LMFDB - Embedded newform 351.2.l.a.199.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [351,2,Mod(127,351)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(351, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 5]))

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

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

chi := DirichletCharacter("351.127");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 351 = 3^{3} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	351.l (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$2.80274911095$
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 117)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			199.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	351.199
Dual form			351.2.l.a.127.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.73205i q^{2} -1.00000 q^{4} +(-1.50000 - 0.866025i) q^{5} +(1.50000 + 0.866025i) q^{7} +1.73205i q^{8} +(1.50000 - 2.59808i) q^{10} +3.46410i q^{11} +(-1.00000 + 3.46410i) q^{13} +(-1.50000 + 2.59808i) q^{14} -5.00000 q^{16} +(1.50000 + 2.59808i) q^{17} +(-1.50000 + 0.866025i) q^{19} +(1.50000 + 0.866025i) q^{20} -6.00000 q^{22} +(-1.50000 - 2.59808i) q^{23} +(-1.00000 - 1.73205i) q^{25} +(-6.00000 - 1.73205i) q^{26} +(-1.50000 - 0.866025i) q^{28} +6.00000 q^{29} +(7.50000 + 4.33013i) q^{31} -5.19615i q^{32} +(-4.50000 + 2.59808i) q^{34} +(-1.50000 - 2.59808i) q^{35} +(-4.50000 - 2.59808i) q^{37} +(-1.50000 - 2.59808i) q^{38} +(1.50000 - 2.59808i) q^{40} +(10.5000 - 6.06218i) q^{41} +(0.500000 - 0.866025i) q^{43} -3.46410i q^{44} +(4.50000 - 2.59808i) q^{46} +(-4.50000 + 2.59808i) q^{47} +(-2.00000 - 3.46410i) q^{49} +(3.00000 - 1.73205i) q^{50} +(1.00000 - 3.46410i) q^{52} -6.00000 q^{53} +(3.00000 - 5.19615i) q^{55} +(-1.50000 + 2.59808i) q^{56} +10.3923i q^{58} +3.46410i q^{59} +(2.50000 - 4.33013i) q^{61} +(-7.50000 + 12.9904i) q^{62} -1.00000 q^{64} +(4.50000 - 4.33013i) q^{65} +(10.5000 - 6.06218i) q^{67} +(-1.50000 - 2.59808i) q^{68} +(4.50000 - 2.59808i) q^{70} +(7.50000 - 4.33013i) q^{71} +6.92820i q^{73} +(4.50000 - 7.79423i) q^{74} +(1.50000 - 0.866025i) q^{76} +(-3.00000 + 5.19615i) q^{77} +(5.50000 + 9.52628i) q^{79} +(7.50000 + 4.33013i) q^{80} +(10.5000 + 18.1865i) q^{82} +(-4.50000 + 2.59808i) q^{83} -5.19615i q^{85} +(1.50000 + 0.866025i) q^{86} -6.00000 q^{88} +(-13.5000 - 7.79423i) q^{89} +(-4.50000 + 4.33013i) q^{91} +(1.50000 + 2.59808i) q^{92} +(-4.50000 - 7.79423i) q^{94} +3.00000 q^{95} +(13.5000 + 7.79423i) q^{97} +(6.00000 - 3.46410i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 3 q^{5} + 3 q^{7} + 3 q^{10} - 2 q^{13} - 3 q^{14} - 10 q^{16} + 3 q^{17} - 3 q^{19} + 3 q^{20} - 12 q^{22} - 3 q^{23} - 2 q^{25} - 12 q^{26} - 3 q^{28} + 12 q^{29} + 15 q^{31} - 9 q^{34}+ \cdots + 12 q^{98}+O(q^{100}) $ 2 * q - 2 * q^4 - 3 * q^5 + 3 * q^7 + 3 * q^10 - 2 * q^13 - 3 * q^14 - 10 * q^16 + 3 * q^17 - 3 * q^19 + 3 * q^20 - 12 * q^22 - 3 * q^23 - 2 * q^25 - 12 * q^26 - 3 * q^28 + 12 * q^29 + 15 * q^31 - 9 * q^34 - 3 * q^35 - 9 * q^37 - 3 * q^38 + 3 * q^40 + 21 * q^41 + q^43 + 9 * q^46 - 9 * q^47 - 4 * q^49 + 6 * q^50 + 2 * q^52 - 12 * q^53 + 6 * q^55 - 3 * q^56 + 5 * q^61 - 15 * q^62 - 2 * q^64 + 9 * q^65 + 21 * q^67 - 3 * q^68 + 9 * q^70 + 15 * q^71 + 9 * q^74 + 3 * q^76 - 6 * q^77 + 11 * q^79 + 15 * q^80 + 21 * q^82 - 9 * q^83 + 3 * q^86 - 12 * q^88 - 27 * q^89 - 9 * q^91 + 3 * q^92 - 9 * q^94 + 6 * q^95 + 27 * q^97 + 12 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/351\mathbb{Z}\right)^\times$.

$n$	$28$	$326$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$

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 $

$2$			1.73205i			1.22474i	0.790569	+	0.612372i	$0.209785\pi$
$2$			1.73205i			1.22474i	−0.790569	+	0.612372i	$0.790215\pi$
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	−1.50000	−	0.866025i	−0.670820	−	0.387298i	0.125567	−	0.992085i	$-0.459925\pi$
$5$	−1.50000	−	0.866025i	−0.670820	−	0.387298i	−0.796387	+	0.604787i	$0.793258\pi$
$6$		0			0
$7$	1.50000	+	0.866025i	0.566947	+	0.327327i	0.755929	−	0.654654i	$-0.227186\pi$
$7$	1.50000	+	0.866025i	0.566947	+	0.327327i	−0.188982	+	0.981981i	$0.560519\pi$
$8$			1.73205i			0.612372i
$9$		0			0
$10$	1.50000	−	2.59808i	0.474342	−	0.821584i
$11$			3.46410i			1.04447i	0.852803	+	0.522233i	$0.174901\pi$
$11$			3.46410i			1.04447i	−0.852803	+	0.522233i	$0.825099\pi$
$12$		0			0
$13$	−1.00000	+	3.46410i	−0.277350	+	0.960769i
$13$	−1.00000	+	3.46410i	−0.277350	+	0.960769i
$14$	−1.50000	+	2.59808i	−0.400892	+	0.694365i
$15$		0			0
$16$	−5.00000			−1.25000
$17$	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
$19$	−1.50000	+	0.866025i	−0.344124	+	0.198680i	−0.662094	−	0.749421i	$-0.730332\pi$
$19$	−1.50000	+	0.866025i	−0.344124	+	0.198680i	0.317970	+	0.948101i	$0.396999\pi$
$20$	1.50000	+	0.866025i	0.335410	+	0.193649i
$21$		0			0
$22$	−6.00000			−1.27920
$23$	−1.50000	−	2.59808i	−0.312772	−	0.541736i	0.666190	−	0.745782i	$-0.267924\pi$
$23$	−1.50000	−	2.59808i	−0.312772	−	0.541736i	−0.978961	+	0.204046i	$0.934591\pi$
$24$		0			0
$25$	−1.00000	−	1.73205i	−0.200000	−	0.346410i
$26$	−6.00000	−	1.73205i	−1.17670	−	0.339683i
$27$		0			0
$28$	−1.50000	−	0.866025i	−0.283473	−	0.163663i
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		0			0
$31$	7.50000	+	4.33013i	1.34704	+	0.777714i	0.987829	−	0.155543i	$-0.0497126\pi$
$31$	7.50000	+	4.33013i	1.34704	+	0.777714i	0.359211	+	0.933257i	$0.383046\pi$
$32$		−	5.19615i		−	0.918559i
$33$		0			0
$34$	−4.50000	+	2.59808i	−0.771744	+	0.445566i
$35$	−1.50000	−	2.59808i	−0.253546	−	0.439155i
$36$		0			0
$37$	−4.50000	−	2.59808i	−0.739795	−	0.427121i	0.0821995	−	0.996616i	$-0.473806\pi$
$37$	−4.50000	−	2.59808i	−0.739795	−	0.427121i	−0.821995	+	0.569495i	$0.807139\pi$
$38$	−1.50000	−	2.59808i	−0.243332	−	0.421464i
$39$		0			0
$40$	1.50000	−	2.59808i	0.237171	−	0.410792i
$41$	10.5000	−	6.06218i	1.63982	−	0.946753i	0.658932	−	0.752202i	$-0.271008\pi$
$41$	10.5000	−	6.06218i	1.63982	−	0.946753i	0.980892	−	0.194551i	$-0.0623249\pi$
$42$		0			0
$43$	0.500000	−	0.866025i	0.0762493	−	0.132068i	−0.825380	−	0.564578i	$-0.809039\pi$
$43$	0.500000	−	0.866025i	0.0762493	−	0.132068i	0.901629	+	0.432511i	$0.142372\pi$
$44$		−	3.46410i		−	0.522233i
$45$		0			0
$46$	4.50000	−	2.59808i	0.663489	−	0.383065i
$47$	−4.50000	+	2.59808i	−0.656392	+	0.378968i	−0.790901	−	0.611944i	$-0.790388\pi$
$47$	−4.50000	+	2.59808i	−0.656392	+	0.378968i	0.134509	+	0.990912i	$0.457054\pi$
$48$		0			0
$49$	−2.00000	−	3.46410i	−0.285714	−	0.494872i
$50$	3.00000	−	1.73205i	0.424264	−	0.244949i
$51$		0			0
$52$	1.00000	−	3.46410i	0.138675	−	0.480384i
$53$	−6.00000			−0.824163			−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000			−0.824163			−0.412082	+	0.911147i	$0.635198\pi$
$54$		0			0
$55$	3.00000	−	5.19615i	0.404520	−	0.700649i
$56$	−1.50000	+	2.59808i	−0.200446	+	0.347183i
$57$		0			0
$58$			10.3923i			1.36458i
$59$			3.46410i			0.450988i	0.974245	+	0.225494i	$0.0723995\pi$
$59$			3.46410i			0.450988i	−0.974245	+	0.225494i	$0.927600\pi$
$60$		0			0
$61$	2.50000	−	4.33013i	0.320092	−	0.554416i	−0.660415	−	0.750901i	$-0.729619\pi$
