LMFDB - Embedded newform 588.2.i.b.361.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [588,2,Mod(361,588)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(588, 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("588.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 588 = 2^{2} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	588.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$4.69520363885$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	588.361
Dual form			588.2.i.b.373.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{3} +(-1.00000 - 1.73205i) q^{5} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})$	\(q+(-0.500000 + 0.866025i) q^{3} +(-1.00000 - 1.73205i) q^{5} +(-0.500000 - 0.866025i) q^{9} +(-1.00000 + 1.73205i) q^{11} +3.00000 q^{13} +2.00000 q^{15} +(4.00000 - 6.92820i) q^{17} +(-0.500000 - 0.866025i) q^{19} +(-4.00000 - 6.92820i) q^{23} +(0.500000 - 0.866025i) q^{25} +1.00000 q^{27} +4.00000 q^{29} +(1.50000 - 2.59808i) q^{31} +(-1.00000 - 1.73205i) q^{33} +(0.500000 + 0.866025i) q^{37} +(-1.50000 + 2.59808i) q^{39} -6.00000 q^{41} +11.0000 q^{43} +(-1.00000 + 1.73205i) q^{45} +(3.00000 + 5.19615i) q^{47} +(4.00000 + 6.92820i) q^{51} +(6.00000 - 10.3923i) q^{53} +4.00000 q^{55} +1.00000 q^{57} +(2.00000 - 3.46410i) q^{59} +(-3.00000 - 5.19615i) q^{61} +(-3.00000 - 5.19615i) q^{65} +(-6.50000 + 11.2583i) q^{67} +8.00000 q^{69} -10.0000 q^{71} +(-5.50000 + 9.52628i) q^{73} +(0.500000 + 0.866025i) q^{75} +(1.50000 + 2.59808i) q^{79} +(-0.500000 + 0.866025i) q^{81} -2.00000 q^{83} -16.0000 q^{85} +(-2.00000 + 3.46410i) q^{87} +(1.50000 + 2.59808i) q^{93} +(-1.00000 + 1.73205i) q^{95} -10.0000 q^{97} +2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 2 q^{5} - q^{9}+O(q^{10}) $ 2 * q - q^3 - 2 * q^5 - q^9	$ 2 q - q^{3} - 2 q^{5} - q^{9} - 2 q^{11} + 6 q^{13} + 4 q^{15} + 8 q^{17} - q^{19} - 8 q^{23} + q^{25} + 2 q^{27} + 8 q^{29} + 3 q^{31} - 2 q^{33} + q^{37} - 3 q^{39} - 12 q^{41} + 22 q^{43} - 2 q^{45} + 6 q^{47} + 8 q^{51} + 12 q^{53} + 8 q^{55} + 2 q^{57} + 4 q^{59} - 6 q^{61} - 6 q^{65} - 13 q^{67} + 16 q^{69} - 20 q^{71} - 11 q^{73} + q^{75} + 3 q^{79} - q^{81} - 4 q^{83} - 32 q^{85} - 4 q^{87} + 3 q^{93} - 2 q^{95} - 20 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q - q^3 - 2 * q^5 - q^9 - 2 * q^11 + 6 * q^13 + 4 * q^15 + 8 * q^17 - q^19 - 8 * q^23 + q^25 + 2 * q^27 + 8 * q^29 + 3 * q^31 - 2 * q^33 + q^37 - 3 * q^39 - 12 * q^41 + 22 * q^43 - 2 * q^45 + 6 * q^47 + 8 * q^51 + 12 * q^53 + 8 * q^55 + 2 * q^57 + 4 * q^59 - 6 * q^61 - 6 * q^65 - 13 * q^67 + 16 * q^69 - 20 * q^71 - 11 * q^73 + q^75 + 3 * q^79 - q^81 - 4 * q^83 - 32 * q^85 - 4 * q^87 + 3 * q^93 - 2 * q^95 - 20 * q^97 + 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/588\mathbb{Z}\right)^\times$.

$n$	$197$	$295$	$493$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$3$	−0.500000	+	0.866025i	−0.288675	+	0.500000i
$3$	−0.500000	+	0.866025i	−0.288675	+	0.500000i
$4$		0			0
$5$	−1.00000	−	1.73205i	−0.447214	−	0.774597i	0.550990	−	0.834512i	$-0.314250\pi$
$5$	−1.00000	−	1.73205i	−0.447214	−	0.774597i	−0.998203	+	0.0599153i	$0.980917\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−0.500000	−	0.866025i	−0.166667	−	0.288675i
$10$		0			0
$11$	−1.00000	+	1.73205i	−0.301511	+	0.522233i	−0.976478	−	0.215615i	$-0.930824\pi$
$11$	−1.00000	+	1.73205i	−0.301511	+	0.522233i	0.674967	+	0.737848i	$0.264158\pi$
$12$		0			0
$13$	3.00000			0.832050			0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000			0.832050			0.416025	+	0.909353i	$0.363423\pi$
$14$		0			0
$15$	2.00000			0.516398
$16$		0			0
$17$	4.00000	−	6.92820i	0.970143	−	1.68034i	0.275029	−	0.961436i	$-0.411312\pi$
$17$	4.00000	−	6.92820i	0.970143	−	1.68034i	0.695113	−	0.718900i	$-0.255354\pi$
$18$		0			0
$19$	−0.500000	−	0.866025i	−0.114708	−	0.198680i	0.802955	−	0.596040i	$-0.203260\pi$
$19$	−0.500000	−	0.866025i	−0.114708	−	0.198680i	−0.917663	+	0.397360i	$0.869927\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.00000	−	6.92820i	−0.834058	−	1.44463i	−0.894795	−	0.446476i	$-0.852679\pi$
$23$	−4.00000	−	6.92820i	−0.834058	−	1.44463i	0.0607377	−	0.998154i	$-0.480655\pi$
$24$		0			0
$25$	0.500000	−	0.866025i	0.100000	−	0.173205i
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	4.00000			0.742781			0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000			0.742781			0.371391	+	0.928477i	$0.378881\pi$
$30$		0			0
$31$	1.50000	−	2.59808i	0.269408	−	0.466628i	−0.699301	−	0.714827i	$-0.746505\pi$
$31$	1.50000	−	2.59808i	0.269408	−	0.466628i	0.968709	+	0.248199i	$0.0798387\pi$
$32$		0			0
$33$	−1.00000	−	1.73205i	−0.174078	−	0.301511i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	0.500000	+	0.866025i	0.0821995	+	0.142374i	0.904194	−	0.427121i	$-0.140472\pi$
$37$	0.500000	+	0.866025i	0.0821995	+	0.142374i	−0.821995	+	0.569495i	$0.807139\pi$
$38$		0			0
