LMFDB - Embedded newform 378.2.k.b.269.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,2,Mod(215,378)]

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([3, 5]))

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

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

chi := DirichletCharacter("378.215");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	378.k (of order $6$, 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:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			269.2
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	378.269
Dual form			378.2.k.b.215.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.866025 - 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(-1.73205 - 3.00000i) q^{5} +(2.00000 + 1.73205i) q^{7} -1.00000i q^{8} +O(q^{10})$	\(q+(0.866025 - 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(-1.73205 - 3.00000i) q^{5} +(2.00000 + 1.73205i) q^{7} -1.00000i q^{8} +(-3.00000 - 1.73205i) q^{10} -5.19615i q^{13} +(2.59808 + 0.500000i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(3.46410 - 6.00000i) q^{17} +(-3.00000 + 1.73205i) q^{19} -3.46410 q^{20} +(-5.19615 + 3.00000i) q^{23} +(-3.50000 + 6.06218i) q^{25} +(-2.59808 - 4.50000i) q^{26} +(2.50000 - 0.866025i) q^{28} +(7.50000 + 4.33013i) q^{31} +(-0.866025 - 0.500000i) q^{32} -6.92820i q^{34} +(1.73205 - 9.00000i) q^{35} +(2.50000 + 4.33013i) q^{37} +(-1.73205 + 3.00000i) q^{38} +(-3.00000 + 1.73205i) q^{40} +6.92820 q^{41} +1.00000 q^{43} +(-3.00000 + 5.19615i) q^{46} +(3.46410 + 6.00000i) q^{47} +(1.00000 + 6.92820i) q^{49} +7.00000i q^{50} +(-4.50000 - 2.59808i) q^{52} +(5.19615 + 3.00000i) q^{53} +(1.73205 - 2.00000i) q^{56} +(-3.46410 + 6.00000i) q^{59} +(-1.50000 + 0.866025i) q^{61} +8.66025 q^{62} -1.00000 q^{64} +(-15.5885 + 9.00000i) q^{65} +(6.50000 - 11.2583i) q^{67} +(-3.46410 - 6.00000i) q^{68} +(-3.00000 - 8.66025i) q^{70} +6.00000i q^{71} +(-6.00000 - 3.46410i) q^{73} +(4.33013 + 2.50000i) q^{74} +3.46410i q^{76} +(3.50000 + 6.06218i) q^{79} +(-1.73205 + 3.00000i) q^{80} +(6.00000 - 3.46410i) q^{82} -3.46410 q^{83} -24.0000 q^{85} +(0.866025 - 0.500000i) q^{86} +(-5.19615 - 9.00000i) q^{89} +(9.00000 - 10.3923i) q^{91} +6.00000i q^{92} +(6.00000 + 3.46410i) q^{94} +(10.3923 + 6.00000i) q^{95} +1.73205i q^{97} +(4.33013 + 5.50000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} + 8 q^{7}+O(q^{10}) $ 4 * q + 2 * q^4 + 8 * q^7	$ 4 q + 2 q^{4} + 8 q^{7} - 12 q^{10} - 2 q^{16} - 12 q^{19} - 14 q^{25} + 10 q^{28} + 30 q^{31} + 10 q^{37} - 12 q^{40} + 4 q^{43} - 12 q^{46} + 4 q^{49} - 18 q^{52} - 6 q^{61} - 4 q^{64} + 26 q^{67} - 12 q^{70} - 24 q^{73} + 14 q^{79} + 24 q^{82} - 96 q^{85} + 36 q^{91} + 24 q^{94}+O(q^{100}) $ 4 * q + 2 * q^4 + 8 * q^7 - 12 * q^10 - 2 * q^16 - 12 * q^19 - 14 * q^25 + 10 * q^28 + 30 * q^31 + 10 * q^37 - 12 * q^40 + 4 * q^43 - 12 * q^46 + 4 * q^49 - 18 * q^52 - 6 * q^61 - 4 * q^64 + 26 * q^67 - 12 * q^70 - 24 * q^73 + 14 * q^79 + 24 * q^82 - 96 * q^85 + 36 * q^91 + 24 * q^94

Display 100 coefficients 10 coefficients

Character values

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

$n$	$29$	$325$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$

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 $

$2$	0.866025	−	0.500000i	0.612372	−	0.353553i
$2$	0.866025	−	0.500000i	0.612372	−	0.353553i
$3$		0			0
$3$		0			0
$4$	0.500000	−	0.866025i	0.250000	−	0.433013i
$5$	−1.73205	−	3.00000i	−0.774597	−	1.34164i	−0.935021	−	0.354593i	$-0.884620\pi$
$5$	−1.73205	−	3.00000i	−0.774597	−	1.34164i	0.160424	−	0.987048i	$-0.448714\pi$
$6$		0			0
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$	−3.00000	−	1.73205i	−0.948683	−	0.547723i
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$		0			0
$13$		−	5.19615i		−	1.44115i	−0.693375	−	0.720577i	$-0.743877\pi$
$13$		−	5.19615i		−	1.44115i	0.693375	−	0.720577i	$-0.256123\pi$
$14$	2.59808	+	0.500000i	0.694365	+	0.133631i
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.125000	−	0.216506i
$17$	3.46410	−	6.00000i	0.840168	−	1.45521i	−0.0495842	−	0.998770i	$-0.515790\pi$
$17$	3.46410	−	6.00000i	0.840168	−	1.45521i	0.889752	−	0.456444i	$-0.150877\pi$
$18$		0			0
$19$	−3.00000	+	1.73205i	−0.688247	+	0.397360i	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−3.00000	+	1.73205i	−0.688247	+	0.397360i	0.114708	+	0.993399i	$0.463407\pi$
$20$	−3.46410			−0.774597
$21$		0			0
$22$		0			0
$23$	−5.19615	+	3.00000i	−1.08347	+	0.625543i	−0.931831	−	0.362892i	$-0.881789\pi$
$23$	−5.19615	+	3.00000i	−1.08347	+	0.625543i	−0.151642	+	0.988436i	$0.548456\pi$
$24$		0			0
$25$	−3.50000	+	6.06218i	−0.700000	+	1.21244i
$26$	−2.59808	−	4.50000i	−0.509525	−	0.882523i
$27$		0			0
$28$	2.50000	−	0.866025i	0.472456	−	0.163663i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	7.50000	+	4.33013i	1.34704	+	0.777714i	0.987829	−	0.155543i	$-0.0497126\pi$
$31$	7.50000	+	4.33013i	1.34704	+	0.777714i	0.359211	+	0.933257i	$0.383046\pi$
$32$	−0.866025	−	0.500000i	−0.153093	−	0.0883883i
$33$		0			0
$34$		−	6.92820i		−	1.18818i
$35$	1.73205	−	9.00000i	0.292770	−	1.52128i
$36$		0			0
$37$	2.50000	+	4.33013i	0.410997	+	0.711868i	0.994999	−	0.0998840i	$-0.0318472\pi$
$37$	2.50000	+	4.33013i	0.410997	+	0.711868i	−0.584002	+	0.811752i	$0.698514\pi$
$38$	−1.73205	+	3.00000i	−0.280976	+	0.486664i
$39$		0			0
$40$	−3.00000	+	1.73205i	−0.474342	+	0.273861i
$41$	6.92820			1.08200			0.541002	−	0.841021i	$-0.318045\pi$
$41$	6.92820			1.08200			0.541002	+	0.841021i	$0.318045\pi$
$42$		0			0
$43$	1.00000			0.152499			0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000			0.152499			0.0762493	+	0.997089i	$0.475706\pi$
$44$		0			0
$45$		0			0
$46$	−3.00000	+	5.19615i	−0.442326	+	0.766131i
$47$	3.46410	+	6.00000i	0.505291	+	0.875190i	0.999981	+	0.00612051i	$0.00194823\pi$
$47$	3.46410	+	6.00000i	0.505291	+	0.875190i	−0.494690	+	0.869069i	$0.664718\pi$
$48$		0			0
$49$	1.00000	+	6.92820i	0.142857	+	0.989743i
$50$			7.00000i			0.989949i
$51$		0			0
$52$	−4.50000	−	2.59808i	−0.624038	−	0.360288i
$53$	5.19615	+	3.00000i	0.713746	+	0.412082i	0.812447	−	0.583036i	$-0.198135\pi$
$53$	5.19615	+	3.00000i	0.713746	+	0.412082i	−0.0987002	+	0.995117i	$0.531468\pi$
$54$		0			0
$55$		0			0
$56$	1.73205	−	2.00000i	0.231455	−	0.267261i
$57$		0			0
$58$		0			0
$59$	−3.46410	+	6.00000i	−0.450988	+	0.781133i	−0.998448	−	0.0556984i	$-0.982261\pi$
$59$	−3.46410	+	6.00000i	−0.450988	+	0.781133i	0.547460	+	0.836832i	$0.315595\pi$
$60$		0			0
$61$	−1.50000	+	0.866025i	−0.192055	+	0.110883i	−0.592944	−	0.805243i	$-0.702035\pi$
$61$	−1.50000	+	0.866025i	−0.192055	+	0.110883i	0.400889	+	0.916127i	$0.368701\pi$
$62$	8.66025			1.09985
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−15.5885	+	9.00000i	−1.93351	+	1.11631i
$66$		0			0
$67$	6.50000	−	11.2583i	0.794101	−	1.37542i	−0.129307	−	0.991605i	$-0.541275\pi$
$67$	6.50000	−	11.2583i	0.794101	−	1.37542i	0.923408	−	0.383819i	$-0.125391\pi$
