LMFDB - Embedded newform 850.2.h.f.251.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [850,2,Mod(251,850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(850, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("850.251");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 850 = 2 \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	850.h (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:	$6.78728417181$
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:	no (minimal twist has level 34)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			251.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	850.251
Dual form			850.2.h.f.701.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +(1.00000 + 1.00000i) q^{3} -1.00000 q^{4} +(-1.00000 + 1.00000i) q^{6} +(2.00000 - 2.00000i) q^{7} -1.00000i q^{8} -1.00000i q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +(1.00000 + 1.00000i) q^{3} -1.00000 q^{4} +(-1.00000 + 1.00000i) q^{6} +(2.00000 - 2.00000i) q^{7} -1.00000i q^{8} -1.00000i q^{9} +(1.00000 - 1.00000i) q^{11} +(-1.00000 - 1.00000i) q^{12} +6.00000 q^{13} +(2.00000 + 2.00000i) q^{14} +1.00000 q^{16} +(-4.00000 - 1.00000i) q^{17} +1.00000 q^{18} -4.00000i q^{19} +4.00000 q^{21} +(1.00000 + 1.00000i) q^{22} +(1.00000 - 1.00000i) q^{24} +6.00000i q^{26} +(4.00000 - 4.00000i) q^{27} +(-2.00000 + 2.00000i) q^{28} +(-2.00000 - 2.00000i) q^{29} +(6.00000 + 6.00000i) q^{31} +1.00000i q^{32} +2.00000 q^{33} +(1.00000 - 4.00000i) q^{34} +1.00000i q^{36} +4.00000 q^{38} +(6.00000 + 6.00000i) q^{39} +(1.00000 - 1.00000i) q^{41} +4.00000i q^{42} +6.00000i q^{43} +(-1.00000 + 1.00000i) q^{44} -8.00000 q^{47} +(1.00000 + 1.00000i) q^{48} -1.00000i q^{49} +(-3.00000 - 5.00000i) q^{51} -6.00000 q^{52} +6.00000i q^{53} +(4.00000 + 4.00000i) q^{54} +(-2.00000 - 2.00000i) q^{56} +(4.00000 - 4.00000i) q^{57} +(2.00000 - 2.00000i) q^{58} +6.00000i q^{59} +(-4.00000 + 4.00000i) q^{61} +(-6.00000 + 6.00000i) q^{62} +(-2.00000 - 2.00000i) q^{63} -1.00000 q^{64} +2.00000i q^{66} +2.00000 q^{67} +(4.00000 + 1.00000i) q^{68} +(-4.00000 - 4.00000i) q^{71} -1.00000 q^{72} +(1.00000 + 1.00000i) q^{73} +4.00000i q^{76} -4.00000i q^{77} +(-6.00000 + 6.00000i) q^{78} +(-8.00000 + 8.00000i) q^{79} +5.00000 q^{81} +(1.00000 + 1.00000i) q^{82} -14.0000i q^{83} -4.00000 q^{84} -6.00000 q^{86} -4.00000i q^{87} +(-1.00000 - 1.00000i) q^{88} +(12.0000 - 12.0000i) q^{91} +12.0000i q^{93} -8.00000i q^{94} +(-1.00000 + 1.00000i) q^{96} +(5.00000 + 5.00000i) q^{97} +1.00000 q^{98} +(-1.00000 - 1.00000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{4} - 2 q^{6} + 4 q^{7}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^4 - 2 * q^6 + 4 * q^7	$ 2 q + 2 q^{3} - 2 q^{4} - 2 q^{6} + 4 q^{7} + 2 q^{11} - 2 q^{12} + 12 q^{13} + 4 q^{14} + 2 q^{16} - 8 q^{17} + 2 q^{18} + 8 q^{21} + 2 q^{22} + 2 q^{24} + 8 q^{27} - 4 q^{28} - 4 q^{29} + 12 q^{31} + 4 q^{33} + 2 q^{34} + 8 q^{38} + 12 q^{39} + 2 q^{41} - 2 q^{44} - 16 q^{47} + 2 q^{48} - 6 q^{51} - 12 q^{52} + 8 q^{54} - 4 q^{56} + 8 q^{57} + 4 q^{58} - 8 q^{61} - 12 q^{62} - 4 q^{63} - 2 q^{64} + 4 q^{67} + 8 q^{68} - 8 q^{71} - 2 q^{72} + 2 q^{73} - 12 q^{78} - 16 q^{79} + 10 q^{81} + 2 q^{82} - 8 q^{84} - 12 q^{86} - 2 q^{88} + 24 q^{91} - 2 q^{96} + 10 q^{97} + 2 q^{98} - 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^4 - 2 * q^6 + 4 * q^7 + 2 * q^11 - 2 * q^12 + 12 * q^13 + 4 * q^14 + 2 * q^16 - 8 * q^17 + 2 * q^18 + 8 * q^21 + 2 * q^22 + 2 * q^24 + 8 * q^27 - 4 * q^28 - 4 * q^29 + 12 * q^31 + 4 * q^33 + 2 * q^34 + 8 * q^38 + 12 * q^39 + 2 * q^41 - 2 * q^44 - 16 * q^47 + 2 * q^48 - 6 * q^51 - 12 * q^52 + 8 * q^54 - 4 * q^56 + 8 * q^57 + 4 * q^58 - 8 * q^61 - 12 * q^62 - 4 * q^63 - 2 * q^64 + 4 * q^67 + 8 * q^68 - 8 * q^71 - 2 * q^72 + 2 * q^73 - 12 * q^78 - 16 * q^79 + 10 * q^81 + 2 * q^82 - 8 * q^84 - 12 * q^86 - 2 * q^88 + 24 * q^91 - 2 * q^96 + 10 * q^97 + 2 * q^98 - 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/850\mathbb{Z}\right)^\times$.

$n$	$477$	$751$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$

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 $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$	1.00000	+	1.00000i	0.577350	+	0.577350i	0.934172	−	0.356822i	$-0.116140\pi$
$3$	1.00000	+	1.00000i	0.577350	+	0.577350i	−0.356822	+	0.934172i	$0.616140\pi$
$4$	−1.00000			−0.500000
$5$		0			0
$5$		0			0
$6$	−1.00000	+	1.00000i	−0.408248	+	0.408248i
$7$	2.00000	−	2.00000i	0.755929	−	0.755929i	−0.219650	−	0.975579i	$-0.570491\pi$
$7$	2.00000	−	2.00000i	0.755929	−	0.755929i	0.975579	+	0.219650i	$0.0704915\pi$
$8$		−	1.00000i		−	0.353553i
$9$		−	1.00000i		−	0.333333i
$10$		0			0
$11$	1.00000	−	1.00000i	0.301511	−	0.301511i	−0.540094	−	0.841605i	$-0.681611\pi$
$11$	1.00000	−	1.00000i	0.301511	−	0.301511i	0.841605	+	0.540094i	$0.181611\pi$
$12$	−1.00000	−	1.00000i	−0.288675	−	0.288675i
$13$	6.00000			1.66410			0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000			1.66410			0.832050	+	0.554700i	$0.187167\pi$
$14$	2.00000	+	2.00000i	0.534522	+	0.534522i
$15$		0			0
$16$	1.00000			0.250000
$17$	−4.00000	−	1.00000i	−0.970143	−	0.242536i
$17$	−4.00000	−	1.00000i	−0.970143	−	0.242536i
$18$	1.00000			0.235702
$19$		−	4.00000i		−	0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$		−	4.00000i		−	0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$		0			0
$21$	4.00000			0.872872
$22$	1.00000	+	1.00000i	0.213201	+	0.213201i
$23$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$23$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$24$	1.00000	−	1.00000i	0.204124	−	0.204124i
$25$		0			0
$26$			6.00000i			1.17670i
$27$	4.00000	−	4.00000i	0.769800	−	0.769800i
$28$	−2.00000	+	2.00000i	−0.377964	+	0.377964i
$29$	−2.00000	−	2.00000i	−0.371391	−	0.371391i	0.496593	−	0.867984i	$-0.334584\pi$
$29$	−2.00000	−	2.00000i	−0.371391	−	0.371391i	−0.867984	+	0.496593i	$0.834584\pi$
$30$		0			0
$31$	6.00000	+	6.00000i	1.07763	+	1.07763i	0.996721	+	0.0809104i	$0.0257828\pi$
$31$	6.00000	+	6.00000i	1.07763	+	1.07763i	0.0809104	+	0.996721i	$0.474217\pi$
$32$			1.00000i			0.176777i
$33$	2.00000			0.348155
$34$	1.00000	−	4.00000i	0.171499	−	0.685994i
$35$		0			0
$36$			1.00000i			0.166667i
$37$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$37$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$38$	4.00000			0.648886
$39$	6.00000	+	6.00000i	0.960769	+	0.960769i
$40$		0			0
$41$	1.00000	−	1.00000i	0.156174	−	0.156174i	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	1.00000	−	1.00000i	0.156174	−	0.156174i	0.780869	+	0.624695i	$0.214777\pi$
$42$			4.00000i			0.617213i
$43$			6.00000i			0.914991i	0.889212	+	0.457496i	$0.151253\pi$
$43$			6.00000i			0.914991i	−0.889212	+	0.457496i	$0.848747\pi$
$44$	−1.00000	+	1.00000i	−0.150756	+	0.150756i
$45$		0			0
$46$		0			0
$47$	−8.00000			−1.16692			−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000			−1.16692			−0.583460	+	0.812142i	$0.698301\pi$
$48$	1.00000	+	1.00000i	0.144338	+	0.144338i
$49$		−	1.00000i		−	0.142857i
$50$		0			0
$51$	−3.00000	−	5.00000i	−0.420084	−	0.700140i
$52$	−6.00000			−0.832050
$53$			6.00000i			0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$			6.00000i			0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	4.00000	+	4.00000i	0.544331	+	0.544331i
$55$		0			0
$56$	−2.00000	−	2.00000i	−0.267261	−	0.267261i
$57$	4.00000	−	4.00000i	0.529813	−	0.529813i
$58$	2.00000	−	2.00000i	0.262613	−	0.262613i
