LMFDB - Embedded newform 2592.2.i.v.1729.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2592,2,Mod(865,2592)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 864)
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.v.865.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.73205i) q^{5} +(1.50000 + 2.59808i) q^{7} +O(q^{10})$	$q+(1.00000 - 1.73205i) q^{5} +(1.50000 + 2.59808i) q^{7} +(-3.00000 - 5.19615i) q^{11} +(1.50000 - 2.59808i) q^{13} +2.00000 q^{17} +3.00000 q^{19} +(-3.00000 + 5.19615i) q^{23} +(0.500000 + 0.866025i) q^{25} +(-4.00000 - 6.92820i) q^{29} +6.00000 q^{35} +7.00000 q^{37} +(4.00000 - 6.92820i) q^{41} +(-6.00000 - 10.3923i) q^{43} +(-3.00000 - 5.19615i) q^{47} +(-1.00000 + 1.73205i) q^{49} -4.00000 q^{53} -12.0000 q^{55} +(-3.00000 + 5.19615i) q^{59} +(0.500000 + 0.866025i) q^{61} +(-3.00000 - 5.19615i) q^{65} +(-1.50000 + 2.59808i) q^{67} +12.0000 q^{71} -15.0000 q^{73} +(9.00000 - 15.5885i) q^{77} +(4.50000 + 7.79423i) q^{79} +(6.00000 + 10.3923i) q^{83} +(2.00000 - 3.46410i) q^{85} +10.0000 q^{89} +9.00000 q^{91} +(3.00000 - 5.19615i) q^{95} +(-4.50000 - 7.79423i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} + 3 q^{7}+O(q^{10}) $ 2 * q + 2 * q^5 + 3 * q^7	$ 2 q + 2 q^{5} + 3 q^{7} - 6 q^{11} + 3 q^{13} + 4 q^{17} + 6 q^{19} - 6 q^{23} + q^{25} - 8 q^{29} + 12 q^{35} + 14 q^{37} + 8 q^{41} - 12 q^{43} - 6 q^{47} - 2 q^{49} - 8 q^{53} - 24 q^{55} - 6 q^{59} + q^{61} - 6 q^{65} - 3 q^{67} + 24 q^{71} - 30 q^{73} + 18 q^{77} + 9 q^{79} + 12 q^{83} + 4 q^{85} + 20 q^{89} + 18 q^{91} + 6 q^{95} - 9 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 + 3 * q^7 - 6 * q^11 + 3 * q^13 + 4 * q^17 + 6 * q^19 - 6 * q^23 + q^25 - 8 * q^29 + 12 * q^35 + 14 * q^37 + 8 * q^41 - 12 * q^43 - 6 * q^47 - 2 * q^49 - 8 * q^53 - 24 * q^55 - 6 * q^59 + q^61 - 6 * q^65 - 3 * q^67 + 24 * q^71 - 30 * q^73 + 18 * q^77 + 9 * q^79 + 12 * q^83 + 4 * q^85 + 20 * q^89 + 18 * q^91 + 6 * q^95 - 9 * q^97

Display 100 coefficients 10 coefficients

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
$5$	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
$7$	1.50000	+	2.59808i	0.566947	+	0.981981i	0.996866	+	0.0791130i	$0.0252088\pi$
$7$	1.50000	+	2.59808i	0.566947	+	0.981981i	−0.429919	+	0.902867i	$0.641458\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.00000	−	5.19615i	−0.904534	−	1.56670i	−0.821541	−	0.570149i	$-0.806886\pi$
$11$	−3.00000	−	5.19615i	−0.904534	−	1.56670i	−0.0829925	−	0.996550i	$-0.526448\pi$
$12$		0			0
$13$	1.50000	−	2.59808i	0.416025	−	0.720577i	−0.579510	−	0.814965i	$-0.696756\pi$
$13$	1.50000	−	2.59808i	0.416025	−	0.720577i	0.995535	+	0.0943882i	$0.0300895\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.00000			0.485071			0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000			0.485071			0.242536	+	0.970143i	$0.422021\pi$
$18$		0			0
$19$	3.00000			0.688247			0.344124	−	0.938924i	$-0.388176\pi$
$19$	3.00000			0.688247			0.344124	+	0.938924i	$0.388176\pi$
$20$		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
$25$	0.500000	+	0.866025i	0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.00000	−	6.92820i	−0.742781	−	1.28654i	−0.951224	−	0.308500i	$-0.900173\pi$
$29$	−4.00000	−	6.92820i	−0.742781	−	1.28654i	0.208443	−	0.978035i	$-0.433160\pi$
$30$		0			0
$31$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$31$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	6.00000			1.01419
$36$		0			0
$37$	7.00000			1.15079			0.575396	−	0.817875i	$-0.304848\pi$
$37$	7.00000			1.15079			0.575396	+	0.817875i	$0.304848\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.00000	−	6.92820i	0.624695	−	1.08200i	−0.363905	−	0.931436i	$-0.618557\pi$
$41$	4.00000	−	6.92820i	0.624695	−	1.08200i	0.988600	−	0.150567i	$-0.0481100\pi$
$42$		0			0
$43$	−6.00000	−	10.3923i	−0.914991	−	1.58481i	−0.806914	−	0.590669i	$-0.798864\pi$
$43$	−6.00000	−	10.3923i	−0.914991	−	1.58481i	−0.108078	−	0.994142i	$-0.534469\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.00000	−	5.19615i	−0.437595	−	0.757937i	0.559908	−	0.828554i	$-0.310836\pi$
$47$	−3.00000	−	5.19615i	−0.437595	−	0.757937i	−0.997503	+	0.0706177i	$0.977503\pi$
$48$		0			0
$49$	−1.00000	+	1.73205i	−0.142857	+	0.247436i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.00000			−0.549442			−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000			−0.549442			−0.274721	+	0.961524i	$0.588586\pi$
$54$		0			0
$55$	−12.0000			−1.61808
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−3.00000	+	5.19615i	−0.390567	+	0.676481i	−0.992524	−	0.122047i	$-0.961054\pi$
$59$	−3.00000	+	5.19615i	−0.390567	+	0.676481i	0.601958	+	0.798528i	$0.294388\pi$
$60$		0			0
$61$	0.500000	+	0.866025i	0.0640184	+	0.110883i	0.896258	−	0.443533i	$-0.146275\pi$
$61$	0.500000	+	0.866025i	0.0640184	+	0.110883i	−0.832240	+	0.554416i	$0.812942\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.00000	−	5.19615i	−0.372104	−	0.644503i
$66$		0			0
$67$	−1.50000	+	2.59808i	−0.183254	+	0.317406i	−0.942987	−	0.332830i	$-0.891996\pi$
$67$	−1.50000	+	2.59808i	−0.183254	+	0.317406i	0.759733	+	0.650236i	$0.225330\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	12.0000			1.42414			0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000			1.42414			0.712069	+	0.702109i	$0.247758\pi$
$72$		0			0
$73$	−15.0000			−1.75562			−0.877809	−	0.479012i	$-0.840995\pi$
$73$	−15.0000			−1.75562			−0.877809	+	0.479012i	$0.840995\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	9.00000	−	15.5885i	1.02565	−	1.77647i
$78$		0			0
$79$	4.50000	+	7.79423i	0.506290	+	0.876919i	0.999974	+	0.00727784i	$0.00231663\pi$
