LMFDB - Embedded newform 351.2.r.a.316.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [351,2,Mod(10,351)]

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([2, 5]))

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

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

chi := DirichletCharacter("351.10");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 351 = 3^{3} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	351.r (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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 117)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			316.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	351.316
Dual form			351.2.r.a.10.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.50000 - 0.866025i) q^{2} +(0.500000 - 0.866025i) q^{4} +(1.50000 - 0.866025i) q^{5} -1.73205i q^{7} +1.73205i q^{8} +(1.50000 - 2.59808i) q^{10} +(3.00000 - 1.73205i) q^{11} +(-2.50000 - 2.59808i) q^{13} +(-1.50000 - 2.59808i) q^{14} +(2.50000 + 4.33013i) q^{16} +(1.50000 + 2.59808i) q^{17} +(-1.50000 + 0.866025i) q^{19} -1.73205i q^{20} +(3.00000 - 5.19615i) q^{22} +3.00000 q^{23} +(-1.00000 + 1.73205i) q^{25} +(-6.00000 - 1.73205i) q^{26} +(-1.50000 - 0.866025i) q^{28} +(-3.00000 - 5.19615i) q^{29} +(-7.50000 + 4.33013i) q^{31} +(4.50000 + 2.59808i) q^{32} +(4.50000 + 2.59808i) q^{34} +(-1.50000 - 2.59808i) q^{35} +(-4.50000 - 2.59808i) q^{37} +(-1.50000 + 2.59808i) q^{38} +(1.50000 + 2.59808i) q^{40} +12.1244i q^{41} -1.00000 q^{43} -3.46410i q^{44} +(4.50000 - 2.59808i) q^{46} +(4.50000 + 2.59808i) q^{47} +4.00000 q^{49} +3.46410i q^{50} +(-3.50000 + 0.866025i) q^{52} -6.00000 q^{53} +(3.00000 - 5.19615i) q^{55} +3.00000 q^{56} +(-9.00000 - 5.19615i) q^{58} +(-3.00000 - 1.73205i) q^{59} -5.00000 q^{61} +(-7.50000 + 12.9904i) q^{62} -1.00000 q^{64} +(-6.00000 - 1.73205i) q^{65} +12.1244i q^{67} +3.00000 q^{68} +(-4.50000 - 2.59808i) q^{70} +(7.50000 - 4.33013i) q^{71} +6.92820i q^{73} -9.00000 q^{74} +1.73205i q^{76} +(-3.00000 - 5.19615i) q^{77} +(5.50000 - 9.52628i) q^{79} +(7.50000 + 4.33013i) q^{80} +(10.5000 + 18.1865i) q^{82} +(4.50000 + 2.59808i) q^{83} +(4.50000 + 2.59808i) q^{85} +(-1.50000 + 0.866025i) q^{86} +(3.00000 + 5.19615i) q^{88} +(-13.5000 - 7.79423i) q^{89} +(-4.50000 + 4.33013i) q^{91} +(1.50000 - 2.59808i) q^{92} +9.00000 q^{94} +(-1.50000 + 2.59808i) q^{95} -15.5885i q^{97} +(6.00000 - 3.46410i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{2} + q^{4} + 3 q^{5} + 3 q^{10} + 6 q^{11} - 5 q^{13} - 3 q^{14} + 5 q^{16} + 3 q^{17} - 3 q^{19} + 6 q^{22} + 6 q^{23} - 2 q^{25} - 12 q^{26} - 3 q^{28} - 6 q^{29} - 15 q^{31} + 9 q^{32} + 9 q^{34}+ \cdots + 12 q^{98}+O(q^{100}) $ 2 * q + 3 * q^2 + q^4 + 3 * q^5 + 3 * q^10 + 6 * q^11 - 5 * q^13 - 3 * q^14 + 5 * q^16 + 3 * q^17 - 3 * q^19 + 6 * q^22 + 6 * q^23 - 2 * q^25 - 12 * q^26 - 3 * q^28 - 6 * q^29 - 15 * q^31 + 9 * q^32 + 9 * q^34 - 3 * q^35 - 9 * q^37 - 3 * q^38 + 3 * q^40 - 2 * q^43 + 9 * q^46 + 9 * q^47 + 8 * q^49 - 7 * q^52 - 12 * q^53 + 6 * q^55 + 6 * q^56 - 18 * q^58 - 6 * q^59 - 10 * q^61 - 15 * q^62 - 2 * q^64 - 12 * q^65 + 6 * q^68 - 9 * q^70 + 15 * q^71 - 18 * q^74 - 6 * q^77 + 11 * q^79 + 15 * q^80 + 21 * q^82 + 9 * q^83 + 9 * q^85 - 3 * q^86 + 6 * q^88 - 27 * q^89 - 9 * q^91 + 3 * q^92 + 18 * q^94 - 3 * q^95 + 12 * q^98

Character values

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

$n$	$28$	$326$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{2}{3}\right)$

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 $

$2$	1.50000	−	0.866025i	1.06066	−	0.612372i	0.135045	−	0.990839i	$-0.456882\pi$
$2$	1.50000	−	0.866025i	1.06066	−	0.612372i	0.925615	+	0.378467i	$0.123549\pi$
$3$		0			0
$3$		0			0
$4$	0.500000	−	0.866025i	0.250000	−	0.433013i
$5$	1.50000	−	0.866025i	0.670820	−	0.387298i	−0.125567	−	0.992085i	$-0.540075\pi$
$5$	1.50000	−	0.866025i	0.670820	−	0.387298i	0.796387	+	0.604787i	$0.206742\pi$
$6$		0			0
$7$		−	1.73205i		−	0.654654i	−0.944911	−	0.327327i	$-0.893852\pi$
$7$		−	1.73205i		−	0.654654i	0.944911	−	0.327327i	$-0.106148\pi$
$8$			1.73205i			0.612372i
$9$		0			0
$10$	1.50000	−	2.59808i	0.474342	−	0.821584i
$11$	3.00000	−	1.73205i	0.904534	−	0.522233i	0.0258656	−	0.999665i	$-0.491766\pi$
$11$	3.00000	−	1.73205i	0.904534	−	0.522233i	0.878668	+	0.477432i	$0.158432\pi$
$12$		0			0
$13$	−2.50000	−	2.59808i	−0.693375	−	0.720577i
$13$	−2.50000	−	2.59808i	−0.693375	−	0.720577i
$14$	−1.50000	−	2.59808i	−0.400892	−	0.694365i
$15$		0			0
$16$	2.50000	+	4.33013i	0.625000	+	1.08253i
$17$	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	$0.396999\pi$
$20$		−	1.73205i		−	0.387298i
$21$		0			0
$22$	3.00000	−	5.19615i	0.639602	−	1.10782i
$23$	3.00000			0.625543			0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000			0.625543			0.312772	+	0.949828i	$0.398743\pi$
$24$		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
$29$	−3.00000	−	5.19615i	−0.557086	−	0.964901i	−0.997738	−	0.0672232i	$-0.978586\pi$
$29$	−3.00000	−	5.19615i	−0.557086	−	0.964901i	0.440652	−	0.897678i	$-0.354747\pi$
$30$		0			0
$31$	−7.50000	+	4.33013i	−1.34704	+	0.777714i	−0.987829	−	0.155543i	$-0.950287\pi$
$31$	−7.50000	+	4.33013i	−1.34704	+	0.777714i	−0.359211	+	0.933257i	$0.616954\pi$
$32$	4.50000	+	2.59808i	0.795495	+	0.459279i
$33$		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
$41$			12.1244i			1.89351i	0.321960	+	0.946753i	$0.395658\pi$
$41$			12.1244i			1.89351i	−0.321960	+	0.946753i	$0.604342\pi$
$42$		0			0
