LMFDB - Embedded newform 1764.2.k.i.361.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,2,Mod(361,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, 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("1764.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1764.k (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:	$14.0856109166$
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:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 588)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1764.361
Dual form			1764.2.k.i.1549.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 + 1.73205i) q^{5} +O(q^{10})$	$q+(1.00000 + 1.73205i) q^{5} +(1.00000 - 1.73205i) q^{11} -4.00000 q^{13} +(3.00000 - 5.19615i) q^{17} +(-4.00000 - 6.92820i) q^{19} +(-3.00000 - 5.19615i) q^{23} +(0.500000 - 0.866025i) q^{25} +10.0000 q^{29} +(-2.00000 + 3.46410i) q^{31} +(-3.00000 - 5.19615i) q^{37} +6.00000 q^{41} +4.00000 q^{43} +(4.00000 + 6.92820i) q^{47} +(1.00000 - 1.73205i) q^{53} +4.00000 q^{55} +(-2.00000 + 3.46410i) q^{59} +(4.00000 + 6.92820i) q^{61} +(-4.00000 - 6.92820i) q^{65} +(4.00000 - 6.92820i) q^{67} +10.0000 q^{71} +(-2.00000 + 3.46410i) q^{73} +(-2.00000 - 3.46410i) q^{79} -12.0000 q^{83} +12.0000 q^{85} +(-7.00000 - 12.1244i) q^{89} +(8.00000 - 13.8564i) q^{95} +4.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5}+O(q^{10}) $ 2 * q + 2 * q^5	$ 2 q + 2 q^{5} + 2 q^{11} - 8 q^{13} + 6 q^{17} - 8 q^{19} - 6 q^{23} + q^{25} + 20 q^{29} - 4 q^{31} - 6 q^{37} + 12 q^{41} + 8 q^{43} + 8 q^{47} + 2 q^{53} + 8 q^{55} - 4 q^{59} + 8 q^{61} - 8 q^{65} + 8 q^{67} + 20 q^{71} - 4 q^{73} - 4 q^{79} - 24 q^{83} + 24 q^{85} - 14 q^{89} + 16 q^{95} + 8 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 + 2 * q^11 - 8 * q^13 + 6 * q^17 - 8 * q^19 - 6 * q^23 + q^25 + 20 * q^29 - 4 * q^31 - 6 * q^37 + 12 * q^41 + 8 * q^43 + 8 * q^47 + 2 * q^53 + 8 * q^55 - 4 * q^59 + 8 * q^61 - 8 * q^65 + 8 * q^67 + 20 * q^71 - 4 * q^73 - 4 * q^79 - 24 * q^83 + 24 * q^85 - 14 * q^89 + 16 * q^95 + 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.00000	+	1.73205i	0.447214	+	0.774597i	0.998203	−	0.0599153i	$-0.0190830\pi$
$5$	1.00000	+	1.73205i	0.447214	+	0.774597i	−0.550990	+	0.834512i	$0.685750\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.00000	−	1.73205i	0.301511	−	0.522233i	−0.674967	−	0.737848i	$-0.735842\pi$
$11$	1.00000	−	1.73205i	0.301511	−	0.522233i	0.976478	+	0.215615i	$0.0691756\pi$
$12$		0			0
$13$	−4.00000			−1.10940			−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000			−1.10940			−0.554700	+	0.832050i	$0.687167\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.00000	−	5.19615i	0.727607	−	1.26025i	−0.230285	−	0.973123i	$-0.573966\pi$
$17$	3.00000	−	5.19615i	0.727607	−	1.26025i	0.957892	−	0.287129i	$-0.0927008\pi$
$18$		0			0
$19$	−4.00000	−	6.92820i	−0.917663	−	1.58944i	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−4.00000	−	6.92820i	−0.917663	−	1.58944i	−0.114708	−	0.993399i	$-0.536593\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.00000	−	5.19615i	−0.625543	−	1.08347i	−0.988436	−	0.151642i	$-0.951544\pi$
$23$	−3.00000	−	5.19615i	−0.625543	−	1.08347i	0.362892	−	0.931831i	$-0.381789\pi$
$24$		0			0
$25$	0.500000	−	0.866025i	0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	10.0000			1.85695			0.928477	−	0.371391i	$-0.121119\pi$
$29$	10.0000			1.85695			0.928477	+	0.371391i	$0.121119\pi$
$30$		0			0
$31$	−2.00000	+	3.46410i	−0.359211	+	0.622171i	−0.987829	−	0.155543i	$-0.950287\pi$
$31$	−2.00000	+	3.46410i	−0.359211	+	0.622171i	0.628619	+	0.777714i	$0.283621\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−3.00000	−	5.19615i	−0.493197	−	0.854242i	0.506772	−	0.862080i	$-0.330838\pi$
$37$	−3.00000	−	5.19615i	−0.493197	−	0.854242i	−0.999969	+	0.00783774i	$0.997505\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	6.00000			0.937043			0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000			0.937043			0.468521	+	0.883452i	$0.344787\pi$
$42$		0			0
$43$	4.00000			0.609994			0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000			0.609994			0.304997	+	0.952353i	$0.401344\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	4.00000	+	6.92820i	0.583460	+	1.01058i	0.995066	+	0.0992202i	$0.0316348\pi$
$47$	4.00000	+	6.92820i	0.583460	+	1.01058i	−0.411606	+	0.911362i	$0.635032\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.00000	−	1.73205i	0.137361	−	0.237915i	−0.789136	−	0.614218i	$-0.789471\pi$
$53$	1.00000	−	1.73205i	0.137361	−	0.237915i	0.926497	+	0.376303i	$0.122805\pi$
$54$		0			0
$55$	4.00000			0.539360
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−2.00000	+	3.46410i	−0.260378	+	0.450988i	−0.966342	−	0.257260i	$-0.917180\pi$
$59$	−2.00000	+	3.46410i	−0.260378	+	0.450988i	0.705965	+	0.708247i	$0.250514\pi$
$60$		0			0
$61$	4.00000	+	6.92820i	0.512148	+	0.887066i	0.999901	+	0.0140840i	$0.00448323\pi$
$61$	4.00000	+	6.92820i	0.512148	+	0.887066i	−0.487753	+	0.872982i	$0.662183\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−4.00000	−	6.92820i	−0.496139	−	0.859338i
$66$		0			0
$67$	4.00000	−	6.92820i	0.488678	−	0.846415i	−0.511237	−	0.859440i	$-0.670813\pi$
$67$	4.00000	−	6.92820i	0.488678	−	0.846415i	0.999915	+	0.0130248i	$0.00414604\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	10.0000			1.18678			0.593391	−	0.804914i	$-0.297789\pi$
$71$	10.0000			1.18678			0.593391	+	0.804914i	$0.297789\pi$
$72$		0			0
