LMFDB - Embedded newform 147.2.e.b.67.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [147,2,Mod(67,147)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(147, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("147.67");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 147 = 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	147.e (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:	$1.17380090971$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 21)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			67.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	147.67
Dual form			147.2.e.b.79.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{3} +(0.500000 - 0.866025i) q^{4} +(1.00000 + 1.73205i) q^{5} -1.00000 q^{6} +3.00000 q^{8} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})$	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{3} +(0.500000 - 0.866025i) q^{4} +(1.00000 + 1.73205i) q^{5} -1.00000 q^{6} +3.00000 q^{8} +(-0.500000 - 0.866025i) q^{9} +(-1.00000 + 1.73205i) q^{10} +(-2.00000 + 3.46410i) q^{11} +(0.500000 + 0.866025i) q^{12} -2.00000 q^{13} -2.00000 q^{15} +(0.500000 + 0.866025i) q^{16} +(3.00000 - 5.19615i) q^{17} +(0.500000 - 0.866025i) q^{18} +(-2.00000 - 3.46410i) q^{19} +2.00000 q^{20} -4.00000 q^{22} +(-1.50000 + 2.59808i) q^{24} +(0.500000 - 0.866025i) q^{25} +(-1.00000 - 1.73205i) q^{26} +1.00000 q^{27} -2.00000 q^{29} +(-1.00000 - 1.73205i) q^{30} +(2.50000 - 4.33013i) q^{32} +(-2.00000 - 3.46410i) q^{33} +6.00000 q^{34} -1.00000 q^{36} +(-3.00000 - 5.19615i) q^{37} +(2.00000 - 3.46410i) q^{38} +(1.00000 - 1.73205i) q^{39} +(3.00000 + 5.19615i) q^{40} +2.00000 q^{41} -4.00000 q^{43} +(2.00000 + 3.46410i) q^{44} +(1.00000 - 1.73205i) q^{45} -1.00000 q^{48} +1.00000 q^{50} +(3.00000 + 5.19615i) q^{51} +(-1.00000 + 1.73205i) q^{52} +(-3.00000 + 5.19615i) q^{53} +(0.500000 + 0.866025i) q^{54} -8.00000 q^{55} +4.00000 q^{57} +(-1.00000 - 1.73205i) q^{58} +(-6.00000 + 10.3923i) q^{59} +(-1.00000 + 1.73205i) q^{60} +(1.00000 + 1.73205i) q^{61} +7.00000 q^{64} +(-2.00000 - 3.46410i) q^{65} +(2.00000 - 3.46410i) q^{66} +(-2.00000 + 3.46410i) q^{67} +(-3.00000 - 5.19615i) q^{68} +(-1.50000 - 2.59808i) q^{72} +(3.00000 - 5.19615i) q^{73} +(3.00000 - 5.19615i) q^{74} +(0.500000 + 0.866025i) q^{75} -4.00000 q^{76} +2.00000 q^{78} +(8.00000 + 13.8564i) q^{79} +(-1.00000 + 1.73205i) q^{80} +(-0.500000 + 0.866025i) q^{81} +(1.00000 + 1.73205i) q^{82} -12.0000 q^{83} +12.0000 q^{85} +(-2.00000 - 3.46410i) q^{86} +(1.00000 - 1.73205i) q^{87} +(-6.00000 + 10.3923i) q^{88} +(7.00000 + 12.1244i) q^{89} +2.00000 q^{90} +(4.00000 - 6.92820i) q^{95} +(2.50000 + 4.33013i) q^{96} +18.0000 q^{97} +4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - q^{3} + q^{4} + 2 q^{5} - 2 q^{6} + 6 q^{8} - q^{9}+O(q^{10}) $ 2 * q + q^2 - q^3 + q^4 + 2 * q^5 - 2 * q^6 + 6 * q^8 - q^9	$ 2 q + q^{2} - q^{3} + q^{4} + 2 q^{5} - 2 q^{6} + 6 q^{8} - q^{9} - 2 q^{10} - 4 q^{11} + q^{12} - 4 q^{13} - 4 q^{15} + q^{16} + 6 q^{17} + q^{18} - 4 q^{19} + 4 q^{20} - 8 q^{22} - 3 q^{24} + q^{25} - 2 q^{26} + 2 q^{27} - 4 q^{29} - 2 q^{30} + 5 q^{32} - 4 q^{33} + 12 q^{34} - 2 q^{36} - 6 q^{37} + 4 q^{38} + 2 q^{39} + 6 q^{40} + 4 q^{41} - 8 q^{43} + 4 q^{44} + 2 q^{45} - 2 q^{48} + 2 q^{50} + 6 q^{51} - 2 q^{52} - 6 q^{53} + q^{54} - 16 q^{55} + 8 q^{57} - 2 q^{58} - 12 q^{59} - 2 q^{60} + 2 q^{61} + 14 q^{64} - 4 q^{65} + 4 q^{66} - 4 q^{67} - 6 q^{68} - 3 q^{72} + 6 q^{73} + 6 q^{74} + q^{75} - 8 q^{76} + 4 q^{78} + 16 q^{79} - 2 q^{80} - q^{81} + 2 q^{82} - 24 q^{83} + 24 q^{85} - 4 q^{86} + 2 q^{87} - 12 q^{88} + 14 q^{89} + 4 q^{90} + 8 q^{95} + 5 q^{96} + 36 q^{97} + 8 q^{99}+O(q^{100}) $ 2 * q + q^2 - q^3 + q^4 + 2 * q^5 - 2 * q^6 + 6 * q^8 - q^9 - 2 * q^10 - 4 * q^11 + q^12 - 4 * q^13 - 4 * q^15 + q^16 + 6 * q^17 + q^18 - 4 * q^19 + 4 * q^20 - 8 * q^22 - 3 * q^24 + q^25 - 2 * q^26 + 2 * q^27 - 4 * q^29 - 2 * q^30 + 5 * q^32 - 4 * q^33 + 12 * q^34 - 2 * q^36 - 6 * q^37 + 4 * q^38 + 2 * q^39 + 6 * q^40 + 4 * q^41 - 8 * q^43 + 4 * q^44 + 2 * q^45 - 2 * q^48 + 2 * q^50 + 6 * q^51 - 2 * q^52 - 6 * q^53 + q^54 - 16 * q^55 + 8 * q^57 - 2 * q^58 - 12 * q^59 - 2 * q^60 + 2 * q^61 + 14 * q^64 - 4 * q^65 + 4 * q^66 - 4 * q^67 - 6 * q^68 - 3 * q^72 + 6 * q^73 + 6 * q^74 + q^75 - 8 * q^76 + 4 * q^78 + 16 * q^79 - 2 * q^80 - q^81 + 2 * q^82 - 24 * q^83 + 24 * q^85 - 4 * q^86 + 2 * q^87 - 12 * q^88 + 14 * q^89 + 4 * q^90 + 8 * q^95 + 5 * q^96 + 36 * q^97 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/147\mathbb{Z}\right)^\times$.

$n$	$50$	$52$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$

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 $

$2$	0.500000	+	0.866025i	0.353553	+	0.612372i	0.986869	−	0.161521i	$-0.0516399\pi$
$2$	0.500000	+	0.866025i	0.353553	+	0.612372i	−0.633316	+	0.773893i	$0.718307\pi$
$3$	−0.500000	+	0.866025i	−0.288675	+	0.500000i
$3$	−0.500000	+	0.866025i	−0.288675	+	0.500000i
$4$	0.500000	−	0.866025i	0.250000	−	0.433013i
$5$	1.00000	+	1.73205i	0.447214	+	0.774597i	0.998203	−	0.0599153i	$-0.0190830\pi$
$5$	1.00000	+	1.73205i	0.447214	+	0.774597i	−0.550990	+	0.834512i	$0.685750\pi$
$6$	−1.00000			−0.408248
$7$		0			0
$7$		0			0
$8$	3.00000			1.06066
$9$	−0.500000	−	0.866025i	−0.166667	−	0.288675i
$10$	−1.00000	+	1.73205i	−0.316228	+	0.547723i
$11$	−2.00000	+	3.46410i	−0.603023	+	1.04447i	0.389338	+	0.921095i	$0.372704\pi$
$11$	−2.00000	+	3.46410i	−0.603023	+	1.04447i	−0.992361	+	0.123371i	$0.960630\pi$
$12$	0.500000	+	0.866025i	0.144338	+	0.250000i
$13$	−2.00000			−0.554700			−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000			−0.554700			−0.277350	+	0.960769i	$0.589456\pi$
$14$		0			0
$15$	−2.00000			−0.516398
$16$	0.500000	+	0.866025i	0.125000	+	0.216506i
$17$	3.00000	−	5.19615i	0.727607	−	1.26025i	−0.230285	−	0.973123i	$-0.573966\pi$
$17$	3.00000	−	5.19615i	0.727607	−	1.26025i	0.957892	−	0.287129i	$-0.0927008\pi$
$18$	0.500000	−	0.866025i	0.117851	−	0.204124i
$19$	−2.00000	−	3.46410i	−0.458831	−	0.794719i	0.540068	−	0.841621i	$-0.318398\pi$
$19$	−2.00000	−	3.46410i	−0.458831	−	0.794719i	−0.998899	+	0.0469020i	$0.985065\pi$
$20$	2.00000			0.447214
$21$		0			0
$22$	−4.00000			−0.852803
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$	−1.50000	+	2.59808i	−0.306186	+	0.530330i
$25$	0.500000	−	0.866025i	0.100000	−	0.173205i
$26$	−1.00000	−	1.73205i	−0.196116	−	0.339683i
$27$	1.00000			0.192450
$28$		0			0
$29$	−2.00000			−0.371391			−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000			−0.371391			−0.185695	+	0.982607i	$0.559454\pi$
$30$	−1.00000	−	1.73205i	−0.182574	−	0.316228i
$31$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$31$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$32$	2.50000	−	4.33013i	0.441942	−	0.765466i
$33$	−2.00000	−	3.46410i	−0.348155	−	0.603023i
$34$	6.00000			1.02899
$35$		0			0
$36$	−1.00000			−0.166667
$37$	−3.00000	−	5.19615i	−0.493197	−	0.854242i	0.506772	−	0.862080i	$-0.330838\pi$
$37$	−3.00000	−	5.19615i	−0.493197	−	0.854242i	−0.999969	+	0.00783774i	$0.997505\pi$
$38$	2.00000	−	3.46410i	0.324443	−	0.561951i
$39$	1.00000	−	1.73205i	0.160128	−	0.277350i
$40$	3.00000	+	5.19615i	0.474342	+	0.821584i
$41$	2.00000			0.312348			0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000			0.312348			0.156174	+	0.987730i	$0.450084\pi$
$42$		0			0
$43$	−4.00000			−0.609994			−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000			−0.609994			−0.304997	+	0.952353i	$0.598656\pi$
$44$	2.00000	+	3.46410i	0.301511	+	0.522233i
$45$	1.00000	−	1.73205i	0.149071	−	0.258199i
$46$		0			0
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$	−1.00000			−0.144338
$49$		0			0
$50$	1.00000			0.141421
$51$	3.00000	+	5.19615i	0.420084	+	0.727607i
$52$	−1.00000	+	1.73205i	−0.138675	+	0.240192i
$53$	−3.00000	+	5.19615i	−0.412082	+	0.713746i	−0.995117	−	0.0987002i	$-0.968532\pi$
$53$	−3.00000	+	5.19615i	−0.412082	+	0.713746i	0.583036	+	0.812447i	$0.301865\pi$
