LMFDB - Embedded newform 78.2.e.a.55.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [78,2,Mod(55,78)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(78, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

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

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

chi := DirichletCharacter("78.55");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 78 = 2 \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	78.e (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:	$0.622833135766$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			55.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	78.55
Dual form			78.2.e.a.61.1

$q$-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} +3.00000 q^{5} +(0.500000 - 0.866025i) q^{6} +(-1.00000 + 1.73205i) q^{7} -1.00000 q^{8} +(-0.500000 + 0.866025i) q^{9} +(1.50000 + 2.59808i) q^{10} +(-3.00000 - 5.19615i) q^{11} +1.00000 q^{12} +(-3.50000 + 0.866025i) q^{13} -2.00000 q^{14} +(-1.50000 - 2.59808i) q^{15} +(-0.500000 - 0.866025i) q^{16} +(1.50000 - 2.59808i) q^{17} -1.00000 q^{18} +(-1.00000 + 1.73205i) q^{19} +(-1.50000 + 2.59808i) q^{20} +2.00000 q^{21} +(3.00000 - 5.19615i) q^{22} +(3.00000 + 5.19615i) q^{23} +(0.500000 + 0.866025i) q^{24} +4.00000 q^{25} +(-2.50000 - 2.59808i) q^{26} +1.00000 q^{27} +(-1.00000 - 1.73205i) q^{28} +(-1.50000 - 2.59808i) q^{29} +(1.50000 - 2.59808i) q^{30} -4.00000 q^{31} +(0.500000 - 0.866025i) q^{32} +(-3.00000 + 5.19615i) q^{33} +3.00000 q^{34} +(-3.00000 + 5.19615i) q^{35} +(-0.500000 - 0.866025i) q^{36} +(3.50000 + 6.06218i) q^{37} -2.00000 q^{38} +(2.50000 + 2.59808i) q^{39} -3.00000 q^{40} +(1.50000 + 2.59808i) q^{41} +(1.00000 + 1.73205i) q^{42} +(5.00000 - 8.66025i) q^{43} +6.00000 q^{44} +(-1.50000 + 2.59808i) q^{45} +(-3.00000 + 5.19615i) q^{46} +6.00000 q^{47} +(-0.500000 + 0.866025i) q^{48} +(1.50000 + 2.59808i) q^{49} +(2.00000 + 3.46410i) q^{50} -3.00000 q^{51} +(1.00000 - 3.46410i) q^{52} +3.00000 q^{53} +(0.500000 + 0.866025i) q^{54} +(-9.00000 - 15.5885i) q^{55} +(1.00000 - 1.73205i) q^{56} +2.00000 q^{57} +(1.50000 - 2.59808i) q^{58} +3.00000 q^{60} +(3.50000 - 6.06218i) q^{61} +(-2.00000 - 3.46410i) q^{62} +(-1.00000 - 1.73205i) q^{63} +1.00000 q^{64} +(-10.5000 + 2.59808i) q^{65} -6.00000 q^{66} +(5.00000 + 8.66025i) q^{67} +(1.50000 + 2.59808i) q^{68} +(3.00000 - 5.19615i) q^{69} -6.00000 q^{70} +(-3.00000 + 5.19615i) q^{71} +(0.500000 - 0.866025i) q^{72} -13.0000 q^{73} +(-3.50000 + 6.06218i) q^{74} +(-2.00000 - 3.46410i) q^{75} +(-1.00000 - 1.73205i) q^{76} +12.0000 q^{77} +(-1.00000 + 3.46410i) q^{78} -4.00000 q^{79} +(-1.50000 - 2.59808i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(-1.50000 + 2.59808i) q^{82} -6.00000 q^{83} +(-1.00000 + 1.73205i) q^{84} +(4.50000 - 7.79423i) q^{85} +10.0000 q^{86} +(-1.50000 + 2.59808i) q^{87} +(3.00000 + 5.19615i) q^{88} +(-9.00000 - 15.5885i) q^{89} -3.00000 q^{90} +(2.00000 - 6.92820i) q^{91} -6.00000 q^{92} +(2.00000 + 3.46410i) q^{93} +(3.00000 + 5.19615i) q^{94} +(-3.00000 + 5.19615i) q^{95} -1.00000 q^{96} +(-7.00000 + 12.1244i) q^{97} +(-1.50000 + 2.59808i) q^{98} +6.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - q^{3} - q^{4} + 6 q^{5} + q^{6} - 2 q^{7} - 2 q^{8} - q^{9} + 3 q^{10} - 6 q^{11} + 2 q^{12} - 7 q^{13} - 4 q^{14} - 3 q^{15} - q^{16} + 3 q^{17} - 2 q^{18} - 2 q^{19} - 3 q^{20} + 4 q^{21}+ \cdots + 12 q^{99}+O(q^{100}) $ 2 * q + q^2 - q^3 - q^4 + 6 * q^5 + q^6 - 2 * q^7 - 2 * q^8 - q^9 + 3 * q^10 - 6 * q^11 + 2 * q^12 - 7 * q^13 - 4 * q^14 - 3 * q^15 - q^16 + 3 * q^17 - 2 * q^18 - 2 * q^19 - 3 * q^20 + 4 * q^21 + 6 * q^22 + 6 * q^23 + q^24 + 8 * q^25 - 5 * q^26 + 2 * q^27 - 2 * q^28 - 3 * q^29 + 3 * q^30 - 8 * q^31 + q^32 - 6 * q^33 + 6 * q^34 - 6 * q^35 - q^36 + 7 * q^37 - 4 * q^38 + 5 * q^39 - 6 * q^40 + 3 * q^41 + 2 * q^42 + 10 * q^43 + 12 * q^44 - 3 * q^45 - 6 * q^46 + 12 * q^47 - q^48 + 3 * q^49 + 4 * q^50 - 6 * q^51 + 2 * q^52 + 6 * q^53 + q^54 - 18 * q^55 + 2 * q^56 + 4 * q^57 + 3 * q^58 + 6 * q^60 + 7 * q^61 - 4 * q^62 - 2 * q^63 + 2 * q^64 - 21 * q^65 - 12 * q^66 + 10 * q^67 + 3 * q^68 + 6 * q^69 - 12 * q^70 - 6 * q^71 + q^72 - 26 * q^73 - 7 * q^74 - 4 * q^75 - 2 * q^76 + 24 * q^77 - 2 * q^78 - 8 * q^79 - 3 * q^80 - q^81 - 3 * q^82 - 12 * q^83 - 2 * q^84 + 9 * q^85 + 20 * q^86 - 3 * q^87 + 6 * q^88 - 18 * q^89 - 6 * q^90 + 4 * q^91 - 12 * q^92 + 4 * q^93 + 6 * q^94 - 6 * q^95 - 2 * q^96 - 14 * q^97 - 3 * q^98 + 12 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/78\mathbb{Z}\right)^\times$.

$n$	$53$	$67$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$

Coefficient data

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

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

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

$2$	0.500000	+	0.866025i	0.353553	+	0.612372i
$2$	0.500000	+	0.866025i	0.353553	+	0.612372i
$3$	−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
$5$	3.00000			1.34164			0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000			1.34164			0.670820	+	0.741620i	$0.265942\pi$
$6$	0.500000	−	0.866025i	0.204124	−	0.353553i
$7$	−1.00000	+	1.73205i	−0.377964	+	0.654654i	−0.990766	−	0.135583i	$-0.956709\pi$
$7$	−1.00000	+	1.73205i	−0.377964	+	0.654654i	0.612801	+	0.790237i	$0.290043\pi$
$8$	−1.00000			−0.353553
$9$	−0.500000	+	0.866025i	−0.166667	+	0.288675i
$10$	1.50000	+	2.59808i	0.474342	+	0.821584i
$11$	−3.00000	−	5.19615i	−0.904534	−	1.56670i	−0.821541	−	0.570149i	$-0.806886\pi$
$11$	−3.00000	−	5.19615i	−0.904534	−	1.56670i	−0.0829925	−	0.996550i	$-0.526448\pi$
$12$	1.00000			0.288675
$13$	−3.50000	+	0.866025i	−0.970725	+	0.240192i
$13$	−3.50000	+	0.866025i	−0.970725	+	0.240192i
$14$	−2.00000			−0.534522
$15$	−1.50000	−	2.59808i	−0.387298	−	0.670820i
$16$	−0.500000	−	0.866025i	−0.125000	−	0.216506i
$17$	1.50000	−	2.59808i	0.363803	−	0.630126i	−0.624780	−	0.780801i	$-0.714811\pi$
$17$	1.50000	−	2.59808i	0.363803	−	0.630126i	0.988583	+	0.150675i	$0.0481447\pi$
$18$	−1.00000			−0.235702
$19$	−1.00000	+	1.73205i	−0.229416	+	0.397360i	−0.957635	−	0.287984i	$-0.907015\pi$
$19$	−1.00000	+	1.73205i	−0.229416	+	0.397360i	0.728219	+	0.685344i	$0.240348\pi$
$20$	−1.50000	+	2.59808i	−0.335410	+	0.580948i
$21$	2.00000			0.436436
$22$	3.00000	−	5.19615i	0.639602	−	1.10782i
$23$	3.00000	+	5.19615i	0.625543	+	1.08347i	0.988436	+	0.151642i	$0.0484560\pi$
$23$	3.00000	+	5.19615i	0.625543	+	1.08347i	−0.362892	+	0.931831i	$0.618211\pi$
$24$	0.500000	+	0.866025i	0.102062	+	0.176777i
$25$	4.00000			0.800000
$26$	−2.50000	−	2.59808i	−0.490290	−	0.509525i
$27$	1.00000			0.192450
$28$	−1.00000	−	1.73205i	−0.188982	−	0.327327i
$29$	−1.50000	−	2.59808i	−0.278543	−	0.482451i	0.692480	−	0.721437i	$-0.256518\pi$
$29$	−1.50000	−	2.59808i	−0.278543	−	0.482451i	−0.971023	+	0.238987i	$0.923185\pi$
