LMFDB - Embedded newform 54.2.c.a.19.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [54,2,Mod(19,54)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(54, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("54.19");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 54 = 2 \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	54.c (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$0.431192170915$
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 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			19.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	54.19
Dual form			54.2.c.a.37.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-1.00000 + 1.73205i) q^{7} -1.00000 q^{8} +O(q^{10})$	\(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-1.00000 + 1.73205i) q^{7} -1.00000 q^{8} +(-1.50000 + 2.59808i) q^{11} +(-1.00000 - 1.73205i) q^{13} +(1.00000 + 1.73205i) q^{14} +(-0.500000 + 0.866025i) q^{16} +3.00000 q^{17} -1.00000 q^{19} +(1.50000 + 2.59808i) q^{22} +(-3.00000 - 5.19615i) q^{23} +(2.50000 - 4.33013i) q^{25} -2.00000 q^{26} +2.00000 q^{28} +(3.00000 - 5.19615i) q^{29} +(2.00000 + 3.46410i) q^{31} +(0.500000 + 0.866025i) q^{32} +(1.50000 - 2.59808i) q^{34} -4.00000 q^{37} +(-0.500000 + 0.866025i) q^{38} +(4.50000 + 7.79423i) q^{41} +(0.500000 - 0.866025i) q^{43} +3.00000 q^{44} -6.00000 q^{46} +(-3.00000 + 5.19615i) q^{47} +(1.50000 + 2.59808i) q^{49} +(-2.50000 - 4.33013i) q^{50} +(-1.00000 + 1.73205i) q^{52} -12.0000 q^{53} +(1.00000 - 1.73205i) q^{56} +(-3.00000 - 5.19615i) q^{58} +(1.50000 + 2.59808i) q^{59} +(-4.00000 + 6.92820i) q^{61} +4.00000 q^{62} +1.00000 q^{64} +(-2.50000 - 4.33013i) q^{67} +(-1.50000 - 2.59808i) q^{68} +12.0000 q^{71} +11.0000 q^{73} +(-2.00000 + 3.46410i) q^{74} +(0.500000 + 0.866025i) q^{76} +(-3.00000 - 5.19615i) q^{77} +(2.00000 - 3.46410i) q^{79} +9.00000 q^{82} +(6.00000 - 10.3923i) q^{83} +(-0.500000 - 0.866025i) q^{86} +(1.50000 - 2.59808i) q^{88} -6.00000 q^{89} +4.00000 q^{91} +(-3.00000 + 5.19615i) q^{92} +(3.00000 + 5.19615i) q^{94} +(-2.50000 + 4.33013i) q^{97} +3.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 3 q^{11} - 2 q^{13} + 2 q^{14} - q^{16} + 6 q^{17} - 2 q^{19} + 3 q^{22} - 6 q^{23} + 5 q^{25} - 4 q^{26} + 4 q^{28} + 6 q^{29} + 4 q^{31} + q^{32} + 3 q^{34} - 8 q^{37} - q^{38} + 9 q^{41} + q^{43} + 6 q^{44} - 12 q^{46} - 6 q^{47} + 3 q^{49} - 5 q^{50} - 2 q^{52} - 24 q^{53} + 2 q^{56} - 6 q^{58} + 3 q^{59} - 8 q^{61} + 8 q^{62} + 2 q^{64} - 5 q^{67} - 3 q^{68} + 24 q^{71} + 22 q^{73} - 4 q^{74} + q^{76} - 6 q^{77} + 4 q^{79} + 18 q^{82} + 12 q^{83} - q^{86} + 3 q^{88} - 12 q^{89} + 8 q^{91} - 6 q^{92} + 6 q^{94} - 5 q^{97} + 6 q^{98}+O(q^{100}) $ 2 * q + q^2 - q^4 - 2 * q^7 - 2 * q^8 - 3 * q^11 - 2 * q^13 + 2 * q^14 - q^16 + 6 * q^17 - 2 * q^19 + 3 * q^22 - 6 * q^23 + 5 * q^25 - 4 * q^26 + 4 * q^28 + 6 * q^29 + 4 * q^31 + q^32 + 3 * q^34 - 8 * q^37 - q^38 + 9 * q^41 + q^43 + 6 * q^44 - 12 * q^46 - 6 * q^47 + 3 * q^49 - 5 * q^50 - 2 * q^52 - 24 * q^53 + 2 * q^56 - 6 * q^58 + 3 * q^59 - 8 * q^61 + 8 * q^62 + 2 * q^64 - 5 * q^67 - 3 * q^68 + 24 * q^71 + 22 * q^73 - 4 * q^74 + q^76 - 6 * q^77 + 4 * q^79 + 18 * q^82 + 12 * q^83 - q^86 + 3 * q^88 - 12 * q^89 + 8 * q^91 - 6 * q^92 + 6 * q^94 - 5 * q^97 + 6 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/54\mathbb{Z}\right)^\times$.

$n$	$29$
$\chi(n)$	$e\left(\frac{2}{3}\right)$

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	$-0.166667\pi$
$5$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$6$		0			0
$7$	−1.00000	+	1.73205i	−0.377964	+	0.654654i	−0.990766	−	0.135583i	$-0.956709\pi$
$7$	−1.00000	+	1.73205i	−0.377964	+	0.654654i	0.612801	+	0.790237i	$0.290043\pi$
$8$	−1.00000			−0.353553
$9$		0			0
$10$		0			0
$11$	−1.50000	+	2.59808i	−0.452267	+	0.783349i	−0.998526	−	0.0542666i	$-0.982718\pi$
$11$	−1.50000	+	2.59808i	−0.452267	+	0.783349i	0.546259	+	0.837616i	$0.316051\pi$
$12$		0			0
$13$	−1.00000	−	1.73205i	−0.277350	−	0.480384i	0.693375	−	0.720577i	$-0.256123\pi$
$13$	−1.00000	−	1.73205i	−0.277350	−	0.480384i	−0.970725	+	0.240192i	$0.922790\pi$
$14$	1.00000	+	1.73205i	0.267261	+	0.462910i
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.125000	+	0.216506i
$17$	3.00000			0.727607			0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000			0.727607			0.363803	+	0.931476i	$0.381478\pi$
$18$		0			0
$19$	−1.00000			−0.229416			−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000			−0.229416			−0.114708	+	0.993399i	$0.536593\pi$
$20$		0			0
$21$		0			0
$22$	1.50000	+	2.59808i	0.319801	+	0.553912i
$23$	−3.00000	−	5.19615i	−0.625543	−	1.08347i	−0.988436	−	0.151642i	$-0.951544\pi$
$23$	−3.00000	−	5.19615i	−0.625543	−	1.08347i	0.362892	−	0.931831i	$-0.381789\pi$
$24$		0			0
$25$	2.50000	−	4.33013i	0.500000	−	0.866025i
$26$	−2.00000			−0.392232
$27$		0			0
$28$	2.00000			0.377964
$29$	3.00000	−	5.19615i	0.557086	−	0.964901i	−0.440652	−	0.897678i	$-0.645253\pi$
$29$	3.00000	−	5.19615i	0.557086	−	0.964901i	0.997738	−	0.0672232i	$-0.0214140\pi$
$30$		0			0
$31$	2.00000	+	3.46410i	0.359211	+	0.622171i	0.987829	−	0.155543i	$-0.0497126\pi$
$31$	2.00000	+	3.46410i	0.359211	+	0.622171i	−0.628619	+	0.777714i	$0.716379\pi$
$32$	0.500000	+	0.866025i	0.0883883	+	0.153093i
$33$		0			0
$34$	1.50000	−	2.59808i	0.257248	−	0.445566i
$35$		0			0
$36$		0			0