$79$	4.50000	+	7.79423i	0.506290	+	0.876919i	−0.493684	+	0.869641i	$0.664350\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	6.00000	+	10.3923i	0.658586	+	1.14070i	0.980982	+	0.194099i	$0.0621783\pi$
$83$	6.00000	+	10.3923i	0.658586	+	1.14070i	−0.322396	+	0.946605i	$0.604488\pi$
$84$		0			0
$85$	2.00000	−	3.46410i	0.216930	−	0.375735i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	10.0000			1.06000			0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000			1.06000			0.529999	+	0.847998i	$0.322192\pi$
$90$		0			0
$91$	9.00000			0.943456
$92$		0			0
$93$		0			0
$94$		0			0
$95$	3.00000	−	5.19615i	0.307794	−	0.533114i
$96$		0			0
$97$	−4.50000	−	7.79423i	−0.456906	−	0.791384i	0.541890	−	0.840450i	$-0.317709\pi$
$97$	−4.50000	−	7.79423i	−0.456906	−	0.791384i	−0.998796	+	0.0490655i	$0.984376\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−8.00000	−	13.8564i	−0.796030	−	1.37876i	−0.922183	−	0.386753i	$-0.873597\pi$
$101$	−8.00000	−	13.8564i	−0.796030	−	1.37876i	0.126153	−	0.992011i	$-0.459737\pi$
$102$		0			0
$103$	7.50000	−	12.9904i	0.738997	−	1.27998i	−0.213950	−	0.976845i	$-0.568633\pi$
$103$	7.50000	−	12.9904i	0.738997	−	1.27998i	0.952947	−	0.303136i	$-0.0980336\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6.00000			−0.580042			−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000			−0.580042			−0.290021	+	0.957020i	$0.593662\pi$
$108$		0			0
$109$	6.00000			0.574696			0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000			0.574696			0.287348	+	0.957826i	$0.407226\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	7.00000	−	12.1244i	0.658505	−	1.14056i	−0.322498	−	0.946570i	$-0.604523\pi$
$113$	7.00000	−	12.1244i	0.658505	−	1.14056i	0.981003	−	0.193993i	$-0.0621440\pi$
$114$		0			0
$115$	6.00000	+	10.3923i	0.559503	+	0.969087i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	3.00000	+	5.19615i	0.275010	+	0.476331i
$120$		0			0
$121$	−12.5000	+	21.6506i	−1.13636	+	1.96824i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	12.0000			1.07331
$126$		0			0
$127$	12.0000			1.06483			0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000			1.06483			0.532414	+	0.846484i	$0.321285\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$131$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$132$		0			0
$133$	4.50000	+	7.79423i	0.390199	+	0.675845i
$134$		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
$139$	−10.5000	+	18.1865i	−0.890598	+	1.54256i	−0.0514389	+	0.998676i	$0.516381\pi$
$139$	−10.5000	+	18.1865i	−0.890598	+	1.54256i	−0.839159	+	0.543885i	$0.816953\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−18.0000			−1.50524
$144$		0			0
$145$	−16.0000			−1.32873
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2.00000	−	3.46410i	0.163846	−	0.283790i	−0.772399	−	0.635138i	$-0.780943\pi$
$149$	2.00000	−	3.46410i	0.163846	−	0.283790i	0.936245	+	0.351348i	$0.114277\pi$
$150$		0			0
$151$	−1.50000	−	2.59808i	−0.122068	−	0.211428i	0.798515	−	0.601975i	$-0.205619\pi$
$151$	−1.50000	−	2.59808i	−0.122068	−	0.211428i	−0.920583	+	0.390547i	$0.872286\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	7.00000	−	12.1244i	0.558661	−	0.967629i	−0.438948	−	0.898513i	$-0.644649\pi$
$157$	7.00000	−	12.1244i	0.558661	−	0.967629i	0.997609	−	0.0691164i	$-0.0220180\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−18.0000			−1.41860
$162$		0			0
$163$	−3.00000			−0.234978			−0.117489	−	0.993074i	$-0.537485\pi$
$163$	−3.00000			−0.234978			−0.117489	+	0.993074i	$0.537485\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	9.00000	−	15.5885i	0.696441	−	1.20627i	−0.273252	−	0.961943i	$-0.588099\pi$
$167$	9.00000	−	15.5885i	0.696441	−	1.20627i	0.969693	−	0.244328i	$-0.0785675\pi$
$168$		0			0
$169$	2.00000	+	3.46410i	0.153846	+	0.266469i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	4.00000	+	6.92820i	0.304114	+	0.526742i	0.977064	−	0.212947i	$-0.0683062\pi$
$173$	4.00000	+	6.92820i	0.304114	+	0.526742i	−0.672949	+	0.739689i	$0.734973\pi$
$174$		0			0
$175$	−1.50000	+	2.59808i	−0.113389	+	0.196396i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−12.0000			−0.896922			−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$		0			0
$181$	9.00000			0.668965			0.334482	−	0.942402i	$-0.391439\pi$
$181$	9.00000			0.668965			0.334482	+	0.942402i	$0.391439\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	7.00000	−	12.1244i	0.514650	−	0.891400i
$186$		0			0
$187$	−6.00000	−	10.3923i	−0.438763	−	0.759961i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	3.00000	+	5.19615i	0.217072	+	0.375980i	0.953912	−	0.300088i	$-0.0970159\pi$
$191$	3.00000	+	5.19615i	0.217072	+	0.375980i	−0.736839	+	0.676068i	$0.763683\pi$
$192$		0			0
$193$	2.50000	−	4.33013i	0.179954	−	0.311689i	−0.761911	−	0.647682i	$-0.775738\pi$
$193$	2.50000	−	4.33013i	0.179954	−	0.311689i	0.941865	+	0.335993i	$0.109072\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2.00000			−0.142494			−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000			−0.142494			−0.0712470	+	0.997459i	$0.522698\pi$
$198$		0			0
$199$	−15.0000			−1.06332			−0.531661	−	0.846957i	$-0.678432\pi$
$199$	−15.0000			−1.06332			−0.531661	+	0.846957i	$0.678432\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	12.0000	−	20.7846i	0.842235	−	1.45879i
$204$		0			0
$205$	−8.00000	−	13.8564i	−0.558744	−	0.967773i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−9.00000	−	15.5885i	−0.622543	−	1.07828i
$210$		0			0
$211$	7.50000	−	12.9904i	0.516321	−	0.894295i	−0.483499	−	0.875345i	$-0.660634\pi$