$68$	−3.46410	−	6.00000i	−0.420084	−	0.727607i
$69$		0			0
$70$	−3.00000	−	8.66025i	−0.358569	−	1.03510i
$71$			6.00000i			0.712069i	0.934473	+	0.356034i	$0.115871\pi$
$71$			6.00000i			0.712069i	−0.934473	+	0.356034i	$0.884129\pi$
$72$		0			0
$73$	−6.00000	−	3.46410i	−0.702247	−	0.405442i	0.105937	−	0.994373i	$-0.466216\pi$
$73$	−6.00000	−	3.46410i	−0.702247	−	0.405442i	−0.808184	+	0.588930i	$0.799549\pi$
$74$	4.33013	+	2.50000i	0.503367	+	0.290619i
$75$		0			0
$76$			3.46410i			0.397360i
$77$		0			0
$78$		0			0
$79$	3.50000	+	6.06218i	0.393781	+	0.682048i	0.992945	−	0.118578i	$-0.0378336\pi$
$79$	3.50000	+	6.06218i	0.393781	+	0.682048i	−0.599164	+	0.800626i	$0.704500\pi$
$80$	−1.73205	+	3.00000i	−0.193649	+	0.335410i
$81$		0			0
$82$	6.00000	−	3.46410i	0.662589	−	0.382546i
$83$	−3.46410			−0.380235			−0.190117	−	0.981761i	$-0.560887\pi$
$83$	−3.46410			−0.380235			−0.190117	+	0.981761i	$0.560887\pi$
$84$		0			0
$85$	−24.0000			−2.60317
$86$	0.866025	−	0.500000i	0.0933859	−	0.0539164i
$87$		0			0
$88$		0			0
$89$	−5.19615	−	9.00000i	−0.550791	−	0.953998i	−0.998218	−	0.0596775i	$-0.980993\pi$
$89$	−5.19615	−	9.00000i	−0.550791	−	0.953998i	0.447427	−	0.894321i	$-0.352341\pi$
$90$		0			0
$91$	9.00000	−	10.3923i	0.943456	−	1.08941i
$92$			6.00000i			0.625543i
$93$		0			0
$94$	6.00000	+	3.46410i	0.618853	+	0.357295i
$95$	10.3923	+	6.00000i	1.06623	+	0.615587i
$96$		0			0
$97$			1.73205i			0.175863i	0.996127	+	0.0879316i	$0.0280257\pi$
$97$			1.73205i			0.175863i	−0.996127	+	0.0879316i	$0.971974\pi$
$98$	4.33013	+	5.50000i	0.437409	+	0.555584i
$99$		0			0
$100$	3.50000	+	6.06218i	0.350000	+	0.606218i
$101$	−5.19615	+	9.00000i	−0.517036	+	0.895533i	0.482768	+	0.875748i	$0.339632\pi$
$101$	−5.19615	+	9.00000i	−0.517036	+	0.895533i	−0.999804	+	0.0197851i	$0.993702\pi$
$102$		0			0
$103$	4.50000	−	2.59808i	0.443398	−	0.255996i	−0.261640	−	0.965166i	$-0.584263\pi$
$103$	4.50000	−	2.59808i	0.443398	−	0.255996i	0.705038	+	0.709170i	$0.250930\pi$
$104$	−5.19615			−0.509525
$105$		0			0
$106$	6.00000			0.582772
$107$	−5.19615	+	3.00000i	−0.502331	+	0.290021i	−0.729676	−	0.683793i	$-0.760329\pi$
$107$	−5.19615	+	3.00000i	−0.502331	+	0.290021i	0.227345	+	0.973814i	$0.426996\pi$
$108$		0			0
$109$	5.50000	−	9.52628i	0.526804	−	0.912452i	−0.472708	−	0.881219i	$-0.656723\pi$
$109$	5.50000	−	9.52628i	0.526804	−	0.912452i	0.999512	−	0.0312328i	$-0.00994332\pi$
$110$		0			0
$111$		0			0
$112$	0.500000	−	2.59808i	0.0472456	−	0.245495i
$113$			18.0000i			1.69330i	0.532152	+	0.846649i	$0.321383\pi$
$113$			18.0000i			1.69330i	−0.532152	+	0.846649i	$0.678617\pi$
$114$		0			0
$115$	18.0000	+	10.3923i	1.67851	+	0.969087i
$116$		0			0
$117$		0			0
$118$			6.92820i			0.637793i
$119$	17.3205	−	6.00000i	1.58777	−	0.550019i
$120$		0			0
$121$	−5.50000	−	9.52628i	−0.500000	−	0.866025i
$122$	−0.866025	+	1.50000i	−0.0784063	+	0.135804i
$123$		0			0
$124$	7.50000	−	4.33013i	0.673520	−	0.388857i
$125$	6.92820			0.619677
$126$		0			0
$127$	−7.00000			−0.621150			−0.310575	−	0.950549i	$-0.600522\pi$
$127$	−7.00000			−0.621150			−0.310575	+	0.950549i	$0.600522\pi$
$128$	−0.866025	+	0.500000i	−0.0765466	+	0.0441942i
$129$		0			0
$130$	−9.00000	+	15.5885i	−0.789352	+	1.36720i
$131$	−10.3923	−	18.0000i	−0.907980	−	1.57267i	−0.816866	−	0.576827i	$-0.804291\pi$
$131$	−10.3923	−	18.0000i	−0.907980	−	1.57267i	−0.0911134	−	0.995841i	$-0.529043\pi$
$132$		0			0
$133$	−9.00000	−	1.73205i	−0.780399	−	0.150188i
$134$		−	13.0000i		−	1.12303i
$135$		0			0
$136$	−6.00000	−	3.46410i	−0.514496	−	0.297044i
$137$	15.5885	+	9.00000i	1.33181	+	0.768922i	0.985577	−	0.169226i	$-0.0541268\pi$
$137$	15.5885	+	9.00000i	1.33181	+	0.768922i	0.346235	+	0.938148i	$0.387460\pi$
$138$		0			0
$139$		−	5.19615i		−	0.440732i	−0.975417	−	0.220366i	$-0.929275\pi$
$139$		−	5.19615i		−	0.440732i	0.975417	−	0.220366i	$-0.0707252\pi$
$140$	−6.92820	−	6.00000i	−0.585540	−	0.507093i
$141$		0			0
$142$	3.00000	+	5.19615i	0.251754	+	0.436051i
$143$		0			0
$144$		0			0
$145$		0			0
$146$	−6.92820			−0.573382
$147$		0			0
$148$	5.00000			0.410997
$149$	5.19615	−	3.00000i	0.425685	−	0.245770i	−0.271821	−	0.962348i	$-0.587626\pi$
$149$	5.19615	−	3.00000i	0.425685	−	0.245770i	0.697507	+	0.716578i	$0.254293\pi$
$150$		0			0
$151$	−0.500000	+	0.866025i	−0.0406894	+	0.0704761i	−0.885653	−	0.464348i	$-0.846289\pi$
$151$	−0.500000	+	0.866025i	−0.0406894	+	0.0704761i	0.844963	+	0.534824i	$0.179622\pi$
$152$	1.73205	+	3.00000i	0.140488	+	0.243332i
$153$		0			0
$154$		0			0
$155$		−	30.0000i		−	2.40966i
$156$		0			0
$157$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$157$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$158$	6.06218	+	3.50000i	0.482281	+	0.278445i
$159$		0			0
$160$			3.46410i			0.273861i
$161$	−15.5885	−	3.00000i	−1.22854	−	0.236433i
$162$		0			0
$163$	−5.50000	−	9.52628i	−0.430793	−	0.746156i	0.566149	−	0.824303i	$-0.308433\pi$
$163$	−5.50000	−	9.52628i	−0.430793	−	0.746156i	−0.996942	+	0.0781474i	$0.975100\pi$
$164$	3.46410	−	6.00000i	0.270501	−	0.468521i
$165$		0			0
$166$	−3.00000	+	1.73205i	−0.232845	+	0.134433i
$167$	3.46410			0.268060			0.134030	−	0.990977i	$-0.457208\pi$
$167$	3.46410			0.268060			0.134030	+	0.990977i	$0.457208\pi$
$168$		0			0
$169$	−14.0000			−1.07692
$170$	−20.7846	+	12.0000i	−1.59411	+	0.920358i
$171$		0			0
$172$	0.500000	−	0.866025i	0.0381246	−	0.0660338i
$173$	−10.3923	−	18.0000i	−0.790112	−	1.36851i	−0.925897	−	0.377776i	$-0.876689\pi$
$173$	−10.3923	−	18.0000i	−0.790112	−	1.36851i	0.135785	−	0.990738i	$-0.456644\pi$
$174$		0			0
$175$	−17.5000	+	6.06218i	−1.32288	+	0.458258i
$176$		0			0
$177$		0			0
$178$	−9.00000	−	5.19615i	−0.674579	−	0.389468i
$179$	5.19615	+	3.00000i	0.388379	+	0.224231i	0.681457	−	0.731858i	$-0.261346\pi$
$179$	5.19615	+	3.00000i	0.388379	+	0.224231i	−0.293079	+	0.956088i	$0.594680\pi$
$180$		0			0
$181$			13.8564i			1.02994i	0.857209	+	0.514969i	$0.172197\pi$
$181$			13.8564i			1.02994i	−0.857209	+	0.514969i	$0.827803\pi$
$182$	2.59808	−	13.5000i	0.192582	−	1.00069i
$183$		0			0
$184$	3.00000	+	5.19615i	0.221163	+	0.383065i
$185$	8.66025	−	15.0000i	0.636715	−	1.10282i
$186$		0			0
$187$		0			0
$188$	6.92820			0.505291
$189$		0			0
$190$	12.0000			0.870572
$191$	−5.19615	+	3.00000i	−0.375980	+	0.217072i	−0.676068	−	0.736839i	$-0.736317\pi$
$191$	−5.19615	+	3.00000i	−0.375980	+	0.217072i	0.300088	+	0.953912i	$0.402984\pi$
$192$		0			0
$193$	−0.500000	+	0.866025i	−0.0359908	+	0.0623379i	−0.883460	−	0.468507i	$-0.844792\pi$
$193$	−0.500000	+	0.866025i	−0.0359908	+	0.0623379i	0.847469	+	0.530845i	$0.178125\pi$
$194$	0.866025	+	1.50000i	0.0621770	+	0.107694i
$195$		0			0
$196$	6.50000	+	2.59808i	0.464286	+	0.185577i
$197$			24.0000i			1.70993i	0.518686	+	0.854965i	$0.326421\pi$
$197$			24.0000i			1.70993i	−0.518686	+	0.854965i	$0.673579\pi$
$198$		0			0
$199$	−1.50000	−	0.866025i	−0.106332	−	0.0613909i	0.445891	−	0.895087i	$-0.352887\pi$
