LMFDB - Embedded newform 364.2.k.a.113.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [364,2,Mod(29,364)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(364, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("364.29");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 364 = 2^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	364.k (of order $3$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$2.90655463357$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			113.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	364.113
Dual form			364.2.k.a.29.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +(0.500000 + 0.866025i) q^{7} +(1.50000 + 2.59808i) q^{9} +(1.00000 - 1.73205i) q^{11} +(3.50000 + 0.866025i) q^{13} +(1.50000 + 2.59808i) q^{17} +(3.00000 + 5.19615i) q^{19} +(2.00000 - 3.46410i) q^{23} -4.00000 q^{25} +(3.50000 - 6.06218i) q^{29} +4.00000 q^{31} +(-0.500000 - 0.866025i) q^{35} +(-4.50000 + 7.79423i) q^{37} +(-4.50000 + 7.79423i) q^{41} +(-5.00000 - 8.66025i) q^{43} +(-1.50000 - 2.59808i) q^{45} -2.00000 q^{47} +(-0.500000 + 0.866025i) q^{49} +9.00000 q^{53} +(-1.00000 + 1.73205i) q^{55} +(-7.00000 - 12.1244i) q^{59} +(2.50000 + 4.33013i) q^{61} +(-1.50000 + 2.59808i) q^{63} +(-3.50000 - 0.866025i) q^{65} +(4.00000 - 6.92820i) q^{67} +(-5.00000 - 8.66025i) q^{71} -7.00000 q^{73} +2.00000 q^{77} +2.00000 q^{79} +(-4.50000 + 7.79423i) q^{81} -6.00000 q^{83} +(-1.50000 - 2.59808i) q^{85} +(-3.00000 + 5.19615i) q^{89} +(1.00000 + 3.46410i) q^{91} +(-3.00000 - 5.19615i) q^{95} +(-1.00000 - 1.73205i) q^{97} +6.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + q^{7} + 3 q^{9} + 2 q^{11} + 7 q^{13} + 3 q^{17} + 6 q^{19} + 4 q^{23} - 8 q^{25} + 7 q^{29} + 8 q^{31} - q^{35} - 9 q^{37} - 9 q^{41} - 10 q^{43} - 3 q^{45} - 4 q^{47} - q^{49} + 18 q^{53}+ \cdots + 12 q^{99}+O(q^{100}) $ 2 * q - 2 * q^5 + q^7 + 3 * q^9 + 2 * q^11 + 7 * q^13 + 3 * q^17 + 6 * q^19 + 4 * q^23 - 8 * q^25 + 7 * q^29 + 8 * q^31 - q^35 - 9 * q^37 - 9 * q^41 - 10 * q^43 - 3 * q^45 - 4 * q^47 - q^49 + 18 * q^53 - 2 * q^55 - 14 * q^59 + 5 * q^61 - 3 * q^63 - 7 * q^65 + 8 * q^67 - 10 * q^71 - 14 * q^73 + 4 * q^77 + 4 * q^79 - 9 * q^81 - 12 * q^83 - 3 * q^85 - 6 * q^89 + 2 * q^91 - 6 * q^95 - 2 * q^97 + 12 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/364\mathbb{Z}\right)^\times$.

$n$	$157$	$183$	$197$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$3$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$4$		0			0
$5$	−1.00000			−0.447214			−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000			−0.447214			−0.223607	+	0.974679i	$0.571783\pi$
$6$		0			0
$7$	0.500000	+	0.866025i	0.188982	+	0.327327i
$7$	0.500000	+	0.866025i	0.188982	+	0.327327i
$8$		0			0
$9$	1.50000	+	2.59808i	0.500000	+	0.866025i
$10$		0			0
$11$	1.00000	−	1.73205i	0.301511	−	0.522233i	−0.674967	−	0.737848i	$-0.735842\pi$
$11$	1.00000	−	1.73205i	0.301511	−	0.522233i	0.976478	+	0.215615i	$0.0691756\pi$
$12$		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
$16$		0			0
$17$	1.50000	+	2.59808i	0.363803	+	0.630126i	0.988583	−	0.150675i	$-0.0481447\pi$
$17$	1.50000	+	2.59808i	0.363803	+	0.630126i	−0.624780	+	0.780801i	$0.714811\pi$
$18$		0			0
$19$	3.00000	+	5.19615i	0.688247	+	1.19208i	0.972404	+	0.233301i	$0.0749529\pi$
$19$	3.00000	+	5.19615i	0.688247	+	1.19208i	−0.284157	+	0.958778i	$0.591714\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.00000	−	3.46410i	0.417029	−	0.722315i	−0.578610	−	0.815604i	$-0.696405\pi$
$23$	2.00000	−	3.46410i	0.417029	−	0.722315i	0.995639	+	0.0932891i	$0.0297381\pi$
$24$		0			0
$25$	−4.00000			−0.800000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.50000	−	6.06218i	0.649934	−	1.12572i	−0.333205	−	0.942855i	$-0.608130\pi$
$29$	3.50000	−	6.06218i	0.649934	−	1.12572i	0.983138	−	0.182864i	$-0.0585367\pi$
$30$		0			0
$31$	4.00000			0.718421			0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000			0.718421			0.359211	+	0.933257i	$0.383046\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.500000	−	0.866025i	−0.0845154	−	0.146385i
$36$		0			0
$37$	−4.50000	+	7.79423i	−0.739795	+	1.28136i	0.212792	+	0.977098i	$0.431744\pi$
$37$	−4.50000	+	7.79423i	−0.739795	+	1.28136i	−0.952587	+	0.304266i	$0.901589\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−4.50000	+	7.79423i	−0.702782	+	1.21725i	0.264704	+	0.964330i	$0.414726\pi$
$41$	−4.50000	+	7.79423i	−0.702782	+	1.21725i	−0.967486	+	0.252924i	$0.918608\pi$
$42$		0			0
$43$	−5.00000	−	8.66025i	−0.762493	−	1.32068i	−0.941562	−	0.336840i	$-0.890642\pi$
$43$	−5.00000	−	8.66025i	−0.762493	−	1.32068i	0.179069	−	0.983836i	$-0.442691\pi$
$44$		0			0
$45$	−1.50000	−	2.59808i	−0.223607	−	0.387298i
$46$		0			0
$47$	−2.00000			−0.291730			−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000			−0.291730			−0.145865	+	0.989305i	$0.546597\pi$
