LMFDB - Embedded newform 2592.2.i.o.1729.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2592,2,Mod(865,2592)] 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("2592.865"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2592, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2592 = 2^{5} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2592.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,1,0,2,0,0,0,-2,0,-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:	$20.6972242039$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1729.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2592.1729
Dual form			2592.2.i.o.865.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{5} +(1.00000 + 1.73205i) q^{7} +(-1.00000 - 1.73205i) q^{11} +(-0.500000 + 0.866025i) q^{13} -3.00000 q^{17} -2.00000 q^{19} +(-3.00000 + 5.19615i) q^{23} +(2.00000 + 3.46410i) q^{25} +(0.500000 + 0.866025i) q^{29} +(-4.00000 + 6.92820i) q^{31} +2.00000 q^{35} +1.00000 q^{37} +(-1.00000 + 1.73205i) q^{41} +(5.00000 + 8.66025i) q^{43} +(-2.00000 - 3.46410i) q^{47} +(1.50000 - 2.59808i) q^{49} +10.0000 q^{53} -2.00000 q^{55} +(-2.00000 + 3.46410i) q^{59} +(-4.50000 - 7.79423i) q^{61} +(0.500000 + 0.866025i) q^{65} +(-7.00000 + 12.1244i) q^{67} +10.0000 q^{71} -9.00000 q^{73} +(2.00000 - 3.46410i) q^{77} +(5.00000 + 8.66025i) q^{79} +(-6.00000 - 10.3923i) q^{83} +(-1.50000 + 2.59808i) q^{85} -11.0000 q^{89} -2.00000 q^{91} +(-1.00000 + 1.73205i) q^{95} +(1.00000 + 1.73205i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{5} + 2 q^{7} - 2 q^{11} - q^{13} - 6 q^{17} - 4 q^{19} - 6 q^{23} + 4 q^{25} + q^{29} - 8 q^{31} + 4 q^{35} + 2 q^{37} - 2 q^{41} + 10 q^{43} - 4 q^{47} + 3 q^{49} + 20 q^{53} - 4 q^{55} - 4 q^{59}+ \cdots + 2 q^{97}+O(q^{100}) $ 2 * q + q^5 + 2 * q^7 - 2 * q^11 - q^13 - 6 * q^17 - 4 * q^19 - 6 * q^23 + 4 * q^25 + q^29 - 8 * q^31 + 4 * q^35 + 2 * q^37 - 2 * q^41 + 10 * q^43 - 4 * q^47 + 3 * q^49 + 20 * q^53 - 4 * q^55 - 4 * q^59 - 9 * q^61 + q^65 - 14 * q^67 + 20 * q^71 - 18 * q^73 + 4 * q^77 + 10 * q^79 - 12 * q^83 - 3 * q^85 - 22 * q^89 - 4 * q^91 - 2 * q^95 + 2 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times$.

$n$	$325$	$1217$	$2431$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i	−0.732294	−	0.680989i	$-0.761550\pi$
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i	0.955901	+	0.293691i	$0.0948835\pi$
$6$		0			0
$7$	1.00000	+	1.73205i	0.377964	+	0.654654i	0.990766	−	0.135583i	$-0.0432908\pi$
$7$	1.00000	+	1.73205i	0.377964	+	0.654654i	−0.612801	+	0.790237i	$0.709957\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.00000	−	1.73205i	−0.301511	−	0.522233i	0.674967	−	0.737848i	$-0.264158\pi$
$11$	−1.00000	−	1.73205i	−0.301511	−	0.522233i	−0.976478	+	0.215615i	$0.930824\pi$
$12$		0			0
$13$	−0.500000	+	0.866025i	−0.138675	+	0.240192i	−0.926995	−	0.375073i	$-0.877618\pi$
$13$	−0.500000	+	0.866025i	−0.138675	+	0.240192i	0.788320	+	0.615265i	$0.210951\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.00000			−0.727607			−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000			−0.727607			−0.363803	+	0.931476i	$0.618522\pi$
$18$		0			0
$19$	−2.00000			−0.458831			−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000			−0.458831			−0.229416	+	0.973329i	$0.573682\pi$
$20$		0			0
$21$		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	+	3.46410i	0.400000	+	0.692820i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	0.500000	+	0.866025i	0.0928477	+	0.160817i	0.908708	−	0.417432i	$-0.137070\pi$
$29$	0.500000	+	0.866025i	0.0928477	+	0.160817i	−0.815861	+	0.578249i	$0.803736\pi$
$30$		0			0
$31$	−4.00000	+	6.92820i	−0.718421	+	1.24434i	0.243204	+	0.969975i	$0.421802\pi$
$31$	−4.00000	+	6.92820i	−0.718421	+	1.24434i	−0.961625	+	0.274367i	$0.911532\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.00000			0.338062
$36$		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
$41$	−1.00000	+	1.73205i	−0.156174	+	0.270501i	−0.933486	−	0.358614i	$-0.883249\pi$
$41$	−1.00000	+	1.73205i	−0.156174	+	0.270501i	0.777312	+	0.629115i	$0.216583\pi$
$42$		0			0
$43$	5.00000	+	8.66025i	0.762493	+	1.32068i	0.941562	+	0.336840i	$0.109358\pi$
$43$	5.00000	+	8.66025i	0.762493	+	1.32068i	−0.179069	+	0.983836i	$0.557309\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.00000	−	3.46410i	−0.291730	−	0.505291i	0.682489	−	0.730896i	$-0.260898\pi$
$47$	−2.00000	−	3.46410i	−0.291730	−	0.505291i	−0.974219	+	0.225605i	$0.927564\pi$
$48$		0			0
$49$	1.50000	−	2.59808i	0.214286	−	0.371154i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	10.0000			1.37361			0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000			1.37361			0.686803	+	0.726844i	$0.259014\pi$
$54$		0			0
$55$	−2.00000			−0.269680
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−2.00000	+	3.46410i	−0.260378	+	0.450988i	−0.966342	−	0.257260i	$-0.917180\pi$
$59$	−2.00000	+	3.46410i	−0.260378	+	0.450988i	0.705965	+	0.708247i	$0.250514\pi$
$60$		0			0
$61$	−4.50000	−	7.79423i	−0.576166	−	0.997949i	−0.995914	−	0.0903080i	$-0.971215\pi$
$61$	−4.50000	−	7.79423i	−0.576166	−	0.997949i	0.419748	−	0.907641i	$-0.362118\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.500000	+	0.866025i	0.0620174	+	0.107417i
$66$		0			0
$67$	−7.00000	+	12.1244i	−0.855186	+	1.48123i	0.0212861	+	0.999773i	$0.493224\pi$
$67$	−7.00000	+	12.1244i	−0.855186	+	1.48123i	−0.876472	+	0.481452i	$0.840109\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	10.0000			1.18678			0.593391	−	0.804914i	$-0.297789\pi$
$71$	10.0000			1.18678			0.593391	+	0.804914i	$0.297789\pi$
$72$		0			0
$73$	−9.00000			−1.05337			−0.526685	−	0.850060i	$-0.676565\pi$
