LMFDB - Embedded newform 2736.2.s.o.1873.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2736,2,Mod(577,2736)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2736.577"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2736, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.s (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,2,0,-6,0,0,0,4,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$21.8470699930$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 342)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1873.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2736.1873
Dual form			2736.2.s.o.577.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.73205i) q^{5} -3.00000 q^{7} +2.00000 q^{11} +(0.500000 + 0.866025i) q^{13} +(3.00000 - 5.19615i) q^{17} +(4.00000 - 1.73205i) q^{19} +(2.00000 + 3.46410i) q^{23} +(0.500000 + 0.866025i) q^{25} +(-1.00000 - 1.73205i) q^{29} -7.00000 q^{31} +(-3.00000 + 5.19615i) q^{35} +1.00000 q^{37} +(-4.00000 + 6.92820i) q^{41} +(3.50000 - 6.06218i) q^{43} +(-4.00000 - 6.92820i) q^{47} +2.00000 q^{49} +(4.00000 + 6.92820i) q^{53} +(2.00000 - 3.46410i) q^{55} +(6.00000 - 10.3923i) q^{59} +(-2.50000 - 4.33013i) q^{61} +2.00000 q^{65} +(-4.50000 - 7.79423i) q^{67} +(-1.00000 + 1.73205i) q^{71} +(7.50000 - 12.9904i) q^{73} -6.00000 q^{77} +(5.50000 - 9.52628i) q^{79} +6.00000 q^{83} +(-6.00000 - 10.3923i) q^{85} +(-1.50000 - 2.59808i) q^{91} +(1.00000 - 8.66025i) q^{95} +(7.00000 - 12.1244i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 6 q^{7} + 4 q^{11} + q^{13} + 6 q^{17} + 8 q^{19} + 4 q^{23} + q^{25} - 2 q^{29} - 14 q^{31} - 6 q^{35} + 2 q^{37} - 8 q^{41} + 7 q^{43} - 8 q^{47} + 4 q^{49} + 8 q^{53} + 4 q^{55} + 12 q^{59}+ \cdots + 14 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 - 6 * q^7 + 4 * q^11 + q^13 + 6 * q^17 + 8 * q^19 + 4 * q^23 + q^25 - 2 * q^29 - 14 * q^31 - 6 * q^35 + 2 * q^37 - 8 * q^41 + 7 * q^43 - 8 * q^47 + 4 * q^49 + 8 * q^53 + 4 * q^55 + 12 * q^59 - 5 * q^61 + 4 * q^65 - 9 * q^67 - 2 * q^71 + 15 * q^73 - 12 * q^77 + 11 * q^79 + 12 * q^83 - 12 * q^85 - 3 * q^91 + 2 * q^95 + 14 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$1$	$1$

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
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.00000	−	1.73205i	0.447214	−	0.774597i	−0.550990	−	0.834512i	$-0.685750\pi$
$5$	1.00000	−	1.73205i	0.447214	−	0.774597i	0.998203	+	0.0599153i	$0.0190830\pi$
$6$		0			0
$7$	−3.00000			−1.13389			−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000			−1.13389			−0.566947	+	0.823754i	$0.691875\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.00000			0.603023			0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000			0.603023			0.301511	+	0.953463i	$0.402509\pi$
$12$		0			0
$13$	0.500000	+	0.866025i	0.138675	+	0.240192i	0.926995	−	0.375073i	$-0.122382\pi$
$13$	0.500000	+	0.866025i	0.138675	+	0.240192i	−0.788320	+	0.615265i	$0.789049\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.00000	−	5.19615i	0.727607	−	1.26025i	−0.230285	−	0.973123i	$-0.573966\pi$
$17$	3.00000	−	5.19615i	0.727607	−	1.26025i	0.957892	−	0.287129i	$-0.0927008\pi$
$18$		0			0
$19$	4.00000	−	1.73205i	0.917663	−	0.397360i
$19$	4.00000	−	1.73205i	0.917663	−	0.397360i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.00000	+	3.46410i	0.417029	+	0.722315i	0.995639	−	0.0932891i	$-0.0297381\pi$
$23$	2.00000	+	3.46410i	0.417029	+	0.722315i	−0.578610	+	0.815604i	$0.696405\pi$
$24$		0			0
$25$	0.500000	+	0.866025i	0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.00000	−	1.73205i	−0.185695	−	0.321634i	0.758115	−	0.652121i	$-0.226120\pi$
$29$	−1.00000	−	1.73205i	−0.185695	−	0.321634i	−0.943811	+	0.330487i	$0.892787\pi$
$30$		0			0
$31$	−7.00000			−1.25724			−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000			−1.25724			−0.628619	+	0.777714i	$0.716379\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−3.00000	+	5.19615i	−0.507093	+	0.878310i
$36$		0			0
$37$	1.00000			0.164399			0.0821995	−	0.996616i	$-0.473806\pi$
$37$	1.00000			0.164399			0.0821995	+	0.996616i	$0.473806\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−4.00000	+	6.92820i	−0.624695	+	1.08200i	0.363905	+	0.931436i	$0.381443\pi$
$41$	−4.00000	+	6.92820i	−0.624695	+	1.08200i	−0.988600	+	0.150567i	$0.951890\pi$
$42$		0			0
$43$	3.50000	−	6.06218i	0.533745	−	0.924473i	−0.465478	−	0.885059i	$-0.654118\pi$
$43$	3.50000	−	6.06218i	0.533745	−	0.924473i	0.999223	−	0.0394140i	$-0.0125491\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.00000	−	6.92820i	−0.583460	−	1.01058i	−0.995066	−	0.0992202i	$-0.968365\pi$
$47$	−4.00000	−	6.92820i	−0.583460	−	1.01058i	0.411606	−	0.911362i	$-0.364968\pi$
$48$		0			0
$49$	2.00000			0.285714
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.00000	+	6.92820i	0.549442	+	0.951662i	0.998313	+	0.0580651i	$0.0184931\pi$
$53$	4.00000	+	6.92820i	0.549442	+	0.951662i	−0.448871	+	0.893597i	$0.648174\pi$
$54$		0			0
$55$	2.00000	−	3.46410i	0.269680	−	0.467099i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	6.00000	−	10.3923i	0.781133	−	1.35296i	−0.150148	−	0.988663i	$-0.547975\pi$
$59$	6.00000	−	10.3923i	0.781133	−	1.35296i	0.931282	−	0.364299i	$-0.118692\pi$
$60$		0			0
$61$	−2.50000	−	4.33013i	−0.320092	−	0.554416i	0.660415	−	0.750901i	$-0.270381\pi$
$61$	−2.50000	−	4.33013i	−0.320092	−	0.554416i	−0.980507	+	0.196485i	$0.937047\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.00000			0.248069
$66$		0			0
$67$	−4.50000	−	7.79423i	−0.549762	−	0.952217i	−0.998290	−	0.0584478i	$-0.981385\pi$
$67$	−4.50000	−	7.79423i	−0.549762	−	0.952217i	0.448528	−	0.893769i	$-0.351948\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−1.00000	+	1.73205i	−0.118678	+	0.205557i	−0.919244	−	0.393688i	$-0.871199\pi$
