LMFDB - Embedded newform 483.2.i.a.415.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [483,2,Mod(277,483)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(483, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("483.277");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 483 = 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	483.i (of order $3$, degree $2$, 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:	$3.85677441763$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			415.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	483.415
Dual form			483.2.i.a.277.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.500000 - 0.866025i) q^{3} +(1.00000 + 1.73205i) q^{4} +(0.500000 + 2.59808i) q^{7} +(-0.500000 + 0.866025i) q^{9} +(1.00000 + 1.73205i) q^{11} +(1.00000 - 1.73205i) q^{12} -3.00000 q^{13} +(-2.00000 + 3.46410i) q^{16} +(-1.00000 - 1.73205i) q^{17} +(-1.50000 + 2.59808i) q^{19} +(2.00000 - 1.73205i) q^{21} +(-0.500000 + 0.866025i) q^{23} +(2.50000 + 4.33013i) q^{25} +1.00000 q^{27} +(-4.00000 + 3.46410i) q^{28} +6.00000 q^{29} +(1.50000 + 2.59808i) q^{31} +(1.00000 - 1.73205i) q^{33} -2.00000 q^{36} +(1.50000 - 2.59808i) q^{37} +(1.50000 + 2.59808i) q^{39} +3.00000 q^{43} +(-2.00000 + 3.46410i) q^{44} +(5.00000 - 8.66025i) q^{47} +4.00000 q^{48} +(-6.50000 + 2.59808i) q^{49} +(-1.00000 + 1.73205i) q^{51} +(-3.00000 - 5.19615i) q^{52} +(2.00000 + 3.46410i) q^{53} +3.00000 q^{57} +(3.00000 + 5.19615i) q^{59} +(1.00000 - 1.73205i) q^{61} +(-2.50000 - 0.866025i) q^{63} -8.00000 q^{64} +(-0.500000 - 0.866025i) q^{67} +(2.00000 - 3.46410i) q^{68} +1.00000 q^{69} -6.00000 q^{71} +(-0.500000 - 0.866025i) q^{73} +(2.50000 - 4.33013i) q^{75} -6.00000 q^{76} +(-4.00000 + 3.46410i) q^{77} +(3.50000 - 6.06218i) q^{79} +(-0.500000 - 0.866025i) q^{81} +4.00000 q^{83} +(5.00000 + 1.73205i) q^{84} +(-3.00000 - 5.19615i) q^{87} +(7.00000 - 12.1244i) q^{89} +(-1.50000 - 7.79423i) q^{91} -2.00000 q^{92} +(1.50000 - 2.59808i) q^{93} -10.0000 q^{97} -2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} + 2 q^{4} + q^{7} - q^{9} + 2 q^{11} + 2 q^{12} - 6 q^{13} - 4 q^{16} - 2 q^{17} - 3 q^{19} + 4 q^{21} - q^{23} + 5 q^{25} + 2 q^{27} - 8 q^{28} + 12 q^{29} + 3 q^{31} + 2 q^{33} - 4 q^{36}+ \cdots - 4 q^{99}+O(q^{100}) $ 2 * q - q^3 + 2 * q^4 + q^7 - q^9 + 2 * q^11 + 2 * q^12 - 6 * q^13 - 4 * q^16 - 2 * q^17 - 3 * q^19 + 4 * q^21 - q^23 + 5 * q^25 + 2 * q^27 - 8 * q^28 + 12 * q^29 + 3 * q^31 + 2 * q^33 - 4 * q^36 + 3 * q^37 + 3 * q^39 + 6 * q^43 - 4 * q^44 + 10 * q^47 + 8 * q^48 - 13 * q^49 - 2 * q^51 - 6 * q^52 + 4 * q^53 + 6 * q^57 + 6 * q^59 + 2 * q^61 - 5 * q^63 - 16 * q^64 - q^67 + 4 * q^68 + 2 * q^69 - 12 * q^71 - q^73 + 5 * q^75 - 12 * q^76 - 8 * q^77 + 7 * q^79 - q^81 + 8 * q^83 + 10 * q^84 - 6 * q^87 + 14 * q^89 - 3 * q^91 - 4 * q^92 + 3 * q^93 - 20 * q^97 - 4 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/483\mathbb{Z}\right)^\times$.

$n$	$323$	$346$	$442$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$2$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$3$	−0.500000	−	0.866025i	−0.288675	−	0.500000i
$3$	−0.500000	−	0.866025i	−0.288675	−	0.500000i
$4$	1.00000	+	1.73205i	0.500000	+	0.866025i
$5$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$5$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$6$		0			0
$7$	0.500000	+	2.59808i	0.188982	+	0.981981i
$7$	0.500000	+	2.59808i	0.188982	+	0.981981i
$8$		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	$-0.0691756\pi$
$11$	1.00000	+	1.73205i	0.301511	+	0.522233i	−0.674967	+	0.737848i	$0.735842\pi$
$12$	1.00000	−	1.73205i	0.288675	−	0.500000i
$13$	−3.00000			−0.832050			−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000			−0.832050			−0.416025	+	0.909353i	$0.636577\pi$
$14$		0			0
$15$		0			0
$16$	−2.00000	+	3.46410i	−0.500000	+	0.866025i
$17$	−1.00000	−	1.73205i	−0.242536	−	0.420084i	0.718900	−	0.695113i	$-0.244646\pi$
$17$	−1.00000	−	1.73205i	−0.242536	−	0.420084i	−0.961436	+	0.275029i	$0.911312\pi$
$18$		0			0
$19$	−1.50000	+	2.59808i	−0.344124	+	0.596040i	−0.985194	−	0.171442i	$-0.945157\pi$
$19$	−1.50000	+	2.59808i	−0.344124	+	0.596040i	0.641071	+	0.767482i	$0.278491\pi$
$20$		0			0
$21$	2.00000	−	1.73205i	0.436436	−	0.377964i
$22$		0			0
$23$	−0.500000	+	0.866025i	−0.104257	+	0.180579i
$23$	−0.500000	+	0.866025i	−0.104257	+	0.180579i
$24$		0			0
$25$	2.50000	+	4.33013i	0.500000	+	0.866025i
$26$		0			0
$27$	1.00000			0.192450
$28$	−4.00000	+	3.46410i	−0.755929	+	0.654654i
$29$	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
$31$	1.50000	+	2.59808i	0.269408	+	0.466628i	0.968709	−	0.248199i	$-0.0798387\pi$
$31$	1.50000	+	2.59808i	0.269408	+	0.466628i	−0.699301	+	0.714827i	$0.746505\pi$
$32$		0			0
$33$	1.00000	−	1.73205i	0.174078	−	0.301511i
$34$		0			0
$35$		0			0
$36$	−2.00000			−0.333333
$37$	1.50000	−	2.59808i	0.246598	−	0.427121i	−0.715981	−	0.698119i	$-0.754020\pi$
$37$	1.50000	−	2.59808i	0.246598	−	0.427121i	0.962580	+	0.270998i	$0.0873538\pi$
$38$		0			0
$39$	1.50000	+	2.59808i	0.240192	+	0.416025i
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	3.00000			0.457496			0.228748	−	0.973486i	$-0.426537\pi$
$43$	3.00000			0.457496			0.228748	+	0.973486i	$0.426537\pi$
$44$	−2.00000	+	3.46410i	−0.301511	+	0.522233i
$45$		0			0
$46$		0			0
$47$	5.00000	−	8.66025i	0.729325	−	1.26323i	−0.227844	−	0.973698i	$-0.573168\pi$