$61$	2.50000	−	4.33013i	0.320092	−	0.554416i	0.980507	+	0.196485i	$0.0629528\pi$
$62$	−7.50000	+	12.9904i	−0.952501	+	1.64978i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	4.50000	−	4.33013i	0.558156	−	0.537086i
$66$		0			0
$67$	10.5000	−	6.06218i	1.28278	−	0.740613i	0.305424	−	0.952217i	$-0.401202\pi$
$67$	10.5000	−	6.06218i	1.28278	−	0.740613i	0.977356	+	0.211604i	$0.0678686\pi$
$68$	−1.50000	−	2.59808i	−0.181902	−	0.315063i
$69$		0			0
$70$	4.50000	−	2.59808i	0.537853	−	0.310530i
$71$	7.50000	−	4.33013i	0.890086	−	0.513892i	0.0161155	−	0.999870i	$-0.494870\pi$
$71$	7.50000	−	4.33013i	0.890086	−	0.513892i	0.873971	+	0.485979i	$0.161537\pi$
$72$		0			0
$73$			6.92820i			0.810885i	0.914121	+	0.405442i	$0.132883\pi$
$73$			6.92820i			0.810885i	−0.914121	+	0.405442i	$0.867117\pi$
$74$	4.50000	−	7.79423i	0.523114	−	0.906061i
$75$		0			0
$76$	1.50000	−	0.866025i	0.172062	−	0.0993399i
$77$	−3.00000	+	5.19615i	−0.341882	+	0.592157i
$78$		0			0
$79$	5.50000	+	9.52628i	0.618798	+	1.07179i	0.989705	+	0.143120i	$0.0457135\pi$
$79$	5.50000	+	9.52628i	0.618798	+	1.07179i	−0.370907	+	0.928670i	$0.620953\pi$
$80$	7.50000	+	4.33013i	0.838525	+	0.484123i
$81$		0			0
$82$	10.5000	+	18.1865i	1.15953	+	2.00837i
$83$	−4.50000	+	2.59808i	−0.493939	+	0.285176i	−0.726207	−	0.687476i	$-0.758719\pi$
$83$	−4.50000	+	2.59808i	−0.493939	+	0.285176i	0.232268	+	0.972652i	$0.425385\pi$
$84$		0			0
$85$		−	5.19615i		−	0.563602i
$86$	1.50000	+	0.866025i	0.161749	+	0.0933859i
$87$		0			0
$88$	−6.00000			−0.639602
$89$	−13.5000	−	7.79423i	−1.43100	−	0.826187i	−0.433800	−	0.901009i	$-0.642828\pi$
$89$	−13.5000	−	7.79423i	−1.43100	−	0.826187i	−0.997197	+	0.0748225i	$0.976161\pi$
$90$		0			0
$91$	−4.50000	+	4.33013i	−0.471728	+	0.453921i
$92$	1.50000	+	2.59808i	0.156386	+	0.270868i
$93$		0			0
$94$	−4.50000	−	7.79423i	−0.464140	−	0.803913i
$95$	3.00000			0.307794
$96$		0			0
$97$	13.5000	+	7.79423i	1.37072	+	0.791384i	0.991018	−	0.133726i	$-0.0426942\pi$
$97$	13.5000	+	7.79423i	1.37072	+	0.791384i	0.379699	+	0.925110i	$0.376028\pi$
$98$	6.00000	−	3.46410i	0.606092	−	0.349927i
$99$		0			0
$100$	1.00000	+	1.73205i	0.100000	+	0.173205i
$101$	6.00000			0.597022			0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000			0.597022			0.298511	+	0.954406i	$0.403510\pi$
$102$		0			0
$103$	6.50000	−	11.2583i	0.640464	−	1.10932i	−0.344865	−	0.938652i	$-0.612075\pi$
$103$	6.50000	−	11.2583i	0.640464	−	1.10932i	0.985329	−	0.170664i	$-0.0545913\pi$
$104$	−6.00000	−	1.73205i	−0.588348	−	0.169842i
$105$		0			0
$106$		−	10.3923i		−	1.00939i
$107$	7.50000	−	12.9904i	0.725052	−	1.25583i	−0.233900	−	0.972261i	$-0.575149\pi$
$107$	7.50000	−	12.9904i	0.725052	−	1.25583i	0.958952	−	0.283567i	$-0.0915178\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$	9.00000	+	5.19615i	0.858116	+	0.495434i
$111$		0			0
$112$	−7.50000	−	4.33013i	−0.708683	−	0.409159i
$113$	−6.00000			−0.564433			−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000			−0.564433			−0.282216	+	0.959351i	$0.591070\pi$
$114$		0			0
$115$			5.19615i			0.484544i
$116$	−6.00000			−0.557086
$117$		0			0
$118$	−6.00000			−0.552345
$119$			5.19615i			0.476331i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$	7.50000	+	4.33013i	0.679018	+	0.392031i
$123$		0			0
$124$	−7.50000	−	4.33013i	−0.673520	−	0.388857i
$125$			12.1244i			1.08444i
$126$		0			0
$127$	2.50000	−	4.33013i	0.221839	−	0.384237i	−0.733527	−	0.679660i	$-0.762127\pi$
$127$	2.50000	−	4.33013i	0.221839	−	0.384237i	0.955366	+	0.295423i	$0.0954607\pi$
$128$		−	12.1244i		−	1.07165i
$129$		0			0
$130$	7.50000	+	7.79423i	0.657794	+	0.683599i
$131$	7.50000	−	12.9904i	0.655278	−	1.13497i	−0.326546	−	0.945181i	$-0.605885\pi$
$131$	7.50000	−	12.9904i	0.655278	−	1.13497i	0.981824	−	0.189794i	$-0.0607819\pi$
$132$		0			0
$133$	−3.00000			−0.260133
$134$	10.5000	+	18.1865i	0.907062	+	1.57108i
$135$		0			0
$136$	−4.50000	+	2.59808i	−0.385872	+	0.222783i
$137$	4.50000	+	2.59808i	0.384461	+	0.221969i	0.679757	−	0.733437i	$-0.262085\pi$
$137$	4.50000	+	2.59808i	0.384461	+	0.221969i	−0.295296	+	0.955406i	$0.595418\pi$
$138$		0			0
$139$	−16.0000			−1.35710			−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000			−1.35710			−0.678551	+	0.734553i	$0.737392\pi$
$140$	1.50000	+	2.59808i	0.126773	+	0.219578i
$141$		0			0
$142$	7.50000	+	12.9904i	0.629386	+	1.09013i
$143$	−12.0000	−	3.46410i	−1.00349	−	0.289683i
$144$		0			0
$145$	−9.00000	−	5.19615i	−0.747409	−	0.431517i
$146$	−12.0000			−0.993127
$147$		0			0
$148$	4.50000	+	2.59808i	0.369898	+	0.213561i
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$	10.5000	−	6.06218i	0.854478	−	0.493333i	−0.00768132	−	0.999970i	$-0.502445\pi$
$151$	10.5000	−	6.06218i	0.854478	−	0.493333i	0.862159	+	0.506637i	$0.169112\pi$
$152$	−1.50000	−	2.59808i	−0.121666	−	0.210732i
$153$		0			0
$154$	−9.00000	−	5.19615i	−0.725241	−	0.418718i
$155$	−7.50000	−	12.9904i	−0.602414	−	1.04341i
$156$		0			0
$157$	−11.5000	+	19.9186i	−0.917800	+	1.58968i	−0.115050	+	0.993360i	$0.536703\pi$
$157$	−11.5000	+	19.9186i	−0.917800	+	1.58968i	−0.802749	+	0.596316i	$0.796630\pi$
$158$	−16.5000	+	9.52628i	−1.31267	+	0.757870i
$159$		0			0
$160$	−4.50000	+	7.79423i	−0.355756	+	0.616188i
$161$		−	5.19615i		−	0.409514i
$162$		0			0
$163$	10.5000	−	6.06218i	0.822423	−	0.474826i	−0.0288280	−	0.999584i	$-0.509178\pi$
$163$	10.5000	−	6.06218i	0.822423	−	0.474826i	0.851251	+	0.524758i	$0.175844\pi$
$164$	−10.5000	+	6.06218i	−0.819912	+	0.473377i
$165$		0			0
$166$	−4.50000	−	7.79423i	−0.349268	−	0.604949i
$167$	−4.50000	+	2.59808i	−0.348220	+	0.201045i	−0.663901	−	0.747820i	$-0.731100\pi$
$167$	−4.50000	+	2.59808i	−0.348220	+	0.201045i	0.315681	+	0.948865i	$0.397767\pi$
$168$		0			0
$169$	−11.0000	−	6.92820i	−0.846154	−	0.532939i
$170$	9.00000			0.690268
$171$		0			0
$172$	−0.500000	+	0.866025i	−0.0381246	+	0.0660338i
$173$	−10.5000	+	18.1865i	−0.798300	+	1.38270i	0.122422	+	0.992478i	$0.460934\pi$
$173$	−10.5000	+	18.1865i	−0.798300	+	1.38270i	−0.920722	+	0.390218i	$0.872399\pi$
$174$		0			0
$175$		−	3.46410i		−	0.261861i
$176$		−	17.3205i		−	1.30558i
$177$		0			0
$178$	13.5000	−	23.3827i	1.01187	−	1.75261i
$179$	1.50000	−	2.59808i	0.112115	−	0.194189i	−0.804508	−	0.593942i	$-0.797571\pi$
$179$	1.50000	−	2.59808i	0.112115	−	0.194189i	0.916623	+	0.399753i	$0.130904\pi$
$180$		0			0
$181$	−22.0000			−1.63525			−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000			−1.63525			−0.817624	+	0.575753i	$0.804709\pi$
$182$	−7.50000	−	7.79423i	−0.555937	−	0.577747i
$183$		0			0
$184$	4.50000	−	2.59808i	0.331744	−	0.191533i
$185$	4.50000	+	7.79423i	0.330847	+	0.573043i
$186$		0			0
$187$	−9.00000	+	5.19615i	−0.658145	+	0.379980i
$188$	4.50000	−	2.59808i	0.328196	−	0.189484i
$189$		0			0
$190$			5.19615i			0.376969i
$191$	1.50000	−	2.59808i	0.108536	−	0.187990i	−0.806641	−	0.591041i	$-0.798717\pi$
$191$	1.50000	−	2.59808i	0.108536	−	0.187990i	0.915177	+	0.403051i	$0.132050\pi$
$192$		0			0
$193$	−4.50000	+	2.59808i	−0.323917	+	0.187014i	−0.653137	−	0.757240i	$-0.726548\pi$
$193$	−4.50000	+	2.59808i	−0.323917	+	0.187014i	0.329220	+	0.944253i	$0.393214\pi$
$194$	−13.5000	+	23.3827i	−0.969244	+	1.67878i
$195$		0			0
