LMFDB - Embedded newform 378.2.g.c.109.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("378.109");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

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

Embedding invariants

Embedding label			109.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	378.109
Dual form			378.2.g.c.163.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(1.00000 + 1.73205i) q^{5} +(0.500000 - 2.59808i) q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(1.00000 + 1.73205i) q^{5} +(0.500000 - 2.59808i) q^{7} +1.00000 q^{8} +(1.00000 - 1.73205i) q^{10} +(-2.50000 + 4.33013i) q^{11} +6.00000 q^{13} +(-2.50000 + 0.866025i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(2.00000 - 3.46410i) q^{17} +(2.00000 + 3.46410i) q^{19} -2.00000 q^{20} +5.00000 q^{22} +(2.00000 + 3.46410i) q^{23} +(0.500000 - 0.866025i) q^{25} +(-3.00000 - 5.19615i) q^{26} +(2.00000 + 1.73205i) q^{28} +7.00000 q^{29} +(-1.50000 + 2.59808i) q^{31} +(-0.500000 + 0.866025i) q^{32} -4.00000 q^{34} +(5.00000 - 1.73205i) q^{35} +(-4.00000 - 6.92820i) q^{37} +(2.00000 - 3.46410i) q^{38} +(1.00000 + 1.73205i) q^{40} -6.00000 q^{41} +8.00000 q^{43} +(-2.50000 - 4.33013i) q^{44} +(2.00000 - 3.46410i) q^{46} +(-3.00000 - 5.19615i) q^{47} +(-6.50000 - 2.59808i) q^{49} -1.00000 q^{50} +(-3.00000 + 5.19615i) q^{52} +(3.00000 - 5.19615i) q^{53} -10.0000 q^{55} +(0.500000 - 2.59808i) q^{56} +(-3.50000 - 6.06218i) q^{58} +(-3.50000 + 6.06218i) q^{59} +3.00000 q^{62} +1.00000 q^{64} +(6.00000 + 10.3923i) q^{65} +(-5.00000 + 8.66025i) q^{67} +(2.00000 + 3.46410i) q^{68} +(-4.00000 - 3.46410i) q^{70} -4.00000 q^{71} +(-6.50000 + 11.2583i) q^{73} +(-4.00000 + 6.92820i) q^{74} -4.00000 q^{76} +(10.0000 + 8.66025i) q^{77} +(1.50000 + 2.59808i) q^{79} +(1.00000 - 1.73205i) q^{80} +(3.00000 + 5.19615i) q^{82} -7.00000 q^{83} +8.00000 q^{85} +(-4.00000 - 6.92820i) q^{86} +(-2.50000 + 4.33013i) q^{88} +(-3.00000 - 5.19615i) q^{89} +(3.00000 - 15.5885i) q^{91} -4.00000 q^{92} +(-3.00000 + 5.19615i) q^{94} +(-4.00000 + 6.92820i) q^{95} -5.00000 q^{97} +(1.00000 + 6.92820i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 2 q^{10} - 5 q^{11} + 12 q^{13} - 5 q^{14} - q^{16} + 4 q^{17} + 4 q^{19} - 4 q^{20} + 10 q^{22} + 4 q^{23} + q^{25} - 6 q^{26} + 4 q^{28} + 14 q^{29} - 3 q^{31} - q^{32} - 8 q^{34} + 10 q^{35} - 8 q^{37} + 4 q^{38} + 2 q^{40} - 12 q^{41} + 16 q^{43} - 5 q^{44} + 4 q^{46} - 6 q^{47} - 13 q^{49} - 2 q^{50} - 6 q^{52} + 6 q^{53} - 20 q^{55} + q^{56} - 7 q^{58} - 7 q^{59} + 6 q^{62} + 2 q^{64} + 12 q^{65} - 10 q^{67} + 4 q^{68} - 8 q^{70} - 8 q^{71} - 13 q^{73} - 8 q^{74} - 8 q^{76} + 20 q^{77} + 3 q^{79} + 2 q^{80} + 6 q^{82} - 14 q^{83} + 16 q^{85} - 8 q^{86} - 5 q^{88} - 6 q^{89} + 6 q^{91} - 8 q^{92} - 6 q^{94} - 8 q^{95} - 10 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q - q^2 - q^4 + 2 * q^5 + q^7 + 2 * q^8 + 2 * q^10 - 5 * q^11 + 12 * q^13 - 5 * q^14 - q^16 + 4 * q^17 + 4 * q^19 - 4 * q^20 + 10 * q^22 + 4 * q^23 + q^25 - 6 * q^26 + 4 * q^28 + 14 * q^29 - 3 * q^31 - q^32 - 8 * q^34 + 10 * q^35 - 8 * q^37 + 4 * q^38 + 2 * q^40 - 12 * q^41 + 16 * q^43 - 5 * q^44 + 4 * q^46 - 6 * q^47 - 13 * q^49 - 2 * q^50 - 6 * q^52 + 6 * q^53 - 20 * q^55 + q^56 - 7 * q^58 - 7 * q^59 + 6 * q^62 + 2 * q^64 + 12 * q^65 - 10 * q^67 + 4 * q^68 - 8 * q^70 - 8 * q^71 - 13 * q^73 - 8 * q^74 - 8 * q^76 + 20 * q^77 + 3 * q^79 + 2 * q^80 + 6 * q^82 - 14 * q^83 + 16 * q^85 - 8 * q^86 - 5 * q^88 - 6 * q^89 + 6 * q^91 - 8 * q^92 - 6 * q^94 - 8 * q^95 - 10 * q^97 + 2 * q^98

Display 100 coefficients 10 coefficients

Character values

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

$n$	$29$	$325$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

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

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

$2$	−0.500000	−	0.866025i	−0.353553	−	0.612372i
$2$	−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	0.998203	−	0.0599153i	$-0.0190830\pi$
$5$	1.00000	+	1.73205i	0.447214	+	0.774597i	−0.550990	+	0.834512i	$0.685750\pi$
$6$		0			0
$7$	0.500000	−	2.59808i	0.188982	−	0.981981i
$7$	0.500000	−	2.59808i	0.188982	−	0.981981i
$8$	1.00000			0.353553
$9$		0			0
$10$	1.00000	−	1.73205i	0.316228	−	0.547723i
$11$	−2.50000	+	4.33013i	−0.753778	+	1.30558i	0.192201	+	0.981356i	$0.438437\pi$
$11$	−2.50000	+	4.33013i	−0.753778	+	1.30558i	−0.945979	+	0.324227i	$0.894896\pi$
$12$		0			0
$13$	6.00000			1.66410			0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000			1.66410			0.832050	+	0.554700i	$0.187167\pi$
$14$	−2.50000	+	0.866025i	−0.668153	+	0.231455i
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.125000	−	0.216506i
$17$	2.00000	−	3.46410i	0.485071	−	0.840168i	−0.514782	−	0.857321i	$-0.672127\pi$
$17$	2.00000	−	3.46410i	0.485071	−	0.840168i	0.999853	+	0.0171533i	$0.00546033\pi$
$18$		0			0
$19$	2.00000	+	3.46410i	0.458831	+	0.794719i	0.998899	−	0.0469020i	$-0.0149348\pi$
$19$	2.00000	+	3.46410i	0.458831	+	0.794719i	−0.540068	+	0.841621i	$0.681602\pi$
$20$	−2.00000			−0.447214
$21$		0			0
$22$	5.00000			1.06600
$23$	2.00000	+	3.46410i	0.417029	+	0.722315i	0.995639	−	0.0932891i	$-0.0297381\pi$
$23$	2.00000	+	3.46410i	0.417029	+	0.722315i	−0.578610	+	0.815604i	$0.696405\pi$
$24$		0			0
$25$	0.500000	−	0.866025i	0.100000	−	0.173205i
$26$	−3.00000	−	5.19615i	−0.588348	−	1.01905i
$27$		0			0
$28$	2.00000	+	1.73205i	0.377964	+	0.327327i
$29$	7.00000			1.29987			0.649934	−	0.759991i	$-0.274797\pi$
$29$	7.00000			1.29987			0.649934	+	0.759991i	$0.274797\pi$
$30$		0			0
$31$	−1.50000	+	2.59808i	−0.269408	+	0.466628i	−0.968709	−	0.248199i	$-0.920161\pi$
$31$	−1.50000	+	2.59808i	−0.269408	+	0.466628i	0.699301	+	0.714827i	$0.253495\pi$
$32$	−0.500000	+	0.866025i	−0.0883883	+	0.153093i
$33$		0			0
$34$	−4.00000			−0.685994
$35$	5.00000	−	1.73205i	0.845154	−	0.292770i
$36$		0			0
$37$	−4.00000	−	6.92820i	−0.657596	−	1.13899i	−0.981236	−	0.192809i	$-0.938240\pi$
$37$	−4.00000	−	6.92820i	−0.657596	−	1.13899i	0.323640	−	0.946180i	$-0.395093\pi$
$38$	2.00000	−	3.46410i	0.324443	−	0.561951i
$39$		0			0
$40$	1.00000	+	1.73205i	0.158114	+	0.273861i
$41$	−6.00000			−0.937043			−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000			−0.937043			−0.468521	+	0.883452i	$0.655213\pi$
$42$		0			0
$43$	8.00000			1.21999			0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000			1.21999			0.609994	+	0.792406i	$0.291172\pi$
$44$	−2.50000	−	4.33013i	−0.376889	−	0.652791i
$45$		0			0
$46$	2.00000	−	3.46410i	0.294884	−	0.510754i
$47$	−3.00000	−	5.19615i	−0.437595	−	0.757937i	0.559908	−	0.828554i	$-0.310836\pi$
$47$	−3.00000	−	5.19615i	−0.437595	−	0.757937i	−0.997503	+	0.0706177i	$0.977503\pi$
$48$		0			0
$49$	−6.50000	−	2.59808i	−0.928571	−	0.371154i
$50$	−1.00000			−0.141421
$51$		0			0
$52$	−3.00000	+	5.19615i	−0.416025	+	0.720577i
$53$	3.00000	−	5.19615i	0.412082	−	0.713746i	−0.583036	−	0.812447i	$-0.698135\pi$
$53$	3.00000	−	5.19615i	0.412082	−	0.713746i	0.995117	+	0.0987002i	$0.0314685\pi$
$54$		0			0
$55$	−10.0000			−1.34840
$56$	0.500000	−	2.59808i	0.0668153	−	0.347183i
$57$		0			0
$58$	−3.50000	−	6.06218i	−0.459573	−	0.796003i
$59$	−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