$30$	1.50000	−	2.59808i	0.273861	−	0.474342i
$31$	−4.00000			−0.718421			−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000			−0.718421			−0.359211	+	0.933257i	$0.616954\pi$
$32$	0.500000	−	0.866025i	0.0883883	−	0.153093i
$33$	−3.00000	+	5.19615i	−0.522233	+	0.904534i
$34$	3.00000			0.514496
$35$	−3.00000	+	5.19615i	−0.507093	+	0.878310i
$36$	−0.500000	−	0.866025i	−0.0833333	−	0.144338i
$37$	3.50000	+	6.06218i	0.575396	+	0.996616i	0.995998	+	0.0893706i	$0.0284856\pi$
$37$	3.50000	+	6.06218i	0.575396	+	0.996616i	−0.420602	+	0.907245i	$0.638181\pi$
$38$	−2.00000			−0.324443
$39$	2.50000	+	2.59808i	0.400320	+	0.416025i
$40$	−3.00000			−0.474342
$41$	1.50000	+	2.59808i	0.234261	+	0.405751i	0.959058	−	0.283211i	$-0.0913998\pi$
$41$	1.50000	+	2.59808i	0.234261	+	0.405751i	−0.724797	+	0.688963i	$0.758066\pi$
$42$	1.00000	+	1.73205i	0.154303	+	0.267261i
$43$	5.00000	−	8.66025i	0.762493	−	1.32068i	−0.179069	−	0.983836i	$-0.557309\pi$
$43$	5.00000	−	8.66025i	0.762493	−	1.32068i	0.941562	−	0.336840i	$-0.109358\pi$
$44$	6.00000			0.904534
$45$	−1.50000	+	2.59808i	−0.223607	+	0.387298i
$46$	−3.00000	+	5.19615i	−0.442326	+	0.766131i
$47$	6.00000			0.875190			0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000			0.875190			0.437595	+	0.899172i	$0.355830\pi$
$48$	−0.500000	+	0.866025i	−0.0721688	+	0.125000i
$49$	1.50000	+	2.59808i	0.214286	+	0.371154i
$50$	2.00000	+	3.46410i	0.282843	+	0.489898i
$51$	−3.00000			−0.420084
$52$	1.00000	−	3.46410i	0.138675	−	0.480384i
$53$	3.00000			0.412082			0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000			0.412082			0.206041	+	0.978543i	$0.433942\pi$
$54$	0.500000	+	0.866025i	0.0680414	+	0.117851i
$55$	−9.00000	−	15.5885i	−1.21356	−	2.10195i
$56$	1.00000	−	1.73205i	0.133631	−	0.231455i
$57$	2.00000			0.264906
$58$	1.50000	−	2.59808i	0.196960	−	0.341144i
$59$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$59$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$60$	3.00000			0.387298
$61$	3.50000	−	6.06218i	0.448129	−	0.776182i	−0.550135	−	0.835076i	$-0.685424\pi$
$61$	3.50000	−	6.06218i	0.448129	−	0.776182i	0.998264	+	0.0588933i	$0.0187572\pi$
$62$	−2.00000	−	3.46410i	−0.254000	−	0.439941i
$63$	−1.00000	−	1.73205i	−0.125988	−	0.218218i
$64$	1.00000			0.125000
$65$	−10.5000	+	2.59808i	−1.30236	+	0.322252i
$66$	−6.00000			−0.738549
$67$	5.00000	+	8.66025i	0.610847	+	1.05802i	0.991098	+	0.133135i	$0.0425044\pi$
$67$	5.00000	+	8.66025i	0.610847	+	1.05802i	−0.380251	+	0.924883i	$0.624162\pi$
$68$	1.50000	+	2.59808i	0.181902	+	0.315063i
$69$	3.00000	−	5.19615i	0.361158	−	0.625543i
$70$	−6.00000			−0.717137
$71$	−3.00000	+	5.19615i	−0.356034	+	0.616670i	−0.987294	−	0.158901i	$-0.949205\pi$
$71$	−3.00000	+	5.19615i	−0.356034	+	0.616670i	0.631260	+	0.775571i	$0.282538\pi$
$72$	0.500000	−	0.866025i	0.0589256	−	0.102062i
$73$	−13.0000			−1.52153			−0.760767	−	0.649025i	$-0.775177\pi$
$73$	−13.0000			−1.52153			−0.760767	+	0.649025i	$0.775177\pi$
$74$	−3.50000	+	6.06218i	−0.406867	+	0.704714i
$75$	−2.00000	−	3.46410i	−0.230940	−	0.400000i
$76$	−1.00000	−	1.73205i	−0.114708	−	0.198680i
$77$	12.0000			1.36753
$78$	−1.00000	+	3.46410i	−0.113228	+	0.392232i
$79$	−4.00000			−0.450035			−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000			−0.450035			−0.225018	+	0.974355i	$0.572244\pi$
$80$	−1.50000	−	2.59808i	−0.167705	−	0.290474i
$81$	−0.500000	−	0.866025i	−0.0555556	−	0.0962250i
$82$	−1.50000	+	2.59808i	−0.165647	+	0.286910i
$83$	−6.00000			−0.658586			−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000			−0.658586			−0.329293	+	0.944228i	$0.606810\pi$
$84$	−1.00000	+	1.73205i	−0.109109	+	0.188982i
$85$	4.50000	−	7.79423i	0.488094	−	0.845403i
$86$	10.0000			1.07833
$87$	−1.50000	+	2.59808i	−0.160817	+	0.278543i
$88$	3.00000	+	5.19615i	0.319801	+	0.553912i
$89$	−9.00000	−	15.5885i	−0.953998	−	1.65237i	−0.736644	−	0.676280i	$-0.763591\pi$
$89$	−9.00000	−	15.5885i	−0.953998	−	1.65237i	−0.217354	−	0.976093i	$-0.569742\pi$
$90$	−3.00000			−0.316228
$91$	2.00000	−	6.92820i	0.209657	−	0.726273i
$92$	−6.00000			−0.625543
$93$	2.00000	+	3.46410i	0.207390	+	0.359211i
$94$	3.00000	+	5.19615i	0.309426	+	0.535942i
$95$	−3.00000	+	5.19615i	−0.307794	+	0.533114i
$96$	−1.00000			−0.102062
$97$	−7.00000	+	12.1244i	−0.710742	+	1.23104i	0.253837	+	0.967247i	$0.418307\pi$
$97$	−7.00000	+	12.1244i	−0.710742	+	1.23104i	−0.964579	+	0.263795i	$0.915026\pi$
$98$	−1.50000	+	2.59808i	−0.151523	+	0.262445i
$99$	6.00000			0.603023
$100$	−2.00000	+	3.46410i	−0.200000	+	0.346410i
$101$	−7.50000	−	12.9904i	−0.746278	−	1.29259i	−0.949595	−	0.313478i	$-0.898506\pi$
$101$	−7.50000	−	12.9904i	−0.746278	−	1.29259i	0.203317	−	0.979113i	$-0.434828\pi$
$102$	−1.50000	−	2.59808i	−0.148522	−	0.257248i
$103$	14.0000			1.37946			0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000			1.37946			0.689730	+	0.724066i	$0.257729\pi$
$104$	3.50000	−	0.866025i	0.343203	−	0.0849208i
$105$	6.00000			0.585540
$106$	1.50000	+	2.59808i	0.145693	+	0.252347i
$107$	3.00000	+	5.19615i	0.290021	+	0.502331i	0.973814	−	0.227345i	$-0.0730044\pi$
$107$	3.00000	+	5.19615i	0.290021	+	0.502331i	−0.683793	+	0.729676i	$0.739671\pi$
$108$	−0.500000	+	0.866025i	−0.0481125	+	0.0833333i
$109$	14.0000			1.34096			0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000			1.34096			0.670478	+	0.741929i	$0.266089\pi$
$110$	9.00000	−	15.5885i	0.858116	−	1.48630i
$111$	3.50000	−	6.06218i	0.332205	−	0.575396i
$112$	2.00000			0.188982
$113$	1.50000	−	2.59808i	0.141108	−	0.244406i	−0.786806	−	0.617200i	$-0.788267\pi$
$113$	1.50000	−	2.59808i	0.141108	−	0.244406i	0.927914	+	0.372794i	$0.121600\pi$
$114$	1.00000	+	1.73205i	0.0936586	+	0.162221i
$115$	9.00000	+	15.5885i	0.839254	+	1.45363i
$116$	3.00000			0.278543
$117$	1.00000	−	3.46410i	0.0924500	−	0.320256i
$118$		0			0
$119$	3.00000	+	5.19615i	0.275010	+	0.476331i
$120$	1.50000	+	2.59808i	0.136931	+	0.237171i
$121$	−12.5000	+	21.6506i	−1.13636	+	1.96824i
$122$	7.00000			0.633750
$123$	1.50000	−	2.59808i	0.135250	−	0.234261i
$124$	2.00000	−	3.46410i	0.179605	−	0.311086i
$125$	−3.00000			−0.268328
$126$	1.00000	−	1.73205i	0.0890871	−	0.154303i
$127$	2.00000	+	3.46410i	0.177471	+	0.307389i	0.941014	−	0.338368i	$-0.109875\pi$
$127$	2.00000	+	3.46410i	0.177471	+	0.307389i	−0.763542	+	0.645758i	$0.776542\pi$
$128$	0.500000	+	0.866025i	0.0441942	+	0.0765466i
$129$	−10.0000			−0.880451
$130$	−7.50000	−	7.79423i	−0.657794	−	0.683599i
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$	−3.00000	−	5.19615i	−0.261116	−	0.452267i
$133$	−2.00000	−	3.46410i	−0.173422	−	0.300376i
$134$	−5.00000	+	8.66025i	−0.431934	+	0.748132i
$135$	3.00000			0.258199
$136$	−1.50000	+	2.59808i	−0.128624	+	0.222783i
$137$	−4.50000	+	7.79423i	−0.384461	+	0.665906i	−0.991694	−	0.128618i	$-0.958946\pi$
$137$	−4.50000	+	7.79423i	−0.384461	+	0.665906i	0.607233	+	0.794524i	$0.292279\pi$
