LMFDB - Embedded newform 2592.2.i.i.865.1

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:	$1$
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:	no (minimal twist has level 864)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			865.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2592.865
Dual form			2592.2.i.i.1729.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{5} +(-1.50000 + 2.59808i) q^{7} +O(q^{10})$	$q+(-0.500000 - 0.866025i) q^{5} +(-1.50000 + 2.59808i) q^{7} +(-1.50000 + 2.59808i) q^{11} -4.00000 q^{17} +6.00000 q^{19} +(3.00000 + 5.19615i) q^{23} +(2.00000 - 3.46410i) q^{25} +(-1.00000 + 1.73205i) q^{29} +(-4.50000 - 7.79423i) q^{31} +3.00000 q^{35} -2.00000 q^{37} +(-5.00000 - 8.66025i) q^{41} +(-3.00000 + 5.19615i) q^{43} +(3.00000 - 5.19615i) q^{47} +(-1.00000 - 1.73205i) q^{49} -13.0000 q^{53} +3.00000 q^{55} +(-6.00000 - 10.3923i) q^{59} +(-4.00000 + 6.92820i) q^{61} +(-3.00000 - 5.19615i) q^{67} -12.0000 q^{71} +9.00000 q^{73} +(-4.50000 - 7.79423i) q^{77} +(-1.50000 + 2.59808i) q^{83} +(2.00000 + 3.46410i) q^{85} -14.0000 q^{89} +(-3.00000 - 5.19615i) q^{95} +(4.50000 - 7.79423i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{5} - 3 q^{7}+O(q^{10}) $ 2 * q - q^5 - 3 * q^7	$ 2 q - q^{5} - 3 q^{7} - 3 q^{11} - 8 q^{17} + 12 q^{19} + 6 q^{23} + 4 q^{25} - 2 q^{29} - 9 q^{31} + 6 q^{35} - 4 q^{37} - 10 q^{41} - 6 q^{43} + 6 q^{47} - 2 q^{49} - 26 q^{53} + 6 q^{55} - 12 q^{59} - 8 q^{61} - 6 q^{67} - 24 q^{71} + 18 q^{73} - 9 q^{77} - 3 q^{83} + 4 q^{85} - 28 q^{89} - 6 q^{95} + 9 q^{97}+O(q^{100}) $ 2 * q - q^5 - 3 * q^7 - 3 * q^11 - 8 * q^17 + 12 * q^19 + 6 * q^23 + 4 * q^25 - 2 * q^29 - 9 * q^31 + 6 * q^35 - 4 * q^37 - 10 * q^41 - 6 * q^43 + 6 * q^47 - 2 * q^49 - 26 * q^53 + 6 * q^55 - 12 * q^59 - 8 * q^61 - 6 * q^67 - 24 * q^71 + 18 * q^73 - 9 * q^77 - 3 * q^83 + 4 * q^85 - 28 * q^89 - 6 * q^95 + 9 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$325$	$1217$	$2431$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

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	$-0.238450\pi$
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i	−0.955901	+	0.293691i	$0.905116\pi$
$6$		0			0
$7$	−1.50000	+	2.59808i	−0.566947	+	0.981981i	0.429919	+	0.902867i	$0.358542\pi$
$7$	−1.50000	+	2.59808i	−0.566947	+	0.981981i	−0.996866	+	0.0791130i	$0.974791\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.50000	+	2.59808i	−0.452267	+	0.783349i	−0.998526	−	0.0542666i	$-0.982718\pi$
$11$	−1.50000	+	2.59808i	−0.452267	+	0.783349i	0.546259	+	0.837616i	$0.316051\pi$
$12$		0			0
$13$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$13$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.00000			−0.970143			−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000			−0.970143			−0.485071	+	0.874475i	$0.661206\pi$
$18$		0			0
$19$	6.00000			1.37649			0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000			1.37649			0.688247	+	0.725476i	$0.258380\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.00000	+	5.19615i	0.625543	+	1.08347i	0.988436	+	0.151642i	$0.0484560\pi$
$23$	3.00000	+	5.19615i	0.625543	+	1.08347i	−0.362892	+	0.931831i	$0.618211\pi$
$24$		0			0
$25$	2.00000	−	3.46410i	0.400000	−	0.692820i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.00000	+	1.73205i	−0.185695	+	0.321634i	−0.943811	−	0.330487i	$-0.892787\pi$
$29$	−1.00000	+	1.73205i	−0.185695	+	0.321634i	0.758115	+	0.652121i	$0.226120\pi$
$30$		0			0
$31$	−4.50000	−	7.79423i	−0.808224	−	1.39988i	−0.914093	−	0.405505i	$-0.867096\pi$
$31$	−4.50000	−	7.79423i	−0.808224	−	1.39988i	0.105869	−	0.994380i	$-0.466238\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.00000			0.507093
$36$		0			0
$37$	−2.00000			−0.328798			−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000			−0.328798			−0.164399	+	0.986394i	$0.552568\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−5.00000	−	8.66025i	−0.780869	−	1.35250i	−0.931436	−	0.363905i	$-0.881443\pi$
$41$	−5.00000	−	8.66025i	−0.780869	−	1.35250i	0.150567	−	0.988600i	$-0.451890\pi$
$42$		0			0
$43$	−3.00000	+	5.19615i	−0.457496	+	0.792406i	−0.998828	−	0.0484030i	$-0.984587\pi$
$43$	−3.00000	+	5.19615i	−0.457496	+	0.792406i	0.541332	+	0.840809i	$0.317920\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.00000	−	5.19615i	0.437595	−	0.757937i	−0.559908	−	0.828554i	$-0.689164\pi$
$47$	3.00000	−	5.19615i	0.437595	−	0.757937i	0.997503	+	0.0706177i	$0.0224970\pi$
$48$		0			0
$49$	−1.00000	−	1.73205i	−0.142857	−	0.247436i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−13.0000			−1.78569			−0.892844	−	0.450367i	$-0.851293\pi$
$53$	−13.0000			−1.78569			−0.892844	+	0.450367i	$0.851293\pi$
$54$		0			0
$55$	3.00000			0.404520
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−6.00000	−	10.3923i	−0.781133	−	1.35296i	−0.931282	−	0.364299i	$-0.881308\pi$
$59$	−6.00000	−	10.3923i	−0.781133	−	1.35296i	0.150148	−	0.988663i	$-0.452025\pi$
$60$		0			0
$61$	−4.00000	+	6.92820i	−0.512148	+	0.887066i	0.487753	+	0.872982i	$0.337817\pi$
$61$	−4.00000	+	6.92820i	−0.512148	+	0.887066i	−0.999901	+	0.0140840i	$0.995517\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−3.00000	−	5.19615i	−0.366508	−	0.634811i	0.622509	−	0.782613i	$-0.286114\pi$
$67$	−3.00000	−	5.19615i	−0.366508	−	0.634811i	−0.989017	+	0.147802i	$0.952780\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−12.0000			−1.42414			−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000			−1.42414			−0.712069	+	0.702109i	$0.752242\pi$
