LMFDB - Embedded newform 50.3.c.b.43.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [50,3,Mod(7,50)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(50, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("50.7");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 50 = 2 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	50.c (of order $4$, 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:	$1.36240132180$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			43.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	50.43
Dual form			50.3.c.b.7.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.00000i) q^{2} +(-3.00000 - 3.00000i) q^{3} -2.00000i q^{4} -6.00000 q^{6} +(3.00000 - 3.00000i) q^{7} +(-2.00000 - 2.00000i) q^{8} +9.00000i q^{9} +O(q^{10})$	\(q+(1.00000 - 1.00000i) q^{2} +(-3.00000 - 3.00000i) q^{3} -2.00000i q^{4} -6.00000 q^{6} +(3.00000 - 3.00000i) q^{7} +(-2.00000 - 2.00000i) q^{8} +9.00000i q^{9} +12.0000 q^{11} +(-6.00000 + 6.00000i) q^{12} +(12.0000 + 12.0000i) q^{13} -6.00000i q^{14} -4.00000 q^{16} +(-12.0000 + 12.0000i) q^{17} +(9.00000 + 9.00000i) q^{18} -20.0000i q^{19} -18.0000 q^{21} +(12.0000 - 12.0000i) q^{22} +(-3.00000 - 3.00000i) q^{23} +12.0000i q^{24} +24.0000 q^{26} +(-6.00000 - 6.00000i) q^{28} +30.0000i q^{29} -8.00000 q^{31} +(-4.00000 + 4.00000i) q^{32} +(-36.0000 - 36.0000i) q^{33} +24.0000i q^{34} +18.0000 q^{36} +(48.0000 - 48.0000i) q^{37} +(-20.0000 - 20.0000i) q^{38} -72.0000i q^{39} -48.0000 q^{41} +(-18.0000 + 18.0000i) q^{42} +(27.0000 + 27.0000i) q^{43} -24.0000i q^{44} -6.00000 q^{46} +(-27.0000 + 27.0000i) q^{47} +(12.0000 + 12.0000i) q^{48} +31.0000i q^{49} +72.0000 q^{51} +(24.0000 - 24.0000i) q^{52} +(12.0000 + 12.0000i) q^{53} -12.0000 q^{56} +(-60.0000 + 60.0000i) q^{57} +(30.0000 + 30.0000i) q^{58} +60.0000i q^{59} +32.0000 q^{61} +(-8.00000 + 8.00000i) q^{62} +(27.0000 + 27.0000i) q^{63} +8.00000i q^{64} -72.0000 q^{66} +(3.00000 - 3.00000i) q^{67} +(24.0000 + 24.0000i) q^{68} +18.0000i q^{69} -48.0000 q^{71} +(18.0000 - 18.0000i) q^{72} +(12.0000 + 12.0000i) q^{73} -96.0000i q^{74} -40.0000 q^{76} +(36.0000 - 36.0000i) q^{77} +(-72.0000 - 72.0000i) q^{78} -40.0000i q^{79} +81.0000 q^{81} +(-48.0000 + 48.0000i) q^{82} +(-93.0000 - 93.0000i) q^{83} +36.0000i q^{84} +54.0000 q^{86} +(90.0000 - 90.0000i) q^{87} +(-24.0000 - 24.0000i) q^{88} -30.0000i q^{89} +72.0000 q^{91} +(-6.00000 + 6.00000i) q^{92} +(24.0000 + 24.0000i) q^{93} +54.0000i q^{94} +24.0000 q^{96} +(-12.0000 + 12.0000i) q^{97} +(31.0000 + 31.0000i) q^{98} +108.000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 6 q^{3} - 12 q^{6} + 6 q^{7} - 4 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 - 6 * q^3 - 12 * q^6 + 6 * q^7 - 4 * q^8	$ 2 q + 2 q^{2} - 6 q^{3} - 12 q^{6} + 6 q^{7} - 4 q^{8} + 24 q^{11} - 12 q^{12} + 24 q^{13} - 8 q^{16} - 24 q^{17} + 18 q^{18} - 36 q^{21} + 24 q^{22} - 6 q^{23} + 48 q^{26} - 12 q^{28} - 16 q^{31} - 8 q^{32} - 72 q^{33} + 36 q^{36} + 96 q^{37} - 40 q^{38} - 96 q^{41} - 36 q^{42} + 54 q^{43} - 12 q^{46} - 54 q^{47} + 24 q^{48} + 144 q^{51} + 48 q^{52} + 24 q^{53} - 24 q^{56} - 120 q^{57} + 60 q^{58} + 64 q^{61} - 16 q^{62} + 54 q^{63} - 144 q^{66} + 6 q^{67} + 48 q^{68} - 96 q^{71} + 36 q^{72} + 24 q^{73} - 80 q^{76} + 72 q^{77} - 144 q^{78} + 162 q^{81} - 96 q^{82} - 186 q^{83} + 108 q^{86} + 180 q^{87} - 48 q^{88} + 144 q^{91} - 12 q^{92} + 48 q^{93} + 48 q^{96} - 24 q^{97} + 62 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 6 * q^3 - 12 * q^6 + 6 * q^7 - 4 * q^8 + 24 * q^11 - 12 * q^12 + 24 * q^13 - 8 * q^16 - 24 * q^17 + 18 * q^18 - 36 * q^21 + 24 * q^22 - 6 * q^23 + 48 * q^26 - 12 * q^28 - 16 * q^31 - 8 * q^32 - 72 * q^33 + 36 * q^36 + 96 * q^37 - 40 * q^38 - 96 * q^41 - 36 * q^42 + 54 * q^43 - 12 * q^46 - 54 * q^47 + 24 * q^48 + 144 * q^51 + 48 * q^52 + 24 * q^53 - 24 * q^56 - 120 * q^57 + 60 * q^58 + 64 * q^61 - 16 * q^62 + 54 * q^63 - 144 * q^66 + 6 * q^67 + 48 * q^68 - 96 * q^71 + 36 * q^72 + 24 * q^73 - 80 * q^76 + 72 * q^77 - 144 * q^78 + 162 * q^81 - 96 * q^82 - 186 * q^83 + 108 * q^86 + 180 * q^87 - 48 * q^88 + 144 * q^91 - 12 * q^92 + 48 * q^93 + 48 * q^96 - 24 * q^97 + 62 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/50\mathbb{Z}\right)^\times$.

$n$	$27$
$\chi(n)$	$e\left(\frac{3}{4}\right)$

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 $

$2$	1.00000	−	1.00000i	0.500000	−	0.500000i
$2$	1.00000	−	1.00000i	0.500000	−	0.500000i
$3$	−3.00000	−	3.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$3$	−3.00000	−	3.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$4$		−	2.00000i		−	0.500000i
$5$		0			0
$5$		0			0
$6$	−6.00000			−1.00000
$7$	3.00000	−	3.00000i	0.428571	−	0.428571i	−0.459570	−	0.888142i	$-0.651996\pi$
$7$	3.00000	−	3.00000i	0.428571	−	0.428571i	0.888142	+	0.459570i	$0.151996\pi$
$8$	−2.00000	−	2.00000i	−0.250000	−	0.250000i
$9$			9.00000i			1.00000i
$10$		0			0
$11$	12.0000			1.09091			0.545455	−	0.838140i	$-0.316357\pi$
$11$	12.0000			1.09091			0.545455	+	0.838140i	$0.316357\pi$
$12$	−6.00000	+	6.00000i	−0.500000	+	0.500000i
$13$	12.0000	+	12.0000i	0.923077	+	0.923077i	0.997246	−	0.0741688i	$-0.0236304\pi$
$13$	12.0000	+	12.0000i	0.923077	+	0.923077i	−0.0741688	+	0.997246i	$0.523630\pi$
$14$		−	6.00000i		−	0.428571i
$15$		0			0
$16$	−4.00000			−0.250000
$17$	−12.0000	+	12.0000i	−0.705882	+	0.705882i	−0.965667	−	0.259784i	$-0.916349\pi$
$17$	−12.0000	+	12.0000i	−0.705882	+	0.705882i	0.259784	+	0.965667i	$0.416349\pi$
$18$	9.00000	+	9.00000i	0.500000	+	0.500000i
$19$		−	20.0000i		−	1.05263i	−0.850289	−	0.526316i	$-0.823573\pi$
$19$		−	20.0000i		−	1.05263i	0.850289	−	0.526316i	$-0.176427\pi$
$20$		0			0
$21$	−18.0000			−0.857143
$22$	12.0000	−	12.0000i	0.545455	−	0.545455i
$23$	−3.00000	−	3.00000i	−0.130435	−	0.130435i	0.638875	−	0.769310i	$-0.279400\pi$
$23$	−3.00000	−	3.00000i	−0.130435	−	0.130435i	−0.769310	+	0.638875i	$0.779400\pi$
$24$			12.0000i			0.500000i
$25$		0			0
$26$	24.0000			0.923077
$27$		0			0
$28$	−6.00000	−	6.00000i	−0.214286	−	0.214286i
$29$			30.0000i			1.03448i	0.855840	+	0.517241i	$0.173041\pi$
$29$			30.0000i			1.03448i	−0.855840	+	0.517241i	$0.826959\pi$
$30$		0			0
$31$	−8.00000			−0.258065			−0.129032	−	0.991640i	$-0.541187\pi$
$31$	−8.00000			−0.258065			−0.129032	+	0.991640i	$0.541187\pi$
$32$	−4.00000	+	4.00000i	−0.125000	+	0.125000i
$33$	−36.0000	−	36.0000i	−1.09091	−	1.09091i
$34$			24.0000i			0.705882i
$35$		0			0
$36$	18.0000			0.500000
$37$	48.0000	−	48.0000i	1.29730	−	1.29730i	0.367126	−	0.930171i	$-0.380342\pi$
$37$	48.0000	−	48.0000i	1.29730	−	1.29730i	0.930171	−	0.367126i	$-0.119658\pi$
$38$	−20.0000	−	20.0000i	−0.526316	−	0.526316i
$39$		−	72.0000i		−	1.84615i
$40$		0			0
$41$	−48.0000			−1.17073			−0.585366	−	0.810769i	$-0.699049\pi$
$41$	−48.0000			−1.17073			−0.585366	+	0.810769i	$0.699049\pi$
$42$	−18.0000	+	18.0000i	−0.428571	+	0.428571i
$43$	27.0000	+	27.0000i	0.627907	+	0.627907i	0.947541	−	0.319634i	$-0.103560\pi$
$43$	27.0000	+	27.0000i	0.627907	+	0.627907i	−0.319634	+	0.947541i	$0.603560\pi$
$44$		−	24.0000i		−	0.545455i
$45$		0			0
$46$	−6.00000			−0.130435
$47$	−27.0000	+	27.0000i	−0.574468	+	0.574468i	−0.933374	−	0.358906i	$-0.883150\pi$
$47$	−27.0000	+	27.0000i	−0.574468	+	0.574468i	0.358906	+	0.933374i	$0.383150\pi$
$48$	12.0000	+	12.0000i	0.250000	+	0.250000i
$49$			31.0000i			0.632653i
$50$		0			0
$51$	72.0000			1.41176
$52$	24.0000	−	24.0000i	0.461538	−	0.461538i