$59$			6.00000i			0.781133i	0.920575	+	0.390567i	$0.127721\pi$
$59$			6.00000i			0.781133i	−0.920575	+	0.390567i	$0.872279\pi$
$60$		0			0
$61$	−4.00000	+	4.00000i	−0.512148	+	0.512148i	−0.915184	−	0.403036i	$-0.867955\pi$
$61$	−4.00000	+	4.00000i	−0.512148	+	0.512148i	0.403036	+	0.915184i	$0.367955\pi$
$62$	−6.00000	+	6.00000i	−0.762001	+	0.762001i
$63$	−2.00000	−	2.00000i	−0.251976	−	0.251976i
$64$	−1.00000			−0.125000
$65$		0			0
$66$			2.00000i			0.246183i
$67$	2.00000			0.244339			0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000			0.244339			0.122169	+	0.992509i	$0.461015\pi$
$68$	4.00000	+	1.00000i	0.485071	+	0.121268i
$69$		0			0
$70$		0			0
$71$	−4.00000	−	4.00000i	−0.474713	−	0.474713i	0.428723	−	0.903436i	$-0.358964\pi$
$71$	−4.00000	−	4.00000i	−0.474713	−	0.474713i	−0.903436	+	0.428723i	$0.858964\pi$
$72$	−1.00000			−0.117851
$73$	1.00000	+	1.00000i	0.117041	+	0.117041i	0.763202	−	0.646160i	$-0.223626\pi$
$73$	1.00000	+	1.00000i	0.117041	+	0.117041i	−0.646160	+	0.763202i	$0.723626\pi$
$74$		0			0
$75$		0			0
$76$			4.00000i			0.458831i
$77$		−	4.00000i		−	0.455842i
$78$	−6.00000	+	6.00000i	−0.679366	+	0.679366i
$79$	−8.00000	+	8.00000i	−0.900070	+	0.900070i	−0.995442	−	0.0953714i	$-0.969596\pi$
$79$	−8.00000	+	8.00000i	−0.900070	+	0.900070i	0.0953714	+	0.995442i	$0.469596\pi$
$80$		0			0
$81$	5.00000			0.555556
$82$	1.00000	+	1.00000i	0.110432	+	0.110432i
$83$		−	14.0000i		−	1.53670i	−0.640030	−	0.768350i	$-0.721078\pi$
$83$		−	14.0000i		−	1.53670i	0.640030	−	0.768350i	$-0.278922\pi$
$84$	−4.00000			−0.436436
$85$		0			0
$86$	−6.00000			−0.646997
$87$		−	4.00000i		−	0.428845i
$88$	−1.00000	−	1.00000i	−0.106600	−	0.106600i
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	12.0000	−	12.0000i	1.25794	−	1.25794i
$92$		0			0
$93$			12.0000i			1.24434i
$94$		−	8.00000i		−	0.825137i
$95$		0			0
$96$	−1.00000	+	1.00000i	−0.102062	+	0.102062i
$97$	5.00000	+	5.00000i	0.507673	+	0.507673i	0.913812	−	0.406138i	$-0.133125\pi$
$97$	5.00000	+	5.00000i	0.507673	+	0.507673i	−0.406138	+	0.913812i	$0.633125\pi$
$98$	1.00000			0.101015
$99$	−1.00000	−	1.00000i	−0.100504	−	0.100504i
$100$		0			0
$101$	2.00000			0.199007			0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000			0.199007			0.0995037	+	0.995037i	$0.468274\pi$
$102$	5.00000	−	3.00000i	0.495074	−	0.297044i
$103$	−4.00000			−0.394132			−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000			−0.394132			−0.197066	+	0.980390i	$0.563141\pi$
$104$		−	6.00000i		−	0.588348i
$105$		0			0
$106$	−6.00000			−0.582772
$107$	5.00000	+	5.00000i	0.483368	+	0.483368i	0.906206	−	0.422837i	$-0.138966\pi$
$107$	5.00000	+	5.00000i	0.483368	+	0.483368i	−0.422837	+	0.906206i	$0.638966\pi$
$108$	−4.00000	+	4.00000i	−0.384900	+	0.384900i
$109$	2.00000	−	2.00000i	0.191565	−	0.191565i	−0.604807	−	0.796372i	$-0.706750\pi$
$109$	2.00000	−	2.00000i	0.191565	−	0.191565i	0.796372	+	0.604807i	$0.206750\pi$
$110$		0			0
$111$		0			0
$112$	2.00000	−	2.00000i	0.188982	−	0.188982i
$113$	5.00000	−	5.00000i	0.470360	−	0.470360i	−0.431671	−	0.902031i	$-0.642076\pi$
$113$	5.00000	−	5.00000i	0.470360	−	0.470360i	0.902031	+	0.431671i	$0.142076\pi$
$114$	4.00000	+	4.00000i	0.374634	+	0.374634i
$115$		0			0
$116$	2.00000	+	2.00000i	0.185695	+	0.185695i
$117$		−	6.00000i		−	0.554700i
$118$	−6.00000			−0.552345
$119$	−10.0000	+	6.00000i	−0.916698	+	0.550019i
$120$		0			0
$121$			9.00000i			0.818182i
$122$	−4.00000	−	4.00000i	−0.362143	−	0.362143i
$123$	2.00000			0.180334
$124$	−6.00000	−	6.00000i	−0.538816	−	0.538816i
$125$		0			0
$126$	2.00000	−	2.00000i	0.178174	−	0.178174i
$127$			8.00000i			0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$			8.00000i			0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$		−	1.00000i		−	0.0883883i
$129$	−6.00000	+	6.00000i	−0.528271	+	0.528271i
$130$		0			0
$131$	1.00000	+	1.00000i	0.0873704	+	0.0873704i	0.749441	−	0.662071i	$-0.230322\pi$
$131$	1.00000	+	1.00000i	0.0873704	+	0.0873704i	−0.662071	+	0.749441i	$0.730322\pi$
$132$	−2.00000			−0.174078
$133$	−8.00000	−	8.00000i	−0.693688	−	0.693688i
$134$			2.00000i			0.172774i
$135$		0			0
$136$	−1.00000	+	4.00000i	−0.0857493	+	0.342997i
$137$	−18.0000			−1.53784			−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000			−1.53784			−0.768922	+	0.639343i	$0.779207\pi$
$138$		0			0
$139$	−7.00000	−	7.00000i	−0.593732	−	0.593732i	0.344905	−	0.938638i	$-0.387911\pi$
$139$	−7.00000	−	7.00000i	−0.593732	−	0.593732i	−0.938638	+	0.344905i	$0.887911\pi$
$140$		0			0
$141$	−8.00000	−	8.00000i	−0.673722	−	0.673722i
$142$	4.00000	−	4.00000i	0.335673	−	0.335673i
$143$	6.00000	−	6.00000i	0.501745	−	0.501745i
$144$		−	1.00000i		−	0.0833333i
$145$		0			0
$146$	−1.00000	+	1.00000i	−0.0827606	+	0.0827606i
$147$	1.00000	−	1.00000i	0.0824786	−	0.0824786i
$148$		0			0
$149$	10.0000			0.819232			0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000			0.819232			0.409616	+	0.912258i	$0.365663\pi$
$150$		0			0
$151$		−	20.0000i		−	1.62758i	−0.581161	−	0.813788i	$-0.697401\pi$
$151$		−	20.0000i		−	1.62758i	0.581161	−	0.813788i	$-0.302599\pi$
$152$	−4.00000			−0.324443
$153$	−1.00000	+	4.00000i	−0.0808452	+	0.323381i
$154$	4.00000			0.322329
$155$		0			0
$156$	−6.00000	−	6.00000i	−0.480384	−	0.480384i
$157$	22.0000			1.75579			0.877896	−	0.478852i	$-0.158947\pi$
$157$	22.0000			1.75579			0.877896	+	0.478852i	$0.158947\pi$
$158$	−8.00000	−	8.00000i	−0.636446	−	0.636446i
$159$	−6.00000	+	6.00000i	−0.475831	+	0.475831i
$160$		0			0
$161$		0			0
$162$			5.00000i			0.392837i
$163$	−15.0000	+	15.0000i	−1.17489	+	1.17489i	−0.193862	+	0.981029i	$0.562101\pi$
$163$	−15.0000	+	15.0000i	−1.17489	+	1.17489i	−0.981029	+	0.193862i	$0.937899\pi$
$164$	−1.00000	+	1.00000i	−0.0780869	+	0.0780869i
$165$		0			0
$166$	14.0000			1.08661
$167$	10.0000	+	10.0000i	0.773823	+	0.773823i	0.978773	−	0.204949i	$-0.0657030\pi$
$167$	10.0000	+	10.0000i	0.773823	+	0.773823i	−0.204949	+	0.978773i	$0.565703\pi$
$168$		−	4.00000i		−	0.308607i
$169$	23.0000			1.76923
$170$		0			0
$171$	−4.00000			−0.305888
$172$		−	6.00000i		−	0.457496i
$173$	−14.0000	−	14.0000i	−1.06440	−	1.06440i	−0.997778	−	0.0666220i	$-0.978778\pi$
$173$	−14.0000	−	14.0000i	−1.06440	−	1.06440i	−0.0666220	−	0.997778i	$-0.521222\pi$
$174$	4.00000			0.303239
$175$		0			0
$176$	1.00000	−	1.00000i	0.0753778	−	0.0753778i
$177$	−6.00000	+	6.00000i	−0.450988	+	0.450988i
$178$		0			0
$179$		−	4.00000i		−	0.298974i	−0.988764	−	0.149487i	$-0.952238\pi$
$179$		−	4.00000i		−	0.298974i	0.988764	−	0.149487i	$-0.0477622\pi$
$180$		0			0
$181$	−4.00000	+	4.00000i	−0.297318	+	0.297318i	−0.839962	−	0.542645i	$-0.817423\pi$
$181$	−4.00000	+	4.00000i	−0.297318	+	0.297318i	0.542645	+	0.839962i	$0.317423\pi$
$182$	12.0000	+	12.0000i	0.889499	+	0.889499i
$183$	−8.00000			−0.591377
$184$		0			0
$185$		0			0
$186$	−12.0000			−0.879883
$187$	−5.00000	+	3.00000i	−0.365636	+	0.219382i
$188$	8.00000			0.583460
$189$		−	16.0000i		−	1.16383i
$190$		0			0
$191$	12.0000			0.868290			0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000			0.868290			0.434145	+	0.900843i	$0.357051\pi$
$192$	−1.00000	−	1.00000i	−0.0721688	−	0.0721688i
$193$	15.0000	−	15.0000i	1.07972	−	1.07972i	0.0831899	−	0.996534i	$-0.473489\pi$
$193$	15.0000	−	15.0000i	1.07972	−	1.07972i	0.996534	−	0.0831899i	$-0.0265108\pi$
$194$	−5.00000	+	5.00000i	−0.358979	+	0.358979i
$195$		0			0
$196$			1.00000i			0.0714286i
$197$	12.0000	−	12.0000i	0.854965	−	0.854965i	−0.135775	−	0.990740i	$-0.543352\pi$