$73$	−9.00000			−1.05337			−0.526685	+	0.850060i	$0.676565\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.00000	−	3.46410i	0.227921	−	0.394771i
$78$		0			0
$79$	5.00000	+	8.66025i	0.562544	+	0.974355i	0.997274	+	0.0737937i	$0.0235106\pi$
$79$	5.00000	+	8.66025i	0.562544	+	0.974355i	−0.434730	+	0.900561i	$0.643156\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−6.00000	−	10.3923i	−0.658586	−	1.14070i	−0.980982	−	0.194099i	$-0.937822\pi$
$83$	−6.00000	−	10.3923i	−0.658586	−	1.14070i	0.322396	−	0.946605i	$-0.395512\pi$
$84$		0			0
$85$	−1.50000	+	2.59808i	−0.162698	+	0.281801i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−11.0000			−1.16600			−0.582999	−	0.812473i	$-0.698121\pi$
$89$	−11.0000			−1.16600			−0.582999	+	0.812473i	$0.698121\pi$
$90$		0			0
$91$	−2.00000			−0.209657
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.00000	+	1.73205i	−0.102598	+	0.177705i
$96$		0			0
$97$	1.00000	+	1.73205i	0.101535	+	0.175863i	0.912317	−	0.409484i	$-0.134291\pi$
$97$	1.00000	+	1.73205i	0.101535	+	0.175863i	−0.810782	+	0.585348i	$0.800958\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−3.00000	−	5.19615i	−0.298511	−	0.517036i	0.677284	−	0.735721i	$-0.263157\pi$
$101$	−3.00000	−	5.19615i	−0.298511	−	0.517036i	−0.975796	+	0.218685i	$0.929823\pi$
$102$		0			0
$103$	−6.00000	+	10.3923i	−0.591198	+	1.02398i	0.402874	+	0.915255i	$0.368011\pi$
$103$	−6.00000	+	10.3923i	−0.591198	+	1.02398i	−0.994071	+	0.108729i	$0.965322\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	16.0000			1.54678			0.773389	−	0.633932i	$-0.218560\pi$
$107$	16.0000			1.54678			0.773389	+	0.633932i	$0.218560\pi$
$108$		0			0
$109$	5.00000			0.478913			0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000			0.478913			0.239457	+	0.970907i	$0.423031\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−8.50000	+	14.7224i	−0.799613	+	1.38497i	0.120256	+	0.992743i	$0.461629\pi$
$113$	−8.50000	+	14.7224i	−0.799613	+	1.38497i	−0.919868	+	0.392227i	$0.871705\pi$
$114$		0			0
$115$	3.00000	+	5.19615i	0.279751	+	0.484544i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−3.00000	−	5.19615i	−0.275010	−	0.476331i
$120$		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
$127$	2.00000			0.177471			0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000			0.177471			0.0887357	+	0.996055i	$0.471717\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−7.00000	+	12.1244i	−0.611593	+	1.05931i	0.379379	+	0.925241i	$0.376138\pi$
$131$	−7.00000	+	12.1244i	−0.611593	+	1.05931i	−0.990972	+	0.134069i	$0.957196\pi$
$132$		0			0
$133$	−2.00000	−	3.46410i	−0.173422	−	0.300376i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.50000	−	7.79423i	−0.384461	−	0.665906i	0.607233	−	0.794524i	$-0.292279\pi$
$137$	−4.50000	−	7.79423i	−0.384461	−	0.665906i	−0.991694	+	0.128618i	$0.958946\pi$
$138$		0			0
$139$	−10.0000	+	17.3205i	−0.848189	+	1.46911i	0.0346338	+	0.999400i	$0.488974\pi$
$139$	−10.0000	+	17.3205i	−0.848189	+	1.46911i	−0.882823	+	0.469706i	$0.844360\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2.00000			0.167248
$144$		0			0
$145$	1.00000			0.0830455
$146$		0			0
$147$		0			0
$148$		0			0
$149$	10.5000	−	18.1865i	0.860194	−	1.48990i	−0.0115483	−	0.999933i	$-0.503676\pi$
$149$	10.5000	−	18.1865i	0.860194	−	1.48990i	0.871742	−	0.489966i	$-0.162991\pi$
$150$		0			0
$151$	6.00000	+	10.3923i	0.488273	+	0.845714i	0.999909	−	0.0134886i	$-0.00429367\pi$
$151$	6.00000	+	10.3923i	0.488273	+	0.845714i	−0.511636	+	0.859202i	$0.670960\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	4.00000	+	6.92820i	0.321288	+	0.556487i
$156$		0			0
$157$	−8.50000	+	14.7224i	−0.678374	+	1.17498i	0.297097	+	0.954847i	$0.403982\pi$
$157$	−8.50000	+	14.7224i	−0.678374	+	1.17498i	−0.975470	+	0.220131i	$0.929352\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−12.0000			−0.945732
$162$		0			0
$163$	4.00000			0.313304			0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000			0.313304			0.156652	+	0.987654i	$0.449930\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.00000	−	1.73205i	0.0773823	−	0.134030i	−0.824737	−	0.565516i	$-0.808677\pi$
$167$	1.00000	−	1.73205i	0.0773823	−	0.134030i	0.902120	+	0.431486i	$0.142010\pi$
$168$		0			0
$169$	6.00000	+	10.3923i	0.461538	+	0.799408i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	10.5000	+	18.1865i	0.798300	+	1.38270i	0.920722	+	0.390218i	$0.127601\pi$
$173$	10.5000	+	18.1865i	0.798300	+	1.38270i	−0.122422	+	0.992478i	$0.539066\pi$
$174$		0			0
$175$	−4.00000	+	6.92820i	−0.302372	+	0.523723i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	14.0000			1.04061			0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000			1.04061			0.520306	+	0.853980i	$0.325818\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.500000	−	0.866025i	0.0367607	−	0.0636715i
$186$		0			0
$187$	3.00000	+	5.19615i	0.219382	+	0.379980i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−9.00000	−	15.5885i	−0.651217	−	1.12794i	−0.982828	−	0.184525i	$-0.940925\pi$
$191$	−9.00000	−	15.5885i	−0.651217	−	1.12794i	0.331611	−	0.943416i	$-0.392408\pi$
$192$		0			0
$193$	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
$197$	11.0000			0.783718			0.391859	−	0.920025i	$-0.371832\pi$
$197$	11.0000			0.783718			0.391859	+	0.920025i	$0.371832\pi$
$198$		0			0
$199$	−16.0000			−1.13421			−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000			−1.13421			−0.567105	+	0.823646i	$0.691937\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.00000	+	1.73205i	−0.0701862	+	0.121566i
$204$		0			0
$205$	1.00000	+	1.73205i	0.0698430	+	0.120972i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2.00000	+	3.46410i	0.138343	+	0.239617i
$210$		0			0