$43$	−1.00000			−0.152499			−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000			−0.152499			−0.0762493	+	0.997089i	$0.524294\pi$
$44$		−	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	$-0.209612\pi$
$47$	4.50000	+	2.59808i	0.656392	+	0.378968i	−0.134509	+	0.990912i	$0.542946\pi$
$48$		0			0
$49$	4.00000			0.571429
$50$			3.46410i			0.489898i
$51$		0			0
$52$	−3.50000	+	0.866025i	−0.485363	+	0.120096i
$53$	−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
$56$	3.00000			0.400892
$57$		0			0
$58$	−9.00000	−	5.19615i	−1.18176	−	0.682288i
$59$	−3.00000	−	1.73205i	−0.390567	−	0.225494i	0.291839	−	0.956467i	$-0.405733\pi$
$59$	−3.00000	−	1.73205i	−0.390567	−	0.225494i	−0.682406	+	0.730974i	$0.739066\pi$
$60$		0			0
$61$	−5.00000			−0.640184			−0.320092	−	0.947386i	$-0.603714\pi$
$61$	−5.00000			−0.640184			−0.320092	+	0.947386i	$0.603714\pi$
$62$	−7.50000	+	12.9904i	−0.952501	+	1.64978i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−6.00000	−	1.73205i	−0.744208	−	0.214834i
$66$		0			0
$67$			12.1244i			1.48123i	0.671932	+	0.740613i	$0.265465\pi$
$67$			12.1244i			1.48123i	−0.671932	+	0.740613i	$0.734535\pi$
$68$	3.00000			0.363803
$69$		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	$0.867117\pi$
$74$	−9.00000			−1.04623
$75$		0			0
$76$			1.73205i			0.198680i
$77$	−3.00000	−	5.19615i	−0.341882	−	0.592157i
$78$		0			0
$79$	5.50000	−	9.52628i	0.618798	−	1.07179i	−0.370907	−	0.928670i	$-0.620953\pi$
$79$	5.50000	−	9.52628i	0.618798	−	1.07179i	0.989705	−	0.143120i	$-0.0457135\pi$
$80$	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	$-0.241281\pi$
$83$	4.50000	+	2.59808i	0.493939	+	0.285176i	−0.232268	+	0.972652i	$0.574615\pi$
$84$		0			0
$85$	4.50000	+	2.59808i	0.488094	+	0.281801i
$86$	−1.50000	+	0.866025i	−0.161749	+	0.0933859i
$87$		0			0
$88$	3.00000	+	5.19615i	0.319801	+	0.553912i
$89$	−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
$94$	9.00000			0.928279
$95$	−1.50000	+	2.59808i	−0.153897	+	0.266557i
$96$		0			0
$97$		−	15.5885i		−	1.58277i	−0.611319	−	0.791384i	$-0.709361\pi$
$97$		−	15.5885i		−	1.58277i	0.611319	−	0.791384i	$-0.290639\pi$
$98$	6.00000	−	3.46410i	0.606092	−	0.349927i
$99$		0			0
$100$	1.00000	+	1.73205i	0.100000	+	0.173205i
$101$	−3.00000	−	5.19615i	−0.298511	−	0.517036i	0.677284	−	0.735721i	$-0.263157\pi$
$101$	−3.00000	−	5.19615i	−0.298511	−	0.517036i	−0.975796	+	0.218685i	$0.929823\pi$
$102$		0			0
$103$	6.50000	+	11.2583i	0.640464	+	1.10932i	0.985329	+	0.170664i	$0.0545913\pi$
$103$	6.50000	+	11.2583i	0.640464	+	1.10932i	−0.344865	+	0.938652i	$0.612075\pi$
$104$	4.50000	−	4.33013i	0.441261	−	0.424604i
$105$		0			0
$106$	−9.00000	+	5.19615i	−0.874157	+	0.504695i
$107$	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			$\pi$
$110$		−	10.3923i		−	0.990867i
$111$		0			0
$112$	7.50000	−	4.33013i	0.708683	−	0.409159i
$113$	3.00000	−	5.19615i	0.282216	−	0.488813i	−0.689714	−	0.724082i	$-0.742264\pi$
$113$	3.00000	−	5.19615i	0.282216	−	0.488813i	0.971930	+	0.235269i	$0.0755971\pi$
$114$		0			0
$115$	4.50000	−	2.59808i	0.419627	−	0.242272i
$116$	−6.00000			−0.557086
$117$		0			0
$118$	−6.00000			−0.552345
$119$	4.50000	−	2.59808i	0.412514	−	0.238165i
$120$		0			0
$121$	0.500000	−	0.866025i	0.0454545	−	0.0787296i
$122$	−7.50000	+	4.33013i	−0.679018	+	0.392031i
$123$		0			0
$124$			8.66025i			0.777714i
$125$			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	$0.0954607\pi$
$128$	−10.5000	+	6.06218i	−0.928078	+	0.535826i
$129$		0			0
$130$	−10.5000	+	2.59808i	−0.920911	+	0.227866i
$131$	7.50000	+	12.9904i	0.655278	+	1.13497i	0.981824	+	0.189794i	$0.0607819\pi$
$131$	7.50000	+	12.9904i	0.655278	+	1.13497i	−0.326546	+	0.945181i	$0.605885\pi$
$132$		0			0
$133$	1.50000	+	2.59808i	0.130066	+	0.225282i
$134$	10.5000	+	18.1865i	0.907062	+	1.57108i
$135$		0			0
$136$	−4.50000	+	2.59808i	−0.385872	+	0.222783i
$137$		−	5.19615i		−	0.443937i	−0.975054	−	0.221969i	$-0.928752\pi$
$137$		−	5.19615i		−	0.443937i	0.975054	−	0.221969i	$-0.0712483\pi$
$138$		0			0
$139$	8.00000	−	13.8564i	0.678551	−	1.17529i	−0.296866	−	0.954919i	$-0.595942\pi$
$139$	8.00000	−	13.8564i	0.678551	−	1.17529i	0.975417	−	0.220366i	$-0.0707252\pi$
$140$	−3.00000			−0.253546
$141$		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
$146$	6.00000	+	10.3923i	0.496564	+	0.860073i
$147$		0			0
$148$	−4.50000	+	2.59808i	−0.369898	+	0.213561i
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$	−10.5000	−	6.06218i	−0.854478	−	0.493333i	0.00768132	−	0.999970i	$-0.497555\pi$
$151$	−10.5000	−	6.06218i	−0.854478	−	0.493333i	−0.862159	+	0.506637i	$0.830888\pi$
$152$	−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	−0.802749	−	0.596316i	$-0.796630\pi$
$157$	−11.5000	−	19.9186i	−0.917800	−	1.58968i	−0.115050	−	0.993360i	$-0.536703\pi$
$158$		−	19.0526i		−	1.51574i
$159$		0			0
$160$	9.00000			0.711512
$161$		−	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
$166$	9.00000			0.698535
$167$		−	5.19615i		−	0.402090i	−0.979582	−	0.201045i	$-0.935566\pi$
$167$		−	5.19615i		−	0.402090i	0.979582	−	0.201045i	$-0.0644338\pi$
$168$		0			0
$169$	−0.500000	+	12.9904i	−0.0384615	+	0.999260i
$170$	9.00000			0.690268
$171$		0			0
$172$	−0.500000	+	0.866025i	−0.0381246	+	0.0660338i
$173$	21.0000			1.59660			0.798300	−	0.602260i	$-0.205733\pi$
$173$	21.0000			1.59660			0.798300	+	0.602260i	$0.205733\pi$
$174$		0			0
$175$	3.00000	+	1.73205i	0.226779	+	0.130931i
$176$	15.0000	+	8.66025i	1.13067	+	0.652791i
$177$		0			0
$178$	−27.0000			−2.02374
