LMFDB - Embedded newform 117.2.q.a.82.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,2,Mod(10,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5]))

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

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

chi := DirichletCharacter("117.10");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	117.q (of order $6$, 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:	$0.934249703649$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			82.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	117.82
Dual form			117.2.q.a.10.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 - 1.73205i) q^{4} -3.46410i q^{5} +(-1.50000 + 0.866025i) q^{7} +O(q^{10})$	$q+(-1.00000 - 1.73205i) q^{4} -3.46410i q^{5} +(-1.50000 + 0.866025i) q^{7} +(3.00000 + 1.73205i) q^{11} +(3.50000 - 0.866025i) q^{13} +(-2.00000 + 3.46410i) q^{16} +(3.00000 - 1.73205i) q^{19} +(-6.00000 + 3.46410i) q^{20} +(-3.00000 + 5.19615i) q^{23} -7.00000 q^{25} +(3.00000 + 1.73205i) q^{28} +(3.00000 - 5.19615i) q^{29} +1.73205i q^{31} +(3.00000 + 5.19615i) q^{35} +(6.00000 + 3.46410i) q^{41} +(0.500000 + 0.866025i) q^{43} -6.92820i q^{44} -3.46410i q^{47} +(-2.00000 + 3.46410i) q^{49} +(-5.00000 - 5.19615i) q^{52} -12.0000 q^{53} +(6.00000 - 10.3923i) q^{55} +(3.00000 - 1.73205i) q^{59} +(-0.500000 - 0.866025i) q^{61} +8.00000 q^{64} +(-3.00000 - 12.1244i) q^{65} +(7.50000 + 4.33013i) q^{67} +(9.00000 - 5.19615i) q^{71} +1.73205i q^{73} +(-6.00000 - 3.46410i) q^{76} -6.00000 q^{77} -11.0000 q^{79} +(12.0000 + 6.92820i) q^{80} +13.8564i q^{83} +(-6.00000 - 3.46410i) q^{89} +(-4.50000 + 4.33013i) q^{91} +12.0000 q^{92} +(-6.00000 - 10.3923i) q^{95} +(-4.50000 + 2.59808i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 3 q^{7}+O(q^{10}) $ 2 * q - 2 * q^4 - 3 * q^7	$ 2 q - 2 q^{4} - 3 q^{7} + 6 q^{11} + 7 q^{13} - 4 q^{16} + 6 q^{19} - 12 q^{20} - 6 q^{23} - 14 q^{25} + 6 q^{28} + 6 q^{29} + 6 q^{35} + 12 q^{41} + q^{43} - 4 q^{49} - 10 q^{52} - 24 q^{53} + 12 q^{55} + 6 q^{59} - q^{61} + 16 q^{64} - 6 q^{65} + 15 q^{67} + 18 q^{71} - 12 q^{76} - 12 q^{77} - 22 q^{79} + 24 q^{80} - 12 q^{89} - 9 q^{91} + 24 q^{92} - 12 q^{95} - 9 q^{97}+O(q^{100}) $ 2 * q - 2 * q^4 - 3 * q^7 + 6 * q^11 + 7 * q^13 - 4 * q^16 + 6 * q^19 - 12 * q^20 - 6 * q^23 - 14 * q^25 + 6 * q^28 + 6 * q^29 + 6 * q^35 + 12 * q^41 + q^43 - 4 * q^49 - 10 * q^52 - 24 * q^53 + 12 * q^55 + 6 * q^59 - q^61 + 16 * q^64 - 6 * q^65 + 15 * q^67 + 18 * q^71 - 12 * q^76 - 12 * q^77 - 22 * q^79 + 24 * q^80 - 12 * q^89 - 9 * q^91 + 24 * q^92 - 12 * q^95 - 9 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$.

$n$	$28$	$92$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$1$

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			0		0.500000	−	0.866025i	$-0.333333\pi$
$2$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$3$		0			0
$3$		0			0
$4$	−1.00000	−	1.73205i	−0.500000	−	0.866025i
$5$		−	3.46410i		−	1.54919i	−0.632456	−	0.774597i	$-0.717953\pi$
$5$		−	3.46410i		−	1.54919i	0.632456	−	0.774597i	$-0.282047\pi$
$6$		0			0
$7$	−1.50000	+	0.866025i	−0.566947	+	0.327327i	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−1.50000	+	0.866025i	−0.566947	+	0.327327i	0.188982	+	0.981981i	$0.439481\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.00000	+	1.73205i	0.904534	+	0.522233i	0.878668	−	0.477432i	$-0.158432\pi$
$11$	3.00000	+	1.73205i	0.904534	+	0.522233i	0.0258656	+	0.999665i	$0.491766\pi$
$12$		0			0
$13$	3.50000	−	0.866025i	0.970725	−	0.240192i
$13$	3.50000	−	0.866025i	0.970725	−	0.240192i
$14$		0			0
$15$		0			0
$16$	−2.00000	+	3.46410i	−0.500000	+	0.866025i
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$		0			0
$19$	3.00000	−	1.73205i	0.688247	−	0.397360i	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	3.00000	−	1.73205i	0.688247	−	0.397360i	0.802955	+	0.596040i	$0.203260\pi$
$20$	−6.00000	+	3.46410i	−1.34164	+	0.774597i
$21$		0			0
$22$		0			0
$23$	−3.00000	+	5.19615i	−0.625543	+	1.08347i	0.362892	+	0.931831i	$0.381789\pi$
$23$	−3.00000	+	5.19615i	−0.625543	+	1.08347i	−0.988436	+	0.151642i	$0.951544\pi$
$24$		0			0
$25$	−7.00000			−1.40000
$26$		0			0
$27$		0			0
$28$	3.00000	+	1.73205i	0.566947	+	0.327327i
$29$	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
$31$			1.73205i			0.311086i	0.987829	+	0.155543i	$0.0497126\pi$
$31$			1.73205i			0.311086i	−0.987829	+	0.155543i	$0.950287\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.00000	+	5.19615i	0.507093	+	0.878310i
$36$		0			0
$37$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$37$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	6.00000	+	3.46410i	0.937043	+	0.541002i	0.889032	−	0.457845i	$-0.151379\pi$
$41$	6.00000	+	3.46410i	0.937043	+	0.541002i	0.0480106	+	0.998847i	$0.484712\pi$
$42$		0			0
$43$	0.500000	+	0.866025i	0.0762493	+	0.132068i	0.901629	−	0.432511i	$-0.142372\pi$