$47$	5.00000	−	8.66025i	0.729325	−	1.26323i	0.957169	−	0.289530i	$-0.0934991\pi$
$48$	4.00000			0.577350
$49$	−6.50000	+	2.59808i	−0.928571	+	0.371154i
$50$		0			0
$51$	−1.00000	+	1.73205i	−0.140028	+	0.242536i
$52$	−3.00000	−	5.19615i	−0.416025	−	0.720577i
$53$	2.00000	+	3.46410i	0.274721	+	0.475831i	0.970065	−	0.242846i	$-0.0780811\pi$
$53$	2.00000	+	3.46410i	0.274721	+	0.475831i	−0.695344	+	0.718677i	$0.744748\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	3.00000			0.397360
$58$		0			0
$59$	3.00000	+	5.19615i	0.390567	+	0.676481i	0.992524	−	0.122047i	$-0.0389457\pi$
$59$	3.00000	+	5.19615i	0.390567	+	0.676481i	−0.601958	+	0.798528i	$0.705612\pi$
$60$		0			0
$61$	1.00000	−	1.73205i	0.128037	−	0.221766i	−0.794879	−	0.606768i	$-0.792466\pi$
$61$	1.00000	−	1.73205i	0.128037	−	0.221766i	0.922916	+	0.385002i	$0.125799\pi$
$62$		0			0
$63$	−2.50000	−	0.866025i	−0.314970	−	0.109109i
$64$	−8.00000			−1.00000
$65$		0			0
$66$		0			0
$67$	−0.500000	−	0.866025i	−0.0610847	−	0.105802i	0.833866	−	0.551967i	$-0.186123\pi$
$67$	−0.500000	−	0.866025i	−0.0610847	−	0.105802i	−0.894951	+	0.446165i	$0.852789\pi$
$68$	2.00000	−	3.46410i	0.242536	−	0.420084i
$69$	1.00000			0.120386
$70$		0			0
$71$	−6.00000			−0.712069			−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000			−0.712069			−0.356034	+	0.934473i	$0.615871\pi$
$72$		0			0
$73$	−0.500000	−	0.866025i	−0.0585206	−	0.101361i	0.835281	−	0.549823i	$-0.185305\pi$
$73$	−0.500000	−	0.866025i	−0.0585206	−	0.101361i	−0.893801	+	0.448463i	$0.851972\pi$
$74$		0			0
$75$	2.50000	−	4.33013i	0.288675	−	0.500000i
$76$	−6.00000			−0.688247
$77$	−4.00000	+	3.46410i	−0.455842	+	0.394771i
$78$		0			0
$79$	3.50000	−	6.06218i	0.393781	−	0.682048i	−0.599164	−	0.800626i	$-0.704500\pi$
$79$	3.50000	−	6.06218i	0.393781	−	0.682048i	0.992945	+	0.118578i	$0.0378336\pi$
$80$		0			0
$81$	−0.500000	−	0.866025i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	4.00000			0.439057			0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000			0.439057			0.219529	+	0.975606i	$0.429548\pi$
$84$	5.00000	+	1.73205i	0.545545	+	0.188982i
$85$		0			0
$86$		0			0
$87$	−3.00000	−	5.19615i	−0.321634	−	0.557086i
$88$		0			0
$89$	7.00000	−	12.1244i	0.741999	−	1.28518i	−0.209585	−	0.977790i	$-0.567211\pi$
$89$	7.00000	−	12.1244i	0.741999	−	1.28518i	0.951584	−	0.307389i	$-0.0994552\pi$
$90$		0			0
$91$	−1.50000	−	7.79423i	−0.157243	−	0.817057i
$92$	−2.00000			−0.208514
$93$	1.50000	−	2.59808i	0.155543	−	0.269408i
$94$		0			0
$95$		0			0
$96$		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
$100$	−5.00000	+	8.66025i	−0.500000	+	0.866025i
$101$	−8.00000	−	13.8564i	−0.796030	−	1.37876i	−0.922183	−	0.386753i	$-0.873597\pi$
$101$	−8.00000	−	13.8564i	−0.796030	−	1.37876i	0.126153	−	0.992011i	$-0.459737\pi$
$102$		0			0
$103$	0.500000	−	0.866025i	0.0492665	−	0.0853320i	−0.840341	−	0.542059i	$-0.817645\pi$
$103$	0.500000	−	0.866025i	0.0492665	−	0.0853320i	0.889607	+	0.456727i	$0.150978\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	8.00000	−	13.8564i	0.773389	−	1.33955i	−0.162306	−	0.986740i	$-0.551893\pi$
$107$	8.00000	−	13.8564i	0.773389	−	1.33955i	0.935695	−	0.352809i	$-0.114773\pi$
$108$	1.00000	+	1.73205i	0.0962250	+	0.166667i
$109$	4.50000	+	7.79423i	0.431022	+	0.746552i	0.996962	−	0.0778949i	$-0.0248199\pi$
$109$	4.50000	+	7.79423i	0.431022	+	0.746552i	−0.565940	+	0.824447i	$0.691487\pi$
$110$		0			0
$111$	−3.00000			−0.284747
$112$	−10.0000	−	3.46410i	−0.944911	−	0.327327i
$113$	−18.0000			−1.69330			−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000			−1.69330			−0.846649	+	0.532152i	$0.821383\pi$
$114$		0			0
$115$		0			0
$116$	6.00000	+	10.3923i	0.557086	+	0.964901i
$117$	1.50000	−	2.59808i	0.138675	−	0.240192i
$118$		0			0
$119$	4.00000	−	3.46410i	0.366679	−	0.317554i
$120$		0			0
$121$	3.50000	−	6.06218i	0.318182	−	0.551107i
$122$		0			0
$123$		0			0
$124$	−3.00000	+	5.19615i	−0.269408	+	0.466628i
$125$		0			0
$126$		0			0
$127$	1.00000			0.0887357			0.0443678	−	0.999015i	$-0.485873\pi$
$127$	1.00000			0.0887357			0.0443678	+	0.999015i	$0.485873\pi$
$128$		0			0
$129$	−1.50000	−	2.59808i	−0.132068	−	0.228748i
$130$		0			0
$131$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$131$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$132$	4.00000			0.348155
$133$	−7.50000	−	2.59808i	−0.650332	−	0.225282i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.00000	−	10.3923i	−0.512615	−	0.887875i	−0.999893	−	0.0146279i	$-0.995344\pi$
$137$	−6.00000	−	10.3923i	−0.512615	−	0.887875i	0.487278	−	0.873247i	$-0.337990\pi$
$138$		0			0
$139$	21.0000			1.78120			0.890598	−	0.454791i	$-0.150286\pi$
$139$	21.0000			1.78120			0.890598	+	0.454791i	$0.150286\pi$
$140$		0			0
$141$	−10.0000			−0.842152
$142$		0			0
$143$	−3.00000	−	5.19615i	−0.250873	−	0.434524i
$144$	−2.00000	−	3.46410i	−0.166667	−	0.288675i
$145$		0			0
$146$		0			0
$147$	5.50000	+	4.33013i	0.453632	+	0.357143i
$148$	6.00000			0.493197
$149$	−5.00000	+	8.66025i	−0.409616	+	0.709476i	−0.994847	−	0.101391i	$-0.967671\pi$
$149$	−5.00000	+	8.66025i	−0.409616	+	0.709476i	0.585231	+	0.810867i	$0.301004\pi$
$150$		0			0
$151$	6.00000	+	10.3923i	0.488273	+	0.845714i	0.999909	−	0.0134886i	$-0.00429367\pi$
$151$	6.00000	+	10.3923i	0.488273	+	0.845714i	−0.511636	+	0.859202i	$0.670960\pi$
