LMFDB - Embedded newform 882.2.g.d.361.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,2,Mod(361,882)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([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("882.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	882.g (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:	$7.04280545828$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 14)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	882.361
Dual form			882.2.g.d.667.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +1.00000 q^{8} +O(q^{10})$	$q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +1.00000 q^{8} +4.00000 q^{13} +(-0.500000 - 0.866025i) q^{16} +(-3.00000 + 5.19615i) q^{17} +(1.00000 + 1.73205i) q^{19} +(2.50000 - 4.33013i) q^{25} +(-2.00000 - 3.46410i) q^{26} +6.00000 q^{29} +(-2.00000 + 3.46410i) q^{31} +(-0.500000 + 0.866025i) q^{32} +6.00000 q^{34} +(-1.00000 - 1.73205i) q^{37} +(1.00000 - 1.73205i) q^{38} +6.00000 q^{41} +8.00000 q^{43} +(6.00000 + 10.3923i) q^{47} -5.00000 q^{50} +(-2.00000 + 3.46410i) q^{52} +(3.00000 - 5.19615i) q^{53} +(-3.00000 - 5.19615i) q^{58} +(3.00000 - 5.19615i) q^{59} +(4.00000 + 6.92820i) q^{61} +4.00000 q^{62} +1.00000 q^{64} +(2.00000 - 3.46410i) q^{67} +(-3.00000 - 5.19615i) q^{68} +(1.00000 - 1.73205i) q^{73} +(-1.00000 + 1.73205i) q^{74} -2.00000 q^{76} +(-4.00000 - 6.92820i) q^{79} +(-3.00000 - 5.19615i) q^{82} -6.00000 q^{83} +(-4.00000 - 6.92820i) q^{86} +(3.00000 + 5.19615i) q^{89} +(6.00000 - 10.3923i) q^{94} +10.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{4} + 2 q^{8}+O(q^{10}) $ 2 * q - q^2 - q^4 + 2 * q^8	$ 2 q - q^{2} - q^{4} + 2 q^{8} + 8 q^{13} - q^{16} - 6 q^{17} + 2 q^{19} + 5 q^{25} - 4 q^{26} + 12 q^{29} - 4 q^{31} - q^{32} + 12 q^{34} - 2 q^{37} + 2 q^{38} + 12 q^{41} + 16 q^{43} + 12 q^{47} - 10 q^{50} - 4 q^{52} + 6 q^{53} - 6 q^{58} + 6 q^{59} + 8 q^{61} + 8 q^{62} + 2 q^{64} + 4 q^{67} - 6 q^{68} + 2 q^{73} - 2 q^{74} - 4 q^{76} - 8 q^{79} - 6 q^{82} - 12 q^{83} - 8 q^{86} + 6 q^{89} + 12 q^{94} + 20 q^{97}+O(q^{100}) $ 2 * q - q^2 - q^4 + 2 * q^8 + 8 * q^13 - q^16 - 6 * q^17 + 2 * q^19 + 5 * q^25 - 4 * q^26 + 12 * q^29 - 4 * q^31 - q^32 + 12 * q^34 - 2 * q^37 + 2 * q^38 + 12 * q^41 + 16 * q^43 + 12 * q^47 - 10 * q^50 - 4 * q^52 + 6 * q^53 - 6 * q^58 + 6 * q^59 + 8 * q^61 + 8 * q^62 + 2 * q^64 + 4 * q^67 - 6 * q^68 + 2 * q^73 - 2 * q^74 - 4 * q^76 - 8 * q^79 - 6 * q^82 - 12 * q^83 - 8 * q^86 + 6 * q^89 + 12 * q^94 + 20 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$.

$n$	$199$	$785$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$

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 $

$2$	−0.500000	−	0.866025i	−0.353553	−	0.612372i
$2$	−0.500000	−	0.866025i	−0.353553	−	0.612372i
$3$		0			0
$3$		0			0
$4$	−0.500000	+	0.866025i	−0.250000	+	0.433013i
$5$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$5$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	1.00000			0.353553
$9$		0			0
$10$		0			0
$11$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$11$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$12$		0			0
$13$	4.00000			1.10940			0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000			1.10940			0.554700	+	0.832050i	$0.312833\pi$
$14$		0			0
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.125000	−	0.216506i
$17$	−3.00000	+	5.19615i	−0.727607	+	1.26025i	0.230285	+	0.973123i	$0.426034\pi$
$17$	−3.00000	+	5.19615i	−0.727607	+	1.26025i	−0.957892	+	0.287129i	$0.907299\pi$
$18$		0			0
$19$	1.00000	+	1.73205i	0.229416	+	0.397360i	0.957635	−	0.287984i	$-0.0929851\pi$
$19$	1.00000	+	1.73205i	0.229416	+	0.397360i	−0.728219	+	0.685344i	$0.759652\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$	2.50000	−	4.33013i	0.500000	−	0.866025i
$26$	−2.00000	−	3.46410i	−0.392232	−	0.679366i
$27$		0			0
$28$		0			0
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		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	$0.283621\pi$
$32$	−0.500000	+	0.866025i	−0.0883883	+	0.153093i
$33$		0			0
$34$	6.00000			1.02899
$35$		0			0
$36$		0			0
$37$	−1.00000	−	1.73205i	−0.164399	−	0.284747i	0.772043	−	0.635571i	$-0.219235\pi$
$37$	−1.00000	−	1.73205i	−0.164399	−	0.284747i	−0.936442	+	0.350823i	$0.885902\pi$
$38$	1.00000	−	1.73205i	0.162221	−	0.280976i
$39$		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
$43$	8.00000			1.21999			0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000			1.21999			0.609994	+	0.792406i	$0.291172\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	6.00000	+	10.3923i	0.875190	+	1.51587i	0.856560	+	0.516047i	$0.172597\pi$
$47$	6.00000	+	10.3923i	0.875190	+	1.51587i	0.0186297	+	0.999826i	$0.494070\pi$
$48$		0			0