$61$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$61$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$62$	3.00000			0.381000
$63$		0			0
$64$	1.00000			0.125000
$65$	6.00000	+	10.3923i	0.744208	+	1.28901i
$66$		0			0
$67$	−5.00000	+	8.66025i	−0.610847	+	1.05802i	0.380251	+	0.924883i	$0.375838\pi$
$67$	−5.00000	+	8.66025i	−0.610847	+	1.05802i	−0.991098	+	0.133135i	$0.957496\pi$
$68$	2.00000	+	3.46410i	0.242536	+	0.420084i
$69$		0			0
$70$	−4.00000	−	3.46410i	−0.478091	−	0.414039i
$71$	−4.00000			−0.474713			−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000			−0.474713			−0.237356	+	0.971423i	$0.576281\pi$
$72$		0			0
$73$	−6.50000	+	11.2583i	−0.760767	+	1.31769i	0.181688	+	0.983356i	$0.441844\pi$
$73$	−6.50000	+	11.2583i	−0.760767	+	1.31769i	−0.942455	+	0.334332i	$0.891489\pi$
$74$	−4.00000	+	6.92820i	−0.464991	+	0.805387i
$75$		0			0
$76$	−4.00000			−0.458831
$77$	10.0000	+	8.66025i	1.13961	+	0.986928i
$78$		0			0
$79$	1.50000	+	2.59808i	0.168763	+	0.292306i	0.937985	−	0.346675i	$-0.112689\pi$
$79$	1.50000	+	2.59808i	0.168763	+	0.292306i	−0.769222	+	0.638982i	$0.779356\pi$
$80$	1.00000	−	1.73205i	0.111803	−	0.193649i
$81$		0			0
$82$	3.00000	+	5.19615i	0.331295	+	0.573819i
$83$	−7.00000			−0.768350			−0.384175	−	0.923260i	$-0.625514\pi$
$83$	−7.00000			−0.768350			−0.384175	+	0.923260i	$0.625514\pi$
$84$		0			0
$85$	8.00000			0.867722
$86$	−4.00000	−	6.92820i	−0.431331	−	0.747087i
$87$		0			0
$88$	−2.50000	+	4.33013i	−0.266501	+	0.461593i
$89$	−3.00000	−	5.19615i	−0.317999	−	0.550791i	0.662071	−	0.749441i	$-0.269678\pi$
$89$	−3.00000	−	5.19615i	−0.317999	−	0.550791i	−0.980071	+	0.198650i	$0.936344\pi$
$90$		0			0
$91$	3.00000	−	15.5885i	0.314485	−	1.63411i
$92$	−4.00000			−0.417029
$93$		0			0
$94$	−3.00000	+	5.19615i	−0.309426	+	0.535942i
$95$	−4.00000	+	6.92820i	−0.410391	+	0.710819i
$96$		0			0
$97$	−5.00000			−0.507673			−0.253837	−	0.967247i	$-0.581693\pi$
$97$	−5.00000			−0.507673			−0.253837	+	0.967247i	$0.581693\pi$
$98$	1.00000	+	6.92820i	0.101015	+	0.699854i
$99$		0			0
$100$	0.500000	+	0.866025i	0.0500000	+	0.0866025i
$101$	2.50000	−	4.33013i	0.248759	−	0.430864i	−0.714423	−	0.699715i	$-0.753311\pi$
$101$	2.50000	−	4.33013i	0.248759	−	0.430864i	0.963182	+	0.268851i	$0.0866439\pi$
$102$		0			0
$103$	−4.00000	−	6.92820i	−0.394132	−	0.682656i	0.598858	−	0.800855i	$-0.295621\pi$
$103$	−4.00000	−	6.92820i	−0.394132	−	0.682656i	−0.992990	+	0.118199i	$0.962288\pi$
$104$	6.00000			0.588348
$105$		0			0
$106$	−6.00000			−0.582772
$107$	−6.00000	−	10.3923i	−0.580042	−	1.00466i	−0.995474	−	0.0950377i	$-0.969703\pi$
$107$	−6.00000	−	10.3923i	−0.580042	−	1.00466i	0.415432	−	0.909624i	$-0.363630\pi$
$108$		0			0
$109$	−8.00000	+	13.8564i	−0.766261	+	1.32720i	0.173316	+	0.984866i	$0.444552\pi$
$109$	−8.00000	+	13.8564i	−0.766261	+	1.32720i	−0.939577	+	0.342337i	$0.888782\pi$
$110$	5.00000	+	8.66025i	0.476731	+	0.825723i
$111$		0			0
$112$	−2.50000	+	0.866025i	−0.236228	+	0.0818317i
$113$	4.00000			0.376288			0.188144	−	0.982141i	$-0.439753\pi$
$113$	4.00000			0.376288			0.188144	+	0.982141i	$0.439753\pi$
$114$		0			0
$115$	−4.00000	+	6.92820i	−0.373002	+	0.646058i
$116$	−3.50000	+	6.06218i	−0.324967	+	0.562859i
$117$		0			0
$118$	7.00000			0.644402
$119$	−8.00000	−	6.92820i	−0.733359	−	0.635107i
$120$		0			0
$121$	−7.00000	−	12.1244i	−0.636364	−	1.10221i
$122$		0			0
$123$		0			0
$124$	−1.50000	−	2.59808i	−0.134704	−	0.233314i
$125$	12.0000			1.07331
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$	−0.500000	−	0.866025i	−0.0441942	−	0.0765466i
$129$		0			0
$130$	6.00000	−	10.3923i	0.526235	−	0.911465i
$131$	−6.50000	−	11.2583i	−0.567908	−	0.983645i	−0.996773	−	0.0802763i	$-0.974420\pi$
$131$	−6.50000	−	11.2583i	−0.567908	−	0.983645i	0.428865	−	0.903369i	$-0.358914\pi$
$132$		0			0
$133$	10.0000	−	3.46410i	0.867110	−	0.300376i
$134$	10.0000			0.863868
$135$		0			0
$136$	2.00000	−	3.46410i	0.171499	−	0.297044i
$137$	4.00000	−	6.92820i	0.341743	−	0.591916i	−0.643013	−	0.765855i	$-0.722316\pi$
$137$	4.00000	−	6.92820i	0.341743	−	0.591916i	0.984757	+	0.173939i	$0.0556494\pi$
$138$		0			0
$139$	−8.00000			−0.678551			−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000			−0.678551			−0.339276	+	0.940687i	$0.610182\pi$
$140$	−1.00000	+	5.19615i	−0.0845154	+	0.439155i
$141$		0			0
$142$	2.00000	+	3.46410i	0.167836	+	0.290701i
$143$	−15.0000	+	25.9808i	−1.25436	+	2.17262i
$144$		0			0
$145$	7.00000	+	12.1244i	0.581318	+	1.00687i
$146$	13.0000			1.07589
$147$		0			0
$148$	8.00000			0.657596
$149$	−4.50000	−	7.79423i	−0.368654	−	0.638528i	0.620701	−	0.784047i	$-0.286848\pi$
$149$	−4.50000	−	7.79423i	−0.368654	−	0.638528i	−0.989355	+	0.145519i	$0.953515\pi$
$150$		0			0
$151$	8.50000	−	14.7224i	0.691720	−	1.19809i	−0.279554	−	0.960130i	$-0.590186\pi$
$151$	8.50000	−	14.7224i	0.691720	−	1.19809i	0.971274	−	0.237964i	$-0.0764802\pi$
$152$	2.00000	+	3.46410i	0.162221	+	0.280976i
$153$		0			0
$154$	2.50000	−	12.9904i	0.201456	−	1.04679i
$155$	−6.00000			−0.481932
$156$		0			0
$157$	−7.00000	+	12.1244i	−0.558661	+	0.967629i	0.438948	+	0.898513i	$0.355351\pi$
$157$	−7.00000	+	12.1244i	−0.558661	+	0.967629i	−0.997609	+	0.0691164i	$0.977982\pi$
$158$	1.50000	−	2.59808i	0.119334	−	0.206692i
$159$		0			0
$160$	−2.00000			−0.158114
$161$	10.0000	−	3.46410i	0.788110	−	0.273009i
$162$		0			0
$163$	−1.00000	−	1.73205i	−0.0783260	−	0.135665i	0.824202	−	0.566296i	$-0.191624\pi$
$163$	−1.00000	−	1.73205i	−0.0783260	−	0.135665i	−0.902528	+	0.430632i	$0.858291\pi$
$164$	3.00000	−	5.19615i	0.234261	−	0.405751i
$165$		0			0
$166$	3.50000	+	6.06218i	0.271653	+	0.470516i
$167$	−14.0000			−1.08335			−0.541676	−	0.840587i	$-0.682210\pi$
$167$	−14.0000			−1.08335			−0.541676	+	0.840587i	$0.682210\pi$
$168$		0			0
$169$	23.0000			1.76923
$170$	−4.00000	−	6.92820i	−0.306786	−	0.531369i
$171$		0			0
$172$	−4.00000	+	6.92820i	−0.304997	+	0.528271i
$173$	−0.500000	−	0.866025i	−0.0380143	−	0.0658427i	0.846392	−	0.532560i	$-0.178770\pi$
$173$	−0.500000	−	0.866025i	−0.0380143	−	0.0658427i	−0.884407	+	0.466717i	$0.845437\pi$
$174$		0			0
$175$	−2.00000	−	1.73205i	−0.151186	−	0.130931i
$176$	5.00000			0.376889
$177$		0			0
$178$	−3.00000	+	5.19615i	−0.224860	+	0.389468i
$179$	7.50000	−	12.9904i	0.560576	−	0.970947i	−0.436870	−	0.899525i	$-0.643913\pi$
$179$	7.50000	−	12.9904i	0.560576	−	0.970947i	0.997446	−	0.0714220i	$-0.0227537\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$	−15.0000	+	5.19615i	−1.11187	+	0.385164i
$183$		0			0
$184$	2.00000	+	3.46410i	0.147442	+	0.255377i
$185$	8.00000	−	13.8564i	0.588172	−	1.01874i
$186$		0			0
$187$	10.0000	+	17.3205i	0.731272	+	1.26660i
$188$	6.00000			0.437595
$189$		0			0
$190$	8.00000			0.580381
$191$	9.00000	+	15.5885i	0.651217	+	1.12794i	0.982828	+	0.184525i	$0.0590746\pi$
$191$	9.00000	+	15.5885i	0.651217	+	1.12794i	−0.331611	+	0.943416i	$0.607592\pi$
$192$		0			0
$193$	9.50000	−	16.4545i	0.683825	−	1.18442i	−0.289980	−	0.957033i	$-0.593649\pi$
$193$	9.50000	−	16.4545i	0.683825	−	1.18442i	0.973805	−	0.227387i	$-0.0730182\pi$
$194$	2.50000	+	4.33013i	0.179490	+	0.310885i
$195$		0			0
$196$	5.50000	−	4.33013i	0.392857	−	0.309295i
$197$	−25.0000			−1.78118			−0.890588	−	0.454811i	$-0.849707\pi$
$197$	−25.0000			−1.78118			−0.890588	+	0.454811i	$0.849707\pi$