$43$	0.500000	+	0.866025i	0.0762493	+	0.132068i	−0.825380	+	0.564578i	$0.809039\pi$
$44$		−	6.92820i		−	1.04447i
$45$		0			0
$46$		0			0
$47$		−	3.46410i		−	0.505291i	−0.967559	−	0.252646i	$-0.918699\pi$
$47$		−	3.46410i		−	0.505291i	0.967559	−	0.252646i	$-0.0813007\pi$
$48$		0			0
$49$	−2.00000	+	3.46410i	−0.285714	+	0.494872i
$50$		0			0
$51$		0			0
$52$	−5.00000	−	5.19615i	−0.693375	−	0.720577i
$53$	−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
$55$	6.00000	−	10.3923i	0.809040	−	1.40130i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.00000	−	1.73205i	0.390567	−	0.225494i	−0.291839	−	0.956467i	$-0.594267\pi$
$59$	3.00000	−	1.73205i	0.390567	−	0.225494i	0.682406	+	0.730974i	$0.260934\pi$
$60$		0			0
$61$	−0.500000	−	0.866025i	−0.0640184	−	0.110883i	0.832240	−	0.554416i	$-0.187058\pi$
$61$	−0.500000	−	0.866025i	−0.0640184	−	0.110883i	−0.896258	+	0.443533i	$0.853725\pi$
$62$		0			0
$63$		0			0
$64$	8.00000			1.00000
$65$	−3.00000	−	12.1244i	−0.372104	−	1.50384i
$66$		0			0
$67$	7.50000	+	4.33013i	0.916271	+	0.529009i	0.882443	−	0.470418i	$-0.155897\pi$
$67$	7.50000	+	4.33013i	0.916271	+	0.529009i	0.0338274	+	0.999428i	$0.489230\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	9.00000	−	5.19615i	1.06810	−	0.616670i	0.140441	−	0.990089i	$-0.455148\pi$
$71$	9.00000	−	5.19615i	1.06810	−	0.616670i	0.927663	+	0.373419i	$0.121815\pi$
$72$		0			0
$73$			1.73205i			0.202721i	0.994850	+	0.101361i	$0.0323196\pi$
$73$			1.73205i			0.202721i	−0.994850	+	0.101361i	$0.967680\pi$
$74$		0			0
$75$		0			0
$76$	−6.00000	−	3.46410i	−0.688247	−	0.397360i
$77$	−6.00000			−0.683763
$78$		0			0
$79$	−11.0000			−1.23760			−0.618798	−	0.785550i	$-0.712380\pi$
$79$	−11.0000			−1.23760			−0.618798	+	0.785550i	$0.712380\pi$
$80$	12.0000	+	6.92820i	1.34164	+	0.774597i
$81$		0			0
$82$		0			0
$83$			13.8564i			1.52094i	0.649374	+	0.760469i	$0.275031\pi$
$83$			13.8564i			1.52094i	−0.649374	+	0.760469i	$0.724969\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.00000	−	3.46410i	−0.635999	−	0.367194i	0.147073	−	0.989126i	$-0.453015\pi$
$89$	−6.00000	−	3.46410i	−0.635999	−	0.367194i	−0.783072	+	0.621932i	$0.786348\pi$
$90$		0			0
$91$	−4.50000	+	4.33013i	−0.471728	+	0.453921i
$92$	12.0000			1.25109
$93$		0			0
$94$		0			0
$95$	−6.00000	−	10.3923i	−0.615587	−	1.06623i
$96$		0			0
$97$	−4.50000	+	2.59808i	−0.456906	+	0.263795i	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−4.50000	+	2.59808i	−0.456906	+	0.263795i	0.253837	+	0.967247i	$0.418307\pi$
$98$		0			0
$99$		0			0
$100$	7.00000	+	12.1244i	0.700000	+	1.21244i
$101$	−9.00000	+	15.5885i	−0.895533	+	1.55111i	−0.0623905	+	0.998052i	$0.519872\pi$
$101$	−9.00000	+	15.5885i	−0.895533	+	1.55111i	−0.833143	+	0.553058i	$0.813461\pi$
$102$		0			0
$103$	−1.00000			−0.0985329			−0.0492665	−	0.998786i	$-0.515688\pi$
$103$	−1.00000			−0.0985329			−0.0492665	+	0.998786i	$0.515688\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−3.00000	+	5.19615i	−0.290021	+	0.502331i	−0.973814	−	0.227345i	$-0.926996\pi$
$107$	−3.00000	+	5.19615i	−0.290021	+	0.502331i	0.683793	+	0.729676i	$0.260329\pi$
$108$		0			0
$109$		−	15.5885i		−	1.49310i	−0.665327	−	0.746552i	$-0.731708\pi$
$109$		−	15.5885i		−	1.49310i	0.665327	−	0.746552i	$-0.268292\pi$
$110$		0			0
$111$		0			0
$112$		−	6.92820i		−	0.654654i
$113$	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
$115$	18.0000	+	10.3923i	1.67851	+	0.969087i
$116$	−12.0000			−1.11417
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.500000	+	0.866025i	0.0454545	+	0.0787296i
$122$		0			0
$123$		0			0
$124$	3.00000	−	1.73205i	0.269408	−	0.155543i
$125$			6.92820i			0.619677i
$126$		0			0
$127$	−6.50000	+	11.2583i	−0.576782	+	0.999015i	0.419064	+	0.907957i	$0.362358\pi$
$127$	−6.50000	+	11.2583i	−0.576782	+	0.999015i	−0.995846	+	0.0910585i	$0.970975\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−6.00000			−0.524222			−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000			−0.524222			−0.262111	+	0.965038i	$0.584419\pi$
$132$		0			0
$133$	−3.00000	+	5.19615i	−0.260133	+	0.450564i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$137$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$138$		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	$0.901346\pi$
$140$	6.00000	−	10.3923i	0.507093	−	0.878310i
$141$		0			0
$142$		0			0
$143$	12.0000	+	3.46410i	1.00349	+	0.289683i
$144$		0			0
$145$	−18.0000	−	10.3923i	−1.49482	−	0.863034i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	6.00000	−	3.46410i	0.491539	−	0.283790i	−0.233674	−	0.972315i	$-0.575075\pi$
$149$	6.00000	−	3.46410i	0.491539	−	0.283790i	0.725213	+	0.688525i	$0.241741\pi$
$150$		0			0
$151$		−	3.46410i		−	0.281905i	−0.990016	−	0.140952i	$-0.954984\pi$
$151$		−	3.46410i		−	0.281905i	0.990016	−	0.140952i	$-0.0450164\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	6.00000			0.481932