$54$	0.500000	+	0.866025i	0.0680414	+	0.117851i
$55$	−8.00000			−1.07872
$56$		0			0
$57$	4.00000			0.529813
$58$	−1.00000	−	1.73205i	−0.131306	−	0.227429i
$59$	−6.00000	+	10.3923i	−0.781133	+	1.35296i	0.150148	+	0.988663i	$0.452025\pi$
$59$	−6.00000	+	10.3923i	−0.781133	+	1.35296i	−0.931282	+	0.364299i	$0.881308\pi$
$60$	−1.00000	+	1.73205i	−0.129099	+	0.223607i
$61$	1.00000	+	1.73205i	0.128037	+	0.221766i	0.922916	−	0.385002i	$-0.125799\pi$
$61$	1.00000	+	1.73205i	0.128037	+	0.221766i	−0.794879	+	0.606768i	$0.792466\pi$
$62$		0			0
$63$		0			0
$64$	7.00000			0.875000
$65$	−2.00000	−	3.46410i	−0.248069	−	0.429669i
$66$	2.00000	−	3.46410i	0.246183	−	0.426401i
$67$	−2.00000	+	3.46410i	−0.244339	+	0.423207i	−0.961946	−	0.273241i	$-0.911904\pi$
$67$	−2.00000	+	3.46410i	−0.244339	+	0.423207i	0.717607	+	0.696449i	$0.245238\pi$
$68$	−3.00000	−	5.19615i	−0.363803	−	0.630126i
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	−1.50000	−	2.59808i	−0.176777	−	0.306186i
$73$	3.00000	−	5.19615i	0.351123	−	0.608164i	−0.635323	−	0.772246i	$-0.719133\pi$
$73$	3.00000	−	5.19615i	0.351123	−	0.608164i	0.986447	+	0.164083i	$0.0524664\pi$
$74$	3.00000	−	5.19615i	0.348743	−	0.604040i
$75$	0.500000	+	0.866025i	0.0577350	+	0.100000i
$76$	−4.00000			−0.458831
$77$		0			0
$78$	2.00000			0.226455
$79$	8.00000	+	13.8564i	0.900070	+	1.55897i	0.827401	+	0.561611i	$0.189818\pi$
$79$	8.00000	+	13.8564i	0.900070	+	1.55897i	0.0726692	+	0.997356i	$0.476848\pi$
$80$	−1.00000	+	1.73205i	−0.111803	+	0.193649i
$81$	−0.500000	+	0.866025i	−0.0555556	+	0.0962250i
$82$	1.00000	+	1.73205i	0.110432	+	0.191273i
$83$	−12.0000			−1.31717			−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000			−1.31717			−0.658586	+	0.752506i	$0.728845\pi$
$84$		0			0
$85$	12.0000			1.30158
$86$	−2.00000	−	3.46410i	−0.215666	−	0.373544i
$87$	1.00000	−	1.73205i	0.107211	−	0.185695i
$88$	−6.00000	+	10.3923i	−0.639602	+	1.10782i
$89$	7.00000	+	12.1244i	0.741999	+	1.28518i	0.951584	+	0.307389i	$0.0994552\pi$
$89$	7.00000	+	12.1244i	0.741999	+	1.28518i	−0.209585	+	0.977790i	$0.567211\pi$
$90$	2.00000			0.210819
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	4.00000	−	6.92820i	0.410391	−	0.710819i
$96$	2.50000	+	4.33013i	0.255155	+	0.441942i
$97$	18.0000			1.82762			0.913812	−	0.406138i	$-0.133125\pi$
$97$	18.0000			1.82762			0.913812	+	0.406138i	$0.133125\pi$
$98$		0			0
$99$	4.00000			0.402015
$100$	−0.500000	−	0.866025i	−0.0500000	−	0.0866025i
$101$	−7.00000	+	12.1244i	−0.696526	+	1.20642i	0.273138	+	0.961975i	$0.411939\pi$
$101$	−7.00000	+	12.1244i	−0.696526	+	1.20642i	−0.969664	+	0.244443i	$0.921395\pi$
$102$	−3.00000	+	5.19615i	−0.297044	+	0.514496i
$103$	−4.00000	−	6.92820i	−0.394132	−	0.682656i	0.598858	−	0.800855i	$-0.295621\pi$
$103$	−4.00000	−	6.92820i	−0.394132	−	0.682656i	−0.992990	+	0.118199i	$0.962288\pi$
$104$	−6.00000			−0.588348
$105$		0			0
$106$	−6.00000			−0.582772
$107$	−2.00000	−	3.46410i	−0.193347	−	0.334887i	0.753010	−	0.658009i	$-0.228601\pi$
$107$	−2.00000	−	3.46410i	−0.193347	−	0.334887i	−0.946357	+	0.323122i	$0.895268\pi$
$108$	0.500000	−	0.866025i	0.0481125	−	0.0833333i
$109$	9.00000	−	15.5885i	0.862044	−	1.49310i	−0.00790932	−	0.999969i	$-0.502518\pi$
$109$	9.00000	−	15.5885i	0.862044	−	1.49310i	0.869953	−	0.493135i	$-0.164149\pi$
$110$	−4.00000	−	6.92820i	−0.381385	−	0.660578i
$111$	6.00000			0.569495
$112$		0			0
$113$	−14.0000			−1.31701			−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000			−1.31701			−0.658505	+	0.752577i	$0.728811\pi$
$114$	2.00000	+	3.46410i	0.187317	+	0.324443i
$115$		0			0
$116$	−1.00000	+	1.73205i	−0.0928477	+	0.160817i
$117$	1.00000	+	1.73205i	0.0924500	+	0.160128i
$118$	−12.0000			−1.10469
$119$		0			0
$120$	−6.00000			−0.547723
$121$	−2.50000	−	4.33013i	−0.227273	−	0.393648i
$122$	−1.00000	+	1.73205i	−0.0905357	+	0.156813i
$123$	−1.00000	+	1.73205i	−0.0901670	+	0.156174i
$124$		0			0
$125$	12.0000			1.07331
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$	−1.50000	−	2.59808i	−0.132583	−	0.229640i
$129$	2.00000	−	3.46410i	0.176090	−	0.304997i
$130$	2.00000	−	3.46410i	0.175412	−	0.303822i
$131$	−2.00000	−	3.46410i	−0.174741	−	0.302660i	0.765331	−	0.643637i	$-0.222575\pi$
$131$	−2.00000	−	3.46410i	−0.174741	−	0.302660i	−0.940072	+	0.340977i	$0.889242\pi$
$132$	−4.00000			−0.348155
$133$		0			0
$134$	−4.00000			−0.345547
$135$	1.00000	+	1.73205i	0.0860663	+	0.149071i
$136$	9.00000	−	15.5885i	0.771744	−	1.33670i
$137$	3.00000	−	5.19615i	0.256307	−	0.443937i	−0.708942	−	0.705266i	$-0.750827\pi$
$137$	3.00000	−	5.19615i	0.256307	−	0.443937i	0.965250	+	0.261329i	$0.0841608\pi$
$138$		0			0
$139$	12.0000			1.01783			0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000			1.01783			0.508913	+	0.860818i	$0.330047\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	4.00000	−	6.92820i	0.334497	−	0.579365i
$144$	0.500000	−	0.866025i	0.0416667	−	0.0721688i
$145$	−2.00000	−	3.46410i	−0.166091	−	0.287678i
$146$	6.00000			0.496564
$147$		0			0
$148$	−6.00000			−0.493197
$149$	−3.00000	−	5.19615i	−0.245770	−	0.425685i	0.716578	−	0.697507i	$-0.245707\pi$
$149$	−3.00000	−	5.19615i	−0.245770	−	0.425685i	−0.962348	+	0.271821i	$0.912374\pi$
$150$	−0.500000	+	0.866025i	−0.0408248	+	0.0707107i
$151$	−4.00000	+	6.92820i	−0.325515	+	0.563809i	−0.981617	−	0.190864i	$-0.938871\pi$
$151$	−4.00000	+	6.92820i	−0.325515	+	0.563809i	0.656101	+	0.754673i	$0.272204\pi$
$152$	−6.00000	−	10.3923i	−0.486664	−	0.842927i
$153$	−6.00000			−0.485071
$154$		0			0
$155$		0			0
$156$	−1.00000	−	1.73205i	−0.0800641	−	0.138675i
$157$	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	$0.141236\pi$
$158$	−8.00000	+	13.8564i	−0.636446	+	1.10236i
$159$	−3.00000	−	5.19615i	−0.237915	−	0.412082i
$160$	10.0000			0.790569
$161$		0			0
$162$	−1.00000			−0.0785674
$163$	−2.00000	−	3.46410i	−0.156652	−	0.271329i	0.777007	−	0.629492i	$-0.216737\pi$
$163$	−2.00000	−	3.46410i	−0.156652	−	0.271329i	−0.933659	+	0.358162i	$0.883403\pi$
$164$	1.00000	−	1.73205i	0.0780869	−	0.135250i
$165$	4.00000	−	6.92820i	0.311400	−	0.539360i
$166$	−6.00000	−	10.3923i	−0.465690	−	0.806599i
$167$	−8.00000			−0.619059			−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000			−0.619059			−0.309529	+	0.950890i	$0.600171\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$	6.00000	+	10.3923i	0.460179	+	0.797053i
$171$	−2.00000	+	3.46410i	−0.152944	+	0.264906i
$172$	−2.00000	+	3.46410i	−0.152499	+	0.264135i
$173$	5.00000	+	8.66025i	0.380143	+	0.658427i	0.991082	−	0.133250i	$-0.0425415\pi$
$173$	5.00000	+	8.66025i	0.380143	+	0.658427i	−0.610939	+	0.791677i	$0.709208\pi$
$174$	2.00000			0.151620
$175$		0			0
$176$	−4.00000			−0.301511
$177$	−6.00000	−	10.3923i	−0.450988	−	0.781133i
$178$	−7.00000	+	12.1244i	−0.524672	+	0.908759i
$179$	2.00000	−	3.46410i	0.149487	−	0.258919i	−0.781551	−	0.623841i	$-0.785571\pi$
$179$	2.00000	−	3.46410i	0.149487	−	0.258919i	0.931038	+	0.364922i	$0.118904\pi$
$180$	−1.00000	−	1.73205i	−0.0745356	−	0.129099i
$181$	−26.0000			−1.93256			−0.966282	−	0.257485i	$-0.917106\pi$
$181$	−26.0000			−1.93256			−0.966282	+	0.257485i	$0.917106\pi$
$182$		0			0
$183$	−2.00000			−0.147844
$184$		0			0
$185$	6.00000	−	10.3923i	0.441129	−	0.764057i
$186$		0			0
$187$	12.0000	+	20.7846i	0.877527	+	1.51992i
$188$		0			0
$189$		0			0
$190$	8.00000			0.580381
$191$	4.00000	+	6.92820i	0.289430	+	0.501307i	0.973674	−	0.227946i	$-0.0732010\pi$
$191$	4.00000	+	6.92820i	0.289430	+	0.501307i	−0.684244	+	0.729253i	$0.739868\pi$
$192$	−3.50000	+	6.06218i	−0.252591	+	0.437500i
$193$	−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	$0.189599\pi$
$194$	9.00000	+	15.5885i	0.646162	+	1.11919i
$195$	4.00000			0.286446
$196$		0			0
$197$	22.0000			1.56744			0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000			1.56744			0.783718	+	0.621117i	$0.213321\pi$
$198$	2.00000	+	3.46410i	0.142134	+	0.246183i
$199$	−12.0000	+	20.7846i	−0.850657	+	1.47338i	0.0299585	+	0.999551i	$0.490462\pi$