$197$	12.0000	−	12.0000i	0.854965	−	0.854965i	0.990740	+	0.135775i	$0.0433525\pi$
$198$	1.00000	−	1.00000i	0.0710669	−	0.0710669i
$199$	8.00000	+	8.00000i	0.567105	+	0.567105i	0.931316	−	0.364211i	$-0.118661\pi$
$199$	8.00000	+	8.00000i	0.567105	+	0.567105i	−0.364211	+	0.931316i	$0.618661\pi$
$200$		0			0
$201$	2.00000	+	2.00000i	0.141069	+	0.141069i
$202$			2.00000i			0.140720i
$203$	−8.00000			−0.561490
$204$	3.00000	+	5.00000i	0.210042	+	0.350070i
$205$		0			0
$206$		−	4.00000i		−	0.278693i
$207$		0			0
$208$	6.00000			0.416025
$209$	−4.00000	−	4.00000i	−0.276686	−	0.276686i
$210$		0			0
$211$	−9.00000	+	9.00000i	−0.619586	+	0.619586i	−0.945425	−	0.325840i	$-0.894353\pi$
$211$	−9.00000	+	9.00000i	−0.619586	+	0.619586i	0.325840	+	0.945425i	$0.394353\pi$
$212$		−	6.00000i		−	0.412082i
$213$		−	8.00000i		−	0.548151i
$214$	−5.00000	+	5.00000i	−0.341793	+	0.341793i
$215$		0			0
$216$	−4.00000	−	4.00000i	−0.272166	−	0.272166i
$217$	24.0000			1.62923
$218$	2.00000	+	2.00000i	0.135457	+	0.135457i
$219$			2.00000i			0.135147i
$220$		0			0
$221$	−24.0000	−	6.00000i	−1.61441	−	0.403604i
$222$		0			0
$223$		−	4.00000i		−	0.267860i	−0.990991	−	0.133930i	$-0.957240\pi$
$223$		−	4.00000i		−	0.267860i	0.990991	−	0.133930i	$-0.0427597\pi$
$224$	2.00000	+	2.00000i	0.133631	+	0.133631i
$225$		0			0
$226$	5.00000	+	5.00000i	0.332595	+	0.332595i
$227$	−13.0000	+	13.0000i	−0.862840	+	0.862840i	−0.991667	−	0.128827i	$-0.958879\pi$
$227$	−13.0000	+	13.0000i	−0.862840	+	0.862840i	0.128827	+	0.991667i	$0.458879\pi$
$228$	−4.00000	+	4.00000i	−0.264906	+	0.264906i
$229$		−	14.0000i		−	0.925146i	−0.886581	−	0.462573i	$-0.846926\pi$
$229$		−	14.0000i		−	0.925146i	0.886581	−	0.462573i	$-0.153074\pi$
$230$		0			0
$231$	4.00000	−	4.00000i	0.263181	−	0.263181i
$232$	−2.00000	+	2.00000i	−0.131306	+	0.131306i
$233$	−9.00000	−	9.00000i	−0.589610	−	0.589610i	0.347916	−	0.937526i	$-0.386889\pi$
$233$	−9.00000	−	9.00000i	−0.589610	−	0.589610i	−0.937526	+	0.347916i	$0.886889\pi$
$234$	6.00000			0.392232
$235$		0			0
$236$		−	6.00000i		−	0.390567i
$237$	−16.0000			−1.03931
$238$	−6.00000	−	10.0000i	−0.388922	−	0.648204i
$239$	−20.0000			−1.29369			−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000			−1.29369			−0.646846	+	0.762620i	$0.723912\pi$
$240$		0			0
$241$	−9.00000	−	9.00000i	−0.579741	−	0.579741i	0.355091	−	0.934832i	$-0.384450\pi$
$241$	−9.00000	−	9.00000i	−0.579741	−	0.579741i	−0.934832	+	0.355091i	$0.884450\pi$
$242$	−9.00000			−0.578542
$243$	−7.00000	−	7.00000i	−0.449050	−	0.449050i
$244$	4.00000	−	4.00000i	0.256074	−	0.256074i
$245$		0			0
$246$			2.00000i			0.127515i
$247$		−	24.0000i		−	1.52708i
$248$	6.00000	−	6.00000i	0.381000	−	0.381000i
$249$	14.0000	−	14.0000i	0.887214	−	0.887214i
$250$		0			0
$251$	−18.0000			−1.13615			−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000			−1.13615			−0.568075	+	0.822977i	$0.692312\pi$
$252$	2.00000	+	2.00000i	0.125988	+	0.125988i
$253$		0			0
$254$	−8.00000			−0.501965
$255$		0			0
$256$	1.00000			0.0625000
$257$			28.0000i			1.74659i	0.487190	+	0.873296i	$0.338022\pi$
$257$			28.0000i			1.74659i	−0.487190	+	0.873296i	$0.661978\pi$
$258$	−6.00000	−	6.00000i	−0.373544	−	0.373544i
$259$		0			0
$260$		0			0
$261$	−2.00000	+	2.00000i	−0.123797	+	0.123797i
$262$	−1.00000	+	1.00000i	−0.0617802	+	0.0617802i
$263$			16.0000i			0.986602i	0.869859	+	0.493301i	$0.164210\pi$
$263$			16.0000i			0.986602i	−0.869859	+	0.493301i	$0.835790\pi$
$264$		−	2.00000i		−	0.123091i
$265$		0			0
$266$	8.00000	−	8.00000i	0.490511	−	0.490511i
$267$		0			0
$268$	−2.00000			−0.122169
$269$	−2.00000	−	2.00000i	−0.121942	−	0.121942i	0.643502	−	0.765444i	$-0.277481\pi$
$269$	−2.00000	−	2.00000i	−0.121942	−	0.121942i	−0.765444	+	0.643502i	$0.777481\pi$
$270$		0			0
$271$	−28.0000			−1.70088			−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000			−1.70088			−0.850439	+	0.526073i	$0.823664\pi$
$272$	−4.00000	−	1.00000i	−0.242536	−	0.0606339i
$273$	24.0000			1.45255
$274$		−	18.0000i		−	1.08742i
$275$		0			0
$276$		0			0
$277$	−10.0000	−	10.0000i	−0.600842	−	0.600842i	0.339694	−	0.940536i	$-0.389676\pi$
$277$	−10.0000	−	10.0000i	−0.600842	−	0.600842i	−0.940536	+	0.339694i	$0.889676\pi$
$278$	7.00000	−	7.00000i	0.419832	−	0.419832i
$279$	6.00000	−	6.00000i	0.359211	−	0.359211i
$280$		0			0
$281$			10.0000i			0.596550i	0.954480	+	0.298275i	$0.0964112\pi$
$281$			10.0000i			0.596550i	−0.954480	+	0.298275i	$0.903589\pi$
$282$	8.00000	−	8.00000i	0.476393	−	0.476393i
$283$	−5.00000	+	5.00000i	−0.297219	+	0.297219i	−0.839924	−	0.542705i	$-0.817400\pi$
$283$	−5.00000	+	5.00000i	−0.297219	+	0.297219i	0.542705	+	0.839924i	$0.317400\pi$
$284$	4.00000	+	4.00000i	0.237356	+	0.237356i
$285$		0			0
$286$	6.00000	+	6.00000i	0.354787	+	0.354787i
$287$		−	4.00000i		−	0.236113i
$288$	1.00000			0.0589256
$289$	15.0000	+	8.00000i	0.882353	+	0.470588i
$290$		0			0
$291$			10.0000i			0.586210i
$292$	−1.00000	−	1.00000i	−0.0585206	−	0.0585206i
$293$	6.00000			0.350524			0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000			0.350524			0.175262	+	0.984522i	$0.443923\pi$
$294$	1.00000	+	1.00000i	0.0583212	+	0.0583212i
$295$		0			0
$296$		0			0
$297$		−	8.00000i		−	0.464207i
$298$			10.0000i			0.579284i
$299$		0			0
$300$		0			0
$301$	12.0000	+	12.0000i	0.691669	+	0.691669i
$302$	20.0000			1.15087
$303$	2.00000	+	2.00000i	0.114897	+	0.114897i
$304$		−	4.00000i		−	0.229416i
$305$		0			0
$306$	−4.00000	−	1.00000i	−0.228665	−	0.0571662i
$307$	−28.0000			−1.59804			−0.799022	−	0.601302i	$-0.794649\pi$
$307$	−28.0000			−1.59804			−0.799022	+	0.601302i	$0.794649\pi$
$308$			4.00000i			0.227921i
$309$	−4.00000	−	4.00000i	−0.227552	−	0.227552i
$310$		0			0
$311$	6.00000	+	6.00000i	0.340229	+	0.340229i	0.856453	−	0.516225i	$-0.172663\pi$
$311$	6.00000	+	6.00000i	0.340229	+	0.340229i	−0.516225	+	0.856453i	$0.672663\pi$
$312$	6.00000	−	6.00000i	0.339683	−	0.339683i
$313$	5.00000	−	5.00000i	0.282617	−	0.282617i	−0.551535	−	0.834152i	$-0.685958\pi$
$313$	5.00000	−	5.00000i	0.282617	−	0.282617i	0.834152	+	0.551535i	$0.185958\pi$
$314$			22.0000i			1.24153i
$315$		0			0
$316$	8.00000	−	8.00000i	0.450035	−	0.450035i
$317$	−8.00000	+	8.00000i	−0.449325	+	0.449325i	−0.895130	−	0.445805i	$-0.852917\pi$
$317$	−8.00000	+	8.00000i	−0.449325	+	0.449325i	0.445805	+	0.895130i	$0.352917\pi$
$318$	−6.00000	−	6.00000i	−0.336463	−	0.336463i
$319$	−4.00000			−0.223957
$320$		0			0
$321$			10.0000i			0.558146i
$322$		0			0
$323$	−4.00000	+	16.0000i	−0.222566	+	0.890264i
$324$	−5.00000			−0.277778
$325$		0			0
$326$	−15.0000	−	15.0000i	−0.830773	−	0.830773i
$327$	4.00000			0.221201
$328$	−1.00000	−	1.00000i	−0.0552158	−	0.0552158i
$329$	−16.0000	+	16.0000i	−0.882109	+	0.882109i
$330$		0			0
$331$			30.0000i			1.64895i	0.565899	+	0.824475i	$0.308529\pi$
$331$			30.0000i			1.64895i	−0.565899	+	0.824475i	$0.691471\pi$
$332$			14.0000i			0.768350i
$333$		0			0
$334$	−10.0000	+	10.0000i	−0.547176	+	0.547176i
$335$		0			0
$336$	4.00000			0.218218
$337$	5.00000	+	5.00000i	0.272367	+	0.272367i	0.830053	−	0.557685i	$-0.188310\pi$
$337$	5.00000	+	5.00000i	0.272367	+	0.272367i	−0.557685	+	0.830053i	$0.688310\pi$
$338$			23.0000i			1.25104i
$339$	10.0000			0.543125
$340$		0			0
$341$	12.0000			0.649836
$342$		−	4.00000i		−	0.216295i
$343$	12.0000	+	12.0000i	0.647939	+	0.647939i
$344$	6.00000			0.323498
$345$		0			0
$346$	14.0000	−	14.0000i	0.752645	−	0.752645i
$347$	−13.0000	+	13.0000i	−0.697877	+	0.697877i	−0.963952	−	0.266076i	$-0.914273\pi$