$49$		0			0
$50$	−5.00000			−0.707107
$51$		0			0
$52$	−2.00000	+	3.46410i	−0.277350	+	0.480384i
$53$	3.00000	−	5.19615i	0.412082	−	0.713746i	−0.583036	−	0.812447i	$-0.698135\pi$
$53$	3.00000	−	5.19615i	0.412082	−	0.713746i	0.995117	+	0.0987002i	$0.0314685\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−3.00000	−	5.19615i	−0.393919	−	0.682288i
$59$	3.00000	−	5.19615i	0.390567	−	0.676481i	−0.601958	−	0.798528i	$-0.705612\pi$
$59$	3.00000	−	5.19615i	0.390567	−	0.676481i	0.992524	+	0.122047i	$0.0389457\pi$
$60$		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	$0.662183\pi$
$62$	4.00000			0.508001
$63$		0			0
$64$	1.00000			0.125000
$65$		0			0
$66$		0			0
$67$	2.00000	−	3.46410i	0.244339	−	0.423207i	−0.717607	−	0.696449i	$-0.754762\pi$
$67$	2.00000	−	3.46410i	0.244339	−	0.423207i	0.961946	+	0.273241i	$0.0880957\pi$
$68$	−3.00000	−	5.19615i	−0.363803	−	0.630126i
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	1.00000	−	1.73205i	0.117041	−	0.202721i	−0.801553	−	0.597924i	$-0.795992\pi$
$73$	1.00000	−	1.73205i	0.117041	−	0.202721i	0.918594	+	0.395203i	$0.129326\pi$
$74$	−1.00000	+	1.73205i	−0.116248	+	0.201347i
$75$		0			0
$76$	−2.00000			−0.229416
$77$		0			0
$78$		0			0
$79$	−4.00000	−	6.92820i	−0.450035	−	0.779484i	0.548352	−	0.836247i	$-0.315255\pi$
$79$	−4.00000	−	6.92820i	−0.450035	−	0.779484i	−0.998388	+	0.0567635i	$0.981922\pi$
$80$		0			0
$81$		0			0
$82$	−3.00000	−	5.19615i	−0.331295	−	0.573819i
$83$	−6.00000			−0.658586			−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000			−0.658586			−0.329293	+	0.944228i	$0.606810\pi$
$84$		0			0
$85$		0			0
$86$	−4.00000	−	6.92820i	−0.431331	−	0.747087i
$87$		0			0
$88$		0			0
$89$	3.00000	+	5.19615i	0.317999	+	0.550791i	0.980071	−	0.198650i	$-0.0636557\pi$
$89$	3.00000	+	5.19615i	0.317999	+	0.550791i	−0.662071	+	0.749441i	$0.730322\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$	6.00000	−	10.3923i	0.618853	−	1.07188i
$95$		0			0
$96$		0			0
$97$	10.0000			1.01535			0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000			1.01535			0.507673	+	0.861550i	$0.330506\pi$
$98$		0			0
$99$		0			0
$100$	2.50000	+	4.33013i	0.250000	+	0.433013i
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$	−2.00000	−	3.46410i	−0.197066	−	0.341328i	0.750510	−	0.660859i	$-0.229808\pi$
$103$	−2.00000	−	3.46410i	−0.197066	−	0.341328i	−0.947576	+	0.319531i	$0.896475\pi$
$104$	4.00000			0.392232
$105$		0			0
$106$	−6.00000			−0.582772
$107$	6.00000	+	10.3923i	0.580042	+	1.00466i	0.995474	+	0.0950377i	$0.0302972\pi$
$107$	6.00000	+	10.3923i	0.580042	+	1.00466i	−0.415432	+	0.909624i	$0.636370\pi$
$108$		0			0
$109$	−1.00000	+	1.73205i	−0.0957826	+	0.165900i	−0.909935	−	0.414751i	$-0.863869\pi$
$109$	−1.00000	+	1.73205i	−0.0957826	+	0.165900i	0.814152	+	0.580651i	$0.197202\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−6.00000			−0.564433			−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000			−0.564433			−0.282216	+	0.959351i	$0.591070\pi$
$114$		0			0
$115$		0			0
$116$	−3.00000	+	5.19615i	−0.278543	+	0.482451i
$117$		0			0
$118$	−6.00000			−0.552345
$119$		0			0
$120$		0			0
$121$	5.50000	+	9.52628i	0.500000	+	0.866025i
$122$	4.00000	−	6.92820i	0.362143	−	0.627250i
$123$		0			0
$124$	−2.00000	−	3.46410i	−0.179605	−	0.311086i
$125$		0			0
$126$		0			0
$127$	−16.0000			−1.41977			−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000			−1.41977			−0.709885	+	0.704317i	$0.751253\pi$
$128$	−0.500000	−	0.866025i	−0.0441942	−	0.0765466i
$129$		0			0
$130$		0			0
$131$	−9.00000	−	15.5885i	−0.786334	−	1.36197i	−0.928199	−	0.372084i	$-0.878643\pi$
$131$	−9.00000	−	15.5885i	−0.786334	−	1.36197i	0.141865	−	0.989886i	$-0.454690\pi$
$132$		0			0
$133$		0			0
$134$	−4.00000			−0.345547
$135$		0			0
$136$	−3.00000	+	5.19615i	−0.257248	+	0.445566i
$137$	9.00000	−	15.5885i	0.768922	−	1.33181i	−0.169226	−	0.985577i	$-0.554127\pi$
$137$	9.00000	−	15.5885i	0.768922	−	1.33181i	0.938148	−	0.346235i	$-0.112540\pi$
$138$		0			0
$139$	−14.0000			−1.18746			−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000			−1.18746			−0.593732	+	0.804663i	$0.702346\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$	−2.00000			−0.165521
$147$		0			0
$148$	2.00000			0.164399
$149$	−9.00000	−	15.5885i	−0.737309	−	1.27706i	−0.953703	−	0.300750i	$-0.902763\pi$
$149$	−9.00000	−	15.5885i	−0.737309	−	1.27706i	0.216394	−	0.976306i	$-0.430570\pi$
$150$		0			0
$151$	−4.00000	+	6.92820i	−0.325515	+	0.563809i	−0.981617	−	0.190864i	$-0.938871\pi$
$151$	−4.00000	+	6.92820i	−0.325515	+	0.563809i	0.656101	+	0.754673i	$0.272204\pi$
$152$	1.00000	+	1.73205i	0.0811107	+	0.140488i
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−2.00000	+	3.46410i	−0.159617	+	0.276465i	−0.934731	−	0.355357i	$-0.884359\pi$
