LMFDB - Embedded newform 2268.2.j.p.757.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2268,2,Mod(757,2268)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2268, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2268.757"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2268 = 2^{2} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2268.j (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,-2,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$18.1100711784$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			757.1
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2268.757
Dual form			2268.2.j.p.1513.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.73205 + 3.00000i) q^{5} +(-0.500000 - 0.866025i) q^{7} +(-0.866025 - 1.50000i) q^{11} +(2.00000 - 3.46410i) q^{13} +3.46410 q^{17} +2.00000 q^{19} +(-1.73205 + 3.00000i) q^{23} +(-3.50000 - 6.06218i) q^{25} +(3.46410 + 6.00000i) q^{29} +(-1.00000 + 1.73205i) q^{31} +3.46410 q^{35} +11.0000 q^{37} +(1.73205 - 3.00000i) q^{41} +(0.500000 + 0.866025i) q^{43} +(-0.500000 + 0.866025i) q^{49} -1.73205 q^{53} +6.00000 q^{55} +(-5.19615 + 9.00000i) q^{59} +(-4.00000 - 6.92820i) q^{61} +(6.92820 + 12.0000i) q^{65} +(-5.50000 + 9.52628i) q^{67} +5.19615 q^{71} -16.0000 q^{73} +(-0.866025 + 1.50000i) q^{77} +(6.50000 + 11.2583i) q^{79} +(8.66025 + 15.0000i) q^{83} +(-6.00000 + 10.3923i) q^{85} +6.92820 q^{89} -4.00000 q^{91} +(-3.46410 + 6.00000i) q^{95} +(-1.00000 - 1.73205i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{7} + 8 q^{13} + 8 q^{19} - 14 q^{25} - 4 q^{31} + 44 q^{37} + 2 q^{43} - 2 q^{49} + 24 q^{55} - 16 q^{61} - 22 q^{67} - 64 q^{73} + 26 q^{79} - 24 q^{85} - 16 q^{91} - 4 q^{97}+O(q^{100}) $ 4 * q - 2 * q^7 + 8 * q^13 + 8 * q^19 - 14 * q^25 - 4 * q^31 + 44 * q^37 + 2 * q^43 - 2 * q^49 + 24 * q^55 - 16 * q^61 - 22 * q^67 - 64 * q^73 + 26 * q^79 - 24 * q^85 - 16 * q^91 - 4 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1541$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	−1.73205	+	3.00000i	−0.774597	+	1.34164i	0.160424	+	0.987048i	$0.448714\pi$
$5$	−1.73205	+	3.00000i	−0.774597	+	1.34164i	−0.935021	+	0.354593i	$0.884620\pi$
$6$		0			0
$7$	−0.500000	−	0.866025i	−0.188982	−	0.327327i
$7$	−0.500000	−	0.866025i	−0.188982	−	0.327327i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.866025	−	1.50000i	−0.261116	−	0.452267i	0.705422	−	0.708787i	$-0.250757\pi$
$11$	−0.866025	−	1.50000i	−0.261116	−	0.452267i	−0.966539	+	0.256520i	$0.917424\pi$
$12$		0			0
$13$	2.00000	−	3.46410i	0.554700	−	0.960769i	−0.443227	−	0.896410i	$-0.646166\pi$
$13$	2.00000	−	3.46410i	0.554700	−	0.960769i	0.997927	−	0.0643593i	$-0.0205004\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.46410			0.840168			0.420084	−	0.907485i	$-0.362001\pi$
$17$	3.46410			0.840168			0.420084	+	0.907485i	$0.362001\pi$
$18$		0			0
$19$	2.00000			0.458831			0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000			0.458831			0.229416	+	0.973329i	$0.426318\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.73205	+	3.00000i	−0.361158	+	0.625543i	−0.988152	−	0.153481i	$-0.950952\pi$
$23$	−1.73205	+	3.00000i	−0.361158	+	0.625543i	0.626994	+	0.779024i	$0.284285\pi$
$24$		0			0
$25$	−3.50000	−	6.06218i	−0.700000	−	1.21244i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.46410	+	6.00000i	0.643268	+	1.11417i	0.984699	+	0.174265i	$0.0557550\pi$
$29$	3.46410	+	6.00000i	0.643268	+	1.11417i	−0.341431	+	0.939907i	$0.610912\pi$
$30$		0			0
$31$	−1.00000	+	1.73205i	−0.179605	+	0.311086i	−0.941745	−	0.336327i	$-0.890815\pi$
$31$	−1.00000	+	1.73205i	−0.179605	+	0.311086i	0.762140	+	0.647412i	$0.224149\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.46410			0.585540
$36$		0			0
$37$	11.0000			1.80839			0.904194	−	0.427121i	$-0.140472\pi$
$37$	11.0000			1.80839			0.904194	+	0.427121i	$0.140472\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.73205	−	3.00000i	0.270501	−	0.468521i	−0.698489	−	0.715621i	$-0.746144\pi$
$41$	1.73205	−	3.00000i	0.270501	−	0.468521i	0.968990	+	0.247099i	$0.0794774\pi$
$42$		0			0
$43$	0.500000	+	0.866025i	0.0762493	+	0.132068i	0.901629	−	0.432511i	$-0.142372\pi$
$43$	0.500000	+	0.866025i	0.0762493	+	0.132068i	−0.825380	+	0.564578i	$0.809039\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$		0			0
$49$	−0.500000	+	0.866025i	−0.0714286	+	0.123718i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.73205			−0.237915			−0.118958	−	0.992899i	$-0.537955\pi$
$53$	−1.73205			−0.237915			−0.118958	+	0.992899i	$0.537955\pi$
$54$		0			0
$55$	6.00000			0.809040
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−5.19615	+	9.00000i	−0.676481	+	1.17170i	0.299552	+	0.954080i	$0.403163\pi$
$59$	−5.19615	+	9.00000i	−0.676481	+	1.17170i	−0.976034	+	0.217620i	$0.930171\pi$
$60$		0			0
$61$	−4.00000	−	6.92820i	−0.512148	−	0.887066i	−0.999901	−	0.0140840i	$-0.995517\pi$
$61$	−4.00000	−	6.92820i	−0.512148	−	0.887066i	0.487753	−	0.872982i	$-0.337817\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	6.92820	+	12.0000i	0.859338	+	1.48842i
$66$		0			0
$67$	−5.50000	+	9.52628i	−0.671932	+	1.16382i	0.305424	+	0.952217i	$0.401202\pi$
$67$	−5.50000	+	9.52628i	−0.671932	+	1.16382i	−0.977356	+	0.211604i	$0.932131\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	5.19615			0.616670			0.308335	−	0.951278i	$-0.400228\pi$
$71$	5.19615			0.616670			0.308335	+	0.951278i	$0.400228\pi$
$72$		0			0
$73$	−16.0000			−1.87266			−0.936329	−	0.351123i	$-0.885800\pi$
$73$	−16.0000			−1.87266			−0.936329	+	0.351123i	$0.885800\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.866025	+	1.50000i	−0.0986928	+	0.170941i
