LMFDB - Embedded newform 342.2.g.c.163.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [342,2,Mod(163,342)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(342, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

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

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

chi := DirichletCharacter("342.163");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 342 = 2 \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	342.g (of order $3$, degree $2$, 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:	$2.73088374913$
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 114)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			163.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	342.163
Dual form			342.2.g.c.235.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +1.00000 q^{7} +1.00000 q^{8} +2.00000 q^{11} +(1.50000 + 2.59808i) q^{13} +(-0.500000 + 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(2.00000 - 3.46410i) q^{17} +(4.00000 + 1.73205i) q^{19} +(-1.00000 + 1.73205i) q^{22} +(2.00000 + 3.46410i) q^{23} +(2.50000 + 4.33013i) q^{25} -3.00000 q^{26} +(-0.500000 - 0.866025i) q^{28} -3.00000 q^{31} +(-0.500000 - 0.866025i) q^{32} +(2.00000 + 3.46410i) q^{34} -5.00000 q^{37} +(-3.50000 + 2.59808i) q^{38} +(2.00000 - 3.46410i) q^{41} +(4.50000 - 7.79423i) q^{43} +(-1.00000 - 1.73205i) q^{44} -4.00000 q^{46} +(5.00000 + 8.66025i) q^{47} -6.00000 q^{49} -5.00000 q^{50} +(1.50000 - 2.59808i) q^{52} +(-2.00000 - 3.46410i) q^{53} +1.00000 q^{56} +(-7.00000 + 12.1244i) q^{59} +(-5.50000 - 9.52628i) q^{61} +(1.50000 - 2.59808i) q^{62} +1.00000 q^{64} +(-1.50000 - 2.59808i) q^{67} -4.00000 q^{68} +(7.00000 - 12.1244i) q^{71} +(5.50000 - 9.52628i) q^{73} +(2.50000 - 4.33013i) q^{74} +(-0.500000 - 4.33013i) q^{76} +2.00000 q^{77} +(-0.500000 + 0.866025i) q^{79} +(2.00000 + 3.46410i) q^{82} -8.00000 q^{83} +(4.50000 + 7.79423i) q^{86} +2.00000 q^{88} +(-7.00000 - 12.1244i) q^{89} +(1.50000 + 2.59808i) q^{91} +(2.00000 - 3.46410i) q^{92} -10.0000 q^{94} +(-1.00000 + 1.73205i) q^{97} +(3.00000 - 5.19615i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{4} + 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q - q^2 - q^4 + 2 * q^7 + 2 * q^8	$ 2 q - q^{2} - q^{4} + 2 q^{7} + 2 q^{8} + 4 q^{11} + 3 q^{13} - q^{14} - q^{16} + 4 q^{17} + 8 q^{19} - 2 q^{22} + 4 q^{23} + 5 q^{25} - 6 q^{26} - q^{28} - 6 q^{31} - q^{32} + 4 q^{34} - 10 q^{37} - 7 q^{38} + 4 q^{41} + 9 q^{43} - 2 q^{44} - 8 q^{46} + 10 q^{47} - 12 q^{49} - 10 q^{50} + 3 q^{52} - 4 q^{53} + 2 q^{56} - 14 q^{59} - 11 q^{61} + 3 q^{62} + 2 q^{64} - 3 q^{67} - 8 q^{68} + 14 q^{71} + 11 q^{73} + 5 q^{74} - q^{76} + 4 q^{77} - q^{79} + 4 q^{82} - 16 q^{83} + 9 q^{86} + 4 q^{88} - 14 q^{89} + 3 q^{91} + 4 q^{92} - 20 q^{94} - 2 q^{97} + 6 q^{98}+O(q^{100}) $ 2 * q - q^2 - q^4 + 2 * q^7 + 2 * q^8 + 4 * q^11 + 3 * q^13 - q^14 - q^16 + 4 * q^17 + 8 * q^19 - 2 * q^22 + 4 * q^23 + 5 * q^25 - 6 * q^26 - q^28 - 6 * q^31 - q^32 + 4 * q^34 - 10 * q^37 - 7 * q^38 + 4 * q^41 + 9 * q^43 - 2 * q^44 - 8 * q^46 + 10 * q^47 - 12 * q^49 - 10 * q^50 + 3 * q^52 - 4 * q^53 + 2 * q^56 - 14 * q^59 - 11 * q^61 + 3 * q^62 + 2 * q^64 - 3 * q^67 - 8 * q^68 + 14 * q^71 + 11 * q^73 + 5 * q^74 - q^76 + 4 * q^77 - q^79 + 4 * q^82 - 16 * q^83 + 9 * q^86 + 4 * q^88 - 14 * q^89 + 3 * q^91 + 4 * q^92 - 20 * q^94 - 2 * q^97 + 6 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/342\mathbb{Z}\right)^\times$.

$n$	$191$	$325$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$

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 $

$2$	−0.500000	+	0.866025i	−0.353553	+	0.612372i
$2$	−0.500000	+	0.866025i	−0.353553	+	0.612372i
$3$		0			0
$3$		0			0
$4$	−0.500000	−	0.866025i	−0.250000	−	0.433013i
$5$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$5$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$6$		0			0
$7$	1.00000			0.377964			0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000			0.377964			0.188982	+	0.981981i	$0.439481\pi$
$8$	1.00000			0.353553
$9$		0			0
$10$		0			0
$11$	2.00000			0.603023			0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000			0.603023			0.301511	+	0.953463i	$0.402509\pi$
$12$		0			0
$13$	1.50000	+	2.59808i	0.416025	+	0.720577i	0.995535	−	0.0943882i	$-0.0300895\pi$
$13$	1.50000	+	2.59808i	0.416025	+	0.720577i	−0.579510	+	0.814965i	$0.696756\pi$
$14$	−0.500000	+	0.866025i	−0.133631	+	0.231455i
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.125000	+	0.216506i
$17$	2.00000	−	3.46410i	0.485071	−	0.840168i	−0.514782	−	0.857321i	$-0.672127\pi$
$17$	2.00000	−	3.46410i	0.485071	−	0.840168i	0.999853	+	0.0171533i	$0.00546033\pi$
$18$		0			0
$19$	4.00000	+	1.73205i	0.917663	+	0.397360i
$19$	4.00000	+	1.73205i	0.917663	+	0.397360i
$20$		0			0
$21$		0			0
$22$	−1.00000	+	1.73205i	−0.213201	+	0.369274i
$23$	2.00000	+	3.46410i	0.417029	+	0.722315i	0.995639	−	0.0932891i	$-0.0297381\pi$
$23$	2.00000	+	3.46410i	0.417029	+	0.722315i	−0.578610	+	0.815604i	$0.696405\pi$
$24$		0			0
$25$	2.50000	+	4.33013i	0.500000	+	0.866025i
$26$	−3.00000			−0.588348
$27$		0			0
$28$	−0.500000	−	0.866025i	−0.0944911	−	0.163663i
$29$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$29$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$30$		0			0
$31$	−3.00000			−0.538816			−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000			−0.538816			−0.269408	+	0.963026i	$0.586828\pi$
$32$	−0.500000	−	0.866025i	−0.0883883	−	0.153093i
$33$		0			0
$34$	2.00000	+	3.46410i	0.342997	+	0.594089i
$35$		0			0
$36$		0			0
$37$	−5.00000			−0.821995			−0.410997	−	0.911636i	$-0.634819\pi$
$37$	−5.00000			−0.821995			−0.410997	+	0.911636i	$0.634819\pi$
$38$	−3.50000	+	2.59808i	−0.567775	+	0.421464i
$39$		0			0
$40$		0			0
$41$	2.00000	−	3.46410i	0.312348	−	0.541002i	−0.666523	−	0.745485i	$-0.732218\pi$
$41$	2.00000	−	3.46410i	0.312348	−	0.541002i	0.978870	+	0.204483i	$0.0655513\pi$
$42$		0			0
$43$	4.50000	−	7.79423i	0.686244	−	1.18861i	−0.286801	−	0.957990i	$-0.592592\pi$
$43$	4.50000	−	7.79423i	0.686244	−	1.18861i	0.973044	−	0.230618i	$-0.0740749\pi$
$44$	−1.00000	−	1.73205i	−0.150756	−	0.261116i
$45$		0			0
$46$	−4.00000			−0.589768
$47$	5.00000	+	8.66025i	0.729325	+	1.26323i	0.957169	+	0.289530i	$0.0934991\pi$
$47$	5.00000	+	8.66025i	0.729325	+	1.26323i	−0.227844	+	0.973698i	$0.573168\pi$
$48$		0			0
$49$	−6.00000			−0.857143
$50$	−5.00000			−0.707107
$51$		0			0
$52$	1.50000	−	2.59808i	0.208013	−	0.360288i
$53$	−2.00000	−	3.46410i	−0.274721	−	0.475831i	0.695344	−	0.718677i	$-0.255252\pi$
$53$	−2.00000	−	3.46410i	−0.274721	−	0.475831i	−0.970065	+	0.242846i	$0.921919\pi$
$54$		0			0
$55$		0			0
$56$	1.00000			0.133631
$57$		0			0
$58$		0			0