$199$	−1.50000	−	0.866025i	−0.106332	−	0.0613909i	−0.552223	+	0.833696i	$0.686220\pi$
$200$	6.06218	+	3.50000i	0.428661	+	0.247487i
$201$		0			0
$202$			10.3923i			0.731200i
$203$		0			0
$204$		0			0
$205$	−12.0000	−	20.7846i	−0.838116	−	1.45166i
$206$	2.59808	−	4.50000i	0.181017	−	0.313530i
$207$		0			0
$208$	−4.50000	+	2.59808i	−0.312019	+	0.180144i
$209$		0			0
$210$		0			0
$211$	25.0000			1.72107			0.860535	−	0.509390i	$-0.170129\pi$
$211$	25.0000			1.72107			0.860535	+	0.509390i	$0.170129\pi$
$212$	5.19615	−	3.00000i	0.356873	−	0.206041i
$213$		0			0
$214$	−3.00000	+	5.19615i	−0.205076	+	0.355202i
$215$	−1.73205	−	3.00000i	−0.118125	−	0.204598i
$216$		0			0
$217$	7.50000	+	21.6506i	0.509133	+	1.46974i
$218$		−	11.0000i		−	0.745014i
$219$		0			0
$220$		0			0
$221$	−31.1769	−	18.0000i	−2.09719	−	1.21081i
$222$		0			0
$223$		−	10.3923i		−	0.695920i	−0.937509	−	0.347960i	$-0.886874\pi$
$223$		−	10.3923i		−	0.695920i	0.937509	−	0.347960i	$-0.113126\pi$
$224$	−0.866025	−	2.50000i	−0.0578638	−	0.167038i
$225$		0			0
$226$	9.00000	+	15.5885i	0.598671	+	1.03693i
$227$	−5.19615	+	9.00000i	−0.344881	+	0.597351i	−0.985332	−	0.170648i	$-0.945414\pi$
$227$	−5.19615	+	9.00000i	−0.344881	+	0.597351i	0.640451	+	0.767999i	$0.278747\pi$
$228$		0			0
$229$	1.50000	−	0.866025i	0.0991228	−	0.0572286i	−0.449619	−	0.893220i	$-0.648440\pi$
$229$	1.50000	−	0.866025i	0.0991228	−	0.0572286i	0.548742	+	0.835992i	$0.315107\pi$
$230$	20.7846			1.37050
$231$		0			0
$232$		0			0
$233$	10.3923	−	6.00000i	0.680823	−	0.393073i	−0.119342	−	0.992853i	$-0.538079\pi$
$233$	10.3923	−	6.00000i	0.680823	−	0.393073i	0.800165	+	0.599780i	$0.204745\pi$
$234$		0			0
$235$	12.0000	−	20.7846i	0.782794	−	1.35584i
$236$	3.46410	+	6.00000i	0.225494	+	0.390567i
$237$		0			0
$238$	12.0000	−	13.8564i	0.777844	−	0.898177i
$239$		−	6.00000i		−	0.388108i	−0.980991	−	0.194054i	$-0.937836\pi$
$239$		−	6.00000i		−	0.388108i	0.980991	−	0.194054i	$-0.0621637\pi$
$240$		0			0
$241$	1.50000	+	0.866025i	0.0966235	+	0.0557856i	0.547533	−	0.836784i	$-0.315567\pi$
$241$	1.50000	+	0.866025i	0.0966235	+	0.0557856i	−0.450910	+	0.892570i	$0.648900\pi$
$242$	−9.52628	−	5.50000i	−0.612372	−	0.353553i
$243$		0			0
$244$			1.73205i			0.110883i
$245$	19.0526	−	15.0000i	1.21722	−	0.958315i
$246$		0			0
$247$	9.00000	+	15.5885i	0.572656	+	0.991870i
$248$	4.33013	−	7.50000i	0.274963	−	0.476250i
$249$		0			0
$250$	6.00000	−	3.46410i	0.379473	−	0.219089i
$251$	−17.3205			−1.09326			−0.546630	−	0.837374i	$-0.684090\pi$
$251$	−17.3205			−1.09326			−0.546630	+	0.837374i	$0.684090\pi$
$252$		0			0
$253$		0			0
$254$	−6.06218	+	3.50000i	−0.380375	+	0.219610i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.0312500	+	0.0541266i
$257$	−5.19615	−	9.00000i	−0.324127	−	0.561405i	0.657208	−	0.753709i	$-0.271737\pi$
$257$	−5.19615	−	9.00000i	−0.324127	−	0.561405i	−0.981335	+	0.192304i	$0.938404\pi$
$258$		0			0
$259$	−2.50000	+	12.9904i	−0.155342	+	0.807183i
$260$			18.0000i			1.11631i
$261$		0			0
$262$	−18.0000	−	10.3923i	−1.11204	−	0.642039i
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$		0			0
$265$		−	20.7846i		−	1.27679i
$266$	−8.66025	+	3.00000i	−0.530994	+	0.183942i
$267$		0			0
$268$	−6.50000	−	11.2583i	−0.397051	−	0.687712i
$269$	−13.8564	+	24.0000i	−0.844840	+	1.46331i	0.0409201	+	0.999162i	$0.486971\pi$
$269$	−13.8564	+	24.0000i	−0.844840	+	1.46331i	−0.885760	+	0.464143i	$0.846362\pi$
$270$		0			0
$271$	−25.5000	+	14.7224i	−1.54901	+	0.894324i	−0.550797	+	0.834639i	$0.685676\pi$
$271$	−25.5000	+	14.7224i	−1.54901	+	0.894324i	−0.998217	+	0.0596851i	$0.980990\pi$
$272$	−6.92820			−0.420084
$273$		0			0
$274$	18.0000			1.08742
$275$		0			0
$276$		0			0
$277$	0.500000	−	0.866025i	0.0300421	−	0.0520344i	−0.850613	−	0.525792i	$-0.823769\pi$
$277$	0.500000	−	0.866025i	0.0300421	−	0.0520344i	0.880656	+	0.473757i	$0.157103\pi$
$278$	−2.59808	−	4.50000i	−0.155822	−	0.269892i
$279$		0			0
$280$	−9.00000	−	1.73205i	−0.537853	−	0.103510i
$281$		−	18.0000i		−	1.07379i	−0.843649	−	0.536895i	$-0.819597\pi$
$281$		−	18.0000i		−	1.07379i	0.843649	−	0.536895i	$-0.180403\pi$
$282$		0			0
$283$	−16.5000	−	9.52628i	−0.980823	−	0.566279i	−0.0783046	−	0.996929i	$-0.524951\pi$
$283$	−16.5000	−	9.52628i	−0.980823	−	0.566279i	−0.902519	+	0.430651i	$0.858284\pi$
$284$	5.19615	+	3.00000i	0.308335	+	0.178017i
$285$		0			0
$286$		0			0
$287$	13.8564	+	12.0000i	0.817918	+	0.708338i
$288$		0			0
$289$	−15.5000	−	26.8468i	−0.911765	−	1.57922i
$290$		0			0
$291$		0			0
$292$	−6.00000	+	3.46410i	−0.351123	+	0.202721i
$293$	27.7128			1.61900			0.809500	−	0.587120i	$-0.199738\pi$
$293$	27.7128			1.61900			0.809500	+	0.587120i	$0.199738\pi$
$294$		0			0
$295$	24.0000			1.39733
$296$	4.33013	−	2.50000i	0.251684	−	0.145310i
$297$		0			0
$298$	3.00000	−	5.19615i	0.173785	−	0.301005i
$299$	15.5885	+	27.0000i	0.901504	+	1.56145i
$300$		0			0
$301$	2.00000	+	1.73205i	0.115278	+	0.0998337i
$302$			1.00000i			0.0575435i
$303$		0			0
$304$	3.00000	+	1.73205i	0.172062	+	0.0993399i
$305$	5.19615	+	3.00000i	0.297531	+	0.171780i
$306$		0			0
$307$		−	8.66025i		−	0.494267i	−0.968981	−	0.247133i	$-0.920511\pi$
$307$		−	8.66025i		−	0.494267i	0.968981	−	0.247133i	$-0.0794886\pi$
$308$		0			0
$309$		0			0
$310$	−15.0000	−	25.9808i	−0.851943	−	1.47561i
$311$	−1.73205	+	3.00000i	−0.0982156	+	0.170114i	−0.910946	−	0.412525i	$-0.864647\pi$
$311$	−1.73205	+	3.00000i	−0.0982156	+	0.170114i	0.812731	+	0.582640i	$0.197980\pi$
$312$		0			0
$313$	−6.00000	+	3.46410i	−0.339140	+	0.195803i	−0.659892	−	0.751361i	$-0.729398\pi$
$313$	−6.00000	+	3.46410i	−0.339140	+	0.195803i	0.320752	+	0.947163i	$0.396065\pi$
$314$		0			0
$315$		0			0
$316$	7.00000			0.393781
$317$	20.7846	−	12.0000i	1.16738	−	0.673987i	0.214318	−	0.976764i	$-0.431247\pi$
$317$	20.7846	−	12.0000i	1.16738	−	0.673987i	0.953062	+	0.302777i	$0.0979136\pi$
$318$		0			0
$319$		0			0
$320$	1.73205	+	3.00000i	0.0968246	+	0.167705i
$321$		0			0
$322$	−15.0000	+	5.19615i	−0.835917	+	0.289570i
$323$			24.0000i			1.33540i
$324$		0			0
$325$	31.5000	+	18.1865i	1.74731	+	1.00881i
$326$	−9.52628	−	5.50000i	−0.527612	−	0.304617i
$327$		0			0
$328$		−	6.92820i		−	0.382546i
$329$	−3.46410	+	18.0000i	−0.190982	+	0.992372i
$330$		0			0
$331$	4.00000	+	6.92820i	0.219860	+	0.380808i	0.954765	−	0.297361i	$-0.0961066\pi$
$331$	4.00000	+	6.92820i	0.219860	+	0.380808i	−0.734905	+	0.678170i	$0.762773\pi$
$332$	−1.73205	+	3.00000i	−0.0950586	+	0.164646i
$333$		0			0
$334$	3.00000	−	1.73205i	0.164153	−	0.0947736i
$335$	−45.0333			−2.46043
$336$		0			0
$337$	22.0000			1.19842			0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000			1.19842			0.599208	+	0.800593i	$0.295482\pi$
$338$	−12.1244	+	7.00000i	−0.659478	+	0.380750i
$339$		0			0
$340$	−12.0000	+	20.7846i	−0.650791	+	1.12720i
$341$		0			0
$342$		0			0
$343$	−10.0000	+	15.5885i	−0.539949	+	0.841698i
$344$		−	1.00000i		−	0.0539164i
$345$		0			0
$346$	−18.0000	−	10.3923i	−0.967686	−	0.558694i
