LMFDB - Embedded newform 117.2.q.c.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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
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.c.10.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.50000 + 0.866025i) q^{2} +(0.500000 + 0.866025i) q^{4} +1.73205i q^{5} -1.73205i q^{8} +O(q^{10})$	\(q+(1.50000 + 0.866025i) q^{2} +(0.500000 + 0.866025i) q^{4} +1.73205i q^{5} -1.73205i q^{8} +(-1.50000 + 2.59808i) q^{10} +(-2.50000 - 2.59808i) q^{13} +(2.50000 - 4.33013i) q^{16} +(-1.50000 - 2.59808i) q^{17} +(-3.00000 + 1.73205i) q^{19} +(-1.50000 + 0.866025i) q^{20} +(-3.00000 + 5.19615i) q^{23} +2.00000 q^{25} +(-1.50000 - 6.06218i) q^{26} +(1.50000 - 2.59808i) q^{29} +3.46410i q^{31} +(4.50000 - 2.59808i) q^{32} -5.19615i q^{34} +(7.50000 + 4.33013i) q^{37} -6.00000 q^{38} +3.00000 q^{40} +(4.50000 + 2.59808i) q^{41} +(-4.00000 - 6.92820i) q^{43} +(-9.00000 + 5.19615i) q^{46} +3.46410i q^{47} +(-3.50000 + 6.06218i) q^{49} +(3.00000 + 1.73205i) q^{50} +(1.00000 - 3.46410i) q^{52} +3.00000 q^{53} +(4.50000 - 2.59808i) q^{58} +(-6.00000 + 3.46410i) q^{59} +(-0.500000 - 0.866025i) q^{61} +(-3.00000 + 5.19615i) q^{62} -1.00000 q^{64} +(4.50000 - 4.33013i) q^{65} +(3.00000 + 1.73205i) q^{67} +(1.50000 - 2.59808i) q^{68} +(-3.00000 + 1.73205i) q^{71} +1.73205i q^{73} +(7.50000 + 12.9904i) q^{74} +(-3.00000 - 1.73205i) q^{76} +4.00000 q^{79} +(7.50000 + 4.33013i) q^{80} +(4.50000 + 7.79423i) q^{82} -13.8564i q^{83} +(4.50000 - 2.59808i) q^{85} -13.8564i q^{86} +(6.00000 + 3.46410i) q^{89} -6.00000 q^{92} +(-3.00000 + 5.19615i) q^{94} +(-3.00000 - 5.19615i) q^{95} +(6.00000 - 3.46410i) q^{97} +(-10.5000 + 6.06218i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{2} + q^{4}+O(q^{10}) $ 2 * q + 3 * q^2 + q^4	$ 2 q + 3 q^{2} + q^{4} - 3 q^{10} - 5 q^{13} + 5 q^{16} - 3 q^{17} - 6 q^{19} - 3 q^{20} - 6 q^{23} + 4 q^{25} - 3 q^{26} + 3 q^{29} + 9 q^{32} + 15 q^{37} - 12 q^{38} + 6 q^{40} + 9 q^{41} - 8 q^{43} - 18 q^{46} - 7 q^{49} + 6 q^{50} + 2 q^{52} + 6 q^{53} + 9 q^{58} - 12 q^{59} - q^{61} - 6 q^{62} - 2 q^{64} + 9 q^{65} + 6 q^{67} + 3 q^{68} - 6 q^{71} + 15 q^{74} - 6 q^{76} + 8 q^{79} + 15 q^{80} + 9 q^{82} + 9 q^{85} + 12 q^{89} - 12 q^{92} - 6 q^{94} - 6 q^{95} + 12 q^{97} - 21 q^{98}+O(q^{100}) $ 2 * q + 3 * q^2 + q^4 - 3 * q^10 - 5 * q^13 + 5 * q^16 - 3 * q^17 - 6 * q^19 - 3 * q^20 - 6 * q^23 + 4 * q^25 - 3 * q^26 + 3 * q^29 + 9 * q^32 + 15 * q^37 - 12 * q^38 + 6 * q^40 + 9 * q^41 - 8 * q^43 - 18 * q^46 - 7 * q^49 + 6 * q^50 + 2 * q^52 + 6 * q^53 + 9 * q^58 - 12 * q^59 - q^61 - 6 * q^62 - 2 * q^64 + 9 * q^65 + 6 * q^67 + 3 * q^68 - 6 * q^71 + 15 * q^74 - 6 * q^76 + 8 * q^79 + 15 * q^80 + 9 * q^82 + 9 * q^85 + 12 * q^89 - 12 * q^92 - 6 * q^94 - 6 * q^95 + 12 * q^97 - 21 * q^98

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 $

$2$	1.50000	+	0.866025i	1.06066	+	0.612372i	0.925615	−	0.378467i	$-0.123549\pi$
$2$	1.50000	+	0.866025i	1.06066	+	0.612372i	0.135045	+	0.990839i	$0.456882\pi$
$3$		0			0
$3$		0			0
$4$	0.500000	+	0.866025i	0.250000	+	0.433013i
$5$			1.73205i			0.774597i	0.921954	+	0.387298i	$0.126592\pi$
$5$			1.73205i			0.774597i	−0.921954	+	0.387298i	$0.873408\pi$
$6$		0			0
$7$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$7$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$8$		−	1.73205i		−	0.612372i
$9$		0			0
$10$	−1.50000	+	2.59808i	−0.474342	+	0.821584i
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$		0			0
$13$	−2.50000	−	2.59808i	−0.693375	−	0.720577i
$13$	−2.50000	−	2.59808i	−0.693375	−	0.720577i
$14$		0			0
$15$		0			0
$16$	2.50000	−	4.33013i	0.625000	−	1.08253i
$17$	−1.50000	−	2.59808i	−0.363803	−	0.630126i	0.624780	−	0.780801i	$-0.285189\pi$
$17$	−1.50000	−	2.59808i	−0.363803	−	0.630126i	−0.988583	+	0.150675i	$0.951855\pi$
$18$		0			0
$19$	−3.00000	+	1.73205i	−0.688247	+	0.397360i	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−3.00000	+	1.73205i	−0.688247	+	0.397360i	0.114708	+	0.993399i	$0.463407\pi$
$20$	−1.50000	+	0.866025i	−0.335410	+	0.193649i
$21$		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
$25$	2.00000			0.400000
$26$	−1.50000	−	6.06218i	−0.294174	−	1.18889i
$27$		0			0
$28$		0			0
$29$	1.50000	−	2.59808i	0.278543	−	0.482451i	−0.692480	−	0.721437i	$-0.743482\pi$
$29$	1.50000	−	2.59808i	0.278543	−	0.482451i	0.971023	+	0.238987i	$0.0768152\pi$
$30$		0			0
$31$			3.46410i			0.622171i	0.950382	+	0.311086i	$0.100693\pi$
$31$			3.46410i			0.622171i	−0.950382	+	0.311086i	$0.899307\pi$
$32$	4.50000	−	2.59808i	0.795495	−	0.459279i
$33$		0			0
$34$		−	5.19615i		−	0.891133i
$35$		0			0
$36$		0			0
$37$	7.50000	+	4.33013i	1.23299	+	0.711868i	0.967653	−	0.252286i	$-0.0811825\pi$
$37$	7.50000	+	4.33013i	1.23299	+	0.711868i	0.265340	+	0.964155i	$0.414516\pi$
$38$	−6.00000			−0.973329
$39$		0			0
$40$	3.00000			0.474342
$41$	4.50000	+	2.59808i	0.702782	+	0.405751i	0.808383	−	0.588657i	$-0.200343\pi$
$41$	4.50000	+	2.59808i	0.702782	+	0.405751i	−0.105601	+	0.994409i	$0.533677\pi$
$42$		0			0
$43$	−4.00000	−	6.92820i	−0.609994	−	1.05654i	−0.991241	−	0.132068i	$-0.957838\pi$
$43$	−4.00000	−	6.92820i	−0.609994	−	1.05654i	0.381246	−	0.924473i	$-0.375495\pi$
$44$		0			0
$45$		0			0
$46$	−9.00000	+	5.19615i	−1.32698	+	0.766131i