$199$	−12.0000	+	20.7846i	−0.850657	+	1.47338i	−0.880616	+	0.473831i	$0.842871\pi$
$200$	1.50000	−	2.59808i	0.106066	−	0.183712i
$201$	−2.00000	−	3.46410i	−0.141069	−	0.244339i
$202$	−14.0000			−0.985037
$203$		0			0
$204$	6.00000			0.420084
$205$	2.00000	+	3.46410i	0.139686	+	0.241943i
$206$	4.00000	−	6.92820i	0.278693	−	0.482711i
$207$		0			0
$208$	−1.00000	−	1.73205i	−0.0693375	−	0.120096i
$209$	16.0000			1.10674
$210$		0			0
$211$	4.00000			0.275371			0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000			0.275371			0.137686	+	0.990476i	$0.456034\pi$
$212$	3.00000	+	5.19615i	0.206041	+	0.356873i
$213$		0			0
$214$	2.00000	−	3.46410i	0.136717	−	0.236801i
$215$	−4.00000	−	6.92820i	−0.272798	−	0.472500i
$216$	3.00000			0.204124
$217$		0			0
$218$	18.0000			1.21911
$219$	3.00000	+	5.19615i	0.202721	+	0.351123i
$220$	−4.00000	+	6.92820i	−0.269680	+	0.467099i
$221$	−6.00000	+	10.3923i	−0.403604	+	0.699062i
$222$	3.00000	+	5.19615i	0.201347	+	0.348743i
$223$	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
$225$	−1.00000			−0.0666667
$226$	−7.00000	−	12.1244i	−0.465633	−	0.806500i
$227$	6.00000	−	10.3923i	0.398234	−	0.689761i	−0.595274	−	0.803523i	$-0.702957\pi$
$227$	6.00000	−	10.3923i	0.398234	−	0.689761i	0.993508	+	0.113761i	$0.0362899\pi$
$228$	2.00000	−	3.46410i	0.132453	−	0.229416i
$229$	5.00000	+	8.66025i	0.330409	+	0.572286i	0.982592	−	0.185776i	$-0.0594799\pi$
$229$	5.00000	+	8.66025i	0.330409	+	0.572286i	−0.652183	+	0.758062i	$0.726147\pi$
$230$		0			0
$231$		0			0
$232$	−6.00000			−0.393919
$233$	3.00000	+	5.19615i	0.196537	+	0.340411i	0.947403	−	0.320043i	$-0.103697\pi$
$233$	3.00000	+	5.19615i	0.196537	+	0.340411i	−0.750867	+	0.660454i	$0.770364\pi$
$234$	−1.00000	+	1.73205i	−0.0653720	+	0.113228i
$235$		0			0
$236$	6.00000	+	10.3923i	0.390567	+	0.676481i
$237$	−16.0000			−1.03931
$238$		0			0
$239$	24.0000			1.55243			0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000			1.55243			0.776215	+	0.630468i	$0.217137\pi$
$240$	−1.00000	−	1.73205i	−0.0645497	−	0.111803i
$241$	−1.00000	+	1.73205i	−0.0644157	+	0.111571i	−0.896435	−	0.443176i	$-0.853852\pi$
$241$	−1.00000	+	1.73205i	−0.0644157	+	0.111571i	0.832019	+	0.554747i	$0.187185\pi$
$242$	2.50000	−	4.33013i	0.160706	−	0.278351i
$243$	−0.500000	−	0.866025i	−0.0320750	−	0.0555556i
$244$	2.00000			0.128037
$245$		0			0
$246$	−2.00000			−0.127515
$247$	4.00000	+	6.92820i	0.254514	+	0.440831i
$248$		0			0
$249$	6.00000	−	10.3923i	0.380235	−	0.658586i
$250$	6.00000	+	10.3923i	0.379473	+	0.657267i
$251$	−20.0000			−1.26239			−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000			−1.26239			−0.631194	+	0.775625i	$0.717435\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	−6.00000	+	10.3923i	−0.375735	+	0.650791i
$256$	8.50000	−	14.7224i	0.531250	−	0.920152i
$257$	−13.0000	−	22.5167i	−0.810918	−	1.40455i	−0.912222	−	0.409695i	$-0.865635\pi$
$257$	−13.0000	−	22.5167i	−0.810918	−	1.40455i	0.101305	−	0.994855i	$-0.467698\pi$
$258$	4.00000			0.249029
$259$		0			0
$260$	−4.00000			−0.248069
$261$	1.00000	+	1.73205i	0.0618984	+	0.107211i
$262$	2.00000	−	3.46410i	0.123560	−	0.214013i
$263$	−8.00000	+	13.8564i	−0.493301	+	0.854423i	−0.999970	−	0.00771799i	$-0.997543\pi$
$263$	−8.00000	+	13.8564i	−0.493301	+	0.854423i	0.506669	+	0.862141i	$0.330877\pi$
$264$	−6.00000	−	10.3923i	−0.369274	−	0.639602i
$265$	−12.0000			−0.737154
$266$		0			0
$267$	−14.0000			−0.856786
$268$	2.00000	+	3.46410i	0.122169	+	0.211604i
$269$	−3.00000	+	5.19615i	−0.182913	+	0.316815i	−0.942871	−	0.333157i	$-0.891886\pi$
$269$	−3.00000	+	5.19615i	−0.182913	+	0.316815i	0.759958	+	0.649972i	$0.225219\pi$
$270$	−1.00000	+	1.73205i	−0.0608581	+	0.105409i
$271$	−8.00000	−	13.8564i	−0.485965	−	0.841717i	0.513905	−	0.857847i	$-0.328199\pi$
$271$	−8.00000	−	13.8564i	−0.485965	−	0.841717i	−0.999870	+	0.0161307i	$0.994865\pi$
$272$	6.00000			0.363803
$273$		0			0
$274$	6.00000			0.362473
$275$	2.00000	+	3.46410i	0.120605	+	0.208893i
$276$		0			0
$277$	−11.0000	+	19.0526i	−0.660926	+	1.14476i	0.319447	+	0.947604i	$0.396503\pi$
$277$	−11.0000	+	19.0526i	−0.660926	+	1.14476i	−0.980373	+	0.197153i	$0.936830\pi$
$278$	6.00000	+	10.3923i	0.359856	+	0.623289i
$279$		0			0
$280$		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
$283$	10.0000	−	17.3205i	0.594438	−	1.02960i	−0.399188	−	0.916869i	$-0.630708\pi$
$283$	10.0000	−	17.3205i	0.594438	−	1.02960i	0.993626	−	0.112728i	$-0.0359589\pi$
$284$		0			0
$285$	4.00000	+	6.92820i	0.236940	+	0.410391i
$286$	8.00000			0.473050
$287$		0			0
$288$	−5.00000			−0.294628
$289$	−9.50000	−	16.4545i	−0.558824	−	0.967911i
$290$	2.00000	−	3.46410i	0.117444	−	0.203419i
$291$	−9.00000	+	15.5885i	−0.527589	+	0.913812i
$292$	−3.00000	−	5.19615i	−0.175562	−	0.304082i
$293$	14.0000			0.817889			0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000			0.817889			0.408944	+	0.912559i	$0.365897\pi$
$294$		0			0
$295$	−24.0000			−1.39733
$296$	−9.00000	−	15.5885i	−0.523114	−	0.906061i
$297$	−2.00000	+	3.46410i	−0.116052	+	0.201008i
$298$	3.00000	−	5.19615i	0.173785	−	0.301005i
$299$		0			0
$300$	1.00000			0.0577350
$301$		0			0
$302$	−8.00000			−0.460348
$303$	−7.00000	−	12.1244i	−0.402139	−	0.696526i
$304$	2.00000	−	3.46410i	0.114708	−	0.198680i
$305$	−2.00000	+	3.46410i	−0.114520	+	0.198354i
$306$	−3.00000	−	5.19615i	−0.171499	−	0.297044i
$307$	4.00000			0.228292			0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000			0.228292			0.114146	+	0.993464i	$0.463587\pi$
$308$		0			0
$309$	8.00000			0.455104
$310$		0			0
$311$	12.0000	−	20.7846i	0.680458	−	1.17859i	−0.294384	−	0.955687i	$-0.595114\pi$
$311$	12.0000	−	20.7846i	0.680458	−	1.17859i	0.974841	−	0.222900i	$-0.0715523\pi$
$312$	3.00000	−	5.19615i	0.169842	−	0.294174i
$313$	−13.0000	−	22.5167i	−0.734803	−	1.27272i	−0.954810	−	0.297218i	$-0.903941\pi$
$313$	−13.0000	−	22.5167i	−0.734803	−	1.27272i	0.220006	−	0.975499i	$-0.429392\pi$
$314$	2.00000			0.112867
$315$		0			0
$316$	16.0000			0.900070
$317$	9.00000	+	15.5885i	0.505490	+	0.875535i	0.999980	+	0.00635137i	$0.00202172\pi$
$317$	9.00000	+	15.5885i	0.505490	+	0.875535i	−0.494489	+	0.869184i	$0.664645\pi$
$318$	3.00000	−	5.19615i	0.168232	−	0.291386i
$319$	4.00000	−	6.92820i	0.223957	−	0.387905i
$320$	7.00000	+	12.1244i	0.391312	+	0.677772i
$321$	4.00000			0.223258
$322$		0			0
$323$	−24.0000			−1.33540
$324$	0.500000	+	0.866025i	0.0277778	+	0.0481125i
$325$	−1.00000	+	1.73205i	−0.0554700	+	0.0960769i
$326$	2.00000	−	3.46410i	0.110770	−	0.191859i
$327$	9.00000	+	15.5885i	0.497701	+	0.862044i
$328$	6.00000			0.331295
$329$		0			0
$330$	8.00000			0.440386
$331$	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	$0.798271\pi$
$332$	−6.00000	+	10.3923i	−0.329293	+	0.570352i
$333$	−3.00000	+	5.19615i	−0.164399	+	0.284747i
$334$	−4.00000	−	6.92820i	−0.218870	−	0.379094i
$335$	−8.00000			−0.437087
$336$		0			0
$337$	−14.0000			−0.762629			−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000			−0.762629			−0.381314	+	0.924445i	$0.624528\pi$
$338$	−4.50000	−	7.79423i	−0.244768	−	0.423950i
$339$	7.00000	−	12.1244i	0.380188	−	0.658505i
$340$	6.00000	−	10.3923i	0.325396	−	0.563602i
$341$		0			0
$342$	−4.00000			−0.216295
$343$		0			0
$344$	−12.0000			−0.646997
$345$		0			0
$346$	−5.00000	+	8.66025i	−0.268802	+	0.465578i
$347$	14.0000	−	24.2487i	0.751559	−	1.30174i	−0.195507	−	0.980702i	$-0.562635\pi$
$347$	14.0000	−	24.2487i	0.751559	−	1.30174i	0.947067	−	0.321037i	$-0.104031\pi$
$348$	−1.00000	−	1.73205i	−0.0536056	−	0.0928477i
$349$	−2.00000			−0.107058			−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000			−0.107058			−0.0535288	+	0.998566i	$0.517047\pi$
$350$		0			0
$351$	−2.00000			−0.106752
$352$	10.0000	+	17.3205i	0.533002	+	0.923186i
$353$	−5.00000	+	8.66025i	−0.266123	+	0.460939i	−0.967857	−	0.251500i	$-0.919076\pi$
$353$	−5.00000	+	8.66025i	−0.266123	+	0.460939i	0.701734	+	0.712439i	$0.252409\pi$