$156$		0			0
$157$	11.0000			0.877896			0.438948	−	0.898513i	$-0.355351\pi$
$157$	11.0000			0.877896			0.438948	+	0.898513i	$0.355351\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	10.3923i		−	0.819028i
$162$		0			0
$163$	−16.5000	+	9.52628i	−1.29238	+	0.746156i	−0.979076	−	0.203497i	$-0.934769\pi$
$163$	−16.5000	+	9.52628i	−1.29238	+	0.746156i	−0.313304	+	0.949653i	$0.601436\pi$
$164$		−	13.8564i		−	1.08200i
$165$		0			0
$166$		0			0
$167$	6.00000	+	3.46410i	0.464294	+	0.268060i	0.713848	−	0.700301i	$-0.246951\pi$
$167$	6.00000	+	3.46410i	0.464294	+	0.268060i	−0.249554	+	0.968361i	$0.580284\pi$
$168$		0			0
$169$	11.5000	−	6.06218i	0.884615	−	0.466321i
$170$		0			0
$171$		0			0
$172$	1.00000	−	1.73205i	0.0762493	−	0.132068i
$173$	−3.00000	−	5.19615i	−0.228086	−	0.395056i	0.729155	−	0.684349i	$-0.239913\pi$
$173$	−3.00000	−	5.19615i	−0.228086	−	0.395056i	−0.957241	+	0.289292i	$0.906580\pi$
$174$		0			0
$175$	10.5000	−	6.06218i	0.793725	−	0.458258i
$176$	−12.0000	+	6.92820i	−0.904534	+	0.522233i
$177$		0			0
$178$		0			0
$179$	6.00000	−	10.3923i	0.448461	−	0.776757i	−0.549825	−	0.835280i	$-0.685306\pi$
$179$	6.00000	−	10.3923i	0.448461	−	0.776757i	0.998286	+	0.0585225i	$0.0186389\pi$
$180$		0			0
$181$	14.0000			1.04061			0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000			1.04061			0.520306	+	0.853980i	$0.325818\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−6.00000	+	3.46410i	−0.437595	+	0.252646i
$189$		0			0
$190$		0			0
$191$	−9.00000	−	15.5885i	−0.651217	−	1.12794i	−0.982828	−	0.184525i	$-0.940925\pi$
$191$	−9.00000	−	15.5885i	−0.651217	−	1.12794i	0.331611	−	0.943416i	$-0.392408\pi$
$192$		0			0
$193$	−13.5000	−	7.79423i	−0.971751	−	0.561041i	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−13.5000	−	7.79423i	−0.971751	−	0.561041i	−0.899770	+	0.436365i	$0.856266\pi$
$194$		0			0
$195$		0			0
$196$	8.00000			0.571429
$197$	−12.0000	−	6.92820i	−0.854965	−	0.493614i	0.00735824	−	0.999973i	$-0.497658\pi$
$197$	−12.0000	−	6.92820i	−0.854965	−	0.493614i	−0.862323	+	0.506359i	$0.830991\pi$
$198$		0			0
$199$	−3.50000	−	6.06218i	−0.248108	−	0.429736i	0.714893	−	0.699234i	$-0.246476\pi$
$199$	−3.50000	−	6.06218i	−0.248108	−	0.429736i	−0.963001	+	0.269498i	$0.913142\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			10.3923i			0.729397i
$204$		0			0
$205$	12.0000	−	20.7846i	0.838116	−	1.45166i
$206$		0			0
$207$		0			0
$208$	−4.00000	+	13.8564i	−0.277350	+	0.960769i
$209$	12.0000			0.830057
$210$		0			0
$211$	−6.50000	+	11.2583i	−0.447478	+	0.775055i	−0.998221	−	0.0596196i	$-0.981011\pi$
$211$	−6.50000	+	11.2583i	−0.447478	+	0.775055i	0.550743	+	0.834675i	$0.314345\pi$
$212$	12.0000	+	20.7846i	0.824163	+	1.42749i
$213$		0			0
$214$		0			0
$215$	3.00000	−	1.73205i	0.204598	−	0.118125i
$216$		0			0
$217$	−1.50000	−	2.59808i	−0.101827	−	0.176369i
$218$		0			0
$219$		0			0
$220$	−24.0000			−1.61808
$221$		0			0
$222$		0			0
$223$	15.0000	+	8.66025i	1.00447	+	0.579934i	0.909569	−	0.415553i	$-0.136412\pi$
$223$	15.0000	+	8.66025i	1.00447	+	0.579934i	0.0949052	+	0.995486i	$0.469745\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−18.0000	+	10.3923i	−1.19470	+	0.689761i	−0.959369	−	0.282153i	$-0.908951\pi$
$227$	−18.0000	+	10.3923i	−1.19470	+	0.689761i	−0.235333	+	0.971915i	$0.575618\pi$
$228$		0			0
$229$			27.7128i			1.83131i	0.401960	+	0.915657i	$0.368329\pi$
$229$			27.7128i			1.83131i	−0.401960	+	0.915657i	$0.631671\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	18.0000			1.17922			0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000			1.17922			0.589610	+	0.807688i	$0.299282\pi$
$234$		0			0
$235$	−12.0000			−0.782794
$236$	−6.00000	−	3.46410i	−0.390567	−	0.225494i
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	18.0000	−	10.3923i	1.15948	−	0.669427i	0.208302	−	0.978065i	$-0.433206\pi$
$241$	18.0000	−	10.3923i	1.15948	−	0.669427i	0.951180	+	0.308637i	$0.0998729\pi$
$242$		0			0
$243$		0			0
$244$	−1.00000	+	1.73205i	−0.0640184	+	0.110883i
$245$	12.0000	+	6.92820i	0.766652	+	0.442627i
$246$		0			0
$247$	9.00000	−	8.66025i	0.572656	−	0.551039i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	6.00000	+	10.3923i	0.378717	+	0.655956i	0.990876	−	0.134778i	$-0.0430322\pi$
$251$	6.00000	+	10.3923i	0.378717	+	0.655956i	−0.612159	+	0.790735i	$0.709699\pi$
$252$		0			0
$253$	−18.0000	+	10.3923i	−1.13165	+	0.653359i
$254$		0			0
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$257$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$258$		0			0
$259$		0			0
$260$	−18.0000	+	17.3205i	−1.11631	+	1.07417i
$261$		0			0
$262$		0			0
$263$	6.00000	−	10.3923i	0.369976	−	0.640817i	−0.619586	−	0.784929i	$-0.712699\pi$
$263$	6.00000	−	10.3923i	0.369976	−	0.640817i	0.989561	+	0.144112i	$0.0460326\pi$
$264$		0			0
$265$			41.5692i			2.55358i