$47$			3.46410i			0.505291i	0.967559	+	0.252646i	$0.0813007\pi$
$47$			3.46410i			0.505291i	−0.967559	+	0.252646i	$0.918699\pi$
$48$		0			0
$49$	−3.50000	+	6.06218i	−0.500000	+	0.866025i
$50$	3.00000	+	1.73205i	0.424264	+	0.244949i
$51$		0			0
$52$	1.00000	−	3.46410i	0.138675	−	0.480384i
$53$	3.00000			0.412082			0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000			0.412082			0.206041	+	0.978543i	$0.433942\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	4.50000	−	2.59808i	0.590879	−	0.341144i
$59$	−6.00000	+	3.46410i	−0.781133	+	0.450988i	−0.836832	−	0.547460i	$-0.815595\pi$
$59$	−6.00000	+	3.46410i	−0.781133	+	0.450988i	0.0556984	+	0.998448i	$0.482261\pi$
$60$		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	$0.853725\pi$
$62$	−3.00000	+	5.19615i	−0.381000	+	0.659912i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	4.50000	−	4.33013i	0.558156	−	0.537086i
$66$		0			0
$67$	3.00000	+	1.73205i	0.366508	+	0.211604i	0.671932	−	0.740613i	$-0.265465\pi$
$67$	3.00000	+	1.73205i	0.366508	+	0.211604i	−0.305424	+	0.952217i	$0.598798\pi$
$68$	1.50000	−	2.59808i	0.181902	−	0.315063i
$69$		0			0
$70$		0			0
$71$	−3.00000	+	1.73205i	−0.356034	+	0.205557i	−0.667340	−	0.744753i	$-0.732567\pi$
$71$	−3.00000	+	1.73205i	−0.356034	+	0.205557i	0.311305	+	0.950310i	$0.399234\pi$
$72$		0			0
$73$			1.73205i			0.202721i	0.994850	+	0.101361i	$0.0323196\pi$
$73$			1.73205i			0.202721i	−0.994850	+	0.101361i	$0.967680\pi$
$74$	7.50000	+	12.9904i	0.871857	+	1.51010i
$75$		0			0
$76$	−3.00000	−	1.73205i	−0.344124	−	0.198680i
$77$		0			0
$78$		0			0
$79$	4.00000			0.450035			0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000			0.450035			0.225018	+	0.974355i	$0.427756\pi$
$80$	7.50000	+	4.33013i	0.838525	+	0.484123i
$81$		0			0
$82$	4.50000	+	7.79423i	0.496942	+	0.860729i
$83$		−	13.8564i		−	1.52094i	−0.649374	−	0.760469i	$-0.724969\pi$
$83$		−	13.8564i		−	1.52094i	0.649374	−	0.760469i	$-0.275031\pi$
$84$		0			0
$85$	4.50000	−	2.59808i	0.488094	−	0.281801i
$86$		−	13.8564i		−	1.49417i
$87$		0			0
$88$		0			0
$89$	6.00000	+	3.46410i	0.635999	+	0.367194i	0.783072	−	0.621932i	$-0.213652\pi$
$89$	6.00000	+	3.46410i	0.635999	+	0.367194i	−0.147073	+	0.989126i	$0.546985\pi$
$90$		0			0
$91$		0			0
$92$	−6.00000			−0.625543
$93$		0			0
$94$	−3.00000	+	5.19615i	−0.309426	+	0.535942i
$95$	−3.00000	−	5.19615i	−0.307794	−	0.533114i
$96$		0			0
$97$	6.00000	−	3.46410i	0.609208	−	0.351726i	−0.163448	−	0.986552i	$-0.552261\pi$
$97$	6.00000	−	3.46410i	0.609208	−	0.351726i	0.772655	+	0.634826i	$0.218928\pi$
$98$	−10.5000	+	6.06218i	−1.06066	+	0.612372i
$99$		0			0
$100$	1.00000	+	1.73205i	0.100000	+	0.173205i
$101$	−1.50000	+	2.59808i	−0.149256	+	0.258518i	−0.930953	−	0.365140i	$-0.881021\pi$
$101$	−1.50000	+	2.59808i	−0.149256	+	0.258518i	0.781697	+	0.623658i	$0.214354\pi$
$102$		0			0
$103$	−10.0000			−0.985329			−0.492665	−	0.870219i	$-0.663977\pi$
$103$	−10.0000			−0.985329			−0.492665	+	0.870219i	$0.663977\pi$
$104$	−4.50000	+	4.33013i	−0.441261	+	0.424604i
$105$		0			0
$106$	4.50000	+	2.59808i	0.437079	+	0.252347i
$107$	3.00000	−	5.19615i	0.290021	−	0.502331i	−0.683793	−	0.729676i	$-0.739671\pi$
$107$	3.00000	−	5.19615i	0.290021	−	0.502331i	0.973814	+	0.227345i	$0.0730044\pi$
$108$		0			0
$109$		−	13.8564i		−	1.32720i	−0.748086	−	0.663602i	$-0.769027\pi$
$109$		−	13.8564i		−	1.32720i	0.748086	−	0.663602i	$-0.230973\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−7.50000	−	12.9904i	−0.705541	−	1.22203i	−0.966496	−	0.256681i	$-0.917371\pi$
$113$	−7.50000	−	12.9904i	−0.705541	−	1.22203i	0.260955	−	0.965351i	$-0.415962\pi$
$114$		0			0
$115$	−9.00000	−	5.19615i	−0.839254	−	0.484544i
$116$	3.00000			0.278543
$117$		0			0
$118$	−12.0000			−1.10469
$119$		0			0
$120$		0			0
$121$	−5.50000	−	9.52628i	−0.500000	−	0.866025i
$122$		−	1.73205i		−	0.156813i
$123$		0			0
$124$	−3.00000	+	1.73205i	−0.269408	+	0.155543i
$125$			12.1244i			1.08444i
$126$		0			0
$127$	1.00000	−	1.73205i	0.0887357	−	0.153695i	−0.818241	−	0.574875i	$-0.805051\pi$
$127$	1.00000	−	1.73205i	0.0887357	−	0.153695i	0.906977	+	0.421180i	$0.138384\pi$
$128$	−10.5000	−	6.06218i	−0.928078	−	0.535826i
$129$		0			0
$130$	10.5000	−	2.59808i	0.920911	−	0.227866i
$131$	−18.0000			−1.57267			−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000			−1.57267			−0.786334	+	0.617802i	$0.788023\pi$
$132$		0			0
$133$		0			0
$134$	3.00000	+	5.19615i	0.259161	+	0.448879i
$135$		0			0
$136$	−4.50000	+	2.59808i	−0.385872	+	0.222783i
$137$	13.5000	−	7.79423i	1.15338	−	0.665906i	0.203674	−	0.979039i	$-0.434712\pi$
$137$	13.5000	−	7.79423i	1.15338	−	0.665906i	0.949709	+	0.313133i	$0.101379\pi$
$138$		0			0
$139$	2.00000	+	3.46410i	0.169638	+	0.293821i	0.938293	−	0.345843i	$-0.112407\pi$
$139$	2.00000	+	3.46410i	0.169638	+	0.293821i	−0.768655	+	0.639664i	$0.779074\pi$
$140$		0			0
$141$		0			0
$142$	−6.00000			−0.503509
$143$		0			0
$144$		0			0
$145$	4.50000	+	2.59808i	0.373705	+	0.215758i
$146$	−1.50000	+	2.59808i	−0.124141	+	0.215018i
$147$		0			0
$148$			8.66025i			0.711868i
$149$	16.5000	−	9.52628i	1.35173	−	0.780423i	0.363241	−	0.931695i	$-0.381670\pi$
$149$	16.5000	−	9.52628i	1.35173	−	0.780423i	0.988492	+	0.151272i	$0.0483370\pi$
$150$		0			0
$151$			17.3205i			1.40952i	0.709444	+	0.704761i	$0.248946\pi$