$39$	−1.50000	+	2.59808i	−0.240192	+	0.416025i
$40$		0			0
$41$	−6.00000			−0.937043			−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000			−0.937043			−0.468521	+	0.883452i	$0.655213\pi$
$42$		0			0
$43$	11.0000			1.67748			0.838742	−	0.544529i	$-0.183292\pi$
$43$	11.0000			1.67748			0.838742	+	0.544529i	$0.183292\pi$
$44$		0			0
$45$	−1.00000	+	1.73205i	−0.149071	+	0.258199i
$46$		0			0
$47$	3.00000	+	5.19615i	0.437595	+	0.757937i	0.997503	−	0.0706177i	$-0.0224970\pi$
$47$	3.00000	+	5.19615i	0.437595	+	0.757937i	−0.559908	+	0.828554i	$0.689164\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	4.00000	+	6.92820i	0.560112	+	0.970143i
$52$		0			0
$53$	6.00000	−	10.3923i	0.824163	−	1.42749i	−0.0783936	−	0.996922i	$-0.524979\pi$
$53$	6.00000	−	10.3923i	0.824163	−	1.42749i	0.902557	−	0.430570i	$-0.141688\pi$
$54$		0			0
$55$	4.00000			0.539360
$56$		0			0
$57$	1.00000			0.132453
$58$		0			0
$59$	2.00000	−	3.46410i	0.260378	−	0.450988i	−0.705965	−	0.708247i	$-0.749486\pi$
$59$	2.00000	−	3.46410i	0.260378	−	0.450988i	0.966342	+	0.257260i	$0.0828195\pi$
$60$		0			0
$61$	−3.00000	−	5.19615i	−0.384111	−	0.665299i	0.607535	−	0.794293i	$-0.292159\pi$
$61$	−3.00000	−	5.19615i	−0.384111	−	0.665299i	−0.991645	+	0.128994i	$0.958825\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.00000	−	5.19615i	−0.372104	−	0.644503i
$66$		0			0
$67$	−6.50000	+	11.2583i	−0.794101	+	1.37542i	0.129307	+	0.991605i	$0.458725\pi$
$67$	−6.50000	+	11.2583i	−0.794101	+	1.37542i	−0.923408	+	0.383819i	$0.874609\pi$
$68$		0			0
$69$	8.00000			0.963087
$70$		0			0
$71$	−10.0000			−1.18678			−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000			−1.18678			−0.593391	+	0.804914i	$0.702211\pi$
$72$		0			0
$73$	−5.50000	+	9.52628i	−0.643726	+	1.11497i	0.340868	+	0.940111i	$0.389279\pi$
$73$	−5.50000	+	9.52628i	−0.643726	+	1.11497i	−0.984594	+	0.174855i	$0.944054\pi$
$74$		0			0
$75$	0.500000	+	0.866025i	0.0577350	+	0.100000i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	1.50000	+	2.59808i	0.168763	+	0.292306i	0.937985	−	0.346675i	$-0.112689\pi$
$79$	1.50000	+	2.59808i	0.168763	+	0.292306i	−0.769222	+	0.638982i	$0.779356\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	−2.00000			−0.219529			−0.109764	−	0.993958i	$-0.535010\pi$
$83$	−2.00000			−0.219529			−0.109764	+	0.993958i	$0.535010\pi$
$84$		0			0
$85$	−16.0000			−1.73544
$86$		0			0
$87$	−2.00000	+	3.46410i	−0.214423	+	0.371391i
$88$		0			0
$89$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$89$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	1.50000	+	2.59808i	0.155543	+	0.269408i
$94$		0			0
$95$	−1.00000	+	1.73205i	−0.102598	+	0.177705i
$96$		0			0
$97$	−10.0000			−1.01535			−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000			−1.01535			−0.507673	+	0.861550i	$0.669494\pi$
$98$		0			0
$99$	2.00000			0.201008
$100$		0			0
$101$	5.00000	−	8.66025i	0.497519	−	0.861727i	−0.502477	−	0.864590i	$-0.667578\pi$
$101$	5.00000	−	8.66025i	0.497519	−	0.861727i	0.999996	+	0.00286291i	$0.000911295\pi$
$102$		0			0
$103$	5.50000	+	9.52628i	0.541931	+	0.938652i	0.998793	+	0.0491146i	$0.0156400\pi$
$103$	5.50000	+	9.52628i	0.541931	+	0.938652i	−0.456862	+	0.889538i	$0.651027\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$107$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$108$		0			0
$109$	5.50000	−	9.52628i	0.526804	−	0.912452i	−0.472708	−	0.881219i	$-0.656723\pi$
$109$	5.50000	−	9.52628i	0.526804	−	0.912452i	0.999512	−	0.0312328i	$-0.00994332\pi$
$110$		0			0
$111$	−1.00000			−0.0949158
$112$		0			0
$113$	−14.0000			−1.31701			−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000			−1.31701			−0.658505	+	0.752577i	$0.728811\pi$
$114$		0			0
$115$	−8.00000	+	13.8564i	−0.746004	+	1.29212i
$116$		0			0
$117$	−1.50000	−	2.59808i	−0.138675	−	0.240192i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	3.50000	+	6.06218i	0.318182	+	0.551107i
$122$		0			0
$123$	3.00000	−	5.19615i	0.270501	−	0.468521i
$124$		0			0
$125$	−12.0000			−1.07331
$126$		0			0
$127$	3.00000			0.266207			0.133103	−	0.991102i	$-0.457506\pi$
$127$	3.00000			0.266207			0.133103	+	0.991102i	$0.457506\pi$
$128$		0			0
$129$	−5.50000	+	9.52628i	−0.484248	+	0.838742i
$130$		0			0
$131$	−1.00000	−	1.73205i	−0.0873704	−	0.151330i	0.819028	−	0.573753i	$-0.194513\pi$
$131$	−1.00000	−	1.73205i	−0.0873704	−	0.151330i	−0.906399	+	0.422423i	$0.861180\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−1.00000	−	1.73205i	−0.0860663	−	0.149071i
$136$		0			0
$137$	−2.00000	+	3.46410i	−0.170872	+	0.295958i	−0.938725	−	0.344668i	$-0.887992\pi$
$137$	−2.00000	+	3.46410i	−0.170872	+	0.295958i	0.767853	+	0.640626i	$0.221325\pi$
$138$		0			0
$139$	5.00000			0.424094			0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000			0.424094			0.212047	+	0.977259i	$0.431987\pi$
$140$		0			0
$141$	−6.00000			−0.505291
$142$		0			0
$143$	−3.00000	+	5.19615i	−0.250873	+	0.434524i
