LMFDB - Embedded newform 378.2.g.a.163.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,2,Mod(109,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, 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("378.109");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	378.g (of order $3$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$3.01834519640$
Analytic rank:	$1$
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			163.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	378.163
Dual form			378.2.g.a.109.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-1.50000 + 2.59808i) q^{5} +(-2.00000 - 1.73205i) q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-1.50000 + 2.59808i) q^{5} +(-2.00000 - 1.73205i) q^{7} +1.00000 q^{8} +(-1.50000 - 2.59808i) q^{10} -4.00000 q^{13} +(2.50000 - 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(-3.00000 - 5.19615i) q^{17} +(2.00000 - 3.46410i) q^{19} +3.00000 q^{20} +(-3.00000 + 5.19615i) q^{23} +(-2.00000 - 3.46410i) q^{25} +(2.00000 - 3.46410i) q^{26} +(-0.500000 + 2.59808i) q^{28} -3.00000 q^{29} +(-4.00000 - 6.92820i) q^{31} +(-0.500000 - 0.866025i) q^{32} +6.00000 q^{34} +(7.50000 - 2.59808i) q^{35} +(-4.00000 + 6.92820i) q^{37} +(2.00000 + 3.46410i) q^{38} +(-1.50000 + 2.59808i) q^{40} -6.00000 q^{41} +8.00000 q^{43} +(-3.00000 - 5.19615i) q^{46} +(-3.00000 + 5.19615i) q^{47} +(1.00000 + 6.92820i) q^{49} +4.00000 q^{50} +(2.00000 + 3.46410i) q^{52} +(-4.50000 - 7.79423i) q^{53} +(-2.00000 - 1.73205i) q^{56} +(1.50000 - 2.59808i) q^{58} +(1.50000 + 2.59808i) q^{59} +(5.00000 - 8.66025i) q^{61} +8.00000 q^{62} +1.00000 q^{64} +(6.00000 - 10.3923i) q^{65} +(5.00000 + 8.66025i) q^{67} +(-3.00000 + 5.19615i) q^{68} +(-1.50000 + 7.79423i) q^{70} +6.00000 q^{71} +(3.50000 + 6.06218i) q^{73} +(-4.00000 - 6.92820i) q^{74} -4.00000 q^{76} +(-8.50000 + 14.7224i) q^{79} +(-1.50000 - 2.59808i) q^{80} +(3.00000 - 5.19615i) q^{82} -12.0000 q^{83} +18.0000 q^{85} +(-4.00000 + 6.92820i) q^{86} +(-3.00000 + 5.19615i) q^{89} +(8.00000 + 6.92820i) q^{91} +6.00000 q^{92} +(-3.00000 - 5.19615i) q^{94} +(6.00000 + 10.3923i) q^{95} -10.0000 q^{97} +(-6.50000 - 2.59808i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{4} - 3 q^{5} - 4 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q - q^2 - q^4 - 3 * q^5 - 4 * q^7 + 2 * q^8	$ 2 q - q^{2} - q^{4} - 3 q^{5} - 4 q^{7} + 2 q^{8} - 3 q^{10} - 8 q^{13} + 5 q^{14} - q^{16} - 6 q^{17} + 4 q^{19} + 6 q^{20} - 6 q^{23} - 4 q^{25} + 4 q^{26} - q^{28} - 6 q^{29} - 8 q^{31} - q^{32} + 12 q^{34} + 15 q^{35} - 8 q^{37} + 4 q^{38} - 3 q^{40} - 12 q^{41} + 16 q^{43} - 6 q^{46} - 6 q^{47} + 2 q^{49} + 8 q^{50} + 4 q^{52} - 9 q^{53} - 4 q^{56} + 3 q^{58} + 3 q^{59} + 10 q^{61} + 16 q^{62} + 2 q^{64} + 12 q^{65} + 10 q^{67} - 6 q^{68} - 3 q^{70} + 12 q^{71} + 7 q^{73} - 8 q^{74} - 8 q^{76} - 17 q^{79} - 3 q^{80} + 6 q^{82} - 24 q^{83} + 36 q^{85} - 8 q^{86} - 6 q^{89} + 16 q^{91} + 12 q^{92} - 6 q^{94} + 12 q^{95} - 20 q^{97} - 13 q^{98}+O(q^{100}) $ 2 * q - q^2 - q^4 - 3 * q^5 - 4 * q^7 + 2 * q^8 - 3 * q^10 - 8 * q^13 + 5 * q^14 - q^16 - 6 * q^17 + 4 * q^19 + 6 * q^20 - 6 * q^23 - 4 * q^25 + 4 * q^26 - q^28 - 6 * q^29 - 8 * q^31 - q^32 + 12 * q^34 + 15 * q^35 - 8 * q^37 + 4 * q^38 - 3 * q^40 - 12 * q^41 + 16 * q^43 - 6 * q^46 - 6 * q^47 + 2 * q^49 + 8 * q^50 + 4 * q^52 - 9 * q^53 - 4 * q^56 + 3 * q^58 + 3 * q^59 + 10 * q^61 + 16 * q^62 + 2 * q^64 + 12 * q^65 + 10 * q^67 - 6 * q^68 - 3 * q^70 + 12 * q^71 + 7 * q^73 - 8 * q^74 - 8 * q^76 - 17 * q^79 - 3 * q^80 + 6 * q^82 - 24 * q^83 + 36 * q^85 - 8 * q^86 - 6 * q^89 + 16 * q^91 + 12 * q^92 - 6 * q^94 + 12 * q^95 - 20 * q^97 - 13 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$

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			0
$3$		0			0
$4$	−0.500000	−	0.866025i	−0.250000	−	0.433013i
$5$	−1.50000	+	2.59808i	−0.670820	+	1.16190i	0.306851	+	0.951757i	$0.400725\pi$
$5$	−1.50000	+	2.59808i	−0.670820	+	1.16190i	−0.977672	+	0.210138i	$0.932609\pi$
$6$		0			0
$7$	−2.00000	−	1.73205i	−0.755929	−	0.654654i
$7$	−2.00000	−	1.73205i	−0.755929	−	0.654654i
$8$	1.00000			0.353553
$9$		0			0
$10$	−1.50000	−	2.59808i	−0.474342	−	0.821584i
$11$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$11$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$12$		0			0
$13$	−4.00000			−1.10940			−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000			−1.10940			−0.554700	+	0.832050i	$0.687167\pi$
$14$	2.50000	−	0.866025i	0.668153	−	0.231455i
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.125000	+	0.216506i
$17$	−3.00000	−	5.19615i	−0.727607	−	1.26025i	−0.957892	−	0.287129i	$-0.907299\pi$
$17$	−3.00000	−	5.19615i	−0.727607	−	1.26025i	0.230285	−	0.973123i	$-0.426034\pi$
$18$		0			0
$19$	2.00000	−	3.46410i	0.458831	−	0.794719i	−0.540068	−	0.841621i	$-0.681602\pi$
$19$	2.00000	−	3.46410i	0.458831	−	0.794719i	0.998899	+	0.0469020i	$0.0149348\pi$
$20$	3.00000			0.670820
$21$		0			0
$22$		0			0
$23$	−3.00000	+	5.19615i	−0.625543	+	1.08347i	0.362892	+	0.931831i	$0.381789\pi$
$23$	−3.00000	+	5.19615i	−0.625543	+	1.08347i	−0.988436	+	0.151642i	$0.951544\pi$
$24$		0			0
$25$	−2.00000	−	3.46410i	−0.400000	−	0.692820i
$26$	2.00000	−	3.46410i	0.392232	−	0.679366i
$27$		0			0
$28$	−0.500000	+	2.59808i	−0.0944911	+	0.490990i
$29$	−3.00000			−0.557086			−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000			−0.557086			−0.278543	+	0.960424i	$0.589851\pi$
$30$		0			0
$31$	−4.00000	−	6.92820i	−0.718421	−	1.24434i	−0.961625	−	0.274367i	$-0.911532\pi$
$31$	−4.00000	−	6.92820i	−0.718421	−	1.24434i	0.243204	−	0.969975i	$-0.421802\pi$
$32$	−0.500000	−	0.866025i	−0.0883883	−	0.153093i
$33$		0			0
$34$	6.00000			1.02899
$35$	7.50000	−	2.59808i	1.26773	−	0.439155i
$36$		0			0
$37$	−4.00000	+	6.92820i	−0.657596	+	1.13899i	0.323640	+	0.946180i	$0.395093\pi$
$37$	−4.00000	+	6.92820i	−0.657596	+	1.13899i	−0.981236	+	0.192809i	$0.938240\pi$
$38$	2.00000	+	3.46410i	0.324443	+	0.561951i
$39$		0			0
$40$	−1.50000	+	2.59808i	−0.237171	+	0.410792i
$41$	−6.00000			−0.937043			−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000			−0.937043			−0.468521	+	0.883452i	$0.655213\pi$
$42$		0			0
$43$	8.00000			1.21999			0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000			1.21999			0.609994	+	0.792406i	$0.291172\pi$
$44$		0			0
$45$		0			0
$46$	−3.00000	−	5.19615i	−0.442326	−	0.766131i
$47$	−3.00000	+	5.19615i	−0.437595	+	0.757937i	−0.997503	−	0.0706177i	$-0.977503\pi$
$47$	−3.00000	+	5.19615i	−0.437595	+	0.757937i	0.559908	+	0.828554i	$0.310836\pi$
$48$		0			0
$49$	1.00000	+	6.92820i	0.142857	+	0.989743i
$50$	4.00000			0.565685
$51$		0			0
$52$	2.00000	+	3.46410i	0.277350	+	0.480384i
$53$	−4.50000	−	7.79423i	−0.618123	−	1.07062i	−0.989828	−	0.142269i	$-0.954560\pi$
$53$	−4.50000	−	7.79423i	−0.618123	−	1.07062i	0.371706	−	0.928351i	$-0.378773\pi$
$54$		0			0
$55$		0			0
$56$	−2.00000	−	1.73205i	−0.267261	−	0.231455i
$57$		0			0
$58$	1.50000	−	2.59808i	0.196960	−	0.341144i
$59$	1.50000	+	2.59808i	0.195283	+	0.338241i	0.946993	−	0.321253i	$-0.104104\pi$
$59$	1.50000	+	2.59808i	0.195283	+	0.338241i	−0.751710	+	0.659494i	$0.770771\pi$
