LMFDB - Embedded newform 2592.2.i.p.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	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.p.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	0.955901	−	0.293691i	$-0.0948835\pi$
$5$	0.500000	+	0.866025i	0.223607	+	0.387298i	−0.732294	+	0.680989i	$0.761550\pi$
$6$		0			0
$7$	1.50000	−	2.59808i	0.566947	−	0.981981i	−0.429919	−	0.902867i	$-0.641458\pi$
$7$	1.50000	−	2.59808i	0.566947	−	0.981981i	0.996866	−	0.0791130i	$-0.0252088\pi$
$8$		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	$-0.338794\pi$
$17$	4.00000			0.970143			0.485071	+	0.874475i	$0.338794\pi$
$18$		0			0
$19$	−6.00000			−1.37649			−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000			−1.37649			−0.688247	+	0.725476i	$0.741620\pi$
$20$		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	−0.758115	−	0.652121i	$-0.773880\pi$
$29$	1.00000	−	1.73205i	0.185695	−	0.321634i	0.943811	+	0.330487i	$0.107213\pi$
$30$		0			0
$31$	4.50000	+	7.79423i	0.808224	+	1.39988i	0.914093	+	0.405505i	$0.132904\pi$
$31$	4.50000	+	7.79423i	0.808224	+	1.39988i	−0.105869	+	0.994380i	$0.533762\pi$
$32$		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	$0.118557\pi$
$41$	5.00000	+	8.66025i	0.780869	+	1.35250i	−0.150567	+	0.988600i	$0.548110\pi$
$42$		0			0
$43$	3.00000	−	5.19615i	0.457496	−	0.792406i	−0.541332	−	0.840809i	$-0.682080\pi$
$43$	3.00000	−	5.19615i	0.457496	−	0.792406i	0.998828	+	0.0484030i	$0.0154132\pi$
$44$		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	$-0.148707\pi$
$53$	13.0000			1.78569			0.892844	+	0.450367i	$0.148707\pi$
$54$		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	0.989017	−	0.147802i	$-0.0472198\pi$
$67$	3.00000	+	5.19615i	0.366508	+	0.634811i	−0.622509	+	0.782613i	$0.713886\pi$
$68$		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	$-0.233878\pi$
$89$	14.0000			1.48400			0.741999	+	0.670402i	$0.233878\pi$
$90$		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	−0.637678	−	0.770303i	$-0.720105\pi$
$101$	3.50000	−	6.06218i	0.348263	−	0.603209i	0.985941	+	0.167094i	$0.0534383\pi$
$102$		0			0
$103$	−6.00000	−	10.3923i	−0.591198	−	1.02398i	−0.994071	−	0.108729i	$-0.965322\pi$
$103$	−6.00000	−	10.3923i	−0.591198	−	1.02398i	0.402874	−	0.915255i	$-0.368011\pi$
$104$		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	0.999425	−	0.0338931i	$-0.0107906\pi$
$113$	5.00000	+	8.66025i	0.470360	+	0.814688i	−0.529065	+	0.848581i	$0.677457\pi$
$114$		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	$-0.457506\pi$
$127$	3.00000			0.266207			0.133103	+	0.991102i	$0.457506\pi$
$128$		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	−0.820141	−	0.572161i	$-0.806105\pi$
$137$	1.00000	−	1.73205i	0.0854358	−	0.147979i	0.905577	+	0.424182i	$0.139438\pi$
$138$		0			0
$139$	−6.00000	−	10.3923i	−0.508913	−	0.881464i	−0.999947	−	0.0103230i	$-0.996714\pi$
$139$	−6.00000	−	10.3923i	−0.508913	−	0.881464i	0.491033	−	0.871141i	$-0.336619\pi$
$140$		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	$0.0785259\pi$
$149$	8.50000	+	14.7224i	0.696347	+	1.20611i	−0.273377	+	0.961907i	$0.588141\pi$
$150$		0			0
$151$	−1.50000	+	2.59808i	−0.122068	+	0.211428i	−0.920583	−	0.390547i	$-0.872286\pi$
$151$	−1.50000	+	2.59808i	−0.122068	+	0.211428i	0.798515	+	0.601975i	$0.205619\pi$
$152$		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	$-0.110906\pi$
$163$	24.0000			1.87983			0.939913	+	0.341415i	$0.110906\pi$
$164$		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	−0.995754	−	0.0920525i	$-0.970657\pi$
$173$	−5.50000	+	9.52628i	−0.418157	+	0.724270i	0.577597	+	0.816322i	$0.303991\pi$