$78$		0			0
$79$	6.50000	+	11.2583i	0.731307	+	1.26666i	0.956325	+	0.292306i	$0.0944227\pi$
$79$	6.50000	+	11.2583i	0.731307	+	1.26666i	−0.225018	+	0.974355i	$0.572244\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	8.66025	+	15.0000i	0.950586	+	1.64646i	0.744160	+	0.668002i	$0.232850\pi$
$83$	8.66025	+	15.0000i	0.950586	+	1.64646i	0.206427	+	0.978462i	$0.433816\pi$
$84$		0			0
$85$	−6.00000	+	10.3923i	−0.650791	+	1.12720i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	6.92820			0.734388			0.367194	−	0.930144i	$-0.380318\pi$
$89$	6.92820			0.734388			0.367194	+	0.930144i	$0.380318\pi$
$90$		0			0
$91$	−4.00000			−0.419314
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−3.46410	+	6.00000i	−0.355409	+	0.615587i
$96$		0			0
$97$	−1.00000	−	1.73205i	−0.101535	−	0.175863i	0.810782	−	0.585348i	$-0.199042\pi$
$97$	−1.00000	−	1.73205i	−0.101535	−	0.175863i	−0.912317	+	0.409484i	$0.865709\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.92820	+	12.0000i	0.689382	+	1.19404i	0.972038	+	0.234823i	$0.0754512\pi$
$101$	6.92820	+	12.0000i	0.689382	+	1.19404i	−0.282656	+	0.959221i	$0.591216\pi$
$102$		0			0
$103$	−7.00000	+	12.1244i	−0.689730	+	1.19465i	0.282194	+	0.959357i	$0.408938\pi$
$103$	−7.00000	+	12.1244i	−0.689730	+	1.19465i	−0.971925	+	0.235291i	$0.924396\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−19.0526			−1.84188			−0.920940	−	0.389704i	$-0.872577\pi$
$107$	−19.0526			−1.84188			−0.920940	+	0.389704i	$0.872577\pi$
$108$		0			0
$109$	2.00000			0.191565			0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000			0.191565			0.0957826	+	0.995402i	$0.469465\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	4.33013	−	7.50000i	0.407344	−	0.705541i	−0.587247	−	0.809408i	$-0.699788\pi$
$113$	4.33013	−	7.50000i	0.407344	−	0.705541i	0.994591	+	0.103867i	$0.0331216\pi$
$114$		0			0
$115$	−6.00000	−	10.3923i	−0.559503	−	0.969087i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.73205	−	3.00000i	−0.158777	−	0.275010i
$120$		0			0
$121$	4.00000	−	6.92820i	0.363636	−	0.629837i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	6.92820			0.619677
$126$		0			0
$127$	17.0000			1.50851			0.754253	−	0.656584i	$-0.227999\pi$
$127$	17.0000			1.50851			0.754253	+	0.656584i	$0.227999\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−6.92820	+	12.0000i	−0.605320	+	1.04844i	0.386681	+	0.922214i	$0.373621\pi$
$131$	−6.92820	+	12.0000i	−0.605320	+	1.04844i	−0.992001	+	0.126231i	$0.959712\pi$
$132$		0			0
$133$	−1.00000	−	1.73205i	−0.0867110	−	0.150188i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	4.33013	+	7.50000i	0.369948	+	0.640768i	0.989557	−	0.144142i	$-0.0460423\pi$
$137$	4.33013	+	7.50000i	0.369948	+	0.640768i	−0.619609	+	0.784910i	$0.712709\pi$
$138$		0			0
$139$	−7.00000	+	12.1244i	−0.593732	+	1.02837i	0.399992	+	0.916519i	$0.369013\pi$
$139$	−7.00000	+	12.1244i	−0.593732	+	1.02837i	−0.993724	+	0.111856i	$0.964321\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−6.92820			−0.579365
$144$		0			0
$145$	−24.0000			−1.99309
$146$		0			0
$147$		0			0
$148$		0			0
$149$	6.06218	−	10.5000i	0.496633	−	0.860194i	−0.503360	−	0.864077i	$-0.667903\pi$
$149$	6.06218	−	10.5000i	0.496633	−	0.860194i	0.999992	+	0.00388357i	$0.00123618\pi$
$150$		0			0
$151$	−5.50000	−	9.52628i	−0.447584	−	0.775238i	0.550645	−	0.834740i	$-0.314382\pi$
$151$	−5.50000	−	9.52628i	−0.447584	−	0.775238i	−0.998228	+	0.0595022i	$0.981049\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−3.46410	−	6.00000i	−0.278243	−	0.481932i
$156$		0			0
$157$	2.00000	−	3.46410i	0.159617	−	0.276465i	−0.775113	−	0.631822i	$-0.782307\pi$
$157$	2.00000	−	3.46410i	0.159617	−	0.276465i	0.934731	+	0.355357i	$0.115641\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	3.46410			0.273009
$162$		0			0
$163$	23.0000			1.80150			0.900750	−	0.434339i	$-0.143018\pi$
$163$	23.0000			1.80150			0.900750	+	0.434339i	$0.143018\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.73205	−	3.00000i	0.134030	−	0.232147i	−0.791196	−	0.611562i	$-0.790541\pi$
$167$	1.73205	−	3.00000i	0.134030	−	0.232147i	0.925227	+	0.379415i	$0.123875\pi$
$168$		0			0
$169$	−1.50000	−	2.59808i	−0.115385	−	0.199852i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$		0			0
$175$	−3.50000	+	6.06218i	−0.264575	+	0.458258i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	24.2487			1.81243			0.906217	−	0.422813i	$-0.138957\pi$
$179$	24.2487			1.81243			0.906217	+	0.422813i	$0.138957\pi$
$180$		0			0
$181$	26.0000			1.93256			0.966282	−	0.257485i	$-0.0828937\pi$
$181$	26.0000			1.93256			0.966282	+	0.257485i	$0.0828937\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−19.0526	+	33.0000i	−1.40077	+	2.42621i
$186$		0			0
$187$	−3.00000	−	5.19615i	−0.219382	−	0.379980i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	7.79423	+	13.5000i	0.563971	+	0.976826i	0.997145	+	0.0755154i	$0.0240602\pi$
$191$	7.79423	+	13.5000i	0.563971	+	0.976826i	−0.433174	+	0.901310i	$0.642606\pi$
$192$		0			0
$193$	−1.00000	+	1.73205i	−0.0719816	+	0.124676i	−0.899770	−	0.436365i	$-0.856266\pi$
$193$	−1.00000	+	1.73205i	−0.0719816	+	0.124676i	0.827788	+	0.561041i	$0.189599\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	5.19615			0.370211			0.185105	−	0.982719i	$-0.440737\pi$
$197$	5.19615			0.370211			0.185105	+	0.982719i	$0.440737\pi$
$198$		0			0
$199$	−4.00000			−0.283552			−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000			−0.283552			−0.141776	+	0.989899i	$0.545281\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	3.46410	−	6.00000i	0.243132	−	0.421117i