$60$		0			0
$61$	5.00000	−	8.66025i	0.640184	−	1.10883i	−0.345207	−	0.938527i	$-0.612191\pi$
$61$	5.00000	−	8.66025i	0.640184	−	1.10883i	0.985391	−	0.170305i	$-0.0544754\pi$
$62$	8.00000			1.01600
$63$		0			0
$64$	1.00000			0.125000
$65$	6.00000	−	10.3923i	0.744208	−	1.28901i
$66$		0			0
$67$	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	$0.624162\pi$
$68$	−3.00000	+	5.19615i	−0.363803	+	0.630126i
$69$		0			0
$70$	−1.50000	+	7.79423i	−0.179284	+	0.931589i
$71$	6.00000			0.712069			0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000			0.712069			0.356034	+	0.934473i	$0.384129\pi$
$72$		0			0
$73$	3.50000	+	6.06218i	0.409644	+	0.709524i	0.994850	−	0.101361i	$-0.0323196\pi$
$73$	3.50000	+	6.06218i	0.409644	+	0.709524i	−0.585206	+	0.810885i	$0.698986\pi$
$74$	−4.00000	−	6.92820i	−0.464991	−	0.805387i
$75$		0			0
$76$	−4.00000			−0.458831
$77$		0			0
$78$		0			0
$79$	−8.50000	+	14.7224i	−0.956325	+	1.65640i	−0.225018	+	0.974355i	$0.572244\pi$
$79$	−8.50000	+	14.7224i	−0.956325	+	1.65640i	−0.731307	+	0.682048i	$0.761089\pi$
$80$	−1.50000	−	2.59808i	−0.167705	−	0.290474i
$81$		0			0
$82$	3.00000	−	5.19615i	0.331295	−	0.573819i
$83$	−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
$85$	18.0000			1.95237
$86$	−4.00000	+	6.92820i	−0.431331	+	0.747087i
$87$		0			0
$88$		0			0
$89$	−3.00000	+	5.19615i	−0.317999	+	0.550791i	−0.980071	−	0.198650i	$-0.936344\pi$
$89$	−3.00000	+	5.19615i	−0.317999	+	0.550791i	0.662071	+	0.749441i	$0.269678\pi$
$90$		0			0
$91$	8.00000	+	6.92820i	0.838628	+	0.726273i
$92$	6.00000			0.625543
$93$		0			0
$94$	−3.00000	−	5.19615i	−0.309426	−	0.535942i
$95$	6.00000	+	10.3923i	0.615587	+	1.06623i
$96$		0			0
$97$	−10.0000			−1.01535			−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000			−1.01535			−0.507673	+	0.861550i	$0.669494\pi$
$98$	−6.50000	−	2.59808i	−0.656599	−	0.262445i
$99$		0			0
$100$	−2.00000	+	3.46410i	−0.200000	+	0.346410i
$101$	7.50000	+	12.9904i	0.746278	+	1.29259i	0.949595	+	0.313478i	$0.101494\pi$
$101$	7.50000	+	12.9904i	0.746278	+	1.29259i	−0.203317	+	0.979113i	$0.565172\pi$
$102$		0			0
$103$	−4.00000	+	6.92820i	−0.394132	+	0.682656i	−0.992990	−	0.118199i	$-0.962288\pi$
$103$	−4.00000	+	6.92820i	−0.394132	+	0.682656i	0.598858	+	0.800855i	$0.295621\pi$
$104$	−4.00000			−0.392232
$105$		0			0
$106$	9.00000			0.874157
$107$	1.50000	−	2.59808i	0.145010	−	0.251166i	−0.784366	−	0.620298i	$-0.787012\pi$
$107$	1.50000	−	2.59808i	0.145010	−	0.251166i	0.929377	+	0.369132i	$0.120345\pi$
$108$		0			0
$109$	2.00000	+	3.46410i	0.191565	+	0.331801i	0.945769	−	0.324840i	$-0.105310\pi$
$109$	2.00000	+	3.46410i	0.191565	+	0.331801i	−0.754204	+	0.656640i	$0.771977\pi$
$110$		0			0
$111$		0			0
$112$	2.50000	−	0.866025i	0.236228	−	0.0818317i
$113$	−6.00000			−0.564433			−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000			−0.564433			−0.282216	+	0.959351i	$0.591070\pi$
$114$		0			0
$115$	−9.00000	−	15.5885i	−0.839254	−	1.45363i
$116$	1.50000	+	2.59808i	0.139272	+	0.241225i
$117$		0			0
$118$	−3.00000			−0.276172
$119$	−3.00000	+	15.5885i	−0.275010	+	1.42899i
$120$		0			0
$121$	5.50000	−	9.52628i	0.500000	−	0.866025i
$122$	5.00000	+	8.66025i	0.452679	+	0.784063i
$123$		0			0
$124$	−4.00000	+	6.92820i	−0.359211	+	0.622171i
$125$	−3.00000			−0.268328
$126$		0			0
$127$	5.00000			0.443678			0.221839	−	0.975083i	$-0.428794\pi$
$127$	5.00000			0.443678			0.221839	+	0.975083i	$0.428794\pi$
$128$	−0.500000	+	0.866025i	−0.0441942	+	0.0765466i
$129$		0			0
$130$	6.00000	+	10.3923i	0.526235	+	0.911465i
$131$	6.00000	−	10.3923i	0.524222	−	0.907980i	−0.475380	−	0.879781i	$-0.657689\pi$
$131$	6.00000	−	10.3923i	0.524222	−	0.907980i	0.999602	−	0.0281993i	$-0.00897729\pi$
$132$		0			0
$133$	−10.0000	+	3.46410i	−0.867110	+	0.300376i
$134$	−10.0000			−0.863868
$135$		0			0
$136$	−3.00000	−	5.19615i	−0.257248	−	0.445566i
$137$	−6.00000	−	10.3923i	−0.512615	−	0.887875i	−0.999893	−	0.0146279i	$-0.995344\pi$
$137$	−6.00000	−	10.3923i	−0.512615	−	0.887875i	0.487278	−	0.873247i	$-0.337990\pi$
$138$		0			0
$139$	2.00000			0.169638			0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000			0.169638			0.0848189	+	0.996396i	$0.472969\pi$
$140$	−6.00000	−	5.19615i	−0.507093	−	0.439155i
$141$		0			0
$142$	−3.00000	+	5.19615i	−0.251754	+	0.436051i
$143$		0			0
$144$		0			0
$145$	4.50000	−	7.79423i	0.373705	−	0.647275i
$146$	−7.00000			−0.579324
$147$		0			0
$148$	8.00000			0.657596
$149$	10.5000	−	18.1865i	0.860194	−	1.48990i	−0.0115483	−	0.999933i	$-0.503676\pi$
$149$	10.5000	−	18.1865i	0.860194	−	1.48990i	0.871742	−	0.489966i	$-0.162991\pi$
$150$		0			0
$151$	−4.00000	−	6.92820i	−0.325515	−	0.563809i	0.656101	−	0.754673i	$-0.272204\pi$
$151$	−4.00000	−	6.92820i	−0.325515	−	0.563809i	−0.981617	+	0.190864i	$0.938871\pi$
$152$	2.00000	−	3.46410i	0.162221	−	0.280976i
$153$		0			0
$154$		0			0
$155$	24.0000			1.92773
$156$		0			0
$157$	−7.00000	−	12.1244i	−0.558661	−	0.967629i	−0.997609	−	0.0691164i	$-0.977982\pi$
$157$	−7.00000	−	12.1244i	−0.558661	−	0.967629i	0.438948	−	0.898513i	$-0.355351\pi$
$158$	−8.50000	−	14.7224i	−0.676224	−	1.17125i
$159$		0			0
$160$	3.00000			0.237171
$161$	15.0000	−	5.19615i	1.18217	−	0.409514i
$162$		0			0
$163$	−1.00000	+	1.73205i	−0.0783260	+	0.135665i	−0.902528	−	0.430632i	$-0.858291\pi$
$163$	−1.00000	+	1.73205i	−0.0783260	+	0.135665i	0.824202	+	0.566296i	$0.191624\pi$
$164$	3.00000	+	5.19615i	0.234261	+	0.405751i
$165$		0			0
$166$	6.00000	−	10.3923i	0.465690	−	0.806599i
$167$	6.00000			0.464294			0.232147	−	0.972681i	$-0.425425\pi$
$167$	6.00000			0.464294			0.232147	+	0.972681i	$0.425425\pi$
$168$		0			0
$169$	3.00000			0.230769
$170$	−9.00000	+	15.5885i	−0.690268	+	1.19558i
$171$		0			0
$172$	−4.00000	−	6.92820i	−0.304997	−	0.528271i
$173$	4.50000	−	7.79423i	0.342129	−	0.592584i	−0.642699	−	0.766119i	$-0.722185\pi$
$173$	4.50000	−	7.79423i	0.342129	−	0.592584i	0.984828	+	0.173534i	$0.0555188\pi$
$174$		0			0
$175$	−2.00000	+	10.3923i	−0.151186	+	0.785584i
$176$		0			0
$177$		0			0
$178$	−3.00000	−	5.19615i	−0.224860	−	0.389468i
$179$	7.50000	+	12.9904i	0.560576	+	0.970947i	0.997446	+	0.0714220i	$0.0227537\pi$
$179$	7.50000	+	12.9904i	0.560576	+	0.970947i	−0.436870	+	0.899525i	$0.643913\pi$
$180$		0			0
$181$	−10.0000			−0.743294			−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000			−0.743294			−0.371647	+	0.928374i	$0.621207\pi$
$182$	−10.0000	+	3.46410i	−0.741249	+	0.256776i
$183$		0			0
$184$	−3.00000	+	5.19615i	−0.221163	+	0.383065i
$185$	−12.0000	−	20.7846i	−0.882258	−	1.52811i
$186$		0			0
$187$		0			0
$188$	6.00000			0.437595
$189$		0			0
$190$	−12.0000			−0.870572
$191$	9.00000	−	15.5885i	0.651217	−	1.12794i	−0.331611	−	0.943416i	$-0.607592\pi$
$191$	9.00000	−	15.5885i	0.651217	−	1.12794i	0.982828	−	0.184525i	$-0.0590746\pi$
$192$		0			0
$193$	−13.0000	−	22.5167i	−0.935760	−	1.62078i	−0.773272	−	0.634074i	$-0.781381\pi$
$193$	−13.0000	−	22.5167i	−0.935760	−	1.62078i	−0.162488	−	0.986710i	$-0.551952\pi$
$194$	5.00000	−	8.66025i	0.358979	−	0.621770i
$195$		0			0
$196$	5.50000	−	4.33013i	0.392857	−	0.309295i