$157$	−2.00000	+	3.46410i	−0.159617	+	0.276465i	0.775113	+	0.631822i	$0.217693\pi$
$158$	−4.00000	+	6.92820i	−0.318223	+	0.551178i
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	8.00000	+	13.8564i	0.626608	+	1.08532i	0.988227	+	0.152992i	$0.0488907\pi$
$163$	8.00000	+	13.8564i	0.626608	+	1.08532i	−0.361619	+	0.932326i	$0.617776\pi$
$164$	−3.00000	+	5.19615i	−0.234261	+	0.405751i
$165$		0			0
$166$	3.00000	+	5.19615i	0.232845	+	0.403300i
$167$	−12.0000			−0.928588			−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000			−0.928588			−0.464294	+	0.885681i	$0.653692\pi$
$168$		0			0
$169$	3.00000			0.230769
$170$		0			0
$171$		0			0
$172$	−4.00000	+	6.92820i	−0.304997	+	0.528271i
$173$	6.00000	+	10.3923i	0.456172	+	0.790112i	0.998755	−	0.0498898i	$-0.0158870\pi$
$173$	6.00000	+	10.3923i	0.456172	+	0.790112i	−0.542583	+	0.840002i	$0.682554\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	3.00000	−	5.19615i	0.224860	−	0.389468i
$179$	−6.00000	+	10.3923i	−0.448461	+	0.776757i	−0.998286	−	0.0585225i	$-0.981361\pi$
$179$	−6.00000	+	10.3923i	−0.448461	+	0.776757i	0.549825	+	0.835280i	$0.314694\pi$
$180$		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
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−12.0000			−0.875190
$189$		0			0
$190$		0			0
$191$	12.0000	+	20.7846i	0.868290	+	1.50392i	0.863743	+	0.503932i	$0.168114\pi$
$191$	12.0000	+	20.7846i	0.868290	+	1.50392i	0.00454614	+	0.999990i	$0.498553\pi$
$192$		0			0
$193$	−7.00000	+	12.1244i	−0.503871	+	0.872730i	0.496119	+	0.868255i	$0.334758\pi$
$193$	−7.00000	+	12.1244i	−0.503871	+	0.872730i	−0.999990	+	0.00447566i	$0.998575\pi$
$194$	−5.00000	−	8.66025i	−0.358979	−	0.621770i
$195$		0			0
$196$		0			0
$197$	18.0000			1.28245			0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000			1.28245			0.641223	+	0.767354i	$0.278427\pi$
$198$		0			0
$199$	10.0000	−	17.3205i	0.708881	−	1.22782i	−0.256391	−	0.966573i	$-0.582534\pi$
$199$	10.0000	−	17.3205i	0.708881	−	1.22782i	0.965272	−	0.261245i	$-0.0841331\pi$
$200$	2.50000	−	4.33013i	0.176777	−	0.306186i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$	−2.00000	+	3.46410i	−0.139347	+	0.241355i
$207$		0			0
$208$	−2.00000	−	3.46410i	−0.138675	−	0.240192i
$209$		0			0
$210$		0			0
$211$	−4.00000			−0.275371			−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000			−0.275371			−0.137686	+	0.990476i	$0.543966\pi$
$212$	3.00000	+	5.19615i	0.206041	+	0.356873i
$213$		0			0
$214$	6.00000	−	10.3923i	0.410152	−	0.710403i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	2.00000			0.135457
$219$		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
$226$	3.00000	+	5.19615i	0.199557	+	0.345643i
$227$	−9.00000	+	15.5885i	−0.597351	+	1.03464i	0.395860	+	0.918311i	$0.370447\pi$
$227$	−9.00000	+	15.5885i	−0.597351	+	1.03464i	−0.993210	+	0.116331i	$0.962887\pi$
$228$		0			0
$229$	−2.00000	−	3.46410i	−0.132164	−	0.228914i	0.792347	−	0.610071i	$-0.208859\pi$
$229$	−2.00000	−	3.46410i	−0.132164	−	0.228914i	−0.924510	+	0.381157i	$0.875526\pi$
$230$		0			0
$231$		0			0
$232$	6.00000			0.393919
$233$	−3.00000	−	5.19615i	−0.196537	−	0.340411i	0.750867	−	0.660454i	$-0.229636\pi$
$233$	−3.00000	−	5.19615i	−0.196537	−	0.340411i	−0.947403	+	0.320043i	$0.896303\pi$
$234$		0			0
$235$		0			0
$236$	3.00000	+	5.19615i	0.195283	+	0.338241i
$237$		0			0
$238$		0			0
$239$	−24.0000			−1.55243			−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000			−1.55243			−0.776215	+	0.630468i	$0.782863\pi$
$240$		0			0
$241$	−5.00000	+	8.66025i	−0.322078	+	0.557856i	−0.980917	−	0.194429i	$-0.937715\pi$
$241$	−5.00000	+	8.66025i	−0.322078	+	0.557856i	0.658838	+	0.752285i	$0.271048\pi$
$242$	5.50000	−	9.52628i	0.353553	−	0.612372i
$243$		0			0
$244$	−8.00000			−0.512148
$245$		0			0
$246$		0			0
$247$	4.00000	+	6.92820i	0.254514	+	0.440831i
$248$	−2.00000	+	3.46410i	−0.127000	+	0.219971i
$249$		0			0
$250$		0			0
$251$	−18.0000			−1.13615			−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000			−1.13615			−0.568075	+	0.822977i	$0.692312\pi$
$252$		0			0
$253$		0			0
$254$	8.00000	+	13.8564i	0.501965	+	0.869428i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.0312500	+	0.0541266i
$257$	−9.00000	−	15.5885i	−0.561405	−	0.972381i	−0.997374	−	0.0724199i	$-0.976928\pi$
$257$	−9.00000	−	15.5885i	−0.561405	−	0.972381i	0.435970	−	0.899961i	$-0.356405\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−9.00000	+	15.5885i	−0.556022	+	0.963058i
$263$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$263$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	2.00000	+	3.46410i	0.122169	+	0.211604i
$269$	6.00000	−	10.3923i	0.365826	−	0.633630i	−0.623082	−	0.782157i	$-0.714120\pi$