$211$	7.50000	−	12.9904i	0.516321	−	0.894295i	0.999820	−	0.0189499i	$-0.00603229\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−24.0000			−1.63679
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	3.00000	−	5.19615i	0.201802	−	0.349531i
$222$		0			0
$223$	−12.0000	−	20.7846i	−0.803579	−	1.39184i	−0.917246	−	0.398321i	$-0.869593\pi$
$223$	−12.0000	−	20.7846i	−0.803579	−	1.39184i	0.113666	−	0.993519i	$-0.463740\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−6.00000	−	10.3923i	−0.398234	−	0.689761i	0.595274	−	0.803523i	$-0.297043\pi$
$227$	−6.00000	−	10.3923i	−0.398234	−	0.689761i	−0.993508	+	0.113761i	$0.963710\pi$
$228$		0			0
$229$	3.00000	−	5.19615i	0.198246	−	0.343371i	−0.749714	−	0.661762i	$-0.769809\pi$
$229$	3.00000	−	5.19615i	0.198246	−	0.343371i	0.947960	+	0.318390i	$0.103142\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	8.00000			0.524097			0.262049	−	0.965055i	$-0.415602\pi$
$233$	8.00000			0.524097			0.262049	+	0.965055i	$0.415602\pi$
$234$		0			0
$235$	−12.0000			−0.782794
$236$		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
$241$	1.50000	+	2.59808i	0.0966235	+	0.167357i	0.910285	−	0.413982i	$-0.135862\pi$
$241$	1.50000	+	2.59808i	0.0966235	+	0.167357i	−0.813662	+	0.581339i	$0.802529\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2.00000	+	3.46410i	0.127775	+	0.221313i
$246$		0			0
$247$	4.50000	−	7.79423i	0.286328	−	0.495935i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$	36.0000			2.26330
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−10.0000	+	17.3205i	−0.623783	+	1.08042i	0.364992	+	0.931011i	$0.381072\pi$
$257$	−10.0000	+	17.3205i	−0.623783	+	1.08042i	−0.988775	+	0.149413i	$0.952262\pi$
$258$		0			0
$259$	10.5000	+	18.1865i	0.652438	+	1.13006i
$260$		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
$265$	−4.00000	+	6.92820i	−0.245718	+	0.425596i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	10.0000			0.609711			0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000			0.609711			0.304855	+	0.952399i	$0.401392\pi$
$270$		0			0
$271$	15.0000			0.911185			0.455593	−	0.890188i	$-0.349427\pi$
$271$	15.0000			0.911185			0.455593	+	0.890188i	$0.349427\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3.00000	−	5.19615i	0.180907	−	0.313340i
$276$		0			0
$277$	9.00000	+	15.5885i	0.540758	+	0.936620i	0.998861	+	0.0477206i	$0.0151957\pi$
$277$	9.00000	+	15.5885i	0.540758	+	0.936620i	−0.458103	+	0.888899i	$0.651471\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	4.00000	+	6.92820i	0.238620	+	0.413302i	0.960319	−	0.278906i	$-0.0899716\pi$
$281$	4.00000	+	6.92820i	0.238620	+	0.413302i	−0.721699	+	0.692207i	$0.756638\pi$
$282$		0			0
$283$	6.00000	−	10.3923i	0.356663	−	0.617758i	−0.630738	−	0.775996i	$-0.717248\pi$
$283$	6.00000	−	10.3923i	0.356663	−	0.617758i	0.987401	+	0.158237i	$0.0505811\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	24.0000			1.41668
$288$		0			0
$289$	−13.0000			−0.764706
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.00000	+	1.73205i	−0.0584206	+	0.101187i	−0.893757	−	0.448552i	$-0.851940\pi$
$293$	−1.00000	+	1.73205i	−0.0584206	+	0.101187i	0.835336	+	0.549740i	$0.185273\pi$
$294$		0			0
$295$	6.00000	+	10.3923i	0.349334	+	0.605063i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	9.00000	+	15.5885i	0.520483	+	0.901504i
$300$		0			0
$301$	18.0000	−	31.1769i	1.03750	−	1.79701i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2.00000			0.114520
$306$		0			0
$307$	−12.0000			−0.684876			−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000			−0.684876			−0.342438	+	0.939540i	$0.611253\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−9.00000	+	15.5885i	−0.510343	+	0.883940i	0.489585	+	0.871956i	$0.337148\pi$
$311$	−9.00000	+	15.5885i	−0.510343	+	0.883940i	−0.999928	+	0.0119847i	$0.996185\pi$
$312$		0			0
$313$	0.500000	+	0.866025i	0.0282617	+	0.0489506i	0.879810	−	0.475325i	$-0.157669\pi$
$313$	0.500000	+	0.866025i	0.0282617	+	0.0489506i	−0.851549	+	0.524276i	$0.824336\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−2.00000	−	3.46410i	−0.112331	−	0.194563i	0.804379	−	0.594117i	$-0.202498\pi$
$317$	−2.00000	−	3.46410i	−0.112331	−	0.194563i	−0.916710	+	0.399554i	$0.869165\pi$
$318$		0			0
$319$	−24.0000	+	41.5692i	−1.34374	+	2.32743i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	6.00000			0.333849
$324$		0			0
$325$	3.00000			0.166410
$326$		0			0
$327$		0			0
$328$		0			0
$329$	9.00000	−	15.5885i	0.496186	−	0.859419i
$330$		0			0
$331$	1.50000	+	2.59808i	0.0824475	+	0.142803i	0.904301	−	0.426896i	$-0.140393\pi$
$331$	1.50000	+	2.59808i	0.0824475	+	0.142803i	−0.821853	+	0.569699i	$0.807060\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	3.00000	+	5.19615i	0.163908	+	0.283896i
$336$		0			0
$337$	−4.50000	+	7.79423i	−0.245131	+	0.424579i	−0.962168	−	0.272456i	$-0.912164\pi$
$337$	−4.50000	+	7.79423i	−0.245131	+	0.424579i	0.717038	+	0.697034i	$0.245498\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	15.0000			0.809924
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−6.00000	+	10.3923i	−0.322097	+	0.557888i	−0.980921	−	0.194409i	$-0.937721\pi$
$347$	−6.00000	+	10.3923i	−0.322097	+	0.557888i	0.658824	+	0.752297i	$0.271054\pi$
$348$		0			0
$349$	0.500000	+	0.866025i	0.0267644	+	0.0463573i	0.879097	−	0.476642i	$-0.158146\pi$
$349$	0.500000	+	0.866025i	0.0267644	+	0.0463573i	−0.852333	+	0.523000i	$0.824813\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	10.0000	+	17.3205i	0.532246	+	0.921878i	0.999291	+	0.0376440i	$0.0119853\pi$
$353$	10.0000	+	17.3205i	0.532246	+	0.921878i	−0.467045	+	0.884234i	$0.654681\pi$
$354$		0			0