$59$	−7.00000	+	12.1244i	−0.911322	+	1.57846i	−0.0991242	+	0.995075i	$0.531604\pi$
$59$	−7.00000	+	12.1244i	−0.911322	+	1.57846i	−0.812198	+	0.583382i	$0.801729\pi$
$60$		0			0
$61$	−5.50000	−	9.52628i	−0.704203	−	1.21972i	−0.966978	−	0.254858i	$-0.917971\pi$
$61$	−5.50000	−	9.52628i	−0.704203	−	1.21972i	0.262776	−	0.964857i	$-0.415362\pi$
$62$	1.50000	−	2.59808i	0.190500	−	0.329956i
$63$		0			0
$64$	1.00000			0.125000
$65$		0			0
$66$		0			0
$67$	−1.50000	−	2.59808i	−0.183254	−	0.317406i	0.759733	−	0.650236i	$-0.225330\pi$
$67$	−1.50000	−	2.59808i	−0.183254	−	0.317406i	−0.942987	+	0.332830i	$0.891996\pi$
$68$	−4.00000			−0.485071
$69$		0			0
$70$		0			0
$71$	7.00000	−	12.1244i	0.830747	−	1.43890i	−0.0666994	−	0.997773i	$-0.521247\pi$
$71$	7.00000	−	12.1244i	0.830747	−	1.43890i	0.897447	−	0.441123i	$-0.145420\pi$
$72$		0			0
$73$	5.50000	−	9.52628i	0.643726	−	1.11497i	−0.340868	−	0.940111i	$-0.610721\pi$
$73$	5.50000	−	9.52628i	0.643726	−	1.11497i	0.984594	−	0.174855i	$-0.0559458\pi$
$74$	2.50000	−	4.33013i	0.290619	−	0.503367i
$75$		0			0
$76$	−0.500000	−	4.33013i	−0.0573539	−	0.496700i
$77$	2.00000			0.227921
$78$		0			0
$79$	−0.500000	+	0.866025i	−0.0562544	+	0.0974355i	−0.892781	−	0.450490i	$-0.851249\pi$
$79$	−0.500000	+	0.866025i	−0.0562544	+	0.0974355i	0.836527	+	0.547926i	$0.184582\pi$
$80$		0			0
$81$		0			0
$82$	2.00000	+	3.46410i	0.220863	+	0.382546i
$83$	−8.00000			−0.878114			−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000			−0.878114			−0.439057	+	0.898459i	$0.644687\pi$
$84$		0			0
$85$		0			0
$86$	4.50000	+	7.79423i	0.485247	+	0.840473i
$87$		0			0
$88$	2.00000			0.213201
$89$	−7.00000	−	12.1244i	−0.741999	−	1.28518i	−0.951584	−	0.307389i	$-0.900545\pi$
$89$	−7.00000	−	12.1244i	−0.741999	−	1.28518i	0.209585	−	0.977790i	$-0.432789\pi$
$90$		0			0
$91$	1.50000	+	2.59808i	0.157243	+	0.272352i
$92$	2.00000	−	3.46410i	0.208514	−	0.361158i
$93$		0			0
$94$	−10.0000			−1.03142
$95$		0			0
$96$		0			0
$97$	−1.00000	+	1.73205i	−0.101535	+	0.175863i	−0.912317	−	0.409484i	$-0.865709\pi$
$97$	−1.00000	+	1.73205i	−0.101535	+	0.175863i	0.810782	+	0.585348i	$0.199042\pi$
$98$	3.00000	−	5.19615i	0.303046	−	0.524891i
$99$		0			0
$100$	2.50000	−	4.33013i	0.250000	−	0.433013i
$101$	−5.00000	−	8.66025i	−0.497519	−	0.861727i	0.502477	−	0.864590i	$-0.332422\pi$
$101$	−5.00000	−	8.66025i	−0.497519	−	0.861727i	−0.999996	+	0.00286291i	$0.999089\pi$
$102$		0			0
$103$	−3.00000			−0.295599			−0.147799	−	0.989017i	$-0.547219\pi$
$103$	−3.00000			−0.295599			−0.147799	+	0.989017i	$0.547219\pi$
$104$	1.50000	+	2.59808i	0.147087	+	0.254762i
$105$		0			0
$106$	4.00000			0.388514
$107$	10.0000			0.966736			0.483368	−	0.875417i	$-0.339413\pi$
$107$	10.0000			0.966736			0.483368	+	0.875417i	$0.339413\pi$
$108$		0			0
$109$	−7.00000	+	12.1244i	−0.670478	+	1.16130i	0.307290	+	0.951616i	$0.400578\pi$
$109$	−7.00000	+	12.1244i	−0.670478	+	1.16130i	−0.977769	+	0.209687i	$0.932756\pi$
$110$		0			0
$111$		0			0
$112$	−0.500000	+	0.866025i	−0.0472456	+	0.0818317i
$113$	−10.0000			−0.940721			−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000			−0.940721			−0.470360	+	0.882474i	$0.655876\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−7.00000	−	12.1244i	−0.644402	−	1.11614i
$119$	2.00000	−	3.46410i	0.183340	−	0.317554i
$120$		0			0
$121$	−7.00000			−0.636364
$122$	11.0000			0.995893
$123$		0			0
$124$	1.50000	+	2.59808i	0.134704	+	0.233314i
$125$		0			0
$126$		0			0
$127$	4.00000	+	6.92820i	0.354943	+	0.614779i	0.987108	−	0.160055i	$-0.0511671\pi$
$127$	4.00000	+	6.92820i	0.354943	+	0.614779i	−0.632166	+	0.774833i	$0.717834\pi$
$128$	−0.500000	+	0.866025i	−0.0441942	+	0.0765466i
$129$		0			0
$130$		0			0
$131$	−3.00000	+	5.19615i	−0.262111	+	0.453990i	−0.966803	−	0.255524i	$-0.917752\pi$
$131$	−3.00000	+	5.19615i	−0.262111	+	0.453990i	0.704692	+	0.709514i	$0.251085\pi$
$132$		0			0
$133$	4.00000	+	1.73205i	0.346844	+	0.150188i
$134$	3.00000			0.259161
$135$		0			0
$136$	2.00000	−	3.46410i	0.171499	−	0.297044i
$137$	−3.00000	−	5.19615i	−0.256307	−	0.443937i	0.708942	−	0.705266i	$-0.249173\pi$
$137$	−3.00000	−	5.19615i	−0.256307	−	0.443937i	−0.965250	+	0.261329i	$0.915839\pi$
$138$		0			0
$139$	9.50000	+	16.4545i	0.805779	+	1.39565i	0.915764	+	0.401718i	$0.131587\pi$
$139$	9.50000	+	16.4545i	0.805779	+	1.39565i	−0.109984	+	0.993933i	$0.535080\pi$
$140$		0			0
$141$		0			0
$142$	7.00000	+	12.1244i	0.587427	+	1.01745i
$143$	3.00000	+	5.19615i	0.250873	+	0.434524i
$144$		0			0
$145$		0			0
$146$	5.50000	+	9.52628i	0.455183	+	0.788400i
$147$		0			0
$148$	2.50000	+	4.33013i	0.205499	+	0.355934i
$149$	−11.0000	+	19.0526i	−0.901155	+	1.56085i	−0.0751583	+	0.997172i	$0.523946\pi$
$149$	−11.0000	+	19.0526i	−0.901155	+	1.56085i	−0.825997	+	0.563675i	$0.809387\pi$
$150$		0			0
$151$	−20.0000			−1.62758			−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000			−1.62758			−0.813788	+	0.581161i	$0.802599\pi$
$152$	4.00000	+	1.73205i	0.324443	+	0.140488i
$153$		0			0
$154$	−1.00000	+	1.73205i	−0.0805823	+	0.139573i
$155$		0			0
$156$		0			0
$157$	10.5000	−	18.1865i	0.837991	−	1.45144i	−0.0535803	−	0.998564i	$-0.517063\pi$
$157$	10.5000	−	18.1865i	0.837991	−	1.45144i	0.891572	−	0.452880i	$-0.149603\pi$
$158$	−0.500000	−	0.866025i	−0.0397779	−	0.0688973i
$159$		0			0
$160$		0			0
$161$	2.00000	+	3.46410i	0.157622	+	0.273009i
$162$		0			0
$163$	11.0000			0.861586			0.430793	−	0.902451i	$-0.358234\pi$
$163$	11.0000			0.861586			0.430793	+	0.902451i	$0.358234\pi$
$164$	−4.00000			−0.312348
$165$		0			0
$166$	4.00000	−	6.92820i	0.310460	−	0.537733i
$167$	3.00000	+	5.19615i	0.232147	+	0.402090i	0.958440	−	0.285295i	$-0.0920916\pi$
$167$	3.00000	+	5.19615i	0.232147	+	0.402090i	−0.726293	+	0.687386i	$0.758758\pi$
$168$		0			0
$169$	2.00000	−	3.46410i	0.153846	−	0.266469i
$170$		0			0
$171$		0			0
$172$	−9.00000			−0.686244
$173$	−12.0000	+	20.7846i	−0.912343	+	1.58022i	−0.101598	+	0.994826i	$0.532395\pi$
$173$	−12.0000	+	20.7846i	−0.912343	+	1.58022i	−0.810745	+	0.585399i	$0.800938\pi$
$174$		0			0
$175$	2.50000	+	4.33013i	0.188982	+	0.327327i
$176$	−1.00000	+	1.73205i	−0.0753778	+	0.130558i
$177$		0			0
$178$	14.0000			1.04934
$179$	−4.00000			−0.298974			−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000			−0.298974			−0.149487	+	0.988764i	$0.547762\pi$
$180$		0			0
$181$	9.00000	+	15.5885i	0.668965	+	1.15868i	0.978194	+	0.207693i	$0.0665956\pi$
$181$	9.00000	+	15.5885i	0.668965	+	1.15868i	−0.309229	+	0.950988i	$0.600071\pi$
$182$	−3.00000			−0.222375
$183$		0			0
$184$	2.00000	+	3.46410i	0.147442	+	0.255377i
$185$		0			0
$186$		0			0
$187$	4.00000	−	6.92820i	0.292509	−	0.506640i
$188$	5.00000	−	8.66025i	0.364662	−	0.631614i
$189$		0			0
$190$		0			0
$191$	8.00000			0.578860			0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000			0.578860			0.289430	+	0.957199i	$0.406534\pi$