$204$		0			0
$205$	6.00000	+	10.3923i	0.419058	+	0.725830i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1.73205	−	3.00000i	−0.119808	−	0.207514i
$210$		0			0
$211$	9.50000	−	16.4545i	0.654007	−	1.13277i	−0.328135	−	0.944631i	$-0.606420\pi$
$211$	9.50000	−	16.4545i	0.654007	−	1.13277i	0.982142	−	0.188142i	$-0.0602466\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−3.46410			−0.236250
$216$		0			0
$217$	2.00000			0.135769
$218$		0			0
$219$		0			0
$220$		0			0
$221$	6.92820	−	12.0000i	0.466041	−	0.807207i
$222$		0			0
$223$	−1.00000	−	1.73205i	−0.0669650	−	0.115987i	0.830599	−	0.556871i	$-0.187998\pi$
$223$	−1.00000	−	1.73205i	−0.0669650	−	0.115987i	−0.897564	+	0.440884i	$0.854665\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−8.66025	−	15.0000i	−0.574801	−	0.995585i	−0.996063	−	0.0886460i	$-0.971746\pi$
$227$	−8.66025	−	15.0000i	−0.574801	−	0.995585i	0.421262	−	0.906939i	$-0.361587\pi$
$228$		0			0
$229$	11.0000	−	19.0526i	0.726900	−	1.25903i	−0.231287	−	0.972886i	$-0.574293\pi$
$229$	11.0000	−	19.0526i	0.726900	−	1.25903i	0.958187	−	0.286143i	$-0.0923732\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	6.92820			0.453882			0.226941	−	0.973909i	$-0.427128\pi$
$233$	6.92820			0.453882			0.226941	+	0.973909i	$0.427128\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−0.866025	+	1.50000i	−0.0560185	+	0.0970269i	−0.892675	−	0.450701i	$-0.851174\pi$
$239$	−0.866025	+	1.50000i	−0.0560185	+	0.0970269i	0.836656	+	0.547728i	$0.184507\pi$
$240$		0			0
$241$	8.00000	+	13.8564i	0.515325	+	0.892570i	0.999842	+	0.0177875i	$0.00566223\pi$
$241$	8.00000	+	13.8564i	0.515325	+	0.892570i	−0.484516	+	0.874782i	$0.661004\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.73205	−	3.00000i	−0.110657	−	0.191663i
$246$		0			0
$247$	4.00000	−	6.92820i	0.254514	−	0.440831i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−20.7846			−1.31191			−0.655956	−	0.754799i	$-0.727735\pi$
$251$	−20.7846			−1.31191			−0.655956	+	0.754799i	$0.727735\pi$
$252$		0			0
$253$	6.00000			0.377217
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−6.92820	+	12.0000i	−0.432169	+	0.748539i	−0.997060	−	0.0766265i	$-0.975585\pi$
$257$	−6.92820	+	12.0000i	−0.432169	+	0.748539i	0.564890	+	0.825166i	$0.308918\pi$
$258$		0			0
$259$	−5.50000	−	9.52628i	−0.341753	−	0.591934i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−11.2583	−	19.5000i	−0.694218	−	1.20242i	−0.970443	−	0.241329i	$-0.922417\pi$
$263$	−11.2583	−	19.5000i	−0.694218	−	1.20242i	0.276225	−	0.961093i	$-0.410916\pi$
$264$		0			0
$265$	3.00000	−	5.19615i	0.184289	−	0.319197i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−6.92820			−0.422420			−0.211210	−	0.977441i	$-0.567740\pi$
$269$	−6.92820			−0.422420			−0.211210	+	0.977441i	$0.567740\pi$
$270$		0			0
$271$	−4.00000			−0.242983			−0.121491	−	0.992592i	$-0.538768\pi$
$271$	−4.00000			−0.242983			−0.121491	+	0.992592i	$0.538768\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−6.06218	+	10.5000i	−0.365563	+	0.633174i
$276$		0			0
$277$	0.500000	+	0.866025i	0.0300421	+	0.0520344i	0.880656	−	0.473757i	$-0.157103\pi$
$277$	0.500000	+	0.866025i	0.0300421	+	0.0520344i	−0.850613	+	0.525792i	$0.823769\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.866025	+	1.50000i	0.0516627	+	0.0894825i	0.890700	−	0.454591i	$-0.150215\pi$
$281$	0.866025	+	1.50000i	0.0516627	+	0.0894825i	−0.839038	+	0.544074i	$0.816881\pi$
$282$		0			0
$283$	8.00000	−	13.8564i	0.475551	−	0.823678i	−0.524057	−	0.851683i	$-0.675582\pi$
$283$	8.00000	−	13.8564i	0.475551	−	0.823678i	0.999608	+	0.0280052i	$0.00891551\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−3.46410			−0.204479
$288$		0			0
$289$	−5.00000			−0.294118
$290$		0			0
$291$		0			0
$292$		0			0
$293$	6.92820	−	12.0000i	0.404750	−	0.701047i	−0.589542	−	0.807737i	$-0.700692\pi$
$293$	6.92820	−	12.0000i	0.404750	−	0.701047i	0.994292	+	0.106690i	$0.0340252\pi$
$294$		0			0
$295$	−18.0000	−	31.1769i	−1.04800	−	1.81519i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	6.92820	+	12.0000i	0.400668	+	0.693978i
$300$		0			0
$301$	0.500000	−	0.866025i	0.0288195	−	0.0499169i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	27.7128			1.58683
$306$		0			0
$307$	−10.0000			−0.570730			−0.285365	−	0.958419i	$-0.592115\pi$
$307$	−10.0000			−0.570730			−0.285365	+	0.958419i	$0.592115\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−10.3923	+	18.0000i	−0.589294	+	1.02069i	0.405032	+	0.914303i	$0.367261\pi$
$311$	−10.3923	+	18.0000i	−0.589294	+	1.02069i	−0.994325	+	0.106384i	$0.966073\pi$
$312$		0			0
$313$	11.0000	+	19.0526i	0.621757	+	1.07691i	0.989158	+	0.146852i	$0.0469141\pi$
$313$	11.0000	+	19.0526i	0.621757	+	1.07691i	−0.367402	+	0.930062i	$0.619753\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$317$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$318$		0			0
$319$	6.00000	−	10.3923i	0.335936	−	0.581857i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	6.92820			0.385496
$324$		0			0
$325$	−28.0000			−1.55316
$326$		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
$335$	−19.0526	−	33.0000i	−1.04095	−	1.80298i
$336$		0			0
$337$	−11.5000	+	19.9186i	−0.626445	+	1.08503i	0.361815	+	0.932250i	$0.382157\pi$
$337$	−11.5000	+	19.9186i	−0.626445	+	1.08503i	−0.988260	+	0.152784i	$0.951176\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	3.46410			0.187592
$342$		0			0
$343$	1.00000			0.0539949
$344$		0			0
$345$		0			0
$346$		0			0
$347$	18.1865	−	31.5000i	0.976304	−	1.69101i	0.300742	−	0.953705i	$-0.402766\pi$
$347$	18.1865	−	31.5000i	0.976304	−	1.69101i	0.675562	−	0.737303i	$-0.263901\pi$
