0760530	−	0.131728i	0.825491	−	0.564416i	$-0.190898\pi$
$389$	−1.50000	−	2.59808i	−0.0760530	−	0.131728i	−0.901544	+	0.432688i	$0.857565\pi$
$390$		0			0
$391$	−9.00000			−0.455150
$392$		0			0
$393$		0			0
$394$	11.0000	−	19.0526i	0.554172	−	0.959854i
$395$	−5.00000	−	8.66025i	−0.251577	−	0.435745i
$396$		0			0
$397$	7.50000	−	12.9904i	0.376414	−	0.651969i	−0.614123	−	0.789210i	$-0.710490\pi$
$397$	7.50000	−	12.9904i	0.376414	−	0.651969i	0.990538	+	0.137241i	$0.0438236\pi$
$398$	54.0000			2.70678
$399$		0			0
$400$	16.0000			0.800000
$401$	−6.00000	+	10.3923i	−0.299626	+	0.518967i	−0.976050	−	0.217545i	$-0.930195\pi$
$401$	−6.00000	+	10.3923i	−0.299626	+	0.518967i	0.676425	+	0.736512i	$0.263528\pi$
$402$		0			0
$403$		0			0
$404$	11.0000	−	19.0526i	0.547270	−	0.947900i
$405$		0			0
$406$	10.0000	+	51.9615i	0.496292	+	2.57881i
$407$	24.0000			1.18964
$408$		0			0
$409$	−16.0000	−	27.7128i	−0.791149	−	1.37031i	−0.925256	−	0.379344i	$-0.876150\pi$
$409$	−16.0000	−	27.7128i	−0.791149	−	1.37031i	0.134107	−	0.990967i	$-0.457183\pi$
$410$	2.00000	+	3.46410i	0.0987730	+	0.171080i
$411$		0			0
$412$	16.0000			0.788263
$413$	30.0000	+	10.3923i	1.47620	+	0.511372i
$414$		0			0
$415$	−1.50000	+	2.59808i	−0.0736321	+	0.127535i
$416$	−16.0000	−	27.7128i	−0.784465	−	1.35873i
$417$		0			0
$418$	4.00000	−	6.92820i	0.195646	−	0.338869i
$419$	9.00000			0.439679			0.219839	−	0.975536i	$-0.429447\pi$
$419$	9.00000			0.439679			0.219839	+	0.975536i	$0.429447\pi$
$420$		0			0
$421$	8.00000			0.389896			0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000			0.389896			0.194948	+	0.980814i	$0.437546\pi$
$422$	4.00000	−	6.92820i	0.194717	−	0.337260i
$423$		0			0
$424$		0			0
$425$	−6.00000	+	10.3923i	−0.291043	+	0.504101i
$426$		0			0
$427$	5.00000	+	25.9808i	0.241967	+	1.25730i
$428$	−20.0000			−0.966736
$429$		0			0
$430$	7.00000	+	12.1244i	0.337570	+	0.584688i
$431$	−6.00000	−	10.3923i	−0.289010	−	0.500580i	0.684564	−	0.728953i	$-0.259993\pi$
$431$	−6.00000	−	10.3923i	−0.289010	−	0.500580i	−0.973574	+	0.228373i	$0.926659\pi$
$432$		0			0
$433$	−14.0000			−0.672797			−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000			−0.672797			−0.336399	+	0.941720i	$0.609209\pi$
$434$		0			0
$435$		0			0
$436$	6.00000	−	10.3923i	0.287348	−	0.497701i
$437$	1.50000	+	2.59808i	0.0717547	+	0.124283i
$438$		0			0
$439$	−15.0000	+	25.9808i	−0.715911	+	1.23999i	0.246696	+	0.969093i	$0.420655\pi$
$439$	−15.0000	+	25.9808i	−0.715911	+	1.23999i	−0.962607	+	0.270901i	$0.912678\pi$
$440$		0			0
$441$		0			0
$442$	24.0000			1.14156
$443$	−10.5000	+	18.1865i	−0.498870	+	0.864068i	−0.999999	−	0.00130426i	$-0.999585\pi$
$443$	−10.5000	+	18.1865i	−0.498870	+	0.864068i	0.501129	+	0.865373i	$0.332918\pi$
$444$		0			0
$445$	7.00000	+	12.1244i	0.331832	+	0.574750i
$446$	10.0000	−	17.3205i	0.473514	−	0.820150i
$447$		0			0
$448$	−16.0000	+	13.8564i	−0.755929	+	0.654654i
$449$	−6.00000			−0.283158			−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000			−0.283158			−0.141579	+	0.989927i	$0.545218\pi$
$450$		0			0
$451$	−4.00000	−	6.92820i	−0.188353	−	0.326236i
$452$	−8.00000	−	13.8564i	−0.376288	−	0.651751i
$453$		0			0
$454$		0			0
$455$	−2.00000	−	10.3923i	−0.0937614	−	0.487199i
$456$		0			0
$457$	−11.0000	+	19.0526i	−0.514558	+	0.891241i	0.485299	+	0.874348i	$0.338711\pi$
$457$	−11.0000	+	19.0526i	−0.514558	+	0.891241i	−0.999857	+	0.0168929i	$0.994623\pi$
$458$	−7.00000	−	12.1244i	−0.327089	−	0.566534i
$459$		0			0
$460$	3.00000	−	5.19615i	0.139876	−	0.242272i
$461$	−26.0000			−1.21094			−0.605470	−	0.795868i	$-0.707015\pi$
$461$	−26.0000			−1.21094			−0.605470	+	0.795868i	$0.707015\pi$
$462$		0			0
$463$	4.00000			0.185896			0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000			0.185896			0.0929479	+	0.995671i	$0.470371\pi$
$464$	−20.0000	+	34.6410i	−0.928477	+	1.60817i
$465$		0			0
$466$	−9.00000	−	15.5885i	−0.416917	−	0.722121i
$467$	18.0000	−	31.1769i	0.832941	−	1.44270i	−0.0627555	−	0.998029i	$-0.519989\pi$