$192$		0			0
$193$	6.50000	−	11.2583i	0.467880	−	0.810392i	−0.531446	−	0.847092i	$-0.678351\pi$
$193$	6.50000	−	11.2583i	0.467880	−	0.810392i	0.999326	+	0.0366998i	$0.0116845\pi$
$194$	−1.00000	−	1.73205i	−0.0717958	−	0.124354i
$195$		0			0
$196$	3.00000	+	5.19615i	0.214286	+	0.371154i
$197$	2.00000			0.142494			0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000			0.142494			0.0712470	+	0.997459i	$0.477302\pi$
$198$		0			0
$199$	−2.50000	−	4.33013i	−0.177220	−	0.306955i	0.763707	−	0.645563i	$-0.223377\pi$
$199$	−2.50000	−	4.33013i	−0.177220	−	0.306955i	−0.940927	+	0.338608i	$0.890044\pi$
$200$	2.50000	+	4.33013i	0.176777	+	0.306186i
$201$		0			0
$202$	10.0000			0.703598
$203$		0			0
$204$		0			0
$205$		0			0
$206$	1.50000	−	2.59808i	0.104510	−	0.181017i
$207$		0			0
$208$	−3.00000			−0.208013
$209$	8.00000	+	3.46410i	0.553372	+	0.239617i
$210$		0			0
$211$	−12.5000	+	21.6506i	−0.860535	+	1.49049i	0.0108774	+	0.999941i	$0.496538\pi$
$211$	−12.5000	+	21.6506i	−0.860535	+	1.49049i	−0.871413	+	0.490550i	$0.836796\pi$
$212$	−2.00000	+	3.46410i	−0.137361	+	0.237915i
$213$		0			0
$214$	−5.00000	+	8.66025i	−0.341793	+	0.592003i
$215$		0			0
$216$		0			0
$217$	−3.00000			−0.203653
$218$	−7.00000	−	12.1244i	−0.474100	−	0.821165i
$219$		0			0
$220$		0			0
$221$	12.0000			0.807207
$222$		0			0
$223$	−4.50000	+	7.79423i	−0.301342	+	0.521940i	−0.976440	−	0.215788i	$-0.930768\pi$
$223$	−4.50000	+	7.79423i	−0.301342	+	0.521940i	0.675098	+	0.737728i	$0.264101\pi$
$224$	−0.500000	−	0.866025i	−0.0334077	−	0.0578638i
$225$		0			0
$226$	5.00000	−	8.66025i	0.332595	−	0.576072i
$227$	−8.00000			−0.530979			−0.265489	−	0.964114i	$-0.585534\pi$
$227$	−8.00000			−0.530979			−0.265489	+	0.964114i	$0.585534\pi$
$228$		0			0
$229$	−11.0000			−0.726900			−0.363450	−	0.931614i	$-0.618401\pi$
$229$	−11.0000			−0.726900			−0.363450	+	0.931614i	$0.618401\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	9.00000	−	15.5885i	0.589610	−	1.02123i	−0.404674	−	0.914461i	$-0.632615\pi$
$233$	9.00000	−	15.5885i	0.589610	−	1.02123i	0.994283	−	0.106773i	$-0.0340517\pi$
$234$		0			0
$235$		0			0
$236$	14.0000			0.911322
$237$		0			0
$238$	2.00000	+	3.46410i	0.129641	+	0.224544i
$239$	12.0000			0.776215			0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000			0.776215			0.388108	+	0.921614i	$0.373129\pi$
$240$		0			0
$241$	−12.5000	−	21.6506i	−0.805196	−	1.39464i	−0.916159	−	0.400815i	$-0.868727\pi$
$241$	−12.5000	−	21.6506i	−0.805196	−	1.39464i	0.110963	−	0.993825i	$-0.464606\pi$
$242$	3.50000	−	6.06218i	0.224989	−	0.389692i
$243$		0			0
$244$	−5.50000	+	9.52628i	−0.352101	+	0.609858i
$245$		0			0
$246$		0			0
$247$	1.50000	+	12.9904i	0.0954427	+	0.826558i
$248$	−3.00000			−0.190500
$249$		0			0
$250$		0			0
$251$	−9.00000	−	15.5885i	−0.568075	−	0.983935i	−0.996756	−	0.0804789i	$-0.974355\pi$
$251$	−9.00000	−	15.5885i	−0.568075	−	0.983935i	0.428681	−	0.903456i	$-0.358978\pi$
$252$		0			0
$253$	4.00000	+	6.92820i	0.251478	+	0.435572i
$254$	−8.00000			−0.501965
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$	−6.00000	−	10.3923i	−0.374270	−	0.648254i	0.615948	−	0.787787i	$-0.288773\pi$
$257$	−6.00000	−	10.3923i	−0.374270	−	0.648254i	−0.990217	+	0.139533i	$0.955440\pi$
$258$		0			0
$259$	−5.00000			−0.310685
$260$		0			0
$261$		0			0
$262$	−3.00000	−	5.19615i	−0.185341	−	0.321019i
$263$	9.00000	−	15.5885i	0.554964	−	0.961225i	−0.442943	−	0.896550i	$-0.646065\pi$
$263$	9.00000	−	15.5885i	0.554964	−	0.961225i	0.997906	−	0.0646755i	$-0.0206012\pi$
$264$		0			0
$265$		0			0
$266$	−3.50000	+	2.59808i	−0.214599	+	0.159298i
$267$		0			0
$268$	−1.50000	+	2.59808i	−0.0916271	+	0.158703i
$269$	1.00000	−	1.73205i	0.0609711	−	0.105605i	−0.833929	−	0.551872i	$-0.813914\pi$
$269$	1.00000	−	1.73205i	0.0609711	−	0.105605i	0.894900	+	0.446267i	$0.147247\pi$
$270$		0			0
$271$	8.00000	−	13.8564i	0.485965	−	0.841717i	−0.513905	−	0.857847i	$-0.671801\pi$
$271$	8.00000	−	13.8564i	0.485965	−	0.841717i	0.999870	+	0.0161307i	$0.00513477\pi$
$272$	2.00000	+	3.46410i	0.121268	+	0.210042i
$273$		0			0
$274$	6.00000			0.362473
$275$	5.00000	+	8.66025i	0.301511	+	0.522233i
$276$		0			0
$277$	−2.00000			−0.120168			−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000			−0.120168			−0.0600842	+	0.998193i	$0.519137\pi$
$278$	−19.0000			−1.13954
$279$		0			0
$280$		0			0
$281$	−4.00000	−	6.92820i	−0.238620	−	0.413302i	0.721699	−	0.692207i	$-0.243362\pi$
$281$	−4.00000	−	6.92820i	−0.238620	−	0.413302i	−0.960319	+	0.278906i	$0.910028\pi$
$282$		0			0
$283$	10.0000	−	17.3205i	0.594438	−	1.02960i	−0.399188	−	0.916869i	$-0.630708\pi$
$283$	10.0000	−	17.3205i	0.594438	−	1.02960i	0.993626	−	0.112728i	$-0.0359589\pi$
$284$	−14.0000			−0.830747
$285$		0			0
$286$	−6.00000			−0.354787
$287$	2.00000	−	3.46410i	0.118056	−	0.204479i
$288$		0			0
$289$	0.500000	+	0.866025i	0.0294118	+	0.0509427i
$290$		0			0
$291$		0			0
$292$	−11.0000			−0.643726
$293$	−14.0000			−0.817889			−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000			−0.817889			−0.408944	+	0.912559i	$0.634103\pi$
$294$		0			0
$295$		0			0
$296$	−5.00000			−0.290619
$297$		0			0
$298$	−11.0000	−	19.0526i	−0.637213	−	1.10369i
$299$	−6.00000	+	10.3923i	−0.346989	+	0.601003i
$300$		0			0
$301$	4.50000	−	7.79423i	0.259376	−	0.449252i
$302$	10.0000	−	17.3205i	0.575435	−	0.996683i
$303$		0			0
$304$	−3.50000	+	2.59808i	−0.200739	+	0.149010i
$305$		0			0
$306$		0			0
$307$	6.00000	−	10.3923i	0.342438	−	0.593120i	−0.642447	−	0.766330i	$-0.722081\pi$
$307$	6.00000	−	10.3923i	0.342438	−	0.593120i	0.984885	+	0.173210i	$0.0554140\pi$
$308$	−1.00000	−	1.73205i	−0.0569803	−	0.0986928i
$309$		0			0
$310$		0			0
$311$	−10.0000			−0.567048			−0.283524	−	0.958965i	$-0.591504\pi$
$311$	−10.0000			−0.567048			−0.283524	+	0.958965i	$0.591504\pi$
$312$		0			0
$313$	−3.00000	−	5.19615i	−0.169570	−	0.293704i	0.768699	−	0.639611i	$-0.220905\pi$
$313$	−3.00000	−	5.19615i	−0.169570	−	0.293704i	−0.938269	+	0.345907i	$0.887571\pi$
$314$	10.5000	+	18.1865i	0.592549	+	1.02633i
$315$		0			0
$316$	1.00000			0.0562544
$317$	6.00000	+	10.3923i	0.336994	+	0.583690i	0.983866	−	0.178908i	$-0.0572566\pi$
$317$	6.00000	+	10.3923i	0.336994	+	0.583690i	−0.646872	+	0.762598i	$0.723923\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$	−4.00000			−0.222911
$323$	14.0000	−	10.3923i	0.778981	−	0.578243i
$324$		0			0
$325$	−7.50000	+	12.9904i	−0.416025	+	0.720577i
$326$	−5.50000	+	9.52628i	−0.304617	+	0.527612i
$327$		0			0
$328$	2.00000	−	3.46410i	0.110432	−	0.191273i
$329$	5.00000	+	8.66025i	0.275659	+	0.477455i
$330$		0			0
$331$	15.0000			0.824475			0.412237	−	0.911077i	$-0.364747\pi$
$331$	15.0000			0.824475			0.412237	+	0.911077i	$0.364747\pi$
$332$	4.00000	+	6.92820i	0.219529	+	0.380235i
