LMFDB - Embedded newform 171.2.f.a.163.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [171,2,Mod(64,171)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(171, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

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

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

chi := DirichletCharacter("171.64");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 171 = 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	171.f (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:	$1.36544187456$
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 57)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			163.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	171.163
Dual form			171.2.f.a.64.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{2} +(0.500000 + 0.866025i) q^{4} +1.00000 q^{7} +3.00000 q^{8} +O(q^{10})$	\(q+(0.500000 - 0.866025i) q^{2} +(0.500000 + 0.866025i) q^{4} +1.00000 q^{7} +3.00000 q^{8} +2.00000 q^{11} +(-2.50000 - 4.33013i) q^{13} +(0.500000 - 0.866025i) q^{14} +(0.500000 - 0.866025i) q^{16} +(-2.00000 + 3.46410i) q^{17} +(-4.00000 + 1.73205i) q^{19} +(1.00000 - 1.73205i) q^{22} +(-2.00000 - 3.46410i) q^{23} +(2.50000 + 4.33013i) q^{25} -5.00000 q^{26} +(0.500000 + 0.866025i) q^{28} +(-4.00000 - 6.92820i) q^{29} -3.00000 q^{31} +(2.50000 + 4.33013i) q^{32} +(2.00000 + 3.46410i) q^{34} +3.00000 q^{37} +(-0.500000 + 4.33013i) q^{38} +(-6.00000 + 10.3923i) q^{41} +(0.500000 - 0.866025i) q^{43} +(1.00000 + 1.73205i) q^{44} -4.00000 q^{46} +(-3.00000 - 5.19615i) q^{47} -6.00000 q^{49} +5.00000 q^{50} +(2.50000 - 4.33013i) q^{52} +(2.00000 + 3.46410i) q^{53} +3.00000 q^{56} -8.00000 q^{58} +(5.00000 - 8.66025i) q^{59} +(6.50000 + 11.2583i) q^{61} +(-1.50000 + 2.59808i) q^{62} +7.00000 q^{64} +(-5.50000 - 9.52628i) q^{67} -4.00000 q^{68} +(3.00000 - 5.19615i) q^{71} +(5.50000 - 9.52628i) q^{73} +(1.50000 - 2.59808i) q^{74} +(-3.50000 - 2.59808i) q^{76} +2.00000 q^{77} +(-0.500000 + 0.866025i) q^{79} +(6.00000 + 10.3923i) q^{82} +(-0.500000 - 0.866025i) q^{86} +6.00000 q^{88} +(-3.00000 - 5.19615i) q^{89} +(-2.50000 - 4.33013i) q^{91} +(2.00000 - 3.46410i) q^{92} -6.00000 q^{94} +(-1.00000 + 1.73205i) q^{97} +(-3.00000 + 5.19615i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + q^{4} + 2 q^{7} + 6 q^{8}+O(q^{10}) $ 2 * q + q^2 + q^4 + 2 * q^7 + 6 * q^8	$ 2 q + q^{2} + q^{4} + 2 q^{7} + 6 q^{8} + 4 q^{11} - 5 q^{13} + q^{14} + q^{16} - 4 q^{17} - 8 q^{19} + 2 q^{22} - 4 q^{23} + 5 q^{25} - 10 q^{26} + q^{28} - 8 q^{29} - 6 q^{31} + 5 q^{32} + 4 q^{34} + 6 q^{37} - q^{38} - 12 q^{41} + q^{43} + 2 q^{44} - 8 q^{46} - 6 q^{47} - 12 q^{49} + 10 q^{50} + 5 q^{52} + 4 q^{53} + 6 q^{56} - 16 q^{58} + 10 q^{59} + 13 q^{61} - 3 q^{62} + 14 q^{64} - 11 q^{67} - 8 q^{68} + 6 q^{71} + 11 q^{73} + 3 q^{74} - 7 q^{76} + 4 q^{77} - q^{79} + 12 q^{82} - q^{86} + 12 q^{88} - 6 q^{89} - 5 q^{91} + 4 q^{92} - 12 q^{94} - 2 q^{97} - 6 q^{98}+O(q^{100}) $ 2 * q + q^2 + q^4 + 2 * q^7 + 6 * q^8 + 4 * q^11 - 5 * q^13 + q^14 + q^16 - 4 * q^17 - 8 * q^19 + 2 * q^22 - 4 * q^23 + 5 * q^25 - 10 * q^26 + q^28 - 8 * q^29 - 6 * q^31 + 5 * q^32 + 4 * q^34 + 6 * q^37 - q^38 - 12 * q^41 + q^43 + 2 * q^44 - 8 * q^46 - 6 * q^47 - 12 * q^49 + 10 * q^50 + 5 * q^52 + 4 * q^53 + 6 * q^56 - 16 * q^58 + 10 * q^59 + 13 * q^61 - 3 * q^62 + 14 * q^64 - 11 * q^67 - 8 * q^68 + 6 * q^71 + 11 * q^73 + 3 * q^74 - 7 * q^76 + 4 * q^77 - q^79 + 12 * q^82 - q^86 + 12 * q^88 - 6 * q^89 - 5 * q^91 + 4 * q^92 - 12 * q^94 - 2 * q^97 - 6 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/171\mathbb{Z}\right)^\times$.

$n$	$20$	$154$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$

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	−0.633316	−	0.773893i	$-0.718307\pi$
$2$	0.500000	−	0.866025i	0.353553	−	0.612372i	0.986869	+	0.161521i	$0.0516399\pi$
$3$		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	$0.439481\pi$
$8$	3.00000			1.06066
$9$		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
$13$	−2.50000	−	4.33013i	−0.693375	−	1.20096i	−0.970725	−	0.240192i	$-0.922790\pi$
$13$	−2.50000	−	4.33013i	−0.693375	−	1.20096i	0.277350	−	0.960769i	$-0.410544\pi$
$14$	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	−0.999853	−	0.0171533i	$-0.994540\pi$
$17$	−2.00000	+	3.46410i	−0.485071	+	0.840168i	0.514782	+	0.857321i	$0.327873\pi$
$18$		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	0.578610	−	0.815604i	$-0.303595\pi$
$23$	−2.00000	−	3.46410i	−0.417029	−	0.722315i	−0.995639	+	0.0932891i	$0.970262\pi$
$24$		0			0
$25$	2.50000	+	4.33013i	0.500000	+	0.866025i
$26$	−5.00000			−0.980581
$27$		0			0
$28$	0.500000	+	0.866025i	0.0944911	+	0.163663i
$29$	−4.00000	−	6.92820i	−0.742781	−	1.28654i	−0.951224	−	0.308500i	$-0.900173\pi$
$29$	−4.00000	−	6.92820i	−0.742781	−	1.28654i	0.208443	−	0.978035i	$-0.433160\pi$
$30$		0			0
$31$	−3.00000			−0.538816			−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000			−0.538816			−0.269408	+	0.963026i	$0.586828\pi$
$32$	2.50000	+	4.33013i	0.441942	+	0.765466i
$33$		0			0
$34$	2.00000	+	3.46410i	0.342997	+	0.594089i