$467$	18.0000	−	31.1769i	0.832941	−	1.44270i	0.895696	−	0.444667i	$-0.146678\pi$
$468$		0			0
$469$	−25.0000	−	8.66025i	−1.15439	−	0.399893i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−14.0000	−	24.2487i	−0.643721	−	1.11496i
$474$		0			0
$475$	4.00000			0.183533
$476$	−3.00000	−	15.5885i	−0.137505	−	0.714496i
$477$		0			0
$478$	−9.00000	+	15.5885i	−0.411650	+	0.712999i
$479$	5.50000	+	9.52628i	0.251301	+	0.435267i	0.963884	−	0.266321i	$-0.0858081\pi$
$479$	5.50000	+	9.52628i	0.251301	+	0.435267i	−0.712583	+	0.701588i	$0.752475\pi$
$480$		0			0
$481$	12.0000	−	20.7846i	0.547153	−	0.947697i
$482$	56.0000			2.55073
$483$		0			0
$484$	10.0000			0.454545
$485$	−6.00000	+	10.3923i	−0.272446	+	0.471890i
$486$		0			0
$487$	16.0000	+	27.7128i	0.725029	+	1.25579i	0.958962	+	0.283535i	$0.0915071\pi$
$487$	16.0000	+	27.7128i	0.725029	+	1.25579i	−0.233933	+	0.972253i	$0.575160\pi$
$488$		0			0
$489$		0			0
$490$	−13.0000	+	5.19615i	−0.587280	+	0.234738i
$491$	1.00000			0.0451294			0.0225647	−	0.999745i	$-0.492817\pi$
$491$	1.00000			0.0451294			0.0225647	+	0.999745i	$0.492817\pi$
$492$		0			0
$493$	−15.0000	−	25.9808i	−0.675566	−	1.17011i
$494$	−4.00000	−	6.92820i	−0.179969	−	0.311715i
$495$		0			0
$496$		0			0
$497$	−12.0000	+	10.3923i	−0.538274	+	0.466159i
$498$		0			0
$499$	−3.50000	+	6.06218i	−0.156682	+	0.271380i	−0.933670	−	0.358134i	$-0.883413\pi$
$499$	−3.50000	+	6.06218i	−0.156682	+	0.271380i	0.776989	+	0.629515i	$0.216746\pi$
$500$	−9.00000	−	15.5885i	−0.402492	−	0.697137i
$501$		0			0
$502$	23.0000	−	39.8372i	1.02654	−	1.77802i
$503$	3.00000			0.133763			0.0668817	−	0.997761i	$-0.478695\pi$
$503$	3.00000			0.133763			0.0668817	+	0.997761i	$0.478695\pi$
$504$		0			0
$505$	11.0000			0.489494
$506$	−12.0000	+	20.7846i	−0.533465	+	0.923989i
$507$		0			0
$508$	−12.0000	−	20.7846i	−0.532414	−	0.922168i
$509$	19.0000	−	32.9090i	0.842160	−	1.45866i	−0.0459045	−	0.998946i	$-0.514617\pi$
$509$	19.0000	−	32.9090i	0.842160	−	1.45866i	0.888065	−	0.459718i	$-0.152050\pi$
$510$		0			0
$511$	−15.0000	−	5.19615i	−0.663561	−	0.229864i
$512$	−32.0000			−1.41421
$513$		0			0
$514$	−2.00000	−	3.46410i	−0.0882162	−	0.152795i
$515$	4.00000	+	6.92820i	0.176261	+	0.305293i
$516$		0			0
$517$		0			0
$518$	−30.0000	−	10.3923i	−1.31812	−	0.456612i
$519$		0			0
$520$		0			0
$521$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$521$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$522$		0			0
$523$	−20.0000	+	34.6410i	−0.874539	+	1.51475i	−0.0172859	+	0.999851i	$0.505503\pi$
$523$	−20.0000	+	34.6410i	−0.874539	+	1.51475i	−0.857253	+	0.514895i	$0.827831\pi$
$524$	10.0000			0.436852
$525$		0			0
$526$	−34.0000			−1.48247
$527$		0			0
$528$		0			0
$529$	7.00000	+	12.1244i	0.304348	+	0.527146i
$530$	−12.0000	+	20.7846i	−0.521247	+	0.902826i
$531$		0			0
$532$	−4.00000	+	3.46410i	−0.173422	+	0.150188i
$533$	−8.00000			−0.346518
$534$		0			0
$535$	−5.00000	−	8.66025i	−0.216169	−	0.374415i
$536$		0			0
$537$		0			0
$538$	−36.0000			−1.55207
$539$	26.0000	−	10.3923i	1.11990	−	0.447628i
$540$		0			0
$541$	−17.5000	+	30.3109i	−0.752384	+	1.30317i	0.194281	+	0.980946i	$0.437763\pi$
$541$	−17.5000	+	30.3109i	−0.752384	+	1.30317i	−0.946664	+	0.322221i	$0.895571\pi$
$542$	−23.0000	−	39.8372i	−0.987935	−	1.71115i
$543$		0			0
$544$	12.0000	−	20.7846i	0.514496	−	0.891133i
$545$	6.00000			0.257012
$546$		0			0
$547$	−2.00000			−0.0855138			−0.0427569	−	0.999086i	$-0.513614\pi$
$547$	−2.00000			−0.0855138			−0.0427569	+	0.999086i	$0.513614\pi$
$548$	−18.0000	+	31.1769i	−0.768922	+	1.33181i
$549$		0			0
$550$	16.0000	+	27.7128i	0.682242	+	1.18168i
$551$	−5.00000	+	8.66025i	−0.213007	+	0.368939i
$552$		0			0
$553$	−5.00000	−	25.9808i	−0.212622	−	1.10481i
$554$	−44.0000			−1.86938
$555$		0			0
$556$	12.0000	+	20.7846i	0.508913	+	0.881464i
$557$	3.50000	+	6.06218i	0.148300	+	0.256863i	0.930599	−	0.366040i	$-0.119287\pi$