$333$		0			0
$334$	−6.00000			−0.328305
$335$		0			0
$336$		0			0
$337$	9.50000	−	16.4545i	0.517498	−	0.896333i	−0.482295	−	0.876009i	$-0.660197\pi$
$337$	9.50000	−	16.4545i	0.517498	−	0.896333i	0.999793	−	0.0203242i	$-0.00646983\pi$
$338$	2.00000	+	3.46410i	0.108786	+	0.188422i
$339$		0			0
$340$		0			0
$341$	−6.00000			−0.324918
$342$		0			0
$343$	−13.0000			−0.701934
$344$	4.50000	−	7.79423i	0.242624	−	0.420237i
$345$		0			0
$346$	−12.0000	−	20.7846i	−0.645124	−	1.11739i
$347$	8.00000	−	13.8564i	0.429463	−	0.743851i	−0.567363	−	0.823468i	$-0.692036\pi$
$347$	8.00000	−	13.8564i	0.429463	−	0.743851i	0.996826	+	0.0796169i	$0.0253697\pi$
$348$		0			0
$349$	−29.0000			−1.55233			−0.776167	−	0.630527i	$-0.782839\pi$
$349$	−29.0000			−1.55233			−0.776167	+	0.630527i	$0.782839\pi$
$350$	−5.00000			−0.267261
$351$		0			0
$352$	−1.00000	−	1.73205i	−0.0533002	−	0.0923186i
$353$	12.0000			0.638696			0.319348	−	0.947638i	$-0.396536\pi$
$353$	12.0000			0.638696			0.319348	+	0.947638i	$0.396536\pi$
$354$		0			0
$355$		0			0
$356$	−7.00000	+	12.1244i	−0.370999	+	0.642590i
$357$		0			0
$358$	2.00000	−	3.46410i	0.105703	−	0.183083i
$359$	−18.0000	+	31.1769i	−0.950004	+	1.64545i	−0.204595	+	0.978847i	$0.565588\pi$
$359$	−18.0000	+	31.1769i	−0.950004	+	1.64545i	−0.745409	+	0.666608i	$0.767746\pi$
$360$		0			0
$361$	13.0000	+	13.8564i	0.684211	+	0.729285i
$362$	−18.0000			−0.946059
$363$		0			0
$364$	1.50000	−	2.59808i	0.0786214	−	0.136176i
$365$		0			0
$366$		0			0
$367$	−11.5000	−	19.9186i	−0.600295	−	1.03974i	−0.992776	−	0.119982i	$-0.961716\pi$
$367$	−11.5000	−	19.9186i	−0.600295	−	1.03974i	0.392481	−	0.919760i	$-0.371617\pi$
$368$	−4.00000			−0.208514
$369$		0			0
$370$		0			0
$371$	−2.00000	−	3.46410i	−0.103835	−	0.179847i
$372$		0			0
$373$	26.0000			1.34623			0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000			1.34623			0.673114	+	0.739538i	$0.264956\pi$
$374$	4.00000	+	6.92820i	0.206835	+	0.358249i
$375$		0			0
$376$	5.00000	+	8.66025i	0.257855	+	0.446619i
$377$		0			0
$378$		0			0
$379$	−13.0000			−0.667765			−0.333883	−	0.942615i	$-0.608359\pi$
$379$	−13.0000			−0.667765			−0.333883	+	0.942615i	$0.608359\pi$
$380$		0			0
$381$		0			0
$382$	−4.00000	+	6.92820i	−0.204658	+	0.354478i
$383$	7.00000	−	12.1244i	0.357683	−	0.619526i	−0.629890	−	0.776684i	$-0.716900\pi$
$383$	7.00000	−	12.1244i	0.357683	−	0.619526i	0.987573	+	0.157159i	$0.0502334\pi$
$384$		0			0
$385$		0			0
$386$	6.50000	+	11.2583i	0.330841	+	0.573034i
$387$		0			0
$388$	2.00000			0.101535
$389$	−8.00000	−	13.8564i	−0.405616	−	0.702548i	0.588777	−	0.808296i	$-0.299610\pi$
$389$	−8.00000	−	13.8564i	−0.405616	−	0.702548i	−0.994393	+	0.105748i	$0.966276\pi$
$390$		0			0
$391$	16.0000			0.809155
$392$	−6.00000			−0.303046
$393$		0			0
$394$	−1.00000	+	1.73205i	−0.0503793	+	0.0872595i
$395$		0			0
$396$		0			0
$397$	−7.50000	+	12.9904i	−0.376414	+	0.651969i	−0.990538	−	0.137241i	$-0.956176\pi$
$397$	−7.50000	+	12.9904i	−0.376414	+	0.651969i	0.614123	+	0.789210i	$0.289510\pi$
$398$	5.00000			0.250627
$399$		0			0
$400$	−5.00000			−0.250000
$401$	9.00000	−	15.5885i	0.449439	−	0.778450i	−0.548911	−	0.835881i	$-0.684957\pi$
$401$	9.00000	−	15.5885i	0.449439	−	0.778450i	0.998350	+	0.0574304i	$0.0182907\pi$
$402$		0			0
$403$	−4.50000	−	7.79423i	−0.224161	−	0.388258i
$404$	−5.00000	+	8.66025i	−0.248759	+	0.430864i
$405$		0			0
$406$		0			0
$407$	−10.0000			−0.495682
$408$		0			0
$409$	15.0000	+	25.9808i	0.741702	+	1.28467i	0.951720	+	0.306968i	$0.0993146\pi$
$409$	15.0000	+	25.9808i	0.741702	+	1.28467i	−0.210017	+	0.977698i	$0.567352\pi$
$410$		0			0
$411$		0			0
$412$	1.50000	+	2.59808i	0.0738997	+	0.127998i
$413$	−7.00000	+	12.1244i	−0.344447	+	0.596601i
$414$		0			0
$415$		0			0
$416$	1.50000	−	2.59808i	0.0735436	−	0.127381i
$417$		0			0
$418$	−7.00000	+	5.19615i	−0.342381	+	0.254152i
$419$	−18.0000			−0.879358			−0.439679	−	0.898155i	$-0.644908\pi$
$419$	−18.0000			−0.879358			−0.439679	+	0.898155i	$0.644908\pi$
$420$		0			0
$421$	−5.00000	+	8.66025i	−0.243685	+	0.422075i	−0.961761	−	0.273890i	$-0.911690\pi$
$421$	−5.00000	+	8.66025i	−0.243685	+	0.422075i	0.718076	+	0.695965i	$0.245023\pi$
$422$	−12.5000	−	21.6506i	−0.608490	−	1.05394i
$423$		0			0
$424$	−2.00000	−	3.46410i	−0.0971286	−	0.168232i
$425$	20.0000			0.970143
$426$		0			0
$427$	−5.50000	−	9.52628i	−0.266164	−	0.461009i
$428$	−5.00000	−	8.66025i	−0.241684	−	0.418609i
$429$		0			0
$430$		0			0
$431$	−5.00000	−	8.66025i	−0.240842	−	0.417150i	0.720113	−	0.693857i	$-0.244090\pi$
$431$	−5.00000	−	8.66025i	−0.240842	−	0.417150i	−0.960954	+	0.276707i	$0.910757\pi$
$432$		0			0
$433$	4.50000	+	7.79423i	0.216256	+	0.374567i	0.953660	−	0.300885i	$-0.0972820\pi$
$433$	4.50000	+	7.79423i	0.216256	+	0.374567i	−0.737404	+	0.675452i	$0.763949\pi$
$434$	1.50000	−	2.59808i	0.0720023	−	0.124712i
$435$		0			0
$436$	14.0000			0.670478
$437$	2.00000	+	17.3205i	0.0956730	+	0.828552i
$438$		0			0
$439$	−9.50000	+	16.4545i	−0.453410	+	0.785330i	−0.998595	−	0.0529862i	$-0.983126\pi$
$439$	−9.50000	+	16.4545i	−0.453410	+	0.785330i	0.545185	+	0.838316i	$0.316459\pi$
$440$		0			0
$441$		0			0
$442$	−6.00000	+	10.3923i	−0.285391	+	0.494312i
$443$	12.0000	+	20.7846i	0.570137	+	0.987507i	0.996551	+	0.0829786i	$0.0264433\pi$
$443$	12.0000	+	20.7846i	0.570137	+	0.987507i	−0.426414	+	0.904528i	$0.640223\pi$
$444$		0			0
$445$		0			0
$446$	−4.50000	−	7.79423i	−0.213081	−	0.369067i
$447$		0			0
$448$	1.00000			0.0472456
$449$	36.0000			1.69895			0.849473	−	0.527633i	$-0.176920\pi$
$449$	36.0000			1.69895			0.849473	+	0.527633i	$0.176920\pi$
$450$		0			0
$451$	4.00000	−	6.92820i	0.188353	−	0.326236i
$452$	5.00000	+	8.66025i	0.235180	+	0.407344i
$453$		0			0
$454$	4.00000	−	6.92820i	0.187729	−	0.325157i
$455$		0			0
$456$		0			0
$457$	5.00000			0.233890			0.116945	−	0.993138i	$-0.462690\pi$
$457$	5.00000			0.233890			0.116945	+	0.993138i	$0.462690\pi$
$458$	5.50000	−	9.52628i	0.256998	−	0.445134i
$459$		0			0
$460$		0			0
$461$	3.00000	−	5.19615i	0.139724	−	0.242009i	−0.787668	−	0.616100i	$-0.788712\pi$
$461$	3.00000	−	5.19615i	0.139724	−	0.242009i	0.927392	+	0.374091i	$0.122045\pi$
$462$		0			0
$463$	−9.00000			−0.418265			−0.209133	−	0.977887i	$-0.567064\pi$
$463$	−9.00000			−0.418265			−0.209133	+	0.977887i	$0.567064\pi$
$464$		0			0
$465$		0			0
$466$	9.00000	+	15.5885i	0.416917	+	0.722121i
$467$	−8.00000			−0.370196			−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000			−0.370196			−0.185098	+	0.982720i	$0.559260\pi$
$468$		0			0
$469$	−1.50000	−	2.59808i	−0.0692636	−	0.119968i
$470$		0			0
$471$		0			0
$472$	−7.00000	+	12.1244i	−0.322201	+	0.558069i
$473$	9.00000	−	15.5885i	0.413820	−	0.716758i
$474$		0			0
$475$	2.50000	+	21.6506i	0.114708	+	0.993399i
$476$	−4.00000			−0.183340