$347$	−13.0000	+	13.0000i	−0.697877	+	0.697877i	0.266076	+	0.963952i	$0.414273\pi$
$348$			4.00000i			0.214423i
$349$		−	14.0000i		−	0.749403i	−0.927146	−	0.374701i	$-0.877745\pi$
$349$		−	14.0000i		−	0.749403i	0.927146	−	0.374701i	$-0.122255\pi$
$350$		0			0
$351$	24.0000	−	24.0000i	1.28103	−	1.28103i
$352$	1.00000	+	1.00000i	0.0533002	+	0.0533002i
$353$	−4.00000			−0.212899			−0.106449	−	0.994318i	$-0.533948\pi$
$353$	−4.00000			−0.212899			−0.106449	+	0.994318i	$0.533948\pi$
$354$	−6.00000	−	6.00000i	−0.318896	−	0.318896i
$355$		0			0
$356$		0			0
$357$	−16.0000	−	4.00000i	−0.846810	−	0.211702i
$358$	4.00000			0.211407
$359$			16.0000i			0.844448i	0.906492	+	0.422224i	$0.138750\pi$
$359$			16.0000i			0.844448i	−0.906492	+	0.422224i	$0.861250\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$	−4.00000	−	4.00000i	−0.210235	−	0.210235i
$363$	−9.00000	+	9.00000i	−0.472377	+	0.472377i
$364$	−12.0000	+	12.0000i	−0.628971	+	0.628971i
$365$		0			0
$366$		−	8.00000i		−	0.418167i
$367$	12.0000	−	12.0000i	0.626395	−	0.626395i	−0.320764	−	0.947159i	$-0.603940\pi$
$367$	12.0000	−	12.0000i	0.626395	−	0.626395i	0.947159	+	0.320764i	$0.103940\pi$
$368$		0			0
$369$	−1.00000	−	1.00000i	−0.0520579	−	0.0520579i
$370$		0			0
$371$	12.0000	+	12.0000i	0.623009	+	0.623009i
$372$		−	12.0000i		−	0.622171i
$373$	−14.0000			−0.724893			−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000			−0.724893			−0.362446	+	0.932005i	$0.618058\pi$
$374$	−3.00000	−	5.00000i	−0.155126	−	0.258544i
$375$		0			0
$376$			8.00000i			0.412568i
$377$	−12.0000	−	12.0000i	−0.618031	−	0.618031i
$378$	16.0000			0.822951
$379$	3.00000	+	3.00000i	0.154100	+	0.154100i	0.779946	−	0.625847i	$-0.215246\pi$
$379$	3.00000	+	3.00000i	0.154100	+	0.154100i	−0.625847	+	0.779946i	$0.715246\pi$
$380$		0			0
$381$	−8.00000	+	8.00000i	−0.409852	+	0.409852i
$382$			12.0000i			0.613973i
$383$		−	4.00000i		−	0.204390i	−0.994764	−	0.102195i	$-0.967413\pi$
$383$		−	4.00000i		−	0.204390i	0.994764	−	0.102195i	$-0.0325866\pi$
$384$	1.00000	−	1.00000i	0.0510310	−	0.0510310i
$385$		0			0
$386$	15.0000	+	15.0000i	0.763480	+	0.763480i
$387$	6.00000			0.304997
$388$	−5.00000	−	5.00000i	−0.253837	−	0.253837i
$389$			6.00000i			0.304212i	0.988364	+	0.152106i	$0.0486055\pi$
$389$			6.00000i			0.304212i	−0.988364	+	0.152106i	$0.951394\pi$
$390$		0			0
$391$		0			0
$392$	−1.00000			−0.0505076
$393$			2.00000i			0.100887i
$394$	12.0000	+	12.0000i	0.604551	+	0.604551i
$395$		0			0
$396$	1.00000	+	1.00000i	0.0502519	+	0.0502519i
$397$	12.0000	−	12.0000i	0.602263	−	0.602263i	−0.338650	−	0.940913i	$-0.609970\pi$
$397$	12.0000	−	12.0000i	0.602263	−	0.602263i	0.940913	+	0.338650i	$0.109970\pi$
$398$	−8.00000	+	8.00000i	−0.401004	+	0.401004i
$399$		−	16.0000i		−	0.801002i
$400$		0			0
$401$	11.0000	−	11.0000i	0.549314	−	0.549314i	−0.376929	−	0.926242i	$-0.623020\pi$
$401$	11.0000	−	11.0000i	0.549314	−	0.549314i	0.926242	+	0.376929i	$0.123020\pi$
$402$	−2.00000	+	2.00000i	−0.0997509	+	0.0997509i
$403$	36.0000	+	36.0000i	1.79329	+	1.79329i
$404$	−2.00000			−0.0995037
$405$		0			0
$406$		−	8.00000i		−	0.397033i
$407$		0			0
$408$	−5.00000	+	3.00000i	−0.247537	+	0.148522i
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$	−18.0000	−	18.0000i	−0.887875	−	0.887875i
$412$	4.00000			0.197066
$413$	12.0000	+	12.0000i	0.590481	+	0.590481i
$414$		0			0
$415$		0			0
$416$			6.00000i			0.294174i
$417$		−	14.0000i		−	0.685583i
$418$	4.00000	−	4.00000i	0.195646	−	0.195646i
$419$	−23.0000	+	23.0000i	−1.12362	+	1.12362i	−0.132431	+	0.991192i	$0.542278\pi$
$419$	−23.0000	+	23.0000i	−1.12362	+	1.12362i	−0.991192	+	0.132431i	$0.957722\pi$
$420$		0			0
$421$	22.0000			1.07221			0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000			1.07221			0.536107	+	0.844150i	$0.319894\pi$
$422$	−9.00000	−	9.00000i	−0.438113	−	0.438113i
$423$			8.00000i			0.388973i
$424$	6.00000			0.291386
$425$		0			0
$426$	8.00000			0.387601
$427$			16.0000i			0.774294i
$428$	−5.00000	−	5.00000i	−0.241684	−	0.241684i
$429$	12.0000			0.579365
$430$		0			0
$431$	16.0000	−	16.0000i	0.770693	−	0.770693i	−0.207535	−	0.978228i	$-0.566544\pi$
$431$	16.0000	−	16.0000i	0.770693	−	0.770693i	0.978228	+	0.207535i	$0.0665440\pi$
$432$	4.00000	−	4.00000i	0.192450	−	0.192450i
$433$		−	14.0000i		−	0.672797i	−0.941720	−	0.336399i	$-0.890791\pi$
$433$		−	14.0000i		−	0.672797i	0.941720	−	0.336399i	$-0.109209\pi$
$434$			24.0000i			1.15204i
$435$		0			0
$436$	−2.00000	+	2.00000i	−0.0957826	+	0.0957826i
$437$		0			0
$438$	−2.00000			−0.0955637
$439$	8.00000	+	8.00000i	0.381819	+	0.381819i	0.871757	−	0.489938i	$-0.162981\pi$
$439$	8.00000	+	8.00000i	0.381819	+	0.381819i	−0.489938	+	0.871757i	$0.662981\pi$
$440$		0			0
$441$	−1.00000			−0.0476190
$442$	6.00000	−	24.0000i	0.285391	−	1.14156i
$443$	6.00000			0.285069			0.142534	−	0.989790i	$-0.454475\pi$
$443$	6.00000			0.285069			0.142534	+	0.989790i	$0.454475\pi$
$444$		0			0
$445$		0			0
$446$	4.00000			0.189405
$447$	10.0000	+	10.0000i	0.472984	+	0.472984i
$448$	−2.00000	+	2.00000i	−0.0944911	+	0.0944911i
$449$	27.0000	−	27.0000i	1.27421	−	1.27421i	0.330350	−	0.943858i	$-0.392833\pi$
$449$	27.0000	−	27.0000i	1.27421	−	1.27421i	0.943858	−	0.330350i	$-0.107167\pi$
$450$		0			0
$451$		−	2.00000i		−	0.0941763i
$452$	−5.00000	+	5.00000i	−0.235180	+	0.235180i
$453$	20.0000	−	20.0000i	0.939682	−	0.939682i
$454$	−13.0000	−	13.0000i	−0.610120	−	0.610120i
$455$		0			0
$456$	−4.00000	−	4.00000i	−0.187317	−	0.187317i
$457$			38.0000i			1.77757i	0.458329	+	0.888783i	$0.348448\pi$
$457$			38.0000i			1.77757i	−0.458329	+	0.888783i	$0.651552\pi$
$458$	14.0000			0.654177
$459$	−20.0000	+	12.0000i	−0.933520	+	0.560112i
$460$		0			0
$461$			30.0000i			1.39724i	0.715493	+	0.698620i	$0.246202\pi$
$461$			30.0000i			1.39724i	−0.715493	+	0.698620i	$0.753798\pi$
$462$	4.00000	+	4.00000i	0.186097	+	0.186097i
$463$	−24.0000			−1.11537			−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000			−1.11537			−0.557687	+	0.830051i	$0.688311\pi$
$464$	−2.00000	−	2.00000i	−0.0928477	−	0.0928477i
$465$		0			0
$466$	9.00000	−	9.00000i	0.416917	−	0.416917i
$467$			28.0000i			1.29569i	0.761774	+	0.647843i	$0.224329\pi$
$467$			28.0000i			1.29569i	−0.761774	+	0.647843i	$0.775671\pi$
$468$			6.00000i			0.277350i
$469$	4.00000	−	4.00000i	0.184703	−	0.184703i
$470$		0			0
$471$	22.0000	+	22.0000i	1.01371	+	1.01371i
$472$	6.00000			0.276172
$473$	6.00000	+	6.00000i	0.275880	+	0.275880i
$474$		−	16.0000i		−	0.734904i
$475$		0			0
$476$	10.0000	−	6.00000i	0.458349	−	0.275010i
$477$	6.00000			0.274721
$478$		−	20.0000i		−	0.914779i
$479$	8.00000	+	8.00000i	0.365529	+	0.365529i	0.865844	−	0.500314i	$-0.166782\pi$
$479$	8.00000	+	8.00000i	0.365529	+	0.365529i	−0.500314	+	0.865844i	$0.666782\pi$
$480$		0			0
$481$		0			0
$482$	9.00000	−	9.00000i	0.409939	−	0.409939i
$483$		0			0
$484$		−	9.00000i		−	0.409091i
$485$		0			0
$486$	7.00000	−	7.00000i	0.317526	−	0.317526i
$487$	12.0000	−	12.0000i	0.543772	−	0.543772i	−0.380861	−	0.924632i	$-0.624372\pi$
$487$	12.0000	−	12.0000i	0.543772	−	0.543772i	0.924632	+	0.380861i	$0.124372\pi$
$488$	4.00000	+	4.00000i	0.181071	+	0.181071i
$489$	−30.0000			−1.35665
$490$		0			0
$491$			20.0000i			0.902587i	0.892375	+	0.451294i	$0.149037\pi$
$491$			20.0000i			0.902587i	−0.892375	+	0.451294i	$0.850963\pi$
$492$	−2.00000			−0.0901670
$493$	6.00000	+	10.0000i	0.270226	+	0.450377i
$494$	24.0000			1.07981
$495$		0			0
$496$	6.00000	+	6.00000i	0.269408	+	0.269408i
$497$	−16.0000			−0.717698