$72$		0			0
$73$	9.00000			1.05337			0.526685	−	0.850060i	$-0.323435\pi$
$73$	9.00000			1.05337			0.526685	+	0.850060i	$0.323435\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−4.50000	−	7.79423i	−0.512823	−	0.888235i
$78$		0			0
$79$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$79$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1.50000	+	2.59808i	−0.164646	+	0.285176i	−0.936530	−	0.350588i	$-0.885982\pi$
$83$	−1.50000	+	2.59808i	−0.164646	+	0.285176i	0.771883	+	0.635764i	$0.219315\pi$
$84$		0			0
$85$	2.00000	+	3.46410i	0.216930	+	0.375735i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−14.0000			−1.48400			−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000			−1.48400			−0.741999	+	0.670402i	$0.766122\pi$
$90$		0			0
$91$		0			0
$92$		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	$-0.682291\pi$
$97$	4.50000	−	7.79423i	0.456906	−	0.791384i	0.998796	+	0.0490655i	$0.0156243\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−3.50000	+	6.06218i	−0.348263	+	0.603209i	−0.985941	−	0.167094i	$-0.946562\pi$
$101$	−3.50000	+	6.06218i	−0.348263	+	0.603209i	0.637678	+	0.770303i	$0.279895\pi$
$102$		0			0
$103$	6.00000	+	10.3923i	0.591198	+	1.02398i	0.994071	+	0.108729i	$0.0346780\pi$
$103$	6.00000	+	10.3923i	0.591198	+	1.02398i	−0.402874	+	0.915255i	$0.631989\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	15.0000			1.45010			0.725052	−	0.688694i	$-0.241816\pi$
$107$	15.0000			1.45010			0.725052	+	0.688694i	$0.241816\pi$
$108$		0			0
$109$	−18.0000			−1.72409			−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000			−1.72409			−0.862044	+	0.506834i	$0.830816\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−5.00000	−	8.66025i	−0.470360	−	0.814688i	0.529065	−	0.848581i	$-0.322543\pi$
$113$	−5.00000	−	8.66025i	−0.470360	−	0.814688i	−0.999425	+	0.0338931i	$0.989209\pi$
$114$		0			0
$115$	3.00000	−	5.19615i	0.279751	−	0.484544i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	6.00000	−	10.3923i	0.550019	−	0.952661i
$120$		0			0
$121$	1.00000	+	1.73205i	0.0909091	+	0.157459i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−9.00000			−0.804984
$126$		0			0
$127$	−3.00000			−0.266207			−0.133103	−	0.991102i	$-0.542494\pi$
$127$	−3.00000			−0.266207			−0.133103	+	0.991102i	$0.542494\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	4.50000	+	7.79423i	0.393167	+	0.680985i	0.992865	−	0.119241i	$-0.0380462\pi$
$131$	4.50000	+	7.79423i	0.393167	+	0.680985i	−0.599699	+	0.800226i	$0.704713\pi$
$132$		0			0
$133$	−9.00000	+	15.5885i	−0.780399	+	1.35169i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.00000	+	1.73205i	−0.0854358	+	0.147979i	−0.905577	−	0.424182i	$-0.860562\pi$
$137$	−1.00000	+	1.73205i	−0.0854358	+	0.147979i	0.820141	+	0.572161i	$0.193895\pi$
$138$		0			0
$139$	6.00000	+	10.3923i	0.508913	+	0.881464i	0.999947	+	0.0103230i	$0.00328598\pi$
$139$	6.00000	+	10.3923i	0.508913	+	0.881464i	−0.491033	+	0.871141i	$0.663381\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	2.00000			0.166091
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−8.50000	−	14.7224i	−0.696347	−	1.20611i	−0.969724	−	0.244202i	$-0.921474\pi$
$149$	−8.50000	−	14.7224i	−0.696347	−	1.20611i	0.273377	−	0.961907i	$-0.411859\pi$
$150$		0			0
$151$	1.50000	−	2.59808i	0.122068	−	0.211428i	−0.798515	−	0.601975i	$-0.794381\pi$
$151$	1.50000	−	2.59808i	0.122068	−	0.211428i	0.920583	+	0.390547i	$0.127714\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.50000	+	7.79423i	−0.361449	+	0.626048i
$156$		0			0
$157$	−2.00000	−	3.46410i	−0.159617	−	0.276465i	0.775113	−	0.631822i	$-0.217693\pi$
$157$	−2.00000	−	3.46410i	−0.159617	−	0.276465i	−0.934731	+	0.355357i	$0.884359\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−18.0000			−1.41860
$162$		0			0
$163$	−24.0000			−1.87983			−0.939913	−	0.341415i	$-0.889094\pi$
$163$	−24.0000			−1.87983			−0.939913	+	0.341415i	$0.889094\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	9.00000	+	15.5885i	0.696441	+	1.20627i	0.969693	+	0.244328i	$0.0785675\pi$
$167$	9.00000	+	15.5885i	0.696441	+	1.20627i	−0.273252	+	0.961943i	$0.588099\pi$
$168$		0			0
$169$	6.50000	−	11.2583i	0.500000	−	0.866025i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	5.50000	−	9.52628i	0.418157	−	0.724270i	−0.577597	−	0.816322i	$-0.696009\pi$
$173$	5.50000	−	9.52628i	0.418157	−	0.724270i	0.995754	+	0.0920525i	$0.0293428\pi$
$174$		0			0
$175$	6.00000	+	10.3923i	0.453557	+	0.785584i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−15.0000			−1.12115			−0.560576	−	0.828103i	$-0.689420\pi$
$179$	−15.0000			−1.12115			−0.560576	+	0.828103i	$0.689420\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.00000	+	1.73205i	0.0735215	+	0.127343i
$186$		0			0
$187$	6.00000	−	10.3923i	0.438763	−	0.759961i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−12.0000	+	20.7846i	−0.868290	+	1.50392i	−0.00454614	+	0.999990i	$0.501447\pi$
$191$	−12.0000	+	20.7846i	−0.868290	+	1.50392i	−0.863743	+	0.503932i	$0.831886\pi$
$192$		0			0
$193$	−6.50000	−	11.2583i	−0.467880	−	0.810392i	0.531446	−	0.847092i	$-0.321649\pi$
$193$	−6.50000	−	11.2583i	−0.467880	−	0.810392i	−0.999326	+	0.0366998i	$0.988315\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−5.00000			−0.356235			−0.178118	−	0.984009i	$-0.557001\pi$
$197$	−5.00000			−0.356235			−0.178118	+	0.984009i	$0.557001\pi$
$198$		0			0
$199$	−3.00000			−0.212664			−0.106332	−	0.994331i	$-0.533911\pi$