$35$		0			0
$36$		0			0
$37$	3.00000			0.493197			0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000			0.493197			0.246598	+	0.969118i	$0.420687\pi$
$38$	−0.500000	+	4.33013i	−0.0811107	+	0.702439i
$39$		0			0
$40$		0			0
$41$	−6.00000	+	10.3923i	−0.937043	+	1.62301i	−0.166092	+	0.986110i	$0.553115\pi$
$41$	−6.00000	+	10.3923i	−0.937043	+	1.62301i	−0.770950	+	0.636895i	$0.780218\pi$
$42$		0			0
$43$	0.500000	−	0.866025i	0.0762493	−	0.132068i	−0.825380	−	0.564578i	$-0.809039\pi$
$43$	0.500000	−	0.866025i	0.0762493	−	0.132068i	0.901629	+	0.432511i	$0.142372\pi$
$44$	1.00000	+	1.73205i	0.150756	+	0.261116i
$45$		0			0
$46$	−4.00000			−0.589768
$47$	−3.00000	−	5.19615i	−0.437595	−	0.757937i	0.559908	−	0.828554i	$-0.310836\pi$
$47$	−3.00000	−	5.19615i	−0.437595	−	0.757937i	−0.997503	+	0.0706177i	$0.977503\pi$
$48$		0			0
$49$	−6.00000			−0.857143
$50$	5.00000			0.707107
$51$		0			0
$52$	2.50000	−	4.33013i	0.346688	−	0.600481i
$53$	2.00000	+	3.46410i	0.274721	+	0.475831i	0.970065	−	0.242846i	$-0.0780811\pi$
$53$	2.00000	+	3.46410i	0.274721	+	0.475831i	−0.695344	+	0.718677i	$0.744748\pi$
$54$		0			0
$55$		0			0
$56$	3.00000			0.400892
$57$		0			0
$58$	−8.00000			−1.05045
$59$	5.00000	−	8.66025i	0.650945	−	1.12747i	−0.331949	−	0.943297i	$-0.607706\pi$
$59$	5.00000	−	8.66025i	0.650945	−	1.12747i	0.982894	−	0.184172i	$-0.0589603\pi$
$60$		0			0
$61$	6.50000	+	11.2583i	0.832240	+	1.44148i	0.896258	+	0.443533i	$0.146275\pi$
$61$	6.50000	+	11.2583i	0.832240	+	1.44148i	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	−1.50000	+	2.59808i	−0.190500	+	0.329956i
$63$		0			0
$64$	7.00000			0.875000
$65$		0			0
$66$		0			0
$67$	−5.50000	−	9.52628i	−0.671932	−	1.16382i	−0.977356	−	0.211604i	$-0.932131\pi$
$67$	−5.50000	−	9.52628i	−0.671932	−	1.16382i	0.305424	−	0.952217i	$-0.401202\pi$
$68$	−4.00000			−0.485071
$69$		0			0
$70$		0			0
$71$	3.00000	−	5.19615i	0.356034	−	0.616670i	−0.631260	−	0.775571i	$-0.717462\pi$
$71$	3.00000	−	5.19615i	0.356034	−	0.616670i	0.987294	+	0.158901i	$0.0507952\pi$
$72$		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	$-0.0559458\pi$
$74$	1.50000	−	2.59808i	0.174371	−	0.302020i
$75$		0			0
$76$	−3.50000	−	2.59808i	−0.401478	−	0.298020i
$77$	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
$82$	6.00000	+	10.3923i	0.662589	+	1.14764i
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$	−0.500000	−	0.866025i	−0.0539164	−	0.0933859i
$87$		0			0
$88$	6.00000			0.639602
$89$	−3.00000	−	5.19615i	−0.317999	−	0.550791i	0.662071	−	0.749441i	$-0.269678\pi$
$89$	−3.00000	−	5.19615i	−0.317999	−	0.550791i	−0.980071	+	0.198650i	$0.936344\pi$
$90$		0			0
$91$	−2.50000	−	4.33013i	−0.262071	−	0.453921i
$92$	2.00000	−	3.46410i	0.208514	−	0.361158i
$93$		0			0
$94$	−6.00000			−0.618853
$95$		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
$101$	7.00000	+	12.1244i	0.696526	+	1.20642i	0.969664	+	0.244443i	$0.0786053\pi$
$101$	7.00000	+	12.1244i	0.696526	+	1.20642i	−0.273138	+	0.961975i	$0.588061\pi$
$102$		0			0
$103$	13.0000			1.28093			0.640464	−	0.767988i	$-0.278742\pi$
$103$	13.0000			1.28093			0.640464	+	0.767988i	$0.278742\pi$
$104$	−7.50000	−	12.9904i	−0.735436	−	1.27381i
$105$		0			0
$106$	4.00000			0.388514
$107$	18.0000			1.74013			0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000			1.74013			0.870063	+	0.492941i	$0.164078\pi$
$108$		0			0
$109$	1.00000	−	1.73205i	0.0957826	−	0.165900i	−0.814152	−	0.580651i	$-0.802798\pi$
$109$	1.00000	−	1.73205i	0.0957826	−	0.165900i	0.909935	+	0.414751i	$0.136131\pi$
$110$		0			0
$111$		0			0
$112$	0.500000	−	0.866025i	0.0472456	−	0.0818317i
$113$	−2.00000			−0.188144			−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000			−0.188144			−0.0940721	+	0.995565i	$0.529988\pi$
$114$		0			0
$115$		0			0
$116$	4.00000	−	6.92820i	0.371391	−	0.643268i
$117$		0			0
$118$	−5.00000	−	8.66025i	−0.460287	−	0.797241i
$119$	−2.00000	+	3.46410i	−0.183340	+	0.317554i
$120$		0			0
$121$	−7.00000			−0.636364
$122$	13.0000			1.17696
$123$		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	$0.717834\pi$
$128$	−1.50000	+	2.59808i	−0.132583	+	0.229640i
$129$		0			0
$130$		0			0
$131$	−7.00000	+	12.1244i	−0.611593	+	1.05931i	0.379379	+	0.925241i	$0.376138\pi$
$131$	−7.00000	+	12.1244i	−0.611593	+	1.05931i	−0.990972	+	0.134069i	$0.957196\pi$
$132$		0			0
$133$	−4.00000	+	1.73205i	−0.346844	+	0.150188i
$134$	−11.0000			−0.950255
$135$		0			0
$136$	−6.00000	+	10.3923i	−0.514496	+	0.891133i
$137$	9.00000	+	15.5885i	0.768922	+	1.33181i	0.938148	+	0.346235i	$0.112540\pi$
$137$	9.00000	+	15.5885i	0.768922	+	1.33181i	−0.169226	+	0.985577i	$0.554127\pi$
$138$		0			0
$139$	−2.50000	−	4.33013i	−0.212047	−	0.367277i	0.740308	−	0.672268i	$-0.234680\pi$
$139$	−2.50000	−	4.33013i	−0.212047	−	0.367277i	−0.952355	+	0.304991i	$0.901346\pi$
$140$		0			0
$141$		0			0
$142$	−3.00000	−	5.19615i	−0.251754	−	0.436051i
$143$	−5.00000	−	8.66025i	−0.418121	−	0.724207i
$144$		0			0
$145$		0			0
$146$	−5.50000	−	9.52628i	−0.455183	−	0.788400i
$147$		0			0
$148$	1.50000	+	2.59808i	0.123299	+	0.213561i
$149$	−3.00000	+	5.19615i	−0.245770	+	0.425685i	−0.962348	−	0.271821i	$-0.912374\pi$