$71$	−1.00000	+	1.73205i	−0.118678	+	0.205557i	0.800566	+	0.599245i	$0.204532\pi$
$72$		0			0
$73$	7.50000	−	12.9904i	0.877809	−	1.52041i	0.0240681	−	0.999710i	$-0.492338\pi$
$73$	7.50000	−	12.9904i	0.877809	−	1.52041i	0.853740	−	0.520699i	$-0.174329\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−6.00000			−0.683763
$78$		0			0
$79$	5.50000	−	9.52628i	0.618798	−	1.07179i	−0.370907	−	0.928670i	$-0.620953\pi$
$79$	5.50000	−	9.52628i	0.618798	−	1.07179i	0.989705	−	0.143120i	$-0.0457135\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	6.00000			0.658586			0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000			0.658586			0.329293	+	0.944228i	$0.393190\pi$
$84$		0			0
$85$	−6.00000	−	10.3923i	−0.650791	−	1.12720i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$89$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$90$		0			0
$91$	−1.50000	−	2.59808i	−0.157243	−	0.272352i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.00000	−	8.66025i	0.102598	−	0.888523i
$96$		0			0
$97$	7.00000	−	12.1244i	0.710742	−	1.23104i	−0.253837	−	0.967247i	$-0.581693\pi$
$97$	7.00000	−	12.1244i	0.710742	−	1.23104i	0.964579	−	0.263795i	$-0.0849741\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	7.00000	+	12.1244i	0.696526	+	1.20642i	0.969664	+	0.244443i	$0.0786053\pi$
$101$	7.00000	+	12.1244i	0.696526	+	1.20642i	−0.273138	+	0.961975i	$0.588061\pi$
$102$		0			0
$103$	−3.00000			−0.295599			−0.147799	−	0.989017i	$-0.547219\pi$
$103$	−3.00000			−0.295599			−0.147799	+	0.989017i	$0.547219\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−10.0000			−0.966736			−0.483368	−	0.875417i	$-0.660587\pi$
$107$	−10.0000			−0.966736			−0.483368	+	0.875417i	$0.660587\pi$
$108$		0			0
$109$	3.00000	−	5.19615i	0.287348	−	0.497701i	−0.685828	−	0.727764i	$-0.740560\pi$
$109$	3.00000	−	5.19615i	0.287348	−	0.497701i	0.973176	+	0.230063i	$0.0738931\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−20.0000			−1.88144			−0.940721	−	0.339182i	$-0.889850\pi$
$113$	−20.0000			−1.88144			−0.940721	+	0.339182i	$0.889850\pi$
$114$		0			0
$115$	8.00000			0.746004
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−9.00000	+	15.5885i	−0.825029	+	1.42899i
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$		0			0
$124$		0			0
$125$	12.0000			1.07331
$126$		0			0
$127$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$127$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	3.00000	−	5.19615i	0.262111	−	0.453990i	−0.704692	−	0.709514i	$-0.748915\pi$
$131$	3.00000	−	5.19615i	0.262111	−	0.453990i	0.966803	+	0.255524i	$0.0822479\pi$
$132$		0			0
$133$	−12.0000	+	5.19615i	−1.04053	+	0.450564i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−7.00000	−	12.1244i	−0.598050	−	1.03585i	−0.993109	−	0.117198i	$-0.962609\pi$
$137$	−7.00000	−	12.1244i	−0.598050	−	1.03585i	0.395058	−	0.918656i	$-0.370724\pi$
$138$		0			0
$139$	−1.50000	−	2.59808i	−0.127228	−	0.220366i	0.795373	−	0.606120i	$-0.207275\pi$
$139$	−1.50000	−	2.59808i	−0.127228	−	0.220366i	−0.922602	+	0.385754i	$0.873941\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1.00000	+	1.73205i	0.0836242	+	0.144841i
$144$		0			0
$145$	−4.00000			−0.332182
$146$		0			0
$147$		0			0
$148$		0			0
$149$	9.00000	−	15.5885i	0.737309	−	1.27706i	−0.216394	−	0.976306i	$-0.569430\pi$
$149$	9.00000	−	15.5885i	0.737309	−	1.27706i	0.953703	−	0.300750i	$-0.0972370\pi$
$150$		0			0
$151$	4.00000			0.325515			0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000			0.325515			0.162758	+	0.986666i	$0.447961\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−7.00000	+	12.1244i	−0.562254	+	0.973852i
$156$		0			0
$157$	−6.50000	+	11.2583i	−0.518756	+	0.898513i	0.481006	+	0.876717i	$0.340272\pi$
$157$	−6.50000	+	11.2583i	−0.518756	+	0.898513i	−0.999762	+	0.0217953i	$0.993062\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−6.00000	−	10.3923i	−0.472866	−	0.819028i
$162$		0			0
$163$	13.0000			1.01824			0.509119	−	0.860696i	$-0.329971\pi$
$163$	13.0000			1.01824			0.509119	+	0.860696i	$0.329971\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.00000	+	10.3923i	0.464294	+	0.804181i	0.999169	−	0.0407502i	$-0.0129748\pi$
$167$	6.00000	+	10.3923i	0.464294	+	0.804181i	−0.534875	+	0.844931i	$0.679641\pi$
$168$		0			0
$169$	6.00000	−	10.3923i	0.461538	−	0.799408i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−9.00000	+	15.5885i	−0.684257	+	1.18517i	0.289412	+	0.957205i	$0.406540\pi$
$173$	−9.00000	+	15.5885i	−0.684257	+	1.18517i	−0.973670	+	0.227964i	$0.926793\pi$
$174$		0			0
$175$	−1.50000	−	2.59808i	−0.113389	−	0.196396i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−24.0000			−1.79384			−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000			−1.79384			−0.896922	+	0.442189i	$0.854202\pi$
$180$		0			0
$181$	−11.0000	−	19.0526i	−0.817624	−	1.41617i	−0.907429	−	0.420206i	$-0.861958\pi$
$181$	−11.0000	−	19.0526i	−0.817624	−	1.41617i	0.0898051	−	0.995959i	$-0.471376\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.00000	−	1.73205i	0.0735215	−	0.127343i
$186$		0			0
$187$	6.00000	−	10.3923i	0.438763	−	0.759961i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	22.0000			1.59186			0.795932	−	0.605386i	$-0.206981\pi$
$191$	22.0000			1.59186			0.795932	+	0.605386i	$0.206981\pi$
$192$		0			0
$193$	4.50000	−	7.79423i	0.323917	−	0.561041i	−0.657376	−	0.753563i	$-0.728333\pi$
$193$	4.50000	−	7.79423i	0.323917	−	0.561041i	0.981293	+	0.192522i	$0.0616668\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−8.00000			−0.569976			−0.284988	−	0.958531i	$-0.591990\pi$
$197$	−8.00000			−0.569976			−0.284988	+	0.958531i	$0.591990\pi$
$198$		0			0
$199$	5.50000	+	9.52628i	0.389885	+	0.675300i	0.992434	−	0.122782i	$-0.0391815\pi$