$199$	−3.00000			−0.212664			−0.106332	+	0.994331i	$0.533911\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−3.00000	−	5.19615i	−0.210559	−	0.364698i
$204$		0			0
$205$	−5.00000	+	8.66025i	−0.349215	+	0.604858i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−9.00000	+	15.5885i	−0.622543	+	1.07828i
$210$		0			0
$211$	−3.00000	−	5.19615i	−0.206529	−	0.357718i	0.744090	−	0.668079i	$-0.232883\pi$
$211$	−3.00000	−	5.19615i	−0.206529	−	0.357718i	−0.950619	+	0.310361i	$0.899550\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	6.00000			0.409197
$216$		0			0
$217$	27.0000			1.83288
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	12.0000	−	20.7846i	0.803579	−	1.39184i	−0.113666	−	0.993519i	$-0.536260\pi$
$223$	12.0000	−	20.7846i	0.803579	−	1.39184i	0.917246	−	0.398321i	$-0.130407\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	6.00000	−	10.3923i	0.398234	−	0.689761i	−0.595274	−	0.803523i	$-0.702957\pi$
$227$	6.00000	−	10.3923i	0.398234	−	0.689761i	0.993508	+	0.113761i	$0.0362899\pi$
$228$		0			0
$229$	9.00000	+	15.5885i	0.594737	+	1.03011i	0.993584	+	0.113097i	$0.0360770\pi$
$229$	9.00000	+	15.5885i	0.594737	+	1.03011i	−0.398847	+	0.917017i	$0.630590\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	14.0000			0.917170			0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000			0.917170			0.458585	+	0.888650i	$0.348356\pi$
$234$		0			0
$235$	−6.00000			−0.391397
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−3.00000	−	5.19615i	−0.194054	−	0.336111i	0.752536	−	0.658551i	$-0.228830\pi$
$239$	−3.00000	−	5.19615i	−0.194054	−	0.336111i	−0.946590	+	0.322440i	$0.895497\pi$
$240$		0			0
$241$	9.00000	−	15.5885i	0.579741	−	1.00414i	−0.415768	−	0.909471i	$-0.636487\pi$
$241$	9.00000	−	15.5885i	0.579741	−	1.00414i	0.995509	−	0.0946700i	$-0.0301796\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.00000	+	1.73205i	−0.0638877	+	0.110657i
$246$		0			0
$247$		0			0
$248$		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
$253$	−18.0000			−1.13165
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−4.00000	−	6.92820i	−0.249513	−	0.432169i	0.713878	−	0.700270i	$-0.246937\pi$
$257$	−4.00000	−	6.92820i	−0.249513	−	0.432169i	−0.963391	+	0.268101i	$0.913604\pi$
$258$		0			0
$259$	3.00000	−	5.19615i	0.186411	−	0.322873i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	3.00000	−	5.19615i	0.184988	−	0.320408i	−0.758585	−	0.651575i	$-0.774109\pi$
$263$	3.00000	−	5.19615i	0.184988	−	0.320408i	0.943572	+	0.331166i	$0.107442\pi$
$264$		0			0
$265$	6.50000	+	11.2583i	0.399292	+	0.691594i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−14.0000			−0.853595			−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000			−0.853595			−0.426798	+	0.904347i	$0.640358\pi$
$270$		0			0
$271$	3.00000			0.182237			0.0911185	−	0.995840i	$-0.470956\pi$
$271$	3.00000			0.182237			0.0911185	+	0.995840i	$0.470956\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	6.00000	+	10.3923i	0.361814	+	0.626680i
$276$		0			0
$277$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$277$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	10.0000	−	17.3205i	0.596550	−	1.03325i	−0.396776	−	0.917915i	$-0.629871\pi$
$281$	10.0000	−	17.3205i	0.596550	−	1.03325i	0.993326	−	0.115339i	$-0.0367956\pi$
$282$		0			0
$283$	3.00000	+	5.19615i	0.178331	+	0.308879i	0.941309	−	0.337546i	$-0.109597\pi$
$283$	3.00000	+	5.19615i	0.178331	+	0.308879i	−0.762978	+	0.646425i	$0.776263\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	30.0000			1.77084
$288$		0			0
$289$	−1.00000			−0.0588235
$290$		0			0
$291$		0			0
$292$		0			0
$293$	5.00000	+	8.66025i	0.292103	+	0.505937i	0.974307	−	0.225225i	$-0.0723116\pi$
$293$	5.00000	+	8.66025i	0.292103	+	0.505937i	−0.682204	+	0.731162i	$0.738978\pi$
$294$		0			0
$295$	−6.00000	+	10.3923i	−0.349334	+	0.605063i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−9.00000	−	15.5885i	−0.518751	−	0.898504i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	8.00000			0.458079
$306$		0			0
$307$	12.0000			0.684876			0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000			0.684876			0.342438	+	0.939540i	$0.388747\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−9.00000	−	15.5885i	−0.510343	−	0.883940i	−0.999928	−	0.0119847i	$-0.996185\pi$
$311$	−9.00000	−	15.5885i	−0.510343	−	0.883940i	0.489585	−	0.871956i	$-0.337148\pi$
$312$		0			0
$313$	9.50000	−	16.4545i	0.536972	−	0.930062i	−0.462093	−	0.886831i	$-0.652902\pi$
$313$	9.50000	−	16.4545i	0.536972	−	0.930062i	0.999065	−	0.0432311i	$-0.0137652\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3.50000	+	6.06218i	−0.196580	+	0.340486i	−0.947417	−	0.320001i	$-0.896317\pi$
$317$	−3.50000	+	6.06218i	−0.196580	+	0.340486i	0.750838	+	0.660487i	$0.229650\pi$
$318$		0			0
$319$	−3.00000	−	5.19615i	−0.167968	−	0.290929i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−24.0000			−1.33540
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	9.00000	+	15.5885i	0.496186	+	0.859419i
$330$		0			0
$331$	−15.0000	+	25.9808i	−0.824475	+	1.42803i	0.0778456	+	0.996965i	$0.475196\pi$
$331$	−15.0000	+	25.9808i	−0.824475	+	1.42803i	−0.902320	+	0.431066i	$0.858137\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3.00000	+	5.19615i	−0.163908	+	0.283896i
$336$		0			0
$337$	−9.00000	−	15.5885i	−0.490261	−	0.849157i	0.509676	−	0.860366i	$-0.329765\pi$
$337$	−9.00000	−	15.5885i	−0.490261	−	0.849157i	−0.999937	+	0.0112091i	$0.996432\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	27.0000			1.46213