$37$	−4.00000			−0.657596			−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000			−0.657596			−0.328798	+	0.944400i	$0.606644\pi$
$38$	−0.500000	+	0.866025i	−0.0811107	+	0.140488i
$39$		0			0
$40$		0			0
$41$	4.50000	+	7.79423i	0.702782	+	1.21725i	0.967486	+	0.252924i	$0.0813924\pi$
$41$	4.50000	+	7.79423i	0.702782	+	1.21725i	−0.264704	+	0.964330i	$0.585274\pi$
$42$		0			0
$43$	0.500000	−	0.866025i	0.0762493	−	0.132068i	−0.825380	−	0.564578i	$-0.809039\pi$
$43$	0.500000	−	0.866025i	0.0762493	−	0.132068i	0.901629	+	0.432511i	$0.142372\pi$
$44$	3.00000			0.452267
$45$		0			0
$46$	−6.00000			−0.884652
$47$	−3.00000	+	5.19615i	−0.437595	+	0.757937i	−0.997503	−	0.0706177i	$-0.977503\pi$
$47$	−3.00000	+	5.19615i	−0.437595	+	0.757937i	0.559908	+	0.828554i	$0.310836\pi$
$48$		0			0
$49$	1.50000	+	2.59808i	0.214286	+	0.371154i
$50$	−2.50000	−	4.33013i	−0.353553	−	0.612372i
$51$		0			0
$52$	−1.00000	+	1.73205i	−0.138675	+	0.240192i
$53$	−12.0000			−1.64833			−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000			−1.64833			−0.824163	+	0.566352i	$0.808354\pi$
$54$		0			0
$55$		0			0
$56$	1.00000	−	1.73205i	0.133631	−	0.231455i
$57$		0			0
$58$	−3.00000	−	5.19615i	−0.393919	−	0.682288i
$59$	1.50000	+	2.59808i	0.195283	+	0.338241i	0.946993	−	0.321253i	$-0.104104\pi$
$59$	1.50000	+	2.59808i	0.195283	+	0.338241i	−0.751710	+	0.659494i	$0.770771\pi$
$60$		0			0
$61$	−4.00000	+	6.92820i	−0.512148	+	0.887066i	0.487753	+	0.872982i	$0.337817\pi$
$61$	−4.00000	+	6.92820i	−0.512148	+	0.887066i	−0.999901	+	0.0140840i	$0.995517\pi$
$62$	4.00000			0.508001
$63$		0			0
$64$	1.00000			0.125000
$65$		0			0
$66$		0			0
$67$	−2.50000	−	4.33013i	−0.305424	−	0.529009i	0.671932	−	0.740613i	$-0.265465\pi$
$67$	−2.50000	−	4.33013i	−0.305424	−	0.529009i	−0.977356	+	0.211604i	$0.932131\pi$
$68$	−1.50000	−	2.59808i	−0.181902	−	0.315063i
$69$		0			0
$70$		0			0
$71$	12.0000			1.42414			0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000			1.42414			0.712069	+	0.702109i	$0.247758\pi$
$72$		0			0
$73$	11.0000			1.28745			0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000			1.28745			0.643726	+	0.765256i	$0.277388\pi$
$74$	−2.00000	+	3.46410i	−0.232495	+	0.402694i
$75$		0			0
$76$	0.500000	+	0.866025i	0.0573539	+	0.0993399i
$77$	−3.00000	−	5.19615i	−0.341882	−	0.592157i
$78$		0			0
$79$	2.00000	−	3.46410i	0.225018	−	0.389742i	−0.731307	−	0.682048i	$-0.761089\pi$
$79$	2.00000	−	3.46410i	0.225018	−	0.389742i	0.956325	+	0.292306i	$0.0944227\pi$
$80$		0			0
$81$		0			0
$82$	9.00000			0.993884
$83$	6.00000	−	10.3923i	0.658586	−	1.14070i	−0.322396	−	0.946605i	$-0.604488\pi$
$83$	6.00000	−	10.3923i	0.658586	−	1.14070i	0.980982	−	0.194099i	$-0.0621783\pi$
$84$		0			0
$85$		0			0
$86$	−0.500000	−	0.866025i	−0.0539164	−	0.0933859i
$87$		0			0
$88$	1.50000	−	2.59808i	0.159901	−	0.276956i
$89$	−6.00000			−0.635999			−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000			−0.635999			−0.317999	+	0.948091i	$0.603011\pi$
$90$		0			0
$91$	4.00000			0.419314
$92$	−3.00000	+	5.19615i	−0.312772	+	0.541736i
$93$		0			0
$94$	3.00000	+	5.19615i	0.309426	+	0.535942i
$95$		0			0
$96$		0			0
$97$	−2.50000	+	4.33013i	−0.253837	+	0.439658i	−0.964579	−	0.263795i	$-0.915026\pi$
$97$	−2.50000	+	4.33013i	−0.253837	+	0.439658i	0.710742	+	0.703452i	$0.248359\pi$
$98$	3.00000			0.303046
$99$		0			0
$100$	−5.00000			−0.500000
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$	−7.00000	−	12.1244i	−0.689730	−	1.19465i	−0.971925	−	0.235291i	$-0.924396\pi$
$103$	−7.00000	−	12.1244i	−0.689730	−	1.19465i	0.282194	−	0.959357i	$-0.408938\pi$
$104$	1.00000	+	1.73205i	0.0980581	+	0.169842i
$105$		0			0
$106$	−6.00000	+	10.3923i	−0.582772	+	1.00939i
$107$	−3.00000			−0.290021			−0.145010	−	0.989430i	$-0.546322\pi$
$107$	−3.00000			−0.290021			−0.145010	+	0.989430i	$0.546322\pi$
$108$		0			0
$109$	−16.0000			−1.53252			−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000			−1.53252			−0.766261	+	0.642529i	$0.777885\pi$
$110$		0			0
$111$		0			0
$112$	−1.00000	−	1.73205i	−0.0944911	−	0.163663i
$113$	3.00000	+	5.19615i	0.282216	+	0.488813i	0.971930	−	0.235269i	$-0.0755971\pi$
$113$	3.00000	+	5.19615i	0.282216	+	0.488813i	−0.689714	+	0.724082i	$0.742264\pi$
$114$		0			0
$115$		0			0
$116$	−6.00000			−0.557086
$117$		0			0
$118$	3.00000			0.276172
$119$	−3.00000	+	5.19615i	−0.275010	+	0.476331i
$120$		0			0
$121$	1.00000	+	1.73205i	0.0909091	+	0.157459i
$122$	4.00000	+	6.92820i	0.362143	+	0.627250i
$123$		0			0
$124$	2.00000	−	3.46410i	0.179605	−	0.311086i
$125$		0			0
$126$		0			0
$127$	2.00000			0.177471			0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000			0.177471			0.0887357	+	0.996055i	$0.471717\pi$
$128$	0.500000	−	0.866025i	0.0441942	−	0.0765466i
$129$		0			0
$130$		0			0
$131$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$131$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$132$		0			0
$133$	1.00000	−	1.73205i	0.0867110	−	0.150188i
$134$	−5.00000			−0.431934
$135$		0			0
$136$	−3.00000			−0.257248
$137$	−1.50000	+	2.59808i	−0.128154	+	0.221969i	−0.922961	−	0.384893i	$-0.874238\pi$
$137$	−1.50000	+	2.59808i	−0.128154	+	0.221969i	0.794808	+	0.606861i	$0.207572\pi$
$138$		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
$142$	6.00000	−	10.3923i	0.503509	−	0.872103i
$143$	6.00000			0.501745
$144$		0			0
$145$		0			0
$146$	5.50000	−	9.52628i	0.455183	−	0.788400i
$147$		0			0
$148$	2.00000	+	3.46410i	0.164399	+	0.284747i
