LMFDB - Embedded newform 378.2.f.b.253.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,2,Mod(127,378)]

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([4, 0]))

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

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

chi := DirichletCharacter("378.127");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	378.f (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$3.01834519640$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			253.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	378.253
Dual form			378.2.f.b.127.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(1.00000 - 1.73205i) q^{5} +(0.500000 + 0.866025i) q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(1.00000 - 1.73205i) q^{5} +(0.500000 + 0.866025i) q^{7} +1.00000 q^{8} -2.00000 q^{10} +(0.500000 + 0.866025i) q^{11} +(3.00000 - 5.19615i) q^{13} +(0.500000 - 0.866025i) q^{14} +(-0.500000 - 0.866025i) q^{16} +5.00000 q^{17} -7.00000 q^{19} +(1.00000 + 1.73205i) q^{20} +(0.500000 - 0.866025i) q^{22} +(2.00000 - 3.46410i) q^{23} +(0.500000 + 0.866025i) q^{25} -6.00000 q^{26} -1.00000 q^{28} +(-2.00000 - 3.46410i) q^{29} +(3.00000 - 5.19615i) q^{31} +(-0.500000 + 0.866025i) q^{32} +(-2.50000 - 4.33013i) q^{34} +2.00000 q^{35} +2.00000 q^{37} +(3.50000 + 6.06218i) q^{38} +(1.00000 - 1.73205i) q^{40} +(1.50000 - 2.59808i) q^{41} +(0.500000 + 0.866025i) q^{43} -1.00000 q^{44} -4.00000 q^{46} +(-0.500000 + 0.866025i) q^{49} +(0.500000 - 0.866025i) q^{50} +(3.00000 + 5.19615i) q^{52} -12.0000 q^{53} +2.00000 q^{55} +(0.500000 + 0.866025i) q^{56} +(-2.00000 + 3.46410i) q^{58} +(-3.50000 + 6.06218i) q^{59} +(6.00000 + 10.3923i) q^{61} -6.00000 q^{62} +1.00000 q^{64} +(-6.00000 - 10.3923i) q^{65} +(-6.50000 + 11.2583i) q^{67} +(-2.50000 + 4.33013i) q^{68} +(-1.00000 - 1.73205i) q^{70} +8.00000 q^{71} +1.00000 q^{73} +(-1.00000 - 1.73205i) q^{74} +(3.50000 - 6.06218i) q^{76} +(-0.500000 + 0.866025i) q^{77} +(3.00000 + 5.19615i) q^{79} -2.00000 q^{80} -3.00000 q^{82} +(8.00000 + 13.8564i) q^{83} +(5.00000 - 8.66025i) q^{85} +(0.500000 - 0.866025i) q^{86} +(0.500000 + 0.866025i) q^{88} +6.00000 q^{89} +6.00000 q^{91} +(2.00000 + 3.46410i) q^{92} +(-7.00000 + 12.1244i) q^{95} +(2.50000 + 4.33013i) q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{4} + 2 q^{5} + q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q - q^2 - q^4 + 2 * q^5 + q^7 + 2 * q^8	$ 2 q - q^{2} - q^{4} + 2 q^{5} + q^{7} + 2 q^{8} - 4 q^{10} + q^{11} + 6 q^{13} + q^{14} - q^{16} + 10 q^{17} - 14 q^{19} + 2 q^{20} + q^{22} + 4 q^{23} + q^{25} - 12 q^{26} - 2 q^{28} - 4 q^{29} + 6 q^{31} - q^{32} - 5 q^{34} + 4 q^{35} + 4 q^{37} + 7 q^{38} + 2 q^{40} + 3 q^{41} + q^{43} - 2 q^{44} - 8 q^{46} - q^{49} + q^{50} + 6 q^{52} - 24 q^{53} + 4 q^{55} + q^{56} - 4 q^{58} - 7 q^{59} + 12 q^{61} - 12 q^{62} + 2 q^{64} - 12 q^{65} - 13 q^{67} - 5 q^{68} - 2 q^{70} + 16 q^{71} + 2 q^{73} - 2 q^{74} + 7 q^{76} - q^{77} + 6 q^{79} - 4 q^{80} - 6 q^{82} + 16 q^{83} + 10 q^{85} + q^{86} + q^{88} + 12 q^{89} + 12 q^{91} + 4 q^{92} - 14 q^{95} + 5 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q - q^2 - q^4 + 2 * q^5 + q^7 + 2 * q^8 - 4 * q^10 + q^11 + 6 * q^13 + q^14 - q^16 + 10 * q^17 - 14 * q^19 + 2 * q^20 + q^22 + 4 * q^23 + q^25 - 12 * q^26 - 2 * q^28 - 4 * q^29 + 6 * q^31 - q^32 - 5 * q^34 + 4 * q^35 + 4 * q^37 + 7 * q^38 + 2 * q^40 + 3 * q^41 + q^43 - 2 * q^44 - 8 * q^46 - q^49 + q^50 + 6 * q^52 - 24 * q^53 + 4 * q^55 + q^56 - 4 * q^58 - 7 * q^59 + 12 * q^61 - 12 * q^62 + 2 * q^64 - 12 * q^65 - 13 * q^67 - 5 * q^68 - 2 * q^70 + 16 * q^71 + 2 * q^73 - 2 * q^74 + 7 * q^76 - q^77 + 6 * q^79 - 4 * q^80 - 6 * q^82 + 16 * q^83 + 10 * q^85 + q^86 + q^88 + 12 * q^89 + 12 * q^91 + 4 * q^92 - 14 * q^95 + 5 * q^97 + 2 * q^98

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$	−0.500000	−	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
$5$	1.00000	−	1.73205i	0.447214	−	0.774597i	−0.550990	−	0.834512i	$-0.685750\pi$
$5$	1.00000	−	1.73205i	0.447214	−	0.774597i	0.998203	+	0.0599153i	$0.0190830\pi$
$6$		0			0
$7$	0.500000	+	0.866025i	0.188982	+	0.327327i
$7$	0.500000	+	0.866025i	0.188982	+	0.327327i
$8$	1.00000			0.353553
$9$		0			0
$10$	−2.00000			−0.632456
$11$	0.500000	+	0.866025i	0.150756	+	0.261116i	0.931505	−	0.363727i	$-0.118496\pi$
$11$	0.500000	+	0.866025i	0.150756	+	0.261116i	−0.780750	+	0.624844i	$0.785163\pi$
$12$		0			0
$13$	3.00000	−	5.19615i	0.832050	−	1.44115i	−0.0643593	−	0.997927i	$-0.520500\pi$
$13$	3.00000	−	5.19615i	0.832050	−	1.44115i	0.896410	−	0.443227i	$-0.146166\pi$
$14$	0.500000	−	0.866025i	0.133631	−	0.231455i
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.125000	−	0.216506i
$17$	5.00000			1.21268			0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000			1.21268			0.606339	+	0.795206i	$0.292637\pi$
$18$		0			0
$19$	−7.00000			−1.60591			−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−7.00000			−1.60591			−0.802955	+	0.596040i	$0.796740\pi$
$20$	1.00000	+	1.73205i	0.223607	+	0.387298i
$21$		0			0
$22$	0.500000	−	0.866025i	0.106600	−	0.184637i
$23$	2.00000	−	3.46410i	0.417029	−	0.722315i	−0.578610	−	0.815604i	$-0.696405\pi$
$23$	2.00000	−	3.46410i	0.417029	−	0.722315i	0.995639	+	0.0932891i	$0.0297381\pi$
$24$		0			0
$25$	0.500000	+	0.866025i	0.100000	+	0.173205i
$26$	−6.00000			−1.17670
$27$		0			0
$28$	−1.00000			−0.188982
$29$	−2.00000	−	3.46410i	−0.371391	−	0.643268i	0.618389	−	0.785872i	$-0.287786\pi$
$29$	−2.00000	−	3.46410i	−0.371391	−	0.643268i	−0.989780	+	0.142605i	$0.954452\pi$
$30$		0			0
$31$	3.00000	−	5.19615i	0.538816	−	0.933257i	−0.460152	−	0.887840i	$-0.652205\pi$
$31$	3.00000	−	5.19615i	0.538816	−	0.933257i	0.998968	−	0.0454165i	$-0.0144615\pi$
$32$	−0.500000	+	0.866025i	−0.0883883	+	0.153093i
$33$		0			0
$34$	−2.50000	−	4.33013i	−0.428746	−	0.742611i
$35$	2.00000			0.338062
$36$		0			0
$37$	2.00000			0.328798			0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000			0.328798			0.164399	+	0.986394i	$0.447432\pi$
$38$	3.50000	+	6.06218i	0.567775	+	0.983415i
$39$		0			0
$40$	1.00000	−	1.73205i	0.158114	−	0.273861i
$41$	1.50000	−	2.59808i	0.234261	−	0.405751i	−0.724797	−	0.688963i	$-0.758066\pi$
$41$	1.50000	−	2.59808i	0.234261	−	0.405751i	0.959058	+	0.283211i	$0.0913998\pi$
$42$		0			0
$43$	0.500000	+	0.866025i	0.0762493	+	0.132068i	0.901629	−	0.432511i	$-0.142372\pi$
$43$	0.500000	+	0.866025i	0.0762493	+	0.132068i	−0.825380	+	0.564578i	$0.809039\pi$
$44$	−1.00000			−0.150756
$45$		0			0
$46$	−4.00000			−0.589768
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$		0			0
$49$	−0.500000	+	0.866025i	−0.0714286	+	0.123718i
$50$	0.500000	−	0.866025i	0.0707107	−	0.122474i
$51$		0			0
$52$	3.00000	+	5.19615i	0.416025	+	0.720577i
$53$	−12.0000			−1.64833			−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000			−1.64833			−0.824163	+	0.566352i	$0.808354\pi$
$54$		0			0
$55$	2.00000			0.269680
$56$	0.500000	+	0.866025i	0.0668153	+	0.115728i
$57$		0			0
$58$	−2.00000	+	3.46410i	−0.262613	+	0.454859i
$59$	−3.50000	+	6.06218i	−0.455661	+	0.789228i	−0.998726	−	0.0504625i	$-0.983930\pi$
$59$	−3.50000	+	6.06218i	−0.455661	+	0.789228i	0.543065	+	0.839691i	$0.317264\pi$
$60$		0			0
$61$	6.00000	+	10.3923i	0.768221	+	1.33060i	0.938527	+	0.345207i	$0.112191\pi$