$152$		0			0
$153$	2.00000			0.161690
$154$		0			0
$155$		0			0
$156$	−3.00000	+	5.19615i	−0.240192	+	0.416025i
$157$	1.00000	+	1.73205i	0.0798087	+	0.138233i	0.903167	−	0.429289i	$-0.141236\pi$
$157$	1.00000	+	1.73205i	0.0798087	+	0.138233i	−0.823359	+	0.567521i	$0.807902\pi$
$158$		0			0
$159$	2.00000	−	3.46410i	0.158610	−	0.274721i
$160$		0			0
$161$	−2.50000	−	0.866025i	−0.197028	−	0.0682524i
$162$		0			0
$163$	10.0000	−	17.3205i	0.783260	−	1.35665i	−0.146772	−	0.989170i	$-0.546888\pi$
$163$	10.0000	−	17.3205i	0.783260	−	1.35665i	0.930033	−	0.367477i	$-0.119778\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	16.0000			1.23812			0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000			1.23812			0.619059	+	0.785345i	$0.287514\pi$
$168$		0			0
$169$	−4.00000			−0.307692
$170$		0			0
$171$	−1.50000	−	2.59808i	−0.114708	−	0.198680i
$172$	3.00000	+	5.19615i	0.228748	+	0.396203i
$173$	−3.00000	+	5.19615i	−0.228086	+	0.395056i	−0.957241	−	0.289292i	$-0.906580\pi$
$173$	−3.00000	+	5.19615i	−0.228086	+	0.395056i	0.729155	+	0.684349i	$0.239913\pi$
$174$		0			0
$175$	−10.0000	+	8.66025i	−0.755929	+	0.654654i
$176$	−8.00000			−0.603023
$177$	3.00000	−	5.19615i	0.225494	−	0.390567i
$178$		0			0
$179$	10.0000	+	17.3205i	0.747435	+	1.29460i	0.949048	+	0.315130i	$0.102048\pi$
$179$	10.0000	+	17.3205i	0.747435	+	1.29460i	−0.201613	+	0.979465i	$0.564618\pi$
$180$		0			0
$181$	−5.00000			−0.371647			−0.185824	−	0.982583i	$-0.559495\pi$
$181$	−5.00000			−0.371647			−0.185824	+	0.982583i	$0.559495\pi$
$182$		0			0
$183$	−2.00000			−0.147844
$184$		0			0
$185$		0			0
$186$		0			0
$187$	2.00000	−	3.46410i	0.146254	−	0.253320i
$188$	20.0000			1.45865
$189$	0.500000	+	2.59808i	0.0363696	+	0.188982i
$190$		0			0
$191$	3.00000	−	5.19615i	0.217072	−	0.375980i	−0.736839	−	0.676068i	$-0.763683\pi$
$191$	3.00000	−	5.19615i	0.217072	−	0.375980i	0.953912	+	0.300088i	$0.0970159\pi$
$192$	4.00000	+	6.92820i	0.288675	+	0.500000i
$193$	2.50000	+	4.33013i	0.179954	+	0.311689i	0.941865	−	0.335993i	$-0.109072\pi$
$193$	2.50000	+	4.33013i	0.179954	+	0.311689i	−0.761911	+	0.647682i	$0.775738\pi$
$194$		0			0
$195$		0			0
$196$	−11.0000	−	8.66025i	−0.785714	−	0.618590i
$197$	4.00000			0.284988			0.142494	−	0.989796i	$-0.454488\pi$
$197$	4.00000			0.284988			0.142494	+	0.989796i	$0.454488\pi$
$198$		0			0
$199$	4.00000	+	6.92820i	0.283552	+	0.491127i	0.972257	−	0.233915i	$-0.0751537\pi$
$199$	4.00000	+	6.92820i	0.283552	+	0.491127i	−0.688705	+	0.725042i	$0.741820\pi$
$200$		0			0
$201$	−0.500000	+	0.866025i	−0.0352673	+	0.0610847i
$202$		0			0
$203$	3.00000	+	15.5885i	0.210559	+	1.09410i
$204$	−4.00000			−0.280056
$205$		0			0
$206$		0			0
$207$	−0.500000	−	0.866025i	−0.0347524	−	0.0601929i
$208$	6.00000	−	10.3923i	0.416025	−	0.720577i
$209$	−6.00000			−0.415029
$210$		0			0
$211$	12.0000			0.826114			0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000			0.826114			0.413057	+	0.910705i	$0.364461\pi$
$212$	−4.00000	+	6.92820i	−0.274721	+	0.475831i
$213$	3.00000	+	5.19615i	0.205557	+	0.356034i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−6.00000	+	5.19615i	−0.407307	+	0.352738i
$218$		0			0
$219$	−0.500000	+	0.866025i	−0.0337869	+	0.0585206i
$220$		0			0
$221$	3.00000	+	5.19615i	0.201802	+	0.349531i
$222$		0			0
$223$	16.0000			1.07144			0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000			1.07144			0.535720	+	0.844396i	$0.320040\pi$
$224$		0			0
$225$	−5.00000			−0.333333
$226$		0			0
$227$	−1.00000	−	1.73205i	−0.0663723	−	0.114960i	0.830930	−	0.556378i	$-0.187809\pi$
$227$	−1.00000	−	1.73205i	−0.0663723	−	0.114960i	−0.897302	+	0.441417i	$0.854476\pi$
$228$	3.00000	+	5.19615i	0.198680	+	0.344124i
$229$	−7.50000	+	12.9904i	−0.495614	+	0.858429i	−0.999987	−	0.00505719i	$-0.998390\pi$
$229$	−7.50000	+	12.9904i	−0.495614	+	0.858429i	0.504373	+	0.863486i	$0.331724\pi$
$230$		0			0
$231$	5.00000	+	1.73205i	0.328976	+	0.113961i
$232$		0			0
$233$	−1.00000	+	1.73205i	−0.0655122	+	0.113470i	−0.896921	−	0.442191i	$-0.854201\pi$
$233$	−1.00000	+	1.73205i	−0.0655122	+	0.113470i	0.831409	+	0.555661i	$0.187535\pi$
$234$		0			0
$235$		0			0
$236$	−6.00000	+	10.3923i	−0.390567	+	0.676481i
$237$	−7.00000			−0.454699
$238$		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
$241$	7.00000	+	12.1244i	0.450910	+	0.780998i	0.998443	−	0.0557856i	$-0.0177663\pi$
$241$	7.00000	+	12.1244i	0.450910	+	0.780998i	−0.547533	+	0.836784i	$0.684433\pi$
$242$		0			0
$243$	−0.500000	+	0.866025i	−0.0320750	+	0.0555556i
$244$	4.00000			0.256074
$245$		0			0
$246$		0			0
$247$	4.50000	−	7.79423i	0.286328	−	0.495935i
$248$		0			0
$249$	−2.00000	−	3.46410i	−0.126745	−	0.219529i
$250$		0			0
$251$	−26.0000			−1.64111			−0.820553	−	0.571571i	$-0.806334\pi$
$251$	−26.0000			−1.64111			−0.820553	+	0.571571i	$0.806334\pi$
$252$	−1.00000	−	5.19615i	−0.0629941	−	0.327327i
$253$	−2.00000			−0.125739
$254$		0			0
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$	10.0000	−	17.3205i	0.623783	−	1.08042i	−0.364992	−	0.931011i	$-0.618928\pi$
$257$	10.0000	−	17.3205i	0.623783	−	1.08042i	0.988775	−	0.149413i	$-0.0477384\pi$
$258$		0			0
$259$	7.50000	+	2.59808i	0.466027	+	0.161437i
$260$		0			0
$261$	−3.00000	+	5.19615i	−0.185695	+	0.321634i
$262$		0			0
$263$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$263$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−14.0000			−0.856786