$149$	−3.00000	−	5.19615i	−0.245770	−	0.425685i	0.716578	−	0.697507i	$-0.245707\pi$
$149$	−3.00000	−	5.19615i	−0.245770	−	0.425685i	−0.962348	+	0.271821i	$0.912374\pi$
$150$		0			0
$151$	5.00000	−	8.66025i	0.406894	−	0.704761i	−0.587646	−	0.809118i	$-0.699945\pi$
$151$	5.00000	−	8.66025i	0.406894	−	0.704761i	0.994540	+	0.104357i	$0.0332784\pi$
$152$	1.00000			0.0811107
$153$		0			0
$154$	−6.00000			−0.483494
$155$		0			0
$156$		0			0
$157$	2.00000	+	3.46410i	0.159617	+	0.276465i	0.934731	−	0.355357i	$-0.115641\pi$
$157$	2.00000	+	3.46410i	0.159617	+	0.276465i	−0.775113	+	0.631822i	$0.782307\pi$
$158$	−2.00000	−	3.46410i	−0.159111	−	0.275589i
$159$		0			0
$160$		0			0
$161$	12.0000			0.945732
$162$		0			0
$163$	−4.00000			−0.313304			−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000			−0.313304			−0.156652	+	0.987654i	$0.550070\pi$
$164$	4.50000	−	7.79423i	0.351391	−	0.608627i
$165$		0			0
$166$	−6.00000	−	10.3923i	−0.465690	−	0.806599i
$167$	−6.00000	−	10.3923i	−0.464294	−	0.804181i	0.534875	−	0.844931i	$-0.320359\pi$
$167$	−6.00000	−	10.3923i	−0.464294	−	0.804181i	−0.999169	+	0.0407502i	$0.987025\pi$
$168$		0			0
$169$	4.50000	−	7.79423i	0.346154	−	0.599556i
$170$		0			0
$171$		0			0
$172$	−1.00000			−0.0762493
$173$	−3.00000	+	5.19615i	−0.228086	+	0.395056i	−0.957241	−	0.289292i	$-0.906580\pi$
$173$	−3.00000	+	5.19615i	−0.228086	+	0.395056i	0.729155	+	0.684349i	$0.239913\pi$
$174$		0			0
$175$	5.00000	+	8.66025i	0.377964	+	0.654654i
$176$	−1.50000	−	2.59808i	−0.113067	−	0.195837i
$177$		0			0
$178$	−3.00000	+	5.19615i	−0.224860	+	0.389468i
$179$	−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
$181$	14.0000			1.04061			0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000			1.04061			0.520306	+	0.853980i	$0.325818\pi$
$182$	2.00000	−	3.46410i	0.148250	−	0.256776i
$183$		0			0
$184$	3.00000	+	5.19615i	0.221163	+	0.383065i
$185$		0			0
$186$		0			0
$187$	−4.50000	+	7.79423i	−0.329073	+	0.569970i
$188$	6.00000			0.437595
$189$		0			0
$190$		0			0
$191$	−9.00000	+	15.5885i	−0.651217	+	1.12794i	0.331611	+	0.943416i	$0.392408\pi$
$191$	−9.00000	+	15.5885i	−0.651217	+	1.12794i	−0.982828	+	0.184525i	$0.940925\pi$
$192$		0			0
$193$	−2.50000	−	4.33013i	−0.179954	−	0.311689i	0.761911	−	0.647682i	$-0.224262\pi$
$193$	−2.50000	−	4.33013i	−0.179954	−	0.311689i	−0.941865	+	0.335993i	$0.890928\pi$
$194$	2.50000	+	4.33013i	0.179490	+	0.310885i
$195$		0			0
$196$	1.50000	−	2.59808i	0.107143	−	0.185577i
$197$	12.0000			0.854965			0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000			0.854965			0.427482	+	0.904024i	$0.359401\pi$
$198$		0			0
$199$	−10.0000			−0.708881			−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000			−0.708881			−0.354441	+	0.935079i	$0.615329\pi$
$200$	−2.50000	+	4.33013i	−0.176777	+	0.306186i
$201$		0			0
$202$		0			0
$203$	6.00000	+	10.3923i	0.421117	+	0.729397i
$204$		0			0
$205$		0			0
$206$	−14.0000			−0.975426
$207$		0			0
$208$	2.00000			0.138675
$209$	1.50000	−	2.59808i	0.103757	−	0.179713i
$210$		0			0
$211$	−10.0000	−	17.3205i	−0.688428	−	1.19239i	−0.972346	−	0.233544i	$-0.924968\pi$
$211$	−10.0000	−	17.3205i	−0.688428	−	1.19239i	0.283918	−	0.958849i	$-0.408366\pi$
$212$	6.00000	+	10.3923i	0.412082	+	0.713746i
$213$		0			0
$214$	−1.50000	+	2.59808i	−0.102538	+	0.177601i
$215$		0			0
$216$		0			0
$217$	−8.00000			−0.543075
$218$	−8.00000	+	13.8564i	−0.541828	+	0.938474i
$219$		0			0
$220$		0			0
$221$	−3.00000	−	5.19615i	−0.201802	−	0.349531i
$222$		0			0
$223$	−13.0000	+	22.5167i	−0.870544	+	1.50783i	−0.00910984	+	0.999959i	$0.502900\pi$
$223$	−13.0000	+	22.5167i	−0.870544	+	1.50783i	−0.861435	+	0.507869i	$0.830434\pi$
$224$	−2.00000			−0.133631
$225$		0			0
$226$	6.00000			0.399114
$227$	10.5000	−	18.1865i	0.696909	−	1.20708i	−0.272623	−	0.962121i	$-0.587891\pi$
$227$	10.5000	−	18.1865i	0.696909	−	1.20708i	0.969533	−	0.244962i	$-0.0787754\pi$
$228$		0			0
$229$	−7.00000	−	12.1244i	−0.462573	−	0.801200i	0.536515	−	0.843891i	$-0.319740\pi$
$229$	−7.00000	−	12.1244i	−0.462573	−	0.801200i	−0.999088	+	0.0426906i	$0.986407\pi$
$230$		0			0
$231$		0			0
$232$	−3.00000	+	5.19615i	−0.196960	+	0.341144i
$233$	−3.00000			−0.196537			−0.0982683	−	0.995160i	$-0.531330\pi$
$233$	−3.00000			−0.196537			−0.0982683	+	0.995160i	$0.531330\pi$
$234$		0			0
$235$		0			0
$236$	1.50000	−	2.59808i	0.0976417	−	0.169120i
$237$		0			0
$238$	3.00000	+	5.19615i	0.194461	+	0.336817i
$239$	3.00000	+	5.19615i	0.194054	+	0.336111i	0.946590	−	0.322440i	$-0.104503\pi$
$239$	3.00000	+	5.19615i	0.194054	+	0.336111i	−0.752536	+	0.658551i	$0.771170\pi$
$240$		0			0
$241$	3.50000	−	6.06218i	0.225455	−	0.390499i	−0.731001	−	0.682376i	$-0.760947\pi$
$241$	3.50000	−	6.06218i	0.225455	−	0.390499i	0.956456	+	0.291877i	$0.0942799\pi$
$242$	2.00000			0.128565
$243$		0			0
$244$	8.00000			0.512148
$245$		0			0
$246$		0			0
$247$	1.00000	+	1.73205i	0.0636285	+	0.110208i
$248$	−2.00000	−	3.46410i	−0.127000	−	0.219971i
$249$		0			0
$250$		0			0
$251$	−21.0000			−1.32551			−0.662754	−	0.748837i	$-0.730613\pi$
$251$	−21.0000			−1.32551			−0.662754	+	0.748837i	$0.730613\pi$
$252$		0			0
$253$	18.0000			1.13165
$254$	1.00000	−	1.73205i	0.0627456	−	0.108679i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$	−10.5000	−	18.1865i	−0.654972	−	1.13444i	−0.981901	−	0.189396i	$-0.939347\pi$
$257$	−10.5000	−	18.1865i	−0.654972	−	1.13444i	0.326929	−	0.945049i	$-0.393986\pi$