$138$	6.00000			0.510754
$139$	2.00000	−	3.46410i	0.169638	−	0.293821i	−0.768655	−	0.639664i	$-0.779074\pi$
$139$	2.00000	−	3.46410i	0.169638	−	0.293821i	0.938293	+	0.345843i	$0.112407\pi$
$140$	−3.00000	−	5.19615i	−0.253546	−	0.439155i
$141$	−3.00000	−	5.19615i	−0.252646	−	0.437595i
$142$	−6.00000			−0.503509
$143$	15.0000	+	15.5885i	1.25436	+	1.30357i
$144$	1.00000			0.0833333
$145$	−4.50000	−	7.79423i	−0.373705	−	0.647275i
$146$	−6.50000	−	11.2583i	−0.537944	−	0.931746i
$147$	1.50000	−	2.59808i	0.123718	−	0.214286i
$148$	−7.00000			−0.575396
$149$	4.50000	−	7.79423i	0.368654	−	0.638528i	−0.620701	−	0.784047i	$-0.713152\pi$
$149$	4.50000	−	7.79423i	0.368654	−	0.638528i	0.989355	+	0.145519i	$0.0464853\pi$
$150$	2.00000	−	3.46410i	0.163299	−	0.282843i
$151$	−10.0000			−0.813788			−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000			−0.813788			−0.406894	+	0.913475i	$0.633388\pi$
$152$	1.00000	−	1.73205i	0.0811107	−	0.140488i
$153$	1.50000	+	2.59808i	0.121268	+	0.210042i
$154$	6.00000	+	10.3923i	0.483494	+	0.837436i
$155$	−12.0000			−0.963863
$156$	−3.50000	+	0.866025i	−0.280224	+	0.0693375i
$157$	5.00000			0.399043			0.199522	−	0.979893i	$-0.436061\pi$
$157$	5.00000			0.399043			0.199522	+	0.979893i	$0.436061\pi$
$158$	−2.00000	−	3.46410i	−0.159111	−	0.275589i
$159$	−1.50000	−	2.59808i	−0.118958	−	0.206041i
$160$	1.50000	−	2.59808i	0.118585	−	0.205396i
$161$	−12.0000			−0.945732
$162$	0.500000	−	0.866025i	0.0392837	−	0.0680414i
$163$	2.00000	−	3.46410i	0.156652	−	0.271329i	−0.777007	−	0.629492i	$-0.783263\pi$
$163$	2.00000	−	3.46410i	0.156652	−	0.271329i	0.933659	+	0.358162i	$0.116597\pi$
$164$	−3.00000			−0.234261
$165$	−9.00000	+	15.5885i	−0.700649	+	1.21356i
$166$	−3.00000	−	5.19615i	−0.232845	−	0.403300i
$167$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$167$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$168$	−2.00000			−0.154303
$169$	11.5000	−	6.06218i	0.884615	−	0.466321i
$170$	9.00000			0.690268
$171$	−1.00000	−	1.73205i	−0.0764719	−	0.132453i
$172$	5.00000	+	8.66025i	0.381246	+	0.660338i
$173$	3.00000	−	5.19615i	0.228086	−	0.395056i	−0.729155	−	0.684349i	$-0.760087\pi$
$173$	3.00000	−	5.19615i	0.228086	−	0.395056i	0.957241	+	0.289292i	$0.0934200\pi$
$174$	−3.00000			−0.227429
$175$	−4.00000	+	6.92820i	−0.302372	+	0.523723i
$176$	−3.00000	+	5.19615i	−0.226134	+	0.391675i
$177$		0			0
$178$	9.00000	−	15.5885i	0.674579	−	1.16840i
$179$	−3.00000	−	5.19615i	−0.224231	−	0.388379i	0.731858	−	0.681457i	$-0.238654\pi$
$179$	−3.00000	−	5.19615i	−0.224231	−	0.388379i	−0.956088	+	0.293079i	$0.905320\pi$
$180$	−1.50000	−	2.59808i	−0.111803	−	0.193649i
$181$	−7.00000			−0.520306			−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000			−0.520306			−0.260153	+	0.965567i	$0.583773\pi$
$182$	7.00000	−	1.73205i	0.518875	−	0.128388i
$183$	−7.00000			−0.517455
$184$	−3.00000	−	5.19615i	−0.221163	−	0.383065i
$185$	10.5000	+	18.1865i	0.771975	+	1.33710i
$186$	−2.00000	+	3.46410i	−0.146647	+	0.254000i
$187$	−18.0000			−1.31629
$188$	−3.00000	+	5.19615i	−0.218797	+	0.378968i
$189$	−1.00000	+	1.73205i	−0.0727393	+	0.125988i
$190$	−6.00000			−0.435286
$191$	6.00000	−	10.3923i	0.434145	−	0.751961i	−0.563081	−	0.826402i	$-0.690384\pi$
$191$	6.00000	−	10.3923i	0.434145	−	0.751961i	0.997225	+	0.0744412i	$0.0237173\pi$
$192$	−0.500000	−	0.866025i	−0.0360844	−	0.0625000i
$193$	−11.5000	−	19.9186i	−0.827788	−	1.43377i	−0.899770	−	0.436365i	$-0.856266\pi$
$193$	−11.5000	−	19.9186i	−0.827788	−	1.43377i	0.0719816	−	0.997406i	$-0.477068\pi$
$194$	−14.0000			−1.00514
$195$	7.50000	+	7.79423i	0.537086	+	0.558156i
$196$	−3.00000			−0.214286
$197$	−3.00000	−	5.19615i	−0.213741	−	0.370211i	0.739141	−	0.673550i	$-0.235232\pi$
$197$	−3.00000	−	5.19615i	−0.213741	−	0.370211i	−0.952882	+	0.303340i	$0.901898\pi$
$198$	3.00000	+	5.19615i	0.213201	+	0.369274i
$199$	5.00000	−	8.66025i	0.354441	−	0.613909i	−0.632581	−	0.774494i	$-0.718005\pi$
$199$	5.00000	−	8.66025i	0.354441	−	0.613909i	0.987022	+	0.160585i	$0.0513380\pi$
$200$	−4.00000			−0.282843
$201$	5.00000	−	8.66025i	0.352673	−	0.610847i
$202$	7.50000	−	12.9904i	0.527698	−	0.914000i
$203$	6.00000			0.421117
$204$	1.50000	−	2.59808i	0.105021	−	0.181902i
$205$	4.50000	+	7.79423i	0.314294	+	0.544373i
$206$	7.00000	+	12.1244i	0.487713	+	0.844744i
$207$	−6.00000			−0.417029
$208$	2.50000	+	2.59808i	0.173344	+	0.180144i
$209$	12.0000			0.830057
$210$	3.00000	+	5.19615i	0.207020	+	0.358569i
$211$	8.00000	+	13.8564i	0.550743	+	0.953914i	0.998221	+	0.0596196i	$0.0189888\pi$
$211$	8.00000	+	13.8564i	0.550743	+	0.953914i	−0.447478	+	0.894295i	$0.647678\pi$
$212$	−1.50000	+	2.59808i	−0.103020	+	0.178437i
$213$	6.00000			0.411113
$214$	−3.00000	+	5.19615i	−0.205076	+	0.355202i
$215$	15.0000	−	25.9808i	1.02299	−	1.77187i
$216$	−1.00000			−0.0680414
$217$	4.00000	−	6.92820i	0.271538	−	0.470317i
$218$	7.00000	+	12.1244i	0.474100	+	0.821165i
$219$	6.50000	+	11.2583i	0.439229	+	0.760767i
$220$	18.0000			1.21356
$221$	−3.00000	+	10.3923i	−0.201802	+	0.699062i
$222$	7.00000			0.469809
$223$	−4.00000	−	6.92820i	−0.267860	−	0.463947i	0.700449	−	0.713702i	$-0.252983\pi$
$223$	−4.00000	−	6.92820i	−0.267860	−	0.463947i	−0.968309	+	0.249756i	$0.919650\pi$
$224$	1.00000	+	1.73205i	0.0668153	+	0.115728i
$225$	−2.00000	+	3.46410i	−0.133333	+	0.230940i
$226$	3.00000			0.199557
$227$	−9.00000	+	15.5885i	−0.597351	+	1.03464i	0.395860	+	0.918311i	$0.370447\pi$
$227$	−9.00000	+	15.5885i	−0.597351	+	1.03464i	−0.993210	+	0.116331i	$0.962887\pi$
$228$	−1.00000	+	1.73205i	−0.0662266	+	0.114708i
$229$	−22.0000			−1.45380			−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000			−1.45380			−0.726900	+	0.686743i	$0.759040\pi$
$230$	−9.00000	+	15.5885i	−0.593442	+	1.02787i
$231$	−6.00000	−	10.3923i	−0.394771	−	0.683763i
$232$	1.50000	+	2.59808i	0.0984798	+	0.170572i
$233$	−6.00000			−0.393073			−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000			−0.393073			−0.196537	+	0.980497i	$0.562969\pi$
$234$	3.50000	−	0.866025i	0.228802	−	0.0566139i
$235$	18.0000			1.17419
$236$		0			0
$237$	2.00000	+	3.46410i	0.129914	+	0.225018i
$238$	−3.00000	+	5.19615i	−0.194461	+	0.336817i
$239$	6.00000			0.388108			0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000			0.388108			0.194054	+	0.980991i	$0.437836\pi$
$240$	−1.50000	+	2.59808i	−0.0968246	+	0.167705i
$241$	0.500000	−	0.866025i	0.0322078	−	0.0557856i	−0.849472	−	0.527633i	$-0.823079\pi$
$241$	0.500000	−	0.866025i	0.0322078	−	0.0557856i	0.881680	+	0.471848i	$0.156413\pi$
$242$	−25.0000			−1.60706
$243$	−0.500000	+	0.866025i	−0.0320750	+	0.0555556i
$244$	3.50000	+	6.06218i	0.224065	+	0.388091i
$245$	4.50000	+	7.79423i	0.287494	+	0.497955i
$246$	3.00000			0.191273
$247$	2.00000	−	6.92820i	0.127257	−	0.440831i
$248$	4.00000			0.254000
$249$	3.00000	+	5.19615i	0.190117	+	0.329293i
$250$	−1.50000	−	2.59808i	−0.0948683	−	0.164317i