$73$	−2.00000	+	3.46410i	−0.234082	+	0.405442i	−0.959006	−	0.283387i	$-0.908542\pi$
$73$	−2.00000	+	3.46410i	−0.234082	+	0.405442i	0.724923	+	0.688830i	$0.241875\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−2.00000	−	3.46410i	−0.225018	−	0.389742i	0.731307	−	0.682048i	$-0.238911\pi$
$79$	−2.00000	−	3.46410i	−0.225018	−	0.389742i	−0.956325	+	0.292306i	$0.905577\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−12.0000			−1.31717			−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000			−1.31717			−0.658586	+	0.752506i	$0.728845\pi$
$84$		0			0
$85$	12.0000			1.30158
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−7.00000	−	12.1244i	−0.741999	−	1.28518i	−0.951584	−	0.307389i	$-0.900545\pi$
$89$	−7.00000	−	12.1244i	−0.741999	−	1.28518i	0.209585	−	0.977790i	$-0.432789\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	8.00000	−	13.8564i	0.820783	−	1.42164i
$96$		0			0
$97$	4.00000			0.406138			0.203069	−	0.979164i	$-0.434908\pi$
$97$	4.00000			0.406138			0.203069	+	0.979164i	$0.434908\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−5.00000	+	8.66025i	−0.497519	+	0.861727i	−0.999996	−	0.00286291i	$-0.999089\pi$
$101$	−5.00000	+	8.66025i	−0.497519	+	0.861727i	0.502477	+	0.864590i	$0.332422\pi$
$102$		0			0
$103$	2.00000	+	3.46410i	0.197066	+	0.341328i	0.947576	−	0.319531i	$-0.103525\pi$
$103$	2.00000	+	3.46410i	0.197066	+	0.341328i	−0.750510	+	0.660859i	$0.770192\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.00000	−	12.1244i	−0.676716	−	1.17211i	−0.975964	−	0.217931i	$-0.930069\pi$
$107$	−7.00000	−	12.1244i	−0.676716	−	1.17211i	0.299249	−	0.954175i	$-0.403264\pi$
$108$		0			0
$109$	−5.00000	+	8.66025i	−0.478913	+	0.829502i	−0.999708	−	0.0241802i	$-0.992302\pi$
$109$	−5.00000	+	8.66025i	−0.478913	+	0.829502i	0.520794	+	0.853682i	$0.325636\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	14.0000			1.31701			0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000			1.31701			0.658505	+	0.752577i	$0.271189\pi$
$114$		0			0
$115$	6.00000	−	10.3923i	0.559503	−	0.969087i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	3.50000	+	6.06218i	0.318182	+	0.551107i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	12.0000			1.07331
$126$		0			0
$127$	−4.00000			−0.354943			−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000			−0.354943			−0.177471	+	0.984126i	$0.556792\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−6.00000	−	10.3923i	−0.524222	−	0.907980i	−0.999602	−	0.0281993i	$-0.991023\pi$
$131$	−6.00000	−	10.3923i	−0.524222	−	0.907980i	0.475380	−	0.879781i	$-0.342311\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−5.00000	+	8.66025i	−0.427179	+	0.739895i	−0.996621	−	0.0821359i	$-0.973826\pi$
$137$	−5.00000	+	8.66025i	−0.427179	+	0.739895i	0.569442	+	0.822031i	$0.307159\pi$
$138$		0			0
$139$	12.0000			1.01783			0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000			1.01783			0.508913	+	0.860818i	$0.330047\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−4.00000	+	6.92820i	−0.334497	+	0.579365i
$144$		0			0
$145$	10.0000	+	17.3205i	0.830455	+	1.43839i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1.00000	+	1.73205i	0.0819232	+	0.141895i	0.904076	−	0.427372i	$-0.140560\pi$
$149$	1.00000	+	1.73205i	0.0819232	+	0.141895i	−0.822153	+	0.569267i	$0.807227\pi$
$150$		0			0
$151$	4.00000	−	6.92820i	0.325515	−	0.563809i	−0.656101	−	0.754673i	$-0.727796\pi$
$151$	4.00000	−	6.92820i	0.325515	−	0.563809i	0.981617	+	0.190864i	$0.0611289\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−8.00000			−0.642575
$156$		0			0
$157$	8.00000	−	13.8564i	0.638470	−	1.10586i	−0.347299	−	0.937754i	$-0.612901\pi$
$157$	8.00000	−	13.8564i	0.638470	−	1.10586i	0.985769	−	0.168107i	$-0.0537655\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−12.0000	−	20.7846i	−0.939913	−	1.62798i	−0.765631	−	0.643280i	$-0.777573\pi$
$163$	−12.0000	−	20.7846i	−0.939913	−	1.62798i	−0.174282	−	0.984696i	$-0.555760\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−16.0000			−1.23812			−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000			−1.23812			−0.619059	+	0.785345i	$0.712486\pi$
$168$		0			0
$169$	3.00000			0.230769
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.00000	−	1.73205i	−0.0760286	−	0.131685i	0.825505	−	0.564396i	$-0.190891\pi$
$173$	−1.00000	−	1.73205i	−0.0760286	−	0.131685i	−0.901533	+	0.432710i	$0.857557\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	3.00000	−	5.19615i	0.224231	−	0.388379i	−0.731858	−	0.681457i	$-0.761346\pi$
$179$	3.00000	−	5.19615i	0.224231	−	0.388379i	0.956088	+	0.293079i	$0.0946798\pi$
$180$		0			0
$181$	−20.0000			−1.48659			−0.743294	−	0.668965i	$-0.766738\pi$
$181$	−20.0000			−1.48659			−0.743294	+	0.668965i	$0.766738\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	6.00000	−	10.3923i	0.441129	−	0.764057i
$186$		0			0
$187$	−6.00000	−	10.3923i	−0.438763	−	0.759961i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	3.00000	+	5.19615i	0.217072	+	0.375980i	0.953912	−	0.300088i	$-0.0970159\pi$
$191$	3.00000	+	5.19615i	0.217072	+	0.375980i	−0.736839	+	0.676068i	$0.763683\pi$
$192$		0			0
$193$	5.00000	−	8.66025i	0.359908	−	0.623379i	−0.628037	−	0.778183i	$-0.716141\pi$
$193$	5.00000	−	8.66025i	0.359908	−	0.623379i	0.987945	+	0.154805i	$0.0494748\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	6.00000			0.427482			0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000			0.427482			0.213741	+	0.976890i	$0.431435\pi$