$61$	6.00000	+	10.3923i	0.768221	+	1.33060i	−0.170305	+	0.985391i	$0.554475\pi$
$62$	−6.00000			−0.762001
$63$		0			0
$64$	1.00000			0.125000
$65$	−6.00000	−	10.3923i	−0.744208	−	1.28901i
$66$		0			0
$67$	−6.50000	+	11.2583i	−0.794101	+	1.37542i	0.129307	+	0.991605i	$0.458725\pi$
$67$	−6.50000	+	11.2583i	−0.794101	+	1.37542i	−0.923408	+	0.383819i	$0.874609\pi$
$68$	−2.50000	+	4.33013i	−0.303170	+	0.525105i
$69$		0			0
$70$	−1.00000	−	1.73205i	−0.119523	−	0.207020i
$71$	8.00000			0.949425			0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000			0.949425			0.474713	+	0.880141i	$0.342552\pi$
$72$		0			0
$73$	1.00000			0.117041			0.0585206	−	0.998286i	$-0.481362\pi$
$73$	1.00000			0.117041			0.0585206	+	0.998286i	$0.481362\pi$
$74$	−1.00000	−	1.73205i	−0.116248	−	0.201347i
$75$		0			0
$76$	3.50000	−	6.06218i	0.401478	−	0.695379i
$77$	−0.500000	+	0.866025i	−0.0569803	+	0.0986928i
$78$		0			0
$79$	3.00000	+	5.19615i	0.337526	+	0.584613i	0.983967	−	0.178352i	$-0.0570765\pi$
$79$	3.00000	+	5.19615i	0.337526	+	0.584613i	−0.646440	+	0.762964i	$0.723743\pi$
$80$	−2.00000			−0.223607
$81$		0			0
$82$	−3.00000			−0.331295
$83$	8.00000	+	13.8564i	0.878114	+	1.52094i	0.853408	+	0.521243i	$0.174532\pi$
$83$	8.00000	+	13.8564i	0.878114	+	1.52094i	0.0247060	+	0.999695i	$0.492135\pi$
$84$		0			0
$85$	5.00000	−	8.66025i	0.542326	−	0.939336i
$86$	0.500000	−	0.866025i	0.0539164	−	0.0933859i
$87$		0			0
$88$	0.500000	+	0.866025i	0.0533002	+	0.0923186i
$89$	6.00000			0.635999			0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000			0.635999			0.317999	+	0.948091i	$0.396989\pi$
$90$		0			0
$91$	6.00000			0.628971
$92$	2.00000	+	3.46410i	0.208514	+	0.361158i
$93$		0			0
$94$		0			0
$95$	−7.00000	+	12.1244i	−0.718185	+	1.24393i
$96$		0			0
$97$	2.50000	+	4.33013i	0.253837	+	0.439658i	0.964579	−	0.263795i	$-0.0849741\pi$
$97$	2.50000	+	4.33013i	0.253837	+	0.439658i	−0.710742	+	0.703452i	$0.751641\pi$
$98$	1.00000			0.101015
$99$		0			0
$100$	−1.00000			−0.100000
$101$	−2.00000	−	3.46410i	−0.199007	−	0.344691i	0.749199	−	0.662344i	$-0.230438\pi$
$101$	−2.00000	−	3.46410i	−0.199007	−	0.344691i	−0.948207	+	0.317653i	$0.897105\pi$
$102$		0			0
$103$	−7.00000	+	12.1244i	−0.689730	+	1.19465i	0.282194	+	0.959357i	$0.408938\pi$
$103$	−7.00000	+	12.1244i	−0.689730	+	1.19465i	−0.971925	+	0.235291i	$0.924396\pi$
$104$	3.00000	−	5.19615i	0.294174	−	0.509525i
$105$		0			0
$106$	6.00000	+	10.3923i	0.582772	+	1.00939i
$107$	−3.00000			−0.290021			−0.145010	−	0.989430i	$-0.546322\pi$
$107$	−3.00000			−0.290021			−0.145010	+	0.989430i	$0.546322\pi$
$108$		0			0
$109$	−2.00000			−0.191565			−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000			−0.191565			−0.0957826	+	0.995402i	$0.530535\pi$
$110$	−1.00000	−	1.73205i	−0.0953463	−	0.165145i
$111$		0			0
$112$	0.500000	−	0.866025i	0.0472456	−	0.0818317i
$113$	−5.00000	+	8.66025i	−0.470360	+	0.814688i	−0.999425	−	0.0338931i	$-0.989209\pi$
$113$	−5.00000	+	8.66025i	−0.470360	+	0.814688i	0.529065	+	0.848581i	$0.322543\pi$
$114$		0			0
$115$	−4.00000	−	6.92820i	−0.373002	−	0.646058i
$116$	4.00000			0.371391
$117$		0			0
$118$	7.00000			0.644402
$119$	2.50000	+	4.33013i	0.229175	+	0.396942i
$120$		0			0
$121$	5.00000	−	8.66025i	0.454545	−	0.787296i
$122$	6.00000	−	10.3923i	0.543214	−	0.940875i
$123$		0			0
$124$	3.00000	+	5.19615i	0.269408	+	0.466628i
$125$	12.0000			1.07331
$126$		0			0
$127$	−12.0000			−1.06483			−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000			−1.06483			−0.532414	+	0.846484i	$0.678715\pi$
$128$	−0.500000	−	0.866025i	−0.0441942	−	0.0765466i
$129$		0			0
$130$	−6.00000	+	10.3923i	−0.526235	+	0.911465i
$131$	−2.00000	+	3.46410i	−0.174741	+	0.302660i	−0.940072	−	0.340977i	$-0.889242\pi$
$131$	−2.00000	+	3.46410i	−0.174741	+	0.302660i	0.765331	+	0.643637i	$0.222575\pi$
$132$		0			0
$133$	−3.50000	−	6.06218i	−0.303488	−	0.525657i
$134$	13.0000			1.12303
$135$		0			0
$136$	5.00000			0.428746
$137$	−9.50000	−	16.4545i	−0.811640	−	1.40580i	−0.911716	−	0.410822i	$-0.865242\pi$
$137$	−9.50000	−	16.4545i	−0.811640	−	1.40580i	0.100076	−	0.994980i	$-0.468091\pi$
$138$		0			0
$139$	2.50000	−	4.33013i	0.212047	−	0.367277i	−0.740308	−	0.672268i	$-0.765320\pi$
$139$	2.50000	−	4.33013i	0.212047	−	0.367277i	0.952355	+	0.304991i	$0.0986536\pi$
$140$	−1.00000	+	1.73205i	−0.0845154	+	0.146385i
$141$		0			0
$142$	−4.00000	−	6.92820i	−0.335673	−	0.581402i
$143$	6.00000			0.501745
$144$		0			0
$145$	−8.00000			−0.664364
$146$	−0.500000	−	0.866025i	−0.0413803	−	0.0716728i
$147$		0			0
$148$	−1.00000	+	1.73205i	−0.0821995	+	0.142374i
$149$	−12.0000	+	20.7846i	−0.983078	+	1.70274i	−0.332896	+	0.942964i	$0.608026\pi$
$149$	−12.0000	+	20.7846i	−0.983078	+	1.70274i	−0.650183	+	0.759778i	$0.725308\pi$
$150$		0			0
$151$	−5.00000	−	8.66025i	−0.406894	−	0.704761i	0.587646	−	0.809118i	$-0.300055\pi$
$151$	−5.00000	−	8.66025i	−0.406894	−	0.704761i	−0.994540	+	0.104357i	$0.966722\pi$
$152$	−7.00000			−0.567775
$153$		0			0
$154$	1.00000			0.0805823
$155$	−6.00000	−	10.3923i	−0.481932	−	0.834730i
$156$		0			0
$157$	−1.00000	+	1.73205i	−0.0798087	+	0.138233i	−0.903167	−	0.429289i	$-0.858764\pi$
$157$	−1.00000	+	1.73205i	−0.0798087	+	0.138233i	0.823359	+	0.567521i	$0.192098\pi$
$158$	3.00000	−	5.19615i	0.238667	−	0.413384i
$159$		0			0
$160$	1.00000	+	1.73205i	0.0790569	+	0.136931i
$161$	4.00000			0.315244
$162$		0			0
$163$	−4.00000			−0.313304			−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000			−0.313304			−0.156652	+	0.987654i	$0.550070\pi$
$164$	1.50000	+	2.59808i	0.117130	+	0.202876i
$165$		0			0
$166$	8.00000	−	13.8564i	0.620920	−	1.07547i
$167$	10.0000	−	17.3205i	0.773823	−	1.34030i	−0.161630	−	0.986851i	$-0.551675\pi$
$167$	10.0000	−	17.3205i	0.773823	−	1.34030i	0.935454	−	0.353450i	$-0.114991\pi$
$168$		0			0
$169$	−11.5000	−	19.9186i	−0.884615	−	1.53220i
$170$	−10.0000			−0.766965
$171$		0			0
$172$	−1.00000			−0.0762493
$173$	1.00000	+	1.73205i	0.0760286	+	0.131685i	0.901533	−	0.432710i	$-0.142443\pi$
$173$	1.00000	+	1.73205i	0.0760286	+	0.131685i	−0.825505	+	0.564396i	$0.809109\pi$
$174$		0			0
$175$	−0.500000	+	0.866025i	−0.0377964	+	0.0654654i
$176$	0.500000	−	0.866025i	0.0376889	−	0.0652791i
$177$		0			0
$178$	−3.00000	−	5.19615i	−0.224860	−	0.389468i
$179$	−24.0000			−1.79384			−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000			−1.79384			−0.896922	+	0.442189i	$0.854202\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$	−3.00000	−	5.19615i	−0.222375	−	0.385164i
$183$		0			0
$184$	2.00000	−	3.46410i	0.147442	−	0.255377i
$185$	2.00000	−	3.46410i	0.147043	−	0.254686i
$186$		0			0
$187$	2.50000	+	4.33013i	0.182818	+	0.316650i
$188$		0			0
$189$		0			0
$190$	14.0000			1.01567
$191$	6.00000	+	10.3923i	0.434145	+	0.751961i	0.997225	−	0.0744412i	$-0.0237173\pi$
$191$	6.00000	+	10.3923i	0.434145	+	0.751961i	−0.563081	+	0.826402i	$0.690384\pi$
$192$		0			0
$193$	−8.50000	+	14.7224i	−0.611843	+	1.05974i	0.379086	+	0.925361i	$0.376238\pi$
$193$	−8.50000	+	14.7224i	−0.611843	+	1.05974i	−0.990930	+	0.134382i	$0.957095\pi$
$194$	2.50000	−	4.33013i	0.179490	−	0.310885i
$195$		0			0
$196$	−0.500000	−	0.866025i	−0.0357143	−	0.0618590i
$197$	−10.0000			−0.712470			−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000			−0.712470			−0.356235	+	0.934396i	$0.615940\pi$
$198$		0			0
$199$	14.0000			0.992434			0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000			0.992434			0.496217	+	0.868199i	$0.334722\pi$