$144$		0			0
$145$	−4.00000	−	6.92820i	−0.332182	−	0.575356i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	6.00000	+	10.3923i	0.491539	+	0.851371i	0.999953	−	0.00974235i	$-0.00310113\pi$
$149$	6.00000	+	10.3923i	0.491539	+	0.851371i	−0.508413	+	0.861113i	$0.669768\pi$
$150$		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
$153$	−8.00000			−0.646762
$154$		0			0
$155$	−6.00000			−0.481932
$156$		0			0
$157$	1.00000	−	1.73205i	0.0798087	−	0.138233i	−0.823359	−	0.567521i	$-0.807902\pi$
$157$	1.00000	−	1.73205i	0.0798087	−	0.138233i	0.903167	+	0.429289i	$0.141236\pi$
$158$		0			0
$159$	6.00000	+	10.3923i	0.475831	+	0.824163i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	2.00000	+	3.46410i	0.156652	+	0.271329i	0.933659	−	0.358162i	$-0.116597\pi$
$163$	2.00000	+	3.46410i	0.156652	+	0.271329i	−0.777007	+	0.629492i	$0.783263\pi$
$164$		0			0
$165$	−2.00000	+	3.46410i	−0.155700	+	0.269680i
$166$		0			0
$167$	2.00000			0.154765			0.0773823	−	0.997001i	$-0.475344\pi$
$167$	2.00000			0.154765			0.0773823	+	0.997001i	$0.475344\pi$
$168$		0			0
$169$	−4.00000			−0.307692
$170$		0			0
$171$	−0.500000	+	0.866025i	−0.0382360	+	0.0662266i
$172$		0			0
$173$	8.00000	+	13.8564i	0.608229	+	1.05348i	0.991532	+	0.129861i	$0.0414530\pi$
$173$	8.00000	+	13.8564i	0.608229	+	1.05348i	−0.383304	+	0.923622i	$0.625214\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	2.00000	+	3.46410i	0.150329	+	0.260378i
$178$		0			0
$179$	−3.00000	+	5.19615i	−0.224231	+	0.388379i	−0.956088	−	0.293079i	$-0.905320\pi$
$179$	−3.00000	+	5.19615i	−0.224231	+	0.388379i	0.731858	+	0.681457i	$0.238654\pi$
$180$		0			0
$181$	15.0000			1.11494			0.557471	−	0.830197i	$-0.311772\pi$
$181$	15.0000			1.11494			0.557471	+	0.830197i	$0.311772\pi$
$182$		0			0
$183$	6.00000			0.443533
$184$		0			0
$185$	1.00000	−	1.73205i	0.0735215	−	0.127343i
$186$		0			0
$187$	8.00000	+	13.8564i	0.585018	+	1.01328i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−3.00000	−	5.19615i	−0.217072	−	0.375980i	0.736839	−	0.676068i	$-0.236317\pi$
$191$	−3.00000	−	5.19615i	−0.217072	−	0.375980i	−0.953912	+	0.300088i	$0.902984\pi$
$192$		0			0
$193$	−5.50000	+	9.52628i	−0.395899	+	0.685717i	−0.993215	−	0.116289i	$-0.962900\pi$
$193$	−5.50000	+	9.52628i	−0.395899	+	0.685717i	0.597317	+	0.802005i	$0.296234\pi$
$194$		0			0
$195$	6.00000			0.429669
$196$		0			0
$197$	8.00000			0.569976			0.284988	−	0.958531i	$-0.408010\pi$
$197$	8.00000			0.569976			0.284988	+	0.958531i	$0.408010\pi$
$198$		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
$201$	−6.50000	−	11.2583i	−0.458475	−	0.794101i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	6.00000	+	10.3923i	0.419058	+	0.725830i
$206$		0			0
$207$	−4.00000	+	6.92820i	−0.278019	+	0.481543i
$208$		0			0
$209$	2.00000			0.138343
$210$		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
$213$	5.00000	−	8.66025i	0.342594	−	0.593391i
$214$		0			0
$215$	−11.0000	−	19.0526i	−0.750194	−	1.29937i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−5.50000	−	9.52628i	−0.371656	−	0.643726i
$220$		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
$225$	−1.00000			−0.0666667
$226$		0			0
$227$	−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
$229$	0.500000	+	0.866025i	0.0330409	+	0.0572286i	0.882073	−	0.471113i	$-0.156147\pi$
$229$	0.500000	+	0.866025i	0.0330409	+	0.0572286i	−0.849032	+	0.528341i	$0.822814\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−7.00000	−	12.1244i	−0.458585	−	0.794293i	0.540301	−	0.841472i	$-0.318310\pi$
$233$	−7.00000	−	12.1244i	−0.458585	−	0.794293i	−0.998886	+	0.0471787i	$0.984977\pi$
$234$		0			0
$235$	6.00000	−	10.3923i	0.391397	−	0.677919i
$236$		0			0
$237$	−3.00000			−0.194871
$238$		0			0
$239$	18.0000			1.16432			0.582162	−	0.813073i	$-0.302207\pi$
$239$	18.0000			1.16432			0.582162	+	0.813073i	$0.302207\pi$
$240$		0			0
$241$	7.00000	−	12.1244i	0.450910	−	0.780998i	−0.547533	−	0.836784i	$-0.684433\pi$
$241$	7.00000	−	12.1244i	0.450910	−	0.780998i	0.998443	+	0.0557856i	$0.0177663\pi$
$242$		0			0
$243$	−0.500000	−	0.866025i	−0.0320750	−	0.0555556i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−1.50000	−	2.59808i	−0.0954427	−	0.165312i
$248$		0			0
$249$	1.00000	−	1.73205i	0.0633724	−	0.109764i
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$	16.0000			1.00591
$254$		0			0
$255$	8.00000	−	13.8564i	0.500979	−	0.867722i
$256$		0			0
$257$	9.00000	+	15.5885i	0.561405	+	0.972381i	0.997374	+	0.0724199i	$0.0230722\pi$
$257$	9.00000	+	15.5885i	0.561405	+	0.972381i	−0.435970	+	0.899961i	$0.643595\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−2.00000	−	3.46410i	−0.123797	−	0.214423i
$262$		0			0
$263$	−6.00000	+	10.3923i	−0.369976	+	0.640817i	−0.989561	−	0.144112i	$-0.953967\pi$
$263$	−6.00000	+	10.3923i	−0.369976	+	0.640817i	0.619586	+	0.784929i	$0.287301\pi$
$264$		0			0
$265$	−24.0000			−1.47431