$53$	12.0000	+	12.0000i	0.226415	+	0.226415i	0.811193	−	0.584778i	$-0.198818\pi$
$53$	12.0000	+	12.0000i	0.226415	+	0.226415i	−0.584778	+	0.811193i	$0.698818\pi$
$54$		0			0
$55$		0			0
$56$	−12.0000			−0.214286
$57$	−60.0000	+	60.0000i	−1.05263	+	1.05263i
$58$	30.0000	+	30.0000i	0.517241	+	0.517241i
$59$			60.0000i			1.01695i	0.861077	+	0.508475i	$0.169790\pi$
$59$			60.0000i			1.01695i	−0.861077	+	0.508475i	$0.830210\pi$
$60$		0			0
$61$	32.0000			0.524590			0.262295	−	0.964988i	$-0.415521\pi$
$61$	32.0000			0.524590			0.262295	+	0.964988i	$0.415521\pi$
$62$	−8.00000	+	8.00000i	−0.129032	+	0.129032i
$63$	27.0000	+	27.0000i	0.428571	+	0.428571i
$64$			8.00000i			0.125000i
$65$		0			0
$66$	−72.0000			−1.09091
$67$	3.00000	−	3.00000i	0.0447761	−	0.0447761i	−0.684364	−	0.729140i	$-0.739920\pi$
$67$	3.00000	−	3.00000i	0.0447761	−	0.0447761i	0.729140	+	0.684364i	$0.239920\pi$
$68$	24.0000	+	24.0000i	0.352941	+	0.352941i
$69$			18.0000i			0.260870i
$70$		0			0
$71$	−48.0000			−0.676056			−0.338028	−	0.941136i	$-0.609760\pi$
$71$	−48.0000			−0.676056			−0.338028	+	0.941136i	$0.609760\pi$
$72$	18.0000	−	18.0000i	0.250000	−	0.250000i
$73$	12.0000	+	12.0000i	0.164384	+	0.164384i	0.784505	−	0.620122i	$-0.212917\pi$
$73$	12.0000	+	12.0000i	0.164384	+	0.164384i	−0.620122	+	0.784505i	$0.712917\pi$
$74$		−	96.0000i		−	1.29730i
$75$		0			0
$76$	−40.0000			−0.526316
$77$	36.0000	−	36.0000i	0.467532	−	0.467532i
$78$	−72.0000	−	72.0000i	−0.923077	−	0.923077i
$79$		−	40.0000i		−	0.506329i	−0.967423	−	0.253165i	$-0.918529\pi$
$79$		−	40.0000i		−	0.506329i	0.967423	−	0.253165i	$-0.0814714\pi$
$80$		0			0
$81$	81.0000			1.00000
$82$	−48.0000	+	48.0000i	−0.585366	+	0.585366i
$83$	−93.0000	−	93.0000i	−1.12048	−	1.12048i	−0.991669	−	0.128813i	$-0.958883\pi$
$83$	−93.0000	−	93.0000i	−1.12048	−	1.12048i	−0.128813	−	0.991669i	$-0.541117\pi$
$84$			36.0000i			0.428571i
$85$		0			0
$86$	54.0000			0.627907
$87$	90.0000	−	90.0000i	1.03448	−	1.03448i
$88$	−24.0000	−	24.0000i	−0.272727	−	0.272727i
$89$		−	30.0000i		−	0.337079i	−0.985695	−	0.168539i	$-0.946095\pi$
$89$		−	30.0000i		−	0.337079i	0.985695	−	0.168539i	$-0.0539050\pi$
$90$		0			0
$91$	72.0000			0.791209
$92$	−6.00000	+	6.00000i	−0.0652174	+	0.0652174i
$93$	24.0000	+	24.0000i	0.258065	+	0.258065i
$94$			54.0000i			0.574468i
$95$		0			0
$96$	24.0000			0.250000
$97$	−12.0000	+	12.0000i	−0.123711	+	0.123711i	−0.766252	−	0.642540i	$-0.777880\pi$
$97$	−12.0000	+	12.0000i	−0.123711	+	0.123711i	0.642540	+	0.766252i	$0.277880\pi$
$98$	31.0000	+	31.0000i	0.316327	+	0.316327i
$99$			108.000i			1.09091i
$100$		0			0
$101$	−78.0000			−0.772277			−0.386139	−	0.922441i	$-0.626191\pi$
$101$	−78.0000			−0.772277			−0.386139	+	0.922441i	$0.626191\pi$
$102$	72.0000	−	72.0000i	0.705882	−	0.705882i
$103$	−93.0000	−	93.0000i	−0.902913	−	0.902913i	0.0927745	−	0.995687i	$-0.470426\pi$
$103$	−93.0000	−	93.0000i	−0.902913	−	0.902913i	−0.995687	+	0.0927745i	$0.970426\pi$
$104$		−	48.0000i		−	0.461538i
$105$		0			0
$106$	24.0000			0.226415
$107$	−27.0000	+	27.0000i	−0.252336	+	0.252336i	−0.821928	−	0.569591i	$-0.807101\pi$
$107$	−27.0000	+	27.0000i	−0.252336	+	0.252336i	0.569591	+	0.821928i	$0.307101\pi$
$108$		0			0
$109$		−	160.000i		−	1.46789i	−0.679209	−	0.733945i	$-0.737677\pi$
$109$		−	160.000i		−	1.46789i	0.679209	−	0.733945i	$-0.262323\pi$
$110$		0			0
$111$	−288.000			−2.59459
$112$	−12.0000	+	12.0000i	−0.107143	+	0.107143i
$113$	72.0000	+	72.0000i	0.637168	+	0.637168i	0.949856	−	0.312688i	$-0.101229\pi$
$113$	72.0000	+	72.0000i	0.637168	+	0.637168i	−0.312688	+	0.949856i	$0.601229\pi$
$114$			120.000i			1.05263i
$115$		0			0
$116$	60.0000			0.517241
$117$	−108.000	+	108.000i	−0.923077	+	0.923077i
$118$	60.0000	+	60.0000i	0.508475	+	0.508475i
$119$			72.0000i			0.605042i
$120$		0			0
$121$	23.0000			0.190083
$122$	32.0000	−	32.0000i	0.262295	−	0.262295i
$123$	144.000	+	144.000i	1.17073	+	1.17073i
$124$			16.0000i			0.129032i
$125$		0			0
$126$	54.0000			0.428571
$127$	−117.000	+	117.000i	−0.921260	+	0.921260i	−0.997119	−	0.0758587i	$-0.975830\pi$
$127$	−117.000	+	117.000i	−0.921260	+	0.921260i	0.0758587	+	0.997119i	$0.475830\pi$
$128$	8.00000	+	8.00000i	0.0625000	+	0.0625000i
$129$		−	162.000i		−	1.25581i
$130$		0			0
$131$	132.000			1.00763			0.503817	−	0.863811i	$-0.331929\pi$
$131$	132.000			1.00763			0.503817	+	0.863811i	$0.331929\pi$
$132$	−72.0000	+	72.0000i	−0.545455	+	0.545455i
$133$	−60.0000	−	60.0000i	−0.451128	−	0.451128i
$134$		−	6.00000i		−	0.0447761i
$135$		0			0
$136$	48.0000			0.352941
$137$	168.000	−	168.000i	1.22628	−	1.22628i	0.260916	−	0.965362i	$-0.415975\pi$
$137$	168.000	−	168.000i	1.22628	−	1.22628i	0.965362	−	0.260916i	$-0.0840245\pi$
$138$	18.0000	+	18.0000i	0.130435	+	0.130435i
$139$			100.000i			0.719424i	0.933063	+	0.359712i	$0.117125\pi$
$139$			100.000i			0.719424i	−0.933063	+	0.359712i	$0.882875\pi$
$140$		0			0
$141$	162.000			1.14894
$142$	−48.0000	+	48.0000i	−0.338028	+	0.338028i
$143$	144.000	+	144.000i	1.00699	+	1.00699i
$144$		−	36.0000i		−	0.250000i
$145$		0			0
$146$	24.0000			0.164384
$147$	93.0000	−	93.0000i	0.632653	−	0.632653i
$148$	−96.0000	−	96.0000i	−0.648649	−	0.648649i
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$	−248.000			−1.64238			−0.821192	−	0.570652i	$-0.806691\pi$
$151$	−248.000			−1.64238			−0.821192	+	0.570652i	$0.806691\pi$
$152$	−40.0000	+	40.0000i	−0.263158	+	0.263158i
$153$	−108.000	−	108.000i	−0.705882	−	0.705882i
$154$		−	72.0000i		−	0.467532i
$155$		0			0
$156$	−144.000			−0.923077
$157$	−72.0000	+	72.0000i	−0.458599	+	0.458599i	−0.898195	−	0.439597i	$-0.855121\pi$
$157$	−72.0000	+	72.0000i	−0.458599	+	0.458599i	0.439597	+	0.898195i	$0.355121\pi$
$158$	−40.0000	−	40.0000i	−0.253165	−	0.253165i
$159$		−	72.0000i		−	0.452830i
$160$		0			0
$161$	−18.0000			−0.111801
$162$	81.0000	−	81.0000i	0.500000	−	0.500000i
$163$	−93.0000	−	93.0000i	−0.570552	−	0.570552i	0.361731	−	0.932283i	$-0.382186\pi$
$163$	−93.0000	−	93.0000i	−0.570552	−	0.570552i	−0.932283	+	0.361731i	$0.882186\pi$
$164$			96.0000i			0.585366i
$165$		0			0
$166$	−186.000			−1.12048
$167$	3.00000	−	3.00000i	0.0179641	−	0.0179641i	−0.698068	−	0.716032i	$-0.745957\pi$
$167$	3.00000	−	3.00000i	0.0179641	−	0.0179641i	0.716032	+	0.698068i	$0.245957\pi$
$168$	36.0000	+	36.0000i	0.214286	+	0.214286i
$169$			119.000i			0.704142i
$170$		0			0
$171$	180.000			1.05263
$172$	54.0000	−	54.0000i	0.313953	−	0.313953i
$173$	−168.000	−	168.000i	−0.971098	−	0.971098i	0.0284957	−	0.999594i	$-0.490928\pi$
$173$	−168.000	−	168.000i	−0.971098	−	0.971098i	−0.999594	+	0.0284957i	$0.990928\pi$
$174$		−	180.000i		−	1.03448i
$175$		0			0
$176$	−48.0000			−0.272727
$177$	180.000	−	180.000i	1.01695	−	1.01695i
$178$	−30.0000	−	30.0000i	−0.168539	−	0.168539i
$179$			300.000i			1.67598i	0.545687	+	0.837989i	$0.316269\pi$
$179$			300.000i			1.67598i	−0.545687	+	0.837989i	$0.683731\pi$
$180$		0			0
$181$	142.000			0.784530			0.392265	−	0.919852i	$-0.371692\pi$
$181$	142.000			0.784530			0.392265	+	0.919852i	$0.371692\pi$
$182$	72.0000	−	72.0000i	0.395604	−	0.395604i
$183$	−96.0000	−	96.0000i	−0.524590	−	0.524590i
$184$			12.0000i			0.0652174i
$185$		0			0
$186$	48.0000			0.258065
$187$	−144.000	+	144.000i	−0.770053	+	0.770053i
$188$	54.0000	+	54.0000i	0.287234	+	0.287234i
$189$		0			0
$190$		0			0
$191$	192.000			1.00524			0.502618	−	0.864509i	$-0.332370\pi$
$191$	192.000			1.00524			0.502618	+	0.864509i	$0.332370\pi$