$48$		0			0
$49$	−0.500000	+	0.866025i	−0.0714286	+	0.123718i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	9.00000			1.23625			0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000			1.23625			0.618123	+	0.786082i	$0.287894\pi$
$54$		0			0
$55$	−1.00000	+	1.73205i	−0.134840	+	0.233550i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−7.00000	−	12.1244i	−0.911322	−	1.57846i	−0.812198	−	0.583382i	$-0.801729\pi$
$59$	−7.00000	−	12.1244i	−0.911322	−	1.57846i	−0.0991242	−	0.995075i	$-0.531604\pi$
$60$		0			0
$61$	2.50000	+	4.33013i	0.320092	+	0.554416i	0.980507	−	0.196485i	$-0.0629528\pi$
$61$	2.50000	+	4.33013i	0.320092	+	0.554416i	−0.660415	+	0.750901i	$0.729619\pi$
$62$		0			0
$63$	−1.50000	+	2.59808i	−0.188982	+	0.327327i
$64$		0			0
$65$	−3.50000	−	0.866025i	−0.434122	−	0.107417i
$66$		0			0
$67$	4.00000	−	6.92820i	0.488678	−	0.846415i	−0.511237	−	0.859440i	$-0.670813\pi$
$67$	4.00000	−	6.92820i	0.488678	−	0.846415i	0.999915	+	0.0130248i	$0.00414604\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−5.00000	−	8.66025i	−0.593391	−	1.02778i	−0.993772	−	0.111434i	$-0.964456\pi$
$71$	−5.00000	−	8.66025i	−0.593391	−	1.02778i	0.400381	−	0.916349i	$-0.368878\pi$
$72$		0			0
$73$	−7.00000			−0.819288			−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000			−0.819288			−0.409644	+	0.912245i	$0.634347\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.00000			0.227921
$78$		0			0
$79$	2.00000			0.225018			0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000			0.225018			0.112509	+	0.993651i	$0.464111\pi$
$80$		0			0
$81$	−4.50000	+	7.79423i	−0.500000	+	0.866025i
$82$		0			0
$83$	−6.00000			−0.658586			−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000			−0.658586			−0.329293	+	0.944228i	$0.606810\pi$
$84$		0			0
$85$	−1.50000	−	2.59808i	−0.162698	−	0.281801i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.00000	+	5.19615i	−0.317999	+	0.550791i	−0.980071	−	0.198650i	$-0.936344\pi$
$89$	−3.00000	+	5.19615i	−0.317999	+	0.550791i	0.662071	+	0.749441i	$0.269678\pi$
$90$		0			0
$91$	1.00000	+	3.46410i	0.104828	+	0.363137i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−3.00000	−	5.19615i	−0.307794	−	0.533114i
$96$		0			0
$97$	−1.00000	−	1.73205i	−0.101535	−	0.175863i	0.810782	−	0.585348i	$-0.199042\pi$
$97$	−1.00000	−	1.73205i	−0.101535	−	0.175863i	−0.912317	+	0.409484i	$0.865709\pi$
$98$		0			0
$99$	6.00000			0.603023
$100$		0			0
$101$	8.50000	−	14.7224i	0.845782	−	1.46494i	−0.0391591	−	0.999233i	$-0.512468\pi$
$101$	8.50000	−	14.7224i	0.845782	−	1.46494i	0.884941	−	0.465704i	$-0.154199\pi$
$102$		0			0
$103$	−14.0000			−1.37946			−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000			−1.37946			−0.689730	+	0.724066i	$0.742271\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.00000	+	1.73205i	−0.0966736	+	0.167444i	−0.910306	−	0.413936i	$-0.864154\pi$
$107$	−1.00000	+	1.73205i	−0.0966736	+	0.167444i	0.813632	+	0.581380i	$0.197487\pi$
$108$		0			0
$109$	−14.0000			−1.34096			−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000			−1.34096			−0.670478	+	0.741929i	$0.733911\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−5.50000	−	9.52628i	−0.517396	−	0.896157i	−0.999796	−	0.0202056i	$-0.993568\pi$
$113$	−5.50000	−	9.52628i	−0.517396	−	0.896157i	0.482399	−	0.875951i	$-0.339765\pi$
$114$		0			0
$115$	−2.00000	+	3.46410i	−0.186501	+	0.323029i
$116$		0			0
$117$	3.00000	+	10.3923i	0.277350	+	0.960769i
$118$		0			0
$119$	−1.50000	+	2.59808i	−0.137505	+	0.238165i
$120$		0			0
$121$	3.50000	+	6.06218i	0.318182	+	0.551107i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	9.00000			0.804984
$126$		0			0
$127$	7.00000	−	12.1244i	0.621150	−	1.07586i	−0.368122	−	0.929777i	$-0.619999\pi$
$127$	7.00000	−	12.1244i	0.621150	−	1.07586i	0.989272	−	0.146085i	$-0.0466674\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	14.0000			1.22319			0.611593	−	0.791173i	$-0.290529\pi$
$131$	14.0000			1.22319			0.611593	+	0.791173i	$0.290529\pi$
$132$		0			0
$133$	−3.00000	+	5.19615i	−0.260133	+	0.450564i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−7.50000	−	12.9904i	−0.640768	−	1.10984i	−0.985262	−	0.171054i	$-0.945283\pi$
$137$	−7.50000	−	12.9904i	−0.640768	−	1.10984i	0.344493	−	0.938789i	$-0.388051\pi$
$138$		0			0
$139$	7.00000	+	12.1244i	0.593732	+	1.02837i	0.993724	+	0.111856i	$0.0356795\pi$
$139$	7.00000	+	12.1244i	0.593732	+	1.02837i	−0.399992	+	0.916519i	$0.630987\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	5.00000	−	5.19615i	0.418121	−	0.434524i
$144$		0			0
$145$	−3.50000	+	6.06218i	−0.290659	+	0.503436i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1.50000	+	2.59808i	0.122885	+	0.212843i	0.920904	−	0.389789i	$-0.127452\pi$