$557$	3.50000	+	6.06218i	0.148300	+	0.256863i	−0.782299	+	0.622903i	$0.785953\pi$
$558$		0			0
$559$	−28.0000			−1.18427
$560$	−10.0000	−	3.46410i	−0.422577	−	0.146385i
$561$		0			0
$562$	−20.0000	+	34.6410i	−0.843649	+	1.46124i
$563$	5.00000	+	8.66025i	0.210725	+	0.364986i	0.951942	−	0.306280i	$-0.0990842\pi$
$563$	5.00000	+	8.66025i	0.210725	+	0.364986i	−0.741217	+	0.671266i	$0.765751\pi$
$564$		0			0
$565$	4.00000	−	6.92820i	0.168281	−	0.291472i
$566$	−2.00000			−0.0840663
$567$		0			0
$568$		0			0
$569$	−10.0000	+	17.3205i	−0.419222	+	0.726113i	−0.995861	−	0.0908852i	$-0.971030\pi$
$569$	−10.0000	+	17.3205i	−0.419222	+	0.726113i	0.576640	+	0.816999i	$0.304364\pi$
$570$		0			0
$571$	−2.50000	−	4.33013i	−0.104622	−	0.181210i	0.808962	−	0.587861i	$-0.200030\pi$
$571$	−2.50000	−	4.33013i	−0.104622	−	0.181210i	−0.913584	+	0.406651i	$0.866697\pi$
$572$	16.0000	−	27.7128i	0.668994	−	1.15873i
$573$		0			0
$574$	2.00000	+	10.3923i	0.0834784	+	0.433766i
$575$	−12.0000			−0.500435
$576$		0			0
$577$	1.00000	+	1.73205i	0.0416305	+	0.0721062i	0.886090	−	0.463513i	$-0.153411\pi$
$577$	1.00000	+	1.73205i	0.0416305	+	0.0721062i	−0.844459	+	0.535620i	$0.820078\pi$
$578$	−8.00000	−	13.8564i	−0.332756	−	0.576351i
$579$		0			0
$580$	20.0000			0.830455
$581$	−6.00000	+	5.19615i	−0.248922	+	0.215573i
$582$		0			0
$583$	24.0000	−	41.5692i	0.993978	−	1.72162i
$584$		0			0
$585$		0			0
$586$	16.0000	−	27.7128i	0.660954	−	1.14481i
$587$	−7.00000			−0.288921			−0.144460	−	0.989511i	$-0.546145\pi$
$587$	−7.00000			−0.288921			−0.144460	+	0.989511i	$0.546145\pi$
$588$		0			0
$589$		0			0
$590$	12.0000	−	20.7846i	0.494032	−	0.855689i
$591$		0			0
$592$	−12.0000	−	20.7846i	−0.493197	−	0.854242i
$593$	−19.5000	+	33.7750i	−0.800769	+	1.38697i	0.118342	+	0.992973i	$0.462242\pi$
$593$	−19.5000	+	33.7750i	−0.800769	+	1.38697i	−0.919111	+	0.394000i	$0.871091\pi$
$594$		0			0
$595$	6.00000	−	5.19615i	0.245976	−	0.213021i
$596$	18.0000			0.737309
$597$		0			0
$598$	12.0000	+	20.7846i	0.490716	+	0.849946i
$599$	6.00000	+	10.3923i	0.245153	+	0.424618i	0.962175	−	0.272433i	$-0.0878284\pi$
$599$	6.00000	+	10.3923i	0.245153	+	0.424618i	−0.717021	+	0.697051i	$0.754495\pi$
$600$		0			0
$601$	−30.0000			−1.22373			−0.611863	−	0.790964i	$-0.709580\pi$
$601$	−30.0000			−1.22373			−0.611863	+	0.790964i	$0.709580\pi$
$602$	7.00000	+	36.3731i	0.285299	+	1.48246i
$603$		0			0
$604$	10.0000	−	17.3205i	0.406894	−	0.704761i
$605$	2.50000	+	4.33013i	0.101639	+	0.176045i
$606$		0			0
$607$	11.0000	−	19.0526i	0.446476	−	0.773320i	−0.551678	−	0.834058i	$-0.686012\pi$
$607$	11.0000	−	19.0526i	0.446476	−	0.773320i	0.998154	+	0.0607380i	$0.0193454\pi$
$608$	−8.00000			−0.324443
$609$		0			0
$610$	20.0000			0.809776
$611$		0			0
$612$		0			0
$613$	−12.5000	−	21.6506i	−0.504870	−	0.874461i	−0.999984	−	0.00563283i	$-0.998207\pi$
$613$	−12.5000	−	21.6506i	−0.504870	−	0.874461i	0.495114	−	0.868828i	$-0.335126\pi$
$614$	22.0000	−	38.1051i	0.887848	−	1.53780i
$615$		0			0
$616$		0			0
$617$	−27.0000			−1.08698			−0.543490	−	0.839416i	$-0.682897\pi$
$617$	−27.0000			−1.08698			−0.543490	+	0.839416i	$0.682897\pi$
$618$		0			0
$619$	17.5000	+	30.3109i	0.703384	+	1.21830i	0.967271	+	0.253744i	$0.0816620\pi$
$619$	17.5000	+	30.3109i	0.703384	+	1.21830i	−0.263887	+	0.964554i	$0.585005\pi$
$620$		0			0
$621$		0			0
$622$	6.00000			0.240578
$623$	7.00000	+	36.3731i	0.280449	+	1.45726i
$624$		0			0
$625$	−5.50000	+	9.52628i	−0.220000	+	0.381051i
$626$	11.0000	+	19.0526i	0.439648	+	0.761493i
$627$		0			0
$628$	−1.00000	+	1.73205i	−0.0399043	+	0.0691164i
$629$	18.0000			0.717707
$630$		0			0
$631$	5.00000			0.199047			0.0995234	−	0.995035i	$-0.468268\pi$
$631$	5.00000			0.199047			0.0995234	+	0.995035i	$0.468268\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	6.00000	−	10.3923i	0.238103	−	0.412406i
$636$		0			0
$637$	4.00000	−	27.7128i	0.158486	−	1.09802i
$638$	−80.0000			−3.16723
$639$		0			0
$640$		0			0