$258$		0			0
$259$	4.00000	−	6.92820i	0.248548	−	0.430498i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	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
$266$	−1.00000	−	1.73205i	−0.0613139	−	0.106199i
$267$		0			0
$268$	−2.50000	+	4.33013i	−0.152712	+	0.264505i
$269$	24.0000			1.46331			0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000			1.46331			0.731653	+	0.681677i	$0.238749\pi$
$270$		0			0
$271$	20.0000			1.21491			0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000			1.21491			0.607457	+	0.794353i	$0.292190\pi$
$272$	−1.50000	+	2.59808i	−0.0909509	+	0.157532i
$273$		0			0
$274$	1.50000	+	2.59808i	0.0906183	+	0.156956i
$275$	7.50000	+	12.9904i	0.452267	+	0.783349i
$276$		0			0
$277$	5.00000	−	8.66025i	0.300421	−	0.520344i	−0.675810	−	0.737075i	$-0.736206\pi$
$277$	5.00000	−	8.66025i	0.300421	−	0.520344i	0.976231	+	0.216731i	$0.0695395\pi$
$278$	19.0000			1.13954
$279$		0			0
$280$		0			0
$281$	3.00000	−	5.19615i	0.178965	−	0.309976i	−0.762561	−	0.646916i	$-0.776058\pi$
$281$	3.00000	−	5.19615i	0.178965	−	0.309976i	0.941526	+	0.336939i	$0.109392\pi$
$282$		0			0
$283$	2.00000	+	3.46410i	0.118888	+	0.205919i	0.919327	−	0.393494i	$-0.128734\pi$
$283$	2.00000	+	3.46410i	0.118888	+	0.205919i	−0.800439	+	0.599414i	$0.795400\pi$
$284$	−6.00000	−	10.3923i	−0.356034	−	0.616670i
$285$		0			0
$286$	3.00000	−	5.19615i	0.177394	−	0.307255i
$287$	−18.0000			−1.06251
$288$		0			0
$289$	−8.00000			−0.470588
$290$		0			0
$291$		0			0
$292$	−5.50000	−	9.52628i	−0.321863	−	0.557483i
$293$	15.0000	+	25.9808i	0.876309	+	1.51781i	0.855361	+	0.518032i	$0.173335\pi$
$293$	15.0000	+	25.9808i	0.876309	+	1.51781i	0.0209480	+	0.999781i	$0.493332\pi$
$294$		0			0
$295$		0			0
$296$	4.00000			0.232495
$297$		0			0
$298$	−6.00000			−0.347571
$299$	−6.00000	+	10.3923i	−0.346989	+	0.601003i
$300$		0			0
$301$	1.00000	+	1.73205i	0.0576390	+	0.0998337i
$302$	−5.00000	−	8.66025i	−0.287718	−	0.498342i
$303$		0			0
$304$	0.500000	−	0.866025i	0.0286770	−	0.0496700i
$305$		0			0
$306$		0			0
$307$	−7.00000			−0.399511			−0.199756	−	0.979846i	$-0.564015\pi$
$307$	−7.00000			−0.399511			−0.199756	+	0.979846i	$0.564015\pi$
$308$	−3.00000	+	5.19615i	−0.170941	+	0.296078i
$309$		0			0
$310$		0			0
$311$	−9.00000	−	15.5885i	−0.510343	−	0.883940i	−0.999928	−	0.0119847i	$-0.996185\pi$
$311$	−9.00000	−	15.5885i	−0.510343	−	0.883940i	0.489585	−	0.871956i	$-0.337148\pi$
$312$		0			0
$313$	−14.5000	+	25.1147i	−0.819588	+	1.41957i	0.0863973	+	0.996261i	$0.472465\pi$
$313$	−14.5000	+	25.1147i	−0.819588	+	1.41957i	−0.905986	+	0.423308i	$0.860869\pi$
$314$	4.00000			0.225733
$315$		0			0
$316$	−4.00000			−0.225018
$317$	−9.00000	+	15.5885i	−0.505490	+	0.875535i	0.494489	+	0.869184i	$0.335355\pi$
$317$	−9.00000	+	15.5885i	−0.505490	+	0.875535i	−0.999980	+	0.00635137i	$0.997978\pi$
$318$		0			0
$319$	9.00000	+	15.5885i	0.503903	+	0.872786i
$320$		0			0
$321$		0			0
$322$	6.00000	−	10.3923i	0.334367	−	0.579141i
$323$	−3.00000			−0.166924
$324$		0			0
$325$	−10.0000			−0.554700
$326$	−2.00000	+	3.46410i	−0.110770	+	0.191859i
$327$		0			0
$328$	−4.50000	−	7.79423i	−0.248471	−	0.430364i
$329$	−6.00000	−	10.3923i	−0.330791	−	0.572946i
$330$		0			0
$331$	2.00000	−	3.46410i	0.109930	−	0.190404i	−0.805812	−	0.592172i	$-0.798271\pi$
$331$	2.00000	−	3.46410i	0.109930	−	0.190404i	0.915742	+	0.401768i	$0.131604\pi$
$332$	−12.0000			−0.658586
$333$		0			0
$334$	−12.0000			−0.656611
$335$		0			0
$336$		0			0
$337$	0.500000	+	0.866025i	0.0272367	+	0.0471754i	0.879322	−	0.476227i	$-0.157996\pi$
$337$	0.500000	+	0.866025i	0.0272367	+	0.0471754i	−0.852086	+	0.523402i	$0.824663\pi$
$338$	−4.50000	−	7.79423i	−0.244768	−	0.423950i
$339$		0			0
$340$		0			0
$341$	−12.0000			−0.649836
$342$		0			0
$343$	−20.0000			−1.07990
$344$	−0.500000	+	0.866025i	−0.0269582	+	0.0466930i
$345$		0			0
$346$	3.00000	+	5.19615i	0.161281	+	0.279347i
$347$	16.5000	+	28.5788i	0.885766	+	1.53419i	0.844833	+	0.535031i	$0.179700\pi$
$347$	16.5000	+	28.5788i	0.885766	+	1.53419i	0.0409337	+	0.999162i	$0.486967\pi$
$348$		0			0
$349$	8.00000	−	13.8564i	0.428230	−	0.741716i	−0.568486	−	0.822693i	$-0.692471\pi$
$349$	8.00000	−	13.8564i	0.428230	−	0.741716i	0.996716	+	0.0809766i	$0.0258039\pi$
$350$	10.0000			0.534522
$351$		0			0
$352$	−3.00000			−0.159901
$353$	−10.5000	+	18.1865i	−0.558859	+	0.967972i	0.438733	+	0.898617i	$0.355427\pi$
$353$	−10.5000	+	18.1865i	−0.558859	+	0.967972i	−0.997592	+	0.0693543i	$0.977906\pi$
$354$		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
$359$	18.0000			0.950004			0.475002	−	0.879985i	$-0.342447\pi$
$359$	18.0000			0.950004			0.475002	+	0.879985i	$0.342447\pi$
$360$		0			0
$361$	−18.0000			−0.947368
$362$	7.00000	−	12.1244i	0.367912	−	0.637242i
$363$		0			0
$364$	−2.00000	−	3.46410i	−0.104828	−	0.181568i
$365$		0			0
$366$		0			0
$367$	14.0000	−	24.2487i	0.730794	−	1.26577i	−0.225750	−	0.974185i	$-0.572483\pi$
$367$	14.0000	−	24.2487i	0.730794	−	1.26577i	0.956544	−	0.291587i	$-0.0941834\pi$
$368$	6.00000			0.312772
$369$		0			0
$370$		0			0
$371$	12.0000	−	20.7846i	0.623009	−	1.07908i
$372$		0			0
$373$	17.0000	+	29.4449i	0.880227	+	1.52460i	0.851089	+	0.525022i	$0.175943\pi$
$373$	17.0000	+	29.4449i	0.880227	+	1.52460i	0.0291379	+	0.999575i	$0.490724\pi$
$374$	4.50000	+	7.79423i	0.232689	+	0.403030i
$375$		0			0
$376$	3.00000	−	5.19615i	0.154713	−	0.267971i
$377$	−12.0000			−0.618031