$266$		0			0
$267$		0			0
$268$		−	17.3205i		−	1.05802i
$269$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$269$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$270$		0			0
$271$	−4.50000	−	2.59808i	−0.273356	−	0.157822i	0.357056	−	0.934083i	$-0.383781\pi$
$271$	−4.50000	−	2.59808i	−0.273356	−	0.157822i	−0.630412	+	0.776261i	$0.717114\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−21.0000	−	12.1244i	−1.26635	−	0.731126i
$276$		0			0
$277$	5.00000	+	8.66025i	0.300421	+	0.520344i	0.976231	−	0.216731i	$-0.0695395\pi$
$277$	5.00000	+	8.66025i	0.300421	+	0.520344i	−0.675810	+	0.737075i	$0.736206\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			24.2487i			1.44656i	0.690557	+	0.723278i	$0.257366\pi$
$281$			24.2487i			1.44656i	−0.690557	+	0.723278i	$0.742634\pi$
$282$		0			0
$283$	5.50000	−	9.52628i	0.326941	−	0.566279i	−0.654962	−	0.755662i	$-0.727315\pi$
$283$	5.50000	−	9.52628i	0.326941	−	0.566279i	0.981903	+	0.189383i	$0.0606488\pi$
$284$	−18.0000	−	10.3923i	−1.06810	−	0.616670i
$285$		0			0
$286$		0			0
$287$	−12.0000			−0.708338
$288$		0			0
$289$	8.50000	−	14.7224i	0.500000	−	0.866025i
$290$		0			0
$291$		0			0
$292$	3.00000	−	1.73205i	0.175562	−	0.101361i
$293$	15.0000	−	8.66025i	0.876309	−	0.505937i	0.00686959	−	0.999976i	$-0.497813\pi$
$293$	15.0000	−	8.66025i	0.876309	−	0.505937i	0.869440	+	0.494039i	$0.164480\pi$
$294$		0			0
$295$	−6.00000	−	10.3923i	−0.349334	−	0.605063i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−6.00000	+	20.7846i	−0.346989	+	1.20201i
$300$		0			0
$301$	−1.50000	−	0.866025i	−0.0864586	−	0.0499169i
$302$		0			0
$303$		0			0
$304$			13.8564i			0.794719i
$305$	−3.00000	+	1.73205i	−0.171780	+	0.0991769i
$306$		0			0
$307$		−	1.73205i		−	0.0988534i	−0.998778	−	0.0494267i	$-0.984261\pi$
$307$		−	1.73205i		−	0.0988534i	0.998778	−	0.0494267i	$-0.0157394\pi$
$308$	6.00000	+	10.3923i	0.341882	+	0.592157i
$309$		0			0
$310$		0			0
$311$	24.0000			1.36092			0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000			1.36092			0.680458	+	0.732787i	$0.261781\pi$
$312$		0			0
$313$	13.0000			0.734803			0.367402	−	0.930062i	$-0.380247\pi$
$313$	13.0000			0.734803			0.367402	+	0.930062i	$0.380247\pi$
$314$		0			0
$315$		0			0
$316$	11.0000	+	19.0526i	0.618798	+	1.07179i
$317$		−	6.92820i		−	0.389127i	−0.980890	−	0.194563i	$-0.937671\pi$
$317$		−	6.92820i		−	0.389127i	0.980890	−	0.194563i	$-0.0623290\pi$
$318$		0			0
$319$	18.0000	−	10.3923i	1.00781	−	0.581857i
$320$		−	27.7128i		−	1.54919i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−24.5000	+	6.06218i	−1.35902	+	0.336269i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	3.00000	+	5.19615i	0.165395	+	0.286473i
$330$		0			0
$331$	4.50000	−	2.59808i	0.247342	−	0.142803i	−0.371204	−	0.928551i	$-0.621055\pi$
$331$	4.50000	−	2.59808i	0.247342	−	0.142803i	0.618547	+	0.785748i	$0.287722\pi$
$332$	24.0000	−	13.8564i	1.31717	−	0.760469i
$333$		0			0
$334$		0			0
$335$	15.0000	−	25.9808i	0.819538	−	1.41948i
$336$		0			0
$337$	−5.00000			−0.272367			−0.136184	−	0.990684i	$-0.543484\pi$
$337$	−5.00000			−0.272367			−0.136184	+	0.990684i	$0.543484\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−3.00000	+	5.19615i	−0.162459	+	0.281387i
$342$		0			0
$343$		−	19.0526i		−	1.02874i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−12.0000	−	20.7846i	−0.644194	−	1.11578i	−0.984487	−	0.175457i	$-0.943860\pi$
$347$	−12.0000	−	20.7846i	−0.644194	−	1.11578i	0.340293	−	0.940319i	$-0.389474\pi$
$348$		0			0
$349$	−16.5000	−	9.52628i	−0.883225	−	0.509930i	−0.0115044	−	0.999934i	$-0.503662\pi$
$349$	−16.5000	−	9.52628i	−0.883225	−	0.509930i	−0.871720	+	0.490004i	$0.836995\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−9.00000	−	5.19615i	−0.479022	−	0.276563i	0.240987	−	0.970528i	$-0.422529\pi$
$353$	−9.00000	−	5.19615i	−0.479022	−	0.276563i	−0.720009	+	0.693965i	$0.755862\pi$
$354$		0			0
$355$	−18.0000	−	31.1769i	−0.955341	−	1.65470i
$356$			13.8564i			0.734388i
$357$		0			0
$358$		0			0
$359$		−	6.92820i		−	0.365657i	−0.983145	−	0.182828i	$-0.941475\pi$
$359$		−	6.92820i		−	0.365657i	0.983145	−	0.182828i	$-0.0585252\pi$
$360$		0			0
$361$	−3.50000	+	6.06218i	−0.184211	+	0.319062i
$362$		0			0
$363$		0			0
$364$	12.0000	+	3.46410i	0.628971	+	0.181568i
$365$	6.00000			0.314054
$366$		0			0
$367$	−8.50000	+	14.7224i	−0.443696	+	0.768505i	−0.997960	−	0.0638362i	$-0.979666\pi$
$367$	−8.50000	+	14.7224i	−0.443696	+	0.768505i	0.554264	+	0.832341i	$0.313000\pi$
$368$	−12.0000	−	20.7846i	−0.625543	−	1.08347i
$369$		0			0
$370$		0			0
$371$	18.0000	−	10.3923i	0.934513	−	0.539542i
$372$		0			0
$373$	5.50000	+	9.52628i	0.284779	+	0.493252i	0.972556	−	0.232671i	$-0.0747464\pi$
$373$	5.50000	+	9.52628i	0.284779	+	0.493252i	−0.687776	+	0.725923i	$0.741413\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	6.00000	−	20.7846i	0.309016	−	1.07046i