$149$	−3.00000	+	5.19615i	−0.245770	+	0.425685i	0.716578	+	0.697507i	$0.245707\pi$
$150$		0			0
$151$	12.0000			0.976546			0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000			0.976546			0.488273	+	0.872691i	$0.337627\pi$
$152$	−12.0000	+	5.19615i	−0.973329	+	0.421464i
$153$		0			0
$154$	1.00000	−	1.73205i	0.0805823	−	0.139573i
$155$		0			0
$156$		0			0
$157$	−1.50000	+	2.59808i	−0.119713	+	0.207349i	−0.919654	−	0.392730i	$-0.871531\pi$
$157$	−1.50000	+	2.59808i	−0.119713	+	0.207349i	0.799941	+	0.600079i	$0.204864\pi$
$158$	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
$163$	3.00000			0.234978			0.117489	−	0.993074i	$-0.462515\pi$
$163$	3.00000			0.234978			0.117489	+	0.993074i	$0.462515\pi$
$164$	−12.0000			−0.937043
$165$		0			0
$166$		0			0
$167$	−1.00000	−	1.73205i	−0.0773823	−	0.134030i	0.824737	−	0.565516i	$-0.191323\pi$
$167$	−1.00000	−	1.73205i	−0.0773823	−	0.134030i	−0.902120	+	0.431486i	$0.857990\pi$
$168$		0			0
$169$	−6.00000	+	10.3923i	−0.461538	+	0.799408i
$170$		0			0
$171$		0			0
$172$	1.00000			0.0762493
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$	2.50000	+	4.33013i	0.188982	+	0.327327i
$176$	1.00000	−	1.73205i	0.0753778	−	0.130558i
$177$		0			0
$178$	−6.00000			−0.449719
$179$	−12.0000			−0.896922			−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$		0			0
$181$	1.00000	+	1.73205i	0.0743294	+	0.128742i	0.900794	−	0.434246i	$-0.142985\pi$
$181$	1.00000	+	1.73205i	0.0743294	+	0.128742i	−0.826465	+	0.562988i	$0.809652\pi$
$182$	−5.00000			−0.370625
$183$		0			0
$184$	−6.00000	−	10.3923i	−0.442326	−	0.766131i
$185$		0			0
$186$		0			0
$187$	−4.00000	+	6.92820i	−0.292509	+	0.506640i
$188$	3.00000	−	5.19615i	0.218797	−	0.378968i
$189$		0			0
$190$		0			0
$191$	−24.0000			−1.73658			−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000			−1.73658			−0.868290	+	0.496058i	$0.834780\pi$
$192$		0			0
$193$	−9.50000	+	16.4545i	−0.683825	+	1.18442i	0.289980	+	0.957033i	$0.406351\pi$
$193$	−9.50000	+	16.4545i	−0.683825	+	1.18442i	−0.973805	+	0.227387i	$0.926982\pi$
$194$	1.00000	+	1.73205i	0.0717958	+	0.124354i
$195$		0			0
$196$	−3.00000	−	5.19615i	−0.214286	−	0.371154i
$197$	26.0000			1.85242			0.926212	−	0.377004i	$-0.123046\pi$
$197$	26.0000			1.85242			0.926212	+	0.377004i	$0.123046\pi$
$198$		0			0
$199$	−10.5000	−	18.1865i	−0.744325	−	1.28921i	−0.950509	−	0.310696i	$-0.899438\pi$
$199$	−10.5000	−	18.1865i	−0.744325	−	1.28921i	0.206184	−	0.978513i	$-0.433895\pi$
$200$	7.50000	+	12.9904i	0.530330	+	0.918559i
$201$		0			0
$202$	14.0000			0.985037
$203$	−4.00000	−	6.92820i	−0.280745	−	0.486265i
$204$		0			0
$205$		0			0
$206$	6.50000	−	11.2583i	0.452876	−	0.784405i
$207$		0			0
$208$	−5.00000			−0.346688
$209$	−8.00000	+	3.46410i	−0.553372	+	0.239617i
$210$		0			0
$211$	7.50000	−	12.9904i	0.516321	−	0.894295i	−0.483499	−	0.875345i	$-0.660634\pi$
$211$	7.50000	−	12.9904i	0.516321	−	0.894295i	0.999820	−	0.0189499i	$-0.00603229\pi$
$212$	−2.00000	+	3.46410i	−0.137361	+	0.237915i
$213$		0			0
$214$	9.00000	−	15.5885i	0.615227	−	1.06561i
$215$		0			0
$216$		0			0
$217$	−3.00000			−0.203653
$218$	−1.00000	−	1.73205i	−0.0677285	−	0.117309i
$219$		0			0
$220$		0			0
$221$	20.0000			1.34535
$222$		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	$0.264101\pi$
$224$	2.50000	+	4.33013i	0.167038	+	0.289319i
$225$		0			0
$226$	−1.00000	+	1.73205i	−0.0665190	+	0.115214i
$227$	24.0000			1.59294			0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000			1.59294			0.796468	+	0.604681i	$0.206699\pi$
$228$		0			0
$229$	13.0000			0.859064			0.429532	−	0.903052i	$-0.358679\pi$
$229$	13.0000			0.859064			0.429532	+	0.903052i	$0.358679\pi$
$230$		0			0
$231$		0			0
$232$	−12.0000	−	20.7846i	−0.787839	−	1.36458i
$233$	−3.00000	+	5.19615i	−0.196537	+	0.340411i	−0.947403	−	0.320043i	$-0.896303\pi$
$233$	−3.00000	+	5.19615i	−0.196537	+	0.340411i	0.750867	+	0.660454i	$0.229636\pi$
$234$		0			0
$235$		0			0
$236$	10.0000			0.650945
$237$		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
$241$	3.50000	+	6.06218i	0.225455	+	0.390499i	0.956456	−	0.291877i	$-0.0942799\pi$
$241$	3.50000	+	6.06218i	0.225455	+	0.390499i	−0.731001	+	0.682376i	$0.760947\pi$
$242$	−3.50000	+	6.06218i	−0.224989	+	0.389692i
$243$		0			0
$244$	−6.50000	+	11.2583i	−0.416120	+	0.720741i
$245$		0			0
$246$		0			0
$247$	17.5000	+	12.9904i	1.11350	+	0.826558i
$248$	−9.00000			−0.571501
$249$		0			0
$250$		0			0
$251$	−1.00000	−	1.73205i	−0.0631194	−	0.109326i	0.832739	−	0.553666i	$-0.186772\pi$
$251$	−1.00000	−	1.73205i	−0.0631194	−	0.109326i	−0.895858	+	0.444340i	$0.853438\pi$
$252$		0			0
$253$	−4.00000	−	6.92820i	−0.251478	−	0.435572i
$254$	8.00000			0.501965
$255$		0			0
$256$	8.50000	+	14.7224i	0.531250	+	0.920152i
$257$	−10.0000	−	17.3205i	−0.623783	−	1.08042i	−0.988775	−	0.149413i	$-0.952262\pi$
$257$	−10.0000	−	17.3205i	−0.623783	−	1.08042i	0.364992	−	0.931011i	$-0.381072\pi$
$258$		0			0
$259$	3.00000			0.186411
$260$		0			0
$261$		0			0
$262$	7.00000	+	12.1244i	0.432461	+	0.749045i
$263$	13.0000	−	22.5167i	0.801614	−	1.38844i	−0.116939	−	0.993139i	$-0.537308\pi$