$192$	24.0000	−	24.0000i	0.125000	−	0.125000i
$193$	132.000	+	132.000i	0.683938	+	0.683938i	0.960885	−	0.276947i	$-0.0893227\pi$
$193$	132.000	+	132.000i	0.683938	+	0.683938i	−0.276947	+	0.960885i	$0.589323\pi$
$194$			24.0000i			0.123711i
$195$		0			0
$196$	62.0000			0.316327
$197$	−132.000	+	132.000i	−0.670051	+	0.670051i	−0.957728	−	0.287677i	$-0.907117\pi$
$197$	−132.000	+	132.000i	−0.670051	+	0.670051i	0.287677	+	0.957728i	$0.407117\pi$
$198$	108.000	+	108.000i	0.545455	+	0.545455i
$199$		−	160.000i		−	0.804020i	−0.915635	−	0.402010i	$-0.868312\pi$
$199$		−	160.000i		−	0.804020i	0.915635	−	0.402010i	$-0.131688\pi$
$200$		0			0
$201$	−18.0000			−0.0895522
$202$	−78.0000	+	78.0000i	−0.386139	+	0.386139i
$203$	90.0000	+	90.0000i	0.443350	+	0.443350i
$204$		−	144.000i		−	0.705882i
$205$		0			0
$206$	−186.000			−0.902913
$207$	27.0000	−	27.0000i	0.130435	−	0.130435i
$208$	−48.0000	−	48.0000i	−0.230769	−	0.230769i
$209$		−	240.000i		−	1.14833i
$210$		0			0
$211$	−28.0000			−0.132701			−0.0663507	−	0.997796i	$-0.521136\pi$
$211$	−28.0000			−0.132701			−0.0663507	+	0.997796i	$0.521136\pi$
$212$	24.0000	−	24.0000i	0.113208	−	0.113208i
$213$	144.000	+	144.000i	0.676056	+	0.676056i
$214$			54.0000i			0.252336i
$215$		0			0
$216$		0			0
$217$	−24.0000	+	24.0000i	−0.110599	+	0.110599i
$218$	−160.000	−	160.000i	−0.733945	−	0.733945i
$219$		−	72.0000i		−	0.328767i
$220$		0			0
$221$	−288.000			−1.30317
$222$	−288.000	+	288.000i	−1.29730	+	1.29730i
$223$	117.000	+	117.000i	0.524664	+	0.524664i	0.918976	−	0.394313i	$-0.129017\pi$
$223$	117.000	+	117.000i	0.524664	+	0.524664i	−0.394313	+	0.918976i	$0.629017\pi$
$224$			24.0000i			0.107143i
$225$		0			0
$226$	144.000			0.637168
$227$	93.0000	−	93.0000i	0.409692	−	0.409692i	−0.471939	−	0.881631i	$-0.656446\pi$
$227$	93.0000	−	93.0000i	0.409692	−	0.409692i	0.881631	+	0.471939i	$0.156446\pi$
$228$	120.000	+	120.000i	0.526316	+	0.526316i
$229$			370.000i			1.61572i	0.589374	+	0.807860i	$0.299374\pi$
$229$			370.000i			1.61572i	−0.589374	+	0.807860i	$0.700626\pi$
$230$		0			0
$231$	−216.000			−0.935065
$232$	60.0000	−	60.0000i	0.258621	−	0.258621i
$233$	252.000	+	252.000i	1.08155	+	1.08155i	0.996366	+	0.0851794i	$0.0271464\pi$
$233$	252.000	+	252.000i	1.08155	+	1.08155i	0.0851794	+	0.996366i	$0.472854\pi$
$234$			216.000i			0.923077i
$235$		0			0
$236$	120.000			0.508475
$237$	−120.000	+	120.000i	−0.506329	+	0.506329i
$238$	72.0000	+	72.0000i	0.302521	+	0.302521i
$239$		−	360.000i		−	1.50628i	−0.657862	−	0.753138i	$-0.728539\pi$
$239$		−	360.000i		−	1.50628i	0.657862	−	0.753138i	$-0.271461\pi$
$240$		0			0
$241$	32.0000			0.132780			0.0663900	−	0.997794i	$-0.478852\pi$
$241$	32.0000			0.132780			0.0663900	+	0.997794i	$0.478852\pi$
$242$	23.0000	−	23.0000i	0.0950413	−	0.0950413i
$243$	−243.000	−	243.000i	−1.00000	−	1.00000i
$244$		−	64.0000i		−	0.262295i
$245$		0			0
$246$	288.000			1.17073
$247$	240.000	−	240.000i	0.971660	−	0.971660i
$248$	16.0000	+	16.0000i	0.0645161	+	0.0645161i
$249$			558.000i			2.24096i
$250$		0			0
$251$	252.000			1.00398			0.501992	−	0.864872i	$-0.332601\pi$
$251$	252.000			1.00398			0.501992	+	0.864872i	$0.332601\pi$
$252$	54.0000	−	54.0000i	0.214286	−	0.214286i
$253$	−36.0000	−	36.0000i	−0.142292	−	0.142292i
$254$			234.000i			0.921260i
$255$		0			0
$256$	16.0000			0.0625000
$257$	−192.000	+	192.000i	−0.747082	+	0.747082i	−0.973930	−	0.226848i	$-0.927158\pi$
$257$	−192.000	+	192.000i	−0.747082	+	0.747082i	0.226848	+	0.973930i	$0.427158\pi$
$258$	−162.000	−	162.000i	−0.627907	−	0.627907i
$259$		−	288.000i		−	1.11197i
$260$		0			0
$261$	−270.000			−1.03448
$262$	132.000	−	132.000i	0.503817	−	0.503817i
$263$	−333.000	−	333.000i	−1.26616	−	1.26616i	−0.948056	−	0.318104i	$-0.896954\pi$
$263$	−333.000	−	333.000i	−1.26616	−	1.26616i	−0.318104	−	0.948056i	$-0.603046\pi$
$264$			144.000i			0.545455i
$265$		0			0
$266$	−120.000			−0.451128
$267$	−90.0000	+	90.0000i	−0.337079	+	0.337079i
$268$	−6.00000	−	6.00000i	−0.0223881	−	0.0223881i
$269$		−	480.000i		−	1.78439i	−0.451654	−	0.892193i	$-0.649166\pi$
$269$		−	480.000i		−	1.78439i	0.451654	−	0.892193i	$-0.350834\pi$
$270$		0			0
$271$	−88.0000			−0.324723			−0.162362	−	0.986731i	$-0.551911\pi$
$271$	−88.0000			−0.324723			−0.162362	+	0.986731i	$0.551911\pi$
$272$	48.0000	−	48.0000i	0.176471	−	0.176471i
$273$	−216.000	−	216.000i	−0.791209	−	0.791209i
$274$		−	336.000i		−	1.22628i
$275$		0			0
$276$	36.0000			0.130435
$277$	288.000	−	288.000i	1.03971	−	1.03971i	0.0405330	−	0.999178i	$-0.487094\pi$
$277$	288.000	−	288.000i	1.03971	−	1.03971i	0.999178	−	0.0405330i	$-0.0129056\pi$
$278$	100.000	+	100.000i	0.359712	+	0.359712i
$279$		−	72.0000i		−	0.258065i
$280$		0			0
$281$	−288.000			−1.02491			−0.512456	−	0.858714i	$-0.671264\pi$
$281$	−288.000			−1.02491			−0.512456	+	0.858714i	$0.671264\pi$
$282$	162.000	−	162.000i	0.574468	−	0.574468i
$283$	117.000	+	117.000i	0.413428	+	0.413428i	0.882931	−	0.469503i	$-0.155567\pi$
$283$	117.000	+	117.000i	0.413428	+	0.413428i	−0.469503	+	0.882931i	$0.655567\pi$
$284$			96.0000i			0.338028i
$285$		0			0
$286$	288.000			1.00699
$287$	−144.000	+	144.000i	−0.501742	+	0.501742i
$288$	−36.0000	−	36.0000i	−0.125000	−	0.125000i
$289$			1.00000i			0.00346021i
$290$		0			0
$291$	72.0000			0.247423
$292$	24.0000	−	24.0000i	0.0821918	−	0.0821918i
$293$	−168.000	−	168.000i	−0.573379	−	0.573379i	0.359692	−	0.933071i	$-0.382882\pi$
$293$	−168.000	−	168.000i	−0.573379	−	0.573379i	−0.933071	+	0.359692i	$0.882882\pi$
$294$		−	186.000i		−	0.632653i
$295$		0			0
$296$	−192.000			−0.648649
$297$		0			0
$298$		0			0
$299$		−	72.0000i		−	0.240803i
$300$		0			0
$301$	162.000			0.538206
$302$	−248.000	+	248.000i	−0.821192	+	0.821192i
$303$	234.000	+	234.000i	0.772277	+	0.772277i
$304$			80.0000i			0.263158i
$305$		0			0
$306$	−216.000			−0.705882
$307$	243.000	−	243.000i	0.791531	−	0.791531i	−0.190212	−	0.981743i	$-0.560918\pi$
$307$	243.000	−	243.000i	0.791531	−	0.791531i	0.981743	+	0.190212i	$0.0609176\pi$
$308$	−72.0000	−	72.0000i	−0.233766	−	0.233766i
$309$			558.000i			1.80583i
$310$		0			0
$311$	552.000			1.77492			0.887460	−	0.460885i	$-0.152468\pi$
$311$	552.000			1.77492			0.887460	+	0.460885i	$0.152468\pi$
$312$	−144.000	+	144.000i	−0.461538	+	0.461538i
$313$	−48.0000	−	48.0000i	−0.153355	−	0.153355i	0.626260	−	0.779614i	$-0.284585\pi$
$313$	−48.0000	−	48.0000i	−0.153355	−	0.153355i	−0.779614	+	0.626260i	$0.784585\pi$
$314$			144.000i			0.458599i
$315$		0			0
$316$	−80.0000			−0.253165
$317$	228.000	−	228.000i	0.719243	−	0.719243i	−0.249207	−	0.968450i	$-0.580170\pi$
$317$	228.000	−	228.000i	0.719243	−	0.719243i	0.968450	+	0.249207i	$0.0801700\pi$
$318$	−72.0000	−	72.0000i	−0.226415	−	0.226415i
$319$			360.000i			1.12853i
$320$		0			0
$321$	162.000			0.504673
$322$	−18.0000	+	18.0000i	−0.0559006	+	0.0559006i
$323$	240.000	+	240.000i	0.743034	+	0.743034i
$324$		−	162.000i		−	0.500000i
$325$		0			0
$326$	−186.000			−0.570552
$327$	−480.000	+	480.000i	−1.46789	+	1.46789i
$328$	96.0000	+	96.0000i	0.292683	+	0.292683i
$329$			162.000i			0.492401i
$330$		0			0
$331$	−148.000			−0.447130			−0.223565	−	0.974689i	$-0.571769\pi$
$331$	−148.000			−0.447130			−0.223565	+	0.974689i	$0.571769\pi$
$332$	−186.000	+	186.000i	−0.560241	+	0.560241i
$333$	432.000	+	432.000i	1.29730	+	1.29730i
$334$		−	6.00000i		−	0.0179641i
$335$		0			0
$336$	72.0000			0.214286
$337$	−192.000	+	192.000i	−0.569733	+	0.569733i	−0.932054	−	0.362321i	$-0.881985\pi$