$149$	1.50000	+	2.59808i	0.122885	+	0.212843i	−0.798019	+	0.602632i	$0.794119\pi$
$150$		0			0
$151$	18.0000			1.46482			0.732410	−	0.680864i	$-0.238396\pi$
$151$	18.0000			1.46482			0.732410	+	0.680864i	$0.238396\pi$
$152$		0			0
$153$	−4.50000	+	7.79423i	−0.363803	+	0.630126i
$154$		0			0
$155$	−4.00000			−0.321288
$156$		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
$161$	4.00000			0.315244
$162$		0			0
$163$	−1.00000	−	1.73205i	−0.0783260	−	0.135665i	0.824202	−	0.566296i	$-0.191624\pi$
$163$	−1.00000	−	1.73205i	−0.0783260	−	0.135665i	−0.902528	+	0.430632i	$0.858291\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−9.00000	+	15.5885i	−0.696441	+	1.20627i	0.273252	+	0.961943i	$0.411901\pi$
$167$	−9.00000	+	15.5885i	−0.696441	+	1.20627i	−0.969693	+	0.244328i	$0.921432\pi$
$168$		0			0
$169$	11.5000	+	6.06218i	0.884615	+	0.466321i
$170$		0			0
$171$	−9.00000	+	15.5885i	−0.688247	+	1.19208i
$172$		0			0
$173$	−1.00000	−	1.73205i	−0.0760286	−	0.131685i	0.825505	−	0.564396i	$-0.190891\pi$
$173$	−1.00000	−	1.73205i	−0.0760286	−	0.131685i	−0.901533	+	0.432710i	$0.857557\pi$
$174$		0			0
$175$	−2.00000	−	3.46410i	−0.151186	−	0.261861i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$179$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$180$		0			0
$181$	15.0000			1.11494			0.557471	−	0.830197i	$-0.311772\pi$
$181$	15.0000			1.11494			0.557471	+	0.830197i	$0.311772\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	4.50000	−	7.79423i	0.330847	−	0.573043i
$186$		0			0
$187$	6.00000			0.438763
$188$		0			0
$189$		0			0
$190$		0			0
$191$	4.00000	+	6.92820i	0.289430	+	0.501307i	0.973674	−	0.227946i	$-0.0732010\pi$
$191$	4.00000	+	6.92820i	0.289430	+	0.501307i	−0.684244	+	0.729253i	$0.739868\pi$
$192$		0			0
$193$	−5.50000	+	9.52628i	−0.395899	+	0.685717i	−0.993215	−	0.116289i	$-0.962900\pi$
$193$	−5.50000	+	9.52628i	−0.395899	+	0.685717i	0.597317	+	0.802005i	$0.296234\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	9.00000	−	15.5885i	0.641223	−	1.11063i	−0.343937	−	0.938993i	$-0.611761\pi$
$197$	9.00000	−	15.5885i	0.641223	−	1.11063i	0.985160	−	0.171639i	$-0.0549062\pi$
$198$		0			0
$199$	−7.00000	−	12.1244i	−0.496217	−	0.859473i	0.503774	−	0.863836i	$-0.331945\pi$
$199$	−7.00000	−	12.1244i	−0.496217	−	0.859473i	−0.999990	+	0.00436292i	$0.998611\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	7.00000			0.491304
$204$		0			0
$205$	4.50000	−	7.79423i	0.314294	−	0.544373i
$206$		0			0
$207$	12.0000			0.834058
$208$		0			0
$209$	12.0000			0.830057
$210$		0			0
$211$	−4.00000	+	6.92820i	−0.275371	+	0.476957i	−0.970229	−	0.242190i	$-0.922134\pi$
$211$	−4.00000	+	6.92820i	−0.275371	+	0.476957i	0.694857	+	0.719148i	$0.255467\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	5.00000	+	8.66025i	0.340997	+	0.590624i
$216$		0			0
$217$	2.00000	+	3.46410i	0.135769	+	0.235159i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	3.00000	+	10.3923i	0.201802	+	0.699062i
$222$		0			0
$223$	−3.00000	+	5.19615i	−0.200895	+	0.347960i	−0.948817	−	0.315826i	$-0.897718\pi$
$223$	−3.00000	+	5.19615i	−0.200895	+	0.347960i	0.747922	+	0.663786i	$0.231052\pi$
$224$		0			0
$225$	−6.00000	−	10.3923i	−0.400000	−	0.692820i
$226$		0			0
$227$	6.00000	+	10.3923i	0.398234	+	0.689761i	0.993508	−	0.113761i	$-0.0362899\pi$
$227$	6.00000	+	10.3923i	0.398234	+	0.689761i	−0.595274	+	0.803523i	$0.702957\pi$
$228$		0			0
$229$	−6.00000			−0.396491			−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000			−0.396491			−0.198246	+	0.980152i	$0.563524\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−26.0000			−1.70332			−0.851658	−	0.524097i	$-0.824403\pi$
$233$	−26.0000			−1.70332			−0.851658	+	0.524097i	$0.824403\pi$
$234$		0			0
$235$	2.00000			0.130466
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−6.00000			−0.388108			−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000			−0.388108			−0.194054	+	0.980991i	$0.562164\pi$
$240$		0			0
$241$	1.50000	+	2.59808i	0.0966235	+	0.167357i	0.910285	−	0.413982i	$-0.135862\pi$
$241$	1.50000	+	2.59808i	0.0966235	+	0.167357i	−0.813662	+	0.581339i	$0.802529\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	0.500000	−	0.866025i	0.0319438	−	0.0553283i
$246$		0			0
$247$	6.00000	+	20.7846i	0.381771	+	1.32249i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−15.0000	−	25.9808i	−0.946792	−	1.63989i	−0.752124	−	0.659022i	$-0.770970\pi$
$251$	−15.0000	−	25.9808i	−0.946792	−	1.63989i	−0.194668	−	0.980869i	$-0.562363\pi$
$252$		0			0
$253$	−4.00000	−	6.92820i	−0.251478	−	0.435572i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	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
$259$	−9.00000			−0.559233
$260$		0			0
$261$	21.0000			1.29987
$262$		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
$265$	−9.00000			−0.552866