$199$	5.50000	+	9.52628i	0.389885	+	0.675300i	−0.602549	+	0.798082i	$0.705848\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	3.00000	+	5.19615i	0.210559	+	0.364698i
$204$		0			0
$205$	8.00000	+	13.8564i	0.558744	+	0.967773i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	8.00000	−	3.46410i	0.553372	−	0.239617i
$210$		0			0
$211$	−7.50000	+	12.9904i	−0.516321	+	0.894295i	0.483499	+	0.875345i	$0.339366\pi$
$211$	−7.50000	+	12.9904i	−0.516321	+	0.894295i	−0.999820	+	0.0189499i	$0.993968\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−7.00000	−	12.1244i	−0.477396	−	0.826874i
$216$		0			0
$217$	21.0000			1.42557
$218$		0			0
$219$		0			0
$220$		0			0
$221$	6.00000			0.403604
$222$		0			0
$223$	5.50000	−	9.52628i	0.368307	−	0.637927i	−0.620994	−	0.783815i	$-0.713271\pi$
$223$	5.50000	−	9.52628i	0.368307	−	0.637927i	0.989301	+	0.145889i	$0.0466041\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	22.0000			1.46019			0.730096	−	0.683345i	$-0.239475\pi$
$227$	22.0000			1.46019			0.730096	+	0.683345i	$0.239475\pi$
$228$		0			0
$229$	−25.0000			−1.65205			−0.826023	−	0.563636i	$-0.809402\pi$
$229$	−25.0000			−1.65205			−0.826023	+	0.563636i	$0.809402\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	9.00000	−	15.5885i	0.589610	−	1.02123i	−0.404674	−	0.914461i	$-0.632615\pi$
$233$	9.00000	−	15.5885i	0.589610	−	1.02123i	0.994283	−	0.106773i	$-0.0340517\pi$
$234$		0			0
$235$	−16.0000			−1.04372
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−18.0000			−1.16432			−0.582162	−	0.813073i	$-0.697793\pi$
$239$	−18.0000			−1.16432			−0.582162	+	0.813073i	$0.697793\pi$
$240$		0			0
$241$	−0.500000	−	0.866025i	−0.0322078	−	0.0557856i	0.849472	−	0.527633i	$-0.176921\pi$
$241$	−0.500000	−	0.866025i	−0.0322078	−	0.0557856i	−0.881680	+	0.471848i	$0.843587\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2.00000	−	3.46410i	0.127775	−	0.221313i
$246$		0			0
$247$	3.50000	+	2.59808i	0.222700	+	0.165312i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	8.00000	+	13.8564i	0.504956	+	0.874609i	0.999984	+	0.00573163i	$0.00182444\pi$
$251$	8.00000	+	13.8564i	0.504956	+	0.874609i	−0.495028	+	0.868877i	$0.664842\pi$
$252$		0			0
$253$	4.00000	+	6.92820i	0.251478	+	0.435572i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	16.0000	+	27.7128i	0.998053	+	1.72868i	0.553047	+	0.833150i	$0.313465\pi$
$257$	16.0000	+	27.7128i	0.998053	+	1.72868i	0.445005	+	0.895528i	$0.353202\pi$
$258$		0			0
$259$	−3.00000			−0.186411
$260$		0			0
$261$		0			0
$262$		0			0
$263$	9.00000	−	15.5885i	0.554964	−	0.961225i	−0.442943	−	0.896550i	$-0.646065\pi$
$263$	9.00000	−	15.5885i	0.554964	−	0.961225i	0.997906	−	0.0646755i	$-0.0206012\pi$
$264$		0			0
$265$	16.0000			0.982872
$266$		0			0
$267$		0			0
$268$		0			0
$269$	6.00000	−	10.3923i	0.365826	−	0.633630i	−0.623082	−	0.782157i	$-0.714120\pi$
$269$	6.00000	−	10.3923i	0.365826	−	0.633630i	0.988908	+	0.148527i	$0.0474530\pi$
$270$		0			0
$271$	−4.00000	+	6.92820i	−0.242983	+	0.420858i	−0.961563	−	0.274586i	$-0.911459\pi$
$271$	−4.00000	+	6.92820i	−0.242983	+	0.420858i	0.718580	+	0.695444i	$0.244792\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.00000	+	1.73205i	0.0603023	+	0.104447i
$276$		0			0
$277$	10.0000			0.600842			0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000			0.600842			0.300421	+	0.953807i	$0.402873\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	11.0000	+	19.0526i	0.656205	+	1.13658i	0.981590	+	0.190999i	$0.0611727\pi$
$281$	11.0000	+	19.0526i	0.656205	+	1.13658i	−0.325385	+	0.945582i	$0.605494\pi$
$282$		0			0
$283$	−6.00000	+	10.3923i	−0.356663	+	0.617758i	−0.987401	−	0.158237i	$-0.949419\pi$
$283$	−6.00000	+	10.3923i	−0.356663	+	0.617758i	0.630738	+	0.775996i	$0.282752\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	12.0000	−	20.7846i	0.708338	−	1.22688i
$288$		0			0
$289$	−9.50000	−	16.4545i	−0.558824	−	0.967911i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−12.0000			−0.701047			−0.350524	−	0.936554i	$-0.613996\pi$
$293$	−12.0000			−0.701047			−0.350524	+	0.936554i	$0.613996\pi$
$294$		0			0
$295$	−12.0000	−	20.7846i	−0.698667	−	1.21013i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.00000	+	3.46410i	−0.115663	+	0.200334i
$300$		0			0
$301$	−10.5000	+	18.1865i	−0.605210	+	1.04825i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−10.0000			−0.572598
$306$		0			0
$307$	6.00000	−	10.3923i	0.342438	−	0.593120i	−0.642447	−	0.766330i	$-0.722081\pi$
$307$	6.00000	−	10.3923i	0.342438	−	0.593120i	0.984885	+	0.173210i	$0.0554140\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	16.0000			0.907277			0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000			0.907277			0.453638	+	0.891186i	$0.350126\pi$
$312$		0			0
$313$	17.0000	+	29.4449i	0.960897	+	1.66432i	0.720257	+	0.693708i	$0.244024\pi$
$313$	17.0000	+	29.4449i	0.960897	+	1.66432i	0.240640	+	0.970614i	$0.422643\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3.00000	−	5.19615i	−0.168497	−	0.291845i	0.769395	−	0.638774i	$-0.220558\pi$
$317$	−3.00000	−	5.19615i	−0.168497	−	0.291845i	−0.937892	+	0.346929i	$0.887225\pi$
$318$		0			0
$319$	−2.00000	−	3.46410i	−0.111979	−	0.193952i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	3.00000	−	25.9808i	0.166924	−	1.44561i
$324$		0			0
$325$	−0.500000	+	0.866025i	−0.0277350	+	0.0480384i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	12.0000	+	20.7846i	0.661581	+	1.14589i
$330$		0			0
$331$	−23.0000			−1.26419			−0.632097	−	0.774889i	$-0.717806\pi$
$331$	−23.0000			−1.26419			−0.632097	+	0.774889i	$0.717806\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−18.0000			−0.983445
$336$		0			0
$337$	11.5000	−	19.9186i	0.626445	−	1.08503i	−0.361815	−	0.932250i	$-0.617843\pi$
$337$	11.5000	−	19.9186i	0.626445	−	1.08503i	0.988260	−	0.152784i	$-0.0488240\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−14.0000			−0.758143