$337$	−192.000	+	192.000i	−0.569733	+	0.569733i	0.362321	+	0.932054i	$0.381985\pi$
$338$	119.000	+	119.000i	0.352071	+	0.352071i
$339$		−	432.000i		−	1.27434i
$340$		0			0
$341$	−96.0000			−0.281525
$342$	180.000	−	180.000i	0.526316	−	0.526316i
$343$	240.000	+	240.000i	0.699708	+	0.699708i
$344$		−	108.000i		−	0.313953i
$345$		0			0
$346$	−336.000			−0.971098
$347$	−117.000	+	117.000i	−0.337176	+	0.337176i	−0.855303	−	0.518128i	$-0.826629\pi$
$347$	−117.000	+	117.000i	−0.337176	+	0.337176i	0.518128	+	0.855303i	$0.326629\pi$
$348$	−180.000	−	180.000i	−0.517241	−	0.517241i
$349$			130.000i			0.372493i	0.982503	+	0.186246i	$0.0596323\pi$
$349$			130.000i			0.372493i	−0.982503	+	0.186246i	$0.940368\pi$
$350$		0			0
$351$		0			0
$352$	−48.0000	+	48.0000i	−0.136364	+	0.136364i
$353$	−288.000	−	288.000i	−0.815864	−	0.815864i	0.169642	−	0.985506i	$-0.445739\pi$
$353$	−288.000	−	288.000i	−0.815864	−	0.815864i	−0.985506	+	0.169642i	$0.945739\pi$
$354$		−	360.000i		−	1.01695i
$355$		0			0
$356$	−60.0000			−0.168539
$357$	216.000	−	216.000i	0.605042	−	0.605042i
$358$	300.000	+	300.000i	0.837989	+	0.837989i
$359$			120.000i			0.334262i	0.985935	+	0.167131i	$0.0534503\pi$
$359$			120.000i			0.334262i	−0.985935	+	0.167131i	$0.946550\pi$
$360$		0			0
$361$	−39.0000			−0.108033
$362$	142.000	−	142.000i	0.392265	−	0.392265i
$363$	−69.0000	−	69.0000i	−0.190083	−	0.190083i
$364$		−	144.000i		−	0.395604i
$365$		0			0
$366$	−192.000			−0.524590
$367$	213.000	−	213.000i	0.580381	−	0.580381i	−0.354627	−	0.935008i	$-0.615392\pi$
$367$	213.000	−	213.000i	0.580381	−	0.580381i	0.935008	+	0.354627i	$0.115392\pi$
$368$	12.0000	+	12.0000i	0.0326087	+	0.0326087i
$369$		−	432.000i		−	1.17073i
$370$		0			0
$371$	72.0000			0.194070
$372$	48.0000	−	48.0000i	0.129032	−	0.129032i
$373$	−168.000	−	168.000i	−0.450402	−	0.450402i	0.445086	−	0.895488i	$-0.353173\pi$
$373$	−168.000	−	168.000i	−0.450402	−	0.450402i	−0.895488	+	0.445086i	$0.853173\pi$
$374$			288.000i			0.770053i
$375$		0			0
$376$	108.000			0.287234
$377$	−360.000	+	360.000i	−0.954907	+	0.954907i
$378$		0			0
$379$		−	20.0000i		−	0.0527704i	−0.999652	−	0.0263852i	$-0.991600\pi$
$379$		−	20.0000i		−	0.0527704i	0.999652	−	0.0263852i	$-0.00839965\pi$
$380$		0			0
$381$	702.000			1.84252
$382$	192.000	−	192.000i	0.502618	−	0.502618i
$383$	−123.000	−	123.000i	−0.321149	−	0.321149i	0.528059	−	0.849208i	$-0.322920\pi$
$383$	−123.000	−	123.000i	−0.321149	−	0.321149i	−0.849208	+	0.528059i	$0.822920\pi$
$384$		−	48.0000i		−	0.125000i
$385$		0			0
$386$	264.000			0.683938
$387$	−243.000	+	243.000i	−0.627907	+	0.627907i
$388$	24.0000	+	24.0000i	0.0618557	+	0.0618557i
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$	72.0000			0.184143
$392$	62.0000	−	62.0000i	0.158163	−	0.158163i
$393$	−396.000	−	396.000i	−1.00763	−	1.00763i
$394$			264.000i			0.670051i
$395$		0			0
$396$	216.000			0.545455
$397$	108.000	−	108.000i	0.272040	−	0.272040i	−0.557881	−	0.829921i	$-0.688385\pi$
$397$	108.000	−	108.000i	0.272040	−	0.272040i	0.829921	+	0.557881i	$0.188385\pi$
$398$	−160.000	−	160.000i	−0.402010	−	0.402010i
$399$			360.000i			0.902256i
$400$		0			0
$401$	−18.0000			−0.0448878			−0.0224439	−	0.999748i	$-0.507145\pi$
$401$	−18.0000			−0.0448878			−0.0224439	+	0.999748i	$0.507145\pi$
$402$	−18.0000	+	18.0000i	−0.0447761	+	0.0447761i
$403$	−96.0000	−	96.0000i	−0.238213	−	0.238213i
$404$			156.000i			0.386139i
$405$		0			0
$406$	180.000			0.443350
$407$	576.000	−	576.000i	1.41523	−	1.41523i
$408$	−144.000	−	144.000i	−0.352941	−	0.352941i
$409$		−	80.0000i		−	0.195599i	−0.995206	−	0.0977995i	$-0.968820\pi$
$409$		−	80.0000i		−	0.195599i	0.995206	−	0.0977995i	$-0.0311804\pi$
$410$		0			0
$411$	−1008.00			−2.45255
$412$	−186.000	+	186.000i	−0.451456	+	0.451456i
$413$	180.000	+	180.000i	0.435835	+	0.435835i
$414$		−	54.0000i		−	0.130435i
$415$		0			0
$416$	−96.0000			−0.230769
$417$	300.000	−	300.000i	0.719424	−	0.719424i
$418$	−240.000	−	240.000i	−0.574163	−	0.574163i
$419$			540.000i			1.28878i	0.764696	+	0.644391i	$0.222889\pi$
$419$			540.000i			1.28878i	−0.764696	+	0.644391i	$0.777111\pi$
$420$		0			0
$421$	−608.000			−1.44418			−0.722090	−	0.691799i	$-0.756818\pi$
$421$	−608.000			−1.44418			−0.722090	+	0.691799i	$0.756818\pi$
$422$	−28.0000	+	28.0000i	−0.0663507	+	0.0663507i
$423$	−243.000	−	243.000i	−0.574468	−	0.574468i
$424$		−	48.0000i		−	0.113208i
$425$		0			0
$426$	288.000			0.676056
$427$	96.0000	−	96.0000i	0.224824	−	0.224824i
$428$	54.0000	+	54.0000i	0.126168	+	0.126168i
$429$		−	864.000i		−	2.01399i
$430$		0			0
$431$	312.000			0.723898			0.361949	−	0.932198i	$-0.382111\pi$
$431$	312.000			0.723898			0.361949	+	0.932198i	$0.382111\pi$
$432$		0			0
$433$	252.000	+	252.000i	0.581986	+	0.581986i	0.935449	−	0.353463i	$-0.114996\pi$
$433$	252.000	+	252.000i	0.581986	+	0.581986i	−0.353463	+	0.935449i	$0.614996\pi$
$434$			48.0000i			0.110599i
$435$		0			0
$436$	−320.000			−0.733945
$437$	−60.0000	+	60.0000i	−0.137300	+	0.137300i
$438$	−72.0000	−	72.0000i	−0.164384	−	0.164384i
$439$		−	40.0000i		−	0.0911162i	−0.998962	−	0.0455581i	$-0.985493\pi$
$439$		−	40.0000i		−	0.0911162i	0.998962	−	0.0455581i	$-0.0145066\pi$
$440$		0			0
$441$	−279.000			−0.632653
$442$	−288.000	+	288.000i	−0.651584	+	0.651584i
$443$	−213.000	−	213.000i	−0.480813	−	0.480813i	0.424578	−	0.905391i	$-0.360422\pi$
$443$	−213.000	−	213.000i	−0.480813	−	0.480813i	−0.905391	+	0.424578i	$0.860422\pi$
$444$			576.000i			1.29730i
$445$		0			0
$446$	234.000			0.524664
$447$		0			0
$448$	24.0000	+	24.0000i	0.0535714	+	0.0535714i
$449$		−	480.000i		−	1.06904i	−0.845155	−	0.534521i	$-0.820492\pi$
$449$		−	480.000i		−	1.06904i	0.845155	−	0.534521i	$-0.179508\pi$
$450$		0			0
$451$	−576.000			−1.27716
$452$	144.000	−	144.000i	0.318584	−	0.318584i
$453$	744.000	+	744.000i	1.64238	+	1.64238i
$454$		−	186.000i		−	0.409692i
$455$		0			0
$456$	240.000			0.526316
$457$	−432.000	+	432.000i	−0.945295	+	0.945295i	−0.998579	−	0.0532840i	$-0.983031\pi$
$457$	−432.000	+	432.000i	−0.945295	+	0.945295i	0.0532840	+	0.998579i	$0.483031\pi$
$458$	370.000	+	370.000i	0.807860	+	0.807860i
$459$		0			0
$460$		0			0
$461$	222.000			0.481562			0.240781	−	0.970579i	$-0.422596\pi$
$461$	222.000			0.481562			0.240781	+	0.970579i	$0.422596\pi$
$462$	−216.000	+	216.000i	−0.467532	+	0.467532i
$463$	−213.000	−	213.000i	−0.460043	−	0.460043i	0.438626	−	0.898670i	$-0.355465\pi$
$463$	−213.000	−	213.000i	−0.460043	−	0.460043i	−0.898670	+	0.438626i	$0.855465\pi$
$464$		−	120.000i		−	0.258621i
$465$		0			0
$466$	504.000			1.08155
$467$	3.00000	−	3.00000i	0.00642398	−	0.00642398i	−0.703887	−	0.710311i	$-0.748554\pi$
$467$	3.00000	−	3.00000i	0.00642398	−	0.00642398i	0.710311	+	0.703887i	$0.248554\pi$
$468$	216.000	+	216.000i	0.461538	+	0.461538i
$469$		−	18.0000i		−	0.0383795i
$470$		0			0
$471$	432.000			0.917197
$472$	120.000	−	120.000i	0.254237	−	0.254237i
$473$	324.000	+	324.000i	0.684989	+	0.684989i
$474$			240.000i			0.506329i
$475$		0			0
$476$	144.000			0.302521
$477$	−108.000	+	108.000i	−0.226415	+	0.226415i
$478$	−360.000	−	360.000i	−0.753138	−	0.753138i
$479$			240.000i			0.501044i	0.968111	+	0.250522i	$0.0806022\pi$
$479$			240.000i			0.501044i	−0.968111	+	0.250522i	$0.919398\pi$
$480$		0			0
$481$	1152.00			2.39501
$482$	32.0000	−	32.0000i	0.0663900	−	0.0663900i
$483$	54.0000	+	54.0000i	0.111801	+	0.111801i
$484$		−	46.0000i		−	0.0950413i
$485$		0			0
$486$	−486.000			−1.00000