$211$	1.00000	−	1.73205i	0.0688428	−	0.119239i	−0.829549	−	0.558433i	$-0.811403\pi$
$211$	1.00000	−	1.73205i	0.0688428	−	0.119239i	0.898392	+	0.439194i	$0.144736\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	10.0000			0.681994
$216$		0			0
$217$	−16.0000			−1.08615
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.50000	−	2.59808i	0.100901	−	0.174766i
$222$		0			0
$223$	1.00000	+	1.73205i	0.0669650	+	0.115987i	0.897564	−	0.440884i	$-0.145335\pi$
$223$	1.00000	+	1.73205i	0.0669650	+	0.115987i	−0.830599	+	0.556871i	$0.812002\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−3.00000	−	5.19615i	−0.199117	−	0.344881i	0.749125	−	0.662428i	$-0.230474\pi$
$227$	−3.00000	−	5.19615i	−0.199117	−	0.344881i	−0.948242	+	0.317547i	$0.897141\pi$
$228$		0			0
$229$	3.50000	−	6.06218i	0.231287	−	0.400600i	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	3.50000	−	6.06218i	0.231287	−	0.400600i	0.958187	+	0.286143i	$0.0923732\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−15.0000			−0.982683			−0.491341	−	0.870967i	$-0.663493\pi$
$233$	−15.0000			−0.982683			−0.491341	+	0.870967i	$0.663493\pi$
$234$		0			0
$235$	−4.00000			−0.260931
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−6.00000	+	10.3923i	−0.388108	+	0.672222i	−0.992195	−	0.124696i	$-0.960204\pi$
$239$	−6.00000	+	10.3923i	−0.388108	+	0.672222i	0.604087	+	0.796918i	$0.293538\pi$
$240$		0			0
$241$	2.50000	+	4.33013i	0.161039	+	0.278928i	0.935242	−	0.354010i	$-0.115182\pi$
$241$	2.50000	+	4.33013i	0.161039	+	0.278928i	−0.774202	+	0.632938i	$0.781849\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.50000	−	2.59808i	−0.0958315	−	0.165985i
$246$		0			0
$247$	1.00000	−	1.73205i	0.0636285	−	0.110208i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−18.0000			−1.13615			−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000			−1.13615			−0.568075	+	0.822977i	$0.692312\pi$
$252$		0			0
$253$	12.0000			0.754434
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−14.5000	+	25.1147i	−0.904485	+	1.56661i	−0.0828783	+	0.996560i	$0.526411\pi$
$257$	−14.5000	+	25.1147i	−0.904485	+	1.56661i	−0.821607	+	0.570055i	$0.806922\pi$
$258$		0			0
$259$	1.00000	+	1.73205i	0.0621370	+	0.107624i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	6.00000	+	10.3923i	0.369976	+	0.640817i	0.989561	−	0.144112i	$-0.0460326\pi$
$263$	6.00000	+	10.3923i	0.369976	+	0.640817i	−0.619586	+	0.784929i	$0.712699\pi$
$264$		0			0
$265$	5.00000	−	8.66025i	0.307148	−	0.531995i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	15.0000			0.914566			0.457283	−	0.889321i	$-0.348823\pi$
$269$	15.0000			0.914566			0.457283	+	0.889321i	$0.348823\pi$
$270$		0			0
$271$	6.00000			0.364474			0.182237	−	0.983255i	$-0.441666\pi$
$271$	6.00000			0.364474			0.182237	+	0.983255i	$0.441666\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	4.00000	−	6.92820i	0.241209	−	0.417786i
$276$		0			0
$277$	1.00000	+	1.73205i	0.0600842	+	0.104069i	0.894503	−	0.447062i	$-0.147530\pi$
$277$	1.00000	+	1.73205i	0.0600842	+	0.104069i	−0.834419	+	0.551131i	$0.814196\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−4.50000	−	7.79423i	−0.268447	−	0.464965i	0.700014	−	0.714130i	$-0.253177\pi$
$281$	−4.50000	−	7.79423i	−0.268447	−	0.464965i	−0.968461	+	0.249165i	$0.919844\pi$
$282$		0			0
$283$	8.00000	−	13.8564i	0.475551	−	0.823678i	−0.524057	−	0.851683i	$-0.675582\pi$
$283$	8.00000	−	13.8564i	0.475551	−	0.823678i	0.999608	+	0.0280052i	$0.00891551\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−4.00000			−0.236113
$288$		0			0
$289$	−8.00000			−0.470588
$290$		0			0
$291$		0			0
$292$		0			0
$293$	10.5000	−	18.1865i	0.613417	−	1.06247i	−0.377244	−	0.926114i	$-0.623128\pi$
$293$	10.5000	−	18.1865i	0.613417	−	1.06247i	0.990660	−	0.136355i	$-0.0435386\pi$
$294$		0			0
$295$	2.00000	+	3.46410i	0.116445	+	0.201688i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3.00000	−	5.19615i	−0.173494	−	0.300501i
$300$		0			0
$301$	−10.0000	+	17.3205i	−0.576390	+	0.998337i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−9.00000			−0.515339
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$	−3.50000	−	6.06218i	−0.197832	−	0.342655i	0.749993	−	0.661445i	$-0.230057\pi$
$313$	−3.50000	−	6.06218i	−0.197832	−	0.342655i	−0.947825	+	0.318791i	$0.896723\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.50000	−	2.59808i	−0.0842484	−	0.145922i	0.820822	−	0.571184i	$-0.193516\pi$
$317$	−1.50000	−	2.59808i	−0.0842484	−	0.145922i	−0.905071	+	0.425261i	$0.860182\pi$
$318$		0			0
$319$	1.00000	−	1.73205i	0.0559893	−	0.0969762i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	6.00000			0.333849
$324$		0			0
$325$	−4.00000			−0.221880
$326$		0			0
$327$		0			0
$328$		0			0
$329$	4.00000	−	6.92820i	0.220527	−	0.381964i
$330$		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
$333$		0			0
$334$		0			0
$335$	7.00000	+	12.1244i	0.382451	+	0.662424i
$336$		0			0
$337$	−7.00000	+	12.1244i	−0.381314	+	0.660456i	−0.991250	−	0.131995i	$-0.957862\pi$
$337$	−7.00000	+	12.1244i	−0.381314	+	0.660456i	0.609936	+	0.792451i	$0.291195\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	16.0000			0.866449
$342$		0			0
$343$	20.0000			1.07990
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−5.00000	+	8.66025i	−0.268414	+	0.464907i	−0.968452	−	0.249198i	$-0.919833\pi$
$347$	−5.00000	+	8.66025i	−0.268414	+	0.464907i	0.700038	+	0.714105i	$0.253166\pi$
$348$		0			0
$349$	−17.0000	−	29.4449i	−0.909989	−	1.57615i	−0.814076	−	0.580758i	$-0.802756\pi$
$349$	−17.0000	−	29.4449i	−0.909989	−	1.57615i	−0.0959126	−	0.995390i	$-0.530577\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.00000	+	1.73205i	0.0532246	+	0.0921878i	0.891410	−	0.453197i	$-0.149717\pi$