$151$			17.3205i			1.40952i	−0.709444	+	0.704761i	$0.751054\pi$
$152$	3.00000	+	5.19615i	0.243332	+	0.421464i
$153$		0			0
$154$		0			0
$155$	−6.00000			−0.481932
$156$		0			0
$157$	−13.0000			−1.03751			−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000			−1.03751			−0.518756	+	0.854922i	$0.673605\pi$
$158$	6.00000	+	3.46410i	0.477334	+	0.275589i
$159$		0			0
$160$	4.50000	+	7.79423i	0.355756	+	0.616188i
$161$		0			0
$162$		0			0
$163$	18.0000	−	10.3923i	1.40987	−	0.813988i	0.414494	−	0.910052i	$-0.363959\pi$
$163$	18.0000	−	10.3923i	1.40987	−	0.813988i	0.995375	+	0.0960641i	$0.0306254\pi$
$164$			5.19615i			0.405751i
$165$		0			0
$166$	12.0000	−	20.7846i	0.931381	−	1.61320i
$167$	12.0000	+	6.92820i	0.928588	+	0.536120i	0.886365	−	0.462988i	$-0.153223\pi$
$167$	12.0000	+	6.92820i	0.928588	+	0.536120i	0.0422232	+	0.999108i	$0.486556\pi$
$168$		0			0
$169$	−0.500000	+	12.9904i	−0.0384615	+	0.999260i
$170$	9.00000			0.690268
$171$		0			0
$172$	4.00000	−	6.92820i	0.304997	−	0.528271i
$173$	3.00000	+	5.19615i	0.228086	+	0.395056i	0.957241	−	0.289292i	$-0.0934200\pi$
$173$	3.00000	+	5.19615i	0.228086	+	0.395056i	−0.729155	+	0.684349i	$0.760087\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	6.00000	+	10.3923i	0.449719	+	0.778936i
$179$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$179$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$180$		0			0
$181$	11.0000			0.817624			0.408812	−	0.912619i	$-0.365943\pi$
$181$	11.0000			0.817624			0.408812	+	0.912619i	$0.365943\pi$
$182$		0			0
$183$		0			0
$184$	9.00000	+	5.19615i	0.663489	+	0.383065i
$185$	−7.50000	+	12.9904i	−0.551411	+	0.955072i
$186$		0			0
$187$		0			0
$188$	−3.00000	+	1.73205i	−0.218797	+	0.126323i
$189$		0			0
$190$		−	10.3923i		−	0.753937i
$191$	9.00000	+	15.5885i	0.651217	+	1.12794i	0.982828	+	0.184525i	$0.0590746\pi$
$191$	9.00000	+	15.5885i	0.651217	+	1.12794i	−0.331611	+	0.943416i	$0.607592\pi$
$192$		0			0
$193$	−4.50000	−	2.59808i	−0.323917	−	0.187014i	0.329220	−	0.944253i	$-0.393214\pi$
$193$	−4.50000	−	2.59808i	−0.323917	−	0.187014i	−0.653137	+	0.757240i	$0.726548\pi$
$194$	12.0000			0.861550
$195$		0			0
$196$	−7.00000			−0.500000
$197$	−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
$199$	1.00000	+	1.73205i	0.0708881	+	0.122782i	0.899291	−	0.437351i	$-0.144083\pi$
$199$	1.00000	+	1.73205i	0.0708881	+	0.122782i	−0.828403	+	0.560133i	$0.810750\pi$
$200$		−	3.46410i		−	0.244949i
$201$		0			0
$202$	−4.50000	+	2.59808i	−0.316619	+	0.182800i
$203$		0			0
$204$		0			0
$205$	−4.50000	+	7.79423i	−0.314294	+	0.544373i
$206$	−15.0000	−	8.66025i	−1.04510	−	0.603388i
$207$		0			0
$208$	−17.5000	+	4.33013i	−1.21341	+	0.300240i
$209$		0			0
$210$		0			0
$211$	−5.00000	+	8.66025i	−0.344214	+	0.596196i	−0.985211	−	0.171347i	$-0.945188\pi$
$211$	−5.00000	+	8.66025i	−0.344214	+	0.596196i	0.640996	+	0.767544i	$0.278521\pi$
$212$	1.50000	+	2.59808i	0.103020	+	0.178437i
$213$		0			0
$214$	9.00000	−	5.19615i	0.615227	−	0.355202i
$215$	12.0000	−	6.92820i	0.818393	−	0.472500i
$216$		0			0
$217$		0			0
$218$	12.0000	−	20.7846i	0.812743	−	1.40771i
$219$		0			0
$220$		0			0
$221$	−3.00000	+	10.3923i	−0.201802	+	0.699062i
$222$		0			0
$223$	−9.00000	−	5.19615i	−0.602685	−	0.347960i	0.167412	−	0.985887i	$-0.446459\pi$
$223$	−9.00000	−	5.19615i	−0.602685	−	0.347960i	−0.770097	+	0.637927i	$0.779792\pi$
$224$		0			0
$225$		0			0
$226$		−	25.9808i		−	1.72821i
$227$	−21.0000	+	12.1244i	−1.39382	+	0.804722i	−0.993736	−	0.111757i	$-0.964352\pi$
$227$	−21.0000	+	12.1244i	−1.39382	+	0.804722i	−0.400083	+	0.916479i	$0.631019\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$	−9.00000	−	15.5885i	−0.593442	−	1.02787i
$231$		0			0
$232$	−4.50000	−	2.59808i	−0.295439	−	0.170572i
$233$	−6.00000			−0.393073			−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000			−0.393073			−0.196537	+	0.980497i	$0.562969\pi$
$234$		0			0
$235$	−6.00000			−0.391397
$236$	−6.00000	−	3.46410i	−0.390567	−	0.225494i
$237$		0			0
$238$		0			0
$239$			20.7846i			1.34444i	0.740349	+	0.672222i	$0.234660\pi$
$239$			20.7846i			1.34444i	−0.740349	+	0.672222i	$0.765340\pi$
$240$		0			0
$241$	−1.50000	+	0.866025i	−0.0966235	+	0.0557856i	−0.547533	−	0.836784i	$-0.684433\pi$
$241$	−1.50000	+	0.866025i	−0.0966235	+	0.0557856i	0.450910	+	0.892570i	$0.351100\pi$
$242$		−	19.0526i		−	1.22474i
$243$		0			0
$244$	0.500000	−	0.866025i	0.0320092	−	0.0554416i
$245$	−10.5000	−	6.06218i	−0.670820	−	0.387298i
$246$		0			0
$247$	12.0000	+	3.46410i	0.763542	+	0.220416i
$248$	6.00000			0.381000
$249$		0			0
$250$	−10.5000	+	18.1865i	−0.664078	+	1.15022i
$251$	−9.00000	−	15.5885i	−0.568075	−	0.983935i	−0.996756	−	0.0804789i	$-0.974355\pi$
$251$	−9.00000	−	15.5885i	−0.568075	−	0.983935i	0.428681	−	0.903456i	$-0.358978\pi$
$252$		0			0
$253$		0			0
$254$	3.00000	−	1.73205i	0.188237	−	0.108679i
$255$		0			0
$256$	−9.50000	−	16.4545i	−0.593750	−	1.02841i
$257$	1.50000	−	2.59808i	0.0935674	−	0.162064i	−0.815442	−	0.578838i	$-0.803506\pi$
$257$	1.50000	−	2.59808i	0.0935674	−	0.162064i	0.909010	+	0.416775i	$0.136840\pi$
$258$		0			0
$259$		0			0
$260$	6.00000	+	1.73205i	0.372104	+	0.107417i
$261$		0			0
$262$	−27.0000	−	15.5885i	−1.66807	−	0.963058i
$263$	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
$265$			5.19615i			0.319197i
$266$		0			0
$267$		0			0