$269$	6.00000	−	10.3923i	0.365826	−	0.633630i	0.988908	+	0.148527i	$0.0474530\pi$
$270$		0			0
$271$	−8.00000	−	13.8564i	−0.485965	−	0.841717i	0.513905	−	0.857847i	$-0.328199\pi$
$271$	−8.00000	−	13.8564i	−0.485965	−	0.841717i	−0.999870	+	0.0161307i	$0.994865\pi$
$272$	6.00000			0.363803
$273$		0			0
$274$	−18.0000			−1.08742
$275$		0			0
$276$		0			0
$277$	5.00000	−	8.66025i	0.300421	−	0.520344i	−0.675810	−	0.737075i	$-0.736206\pi$
$277$	5.00000	−	8.66025i	0.300421	−	0.520344i	0.976231	+	0.216731i	$0.0695395\pi$
$278$	7.00000	+	12.1244i	0.419832	+	0.727171i
$279$		0			0
$280$		0			0
$281$	6.00000			0.357930			0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000			0.357930			0.178965	+	0.983855i	$0.442725\pi$
$282$		0			0
$283$	−11.0000	+	19.0526i	−0.653882	+	1.13256i	0.328291	+	0.944577i	$0.393527\pi$
$283$	−11.0000	+	19.0526i	−0.653882	+	1.13256i	−0.982173	+	0.187980i	$0.939806\pi$
$284$		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
$292$	1.00000	+	1.73205i	0.0585206	+	0.101361i
$293$	24.0000			1.40209			0.701047	−	0.713115i	$-0.252716\pi$
$293$	24.0000			1.40209			0.701047	+	0.713115i	$0.252716\pi$
$294$		0			0
$295$		0			0
$296$	−1.00000	−	1.73205i	−0.0581238	−	0.100673i
$297$		0			0
$298$	−9.00000	+	15.5885i	−0.521356	+	0.903015i
$299$		0			0
$300$		0			0
$301$		0			0
$302$	8.00000			0.460348
$303$		0			0
$304$	1.00000	−	1.73205i	0.0573539	−	0.0993399i
$305$		0			0
$306$		0			0
$307$	−2.00000			−0.114146			−0.0570730	−	0.998370i	$-0.518177\pi$
$307$	−2.00000			−0.114146			−0.0570730	+	0.998370i	$0.518177\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	12.0000	−	20.7846i	0.680458	−	1.17859i	−0.294384	−	0.955687i	$-0.595114\pi$
$311$	12.0000	−	20.7846i	0.680458	−	1.17859i	0.974841	−	0.222900i	$-0.0715523\pi$
$312$		0			0
$313$	−5.00000	−	8.66025i	−0.282617	−	0.489506i	0.689412	−	0.724370i	$-0.257869\pi$
$313$	−5.00000	−	8.66025i	−0.282617	−	0.489506i	−0.972028	+	0.234863i	$0.924536\pi$
$314$	4.00000			0.225733
$315$		0			0
$316$	8.00000			0.450035
$317$	3.00000	+	5.19615i	0.168497	+	0.291845i	0.937892	−	0.346929i	$-0.112775\pi$
$317$	3.00000	+	5.19615i	0.168497	+	0.291845i	−0.769395	+	0.638774i	$0.779442\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−12.0000			−0.667698
$324$		0			0
$325$	10.0000	−	17.3205i	0.554700	−	0.960769i
$326$	8.00000	−	13.8564i	0.443079	−	0.767435i
$327$		0			0
$328$	6.00000			0.331295
$329$		0			0
$330$		0			0
$331$	−4.00000	−	6.92820i	−0.219860	−	0.380808i	0.734905	−	0.678170i	$-0.237227\pi$
$331$	−4.00000	−	6.92820i	−0.219860	−	0.380808i	−0.954765	+	0.297361i	$0.903893\pi$
$332$	3.00000	−	5.19615i	0.164646	−	0.285176i
$333$		0			0
$334$	6.00000	+	10.3923i	0.328305	+	0.568642i
$335$		0			0
$336$		0			0
$337$	14.0000			0.762629			0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000			0.762629			0.381314	+	0.924445i	$0.375472\pi$
$338$	−1.50000	−	2.59808i	−0.0815892	−	0.141317i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	8.00000			0.431331
$345$		0			0
$346$	6.00000	−	10.3923i	0.322562	−	0.558694i
$347$	−12.0000	+	20.7846i	−0.644194	+	1.11578i	0.340293	+	0.940319i	$0.389474\pi$
$347$	−12.0000	+	20.7846i	−0.644194	+	1.11578i	−0.984487	+	0.175457i	$0.943860\pi$
$348$		0			0
$349$	28.0000			1.49881			0.749403	−	0.662114i	$-0.230341\pi$
$349$	28.0000			1.49881			0.749403	+	0.662114i	$0.230341\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−9.00000	+	15.5885i	−0.479022	+	0.829690i	−0.999711	−	0.0240566i	$-0.992342\pi$
$353$	−9.00000	+	15.5885i	−0.479022	+	0.829690i	0.520689	+	0.853746i	$0.325675\pi$
$354$		0			0
$355$		0			0
$356$	−6.00000			−0.317999
$357$		0			0
$358$	12.0000			0.634220
$359$	−12.0000	−	20.7846i	−0.633336	−	1.09697i	−0.986865	−	0.161546i	$-0.948352\pi$
$359$	−12.0000	−	20.7846i	−0.633336	−	1.09697i	0.353529	−	0.935423i	$-0.384981\pi$
$360$		0			0
$361$	7.50000	−	12.9904i	0.394737	−	0.683704i
$362$	10.0000	+	17.3205i	0.525588	+	0.910346i
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	4.00000	−	6.92820i	0.208798	−	0.361649i	−0.742538	−	0.669804i	$-0.766378\pi$
$367$	4.00000	−	6.92820i	0.208798	−	0.361649i	0.951336	+	0.308155i	$0.0997115\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−7.00000	−	12.1244i	−0.362446	−	0.627775i	0.625917	−	0.779890i	$-0.284725\pi$
$373$	−7.00000	−	12.1244i	−0.362446	−	0.627775i	−0.988363	+	0.152115i	$0.951392\pi$
$374$		0			0
$375$		0			0
$376$	6.00000	+	10.3923i	0.309426	+	0.535942i
$377$	24.0000			1.23606
$378$		0			0
$379$	−16.0000			−0.821865			−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000			−0.821865			−0.410932	+	0.911666i	$0.634797\pi$
$380$		0			0
$381$		0			0
$382$	12.0000	−	20.7846i	0.613973	−	1.06343i