$251$	6.00000	−	10.3923i	0.378717	−	0.655956i	−0.612159	−	0.790735i	$-0.709699\pi$
$251$	6.00000	−	10.3923i	0.378717	−	0.655956i	0.990876	+	0.134778i	$0.0430322\pi$
$252$	2.00000			0.125988
$253$	18.0000	−	31.1769i	1.13165	−	1.96008i
$254$	−2.00000	+	3.46410i	−0.125491	+	0.217357i
$255$	−9.00000			−0.563602
$256$	−0.500000	+	0.866025i	−0.0312500	+	0.0541266i
$257$	1.50000	+	2.59808i	0.0935674	+	0.162064i	0.909010	−	0.416775i	$-0.136840\pi$
$257$	1.50000	+	2.59808i	0.0935674	+	0.162064i	−0.815442	+	0.578838i	$0.803506\pi$
$258$	−5.00000	−	8.66025i	−0.311286	−	0.539164i
$259$	−14.0000			−0.869918
$260$	3.00000	−	10.3923i	0.186052	−	0.644503i
$261$	3.00000			0.185695
$262$		0			0
$263$	3.00000	+	5.19615i	0.184988	+	0.320408i	0.943572	−	0.331166i	$-0.107442\pi$
$263$	3.00000	+	5.19615i	0.184988	+	0.320408i	−0.758585	+	0.651575i	$0.774109\pi$
$264$	3.00000	−	5.19615i	0.184637	−	0.319801i
$265$	9.00000			0.552866
$266$	2.00000	−	3.46410i	0.122628	−	0.212398i
$267$	−9.00000	+	15.5885i	−0.550791	+	0.953998i
$268$	−10.0000			−0.610847
$269$	−9.00000	+	15.5885i	−0.548740	+	0.950445i	0.449622	+	0.893219i	$0.351559\pi$
$269$	−9.00000	+	15.5885i	−0.548740	+	0.950445i	−0.998361	+	0.0572259i	$0.981774\pi$
$270$	1.50000	+	2.59808i	0.0912871	+	0.158114i
$271$	8.00000	+	13.8564i	0.485965	+	0.841717i	0.999870	−	0.0161307i	$-0.00513477\pi$
$271$	8.00000	+	13.8564i	0.485965	+	0.841717i	−0.513905	+	0.857847i	$0.671801\pi$
$272$	−3.00000			−0.181902
$273$	−7.00000	+	1.73205i	−0.423659	+	0.104828i
$274$	−9.00000			−0.543710
$275$	−12.0000	−	20.7846i	−0.723627	−	1.25336i
$276$	3.00000	+	5.19615i	0.180579	+	0.312772i
$277$	−8.50000	+	14.7224i	−0.510716	+	0.884585i	0.489207	+	0.872167i	$0.337286\pi$
$277$	−8.50000	+	14.7224i	−0.510716	+	0.884585i	−0.999923	+	0.0124177i	$0.996047\pi$
$278$	4.00000			0.239904
$279$	2.00000	−	3.46410i	0.119737	−	0.207390i
$280$	3.00000	−	5.19615i	0.179284	−	0.310530i
$281$	9.00000			0.536895			0.268447	−	0.963294i	$-0.413489\pi$
$281$	9.00000			0.536895			0.268447	+	0.963294i	$0.413489\pi$
$282$	3.00000	−	5.19615i	0.178647	−	0.309426i
$283$	−7.00000	−	12.1244i	−0.416107	−	0.720718i	0.579437	−	0.815017i	$-0.303272\pi$
$283$	−7.00000	−	12.1244i	−0.416107	−	0.720718i	−0.995544	+	0.0942988i	$0.969939\pi$
$284$	−3.00000	−	5.19615i	−0.178017	−	0.308335i
$285$	6.00000			0.355409
$286$	−6.00000	+	20.7846i	−0.354787	+	1.22902i
$287$	−6.00000			−0.354169
$288$	0.500000	+	0.866025i	0.0294628	+	0.0510310i
$289$	4.00000	+	6.92820i	0.235294	+	0.407541i
$290$	4.50000	−	7.79423i	0.264249	−	0.457693i
$291$	14.0000			0.820695
$292$	6.50000	−	11.2583i	0.380384	−	0.658844i
$293$	10.5000	−	18.1865i	0.613417	−	1.06247i	−0.377244	−	0.926114i	$-0.623128\pi$
$293$	10.5000	−	18.1865i	0.613417	−	1.06247i	0.990660	−	0.136355i	$-0.0435386\pi$
$294$	3.00000			0.174964
$295$		0			0
$296$	−3.50000	−	6.06218i	−0.203433	−	0.352357i
$297$	−3.00000	−	5.19615i	−0.174078	−	0.301511i
$298$	9.00000			0.521356
$299$	−15.0000	−	15.5885i	−0.867472	−	0.901504i
$300$	4.00000			0.230940
$301$	10.0000	+	17.3205i	0.576390	+	0.998337i
$302$	−5.00000	−	8.66025i	−0.287718	−	0.498342i
$303$	−7.50000	+	12.9904i	−0.430864	+	0.746278i
$304$	2.00000			0.114708
$305$	10.5000	−	18.1865i	0.601228	−	1.04136i
$306$	−1.50000	+	2.59808i	−0.0857493	+	0.148522i
$307$	−10.0000			−0.570730			−0.285365	−	0.958419i	$-0.592115\pi$
$307$	−10.0000			−0.570730			−0.285365	+	0.958419i	$0.592115\pi$
$308$	−6.00000	+	10.3923i	−0.341882	+	0.592157i
$309$	−7.00000	−	12.1244i	−0.398216	−	0.689730i
$310$	−6.00000	−	10.3923i	−0.340777	−	0.590243i
$311$	−30.0000			−1.70114			−0.850572	−	0.525859i	$-0.823744\pi$
$311$	−30.0000			−1.70114			−0.850572	+	0.525859i	$0.823744\pi$
$312$	−2.50000	−	2.59808i	−0.141535	−	0.147087i
$313$	−10.0000			−0.565233			−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000			−0.565233			−0.282617	+	0.959233i	$0.591202\pi$
$314$	2.50000	+	4.33013i	0.141083	+	0.244363i
$315$	−3.00000	−	5.19615i	−0.169031	−	0.292770i
$316$	2.00000	−	3.46410i	0.112509	−	0.194871i
$317$	3.00000			0.168497			0.0842484	−	0.996445i	$-0.473151\pi$
$317$	3.00000			0.168497			0.0842484	+	0.996445i	$0.473151\pi$
$318$	1.50000	−	2.59808i	0.0841158	−	0.145693i
$319$	−9.00000	+	15.5885i	−0.503903	+	0.872786i
$320$	3.00000			0.167705
$321$	3.00000	−	5.19615i	0.167444	−	0.290021i
$322$	−6.00000	−	10.3923i	−0.334367	−	0.579141i
$323$	3.00000	+	5.19615i	0.166924	+	0.289122i
$324$	1.00000			0.0555556
$325$	−14.0000	+	3.46410i	−0.776580	+	0.192154i
$326$	4.00000			0.221540
$327$	−7.00000	−	12.1244i	−0.387101	−	0.670478i
$328$	−1.50000	−	2.59808i	−0.0828236	−	0.143455i
$329$	−6.00000	+	10.3923i	−0.330791	+	0.572946i
$330$	−18.0000			−0.990867
$331$	2.00000	−	3.46410i	0.109930	−	0.190404i	−0.805812	−	0.592172i	$-0.798271\pi$
$331$	2.00000	−	3.46410i	0.109930	−	0.190404i	0.915742	+	0.401768i	$0.131604\pi$
$332$	3.00000	−	5.19615i	0.164646	−	0.285176i
$333$	−7.00000			−0.383598
$334$		0			0
$335$	15.0000	+	25.9808i	0.819538	+	1.41948i
$336$	−1.00000	−	1.73205i	−0.0545545	−	0.0944911i
$337$	23.0000			1.25289			0.626445	−	0.779466i	$-0.284509\pi$
$337$	23.0000			1.25289			0.626445	+	0.779466i	$0.284509\pi$
$338$	11.0000	+	6.92820i	0.598321	+	0.376845i
$339$	−3.00000			−0.162938
$340$	4.50000	+	7.79423i	0.244047	+	0.422701i
$341$	12.0000	+	20.7846i	0.649836	+	1.12555i
$342$	1.00000	−	1.73205i	0.0540738	−	0.0936586i
$343$	−20.0000			−1.07990
$344$	−5.00000	+	8.66025i	−0.269582	+	0.466930i
$345$	9.00000	−	15.5885i	0.484544	−	0.839254i
$346$	6.00000			0.322562
$347$	15.0000	−	25.9808i	0.805242	−	1.39472i	−0.110885	−	0.993833i	$-0.535369\pi$
$347$	15.0000	−	25.9808i	0.805242	−	1.39472i	0.916127	−	0.400887i	$-0.131298\pi$
$348$	−1.50000	−	2.59808i	−0.0804084	−	0.139272i
$349$	5.00000	+	8.66025i	0.267644	+	0.463573i	0.968253	−	0.249973i	$-0.0804216\pi$
$349$	5.00000	+	8.66025i	0.267644	+	0.463573i	−0.700609	+	0.713545i	$0.747088\pi$
$350$	−8.00000			−0.427618
$351$	−3.50000	+	0.866025i	−0.186816	+	0.0462250i
$352$	−6.00000			−0.319801
$353$	7.50000	+	12.9904i	0.399185	+	0.691408i	0.993626	−	0.112731i	$-0.0359599\pi$
$353$	7.50000	+	12.9904i	0.399185	+	0.691408i	−0.594441	+	0.804139i	$0.702627\pi$
$354$		0			0
$355$	−9.00000	+	15.5885i	−0.477670	+	0.827349i
$356$	18.0000			0.953998
$357$	3.00000	−	5.19615i	0.158777	−	0.275010i
$358$	3.00000	−	5.19615i	0.158555	−	0.274625i
$359$	6.00000			0.316668			0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000			0.316668			0.158334	+	0.987386i	$0.449388\pi$
$360$	1.50000	−	2.59808i	0.0790569	−	0.136931i
$361$	7.50000	+	12.9904i	0.394737	+	0.683704i
$362$	−3.50000	−	6.06218i	−0.183956	−	0.318621i
$363$	25.0000			1.31216
$364$	5.00000	+	5.19615i	0.262071	+	0.272352i
$365$	−39.0000			−2.04135
$366$	−3.50000	−	6.06218i	−0.182948	−	0.316875i
$367$	−1.00000	−	1.73205i	−0.0521996	−	0.0904123i	0.838745	−	0.544524i	$-0.183290\pi$