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.00000	+	1.73205i	−0.0609711	+	0.105605i	−0.894900	−	0.446267i	$-0.852753\pi$
$269$	−1.00000	+	1.73205i	−0.0609711	+	0.105605i	0.833929	+	0.551872i	$0.186086\pi$
$270$		0			0
$271$	−12.0000	−	20.7846i	−0.728948	−	1.26258i	−0.957328	−	0.289003i	$-0.906676\pi$
$271$	−12.0000	−	20.7846i	−0.728948	−	1.26258i	0.228380	−	0.973572i	$-0.426657\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.00000	+	1.73205i	0.0603023	+	0.104447i
$276$		0			0
$277$	−8.50000	+	14.7224i	−0.510716	+	0.884585i	0.489207	+	0.872167i	$0.337286\pi$
$277$	−8.50000	+	14.7224i	−0.510716	+	0.884585i	−0.999923	+	0.0124177i	$0.996047\pi$
$278$		0			0
$279$	−3.00000			−0.179605
$280$		0			0
$281$	20.0000			1.19310			0.596550	−	0.802576i	$-0.296538\pi$
$281$	20.0000			1.19310			0.596550	+	0.802576i	$0.296538\pi$
$282$		0			0
$283$	9.50000	−	16.4545i	0.564716	−	0.978117i	−0.432360	−	0.901701i	$-0.642319\pi$
$283$	9.50000	−	16.4545i	0.564716	−	0.978117i	0.997076	−	0.0764162i	$-0.0243478\pi$
$284$		0			0
$285$	−1.00000	−	1.73205i	−0.0592349	−	0.102598i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−23.5000	−	40.7032i	−1.38235	−	2.39431i
$290$		0			0
$291$	5.00000	−	8.66025i	0.293105	−	0.507673i
$292$		0			0
$293$	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
$295$	−8.00000			−0.465778
$296$		0			0
$297$	−1.00000	+	1.73205i	−0.0580259	+	0.100504i
$298$		0			0
$299$	−12.0000	−	20.7846i	−0.693978	−	1.20201i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	5.00000	+	8.66025i	0.287242	+	0.497519i
$304$		0			0
$305$	−6.00000	+	10.3923i	−0.343559	+	0.595062i
$306$		0			0
$307$	23.0000			1.31268			0.656340	−	0.754466i	$-0.272104\pi$
$307$	23.0000			1.31268			0.656340	+	0.754466i	$0.272104\pi$
$308$		0			0
$309$	−11.0000			−0.625768
$310$		0			0
$311$	−1.00000	+	1.73205i	−0.0567048	+	0.0982156i	−0.892984	−	0.450088i	$-0.851393\pi$
$311$	−1.00000	+	1.73205i	−0.0567048	+	0.0982156i	0.836280	+	0.548303i	$0.184726\pi$
$312$		0			0
$313$	−8.50000	−	14.7224i	−0.480448	−	0.832161i	0.519300	−	0.854592i	$-0.326193\pi$
$313$	−8.50000	−	14.7224i	−0.480448	−	0.832161i	−0.999748	+	0.0224310i	$0.992859\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	12.0000	+	20.7846i	0.673987	+	1.16738i	0.976764	+	0.214318i	$0.0687530\pi$
$317$	12.0000	+	20.7846i	0.673987	+	1.16738i	−0.302777	+	0.953062i	$0.597914\pi$
$318$		0			0
$319$	−4.00000	+	6.92820i	−0.223957	+	0.387905i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−8.00000			−0.445132
$324$		0			0
$325$	1.50000	−	2.59808i	0.0832050	−	0.144115i
$326$		0			0
$327$	5.50000	+	9.52628i	0.304151	+	0.526804i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−8.50000	−	14.7224i	−0.467202	−	0.809218i	0.532096	−	0.846684i	$-0.321405\pi$
$331$	−8.50000	−	14.7224i	−0.467202	−	0.809218i	−0.999298	+	0.0374662i	$0.988071\pi$
$332$		0			0
$333$	0.500000	−	0.866025i	0.0273998	−	0.0474579i
$334$		0			0
$335$	26.0000			1.42053
$336$		0			0
$337$	21.0000			1.14394			0.571971	−	0.820274i	$-0.306179\pi$
$337$	21.0000			1.14394			0.571971	+	0.820274i	$0.306179\pi$
$338$		0			0
$339$	7.00000	−	12.1244i	0.380188	−	0.658505i
$340$		0			0
$341$	3.00000	+	5.19615i	0.162459	+	0.281387i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−8.00000	−	13.8564i	−0.430706	−	0.746004i
$346$		0			0
$347$	−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
$349$	14.0000			0.749403			0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000			0.749403			0.374701	+	0.927146i	$0.377745\pi$
$350$		0			0
$351$	3.00000			0.160128
$352$		0			0
$353$	−3.00000	+	5.19615i	−0.159674	+	0.276563i	−0.934751	−	0.355303i	$-0.884378\pi$
$353$	−3.00000	+	5.19615i	−0.159674	+	0.276563i	0.775077	+	0.631867i	$0.217711\pi$
$354$		0			0
$355$	10.0000	+	17.3205i	0.530745	+	0.919277i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	10.0000	+	17.3205i	0.527780	+	0.914141i	0.999476	+	0.0323801i	$0.0103087\pi$
$359$	10.0000	+	17.3205i	0.527780	+	0.914141i	−0.471696	+	0.881761i	$0.656358\pi$
$360$		0			0
$361$	9.00000	−	15.5885i	0.473684	−	0.820445i
$362$		0			0
$363$	−7.00000			−0.367405
$364$		0			0
$365$	22.0000			1.15153
$366$		0			0
$367$	2.50000	−	4.33013i	0.130499	−	0.226031i	−0.793370	−	0.608740i	$-0.791675\pi$
$367$	2.50000	−	4.33013i	0.130499	−	0.226031i	0.923869	+	0.382709i	$0.125009\pi$
$368$		0			0
$369$	3.00000	+	5.19615i	0.156174	+	0.270501i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	2.50000	+	4.33013i	0.129445	+	0.224205i	0.923462	−	0.383691i	$-0.125347\pi$
$373$	2.50000	+	4.33013i	0.129445	+	0.224205i	−0.794017	+	0.607896i	$0.792014\pi$
$374$		0			0
$375$	6.00000	−	10.3923i	0.309839	−	0.536656i
$376$		0			0
$377$	12.0000			0.618031
$378$		0			0
$379$	13.0000			0.667765			0.333883	−	0.942615i	$-0.391641\pi$
$379$	13.0000			0.667765			0.333883	+	0.942615i	$0.391641\pi$
$380$		0			0
$381$	−1.50000	+	2.59808i	−0.0768473	+	0.133103i