$263$	13.0000	−	22.5167i	0.801614	−	1.38844i	0.918553	−	0.395298i	$-0.129359\pi$
$264$		0			0
$265$		0			0
$266$	−0.500000	+	4.33013i	−0.0306570	+	0.265497i
$267$		0			0
$268$	5.50000	−	9.52628i	0.335966	−	0.581910i
$269$	5.00000	−	8.66025i	0.304855	−	0.528025i	−0.672374	−	0.740212i	$-0.734725\pi$
$269$	5.00000	−	8.66025i	0.304855	−	0.528025i	0.977229	+	0.212187i	$0.0680585\pi$
$270$		0			0
$271$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$271$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$272$	2.00000	+	3.46410i	0.121268	+	0.210042i
$273$		0			0
$274$	18.0000			1.08742
$275$	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	$0.519137\pi$
$278$	−5.00000			−0.299880
$279$		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
$283$	2.00000	−	3.46410i	0.118888	−	0.205919i	−0.800439	−	0.599414i	$-0.795400\pi$
$283$	2.00000	−	3.46410i	0.118888	−	0.205919i	0.919327	+	0.393494i	$0.128734\pi$
$284$	6.00000			0.356034
$285$		0			0
$286$	−10.0000			−0.591312
$287$	−6.00000	+	10.3923i	−0.354169	+	0.613438i
$288$		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
$296$	9.00000			0.523114
$297$		0			0
$298$	3.00000	+	5.19615i	0.173785	+	0.301005i
$299$	−10.0000	+	17.3205i	−0.578315	+	1.00167i
$300$		0			0
$301$	0.500000	−	0.866025i	0.0288195	−	0.0499169i
$302$	6.00000	−	10.3923i	0.345261	−	0.598010i
$303$		0			0
$304$	−0.500000	+	4.33013i	−0.0286770	+	0.248350i
$305$		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
$311$	−18.0000			−1.02069			−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000			−1.02069			−0.510343	+	0.859971i	$0.670482\pi$
$312$		0			0
$313$	−11.0000	−	19.0526i	−0.621757	−	1.07691i	−0.989158	−	0.146852i	$-0.953086\pi$
$313$	−11.0000	−	19.0526i	−0.621757	−	1.07691i	0.367402	−	0.930062i	$-0.380247\pi$
$314$	1.50000	+	2.59808i	0.0846499	+	0.146618i
$315$		0			0
$316$	−1.00000			−0.0562544
$317$	−2.00000	−	3.46410i	−0.112331	−	0.194563i	0.804379	−	0.594117i	$-0.202498\pi$
$317$	−2.00000	−	3.46410i	−0.112331	−	0.194563i	−0.916710	+	0.399554i	$0.869165\pi$
$318$		0			0
$319$	−8.00000	−	13.8564i	−0.447914	−	0.775810i
$320$		0			0
$321$		0			0
$322$	−4.00000			−0.222911
$323$	2.00000	−	17.3205i	0.111283	−	0.963739i
$324$		0			0
$325$	12.5000	−	21.6506i	0.693375	−	1.20096i
$326$	1.50000	−	2.59808i	0.0830773	−	0.143894i
$327$		0			0
$328$	−18.0000	+	31.1769i	−0.993884	+	1.72146i
$329$	−3.00000	−	5.19615i	−0.165395	−	0.286473i
$330$		0			0
$331$	−25.0000			−1.37412			−0.687062	−	0.726599i	$-0.741100\pi$
$331$	−25.0000			−1.37412			−0.687062	+	0.726599i	$0.741100\pi$
$332$		0			0
$333$		0			0
$334$	−2.00000			−0.109435
$335$		0			0
$336$		0			0
$337$	−6.50000	+	11.2583i	−0.354078	+	0.613280i	−0.986960	−	0.160968i	$-0.948538\pi$
$337$	−6.50000	+	11.2583i	−0.354078	+	0.613280i	0.632882	+	0.774248i	$0.281872\pi$
$338$	6.00000	+	10.3923i	0.326357	+	0.565267i
$339$		0			0
$340$		0			0
$341$	−6.00000			−0.324918
$342$		0			0
$343$	−13.0000			−0.701934
$344$	1.50000	−	2.59808i	0.0808746	−	0.140079i
$345$		0			0
$346$		0			0
$347$	−8.00000	+	13.8564i	−0.429463	+	0.743851i	−0.996826	−	0.0796169i	$-0.974630\pi$
$347$	−8.00000	+	13.8564i	−0.429463	+	0.743851i	0.567363	+	0.823468i	$0.307964\pi$
$348$		0			0
$349$	−21.0000			−1.12410			−0.562052	−	0.827102i	$-0.689988\pi$
$349$	−21.0000			−1.12410			−0.562052	+	0.827102i	$0.689988\pi$
$350$	5.00000			0.267261
$351$		0			0
$352$	5.00000	+	8.66025i	0.266501	+	0.461593i
$353$	−4.00000			−0.212899			−0.106449	−	0.994318i	$-0.533948\pi$
$353$	−4.00000			−0.212899			−0.106449	+	0.994318i	$0.533948\pi$
$354$		0			0
$355$		0			0
$356$	3.00000	−	5.19615i	0.159000	−	0.275396i
$357$		0			0
$358$	−6.00000	+	10.3923i	−0.317110	+	0.549250i
$359$	−10.0000	+	17.3205i	−0.527780	+	0.914141i	0.471696	+	0.881761i	$0.343642\pi$
$359$	−10.0000	+	17.3205i	−0.527780	+	0.914141i	−0.999476	+	0.0323801i	$0.989691\pi$
$360$		0			0
$361$	13.0000	−	13.8564i	0.684211	−	0.729285i
$362$	2.00000			0.105118
$363$		0			0
$364$	2.50000	−	4.33013i	0.131036	−	0.226960i
$365$		0			0
$366$		0			0
$367$	−3.50000	−	6.06218i	−0.182699	−	0.316443i	0.760100	−	0.649806i	$-0.225150\pi$
$367$	−3.50000	−	6.06218i	−0.182699	−	0.316443i	−0.942799	+	0.333363i	$0.891817\pi$
$368$	−4.00000			−0.208514
$369$		0			0
$370$		0			0
$371$	2.00000	+	3.46410i	0.103835	+	0.179847i
$372$		0			0
$373$	10.0000			0.517780			0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000			0.517780			0.258890	+	0.965907i	$0.416643\pi$
$374$	4.00000	+	6.92820i	0.206835	+	0.358249i
$375$		0			0
$376$	−9.00000	−	15.5885i	−0.464140	−	0.803913i
$377$	−20.0000	+	34.6410i	−1.03005	+	1.78410i
$378$		0			0
$379$	−5.00000			−0.256833			−0.128416	−	0.991720i	$-0.540989\pi$
$379$	−5.00000			−0.256833			−0.128416	+	0.991720i	$0.540989\pi$
$380$		0			0
$381$		0			0
$382$	−12.0000	+	20.7846i	−0.613973	+	1.06343i
$383$	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
$386$	9.50000	+	16.4545i	0.483537	+	0.837511i
$387$		0			0
$388$	−2.00000			−0.101535
$389$	−12.0000	−	20.7846i	−0.608424	−	1.05382i	−0.991500	−	0.130105i	$-0.958469\pi$
$389$	−12.0000	−	20.7846i	−0.608424	−	1.05382i	0.383076	−	0.923717i	$-0.374865\pi$