$367$	−1.00000	−	1.73205i	−0.0521996	−	0.0904123i	−0.890945	+	0.454112i	$0.849957\pi$
$368$	3.00000	−	5.19615i	0.156386	−	0.270868i
$369$	−3.00000			−0.156174
$370$	−10.5000	+	18.1865i	−0.545869	+	0.945473i
$371$	−3.00000	+	5.19615i	−0.155752	+	0.269771i
$372$	−4.00000			−0.207390
$373$	−14.5000	+	25.1147i	−0.750782	+	1.30039i	0.196663	+	0.980471i	$0.436990\pi$
$373$	−14.5000	+	25.1147i	−0.750782	+	1.30039i	−0.947444	+	0.319921i	$0.896344\pi$
$374$	−9.00000	−	15.5885i	−0.465379	−	0.806060i
$375$	1.50000	+	2.59808i	0.0774597	+	0.134164i
$376$	−6.00000			−0.309426
$377$	7.50000	+	7.79423i	0.386270	+	0.401423i
$378$	−2.00000			−0.102869
$379$	−10.0000	−	17.3205i	−0.513665	−	0.889695i	−0.999874	−	0.0158521i	$-0.994954\pi$
$379$	−10.0000	−	17.3205i	−0.513665	−	0.889695i	0.486209	−	0.873843i	$-0.338379\pi$
$380$	−3.00000	−	5.19615i	−0.153897	−	0.266557i
$381$	2.00000	−	3.46410i	0.102463	−	0.177471i
$382$	12.0000			0.613973
$383$	−12.0000	+	20.7846i	−0.613171	+	1.06204i	0.377531	+	0.925997i	$0.376773\pi$
$383$	−12.0000	+	20.7846i	−0.613171	+	1.06204i	−0.990702	+	0.136047i	$0.956560\pi$
$384$	0.500000	−	0.866025i	0.0255155	−	0.0441942i
$385$	36.0000			1.83473
$386$	11.5000	−	19.9186i	0.585335	−	1.01383i
$387$	5.00000	+	8.66025i	0.254164	+	0.440225i
$388$	−7.00000	−	12.1244i	−0.355371	−	0.615521i
$389$	39.0000			1.97738			0.988689	−	0.149979i	$-0.0479205\pi$
$389$	39.0000			1.97738			0.988689	+	0.149979i	$0.0479205\pi$
$390$	−3.00000	+	10.3923i	−0.151911	+	0.526235i
$391$	18.0000			0.910299
$392$	−1.50000	−	2.59808i	−0.0757614	−	0.131223i
$393$		0			0
$394$	3.00000	−	5.19615i	0.151138	−	0.261778i
$395$	−12.0000			−0.603786
$396$	−3.00000	+	5.19615i	−0.150756	+	0.261116i
$397$	−7.00000	+	12.1244i	−0.351320	+	0.608504i	−0.986481	−	0.163876i	$-0.947600\pi$
$397$	−7.00000	+	12.1244i	−0.351320	+	0.608504i	0.635161	+	0.772380i	$0.280934\pi$
$398$	10.0000			0.501255
$399$	−2.00000	+	3.46410i	−0.100125	+	0.173422i
$400$	−2.00000	−	3.46410i	−0.100000	−	0.173205i
$401$	1.50000	+	2.59808i	0.0749064	+	0.129742i	0.901046	−	0.433724i	$-0.142801\pi$
$401$	1.50000	+	2.59808i	0.0749064	+	0.129742i	−0.826139	+	0.563466i	$0.809468\pi$
$402$	10.0000			0.498755
$403$	14.0000	−	3.46410i	0.697390	−	0.172559i
$404$	15.0000			0.746278
$405$	−1.50000	−	2.59808i	−0.0745356	−	0.129099i
$406$	3.00000	+	5.19615i	0.148888	+	0.257881i
$407$	21.0000	−	36.3731i	1.04093	−	1.80295i
$408$	3.00000			0.148522
$409$	0.500000	−	0.866025i	0.0247234	−	0.0428222i	−0.853399	−	0.521258i	$-0.825463\pi$
$409$	0.500000	−	0.866025i	0.0247234	−	0.0428222i	0.878122	+	0.478436i	$0.158796\pi$
$410$	−4.50000	+	7.79423i	−0.222239	+	0.384930i
$411$	9.00000			0.443937
$412$	−7.00000	+	12.1244i	−0.344865	+	0.597324i
$413$		0			0
$414$	−3.00000	−	5.19615i	−0.147442	−	0.255377i
$415$	−18.0000			−0.883585
$416$	−1.00000	+	3.46410i	−0.0490290	+	0.169842i
$417$	−4.00000			−0.195881
$418$	6.00000	+	10.3923i	0.293470	+	0.508304i
$419$	−12.0000	−	20.7846i	−0.586238	−	1.01539i	−0.994720	−	0.102628i	$-0.967275\pi$
$419$	−12.0000	−	20.7846i	−0.586238	−	1.01539i	0.408481	−	0.912767i	$-0.366058\pi$
$420$	−3.00000	+	5.19615i	−0.146385	+	0.253546i
$421$	29.0000			1.41337			0.706687	−	0.707527i	$-0.250189\pi$
$421$	29.0000			1.41337			0.706687	+	0.707527i	$0.250189\pi$
$422$	−8.00000	+	13.8564i	−0.389434	+	0.674519i
$423$	−3.00000	+	5.19615i	−0.145865	+	0.252646i
$424$	−3.00000			−0.145693
$425$	6.00000	−	10.3923i	0.291043	−	0.504101i
$426$	3.00000	+	5.19615i	0.145350	+	0.251754i
$427$	7.00000	+	12.1244i	0.338754	+	0.586739i
$428$	−6.00000			−0.290021
$429$	6.00000	−	20.7846i	0.289683	−	1.00349i
$430$	30.0000			1.44673
$431$	3.00000	+	5.19615i	0.144505	+	0.250290i	0.929188	−	0.369607i	$-0.120508\pi$
$431$	3.00000	+	5.19615i	0.144505	+	0.250290i	−0.784683	+	0.619897i	$0.787174\pi$
$432$	−0.500000	−	0.866025i	−0.0240563	−	0.0416667i
$433$	6.50000	−	11.2583i	0.312370	−	0.541041i	−0.666505	−	0.745501i	$-0.732210\pi$
$433$	6.50000	−	11.2583i	0.312370	−	0.541041i	0.978875	+	0.204460i	$0.0655438\pi$
$434$	8.00000			0.384012
$435$	−4.50000	+	7.79423i	−0.215758	+	0.373705i
$436$	−7.00000	+	12.1244i	−0.335239	+	0.580651i
$437$	−12.0000			−0.574038
$438$	−6.50000	+	11.2583i	−0.310582	+	0.537944i
$439$	−7.00000	−	12.1244i	−0.334092	−	0.578664i	0.649218	−	0.760602i	$-0.275096\pi$
$439$	−7.00000	−	12.1244i	−0.334092	−	0.578664i	−0.983310	+	0.181938i	$0.941763\pi$
$440$	9.00000	+	15.5885i	0.429058	+	0.743151i
$441$	−3.00000			−0.142857
$442$	−10.5000	+	2.59808i	−0.499434	+	0.123578i
$443$	−36.0000			−1.71041			−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000			−1.71041			−0.855206	+	0.518289i	$0.826569\pi$
$444$	3.50000	+	6.06218i	0.166103	+	0.287698i
$445$	−27.0000	−	46.7654i	−1.27992	−	2.21689i
$446$	4.00000	−	6.92820i	0.189405	−	0.328060i
$447$	−9.00000			−0.425685
$448$	−1.00000	+	1.73205i	−0.0472456	+	0.0818317i
$449$	−9.00000	+	15.5885i	−0.424736	+	0.735665i	−0.996396	−	0.0848262i	$-0.972967\pi$
$449$	−9.00000	+	15.5885i	−0.424736	+	0.735665i	0.571660	+	0.820491i	$0.306300\pi$
$450$	−4.00000			−0.188562
$451$	9.00000	−	15.5885i	0.423793	−	0.734032i
$452$	1.50000	+	2.59808i	0.0705541	+	0.122203i
$453$	5.00000	+	8.66025i	0.234920	+	0.406894i
$454$	−18.0000			−0.844782
$455$	6.00000	−	20.7846i	0.281284	−	0.974398i
$456$	−2.00000			−0.0936586
$457$	−5.50000	−	9.52628i	−0.257279	−	0.445621i	0.708233	−	0.705979i	$-0.249493\pi$
$457$	−5.50000	−	9.52628i	−0.257279	−	0.445621i	−0.965512	+	0.260358i	$0.916159\pi$
$458$	−11.0000	−	19.0526i	−0.513996	−	0.890268i
$459$	1.50000	−	2.59808i	0.0700140	−	0.121268i
$460$	−18.0000			−0.839254
$461$	−7.50000	+	12.9904i	−0.349310	+	0.605022i	−0.986127	−	0.165992i	$-0.946917\pi$
$461$	−7.50000	+	12.9904i	−0.349310	+	0.605022i	0.636817	+	0.771015i	$0.280251\pi$
$462$	6.00000	−	10.3923i	0.279145	−	0.483494i
$463$	38.0000			1.76601			0.883005	−	0.469364i	$-0.155517\pi$
$463$	38.0000			1.76601			0.883005	+	0.469364i	$0.155517\pi$
$464$	−1.50000	+	2.59808i	−0.0696358	+	0.120613i
$465$	6.00000	+	10.3923i	0.278243	+	0.481932i
$466$	−3.00000	−	5.19615i	−0.138972	−	0.240707i
$467$	−18.0000			−0.832941			−0.416470	−	0.909149i	$-0.636733\pi$
$467$	−18.0000			−0.832941			−0.416470	+	0.909149i	$0.636733\pi$
$468$	2.50000	+	2.59808i	0.115563	+	0.120096i
$469$	−20.0000			−0.923514
$470$	9.00000	+	15.5885i	0.415139	+	0.719042i
$471$	−2.50000	−	4.33013i	−0.115194	−	0.199522i
$472$		0			0
$473$	−60.0000			−2.75880
$474$	−2.00000	+	3.46410i	−0.0918630	+	0.159111i
$475$	−4.00000	+	6.92820i	−0.183533	+	0.317888i
$476$	−6.00000			−0.275010
$477$	−1.50000	+	2.59808i	−0.0686803	+	0.118958i
$478$	3.00000	+	5.19615i	0.137217	+	0.237666i
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$	−3.00000			−0.136931
$481$	−17.5000	−	18.1865i	−0.797931	−	0.829235i
$482$	1.00000			0.0455488
$483$	6.00000	+	10.3923i	0.273009	+	0.472866i