$268$	1.00000	−	1.73205i	0.0610847	−	0.105802i
$269$	12.0000	+	20.7846i	0.731653	+	1.26726i	0.956176	+	0.292791i	$0.0945841\pi$
$269$	12.0000	+	20.7846i	0.731653	+	1.26726i	−0.224523	+	0.974469i	$0.572083\pi$
$270$		0			0
$271$	−10.0000	+	17.3205i	−0.607457	+	1.05215i	0.384201	+	0.923249i	$0.374477\pi$
$271$	−10.0000	+	17.3205i	−0.607457	+	1.05215i	−0.991658	+	0.128897i	$0.958856\pi$
$272$	8.00000			0.485071
$273$	−6.00000	+	5.19615i	−0.363137	+	0.314485i
$274$		0			0
$275$	−5.00000	+	8.66025i	−0.301511	+	0.522233i
$276$	1.00000	+	1.73205i	0.0601929	+	0.104257i
$277$	3.50000	+	6.06218i	0.210295	+	0.364241i	0.951807	−	0.306699i	$-0.0992243\pi$
$277$	3.50000	+	6.06218i	0.210295	+	0.364241i	−0.741512	+	0.670940i	$0.765891\pi$
$278$		0			0
$279$	−3.00000			−0.179605
$280$		0			0
$281$	−22.0000			−1.31241			−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000			−1.31241			−0.656205	+	0.754583i	$0.727839\pi$
$282$		0			0
$283$	−7.50000	−	12.9904i	−0.445829	−	0.772198i	0.552281	−	0.833658i	$-0.313758\pi$
$283$	−7.50000	−	12.9904i	−0.445829	−	0.772198i	−0.998110	+	0.0614601i	$0.980424\pi$
$284$	−6.00000	−	10.3923i	−0.356034	−	0.616670i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	6.50000	−	11.2583i	0.382353	−	0.662255i
$290$		0			0
$291$	5.00000	+	8.66025i	0.293105	+	0.507673i
$292$	1.00000	−	1.73205i	0.0585206	−	0.101361i
$293$	−6.00000			−0.350524			−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000			−0.350524			−0.175262	+	0.984522i	$0.556077\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	1.00000	+	1.73205i	0.0580259	+	0.100504i
$298$		0			0
$299$	1.50000	−	2.59808i	0.0867472	−	0.150251i
$300$	10.0000			0.577350
$301$	1.50000	+	7.79423i	0.0864586	+	0.449252i
$302$		0			0
$303$	−8.00000	+	13.8564i	−0.459588	+	0.796030i
$304$	−6.00000	−	10.3923i	−0.344124	−	0.596040i
$305$		0			0
$306$		0			0
$307$	27.0000			1.54097			0.770486	−	0.637457i	$-0.220014\pi$
$307$	27.0000			1.54097			0.770486	+	0.637457i	$0.220014\pi$
$308$	−10.0000	−	3.46410i	−0.569803	−	0.197386i
$309$	−1.00000			−0.0568880
$310$		0			0
$311$	11.0000	+	19.0526i	0.623753	+	1.08037i	0.988781	+	0.149375i	$0.0477261\pi$
$311$	11.0000	+	19.0526i	0.623753	+	1.08037i	−0.365028	+	0.930997i	$0.618941\pi$
$312$		0			0
$313$	−0.500000	+	0.866025i	−0.0282617	+	0.0489506i	−0.879810	−	0.475325i	$-0.842331\pi$
$313$	−0.500000	+	0.866025i	−0.0282617	+	0.0489506i	0.851549	+	0.524276i	$0.175664\pi$
$314$		0			0
$315$		0			0
$316$	14.0000			0.787562
$317$	−1.00000	+	1.73205i	−0.0561656	+	0.0972817i	−0.892741	−	0.450570i	$-0.851221\pi$
$317$	−1.00000	+	1.73205i	−0.0561656	+	0.0972817i	0.836576	+	0.547852i	$0.184554\pi$
$318$		0			0
$319$	6.00000	+	10.3923i	0.335936	+	0.581857i
$320$		0			0
$321$	−16.0000			−0.893033
$322$		0			0
$323$	6.00000			0.333849
$324$	1.00000	−	1.73205i	0.0555556	−	0.0962250i
$325$	−7.50000	−	12.9904i	−0.416025	−	0.720577i
$326$		0			0
$327$	4.50000	−	7.79423i	0.248851	−	0.431022i
$328$		0			0
$329$	25.0000	+	8.66025i	1.37829	+	0.477455i
$330$		0			0
$331$	6.50000	−	11.2583i	0.357272	−	0.618814i	−0.630232	−	0.776407i	$-0.717040\pi$
$331$	6.50000	−	11.2583i	0.357272	−	0.618814i	0.987504	+	0.157593i	$0.0503735\pi$
$332$	4.00000	+	6.92820i	0.219529	+	0.380235i
$333$	1.50000	+	2.59808i	0.0821995	+	0.142374i
$334$		0			0
$335$		0			0
$336$	2.00000	+	10.3923i	0.109109	+	0.566947i
$337$	27.0000			1.47078			0.735392	−	0.677642i	$-0.236998\pi$
$337$	27.0000			1.47078			0.735392	+	0.677642i	$0.236998\pi$
$338$		0			0
$339$	9.00000	+	15.5885i	0.488813	+	0.846649i
$340$		0			0
$341$	−3.00000	+	5.19615i	−0.162459	+	0.281387i
$342$		0			0
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	14.0000	+	24.2487i	0.751559	+	1.30174i	0.947067	+	0.321037i	$0.104031\pi$
$347$	14.0000	+	24.2487i	0.751559	+	1.30174i	−0.195507	+	0.980702i	$0.562635\pi$
$348$	6.00000	−	10.3923i	0.321634	−	0.557086i
$349$	10.0000			0.535288			0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000			0.535288			0.267644	+	0.963518i	$0.413755\pi$
$350$		0			0
$351$	−3.00000			−0.160128
$352$		0			0
$353$	−5.00000	−	8.66025i	−0.266123	−	0.460939i	0.701734	−	0.712439i	$-0.252409\pi$
$353$	−5.00000	−	8.66025i	−0.266123	−	0.460939i	−0.967857	+	0.251500i	$0.919076\pi$
$354$		0			0
$355$		0			0
$356$	28.0000			1.48400
$357$	−5.00000	−	1.73205i	−0.264628	−	0.0916698i
$358$		0			0
$359$	5.00000	−	8.66025i	0.263890	−	0.457071i	−0.703382	−	0.710812i	$-0.748328\pi$
$359$	5.00000	−	8.66025i	0.263890	−	0.457071i	0.967272	+	0.253741i	$0.0816611\pi$
$360$		0			0
$361$	5.00000	+	8.66025i	0.263158	+	0.455803i
$362$		0			0
$363$	−7.00000			−0.367405
$364$	12.0000	−	10.3923i	0.628971	−	0.544705i
$365$		0			0
$366$		0			0
$367$	−18.5000	−	32.0429i	−0.965692	−	1.67263i	−0.707744	−	0.706469i	$-0.750287\pi$
$367$	−18.5000	−	32.0429i	−0.965692	−	1.67263i	−0.257948	−	0.966159i	$-0.583046\pi$
$368$	−2.00000	−	3.46410i	−0.104257	−	0.180579i
$369$		0			0
$370$		0			0
$371$	−8.00000	+	6.92820i	−0.415339	+	0.359694i
$372$	6.00000			0.311086
$373$	5.50000	−	9.52628i	0.284779	−	0.493252i	−0.687776	−	0.725923i	$-0.741413\pi$
$373$	5.50000	−	9.52628i	0.284779	−	0.493252i	0.972556	+	0.232671i	$0.0747464\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−18.0000			−0.927047
$378$		0			0
$379$	29.0000			1.48963			0.744815	−	0.667271i	$-0.232538\pi$