$179$	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	$0.804709\pi$
$182$	−3.00000	+	10.3923i	−0.222375	+	0.770329i
$183$		0			0
$184$			5.19615i			0.383065i
$185$	−9.00000			−0.661693
$186$		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
$191$	−3.00000			−0.217072			−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000			−0.217072			−0.108536	+	0.994092i	$0.534616\pi$
$192$		0			0
$193$		−	5.19615i		−	0.374027i	−0.982357	−	0.187014i	$-0.940119\pi$
$193$		−	5.19615i		−	0.374027i	0.982357	−	0.187014i	$-0.0598809\pi$
$194$	−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
$202$	−9.00000	−	5.19615i	−0.633238	−	0.365600i
$203$	−9.00000	+	5.19615i	−0.631676	+	0.364698i
$204$		0			0
$205$	10.5000	+	18.1865i	0.733352	+	1.27020i
$206$	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
$211$	13.0000			0.894957			0.447478	−	0.894295i	$-0.352322\pi$
$211$	13.0000			0.894957			0.447478	+	0.894295i	$0.352322\pi$
$212$	−3.00000	+	5.19615i	−0.206041	+	0.356873i
$213$		0			0
$214$		−	25.9808i		−	1.77601i
$215$	−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
$221$	3.00000	−	10.3923i	0.201802	−	0.699062i
$222$		0			0
$223$	3.00000	−	1.73205i	0.200895	−	0.115987i	−0.396178	−	0.918174i	$-0.629664\pi$
$223$	3.00000	−	1.73205i	0.200895	−	0.115987i	0.597073	+	0.802187i	$0.296330\pi$
$224$	4.50000	−	7.79423i	0.300669	−	0.520774i
$225$		0			0
$226$		−	10.3923i		−	0.691286i
$227$		−	12.1244i		−	0.804722i	−0.915481	−	0.402361i	$-0.868190\pi$
$227$		−	12.1244i		−	0.804722i	0.915481	−	0.402361i	$-0.131810\pi$
$228$		0			0
$229$	−7.50000	+	4.33013i	−0.495614	+	0.286143i	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−7.50000	+	4.33013i	−0.495614	+	0.286143i	0.231287	+	0.972886i	$0.425707\pi$
$230$	4.50000	−	7.79423i	0.296721	−	0.513936i
$231$		0			0
$232$	9.00000	−	5.19615i	0.590879	−	0.341144i
$233$	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
$236$	−3.00000	+	1.73205i	−0.195283	+	0.112747i
$237$		0			0
$238$	4.50000	−	7.79423i	0.291692	−	0.505225i
$239$	−4.50000	+	2.59808i	−0.291081	+	0.168056i	−0.638429	−	0.769681i	$-0.720415\pi$
$239$	−4.50000	+	2.59808i	−0.291081	+	0.168056i	0.347348	+	0.937736i	$0.387082\pi$
$240$		0			0
$241$			5.19615i			0.334714i	0.985896	+	0.167357i	$0.0535232\pi$
$241$			5.19615i			0.334714i	−0.985896	+	0.167357i	$0.946477\pi$
$242$		−	1.73205i		−	0.111340i
$243$		0			0
$244$	−2.50000	+	4.33013i	−0.160046	+	0.277208i
$245$	6.00000	−	3.46410i	0.383326	−	0.221313i
$246$		0			0
$247$	6.00000	+	1.73205i	0.381771	+	0.110208i
$248$	−7.50000	−	12.9904i	−0.476250	−	0.824890i
$249$		0			0
$250$	10.5000	+	18.1865i	0.664078	+	1.15022i
$251$	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
$254$		−	8.66025i		−	0.543393i
$255$		0			0
$256$	−9.50000	+	16.4545i	−0.593750	+	1.02841i
$257$	−3.00000			−0.187135			−0.0935674	−	0.995613i	$-0.529827\pi$
$257$	−3.00000			−0.187135			−0.0935674	+	0.995613i	$0.529827\pi$
$258$		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
$263$	12.0000	+	20.7846i	0.739952	+	1.28163i	0.952517	+	0.304487i	$0.0984850\pi$
$263$	12.0000	+	20.7846i	0.739952	+	1.28163i	−0.212565	+	0.977147i	$0.568182\pi$
$264$		0			0
$265$	−9.00000	+	5.19615i	−0.552866	+	0.319197i
$266$	4.50000	+	2.59808i	0.275913	+	0.159298i
$267$		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
$275$			6.92820i			0.417786i
$276$		0			0
$277$	23.0000			1.38194			0.690968	−	0.722885i	$-0.257185\pi$
$277$	23.0000			1.38194			0.690968	+	0.722885i	$0.257185\pi$
$278$		−	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	$-0.382867\pi$
$281$	−4.50000	−	2.59808i	−0.268447	−	0.154988i	−0.628182	+	0.778067i	$0.716201\pi$
$282$		0			0
$283$	−23.0000			−1.36721			−0.683604	−	0.729853i	$-0.739588\pi$
$283$	−23.0000			−1.36721			−0.683604	+	0.729853i	$0.739588\pi$
$284$		−	8.66025i		−	0.513892i
$285$		0			0
$286$	−21.0000	+	5.19615i	−1.24176	+	0.307255i
$287$	21.0000			1.23959
$288$		0			0
$289$	4.00000	−	6.92820i	0.235294	−	0.407541i
$290$	−18.0000			−1.05700
$291$		0			0
$292$	6.00000	+	3.46410i	0.351123	+	0.202721i
$293$	−12.0000	−	6.92820i	−0.701047	−	0.404750i	0.106690	−	0.994292i	$-0.465975\pi$
$293$	−12.0000	−	6.92820i	−0.701047	−	0.404750i	−0.807737	+	0.589542i	$0.799308\pi$
$294$		0			0
$295$	−6.00000			−0.349334
$296$	4.50000	−	7.79423i	0.261557	−	0.453030i
$297$		0			0
$298$		0			0
$299$	−7.50000	−	7.79423i	−0.433736	−	0.450752i
$300$		0			0
$301$			1.73205i			0.0998337i
$302$	−21.0000			−1.20841
$303$		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	$0.756741\pi$
$308$	−6.00000			−0.341882
$309$		0			0
$310$			25.9808i			1.47561i
$311$	13.5000	+	23.3827i	0.765515	+	1.32591i	0.939974	+	0.341246i	$0.110849\pi$
$311$	13.5000	+	23.3827i	0.765515	+	1.32591i	−0.174459	+	0.984664i	$0.555818\pi$
$312$		0			0
$313$	−9.50000	+	16.4545i	−0.536972	+	0.930062i	0.462093	+	0.886831i	$0.347098\pi$
$313$	−9.50000	+	16.4545i	−0.536972	+	0.930062i	−0.999065	+	0.0432311i	$0.986235\pi$
$314$	−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	$-0.255135\pi$
$317$	7.50000	+	4.33013i	0.421242	+	0.243204i	−0.274367	+	0.961625i	$0.588468\pi$
$318$		0			0
$319$	−18.0000	−	10.3923i	−1.00781	−	0.581857i
$320$	−1.50000	+	0.866025i	−0.0838525	+	0.0484123i
$321$		0			0
$322$	−4.50000	−	7.79423i	−0.250775	−	0.434355i
$323$	−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
$328$	−21.0000			−1.15953
$329$	4.50000	−	7.79423i	0.248093	−	0.429710i
$330$		0			0
$331$		−	15.5885i		−	0.856819i	−0.903585	−	0.428410i	$-0.859074\pi$