$353$	1.00000	+	1.73205i	0.0532246	+	0.0921878i	−0.838186	+	0.545385i	$0.816383\pi$
$354$		0			0
$355$	5.00000	−	8.66025i	0.265372	−	0.459639i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−34.0000			−1.79445			−0.897226	−	0.441572i	$-0.854421\pi$
$359$	−34.0000			−1.79445			−0.897226	+	0.441572i	$0.854421\pi$
$360$		0			0
$361$	−15.0000			−0.789474
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−4.50000	+	7.79423i	−0.235541	+	0.407969i
$366$		0			0
$367$	−12.0000	−	20.7846i	−0.626395	−	1.08495i	−0.988269	−	0.152721i	$-0.951196\pi$
$367$	−12.0000	−	20.7846i	−0.626395	−	1.08495i	0.361874	−	0.932227i	$-0.382137\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	10.0000	+	17.3205i	0.519174	+	0.899236i
$372$		0			0
$373$	5.00000	−	8.66025i	0.258890	−	0.448411i	−0.707055	−	0.707159i	$-0.749977\pi$
$373$	5.00000	−	8.66025i	0.258890	−	0.448411i	0.965945	+	0.258748i	$0.0833099\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.00000			−0.0515026
$378$		0			0
$379$	−8.00000			−0.410932			−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000			−0.410932			−0.205466	+	0.978664i	$0.565871\pi$
$380$		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
$385$	−2.00000	−	3.46410i	−0.101929	−	0.176547i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−9.00000	−	15.5885i	−0.456318	−	0.790366i	0.542445	−	0.840091i	$-0.317499\pi$
$389$	−9.00000	−	15.5885i	−0.456318	−	0.790366i	−0.998763	+	0.0497253i	$0.984165\pi$
$390$		0			0
$391$	9.00000	−	15.5885i	0.455150	−	0.788342i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	10.0000			0.503155
$396$		0			0
$397$	13.0000			0.652451			0.326226	−	0.945292i	$-0.394223\pi$
$397$	13.0000			0.652451			0.326226	+	0.945292i	$0.394223\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−12.5000	+	21.6506i	−0.624220	+	1.08118i	0.364471	+	0.931215i	$0.381250\pi$
$401$	−12.5000	+	21.6506i	−0.624220	+	1.08118i	−0.988691	+	0.149966i	$0.952083\pi$
$402$		0			0
$403$	−4.00000	−	6.92820i	−0.199254	−	0.345118i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1.00000	−	1.73205i	−0.0495682	−	0.0858546i
$408$		0			0
$409$	14.5000	−	25.1147i	0.716979	−	1.24184i	−0.245212	−	0.969469i	$-0.578858\pi$
$409$	14.5000	−	25.1147i	0.716979	−	1.24184i	0.962191	−	0.272374i	$-0.0878089\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−8.00000			−0.393654
$414$		0			0
$415$	−12.0000			−0.589057
$416$		0			0
$417$		0			0
$418$		0			0
$419$	20.0000	−	34.6410i	0.977064	−	1.69232i	0.304115	−	0.952635i	$-0.401639\pi$
$419$	20.0000	−	34.6410i	0.977064	−	1.69232i	0.672949	−	0.739689i	$-0.265027\pi$
$420$		0			0
$421$	3.50000	+	6.06218i	0.170580	+	0.295452i	0.938623	−	0.344946i	$-0.112103\pi$
$421$	3.50000	+	6.06218i	0.170580	+	0.295452i	−0.768043	+	0.640398i	$0.778769\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−6.00000	−	10.3923i	−0.291043	−	0.504101i
$426$		0			0
$427$	9.00000	−	15.5885i	0.435541	−	0.754378i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	12.0000			0.578020			0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000			0.578020			0.289010	+	0.957326i	$0.406674\pi$
$432$		0			0
$433$	19.0000			0.913082			0.456541	−	0.889702i	$-0.349088\pi$
$433$	19.0000			0.913082			0.456541	+	0.889702i	$0.349088\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	6.00000	−	10.3923i	0.287019	−	0.497131i
$438$		0			0
$439$	−14.0000	−	24.2487i	−0.668184	−	1.15733i	−0.978412	−	0.206666i	$-0.933739\pi$
$439$	−14.0000	−	24.2487i	−0.668184	−	1.15733i	0.310228	−	0.950662i	$-0.399595\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−6.00000	−	10.3923i	−0.285069	−	0.493753i	0.687557	−	0.726130i	$-0.258683\pi$
$443$	−6.00000	−	10.3923i	−0.285069	−	0.493753i	−0.972626	+	0.232377i	$0.925350\pi$
$444$		0			0
$445$	−5.50000	+	9.52628i	−0.260725	+	0.451589i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−14.0000			−0.660701			−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000			−0.660701			−0.330350	+	0.943858i	$0.607167\pi$
$450$		0			0
$451$	4.00000			0.188353
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.00000	+	1.73205i	−0.0468807	+	0.0811998i
$456$		0			0
$457$	18.5000	+	32.0429i	0.865393	+	1.49891i	0.866656	+	0.498906i	$0.166265\pi$
$457$	18.5000	+	32.0429i	0.865393	+	1.49891i	−0.00126243	+	0.999999i	$0.500402\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	15.0000	+	25.9808i	0.698620	+	1.21004i	0.968945	+	0.247276i	$0.0795353\pi$
$461$	15.0000	+	25.9808i	0.698620	+	1.21004i	−0.270326	+	0.962769i	$0.587131\pi$
$462$		0			0
$463$	14.0000	−	24.2487i	0.650635	−	1.12693i	−0.332334	−	0.943162i	$-0.607836\pi$
$463$	14.0000	−	24.2487i	0.650635	−	1.12693i	0.982969	−	0.183771i	$-0.0588306\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−28.0000			−1.29569			−0.647843	−	0.761774i	$-0.724329\pi$
$467$	−28.0000			−1.29569			−0.647843	+	0.761774i	$0.724329\pi$
$468$		0			0
$469$	−28.0000			−1.29292
$470$		0			0
$471$		0			0
$472$		0			0
$473$	10.0000	−	17.3205i	0.459800	−	0.796398i
$474$		0			0
$475$	−4.00000	−	6.92820i	−0.183533	−	0.317888i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	21.0000	+	36.3731i	0.959514	+	1.66193i	0.723681	+	0.690134i	$0.242449\pi$
$479$	21.0000	+	36.3731i	0.959514	+	1.66193i	0.235833	+	0.971794i	$0.424218\pi$
$480$		0			0
$481$	−0.500000	+	0.866025i	−0.0227980	+	0.0394874i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2.00000			0.0908153
$486$		0			0
$487$	16.0000			0.725029			0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000			0.725029			0.362515	+	0.931978i	$0.381918\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	17.0000	−	29.4449i	0.767199	−	1.32883i	−0.171877	−	0.985118i	$-0.554983\pi$
$491$	17.0000	−	29.4449i	0.767199	−	1.32883i	0.939076	−	0.343710i	$-0.111684\pi$