$354$	6.00000	−	10.3923i	0.318896	−	0.552345i
$355$		0			0
$356$	14.0000			0.741999
$357$		0			0
$358$	4.00000			0.211407
$359$	−16.0000	−	27.7128i	−0.844448	−	1.46263i	−0.886100	−	0.463494i	$-0.846596\pi$
$359$	−16.0000	−	27.7128i	−0.844448	−	1.46263i	0.0416523	−	0.999132i	$-0.486738\pi$
$360$	3.00000	−	5.19615i	0.158114	−	0.273861i
$361$	1.50000	−	2.59808i	0.0789474	−	0.136741i
$362$	−13.0000	−	22.5167i	−0.683265	−	1.18345i
$363$	5.00000			0.262432
$364$		0			0
$365$	12.0000			0.628109
$366$	−1.00000	−	1.73205i	−0.0522708	−	0.0905357i
$367$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$367$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$368$		0			0
$369$	−1.00000	−	1.73205i	−0.0520579	−	0.0901670i
$370$	12.0000			0.623850
$371$		0			0
$372$		0			0
$373$	5.00000	+	8.66025i	0.258890	+	0.448411i	0.965945	−	0.258748i	$-0.0833099\pi$
$373$	5.00000	+	8.66025i	0.258890	+	0.448411i	−0.707055	+	0.707159i	$0.749977\pi$
$374$	−12.0000	+	20.7846i	−0.620505	+	1.07475i
$375$	−6.00000	+	10.3923i	−0.309839	+	0.536656i
$376$		0			0
$377$	4.00000			0.206010
$378$		0			0
$379$	12.0000			0.616399			0.308199	−	0.951322i	$-0.400274\pi$
$379$	12.0000			0.616399			0.308199	+	0.951322i	$0.400274\pi$
$380$	−4.00000	−	6.92820i	−0.205196	−	0.355409i
$381$		0			0
$382$	−4.00000	+	6.92820i	−0.204658	+	0.354478i
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$	3.00000			0.153093
$385$		0			0
$386$	−2.00000			−0.101797
$387$	2.00000	+	3.46410i	0.101666	+	0.176090i
$388$	9.00000	−	15.5885i	0.456906	−	0.791384i
$389$	−3.00000	+	5.19615i	−0.152106	+	0.263455i	−0.932002	−	0.362454i	$-0.881939\pi$
$389$	−3.00000	+	5.19615i	−0.152106	+	0.263455i	0.779895	+	0.625910i	$0.215272\pi$
$390$	2.00000	+	3.46410i	0.101274	+	0.175412i
$391$		0			0
$392$		0			0
$393$	4.00000			0.201773
$394$	11.0000	+	19.0526i	0.554172	+	0.959854i
$395$	−16.0000	+	27.7128i	−0.805047	+	1.39438i
$396$	2.00000	−	3.46410i	0.100504	−	0.174078i
$397$	9.00000	+	15.5885i	0.451697	+	0.782362i	0.998492	−	0.0549046i	$-0.0174855\pi$
$397$	9.00000	+	15.5885i	0.451697	+	0.782362i	−0.546795	+	0.837267i	$0.684152\pi$
$398$	−24.0000			−1.20301
$399$		0			0
$400$	1.00000			0.0500000
$401$	15.0000	+	25.9808i	0.749064	+	1.29742i	0.948272	+	0.317460i	$0.102830\pi$
$401$	15.0000	+	25.9808i	0.749064	+	1.29742i	−0.199207	+	0.979957i	$0.563837\pi$
$402$	2.00000	−	3.46410i	0.0997509	−	0.172774i
$403$		0			0
$404$	7.00000	+	12.1244i	0.348263	+	0.603209i
$405$	−2.00000			−0.0993808
$406$		0			0
$407$	24.0000			1.18964
$408$	9.00000	+	15.5885i	0.445566	+	0.771744i
$409$	11.0000	−	19.0526i	0.543915	−	0.942088i	−0.454759	−	0.890614i	$-0.650275\pi$
$409$	11.0000	−	19.0526i	0.543915	−	0.942088i	0.998674	−	0.0514740i	$-0.0163919\pi$
$410$	−2.00000	+	3.46410i	−0.0987730	+	0.171080i
$411$	3.00000	+	5.19615i	0.147979	+	0.256307i
$412$	−8.00000			−0.394132
$413$		0			0
$414$		0			0
$415$	−12.0000	−	20.7846i	−0.589057	−	1.02028i
$416$	−5.00000	+	8.66025i	−0.245145	+	0.424604i
$417$	−6.00000	+	10.3923i	−0.293821	+	0.508913i
$418$	8.00000	+	13.8564i	0.391293	+	0.677739i
$419$	−12.0000			−0.586238			−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000			−0.586238			−0.293119	+	0.956076i	$0.594693\pi$
$420$		0			0
$421$	38.0000			1.85201			0.926003	−	0.377515i	$-0.123221\pi$
$421$	38.0000			1.85201			0.926003	+	0.377515i	$0.123221\pi$
$422$	2.00000	+	3.46410i	0.0973585	+	0.168630i
$423$		0			0
$424$	−9.00000	+	15.5885i	−0.437079	+	0.757042i
$425$	−3.00000	−	5.19615i	−0.145521	−	0.252050i
$426$		0			0
$427$		0			0
$428$	−4.00000			−0.193347
$429$	4.00000	+	6.92820i	0.193122	+	0.334497i
$430$	4.00000	−	6.92820i	0.192897	−	0.334108i
$431$	12.0000	−	20.7846i	0.578020	−	1.00116i	−0.417687	−	0.908591i	$-0.637159\pi$
$431$	12.0000	−	20.7846i	0.578020	−	1.00116i	0.995706	−	0.0925683i	$-0.0295076\pi$
$432$	0.500000	+	0.866025i	0.0240563	+	0.0416667i
$433$	−14.0000			−0.672797			−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000			−0.672797			−0.336399	+	0.941720i	$0.609209\pi$
$434$		0			0
$435$	4.00000			0.191785
$436$	−9.00000	−	15.5885i	−0.431022	−	0.746552i
$437$		0			0
$438$	−3.00000	+	5.19615i	−0.143346	+	0.248282i
$439$	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	$0.639218\pi$
$440$	−24.0000			−1.14416
$441$		0			0
$442$	−12.0000			−0.570782
$443$	−18.0000	−	31.1769i	−0.855206	−	1.48126i	−0.876454	−	0.481486i	$-0.840097\pi$
$443$	−18.0000	−	31.1769i	−0.855206	−	1.48126i	0.0212481	−	0.999774i	$-0.493236\pi$
$444$	3.00000	−	5.19615i	0.142374	−	0.246598i
$445$	−14.0000	+	24.2487i	−0.663664	+	1.14950i
$446$	8.00000	+	13.8564i	0.378811	+	0.656120i
$447$	6.00000			0.283790
$448$		0			0
$449$	−30.0000			−1.41579			−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000			−1.41579			−0.707894	+	0.706319i	$0.750354\pi$
$450$	−0.500000	−	0.866025i	−0.0235702	−	0.0408248i
$451$	−4.00000	+	6.92820i	−0.188353	+	0.326236i
$452$	−7.00000	+	12.1244i	−0.329252	+	0.570282i
$453$	−4.00000	−	6.92820i	−0.187936	−	0.325515i
$454$	12.0000			0.563188
$455$		0			0
$456$	12.0000			0.561951
$457$	−5.00000	−	8.66025i	−0.233890	−	0.405110i	0.725059	−	0.688686i	$-0.241812\pi$
$457$	−5.00000	−	8.66025i	−0.233890	−	0.405110i	−0.958950	+	0.283577i	$0.908479\pi$
$458$	−5.00000	+	8.66025i	−0.233635	+	0.404667i
$459$	3.00000	−	5.19615i	0.140028	−	0.242536i
$460$		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
$463$	16.0000			0.743583			0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000			0.743583			0.371792	+	0.928316i	$0.378744\pi$
$464$	−1.00000	−	1.73205i	−0.0464238	−	0.0804084i
$465$		0			0
$466$	−3.00000	+	5.19615i	−0.138972	+	0.240707i
$467$	−18.0000	−	31.1769i	−0.832941	−	1.44270i	−0.895696	−	0.444667i	$-0.853322\pi$
$467$	−18.0000	−	31.1769i	−0.832941	−	1.44270i	0.0627555	−	0.998029i	$-0.480011\pi$
$468$	2.00000			0.0924500
$469$		0			0
$470$		0			0
$471$	1.00000	+	1.73205i	0.0460776	+	0.0798087i
$472$	−18.0000	+	31.1769i	−0.828517	+	1.43503i
$473$	8.00000	−	13.8564i	0.367840	−	0.637118i
$474$	−8.00000	−	13.8564i	−0.367452	−	0.636446i
$475$	−4.00000			−0.183533
$476$		0			0
$477$	6.00000			0.274721
$478$	12.0000	+	20.7846i	0.548867	+	0.950666i
$479$	8.00000	−	13.8564i	0.365529	−	0.633115i	−0.623332	−	0.781958i	$-0.714221\pi$
$479$	8.00000	−	13.8564i	0.365529	−	0.633115i	0.988861	+	0.148842i	$0.0475547\pi$
$480$	−5.00000	+	8.66025i	−0.228218	+	0.395285i
$481$	6.00000	+	10.3923i	0.273576	+	0.473848i
$482$	−2.00000			−0.0910975
$483$		0			0
$484$	−5.00000			−0.227273
$485$	18.0000	+	31.1769i	0.817338	+	1.41567i
$486$	0.500000	−	0.866025i	0.0226805	−	0.0392837i
$487$	4.00000	−	6.92820i	0.181257	−	0.313947i	−0.761052	−	0.648691i	$-0.775317\pi$
$487$	4.00000	−	6.92820i	0.181257	−	0.313947i	0.942309	+	0.334744i	$0.108650\pi$
$488$	3.00000	+	5.19615i	0.135804	+	0.235219i
$489$	4.00000			0.180886
$490$		0			0
$491$	20.0000			0.902587			0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000			0.902587			0.451294	+	0.892375i	$0.350963\pi$
$492$	1.00000	+	1.73205i	0.0450835	+	0.0780869i
$493$	−6.00000	+	10.3923i	−0.270226	+	0.468046i
$494$	−4.00000	+	6.92820i	−0.179969	+	0.311715i
$495$	4.00000	+	6.92820i	0.179787	+	0.311400i
$496$		0			0
$497$		0			0
$498$	12.0000			0.537733
$499$	−2.00000	−	3.46410i	−0.0895323	−	0.155074i	0.817781	−	0.575529i	$-0.195204\pi$
$499$	−2.00000	−	3.46410i	−0.0895323	−	0.155074i	−0.907314	+	0.420455i	$0.861871\pi$
$500$	6.00000	−	10.3923i	0.268328	−	0.464758i
$501$	4.00000	−	6.92820i	0.178707	−	0.309529i
$502$	−10.0000	−	17.3205i	−0.446322	−	0.773052i
$503$	24.0000			1.07011			0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000			1.07011			0.535054	+	0.844818i	$0.320291\pi$
$504$		0			0
$505$	−28.0000			−1.24598
$506$		0			0
$507$	4.50000	−	7.79423i	0.199852	−	0.346154i
$508$		0			0
$509$	5.00000	+	8.66025i	0.221621	+	0.383859i	0.955300	−	0.295637i	$-0.0955319\pi$
$509$	5.00000	+	8.66025i	0.221621	+	0.383859i	−0.733679	+	0.679496i	$0.762199\pi$
$510$	−12.0000			−0.531369
$511$		0			0
