LMFDB - Embedded newform 425.2.e.a.251.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [425,2,Mod(251,425)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("425.251"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 425 = 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	425.e (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$3.39364208590$
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 85)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			251.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	425.251
Dual form			425.2.e.a.276.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} +(-1.00000 - 1.00000i) q^{3} +1.00000 q^{4} +(1.00000 - 1.00000i) q^{6} +(-3.00000 + 3.00000i) q^{7} +3.00000i q^{8} -1.00000i q^{9} +(-3.00000 + 3.00000i) q^{11} +(-1.00000 - 1.00000i) q^{12} +(-3.00000 - 3.00000i) q^{14} -1.00000 q^{16} +(4.00000 + 1.00000i) q^{17} +1.00000 q^{18} +6.00000i q^{19} +6.00000 q^{21} +(-3.00000 - 3.00000i) q^{22} +(-1.00000 + 1.00000i) q^{23} +(3.00000 - 3.00000i) q^{24} +(-4.00000 + 4.00000i) q^{27} +(-3.00000 + 3.00000i) q^{28} +(3.00000 + 3.00000i) q^{29} +(-1.00000 - 1.00000i) q^{31} +5.00000i q^{32} +6.00000 q^{33} +(-1.00000 + 4.00000i) q^{34} -1.00000i q^{36} +(-3.00000 - 3.00000i) q^{37} -6.00000 q^{38} +(-3.00000 + 3.00000i) q^{41} +6.00000i q^{42} -12.0000i q^{43} +(-3.00000 + 3.00000i) q^{44} +(-1.00000 - 1.00000i) q^{46} +2.00000 q^{47} +(1.00000 + 1.00000i) q^{48} -11.0000i q^{49} +(-3.00000 - 5.00000i) q^{51} +2.00000i q^{53} +(-4.00000 - 4.00000i) q^{54} +(-9.00000 - 9.00000i) q^{56} +(6.00000 - 6.00000i) q^{57} +(-3.00000 + 3.00000i) q^{58} +6.00000i q^{59} +(1.00000 - 1.00000i) q^{61} +(1.00000 - 1.00000i) q^{62} +(3.00000 + 3.00000i) q^{63} -7.00000 q^{64} +6.00000i q^{66} +6.00000 q^{67} +(4.00000 + 1.00000i) q^{68} +2.00000 q^{69} +(3.00000 + 3.00000i) q^{71} +3.00000 q^{72} +(3.00000 + 3.00000i) q^{73} +(3.00000 - 3.00000i) q^{74} +6.00000i q^{76} -18.0000i q^{77} +(7.00000 - 7.00000i) q^{79} +5.00000 q^{81} +(-3.00000 - 3.00000i) q^{82} -4.00000i q^{83} +6.00000 q^{84} +12.0000 q^{86} -6.00000i q^{87} +(-9.00000 - 9.00000i) q^{88} -6.00000 q^{89} +(-1.00000 + 1.00000i) q^{92} +2.00000i q^{93} +2.00000i q^{94} +(5.00000 - 5.00000i) q^{96} +(-3.00000 - 3.00000i) q^{97} +11.0000 q^{98} +(3.00000 + 3.00000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{4} + 2 q^{6} - 6 q^{7} - 6 q^{11} - 2 q^{12} - 6 q^{14} - 2 q^{16} + 8 q^{17} + 2 q^{18} + 12 q^{21} - 6 q^{22} - 2 q^{23} + 6 q^{24} - 8 q^{27} - 6 q^{28} + 6 q^{29} - 2 q^{31} + 12 q^{33}+ \cdots + 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^4 + 2 * q^6 - 6 * q^7 - 6 * q^11 - 2 * q^12 - 6 * q^14 - 2 * q^16 + 8 * q^17 + 2 * q^18 + 12 * q^21 - 6 * q^22 - 2 * q^23 + 6 * q^24 - 8 * q^27 - 6 * q^28 + 6 * q^29 - 2 * q^31 + 12 * q^33 - 2 * q^34 - 6 * q^37 - 12 * q^38 - 6 * q^41 - 6 * q^44 - 2 * q^46 + 4 * q^47 + 2 * q^48 - 6 * q^51 - 8 * q^54 - 18 * q^56 + 12 * q^57 - 6 * q^58 + 2 * q^61 + 2 * q^62 + 6 * q^63 - 14 * q^64 + 12 * q^67 + 8 * q^68 + 4 * q^69 + 6 * q^71 + 6 * q^72 + 6 * q^73 + 6 * q^74 + 14 * q^79 + 10 * q^81 - 6 * q^82 + 12 * q^84 + 24 * q^86 - 18 * q^88 - 12 * q^89 - 2 * q^92 + 10 * q^96 - 6 * q^97 + 22 * q^98 + 6 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/425\mathbb{Z}\right)^\times$.