$390$		0			0
$391$	16.0000			0.809155
$392$	−18.0000			−0.909137
$393$		0			0
$394$	13.0000	−	22.5167i	0.654931	−	1.13437i
$395$		0			0
$396$		0			0
$397$	−3.50000	+	6.06218i	−0.175660	+	0.304252i	−0.940389	−	0.340099i	$-0.889539\pi$
$397$	−3.50000	+	6.06218i	−0.175660	+	0.304252i	0.764730	+	0.644351i	$0.222873\pi$
$398$	−21.0000			−1.05263
$399$		0			0
$400$	5.00000			0.250000
$401$	−3.00000	+	5.19615i	−0.149813	+	0.259483i	−0.931158	−	0.364615i	$-0.881200\pi$
$401$	−3.00000	+	5.19615i	−0.149813	+	0.259483i	0.781345	+	0.624099i	$0.214534\pi$
$402$		0			0
$403$	7.50000	+	12.9904i	0.373602	+	0.647097i
$404$	−7.00000	+	12.1244i	−0.348263	+	0.603209i
$405$		0			0
$406$	−8.00000			−0.397033
$407$	6.00000			0.297409
$408$		0			0
$409$	7.00000	+	12.1244i	0.346128	+	0.599511i	0.985558	−	0.169338i	$-0.0541630\pi$
$409$	7.00000	+	12.1244i	0.346128	+	0.599511i	−0.639430	+	0.768849i	$0.720830\pi$
$410$		0			0
$411$		0			0
$412$	6.50000	+	11.2583i	0.320232	+	0.554658i
$413$	5.00000	−	8.66025i	0.246034	−	0.426143i
$414$		0			0
$415$		0			0
$416$	12.5000	−	21.6506i	0.612863	−	1.06151i
$417$		0			0
$418$	−1.00000	+	8.66025i	−0.0489116	+	0.423587i
$419$	14.0000			0.683945			0.341972	−	0.939710i	$-0.388905\pi$
$419$	14.0000			0.683945			0.341972	+	0.939710i	$0.388905\pi$
$420$		0			0
$421$	11.0000	−	19.0526i	0.536107	−	0.928565i	−0.463002	−	0.886357i	$-0.653228\pi$
$421$	11.0000	−	19.0526i	0.536107	−	0.928565i	0.999109	−	0.0422075i	$-0.0134391\pi$
$422$	−7.50000	−	12.9904i	−0.365094	−	0.632362i
$423$		0			0
$424$	6.00000	+	10.3923i	0.291386	+	0.504695i
$425$	−20.0000			−0.970143
$426$		0			0
$427$	6.50000	+	11.2583i	0.314557	+	0.544829i
$428$	9.00000	+	15.5885i	0.435031	+	0.753497i
$429$		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
$436$	2.00000			0.0957826
$437$	14.0000	+	10.3923i	0.669711	+	0.497131i
$438$		0			0
$439$	−17.5000	+	30.3109i	−0.835229	+	1.44666i	0.0586141	+	0.998281i	$0.481332\pi$
$439$	−17.5000	+	30.3109i	−0.835229	+	1.44666i	−0.893843	+	0.448379i	$0.852001\pi$
$440$		0			0
$441$		0			0
$442$	10.0000	−	17.3205i	0.475651	−	0.823853i
$443$	−16.0000	−	27.7128i	−0.760183	−	1.31668i	−0.942756	−	0.333483i	$-0.891776\pi$
$443$	−16.0000	−	27.7128i	−0.760183	−	1.31668i	0.182573	−	0.983192i	$-0.441557\pi$
$444$		0			0
$445$		0			0
$446$	4.50000	+	7.79423i	0.213081	+	0.369067i
$447$		0			0
$448$	7.00000			0.330719
$449$	12.0000			0.566315			0.283158	−	0.959073i	$-0.408618\pi$
$449$	12.0000			0.566315			0.283158	+	0.959073i	$0.408618\pi$
$450$		0			0
$451$	−12.0000	+	20.7846i	−0.565058	+	0.978709i
$452$	−1.00000	−	1.73205i	−0.0470360	−	0.0814688i
$453$		0			0
$454$	12.0000	−	20.7846i	0.563188	−	0.975470i
$455$		0			0
$456$		0			0
$457$	−11.0000			−0.514558			−0.257279	−	0.966337i	$-0.582826\pi$
$457$	−11.0000			−0.514558			−0.257279	+	0.966337i	$0.582826\pi$
$458$	6.50000	−	11.2583i	0.303725	−	0.526067i
$459$		0			0
$460$		0			0
$461$	15.0000	−	25.9808i	0.698620	−	1.21004i	−0.270326	−	0.962769i	$-0.587131\pi$
$461$	15.0000	−	25.9808i	0.698620	−	1.21004i	0.968945	−	0.247276i	$-0.0795353\pi$
$462$		0			0
$463$	23.0000			1.06890			0.534450	−	0.845200i	$-0.320519\pi$
$463$	23.0000			1.06890			0.534450	+	0.845200i	$0.320519\pi$
$464$	−8.00000			−0.371391
$465$		0			0
$466$	3.00000	+	5.19615i	0.138972	+	0.240707i
$467$	8.00000			0.370196			0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000			0.370196			0.185098	+	0.982720i	$0.440740\pi$
$468$		0			0
$469$	−5.50000	−	9.52628i	−0.253966	−	0.439883i
$470$		0			0
$471$		0			0
$472$	15.0000	−	25.9808i	0.690431	−	1.19586i
$473$	1.00000	−	1.73205i	0.0459800	−	0.0796398i
$474$		0			0
$475$	−17.5000	−	12.9904i	−0.802955	−	0.596040i
$476$	−4.00000			−0.183340
$477$		0			0
$478$	6.00000	−	10.3923i	0.274434	−	0.475333i
$479$	7.00000	+	12.1244i	0.319838	+	0.553976i	0.980454	−	0.196748i	$-0.0630381\pi$
$479$	7.00000	+	12.1244i	0.319838	+	0.553976i	−0.660616	+	0.750724i	$0.729705\pi$
$480$		0			0
$481$	−7.50000	−	12.9904i	−0.341971	−	0.592310i
$482$	7.00000			0.318841
$483$		0			0
$484$	−3.50000	−	6.06218i	−0.159091	−	0.275554i
$485$		0			0
$486$		0			0
$487$	−16.0000			−0.725029			−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000			−0.725029			−0.362515	+	0.931978i	$0.618082\pi$
$488$	19.5000	+	33.7750i	0.882724	+	1.52892i
$489$		0			0
$490$		0			0
$491$	3.00000	−	5.19615i	0.135388	−	0.234499i	−0.790358	−	0.612646i	$-0.790105\pi$
$491$	3.00000	−	5.19615i	0.135388	−	0.234499i	0.925746	+	0.378147i	$0.123439\pi$
$492$		0			0
$493$	32.0000			1.44121
$494$	20.0000	−	8.66025i	0.899843	−	0.389643i
$495$		0			0
$496$	−1.50000	+	2.59808i	−0.0673520	+	0.116657i
$497$	3.00000	−	5.19615i	0.134568	−	0.233079i
$498$		0			0
$499$	−14.5000	+	25.1147i	−0.649109	+	1.12429i	0.334227	+	0.942493i	$0.391525\pi$
$499$	−14.5000	+	25.1147i	−0.649109	+	1.12429i	−0.983336	+	0.181797i	$0.941809\pi$
$500$		0			0
$501$		0			0
$502$	−2.00000			−0.0892644
$503$	20.0000	+	34.6410i	0.891756	+	1.54457i	0.837769	+	0.546025i	$0.183860\pi$
$503$	20.0000	+	34.6410i	0.891756	+	1.54457i	0.0539870	+	0.998542i	$0.482807\pi$
$504$		0			0
$505$		0			0
$506$	−8.00000			−0.355643