$266$		0			0
$267$		0			0
$268$		0			0
$269$	3.00000	+	5.19615i	0.182913	+	0.316815i	0.942871	−	0.333157i	$-0.108114\pi$
$269$	3.00000	+	5.19615i	0.182913	+	0.316815i	−0.759958	+	0.649972i	$0.774781\pi$
$270$		0			0
$271$	−5.00000	+	8.66025i	−0.303728	+	0.526073i	−0.976977	−	0.213343i	$-0.931565\pi$
$271$	−5.00000	+	8.66025i	−0.303728	+	0.526073i	0.673249	+	0.739416i	$0.264898\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−4.00000	+	6.92820i	−0.241209	+	0.417786i
$276$		0			0
$277$	1.50000	+	2.59808i	0.0901263	+	0.156103i	0.907564	−	0.419914i	$-0.137940\pi$
$277$	1.50000	+	2.59808i	0.0901263	+	0.156103i	−0.817438	+	0.576017i	$0.804606\pi$
$278$		0			0
$279$	6.00000	+	10.3923i	0.359211	+	0.622171i
$280$		0			0
$281$	7.00000			0.417585			0.208792	−	0.977960i	$-0.433047\pi$
$281$	7.00000			0.417585			0.208792	+	0.977960i	$0.433047\pi$
$282$		0			0
$283$	1.00000	−	1.73205i	0.0594438	−	0.102960i	−0.834772	−	0.550596i	$-0.814401\pi$
$283$	1.00000	−	1.73205i	0.0594438	−	0.102960i	0.894216	+	0.447636i	$0.147734\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−9.00000			−0.531253
$288$		0			0
$289$	4.00000	−	6.92820i	0.235294	−	0.407541i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−15.5000	−	26.8468i	−0.905520	−	1.56841i	−0.820218	−	0.572051i	$-0.806148\pi$
$293$	−15.5000	−	26.8468i	−0.905520	−	1.56841i	−0.0853015	−	0.996355i	$-0.527185\pi$
$294$		0			0
$295$	7.00000	+	12.1244i	0.407556	+	0.705907i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	10.0000	−	10.3923i	0.578315	−	0.601003i
$300$		0			0
$301$	5.00000	−	8.66025i	0.288195	−	0.499169i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2.50000	−	4.33013i	−0.143150	−	0.247942i
$306$		0			0
$307$	−4.00000			−0.228292			−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000			−0.228292			−0.114146	+	0.993464i	$0.536413\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−12.0000			−0.680458			−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000			−0.680458			−0.340229	+	0.940343i	$0.610505\pi$
$312$		0			0
$313$	2.00000			0.113047			0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000			0.113047			0.0565233	+	0.998401i	$0.481998\pi$
$314$		0			0
$315$	1.50000	−	2.59808i	0.0845154	−	0.146385i
$316$		0			0
$317$	−3.00000			−0.168497			−0.0842484	−	0.996445i	$-0.526849\pi$
$317$	−3.00000			−0.168497			−0.0842484	+	0.996445i	$0.526849\pi$
$318$		0			0
$319$	−7.00000	−	12.1244i	−0.391925	−	0.678834i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−9.00000	+	15.5885i	−0.500773	+	0.867365i
$324$		0			0
$325$	−14.0000	−	3.46410i	−0.776580	−	0.192154i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.00000	−	1.73205i	−0.0551318	−	0.0954911i
$330$		0			0
$331$	−7.00000	−	12.1244i	−0.384755	−	0.666415i	0.606980	−	0.794717i	$-0.292381\pi$
$331$	−7.00000	−	12.1244i	−0.384755	−	0.666415i	−0.991735	+	0.128302i	$0.959047\pi$
$332$		0			0
$333$	−27.0000			−1.47959
$334$		0			0
$335$	−4.00000	+	6.92820i	−0.218543	+	0.378528i
$336$		0			0
$337$	−25.0000			−1.36184			−0.680918	−	0.732359i	$-0.738419\pi$
$337$	−25.0000			−1.36184			−0.680918	+	0.732359i	$0.738419\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	4.00000	−	6.92820i	0.216612	−	0.375183i
$342$		0			0
$343$	−1.00000			−0.0539949
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$347$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$348$		0			0
$349$	7.00000	−	12.1244i	0.374701	−	0.649002i	−0.615581	−	0.788074i	$-0.711079\pi$
$349$	7.00000	−	12.1244i	0.374701	−	0.649002i	0.990282	+	0.139072i	$0.0444119\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−0.500000	+	0.866025i	−0.0266123	+	0.0460939i	−0.879025	−	0.476776i	$-0.841805\pi$
$353$	−0.500000	+	0.866025i	−0.0266123	+	0.0460939i	0.852413	+	0.522870i	$0.175139\pi$
$354$		0			0
$355$	5.00000	+	8.66025i	0.265372	+	0.459639i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−8.50000	+	14.7224i	−0.447368	+	0.774865i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	7.00000			0.366397
$366$		0			0
$367$	−10.0000	+	17.3205i	−0.521996	+	0.904123i	0.477677	+	0.878536i	$0.341479\pi$
$367$	−10.0000	+	17.3205i	−0.521996	+	0.904123i	−0.999673	+	0.0255875i	$0.991854\pi$
$368$		0			0
$369$	−27.0000			−1.40556
$370$		0			0
$371$	4.50000	+	7.79423i	0.233628	+	0.404656i
$372$		0			0
$373$	−14.5000	−	25.1147i	−0.750782	−	1.30039i	−0.947444	−	0.319921i	$-0.896344\pi$
$373$	−14.5000	−	25.1147i	−0.750782	−	1.30039i	0.196663	−	0.980471i	$-0.436990\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	17.5000	−	18.1865i	0.901296	−	0.936654i
$378$		0			0
$379$	−8.00000	+	13.8564i	−0.410932	+	0.711756i	−0.994992	−	0.0999550i	$-0.968130\pi$
$379$	−8.00000	+	13.8564i	−0.410932	+	0.711756i	0.584060	+	0.811711i	$0.301463\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−15.0000	−	25.9808i	−0.766464	−	1.32755i	−0.939469	−	0.342634i	$-0.888681\pi$