$200$	0.500000	+	0.866025i	0.0353553	+	0.0612372i
$201$		0			0
$202$	−2.00000	+	3.46410i	−0.140720	+	0.243733i
$203$	2.00000	−	3.46410i	0.140372	−	0.243132i
$204$		0			0
$205$	−3.00000	−	5.19615i	−0.209529	−	0.362915i
$206$	14.0000			0.975426
$207$		0			0
$208$	−6.00000			−0.416025
$209$	−3.50000	−	6.06218i	−0.242100	−	0.419330i
$210$		0			0
$211$	8.00000	−	13.8564i	0.550743	−	0.953914i	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	8.00000	−	13.8564i	0.550743	−	0.953914i	0.998221	−	0.0596196i	$-0.0189888\pi$
$212$	6.00000	−	10.3923i	0.412082	−	0.713746i
$213$		0			0
$214$	1.50000	+	2.59808i	0.102538	+	0.177601i
$215$	2.00000			0.136399
$216$		0			0
$217$	6.00000			0.407307
$218$	1.00000	+	1.73205i	0.0677285	+	0.117309i
$219$		0			0
$220$	−1.00000	+	1.73205i	−0.0674200	+	0.116775i
$221$	15.0000	−	25.9808i	1.00901	−	1.74766i
$222$		0			0
$223$	2.00000	+	3.46410i	0.133930	+	0.231973i	0.925188	−	0.379509i	$-0.123907\pi$
$223$	2.00000	+	3.46410i	0.133930	+	0.231973i	−0.791258	+	0.611482i	$0.790574\pi$
$224$	−1.00000			−0.0668153
$225$		0			0
$226$	10.0000			0.665190
$227$	1.50000	+	2.59808i	0.0995585	+	0.172440i	0.911502	−	0.411296i	$-0.134924\pi$
$227$	1.50000	+	2.59808i	0.0995585	+	0.172440i	−0.811943	+	0.583736i	$0.801590\pi$
$228$		0			0
$229$	13.0000	−	22.5167i	0.859064	−	1.48794i	−0.0137585	−	0.999905i	$-0.504380\pi$
$229$	13.0000	−	22.5167i	0.859064	−	1.48794i	0.872823	−	0.488037i	$-0.162287\pi$
$230$	−4.00000	+	6.92820i	−0.263752	+	0.456832i
$231$		0			0
$232$	−2.00000	−	3.46410i	−0.131306	−	0.227429i
$233$	29.0000			1.89985			0.949927	−	0.312473i	$-0.101157\pi$
$233$	29.0000			1.89985			0.949927	+	0.312473i	$0.101157\pi$
$234$		0			0
$235$		0			0
$236$	−3.50000	−	6.06218i	−0.227831	−	0.394614i
$237$		0			0
$238$	2.50000	−	4.33013i	0.162051	−	0.280680i
$239$	3.00000	−	5.19615i	0.194054	−	0.336111i	−0.752536	−	0.658551i	$-0.771170\pi$
$239$	3.00000	−	5.19615i	0.194054	−	0.336111i	0.946590	+	0.322440i	$0.104503\pi$
$240$		0			0
$241$	−11.5000	−	19.9186i	−0.740780	−	1.28307i	−0.952141	−	0.305661i	$-0.901123\pi$
$241$	−11.5000	−	19.9186i	−0.740780	−	1.28307i	0.211360	−	0.977408i	$-0.432211\pi$
$242$	−10.0000			−0.642824
$243$		0			0
$244$	−12.0000			−0.768221
$245$	1.00000	+	1.73205i	0.0638877	+	0.110657i
$246$		0			0
$247$	−21.0000	+	36.3731i	−1.33620	+	2.31436i
$248$	3.00000	−	5.19615i	0.190500	−	0.329956i
$249$		0			0
$250$	−6.00000	−	10.3923i	−0.379473	−	0.657267i
$251$	−3.00000			−0.189358			−0.0946792	−	0.995508i	$-0.530183\pi$
$251$	−3.00000			−0.189358			−0.0946792	+	0.995508i	$0.530183\pi$
$252$		0			0
$253$	4.00000			0.251478
$254$	6.00000	+	10.3923i	0.376473	+	0.652071i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.0312500	+	0.0541266i
$257$	−7.50000	+	12.9904i	−0.467837	+	0.810318i	−0.999325	−	0.0367485i	$-0.988300\pi$
$257$	−7.50000	+	12.9904i	−0.467837	+	0.810318i	0.531487	+	0.847066i	$0.321633\pi$
$258$		0			0
$259$	1.00000	+	1.73205i	0.0621370	+	0.107624i
$260$	12.0000			0.744208
$261$		0			0
$262$	4.00000			0.247121
$263$	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
$265$	−12.0000	+	20.7846i	−0.737154	+	1.27679i
$266$	−3.50000	+	6.06218i	−0.214599	+	0.371696i
$267$		0			0
$268$	−6.50000	−	11.2583i	−0.397051	−	0.687712i
$269$	−20.0000			−1.21942			−0.609711	−	0.792624i	$-0.708714\pi$
$269$	−20.0000			−1.21942			−0.609711	+	0.792624i	$0.708714\pi$
$270$		0			0
$271$	−6.00000			−0.364474			−0.182237	−	0.983255i	$-0.558334\pi$
$271$	−6.00000			−0.364474			−0.182237	+	0.983255i	$0.558334\pi$
$272$	−2.50000	−	4.33013i	−0.151585	−	0.262553i
$273$		0			0
$274$	−9.50000	+	16.4545i	−0.573916	+	0.994052i
$275$	−0.500000	+	0.866025i	−0.0301511	+	0.0522233i
$276$		0			0
$277$	−1.00000	−	1.73205i	−0.0600842	−	0.104069i	0.834419	−	0.551131i	$-0.185804\pi$
$277$	−1.00000	−	1.73205i	−0.0600842	−	0.104069i	−0.894503	+	0.447062i	$0.852470\pi$
$278$	−5.00000			−0.299880
$279$		0			0
$280$	2.00000			0.119523
$281$	11.0000	+	19.0526i	0.656205	+	1.13658i	0.981590	+	0.190999i	$0.0611727\pi$
$281$	11.0000	+	19.0526i	0.656205	+	1.13658i	−0.325385	+	0.945582i	$0.605494\pi$
$282$		0			0
$283$	−2.00000	+	3.46410i	−0.118888	+	0.205919i	−0.919327	−	0.393494i	$-0.871266\pi$
$283$	−2.00000	+	3.46410i	−0.118888	+	0.205919i	0.800439	+	0.599414i	$0.204600\pi$
$284$	−4.00000	+	6.92820i	−0.237356	+	0.411113i
$285$		0			0
$286$	−3.00000	−	5.19615i	−0.177394	−	0.307255i
$287$	3.00000			0.177084
$288$		0			0
$289$	8.00000			0.470588
$290$	4.00000	+	6.92820i	0.234888	+	0.406838i
$291$		0			0
$292$	−0.500000	+	0.866025i	−0.0292603	+	0.0506803i
$293$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$293$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$294$		0			0
$295$	7.00000	+	12.1244i	0.407556	+	0.705907i
$296$	2.00000			0.116248
$297$		0			0
$298$	24.0000			1.39028
$299$	−12.0000	−	20.7846i	−0.693978	−	1.20201i
$300$		0			0
$301$	−0.500000	+	0.866025i	−0.0288195	+	0.0499169i
$302$	−5.00000	+	8.66025i	−0.287718	+	0.498342i
$303$		0			0
$304$	3.50000	+	6.06218i	0.200739	+	0.347690i
$305$	24.0000			1.37424
$306$		0			0
$307$	7.00000			0.399511			0.199756	−	0.979846i	$-0.435985\pi$
$307$	7.00000			0.399511			0.199756	+	0.979846i	$0.435985\pi$
$308$	−0.500000	−	0.866025i	−0.0284901	−	0.0493464i
$309$		0			0
$310$	−6.00000	+	10.3923i	−0.340777	+	0.590243i
$311$	1.00000	−	1.73205i	0.0567048	−	0.0982156i	−0.836280	−	0.548303i	$-0.815274\pi$
$311$	1.00000	−	1.73205i	0.0567048	−	0.0982156i	0.892984	+	0.450088i	$0.148607\pi$
$312$		0			0
$313$	8.50000	+	14.7224i	0.480448	+	0.832161i	0.999748	−	0.0224310i	$-0.00714060\pi$
$313$	8.50000	+	14.7224i	0.480448	+	0.832161i	−0.519300	+	0.854592i	$0.673807\pi$
$314$	2.00000			0.112867
$315$		0			0
$316$	−6.00000			−0.337526
$317$	−3.00000	−	5.19615i	−0.168497	−	0.291845i	0.769395	−	0.638774i	$-0.220558\pi$
$317$	−3.00000	−	5.19615i	−0.168497	−	0.291845i	−0.937892	+	0.346929i	$0.887225\pi$
$318$		0			0
$319$	2.00000	−	3.46410i	0.111979	−	0.193952i
$320$	1.00000	−	1.73205i	0.0559017	−	0.0968246i
$321$		0			0
$322$	−2.00000	−	3.46410i	−0.111456	−	0.193047i
$323$	−35.0000			−1.94745
$324$		0			0
$325$	6.00000			0.332820
$326$	2.00000	+	3.46410i	0.110770	+	0.191859i
$327$		0			0
$328$	1.50000	−	2.59808i	0.0828236	−	0.143455i
$329$		0			0
$330$		0			0
$331$	−4.00000	−	6.92820i	−0.219860	−	0.380808i	0.734905	−	0.678170i	$-0.237227\pi$
$331$	−4.00000	−	6.92820i	−0.219860	−	0.380808i	−0.954765	+	0.297361i	$0.903893\pi$
$332$	−16.0000			−0.878114
$333$		0			0
$334$	−20.0000			−1.09435
$335$	13.0000	+	22.5167i	0.710266	+	1.23022i
$336$		0			0
$337$	4.50000	−	7.79423i	0.245131	−	0.424579i	−0.717038	−	0.697034i	$-0.754502\pi$
$337$	4.50000	−	7.79423i	0.245131	−	0.424579i	0.962168	+	0.272456i	$0.0878358\pi$
$338$	−11.5000	+	19.9186i	−0.625518	+	1.08343i
$339$		0			0
$340$	5.00000	+	8.66025i	0.271163	+	0.469668i
$341$	6.00000			0.324918
$342$		0			0
$343$	−1.00000			−0.0539949
$344$	0.500000	+	0.866025i	0.0269582	+	0.0466930i
$345$		0			0
$346$	1.00000	−	1.73205i	0.0537603	−	0.0931156i
$347$	−1.50000	+	2.59808i	−0.0805242	+	0.139472i	−0.903475	−	0.428640i	$-0.858993\pi$
$347$	−1.50000	+	2.59808i	−0.0805242	+	0.139472i	0.822951	+	0.568112i	$0.192326\pi$
$348$		0			0
$349$	7.00000	+	12.1244i	0.374701	+	0.649002i	0.990282	−	0.139072i	$-0.0444119\pi$