$487$	−627.000	+	627.000i	−1.28747	+	1.28747i	−0.351158	+	0.936316i	$0.614212\pi$
$487$	−627.000	+	627.000i	−1.28747	+	1.28747i	−0.936316	+	0.351158i	$0.885788\pi$
$488$	−64.0000	−	64.0000i	−0.131148	−	0.131148i
$489$			558.000i			1.14110i
$490$		0			0
$491$	−588.000			−1.19756			−0.598778	−	0.800915i	$-0.704347\pi$
$491$	−588.000			−1.19756			−0.598778	+	0.800915i	$0.704347\pi$
$492$	288.000	−	288.000i	0.585366	−	0.585366i
$493$	−360.000	−	360.000i	−0.730223	−	0.730223i
$494$		−	480.000i		−	0.971660i
$495$		0			0
$496$	32.0000			0.0645161
$497$	−144.000	+	144.000i	−0.289738	+	0.289738i
$498$	558.000	+	558.000i	1.12048	+	1.12048i
$499$		−	460.000i		−	0.921844i	−0.887441	−	0.460922i	$-0.847519\pi$
$499$		−	460.000i		−	0.921844i	0.887441	−	0.460922i	$-0.152481\pi$
$500$		0			0
$501$	−18.0000			−0.0359281
$502$	252.000	−	252.000i	0.501992	−	0.501992i
$503$	627.000	+	627.000i	1.24652	+	1.24652i	0.957246	+	0.289275i	$0.0934141\pi$
$503$	627.000	+	627.000i	1.24652	+	1.24652i	0.289275	+	0.957246i	$0.406586\pi$
$504$		−	108.000i		−	0.214286i
$505$		0			0
$506$	−72.0000			−0.142292
$507$	357.000	−	357.000i	0.704142	−	0.704142i
$508$	234.000	+	234.000i	0.460630	+	0.460630i
$509$		−	450.000i		−	0.884086i	−0.896994	−	0.442043i	$-0.854254\pi$
$509$		−	450.000i		−	0.884086i	0.896994	−	0.442043i	$-0.145746\pi$
$510$		0			0
$511$	72.0000			0.140900
$512$	16.0000	−	16.0000i	0.0312500	−	0.0312500i
$513$		0			0
$514$			384.000i			0.747082i
$515$		0			0
$516$	−324.000			−0.627907
$517$	−324.000	+	324.000i	−0.626692	+	0.626692i
$518$	−288.000	−	288.000i	−0.555985	−	0.555985i
$519$			1008.00i			1.94220i
$520$		0			0
$521$	−558.000			−1.07102			−0.535509	−	0.844530i	$-0.679880\pi$
$521$	−558.000			−1.07102			−0.535509	+	0.844530i	$0.679880\pi$
$522$	−270.000	+	270.000i	−0.517241	+	0.517241i
$523$	−123.000	−	123.000i	−0.235182	−	0.235182i	0.579670	−	0.814851i	$-0.303182\pi$
$523$	−123.000	−	123.000i	−0.235182	−	0.235182i	−0.814851	+	0.579670i	$0.803182\pi$
$524$		−	264.000i		−	0.503817i
$525$		0			0
$526$	−666.000			−1.26616
$527$	96.0000	−	96.0000i	0.182163	−	0.182163i
$528$	144.000	+	144.000i	0.272727	+	0.272727i
$529$		−	511.000i		−	0.965974i
$530$		0			0
$531$	−540.000			−1.01695
$532$	−120.000	+	120.000i	−0.225564	+	0.225564i
$533$	−576.000	−	576.000i	−1.08068	−	1.08068i
$534$			180.000i			0.337079i
$535$		0			0
$536$	−12.0000			−0.0223881
$537$	900.000	−	900.000i	1.67598	−	1.67598i
$538$	−480.000	−	480.000i	−0.892193	−	0.892193i
$539$			372.000i			0.690167i
$540$		0			0
$541$	542.000			1.00185			0.500924	−	0.865491i	$-0.332994\pi$
$541$	542.000			1.00185			0.500924	+	0.865491i	$0.332994\pi$
$542$	−88.0000	+	88.0000i	−0.162362	+	0.162362i
$543$	−426.000	−	426.000i	−0.784530	−	0.784530i
$544$		−	96.0000i		−	0.176471i
$545$		0			0
$546$	−432.000			−0.791209
$547$	−147.000	+	147.000i	−0.268739	+	0.268739i	−0.828592	−	0.559853i	$-0.810858\pi$
$547$	−147.000	+	147.000i	−0.268739	+	0.268739i	0.559853	+	0.828592i	$0.310858\pi$
$548$	−336.000	−	336.000i	−0.613139	−	0.613139i
$549$			288.000i			0.524590i
$550$		0			0
$551$	600.000			1.08893
$552$	36.0000	−	36.0000i	0.0652174	−	0.0652174i
$553$	−120.000	−	120.000i	−0.216998	−	0.216998i
$554$		−	576.000i		−	1.03971i
$555$		0			0
$556$	200.000			0.359712
$557$	288.000	−	288.000i	0.517056	−	0.517056i	−0.399624	−	0.916679i	$-0.630859\pi$
$557$	288.000	−	288.000i	0.517056	−	0.517056i	0.916679	+	0.399624i	$0.130859\pi$
$558$	−72.0000	−	72.0000i	−0.129032	−	0.129032i
$559$			648.000i			1.15921i
$560$		0			0
$561$	864.000			1.54011
$562$	−288.000	+	288.000i	−0.512456	+	0.512456i
$563$	477.000	+	477.000i	0.847247	+	0.847247i	0.989789	−	0.142542i	$-0.0455276\pi$
$563$	477.000	+	477.000i	0.847247	+	0.847247i	−0.142542	+	0.989789i	$0.545528\pi$
$564$		−	324.000i		−	0.574468i
$565$		0			0
$566$	234.000			0.413428
$567$	243.000	−	243.000i	0.428571	−	0.428571i
$568$	96.0000	+	96.0000i	0.169014	+	0.169014i
$569$			240.000i			0.421793i	0.977508	+	0.210896i	$0.0676382\pi$
$569$			240.000i			0.421793i	−0.977508	+	0.210896i	$0.932362\pi$
$570$		0			0
$571$	692.000			1.21191			0.605954	−	0.795499i	$-0.292791\pi$
$571$	692.000			1.21191			0.605954	+	0.795499i	$0.292791\pi$
$572$	288.000	−	288.000i	0.503497	−	0.503497i
$573$	−576.000	−	576.000i	−1.00524	−	1.00524i
$574$			288.000i			0.501742i
$575$		0			0
$576$	−72.0000			−0.125000
$577$	168.000	−	168.000i	0.291161	−	0.291161i	−0.546378	−	0.837539i	$-0.683994\pi$
$577$	168.000	−	168.000i	0.291161	−	0.291161i	0.837539	+	0.546378i	$0.183994\pi$
$578$	1.00000	+	1.00000i	0.00173010	+	0.00173010i
$579$		−	792.000i		−	1.36788i
$580$		0			0
$581$	−558.000			−0.960413
$582$	72.0000	−	72.0000i	0.123711	−	0.123711i
$583$	144.000	+	144.000i	0.246998	+	0.246998i
$584$		−	48.0000i		−	0.0821918i
$585$		0			0
$586$	−336.000			−0.573379
$587$	213.000	−	213.000i	0.362862	−	0.362862i	−0.502004	−	0.864866i	$-0.667404\pi$
$587$	213.000	−	213.000i	0.362862	−	0.362862i	0.864866	+	0.502004i	$0.167404\pi$
$588$	−186.000	−	186.000i	−0.316327	−	0.316327i
$589$			160.000i			0.271647i
$590$		0			0
$591$	792.000			1.34010
$592$	−192.000	+	192.000i	−0.324324	+	0.324324i
$593$	312.000	+	312.000i	0.526138	+	0.526138i	0.919419	−	0.393280i	$-0.128660\pi$
$593$	312.000	+	312.000i	0.526138	+	0.526138i	−0.393280	+	0.919419i	$0.628660\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−480.000	+	480.000i	−0.804020	+	0.804020i
$598$	−72.0000	−	72.0000i	−0.120401	−	0.120401i
$599$		−	240.000i		−	0.400668i	−0.979728	−	0.200334i	$-0.935797\pi$
$599$		−	240.000i		−	0.400668i	0.979728	−	0.200334i	$-0.0642027\pi$
$600$		0			0
$601$	−608.000			−1.01165			−0.505824	−	0.862637i	$-0.668811\pi$
$601$	−608.000			−1.01165			−0.505824	+	0.862637i	$0.668811\pi$
$602$	162.000	−	162.000i	0.269103	−	0.269103i
$603$	27.0000	+	27.0000i	0.0447761	+	0.0447761i
$604$			496.000i			0.821192i
$605$		0			0
$606$	468.000			0.772277
$607$	−267.000	+	267.000i	−0.439868	+	0.439868i	−0.891968	−	0.452099i	$-0.850675\pi$
$607$	−267.000	+	267.000i	−0.439868	+	0.439868i	0.452099	+	0.891968i	$0.350675\pi$
$608$	80.0000	+	80.0000i	0.131579	+	0.131579i
$609$		−	540.000i		−	0.886700i
$610$		0			0
$611$	−648.000			−1.06056
$612$	−216.000	+	216.000i	−0.352941	+	0.352941i
$613$	−228.000	−	228.000i	−0.371941	−	0.371941i	0.496243	−	0.868184i	$-0.334713\pi$
$613$	−228.000	−	228.000i	−0.371941	−	0.371941i	−0.868184	+	0.496243i	$0.834713\pi$
$614$		−	486.000i		−	0.791531i
$615$		0			0
$616$	−144.000			−0.233766
$617$	348.000	−	348.000i	0.564019	−	0.564019i	−0.366427	−	0.930447i	$-0.619419\pi$
$617$	348.000	−	348.000i	0.564019	−	0.564019i	0.930447	+	0.366427i	$0.119419\pi$
$618$	558.000	+	558.000i	0.902913	+	0.902913i
$619$		−	940.000i		−	1.51858i	−0.650753	−	0.759289i	$-0.725547\pi$
$619$		−	940.000i		−	1.51858i	0.650753	−	0.759289i	$-0.274453\pi$
$620$		0			0
$621$		0			0
$622$	552.000	−	552.000i	0.887460	−	0.887460i
$623$	−90.0000	−	90.0000i	−0.144462	−	0.144462i
$624$			288.000i			0.461538i
$625$		0			0
$626$	−96.0000			−0.153355
$627$	−720.000	+	720.000i	−1.14833	+	1.14833i
$628$	144.000	+	144.000i	0.229299	+	0.229299i
$629$			1152.00i			1.83148i
$630$		0			0
$631$	−808.000			−1.28051			−0.640254	−	0.768164i	$-0.721171\pi$
$631$	−808.000			−1.28051			−0.640254	+	0.768164i	$0.721171\pi$
$632$	−80.0000	+	80.0000i	−0.126582	+	0.126582i
$633$	84.0000	+	84.0000i	0.132701	+	0.132701i
$634$		−	456.000i		−	0.719243i
$635$		0			0
$636$	−144.000			−0.226415