$197$	−15.0000			−1.06871			−0.534353	−	0.845262i	$-0.679445\pi$
$197$	−15.0000			−1.06871			−0.534353	+	0.845262i	$0.679445\pi$
$198$		0			0
$199$	−5.50000	−	9.52628i	−0.389885	−	0.675300i	0.602549	−	0.798082i	$-0.294152\pi$
$199$	−5.50000	−	9.52628i	−0.389885	−	0.675300i	−0.992434	+	0.122782i	$0.960818\pi$
$200$	−2.00000	−	3.46410i	−0.141421	−	0.244949i
$201$		0			0
$202$	−15.0000			−1.05540
$203$	6.00000	+	5.19615i	0.421117	+	0.364698i
$204$		0			0
$205$	9.00000	−	15.5885i	0.628587	−	1.08875i
$206$	−4.00000	−	6.92820i	−0.278693	−	0.482711i
$207$		0			0
$208$	2.00000	−	3.46410i	0.138675	−	0.240192i
$209$		0			0
$210$		0			0
$211$	−4.00000			−0.275371			−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000			−0.275371			−0.137686	+	0.990476i	$0.543966\pi$
$212$	−4.50000	+	7.79423i	−0.309061	+	0.535310i
$213$		0			0
$214$	1.50000	+	2.59808i	0.102538	+	0.177601i
$215$	−12.0000	+	20.7846i	−0.818393	+	1.41750i
$216$		0			0
$217$	−4.00000	+	20.7846i	−0.271538	+	1.41095i
$218$	−4.00000			−0.270914
$219$		0			0
$220$		0			0
$221$	12.0000	+	20.7846i	0.807207	+	1.39812i
$222$		0			0
$223$	−1.00000			−0.0669650			−0.0334825	−	0.999439i	$-0.510660\pi$
$223$	−1.00000			−0.0669650			−0.0334825	+	0.999439i	$0.510660\pi$
$224$	−0.500000	+	2.59808i	−0.0334077	+	0.173591i
$225$		0			0
$226$	3.00000	−	5.19615i	0.199557	−	0.345643i
$227$	−1.50000	−	2.59808i	−0.0995585	−	0.172440i	0.811943	−	0.583736i	$-0.198410\pi$
$227$	−1.50000	−	2.59808i	−0.0995585	−	0.172440i	−0.911502	+	0.411296i	$0.865076\pi$
$228$		0			0
$229$	−7.00000	+	12.1244i	−0.462573	+	0.801200i	−0.999088	−	0.0426906i	$-0.986407\pi$
$229$	−7.00000	+	12.1244i	−0.462573	+	0.801200i	0.536515	+	0.843891i	$0.319740\pi$
$230$	18.0000			1.18688
$231$		0			0
$232$	−3.00000			−0.196960
$233$	−9.00000	+	15.5885i	−0.589610	+	1.02123i	0.404674	+	0.914461i	$0.367385\pi$
$233$	−9.00000	+	15.5885i	−0.589610	+	1.02123i	−0.994283	+	0.106773i	$0.965948\pi$
$234$		0			0
$235$	−9.00000	−	15.5885i	−0.587095	−	1.01688i
$236$	1.50000	−	2.59808i	0.0976417	−	0.169120i
$237$		0			0
$238$	−12.0000	−	10.3923i	−0.777844	−	0.673633i
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	0.500000	+	0.866025i	0.0322078	+	0.0557856i	0.881680	−	0.471848i	$-0.156413\pi$
$241$	0.500000	+	0.866025i	0.0322078	+	0.0557856i	−0.849472	+	0.527633i	$0.823079\pi$
$242$	5.50000	+	9.52628i	0.353553	+	0.612372i
$243$		0			0
$244$	−10.0000			−0.640184
$245$	−19.5000	−	7.79423i	−1.24581	−	0.497955i
$246$		0			0
$247$	−8.00000	+	13.8564i	−0.509028	+	0.881662i
$248$	−4.00000	−	6.92820i	−0.254000	−	0.439941i
$249$		0			0
$250$	1.50000	−	2.59808i	0.0948683	−	0.164317i
$251$	9.00000			0.568075			0.284037	−	0.958813i	$-0.408326\pi$
$251$	9.00000			0.568075			0.284037	+	0.958813i	$0.408326\pi$
$252$		0			0
$253$		0			0
$254$	−2.50000	+	4.33013i	−0.156864	+	0.271696i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$257$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$258$		0			0
$259$	20.0000	−	6.92820i	1.24274	−	0.430498i
$260$	−12.0000			−0.744208
$261$		0			0
$262$	6.00000	+	10.3923i	0.370681	+	0.642039i
$263$	9.00000	+	15.5885i	0.554964	+	0.961225i	0.997906	+	0.0646755i	$0.0206012\pi$
$263$	9.00000	+	15.5885i	0.554964	+	0.961225i	−0.442943	+	0.896550i	$0.646065\pi$
$264$		0			0
$265$	27.0000			1.65860
$266$	2.00000	−	10.3923i	0.122628	−	0.637193i
$267$		0			0
$268$	5.00000	−	8.66025i	0.305424	−	0.529009i
$269$	−10.5000	−	18.1865i	−0.640196	−	1.10885i	−0.985389	−	0.170321i	$-0.945520\pi$
$269$	−10.5000	−	18.1865i	−0.640196	−	1.10885i	0.345192	−	0.938532i	$-0.387814\pi$
$270$		0			0
$271$	12.5000	−	21.6506i	0.759321	−	1.31518i	−0.183876	−	0.982949i	$-0.558865\pi$
$271$	12.5000	−	21.6506i	0.759321	−	1.31518i	0.943197	−	0.332233i	$-0.107802\pi$
$272$	6.00000			0.363803
$273$		0			0
$274$	12.0000			0.724947
$275$		0			0
$276$		0			0
$277$	14.0000	+	24.2487i	0.841178	+	1.45696i	0.888899	+	0.458103i	$0.151471\pi$
$277$	14.0000	+	24.2487i	0.841178	+	1.45696i	−0.0477206	+	0.998861i	$0.515196\pi$
$278$	−1.00000	+	1.73205i	−0.0599760	+	0.103882i
$279$		0			0
$280$	7.50000	−	2.59808i	0.448211	−	0.155265i
$281$	−18.0000			−1.07379			−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000			−1.07379			−0.536895	+	0.843649i	$0.680403\pi$
$282$		0			0
$283$	2.00000	+	3.46410i	0.118888	+	0.205919i	0.919327	−	0.393494i	$-0.128734\pi$
$283$	2.00000	+	3.46410i	0.118888	+	0.205919i	−0.800439	+	0.599414i	$0.795400\pi$
$284$	−3.00000	−	5.19615i	−0.178017	−	0.308335i
$285$		0			0
$286$		0			0
$287$	12.0000	+	10.3923i	0.708338	+	0.613438i
$288$		0			0
$289$	−9.50000	+	16.4545i	−0.558824	+	0.967911i
$290$	4.50000	+	7.79423i	0.264249	+	0.457693i
$291$		0			0
$292$	3.50000	−	6.06218i	0.204822	−	0.354762i
$293$	−3.00000			−0.175262			−0.0876309	−	0.996153i	$-0.527930\pi$
$293$	−3.00000			−0.175262			−0.0876309	+	0.996153i	$0.527930\pi$
$294$		0			0
$295$	−9.00000			−0.524000
$296$	−4.00000	+	6.92820i	−0.232495	+	0.402694i
$297$		0			0
$298$	10.5000	+	18.1865i	0.608249	+	1.05352i
$299$	12.0000	−	20.7846i	0.693978	−	1.20201i
$300$		0			0
$301$	−16.0000	−	13.8564i	−0.922225	−	0.798670i
$302$	8.00000			0.460348
$303$		0			0
$304$	2.00000	+	3.46410i	0.114708	+	0.198680i
$305$	15.0000	+	25.9808i	0.858898	+	1.48765i
$306$		0			0
$307$	8.00000			0.456584			0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000			0.456584			0.228292	+	0.973593i	$0.426686\pi$
$308$		0			0
$309$		0			0
$310$	−12.0000	+	20.7846i	−0.681554	+	1.18049i
$311$	−6.00000	−	10.3923i	−0.340229	−	0.589294i	0.644246	−	0.764818i	$-0.277171\pi$
$311$	−6.00000	−	10.3923i	−0.340229	−	0.589294i	−0.984475	+	0.175525i	$0.943838\pi$
$312$		0			0
$313$	−2.50000	+	4.33013i	−0.141308	+	0.244753i	−0.927990	−	0.372606i	$-0.878464\pi$
$313$	−2.50000	+	4.33013i	−0.141308	+	0.244753i	0.786681	+	0.617359i	$0.211798\pi$
$314$	14.0000			0.790066
$315$		0			0
$316$	17.0000			0.956325
$317$	9.00000	−	15.5885i	0.505490	−	0.875535i	−0.494489	−	0.869184i	$-0.664645\pi$
$317$	9.00000	−	15.5885i	0.505490	−	0.875535i	0.999980	−	0.00635137i	$-0.00202172\pi$
$318$		0			0
$319$		0			0
$320$	−1.50000	+	2.59808i	−0.0838525	+	0.145237i
$321$		0			0
$322$	−3.00000	+	15.5885i	−0.167183	+	0.868711i
$323$	−24.0000			−1.33540
$324$		0			0
$325$	8.00000	+	13.8564i	0.443760	+	0.768615i
$326$	−1.00000	−	1.73205i	−0.0553849	−	0.0959294i
$327$		0			0
$328$	−6.00000			−0.331295
$329$	15.0000	−	5.19615i	0.826977	−	0.286473i
$330$		0			0
$331$	−1.00000	+	1.73205i	−0.0549650	+	0.0952021i	−0.892199	−	0.451643i	$-0.850838\pi$
$331$	−1.00000	+	1.73205i	−0.0549650	+	0.0952021i	0.837234	+	0.546845i	$0.184171\pi$
$332$	6.00000	+	10.3923i	0.329293	+	0.570352i
$333$		0			0
$334$	−3.00000	+	5.19615i	−0.164153	+	0.284321i
$335$	−30.0000			−1.63908
$336$		0			0
$337$	−7.00000			−0.381314			−0.190657	−	0.981657i	$-0.561062\pi$
$337$	−7.00000			−0.381314			−0.190657	+	0.981657i	$0.561062\pi$
$338$	−1.50000	+	2.59808i	−0.0815892	+	0.141317i
$339$		0			0
$340$	−9.00000	−	15.5885i	−0.488094	−	0.845403i
$341$		0			0
$342$		0			0
$343$	10.0000	−	15.5885i	0.539949	−	0.841698i
$344$	8.00000			0.431331
$345$		0			0
$346$	4.50000	+	7.79423i	0.241921	+	0.419020i