$378$		0			0
$379$	23.0000			1.18143			0.590715	−	0.806880i	$-0.298846\pi$
$379$	23.0000			1.18143			0.590715	+	0.806880i	$0.298846\pi$
$380$		0			0
$381$		0			0
$382$	9.00000	+	15.5885i	0.460480	+	0.797575i
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$		0			0
$386$	−5.00000			−0.254493
$387$		0			0
$388$	5.00000			0.253837
$389$	9.00000	−	15.5885i	0.456318	−	0.790366i	−0.542445	−	0.840091i	$-0.682501\pi$
$389$	9.00000	−	15.5885i	0.456318	−	0.790366i	0.998763	+	0.0497253i	$0.0158346\pi$
$390$		0			0
$391$	−9.00000	−	15.5885i	−0.455150	−	0.788342i
$392$	−1.50000	−	2.59808i	−0.0757614	−	0.131223i
$393$		0			0
$394$	6.00000	−	10.3923i	0.302276	−	0.523557i
$395$		0			0
$396$		0			0
$397$	20.0000			1.00377			0.501886	−	0.864934i	$-0.332640\pi$
$397$	20.0000			1.00377			0.501886	+	0.864934i	$0.332640\pi$
$398$	−5.00000	+	8.66025i	−0.250627	+	0.434099i
$399$		0			0
$400$	2.50000	+	4.33013i	0.125000	+	0.216506i
$401$	−13.5000	−	23.3827i	−0.674158	−	1.16768i	−0.976714	−	0.214544i	$-0.931173\pi$
$401$	−13.5000	−	23.3827i	−0.674158	−	1.16768i	0.302556	−	0.953131i	$-0.402160\pi$
$402$		0			0
$403$	4.00000	−	6.92820i	0.199254	−	0.345118i
$404$		0			0
$405$		0			0
$406$	12.0000			0.595550
$407$	6.00000	−	10.3923i	0.297409	−	0.515127i
$408$		0			0
$409$	−8.50000	−	14.7224i	−0.420298	−	0.727977i	0.575670	−	0.817682i	$-0.304741\pi$
$409$	−8.50000	−	14.7224i	−0.420298	−	0.727977i	−0.995968	+	0.0897044i	$0.971408\pi$
$410$		0			0
$411$		0			0
$412$	−7.00000	+	12.1244i	−0.344865	+	0.597324i
$413$	−6.00000			−0.295241
$414$		0			0
$415$		0			0
$416$	1.00000	−	1.73205i	0.0490290	−	0.0849208i
$417$		0			0
$418$	−1.50000	−	2.59808i	−0.0733674	−	0.127076i
$419$	−6.00000	−	10.3923i	−0.293119	−	0.507697i	0.681426	−	0.731887i	$-0.261360\pi$
$419$	−6.00000	−	10.3923i	−0.293119	−	0.507697i	−0.974546	+	0.224189i	$0.928027\pi$
$420$		0			0
$421$	−10.0000	+	17.3205i	−0.487370	+	0.844150i	−0.999895	−	0.0145228i	$-0.995377\pi$
$421$	−10.0000	+	17.3205i	−0.487370	+	0.844150i	0.512524	+	0.858673i	$0.328710\pi$
$422$	−20.0000			−0.973585
$423$		0			0
$424$	12.0000			0.582772
$425$	7.50000	−	12.9904i	0.363803	−	0.630126i
$426$		0			0
$427$	−8.00000	−	13.8564i	−0.387147	−	0.670559i
$428$	1.50000	+	2.59808i	0.0725052	+	0.125583i
$429$		0			0
$430$		0			0
$431$	30.0000			1.44505			0.722525	−	0.691345i	$-0.242982\pi$
$431$	30.0000			1.44505			0.722525	+	0.691345i	$0.242982\pi$
$432$		0			0
$433$	−7.00000			−0.336399			−0.168199	−	0.985753i	$-0.553795\pi$
$433$	−7.00000			−0.336399			−0.168199	+	0.985753i	$0.553795\pi$
$434$	−4.00000	+	6.92820i	−0.192006	+	0.332564i
$435$		0			0
$436$	8.00000	+	13.8564i	0.383131	+	0.663602i
$437$	3.00000	+	5.19615i	0.143509	+	0.248566i
$438$		0			0
$439$	−4.00000	+	6.92820i	−0.190910	+	0.330665i	−0.945552	−	0.325471i	$-0.894477\pi$
$439$	−4.00000	+	6.92820i	−0.190910	+	0.330665i	0.754642	+	0.656136i	$0.227810\pi$
$440$		0			0
$441$		0			0
$442$	−6.00000			−0.285391
$443$	1.50000	−	2.59808i	0.0712672	−	0.123438i	−0.828190	−	0.560448i	$-0.810629\pi$
$443$	1.50000	−	2.59808i	0.0712672	−	0.123438i	0.899457	+	0.437009i	$0.143962\pi$
$444$		0			0
$445$		0			0
$446$	13.0000	+	22.5167i	0.615568	+	1.06619i
$447$		0			0
$448$	−1.00000	+	1.73205i	−0.0472456	+	0.0818317i
$449$	−9.00000			−0.424736			−0.212368	−	0.977190i	$-0.568118\pi$
$449$	−9.00000			−0.424736			−0.212368	+	0.977190i	$0.568118\pi$
$450$		0			0
$451$	−27.0000			−1.27138
$452$	3.00000	−	5.19615i	0.141108	−	0.244406i
$453$		0			0
$454$	−10.5000	−	18.1865i	−0.492789	−	0.853536i
$455$		0			0
$456$		0			0
$457$	−8.50000	+	14.7224i	−0.397613	+	0.688686i	−0.993431	−	0.114433i	$-0.963495\pi$
$457$	−8.50000	+	14.7224i	−0.397613	+	0.688686i	0.595818	+	0.803120i	$0.296828\pi$
$458$	−14.0000			−0.654177
$459$		0			0
$460$		0			0
$461$	−15.0000	+	25.9808i	−0.698620	+	1.21004i	0.270326	+	0.962769i	$0.412869\pi$
$461$	−15.0000	+	25.9808i	−0.698620	+	1.21004i	−0.968945	+	0.247276i	$0.920465\pi$
$462$		0			0
$463$	−10.0000	−	17.3205i	−0.464739	−	0.804952i	0.534450	−	0.845200i	$-0.320519\pi$
$463$	−10.0000	−	17.3205i	−0.464739	−	0.804952i	−0.999190	+	0.0402476i	$0.987185\pi$
$464$	3.00000	+	5.19615i	0.139272	+	0.241225i
$465$		0			0
$466$	−1.50000	+	2.59808i	−0.0694862	+	0.120354i
$467$	15.0000			0.694117			0.347059	−	0.937843i	$-0.387180\pi$
$467$	15.0000			0.694117			0.347059	+	0.937843i	$0.387180\pi$
$468$		0			0
$469$	10.0000			0.461757
$470$		0			0
$471$		0			0
$472$	−1.50000	−	2.59808i	−0.0690431	−	0.119586i
$473$	1.50000	+	2.59808i	0.0689701	+	0.119460i
$474$		0			0
$475$	−2.50000	+	4.33013i	−0.114708	+	0.198680i
$476$	6.00000			0.275010
$477$		0			0
$478$	6.00000			0.274434
$479$	−21.0000	+	36.3731i	−0.959514	+	1.66193i	−0.235833	+	0.971794i	$0.575782\pi$
$479$	−21.0000	+	36.3731i	−0.959514	+	1.66193i	−0.723681	+	0.690134i	$0.757551\pi$
$480$		0			0
$481$	4.00000	+	6.92820i	0.182384	+	0.315899i
$482$	−3.50000	−	6.06218i	−0.159421	−	0.276125i
$483$		0			0
$484$	1.00000	−	1.73205i	0.0454545	−	0.0787296i
$485$		0			0
$486$		0			0
$487$	26.0000			1.17817			0.589086	−	0.808070i	$-0.299488\pi$
$487$	26.0000			1.17817			0.589086	+	0.808070i	$0.299488\pi$
$488$	4.00000	−	6.92820i	0.181071	−	0.313625i
$489$		0			0
$490$		0			0
$491$	−7.50000	−	12.9904i	−0.338470	−	0.586248i	0.645675	−	0.763612i	$-0.276576\pi$
$491$	−7.50000	−	12.9904i	−0.338470	−	0.586248i	−0.984145	+	0.177365i	$0.943243\pi$
$492$		0			0