$348$		0			0
$349$	8.00000	+	13.8564i	0.428230	+	0.741716i	0.996716	−	0.0809766i	$-0.0258039\pi$
$349$	8.00000	+	13.8564i	0.428230	+	0.741716i	−0.568486	+	0.822693i	$0.692471\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.73205	+	3.00000i	0.0921878	+	0.159674i	0.908431	−	0.418034i	$-0.137281\pi$
$353$	1.73205	+	3.00000i	0.0921878	+	0.159674i	−0.816244	+	0.577708i	$0.803947\pi$
$354$		0			0
$355$	−9.00000	+	15.5885i	−0.477670	+	0.827349i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−29.4449			−1.55404			−0.777020	−	0.629476i	$-0.783270\pi$
$359$	−29.4449			−1.55404			−0.777020	+	0.629476i	$0.783270\pi$
$360$		0			0
$361$	−15.0000			−0.789474
$362$		0			0
$363$		0			0
$364$		0			0
$365$	27.7128	−	48.0000i	1.45055	−	2.51243i
$366$		0			0
$367$	14.0000	+	24.2487i	0.730794	+	1.26577i	0.956544	+	0.291587i	$0.0941834\pi$
$367$	14.0000	+	24.2487i	0.730794	+	1.26577i	−0.225750	+	0.974185i	$0.572483\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	0.866025	+	1.50000i	0.0449618	+	0.0778761i
$372$		0			0
$373$	6.50000	−	11.2583i	0.336557	−	0.582934i	−0.647225	−	0.762299i	$-0.724071\pi$
$373$	6.50000	−	11.2583i	0.336557	−	0.582934i	0.983783	+	0.179364i	$0.0574041\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	27.7128			1.42728
$378$		0			0
$379$	17.0000			0.873231			0.436616	−	0.899648i	$-0.356177\pi$
$379$	17.0000			0.873231			0.436616	+	0.899648i	$0.356177\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	13.8564	−	24.0000i	0.708029	−	1.22634i	−0.257558	−	0.966263i	$-0.582918\pi$
$383$	13.8564	−	24.0000i	0.708029	−	1.22634i	0.965587	−	0.260080i	$-0.0837489\pi$
$384$		0			0
$385$	−3.00000	−	5.19615i	−0.152894	−	0.264820i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	6.92820	+	12.0000i	0.351274	+	0.608424i	0.986473	−	0.163924i	$-0.0524153\pi$
$389$	6.92820	+	12.0000i	0.351274	+	0.608424i	−0.635199	+	0.772348i	$0.719082\pi$
$390$		0			0
$391$	−6.00000	+	10.3923i	−0.303433	+	0.525561i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−45.0333			−2.26587
$396$		0			0
$397$	20.0000			1.00377			0.501886	−	0.864934i	$-0.332640\pi$
$397$	20.0000			1.00377			0.501886	+	0.864934i	$0.332640\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−14.7224	+	25.5000i	−0.735203	+	1.27341i	0.219431	+	0.975628i	$0.429580\pi$
$401$	−14.7224	+	25.5000i	−0.735203	+	1.27341i	−0.954634	+	0.297781i	$0.903753\pi$
$402$		0			0
$403$	4.00000	+	6.92820i	0.199254	+	0.345118i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−9.52628	−	16.5000i	−0.472200	−	0.817875i
$408$		0			0
$409$	−1.00000	+	1.73205i	−0.0494468	+	0.0856444i	−0.889689	−	0.456566i	$-0.849079\pi$
$409$	−1.00000	+	1.73205i	−0.0494468	+	0.0856444i	0.840243	+	0.542211i	$0.182412\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	10.3923			0.511372
$414$		0			0
$415$	−60.0000			−2.94528
$416$		0			0
$417$		0			0
$418$		0			0
$419$	12.1244	−	21.0000i	0.592314	−	1.02592i	−0.401606	−	0.915812i	$-0.631548\pi$
$419$	12.1244	−	21.0000i	0.592314	−	1.02592i	0.993920	−	0.110105i	$-0.0351186\pi$
$420$		0			0
$421$	18.5000	+	32.0429i	0.901635	+	1.56168i	0.825372	+	0.564590i	$0.190966\pi$
$421$	18.5000	+	32.0429i	0.901635	+	1.56168i	0.0762630	+	0.997088i	$0.475701\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−12.1244	−	21.0000i	−0.588118	−	1.01865i
$426$		0			0
$427$	−4.00000	+	6.92820i	−0.193574	+	0.335279i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−38.1051			−1.83546			−0.917729	−	0.397206i	$-0.869980\pi$
$431$	−38.1051			−1.83546			−0.917729	+	0.397206i	$0.869980\pi$
$432$		0			0
$433$	−10.0000			−0.480569			−0.240285	−	0.970702i	$-0.577241\pi$
$433$	−10.0000			−0.480569			−0.240285	+	0.970702i	$0.577241\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−3.46410	+	6.00000i	−0.165710	+	0.287019i
$438$		0			0
$439$	−4.00000	−	6.92820i	−0.190910	−	0.330665i	0.754642	−	0.656136i	$-0.227810\pi$
$439$	−4.00000	−	6.92820i	−0.190910	−	0.330665i	−0.945552	+	0.325471i	$0.894477\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−5.19615	−	9.00000i	−0.246877	−	0.427603i	0.715781	−	0.698325i	$-0.246071\pi$
$443$	−5.19615	−	9.00000i	−0.246877	−	0.427603i	−0.962658	+	0.270722i	$0.912738\pi$
$444$		0			0
$445$	−12.0000	+	20.7846i	−0.568855	+	0.985285i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−5.19615			−0.245222			−0.122611	−	0.992455i	$-0.539127\pi$
$449$	−5.19615			−0.245222			−0.122611	+	0.992455i	$0.539127\pi$
$450$		0			0
$451$	−6.00000			−0.282529
$452$		0			0
$453$		0			0
$454$		0			0
$455$	6.92820	−	12.0000i	0.324799	−	0.562569i
$456$		0			0
$457$	−2.50000	−	4.33013i	−0.116945	−	0.202555i	0.801611	−	0.597847i	$-0.203977\pi$
$457$	−2.50000	−	4.33013i	−0.116945	−	0.202555i	−0.918556	+	0.395292i	$0.870643\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−6.92820	−	12.0000i	−0.322679	−	0.558896i	0.658361	−	0.752702i	$-0.271250\pi$
$461$	−6.92820	−	12.0000i	−0.322679	−	0.558896i	−0.981040	+	0.193806i	$0.937917\pi$
$462$		0			0
$463$	−11.5000	+	19.9186i	−0.534450	+	0.925695i	0.464739	+	0.885448i	$0.346148\pi$
$463$	−11.5000	+	19.9186i	−0.534450	+	0.925695i	−0.999190	+	0.0402476i	$0.987185\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−24.2487			−1.12210			−0.561048	−	0.827783i	$-0.689602\pi$
$467$	−24.2487			−1.12210			−0.561048	+	0.827783i	$0.689602\pi$
$468$		0			0
$469$	11.0000			0.507933
$470$		0			0
$471$		0			0
$472$		0			0
$473$	0.866025	−	1.50000i	0.0398199	−	0.0689701i
$474$		0			0
$475$	−7.00000	−	12.1244i	−0.321182	−	0.556304i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	12.1244	+	21.0000i	0.553976	+	0.959514i	0.997982	+	0.0634909i	$0.0202234\pi$