$383$	−15.0000	−	25.9808i	−0.766464	−	1.32755i	0.173005	−	0.984921i	$-0.444652\pi$
$384$		0			0
$385$	−2.00000			−0.101929
$386$		0			0
$387$	15.0000	−	25.9808i	0.762493	−	1.32068i
$388$		0			0
$389$	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
$391$	12.0000			0.606866
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2.00000			−0.100631
$396$		0			0
$397$	−7.00000	−	12.1244i	−0.351320	−	0.608504i	0.635161	−	0.772380i	$-0.280934\pi$
$397$	−7.00000	−	12.1244i	−0.351320	−	0.608504i	−0.986481	+	0.163876i	$0.947600\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−7.50000	+	12.9904i	−0.374532	+	0.648709i	−0.990257	−	0.139253i	$-0.955530\pi$
$401$	−7.50000	+	12.9904i	−0.374532	+	0.648709i	0.615725	+	0.787961i	$0.288863\pi$
$402$		0			0
$403$	14.0000	+	3.46410i	0.697390	+	0.172559i
$404$		0			0
$405$	4.50000	−	7.79423i	0.223607	−	0.387298i
$406$		0			0
$407$	9.00000	+	15.5885i	0.446113	+	0.772691i
$408$		0			0
$409$	−0.500000	−	0.866025i	−0.0247234	−	0.0428222i	0.853399	−	0.521258i	$-0.174537\pi$
$409$	−0.500000	−	0.866025i	−0.0247234	−	0.0428222i	−0.878122	+	0.478436i	$0.841204\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	7.00000	−	12.1244i	0.344447	−	0.596601i
$414$		0			0
$415$	6.00000			0.294528
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−18.0000	+	31.1769i	−0.879358	+	1.52309i	−0.0273103	+	0.999627i	$0.508694\pi$
$419$	−18.0000	+	31.1769i	−0.879358	+	1.52309i	−0.852047	+	0.523465i	$0.824639\pi$
$420$		0			0
$421$	25.0000			1.21843			0.609213	−	0.793007i	$-0.291486\pi$
$421$	25.0000			1.21843			0.609213	+	0.793007i	$0.291486\pi$
$422$		0			0
$423$	−3.00000	−	5.19615i	−0.145865	−	0.252646i
$424$		0			0
$425$	−6.00000	−	10.3923i	−0.291043	−	0.504101i
$426$		0			0
$427$	−2.50000	+	4.33013i	−0.120983	+	0.209550i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−12.0000	+	20.7846i	−0.578020	+	1.00116i	0.417687	+	0.908591i	$0.362841\pi$
$431$	−12.0000	+	20.7846i	−0.578020	+	1.00116i	−0.995706	+	0.0925683i	$0.970492\pi$
$432$		0			0
$433$	−0.500000	−	0.866025i	−0.0240285	−	0.0416185i	0.853761	−	0.520665i	$-0.174316\pi$
$433$	−0.500000	−	0.866025i	−0.0240285	−	0.0416185i	−0.877790	+	0.479046i	$0.840983\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	24.0000			1.14808
$438$		0			0
$439$	17.0000	−	29.4449i	0.811366	−	1.40533i	−0.100543	−	0.994933i	$-0.532058\pi$
$439$	17.0000	−	29.4449i	0.811366	−	1.40533i	0.911908	−	0.410394i	$-0.134609\pi$
$440$		0			0
$441$	−3.00000			−0.142857
$442$		0			0
$443$	−36.0000			−1.71041			−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000			−1.71041			−0.855206	+	0.518289i	$0.826569\pi$
$444$		0			0
$445$	3.00000	−	5.19615i	0.142214	−	0.246321i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	13.0000	+	22.5167i	0.613508	+	1.06263i	0.990644	+	0.136469i	$0.0435755\pi$
$449$	13.0000	+	22.5167i	0.613508	+	1.06263i	−0.377136	+	0.926158i	$0.623091\pi$
$450$		0			0
$451$	9.00000	+	15.5885i	0.423793	+	0.734032i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.00000	−	3.46410i	−0.0468807	−	0.162400i
$456$		0			0
$457$	8.50000	−	14.7224i	0.397613	−	0.688686i	−0.595818	−	0.803120i	$-0.703172\pi$
$457$	8.50000	−	14.7224i	0.397613	−	0.688686i	0.993431	+	0.114433i	$0.0365053\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	10.5000	+	18.1865i	0.489034	+	0.847031i	0.999920	−	0.0126168i	$-0.00401615\pi$
$461$	10.5000	+	18.1865i	0.489034	+	0.847031i	−0.510887	+	0.859648i	$0.670683\pi$
$462$		0			0
$463$	−14.0000			−0.650635			−0.325318	−	0.945605i	$-0.605471\pi$
$463$	−14.0000			−0.650635			−0.325318	+	0.945605i	$0.605471\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	14.0000			0.647843			0.323921	−	0.946084i	$-0.394999\pi$
$467$	14.0000			0.647843			0.323921	+	0.946084i	$0.394999\pi$
$468$		0			0
$469$	8.00000			0.369406
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−20.0000			−0.919601
$474$		0			0
$475$	−12.0000	−	20.7846i	−0.550598	−	0.953663i
$476$		0			0
$477$	13.5000	+	23.3827i	0.618123	+	1.07062i
$478$		0			0
$479$	−6.00000	+	10.3923i	−0.274147	+	0.474837i	−0.969920	−	0.243426i	$-0.921729\pi$
$479$	−6.00000	+	10.3923i	−0.274147	+	0.474837i	0.695773	+	0.718262i	$0.255062\pi$
$480$		0			0
$481$	−22.5000	+	23.3827i	−1.02591	+	1.06616i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.00000	+	1.73205i	0.0454077	+	0.0786484i
$486$		0			0
$487$	8.00000	+	13.8564i	0.362515	+	0.627894i	0.988374	−	0.152042i	$-0.0485850\pi$
$487$	8.00000	+	13.8564i	0.362515	+	0.627894i	−0.625859	+	0.779936i	$0.715252\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−8.00000	+	13.8564i	−0.361035	+	0.625331i	−0.988131	−	0.153611i	$-0.950910\pi$
$491$	−8.00000	+	13.8564i	−0.361035	+	0.625331i	0.627096	+	0.778942i	$0.284243\pi$