$383$	−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
$385$		0			0
$386$	14.0000			0.712581
$387$		0			0
$388$	−5.00000	+	8.66025i	−0.253837	+	0.439658i
$389$	9.00000	−	15.5885i	0.456318	−	0.790366i	−0.542445	−	0.840091i	$-0.682501\pi$
$389$	9.00000	−	15.5885i	0.456318	−	0.790366i	0.998763	+	0.0497253i	$0.0158346\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	−9.00000	−	15.5885i	−0.453413	−	0.785335i
$395$		0			0
$396$		0			0
$397$	10.0000	+	17.3205i	0.501886	+	0.869291i	0.999998	+	0.00217869i	$0.000693499\pi$
$397$	10.0000	+	17.3205i	0.501886	+	0.869291i	−0.498112	+	0.867113i	$0.665973\pi$
$398$	−20.0000			−1.00251
$399$		0			0
$400$	−5.00000			−0.250000
$401$	−9.00000	−	15.5885i	−0.449439	−	0.778450i	0.548911	−	0.835881i	$-0.315043\pi$
$401$	−9.00000	−	15.5885i	−0.449439	−	0.778450i	−0.998350	+	0.0574304i	$0.981709\pi$
$402$		0			0
$403$	−8.00000	+	13.8564i	−0.398508	+	0.690237i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	7.00000	−	12.1244i	0.346128	−	0.599511i	−0.639430	−	0.768849i	$-0.720830\pi$
$409$	7.00000	−	12.1244i	0.346128	−	0.599511i	0.985558	+	0.169338i	$0.0541630\pi$
$410$		0			0
$411$		0			0
$412$	4.00000			0.197066
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−2.00000	+	3.46410i	−0.0980581	+	0.169842i
$417$		0			0
$418$		0			0
$419$	6.00000			0.293119			0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000			0.293119			0.146560	+	0.989202i	$0.453180\pi$
$420$		0			0
$421$	−10.0000			−0.487370			−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000			−0.487370			−0.243685	+	0.969854i	$0.578356\pi$
$422$	2.00000	+	3.46410i	0.0973585	+	0.168630i
$423$		0			0
$424$	3.00000	−	5.19615i	0.145693	−	0.252347i
$425$	15.0000	+	25.9808i	0.727607	+	1.26025i
$426$		0			0
$427$		0			0
$428$	−12.0000			−0.580042
$429$		0			0
$430$		0			0
$431$	12.0000	−	20.7846i	0.578020	−	1.00116i	−0.417687	−	0.908591i	$-0.637159\pi$
$431$	12.0000	−	20.7846i	0.578020	−	1.00116i	0.995706	−	0.0925683i	$-0.0295076\pi$
$432$		0			0
$433$	34.0000			1.63394			0.816968	−	0.576683i	$-0.195653\pi$
$433$	34.0000			1.63394			0.816968	+	0.576683i	$0.195653\pi$
$434$		0			0
$435$		0			0
$436$	−1.00000	−	1.73205i	−0.0478913	−	0.0829502i
$437$		0			0
$438$		0			0
$439$	4.00000	+	6.92820i	0.190910	+	0.330665i	0.945552	−	0.325471i	$-0.105523\pi$
$439$	4.00000	+	6.92820i	0.190910	+	0.330665i	−0.754642	+	0.656136i	$0.772190\pi$
$440$		0			0
$441$		0			0
$442$	24.0000			1.14156
$443$	−6.00000	−	10.3923i	−0.285069	−	0.493753i	0.687557	−	0.726130i	$-0.258683\pi$
$443$	−6.00000	−	10.3923i	−0.285069	−	0.493753i	−0.972626	+	0.232377i	$0.925350\pi$
$444$		0			0
$445$		0			0
$446$	4.00000	+	6.92820i	0.189405	+	0.328060i
$447$		0			0
$448$		0			0
$449$	−18.0000			−0.849473			−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000			−0.849473			−0.424736	+	0.905317i	$0.639633\pi$
$450$		0			0
$451$		0			0
$452$	3.00000	−	5.19615i	0.141108	−	0.244406i
$453$		0			0
$454$	18.0000			0.844782
$455$		0			0
$456$		0			0
$457$	5.00000	+	8.66025i	0.233890	+	0.405110i	0.958950	−	0.283577i	$-0.0915211\pi$
$457$	5.00000	+	8.66025i	0.233890	+	0.405110i	−0.725059	+	0.688686i	$0.758188\pi$
$458$	−2.00000	+	3.46410i	−0.0934539	+	0.161867i
$459$		0			0
$460$		0			0
$461$	12.0000			0.558896			0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000			0.558896			0.279448	+	0.960161i	$0.409849\pi$
$462$		0			0
$463$	32.0000			1.48717			0.743583	−	0.668644i	$-0.233125\pi$
$463$	32.0000			1.48717			0.743583	+	0.668644i	$0.233125\pi$
$464$	−3.00000	−	5.19615i	−0.139272	−	0.241225i
$465$		0			0
$466$	−3.00000	+	5.19615i	−0.138972	+	0.240707i
$467$	3.00000	+	5.19615i	0.138823	+	0.240449i	0.927052	−	0.374934i	$-0.122335\pi$
$467$	3.00000	+	5.19615i	0.138823	+	0.240449i	−0.788228	+	0.615383i	$0.789001\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	3.00000	−	5.19615i	0.138086	−	0.239172i
$473$		0			0
$474$		0			0
$475$	10.0000			0.458831
$476$		0			0
$477$		0			0
$478$	12.0000	+	20.7846i	0.548867	+	0.950666i
$479$	18.0000	−	31.1769i	0.822441	−	1.42451i	−0.0814184	−	0.996680i	$-0.525945\pi$
$479$	18.0000	−	31.1769i	0.822441	−	1.42451i	0.903859	−	0.427830i	$-0.140722\pi$
$480$		0			0
$481$	−4.00000	−	6.92820i	−0.182384	−	0.315899i
$482$	10.0000			0.455488
$483$		0			0
$484$	−11.0000			−0.500000
$485$		0			0
$486$		0			0
$487$	8.00000	−	13.8564i	0.362515	−	0.627894i	−0.625859	−	0.779936i	$-0.715252\pi$
$487$	8.00000	−	13.8564i	0.362515	−	0.627894i	0.988374	+	0.152042i	$0.0485850\pi$
$488$	4.00000	+	6.92820i	0.181071	+	0.313625i
$489$		0			0
$490$		0			0
$491$	12.0000			0.541552			0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000			0.541552			0.270776	+	0.962642i	$0.412720\pi$
$492$		0			0