$268$			3.46410i			0.211604i
$269$	−3.00000	−	5.19615i	−0.182913	−	0.316815i	0.759958	−	0.649972i	$-0.225219\pi$
$269$	−3.00000	−	5.19615i	−0.182913	−	0.316815i	−0.942871	+	0.333157i	$0.891886\pi$
$270$		0			0
$271$	18.0000	+	10.3923i	1.09342	+	0.631288i	0.934485	−	0.356001i	$-0.115860\pi$
$271$	18.0000	+	10.3923i	1.09342	+	0.631288i	0.158937	+	0.987289i	$0.449193\pi$
$272$	−15.0000			−0.909509
$273$		0			0
$274$	27.0000			1.63113
$275$		0			0
$276$		0			0
$277$	3.50000	+	6.06218i	0.210295	+	0.364241i	0.951807	−	0.306699i	$-0.0992243\pi$
$277$	3.50000	+	6.06218i	0.210295	+	0.364241i	−0.741512	+	0.670940i	$0.765891\pi$
$278$			6.92820i			0.415526i
$279$		0			0
$280$		0			0
$281$			22.5167i			1.34323i	0.740900	+	0.671616i	$0.234399\pi$
$281$			22.5167i			1.34323i	−0.740900	+	0.671616i	$0.765601\pi$
$282$		0			0
$283$	−2.00000	+	3.46410i	−0.118888	+	0.205919i	−0.919327	−	0.393494i	$-0.871266\pi$
$283$	−2.00000	+	3.46410i	−0.118888	+	0.205919i	0.800439	+	0.599414i	$0.204600\pi$
$284$	−3.00000	−	1.73205i	−0.178017	−	0.102778i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	4.00000	−	6.92820i	0.235294	−	0.407541i
$290$	4.50000	+	7.79423i	0.264249	+	0.457693i
$291$		0			0
$292$	−1.50000	+	0.866025i	−0.0877809	+	0.0506803i
$293$	−4.50000	+	2.59808i	−0.262893	+	0.151781i	−0.625653	−	0.780101i	$-0.715168\pi$
$293$	−4.50000	+	2.59808i	−0.262893	+	0.151781i	0.362761	+	0.931882i	$0.381834\pi$
$294$		0			0
$295$	−6.00000	−	10.3923i	−0.349334	−	0.605063i
$296$	7.50000	−	12.9904i	0.435929	−	0.755051i
$297$		0			0
$298$	33.0000			1.91164
$299$	21.0000	−	5.19615i	1.21446	−	0.300501i
$300$		0			0
$301$		0			0
$302$	−15.0000	+	25.9808i	−0.863153	+	1.49502i
$303$		0			0
$304$			17.3205i			0.993399i
$305$	1.50000	−	0.866025i	0.0858898	−	0.0495885i
$306$		0			0
$307$		−	17.3205i		−	0.988534i	−0.869310	−	0.494267i	$-0.835437\pi$
$307$		−	17.3205i		−	0.988534i	0.869310	−	0.494267i	$-0.164563\pi$
$308$		0			0
$309$		0			0
$310$	−9.00000	−	5.19615i	−0.511166	−	0.295122i
$311$	30.0000			1.70114			0.850572	−	0.525859i	$-0.176256\pi$
$311$	30.0000			1.70114			0.850572	+	0.525859i	$0.176256\pi$
$312$		0			0
$313$	10.0000			0.565233			0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000			0.565233			0.282617	+	0.959233i	$0.408798\pi$
$314$	−19.5000	−	11.2583i	−1.10045	−	0.635344i
$315$		0			0
$316$	2.00000	+	3.46410i	0.112509	+	0.194871i
$317$			5.19615i			0.291845i	0.989296	+	0.145922i	$0.0466150\pi$
$317$			5.19615i			0.291845i	−0.989296	+	0.145922i	$0.953385\pi$
$318$		0			0
$319$		0			0
$320$		−	1.73205i		−	0.0968246i
$321$		0			0
$322$		0			0
$323$	9.00000	+	5.19615i	0.500773	+	0.289122i
$324$		0			0
$325$	−5.00000	−	5.19615i	−0.277350	−	0.288231i
$326$	36.0000			1.99386
$327$		0			0
$328$	4.50000	−	7.79423i	0.248471	−	0.430364i
$329$		0			0
$330$		0			0
$331$	−24.0000	+	13.8564i	−1.31916	+	0.761617i	−0.983593	−	0.180400i	$-0.942261\pi$
$331$	−24.0000	+	13.8564i	−1.31916	+	0.761617i	−0.335566	+	0.942017i	$0.608928\pi$
$332$	12.0000	−	6.92820i	0.658586	−	0.380235i
$333$		0			0
$334$	12.0000	+	20.7846i	0.656611	+	1.13728i
$335$	−3.00000	+	5.19615i	−0.163908	+	0.283896i
$336$		0			0
$337$	−23.0000			−1.25289			−0.626445	−	0.779466i	$-0.715491\pi$
$337$	−23.0000			−1.25289			−0.626445	+	0.779466i	$0.715491\pi$
$338$	−12.0000	+	19.0526i	−0.652714	+	1.03632i
$339$		0			0
$340$	4.50000	+	2.59808i	0.244047	+	0.140900i
$341$		0			0
$342$		0			0
$343$		0			0
$344$	−12.0000	+	6.92820i	−0.646997	+	0.373544i
$345$		0			0
$346$			10.3923i			0.558694i
$347$	−15.0000	−	25.9808i	−0.805242	−	1.39472i	−0.916127	−	0.400887i	$-0.868702\pi$
$347$	−15.0000	−	25.9808i	−0.805242	−	1.39472i	0.110885	−	0.993833i	$-0.464631\pi$
$348$		0			0
$349$	12.0000	+	6.92820i	0.642345	+	0.370858i	0.785517	−	0.618840i	$-0.212397\pi$
$349$	12.0000	+	6.92820i	0.642345	+	0.370858i	−0.143172	+	0.989698i	$0.545730\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−28.5000	−	16.4545i	−1.51690	−	0.875784i	−0.999803	−	0.0198582i	$-0.993679\pi$
$353$	−28.5000	−	16.4545i	−1.51690	−	0.875784i	−0.517099	−	0.855926i	$-0.672988\pi$
$354$		0			0
$355$	−3.00000	−	5.19615i	−0.159223	−	0.275783i
$356$			6.92820i			0.367194i
$357$		0			0
$358$		0			0
$359$			6.92820i			0.365657i	0.983145	+	0.182828i	$0.0585252\pi$
$359$			6.92820i			0.365657i	−0.983145	+	0.182828i	$0.941475\pi$
$360$		0			0
$361$	−3.50000	+	6.06218i	−0.184211	+	0.319062i
$362$	16.5000	+	9.52628i	0.867221	+	0.500690i
$363$		0			0
$364$		0			0
$365$	−3.00000			−0.157027
$366$		0			0
$367$	11.0000	−	19.0526i	0.574195	−	0.994535i	−0.421933	−	0.906627i	$-0.638648\pi$
$367$	11.0000	−	19.0526i	0.574195	−	0.994535i	0.996129	−	0.0879086i	$-0.0280183\pi$
$368$	15.0000	+	25.9808i	0.781929	+	1.35434i
$369$		0			0
$370$	−22.5000	+	12.9904i	−1.16972	+	0.675338i
$371$		0			0
$372$		0			0
$373$	−9.50000	−	16.4545i	−0.491891	−	0.851981i	0.508065	−	0.861319i	$-0.330361\pi$
$373$	−9.50000	−	16.4545i	−0.491891	−	0.851981i	−0.999956	+	0.00933789i	$0.997028\pi$
$374$		0			0
$375$		0			0
$376$	6.00000			0.309426
$377$	−10.5000	+	2.59808i	−0.540778	+	0.133808i
$378$		0			0
$379$	−21.0000	−	12.1244i	−1.07870	−	0.622786i	−0.148153	−	0.988964i	$-0.547333\pi$
$379$	−21.0000	−	12.1244i	−1.07870	−	0.622786i	−0.930545	+	0.366178i	$0.880666\pi$
$380$	3.00000	−	5.19615i	0.153897	−	0.266557i