$507$		0			0
$508$	−4.00000	+	6.92820i	−0.177471	+	0.307389i
$509$	−3.00000	−	5.19615i	−0.132973	−	0.230315i	0.791849	−	0.610718i	$-0.209119\pi$
$509$	−3.00000	−	5.19615i	−0.132973	−	0.230315i	−0.924821	+	0.380402i	$0.875786\pi$
$510$		0			0
$511$	5.50000	−	9.52628i	0.243306	−	0.421418i
$512$	11.0000			0.486136
$513$		0			0
$514$	−20.0000			−0.882162
$515$		0			0
$516$		0			0
$517$	−6.00000	−	10.3923i	−0.263880	−	0.457053i
$518$	1.50000	−	2.59808i	0.0659062	−	0.114153i
$519$		0			0
$520$		0			0
$521$	−18.0000			−0.788594			−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000			−0.788594			−0.394297	+	0.918983i	$0.629012\pi$
$522$		0			0
$523$	−14.5000	−	25.1147i	−0.634041	−	1.09819i	−0.986718	−	0.162446i	$-0.948062\pi$
$523$	−14.5000	−	25.1147i	−0.634041	−	1.09819i	0.352677	−	0.935745i	$-0.385272\pi$
$524$	−14.0000			−0.611593
$525$		0			0
$526$	−13.0000	−	22.5167i	−0.566827	−	0.981773i
$527$	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
$532$	−3.50000	−	2.59808i	−0.151744	−	0.112641i
$533$	60.0000			2.59889
$534$		0			0
$535$		0			0
$536$	−16.5000	−	28.5788i	−0.712691	−	1.23442i
$537$		0			0
$538$	−5.00000	−	8.66025i	−0.215565	−	0.373370i
$539$	−12.0000			−0.516877
$540$		0			0
$541$	19.5000	+	33.7750i	0.838370	+	1.45210i	0.891256	+	0.453500i	$0.149825\pi$
$541$	19.5000	+	33.7750i	0.838370	+	1.45210i	−0.0528859	+	0.998601i	$0.516842\pi$
$542$		0			0
$543$		0			0
$544$	−20.0000			−0.857493
$545$		0			0
$546$		0			0
$547$	14.5000	+	25.1147i	0.619975	+	1.07383i	0.989490	+	0.144604i	$0.0461907\pi$
$547$	14.5000	+	25.1147i	0.619975	+	1.07383i	−0.369514	+	0.929225i	$0.620476\pi$
$548$	−9.00000	+	15.5885i	−0.384461	+	0.665906i
$549$		0			0
$550$	10.0000			0.426401
$551$	28.0000	+	20.7846i	1.19284	+	0.885454i
$552$		0			0
$553$	−0.500000	+	0.866025i	−0.0212622	+	0.0368271i
$554$	−1.00000	+	1.73205i	−0.0424859	+	0.0735878i
$555$		0			0
$556$	2.50000	−	4.33013i	0.106024	−	0.183638i
$557$	−19.0000	−	32.9090i	−0.805056	−	1.39440i	−0.916253	−	0.400599i	$-0.868802\pi$
$557$	−19.0000	−	32.9090i	−0.805056	−	1.39440i	0.111198	−	0.993798i	$-0.464531\pi$
$558$		0			0
$559$	−5.00000			−0.211477
$560$		0			0
$561$		0			0
$562$	−8.00000			−0.337460
$563$	−18.0000			−0.758610			−0.379305	−	0.925272i	$-0.623837\pi$
$563$	−18.0000			−0.758610			−0.379305	+	0.925272i	$0.623837\pi$
$564$		0			0
$565$		0			0
$566$	−2.00000	−	3.46410i	−0.0840663	−	0.145607i
$567$		0			0
$568$	9.00000	−	15.5885i	0.377632	−	0.654077i
$569$	−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
$571$	33.0000			1.38101			0.690504	−	0.723329i	$-0.257389\pi$
$571$	33.0000			1.38101			0.690504	+	0.723329i	$0.257389\pi$
$572$	5.00000	−	8.66025i	0.209061	−	0.362103i
$573$		0			0
$574$	6.00000	+	10.3923i	0.250435	+	0.433766i
$575$	10.0000	−	17.3205i	0.417029	−	0.722315i
$576$		0			0
$577$	18.0000			0.749350			0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000			0.749350			0.374675	+	0.927156i	$0.377754\pi$
$578$	1.00000			0.0415945
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	4.00000	+	6.92820i	0.165663	+	0.286937i
$584$	16.5000	−	28.5788i	0.682775	−	1.18260i
$585$		0			0
$586$	−7.00000	+	12.1244i	−0.289167	+	0.500853i
$587$	−13.0000	+	22.5167i	−0.536567	+	0.929362i	0.462518	+	0.886610i	$0.346946\pi$
$587$	−13.0000	+	22.5167i	−0.536567	+	0.929362i	−0.999086	+	0.0427523i	$0.986387\pi$
$588$		0			0
$589$	12.0000	−	5.19615i	0.494451	−	0.214104i
$590$		0			0
$591$		0			0
$592$	1.50000	−	2.59808i	0.0616496	−	0.106780i
$593$	−3.00000	−	5.19615i	−0.123195	−	0.213380i	0.797831	−	0.602881i	$-0.205981\pi$
$593$	−3.00000	−	5.19615i	−0.123195	−	0.213380i	−0.921026	+	0.389501i	$0.872647\pi$
$594$		0			0
$595$		0			0
$596$	−6.00000			−0.245770
$597$		0			0
$598$	10.0000	+	17.3205i	0.408930	+	0.708288i
$599$	9.00000	+	15.5885i	0.367730	+	0.636927i	0.989210	−	0.146503i	$-0.0468017\pi$
$599$	9.00000	+	15.5885i	0.367730	+	0.636927i	−0.621480	+	0.783430i	$0.713468\pi$
$600$		0			0
$601$	−21.0000			−0.856608			−0.428304	−	0.903635i	$-0.640889\pi$
$601$	−21.0000			−0.856608			−0.428304	+	0.903635i	$0.640889\pi$
$602$	−0.500000	−	0.866025i	−0.0203785	−	0.0352966i
$603$		0			0
$604$	6.00000	+	10.3923i	0.244137	+	0.422857i
$605$		0			0
$606$		0			0
$607$	−43.0000			−1.74532			−0.872658	−	0.488332i	$-0.837606\pi$
$607$	−43.0000			−1.74532			−0.872658	+	0.488332i	$0.837606\pi$
$608$	−17.5000	−	12.9904i	−0.709719	−	0.526830i
$609$		0			0
$610$		0			0
$611$	−15.0000	+	25.9808i	−0.606835	+	1.05107i
$612$		0			0
$613$	15.0000	−	25.9808i	0.605844	−	1.04935i	−0.386073	−	0.922468i	$-0.626169\pi$
$613$	15.0000	−	25.9808i	0.605844	−	1.04935i	0.991917	−	0.126885i	$-0.0404979\pi$
$614$	−6.00000	−	10.3923i	−0.242140	−	0.419399i
$615$		0			0
$616$	6.00000			0.241747
$617$	3.00000	+	5.19615i	0.120775	+	0.209189i	0.920074	−	0.391745i	$-0.128129\pi$
$617$	3.00000	+	5.19615i	0.120775	+	0.209189i	−0.799298	+	0.600935i	$0.794795\pi$
$618$		0			0
$619$	25.0000			1.00483			0.502417	−	0.864625i	$-0.332444\pi$
$619$	25.0000			1.00483			0.502417	+	0.864625i	$0.332444\pi$
$620$		0			0
$621$		0			0
$622$	−9.00000	+	15.5885i	−0.360867	+	0.625040i