$355$	12.0000	−	20.7846i	0.636894	−	1.10313i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	18.0000			0.950004			0.475002	−	0.879985i	$-0.342447\pi$
$359$	18.0000			0.950004			0.475002	+	0.879985i	$0.342447\pi$
$360$		0			0
$361$	−10.0000			−0.526316
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−15.0000	+	25.9808i	−0.785136	+	1.35990i
$366$		0			0
$367$	1.50000	+	2.59808i	0.0782994	+	0.135618i	0.902516	−	0.430656i	$-0.141718\pi$
$367$	1.50000	+	2.59808i	0.0782994	+	0.135618i	−0.824217	+	0.566274i	$0.808384\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−6.00000	−	10.3923i	−0.311504	−	0.539542i
$372$		0			0
$373$	−11.5000	+	19.9186i	−0.595447	+	1.03135i	0.398036	+	0.917370i	$0.369692\pi$
$373$	−11.5000	+	19.9186i	−0.595447	+	1.03135i	−0.993484	+	0.113975i	$0.963641\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−24.0000			−1.23606
$378$		0			0
$379$	27.0000			1.38690			0.693448	−	0.720506i	$-0.256091\pi$
$379$	27.0000			1.38690			0.693448	+	0.720506i	$0.256091\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$		0			0
$385$	−18.0000	−	31.1769i	−0.917365	−	1.58892i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	17.0000	+	29.4449i	0.861934	+	1.49291i	0.870059	+	0.492947i	$0.164080\pi$
$389$	17.0000	+	29.4449i	0.861934	+	1.49291i	−0.00812520	+	0.999967i	$0.502586\pi$
$390$		0			0
$391$	−6.00000	+	10.3923i	−0.303433	+	0.525561i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	18.0000			0.905678
$396$		0			0
$397$	2.00000			0.100377			0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000			0.100377			0.0501886	+	0.998740i	$0.484018\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−10.0000	+	17.3205i	−0.499376	+	0.864945i	−1.00000		0.000720188i	$-0.999771\pi$
$401$	−10.0000	+	17.3205i	−0.499376	+	0.864945i	0.500624	+	0.865665i	$0.333104\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−21.0000	−	36.3731i	−1.04093	−	1.80295i
$408$		0			0
$409$	−10.5000	+	18.1865i	−0.519192	+	0.899266i	0.480560	+	0.876962i	$0.340434\pi$
$409$	−10.5000	+	18.1865i	−0.519192	+	0.899266i	−0.999751	+	0.0223042i	$0.992900\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−18.0000			−0.885722
$414$		0			0
$415$	24.0000			1.17811
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−15.0000	+	25.9808i	−0.732798	+	1.26924i	0.222885	+	0.974845i	$0.428453\pi$
$419$	−15.0000	+	25.9808i	−0.732798	+	1.26924i	−0.955683	+	0.294398i	$0.904881\pi$
$420$		0			0
$421$	7.50000	+	12.9904i	0.365528	+	0.633112i	0.988861	−	0.148844i	$-0.0475552\pi$
$421$	7.50000	+	12.9904i	0.365528	+	0.633112i	−0.623333	+	0.781956i	$0.714222\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.00000	+	1.73205i	0.0485071	+	0.0840168i
$426$		0			0
$427$	−1.50000	+	2.59808i	−0.0725901	+	0.125730i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	30.0000			1.44505			0.722525	−	0.691345i	$-0.242982\pi$
$431$	30.0000			1.44505			0.722525	+	0.691345i	$0.242982\pi$
$432$		0			0
$433$	−2.00000			−0.0961139			−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000			−0.0961139			−0.0480569	+	0.998845i	$0.515303\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−9.00000	+	15.5885i	−0.430528	+	0.745697i
$438$		0			0
$439$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$439$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	6.00000	+	10.3923i	0.285069	+	0.493753i	0.972626	−	0.232377i	$-0.0746503\pi$
$443$	6.00000	+	10.3923i	0.285069	+	0.493753i	−0.687557	+	0.726130i	$0.741317\pi$
$444$		0			0
$445$	10.0000	−	17.3205i	0.474045	−	0.821071i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	22.0000			1.03824			0.519122	−	0.854700i	$-0.326259\pi$
$449$	22.0000			1.03824			0.519122	+	0.854700i	$0.326259\pi$
$450$		0			0
$451$	−48.0000			−2.26023
$452$		0			0
$453$		0			0
$454$		0			0
$455$	9.00000	−	15.5885i	0.421927	−	0.730798i
$456$		0			0
$457$	15.0000	+	25.9808i	0.701670	+	1.21533i	0.967880	+	0.251414i	$0.0808954\pi$
$457$	15.0000	+	25.9808i	0.701670	+	1.21533i	−0.266209	+	0.963915i	$0.585771\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−1.00000	−	1.73205i	−0.0465746	−	0.0806696i	0.841798	−	0.539792i	$-0.181497\pi$
$461$	−1.00000	−	1.73205i	−0.0465746	−	0.0806696i	−0.888373	+	0.459123i	$0.848164\pi$
$462$		0			0
$463$	7.50000	−	12.9904i	0.348555	−	0.603714i	−0.637438	−	0.770501i	$-0.720006\pi$
$463$	7.50000	−	12.9904i	0.348555	−	0.603714i	0.985993	+	0.166787i	$0.0533393\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	6.00000			0.277647			0.138823	−	0.990317i	$-0.455668\pi$
$467$	6.00000			0.277647			0.138823	+	0.990317i	$0.455668\pi$
$468$		0			0
$469$	−9.00000			−0.415581
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−36.0000	+	62.3538i	−1.65528	+	2.86703i
$474$		0			0
$475$	1.50000	+	2.59808i	0.0688247	+	0.119208i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−6.00000	−	10.3923i	−0.274147	−	0.474837i	0.695773	−	0.718262i	$-0.255062\pi$
$479$	−6.00000	−	10.3923i	−0.274147	−	0.474837i	−0.969920	+	0.243426i	$0.921729\pi$
$480$		0			0
$481$	10.5000	−	18.1865i	0.478759	−	0.829235i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−18.0000			−0.817338
$486$		0			0
$487$	−15.0000			−0.679715			−0.339857	−	0.940477i	$-0.610379\pi$
$487$	−15.0000			−0.679715			−0.339857	+	0.940477i	$0.610379\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−21.0000	+	36.3731i	−0.947717	+	1.64149i	−0.197499	+	0.980303i	$0.563282\pi$
$491$	−21.0000	+	36.3731i	−0.947717	+	1.64149i	−0.750218	+	0.661190i	$0.770052\pi$
$492$		0			0
$493$	−8.00000	−	13.8564i	−0.360302	−	0.624061i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	18.0000	+	31.1769i	0.807410	+	1.39848i
$498$		0			0