$342$		0			0
$343$	−15.0000			−0.809924
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1.50000	+	2.59808i	0.0805242	+	0.139472i	0.903475	−	0.428640i	$-0.141007\pi$
$347$	1.50000	+	2.59808i	0.0805242	+	0.139472i	−0.822951	+	0.568112i	$0.807674\pi$
$348$		0			0
$349$	−13.0000	+	22.5167i	−0.695874	+	1.20529i	0.274011	+	0.961727i	$0.411649\pi$
$349$	−13.0000	+	22.5167i	−0.695874	+	1.20529i	−0.969885	+	0.243563i	$0.921684\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.00000	−	1.73205i	0.0532246	−	0.0921878i	−0.838186	−	0.545385i	$-0.816383\pi$
$353$	1.00000	−	1.73205i	0.0532246	−	0.0921878i	0.891410	+	0.453197i	$0.149717\pi$
$354$		0			0
$355$	6.00000	+	10.3923i	0.318447	+	0.551566i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−18.0000			−0.950004			−0.475002	−	0.879985i	$-0.657553\pi$
$359$	−18.0000			−0.950004			−0.475002	+	0.879985i	$0.657553\pi$
$360$		0			0
$361$	17.0000			0.894737
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−4.50000	−	7.79423i	−0.235541	−	0.407969i
$366$		0			0
$367$	−10.5000	+	18.1865i	−0.548096	+	0.949329i	0.450310	+	0.892873i	$0.351314\pi$
$367$	−10.5000	+	18.1865i	−0.548096	+	0.949329i	−0.998405	+	0.0564568i	$0.982020\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	19.5000	−	33.7750i	1.01239	−	1.75351i
$372$		0			0
$373$	2.00000	+	3.46410i	0.103556	+	0.179364i	0.913147	−	0.407630i	$-0.133645\pi$
$373$	2.00000	+	3.46410i	0.103556	+	0.179364i	−0.809591	+	0.586994i	$0.800311\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	36.0000			1.84920			0.924598	−	0.380945i	$-0.124401\pi$
$379$	36.0000			1.84920			0.924598	+	0.380945i	$0.124401\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−18.0000	−	31.1769i	−0.919757	−	1.59307i	−0.799783	−	0.600289i	$-0.795052\pi$
$383$	−18.0000	−	31.1769i	−0.919757	−	1.59307i	−0.119974	−	0.992777i	$-0.538281\pi$
$384$		0			0
$385$	−4.50000	+	7.79423i	−0.229341	+	0.397231i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3.50000	−	6.06218i	0.177457	−	0.307365i	−0.763552	−	0.645747i	$-0.776546\pi$
$389$	3.50000	−	6.06218i	0.177457	−	0.307365i	0.941009	+	0.338382i	$0.109880\pi$
$390$		0			0
$391$	−12.0000	−	20.7846i	−0.606866	−	1.05112i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−16.0000			−0.803017			−0.401508	−	0.915855i	$-0.631514\pi$
$397$	−16.0000			−0.803017			−0.401508	+	0.915855i	$0.631514\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	14.0000	+	24.2487i	0.699127	+	1.21092i	0.968770	+	0.247962i	$0.0797610\pi$
$401$	14.0000	+	24.2487i	0.699127	+	1.21092i	−0.269643	+	0.962960i	$0.586906\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	3.00000	−	5.19615i	0.148704	−	0.257564i
$408$		0			0
$409$	4.50000	+	7.79423i	0.222511	+	0.385400i	0.955570	−	0.294765i	$-0.0952414\pi$
$409$	4.50000	+	7.79423i	0.222511	+	0.385400i	−0.733059	+	0.680165i	$0.761908\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	36.0000			1.77144
$414$		0			0
$415$	3.00000			0.147264
$416$		0			0
$417$		0			0
$418$		0			0
$419$	6.00000	+	10.3923i	0.293119	+	0.507697i	0.974546	−	0.224189i	$-0.0719734\pi$
$419$	6.00000	+	10.3923i	0.293119	+	0.507697i	−0.681426	+	0.731887i	$0.738640\pi$
$420$		0			0
$421$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$421$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−8.00000	+	13.8564i	−0.388057	+	0.672134i
$426$		0			0
$427$	−12.0000	−	20.7846i	−0.580721	−	1.00584i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	6.00000			0.289010			0.144505	−	0.989504i	$-0.453841\pi$
$431$	6.00000			0.289010			0.144505	+	0.989504i	$0.453841\pi$
$432$		0			0
$433$	−11.0000			−0.528626			−0.264313	−	0.964437i	$-0.585145\pi$
$433$	−11.0000			−0.528626			−0.264313	+	0.964437i	$0.585145\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	18.0000	+	31.1769i	0.861057	+	1.49139i
$438$		0			0
$439$	−4.50000	+	7.79423i	−0.214773	+	0.371998i	−0.953202	−	0.302333i	$-0.902235\pi$
$439$	−4.50000	+	7.79423i	−0.214773	+	0.371998i	0.738429	+	0.674331i	$0.235568\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−6.00000	+	10.3923i	−0.285069	+	0.493753i	−0.972626	−	0.232377i	$-0.925350\pi$
$443$	−6.00000	+	10.3923i	−0.285069	+	0.493753i	0.687557	+	0.726130i	$0.258683\pi$
$444$		0			0
$445$	7.00000	+	12.1244i	0.331832	+	0.574750i
$446$		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
$451$	30.0000			1.41264
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	4.50000	−	7.79423i	0.210501	−	0.364599i	−0.741370	−	0.671096i	$-0.765824\pi$
$457$	4.50000	−	7.79423i	0.210501	−	0.364599i	0.951871	+	0.306497i	$0.0991571\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	3.50000	−	6.06218i	0.163011	−	0.282344i	−0.772936	−	0.634484i	$-0.781213\pi$
$461$	3.50000	−	6.06218i	0.163011	−	0.282344i	0.935947	+	0.352140i	$0.114546\pi$
$462$		0			0
$463$	−7.50000	−	12.9904i	−0.348555	−	0.603714i	0.637438	−	0.770501i	$-0.279994\pi$
$463$	−7.50000	−	12.9904i	−0.348555	−	0.603714i	−0.985993	+	0.166787i	$0.946661\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	3.00000			0.138823			0.0694117	−	0.997588i	$-0.477888\pi$
$467$	3.00000			0.138823			0.0694117	+	0.997588i	$0.477888\pi$
$468$		0			0
$469$	18.0000			0.831163
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−9.00000	−	15.5885i	−0.413820	−	0.716758i
$474$		0			0
$475$	12.0000	−	20.7846i	0.550598	−	0.953663i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−21.0000	+	36.3731i	−0.959514	+	1.66193i	−0.235833	+	0.971794i	$0.575782\pi$