$382$		0			0
$383$	−14.0000	−	24.2487i	−0.715367	−	1.23905i	−0.962818	−	0.270151i	$-0.912926\pi$
$383$	−14.0000	−	24.2487i	−0.715367	−	1.23905i	0.247451	−	0.968900i	$-0.420407\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−5.50000	−	9.52628i	−0.279581	−	0.484248i
$388$		0			0
$389$	−5.00000	+	8.66025i	−0.253510	+	0.439092i	−0.964490	−	0.264120i	$-0.914918\pi$
$389$	−5.00000	+	8.66025i	−0.253510	+	0.439092i	0.710980	+	0.703213i	$0.248252\pi$
$390$		0			0
$391$	−64.0000			−3.23662
$392$		0			0
$393$	2.00000			0.100887
$394$		0			0
$395$	3.00000	−	5.19615i	0.150946	−	0.261447i
$396$		0			0
$397$	1.50000	+	2.59808i	0.0752828	+	0.130394i	0.901209	−	0.433384i	$-0.142681\pi$
$397$	1.50000	+	2.59808i	0.0752828	+	0.130394i	−0.825926	+	0.563778i	$0.809347\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	6.00000	+	10.3923i	0.299626	+	0.518967i	0.976050	−	0.217545i	$-0.0698049\pi$
$401$	6.00000	+	10.3923i	0.299626	+	0.518967i	−0.676425	+	0.736512i	$0.736472\pi$
$402$		0			0
$403$	4.50000	−	7.79423i	0.224161	−	0.388258i
$404$		0			0
$405$	2.00000			0.0993808
$406$		0			0
$407$	−2.00000			−0.0991363
$408$		0			0
$409$	−9.50000	+	16.4545i	−0.469745	+	0.813622i	−0.999402	−	0.0345902i	$-0.988987\pi$
$409$	−9.50000	+	16.4545i	−0.469745	+	0.813622i	0.529657	+	0.848212i	$0.322321\pi$
$410$		0			0
$411$	−2.00000	−	3.46410i	−0.0986527	−	0.170872i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	2.00000	+	3.46410i	0.0981761	+	0.170046i
$416$		0			0
$417$	−2.50000	+	4.33013i	−0.122426	+	0.212047i
$418$		0			0
$419$	−18.0000			−0.879358			−0.439679	−	0.898155i	$-0.644908\pi$
$419$	−18.0000			−0.879358			−0.439679	+	0.898155i	$0.644908\pi$
$420$		0			0
$421$	−27.0000			−1.31590			−0.657950	−	0.753062i	$-0.728576\pi$
$421$	−27.0000			−1.31590			−0.657950	+	0.753062i	$0.728576\pi$
$422$		0			0
$423$	3.00000	−	5.19615i	0.145865	−	0.252646i
$424$		0			0
$425$	−4.00000	−	6.92820i	−0.194029	−	0.336067i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−3.00000	−	5.19615i	−0.144841	−	0.250873i
$430$		0			0
$431$	15.0000	−	25.9808i	0.722525	−	1.25145i	−0.237460	−	0.971397i	$-0.576315\pi$
$431$	15.0000	−	25.9808i	0.722525	−	1.25145i	0.959985	−	0.280052i	$-0.0903517\pi$
$432$		0			0
$433$	25.0000			1.20142			0.600712	−	0.799466i	$-0.294884\pi$
$433$	25.0000			1.20142			0.600712	+	0.799466i	$0.294884\pi$
$434$		0			0
$435$	8.00000			0.383571
$436$		0			0
$437$	−4.00000	+	6.92820i	−0.191346	+	0.331421i
$438$		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
$443$	2.00000	+	3.46410i	0.0950229	+	0.164584i	0.909618	−	0.415445i	$-0.136374\pi$
$443$	2.00000	+	3.46410i	0.0950229	+	0.164584i	−0.814595	+	0.580030i	$0.803041\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−12.0000			−0.567581
$448$		0			0
$449$	22.0000			1.03824			0.519122	−	0.854700i	$-0.326259\pi$
$449$	22.0000			1.03824			0.519122	+	0.854700i	$0.326259\pi$
$450$		0			0
$451$	6.00000	−	10.3923i	0.282529	−	0.489355i
$452$		0			0
$453$	4.00000	+	6.92820i	0.187936	+	0.325515i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−6.50000	−	11.2583i	−0.304057	−	0.526642i	0.672994	−	0.739648i	$-0.265008\pi$
$457$	−6.50000	−	11.2583i	−0.304057	−	0.526642i	−0.977051	+	0.213006i	$0.931675\pi$
$458$		0			0
$459$	4.00000	−	6.92820i	0.186704	−	0.323381i
$460$		0			0
$461$	−4.00000			−0.186299			−0.0931493	−	0.995652i	$-0.529693\pi$
$461$	−4.00000			−0.186299			−0.0931493	+	0.995652i	$0.529693\pi$
$462$		0			0
$463$	−11.0000			−0.511213			−0.255607	−	0.966781i	$-0.582275\pi$
$463$	−11.0000			−0.511213			−0.255607	+	0.966781i	$0.582275\pi$
$464$		0			0
$465$	3.00000	−	5.19615i	0.139122	−	0.240966i
$466$		0			0
$467$	17.0000	+	29.4449i	0.786666	+	1.36255i	0.927999	+	0.372584i	$0.121528\pi$
$467$	17.0000	+	29.4449i	0.786666	+	1.36255i	−0.141332	+	0.989962i	$0.545139\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	1.00000	+	1.73205i	0.0460776	+	0.0798087i
$472$		0			0
$473$	−11.0000	+	19.0526i	−0.505781	+	0.876038i
$474$		0			0
$475$	−1.00000			−0.0458831
$476$		0			0
$477$	−12.0000			−0.549442
$478$		0			0
$479$	−14.0000	+	24.2487i	−0.639676	+	1.10795i	0.345827	+	0.938298i	$0.387598\pi$
$479$	−14.0000	+	24.2487i	−0.639676	+	1.10795i	−0.985504	+	0.169654i	$0.945735\pi$
$480$		0			0
$481$	1.50000	+	2.59808i	0.0683941	+	0.118462i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	10.0000	+	17.3205i	0.454077	+	0.786484i
$486$		0			0
$487$	9.50000	−	16.4545i	0.430486	−	0.745624i	−0.566429	−	0.824110i	$-0.691675\pi$
$487$	9.50000	−	16.4545i	0.430486	−	0.745624i	0.996915	+	0.0784867i	$0.0250088\pi$
$488$		0			0
$489$	−4.00000			−0.180886
$490$		0			0
$491$	−36.0000			−1.62466			−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000			−1.62466			−0.812329	+	0.583200i	$0.801800\pi$
$492$		0			0
$493$	16.0000	−	27.7128i	0.720604	−	1.24812i
$494$		0			0