$641$	−15.0000	−	25.9808i	−0.592464	−	1.02618i	−0.993899	−	0.110291i	$-0.964822\pi$
$641$	−15.0000	−	25.9808i	−0.592464	−	1.02618i	0.401435	−	0.915888i	$-0.368512\pi$
$642$		0			0
$643$	11.0000			0.433798			0.216899	−	0.976194i	$-0.430406\pi$
$643$	11.0000			0.433798			0.216899	+	0.976194i	$0.430406\pi$
$644$	12.0000	−	10.3923i	0.472866	−	0.409514i
$645$		0			0
$646$	3.00000	−	5.19615i	0.118033	−	0.204440i
$647$	11.5000	+	19.9186i	0.452112	+	0.783080i	0.998517	−	0.0544405i	$-0.0173375\pi$
$647$	11.5000	+	19.9186i	0.452112	+	0.783080i	−0.546405	+	0.837521i	$0.684004\pi$
$648$		0			0
$649$	−24.0000	+	41.5692i	−0.942082	+	1.63173i
$650$	32.0000			1.25514
$651$		0			0
$652$	38.0000			1.48819
$653$	−10.5000	+	18.1865i	−0.410897	+	0.711694i	−0.994988	−	0.0999939i	$-0.968118\pi$
$653$	−10.5000	+	18.1865i	−0.410897	+	0.711694i	0.584091	+	0.811688i	$0.301451\pi$
$654$		0			0
$655$	2.50000	+	4.33013i	0.0976831	+	0.169192i
$656$	−4.00000	+	6.92820i	−0.156174	+	0.270501i
$657$		0			0
$658$		0			0
$659$	30.0000			1.16863			0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000			1.16863			0.584317	+	0.811525i	$0.301362\pi$
$660$		0			0
$661$	−20.0000	−	34.6410i	−0.777910	−	1.34738i	−0.933144	−	0.359502i	$-0.882947\pi$
$661$	−20.0000	−	34.6410i	−0.777910	−	1.34738i	0.155235	−	0.987878i	$-0.450387\pi$
$662$	−22.0000	−	38.1051i	−0.855054	−	1.48100i
$663$		0			0
$664$		0			0
$665$	−2.50000	−	0.866025i	−0.0969458	−	0.0335830i
$666$		0			0
$667$	15.0000	−	25.9808i	0.580802	−	1.00598i
$668$	−2.00000	−	3.46410i	−0.0773823	−	0.134030i
$669$		0			0
$670$	−10.0000	+	17.3205i	−0.386334	+	0.669150i
$671$	−40.0000			−1.54418
$672$		0			0
$673$	16.0000			0.616755			0.308377	−	0.951264i	$-0.400214\pi$
$673$	16.0000			0.616755			0.308377	+	0.951264i	$0.400214\pi$
$674$	−8.00000	+	13.8564i	−0.308148	+	0.533729i
$675$		0			0
$676$	−3.00000	−	5.19615i	−0.115385	−	0.199852i
$677$	15.0000	−	25.9808i	0.576497	−	0.998522i	−0.419380	−	0.907811i	$-0.637753\pi$
$677$	15.0000	−	25.9808i	0.576497	−	0.998522i	0.995877	−	0.0907112i	$-0.0289140\pi$
$678$		0			0
$679$	−24.0000	+	20.7846i	−0.921035	+	0.797640i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−18.0000	−	31.1769i	−0.688751	−	1.19295i	−0.972242	−	0.233977i	$-0.924826\pi$
$683$	−18.0000	−	31.1769i	−0.688751	−	1.19295i	0.283491	−	0.958975i	$-0.408507\pi$
$684$		0			0
$685$	−18.0000			−0.687745
$686$	−37.0000	+	1.73205i	−1.41267	+	0.0661300i
$687$		0			0
$688$	−14.0000	+	24.2487i	−0.533745	+	0.924473i
$689$	−24.0000	−	41.5692i	−0.914327	−	1.58366i
$690$		0			0
$691$	−20.5000	+	35.5070i	−0.779857	+	1.35075i	0.152167	+	0.988355i	$0.451375\pi$
$691$	−20.5000	+	35.5070i	−0.779857	+	1.35075i	−0.932024	+	0.362397i	$0.881959\pi$
$692$	−28.0000			−1.06440
$693$		0			0
$694$	58.0000			2.20165
$695$	−6.00000	+	10.3923i	−0.227593	+	0.394203i
$696$		0			0
$697$	−3.00000	−	5.19615i	−0.113633	−	0.196818i
$698$	13.0000	−	22.5167i	0.492057	−	0.852268i
$699$		0			0
$700$	−4.00000	−	20.7846i	−0.151186	−	0.785584i
$701$	21.0000			0.793159			0.396580	−	0.918000i	$-0.370197\pi$
$701$	21.0000			0.793159			0.396580	+	0.918000i	$0.370197\pi$
$702$		0			0
$703$	−3.00000	−	5.19615i	−0.113147	−	0.195977i
$704$	−16.0000	−	27.7128i	−0.603023	−	1.04447i
$705$		0			0
$706$	−52.0000			−1.95705
$707$	27.5000	+	9.52628i	1.03424	+	0.358273i
$708$		0			0
$709$	−10.5000	+	18.1865i	−0.394336	+	0.683010i	−0.993016	−	0.117978i	$-0.962359\pi$
$709$	−10.5000	+	18.1865i	−0.394336	+	0.683010i	0.598680	+	0.800988i	$0.295692\pi$
$710$	6.00000	+	10.3923i	0.225176	+	0.390016i
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	16.0000			0.598366
$716$	−16.0000	+	27.7128i	−0.597948	+	1.03568i
$717$		0			0
$718$	−20.0000	−	34.6410i	−0.746393	−	1.29279i
$719$	10.5000	−	18.1865i	0.391584	−	0.678243i	−0.601075	−	0.799193i	$-0.705261\pi$
$719$	10.5000	−	18.1865i	0.391584	−	0.678243i	0.992659	+	0.120950i	$0.0385939\pi$
$720$		0			0
$721$	4.00000	+	20.7846i	0.148968	+	0.774059i
$722$	−2.00000			−0.0744323