$479$	−21.0000	+	36.3731i	−0.959514	+	1.66193i	−0.723681	+	0.690134i	$0.757551\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−9.00000			−0.408669
$486$		0			0
$487$	24.0000			1.08754			0.543772	−	0.839233i	$-0.316996\pi$
$487$	24.0000			1.08754			0.543772	+	0.839233i	$0.316996\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	7.50000	+	12.9904i	0.338470	+	0.586248i	0.984145	−	0.177365i	$-0.0567572\pi$
$491$	7.50000	+	12.9904i	0.338470	+	0.586248i	−0.645675	+	0.763612i	$0.723424\pi$
$492$		0			0
$493$	4.00000	−	6.92820i	0.180151	−	0.312031i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	18.0000	−	31.1769i	0.807410	−	1.39848i
$498$		0			0
$499$	−3.00000	−	5.19615i	−0.134298	−	0.232612i	0.791031	−	0.611776i	$-0.209545\pi$
$499$	−3.00000	−	5.19615i	−0.134298	−	0.232612i	−0.925329	+	0.379165i	$0.876211\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	6.00000			0.267527			0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000			0.267527			0.133763	+	0.991013i	$0.457294\pi$
$504$		0			0
$505$	7.00000			0.311496
$506$		0			0
$507$		0			0
$508$		0			0
$509$	18.5000	+	32.0429i	0.819998	+	1.42028i	0.905683	+	0.423956i	$0.139359\pi$
$509$	18.5000	+	32.0429i	0.819998	+	1.42028i	−0.0856847	+	0.996322i	$0.527308\pi$
$510$		0			0
$511$	−13.5000	+	23.3827i	−0.597205	+	1.03439i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	6.00000	−	10.3923i	0.264392	−	0.457940i
$516$		0			0
$517$	9.00000	+	15.5885i	0.395820	+	0.685580i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	32.0000			1.40195			0.700973	−	0.713188i	$-0.252749\pi$
$521$	32.0000			1.40195			0.700973	+	0.713188i	$0.252749\pi$
$522$		0			0
$523$	−36.0000			−1.57417			−0.787085	−	0.616844i	$-0.788411\pi$
$523$	−36.0000			−1.57417			−0.787085	+	0.616844i	$0.788411\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	18.0000	+	31.1769i	0.784092	+	1.35809i
$528$		0			0
$529$	−6.50000	+	11.2583i	−0.282609	+	0.489493i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−7.50000	−	12.9904i	−0.324253	−	0.561623i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	6.00000			0.258438
$540$		0			0
$541$	−36.0000			−1.54776			−0.773880	−	0.633332i	$-0.781687\pi$
$541$	−36.0000			−1.54776			−0.773880	+	0.633332i	$0.781687\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	9.00000	+	15.5885i	0.385518	+	0.667736i
$546$		0			0
$547$	−18.0000	+	31.1769i	−0.769624	+	1.33303i	0.168142	+	0.985763i	$0.446223\pi$
$547$	−18.0000	+	31.1769i	−0.769624	+	1.33303i	−0.937767	+	0.347266i	$0.887110\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−6.00000	+	10.3923i	−0.255609	+	0.442727i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	41.0000			1.73723			0.868613	−	0.495491i	$-0.165012\pi$
$557$	41.0000			1.73723			0.868613	+	0.495491i	$0.165012\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1.50000	+	2.59808i	0.0632175	+	0.109496i	0.895902	−	0.444252i	$-0.146530\pi$
$563$	1.50000	+	2.59808i	0.0632175	+	0.109496i	−0.832684	+	0.553748i	$0.813197\pi$
$564$		0			0
$565$	−5.00000	+	8.66025i	−0.210352	+	0.364340i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−10.0000	+	17.3205i	−0.419222	+	0.726113i	−0.995861	−	0.0908852i	$-0.971030\pi$
$569$	−10.0000	+	17.3205i	−0.419222	+	0.726113i	0.576640	+	0.816999i	$0.304364\pi$
$570$		0			0
$571$	−12.0000	−	20.7846i	−0.502184	−	0.869809i	−0.999997	−	0.00252413i	$-0.999197\pi$
$571$	−12.0000	−	20.7846i	−0.502184	−	0.869809i	0.497812	−	0.867285i	$-0.334137\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	24.0000			1.00087
$576$		0			0
$577$	−34.0000			−1.41544			−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000			−1.41544			−0.707719	+	0.706494i	$0.750276\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−4.50000	−	7.79423i	−0.186691	−	0.323359i
$582$		0			0
$583$	19.5000	−	33.7750i	0.807607	−	1.39882i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−13.5000	+	23.3827i	−0.557205	+	0.965107i	0.440524	+	0.897741i	$0.354793\pi$
$587$	−13.5000	+	23.3827i	−0.557205	+	0.965107i	−0.997728	+	0.0673658i	$0.978541\pi$
$588$		0			0
$589$	−27.0000	−	46.7654i	−1.11252	−	1.92693i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	14.0000			0.574911			0.287456	−	0.957794i	$-0.407191\pi$
$593$	14.0000			0.574911			0.287456	+	0.957794i	$0.407191\pi$
$594$		0			0
$595$	−12.0000			−0.491952
$596$		0			0
$597$		0			0
$598$		0			0
$599$	3.00000	+	5.19615i	0.122577	+	0.212309i	0.920783	−	0.390075i	$-0.127551\pi$
$599$	3.00000	+	5.19615i	0.122577	+	0.212309i	−0.798206	+	0.602384i	$0.794218\pi$
$600$		0			0
$601$	−9.50000	+	16.4545i	−0.387513	+	0.671192i	−0.992114	−	0.125336i	$-0.959999\pi$
$601$	−9.50000	+	16.4545i	−0.387513	+	0.671192i	0.604601	+	0.796528i	$0.293332\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.00000	−	1.73205i	0.0406558	−	0.0704179i
$606$		0			0
$607$	12.0000	+	20.7846i	0.487065	+	0.843621i	0.999889	−	0.0148722i	$-0.00473415\pi$
$607$	12.0000	+	20.7846i	0.487065	+	0.843621i	−0.512824	+	0.858494i	$0.671401\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	38.0000			1.53481			0.767403	−	0.641165i	$-0.221549\pi$
$613$	38.0000			1.53481			0.767403	+	0.641165i	$0.221549\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	13.0000	+	22.5167i	0.523360	+	0.906487i	0.999630	+	0.0271876i	$0.00865514\pi$
$617$	13.0000	+	22.5167i	0.523360	+	0.906487i	−0.476270	+	0.879299i	$0.658012\pi$
$618$		0			0