$512$	11.0000			0.486136
$513$	−2.00000	−	3.46410i	−0.0883022	−	0.152944i
$514$	13.0000	−	22.5167i	0.573405	−	0.993167i
$515$	8.00000	−	13.8564i	0.352522	−	0.610586i
$516$	−2.00000	−	3.46410i	−0.0880451	−	0.152499i
$517$		0			0
$518$		0			0
$519$	−10.0000			−0.438951
$520$	−6.00000	−	10.3923i	−0.263117	−	0.455733i
$521$	−9.00000	+	15.5885i	−0.394297	+	0.682943i	−0.993011	−	0.118020i	$-0.962345\pi$
$521$	−9.00000	+	15.5885i	−0.394297	+	0.682943i	0.598714	+	0.800963i	$0.295679\pi$
$522$	−1.00000	+	1.73205i	−0.0437688	+	0.0758098i
$523$	10.0000	+	17.3205i	0.437269	+	0.757373i	0.997478	−	0.0709788i	$-0.0226123\pi$
$523$	10.0000	+	17.3205i	0.437269	+	0.757373i	−0.560208	+	0.828352i	$0.689279\pi$
$524$	−4.00000			−0.174741
$525$		0			0
$526$	−16.0000			−0.697633
$527$		0			0
$528$	2.00000	−	3.46410i	0.0870388	−	0.150756i
$529$	11.5000	−	19.9186i	0.500000	−	0.866025i
$530$	−6.00000	−	10.3923i	−0.260623	−	0.451413i
$531$	12.0000			0.520756
$532$		0			0
$533$	−4.00000			−0.173259
$534$	−7.00000	−	12.1244i	−0.302920	−	0.524672i
$535$	4.00000	−	6.92820i	0.172935	−	0.299532i
$536$	−6.00000	+	10.3923i	−0.259161	+	0.448879i
$537$	2.00000	+	3.46410i	0.0863064	+	0.149487i
$538$	−6.00000			−0.258678
$539$		0			0
$540$	2.00000			0.0860663
$541$	17.0000	+	29.4449i	0.730887	+	1.26593i	0.956504	+	0.291718i	$0.0942267\pi$
$541$	17.0000	+	29.4449i	0.730887	+	1.26593i	−0.225617	+	0.974216i	$0.572440\pi$
$542$	8.00000	−	13.8564i	0.343629	−	0.595184i
$543$	13.0000	−	22.5167i	0.557883	−	0.966282i
$544$	−15.0000	−	25.9808i	−0.643120	−	1.11392i
$545$	36.0000			1.54207
$546$		0			0
$547$	4.00000			0.171028			0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000			0.171028			0.0855138	+	0.996337i	$0.472747\pi$
$548$	−3.00000	−	5.19615i	−0.128154	−	0.221969i
$549$	1.00000	−	1.73205i	0.0426790	−	0.0739221i
$550$	−2.00000	+	3.46410i	−0.0852803	+	0.147710i
$551$	4.00000	+	6.92820i	0.170406	+	0.295151i
$552$		0			0
$553$		0			0
$554$	−22.0000			−0.934690
$555$	6.00000	+	10.3923i	0.254686	+	0.441129i
$556$	6.00000	−	10.3923i	0.254457	−	0.440732i
$557$	1.00000	−	1.73205i	0.0423714	−	0.0733893i	−0.844062	−	0.536246i	$-0.819842\pi$
$557$	1.00000	−	1.73205i	0.0423714	−	0.0733893i	0.886433	+	0.462856i	$0.153175\pi$
$558$		0			0
$559$	8.00000			0.338364
$560$		0			0
$561$	−24.0000			−1.01328
$562$	−11.0000	−	19.0526i	−0.464007	−	0.803684i
$563$	−2.00000	+	3.46410i	−0.0842900	+	0.145994i	−0.905088	−	0.425223i	$-0.860196\pi$
$563$	−2.00000	+	3.46410i	−0.0842900	+	0.145994i	0.820798	+	0.571218i	$0.193529\pi$
$564$		0			0
$565$	−14.0000	−	24.2487i	−0.588984	−	1.02015i
$566$	20.0000			0.840663
$567$		0			0
$568$		0			0
$569$	−5.00000	−	8.66025i	−0.209611	−	0.363057i	0.741981	−	0.670421i	$-0.233886\pi$
$569$	−5.00000	−	8.66025i	−0.209611	−	0.363057i	−0.951592	+	0.307364i	$0.900553\pi$
$570$	−4.00000	+	6.92820i	−0.167542	+	0.290191i
$571$	2.00000	−	3.46410i	0.0836974	−	0.144968i	−0.821138	−	0.570730i	$-0.806660\pi$
$571$	2.00000	−	3.46410i	0.0836974	−	0.144968i	0.904835	+	0.425762i	$0.139994\pi$
$572$	−4.00000	−	6.92820i	−0.167248	−	0.289683i
$573$	−8.00000			−0.334205
$574$		0			0
$575$		0			0
$576$	−3.50000	−	6.06218i	−0.145833	−	0.252591i
$577$	−17.0000	+	29.4449i	−0.707719	+	1.22581i	0.257982	+	0.966150i	$0.416942\pi$
$577$	−17.0000	+	29.4449i	−0.707719	+	1.22581i	−0.965701	+	0.259656i	$0.916391\pi$
$578$	9.50000	−	16.4545i	0.395148	−	0.684416i
$579$	−1.00000	−	1.73205i	−0.0415586	−	0.0719816i
$580$	−4.00000			−0.166091
$581$		0			0
$582$	−18.0000			−0.746124
$583$	−12.0000	−	20.7846i	−0.496989	−	0.860811i
$584$	9.00000	−	15.5885i	0.372423	−	0.645055i
$585$	−2.00000	+	3.46410i	−0.0826898	+	0.143223i
$586$	7.00000	+	12.1244i	0.289167	+	0.500853i
$587$	28.0000			1.15568			0.577842	−	0.816149i	$-0.303895\pi$
$587$	28.0000			1.15568			0.577842	+	0.816149i	$0.303895\pi$
$588$		0			0
$589$		0			0
$590$	−12.0000	−	20.7846i	−0.494032	−	0.855689i
$591$	−11.0000	+	19.0526i	−0.452480	+	0.783718i
$592$	3.00000	−	5.19615i	0.123299	−	0.213561i
$593$	3.00000	+	5.19615i	0.123195	+	0.213380i	0.921026	−	0.389501i	$-0.127353\pi$
$593$	3.00000	+	5.19615i	0.123195	+	0.213380i	−0.797831	+	0.602881i	$0.794019\pi$
$594$	−4.00000			−0.164122
$595$		0			0
$596$	−6.00000			−0.245770
$597$	−12.0000	−	20.7846i	−0.491127	−	0.850657i
$598$		0			0
$599$	−24.0000	+	41.5692i	−0.980613	+	1.69847i	−0.320607	+	0.947212i	$0.603887\pi$
$599$	−24.0000	+	41.5692i	−0.980613	+	1.69847i	−0.660006	+	0.751260i	$0.729446\pi$
$600$	1.50000	+	2.59808i	0.0612372	+	0.106066i
$601$	−6.00000			−0.244745			−0.122373	−	0.992484i	$-0.539050\pi$
$601$	−6.00000			−0.244745			−0.122373	+	0.992484i	$0.539050\pi$
$602$		0			0
$603$	4.00000			0.162893
$604$	4.00000	+	6.92820i	0.162758	+	0.281905i
$605$	5.00000	−	8.66025i	0.203279	−	0.352089i
$606$	7.00000	−	12.1244i	0.284356	−	0.492518i
$607$	8.00000	+	13.8564i	0.324710	+	0.562414i	0.981454	−	0.191700i	$-0.0614000\pi$
$607$	8.00000	+	13.8564i	0.324710	+	0.562414i	−0.656744	+	0.754114i	$0.728067\pi$
$608$	−20.0000			−0.811107
$609$		0			0
$610$	−4.00000			−0.161955
$611$		0			0
$612$	−3.00000	+	5.19615i	−0.121268	+	0.210042i
$613$	13.0000	−	22.5167i	0.525065	−	0.909439i	−0.474509	−	0.880251i	$-0.657374\pi$
$613$	13.0000	−	22.5167i	0.525065	−	0.909439i	0.999574	−	0.0291886i	$-0.00929235\pi$
$614$	2.00000	+	3.46410i	0.0807134	+	0.139800i
$615$	−4.00000			−0.161296
$616$		0			0
$617$	−6.00000			−0.241551			−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000			−0.241551			−0.120775	+	0.992680i	$0.538538\pi$
$618$	4.00000	+	6.92820i	0.160904	+	0.278693i
$619$	10.0000	−	17.3205i	0.401934	−	0.696170i	−0.592025	−	0.805919i	$-0.701671\pi$
$619$	10.0000	−	17.3205i	0.401934	−	0.696170i	0.993959	+	0.109749i	$0.0350048\pi$
$620$		0			0
$621$		0			0
$622$	24.0000			0.962312
$623$		0			0
$624$	2.00000			0.0800641
$625$	9.50000	+	16.4545i	0.380000	+	0.658179i
$626$	13.0000	−	22.5167i	0.519584	−	0.899947i
$627$	−8.00000	+	13.8564i	−0.319489	+	0.553372i
$628$	−1.00000	−	1.73205i	−0.0399043	−	0.0691164i
$629$	−36.0000			−1.43541
$630$		0			0
$631$	−40.0000			−1.59237			−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000			−1.59237			−0.796187	+	0.605050i	$0.793153\pi$
$632$	24.0000	+	41.5692i	0.954669	+	1.65353i
$633$	−2.00000	+	3.46410i	−0.0794929	+	0.137686i
$634$	−9.00000	+	15.5885i	−0.357436	+	0.619097i
$635$		0			0
$636$	−6.00000			−0.237915
$637$		0			0
$638$	8.00000			0.316723
$639$		0			0
$640$	3.00000	−	5.19615i	0.118585	−	0.205396i
$641$	−9.00000	+	15.5885i	−0.355479	+	0.615707i	−0.987200	−	0.159489i	$-0.949015\pi$
$641$	−9.00000	+	15.5885i	−0.355479	+	0.615707i	0.631721	+	0.775196i	$0.282349\pi$
$642$	2.00000	+	3.46410i	0.0789337	+	0.136717i
$643$	20.0000			0.788723			0.394362	−	0.918955i	$-0.370966\pi$
$643$	20.0000			0.788723			0.394362	+	0.918955i	$0.370966\pi$
$644$		0			0
$645$	8.00000			0.315000
$646$	−12.0000	−	20.7846i	−0.472134	−	0.817760i
$647$	20.0000	−	34.6410i	0.786281	−	1.36188i	−0.141950	−	0.989874i	$-0.545337\pi$
$647$	20.0000	−	34.6410i	0.786281	−	1.36188i	0.928231	−	0.372005i	$-0.121330\pi$
$648$	−1.50000	+	2.59808i	−0.0589256	+	0.102062i
$649$	−24.0000	−	41.5692i	−0.942082	−	1.63173i
$650$	−2.00000			−0.0784465
$651$		0			0
$652$	−4.00000			−0.156652
$653$	9.00000	+	15.5885i	0.352197	+	0.610023i	0.986634	−	0.162951i	$-0.0521013\pi$
$653$	9.00000	+	15.5885i	0.352197	+	0.610023i	−0.634437	+	0.772975i	$0.718768\pi$
$654$	−9.00000	+	15.5885i	−0.351928	+	0.609557i
$655$	4.00000	−	6.92820i	0.156293	−	0.270707i
$656$	1.00000	+	1.73205i	0.0390434	+	0.0676252i
$657$	−6.00000			−0.234082
$658$		0			0
$659$	12.0000			0.467454			0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000			0.467454			0.233727	+	0.972302i	$0.424908\pi$
$660$	−4.00000	−	6.92820i	−0.155700	−	0.269680i
$661$	−11.0000	+	19.0526i	−0.427850	+	0.741059i	−0.996682	−	0.0813955i	$-0.974062\pi$
$661$	−11.0000	+	19.0526i	−0.427850	+	0.741059i	0.568831	+	0.822454i	$0.307396\pi$