$477$		0			0
$478$	−6.00000	+	10.3923i	−0.274434	+	0.475333i
$479$	3.00000	+	5.19615i	0.137073	+	0.237418i	0.926388	−	0.376571i	$-0.122897\pi$
$479$	3.00000	+	5.19615i	0.137073	+	0.237418i	−0.789314	+	0.613990i	$0.789564\pi$
$480$		0			0
$481$	−7.50000	−	12.9904i	−0.341971	−	0.592310i
$482$	25.0000			1.13872
$483$		0			0
$484$	3.50000	+	6.06218i	0.159091	+	0.275554i
$485$		0			0
$486$		0			0
$487$	32.0000			1.45006			0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000			1.45006			0.725029	+	0.688718i	$0.241826\pi$
$488$	−5.50000	−	9.52628i	−0.248973	−	0.431234i
$489$		0			0
$490$		0			0
$491$	15.0000	−	25.9808i	0.676941	−	1.17250i	−0.298957	−	0.954267i	$-0.596639\pi$
$491$	15.0000	−	25.9808i	0.676941	−	1.17250i	0.975898	−	0.218229i	$-0.0700279\pi$
$492$		0			0
$493$		0			0
$494$	−12.0000	−	5.19615i	−0.539906	−	0.233786i
$495$		0			0
$496$	1.50000	−	2.59808i	0.0673520	−	0.116657i
$497$	7.00000	−	12.1244i	0.313993	−	0.543852i
$498$		0			0
$499$	−2.50000	+	4.33013i	−0.111915	+	0.193843i	−0.916542	−	0.399937i	$-0.869032\pi$
$499$	−2.50000	+	4.33013i	−0.111915	+	0.193843i	0.804627	+	0.593780i	$0.202365\pi$
$500$		0			0
$501$		0			0
$502$	18.0000			0.803379
$503$	8.00000	+	13.8564i	0.356702	+	0.617827i	0.987408	−	0.158196i	$-0.0505677\pi$
$503$	8.00000	+	13.8564i	0.356702	+	0.617827i	−0.630705	+	0.776022i	$0.717234\pi$
$504$		0			0
$505$		0			0
$506$	−8.00000			−0.355643
$507$		0			0
$508$	4.00000	−	6.92820i	0.177471	−	0.307389i
$509$	13.0000	+	22.5167i	0.576215	+	0.998033i	0.995908	+	0.0903676i	$0.0288042\pi$
$509$	13.0000	+	22.5167i	0.576215	+	0.998033i	−0.419694	+	0.907666i	$0.637862\pi$
$510$		0			0
$511$	5.50000	−	9.52628i	0.243306	−	0.421418i
$512$	1.00000			0.0441942
$513$		0			0
$514$	12.0000			0.529297
$515$		0			0
$516$		0			0
$517$	10.0000	+	17.3205i	0.439799	+	0.761755i
$518$	2.50000	−	4.33013i	0.109844	−	0.190255i
$519$		0			0
$520$		0			0
$521$	−34.0000			−1.48957			−0.744784	−	0.667306i	$-0.767447\pi$
$521$	−34.0000			−1.48957			−0.744784	+	0.667306i	$0.767447\pi$
$522$		0			0
$523$	−18.5000	−	32.0429i	−0.808949	−	1.40114i	−0.913593	−	0.406630i	$-0.866704\pi$
$523$	−18.5000	−	32.0429i	−0.808949	−	1.40114i	0.104644	−	0.994510i	$-0.466630\pi$
$524$	6.00000			0.262111
$525$		0			0
$526$	9.00000	+	15.5885i	0.392419	+	0.679689i
$527$	−6.00000	+	10.3923i	−0.261364	+	0.452696i
$528$		0			0
$529$	3.50000	−	6.06218i	0.152174	−	0.263573i
$530$		0			0
$531$		0			0
$532$	−0.500000	−	4.33013i	−0.0216777	−	0.187735i
$533$	12.0000			0.519778
$534$		0			0
$535$		0			0
$536$	−1.50000	−	2.59808i	−0.0647901	−	0.112220i
$537$		0			0
$538$	1.00000	+	1.73205i	0.0431131	+	0.0746740i
$539$	−12.0000			−0.516877
$540$		0			0
$541$	15.5000	+	26.8468i	0.666397	+	1.15423i	0.978905	+	0.204318i	$0.0654977\pi$
$541$	15.5000	+	26.8468i	0.666397	+	1.15423i	−0.312507	+	0.949915i	$0.601169\pi$
$542$	8.00000	+	13.8564i	0.343629	+	0.595184i
$543$		0			0
$544$	−4.00000			−0.171499
$545$		0			0
$546$		0			0
$547$	−5.50000	−	9.52628i	−0.235163	−	0.407314i	0.724157	−	0.689635i	$-0.242229\pi$
$547$	−5.50000	−	9.52628i	−0.235163	−	0.407314i	−0.959320	+	0.282321i	$0.908896\pi$
$548$	−3.00000	+	5.19615i	−0.128154	+	0.221969i
$549$		0			0
$550$	−10.0000			−0.426401
$551$		0			0
$552$		0			0
$553$	−0.500000	+	0.866025i	−0.0212622	+	0.0368271i
$554$	1.00000	−	1.73205i	0.0424859	−	0.0735878i
$555$		0			0
$556$	9.50000	−	16.4545i	0.402890	−	0.697826i
$557$	17.0000	+	29.4449i	0.720313	+	1.24762i	0.960874	+	0.276985i	$0.0893352\pi$
$557$	17.0000	+	29.4449i	0.720313	+	1.24762i	−0.240561	+	0.970634i	$0.577331\pi$
$558$		0			0
$559$	27.0000			1.14198
$560$		0			0
$561$		0			0
$562$	8.00000			0.337460
$563$	6.00000			0.252870			0.126435	−	0.991975i	$-0.459647\pi$
$563$	6.00000			0.252870			0.126435	+	0.991975i	$0.459647\pi$
$564$		0			0
$565$		0			0
$566$	10.0000	+	17.3205i	0.420331	+	0.728035i
$567$		0			0
$568$	7.00000	−	12.1244i	0.293713	−	0.508727i
$569$	−24.0000			−1.00613			−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000			−1.00613			−0.503066	+	0.864248i	$0.667795\pi$
$570$		0			0
$571$	−7.00000			−0.292941			−0.146470	−	0.989215i	$-0.546791\pi$
$571$	−7.00000			−0.292941			−0.146470	+	0.989215i	$0.546791\pi$
$572$	3.00000	−	5.19615i	0.125436	−	0.217262i
$573$		0			0
$574$	2.00000	+	3.46410i	0.0834784	+	0.144589i
$575$	−10.0000	+	17.3205i	−0.417029	+	0.722315i
$576$		0			0
$577$	−14.0000			−0.582828			−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000			−0.582828			−0.291414	+	0.956597i	$0.594126\pi$
$578$	−1.00000			−0.0415945
$579$		0			0
$580$		0			0
$581$	−8.00000			−0.331896
$582$		0			0
$583$	−4.00000	−	6.92820i	−0.165663	−	0.286937i
$584$	5.50000	−	9.52628i	0.227592	−	0.394200i
$585$		0			0
$586$	7.00000	−	12.1244i	0.289167	−	0.500853i
$587$	−1.00000	+	1.73205i	−0.0412744	+	0.0714894i	−0.885925	−	0.463829i	$-0.846475\pi$
$587$	−1.00000	+	1.73205i	−0.0412744	+	0.0714894i	0.844650	+	0.535319i	$0.179808\pi$
$588$		0			0
$589$	−12.0000	−	5.19615i	−0.494451	−	0.214104i
$590$		0			0
$591$		0			0
$592$	2.50000	−	4.33013i	0.102749	−	0.177967i
$593$	17.0000	+	29.4449i	0.698106	+	1.20916i	0.969122	+	0.246581i	$0.0793071\pi$
$593$	17.0000	+	29.4449i	0.698106	+	1.20916i	−0.271016	+	0.962575i	$0.587360\pi$
$594$		0			0
$595$		0			0
$596$	22.0000			0.901155
$597$		0			0
$598$	−6.00000	−	10.3923i	−0.245358	−	0.424973i
$599$	−23.0000	−	39.8372i	−0.939755	−	1.62770i	−0.765928	−	0.642926i	$-0.777720\pi$
$599$	−23.0000	−	39.8372i	−0.939755	−	1.62770i	−0.173826	−	0.984776i	$-0.555613\pi$
$600$		0			0
$601$	27.0000			1.10135			0.550676	−	0.834719i	$-0.314370\pi$
$601$	27.0000			1.10135			0.550676	+	0.834719i	$0.314370\pi$
$602$	4.50000	+	7.79423i	0.183406	+	0.317669i
$603$		0			0
$604$	10.0000	+	17.3205i	0.406894	+	0.704761i
$605$		0			0
$606$		0			0
$607$	5.00000			0.202944			0.101472	−	0.994838i	$-0.467645\pi$
$607$	5.00000			0.202944			0.101472	+	0.994838i	$0.467645\pi$
$608$	−0.500000	−	4.33013i	−0.0202777	−	0.175610i
$609$		0			0
$610$		0			0
$611$	−15.0000	+	25.9808i	−0.606835	+	1.05107i
$612$		0			0
$613$	−1.00000	+	1.73205i	−0.0403896	+	0.0699569i	−0.885514	−	0.464614i	$-0.846193\pi$
$613$	−1.00000	+	1.73205i	−0.0403896	+	0.0699569i	0.845124	+	0.534570i	$0.179527\pi$
$614$	6.00000	+	10.3923i	0.242140	+	0.419399i
$615$		0			0
$616$	2.00000			0.0805823
$617$	−21.0000	−	36.3731i	−0.845428	−	1.46432i	−0.885249	−	0.465118i	$-0.846012\pi$
$617$	−21.0000	−	36.3731i	−0.845428	−	1.46432i	0.0398207	−	0.999207i	$-0.487321\pi$
$618$		0			0
$619$	1.00000			0.0401934			0.0200967	−	0.999798i	$-0.493603\pi$
$619$	1.00000			0.0401934			0.0200967	+	0.999798i	$0.493603\pi$
$620$		0			0
$621$		0			0
$622$	5.00000	−	8.66025i	0.200482	−	0.347245i
$623$	−7.00000	−	12.1244i	−0.280449	−	0.485752i
$624$		0			0