$331$		−	15.5885i		−	0.856819i	0.903585	−	0.428410i	$-0.140926\pi$
$332$	4.50000	−	2.59808i	0.246970	−	0.142588i
$333$		0			0
$334$	−4.50000	−	7.79423i	−0.246229	−	0.426481i
$335$	10.5000	+	18.1865i	0.573676	+	0.993636i
$336$		0			0
$337$	14.5000	+	25.1147i	0.789865	+	1.36809i	0.926049	+	0.377403i	$0.123183\pi$
$337$	14.5000	+	25.1147i	0.789865	+	1.36809i	−0.136184	+	0.990684i	$0.543484\pi$
$338$	10.5000	+	19.9186i	0.571125	+	1.08343i
$339$		0			0
$340$	4.50000	−	2.59808i	0.244047	−	0.140900i
$341$	−15.0000	+	25.9808i	−0.812296	+	1.40694i
$342$		0			0
$343$		−	19.0526i		−	1.02874i
$344$		−	1.73205i		−	0.0933859i
$345$		0			0
$346$	31.5000	−	18.1865i	1.69345	−	0.977714i
$347$	−6.00000	+	10.3923i	−0.322097	+	0.557888i	−0.980921	−	0.194409i	$-0.937721\pi$
$347$	−6.00000	+	10.3923i	−0.322097	+	0.557888i	0.658824	+	0.752297i	$0.271054\pi$
$348$		0			0
$349$	−24.0000	+	13.8564i	−1.28469	+	0.741716i	−0.977702	−	0.209997i	$-0.932655\pi$
$349$	−24.0000	+	13.8564i	−1.28469	+	0.741716i	−0.306988	+	0.951713i	$0.599321\pi$
$350$	6.00000			0.320713
$351$		0			0
$352$	18.0000			0.959403
$353$	−6.00000	+	3.46410i	−0.319348	+	0.184376i	−0.651102	−	0.758990i	$-0.725693\pi$
$353$	−6.00000	+	3.46410i	−0.319348	+	0.184376i	0.331754	+	0.943366i	$0.392360\pi$
$354$		0			0
$355$	7.50000	−	12.9904i	0.398059	−	0.689458i
$356$	−13.5000	+	7.79423i	−0.715499	+	0.413093i
$357$		0			0
$358$		−	5.19615i		−	0.274625i
$359$			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
$362$	−33.0000	+	19.0526i	−1.73444	+	1.00138i
$363$		0			0
$364$	1.50000	+	6.06218i	0.0786214	+	0.317744i
$365$	6.00000	+	10.3923i	0.314054	+	0.543958i
$366$		0			0
$367$	−4.00000	−	6.92820i	−0.208798	−	0.361649i	0.742538	−	0.669804i	$-0.233622\pi$
$367$	−4.00000	−	6.92820i	−0.208798	−	0.361649i	−0.951336	+	0.308155i	$0.900289\pi$
$368$	7.50000	+	12.9904i	0.390965	+	0.677170i
$369$		0			0
$370$	−13.5000	+	7.79423i	−0.701832	+	0.405203i
$371$			10.3923i			0.539542i
$372$		0			0
$373$	7.00000	−	12.1244i	0.362446	−	0.627775i	−0.625917	−	0.779890i	$-0.715275\pi$
$373$	7.00000	−	12.1244i	0.362446	−	0.627775i	0.988363	+	0.152115i	$0.0486083\pi$
$374$	18.0000			0.930758
$375$		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	$0.767462\pi$
$380$	1.50000	+	2.59808i	0.0769484	+	0.133278i
$381$		0			0
$382$	−4.50000	+	2.59808i	−0.230240	+	0.132929i
$383$	3.00000	+	1.73205i	0.153293	+	0.0885037i	0.574684	−	0.818375i	$-0.305125\pi$
$383$	3.00000	+	1.73205i	0.153293	+	0.0885037i	−0.421392	+	0.906879i	$0.638458\pi$
$384$		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	−0.825491	−	0.564416i	$-0.809102\pi$
$389$	1.50000	−	2.59808i	0.0760530	−	0.131728i	0.901544	+	0.432688i	$0.142435\pi$
$390$		0			0
$391$	4.50000	+	7.79423i	0.227575	+	0.394171i
$392$			6.92820i			0.349927i
$393$		0			0
$394$	−15.0000			−0.755689
$395$		−	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
$400$	−10.0000			−0.500000
$401$		−	29.4449i		−	1.47041i	−0.677847	−	0.735203i	$-0.737087\pi$
$401$		−	29.4449i		−	1.47041i	0.677847	−	0.735203i	$-0.262913\pi$
$402$		0			0
$403$	30.0000	+	8.66025i	1.49441	+	0.431398i
$404$	−6.00000			−0.298511
$405$		0			0
$406$	−9.00000	+	15.5885i	−0.446663	+	0.773642i
$407$	−18.0000			−0.892227
$408$		0			0
$409$	6.00000	+	3.46410i	0.296681	+	0.171289i	0.640951	−	0.767582i	$-0.278540\pi$
$409$	6.00000	+	3.46410i	0.296681	+	0.171289i	−0.344270	+	0.938871i	$0.611874\pi$
$410$	31.5000	+	18.1865i	1.55567	+	0.898169i
$411$		0			0
$412$	13.0000			0.640464
$413$	−3.00000	+	5.19615i	−0.147620	+	0.255686i
$414$		0			0
$415$	9.00000			0.441793
$416$	−4.50000	−	18.1865i	−0.220631	−	0.891668i
$417$		0			0
$418$			10.3923i			0.508304i
$419$	−9.00000			−0.439679			−0.219839	−	0.975536i	$-0.570553\pi$
$419$	−9.00000			−0.439679			−0.219839	+	0.975536i	$0.570553\pi$
$420$		0			0
$421$	−7.50000	−	4.33013i	−0.365528	−	0.211037i	0.305975	−	0.952039i	$-0.401018\pi$
$421$	−7.50000	−	4.33013i	−0.365528	−	0.211037i	−0.671503	+	0.741002i	$0.734351\pi$
$422$	19.5000	−	11.2583i	0.949245	−	0.548047i
$423$		0			0
$424$		−	10.3923i		−	0.504695i
$425$	−6.00000			−0.291043
$426$		0			0
$427$			8.66025i			0.419099i
$428$	−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
$439$	−4.00000	−	6.92820i	−0.190910	−	0.330665i	0.754642	−	0.656136i	$-0.227810\pi$
$439$	−4.00000	−	6.92820i	−0.190910	−	0.330665i	−0.945552	+	0.325471i	$0.894477\pi$
$440$	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	−0.501129	−	0.865373i	$-0.667082\pi$
$443$	10.5000	−	18.1865i	0.498870	−	0.864068i	0.999999	+	0.00130426i	$0.000415158\pi$
$444$		0			0
$445$	−27.0000			−1.27992
$446$	3.00000	−	5.19615i	0.142054	−	0.246045i
$447$		0			0
$448$			1.73205i			0.0818317i
$449$	−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
$452$	−3.00000	−	5.19615i	−0.141108	−	0.244406i
$453$		0			0
$454$	−10.5000	−	18.1865i	−0.492789	−	0.853536i
$455$	−3.00000	+	10.3923i	−0.140642	+	0.487199i
$456$		0			0
$457$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$457$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$458$	−7.50000	+	12.9904i	−0.350452	+	0.607001i
$459$		0			0
$460$		−	5.19615i		−	0.242272i
$461$			1.73205i			0.0806696i	0.999186	+	0.0403348i	$0.0128425\pi$
$461$			1.73205i			0.0806696i	−0.999186	+	0.0403348i	$0.987158\pi$
$462$		0			0
$463$	22.5000	−	12.9904i	1.04566	−	0.603714i	0.124231	−	0.992253i	$-0.460353\pi$
$463$	22.5000	−	12.9904i	1.04566	−	0.603714i	0.921432	+	0.388539i	$0.127020\pi$
$464$	15.0000	−	25.9808i	0.696358	−	1.20613i
$465$		0			0
$466$	27.0000	−	15.5885i	1.25075	−	0.722121i