$349$	7.00000	+	12.1244i	0.374701	+	0.649002i	−0.615581	+	0.788074i	$0.711079\pi$
$350$	1.00000			0.0534522
$351$		0			0
$352$	−1.00000			−0.0533002
$353$	−7.50000	−	12.9904i	−0.399185	−	0.691408i	0.594441	−	0.804139i	$-0.297373\pi$
$353$	−7.50000	−	12.9904i	−0.399185	−	0.691408i	−0.993626	+	0.112731i	$0.964040\pi$
$354$		0			0
$355$	8.00000	−	13.8564i	0.424596	−	0.735422i
$356$	−3.00000	+	5.19615i	−0.159000	+	0.275396i
$357$		0			0
$358$	12.0000	+	20.7846i	0.634220	+	1.09850i
$359$	−2.00000			−0.105556			−0.0527780	−	0.998606i	$-0.516808\pi$
$359$	−2.00000			−0.105556			−0.0527780	+	0.998606i	$0.516808\pi$
$360$		0			0
$361$	30.0000			1.57895
$362$		0			0
$363$		0			0
$364$	−3.00000	+	5.19615i	−0.157243	+	0.272352i
$365$	1.00000	−	1.73205i	0.0523424	−	0.0906597i
$366$		0			0
$367$	11.0000	+	19.0526i	0.574195	+	0.994535i	0.996129	+	0.0879086i	$0.0280183\pi$
$367$	11.0000	+	19.0526i	0.574195	+	0.994535i	−0.421933	+	0.906627i	$0.638648\pi$
$368$	−4.00000			−0.208514
$369$		0			0
$370$	−4.00000			−0.207950
$371$	−6.00000	−	10.3923i	−0.311504	−	0.539542i
$372$		0			0
$373$	−11.0000	+	19.0526i	−0.569558	+	0.986504i	0.427051	+	0.904227i	$0.359552\pi$
$373$	−11.0000	+	19.0526i	−0.569558	+	0.986504i	−0.996610	+	0.0822766i	$0.973781\pi$
$374$	2.50000	−	4.33013i	0.129272	−	0.223906i
$375$		0			0
$376$		0			0
$377$	−24.0000			−1.23606
$378$		0			0
$379$	−17.0000			−0.873231			−0.436616	−	0.899648i	$-0.643823\pi$
$379$	−17.0000			−0.873231			−0.436616	+	0.899648i	$0.643823\pi$
$380$	−7.00000	−	12.1244i	−0.359092	−	0.621966i
$381$		0			0
$382$	6.00000	−	10.3923i	0.306987	−	0.531717i
$383$	2.00000	−	3.46410i	0.102195	−	0.177007i	−0.810394	−	0.585886i	$-0.800747\pi$
$383$	2.00000	−	3.46410i	0.102195	−	0.177007i	0.912589	+	0.408879i	$0.134080\pi$
$384$		0			0
$385$	1.00000	+	1.73205i	0.0509647	+	0.0882735i
$386$	17.0000			0.865277
$387$		0			0
$388$	−5.00000			−0.253837
$389$	4.00000	+	6.92820i	0.202808	+	0.351274i	0.949432	−	0.313972i	$-0.101660\pi$
$389$	4.00000	+	6.92820i	0.202808	+	0.351274i	−0.746624	+	0.665246i	$0.768327\pi$
$390$		0			0
$391$	10.0000	−	17.3205i	0.505722	−	0.875936i
$392$	−0.500000	+	0.866025i	−0.0252538	+	0.0437409i
$393$		0			0
$394$	5.00000	+	8.66025i	0.251896	+	0.436297i
$395$	12.0000			0.603786
$396$		0			0
$397$	18.0000			0.903394			0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000			0.903394			0.451697	+	0.892171i	$0.350819\pi$
$398$	−7.00000	−	12.1244i	−0.350878	−	0.607739i
$399$		0			0
$400$	0.500000	−	0.866025i	0.0250000	−	0.0433013i
$401$	4.50000	−	7.79423i	0.224719	−	0.389225i	−0.731516	−	0.681824i	$-0.761187\pi$
$401$	4.50000	−	7.79423i	0.224719	−	0.389225i	0.956235	+	0.292599i	$0.0945202\pi$
$402$		0			0
$403$	−18.0000	−	31.1769i	−0.896644	−	1.55303i
$404$	4.00000			0.199007
$405$		0			0
$406$	−4.00000			−0.198517
$407$	1.00000	+	1.73205i	0.0495682	+	0.0858546i
$408$		0			0
$409$	−5.50000	+	9.52628i	−0.271957	+	0.471044i	−0.969363	−	0.245633i	$-0.921004\pi$
$409$	−5.50000	+	9.52628i	−0.271957	+	0.471044i	0.697406	+	0.716677i	$0.254338\pi$
$410$	−3.00000	+	5.19615i	−0.148159	+	0.256620i
$411$		0			0
$412$	−7.00000	−	12.1244i	−0.344865	−	0.597324i
$413$	−7.00000			−0.344447
$414$		0			0
$415$	32.0000			1.57082
$416$	3.00000	+	5.19615i	0.147087	+	0.254762i
$417$		0			0
$418$	−3.50000	+	6.06218i	−0.171191	+	0.296511i
$419$	−6.00000	+	10.3923i	−0.293119	+	0.507697i	−0.974546	−	0.224189i	$-0.928027\pi$
$419$	−6.00000	+	10.3923i	−0.293119	+	0.507697i	0.681426	+	0.731887i	$0.261360\pi$
$420$		0			0
$421$	6.00000	+	10.3923i	0.292422	+	0.506490i	0.974382	−	0.224900i	$-0.0722054\pi$
$421$	6.00000	+	10.3923i	0.292422	+	0.506490i	−0.681960	+	0.731390i	$0.738872\pi$
$422$	−16.0000			−0.778868
$423$		0			0
$424$	−12.0000			−0.582772
$425$	2.50000	+	4.33013i	0.121268	+	0.210042i
$426$		0			0
$427$	−6.00000	+	10.3923i	−0.290360	+	0.502919i
$428$	1.50000	−	2.59808i	0.0725052	−	0.125583i
$429$		0			0
$430$	−1.00000	−	1.73205i	−0.0482243	−	0.0835269i
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	−25.0000			−1.20142			−0.600712	−	0.799466i	$-0.705116\pi$
$433$	−25.0000			−1.20142			−0.600712	+	0.799466i	$0.705116\pi$
$434$	−3.00000	−	5.19615i	−0.144005	−	0.249423i
$435$		0			0
$436$	1.00000	−	1.73205i	0.0478913	−	0.0829502i
$437$	−14.0000	+	24.2487i	−0.669711	+	1.15997i
$438$		0			0
$439$	12.0000	+	20.7846i	0.572729	+	0.991995i	0.996284	+	0.0861252i	$0.0274485\pi$
$439$	12.0000	+	20.7846i	0.572729	+	0.991995i	−0.423556	+	0.905870i	$0.639218\pi$
$440$	2.00000			0.0953463
$441$		0			0
$442$	−30.0000			−1.42695
$443$	3.50000	+	6.06218i	0.166290	+	0.288023i	0.937113	−	0.349027i	$-0.113488\pi$
$443$	3.50000	+	6.06218i	0.166290	+	0.288023i	−0.770823	+	0.637050i	$0.780155\pi$
$444$		0			0
$445$	6.00000	−	10.3923i	0.284427	−	0.492642i
$446$	2.00000	−	3.46410i	0.0947027	−	0.164030i
$447$		0			0
$448$	0.500000	+	0.866025i	0.0236228	+	0.0409159i
$449$	−17.0000			−0.802280			−0.401140	−	0.916017i	$-0.631386\pi$
$449$	−17.0000			−0.802280			−0.401140	+	0.916017i	$0.631386\pi$
$450$		0			0
$451$	3.00000			0.141264
$452$	−5.00000	−	8.66025i	−0.235180	−	0.407344i
$453$		0			0
$454$	1.50000	−	2.59808i	0.0703985	−	0.121934i
$455$	6.00000	−	10.3923i	0.281284	−	0.487199i
$456$		0			0
$457$	−0.500000	−	0.866025i	−0.0233890	−	0.0405110i	0.854094	−	0.520119i	$-0.174112\pi$
$457$	−0.500000	−	0.866025i	−0.0233890	−	0.0405110i	−0.877483	+	0.479608i	$0.840779\pi$
$458$	−26.0000			−1.21490
$459$		0			0
$460$	8.00000			0.373002
$461$	7.00000	+	12.1244i	0.326023	+	0.564688i	0.981719	−	0.190337i	$-0.0609581\pi$
$461$	7.00000	+	12.1244i	0.326023	+	0.564688i	−0.655696	+	0.755025i	$0.727625\pi$
$462$		0			0
$463$	4.00000	−	6.92820i	0.185896	−	0.321981i	−0.757982	−	0.652275i	$-0.773815\pi$
$463$	4.00000	−	6.92820i	0.185896	−	0.321981i	0.943878	+	0.330294i	$0.107148\pi$
$464$	−2.00000	+	3.46410i	−0.0928477	+	0.160817i
$465$		0			0
$466$	−14.5000	−	25.1147i	−0.671700	−	1.16342i
$467$	13.0000			0.601568			0.300784	−	0.953692i	$-0.402752\pi$
$467$	13.0000			0.601568			0.300784	+	0.953692i	$0.402752\pi$
$468$		0			0
$469$	−13.0000			−0.600284
$470$		0			0
$471$		0			0
$472$	−3.50000	+	6.06218i	−0.161101	+	0.279034i
$473$	−0.500000	+	0.866025i	−0.0229900	+	0.0398199i
$474$		0			0
$475$	−3.50000	−	6.06218i	−0.160591	−	0.278152i
$476$	−5.00000			−0.229175
$477$		0			0
$478$	−6.00000			−0.274434
$479$	−10.0000	−	17.3205i	−0.456912	−	0.791394i	0.541884	−	0.840453i	$-0.317711\pi$
$479$	−10.0000	−	17.3205i	−0.456912	−	0.791394i	−0.998796	+	0.0490589i	$0.984378\pi$
$480$		0			0
$481$	6.00000	−	10.3923i	0.273576	−	0.473848i
$482$	−11.5000	+	19.9186i	−0.523811	+	0.907267i
$483$		0			0
$484$	5.00000	+	8.66025i	0.227273	+	0.393648i
$485$	10.0000			0.454077
$486$		0			0
$487$	−10.0000			−0.453143			−0.226572	−	0.973995i	$-0.572752\pi$
$487$	−10.0000			−0.453143			−0.226572	+	0.973995i	$0.572752\pi$
$488$	6.00000	+	10.3923i	0.271607	+	0.470438i
$489$		0			0
$490$	1.00000	−	1.73205i	0.0451754	−	0.0782461i
$491$	16.5000	−	28.5788i	0.744635	−	1.28974i	−0.205731	−	0.978609i	$-0.565957\pi$
$491$	16.5000	−	28.5788i	0.744635	−	1.28974i	0.950365	−	0.311136i	$-0.100710\pi$
$492$		0			0
$493$	−10.0000	−	17.3205i	−0.450377	−	0.780076i
$494$	42.0000			1.88967
$495$		0			0
$496$	−6.00000			−0.269408
$497$	4.00000	+	6.92820i	0.179425	+	0.310772i
$498$		0			0
$499$	14.5000	−	25.1147i	0.649109	−	1.12429i	−0.334227	−	0.942493i	$-0.608475\pi$