$484$	−12.5000	−	21.6506i	−0.568182	−	0.984120i
$485$	−21.0000	+	36.3731i	−0.953561	+	1.65162i
$486$	−1.00000			−0.0453609
$487$	−1.00000	+	1.73205i	−0.0453143	+	0.0784867i	−0.887793	−	0.460243i	$-0.847762\pi$
$487$	−1.00000	+	1.73205i	−0.0453143	+	0.0784867i	0.842479	+	0.538730i	$0.181096\pi$
$488$	−3.50000	+	6.06218i	−0.158438	+	0.274422i
$489$	−4.00000			−0.180886
$490$	−4.50000	+	7.79423i	−0.203289	+	0.352107i
$491$	9.00000	+	15.5885i	0.406164	+	0.703497i	0.994456	−	0.105151i	$-0.0335327\pi$
$491$	9.00000	+	15.5885i	0.406164	+	0.703497i	−0.588292	+	0.808649i	$0.700199\pi$
$492$	1.50000	+	2.59808i	0.0676252	+	0.117130i
$493$	−9.00000			−0.405340
$494$	7.00000	−	1.73205i	0.314945	−	0.0779287i
$495$	18.0000			0.809040
$496$	2.00000	+	3.46410i	0.0898027	+	0.155543i
$497$	−6.00000	−	10.3923i	−0.269137	−	0.466159i
$498$	−3.00000	+	5.19615i	−0.134433	+	0.232845i
$499$	32.0000			1.43252			0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000			1.43252			0.716258	+	0.697835i	$0.245853\pi$
$500$	1.50000	−	2.59808i	0.0670820	−	0.116190i
$501$		0			0
$502$	12.0000			0.535586
$503$	−3.00000	+	5.19615i	−0.133763	+	0.231685i	−0.925124	−	0.379664i	$-0.876040\pi$
$503$	−3.00000	+	5.19615i	−0.133763	+	0.231685i	0.791361	+	0.611349i	$0.209373\pi$
$504$	1.00000	+	1.73205i	0.0445435	+	0.0771517i
$505$	−22.5000	−	38.9711i	−1.00124	−	1.73419i
$506$	36.0000			1.60040
$507$	−11.0000	−	6.92820i	−0.488527	−	0.307692i
$508$	−4.00000			−0.177471
$509$	−1.50000	−	2.59808i	−0.0664863	−	0.115158i	0.830866	−	0.556473i	$-0.187846\pi$
$509$	−1.50000	−	2.59808i	−0.0664863	−	0.115158i	−0.897352	+	0.441315i	$0.854512\pi$
$510$	−4.50000	−	7.79423i	−0.199263	−	0.345134i
$511$	13.0000	−	22.5167i	0.575086	−	0.996078i
$512$	−1.00000			−0.0441942
$513$	−1.00000	+	1.73205i	−0.0441511	+	0.0764719i
$514$	−1.50000	+	2.59808i	−0.0661622	+	0.114596i
$515$	42.0000			1.85074
$516$	5.00000	−	8.66025i	0.220113	−	0.381246i
$517$	−18.0000	−	31.1769i	−0.791639	−	1.37116i
$518$	−7.00000	−	12.1244i	−0.307562	−	0.532714i
$519$	−6.00000			−0.263371
$520$	10.5000	−	2.59808i	0.460455	−	0.113933i
$521$	33.0000			1.44576			0.722878	−	0.690976i	$-0.242819\pi$
$521$	33.0000			1.44576			0.722878	+	0.690976i	$0.242819\pi$
$522$	1.50000	+	2.59808i	0.0656532	+	0.113715i
$523$	17.0000	+	29.4449i	0.743358	+	1.28753i	0.950958	+	0.309320i	$0.100101\pi$
$523$	17.0000	+	29.4449i	0.743358	+	1.28753i	−0.207600	+	0.978214i	$0.566565\pi$
$524$		0			0
$525$	8.00000			0.349149
$526$	−3.00000	+	5.19615i	−0.130806	+	0.226563i
$527$	−6.00000	+	10.3923i	−0.261364	+	0.452696i
$528$	6.00000			0.261116
$529$	−6.50000	+	11.2583i	−0.282609	+	0.489493i
$530$	4.50000	+	7.79423i	0.195468	+	0.338560i
$531$		0			0
$532$	4.00000			0.173422
$533$	−7.50000	−	7.79423i	−0.324861	−	0.337606i
$534$	−18.0000			−0.778936
$535$	9.00000	+	15.5885i	0.389104	+	0.673948i
$536$	−5.00000	−	8.66025i	−0.215967	−	0.374066i
$537$	−3.00000	+	5.19615i	−0.129460	+	0.224231i
$538$	−18.0000			−0.776035
$539$	9.00000	−	15.5885i	0.387657	−	0.671442i
$540$	−1.50000	+	2.59808i	−0.0645497	+	0.111803i
$541$	29.0000			1.24681			0.623404	−	0.781900i	$-0.285749\pi$
$541$	29.0000			1.24681			0.623404	+	0.781900i	$0.285749\pi$
$542$	−8.00000	+	13.8564i	−0.343629	+	0.595184i
$543$	3.50000	+	6.06218i	0.150199	+	0.260153i
$544$	−1.50000	−	2.59808i	−0.0643120	−	0.111392i
$545$	42.0000			1.79908
$546$	−5.00000	−	5.19615i	−0.213980	−	0.222375i
$547$	−34.0000			−1.45374			−0.726868	−	0.686778i	$-0.759025\pi$
$547$	−34.0000			−1.45374			−0.726868	+	0.686778i	$0.759025\pi$
$548$	−4.50000	−	7.79423i	−0.192230	−	0.332953i
$549$	3.50000	+	6.06218i	0.149376	+	0.258727i
$550$	12.0000	−	20.7846i	0.511682	−	0.886259i
$551$	6.00000			0.255609
$552$	−3.00000	+	5.19615i	−0.127688	+	0.221163i
$553$	4.00000	−	6.92820i	0.170097	−	0.294617i
$554$	−17.0000			−0.722261
$555$	10.5000	−	18.1865i	0.445700	−	0.771975i
$556$	2.00000	+	3.46410i	0.0848189	+	0.146911i
$557$	−1.50000	−	2.59808i	−0.0635570	−	0.110084i	0.832496	−	0.554031i	$-0.186911\pi$
$557$	−1.50000	−	2.59808i	−0.0635570	−	0.110084i	−0.896053	+	0.443947i	$0.853578\pi$
$558$	4.00000			0.169334
$559$	−10.0000	+	34.6410i	−0.422955	+	1.46516i
$560$	6.00000			0.253546
$561$	9.00000	+	15.5885i	0.379980	+	0.658145i
$562$	4.50000	+	7.79423i	0.189821	+	0.328780i
$563$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$563$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$564$	6.00000			0.252646
$565$	4.50000	−	7.79423i	0.189316	−	0.327906i
$566$	7.00000	−	12.1244i	0.294232	−	0.509625i
$567$	2.00000			0.0839921
$568$	3.00000	−	5.19615i	0.125877	−	0.218026i
$569$	−3.00000	−	5.19615i	−0.125767	−	0.217834i	0.796266	−	0.604947i	$-0.206806\pi$
$569$	−3.00000	−	5.19615i	−0.125767	−	0.217834i	−0.922032	+	0.387113i	$0.873472\pi$
$570$	3.00000	+	5.19615i	0.125656	+	0.217643i
$571$	−22.0000			−0.920671			−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000			−0.920671			−0.460336	+	0.887745i	$0.652271\pi$
$572$	−21.0000	+	5.19615i	−0.878054	+	0.217262i
$573$	−12.0000			−0.501307
$574$	−3.00000	−	5.19615i	−0.125218	−	0.216883i
$575$	12.0000	+	20.7846i	0.500435	+	0.866778i
$576$	−0.500000	+	0.866025i	−0.0208333	+	0.0360844i
$577$	11.0000			0.457936			0.228968	−	0.973434i	$-0.426465\pi$
$577$	11.0000			0.457936			0.228968	+	0.973434i	$0.426465\pi$
$578$	−4.00000	+	6.92820i	−0.166378	+	0.288175i
$579$	−11.5000	+	19.9186i	−0.477924	+	0.827788i
$580$	9.00000			0.373705
$581$	6.00000	−	10.3923i	0.248922	−	0.431145i
$582$	7.00000	+	12.1244i	0.290159	+	0.502571i
$583$	−9.00000	−	15.5885i	−0.372742	−	0.645608i
$584$	13.0000			0.537944
$585$	3.00000	−	10.3923i	0.124035	−	0.429669i
$586$	21.0000			0.867502
$587$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$587$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$588$	1.50000	+	2.59808i	0.0618590	+	0.107143i
$589$	4.00000	−	6.92820i	0.164817	−	0.285472i
$590$		0			0
$591$	−3.00000	+	5.19615i	−0.123404	+	0.213741i
$592$	3.50000	−	6.06218i	0.143849	−	0.249154i
$593$	9.00000			0.369586			0.184793	−	0.982777i	$-0.440839\pi$
$593$	9.00000			0.369586			0.184793	+	0.982777i	$0.440839\pi$
$594$	3.00000	−	5.19615i	0.123091	−	0.213201i
$595$	9.00000	+	15.5885i	0.368964	+	0.639064i
$596$	4.50000	+	7.79423i	0.184327	+	0.319264i
$597$	−10.0000			−0.409273
$598$	6.00000	−	20.7846i	0.245358	−	0.849946i
$599$	24.0000			0.980613			0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000			0.980613			0.490307	+	0.871550i	$0.336885\pi$
$600$	2.00000	+	3.46410i	0.0816497	+	0.141421i
$601$	18.5000	+	32.0429i	0.754631	+	1.30706i	0.945558	+	0.325455i	$0.105517\pi$
$601$	18.5000	+	32.0429i	0.754631	+	1.30706i	−0.190927	+	0.981604i	$0.561149\pi$
$602$	−10.0000	+	17.3205i	−0.407570	+	0.705931i
$603$	−10.0000			−0.407231
$604$	5.00000	−	8.66025i	0.203447	−	0.352381i
$605$	−37.5000	+	64.9519i	−1.52459	+	2.64067i
$606$	−15.0000			−0.609333