$198$		0			0
$199$	9.50000	−	16.4545i	0.673437	−	1.16643i	−0.303486	−	0.952836i	$-0.598151\pi$
$199$	9.50000	−	16.4545i	0.673437	−	1.16643i	0.976923	−	0.213591i	$-0.0685161\pi$
$200$	0.500000	−	0.866025i	0.0353553	−	0.0612372i
$201$		0			0
$202$	−5.00000			−0.351799
$203$	3.50000	−	18.1865i	0.245652	−	1.27644i
$204$		0			0
$205$	−6.00000	−	10.3923i	−0.419058	−	0.725830i
$206$	−4.00000	+	6.92820i	−0.278693	+	0.482711i
$207$		0			0
$208$	−3.00000	−	5.19615i	−0.208013	−	0.360288i
$209$	−20.0000			−1.38343
$210$		0			0
$211$	26.0000			1.78991			0.894957	−	0.446153i	$-0.147206\pi$
$211$	26.0000			1.78991			0.894957	+	0.446153i	$0.147206\pi$
$212$	3.00000	+	5.19615i	0.206041	+	0.356873i
$213$		0			0
$214$	−6.00000	+	10.3923i	−0.410152	+	0.710403i
$215$	8.00000	+	13.8564i	0.545595	+	0.944999i
$216$		0			0
$217$	6.00000	+	5.19615i	0.407307	+	0.352738i
$218$	16.0000			1.08366
$219$		0			0
$220$	5.00000	−	8.66025i	0.337100	−	0.583874i
$221$	12.0000	−	20.7846i	0.807207	−	1.39812i
$222$		0			0
$223$	−1.00000			−0.0669650			−0.0334825	−	0.999439i	$-0.510660\pi$
$223$	−1.00000			−0.0669650			−0.0334825	+	0.999439i	$0.510660\pi$
$224$	2.00000	+	1.73205i	0.133631	+	0.115728i
$225$		0			0
$226$	−2.00000	−	3.46410i	−0.133038	−	0.230429i
$227$	13.5000	−	23.3827i	0.896026	−	1.55196i	0.0634974	−	0.997982i	$-0.479775\pi$
$227$	13.5000	−	23.3827i	0.896026	−	1.55196i	0.832529	−	0.553981i	$-0.186892\pi$
$228$		0			0
$229$	−2.00000	−	3.46410i	−0.132164	−	0.228914i	0.792347	−	0.610071i	$-0.208859\pi$
$229$	−2.00000	−	3.46410i	−0.132164	−	0.228914i	−0.924510	+	0.381157i	$0.875526\pi$
$230$	8.00000			0.527504
$231$		0			0
$232$	7.00000			0.459573
$233$	11.0000	+	19.0526i	0.720634	+	1.24817i	0.960746	+	0.277429i	$0.0894825\pi$
$233$	11.0000	+	19.0526i	0.720634	+	1.24817i	−0.240112	+	0.970745i	$0.577184\pi$
$234$		0			0
$235$	6.00000	−	10.3923i	0.391397	−	0.677919i
$236$	−3.50000	−	6.06218i	−0.227831	−	0.394614i
$237$		0			0
$238$	−2.00000	+	10.3923i	−0.129641	+	0.673633i
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	0.500000	−	0.866025i	0.0322078	−	0.0557856i	−0.849472	−	0.527633i	$-0.823079\pi$
$241$	0.500000	−	0.866025i	0.0322078	−	0.0557856i	0.881680	+	0.471848i	$0.156413\pi$
$242$	−7.00000	+	12.1244i	−0.449977	+	0.779383i
$243$		0			0
$244$		0			0
$245$	−2.00000	−	13.8564i	−0.127775	−	0.885253i
$246$		0			0
$247$	12.0000	+	20.7846i	0.763542	+	1.32249i
$248$	−1.50000	+	2.59808i	−0.0952501	+	0.164978i
$249$		0			0
$250$	−6.00000	−	10.3923i	−0.379473	−	0.657267i
$251$	−21.0000			−1.32551			−0.662754	−	0.748837i	$-0.730613\pi$
$251$	−21.0000			−1.32551			−0.662754	+	0.748837i	$0.730613\pi$
$252$		0			0
$253$	−20.0000			−1.25739
$254$		0			0
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.0312500	+	0.0541266i
$257$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$257$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$258$		0			0
$259$	−20.0000	+	6.92820i	−1.24274	+	0.430498i
$260$	−12.0000			−0.744208
$261$		0			0
$262$	−6.50000	+	11.2583i	−0.401571	+	0.695542i
$263$	−6.00000	+	10.3923i	−0.369976	+	0.640817i	−0.989561	−	0.144112i	$-0.953967\pi$
$263$	−6.00000	+	10.3923i	−0.369976	+	0.640817i	0.619586	+	0.784929i	$0.287301\pi$
$264$		0			0
$265$	12.0000			0.737154
$266$	−8.00000	−	6.92820i	−0.490511	−	0.424795i
$267$		0			0
$268$	−5.00000	−	8.66025i	−0.305424	−	0.529009i
$269$	−15.5000	+	26.8468i	−0.945052	+	1.63688i	−0.189404	+	0.981899i	$0.560656\pi$
$269$	−15.5000	+	26.8468i	−0.945052	+	1.63688i	−0.755648	+	0.654978i	$0.772678\pi$
$270$		0			0
$271$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$271$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$272$	−4.00000			−0.242536
$273$		0			0
$274$	−8.00000			−0.483298
$275$	2.50000	+	4.33013i	0.150756	+	0.261116i
$276$		0			0
$277$	−1.00000	+	1.73205i	−0.0600842	+	0.104069i	−0.894503	−	0.447062i	$-0.852470\pi$
$277$	−1.00000	+	1.73205i	−0.0600842	+	0.104069i	0.834419	+	0.551131i	$0.185804\pi$
$278$	4.00000	+	6.92820i	0.239904	+	0.415526i
$279$		0			0
$280$	5.00000	−	1.73205i	0.298807	−	0.103510i
$281$	2.00000			0.119310			0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000			0.119310			0.0596550	+	0.998219i	$0.481000\pi$
$282$		0			0
$283$	−8.00000	+	13.8564i	−0.475551	+	0.823678i	−0.999608	−	0.0280052i	$-0.991084\pi$
$283$	−8.00000	+	13.8564i	−0.475551	+	0.823678i	0.524057	+	0.851683i	$0.324418\pi$
$284$	2.00000	−	3.46410i	0.118678	−	0.205557i
$285$		0			0
$286$	30.0000			1.77394
$287$	−3.00000	+	15.5885i	−0.177084	+	0.920158i
$288$		0			0
$289$	0.500000	+	0.866025i	0.0294118	+	0.0509427i
$290$	7.00000	−	12.1244i	0.411054	−	0.711967i
$291$		0			0
$292$	−6.50000	−	11.2583i	−0.380384	−	0.658844i
$293$	27.0000			1.57736			0.788678	−	0.614806i	$-0.210766\pi$
$293$	27.0000			1.57736			0.788678	+	0.614806i	$0.210766\pi$
$294$		0			0
$295$	−14.0000			−0.815112
$296$	−4.00000	−	6.92820i	−0.232495	−	0.402694i
$297$		0			0
$298$	−4.50000	+	7.79423i	−0.260678	+	0.451508i
$299$	12.0000	+	20.7846i	0.693978	+	1.20201i
$300$		0			0
$301$	4.00000	−	20.7846i	0.230556	−	1.19800i
$302$	−17.0000			−0.978240
$303$		0			0
$304$	2.00000	−	3.46410i	0.114708	−	0.198680i
$305$		0			0
$306$		0			0
$307$	−2.00000			−0.114146			−0.0570730	−	0.998370i	$-0.518177\pi$
$307$	−2.00000			−0.114146			−0.0570730	+	0.998370i	$0.518177\pi$
$308$	−12.5000	+	4.33013i	−0.712254	+	0.246732i
$309$		0			0
$310$	3.00000	+	5.19615i	0.170389	+	0.295122i
$311$	−11.0000	+	19.0526i	−0.623753	+	1.08037i	0.365028	+	0.930997i	$0.381059\pi$
$311$	−11.0000	+	19.0526i	−0.623753	+	1.08037i	−0.988781	+	0.149375i	$0.952274\pi$
$312$		0			0
$313$	−5.00000	−	8.66025i	−0.282617	−	0.489506i	0.689412	−	0.724370i	$-0.257869\pi$
$313$	−5.00000	−	8.66025i	−0.282617	−	0.489506i	−0.972028	+	0.234863i	$0.924536\pi$
$314$	14.0000			0.790066
$315$		0			0
$316$	−3.00000			−0.168763
$317$	16.5000	+	28.5788i	0.926732	+	1.60515i	0.788751	+	0.614713i	$0.210728\pi$
$317$	16.5000	+	28.5788i	0.926732	+	1.60515i	0.137981	+	0.990435i	$0.455939\pi$
$318$		0			0
$319$	−17.5000	+	30.3109i	−0.979812	+	1.69708i
$320$	1.00000	+	1.73205i	0.0559017	+	0.0968246i
$321$		0			0
$322$	−8.00000	−	6.92820i	−0.445823	−	0.386094i
$323$	16.0000			0.890264
$324$		0			0
$325$	3.00000	−	5.19615i	0.166410	−	0.288231i
$326$	−1.00000	+	1.73205i	−0.0553849	+	0.0959294i
$327$		0			0
$328$	−6.00000			−0.331295
$329$	−15.0000	+	5.19615i	−0.826977	+	0.286473i
$330$		0			0
$331$	−16.0000	−	27.7128i	−0.879440	−	1.52323i	−0.851957	−	0.523612i	$-0.824584\pi$
$331$	−16.0000	−	27.7128i	−0.879440	−	1.52323i	−0.0274825	−	0.999622i	$-0.508749\pi$
$332$	3.50000	−	6.06218i	0.192087	−	0.332705i
$333$		0			0
$334$	7.00000	+	12.1244i	0.383023	+	0.663415i
$335$	−20.0000			−1.09272
$336$		0			0
$337$	−27.0000			−1.47078			−0.735392	−	0.677642i	$-0.763002\pi$
$337$	−27.0000			−1.47078			−0.735392	+	0.677642i	$0.763002\pi$
$338$	−11.5000	−	19.9186i	−0.625518	−	1.08343i
$339$		0			0
$340$	−4.00000	+	6.92820i	−0.216930	+	0.375735i
$341$	−7.50000	−	12.9904i	−0.406148	−	0.703469i
$342$		0			0
$343$	−10.0000	+	15.5885i	−0.539949	+	0.841698i
$344$	8.00000			0.431331
$345$		0			0
$346$	−0.500000	+	0.866025i	−0.0268802	+	0.0465578i
$347$	13.5000	−	23.3827i	0.724718	−	1.25525i	−0.234372	−	0.972147i	$-0.575303\pi$