$381$		0			0
$382$			31.1769i			1.59515i
$383$	18.0000	−	10.3923i	0.919757	−	0.531022i	0.0361995	−	0.999345i	$-0.488475\pi$
$383$	18.0000	−	10.3923i	0.919757	−	0.531022i	0.883558	+	0.468323i	$0.155141\pi$
$384$		0			0
$385$		0			0
$386$	−4.50000	−	7.79423i	−0.229044	−	0.396716i
$387$		0			0
$388$	6.00000	+	3.46410i	0.304604	+	0.175863i
$389$	9.00000			0.456318			0.228159	−	0.973624i	$-0.426729\pi$
$389$	9.00000			0.456318			0.228159	+	0.973624i	$0.426729\pi$
$390$		0			0
$391$	18.0000			0.910299
$392$	10.5000	+	6.06218i	0.530330	+	0.306186i
$393$		0			0
$394$	−12.0000	−	20.7846i	−0.604551	−	1.04711i
$395$			6.92820i			0.348596i
$396$		0			0
$397$	−12.0000	+	6.92820i	−0.602263	+	0.347717i	−0.769931	−	0.638127i	$-0.779710\pi$
$397$	−12.0000	+	6.92820i	−0.602263	+	0.347717i	0.167668	+	0.985843i	$0.446376\pi$
$398$			3.46410i			0.173640i
$399$		0			0
$400$	5.00000	−	8.66025i	0.250000	−	0.433013i
$401$	1.50000	+	0.866025i	0.0749064	+	0.0432472i	0.536985	−	0.843592i	$-0.319563\pi$
$401$	1.50000	+	0.866025i	0.0749064	+	0.0432472i	−0.462079	+	0.886839i	$0.652896\pi$
$402$		0			0
$403$	9.00000	−	8.66025i	0.448322	−	0.431398i
$404$	−3.00000			−0.149256
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	13.5000	−	7.79423i	0.667532	−	0.385400i	−0.127609	−	0.991825i	$-0.540730\pi$
$409$	13.5000	−	7.79423i	0.667532	−	0.385400i	0.795141	+	0.606425i	$0.207397\pi$
$410$	−13.5000	+	7.79423i	−0.666717	+	0.384930i
$411$		0			0
$412$	−5.00000	−	8.66025i	−0.246332	−	0.426660i
$413$		0			0
$414$		0			0
$415$	24.0000			1.17811
$416$	−18.0000	−	5.19615i	−0.882523	−	0.254762i
$417$		0			0
$418$		0			0
$419$	9.00000	−	15.5885i	0.439679	−	0.761546i	−0.557986	−	0.829851i	$-0.688426\pi$
$419$	9.00000	−	15.5885i	0.439679	−	0.761546i	0.997665	+	0.0683046i	$0.0217590\pi$
$420$		0			0
$421$			15.5885i			0.759735i	0.925041	+	0.379867i	$0.124030\pi$
$421$			15.5885i			0.759735i	−0.925041	+	0.379867i	$0.875970\pi$
$422$	−15.0000	+	8.66025i	−0.730189	+	0.421575i
$423$		0			0
$424$		−	5.19615i		−	0.252347i
$425$	−3.00000	−	5.19615i	−0.145521	−	0.252050i
$426$		0			0
$427$		0			0
$428$	6.00000			0.290021
$429$		0			0
$430$	24.0000			1.15738
$431$	6.00000	+	3.46410i	0.289010	+	0.166860i	0.637495	−	0.770454i	$-0.279971\pi$
$431$	6.00000	+	3.46410i	0.289010	+	0.166860i	−0.348485	+	0.937314i	$0.613304\pi$
$432$		0			0
$433$	8.50000	+	14.7224i	0.408484	+	0.707515i	0.994720	−	0.102625i	$-0.0327243\pi$
$433$	8.50000	+	14.7224i	0.408484	+	0.707515i	−0.586236	+	0.810140i	$0.699391\pi$
$434$		0			0
$435$		0			0
$436$	12.0000	−	6.92820i	0.574696	−	0.331801i
$437$		−	20.7846i		−	0.994263i
$438$		0			0
$439$	14.0000	−	24.2487i	0.668184	−	1.15733i	−0.310228	−	0.950662i	$-0.600405\pi$
$439$	14.0000	−	24.2487i	0.668184	−	1.15733i	0.978412	−	0.206666i	$-0.0662612\pi$
$440$		0			0
$441$		0			0
$442$	−13.5000	+	12.9904i	−0.642130	+	0.617889i
$443$	12.0000			0.570137			0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000			0.570137			0.285069	+	0.958507i	$0.407984\pi$
$444$		0			0
$445$	−6.00000	+	10.3923i	−0.284427	+	0.492642i
$446$	−9.00000	−	15.5885i	−0.426162	−	0.738135i
$447$		0			0
$448$		0			0
$449$	6.00000	−	3.46410i	0.283158	−	0.163481i	−0.351694	−	0.936115i	$-0.614394\pi$
$449$	6.00000	−	3.46410i	0.283158	−	0.163481i	0.634852	+	0.772634i	$0.281061\pi$
$450$		0			0
$451$		0			0
$452$	7.50000	−	12.9904i	0.352770	−	0.611016i
$453$		0			0
$454$	−42.0000			−1.97116
$455$		0			0
$456$		0			0
$457$	1.50000	+	0.866025i	0.0701670	+	0.0405110i	0.534673	−	0.845059i	$-0.320435\pi$
$457$	1.50000	+	0.866025i	0.0701670	+	0.0405110i	−0.464506	+	0.885570i	$0.653768\pi$
$458$		0			0
$459$		0			0
$460$		−	10.3923i		−	0.484544i
$461$	−19.5000	+	11.2583i	−0.908206	+	0.524353i	−0.879853	−	0.475245i	$-0.842359\pi$
$461$	−19.5000	+	11.2583i	−0.908206	+	0.524353i	−0.0283522	+	0.999598i	$0.509026\pi$
$462$		0			0
$463$			13.8564i			0.643962i	0.946746	+	0.321981i	$0.104349\pi$
$463$			13.8564i			0.643962i	−0.946746	+	0.321981i	$0.895651\pi$
$464$	−7.50000	−	12.9904i	−0.348179	−	0.603063i
$465$		0			0
$466$	−9.00000	−	5.19615i	−0.416917	−	0.240707i
$467$	−12.0000			−0.555294			−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000			−0.555294			−0.277647	+	0.960683i	$0.589555\pi$
$468$		0			0
$469$		0			0
$470$	−9.00000	−	5.19615i	−0.415139	−	0.239681i
$471$		0			0
$472$	6.00000	+	10.3923i	0.276172	+	0.478345i
$473$		0			0
$474$		0			0
$475$	−6.00000	+	3.46410i	−0.275299	+	0.158944i
$476$		0			0
$477$		0			0
$478$	−18.0000	+	31.1769i	−0.823301	+	1.42600i
$479$	−21.0000	−	12.1244i	−0.959514	−	0.553976i	−0.0634909	−	0.997982i	$-0.520223\pi$
$479$	−21.0000	−	12.1244i	−0.959514	−	0.553976i	−0.896024	+	0.444006i	$0.853557\pi$
$480$		0			0
$481$	−7.50000	−	30.3109i	−0.341971	−	1.38206i
$482$	−3.00000			−0.136646
$483$		0			0
$484$	5.50000	−	9.52628i	0.250000	−	0.433013i
$485$	6.00000	+	10.3923i	0.272446	+	0.471890i
$486$		0			0
$487$	−6.00000	+	3.46410i	−0.271886	+	0.156973i	−0.629744	−	0.776802i	$-0.716840\pi$
$487$	−6.00000	+	3.46410i	−0.271886	+	0.156973i	0.357858	+	0.933776i	$0.383507\pi$
$488$	−1.50000	+	0.866025i	−0.0679018	+	0.0392031i
$489$		0			0
$490$	−10.5000	−	18.1865i	−0.474342	−	0.821584i
$491$	6.00000	−	10.3923i	0.270776	−	0.468998i	−0.698285	−	0.715820i	$-0.746053\pi$