$493$	9.00000	−	15.5885i	0.405340	−	0.702069i
$494$	2.00000			0.0899843
$495$		0			0
$496$	−4.00000			−0.179605
$497$	−12.0000	+	20.7846i	−0.538274	+	0.932317i
$498$		0			0
$499$	6.50000	+	11.2583i	0.290980	+	0.503992i	0.974042	−	0.226369i	$-0.0726854\pi$
$499$	6.50000	+	11.2583i	0.290980	+	0.503992i	−0.683062	+	0.730361i	$0.739352\pi$
$500$		0			0
$501$		0			0
$502$	−10.5000	+	18.1865i	−0.468638	+	0.811705i
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$	9.00000	−	15.5885i	0.400099	−	0.692991i
$507$		0			0
$508$	−1.00000	−	1.73205i	−0.0443678	−	0.0768473i
$509$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$	−11.0000	+	19.0526i	−0.486611	+	0.842836i
$512$	−1.00000			−0.0441942
$513$		0			0
$514$	−21.0000			−0.926270
$515$		0			0
$516$		0			0
$517$	−9.00000	−	15.5885i	−0.395820	−	0.685580i
$518$	−4.00000	−	6.92820i	−0.175750	−	0.304408i
$519$		0			0
$520$		0			0
$521$	3.00000			0.131432			0.0657162	−	0.997838i	$-0.479067\pi$
$521$	3.00000			0.131432			0.0657162	+	0.997838i	$0.479067\pi$
$522$		0			0
$523$	20.0000			0.874539			0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000			0.874539			0.437269	+	0.899331i	$0.355946\pi$
$524$		0			0
$525$		0			0
$526$	−9.00000	−	15.5885i	−0.392419	−	0.679689i
$527$	6.00000	+	10.3923i	0.261364	+	0.452696i
$528$		0			0
$529$	−6.50000	+	11.2583i	−0.282609	+	0.489493i
$530$		0			0
$531$		0			0
$532$	−2.00000			−0.0867110
$533$	9.00000	−	15.5885i	0.389833	−	0.675211i
$534$		0			0
$535$		0			0
$536$	2.50000	+	4.33013i	0.107984	+	0.187033i
$537$		0			0
$538$	12.0000	−	20.7846i	0.517357	−	0.896088i
$539$	−9.00000			−0.387657
$540$		0			0
$541$	−4.00000			−0.171973			−0.0859867	−	0.996296i	$-0.527404\pi$
$541$	−4.00000			−0.171973			−0.0859867	+	0.996296i	$0.527404\pi$
$542$	10.0000	−	17.3205i	0.429537	−	0.743980i
$543$		0			0
$544$	1.50000	+	2.59808i	0.0643120	+	0.111392i
$545$		0			0
$546$		0			0
$547$	0.500000	−	0.866025i	0.0213785	−	0.0370286i	−0.855138	−	0.518400i	$-0.826528\pi$
$547$	0.500000	−	0.866025i	0.0213785	−	0.0370286i	0.876517	+	0.481371i	$0.159861\pi$
$548$	3.00000			0.128154
$549$		0			0
$550$	15.0000			0.639602
$551$	−3.00000	+	5.19615i	−0.127804	+	0.221364i
$552$		0			0
$553$	4.00000	+	6.92820i	0.170097	+	0.294617i
$554$	−5.00000	−	8.66025i	−0.212430	−	0.367939i
$555$		0			0
$556$	9.50000	−	16.4545i	0.402890	−	0.697826i
$557$	30.0000			1.27114			0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000			1.27114			0.635570	+	0.772043i	$0.280765\pi$
$558$		0			0
$559$	−2.00000			−0.0845910
$560$		0			0
$561$		0			0
$562$	−3.00000	−	5.19615i	−0.126547	−	0.219186i
$563$	−19.5000	−	33.7750i	−0.821827	−	1.42345i	−0.904320	−	0.426855i	$-0.859622\pi$
$563$	−19.5000	−	33.7750i	−0.821827	−	1.42345i	0.0824933	−	0.996592i	$-0.473712\pi$
$564$		0			0
$565$		0			0
$566$	4.00000			0.168133
$567$		0			0
$568$	−12.0000			−0.503509
$569$	22.5000	−	38.9711i	0.943249	−	1.63376i	0.184030	−	0.982921i	$-0.441086\pi$
$569$	22.5000	−	38.9711i	0.943249	−	1.63376i	0.759220	−	0.650835i	$-0.225581\pi$
$570$		0			0
$571$	18.5000	+	32.0429i	0.774201	+	1.34096i	0.935243	+	0.354008i	$0.115181\pi$
$571$	18.5000	+	32.0429i	0.774201	+	1.34096i	−0.161042	+	0.986948i	$0.551485\pi$
$572$	−3.00000	−	5.19615i	−0.125436	−	0.217262i
$573$		0			0
$574$	−9.00000	+	15.5885i	−0.375653	+	0.650650i
$575$	−30.0000			−1.25109
$576$		0			0
$577$	11.0000			0.457936			0.228968	−	0.973434i	$-0.426465\pi$
$577$	11.0000			0.457936			0.228968	+	0.973434i	$0.426465\pi$
$578$	−4.00000	+	6.92820i	−0.166378	+	0.288175i
$579$		0			0
$580$		0			0
$581$	12.0000	+	20.7846i	0.497844	+	0.862291i
$582$		0			0
$583$	18.0000	−	31.1769i	0.745484	−	1.29122i
$584$	−11.0000			−0.455183
$585$		0			0
$586$	30.0000			1.23929
$587$	−4.50000	+	7.79423i	−0.185735	+	0.321702i	−0.943824	−	0.330449i	$-0.892800\pi$
$587$	−4.50000	+	7.79423i	−0.185735	+	0.321702i	0.758089	+	0.652151i	$0.226133\pi$
$588$		0			0
$589$	−2.00000	−	3.46410i	−0.0824086	−	0.142736i
$590$		0			0
$591$		0			0
$592$	2.00000	−	3.46410i	0.0821995	−	0.142374i
$593$	−6.00000			−0.246390			−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000			−0.246390			−0.123195	+	0.992382i	$0.539314\pi$
$594$		0			0
$595$		0			0
$596$	−3.00000	+	5.19615i	−0.122885	+	0.212843i
$597$		0			0
$598$	6.00000	+	10.3923i	0.245358	+	0.424973i
$599$	6.00000	+	10.3923i	0.245153	+	0.424618i	0.962175	−	0.272433i	$-0.0878284\pi$
$599$	6.00000	+	10.3923i	0.245153	+	0.424618i	−0.717021	+	0.697051i	$0.754495\pi$
$600$		0			0
$601$	18.5000	−	32.0429i	0.754631	−	1.30706i	−0.190927	−	0.981604i	$-0.561149\pi$
$601$	18.5000	−	32.0429i	0.754631	−	1.30706i	0.945558	−	0.325455i	$-0.105517\pi$
$602$	2.00000			0.0815139
$603$		0			0
$604$	−10.0000			−0.406894
$605$		0			0
$606$		0			0
$607$	14.0000	+	24.2487i	0.568242	+	0.984225i	0.996740	+	0.0806818i	$0.0257098\pi$
$607$	14.0000	+	24.2487i	0.568242	+	0.984225i	−0.428497	+	0.903543i	$0.640957\pi$
$608$	−0.500000	−	0.866025i	−0.0202777	−	0.0351220i
$609$		0			0
$610$		0			0
$611$	12.0000			0.485468
$612$		0			0
$613$	−16.0000			−0.646234			−0.323117	−	0.946359i	$-0.604731\pi$
$613$	−16.0000			−0.646234			−0.323117	+	0.946359i	$0.604731\pi$
$614$	−3.50000	+	6.06218i	−0.141249	+	0.244650i
$615$		0			0
$616$	3.00000	+	5.19615i	0.120873	+	0.209359i
$617$	13.5000	+	23.3827i	0.543490	+	0.941351i	0.998700	+	0.0509678i	$0.0162306\pi$
$617$	13.5000	+	23.3827i	0.543490	+	0.941351i	−0.455211	+	0.890384i	$0.650436\pi$