$347$	20.7846	+	12.0000i	1.11578	+	0.644194i	0.940319	−	0.340293i	$-0.110526\pi$
$347$	20.7846	+	12.0000i	1.11578	+	0.644194i	0.175457	+	0.984487i	$0.443860\pi$
$348$		0			0
$349$			19.0526i			1.01986i	0.860216	+	0.509930i	$0.170329\pi$
$349$			19.0526i			1.01986i	−0.860216	+	0.509930i	$0.829671\pi$
$350$	−12.1244	+	14.0000i	−0.648074	+	0.748331i
$351$		0			0
$352$		0			0
$353$	3.46410	−	6.00000i	0.184376	−	0.319348i	−0.758990	−	0.651102i	$-0.774307\pi$
$353$	3.46410	−	6.00000i	0.184376	−	0.319348i	0.943366	+	0.331754i	$0.107640\pi$
$354$		0			0
$355$	18.0000	−	10.3923i	0.955341	−	0.551566i
$356$	−10.3923			−0.550791
$357$		0			0
$358$	6.00000			0.317110
$359$	−5.19615	+	3.00000i	−0.274242	+	0.158334i	−0.630814	−	0.775934i	$-0.717279\pi$
$359$	−5.19615	+	3.00000i	−0.274242	+	0.158334i	0.356572	+	0.934268i	$0.383946\pi$
$360$		0			0
$361$	−3.50000	+	6.06218i	−0.184211	+	0.319062i
$362$	6.92820	+	12.0000i	0.364138	+	0.630706i
$363$		0			0
$364$	−4.50000	−	12.9904i	−0.235864	−	0.680881i
$365$			24.0000i			1.25622i
$366$		0			0
$367$	15.0000	+	8.66025i	0.782994	+	0.452062i	0.837490	−	0.546452i	$-0.184022\pi$
$367$	15.0000	+	8.66025i	0.782994	+	0.452062i	−0.0544966	+	0.998514i	$0.517355\pi$
$368$	5.19615	+	3.00000i	0.270868	+	0.156386i
$369$		0			0
$370$		−	17.3205i		−	0.900450i
$371$	5.19615	+	15.0000i	0.269771	+	0.778761i
$372$		0			0
$373$	11.0000	+	19.0526i	0.569558	+	0.986504i	0.996610	+	0.0822766i	$0.0262191\pi$
$373$	11.0000	+	19.0526i	0.569558	+	0.986504i	−0.427051	+	0.904227i	$0.640448\pi$
$374$		0			0
$375$		0			0
$376$	6.00000	−	3.46410i	0.309426	−	0.178647i
$377$		0			0
$378$		0			0
$379$	−23.0000			−1.18143			−0.590715	−	0.806880i	$-0.701154\pi$
$379$	−23.0000			−1.18143			−0.590715	+	0.806880i	$0.701154\pi$
$380$	10.3923	−	6.00000i	0.533114	−	0.307794i
$381$		0			0
$382$	−3.00000	+	5.19615i	−0.153493	+	0.265858i
$383$	1.73205	+	3.00000i	0.0885037	+	0.153293i	0.906879	−	0.421392i	$-0.138458\pi$
$383$	1.73205	+	3.00000i	0.0885037	+	0.153293i	−0.818375	+	0.574684i	$0.805125\pi$
$384$		0			0
$385$		0			0
$386$			1.00000i			0.0508987i
$387$		0			0
$388$	1.50000	+	0.866025i	0.0761510	+	0.0439658i
$389$	−10.3923	−	6.00000i	−0.526911	−	0.304212i	0.212847	−	0.977086i	$-0.431726\pi$
$389$	−10.3923	−	6.00000i	−0.526911	−	0.304212i	−0.739758	+	0.672874i	$0.765060\pi$
$390$		0			0
$391$			41.5692i			2.10225i
$392$	6.92820	−	1.00000i	0.349927	−	0.0505076i
$393$		0			0
$394$	12.0000	+	20.7846i	0.604551	+	1.04711i
$395$	12.1244	−	21.0000i	0.610043	−	1.05662i
$396$		0			0
$397$	7.50000	−	4.33013i	0.376414	−	0.217323i	−0.299843	−	0.953989i	$-0.596934\pi$
$397$	7.50000	−	4.33013i	0.376414	−	0.217323i	0.676257	+	0.736666i	$0.263601\pi$
$398$	−1.73205			−0.0868199
$399$		0			0
$400$	7.00000			0.350000
$401$	−20.7846	+	12.0000i	−1.03793	+	0.599251i	−0.919247	−	0.393680i	$-0.871202\pi$
$401$	−20.7846	+	12.0000i	−1.03793	+	0.599251i	−0.118686	+	0.992932i	$0.537868\pi$
$402$		0			0
$403$	22.5000	−	38.9711i	1.12080	−	1.94129i
$404$	5.19615	+	9.00000i	0.258518	+	0.447767i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−13.5000	−	7.79423i	−0.667532	−	0.385400i	0.127609	−	0.991825i	$-0.459270\pi$
$409$	−13.5000	−	7.79423i	−0.667532	−	0.385400i	−0.795141	+	0.606425i	$0.792603\pi$
$410$	−20.7846	−	12.0000i	−1.02648	−	0.592638i
$411$		0			0
$412$		−	5.19615i		−	0.255996i
$413$	−17.3205	+	6.00000i	−0.852286	+	0.295241i
$414$		0			0
$415$	6.00000	+	10.3923i	0.294528	+	0.510138i
$416$	−2.59808	+	4.50000i	−0.127381	+	0.220631i
$417$		0			0
$418$		0			0
$419$	24.2487			1.18463			0.592314	−	0.805708i	$-0.298215\pi$
$419$	24.2487			1.18463			0.592314	+	0.805708i	$0.298215\pi$
$420$		0			0
$421$	−22.0000			−1.07221			−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000			−1.07221			−0.536107	+	0.844150i	$0.680106\pi$
$422$	21.6506	−	12.5000i	1.05394	−	0.608490i
$423$		0			0
$424$	3.00000	−	5.19615i	0.145693	−	0.252347i
$425$	24.2487	+	42.0000i	1.17624	+	2.03730i
$426$		0			0
$427$	−4.50000	−	0.866025i	−0.217770	−	0.0419099i
$428$			6.00000i			0.290021i
$429$		0			0
$430$	−3.00000	−	1.73205i	−0.144673	−	0.0835269i
$431$	−31.1769	−	18.0000i	−1.50174	−	0.867029i	−0.999998	−	0.00201168i	$-0.999360\pi$
$431$	−31.1769	−	18.0000i	−1.50174	−	0.867029i	−0.501741	−	0.865018i	$-0.667307\pi$
$432$		0			0
$433$			39.8372i			1.91445i	0.289341	+	0.957226i	$0.406564\pi$
$433$			39.8372i			1.91445i	−0.289341	+	0.957226i	$0.593436\pi$
$434$	17.3205	+	15.0000i	0.831411	+	0.720023i
$435$		0			0
$436$	−5.50000	−	9.52628i	−0.263402	−	0.456226i
$437$	10.3923	−	18.0000i	0.497131	−	0.861057i
$438$		0			0
$439$	−27.0000	+	15.5885i	−1.28864	+	0.743996i	−0.978412	−	0.206666i	$-0.933739\pi$
$439$	−27.0000	+	15.5885i	−1.28864	+	0.743996i	−0.310228	+	0.950662i	$0.600405\pi$
$440$		0			0
$441$		0			0
$442$	−36.0000			−1.71235
$443$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$443$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$444$		0			0
$445$	−18.0000	+	31.1769i	−0.853282	+	1.47793i
$446$	−5.19615	−	9.00000i	−0.246045	−	0.426162i
$447$		0			0
$448$	−2.00000	−	1.73205i	−0.0944911	−	0.0818317i
$449$		−	30.0000i		−	1.41579i	−0.706319	−	0.707894i	$-0.749646\pi$
$449$		−	30.0000i		−	1.41579i	0.706319	−	0.707894i	$-0.250354\pi$
$450$		0			0
$451$		0			0
$452$	15.5885	+	9.00000i	0.733219	+	0.423324i
$453$		0			0
$454$			10.3923i			0.487735i
$455$	−46.7654	−	9.00000i	−2.19239	−	0.421927i
$456$		0			0
$457$	−15.5000	−	26.8468i	−0.725059	−	1.25584i	−0.958950	−	0.283577i	$-0.908479\pi$
$457$	−15.5000	−	26.8468i	−0.725059	−	1.25584i	0.233890	−	0.972263i	$-0.424854\pi$
$458$	0.866025	−	1.50000i	0.0404667	−	0.0700904i
$459$		0			0
$460$	18.0000	−	10.3923i	0.839254	−	0.484544i
$461$	−13.8564			−0.645357			−0.322679	−	0.946509i	$-0.604583\pi$
$461$	−13.8564			−0.645357			−0.322679	+	0.946509i	$0.604583\pi$
$462$		0			0
$463$	−40.0000			−1.85896			−0.929479	−	0.368875i	$-0.879743\pi$
$463$	−40.0000			−1.85896			−0.929479	+	0.368875i	$0.879743\pi$
$464$		0			0
$465$		0			0
$466$	6.00000	−	10.3923i	0.277945	−	0.481414i
$467$	5.19615	+	9.00000i	0.240449	+	0.416470i	0.960842	−	0.277096i	$-0.0893719\pi$
$467$	5.19615	+	9.00000i	0.240449	+	0.416470i	−0.720393	+	0.693566i	$0.756039\pi$
$468$		0			0
$469$	32.5000	−	11.2583i	1.50071	−	0.519861i
$470$		−	24.0000i		−	1.10704i
$471$		0			0
$472$	6.00000	+	3.46410i	0.276172	+	0.159448i
$473$		0			0
$474$		0			0
$475$		−	24.2487i		−	1.11261i
$476$	3.46410	−	18.0000i	0.158777	−	0.825029i
$477$		0			0
$478$	−3.00000	−	5.19615i	−0.137217	−	0.237666i
$479$	1.73205	−	3.00000i	0.0791394	−	0.137073i	−0.823739	−	0.566969i	$-0.808116\pi$
$479$	1.73205	−	3.00000i	0.0791394	−	0.137073i	0.902879	+	0.429895i	$0.141449\pi$
$480$		0			0
$481$	22.5000	−	12.9904i	1.02591	−	0.592310i
$482$	1.73205			0.0788928
$483$		0			0
$484$	−11.0000			−0.500000
$485$	5.19615	−	3.00000i	0.235945	−	0.136223i
$486$		0			0
$487$	4.00000	−	6.92820i	0.181257	−	0.313947i	−0.761052	−	0.648691i	$-0.775317\pi$