$378$		0			0
$379$	−19.5000	−	11.2583i	−1.00165	−	0.578302i	−0.0929123	−	0.995674i	$-0.529618\pi$
$379$	−19.5000	−	11.2583i	−1.00165	−	0.578302i	−0.908735	+	0.417373i	$0.862951\pi$
$380$	−12.0000	+	20.7846i	−0.615587	+	1.06623i
$381$		0			0
$382$		0			0
$383$	−24.0000	+	13.8564i	−1.22634	+	0.708029i	−0.966263	−	0.257558i	$-0.917082\pi$
$383$	−24.0000	+	13.8564i	−1.22634	+	0.708029i	−0.260080	+	0.965587i	$0.583749\pi$
$384$		0			0
$385$			20.7846i			1.05928i
$386$		0			0
$387$		0			0
$388$	9.00000	+	5.19615i	0.456906	+	0.263795i
$389$	−12.0000			−0.608424			−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000			−0.608424			−0.304212	+	0.952604i	$0.598393\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$			38.1051i			1.91728i
$396$		0			0
$397$	13.5000	−	7.79423i	0.677546	−	0.391181i	−0.121384	−	0.992606i	$-0.538733\pi$
$397$	13.5000	−	7.79423i	0.677546	−	0.391181i	0.798930	+	0.601424i	$0.205400\pi$
$398$		0			0
$399$		0			0
$400$	14.0000	−	24.2487i	0.700000	−	1.21244i
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$		0			0
$403$	1.50000	+	6.06218i	0.0747203	+	0.301979i
$404$	36.0000			1.79107
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	7.50000	−	4.33013i	0.370851	−	0.214111i	−0.302979	−	0.952997i	$-0.597981\pi$
$409$	7.50000	−	4.33013i	0.370851	−	0.214111i	0.673830	+	0.738886i	$0.264648\pi$
$410$		0			0
$411$		0			0
$412$	1.00000	+	1.73205i	0.0492665	+	0.0853320i
$413$	−3.00000	+	5.19615i	−0.147620	+	0.255686i
$414$		0			0
$415$	48.0000			2.35623
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−6.00000	+	10.3923i	−0.293119	+	0.507697i	−0.974546	−	0.224189i	$-0.928027\pi$
$419$	−6.00000	+	10.3923i	−0.293119	+	0.507697i	0.681426	+	0.731887i	$0.261360\pi$
$420$		0			0
$421$		−	12.1244i		−	0.590905i	−0.955357	−	0.295452i	$-0.904530\pi$
$421$		−	12.1244i		−	0.590905i	0.955357	−	0.295452i	$-0.0954704\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	1.50000	+	0.866025i	0.0725901	+	0.0419099i
$428$	12.0000			0.580042
$429$		0			0
$430$		0			0
$431$	−18.0000	−	10.3923i	−0.867029	−	0.500580i	−0.000669521	−	1.00000i	$-0.500213\pi$
$431$	−18.0000	−	10.3923i	−0.867029	−	0.500580i	−0.866360	+	0.499420i	$0.833546\pi$
$432$		0			0
$433$	11.5000	+	19.9186i	0.552655	+	0.957226i	0.998082	+	0.0619079i	$0.0197185\pi$
$433$	11.5000	+	19.9186i	0.552655	+	0.957226i	−0.445427	+	0.895318i	$0.646948\pi$
$434$		0			0
$435$		0			0
$436$	−27.0000	+	15.5885i	−1.29307	+	0.746552i
$437$			20.7846i			0.994263i
$438$		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
$442$		0			0
$443$	−24.0000			−1.14027			−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000			−1.14027			−0.570137	+	0.821549i	$0.693110\pi$
$444$		0			0
$445$	−12.0000	+	20.7846i	−0.568855	+	0.985285i
$446$		0			0
$447$		0			0
$448$	−12.0000	+	6.92820i	−0.566947	+	0.327327i
$449$	−33.0000	+	19.0526i	−1.55737	+	0.899146i	−0.559859	+	0.828588i	$0.689145\pi$
$449$	−33.0000	+	19.0526i	−1.55737	+	0.899146i	−0.997508	+	0.0705577i	$0.977522\pi$
$450$		0			0
$451$	12.0000	+	20.7846i	0.565058	+	0.978709i
$452$	6.00000	−	10.3923i	0.282216	−	0.488813i
$453$		0			0
$454$		0			0
$455$	15.0000	+	15.5885i	0.703211	+	0.730798i
$456$		0			0
$457$	−31.5000	−	18.1865i	−1.47351	−	0.850730i	−0.473953	−	0.880550i	$-0.657173\pi$
$457$	−31.5000	−	18.1865i	−1.47351	−	0.850730i	−0.999555	+	0.0298202i	$0.990507\pi$
$458$		0			0
$459$		0			0
$460$		−	41.5692i		−	1.93817i
$461$	36.0000	−	20.7846i	1.67669	−	0.968036i	0.712938	−	0.701228i	$-0.247364\pi$
$461$	36.0000	−	20.7846i	1.67669	−	0.968036i	0.963750	−	0.266808i	$-0.0859690\pi$
$462$		0			0
$463$		−	36.3731i		−	1.69040i	−0.534450	−	0.845200i	$-0.679481\pi$
$463$		−	36.3731i		−	1.69040i	0.534450	−	0.845200i	$-0.320519\pi$
$464$	12.0000	+	20.7846i	0.557086	+	0.964901i
$465$		0			0
$466$		0			0
$467$	−6.00000			−0.277647			−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000			−0.277647			−0.138823	+	0.990317i	$0.544332\pi$
$468$		0			0
$469$	−15.0000			−0.692636
$470$		0			0
$471$		0			0
$472$		0			0
$473$			3.46410i			0.159280i
$474$		0			0
$475$	−21.0000	+	12.1244i	−0.963546	+	0.556304i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−30.0000	−	17.3205i	−1.37073	−	0.791394i	−0.379714	−	0.925104i	$-0.623978\pi$
$479$	−30.0000	−	17.3205i	−1.37073	−	0.791394i	−0.991021	+	0.133710i	$0.957311\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	1.00000	−	1.73205i	0.0454545	−	0.0787296i
$485$	9.00000	+	15.5885i	0.408669	+	0.707835i
$486$		0			0
$487$	−21.0000	+	12.1244i	−0.951601	+	0.549407i	−0.893578	−	0.448908i	$-0.851813\pi$
$487$	−21.0000	+	12.1244i	−0.951601	+	0.549407i	−0.0580230	+	0.998315i	$0.518480\pi$
$488$		0			0
$489$		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
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−6.00000	−	3.46410i	−0.269408	−	0.155543i
$497$	−9.00000	+	15.5885i	−0.403705	+	0.699238i