$198$		0			0
$199$	4.00000	−	6.92820i	0.283552	−	0.491127i	−0.688705	−	0.725042i	$-0.741820\pi$
$199$	4.00000	−	6.92820i	0.283552	−	0.491127i	0.972257	+	0.233915i	$0.0751537\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	6.00000	+	10.3923i	0.419058	+	0.725830i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−16.0000			−1.10674
$210$		0			0
$211$	−4.00000			−0.275371			−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000			−0.275371			−0.137686	+	0.990476i	$0.543966\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4.00000	+	6.92820i	0.272798	+	0.472500i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−12.0000	+	20.7846i	−0.807207	+	1.39812i
$222$		0			0
$223$	−8.00000			−0.535720			−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000			−0.535720			−0.267860	+	0.963458i	$0.586316\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2.00000	−	3.46410i	0.132745	−	0.229920i	−0.791989	−	0.610535i	$-0.790954\pi$
$227$	2.00000	−	3.46410i	0.132745	−	0.229920i	0.924734	+	0.380615i	$0.124288\pi$
$228$		0			0
$229$	−10.0000	−	17.3205i	−0.660819	−	1.14457i	−0.980401	−	0.197013i	$-0.936876\pi$
$229$	−10.0000	−	17.3205i	−0.660819	−	1.14457i	0.319582	−	0.947559i	$-0.396457\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	7.00000	+	12.1244i	0.458585	+	0.794293i	0.998886	−	0.0471787i	$-0.0150230\pi$
$233$	7.00000	+	12.1244i	0.458585	+	0.794293i	−0.540301	+	0.841472i	$0.681690\pi$
$234$		0			0
$235$	−8.00000	+	13.8564i	−0.521862	+	0.903892i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−18.0000			−1.16432			−0.582162	−	0.813073i	$-0.697793\pi$
$239$	−18.0000			−1.16432			−0.582162	+	0.813073i	$0.697793\pi$
$240$		0			0
$241$	−14.0000	+	24.2487i	−0.901819	+	1.56200i	−0.0766885	+	0.997055i	$0.524435\pi$
$241$	−14.0000	+	24.2487i	−0.901819	+	1.56200i	−0.825131	+	0.564942i	$0.808899\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	16.0000	+	27.7128i	1.01806	+	1.76332i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	28.0000			1.76734			0.883672	−	0.468106i	$-0.155064\pi$
$251$	28.0000			1.76734			0.883672	+	0.468106i	$0.155064\pi$
$252$		0			0
$253$	−12.0000			−0.754434
$254$		0			0
$255$		0			0
$256$		0			0
$257$	5.00000	+	8.66025i	0.311891	+	0.540212i	0.978772	−	0.204953i	$-0.0657041\pi$
$257$	5.00000	+	8.66025i	0.311891	+	0.540212i	−0.666880	+	0.745165i	$0.732371\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	13.0000	−	22.5167i	0.801614	−	1.38844i	−0.116939	−	0.993139i	$-0.537308\pi$
$263$	13.0000	−	22.5167i	0.801614	−	1.38844i	0.918553	−	0.395298i	$-0.129359\pi$
$264$		0			0
$265$	4.00000			0.245718
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1.00000	−	1.73205i	0.0609711	−	0.105605i	−0.833929	−	0.551872i	$-0.813914\pi$
$269$	1.00000	−	1.73205i	0.0609711	−	0.105605i	0.894900	+	0.446267i	$0.147247\pi$
$270$		0			0
$271$	2.00000	+	3.46410i	0.121491	+	0.210429i	0.920356	−	0.391082i	$-0.127899\pi$
$271$	2.00000	+	3.46410i	0.121491	+	0.210429i	−0.798865	+	0.601511i	$0.794566\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.00000	−	1.73205i	−0.0603023	−	0.104447i
$276$		0			0
$277$	−5.00000	+	8.66025i	−0.300421	+	0.520344i	−0.976231	−	0.216731i	$-0.930460\pi$
$277$	−5.00000	+	8.66025i	−0.300421	+	0.520344i	0.675810	+	0.737075i	$0.263794\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−6.00000			−0.357930			−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000			−0.357930			−0.178965	+	0.983855i	$0.557275\pi$
$282$		0			0
$283$	−8.00000	+	13.8564i	−0.475551	+	0.823678i	−0.999608	−	0.0280052i	$-0.991084\pi$
$283$	−8.00000	+	13.8564i	−0.475551	+	0.823678i	0.524057	+	0.851683i	$0.324418\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−9.50000	−	16.4545i	−0.558824	−	0.967911i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	18.0000			1.05157			0.525786	−	0.850617i	$-0.323771\pi$
$293$	18.0000			1.05157			0.525786	+	0.850617i	$0.323771\pi$
$294$		0			0
$295$	−8.00000			−0.465778
$296$		0			0
$297$		0			0
$298$		0			0
$299$	12.0000	+	20.7846i	0.693978	+	1.20201i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−8.00000	+	13.8564i	−0.458079	+	0.793416i
$306$		0			0
$307$	16.0000			0.913168			0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000			0.913168			0.456584	+	0.889680i	$0.349073\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	8.00000	−	13.8564i	0.453638	−	0.785725i	−0.544970	−	0.838455i	$-0.683459\pi$
$311$	8.00000	−	13.8564i	0.453638	−	0.785725i	0.998609	+	0.0527306i	$0.0167924\pi$
$312$		0			0
$313$	−12.0000	−	20.7846i	−0.678280	−	1.17482i	−0.975499	−	0.220006i	$-0.929392\pi$
$313$	−12.0000	−	20.7846i	−0.678280	−	1.17482i	0.297218	−	0.954810i	$-0.403941\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	9.00000	+	15.5885i	0.505490	+	0.875535i	0.999980	+	0.00635137i	$0.00202172\pi$
$317$	9.00000	+	15.5885i	0.505490	+	0.875535i	−0.494489	+	0.869184i	$0.664645\pi$
$318$		0			0
$319$	10.0000	−	17.3205i	0.559893	−	0.969762i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−48.0000			−2.67079
$324$		0			0
$325$	−2.00000	+	3.46410i	−0.110940	+	0.192154i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	2.00000	+	3.46410i	0.109930	+	0.190404i	0.915742	−	0.401768i	$-0.131604\pi$
$331$	2.00000	+	3.46410i	0.109930	+	0.190404i	−0.805812	+	0.592172i	$0.798271\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	16.0000			0.874173
$336$		0			0