$495$	−2.00000	−	3.46410i	−0.0898933	−	0.155700i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	14.5000	+	25.1147i	0.649109	+	1.12429i	0.983336	+	0.181797i	$0.0581915\pi$
$499$	14.5000	+	25.1147i	0.649109	+	1.12429i	−0.334227	+	0.942493i	$0.608475\pi$
$500$		0			0
$501$	−1.00000	+	1.73205i	−0.0446767	+	0.0773823i
$502$		0			0
$503$	30.0000			1.33763			0.668817	−	0.743427i	$-0.266801\pi$
$503$	30.0000			1.33763			0.668817	+	0.743427i	$0.266801\pi$
$504$		0			0
$505$	−20.0000			−0.889988
$506$		0			0
$507$	2.00000	−	3.46410i	0.0888231	−	0.153846i
$508$		0			0
$509$	9.00000	+	15.5885i	0.398918	+	0.690946i	0.993593	−	0.113020i	$-0.0360525\pi$
$509$	9.00000	+	15.5885i	0.398918	+	0.690946i	−0.594675	+	0.803966i	$0.702719\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−0.500000	−	0.866025i	−0.0220755	−	0.0382360i
$514$		0			0
$515$	11.0000	−	19.0526i	0.484718	−	0.839556i
$516$		0			0
$517$	−12.0000			−0.527759
$518$		0			0
$519$	−16.0000			−0.702322
$520$		0			0
$521$	−18.0000	+	31.1769i	−0.788594	+	1.36589i	0.138234	+	0.990400i	$0.455857\pi$
$521$	−18.0000	+	31.1769i	−0.788594	+	1.36589i	−0.926828	+	0.375486i	$0.877476\pi$
$522$		0			0
$523$	−15.5000	−	26.8468i	−0.677768	−	1.17393i	−0.975652	−	0.219326i	$-0.929614\pi$
$523$	−15.5000	−	26.8468i	−0.677768	−	1.17393i	0.297884	−	0.954602i	$-0.403719\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−12.0000	−	20.7846i	−0.522728	−	0.905392i
$528$		0			0
$529$	−20.5000	+	35.5070i	−0.891304	+	1.54378i
$530$		0			0
$531$	−4.00000			−0.173585
$532$		0			0
$533$	−18.0000			−0.779667
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−3.00000	−	5.19615i	−0.129460	−	0.224231i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	7.50000	+	12.9904i	0.322450	+	0.558500i	0.980993	−	0.194043i	$-0.0621602\pi$
$541$	7.50000	+	12.9904i	0.322450	+	0.558500i	−0.658543	+	0.752543i	$0.728827\pi$
$542$		0			0
$543$	−7.50000	+	12.9904i	−0.321856	+	0.557471i
$544$		0			0
$545$	−22.0000			−0.942376
$546$		0			0
$547$	−12.0000			−0.513083			−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000			−0.513083			−0.256541	+	0.966533i	$0.582583\pi$
$548$		0			0
$549$	−3.00000	+	5.19615i	−0.128037	+	0.221766i
$550$		0			0
$551$	−2.00000	−	3.46410i	−0.0852029	−	0.147576i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	1.00000	+	1.73205i	0.0424476	+	0.0735215i
$556$		0			0
$557$	−11.0000	+	19.0526i	−0.466085	+	0.807283i	−0.999250	−	0.0387286i	$-0.987669\pi$
$557$	−11.0000	+	19.0526i	−0.466085	+	0.807283i	0.533165	+	0.846011i	$0.321003\pi$
$558$		0			0
$559$	33.0000			1.39575
$560$		0			0
$561$	−16.0000			−0.675521
$562$		0			0
$563$	−23.0000	+	39.8372i	−0.969334	+	1.67894i	−0.271846	+	0.962341i	$0.587634\pi$
$563$	−23.0000	+	39.8372i	−0.969334	+	1.67894i	−0.697489	+	0.716596i	$0.745699\pi$
$564$		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	0.796266	−	0.604947i	$-0.206806\pi$
$569$	−3.00000	−	5.19615i	−0.125767	−	0.217834i	−0.922032	+	0.387113i	$0.873472\pi$
$570$		0			0
$571$	10.5000	−	18.1865i	0.439411	−	0.761083i	−0.558233	−	0.829684i	$-0.688520\pi$
$571$	10.5000	−	18.1865i	0.439411	−	0.761083i	0.997644	+	0.0686016i	$0.0218537\pi$
$572$		0			0
$573$	6.00000			0.250654
$574$		0			0
$575$	−8.00000			−0.333623
$576$		0			0
$577$	−20.5000	+	35.5070i	−0.853426	+	1.47818i	0.0246713	+	0.999696i	$0.492146\pi$
$577$	−20.5000	+	35.5070i	−0.853426	+	1.47818i	−0.878097	+	0.478482i	$0.841187\pi$
$578$		0			0
$579$	−5.50000	−	9.52628i	−0.228572	−	0.395899i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	12.0000	+	20.7846i	0.496989	+	0.860811i
$584$		0			0
$585$	−3.00000	+	5.19615i	−0.124035	+	0.214834i
$586$		0			0
$587$	−32.0000			−1.32078			−0.660391	−	0.750922i	$-0.729609\pi$
$587$	−32.0000			−1.32078			−0.660391	+	0.750922i	$0.729609\pi$
$588$		0			0
$589$	−3.00000			−0.123613
$590$		0			0
$591$	−4.00000	+	6.92820i	−0.164538	+	0.284988i
$592$		0			0
$593$	−3.00000	−	5.19615i	−0.123195	−	0.213380i	0.797831	−	0.602881i	$-0.205981\pi$
$593$	−3.00000	−	5.19615i	−0.123195	−	0.213380i	−0.921026	+	0.389501i	$0.872647\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	4.00000	+	6.92820i	0.163709	+	0.283552i
$598$		0			0
$599$	6.00000	−	10.3923i	0.245153	−	0.424618i	−0.717021	−	0.697051i	$-0.754495\pi$
$599$	6.00000	−	10.3923i	0.245153	−	0.424618i	0.962175	+	0.272433i	$0.0878284\pi$
$600$		0			0
$601$	1.00000			0.0407909			0.0203954	−	0.999792i	$-0.493507\pi$
$601$	1.00000			0.0407909			0.0203954	+	0.999792i	$0.493507\pi$
$602$		0			0
$603$	13.0000			0.529401
$604$		0			0
$605$	7.00000	−	12.1244i	0.284590	−	0.492925i
$606$		0			0
$607$	−1.50000	−	2.59808i	−0.0608831	−	0.105453i	0.833977	−	0.551799i	$-0.186058\pi$
$607$	−1.50000	−	2.59808i	−0.0608831	−	0.105453i	−0.894860	+	0.446346i	$0.852725\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	9.00000	+	15.5885i	0.364101	+	0.630641i
$612$		0			0
$613$	15.0000	−	25.9808i	0.605844	−	1.04935i	−0.386073	−	0.922468i	$-0.626169\pi$