$618$		0			0
$619$	−17.5000	+	30.3109i	−0.703384	+	1.21830i	0.263887	+	0.964554i	$0.414995\pi$
$619$	−17.5000	+	30.3109i	−0.703384	+	1.21830i	−0.967271	+	0.253744i	$0.918338\pi$
$620$		0			0
$621$		0			0
$622$	−18.0000			−0.721734
$623$	6.00000	−	10.3923i	0.240385	−	0.416359i
$624$		0			0
$625$	−12.5000	−	21.6506i	−0.500000	−	0.866025i
$626$	14.5000	+	25.1147i	0.579537	+	1.00379i
$627$		0			0
$628$	2.00000	−	3.46410i	0.0798087	−	0.138233i
$629$	−12.0000			−0.478471
$630$		0			0
$631$	−40.0000			−1.59237			−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000			−1.59237			−0.796187	+	0.605050i	$0.793153\pi$
$632$	−2.00000	+	3.46410i	−0.0795557	+	0.137795i
$633$		0			0
$634$	9.00000	+	15.5885i	0.357436	+	0.619097i
$635$		0			0
$636$		0			0
$637$	3.00000	−	5.19615i	0.118864	−	0.205879i
$638$	18.0000			0.712627
$639$		0			0
$640$		0			0
$641$	−1.50000	+	2.59808i	−0.0592464	+	0.102618i	−0.894127	−	0.447813i	$-0.852203\pi$
$641$	−1.50000	+	2.59808i	−0.0592464	+	0.102618i	0.834881	+	0.550431i	$0.185536\pi$
$642$		0			0
$643$	−11.5000	−	19.9186i	−0.453516	−	0.785512i	0.545086	−	0.838380i	$-0.316497\pi$
$643$	−11.5000	−	19.9186i	−0.453516	−	0.785512i	−0.998602	+	0.0528680i	$0.983164\pi$
$644$	−6.00000	−	10.3923i	−0.236433	−	0.409514i
$645$		0			0
$646$	−1.50000	+	2.59808i	−0.0590167	+	0.102220i
$647$	−18.0000			−0.707653			−0.353827	−	0.935311i	$-0.615120\pi$
$647$	−18.0000			−0.707653			−0.353827	+	0.935311i	$0.615120\pi$
$648$		0			0
$649$	−9.00000			−0.353281
$650$	−5.00000	+	8.66025i	−0.196116	+	0.339683i
$651$		0			0
$652$	2.00000	+	3.46410i	0.0783260	+	0.135665i
$653$	−3.00000	−	5.19615i	−0.117399	−	0.203341i	0.801337	−	0.598213i	$-0.204122\pi$
$653$	−3.00000	−	5.19615i	−0.117399	−	0.203341i	−0.918736	+	0.394872i	$0.870789\pi$
$654$		0			0
$655$		0			0
$656$	−9.00000			−0.351391
$657$		0			0
$658$	−12.0000			−0.467809
$659$	−18.0000	+	31.1769i	−0.701180	+	1.21448i	0.266872	+	0.963732i	$0.414010\pi$
$659$	−18.0000	+	31.1769i	−0.701180	+	1.21448i	−0.968052	+	0.250748i	$0.919323\pi$
$660$		0			0
$661$	2.00000	+	3.46410i	0.0777910	+	0.134738i	0.902297	−	0.431116i	$-0.141880\pi$
$661$	2.00000	+	3.46410i	0.0777910	+	0.134738i	−0.824506	+	0.565854i	$0.808547\pi$
$662$	−2.00000	−	3.46410i	−0.0777322	−	0.134636i
$663$		0			0
$664$	−6.00000	+	10.3923i	−0.232845	+	0.403300i
$665$		0			0
$666$		0			0
$667$	−36.0000			−1.39393
$668$	−6.00000	+	10.3923i	−0.232147	+	0.402090i
$669$		0			0
$670$		0			0
$671$	−12.0000	−	20.7846i	−0.463255	−	0.802381i
$672$		0			0
$673$	11.0000	−	19.0526i	0.424019	−	0.734422i	−0.572309	−	0.820038i	$-0.693952\pi$
$673$	11.0000	−	19.0526i	0.424019	−	0.734422i	0.996328	+	0.0856156i	$0.0272857\pi$
$674$	1.00000			0.0385186
$675$		0			0
$676$	−9.00000			−0.346154
$677$	18.0000	−	31.1769i	0.691796	−	1.19823i	−0.279453	−	0.960159i	$-0.590153\pi$
$677$	18.0000	−	31.1769i	0.691796	−	1.19823i	0.971249	−	0.238067i	$-0.0765137\pi$
$678$		0			0
$679$	−5.00000	−	8.66025i	−0.191882	−	0.332350i
$680$		0			0
$681$		0			0
$682$	−6.00000	+	10.3923i	−0.229752	+	0.397942i
$683$	9.00000			0.344375			0.172188	−	0.985064i	$-0.444916\pi$
$683$	9.00000			0.344375			0.172188	+	0.985064i	$0.444916\pi$
$684$		0			0
$685$		0			0
$686$	−10.0000	+	17.3205i	−0.381802	+	0.661300i
$687$		0			0
$688$	0.500000	+	0.866025i	0.0190623	+	0.0330169i
$689$	12.0000	+	20.7846i	0.457164	+	0.791831i
$690$		0			0
$691$	−4.00000	+	6.92820i	−0.152167	+	0.263561i	−0.932024	−	0.362397i	$-0.881959\pi$
$691$	−4.00000	+	6.92820i	−0.152167	+	0.263561i	0.779857	+	0.625958i	$0.215292\pi$
$692$	6.00000			0.228086
$693$		0			0
$694$	33.0000			1.25266
$695$		0			0
$696$		0			0
$697$	13.5000	+	23.3827i	0.511349	+	0.885682i
$698$	−8.00000	−	13.8564i	−0.302804	−	0.524473i
$699$		0			0
$700$	5.00000	−	8.66025i	0.188982	−	0.327327i
$701$	−30.0000			−1.13308			−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000			−1.13308			−0.566542	+	0.824033i	$0.691719\pi$
$702$		0			0
$703$	4.00000			0.150863
$704$	−1.50000	+	2.59808i	−0.0565334	+	0.0979187i
$705$		0			0
$706$	10.5000	+	18.1865i	0.395173	+	0.684459i
$707$		0			0
$708$		0			0
$709$	2.00000	−	3.46410i	0.0751116	−	0.130097i	−0.826023	−	0.563636i	$-0.809402\pi$
$709$	2.00000	−	3.46410i	0.0751116	−	0.130097i	0.901135	+	0.433539i	$0.142735\pi$
$710$		0			0
$711$		0			0
$712$	6.00000			0.224860
$713$	12.0000	−	20.7846i	0.449404	−	0.778390i
$714$		0			0
$715$		0			0
$716$	6.00000	+	10.3923i	0.224231	+	0.388379i
$717$		0			0
$718$	9.00000	−	15.5885i	0.335877	−	0.581756i
$719$	36.0000			1.34257			0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000			1.34257			0.671287	+	0.741198i	$0.265742\pi$
$720$		0			0
$721$	28.0000			1.04277
$722$	−9.00000	+	15.5885i	−0.334945	+	0.580142i
$723$		0			0
$724$	−7.00000	−	12.1244i	−0.260153	−	0.450598i
$725$	−15.0000	−	25.9808i	−0.557086	−	0.964901i
$726$		0			0
$727$	−13.0000	+	22.5167i	−0.482143	+	0.835097i	−0.999790	−	0.0204978i	$-0.993475\pi$
$727$	−13.0000	+	22.5167i	−0.482143	+	0.835097i	0.517647	+	0.855595i	$0.326808\pi$
$728$	−4.00000			−0.148250
$729$		0			0
$730$		0			0
$731$	1.50000	−	2.59808i	0.0554795	−	0.0960933i
$732$		0			0
$733$	−7.00000	−	12.1244i	−0.258551	−	0.447823i	0.707303	−	0.706910i	$-0.249912\pi$
$733$	−7.00000	−	12.1244i	−0.258551	−	0.447823i	−0.965854	+	0.259087i	$0.916578\pi$
$734$	−14.0000	−	24.2487i	−0.516749	−	0.895036i
$735$		0			0
$736$	3.00000	−	5.19615i	0.110581	−	0.191533i