$625$	−12.5000	+	21.6506i	−0.500000	+	0.866025i
$626$	6.00000			0.239808
$627$		0			0
$628$	−21.0000			−0.837991
$629$	−10.0000	+	17.3205i	−0.398726	+	0.690614i
$630$		0			0
$631$	−8.50000	−	14.7224i	−0.338380	−	0.586091i	0.645748	−	0.763550i	$-0.276545\pi$
$631$	−8.50000	−	14.7224i	−0.338380	−	0.586091i	−0.984128	+	0.177459i	$0.943212\pi$
$632$	−0.500000	+	0.866025i	−0.0198889	+	0.0344486i
$633$		0			0
$634$	−12.0000			−0.476581
$635$		0			0
$636$		0			0
$637$	−9.00000	−	15.5885i	−0.356593	−	0.617637i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−7.00000	+	12.1244i	−0.276483	+	0.478883i	−0.970508	−	0.241068i	$-0.922502\pi$
$641$	−7.00000	+	12.1244i	−0.276483	+	0.478883i	0.694025	+	0.719951i	$0.255836\pi$
$642$		0			0
$643$	−18.5000	+	32.0429i	−0.729569	+	1.26365i	0.227497	+	0.973779i	$0.426946\pi$
$643$	−18.5000	+	32.0429i	−0.729569	+	1.26365i	−0.957066	+	0.289871i	$0.906387\pi$
$644$	2.00000	−	3.46410i	0.0788110	−	0.136505i
$645$		0			0
$646$	2.00000	+	17.3205i	0.0786889	+	0.681466i
$647$	−48.0000			−1.88707			−0.943537	−	0.331266i	$-0.892524\pi$
$647$	−48.0000			−1.88707			−0.943537	+	0.331266i	$0.892524\pi$
$648$		0			0
$649$	−14.0000	+	24.2487i	−0.549548	+	0.951845i
$650$	−7.50000	−	12.9904i	−0.294174	−	0.509525i
$651$		0			0
$652$	−5.50000	−	9.52628i	−0.215397	−	0.373078i
$653$	42.0000			1.64359			0.821794	−	0.569785i	$-0.192974\pi$
$653$	42.0000			1.64359			0.821794	+	0.569785i	$0.192974\pi$
$654$		0			0
$655$		0			0
$656$	2.00000	+	3.46410i	0.0780869	+	0.135250i
$657$		0			0
$658$	−10.0000			−0.389841
$659$	−7.00000	−	12.1244i	−0.272681	−	0.472298i	0.696866	−	0.717201i	$-0.254577\pi$
$659$	−7.00000	−	12.1244i	−0.272681	−	0.472298i	−0.969548	+	0.244903i	$0.921244\pi$
$660$		0			0
$661$	−5.00000	−	8.66025i	−0.194477	−	0.336845i	0.752252	−	0.658876i	$-0.228968\pi$
$661$	−5.00000	−	8.66025i	−0.194477	−	0.336845i	−0.946729	+	0.322031i	$0.895634\pi$
$662$	−7.50000	+	12.9904i	−0.291496	+	0.504885i
$663$		0			0
$664$	−8.00000			−0.310460
$665$		0			0
$666$		0			0
$667$		0			0
$668$	3.00000	−	5.19615i	0.116073	−	0.201045i
$669$		0			0
$670$		0			0
$671$	−11.0000	−	19.0526i	−0.424650	−	0.735516i
$672$		0			0
$673$	−1.00000			−0.0385472			−0.0192736	−	0.999814i	$-0.506135\pi$
$673$	−1.00000			−0.0385472			−0.0192736	+	0.999814i	$0.506135\pi$
$674$	9.50000	+	16.4545i	0.365926	+	0.633803i
$675$		0			0
$676$	−4.00000			−0.153846
$677$	−18.0000			−0.691796			−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000			−0.691796			−0.345898	+	0.938272i	$0.612426\pi$
$678$		0			0
$679$	−1.00000	+	1.73205i	−0.0383765	+	0.0664700i
$680$		0			0
$681$		0			0
$682$	3.00000	−	5.19615i	0.114876	−	0.198971i
$683$	36.0000			1.37750			0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000			1.37750			0.688751	+	0.724998i	$0.258159\pi$
$684$		0			0
$685$		0			0
$686$	6.50000	−	11.2583i	0.248171	−	0.429845i
$687$		0			0
$688$	4.50000	+	7.79423i	0.171561	+	0.297152i
$689$	6.00000	−	10.3923i	0.228582	−	0.395915i
$690$		0			0
$691$	4.00000			0.152167			0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000			0.152167			0.0760836	+	0.997101i	$0.475758\pi$
$692$	24.0000			0.912343
$693$		0			0
$694$	8.00000	+	13.8564i	0.303676	+	0.525982i
$695$		0			0
$696$		0			0
$697$	−8.00000	−	13.8564i	−0.303022	−	0.524849i
$698$	14.5000	−	25.1147i	0.548833	−	0.950607i
$699$		0			0
$700$	2.50000	−	4.33013i	0.0944911	−	0.163663i
$701$	−22.0000	+	38.1051i	−0.830929	+	1.43921i	0.0663742	+	0.997795i	$0.478857\pi$
$701$	−22.0000	+	38.1051i	−0.830929	+	1.43921i	−0.897303	+	0.441416i	$0.854476\pi$
$702$		0			0
$703$	−20.0000	−	8.66025i	−0.754314	−	0.326628i
$704$	2.00000			0.0753778
$705$		0			0
$706$	−6.00000	+	10.3923i	−0.225813	+	0.391120i
$707$	−5.00000	−	8.66025i	−0.188044	−	0.325702i
$708$		0			0
$709$	15.5000	+	26.8468i	0.582115	+	1.00825i	0.995228	+	0.0975728i	$0.0311079\pi$
$709$	15.5000	+	26.8468i	0.582115	+	1.00825i	−0.413114	+	0.910679i	$0.635559\pi$
$710$		0			0
$711$		0			0
$712$	−7.00000	−	12.1244i	−0.262336	−	0.454379i
$713$	−6.00000	−	10.3923i	−0.224702	−	0.389195i
$714$		0			0
$715$		0			0
$716$	2.00000	+	3.46410i	0.0747435	+	0.129460i
$717$		0			0
$718$	−18.0000	−	31.1769i	−0.671754	−	1.16351i
$719$	20.0000	−	34.6410i	0.745874	−	1.29189i	−0.203911	−	0.978989i	$-0.565365\pi$
$719$	20.0000	−	34.6410i	0.745874	−	1.29189i	0.949785	−	0.312903i	$-0.101301\pi$
$720$		0			0
$721$	−3.00000			−0.111726
$722$	−18.5000	+	4.33013i	−0.688499	+	0.161151i
$723$		0			0
$724$	9.00000	−	15.5885i	0.334482	−	0.579340i
$725$		0			0
$726$		0			0
$727$	−8.50000	+	14.7224i	−0.315248	+	0.546025i	−0.979490	−	0.201492i	$-0.935421\pi$
$727$	−8.50000	+	14.7224i	−0.315248	+	0.546025i	0.664243	+	0.747517i	$0.268754\pi$
$728$	1.50000	+	2.59808i	0.0555937	+	0.0962911i
$729$		0			0
$730$		0			0
$731$	−18.0000	−	31.1769i	−0.665754	−	1.15312i
$732$		0			0
$733$	−2.00000			−0.0738717			−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000			−0.0738717			−0.0369358	+	0.999318i	$0.511760\pi$
$734$	23.0000			0.848945
$735$		0			0
$736$	2.00000	−	3.46410i	0.0737210	−	0.127688i
$737$	−3.00000	−	5.19615i	−0.110506	−	0.191403i
$738$		0			0
$739$	2.50000	−	4.33013i	0.0919640	−	0.159286i	−0.816373	−	0.577524i	$-0.804019\pi$
$739$	2.50000	−	4.33013i	0.0919640	−	0.159286i	0.908337	+	0.418238i	$0.137352\pi$
$740$		0			0
$741$		0			0
$742$	4.00000			0.146845
$743$	−9.00000	+	15.5885i	−0.330178	+	0.571885i	−0.982547	−	0.186017i	$-0.940442\pi$
$743$	−9.00000	+	15.5885i	−0.330178	+	0.571885i	0.652369	+	0.757902i	$0.273775\pi$
$744$		0			0
$745$		0			0
$746$	−13.0000	+	22.5167i	−0.475964	+	0.824394i
$747$		0			0
$748$	−8.00000			−0.292509
$749$	10.0000			0.365392
$750$		0			0
$751$	−6.50000	−	11.2583i	−0.237188	−	0.410822i	0.722718	−	0.691143i	$-0.242893\pi$
$751$	−6.50000	−	11.2583i	−0.237188	−	0.410822i	−0.959906	+	0.280321i	$0.909559\pi$
$752$	−10.0000			−0.364662
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−2.50000	+	4.33013i	−0.0908640	+	0.157381i	−0.907875	−	0.419241i	$-0.862296\pi$
$757$	−2.50000	+	4.33013i	−0.0908640	+	0.157381i	0.817011	+	0.576622i	$0.195630\pi$
$758$	6.50000	−	11.2583i	0.236091	−	0.408921i
$759$		0			0
$760$		0			0
$761$	34.0000			1.23250			0.616250	−	0.787551i	$-0.288651\pi$
$761$	34.0000			1.23250			0.616250	+	0.787551i	$0.288651\pi$
$762$		0			0
$763$	−7.00000	+	12.1244i	−0.253417	+	0.438931i
$764$	−4.00000	−	6.92820i	−0.144715	−	0.250654i
$765$		0			0
$766$	7.00000	+	12.1244i	0.252920	+	0.438071i
$767$	−42.0000			−1.51653
$768$		0			0
$769$	−13.5000	−	23.3827i	−0.486822	−	0.843201i	0.513063	−	0.858351i	$-0.328511\pi$
$769$	−13.5000	−	23.3827i	−0.486822	−	0.843201i	−0.999885	+	0.0151499i	$0.995177\pi$
$770$		0			0
$771$		0			0
$772$	−13.0000			−0.467880
$773$	−9.00000	−	15.5885i	−0.323708	−	0.560678i	0.657542	−	0.753418i	$-0.271596\pi$