$379$	29.0000			1.48963			0.744815	+	0.667271i	$0.232538\pi$
$380$		0			0
$381$	−0.500000	−	0.866025i	−0.0256158	−	0.0443678i
$382$		0			0
$383$	−9.00000	+	15.5885i	−0.459879	+	0.796533i	−0.998954	−	0.0457244i	$-0.985440\pi$
$383$	−9.00000	+	15.5885i	−0.459879	+	0.796533i	0.539076	+	0.842257i	$0.318774\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−1.50000	+	2.59808i	−0.0762493	+	0.132068i
$388$	−10.0000	−	17.3205i	−0.507673	−	0.879316i
$389$	−18.0000	−	31.1769i	−0.912636	−	1.58073i	−0.810326	−	0.585980i	$-0.800710\pi$
$389$	−18.0000	−	31.1769i	−0.912636	−	1.58073i	−0.102311	−	0.994753i	$-0.532624\pi$
$390$		0			0
$391$	2.00000			0.101144
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	−2.00000	−	3.46410i	−0.100504	−	0.174078i
$397$	−3.50000	+	6.06218i	−0.175660	+	0.304252i	−0.940389	−	0.340099i	$-0.889539\pi$
$397$	−3.50000	+	6.06218i	−0.175660	+	0.304252i	0.764730	+	0.644351i	$0.222873\pi$
$398$		0			0
$399$	1.50000	+	7.79423i	0.0750939	+	0.390199i
$400$	−20.0000			−1.00000
$401$	−16.0000	+	27.7128i	−0.799002	+	1.38391i	0.121265	+	0.992620i	$0.461305\pi$
$401$	−16.0000	+	27.7128i	−0.799002	+	1.38391i	−0.920267	+	0.391292i	$0.872028\pi$
$402$		0			0
$403$	−4.50000	−	7.79423i	−0.224161	−	0.388258i
$404$	16.0000	−	27.7128i	0.796030	−	1.37876i
$405$		0			0
$406$		0			0
$407$	6.00000			0.297409
$408$		0			0
$409$	−18.5000	−	32.0429i	−0.914766	−	1.58442i	−0.807243	−	0.590219i	$-0.799041\pi$
$409$	−18.5000	−	32.0429i	−0.914766	−	1.58442i	−0.107523	−	0.994203i	$-0.534292\pi$
$410$		0			0
$411$	−6.00000	+	10.3923i	−0.295958	+	0.512615i
$412$	2.00000			0.0985329
$413$	−12.0000	+	10.3923i	−0.590481	+	0.511372i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−10.5000	−	18.1865i	−0.514187	−	0.890598i
$418$		0			0
$419$	−28.0000			−1.36789			−0.683945	−	0.729534i	$-0.739737\pi$
$419$	−28.0000			−1.36789			−0.683945	+	0.729534i	$0.739737\pi$
$420$		0			0
$421$	15.0000			0.731055			0.365528	−	0.930800i	$-0.380889\pi$
$421$	15.0000			0.731055			0.365528	+	0.930800i	$0.380889\pi$
$422$		0			0
$423$	5.00000	+	8.66025i	0.243108	+	0.421076i
$424$		0			0
$425$	5.00000	−	8.66025i	0.242536	−	0.420084i
$426$		0			0
$427$	5.00000	+	1.73205i	0.241967	+	0.0838198i
$428$	32.0000			1.54678
$429$	−3.00000	+	5.19615i	−0.144841	+	0.250873i
$430$		0			0
$431$	−6.00000	−	10.3923i	−0.289010	−	0.500580i	0.684564	−	0.728953i	$-0.259993\pi$
$431$	−6.00000	−	10.3923i	−0.289010	−	0.500580i	−0.973574	+	0.228373i	$0.926659\pi$
$432$	−2.00000	+	3.46410i	−0.0962250	+	0.166667i
$433$	−7.00000			−0.336399			−0.168199	−	0.985753i	$-0.553795\pi$
$433$	−7.00000			−0.336399			−0.168199	+	0.985753i	$0.553795\pi$
$434$		0			0
$435$		0			0
$436$	−9.00000	+	15.5885i	−0.431022	+	0.746552i
$437$	−1.50000	−	2.59808i	−0.0717547	−	0.124283i
$438$		0			0
$439$	−4.00000	+	6.92820i	−0.190910	+	0.330665i	−0.945552	−	0.325471i	$-0.894477\pi$
$439$	−4.00000	+	6.92820i	−0.190910	+	0.330665i	0.754642	+	0.656136i	$0.227810\pi$
$440$		0			0
$441$	1.00000	−	6.92820i	0.0476190	−	0.329914i
$442$		0			0
$443$	7.00000	−	12.1244i	0.332580	−	0.576046i	−0.650437	−	0.759560i	$-0.725414\pi$
$443$	7.00000	−	12.1244i	0.332580	−	0.576046i	0.983017	+	0.183515i	$0.0587475\pi$
$444$	−3.00000	−	5.19615i	−0.142374	−	0.246598i
$445$		0			0
$446$		0			0
$447$	10.0000			0.472984
$448$	−4.00000	−	20.7846i	−0.188982	−	0.981981i
$449$	−6.00000			−0.283158			−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000			−0.283158			−0.141579	+	0.989927i	$0.545218\pi$
$450$		0			0
$451$		0			0
$452$	−18.0000	−	31.1769i	−0.846649	−	1.46644i
$453$	6.00000	−	10.3923i	0.281905	−	0.488273i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−13.5000	+	23.3827i	−0.631503	+	1.09380i	0.355741	+	0.934585i	$0.384228\pi$
$457$	−13.5000	+	23.3827i	−0.631503	+	1.09380i	−0.987245	+	0.159211i	$0.949105\pi$
$458$		0			0
$459$	−1.00000	−	1.73205i	−0.0466760	−	0.0808452i
$460$		0			0
$461$	−10.0000			−0.465746			−0.232873	−	0.972507i	$-0.574813\pi$
$461$	−10.0000			−0.465746			−0.232873	+	0.972507i	$0.574813\pi$
$462$		0			0
$463$	−29.0000			−1.34774			−0.673872	−	0.738848i	$-0.735370\pi$
$463$	−29.0000			−1.34774			−0.673872	+	0.738848i	$0.735370\pi$
$464$	−12.0000	+	20.7846i	−0.557086	+	0.964901i
$465$		0			0
$466$		0			0
$467$	13.0000	−	22.5167i	0.601568	−	1.04195i	−0.391015	−	0.920384i	$-0.627876\pi$
$467$	13.0000	−	22.5167i	0.601568	−	1.04195i	0.992584	−	0.121563i	$-0.0387905\pi$
$468$	6.00000			0.277350
$469$	2.00000	−	1.73205i	0.0923514	−	0.0799787i
$470$		0			0
$471$	1.00000	−	1.73205i	0.0460776	−	0.0798087i
$472$		0			0
$473$	3.00000	+	5.19615i	0.137940	+	0.238919i
$474$		0			0
$475$	−15.0000			−0.688247
$476$	10.0000	+	3.46410i	0.458349	+	0.158777i
$477$	−4.00000			−0.183147
$478$		0			0
$479$	−9.00000	−	15.5885i	−0.411220	−	0.712255i	0.583803	−	0.811895i	$-0.301564\pi$
$479$	−9.00000	−	15.5885i	−0.411220	−	0.712255i	−0.995023	+	0.0996406i	$0.968231\pi$
$480$		0			0
$481$	−4.50000	+	7.79423i	−0.205182	+	0.355386i
$482$		0			0
$483$	0.500000	+	2.59808i	0.0227508	+	0.118217i
$484$	14.0000			0.636364
$485$		0			0
$486$		0			0
$487$	−9.50000	−	16.4545i	−0.430486	−	0.745624i	0.566429	−	0.824110i	$-0.308325\pi$
$487$	−9.50000	−	16.4545i	−0.430486	−	0.745624i	−0.996915	+	0.0784867i	$0.974991\pi$
$488$		0			0
$489$	−20.0000			−0.904431
$490$		0			0
$491$	30.0000			1.35388			0.676941	−	0.736038i	$-0.263305\pi$