$662$	−2.00000	+	3.46410i	−0.0777322	+	0.134636i
$663$	−6.00000	−	10.3923i	−0.233021	−	0.403604i
$664$	−36.0000			−1.39707
$665$		0			0
$666$	−6.00000			−0.232495
$667$		0			0
$668$	−4.00000	+	6.92820i	−0.154765	+	0.268060i
$669$	−8.00000	+	13.8564i	−0.309298	+	0.535720i
$670$	−4.00000	−	6.92820i	−0.154533	−	0.267660i
$671$	−8.00000			−0.308837
$672$		0			0
$673$	34.0000			1.31060			0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000			1.31060			0.655302	+	0.755367i	$0.272541\pi$
$674$	−7.00000	−	12.1244i	−0.269630	−	0.467013i
$675$	0.500000	−	0.866025i	0.0192450	−	0.0333333i
$676$	−4.50000	+	7.79423i	−0.173077	+	0.299778i
$677$	9.00000	+	15.5885i	0.345898	+	0.599113i	0.985517	−	0.169580i	$-0.0542410\pi$
$677$	9.00000	+	15.5885i	0.345898	+	0.599113i	−0.639618	+	0.768693i	$0.720908\pi$
$678$	14.0000			0.537667
$679$		0			0
$680$	36.0000			1.38054
$681$	6.00000	+	10.3923i	0.229920	+	0.398234i
$682$		0			0
$683$	6.00000	−	10.3923i	0.229584	−	0.397650i	−0.728101	−	0.685470i	$-0.759597\pi$
$683$	6.00000	−	10.3923i	0.229584	−	0.397650i	0.957685	+	0.287819i	$0.0929302\pi$
$684$	2.00000	+	3.46410i	0.0764719	+	0.132453i
$685$	12.0000			0.458496
$686$		0			0
$687$	−10.0000			−0.381524
$688$	−2.00000	−	3.46410i	−0.0762493	−	0.132068i
$689$	6.00000	−	10.3923i	0.228582	−	0.395915i
$690$		0			0
$691$	−10.0000	−	17.3205i	−0.380418	−	0.658903i	0.610704	−	0.791859i	$-0.290887\pi$
$691$	−10.0000	−	17.3205i	−0.380418	−	0.658903i	−0.991122	+	0.132956i	$0.957553\pi$
$692$	10.0000			0.380143
$693$		0			0
$694$	28.0000			1.06287
$695$	12.0000	+	20.7846i	0.455186	+	0.788405i
$696$	3.00000	−	5.19615i	0.113715	−	0.196960i
$697$	6.00000	−	10.3923i	0.227266	−	0.393637i
$698$	−1.00000	−	1.73205i	−0.0378506	−	0.0655591i
$699$	−6.00000			−0.226941
$700$		0			0
$701$	30.0000			1.13308			0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000			1.13308			0.566542	+	0.824033i	$0.308281\pi$
$702$	−1.00000	−	1.73205i	−0.0377426	−	0.0653720i
$703$	−12.0000	+	20.7846i	−0.452589	+	0.783906i
$704$	−14.0000	+	24.2487i	−0.527645	+	0.913908i
$705$		0			0
$706$	−10.0000			−0.376355
$707$		0			0
$708$	−12.0000			−0.450988
$709$	−3.00000	−	5.19615i	−0.112667	−	0.195146i	0.804178	−	0.594389i	$-0.202606\pi$
$709$	−3.00000	−	5.19615i	−0.112667	−	0.195146i	−0.916845	+	0.399244i	$0.869273\pi$
$710$		0			0
$711$	8.00000	−	13.8564i	0.300023	−	0.519656i
$712$	21.0000	+	36.3731i	0.787008	+	1.36314i
$713$		0			0
$714$		0			0
$715$	16.0000			0.598366
$716$	−2.00000	−	3.46410i	−0.0747435	−	0.129460i
$717$	−12.0000	+	20.7846i	−0.448148	+	0.776215i
$718$	16.0000	−	27.7128i	0.597115	−	1.03423i
$719$	24.0000	+	41.5692i	0.895049	+	1.55027i	0.833744	+	0.552151i	$0.186193\pi$
$719$	24.0000	+	41.5692i	0.895049	+	1.55027i	0.0613050	+	0.998119i	$0.480474\pi$
$720$	2.00000			0.0745356
$721$		0			0
$722$	3.00000			0.111648
$723$	−1.00000	−	1.73205i	−0.0371904	−	0.0644157i
$724$	−13.0000	+	22.5167i	−0.483141	+	0.836825i
$725$	−1.00000	+	1.73205i	−0.0371391	+	0.0643268i
$726$	2.50000	+	4.33013i	0.0927837	+	0.160706i
$727$	−40.0000			−1.48352			−0.741759	−	0.670667i	$-0.766008\pi$
$727$	−40.0000			−1.48352			−0.741759	+	0.670667i	$0.766008\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$	6.00000	+	10.3923i	0.222070	+	0.384636i
$731$	−12.0000	+	20.7846i	−0.443836	+	0.768747i
$732$	−1.00000	+	1.73205i	−0.0369611	+	0.0640184i
$733$	9.00000	+	15.5885i	0.332423	+	0.575773i	0.982986	−	0.183679i	$-0.0588007\pi$
$733$	9.00000	+	15.5885i	0.332423	+	0.575773i	−0.650564	+	0.759452i	$0.725467\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−8.00000	−	13.8564i	−0.294684	−	0.510407i
$738$	1.00000	−	1.73205i	0.0368105	−	0.0637577i
$739$	−18.0000	+	31.1769i	−0.662141	+	1.14686i	0.317911	+	0.948120i	$0.397019\pi$
$739$	−18.0000	+	31.1769i	−0.662141	+	1.14686i	−0.980052	+	0.198741i	$0.936315\pi$
$740$	−6.00000	−	10.3923i	−0.220564	−	0.382029i
$741$	−8.00000			−0.293887
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$	6.00000	−	10.3923i	0.219823	−	0.380745i
$746$	−5.00000	+	8.66025i	−0.183063	+	0.317074i
$747$	6.00000	+	10.3923i	0.219529	+	0.380235i
$748$	24.0000			0.877527
$749$		0			0
$750$	−12.0000			−0.438178
$751$	16.0000	+	27.7128i	0.583848	+	1.01125i	0.995018	+	0.0996961i	$0.0317870\pi$
$751$	16.0000	+	27.7128i	0.583848	+	1.01125i	−0.411170	+	0.911559i	$0.634880\pi$
$752$		0			0
$753$	10.0000	−	17.3205i	0.364420	−	0.631194i
$754$	2.00000	+	3.46410i	0.0728357	+	0.126155i
$755$	−16.0000			−0.582300
$756$		0			0
$757$	−10.0000			−0.363456			−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000			−0.363456			−0.181728	+	0.983349i	$0.558169\pi$
$758$	6.00000	+	10.3923i	0.217930	+	0.377466i
$759$		0			0
$760$	12.0000	−	20.7846i	0.435286	−	0.753937i
$761$	−9.00000	−	15.5885i	−0.326250	−	0.565081i	0.655515	−	0.755182i	$-0.272452\pi$
$761$	−9.00000	−	15.5885i	−0.326250	−	0.565081i	−0.981764	+	0.190101i	$0.939118\pi$
$762$		0			0
$763$		0			0
$764$	8.00000			0.289430
$765$	−6.00000	−	10.3923i	−0.216930	−	0.375735i
$766$		0			0
$767$	12.0000	−	20.7846i	0.433295	−	0.750489i
$768$	8.50000	+	14.7224i	0.306717	+	0.531250i
$769$	2.00000			0.0721218			0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000			0.0721218			0.0360609	+	0.999350i	$0.488519\pi$
$770$		0			0
$771$	26.0000			0.936367
$772$	1.00000	+	1.73205i	0.0359908	+	0.0623379i
$773$	−7.00000	+	12.1244i	−0.251773	+	0.436083i	−0.964014	−	0.265852i	$-0.914347\pi$
$773$	−7.00000	+	12.1244i	−0.251773	+	0.436083i	0.712241	+	0.701935i	$0.247680\pi$
$774$	−2.00000	+	3.46410i	−0.0718885	+	0.124515i
$775$		0			0
$776$	54.0000			1.93849
$777$		0			0
$778$	−6.00000			−0.215110
$779$	−4.00000	−	6.92820i	−0.143315	−	0.248229i
$780$	2.00000	−	3.46410i	0.0716115	−	0.124035i
$781$		0			0
$782$		0			0
$783$	−2.00000			−0.0714742
$784$		0			0
$785$	4.00000			0.142766
$786$	2.00000	+	3.46410i	0.0713376	+	0.123560i
$787$	22.0000	−	38.1051i	0.784215	−	1.35830i	−0.145251	−	0.989395i	$-0.546399\pi$
$787$	22.0000	−	38.1051i	0.784215	−	1.35830i	0.929467	−	0.368906i	$-0.120268\pi$
$788$	11.0000	−	19.0526i	0.391859	−	0.678719i
$789$	−8.00000	−	13.8564i	−0.284808	−	0.493301i
$790$	−32.0000			−1.13851
$791$		0			0
$792$	12.0000			0.426401
$793$	−2.00000	−	3.46410i	−0.0710221	−	0.123014i
$794$	−9.00000	+	15.5885i	−0.319398	+	0.553214i
$795$	6.00000	−	10.3923i	0.212798	−	0.368577i
$796$	12.0000	+	20.7846i	0.425329	+	0.736691i
$797$	−26.0000			−0.920967			−0.460484	−	0.887668i	$-0.652324\pi$
$797$	−26.0000			−0.920967			−0.460484	+	0.887668i	$0.652324\pi$
$798$		0			0
$799$		0			0
$800$	−2.50000	−	4.33013i	−0.0883883	−	0.153093i
$801$	7.00000	−	12.1244i	0.247333	−	0.428393i
$802$	−15.0000	+	25.9808i	−0.529668	+	0.917413i
$803$	12.0000	+	20.7846i	0.423471	+	0.733473i
$804$	−4.00000			−0.141069
$805$		0			0
$806$		0			0
$807$	−3.00000	−	5.19615i	−0.105605	−	0.182913i
$808$	−21.0000	+	36.3731i	−0.738777	+	1.27960i
$809$	−21.0000	+	36.3731i	−0.738321	+	1.27881i	0.214930	+	0.976629i	$0.431048\pi$
$809$	−21.0000	+	36.3731i	−0.738321	+	1.27881i	−0.953251	+	0.302180i	$0.902286\pi$
$810$	−1.00000	−	1.73205i	−0.0351364	−	0.0608581i
$811$	44.0000			1.54505			0.772524	−	0.634985i	$-0.218994\pi$
$811$	44.0000			1.54505			0.772524	+	0.634985i	$0.218994\pi$
$812$		0			0
$813$	16.0000			0.561144
$814$	12.0000	+	20.7846i	0.420600	+	0.728500i
$815$	4.00000	−	6.92820i	0.140114	−	0.242684i
$816$	−3.00000	+	5.19615i	−0.105021	+	0.181902i
$817$	8.00000	+	13.8564i	0.279885	+	0.484774i
$818$	22.0000			0.769212
$819$		0			0
$820$	4.00000			0.139686
$821$	−19.0000	−	32.9090i	−0.663105	−	1.14853i	−0.979795	−	0.200002i	$-0.935905\pi$
$821$	−19.0000	−	32.9090i	−0.663105	−	1.14853i	0.316691	−	0.948529i	$-0.397428\pi$
$822$	−3.00000	+	5.19615i	−0.104637	+	0.181237i
$823$	−12.0000	+	20.7846i	−0.418294	+	0.724506i	−0.995768	−	0.0919029i	$-0.970705\pi$
$823$	−12.0000	+	20.7846i	−0.418294	+	0.724506i	0.577474	+	0.816409i	$0.304038\pi$
$824$	−12.0000	−	20.7846i	−0.418040	−	0.724066i
$825$	−4.00000			−0.139262