$773$	−9.00000	−	15.5885i	−0.323708	−	0.560678i	−0.981250	+	0.192740i	$0.938263\pi$
$774$		0			0
$775$	−7.50000	−	12.9904i	−0.269408	−	0.466628i
$776$	−1.00000	+	1.73205i	−0.0358979	+	0.0621770i
$777$		0			0
$778$	16.0000			0.573628
$779$	14.0000	−	10.3923i	0.501602	−	0.372343i
$780$		0			0
$781$	14.0000	−	24.2487i	0.500959	−	0.867687i
$782$	−8.00000	+	13.8564i	−0.286079	+	0.495504i
$783$		0			0
$784$	3.00000	−	5.19615i	0.107143	−	0.185577i
$785$		0			0
$786$		0			0
$787$	−47.0000			−1.67537			−0.837685	−	0.546154i	$-0.816091\pi$
$787$	−47.0000			−1.67537			−0.837685	+	0.546154i	$0.816091\pi$
$788$	−1.00000	−	1.73205i	−0.0356235	−	0.0617018i
$789$		0			0
$790$		0			0
$791$	−10.0000			−0.355559
$792$		0			0
$793$	16.5000	−	28.5788i	0.585932	−	1.01486i
$794$	−7.50000	−	12.9904i	−0.266165	−	0.461011i
$795$		0			0
$796$	−2.50000	+	4.33013i	−0.0886102	+	0.153477i
$797$	−26.0000			−0.920967			−0.460484	−	0.887668i	$-0.652324\pi$
$797$	−26.0000			−0.920967			−0.460484	+	0.887668i	$0.652324\pi$
$798$		0			0
$799$	40.0000			1.41510
$800$	2.50000	−	4.33013i	0.0883883	−	0.153093i
$801$		0			0
$802$	9.00000	+	15.5885i	0.317801	+	0.550448i
$803$	11.0000	−	19.0526i	0.388182	−	0.672350i
$804$		0			0
$805$		0			0
$806$	9.00000			0.317011
$807$		0			0
$808$	−5.00000	−	8.66025i	−0.175899	−	0.304667i
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$	−2.00000	−	3.46410i	−0.0702295	−	0.121641i	0.828772	−	0.559586i	$-0.189040\pi$
$811$	−2.00000	−	3.46410i	−0.0702295	−	0.121641i	−0.899002	+	0.437945i	$0.855706\pi$
$812$		0			0
$813$		0			0
$814$	5.00000	−	8.66025i	0.175250	−	0.303542i
$815$		0			0
$816$		0			0
$817$	31.5000	−	23.3827i	1.10205	−	0.818057i
$818$	−30.0000			−1.04893
$819$		0			0
$820$		0			0
$821$	24.0000	+	41.5692i	0.837606	+	1.45078i	0.891891	+	0.452250i	$0.149379\pi$
$821$	24.0000	+	41.5692i	0.837606	+	1.45078i	−0.0542853	+	0.998525i	$0.517288\pi$
$822$		0			0
$823$	−2.00000	−	3.46410i	−0.0697156	−	0.120751i	0.829060	−	0.559159i	$-0.188876\pi$
$823$	−2.00000	−	3.46410i	−0.0697156	−	0.120751i	−0.898776	+	0.438408i	$0.855543\pi$
$824$	−3.00000			−0.104510
$825$		0			0
$826$	−7.00000	−	12.1244i	−0.243561	−	0.421860i
$827$	18.0000	+	31.1769i	0.625921	+	1.08413i	0.988362	+	0.152121i	$0.0486102\pi$
$827$	18.0000	+	31.1769i	0.625921	+	1.08413i	−0.362441	+	0.932007i	$0.618056\pi$
$828$		0			0
$829$	−31.0000			−1.07667			−0.538337	−	0.842729i	$-0.680947\pi$
$829$	−31.0000			−1.07667			−0.538337	+	0.842729i	$0.680947\pi$
$830$		0			0
$831$		0			0
$832$	1.50000	+	2.59808i	0.0520031	+	0.0900721i
$833$	−12.0000	+	20.7846i	−0.415775	+	0.720144i
$834$		0			0
$835$		0			0
$836$	−1.00000	−	8.66025i	−0.0345857	−	0.299521i
$837$		0			0
$838$	9.00000	−	15.5885i	0.310900	−	0.538494i
$839$	−6.00000	+	10.3923i	−0.207143	+	0.358782i	−0.950813	−	0.309764i	$-0.899750\pi$
$839$	−6.00000	+	10.3923i	−0.207143	+	0.358782i	0.743670	+	0.668546i	$0.233083\pi$
$840$		0			0
$841$	14.5000	−	25.1147i	0.500000	−	0.866025i
$842$	−5.00000	−	8.66025i	−0.172311	−	0.298452i
$843$		0			0
$844$	25.0000			0.860535
$845$		0			0
$846$		0			0
$847$	−7.00000			−0.240523
$848$	4.00000			0.137361
$849$		0			0
$850$	−10.0000	+	17.3205i	−0.342997	+	0.594089i
$851$	−10.0000	−	17.3205i	−0.342796	−	0.593739i
$852$		0			0
$853$	−1.50000	+	2.59808i	−0.0513590	+	0.0889564i	−0.890562	−	0.454862i	$-0.849689\pi$
$853$	−1.50000	+	2.59808i	−0.0513590	+	0.0889564i	0.839203	+	0.543818i	$0.183022\pi$
$854$	11.0000			0.376412
$855$		0			0
$856$	10.0000			0.341793
$857$	21.0000	−	36.3731i	0.717346	−	1.24248i	−0.244701	−	0.969599i	$-0.578690\pi$
$857$	21.0000	−	36.3731i	0.717346	−	1.24248i	0.962048	−	0.272882i	$-0.0879768\pi$
$858$		0			0
$859$	−3.50000	−	6.06218i	−0.119418	−	0.206839i	0.800119	−	0.599841i	$-0.204770\pi$
$859$	−3.50000	−	6.06218i	−0.119418	−	0.206839i	−0.919537	+	0.393003i	$0.871436\pi$
$860$		0			0
$861$		0			0
$862$	10.0000			0.340601
$863$	22.0000			0.748889			0.374444	−	0.927249i	$-0.377833\pi$
$863$	22.0000			0.748889			0.374444	+	0.927249i	$0.377833\pi$
$864$		0			0
$865$		0			0
$866$	−9.00000			−0.305832
$867$		0			0
$868$	1.50000	+	2.59808i	0.0509133	+	0.0881845i
$869$	−1.00000	+	1.73205i	−0.0339227	+	0.0587558i
$870$		0			0
$871$	4.50000	−	7.79423i	0.152477	−	0.264097i
$872$	−7.00000	+	12.1244i	−0.237050	+	0.410582i
$873$		0			0
$874$	−16.0000	−	6.92820i	−0.541208	−	0.234350i
$875$		0			0
$876$		0			0
$877$	12.5000	−	21.6506i	0.422095	−	0.731090i	−0.574049	−	0.818821i	$-0.694628\pi$
$877$	12.5000	−	21.6506i	0.422095	−	0.731090i	0.996144	+	0.0877308i	$0.0279615\pi$
$878$	−9.50000	−	16.4545i	−0.320609	−	0.555312i
$879$		0			0
$880$		0			0
$881$	−36.0000			−1.21287			−0.606435	−	0.795133i	$-0.707401\pi$
$881$	−36.0000			−1.21287			−0.606435	+	0.795133i	$0.707401\pi$
$882$		0			0
$883$	4.50000	+	7.79423i	0.151437	+	0.262297i	0.931756	−	0.363085i	$-0.118277\pi$
$883$	4.50000	+	7.79423i	0.151437	+	0.262297i	−0.780319	+	0.625382i	$0.784943\pi$
$884$	−6.00000	−	10.3923i	−0.201802	−	0.349531i
$885$		0			0
$886$	−24.0000			−0.806296
$887$	9.00000	+	15.5885i	0.302190	+	0.523409i	0.976632	−	0.214919i	$-0.0689488\pi$
$887$	9.00000	+	15.5885i	0.302190	+	0.523409i	−0.674441	+	0.738328i	$0.735615\pi$
$888$		0			0
$889$	4.00000	+	6.92820i	0.134156	+	0.232364i
$890$		0			0
$891$		0			0
$892$	9.00000			0.301342
$893$	5.00000	+	43.3013i	0.167319	+	1.44902i
$894$		0			0
$895$		0			0
$896$	−0.500000	+	0.866025i	−0.0167038	+	0.0289319i
$897$		0			0
$898$	−18.0000	+	31.1769i	−0.600668	+	1.04039i
$899$		0			0
$900$		0			0
$901$	−16.0000			−0.533037
$902$	4.00000	+	6.92820i	0.133185	+	0.230684i
$903$		0			0
$904$	−10.0000			−0.332595
$905$		0			0
$906$		0			0
$907$	−26.0000	+	45.0333i	−0.863316	+	1.49531i	0.00539395	+	0.999985i	$0.498283\pi$
$907$	−26.0000	+	45.0333i	−0.863316	+	1.49531i	−0.868710	+	0.495321i	$0.835050\pi$
$908$	4.00000	+	6.92820i	0.132745	+	0.229920i
$909$		0			0
$910$		0			0
$911$	60.0000			1.98789			0.993944	−	0.109885i	$-0.0350482\pi$
$911$	60.0000			1.98789			0.993944	+	0.109885i	$0.0350482\pi$
$912$		0			0
$913$	−16.0000			−0.529523
$914$	−2.50000	+	4.33013i	−0.0826927	+	0.143228i
$915$		0			0
$916$	5.50000	+	9.52628i	0.181725	+	0.314757i
$917$	−3.00000	+	5.19615i	−0.0990687	+	0.171592i
$918$		0			0
$919$	55.0000			1.81428			0.907141	−	0.420826i	$-0.138260\pi$
$919$	55.0000			1.81428			0.907141	+	0.420826i	$0.138260\pi$
$920$		0			0
$921$		0			0
$922$	3.00000	+	5.19615i	0.0987997	+	0.171126i
$923$	42.0000			1.38245
$924$		0			0
$925$	−12.5000	−	21.6506i	−0.410997	−	0.711868i
$926$	4.50000	−	7.79423i	0.147879	−	0.256134i
$927$		0			0
$928$		0			0
$929$	6.00000	−	10.3923i	0.196854	−	0.340960i	−0.750653	−	0.660697i	$-0.770261\pi$
$929$	6.00000	−	10.3923i	0.196854	−	0.340960i	0.947507	+	0.319736i	$0.103594\pi$
$930$		0			0