$n$	$52$	$326$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$

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	0.935414	+	0.353553i	$0.115027\pi$
$2$			1.00000i			0.707107i	−0.935414	+	0.353553i	$0.884973\pi$
$3$	−1.00000	−	1.00000i	−0.577350	−	0.577350i	0.356822	−	0.934172i	$-0.383860\pi$
$3$	−1.00000	−	1.00000i	−0.577350	−	0.577350i	−0.934172	+	0.356822i	$0.883860\pi$
$4$	1.00000			0.500000
$5$		0			0
$5$		0			0
$6$	1.00000	−	1.00000i	0.408248	−	0.408248i
$7$	−3.00000	+	3.00000i	−1.13389	+	1.13389i	−0.144370	+	0.989524i	$0.546115\pi$
$7$	−3.00000	+	3.00000i	−1.13389	+	1.13389i	−0.989524	+	0.144370i	$0.953885\pi$
$8$			3.00000i			1.06066i
$9$		−	1.00000i		−	0.333333i
$10$		0			0
$11$	−3.00000	+	3.00000i	−0.904534	+	0.904534i	−0.995824	−	0.0912903i	$-0.970901\pi$
$11$	−3.00000	+	3.00000i	−0.904534	+	0.904534i	0.0912903	+	0.995824i	$0.470901\pi$
$12$	−1.00000	−	1.00000i	−0.288675	−	0.288675i
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$	−3.00000	−	3.00000i	−0.801784	−	0.801784i
$15$		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
$19$			6.00000i			1.37649i	0.725476	+	0.688247i	$0.241620\pi$
$19$			6.00000i			1.37649i	−0.725476	+	0.688247i	$0.758380\pi$
$20$		0			0
$21$	6.00000			1.30931
$22$	−3.00000	−	3.00000i	−0.639602	−	0.639602i
$23$	−1.00000	+	1.00000i	−0.208514	+	0.208514i	−0.803636	−	0.595121i	$-0.797104\pi$
$23$	−1.00000	+	1.00000i	−0.208514	+	0.208514i	0.595121	+	0.803636i	$0.297104\pi$
$24$	3.00000	−	3.00000i	0.612372	−	0.612372i
$25$		0			0
$26$		0			0
$27$	−4.00000	+	4.00000i	−0.769800	+	0.769800i
$28$	−3.00000	+	3.00000i	−0.566947	+	0.566947i
$29$	3.00000	+	3.00000i	0.557086	+	0.557086i	0.928477	−	0.371391i	$-0.121119\pi$
$29$	3.00000	+	3.00000i	0.557086	+	0.557086i	−0.371391	+	0.928477i	$0.621119\pi$
$30$		0			0
$31$	−1.00000	−	1.00000i	−0.179605	−	0.179605i	0.611578	−	0.791184i	$-0.290535\pi$
$31$	−1.00000	−	1.00000i	−0.179605	−	0.179605i	−0.791184	+	0.611578i	$0.790535\pi$
$32$			5.00000i			0.883883i
$33$	6.00000			1.04447
$34$	−1.00000	+	4.00000i	−0.171499	+	0.685994i
$35$		0			0
$36$		−	1.00000i		−	0.166667i
$37$	−3.00000	−	3.00000i	−0.493197	−	0.493197i	0.416115	−	0.909312i	$-0.363391\pi$
$37$	−3.00000	−	3.00000i	−0.493197	−	0.493197i	−0.909312	+	0.416115i	$0.863391\pi$
$38$	−6.00000			−0.973329
$39$		0			0
$40$		0			0
$41$	−3.00000	+	3.00000i	−0.468521	+	0.468521i	−0.901435	−	0.432914i	$-0.857485\pi$
$41$	−3.00000	+	3.00000i	−0.468521	+	0.468521i	0.432914	+	0.901435i	$0.357485\pi$
$42$			6.00000i			0.925820i
$43$		−	12.0000i		−	1.82998i	−0.403473	−	0.914991i	$-0.632197\pi$