$637$	−372.000	+	372.000i	−0.583987	+	0.583987i
$638$	360.000	+	360.000i	0.564263	+	0.564263i
$639$		−	432.000i		−	0.676056i
$640$		0			0
$641$	−768.000			−1.19813			−0.599064	−	0.800701i	$-0.704460\pi$
$641$	−768.000			−1.19813			−0.599064	+	0.800701i	$0.704460\pi$
$642$	162.000	−	162.000i	0.252336	−	0.252336i
$643$	477.000	+	477.000i	0.741835	+	0.741835i	0.972931	−	0.231096i	$-0.0742311\pi$
$643$	477.000	+	477.000i	0.741835	+	0.741835i	−0.231096	+	0.972931i	$0.574231\pi$
$644$			36.0000i			0.0559006i
$645$		0			0
$646$	480.000			0.743034
$647$	−627.000	+	627.000i	−0.969088	+	0.969088i	−0.999536	−	0.0304482i	$-0.990307\pi$
$647$	−627.000	+	627.000i	−0.969088	+	0.969088i	0.0304482	+	0.999536i	$0.490307\pi$
$648$	−162.000	−	162.000i	−0.250000	−	0.250000i
$649$			720.000i			1.10940i
$650$		0			0
$651$	144.000			0.221198
$652$	−186.000	+	186.000i	−0.285276	+	0.285276i
$653$	12.0000	+	12.0000i	0.0183767	+	0.0183767i	0.716235	−	0.697859i	$-0.245864\pi$
$653$	12.0000	+	12.0000i	0.0183767	+	0.0183767i	−0.697859	+	0.716235i	$0.745864\pi$
$654$			960.000i			1.46789i
$655$		0			0
$656$	192.000			0.292683
$657$	−108.000	+	108.000i	−0.164384	+	0.164384i
$658$	162.000	+	162.000i	0.246201	+	0.246201i
$659$			540.000i			0.819423i	0.912215	+	0.409712i	$0.134371\pi$
$659$			540.000i			0.819423i	−0.912215	+	0.409712i	$0.865629\pi$
$660$		0			0
$661$	352.000			0.532526			0.266263	−	0.963900i	$-0.414211\pi$
$661$	352.000			0.532526			0.266263	+	0.963900i	$0.414211\pi$
$662$	−148.000	+	148.000i	−0.223565	+	0.223565i
$663$	864.000	+	864.000i	1.30317	+	1.30317i
$664$			372.000i			0.560241i
$665$		0			0
$666$	864.000			1.29730
$667$	90.0000	−	90.0000i	0.134933	−	0.134933i
$668$	−6.00000	−	6.00000i	−0.00898204	−	0.00898204i
$669$		−	702.000i		−	1.04933i
$670$		0			0
$671$	384.000			0.572280
$672$	72.0000	−	72.0000i	0.107143	−	0.107143i
$673$	732.000	+	732.000i	1.08767	+	1.08767i	0.995768	+	0.0918988i	$0.0292936\pi$
$673$	732.000	+	732.000i	1.08767	+	1.08767i	0.0918988	+	0.995768i	$0.470706\pi$
$674$			384.000i			0.569733i
$675$		0			0
$676$	238.000			0.352071
$677$	108.000	−	108.000i	0.159527	−	0.159527i	−0.622830	−	0.782357i	$-0.714017\pi$
$677$	108.000	−	108.000i	0.159527	−	0.159527i	0.782357	+	0.622830i	$0.214017\pi$
$678$	−432.000	−	432.000i	−0.637168	−	0.637168i
$679$			72.0000i			0.106038i
$680$		0			0
$681$	−558.000			−0.819383
$682$	−96.0000	+	96.0000i	−0.140762	+	0.140762i
$683$	−933.000	−	933.000i	−1.36603	−	1.36603i	−0.866016	−	0.500016i	$-0.833327\pi$
$683$	−933.000	−	933.000i	−1.36603	−	1.36603i	−0.500016	−	0.866016i	$-0.666673\pi$
$684$		−	360.000i		−	0.526316i
$685$		0			0
$686$	480.000			0.699708
$687$	1110.00	−	1110.00i	1.61572	−	1.61572i
$688$	−108.000	−	108.000i	−0.156977	−	0.156977i
$689$			288.000i			0.417997i
$690$		0			0
$691$	−68.0000			−0.0984081			−0.0492041	−	0.998789i	$-0.515668\pi$
$691$	−68.0000			−0.0984081			−0.0492041	+	0.998789i	$0.515668\pi$
$692$	−336.000	+	336.000i	−0.485549	+	0.485549i
$693$	324.000	+	324.000i	0.467532	+	0.467532i
$694$			234.000i			0.337176i
$695$		0			0
$696$	−360.000			−0.517241
$697$	576.000	−	576.000i	0.826399	−	0.826399i
$698$	130.000	+	130.000i	0.186246	+	0.186246i
$699$		−	1512.00i		−	2.16309i
$700$		0			0
$701$	192.000			0.273894			0.136947	−	0.990578i	$-0.456271\pi$
$701$	192.000			0.273894			0.136947	+	0.990578i	$0.456271\pi$
$702$		0			0
$703$	−960.000	−	960.000i	−1.36558	−	1.36558i
$704$			96.0000i			0.136364i
$705$		0			0
$706$	−576.000			−0.815864
$707$	−234.000	+	234.000i	−0.330976	+	0.330976i
$708$	−360.000	−	360.000i	−0.508475	−	0.508475i
$709$		−	50.0000i		−	0.0705219i	−0.999378	−	0.0352609i	$-0.988774\pi$
$709$		−	50.0000i		−	0.0705219i	0.999378	−	0.0352609i	$-0.0112262\pi$
$710$		0			0
$711$	360.000			0.506329
$712$	−60.0000	+	60.0000i	−0.0842697	+	0.0842697i
$713$	24.0000	+	24.0000i	0.0336606	+	0.0336606i
$714$		−	432.000i		−	0.605042i
$715$		0			0
$716$	600.000			0.837989
$717$	−1080.00	+	1080.00i	−1.50628	+	1.50628i
$718$	120.000	+	120.000i	0.167131	+	0.167131i
$719$		−	840.000i		−	1.16829i	−0.811650	−	0.584145i	$-0.801430\pi$
$719$		−	840.000i		−	1.16829i	0.811650	−	0.584145i	$-0.198570\pi$
$720$		0			0
$721$	−558.000			−0.773925
$722$	−39.0000	+	39.0000i	−0.0540166	+	0.0540166i
$723$	−96.0000	−	96.0000i	−0.132780	−	0.132780i
$724$		−	284.000i		−	0.392265i
$725$		0			0
$726$	−138.000			−0.190083
$727$	963.000	−	963.000i	1.32462	−	1.32462i	0.414633	−	0.909989i	$-0.363910\pi$
$727$	963.000	−	963.000i	1.32462	−	1.32462i	0.909989	−	0.414633i	$-0.136090\pi$
$728$	−144.000	−	144.000i	−0.197802	−	0.197802i
$729$			729.000i			1.00000i
$730$		0			0
$731$	−648.000			−0.886457
$732$	−192.000	+	192.000i	−0.262295	+	0.262295i
$733$	72.0000	+	72.0000i	0.0982265	+	0.0982265i	0.754512	−	0.656286i	$-0.227873\pi$
$733$	72.0000	+	72.0000i	0.0982265	+	0.0982265i	−0.656286	+	0.754512i	$0.727873\pi$
$734$		−	426.000i		−	0.580381i
$735$		0			0
$736$	24.0000			0.0326087
$737$	36.0000	−	36.0000i	0.0488467	−	0.0488467i
$738$	−432.000	−	432.000i	−0.585366	−	0.585366i
$739$		−	20.0000i		−	0.0270636i	−0.999908	−	0.0135318i	$-0.995693\pi$
$739$		−	20.0000i		−	0.0270636i	0.999908	−	0.0135318i	$-0.00430744\pi$
$740$		0			0
$741$	−1440.00			−1.94332
$742$	72.0000	−	72.0000i	0.0970350	−	0.0970350i
$743$	−243.000	−	243.000i	−0.327052	−	0.327052i	0.524412	−	0.851465i	$-0.324285\pi$
$743$	−243.000	−	243.000i	−0.327052	−	0.327052i	−0.851465	+	0.524412i	$0.824285\pi$
$744$		−	96.0000i		−	0.129032i
$745$		0			0
$746$	−336.000			−0.450402
$747$	837.000	−	837.000i	1.12048	−	1.12048i
$748$	288.000	+	288.000i	0.385027	+	0.385027i
$749$			162.000i			0.216288i
$750$		0			0
$751$	1072.00			1.42743			0.713715	−	0.700436i	$-0.247011\pi$
$751$	1072.00			1.42743			0.713715	+	0.700436i	$0.247011\pi$
$752$	108.000	−	108.000i	0.143617	−	0.143617i
$753$	−756.000	−	756.000i	−1.00398	−	1.00398i
$754$			720.000i			0.954907i
$755$		0			0
$756$		0			0
$757$	408.000	−	408.000i	0.538970	−	0.538970i	−0.384257	−	0.923226i	$-0.625542\pi$
$757$	408.000	−	408.000i	0.538970	−	0.538970i	0.923226	+	0.384257i	$0.125542\pi$
$758$	−20.0000	−	20.0000i	−0.0263852	−	0.0263852i
$759$			216.000i			0.284585i
$760$		0			0
$761$	1362.00			1.78975			0.894875	−	0.446317i	$-0.147264\pi$
$761$	1362.00			1.78975			0.894875	+	0.446317i	$0.147264\pi$
$762$	702.000	−	702.000i	0.921260	−	0.921260i
$763$	−480.000	−	480.000i	−0.629096	−	0.629096i
$764$		−	384.000i		−	0.502618i
$765$		0			0
$766$	−246.000			−0.321149
$767$	−720.000	+	720.000i	−0.938722	+	0.938722i
$768$	−48.0000	−	48.0000i	−0.0625000	−	0.0625000i
$769$		−	370.000i		−	0.481144i	−0.970631	−	0.240572i	$-0.922665\pi$
$769$		−	370.000i		−	0.481144i	0.970631	−	0.240572i	$-0.0773351\pi$
$770$		0			0
$771$	1152.00			1.49416
$772$	264.000	−	264.000i	0.341969	−	0.341969i
$773$	132.000	+	132.000i	0.170763	+	0.170763i	0.787315	−	0.616551i	$-0.211471\pi$
$773$	132.000	+	132.000i	0.170763	+	0.170763i	−0.616551	+	0.787315i	$0.711471\pi$
$774$			486.000i			0.627907i
$775$		0			0
$776$	48.0000			0.0618557
$777$	−864.000	+	864.000i	−1.11197	+	1.11197i
$778$		0			0
$779$			960.000i			1.23235i
$780$		0			0
$781$	−576.000			−0.737516
$782$	72.0000	−	72.0000i	0.0920716	−	0.0920716i
$783$		0			0
$784$		−	124.000i		−	0.158163i
$785$		0			0
$786$	−792.000			−1.00763
$787$	93.0000	−	93.0000i	0.118170	−	0.118170i	−0.645549	−	0.763719i	$-0.723371\pi$
$787$	93.0000	−	93.0000i	0.118170	−	0.118170i	0.763719	+	0.645549i	$0.223371\pi$