$347$	−16.5000	−	28.5788i	−0.885766	−	1.53419i	−0.844833	−	0.535031i	$-0.820300\pi$
$347$	−16.5000	−	28.5788i	−0.885766	−	1.53419i	−0.0409337	−	0.999162i	$-0.513033\pi$
$348$		0			0
$349$	−10.0000			−0.535288			−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000			−0.535288			−0.267644	+	0.963518i	$0.586245\pi$
$350$	−8.00000	−	6.92820i	−0.427618	−	0.370328i
$351$		0			0
$352$		0			0
$353$	15.0000	+	25.9808i	0.798369	+	1.38282i	0.920677	+	0.390324i	$0.127637\pi$
$353$	15.0000	+	25.9808i	0.798369	+	1.38282i	−0.122308	+	0.992492i	$0.539030\pi$
$354$		0			0
$355$	−9.00000	+	15.5885i	−0.477670	+	0.827349i
$356$	6.00000			0.317999
$357$		0			0
$358$	−15.0000			−0.792775
$359$	−18.0000	+	31.1769i	−0.950004	+	1.64545i	−0.204595	+	0.978847i	$0.565588\pi$
$359$	−18.0000	+	31.1769i	−0.950004	+	1.64545i	−0.745409	+	0.666608i	$0.767746\pi$
$360$		0			0
$361$	1.50000	+	2.59808i	0.0789474	+	0.136741i
$362$	5.00000	−	8.66025i	0.262794	−	0.455173i
$363$		0			0
$364$	2.00000	−	10.3923i	0.104828	−	0.544705i
$365$	−21.0000			−1.09919
$366$		0			0
$367$	9.50000	+	16.4545i	0.495896	+	0.858917i	0.999989	−	0.00473247i	$-0.00150640\pi$
$367$	9.50000	+	16.4545i	0.495896	+	0.858917i	−0.504093	+	0.863649i	$0.668173\pi$
$368$	−3.00000	−	5.19615i	−0.156386	−	0.270868i
$369$		0			0
$370$	24.0000			1.24770
$371$	−4.50000	+	23.3827i	−0.233628	+	1.21397i
$372$		0			0
$373$	5.00000	−	8.66025i	0.258890	−	0.448411i	−0.707055	−	0.707159i	$-0.749977\pi$
$373$	5.00000	−	8.66025i	0.258890	−	0.448411i	0.965945	+	0.258748i	$0.0833099\pi$
$374$		0			0
$375$		0			0
$376$	−3.00000	+	5.19615i	−0.154713	+	0.267971i
$377$	12.0000			0.618031
$378$		0			0
$379$	−10.0000			−0.513665			−0.256833	−	0.966456i	$-0.582679\pi$
$379$	−10.0000			−0.513665			−0.256833	+	0.966456i	$0.582679\pi$
$380$	6.00000	−	10.3923i	0.307794	−	0.533114i
$381$		0			0
$382$	9.00000	+	15.5885i	0.460480	+	0.797575i
$383$	−3.00000	+	5.19615i	−0.153293	+	0.265511i	−0.932436	−	0.361335i	$-0.882321\pi$
$383$	−3.00000	+	5.19615i	−0.153293	+	0.265511i	0.779143	+	0.626846i	$0.215654\pi$
$384$		0			0
$385$		0			0
$386$	26.0000			1.32337
$387$		0			0
$388$	5.00000	+	8.66025i	0.253837	+	0.439658i
$389$	4.50000	+	7.79423i	0.228159	+	0.395183i	0.957263	−	0.289220i	$-0.0933960\pi$
$389$	4.50000	+	7.79423i	0.228159	+	0.395183i	−0.729103	+	0.684403i	$0.760063\pi$
$390$		0			0
$391$	36.0000			1.82060
$392$	1.00000	+	6.92820i	0.0505076	+	0.349927i
$393$		0			0
$394$	7.50000	−	12.9904i	0.377845	−	0.654446i
$395$	−25.5000	−	44.1673i	−1.28304	−	2.22230i
$396$		0			0
$397$	−16.0000	+	27.7128i	−0.803017	+	1.39087i	0.114605	+	0.993411i	$0.463440\pi$
$397$	−16.0000	+	27.7128i	−0.803017	+	1.39087i	−0.917622	+	0.397455i	$0.869893\pi$
$398$	11.0000			0.551380
$399$		0			0
$400$	4.00000			0.200000
$401$	9.00000	−	15.5885i	0.449439	−	0.778450i	−0.548911	−	0.835881i	$-0.684957\pi$
$401$	9.00000	−	15.5885i	0.449439	−	0.778450i	0.998350	+	0.0574304i	$0.0182907\pi$
$402$		0			0
$403$	16.0000	+	27.7128i	0.797017	+	1.38047i
$404$	7.50000	−	12.9904i	0.373139	−	0.646296i
$405$		0			0
$406$	−7.50000	+	2.59808i	−0.372219	+	0.128940i
$407$		0			0
$408$		0			0
$409$	12.5000	+	21.6506i	0.618085	+	1.07056i	0.989835	+	0.142222i	$0.0454247\pi$
$409$	12.5000	+	21.6506i	0.618085	+	1.07056i	−0.371750	+	0.928333i	$0.621242\pi$
$410$	9.00000	+	15.5885i	0.444478	+	0.769859i
$411$		0			0
$412$	8.00000			0.394132
$413$	1.50000	−	7.79423i	0.0738102	−	0.383529i
$414$		0			0
$415$	18.0000	−	31.1769i	0.883585	−	1.53041i
$416$	2.00000	+	3.46410i	0.0980581	+	0.169842i
$417$		0			0
$418$		0			0
$419$	12.0000			0.586238			0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000			0.586238			0.293119	+	0.956076i	$0.405307\pi$
$420$		0			0
$421$	32.0000			1.55958			0.779792	−	0.626038i	$-0.215325\pi$
$421$	32.0000			1.55958			0.779792	+	0.626038i	$0.215325\pi$
$422$	2.00000	−	3.46410i	0.0973585	−	0.168630i
$423$		0			0
$424$	−4.50000	−	7.79423i	−0.218539	−	0.378521i
$425$	−12.0000	+	20.7846i	−0.582086	+	1.00820i
$426$		0			0
$427$	−25.0000	+	8.66025i	−1.20983	+	0.419099i
$428$	−3.00000			−0.145010
$429$		0			0
$430$	−12.0000	−	20.7846i	−0.578691	−	1.00232i
$431$	−6.00000	−	10.3923i	−0.289010	−	0.500580i	0.684564	−	0.728953i	$-0.259993\pi$
$431$	−6.00000	−	10.3923i	−0.289010	−	0.500580i	−0.973574	+	0.228373i	$0.926659\pi$
$432$		0			0
$433$	−7.00000			−0.336399			−0.168199	−	0.985753i	$-0.553795\pi$
$433$	−7.00000			−0.336399			−0.168199	+	0.985753i	$0.553795\pi$
$434$	−16.0000	−	13.8564i	−0.768025	−	0.665129i
$435$		0			0
$436$	2.00000	−	3.46410i	0.0957826	−	0.165900i
$437$	12.0000	+	20.7846i	0.574038	+	0.994263i
$438$		0			0
$439$	−4.00000	+	6.92820i	−0.190910	+	0.330665i	−0.945552	−	0.325471i	$-0.894477\pi$
$439$	−4.00000	+	6.92820i	−0.190910	+	0.330665i	0.754642	+	0.656136i	$0.227810\pi$
$440$		0			0
$441$		0			0
$442$	−24.0000			−1.14156
$443$	19.5000	−	33.7750i	0.926473	−	1.60470i	0.137298	−	0.990530i	$-0.456158\pi$
$443$	19.5000	−	33.7750i	0.926473	−	1.60470i	0.789175	−	0.614168i	$-0.210508\pi$
$444$		0			0
$445$	−9.00000	−	15.5885i	−0.426641	−	0.738964i
$446$	0.500000	−	0.866025i	0.0236757	−	0.0410075i
$447$		0			0
$448$	−2.00000	−	1.73205i	−0.0944911	−	0.0818317i
$449$	6.00000			0.283158			0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000			0.283158			0.141579	+	0.989927i	$0.454782\pi$
$450$		0			0
$451$		0			0
$452$	3.00000	+	5.19615i	0.141108	+	0.244406i
$453$		0			0
$454$	3.00000			0.140797
$455$	−30.0000	+	10.3923i	−1.40642	+	0.487199i
$456$		0			0
$457$	0.500000	−	0.866025i	0.0233890	−	0.0405110i	−0.854094	−	0.520119i	$-0.825888\pi$
$457$	0.500000	−	0.866025i	0.0233890	−	0.0405110i	0.877483	+	0.479608i	$0.159221\pi$
$458$	−7.00000	−	12.1244i	−0.327089	−	0.566534i
$459$		0			0
$460$	−9.00000	+	15.5885i	−0.419627	+	0.726816i
$461$	−33.0000			−1.53696			−0.768482	−	0.639872i	$-0.778987\pi$
$461$	−33.0000			−1.53696			−0.768482	+	0.639872i	$0.778987\pi$
$462$		0			0
$463$	−19.0000			−0.883005			−0.441502	−	0.897260i	$-0.645554\pi$
$463$	−19.0000			−0.883005			−0.441502	+	0.897260i	$0.645554\pi$
$464$	1.50000	−	2.59808i	0.0696358	−	0.120613i
$465$		0			0
$466$	−9.00000	−	15.5885i	−0.416917	−	0.722121i
$467$	16.5000	−	28.5788i	0.763529	−	1.32247i	−0.177492	−	0.984122i	$-0.556798\pi$
$467$	16.5000	−	28.5788i	0.763529	−	1.32247i	0.941021	−	0.338349i	$-0.109868\pi$
$468$		0			0
$469$	5.00000	−	25.9808i	0.230879	−	1.19968i
$470$	18.0000			0.830278
$471$		0			0
$472$	1.50000	+	2.59808i	0.0690431	+	0.119586i
$473$		0			0
$474$		0			0
$475$	−16.0000			−0.734130
$476$	15.0000	−	5.19615i	0.687524	−	0.238165i
$477$		0			0
$478$		0			0
$479$	12.0000	+	20.7846i	0.548294	+	0.949673i	0.998392	+	0.0566937i	$0.0180558\pi$
$479$	12.0000	+	20.7846i	0.548294	+	0.949673i	−0.450098	+	0.892979i	$0.648611\pi$
$480$		0			0
$481$	16.0000	−	27.7128i	0.729537	−	1.26360i
$482$	−1.00000			−0.0455488
$483$		0			0
$484$	−11.0000			−0.500000
$485$	15.0000	−	25.9808i	0.681115	−	1.17973i
$486$		0			0
$487$	6.50000	+	11.2583i	0.294543	+	0.510164i	0.974879	−	0.222737i	$-0.0714992\pi$
$487$	6.50000	+	11.2583i	0.294543	+	0.510164i	−0.680335	+	0.732901i	$0.738166\pi$
$488$	5.00000	−	8.66025i	0.226339	−	0.392031i