$498$	14.0000	+	14.0000i	0.627355	+	0.627355i
$499$	−3.00000	+	3.00000i	−0.134298	+	0.134298i	−0.771060	−	0.636762i	$-0.780273\pi$
$499$	−3.00000	+	3.00000i	−0.134298	+	0.134298i	0.636762	+	0.771060i	$0.280273\pi$
$500$		0			0
$501$			20.0000i			0.893534i
$502$		−	18.0000i		−	0.803379i
$503$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$503$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$504$	−2.00000	+	2.00000i	−0.0890871	+	0.0890871i
$505$		0			0
$506$		0			0
$507$	23.0000	+	23.0000i	1.02147	+	1.02147i
$508$		−	8.00000i		−	0.354943i
$509$	−10.0000			−0.443242			−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000			−0.443242			−0.221621	+	0.975133i	$0.571135\pi$
$510$		0			0
$511$	4.00000			0.176950
$512$			1.00000i			0.0441942i
$513$	−16.0000	−	16.0000i	−0.706417	−	0.706417i
$514$	−28.0000			−1.23503
$515$		0			0
$516$	6.00000	−	6.00000i	0.264135	−	0.264135i
$517$	−8.00000	+	8.00000i	−0.351840	+	0.351840i
$518$		0			0
$519$		−	28.0000i		−	1.22906i
$520$		0			0
$521$	−29.0000	+	29.0000i	−1.27051	+	1.27051i	−0.324694	+	0.945819i	$0.605261\pi$
$521$	−29.0000	+	29.0000i	−1.27051	+	1.27051i	−0.945819	+	0.324694i	$0.894739\pi$
$522$	−2.00000	−	2.00000i	−0.0875376	−	0.0875376i
$523$	6.00000			0.262362			0.131181	−	0.991358i	$-0.458123\pi$
$523$	6.00000			0.262362			0.131181	+	0.991358i	$0.458123\pi$
$524$	−1.00000	−	1.00000i	−0.0436852	−	0.0436852i
$525$		0			0
$526$	−16.0000			−0.697633
$527$	−18.0000	−	30.0000i	−0.784092	−	1.30682i
$528$	2.00000			0.0870388
$529$			23.0000i			1.00000i
$530$		0			0
$531$	6.00000			0.260378
$532$	8.00000	+	8.00000i	0.346844	+	0.346844i
$533$	6.00000	−	6.00000i	0.259889	−	0.259889i
$534$		0			0
$535$		0			0
$536$		−	2.00000i		−	0.0863868i
$537$	4.00000	−	4.00000i	0.172613	−	0.172613i
$538$	2.00000	−	2.00000i	0.0862261	−	0.0862261i
$539$	−1.00000	−	1.00000i	−0.0430730	−	0.0430730i
$540$		0			0
$541$	−24.0000	−	24.0000i	−1.03184	−	1.03184i	−0.999476	−	0.0323645i	$-0.989696\pi$
$541$	−24.0000	−	24.0000i	−1.03184	−	1.03184i	−0.0323645	−	0.999476i	$-0.510304\pi$
$542$		−	28.0000i		−	1.20270i
$543$	−8.00000			−0.343313
$544$	1.00000	−	4.00000i	0.0428746	−	0.171499i
$545$		0			0
$546$			24.0000i			1.02711i
$547$	−25.0000	−	25.0000i	−1.06892	−	1.06892i	−0.997442	−	0.0714808i	$-0.977228\pi$
$547$	−25.0000	−	25.0000i	−1.06892	−	1.06892i	−0.0714808	−	0.997442i	$-0.522772\pi$
$548$	18.0000			0.768922
$549$	4.00000	+	4.00000i	0.170716	+	0.170716i
$550$		0			0
$551$	−8.00000	+	8.00000i	−0.340811	+	0.340811i
$552$		0			0
$553$			32.0000i			1.36078i
$554$	10.0000	−	10.0000i	0.424859	−	0.424859i
$555$		0			0
$556$	7.00000	+	7.00000i	0.296866	+	0.296866i
$557$	−18.0000			−0.762684			−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000			−0.762684			−0.381342	+	0.924434i	$0.624538\pi$
$558$	6.00000	+	6.00000i	0.254000	+	0.254000i
$559$			36.0000i			1.52264i
$560$		0			0
$561$	−8.00000	−	2.00000i	−0.337760	−	0.0844401i
$562$	−10.0000			−0.421825
$563$			6.00000i			0.252870i	0.991975	+	0.126435i	$0.0403535\pi$
$563$			6.00000i			0.252870i	−0.991975	+	0.126435i	$0.959647\pi$
$564$	8.00000	+	8.00000i	0.336861	+	0.336861i
$565$		0			0
$566$	−5.00000	−	5.00000i	−0.210166	−	0.210166i
$567$	10.0000	−	10.0000i	0.419961	−	0.419961i
$568$	−4.00000	+	4.00000i	−0.167836	+	0.167836i
$569$		−	4.00000i		−	0.167689i	−0.996479	−	0.0838444i	$-0.973280\pi$
$569$		−	4.00000i		−	0.167689i	0.996479	−	0.0838444i	$-0.0267199\pi$
$570$		0			0
$571$	1.00000	−	1.00000i	0.0418487	−	0.0418487i	−0.685873	−	0.727721i	$-0.740579\pi$
$571$	1.00000	−	1.00000i	0.0418487	−	0.0418487i	0.727721	+	0.685873i	$0.240579\pi$
$572$	−6.00000	+	6.00000i	−0.250873	+	0.250873i
$573$	12.0000	+	12.0000i	0.501307	+	0.501307i
$574$	4.00000			0.166957
$575$		0			0
$576$			1.00000i			0.0416667i
$577$	2.00000			0.0832611			0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000			0.0832611			0.0416305	+	0.999133i	$0.486745\pi$
$578$	−8.00000	+	15.0000i	−0.332756	+	0.623918i
$579$	30.0000			1.24676
$580$		0			0
$581$	−28.0000	−	28.0000i	−1.16164	−	1.16164i
$582$	−10.0000			−0.414513
$583$	6.00000	+	6.00000i	0.248495	+	0.248495i
$584$	1.00000	−	1.00000i	0.0413803	−	0.0413803i
$585$		0			0
$586$			6.00000i			0.247858i
$587$			28.0000i			1.15568i	0.816149	+	0.577842i	$0.196105\pi$
$587$			28.0000i			1.15568i	−0.816149	+	0.577842i	$0.803895\pi$
$588$	−1.00000	+	1.00000i	−0.0412393	+	0.0412393i
$589$	24.0000	−	24.0000i	0.988903	−	0.988903i
$590$		0			0
$591$	24.0000			0.987228
$592$		0			0
$593$		−	24.0000i		−	0.985562i	−0.870153	−	0.492781i	$-0.835980\pi$
$593$		−	24.0000i		−	0.985562i	0.870153	−	0.492781i	$-0.164020\pi$
$594$	8.00000			0.328244
$595$		0			0
$596$	−10.0000			−0.409616
$597$			16.0000i			0.654836i
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	31.0000	−	31.0000i	1.26452	−	1.26452i	0.315636	−	0.948880i	$-0.397782\pi$
$601$	31.0000	−	31.0000i	1.26452	−	1.26452i	0.948880	−	0.315636i	$-0.102218\pi$
$602$	−12.0000	+	12.0000i	−0.489083	+	0.489083i
$603$		−	2.00000i		−	0.0814463i
$604$			20.0000i			0.813788i
$605$		0			0
$606$	−2.00000	+	2.00000i	−0.0812444	+	0.0812444i
$607$	−30.0000	−	30.0000i	−1.21766	−	1.21766i	−0.968448	−	0.249214i	$-0.919828\pi$
$607$	−30.0000	−	30.0000i	−1.21766	−	1.21766i	−0.249214	−	0.968448i	$-0.580172\pi$
$608$	4.00000			0.162221
$609$	−8.00000	−	8.00000i	−0.324176	−	0.324176i
$610$		0			0
$611$	−48.0000			−1.94187
$612$	1.00000	−	4.00000i	0.0404226	−	0.161690i
$613$	6.00000			0.242338			0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000			0.242338			0.121169	+	0.992632i	$0.461336\pi$
$614$		−	28.0000i		−	1.12999i
$615$		0			0
$616$	−4.00000			−0.161165
$617$	25.0000	+	25.0000i	1.00646	+	1.00646i	0.999979	+	0.00648312i	$0.00206366\pi$
$617$	25.0000	+	25.0000i	1.00646	+	1.00646i	0.00648312	+	0.999979i	$0.497936\pi$
$618$	4.00000	−	4.00000i	0.160904	−	0.160904i
$619$	27.0000	−	27.0000i	1.08522	−	1.08522i	0.0892087	−	0.996013i	$-0.471566\pi$
$619$	27.0000	−	27.0000i	1.08522	−	1.08522i	0.996013	−	0.0892087i	$-0.0284338\pi$
$620$		0			0
$621$		0			0
$622$	−6.00000	+	6.00000i	−0.240578	+	0.240578i
$623$		0			0
$624$	6.00000	+	6.00000i	0.240192	+	0.240192i
$625$		0			0
$626$	5.00000	+	5.00000i	0.199840	+	0.199840i
$627$		−	8.00000i		−	0.319489i
$628$	−22.0000			−0.877896
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$	8.00000	+	8.00000i	0.318223	+	0.318223i
$633$	−18.0000			−0.715436
$634$	−8.00000	−	8.00000i	−0.317721	−	0.317721i
$635$		0			0
$636$	6.00000	−	6.00000i	0.237915	−	0.237915i
$637$		−	6.00000i		−	0.237729i
$638$		−	4.00000i		−	0.158362i
$639$	−4.00000	+	4.00000i	−0.158238	+	0.158238i
$640$		0			0
$641$	−19.0000	−	19.0000i	−0.750455	−	0.750455i	0.224109	−	0.974564i	$-0.428053\pi$
$641$	−19.0000	−	19.0000i	−0.750455	−	0.750455i	−0.974564	+	0.224109i	$0.928053\pi$
$642$	−10.0000			−0.394669
$643$	11.0000	+	11.0000i	0.433798	+	0.433798i	0.889918	−	0.456120i	$-0.150761\pi$
$643$	11.0000	+	11.0000i	0.433798	+	0.433798i	−0.456120	+	0.889918i	$0.650761\pi$
$644$		0			0
$645$		0			0
$646$	−16.0000	−	4.00000i	−0.629512	−	0.157378i
$647$	12.0000			0.471769			0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000			0.471769			0.235884	+	0.971781i	$0.424201\pi$
$648$		−	5.00000i		−	0.196419i
$649$	6.00000	+	6.00000i	0.235521	+	0.235521i
$650$		0			0
$651$	24.0000	+	24.0000i	0.940634	+	0.940634i
$652$	15.0000	−	15.0000i	0.587445	−	0.587445i
$653$	10.0000	−	10.0000i	0.391330	−	0.391330i	−0.483831	−	0.875161i	$-0.660755\pi$