$491$	6.00000	−	10.3923i	0.270776	−	0.468998i	0.969061	+	0.246822i	$0.0793863\pi$
$492$		0			0
$493$	−9.00000			−0.405340
$494$	15.0000	+	15.5885i	0.674882	+	0.701358i
$495$		0			0
$496$	15.0000	+	8.66025i	0.673520	+	0.388857i
$497$		0			0
$498$		0			0
$499$			31.1769i			1.39567i	0.716258	+	0.697835i	$0.245853\pi$
$499$			31.1769i			1.39567i	−0.716258	+	0.697835i	$0.754147\pi$
$500$	−10.5000	+	6.06218i	−0.469574	+	0.271109i
$501$		0			0
$502$		−	31.1769i		−	1.39149i
$503$	18.0000	+	31.1769i	0.802580	+	1.39011i	0.917912	+	0.396783i	$0.129873\pi$
$503$	18.0000	+	31.1769i	0.802580	+	1.39011i	−0.115332	+	0.993327i	$0.536793\pi$
$504$		0			0
$505$	−4.50000	−	2.59808i	−0.200247	−	0.115613i
$506$		0			0
$507$		0			0
$508$	2.00000			0.0887357
$509$	−16.5000	−	9.52628i	−0.731350	−	0.422245i	0.0875661	−	0.996159i	$-0.472091\pi$
$509$	−16.5000	−	9.52628i	−0.731350	−	0.422245i	−0.818916	+	0.573914i	$0.805424\pi$
$510$		0			0
$511$		0			0
$512$		−	8.66025i		−	0.382733i
$513$		0			0
$514$	4.50000	−	2.59808i	0.198486	−	0.114596i
$515$		−	17.3205i		−	0.763233i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−7.50000	−	7.79423i	−0.328897	−	0.341800i
$521$	−9.00000			−0.394297			−0.197149	−	0.980374i	$-0.563168\pi$
$521$	−9.00000			−0.394297			−0.197149	+	0.980374i	$0.563168\pi$
$522$		0			0
$523$	8.00000	−	13.8564i	0.349816	−	0.605898i	−0.636401	−	0.771358i	$-0.719578\pi$
$523$	8.00000	−	13.8564i	0.349816	−	0.605898i	0.986216	+	0.165460i	$0.0529109\pi$
$524$	−9.00000	−	15.5885i	−0.393167	−	0.680985i
$525$		0			0
$526$	18.0000	−	10.3923i	0.784837	−	0.453126i
$527$	9.00000	−	5.19615i	0.392046	−	0.226348i
$528$		0			0
$529$	−6.50000	−	11.2583i	−0.282609	−	0.489493i
$530$	−4.50000	+	7.79423i	−0.195468	+	0.338560i
$531$		0			0
$532$		0			0
$533$	−4.50000	−	18.1865i	−0.194917	−	0.787746i
$534$		0			0
$535$	9.00000	+	5.19615i	0.389104	+	0.224649i
$536$	3.00000	−	5.19615i	0.129580	−	0.224440i
$537$		0			0
$538$		−	10.3923i		−	0.448044i
$539$		0			0
$540$		0			0
$541$			29.4449i			1.26593i	0.774179	+	0.632967i	$0.218163\pi$
$541$			29.4449i			1.26593i	−0.774179	+	0.632967i	$0.781837\pi$
$542$	18.0000	+	31.1769i	0.773166	+	1.33916i
$543$		0			0
$544$	−13.5000	−	7.79423i	−0.578808	−	0.334175i
$545$	24.0000			1.02805
$546$		0			0
$547$	−22.0000			−0.940652			−0.470326	−	0.882493i	$-0.655864\pi$
$547$	−22.0000			−0.940652			−0.470326	+	0.882493i	$0.655864\pi$
$548$	13.5000	+	7.79423i	0.576691	+	0.332953i
$549$		0			0
$550$		0			0
$551$			10.3923i			0.442727i
$552$		0			0
$553$		0			0
$554$			12.1244i			0.515115i
$555$		0			0
$556$	−2.00000	+	3.46410i	−0.0848189	+	0.146911i
$557$	13.5000	+	7.79423i	0.572013	+	0.330252i	0.757953	−	0.652309i	$-0.226200\pi$
$557$	13.5000	+	7.79423i	0.572013	+	0.330252i	−0.185940	+	0.982561i	$0.559533\pi$
$558$		0			0
$559$	−8.00000	+	27.7128i	−0.338364	+	1.17213i
$560$		0			0
$561$		0			0
$562$	−19.5000	+	33.7750i	−0.822558	+	1.42471i
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$	22.5000	−	12.9904i	0.946582	−	0.546509i
$566$	−6.00000	+	3.46410i	−0.252199	+	0.145607i
$567$		0			0
$568$	3.00000	+	5.19615i	0.125877	+	0.218026i
$569$	−21.0000	+	36.3731i	−0.880366	+	1.52484i	−0.0294311	+	0.999567i	$0.509370\pi$
$569$	−21.0000	+	36.3731i	−0.880366	+	1.52484i	−0.850935	+	0.525271i	$0.823964\pi$
$570$		0			0
$571$	40.0000			1.67395			0.836974	−	0.547243i	$-0.184323\pi$
$571$	40.0000			1.67395			0.836974	+	0.547243i	$0.184323\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−6.00000	+	10.3923i	−0.250217	+	0.433389i
$576$		0			0
$577$		−	19.0526i		−	0.793168i	−0.917998	−	0.396584i	$-0.870195\pi$
$577$		−	19.0526i		−	0.793168i	0.917998	−	0.396584i	$-0.129805\pi$
$578$	12.0000	−	6.92820i	0.499134	−	0.288175i
$579$		0			0
$580$			5.19615i			0.215758i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	3.00000			0.124141
$585$		0			0
$586$	−9.00000			−0.371787
$587$	18.0000	+	10.3923i	0.742940	+	0.428936i	0.823137	−	0.567843i	$-0.192222\pi$
$587$	18.0000	+	10.3923i	0.742940	+	0.428936i	−0.0801976	+	0.996779i	$0.525555\pi$
$588$		0			0
$589$	−6.00000	−	10.3923i	−0.247226	−	0.428207i
$590$		−	20.7846i		−	0.855689i
$591$		0			0
$592$	37.5000	−	21.6506i	1.54124	−	0.889836i
$593$			25.9808i			1.06690i	0.845831	+	0.533451i	$0.179105\pi$
$593$			25.9808i			1.06690i	−0.845831	+	0.533451i	$0.820895\pi$
$594$		0			0
$595$		0			0
$596$	16.5000	+	9.52628i	0.675866	+	0.390212i
$597$		0			0
$598$	36.0000	+	10.3923i	1.47215	+	0.424973i
$599$	−30.0000			−1.22577			−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000			−1.22577			−0.612883	+	0.790173i	$0.709990\pi$
$600$		0			0
$601$	−12.5000	+	21.6506i	−0.509886	+	0.883148i	0.490049	+	0.871695i	$0.336979\pi$
$601$	−12.5000	+	21.6506i	−0.509886	+	0.883148i	−0.999934	+	0.0114528i	$0.996354\pi$
$602$		0			0
$603$		0			0
$604$	−15.0000	+	8.66025i	−0.610341	+	0.352381i
$605$	16.5000	−	9.52628i	0.670820	−	0.387298i
$606$		0			0
$607$	17.0000	+	29.4449i	0.690009	+	1.19513i	0.971834	+	0.235665i	$0.0757267\pi$
$607$	17.0000	+	29.4449i	0.690009	+	1.19513i	−0.281826	+	0.959466i	$0.590940\pi$
$608$	−9.00000	+	15.5885i	−0.364998	+	0.632195i
$609$		0			0
$610$	3.00000			0.121466
$611$	9.00000	−	8.66025i	0.364101	−	0.350356i
$612$		0			0
$613$	−10.5000	−	6.06218i	−0.424091	−	0.244849i	0.272735	−	0.962089i	$-0.412072\pi$