$619$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$619$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	21.0000	−	36.3731i	0.841347	−	1.45726i
$624$		0			0
$625$	−5.50000	−	9.52628i	−0.220000	−	0.381051i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	8.00000			0.318981
$630$		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
$635$	1.50000	+	2.59808i	0.0595257	+	0.103102i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1.00000	+	1.73205i	−0.0394976	+	0.0684119i	−0.885098	−	0.465404i	$-0.845909\pi$
$641$	−1.00000	+	1.73205i	−0.0394976	+	0.0684119i	0.845601	+	0.533816i	$0.179242\pi$
$642$		0			0
$643$	18.0000	+	31.1769i	0.709851	+	1.22950i	0.964912	+	0.262573i	$0.0845709\pi$
$643$	18.0000	+	31.1769i	0.709851	+	1.22950i	−0.255062	+	0.966925i	$0.582096\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−24.0000			−0.943537			−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000			−0.943537			−0.471769	+	0.881722i	$0.656384\pi$
$648$		0			0
$649$	36.0000			1.41312
$650$		0			0
$651$		0			0
$652$		0			0
$653$	17.5000	+	30.3109i	0.684828	+	1.18616i	0.973491	+	0.228726i	$0.0734560\pi$
$653$	17.5000	+	30.3109i	0.684828	+	1.18616i	−0.288663	+	0.957431i	$0.593211\pi$
$654$		0			0
$655$	4.50000	−	7.79423i	0.175830	−	0.304546i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−10.5000	+	18.1865i	−0.409022	+	0.708447i	−0.994780	−	0.102039i	$-0.967463\pi$
$659$	−10.5000	+	18.1865i	−0.409022	+	0.708447i	0.585758	+	0.810486i	$0.300797\pi$
$660$		0			0
$661$	7.00000	+	12.1244i	0.272268	+	0.471583i	0.969442	−	0.245319i	$-0.0788928\pi$
$661$	7.00000	+	12.1244i	0.272268	+	0.471583i	−0.697174	+	0.716902i	$0.745559\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	18.0000			0.698010
$666$		0			0
$667$	−12.0000			−0.464642
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−12.0000	−	20.7846i	−0.463255	−	0.802381i
$672$		0			0
$673$	9.50000	−	16.4545i	0.366198	−	0.634274i	−0.622770	−	0.782405i	$-0.713993\pi$
$673$	9.50000	−	16.4545i	0.366198	−	0.634274i	0.988968	+	0.148132i	$0.0473259\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	1.00000	−	1.73205i	0.0384331	−	0.0665681i	−0.846169	−	0.532915i	$-0.821097\pi$
$677$	1.00000	−	1.73205i	0.0384331	−	0.0665681i	0.884602	+	0.466347i	$0.154430\pi$
$678$		0			0
$679$	13.5000	+	23.3827i	0.518082	+	0.897345i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−12.0000			−0.459167			−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000			−0.459167			−0.229584	+	0.973289i	$0.573736\pi$
$684$		0			0
$685$	2.00000			0.0764161
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−6.00000	+	10.3923i	−0.228251	+	0.395342i	−0.957290	−	0.289130i	$-0.906634\pi$
$691$	−6.00000	+	10.3923i	−0.228251	+	0.395342i	0.729039	+	0.684472i	$0.239967\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	6.00000	−	10.3923i	0.227593	−	0.394203i
$696$		0			0
$697$	20.0000	+	34.6410i	0.757554	+	1.31212i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−13.0000			−0.491003			−0.245502	−	0.969396i	$-0.578953\pi$
$701$	−13.0000			−0.491003			−0.245502	+	0.969396i	$0.578953\pi$
$702$		0			0
$703$	−12.0000			−0.452589
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−10.5000	−	18.1865i	−0.394893	−	0.683975i
$708$		0			0
$709$	18.0000	−	31.1769i	0.676004	−	1.17087i	−0.300170	−	0.953886i	$-0.597043\pi$
$709$	18.0000	−	31.1769i	0.676004	−	1.17087i	0.976174	−	0.216988i	$-0.0696232\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	27.0000	−	46.7654i	1.01116	−	1.75138i
$714$		0			0
$715$		0			0
$716$		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
$721$	−36.0000			−1.34071
$722$		0			0
$723$		0			0
$724$		0			0
$725$	4.00000	+	6.92820i	0.148556	+	0.257307i
$726$		0			0
$727$	10.5000	−	18.1865i	0.389423	−	0.674501i	−0.602949	−	0.797780i	$-0.706008\pi$
$727$	10.5000	−	18.1865i	0.389423	−	0.674501i	0.992372	+	0.123279i	$0.0393409\pi$
$728$		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	0.982986	−	0.183679i	$-0.0588007\pi$
$733$	9.00000	+	15.5885i	0.332423	+	0.575773i	−0.650564	+	0.759452i	$0.725467\pi$
$734$		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	$-0.570839\pi$
$739$	−12.0000			−0.441427			−0.220714	+	0.975339i	$0.570839\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	12.0000	+	20.7846i	0.440237	+	0.762513i	0.997707	−	0.0676840i	$-0.0215610\pi$
$743$	12.0000	+	20.7846i	0.440237	+	0.762513i	−0.557470	+	0.830197i	$0.688228\pi$
$744$		0			0
$745$	−8.50000	+	14.7224i	−0.311416	+	0.539388i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−22.5000	+	38.9711i	−0.822132	+	1.42397i
$750$		0			0
$751$	13.5000	+	23.3827i	0.492622	+	0.853246i	0.999964	−	0.00849853i	$-0.00270520\pi$
$751$	13.5000	+	23.3827i	0.492622	+	0.853246i	−0.507342	+	0.861745i	$0.669372\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−3.00000			−0.109181
$756$		0			0
$757$	18.0000			0.654221			0.327111	−	0.944986i	$-0.393925\pi$
$757$	18.0000			0.654221			0.327111	+	0.944986i	$0.393925\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	4.00000	+	6.92820i	0.145000	+	0.251147i	0.929373	−	0.369142i	$-0.120348\pi$
$761$	4.00000	+	6.92820i	0.145000	+	0.251147i	−0.784373	+	0.620289i	$0.787015\pi$
$762$		0			0
$763$	27.0000	−	46.7654i	0.977466	−	1.69302i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−20.5000	−	35.5070i	−0.739249	−	1.28042i	−0.952834	−	0.303492i	$-0.901847\pi$
$769$	−20.5000	−	35.5070i	−0.739249	−	1.28042i	0.213585	−	0.976924i	$-0.431486\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	50.0000			1.79838			0.899188	−	0.437564i	$-0.144158\pi$