$493$	−18.0000	+	31.1769i	−0.810679	+	1.40414i
$494$	4.00000	−	6.92820i	0.179969	−	0.311715i
$495$		0			0
$496$	4.00000			0.179605
$497$		0			0
$498$		0			0
$499$	2.00000	+	3.46410i	0.0895323	+	0.155074i	0.907314	−	0.420455i	$-0.138129\pi$
$499$	2.00000	+	3.46410i	0.0895323	+	0.155074i	−0.817781	+	0.575529i	$0.804796\pi$
$500$		0			0
$501$		0			0
$502$	9.00000	+	15.5885i	0.401690	+	0.695747i
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	8.00000	−	13.8564i	0.354943	−	0.614779i
$509$	−18.0000	−	31.1769i	−0.797836	−	1.38189i	−0.921023	−	0.389509i	$-0.872645\pi$
$509$	−18.0000	−	31.1769i	−0.797836	−	1.38189i	0.123187	−	0.992384i	$-0.460689\pi$
$510$		0			0
$511$		0			0
$512$	1.00000			0.0441942
$513$		0			0
$514$	−9.00000	+	15.5885i	−0.396973	+	0.687577i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−3.00000	+	5.19615i	−0.131432	+	0.227648i	−0.924229	−	0.381839i	$-0.875291\pi$
$521$	−3.00000	+	5.19615i	−0.131432	+	0.227648i	0.792797	+	0.609486i	$0.208624\pi$
$522$		0			0
$523$	1.00000	+	1.73205i	0.0437269	+	0.0757373i	0.887061	−	0.461653i	$-0.152744\pi$
$523$	1.00000	+	1.73205i	0.0437269	+	0.0757373i	−0.843334	+	0.537390i	$0.819410\pi$
$524$	18.0000			0.786334
$525$		0			0
$526$		0			0
$527$	−12.0000	−	20.7846i	−0.522728	−	0.905392i
$528$		0			0
$529$	11.5000	−	19.9186i	0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	24.0000			1.03956
$534$		0			0
$535$		0			0
$536$	2.00000	−	3.46410i	0.0863868	−	0.149626i
$537$		0			0
$538$	−12.0000			−0.517357
$539$		0			0
$540$		0			0
$541$	−19.0000	−	32.9090i	−0.816874	−	1.41487i	−0.907975	−	0.419025i	$-0.862372\pi$
$541$	−19.0000	−	32.9090i	−0.816874	−	1.41487i	0.0911008	−	0.995842i	$-0.470961\pi$
$542$	−8.00000	+	13.8564i	−0.343629	+	0.595184i
$543$		0			0
$544$	−3.00000	−	5.19615i	−0.128624	−	0.222783i
$545$		0			0
$546$		0			0
$547$	8.00000			0.342055			0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000			0.342055			0.171028	+	0.985266i	$0.445291\pi$
$548$	9.00000	+	15.5885i	0.384461	+	0.665906i
$549$		0			0
$550$		0			0
$551$	6.00000	+	10.3923i	0.255609	+	0.442727i
$552$		0			0
$553$		0			0
$554$	−10.0000			−0.424859
$555$		0			0
$556$	7.00000	−	12.1244i	0.296866	−	0.514187i
$557$	3.00000	−	5.19615i	0.127114	−	0.220168i	−0.795443	−	0.606028i	$-0.792762\pi$
$557$	3.00000	−	5.19615i	0.127114	−	0.220168i	0.922557	+	0.385860i	$0.126095\pi$
$558$		0			0
$559$	32.0000			1.35346
$560$		0			0
$561$		0			0
$562$	−3.00000	−	5.19615i	−0.126547	−	0.219186i
$563$	−15.0000	+	25.9808i	−0.632175	+	1.09496i	0.354932	+	0.934892i	$0.384504\pi$
$563$	−15.0000	+	25.9808i	−0.632175	+	1.09496i	−0.987106	+	0.160066i	$0.948829\pi$
$564$		0			0
$565$		0			0
$566$	22.0000			0.924729
$567$		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
$571$	−16.0000	+	27.7128i	−0.669579	+	1.15975i	0.308443	+	0.951243i	$0.400192\pi$
$571$	−16.0000	+	27.7128i	−0.669579	+	1.15975i	−0.978022	+	0.208502i	$0.933141\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	1.00000	−	1.73205i	0.0416305	−	0.0721062i	−0.844459	−	0.535620i	$-0.820078\pi$
$577$	1.00000	−	1.73205i	0.0416305	−	0.0721062i	0.886090	+	0.463513i	$0.153411\pi$
$578$	−9.50000	+	16.4545i	−0.395148	+	0.684416i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	1.00000	−	1.73205i	0.0413803	−	0.0716728i
$585$		0			0
$586$	−12.0000	−	20.7846i	−0.495715	−	0.858604i
$587$	−42.0000			−1.73353			−0.866763	−	0.498721i	$-0.833803\pi$
$587$	−42.0000			−1.73353			−0.866763	+	0.498721i	$0.833803\pi$
$588$		0			0
$589$	−8.00000			−0.329634
$590$		0			0
$591$		0			0
$592$	−1.00000	+	1.73205i	−0.0410997	+	0.0711868i
$593$	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
$596$	18.0000			0.737309
$597$		0			0
$598$		0			0
$599$	−12.0000	+	20.7846i	−0.490307	+	0.849236i	−0.999938	−	0.0111569i	$-0.996449\pi$
$599$	−12.0000	+	20.7846i	−0.490307	+	0.849236i	0.509631	+	0.860393i	$0.329782\pi$
$600$		0			0
$601$	−26.0000			−1.06056			−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000			−1.06056			−0.530281	+	0.847822i	$0.677914\pi$
$602$		0			0
$603$		0			0
$604$	−4.00000	−	6.92820i	−0.162758	−	0.281905i
$605$		0			0
$606$		0			0
$607$	16.0000	+	27.7128i	0.649420	+	1.12483i	0.983262	+	0.182199i	$0.0583216\pi$
$607$	16.0000	+	27.7128i	0.649420	+	1.12483i	−0.333842	+	0.942629i	$0.608345\pi$
$608$	−2.00000			−0.0811107
$609$		0			0
$610$		0			0
$611$	24.0000	+	41.5692i	0.970936	+	1.68171i
$612$		0			0
$613$	−1.00000	+	1.73205i	−0.0403896	+	0.0699569i	−0.885514	−	0.464614i	$-0.846193\pi$
$613$	−1.00000	+	1.73205i	−0.0403896	+	0.0699569i	0.845124	+	0.534570i	$0.179527\pi$
$614$	1.00000	+	1.73205i	0.0403567	+	0.0698999i
$615$		0			0
$616$		0			0
$617$	−6.00000			−0.241551			−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000			−0.241551			−0.120775	+	0.992680i	$0.538538\pi$