$337$	14.0000			0.762629			0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000			0.762629			0.381314	+	0.924445i	$0.375472\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	4.00000	+	6.92820i	0.216612	+	0.375183i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−9.00000	+	15.5885i	−0.483145	+	0.836832i	−0.999813	−	0.0193540i	$-0.993839\pi$
$347$	−9.00000	+	15.5885i	−0.483145	+	0.836832i	0.516667	+	0.856186i	$0.327172\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	3.00000	−	5.19615i	0.159674	−	0.276563i	−0.775077	−	0.631867i	$-0.782289\pi$
$353$	3.00000	−	5.19615i	0.159674	−	0.276563i	0.934751	+	0.355303i	$0.115622\pi$
$354$		0			0
$355$	10.0000	+	17.3205i	0.530745	+	0.919277i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	11.0000	+	19.0526i	0.580558	+	1.00556i	0.995413	+	0.0956683i	$0.0304988\pi$
$359$	11.0000	+	19.0526i	0.580558	+	1.00556i	−0.414855	+	0.909887i	$0.636168\pi$
$360$		0			0
$361$	−22.5000	+	38.9711i	−1.18421	+	2.05111i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−8.00000			−0.418739
$366$		0			0
$367$	−8.00000	+	13.8564i	−0.417597	+	0.723299i	−0.995697	−	0.0926670i	$-0.970461\pi$
$367$	−8.00000	+	13.8564i	−0.417597	+	0.723299i	0.578101	+	0.815966i	$0.303794\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	13.0000	+	22.5167i	0.673114	+	1.16587i	0.977016	+	0.213165i	$0.0683772\pi$
$373$	13.0000	+	22.5167i	0.673114	+	1.16587i	−0.303902	+	0.952703i	$0.598289\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−40.0000			−2.06010
$378$		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
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−9.00000	+	15.5885i	−0.456318	+	0.790366i	−0.998763	−	0.0497253i	$-0.984165\pi$
$389$	−9.00000	+	15.5885i	−0.456318	+	0.790366i	0.542445	+	0.840091i	$0.317499\pi$
$390$		0			0
$391$	−36.0000			−1.82060
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4.00000	−	6.92820i	0.201262	−	0.348596i
$396$		0			0
$397$	−16.0000	−	27.7128i	−0.803017	−	1.39087i	−0.917622	−	0.397455i	$-0.869893\pi$
$397$	−16.0000	−	27.7128i	−0.803017	−	1.39087i	0.114605	−	0.993411i	$-0.463440\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	15.0000	+	25.9808i	0.749064	+	1.29742i	0.948272	+	0.317460i	$0.102830\pi$
$401$	15.0000	+	25.9808i	0.749064	+	1.29742i	−0.199207	+	0.979957i	$0.563837\pi$
$402$		0			0
$403$	8.00000	−	13.8564i	0.398508	−	0.690237i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−12.0000			−0.594818
$408$		0			0
$409$	−6.00000	+	10.3923i	−0.296681	+	0.513866i	−0.975375	−	0.220555i	$-0.929213\pi$
$409$	−6.00000	+	10.3923i	−0.296681	+	0.513866i	0.678694	+	0.734422i	$0.262546\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−12.0000	−	20.7846i	−0.589057	−	1.02028i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	4.00000			0.195413			0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000			0.195413			0.0977064	+	0.995215i	$0.468849\pi$
$420$		0			0
$421$	−6.00000			−0.292422			−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000			−0.292422			−0.146211	+	0.989253i	$0.546708\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3.00000	−	5.19615i	−0.145521	−	0.252050i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−1.00000	+	1.73205i	−0.0481683	+	0.0834300i	−0.889104	−	0.457705i	$-0.848672\pi$
$431$	−1.00000	+	1.73205i	−0.0481683	+	0.0834300i	0.840936	+	0.541135i	$0.182005\pi$
$432$		0			0
$433$	32.0000			1.53782			0.768911	−	0.639356i	$-0.220799\pi$
$433$	32.0000			1.53782			0.768911	+	0.639356i	$0.220799\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−24.0000	+	41.5692i	−1.14808	+	1.98853i
$438$		0			0
$439$	12.0000	+	20.7846i	0.572729	+	0.991995i	0.996284	+	0.0861252i	$0.0274485\pi$
$439$	12.0000	+	20.7846i	0.572729	+	0.991995i	−0.423556	+	0.905870i	$0.639218\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−9.00000	−	15.5885i	−0.427603	−	0.740630i	0.569057	−	0.822298i	$-0.307309\pi$
$443$	−9.00000	−	15.5885i	−0.427603	−	0.740630i	−0.996660	+	0.0816684i	$0.973975\pi$
$444$		0			0
$445$	14.0000	−	24.2487i	0.663664	−	1.14950i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	6.00000			0.283158			0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000			0.283158			0.141579	+	0.989927i	$0.454782\pi$
$450$		0			0
$451$	6.00000	−	10.3923i	0.282529	−	0.489355i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	11.0000	+	19.0526i	0.514558	+	0.891241i	0.999857	+	0.0168929i	$0.00537742\pi$
$457$	11.0000	+	19.0526i	0.514558	+	0.891241i	−0.485299	+	0.874348i	$0.661289\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	18.0000			0.838344			0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000			0.838344			0.419172	+	0.907907i	$0.362320\pi$
$462$		0			0
$463$	−4.00000			−0.185896			−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000			−0.185896			−0.0929479	+	0.995671i	$0.529629\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−10.0000	−	17.3205i	−0.462745	−	0.801498i	0.536352	−	0.843995i	$-0.319802\pi$
$467$	−10.0000	−	17.3205i	−0.462745	−	0.801498i	−0.999097	+	0.0424970i	$0.986469\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	4.00000	−	6.92820i	0.183920	−	0.318559i
$474$		0			0
$475$	−8.00000			−0.367065
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$	12.0000	+	20.7846i	0.547153	+	0.947697i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	4.00000	+	6.92820i	0.181631	+	0.314594i
$486$		0			0
$487$	−8.00000	+	13.8564i	−0.362515	+	0.627894i	−0.988374	−	0.152042i	$-0.951415\pi$