$499$	12.0000	−	20.7846i	0.537194	−	0.930447i	−0.461860	−	0.886953i	$-0.652818\pi$
$499$	12.0000	−	20.7846i	0.537194	−	0.930447i	0.999054	−	0.0434940i	$-0.0138489\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−42.0000			−1.87269			−0.936344	−	0.351085i	$-0.885813\pi$
$503$	−42.0000			−1.87269			−0.936344	+	0.351085i	$0.885813\pi$
$504$		0			0
$505$	−32.0000			−1.42398
$506$		0			0
$507$		0			0
$508$		0			0
$509$	5.00000	−	8.66025i	0.221621	−	0.383859i	−0.733679	−	0.679496i	$-0.762199\pi$
$509$	5.00000	−	8.66025i	0.221621	−	0.383859i	0.955300	+	0.295637i	$0.0955319\pi$
$510$		0			0
$511$	−22.5000	−	38.9711i	−0.995341	−	1.72398i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−15.0000	−	25.9808i	−0.660979	−	1.14485i
$516$		0			0
$517$	−18.0000	+	31.1769i	−0.791639	+	1.37116i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	14.0000			0.613351			0.306676	−	0.951814i	$-0.400783\pi$
$521$	14.0000			0.613351			0.306676	+	0.951814i	$0.400783\pi$
$522$		0			0
$523$	−9.00000			−0.393543			−0.196771	−	0.980449i	$-0.563046\pi$
$523$	−9.00000			−0.393543			−0.196771	+	0.980449i	$0.563046\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−6.50000	−	11.2583i	−0.282609	−	0.489493i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−12.0000	−	20.7846i	−0.519778	−	0.900281i
$534$		0			0
$535$	−6.00000	+	10.3923i	−0.259403	+	0.449299i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	12.0000			0.516877
$540$		0			0
$541$	−15.0000			−0.644900			−0.322450	−	0.946586i	$-0.604506\pi$
$541$	−15.0000			−0.644900			−0.322450	+	0.946586i	$0.604506\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	6.00000	−	10.3923i	0.257012	−	0.445157i
$546$		0			0
$547$	−4.50000	−	7.79423i	−0.192406	−	0.333257i	0.753641	−	0.657286i	$-0.228296\pi$
$547$	−4.50000	−	7.79423i	−0.192406	−	0.333257i	−0.946047	+	0.324029i	$0.894962\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−12.0000	−	20.7846i	−0.511217	−	0.885454i
$552$		0			0
$553$	−13.5000	+	23.3827i	−0.574078	+	0.994333i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	2.00000			0.0847427			0.0423714	−	0.999102i	$-0.486509\pi$
$557$	2.00000			0.0847427			0.0423714	+	0.999102i	$0.486509\pi$
$558$		0			0
$559$	−36.0000			−1.52264
$560$		0			0
$561$		0			0
$562$		0			0
$563$	12.0000	−	20.7846i	0.505740	−	0.875967i	−0.494238	−	0.869326i	$-0.664553\pi$
$563$	12.0000	−	20.7846i	0.505740	−	0.875967i	0.999978	−	0.00664037i	$-0.00211371\pi$
$564$		0			0
$565$	−14.0000	−	24.2487i	−0.588984	−	1.02015i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	23.0000	+	39.8372i	0.964210	+	1.67006i	0.711722	+	0.702461i	$0.247915\pi$
$569$	23.0000	+	39.8372i	0.964210	+	1.67006i	0.252488	+	0.967600i	$0.418751\pi$
$570$		0			0
$571$	−1.50000	+	2.59808i	−0.0627730	+	0.108726i	−0.895704	−	0.444651i	$-0.853328\pi$
$571$	−1.50000	+	2.59808i	−0.0627730	+	0.108726i	0.832931	+	0.553377i	$0.186661\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−6.00000			−0.250217
$576$		0			0
$577$	11.0000			0.457936			0.228968	−	0.973434i	$-0.426465\pi$
$577$	11.0000			0.457936			0.228968	+	0.973434i	$0.426465\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−18.0000	+	31.1769i	−0.746766	+	1.29344i
$582$		0			0
$583$	12.0000	+	20.7846i	0.496989	+	0.860811i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−9.00000	−	15.5885i	−0.371470	−	0.643404i	0.618322	−	0.785925i	$-0.287813\pi$
$587$	−9.00000	−	15.5885i	−0.371470	−	0.643404i	−0.989792	+	0.142520i	$0.954479\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−28.0000			−1.14982			−0.574911	−	0.818216i	$-0.694963\pi$
$593$	−28.0000			−1.14982			−0.574911	+	0.818216i	$0.694963\pi$
$594$		0			0
$595$	12.0000			0.491952
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−12.0000	+	20.7846i	−0.490307	+	0.849236i	−0.999938	−	0.0111569i	$-0.996449\pi$
$599$	−12.0000	+	20.7846i	−0.490307	+	0.849236i	0.509631	+	0.860393i	$0.329782\pi$
$600$		0			0
$601$	13.0000	+	22.5167i	0.530281	+	0.918474i	0.999376	+	0.0353259i	$0.0112469\pi$
$601$	13.0000	+	22.5167i	0.530281	+	0.918474i	−0.469095	+	0.883148i	$0.655420\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	25.0000	+	43.3013i	1.01639	+	1.76045i
$606$		0			0
$607$	1.50000	−	2.59808i	0.0608831	−	0.105453i	−0.833977	−	0.551799i	$-0.813942\pi$
$607$	1.50000	−	2.59808i	0.0608831	−	0.105453i	0.894860	+	0.446346i	$0.147275\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−18.0000			−0.728202
$612$		0			0
$613$	11.0000			0.444286			0.222143	−	0.975014i	$-0.428695\pi$
$613$	11.0000			0.444286			0.222143	+	0.975014i	$0.428695\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.00000	−	1.73205i	0.0402585	−	0.0697297i	−0.845194	−	0.534460i	$-0.820515\pi$
$617$	1.00000	−	1.73205i	0.0402585	−	0.0697297i	0.885453	+	0.464730i	$0.153849\pi$
$618$		0			0
$619$	4.50000	+	7.79423i	0.180870	+	0.313276i	0.942177	−	0.335115i	$-0.108775\pi$
$619$	4.50000	+	7.79423i	0.180870	+	0.313276i	−0.761307	+	0.648392i	$0.775442\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	15.0000	+	25.9808i	0.600962	+	1.04090i
$624$		0			0
$625$	9.50000	−	16.4545i	0.380000	−	0.658179i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	14.0000			0.558217
$630$		0			0
$631$	27.0000			1.07485			0.537427	−	0.843311i	$-0.319397\pi$
$631$	27.0000			1.07485			0.537427	+	0.843311i	$0.319397\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	12.0000	−	20.7846i	0.476205	−	0.824812i
$636$		0			0
$637$	3.00000	+	5.19615i	0.118864	+	0.205879i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	2.00000	+	3.46410i	0.0789953	+	0.136824i	0.902817	−	0.430026i	$-0.141495\pi$