$613$	−10.5000	−	6.06218i	−0.424091	−	0.244849i	−0.696826	+	0.717240i	$0.745405\pi$
$614$	15.0000	−	25.9808i	0.605351	−	1.04850i
$615$		0			0
$616$		0			0
$617$	19.5000	−	11.2583i	0.785040	−	0.453243i	−0.0531732	−	0.998585i	$-0.516934\pi$
$617$	19.5000	−	11.2583i	0.785040	−	0.453243i	0.838214	+	0.545342i	$0.183600\pi$
$618$		0			0
$619$		−	20.7846i		−	0.835404i	−0.908584	−	0.417702i	$-0.862836\pi$
$619$		−	20.7846i		−	0.835404i	0.908584	−	0.417702i	$-0.137164\pi$
$620$	−3.00000	−	5.19615i	−0.120483	−	0.208683i
$621$		0			0
$622$	45.0000	+	25.9808i	1.80434	+	1.04173i
$623$		0			0
$624$		0			0
$625$	−11.0000			−0.440000
$626$	15.0000	+	8.66025i	0.599521	+	0.346133i
$627$		0			0
$628$	−6.50000	−	11.2583i	−0.259378	−	0.449256i
$629$		−	25.9808i		−	1.03592i
$630$		0			0
$631$	−42.0000	+	24.2487i	−1.67199	+	0.965326i	−0.705473	+	0.708737i	$0.749265\pi$
$631$	−42.0000	+	24.2487i	−1.67199	+	0.965326i	−0.966521	+	0.256589i	$0.917401\pi$
$632$		−	6.92820i		−	0.275589i
$633$		0			0
$634$	−4.50000	+	7.79423i	−0.178718	+	0.309548i
$635$	3.00000	+	1.73205i	0.119051	+	0.0687343i
$636$		0			0
$637$	24.5000	−	6.06218i	0.970725	−	0.240192i
$638$		0			0
$639$		0			0
$640$	10.5000	−	18.1865i	0.415049	−	0.718886i
$641$	16.5000	+	28.5788i	0.651711	+	1.12880i	0.982708	+	0.185164i	$0.0592817\pi$
$641$	16.5000	+	28.5788i	0.651711	+	1.12880i	−0.330997	+	0.943632i	$0.607385\pi$
$642$		0			0
$643$	12.0000	−	6.92820i	0.473234	−	0.273222i	−0.244359	−	0.969685i	$-0.578577\pi$
$643$	12.0000	−	6.92820i	0.473234	−	0.273222i	0.717592	+	0.696463i	$0.245244\pi$
$644$		0			0
$645$		0			0
$646$	9.00000	+	15.5885i	0.354100	+	0.613320i
$647$	9.00000	−	15.5885i	0.353827	−	0.612845i	−0.633090	−	0.774078i	$-0.718214\pi$
$647$	9.00000	−	15.5885i	0.353827	−	0.612845i	0.986916	+	0.161233i	$0.0515470\pi$
$648$		0			0
$649$		0			0
$650$	−3.00000	−	12.1244i	−0.117670	−	0.475556i
$651$		0			0
$652$	18.0000	+	10.3923i	0.704934	+	0.406994i
$653$	−15.0000	+	25.9808i	−0.586995	+	1.01671i	0.407628	+	0.913148i	$0.366356\pi$
$653$	−15.0000	+	25.9808i	−0.586995	+	1.01671i	−0.994623	+	0.103558i	$0.966977\pi$
$654$		0			0
$655$		−	31.1769i		−	1.21818i
$656$	22.5000	−	12.9904i	0.878477	−	0.507189i
$657$		0			0
$658$		0			0
$659$	−6.00000	−	10.3923i	−0.233727	−	0.404827i	0.725175	−	0.688565i	$-0.241759\pi$
$659$	−6.00000	−	10.3923i	−0.233727	−	0.404827i	−0.958902	+	0.283738i	$0.908425\pi$
$660$		0			0
$661$	−40.5000	−	23.3827i	−1.57527	−	0.909481i	−0.995506	−	0.0946945i	$-0.969813\pi$
$661$	−40.5000	−	23.3827i	−1.57527	−	0.909481i	−0.579761	−	0.814787i	$-0.696854\pi$
$662$	−48.0000			−1.86557
$663$		0			0
$664$	−24.0000			−0.931381
$665$		0			0
$666$		0			0
$667$	9.00000	+	15.5885i	0.348481	+	0.603587i
$668$			13.8564i			0.536120i
$669$		0			0
$670$	−9.00000	+	5.19615i	−0.347700	+	0.200745i
$671$		0			0
$672$		0			0
$673$	9.50000	−	16.4545i	0.366198	−	0.634274i	−0.622770	−	0.782405i	$-0.713993\pi$
$673$	9.50000	−	16.4545i	0.366198	−	0.634274i	0.988968	+	0.148132i	$0.0473259\pi$
$674$	−34.5000	−	19.9186i	−1.32889	−	0.767235i
$675$		0			0
$676$	−11.5000	+	6.06218i	−0.442308	+	0.233161i
$677$	6.00000			0.230599			0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000			0.230599			0.115299	+	0.993331i	$0.463217\pi$
$678$		0			0
$679$		0			0
$680$	−4.50000	−	7.79423i	−0.172567	−	0.298895i
$681$		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
$685$	13.5000	+	23.3827i	0.515808	+	0.893407i
$686$		0			0
$687$		0			0
$688$	−40.0000			−1.52499
$689$	−7.50000	−	7.79423i	−0.285727	−	0.296936i
$690$		0			0
$691$	12.0000	+	6.92820i	0.456502	+	0.263561i	0.710572	−	0.703624i	$-0.248436\pi$
$691$	12.0000	+	6.92820i	0.456502	+	0.263561i	−0.254071	+	0.967186i	$0.581770\pi$
$692$	−3.00000	+	5.19615i	−0.114043	+	0.197528i
$693$		0			0
$694$		−	51.9615i		−	1.97243i
$695$	−6.00000	+	3.46410i	−0.227593	+	0.131401i
$696$		0			0
$697$		−	15.5885i		−	0.590455i
$698$	12.0000	+	20.7846i	0.454207	+	0.786709i
$699$		0			0
$700$		0			0
$701$	−18.0000			−0.679851			−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000			−0.679851			−0.339925	+	0.940452i	$0.610402\pi$
$702$		0			0
$703$	−30.0000			−1.13147
$704$		0			0
$705$		0			0
$706$	−28.5000	−	49.3634i	−1.07261	−	1.85782i
$707$		0			0
$708$		0			0
$709$	−4.50000	+	2.59808i	−0.169001	+	0.0975728i	−0.582115	−	0.813107i	$-0.697775\pi$
$709$	−4.50000	+	2.59808i	−0.169001	+	0.0975728i	0.413114	+	0.910679i	$0.364441\pi$
$710$		−	10.3923i		−	0.390016i
$711$		0			0
$712$	6.00000	−	10.3923i	0.224860	−	0.389468i
$713$	−18.0000	−	10.3923i	−0.674105	−	0.389195i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$	−6.00000	+	10.3923i	−0.223918	+	0.387837i
$719$	−24.0000	−	41.5692i	−0.895049	−	1.55027i	−0.833744	−	0.552151i	$-0.813807\pi$
$719$	−24.0000	−	41.5692i	−0.895049	−	1.55027i	−0.0613050	−	0.998119i	$-0.519526\pi$
$720$		0			0
$721$		0			0
$722$	−10.5000	+	6.06218i	−0.390770	+	0.225611i
$723$		0			0
$724$	5.50000	+	9.52628i	0.204406	+	0.354041i
$725$	3.00000	−	5.19615i	0.111417	−	0.192980i
$726$		0			0
$727$	−32.0000			−1.18681			−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000			−1.18681			−0.593407	+	0.804902i	$0.702218\pi$
$728$		0			0
$729$		0			0
$730$	−4.50000	−	2.59808i	−0.166552	−	0.0961591i
$731$	−12.0000	+	20.7846i	−0.443836	+	0.768747i
$732$		0			0
$733$			12.1244i			0.447823i	0.974609	+	0.223912i	$0.0718827\pi$