$931$	−24.0000	−	10.3923i	−0.786568	−	0.340594i
$932$	−18.0000			−0.589610
$933$		0			0
$934$	4.00000	−	6.92820i	0.130884	−	0.226698i
$935$		0			0
$936$		0			0
$937$	12.5000	+	21.6506i	0.408357	+	0.707295i	0.994706	−	0.102763i	$-0.0327685\pi$
$937$	12.5000	+	21.6506i	0.408357	+	0.707295i	−0.586349	+	0.810059i	$0.699435\pi$
$938$	3.00000			0.0979535
$939$		0			0
$940$		0			0
$941$	−16.0000	−	27.7128i	−0.521585	−	0.903412i	−0.999685	−	0.0251063i	$-0.992008\pi$
$941$	−16.0000	−	27.7128i	−0.521585	−	0.903412i	0.478100	−	0.878306i	$-0.341326\pi$
$942$		0			0
$943$	16.0000			0.521032
$944$	−7.00000	−	12.1244i	−0.227831	−	0.394614i
$945$		0			0
$946$	9.00000	+	15.5885i	0.292615	+	0.506824i
$947$	5.00000	−	8.66025i	0.162478	−	0.281420i	−0.773279	−	0.634066i	$-0.781385\pi$
$947$	5.00000	−	8.66025i	0.162478	−	0.281420i	0.935757	+	0.352646i	$0.114718\pi$
$948$		0			0
$949$	33.0000			1.07123
$950$	−20.0000	−	8.66025i	−0.648886	−	0.280976i
$951$		0			0
$952$	2.00000	−	3.46410i	0.0648204	−	0.112272i
$953$	−15.0000	+	25.9808i	−0.485898	+	0.841599i	−0.999869	−	0.0162081i	$-0.994841\pi$
$953$	−15.0000	+	25.9808i	−0.485898	+	0.841599i	0.513971	+	0.857808i	$0.328174\pi$
$954$		0			0
$955$		0			0
$956$	−6.00000	−	10.3923i	−0.194054	−	0.336111i
$957$		0			0
$958$	−6.00000			−0.193851
$959$	−3.00000	−	5.19615i	−0.0968751	−	0.167793i
$960$		0			0
$961$	−22.0000			−0.709677
$962$	15.0000			0.483619
$963$		0			0
$964$	−12.5000	+	21.6506i	−0.402598	+	0.697320i
$965$		0			0
$966$		0			0
$967$	−25.5000	+	44.1673i	−0.820025	+	1.42032i	0.0856383	+	0.996326i	$0.472707\pi$
$967$	−25.5000	+	44.1673i	−0.820025	+	1.42032i	−0.905663	+	0.423998i	$0.860626\pi$
$968$	−7.00000			−0.224989
$969$		0			0
$970$		0			0
$971$	3.00000	−	5.19615i	0.0962746	−	0.166752i	−0.813865	−	0.581054i	$-0.802641\pi$
$971$	3.00000	−	5.19615i	0.0962746	−	0.166752i	0.910140	+	0.414301i	$0.135974\pi$
$972$		0			0
$973$	9.50000	+	16.4545i	0.304556	+	0.527506i
$974$	−16.0000	+	27.7128i	−0.512673	+	0.887976i
$975$		0			0
$976$	11.0000			0.352101
$977$	42.0000			1.34370			0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000			1.34370			0.671850	+	0.740688i	$0.265500\pi$
$978$		0			0
$979$	−14.0000	−	24.2487i	−0.447442	−	0.774992i
$980$		0			0
$981$		0			0
$982$	15.0000	+	25.9808i	0.478669	+	0.829079i
$983$	−3.00000	+	5.19615i	−0.0956851	+	0.165732i	−0.909894	−	0.414840i	$-0.863838\pi$
$983$	−3.00000	+	5.19615i	−0.0956851	+	0.165732i	0.814209	+	0.580572i	$0.197171\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	10.5000	−	7.79423i	0.334050	−	0.247967i
$989$	36.0000			1.14473
$990$		0			0
$991$	8.50000	−	14.7224i	0.270011	−	0.467673i	−0.698853	−	0.715265i	$-0.746306\pi$
$991$	8.50000	−	14.7224i	0.270011	−	0.467673i	0.968864	+	0.247592i	$0.0796392\pi$
$992$	1.50000	+	2.59808i	0.0476250	+	0.0824890i
$993$		0			0
$994$	7.00000	+	12.1244i	0.222027	+	0.384561i
$995$		0			0
$996$		0			0
$997$	24.5000	+	42.4352i	0.775923	+	1.34394i	0.934274	+	0.356555i	$0.116049\pi$
$997$	24.5000	+	42.4352i	0.775923	+	1.34394i	−0.158352	+	0.987383i	$0.550618\pi$
$998$	−2.50000	−	4.33013i	−0.0791361	−	0.137068i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.2.g.c.163.1		2
3.2	odd	2		114.2.e.b.49.1	yes	2
4.3	odd	2		2736.2.s.k.1873.1		2
12.11	even	2		912.2.q.b.49.1		2
19.7	even	3	inner	342.2.g.c.235.1		2
19.8	odd	6		6498.2.a.g.1.1		1
19.11	even	3		6498.2.a.u.1.1		1
57.8	even	6		2166.2.a.h.1.1		1
57.11	odd	6		2166.2.a.b.1.1		1
57.26	odd	6		114.2.e.b.7.1	✓	2
76.7	odd	6		2736.2.s.k.577.1		2
228.83	even	6		912.2.q.b.577.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.2.e.b.7.1	✓	2	57.26	odd	6
114.2.e.b.49.1	yes	2	3.2	odd	2
342.2.g.c.163.1		2	1.1	even	1	trivial
342.2.g.c.235.1		2	19.7	even	3	inner
912.2.q.b.49.1		2	12.11	even	2
912.2.q.b.577.1		2	228.83	even	6
2166.2.a.b.1.1		1	57.11	odd	6
2166.2.a.h.1.1		1	57.8	even	6
2736.2.s.k.577.1		2	76.7	odd	6
2736.2.s.k.1873.1		2	4.3	odd	2
6498.2.a.g.1.1		1	19.8	odd	6
6498.2.a.u.1.1		1	19.11	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:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	342.g (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +1.00000 q^{7} +1.00000 q^{8} +2.00000 q^{11} +(1.50000 + 2.59808i) q^{13} +(-0.500000 + 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(2.00000 - 3.46410i) q^{17} +(4.00000 + 1.73205i) q^{19} +(-1.00000 + 1.73205i) q^{22} +(2.00000 + 3.46410i) q^{23} +(2.50000 + 4.33013i) q^{25} -3.00000 q^{26} +(-0.500000 - 0.866025i) q^{28} -3.00000 q^{31} +(-0.500000 - 0.866025i) q^{32} +(2.00000 + 3.46410i) q^{34} -5.00000 q^{37} +(-3.50000 + 2.59808i) q^{38} +(2.00000 - 3.46410i) q^{41} +(4.50000 - 7.79423i) q^{43} +(-1.00000 - 1.73205i) q^{44} -4.00000 q^{46} +(5.00000 + 8.66025i) q^{47} -6.00000 q^{49} -5.00000 q^{50} +(1.50000 - 2.59808i) q^{52} +(-2.00000 - 3.46410i) q^{53} +1.00000 q^{56} +(-7.00000 + 12.1244i) q^{59} +(-5.50000 - 9.52628i) q^{61} +(1.50000 - 2.59808i) q^{62} +1.00000 q^{64} +(-1.50000 - 2.59808i) q^{67} -4.00000 q^{68} +(7.00000 - 12.1244i) q^{71} +(5.50000 - 9.52628i) q^{73} +(2.50000 - 4.33013i) q^{74} +(-0.500000 - 4.33013i) q^{76} +2.00000 q^{77} +(-0.500000 + 0.866025i) q^{79} +(2.00000 + 3.46410i) q^{82} -8.00000 q^{83} +(4.50000 + 7.79423i) q^{86} +2.00000 q^{88} +(-7.00000 - 12.1244i) q^{89} +(1.50000 + 2.59808i) q^{91} +(2.00000 - 3.46410i) q^{92} -10.0000 q^{94} +(-1.00000 + 1.73205i) q^{97} +(3.00000 - 5.19615i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} + 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q - q^2 - q^4 + 2 * q^7 + 2 * q^8	\( 2 q - q^{2} - q^{4} + 2 q^{7} + 2 q^{8} + 4 q^{11} + 3 q^{13} - q^{14} - q^{16} + 4 q^{17} + 8 q^{19} - 2 q^{22} + 4 q^{23} + 5 q^{25} - 6 q^{26} - q^{28} - 6 q^{31} - q^{32} + 4 q^{34} - 10 q^{37} - 7 q^{38} + 4 q^{41} + 9 q^{43} - 2 q^{44} - 8 q^{46} + 10 q^{47} - 12 q^{49} - 10 q^{50} + 3 q^{52} - 4 q^{53} + 2 q^{56} - 14 q^{59} - 11 q^{61} + 3 q^{62} + 2 q^{64} - 3 q^{67} - 8 q^{68} + 14 q^{71} + 11 q^{73} + 5 q^{74} - q^{76} + 4 q^{77} - q^{79} + 4 q^{82} - 16 q^{83} + 9 q^{86} + 4 q^{88} - 14 q^{89} + 3 q^{91} + 4 q^{92} - 20 q^{94} - 2 q^{97} + 6 q^{98}+O(q^{100}) \) 2 * q - q^2 - q^4 + 2 * q^7 + 2 * q^8 + 4 * q^11 + 3 * q^13 - q^14 - q^16 + 4 * q^17 + 8 * q^19 - 2 * q^22 + 4 * q^23 + 5 * q^25 - 6 * q^26 - q^28 - 6 * q^31 - q^32 + 4 * q^34 - 10 * q^37 - 7 * q^38 + 4 * q^41 + 9 * q^43 - 2 * q^44 - 8 * q^46 + 10 * q^47 - 12 * q^49 - 10 * q^50 + 3 * q^52 - 4 * q^53 + 2 * q^56 - 14 * q^59 - 11 * q^61 + 3 * q^62 + 2 * q^64 - 3 * q^67 - 8 * q^68 + 14 * q^71 + 11 * q^73 + 5 * q^74 - q^76 + 4 * q^77 - q^79 + 4 * q^82 - 16 * q^83 + 9 * q^86 + 4 * q^88 - 14 * q^89 + 3 * q^91 + 4 * q^92 - 20 * q^94 - 2 * q^97 + 6 * q^98

Introduction

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