$196$	2.00000	+	3.46410i	0.142857	+	0.247436i
$197$	−7.50000	−	4.33013i	−0.534353	−	0.308509i	0.208434	−	0.978036i	$-0.433163\pi$
$197$	−7.50000	−	4.33013i	−0.534353	−	0.308509i	−0.742787	+	0.669528i	$0.766497\pi$
$198$		0			0
$199$	−6.50000	−	11.2583i	−0.460773	−	0.798082i	0.538227	−	0.842800i	$-0.319094\pi$
$199$	−6.50000	−	11.2583i	−0.460773	−	0.798082i	−0.999000	+	0.0447181i	$0.985761\pi$
$200$	3.00000	−	1.73205i	0.212132	−	0.122474i
$201$		0			0
$202$			10.3923i			0.731200i
$203$	9.00000	+	5.19615i	0.631676	+	0.364698i
$204$		0			0
$205$	−21.0000			−1.46670
$206$	19.5000	+	11.2583i	1.35863	+	0.784405i
$207$		0			0
$208$	5.00000	−	17.3205i	0.346688	−	1.20096i
$209$	−3.00000	−	5.19615i	−0.207514	−	0.359425i
$210$		0			0
$211$	−6.50000	−	11.2583i	−0.447478	−	0.775055i	0.550743	−	0.834675i	$-0.314345\pi$
$211$	−6.50000	−	11.2583i	−0.447478	−	0.775055i	−0.998221	+	0.0596196i	$0.981011\pi$
$212$	6.00000			0.412082
$213$		0			0
$214$	22.5000	+	12.9904i	1.53807	+	0.888004i
$215$	−1.50000	+	0.866025i	−0.102299	+	0.0590624i
$216$		0			0
$217$	7.50000	+	12.9904i	0.509133	+	0.881845i
$218$		0			0
$219$		0			0
$220$	−3.00000	+	5.19615i	−0.202260	+	0.350325i
$221$	−10.5000	+	2.59808i	−0.706306	+	0.174766i
$222$		0			0
$223$			3.46410i			0.231973i	0.993251	+	0.115987i	$0.0370030\pi$
$223$			3.46410i			0.231973i	−0.993251	+	0.115987i	$0.962997\pi$
$224$	4.50000	−	7.79423i	0.300669	−	0.520774i
$225$		0			0
$226$		−	10.3923i		−	0.691286i
$227$	10.5000	+	6.06218i	0.696909	+	0.402361i	0.806195	−	0.591649i	$-0.201523\pi$
$227$	10.5000	+	6.06218i	0.696909	+	0.402361i	−0.109286	+	0.994010i	$0.534856\pi$
$228$		0			0
$229$	7.50000	+	4.33013i	0.495614	+	0.286143i	0.726900	−	0.686743i	$-0.240960\pi$
$229$	7.50000	+	4.33013i	0.495614	+	0.286143i	−0.231287	+	0.972886i	$0.574293\pi$
$230$	−9.00000			−0.593442
$231$		0			0
$232$			10.3923i			0.682288i
$233$	18.0000			1.17922			0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000			1.17922			0.589610	+	0.807688i	$0.299282\pi$
$234$		0			0
$235$	9.00000			0.587095
$236$		−	3.46410i		−	0.225494i
$237$		0			0
$238$	−9.00000			−0.583383
$239$	4.50000	+	2.59808i	0.291081	+	0.168056i	0.638429	−	0.769681i	$-0.279585\pi$
$239$	4.50000	+	2.59808i	0.291081	+	0.168056i	−0.347348	+	0.937736i	$0.612918\pi$
$240$		0			0
$241$	−4.50000	−	2.59808i	−0.289870	−	0.167357i	0.348013	−	0.937490i	$-0.386857\pi$
$241$	−4.50000	−	2.59808i	−0.289870	−	0.167357i	−0.637883	+	0.770133i	$0.720190\pi$
$242$		−	1.73205i		−	0.111340i
$243$		0			0
$244$	−2.50000	+	4.33013i	−0.160046	+	0.277208i
$245$			6.92820i			0.442627i
$246$		0			0
$247$	−1.50000	−	6.06218i	−0.0954427	−	0.385727i
$248$	−7.50000	+	12.9904i	−0.476250	+	0.824890i
$249$		0			0
$250$	−21.0000			−1.32816
$251$	4.50000	+	7.79423i	0.284037	+	0.491967i	0.972375	−	0.233423i	$-0.0749927\pi$
$251$	4.50000	+	7.79423i	0.284037	+	0.491967i	−0.688338	+	0.725390i	$0.741659\pi$
$252$		0			0
$253$	9.00000	−	5.19615i	0.565825	−	0.326679i
$254$	7.50000	+	4.33013i	0.470592	+	0.271696i
$255$		0			0
$256$	19.0000			1.18750
$257$	1.50000	+	2.59808i	0.0935674	+	0.162064i	0.909010	−	0.416775i	$-0.136840\pi$
$257$	1.50000	+	2.59808i	0.0935674	+	0.162064i	−0.815442	+	0.578838i	$0.803506\pi$
$258$		0			0
$259$	−4.50000	−	7.79423i	−0.279616	−	0.484310i
$260$	−4.50000	+	4.33013i	−0.279078	+	0.268543i
$261$		0			0
$262$	22.5000	+	12.9904i	1.39005	+	0.802548i
$263$	−24.0000			−1.47990			−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000			−1.47990			−0.739952	+	0.672660i	$0.765152\pi$
$264$		0			0
$265$	9.00000	+	5.19615i	0.552866	+	0.319197i
$266$		−	5.19615i		−	0.318597i
$267$		0			0
$268$	−10.5000	+	6.06218i	−0.641390	+	0.370306i
$269$	1.50000	+	2.59808i	0.0914566	+	0.158408i	0.908124	−	0.418701i	$-0.137514\pi$
$269$	1.50000	+	2.59808i	0.0914566	+	0.158408i	−0.816668	+	0.577108i	$0.804181\pi$
$270$		0			0
$271$	13.5000	+	7.79423i	0.820067	+	0.473466i	0.850439	−	0.526073i	$-0.176336\pi$
$271$	13.5000	+	7.79423i	0.820067	+	0.473466i	−0.0303728	+	0.999539i	$0.509669\pi$
$272$	−7.50000	−	12.9904i	−0.454754	−	0.787658i
$273$		0			0
$274$	−4.50000	+	7.79423i	−0.271855	+	0.470867i
$275$	6.00000	−	3.46410i	0.361814	−	0.208893i
$276$		0			0
$277$	−11.5000	+	19.9186i	−0.690968	+	1.19679i	0.280553	+	0.959839i	$0.409482\pi$
$277$	−11.5000	+	19.9186i	−0.690968	+	1.19679i	−0.971521	+	0.236953i	$0.923851\pi$
$278$		−	27.7128i		−	1.66210i
$279$		0			0
$280$	4.50000	−	2.59808i	0.268926	−	0.155265i
$281$	4.50000	−	2.59808i	0.268447	−	0.154988i	−0.359734	−	0.933055i	$-0.617133\pi$
$281$	4.50000	−	2.59808i	0.268447	−	0.154988i	0.628182	+	0.778067i	$0.283799\pi$
$282$		0			0
$283$	11.5000	+	19.9186i	0.683604	+	1.18404i	0.973873	+	0.227092i	$0.0729218\pi$
$283$	11.5000	+	19.9186i	0.683604	+	1.18404i	−0.290269	+	0.956945i	$0.593745\pi$
$284$	−7.50000	+	4.33013i	−0.445043	+	0.256946i
$285$		0			0
$286$	6.00000	−	20.7846i	0.354787	−	1.22902i
$287$	21.0000			1.23959
$288$		0			0
$289$	4.00000	−	6.92820i	0.235294	−	0.407541i
$290$	9.00000	−	15.5885i	0.528498	−	0.915386i
$291$		0			0
$292$		−	6.92820i		−	0.405442i
$293$			13.8564i			0.809500i	0.914427	+	0.404750i	$0.132641\pi$
$293$			13.8564i			0.809500i	−0.914427	+	0.404750i	$0.867359\pi$
$294$		0			0
$295$	3.00000	−	5.19615i	0.174667	−	0.302532i
$296$	4.50000	−	7.79423i	0.261557	−	0.453030i
$297$		0			0
$298$		0			0
$299$	10.5000	−	2.59808i	0.607231	−	0.150251i
$300$		0			0
$301$	1.50000	−	0.866025i	0.0864586	−	0.0499169i
$302$	10.5000	+	18.1865i	0.604207	+	1.04652i
$303$		0			0
$304$	7.50000	−	4.33013i	0.430155	−	0.248350i
$305$	−7.50000	+	4.33013i	−0.429449	+	0.247942i
$306$		0			0
$307$			24.2487i			1.38395i	0.721923	+	0.691974i	$0.243259\pi$
$307$			24.2487i			1.38395i	−0.721923	+	0.691974i	$0.756741\pi$
$308$	3.00000	−	5.19615i	0.170941	−	0.296078i
$309$		0			0
$310$	22.5000	−	12.9904i	1.27791	−	0.737804i
$311$	13.5000	−	23.3827i	0.765515	−	1.32591i	−0.174459	−	0.984664i	$-0.555818\pi$
$311$	13.5000	−	23.3827i	0.765515	−	1.32591i	0.939974	−	0.341246i	$-0.110849\pi$
$312$		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	$-0.347098\pi$
$314$	−34.5000	−	19.9186i	−1.94695	−	1.12407i
$315$		0			0
$316$	−5.50000	−	9.52628i	−0.309399	−	0.535895i
$317$	−7.50000	+	4.33013i	−0.421242	+	0.243204i	−0.695609	−	0.718421i	$-0.744865\pi$
$317$	−7.50000	+	4.33013i	−0.421242	+	0.243204i	0.274367	+	0.961625i	$0.411532\pi$
$318$		0			0
$319$			20.7846i			1.16371i
$320$	1.50000	+	0.866025i	0.0838525	+	0.0484123i
$321$		0			0
$322$	9.00000			0.501550
$323$	−4.50000	−	2.59808i	−0.250387	−	0.144561i
$324$		0			0
$325$	7.00000	−	1.73205i	0.388290	−	0.0960769i
$326$	10.5000	+	18.1865i	0.581541	+	1.00726i
$327$		0			0
$328$	10.5000	+	18.1865i	0.579766	+	1.00418i
$329$	−9.00000			−0.496186
$330$		0			0
$331$	13.5000	+	7.79423i	0.742027	+	0.428410i	0.822806	−	0.568323i	$-0.192407\pi$
$331$	13.5000	+	7.79423i	0.742027	+	0.428410i	−0.0807788	+	0.996732i	$0.525741\pi$
$332$	4.50000	−	2.59808i	0.246970	−	0.142588i
$333$		0			0
$334$	−4.50000	−	7.79423i	−0.246229	−	0.426481i
$335$	−21.0000			−1.14735
$336$		0			0
$337$	14.5000	−	25.1147i	0.789865	−	1.36809i	−0.136184	−	0.990684i	$-0.543484\pi$
$337$	14.5000	−	25.1147i	0.789865	−	1.36809i	0.926049	−	0.377403i	$-0.123183\pi$