$788$	264.000	+	264.000i	0.335025	+	0.335025i
$789$			1998.00i			2.53232i
$790$		0			0
$791$	432.000			0.546144
$792$	216.000	−	216.000i	0.272727	−	0.272727i
$793$	384.000	+	384.000i	0.484237	+	0.484237i
$794$		−	216.000i		−	0.272040i
$795$		0			0
$796$	−320.000			−0.402010
$797$	228.000	−	228.000i	0.286073	−	0.286073i	−0.549452	−	0.835525i	$-0.685164\pi$
$797$	228.000	−	228.000i	0.286073	−	0.286073i	0.835525	+	0.549452i	$0.185164\pi$
$798$	360.000	+	360.000i	0.451128	+	0.451128i
$799$		−	648.000i		−	0.811014i
$800$		0			0
$801$	270.000			0.337079
$802$	−18.0000	+	18.0000i	−0.0224439	+	0.0224439i
$803$	144.000	+	144.000i	0.179328	+	0.179328i
$804$			36.0000i			0.0447761i
$805$		0			0
$806$	−192.000			−0.238213
$807$	−1440.00	+	1440.00i	−1.78439	+	1.78439i
$808$	156.000	+	156.000i	0.193069	+	0.193069i
$809$			750.000i			0.927070i	0.886078	+	0.463535i	$0.153419\pi$
$809$			750.000i			0.927070i	−0.886078	+	0.463535i	$0.846581\pi$
$810$		0			0
$811$	412.000			0.508015			0.254007	−	0.967202i	$-0.418251\pi$
$811$	412.000			0.508015			0.254007	+	0.967202i	$0.418251\pi$
$812$	180.000	−	180.000i	0.221675	−	0.221675i
$813$	264.000	+	264.000i	0.324723	+	0.324723i
$814$		−	1152.00i		−	1.41523i
$815$		0			0
$816$	−288.000			−0.352941
$817$	540.000	−	540.000i	0.660955	−	0.660955i
$818$	−80.0000	−	80.0000i	−0.0977995	−	0.0977995i
$819$			648.000i			0.791209i
$820$		0			0
$821$	672.000			0.818514			0.409257	−	0.912419i	$-0.365788\pi$
$821$	672.000			0.818514			0.409257	+	0.912419i	$0.365788\pi$
$822$	−1008.00	+	1008.00i	−1.22628	+	1.22628i
$823$	717.000	+	717.000i	0.871203	+	0.871203i	0.992604	−	0.121401i	$-0.0387386\pi$
$823$	717.000	+	717.000i	0.871203	+	0.871203i	−0.121401	+	0.992604i	$0.538739\pi$
$824$			372.000i			0.451456i
$825$		0			0
$826$	360.000			0.435835
$827$	123.000	−	123.000i	0.148730	−	0.148730i	−0.628820	−	0.777551i	$-0.716462\pi$
$827$	123.000	−	123.000i	0.148730	−	0.148730i	0.777551	+	0.628820i	$0.216462\pi$
$828$	−54.0000	−	54.0000i	−0.0652174	−	0.0652174i
$829$		−	1280.00i		−	1.54403i	−0.635605	−	0.772014i	$-0.719249\pi$
$829$		−	1280.00i		−	1.54403i	0.635605	−	0.772014i	$-0.280751\pi$
$830$		0			0
$831$	−1728.00			−2.07942
$832$	−96.0000	+	96.0000i	−0.115385	+	0.115385i
$833$	−372.000	−	372.000i	−0.446579	−	0.446579i
$834$		−	600.000i		−	0.719424i
$835$		0			0
$836$	−480.000			−0.574163
$837$		0			0
$838$	540.000	+	540.000i	0.644391	+	0.644391i
$839$		−	1560.00i		−	1.85936i	−0.368373	−	0.929678i	$-0.620085\pi$
$839$		−	1560.00i		−	1.85936i	0.368373	−	0.929678i	$-0.379915\pi$
$840$		0			0
$841$	−59.0000			−0.0701546
$842$	−608.000	+	608.000i	−0.722090	+	0.722090i
$843$	864.000	+	864.000i	1.02491	+	1.02491i
$844$			56.0000i			0.0663507i
$845$		0			0
$846$	−486.000			−0.574468
$847$	69.0000	−	69.0000i	0.0814640	−	0.0814640i
$848$	−48.0000	−	48.0000i	−0.0566038	−	0.0566038i
$849$		−	702.000i		−	0.826855i
$850$		0			0
$851$	−288.000			−0.338425
$852$	288.000	−	288.000i	0.338028	−	0.338028i
$853$	372.000	+	372.000i	0.436108	+	0.436108i	0.890700	−	0.454592i	$-0.150215\pi$
$853$	372.000	+	372.000i	0.436108	+	0.436108i	−0.454592	+	0.890700i	$0.650215\pi$
$854$		−	192.000i		−	0.224824i
$855$		0			0
$856$	108.000			0.126168
$857$	−552.000	+	552.000i	−0.644107	+	0.644107i	−0.951563	−	0.307455i	$-0.900523\pi$
$857$	−552.000	+	552.000i	−0.644107	+	0.644107i	0.307455	+	0.951563i	$0.400523\pi$
$858$	−864.000	−	864.000i	−1.00699	−	1.00699i
$859$			620.000i			0.721769i	0.932610	+	0.360885i	$0.117525\pi$
$859$			620.000i			0.721769i	−0.932610	+	0.360885i	$0.882475\pi$
$860$		0			0
$861$	864.000			1.00348
$862$	312.000	−	312.000i	0.361949	−	0.361949i
$863$	−123.000	−	123.000i	−0.142526	−	0.142526i	0.632244	−	0.774770i	$-0.282134\pi$
$863$	−123.000	−	123.000i	−0.142526	−	0.142526i	−0.774770	+	0.632244i	$0.782134\pi$
$864$		0			0
$865$		0			0
$866$	504.000			0.581986
$867$	3.00000	−	3.00000i	0.00346021	−	0.00346021i
$868$	48.0000	+	48.0000i	0.0552995	+	0.0552995i
$869$		−	480.000i		−	0.552359i
$870$		0			0
$871$	72.0000			0.0826636
$872$	−320.000	+	320.000i	−0.366972	+	0.366972i
$873$	−108.000	−	108.000i	−0.123711	−	0.123711i
$874$			120.000i			0.137300i
$875$		0			0
$876$	−144.000			−0.164384
$877$	1128.00	−	1128.00i	1.28620	−	1.28620i	0.349128	−	0.937075i	$-0.386478\pi$
$877$	1128.00	−	1128.00i	1.28620	−	1.28620i	0.937075	−	0.349128i	$-0.113522\pi$
$878$	−40.0000	−	40.0000i	−0.0455581	−	0.0455581i
$879$			1008.00i			1.14676i
$880$		0			0
$881$	912.000			1.03519			0.517594	−	0.855627i	$-0.326828\pi$
$881$	912.000			1.03519			0.517594	+	0.855627i	$0.326828\pi$
$882$	−279.000	+	279.000i	−0.316327	+	0.316327i
$883$	957.000	+	957.000i	1.08381	+	1.08381i	0.996151	+	0.0876543i	$0.0279371\pi$
$883$	957.000	+	957.000i	1.08381	+	1.08381i	0.0876543	+	0.996151i	$0.472063\pi$
$884$			576.000i			0.651584i
$885$		0			0
$886$	−426.000			−0.480813
$887$	483.000	−	483.000i	0.544532	−	0.544532i	−0.380322	−	0.924854i	$-0.624187\pi$
$887$	483.000	−	483.000i	0.544532	−	0.544532i	0.924854	+	0.380322i	$0.124187\pi$
$888$	576.000	+	576.000i	0.648649	+	0.648649i
$889$			702.000i			0.789651i
$890$		0			0
$891$	972.000			1.09091
$892$	234.000	−	234.000i	0.262332	−	0.262332i
$893$	540.000	+	540.000i	0.604703	+	0.604703i
$894$		0			0
$895$		0			0
$896$	48.0000			0.0535714
$897$	−216.000	+	216.000i	−0.240803	+	0.240803i
$898$	−480.000	−	480.000i	−0.534521	−	0.534521i
$899$		−	240.000i		−	0.266963i
$900$		0			0
$901$	−288.000			−0.319645
$902$	−576.000	+	576.000i	−0.638581	+	0.638581i
$903$	−486.000	−	486.000i	−0.538206	−	0.538206i
$904$		−	288.000i		−	0.318584i
$905$		0			0
$906$	1488.00			1.64238
$907$	−1077.00	+	1077.00i	−1.18743	+	1.18743i	−0.209656	+	0.977775i	$0.567234\pi$
$907$	−1077.00	+	1077.00i	−1.18743	+	1.18743i	−0.977775	+	0.209656i	$0.932766\pi$
$908$	−186.000	−	186.000i	−0.204846	−	0.204846i
$909$		−	702.000i		−	0.772277i
$910$		0			0
$911$	−1128.00			−1.23820			−0.619100	−	0.785312i	$-0.712502\pi$
$911$	−1128.00			−1.23820			−0.619100	+	0.785312i	$0.712502\pi$
$912$	240.000	−	240.000i	0.263158	−	0.263158i
$913$	−1116.00	−	1116.00i	−1.22234	−	1.22234i
$914$			864.000i			0.945295i
$915$		0			0
$916$	740.000			0.807860
$917$	396.000	−	396.000i	0.431843	−	0.431843i
$918$		0			0
$919$			1600.00i			1.74102i	0.492148	+	0.870511i	$0.336212\pi$
$919$			1600.00i			1.74102i	−0.492148	+	0.870511i	$0.663788\pi$
$920$		0			0
$921$	−1458.00			−1.58306
$922$	222.000	−	222.000i	0.240781	−	0.240781i
$923$	−576.000	−	576.000i	−0.624052	−	0.624052i
$924$			432.000i			0.467532i
$925$		0			0
$926$	−426.000			−0.460043
$927$	837.000	−	837.000i	0.902913	−	0.902913i
$928$	−120.000	−	120.000i	−0.129310	−	0.129310i
$929$		−	960.000i		−	1.03337i	−0.856176	−	0.516685i	$-0.827166\pi$
$929$		−	960.000i		−	1.03337i	0.856176	−	0.516685i	$-0.172834\pi$
$930$		0			0
$931$	620.000			0.665951
$932$	504.000	−	504.000i	0.540773	−	0.540773i
$933$	−1656.00	−	1656.00i	−1.77492	−	1.77492i
$934$		−	6.00000i		−	0.00642398i
$935$		0			0
$936$	432.000			0.461538
$937$	−492.000	+	492.000i	−0.525080	+	0.525080i	−0.919101	−	0.394021i	$-0.871084\pi$
$937$	−492.000	+	492.000i	−0.525080	+	0.525080i	0.394021	+	0.919101i	$0.371084\pi$
$938$	−18.0000	−	18.0000i	−0.0191898	−	0.0191898i
$939$			288.000i			0.306709i
$940$		0			0
$941$	−738.000			−0.784272			−0.392136	−	0.919907i	$-0.628264\pi$
$941$	−738.000			−0.784272			−0.392136	+	0.919907i	$0.628264\pi$
$942$	432.000	−	432.000i	0.458599	−	0.458599i
$943$	144.000	+	144.000i	0.152704	+	0.152704i