$479$	12.1244	+	21.0000i	0.553976	+	0.959514i	−0.444006	+	0.896024i	$0.646443\pi$
$480$		0			0
$481$	22.0000	−	38.1051i	1.00311	−	1.73744i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	6.92820			0.314594
$486$		0			0
$487$	29.0000			1.31412			0.657058	−	0.753840i	$-0.271801\pi$
$487$	29.0000			1.31412			0.657058	+	0.753840i	$0.271801\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	1.73205	−	3.00000i	0.0781664	−	0.135388i	−0.824292	−	0.566164i	$-0.808427\pi$
$491$	1.73205	−	3.00000i	0.0781664	−	0.135388i	0.902459	+	0.430776i	$0.141760\pi$
$492$		0			0
$493$	12.0000	+	20.7846i	0.540453	+	0.936092i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−2.59808	−	4.50000i	−0.116540	−	0.201853i
$498$		0			0
$499$	2.00000	−	3.46410i	0.0895323	−	0.155074i	−0.817781	−	0.575529i	$-0.804796\pi$
$499$	2.00000	−	3.46410i	0.0895323	−	0.155074i	0.907314	+	0.420455i	$0.138129\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$	−48.0000			−2.13597
$506$		0			0
$507$		0			0
$508$		0			0
$509$	8.66025	−	15.0000i	0.383859	−	0.664863i	−0.607751	−	0.794128i	$-0.707928\pi$
$509$	8.66025	−	15.0000i	0.383859	−	0.664863i	0.991610	+	0.129264i	$0.0412615\pi$
$510$		0			0
$511$	8.00000	+	13.8564i	0.353899	+	0.612971i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−24.2487	−	42.0000i	−1.06853	−	1.85074i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−38.1051			−1.66942			−0.834708	−	0.550693i	$-0.814363\pi$
$521$	−38.1051			−1.66942			−0.834708	+	0.550693i	$0.814363\pi$
$522$		0			0
$523$	−16.0000			−0.699631			−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000			−0.699631			−0.349816	+	0.936819i	$0.613756\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−3.46410	+	6.00000i	−0.150899	+	0.261364i
$528$		0			0
$529$	5.50000	+	9.52628i	0.239130	+	0.414186i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−6.92820	−	12.0000i	−0.300094	−	0.519778i
$534$		0			0
$535$	33.0000	−	57.1577i	1.42671	−	2.47114i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1.73205			0.0746047
$540$		0			0
$541$	−25.0000			−1.07483			−0.537417	−	0.843317i	$-0.680600\pi$
$541$	−25.0000			−1.07483			−0.537417	+	0.843317i	$0.680600\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−3.46410	+	6.00000i	−0.148386	+	0.257012i
$546$		0			0
$547$	−14.5000	−	25.1147i	−0.619975	−	1.07383i	−0.989490	−	0.144604i	$-0.953809\pi$
$547$	−14.5000	−	25.1147i	−0.619975	−	1.07383i	0.369514	−	0.929225i	$-0.379524\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	6.92820	+	12.0000i	0.295151	+	0.511217i
$552$		0			0
$553$	6.50000	−	11.2583i	0.276408	−	0.478753i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	19.0526			0.807283			0.403641	−	0.914917i	$-0.367744\pi$
$557$	19.0526			0.807283			0.403641	+	0.914917i	$0.367744\pi$
$558$		0			0
$559$	4.00000			0.169182
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−5.19615	+	9.00000i	−0.218992	+	0.379305i	−0.954500	−	0.298211i	$-0.903610\pi$
$563$	−5.19615	+	9.00000i	−0.218992	+	0.379305i	0.735508	+	0.677516i	$0.236943\pi$
$564$		0			0
$565$	15.0000	+	25.9808i	0.631055	+	1.09302i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−18.1865	−	31.5000i	−0.762419	−	1.32055i	−0.941600	−	0.336733i	$-0.890678\pi$
$569$	−18.1865	−	31.5000i	−0.762419	−	1.32055i	0.179181	−	0.983816i	$-0.442655\pi$
$570$		0			0
$571$	−10.0000	+	17.3205i	−0.418487	+	0.724841i	−0.995788	−	0.0916910i	$-0.970773\pi$
$571$	−10.0000	+	17.3205i	−0.418487	+	0.724841i	0.577301	+	0.816532i	$0.304106\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	24.2487			1.01124
$576$		0			0
$577$	−16.0000			−0.666089			−0.333044	−	0.942911i	$-0.608076\pi$
$577$	−16.0000			−0.666089			−0.333044	+	0.942911i	$0.608076\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	8.66025	−	15.0000i	0.359288	−	0.622305i
$582$		0			0
$583$	1.50000	+	2.59808i	0.0621237	+	0.107601i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	8.66025	+	15.0000i	0.357447	+	0.619116i	0.987534	−	0.157409i	$-0.0503140\pi$
$587$	8.66025	+	15.0000i	0.357447	+	0.619116i	−0.630087	+	0.776525i	$0.716981\pi$
$588$		0			0
$589$	−2.00000	+	3.46410i	−0.0824086	+	0.142736i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−24.2487			−0.995775			−0.497888	−	0.867242i	$-0.665891\pi$
$593$	−24.2487			−0.995775			−0.497888	+	0.867242i	$0.665891\pi$
$594$		0			0
$595$	12.0000			0.491952
$596$		0			0
$597$		0			0
$598$		0			0
$599$	18.1865	−	31.5000i	0.743082	−	1.28706i	−0.208004	−	0.978128i	$-0.566697\pi$
$599$	18.1865	−	31.5000i	0.743082	−	1.28706i	0.951086	−	0.308927i	$-0.0999699\pi$
$600$		0			0
$601$	−13.0000	−	22.5167i	−0.530281	−	0.918474i	−0.999376	−	0.0353259i	$-0.988753\pi$
$601$	−13.0000	−	22.5167i	−0.530281	−	0.918474i	0.469095	−	0.883148i	$-0.344580\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	13.8564	+	24.0000i	0.563343	+	0.975739i
$606$		0			0
$607$	2.00000	−	3.46410i	0.0811775	−	0.140604i	−0.822578	−	0.568652i	$-0.807465\pi$
$607$	2.00000	−	3.46410i	0.0811775	−	0.140604i	0.903756	+	0.428048i	$0.140799\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	41.0000			1.65597			0.827987	−	0.560747i	$-0.189486\pi$
$613$	41.0000			1.65597			0.827987	+	0.560747i	$0.189486\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	17.3205	−	30.0000i	0.697297	−	1.20775i	−0.272103	−	0.962268i	$-0.587719\pi$
$617$	17.3205	−	30.0000i	0.697297	−	1.20775i	0.969400	−	0.245486i	$-0.0789476\pi$
$618$		0			0
$619$	14.0000	+	24.2487i	0.562708	+	0.974638i	0.997259	+	0.0739910i	$0.0235736\pi$