$653$	10.0000	−	10.0000i	0.391330	−	0.391330i	0.875161	+	0.483831i	$0.160755\pi$
$654$			4.00000i			0.156412i
$655$		0			0
$656$	1.00000	−	1.00000i	0.0390434	−	0.0390434i
$657$	1.00000	−	1.00000i	0.0390137	−	0.0390137i
$658$	−16.0000	−	16.0000i	−0.623745	−	0.623745i
$659$	−20.0000			−0.779089			−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000			−0.779089			−0.389545	+	0.921008i	$0.627368\pi$
$660$		0			0
$661$		−	10.0000i		−	0.388955i	−0.980907	−	0.194477i	$-0.937699\pi$
$661$		−	10.0000i		−	0.388955i	0.980907	−	0.194477i	$-0.0623011\pi$
$662$	−30.0000			−1.16598
$663$	−18.0000	−	30.0000i	−0.699062	−	1.16510i
$664$	−14.0000			−0.543305
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−10.0000	−	10.0000i	−0.386912	−	0.386912i
$669$	4.00000	−	4.00000i	0.154649	−	0.154649i
$670$		0			0
$671$			8.00000i			0.308837i
$672$			4.00000i			0.154303i
$673$	15.0000	−	15.0000i	0.578208	−	0.578208i	−0.356202	−	0.934409i	$-0.615928\pi$
$673$	15.0000	−	15.0000i	0.578208	−	0.578208i	0.934409	+	0.356202i	$0.115928\pi$
$674$	−5.00000	+	5.00000i	−0.192593	+	0.192593i
$675$		0			0
$676$	−23.0000			−0.884615
$677$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$677$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$678$			10.0000i			0.384048i
$679$	20.0000			0.767530
$680$		0			0
$681$	−26.0000			−0.996322
$682$			12.0000i			0.459504i
$683$	−29.0000	−	29.0000i	−1.10965	−	1.10965i	−0.993196	−	0.116459i	$-0.962846\pi$
$683$	−29.0000	−	29.0000i	−1.10965	−	1.10965i	−0.116459	−	0.993196i	$-0.537154\pi$
$684$	4.00000			0.152944
$685$		0			0
$686$	−12.0000	+	12.0000i	−0.458162	+	0.458162i
$687$	14.0000	−	14.0000i	0.534133	−	0.534133i
$688$			6.00000i			0.228748i
$689$			36.0000i			1.37149i
$690$		0			0
$691$	1.00000	−	1.00000i	0.0380418	−	0.0380418i	−0.687830	−	0.725872i	$-0.741437\pi$
$691$	1.00000	−	1.00000i	0.0380418	−	0.0380418i	0.725872	+	0.687830i	$0.241437\pi$
$692$	14.0000	+	14.0000i	0.532200	+	0.532200i
$693$	−4.00000			−0.151947
$694$	−13.0000	−	13.0000i	−0.493473	−	0.493473i
$695$		0			0
$696$	−4.00000			−0.151620
$697$	−5.00000	+	3.00000i	−0.189389	+	0.113633i
$698$	14.0000			0.529908
$699$		−	18.0000i		−	0.680823i
$700$		0			0
$701$	42.0000			1.58632			0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000			1.58632			0.793159	+	0.609015i	$0.208435\pi$
$702$	24.0000	+	24.0000i	0.905822	+	0.905822i
$703$		0			0
$704$	−1.00000	+	1.00000i	−0.0376889	+	0.0376889i
$705$		0			0
$706$		−	4.00000i		−	0.150542i
$707$	4.00000	−	4.00000i	0.150435	−	0.150435i
$708$	6.00000	−	6.00000i	0.225494	−	0.225494i
$709$	−22.0000	−	22.0000i	−0.826227	−	0.826227i	0.160765	−	0.986993i	$-0.448604\pi$
$709$	−22.0000	−	22.0000i	−0.826227	−	0.826227i	−0.986993	+	0.160765i	$0.948604\pi$
$710$		0			0
$711$	8.00000	+	8.00000i	0.300023	+	0.300023i
$712$		0			0
$713$		0			0
$714$	4.00000	−	16.0000i	0.149696	−	0.598785i
$715$		0			0
$716$			4.00000i			0.149487i
$717$	−20.0000	−	20.0000i	−0.746914	−	0.746914i
$718$	−16.0000			−0.597115
$719$	−32.0000	−	32.0000i	−1.19340	−	1.19340i	−0.976107	−	0.217292i	$-0.930278\pi$
$719$	−32.0000	−	32.0000i	−1.19340	−	1.19340i	−0.217292	−	0.976107i	$-0.569722\pi$
$720$		0			0
$721$	−8.00000	+	8.00000i	−0.297936	+	0.297936i
$722$			3.00000i			0.111648i
$723$		−	18.0000i		−	0.669427i
$724$	4.00000	−	4.00000i	0.148659	−	0.148659i
$725$		0			0
$726$	−9.00000	−	9.00000i	−0.334021	−	0.334021i
$727$	32.0000			1.18681			0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000			1.18681			0.593407	+	0.804902i	$0.297782\pi$
$728$	−12.0000	−	12.0000i	−0.444750	−	0.444750i
$729$		−	29.0000i		−	1.07407i
$730$		0			0
$731$	6.00000	−	24.0000i	0.221918	−	0.887672i
$732$	8.00000			0.295689
$733$		−	14.0000i		−	0.517102i	−0.965998	−	0.258551i	$-0.916755\pi$
$733$		−	14.0000i		−	0.517102i	0.965998	−	0.258551i	$-0.0832450\pi$
$734$	12.0000	+	12.0000i	0.442928	+	0.442928i
$735$		0			0
$736$		0			0
$737$	2.00000	−	2.00000i	0.0736709	−	0.0736709i
$738$	1.00000	−	1.00000i	0.0368105	−	0.0368105i
$739$			36.0000i			1.32428i	0.749380	+	0.662141i	$0.230352\pi$
$739$			36.0000i			1.32428i	−0.749380	+	0.662141i	$0.769648\pi$
$740$		0			0
$741$	24.0000	−	24.0000i	0.881662	−	0.881662i
$742$	−12.0000	+	12.0000i	−0.440534	+	0.440534i
$743$	6.00000	+	6.00000i	0.220119	+	0.220119i	0.808548	−	0.588430i	$-0.200254\pi$
$743$	6.00000	+	6.00000i	0.220119	+	0.220119i	−0.588430	+	0.808548i	$0.700254\pi$
$744$	12.0000			0.439941
$745$		0			0
$746$		−	14.0000i		−	0.512576i
$747$	−14.0000			−0.512233
$748$	5.00000	−	3.00000i	0.182818	−	0.109691i
$749$	20.0000			0.730784
$750$		0			0
$751$	−24.0000	−	24.0000i	−0.875772	−	0.875772i	0.117322	−	0.993094i	$-0.462569\pi$
$751$	−24.0000	−	24.0000i	−0.875772	−	0.875772i	−0.993094	+	0.117322i	$0.962569\pi$
$752$	−8.00000			−0.291730
$753$	−18.0000	−	18.0000i	−0.655956	−	0.655956i
$754$	12.0000	−	12.0000i	0.437014	−	0.437014i
$755$		0			0
$756$			16.0000i			0.581914i
$757$		−	22.0000i		−	0.799604i	−0.916602	−	0.399802i	$-0.869079\pi$
$757$		−	22.0000i		−	0.799604i	0.916602	−	0.399802i	$-0.130921\pi$
$758$	−3.00000	+	3.00000i	−0.108965	+	0.108965i
$759$		0			0
$760$		0			0
$761$	22.0000			0.797499			0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000			0.797499			0.398750	+	0.917060i	$0.369444\pi$
$762$	−8.00000	−	8.00000i	−0.289809	−	0.289809i
$763$		−	8.00000i		−	0.289619i
$764$	−12.0000			−0.434145
$765$		0			0
$766$	4.00000			0.144526
$767$			36.0000i			1.29988i
$768$	1.00000	+	1.00000i	0.0360844	+	0.0360844i
$769$	−10.0000			−0.360609			−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000			−0.360609			−0.180305	+	0.983611i	$0.557708\pi$
$770$		0			0
$771$	−28.0000	+	28.0000i	−1.00840	+	1.00840i
$772$	−15.0000	+	15.0000i	−0.539862	+	0.539862i
$773$		−	14.0000i		−	0.503545i	−0.967786	−	0.251773i	$-0.918987\pi$
$773$		−	14.0000i		−	0.503545i	0.967786	−	0.251773i	$-0.0810135\pi$
$774$			6.00000i			0.215666i
$775$		0			0
$776$	5.00000	−	5.00000i	0.179490	−	0.179490i
$777$		0			0
$778$	−6.00000			−0.215110
$779$	−4.00000	−	4.00000i	−0.143315	−	0.143315i
$780$		0			0
$781$	−8.00000			−0.286263
$782$		0			0
$783$	−16.0000			−0.571793
$784$		−	1.00000i		−	0.0357143i
$785$		0			0
$786$	−2.00000			−0.0713376
$787$	−15.0000	−	15.0000i	−0.534692	−	0.534692i	0.387273	−	0.921965i	$-0.373417\pi$
$787$	−15.0000	−	15.0000i	−0.534692	−	0.534692i	−0.921965	+	0.387273i	$0.873417\pi$
$788$	−12.0000	+	12.0000i	−0.427482	+	0.427482i
$789$	−16.0000	+	16.0000i	−0.569615	+	0.569615i
$790$		0			0
$791$		−	20.0000i		−	0.711118i
$792$	−1.00000	+	1.00000i	−0.0355335	+	0.0355335i
$793$	−24.0000	+	24.0000i	−0.852265	+	0.852265i
$794$	12.0000	+	12.0000i	0.425864	+	0.425864i
$795$		0			0
$796$	−8.00000	−	8.00000i	−0.283552	−	0.283552i
$797$			18.0000i			0.637593i	0.947823	+	0.318796i	$0.103279\pi$
$797$			18.0000i			0.637593i	−0.947823	+	0.318796i	$0.896721\pi$
$798$	16.0000			0.566394
$799$	32.0000	+	8.00000i	1.13208	+	0.283020i
$800$		0			0
$801$		0			0
$802$	11.0000	+	11.0000i	0.388424	+	0.388424i
$803$	2.00000			0.0705785
$804$	−2.00000	−	2.00000i	−0.0705346	−	0.0705346i
$805$		0			0
$806$	−36.0000	+	36.0000i	−1.26805	+	1.26805i
$807$		−	4.00000i		−	0.140807i
$808$		−	2.00000i		−	0.0703598i
$809$	−13.0000	+	13.0000i	−0.457056	+	0.457056i	−0.897688	−	0.440632i	$-0.854754\pi$
$809$	−13.0000	+	13.0000i	−0.457056	+	0.457056i	0.440632	+	0.897688i	$0.354754\pi$
$810$		0			0
$811$	−19.0000	−	19.0000i	−0.667180	−	0.667180i	0.289882	−	0.957062i	$-0.406384\pi$
$811$	−19.0000	−	19.0000i	−0.667180	−	0.667180i	−0.957062	+	0.289882i	$0.906384\pi$