$467$	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
$470$	13.5000	−	7.79423i	0.622709	−	0.359521i
$471$		0			0
$472$	3.00000	−	5.19615i	0.138086	−	0.239172i
$473$	−3.00000	+	1.73205i	−0.137940	+	0.0796398i
$474$		0			0
$475$		−	3.46410i		−	0.158944i
$476$		−	5.19615i		−	0.238165i
$477$		0			0
$478$	−4.50000	+	7.79423i	−0.205825	+	0.356500i
$479$	21.0000	−	12.1244i	0.959514	−	0.553976i	0.0634909	−	0.997982i	$-0.479777\pi$
$479$	21.0000	−	12.1244i	0.959514	−	0.553976i	0.896024	+	0.444006i	$0.146443\pi$
$480$		0			0
$481$	4.50000	+	18.1865i	0.205182	+	0.829235i
$482$	4.50000	+	7.79423i	0.204969	+	0.355017i
$483$		0			0
$484$	−0.500000	−	0.866025i	−0.0227273	−	0.0393648i
$485$	−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	$0.396199\pi$
$488$		−	8.66025i		−	0.392031i
$489$		0			0
$490$	6.00000	−	10.3923i	0.271052	−	0.469476i
$491$	3.00000			0.135388			0.0676941	−	0.997706i	$-0.478436\pi$
$491$	3.00000			0.135388			0.0676941	+	0.997706i	$0.478436\pi$
$492$		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
$497$	−7.50000	−	12.9904i	−0.336421	−	0.582698i
$498$		0			0
$499$	22.5000	−	12.9904i	1.00724	−	0.581529i	0.0968564	−	0.995298i	$-0.469121\pi$
$499$	22.5000	−	12.9904i	1.00724	−	0.581529i	0.910382	+	0.413769i	$0.135788\pi$
$500$	10.5000	+	6.06218i	0.469574	+	0.271109i
$501$		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
$509$		−	1.73205i		−	0.0767718i	−0.999263	−	0.0383859i	$-0.987778\pi$
$509$		−	1.73205i		−	0.0767718i	0.999263	−	0.0383859i	$-0.0122216\pi$
$510$		0			0
$511$	12.0000			0.530849
$512$			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
$517$	18.0000			0.791639
$518$			15.5885i			0.684917i
$519$		0			0
$520$	3.00000	−	10.3923i	0.131559	−	0.455733i
$521$	−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	$-0.0174065\pi$
$524$	15.0000			0.655278
$525$		0			0
$526$	36.0000	+	20.7846i	1.56967	+	0.906252i
$527$	−22.5000	−	12.9904i	−0.980115	−	0.565870i
$528$		0			0
$529$	−14.0000			−0.608696
$530$	−9.00000	+	15.5885i	−0.390935	+	0.677119i
$531$		0			0
$532$	3.00000			0.130066
$533$	31.5000	−	30.3109i	1.36442	−	1.31291i
$534$		0			0
$535$		−	25.9808i		−	1.12325i
$536$	−21.0000			−0.907062
$537$		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	$-0.0962751\pi$
$542$	27.0000			1.15975
$543$		0			0
$544$			15.5885i			0.668350i
$545$		0			0
$546$		0			0
$547$	−8.50000	+	14.7224i	−0.363434	+	0.629486i	−0.988524	−	0.151067i	$-0.951729\pi$
$547$	−8.50000	+	14.7224i	−0.363434	+	0.629486i	0.625090	+	0.780553i	$0.285062\pi$
$548$	−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
$553$	−16.5000	−	9.52628i	−0.701651	−	0.405099i
$554$	34.5000	−	19.9186i	1.46576	−	0.846260i
$555$		0			0
$556$	−8.00000	−	13.8564i	−0.339276	−	0.587643i
$557$	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
$562$	−9.00000			−0.379642
$563$	12.0000	−	20.7846i	0.505740	−	0.875967i	−0.494238	−	0.869326i	$-0.664553\pi$
$563$	12.0000	−	20.7846i	0.505740	−	0.875967i	0.999978	−	0.00664037i	$-0.00211371\pi$
$564$		0			0
$565$		−	10.3923i		−	0.437208i
$566$	−34.5000	+	19.9186i	−1.45014	+	0.837241i
$567$		0			0
$568$	7.50000	+	12.9904i	0.314693	+	0.545064i
$569$	−9.00000	−	15.5885i	−0.377300	−	0.653502i	0.613369	−	0.789797i	$-0.289814\pi$
$569$	−9.00000	−	15.5885i	−0.377300	−	0.653502i	−0.990668	+	0.136295i	$0.956481\pi$
$570$		0			0
$571$	2.50000	+	4.33013i	0.104622	+	0.181210i	0.913584	−	0.406651i	$-0.133303\pi$
$571$	2.50000	+	4.33013i	0.104622	+	0.181210i	−0.808962	+	0.587861i	$0.799970\pi$
$572$	−9.00000	+	8.66025i	−0.376309	+	0.362103i
$573$		0			0
$574$	31.5000	−	18.1865i	1.31478	−	0.759091i
$575$	−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	$0.953935\pi$
$578$		−	13.8564i		−	0.576351i
$579$		0			0
$580$	−9.00000	+	5.19615i	−0.373705	+	0.215758i
$581$	4.50000	−	7.79423i	0.186691	−	0.323359i
$582$		0			0
$583$	−18.0000	+	10.3923i	−0.745484	+	0.430405i
$584$	−12.0000			−0.496564
$585$		0			0
$586$	−24.0000			−0.991431
$587$	−9.00000	+	5.19615i	−0.371470	+	0.214468i	−0.674100	−	0.738640i	$-0.735468\pi$
$587$	−9.00000	+	5.19615i	−0.371470	+	0.214468i	0.302631	+	0.953108i	$0.402135\pi$
$588$		0			0
$589$	7.50000	−	12.9904i	0.309032	−	0.535259i
$590$	−9.00000	+	5.19615i	−0.370524	+	0.213922i
$591$		0			0
$592$		−	25.9808i		−	1.06780i
$593$		−	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
$598$	−18.0000	−	5.19615i	−0.736075	−	0.212486i
$599$	13.5000	+	23.3827i	0.551595	+	0.955391i	0.998160	+	0.0606393i	$0.0193139\pi$
$599$	13.5000	+	23.3827i	0.551595	+	0.955391i	−0.446565	+	0.894751i	$0.647353\pi$
$600$		0			0
$601$	−11.0000	−	19.0526i	−0.448699	−	0.777170i	0.549602	−	0.835426i	$-0.314779\pi$
$601$	−11.0000	−	19.0526i	−0.448699	−	0.777170i	−0.998302	+	0.0582563i	$0.981446\pi$
$602$	1.50000	+	2.59808i	0.0611354	+	0.105890i
$603$		0			0
$604$	−10.5000	+	6.06218i	−0.427239	+	0.246667i
$605$		−	1.73205i		−	0.0704179i
$606$		0			0
$607$	−4.00000	+	6.92820i	−0.162355	+	0.281207i	−0.935713	−	0.352763i	$-0.885242\pi$
$607$	−4.00000	+	6.92820i	−0.162355	+	0.281207i	0.773358	+	0.633970i	$0.218576\pi$
$608$	−9.00000			−0.364998
$609$		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	$0.745405\pi$
$614$	21.0000	+	36.3731i	0.847491	+	1.46790i
$615$		0			0
$616$	9.00000	−	5.19615i	0.362620	−	0.209359i
$617$	24.0000	+	13.8564i	0.966204	+	0.557838i	0.898077	−	0.439839i	$-0.144964\pi$
$617$	24.0000	+	13.8564i	0.966204	+	0.557838i	0.0681269	+	0.997677i	$0.478298\pi$
$618$		0			0
$619$	−22.5000	−	12.9904i	−0.904351	−	0.522127i	−0.0257420	−	0.999669i	$-0.508195\pi$