$43$		−	12.0000i		−	1.82998i	0.403473	−	0.914991i	$-0.367803\pi$
$44$	−3.00000	+	3.00000i	−0.452267	+	0.452267i
$45$		0			0
$46$	−1.00000	−	1.00000i	−0.147442	−	0.147442i
$47$	2.00000			0.291730			0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000			0.291730			0.145865	+	0.989305i	$0.453403\pi$
$48$	1.00000	+	1.00000i	0.144338	+	0.144338i
$49$		−	11.0000i		−	1.57143i
$50$		0			0
$51$	−3.00000	−	5.00000i	−0.420084	−	0.700140i
$52$		0			0
$53$			2.00000i			0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$			2.00000i			0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$	−4.00000	−	4.00000i	−0.544331	−	0.544331i
$55$		0			0
$56$	−9.00000	−	9.00000i	−1.20268	−	1.20268i
$57$	6.00000	−	6.00000i	0.794719	−	0.794719i
$58$	−3.00000	+	3.00000i	−0.393919	+	0.393919i
$59$			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
$61$	1.00000	−	1.00000i	0.128037	−	0.128037i	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	1.00000	−	1.00000i	0.128037	−	0.128037i	0.768221	+	0.640184i	$0.221142\pi$
$62$	1.00000	−	1.00000i	0.127000	−	0.127000i
$63$	3.00000	+	3.00000i	0.377964	+	0.377964i
$64$	−7.00000			−0.875000
$65$		0			0
$66$			6.00000i			0.738549i
$67$	6.00000			0.733017			0.366508	−	0.930415i	$-0.380553\pi$
$67$	6.00000			0.733017			0.366508	+	0.930415i	$0.380553\pi$
$68$	4.00000	+	1.00000i	0.485071	+	0.121268i
$69$	2.00000			0.240772
$70$		0			0
$71$	3.00000	+	3.00000i	0.356034	+	0.356034i	0.862349	−	0.506314i	$-0.168992\pi$
$71$	3.00000	+	3.00000i	0.356034	+	0.356034i	−0.506314	+	0.862349i	$0.668992\pi$
$72$	3.00000			0.353553
$73$	3.00000	+	3.00000i	0.351123	+	0.351123i	0.860527	−	0.509404i	$-0.170134\pi$
$73$	3.00000	+	3.00000i	0.351123	+	0.351123i	−0.509404	+	0.860527i	$0.670134\pi$
$74$	3.00000	−	3.00000i	0.348743	−	0.348743i
$75$		0			0
$76$			6.00000i			0.688247i
$77$		−	18.0000i		−	2.05129i
$78$		0			0
$79$	7.00000	−	7.00000i	0.787562	−	0.787562i	−0.193532	−	0.981094i	$-0.561994\pi$
$79$	7.00000	−	7.00000i	0.787562	−	0.787562i	0.981094	+	0.193532i	$0.0619944\pi$
$80$		0			0
$81$	5.00000			0.555556
$82$	−3.00000	−	3.00000i	−0.331295	−	0.331295i
$83$		−	4.00000i		−	0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$		−	4.00000i		−	0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$	6.00000			0.654654
$85$		0			0
$86$	12.0000			1.29399
$87$		−	6.00000i		−	0.643268i
$88$	−9.00000	−	9.00000i	−0.959403	−	0.959403i
$89$	−6.00000			−0.635999			−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000			−0.635999			−0.317999	+	0.948091i	$0.603011\pi$
$90$		0			0
$91$		0			0