$487$	−8.00000	+	13.8564i	−0.362515	+	0.627894i	0.625859	+	0.779936i	$0.284748\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−6.00000			−0.270776			−0.135388	−	0.990793i	$-0.543228\pi$
$491$	−6.00000			−0.270776			−0.135388	+	0.990793i	$0.543228\pi$
$492$		0			0
$493$	30.0000	−	51.9615i	1.35113	−	2.34023i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	18.0000	+	31.1769i	0.805791	+	1.39567i	0.915756	+	0.401735i	$0.131593\pi$
$499$	18.0000	+	31.1769i	0.805791	+	1.39567i	−0.109965	+	0.993935i	$0.535074\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−16.0000			−0.713405			−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000			−0.713405			−0.356702	+	0.934218i	$0.616099\pi$
$504$		0			0
$505$	−20.0000			−0.889988
$506$		0			0
$507$		0			0
$508$		0			0
$509$	5.00000	+	8.66025i	0.221621	+	0.383859i	0.955300	−	0.295637i	$-0.0955319\pi$
$509$	5.00000	+	8.66025i	0.221621	+	0.383859i	−0.733679	+	0.679496i	$0.762199\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−4.00000	+	6.92820i	−0.176261	+	0.305293i
$516$		0			0
$517$	16.0000			0.703679
$518$		0			0
$519$		0			0
$520$		0			0
$521$	11.0000	−	19.0526i	0.481919	−	0.834708i	−0.517866	−	0.855462i	$-0.673273\pi$
$521$	11.0000	−	19.0526i	0.481919	−	0.834708i	0.999785	+	0.0207541i	$0.00660670\pi$
$522$		0			0
$523$	2.00000	+	3.46410i	0.0874539	+	0.151475i	0.906434	−	0.422347i	$-0.138794\pi$
$523$	2.00000	+	3.46410i	0.0874539	+	0.151475i	−0.818980	+	0.573822i	$0.805460\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	12.0000	+	20.7846i	0.522728	+	0.905392i
$528$		0			0
$529$	−6.50000	+	11.2583i	−0.282609	+	0.489493i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−24.0000			−1.03956
$534$		0			0
$535$	14.0000	−	24.2487i	0.605273	−	1.04836i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−17.0000	−	29.4449i	−0.730887	−	1.26593i	−0.956504	−	0.291718i	$-0.905773\pi$
$541$	−17.0000	−	29.4449i	−0.730887	−	1.26593i	0.225617	−	0.974216i	$-0.427560\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−20.0000			−0.856706
$546$		0			0
$547$	16.0000			0.684111			0.342055	−	0.939680i	$-0.388877\pi$
$547$	16.0000			0.684111			0.342055	+	0.939680i	$0.388877\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−40.0000	−	69.2820i	−1.70406	−	2.95151i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−3.00000	+	5.19615i	−0.127114	+	0.220168i	−0.922557	−	0.385860i	$-0.873905\pi$
$557$	−3.00000	+	5.19615i	−0.127114	+	0.220168i	0.795443	+	0.606028i	$0.207238\pi$
$558$		0			0
$559$	−16.0000			−0.676728
$560$		0			0
$561$		0			0
$562$		0			0
$563$	2.00000	−	3.46410i	0.0842900	−	0.145994i	−0.820798	−	0.571218i	$-0.806471\pi$
$563$	2.00000	−	3.46410i	0.0842900	−	0.145994i	0.905088	+	0.425223i	$0.139804\pi$
$564$		0			0
$565$	14.0000	+	24.2487i	0.588984	+	1.02015i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	3.00000	+	5.19615i	0.125767	+	0.217834i	0.922032	−	0.387113i	$-0.126528\pi$
$569$	3.00000	+	5.19615i	0.125767	+	0.217834i	−0.796266	+	0.604947i	$0.793194\pi$
$570$		0			0
$571$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$571$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−6.00000			−0.250217
$576$		0			0
$577$	4.00000	−	6.92820i	0.166522	−	0.288425i	−0.770673	−	0.637231i	$-0.780080\pi$
$577$	4.00000	−	6.92820i	0.166522	−	0.288425i	0.937195	+	0.348806i	$0.113413\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−2.00000	−	3.46410i	−0.0828315	−	0.143468i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	4.00000			0.165098			0.0825488	−	0.996587i	$-0.473694\pi$
$587$	4.00000			0.165098			0.0825488	+	0.996587i	$0.473694\pi$
$588$		0			0
$589$	32.0000			1.31854
$590$		0			0
$591$		0			0
$592$		0			0
$593$	3.00000	+	5.19615i	0.123195	+	0.213380i	0.921026	−	0.389501i	$-0.127353\pi$
$593$	3.00000	+	5.19615i	0.123195	+	0.213380i	−0.797831	+	0.602881i	$0.794019\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	15.0000	−	25.9808i	0.612883	−	1.06155i	−0.377869	−	0.925859i	$-0.623343\pi$
$599$	15.0000	−	25.9808i	0.612883	−	1.06155i	0.990752	−	0.135686i	$-0.0433238\pi$
$600$		0			0
$601$	8.00000			0.326327			0.163163	−	0.986599i	$-0.447830\pi$
$601$	8.00000			0.326327			0.163163	+	0.986599i	$0.447830\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−7.00000	+	12.1244i	−0.284590	+	0.492925i
$606$		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
$611$	−16.0000	−	27.7128i	−0.647291	−	1.12114i
$612$		0			0
$613$	−13.0000	+	22.5167i	−0.525065	+	0.909439i	0.474509	+	0.880251i	$0.342626\pi$
$613$	−13.0000	+	22.5167i	−0.525065	+	0.909439i	−0.999574	+	0.0291886i	$0.990708\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	2.00000			0.0805170			0.0402585	−	0.999189i	$-0.487182\pi$
$617$	2.00000			0.0805170			0.0402585	+	0.999189i	$0.487182\pi$
$618$		0			0
$619$	−2.00000	+	3.46410i	−0.0803868	+	0.139234i	−0.903416	−	0.428765i	$-0.858949\pi$
$619$	−2.00000	+	3.46410i	−0.0803868	+	0.139234i	0.823029	+	0.567999i	$0.192282\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	9.50000	+	16.4545i	0.380000	+	0.658179i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−36.0000			−1.43541
$630$		0			0
$631$	12.0000			0.477712			0.238856	−	0.971055i	$-0.423228\pi$
$631$	12.0000			0.477712			0.238856	+	0.971055i	$0.423228\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−4.00000	−	6.92820i	−0.158735	−	0.274937i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−13.0000	+	22.5167i	−0.513469	+	0.889355i	0.486409	+	0.873731i	$0.338307\pi$