$338$	12.0000	−	19.0526i	0.652714	−	1.03632i
$339$		0			0
$340$			5.19615i			0.281801i
$341$	−15.0000	+	25.9808i	−0.812296	+	1.40694i
$342$		0			0
$343$		−	19.0526i		−	1.02874i
$344$	1.50000	+	0.866025i	0.0808746	+	0.0466930i
$345$		0			0
$346$	−31.5000	−	18.1865i	−1.69345	−	0.977714i
$347$	12.0000			0.644194			0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000			0.644194			0.322097	+	0.946707i	$0.395612\pi$
$348$		0			0
$349$		−	27.7128i		−	1.48343i	−0.670714	−	0.741716i	$-0.734012\pi$
$349$		−	27.7128i		−	1.48343i	0.670714	−	0.741716i	$-0.265988\pi$
$350$	6.00000			0.320713
$351$		0			0
$352$	18.0000			0.959403
$353$		−	6.92820i		−	0.368751i	−0.982856	−	0.184376i	$-0.940974\pi$
$353$		−	6.92820i		−	0.368751i	0.982856	−	0.184376i	$-0.0590263\pi$
$354$		0			0
$355$	−15.0000			−0.796117
$356$	13.5000	+	7.79423i	0.715499	+	0.413093i
$357$		0			0
$358$	4.50000	+	2.59808i	0.237832	+	0.137313i
$359$			10.3923i			0.548485i	0.961661	+	0.274242i	$0.0884271\pi$
$359$			10.3923i			0.548485i	−0.961661	+	0.274242i	$0.911573\pi$
$360$		0			0
$361$	−8.00000	+	13.8564i	−0.421053	+	0.729285i
$362$		−	38.1051i		−	2.00276i
$363$		0			0
$364$	4.50000	−	4.33013i	0.235864	−	0.226960i
$365$	6.00000	−	10.3923i	0.314054	−	0.543958i
$366$		0			0
$367$	8.00000			0.417597			0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000			0.417597			0.208798	+	0.977959i	$0.433045\pi$
$368$	7.50000	+	12.9904i	0.390965	+	0.677170i
$369$		0			0
$370$	−13.5000	+	7.79423i	−0.701832	+	0.405203i
$371$	−9.00000	−	5.19615i	−0.467257	−	0.269771i
$372$		0			0
$373$	−14.0000			−0.724893			−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000			−0.724893			−0.362446	+	0.932005i	$0.618058\pi$
$374$	−9.00000	−	15.5885i	−0.465379	−	0.806060i
$375$		0			0
$376$	−4.50000	−	7.79423i	−0.232070	−	0.401957i
$377$	−6.00000	+	20.7846i	−0.309016	+	1.07046i
$378$		0			0
$379$	−10.5000	−	6.06218i	−0.539349	−	0.311393i	0.205466	−	0.978664i	$-0.434129\pi$
$379$	−10.5000	−	6.06218i	−0.539349	−	0.311393i	−0.744815	+	0.667271i	$0.767462\pi$
$380$	−3.00000			−0.153897
$381$		0			0
$382$	4.50000	+	2.59808i	0.230240	+	0.132929i
$383$		−	3.46410i		−	0.177007i	−0.996076	−	0.0885037i	$-0.971792\pi$
$383$		−	3.46410i		−	0.177007i	0.996076	−	0.0885037i	$-0.0282085\pi$
$384$		0			0
$385$	9.00000	−	5.19615i	0.458682	−	0.264820i
$386$	−4.50000	−	7.79423i	−0.229044	−	0.396716i
$387$		0			0
$388$	−13.5000	−	7.79423i	−0.685359	−	0.395692i
$389$	1.50000	+	2.59808i	0.0760530	+	0.131728i	0.901544	−	0.432688i	$-0.142435\pi$
$389$	1.50000	+	2.59808i	0.0760530	+	0.131728i	−0.825491	+	0.564416i	$0.809102\pi$
$390$		0			0
$391$	4.50000	−	7.79423i	0.227575	−	0.394171i
$392$	6.00000	−	3.46410i	0.303046	−	0.174964i
$393$		0			0
$394$	7.50000	−	12.9904i	0.377845	−	0.654446i
$395$		−	19.0526i		−	0.958638i
$396$		0			0
$397$	−22.5000	+	12.9904i	−1.12924	+	0.651969i	−0.943744	−	0.330676i	$-0.892723\pi$
$397$	−22.5000	+	12.9904i	−1.12924	+	0.651969i	−0.185498	+	0.982645i	$0.559390\pi$
$398$	19.5000	−	11.2583i	0.977447	−	0.564329i
$399$		0			0
$400$	5.00000	+	8.66025i	0.250000	+	0.433013i
$401$	−25.5000	+	14.7224i	−1.27341	+	0.735203i	−0.975628	−	0.219431i	$-0.929580\pi$
$401$	−25.5000	+	14.7224i	−1.27341	+	0.735203i	−0.297781	+	0.954634i	$0.596247\pi$
$402$		0			0
$403$	−22.5000	+	21.6506i	−1.12080	+	1.07849i
$404$	−6.00000			−0.298511
$405$		0			0
$406$	−9.00000	+	15.5885i	−0.446663	+	0.773642i
$407$	9.00000	−	15.5885i	0.446113	−	0.772691i
$408$		0			0
$409$		−	6.92820i		−	0.342578i	−0.985221	−	0.171289i	$-0.945207\pi$
$409$		−	6.92820i		−	0.342578i	0.985221	−	0.171289i	$-0.0547931\pi$
$410$		−	36.3731i		−	1.79634i
$411$		0			0
$412$	−6.50000	+	11.2583i	−0.320232	+	0.554658i
$413$	−3.00000	+	5.19615i	−0.147620	+	0.255686i
$414$		0			0
$415$	9.00000			0.441793
$416$	18.0000	+	5.19615i	0.882523	+	0.254762i
$417$		0			0
$418$	9.00000	−	5.19615i	0.440204	−	0.254152i
$419$	4.50000	+	7.79423i	0.219839	+	0.380773i	0.954759	−	0.297382i	$-0.0961133\pi$
$419$	4.50000	+	7.79423i	0.219839	+	0.380773i	−0.734919	+	0.678155i	$0.762780\pi$
$420$		0			0
$421$	7.50000	−	4.33013i	0.365528	−	0.211037i	−0.305975	−	0.952039i	$-0.598982\pi$
$421$	7.50000	−	4.33013i	0.365528	−	0.211037i	0.671503	+	0.741002i	$0.265649\pi$
$422$	19.5000	−	11.2583i	0.949245	−	0.548047i
$423$		0			0
$424$		−	10.3923i		−	0.504695i
$425$	3.00000	−	5.19615i	0.145521	−	0.252050i
$426$		0			0
$427$	7.50000	−	4.33013i	0.362950	−	0.209550i
$428$	−7.50000	+	12.9904i	−0.362526	+	0.627914i
$429$		0			0
$430$	−1.50000	−	2.59808i	−0.0723364	−	0.125290i
$431$	−19.5000	−	11.2583i	−0.939282	−	0.542295i	−0.0495468	−	0.998772i	$-0.515778\pi$
$431$	−19.5000	−	11.2583i	−0.939282	−	0.542295i	−0.889735	+	0.456477i	$0.849111\pi$
$432$		0			0
$433$	2.50000	+	4.33013i	0.120142	+	0.208093i	0.919824	−	0.392332i	$-0.128332\pi$
$433$	2.50000	+	4.33013i	0.120142	+	0.208093i	−0.799681	+	0.600425i	$0.794998\pi$
$434$	−22.5000	+	12.9904i	−1.08003	+	0.623558i
$435$		0			0
$436$		0			0
$437$	4.50000	+	2.59808i	0.215264	+	0.124283i
$438$		0			0
$439$	8.00000			0.381819			0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000			0.381819			0.190910	+	0.981608i	$0.438856\pi$
$440$	9.00000	+	5.19615i	0.429058	+	0.247717i
$441$		0			0
$442$	−4.50000	−	18.1865i	−0.214043	−	0.865045i
$443$	10.5000	+	18.1865i	0.498870	+	0.864068i	0.999999	−	0.00130426i	$-0.000415158\pi$
$443$	10.5000	+	18.1865i	0.498870	+	0.864068i	−0.501129	+	0.865373i	$0.667082\pi$
$444$		0			0
$445$	13.5000	+	23.3827i	0.639961	+	1.10845i
$446$	−6.00000			−0.284108
$447$		0			0
$448$	−1.50000	−	0.866025i	−0.0708683	−	0.0409159i
$449$	−13.5000	+	7.79423i	−0.637104	+	0.367832i	−0.783498	−	0.621394i	$-0.786567\pi$
$449$	−13.5000	+	7.79423i	−0.637104	+	0.367832i	0.146394	+	0.989226i	$0.453233\pi$
$450$		0			0
$451$	21.0000	+	36.3731i	0.988851	+	1.71274i
$452$	6.00000			0.282216
$453$		0			0
$454$	−10.5000	+	18.1865i	−0.492789	+	0.853536i
$455$	10.5000	−	2.59808i	0.492248	−	0.121800i
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$	−7.50000	+	12.9904i	−0.350452	+	0.607001i
$459$		0			0
$460$		−	5.19615i		−	0.242272i
$461$	−1.50000	−	0.866025i	−0.0698620	−	0.0403348i	0.464662	−	0.885488i	$-0.346176\pi$
$461$	−1.50000	−	0.866025i	−0.0698620	−	0.0403348i	−0.534524	+	0.845153i	$0.679509\pi$
$462$		0			0
$463$	−22.5000	−	12.9904i	−1.04566	−	0.603714i	−0.124231	−	0.992253i	$-0.539647\pi$
$463$	−22.5000	−	12.9904i	−1.04566	−	0.603714i	−0.921432	+	0.388539i	$0.872980\pi$
$464$	−30.0000			−1.39272
$465$		0			0
$466$			31.1769i			1.44424i
$467$	12.0000			0.555294			0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000			0.555294			0.277647	+	0.960683i	$0.410445\pi$
$468$		0			0
$469$	21.0000			0.969690
$470$			15.5885i			0.719042i
$471$		0			0
$472$	−6.00000			−0.276172
$473$	3.00000	+	1.73205i	0.137940	+	0.0796398i
$474$		0			0
$475$	3.00000	+	1.73205i	0.137649	+	0.0794719i
$476$		−	5.19615i		−	0.238165i
$477$		0			0
$478$	−4.50000	+	7.79423i	−0.205825	+	0.356500i
$479$			24.2487i			1.10795i	0.832533	+	0.553976i	$0.186890\pi$
$479$			24.2487i			1.10795i	−0.832533	+	0.553976i	$0.813110\pi$
$480$		0			0
$481$	13.5000	−	12.9904i	0.615547	−	0.592310i