$92$	−1.00000	+	1.00000i	−0.104257	+	0.104257i
$93$			2.00000i			0.207390i
$94$			2.00000i			0.206284i
$95$		0			0
$96$	5.00000	−	5.00000i	0.510310	−	0.510310i
$97$	−3.00000	−	3.00000i	−0.304604	−	0.304604i	0.538208	−	0.842812i	$-0.319101\pi$
$97$	−3.00000	−	3.00000i	−0.304604	−	0.304604i	−0.842812	+	0.538208i	$0.819101\pi$
$98$	11.0000			1.11117
$99$	3.00000	+	3.00000i	0.301511	+	0.301511i
$100$		0			0
$101$	−6.00000			−0.597022			−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000			−0.597022			−0.298511	+	0.954406i	$0.596490\pi$
$102$	5.00000	−	3.00000i	0.495074	−	0.297044i
$103$	6.00000			0.591198			0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000			0.591198			0.295599	+	0.955312i	$0.404481\pi$
$104$		0			0
$105$		0			0
$106$	−2.00000			−0.194257
$107$	9.00000	+	9.00000i	0.870063	+	0.870063i	0.992479	−	0.122416i	$-0.0390642\pi$
$107$	9.00000	+	9.00000i	0.870063	+	0.870063i	−0.122416	+	0.992479i	$0.539064\pi$
$108$	−4.00000	+	4.00000i	−0.384900	+	0.384900i
$109$	7.00000	−	7.00000i	0.670478	−	0.670478i	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	7.00000	−	7.00000i	0.670478	−	0.670478i	0.957826	+	0.287348i	$0.0927736\pi$
$110$		0			0
$111$			6.00000i			0.569495i
$112$	3.00000	−	3.00000i	0.283473	−	0.283473i
$113$	9.00000	−	9.00000i	0.846649	−	0.846649i	−0.143065	−	0.989713i	$-0.545696\pi$
$113$	9.00000	−	9.00000i	0.846649	−	0.846649i	0.989713	+	0.143065i	$0.0456957\pi$
$114$	6.00000	+	6.00000i	0.561951	+	0.561951i
$115$		0			0
$116$	3.00000	+	3.00000i	0.278543	+	0.278543i
$117$		0			0
$118$	−6.00000			−0.552345
$119$	−15.0000	+	9.00000i	−1.37505	+	0.825029i
$120$		0			0
$121$		−	7.00000i		−	0.636364i
$122$	1.00000	+	1.00000i	0.0905357	+	0.0905357i
$123$	6.00000			0.541002
$124$	−1.00000	−	1.00000i	−0.0898027	−	0.0898027i
$125$		0			0
$126$	−3.00000	+	3.00000i	−0.267261	+	0.267261i
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$			3.00000i			0.265165i
$129$	−12.0000	+	12.0000i	−1.05654	+	1.05654i
$130$		0			0
$131$	3.00000	+	3.00000i	0.262111	+	0.262111i	0.825911	−	0.563800i	$-0.190661\pi$
$131$	3.00000	+	3.00000i	0.262111	+	0.262111i	−0.563800	+	0.825911i	$0.690661\pi$
$132$	6.00000			0.522233
$133$	−18.0000	−	18.0000i	−1.56080	−	1.56080i
$134$			6.00000i			0.518321i
$135$		0			0
$136$	−3.00000	+	12.0000i	−0.257248	+	1.02899i
$137$	4.00000			0.341743			0.170872	−	0.985293i	$-0.445342\pi$
$137$	4.00000			0.341743			0.170872	+	0.985293i	$0.445342\pi$
$138$			2.00000i			0.170251i
$139$	−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
$141$	−2.00000	−	2.00000i	−0.168430	−	0.168430i