$487$	4.00000	−	6.92820i	0.181257	−	0.313947i	0.942309	+	0.334744i	$0.108650\pi$
$488$	0.866025	+	1.50000i	0.0392031	+	0.0679018i
$489$		0			0
$490$	9.00000	−	22.5167i	0.406579	−	1.01720i
$491$		−	6.00000i		−	0.270776i	−0.990793	−	0.135388i	$-0.956772\pi$
$491$		−	6.00000i		−	0.270776i	0.990793	−	0.135388i	$-0.0432281\pi$
$492$		0			0
$493$		0			0
$494$	15.5885	+	9.00000i	0.701358	+	0.404929i
$495$		0			0
$496$		−	8.66025i		−	0.388857i
$497$	−10.3923	+	12.0000i	−0.466159	+	0.538274i
$498$		0			0
$499$	11.5000	+	19.9186i	0.514811	+	0.891678i	0.999852	+	0.0171872i	$0.00547113\pi$
$499$	11.5000	+	19.9186i	0.514811	+	0.891678i	−0.485042	+	0.874491i	$0.661196\pi$
$500$	3.46410	−	6.00000i	0.154919	−	0.268328i
$501$		0			0
$502$	−15.0000	+	8.66025i	−0.669483	+	0.386526i
$503$	38.1051			1.69902			0.849512	−	0.527570i	$-0.176897\pi$
$503$	38.1051			1.69902			0.849512	+	0.527570i	$0.176897\pi$
$504$		0			0
$505$	36.0000			1.60198
$506$		0			0
$507$		0			0
$508$	−3.50000	+	6.06218i	−0.155287	+	0.268966i
$509$	12.1244	+	21.0000i	0.537403	+	0.930809i	0.999043	+	0.0437414i	$0.0139278\pi$
$509$	12.1244	+	21.0000i	0.537403	+	0.930809i	−0.461640	+	0.887067i	$0.652739\pi$
$510$		0			0
$511$	−6.00000	−	17.3205i	−0.265424	−	0.766214i
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	−9.00000	−	5.19615i	−0.396973	−	0.229192i
$515$	−15.5885	−	9.00000i	−0.686909	−	0.396587i
$516$		0			0
$517$		0			0
$518$	4.33013	+	12.5000i	0.190255	+	0.549218i
$519$		0			0
$520$	9.00000	+	15.5885i	0.394676	+	0.683599i
$521$	12.1244	−	21.0000i	0.531178	−	0.920027i	−0.468160	−	0.883644i	$-0.655083\pi$
$521$	12.1244	−	21.0000i	0.531178	−	0.920027i	0.999338	−	0.0363831i	$-0.0115837\pi$
$522$		0			0
$523$	−4.50000	+	2.59808i	−0.196771	+	0.113606i	−0.595149	−	0.803616i	$-0.702907\pi$
$523$	−4.50000	+	2.59808i	−0.196771	+	0.113606i	0.398377	+	0.917222i	$0.369573\pi$
$524$	−20.7846			−0.907980
$525$		0			0
$526$		0			0
$527$	51.9615	−	30.0000i	2.26348	−	1.30682i
$528$		0			0
$529$	6.50000	−	11.2583i	0.282609	−	0.489493i
$530$	−10.3923	−	18.0000i	−0.451413	−	0.781870i
$531$		0			0
$532$	−6.00000	+	6.92820i	−0.260133	+	0.300376i
$533$		−	36.0000i		−	1.55933i
$534$		0			0
$535$	18.0000	+	10.3923i	0.778208	+	0.449299i
$536$	−11.2583	−	6.50000i	−0.486286	−	0.280757i
$537$		0			0
$538$			27.7128i			1.19478i
$539$		0			0
$540$		0			0
$541$	−1.00000	−	1.73205i	−0.0429934	−	0.0744667i	0.843728	−	0.536771i	$-0.180356\pi$
$541$	−1.00000	−	1.73205i	−0.0429934	−	0.0744667i	−0.886721	+	0.462304i	$0.847023\pi$
$542$	−14.7224	+	25.5000i	−0.632383	+	1.09532i
$543$		0			0
$544$	−6.00000	+	3.46410i	−0.257248	+	0.148522i
$545$	−38.1051			−1.63224
$546$		0			0
$547$	−1.00000			−0.0427569			−0.0213785	−	0.999771i	$-0.506805\pi$
$547$	−1.00000			−0.0427569			−0.0213785	+	0.999771i	$0.506805\pi$
$548$	15.5885	−	9.00000i	0.665906	−	0.384461i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−3.50000	+	18.1865i	−0.148835	+	0.773370i
$554$		−	1.00000i		−	0.0424859i
$555$		0			0
$556$	−4.50000	−	2.59808i	−0.190843	−	0.110183i
$557$	−15.5885	−	9.00000i	−0.660504	−	0.381342i	0.131965	−	0.991254i	$-0.457871\pi$
$557$	−15.5885	−	9.00000i	−0.660504	−	0.381342i	−0.792469	+	0.609912i	$0.791205\pi$
$558$		0			0
$559$		−	5.19615i		−	0.219774i
$560$	−8.66025	+	3.00000i	−0.365963	+	0.126773i
$561$		0			0
$562$	−9.00000	−	15.5885i	−0.379642	−	0.657559i
$563$	−5.19615	+	9.00000i	−0.218992	+	0.379305i	−0.954500	−	0.298211i	$-0.903610\pi$
$563$	−5.19615	+	9.00000i	−0.218992	+	0.379305i	0.735508	+	0.677516i	$0.236943\pi$
$564$		0			0
$565$	54.0000	−	31.1769i	2.27180	−	1.31162i
$566$	−19.0526			−0.800839
$567$		0			0
$568$	6.00000			0.251754
$569$	5.19615	−	3.00000i	0.217834	−	0.125767i	−0.387113	−	0.922032i	$-0.626528\pi$
$569$	5.19615	−	3.00000i	0.217834	−	0.125767i	0.604947	+	0.796266i	$0.293194\pi$
$570$		0			0
$571$	2.00000	−	3.46410i	0.0836974	−	0.144968i	−0.821138	−	0.570730i	$-0.806660\pi$
$571$	2.00000	−	3.46410i	0.0836974	−	0.144968i	0.904835	+	0.425762i	$0.139994\pi$
$572$		0			0
$573$		0			0
$574$	18.0000	+	3.46410i	0.751305	+	0.144589i
$575$		−	42.0000i		−	1.75152i
$576$		0			0
$577$	31.5000	+	18.1865i	1.31136	+	0.757115i	0.982322	−	0.187201i	$-0.0599414\pi$
$577$	31.5000	+	18.1865i	1.31136	+	0.757115i	0.329040	+	0.944316i	$0.393275\pi$
$578$	−26.8468	−	15.5000i	−1.11668	−	0.644715i
$579$		0			0
$580$		0			0
$581$	−6.92820	−	6.00000i	−0.287430	−	0.248922i
$582$		0			0
$583$		0			0
$584$	−3.46410	+	6.00000i	−0.143346	+	0.248282i
$585$		0			0
$586$	24.0000	−	13.8564i	0.991431	−	0.572403i
$587$	−10.3923			−0.428936			−0.214468	−	0.976731i	$-0.568802\pi$
$587$	−10.3923			−0.428936			−0.214468	+	0.976731i	$0.568802\pi$
$588$		0			0
$589$	−30.0000			−1.23613
$590$	20.7846	−	12.0000i	0.855689	−	0.494032i
$591$		0			0
$592$	2.50000	−	4.33013i	0.102749	−	0.177967i
$593$	−8.66025	−	15.0000i	−0.355634	−	0.615976i	0.631592	−	0.775301i	$-0.282402\pi$
$593$	−8.66025	−	15.0000i	−0.355634	−	0.615976i	−0.987226	+	0.159325i	$0.949068\pi$
$594$		0			0
$595$	−48.0000	−	41.5692i	−1.96781	−	1.70417i
$596$		−	6.00000i		−	0.245770i
$597$		0			0
$598$	27.0000	+	15.5885i	1.10411	+	0.637459i
$599$	−20.7846	−	12.0000i	−0.849236	−	0.490307i	0.0111569	−	0.999938i	$-0.496449\pi$
$599$	−20.7846	−	12.0000i	−0.849236	−	0.490307i	−0.860393	+	0.509631i	$0.829782\pi$
$600$		0			0
$601$		−	19.0526i		−	0.777170i	−0.921413	−	0.388585i	$-0.872964\pi$
$601$		−	19.0526i		−	0.777170i	0.921413	−	0.388585i	$-0.127036\pi$
$602$	2.59808	+	0.500000i	0.105890	+	0.0203785i
$603$		0			0
$604$	0.500000	+	0.866025i	0.0203447	+	0.0352381i
$605$	−19.0526	+	33.0000i	−0.774597	+	1.34164i
$606$		0			0
$607$	−15.0000	+	8.66025i	−0.608831	+	0.351509i	−0.772508	−	0.635005i	$-0.780998\pi$
$607$	−15.0000	+	8.66025i	−0.608831	+	0.351509i	0.163677	+	0.986514i	$0.447665\pi$
$608$	3.46410			0.140488
$609$		0			0
$610$	6.00000			0.242933
$611$	31.1769	−	18.0000i	1.26128	−	0.728202i
$612$		0			0
$613$	−8.50000	+	14.7224i	−0.343312	+	0.594633i	−0.985046	−	0.172294i	$-0.944882\pi$
$613$	−8.50000	+	14.7224i	−0.343312	+	0.594633i	0.641734	+	0.766927i	$0.278215\pi$
$614$	−4.33013	−	7.50000i	−0.174750	−	0.302675i
$615$		0			0
$616$		0			0
$617$			12.0000i			0.483102i	0.970388	+	0.241551i	$0.0776561\pi$
$617$			12.0000i			0.483102i	−0.970388	+	0.241551i	$0.922344\pi$
$618$		0			0
$619$	−13.5000	−	7.79423i	−0.542611	−	0.313276i	0.203526	−	0.979070i	$-0.434760\pi$
$619$	−13.5000	−	7.79423i	−0.542611	−	0.313276i	−0.746136	+	0.665793i	$0.768093\pi$
$620$	−25.9808	−	15.0000i	−1.04341	−	0.602414i
$621$		0			0
$622$			3.46410i			0.138898i
$623$	5.19615	−	27.0000i	0.208179	−	1.08173i
$624$		0			0
$625$	5.50000	+	9.52628i	0.220000	+	0.381051i
$626$	−3.46410	+	6.00000i	−0.138453	+	0.239808i
$627$		0			0
$628$		0			0
$629$	34.6410			1.38123
$630$		0			0
$631$	−25.0000			−0.995234			−0.497617	−	0.867397i	$-0.665792\pi$
$631$	−25.0000			−0.995234			−0.497617	+	0.867397i	$0.665792\pi$