$492$		0			0
$493$	−1.50000	−	2.59808i	−0.0675566	−	0.117011i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	10.0000	+	17.3205i	0.448561	+	0.776931i
$498$		0			0
$499$	3.00000	−	5.19615i	0.134298	−	0.232612i	−0.791031	−	0.611776i	$-0.790455\pi$
$499$	3.00000	−	5.19615i	0.134298	−	0.232612i	0.925329	+	0.379165i	$0.123789\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$	−6.00000			−0.266996
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−9.00000	+	15.5885i	−0.398918	+	0.690946i	−0.993593	−	0.113020i	$-0.963948\pi$
$509$	−9.00000	+	15.5885i	−0.398918	+	0.690946i	0.594675	+	0.803966i	$0.297281\pi$
$510$		0			0
$511$	−9.00000	−	15.5885i	−0.398137	−	0.689593i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	6.00000	+	10.3923i	0.264392	+	0.457940i
$516$		0			0
$517$	−4.00000	+	6.92820i	−0.175920	+	0.304702i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	18.0000			0.788594			0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000			0.788594			0.394297	+	0.918983i	$0.370988\pi$
$522$		0			0
$523$	30.0000			1.31181			0.655904	−	0.754844i	$-0.272288\pi$
$523$	30.0000			1.31181			0.655904	+	0.754844i	$0.272288\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	12.0000	−	20.7846i	0.522728	−	0.905392i
$528$		0			0
$529$	−6.50000	−	11.2583i	−0.282609	−	0.489493i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1.00000	−	1.73205i	−0.0433148	−	0.0750234i
$534$		0			0
$535$	8.00000	−	13.8564i	0.345870	−	0.599065i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−6.00000			−0.258438
$540$		0			0
$541$	13.0000			0.558914			0.279457	−	0.960158i	$-0.409846\pi$
$541$	13.0000			0.558914			0.279457	+	0.960158i	$0.409846\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	2.50000	−	4.33013i	0.107088	−	0.185482i
$546$		0			0
$547$	−6.00000	−	10.3923i	−0.256541	−	0.444343i	0.708772	−	0.705438i	$-0.249250\pi$
$547$	−6.00000	−	10.3923i	−0.256541	−	0.444343i	−0.965313	+	0.261095i	$0.915916\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1.00000	−	1.73205i	−0.0426014	−	0.0737878i
$552$		0			0
$553$	−10.0000	+	17.3205i	−0.425243	+	0.736543i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	11.0000			0.466085			0.233042	−	0.972467i	$-0.425132\pi$
$557$	11.0000			0.466085			0.233042	+	0.972467i	$0.425132\pi$
$558$		0			0
$559$	−10.0000			−0.422955
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−18.0000	+	31.1769i	−0.758610	+	1.31395i	0.184950	+	0.982748i	$0.440788\pi$
$563$	−18.0000	+	31.1769i	−0.758610	+	1.31395i	−0.943560	+	0.331202i	$0.892546\pi$
$564$		0			0
$565$	8.50000	+	14.7224i	0.357598	+	0.619377i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−2.50000	−	4.33013i	−0.104805	−	0.181528i	0.808853	−	0.588011i	$-0.200089\pi$
$569$	−2.50000	−	4.33013i	−0.104805	−	0.181528i	−0.913659	+	0.406482i	$0.866755\pi$
$570$		0			0
$571$	10.0000	−	17.3205i	0.418487	−	0.724841i	−0.577301	−	0.816532i	$-0.695894\pi$
$571$	10.0000	−	17.3205i	0.418487	−	0.724841i	0.995788	+	0.0916910i	$0.0292272\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−24.0000			−1.00087
$576$		0			0
$577$	27.0000			1.12402			0.562012	−	0.827129i	$-0.310027\pi$
$577$	27.0000			1.12402			0.562012	+	0.827129i	$0.310027\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	12.0000	−	20.7846i	0.497844	−	0.862291i
$582$		0			0
$583$	−10.0000	−	17.3205i	−0.414158	−	0.717342i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	15.0000	+	25.9808i	0.619116	+	1.07234i	0.989647	+	0.143521i	$0.0458424\pi$
$587$	15.0000	+	25.9808i	0.619116	+	1.07234i	−0.370531	+	0.928820i	$0.620824\pi$
$588$		0			0
$589$	8.00000	−	13.8564i	0.329634	−	0.570943i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−15.0000			−0.615976			−0.307988	−	0.951390i	$-0.599656\pi$
$593$	−15.0000			−0.615976			−0.307988	+	0.951390i	$0.599656\pi$
$594$		0			0
$595$	−6.00000			−0.245976
$596$		0			0
$597$		0			0
$598$		0			0
$599$	16.0000	−	27.7128i	0.653742	−	1.13231i	−0.328465	−	0.944516i	$-0.606531\pi$
$599$	16.0000	−	27.7128i	0.653742	−	1.13231i	0.982208	−	0.187799i	$-0.0601353\pi$
$600$		0			0
$601$	8.50000	+	14.7224i	0.346722	+	0.600541i	0.985665	−	0.168714i	$-0.0539613\pi$
$601$	8.50000	+	14.7224i	0.346722	+	0.600541i	−0.638943	+	0.769254i	$0.720628\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−3.50000	−	6.06218i	−0.142295	−	0.246463i
$606$		0			0
$607$	11.0000	−	19.0526i	0.446476	−	0.773320i	−0.551678	−	0.834058i	$-0.686012\pi$
$607$	11.0000	−	19.0526i	0.446476	−	0.773320i	0.998154	+	0.0607380i	$0.0193454\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	4.00000			0.161823
$612$		0			0
$613$	6.00000			0.242338			0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000			0.242338			0.121169	+	0.992632i	$0.461336\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	7.50000	−	12.9904i	0.301939	−	0.522973i	−0.674636	−	0.738150i	$-0.735700\pi$
$617$	7.50000	−	12.9904i	0.301939	−	0.522973i	0.976575	+	0.215177i	$0.0690329\pi$
$618$		0			0
$619$	−24.0000	−	41.5692i	−0.964641	−	1.67081i	−0.710575	−	0.703621i	$-0.751565\pi$
$619$	−24.0000	−	41.5692i	−0.964641	−	1.67081i	−0.254066	−	0.967187i	$-0.581768\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−11.0000	−	19.0526i	−0.440706	−	0.763325i
$624$		0			0
$625$	−5.50000	+	9.52628i	−0.220000	+	0.381051i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−3.00000			−0.119618
$630$		0			0
$631$	20.0000			0.796187			0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000			0.796187			0.398094	+	0.917345i	$0.369672\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1.00000	−	1.73205i	0.0396838	−	0.0687343i
$636$		0			0
$637$	1.50000	+	2.59808i	0.0594322	+	0.102940i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	17.5000	+	30.3109i	0.691208	+	1.19721i	0.971442	+	0.237276i	$0.0762547\pi$