$342$		0			0
$343$	15.0000			0.809924
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1.00000	−	1.73205i	0.0536828	−	0.0929814i	−0.837935	−	0.545770i	$-0.816237\pi$
$347$	1.00000	−	1.73205i	0.0536828	−	0.0929814i	0.891618	+	0.452788i	$0.149571\pi$
$348$		0			0
$349$	−11.0000			−0.588817			−0.294408	−	0.955680i	$-0.595123\pi$
$349$	−11.0000			−0.588817			−0.294408	+	0.955680i	$0.595123\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−6.00000			−0.319348			−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000			−0.319348			−0.159674	+	0.987170i	$0.551044\pi$
$354$		0			0
$355$	2.00000	+	3.46410i	0.106149	+	0.183855i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−18.0000	+	31.1769i	−0.950004	+	1.64545i	−0.204595	+	0.978847i	$0.565588\pi$
$359$	−18.0000	+	31.1769i	−0.950004	+	1.64545i	−0.745409	+	0.666608i	$0.767746\pi$
$360$		0			0
$361$	13.0000	−	13.8564i	0.684211	−	0.729285i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−15.0000	−	25.9808i	−0.785136	−	1.35990i
$366$		0			0
$367$	−5.50000	−	9.52628i	−0.287098	−	0.497268i	0.686018	−	0.727585i	$-0.259357\pi$
$367$	−5.50000	−	9.52628i	−0.287098	−	0.497268i	−0.973116	+	0.230317i	$0.926024\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−12.0000	−	20.7846i	−0.623009	−	1.07908i
$372$		0			0
$373$	−22.0000			−1.13912			−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000			−1.13912			−0.569558	+	0.821951i	$0.692886\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1.00000	−	1.73205i	0.0515026	−	0.0892052i
$378$		0			0
$379$	−3.00000			−0.154100			−0.0770498	−	0.997027i	$-0.524550\pi$
$379$	−3.00000			−0.154100			−0.0770498	+	0.997027i	$0.524550\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	7.00000	−	12.1244i	0.357683	−	0.619526i	−0.629890	−	0.776684i	$-0.716900\pi$
$383$	7.00000	−	12.1244i	0.357683	−	0.619526i	0.987573	+	0.157159i	$0.0502334\pi$
$384$		0			0
$385$	−6.00000	+	10.3923i	−0.305788	+	0.529641i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	6.00000	+	10.3923i	0.304212	+	0.526911i	0.977086	−	0.212847i	$-0.0682735\pi$
$389$	6.00000	+	10.3923i	0.304212	+	0.526911i	−0.672874	+	0.739758i	$0.734940\pi$
$390$		0			0
$391$	24.0000			1.21373
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−11.0000	−	19.0526i	−0.553470	−	0.958638i
$396$		0			0
$397$	3.50000	−	6.06218i	0.175660	−	0.304252i	−0.764730	−	0.644351i	$-0.777127\pi$
$397$	3.50000	−	6.06218i	0.175660	−	0.304252i	0.940389	+	0.340099i	$0.110461\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	2.00000	−	3.46410i	0.0998752	−	0.172989i	−0.811758	−	0.583994i	$-0.801489\pi$
$401$	2.00000	−	3.46410i	0.0998752	−	0.172989i	0.911633	+	0.411005i	$0.134822\pi$
$402$		0			0
$403$	−3.50000	−	6.06218i	−0.174347	−	0.301979i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	2.00000			0.0991363
$408$		0			0
$409$	7.00000	+	12.1244i	0.346128	+	0.599511i	0.985558	−	0.169338i	$-0.0541630\pi$
$409$	7.00000	+	12.1244i	0.346128	+	0.599511i	−0.639430	+	0.768849i	$0.720830\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−18.0000	+	31.1769i	−0.885722	+	1.53412i
$414$		0			0
$415$	6.00000	−	10.3923i	0.294528	−	0.510138i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	6.00000			0.293119			0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000			0.293119			0.146560	+	0.989202i	$0.453180\pi$
$420$		0			0
$421$	−13.0000	+	22.5167i	−0.633581	+	1.09739i	0.353233	+	0.935536i	$0.385082\pi$
$421$	−13.0000	+	22.5167i	−0.633581	+	1.09739i	−0.986814	+	0.161859i	$0.948251\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	6.00000			0.291043
$426$		0			0
$427$	7.50000	+	12.9904i	0.362950	+	0.628649i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	6.00000	+	10.3923i	0.289010	+	0.500580i	0.973574	−	0.228373i	$-0.0733406\pi$
$431$	6.00000	+	10.3923i	0.289010	+	0.500580i	−0.684564	+	0.728953i	$0.740007\pi$
$432$		0			0
$433$	6.50000	+	11.2583i	0.312370	+	0.541041i	0.978875	−	0.204460i	$-0.0655438\pi$
$433$	6.50000	+	11.2583i	0.312370	+	0.541041i	−0.666505	+	0.745501i	$0.732210\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	14.0000	+	10.3923i	0.669711	+	0.497131i
$438$		0			0
$439$	−9.50000	+	16.4545i	−0.453410	+	0.785330i	−0.998595	−	0.0529862i	$-0.983126\pi$
$439$	−9.50000	+	16.4545i	−0.453410	+	0.785330i	0.545185	+	0.838316i	$0.316459\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	13.0000	+	22.5167i	0.617649	+	1.06980i	0.989914	+	0.141672i	$0.0452479\pi$
$443$	13.0000	+	22.5167i	0.617649	+	1.06980i	−0.372265	+	0.928126i	$0.621419\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	14.0000			0.660701			0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000			0.660701			0.330350	+	0.943858i	$0.392833\pi$
$450$		0			0
$451$	−8.00000	+	13.8564i	−0.376705	+	0.652473i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−6.00000			−0.281284
$456$		0			0
$457$	17.0000			0.795226			0.397613	−	0.917553i	$-0.369839\pi$
$457$	17.0000			0.795226			0.397613	+	0.917553i	$0.369839\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	16.0000	−	27.7128i	0.745194	−	1.29071i	−0.204910	−	0.978781i	$-0.565690\pi$
$461$	16.0000	−	27.7128i	0.745194	−	1.29071i	0.950104	−	0.311933i	$-0.100977\pi$
$462$		0			0
$463$	−25.0000			−1.16185			−0.580924	−	0.813958i	$-0.697309\pi$
$463$	−25.0000			−1.16185			−0.580924	+	0.813958i	$0.697309\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	16.0000			0.740392			0.370196	−	0.928954i	$-0.379291\pi$
$467$	16.0000			0.740392			0.370196	+	0.928954i	$0.379291\pi$
$468$		0			0
$469$	13.5000	+	23.3827i	0.623372	+	1.07971i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	7.00000	−	12.1244i	0.321860	−	0.557478i
$474$		0			0