$723$		0			0
$724$	14.0000	+	24.2487i	0.520306	+	0.901196i
$725$	−20.0000	−	34.6410i	−0.742781	−	1.28654i
$726$		0			0
$727$	47.0000			1.74313			0.871567	−	0.490277i	$-0.163104\pi$
$727$	47.0000			1.74313			0.871567	+	0.490277i	$0.163104\pi$
$728$		0			0
$729$		0			0
$730$	−6.00000	+	10.3923i	−0.222070	+	0.384636i
$731$	−10.5000	−	18.1865i	−0.388357	−	0.672653i
$732$		0			0
$733$	15.0000	−	25.9808i	0.554038	−	0.959621i	−0.443940	−	0.896056i	$-0.646420\pi$
$733$	15.0000	−	25.9808i	0.554038	−	0.959621i	0.997978	−	0.0635649i	$-0.0202470\pi$
$734$		0			0
$735$		0			0
$736$	24.0000			0.884652
$737$	20.0000	−	34.6410i	0.736709	−	1.27602i
$738$		0			0
$739$	−25.5000	−	44.1673i	−0.938033	−	1.62472i	−0.769135	−	0.639087i	$-0.779313\pi$
$739$	−25.5000	−	44.1673i	−0.938033	−	1.62472i	−0.168898	−	0.985634i	$-0.554021\pi$
$740$	−6.00000	+	10.3923i	−0.220564	+	0.382029i
$741$		0			0
$742$	−48.0000	+	41.5692i	−1.76214	+	1.52605i
$743$	24.0000			0.880475			0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000			0.880475			0.440237	+	0.897881i	$0.354894\pi$
$744$		0			0
$745$	4.50000	+	7.79423i	0.164867	+	0.285558i
$746$	32.0000	+	55.4256i	1.17160	+	2.02928i
$747$		0			0
$748$	24.0000			0.877527
$749$	−5.00000	−	25.9808i	−0.182696	−	0.949316i
$750$		0			0
$751$	7.00000	−	12.1244i	0.255434	−	0.442424i	−0.709580	−	0.704625i	$-0.751115\pi$
$751$	7.00000	−	12.1244i	0.255434	−	0.442424i	0.965013	+	0.262201i	$0.0844484\pi$
$752$		0			0
$753$		0			0
$754$	−40.0000	+	69.2820i	−1.45671	+	2.52310i
$755$	10.0000			0.363937
$756$		0			0
$757$	2.00000			0.0726912			0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000			0.0726912			0.0363456	+	0.999339i	$0.488428\pi$
$758$	12.0000	−	20.7846i	0.435860	−	0.754931i
$759$		0			0
$760$		0			0
$761$	−10.5000	+	18.1865i	−0.380625	+	0.659261i	−0.991152	−	0.132734i	$-0.957624\pi$
$761$	−10.5000	+	18.1865i	−0.380625	+	0.659261i	0.610527	+	0.791995i	$0.290958\pi$
$762$		0			0
$763$	15.0000	+	5.19615i	0.543036	+	0.188113i
$764$	34.0000			1.23008
$765$		0			0
$766$	−12.0000	−	20.7846i	−0.433578	−	0.750978i
$767$	24.0000	+	41.5692i	0.866590	+	1.50098i
$768$		0			0
$769$	−37.0000			−1.33425			−0.667127	−	0.744944i	$-0.732476\pi$
$769$	−37.0000			−1.33425			−0.667127	+	0.744944i	$0.732476\pi$
$770$	−4.00000	−	20.7846i	−0.144150	−	0.749025i
$771$		0			0
$772$	16.0000	−	27.7128i	0.575853	−	0.997406i
$773$	−14.0000	−	24.2487i	−0.503545	−	0.872166i	−0.999992	−	0.00409826i	$-0.998695\pi$
$773$	−14.0000	−	24.2487i	−0.503545	−	0.872166i	0.496447	−	0.868067i	$-0.334638\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−6.00000			−0.215110
$779$	−1.00000	+	1.73205i	−0.0358287	+	0.0620572i
$780$		0			0
$781$	−12.0000	−	20.7846i	−0.429394	−	0.743732i
$782$	−9.00000	+	15.5885i	−0.321839	+	0.557442i
$783$		0			0
$784$	−22.0000	−	17.3205i	−0.785714	−	0.618590i
$785$	−1.00000			−0.0356915
$786$		0			0
$787$	−23.0000	−	39.8372i	−0.819861	−	1.42004i	−0.905784	−	0.423740i	$-0.860717\pi$
$787$	−23.0000	−	39.8372i	−0.819861	−	1.42004i	0.0859225	−	0.996302i	$-0.472616\pi$
$788$	−11.0000	−	19.0526i	−0.391859	−	0.678719i
$789$		0			0
$790$	−20.0000			−0.711568
$791$	16.0000	−	13.8564i	0.568895	−	0.492677i
$792$		0			0
$793$	−20.0000	+	34.6410i	−0.710221	+	1.23014i
$794$	−15.0000	−	25.9808i	−0.532330	−	0.922023i
$795$		0			0
$796$	27.0000	−	46.7654i	0.956990	−	1.65755i
$797$	−4.00000			−0.141687			−0.0708436	−	0.997487i	$-0.522569\pi$
$797$	−4.00000			−0.141687			−0.0708436	+	0.997487i	$0.522569\pi$
$798$		0			0
$799$		0			0
$800$	16.0000	−	27.7128i	0.565685	−	0.979796i
$801$		0			0
$802$	12.0000	+	20.7846i	0.423735	+	0.733930i
$803$	12.0000	−	20.7846i	0.423471	−	0.733473i
$804$		0			0
$805$	7.50000	+	2.59808i	0.264340	+	0.0915702i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−13.5000	−	23.3827i	−0.474635	−	0.822091i	0.524943	−	0.851137i	$-0.324086\pi$
$809$	−13.5000	−	23.3827i	−0.474635	−	0.822091i	−0.999578	+	0.0290457i	$0.990753\pi$
$810$		0			0