$491$	30.0000			1.35388			0.676941	+	0.736038i	$0.263305\pi$
$492$		0			0
$493$	−6.00000	−	10.3923i	−0.270226	−	0.468046i
$494$		0			0
$495$		0			0
$496$	−12.0000			−0.538816
$497$	−3.00000	−	15.5885i	−0.134568	−	0.699238i
$498$		0			0
$499$	17.5000	−	30.3109i	0.783408	−	1.35690i	−0.146538	−	0.989205i	$-0.546813\pi$
$499$	17.5000	−	30.3109i	0.783408	−	1.35690i	0.929946	−	0.367697i	$-0.119854\pi$
$500$		0			0
$501$	−8.00000	−	13.8564i	−0.357414	−	0.619059i
$502$		0			0
$503$	−36.0000			−1.60516			−0.802580	−	0.596544i	$-0.796540\pi$
$503$	−36.0000			−1.60516			−0.802580	+	0.596544i	$0.796540\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	2.00000	+	3.46410i	0.0888231	+	0.153846i
$508$	1.00000	+	1.73205i	0.0443678	+	0.0768473i
$509$	−10.0000	+	17.3205i	−0.443242	+	0.767718i	−0.997928	−	0.0643419i	$-0.979505\pi$
$509$	−10.0000	+	17.3205i	−0.443242	+	0.767718i	0.554686	+	0.832060i	$0.312839\pi$
$510$		0			0
$511$	2.00000	−	1.73205i	0.0884748	−	0.0766214i
$512$		0			0
$513$	−1.50000	+	2.59808i	−0.0662266	+	0.114708i
$514$		0			0
$515$		0			0
$516$	3.00000	−	5.19615i	0.132068	−	0.228748i
$517$	20.0000			0.879599
$518$		0			0
$519$	6.00000			0.263371
$520$		0			0
$521$	−3.00000	−	5.19615i	−0.131432	−	0.227648i	0.792797	−	0.609486i	$-0.208624\pi$
$521$	−3.00000	−	5.19615i	−0.131432	−	0.227648i	−0.924229	+	0.381839i	$0.875291\pi$
$522$		0			0
$523$	−10.5000	+	18.1865i	−0.459133	+	0.795242i	−0.998915	−	0.0465630i	$-0.985173\pi$
$523$	−10.5000	+	18.1865i	−0.459133	+	0.795242i	0.539782	+	0.841805i	$0.318507\pi$
$524$		0			0
$525$	12.5000	+	4.33013i	0.545545	+	0.188982i
$526$		0			0
$527$	3.00000	−	5.19615i	0.130682	−	0.226348i
$528$	4.00000	+	6.92820i	0.174078	+	0.301511i
$529$	−0.500000	−	0.866025i	−0.0217391	−	0.0376533i
$530$		0			0
$531$	−6.00000			−0.260378
$532$	−3.00000	−	15.5885i	−0.130066	−	0.675845i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	10.0000	−	17.3205i	0.431532	−	0.747435i
$538$		0			0
$539$	−11.0000	−	8.66025i	−0.473804	−	0.373024i
$540$		0			0
$541$	−8.50000	+	14.7224i	−0.365444	+	0.632967i	−0.988847	−	0.148933i	$-0.952416\pi$
$541$	−8.50000	+	14.7224i	−0.365444	+	0.632967i	0.623404	+	0.781900i	$0.285749\pi$
$542$		0			0
$543$	2.50000	+	4.33013i	0.107285	+	0.185824i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	28.0000			1.19719			0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000			1.19719			0.598597	+	0.801050i	$0.295725\pi$
$548$	12.0000	−	20.7846i	0.512615	−	0.887875i
$549$	1.00000	+	1.73205i	0.0426790	+	0.0739221i
$550$		0			0
$551$	−9.00000	+	15.5885i	−0.383413	+	0.664091i
$552$		0			0
$553$	17.5000	+	6.06218i	0.744176	+	0.257790i
$554$		0			0
$555$		0			0
$556$	21.0000	+	36.3731i	0.890598	+	1.54256i
$557$	−9.00000	−	15.5885i	−0.381342	−	0.660504i	0.609912	−	0.792469i	$-0.291205\pi$
$557$	−9.00000	−	15.5885i	−0.381342	−	0.660504i	−0.991254	+	0.131965i	$0.957871\pi$
$558$		0			0
$559$	−9.00000			−0.380659
$560$		0			0
$561$	−4.00000			−0.168880
$562$		0			0
$563$	18.0000	+	31.1769i	0.758610	+	1.31395i	0.943560	+	0.331202i	$0.107454\pi$
$563$	18.0000	+	31.1769i	0.758610	+	1.31395i	−0.184950	+	0.982748i	$0.559212\pi$
$564$	−10.0000	−	17.3205i	−0.421076	−	0.729325i
$565$		0			0
$566$		0			0
$567$	2.00000	−	1.73205i	0.0839921	−	0.0727393i
$568$		0			0
$569$	20.0000	−	34.6410i	0.838444	−	1.45223i	−0.0527519	−	0.998608i	$-0.516799\pi$
$569$	20.0000	−	34.6410i	0.838444	−	1.45223i	0.891196	−	0.453619i	$-0.149867\pi$
$570$		0			0
$571$	8.50000	+	14.7224i	0.355714	+	0.616115i	0.987240	−	0.159240i	$-0.0509044\pi$
$571$	8.50000	+	14.7224i	0.355714	+	0.616115i	−0.631526	+	0.775355i	$0.717571\pi$
$572$	6.00000	−	10.3923i	0.250873	−	0.434524i
$573$	−6.00000			−0.250654
$574$		0			0
$575$	−5.00000			−0.208514
$576$	4.00000	−	6.92820i	0.166667	−	0.288675i
$577$	−1.50000	−	2.59808i	−0.0624458	−	0.108159i	0.833112	−	0.553104i	$-0.186557\pi$
$577$	−1.50000	−	2.59808i	−0.0624458	−	0.108159i	−0.895558	+	0.444945i	$0.853223\pi$
$578$		0			0
$579$	2.50000	−	4.33013i	0.103896	−	0.179954i
$580$		0			0
$581$	2.00000	+	10.3923i	0.0829740	+	0.431145i
$582$		0			0
$583$	−4.00000	+	6.92820i	−0.165663	+	0.286937i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	18.0000			0.742940			0.371470	−	0.928445i	$-0.378854\pi$
$587$	18.0000			0.742940			0.371470	+	0.928445i	$0.378854\pi$
$588$	−2.00000	+	13.8564i	−0.0824786	+	0.571429i
$589$	−9.00000			−0.370839
$590$		0			0
$591$	−2.00000	−	3.46410i	−0.0822690	−	0.142494i
$592$	6.00000	+	10.3923i	0.246598	+	0.427121i
$593$	17.0000	−	29.4449i	0.698106	−	1.20916i	−0.271016	−	0.962575i	$-0.587360\pi$
$593$	17.0000	−	29.4449i	0.698106	−	1.20916i	0.969122	−	0.246581i	$-0.0793071\pi$
$594$		0			0
$595$		0			0
$596$	−20.0000			−0.819232
$597$	4.00000	−	6.92820i	0.163709	−	0.283552i
$598$		0			0
$599$	20.0000	+	34.6410i	0.817178	+	1.41539i	0.907754	+	0.419504i	$0.137796\pi$
$599$	20.0000	+	34.6410i	0.817178	+	1.41539i	−0.0905757	+	0.995890i	$0.528871\pi$
$600$		0			0
$601$	−13.0000			−0.530281			−0.265141	−	0.964210i	$-0.585418\pi$
$601$	−13.0000			−0.530281			−0.265141	+	0.964210i	$0.585418\pi$
$602$		0			0
$603$	1.00000			0.0407231
$604$	−12.0000	+	20.7846i	−0.488273	+	0.845714i
$605$		0			0
$606$		0			0
$607$	−19.5000	+	33.7750i	−0.791481	+	1.37088i	0.133570	+	0.991039i	$0.457356\pi$
$607$	−19.5000	+	33.7750i	−0.791481	+	1.37088i	−0.925050	+	0.379845i	$0.875977\pi$