$498$		0			0
$499$			31.1769i			1.39567i	0.716258	+	0.697835i	$0.245853\pi$
$499$			31.1769i			1.39567i	−0.716258	+	0.697835i	$0.754147\pi$
$500$	12.0000	−	6.92820i	0.536656	−	0.309839i
$501$		0			0
$502$		0			0
$503$	−15.0000	−	25.9808i	−0.668817	−	1.15842i	−0.978235	−	0.207499i	$-0.933468\pi$
$503$	−15.0000	−	25.9808i	−0.668817	−	1.15842i	0.309418	−	0.950926i	$-0.399866\pi$
$504$		0			0
$505$	54.0000	+	31.1769i	2.40297	+	1.38735i
$506$		0			0
$507$		0			0
$508$	26.0000			1.15356
$509$	15.0000	+	8.66025i	0.664863	+	0.383859i	0.794128	−	0.607751i	$-0.207928\pi$
$509$	15.0000	+	8.66025i	0.664863	+	0.383859i	−0.129264	+	0.991610i	$0.541262\pi$
$510$		0			0
$511$	−1.50000	−	2.59808i	−0.0663561	−	0.114932i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			3.46410i			0.152647i
$516$		0			0
$517$	6.00000	−	10.3923i	0.263880	−	0.457053i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	24.0000			1.05146			0.525730	−	0.850652i	$-0.323792\pi$
$521$	24.0000			1.05146			0.525730	+	0.850652i	$0.323792\pi$
$522$		0			0
$523$	14.0000	−	24.2487i	0.612177	−	1.06032i	−0.378695	−	0.925521i	$-0.623627\pi$
$523$	14.0000	−	24.2487i	0.612177	−	1.06032i	0.990873	−	0.134801i	$-0.0430394\pi$
$524$	6.00000	+	10.3923i	0.262111	+	0.453990i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−6.50000	−	11.2583i	−0.282609	−	0.489493i
$530$		0			0
$531$		0			0
$532$	12.0000			0.520266
$533$	24.0000	+	6.92820i	1.03956	+	0.300094i
$534$		0			0
$535$	18.0000	+	10.3923i	0.778208	+	0.449299i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−12.0000	+	6.92820i	−0.516877	+	0.298419i
$540$		0			0
$541$		−	29.4449i		−	1.26593i	−0.774179	−	0.632967i	$-0.781837\pi$
$541$		−	29.4449i		−	1.26593i	0.774179	−	0.632967i	$-0.218163\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−54.0000			−2.31311
$546$		0			0
$547$	−19.0000			−0.812381			−0.406191	−	0.913788i	$-0.633143\pi$
$547$	−19.0000			−0.812381			−0.406191	+	0.913788i	$0.633143\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	20.7846i		−	0.885454i
$552$		0			0
$553$	16.5000	−	9.52628i	0.701651	−	0.405099i
$554$		0			0
$555$		0			0
$556$	−5.00000	+	8.66025i	−0.212047	+	0.367277i
$557$	24.0000	+	13.8564i	1.01691	+	0.587115i	0.913208	−	0.407493i	$-0.133597\pi$
$557$	24.0000	+	13.8564i	1.01691	+	0.587115i	0.103704	+	0.994608i	$0.466930\pi$
$558$		0			0
$559$	2.50000	+	2.59808i	0.105739	+	0.109887i
$560$	−24.0000			−1.01419
$561$		0			0
$562$		0			0
$563$	−21.0000	−	36.3731i	−0.885044	−	1.53294i	−0.845663	−	0.533718i	$-0.820794\pi$
$563$	−21.0000	−	36.3731i	−0.885044	−	1.53294i	−0.0393818	−	0.999224i	$-0.512539\pi$
$564$		0			0
$565$	18.0000	−	10.3923i	0.757266	−	0.437208i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−3.00000	+	5.19615i	−0.125767	+	0.217834i	−0.922032	−	0.387113i	$-0.873472\pi$
$569$	−3.00000	+	5.19615i	−0.125767	+	0.217834i	0.796266	+	0.604947i	$0.206806\pi$
$570$		0			0
$571$	−44.0000			−1.84134			−0.920671	−	0.390339i	$-0.872358\pi$
$571$	−44.0000			−1.84134			−0.920671	+	0.390339i	$0.872358\pi$
$572$	−6.00000	−	24.2487i	−0.250873	−	1.01389i
$573$		0			0
$574$		0			0
$575$	21.0000	−	36.3731i	0.875761	−	1.51686i
$576$		0			0
$577$		−	34.6410i		−	1.44212i	−0.692870	−	0.721062i	$-0.743654\pi$
$577$		−	34.6410i		−	1.44212i	0.692870	−	0.721062i	$-0.256346\pi$
$578$		0			0
$579$		0			0
$580$			41.5692i			1.72607i
$581$	−12.0000	−	20.7846i	−0.497844	−	0.862291i
$582$		0			0
$583$	−36.0000	−	20.7846i	−1.49097	−	0.860811i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	27.0000	+	15.5885i	1.11441	+	0.643404i	0.939968	−	0.341263i	$-0.110855\pi$
$587$	27.0000	+	15.5885i	1.11441	+	0.643404i	0.174441	+	0.984668i	$0.444188\pi$
$588$		0			0
$589$	3.00000	+	5.19615i	0.123613	+	0.214104i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	3.46410i		−	0.142254i	−0.997467	−	0.0711268i	$-0.977341\pi$
$593$		−	3.46410i		−	0.142254i	0.997467	−	0.0711268i	$-0.0226595\pi$
$594$		0			0
$595$		0			0
$596$	−12.0000	−	6.92820i	−0.491539	−	0.283790i
$597$		0			0
$598$		0			0
$599$	30.0000			1.22577			0.612883	−	0.790173i	$-0.290010\pi$
$599$	30.0000			1.22577			0.612883	+	0.790173i	$0.290010\pi$
$600$		0			0
$601$	13.0000	−	22.5167i	0.530281	−	0.918474i	−0.469095	−	0.883148i	$-0.655420\pi$
$601$	13.0000	−	22.5167i	0.530281	−	0.918474i	0.999376	−	0.0353259i	$-0.0112469\pi$
$602$		0			0
$603$		0			0
$604$	−6.00000	+	3.46410i	−0.244137	+	0.140952i
$605$	3.00000	−	1.73205i	0.121967	−	0.0704179i
$606$		0			0
$607$	−4.00000	−	6.92820i	−0.162355	−	0.281207i	0.773358	−	0.633970i	$-0.218576\pi$
$607$	−4.00000	−	6.92820i	−0.162355	−	0.281207i	−0.935713	+	0.352763i	$0.885242\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−3.00000	−	12.1244i	−0.121367	−	0.490499i
$612$		0			0
$613$	7.50000	+	4.33013i	0.302922	+	0.174892i	0.643755	−	0.765232i	$-0.277376\pi$