$142$	−3.00000	+	3.00000i	−0.251754	+	0.251754i
$143$		0			0
$144$			1.00000i			0.0833333i
$145$		0			0
$146$	−3.00000	+	3.00000i	−0.248282	+	0.248282i
$147$	−11.0000	+	11.0000i	−0.907265	+	0.907265i
$148$	−3.00000	−	3.00000i	−0.246598	−	0.246598i
$149$	18.0000			1.47462			0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000			1.47462			0.737309	+	0.675556i	$0.236096\pi$
$150$		0			0
$151$		−	10.0000i		−	0.813788i	−0.913475	−	0.406894i	$-0.866612\pi$
$151$		−	10.0000i		−	0.813788i	0.913475	−	0.406894i	$-0.133388\pi$
$152$	−18.0000			−1.45999
$153$	1.00000	−	4.00000i	0.0808452	−	0.323381i
$154$	18.0000			1.45048
$155$		0			0
$156$		0			0
$157$	12.0000			0.957704			0.478852	−	0.877896i	$-0.341053\pi$
$157$	12.0000			0.957704			0.478852	+	0.877896i	$0.341053\pi$
$158$	7.00000	+	7.00000i	0.556890	+	0.556890i
$159$	2.00000	−	2.00000i	0.158610	−	0.158610i
$160$		0			0
$161$		−	6.00000i		−	0.472866i
$162$			5.00000i			0.392837i
$163$	−9.00000	+	9.00000i	−0.704934	+	0.704934i	−0.965465	−	0.260531i	$-0.916102\pi$
$163$	−9.00000	+	9.00000i	−0.704934	+	0.704934i	0.260531	+	0.965465i	$0.416102\pi$
$164$	−3.00000	+	3.00000i	−0.234261	+	0.234261i
$165$		0			0
$166$	4.00000			0.310460
$167$	−3.00000	−	3.00000i	−0.232147	−	0.232147i	0.581441	−	0.813588i	$-0.302489\pi$
$167$	−3.00000	−	3.00000i	−0.232147	−	0.232147i	−0.813588	+	0.581441i	$0.802489\pi$
$168$			18.0000i			1.38873i
$169$	−13.0000			−1.00000
$170$		0			0
$171$	6.00000			0.458831
$172$		−	12.0000i		−	0.914991i
$173$	15.0000	+	15.0000i	1.14043	+	1.14043i	0.988372	+	0.152057i	$0.0485898\pi$
$173$	15.0000	+	15.0000i	1.14043	+	1.14043i	0.152057	+	0.988372i	$0.451410\pi$
$174$	6.00000			0.454859
$175$		0			0
$176$	3.00000	−	3.00000i	0.226134	−	0.226134i
$177$	6.00000	−	6.00000i	0.450988	−	0.450988i
$178$		−	6.00000i		−	0.449719i
$179$			6.00000i			0.448461i	0.974536	+	0.224231i	$0.0719869\pi$
$179$			6.00000i			0.448461i	−0.974536	+	0.224231i	$0.928013\pi$
$180$		0			0
$181$	−11.0000	+	11.0000i	−0.817624	+	0.817624i	−0.985763	−	0.168140i	$-0.946224\pi$
$181$	−11.0000	+	11.0000i	−0.817624	+	0.817624i	0.168140	+	0.985763i	$0.446224\pi$
$182$		0			0
$183$	−2.00000			−0.147844
$184$	−3.00000	−	3.00000i	−0.221163	−	0.221163i
$185$		0			0
$186$	−2.00000			−0.146647
$187$	−15.0000	+	9.00000i	−1.09691	+	0.658145i
$188$	2.00000			0.145865
$189$		−	24.0000i		−	1.74574i
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	7.00000	+	7.00000i	0.505181	+	0.505181i
$193$	−3.00000	+	3.00000i	−0.215945	+	0.215945i	−0.806787	−	0.590842i	$-0.798796\pi$
$193$	−3.00000	+	3.00000i	−0.215945	+	0.215945i	0.590842	+	0.806787i	$0.298796\pi$