$773$	50.0000			1.79838			0.899188	+	0.437564i	$0.144158\pi$
$774$		0			0
$775$	−36.0000			−1.29316
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−30.0000	−	51.9615i	−1.07486	−	1.86171i
$780$		0			0
$781$	18.0000	−	31.1769i	0.644091	−	1.11560i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−2.00000	+	3.46410i	−0.0713831	+	0.123639i
$786$		0			0
$787$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$787$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	30.0000			1.06668
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−18.5000	−	32.0429i	−0.655304	−	1.13502i	−0.981818	−	0.189827i	$-0.939207\pi$
$797$	−18.5000	−	32.0429i	−0.655304	−	1.13502i	0.326514	−	0.945192i	$-0.394126\pi$
$798$		0			0
$799$	−12.0000	+	20.7846i	−0.424529	+	0.735307i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−13.5000	+	23.3827i	−0.476405	+	0.825157i
$804$		0			0
$805$	9.00000	+	15.5885i	0.317208	+	0.549421i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−50.0000			−1.75791			−0.878953	−	0.476908i	$-0.841757\pi$
$809$	−50.0000			−1.75791			−0.878953	+	0.476908i	$0.841757\pi$
$810$		0			0
$811$	−18.0000			−0.632065			−0.316033	−	0.948748i	$-0.602351\pi$
$811$	−18.0000			−0.632065			−0.316033	+	0.948748i	$0.602351\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	12.0000	+	20.7846i	0.420342	+	0.728053i
$816$		0			0
$817$	−18.0000	+	31.1769i	−0.629740	+	1.09074i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	19.0000	−	32.9090i	0.663105	−	1.14853i	−0.316691	−	0.948529i	$-0.602572\pi$
$821$	19.0000	−	32.9090i	0.663105	−	1.14853i	0.979795	−	0.200002i	$-0.0640949\pi$
$822$		0			0
$823$	13.5000	+	23.3827i	0.470580	+	0.815069i	0.999434	−	0.0336440i	$-0.0107112\pi$
$823$	13.5000	+	23.3827i	0.470580	+	0.815069i	−0.528853	+	0.848713i	$0.677378\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	48.0000			1.66912			0.834562	−	0.550914i	$-0.185721\pi$
$827$	48.0000			1.66912			0.834562	+	0.550914i	$0.185721\pi$
$828$		0			0
$829$	18.0000			0.625166			0.312583	−	0.949890i	$-0.398806\pi$
$829$	18.0000			0.625166			0.312583	+	0.949890i	$0.398806\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	4.00000	+	6.92820i	0.138592	+	0.240048i
$834$		0			0
$835$	9.00000	−	15.5885i	0.311458	−	0.539461i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	18.0000	−	31.1769i	0.621429	−	1.07635i	−0.367791	−	0.929909i	$-0.619886\pi$
$839$	18.0000	−	31.1769i	0.621429	−	1.07635i	0.989220	−	0.146438i	$-0.0467809\pi$
$840$		0			0
$841$	12.5000	+	21.6506i	0.431034	+	0.746574i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−13.0000			−0.447214
$846$		0			0
$847$	−6.00000			−0.206162
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−6.00000	−	10.3923i	−0.205677	−	0.356244i
$852$		0			0
$853$	13.0000	−	22.5167i	0.445112	−	0.770956i	−0.552948	−	0.833215i	$-0.686497\pi$
$853$	13.0000	−	22.5167i	0.445112	−	0.770956i	0.998060	+	0.0622597i	$0.0198307\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−26.0000	+	45.0333i	−0.888143	+	1.53831i	−0.0460748	+	0.998938i	$0.514671\pi$
$857$	−26.0000	+	45.0333i	−0.888143	+	1.53831i	−0.842068	+	0.539371i	$0.818662\pi$
$858$		0			0
$859$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$859$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−12.0000			−0.408485			−0.204242	−	0.978920i	$-0.565473\pi$
$863$	−12.0000			−0.408485			−0.204242	+	0.978920i	$0.565473\pi$
$864$		0			0
$865$	−11.0000			−0.374011
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	13.5000	−	23.3827i	0.456383	−	0.790479i
$876$		0			0
$877$	−25.0000	−	43.3013i	−0.844190	−	1.46218i	−0.886323	−	0.463068i	$-0.846749\pi$
$877$	−25.0000	−	43.3013i	−0.844190	−	1.46218i	0.0421327	−	0.999112i	$-0.486585\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	32.0000			1.07811			0.539054	−	0.842271i	$-0.318782\pi$
$881$	32.0000			1.07811			0.539054	+	0.842271i	$0.318782\pi$
$882$		0			0
$883$	−18.0000			−0.605748			−0.302874	−	0.953031i	$-0.597946\pi$
$883$	−18.0000			−0.605748			−0.302874	+	0.953031i	$0.597946\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$887$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$	4.50000	−	7.79423i	0.150925	−	0.261410i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	18.0000	−	31.1769i	0.602347	−	1.04330i
$894$		0			0
$895$	7.50000	+	12.9904i	0.250697	+	0.434221i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	18.0000			0.600334
$900$		0			0
$901$	52.0000			1.73237
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−3.00000	+	5.19615i	−0.0996134	+	0.172535i	−0.911525	−	0.411245i	$-0.865094\pi$
$907$	−3.00000	+	5.19615i	−0.0996134	+	0.172535i	0.811911	+	0.583781i	$0.198427\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	15.0000	−	25.9808i	0.496972	−	0.860781i	−0.503022	−	0.864274i	$-0.667778\pi$
$911$	15.0000	−	25.9808i	0.496972	−	0.860781i	0.999994	+	0.00349271i	$0.00111177\pi$
$912$		0			0
$913$	−4.50000	−	7.79423i	−0.148928	−	0.257951i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−27.0000			−0.891619
$918$		0			0
$919$	−39.0000			−1.28649			−0.643246	−	0.765660i	$-0.722413\pi$
$919$	−39.0000			−1.28649			−0.643246	+	0.765660i	$0.722413\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−4.00000	+	6.92820i	−0.131519	+	0.227798i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−10.0000	+	17.3205i	−0.328089	+	0.568267i	−0.982133	−	0.188190i	$-0.939738\pi$
$929$	−10.0000	+	17.3205i	−0.328089	+	0.568267i	0.654043	+	0.756457i	$0.273071\pi$