$174$		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	$-0.442999\pi$
$197$	5.00000			0.356235			0.178118	+	0.984009i	$0.442999\pi$
$198$		0			0
$199$	3.00000			0.212664			0.106332	−	0.994331i	$-0.466089\pi$
$199$	3.00000			0.212664			0.106332	+	0.994331i	$0.466089\pi$
$200$		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	0.950619	−	0.310361i	$-0.100450\pi$
$211$	3.00000	+	5.19615i	0.206529	+	0.357718i	−0.744090	+	0.668079i	$0.767117\pi$
$212$		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	$0.463740\pi$
$223$	−12.0000	+	20.7846i	−0.803579	+	1.39184i	−0.917246	+	0.398321i	$0.869593\pi$
$224$		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	$-0.651644\pi$
$233$	−14.0000			−0.917170			−0.458585	+	0.888650i	$0.651644\pi$
$234$		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	0.963391	−	0.268101i	$-0.0863961\pi$
$257$	4.00000	+	6.92820i	0.249513	+	0.432169i	−0.713878	+	0.700270i	$0.753063\pi$
$258$		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	$-0.359642\pi$
$269$	14.0000			0.853595			0.426798	+	0.904347i	$0.359642\pi$
$270$		0			0
$271$	−3.00000			−0.182237			−0.0911185	−	0.995840i	$-0.529044\pi$
$271$	−3.00000			−0.182237			−0.0911185	+	0.995840i	$0.529044\pi$
$272$		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	$0.370129\pi$
$281$	−10.0000	+	17.3205i	−0.596550	+	1.03325i	−0.993326	+	0.115339i	$0.963204\pi$
$282$		0			0
$283$	−3.00000	−	5.19615i	−0.178331	−	0.308879i	0.762978	−	0.646425i	$-0.223737\pi$
$283$	−3.00000	−	5.19615i	−0.178331	−	0.308879i	−0.941309	+	0.337546i	$0.890403\pi$
$284$		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	0.682204	−	0.731162i	$-0.261022\pi$
$293$	−5.00000	−	8.66025i	−0.292103	−	0.505937i	−0.974307	+	0.225225i	$0.927688\pi$
$294$		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	$-0.611253\pi$
$307$	−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	−0.999928	−	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	−0.750838	−	0.660487i	$-0.770350\pi$
$317$	3.50000	−	6.06218i	0.196580	−	0.340486i	0.947417	+	0.320001i	$0.103683\pi$
$318$		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	$-0.524804\pi$
$331$	15.0000	−	25.9808i	0.824475	−	1.42803i	0.902320	−	0.431066i	$-0.141863\pi$
$332$		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	−0.891410	−	0.453197i	$-0.850283\pi$
$353$	−1.00000	+	1.73205i	−0.0532246	+	0.0921878i	0.838186	+	0.545385i	$0.183617\pi$
$354$		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	$-0.648686\pi$
$367$	10.5000	−	18.1865i	0.548096	−	0.949329i	0.998405	−	0.0564568i	$-0.0179803\pi$
$368$		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	$-0.875599\pi$
$379$	−36.0000			−1.84920			−0.924598	+	0.380945i	$0.875599\pi$
$380$		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	−0.941009	−	0.338382i	$-0.890120\pi$
$389$	−3.50000	+	6.06218i	−0.177457	+	0.307365i	0.763552	+	0.645747i	$0.223454\pi$
$390$		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	$-0.920239\pi$
$401$	−14.0000	−	24.2487i	−0.699127	−	1.21092i	0.269643	−	0.962960i	$-0.413094\pi$
$402$		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	−0.738429	−	0.674331i	$-0.764432\pi$
$439$	4.50000	−	7.79423i	0.214773	−	0.371998i	0.953202	+	0.302333i	$0.0977654\pi$
$440$		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	$-0.673741\pi$
$449$	−22.0000			−1.03824			−0.519122	+	0.854700i	$0.673741\pi$
$450$		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	−0.935947	−	0.352140i	$-0.885454\pi$
$461$	−3.50000	+	6.06218i	−0.163011	+	0.282344i	0.772936	+	0.634484i	$0.218787\pi$
$462$		0			0
$463$	7.50000	+	12.9904i	0.348555	+	0.603714i	0.985993	−	0.166787i	$-0.0533393\pi$