$619$	−22.5000	−	12.9904i	−0.904351	−	0.522127i	−0.878609	+	0.477541i	$0.841528\pi$
$620$	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
$626$			32.9090i			1.31531i
$627$		0			0
$628$	−23.0000			−0.917800
$629$		−	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
$634$	15.0000			0.595726
$635$		−	8.66025i		−	0.343672i
$636$		0			0
$637$	−10.0000	−	10.3923i	−0.396214	−	0.411758i
$638$	−36.0000			−1.42525
$639$		0			0
$640$	−10.5000	+	18.1865i	−0.415049	+	0.718886i
$641$	−39.0000			−1.54041			−0.770204	−	0.637798i	$-0.779845\pi$
$641$	−39.0000			−1.54041			−0.770204	+	0.637798i	$0.779845\pi$
$642$		0			0
$643$	−3.00000	−	1.73205i	−0.118308	−	0.0683054i	0.439678	−	0.898155i	$-0.355093\pi$
$643$	−3.00000	−	1.73205i	−0.118308	−	0.0683054i	−0.557986	+	0.829850i	$0.688426\pi$
$644$	−4.50000	−	2.59808i	−0.177325	−	0.102379i
$645$		0			0
$646$	−9.00000			−0.354100
$647$	−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
$650$	9.00000	−	8.66025i	0.353009	−	0.339683i
$651$		0			0
$652$		−	12.1244i		−	0.474826i
$653$	45.0000			1.76099			0.880493	−	0.474059i	$-0.157212\pi$
$653$	45.0000			1.76099			0.880493	+	0.474059i	$0.157212\pi$
$654$		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
$659$	−39.0000			−1.51922			−0.759612	−	0.650376i	$-0.774611\pi$
$659$	−39.0000			−1.51922			−0.759612	+	0.650376i	$0.774611\pi$
$660$		0			0
$661$		−	46.7654i		−	1.81896i	−0.415745	−	0.909481i	$-0.636479\pi$
$661$		−	46.7654i		−	1.81896i	0.415745	−	0.909481i	$-0.363521\pi$
$662$	−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
$670$	31.5000	+	18.1865i	1.21695	+	0.702607i
$671$	−15.0000	+	8.66025i	−0.579069	+	0.334325i
$672$		0			0
$673$	17.0000	+	29.4449i	0.655302	+	1.13502i	0.981818	+	0.189824i	$0.0607919\pi$
$673$	17.0000	+	29.4449i	0.655302	+	1.13502i	−0.326516	+	0.945192i	$0.605875\pi$
$674$	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	−0.994151	−	0.107997i	$-0.965556\pi$
$677$	−10.5000	+	18.1865i	−0.403548	+	0.698965i	0.590603	+	0.806962i	$0.298890\pi$
$678$		0			0
$679$	−27.0000			−1.03616
$680$	−4.50000	+	7.79423i	−0.172567	+	0.298895i
$681$		0			0
$682$			51.9615i			1.98971i
$683$	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
$686$	−16.5000	−	28.5788i	−0.629973	−	1.09115i
$687$		0			0
$688$	−2.50000	−	4.33013i	−0.0953116	−	0.165085i
$689$	15.0000	+	15.5885i	0.571454	+	0.593873i
$690$		0			0
$691$	−15.0000	+	8.66025i	−0.570627	+	0.329452i	−0.757400	−	0.652952i	$-0.773531\pi$
$691$	−15.0000	+	8.66025i	−0.570627	+	0.329452i	0.186773	+	0.982403i	$0.440197\pi$
$692$	10.5000	−	18.1865i	0.399150	−	0.691348i
$693$		0			0
$694$			20.7846i			0.788973i
$695$		−	27.7128i		−	1.05121i
$696$		0			0
$697$	−31.5000	+	18.1865i	−1.19315	+	0.688864i
$698$	−24.0000	+	41.5692i	−0.908413	+	1.57342i
$699$		0			0
$700$	3.00000	−	1.73205i	0.113389	−	0.0654654i
$701$	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
$704$	−3.00000	+	1.73205i	−0.113067	+	0.0652791i
$705$		0			0
$706$	−6.00000	+	10.3923i	−0.225813	+	0.391120i
$707$	−9.00000	+	5.19615i	−0.338480	+	0.195421i
$708$		0			0
$709$			12.1244i			0.455340i	0.973738	+	0.227670i	$0.0731107\pi$
$709$			12.1244i			0.455340i	−0.973738	+	0.227670i	$0.926889\pi$
$710$		−	25.9808i		−	0.975041i
$711$		0			0
$712$	13.5000	−	23.3827i	0.505934	−	0.876303i
$713$	−22.5000	+	12.9904i	−0.842632	+	0.486494i
$714$		0			0
$715$	−21.0000	+	5.19615i	−0.785355	+	0.194325i
$716$	−1.50000	−	2.59808i	−0.0560576	−	0.0970947i
$717$		0			0
$718$	9.00000	+	15.5885i	0.335877	+	0.581756i
$719$	−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
$722$			27.7128i			1.03136i
$723$		0			0
$724$	−11.0000	+	19.0526i	−0.408812	+	0.708083i
$725$	12.0000			0.445669
$726$		0			0
$727$	−0.500000	+	0.866025i	−0.0185440	+	0.0321191i	−0.875148	−	0.483854i	$-0.839236\pi$
$727$	−0.500000	+	0.866025i	−0.0185440	+	0.0321191i	0.856605	+	0.515974i	$0.172570\pi$
$728$	−7.50000	−	7.79423i	−0.277968	−	0.288873i
$729$		0			0
$730$	18.0000	+	10.3923i	0.666210	+	0.384636i
$731$	−1.50000	−	2.59808i	−0.0554795	−	0.0960933i
$732$		0			0
$733$	16.5000	−	9.52628i	0.609441	−	0.351861i	−0.163305	−	0.986576i	$-0.552216\pi$
$733$	16.5000	−	9.52628i	0.609441	−	0.351861i	0.772747	+	0.634714i	$0.218882\pi$
$734$	−12.0000	−	6.92820i	−0.442928	−	0.255725i
$735$		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
$743$		−	12.1244i		−	0.444799i	−0.974956	−	0.222400i	$-0.928611\pi$
$743$		−	12.1244i		−	0.444799i	0.974956	−	0.222400i	$-0.0713890\pi$
$744$		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
$751$	−19.0000			−0.693320			−0.346660	−	0.937991i	$-0.612684\pi$
$751$	−19.0000			−0.693320			−0.346660	+	0.937991i	$0.612684\pi$
$752$			25.9808i			0.947421i
$753$		0			0
$754$	9.00000	+	36.3731i	0.327761	+	1.32463i
$755$	−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	$0.207269\pi$
$758$	−21.0000			−0.762754
$759$		0			0
$760$	−4.50000	−	2.59808i	−0.163232	−	0.0942421i
$761$	−12.0000	−	6.92820i	−0.435000	−	0.251147i	0.266475	−	0.963842i	$-0.414141\pi$
$761$	−12.0000	−	6.92820i	−0.435000	−	0.251147i	−0.701474	+	0.712695i	$0.747474\pi$
$762$		0			0
$763$		0			0
$764$	−1.50000	+	2.59808i	−0.0542681	+	0.0939951i
$765$		0			0
$766$	6.00000			0.216789
$767$	3.00000	+	12.1244i	0.108324	+	0.437785i
$768$		0			0
$769$			22.5167i			0.811972i	0.913879	+	0.405986i	$0.133072\pi$
$769$			22.5167i			0.811972i	−0.913879	+	0.405986i	$0.866928\pi$
$770$	−18.0000			−0.648675
$771$		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