$641$	17.5000	+	30.3109i	0.691208	+	1.19721i	−0.280234	+	0.959932i	$0.590412\pi$
$642$		0			0
$643$	18.0000	−	31.1769i	0.709851	−	1.22950i	−0.255062	−	0.966925i	$-0.582096\pi$
$643$	18.0000	−	31.1769i	0.709851	−	1.22950i	0.964912	−	0.262573i	$-0.0845709\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	6.00000			0.235884			0.117942	−	0.993020i	$-0.462370\pi$
$647$	6.00000			0.235884			0.117942	+	0.993020i	$0.462370\pi$
$648$		0			0
$649$	8.00000			0.314027
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−17.0000	+	29.4449i	−0.665261	+	1.15227i	0.313953	+	0.949439i	$0.398347\pi$
$653$	−17.0000	+	29.4449i	−0.665261	+	1.15227i	−0.979214	+	0.202828i	$0.934987\pi$
$654$		0			0
$655$	7.00000	+	12.1244i	0.273513	+	0.473738i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	15.0000	+	25.9808i	0.584317	+	1.01207i	0.994960	+	0.100271i	$0.0319709\pi$
$659$	15.0000	+	25.9808i	0.584317	+	1.01207i	−0.410643	+	0.911796i	$0.634696\pi$
$660$		0			0
$661$	15.5000	−	26.8468i	0.602880	−	1.04422i	−0.389503	−	0.921025i	$-0.627353\pi$
$661$	15.5000	−	26.8468i	0.602880	−	1.04422i	0.992383	−	0.123194i	$-0.0393136\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−4.00000			−0.155113
$666$		0			0
$667$	−6.00000			−0.232321
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−9.00000	+	15.5885i	−0.347441	+	0.601786i
$672$		0			0
$673$	−11.5000	−	19.9186i	−0.443292	−	0.767805i	0.554639	−	0.832091i	$-0.312856\pi$
$673$	−11.5000	−	19.9186i	−0.443292	−	0.767805i	−0.997932	+	0.0642860i	$0.979523\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−17.0000	−	29.4449i	−0.653363	−	1.13166i	−0.982301	−	0.187307i	$-0.940024\pi$
$677$	−17.0000	−	29.4449i	−0.653363	−	1.13166i	0.328938	−	0.944351i	$-0.393309\pi$
$678$		0			0
$679$	−2.00000	+	3.46410i	−0.0767530	+	0.132940i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−36.0000			−1.37750			−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000			−1.37750			−0.688751	+	0.724998i	$0.741841\pi$
$684$		0			0
$685$	−9.00000			−0.343872
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−5.00000	+	8.66025i	−0.190485	+	0.329929i
$690$		0			0
$691$	−5.00000	−	8.66025i	−0.190209	−	0.329452i	0.755110	−	0.655598i	$-0.227583\pi$
$691$	−5.00000	−	8.66025i	−0.190209	−	0.329452i	−0.945319	+	0.326146i	$0.894250\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	10.0000	+	17.3205i	0.379322	+	0.657004i
$696$		0			0
$697$	3.00000	−	5.19615i	0.113633	−	0.196818i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−45.0000			−1.69963			−0.849813	−	0.527084i	$-0.823285\pi$
$701$	−45.0000			−1.69963			−0.849813	+	0.527084i	$0.823285\pi$
$702$		0			0
$703$	−2.00000			−0.0754314
$704$		0			0
$705$		0			0
$706$		0			0
$707$	6.00000	−	10.3923i	0.225653	−	0.390843i
$708$		0			0
$709$	−0.500000	−	0.866025i	−0.0187779	−	0.0325243i	0.856484	−	0.516174i	$-0.172644\pi$
$709$	−0.500000	−	0.866025i	−0.0187779	−	0.0325243i	−0.875262	+	0.483650i	$0.839311\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−24.0000	−	41.5692i	−0.898807	−	1.55678i
$714$		0			0
$715$	1.00000	−	1.73205i	0.0373979	−	0.0647750i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−14.0000			−0.522112			−0.261056	−	0.965324i	$-0.584071\pi$
$719$	−14.0000			−0.522112			−0.261056	+	0.965324i	$0.584071\pi$
$720$		0			0
$721$	−24.0000			−0.893807
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2.00000	+	3.46410i	−0.0742781	+	0.128654i
$726$		0			0
$727$	3.00000	+	5.19615i	0.111264	+	0.192715i	0.916280	−	0.400538i	$-0.131177\pi$
$727$	3.00000	+	5.19615i	0.111264	+	0.192715i	−0.805016	+	0.593253i	$0.797843\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−15.0000	−	25.9808i	−0.554795	−	0.960933i
$732$		0			0
$733$	−7.00000	+	12.1244i	−0.258551	+	0.447823i	−0.965854	−	0.259087i	$-0.916578\pi$
$733$	−7.00000	+	12.1244i	−0.258551	+	0.447823i	0.707303	+	0.706910i	$0.249912\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	28.0000			1.03139
$738$		0			0
$739$	−8.00000			−0.294285			−0.147142	−	0.989115i	$-0.547008\pi$
$739$	−8.00000			−0.294285			−0.147142	+	0.989115i	$0.547008\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	12.0000	−	20.7846i	0.440237	−	0.762513i	−0.557470	−	0.830197i	$-0.688228\pi$
$743$	12.0000	−	20.7846i	0.440237	−	0.762513i	0.997707	+	0.0676840i	$0.0215610\pi$
$744$		0			0
$745$	−10.5000	−	18.1865i	−0.384690	−	0.666303i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	16.0000	+	27.7128i	0.584627	+	1.01260i
$750$		0			0
$751$	11.0000	−	19.0526i	0.401396	−	0.695238i	−0.592499	−	0.805571i	$-0.701859\pi$
$751$	11.0000	−	19.0526i	0.401396	−	0.695238i	0.993895	+	0.110333i	$0.0351919\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	12.0000			0.436725
$756$		0			0
$757$	−42.0000			−1.52652			−0.763258	−	0.646094i	$-0.776401\pi$
$757$	−42.0000			−1.52652			−0.763258	+	0.646094i	$0.776401\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	19.5000	−	33.7750i	0.706874	−	1.22434i	−0.259136	−	0.965841i	$-0.583438\pi$
$761$	19.5000	−	33.7750i	0.706874	−	1.22434i	0.966011	−	0.258502i	$-0.0832288\pi$
$762$		0			0
$763$	5.00000	+	8.66025i	0.181012	+	0.313522i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2.00000	−	3.46410i	−0.0722158	−	0.125081i
$768$		0			0
$769$	−15.5000	+	26.8468i	−0.558944	+	0.968120i	0.438641	+	0.898663i	$0.355460\pi$
$769$	−15.5000	+	26.8468i	−0.558944	+	0.968120i	−0.997585	+	0.0694574i	$0.977873\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−9.00000			−0.323708			−0.161854	−	0.986815i	$-0.551747\pi$
$773$	−9.00000			−0.323708			−0.161854	+	0.986815i	$0.551747\pi$
$774$		0			0
$775$	−32.0000			−1.14947
$776$		0			0
$777$		0			0