$944$		−	240.000i		−	0.254237i
$945$		0			0
$946$	648.000			0.684989
$947$	−237.000	+	237.000i	−0.250264	+	0.250264i	−0.821079	−	0.570815i	$-0.806627\pi$
$947$	−237.000	+	237.000i	−0.250264	+	0.250264i	0.570815	+	0.821079i	$0.306627\pi$
$948$	240.000	+	240.000i	0.253165	+	0.253165i
$949$			288.000i			0.303477i
$950$		0			0
$951$	−1368.00			−1.43849
$952$	144.000	−	144.000i	0.151261	−	0.151261i
$953$	−648.000	−	648.000i	−0.679958	−	0.679958i	0.280032	−	0.959991i	$-0.409655\pi$
$953$	−648.000	−	648.000i	−0.679958	−	0.679958i	−0.959991	+	0.280032i	$0.909655\pi$
$954$			216.000i			0.226415i
$955$		0			0
$956$	−720.000			−0.753138
$957$	1080.00	−	1080.00i	1.12853	−	1.12853i
$958$	240.000	+	240.000i	0.250522	+	0.250522i
$959$		−	1008.00i		−	1.05109i
$960$		0			0
$961$	−897.000			−0.933403
$962$	1152.00	−	1152.00i	1.19751	−	1.19751i
$963$	−243.000	−	243.000i	−0.252336	−	0.252336i
$964$		−	64.0000i		−	0.0663900i
$965$		0			0
$966$	108.000			0.111801
$967$	−627.000	+	627.000i	−0.648397	+	0.648397i	−0.952606	−	0.304208i	$-0.901608\pi$
$967$	−627.000	+	627.000i	−0.648397	+	0.648397i	0.304208	+	0.952606i	$0.401608\pi$
$968$	−46.0000	−	46.0000i	−0.0475207	−	0.0475207i
$969$		−	1440.00i		−	1.48607i
$970$		0			0
$971$	−708.000			−0.729145			−0.364573	−	0.931175i	$-0.618785\pi$
$971$	−708.000			−0.729145			−0.364573	+	0.931175i	$0.618785\pi$
$972$	−486.000	+	486.000i	−0.500000	+	0.500000i
$973$	300.000	+	300.000i	0.308325	+	0.308325i
$974$			1254.00i			1.28747i
$975$		0			0
$976$	−128.000			−0.131148
$977$	−612.000	+	612.000i	−0.626407	+	0.626407i	−0.947162	−	0.320755i	$-0.896063\pi$
$977$	−612.000	+	612.000i	−0.626407	+	0.626407i	0.320755	+	0.947162i	$0.396063\pi$
$978$	558.000	+	558.000i	0.570552	+	0.570552i
$979$		−	360.000i		−	0.367722i
$980$		0			0
$981$	1440.00			1.46789
$982$	−588.000	+	588.000i	−0.598778	+	0.598778i
$983$	627.000	+	627.000i	0.637843	+	0.637843i	0.950023	−	0.312180i	$-0.101059\pi$
$983$	627.000	+	627.000i	0.637843	+	0.637843i	−0.312180	+	0.950023i	$0.601059\pi$
$984$		−	576.000i		−	0.585366i
$985$		0			0
$986$	−720.000			−0.730223
$987$	486.000	−	486.000i	0.492401	−	0.492401i
$988$	−480.000	−	480.000i	−0.485830	−	0.485830i
$989$		−	162.000i		−	0.163802i
$990$		0			0
$991$	−1168.00			−1.17861			−0.589304	−	0.807912i	$-0.700598\pi$
$991$	−1168.00			−1.17861			−0.589304	+	0.807912i	$0.700598\pi$
$992$	32.0000	−	32.0000i	0.0322581	−	0.0322581i
$993$	444.000	+	444.000i	0.447130	+	0.447130i
$994$			288.000i			0.289738i
$995$		0			0
$996$	1116.00			1.12048
$997$	108.000	−	108.000i	0.108325	−	0.108325i	−0.650867	−	0.759192i	$-0.725594\pi$
$997$	108.000	−	108.000i	0.108325	−	0.108325i	0.759192	+	0.650867i	$0.225594\pi$
$998$	−460.000	−	460.000i	−0.460922	−	0.460922i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.3.c.b.43.1	yes	2
3.2	odd	2		450.3.g.c.343.1		2
4.3	odd	2		400.3.p.g.193.1		2
5.2	odd	4	inner	50.3.c.b.7.1	yes	2
5.3	odd	4		50.3.c.a.7.1	✓	2
5.4	even	2		50.3.c.a.43.1	yes	2
15.2	even	4		450.3.g.c.307.1		2
15.8	even	4		450.3.g.e.307.1		2
15.14	odd	2		450.3.g.e.343.1		2
20.3	even	4		400.3.p.a.257.1		2
20.7	even	4		400.3.p.g.257.1		2
20.19	odd	2		400.3.p.a.193.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.3.c.a.7.1	✓	2	5.3	odd	4
50.3.c.a.43.1	yes	2	5.4	even	2
50.3.c.b.7.1	yes	2	5.2	odd	4	inner
50.3.c.b.43.1	yes	2	1.1	even	1	trivial
400.3.p.a.193.1		2	20.19	odd	2
400.3.p.a.257.1		2	20.3	even	4
400.3.p.g.193.1		2	4.3	odd	2
400.3.p.g.257.1		2	20.7	even	4
450.3.g.c.307.1		2	15.2	even	4
450.3.g.c.343.1		2	3.2	odd	2
450.3.g.e.307.1		2	15.8	even	4
450.3.g.e.343.1		2	15.14	odd	2

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:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	50.c (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.00000i) q^{2} +(-3.00000 - 3.00000i) q^{3} -2.00000i q^{4} -6.00000 q^{6} +(3.00000 - 3.00000i) q^{7} +(-2.00000 - 2.00000i) q^{8} +9.00000i q^{9} +O(q^{10})\)	\(q+(1.00000 - 1.00000i) q^{2} +(-3.00000 - 3.00000i) q^{3} -2.00000i q^{4} -6.00000 q^{6} +(3.00000 - 3.00000i) q^{7} +(-2.00000 - 2.00000i) q^{8} +9.00000i q^{9} +12.0000 q^{11} +(-6.00000 + 6.00000i) q^{12} +(12.0000 + 12.0000i) q^{13} -6.00000i q^{14} -4.00000 q^{16} +(-12.0000 + 12.0000i) q^{17} +(9.00000 + 9.00000i) q^{18} -20.0000i q^{19} -18.0000 q^{21} +(12.0000 - 12.0000i) q^{22} +(-3.00000 - 3.00000i) q^{23} +12.0000i q^{24} +24.0000 q^{26} +(-6.00000 - 6.00000i) q^{28} +30.0000i q^{29} -8.00000 q^{31} +(-4.00000 + 4.00000i) q^{32} +(-36.0000 - 36.0000i) q^{33} +24.0000i q^{34} +18.0000 q^{36} +(48.0000 - 48.0000i) q^{37} +(-20.0000 - 20.0000i) q^{38} -72.0000i q^{39} -48.0000 q^{41} +(-18.0000 + 18.0000i) q^{42} +(27.0000 + 27.0000i) q^{43} -24.0000i q^{44} -6.00000 q^{46} +(-27.0000 + 27.0000i) q^{47} +(12.0000 + 12.0000i) q^{48} +31.0000i q^{49} +72.0000 q^{51} +(24.0000 - 24.0000i) q^{52} +(12.0000 + 12.0000i) q^{53} -12.0000 q^{56} +(-60.0000 + 60.0000i) q^{57} +(30.0000 + 30.0000i) q^{58} +60.0000i q^{59} +32.0000 q^{61} +(-8.00000 + 8.00000i) q^{62} +(27.0000 + 27.0000i) q^{63} +8.00000i q^{64} -72.0000 q^{66} +(3.00000 - 3.00000i) q^{67} +(24.0000 + 24.0000i) q^{68} +18.0000i q^{69} -48.0000 q^{71} +(18.0000 - 18.0000i) q^{72} +(12.0000 + 12.0000i) q^{73} -96.0000i q^{74} -40.0000 q^{76} +(36.0000 - 36.0000i) q^{77} +(-72.0000 - 72.0000i) q^{78} -40.0000i q^{79} +81.0000 q^{81} +(-48.0000 + 48.0000i) q^{82} +(-93.0000 - 93.0000i) q^{83} +36.0000i q^{84} +54.0000 q^{86} +(90.0000 - 90.0000i) q^{87} +(-24.0000 - 24.0000i) q^{88} -30.0000i q^{89} +72.0000 q^{91} +(-6.00000 + 6.00000i) q^{92} +(24.0000 + 24.0000i) q^{93} +54.0000i q^{94} +24.0000 q^{96} +(-12.0000 + 12.0000i) q^{97} +(31.0000 + 31.0000i) q^{98} +108.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 6 q^{3} - 12 q^{6} + 6 q^{7} - 4 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 - 6 * q^3 - 12 * q^6 + 6 * q^7 - 4 * q^8	\( 2 q + 2 q^{2} - 6 q^{3} - 12 q^{6} + 6 q^{7} - 4 q^{8} + 24 q^{11} - 12 q^{12} + 24 q^{13} - 8 q^{16} - 24 q^{17} + 18 q^{18} - 36 q^{21} + 24 q^{22} - 6 q^{23} + 48 q^{26} - 12 q^{28} - 16 q^{31} - 8 q^{32} - 72 q^{33} + 36 q^{36} + 96 q^{37} - 40 q^{38} - 96 q^{41} - 36 q^{42} + 54 q^{43} - 12 q^{46} - 54 q^{47} + 24 q^{48} + 144 q^{51} + 48 q^{52} + 24 q^{53} - 24 q^{56} - 120 q^{57} + 60 q^{58} + 64 q^{61} - 16 q^{62} + 54 q^{63} - 144 q^{66} + 6 q^{67} + 48 q^{68} - 96 q^{71} + 36 q^{72} + 24 q^{73} - 80 q^{76} + 72 q^{77} - 144 q^{78} + 162 q^{81} - 96 q^{82} - 186 q^{83} + 108 q^{86} + 180 q^{87} - 48 q^{88} + 144 q^{91} - 12 q^{92} + 48 q^{93} + 48 q^{96} - 24 q^{97} + 62 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 6 * q^3 - 12 * q^6 + 6 * q^7 - 4 * q^8 + 24 * q^11 - 12 * q^12 + 24 * q^13 - 8 * q^16 - 24 * q^17 + 18 * q^18 - 36 * q^21 + 24 * q^22 - 6 * q^23 + 48 * q^26 - 12 * q^28 - 16 * q^31 - 8 * q^32 - 72 * q^33 + 36 * q^36 + 96 * q^37 - 40 * q^38 - 96 * q^41 - 36 * q^42 + 54 * q^43 - 12 * q^46 - 54 * q^47 + 24 * q^48 + 144 * q^51 + 48 * q^52 + 24 * q^53 - 24 * q^56 - 120 * q^57 + 60 * q^58 + 64 * q^61 - 16 * q^62 + 54 * q^63 - 144 * q^66 + 6 * q^67 + 48 * q^68 - 96 * q^71 + 36 * q^72 + 24 * q^73 - 80 * q^76 + 72 * q^77 - 144 * q^78 + 162 * q^81 - 96 * q^82 - 186 * q^83 + 108 * q^86 + 180 * q^87 - 48 * q^88 + 144 * q^91 - 12 * q^92 + 48 * q^93 + 48 * q^96 - 24 * q^97 + 62 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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