$613$	15.0000	−	25.9808i	0.605844	−	1.04935i	0.991917	−	0.126885i	$-0.0404979\pi$
$614$		0			0
$615$	−12.0000			−0.483887
$616$		0			0
$617$	26.0000			1.04672			0.523360	−	0.852111i	$-0.324678\pi$
$617$	26.0000			1.04672			0.523360	+	0.852111i	$0.324678\pi$
$618$		0			0
$619$	−5.50000	+	9.52628i	−0.221064	+	0.382893i	−0.955131	−	0.296183i	$-0.904286\pi$
$619$	−5.50000	+	9.52628i	−0.221064	+	0.382893i	0.734068	+	0.679076i	$0.237620\pi$
$620$		0			0
$621$	−4.00000	−	6.92820i	−0.160514	−	0.278019i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	9.50000	+	16.4545i	0.380000	+	0.658179i
$626$		0			0
$627$	−1.00000	+	1.73205i	−0.0399362	+	0.0691714i
$628$		0			0
$629$	8.00000			0.318981
$630$		0			0
$631$	−16.0000			−0.636950			−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000			−0.636950			−0.318475	+	0.947931i	$0.603171\pi$
$632$		0			0
$633$	2.00000	−	3.46410i	0.0794929	−	0.137686i
$634$		0			0
$635$	−3.00000	−	5.19615i	−0.119051	−	0.206203i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	5.00000	+	8.66025i	0.197797	+	0.342594i
$640$		0			0
$641$	20.0000	−	34.6410i	0.789953	−	1.36824i	−0.136043	−	0.990703i	$-0.543438\pi$
$641$	20.0000	−	34.6410i	0.789953	−	1.36824i	0.925995	−	0.377535i	$-0.123228\pi$
$642$		0			0
$643$	−35.0000			−1.38027			−0.690133	−	0.723683i	$-0.742448\pi$
$643$	−35.0000			−1.38027			−0.690133	+	0.723683i	$0.742448\pi$
$644$		0			0
$645$	22.0000			0.866249
$646$		0			0
$647$	3.00000	−	5.19615i	0.117942	−	0.204282i	−0.801010	−	0.598651i	$-0.795704\pi$
$647$	3.00000	−	5.19615i	0.117942	−	0.204282i	0.918952	+	0.394369i	$0.129037\pi$
$648$		0			0
$649$	4.00000	+	6.92820i	0.157014	+	0.271956i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	3.00000	+	5.19615i	0.117399	+	0.203341i	0.918736	−	0.394872i	$-0.129211\pi$
$653$	3.00000	+	5.19615i	0.117399	+	0.203341i	−0.801337	+	0.598213i	$0.795878\pi$
$654$		0			0
$655$	−2.00000	+	3.46410i	−0.0781465	+	0.135354i
$656$		0			0
$657$	11.0000			0.429151
$658$		0			0
$659$	−28.0000			−1.09073			−0.545363	−	0.838200i	$-0.683608\pi$
$659$	−28.0000			−1.09073			−0.545363	+	0.838200i	$0.683608\pi$
$660$		0			0
$661$	−14.5000	+	25.1147i	−0.563985	+	0.976850i	0.433159	+	0.901318i	$0.357399\pi$
$661$	−14.5000	+	25.1147i	−0.563985	+	0.976850i	−0.997143	+	0.0755324i	$0.975934\pi$
$662$		0			0
$663$	12.0000	+	20.7846i	0.466041	+	0.807207i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−16.0000	−	27.7128i	−0.619522	−	1.07304i
$668$		0			0
$669$	4.00000	−	6.92820i	0.154649	−	0.267860i
$670$		0			0
$671$	12.0000			0.463255
$672$		0			0
$673$	−1.00000			−0.0385472			−0.0192736	−	0.999814i	$-0.506135\pi$
$673$	−1.00000			−0.0385472			−0.0192736	+	0.999814i	$0.506135\pi$
$674$		0			0
$675$	0.500000	−	0.866025i	0.0192450	−	0.0333333i
$676$		0			0
$677$	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
$681$	−9.00000	−	15.5885i	−0.344881	−	0.597351i
$682$		0			0
$683$	−18.0000	+	31.1769i	−0.688751	+	1.19295i	0.283491	+	0.958975i	$0.408507\pi$
$683$	−18.0000	+	31.1769i	−0.688751	+	1.19295i	−0.972242	+	0.233977i	$0.924826\pi$
$684$		0			0
$685$	8.00000			0.305664
$686$		0			0
$687$	−1.00000			−0.0381524
$688$		0			0
$689$	18.0000	−	31.1769i	0.685745	−	1.18775i
$690$		0			0
$691$	−21.5000	−	37.2391i	−0.817899	−	1.41664i	−0.907228	−	0.420640i	$-0.861806\pi$
$691$	−21.5000	−	37.2391i	−0.817899	−	1.41664i	0.0893292	−	0.996002i	$-0.471528\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−5.00000	−	8.66025i	−0.189661	−	0.328502i
$696$		0			0
$697$	−24.0000	+	41.5692i	−0.909065	+	1.57455i
$698$		0			0
$699$	14.0000			0.529529
$700$		0			0
$701$	−8.00000			−0.302156			−0.151078	−	0.988522i	$-0.548274\pi$
$701$	−8.00000			−0.302156			−0.151078	+	0.988522i	$0.548274\pi$
$702$		0			0
$703$	0.500000	−	0.866025i	0.0188579	−	0.0326628i
$704$		0			0
$705$	6.00000	+	10.3923i	0.225973	+	0.391397i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−7.00000	−	12.1244i	−0.262891	−	0.455340i	0.704118	−	0.710083i	$-0.251342\pi$
$709$	−7.00000	−	12.1244i	−0.262891	−	0.455340i	−0.967009	+	0.254743i	$0.918009\pi$
$710$		0			0
$711$	1.50000	−	2.59808i	0.0562544	−	0.0974355i
$712$		0			0
$713$	−24.0000			−0.898807
$714$		0			0
$715$	12.0000			0.448775
$716$		0			0
$717$	−9.00000	+	15.5885i	−0.336111	+	0.582162i
$718$		0			0
$719$	−3.00000	−	5.19615i	−0.111881	−	0.193784i	0.804648	−	0.593753i	$-0.202354\pi$
$719$	−3.00000	−	5.19615i	−0.111881	−	0.193784i	−0.916529	+	0.399969i	$0.869021\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	7.00000	+	12.1244i	0.260333	+	0.450910i
$724$		0			0
$725$	2.00000	−	3.46410i	0.0742781	−	0.128654i
$726$		0			0
$727$	23.0000			0.853023			0.426511	−	0.904482i	$-0.359742\pi$
$727$	23.0000			0.853023			0.426511	+	0.904482i	$0.359742\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	44.0000	−	76.2102i	1.62740	−	2.81874i
$732$		0			0
$733$	22.5000	+	38.9711i	0.831056	+	1.43943i	0.897201	+	0.441622i	$0.145597\pi$