$194$	3.00000	−	3.00000i	0.215387	−	0.215387i
$195$		0			0
$196$		−	11.0000i		−	0.785714i
$197$	−13.0000	+	13.0000i	−0.926212	+	0.926212i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$	−13.0000	+	13.0000i	−0.926212	+	0.926212i	0.0712470	+	0.997459i	$0.477302\pi$
$198$	−3.00000	+	3.00000i	−0.213201	+	0.213201i
$199$	1.00000	+	1.00000i	0.0708881	+	0.0708881i	0.741662	−	0.670774i	$-0.234038\pi$
$199$	1.00000	+	1.00000i	0.0708881	+	0.0708881i	−0.670774	+	0.741662i	$0.734038\pi$
$200$		0			0
$201$	−6.00000	−	6.00000i	−0.423207	−	0.423207i
$202$		−	6.00000i		−	0.422159i
$203$	−18.0000			−1.26335
$204$	−3.00000	−	5.00000i	−0.210042	−	0.350070i
$205$		0			0
$206$			6.00000i			0.418040i
$207$	1.00000	+	1.00000i	0.0695048	+	0.0695048i
$208$		0			0
$209$	−18.0000	−	18.0000i	−1.24509	−	1.24509i
$210$		0			0
$211$	17.0000	−	17.0000i	1.17033	−	1.17033i	0.188197	−	0.982131i	$-0.439736\pi$
$211$	17.0000	−	17.0000i	1.17033	−	1.17033i	0.982131	−	0.188197i	$-0.0602643\pi$
$212$			2.00000i			0.137361i
$213$		−	6.00000i		−	0.411113i
$214$	−9.00000	+	9.00000i	−0.615227	+	0.615227i
$215$		0			0
$216$	−12.0000	−	12.0000i	−0.816497	−	0.816497i
$217$	6.00000			0.407307
$218$	7.00000	+	7.00000i	0.474100	+	0.474100i
$219$		−	6.00000i		−	0.405442i
$220$		0			0
$221$		0			0
$222$	−6.00000			−0.402694
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$	−15.0000	−	15.0000i	−1.00223	−	1.00223i
$225$		0			0
$226$	9.00000	+	9.00000i	0.598671	+	0.598671i
$227$	−15.0000	+	15.0000i	−0.995585	+	0.995585i	−0.999990	−	0.00440533i	$-0.998598\pi$
$227$	−15.0000	+	15.0000i	−0.995585	+	0.995585i	0.00440533	+	0.999990i	$0.498598\pi$
$228$	6.00000	−	6.00000i	0.397360	−	0.397360i
$229$			24.0000i			1.58596i	0.609245	+	0.792982i	$0.291473\pi$
$229$			24.0000i			1.58596i	−0.609245	+	0.792982i	$0.708527\pi$
$230$		0			0
$231$	−18.0000	+	18.0000i	−1.18431	+	1.18431i
$232$	−9.00000	+	9.00000i	−0.590879	+	0.590879i
$233$	−5.00000	−	5.00000i	−0.327561	−	0.327561i	0.524097	−	0.851658i	$-0.324403\pi$
$233$	−5.00000	−	5.00000i	−0.327561	−	0.327561i	−0.851658	+	0.524097i	$0.824403\pi$
$234$		0			0
$235$		0			0
$236$			6.00000i			0.390567i
$237$	−14.0000			−0.909398
$238$	−9.00000	−	15.0000i	−0.583383	−	0.972306i
$239$	24.0000			1.55243			0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000			1.55243			0.776215	+	0.630468i	$0.217137\pi$
$240$		0			0
$241$	−19.0000	−	19.0000i	−1.22390	−	1.22390i	−0.966235	−	0.257663i	$-0.917048\pi$
$241$	−19.0000	−	19.0000i	−1.22390	−	1.22390i	−0.257663	−	0.966235i	$-0.582952\pi$
$242$	7.00000			0.449977