$812$	8.00000			0.280745
$813$	−28.0000	−	28.0000i	−0.982003	−	0.982003i
$814$		0			0
$815$		0			0
$816$	−3.00000	−	5.00000i	−0.105021	−	0.175035i
$817$	24.0000			0.839654
$818$		0			0
$819$	−12.0000	−	12.0000i	−0.419314	−	0.419314i
$820$		0			0
$821$	−4.00000	−	4.00000i	−0.139601	−	0.139601i	0.633853	−	0.773454i	$-0.281473\pi$
$821$	−4.00000	−	4.00000i	−0.139601	−	0.139601i	−0.773454	+	0.633853i	$0.781473\pi$
$822$	18.0000	−	18.0000i	0.627822	−	0.627822i
$823$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$823$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$824$			4.00000i			0.139347i
$825$		0			0
$826$	−12.0000	+	12.0000i	−0.417533	+	0.417533i
$827$	37.0000	−	37.0000i	1.28662	−	1.28662i	0.349787	−	0.936829i	$-0.386254\pi$
$827$	37.0000	−	37.0000i	1.28662	−	1.28662i	0.936829	−	0.349787i	$-0.113746\pi$
$828$		0			0
$829$	−10.0000			−0.347314			−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000			−0.347314			−0.173657	+	0.984806i	$0.555558\pi$
$830$		0			0
$831$		−	20.0000i		−	0.693792i
$832$	−6.00000			−0.208013
$833$	−1.00000	+	4.00000i	−0.0346479	+	0.138592i
$834$	14.0000			0.484780
$835$		0			0
$836$	4.00000	+	4.00000i	0.138343	+	0.138343i
$837$	48.0000			1.65912
$838$	−23.0000	−	23.0000i	−0.794522	−	0.794522i
$839$	−38.0000	+	38.0000i	−1.31191	+	1.31191i	−0.391896	+	0.920009i	$0.628181\pi$
$839$	−38.0000	+	38.0000i	−1.31191	+	1.31191i	−0.920009	+	0.391896i	$0.871819\pi$
$840$		0			0
$841$		−	21.0000i		−	0.724138i
$842$			22.0000i			0.758170i
$843$	−10.0000	+	10.0000i	−0.344418	+	0.344418i
$844$	9.00000	−	9.00000i	0.309793	−	0.309793i
$845$		0			0
$846$	−8.00000			−0.275046
$847$	18.0000	+	18.0000i	0.618487	+	0.618487i
$848$			6.00000i			0.206041i
$849$	−10.0000			−0.343199
$850$		0			0
$851$		0			0
$852$			8.00000i			0.274075i
$853$	26.0000	+	26.0000i	0.890223	+	0.890223i	0.994544	−	0.104321i	$-0.0332668\pi$
$853$	26.0000	+	26.0000i	0.890223	+	0.890223i	−0.104321	+	0.994544i	$0.533267\pi$
$854$	−16.0000			−0.547509
$855$		0			0
$856$	5.00000	−	5.00000i	0.170896	−	0.170896i
$857$	27.0000	−	27.0000i	0.922302	−	0.922302i	−0.0748894	−	0.997192i	$-0.523860\pi$
$857$	27.0000	−	27.0000i	0.922302	−	0.922302i	0.997192	+	0.0748894i	$0.0238604\pi$
$858$			12.0000i			0.409673i
$859$			6.00000i			0.204717i	0.994748	+	0.102359i	$0.0326389\pi$
$859$			6.00000i			0.204717i	−0.994748	+	0.102359i	$0.967361\pi$
$860$		0			0
$861$	4.00000	−	4.00000i	0.136320	−	0.136320i
$862$	16.0000	+	16.0000i	0.544962	+	0.544962i
$863$	16.0000			0.544646			0.272323	−	0.962206i	$-0.412208\pi$
$863$	16.0000			0.544646			0.272323	+	0.962206i	$0.412208\pi$
$864$	4.00000	+	4.00000i	0.136083	+	0.136083i
$865$		0			0
$866$	14.0000			0.475739
$867$	7.00000	+	23.0000i	0.237732	+	0.781121i
$868$	−24.0000			−0.814613
$869$			16.0000i			0.542763i
$870$		0			0
$871$	12.0000			0.406604
$872$	−2.00000	−	2.00000i	−0.0677285	−	0.0677285i
$873$	5.00000	−	5.00000i	0.169224	−	0.169224i
$874$		0			0
$875$		0			0
$876$		−	2.00000i		−	0.0675737i
$877$	2.00000	−	2.00000i	0.0675352	−	0.0675352i	−0.672532	−	0.740068i	$-0.734793\pi$
$877$	2.00000	−	2.00000i	0.0675352	−	0.0675352i	0.740068	+	0.672532i	$0.234793\pi$
$878$	−8.00000	+	8.00000i	−0.269987	+	0.269987i
$879$	6.00000	+	6.00000i	0.202375	+	0.202375i
$880$		0			0
$881$	41.0000	+	41.0000i	1.38133	+	1.38133i	0.842271	+	0.539054i	$0.181218\pi$
$881$	41.0000	+	41.0000i	1.38133	+	1.38133i	0.539054	+	0.842271i	$0.318782\pi$
$882$		−	1.00000i		−	0.0336718i
$883$	6.00000			0.201916			0.100958	−	0.994891i	$-0.467809\pi$
$883$	6.00000			0.201916			0.100958	+	0.994891i	$0.467809\pi$
$884$	24.0000	+	6.00000i	0.807207	+	0.201802i
$885$		0			0
$886$			6.00000i			0.201574i
$887$	−20.0000	−	20.0000i	−0.671534	−	0.671534i	0.286535	−	0.958070i	$-0.407496\pi$
$887$	−20.0000	−	20.0000i	−0.671534	−	0.671534i	−0.958070	+	0.286535i	$0.907496\pi$
$888$		0			0
$889$	16.0000	+	16.0000i	0.536623	+	0.536623i
$890$		0			0
$891$	5.00000	−	5.00000i	0.167506	−	0.167506i
$892$			4.00000i			0.133930i
$893$			32.0000i			1.07084i
$894$	−10.0000	+	10.0000i	−0.334450	+	0.334450i
$895$		0			0
$896$	−2.00000	−	2.00000i	−0.0668153	−	0.0668153i
$897$		0			0
$898$	27.0000	+	27.0000i	0.901002	+	0.901002i
$899$		−	24.0000i		−	0.800445i
$900$		0			0
$901$	6.00000	−	24.0000i	0.199889	−	0.799556i
$902$	2.00000			0.0665927
$903$			24.0000i			0.798670i
$904$	−5.00000	−	5.00000i	−0.166298	−	0.166298i
$905$		0			0
$906$	20.0000	+	20.0000i	0.664455	+	0.664455i
$907$	7.00000	−	7.00000i	0.232431	−	0.232431i	−0.581276	−	0.813707i	$-0.697446\pi$
$907$	7.00000	−	7.00000i	0.232431	−	0.232431i	0.813707	+	0.581276i	$0.197446\pi$
$908$	13.0000	−	13.0000i	0.431420	−	0.431420i
$909$		−	2.00000i		−	0.0663358i
$910$		0			0
$911$	26.0000	−	26.0000i	0.861418	−	0.861418i	−0.130084	−	0.991503i	$-0.541525\pi$
$911$	26.0000	−	26.0000i	0.861418	−	0.861418i	0.991503	+	0.130084i	$0.0415249\pi$
$912$	4.00000	−	4.00000i	0.132453	−	0.132453i
$913$	−14.0000	−	14.0000i	−0.463332	−	0.463332i
$914$	−38.0000			−1.25693
$915$		0			0
$916$			14.0000i			0.462573i
$917$	4.00000			0.132092
$918$	−12.0000	−	20.0000i	−0.396059	−	0.660098i
$919$	−40.0000			−1.31948			−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000			−1.31948			−0.659739	+	0.751495i	$0.729333\pi$
$920$		0			0
$921$	−28.0000	−	28.0000i	−0.922631	−	0.922631i
$922$	−30.0000			−0.987997
$923$	−24.0000	−	24.0000i	−0.789970	−	0.789970i
$924$	−4.00000	+	4.00000i	−0.131590	+	0.131590i
$925$		0			0
$926$		−	24.0000i		−	0.788689i
$927$			4.00000i			0.131377i
$928$	2.00000	−	2.00000i	0.0656532	−	0.0656532i
$929$	37.0000	−	37.0000i	1.21393	−	1.21393i	0.244208	−	0.969723i	$-0.421472\pi$
$929$	37.0000	−	37.0000i	1.21393	−	1.21393i	0.969723	−	0.244208i	$-0.0785279\pi$
$930$		0			0
$931$	−4.00000			−0.131095
$932$	9.00000	+	9.00000i	0.294805	+	0.294805i
$933$			12.0000i			0.392862i
$934$	−28.0000			−0.916188
$935$		0			0
$936$	−6.00000			−0.196116
$937$			38.0000i			1.24141i	0.784046	+	0.620703i	$0.213153\pi$
$937$			38.0000i			1.24141i	−0.784046	+	0.620703i	$0.786847\pi$
$938$	4.00000	+	4.00000i	0.130605	+	0.130605i
$939$	10.0000			0.326338
$940$		0			0
$941$	36.0000	−	36.0000i	1.17357	−	1.17357i	0.192213	−	0.981353i	$-0.438433\pi$
$941$	36.0000	−	36.0000i	1.17357	−	1.17357i	0.981353	−	0.192213i	$-0.0615665\pi$
$942$	−22.0000	+	22.0000i	−0.716799	+	0.716799i
$943$		0			0
$944$			6.00000i			0.195283i
$945$		0			0
$946$	−6.00000	+	6.00000i	−0.195077	+	0.195077i
$947$	−15.0000	−	15.0000i	−0.487435	−	0.487435i	0.420061	−	0.907496i	$-0.362009\pi$
$947$	−15.0000	−	15.0000i	−0.487435	−	0.487435i	−0.907496	+	0.420061i	$0.862009\pi$
$948$	16.0000			0.519656
$949$	6.00000	+	6.00000i	0.194768	+	0.194768i
$950$		0			0
$951$	−16.0000			−0.518836
$952$	6.00000	+	10.0000i	0.194461	+	0.324102i
$953$	−44.0000			−1.42530			−0.712650	−	0.701520i	$-0.752505\pi$
$953$	−44.0000			−1.42530			−0.712650	+	0.701520i	$0.752505\pi$
$954$			6.00000i			0.194257i
$955$		0			0
$956$	20.0000			0.646846
$957$	−4.00000	−	4.00000i	−0.129302	−	0.129302i
$958$	−8.00000	+	8.00000i	−0.258468	+	0.258468i
$959$	−36.0000	+	36.0000i	−1.16250	+	1.16250i
$960$		0			0
$961$			41.0000i			1.32258i
$962$		0			0
$963$	5.00000	−	5.00000i	0.161123	−	0.161123i
$964$	9.00000	+	9.00000i	0.289870	+	0.289870i
$965$		0			0
$966$		0			0
$967$			8.00000i			0.257263i	0.991692	+	0.128631i	$0.0410584\pi$
$967$			8.00000i			0.257263i	−0.991692	+	0.128631i	$0.958942\pi$
$968$	9.00000			0.289271