$608$		0			0
$609$	12.0000	−	10.3923i	0.486265	−	0.421117i
$610$		0			0
$611$	−15.0000	+	25.9808i	−0.606835	+	1.05107i
$612$	2.00000	+	3.46410i	0.0808452	+	0.140028i
$613$	19.0000	+	32.9090i	0.767403	+	1.32918i	0.938967	+	0.344008i	$0.111785\pi$
$613$	19.0000	+	32.9090i	0.767403	+	1.32918i	−0.171564	+	0.985173i	$0.554882\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	6.00000			0.241551			0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000			0.241551			0.120775	+	0.992680i	$0.461462\pi$
$618$		0			0
$619$	3.50000	+	6.06218i	0.140677	+	0.243659i	0.927752	−	0.373198i	$-0.121739\pi$
$619$	3.50000	+	6.06218i	0.140677	+	0.243659i	−0.787075	+	0.616858i	$0.788405\pi$
$620$		0			0
$621$	−0.500000	+	0.866025i	−0.0200643	+	0.0347524i
$622$		0			0
$623$	35.0000	+	12.1244i	1.40225	+	0.485752i
$624$	−12.0000			−0.480384
$625$	−12.5000	+	21.6506i	−0.500000	+	0.866025i
$626$		0			0
$627$	3.00000	+	5.19615i	0.119808	+	0.207514i
$628$	−2.00000	+	3.46410i	−0.0798087	+	0.138233i
$629$	−6.00000			−0.239236
$630$		0			0
$631$	16.0000			0.636950			0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000			0.636950			0.318475	+	0.947931i	$0.396829\pi$
$632$		0			0
$633$	−6.00000	−	10.3923i	−0.238479	−	0.413057i
$634$		0			0
$635$		0			0
$636$	8.00000			0.317221
$637$	19.5000	−	7.79423i	0.772618	−	0.308819i
$638$		0			0
$639$	3.00000	−	5.19615i	0.118678	−	0.205557i
$640$		0			0
$641$	−24.0000	−	41.5692i	−0.947943	−	1.64189i	−0.749749	−	0.661723i	$-0.769826\pi$
$641$	−24.0000	−	41.5692i	−0.947943	−	1.64189i	−0.198194	−	0.980163i	$-0.563508\pi$
$642$		0			0
$643$	−41.0000			−1.61688			−0.808441	−	0.588577i	$-0.799688\pi$
$643$	−41.0000			−1.61688			−0.808441	+	0.588577i	$0.799688\pi$
$644$	−1.00000	−	5.19615i	−0.0394055	−	0.204757i
$645$		0			0
$646$		0			0
$647$	15.0000	+	25.9808i	0.589711	+	1.02141i	0.994270	+	0.106897i	$0.0340916\pi$
$647$	15.0000	+	25.9808i	0.589711	+	1.02141i	−0.404559	+	0.914512i	$0.632575\pi$
$648$		0			0
$649$	−6.00000	+	10.3923i	−0.235521	+	0.407934i
$650$		0			0
$651$	7.50000	+	2.59808i	0.293948	+	0.101827i
$652$	40.0000			1.56652
$653$	13.0000	−	22.5167i	0.508729	−	0.881145i	−0.491220	−	0.871036i	$-0.663449\pi$
$653$	13.0000	−	22.5167i	0.508729	−	0.881145i	0.999949	−	0.0101092i	$-0.00321793\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	1.00000			0.0390137
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	5.50000	+	9.52628i	0.213925	+	0.370529i	0.952940	−	0.303160i	$-0.0980418\pi$
$661$	5.50000	+	9.52628i	0.213925	+	0.370529i	−0.739014	+	0.673690i	$0.764708\pi$
$662$		0			0
$663$	3.00000	−	5.19615i	0.116510	−	0.201802i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−3.00000	+	5.19615i	−0.116160	+	0.201196i
$668$	16.0000	+	27.7128i	0.619059	+	1.07224i
$669$	−8.00000	−	13.8564i	−0.309298	−	0.535720i
$670$		0			0
$671$	4.00000			0.154418
$672$		0			0
$673$	7.00000			0.269830			0.134915	−	0.990857i	$-0.456924\pi$
$673$	7.00000			0.269830			0.134915	+	0.990857i	$0.456924\pi$
$674$		0			0
$675$	2.50000	+	4.33013i	0.0962250	+	0.166667i
$676$	−4.00000	−	6.92820i	−0.153846	−	0.266469i
$677$	−11.0000	+	19.0526i	−0.422764	+	0.732249i	−0.996209	−	0.0869952i	$-0.972274\pi$
$677$	−11.0000	+	19.0526i	−0.422764	+	0.732249i	0.573444	+	0.819244i	$0.305607\pi$
$678$		0			0
$679$	−5.00000	−	25.9808i	−0.191882	−	0.997050i
$680$		0			0
$681$	−1.00000	+	1.73205i	−0.0383201	+	0.0663723i
$682$		0			0
$683$	2.00000	+	3.46410i	0.0765279	+	0.132550i	0.901750	−	0.432259i	$-0.142283\pi$
$683$	2.00000	+	3.46410i	0.0765279	+	0.132550i	−0.825222	+	0.564809i	$0.808950\pi$
$684$	3.00000	−	5.19615i	0.114708	−	0.198680i
$685$		0			0
$686$		0			0
$687$	15.0000			0.572286
$688$	−6.00000	+	10.3923i	−0.228748	+	0.396203i
$689$	−6.00000	−	10.3923i	−0.228582	−	0.395915i
$690$		0			0
$691$	−21.5000	+	37.2391i	−0.817899	+	1.41664i	0.0893292	+	0.996002i	$0.471528\pi$
$691$	−21.5000	+	37.2391i	−0.817899	+	1.41664i	−0.907228	+	0.420640i	$0.861806\pi$
$692$	−12.0000			−0.456172
$693$	−1.00000	−	5.19615i	−0.0379869	−	0.197386i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$	2.00000			0.0756469
$700$	−25.0000	−	8.66025i	−0.944911	−	0.327327i
$701$	6.00000			0.226617			0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000			0.226617			0.113308	+	0.993560i	$0.463855\pi$
$702$		0			0
$703$	4.50000	+	7.79423i	0.169721	+	0.293965i
$704$	−8.00000	−	13.8564i	−0.301511	−	0.522233i
$705$		0			0
$706$		0			0
$707$	32.0000	−	27.7128i	1.20348	−	1.04225i
$708$	12.0000			0.450988
$709$	5.00000	−	8.66025i	0.187779	−	0.325243i	−0.756730	−	0.653727i	$-0.773204\pi$
$709$	5.00000	−	8.66025i	0.187779	−	0.325243i	0.944509	+	0.328484i	$0.106538\pi$
$710$		0			0
$711$	3.50000	+	6.06218i	0.131260	+	0.227349i
$712$		0			0
$713$	−3.00000			−0.112351
$714$		0			0
$715$		0			0
$716$	−20.0000	+	34.6410i	−0.747435	+	1.29460i
$717$	12.0000	+	20.7846i	0.448148	+	0.776215i
$718$		0			0
$719$	−13.0000	+	22.5167i	−0.484818	+	0.839730i	−0.999848	−	0.0174426i	$-0.994448\pi$
$719$	−13.0000	+	22.5167i	−0.484818	+	0.839730i	0.515030	+	0.857172i	$0.327781\pi$
$720$		0			0
$721$	2.50000	+	0.866025i	0.0931049	+	0.0322525i
$722$		0			0
$723$	7.00000	−	12.1244i	0.260333	−	0.450910i
$724$	−5.00000	−	8.66025i	−0.185824	−	0.321856i
$725$	15.0000	+	25.9808i	0.557086	+	0.964901i
$726$		0			0
$727$	1.00000			0.0370879			0.0185440	−	0.999828i	$-0.494097\pi$