$499$	14.5000	−	25.1147i	0.649109	−	1.12429i	0.983336	−	0.181797i	$-0.0581915\pi$
$500$	−6.00000	+	10.3923i	−0.268328	+	0.464758i
$501$		0			0
$502$	1.50000	+	2.59808i	0.0669483	+	0.115958i
$503$	−6.00000			−0.267527			−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000			−0.267527			−0.133763	+	0.991013i	$0.542706\pi$
$504$		0			0
$505$	−8.00000			−0.355995
$506$	−2.00000	−	3.46410i	−0.0889108	−	0.153998i
$507$		0			0
$508$	6.00000	−	10.3923i	0.266207	−	0.461084i
$509$	15.0000	−	25.9808i	0.664863	−	1.15158i	−0.314459	−	0.949271i	$-0.601823\pi$
$509$	15.0000	−	25.9808i	0.664863	−	1.15158i	0.979322	−	0.202306i	$-0.0648436\pi$
$510$		0			0
$511$	0.500000	+	0.866025i	0.0221187	+	0.0383107i
$512$	1.00000			0.0441942
$513$		0			0
$514$	15.0000			0.661622
$515$	14.0000	+	24.2487i	0.616914	+	1.06853i
$516$		0			0
$517$		0			0
$518$	1.00000	−	1.73205i	0.0439375	−	0.0761019i
$519$		0			0
$520$	−6.00000	−	10.3923i	−0.263117	−	0.455733i
$521$	9.00000			0.394297			0.197149	−	0.980374i	$-0.436832\pi$
$521$	9.00000			0.394297			0.197149	+	0.980374i	$0.436832\pi$
$522$		0			0
$523$	−28.0000			−1.22435			−0.612177	−	0.790721i	$-0.709706\pi$
$523$	−28.0000			−1.22435			−0.612177	+	0.790721i	$0.709706\pi$
$524$	−2.00000	−	3.46410i	−0.0873704	−	0.151330i
$525$		0			0
$526$	9.00000	−	15.5885i	0.392419	−	0.679689i
$527$	15.0000	−	25.9808i	0.653410	−	1.13174i
$528$		0			0
$529$	3.50000	+	6.06218i	0.152174	+	0.263573i
$530$	24.0000			1.04249
$531$		0			0
$532$	7.00000			0.303488
$533$	−9.00000	−	15.5885i	−0.389833	−	0.675211i
$534$		0			0
$535$	−3.00000	+	5.19615i	−0.129701	+	0.224649i
$536$	−6.50000	+	11.2583i	−0.280757	+	0.486286i
$537$		0			0
$538$	10.0000	+	17.3205i	0.431131	+	0.746740i
$539$	−1.00000			−0.0430730
$540$		0			0
$541$	−24.0000			−1.03184			−0.515920	−	0.856637i	$-0.672550\pi$
$541$	−24.0000			−1.03184			−0.515920	+	0.856637i	$0.672550\pi$
$542$	3.00000	+	5.19615i	0.128861	+	0.223194i
$543$		0			0
$544$	−2.50000	+	4.33013i	−0.107187	+	0.185653i
$545$	−2.00000	+	3.46410i	−0.0856706	+	0.148386i
$546$		0			0
$547$	10.5000	+	18.1865i	0.448948	+	0.777600i	0.998318	−	0.0579790i	$-0.0184657\pi$
$547$	10.5000	+	18.1865i	0.448948	+	0.777600i	−0.549370	+	0.835579i	$0.685132\pi$
$548$	19.0000			0.811640
$549$		0			0
$550$	1.00000			0.0426401
$551$	14.0000	+	24.2487i	0.596420	+	1.03303i
$552$		0			0
$553$	−3.00000	+	5.19615i	−0.127573	+	0.220963i
$554$	−1.00000	+	1.73205i	−0.0424859	+	0.0735878i
$555$		0			0
$556$	2.50000	+	4.33013i	0.106024	+	0.183638i
$557$	28.0000			1.18640			0.593199	−	0.805056i	$-0.297865\pi$
$557$	28.0000			1.18640			0.593199	+	0.805056i	$0.297865\pi$
$558$		0			0
$559$	6.00000			0.253773
$560$	−1.00000	−	1.73205i	−0.0422577	−	0.0731925i
$561$		0			0
$562$	11.0000	−	19.0526i	0.464007	−	0.803684i
$563$	15.5000	−	26.8468i	0.653247	−	1.13146i	−0.329083	−	0.944301i	$-0.606740\pi$
$563$	15.5000	−	26.8468i	0.653247	−	1.13146i	0.982330	−	0.187156i	$-0.0599271\pi$
$564$		0			0
$565$	10.0000	+	17.3205i	0.420703	+	0.728679i
$566$	4.00000			0.168133
$567$		0			0
$568$	8.00000			0.335673
$569$	−7.50000	−	12.9904i	−0.314416	−	0.544585i	0.664897	−	0.746935i	$-0.268475\pi$
$569$	−7.50000	−	12.9904i	−0.314416	−	0.544585i	−0.979313	+	0.202350i	$0.935142\pi$
$570$		0			0
$571$	16.5000	−	28.5788i	0.690504	−	1.19599i	−0.281170	−	0.959658i	$-0.590722\pi$
$571$	16.5000	−	28.5788i	0.690504	−	1.19599i	0.971673	−	0.236329i	$-0.0759443\pi$
$572$	−3.00000	+	5.19615i	−0.125436	+	0.217262i
$573$		0			0
$574$	−1.50000	−	2.59808i	−0.0626088	−	0.108442i
$575$	4.00000			0.166812
$576$		0			0
$577$	−35.0000			−1.45707			−0.728535	−	0.685009i	$-0.759798\pi$
$577$	−35.0000			−1.45707			−0.728535	+	0.685009i	$0.759798\pi$
$578$	−4.00000	−	6.92820i	−0.166378	−	0.288175i
$579$		0			0
$580$	4.00000	−	6.92820i	0.166091	−	0.287678i
$581$	−8.00000	+	13.8564i	−0.331896	+	0.574861i
$582$		0			0
$583$	−6.00000	−	10.3923i	−0.248495	−	0.430405i
$584$	1.00000			0.0413803
$585$		0			0
$586$		0			0
$587$	−23.5000	−	40.7032i	−0.969949	−	1.68000i	−0.695686	−	0.718346i	$-0.744900\pi$
$587$	−23.5000	−	40.7032i	−0.969949	−	1.68000i	−0.274263	−	0.961655i	$-0.588434\pi$
$588$		0			0
$589$	−21.0000	+	36.3731i	−0.865290	+	1.49873i
$590$	7.00000	−	12.1244i	0.288185	−	0.499152i
$591$		0			0
$592$	−1.00000	−	1.73205i	−0.0410997	−	0.0711868i
$593$	6.00000			0.246390			0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000			0.246390			0.123195	+	0.992382i	$0.460686\pi$
$594$		0			0
$595$	10.0000			0.409960
$596$	−12.0000	−	20.7846i	−0.491539	−	0.851371i
$597$		0			0
$598$	−12.0000	+	20.7846i	−0.490716	+	0.849946i
$599$	12.0000	−	20.7846i	0.490307	−	0.849236i	−0.509631	−	0.860393i	$-0.670218\pi$
$599$	12.0000	−	20.7846i	0.490307	−	0.849236i	0.999938	+	0.0111569i	$0.00355143\pi$
$600$		0			0
$601$	9.50000	+	16.4545i	0.387513	+	0.671192i	0.992114	−	0.125336i	$-0.0400009\pi$
$601$	9.50000	+	16.4545i	0.387513	+	0.671192i	−0.604601	+	0.796528i	$0.706668\pi$
$602$	1.00000			0.0407570
$603$		0			0
$604$	10.0000			0.406894
$605$	−10.0000	−	17.3205i	−0.406558	−	0.704179i
$606$		0			0
$607$	12.0000	−	20.7846i	0.487065	−	0.843621i	−0.512824	−	0.858494i	$-0.671401\pi$
$607$	12.0000	−	20.7846i	0.487065	−	0.843621i	0.999889	+	0.0148722i	$0.00473415\pi$
$608$	3.50000	−	6.06218i	0.141944	−	0.245854i
$609$		0			0
$610$	−12.0000	−	20.7846i	−0.485866	−	0.841544i
$611$		0			0
$612$		0			0
$613$	42.0000			1.69636			0.848182	−	0.529705i	$-0.177697\pi$
$613$	42.0000			1.69636			0.848182	+	0.529705i	$0.177697\pi$
$614$	−3.50000	−	6.06218i	−0.141249	−	0.244650i
$615$		0			0
$616$	−0.500000	+	0.866025i	−0.0201456	+	0.0348932i
$617$	−8.50000	+	14.7224i	−0.342197	+	0.592703i	−0.984840	−	0.173463i	$-0.944504\pi$
$617$	−8.50000	+	14.7224i	−0.342197	+	0.592703i	0.642643	+	0.766165i	$0.277838\pi$
$618$		0			0
$619$	−18.5000	−	32.0429i	−0.743578	−	1.28791i	−0.950856	−	0.309633i	$-0.899794\pi$
$619$	−18.5000	−	32.0429i	−0.743578	−	1.28791i	0.207279	−	0.978282i	$-0.433539\pi$
$620$	12.0000			0.481932
$621$		0			0
$622$	−2.00000			−0.0801927
$623$	3.00000	+	5.19615i	0.120192	+	0.208179i
$624$		0			0
$625$	9.50000	−	16.4545i	0.380000	−	0.658179i
$626$	8.50000	−	14.7224i	0.339728	−	0.588427i
$627$		0			0
$628$	−1.00000	−	1.73205i	−0.0399043	−	0.0691164i
$629$	10.0000			0.398726
$630$		0			0
$631$	8.00000			0.318475			0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000			0.318475			0.159237	+	0.987240i	$0.449096\pi$
$632$	3.00000	+	5.19615i	0.119334	+	0.206692i
$633$		0			0
$634$	−3.00000	+	5.19615i	−0.119145	+	0.206366i
$635$	−12.0000	+	20.7846i	−0.476205	+	0.824812i
$636$		0			0
$637$	3.00000	+	5.19615i	0.118864	+	0.205879i
$638$	−4.00000			−0.158362
$639$		0			0
$640$	−2.00000			−0.0790569
$641$	0.500000	+	0.866025i	0.0197488	+	0.0342059i	0.875731	−	0.482800i	$-0.160380\pi$
$641$	0.500000	+	0.866025i	0.0197488	+	0.0342059i	−0.855982	+	0.517005i	$0.827047\pi$
$642$		0			0
$643$	3.50000	−	6.06218i	0.138027	−	0.239069i	−0.788723	−	0.614749i	$-0.789257\pi$
$643$	3.50000	−	6.06218i	0.138027	−	0.239069i	0.926750	+	0.375680i	$0.122591\pi$
$644$	−2.00000	+	3.46410i	−0.0788110	+	0.136505i
$645$		0			0
$646$	17.5000	+	30.3109i	0.688528	+	1.19257i
$647$	−12.0000			−0.471769			−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000			−0.471769			−0.235884	+	0.971781i	$0.575799\pi$
$648$		0			0