$492$		0			0
$493$	21.0000			0.945792
$494$		0			0
$495$	−6.00000			−0.269680
$496$		0			0
$497$	5.00000	−	8.66025i	0.224281	−	0.388465i
$498$		0			0
$499$	40.0000			1.79065			0.895323	−	0.445418i	$-0.146945\pi$
$499$	40.0000			1.79065			0.895323	+	0.445418i	$0.146945\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	1.00000	+	1.73205i	0.0445878	+	0.0772283i	0.887458	−	0.460889i	$-0.152469\pi$
$503$	1.00000	+	1.73205i	0.0445878	+	0.0772283i	−0.842870	+	0.538117i	$0.819136\pi$
$504$		0			0
$505$	−8.50000	+	14.7224i	−0.378245	+	0.655140i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	14.5000	−	25.1147i	0.642701	−	1.11319i	−0.342126	−	0.939654i	$-0.611147\pi$
$509$	14.5000	−	25.1147i	0.642701	−	1.11319i	0.984827	−	0.173537i	$-0.0555197\pi$
$510$		0			0
$511$	−3.50000	−	6.06218i	−0.154831	−	0.268175i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	14.0000			0.616914
$516$		0			0
$517$	−2.00000	+	3.46410i	−0.0879599	+	0.152351i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	21.0000			0.920027			0.460013	−	0.887912i	$-0.347845\pi$
$521$	21.0000			0.920027			0.460013	+	0.887912i	$0.347845\pi$
$522$		0			0
$523$	6.00000	−	10.3923i	0.262362	−	0.454424i	−0.704507	−	0.709697i	$-0.748832\pi$
$523$	6.00000	−	10.3923i	0.262362	−	0.454424i	0.966869	+	0.255273i	$0.0821653\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	6.00000	+	10.3923i	0.261364	+	0.452696i
$528$		0			0
$529$	3.50000	+	6.06218i	0.152174	+	0.263573i
$530$		0			0
$531$	21.0000	−	36.3731i	0.911322	−	1.57846i
$532$		0			0
$533$	−22.5000	+	23.3827i	−0.974583	+	1.01282i
$534$		0			0
$535$	1.00000	−	1.73205i	0.0432338	−	0.0748831i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1.00000	+	1.73205i	0.0430730	+	0.0746047i
$540$		0			0
$541$	−7.00000			−0.300954			−0.150477	−	0.988614i	$-0.548081\pi$
$541$	−7.00000			−0.300954			−0.150477	+	0.988614i	$0.548081\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	14.0000			0.599694
$546$		0			0
$547$	−8.00000			−0.342055			−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000			−0.342055			−0.171028	+	0.985266i	$0.554709\pi$
$548$		0			0
$549$	−7.50000	+	12.9904i	−0.320092	+	0.554416i
$550$		0			0
$551$	42.0000			1.78926
$552$		0			0
$553$	1.00000	+	1.73205i	0.0425243	+	0.0736543i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1.50000	−	2.59808i	0.0635570	−	0.110084i	−0.832496	−	0.554031i	$-0.813089\pi$
$557$	1.50000	−	2.59808i	0.0635570	−	0.110084i	0.896053	+	0.443947i	$0.146422\pi$
$558$		0			0
$559$	−10.0000	−	34.6410i	−0.422955	−	1.46516i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	5.00000	+	8.66025i	0.210725	+	0.364986i	0.951942	−	0.306280i	$-0.0990842\pi$
$563$	5.00000	+	8.66025i	0.210725	+	0.364986i	−0.741217	+	0.671266i	$0.765751\pi$
$564$		0			0
$565$	5.50000	+	9.52628i	0.231387	+	0.400774i
$566$		0			0
$567$	−9.00000			−0.377964
$568$		0			0
$569$	−11.0000	+	19.0526i	−0.461144	+	0.798725i	−0.999018	−	0.0443003i	$-0.985894\pi$
$569$	−11.0000	+	19.0526i	−0.461144	+	0.798725i	0.537874	+	0.843025i	$0.319228\pi$
$570$		0			0
$571$	12.0000			0.502184			0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000			0.502184			0.251092	+	0.967963i	$0.419210\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−8.00000	+	13.8564i	−0.333623	+	0.577852i
$576$		0			0
$577$	−27.0000			−1.12402			−0.562012	−	0.827129i	$-0.689973\pi$
$577$	−27.0000			−1.12402			−0.562012	+	0.827129i	$0.689973\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−3.00000	−	5.19615i	−0.124461	−	0.215573i
$582$		0			0
$583$	9.00000	−	15.5885i	0.372742	−	0.645608i
$584$		0			0
$585$	−3.00000	−	10.3923i	−0.124035	−	0.429669i
$586$		0			0
$587$	−22.0000	+	38.1051i	−0.908037	+	1.57277i	−0.0912496	+	0.995828i	$0.529086\pi$
$587$	−22.0000	+	38.1051i	−0.908037	+	1.57277i	−0.816788	+	0.576938i	$0.804247\pi$
$588$		0			0
$589$	12.0000	+	20.7846i	0.494451	+	0.856415i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	29.0000			1.19089			0.595444	−	0.803397i	$-0.296976\pi$
$593$	29.0000			1.19089			0.595444	+	0.803397i	$0.296976\pi$
$594$		0			0
$595$	1.50000	−	2.59808i	0.0614940	−	0.106511i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	34.0000			1.38920			0.694601	−	0.719395i	$-0.255581\pi$
$599$	34.0000			1.38920			0.694601	+	0.719395i	$0.255581\pi$
$600$		0			0
$601$	17.5000	−	30.3109i	0.713840	−	1.23641i	−0.249565	−	0.968358i	$-0.580288\pi$
$601$	17.5000	−	30.3109i	0.713840	−	1.23641i	0.963405	−	0.268049i	$-0.0863789\pi$
$602$		0			0
$603$	24.0000			0.977356
$604$		0			0
$605$	−3.50000	−	6.06218i	−0.142295	−	0.246463i
$606$		0			0
$607$	−9.00000	−	15.5885i	−0.365299	−	0.632716i	0.623525	−	0.781803i	$-0.285700\pi$
$607$	−9.00000	−	15.5885i	−0.365299	−	0.632716i	−0.988824	+	0.149087i	$0.952366\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−7.00000	−	1.73205i	−0.283190	−	0.0700713i