$482$	4.50000	−	7.79423i	0.204969	−	0.355017i
$483$		0			0
$484$	1.00000			0.0454545
$485$	−13.5000	−	23.3827i	−0.613003	−	1.06175i
$486$		0			0
$487$	−7.50000	+	4.33013i	−0.339857	+	0.196217i	−0.660209	−	0.751082i	$-0.729532\pi$
$487$	−7.50000	+	4.33013i	−0.339857	+	0.196217i	0.320352	+	0.947299i	$0.396199\pi$
$488$	7.50000	+	4.33013i	0.339509	+	0.196016i
$489$		0			0
$490$	−12.0000			−0.542105
$491$	−1.50000	−	2.59808i	−0.0676941	−	0.117250i	0.830192	−	0.557478i	$-0.188231\pi$
$491$	−1.50000	−	2.59808i	−0.0676941	−	0.117250i	−0.897886	+	0.440228i	$0.854898\pi$
$492$		0			0
$493$	9.00000	+	15.5885i	0.405340	+	0.702069i
$494$	10.5000	−	2.59808i	0.472417	−	0.116893i
$495$		0			0
$496$	−37.5000	−	21.6506i	−1.68380	−	0.972142i
$497$	15.0000			0.672842
$498$		0			0
$499$	−22.5000	−	12.9904i	−1.00724	−	0.581529i	−0.0968564	−	0.995298i	$-0.530879\pi$
$499$	−22.5000	−	12.9904i	−1.00724	−	0.581529i	−0.910382	+	0.413769i	$0.864212\pi$
$500$		−	12.1244i		−	0.542218i
$501$		0			0
$502$	−13.5000	+	7.79423i	−0.602534	+	0.347873i
$503$	10.5000	+	18.1865i	0.468172	+	0.810897i	0.999338	−	0.0363700i	$-0.0115795\pi$
$503$	10.5000	+	18.1865i	0.468172	+	0.810897i	−0.531167	+	0.847267i	$0.678246\pi$
$504$		0			0
$505$	−9.00000	−	5.19615i	−0.400495	−	0.231226i
$506$	9.00000	+	15.5885i	0.400099	+	0.692991i
$507$		0			0
$508$	−2.50000	+	4.33013i	−0.110920	+	0.192118i
$509$	−1.50000	+	0.866025i	−0.0664863	+	0.0383859i	−0.532875	−	0.846194i	$-0.678888\pi$
$509$	−1.50000	+	0.866025i	−0.0664863	+	0.0383859i	0.466388	+	0.884580i	$0.345555\pi$
$510$		0			0
$511$	−6.00000	+	10.3923i	−0.265424	+	0.459728i
$512$			8.66025i			0.382733i
$513$		0			0
$514$	−4.50000	+	2.59808i	−0.198486	+	0.114596i
$515$	−19.5000	+	11.2583i	−0.859273	+	0.496101i
$516$		0			0
$517$	−9.00000	−	15.5885i	−0.395820	−	0.685580i
$518$	13.5000	−	7.79423i	0.593156	−	0.342459i
$519$		0			0
$520$	7.50000	+	7.79423i	0.328897	+	0.341800i
$521$	−6.00000			−0.262865			−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000			−0.262865			−0.131432	+	0.991325i	$0.541958\pi$
$522$		0			0
$523$	12.5000	−	21.6506i	0.546587	−	0.946716i	−0.451918	−	0.892059i	$-0.649260\pi$
$523$	12.5000	−	21.6506i	0.546587	−	0.946716i	0.998505	−	0.0546569i	$-0.0174065\pi$
$524$	−7.50000	+	12.9904i	−0.327639	+	0.567487i
$525$		0			0
$526$		−	41.5692i		−	1.81250i
$527$			25.9808i			1.13174i
$528$		0			0
$529$	7.00000	−	12.1244i	0.304348	−	0.527146i
$530$	−9.00000	+	15.5885i	−0.390935	+	0.677119i
$531$		0			0
$532$	3.00000			0.130066
$533$	10.5000	+	42.4352i	0.454805	+	1.83807i
$534$		0			0
$535$	−22.5000	+	12.9904i	−0.972760	+	0.561623i
$536$	10.5000	+	18.1865i	0.453531	+	0.785539i
$537$		0			0
$538$	−4.50000	+	2.59808i	−0.194009	+	0.112011i
$539$	12.0000	−	6.92820i	0.516877	−	0.298419i
$540$		0			0
$541$		−	13.8564i		−	0.595733i	−0.954607	−	0.297867i	$-0.903725\pi$
$541$		−	13.8564i		−	0.595733i	0.954607	−	0.297867i	$-0.0962751\pi$
$542$	−13.5000	+	23.3827i	−0.579875	+	1.00437i
$543$		0			0
$544$	13.5000	−	7.79423i	0.578808	−	0.334175i
$545$		0			0
$546$		0			0
$547$	−8.50000	−	14.7224i	−0.363434	−	0.629486i	0.625090	−	0.780553i	$-0.285062\pi$
$547$	−8.50000	−	14.7224i	−0.363434	−	0.629486i	−0.988524	+	0.151067i	$0.951729\pi$
$548$	−4.50000	−	2.59808i	−0.192230	−	0.110984i
$549$		0			0
$550$	6.00000	+	10.3923i	0.255841	+	0.443129i
$551$	−9.00000	+	5.19615i	−0.383413	+	0.221364i
$552$		0			0
$553$			19.0526i			0.810197i
$554$	−34.5000	−	19.9186i	−1.46576	−	0.846260i
$555$		0			0
$556$	16.0000			0.678551
$557$	16.5000	+	9.52628i	0.699127	+	0.403641i	0.807022	−	0.590521i	$-0.201078\pi$
$557$	16.5000	+	9.52628i	0.699127	+	0.403641i	−0.107895	+	0.994162i	$0.534411\pi$
$558$		0			0
$559$	2.50000	+	2.59808i	0.105739	+	0.109887i
$560$	7.50000	+	12.9904i	0.316933	+	0.548944i
$561$		0			0
$562$	4.50000	+	7.79423i	0.189821	+	0.328780i
$563$	−24.0000			−1.01148			−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000			−1.01148			−0.505740	+	0.862686i	$0.668780\pi$
$564$		0			0
$565$	9.00000	+	5.19615i	0.378633	+	0.218604i
$566$	−34.5000	+	19.9186i	−1.45014	+	0.837241i
$567$		0			0
$568$	7.50000	+	12.9904i	0.314693	+	0.545064i
$569$	18.0000			0.754599			0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000			0.754599			0.377300	+	0.926091i	$0.376853\pi$
$570$		0			0
$571$	2.50000	−	4.33013i	0.104622	−	0.181210i	−0.808962	−	0.587861i	$-0.799970\pi$
$571$	2.50000	−	4.33013i	0.104622	−	0.181210i	0.913584	+	0.406651i	$0.133303\pi$
$572$	12.0000	+	3.46410i	0.501745	+	0.144841i
$573$		0			0
$574$			36.3731i			1.51818i
$575$	−3.00000	+	5.19615i	−0.125109	+	0.216695i
$576$		0			0
$577$			6.92820i			0.288425i	0.989547	+	0.144212i	$0.0460649\pi$
$577$			6.92820i			0.288425i	−0.989547	+	0.144212i	$0.953935\pi$
$578$	12.0000	+	6.92820i	0.499134	+	0.288175i
$579$		0			0
$580$	9.00000	+	5.19615i	0.373705	+	0.215758i
$581$	−9.00000			−0.373383
$582$		0			0
$583$		−	20.7846i		−	0.860811i
$584$	−12.0000			−0.496564
$585$		0			0
$586$	−24.0000			−0.991431
$587$		−	10.3923i		−	0.428936i	−0.976731	−	0.214468i	$-0.931198\pi$
$587$		−	10.3923i		−	0.428936i	0.976731	−	0.214468i	$-0.0688018\pi$
$588$		0			0
$589$	−15.0000			−0.618064
$590$	9.00000	+	5.19615i	0.370524	+	0.213922i
$591$		0			0
$592$	22.5000	+	12.9904i	0.924744	+	0.533901i
$593$		−	27.7128i		−	1.13803i	−0.822328	−	0.569014i	$-0.807325\pi$
$593$		−	27.7128i		−	1.13803i	0.822328	−	0.569014i	$-0.192675\pi$
$594$		0			0
$595$	4.50000	−	7.79423i	0.184482	−	0.319532i
$596$		0			0
$597$		0			0
$598$	4.50000	+	18.1865i	0.184019	+	0.743703i
$599$	13.5000	−	23.3827i	0.551595	−	0.955391i	−0.446565	−	0.894751i	$-0.647353\pi$
$599$	13.5000	−	23.3827i	0.551595	−	0.955391i	0.998160	−	0.0606393i	$-0.0193139\pi$
$600$		0			0
$601$	22.0000			0.897399			0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000			0.897399			0.448699	+	0.893683i	$0.351887\pi$
$602$	1.50000	+	2.59808i	0.0611354	+	0.105890i
$603$		0			0
$604$	−10.5000	+	6.06218i	−0.427239	+	0.246667i
$605$	1.50000	+	0.866025i	0.0609837	+	0.0352089i
$606$		0			0
$607$	8.00000			0.324710			0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000			0.324710			0.162355	+	0.986732i	$0.448091\pi$
$608$	4.50000	+	7.79423i	0.182499	+	0.316098i
$609$		0			0
$610$	−7.50000	−	12.9904i	−0.303666	−	0.525965i
$611$	−4.50000	−	18.1865i	−0.182051	−	0.735748i
$612$		0			0
$613$	−10.5000	−	6.06218i	−0.424091	−	0.244849i	0.272735	−	0.962089i	$-0.412072\pi$
$613$	−10.5000	−	6.06218i	−0.424091	−	0.244849i	−0.696826	+	0.717240i	$0.745405\pi$
$614$	−42.0000			−1.69498
$615$		0			0
$616$	−9.00000	−	5.19615i	−0.362620	−	0.209359i
$617$		−	27.7128i		−	1.11568i	−0.829950	−	0.557838i	$-0.811631\pi$
$617$		−	27.7128i		−	1.11568i	0.829950	−	0.557838i	$-0.188369\pi$
$618$		0			0
$619$	22.5000	−	12.9904i	0.904351	−	0.522127i	0.0257420	−	0.999669i	$-0.491805\pi$
$619$	22.5000	−	12.9904i	0.904351	−	0.522127i	0.878609	+	0.477541i	$0.158472\pi$
$620$	7.50000	+	12.9904i	0.301207	+	0.521706i
$621$		0			0
$622$	40.5000	+	23.3827i	1.62390	+	0.937560i
$623$	−13.5000	−	23.3827i	−0.540866	−	0.936808i
$624$		0			0
$625$	5.50000	−	9.52628i	0.220000	−	0.381051i
$626$	28.5000	−	16.4545i	1.13909	−	0.657653i