$347$	13.5000	−	23.3827i	0.724718	−	1.25525i	0.959090	−	0.283101i	$-0.0913633\pi$
$348$		0			0
$349$	−20.0000			−1.07058			−0.535288	−	0.844670i	$-0.679797\pi$
$349$	−20.0000			−1.07058			−0.535288	+	0.844670i	$0.679797\pi$
$350$	−0.500000	+	2.59808i	−0.0267261	+	0.138873i
$351$		0			0
$352$	−2.50000	−	4.33013i	−0.133250	−	0.230797i
$353$	15.0000	−	25.9808i	0.798369	−	1.38282i	−0.122308	−	0.992492i	$-0.539030\pi$
$353$	15.0000	−	25.9808i	0.798369	−	1.38282i	0.920677	−	0.390324i	$-0.127637\pi$
$354$		0			0
$355$	−4.00000	−	6.92820i	−0.212298	−	0.367711i
$356$	6.00000			0.317999
$357$		0			0
$358$	−15.0000			−0.792775
$359$	−8.00000	−	13.8564i	−0.422224	−	0.731313i	0.573933	−	0.818902i	$-0.305417\pi$
$359$	−8.00000	−	13.8564i	−0.422224	−	0.731313i	−0.996157	+	0.0875892i	$0.972084\pi$
$360$		0			0
$361$	1.50000	−	2.59808i	0.0789474	−	0.136741i
$362$		0			0
$363$		0			0
$364$	12.0000	+	10.3923i	0.628971	+	0.544705i
$365$	−26.0000			−1.36090
$366$		0			0
$367$	2.00000	−	3.46410i	0.104399	−	0.180825i	−0.809093	−	0.587680i	$-0.800041\pi$
$367$	2.00000	−	3.46410i	0.104399	−	0.180825i	0.913493	+	0.406855i	$0.133375\pi$
$368$	2.00000	−	3.46410i	0.104257	−	0.180579i
$369$		0			0
$370$	−16.0000			−0.831800
$371$	−12.0000	−	10.3923i	−0.623009	−	0.539542i
$372$		0			0
$373$	10.0000	+	17.3205i	0.517780	+	0.896822i	0.999787	+	0.0206542i	$0.00657489\pi$
$373$	10.0000	+	17.3205i	0.517780	+	0.896822i	−0.482006	+	0.876168i	$0.660092\pi$
$374$	10.0000	−	17.3205i	0.517088	−	0.895622i
$375$		0			0
$376$	−3.00000	−	5.19615i	−0.154713	−	0.267971i
$377$	42.0000			2.16311
$378$		0			0
$379$	10.0000			0.513665			0.256833	−	0.966456i	$-0.417321\pi$
$379$	10.0000			0.513665			0.256833	+	0.966456i	$0.417321\pi$
$380$	−4.00000	−	6.92820i	−0.205196	−	0.355409i
$381$		0			0
$382$	9.00000	−	15.5885i	0.460480	−	0.797575i
$383$	2.00000	+	3.46410i	0.102195	+	0.177007i	0.912589	−	0.408879i	$-0.134080\pi$
$383$	2.00000	+	3.46410i	0.102195	+	0.177007i	−0.810394	+	0.585886i	$0.800747\pi$
$384$		0			0
$385$	−5.00000	+	25.9808i	−0.254824	+	1.32410i
$386$	−19.0000			−0.967075
$387$		0			0
$388$	2.50000	−	4.33013i	0.126918	−	0.219829i
$389$	−0.500000	+	0.866025i	−0.0253510	+	0.0439092i	−0.878423	−	0.477885i	$-0.841404\pi$
$389$	−0.500000	+	0.866025i	−0.0253510	+	0.0439092i	0.853072	+	0.521794i	$0.174737\pi$
$390$		0			0
$391$	16.0000			0.809155
$392$	−6.50000	−	2.59808i	−0.328300	−	0.131223i
$393$		0			0
$394$	12.5000	+	21.6506i	0.629741	+	1.09074i
$395$	−3.00000	+	5.19615i	−0.150946	+	0.261447i
$396$		0			0
$397$	9.00000	+	15.5885i	0.451697	+	0.782362i	0.998492	−	0.0549046i	$-0.0174855\pi$
$397$	9.00000	+	15.5885i	0.451697	+	0.782362i	−0.546795	+	0.837267i	$0.684152\pi$
$398$	−19.0000			−0.952384
$399$		0			0
$400$	−1.00000			−0.0500000
$401$	9.00000	+	15.5885i	0.449439	+	0.778450i	0.998350	−	0.0574304i	$-0.0182907\pi$
$401$	9.00000	+	15.5885i	0.449439	+	0.778450i	−0.548911	+	0.835881i	$0.684957\pi$
$402$		0			0
$403$	−9.00000	+	15.5885i	−0.448322	+	0.776516i
$404$	2.50000	+	4.33013i	0.124380	+	0.215432i
$405$		0			0
$406$	−17.5000	+	6.06218i	−0.868510	+	0.300861i
$407$	40.0000			1.98273
$408$		0			0
$409$	5.00000	−	8.66025i	0.247234	−	0.428222i	−0.715523	−	0.698589i	$-0.753812\pi$
$409$	5.00000	−	8.66025i	0.247234	−	0.428222i	0.962757	+	0.270367i	$0.0871450\pi$
$410$	−6.00000	+	10.3923i	−0.296319	+	0.513239i
$411$		0			0
$412$	8.00000			0.394132
$413$	14.0000	+	12.1244i	0.688895	+	0.596601i
$414$		0			0
$415$	−7.00000	−	12.1244i	−0.343616	−	0.595161i
$416$	−3.00000	+	5.19615i	−0.147087	+	0.254762i
$417$		0			0
$418$	10.0000	+	17.3205i	0.489116	+	0.847174i
$419$	12.0000			0.586238			0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000			0.586238			0.293119	+	0.956076i	$0.405307\pi$
$420$		0			0
$421$	−18.0000			−0.877266			−0.438633	−	0.898666i	$-0.644537\pi$
$421$	−18.0000			−0.877266			−0.438633	+	0.898666i	$0.644537\pi$
$422$	−13.0000	−	22.5167i	−0.632830	−	1.09609i
$423$		0			0
$424$	3.00000	−	5.19615i	0.145693	−	0.252347i
$425$	−2.00000	−	3.46410i	−0.0970143	−	0.168034i
$426$		0			0
$427$		0			0
$428$	12.0000			0.580042
$429$		0			0
$430$	8.00000	−	13.8564i	0.385794	−	0.668215i
$431$	9.00000	−	15.5885i	0.433515	−	0.750870i	−0.563658	−	0.826008i	$-0.690607\pi$
$431$	9.00000	−	15.5885i	0.433515	−	0.750870i	0.997173	+	0.0751385i	$0.0239399\pi$
$432$		0			0
$433$	−7.00000			−0.336399			−0.168199	−	0.985753i	$-0.553795\pi$
$433$	−7.00000			−0.336399			−0.168199	+	0.985753i	$0.553795\pi$
$434$	1.50000	−	7.79423i	0.0720023	−	0.374135i
$435$		0			0
$436$	−8.00000	−	13.8564i	−0.383131	−	0.663602i
$437$	−8.00000	+	13.8564i	−0.382692	+	0.662842i
$438$		0			0
$439$	−1.50000	−	2.59808i	−0.0715911	−	0.123999i	0.828008	−	0.560717i	$-0.189474\pi$
$439$	−1.50000	−	2.59808i	−0.0715911	−	0.123999i	−0.899599	+	0.436717i	$0.856141\pi$
$440$	−10.0000			−0.476731
$441$		0			0
$442$	−24.0000			−1.14156
$443$	−5.50000	−	9.52628i	−0.261313	−	0.452607i	0.705278	−	0.708931i	$-0.250822\pi$
$443$	−5.50000	−	9.52628i	−0.261313	−	0.452607i	−0.966591	+	0.256323i	$0.917489\pi$
$444$		0			0
$445$	6.00000	−	10.3923i	0.284427	−	0.492642i
$446$	0.500000	+	0.866025i	0.0236757	+	0.0410075i
$447$		0			0
$448$	0.500000	−	2.59808i	0.0236228	−	0.122748i
$449$	−14.0000			−0.660701			−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000			−0.660701			−0.330350	+	0.943858i	$0.607167\pi$
$450$		0			0
$451$	15.0000	−	25.9808i	0.706322	−	1.22339i
$452$	−2.00000	+	3.46410i	−0.0940721	+	0.162938i
$453$		0			0
$454$	−27.0000			−1.26717
$455$	30.0000	−	10.3923i	1.40642	−	0.487199i
$456$		0			0
$457$	13.0000	+	22.5167i	0.608114	+	1.05328i	0.991551	+	0.129718i	$0.0414071\pi$
$457$	13.0000	+	22.5167i	0.608114	+	1.05328i	−0.383437	+	0.923567i	$0.625260\pi$
$458$	−2.00000	+	3.46410i	−0.0934539	+	0.161867i
$459$		0			0
$460$	−4.00000	−	6.92820i	−0.186501	−	0.323029i
$461$	−23.0000			−1.07122			−0.535608	−	0.844466i	$-0.679918\pi$
$461$	−23.0000			−1.07122			−0.535608	+	0.844466i	$0.679918\pi$
$462$		0			0
$463$	−29.0000			−1.34774			−0.673872	−	0.738848i	$-0.735370\pi$
$463$	−29.0000			−1.34774			−0.673872	+	0.738848i	$0.735370\pi$
$464$	−3.50000	−	6.06218i	−0.162483	−	0.281430i
$465$		0			0
$466$	11.0000	−	19.0526i	0.509565	−	0.882593i
$467$	−3.50000	−	6.06218i	−0.161961	−	0.280524i	0.773611	−	0.633661i	$-0.218448\pi$
$467$	−3.50000	−	6.06218i	−0.161961	−	0.280524i	−0.935572	+	0.353137i	$0.885115\pi$
$468$		0			0
$469$	20.0000	+	17.3205i	0.923514	+	0.799787i
$470$	−12.0000			−0.553519
$471$		0			0
$472$	−3.50000	+	6.06218i	−0.161101	+	0.279034i
$473$	−20.0000	+	34.6410i	−0.919601	+	1.59280i
$474$		0			0
$475$	4.00000			0.183533
$476$	10.0000	−	3.46410i	0.458349	−	0.158777i
$477$		0			0
$478$		0			0
$479$	−13.0000	+	22.5167i	−0.593985	+	1.02881i	0.399704	+	0.916644i	$0.369113\pi$
$479$	−13.0000	+	22.5167i	−0.593985	+	1.02881i	−0.993689	+	0.112168i	$0.964220\pi$
$480$		0			0
$481$	−24.0000	−	41.5692i	−1.09431	−	1.89539i
$482$	−1.00000			−0.0455488
$483$		0			0
$484$	14.0000			0.636364
$485$	−5.00000	−	8.66025i	−0.227038	−	0.393242i
$486$		0			0
$487$	6.50000	−	11.2583i	0.294543	−	0.510164i	−0.680335	−	0.732901i	$-0.738166\pi$
$487$	6.50000	−	11.2583i	0.294543	−	0.510164i	0.974879	+	0.222737i	$0.0714992\pi$