$613$	7.50000	+	4.33013i	0.302922	+	0.174892i	−0.340833	+	0.940124i	$0.610709\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	9.00000	−	5.19615i	0.362326	−	0.209189i	−0.307774	−	0.951459i	$-0.599584\pi$
$617$	9.00000	−	5.19615i	0.362326	−	0.209189i	0.670101	+	0.742270i	$0.266251\pi$
$618$		0			0
$619$			25.9808i			1.04425i	0.852867	+	0.522127i	$0.174861\pi$
$619$			25.9808i			1.04425i	−0.852867	+	0.522127i	$0.825139\pi$
$620$	−6.00000	−	10.3923i	−0.240966	−	0.417365i
$621$		0			0
$622$		0			0
$623$	12.0000			0.480770
$624$		0			0
$625$	−11.0000			−0.440000
$626$		0			0
$627$		0			0
$628$	−11.0000	−	19.0526i	−0.438948	−	0.760280i
$629$		0			0
$630$		0			0
$631$	1.50000	−	0.866025i	0.0597141	−	0.0344759i	−0.469846	−	0.882749i	$-0.655690\pi$
$631$	1.50000	−	0.866025i	0.0597141	−	0.0344759i	0.529560	+	0.848273i	$0.322357\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	39.0000	+	22.5167i	1.54767	+	0.893546i
$636$		0			0
$637$	−4.00000	+	13.8564i	−0.158486	+	0.549011i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	9.00000	+	15.5885i	0.355479	+	0.615707i	0.987200	−	0.159489i	$-0.0509845\pi$
$641$	9.00000	+	15.5885i	0.355479	+	0.615707i	−0.631721	+	0.775196i	$0.717651\pi$
$642$		0			0
$643$	16.5000	−	9.52628i	0.650696	−	0.375680i	−0.138027	−	0.990429i	$-0.544076\pi$
$643$	16.5000	−	9.52628i	0.650696	−	0.375680i	0.788723	+	0.614749i	$0.210743\pi$
$644$	−18.0000	+	10.3923i	−0.709299	+	0.409514i
$645$		0			0
$646$		0			0
$647$	−9.00000	+	15.5885i	−0.353827	+	0.612845i	−0.986916	−	0.161233i	$-0.948453\pi$
$647$	−9.00000	+	15.5885i	−0.353827	+	0.612845i	0.633090	+	0.774078i	$0.281786\pi$
$648$		0			0
$649$	12.0000			0.471041
$650$		0			0
$651$		0			0
$652$	33.0000	+	19.0526i	1.29238	+	0.746156i
$653$	18.0000	−	31.1769i	0.704394	−	1.22005i	−0.262515	−	0.964928i	$-0.584552\pi$
$653$	18.0000	−	31.1769i	0.704394	−	1.22005i	0.966910	−	0.255119i	$-0.0821147\pi$
$654$		0			0
$655$			20.7846i			0.812122i
$656$	−24.0000	+	13.8564i	−0.937043	+	0.541002i
$657$		0			0
$658$		0			0
$659$	24.0000	+	41.5692i	0.934907	+	1.61931i	0.774799	+	0.632207i	$0.217851\pi$
$659$	24.0000	+	41.5692i	0.934907	+	1.61931i	0.160108	+	0.987099i	$0.448816\pi$
$660$		0			0
$661$	−22.5000	−	12.9904i	−0.875149	−	0.505267i	−0.00609283	−	0.999981i	$-0.501939\pi$
$661$	−22.5000	−	12.9904i	−0.875149	−	0.505267i	−0.869056	+	0.494714i	$0.835273\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	18.0000	+	10.3923i	0.698010	+	0.402996i
$666$		0			0
$667$	18.0000	+	31.1769i	0.696963	+	1.20717i
$668$		−	13.8564i		−	0.536120i
$669$		0			0
$670$		0			0
$671$		−	3.46410i		−	0.133730i
$672$		0			0
$673$	0.500000	−	0.866025i	0.0192736	−	0.0333828i	−0.856228	−	0.516599i	$-0.827198\pi$
$673$	0.500000	−	0.866025i	0.0192736	−	0.0333828i	0.875501	+	0.483216i	$0.160531\pi$
$674$		0			0
$675$		0			0
$676$	−22.0000	−	13.8564i	−0.846154	−	0.532939i
$677$	−48.0000			−1.84479			−0.922395	−	0.386248i	$-0.873771\pi$
$677$	−48.0000			−1.84479			−0.922395	+	0.386248i	$0.873771\pi$
$678$		0			0
$679$	4.50000	−	7.79423i	0.172694	−	0.299115i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−21.0000	+	12.1244i	−0.803543	+	0.463926i	−0.844708	−	0.535227i	$-0.820226\pi$
$683$	−21.0000	+	12.1244i	−0.803543	+	0.463926i	0.0411658	+	0.999152i	$0.486893\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−4.00000			−0.152499
$689$	−42.0000	+	10.3923i	−1.60007	+	0.395915i
$690$		0			0
$691$	37.5000	+	21.6506i	1.42657	+	0.823629i	0.996848	−	0.0793336i	$-0.0252792\pi$
$691$	37.5000	+	21.6506i	1.42657	+	0.823629i	0.429719	+	0.902963i	$0.358613\pi$
$692$	−6.00000	+	10.3923i	−0.228086	+	0.395056i
$693$		0			0
$694$		0			0
$695$	−15.0000	+	8.66025i	−0.568982	+	0.328502i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−21.0000	−	12.1244i	−0.793725	−	0.458258i
$701$	12.0000			0.453234			0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000			0.453234			0.226617	+	0.973984i	$0.427233\pi$
$702$		0			0
$703$		0			0
$704$	24.0000	+	13.8564i	0.904534	+	0.522233i
$705$		0			0
$706$		0			0
$707$		−	31.1769i		−	1.17253i
$708$		0			0
$709$	−16.5000	+	9.52628i	−0.619671	+	0.357767i	−0.776741	−	0.629821i	$-0.783128\pi$
$709$	−16.5000	+	9.52628i	−0.619671	+	0.357767i	0.157070	+	0.987587i	$0.449795\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−9.00000	−	5.19615i	−0.337053	−	0.194597i
$714$		0			0
$715$	12.0000	−	41.5692i	0.448775	−	1.55460i
$716$	−24.0000			−0.896922
$717$		0			0
$718$		0			0
$719$	−3.00000	−	5.19615i	−0.111881	−	0.193784i	0.804648	−	0.593753i	$-0.202354\pi$
$719$	−3.00000	−	5.19615i	−0.111881	−	0.193784i	−0.916529	+	0.399969i	$0.869021\pi$
$720$		0			0
$721$	1.50000	−	0.866025i	0.0558629	−	0.0322525i
$722$		0			0
$723$		0			0
$724$	−14.0000	−	24.2487i	−0.520306	−	0.901196i
$725$	−21.0000	+	36.3731i	−0.779920	+	1.35086i
$726$		0			0
$727$	−35.0000			−1.29808			−0.649039	−	0.760755i	$-0.724829\pi$