$475$	3.50000	+	2.59808i	0.160591	+	0.119208i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	1.00000	+	1.73205i	0.0456912	+	0.0791394i	0.887967	−	0.459908i	$-0.152118\pi$
$479$	1.00000	+	1.73205i	0.0456912	+	0.0791394i	−0.842275	+	0.539048i	$0.818784\pi$
$480$		0			0
$481$	0.500000	+	0.866025i	0.0227980	+	0.0394874i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−14.0000	−	24.2487i	−0.635707	−	1.10108i
$486$		0			0
$487$	32.0000			1.45006			0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000			1.45006			0.725029	+	0.688718i	$0.241826\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−19.0000	+	32.9090i	−0.857458	+	1.48516i	0.0168878	+	0.999857i	$0.494624\pi$
$491$	−19.0000	+	32.9090i	−0.857458	+	1.48516i	−0.874346	+	0.485303i	$0.838709\pi$
$492$		0			0
$493$	−12.0000			−0.540453
$494$		0			0
$495$		0			0
$496$		0			0
$497$	3.00000	−	5.19615i	0.134568	−	0.233079i
$498$		0			0
$499$	−11.5000	+	19.9186i	−0.514811	+	0.891678i	0.485042	+	0.874491i	$0.338804\pi$
$499$	−11.5000	+	19.9186i	−0.514811	+	0.891678i	−0.999852	+	0.0171872i	$0.994529\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	9.00000	+	15.5885i	0.401290	+	0.695055i	0.993882	−	0.110448i	$-0.0352286\pi$
$503$	9.00000	+	15.5885i	0.401290	+	0.695055i	−0.592592	+	0.805503i	$0.701895\pi$
$504$		0			0
$505$	28.0000			1.24598
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−6.00000	−	10.3923i	−0.265945	−	0.460631i	0.701866	−	0.712309i	$-0.252351\pi$
$509$	−6.00000	−	10.3923i	−0.265945	−	0.460631i	−0.967811	+	0.251679i	$0.919017\pi$
$510$		0			0
$511$	−22.5000	+	38.9711i	−0.995341	+	1.72398i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−3.00000	+	5.19615i	−0.132196	+	0.228970i
$516$		0			0
$517$	−8.00000	−	13.8564i	−0.351840	−	0.609404i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−26.0000			−1.13908			−0.569540	−	0.821963i	$-0.692879\pi$
$521$	−26.0000			−1.13908			−0.569540	+	0.821963i	$0.692879\pi$
$522$		0			0
$523$	−15.5000	−	26.8468i	−0.677768	−	1.17393i	−0.975652	−	0.219326i	$-0.929614\pi$
$523$	−15.5000	−	26.8468i	−0.677768	−	1.17393i	0.297884	−	0.954602i	$-0.403719\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−21.0000	+	36.3731i	−0.914774	+	1.58444i
$528$		0			0
$529$	3.50000	−	6.06218i	0.152174	−	0.263573i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−8.00000			−0.346518
$534$		0			0
$535$	−10.0000	+	17.3205i	−0.432338	+	0.748831i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	4.00000			0.172292
$540$		0			0
$541$	0.500000	+	0.866025i	0.0214967	+	0.0372333i	0.876574	−	0.481268i	$-0.159824\pi$
$541$	0.500000	+	0.866025i	0.0214967	+	0.0372333i	−0.855077	+	0.518501i	$0.826490\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−6.00000	−	10.3923i	−0.257012	−	0.445157i
$546$		0			0
$547$	21.5000	+	37.2391i	0.919274	+	1.59223i	0.800521	+	0.599305i	$0.204556\pi$
$547$	21.5000	+	37.2391i	0.919274	+	1.59223i	0.118753	+	0.992924i	$0.462110\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−7.00000	−	5.19615i	−0.298210	−	0.221364i
$552$		0			0
$553$	−16.5000	+	28.5788i	−0.701651	+	1.21530i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	5.00000	+	8.66025i	0.211857	+	0.366947i	0.952296	−	0.305177i	$-0.0987156\pi$
$557$	5.00000	+	8.66025i	0.211857	+	0.366947i	−0.740439	+	0.672124i	$0.765382\pi$
$558$		0			0
$559$	7.00000			0.296068
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−18.0000			−0.758610			−0.379305	−	0.925272i	$-0.623837\pi$
$563$	−18.0000			−0.758610			−0.379305	+	0.925272i	$0.623837\pi$
$564$		0			0
$565$	−20.0000	+	34.6410i	−0.841406	+	1.45736i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−4.00000			−0.167689			−0.0838444	−	0.996479i	$-0.526720\pi$
$569$	−4.00000			−0.167689			−0.0838444	+	0.996479i	$0.526720\pi$
$570$		0			0
$571$	−29.0000			−1.21361			−0.606806	−	0.794850i	$-0.707550\pi$
$571$	−29.0000			−1.21361			−0.606806	+	0.794850i	$0.707550\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−2.00000	+	3.46410i	−0.0834058	+	0.144463i
$576$		0			0
$577$	14.0000			0.582828			0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000			0.582828			0.291414	+	0.956597i	$0.405874\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−18.0000			−0.746766
$582$		0			0
$583$	8.00000	+	13.8564i	0.331326	+	0.573874i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$587$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$588$		0			0
$589$	−28.0000	+	12.1244i	−1.15372	+	0.499575i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	8.00000	+	13.8564i	0.328521	+	0.569014i	0.982219	−	0.187741i	$-0.0601166\pi$
$593$	8.00000	+	13.8564i	0.328521	+	0.569014i	−0.653698	+	0.756756i	$0.726783\pi$
$594$		0			0
$595$	18.0000	+	31.1769i	0.737928	+	1.27813i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−15.0000	−	25.9808i	−0.612883	−	1.06155i	−0.990752	−	0.135686i	$-0.956676\pi$
$599$	−15.0000	−	25.9808i	−0.612883	−	1.06155i	0.377869	−	0.925859i	$-0.376657\pi$
$600$		0			0
$601$	−41.0000			−1.67242			−0.836212	−	0.548406i	$-0.815235\pi$
$601$	−41.0000			−1.67242			−0.836212	+	0.548406i	$0.815235\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−7.00000	+	12.1244i	−0.284590	+	0.492925i
$606$		0			0
$607$	13.0000			0.527654			0.263827	−	0.964570i	$-0.415015\pi$
$607$	13.0000			0.527654			0.263827	+	0.964570i	$0.415015\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	4.00000	−	6.92820i	0.161823	−	0.280285i
$612$		0			0
$613$	−23.0000	+	39.8372i	−0.928961	+	1.60901i	−0.143898	+	0.989593i	$0.545964\pi$
$613$	−23.0000	+	39.8372i	−0.928961	+	1.60901i	−0.785063	+	0.619416i	$0.787370\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	3.00000	+	5.19615i	0.120775	+	0.209189i	0.920074	−	0.391745i	$-0.128129\pi$
$617$	3.00000	+	5.19615i	0.120775	+	0.209189i	−0.799298	+	0.600935i	$0.794795\pi$
$618$		0			0
$619$	23.0000			0.924448			0.462224	−	0.886763i	$-0.347052\pi$