$776$	27.0000			0.969244
$777$		0			0
$778$		−	5.19615i		−	0.186291i
$779$	−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
$787$	3.00000	+	1.73205i	0.106938	+	0.0617409i	0.552515	−	0.833503i	$-0.313668\pi$
$787$	3.00000	+	1.73205i	0.106938	+	0.0617409i	−0.445577	+	0.895244i	$0.647001\pi$
$788$	−7.50000	+	4.33013i	−0.267176	+	0.154254i
$789$		0			0
$790$	−16.5000	−	28.5788i	−0.587044	−	1.01679i
$791$	−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
$796$	−13.0000			−0.460773
$797$	9.00000	−	15.5885i	0.318796	−	0.552171i	−0.661441	−	0.749997i	$-0.730055\pi$
$797$	9.00000	−	15.5885i	0.318796	−	0.552171i	0.980237	+	0.197826i	$0.0633881\pi$
$798$		0			0
$799$			15.5885i			0.551480i
$800$	−9.00000	+	5.19615i	−0.318198	+	0.183712i
$801$		0			0
$802$	−25.5000	−	44.1673i	−0.900436	−	1.55960i
$803$	12.0000	+	20.7846i	0.423471	+	0.733473i
$804$		0			0
$805$	−4.50000	−	7.79423i	−0.158604	−	0.274710i
$806$	52.5000	−	12.9904i	1.84923	−	0.457567i
$807$		0			0
$808$	9.00000	−	5.19615i	0.316619	−	0.182800i
$809$	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	$0.634299\pi$
$812$			10.3923i			0.364698i
$813$		0			0
$814$	−27.0000	+	15.5885i	−0.946350	+	0.546375i
$815$	10.5000	−	18.1865i	0.367799	−	0.637046i
$816$		0			0
$817$	1.50000	−	0.866025i	0.0524784	−	0.0302984i
$818$	12.0000			0.419570
$819$		0			0
$820$	21.0000			0.733352
$821$	24.0000	−	13.8564i	0.837606	−	0.483592i	−0.0188439	−	0.999822i	$-0.505999\pi$
$821$	24.0000	−	13.8564i	0.837606	−	0.483592i	0.856450	+	0.516231i	$0.172665\pi$
$822$		0			0
$823$	−28.0000	+	48.4974i	−0.976019	+	1.69051i	−0.299487	+	0.954100i	$0.596815\pi$
$823$	−28.0000	+	48.4974i	−0.976019	+	1.69051i	−0.676532	+	0.736413i	$0.736518\pi$
$824$	−19.5000	+	11.2583i	−0.679315	+	0.392203i
$825$		0			0
$826$			10.3923i			0.361595i
$827$		−	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	$-0.00132693\pi$
$830$	13.5000	−	7.79423i	0.468592	−	0.270542i
$831$		0			0
$832$	2.50000	+	2.59808i	0.0866719	+	0.0900721i
$833$	6.00000	+	10.3923i	0.207888	+	0.360072i
$834$		0			0
$835$	−4.50000	−	7.79423i	−0.155729	−	0.269730i
$836$	3.00000	+	5.19615i	0.103757	+	0.179713i
$837$		0			0
$838$	−13.5000	+	7.79423i	−0.466350	+	0.269247i
$839$		−	12.1244i		−	0.418579i	−0.977854	−	0.209290i	$-0.932885\pi$
$839$		−	12.1244i		−	0.418579i	0.977854	−	0.209290i	$-0.0671151\pi$
$840$		0			0
$841$	−3.50000	+	6.06218i	−0.120690	+	0.209041i
$842$	−15.0000			−0.516934
$843$		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
$848$	−15.0000	−	25.9808i	−0.515102	−	0.892183i
$849$		0			0
$850$	−9.00000	+	5.19615i	−0.308697	+	0.178227i
$851$	−13.5000	−	7.79423i	−0.462774	−	0.267183i
$852$		0			0
$853$	4.50000	+	2.59808i	0.154077	+	0.0889564i	0.575056	−	0.818114i	$-0.304980\pi$
$853$	4.50000	+	2.59808i	0.154077	+	0.0889564i	−0.420979	+	0.907070i	$0.638313\pi$
$854$	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	−0.839268	−	0.543718i	$-0.817016\pi$
$857$	1.50000	−	2.59808i	0.0512390	−	0.0887486i	0.890507	+	0.454969i	$0.150350\pi$
$858$		0			0
$859$	2.50000	+	4.33013i	0.0852989	+	0.147742i	0.905519	−	0.424307i	$-0.139482\pi$
$859$	2.50000	+	4.33013i	0.0852989	+	0.147742i	−0.820220	+	0.572049i	$0.806149\pi$
$860$			1.73205i			0.0590624i
$861$		0			0
$862$	−39.0000			−1.32835
$863$			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
$868$	15.0000			0.509133
$869$		−	38.1051i		−	1.29263i
$870$		0			0
$871$	31.5000	−	30.3109i	1.06734	−	1.02705i
$872$		0			0
$873$		0			0
$874$	−4.50000	+	7.79423i	−0.152215	+	0.263644i
$875$	21.0000			0.709930
$876$		0			0
$877$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$877$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$878$	−12.0000	−	6.92820i	−0.404980	−	0.233816i
$879$		0			0
$880$	30.0000			1.01130
$881$	−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	$0.264982\pi$
$884$	−7.50000	−	7.79423i	−0.252252	−	0.262148i
$885$		0			0
$886$		−	36.3731i		−	1.22198i
$887$	15.0000			0.503651			0.251825	−	0.967773i	$-0.418969\pi$
$887$	15.0000			0.503651			0.251825	+	0.967773i	$0.418969\pi$
$888$		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
$893$	−9.00000			−0.301174
$894$		0			0
$895$		−	5.19615i		−	0.173688i
$896$	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
$904$	9.00000	+	5.19615i	0.299336	+	0.172821i
$905$	−33.0000	+	19.0526i	−1.09696	+	0.633328i
$906$		0			0
$907$	−2.00000	−	3.46410i	−0.0664089	−	0.115024i	0.830909	−	0.556408i	$-0.187821\pi$
$907$	−2.00000	−	3.46410i	−0.0664089	−	0.115024i	−0.897318	+	0.441384i	$0.854488\pi$
$908$	−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	−0.204522	−	0.978862i	$-0.565564\pi$
$911$	22.5000	−	38.9711i	0.745458	−	1.29117i	0.949980	−	0.312310i	$-0.101103\pi$
$912$		0			0
$913$	18.0000			0.595713
$914$		0			0
$915$		0			0
$916$			8.66025i			0.286143i
$917$	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	$0.968621\pi$
$920$	4.50000	+	7.79423i	0.148361	+	0.256968i
$921$		0			0
$922$	1.50000	+	2.59808i	0.0493999	+	0.0855631i
$923$	−30.0000	−	8.66025i	−0.987462	−	0.285056i
$924$		0			0
$925$	9.00000	−	5.19615i	0.295918	−	0.170848i
$926$	22.5000	−	38.9711i	0.739396	−	1.28067i
$927$		0			0
$928$		−	31.1769i		−	1.02343i
$929$			57.1577i			1.87528i	0.347604	+	0.937641i	$0.386995\pi$
$929$			57.1577i			1.87528i	−0.347604	+	0.937641i	$0.613005\pi$
$930$		0			0
$931$	−6.00000	+	3.46410i	−0.196642	+	0.113531i
$932$	9.00000	−	15.5885i	0.294805	−	0.510617i
$933$		0			0
$934$	18.0000	−	10.3923i	0.588978	−	0.340047i