$618$		0			0
$619$	13.0000	−	22.5167i	0.522514	−	0.905021i	−0.477143	−	0.878826i	$-0.658328\pi$
$619$	13.0000	−	22.5167i	0.522514	−	0.905021i	0.999657	−	0.0261952i	$-0.00833914\pi$
$620$		0			0
$621$		0			0
$622$	−24.0000			−0.962312
$623$		0			0
$624$		0			0
$625$	−12.5000	−	21.6506i	−0.500000	−	0.866025i
$626$	−5.00000	+	8.66025i	−0.199840	+	0.346133i
$627$		0			0
$628$	−2.00000	−	3.46410i	−0.0798087	−	0.138233i
$629$	12.0000			0.478471
$630$		0			0
$631$	−16.0000			−0.636950			−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000			−0.636950			−0.318475	+	0.947931i	$0.603171\pi$
$632$	−4.00000	−	6.92820i	−0.159111	−	0.275589i
$633$		0			0
$634$	3.00000	−	5.19615i	0.119145	−	0.206366i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−9.00000	+	15.5885i	−0.355479	+	0.615707i	−0.987200	−	0.159489i	$-0.949015\pi$
$641$	−9.00000	+	15.5885i	−0.355479	+	0.615707i	0.631721	+	0.775196i	$0.282349\pi$
$642$		0			0
$643$	−14.0000			−0.552106			−0.276053	−	0.961142i	$-0.589027\pi$
$643$	−14.0000			−0.552106			−0.276053	+	0.961142i	$0.589027\pi$
$644$		0			0
$645$		0			0
$646$	6.00000	+	10.3923i	0.236067	+	0.408880i
$647$	6.00000	−	10.3923i	0.235884	−	0.408564i	−0.723645	−	0.690172i	$-0.757535\pi$
$647$	6.00000	−	10.3923i	0.235884	−	0.408564i	0.959529	+	0.281609i	$0.0908680\pi$
$648$		0			0
$649$		0			0
$650$	−20.0000			−0.784465
$651$		0			0
$652$	−16.0000			−0.626608
$653$	9.00000	+	15.5885i	0.352197	+	0.610023i	0.986634	−	0.162951i	$-0.0521013\pi$
$653$	9.00000	+	15.5885i	0.352197	+	0.610023i	−0.634437	+	0.772975i	$0.718768\pi$
$654$		0			0
$655$		0			0
$656$	−3.00000	−	5.19615i	−0.117130	−	0.202876i
$657$		0			0
$658$		0			0
$659$	24.0000			0.934907			0.467454	−	0.884018i	$-0.345171\pi$
$659$	24.0000			0.934907			0.467454	+	0.884018i	$0.345171\pi$
$660$		0			0
$661$	−20.0000	+	34.6410i	−0.777910	+	1.34738i	0.155235	+	0.987878i	$0.450387\pi$
$661$	−20.0000	+	34.6410i	−0.777910	+	1.34738i	−0.933144	+	0.359502i	$0.882947\pi$
$662$	−4.00000	+	6.92820i	−0.155464	+	0.269272i
$663$		0			0
$664$	−6.00000			−0.232845
$665$		0			0
$666$		0			0
$667$		0			0
$668$	6.00000	−	10.3923i	0.232147	−	0.402090i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	26.0000			1.00223			0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000			1.00223			0.501113	+	0.865382i	$0.332924\pi$
$674$	−7.00000	−	12.1244i	−0.269630	−	0.467013i
$675$		0			0
$676$	−1.50000	+	2.59808i	−0.0576923	+	0.0999260i
$677$	6.00000	+	10.3923i	0.230599	+	0.399409i	0.957984	−	0.286820i	$-0.0925982\pi$
$677$	6.00000	+	10.3923i	0.230599	+	0.399409i	−0.727386	+	0.686229i	$0.759265\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−6.00000	+	10.3923i	−0.229584	+	0.397650i	−0.957685	−	0.287819i	$-0.907070\pi$
$683$	−6.00000	+	10.3923i	−0.229584	+	0.397650i	0.728101	+	0.685470i	$0.240403\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−4.00000	−	6.92820i	−0.152499	−	0.264135i
$689$	12.0000	−	20.7846i	0.457164	−	0.791831i
$690$		0			0
$691$	−23.0000	−	39.8372i	−0.874961	−	1.51548i	−0.856804	−	0.515642i	$-0.827553\pi$
$691$	−23.0000	−	39.8372i	−0.874961	−	1.51548i	−0.0181572	−	0.999835i	$-0.505780\pi$
$692$	−12.0000			−0.456172
$693$		0			0
$694$	24.0000			0.911028
$695$		0			0
$696$		0			0
$697$	−18.0000	+	31.1769i	−0.681799	+	1.18091i
$698$	−14.0000	−	24.2487i	−0.529908	−	0.917827i
$699$		0			0
$700$		0			0
$701$	−18.0000			−0.679851			−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000			−0.679851			−0.339925	+	0.940452i	$0.610402\pi$
$702$		0			0
$703$	2.00000	−	3.46410i	0.0754314	−	0.130651i
$704$		0			0
$705$		0			0
$706$	18.0000			0.677439
$707$		0			0
$708$		0			0
$709$	23.0000	+	39.8372i	0.863783	+	1.49612i	0.868250	+	0.496126i	$0.165245\pi$
$709$	23.0000	+	39.8372i	0.863783	+	1.49612i	−0.00446726	+	0.999990i	$0.501422\pi$
$710$		0			0
$711$		0			0
$712$	3.00000	+	5.19615i	0.112430	+	0.194734i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−6.00000	−	10.3923i	−0.224231	−	0.388379i
$717$		0			0
$718$	−12.0000	+	20.7846i	−0.447836	+	0.775675i
$719$	−6.00000	−	10.3923i	−0.223762	−	0.387568i	0.732185	−	0.681106i	$-0.238501\pi$
$719$	−6.00000	−	10.3923i	−0.223762	−	0.387568i	−0.955947	+	0.293538i	$0.905167\pi$
$720$		0			0
$721$		0			0
$722$	−15.0000			−0.558242
$723$		0			0
$724$	10.0000	−	17.3205i	0.371647	−	0.643712i
$725$	15.0000	−	25.9808i	0.557086	−	0.964901i
$726$		0			0
$727$	−44.0000			−1.63187			−0.815935	−	0.578144i	$-0.803777\pi$
$727$	−44.0000			−1.63187			−0.815935	+	0.578144i	$0.803777\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−24.0000	+	41.5692i	−0.887672	+	1.53749i
$732$		0			0
$733$	−20.0000	−	34.6410i	−0.738717	−	1.27950i	−0.953073	−	0.302740i	$-0.902099\pi$