$612$		0			0
$613$	−8.50000	+	14.7224i	−0.343312	+	0.594633i	−0.985046	−	0.172294i	$-0.944882\pi$
$613$	−8.50000	+	14.7224i	−0.343312	+	0.594633i	0.641734	+	0.766927i	$0.278215\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	20.5000	+	35.5070i	0.825299	+	1.42946i	0.901691	+	0.432382i	$0.142327\pi$
$617$	20.5000	+	35.5070i	0.825299	+	1.42946i	−0.0763917	+	0.997078i	$0.524340\pi$
$618$		0			0
$619$	−2.00000			−0.0803868			−0.0401934	−	0.999192i	$-0.512797\pi$
$619$	−2.00000			−0.0803868			−0.0401934	+	0.999192i	$0.512797\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−6.00000			−0.240385
$624$		0			0
$625$	11.0000			0.440000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−27.0000			−1.07656
$630$		0			0
$631$	16.0000	+	27.7128i	0.636950	+	1.10323i	0.986098	+	0.166162i	$0.0531375\pi$
$631$	16.0000	+	27.7128i	0.636950	+	1.10323i	−0.349148	+	0.937067i	$0.613529\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−7.00000	+	12.1244i	−0.277787	+	0.481140i
$636$		0			0
$637$	−2.50000	+	2.59808i	−0.0990536	+	0.102940i
$638$		0			0
$639$	15.0000	−	25.9808i	0.593391	−	1.02778i
$640$		0			0
$641$	20.5000	+	35.5070i	0.809701	+	1.40244i	0.913071	+	0.407801i	$0.133704\pi$
$641$	20.5000	+	35.5070i	0.809701	+	1.40244i	−0.103370	+	0.994643i	$0.532962\pi$
$642$		0			0
$643$	−10.0000	−	17.3205i	−0.394362	−	0.683054i	0.598658	−	0.801005i	$-0.295701\pi$
$643$	−10.0000	−	17.3205i	−0.394362	−	0.683054i	−0.993019	+	0.117951i	$0.962368\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−23.0000	+	39.8372i	−0.904223	+	1.56616i	−0.0822669	+	0.996610i	$0.526216\pi$
$647$	−23.0000	+	39.8372i	−0.904223	+	1.56616i	−0.821956	+	0.569550i	$0.807117\pi$
$648$		0			0
$649$	−28.0000			−1.09910
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−3.00000	+	5.19615i	−0.117399	+	0.203341i	−0.918736	−	0.394872i	$-0.870789\pi$
$653$	−3.00000	+	5.19615i	−0.117399	+	0.203341i	0.801337	+	0.598213i	$0.204122\pi$
$654$		0			0
$655$	−14.0000			−0.547025
$656$		0			0
$657$	−10.5000	−	18.1865i	−0.409644	−	0.709524i
$658$		0			0
$659$	−18.0000	−	31.1769i	−0.701180	−	1.21448i	−0.968052	−	0.250748i	$-0.919323\pi$
$659$	−18.0000	−	31.1769i	−0.701180	−	1.21448i	0.266872	−	0.963732i	$-0.414010\pi$
$660$		0			0
$661$	6.50000	−	11.2583i	0.252821	−	0.437898i	−0.711481	−	0.702706i	$-0.751975\pi$
$661$	6.50000	−	11.2583i	0.252821	−	0.437898i	0.964301	+	0.264807i	$0.0853084\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	3.00000	−	5.19615i	0.116335	−	0.201498i
$666$		0			0
$667$	−14.0000	−	24.2487i	−0.542082	−	0.938914i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	10.0000			0.386046
$672$		0			0
$673$	20.5000	−	35.5070i	0.790217	−	1.36870i	−0.135615	−	0.990762i	$-0.543301\pi$
$673$	20.5000	−	35.5070i	0.790217	−	1.36870i	0.925832	−	0.377934i	$-0.123365\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−30.0000			−1.15299			−0.576497	−	0.817099i	$-0.695581\pi$
$677$	−30.0000			−1.15299			−0.576497	+	0.817099i	$0.695581\pi$
$678$		0			0
$679$	1.00000	−	1.73205i	0.0383765	−	0.0664700i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.00000	+	1.73205i	0.0382639	+	0.0662751i	0.884523	−	0.466496i	$-0.154484\pi$
$683$	1.00000	+	1.73205i	0.0382639	+	0.0662751i	−0.846259	+	0.532771i	$0.821151\pi$
$684$		0			0
$685$	7.50000	+	12.9904i	0.286560	+	0.496337i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	31.5000	+	7.79423i	1.20005	+	0.296936i
$690$		0			0
$691$	10.0000	−	17.3205i	0.380418	−	0.658903i	−0.610704	−	0.791859i	$-0.709113\pi$
$691$	10.0000	−	17.3205i	0.380418	−	0.658903i	0.991122	+	0.132956i	$0.0424468\pi$
$692$		0			0
$693$	3.00000	+	5.19615i	0.113961	+	0.197386i
$694$		0			0
$695$	−7.00000	−	12.1244i	−0.265525	−	0.459903i
$696$		0			0
$697$	−27.0000			−1.02270
$698$		0			0
$699$		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
$703$	−54.0000			−2.03665
$704$		0			0
$705$		0			0
$706$		0			0
$707$	17.0000			0.639351
$708$		0			0
$709$	21.5000	+	37.2391i	0.807449	+	1.39854i	0.914625	+	0.404303i	$0.132486\pi$
$709$	21.5000	+	37.2391i	0.807449	+	1.39854i	−0.107176	+	0.994240i	$0.534181\pi$
$710$		0			0
$711$	3.00000	+	5.19615i	0.112509	+	0.194871i
$712$		0			0
$713$	8.00000	−	13.8564i	0.299602	−	0.518927i
$714$		0			0
$715$	−5.00000	+	5.19615i	−0.186989	+	0.194325i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−7.00000	−	12.1244i	−0.261056	−	0.452162i	0.705467	−	0.708743i	$-0.250737\pi$
$719$	−7.00000	−	12.1244i	−0.261056	−	0.452162i	−0.966523	+	0.256581i	$0.917404\pi$
$720$		0			0
$721$	−7.00000	−	12.1244i	−0.260694	−	0.451535i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−14.0000	+	24.2487i	−0.519947	+	0.900575i
$726$		0			0
$727$	52.0000			1.92857			0.964287	−	0.264861i	$-0.0853260\pi$