$930$		0			0
$931$	−6.00000	−	10.3923i	−0.196642	−	0.340594i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−12.0000			−0.392442
$936$		0			0
$937$	−29.0000			−0.947389			−0.473694	−	0.880689i	$-0.657080\pi$
$937$	−29.0000			−0.947389			−0.473694	+	0.880689i	$0.657080\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	9.50000	+	16.4545i	0.309691	+	0.536401i	0.978295	−	0.207218i	$-0.0664410\pi$
$941$	9.50000	+	16.4545i	0.309691	+	0.536401i	−0.668604	+	0.743619i	$0.733108\pi$
$942$		0			0
$943$	30.0000	−	51.9615i	0.976934	−	1.69210i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−22.5000	+	38.9711i	−0.731152	+	1.26639i	0.225240	+	0.974303i	$0.427684\pi$
$947$	−22.5000	+	38.9711i	−0.731152	+	1.26639i	−0.956391	+	0.292089i	$0.905650\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−4.00000			−0.129573			−0.0647864	−	0.997899i	$-0.520637\pi$
$953$	−4.00000			−0.129573			−0.0647864	+	0.997899i	$0.520637\pi$
$954$		0			0
$955$	24.0000			0.776622
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−3.00000	−	5.19615i	−0.0968751	−	0.167793i
$960$		0			0
$961$	−25.0000	+	43.3013i	−0.806452	+	1.39682i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−6.50000	+	11.2583i	−0.209242	+	0.362418i
$966$		0			0
$967$	−25.5000	−	44.1673i	−0.820025	−	1.42032i	−0.905663	−	0.423998i	$-0.860626\pi$
$967$	−25.5000	−	44.1673i	−0.820025	−	1.42032i	0.0856383	−	0.996326i	$-0.472707\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−21.0000			−0.673922			−0.336961	−	0.941519i	$-0.609399\pi$
$971$	−21.0000			−0.673922			−0.336961	+	0.941519i	$0.609399\pi$
$972$		0			0
$973$	−36.0000			−1.15411
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−5.00000	−	8.66025i	−0.159964	−	0.277066i	0.774891	−	0.632094i	$-0.217805\pi$
$977$	−5.00000	−	8.66025i	−0.159964	−	0.277066i	−0.934856	+	0.355028i	$0.884471\pi$
$978$		0			0
$979$	21.0000	−	36.3731i	0.671163	−	1.16249i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−21.0000	+	36.3731i	−0.669796	+	1.16012i	0.308165	+	0.951333i	$0.400285\pi$
$983$	−21.0000	+	36.3731i	−0.669796	+	1.16012i	−0.977961	+	0.208788i	$0.933048\pi$
$984$		0			0
$985$	2.50000	+	4.33013i	0.0796566	+	0.137969i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−36.0000			−1.14473
$990$		0			0
$991$	51.0000			1.62007			0.810034	−	0.586383i	$-0.199448\pi$
$991$	51.0000			1.62007			0.810034	+	0.586383i	$0.199448\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1.50000	+	2.59808i	0.0475532	+	0.0823646i
$996$		0			0
$997$	−4.00000	+	6.92820i	−0.126681	+	0.219418i	−0.922389	−	0.386263i	$-0.873766\pi$
$997$	−4.00000	+	6.92820i	−0.126681	+	0.219418i	0.795708	+	0.605681i	$0.207099\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2592.2.i.i.865.1		2
3.2	odd	2		2592.2.i.m.865.1		2
4.3	odd	2		2592.2.i.l.865.1		2
9.2	odd	6		864.2.a.f.1.1	yes	1
9.4	even	3	inner	2592.2.i.i.1729.1		2
9.5	odd	6		2592.2.i.m.1729.1		2
9.7	even	3		864.2.a.h.1.1	yes	1
12.11	even	2		2592.2.i.p.865.1		2
36.7	odd	6		864.2.a.g.1.1	yes	1
36.11	even	6		864.2.a.e.1.1	✓	1
36.23	even	6		2592.2.i.p.1729.1		2
36.31	odd	6		2592.2.i.l.1729.1		2
72.11	even	6		1728.2.a.q.1.1		1
72.29	odd	6		1728.2.a.t.1.1		1
72.43	odd	6		1728.2.a.j.1.1		1
72.61	even	6		1728.2.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.2.a.e.1.1	✓	1	36.11	even	6
864.2.a.f.1.1	yes	1	9.2	odd	6
864.2.a.g.1.1	yes	1	36.7	odd	6
864.2.a.h.1.1	yes	1	9.7	even	3
1728.2.a.j.1.1		1	72.43	odd	6
1728.2.a.k.1.1		1	72.61	even	6
1728.2.a.q.1.1		1	72.11	even	6
1728.2.a.t.1.1		1	72.29	odd	6
2592.2.i.i.865.1		2	1.1	even	1	trivial
2592.2.i.i.1729.1		2	9.4	even	3	inner
2592.2.i.l.865.1		2	4.3	odd	2
2592.2.i.l.1729.1		2	36.31	odd	6
2592.2.i.m.865.1		2	3.2	odd	2
2592.2.i.m.1729.1		2	9.5	odd	6
2592.2.i.p.865.1		2	12.11	even	2
2592.2.i.p.1729.1		2	36.23	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{5} +(-1.50000 + 2.59808i) q^{7} +O(q^{10})\)	\(q+(-0.500000 - 0.866025i) q^{5} +(-1.50000 + 2.59808i) q^{7} +(-1.50000 + 2.59808i) q^{11} -4.00000 q^{17} +6.00000 q^{19} +(3.00000 + 5.19615i) q^{23} +(2.00000 - 3.46410i) q^{25} +(-1.00000 + 1.73205i) q^{29} +(-4.50000 - 7.79423i) q^{31} +3.00000 q^{35} -2.00000 q^{37} +(-5.00000 - 8.66025i) q^{41} +(-3.00000 + 5.19615i) q^{43} +(3.00000 - 5.19615i) q^{47} +(-1.00000 - 1.73205i) q^{49} -13.0000 q^{53} +3.00000 q^{55} +(-6.00000 - 10.3923i) q^{59} +(-4.00000 + 6.92820i) q^{61} +(-3.00000 - 5.19615i) q^{67} -12.0000 q^{71} +9.00000 q^{73} +(-4.50000 - 7.79423i) q^{77} +(-1.50000 + 2.59808i) q^{83} +(2.00000 + 3.46410i) q^{85} -14.0000 q^{89} +(-3.00000 - 5.19615i) q^{95} +(4.50000 - 7.79423i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{5} - 3 q^{7}+O(q^{10}) \) 2 * q - q^5 - 3 * q^7	\( 2 q - q^{5} - 3 q^{7} - 3 q^{11} - 8 q^{17} + 12 q^{19} + 6 q^{23} + 4 q^{25} - 2 q^{29} - 9 q^{31} + 6 q^{35} - 4 q^{37} - 10 q^{41} - 6 q^{43} + 6 q^{47} - 2 q^{49} - 26 q^{53} + 6 q^{55} - 12 q^{59} - 8 q^{61} - 6 q^{67} - 24 q^{71} + 18 q^{73} - 9 q^{77} - 3 q^{83} + 4 q^{85} - 28 q^{89} - 6 q^{95} + 9 q^{97}+O(q^{100}) \) 2 * q - q^5 - 3 * q^7 - 3 * q^11 - 8 * q^17 + 12 * q^19 + 6 * q^23 + 4 * q^25 - 2 * q^29 - 9 * q^31 + 6 * q^35 - 4 * q^37 - 10 * q^41 - 6 * q^43 + 6 * q^47 - 2 * q^49 - 26 * q^53 + 6 * q^55 - 12 * q^59 - 8 * q^61 - 6 * q^67 - 24 * q^71 + 18 * q^73 - 9 * q^77 - 3 * q^83 + 4 * q^85 - 28 * q^89 - 6 * q^95 + 9 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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