$811$	32.0000			1.12367			0.561836	−	0.827249i	$-0.310095\pi$
$811$	32.0000			1.12367			0.561836	+	0.827249i	$0.310095\pi$
$812$	50.0000	+	17.3205i	1.75466	+	0.607831i
$813$		0			0
$814$	24.0000	−	41.5692i	0.841200	−	1.45700i
$815$	9.50000	+	16.4545i	0.332770	+	0.576375i
$816$		0			0
$817$	−3.50000	+	6.06218i	−0.122449	+	0.212089i
$818$	−64.0000			−2.23771
$819$		0			0
$820$	4.00000			0.139686
$821$	−5.00000	+	8.66025i	−0.174501	+	0.302245i	−0.939989	−	0.341206i	$-0.889165\pi$
$821$	−5.00000	+	8.66025i	−0.174501	+	0.302245i	0.765487	+	0.643451i	$0.222498\pi$
$822$		0			0
$823$	12.0000	+	20.7846i	0.418294	+	0.724506i	0.995768	−	0.0919029i	$-0.0292950\pi$
$823$	12.0000	+	20.7846i	0.418294	+	0.724506i	−0.577474	+	0.816409i	$0.695962\pi$
$824$		0			0
$825$		0			0
$826$	48.0000	−	41.5692i	1.67013	−	1.44638i
$827$	6.00000			0.208640			0.104320	−	0.994544i	$-0.466733\pi$
$827$	6.00000			0.208640			0.104320	+	0.994544i	$0.466733\pi$
$828$		0			0
$829$	26.0000	+	45.0333i	0.903017	+	1.56407i	0.823557	+	0.567234i	$0.191986\pi$
$829$	26.0000	+	45.0333i	0.903017	+	1.56407i	0.0794606	+	0.996838i	$0.474680\pi$
$830$	3.00000	+	5.19615i	0.104132	+	0.180361i
$831$		0			0
$832$	−32.0000			−1.10940
$833$	19.5000	−	7.79423i	0.675635	−	0.270054i
$834$		0			0
$835$	1.00000	−	1.73205i	0.0346064	−	0.0599401i
$836$	−4.00000	−	6.92820i	−0.138343	−	0.239617i
$837$		0			0
$838$	9.00000	−	15.5885i	0.310900	−	0.538494i
$839$	56.0000			1.93333			0.966667	−	0.256036i	$-0.0824164\pi$
$839$	56.0000			1.93333			0.966667	+	0.256036i	$0.0824164\pi$
$840$		0			0
$841$	71.0000			2.44828
$842$	8.00000	−	13.8564i	0.275698	−	0.477523i
$843$		0			0
$844$	−4.00000	−	6.92820i	−0.137686	−	0.238479i
$845$	1.50000	−	2.59808i	0.0516016	−	0.0893765i
$846$		0			0
$847$	2.50000	+	12.9904i	0.0859010	+	0.446355i
$848$	−48.0000			−1.64833
$849$		0			0
$850$	12.0000	+	20.7846i	0.411597	+	0.712906i
$851$	9.00000	+	15.5885i	0.308516	+	0.534365i
$852$		0			0
$853$	−37.0000			−1.26686			−0.633428	−	0.773802i	$-0.718353\pi$
$853$	−37.0000			−1.26686			−0.633428	+	0.773802i	$0.718353\pi$
$854$	50.0000	+	17.3205i	1.71096	+	0.592696i
$855$		0			0
$856$		0			0
$857$	20.0000	+	34.6410i	0.683187	+	1.18331i	0.974003	+	0.226536i	$0.0727399\pi$
$857$	20.0000	+	34.6410i	0.683187	+	1.18331i	−0.290816	+	0.956779i	$0.593927\pi$
$858$		0			0
$859$	−8.50000	+	14.7224i	−0.290016	+	0.502323i	−0.973813	−	0.227349i	$-0.926994\pi$
$859$	−8.50000	+	14.7224i	−0.290016	+	0.502323i	0.683797	+	0.729672i	$0.260327\pi$
$860$	14.0000			0.477396
$861$		0			0
$862$	−24.0000			−0.817443
$863$	−6.00000	+	10.3923i	−0.204242	+	0.353758i	−0.949891	−	0.312581i	$-0.898806\pi$
$863$	−6.00000	+	10.3923i	−0.204242	+	0.353758i	0.745649	+	0.666339i	$0.232140\pi$
$864$		0			0
$865$	−7.00000	−	12.1244i	−0.238007	−	0.412240i
$866$	−14.0000	+	24.2487i	−0.475739	+	0.824005i
$867$		0			0
$868$		0			0
$869$	40.0000			1.35691
$870$		0			0
$871$	−20.0000	−	34.6410i	−0.677674	−	1.17377i
$872$		0			0
$873$		0			0
$874$	6.00000			0.202953
$875$	18.0000	−	15.5885i	0.608511	−	0.526986i
$876$		0			0
$877$	16.0000	−	27.7128i	0.540282	−	0.935795i	−0.458606	−	0.888640i	$-0.651651\pi$
$877$	16.0000	−	27.7128i	0.540282	−	0.935795i	0.998888	−	0.0471555i	$-0.0150156\pi$
$878$	30.0000	+	51.9615i	1.01245	+	1.75362i
$879$		0			0
$880$	8.00000	−	13.8564i	0.269680	−	0.467099i
$881$	−15.0000			−0.505363			−0.252681	−	0.967550i	$-0.581312\pi$
$881$	−15.0000			−0.505363			−0.252681	+	0.967550i	$0.581312\pi$
$882$		0			0
$883$	47.0000			1.58168			0.790838	−	0.612026i	$-0.209645\pi$
$883$	47.0000			1.58168			0.790838	+	0.612026i	$0.209645\pi$
$884$	12.0000	−	20.7846i	0.403604	−	0.699062i
$885$		0			0
$886$	21.0000	+	36.3731i	0.705509	+	1.22198i
$887$	7.00000	−	12.1244i	0.235037	−	0.407096i	−0.724246	−	0.689541i	$-0.757812\pi$
$887$	7.00000	−	12.1244i	0.235037	−	0.407096i	0.959283	+	0.282445i	$0.0911455\pi$
$888$		0			0
$889$	24.0000	−	20.7846i	0.804934	−	0.697093i