$619$	14.0000	+	24.2487i	0.562708	+	0.974638i	−0.434551	+	0.900647i	$0.643093\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−3.46410	−	6.00000i	−0.138786	−	0.240385i
$624$		0			0
$625$	5.50000	−	9.52628i	0.220000	−	0.381051i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	38.1051			1.51935
$630$		0			0
$631$	20.0000			0.796187			0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000			0.796187			0.398094	+	0.917345i	$0.369672\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−29.4449	+	51.0000i	−1.16848	+	2.02387i
$636$		0			0
$637$	2.00000	+	3.46410i	0.0792429	+	0.137253i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−6.06218	−	10.5000i	−0.239442	−	0.414725i	0.721113	−	0.692818i	$-0.243631\pi$
$641$	−6.06218	−	10.5000i	−0.239442	−	0.414725i	−0.960554	+	0.278093i	$0.910298\pi$
$642$		0			0
$643$	2.00000	−	3.46410i	0.0788723	−	0.136611i	−0.823891	−	0.566748i	$-0.808201\pi$
$643$	2.00000	−	3.46410i	0.0788723	−	0.136611i	0.902764	+	0.430137i	$0.141535\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	3.46410			0.136188			0.0680939	−	0.997679i	$-0.478308\pi$
$647$	3.46410			0.136188			0.0680939	+	0.997679i	$0.478308\pi$
$648$		0			0
$649$	18.0000			0.706562
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−19.9186	+	34.5000i	−0.779474	+	1.35009i	0.152771	+	0.988262i	$0.451180\pi$
$653$	−19.9186	+	34.5000i	−0.779474	+	1.35009i	−0.932245	+	0.361828i	$0.882153\pi$
$654$		0			0
$655$	−24.0000	−	41.5692i	−0.937758	−	1.62424i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−25.1147	−	43.5000i	−0.978331	−	1.69452i	−0.668473	−	0.743737i	$-0.733052\pi$
$659$	−25.1147	−	43.5000i	−0.978331	−	1.69452i	−0.309859	−	0.950783i	$-0.600282\pi$
$660$		0			0
$661$	8.00000	−	13.8564i	0.311164	−	0.538952i	−0.667451	−	0.744654i	$-0.732615\pi$
$661$	8.00000	−	13.8564i	0.311164	−	0.538952i	0.978615	+	0.205702i	$0.0659478\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	6.92820			0.268664
$666$		0			0
$667$	−24.0000			−0.929284
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−6.92820	+	12.0000i	−0.267460	+	0.463255i
$672$		0			0
$673$	0.500000	+	0.866025i	0.0192736	+	0.0333828i	0.875501	−	0.483216i	$-0.160531\pi$
$673$	0.500000	+	0.866025i	0.0192736	+	0.0333828i	−0.856228	+	0.516599i	$0.827198\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−15.5885	−	27.0000i	−0.599113	−	1.03769i	−0.992952	−	0.118515i	$-0.962187\pi$
$677$	−15.5885	−	27.0000i	−0.599113	−	1.03769i	0.393839	−	0.919179i	$-0.371147\pi$
$678$		0			0
$679$	−1.00000	+	1.73205i	−0.0383765	+	0.0664700i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.73205			−0.0662751			−0.0331375	−	0.999451i	$-0.510550\pi$
$683$	−1.73205			−0.0662751			−0.0331375	+	0.999451i	$0.510550\pi$
$684$		0			0
$685$	−30.0000			−1.14624
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−3.46410	+	6.00000i	−0.131972	+	0.228582i
$690$		0			0
$691$	11.0000	+	19.0526i	0.418460	+	0.724793i	0.995785	−	0.0917209i	$-0.0292368\pi$
$691$	11.0000	+	19.0526i	0.418460	+	0.724793i	−0.577325	+	0.816514i	$0.695903\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−24.2487	−	42.0000i	−0.919806	−	1.59315i
$696$		0			0
$697$	6.00000	−	10.3923i	0.227266	−	0.393637i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	46.7654			1.76630			0.883152	−	0.469087i	$-0.155417\pi$
$701$	46.7654			1.76630			0.883152	+	0.469087i	$0.155417\pi$
$702$		0			0
$703$	22.0000			0.829746
$704$		0			0
$705$		0			0
$706$		0			0
$707$	6.92820	−	12.0000i	0.260562	−	0.451306i
$708$		0			0
$709$	−17.5000	−	30.3109i	−0.657226	−	1.13835i	−0.981331	−	0.192328i	$-0.938396\pi$
$709$	−17.5000	−	30.3109i	−0.657226	−	1.13835i	0.324104	−	0.946021i	$-0.394937\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−3.46410	−	6.00000i	−0.129732	−	0.224702i
$714$		0			0
$715$	12.0000	−	20.7846i	0.448775	−	0.777300i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−24.2487			−0.904324			−0.452162	−	0.891936i	$-0.649347\pi$
$719$	−24.2487			−0.904324			−0.452162	+	0.891936i	$0.649347\pi$
$720$		0			0
$721$	14.0000			0.521387
$722$		0			0
$723$		0			0
$724$		0			0
$725$	24.2487	−	42.0000i	0.900575	−	1.55984i
$726$		0			0
$727$	−4.00000	−	6.92820i	−0.148352	−	0.256953i	0.782267	−	0.622944i	$-0.214063\pi$
$727$	−4.00000	−	6.92820i	−0.148352	−	0.256953i	−0.930618	+	0.365991i	$0.880730\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1.73205	+	3.00000i	0.0640622	+	0.110959i
$732$		0			0
$733$	11.0000	−	19.0526i	0.406294	−	0.703722i	−0.588177	−	0.808732i	$-0.700154\pi$
$733$	11.0000	−	19.0526i	0.406294	−	0.703722i	0.994471	+	0.105010i	$0.0334875\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	19.0526			0.701810
$738$		0			0
$739$	29.0000			1.06678			0.533391	−	0.845869i	$-0.320917\pi$
$739$	29.0000			1.06678			0.533391	+	0.845869i	$0.320917\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−6.06218	+	10.5000i	−0.222400	+	0.385208i	−0.955536	−	0.294874i	$-0.904722\pi$
$743$	−6.06218	+	10.5000i	−0.222400	+	0.385208i	0.733136	+	0.680082i	$0.238056\pi$
$744$		0			0
$745$	21.0000	+	36.3731i	0.769380	+	1.33261i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	9.52628	+	16.5000i	0.348083	+	0.602897i
$750$		0			0
$751$	−11.5000	+	19.9186i	−0.419641	+	0.726839i	−0.995903	−	0.0904254i	$-0.971177\pi$
$751$	−11.5000	+	19.9186i	−0.419641	+	0.726839i	0.576262	+	0.817265i	$0.304511\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	38.1051			1.38679
$756$		0			0
$757$	2.00000			0.0726912			0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000			0.0726912			0.0363456	+	0.999339i	$0.488428\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	3.46410	−	6.00000i	0.125574	−	0.217500i	−0.796383	−	0.604792i	$-0.793256\pi$