$619$	23.0000			0.924448			0.462224	+	0.886763i	$0.347052\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	9.50000	−	16.4545i	0.380000	−	0.658179i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	3.00000	−	5.19615i	0.119618	−	0.207184i
$630$		0			0
$631$	23.5000	+	40.7032i	0.935520	+	1.62037i	0.773704	+	0.633548i	$0.218402\pi$
$631$	23.5000	+	40.7032i	0.935520	+	1.62037i	0.161817	+	0.986821i	$0.448265\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	1.00000	+	1.73205i	0.0396214	+	0.0686264i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$641$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$642$		0			0
$643$	6.50000	−	11.2583i	0.256335	−	0.443985i	−0.708922	−	0.705287i	$-0.750818\pi$
$643$	6.50000	−	11.2583i	0.256335	−	0.443985i	0.965257	+	0.261301i	$0.0841516\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−10.0000			−0.393141			−0.196570	−	0.980490i	$-0.562980\pi$
$647$	−10.0000			−0.393141			−0.196570	+	0.980490i	$0.562980\pi$
$648$		0			0
$649$	12.0000	−	20.7846i	0.471041	−	0.815867i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	18.0000			0.704394			0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000			0.704394			0.352197	+	0.935926i	$0.385435\pi$
$654$		0			0
$655$	−6.00000	−	10.3923i	−0.234439	−	0.406061i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	11.0000	+	19.0526i	0.428499	+	0.742182i	0.996740	−	0.0806799i	$-0.0257092\pi$
$659$	11.0000	+	19.0526i	0.428499	+	0.742182i	−0.568241	+	0.822862i	$0.692376\pi$
$660$		0			0
$661$	1.00000	+	1.73205i	0.0388955	+	0.0673690i	0.884818	−	0.465937i	$-0.154283\pi$
$661$	1.00000	+	1.73205i	0.0388955	+	0.0673690i	−0.845922	+	0.533306i	$0.820949\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−3.00000	+	25.9808i	−0.116335	+	1.00749i
$666$		0			0
$667$	4.00000	−	6.92820i	0.154881	−	0.268261i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−5.00000	−	8.66025i	−0.193023	−	0.334325i
$672$		0			0
$673$	−1.00000			−0.0385472			−0.0192736	−	0.999814i	$-0.506135\pi$
$673$	−1.00000			−0.0385472			−0.0192736	+	0.999814i	$0.506135\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	8.00000			0.307465			0.153732	−	0.988113i	$-0.450871\pi$
$677$	8.00000			0.307465			0.153732	+	0.988113i	$0.450871\pi$
$678$		0			0
$679$	−21.0000	+	36.3731i	−0.805906	+	1.39587i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	46.0000			1.76014			0.880071	−	0.474843i	$-0.157495\pi$
$683$	46.0000			1.76014			0.880071	+	0.474843i	$0.157495\pi$
$684$		0			0
$685$	−28.0000			−1.06983
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−4.00000	+	6.92820i	−0.152388	+	0.263944i
$690$		0			0
$691$	12.0000			0.456502			0.228251	−	0.973602i	$-0.426699\pi$
$691$	12.0000			0.456502			0.228251	+	0.973602i	$0.426699\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−6.00000			−0.227593
$696$		0			0
$697$	24.0000	+	41.5692i	0.909065	+	1.57455i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−17.0000	+	29.4449i	−0.642081	+	1.11212i	0.342886	+	0.939377i	$0.388595\pi$
$701$	−17.0000	+	29.4449i	−0.642081	+	1.11212i	−0.984967	+	0.172740i	$0.944738\pi$
$702$		0			0
$703$	4.00000	−	1.73205i	0.150863	−	0.0653255i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−21.0000	−	36.3731i	−0.789786	−	1.36795i
$708$		0			0
$709$	4.50000	+	7.79423i	0.169001	+	0.292718i	0.938069	−	0.346449i	$-0.112613\pi$
$709$	4.50000	+	7.79423i	0.169001	+	0.292718i	−0.769068	+	0.639167i	$0.779279\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−14.0000	−	24.2487i	−0.524304	−	0.908121i
$714$		0			0
$715$	4.00000			0.149592
$716$		0			0
$717$		0			0
$718$		0			0
$719$	22.0000	−	38.1051i	0.820462	−	1.42108i	−0.0848774	−	0.996391i	$-0.527050\pi$
$719$	22.0000	−	38.1051i	0.820462	−	1.42108i	0.905339	−	0.424690i	$-0.139617\pi$
$720$		0			0
$721$	9.00000			0.335178
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.00000	−	1.73205i	0.0371391	−	0.0643268i
$726$		0			0
$727$	3.50000	−	6.06218i	0.129808	−	0.224834i	−0.793794	−	0.608186i	$-0.791897\pi$
$727$	3.50000	−	6.06218i	0.129808	−	0.224834i	0.923602	+	0.383353i	$0.125231\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−21.0000	−	36.3731i	−0.776713	−	1.34531i
$732$		0			0
$733$	−6.00000			−0.221615			−0.110808	−	0.993842i	$-0.535344\pi$
$733$	−6.00000			−0.221615			−0.110808	+	0.993842i	$0.535344\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−9.00000	−	15.5885i	−0.331519	−	0.574208i
$738$		0			0
$739$	−0.500000	+	0.866025i	−0.0183928	+	0.0318573i	−0.875075	−	0.483987i	$-0.839188\pi$
$739$	−0.500000	+	0.866025i	−0.0183928	+	0.0318573i	0.856683	+	0.515844i	$0.172522\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	5.00000	−	8.66025i	0.183432	−	0.317714i	−0.759615	−	0.650373i	$-0.774613\pi$
$743$	5.00000	−	8.66025i	0.183432	−	0.317714i	0.943047	+	0.332659i	$0.107946\pi$
$744$		0			0
$745$	−18.0000	−	31.1769i	−0.659469	−	1.14223i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	30.0000			1.09618
$750$		0			0
$751$	−18.5000	−	32.0429i	−0.675075	−	1.16926i	−0.976447	−	0.215757i	$-0.930778\pi$
$751$	−18.5000	−	32.0429i	−0.675075	−	1.16926i	0.301373	−	0.953506i	$-0.402555\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	4.00000	−	6.92820i	0.145575	−	0.252143i
$756$		0			0
$757$	20.5000	−	35.5070i	0.745085	−	1.29053i	−0.205070	−	0.978747i	$-0.565742\pi$
$757$	20.5000	−	35.5070i	0.745085	−	1.29053i	0.950155	−	0.311778i	$-0.100925\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	12.0000			0.435000			0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000			0.435000			0.217500	+	0.976060i	$0.430210\pi$
$762$		0			0
$763$	−9.00000	+	15.5885i	−0.325822	+	0.564340i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	12.0000			0.433295
$768$		0			0
$769$	14.5000	+	25.1147i	0.522883	+	0.905661i	0.999645	+	0.0266282i	$0.00847701\pi$