$488$		0			0
$489$		0			0
$490$	−11.0000	+	8.66025i	−0.496929	+	0.391230i
$491$	12.0000			0.541552			0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000			0.541552			0.270776	+	0.962642i	$0.412720\pi$
$492$		0			0
$493$	14.0000	−	24.2487i	0.630528	−	1.09211i
$494$	12.0000	−	20.7846i	0.539906	−	0.935144i
$495$		0			0
$496$	3.00000			0.134704
$497$	−2.00000	+	10.3923i	−0.0897123	+	0.466159i
$498$		0			0
$499$	4.00000	+	6.92820i	0.179065	+	0.310149i	0.941560	−	0.336844i	$-0.109360\pi$
$499$	4.00000	+	6.92820i	0.179065	+	0.310149i	−0.762496	+	0.646993i	$0.776026\pi$
$500$	−6.00000	+	10.3923i	−0.268328	+	0.464758i
$501$		0			0
$502$	10.5000	+	18.1865i	0.468638	+	0.811705i
$503$	6.00000			0.267527			0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000			0.267527			0.133763	+	0.991013i	$0.457294\pi$
$504$		0			0
$505$	10.0000			0.444994
$506$	10.0000	+	17.3205i	0.444554	+	0.769991i
$507$		0			0
$508$		0			0
$509$	−1.50000	−	2.59808i	−0.0664863	−	0.115158i	0.830866	−	0.556473i	$-0.187846\pi$
$509$	−1.50000	−	2.59808i	−0.0664863	−	0.115158i	−0.897352	+	0.441315i	$0.854512\pi$
$510$		0			0
$511$	26.0000	+	22.5167i	1.15017	+	0.996078i
$512$	1.00000			0.0441942
$513$		0			0
$514$		0			0
$515$	8.00000	−	13.8564i	0.352522	−	0.610586i
$516$		0			0
$517$	30.0000			1.31940
$518$	16.0000	+	13.8564i	0.703000	+	0.608816i
$519$		0			0
$520$	6.00000	+	10.3923i	0.263117	+	0.455733i
$521$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$521$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$522$		0			0
$523$	−13.0000	−	22.5167i	−0.568450	−	0.984585i	−0.996719	−	0.0809336i	$-0.974210\pi$
$523$	−13.0000	−	22.5167i	−0.568450	−	0.984585i	0.428269	−	0.903651i	$-0.359124\pi$
$524$	13.0000			0.567908
$525$		0			0
$526$	12.0000			0.523225
$527$	6.00000	+	10.3923i	0.261364	+	0.452696i
$528$		0			0
$529$	3.50000	−	6.06218i	0.152174	−	0.263573i
$530$	−6.00000	−	10.3923i	−0.260623	−	0.451413i
$531$		0			0
$532$	−2.00000	+	10.3923i	−0.0867110	+	0.450564i
$533$	−36.0000			−1.55933
$534$		0			0
$535$	12.0000	−	20.7846i	0.518805	−	0.898597i
$536$	−5.00000	+	8.66025i	−0.215967	+	0.374066i
$537$		0			0
$538$	31.0000			1.33650
$539$	27.5000	−	21.6506i	1.18451	−	0.932559i
$540$		0			0
$541$	15.0000	+	25.9808i	0.644900	+	1.11700i	0.984325	+	0.176367i	$0.0564345\pi$
$541$	15.0000	+	25.9808i	0.644900	+	1.11700i	−0.339424	+	0.940633i	$0.610232\pi$
$542$		0			0
$543$		0			0
$544$	2.00000	+	3.46410i	0.0857493	+	0.148522i
$545$	−32.0000			−1.37073
$546$		0			0
$547$	−18.0000			−0.769624			−0.384812	−	0.922995i	$-0.625734\pi$
$547$	−18.0000			−0.769624			−0.384812	+	0.922995i	$0.625734\pi$
$548$	4.00000	+	6.92820i	0.170872	+	0.295958i
$549$		0			0
$550$	2.50000	−	4.33013i	0.106600	−	0.184637i
$551$	14.0000	+	24.2487i	0.596420	+	1.03303i
$552$		0			0
$553$	7.50000	−	2.59808i	0.318932	−	0.110481i
$554$	2.00000			0.0849719
$555$		0			0
$556$	4.00000	−	6.92820i	0.169638	−	0.293821i
$557$	5.50000	−	9.52628i	0.233042	−	0.403641i	−0.725660	−	0.688054i	$-0.758465\pi$
$557$	5.50000	−	9.52628i	0.233042	−	0.403641i	0.958702	+	0.284413i	$0.0917985\pi$
$558$		0			0
$559$	48.0000			2.03018
$560$	−4.00000	−	3.46410i	−0.169031	−	0.146385i
$561$		0			0
$562$	−1.00000	−	1.73205i	−0.0421825	−	0.0730622i
$563$	−10.0000	+	17.3205i	−0.421450	+	0.729972i	−0.996082	−	0.0884397i	$-0.971812\pi$
$563$	−10.0000	+	17.3205i	−0.421450	+	0.729972i	0.574632	+	0.818412i	$0.305145\pi$
$564$		0			0
$565$	4.00000	+	6.92820i	0.168281	+	0.291472i
$566$	16.0000			0.672530
$567$		0			0
$568$	−4.00000			−0.167836
$569$	6.00000	+	10.3923i	0.251533	+	0.435668i	0.963948	−	0.266090i	$-0.0857319\pi$
$569$	6.00000	+	10.3923i	0.251533	+	0.435668i	−0.712415	+	0.701758i	$0.752399\pi$
$570$		0			0
$571$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$571$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$572$	−15.0000	−	25.9808i	−0.627182	−	1.08631i
$573$		0			0
$574$	15.0000	−	5.19615i	0.626088	−	0.216883i
$575$	4.00000			0.166812
$576$		0			0
$577$	−21.5000	+	37.2391i	−0.895057	+	1.55028i	−0.0613223	+	0.998118i	$0.519532\pi$
$577$	−21.5000	+	37.2391i	−0.895057	+	1.55028i	−0.833734	+	0.552166i	$0.813802\pi$
$578$	0.500000	−	0.866025i	0.0207973	−	0.0360219i
$579$		0			0
$580$	−14.0000			−0.581318
$581$	−3.50000	+	18.1865i	−0.145204	+	0.754505i
$582$		0			0
$583$	15.0000	+	25.9808i	0.621237	+	1.07601i
$584$	−6.50000	+	11.2583i	−0.268972	+	0.465873i
$585$		0			0
$586$	−13.5000	−	23.3827i	−0.557680	−	0.965930i
$587$	20.0000			0.825488			0.412744	−	0.910847i	$-0.364570\pi$
$587$	20.0000			0.825488			0.412744	+	0.910847i	$0.364570\pi$
$588$		0			0
$589$	−12.0000			−0.494451
$590$	7.00000	+	12.1244i	0.288185	+	0.499152i
$591$		0			0
$592$	−4.00000	+	6.92820i	−0.164399	+	0.284747i
$593$	−18.0000	−	31.1769i	−0.739171	−	1.28028i	−0.952869	−	0.303383i	$-0.901884\pi$
$593$	−18.0000	−	31.1769i	−0.739171	−	1.28028i	0.213697	−	0.976900i	$-0.431449\pi$
$594$		0			0
$595$	4.00000	−	20.7846i	0.163984	−	0.852086i
$596$	9.00000			0.368654
$597$		0			0
$598$	12.0000	−	20.7846i	0.490716	−	0.849946i
$599$	−15.0000	+	25.9808i	−0.612883	+	1.06155i	0.377869	+	0.925859i	$0.376657\pi$
$599$	−15.0000	+	25.9808i	−0.612883	+	1.06155i	−0.990752	+	0.135686i	$0.956676\pi$
$600$		0			0
$601$	−10.0000			−0.407909			−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000			−0.407909			−0.203954	+	0.978980i	$0.565379\pi$
$602$	−20.0000	+	6.92820i	−0.815139	+	0.282372i
$603$		0			0
$604$	8.50000	+	14.7224i	0.345860	+	0.599047i
$605$	14.0000	−	24.2487i	0.569181	−	0.985850i
$606$		0			0
$607$	4.50000	+	7.79423i	0.182649	+	0.316358i	0.942782	−	0.333410i	$-0.108199\pi$
$607$	4.50000	+	7.79423i	0.182649	+	0.316358i	−0.760133	+	0.649768i	$0.774866\pi$
$608$	−4.00000			−0.162221
$609$		0			0
$610$		0			0
$611$	−18.0000	−	31.1769i	−0.728202	−	1.26128i
$612$		0			0
$613$	3.00000	−	5.19615i	0.121169	−	0.209871i	−0.799060	−	0.601251i	$-0.794669\pi$
$613$	3.00000	−	5.19615i	0.121169	−	0.209871i	0.920229	+	0.391381i	$0.128002\pi$
$614$	1.00000	+	1.73205i	0.0403567	+	0.0698999i
$615$		0			0
$616$	10.0000	+	8.66025i	0.402911	+	0.348932i
$617$	2.00000			0.0805170			0.0402585	−	0.999189i	$-0.487182\pi$
$617$	2.00000			0.0805170			0.0402585	+	0.999189i	$0.487182\pi$
$618$		0			0
$619$	−2.00000	+	3.46410i	−0.0803868	+	0.139234i	−0.903416	−	0.428765i	$-0.858949\pi$
$619$	−2.00000	+	3.46410i	−0.0803868	+	0.139234i	0.823029	+	0.567999i	$0.192282\pi$
$620$	3.00000	−	5.19615i	0.120483	−	0.208683i
$621$		0			0
$622$	22.0000			0.882120
$623$	−15.0000	+	5.19615i	−0.600962	+	0.208179i
$624$		0			0
$625$	9.50000	+	16.4545i	0.380000	+	0.658179i
$626$	−5.00000	+	8.66025i	−0.199840	+	0.346133i
$627$		0			0
$628$	−7.00000	−	12.1244i	−0.279330	−	0.483814i
$629$	−32.0000			−1.27592
$630$		0			0
$631$	5.00000			0.199047			0.0995234	−	0.995035i	$-0.468268\pi$
$631$	5.00000			0.199047			0.0995234	+	0.995035i	$0.468268\pi$
$632$	1.50000	+	2.59808i	0.0596668	+	0.103346i
$633$		0			0
$634$	16.5000	−	28.5788i	0.655299	−	1.13501i
$635$		0			0
$636$		0			0
$637$	−39.0000	−	15.5885i	−1.54524	−	0.617637i
$638$	35.0000			1.38566
$639$		0			0
$640$	1.00000	−	1.73205i	0.0395285	−	0.0684653i
$641$	−7.00000	+	12.1244i	−0.276483	+	0.478883i	−0.970508	−	0.241068i	$-0.922502\pi$