$641$	2.00000	+	3.46410i	0.0789953	+	0.136824i	−0.823821	+	0.566849i	$0.808162\pi$
$642$		0			0
$643$	−18.0000	+	31.1769i	−0.709851	+	1.22950i	0.255062	+	0.966925i	$0.417904\pi$
$643$	−18.0000	+	31.1769i	−0.709851	+	1.22950i	−0.964912	+	0.262573i	$0.915429\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	24.0000			0.943537			0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000			0.943537			0.471769	+	0.881722i	$0.343616\pi$
$648$		0			0
$649$	36.0000			1.41312
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−14.0000	+	24.2487i	−0.547862	+	0.948925i	0.450558	+	0.892747i	$0.351225\pi$
$653$	−14.0000	+	24.2487i	−0.547862	+	0.948925i	−0.998421	+	0.0561784i	$0.982108\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	6.00000	+	10.3923i	0.233727	+	0.404827i	0.958902	−	0.283738i	$-0.0915745\pi$
$659$	6.00000	+	10.3923i	0.233727	+	0.404827i	−0.725175	+	0.688565i	$0.758241\pi$
$660$		0			0
$661$	2.50000	−	4.33013i	0.0972387	−	0.168422i	−0.813302	−	0.581842i	$-0.802332\pi$
$661$	2.50000	−	4.33013i	0.0972387	−	0.168422i	0.910541	+	0.413419i	$0.135666\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	18.0000			0.698010
$666$		0			0
$667$	48.0000			1.85857
$668$		0			0
$669$		0			0
$670$		0			0
$671$	3.00000	−	5.19615i	0.115814	−	0.200595i
$672$		0			0
$673$	0.500000	+	0.866025i	0.0192736	+	0.0333828i	0.875501	−	0.483216i	$-0.160531\pi$
$673$	0.500000	+	0.866025i	0.0192736	+	0.0333828i	−0.856228	+	0.516599i	$0.827198\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	1.00000	+	1.73205i	0.0384331	+	0.0665681i	0.884602	−	0.466347i	$-0.154430\pi$
$677$	1.00000	+	1.73205i	0.0384331	+	0.0665681i	−0.846169	+	0.532915i	$0.821097\pi$
$678$		0			0
$679$	13.5000	−	23.3827i	0.518082	−	0.897345i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−24.0000			−0.918334			−0.459167	−	0.888350i	$-0.651852\pi$
$683$	−24.0000			−0.918334			−0.459167	+	0.888350i	$0.651852\pi$
$684$		0			0
$685$	−28.0000			−1.06983
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−6.00000	+	10.3923i	−0.228582	+	0.395915i
$690$		0			0
$691$	6.00000	+	10.3923i	0.228251	+	0.395342i	0.957290	−	0.289130i	$-0.0933661\pi$
$691$	6.00000	+	10.3923i	0.228251	+	0.395342i	−0.729039	+	0.684472i	$0.760033\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	21.0000	+	36.3731i	0.796575	+	1.37971i
$696$		0			0
$697$	8.00000	−	13.8564i	0.303022	−	0.524849i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−46.0000			−1.73740			−0.868698	−	0.495342i	$-0.835043\pi$
$701$	−46.0000			−1.73740			−0.868698	+	0.495342i	$0.835043\pi$
$702$		0			0
$703$	21.0000			0.792030
$704$		0			0
$705$		0			0
$706$		0			0
$707$	24.0000	−	41.5692i	0.902613	−	1.56337i
$708$		0			0
$709$	25.5000	+	44.1673i	0.957673	+	1.65874i	0.728130	+	0.685439i	$0.240390\pi$
$709$	25.5000	+	44.1673i	0.957673	+	1.65874i	0.229543	+	0.973299i	$0.426277\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−18.0000	+	31.1769i	−0.673162	+	1.16595i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−36.0000			−1.34257			−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000			−1.34257			−0.671287	+	0.741198i	$0.734258\pi$
$720$		0			0
$721$	45.0000			1.67589
$722$		0			0
$723$		0			0
$724$		0			0
$725$	4.00000	−	6.92820i	0.148556	−	0.257307i
$726$		0			0
$727$	12.0000	+	20.7846i	0.445055	+	0.770859i	0.998056	−	0.0623223i	$-0.0198507\pi$
$727$	12.0000	+	20.7846i	0.445055	+	0.770859i	−0.553001	+	0.833181i	$0.686517\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−12.0000	−	20.7846i	−0.443836	−	0.768747i
$732$		0			0
$733$	9.00000	−	15.5885i	0.332423	−	0.575773i	−0.650564	−	0.759452i	$-0.725467\pi$
$733$	9.00000	−	15.5885i	0.332423	−	0.575773i	0.982986	+	0.183679i	$0.0588007\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	18.0000			0.663039
$738$		0			0
$739$	12.0000			0.441427			0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000			0.441427			0.220714	+	0.975339i	$0.429161\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−3.00000	+	5.19615i	−0.110059	+	0.190628i	−0.915794	−	0.401648i	$-0.868437\pi$
$743$	−3.00000	+	5.19615i	−0.110059	+	0.190628i	0.805735	+	0.592277i	$0.201771\pi$
$744$		0			0
$745$	−4.00000	−	6.92820i	−0.146549	−	0.253830i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−9.00000	−	15.5885i	−0.328853	−	0.569590i
$750$		0			0
$751$	22.5000	−	38.9711i	0.821037	−	1.42208i	−0.0838743	−	0.996476i	$-0.526729\pi$
$751$	22.5000	−	38.9711i	0.821037	−	1.42208i	0.904911	−	0.425601i	$-0.139937\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−6.00000			−0.218362
$756$		0			0
$757$	−27.0000			−0.981332			−0.490666	−	0.871348i	$-0.663246\pi$
$757$	−27.0000			−0.981332			−0.490666	+	0.871348i	$0.663246\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−17.0000	+	29.4449i	−0.616250	+	1.06738i	0.373914	+	0.927463i	$0.378015\pi$
$761$	−17.0000	+	29.4449i	−0.616250	+	1.06738i	−0.990164	+	0.139912i	$0.955318\pi$
$762$		0			0
$763$	9.00000	+	15.5885i	0.325822	+	0.564340i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	9.00000	+	15.5885i	0.324971	+	0.562867i
$768$		0			0
$769$	24.5000	−	42.4352i	0.883493	−	1.53025i	0.0360609	−	0.999350i	$-0.488519\pi$
$769$	24.5000	−	42.4352i	0.883493	−	1.53025i	0.847432	−	0.530904i	$-0.178148\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	20.0000			0.719350			0.359675	−	0.933078i	$-0.382888\pi$
$773$	20.0000			0.719350			0.359675	+	0.933078i	$0.382888\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	12.0000	−	20.7846i	0.429945	−	0.744686i
$780$		0			0
$781$	−36.0000	−	62.3538i	−1.28818	−	2.23120i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−14.0000	−	24.2487i	−0.499681	−	0.865474i