$761$	3.46410	−	6.00000i	0.125574	−	0.217500i	0.921957	+	0.387292i	$0.126590\pi$
$762$		0			0
$763$	−1.00000	−	1.73205i	−0.0362024	−	0.0627044i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	20.7846	+	36.0000i	0.750489	+	1.29988i
$768$		0			0
$769$	11.0000	−	19.0526i	0.396670	−	0.687053i	−0.596643	−	0.802507i	$-0.703499\pi$
$769$	11.0000	−	19.0526i	0.396670	−	0.687053i	0.993313	+	0.115454i	$0.0368323\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−38.1051			−1.37055			−0.685273	−	0.728286i	$-0.740317\pi$
$773$	−38.1051			−1.37055			−0.685273	+	0.728286i	$0.740317\pi$
$774$		0			0
$775$	14.0000			0.502895
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3.46410	−	6.00000i	0.124114	−	0.214972i
$780$		0			0
$781$	−4.50000	−	7.79423i	−0.161023	−	0.278899i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	6.92820	+	12.0000i	0.247278	+	0.428298i
$786$		0			0
$787$	11.0000	−	19.0526i	0.392108	−	0.679150i	−0.600620	−	0.799535i	$-0.705079\pi$
$787$	11.0000	−	19.0526i	0.392108	−	0.679150i	0.992727	+	0.120384i	$0.0384127\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−8.66025			−0.307923
$792$		0			0
$793$	−32.0000			−1.13635
$794$		0			0
$795$		0			0
$796$		0			0
$797$	27.7128	−	48.0000i	0.981638	−	1.70025i	0.325623	−	0.945500i	$-0.394426\pi$
$797$	27.7128	−	48.0000i	0.981638	−	1.70025i	0.656015	−	0.754747i	$-0.272241\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	13.8564	+	24.0000i	0.488982	+	0.846942i
$804$		0			0
$805$	−6.00000	+	10.3923i	−0.211472	+	0.366281i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−32.9090			−1.15702			−0.578509	−	0.815676i	$-0.696365\pi$
$809$	−32.9090			−1.15702			−0.578509	+	0.815676i	$0.696365\pi$
$810$		0			0
$811$	−16.0000			−0.561836			−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000			−0.561836			−0.280918	+	0.959732i	$0.590639\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−39.8372	+	69.0000i	−1.39544	+	2.41696i
$816$		0			0
$817$	1.00000	+	1.73205i	0.0349856	+	0.0605968i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	18.1865	+	31.5000i	0.634714	+	1.09936i	0.986576	+	0.163305i	$0.0522156\pi$
$821$	18.1865	+	31.5000i	0.634714	+	1.09936i	−0.351861	+	0.936052i	$0.614451\pi$
$822$		0			0
$823$	20.0000	−	34.6410i	0.697156	−	1.20751i	−0.272292	−	0.962215i	$-0.587782\pi$
$823$	20.0000	−	34.6410i	0.697156	−	1.20751i	0.969448	−	0.245295i	$-0.0788849\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	15.5885			0.542064			0.271032	−	0.962570i	$-0.412635\pi$
$827$	15.5885			0.542064			0.271032	+	0.962570i	$0.412635\pi$
$828$		0			0
$829$	20.0000			0.694629			0.347314	−	0.937749i	$-0.387094\pi$
$829$	20.0000			0.694629			0.347314	+	0.937749i	$0.387094\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.73205	+	3.00000i	−0.0600120	+	0.103944i
$834$		0			0
$835$	6.00000	+	10.3923i	0.207639	+	0.359641i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−12.1244	−	21.0000i	−0.418579	−	0.725001i	0.577218	−	0.816590i	$-0.304138\pi$
$839$	−12.1244	−	21.0000i	−0.418579	−	0.725001i	−0.995797	+	0.0915899i	$0.970805\pi$
$840$		0			0
$841$	−9.50000	+	16.4545i	−0.327586	+	0.567396i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	10.3923			0.357506
$846$		0			0
$847$	−8.00000			−0.274883
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−19.0526	+	33.0000i	−0.653113	+	1.13123i
$852$		0			0
$853$	−22.0000	−	38.1051i	−0.753266	−	1.30469i	−0.946232	−	0.323489i	$-0.895144\pi$
$853$	−22.0000	−	38.1051i	−0.753266	−	1.30469i	0.192966	−	0.981205i	$-0.438189\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	22.5167	+	39.0000i	0.769154	+	1.33221i	0.938022	+	0.346576i	$0.112656\pi$
$857$	22.5167	+	39.0000i	0.769154	+	1.33221i	−0.168867	+	0.985639i	$0.554011\pi$
$858$		0			0
$859$	−22.0000	+	38.1051i	−0.750630	+	1.30013i	0.196887	+	0.980426i	$0.436917\pi$
$859$	−22.0000	+	38.1051i	−0.750630	+	1.30013i	−0.947518	+	0.319704i	$0.896417\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1.73205			0.0589597			0.0294798	−	0.999565i	$-0.490615\pi$
$863$	1.73205			0.0589597			0.0294798	+	0.999565i	$0.490615\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	11.2583	−	19.5000i	0.381913	−	0.661492i
$870$		0			0
$871$	22.0000	+	38.1051i	0.745442	+	1.29114i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−3.46410	−	6.00000i	−0.117108	−	0.202837i
$876$		0			0
$877$	−8.50000	+	14.7224i	−0.287025	+	0.497141i	−0.973098	−	0.230391i	$-0.925999\pi$
$877$	−8.50000	+	14.7224i	−0.287025	+	0.497141i	0.686074	+	0.727532i	$0.259333\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−51.9615			−1.75063			−0.875314	−	0.483555i	$-0.839345\pi$
$881$	−51.9615			−1.75063			−0.875314	+	0.483555i	$0.839345\pi$
$882$		0			0
$883$	35.0000			1.17784			0.588922	−	0.808190i	$-0.299553\pi$
$883$	35.0000			1.17784			0.588922	+	0.808190i	$0.299553\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	13.8564	−	24.0000i	0.465253	−	0.805841i	−0.533960	−	0.845510i	$-0.679297\pi$
$887$	13.8564	−	24.0000i	0.465253	−	0.805841i	0.999213	+	0.0396684i	$0.0126302\pi$
$888$		0			0
$889$	−8.50000	−	14.7224i	−0.285081	−	0.493775i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−42.0000	+	72.7461i	−1.40391	+	2.43164i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−13.8564			−0.462137
$900$		0			0
$901$	−6.00000			−0.199889
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−45.0333	+	78.0000i	−1.49696	+	2.59281i
$906$		0			0
$907$	18.5000	+	32.0429i	0.614282	+	1.06397i	0.990510	+	0.137441i	$0.0438878\pi$