$641$	−7.00000	+	12.1244i	−0.276483	+	0.478883i	0.694025	+	0.719951i	$0.255836\pi$
$642$		0			0
$643$	14.0000			0.552106			0.276053	−	0.961142i	$-0.410973\pi$
$643$	14.0000			0.552106			0.276053	+	0.961142i	$0.410973\pi$
$644$	−2.00000	+	10.3923i	−0.0788110	+	0.409514i
$645$		0			0
$646$	−8.00000	−	13.8564i	−0.314756	−	0.545173i
$647$	9.00000	−	15.5885i	0.353827	−	0.612845i	−0.633090	−	0.774078i	$-0.718214\pi$
$647$	9.00000	−	15.5885i	0.353827	−	0.612845i	0.986916	+	0.161233i	$0.0515470\pi$
$648$		0			0
$649$	−17.5000	−	30.3109i	−0.686935	−	1.18981i
$650$	−6.00000			−0.235339
$651$		0			0
$652$	2.00000			0.0783260
$653$	−9.00000	−	15.5885i	−0.352197	−	0.610023i	0.634437	−	0.772975i	$-0.281232\pi$
$653$	−9.00000	−	15.5885i	−0.352197	−	0.610023i	−0.986634	+	0.162951i	$0.947899\pi$
$654$		0			0
$655$	13.0000	−	22.5167i	0.507952	−	0.879799i
$656$	3.00000	+	5.19615i	0.117130	+	0.202876i
$657$		0			0
$658$	12.0000	+	10.3923i	0.467809	+	0.405134i
$659$	41.0000			1.59713			0.798567	−	0.601906i	$-0.205592\pi$
$659$	41.0000			1.59713			0.798567	+	0.601906i	$0.205592\pi$
$660$		0			0
$661$	19.0000	−	32.9090i	0.739014	−	1.28001i	−0.213925	−	0.976850i	$-0.568625\pi$
$661$	19.0000	−	32.9090i	0.739014	−	1.28001i	0.952940	−	0.303160i	$-0.0980418\pi$
$662$	−16.0000	+	27.7128i	−0.621858	+	1.07709i
$663$		0			0
$664$	−7.00000			−0.271653
$665$	16.0000	+	13.8564i	0.620453	+	0.537328i
$666$		0			0
$667$	14.0000	+	24.2487i	0.542082	+	0.938914i
$668$	7.00000	−	12.1244i	0.270838	−	0.469105i
$669$		0			0
$670$	10.0000	+	17.3205i	0.386334	+	0.669150i
$671$		0			0
$672$		0			0
$673$	2.00000			0.0770943			0.0385472	−	0.999257i	$-0.487727\pi$
$673$	2.00000			0.0770943			0.0385472	+	0.999257i	$0.487727\pi$
$674$	13.5000	+	23.3827i	0.520001	+	0.900667i
$675$		0			0
$676$	−11.5000	+	19.9186i	−0.442308	+	0.766099i
$677$	−4.50000	−	7.79423i	−0.172949	−	0.299557i	0.766501	−	0.642244i	$-0.221996\pi$
$677$	−4.50000	−	7.79423i	−0.172949	−	0.299557i	−0.939450	+	0.342687i	$0.888663\pi$
$678$		0			0
$679$	−2.50000	+	12.9904i	−0.0959412	+	0.498525i
$680$	8.00000			0.306786
$681$		0			0
$682$	−7.50000	+	12.9904i	−0.287190	+	0.497427i
$683$	−10.5000	+	18.1865i	−0.401771	+	0.695888i	−0.993940	−	0.109926i	$-0.964939\pi$
$683$	−10.5000	+	18.1865i	−0.401771	+	0.695888i	0.592168	+	0.805814i	$0.298272\pi$
$684$		0			0
$685$	16.0000			0.611329
$686$	18.5000	+	0.866025i	0.706333	+	0.0330650i
$687$		0			0
$688$	−4.00000	−	6.92820i	−0.152499	−	0.264135i
$689$	18.0000	−	31.1769i	0.685745	−	1.18775i
$690$		0			0
$691$	−4.00000	−	6.92820i	−0.152167	−	0.263561i	0.779857	−	0.625958i	$-0.215292\pi$
$691$	−4.00000	−	6.92820i	−0.152167	−	0.263561i	−0.932024	+	0.362397i	$0.881959\pi$
$692$	1.00000			0.0380143
$693$		0			0
$694$	−27.0000			−1.02491
$695$	−8.00000	−	13.8564i	−0.303457	−	0.525603i
$696$		0			0
$697$	−12.0000	+	20.7846i	−0.454532	+	0.787273i
$698$	10.0000	+	17.3205i	0.378506	+	0.655591i
$699$		0			0
$700$	2.50000	−	0.866025i	0.0944911	−	0.0327327i
$701$	−14.0000			−0.528773			−0.264386	−	0.964417i	$-0.585169\pi$
$701$	−14.0000			−0.528773			−0.264386	+	0.964417i	$0.585169\pi$
$702$		0			0
$703$	16.0000	−	27.7128i	0.603451	−	1.04521i
$704$	−2.50000	+	4.33013i	−0.0942223	+	0.163198i
$705$		0			0
$706$	−30.0000			−1.12906
$707$	−10.0000	−	8.66025i	−0.376089	−	0.325702i
$708$		0			0
$709$	−4.00000	−	6.92820i	−0.150223	−	0.260194i	0.781086	−	0.624423i	$-0.214666\pi$
$709$	−4.00000	−	6.92820i	−0.150223	−	0.260194i	−0.931309	+	0.364229i	$0.881333\pi$
$710$	−4.00000	+	6.92820i	−0.150117	+	0.260011i
$711$		0			0
$712$	−3.00000	−	5.19615i	−0.112430	−	0.194734i
$713$	−12.0000			−0.449404
$714$		0			0
$715$	−60.0000			−2.24387
$716$	7.50000	+	12.9904i	0.280288	+	0.485473i
$717$		0			0
$718$	−8.00000	+	13.8564i	−0.298557	+	0.517116i
$719$	−3.00000	−	5.19615i	−0.111881	−	0.193784i	0.804648	−	0.593753i	$-0.202354\pi$
$719$	−3.00000	−	5.19615i	−0.111881	−	0.193784i	−0.916529	+	0.399969i	$0.869021\pi$
$720$		0			0
$721$	−20.0000	+	6.92820i	−0.744839	+	0.258020i
$722$	−3.00000			−0.111648
$723$		0			0
$724$		0			0
$725$	3.50000	−	6.06218i	0.129987	−	0.225144i
$726$		0			0
$727$	−32.0000			−1.18681			−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000			−1.18681			−0.593407	+	0.804902i	$0.702218\pi$
$728$	3.00000	−	15.5885i	0.111187	−	0.577747i
$729$		0			0
$730$	13.0000	+	22.5167i	0.481152	+	0.833379i
$731$	16.0000	−	27.7128i	0.591781	−	1.02500i
$732$		0			0
$733$	18.0000	+	31.1769i	0.664845	+	1.15155i	0.979327	+	0.202282i	$0.0648358\pi$
$733$	18.0000	+	31.1769i	0.664845	+	1.15155i	−0.314482	+	0.949263i	$0.601831\pi$
$734$	−4.00000			−0.147643
$735$		0			0
$736$	−4.00000			−0.147442
$737$	−25.0000	−	43.3013i	−0.920887	−	1.59502i
$738$		0			0
$739$	3.00000	−	5.19615i	0.110357	−	0.191144i	−0.805557	−	0.592518i	$-0.798134\pi$
$739$	3.00000	−	5.19615i	0.110357	−	0.191144i	0.915914	+	0.401374i	$0.131467\pi$
$740$	8.00000	+	13.8564i	0.294086	+	0.509372i
$741$		0			0
$742$	−3.00000	+	15.5885i	−0.110133	+	0.572270i
$743$	−6.00000			−0.220119			−0.110059	−	0.993925i	$-0.535104\pi$
$743$	−6.00000			−0.220119			−0.110059	+	0.993925i	$0.535104\pi$
$744$		0			0
$745$	9.00000	−	15.5885i	0.329734	−	0.571117i
$746$	10.0000	−	17.3205i	0.366126	−	0.634149i
$747$		0			0
$748$	−20.0000			−0.731272
$749$	−30.0000	+	10.3923i	−1.09618	+	0.379727i
$750$		0			0
$751$	−6.00000	−	10.3923i	−0.218943	−	0.379221i	0.735542	−	0.677479i	$-0.236928\pi$
$751$	−6.00000	−	10.3923i	−0.218943	−	0.379221i	−0.954485	+	0.298259i	$0.903594\pi$
$752$	−3.00000	+	5.19615i	−0.109399	+	0.189484i
$753$		0			0
$754$	−21.0000	−	36.3731i	−0.764775	−	1.32463i
$755$	34.0000			1.23739
$756$		0			0
$757$	−18.0000			−0.654221			−0.327111	−	0.944986i	$-0.606075\pi$
$757$	−18.0000			−0.654221			−0.327111	+	0.944986i	$0.606075\pi$
$758$	−5.00000	−	8.66025i	−0.181608	−	0.314555i
$759$		0			0
$760$	−4.00000	+	6.92820i	−0.145095	+	0.251312i
$761$	−4.00000	−	6.92820i	−0.145000	−	0.251147i	0.784373	−	0.620289i	$-0.212985\pi$
$761$	−4.00000	−	6.92820i	−0.145000	−	0.251147i	−0.929373	+	0.369142i	$0.879652\pi$
$762$		0			0
$763$	32.0000	+	27.7128i	1.15848	+	1.00327i
$764$	−18.0000			−0.651217
$765$		0			0
$766$	2.00000	−	3.46410i	0.0722629	−	0.125163i
$767$	−21.0000	+	36.3731i	−0.758266	+	1.31336i
$768$		0			0
$769$	1.00000			0.0360609			0.0180305	−	0.999837i	$-0.494260\pi$
$769$	1.00000			0.0360609			0.0180305	+	0.999837i	$0.494260\pi$
$770$	25.0000	−	8.66025i	0.900937	−	0.312094i
$771$		0			0
$772$	9.50000	+	16.4545i	0.341912	+	0.592210i
$773$	19.0000	−	32.9090i	0.683383	−	1.18365i	−0.290560	−	0.956857i	$-0.593841\pi$
$773$	19.0000	−	32.9090i	0.683383	−	1.18365i	0.973942	−	0.226796i	$-0.0728252\pi$
$774$		0			0
$775$	1.50000	+	2.59808i	0.0538816	+	0.0933257i
$776$	−5.00000			−0.179490
$777$		0			0
$778$	1.00000			0.0358517
$779$	−12.0000	−	20.7846i	−0.429945	−	0.744686i
$780$		0			0
$781$	10.0000	−	17.3205i	0.357828	−	0.619777i
$782$	−8.00000	−	13.8564i	−0.286079	−	0.495504i
$783$		0			0
$784$	1.00000	+	6.92820i	0.0357143	+	0.247436i
$785$	−28.0000			−0.999363
$786$		0			0
$787$	−9.00000	+	15.5885i	−0.320815	+	0.555668i	−0.980656	−	0.195737i	$-0.937290\pi$
$787$	−9.00000	+	15.5885i	−0.320815	+	0.555668i	0.659841	+	0.751405i	$0.270624\pi$
$788$	12.5000	−	21.6506i	0.445294	−	0.771272i