$463$	7.50000	+	12.9904i	0.348555	+	0.603714i	−0.637438	+	0.770501i	$0.720006\pi$
$464$		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	$-0.683004\pi$
$487$	−24.0000			−1.08754			−0.543772	+	0.839233i	$0.683004\pi$
$488$		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	0.925329	−	0.379165i	$-0.123789\pi$
$499$	3.00000	+	5.19615i	0.134298	+	0.232612i	−0.791031	+	0.611776i	$0.790455\pi$
$500$		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	$-0.860641\pi$
$509$	−18.5000	−	32.0429i	−0.819998	−	1.42028i	0.0856847	−	0.996322i	$-0.472692\pi$
$510$		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	$-0.747251\pi$
$521$	−32.0000			−1.40195			−0.700973	+	0.713188i	$0.747251\pi$
$522$		0			0
$523$	36.0000			1.57417			0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000			1.57417			0.787085	+	0.616844i	$0.211589\pi$
$524$		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	$-0.553777\pi$
$547$	18.0000	−	31.1769i	0.769624	−	1.33303i	0.937767	−	0.347266i	$-0.112890\pi$
$548$		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	$-0.834988\pi$
$557$	−41.0000			−1.73723			−0.868613	+	0.495491i	$0.834988\pi$
$558$		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	−0.576640	−	0.816999i	$-0.695636\pi$
$569$	10.0000	−	17.3205i	0.419222	−	0.726113i	0.995861	+	0.0908852i	$0.0289696\pi$
$570$		0			0
$571$	12.0000	+	20.7846i	0.502184	+	0.869809i	0.999997	+	0.00252413i	$0.000803457\pi$
$571$	12.0000	+	20.7846i	0.502184	+	0.869809i	−0.497812	+	0.867285i	$0.665863\pi$
$572$		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	$-0.592809\pi$
$593$	−14.0000			−0.574911			−0.287456	+	0.957794i	$0.592809\pi$
$594$		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	0.512824	−	0.858494i	$-0.328599\pi$
$607$	−12.0000	−	20.7846i	−0.487065	−	0.843621i	−0.999889	+	0.0148722i	$0.995266\pi$
$608$		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	$-0.991345\pi$
$617$	−13.0000	−	22.5167i	−0.523360	−	0.906487i	0.476270	−	0.879299i	$-0.341988\pi$
$618$		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	$-0.680603\pi$
$631$	−27.0000			−1.07485			−0.537427	+	0.843311i	$0.680603\pi$
$632$		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	−0.845601	−	0.533816i	$-0.820758\pi$
$641$	1.00000	−	1.73205i	0.0394976	−	0.0684119i	0.885098	+	0.465404i	$0.154091\pi$
$642$		0			0
$643$	−18.0000	−	31.1769i	−0.709851	−	1.22950i	−0.964912	−	0.262573i	$-0.915429\pi$
$643$	−18.0000	−	31.1769i	−0.709851	−	1.22950i	0.255062	−	0.966925i	$-0.417904\pi$
$644$		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	$-0.926544\pi$
$653$	−17.5000	−	30.3109i	−0.684828	−	1.18616i	0.288663	−	0.957431i	$-0.406789\pi$
$654$		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	−0.884602	−	0.466347i	$-0.845570\pi$
$677$	−1.00000	+	1.73205i	−0.0384331	+	0.0665681i	0.846169	+	0.532915i	$0.178903\pi$
$678$		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	−0.729039	−	0.684472i	$-0.760033\pi$
$691$	6.00000	−	10.3923i	0.228251	−	0.395342i	0.957290	+	0.289130i	$0.0933661\pi$
$692$		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	$-0.421047\pi$
$701$	13.0000			0.491003			0.245502	+	0.969396i	$0.421047\pi$
$702$		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	−0.992372	−	0.123279i	$-0.960659\pi$
$727$	−10.5000	+	18.1865i	−0.389423	+	0.674501i	0.602949	+	0.797780i	$0.293992\pi$
$728$		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	$-0.429161\pi$
$739$	12.0000			0.441427			0.220714	+	0.975339i	$0.429161\pi$
$740$		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	0.507342	−	0.861745i	$-0.330628\pi$