$641$	−13.0000	+	22.5167i	−0.513469	+	0.889355i	−0.999878	+	0.0156233i	$0.995027\pi$
$642$		0			0
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−24.0000	+	41.5692i	−0.943537	+	1.63425i	−0.184884	+	0.982760i	$0.559191\pi$
$647$	−24.0000	+	41.5692i	−0.943537	+	1.63425i	−0.758654	+	0.651494i	$0.774142\pi$
$648$		0			0
$649$	4.00000	+	6.92820i	0.157014	+	0.271956i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−17.0000	−	29.4449i	−0.665261	−	1.15227i	−0.979214	−	0.202828i	$-0.934987\pi$
$653$	−17.0000	−	29.4449i	−0.665261	−	1.15227i	0.313953	−	0.949439i	$-0.398347\pi$
$654$		0			0
$655$	12.0000	−	20.7846i	0.468879	−	0.812122i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	14.0000			0.545363			0.272681	−	0.962104i	$-0.412090\pi$
$659$	14.0000			0.545363			0.272681	+	0.962104i	$0.412090\pi$
$660$		0			0
$661$	−4.00000	+	6.92820i	−0.155582	+	0.269476i	−0.933271	−	0.359174i	$-0.883059\pi$
$661$	−4.00000	+	6.92820i	−0.155582	+	0.269476i	0.777689	+	0.628649i	$0.216392\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−30.0000	−	51.9615i	−1.16160	−	2.01196i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	16.0000			0.617673
$672$		0			0
$673$	−22.0000			−0.848038			−0.424019	−	0.905653i	$-0.639381\pi$
$673$	−22.0000			−0.848038			−0.424019	+	0.905653i	$0.639381\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	15.0000	+	25.9808i	0.576497	+	0.998522i	0.995877	+	0.0907112i	$0.0289140\pi$
$677$	15.0000	+	25.9808i	0.576497	+	0.998522i	−0.419380	+	0.907811i	$0.637753\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	11.0000	−	19.0526i	0.420903	−	0.729026i	−0.575125	−	0.818066i	$-0.695047\pi$
$683$	11.0000	−	19.0526i	0.420903	−	0.729026i	0.996028	+	0.0890398i	$0.0283798\pi$
$684$		0			0
$685$	−20.0000			−0.764161
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−4.00000	+	6.92820i	−0.152388	+	0.263944i
$690$		0			0
$691$	10.0000	+	17.3205i	0.380418	+	0.658903i	0.991122	−	0.132956i	$-0.0424468\pi$
$691$	10.0000	+	17.3205i	0.380418	+	0.658903i	−0.610704	+	0.791859i	$0.709113\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	12.0000	+	20.7846i	0.455186	+	0.788405i
$696$		0			0
$697$	18.0000	−	31.1769i	0.681799	−	1.18091i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	50.0000			1.88847			0.944237	−	0.329267i	$-0.106802\pi$
$701$	50.0000			1.88847			0.944237	+	0.329267i	$0.106802\pi$
$702$		0			0
$703$	−24.0000	+	41.5692i	−0.905177	+	1.56781i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	7.00000	+	12.1244i	0.262891	+	0.455340i	0.967009	−	0.254743i	$-0.0819909\pi$
$709$	7.00000	+	12.1244i	0.262891	+	0.455340i	−0.704118	+	0.710083i	$0.748658\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	24.0000			0.898807
$714$		0			0
$715$	−16.0000			−0.598366
$716$		0			0
$717$		0			0
$718$		0			0
$719$	24.0000	+	41.5692i	0.895049	+	1.55027i	0.833744	+	0.552151i	$0.186193\pi$
$719$	24.0000	+	41.5692i	0.895049	+	1.55027i	0.0613050	+	0.998119i	$0.480474\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	5.00000	−	8.66025i	0.185695	−	0.321634i
$726$		0			0
$727$	44.0000			1.63187			0.815935	−	0.578144i	$-0.196223\pi$
$727$	44.0000			1.63187			0.815935	+	0.578144i	$0.196223\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	12.0000	−	20.7846i	0.443836	−	0.768747i
$732$		0			0
$733$	−2.00000	−	3.46410i	−0.0738717	−	0.127950i	0.826723	−	0.562609i	$-0.190202\pi$
$733$	−2.00000	−	3.46410i	−0.0738717	−	0.127950i	−0.900595	+	0.434659i	$0.856869\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−8.00000	−	13.8564i	−0.294684	−	0.510407i
$738$		0			0
$739$	−20.0000	+	34.6410i	−0.735712	+	1.27429i	0.218698	+	0.975793i	$0.429819\pi$
$739$	−20.0000	+	34.6410i	−0.735712	+	1.27429i	−0.954410	+	0.298498i	$0.903514\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	18.0000			0.660356			0.330178	−	0.943919i	$-0.392891\pi$
$743$	18.0000			0.660356			0.330178	+	0.943919i	$0.392891\pi$
$744$		0			0
$745$	−2.00000	+	3.46410i	−0.0732743	+	0.126915i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	24.0000	+	41.5692i	0.875772	+	1.51688i	0.855938	+	0.517079i	$0.172981\pi$
$751$	24.0000	+	41.5692i	0.875772	+	1.51688i	0.0198348	+	0.999803i	$0.493686\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	16.0000			0.582300
$756$		0			0
$757$	−14.0000			−0.508839			−0.254419	−	0.967094i	$-0.581884\pi$
$757$	−14.0000			−0.508839			−0.254419	+	0.967094i	$0.581884\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−11.0000	−	19.0526i	−0.398750	−	0.690655i	0.594822	−	0.803857i	$-0.297222\pi$
$761$	−11.0000	−	19.0526i	−0.398750	−	0.690655i	−0.993572	+	0.113203i	$0.963889\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	8.00000	−	13.8564i	0.288863	−	0.500326i
$768$		0			0
$769$	32.0000			1.15395			0.576975	−	0.816762i	$-0.304233\pi$
$769$	32.0000			1.15395			0.576975	+	0.816762i	$0.304233\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	3.00000	−	5.19615i	0.107903	−	0.186893i	−0.807018	−	0.590527i	$-0.798920\pi$
$773$	3.00000	−	5.19615i	0.107903	−	0.186893i	0.914920	+	0.403634i	$0.132253\pi$
$774$		0			0
$775$	2.00000	+	3.46410i	0.0718421	+	0.124434i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−24.0000	−	41.5692i	−0.859889	−	1.48937i
$780$		0			0
$781$	10.0000	−	17.3205i	0.357828	−	0.619777i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	32.0000			1.14213
$786$		0			0
$787$	−2.00000	+	3.46410i	−0.0712923	+	0.123482i	−0.899468	−	0.436987i	$-0.856046\pi$