$627$		0			0
$628$	11.5000	−	19.9186i	0.458900	−	0.794838i
$629$		−	15.5885i		−	0.621552i
$630$		0			0
$631$	−7.50000	+	4.33013i	−0.298570	+	0.172380i	−0.641800	−	0.766872i	$-0.721812\pi$
$631$	−7.50000	+	4.33013i	−0.298570	+	0.172380i	0.343230	+	0.939251i	$0.388479\pi$
$632$	−16.5000	+	9.52628i	−0.656335	+	0.378935i
$633$		0			0
$634$	−7.50000	−	12.9904i	−0.297863	−	0.515914i
$635$	−7.50000	+	4.33013i	−0.297628	+	0.171836i
$636$		0			0
$637$	14.0000	−	3.46410i	0.554700	−	0.137253i
$638$	−36.0000			−1.42525
$639$		0			0
$640$	−10.5000	+	18.1865i	−0.415049	+	0.718886i
$641$	19.5000	−	33.7750i	0.770204	−	1.33403i	−0.167247	−	0.985915i	$-0.553488\pi$
$641$	19.5000	−	33.7750i	0.770204	−	1.33403i	0.937451	−	0.348117i	$-0.113179\pi$
$642$		0			0
$643$			3.46410i			0.136611i	0.997664	+	0.0683054i	$0.0217592\pi$
$643$			3.46410i			0.136611i	−0.997664	+	0.0683054i	$0.978241\pi$
$644$			5.19615i			0.204757i
$645$		0			0
$646$	4.50000	−	7.79423i	0.177050	−	0.306660i
$647$	−4.50000	+	7.79423i	−0.176913	+	0.306423i	−0.940822	−	0.338902i	$-0.889945\pi$
$647$	−4.50000	+	7.79423i	−0.176913	+	0.306423i	0.763908	+	0.645325i	$0.223278\pi$
$648$		0			0
$649$	−12.0000			−0.471041
$650$	3.00000	+	12.1244i	0.117670	+	0.475556i
$651$		0			0
$652$	−10.5000	+	6.06218i	−0.411212	+	0.237413i
$653$	−22.5000	−	38.9711i	−0.880493	−	1.52506i	−0.850794	−	0.525500i	$-0.823878\pi$
$653$	−22.5000	−	38.9711i	−0.880493	−	1.52506i	−0.0296993	−	0.999559i	$-0.509455\pi$
$654$		0			0
$655$	−22.5000	+	12.9904i	−0.879148	+	0.507576i
$656$	−52.5000	+	30.3109i	−2.04978	+	1.18344i
$657$		0			0
$658$		−	15.5885i		−	0.607701i
$659$	19.5000	−	33.7750i	0.759612	−	1.31569i	−0.183436	−	0.983032i	$-0.558722\pi$
$659$	19.5000	−	33.7750i	0.759612	−	1.31569i	0.943049	−	0.332655i	$-0.107945\pi$
$660$		0			0
$661$	−40.5000	+	23.3827i	−1.57527	+	0.909481i	−0.579761	+	0.814787i	$0.696854\pi$
$661$	−40.5000	+	23.3827i	−1.57527	+	0.909481i	−0.995506	+	0.0946945i	$0.969813\pi$
$662$	−13.5000	+	23.3827i	−0.524692	+	0.908794i
$663$		0			0
$664$	−4.50000	−	7.79423i	−0.174634	−	0.302475i
$665$	4.50000	+	2.59808i	0.174503	+	0.100749i
$666$		0			0
$667$	−9.00000	−	15.5885i	−0.348481	−	0.603587i
$668$	4.50000	−	2.59808i	0.174110	−	0.100523i
$669$		0			0
$670$		−	36.3731i		−	1.40521i
$671$	15.0000	+	8.66025i	0.579069	+	0.334325i
$672$		0			0
$673$	−34.0000			−1.31060			−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000			−1.31060			−0.655302	+	0.755367i	$0.727459\pi$
$674$	43.5000	+	25.1147i	1.67556	+	0.967384i
$675$		0			0
$676$	11.0000	+	6.92820i	0.423077	+	0.266469i
$677$	−10.5000	−	18.1865i	−0.403548	−	0.698965i	0.590603	−	0.806962i	$-0.298890\pi$
$677$	−10.5000	−	18.1865i	−0.403548	−	0.698965i	−0.994151	+	0.107997i	$0.965556\pi$
$678$		0			0
$679$	13.5000	+	23.3827i	0.518082	+	0.897345i
$680$	9.00000			0.345134
$681$		0			0
$682$	−45.0000	−	25.9808i	−1.72314	−	0.994855i
$683$	19.5000	−	11.2583i	0.746147	−	0.430788i	−0.0781532	−	0.996941i	$-0.524902\pi$
$683$	19.5000	−	11.2583i	0.746147	−	0.430788i	0.824300	+	0.566153i	$0.191569\pi$
$684$		0			0
$685$	−4.50000	−	7.79423i	−0.171936	−	0.297802i
$686$	33.0000			1.25995
$687$		0			0
$688$	−2.50000	+	4.33013i	−0.0953116	+	0.165085i
$689$	6.00000	−	20.7846i	0.228582	−	0.791831i
$690$		0			0
$691$		−	17.3205i		−	0.658903i	−0.944172	−	0.329452i	$-0.893136\pi$
$691$		−	17.3205i		−	0.658903i	0.944172	−	0.329452i	$-0.106864\pi$
$692$	10.5000	−	18.1865i	0.399150	−	0.691348i
$693$		0			0
$694$			20.7846i			0.788973i
$695$	24.0000	+	13.8564i	0.910372	+	0.525603i
$696$		0			0
$697$	31.5000	+	18.1865i	1.19315	+	0.688864i
$698$	48.0000			1.81683
$699$		0			0
$700$			3.46410i			0.130931i
$701$	30.0000			1.13308			0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000			1.13308			0.566542	+	0.824033i	$0.308281\pi$
$702$		0			0
$703$	9.00000			0.339441
$704$		−	3.46410i		−	0.130558i
$705$		0			0
$706$	12.0000			0.451626
$707$	9.00000	+	5.19615i	0.338480	+	0.195421i
$708$		0			0
$709$	−10.5000	−	6.06218i	−0.394336	−	0.227670i	0.289701	−	0.957117i	$-0.406444\pi$
$709$	−10.5000	−	6.06218i	−0.394336	−	0.227670i	−0.684037	+	0.729447i	$0.739777\pi$
$710$		−	25.9808i		−	0.975041i
$711$		0			0
$712$	13.5000	−	23.3827i	0.505934	−	0.876303i
$713$		−	25.9808i		−	0.972987i
$714$		0			0
$715$	15.0000	+	15.5885i	0.560968	+	0.582975i
$716$	−1.50000	+	2.59808i	−0.0560576	+	0.0970947i
$717$		0			0
$718$	−18.0000			−0.671754
$719$	−19.5000	−	33.7750i	−0.727227	−	1.25959i	−0.958051	−	0.286599i	$-0.907475\pi$
$719$	−19.5000	−	33.7750i	−0.727227	−	1.25959i	0.230823	−	0.972996i	$-0.425858\pi$
$720$		0			0
$721$	19.5000	−	11.2583i	0.726218	−	0.419282i
$722$	−24.0000	−	13.8564i	−0.893188	−	0.515682i
$723$		0			0
$724$	22.0000			0.817624
$725$	−6.00000	−	10.3923i	−0.222834	−	0.385961i
$726$		0			0
$727$	−0.500000	−	0.866025i	−0.0185440	−	0.0321191i	0.856605	−	0.515974i	$-0.172570\pi$
$727$	−0.500000	−	0.866025i	−0.0185440	−	0.0321191i	−0.875148	+	0.483854i	$0.839236\pi$
$728$	−7.50000	−	7.79423i	−0.277968	−	0.288873i
$729$		0			0
$730$	18.0000	+	10.3923i	0.666210	+	0.384636i
$731$	3.00000			0.110959
$732$		0			0
$733$	−16.5000	−	9.52628i	−0.609441	−	0.351861i	0.163305	−	0.986576i	$-0.447784\pi$
$733$	−16.5000	−	9.52628i	−0.609441	−	0.351861i	−0.772747	+	0.634714i	$0.781118\pi$
$734$			13.8564i			0.511449i
$735$		0			0
$736$	−13.5000	+	7.79423i	−0.497617	+	0.287299i
$737$	21.0000	+	36.3731i	0.773545	+	1.33982i
$738$		0			0
$739$	−28.5000	−	16.4545i	−1.04839	−	0.605288i	−0.126191	−	0.992006i	$-0.540275\pi$
$739$	−28.5000	−	16.4545i	−1.04839	−	0.605288i	−0.922198	+	0.386718i	$0.873609\pi$
$740$	−4.50000	−	7.79423i	−0.165423	−	0.286522i
$741$		0			0
$742$	9.00000	−	15.5885i	0.330400	−	0.572270i
$743$	−10.5000	+	6.06218i	−0.385208	+	0.222400i	−0.680082	−	0.733136i	$-0.738056\pi$
$743$	−10.5000	+	6.06218i	−0.385208	+	0.222400i	0.294874	+	0.955536i	$0.404722\pi$
$744$		0			0
$745$		0			0
$746$		−	24.2487i		−	0.887808i
$747$		0			0
$748$	9.00000	−	5.19615i	0.329073	−	0.189990i
$749$	22.5000	−	12.9904i	0.822132	−	0.474658i
$750$		0			0
$751$	9.50000	+	16.4545i	0.346660	+	0.600433i	0.985654	−	0.168779i	$-0.0539825\pi$
$751$	9.50000	+	16.4545i	0.346660	+	0.600433i	−0.638994	+	0.769212i	$0.720649\pi$
$752$	22.5000	−	12.9904i	0.820491	−	0.473710i
$753$		0			0
$754$	−36.0000	−	10.3923i	−1.31104	−	0.378465i
$755$	−21.0000			−0.764268
$756$		0			0
$757$	−3.50000	+	6.06218i	−0.127210	+	0.220334i	−0.922595	−	0.385771i	$-0.873935\pi$
$757$	−3.50000	+	6.06218i	−0.127210	+	0.220334i	0.795385	+	0.606105i	$0.207269\pi$
$758$	10.5000	−	18.1865i	0.381377	−	0.660565i
$759$		0			0
$760$			5.19615i			0.188484i
$761$			13.8564i			0.502294i	0.967949	+	0.251147i	$0.0808078\pi$
$761$			13.8564i			0.502294i	−0.967949	+	0.251147i	$0.919192\pi$
$762$		0			0
$763$		0			0
$764$	−1.50000	+	2.59808i	−0.0542681	+	0.0939951i
$765$		0			0
$766$	6.00000			0.216789
$767$	−12.0000	−	3.46410i	−0.433295	−	0.125081i
$768$		0			0
$769$	19.5000	−	11.2583i	0.703188	−	0.405986i	−0.105346	−	0.994436i	$-0.533595\pi$
$769$	19.5000	−	11.2583i	0.703188	−	0.405986i	0.808534	+	0.588450i	$0.200262\pi$
$770$	9.00000	+	15.5885i	0.324337	+	0.561769i
$771$		0			0
$772$	4.50000	−	2.59808i	0.161959	−	0.0935068i