$727$	−35.0000			−1.29808			−0.649039	+	0.760755i	$0.724829\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$			39.8372i			1.47142i	0.677297	+	0.735710i	$0.263151\pi$
$733$			39.8372i			1.47142i	−0.677297	+	0.735710i	$0.736849\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	15.0000	+	25.9808i	0.552532	+	0.957014i
$738$		0			0
$739$	39.0000	+	22.5167i	1.43464	+	0.828289i	0.997470	−	0.0710909i	$-0.0226481\pi$
$739$	39.0000	+	22.5167i	1.43464	+	0.828289i	0.437168	+	0.899380i	$0.355981\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	9.00000	+	5.19615i	0.330178	+	0.190628i	0.655920	−	0.754830i	$-0.272281\pi$
$743$	9.00000	+	5.19615i	0.330178	+	0.190628i	−0.325742	+	0.945459i	$0.605614\pi$
$744$		0			0
$745$	−12.0000	−	20.7846i	−0.439646	−	0.761489i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	10.3923i		−	0.379727i
$750$		0			0
$751$	−4.00000	+	6.92820i	−0.145962	+	0.252814i	−0.929731	−	0.368238i	$-0.879961\pi$
$751$	−4.00000	+	6.92820i	−0.145962	+	0.252814i	0.783769	+	0.621052i	$0.213294\pi$
$752$	12.0000	+	6.92820i	0.437595	+	0.252646i
$753$		0			0
$754$		0			0
$755$	−12.0000			−0.436725
$756$		0			0
$757$	−17.0000	+	29.4449i	−0.617876	+	1.07019i	0.371997	+	0.928234i	$0.378673\pi$
$757$	−17.0000	+	29.4449i	−0.617876	+	1.07019i	−0.989873	+	0.141958i	$0.954660\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	18.0000	−	10.3923i	0.652499	−	0.376721i	−0.136914	−	0.990583i	$-0.543718\pi$
$761$	18.0000	−	10.3923i	0.652499	−	0.376721i	0.789413	+	0.613862i	$0.210385\pi$
$762$		0			0
$763$	13.5000	+	23.3827i	0.488733	+	0.846510i
$764$	−18.0000	+	31.1769i	−0.651217	+	1.12794i
$765$		0			0
$766$		0			0
$767$	9.00000	−	8.66025i	0.324971	−	0.312704i
$768$		0			0
$769$	−6.00000	−	3.46410i	−0.216366	−	0.124919i	0.387901	−	0.921701i	$-0.373200\pi$
$769$	−6.00000	−	3.46410i	−0.216366	−	0.124919i	−0.604266	+	0.796782i	$0.706534\pi$
$770$		0			0
$771$		0			0
$772$			31.1769i			1.12208i
$773$	45.0000	−	25.9808i	1.61854	−	0.934463i	0.631239	−	0.775589i	$-0.282547\pi$
$773$	45.0000	−	25.9808i	1.61854	−	0.934463i	0.987299	−	0.158874i	$-0.0507865\pi$
$774$		0			0
$775$		−	12.1244i		−	0.435520i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	24.0000			0.859889
$780$		0			0
$781$	36.0000			1.28818
$782$		0			0
$783$		0			0
$784$	−8.00000	−	13.8564i	−0.285714	−	0.494872i
$785$		−	38.1051i		−	1.36003i
$786$		0			0
$787$	28.5000	−	16.4545i	1.01592	−	0.586539i	0.102997	−	0.994682i	$-0.467157\pi$
$787$	28.5000	−	16.4545i	1.01592	−	0.586539i	0.912918	+	0.408143i	$0.133823\pi$
$788$			27.7128i			0.987228i
$789$		0			0
$790$		0			0
$791$	−9.00000	−	5.19615i	−0.320003	−	0.184754i
$792$		0			0

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:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	117.q (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.73205i) q^{4} -3.46410i q^{5} +(-1.50000 + 0.866025i) q^{7} +O(q^{10})\)	\(q+(-1.00000 - 1.73205i) q^{4} -3.46410i q^{5} +(-1.50000 + 0.866025i) q^{7} +(3.00000 + 1.73205i) q^{11} +(3.50000 - 0.866025i) q^{13} +(-2.00000 + 3.46410i) q^{16} +(3.00000 - 1.73205i) q^{19} +(-6.00000 + 3.46410i) q^{20} +(-3.00000 + 5.19615i) q^{23} -7.00000 q^{25} +(3.00000 + 1.73205i) q^{28} +(3.00000 - 5.19615i) q^{29} +1.73205i q^{31} +(3.00000 + 5.19615i) q^{35} +(6.00000 + 3.46410i) q^{41} +(0.500000 + 0.866025i) q^{43} -6.92820i q^{44} -3.46410i q^{47} +(-2.00000 + 3.46410i) q^{49} +(-5.00000 - 5.19615i) q^{52} -12.0000 q^{53} +(6.00000 - 10.3923i) q^{55} +(3.00000 - 1.73205i) q^{59} +(-0.500000 - 0.866025i) q^{61} +8.00000 q^{64} +(-3.00000 - 12.1244i) q^{65} +(7.50000 + 4.33013i) q^{67} +(9.00000 - 5.19615i) q^{71} +1.73205i q^{73} +(-6.00000 - 3.46410i) q^{76} -6.00000 q^{77} -11.0000 q^{79} +(12.0000 + 6.92820i) q^{80} +13.8564i q^{83} +(-6.00000 - 3.46410i) q^{89} +(-4.50000 + 4.33013i) q^{91} +12.0000 q^{92} +(-6.00000 - 10.3923i) q^{95} +(-4.50000 + 2.59808i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 3 q^{7}+O(q^{10}) \) 2 * q - 2 * q^4 - 3 * q^7	\( 2 q - 2 q^{4} - 3 q^{7} + 6 q^{11} + 7 q^{13} - 4 q^{16} + 6 q^{19} - 12 q^{20} - 6 q^{23} - 14 q^{25} + 6 q^{28} + 6 q^{29} + 6 q^{35} + 12 q^{41} + q^{43} - 4 q^{49} - 10 q^{52} - 24 q^{53} + 12 q^{55} + 6 q^{59} - q^{61} + 16 q^{64} - 6 q^{65} + 15 q^{67} + 18 q^{71} - 12 q^{76} - 12 q^{77} - 22 q^{79} + 24 q^{80} - 12 q^{89} - 9 q^{91} + 24 q^{92} - 12 q^{95} - 9 q^{97}+O(q^{100}) \) 2 * q - 2 * q^4 - 3 * q^7 + 6 * q^11 + 7 * q^13 - 4 * q^16 + 6 * q^19 - 12 * q^20 - 6 * q^23 - 14 * q^25 + 6 * q^28 + 6 * q^29 + 6 * q^35 + 12 * q^41 + q^43 - 4 * q^49 - 10 * q^52 - 24 * q^53 + 12 * q^55 + 6 * q^59 - q^61 + 16 * q^64 - 6 * q^65 + 15 * q^67 + 18 * q^71 - 12 * q^76 - 12 * q^77 - 22 * q^79 + 24 * q^80 - 12 * q^89 - 9 * q^91 + 24 * q^92 - 12 * q^95 - 9 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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