$727$	1.00000			0.0370879			0.0185440	+	0.999828i	$0.494097\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−3.00000	−	5.19615i	−0.110959	−	0.192187i
$732$	−2.00000	−	3.46410i	−0.0739221	−	0.128037i
$733$	−17.5000	+	30.3109i	−0.646377	+	1.11956i	0.337604	+	0.941288i	$0.390383\pi$
$733$	−17.5000	+	30.3109i	−0.646377	+	1.11956i	−0.983982	+	0.178270i	$0.942950\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1.00000	−	1.73205i	0.0368355	−	0.0638009i
$738$		0			0
$739$	−14.5000	−	25.1147i	−0.533391	−	0.923861i	−0.999239	−	0.0389959i	$-0.987584\pi$
$739$	−14.5000	−	25.1147i	−0.533391	−	0.923861i	0.465848	−	0.884865i	$-0.345749\pi$
$740$		0			0
$741$	−9.00000			−0.330623
$742$		0			0
$743$	−8.00000			−0.293492			−0.146746	−	0.989174i	$-0.546880\pi$
$743$	−8.00000			−0.293492			−0.146746	+	0.989174i	$0.546880\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−2.00000	+	3.46410i	−0.0731762	+	0.126745i
$748$	8.00000			0.292509
$749$	40.0000	+	13.8564i	1.46157	+	0.506302i
$750$		0			0
$751$	−5.50000	+	9.52628i	−0.200698	+	0.347619i	−0.948753	−	0.316017i	$-0.897654\pi$
$751$	−5.50000	+	9.52628i	−0.200698	+	0.347619i	0.748056	+	0.663636i	$0.230988\pi$
$752$	20.0000	+	34.6410i	0.729325	+	1.26323i
$753$	13.0000	+	22.5167i	0.473746	+	0.820553i
$754$		0			0
$755$		0			0
$756$	−4.00000	+	3.46410i	−0.145479	+	0.125988i
$757$	−46.0000			−1.67190			−0.835949	−	0.548807i	$-0.815082\pi$
$757$	−46.0000			−1.67190			−0.835949	+	0.548807i	$0.815082\pi$
$758$		0			0
$759$	1.00000	+	1.73205i	0.0362977	+	0.0628695i
$760$		0			0
$761$	−15.0000	+	25.9808i	−0.543750	+	0.941802i	0.454935	+	0.890525i	$0.349663\pi$
$761$	−15.0000	+	25.9808i	−0.543750	+	0.941802i	−0.998684	+	0.0512772i	$0.983671\pi$
$762$		0			0
$763$	−18.0000	+	15.5885i	−0.651644	+	0.564340i
$764$	12.0000			0.434145
$765$		0			0
$766$		0			0
$767$	−9.00000	−	15.5885i	−0.324971	−	0.562867i
$768$	−8.00000	+	13.8564i	−0.288675	+	0.500000i
$769$	29.0000			1.04577			0.522883	−	0.852404i	$-0.324856\pi$
$769$	29.0000			1.04577			0.522883	+	0.852404i	$0.324856\pi$
$770$		0			0
$771$	−20.0000			−0.720282
$772$	−5.00000	+	8.66025i	−0.179954	+	0.311689i
$773$	−5.00000	−	8.66025i	−0.179838	−	0.311488i	0.761987	−	0.647592i	$-0.224224\pi$
$773$	−5.00000	−	8.66025i	−0.179838	−	0.311488i	−0.941825	+	0.336104i	$0.890891\pi$
$774$		0			0
$775$	−7.50000	+	12.9904i	−0.269408	+	0.466628i
$776$		0			0
$777$	−1.50000	−	7.79423i	−0.0538122	−	0.279616i
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−6.00000	−	10.3923i	−0.214697	−	0.371866i
$782$		0			0
$783$	6.00000			0.214423
$784$	4.00000	−	27.7128i	0.142857	−	0.989743i
$785$		0			0
$786$		0			0
$787$	−4.00000	−	6.92820i	−0.142585	−	0.246964i	0.785885	−	0.618373i	$-0.212208\pi$
$787$	−4.00000	−	6.92820i	−0.142585	−	0.246964i	−0.928469	+	0.371409i	$0.878875\pi$
$788$	4.00000	+	6.92820i	0.142494	+	0.246807i
$789$		0

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:	\( N \)	\(=\)	\( 483 = 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	483.i (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{3} +(1.00000 + 1.73205i) q^{4} +(0.500000 + 2.59808i) q^{7} +(-0.500000 + 0.866025i) q^{9} +(1.00000 + 1.73205i) q^{11} +(1.00000 - 1.73205i) q^{12} -3.00000 q^{13} +(-2.00000 + 3.46410i) q^{16} +(-1.00000 - 1.73205i) q^{17} +(-1.50000 + 2.59808i) q^{19} +(2.00000 - 1.73205i) q^{21} +(-0.500000 + 0.866025i) q^{23} +(2.50000 + 4.33013i) q^{25} +1.00000 q^{27} +(-4.00000 + 3.46410i) q^{28} +6.00000 q^{29} +(1.50000 + 2.59808i) q^{31} +(1.00000 - 1.73205i) q^{33} -2.00000 q^{36} +(1.50000 - 2.59808i) q^{37} +(1.50000 + 2.59808i) q^{39} +3.00000 q^{43} +(-2.00000 + 3.46410i) q^{44} +(5.00000 - 8.66025i) q^{47} +4.00000 q^{48} +(-6.50000 + 2.59808i) q^{49} +(-1.00000 + 1.73205i) q^{51} +(-3.00000 - 5.19615i) q^{52} +(2.00000 + 3.46410i) q^{53} +3.00000 q^{57} +(3.00000 + 5.19615i) q^{59} +(1.00000 - 1.73205i) q^{61} +(-2.50000 - 0.866025i) q^{63} -8.00000 q^{64} +(-0.500000 - 0.866025i) q^{67} +(2.00000 - 3.46410i) q^{68} +1.00000 q^{69} -6.00000 q^{71} +(-0.500000 - 0.866025i) q^{73} +(2.50000 - 4.33013i) q^{75} -6.00000 q^{76} +(-4.00000 + 3.46410i) q^{77} +(3.50000 - 6.06218i) q^{79} +(-0.500000 - 0.866025i) q^{81} +4.00000 q^{83} +(5.00000 + 1.73205i) q^{84} +(-3.00000 - 5.19615i) q^{87} +(7.00000 - 12.1244i) q^{89} +(-1.50000 - 7.79423i) q^{91} -2.00000 q^{92} +(1.50000 - 2.59808i) q^{93} -10.0000 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + 2 q^{4} + q^{7} - q^{9} + 2 q^{11} + 2 q^{12} - 6 q^{13} - 4 q^{16} - 2 q^{17} - 3 q^{19} + 4 q^{21} - q^{23} + 5 q^{25} + 2 q^{27} - 8 q^{28} + 12 q^{29} + 3 q^{31} + 2 q^{33} - 4 q^{36}+ \cdots - 4 q^{99}+O(q^{100}) \) 2 * q - q^3 + 2 * q^4 + q^7 - q^9 + 2 * q^11 + 2 * q^12 - 6 * q^13 - 4 * q^16 - 2 * q^17 - 3 * q^19 + 4 * q^21 - q^23 + 5 * q^25 + 2 * q^27 - 8 * q^28 + 12 * q^29 + 3 * q^31 + 2 * q^33 - 4 * q^36 + 3 * q^37 + 3 * q^39 + 6 * q^43 - 4 * q^44 + 10 * q^47 + 8 * q^48 - 13 * q^49 - 2 * q^51 - 6 * q^52 + 4 * q^53 + 6 * q^57 + 6 * q^59 + 2 * q^61 - 5 * q^63 - 16 * q^64 - q^67 + 4 * q^68 + 2 * q^69 - 12 * q^71 - q^73 + 5 * q^75 - 12 * q^76 - 8 * q^77 + 7 * q^79 - q^81 + 8 * q^83 + 10 * q^84 - 6 * q^87 + 14 * q^89 - 3 * q^91 - 4 * q^92 + 3 * q^93 - 20 * q^97 - 4 * q^99

Introduction

L-functions

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