$789$		0			0
$790$	6.00000			0.213470
$791$	2.00000	−	10.3923i	0.0711118	−	0.369508i
$792$		0			0
$793$		0			0
$794$	9.00000	−	15.5885i	0.319398	−	0.553214i
$795$		0			0
$796$	9.50000	+	16.4545i	0.336719	+	0.583214i
$797$	33.0000			1.16892			0.584460	−	0.811423i	$-0.301306\pi$
$797$	33.0000			1.16892			0.584460	+	0.811423i	$0.301306\pi$
$798$		0			0
$799$	−24.0000			−0.849059
$800$	0.500000	+	0.866025i	0.0176777	+	0.0306186i
$801$		0			0
$802$	9.00000	−	15.5885i	0.317801	−	0.550448i
$803$	−32.5000	−	56.2917i	−1.14690	−	1.98649i
$804$		0			0
$805$	16.0000	+	13.8564i	0.563926	+	0.488374i
$806$	18.0000			0.634023
$807$		0			0
$808$	2.50000	−	4.33013i	0.0879497	−	0.152333i
$809$	−17.0000	+	29.4449i	−0.597688	+	1.03523i	0.395473	+	0.918477i	$0.370581\pi$
$809$	−17.0000	+	29.4449i	−0.597688	+	1.03523i	−0.993161	+	0.116749i	$0.962753\pi$
$810$		0			0
$811$	−50.0000			−1.75574			−0.877869	−	0.478901i	$-0.841035\pi$
$811$	−50.0000			−1.75574			−0.877869	+	0.478901i	$0.841035\pi$
$812$	14.0000	+	12.1244i	0.491304	+	0.425481i
$813$		0			0
$814$	−20.0000	−	34.6410i	−0.701000	−	1.21417i
$815$	2.00000	−	3.46410i	0.0700569	−	0.121342i
$816$		0			0
$817$	16.0000	+	27.7128i	0.559769	+	0.969549i
$818$	−10.0000			−0.349642
$819$		0			0
$820$	12.0000			0.419058
$821$	24.5000	+	42.4352i	0.855056	+	1.48100i	0.876593	+	0.481232i	$0.159811\pi$
$821$	24.5000	+	42.4352i	0.855056	+	1.48100i	−0.0215373	+	0.999768i	$0.506856\pi$
$822$		0			0
$823$	−6.50000	+	11.2583i	−0.226576	+	0.392441i	−0.956791	−	0.290776i	$-0.906086\pi$
$823$	−6.50000	+	11.2583i	−0.226576	+	0.392441i	0.730215	+	0.683217i	$0.239420\pi$
$824$	−4.00000	−	6.92820i	−0.139347	−	0.241355i
$825$		0			0
$826$	3.50000	−	18.1865i	0.121781	−	0.632790i
$827$	51.0000			1.77344			0.886722	−	0.462303i	$-0.152977\pi$
$827$	51.0000			1.77344			0.886722	+	0.462303i	$0.152977\pi$
$828$		0			0
$829$	−11.0000	+	19.0526i	−0.382046	+	0.661723i	−0.991355	−	0.131210i	$-0.958114\pi$
$829$	−11.0000	+	19.0526i	−0.382046	+	0.661723i	0.609309	+	0.792933i	$0.291447\pi$
$830$	−7.00000	+	12.1244i	−0.242974	+	0.420843i
$831$		0			0
$832$	6.00000			0.208013
$833$	−22.0000	+	17.3205i	−0.762255	+	0.600120i
$834$		0			0
$835$	−14.0000	−	24.2487i	−0.484490	−	0.839161i
$836$	10.0000	−	17.3205i	0.345857	−	0.599042i
$837$		0			0
$838$	−6.00000	−	10.3923i	−0.207267	−	0.358996i
$839$	−4.00000			−0.138095			−0.0690477	−	0.997613i	$-0.521996\pi$
$839$	−4.00000			−0.138095			−0.0690477	+	0.997613i	$0.521996\pi$
$840$		0			0
$841$	20.0000			0.689655
$842$	9.00000	+	15.5885i	0.310160	+	0.537214i
$843$		0			0
$844$	−13.0000	+	22.5167i	−0.447478	+	0.775055i
$845$	23.0000	+	39.8372i	0.791224	+	1.37044i
$846$		0			0
$847$	−35.0000	+	12.1244i	−1.20261	+	0.416598i
$848$	−6.00000			−0.206041
$849$		0			0
$850$	−2.00000	+	3.46410i	−0.0685994	+	0.118818i
$851$	16.0000	−	27.7128i	0.548473	−	0.949983i
$852$		0			0
$853$	−46.0000			−1.57501			−0.787505	−	0.616308i	$-0.788628\pi$
$853$	−46.0000			−1.57501			−0.787505	+	0.616308i	$0.788628\pi$
$854$		0			0
$855$		0			0
$856$	−6.00000	−	10.3923i	−0.205076	−	0.355202i
$857$	−15.0000	+	25.9808i	−0.512390	+	0.887486i	0.487507	+	0.873119i	$0.337907\pi$
$857$	−15.0000	+	25.9808i	−0.512390	+	0.887486i	−0.999897	+	0.0143666i	$0.995427\pi$
$858$		0			0
$859$	8.00000	+	13.8564i	0.272956	+	0.472774i	0.969618	−	0.244626i	$-0.0786652\pi$
$859$	8.00000	+	13.8564i	0.272956	+	0.472774i	−0.696661	+	0.717400i	$0.745332\pi$
$860$	−16.0000			−0.545595
$861$		0			0
$862$	−18.0000			−0.613082
$863$	19.0000	+	32.9090i	0.646768	+	1.12023i	0.983890	+	0.178774i	$0.0572129\pi$
$863$	19.0000	+	32.9090i	0.646768	+	1.12023i	−0.337123	+	0.941461i	$0.609454\pi$
$864$		0			0
$865$	1.00000	−	1.73205i	0.0340010	−	0.0588915i
$866$	3.50000	+	6.06218i	0.118935	+	0.206001i
$867$		0			0
$868$	−7.50000	+	2.59808i	−0.254567	+	0.0881845i
$869$	−15.0000			−0.508840
$870$		0			0
$871$	−30.0000	+	51.9615i	−1.01651	+	1.76065i
$872$	−8.00000	+	13.8564i	−0.270914	+	0.469237i
$873$		0			0
$874$	16.0000			0.541208
$875$	6.00000	−	31.1769i	0.202837	−	1.05397i
$876$		0			0
$877$	−17.0000	−	29.4449i	−0.574049	−	0.994282i	−0.996144	−	0.0877308i	$-0.972038\pi$
$877$	−17.0000	−	29.4449i	−0.574049	−	0.994282i	0.422095	−	0.906552i	$-0.361295\pi$
$878$	−1.50000	+	2.59808i	−0.0506225	+	0.0876808i
$879$		0			0
$880$	5.00000	+	8.66025i	0.168550	+	0.291937i
$881$	−54.0000			−1.81931			−0.909653	−	0.415369i	$-0.863653\pi$
$881$	−54.0000			−1.81931			−0.909653	+	0.415369i	$0.863653\pi$
$882$		0			0
$883$	2.00000			0.0673054			0.0336527	−	0.999434i	$-0.489286\pi$
$883$	2.00000			0.0673054			0.0336527	+	0.999434i	$0.489286\pi$
$884$	12.0000	+	20.7846i	0.403604	+	0.699062i
$885$		0			0
$886$	−5.50000	+	9.52628i	−0.184776	+	0.320042i
$887$	21.0000	+	36.3731i	0.705111	+	1.22129i	0.966651	+	0.256096i	$0.0824362\pi$
$887$	21.0000	+	36.3731i	0.705111	+	1.22129i	−0.261540	+	0.965193i	$0.584230\pi$
$888$		0			0
$889$		0			0
$890$	−12.0000			−0.402241
$891$		0			0
$892$	0.500000	−	0.866025i	0.0167412	−	0.0289967i
$893$	12.0000	−	20.7846i	0.401565	−	0.695530i
$894$		0			0
$895$	30.0000			1.00279
$896$	−2.50000	+	0.866025i	−0.0835191	+	0.0289319i
$897$		0			0
$898$	7.00000	+	12.1244i	0.233593	+	0.404595i
$899$	−10.5000	+	18.1865i	−0.350195	+	0.606555i
$900$		0			0
$901$	−12.0000	−	20.7846i	−0.399778	−	0.692436i
$902$	−30.0000			−0.998891
$903$		0			0
$904$	4.00000			0.133038
$905$		0			0
$906$		0			0
$907$	21.0000	−	36.3731i	0.697294	−	1.20775i	−0.272108	−	0.962267i	$-0.587721\pi$
$907$	21.0000	−	36.3731i	0.697294	−	1.20775i	0.969401	−	0.245481i	$-0.0789459\pi$
$908$	13.5000	+	23.3827i	0.448013	+	0.775982i
$909$		0			0
$910$	−24.0000	−	20.7846i	−0.795592	−	0.689003i
$911$	36.0000			1.19273			0.596367	−	0.802712i	$-0.296610\pi$
$911$	36.0000			1.19273			0.596367	+	0.802712i	$0.296610\pi$
$912$		0			0
$913$	17.5000	−	30.3109i	0.579165	−	1.00314i
$914$	13.0000	−	22.5167i	0.430002	−	0.744785i
$915$		0			0
$916$	4.00000			0.132164
$917$	−32.5000	+	11.2583i	−1.07324	+	0.371783i
$918$		0			0
$919$	−6.50000	−	11.2583i	−0.214415	−	0.371378i	0.738676	−	0.674060i	$-0.235451\pi$
$919$	−6.50000	−	11.2583i	−0.214415	−	0.371378i	−0.953092	+	0.302682i	$0.902118\pi$
$920$	−4.00000	+	6.92820i	−0.131876	+	0.228416i
$921$		0			0
$922$	11.5000	+	19.9186i	0.378732	+	0.655984i
$923$	−24.0000			−0.789970
$924$		0			0
$925$	−8.00000			−0.263038
$926$	14.5000	+	25.1147i	0.476500	+	0.825321i
$927$		0			0
$928$	−3.50000	+	6.06218i	−0.114893	+	0.199001i
$929$	−18.0000	−	31.1769i	−0.590561	−	1.02288i	−0.994157	−	0.107944i	$-0.965573\pi$
$929$	−18.0000	−	31.1769i	−0.590561	−	1.02288i	0.403596	−	0.914937i	$-0.367760\pi$
$930$		0			0
$931$	−4.00000	−	27.7128i	−0.131095	−	0.908251i
$932$	−22.0000			−0.720634
$933$		0			0
$934$	−3.50000	+	6.06218i	−0.114523	+	0.198361i
$935$	−20.0000	+	34.6410i	−0.654070	+	1.13288i
$936$		0			0
$937$	14.0000			0.457360			0.228680	−	0.973502i	$-0.426559\pi$
$937$	14.0000			0.457360			0.228680	+	0.973502i	$0.426559\pi$
$938$	5.00000	−	25.9808i	0.163256	−	0.848302i
$939$		0			0
$940$	6.00000	+	10.3923i	0.195698	+	0.338960i
$941$	−6.50000	+	11.2583i	−0.211894	+	0.367011i	−0.952307	−	0.305141i	$-0.901296\pi$
$941$	−6.50000	+	11.2583i	−0.211894	+	0.367011i	0.740413	+	0.672152i	$0.234630\pi$