$773$	−25.5000	+	14.7224i	−0.917171	+	0.529529i	−0.882732	−	0.469878i	$-0.844298\pi$
$773$	−25.5000	+	14.7224i	−0.917171	+	0.529529i	−0.0344397	+	0.999407i	$0.510965\pi$
$774$		0			0
$775$		−	17.3205i		−	0.622171i
$776$	−13.5000	+	23.3827i	−0.484622	+	0.839390i
$777$		0			0
$778$	−4.50000	+	2.59808i	−0.161333	+	0.0931455i
$779$	−10.5000	+	18.1865i	−0.376202	+	0.651600i
$780$		0			0
$781$	15.0000	+	25.9808i	0.536742	+	0.929665i
$782$	13.5000	+	7.79423i	0.482759	+	0.278721i
$783$		0			0
$784$	10.0000	+	17.3205i	0.357143	+	0.618590i
$785$	34.5000	−	19.9186i	1.23136	−	0.710925i
$786$		0			0
$787$		−	3.46410i		−	0.123482i	−0.998092	−	0.0617409i	$-0.980335\pi$
$787$		−	3.46410i		−	0.123482i	0.998092	−	0.0617409i	$-0.0196653\pi$
$788$	7.50000	+	4.33013i	0.267176	+	0.154254i
$789$		0			0
$790$	33.0000			1.17409
$791$	−9.00000	−	5.19615i	−0.320003	−	0.184754i
$792$		0			0
$793$	12.5000	+	12.9904i	0.443888	+	0.461302i
$794$	−22.5000	−	38.9711i	−0.798495	−	1.38303i
$795$		0			0
$796$	6.50000	+	11.2583i	0.230386	+	0.399041i
$797$	−18.0000			−0.637593			−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000			−0.637593			−0.318796	+	0.947823i	$0.603279\pi$
$798$		0			0
$799$	−13.5000	−	7.79423i	−0.477596	−	0.275740i
$800$	−9.00000	+	5.19615i	−0.318198	+	0.183712i
$801$		0			0
$802$	−25.5000	−	44.1673i	−0.900436	−	1.55960i
$803$	−24.0000			−0.846942
$804$		0			0
$805$	−4.50000	+	7.79423i	−0.158604	+	0.274710i
$806$	−37.5000	−	38.9711i	−1.32088	−	1.37270i
$807$		0			0
$808$			10.3923i			0.365600i
$809$	13.5000	−	23.3827i	0.474635	−	0.822091i	−0.524943	−	0.851137i	$-0.675914\pi$
$809$	13.5000	−	23.3827i	0.474635	−	0.822091i	0.999578	+	0.0290457i	$0.00924684\pi$
$810$		0			0
$811$			51.9615i			1.82462i	0.409505	+	0.912308i	$0.365701\pi$
$811$			51.9615i			1.82462i	−0.409505	+	0.912308i	$0.634299\pi$
$812$	−9.00000	−	5.19615i	−0.315838	−	0.182349i
$813$		0			0
$814$	27.0000	+	15.5885i	0.946350	+	0.546375i
$815$	−21.0000			−0.735598
$816$		0			0
$817$			1.73205i			0.0605968i
$818$	12.0000			0.419570
$819$		0			0
$820$	21.0000			0.733352
$821$			27.7128i			0.967184i	0.875294	+	0.483592i	$0.160668\pi$
$821$			27.7128i			0.967184i	−0.875294	+	0.483592i	$0.839332\pi$
$822$		0			0
$823$	56.0000			1.95204			0.976019	−	0.217687i	$-0.0698512\pi$
$823$	56.0000			1.95204			0.976019	+	0.217687i	$0.0698512\pi$
$824$	19.5000	+	11.2583i	0.679315	+	0.392203i
$825$		0			0
$826$	−9.00000	−	5.19615i	−0.313150	−	0.180797i
$827$		−	24.2487i		−	0.843210i	−0.906780	−	0.421605i	$-0.861467\pi$
$827$		−	24.2487i		−	0.843210i	0.906780	−	0.421605i	$-0.138533\pi$
$828$		0			0
$829$	14.5000	−	25.1147i	0.503606	−	0.872271i	−0.496385	−	0.868102i	$-0.665340\pi$
$829$	14.5000	−	25.1147i	0.503606	−	0.872271i	0.999991	−	0.00416865i	$-0.00132693\pi$
$830$			15.5885i			0.541083i
$831$		0			0
$832$	1.00000	−	3.46410i	0.0346688	−	0.120096i
$833$	6.00000	−	10.3923i	0.207888	−	0.360072i
$834$		0			0
$835$	9.00000			0.311458
$836$	3.00000	+	5.19615i	0.103757	+	0.179713i
$837$		0			0
$838$	−13.5000	+	7.79423i	−0.466350	+	0.269247i
$839$	10.5000	+	6.06218i	0.362500	+	0.209290i	0.670177	−	0.742201i	$-0.266218\pi$
$839$	10.5000	+	6.06218i	0.362500	+	0.209290i	−0.307677	+	0.951491i	$0.599552\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$	7.50000	+	12.9904i	0.258467	+	0.447678i
$843$		0			0
$844$	6.50000	+	11.2583i	0.223739	+	0.387528i
$845$	10.5000	+	19.9186i	0.361211	+	0.685220i
$846$		0			0
$847$	−1.50000	−	0.866025i	−0.0515406	−	0.0297570i
$848$	30.0000			1.03020
$849$		0			0
$850$	9.00000	+	5.19615i	0.308697	+	0.178227i
$851$			15.5885i			0.534365i
$852$		0			0
$853$	−4.50000	+	2.59808i	−0.154077	+	0.0889564i	−0.575056	−	0.818114i	$-0.695020\pi$
$853$	−4.50000	+	2.59808i	−0.154077	+	0.0889564i	0.420979	+	0.907070i	$0.361687\pi$
$854$	7.50000	+	12.9904i	0.256645	+	0.444522i
$855$		0			0
$856$	22.5000	+	12.9904i	0.769034	+	0.444002i
$857$	1.50000	+	2.59808i	0.0512390	+	0.0887486i	0.890507	−	0.454969i	$-0.150350\pi$
$857$	1.50000	+	2.59808i	0.0512390	+	0.0887486i	−0.839268	+	0.543718i	$0.817016\pi$
$858$		0			0
$859$	2.50000	−	4.33013i	0.0852989	−	0.147742i	−0.820220	−	0.572049i	$-0.806149\pi$
$859$	2.50000	−	4.33013i	0.0852989	−	0.147742i	0.905519	+	0.424307i	$0.139482\pi$
$860$	1.50000	−	0.866025i	0.0511496	−	0.0295312i
$861$		0			0
$862$	19.5000	−	33.7750i	0.664173	−	1.15038i
$863$			17.3205i			0.589597i	0.955559	+	0.294798i	$0.0952525\pi$
$863$			17.3205i			0.589597i	−0.955559	+	0.294798i	$0.904747\pi$
$864$		0			0
$865$	31.5000	−	18.1865i	1.07103	−	0.618361i
$866$	−7.50000	+	4.33013i	−0.254860	+	0.147144i
$867$		0			0
$868$	−7.50000	−	12.9904i	−0.254567	−	0.440922i
$869$	−33.0000	+	19.0526i	−1.11945	+	0.646314i
$870$		0			0
$871$	10.5000	+	42.4352i	0.355779	+	1.43786i
$872$		0			0
$873$		0			0
$874$	−4.50000	+	7.79423i	−0.152215	+	0.263644i
$875$	−10.5000	+	18.1865i	−0.354965	+	0.614817i
$876$		0			0
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$			13.8564i			0.467631i
$879$		0			0
$880$	−15.0000	+	25.9808i	−0.505650	+	0.875811i
$881$	−4.50000	+	7.79423i	−0.151609	+	0.262594i	−0.931819	−	0.362923i	$-0.881779\pi$
$881$	−4.50000	+	7.79423i	−0.151609	+	0.262594i	0.780210	+	0.625517i	$0.215112\pi$
$882$		0			0
$883$	40.0000			1.34611			0.673054	−	0.739594i	$-0.264982\pi$
$883$	40.0000			1.34611			0.673054	+	0.739594i	$0.264982\pi$
$884$	10.5000	−	2.59808i	0.353153	−	0.0873828i
$885$		0			0
$886$	−31.5000	+	18.1865i	−1.05826	+	0.610989i
$887$	−7.50000	−	12.9904i	−0.251825	−	0.436174i	0.712203	−	0.701974i	$-0.247698\pi$
$887$	−7.50000	−	12.9904i	−0.251825	−	0.436174i	−0.964028	+	0.265799i	$0.914364\pi$
$888$		0			0
$889$	7.50000	−	4.33013i	0.251542	−	0.145228i
$890$	−40.5000	+	23.3827i	−1.35756	+	0.783789i
$891$		0			0
$892$		−	3.46410i		−	0.115987i
$893$	4.50000	−	7.79423i	0.150587	−	0.260824i
$894$		0			0
$895$	−4.50000	+	2.59808i	−0.150418	+	0.0868441i
$896$	10.5000	−	18.1865i	0.350780	−	0.607569i
$897$		0			0
$898$	−13.5000	−	23.3827i	−0.450501	−	0.780290i
$899$	45.0000	+	25.9808i	1.50083	+	0.866507i
$900$		0			0
$901$	−9.00000	−	15.5885i	−0.299833	−	0.519327i
$902$	−63.0000	+	36.3731i	−2.09767	+	1.21109i
$903$		0			0
$904$		−	10.3923i		−	0.345643i
$905$	33.0000	+	19.0526i	1.09696	+	0.633328i
$906$		0			0
$907$	4.00000			0.132818			0.0664089	−	0.997792i	$-0.478846\pi$
$907$	4.00000			0.132818			0.0664089	+	0.997792i	$0.478846\pi$
$908$	−10.5000	−	6.06218i	−0.348455	−	0.201180i
$909$		0			0
$910$	4.50000	+	18.1865i	0.149174	+	0.602878i
$911$	22.5000	+	38.9711i	0.745458	+	1.29117i	0.949980	+	0.312310i	$0.101103\pi$
$911$	22.5000	+	38.9711i	0.745458	+	1.29117i	−0.204522	+	0.978862i	$0.565564\pi$
$912$		0			0
$913$	−9.00000	−	15.5885i	−0.297857	−	0.515903i
$914$		0			0
$915$		0			0
$916$	−7.50000	−	4.33013i	−0.247807	−	0.143071i
$917$	22.5000	−	12.9904i	0.743015	−	0.428980i
$918$		0			0
$919$	−12.5000	−	21.6506i	−0.412337	−	0.714189i	0.582808	−	0.812610i	$-0.301954\pi$
$919$	−12.5000	−	21.6506i	−0.412337	−	0.714189i	−0.995145	+	0.0984214i	$0.968621\pi$
$920$	−9.00000			−0.296721
$921$		0			0
$922$	1.50000	−	2.59808i	0.0493999	−	0.0855631i
$923$	7.50000	+	30.3109i	0.246866	+	0.997695i
$924$		0			0