$786$		0			0
$787$	−13.5000	+	23.3827i	−0.481223	+	0.833503i	−0.999768	−	0.0215477i	$-0.993141\pi$
$787$	−13.5000	+	23.3827i	−0.481223	+	0.833503i	0.518545	+	0.855050i	$0.326474\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	42.0000			1.49335
$792$		0			0
$793$	3.00000			0.106533
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−20.0000	+	34.6410i	−0.708436	+	1.22705i	0.257001	+	0.966411i	$0.417266\pi$
$797$	−20.0000	+	34.6410i	−0.708436	+	1.22705i	−0.965437	+	0.260637i	$0.916068\pi$
$798$		0			0
$799$	−6.00000	−	10.3923i	−0.212265	−	0.367653i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	45.0000	+	77.9423i	1.58802	+	2.75052i
$804$		0			0
$805$	−18.0000	+	31.1769i	−0.634417	+	1.09884i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−44.0000			−1.54696			−0.773479	−	0.633822i	$-0.781485\pi$
$809$	−44.0000			−1.54696			−0.773479	+	0.633822i	$0.781485\pi$
$810$		0			0
$811$	−36.0000			−1.26413			−0.632065	−	0.774915i	$-0.717793\pi$
$811$	−36.0000			−1.26413			−0.632065	+	0.774915i	$0.717793\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−3.00000	+	5.19615i	−0.105085	+	0.182013i
$816$		0			0
$817$	−18.0000	−	31.1769i	−0.629740	−	1.09074i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	25.0000	+	43.3013i	0.872506	+	1.51122i	0.859396	+	0.511311i	$0.170840\pi$
$821$	25.0000	+	43.3013i	0.872506	+	1.51122i	0.0131101	+	0.999914i	$0.495827\pi$
$822$		0			0
$823$	4.50000	−	7.79423i	0.156860	−	0.271690i	−0.776875	−	0.629655i	$-0.783196\pi$
$823$	4.50000	−	7.79423i	0.156860	−	0.271690i	0.933735	+	0.357966i	$0.116529\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	6.00000			0.208640			0.104320	−	0.994544i	$-0.466733\pi$
$827$	6.00000			0.208640			0.104320	+	0.994544i	$0.466733\pi$
$828$		0			0
$829$	45.0000			1.56291			0.781457	−	0.623959i	$-0.214477\pi$
$829$	45.0000			1.56291			0.781457	+	0.623959i	$0.214477\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−2.00000	+	3.46410i	−0.0692959	+	0.120024i
$834$		0			0
$835$	−18.0000	−	31.1769i	−0.622916	−	1.07892i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	18.0000	+	31.1769i	0.621429	+	1.07635i	0.989220	+	0.146438i	$0.0467809\pi$
$839$	18.0000	+	31.1769i	0.621429	+	1.07635i	−0.367791	+	0.929909i	$0.619886\pi$
$840$		0			0
$841$	−17.5000	+	30.3109i	−0.603448	+	1.04520i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	8.00000			0.275208
$846$		0			0
$847$	−75.0000			−2.57703
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−21.0000	+	36.3731i	−0.719871	+	1.24685i
$852$		0			0
$853$	26.5000	+	45.8993i	0.907343	+	1.57156i	0.817741	+	0.575586i	$0.195226\pi$
$853$	26.5000	+	45.8993i	0.907343	+	1.57156i	0.0896015	+	0.995978i	$0.471441\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−2.00000	−	3.46410i	−0.0683187	−	0.118331i	0.829843	−	0.557998i	$-0.188430\pi$
$857$	−2.00000	−	3.46410i	−0.0683187	−	0.118331i	−0.898161	+	0.439666i	$0.855097\pi$
$858$		0			0
$859$	13.5000	−	23.3827i	0.460614	−	0.797807i	−0.538378	−	0.842704i	$-0.680963\pi$
$859$	13.5000	−	23.3827i	0.460614	−	0.797807i	0.998992	+	0.0448968i	$0.0142959\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−42.0000			−1.42970			−0.714848	−	0.699280i	$-0.753504\pi$
$863$	−42.0000			−1.42970			−0.714848	+	0.699280i	$0.753504\pi$
$864$		0			0
$865$	16.0000			0.544016
$866$		0			0
$867$		0			0
$868$		0			0
$869$	27.0000	−	46.7654i	0.915912	−	1.58641i
$870$		0			0
$871$	4.50000	+	7.79423i	0.152477	+	0.264097i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	18.0000	+	31.1769i	0.608511	+	1.05397i
$876$		0			0
$877$	6.50000	−	11.2583i	0.219489	−	0.380167i	−0.735163	−	0.677891i	$-0.762894\pi$
$877$	6.50000	−	11.2583i	0.219489	−	0.380167i	0.954652	+	0.297724i	$0.0962275\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	14.0000			0.471672			0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000			0.471672			0.235836	+	0.971793i	$0.424217\pi$
$882$		0			0
$883$	9.00000			0.302874			0.151437	−	0.988467i	$-0.451610\pi$
$883$	9.00000			0.302874			0.151437	+	0.988467i	$0.451610\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	18.0000	−	31.1769i	0.604381	−	1.04682i	−0.387768	−	0.921757i	$-0.626754\pi$
$887$	18.0000	−	31.1769i	0.604381	−	1.04682i	0.992149	−	0.125061i	$-0.0399128\pi$
$888$		0			0
$889$	18.0000	+	31.1769i	0.603701	+	1.04564i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−9.00000	−	15.5885i	−0.301174	−	0.521648i
$894$		0			0
$895$	−12.0000	+	20.7846i	−0.401116	+	0.694753i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−8.00000			−0.266519
$902$		0			0
$903$		0			0
$904$		0			0
$905$	9.00000	−	15.5885i	0.299170	−	0.518178i
$906$		0			0
$907$	−19.5000	−	33.7750i	−0.647487	−	1.12148i	−0.983721	−	0.179702i	$-0.942487\pi$
$907$	−19.5000	−	33.7750i	−0.647487	−	1.12148i	0.336234	−	0.941778i	$-0.390847\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−24.0000	−	41.5692i	−0.795155	−	1.37725i	−0.922740	−	0.385422i	$-0.874056\pi$
$911$	−24.0000	−	41.5692i	−0.795155	−	1.37725i	0.127585	−	0.991828i	$-0.459277\pi$
$912$		0			0
$913$	36.0000	−	62.3538i	1.19143	−	2.06361i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−24.0000			−0.791687			−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000			−0.791687			−0.395843	+	0.918318i	$0.629548\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	18.0000	−	31.1769i	0.592477	−	1.02620i
$924$		0			0
$925$	3.50000	+	6.06218i	0.115079	+	0.199323i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	14.0000	+	24.2487i	0.459325	+	0.795574i	0.998925	−	0.0463469i	$-0.0147580\pi$
$929$	14.0000	+	24.2487i	0.459325	+	0.795574i	−0.539600	+	0.841921i	$0.681425\pi$