$733$	−20.0000	−	34.6410i	−0.738717	−	1.27950i	0.214356	−	0.976756i	$-0.431235\pi$
$734$	−8.00000			−0.295285
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	8.00000	−	13.8564i	0.294285	−	0.509716i	−0.680534	−	0.732717i	$-0.738252\pi$
$739$	8.00000	−	13.8564i	0.294285	−	0.509716i	0.974818	+	0.223001i	$0.0715853\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−24.0000			−0.880475			−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000			−0.880475			−0.440237	+	0.897881i	$0.645106\pi$
$744$		0			0
$745$		0			0
$746$	−7.00000	+	12.1244i	−0.256288	+	0.443904i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	20.0000	+	34.6410i	0.729810	+	1.26407i	0.956963	+	0.290209i	$0.0937250\pi$
$751$	20.0000	+	34.6410i	0.729810	+	1.26407i	−0.227153	+	0.973859i	$0.572942\pi$
$752$	6.00000	−	10.3923i	0.218797	−	0.378968i
$753$		0			0
$754$	−12.0000	−	20.7846i	−0.437014	−	0.756931i
$755$		0			0
$756$		0			0
$757$	2.00000			0.0726912			0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000			0.0726912			0.0363456	+	0.999339i	$0.488428\pi$
$758$	8.00000	+	13.8564i	0.290573	+	0.503287i
$759$		0			0
$760$		0			0
$761$	9.00000	+	15.5885i	0.326250	+	0.565081i	0.981764	−	0.190101i	$-0.0608816\pi$
$761$	9.00000	+	15.5885i	0.326250	+	0.565081i	−0.655515	+	0.755182i	$0.727548\pi$
$762$		0			0
$763$		0			0
$764$	−24.0000			−0.868290
$765$		0			0
$766$	−18.0000	+	31.1769i	−0.650366	+	1.12647i
$767$	12.0000	−	20.7846i	0.433295	−	0.750489i
$768$		0			0
$769$	−14.0000			−0.504853			−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000			−0.504853			−0.252426	+	0.967616i	$0.581229\pi$
$770$		0			0
$771$		0			0
$772$	−7.00000	−	12.1244i	−0.251936	−	0.436365i
$773$	−12.0000	+	20.7846i	−0.431610	+	0.747570i	−0.997012	−	0.0772449i	$-0.975388\pi$
$773$	−12.0000	+	20.7846i	−0.431610	+	0.747570i	0.565402	+	0.824815i	$0.308721\pi$
$774$		0			0
$775$	10.0000	+	17.3205i	0.359211	+	0.622171i
$776$	10.0000			0.358979
$777$		0			0
$778$	−18.0000			−0.645331
$779$	6.00000	+	10.3923i	0.214972	+	0.372343i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−11.0000	+	19.0526i	−0.392108	+	0.679150i	−0.992727	−	0.120384i	$-0.961587\pi$
$787$	−11.0000	+	19.0526i	−0.392108	+	0.679150i	0.600620	+	0.799535i	$0.294921\pi$
$788$	−9.00000	+	15.5885i	−0.320612	+	0.555316i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	16.0000	+	27.7128i	0.568177	+	0.984111i
$794$	10.0000	−	17.3205i	0.354887	−	0.614682i
$795$		0			0
$796$

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:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	882.g (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +1.00000 q^{8} +O(q^{10})\)	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +1.00000 q^{8} +4.00000 q^{13} +(-0.500000 - 0.866025i) q^{16} +(-3.00000 + 5.19615i) q^{17} +(1.00000 + 1.73205i) q^{19} +(2.50000 - 4.33013i) q^{25} +(-2.00000 - 3.46410i) q^{26} +6.00000 q^{29} +(-2.00000 + 3.46410i) q^{31} +(-0.500000 + 0.866025i) q^{32} +6.00000 q^{34} +(-1.00000 - 1.73205i) q^{37} +(1.00000 - 1.73205i) q^{38} +6.00000 q^{41} +8.00000 q^{43} +(6.00000 + 10.3923i) q^{47} -5.00000 q^{50} +(-2.00000 + 3.46410i) q^{52} +(3.00000 - 5.19615i) q^{53} +(-3.00000 - 5.19615i) q^{58} +(3.00000 - 5.19615i) q^{59} +(4.00000 + 6.92820i) q^{61} +4.00000 q^{62} +1.00000 q^{64} +(2.00000 - 3.46410i) q^{67} +(-3.00000 - 5.19615i) q^{68} +(1.00000 - 1.73205i) q^{73} +(-1.00000 + 1.73205i) q^{74} -2.00000 q^{76} +(-4.00000 - 6.92820i) q^{79} +(-3.00000 - 5.19615i) q^{82} -6.00000 q^{83} +(-4.00000 - 6.92820i) q^{86} +(3.00000 + 5.19615i) q^{89} +(6.00000 - 10.3923i) q^{94} +10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} + 2 q^{8}+O(q^{10}) \) 2 * q - q^2 - q^4 + 2 * q^8	\( 2 q - q^{2} - q^{4} + 2 q^{8} + 8 q^{13} - q^{16} - 6 q^{17} + 2 q^{19} + 5 q^{25} - 4 q^{26} + 12 q^{29} - 4 q^{31} - q^{32} + 12 q^{34} - 2 q^{37} + 2 q^{38} + 12 q^{41} + 16 q^{43} + 12 q^{47} - 10 q^{50} - 4 q^{52} + 6 q^{53} - 6 q^{58} + 6 q^{59} + 8 q^{61} + 8 q^{62} + 2 q^{64} + 4 q^{67} - 6 q^{68} + 2 q^{73} - 2 q^{74} - 4 q^{76} - 8 q^{79} - 6 q^{82} - 12 q^{83} - 8 q^{86} + 6 q^{89} + 12 q^{94} + 20 q^{97}+O(q^{100}) \) 2 * q - q^2 - q^4 + 2 * q^8 + 8 * q^13 - q^16 - 6 * q^17 + 2 * q^19 + 5 * q^25 - 4 * q^26 + 12 * q^29 - 4 * q^31 - q^32 + 12 * q^34 - 2 * q^37 + 2 * q^38 + 12 * q^41 + 16 * q^43 + 12 * q^47 - 10 * q^50 - 4 * q^52 + 6 * q^53 - 6 * q^58 + 6 * q^59 + 8 * q^61 + 8 * q^62 + 2 * q^64 + 4 * q^67 - 6 * q^68 + 2 * q^73 - 2 * q^74 - 4 * q^76 - 8 * q^79 - 6 * q^82 - 12 * q^83 - 8 * q^86 + 6 * q^89 + 12 * q^94 + 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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