$489$		0			0
$490$	16.5000	−	12.9904i	0.745394	−	0.586846i
$491$	27.0000			1.21849			0.609246	−	0.792981i	$-0.291472\pi$
$491$	27.0000			1.21849			0.609246	+	0.792981i	$0.291472\pi$
$492$		0			0
$493$	9.00000	+	15.5885i	0.405340	+	0.702069i
$494$	−8.00000	−	13.8564i	−0.359937	−	0.623429i
$495$		0			0
$496$	8.00000			0.359211
$497$	−12.0000	−	10.3923i	−0.538274	−	0.466159i
$498$		0			0
$499$	−1.00000	+	1.73205i	−0.0447661	+	0.0775372i	−0.887540	−	0.460730i	$-0.847588\pi$
$499$	−1.00000	+	1.73205i	−0.0447661	+	0.0775372i	0.842774	+	0.538267i	$0.180921\pi$
$500$	1.50000	+	2.59808i	0.0670820	+	0.116190i
$501$		0			0
$502$	−4.50000	+	7.79423i	−0.200845	+	0.347873i
$503$	−24.0000			−1.07011			−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000			−1.07011			−0.535054	+	0.844818i	$0.679709\pi$
$504$		0			0
$505$	−45.0000			−2.00247
$506$		0			0
$507$		0			0
$508$	−2.50000	−	4.33013i	−0.110920	−	0.192118i
$509$	−9.00000	+	15.5885i	−0.398918	+	0.690946i	−0.993593	−	0.113020i	$-0.963948\pi$
$509$	−9.00000	+	15.5885i	−0.398918	+	0.690946i	0.594675	+	0.803966i	$0.297281\pi$
$510$		0			0
$511$	3.50000	−	18.1865i	0.154831	−	0.804525i
$512$	1.00000			0.0441942
$513$		0			0
$514$		0			0
$515$	−12.0000	−	20.7846i	−0.528783	−	0.915879i
$516$		0			0
$517$		0			0
$518$	−4.00000	+	20.7846i	−0.175750	+	0.913223i
$519$		0			0
$520$	6.00000	−	10.3923i	0.263117	−	0.455733i
$521$	15.0000	+	25.9808i	0.657162	+	1.13824i	0.981347	+	0.192244i	$0.0615766\pi$
$521$	15.0000	+	25.9808i	0.657162	+	1.13824i	−0.324185	+	0.945994i	$0.605090\pi$
$522$		0			0
$523$	2.00000	−	3.46410i	0.0874539	−	0.151475i	−0.818980	−	0.573822i	$-0.805460\pi$
$523$	2.00000	−	3.46410i	0.0874539	−	0.151475i	0.906434	+	0.422347i	$0.138794\pi$
$524$	−12.0000			−0.524222
$525$		0			0
$526$	−18.0000			−0.784837
$527$	−24.0000	+	41.5692i	−1.04546	+	1.81078i
$528$		0			0
$529$	−6.50000	−	11.2583i	−0.282609	−	0.489493i
$530$	−13.5000	+	23.3827i	−0.586403	+	1.01568i
$531$		0			0
$532$	8.00000	+	6.92820i	0.346844	+	0.300376i
$533$	24.0000			1.03956
$534$		0			0
$535$	4.50000	+	7.79423i	0.194552	+	0.336974i
$536$	5.00000	+	8.66025i	0.215967	+	0.374066i
$537$		0			0
$538$	21.0000			0.905374
$539$		0			0
$540$		0			0
$541$	5.00000	−	8.66025i	0.214967	−	0.372333i	−0.738296	−	0.674477i	$-0.764369\pi$
$541$	5.00000	−	8.66025i	0.214967	−	0.372333i	0.953262	+	0.302144i	$0.0977023\pi$
$542$	12.5000	+	21.6506i	0.536921	+	0.929974i
$543$		0			0
$544$	−3.00000	+	5.19615i	−0.128624	+	0.222783i
$545$	−12.0000			−0.514024
$546$		0			0
$547$	−28.0000			−1.19719			−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000			−1.19719			−0.598597	+	0.801050i	$0.704275\pi$
$548$	−6.00000	+	10.3923i	−0.256307	+	0.443937i
$549$		0			0
$550$		0			0
$551$	−6.00000	+	10.3923i	−0.255609	+	0.442727i
$552$		0			0
$553$	42.5000	−	14.7224i	1.80728	−	0.626061i
$554$	−28.0000			−1.18961
$555$		0			0
$556$	−1.00000	−	1.73205i	−0.0424094	−	0.0734553i
$557$	3.00000	+	5.19615i	0.127114	+	0.220168i	0.922557	−	0.385860i	$-0.126095\pi$
$557$	3.00000	+	5.19615i	0.127114	+	0.220168i	−0.795443	+	0.606028i	$0.792762\pi$
$558$		0			0
$559$	−32.0000			−1.35346
$560$	−1.50000	+	7.79423i	−0.0633866	+	0.329366i
$561$		0			0
$562$	9.00000	−	15.5885i	0.379642	−	0.657559i
$563$	−7.50000	−	12.9904i	−0.316087	−	0.547479i	0.663581	−	0.748105i	$-0.269036\pi$
$563$	−7.50000	−	12.9904i	−0.316087	−	0.547479i	−0.979668	+	0.200625i	$0.935703\pi$
$564$		0			0
$565$	9.00000	−	15.5885i	0.378633	−	0.655811i
$566$	−4.00000			−0.168133
$567$		0			0
$568$	6.00000			0.251754
$569$	−9.00000	+	15.5885i	−0.377300	+	0.653502i	−0.990668	−	0.136295i	$-0.956481\pi$
$569$	−9.00000	+	15.5885i	−0.377300	+	0.653502i	0.613369	+	0.789797i	$0.289814\pi$
$570$		0			0
$571$	−10.0000	−	17.3205i	−0.418487	−	0.724841i	0.577301	−	0.816532i	$-0.304106\pi$
$571$	−10.0000	−	17.3205i	−0.418487	−	0.724841i	−0.995788	+	0.0916910i	$0.970773\pi$
$572$		0			0
$573$		0			0
$574$	−15.0000	+	5.19615i	−0.626088	+	0.216883i
$575$	24.0000			1.00087
$576$		0			0
$577$	−11.5000	−	19.9186i	−0.478751	−	0.829222i	0.520952	−	0.853586i	$-0.325577\pi$
$577$	−11.5000	−	19.9186i	−0.478751	−	0.829222i	−0.999703	+	0.0243645i	$0.992244\pi$
$578$	−9.50000	−	16.4545i	−0.395148	−	0.684416i
$579$		0			0
$580$	−9.00000			−0.373705
$581$	24.0000	+	20.7846i	0.995688	+	0.862291i
$582$		0			0
$583$		0			0
$584$	3.50000	+	6.06218i	0.144831	+	0.250855i
$585$		0			0
$586$	1.50000	−	2.59808i	0.0619644	−	0.107326i
$587$	−15.0000			−0.619116			−0.309558	−	0.950881i	$-0.600181\pi$
$587$	−15.0000			−0.619116			−0.309558	+	0.950881i	$0.600181\pi$
$588$		0			0
$589$	−32.0000			−1.31854
$590$	4.50000	−	7.79423i	0.185262	−	0.320883i
$591$		0			0
$592$	−4.00000	−	6.92820i	−0.164399	−	0.284747i
$593$	−3.00000	+	5.19615i	−0.123195	+	0.213380i	−0.921026	−	0.389501i	$-0.872647\pi$
$593$	−3.00000	+	5.19615i	−0.123195	+	0.213380i	0.797831	+	0.602881i	$0.205981\pi$
$594$		0			0
$595$	−36.0000	−	31.1769i	−1.47586	−	1.27813i
$596$	−21.0000			−0.860194
$597$		0			0
$598$	12.0000	+	20.7846i	0.490716	+	0.849946i
$599$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$599$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$600$		0			0
$601$	−25.0000			−1.01977			−0.509886	−	0.860242i	$-0.670312\pi$
$601$	−25.0000			−1.01977			−0.509886	+	0.860242i	$0.670312\pi$
$602$	20.0000	−	6.92820i	0.815139	−	0.282372i
$603$		0			0
$604$	−4.00000	+	6.92820i	−0.162758	+	0.281905i
$605$	16.5000	+	28.5788i	0.670820	+	1.16190i
$606$		0			0
$607$	−5.50000	+	9.52628i	−0.223238	+	0.386660i	−0.955789	−	0.294052i	$-0.904996\pi$
$607$	−5.50000	+	9.52628i	−0.223238	+	0.386660i	0.732551	+	0.680712i	$0.238329\pi$
$608$	−4.00000			−0.162221
$609$		0			0
$610$	−30.0000			−1.21466
$611$	12.0000	−	20.7846i	0.485468	−	0.840855i
$612$		0			0
$613$	8.00000	+	13.8564i	0.323117	+	0.559655i	0.981129	−	0.193352i	$-0.0619359\pi$
$613$	8.00000	+	13.8564i	0.323117	+	0.559655i	−0.658012	+	0.753007i	$0.728603\pi$
$614$	−4.00000	+	6.92820i	−0.161427	+	0.279600i
$615$		0			0
$616$		0			0
$617$	−18.0000			−0.724653			−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000			−0.724653			−0.362326	+	0.932051i	$0.618017\pi$
$618$		0			0
$619$	8.00000	+	13.8564i	0.321547	+	0.556936i	0.980807	−	0.194979i	$-0.0624638\pi$
$619$	8.00000	+	13.8564i	0.321547	+	0.556936i	−0.659260	+	0.751915i	$0.729130\pi$
$620$	−12.0000	−	20.7846i	−0.481932	−	0.834730i
$621$		0			0
$622$	12.0000			0.481156
$623$	15.0000	−	5.19615i	0.600962	−	0.208179i
$624$		0			0
$625$	14.5000	−	25.1147i	0.580000	−	1.00459i
$626$	−2.50000	−	4.33013i	−0.0999201	−	0.173067i
$627$		0			0
$628$	−7.00000	+	12.1244i	−0.279330	+	0.483814i
$629$	48.0000			1.91389
$630$		0			0
$631$	20.0000			0.796187			0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000			0.796187			0.398094	+	0.917345i	$0.369672\pi$
$632$	−8.50000	+	14.7224i	−0.338112	+	0.585627i
$633$		0			0
$634$	9.00000	+	15.5885i	0.357436	+	0.619097i
$635$	−7.50000	+	12.9904i	−0.297628	+	0.515508i
$636$		0			0
$637$	−4.00000	−	27.7128i	−0.158486	−	1.09802i
$638$		0			0
$639$		0			0
$640$	−1.50000	−	2.59808i	−0.0592927	−	0.102698i