$778$		0			0
$779$	2.00000	−	3.46410i	0.0716574	−	0.124114i
$780$		0			0
$781$	−10.0000	−	17.3205i	−0.357828	−	0.619777i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	8.50000	+	14.7224i	0.303378	+	0.525466i
$786$		0			0
$787$	9.00000	−	15.5885i	0.320815	−	0.555668i	−0.659841	−	0.751405i	$-0.729376\pi$
$787$	9.00000	−	15.5885i	0.320815	−	0.555668i	0.980656	+	0.195737i	$0.0627098\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−34.0000			−1.20890
$792$		0			0
$793$	9.00000			0.319599
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−19.5000	+	33.7750i	−0.690725	+	1.19637i	0.280875	+	0.959744i	$0.409375\pi$
$797$	−19.5000	+	33.7750i	−0.690725	+	1.19637i	−0.971601	+	0.236627i	$0.923958\pi$
$798$		0			0
$799$	6.00000	+	10.3923i	0.212265	+	0.367653i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	9.00000	+	15.5885i	0.317603	+	0.550105i
$804$		0			0
$805$	−6.00000	+	10.3923i	−0.211472	+	0.366281i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	37.0000			1.30085			0.650425	−	0.759570i	$-0.274591\pi$
$809$	37.0000			1.30085			0.650425	+	0.759570i	$0.274591\pi$
$810$		0			0
$811$	16.0000			0.561836			0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000			0.561836			0.280918	+	0.959732i	$0.409361\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	2.00000	−	3.46410i	0.0700569	−	0.121342i
$816$		0			0
$817$	−10.0000	−	17.3205i	−0.349856	−	0.605968i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−11.5000	−	19.9186i	−0.401353	−	0.695163i	0.592537	−	0.805543i	$-0.298127\pi$
$821$	−11.5000	−	19.9186i	−0.401353	−	0.695163i	−0.993889	+	0.110380i	$0.964793\pi$
$822$		0			0
$823$	4.00000	−	6.92820i	0.139431	−	0.241502i	−0.787850	−	0.615867i	$-0.788806\pi$
$823$	4.00000	−	6.92820i	0.139431	−	0.241502i	0.927281	+	0.374365i	$0.122139\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−22.0000			−0.765015			−0.382507	−	0.923952i	$-0.624939\pi$
$827$	−22.0000			−0.765015			−0.382507	+	0.923952i	$0.624939\pi$
$828$		0			0
$829$	38.0000			1.31979			0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000			1.31979			0.659897	+	0.751356i	$0.270600\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−4.50000	+	7.79423i	−0.155916	+	0.270054i
$834$		0			0
$835$	−1.00000	−	1.73205i	−0.0346064	−	0.0599401i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−11.0000	−	19.0526i	−0.379762	−	0.657767i	0.611265	−	0.791426i	$-0.290661\pi$
$839$	−11.0000	−	19.0526i	−0.379762	−	0.657767i	−0.991027	+	0.133658i	$0.957328\pi$
$840$		0			0
$841$	14.0000	−	24.2487i	0.482759	−	0.836162i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	12.0000			0.412813
$846$		0			0
$847$	14.0000			0.481046
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−3.00000	+	5.19615i	−0.102839	+	0.178122i
$852$		0			0
$853$	21.0000	+	36.3731i	0.719026	+	1.24539i	0.961386	+	0.275204i	$0.0887453\pi$
$853$	21.0000	+	36.3731i	0.719026	+	1.24539i	−0.242360	+	0.970186i	$0.577921\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−6.50000	−	11.2583i	−0.222036	−	0.384577i	0.733390	−	0.679808i	$-0.237937\pi$
$857$	−6.50000	−	11.2583i	−0.222036	−	0.384577i	−0.955426	+	0.295231i	$0.904604\pi$
$858$		0			0
$859$	−22.0000	+	38.1051i	−0.750630	+	1.30013i	0.196887	+	0.980426i	$0.436917\pi$
$859$	−22.0000	+	38.1051i	−0.750630	+	1.30013i	−0.947518	+	0.319704i	$0.896417\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	46.0000			1.56586			0.782929	−	0.622111i	$-0.213725\pi$
$863$	46.0000			1.56586			0.782929	+	0.622111i	$0.213725\pi$
$864$		0			0
$865$	21.0000			0.714021
$866$		0			0
$867$		0			0
$868$		0			0
$869$	10.0000	−	17.3205i	0.339227	−	0.587558i
$870$		0			0
$871$	−7.00000	−	12.1244i	−0.237186	−	0.410818i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	9.00000	+	15.5885i	0.304256	+	0.526986i
$876$		0			0
$877$	17.5000	−	30.3109i	0.590933	−	1.02353i	−0.403174	−	0.915123i	$-0.632093\pi$
$877$	17.5000	−	30.3109i	0.590933	−	1.02353i	0.994107	−	0.108403i	$-0.0345736\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	42.0000			1.41502			0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000			1.41502			0.707508	+	0.706705i	$0.249819\pi$
$882$		0			0
$883$	20.0000			0.673054			0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000			0.673054			0.336527	+	0.941674i	$0.390748\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−27.0000	+	46.7654i	−0.906571	+	1.57023i	−0.0877772	+	0.996140i	$0.527976\pi$
$887$	−27.0000	+	46.7654i	−0.906571	+	1.57023i	−0.818794	+	0.574087i	$0.805357\pi$
$888$		0			0
$889$	2.00000	+	3.46410i	0.0670778	+	0.116182i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	4.00000	+	6.92820i	0.133855	+	0.231843i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−8.00000			−0.266815
$900$		0			0
$901$	−30.0000			−0.999445
$902$		0			0
$903$		0			0
$904$		0			0
$905$	7.00000	−	12.1244i	0.232688	−	0.403027i
$906$		0			0
$907$	4.00000	+	6.92820i	0.132818	+	0.230047i	0.924762	−	0.380547i	$-0.124264\pi$
$907$	4.00000	+	6.92820i	0.132818	+	0.230047i	−0.791944	+	0.610594i	$0.790931\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	15.0000	+	25.9808i	0.496972	+	0.860781i	0.999994	−	0.00349271i	$-0.00111177\pi$
$911$	15.0000	+	25.9808i	0.496972	+	0.860781i	−0.503022	+	0.864274i	$0.667778\pi$
$912$		0			0
$913$	−12.0000	+	20.7846i	−0.397142	+	0.687870i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−28.0000			−0.924641
$918$		0			0
$919$	−42.0000			−1.38545			−0.692726	−	0.721201i	$-0.743591\pi$
$919$	−42.0000			−1.38545			−0.692726	+	0.721201i	$0.743591\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−5.00000	+	8.66025i	−0.164577	+	0.285056i
$924$		0			0