$925$			10.3923i			0.341697i
$926$	22.5000	−	38.9711i	0.739396	−	1.28067i
$927$		0			0
$928$		−	31.1769i		−	1.02343i
$929$	−49.5000	−	28.5788i	−1.62404	−	0.937641i	−0.985823	−	0.167786i	$-0.946338\pi$
$929$	−49.5000	−	28.5788i	−1.62404	−	0.937641i	−0.638219	−	0.769855i	$-0.720329\pi$
$930$		0			0
$931$	6.00000	+	3.46410i	0.196642	+	0.113531i
$932$	−18.0000			−0.589610
$933$		0			0
$934$			20.7846i			0.680093i
$935$	18.0000			0.588663
$936$		0			0
$937$	−2.00000			−0.0653372			−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000			−0.0653372			−0.0326686	+	0.999466i	$0.510401\pi$
$938$			36.3731i			1.18762i
$939$		0			0
$940$	−9.00000			−0.293548
$941$	−13.5000	−	7.79423i	−0.440087	−	0.254085i	0.263547	−	0.964646i	$-0.415107\pi$
$941$	−13.5000	−	7.79423i	−0.440087	−	0.254085i	−0.703635	+	0.710562i	$0.748441\pi$
$942$		0			0
$943$	−31.5000	−	18.1865i	−1.02578	−	0.592235i
$944$		−	17.3205i		−	0.563735i
$945$		0			0
$946$	−3.00000	+	5.19615i	−0.0975384	+	0.168941i
$947$		−	38.1051i		−	1.23825i	−0.785292	−	0.619125i	$-0.787487\pi$
$947$		−	38.1051i		−	1.23825i	0.785292	−	0.619125i	$-0.212513\pi$
$948$		0			0
$949$	−24.0000	−	6.92820i	−0.779073	−	0.224899i
$950$	−3.00000	+	5.19615i	−0.0973329	+	0.168585i
$951$		0			0
$952$	−9.00000			−0.291692
$953$	13.5000	+	23.3827i	0.437308	+	0.757439i	0.997481	−	0.0709362i	$-0.0225987\pi$
$953$	13.5000	+	23.3827i	0.437308	+	0.757439i	−0.560173	+	0.828376i	$0.689265\pi$
$954$		0			0
$955$	−4.50000	+	2.59808i	−0.145617	+	0.0840718i
$956$	−4.50000	−	2.59808i	−0.145540	−	0.0840278i
$957$		0			0
$958$	−42.0000			−1.35696
$959$	4.50000	+	7.79423i	0.145313	+	0.251689i
$960$		0			0
$961$	22.0000	+	38.1051i	0.709677	+	1.22920i
$962$	22.5000	+	23.3827i	0.725429	+	0.753888i
$963$		0			0
$964$	4.50000	+	2.59808i	0.144935	+	0.0836784i
$965$	9.00000			0.289720
$966$		0			0
$967$	−10.5000	−	6.06218i	−0.337657	−	0.194946i	0.321578	−	0.946883i	$-0.395787\pi$
$967$	−10.5000	−	6.06218i	−0.337657	−	0.194946i	−0.659236	+	0.751936i	$0.729120\pi$
$968$		−	1.73205i		−	0.0556702i
$969$		0			0
$970$	40.5000	−	23.3827i	1.30038	−	0.750773i
$971$	−13.5000	−	23.3827i	−0.433236	−	0.750386i	0.563914	−	0.825833i	$-0.309295\pi$
$971$	−13.5000	−	23.3827i	−0.433236	−	0.750386i	−0.997150	+	0.0754473i	$0.975962\pi$
$972$		0			0
$973$	−24.0000	−	13.8564i	−0.769405	−	0.444216i
$974$	−7.50000	−	12.9904i	−0.240316	−	0.416239i
$975$		0			0
$976$	−12.5000	+	21.6506i	−0.400115	+	0.693020i
$977$	22.5000	−	12.9904i	0.719839	−	0.415599i	−0.0948546	−	0.995491i	$-0.530239\pi$
$977$	22.5000	−	12.9904i	0.719839	−	0.415599i	0.814693	+	0.579892i	$0.196905\pi$
$978$		0			0
$979$	27.0000	−	46.7654i	0.862924	−	1.49463i
$980$		−	6.92820i		−	0.221313i
$981$		0			0
$982$	4.50000	−	2.59808i	0.143601	−	0.0829079i
$983$	−4.50000	+	2.59808i	−0.143528	+	0.0828658i	−0.570044	−	0.821614i	$-0.693074\pi$
$983$	−4.50000	+	2.59808i	−0.143528	+	0.0828658i	0.426517	+	0.904480i	$0.359741\pi$
$984$		0			0
$985$	7.50000	+	12.9904i	0.238970	+	0.413908i
$986$	−27.0000	+	15.5885i	−0.859855	+	0.496438i
$987$		0			0
$988$	1.50000	+	6.06218i	0.0477214	+	0.192864i
$989$	−3.00000			−0.0953945
$990$		0			0
$991$	18.5000	−	32.0429i	0.587672	−	1.01788i	−0.406865	−	0.913488i	$-0.633378\pi$
$991$	18.5000	−	32.0429i	0.587672	−	1.01788i	0.994537	−	0.104389i	$-0.0332887\pi$
$992$	22.5000	−	38.9711i	0.714376	−	1.23734i
$993$		0			0
$994$			25.9808i			0.824060i
$995$			22.5167i			0.713826i
$996$		0			0
$997$	−11.5000	+	19.9186i	−0.364209	+	0.630828i	−0.988649	−	0.150245i	$-0.951994\pi$
$997$	−11.5000	+	19.9186i	−0.364209	+	0.630828i	0.624440	+	0.781073i	$0.285327\pi$
$998$	22.5000	−	38.9711i	0.712225	−	1.23361i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	351.2.l.a.199.1		2
3.2	odd	2		117.2.l.a.4.1	✓	2
9.2	odd	6		117.2.r.a.43.1	yes	2
9.7	even	3		351.2.r.a.316.1		2
13.10	even	6		351.2.r.a.10.1		2
39.23	odd	6		117.2.r.a.49.1	yes	2
117.88	even	6	inner	351.2.l.a.127.1		2
117.101	odd	6		117.2.l.a.88.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.2.l.a.4.1	✓	2	3.2	odd	2
117.2.l.a.88.1	yes	2	117.101	odd	6
117.2.r.a.43.1	yes	2	9.2	odd	6
117.2.r.a.49.1	yes	2	39.23	odd	6
351.2.l.a.127.1		2	117.88	even	6	inner
351.2.l.a.199.1		2	1.1	even	1	trivial
351.2.r.a.10.1		2	13.10	even	6
351.2.r.a.316.1		2	9.7	even	3

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 351 = 3^{3} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	351.l (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{2} -1.00000 q^{4} +(-1.50000 - 0.866025i) q^{5} +(1.50000 + 0.866025i) q^{7} +1.73205i q^{8} +(1.50000 - 2.59808i) q^{10} +3.46410i q^{11} +(-1.00000 + 3.46410i) q^{13} +(-1.50000 + 2.59808i) q^{14} -5.00000 q^{16} +(1.50000 + 2.59808i) q^{17} +(-1.50000 + 0.866025i) q^{19} +(1.50000 + 0.866025i) q^{20} -6.00000 q^{22} +(-1.50000 - 2.59808i) q^{23} +(-1.00000 - 1.73205i) q^{25} +(-6.00000 - 1.73205i) q^{26} +(-1.50000 - 0.866025i) q^{28} +6.00000 q^{29} +(7.50000 + 4.33013i) q^{31} -5.19615i q^{32} +(-4.50000 + 2.59808i) q^{34} +(-1.50000 - 2.59808i) q^{35} +(-4.50000 - 2.59808i) q^{37} +(-1.50000 - 2.59808i) q^{38} +(1.50000 - 2.59808i) q^{40} +(10.5000 - 6.06218i) q^{41} +(0.500000 - 0.866025i) q^{43} -3.46410i q^{44} +(4.50000 - 2.59808i) q^{46} +(-4.50000 + 2.59808i) q^{47} +(-2.00000 - 3.46410i) q^{49} +(3.00000 - 1.73205i) q^{50} +(1.00000 - 3.46410i) q^{52} -6.00000 q^{53} +(3.00000 - 5.19615i) q^{55} +(-1.50000 + 2.59808i) q^{56} +10.3923i q^{58} +3.46410i q^{59} +(2.50000 - 4.33013i) q^{61} +(-7.50000 + 12.9904i) q^{62} -1.00000 q^{64} +(4.50000 - 4.33013i) q^{65} +(10.5000 - 6.06218i) q^{67} +(-1.50000 - 2.59808i) q^{68} +(4.50000 - 2.59808i) q^{70} +(7.50000 - 4.33013i) q^{71} +6.92820i q^{73} +(4.50000 - 7.79423i) q^{74} +(1.50000 - 0.866025i) q^{76} +(-3.00000 + 5.19615i) q^{77} +(5.50000 + 9.52628i) q^{79} +(7.50000 + 4.33013i) q^{80} +(10.5000 + 18.1865i) q^{82} +(-4.50000 + 2.59808i) q^{83} -5.19615i q^{85} +(1.50000 + 0.866025i) q^{86} -6.00000 q^{88} +(-13.5000 - 7.79423i) q^{89} +(-4.50000 + 4.33013i) q^{91} +(1.50000 + 2.59808i) q^{92} +(-4.50000 - 7.79423i) q^{94} +3.00000 q^{95} +(13.5000 + 7.79423i) q^{97} +(6.00000 - 3.46410i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 3 q^{5} + 3 q^{7} + 3 q^{10} - 2 q^{13} - 3 q^{14} - 10 q^{16} + 3 q^{17} - 3 q^{19} + 3 q^{20} - 12 q^{22} - 3 q^{23} - 2 q^{25} - 12 q^{26} - 3 q^{28} + 12 q^{29} + 15 q^{31} - 9 q^{34}+ \cdots + 12 q^{98}+O(q^{100}) \) 2 * q - 2 * q^4 - 3 * q^5 + 3 * q^7 + 3 * q^10 - 2 * q^13 - 3 * q^14 - 10 * q^16 + 3 * q^17 - 3 * q^19 + 3 * q^20 - 12 * q^22 - 3 * q^23 - 2 * q^25 - 12 * q^26 - 3 * q^28 + 12 * q^29 + 15 * q^31 - 9 * q^34 - 3 * q^35 - 9 * q^37 - 3 * q^38 + 3 * q^40 + 21 * q^41 + q^43 + 9 * q^46 - 9 * q^47 - 4 * q^49 + 6 * q^50 + 2 * q^52 - 12 * q^53 + 6 * q^55 - 3 * q^56 + 5 * q^61 - 15 * q^62 - 2 * q^64 + 9 * q^65 + 21 * q^67 - 3 * q^68 + 9 * q^70 + 15 * q^71 + 9 * q^74 + 3 * q^76 - 6 * q^77 + 11 * q^79 + 15 * q^80 + 21 * q^82 - 9 * q^83 + 3 * q^86 - 12 * q^88 - 27 * q^89 - 9 * q^91 + 3 * q^92 - 9 * q^94 + 6 * q^95 + 27 * q^97 + 12 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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