$787$	−2.00000	+	3.46410i	−0.0712923	+	0.123482i	0.828176	+	0.560469i	$0.189379\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−16.0000	−	27.7128i	−0.568177	−	0.984111i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−6.00000			−0.212531			−0.106265	−	0.994338i	$-0.533889\pi$
$797$	−6.00000			−0.212531			−0.106265	+	0.994338i	$0.533889\pi$
$798$		0			0
$799$	48.0000			1.69812
$800$		0			0
$801$		0			0
$802$		0			0
$803$	4.00000	+	6.92820i	0.141157	+	0.244491i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−3.00000	+	5.19615i	−0.105474	+	0.182687i	−0.913932	−	0.405868i	$-0.866969\pi$
$809$	−3.00000	+	5.19615i	−0.105474	+	0.182687i	0.808458	+	0.588555i	$0.200303\pi$
$810$		0			0
$811$	−52.0000			−1.82597			−0.912983	−	0.407997i	$-0.866228\pi$
$811$	−52.0000			−1.82597			−0.912983	+	0.407997i	$0.866228\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	24.0000	−	41.5692i	0.840683	−	1.45611i
$816$		0			0
$817$	−16.0000	−	27.7128i	−0.559769	−	0.969549i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.00000	+	1.73205i	0.0349002	+	0.0604490i	0.882948	−	0.469471i	$-0.155555\pi$
$821$	1.00000	+	1.73205i	0.0349002	+	0.0604490i	−0.848048	+	0.529920i	$0.822222\pi$
$822$		0			0
$823$	22.0000	−	38.1051i	0.766872	−	1.32826i	−0.172379	−	0.985031i	$-0.555146\pi$
$823$	22.0000	−	38.1051i	0.766872	−	1.32826i	0.939251	−	0.343230i	$-0.111521\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	30.0000			1.04320			0.521601	−	0.853189i	$-0.325335\pi$
$827$	30.0000			1.04320			0.521601	+	0.853189i	$0.325335\pi$
$828$		0			0
$829$	−2.00000	+	3.46410i	−0.0694629	+	0.120313i	−0.898665	−	0.438636i	$-0.855462\pi$
$829$	−2.00000	+	3.46410i	−0.0694629	+	0.120313i	0.829202	+	0.558949i	$0.188795\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−16.0000	−	27.7128i	−0.553703	−	0.959041i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	24.0000			0.828572			0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000			0.828572			0.414286	+	0.910147i	$0.364031\pi$
$840$		0			0
$841$	71.0000			2.44828
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3.00000	+	5.19615i	0.103203	+	0.178753i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−18.0000	+	31.1769i	−0.617032	+	1.06873i
$852$		0			0
$853$	−16.0000			−0.547830			−0.273915	−	0.961754i	$-0.588319\pi$
$853$	−16.0000			−0.547830			−0.273915	+	0.961754i	$0.588319\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	21.0000	−	36.3731i	0.717346	−	1.24248i	−0.244701	−	0.969599i	$-0.578690\pi$
$857$	21.0000	−	36.3731i	0.717346	−	1.24248i	0.962048	−	0.272882i	$-0.0879768\pi$
$858$		0			0
$859$	−24.0000	−	41.5692i	−0.818869	−	1.41832i	−0.906516	−	0.422172i	$-0.861268\pi$
$859$	−24.0000	−	41.5692i	−0.818869	−	1.41832i	0.0876464	−	0.996152i	$-0.472065\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	9.00000	+	15.5885i	0.306364	+	0.530637i	0.977564	−	0.210639i	$-0.0675543\pi$
$863$	9.00000	+	15.5885i	0.306364	+	0.530637i	−0.671200	+	0.741276i	$0.734221\pi$
$864$		0			0
$865$	2.00000	−	3.46410i	0.0680020	−	0.117783i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−8.00000			−0.271381
$870$		0			0
$871$	−16.0000	+	27.7128i	−0.542139	+	0.939013i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−9.00000	−	15.5885i	−0.303908	−	0.526385i	0.673109	−	0.739543i	$-0.264958\pi$
$877$	−9.00000	−	15.5885i	−0.303908	−	0.526385i	−0.977018	+	0.213158i	$0.931625\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	34.0000			1.14549			0.572745	−	0.819734i	$-0.305879\pi$
$881$	34.0000			1.14549			0.572745	+	0.819734i	$0.305879\pi$
$882$		0			0
$883$	−20.0000			−0.673054			−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000			−0.673054			−0.336527	+	0.941674i	$0.609252\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	24.0000	+	41.5692i	0.805841	+	1.39576i	0.915722	+	0.401813i	$0.131620\pi$
$887$	24.0000	+	41.5692i	0.805841	+	1.39576i	−0.109881	+	0.993945i	$0.535047\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	32.0000	−	55.4256i	1.07084	−	1.85475i
$894$		0			0
$895$	12.0000			0.401116
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−20.0000	+	34.6410i	−0.667037	+	1.15534i
$900$		0			0
$901$	−6.00000	−	10.3923i	−0.199889	−	0.346218i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−20.0000	−	34.6410i	−0.664822	−	1.15151i
$906$		0			0
$907$	−14.0000	+	24.2487i	−0.464862	+	0.805165i	−0.999195	−	0.0401089i	$-0.987230\pi$
$907$	−14.0000	+	24.2487i	−0.464862	+	0.805165i	0.534333	+	0.845274i	$0.320563\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	50.0000			1.65657			0.828287	−	0.560304i	$-0.189316\pi$
$911$	50.0000			1.65657			0.828287	+	0.560304i	$0.189316\pi$
$912$		0			0
$913$	−12.0000	+	20.7846i	−0.397142	+	0.687870i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−12.0000	−	20.7846i	−0.395843	−	0.685621i	0.597365	−	0.801970i	$-0.296214\pi$
$919$	−12.0000	−	20.7846i	−0.395843	−	0.685621i	−0.993208	+	0.116348i	$0.962881\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−40.0000			−1.31662
$924$		0			0
$925$	−6.00000			−0.197279
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−3.00000	−	5.19615i	−0.0984268	−	0.170480i	0.812607	−	0.582812i	$-0.198048\pi$
$929$	−3.00000	−	5.19615i	−0.0984268	−	0.170480i	−0.911034	+	0.412332i	$0.864714\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	12.0000	−	20.7846i	0.392442	−	0.679729i
$936$		0			0