$925$	2.00000	+	3.46410i	0.0657596	+	0.113899i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	13.5000	+	23.3827i	0.442921	+	0.767161i	0.997905	−	0.0646999i	$-0.0206090\pi$
$929$	13.5000	+	23.3827i	0.442921	+	0.767161i	−0.554984	+	0.831861i	$0.687276\pi$
$930$		0			0
$931$	−3.00000	+	5.19615i	−0.0983210	+	0.170297i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	6.00000			0.196221
$936$		0			0
$937$	−25.0000			−0.816714			−0.408357	−	0.912822i	$-0.633898\pi$
$937$	−25.0000			−0.816714			−0.408357	+	0.912822i	$0.633898\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−21.5000	+	37.2391i	−0.700880	+	1.21396i	0.267278	+	0.963619i	$0.413876\pi$
$941$	−21.5000	+	37.2391i	−0.700880	+	1.21396i	−0.968158	+	0.250340i	$0.919458\pi$
$942$		0			0
$943$	−6.00000	−	10.3923i	−0.195387	−	0.338420i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−27.0000	−	46.7654i	−0.877382	−	1.51967i	−0.854203	−	0.519939i	$-0.825955\pi$
$947$	−27.0000	−	46.7654i	−0.877382	−	1.51967i	−0.0231788	−	0.999731i	$-0.507379\pi$
$948$		0			0
$949$	4.50000	−	7.79423i	0.146076	−	0.253011i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	21.0000			0.680257			0.340128	−	0.940379i	$-0.389529\pi$
$953$	21.0000			0.680257			0.340128	+	0.940379i	$0.389529\pi$
$954$		0			0
$955$	−18.0000			−0.582466
$956$		0			0
$957$		0			0
$958$		0			0
$959$	9.00000	−	15.5885i	0.290625	−	0.503378i
$960$		0			0
$961$	−16.5000	−	28.5788i	−0.532258	−	0.921898i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−4.50000	−	7.79423i	−0.144860	−	0.250905i
$966$		0			0
$967$	−9.00000	+	15.5885i	−0.289420	+	0.501291i	−0.973672	−	0.227956i	$-0.926796\pi$
$967$	−9.00000	+	15.5885i	−0.289420	+	0.501291i	0.684251	+	0.729247i	$0.260129\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	38.0000			1.21948			0.609739	−	0.792602i	$-0.291274\pi$
$971$	38.0000			1.21948			0.609739	+	0.792602i	$0.291274\pi$
$972$		0			0
$973$	−40.0000			−1.28234
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1.00000	+	1.73205i	−0.0319928	+	0.0554132i	−0.881579	−	0.472037i	$-0.843519\pi$
$977$	−1.00000	+	1.73205i	−0.0319928	+	0.0554132i	0.849586	+	0.527451i	$0.176852\pi$
$978$		0			0
$979$	11.0000	+	19.0526i	0.351562	+	0.608922i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−12.0000	−	20.7846i	−0.382741	−	0.662926i	0.608712	−	0.793391i	$-0.291686\pi$
$983$	−12.0000	−	20.7846i	−0.382741	−	0.662926i	−0.991453	+	0.130465i	$0.958353\pi$
$984$		0			0
$985$	5.50000	−	9.52628i	0.175245	−	0.303533i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−60.0000			−1.90789
$990$		0			0
$991$	38.0000			1.20711			0.603555	−	0.797321i	$-0.293750\pi$
$991$	38.0000			1.20711			0.603555	+	0.797321i	$0.293750\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−8.00000	+	13.8564i	−0.253617	+	0.439278i
$996$		0			0
$997$	7.50000	+	12.9904i	0.237527	+	0.411409i	0.960004	−	0.279986i	$-0.0903297\pi$
$997$	7.50000	+	12.9904i	0.237527	+	0.411409i	−0.722477	+	0.691395i	$0.756996\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2592.2.i.o.1729.1		2
3.2	odd	2		2592.2.i.k.1729.1		2
4.3	odd	2		2592.2.i.n.1729.1		2
9.2	odd	6		2592.2.i.k.865.1		2
9.4	even	3		2592.2.a.c.1.1	✓	1
9.5	odd	6		2592.2.a.e.1.1	yes	1
9.7	even	3	inner	2592.2.i.o.865.1		2
12.11	even	2		2592.2.i.j.1729.1		2
36.7	odd	6		2592.2.i.n.865.1		2
36.11	even	6		2592.2.i.j.865.1		2
36.23	even	6		2592.2.a.f.1.1	yes	1
36.31	odd	6		2592.2.a.d.1.1	yes	1
72.5	odd	6		5184.2.a.j.1.1		1
72.13	even	6		5184.2.a.t.1.1		1
72.59	even	6		5184.2.a.m.1.1		1
72.67	odd	6		5184.2.a.w.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2592.2.a.c.1.1	✓	1	9.4	even	3
2592.2.a.d.1.1	yes	1	36.31	odd	6
2592.2.a.e.1.1	yes	1	9.5	odd	6
2592.2.a.f.1.1	yes	1	36.23	even	6
2592.2.i.j.865.1		2	36.11	even	6
2592.2.i.j.1729.1		2	12.11	even	2
2592.2.i.k.865.1		2	9.2	odd	6
2592.2.i.k.1729.1		2	3.2	odd	2
2592.2.i.n.865.1		2	36.7	odd	6
2592.2.i.n.1729.1		2	4.3	odd	2
2592.2.i.o.865.1		2	9.7	even	3	inner
2592.2.i.o.1729.1		2	1.1	even	1	trivial
5184.2.a.j.1.1		1	72.5	odd	6
5184.2.a.m.1.1		1	72.59	even	6
5184.2.a.t.1.1		1	72.13	even	6
5184.2.a.w.1.1		1	72.67	odd	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:	\( N \)	\(=\)	\( 2592 = 2^{5} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2592.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{5} +(1.00000 + 1.73205i) q^{7} +(-1.00000 - 1.73205i) q^{11} +(-0.500000 + 0.866025i) q^{13} -3.00000 q^{17} -2.00000 q^{19} +(-3.00000 + 5.19615i) q^{23} +(2.00000 + 3.46410i) q^{25} +(0.500000 + 0.866025i) q^{29} +(-4.00000 + 6.92820i) q^{31} +2.00000 q^{35} +1.00000 q^{37} +(-1.00000 + 1.73205i) q^{41} +(5.00000 + 8.66025i) q^{43} +(-2.00000 - 3.46410i) q^{47} +(1.50000 - 2.59808i) q^{49} +10.0000 q^{53} -2.00000 q^{55} +(-2.00000 + 3.46410i) q^{59} +(-4.50000 - 7.79423i) q^{61} +(0.500000 + 0.866025i) q^{65} +(-7.00000 + 12.1244i) q^{67} +10.0000 q^{71} -9.00000 q^{73} +(2.00000 - 3.46410i) q^{77} +(5.00000 + 8.66025i) q^{79} +(-6.00000 - 10.3923i) q^{83} +(-1.50000 + 2.59808i) q^{85} -11.0000 q^{89} -2.00000 q^{91} +(-1.00000 + 1.73205i) q^{95} +(1.00000 + 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{5} + 2 q^{7} - 2 q^{11} - q^{13} - 6 q^{17} - 4 q^{19} - 6 q^{23} + 4 q^{25} + q^{29} - 8 q^{31} + 4 q^{35} + 2 q^{37} - 2 q^{41} + 10 q^{43} - 4 q^{47} + 3 q^{49} + 20 q^{53} - 4 q^{55} - 4 q^{59}+ \cdots + 2 q^{97}+O(q^{100}) \) 2 * q + q^5 + 2 * q^7 - 2 * q^11 - q^13 - 6 * q^17 - 4 * q^19 - 6 * q^23 + 4 * q^25 + q^29 - 8 * q^31 + 4 * q^35 + 2 * q^37 - 2 * q^41 + 10 * q^43 - 4 * q^47 + 3 * q^49 + 20 * q^53 - 4 * q^55 - 4 * q^59 - 9 * q^61 + q^65 - 14 * q^67 + 20 * q^71 - 18 * q^73 + 4 * q^77 + 10 * q^79 - 12 * q^83 - 3 * q^85 - 22 * q^89 - 4 * q^91 - 2 * q^95 + 2 * q^97

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