$641$	−12.0000	−	20.7846i	−0.473972	−	0.820943i	0.525584	−	0.850741i	$-0.323847\pi$
$641$	−12.0000	−	20.7846i	−0.473972	−	0.820943i	−0.999556	+	0.0297987i	$0.990513\pi$
$642$		0			0
$643$	−16.0000			−0.630978			−0.315489	−	0.948929i	$-0.602169\pi$
$643$	−16.0000			−0.630978			−0.315489	+	0.948929i	$0.602169\pi$
$644$	−12.0000	−	10.3923i	−0.472866	−	0.409514i
$645$		0			0
$646$	12.0000	−	20.7846i	0.472134	−	0.817760i
$647$	−21.0000	−	36.3731i	−0.825595	−	1.42997i	−0.901464	−	0.432855i	$-0.857506\pi$
$647$	−21.0000	−	36.3731i	−0.825595	−	1.42997i	0.0758684	−	0.997118i	$-0.475827\pi$
$648$		0			0
$649$		0			0
$650$	−16.0000			−0.627572
$651$		0			0
$652$	2.00000			0.0783260
$653$	−1.50000	+	2.59808i	−0.0586995	+	0.101671i	−0.893882	−	0.448303i	$-0.852029\pi$
$653$	−1.50000	+	2.59808i	−0.0586995	+	0.101671i	0.835182	+	0.549973i	$0.185362\pi$
$654$		0			0
$655$	18.0000	+	31.1769i	0.703318	+	1.21818i
$656$	3.00000	−	5.19615i	0.117130	−	0.202876i
$657$		0			0
$658$	−3.00000	+	15.5885i	−0.116952	+	0.607701i
$659$	21.0000			0.818044			0.409022	−	0.912525i	$-0.365870\pi$
$659$	21.0000			0.818044			0.409022	+	0.912525i	$0.365870\pi$
$660$		0			0
$661$	−16.0000	−	27.7128i	−0.622328	−	1.07790i	−0.989051	−	0.147573i	$-0.952854\pi$
$661$	−16.0000	−	27.7128i	−0.622328	−	1.07790i	0.366723	−	0.930330i	$-0.380480\pi$
$662$	−1.00000	−	1.73205i	−0.0388661	−	0.0673181i
$663$		0			0
$664$	−12.0000			−0.465690
$665$	6.00000	−	31.1769i	0.232670	−	1.20899i
$666$		0			0
$667$	9.00000	−	15.5885i	0.348481	−	0.603587i
$668$	−3.00000	−	5.19615i	−0.116073	−	0.201045i
$669$		0			0
$670$	15.0000	−	25.9808i	0.579501	−	1.00372i
$671$		0			0
$672$		0			0
$673$	−43.0000			−1.65753			−0.828764	−	0.559598i	$-0.810955\pi$
$673$	−43.0000			−1.65753			−0.828764	+	0.559598i	$0.810955\pi$
$674$	3.50000	−	6.06218i	0.134815	−	0.233506i
$675$		0			0
$676$	−1.50000	−	2.59808i	−0.0576923	−	0.0999260i
$677$	10.5000	−	18.1865i	0.403548	−	0.698965i	−0.590603	−	0.806962i	$-0.701110\pi$
$677$	10.5000	−	18.1865i	0.403548	−	0.698965i	0.994151	+	0.107997i	$0.0344436\pi$
$678$		0			0
$679$	20.0000	+	17.3205i	0.767530	+	0.664700i
$680$	18.0000			0.690268
$681$		0			0
$682$		0			0
$683$	4.50000	+	7.79423i	0.172188	+	0.298238i	0.939184	−	0.343413i	$-0.111583\pi$
$683$	4.50000	+	7.79423i	0.172188	+	0.298238i	−0.766997	+	0.641651i	$0.778250\pi$
$684$		0			0
$685$	36.0000			1.37549
$686$	8.50000	+	16.4545i	0.324532	+	0.628235i
$687$		0			0
$688$	−4.00000	+	6.92820i	−0.152499	+	0.264135i
$689$	18.0000	+	31.1769i	0.685745	+	1.18775i
$690$		0			0
$691$	−4.00000	+	6.92820i	−0.152167	+	0.263561i	−0.932024	−	0.362397i	$-0.881959\pi$
$691$	−4.00000	+	6.92820i	−0.152167	+	0.263561i	0.779857	+	0.625958i	$0.215292\pi$
$692$	−9.00000			−0.342129
$693$		0			0
$694$	33.0000			1.25266
$695$	−3.00000	+	5.19615i	−0.113796	+	0.197101i
$696$		0			0
$697$	18.0000	+	31.1769i	0.681799	+	1.18091i
$698$	5.00000	−	8.66025i	0.189253	−	0.327795i
$699$		0			0
$700$	10.0000	−	3.46410i	0.377964	−	0.130931i
$701$	−39.0000			−1.47301			−0.736505	−	0.676432i	$-0.763525\pi$
$701$	−39.0000			−1.47301			−0.736505	+	0.676432i	$0.763525\pi$
$702$		0			0
$703$	16.0000	+	27.7128i	0.603451	+	1.04521i
$704$		0			0
$705$		0			0
$706$	−30.0000			−1.12906
$707$	7.50000	−	38.9711i	0.282067	−	1.46566i
$708$		0			0
$709$	−4.00000	+	6.92820i	−0.150223	+	0.260194i	−0.931309	−	0.364229i	$-0.881333\pi$
$709$	−4.00000	+	6.92820i	−0.150223	+	0.260194i	0.781086	+	0.624423i	$0.214666\pi$
$710$	−9.00000	−	15.5885i	−0.337764	−	0.585024i
$711$		0			0
$712$	−3.00000	+	5.19615i	−0.112430	+	0.194734i
$713$	48.0000			1.79761
$714$		0			0
$715$		0			0
$716$	7.50000	−	12.9904i	0.280288	−	0.485473i
$717$		0			0
$718$	−18.0000	−	31.1769i	−0.671754	−	1.16351i
$719$	12.0000	−	20.7846i	0.447524	−	0.775135i	−0.550700	−	0.834703i	$-0.685639\pi$
$719$	12.0000	−	20.7846i	0.447524	−	0.775135i	0.998224	+	0.0595683i	$0.0189724\pi$
$720$		0			0
$721$	20.0000	−	6.92820i	0.744839	−	0.258020i
$722$	−3.00000			−0.111648
$723$		0			0
$724$	5.00000	+	8.66025i	0.185824	+	0.321856i
$725$	6.00000	+	10.3923i	0.222834	+	0.385961i
$726$		0			0
$727$	23.0000			0.853023			0.426511	−	0.904482i	$-0.359742\pi$
$727$	23.0000			0.853023			0.426511	+	0.904482i	$0.359742\pi$
$728$	8.00000	+	6.92820i	0.296500	+	0.256776i
$729$		0			0
$730$	10.5000	−	18.1865i	0.388622	−	0.673114i
$731$	−24.0000	−	41.5692i	−0.887672	−	1.53749i
$732$		0			0
$733$	−22.0000	+	38.1051i	−0.812589	+	1.40744i	0.0984580	+	0.995141i	$0.468609\pi$
$733$	−22.0000	+	38.1051i	−0.812589	+	1.40744i	−0.911047	+	0.412303i	$0.864724\pi$
$734$	−19.0000			−0.701303
$735$		0			0
$736$	6.00000			0.221163
$737$		0			0
$738$		0			0
$739$	−7.00000	−	12.1244i	−0.257499	−	0.446002i	0.708072	−	0.706140i	$-0.249565\pi$
$739$	−7.00000	−	12.1244i	−0.257499	−	0.446002i	−0.965571	+	0.260138i	$0.916232\pi$
$740$	−12.0000	+	20.7846i	−0.441129	+	0.764057i
$741$		0			0
$742$	−18.0000	−	15.5885i	−0.660801	−	0.572270i
$743$	24.0000			0.880475			0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000			0.880475			0.440237	+	0.897881i	$0.354894\pi$
$744$		0			0
$745$	31.5000	+	54.5596i	1.15407	+	1.99891i
$746$	5.00000	+	8.66025i	0.183063	+	0.317074i
$747$		0			0
$748$		0			0
$749$	−7.50000	+	2.59808i	−0.274044	+	0.0949316i
$750$		0			0
$751$	21.5000	−	37.2391i	0.784546	−	1.35887i	−0.144724	−	0.989472i	$-0.546229\pi$
$751$	21.5000	−	37.2391i	0.784546	−	1.35887i	0.929270	−	0.369402i	$-0.120437\pi$
$752$	−3.00000	−	5.19615i	−0.109399	−	0.189484i
$753$		0			0
$754$	−6.00000	+	10.3923i	−0.218507	+	0.378465i
$755$	24.0000			0.873449
$756$		0			0
$757$	2.00000			0.0726912			0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000			0.0726912			0.0363456	+	0.999339i	$0.488428\pi$
$758$	5.00000	−	8.66025i	0.181608	−	0.314555i
$759$		0			0
$760$	6.00000	+	10.3923i	0.217643	+	0.376969i
$761$	−24.0000	+	41.5692i	−0.869999	+	1.50688i	−0.00800331	+	0.999968i	$0.502548\pi$
$761$	−24.0000	+	41.5692i	−0.869999	+	1.50688i	−0.861996	+	0.506915i	$0.830786\pi$
$762$		0			0
$763$	2.00000	−	10.3923i	0.0724049	−	0.376227i
$764$	−18.0000			−0.651217
$765$		0			0
$766$	−3.00000	−	5.19615i	−0.108394	−	0.187745i
$767$	−6.00000	−	10.3923i	−0.216647	−	0.375244i
$768$		0			0
$769$	26.0000			0.937584			0.468792	−	0.883309i	$-0.344689\pi$
$769$	26.0000			0.937584			0.468792	+	0.883309i	$0.344689\pi$
$770$		0			0
$771$		0			0
$772$	−13.0000	+	22.5167i	−0.467880	+	0.810392i
$773$	9.00000	+	15.5885i	0.323708	+	0.560678i	0.981250	−	0.192740i	$-0.0617373\pi$
$773$	9.00000	+	15.5885i	0.323708	+	0.560678i	−0.657542	+	0.753418i	$0.728404\pi$
$774$		0			0
$775$	−16.0000	+	27.7128i	−0.574737	+	0.995474i
$776$	−10.0000			−0.358979
$777$		0			0
$778$	−9.00000			−0.322666
$779$	−12.0000	+	20.7846i	−0.429945	+	0.744686i
$780$		0			0
$781$		0			0
$782$	−18.0000	+	31.1769i	−0.643679	+	1.11488i
$783$		0			0
$784$	−6.50000	−	2.59808i	−0.232143	−	0.0927884i
$785$	42.0000			1.49904
$786$		0			0
$787$	11.0000	+	19.0526i	0.392108	+	0.679150i	0.992727	−	0.120384i	$-0.0384127\pi$
$787$	11.0000	+	19.0526i	0.392108	+	0.679150i	−0.600620	+	0.799535i	$0.705079\pi$
$788$	7.50000	+	12.9904i	0.267176	+	0.462763i
$789$		0			0