$769$	14.5000	+	25.1147i	0.522883	+	0.905661i	−0.476762	+	0.879032i	$0.658190\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	24.0000	+	41.5692i	0.863220	+	1.49514i	0.868804	+	0.495156i	$0.164889\pi$
$773$	24.0000	+	41.5692i	0.863220	+	1.49514i	−0.00558380	+	0.999984i	$0.501777\pi$
$774$		0			0
$775$	−3.50000	−	6.06218i	−0.125724	−	0.217760i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−4.00000	+	34.6410i	−0.143315	+	1.24114i
$780$		0			0
$781$	−2.00000	+	3.46410i	−0.0715656	+	0.123955i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	13.0000	+	22.5167i	0.463990	+	0.803654i
$786$		0			0
$787$	35.0000			1.24762			0.623808	−	0.781578i	$-0.285585\pi$
$787$	35.0000			1.24762			0.623808	+	0.781578i	$0.285585\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	60.0000			2.13335
$792$		0			0
$793$	2.50000	−	4.33013i	0.0887776	−	0.153767i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	18.0000			0.637593			0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000			0.637593			0.318796	+	0.947823i	$0.396721\pi$
$798$		0			0
$799$	−48.0000			−1.69812
$800$		0			0
$801$		0			0
$802$		0			0
$803$	15.0000	−	25.9808i	0.529339	−	0.916841i
$804$		0			0
$805$	−24.0000			−0.845889
$806$		0			0
$807$		0			0
$808$		0			0
$809$	6.00000			0.210949			0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000			0.210949			0.105474	+	0.994422i	$0.466364\pi$
$810$		0			0
$811$	6.00000	+	10.3923i	0.210688	+	0.364923i	0.951930	−	0.306315i	$-0.0990961\pi$
$811$	6.00000	+	10.3923i	0.210688	+	0.364923i	−0.741242	+	0.671238i	$0.765763\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	13.0000	−	22.5167i	0.455370	−	0.788724i
$816$		0			0
$817$	3.50000	−	30.3109i	0.122449	−	1.06044i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−15.0000	−	25.9808i	−0.523504	−	0.906735i	−0.999626	−	0.0273557i	$-0.991291\pi$
$821$	−15.0000	−	25.9808i	−0.523504	−	0.906735i	0.476122	−	0.879379i	$-0.342042\pi$
$822$		0			0
$823$	−8.00000	−	13.8564i	−0.278862	−	0.483004i	0.692240	−	0.721668i	$-0.256624\pi$
$823$	−8.00000	−	13.8564i	−0.278862	−	0.483004i	−0.971102	+	0.238664i	$0.923291\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	11.0000	+	19.0526i	0.382507	+	0.662522i	0.991420	−	0.130715i	$-0.0417273\pi$
$827$	11.0000	+	19.0526i	0.382507	+	0.662522i	−0.608913	+	0.793237i	$0.708394\pi$
$828$		0			0
$829$	−9.00000			−0.312583			−0.156291	−	0.987711i	$-0.549954\pi$
$829$	−9.00000			−0.312583			−0.156291	+	0.987711i	$0.549954\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	6.00000	−	10.3923i	0.207888	−	0.360072i
$834$		0			0
$835$	24.0000			0.830554
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−7.00000	+	12.1244i	−0.241667	+	0.418579i	−0.961189	−	0.275890i	$-0.911027\pi$
$839$	−7.00000	+	12.1244i	−0.241667	+	0.418579i	0.719522	+	0.694469i	$0.244361\pi$
$840$		0			0
$841$	12.5000	−	21.6506i	0.431034	−	0.746574i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−12.0000	−	20.7846i	−0.412813	−	0.715012i
$846$		0			0
$847$	21.0000			0.721569
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2.00000	+	3.46410i	0.0685591	+	0.118748i
$852$		0			0
$853$	−4.50000	+	7.79423i	−0.154077	+	0.266869i	−0.932723	−	0.360595i	$-0.882574\pi$
$853$	−4.50000	+	7.79423i	−0.154077	+	0.266869i	0.778646	+	0.627464i	$0.215907\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	12.0000	−	20.7846i	0.409912	−	0.709989i	−0.584967	−	0.811057i	$-0.698893\pi$
$857$	12.0000	−	20.7846i	0.409912	−	0.709989i	0.994880	+	0.101068i	$0.0322260\pi$
$858$		0			0
$859$	−2.50000	−	4.33013i	−0.0852989	−	0.147742i	0.820220	−	0.572049i	$-0.193851\pi$
$859$	−2.50000	−	4.33013i	−0.0852989	−	0.147742i	−0.905519	+	0.424307i	$0.860518\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−12.0000			−0.408485			−0.204242	−	0.978920i	$-0.565473\pi$
$863$	−12.0000			−0.408485			−0.204242	+	0.978920i	$0.565473\pi$
$864$		0			0
$865$	18.0000	+	31.1769i	0.612018	+	1.06005i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	11.0000	−	19.0526i	0.373149	−	0.646314i
$870$		0			0
$871$	4.50000	−	7.79423i	0.152477	−	0.264097i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−36.0000			−1.21702
$876$		0			0
$877$	−12.5000	+	21.6506i	−0.422095	+	0.731090i	−0.996144	−	0.0877308i	$-0.972038\pi$
$877$	−12.5000	+	21.6506i	−0.422095	+	0.731090i	0.574049	+	0.818821i	$0.305372\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	30.0000			1.01073			0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000			1.01073			0.505363	+	0.862907i	$0.331359\pi$
$882$		0			0
$883$	−14.5000	−	25.1147i	−0.487964	−	0.845178i	0.511940	−	0.859021i	$-0.328927\pi$
$883$	−14.5000	−	25.1147i	−0.487964	−	0.845178i	−0.999904	+	0.0138428i	$0.995594\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	4.00000	+	6.92820i	0.134307	+	0.232626i	0.925332	−	0.379157i	$-0.123786\pi$
$887$	4.00000	+	6.92820i	0.134307	+	0.232626i	−0.791026	+	0.611783i	$0.790453\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−28.0000	−	20.7846i	−0.936984	−	0.695530i
$894$		0			0
$895$	−24.0000	+	41.5692i	−0.802232	+	1.38951i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	7.00000	+	12.1244i	0.233463	+	0.404370i
$900$		0			0
$901$	48.0000			1.59911
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−44.0000			−1.46261
$906$		0			0
$907$	−14.0000	+	24.2487i	−0.464862	+	0.805165i	−0.999195	−	0.0401089i	$-0.987230\pi$
$907$	−14.0000	+	24.2487i	−0.464862	+	0.805165i	0.534333	+	0.845274i	$0.320563\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−48.0000			−1.59031			−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000			−1.59031			−0.795155	+	0.606406i	$0.792611\pi$
$912$		0			0
$913$	12.0000			0.397142
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−9.00000	+	15.5885i	−0.297206	+	0.514776i
$918$		0			0