$649$	−7.00000			−0.274774
$650$	−3.00000	−	5.19615i	−0.117670	−	0.203810i
$651$		0			0
$652$	2.00000	−	3.46410i	0.0783260	−	0.135665i
$653$	−3.00000	+	5.19615i	−0.117399	+	0.203341i	−0.918736	−	0.394872i	$-0.870789\pi$
$653$	−3.00000	+	5.19615i	−0.117399	+	0.203341i	0.801337	+	0.598213i	$0.204122\pi$
$654$		0			0
$655$	4.00000	+	6.92820i	0.156293	+	0.270707i
$656$	−3.00000			−0.117130
$657$		0			0
$658$		0			0
$659$	8.00000	+	13.8564i	0.311636	+	0.539769i	0.978717	−	0.205216i	$-0.0657898\pi$
$659$	8.00000	+	13.8564i	0.311636	+	0.539769i	−0.667081	+	0.744985i	$0.732456\pi$
$660$		0			0
$661$	−14.0000	+	24.2487i	−0.544537	+	0.943166i	0.454099	+	0.890951i	$0.349961\pi$
$661$	−14.0000	+	24.2487i	−0.544537	+	0.943166i	−0.998636	+	0.0522143i	$0.983372\pi$
$662$	−4.00000	+	6.92820i	−0.155464	+	0.269272i
$663$		0			0
$664$	8.00000	+	13.8564i	0.310460	+	0.537733i
$665$	−14.0000			−0.542897
$666$		0			0
$667$	−16.0000			−0.619522
$668$	10.0000	+	17.3205i	0.386912	+	0.670151i
$669$		0			0
$670$	13.0000	−	22.5167i	0.502234	−	0.869894i
$671$	−6.00000	+	10.3923i	−0.231627	+	0.401190i
$672$		0			0
$673$	−7.00000	−	12.1244i	−0.269830	−	0.467360i	0.698988	−	0.715134i	$-0.253634\pi$
$673$	−7.00000	−	12.1244i	−0.269830	−	0.467360i	−0.968818	+	0.247774i	$0.920301\pi$
$674$	−9.00000			−0.346667
$675$		0			0
$676$	23.0000			0.884615
$677$	15.0000	+	25.9808i	0.576497	+	0.998522i	0.995877	+	0.0907112i	$0.0289140\pi$
$677$	15.0000	+	25.9808i	0.576497	+	0.998522i	−0.419380	+	0.907811i	$0.637753\pi$
$678$		0			0
$679$	−2.50000	+	4.33013i	−0.0959412	+	0.166175i
$680$	5.00000	−	8.66025i	0.191741	−	0.332106i
$681$		0			0
$682$	−3.00000	−	5.19615i	−0.114876	−	0.198971i
$683$	−39.0000			−1.49229			−0.746147	−	0.665782i	$-0.768098\pi$
$683$	−39.0000			−1.49229			−0.746147	+	0.665782i	$0.768098\pi$
$684$		0			0
$685$	−38.0000			−1.45191
$686$	0.500000	+	0.866025i	0.0190901	+	0.0330650i
$687$		0			0
$688$	0.500000	−	0.866025i	0.0190623	−	0.0330169i
$689$	−36.0000	+	62.3538i	−1.37149	+	2.37549i
$690$		0			0
$691$	−16.0000	−	27.7128i	−0.608669	−	1.05425i	−0.991460	−	0.130410i	$-0.958371\pi$
$691$	−16.0000	−	27.7128i	−0.608669	−	1.05425i	0.382791	−	0.923835i	$-0.374963\pi$
$692$	−2.00000			−0.0760286
$693$		0			0
$694$	3.00000			0.113878
$695$	−5.00000	−	8.66025i	−0.189661	−	0.328502i
$696$		0			0
$697$	7.50000	−	12.9904i	0.284083	−	0.492046i
$698$	7.00000	−	12.1244i	0.264954	−	0.458914i
$699$		0			0
$700$	−0.500000	−	0.866025i	−0.0188982	−	0.0327327i
$701$	−8.00000			−0.302156			−0.151078	−	0.988522i	$-0.548274\pi$
$701$	−8.00000			−0.302156			−0.151078	+	0.988522i	$0.548274\pi$
$702$		0			0
$703$	−14.0000			−0.528020
$704$	0.500000	+	0.866025i	0.0188445	+	0.0326396i
$705$		0			0
$706$	−7.50000	+	12.9904i	−0.282266	+	0.488899i
$707$	2.00000	−	3.46410i	0.0752177	−	0.130281i
$708$		0			0
$709$	2.00000	+	3.46410i	0.0751116	+	0.130097i	0.901135	−	0.433539i	$-0.142735\pi$
$709$	2.00000	+	3.46410i	0.0751116	+	0.130097i	−0.826023	+	0.563636i	$0.809402\pi$
$710$	−16.0000			−0.600469
$711$		0			0
$712$	6.00000			0.224860
$713$	−12.0000	−	20.7846i	−0.449404	−	0.778390i
$714$		0			0
$715$	6.00000	−	10.3923i	0.224387	−	0.388650i
$716$	12.0000	−	20.7846i	0.448461	−	0.776757i
$717$		0			0
$718$	1.00000	+	1.73205i	0.0373197	+	0.0646396i
$719$	−6.00000			−0.223762			−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000			−0.223762			−0.111881	+	0.993722i	$0.535688\pi$
$720$		0			0
$721$	−14.0000			−0.521387
$722$	−15.0000	−	25.9808i	−0.558242	−	0.966904i
$723$		0			0
$724$		0			0
$725$	2.00000	−	3.46410i	0.0742781	−	0.128654i
$726$		0			0
$727$	7.00000	+	12.1244i	0.259616	+	0.449667i	0.966139	−	0.258022i	$-0.0830708\pi$
$727$	7.00000	+	12.1244i	0.259616	+	0.449667i	−0.706523	+	0.707690i	$0.749737\pi$
$728$	6.00000			0.222375
$729$		0			0
$730$	−2.00000			−0.0740233
$731$	2.50000	+	4.33013i	0.0924658	+	0.160156i
$732$		0			0
$733$	9.00000	−	15.5885i	0.332423	−	0.575773i	−0.650564	−	0.759452i	$-0.725467\pi$
$733$	9.00000	−	15.5885i	0.332423	−	0.575773i	0.982986	+	0.183679i	$0.0588007\pi$
$734$	11.0000	−	19.0526i	0.406017	−	0.703243i
$735$		0			0
$736$	2.00000	+	3.46410i	0.0737210	+	0.127688i
$737$	−13.0000			−0.478861
$738$		0			0
$739$	−33.0000			−1.21392			−0.606962	−	0.794731i	$-0.707612\pi$
$739$	−33.0000			−1.21392			−0.606962	+	0.794731i	$0.707612\pi$
$740$	2.00000	+	3.46410i	0.0735215	+	0.127343i
$741$		0			0
$742$	−6.00000	+	10.3923i	−0.220267	+	0.381514i
$743$	−3.00000	+	5.19615i	−0.110059	+	0.190628i	−0.915794	−	0.401648i	$-0.868437\pi$
$743$	−3.00000	+	5.19615i	−0.110059	+	0.190628i	0.805735	+	0.592277i	$0.201771\pi$
$744$		0			0
$745$	24.0000	+	41.5692i	0.879292	+	1.52298i
$746$	22.0000			0.805477
$747$		0			0
$748$	−5.00000			−0.182818
$749$	−1.50000	−	2.59808i	−0.0548088	−	0.0949316i
$750$		0			0
$751$	9.00000	−	15.5885i	0.328415	−	0.568831i	−0.653783	−	0.756682i	$-0.726819\pi$
$751$	9.00000	−	15.5885i	0.328415	−	0.568831i	0.982197	+	0.187851i	$0.0601523\pi$
$752$		0			0
$753$		0			0
$754$	12.0000	+	20.7846i	0.437014	+	0.756931i
$755$	−20.0000			−0.727875
$756$		0			0
$757$	−48.0000			−1.74459			−0.872295	−	0.488980i	$-0.837369\pi$
$757$	−48.0000			−1.74459			−0.872295	+	0.488980i	$0.837369\pi$
$758$	8.50000	+	14.7224i	0.308734	+	0.534743i
$759$		0			0
$760$	−7.00000	+	12.1244i	−0.253917	+	0.439797i
$761$	5.00000	−	8.66025i	0.181250	−	0.313934i	−0.761057	−	0.648686i	$-0.775319\pi$
$761$	5.00000	−	8.66025i	0.181250	−	0.313934i	0.942306	+	0.334752i	$0.108652\pi$
$762$		0			0
$763$	−1.00000	−	1.73205i	−0.0362024	−	0.0627044i
$764$	−12.0000			−0.434145
$765$		0			0
$766$	−4.00000			−0.144526
$767$	21.0000	+	36.3731i	0.758266	+	1.31336i
$768$		0			0
$769$	−11.0000	+	19.0526i	−0.396670	+	0.687053i	−0.993313	−	0.115454i	$-0.963168\pi$
$769$	−11.0000	+	19.0526i	−0.396670	+	0.687053i	0.596643	+	0.802507i	$0.296501\pi$
$770$	1.00000	−	1.73205i	0.0360375	−	0.0624188i
$771$		0			0
$772$	−8.50000	−	14.7224i	−0.305922	−	0.529872i
$773$	52.0000			1.87031			0.935155	−	0.354239i	$-0.115260\pi$
$773$	52.0000			1.87031			0.935155	+	0.354239i	$0.115260\pi$
$774$		0			0
$775$	6.00000			0.215526
$776$	2.50000	+	4.33013i	0.0897448	+	0.155443i
$777$		0			0
$778$	4.00000	−	6.92820i	0.143407	−	0.248388i
$779$	−10.5000	+	18.1865i	−0.376202	+	0.651600i
$780$		0			0
$781$	4.00000	+	6.92820i	0.143131	+	0.247911i
$782$	−20.0000			−0.715199
$783$		0			0
$784$	1.00000			0.0357143
$785$	2.00000	+	3.46410i	0.0713831	+	0.123639i
$786$		0			0
$787$	6.00000	−	10.3923i	0.213877	−	0.370446i	−0.739048	−	0.673653i	$-0.764724\pi$
$787$	6.00000	−	10.3923i	0.213877	−	0.370446i	0.952925	+	0.303207i	$0.0980575\pi$
$788$	5.00000	−	8.66025i	0.178118	−	0.308509i
$789$		0			0
$790$	−6.00000	−	10.3923i	−0.213470	−	0.369742i
$791$	−10.0000			−0.355559
$792$		0			0
$793$	72.0000			2.55679
$794$	−9.00000	−	15.5885i	−0.319398	−	0.553214i
$795$		0			0
$796$	−7.00000	+	12.1244i	−0.248108	+	0.429736i
$797$	−6.00000	+	10.3923i	−0.212531	+	0.368114i	−0.952506	−	0.304520i	$-0.901504\pi$
$797$	−6.00000	+	10.3923i	−0.212531	+	0.368114i	0.739975	+	0.672634i	$0.234837\pi$
$798$		0			0
$799$		0			0
$800$	−1.00000			−0.0353553
$801$		0			0
$802$	−9.00000			−0.317801
$803$	0.500000	+	0.866025i	0.0176446	+	0.0305614i
$804$		0			0
$805$	4.00000	−	6.92820i	0.140981	−	0.244187i
$806$	−18.0000	+	31.1769i	−0.634023	+	1.09816i
$807$		0			0