$737$	15.0000			0.552532
$738$		0			0
$739$	47.0000			1.72892			0.864461	−	0.502699i	$-0.167660\pi$
$739$	47.0000			1.72892			0.864461	+	0.502699i	$0.167660\pi$
$740$		0			0
$741$		0			0
$742$	−12.0000	−	20.7846i	−0.440534	−	0.763027i
$743$	3.00000	+	5.19615i	0.110059	+	0.190628i	0.915794	−	0.401648i	$-0.131563\pi$
$743$	3.00000	+	5.19615i	0.110059	+	0.190628i	−0.805735	+	0.592277i	$0.798229\pi$
$744$		0			0
$745$		0			0
$746$	34.0000			1.24483
$747$		0			0
$748$	9.00000			0.329073
$749$	3.00000	−	5.19615i	0.109618	−	0.189863i
$750$		0			0
$751$	−4.00000	−	6.92820i	−0.145962	−	0.252814i	0.783769	−	0.621052i	$-0.213294\pi$
$751$	−4.00000	−	6.92820i	−0.145962	−	0.252814i	−0.929731	+	0.368238i	$0.879961\pi$
$752$	−3.00000	−	5.19615i	−0.109399	−	0.189484i
$753$		0			0
$754$	−6.00000	+	10.3923i	−0.218507	+	0.378465i
$755$		0			0
$756$		0			0
$757$	2.00000			0.0726912			0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000			0.0726912			0.0363456	+	0.999339i	$0.488428\pi$
$758$	11.5000	−	19.9186i	0.417699	−	0.723476i
$759$		0			0
$760$		0			0
$761$	21.0000	+	36.3731i	0.761249	+	1.31852i	0.942207	+	0.335032i	$0.108747\pi$
$761$	21.0000	+	36.3731i	0.761249	+	1.31852i	−0.180957	+	0.983491i	$0.557920\pi$
$762$		0			0
$763$	16.0000	−	27.7128i	0.579239	−	1.00327i
$764$	18.0000			0.651217
$765$		0			0
$766$		0			0
$767$	3.00000	−	5.19615i	0.108324	−	0.187622i
$768$		0			0
$769$	−1.00000	−	1.73205i	−0.0360609	−	0.0624593i	0.847432	−	0.530904i	$-0.178148\pi$
$769$	−1.00000	−	1.73205i	−0.0360609	−	0.0624593i	−0.883493	+	0.468445i	$0.844814\pi$
$770$		0			0
$771$		0			0
$772$	−2.50000	+	4.33013i	−0.0899770	+	0.155845i
$773$	−18.0000			−0.647415			−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000			−0.647415			−0.323708	+	0.946157i	$0.604929\pi$
$774$		0			0
$775$	20.0000			0.718421
$776$	2.50000	−	4.33013i	0.0897448	−	0.155443i
$777$		0			0
$778$	−9.00000	−	15.5885i	−0.322666	−	0.558873i
$779$	−4.50000	−	7.79423i	−0.161229	−	0.279257i
$780$		0			0
$781$	−18.0000	+	31.1769i	−0.644091	+	1.11560i
$782$	−18.0000			−0.643679
$783$		0			0
$784$	−3.00000			−0.107143
$785$		0			0
$786$		0			0
$787$	2.00000	+	3.46410i	0.0712923	+	0.123482i	0.899468	−	0.436987i	$-0.143954\pi$
$787$	2.00000	+	3.46410i	0.0712923	+	0.123482i	−0.828176	+	0.560469i	$0.810621\pi$
$788$	−6.00000	−	10.3923i	−0.213741	−	0.370211i
$789$

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:	\( N \)	\(=\)	\( 54 = 2 \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	54.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-1.00000 + 1.73205i) q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-1.00000 + 1.73205i) q^{7} -1.00000 q^{8} +(-1.50000 + 2.59808i) q^{11} +(-1.00000 - 1.73205i) q^{13} +(1.00000 + 1.73205i) q^{14} +(-0.500000 + 0.866025i) q^{16} +3.00000 q^{17} -1.00000 q^{19} +(1.50000 + 2.59808i) q^{22} +(-3.00000 - 5.19615i) q^{23} +(2.50000 - 4.33013i) q^{25} -2.00000 q^{26} +2.00000 q^{28} +(3.00000 - 5.19615i) q^{29} +(2.00000 + 3.46410i) q^{31} +(0.500000 + 0.866025i) q^{32} +(1.50000 - 2.59808i) q^{34} -4.00000 q^{37} +(-0.500000 + 0.866025i) q^{38} +(4.50000 + 7.79423i) q^{41} +(0.500000 - 0.866025i) q^{43} +3.00000 q^{44} -6.00000 q^{46} +(-3.00000 + 5.19615i) q^{47} +(1.50000 + 2.59808i) q^{49} +(-2.50000 - 4.33013i) q^{50} +(-1.00000 + 1.73205i) q^{52} -12.0000 q^{53} +(1.00000 - 1.73205i) q^{56} +(-3.00000 - 5.19615i) q^{58} +(1.50000 + 2.59808i) q^{59} +(-4.00000 + 6.92820i) q^{61} +4.00000 q^{62} +1.00000 q^{64} +(-2.50000 - 4.33013i) q^{67} +(-1.50000 - 2.59808i) q^{68} +12.0000 q^{71} +11.0000 q^{73} +(-2.00000 + 3.46410i) q^{74} +(0.500000 + 0.866025i) q^{76} +(-3.00000 - 5.19615i) q^{77} +(2.00000 - 3.46410i) q^{79} +9.00000 q^{82} +(6.00000 - 10.3923i) q^{83} +(-0.500000 - 0.866025i) q^{86} +(1.50000 - 2.59808i) q^{88} -6.00000 q^{89} +4.00000 q^{91} +(-3.00000 + 5.19615i) q^{92} +(3.00000 + 5.19615i) q^{94} +(-2.50000 + 4.33013i) q^{97} +3.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} - 2 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q + q^2 - q^4 - 2 * q^7 - 2 * q^8	\( 2 q + q^{2} - q^{4} - 2 q^{7} - 2 q^{8} - 3 q^{11} - 2 q^{13} + 2 q^{14} - q^{16} + 6 q^{17} - 2 q^{19} + 3 q^{22} - 6 q^{23} + 5 q^{25} - 4 q^{26} + 4 q^{28} + 6 q^{29} + 4 q^{31} + q^{32} + 3 q^{34} - 8 q^{37} - q^{38} + 9 q^{41} + q^{43} + 6 q^{44} - 12 q^{46} - 6 q^{47} + 3 q^{49} - 5 q^{50} - 2 q^{52} - 24 q^{53} + 2 q^{56} - 6 q^{58} + 3 q^{59} - 8 q^{61} + 8 q^{62} + 2 q^{64} - 5 q^{67} - 3 q^{68} + 24 q^{71} + 22 q^{73} - 4 q^{74} + q^{76} - 6 q^{77} + 4 q^{79} + 18 q^{82} + 12 q^{83} - q^{86} + 3 q^{88} - 12 q^{89} + 8 q^{91} - 6 q^{92} + 6 q^{94} - 5 q^{97} + 6 q^{98}+O(q^{100}) \) 2 * q + q^2 - q^4 - 2 * q^7 - 2 * q^8 - 3 * q^11 - 2 * q^13 + 2 * q^14 - q^16 + 6 * q^17 - 2 * q^19 + 3 * q^22 - 6 * q^23 + 5 * q^25 - 4 * q^26 + 4 * q^28 + 6 * q^29 + 4 * q^31 + q^32 + 3 * q^34 - 8 * q^37 - q^38 + 9 * q^41 + q^43 + 6 * q^44 - 12 * q^46 - 6 * q^47 + 3 * q^49 - 5 * q^50 - 2 * q^52 - 24 * q^53 + 2 * q^56 - 6 * q^58 + 3 * q^59 - 8 * q^61 + 8 * q^62 + 2 * q^64 - 5 * q^67 - 3 * q^68 + 24 * q^71 + 22 * q^73 - 4 * q^74 + q^76 - 6 * q^77 + 4 * q^79 + 18 * q^82 + 12 * q^83 - q^86 + 3 * q^88 - 12 * q^89 + 8 * q^91 - 6 * q^92 + 6 * q^94 - 5 * q^97 + 6 * q^98

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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