$243$	7.00000	+	7.00000i	0.449050	+	0.449050i
$244$	1.00000	−	1.00000i	0.0640184	−	0.0640184i
$245$		0			0
$246$			6.00000i			0.382546i
$247$		0			0
$248$	3.00000	−	3.00000i	0.190500	−	0.190500i
$249$	−4.00000	+	4.00000i	−0.253490	+	0.253490i
$250$		0			0
$251$	−12.0000			−0.757433			−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000			−0.757433			−0.378717	+	0.925513i	$0.623635\pi$
$252$	3.00000	+	3.00000i	0.188982	+	0.188982i
$253$		−	6.00000i		−	0.377217i
$254$		0			0
$255$		0			0
$256$	−17.0000			−1.06250
$257$			14.0000i			0.873296i	0.899632	+	0.436648i	$0.143834\pi$
$257$			14.0000i			0.873296i	−0.899632	+	0.436648i	$0.856166\pi$
$258$	−12.0000	−	12.0000i	−0.747087	−	0.747087i
$259$	18.0000			1.11847
$260$		0			0
$261$	3.00000	−	3.00000i	0.185695	−	0.185695i
$262$	−3.00000	+	3.00000i	−0.185341	+	0.185341i
$263$			16.0000i			0.986602i	0.869859	+	0.493301i	$0.164210\pi$
$263$			16.0000i			0.986602i	−0.869859	+	0.493301i	$0.835790\pi$
$264$			18.0000i			1.10782i
$265$		0			0
$266$	18.0000	−	18.0000i	1.10365	−	1.10365i
$267$	6.00000	+	6.00000i	0.367194	+	0.367194i
$268$	6.00000			0.366508
$269$	3.00000	+	3.00000i	0.182913	+	0.182913i	0.792624	−	0.609711i	$-0.208714\pi$
$269$	3.00000	+	3.00000i	0.182913	+	0.182913i	−0.609711	+	0.792624i	$0.708714\pi$
$270$		0			0
$271$	24.0000			1.45790			0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000			1.45790			0.728948	+	0.684569i	$0.240010\pi$
$272$	−4.00000	−	1.00000i	−0.242536	−	0.0606339i
$273$		0			0
$274$			4.00000i			0.241649i
$275$		0			0
$276$	2.00000			0.120386
$277$	21.0000	+	21.0000i	1.26177	+	1.26177i	0.950236	+	0.311532i	$0.100842\pi$
$277$	21.0000	+	21.0000i	1.26177	+	1.26177i	0.311532	+	0.950236i	$0.399158\pi$
$278$	7.00000	−	7.00000i	0.419832	−	0.419832i
$279$	−1.00000	+	1.00000i	−0.0598684	+	0.0598684i
$280$		0			0
$281$			24.0000i			1.43172i	0.698244	+	0.715860i	$0.253965\pi$
$281$			24.0000i			1.43172i	−0.698244	+	0.715860i	$0.746035\pi$
$282$	2.00000	−	2.00000i	0.119098	−	0.119098i
$283$	−9.00000	+	9.00000i	−0.534994	+	0.534994i	−0.922055	−	0.387060i	$-0.873491\pi$
$283$	−9.00000	+	9.00000i	−0.534994	+	0.534994i	0.387060	+	0.922055i	$0.373491\pi$
$284$	3.00000	+	3.00000i	0.178017	+	0.178017i
$285$		0			0
$286$		0			0
$287$		−	18.0000i		−	1.06251i
$288$	5.00000			0.294628
$289$	15.0000	+	8.00000i	0.882353	+	0.470588i
$290$		0			0
$291$			6.00000i			0.351726i
$292$	3.00000	+	3.00000i	0.175562	+	0.175562i
$293$	4.00000			0.233682			0.116841	−	0.993151i	$-0.462723\pi$
$293$	4.00000			0.233682			0.116841	+	0.993151i	$0.462723\pi$