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	9.00000	−	15.5885i	0.293392	−	0.508169i	−0.681218	−	0.732081i	$-0.738549\pi$
$941$	9.00000	−	15.5885i	0.293392	−	0.508169i	0.974609	+	0.223912i	$0.0718827\pi$
$942$		0			0
$943$	−18.0000	−	31.1769i	−0.586161	−	1.01526i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−23.0000	−	39.8372i	−0.747400	−	1.29453i	−0.949065	−	0.315080i	$-0.897969\pi$
$947$	−23.0000	−	39.8372i	−0.747400	−	1.29453i	0.201666	−	0.979454i	$-0.435365\pi$
$948$		0			0
$949$	8.00000	−	13.8564i	0.259691	−	0.449798i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−42.0000			−1.36051			−0.680257	−	0.732974i	$-0.738132\pi$
$953$	−42.0000			−1.36051			−0.680257	+	0.732974i	$0.738132\pi$
$954$		0			0
$955$	−6.00000	+	10.3923i	−0.194155	+	0.336287i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	7.50000	+	12.9904i	0.241935	+	0.419045i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	20.0000			0.643823
$966$		0			0
$967$	4.00000			0.128631			0.0643157	−	0.997930i	$-0.479514\pi$
$967$	4.00000			0.128631			0.0643157	+	0.997930i	$0.479514\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	10.0000	+	17.3205i	0.320915	+	0.555842i	0.980677	−	0.195633i	$-0.0626762\pi$
$971$	10.0000	+	17.3205i	0.320915	+	0.555842i	−0.659762	+	0.751475i	$0.729343\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	19.0000	−	32.9090i	0.607864	−	1.05285i	−0.383728	−	0.923446i	$-0.625360\pi$
$977$	19.0000	−	32.9090i	0.607864	−	1.05285i	0.991592	−	0.129405i	$-0.0413067\pi$
$978$		0			0
$979$	−28.0000			−0.894884
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−8.00000	+	13.8564i	−0.255160	+	0.441951i	−0.964939	−	0.262474i	$-0.915462\pi$
$983$	−8.00000	+	13.8564i	−0.255160	+	0.441951i	0.709779	+	0.704425i	$0.248795\pi$
$984$		0			0
$985$	6.00000	+	10.3923i	0.191176	+	0.331126i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−12.0000	−	20.7846i	−0.381578	−	0.660912i
$990$		0			0
$991$	−16.0000	+	27.7128i	−0.508257	+	0.880327i	0.491698	+	0.870766i	$0.336377\pi$
$991$	−16.0000	+	27.7128i	−0.508257	+	0.880327i	−0.999954	+	0.00956046i	$0.996957\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	16.0000			0.507234
$996$		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	1764.2.k.i.361.1		2
3.2	odd	2		588.2.i.a.361.1		2
7.2	even	3	inner	1764.2.k.i.1549.1		2
7.3	odd	6		1764.2.a.i.1.1		1
7.4	even	3		1764.2.a.b.1.1		1
7.5	odd	6		1764.2.k.c.1549.1		2
7.6	odd	2		1764.2.k.c.361.1		2
12.11	even	2		2352.2.q.p.1537.1		2
21.2	odd	6		588.2.i.a.373.1		2
21.5	even	6		588.2.i.g.373.1		2
21.11	odd	6		588.2.a.e.1.1	yes	1
21.17	even	6		588.2.a.b.1.1	✓	1
21.20	even	2		588.2.i.g.361.1		2
28.3	even	6		7056.2.a.bu.1.1		1
28.11	odd	6		7056.2.a.n.1.1		1
84.11	even	6		2352.2.a.j.1.1		1
84.23	even	6		2352.2.q.p.961.1		2
84.47	odd	6		2352.2.q.k.961.1		2
84.59	odd	6		2352.2.a.p.1.1		1
84.83	odd	2		2352.2.q.k.1537.1		2
168.11	even	6		9408.2.a.ca.1.1		1
168.53	odd	6		9408.2.a.l.1.1		1
168.59	odd	6		9408.2.a.bf.1.1		1
168.101	even	6		9408.2.a.cu.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.2.a.b.1.1	✓	1	21.17	even	6
588.2.a.e.1.1	yes	1	21.11	odd	6
588.2.i.a.361.1		2	3.2	odd	2
588.2.i.a.373.1		2	21.2	odd	6
588.2.i.g.361.1		2	21.20	even	2
588.2.i.g.373.1		2	21.5	even	6
1764.2.a.b.1.1		1	7.4	even	3
1764.2.a.i.1.1		1	7.3	odd	6
1764.2.k.c.361.1		2	7.6	odd	2
1764.2.k.c.1549.1		2	7.5	odd	6
1764.2.k.i.361.1		2	1.1	even	1	trivial
1764.2.k.i.1549.1		2	7.2	even	3	inner
2352.2.a.j.1.1		1	84.11	even	6
2352.2.a.p.1.1		1	84.59	odd	6
2352.2.q.k.961.1		2	84.47	odd	6
2352.2.q.k.1537.1		2	84.83	odd	2
2352.2.q.p.961.1		2	84.23	even	6
2352.2.q.p.1537.1		2	12.11	even	2
7056.2.a.n.1.1		1	28.11	odd	6
7056.2.a.bu.1.1		1	28.3	even	6
9408.2.a.l.1.1		1	168.53	odd	6
9408.2.a.bf.1.1		1	168.59	odd	6
9408.2.a.ca.1.1		1	168.11	even	6
9408.2.a.cu.1.1		1	168.101	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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1764.k (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.73205i) q^{5} +O(q^{10})\)	\(q+(1.00000 + 1.73205i) q^{5} +(1.00000 - 1.73205i) q^{11} -4.00000 q^{13} +(3.00000 - 5.19615i) q^{17} +(-4.00000 - 6.92820i) q^{19} +(-3.00000 - 5.19615i) q^{23} +(0.500000 - 0.866025i) q^{25} +10.0000 q^{29} +(-2.00000 + 3.46410i) q^{31} +(-3.00000 - 5.19615i) q^{37} +6.00000 q^{41} +4.00000 q^{43} +(4.00000 + 6.92820i) q^{47} +(1.00000 - 1.73205i) q^{53} +4.00000 q^{55} +(-2.00000 + 3.46410i) q^{59} +(4.00000 + 6.92820i) q^{61} +(-4.00000 - 6.92820i) q^{65} +(4.00000 - 6.92820i) q^{67} +10.0000 q^{71} +(-2.00000 + 3.46410i) q^{73} +(-2.00000 - 3.46410i) q^{79} -12.0000 q^{83} +12.0000 q^{85} +(-7.00000 - 12.1244i) q^{89} +(8.00000 - 13.8564i) q^{95} +4.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5}+O(q^{10}) \) 2 * q + 2 * q^5	\( 2 q + 2 q^{5} + 2 q^{11} - 8 q^{13} + 6 q^{17} - 8 q^{19} - 6 q^{23} + q^{25} + 20 q^{29} - 4 q^{31} - 6 q^{37} + 12 q^{41} + 8 q^{43} + 8 q^{47} + 2 q^{53} + 8 q^{55} - 4 q^{59} + 8 q^{61} - 8 q^{65} + 8 q^{67} + 20 q^{71} - 4 q^{73} - 4 q^{79} - 24 q^{83} + 24 q^{85} - 14 q^{89} + 16 q^{95} + 8 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 + 2 * q^11 - 8 * q^13 + 6 * q^17 - 8 * q^19 - 6 * q^23 + q^25 + 20 * q^29 - 4 * q^31 - 6 * q^37 + 12 * q^41 + 8 * q^43 + 8 * q^47 + 2 * q^53 + 8 * q^55 - 4 * q^59 + 8 * q^61 - 8 * q^65 + 8 * q^67 + 20 * q^71 - 4 * q^73 - 4 * q^79 - 24 * q^83 + 24 * q^85 - 14 * q^89 + 16 * q^95 + 8 * q^97

Introduction

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