$969$	−20.0000	+	12.0000i	−0.642493	+	0.385496i
$970$		0			0
$971$			30.0000i			0.962746i	0.876516	+	0.481373i	$0.159862\pi$
$971$			30.0000i			0.962746i	−0.876516	+	0.481373i	$0.840138\pi$
$972$	7.00000	+	7.00000i	0.224525	+	0.224525i
$973$	−28.0000			−0.897639
$974$	12.0000	+	12.0000i	0.384505	+	0.384505i
$975$		0			0
$976$	−4.00000	+	4.00000i	−0.128037	+	0.128037i
$977$		−	52.0000i		−	1.66363i	−0.555055	−	0.831814i	$-0.687303\pi$
$977$		−	52.0000i		−	1.66363i	0.555055	−	0.831814i	$-0.312697\pi$
$978$		−	30.0000i		−	0.959294i
$979$		0			0
$980$		0			0
$981$	−2.00000	−	2.00000i	−0.0638551	−	0.0638551i
$982$	−20.0000			−0.638226
$983$	16.0000	+	16.0000i	0.510321	+	0.510321i	0.914625	−	0.404304i	$-0.132486\pi$
$983$	16.0000	+	16.0000i	0.510321	+	0.510321i	−0.404304	+	0.914625i	$0.632486\pi$
$984$		−	2.00000i		−	0.0637577i
$985$		0			0
$986$	−10.0000	+	6.00000i	−0.318465	+	0.191079i
$987$	−32.0000			−1.01857
$988$			24.0000i			0.763542i
$989$		0			0
$990$		0			0
$991$	−44.0000	−	44.0000i	−1.39771	−	1.39771i	−0.806577	−	0.591129i	$-0.798682\pi$
$991$	−44.0000	−	44.0000i	−1.39771	−	1.39771i	−0.591129	−	0.806577i	$-0.701318\pi$
$992$	−6.00000	+	6.00000i	−0.190500	+	0.190500i
$993$	−30.0000	+	30.0000i	−0.952021	+	0.952021i
$994$		−	16.0000i		−	0.507489i
$995$		0			0
$996$	−14.0000	+	14.0000i	−0.443607	+	0.443607i
$997$	12.0000	−	12.0000i	0.380044	−	0.380044i	−0.491074	−	0.871118i	$-0.663396\pi$
$997$	12.0000	−	12.0000i	0.380044	−	0.380044i	0.871118	+	0.491074i	$0.163396\pi$
$998$	−3.00000	−	3.00000i	−0.0949633	−	0.0949633i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	850.2.h.f.251.1		2
5.2	odd	4		850.2.g.b.149.1		2
5.3	odd	4		850.2.g.c.149.1		2
5.4	even	2		34.2.c.a.13.1	✓	2
15.14	odd	2		306.2.g.b.217.1		2
17.4	even	4	inner	850.2.h.f.701.1		2
20.19	odd	2		272.2.o.f.81.1		2
40.19	odd	2		1088.2.o.d.897.1		2
40.29	even	2		1088.2.o.l.897.1		2
60.59	even	2		2448.2.be.c.1441.1		2
85.4	even	4		34.2.c.a.21.1	yes	2
85.9	even	8		578.2.b.b.577.1		2
85.14	odd	16		578.2.d.d.423.2		8
85.19	even	8		578.2.a.c.1.1		2
85.24	odd	16		578.2.d.d.399.1		8
85.29	odd	16		578.2.d.d.155.1		8
85.38	odd	4		850.2.g.b.599.1		2
85.39	odd	16		578.2.d.d.155.2		8
85.44	odd	16		578.2.d.d.399.2		8
85.49	even	8		578.2.a.c.1.2		2
85.54	odd	16		578.2.d.d.423.1		8
85.59	even	8		578.2.b.b.577.2		2
85.64	even	4		578.2.c.c.327.1		2
85.72	odd	4		850.2.g.c.599.1		2
85.74	odd	16		578.2.d.d.179.1		8
85.79	odd	16		578.2.d.d.179.2		8
85.84	even	2		578.2.c.c.251.1		2
255.89	odd	4		306.2.g.b.55.1		2
255.104	odd	8		5202.2.a.v.1.2		2
255.134	odd	8		5202.2.a.v.1.1		2
340.19	odd	8		4624.2.a.r.1.2		2
340.219	odd	8		4624.2.a.r.1.1		2
340.259	odd	4		272.2.o.f.225.1		2
680.259	odd	4		1088.2.o.d.769.1		2
680.429	even	4		1088.2.o.l.769.1		2
1020.599	even	4		2448.2.be.c.1585.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
34.2.c.a.13.1	✓	2	5.4	even	2
34.2.c.a.21.1	yes	2	85.4	even	4
272.2.o.f.81.1		2	20.19	odd	2
272.2.o.f.225.1		2	340.259	odd	4
306.2.g.b.55.1		2	255.89	odd	4
306.2.g.b.217.1		2	15.14	odd	2
578.2.a.c.1.1		2	85.19	even	8
578.2.a.c.1.2		2	85.49	even	8
578.2.b.b.577.1		2	85.9	even	8
578.2.b.b.577.2		2	85.59	even	8
578.2.c.c.251.1		2	85.84	even	2
578.2.c.c.327.1		2	85.64	even	4
578.2.d.d.155.1		8	85.29	odd	16
578.2.d.d.155.2		8	85.39	odd	16
578.2.d.d.179.1		8	85.74	odd	16
578.2.d.d.179.2		8	85.79	odd	16
578.2.d.d.399.1		8	85.24	odd	16
578.2.d.d.399.2		8	85.44	odd	16
578.2.d.d.423.1		8	85.54	odd	16
578.2.d.d.423.2		8	85.14	odd	16
850.2.g.b.149.1		2	5.2	odd	4
850.2.g.b.599.1		2	85.38	odd	4
850.2.g.c.149.1		2	5.3	odd	4
850.2.g.c.599.1		2	85.72	odd	4
850.2.h.f.251.1		2	1.1	even	1	trivial
850.2.h.f.701.1		2	17.4	even	4	inner
1088.2.o.d.769.1		2	680.259	odd	4
1088.2.o.d.897.1		2	40.19	odd	2
1088.2.o.l.769.1		2	680.429	even	4
1088.2.o.l.897.1		2	40.29	even	2
2448.2.be.c.1441.1		2	60.59	even	2
2448.2.be.c.1585.1		2	1020.599	even	4
4624.2.a.r.1.1		2	340.219	odd	8
4624.2.a.r.1.2		2	340.19	odd	8
5202.2.a.v.1.1		2	255.134	odd	8
5202.2.a.v.1.2		2	255.104	odd	8

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:	\( N \)	\(=\)	\( 850 = 2 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	850.h (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +(1.00000 + 1.00000i) q^{3} -1.00000 q^{4} +(-1.00000 + 1.00000i) q^{6} +(2.00000 - 2.00000i) q^{7} -1.00000i q^{8} -1.00000i q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +(1.00000 + 1.00000i) q^{3} -1.00000 q^{4} +(-1.00000 + 1.00000i) q^{6} +(2.00000 - 2.00000i) q^{7} -1.00000i q^{8} -1.00000i q^{9} +(1.00000 - 1.00000i) q^{11} +(-1.00000 - 1.00000i) q^{12} +6.00000 q^{13} +(2.00000 + 2.00000i) q^{14} +1.00000 q^{16} +(-4.00000 - 1.00000i) q^{17} +1.00000 q^{18} -4.00000i q^{19} +4.00000 q^{21} +(1.00000 + 1.00000i) q^{22} +(1.00000 - 1.00000i) q^{24} +6.00000i q^{26} +(4.00000 - 4.00000i) q^{27} +(-2.00000 + 2.00000i) q^{28} +(-2.00000 - 2.00000i) q^{29} +(6.00000 + 6.00000i) q^{31} +1.00000i q^{32} +2.00000 q^{33} +(1.00000 - 4.00000i) q^{34} +1.00000i q^{36} +4.00000 q^{38} +(6.00000 + 6.00000i) q^{39} +(1.00000 - 1.00000i) q^{41} +4.00000i q^{42} +6.00000i q^{43} +(-1.00000 + 1.00000i) q^{44} -8.00000 q^{47} +(1.00000 + 1.00000i) q^{48} -1.00000i q^{49} +(-3.00000 - 5.00000i) q^{51} -6.00000 q^{52} +6.00000i q^{53} +(4.00000 + 4.00000i) q^{54} +(-2.00000 - 2.00000i) q^{56} +(4.00000 - 4.00000i) q^{57} +(2.00000 - 2.00000i) q^{58} +6.00000i q^{59} +(-4.00000 + 4.00000i) q^{61} +(-6.00000 + 6.00000i) q^{62} +(-2.00000 - 2.00000i) q^{63} -1.00000 q^{64} +2.00000i q^{66} +2.00000 q^{67} +(4.00000 + 1.00000i) q^{68} +(-4.00000 - 4.00000i) q^{71} -1.00000 q^{72} +(1.00000 + 1.00000i) q^{73} +4.00000i q^{76} -4.00000i q^{77} +(-6.00000 + 6.00000i) q^{78} +(-8.00000 + 8.00000i) q^{79} +5.00000 q^{81} +(1.00000 + 1.00000i) q^{82} -14.0000i q^{83} -4.00000 q^{84} -6.00000 q^{86} -4.00000i q^{87} +(-1.00000 - 1.00000i) q^{88} +(12.0000 - 12.0000i) q^{91} +12.0000i q^{93} -8.00000i q^{94} +(-1.00000 + 1.00000i) q^{96} +(5.00000 + 5.00000i) q^{97} +1.00000 q^{98} +(-1.00000 - 1.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{4} - 2 q^{6} + 4 q^{7}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^4 - 2 * q^6 + 4 * q^7	\( 2 q + 2 q^{3} - 2 q^{4} - 2 q^{6} + 4 q^{7} + 2 q^{11} - 2 q^{12} + 12 q^{13} + 4 q^{14} + 2 q^{16} - 8 q^{17} + 2 q^{18} + 8 q^{21} + 2 q^{22} + 2 q^{24} + 8 q^{27} - 4 q^{28} - 4 q^{29} + 12 q^{31} + 4 q^{33} + 2 q^{34} + 8 q^{38} + 12 q^{39} + 2 q^{41} - 2 q^{44} - 16 q^{47} + 2 q^{48} - 6 q^{51} - 12 q^{52} + 8 q^{54} - 4 q^{56} + 8 q^{57} + 4 q^{58} - 8 q^{61} - 12 q^{62} - 4 q^{63} - 2 q^{64} + 4 q^{67} + 8 q^{68} - 8 q^{71} - 2 q^{72} + 2 q^{73} - 12 q^{78} - 16 q^{79} + 10 q^{81} + 2 q^{82} - 8 q^{84} - 12 q^{86} - 2 q^{88} + 24 q^{91} - 2 q^{96} + 10 q^{97} + 2 q^{98} - 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^4 - 2 * q^6 + 4 * q^7 + 2 * q^11 - 2 * q^12 + 12 * q^13 + 4 * q^14 + 2 * q^16 - 8 * q^17 + 2 * q^18 + 8 * q^21 + 2 * q^22 + 2 * q^24 + 8 * q^27 - 4 * q^28 - 4 * q^29 + 12 * q^31 + 4 * q^33 + 2 * q^34 + 8 * q^38 + 12 * q^39 + 2 * q^41 - 2 * q^44 - 16 * q^47 + 2 * q^48 - 6 * q^51 - 12 * q^52 + 8 * q^54 - 4 * q^56 + 8 * q^57 + 4 * q^58 - 8 * q^61 - 12 * q^62 - 4 * q^63 - 2 * q^64 + 4 * q^67 + 8 * q^68 - 8 * q^71 - 2 * q^72 + 2 * q^73 - 12 * q^78 - 16 * q^79 + 10 * q^81 + 2 * q^82 - 8 * q^84 - 12 * q^86 - 2 * q^88 + 24 * q^91 - 2 * q^96 + 10 * q^97 + 2 * q^98 - 2 * q^99

Introduction

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