$808$	−2.00000	−	3.46410i	−0.0703598	−	0.121867i
$809$	43.0000			1.51180			0.755900	−	0.654687i	$-0.227200\pi$
$809$	43.0000			1.51180			0.755900	+	0.654687i	$0.227200\pi$
$810$		0			0
$811$	31.0000			1.08856			0.544279	−	0.838905i	$-0.316803\pi$
$811$	31.0000			1.08856			0.544279	+	0.838905i	$0.316803\pi$
$812$	2.00000	+	3.46410i	0.0701862	+	0.121566i
$813$		0			0
$814$	1.00000	−	1.73205i	0.0350500	−	0.0607083i
$815$	−4.00000	+	6.92820i	−0.140114	+	0.242684i
$816$		0			0
$817$	−3.50000	−	6.06218i	−0.122449	−	0.212089i
$818$	11.0000			0.384606
$819$		0			0
$820$	6.00000			0.209529
$821$	23.0000	+	39.8372i	0.802706	+	1.39033i	0.917829	+	0.396976i	$0.129940\pi$
$821$	23.0000	+	39.8372i	0.802706	+	1.39033i	−0.115124	+	0.993351i	$0.536726\pi$
$822$		0			0
$823$	−17.0000	+	29.4449i	−0.592583	+	1.02638i	0.401300	+	0.915947i	$0.368558\pi$
$823$	−17.0000	+	29.4449i	−0.592583	+	1.02638i	−0.993883	+	0.110437i	$0.964775\pi$
$824$	−7.00000	+	12.1244i	−0.243857	+	0.422372i
$825$		0			0
$826$	3.50000	+	6.06218i	0.121781	+	0.210930i
$827$	−12.0000			−0.417281			−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000			−0.417281			−0.208640	+	0.977992i	$0.566904\pi$
$828$		0			0
$829$	4.00000			0.138926			0.0694629	−	0.997585i	$-0.477871\pi$
$829$	4.00000			0.138926			0.0694629	+	0.997585i	$0.477871\pi$
$830$	−16.0000	−	27.7128i	−0.555368	−	0.961926i
$831$		0			0
$832$	3.00000	−	5.19615i	0.104006	−	0.180144i
$833$	−2.50000	+	4.33013i	−0.0866199	+	0.150030i
$834$		0			0
$835$	−20.0000	−	34.6410i	−0.692129	−	1.19880i
$836$	7.00000			0.242100
$837$		0			0
$838$	12.0000			0.414533
$839$	−10.0000	−	17.3205i	−0.345238	−	0.597970i	0.640159	−	0.768243i	$-0.278869\pi$
$839$	−10.0000	−	17.3205i	−0.345238	−	0.597970i	−0.985397	+	0.170272i	$0.945535\pi$
$840$		0			0
$841$	6.50000	−	11.2583i	0.224138	−	0.388218i
$842$	6.00000	−	10.3923i	0.206774	−	0.358142i
$843$		0			0
$844$	8.00000	+	13.8564i	0.275371	+	0.476957i
$845$	−46.0000			−1.58245
$846$		0			0
$847$	10.0000			0.343604
$848$	6.00000	+	10.3923i	0.206041	+	0.356873i
$849$		0			0
$850$	2.50000	−	4.33013i	0.0857493	−	0.148522i
$851$	4.00000	−	6.92820i	0.137118	−	0.237496i
$852$		0			0
$853$	−22.0000	−	38.1051i	−0.753266	−	1.30469i	−0.946232	−	0.323489i	$-0.895144\pi$
$853$	−22.0000	−	38.1051i	−0.753266	−	1.30469i	0.192966	−	0.981205i	$-0.438189\pi$
$854$	12.0000			0.410632
$855$		0			0
$856$	−3.00000			−0.102538
$857$	15.0000	+	25.9808i	0.512390	+	0.887486i	0.999897	+	0.0143666i	$0.00457319\pi$
$857$	15.0000	+	25.9808i	0.512390	+	0.887486i	−0.487507	+	0.873119i	$0.662093\pi$
$858$		0			0
$859$	−14.5000	+	25.1147i	−0.494734	+	0.856904i	−0.999982	−	0.00607046i	$-0.998068\pi$
$859$	−14.5000	+	25.1147i	−0.494734	+	0.856904i	0.505248	+	0.862974i	$0.331401\pi$
$860$	−1.00000	+	1.73205i	−0.0340997	+	0.0590624i
$861$		0			0
$862$		0			0
$863$	−38.0000			−1.29354			−0.646768	−	0.762687i	$-0.723880\pi$
$863$	−38.0000			−1.29354			−0.646768	+	0.762687i	$0.723880\pi$
$864$		0			0
$865$	4.00000			0.136004
$866$	12.5000	+	21.6506i	0.424767	+	0.735719i
$867$		0			0
$868$	−3.00000	+	5.19615i	−0.101827	+	0.176369i
$869$	−3.00000	+	5.19615i	−0.101768	+	0.176267i
$870$		0			0
$871$	39.0000	+	67.5500i	1.32146	+	2.28884i
$872$	−2.00000			−0.0677285
$873$		0			0
$874$	28.0000			0.947114
$875$	6.00000	+	10.3923i	0.202837	+	0.351324i
$876$		0			0
$877$	−8.00000	+	13.8564i	−0.270141	+	0.467898i	−0.968898	−	0.247462i	$-0.920404\pi$
$877$	−8.00000	+	13.8564i	−0.270141	+	0.467898i	0.698757	+	0.715359i	$0.253737\pi$
$878$	12.0000	−	20.7846i	0.404980	−	0.701447i
$879$		0			0
$880$	−1.00000	−	1.73205i	−0.0337100	−	0.0583874i
$881$	−42.0000			−1.41502			−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000			−1.41502			−0.707508	+	0.706705i	$0.750181\pi$
$882$		0			0
$883$	53.0000			1.78359			0.891796	−	0.452438i	$-0.149446\pi$
$883$	53.0000			1.78359			0.891796	+	0.452438i	$0.149446\pi$
$884$	15.0000	+	25.9808i	0.504505	+	0.873828i
$885$		0			0
$886$	3.50000	−	6.06218i	0.117585	−	0.203663i
$887$	−3.00000	+	5.19615i	−0.100730	+	0.174470i	−0.911986	−	0.410222i	$-0.865451\pi$
$887$	−3.00000	+	5.19615i	−0.100730	+	0.174470i	0.811256	+	0.584692i	$0.198785\pi$
$888$		0			0
$889$	−6.00000	−	10.3923i	−0.201234	−	0.348547i
$890$	−12.0000			−0.402241
$891$		0			0
$892$	−4.00000			−0.133930
$893$		0			0
$894$		0			0
$895$	−24.0000	+	41.5692i	−0.802232	+	1.38951i
$896$	0.500000	−	0.866025i	0.0167038	−	0.0289319i
$897$		0			0
$898$	8.50000	+	14.7224i	0.283649	+	0.491294i
$899$	−24.0000			−0.800445
$900$		0			0
$901$	−60.0000			−1.99889
$902$	−1.50000	−	2.59808i	−0.0499445	−	0.0865065i
$903$		0			0
$904$	−5.00000	+	8.66025i	−0.166298	+	0.288036i
$905$		0			0
$906$		0			0
$907$	13.5000	+	23.3827i	0.448260	+	0.776409i	0.998273	−	0.0587469i	$-0.0187105\pi$
$907$	13.5000	+	23.3827i	0.448260	+	0.776409i	−0.550013	+	0.835156i	$0.685377\pi$
$908$	−3.00000			−0.0995585
$909$		0			0
$910$	−12.0000			−0.397796
$911$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$911$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$912$		0			0
$913$	−8.00000	+	13.8564i	−0.264761	+	0.458580i
$914$	−0.500000	+	0.866025i	−0.0165385	+	0.0286456i
$915$		0			0
$916$	13.0000	+	22.5167i	0.429532	+	0.743971i
$917$	−4.00000			−0.132092
$918$		0			0
$919$	16.0000			0.527791			0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000			0.527791			0.263896	+	0.964551i	$0.414993\pi$
$920$	−4.00000	−	6.92820i	−0.131876	−	0.228416i
$921$		0			0
$922$	7.00000	−	12.1244i	0.230533	−	0.399294i
$923$	24.0000	−	41.5692i	0.789970	−	1.36827i
$924$		0			0
$925$	1.00000	+	1.73205i	0.0328798	+	0.0569495i
$926$	−8.00000			−0.262896
$927$		0			0
$928$	4.00000			0.131306
$929$	−9.00000	−	15.5885i	−0.295280	−	0.511441i	0.679770	−	0.733426i	$-0.262080\pi$
$929$	−9.00000	−	15.5885i	−0.295280	−	0.511441i	−0.975050	+	0.221985i	$0.928746\pi$
$930$		0			0
$931$	3.50000	−	6.06218i	0.114708	−	0.198680i
$932$	−14.5000	+	25.1147i	−0.474963	+	0.822661i
$933$		0			0
$934$	−6.50000	−	11.2583i	−0.212686	−	0.368384i
$935$	10.0000			0.327035
$936$		0			0
$937$	2.00000			0.0653372			0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000			0.0653372			0.0326686	+	0.999466i	$0.489599\pi$
$938$	6.50000	+	11.2583i	0.212233	+	0.367598i
$939$		0			0
$940$		0			0
$941$	10.0000	−	17.3205i	0.325991	−	0.564632i	−0.655722	−	0.755003i	$-0.727636\pi$
$941$	10.0000	−	17.3205i	0.325991	−	0.564632i	0.981712	+	0.190370i	$0.0609689\pi$
$942$		0			0
$943$	−6.00000	−	10.3923i	−0.195387	−	0.338420i
$944$	7.00000			0.227831
$945$		0			0
$946$	1.00000			0.0325128
$947$	−18.5000	−	32.0429i	−0.601169	−	1.04126i	−0.992644	−	0.121067i	$-0.961368\pi$
$947$	−18.5000	−	32.0429i	−0.601169	−	1.04126i	0.391475	−	0.920189i	$-0.371965\pi$
$948$		0			0
$949$	3.00000	−	5.19615i	0.0973841	−	0.168674i
$950$	−3.50000	+	6.06218i	−0.113555	+	0.196683i
$951$		0			0
$952$	2.50000	+	4.33013i	0.0810255	+	0.140340i
$953$	35.0000			1.13376			0.566881	−	0.823800i	$-0.308150\pi$
$953$	35.0000			1.13376			0.566881	+	0.823800i	$0.308150\pi$
$954$		0			0
$955$	24.0000			0.776622
$956$	3.00000	+	5.19615i	0.0970269	+	0.168056i
$957$		0			0
$958$	−10.0000	+	17.3205i	−0.323085	+	0.559600i
$959$	9.50000	−	16.4545i	0.306771	−	0.531343i
$960$		0			0
$961$	−2.50000	−	4.33013i	−0.0806452	−	0.139682i
$962$	−12.0000			−0.386896
$963$		0			0
$964$	23.0000			0.740780
$965$	17.0000	+	29.4449i	0.547249	+	0.947864i