$733$	22.5000	+	38.9711i	0.831056	+	1.43943i	−0.0661448	+	0.997810i	$0.521070\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−13.0000	−	22.5167i	−0.478861	−	0.829412i
$738$		0			0
$739$	4.50000	−	7.79423i	0.165535	−	0.286715i	−0.771310	−	0.636460i	$-0.780398\pi$
$739$	4.50000	−	7.79423i	0.165535	−	0.286715i	0.936845	+	0.349744i	$0.113732\pi$
$740$		0			0
$741$	3.00000			0.110208
$742$		0			0
$743$	−18.0000			−0.660356			−0.330178	−	0.943919i	$-0.607109\pi$
$743$	−18.0000			−0.660356			−0.330178	+	0.943919i	$0.607109\pi$
$744$		0			0
$745$	12.0000	−	20.7846i	0.439646	−	0.761489i
$746$		0			0
$747$	1.00000	+	1.73205i	0.0365881	+	0.0633724i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−7.50000	−	12.9904i	−0.273679	−	0.474026i	0.696122	−	0.717923i	$-0.254907\pi$
$751$	−7.50000	−	12.9904i	−0.273679	−	0.474026i	−0.969801	+	0.243898i	$0.921574\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−16.0000			−0.582300
$756$		0			0
$757$	42.0000			1.52652			0.763258	−	0.646094i	$-0.223599\pi$
$757$	42.0000			1.52652			0.763258	+	0.646094i	$0.223599\pi$
$758$		0			0
$759$	−8.00000	+	13.8564i	−0.290382	+	0.502956i
$760$		0			0
$761$	4.00000	+	6.92820i	0.145000	+	0.251147i	0.929373	−	0.369142i	$-0.120348\pi$
$761$	4.00000	+	6.92820i	0.145000	+	0.251147i	−0.784373	+	0.620289i	$0.787015\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	8.00000	+	13.8564i	0.289241	+	0.500979i
$766$		0			0
$767$	6.00000	−	10.3923i	0.216647	−	0.375244i
$768$		0			0
$769$	−31.0000			−1.11789			−0.558944	−	0.829205i	$-0.688793\pi$
$769$	−31.0000			−1.11789			−0.558944	+	0.829205i	$0.688793\pi$
$770$		0			0
$771$	−18.0000			−0.648254
$772$		0			0
$773$	11.0000	−	19.0526i	0.395643	−	0.685273i	−0.597540	−	0.801839i	$-0.703855\pi$
$773$	11.0000	−	19.0526i	0.395643	−	0.685273i	0.993183	+	0.116566i	$0.0371886\pi$
$774$		0			0
$775$	−1.50000	−	2.59808i	−0.0538816	−	0.0933257i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3.00000	+	5.19615i	0.107486	+	0.186171i
$780$		0			0
$781$	10.0000	−	17.3205i	0.357828	−	0.619777i
$782$		0			0
$783$	4.00000			0.142948
$784$		0			0
$785$	−4.00000			−0.142766
$786$		0			0
$787$	12.0000	−	20.7846i	0.427754	−	0.740891i	−0.568919	−	0.822393i	$-0.692638\pi$
$787$	12.0000	−	20.7846i	0.427754	−	0.740891i	0.996673	+	0.0815020i	$0.0259717\pi$
$788$		0			0
$789$	−6.00000	−	10.3923i	−0.213606	−	0.369976i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−9.00000	−	15.5885i

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:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	588.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{3} +(-1.00000 - 1.73205i) q^{5} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\)	\(q+(-0.500000 + 0.866025i) q^{3} +(-1.00000 - 1.73205i) q^{5} +(-0.500000 - 0.866025i) q^{9} +(-1.00000 + 1.73205i) q^{11} +3.00000 q^{13} +2.00000 q^{15} +(4.00000 - 6.92820i) q^{17} +(-0.500000 - 0.866025i) q^{19} +(-4.00000 - 6.92820i) q^{23} +(0.500000 - 0.866025i) q^{25} +1.00000 q^{27} +4.00000 q^{29} +(1.50000 - 2.59808i) q^{31} +(-1.00000 - 1.73205i) q^{33} +(0.500000 + 0.866025i) q^{37} +(-1.50000 + 2.59808i) q^{39} -6.00000 q^{41} +11.0000 q^{43} +(-1.00000 + 1.73205i) q^{45} +(3.00000 + 5.19615i) q^{47} +(4.00000 + 6.92820i) q^{51} +(6.00000 - 10.3923i) q^{53} +4.00000 q^{55} +1.00000 q^{57} +(2.00000 - 3.46410i) q^{59} +(-3.00000 - 5.19615i) q^{61} +(-3.00000 - 5.19615i) q^{65} +(-6.50000 + 11.2583i) q^{67} +8.00000 q^{69} -10.0000 q^{71} +(-5.50000 + 9.52628i) q^{73} +(0.500000 + 0.866025i) q^{75} +(1.50000 + 2.59808i) q^{79} +(-0.500000 + 0.866025i) q^{81} -2.00000 q^{83} -16.0000 q^{85} +(-2.00000 + 3.46410i) q^{87} +(1.50000 + 2.59808i) q^{93} +(-1.00000 + 1.73205i) q^{95} -10.0000 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 2 q^{5} - q^{9}+O(q^{10}) \) 2 * q - q^3 - 2 * q^5 - q^9	\( 2 q - q^{3} - 2 q^{5} - q^{9} - 2 q^{11} + 6 q^{13} + 4 q^{15} + 8 q^{17} - q^{19} - 8 q^{23} + q^{25} + 2 q^{27} + 8 q^{29} + 3 q^{31} - 2 q^{33} + q^{37} - 3 q^{39} - 12 q^{41} + 22 q^{43} - 2 q^{45} + 6 q^{47} + 8 q^{51} + 12 q^{53} + 8 q^{55} + 2 q^{57} + 4 q^{59} - 6 q^{61} - 6 q^{65} - 13 q^{67} + 16 q^{69} - 20 q^{71} - 11 q^{73} + q^{75} + 3 q^{79} - q^{81} - 4 q^{83} - 32 q^{85} - 4 q^{87} + 3 q^{93} - 2 q^{95} - 20 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q - q^3 - 2 * q^5 - q^9 - 2 * q^11 + 6 * q^13 + 4 * q^15 + 8 * q^17 - q^19 - 8 * q^23 + q^25 + 2 * q^27 + 8 * q^29 + 3 * q^31 - 2 * q^33 + q^37 - 3 * q^39 - 12 * q^41 + 22 * q^43 - 2 * q^45 + 6 * q^47 + 8 * q^51 + 12 * q^53 + 8 * q^55 + 2 * q^57 + 4 * q^59 - 6 * q^61 - 6 * q^65 - 13 * q^67 + 16 * q^69 - 20 * q^71 - 11 * q^73 + q^75 + 3 * q^79 - q^81 - 4 * q^83 - 32 * q^85 - 4 * q^87 + 3 * q^93 - 2 * q^95 - 20 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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