$826$		0			0
$827$	−12.0000			−0.417281			−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000			−0.417281			−0.208640	+	0.977992i	$0.566904\pi$
$828$		0			0
$829$	−7.00000	+	12.1244i	−0.243120	+	0.421096i	−0.961601	−	0.274450i	$-0.911504\pi$
$829$	−7.00000	+	12.1244i	−0.243120	+	0.421096i	0.718481	+	0.695546i	$0.244838\pi$
$830$	12.0000	−	20.7846i	0.416526	−	0.721444i
$831$	−11.0000	−	19.0526i	−0.381586	−	0.660926i
$832$	−14.0000			−0.485363
$833$		0			0
$834$	−12.0000			−0.415526
$835$	−8.00000	−	13.8564i	−0.276851	−	0.479521i
$836$	8.00000	−	13.8564i	0.276686	−	0.479234i
$837$		0			0
$838$	−6.00000	−	10.3923i	−0.207267	−	0.358996i
$839$	−8.00000			−0.276191			−0.138095	−	0.990419i	$-0.544098\pi$
$839$	−8.00000			−0.276191			−0.138095	+	0.990419i	$0.544098\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$	19.0000	+	32.9090i	0.654783	+	1.13412i
$843$	11.0000	−	19.0526i	0.378860	−	0.656205i
$844$	2.00000	−	3.46410i	0.0688428	−	0.119239i
$845$	−9.00000	−	15.5885i	−0.309609	−	0.536259i
$846$		0			0
$847$		0			0
$848$	−6.00000			−0.206041
$849$	10.0000	+	17.3205i	0.343199	+	0.594438i
$850$	3.00000	−	5.19615i	0.102899	−	0.178227i
$851$		0			0
$852$		0			0
$853$	−10.0000			−0.342393			−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000			−0.342393			−0.171197	+	0.985237i	$0.554763\pi$
$854$		0			0
$855$	−8.00000			−0.273594
$856$	−6.00000	−	10.3923i	−0.205076	−	0.355202i
$857$	7.00000	−	12.1244i	0.239115	−	0.414160i	−0.721345	−	0.692576i	$-0.756476\pi$
$857$	7.00000	−	12.1244i	0.239115	−	0.414160i	0.960461	+	0.278416i	$0.0898092\pi$
$858$	−4.00000	+	6.92820i	−0.136558	+	0.236525i
$859$	−22.0000	−	38.1051i	−0.750630	−	1.30013i	−0.947518	−	0.319704i	$-0.896417\pi$
$859$	−22.0000	−	38.1051i	−0.750630	−	1.30013i	0.196887	−	0.980426i	$-0.436917\pi$
$860$	−8.00000			−0.272798
$861$		0			0
$862$	24.0000			0.817443
$863$	12.0000	+	20.7846i	0.408485	+	0.707516i	0.994720	−	0.102624i	$-0.0327240\pi$
$863$	12.0000	+	20.7846i	0.408485	+	0.707516i	−0.586235	+	0.810141i	$0.699391\pi$
$864$	2.50000	−	4.33013i	0.0850517	−	0.147314i
$865$	−10.0000	+	17.3205i	−0.340010	+	0.588915i
$866$	−7.00000	−	12.1244i	−0.237870	−	0.412002i
$867$	19.0000			0.645274
$868$		0			0
$869$	−64.0000			−2.17105
$870$	2.00000	+	3.46410i	0.0678064	+	0.117444i
$871$	4.00000	−	6.92820i	0.135535	−	0.234753i
$872$	27.0000	−	46.7654i	0.914335	−	1.58368i
$873$	−9.00000	−	15.5885i	−0.304604	−	0.527589i
$874$		0			0
$875$		0			0
$876$	6.00000			0.202721
$877$	−23.0000	−	39.8372i	−0.776655	−	1.34521i	−0.933860	−	0.357640i	$-0.883582\pi$
$877$	−23.0000	−	39.8372i	−0.776655	−	1.34521i	0.157205	−	0.987566i	$-0.449752\pi$
$878$	−12.0000	+	20.7846i	−0.404980	+	0.701447i
$879$	−7.00000	+	12.1244i	−0.236104	+	0.408944i
$880$	−4.00000	−	6.92820i	−0.134840	−	0.233550i
$881$	−6.00000			−0.202145			−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000			−0.202145			−0.101073	+	0.994879i	$0.532227\pi$
$882$		0			0
$883$	−28.0000			−0.942275			−0.471138	−	0.882060i	$-0.656156\pi$
$883$	−28.0000			−0.942275			−0.471138	+	0.882060i	$0.656156\pi$
$884$	6.00000	+	10.3923i	0.201802	+	0.349531i
$885$	12.0000	−	20.7846i	0.403376	−	0.698667i
$886$	18.0000	−	31.1769i	0.604722	−	1.04741i
$887$	−4.00000	−	6.92820i	−0.134307	−	0.232626i	0.791026	−	0.611783i	$-0.209547\pi$
$887$	−4.00000	−	6.92820i	−0.134307	−	0.232626i	−0.925332	+	0.379157i	$0.876214\pi$
$888$	18.0000			0.604040
$889$		0			0
$890$	−28.0000			−0.938562
$891$	−2.00000	−	3.46410i	−0.0670025	−	0.116052i
$892$	8.00000	−	13.8564i	0.267860	−	0.463947i
$893$		0			0
$894$	3.00000	+	5.19615i	0.100335	+	0.173785i
$895$	8.00000			0.267411
$896$		0			0
$897$		0			0
$898$	−15.0000	−	25.9808i	−0.500556	−	0.866989i
$899$		0			0
$900$	−0.500000	+	0.866025i	−0.0166667	+	0.0288675i
$901$	18.0000	+	31.1769i	0.599667	+	1.03865i
$902$	−8.00000			−0.266371
$903$		0			0
$904$	−42.0000			−1.39690
$905$	−26.0000	−	45.0333i	−0.864269	−	1.49696i
$906$	4.00000	−	6.92820i	0.132891	−	0.230174i
$907$	2.00000	−	3.46410i	0.0664089	−	0.115024i	−0.830909	−	0.556408i	$-0.812179\pi$
$907$	2.00000	−	3.46410i	0.0664089	−	0.115024i	0.897318	+	0.441384i	$0.145512\pi$
$908$	−6.00000	−	10.3923i	−0.199117	−	0.344881i
$909$	14.0000			0.464351
$910$		0			0
$911$	−24.0000			−0.795155			−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000			−0.795155			−0.397578	+	0.917568i	$0.630149\pi$
$912$	2.00000	+	3.46410i	0.0662266	+	0.114708i
$913$	24.0000	−	41.5692i	0.794284	−	1.37574i
$914$	5.00000	−	8.66025i	0.165385	−	0.286456i
$915$	−2.00000	−	3.46410i	−0.0661180	−	0.114520i
$916$	10.0000			0.330409
$917$		0			0
$918$	6.00000			0.198030
$919$	−4.00000	−	6.92820i	−0.131948	−	0.228540i	0.792480	−	0.609898i	$-0.208790\pi$
$919$	−4.00000	−	6.92820i	−0.131948	−	0.228540i	−0.924427	+	0.381358i	$0.875456\pi$
$920$		0			0
$921$	−2.00000	+	3.46410i	−0.0659022	+	0.114146i
$922$	−5.00000	−	8.66025i	−0.164666	−	0.285210i
$923$		0			0
$924$		0			0
$925$	−6.00000			−0.197279
$926$	8.00000	+	13.8564i	0.262896	+	0.455350i
$927$	−4.00000	+	6.92820i	−0.131377	+	0.227552i
$928$	−5.00000	+	8.66025i	−0.164133	+	0.284287i
$929$	−13.0000	−	22.5167i	−0.426516	−	0.738748i	0.570045	−	0.821614i	$-0.306926\pi$
$929$	−13.0000	−	22.5167i	−0.426516	−	0.738748i	−0.996561	+	0.0828661i	$0.973593\pi$
$930$		0			0
$931$		0			0
$932$	6.00000			0.196537
$933$	12.0000	+	20.7846i	0.392862	+	0.680458i
$934$	18.0000	−	31.1769i	0.588978	−	1.02014i
$935$	−24.0000	+	41.5692i	−0.784884	+	1.35946i
$936$	3.00000	+	5.19615i	0.0980581	+	0.169842i
$937$	42.0000			1.37208			0.686040	−	0.727564i	$-0.259347\pi$
$937$	42.0000			1.37208			0.686040	+	0.727564i	$0.259347\pi$
$938$		0			0
$939$	26.0000			0.848478
$940$		0			0
$941$	−19.0000	+	32.9090i	−0.619382	+	1.07280i	0.370216	+	0.928946i	$0.379284\pi$
$941$	−19.0000	+	32.9090i	−0.619382	+	1.07280i	−0.989599	+	0.143856i	$0.954050\pi$
$942$	−1.00000	+	1.73205i	−0.0325818	+	0.0564333i
$943$		0			0
$944$	−12.0000			−0.390567
$945$		0			0
$946$	16.0000			0.520205
$947$	−22.0000	−	38.1051i	−0.714904	−	1.23825i	−0.962997	−	0.269514i	$-0.913137\pi$
$947$	−22.0000	−	38.1051i	−0.714904	−	1.23825i	0.248093	−	0.968736i	$-0.420196\pi$
$948$	−8.00000	+	13.8564i	−0.259828	+	0.450035i
$949$	−6.00000	+	10.3923i	−0.194768	+	0.337348i
$950$	−2.00000	−	3.46410i	−0.0648886	−	0.112390i
$951$	−18.0000			−0.583690
$952$		0			0
$953$	26.0000			0.842223			0.421111	−	0.907009i	$-0.361640\pi$
$953$	26.0000			0.842223			0.421111	+	0.907009i	$0.361640\pi$
$954$	3.00000	+	5.19615i	0.0971286	+	0.168232i
$955$	−8.00000	+	13.8564i	−0.258874	+	0.448383i
$956$	12.0000	−	20.7846i	0.388108	−	0.672222i
$957$	4.00000	+	6.92820i	0.129302	+	0.223957i
$958$	16.0000			0.516937
$959$		0			0
$960$	−14.0000			−0.451848
$961$	15.5000	+	26.8468i	0.500000	+	0.866025i
$962$	−6.00000	+	10.3923i	−0.193448	+	0.335061i
$963$	−2.00000	+	3.46410i	−0.0644491	+	0.111629i
$964$	1.00000	+	1.73205i	0.0322078	+	0.0557856i
$965$	−4.00000			−0.128765
$966$		0			0
$967$	40.0000			1.28631			0.643157	−	0.765735i	$-0.277624\pi$
$967$	40.0000			1.28631			0.643157	+	0.765735i	$0.277624\pi$
$968$	−7.50000	−	12.9904i	−0.241059	−	0.417527i
$969$	12.0000	−	20.7846i	0.385496	−	0.667698i
$970$	−18.0000	+	31.1769i	−0.577945	+	1.00103i
$971$	−6.00000	−	10.3923i	−0.192549	−	0.333505i	0.753545	−	0.657396i	$-0.228342\pi$
$971$	−6.00000	−	10.3923i	−0.192549	−	0.333505i	−0.946094	+	0.323891i	$0.895009\pi$
$972$	−1.00000			−0.0320750
$973$		0			0
$974$	8.00000			0.256337
$975$	−1.00000	−	1.73205i	−0.0320256	−	0.0554700i
$976$	−1.00000	+	1.73205i	−0.0320092	+	0.0554416i
$977$	15.0000	−	25.9808i	0.479893	−	0.831198i	−0.519841	−	0.854263i	$-0.674009\pi$
$977$	15.0000	−	25.9808i	0.479893	−	0.831198i	0.999734	+	0.0230645i	$0.00734232\pi$
$978$	2.00000	+	3.46410i	0.0639529	+	0.110770i
$979$	−56.0000			−1.78977
$980$		0			0
$981$	−18.0000			−0.574696
$982$	10.0000	+	17.3205i	0.319113	+	0.552720i
$983$	−12.0000	+	20.7846i	−0.382741	+	0.662926i	−0.991453	−	0.130465i	$-0.958353\pi$