$733$			12.1244i			0.447823i	−0.974609	+	0.223912i	$0.928117\pi$
$734$	33.0000	−	19.0526i	1.21805	−	0.703243i
$735$		0			0
$736$			31.1769i			1.14920i
$737$		0			0
$738$		0			0
$739$	−18.0000	−	10.3923i	−0.662141	−	0.382287i	0.130951	−	0.991389i	$-0.458197\pi$
$739$	−18.0000	−	10.3923i	−0.662141	−	0.382287i	−0.793092	+	0.609102i	$0.791530\pi$
$740$	−15.0000			−0.551411
$741$		0			0
$742$		0			0
$743$	30.0000	+	17.3205i	1.10059	+	0.635428i	0.936377	−	0.350997i	$-0.114157\pi$
$743$	30.0000	+	17.3205i	1.10059	+	0.635428i	0.164216	+	0.986424i	$0.447490\pi$
$744$		0			0
$745$	16.5000	+	28.5788i	0.604513	+	1.04705i
$746$		−	32.9090i		−	1.20488i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	8.00000	−	13.8564i	0.291924	−	0.505627i	−0.682341	−	0.731034i	$-0.739038\pi$
$751$	8.00000	−	13.8564i	0.291924	−	0.505627i	0.974265	+	0.225407i	$0.0723712\pi$
$752$	15.0000	+	8.66025i	0.546994	+	0.315807i
$753$		0			0
$754$	−18.0000	−	5.19615i	−0.655521	−	0.189233i
$755$	−30.0000			−1.09181
$756$		0			0
$757$	13.0000	−	22.5167i	0.472493	−	0.818382i	−0.527011	−	0.849858i	$-0.676688\pi$
$757$	13.0000	−	22.5167i	0.472493	−	0.818382i	0.999505	+	0.0314762i	$0.0100208\pi$
$758$	−21.0000	−	36.3731i	−0.762754	−	1.32113i
$759$		0			0
$760$	−9.00000	+	5.19615i	−0.326464	+	0.188484i
$761$	−30.0000	+	17.3205i	−1.08750	+	0.627868i	−0.932910	−	0.360111i	$-0.882739\pi$
$761$	−30.0000	+	17.3205i	−1.08750	+	0.627868i	−0.154590	+	0.987979i	$0.549406\pi$
$762$		0			0
$763$		0			0
$764$	−9.00000	+	15.5885i	−0.325609	+	0.563971i
$765$		0			0
$766$	36.0000			1.30073
$767$	24.0000	+	6.92820i	0.866590	+	0.250163i
$768$		0			0
$769$	6.00000	+	3.46410i	0.216366	+	0.124919i	0.604266	−	0.796782i	$-0.293466\pi$
$769$	6.00000	+	3.46410i	0.216366	+	0.124919i	−0.387901	+	0.921701i	$0.626800\pi$
$770$		0			0
$771$		0			0
$772$		−	5.19615i		−	0.187014i
$773$	30.0000	−	17.3205i	1.07903	−	0.622975i	0.148392	−	0.988929i	$-0.452590\pi$
$773$	30.0000	−	17.3205i	1.07903	−	0.622975i	0.930633	+	0.365953i	$0.119257\pi$
$774$		0			0
$775$			6.92820i			0.248868i
$776$	−6.00000	−	10.3923i	−0.215387	−	0.373062i
$777$		0			0
$778$	13.5000	+	7.79423i	0.483998	+	0.279437i
$779$	−18.0000			−0.644917
$780$		0			0
$781$		0			0
$782$	27.0000	+	15.5885i	0.965518	+	0.557442i
$783$		0			0
$784$	17.5000	+	30.3109i	0.625000	+	1.08253i
$785$		−	22.5167i		−	0.803654i
$786$		0			0
$787$	33.0000	−	19.0526i	1.17632	−	0.679150i	0.221162	−	0.975237i	$-0.429015\pi$
$787$	33.0000	−	19.0526i	1.17632	−	0.679150i	0.955161	+	0.296087i	$0.0956817\pi$
$788$		&mi

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)

\(f(q)\)	\(=\)	\(q+(1.50000 + 0.866025i) q^{2} +(0.500000 + 0.866025i) q^{4} +1.73205i q^{5} -1.73205i q^{8} +O(q^{10})\)	\(q+(1.50000 + 0.866025i) q^{2} +(0.500000 + 0.866025i) q^{4} +1.73205i q^{5} -1.73205i q^{8} +(-1.50000 + 2.59808i) q^{10} +(-2.50000 - 2.59808i) q^{13} +(2.50000 - 4.33013i) q^{16} +(-1.50000 - 2.59808i) q^{17} +(-3.00000 + 1.73205i) q^{19} +(-1.50000 + 0.866025i) q^{20} +(-3.00000 + 5.19615i) q^{23} +2.00000 q^{25} +(-1.50000 - 6.06218i) q^{26} +(1.50000 - 2.59808i) q^{29} +3.46410i q^{31} +(4.50000 - 2.59808i) q^{32} -5.19615i q^{34} +(7.50000 + 4.33013i) q^{37} -6.00000 q^{38} +3.00000 q^{40} +(4.50000 + 2.59808i) q^{41} +(-4.00000 - 6.92820i) q^{43} +(-9.00000 + 5.19615i) q^{46} +3.46410i q^{47} +(-3.50000 + 6.06218i) q^{49} +(3.00000 + 1.73205i) q^{50} +(1.00000 - 3.46410i) q^{52} +3.00000 q^{53} +(4.50000 - 2.59808i) q^{58} +(-6.00000 + 3.46410i) q^{59} +(-0.500000 - 0.866025i) q^{61} +(-3.00000 + 5.19615i) q^{62} -1.00000 q^{64} +(4.50000 - 4.33013i) q^{65} +(3.00000 + 1.73205i) q^{67} +(1.50000 - 2.59808i) q^{68} +(-3.00000 + 1.73205i) q^{71} +1.73205i q^{73} +(7.50000 + 12.9904i) q^{74} +(-3.00000 - 1.73205i) q^{76} +4.00000 q^{79} +(7.50000 + 4.33013i) q^{80} +(4.50000 + 7.79423i) q^{82} -13.8564i q^{83} +(4.50000 - 2.59808i) q^{85} -13.8564i q^{86} +(6.00000 + 3.46410i) q^{89} -6.00000 q^{92} +(-3.00000 + 5.19615i) q^{94} +(-3.00000 - 5.19615i) q^{95} +(6.00000 - 3.46410i) q^{97} +(-10.5000 + 6.06218i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{2} + q^{4}+O(q^{10}) \) 2 * q + 3 * q^2 + q^4	\( 2 q + 3 q^{2} + q^{4} - 3 q^{10} - 5 q^{13} + 5 q^{16} - 3 q^{17} - 6 q^{19} - 3 q^{20} - 6 q^{23} + 4 q^{25} - 3 q^{26} + 3 q^{29} + 9 q^{32} + 15 q^{37} - 12 q^{38} + 6 q^{40} + 9 q^{41} - 8 q^{43} - 18 q^{46} - 7 q^{49} + 6 q^{50} + 2 q^{52} + 6 q^{53} + 9 q^{58} - 12 q^{59} - q^{61} - 6 q^{62} - 2 q^{64} + 9 q^{65} + 6 q^{67} + 3 q^{68} - 6 q^{71} + 15 q^{74} - 6 q^{76} + 8 q^{79} + 15 q^{80} + 9 q^{82} + 9 q^{85} + 12 q^{89} - 12 q^{92} - 6 q^{94} - 6 q^{95} + 12 q^{97} - 21 q^{98}+O(q^{100}) \) 2 * q + 3 * q^2 + q^4 - 3 * q^10 - 5 * q^13 + 5 * q^16 - 3 * q^17 - 6 * q^19 - 3 * q^20 - 6 * q^23 + 4 * q^25 - 3 * q^26 + 3 * q^29 + 9 * q^32 + 15 * q^37 - 12 * q^38 + 6 * q^40 + 9 * q^41 - 8 * q^43 - 18 * q^46 - 7 * q^49 + 6 * q^50 + 2 * q^52 + 6 * q^53 + 9 * q^58 - 12 * q^59 - q^61 - 6 * q^62 - 2 * q^64 + 9 * q^65 + 6 * q^67 + 3 * q^68 - 6 * q^71 + 15 * q^74 - 6 * q^76 + 8 * q^79 + 15 * q^80 + 9 * q^82 + 9 * q^85 + 12 * q^89 - 12 * q^92 - 6 * q^94 - 6 * q^95 + 12 * q^97 - 21 * q^98

Introduction

L-functions

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