$890$	28.0000			0.938562
$891$		0			0
$892$	−10.0000	−	17.3205i	−0.334825	−	0.579934i
$893$		0			0
$894$		0			0
$895$	−16.0000			−0.534821
$896$		0			0
$897$		0			0
$898$	−6.00000	+	10.3923i	−0.200223	+	0.346796i
$899$		0			0
$900$		0			0
$901$	18.0000	−	31.1769i	0.599667	−	1.03865i
$902$	−16.0000			−0.532742
$903$		0			0
$904$		0			0
$905$	−7.00000	+	12.1244i	−0.232688	+	0.403027i
$906$		0			0
$907$	−10.0000	−	17.3205i	−0.332045	−	0.575118i	0.650868	−	0.759191i	$-0.274405\pi$
$907$	−10.0000	−	17.3205i	−0.332045	−	0.575118i	−0.982913	+	0.184073i	$0.941072\pi$
$908$		0			0
$909$		0			0
$910$	−20.0000	−	6.92820i	−0.662994	−	0.229668i
$911$	42.0000			1.39152			0.695761	−	0.718273i	$-0.255067\pi$
$911$	42.0000			1.39152			0.695761	+	0.718273i	$0.255067\pi$
$912$		0			0
$913$	−6.00000	−	10.3923i	−0.198571	−	0.343935i
$914$	22.0000	+	38.1051i	0.727695	+	1.26041i
$915$		0			0
$916$	−14.0000			−0.462573
$917$	2.50000	+	12.9904i	0.0825573	+	0.428980i
$918$		0			0
$919$	3.50000	−	6.06218i	0.115454	−	0.199973i	−0.802507	−	0.596643i	$-0.796501\pi$
$919$	3.50000	−	6.06218i	0.115454	−	0.199973i	0.917961	+	0.396670i	$0.129834\pi$
$920$		0			0
$921$		0			0
$922$	−26.0000	+	45.0333i	−0.856264	+	1.48309i
$923$	−24.0000			−0.789970
$924$		0			0
$925$	24.0000			0.789115
$926$	4.00000	−	6.92820i	0.131448	−	0.227675i
$927$		0			0
$928$	40.0000	+	69.2820i	1.31306	+	2.27429i
$929$	−27.5000	+	47.6314i	−0.902246	+	1.56274i	−0.0776734	+	0.996979i	$0.524749\pi$
$929$	−27.5000	+	47.6314i	−0.902246	+	1.56274i	−0.824572	+	0.565757i	$0.808584\pi$
$930$		0			0
$931$	−5.50000	−	4.33013i	−0.180255	−	0.141914i
$932$	−18.0000			−0.589610
$933$		0			0
$934$	−36.0000	−	62.3538i	−1.17796	−	2.04028i
$935$	6.00000	+	10.3923i	0.196221	+	0.339865i
$936$		0			0
$937$	57.0000			1.86211			0.931054	−	0.364880i	$-0.118890\pi$
$937$	57.0000			1.86211			0.931054	+	0.364880i	$0.118890\pi$
$938$	−40.0000	+	34.6410i	−1.30605	+	1.13107i
$939$		0			0
$940$		0			0
$941$	10.0000	+	17.3205i	0.325991	+	0.564632i	0.981712	−	0.190370i	$-0.0609689\pi$
$941$	10.0000	+	17.3205i	0.325991	+	0.564632i	−0.655722	+	0.755003i	$0.727636\pi$
$942$		0			0
$943$	3.00000	−	5.19615i	0.0976934	−	0.169210i
$944$	48.0000			1.56227
$945$		0			0
$946$	−56.0000			−1.82072
$947$	−14.0000	+	24.2487i	−0.454939	+	0.787977i	−0.998685	−	0.0512727i	$-0.983672\pi$
$947$	−14.0000	+	24.2487i	−0.454939	+	0.787977i	0.543746	+	0.839250i	$0.317006\pi$
$948$		0			0
$949$	−12.0000	−	20.7846i	−0.389536	−	0.674697i
$950$	4.00000	−	6.92820i	0.129777	−	0.224781i
$951$		0			0
$952$		0			0
$953$	34.0000			1.10137			0.550684	−	0.834714i	$-0.314367\pi$
$953$	34.0000			1.10137			0.550684	+	0.834714i	$0.314367\pi$
$954$		0			0
$955$	8.50000	+	14.7224i	0.275054	+	0.476407i
$956$	9.00000	+	15.5885i	0.291081	+	0.504167i
$957$		0			0
$958$	22.0000			0.710788
$959$	−45.0000	−	15.5885i	−1.45313	−	0.503378i
$960$		0			0
$961$	15.5000	−	26.8468i	0.500000	−	0.866025i
$962$	−24.0000	−	41.5692i	−0.773791	−	1.34025i
$963$		0			0
$964$	28.0000	−	48.4974i	0.901819	−	1.56200i
$965$	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
$969$		0			0
$970$	12.0000	+	20.7846i	0.385297	+	0.667354i
$971$	24.0000	−	41.5692i	0.770197	−	1.33402i	−0.167258	−	0.985913i	$-0.553491\pi$
$971$	24.0000	−	41.5692i	0.770197	−	1.33402i	0.937455	−	0.348107i	$-0.113175\pi$
$972$		0			0
$973$	−24.0000	+	20.7846i	−0.769405	+	0.666324i
$974$	64.0000			2.05069
$975$		0			0
$976$	20.0000	+	34.6410i	0.640184	+	1.10883i
$977$	−21.0000	−	36.3731i	−0.671850	−	1.16368i	−0.977379	−	0.211495i	$-0.932167\pi$
$977$	−21.0000	−	36.3731i	−0.671850	−	1.16368i	0.305530	−	0.952183i	$-0.401167\pi$
$978$		0			0
$979$	−56.0000			−1.78977
$980$	−2.00000	+	13.8564i	−0.0638877	+	0.442627i
$981$		0			0
$982$	1.00000	−	1.73205i	0.0319113	−	0.0552720i
$983$	12.0000	+	20.7846i	0.382741	+	0.662926i	0.991453	−	0.130465i	$-0.0416470\pi$
$983$	12.0000	+	20.7846i	0.382741	+	0.662926i	−0.608712	+	0.793391i	$0.708314\pi$
$984$		0			0