$919$	−17.0000			−0.560778			−0.280389	−	0.959886i	$-0.590464\pi$
$919$	−17.0000			−0.560778			−0.280389	+	0.959886i	$0.590464\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−2.00000			−0.0658308
$924$		0			0
$925$	0.500000	+	0.866025i	0.0164399	+	0.0284747i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−17.0000	+	29.4449i	−0.557752	+	0.966055i	0.439932	+	0.898031i	$0.355003\pi$
$929$	−17.0000	+	29.4449i	−0.557752	+	0.966055i	−0.997684	+	0.0680235i	$0.978331\pi$
$930$		0			0
$931$	8.00000	−	3.46410i	0.262189	−	0.113531i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−12.0000	−	20.7846i	−0.392442	−	0.679729i
$936$		0			0
$937$	−23.5000	−	40.7032i	−0.767712	−	1.32972i	−0.938801	−	0.344460i	$-0.888062\pi$
$937$	−23.5000	−	40.7032i	−0.767712	−	1.32972i	0.171089	−	0.985255i	$-0.445271\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	20.0000	+	34.6410i	0.651981	+	1.12926i	0.982641	+	0.185515i	$0.0593953\pi$
$941$	20.0000	+	34.6410i	0.651981	+	1.12926i	−0.330660	+	0.943750i	$0.607271\pi$
$942$		0			0
$943$	−32.0000			−1.04206
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−3.00000	+	5.19615i	−0.0974869	+	0.168852i	−0.910644	−	0.413192i	$-0.864414\pi$
$947$	−3.00000	+	5.19615i	−0.0974869	+	0.168852i	0.813157	+	0.582045i	$0.197747\pi$
$948$		0			0
$949$	15.0000			0.486921
$950$		0			0
$951$		0			0
$952$		0			0
$953$	13.0000	−	22.5167i	0.421111	−	0.729386i	−0.574937	−	0.818198i	$-0.694974\pi$
$953$	13.0000	−	22.5167i	0.421111	−	0.729386i	0.996048	+	0.0888114i	$0.0283068\pi$
$954$		0			0
$955$	22.0000	−	38.1051i	0.711903	−	1.23305i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	21.0000	+	36.3731i	0.678125	+	1.17455i
$960$		0			0
$961$	18.0000			0.580645
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−9.00000	−	15.5885i	−0.289720	−	0.501810i
$966$		0			0
$967$	4.50000	−	7.79423i	0.144710	−	0.250645i	−0.784555	−	0.620060i	$-0.787108\pi$
$967$	4.50000	−	7.79423i	0.144710	−	0.250645i	0.929265	+	0.369414i	$0.120442\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−10.0000	+	17.3205i	−0.320915	+	0.555842i	−0.980677	−	0.195633i	$-0.937324\pi$
$971$	−10.0000	+	17.3205i	−0.320915	+	0.555842i	0.659762	+	0.751475i	$0.270657\pi$
$972$		0			0
$973$	4.50000	+	7.79423i	0.144263	+	0.249871i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	4.00000			0.127971			0.0639857	−	0.997951i	$-0.479619\pi$
$977$	4.00000			0.127971			0.0639857	+	0.997951i	$0.479619\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−3.00000	+	5.19615i	−0.0956851	+	0.165732i	−0.909894	−	0.414840i	$-0.863838\pi$
$983$	−3.00000	+	5.19615i	−0.0956851	+	0.165732i	0.814209	+	0.580572i	$0.197171\pi$
$984$		0			0
$985$	−8.00000	+	13.8564i	−0.254901	+	0.441502i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	28.0000			0.890348
$990$		0			0
$991$	−5.50000	+	9.52628i	−0.174713	+	0.302612i	−0.940062	−	0.341004i	$-0.889233\pi$
$991$	−5.50000	+	9.52628i	−0.174713	+	0.302612i	0.765349	+	0.643616i	$0.222567\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	22.0000			0.697447
$996$		0			0
$997$	−0.500000	−	0.866025i	−0.0158352	−	0.0274273i	0.857999	−	0.513651i	$-0.171707\pi$
$997$	−0.500000	−	0.866025i	−0.0158352	−	0.0274273i	−0.873834	+	0.486224i	$0.838374\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.s.o.1873.1		2
3.2	odd	2		2736.2.s.d.1873.1		2
4.3	odd	2		342.2.g.e.163.1	yes	2
12.11	even	2		342.2.g.a.163.1	✓	2
19.7	even	3	inner	2736.2.s.o.577.1		2
57.26	odd	6		2736.2.s.d.577.1		2
76.7	odd	6		342.2.g.e.235.1	yes	2
76.11	odd	6		6498.2.a.c.1.1		1
76.27	even	6		6498.2.a.q.1.1		1
228.11	even	6		6498.2.a.w.1.1		1
228.83	even	6		342.2.g.a.235.1	yes	2
228.179	odd	6		6498.2.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
342.2.g.a.163.1	✓	2	12.11	even	2
342.2.g.a.235.1	yes	2	228.83	even	6
342.2.g.e.163.1	yes	2	4.3	odd	2
342.2.g.e.235.1	yes	2	76.7	odd	6
2736.2.s.d.577.1		2	57.26	odd	6
2736.2.s.d.1873.1		2	3.2	odd	2
2736.2.s.o.577.1		2	19.7	even	3	inner
2736.2.s.o.1873.1		2	1.1	even	1	trivial
6498.2.a.c.1.1		1	76.11	odd	6
6498.2.a.k.1.1		1	228.179	odd	6
6498.2.a.q.1.1		1	76.27	even	6
6498.2.a.w.1.1		1	228.11	even	6

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2736.s (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.73205i) q^{5} -3.00000 q^{7} +2.00000 q^{11} +(0.500000 + 0.866025i) q^{13} +(3.00000 - 5.19615i) q^{17} +(4.00000 - 1.73205i) q^{19} +(2.00000 + 3.46410i) q^{23} +(0.500000 + 0.866025i) q^{25} +(-1.00000 - 1.73205i) q^{29} -7.00000 q^{31} +(-3.00000 + 5.19615i) q^{35} +1.00000 q^{37} +(-4.00000 + 6.92820i) q^{41} +(3.50000 - 6.06218i) q^{43} +(-4.00000 - 6.92820i) q^{47} +2.00000 q^{49} +(4.00000 + 6.92820i) q^{53} +(2.00000 - 3.46410i) q^{55} +(6.00000 - 10.3923i) q^{59} +(-2.50000 - 4.33013i) q^{61} +2.00000 q^{65} +(-4.50000 - 7.79423i) q^{67} +(-1.00000 + 1.73205i) q^{71} +(7.50000 - 12.9904i) q^{73} -6.00000 q^{77} +(5.50000 - 9.52628i) q^{79} +6.00000 q^{83} +(-6.00000 - 10.3923i) q^{85} +(-1.50000 - 2.59808i) q^{91} +(1.00000 - 8.66025i) q^{95} +(7.00000 - 12.1244i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 6 q^{7} + 4 q^{11} + q^{13} + 6 q^{17} + 8 q^{19} + 4 q^{23} + q^{25} - 2 q^{29} - 14 q^{31} - 6 q^{35} + 2 q^{37} - 8 q^{41} + 7 q^{43} - 8 q^{47} + 4 q^{49} + 8 q^{53} + 4 q^{55} + 12 q^{59}+ \cdots + 14 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 - 6 * q^7 + 4 * q^11 + q^13 + 6 * q^17 + 8 * q^19 + 4 * q^23 + q^25 - 2 * q^29 - 14 * q^31 - 6 * q^35 + 2 * q^37 - 8 * q^41 + 7 * q^43 - 8 * q^47 + 4 * q^49 + 8 * q^53 + 4 * q^55 + 12 * q^59 - 5 * q^61 + 4 * q^65 - 9 * q^67 - 2 * q^71 + 15 * q^73 - 12 * q^77 + 11 * q^79 + 12 * q^83 - 12 * q^85 - 3 * q^91 + 2 * q^95 + 14 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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