$942$		0			0
$943$	−12.0000	−	20.7846i	−0.390774	−	0.676840i
$944$	7.00000			0.227831
$945$		0			0
$946$	40.0000			1.30051
$947$	−12.5000	−	21.6506i	−0.406195	−	0.703551i	0.588264	−	0.808669i	$-0.299811\pi$
$947$	−12.5000	−	21.6506i	−0.406195	−	0.703551i	−0.994460	+	0.105118i	$0.966478\pi$
$948$		0			0
$949$	−39.0000	+	67.5500i	−1.26599	+	2.19277i
$950$	−2.00000	−	3.46410i	−0.0648886	−	0.112390i
$951$		0			0
$952$	−8.00000	−	6.92820i	−0.259281	−	0.224544i
$953$	2.00000			0.0647864			0.0323932	−	0.999475i	$-0.489687\pi$
$953$	2.00000			0.0647864			0.0323932	+	0.999475i	$0.489687\pi$
$954$		0			0
$955$	−18.0000	+	31.1769i	−0.582466	+	1.00886i
$956$		0			0
$957$		0			0
$958$	26.0000			0.840022
$959$	−16.0000	−	13.8564i	−0.516667	−	0.447447i
$960$		0			0
$961$	11.0000	+	19.0526i	0.354839	+	0.614599i
$962$	−24.0000	+	41.5692i	−0.773791	+	1.34025i
$963$		0			0
$964$	0.500000	+	0.866025i	0.0161039	+	0.0278928i
$965$	38.0000			1.22326
$966$		0			0
$967$	8.00000			0.257263			0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000			0.257263			0.128631	+	0.991692i	$0.458942\pi$
$968$	−7.00000	−	12.1244i	−0.224989	−	0.389692i
$969$		0			0
$970$	−5.00000	+	8.66025i	−0.160540	+	0.278064i
$971$	−6.00000	−	10.3923i	−0.192549	−	0.333505i	0.753545	−	0.657396i	$-0.228342\pi$
$971$	−6.00000	−	10.3923i	−0.192549	−	0.333505i	−0.946094	+	0.323891i	$0.895009\pi$
$972$		0			0
$973$	−4.00000	+	20.7846i	−0.128234	+	0.666324i
$974$	−13.0000			−0.416547
$975$		0			0
$976$		0			0
$977$	9.00000	−	15.5885i	0.287936	−	0.498719i	−0.685381	−	0.728184i	$-0.740364\pi$
$977$	9.00000	−	15.5885i	0.287936	−	0.498719i	0.973317	+	0.229465i	$0.0736978\pi$
$978$		0			0
$979$	30.0000			0.958804
$980$	13.0000	+	5.19615i	0.415270	+	0.165985i
$981$		0			0
$982$	−6.00000	−	10.3923i	−0.191468	−	0.331632i
$983$	−18.0000	+	31.1769i	−0.574111	+	0.994389i	0.422027	+	0.906583i	$0.361319\pi$
$983$	−18.0000	+	31.1769i	−0.574111	+	0.994389i	−0.996138	+	0.0878058i	$0.972015\pi$
$984$		0			0
$985$	−25.0000	−	43.3013i	−0.796566	−	1.37969i
$986$	−28.0000			−0.891702
$987$		0			0
$988$	−24.0000			−0.763542
$989$	16.0000	+	27.7128i	0.508770	+	0.881216i
$990$		0			0
$991$	−4.00000	+	6.92820i	−0.127064	+	0.220082i	−0.922538	−	0.385906i	$-0.873889\pi$
$991$	−4.00000	+	6.92820i	−0.127064	+	0.220082i	0.795474	+	0.605988i	$0.207222\pi$
$992$	−1.50000	−	2.59808i	−0.0476250	−	0.0824890i
$993$		0			0
$994$	10.0000	−	3.46410i	0.317181	−	0.109875i
$995$	38.0000			1.20468
$996$		0			0
$997$	29.0000	−	50.2295i	0.918439	−	1.59078i	0.116653	−	0.993173i	$-0.462784\pi$
$997$	29.0000	−	50.2295i	0.918439	−	1.59078i	0.801786	−	0.597611i	$-0.203883\pi$
$998$	4.00000	−	6.92820i	0.126618	−	0.219308i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.2.g.c.109.1	✓	2
3.2	odd	2		378.2.g.d.109.1	yes	2
7.2	even	3	inner	378.2.g.c.163.1	yes	2
7.3	odd	6		2646.2.a.bb.1.1		1
7.4	even	3		2646.2.a.t.1.1		1
9.2	odd	6		1134.2.h.o.109.1		2
9.4	even	3		1134.2.e.o.865.1		2
9.5	odd	6		1134.2.e.b.865.1		2
9.7	even	3		1134.2.h.b.109.1		2
21.2	odd	6		378.2.g.d.163.1	yes	2
21.11	odd	6		2646.2.a.k.1.1		1
21.17	even	6		2646.2.a.c.1.1		1
63.2	odd	6		1134.2.e.b.919.1		2
63.16	even	3		1134.2.e.o.919.1		2
63.23	odd	6		1134.2.h.o.541.1		2
63.58	even	3		1134.2.h.b.541.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.2.g.c.109.1	✓	2	1.1	even	1	trivial
378.2.g.c.163.1	yes	2	7.2	even	3	inner
378.2.g.d.109.1	yes	2	3.2	odd	2
378.2.g.d.163.1	yes	2	21.2	odd	6
1134.2.e.b.865.1		2	9.5	odd	6
1134.2.e.b.919.1		2	63.2	odd	6
1134.2.e.o.865.1		2	9.4	even	3
1134.2.e.o.919.1		2	63.16	even	3
1134.2.h.b.109.1		2	9.7	even	3
1134.2.h.b.541.1		2	63.58	even	3
1134.2.h.o.109.1		2	9.2	odd	6
1134.2.h.o.541.1		2	63.23	odd	6
2646.2.a.c.1.1		1	21.17	even	6
2646.2.a.k.1.1		1	21.11	odd	6
2646.2.a.t.1.1		1	7.4	even	3
2646.2.a.bb.1.1		1	7.3	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(1.00000 + 1.73205i) q^{5} +(0.500000 - 2.59808i) q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(1.00000 + 1.73205i) q^{5} +(0.500000 - 2.59808i) q^{7} +1.00000 q^{8} +(1.00000 - 1.73205i) q^{10} +(-2.50000 + 4.33013i) q^{11} +6.00000 q^{13} +(-2.50000 + 0.866025i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(2.00000 - 3.46410i) q^{17} +(2.00000 + 3.46410i) q^{19} -2.00000 q^{20} +5.00000 q^{22} +(2.00000 + 3.46410i) q^{23} +(0.500000 - 0.866025i) q^{25} +(-3.00000 - 5.19615i) q^{26} +(2.00000 + 1.73205i) q^{28} +7.00000 q^{29} +(-1.50000 + 2.59808i) q^{31} +(-0.500000 + 0.866025i) q^{32} -4.00000 q^{34} +(5.00000 - 1.73205i) q^{35} +(-4.00000 - 6.92820i) q^{37} +(2.00000 - 3.46410i) q^{38} +(1.00000 + 1.73205i) q^{40} -6.00000 q^{41} +8.00000 q^{43} +(-2.50000 - 4.33013i) q^{44} +(2.00000 - 3.46410i) q^{46} +(-3.00000 - 5.19615i) q^{47} +(-6.50000 - 2.59808i) q^{49} -1.00000 q^{50} +(-3.00000 + 5.19615i) q^{52} +(3.00000 - 5.19615i) q^{53} -10.0000 q^{55} +(0.500000 - 2.59808i) q^{56} +(-3.50000 - 6.06218i) q^{58} +(-3.50000 + 6.06218i) q^{59} +3.00000 q^{62} +1.00000 q^{64} +(6.00000 + 10.3923i) q^{65} +(-5.00000 + 8.66025i) q^{67} +(2.00000 + 3.46410i) q^{68} +(-4.00000 - 3.46410i) q^{70} -4.00000 q^{71} +(-6.50000 + 11.2583i) q^{73} +(-4.00000 + 6.92820i) q^{74} -4.00000 q^{76} +(10.0000 + 8.66025i) q^{77} +(1.50000 + 2.59808i) q^{79} +(1.00000 - 1.73205i) q^{80} +(3.00000 + 5.19615i) q^{82} -7.00000 q^{83} +8.00000 q^{85} +(-4.00000 - 6.92820i) q^{86} +(-2.50000 + 4.33013i) q^{88} +(-3.00000 - 5.19615i) q^{89} +(3.00000 - 15.5885i) q^{91} -4.00000 q^{92} +(-3.00000 + 5.19615i) q^{94} +(-4.00000 + 6.92820i) q^{95} -5.00000 q^{97} +(1.00000 + 6.92820i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} + 2 q^{5} + q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q - q^2 - q^4 + 2 * q^5 + q^7 + 2 * q^8	\( 2 q - q^{2} - q^{4} + 2 q^{5} + q^{7} + 2 q^{8} + 2 q^{10} - 5 q^{11} + 12 q^{13} - 5 q^{14} - q^{16} + 4 q^{17} + 4 q^{19} - 4 q^{20} + 10 q^{22} + 4 q^{23} + q^{25} - 6 q^{26} + 4 q^{28} + 14 q^{29} - 3 q^{31} - q^{32} - 8 q^{34} + 10 q^{35} - 8 q^{37} + 4 q^{38} + 2 q^{40} - 12 q^{41} + 16 q^{43} - 5 q^{44} + 4 q^{46} - 6 q^{47} - 13 q^{49} - 2 q^{50} - 6 q^{52} + 6 q^{53} - 20 q^{55} + q^{56} - 7 q^{58} - 7 q^{59} + 6 q^{62} + 2 q^{64} + 12 q^{65} - 10 q^{67} + 4 q^{68} - 8 q^{70} - 8 q^{71} - 13 q^{73} - 8 q^{74} - 8 q^{76} + 20 q^{77} + 3 q^{79} + 2 q^{80} + 6 q^{82} - 14 q^{83} + 16 q^{85} - 8 q^{86} - 5 q^{88} - 6 q^{89} + 6 q^{91} - 8 q^{92} - 6 q^{94} - 8 q^{95} - 10 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q - q^2 - q^4 + 2 * q^5 + q^7 + 2 * q^8 + 2 * q^10 - 5 * q^11 + 12 * q^13 - 5 * q^14 - q^16 + 4 * q^17 + 4 * q^19 - 4 * q^20 + 10 * q^22 + 4 * q^23 + q^25 - 6 * q^26 + 4 * q^28 + 14 * q^29 - 3 * q^31 - q^32 - 8 * q^34 + 10 * q^35 - 8 * q^37 + 4 * q^38 + 2 * q^40 - 12 * q^41 + 16 * q^43 - 5 * q^44 + 4 * q^46 - 6 * q^47 - 13 * q^49 - 2 * q^50 - 6 * q^52 + 6 * q^53 - 20 * q^55 + q^56 - 7 * q^58 - 7 * q^59 + 6 * q^62 + 2 * q^64 + 12 * q^65 - 10 * q^67 + 4 * q^68 - 8 * q^70 - 8 * q^71 - 13 * q^73 - 8 * q^74 - 8 * q^76 + 20 * q^77 + 3 * q^79 + 2 * q^80 + 6 * q^82 - 14 * q^83 + 16 * q^85 - 8 * q^86 - 5 * q^88 - 6 * q^89 + 6 * q^91 - 8 * q^92 - 6 * q^94 - 8 * q^95 - 10 * q^97 + 2 * q^98

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