$966$		0			0
$967$	−7.00000	+	12.1244i	−0.225105	+	0.389893i	−0.956351	−	0.292221i	$-0.905606\pi$
$967$	−7.00000	+	12.1244i	−0.225105	+	0.389893i	0.731246	+	0.682114i	$0.238939\pi$
$968$	5.00000	−	8.66025i	0.160706	−	0.278351i
$969$		0			0
$970$	−5.00000	−	8.66025i	−0.160540	−	0.278064i
$971$	12.0000			0.385098			0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000			0.385098			0.192549	+	0.981287i	$0.438325\pi$
$972$		0			0
$973$	5.00000			0.160293
$974$	5.00000	+	8.66025i	0.160210	+	0.277492i
$975$		0			0
$976$	6.00000	−	10.3923i	0.192055	−	0.332650i
$977$	−7.50000	+	12.9904i	−0.239946	+	0.415599i	−0.960699	−	0.277594i	$-0.910463\pi$
$977$	−7.50000	+	12.9904i	−0.239946	+	0.415599i	0.720752	+	0.693193i	$0.243796\pi$
$978$		0			0
$979$	3.00000	+	5.19615i	0.0958804	+	0.166070i
$980$	−2.00000			−0.0638877
$981$		0			0
$982$	−33.0000			−1.05307
$983$	−30.0000	−	51.9615i	−0.956851	−	1.65732i	−0.730073	−	0.683369i	$-0.760514\pi$
$983$	−30.0000	−	51.9615i	−0.956851	−	1.65732i	−0.226778	−	0.973946i	$-0.572819\pi$
$984$		0			0
$985$	−10.0000	+	17.3205i	−0.318626	+	0.551877i
$986$	−10.0000	+	17.3205i	−0.318465	+	0.551597i
$987$		0			0
$988$	−21.0000	−	36.3731i	−0.668099	−	1.15718i
$989$	4.00000			0.127193
$990$		0			0
$991$	−28.0000			−0.889449			−0.444725	−	0.895667i	$-0.646698\pi$
$991$	−28.0000			−0.889449			−0.444725	+	0.895667i	$0.646698\pi$
$992$	3.00000	+	5.19615i	0.0952501	+	0.164978i
$993$		0			0
$994$	4.00000	−	6.92820i	0.126872	−	0.219749i
$995$	14.0000	−	24.2487i	0.443830	−	0.768736i
$996$		0			0
$997$	−1.00000	−	1.73205i	−0.0316703	−	0.0548546i	0.849756	−	0.527176i	$-0.176749\pi$
$997$	−1.00000	−	1.73205i	−0.0316703	−	0.0548546i	−0.881426	+	0.472322i	$0.843416\pi$
$998$	−29.0000			−0.917979
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.2.f.b.253.1		2
3.2	odd	2		126.2.f.b.85.1	yes	2
4.3	odd	2		3024.2.r.c.1009.1		2
7.2	even	3		2646.2.h.b.361.1		2
7.3	odd	6		2646.2.e.h.1549.1		2
7.4	even	3		2646.2.e.i.1549.1		2
7.5	odd	6		2646.2.h.c.361.1		2
7.6	odd	2		2646.2.f.b.1765.1		2
9.2	odd	6		126.2.f.b.43.1	✓	2
9.4	even	3		1134.2.a.f.1.1		1
9.5	odd	6		1134.2.a.c.1.1		1
9.7	even	3	inner	378.2.f.b.127.1		2
12.11	even	2		1008.2.r.a.337.1		2
21.2	odd	6		882.2.h.h.67.1		2
21.5	even	6		882.2.h.g.67.1		2
21.11	odd	6		882.2.e.a.373.1		2
21.17	even	6		882.2.e.e.373.1		2
21.20	even	2		882.2.f.f.589.1		2
36.7	odd	6		3024.2.r.c.2017.1		2
36.11	even	6		1008.2.r.a.673.1		2
36.23	even	6		9072.2.a.t.1.1		1
36.31	odd	6		9072.2.a.f.1.1		1
63.2	odd	6		882.2.e.a.655.1		2
63.11	odd	6		882.2.h.h.79.1		2
63.13	odd	6		7938.2.a.bb.1.1		1
63.16	even	3		2646.2.e.i.2125.1		2
63.20	even	6		882.2.f.f.295.1		2
63.25	even	3		2646.2.h.b.667.1		2
63.34	odd	6		2646.2.f.b.883.1		2
63.38	even	6		882.2.h.g.79.1		2
63.41	even	6		7938.2.a.e.1.1		1
63.47	even	6		882.2.e.e.655.1		2
63.52	odd	6		2646.2.h.c.667.1		2
63.61	odd	6		2646.2.e.h.2125.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.2.f.b.43.1	✓	2	9.2	odd	6
126.2.f.b.85.1	yes	2	3.2	odd	2
378.2.f.b.127.1		2	9.7	even	3	inner
378.2.f.b.253.1		2	1.1	even	1	trivial
882.2.e.a.373.1		2	21.11	odd	6
882.2.e.a.655.1		2	63.2	odd	6
882.2.e.e.373.1		2	21.17	even	6
882.2.e.e.655.1		2	63.47	even	6
882.2.f.f.295.1		2	63.20	even	6
882.2.f.f.589.1		2	21.20	even	2
882.2.h.g.67.1		2	21.5	even	6
882.2.h.g.79.1		2	63.38	even	6
882.2.h.h.67.1		2	21.2	odd	6
882.2.h.h.79.1		2	63.11	odd	6
1008.2.r.a.337.1		2	12.11	even	2
1008.2.r.a.673.1		2	36.11	even	6
1134.2.a.c.1.1		1	9.5	odd	6
1134.2.a.f.1.1		1	9.4	even	3
2646.2.e.h.1549.1		2	7.3	odd	6
2646.2.e.h.2125.1		2	63.61	odd	6
2646.2.e.i.1549.1		2	7.4	even	3
2646.2.e.i.2125.1		2	63.16	even	3
2646.2.f.b.883.1		2	63.34	odd	6
2646.2.f.b.1765.1		2	7.6	odd	2
2646.2.h.b.361.1		2	7.2	even	3
2646.2.h.b.667.1		2	63.25	even	3
2646.2.h.c.361.1		2	7.5	odd	6
2646.2.h.c.667.1		2	63.52	odd	6
3024.2.r.c.1009.1		2	4.3	odd	2
3024.2.r.c.2017.1		2	36.7	odd	6
7938.2.a.e.1.1		1	63.41	even	6
7938.2.a.bb.1.1		1	63.13	odd	6
9072.2.a.f.1.1		1	36.31	odd	6
9072.2.a.t.1.1		1	36.23	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	378.f (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(1.00000 - 1.73205i) q^{5} +(0.500000 + 0.866025i) q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(1.00000 - 1.73205i) q^{5} +(0.500000 + 0.866025i) q^{7} +1.00000 q^{8} -2.00000 q^{10} +(0.500000 + 0.866025i) q^{11} +(3.00000 - 5.19615i) q^{13} +(0.500000 - 0.866025i) q^{14} +(-0.500000 - 0.866025i) q^{16} +5.00000 q^{17} -7.00000 q^{19} +(1.00000 + 1.73205i) q^{20} +(0.500000 - 0.866025i) q^{22} +(2.00000 - 3.46410i) q^{23} +(0.500000 + 0.866025i) q^{25} -6.00000 q^{26} -1.00000 q^{28} +(-2.00000 - 3.46410i) q^{29} +(3.00000 - 5.19615i) q^{31} +(-0.500000 + 0.866025i) q^{32} +(-2.50000 - 4.33013i) q^{34} +2.00000 q^{35} +2.00000 q^{37} +(3.50000 + 6.06218i) q^{38} +(1.00000 - 1.73205i) q^{40} +(1.50000 - 2.59808i) q^{41} +(0.500000 + 0.866025i) q^{43} -1.00000 q^{44} -4.00000 q^{46} +(-0.500000 + 0.866025i) q^{49} +(0.500000 - 0.866025i) q^{50} +(3.00000 + 5.19615i) q^{52} -12.0000 q^{53} +2.00000 q^{55} +(0.500000 + 0.866025i) q^{56} +(-2.00000 + 3.46410i) q^{58} +(-3.50000 + 6.06218i) q^{59} +(6.00000 + 10.3923i) q^{61} -6.00000 q^{62} +1.00000 q^{64} +(-6.00000 - 10.3923i) q^{65} +(-6.50000 + 11.2583i) q^{67} +(-2.50000 + 4.33013i) q^{68} +(-1.00000 - 1.73205i) q^{70} +8.00000 q^{71} +1.00000 q^{73} +(-1.00000 - 1.73205i) q^{74} +(3.50000 - 6.06218i) q^{76} +(-0.500000 + 0.866025i) q^{77} +(3.00000 + 5.19615i) q^{79} -2.00000 q^{80} -3.00000 q^{82} +(8.00000 + 13.8564i) q^{83} +(5.00000 - 8.66025i) q^{85} +(0.500000 - 0.866025i) q^{86} +(0.500000 + 0.866025i) q^{88} +6.00000 q^{89} +6.00000 q^{91} +(2.00000 + 3.46410i) q^{92} +(-7.00000 + 12.1244i) q^{95} +(2.50000 + 4.33013i) q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} + 2 q^{5} + q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q - q^2 - q^4 + 2 * q^5 + q^7 + 2 * q^8	\( 2 q - q^{2} - q^{4} + 2 q^{5} + q^{7} + 2 q^{8} - 4 q^{10} + q^{11} + 6 q^{13} + q^{14} - q^{16} + 10 q^{17} - 14 q^{19} + 2 q^{20} + q^{22} + 4 q^{23} + q^{25} - 12 q^{26} - 2 q^{28} - 4 q^{29} + 6 q^{31} - q^{32} - 5 q^{34} + 4 q^{35} + 4 q^{37} + 7 q^{38} + 2 q^{40} + 3 q^{41} + q^{43} - 2 q^{44} - 8 q^{46} - q^{49} + q^{50} + 6 q^{52} - 24 q^{53} + 4 q^{55} + q^{56} - 4 q^{58} - 7 q^{59} + 12 q^{61} - 12 q^{62} + 2 q^{64} - 12 q^{65} - 13 q^{67} - 5 q^{68} - 2 q^{70} + 16 q^{71} + 2 q^{73} - 2 q^{74} + 7 q^{76} - q^{77} + 6 q^{79} - 4 q^{80} - 6 q^{82} + 16 q^{83} + 10 q^{85} + q^{86} + q^{88} + 12 q^{89} + 12 q^{91} + 4 q^{92} - 14 q^{95} + 5 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q - q^2 - q^4 + 2 * q^5 + q^7 + 2 * q^8 - 4 * q^10 + q^11 + 6 * q^13 + q^14 - q^16 + 10 * q^17 - 14 * q^19 + 2 * q^20 + q^22 + 4 * q^23 + q^25 - 12 * q^26 - 2 * q^28 - 4 * q^29 + 6 * q^31 - q^32 - 5 * q^34 + 4 * q^35 + 4 * q^37 + 7 * q^38 + 2 * q^40 + 3 * q^41 + q^43 - 2 * q^44 - 8 * q^46 - q^49 + q^50 + 6 * q^52 - 24 * q^53 + 4 * q^55 + q^56 - 4 * q^58 - 7 * q^59 + 12 * q^61 - 12 * q^62 + 2 * q^64 - 12 * q^65 - 13 * q^67 - 5 * q^68 - 2 * q^70 + 16 * q^71 + 2 * q^73 - 2 * q^74 + 7 * q^76 - q^77 + 6 * q^79 - 4 * q^80 - 6 * q^82 + 16 * q^83 + 10 * q^85 + q^86 + q^88 + 12 * q^89 + 12 * q^91 + 4 * q^92 - 14 * q^95 + 5 * q^97 + 2 * q^98

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