$790$	51.0000			1.81450
$791$	12.0000	+	10.3923i	0.426671	+	0.369508i
$792$		0			0
$793$	−20.0000	+	34.6410i	−0.710221	+	1.23014i
$794$	−16.0000	−	27.7128i	−0.567819	−	0.983491i
$795$		0			0
$796$	−5.50000	+	9.52628i	−0.194942	+	0.337650i
$797$	18.0000			0.637593			0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000			0.637593			0.318796	+	0.947823i	$0.396721\pi$
$798$		0			0
$799$	36.0000			1.27359
$800$	−2.00000	+	3.46410i	−0.0707107	+	0.122474i
$801$		0			0
$802$	9.00000	+	15.5885i	0.317801	+	0.550448i
$803$		0			0
$804$		0			0
$805$	−9.00000	+	46.7654i	−0.317208	+	1.64826i
$806$	−32.0000			−1.12715
$807$		0			0
$808$	7.50000	+	12.9904i	0.263849	+	0.457000i
$809$	−12.0000	−	20.7846i	−0.421898	−	0.730748i	0.574228	−	0.818696i	$-0.305302\pi$
$809$	−12.0000	−	20.7846i	−0.421898	−	0.730748i	−0.996125	+	0.0879478i	$0.971969\pi$
$810$		0			0
$811$	20.0000			0.702295			0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000			0.702295			0.351147	+	0.936320i	$0.385792\pi$
$812$	1.50000	−	7.79423i	0.0526397	−	0.273524i
$813$		0			0
$814$		0			0
$815$	−3.00000	−	5.19615i	−0.105085	−	0.182013i
$816$		0			0
$817$	16.0000	−	27.7128i	0.559769	−	0.969549i
$818$	−25.0000			−0.874105
$819$		0			0
$820$	−18.0000			−0.628587
$821$	−10.5000	+	18.1865i	−0.366453	+	0.634714i	−0.989008	−	0.147861i	$-0.952761\pi$
$821$	−10.5000	+	18.1865i	−0.366453	+	0.634714i	0.622556	+	0.782576i	$0.286094\pi$
$822$		0			0
$823$	−11.5000	−	19.9186i	−0.400865	−	0.694318i	0.592966	−	0.805228i	$-0.297957\pi$
$823$	−11.5000	−	19.9186i	−0.400865	−	0.694318i	−0.993831	+	0.110910i	$0.964624\pi$
$824$	−4.00000	+	6.92820i	−0.139347	+	0.241355i
$825$		0			0
$826$	6.00000	+	5.19615i	0.208767	+	0.180797i
$827$	21.0000			0.730242			0.365121	−	0.930960i	$-0.381028\pi$
$827$	21.0000			0.730242			0.365121	+	0.930960i	$0.381028\pi$
$828$		0			0
$829$	−16.0000	−	27.7128i	−0.555703	−	0.962506i	−0.997848	−	0.0655624i	$-0.979116\pi$
$829$	−16.0000	−	27.7128i	−0.555703	−	0.962506i	0.442145	−	0.896943i	$-0.354217\pi$
$830$	18.0000	+	31.1769i	0.624789	+	1.08217i
$831$		0			0
$832$	−4.00000			−0.138675
$833$	33.0000	−	25.9808i	1.14338	−	0.900180i
$834$		0			0
$835$	−9.00000	+	15.5885i	−0.311458	+	0.539461i
$836$		0			0
$837$		0			0
$838$	−6.00000	+	10.3923i	−0.207267	+	0.358996i
$839$	−24.0000			−0.828572			−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000			−0.828572			−0.414286	+	0.910147i	$0.635969\pi$
$840$		0			0
$841$	−20.0000			−0.689655
$842$	−16.0000	+	27.7128i	−0.551396	+	0.955047i
$843$		0			0
$844$	2.00000	+	3.46410i	0.0688428	+	0.119239i
$845$	−4.50000	+	7.79423i	−0.154805	+	0.268130i
$846$		0			0
$847$	−27.5000	+	9.52628i	−0.944911	+	0.327327i
$848$	9.00000			0.309061
$849$		0			0
$850$	−12.0000	−	20.7846i	−0.411597	−	0.712906i
$851$	−24.0000	−	41.5692i	−0.822709	−	1.42497i
$852$		0			0
$853$	44.0000			1.50653			0.753266	−	0.657716i	$-0.228477\pi$
$853$	44.0000			1.50653			0.753266	+	0.657716i	$0.228477\pi$
$854$	5.00000	−	25.9808i	0.171096	−	0.889043i
$855$		0			0
$856$	1.50000	−	2.59808i	0.0512689	−	0.0888004i
$857$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$857$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$858$		0			0
$859$	−7.00000	+	12.1244i	−0.238837	+	0.413678i	−0.960381	−	0.278691i	$-0.910099\pi$
$859$	−7.00000	+	12.1244i	−0.238837	+	0.413678i	0.721544	+	0.692369i	$0.243433\pi$
$860$	24.0000			0.818393
$861$		0			0
$862$	12.0000			0.408722
$863$	−21.0000	+	36.3731i	−0.714848	+	1.23815i	0.248170	+	0.968717i	$0.420171\pi$
$863$	−21.0000	+	36.3731i	−0.714848	+	1.23815i	−0.963018	+	0.269437i	$0.913162\pi$
$864$		0			0
$865$	13.5000	+	23.3827i	0.459014	+	0.795035i
$866$	3.50000	−	6.06218i	0.118935	−	0.206001i
$867$		0			0
$868$	20.0000	−	6.92820i	0.678844	−	0.235159i
$869$		0			0
$870$		0			0
$871$	−20.0000	−	34.6410i	−0.677674	−	1.17377i
$872$	2.00000	+	3.46410i	0.0677285	+	0.117309i
$873$		0			0
$874$	−24.0000			−0.811812
$875$	6.00000	+	5.19615i	0.202837	+	0.175662i
$876$		0			0
$877$	−7.00000	+	12.1244i	−0.236373	+	0.409410i	−0.959671	−	0.281126i	$-0.909292\pi$
$877$	−7.00000	+	12.1244i	−0.236373	+	0.409410i	0.723298	+	0.690536i	$0.242625\pi$
$878$	−4.00000	−	6.92820i	−0.134993	−	0.233816i
$879$		0			0
$880$		0			0
$881$	6.00000			0.202145			0.101073	−	0.994879i	$-0.467773\pi$
$881$	6.00000			0.202145			0.101073	+	0.994879i	$0.467773\pi$
$882$		0			0
$883$	32.0000			1.07689			0.538443	−	0.842662i	$-0.319013\pi$
$883$	32.0000			1.07689			0.538443	+	0.842662i	$0.319013\pi$
$884$	12.0000	−	20.7846i	0.403604	−	0.699062i
$885$		0			0
$886$	19.5000	+	33.7750i	0.655115	+	1.13469i
$887$	21.0000	−	36.3731i	0.705111	−	1.22129i	−0.261540	−	0.965193i	$-0.584230\pi$
$887$	21.0000	−	36.3731i	0.705111	−	1.22129i	0.966651	−	0.256096i	$-0.0824362\pi$
$888$		0			0
$889$	−10.0000	−	8.66025i	−0.335389	−	0.290456i
$890$	18.0000			0.603361
$891$		0			0
$892$	0.500000	+	0.866025i	0.0167412	+	0.0289967i
$893$	12.0000	+	20.7846i	0.401565	+	0.695530i
$894$		0			0
$895$	−45.0000			−1.50418
$896$	2.50000	−	0.866025i	0.0835191	−	0.0289319i
$897$		0			0
$898$	−3.00000	+	5.19615i	−0.100111	+	0.173398i
$899$	12.0000	+	20.7846i	0.400222	+	0.693206i
$900$		0			0
$901$	−27.0000	+	46.7654i	−0.899500	+	1.55798i
$902$		0			0
$903$		0			0
$904$	−6.00000			−0.199557
$905$	15.0000	−	25.9808i	0.498617	−	0.863630i
$906$		0			0
$907$	−4.00000	−	6.92820i	−0.132818	−	0.230047i	0.791944	−	0.610594i	$-0.209069\pi$
$907$	−4.00000	−	6.92820i	−0.132818	−	0.230047i	−0.924762	+	0.380547i	$0.875736\pi$
$908$	−1.50000	+	2.59808i	−0.0497792	+	0.0862202i
$909$		0			0
$910$	6.00000	−	31.1769i	0.198898	−	1.03350i
$911$	−54.0000			−1.78910			−0.894550	−	0.446968i	$-0.852504\pi$
$911$	−54.0000			−1.78910			−0.894550	+	0.446968i	$0.852504\pi$
$912$		0			0
$913$		0			0
$914$	0.500000	+	0.866025i	0.0165385	+	0.0286456i
$915$		0			0
$916$	14.0000			0.462573
$917$	−30.0000	+	10.3923i	−0.990687	+	0.343184i
$918$		0			0
$919$	18.5000	−	32.0429i	0.610259	−	1.05700i	−0.380938	−	0.924601i	$-0.624399\pi$
$919$	18.5000	−	32.0429i	0.610259	−	1.05700i	0.991197	−	0.132398i	$-0.0422678\pi$
$920$	−9.00000	−	15.5885i	−0.296721	−	0.513936i
$921$		0			0
$922$	16.5000	−	28.5788i	0.543399	−	0.941194i
$923$	−24.0000			−0.789970
$924$		0			0
$925$	32.0000			1.05215
$926$	9.50000	−	16.4545i	0.312189	−	0.540728i
$927$		0			0
$928$	1.50000	+	2.59808i	0.0492399	+	0.0852860i
$929$	−3.00000	+	5.19615i	−0.0984268	+	0.170480i	−0.911034	−	0.412332i	$-0.864714\pi$
$929$	−3.00000	+	5.19615i	−0.0984268	+	0.170480i	0.812607	+	0.582812i	$0.198048\pi$
$930$		0			0
$931$	26.0000	+	10.3923i	0.852116	+	0.340594i
$932$	18.0000			0.589610
$933$		0			0
$934$	16.5000	+	28.5788i	0.539896	+	0.935128i
$935$		0			0
$936$		0			0
$937$	−1.00000			−0.0326686			−0.0163343	−	0.999867i	$-0.505200\pi$
$937$	−1.00000			−0.0326686			−0.0163343	+	0.999867i	$0.505200\pi$
$938$	20.0000	+	17.3205i	0.653023	+	0.565535i
$939$		0			0
$940$	−9.00000	+	15.5885i	−0.293548	+	0.508439i
$941$	−9.00000	−	15.5885i	−0.293392	−	0.508169i	0.681218	−	0.732081i	$-0.261451\pi$
$941$	−9.00000	−	15.5885i	−0.293392	−	0.508169i	−0.974609	+	0.223912i	$0.928117\pi$