$632$	6.06218	−	3.50000i	0.241140	−	0.139223i
$633$		0			0
$634$	12.0000	−	20.7846i	0.476581	−	0.825462i
$635$	12.1244	+	21.0000i	0.481140	+	0.833360i
$636$		0			0
$637$	36.0000	−	5.19615i	1.42637	−	0.205879i
$638$		0			0
$639$		0			0
$640$	3.00000	+	1.73205i	0.118585	+	0.0684653i
$641$	20.7846	+	12.0000i	0.820943	+	0.473972i	0.850741	−	0.525584i	$-0.176153\pi$
$641$	20.7846	+	12.0000i	0.820943	+	0.473972i	−0.0297987	+	0.999556i	$0.509487\pi$
$642$		0			0
$643$		−	43.3013i		−	1.70764i	−0.520572	−	0.853818i	$-0.674281\pi$
$643$		−	43.3013i		−	1.70764i	0.520572	−	0.853818i	$-0.325719\pi$
$644$	−10.3923	+	12.0000i	−0.409514	+	0.472866i
$645$		0			0
$646$	12.0000	+	20.7846i	0.472134	+	0.817760i
$647$	10.3923	−	18.0000i	0.408564	−	0.707653i	−0.586165	−	0.810191i	$-0.699363\pi$
$647$	10.3923	−	18.0000i	0.408564	−	0.707653i	0.994729	+	0.102538i	$0.0326965\pi$
$648$		0			0
$649$		0			0
$650$	36.3731			1.42667
$651$		0			0
$652$	−11.0000			−0.430793
$653$	−5.19615	+	3.00000i	−0.203341	+	0.117399i	−0.598213	−	0.801337i	$-0.704122\pi$
$653$	−5.19615	+	3.00000i	−0.203341	+	0.117399i	0.394872	+	0.918736i	$0.370789\pi$
$654$		0			0
$655$	−36.0000	+	62.3538i	−1.40664	+	2.43637i
$656$	−3.46410	−	6.00000i	−0.135250	−	0.234261i
$657$		0			0
$658$	6.00000	+	17.3205i	0.233904	+	0.675224i
$659$			18.0000i			0.701180i	0.936529	+	0.350590i	$0.114019\pi$
$659$			18.0000i			0.701180i	−0.936529	+	0.350590i	$0.885981\pi$
$660$		0			0
$661$	−30.0000	−	17.3205i	−1.16686	−	0.673690i	−0.213925	−	0.976850i	$-0.568625\pi$
$661$	−30.0000	−	17.3205i	−1.16686	−	0.673690i	−0.952940	+	0.303160i	$0.901958\pi$
$662$	6.92820	+	4.00000i	0.269272	+	0.155464i
$663$		0			0
$664$			3.46410i			0.134433i
$665$	10.3923	+	30.0000i	0.402996	+	1.16335i
$666$		0			0
$667$		0			0
$668$	1.73205	−	3.00000i	0.0670151	−	0.116073i
$669$		0			0
$670$	−39.0000	+	22.5167i	−1.50670	+	0.869894i
$671$		0			0
$672$		0			0
$673$	−14.0000			−0.539660			−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000			−0.539660			−0.269830	+	0.962908i	$0.586968\pi$
$674$	19.0526	−	11.0000i	0.733877	−	0.423704i
$675$		0			0
$676$	−7.00000	+	12.1244i	−0.269231	+	0.466321i
$677$	20.7846	+	36.0000i	0.798817	+	1.38359i	0.920387	+	0.391009i	$0.127874\pi$
$677$	20.7846	+	36.0000i	0.798817	+	1.38359i	−0.121569	+	0.992583i	$0.538793\pi$
$678$		0			0
$679$	−3.00000	+	3.46410i	−0.115129	+	0.132940i
$680$			24.0000i			0.920358i
$681$		0			0
$682$		0			0
$683$	−15.5885	−	9.00000i	−0.596476	−	0.344375i	0.171178	−	0.985240i	$-0.445243\pi$
$683$	−15.5885	−	9.00000i	−0.596476	−	0.344375i	−0.767654	+	0.640865i	$0.778576\pi$
$684$		0			0
$685$		−	62.3538i		−	2.38242i
$686$	−0.866025	+	18.5000i	−0.0330650	+	0.706333i
$687$		0			0
$688$	−0.500000	−	0.866025i	−0.0190623	−	0.0330169i
$689$	15.5885	−	27.0000i	0.593873	−	1.02862i
$690$		0			0
$691$	−28.5000	+	16.4545i	−1.08419	+	0.625958i	−0.932024	−	0.362397i	$-0.881959\pi$
$691$	−28.5000	+	16.4545i	−1.08419	+	0.625958i	−0.152167	+	0.988355i	$0.548625\pi$
$692$	−20.7846			−0.790112
$693$		0			0
$694$	24.0000			0.911028
$695$	−15.5885	+	9.00000i	−0.591304	+	0.341389i
$696$		0			0
$697$	24.0000	−	41.5692i	0.909065	−	1.57455i
$698$	9.52628	+	16.5000i	0.360575	+	0.624534i
$699$		0			0
$700$	−3.50000	+	18.1865i	−0.132288	+	0.687386i
$701$		−	36.0000i		−	1.35970i	−0.733351	−	0.679851i	$-0.762045\pi$
$701$		−	36.0000i		−	1.35970i	0.733351	−	0.679851i	$-0.237955\pi$
$702$		0			0
$703$	−15.0000	−	8.66025i	−0.565736	−	0.326628i
$704$		0			0
$705$		0			0
$706$		−	6.92820i		−	0.260746i
$707$	−25.9808	+	9.00000i	−0.977107	+	0.338480i
$708$		0			0
$709$	−0.500000	−	0.866025i	−0.0187779	−	0.0325243i	0.856484	−	0.516174i	$-0.172644\pi$
$709$	−0.500000	−	0.866025i	−0.0187779	−	0.0325243i	−0.875262	+	0.483650i	$0.839311\pi$
$710$	10.3923	−	18.0000i	0.390016	−	0.675528i
$711$		0			0
$712$	−9.00000	+	5.19615i	−0.337289	+	0.194734i
$713$	−51.9615			−1.94597
$714$		0			0
$715$		0			0
$716$	5.19615	−	3.00000i	0.194189	−	0.112115i
$717$		0			0
$718$	−3.00000	+	5.19615i	−0.111959	+	0.193919i
$719$	1.73205	+	3.00000i	0.0645946	+	0.111881i	0.896514	−	0.443015i	$-0.146091\pi$
$719$	1.73205	+	3.00000i	0.0645946	+	0.111881i	−0.831919	+	0.554896i	$0.812758\pi$
$720$		0			0
$721$	13.5000	+	2.59808i	0.502766	+	0.0967574i
$722$			7.00000i			0.260513i
$723$		0			0
$724$	12.0000	+	6.92820i	0.445976	+	0.257485i
$725$		0			0
$726$		0			0
$727$			19.0526i			0.706620i	0.935506	+	0.353310i	$0.114944\pi$
$727$			19.0526i			0.706620i	−0.935506	+	0.353310i	$0.885056\pi$
$728$	−10.3923	−	9.00000i	−0.385164	−	0.333562i
$729$		0			0
$730$	12.0000	+	20.7846i	0.444140	+	0.769273i
$731$	3.46410	−	6.00000i	0.128124	−	0.221918i
$732$		0			0
$733$	−1.50000	+	0.866025i	−0.0554038	+	0.0319874i	−0.527446	−	0.849589i	$-0.676850\pi$
$733$	−1.50000	+	0.866025i	−0.0554038	+	0.0319874i	0.472042	+	0.881576i	$0.343517\pi$
$734$	17.3205			0.639312
$735$		0			0
$736$	6.00000			0.221163
$737$		0			0
$738$		0			0
$739$	−17.5000	+	30.3109i	−0.643748	+	1.11500i	0.340841	+	0.940121i	$0.389288\pi$
$739$	−17.5000	+	30.3109i	−0.643748	+	1.11500i	−0.984589	+	0.174883i	$0.944045\pi$
$740$	−8.66025	−	15.0000i	−0.318357	−	0.551411i
$741$		0			0
$742$	12.0000	+	10.3923i	0.440534	+	0.381514i
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$	−18.0000	−	10.3923i	−0.659469	−	0.380745i
$746$	19.0526	+	11.0000i	0.697564	+	0.402739i
$747$		0			0
$748$		0			0
$749$	−15.5885	−	3.00000i	−0.569590	−	0.109618i
$750$		0			0
$751$	10.0000	+	17.3205i	0.364905	+	0.632034i	0.988761	−	0.149505i	$-0.0477681\pi$
$751$	10.0000	+	17.3205i	0.364905	+	0.632034i	−0.623856	+	0.781540i	$0.714435\pi$
$752$	3.46410	−	6.00000i	0.126323	−	0.218797i
$753$		0			0
$754$		0			0
$755$	3.46410			0.126072
$756$		0			0
$757$	25.0000			0.908640			0.454320	−	0.890838i	$-0.349882\pi$
$757$	25.0000			0.908640			0.454320	+	0.890838i	$0.349882\pi$
$758$	−19.9186	+	11.5000i	−0.723476	+	0.417699i
$759$		0			0
$760$	6.00000	−	10.3923i	0.217643	−	0.376969i
$761$	−3.46410	−	6.00000i	−0.125574	−	0.217500i	0.796383	−	0.604792i	$-0.206744\pi$
$761$	−3.46410	−	6.00000i	−0.125574	−	0.217500i	−0.921957	+	0.387292i	$0.873410\pi$
$762$		0			0
$763$	27.5000	−	9.52628i	0.995567	−	0.344874i
$764$			6.00000i			0.217072i
$765$		0			0
$766$	3.00000	+	1.73205i	0.108394	+	0.0625815i
$767$	31.1769	+	18.0000i	1.12573	+	0.649942i
$768$		0			0
$769$			6.92820i			0.249837i	0.992167	+	0.124919i	$0.0398670\pi$
$769$			6.92820i			0.249837i	−0.992167	+	0.124919i	$0.960133\pi$
$770$		0			0
$771$		0			0
$772$	0.500000	+	0.866025i	0.0179954	+	0.0311689i
$773$	−17.3205	+	30.0000i	−0.622975	+	1.07903i	0.365953	+	0.930633i	$0.380743\pi$
$773$	−17.3205	+	30.0000i	−0.622975	+	1.07903i	−0.988929	+	0.148392i	$0.952590\pi$
$774$		0			0
$775$	−52.5000	+	30.3109i	−1.88586	+	1.08880i
$776$	1.73205			0.0621770
$777$		0			0
$778$	−12.0000			−0.430221
$779$	−20.7846	+	12.0000i	−0.744686	+	0.429945i
$780$		0			0
$781$		0			0