$930$		0			0
$931$	−3.00000	+	5.19615i	−0.0983210	+	0.170297i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−24.0000			−0.784884
$936$		0			0
$937$	7.00000			0.228680			0.114340	−	0.993442i	$-0.463525\pi$
$937$	7.00000			0.228680			0.114340	+	0.993442i	$0.463525\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	11.0000	−	19.0526i	0.358590	−	0.621096i	−0.629136	−	0.777295i	$-0.716591\pi$
$941$	11.0000	−	19.0526i	0.358590	−	0.621096i	0.987725	+	0.156200i	$0.0499244\pi$
$942$		0			0
$943$	24.0000	+	41.5692i	0.781548	+	1.35368i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−9.00000	−	15.5885i	−0.292461	−	0.506557i	0.681930	−	0.731417i	$-0.261141\pi$
$947$	−9.00000	−	15.5885i	−0.292461	−	0.506557i	−0.974391	+	0.224860i	$0.927807\pi$
$948$		0			0
$949$	−22.5000	+	38.9711i	−0.730381	+	1.26506i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−46.0000			−1.49009			−0.745043	−	0.667016i	$-0.767571\pi$
$953$	−46.0000			−1.49009			−0.745043	+	0.667016i	$0.767571\pi$
$954$		0			0
$955$	12.0000			0.388311
$956$		0			0
$957$		0			0
$958$		0			0
$959$	21.0000	−	36.3731i	0.678125	−	1.17455i
$960$		0			0
$961$	15.5000	+	26.8468i	0.500000	+	0.866025i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−5.00000	−	8.66025i	−0.160956	−	0.278783i
$966$		0			0
$967$	16.5000	−	28.5788i	0.530604	−	0.919033i	−0.468758	−	0.883327i	$-0.655298\pi$
$967$	16.5000	−	28.5788i	0.530604	−	0.919033i	0.999362	−	0.0357069i	$-0.0113683\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−6.00000			−0.192549			−0.0962746	−	0.995355i	$-0.530693\pi$
$971$	−6.00000			−0.192549			−0.0962746	+	0.995355i	$0.530693\pi$
$972$		0			0
$973$	−63.0000			−2.01969
$974$		0			0
$975$		0			0
$976$		0			0
$977$	22.0000	−	38.1051i	0.703842	−	1.21909i	−0.263265	−	0.964723i	$-0.584799\pi$
$977$	22.0000	−	38.1051i	0.703842	−	1.21909i	0.967108	−	0.254367i	$-0.0818672\pi$
$978$		0			0
$979$	−30.0000	−	51.9615i	−0.958804	−	1.66070i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	3.00000	+	5.19615i	0.0956851	+	0.165732i	0.909894	−	0.414840i	$-0.136162\pi$
$983$	3.00000	+	5.19615i	0.0956851	+	0.165732i	−0.814209	+	0.580572i	$0.802829\pi$
$984$		0			0
$985$	−2.00000	+	3.46410i	−0.0637253	+	0.110375i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	72.0000			2.28947
$990$		0			0
$991$	21.0000			0.667087			0.333543	−	0.942735i	$-0.391756\pi$
$991$	21.0000			0.667087			0.333543	+	0.942735i	$0.391756\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−15.0000	+	25.9808i	−0.475532	+	0.823646i
$996$		0			0
$997$	−13.0000	−	22.5167i	−0.411714	−	0.713110i	0.583363	−	0.812211i	$-0.301736\pi$
$997$	−13.0000	−	22.5167i	−0.411714	−	0.713110i	−0.995077	+	0.0991016i	$0.968403\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2592.2.i.v.1729.1		2
3.2	odd	2		2592.2.i.g.1729.1		2
4.3	odd	2		2592.2.i.r.1729.1		2
9.2	odd	6		2592.2.i.g.865.1		2
9.4	even	3		864.2.a.a.1.1	✓	1
9.5	odd	6		864.2.a.i.1.1	yes	1
9.7	even	3	inner	2592.2.i.v.865.1		2
12.11	even	2		2592.2.i.c.1729.1		2
36.7	odd	6		2592.2.i.r.865.1		2
36.11	even	6		2592.2.i.c.865.1		2
36.23	even	6		864.2.a.l.1.1	yes	1
36.31	odd	6		864.2.a.d.1.1	yes	1
72.5	odd	6		1728.2.a.e.1.1		1
72.13	even	6		1728.2.a.u.1.1		1
72.59	even	6		1728.2.a.h.1.1		1
72.67	odd	6		1728.2.a.x.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.2.a.a.1.1	✓	1	9.4	even	3
864.2.a.d.1.1	yes	1	36.31	odd	6
864.2.a.i.1.1	yes	1	9.5	odd	6
864.2.a.l.1.1	yes	1	36.23	even	6
1728.2.a.e.1.1		1	72.5	odd	6
1728.2.a.h.1.1		1	72.59	even	6
1728.2.a.u.1.1		1	72.13	even	6
1728.2.a.x.1.1		1	72.67	odd	6
2592.2.i.c.865.1		2	36.11	even	6
2592.2.i.c.1729.1		2	12.11	even	2
2592.2.i.g.865.1		2	9.2	odd	6
2592.2.i.g.1729.1		2	3.2	odd	2
2592.2.i.r.865.1		2	36.7	odd	6
2592.2.i.r.1729.1		2	4.3	odd	2
2592.2.i.v.865.1		2	9.7	even	3	inner
2592.2.i.v.1729.1		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.73205i) q^{5} +(1.50000 + 2.59808i) q^{7} +O(q^{10})\)	\(q+(1.00000 - 1.73205i) q^{5} +(1.50000 + 2.59808i) q^{7} +(-3.00000 - 5.19615i) q^{11} +(1.50000 - 2.59808i) q^{13} +2.00000 q^{17} +3.00000 q^{19} +(-3.00000 + 5.19615i) q^{23} +(0.500000 + 0.866025i) q^{25} +(-4.00000 - 6.92820i) q^{29} +6.00000 q^{35} +7.00000 q^{37} +(4.00000 - 6.92820i) q^{41} +(-6.00000 - 10.3923i) q^{43} +(-3.00000 - 5.19615i) q^{47} +(-1.00000 + 1.73205i) q^{49} -4.00000 q^{53} -12.0000 q^{55} +(-3.00000 + 5.19615i) q^{59} +(0.500000 + 0.866025i) q^{61} +(-3.00000 - 5.19615i) q^{65} +(-1.50000 + 2.59808i) q^{67} +12.0000 q^{71} -15.0000 q^{73} +(9.00000 - 15.5885i) q^{77} +(4.50000 + 7.79423i) q^{79} +(6.00000 + 10.3923i) q^{83} +(2.00000 - 3.46410i) q^{85} +10.0000 q^{89} +9.00000 q^{91} +(3.00000 - 5.19615i) q^{95} +(-4.50000 - 7.79423i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} + 3 q^{7}+O(q^{10}) \) 2 * q + 2 * q^5 + 3 * q^7	\( 2 q + 2 q^{5} + 3 q^{7} - 6 q^{11} + 3 q^{13} + 4 q^{17} + 6 q^{19} - 6 q^{23} + q^{25} - 8 q^{29} + 12 q^{35} + 14 q^{37} + 8 q^{41} - 12 q^{43} - 6 q^{47} - 2 q^{49} - 8 q^{53} - 24 q^{55} - 6 q^{59} + q^{61} - 6 q^{65} - 3 q^{67} + 24 q^{71} - 30 q^{73} + 18 q^{77} + 9 q^{79} + 12 q^{83} + 4 q^{85} + 20 q^{89} + 18 q^{91} + 6 q^{95} - 9 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 + 3 * q^7 - 6 * q^11 + 3 * q^13 + 4 * q^17 + 6 * q^19 - 6 * q^23 + q^25 - 8 * q^29 + 12 * q^35 + 14 * q^37 + 8 * q^41 - 12 * q^43 - 6 * q^47 - 2 * q^49 - 8 * q^53 - 24 * q^55 - 6 * q^59 + q^61 - 6 * q^65 - 3 * q^67 + 24 * q^71 - 30 * q^73 + 18 * q^77 + 9 * q^79 + 12 * q^83 + 4 * q^85 + 20 * q^89 + 18 * q^91 + 6 * q^95 - 9 * q^97

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