$942$		0			0
$943$	18.0000	−	31.1769i	0.586161	−	1.01526i
$944$	−3.00000			−0.0976417
$945$		0			0
$946$		0			0
$947$	−22.5000	+	38.9711i	−0.731152	+	1.26639i	0.225240	+	0.974303i	$0.427684\pi$
$947$	−22.5000	+	38.9711i	−0.731152	+	1.26639i	−0.956391	+	0.292089i	$0.905650\pi$
$948$		0			0
$949$	−14.0000	−	24.2487i	−0.454459	−	0.787146i
$950$	8.00000	−	13.8564i	0.259554	−	0.449561i
$951$		0			0
$952$	−3.00000	+	15.5885i	−0.0972306	+	0.505225i
$953$	−48.0000			−1.55487			−0.777436	−	0.628962i	$-0.783480\pi$
$953$	−48.0000			−1.55487			−0.777436	+	0.628962i	$0.783480\pi$
$954$		0			0
$955$	27.0000	+	46.7654i	0.873699	+	1.51329i
$956$		0			0
$957$		0			0
$958$	−24.0000			−0.775405
$959$	−6.00000	+	31.1769i	−0.193750	+	1.00676i
$960$		0			0
$961$	−16.5000	+	28.5788i	−0.532258	+	0.921898i
$962$	16.0000	+	27.7128i	0.515861	+	0.893497i
$963$		0			0
$964$	0.500000	−	0.866025i	0.0161039	−	0.0278928i
$965$	78.0000			2.51091
$966$		0			0
$967$	−37.0000			−1.18984			−0.594920	−	0.803785i	$-0.702816\pi$
$967$	−37.0000			−1.18984			−0.594920	+	0.803785i	$0.702816\pi$
$968$	5.50000	−	9.52628i	0.176777	−	0.306186i
$969$		0			0
$970$	15.0000	+	25.9808i	0.481621	+	0.834192i
$971$	1.50000	−	2.59808i	0.0481373	−	0.0833762i	−0.840953	−	0.541108i	$-0.818005\pi$
$971$	1.50000	−	2.59808i	0.0481373	−	0.0833762i	0.889090	+	0.457732i	$0.151338\pi$
$972$		0			0
$973$	−4.00000	−	3.46410i	−0.128234	−	0.111054i
$974$	−13.0000			−0.416547
$975$		0			0
$976$	5.00000	+	8.66025i	0.160046	+	0.277208i
$977$	−6.00000	−	10.3923i	−0.191957	−	0.332479i	0.753942	−	0.656941i	$-0.228150\pi$
$977$	−6.00000	−	10.3923i	−0.191957	−	0.332479i	−0.945899	+	0.324462i	$0.894817\pi$
$978$		0			0
$979$		0			0
$980$	3.00000	+	20.7846i	0.0958315	+	0.663940i
$981$		0			0
$982$	−13.5000	+	23.3827i	−0.430802	+	0.746171i
$983$	12.0000	+	20.7846i	0.382741	+	0.662926i	0.991453	−	0.130465i	$-0.0416470\pi$
$983$	12.0000	+	20.7846i	0.382741	+	0.662926i	−0.608712	+	0.793391i	$0.708314\pi$
$984$		0			0
$985$	22.5000	−	38.9711i	0.716910	−	1.24172i
$986$	−18.0000			−0.573237
$987$		0			0
$988$	16.0000			0.509028
$989$	−24.0000	+	41.5692i	−0.763156	+	1.32182i
$990$		0			0
$991$	−11.5000	−	19.9186i	−0.365310	−	0.632735i	0.623516	−	0.781810i	$-0.285704\pi$
$991$	−11.5000	−	19.9186i	−0.365310	−	0.632735i	−0.988826	+	0.149076i	$0.952370\pi$
$992$	−4.00000	+	6.92820i	−0.127000	+	0.219971i
$993$		0			0
$994$	15.0000	−	5.19615i	0.475771	−	0.164812i
$995$	33.0000			1.04617
$996$		0			0
$997$	−16.0000	−	27.7128i	−0.506725	−	0.877674i	−0.999970	−	0.00778294i	$-0.997523\pi$
$997$	−16.0000	−	27.7128i	−0.506725	−	0.877674i	0.493245	−	0.869891i	$-0.335811\pi$
$998$	−1.00000	−	1.73205i	−0.0316544	−	0.0548271i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.2.g.a.163.1	yes	2
3.2	odd	2		378.2.g.f.163.1	yes	2
7.2	even	3		2646.2.a.bc.1.1		1
7.4	even	3	inner	378.2.g.a.109.1	✓	2
7.5	odd	6		2646.2.a.r.1.1		1
9.2	odd	6		1134.2.e.f.919.1		2
9.4	even	3		1134.2.h.g.541.1		2
9.5	odd	6		1134.2.h.k.541.1		2
9.7	even	3		1134.2.e.j.919.1		2
21.2	odd	6		2646.2.a.b.1.1		1
21.5	even	6		2646.2.a.m.1.1		1
21.11	odd	6		378.2.g.f.109.1	yes	2
63.4	even	3		1134.2.e.j.865.1		2
63.11	odd	6		1134.2.h.k.109.1		2
63.25	even	3		1134.2.h.g.109.1		2
63.32	odd	6		1134.2.e.f.865.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.2.g.a.109.1	✓	2	7.4	even	3	inner
378.2.g.a.163.1	yes	2	1.1	even	1	trivial
378.2.g.f.109.1	yes	2	21.11	odd	6
378.2.g.f.163.1	yes	2	3.2	odd	2
1134.2.e.f.865.1		2	63.32	odd	6
1134.2.e.f.919.1		2	9.2	odd	6
1134.2.e.j.865.1		2	63.4	even	3
1134.2.e.j.919.1		2	9.7	even	3
1134.2.h.g.109.1		2	63.25	even	3
1134.2.h.g.541.1		2	9.4	even	3
1134.2.h.k.109.1		2	63.11	odd	6
1134.2.h.k.541.1		2	9.5	odd	6
2646.2.a.b.1.1		1	21.2	odd	6
2646.2.a.m.1.1		1	21.5	even	6
2646.2.a.r.1.1		1	7.5	odd	6
2646.2.a.bc.1.1		1	7.2	even	3

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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	378.g (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-1.50000 + 2.59808i) q^{5} +(-2.00000 - 1.73205i) q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-1.50000 + 2.59808i) q^{5} +(-2.00000 - 1.73205i) q^{7} +1.00000 q^{8} +(-1.50000 - 2.59808i) q^{10} -4.00000 q^{13} +(2.50000 - 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(-3.00000 - 5.19615i) q^{17} +(2.00000 - 3.46410i) q^{19} +3.00000 q^{20} +(-3.00000 + 5.19615i) q^{23} +(-2.00000 - 3.46410i) q^{25} +(2.00000 - 3.46410i) q^{26} +(-0.500000 + 2.59808i) q^{28} -3.00000 q^{29} +(-4.00000 - 6.92820i) q^{31} +(-0.500000 - 0.866025i) q^{32} +6.00000 q^{34} +(7.50000 - 2.59808i) q^{35} +(-4.00000 + 6.92820i) q^{37} +(2.00000 + 3.46410i) q^{38} +(-1.50000 + 2.59808i) q^{40} -6.00000 q^{41} +8.00000 q^{43} +(-3.00000 - 5.19615i) q^{46} +(-3.00000 + 5.19615i) q^{47} +(1.00000 + 6.92820i) q^{49} +4.00000 q^{50} +(2.00000 + 3.46410i) q^{52} +(-4.50000 - 7.79423i) q^{53} +(-2.00000 - 1.73205i) q^{56} +(1.50000 - 2.59808i) q^{58} +(1.50000 + 2.59808i) q^{59} +(5.00000 - 8.66025i) q^{61} +8.00000 q^{62} +1.00000 q^{64} +(6.00000 - 10.3923i) q^{65} +(5.00000 + 8.66025i) q^{67} +(-3.00000 + 5.19615i) q^{68} +(-1.50000 + 7.79423i) q^{70} +6.00000 q^{71} +(3.50000 + 6.06218i) q^{73} +(-4.00000 - 6.92820i) q^{74} -4.00000 q^{76} +(-8.50000 + 14.7224i) q^{79} +(-1.50000 - 2.59808i) q^{80} +(3.00000 - 5.19615i) q^{82} -12.0000 q^{83} +18.0000 q^{85} +(-4.00000 + 6.92820i) q^{86} +(-3.00000 + 5.19615i) q^{89} +(8.00000 + 6.92820i) q^{91} +6.00000 q^{92} +(-3.00000 - 5.19615i) q^{94} +(6.00000 + 10.3923i) q^{95} -10.0000 q^{97} +(-6.50000 - 2.59808i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} - 3 q^{5} - 4 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q - q^2 - q^4 - 3 * q^5 - 4 * q^7 + 2 * q^8	\( 2 q - q^{2} - q^{4} - 3 q^{5} - 4 q^{7} + 2 q^{8} - 3 q^{10} - 8 q^{13} + 5 q^{14} - q^{16} - 6 q^{17} + 4 q^{19} + 6 q^{20} - 6 q^{23} - 4 q^{25} + 4 q^{26} - q^{28} - 6 q^{29} - 8 q^{31} - q^{32} + 12 q^{34} + 15 q^{35} - 8 q^{37} + 4 q^{38} - 3 q^{40} - 12 q^{41} + 16 q^{43} - 6 q^{46} - 6 q^{47} + 2 q^{49} + 8 q^{50} + 4 q^{52} - 9 q^{53} - 4 q^{56} + 3 q^{58} + 3 q^{59} + 10 q^{61} + 16 q^{62} + 2 q^{64} + 12 q^{65} + 10 q^{67} - 6 q^{68} - 3 q^{70} + 12 q^{71} + 7 q^{73} - 8 q^{74} - 8 q^{76} - 17 q^{79} - 3 q^{80} + 6 q^{82} - 24 q^{83} + 36 q^{85} - 8 q^{86} - 6 q^{89} + 16 q^{91} + 12 q^{92} - 6 q^{94} + 12 q^{95} - 20 q^{97} - 13 q^{98}+O(q^{100}) \) 2 * q - q^2 - q^4 - 3 * q^5 - 4 * q^7 + 2 * q^8 - 3 * q^10 - 8 * q^13 + 5 * q^14 - q^16 - 6 * q^17 + 4 * q^19 + 6 * q^20 - 6 * q^23 - 4 * q^25 + 4 * q^26 - q^28 - 6 * q^29 - 8 * q^31 - q^32 + 12 * q^34 + 15 * q^35 - 8 * q^37 + 4 * q^38 - 3 * q^40 - 12 * q^41 + 16 * q^43 - 6 * q^46 - 6 * q^47 + 2 * q^49 + 8 * q^50 + 4 * q^52 - 9 * q^53 - 4 * q^56 + 3 * q^58 + 3 * q^59 + 10 * q^61 + 16 * q^62 + 2 * q^64 + 12 * q^65 + 10 * q^67 - 6 * q^68 - 3 * q^70 + 12 * q^71 + 7 * q^73 - 8 * q^74 - 8 * q^76 - 17 * q^79 - 3 * q^80 + 6 * q^82 - 24 * q^83 + 36 * q^85 - 8 * q^86 - 6 * q^89 + 16 * q^91 + 12 * q^92 - 6 * q^94 + 12 * q^95 - 20 * q^97 - 13 * q^98

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