$623$	−3.00000	−	5.19615i	−0.120192	−	0.208179i
$624$		0			0
$625$	−12.5000	+	21.6506i	−0.500000	+	0.866025i
$626$	−22.0000			−0.879297
$627$		0			0
$628$	−3.00000			−0.119713
$629$	−6.00000	+	10.3923i	−0.239236	+	0.414368i
$630$		0			0
$631$	7.50000	+	12.9904i	0.298570	+	0.517139i	0.975809	−	0.218624i	$-0.0701569\pi$
$631$	7.50000	+	12.9904i	0.298570	+	0.517139i	−0.677239	+	0.735763i	$0.736824\pi$
$632$	−1.50000	+	2.59808i	−0.0596668	+	0.103346i
$633$		0			0
$634$	−4.00000			−0.158860
$635$		0			0
$636$		0			0
$637$	15.0000	+	25.9808i	0.594322	+	1.02940i
$638$	−16.0000			−0.633446
$639$		0			0
$640$		0			0
$641$	−15.0000	+	25.9808i	−0.592464	+	1.02618i	0.401435	+	0.915888i	$0.368512\pi$
$641$	−15.0000	+	25.9808i	−0.592464	+	1.02618i	−0.993899	+	0.110291i	$0.964822\pi$
$642$		0			0
$643$	9.50000	−	16.4545i	0.374643	−	0.648901i	−0.615630	−	0.788035i	$-0.711098\pi$
$643$	9.50000	−	16.4545i	0.374643	−	0.648901i	0.990274	+	0.139134i	$0.0444318\pi$
$644$	2.00000	−	3.46410i	0.0788110	−	0.136505i
$645$		0			0
$646$	−14.0000	−	10.3923i	−0.550823	−	0.408880i
$647$	24.0000			0.943537			0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000			0.943537			0.471769	+	0.881722i	$0.343616\pi$
$648$		0			0
$649$	10.0000	−	17.3205i	0.392534	−	0.679889i
$650$	−12.5000	−	21.6506i	−0.490290	−	0.849208i
$651$		0			0
$652$	1.50000	+	2.59808i	0.0587445	+	0.101749i
$653$	18.0000			0.704394			0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000			0.704394			0.352197	+	0.935926i	$0.385435\pi$
$654$		0			0
$655$		0			0
$656$	6.00000	+	10.3923i	0.234261	+	0.405751i
$657$		0			0
$658$	−6.00000			−0.233904
$659$	17.0000	+	29.4449i	0.662226	+	1.14701i	0.980029	+	0.198852i	$0.0637214\pi$
$659$	17.0000	+	29.4449i	0.662226	+	1.14701i	−0.317803	+	0.948157i	$0.602945\pi$
$660$		0			0
$661$	−21.0000	−	36.3731i	−0.816805	−	1.41475i	−0.908024	−	0.418917i	$-0.862410\pi$
$661$	−21.0000	−	36.3731i	−0.816805	−	1.41475i	0.0912190	−	0.995831i	$-0.470924\pi$
$662$	−12.5000	+	21.6506i	−0.485826	+	0.841476i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−16.0000	+	27.7128i	−0.619522	+	1.07304i
$668$	1.00000	−	1.73205i	0.0386912	−	0.0670151i
$669$		0			0
$670$		0			0
$671$	13.0000	+	22.5167i	0.501859	+	0.869246i
$672$		0			0
$673$	−33.0000			−1.27206			−0.636028	−	0.771666i	$-0.719424\pi$
$673$	−33.0000			−1.27206			−0.636028	+	0.771666i	$0.719424\pi$
$674$	6.50000	+	11.2583i	0.250371	+	0.433655i
$675$		0			0
$676$	−12.0000			−0.461538
$677$	−34.0000			−1.30673			−0.653363	−	0.757045i	$-0.726642\pi$
$677$	−34.0000			−1.30673			−0.653363	+	0.757045i	$0.726642\pi$
$678$		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
$683$	−12.0000			−0.459167			−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000			−0.459167			−0.229584	+	0.973289i	$0.573736\pi$
$684$		0			0
$685$		0			0
$686$	−6.50000	+	11.2583i	−0.248171	+	0.429845i
$687$		0			0
$688$	−0.500000	−	0.866025i	−0.0190623	−	0.0330169i
$689$	10.0000	−	17.3205i	0.380970	−	0.659859i
$690$		0			0
$691$	20.0000			0.760836			0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000			0.760836			0.380418	+	0.924815i	$0.375780\pi$
$692$		0			0
$693$		0			0
$694$	8.00000	+	13.8564i	0.303676	+	0.525982i
$695$		0			0
$696$		0			0
$697$	−24.0000	−	41.5692i	−0.909065	−	1.57455i
$698$	−10.5000	+	18.1865i	−0.397431	+	0.688370i
$699$		0			0
$700$	−2.50000	+	4.33013i	−0.0944911	+	0.163663i
$701$	10.0000	−	17.3205i	0.377695	−	0.654187i	−0.613032	−	0.790058i	$-0.710050\pi$
$701$	10.0000	−	17.3205i	0.377695	−	0.654187i	0.990726	+	0.135872i	$0.0433835\pi$
$702$		0			0
$703$	−12.0000	+	5.19615i	−0.452589	+	0.195977i
$704$	14.0000			0.527645
$705$		0			0
$706$	−2.00000	+	3.46410i	−0.0752710	+	0.130373i
$707$	7.00000	+	12.1244i	0.263262	+	0.455983i
$708$		0			0
$709$	19.5000	+	33.7750i	0.732338	+	1.26845i	0.955882	+	0.293752i	$0.0949041\pi$
$709$	19.5000	+	33.7750i	0.732338	+	1.26845i	−0.223544	+	0.974694i	$0.571763\pi$
$710$		0			0
$711$		0			0
$712$	−9.00000	−	15.5885i	−0.337289	−	0.584202i
$713$	6.00000	+	10.3923i	0.224702	+	0.389195i
$714$		0			0
$715$		0			0
$716$	−6.00000	−	10.3923i	−0.224231	−	0.388379i
$717$		0			0
$718$	10.0000	+	17.3205i	0.373197	+	0.646396i
$719$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$		0			0
$721$	13.0000			0.484145
$722$	−5.50000	−	18.1865i	−0.204689	−	0.676833i
$723$		0			0
$724$	−1.00000	+	1.73205i	−0.0371647	+	0.0643712i
$725$	20.0000	−	34.6410i	0.742781	−	1.28654i
$726$		0			0
$727$	23.5000	−	40.7032i	0.871567	−	1.50960i	0.0111912	−	0.999937i	$-0.496438\pi$
$727$	23.5000	−	40.7032i	0.871567	−	1.50960i	0.860376	−	0.509661i	$-0.170229\pi$
$728$	−7.50000	−	12.9904i	−0.277968	−	0.481456i
$729$		0			0
$730$		0			0
$731$	2.00000	+	3.46410i	0.0739727	+	0.128124i
$732$		0			0
$733$	−50.0000			−1.84679			−0.923396	−	0.383849i	$-0.874598\pi$
$733$	−50.0000			−1.84679			−0.923396	+	0.383849i	$0.874598\pi$
$734$	−7.00000			−0.258375
$735$		0			0
$736$	10.0000	−	17.3205i	0.368605	−	0.638442i
$737$	−11.0000	−	19.0526i	−0.405190	−	0.701810i
$738$		0			0
$739$	−9.50000	+	16.4545i	−0.349463	+	0.605288i	−0.986154	−	0.165831i	$-0.946969\pi$