$782$	20.7846	+	36.0000i	0.743256	+	1.28736i
$783$		0			0
$784$	5.50000	−	4.33013i	0.196429	−	0.154647i
$785$		0			0
$786$		0			0
$787$	−1.50000	−	0.866025i	−0.0534692	−	0.0308705i	0.473027	−	0.881048i	$-0.343161\pi$
$787$	−1.50000	−	0.866025i	−0.0534692	−	0.0308705i	−0.526496	+	0.850177i	$0.676495\pi$
$788$	20.7846	+	12.0000i	0.740421	+	0.427482i
$789$		0			0
$790$		−	24.2487i		−	0.862730i
$791$	−31.1769	+	36.0000i	−1.10852	+	1.28001i
$792$		0			0
$793$	4.50000	+	7.79423i	0.159800	+	0.276781i
$794$	4.33013	−	7.50000i	0.153670	−	0.266165i
$795$		0			0
$796$	−1.50000	+	0.866025i	−0.0531661	+	0.0306955i
$797$	−20.7846			−0.736229			−0.368114	−	0.929781i	$-0.619996\pi$
$797$	−20.7846			−0.736229			−0.368114	+	0.929781i	$0.619996\pi$
$798$		0			0
$799$	48.0000			1.69812
$800$	6.06218	−	3.50000i	0.214330	−	0.123744i
$801$		0			0
$802$	−12.0000	+	20.7846i	−0.423735	+	0.733930i
$803$		0			0
$804$		0			0
$805$	18.0000	+	51.9615i	0.634417	+	1.83140i
$806$		−	45.0000i		−	1.58506i
$807$		0			0
$808$	9.00000	+	5.19615i	0.316619	+	0.182800i
$809$	25.9808	+	15.0000i	0.913435	+	0.527372i	0.881535	−	0.472119i	$-0.156511\pi$
$809$	25.9808	+	15.0000i	0.913435	+	0.527372i	0.0319002	+	0.999491i	$0.489844\pi$
$810$		0			0
$811$		−	10.3923i		−	0.364923i	−0.983213	−	0.182462i	$-0.941593\pi$
$811$		−	10.3923i		−	0.364923i	0.983213	−	0.182462i	$-0.0584065\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−19.0526	+	33.0000i	−0.667382	+	1.15594i
$816$		0			0
$817$	−3.00000	+	1.73205i	−0.104957	+	0.0605968i
$818$	−15.5885			−0.545038
$819$		0			0
$820$	−24.0000			−0.838116
$821$	15.5885	−	9.00000i	0.544041	−	0.314102i	−0.202674	−	0.979246i	$-0.564963\pi$
$821$	15.5885	−	9.00000i	0.544041	−	0.314102i	0.746715	+	0.665144i	$0.231630\pi$
$822$		0			0
$823$	24.5000	−	42.4352i	0.854016	−	1.47920i	−0.0235383	−	0.999723i	$-0.507493\pi$
$823$	24.5000	−	42.4352i	0.854016	−	1.47920i	0.877555	−	0.479477i	$-0.159174\pi$
$824$	−2.59808	−	4.50000i	−0.0905083	−	0.156765i
$825$		0			0
$826$	−12.0000	+	13.8564i	−0.417533	+	0.482126i
$827$			48.0000i			1.66912i	0.550914	+	0.834562i	$0.314279\pi$
$827$			48.0000i			1.66912i	−0.550914	+	0.834562i	$0.685721\pi$
$828$		0			0
$829$	−12.0000	−	6.92820i	−0.416777	−	0.240626i	0.276920	−	0.960893i	$-0.410686\pi$
$829$	−12.0000	−	6.92820i	−0.416777	−	0.240626i	−0.693698	+	0.720266i	$0.744020\pi$
$830$	10.3923	+	6.00000i	0.360722	+	0.208263i
$831$		0			0
$832$			5.19615i			0.180144i
$833$	45.0333	+	18.0000i	1.56031	+	0.623663i
$834$		0			0
$835$	−6.00000	−	10.3923i	−0.207639	−	0.359641i
$836$		0			0
$837$		0			0
$838$	21.0000	−	12.1244i	0.725433	−	0.418829i
$839$	45.0333			1.55472			0.777361	−	0.629054i	$-0.216558\pi$
$839$	45.0333			1.55472			0.777361	+	0.629054i	$0.216558\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$	−19.0526	+	11.0000i	−0.656595	+	0.379085i
$843$		0			0
$844$	12.5000	−	21.6506i	0.430268	−	0.745246i
$845$	24.2487	+	42.0000i	0.834181	+	1.44484i
$846$		0			0
$847$	5.50000	−	28.5788i	0.188982	−	0.981981i
$848$		−	6.00000i		−	0.206041i
$849$		0			0
$850$	42.0000	+	24.2487i	1.44059	+	0.831724i
$851$	−25.9808	−	15.0000i	−0.890609	−	0.514193i
$852$		0			0
$853$			13.8564i			0.474434i	0.971457	+	0.237217i	$0.0762353\pi$
$853$			13.8564i			0.474434i	−0.971457	+	0.237217i	$0.923765\pi$
$854$	−4.33013	+	1.50000i	−0.148174	+	0.0513289i
$855$		0			0
$856$	3.00000	+	5.19615i	0.102538	+	0.177601i
$857$	8.66025	−	15.0000i	0.295829	−	0.512390i	−0.679349	−	0.733816i	$-0.737738\pi$
$857$	8.66025	−	15.0000i	0.295829	−	0.512390i	0.975177	+	0.221425i	$0.0710709\pi$
$858$		0			0
$859$	−7.50000	+	4.33013i	−0.255897	+	0.147742i	−0.622461	−	0.782651i	$-0.713867\pi$
$859$	−7.50000	+	4.33013i	−0.255897	+	0.147742i	0.366565	+	0.930393i	$0.380534\pi$
$860$	−3.46410			−0.118125
$861$		0			0
$862$	−36.0000			−1.22616
$863$	−31.1769	+	18.0000i	−1.06127	+	0.612727i	−0.925785	−	0.378052i	$-0.876594\pi$
$863$	−31.1769	+	18.0000i	−1.06127	+	0.612727i	−0.135490	+	0.990779i	$0.543261\pi$
$864$		0			0
$865$	−36.0000	+	62.3538i	−1.22404	+	2.12009i
$866$	19.9186	+	34.5000i	0.676861	+	1.17236i
$867$		0			0
$868$	22.5000	+	4.33013i	0.763700	+	0.146974i
$869$		0			0
$870$		0			0
$871$	−58.5000	−	33.7750i	−1.98220	−	1.14442i
$872$	−9.52628	−	5.50000i	−0.322601	−	0.186254i
$873$		0			0
$874$		−	20.7846i		−	0.703050i
$875$	13.8564	+	12.0000i	0.468432	+	0.405674i
$876$		0			0
$877$	−20.5000	−	35.5070i	−0.692236	−	1.19899i	−0.971104	−	0.238658i	$-0.923292\pi$
$877$	−20.5000	−	35.5070i	−0.692236	−	1.19899i	0.278868	−	0.960329i	$-0.410041\pi$
$878$	−15.5885	+	27.0000i	−0.526085	+	0.911206i
$879$		0			0
$880$		0			0
$881$	−31.1769			−1.05038			−0.525188	−	0.850986i	$-0.676005\pi$
$881$	−31.1769			−1.05038			−0.525188	+	0.850986i	$0.676005\pi$
$882$		0			0
$883$	52.0000			1.74994			0.874970	−	0.484178i	$-0.160881\pi$
$883$	52.0000			1.74994			0.874970	+	0.484178i	$0.160881\pi$
$884$	−31.1769	+	18.0000i	−1.04859	+	0.605406i
$885$		0			0
$886$		0			0
$887$	−1.73205	−	3.00000i	−0.0581566	−	0.100730i	0.835481	−	0.549519i	$-0.185189\pi$
$887$	−1.73205	−	3.00000i	−0.0581566	−	0.100730i	−0.893638	+	0.448789i	$0.851856\pi$
$888$		0			0
$889$	−14.0000	−	12.1244i	−0.469545	−	0.406638i
$890$			36.0000i			1.20672i
$891$		0			0
$892$	−9.00000	−	5.19615i	−0.301342	−	0.173980i
$893$	−20.7846	−	12.0000i	−0.695530	−	0.401565i
$894$		0			0
$895$		−	20.7846i		−	0.694753i
$896$	−2.59808	−	0.500000i	−0.0867956	−	0.0167038i
$897$		0			0
$898$	−15.0000	−	25.9808i	−0.500556	−	0.866989i
$899$		0			0
$900$		0			0
$901$	36.0000	−	20.7846i	1.19933	−	0.692436i
$902$		0			0
$903$		0			0
$904$	18.0000			0.598671
$905$	41.5692	−	24.0000i	1.38181	−	0.797787i
$906$		0			0
$907$	23.5000	−	40.7032i	0.780305	−	1.35153i	−0.151460	−	0.988463i	$-0.548397\pi$
$907$	23.5000	−	40.7032i	0.780305	−	1.35153i	0.931764	−	0.363064i	$-0.118269\pi$
$908$	5.19615	+	9.00000i	0.172440	+	0.298675i
$909$		0			0
$910$	−45.0000	+	15.5885i	−1.49174	+	0.516752i
$911$			48.0000i			1.59031i	0.606406	+	0.795155i	$0.292611\pi$
$911$			48.0000i			1.59031i	−0.606406	+	0.795155i	$0.707389\pi$
$912$		0			0
$913$		0			0
$914$	−26.8468	−	15.5000i	−0.888013	−	0.512694i
$915$		0			0
$916$		−	1.73205i		−	0.0572286i
$917$	10.3923	−	54.0000i	0.343184	−	1.78324i
$918$		0			0
$919$	−12.5000	−	21.6506i	−0.412337	−	0.714189i	0.582808	−	0.812610i	$-0.301954\pi$
$919$	−12.5000	−	21.6506i	−0.412337	−	0.714189i	−0.995145	+	0.0984214i	$0.968621\pi$
$920$	10.3923	−	18.0000i	0.342624	−	0.593442i
$921$		0			0
$922$	−12.0000	+	6.92820i	−0.395199	+	0.228168i
$923$	31.1769			1.02620
$924$		0			0
$925$	−35.0000			−1.15079
$926$	−34.6410	+	20.0000i	−1.13837	+	0.657241i
$927$		0			0
$928$		0			0
$929$	−22.5167	−	39.0000i	−0.738748	−	1.27955i	−0.953059	−	0.302783i	$-0.902084\pi$
$929$	−22.5167	−	39.0000i	−0.738748	−	1.27955i	0.214312	−	0.976765i	$-0.431249\pi$
$930$		0			0
$931$	−15.0000	−	19.0526i	−0.491605	−	0.624422i
$932$		−	12.0000i		−	0.393073i