$985$	5.50000	−	9.52628i	0.175245	−	0.303533i
$986$	−60.0000			−1.91079
$987$		0			0
$988$	−8.00000			−0.254514
$989$	10.5000	−	18.1865i	0.333881	−	0.578298i
$990$		0			0
$991$	8.00000	+	13.8564i	0.254128	+	0.440163i	0.964658	−	0.263504i	$-0.0848781\pi$
$991$	8.00000	+	13.8564i	0.254128	+	0.440163i	−0.710530	+	0.703667i	$0.751545\pi$
$992$		0			0
$993$		0			0
$994$	6.00000	+	31.1769i	0.190308	+	0.988872i
$995$	27.0000			0.855958
$996$		0			0
$997$	−12.5000	−	21.6506i	−0.395879	−	0.685682i	0.597334	−	0.801993i	$-0.296227\pi$
$997$	−12.5000	−	21.6506i	−0.395879	−	0.685682i	−0.993213	+	0.116310i	$0.962893\pi$
$998$	7.00000	+	12.1244i	0.221581	+	0.383790i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1197.2.j.b.856.1		2
3.2	odd	2		399.2.j.a.58.1	✓	2
7.2	even	3		8379.2.a.b.1.1		1
7.4	even	3	inner	1197.2.j.b.172.1		2
7.5	odd	6		8379.2.a.c.1.1		1
21.2	odd	6		2793.2.a.l.1.1		1
21.5	even	6		2793.2.a.k.1.1		1
21.11	odd	6		399.2.j.a.172.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
399.2.j.a.58.1	✓	2	3.2	odd	2
399.2.j.a.172.1	yes	2	21.11	odd	6
1197.2.j.b.172.1		2	7.4	even	3	inner
1197.2.j.b.856.1		2	1.1	even	1	trivial
2793.2.a.k.1.1		1	21.5	even	6
2793.2.a.l.1.1		1	21.2	odd	6
8379.2.a.b.1.1		1	7.2	even	3
8379.2.a.c.1.1		1	7.5	odd	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:	\( N \)	\(=\)	\( 1197 = 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1197.j (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.73205i) q^{2} +(-1.00000 - 1.73205i) q^{4} +(0.500000 - 0.866025i) q^{5} +(2.00000 - 1.73205i) q^{7} +(-1.00000 - 1.73205i) q^{10} +(2.00000 + 3.46410i) q^{11} +4.00000 q^{13} +(-1.00000 - 5.19615i) q^{14} +(2.00000 - 3.46410i) q^{16} +(1.50000 + 2.59808i) q^{17} +(0.500000 - 0.866025i) q^{19} -2.00000 q^{20} +8.00000 q^{22} +(-1.50000 + 2.59808i) q^{23} +(2.00000 + 3.46410i) q^{25} +(4.00000 - 6.92820i) q^{26} +(-5.00000 - 1.73205i) q^{28} -10.0000 q^{29} +(-4.00000 - 6.92820i) q^{32} +6.00000 q^{34} +(-0.500000 - 2.59808i) q^{35} +(3.00000 - 5.19615i) q^{37} +(-1.00000 - 1.73205i) q^{38} -2.00000 q^{41} -7.00000 q^{43} +(4.00000 - 6.92820i) q^{44} +(3.00000 + 5.19615i) q^{46} +(1.00000 - 6.92820i) q^{49} +8.00000 q^{50} +(-4.00000 - 6.92820i) q^{52} +(-6.00000 - 10.3923i) q^{53} +4.00000 q^{55} +(-10.0000 + 17.3205i) q^{58} +(6.00000 + 10.3923i) q^{59} +(-5.00000 + 8.66025i) q^{61} -8.00000 q^{64} +(2.00000 - 3.46410i) q^{65} +(-5.00000 - 8.66025i) q^{67} +(3.00000 - 5.19615i) q^{68} +(-5.00000 - 1.73205i) q^{70} -6.00000 q^{71} +(-3.00000 - 5.19615i) q^{73} +(-6.00000 - 10.3923i) q^{74} -2.00000 q^{76} +(10.0000 + 3.46410i) q^{77} +(5.00000 - 8.66025i) q^{79} +(-2.00000 - 3.46410i) q^{80} +(-2.00000 + 3.46410i) q^{82} -3.00000 q^{83} +3.00000 q^{85} +(-7.00000 + 12.1244i) q^{86} +(-7.00000 + 12.1244i) q^{89} +(8.00000 - 6.92820i) q^{91} +6.00000 q^{92} +(-0.500000 - 0.866025i) q^{95} -12.0000 q^{97} +(-11.0000 - 8.66025i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{4} + q^{5} + 4 q^{7} - 2 q^{10} + 4 q^{11} + 8 q^{13} - 2 q^{14} + 4 q^{16} + 3 q^{17} + q^{19} - 4 q^{20} + 16 q^{22} - 3 q^{23} + 4 q^{25} + 8 q^{26} - 10 q^{28} - 20 q^{29} - 8 q^{32}+ \cdots - 22 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^4 + q^5 + 4 * q^7 - 2 * q^10 + 4 * q^11 + 8 * q^13 - 2 * q^14 + 4 * q^16 + 3 * q^17 + q^19 - 4 * q^20 + 16 * q^22 - 3 * q^23 + 4 * q^25 + 8 * q^26 - 10 * q^28 - 20 * q^29 - 8 * q^32 + 12 * q^34 - q^35 + 6 * q^37 - 2 * q^38 - 4 * q^41 - 14 * q^43 + 8 * q^44 + 6 * q^46 + 2 * q^49 + 16 * q^50 - 8 * q^52 - 12 * q^53 + 8 * q^55 - 20 * q^58 + 12 * q^59 - 10 * q^61 - 16 * q^64 + 4 * q^65 - 10 * q^67 + 6 * q^68 - 10 * q^70 - 12 * q^71 - 6 * q^73 - 12 * q^74 - 4 * q^76 + 20 * q^77 + 10 * q^79 - 4 * q^80 - 4 * q^82 - 6 * q^83 + 6 * q^85 - 14 * q^86 - 14 * q^89 + 16 * q^91 + 12 * q^92 - q^95 - 24 * q^97 - 22 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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