$727$	52.0000			1.92857			0.964287	+	0.264861i	$0.0853260\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$	15.0000	−	25.9808i	0.554795	−	0.960933i
$732$		0			0
$733$	−1.00000			−0.0369358			−0.0184679	−	0.999829i	$-0.505879\pi$
$733$	−1.00000			−0.0369358			−0.0184679	+	0.999829i	$0.505879\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−8.00000	−	13.8564i	−0.294684	−	0.510407i
$738$		0			0
$739$	18.0000	−	31.1769i	0.662141	−	1.14686i	−0.317911	−	0.948120i	$-0.602981\pi$
$739$	18.0000	−	31.1769i	0.662141	−	1.14686i	0.980052	−	0.198741i	$-0.0636852\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−8.00000	+	13.8564i	−0.293492	+	0.508342i	−0.974633	−	0.223810i	$-0.928151\pi$
$743$	−8.00000	+	13.8564i	−0.293492	+	0.508342i	0.681141	+	0.732152i	$0.261484\pi$
$744$		0			0
$745$	−1.50000	−	2.59808i	−0.0549557	−	0.0951861i
$746$		0			0
$747$	−9.00000	−	15.5885i	−0.329293	−	0.570352i
$748$		0			0
$749$	−2.00000			−0.0730784
$750$		0			0
$751$	3.00000	−	5.19615i	0.109472	−	0.189610i	−0.806085	−	0.591800i	$-0.798417\pi$
$751$	3.00000	−	5.19615i	0.109472	−	0.189610i	0.915556	+	0.402190i	$0.131751\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−18.0000			−0.655087
$756$		0			0
$757$	−3.00000	+	5.19615i	−0.109037	+	0.188857i	−0.915380	−	0.402590i	$-0.868110\pi$
$757$	−3.00000	+	5.19615i	−0.109037	+	0.188857i	0.806343	+	0.591448i	$0.201443\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	3.00000	+	5.19615i	0.108750	+	0.188360i	0.915264	−	0.402854i	$-0.131982\pi$
$761$	3.00000	+	5.19615i	0.108750	+	0.188360i	−0.806514	+	0.591215i	$0.798649\pi$
$762$		0			0
$763$	−7.00000	−	12.1244i	−0.253417	−	0.438931i
$764$		0			0
$765$	4.50000	−	7.79423i	0.162698	−	0.281801i
$766$		0			0
$767$	−14.0000	−	48.4974i	−0.505511	−	1.75114i
$768$		0			0
$769$	−9.00000	+	15.5885i	−0.324548	+	0.562134i	−0.981421	−	0.191867i	$-0.938546\pi$
$769$	−9.00000	+	15.5885i	−0.324548	+	0.562134i	0.656873	+	0.754002i	$0.271879\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1.00000	−	1.73205i	−0.0359675	−	0.0622975i	0.847481	−	0.530825i	$-0.178118\pi$
$773$	−1.00000	−	1.73205i	−0.0359675	−	0.0622975i	−0.883449	+	0.468528i	$0.844785\pi$
$774$		0			0
$775$	−16.0000			−0.574737
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−54.0000			−1.93475
$780$		0			0
$781$	−20.0000			−0.715656
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−11.0000			−0.392607
$786$		0			0
$787$	26.0000	+	45.0333i	0.926800	+	1.60526i	0.788641	+	0.614855i	$0.210785\pi$
$787$	26.0000	+	45.0333i	0.926800	+	1.60526i	0.138159	+	0.990410i	$0.455881\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	5.50000	−	9.52628i	0.195557	−	0.338716i
$792$		0			0
$793$	5.00000	+	17.3205i	0.177555	+	0.615069i
$794$		0			0
$795$		0			0
$796$		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:	\( N \)	\(=\)	\( 364 = 2^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	364.k (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +(0.500000 + 0.866025i) q^{7} +(1.50000 + 2.59808i) q^{9} +(1.00000 - 1.73205i) q^{11} +(3.50000 + 0.866025i) q^{13} +(1.50000 + 2.59808i) q^{17} +(3.00000 + 5.19615i) q^{19} +(2.00000 - 3.46410i) q^{23} -4.00000 q^{25} +(3.50000 - 6.06218i) q^{29} +4.00000 q^{31} +(-0.500000 - 0.866025i) q^{35} +(-4.50000 + 7.79423i) q^{37} +(-4.50000 + 7.79423i) q^{41} +(-5.00000 - 8.66025i) q^{43} +(-1.50000 - 2.59808i) q^{45} -2.00000 q^{47} +(-0.500000 + 0.866025i) q^{49} +9.00000 q^{53} +(-1.00000 + 1.73205i) q^{55} +(-7.00000 - 12.1244i) q^{59} +(2.50000 + 4.33013i) q^{61} +(-1.50000 + 2.59808i) q^{63} +(-3.50000 - 0.866025i) q^{65} +(4.00000 - 6.92820i) q^{67} +(-5.00000 - 8.66025i) q^{71} -7.00000 q^{73} +2.00000 q^{77} +2.00000 q^{79} +(-4.50000 + 7.79423i) q^{81} -6.00000 q^{83} +(-1.50000 - 2.59808i) q^{85} +(-3.00000 + 5.19615i) q^{89} +(1.00000 + 3.46410i) q^{91} +(-3.00000 - 5.19615i) q^{95} +(-1.00000 - 1.73205i) q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + q^{7} + 3 q^{9} + 2 q^{11} + 7 q^{13} + 3 q^{17} + 6 q^{19} + 4 q^{23} - 8 q^{25} + 7 q^{29} + 8 q^{31} - q^{35} - 9 q^{37} - 9 q^{41} - 10 q^{43} - 3 q^{45} - 4 q^{47} - q^{49} + 18 q^{53}+ \cdots + 12 q^{99}+O(q^{100}) \) 2 * q - 2 * q^5 + q^7 + 3 * q^9 + 2 * q^11 + 7 * q^13 + 3 * q^17 + 6 * q^19 + 4 * q^23 - 8 * q^25 + 7 * q^29 + 8 * q^31 - q^35 - 9 * q^37 - 9 * q^41 - 10 * q^43 - 3 * q^45 - 4 * q^47 - q^49 + 18 * q^53 - 2 * q^55 - 14 * q^59 + 5 * q^61 - 3 * q^63 - 7 * q^65 + 8 * q^67 - 10 * q^71 - 14 * q^73 + 4 * q^77 + 4 * q^79 - 9 * q^81 - 12 * q^83 - 3 * q^85 - 6 * q^89 + 2 * q^91 - 6 * q^95 - 2 * q^97 + 12 * q^99

Introduction

L-functions

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Varieties

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Database

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