$607$	−16.0000	+	27.7128i	−0.649420	+	1.12483i	0.333842	+	0.942629i	$0.391655\pi$
$607$	−16.0000	+	27.7128i	−0.649420	+	1.12483i	−0.983262	+	0.182199i	$0.941678\pi$
$608$	1.00000	+	1.73205i	0.0405554	+	0.0702439i
$609$	−3.00000	−	5.19615i	−0.121566	−	0.210559i
$610$	21.0000			0.850265
$611$	−21.0000	+	5.19615i	−0.849569	+	0.210214i
$612$	−3.00000			−0.121268
$613$	15.5000	+	26.8468i	0.626039	+	1.08433i	0.988339	+	0.152270i	$0.0486583\pi$
$613$	15.5000	+	26.8468i	0.626039	+	1.08433i	−0.362300	+	0.932062i	$0.618008\pi$
$614$	−5.00000	−	8.66025i	−0.201784	−	0.349499i
$615$	4.50000	−	7.79423i	0.181458	−	0.314294i
$616$	−12.0000			−0.483494
$617$	7.50000	−	12.9904i	0.301939	−	0.522973i	−0.674636	−	0.738150i	$-0.735700\pi$
$617$	7.50000	−	12.9904i	0.301939	−	0.522973i	0.976575	+	0.215177i	$0.0690329\pi$
$618$	7.00000	−	12.1244i	0.281581	−	0.487713i
$619$	8.00000			0.321547			0.160774	−	0.986991i	$-0.448601\pi$
$619$	8.00000			0.321547			0.160774	+	0.986991i	$0.448601\pi$
$620$	6.00000	−	10.3923i	0.240966	−	0.417365i
$621$	3.00000	+	5.19615i	0.120386	+	0.208514i
$622$	−15.0000	−	25.9808i	−0.601445	−	1.04173i
$623$	36.0000			1.44231
$624$	1.00000	−	3.46410i	0.0400320	−	0.138675i
$625$	−29.0000			−1.16000
$626$	−5.00000	−	8.66025i	−0.199840	−	0.346133i
$627$	−6.00000	−	10.3923i	−0.239617	−	0.415029i
$628$	−2.50000	+	4.33013i	−0.0997609	+	0.172791i
$629$	21.0000			0.837325
$630$	3.00000	−	5.19615i	0.119523	−	0.207020i
$631$	−10.0000	+	17.3205i	−0.398094	+	0.689519i	−0.993491	−	0.113913i	$-0.963661\pi$
$631$	−10.0000	+	17.3205i	−0.398094	+	0.689519i	0.595397	+	0.803432i	$0.296995\pi$
$632$	4.00000			0.159111
$633$	8.00000	−	13.8564i	0.317971	−	0.550743i
$634$	1.50000	+	2.59808i	0.0595726	+	0.103183i
$635$	6.00000	+	10.3923i	0.238103	+	0.412406i
$636$	3.00000			0.118958
$637$	−7.50000	−	7.79423i	−0.297161	−	0.308819i
$638$	−18.0000			−0.712627
$639$	−3.00000	−	5.19615i	−0.118678	−	0.205557i
$640$	1.50000	+	2.59808i	0.0592927	+	0.102698i
$641$	1.50000	−	2.59808i	0.0592464	−	0.102618i	−0.834881	−	0.550431i	$-0.814464\pi$
$641$	1.50000	−	2.59808i	0.0592464	−	0.102618i	0.894127	+	0.447813i	$0.147797\pi$
$642$	6.00000			0.236801
$643$	8.00000	−	13.8564i	0.315489	−	0.546443i	−0.664052	−	0.747686i	$-0.731165\pi$
$643$	8.00000	−	13.8564i	0.315489	−	0.546443i	0.979541	+	0.201243i	$0.0644981\pi$
$644$	6.00000	−	10.3923i	0.236433	−	0.409514i
$645$	−30.0000			−1.18125
$646$	−3.00000	+	5.19615i	−0.118033	+	0.204440i
$647$	−12.0000	−	20.7846i	−0.471769	−	0.817127i	0.527710	−	0.849425i	$-0.323051\pi$
$647$	−12.0000	−	20.7846i	−0.471769	−	0.817127i	−0.999478	+	0.0322975i	$0.989718\pi$
$648$	0.500000	+	0.866025i	0.0196419	+	0.0340207i
$649$		0			0
$650$	−10.0000	−	10.3923i	−0.392232	−	0.407620i
$651$	−8.00000			−0.313545
$652$	2.00000	+	3.46410i	0.0783260	+	0.135665i
$653$	−21.0000	−	36.3731i	−0.821794	−	1.42339i	−0.904345	−	0.426801i	$-0.859640\pi$
$653$	−21.0000	−	36.3731i	−0.821794	−	1.42339i	0.0825519	−	0.996587i	$-0.473693\pi$
$654$	7.00000	−	12.1244i	0.273722	−	0.474100i
$655$		0			0
$656$	1.50000	−	2.59808i	0.0585652	−	0.101438i
$657$	6.50000	−	11.2583i	0.253589	−	0.439229i
$658$	−12.0000			−0.467809
$659$	−12.0000	+	20.7846i	−0.467454	+	0.809653i	−0.999309	−	0.0371821i	$-0.988162\pi$
$659$	−12.0000	+	20.7846i	−0.467454	+	0.809653i	0.531855	+	0.846836i	$0.321495\pi$
$660$	−9.00000	−	15.5885i	−0.350325	−	0.606780i
$661$	−2.50000	−	4.33013i	−0.0972387	−	0.168422i	0.813302	−	0.581842i	$-0.197668\pi$
$661$	−2.50000	−	4.33013i	−0.0972387	−	0.168422i	−0.910541	+	0.413419i	$0.864334\pi$
$662$	4.00000			0.155464
$663$	10.5000	−	2.59808i	0.407786	−	0.100901i
$664$	6.00000			0.232845
$665$	−6.00000	−	10.3923i	−0.232670	−	0.402996i
$666$	−3.50000	−	6.06218i	−0.135622	−	0.234905i
$667$	9.00000	−	15.5885i	0.348481	−	0.603587i
$668$		0			0
$669$	−4.00000	+	6.92820i	−0.154649	+	0.267860i
$670$	−15.0000	+	25.9808i	−0.579501	+	1.00372i
$671$	−42.0000			−1.62139
$672$	1.00000	−	1.73205i	0.0385758	−	0.0668153i
$673$	6.50000	+	11.2583i	0.250557	+	0.433977i	0.963679	−	0.267063i	$-0.0860531\pi$
$673$	6.50000	+	11.2583i	0.250557	+	0.433977i	−0.713123	+	0.701039i	$0.752720\pi$
$674$	11.5000	+	19.9186i	0.442963	+	0.767235i
$675$	4.00000			0.153960
$676$	−0.500000	+	12.9904i	−0.0192308	+	0.499630i
$677$	18.0000			0.691796			0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000			0.691796			0.345898	+	0.938272i	$0.387574\pi$
$678$	−1.50000	−	2.59808i	−0.0576072	−	0.0997785i
$679$	−14.0000	−	24.2487i	−0.537271	−	0.930580i
$680$	−4.50000	+	7.79423i	−0.172567	+	0.298895i
$681$	18.0000			0.689761
$682$	−12.0000	+	20.7846i	−0.459504	+	0.795884i
$683$	24.0000	−	41.5692i	0.918334	−	1.59060i	0.116390	−	0.993204i	$-0.462868\pi$
$683$	24.0000	−	41.5692i	0.918334	−	1.59060i	0.801945	−	0.597398i	$-0.203799\pi$
$684$	2.00000			0.0764719
$685$	−13.5000	+	23.3827i	−0.515808	+	0.893407i
$686$	−10.0000	−	17.3205i	−0.381802	−	0.661300i
$687$	11.0000	+	19.0526i	0.419676	+	0.726900i
$688$	−10.0000			−0.381246
$689$	−10.5000	+	2.59808i	−0.400018	+	0.0989788i
$690$	18.0000			0.685248
$691$	−13.0000	−	22.5167i	−0.494543	−	0.856574i	0.505437	−	0.862864i	$-0.331331\pi$
$691$	−13.0000	−	22.5167i	−0.494543	−	0.856574i	−0.999980	+	0.00628943i	$0.997998\pi$
$692$	3.00000	+	5.19615i	0.114043	+	0.197528i
$693$	−6.00000	+	10.3923i	−0.227921	+	0.394771i
$694$	30.0000			1.13878
$695$	6.00000	−	10.3923i	0.227593	−	0.394203i
$696$	1.50000	−	2.59808i	0.0568574	−	0.0984798i
$697$	9.00000			0.340899
$698$	−5.00000	+	8.66025i	−0.189253	+	0.327795i
$699$	3.00000	+	5.19615i	0.113470	+	0.196537i
$700$	−4.00000	−	6.92820i	−0.151186	−	0.261861i
$701$	−30.0000			−1.13308			−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000			−1.13308			−0.566542	+	0.824033i	$0.691719\pi$
$702$	−2.50000	−	2.59808i	−0.0943564	−	0.0980581i
$703$	−14.0000			−0.528020
$704$	−3.00000	−	5.19615i	−0.113067	−	0.195837i
$705$	−9.00000	−	15.5885i	−0.338960	−	0.587095i
$706$	−7.50000	+	12.9904i	−0.282266	+	0.488899i
$707$	30.0000			1.12827
$708$		0			0
$709$	−2.50000	+	4.33013i	−0.0938895	+	0.162621i	−0.909145	−	0.416481i	$-0.863263\pi$
$709$	−2.50000	+	4.33013i	−0.0938895	+	0.162621i	0.815255	+	0.579102i	$0.196597\pi$
$710$	−18.0000			−0.675528
$711$	2.00000	−	3.46410i	0.0750059	−	0.129914i
$712$	9.00000	+	15.5885i	0.337289	+	0.584202i
$713$	−12.0000	−	20.7846i	−0.449404	−	0.778390i
$714$	6.00000			0.224544
$715$	45.0000	+	46.7654i	1.68290	+	1.74893i
$716$	6.00000			0.224231
$717$	−3.00000	−	5.19615i	−0.112037	−	0.194054i
$718$	3.00000	+	5.19615i	0.111959	+	0.193919i
$719$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$	3.00000			0.111803
$721$	−14.0000	+	24.2487i	−0.521387	+	0.903069i
$722$	−7.50000	+	12.9904i	−0.279121	+	0.483452i
$723$	−1.00000			−0.0371904
$724$	3.50000	−	6.06218i	0.130076	−	0.225299i
$725$	−6.00000	−	10.3923i	−0.222834	−	0.385961i