$935$	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	$0.510401\pi$
$938$	31.5000	−	18.1865i	1.02851	−	0.593811i
$939$		0			0
$940$	4.50000	−	7.79423i	0.146774	−	0.254220i
$941$	13.5000	−	7.79423i	0.440087	−	0.254085i	−0.263547	−	0.964646i	$-0.584893\pi$
$941$	13.5000	−	7.79423i	0.440087	−	0.254085i	0.703635	+	0.710562i	$0.251559\pi$
$942$		0			0
$943$			36.3731i			1.18447i
$944$		−	17.3205i		−	0.563735i
$945$		0			0
$946$	−3.00000	+	5.19615i	−0.0975384	+	0.168941i
$947$	−33.0000	+	19.0526i	−1.07236	+	0.619125i	−0.928824	−	0.370521i	$-0.879179\pi$
$947$	−33.0000	+	19.0526i	−1.07236	+	0.619125i	−0.143532	+	0.989646i	$0.545846\pi$
$948$		0			0
$949$	18.0000	−	17.3205i	0.584305	−	0.562247i
$950$	−3.00000	−	5.19615i	−0.0973329	−	0.168585i
$951$		0			0
$952$	4.50000	+	7.79423i	0.145846	+	0.252612i
$953$	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
$956$			5.19615i			0.168056i
$957$		0			0
$958$	21.0000	−	36.3731i	0.678479	−	1.17516i
$959$	−9.00000			−0.290625
$960$		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
$965$	−4.50000	−	7.79423i	−0.144860	−	0.250905i
$966$		0			0
$967$	10.5000	−	6.06218i	0.337657	−	0.194946i	−0.321578	−	0.946883i	$-0.604213\pi$
$967$	10.5000	−	6.06218i	0.337657	−	0.194946i	0.659236	+	0.751936i	$0.270880\pi$
$968$	1.50000	+	0.866025i	0.0482118	+	0.0278351i
$969$		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
$977$			25.9808i			0.831198i	0.909548	+	0.415599i	$0.136428\pi$
$977$			25.9808i			0.831198i	−0.909548	+	0.415599i	$0.863572\pi$
$978$		0			0
$979$	−54.0000			−1.72585
$980$		−	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	$-0.306926\pi$
$983$	4.50000	+	2.59808i	0.143528	+	0.0828658i	−0.426517	+	0.904480i	$0.640259\pi$
$984$		0			0
$985$	−15.0000			−0.477940
$986$		−	31.1769i		−	0.992875i
$987$		0			0
$988$	4.50000	−	4.33013i	0.143164	−	0.137760i
$989$	−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	$-0.0332887\pi$
$992$	−45.0000			−1.42875
$993$		0			0
$994$	−22.5000	−	12.9904i	−0.713657	−	0.412030i
$995$	−19.5000	−	11.2583i	−0.618192	−	0.356913i
$996$		0			0
$997$	23.0000			0.728417			0.364209	−	0.931317i	$-0.381339\pi$
$997$	23.0000			0.728417			0.364209	+	0.931317i	$0.381339\pi$
$998$	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.r.a.316.1		2
3.2	odd	2		117.2.r.a.43.1	yes	2
9.4	even	3		351.2.l.a.199.1		2
9.5	odd	6		117.2.l.a.4.1	✓	2
13.10	even	6		351.2.l.a.127.1		2
39.23	odd	6		117.2.l.a.88.1	yes	2
117.23	odd	6		117.2.r.a.49.1	yes	2
117.49	even	6	inner	351.2.r.a.10.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.2.l.a.4.1	✓	2	9.5	odd	6
117.2.l.a.88.1	yes	2	39.23	odd	6
117.2.r.a.43.1	yes	2	3.2	odd	2
117.2.r.a.49.1	yes	2	117.23	odd	6
351.2.l.a.127.1		2	13.10	even	6
351.2.l.a.199.1		2	9.4	even	3
351.2.r.a.10.1		2	117.49	even	6	inner
351.2.r.a.316.1		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(1.50000 - 0.866025i) q^{2} +(0.500000 - 0.866025i) q^{4} +(1.50000 - 0.866025i) q^{5} -1.73205i q^{7} +1.73205i q^{8} +(1.50000 - 2.59808i) q^{10} +(3.00000 - 1.73205i) q^{11} +(-2.50000 - 2.59808i) q^{13} +(-1.50000 - 2.59808i) q^{14} +(2.50000 + 4.33013i) q^{16} +(1.50000 + 2.59808i) q^{17} +(-1.50000 + 0.866025i) q^{19} -1.73205i q^{20} +(3.00000 - 5.19615i) q^{22} +3.00000 q^{23} +(-1.00000 + 1.73205i) q^{25} +(-6.00000 - 1.73205i) q^{26} +(-1.50000 - 0.866025i) q^{28} +(-3.00000 - 5.19615i) q^{29} +(-7.50000 + 4.33013i) q^{31} +(4.50000 + 2.59808i) q^{32} +(4.50000 + 2.59808i) q^{34} +(-1.50000 - 2.59808i) q^{35} +(-4.50000 - 2.59808i) q^{37} +(-1.50000 + 2.59808i) q^{38} +(1.50000 + 2.59808i) q^{40} +12.1244i q^{41} -1.00000 q^{43} -3.46410i q^{44} +(4.50000 - 2.59808i) q^{46} +(4.50000 + 2.59808i) q^{47} +4.00000 q^{49} +3.46410i q^{50} +(-3.50000 + 0.866025i) q^{52} -6.00000 q^{53} +(3.00000 - 5.19615i) q^{55} +3.00000 q^{56} +(-9.00000 - 5.19615i) q^{58} +(-3.00000 - 1.73205i) q^{59} -5.00000 q^{61} +(-7.50000 + 12.9904i) q^{62} -1.00000 q^{64} +(-6.00000 - 1.73205i) q^{65} +12.1244i q^{67} +3.00000 q^{68} +(-4.50000 - 2.59808i) q^{70} +(7.50000 - 4.33013i) q^{71} +6.92820i q^{73} -9.00000 q^{74} +1.73205i q^{76} +(-3.00000 - 5.19615i) q^{77} +(5.50000 - 9.52628i) q^{79} +(7.50000 + 4.33013i) q^{80} +(10.5000 + 18.1865i) q^{82} +(4.50000 + 2.59808i) q^{83} +(4.50000 + 2.59808i) q^{85} +(-1.50000 + 0.866025i) q^{86} +(3.00000 + 5.19615i) q^{88} +(-13.5000 - 7.79423i) q^{89} +(-4.50000 + 4.33013i) q^{91} +(1.50000 - 2.59808i) q^{92} +9.00000 q^{94} +(-1.50000 + 2.59808i) q^{95} -15.5885i q^{97} +(6.00000 - 3.46410i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{2} + q^{4} + 3 q^{5} + 3 q^{10} + 6 q^{11} - 5 q^{13} - 3 q^{14} + 5 q^{16} + 3 q^{17} - 3 q^{19} + 6 q^{22} + 6 q^{23} - 2 q^{25} - 12 q^{26} - 3 q^{28} - 6 q^{29} - 15 q^{31} + 9 q^{32} + 9 q^{34}+ \cdots + 12 q^{98}+O(q^{100}) \) 2 * q + 3 * q^2 + q^4 + 3 * q^5 + 3 * q^10 + 6 * q^11 - 5 * q^13 - 3 * q^14 + 5 * q^16 + 3 * q^17 - 3 * q^19 + 6 * q^22 + 6 * q^23 - 2 * q^25 - 12 * q^26 - 3 * q^28 - 6 * q^29 - 15 * q^31 + 9 * q^32 + 9 * q^34 - 3 * q^35 - 9 * q^37 - 3 * q^38 + 3 * q^40 - 2 * q^43 + 9 * q^46 + 9 * q^47 + 8 * q^49 - 7 * q^52 - 12 * q^53 + 6 * q^55 + 6 * q^56 - 18 * q^58 - 6 * q^59 - 10 * q^61 - 15 * q^62 - 2 * q^64 - 12 * q^65 + 6 * q^68 - 9 * q^70 + 15 * q^71 - 18 * q^74 - 6 * q^77 + 11 * q^79 + 15 * q^80 + 21 * q^82 + 9 * q^83 + 9 * q^85 - 3 * q^86 + 6 * q^88 - 27 * q^89 - 9 * q^91 + 3 * q^92 + 18 * q^94 - 3 * q^95 + 12 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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