$751$	−13.5000	−	23.3827i	−0.492622	−	0.853246i	−0.999964	+	0.00849853i	$0.997295\pi$
$752$		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	0.784373	−	0.620289i	$-0.212985\pi$
$761$	−4.00000	−	6.92820i	−0.145000	−	0.251147i	−0.929373	+	0.369142i	$0.879652\pi$
$762$		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	$-0.855842\pi$
$773$	−50.0000			−1.79838			−0.899188	+	0.437564i	$0.855842\pi$
$774$		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	$0.0607926\pi$
$797$	18.5000	+	32.0429i	0.655304	+	1.13502i	−0.326514	+	0.945192i	$0.605874\pi$
$798$		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	$-0.158243\pi$
$809$	50.0000			1.75791			0.878953	+	0.476908i	$0.158243\pi$
$810$		0			0
$811$	18.0000			0.632065			0.316033	−	0.948748i	$-0.397649\pi$
$811$	18.0000			0.632065			0.316033	+	0.948748i	$0.397649\pi$
$812$		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	$0.397428\pi$
$821$	−19.0000	+	32.9090i	−0.663105	+	1.14853i	−0.979795	+	0.200002i	$0.935905\pi$
$822$		0			0
$823$	−13.5000	−	23.3827i	−0.470580	−	0.815069i	0.528853	−	0.848713i	$-0.322622\pi$
$823$	−13.5000	−	23.3827i	−0.470580	−	0.815069i	−0.999434	+	0.0336440i	$0.989289\pi$
$824$		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	$-0.485329\pi$
$857$	26.0000	−	45.0333i	0.888143	−	1.53831i	0.842068	−	0.539371i	$-0.181338\pi$
$858$		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	$-0.681218\pi$
$881$	−32.0000			−1.07811			−0.539054	+	0.842271i	$0.681218\pi$
$882$		0			0
$883$	18.0000			0.605748			0.302874	−	0.953031i	$-0.402054\pi$
$883$	18.0000			0.605748			0.302874	+	0.953031i	$0.402054\pi$
$884$		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	−0.811911	−	0.583781i	$-0.801573\pi$
$907$	3.00000	−	5.19615i	0.0996134	−	0.172535i	0.911525	+	0.411245i	$0.134906\pi$
$908$		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	$-0.277587\pi$
$919$	39.0000			1.28649			0.643246	+	0.765660i	$0.277587\pi$
$920$		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	−0.654043	−	0.756457i	$-0.726929\pi$
$929$	10.0000	−	17.3205i	0.328089	−	0.568267i	0.982133	+	0.188190i	$0.0602620\pi$
$930$		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	0.668604	−	0.743619i	$-0.266892\pi$
$941$	−9.50000	−	16.4545i	−0.309691	−	0.536401i	−0.978295	+	0.207218i	$0.933559\pi$
$942$		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	$-0.479363\pi$
$953$	4.00000			0.129573			0.0647864	+	0.997899i	$0.479363\pi$
$954$		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	$0.139374\pi$
$967$	25.5000	+	44.1673i	0.820025	+	1.42032i	−0.0856383	+	0.996326i	$0.527293\pi$
$968$		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	0.934856	−	0.355028i	$-0.115529\pi$
$977$	5.00000	+	8.66025i	0.159964	+	0.277066i	−0.774891	+	0.632094i	$0.782195\pi$
$978$		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	$-0.800552\pi$
$991$	−51.0000			−1.62007			−0.810034	+	0.586383i	$0.800552\pi$
$992$		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.p.865.1		2
3.2	odd	2		2592.2.i.l.865.1		2
4.3	odd	2		2592.2.i.m.865.1		2
9.2	odd	6		864.2.a.g.1.1	yes	1
9.4	even	3	inner	2592.2.i.p.1729.1		2
9.5	odd	6		2592.2.i.l.1729.1		2
9.7	even	3		864.2.a.e.1.1	✓	1
12.11	even	2		2592.2.i.i.865.1		2
36.7	odd	6		864.2.a.f.1.1	yes	1
36.11	even	6		864.2.a.h.1.1	yes	1
36.23	even	6		2592.2.i.i.1729.1		2
36.31	odd	6		2592.2.i.m.1729.1		2
72.11	even	6		1728.2.a.k.1.1		1
72.29	odd	6		1728.2.a.j.1.1		1
72.43	odd	6		1728.2.a.t.1.1		1
72.61	even	6		1728.2.a.q.1.1		1

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

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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