$739$	−9.50000	+	16.4545i	−0.349463	+	0.605288i	0.636691	+	0.771119i	$0.280303\pi$
$740$		0			0
$741$		0			0
$742$	4.00000			0.146845
$743$	−25.0000	+	43.3013i	−0.917161	+	1.58857i	−0.113455	+	0.993543i	$0.536192\pi$
$743$	−25.0000	+	43.3013i	−0.917161	+	1.58857i	−0.803706	+	0.595026i	$0.797142\pi$
$744$		0			0
$745$		0			0
$746$	5.00000	−	8.66025i	0.183063	−	0.317074i
$747$		0			0
$748$	−8.00000			−0.292509
$749$	18.0000			0.657706
$750$		0			0
$751$	−22.5000	−	38.9711i	−0.821037	−	1.42208i	−0.904911	−	0.425601i	$-0.860063\pi$
$751$	−22.5000	−	38.9711i	−0.821037	−	1.42208i	0.0838743	−	0.996476i	$-0.473271\pi$
$752$	−6.00000			−0.218797
$753$		0			0
$754$	20.0000	+	34.6410i	0.728357	+	1.26155i
$755$		0			0
$756$		0			0
$757$	−6.50000	+	11.2583i	−0.236247	+	0.409191i	−0.959634	−	0.281251i	$-0.909251\pi$
$757$	−6.50000	+	11.2583i	−0.236247	+	0.409191i	0.723388	+	0.690442i	$0.242584\pi$
$758$	−2.50000	+	4.33013i	−0.0908041	+	0.157277i
$759$		0			0
$760$		0			0
$761$	−6.00000			−0.217500			−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000			−0.217500			−0.108750	+	0.994069i	$0.534685\pi$
$762$		0			0
$763$	1.00000	−	1.73205i	0.0362024	−	0.0627044i
$764$	−12.0000	−	20.7846i	−0.434145	−	0.751961i
$765$		0			0
$766$	−7.00000	−	12.1244i	−0.252920	−	0.438071i
$767$	−50.0000			−1.80540
$768$		0			0
$769$	18.5000	+	32.0429i	0.667127	+	1.15550i	0.978704	+	0.205277i	$0.0658095\pi$
$769$	18.5000	+	32.0429i	0.667127	+	1.15550i	−0.311577	+	0.950221i	$0.600857\pi$
$770$		0			0
$771$		0			0
$772$	−19.0000			−0.683825
$773$	7.00000	+	12.1244i	0.251773	+	0.436083i	0.964014	−	0.265852i	$-0.0856532\pi$
$773$	7.00000	+	12.1244i	0.251773	+	0.436083i	−0.712241	+	0.701935i	$0.752320\pi$
$774$		0			0
$775$	−7.50000	−	12.9904i	−0.269408	−	0.466628i
$776$	−3.00000	+	5.19615i	−0.107694	+	0.186531i
$777$		0			0
$778$	−24.0000			−0.860442
$779$	6.00000	−	51.9615i	0.214972	−	1.86171i
$780$		0			0
$781$	6.00000	−	10.3923i	0.214697	−	0.371866i
$782$	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
$787$	25.0000			0.891154			0.445577	−	0.895244i	$-0.352999\pi$
$787$	25.0000			0.891154			0.445577	+	0.895244i	$0.352999\pi$

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:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	171.f (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{2} +(0.500000 + 0.866025i) q^{4} +1.00000 q^{7} +3.00000 q^{8} +O(q^{10})\)	\(q+(0.500000 - 0.866025i) q^{2} +(0.500000 + 0.866025i) q^{4} +1.00000 q^{7} +3.00000 q^{8} +2.00000 q^{11} +(-2.50000 - 4.33013i) q^{13} +(0.500000 - 0.866025i) q^{14} +(0.500000 - 0.866025i) q^{16} +(-2.00000 + 3.46410i) q^{17} +(-4.00000 + 1.73205i) q^{19} +(1.00000 - 1.73205i) q^{22} +(-2.00000 - 3.46410i) q^{23} +(2.50000 + 4.33013i) q^{25} -5.00000 q^{26} +(0.500000 + 0.866025i) q^{28} +(-4.00000 - 6.92820i) q^{29} -3.00000 q^{31} +(2.50000 + 4.33013i) q^{32} +(2.00000 + 3.46410i) q^{34} +3.00000 q^{37} +(-0.500000 + 4.33013i) q^{38} +(-6.00000 + 10.3923i) q^{41} +(0.500000 - 0.866025i) q^{43} +(1.00000 + 1.73205i) q^{44} -4.00000 q^{46} +(-3.00000 - 5.19615i) q^{47} -6.00000 q^{49} +5.00000 q^{50} +(2.50000 - 4.33013i) q^{52} +(2.00000 + 3.46410i) q^{53} +3.00000 q^{56} -8.00000 q^{58} +(5.00000 - 8.66025i) q^{59} +(6.50000 + 11.2583i) q^{61} +(-1.50000 + 2.59808i) q^{62} +7.00000 q^{64} +(-5.50000 - 9.52628i) q^{67} -4.00000 q^{68} +(3.00000 - 5.19615i) q^{71} +(5.50000 - 9.52628i) q^{73} +(1.50000 - 2.59808i) q^{74} +(-3.50000 - 2.59808i) q^{76} +2.00000 q^{77} +(-0.500000 + 0.866025i) q^{79} +(6.00000 + 10.3923i) q^{82} +(-0.500000 - 0.866025i) q^{86} +6.00000 q^{88} +(-3.00000 - 5.19615i) q^{89} +(-2.50000 - 4.33013i) q^{91} +(2.00000 - 3.46410i) q^{92} -6.00000 q^{94} +(-1.00000 + 1.73205i) q^{97} +(-3.00000 + 5.19615i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + q^{4} + 2 q^{7} + 6 q^{8}+O(q^{10}) \) 2 * q + q^2 + q^4 + 2 * q^7 + 6 * q^8	\( 2 q + q^{2} + q^{4} + 2 q^{7} + 6 q^{8} + 4 q^{11} - 5 q^{13} + q^{14} + q^{16} - 4 q^{17} - 8 q^{19} + 2 q^{22} - 4 q^{23} + 5 q^{25} - 10 q^{26} + q^{28} - 8 q^{29} - 6 q^{31} + 5 q^{32} + 4 q^{34} + 6 q^{37} - q^{38} - 12 q^{41} + q^{43} + 2 q^{44} - 8 q^{46} - 6 q^{47} - 12 q^{49} + 10 q^{50} + 5 q^{52} + 4 q^{53} + 6 q^{56} - 16 q^{58} + 10 q^{59} + 13 q^{61} - 3 q^{62} + 14 q^{64} - 11 q^{67} - 8 q^{68} + 6 q^{71} + 11 q^{73} + 3 q^{74} - 7 q^{76} + 4 q^{77} - q^{79} + 12 q^{82} - q^{86} + 12 q^{88} - 6 q^{89} - 5 q^{91} + 4 q^{92} - 12 q^{94} - 2 q^{97} - 6 q^{98}+O(q^{100}) \) 2 * q + q^2 + q^4 + 2 * q^7 + 6 * q^8 + 4 * q^11 - 5 * q^13 + q^14 + q^16 - 4 * q^17 - 8 * q^19 + 2 * q^22 - 4 * q^23 + 5 * q^25 - 10 * q^26 + q^28 - 8 * q^29 - 6 * q^31 + 5 * q^32 + 4 * q^34 + 6 * q^37 - q^38 - 12 * q^41 + q^43 + 2 * q^44 - 8 * q^46 - 6 * q^47 - 12 * q^49 + 10 * q^50 + 5 * q^52 + 4 * q^53 + 6 * q^56 - 16 * q^58 + 10 * q^59 + 13 * q^61 - 3 * q^62 + 14 * q^64 - 11 * q^67 - 8 * q^68 + 6 * q^71 + 11 * q^73 + 3 * q^74 - 7 * q^76 + 4 * q^77 - q^79 + 12 * q^82 - q^86 + 12 * q^88 - 6 * q^89 - 5 * q^91 + 4 * q^92 - 12 * q^94 - 2 * q^97 - 6 * q^98

Introduction

L-functions

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