$907$	18.5000	+	32.0429i	0.614282	+	1.06397i	−0.376228	+	0.926527i	$0.622779\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	19.0526	+	33.0000i	0.631239	+	1.09334i	0.987299	+	0.158875i	$0.0507868\pi$
$911$	19.0526	+	33.0000i	0.631239	+	1.09334i	−0.356059	+	0.934463i	$0.615880\pi$
$912$		0			0
$913$	15.0000	−	25.9808i	0.496428	−	0.859838i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	13.8564			0.457579
$918$		0			0
$919$	−25.0000			−0.824674			−0.412337	−	0.911031i	$-0.635287\pi$
$919$	−25.0000			−0.824674			−0.412337	+	0.911031i	$0.635287\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	10.3923	−	18.0000i	0.342067	−	0.592477i
$924$		0			0
$925$	−38.5000	−	66.6840i	−1.26587	−	2.19255i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	20.7846	+	36.0000i	0.681921	+	1.18112i	0.974394	+	0.224848i	$0.0721885\pi$
$929$	20.7846	+	36.0000i	0.681921	+	1.18112i	−0.292473	+	0.956274i	$0.594478\pi$
$930$		0			0
$931$	−1.00000	+	1.73205i	−0.0327737	+	0.0567657i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	20.7846			0.679729
$936$		0			0
$937$	8.00000			0.261349			0.130674	−	0.991425i	$-0.458286\pi$
$937$	8.00000			0.261349			0.130674	+	0.991425i	$0.458286\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−10.3923	+	18.0000i	−0.338779	+	0.586783i	−0.984203	−	0.177041i	$-0.943347\pi$
$941$	−10.3923	+	18.0000i	−0.338779	+	0.586783i	0.645424	+	0.763825i	$0.276681\pi$
$942$		0			0
$943$	6.00000	+	10.3923i	0.195387	+	0.338420i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	5.19615	+	9.00000i	0.168852	+	0.292461i	0.938017	−	0.346590i	$-0.112661\pi$
$947$	5.19615	+	9.00000i	0.168852	+	0.292461i	−0.769164	+	0.639051i	$0.779327\pi$
$948$		0			0
$949$	−32.0000	+	55.4256i	−1.03876	+	1.79919i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−20.7846			−0.673280			−0.336640	−	0.941634i	$-0.609290\pi$
$953$	−20.7846			−0.673280			−0.336640	+	0.941634i	$0.609290\pi$
$954$		0			0
$955$	−54.0000			−1.74740
$956$		0			0
$957$		0			0
$958$		0			0
$959$	4.33013	−	7.50000i	0.139827	−	0.242188i
$960$		0			0
$961$	13.5000	+	23.3827i	0.435484	+	0.754280i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−3.46410	−	6.00000i	−0.111513	−	0.193147i
$966$		0			0
$967$	−16.0000	+	27.7128i	−0.514525	+	0.891184i	0.485333	+	0.874330i	$0.338699\pi$
$967$	−16.0000	+	27.7128i	−0.514525	+	0.891184i	−0.999858	+	0.0168544i	$0.994635\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−3.46410			−0.111168			−0.0555842	−	0.998454i	$-0.517702\pi$
$971$	−3.46410			−0.111168			−0.0555842	+	0.998454i	$0.517702\pi$
$972$		0			0
$973$	14.0000			0.448819
$974$		0			0
$975$		0			0
$976$		0			0
$977$	10.3923	−	18.0000i	0.332479	−	0.575871i	−0.650518	−	0.759491i	$-0.725448\pi$
$977$	10.3923	−	18.0000i	0.332479	−	0.575871i	0.982997	+	0.183620i	$0.0587815\pi$
$978$		0			0
$979$	−6.00000	−	10.3923i	−0.191761	−	0.332140i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−13.8564	−	24.0000i	−0.441951	−	0.765481i	0.555883	−	0.831260i	$-0.312380\pi$
$983$	−13.8564	−	24.0000i	−0.441951	−	0.765481i	−0.997834	+	0.0657791i	$0.979047\pi$
$984$		0			0
$985$	−9.00000	+	15.5885i	−0.286764	+	0.496690i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−3.46410			−0.110152
$990$		0			0
$991$	−25.0000			−0.794151			−0.397076	−	0.917786i	$-0.629975\pi$
$991$	−25.0000			−0.794151			−0.397076	+	0.917786i	$0.629975\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	6.92820	−	12.0000i	0.219639	−	0.380426i
$996$		0			0
$997$	−13.0000	−	22.5167i	−0.411714	−	0.713110i	0.583363	−	0.812211i	$-0.301736\pi$
$997$	−13.0000	−	22.5167i	−0.411714	−	0.713110i	−0.995077	+	0.0991016i	$0.968403\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2268.2.j.p.757.1		4
3.2	odd	2	inner	2268.2.j.p.757.2		4
9.2	odd	6	inner	2268.2.j.p.1513.2		4
9.4	even	3		2268.2.a.e.1.2	yes	2
9.5	odd	6		2268.2.a.e.1.1	✓	2
9.7	even	3	inner	2268.2.j.p.1513.1		4
36.23	even	6		9072.2.a.bg.1.1		2
36.31	odd	6		9072.2.a.bg.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2268.2.a.e.1.1	✓	2	9.5	odd	6
2268.2.a.e.1.2	yes	2	9.4	even	3
2268.2.j.p.757.1		4	1.1	even	1	trivial
2268.2.j.p.757.2		4	3.2	odd	2	inner
2268.2.j.p.1513.1		4	9.7	even	3	inner
2268.2.j.p.1513.2		4	9.2	odd	6	inner
9072.2.a.bg.1.1		2	36.23	even	6
9072.2.a.bg.1.2		2	36.31	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2268.j (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.73205 + 3.00000i) q^{5} +(-0.500000 - 0.866025i) q^{7} +(-0.866025 - 1.50000i) q^{11} +(2.00000 - 3.46410i) q^{13} +3.46410 q^{17} +2.00000 q^{19} +(-1.73205 + 3.00000i) q^{23} +(-3.50000 - 6.06218i) q^{25} +(3.46410 + 6.00000i) q^{29} +(-1.00000 + 1.73205i) q^{31} +3.46410 q^{35} +11.0000 q^{37} +(1.73205 - 3.00000i) q^{41} +(0.500000 + 0.866025i) q^{43} +(-0.500000 + 0.866025i) q^{49} -1.73205 q^{53} +6.00000 q^{55} +(-5.19615 + 9.00000i) q^{59} +(-4.00000 - 6.92820i) q^{61} +(6.92820 + 12.0000i) q^{65} +(-5.50000 + 9.52628i) q^{67} +5.19615 q^{71} -16.0000 q^{73} +(-0.866025 + 1.50000i) q^{77} +(6.50000 + 11.2583i) q^{79} +(8.66025 + 15.0000i) q^{83} +(-6.00000 + 10.3923i) q^{85} +6.92820 q^{89} -4.00000 q^{91} +(-3.46410 + 6.00000i) q^{95} +(-1.00000 - 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{7} + 8 q^{13} + 8 q^{19} - 14 q^{25} - 4 q^{31} + 44 q^{37} + 2 q^{43} - 2 q^{49} + 24 q^{55} - 16 q^{61} - 22 q^{67} - 64 q^{73} + 26 q^{79} - 24 q^{85} - 16 q^{91} - 4 q^{97}+O(q^{100}) \) 4 * q - 2 * q^7 + 8 * q^13 + 8 * q^19 - 14 * q^25 - 4 * q^31 + 44 * q^37 + 2 * q^43 - 2 * q^49 + 24 * q^55 - 16 * q^61 - 22 * q^67 - 64 * q^73 + 26 * q^79 - 24 * q^85 - 16 * q^91 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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