$726$	12.5000	+	21.6506i	0.463919	+	0.803530i
$727$	14.0000			0.519231			0.259616	−	0.965712i	$-0.416404\pi$
$727$	14.0000			0.519231			0.259616	+	0.965712i	$0.416404\pi$
$728$	−2.00000	+	6.92820i	−0.0741249	+	0.256776i
$729$	1.00000			0.0370370
$730$	−19.5000	−	33.7750i	−0.721727	−	1.25007i
$731$	−15.0000	−	25.9808i	−0.554795	−	0.960933i
$732$	3.50000	−	6.06218i	0.129364	−	0.224065i
$733$	−31.0000			−1.14501			−0.572506	−	0.819901i	$-0.694029\pi$
$733$	−31.0000			−1.14501			−0.572506	+	0.819901i	$0.694029\pi$
$734$	1.00000	−	1.73205i	0.0369107	−	0.0639312i
$735$	4.50000	−	7.79423i	0.165985	−	0.287494i
$736$	6.00000			0.221163
$737$	30.0000	−	51.9615i	1.10506	−	1.91403i
$738$	−1.50000	−	2.59808i	−0.0552158	−	0.0956365i
$739$	8.00000	+	13.8564i	0.294285	+	0.509716i	0.974818	−	0.223001i	$-0.0715853\pi$
$739$	8.00000	+	13.8564i	0.294285	+	0.509716i	−0.680534	+	0.732717i	$0.738252\pi$
$740$	−21.0000			−0.771975
$741$	−7.00000	+	1.73205i	−0.257151	+	0.0636285i
$742$	−6.00000			−0.220267
$743$	18.0000	+	31.1769i	0.660356	+	1.14377i	0.980522	+	0.196409i	$0.0629279\pi$
$743$	18.0000	+	31.1769i	0.660356	+	1.14377i	−0.320166	+	0.947361i	$0.603739\pi$
$744$	−2.00000	−	3.46410i	−0.0733236	−	0.127000i
$745$	13.5000	−	23.3827i	0.494602	−	0.856675i
$746$	−29.0000			−1.06177
$747$	3.00000	−	5.19615i	0.109764	−	0.190117i
$748$	9.00000	−	15.5885i	0.329073	−	0.569970i
$749$	−12.0000			−0.438470
$750$	−1.50000	+	2.59808i	−0.0547723	+	0.0948683i
$751$	−7.00000	−	12.1244i	−0.255434	−	0.442424i	0.709580	−	0.704625i	$-0.248885\pi$
$751$	−7.00000	−	12.1244i	−0.255434	−	0.442424i	−0.965013	+	0.262201i	$0.915552\pi$
$752$	−3.00000	−	5.19615i	−0.109399	−	0.189484i
$753$	−12.0000			−0.437304
$754$	−3.00000	+	10.3923i	−0.109254	+	0.378465i
$755$	−30.0000			−1.09181
$756$	−1.00000	−	1.73205i	−0.0363696	−	0.0629941i
$757$	17.0000	+	29.4449i	0.617876	+	1.07019i	0.989873	+	0.141958i	$0.0453398\pi$
$757$	17.0000	+	29.4449i	0.617876	+	1.07019i	−0.371997	+	0.928234i	$0.621327\pi$
$758$	10.0000	−	17.3205i	0.363216	−	0.629109i
$759$	−36.0000			−1.30672
$760$	3.00000	−	5.19615i	0.108821	−	0.188484i
$761$	15.0000	−	25.9808i	0.543750	−	0.941802i	−0.454935	−	0.890525i	$-0.650337\pi$
$761$	15.0000	−	25.9808i	0.543750	−	0.941802i	0.998684	−	0.0512772i	$-0.0163292\pi$
$762$	4.00000			0.144905
$763$	−14.0000	+	24.2487i	−0.506834	+	0.877862i
$764$	6.00000	+	10.3923i	0.217072	+	0.375980i
$765$	4.50000	+	7.79423i	0.162698	+	0.281801i
$766$	−24.0000			−0.867155
$767$		0			0
$768$	1.00000			0.0360844
$769$	−7.00000	−	12.1244i	−0.252426	−	0.437215i	0.711767	−	0.702416i	$-0.247895\pi$
$769$	−7.00000	−	12.1244i	−0.252426	−	0.437215i	−0.964193	+	0.265200i	$0.914562\pi$
$770$	18.0000	+	31.1769i	0.648675	+	1.12354i
$771$	1.50000	−	2.59808i	0.0540212	−	0.0935674i
$772$	23.0000			0.827788
$773$	15.0000	−	25.9808i	0.539513	−

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:	\( N \)	\(=\)	\( 78 = 2 \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	78.e (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{4} +3.00000 q^{5} +(0.500000 - 0.866025i) q^{6} +(-1.00000 + 1.73205i) q^{7} -1.00000 q^{8} +(-0.500000 + 0.866025i) q^{9} +(1.50000 + 2.59808i) q^{10} +(-3.00000 - 5.19615i) q^{11} +1.00000 q^{12} +(-3.50000 + 0.866025i) q^{13} -2.00000 q^{14} +(-1.50000 - 2.59808i) q^{15} +(-0.500000 - 0.866025i) q^{16} +(1.50000 - 2.59808i) q^{17} -1.00000 q^{18} +(-1.00000 + 1.73205i) q^{19} +(-1.50000 + 2.59808i) q^{20} +2.00000 q^{21} +(3.00000 - 5.19615i) q^{22} +(3.00000 + 5.19615i) q^{23} +(0.500000 + 0.866025i) q^{24} +4.00000 q^{25} +(-2.50000 - 2.59808i) q^{26} +1.00000 q^{27} +(-1.00000 - 1.73205i) q^{28} +(-1.50000 - 2.59808i) q^{29} +(1.50000 - 2.59808i) q^{30} -4.00000 q^{31} +(0.500000 - 0.866025i) q^{32} +(-3.00000 + 5.19615i) q^{33} +3.00000 q^{34} +(-3.00000 + 5.19615i) q^{35} +(-0.500000 - 0.866025i) q^{36} +(3.50000 + 6.06218i) q^{37} -2.00000 q^{38} +(2.50000 + 2.59808i) q^{39} -3.00000 q^{40} +(1.50000 + 2.59808i) q^{41} +(1.00000 + 1.73205i) q^{42} +(5.00000 - 8.66025i) q^{43} +6.00000 q^{44} +(-1.50000 + 2.59808i) q^{45} +(-3.00000 + 5.19615i) q^{46} +6.00000 q^{47} +(-0.500000 + 0.866025i) q^{48} +(1.50000 + 2.59808i) q^{49} +(2.00000 + 3.46410i) q^{50} -3.00000 q^{51} +(1.00000 - 3.46410i) q^{52} +3.00000 q^{53} +(0.500000 + 0.866025i) q^{54} +(-9.00000 - 15.5885i) q^{55} +(1.00000 - 1.73205i) q^{56} +2.00000 q^{57} +(1.50000 - 2.59808i) q^{58} +3.00000 q^{60} +(3.50000 - 6.06218i) q^{61} +(-2.00000 - 3.46410i) q^{62} +(-1.00000 - 1.73205i) q^{63} +1.00000 q^{64} +(-10.5000 + 2.59808i) q^{65} -6.00000 q^{66} +(5.00000 + 8.66025i) q^{67} +(1.50000 + 2.59808i) q^{68} +(3.00000 - 5.19615i) q^{69} -6.00000 q^{70} +(-3.00000 + 5.19615i) q^{71} +(0.500000 - 0.866025i) q^{72} -13.0000 q^{73} +(-3.50000 + 6.06218i) q^{74} +(-2.00000 - 3.46410i) q^{75} +(-1.00000 - 1.73205i) q^{76} +12.0000 q^{77} +(-1.00000 + 3.46410i) q^{78} -4.00000 q^{79} +(-1.50000 - 2.59808i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(-1.50000 + 2.59808i) q^{82} -6.00000 q^{83} +(-1.00000 + 1.73205i) q^{84} +(4.50000 - 7.79423i) q^{85} +10.0000 q^{86} +(-1.50000 + 2.59808i) q^{87} +(3.00000 + 5.19615i) q^{88} +(-9.00000 - 15.5885i) q^{89} -3.00000 q^{90} +(2.00000 - 6.92820i) q^{91} -6.00000 q^{92} +(2.00000 + 3.46410i) q^{93} +(3.00000 + 5.19615i) q^{94} +(-3.00000 + 5.19615i) q^{95} -1.00000 q^{96} +(-7.00000 + 12.1244i) q^{97} +(-1.50000 + 2.59808i) q^{98} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{3} - q^{4} + 6 q^{5} + q^{6} - 2 q^{7} - 2 q^{8} - q^{9} + 3 q^{10} - 6 q^{11} + 2 q^{12} - 7 q^{13} - 4 q^{14} - 3 q^{15} - q^{16} + 3 q^{17} - 2 q^{18} - 2 q^{19} - 3 q^{20} + 4 q^{21}+ \cdots + 12 q^{99}+O(q^{100}) \) 2 * q + q^2 - q^3 - q^4 + 6 * q^5 + q^6 - 2 * q^7 - 2 * q^8 - q^9 + 3 * q^10 - 6 * q^11 + 2 * q^12 - 7 * q^13 - 4 * q^14 - 3 * q^15 - q^16 + 3 * q^17 - 2 * q^18 - 2 * q^19 - 3 * q^20 + 4 * q^21 + 6 * q^22 + 6 * q^23 + q^24 + 8 * q^25 - 5 * q^26 + 2 * q^27 - 2 * q^28 - 3 * q^29 + 3 * q^30 - 8 * q^31 + q^32 - 6 * q^33 + 6 * q^34 - 6 * q^35 - q^36 + 7 * q^37 - 4 * q^38 + 5 * q^39 - 6 * q^40 + 3 * q^41 + 2 * q^42 + 10 * q^43 + 12 * q^44 - 3 * q^45 - 6 * q^46 + 12 * q^47 - q^48 + 3 * q^49 + 4 * q^50 - 6 * q^51 + 2 * q^52 + 6 * q^53 + q^54 - 18 * q^55 + 2 * q^56 + 4 * q^57 + 3 * q^58 + 6 * q^60 + 7 * q^61 - 4 * q^62 - 2 * q^63 + 2 * q^64 - 21 * q^65 - 12 * q^66 + 10 * q^67 + 3 * q^68 + 6 * q^69 - 12 * q^70 - 6 * q^71 + q^72 - 26 * q^73 - 7 * q^74 - 4 * q^75 - 2 * q^76 + 24 * q^77 - 2 * q^78 - 8 * q^79 - 3 * q^80 - q^81 - 3 * q^82 - 12 * q^83 - 2 * q^84 + 9 * q^85 + 20 * q^86 - 3 * q^87 + 6 * q^88 - 18 * q^89 - 6 * q^90 + 4 * q^91 - 12 * q^92 + 4 * q^93 + 6 * q^94 - 6 * q^95 - 2 * q^96 - 14 * q^97 - 3 * q^98 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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