$983$	−12.0000	+	20.7846i	−0.382741	+	0.662926i	0.608712	+	0.793391i	$0.291686\pi$
$984$	−3.00000	+	5.19615i	−0.0956365	+	0.165647i
$985$	22.0000	+	38.1051i	0.700978	+	1.21413i
$986$	−12.0000			−0.382158
$987$		0			0
$988$	8.00000			0.254514
$989$		0			0
$990$	−4.00000	+	6.92820i	−0.127128	+	0.220193i
$991$	8.00000	−	13.8564i	0.254128	−	0.440163i	−0.710530	−	0.703667i	$-0.751545\pi$
$991$	8.00000	−	13.8564i	0.254128	−	0.440163i	0.964658	+	0.263504i	$0.0848781\pi$
$992$		0			0
$993$	−4.00000			−0.126936
$994$		0			0
$995$	−48.0000			−1.52170
$996$	−6.00000	−	10.3923i	−0.190117	−	0.329293i
$997$	13.0000	−	22.5167i	0.411714	−	0.713110i	−0.583363	−	0.812211i	$-0.698264\pi$
$997$	13.0000	−	22.5167i	0.411714	−	0.713110i	0.995077	+	0.0991016i	$0.0315969\pi$
$998$	2.00000	−	3.46410i	0.0633089	−	0.109654i
$999$	−3.00000	−	5.19615i	−0.0949158	−	0.164399i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.2.e.b.67.1		2
3.2	odd	2		441.2.e.a.361.1		2
4.3	odd	2		2352.2.q.x.1537.1		2
7.2	even	3	inner	147.2.e.b.79.1		2
7.3	odd	6		147.2.a.a.1.1		1
7.4	even	3		21.2.a.a.1.1	✓	1
7.5	odd	6		147.2.e.c.79.1		2
7.6	odd	2		147.2.e.c.67.1		2
21.2	odd	6		441.2.e.a.226.1		2
21.5	even	6		441.2.e.b.226.1		2
21.11	odd	6		63.2.a.a.1.1		1
21.17	even	6		441.2.a.f.1.1		1
21.20	even	2		441.2.e.b.361.1		2
28.3	even	6		2352.2.a.v.1.1		1
28.11	odd	6		336.2.a.a.1.1		1
28.19	even	6		2352.2.q.e.961.1		2
28.23	odd	6		2352.2.q.x.961.1		2
28.27	even	2		2352.2.q.e.1537.1		2
35.4	even	6		525.2.a.d.1.1		1
35.18	odd	12		525.2.d.a.274.2		2
35.24	odd	6		3675.2.a.n.1.1		1
35.32	odd	12		525.2.d.a.274.1		2
56.3	even	6		9408.2.a.m.1.1		1
56.11	odd	6		1344.2.a.s.1.1		1
56.45	odd	6		9408.2.a.bv.1.1		1
56.53	even	6		1344.2.a.g.1.1		1
63.4	even	3		567.2.f.g.379.1		2
63.11	odd	6		567.2.f.b.190.1		2
63.25	even	3		567.2.f.g.190.1		2
63.32	odd	6		567.2.f.b.379.1		2
77.32	odd	6		2541.2.a.j.1.1		1
84.11	even	6		1008.2.a.l.1.1		1
84.59	odd	6		7056.2.a.p.1.1		1
91.25	even	6		3549.2.a.c.1.1		1
105.32	even	12		1575.2.d.a.1324.2		2
105.53	even	12		1575.2.d.a.1324.1		2
105.74	odd	6		1575.2.a.c.1.1		1
112.11	odd	12		5376.2.c.l.2689.2		2
112.53	even	12		5376.2.c.r.2689.1		2
112.67	odd	12		5376.2.c.l.2689.1		2
112.109	even	12		5376.2.c.r.2689.2		2
119.67	even	6		6069.2.a.b.1.1		1
133.18	odd	6		7581.2.a.d.1.1		1
140.39	odd	6		8400.2.a.bn.1.1		1
168.11	even	6		4032.2.a.k.1.1		1
168.53	odd	6		4032.2.a.h.1.1		1
231.32	even	6		7623.2.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.2.a.a.1.1	✓	1	7.4	even	3
63.2.a.a.1.1		1	21.11	odd	6
147.2.a.a.1.1		1	7.3	odd	6
147.2.e.b.67.1		2	1.1	even	1	trivial
147.2.e.b.79.1		2	7.2	even	3	inner
147.2.e.c.67.1		2	7.6	odd	2
147.2.e.c.79.1		2	7.5	odd	6
336.2.a.a.1.1		1	28.11	odd	6
441.2.a.f.1.1		1	21.17	even	6
441.2.e.a.226.1		2	21.2	odd	6
441.2.e.a.361.1		2	3.2	odd	2
441.2.e.b.226.1		2	21.5	even	6
441.2.e.b.361.1		2	21.20	even	2
525.2.a.d.1.1		1	35.4	even	6
525.2.d.a.274.1		2	35.32	odd	12
525.2.d.a.274.2		2	35.18	odd	12
567.2.f.b.190.1		2	63.11	odd	6
567.2.f.b.379.1		2	63.32	odd	6
567.2.f.g.190.1		2	63.25	even	3
567.2.f.g.379.1		2	63.4	even	3
1008.2.a.l.1.1		1	84.11	even	6
1344.2.a.g.1.1		1	56.53	even	6
1344.2.a.s.1.1		1	56.11	odd	6
1575.2.a.c.1.1		1	105.74	odd	6
1575.2.d.a.1324.1		2	105.53	even	12
1575.2.d.a.1324.2		2	105.32	even	12
2352.2.a.v.1.1		1	28.3	even	6
2352.2.q.e.961.1		2	28.19	even	6
2352.2.q.e.1537.1		2	28.27	even	2
2352.2.q.x.961.1		2	28.23	odd	6
2352.2.q.x.1537.1		2	4.3	odd	2
2541.2.a.j.1.1		1	77.32	odd	6
3549.2.a.c.1.1		1	91.25	even	6
3675.2.a.n.1.1		1	35.24	odd	6
4032.2.a.h.1.1		1	168.53	odd	6
4032.2.a.k.1.1		1	168.11	even	6
5376.2.c.l.2689.1		2	112.67	odd	12
5376.2.c.l.2689.2		2	112.11	odd	12
5376.2.c.r.2689.1		2	112.53	even	12
5376.2.c.r.2689.2		2	112.109	even	12
6069.2.a.b.1.1		1	119.67	even	6
7056.2.a.p.1.1		1	84.59	odd	6
7581.2.a.d.1.1		1	133.18	odd	6
7623.2.a.g.1.1		1	231.32	even	6
8400.2.a.bn.1.1		1	140.39	odd	6
9408.2.a.m.1.1		1	56.3	even	6
9408.2.a.bv.1.1		1	56.45	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	147.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{3} +(0.500000 - 0.866025i) q^{4} +(1.00000 + 1.73205i) q^{5} -1.00000 q^{6} +3.00000 q^{8} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\)	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{3} +(0.500000 - 0.866025i) q^{4} +(1.00000 + 1.73205i) q^{5} -1.00000 q^{6} +3.00000 q^{8} +(-0.500000 - 0.866025i) q^{9} +(-1.00000 + 1.73205i) q^{10} +(-2.00000 + 3.46410i) q^{11} +(0.500000 + 0.866025i) q^{12} -2.00000 q^{13} -2.00000 q^{15} +(0.500000 + 0.866025i) q^{16} +(3.00000 - 5.19615i) q^{17} +(0.500000 - 0.866025i) q^{18} +(-2.00000 - 3.46410i) q^{19} +2.00000 q^{20} -4.00000 q^{22} +(-1.50000 + 2.59808i) q^{24} +(0.500000 - 0.866025i) q^{25} +(-1.00000 - 1.73205i) q^{26} +1.00000 q^{27} -2.00000 q^{29} +(-1.00000 - 1.73205i) q^{30} +(2.50000 - 4.33013i) q^{32} +(-2.00000 - 3.46410i) q^{33} +6.00000 q^{34} -1.00000 q^{36} +(-3.00000 - 5.19615i) q^{37} +(2.00000 - 3.46410i) q^{38} +(1.00000 - 1.73205i) q^{39} +(3.00000 + 5.19615i) q^{40} +2.00000 q^{41} -4.00000 q^{43} +(2.00000 + 3.46410i) q^{44} +(1.00000 - 1.73205i) q^{45} -1.00000 q^{48} +1.00000 q^{50} +(3.00000 + 5.19615i) q^{51} +(-1.00000 + 1.73205i) q^{52} +(-3.00000 + 5.19615i) q^{53} +(0.500000 + 0.866025i) q^{54} -8.00000 q^{55} +4.00000 q^{57} +(-1.00000 - 1.73205i) q^{58} +(-6.00000 + 10.3923i) q^{59} +(-1.00000 + 1.73205i) q^{60} +(1.00000 + 1.73205i) q^{61} +7.00000 q^{64} +(-2.00000 - 3.46410i) q^{65} +(2.00000 - 3.46410i) q^{66} +(-2.00000 + 3.46410i) q^{67} +(-3.00000 - 5.19615i) q^{68} +(-1.50000 - 2.59808i) q^{72} +(3.00000 - 5.19615i) q^{73} +(3.00000 - 5.19615i) q^{74} +(0.500000 + 0.866025i) q^{75} -4.00000 q^{76} +2.00000 q^{78} +(8.00000 + 13.8564i) q^{79} +(-1.00000 + 1.73205i) q^{80} +(-0.500000 + 0.866025i) q^{81} +(1.00000 + 1.73205i) q^{82} -12.0000 q^{83} +12.0000 q^{85} +(-2.00000 - 3.46410i) q^{86} +(1.00000 - 1.73205i) q^{87} +(-6.00000 + 10.3923i) q^{88} +(7.00000 + 12.1244i) q^{89} +2.00000 q^{90} +(4.00000 - 6.92820i) q^{95} +(2.50000 + 4.33013i) q^{96} +18.0000 q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{3} + q^{4} + 2 q^{5} - 2 q^{6} + 6 q^{8} - q^{9}+O(q^{10}) \) 2 * q + q^2 - q^3 + q^4 + 2 * q^5 - 2 * q^6 + 6 * q^8 - q^9	\( 2 q + q^{2} - q^{3} + q^{4} + 2 q^{5} - 2 q^{6} + 6 q^{8} - q^{9} - 2 q^{10} - 4 q^{11} + q^{12} - 4 q^{13} - 4 q^{15} + q^{16} + 6 q^{17} + q^{18} - 4 q^{19} + 4 q^{20} - 8 q^{22} - 3 q^{24} + q^{25} - 2 q^{26} + 2 q^{27} - 4 q^{29} - 2 q^{30} + 5 q^{32} - 4 q^{33} + 12 q^{34} - 2 q^{36} - 6 q^{37} + 4 q^{38} + 2 q^{39} + 6 q^{40} + 4 q^{41} - 8 q^{43} + 4 q^{44} + 2 q^{45} - 2 q^{48} + 2 q^{50} + 6 q^{51} - 2 q^{52} - 6 q^{53} + q^{54} - 16 q^{55} + 8 q^{57} - 2 q^{58} - 12 q^{59} - 2 q^{60} + 2 q^{61} + 14 q^{64} - 4 q^{65} + 4 q^{66} - 4 q^{67} - 6 q^{68} - 3 q^{72} + 6 q^{73} + 6 q^{74} + q^{75} - 8 q^{76} + 4 q^{78} + 16 q^{79} - 2 q^{80} - q^{81} + 2 q^{82} - 24 q^{83} + 24 q^{85} - 4 q^{86} + 2 q^{87} - 12 q^{88} + 14 q^{89} + 4 q^{90} + 8 q^{95} + 5 q^{96} + 36 q^{97} + 8 q^{99}+O(q^{100}) \) 2 * q + q^2 - q^3 + q^4 + 2 * q^5 - 2 * q^6 + 6 * q^8 - q^9 - 2 * q^10 - 4 * q^11 + q^12 - 4 * q^13 - 4 * q^15 + q^16 + 6 * q^17 + q^18 - 4 * q^19 + 4 * q^20 - 8 * q^22 - 3 * q^24 + q^25 - 2 * q^26 + 2 * q^27 - 4 * q^29 - 2 * q^30 + 5 * q^32 - 4 * q^33 + 12 * q^34 - 2 * q^36 - 6 * q^37 + 4 * q^38 + 2 * q^39 + 6 * q^40 + 4 * q^41 - 8 * q^43 + 4 * q^44 + 2 * q^45 - 2 * q^48 + 2 * q^50 + 6 * q^51 - 2 * q^52 - 6 * q^53 + q^54 - 16 * q^55 + 8 * q^57 - 2 * q^58 - 12 * q^59 - 2 * q^60 + 2 * q^61 + 14 * q^64 - 4 * q^65 + 4 * q^66 - 4 * q^67 - 6 * q^68 - 3 * q^72 + 6 * q^73 + 6 * q^74 + q^75 - 8 * q^76 + 4 * q^78 + 16 * q^79 - 2 * q^80 - q^81 + 2 * q^82 - 24 * q^83 + 24 * q^85 - 4 * q^86 + 2 * q^87 - 12 * q^88 + 14 * q^89 + 4 * q^90 + 8 * q^95 + 5 * q^96 + 36 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more