$933$		0			0
$934$	9.00000	+	5.19615i	0.294489	+	0.170023i
$935$		0			0
$936$		0			0
$937$			5.19615i			0.169751i	0.996392	+	0.0848755i	$0.0270492\pi$
$937$			5.19615i			0.169751i	−0.996392	+	0.0848755i	$0.972951\pi$
$938$	22.5167	−	26.0000i	0.735195	−	0.848930i
$939$		0			0
$940$	−12.0000	−	20.7846i	−0.391397	−	0.677919i
$941$	6.92820	−	12.0000i	0.225853	−	0.391189i	−0.730722	−	0.682675i	$-0.760816\pi$
$941$	6.92820	−	12.0000i	0.225853	−	0.391189i	0.956575	+	0.291486i	$0.0941498\pi$
$942$		0			0
$943$	−36.0000	+	20.7846i	−1.17232	+	0.676840i
$944$	6.92820			0.225494
$945$		0			0
$946$		0			0
$947$	31.1769	−	18.0000i	1.01311	−	0.584921i	0.101012	−	0.994885i	$-0.467792\pi$
$947$	31.1769	−	18.0000i	1.01311	−	0.584921i	0.912102	+	0.409964i	$0.134459\pi$
$948$		0			0
$949$	−18.0000	+	31.1769i	−0.584305	+	1.01205i
$950$	−12.1244	−	21.0000i	−0.393366	−	0.681330i
$951$		0			0
$952$	−6.00000	−	17.3205i	−0.194461	−	0.561361i
$953$		−	6.00000i		−	0.194359i	−0.995267	−	0.0971795i	$-0.969018\pi$
$953$		−	6.00000i		−	0.194359i	0.995267	−	0.0971795i	$-0.0309821\pi$
$954$		0			0
$955$	18.0000	+	10.3923i	0.582466	+	0.336287i
$956$	−5.19615	−	3.00000i	−0.168056	−	0.0970269i
$957$		0			0
$958$		−	3.46410i		−	0.111920i
$959$	15.5885	+	45.0000i	0.503378	+	1.45313i
$960$		0			0
$961$	22.0000	+	38.1051i	0.709677	+	1.22920i
$962$	12.9904	−	22.5000i	0.418827	−	0.725429i
$963$		0			0
$964$	1.50000	−	0.866025i	0.0483117	−	0.0278928i
$965$	3.46410			0.111513
$966$		0			0
$967$	−31.0000			−0.996893			−0.498446	−	0.866921i	$-0.666096\pi$
$967$	−31.0000			−0.996893			−0.498446	+	0.866921i	$0.666096\pi$
$968$	−9.52628	+	5.50000i	−0.306186	+	0.176777i
$969$		0			0
$970$	3.00000	−	5.19615i	0.0963242	−	0.166838i
$971$	−13.8564	−	24.0000i	−0.444673	−	0.770197i	0.553356	−	0.832945i	$-0.313347\pi$
$971$	−13.8564	−	24.0000i	−0.444673	−	0.770197i	−0.998029	+	0.0627481i	$0.980014\pi$
$972$		0			0
$973$	9.00000	−	10.3923i	0.288527	−	0.333162i
$974$		−	8.00000i		−	0.256337i
$975$		0			0
$976$	1.50000	+	0.866025i	0.0480138	+	0.0277208i
$977$	5.19615	+	3.00000i	0.166240	+	0.0959785i	0.580812	−	0.814038i	$-0.302735\pi$
$977$	5.19615	+	3.00000i	0.166240	+	0.0959785i	−0.414572	+	0.910017i	$0.636069\pi$
$978$		0			0
$979$		0			0
$980$	−3.46410	−	24.0000i	−0.110657	−	0.766652i
$981$		0			0
$982$	−3.00000	−	5.19615i	−0.0957338	−	0.165816i
$983$	12.1244	−	21.0000i	0.386707	−	0.669796i	−0.605298	−	0.795999i	$-0.706946\pi$
$983$	12.1244	−	21.0000i	0.386707	−	0.669796i	0.992004	+	0.126203i	$0.0402792\pi$
$984$		0			0
$985$	72.0000	−	41.5692i	2.29411	−	1.32451i
$986$		0			0
$987$		0			0
$988$	18.0000			0.572656
$989$	−5.19615	+	3.00000i	−0.165228	+	0.0953945i
$990$		0			0
$991$	−6.50000	+	11.2583i	−0.206479	+	0.357633i	−0.950603	−	0.310409i	$-0.899534\pi$
$991$	−6.50000	+	11.2583i	−0.206479	+	0.357633i	0.744124	+	0.668042i	$0.232867\pi$
$992$	−4.33013	−	7.50000i	−0.137482	−	0.238125i
$993$		0			0
$994$	−3.00000	+	15.5885i	−0.0951542	+	0.494436i
$995$			6.00000i			0.190213i
$996$		0			0
$997$	28.5000	+	16.4545i	0.902604	+	0.521119i	0.878044	−	0.478580i	$-0.158848\pi$
$997$	28.5000	+	16.4545i	0.902604	+	0.521119i	0.0245599	+	0.999698i	$0.492182\pi$
$998$	19.9186	+	11.5000i	0.630512	+	0.364026i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.2.k.b.269.2	yes	4
3.2	odd	2	inner	378.2.k.b.269.1	yes	4
7.3	odd	6		2646.2.d.b.2645.3		4
7.4	even	3		2646.2.d.b.2645.4		4
7.5	odd	6	inner	378.2.k.b.215.1	✓	4
9.2	odd	6		1134.2.t.b.1025.2		4
9.4	even	3		1134.2.l.a.269.2		4
9.5	odd	6		1134.2.l.a.269.1		4
9.7	even	3		1134.2.t.b.1025.1		4
21.5	even	6	inner	378.2.k.b.215.2	yes	4
21.11	odd	6		2646.2.d.b.2645.1		4
21.17	even	6		2646.2.d.b.2645.2		4
63.5	even	6		1134.2.t.b.593.1		4
63.40	odd	6		1134.2.t.b.593.2		4
63.47	even	6		1134.2.l.a.215.1		4
63.61	odd	6		1134.2.l.a.215.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.2.k.b.215.1	✓	4	7.5	odd	6	inner
378.2.k.b.215.2	yes	4	21.5	even	6	inner
378.2.k.b.269.1	yes	4	3.2	odd	2	inner
378.2.k.b.269.2	yes	4	1.1	even	1	trivial
1134.2.l.a.215.1		4	63.47	even	6
1134.2.l.a.215.2		4	63.61	odd	6
1134.2.l.a.269.1		4	9.5	odd	6
1134.2.l.a.269.2		4	9.4	even	3
1134.2.t.b.593.1		4	63.5	even	6
1134.2.t.b.593.2		4	63.40	odd	6
1134.2.t.b.1025.1		4	9.7	even	3
1134.2.t.b.1025.2		4	9.2	odd	6
2646.2.d.b.2645.1		4	21.11	odd	6
2646.2.d.b.2645.2		4	21.17	even	6
2646.2.d.b.2645.3		4	7.3	odd	6
2646.2.d.b.2645.4		4	7.4	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.k (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.866025 - 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(-1.73205 - 3.00000i) q^{5} +(2.00000 + 1.73205i) q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+(0.866025 - 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(-1.73205 - 3.00000i) q^{5} +(2.00000 + 1.73205i) q^{7} -1.00000i q^{8} +(-3.00000 - 1.73205i) q^{10} -5.19615i q^{13} +(2.59808 + 0.500000i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(3.46410 - 6.00000i) q^{17} +(-3.00000 + 1.73205i) q^{19} -3.46410 q^{20} +(-5.19615 + 3.00000i) q^{23} +(-3.50000 + 6.06218i) q^{25} +(-2.59808 - 4.50000i) q^{26} +(2.50000 - 0.866025i) q^{28} +(7.50000 + 4.33013i) q^{31} +(-0.866025 - 0.500000i) q^{32} -6.92820i q^{34} +(1.73205 - 9.00000i) q^{35} +(2.50000 + 4.33013i) q^{37} +(-1.73205 + 3.00000i) q^{38} +(-3.00000 + 1.73205i) q^{40} +6.92820 q^{41} +1.00000 q^{43} +(-3.00000 + 5.19615i) q^{46} +(3.46410 + 6.00000i) q^{47} +(1.00000 + 6.92820i) q^{49} +7.00000i q^{50} +(-4.50000 - 2.59808i) q^{52} +(5.19615 + 3.00000i) q^{53} +(1.73205 - 2.00000i) q^{56} +(-3.46410 + 6.00000i) q^{59} +(-1.50000 + 0.866025i) q^{61} +8.66025 q^{62} -1.00000 q^{64} +(-15.5885 + 9.00000i) q^{65} +(6.50000 - 11.2583i) q^{67} +(-3.46410 - 6.00000i) q^{68} +(-3.00000 - 8.66025i) q^{70} +6.00000i q^{71} +(-6.00000 - 3.46410i) q^{73} +(4.33013 + 2.50000i) q^{74} +3.46410i q^{76} +(3.50000 + 6.06218i) q^{79} +(-1.73205 + 3.00000i) q^{80} +(6.00000 - 3.46410i) q^{82} -3.46410 q^{83} -24.0000 q^{85} +(0.866025 - 0.500000i) q^{86} +(-5.19615 - 9.00000i) q^{89} +(9.00000 - 10.3923i) q^{91} +6.00000i q^{92} +(6.00000 + 3.46410i) q^{94} +(10.3923 + 6.00000i) q^{95} +1.73205i q^{97} +(4.33013 + 5.50000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} + 8 q^{7}+O(q^{10}) \) 4 * q + 2 * q^4 + 8 * q^7	\( 4 q + 2 q^{4} + 8 q^{7} - 12 q^{10} - 2 q^{16} - 12 q^{19} - 14 q^{25} + 10 q^{28} + 30 q^{31} + 10 q^{37} - 12 q^{40} + 4 q^{43} - 12 q^{46} + 4 q^{49} - 18 q^{52} - 6 q^{61} - 4 q^{64} + 26 q^{67} - 12 q^{70} - 24 q^{73} + 14 q^{79} + 24 q^{82} - 96 q^{85} + 36 q^{91} + 24 q^{94}+O(q^{100}) \) 4 * q + 2 * q^4 + 8 * q^7 - 12 * q^10 - 2 * q^16 - 12 * q^19 - 14 * q^25 + 10 * q^28 + 30 * q^31 + 10 * q^37 - 12 * q^40 + 4 * q^43 - 12 * q^46 + 4 * q^49 - 18 * q^52 - 6 * q^61 - 4 * q^64 + 26 * q^67 - 12 * q^70 - 24 * q^73 + 14 * q^79 + 24 * q^82 - 96 * q^85 + 36 * q^91 + 24 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more