$294$	−11.0000	−	11.0000i	−0.641533	−	0.641533i
$295$		0			0
$296$	9.00000	−	9.00000i	0.523114	−	0.523114i
$297$		−	24.0000i		−	1.39262i
$298$			18.0000i			1.04271i
$299$		0			0
$300$		0			0
$301$	36.0000	+	36.0000i	2.07501	+	2.07501i
$302$	10.0000			0.575435
$303$	6.00000	+	6.00000i	0.344691	+	0.344691i
$304$		−	6.00000i		−	0.344124i
$305$		0			0
$306$	4.00000	+	1.00000i	0.228665	+	0.0571662i
$307$	−18.0000			−1.02731			−0.513657	−	0.857996i	$-0.671710\pi$
$307$	−18.0000			−1.02731			−0.513657	+	0.857996i	$0.671710\pi$
$308$		−	18.0000i		−	1.02565i
$309$	−6.00000	−	6.00000i	−0.341328	−	0.341328i
$310$		0			0
$311$	3.00000	+	3.00000i	0.170114	+	0.170114i	0.787030	−	0.616915i	$-0.211618\pi$
$311$	3.00000	+	3.00000i	0.170114	+	0.170114i	−0.616915	+	0.787030i	$0.711618\pi$
$312$		0			0
$313$	−3.00000	+	3.00000i	−0.169570	+	0.169570i	−0.786790	−	0.617220i	$-0.788259\pi$
$313$	−3.00000	+	3.00000i	−0.169570	+	0.169570i	0.617220	+	0.786790i	$0.288259\pi$
$314$			12.0000i			0.677199i
$315$		0			0
$316$	7.00000	−	7.00000i	0.393781	−	0.393781i
$317$	15.0000	−	15.0000i	0.842484	−	0.842484i	−0.146697	−	0.989181i	$-0.546864\pi$
$317$	15.0000	−	15.0000i	0.842484	−	0.842484i	0.989181	+	0.146697i	$0.0468644\pi$
$318$	2.00000	+	2.00000i	0.112154	+	0.112154i
$319$	−18.0000			−1.00781
$320$		0			0
$321$		−	18.0000i		−	1.00466i
$322$	6.00000			0.334367
$323$	−6.00000	+	24.0000i	−0.333849	+	1.33540i
$324$	5.00000			0.277778
$325$		0			0
$326$	−9.00000	−	9.00000i	−0.498464	−	0.498464i
$327$	−14.0000			−0.774202
$328$	−9.00000	−	9.00000i	−0.496942	−	0.496942i
$329$	−6.00000	+	6.00000i	−0.330791	+	0.330791i
$330$		0			0
$331$			18.0000i			0.989369i	0.869072	+	0.494685i	$0.164716\pi$
$331$			18.0000i			0.989369i	−0.869072	+	0.494685i	$0.835284\pi$
$332$		−	4.00000i		−	0.219529i
$333$	−3.00000	+	3.00000i	−0.164399	+	0.164399i
$334$	3.00000	−	3.00000i	0.164153	−	0.164153i
$335$		0			0
$336$	−6.00000			−0.327327
$337$	−15.0000	−	15.0000i	−0.817102	−	0.817102i	0.168585	−	0.985687i	$-0.446080\pi$
$337$	−15.0000	−	15.0000i	−0.817102	−	0.817102i	−0.985687	+	0.168585i	$0.946080\pi$
$338$		−	13.0000i		−	0.707107i
$339$	−18.0000			−0.977626
$340$		0			0
$341$	6.00000			0.324918
$342$			6.00000i			0.324443i
$343$	12.0000	+	12.0000i	0.647939	+	0.647939i
$344$	36.0000			1.94099
$345$		0			0
$346$	−15.0000	+	15.0000i	−0.806405	+	0.806405i
$347$	9.00000	−	9.00000i	0.483145	−	0.483145i	−0.422989	−	0.906135i	$-0.639019\pi$
$347$	9.00000	−	9.00000i	0.483145	−	0.483145i	0.906135	+	0.422989i	$0.139019\pi$
$348$		−	6.00000i		−	0.321634i
$349$			28.0000i			1.49881i	0.662114	+	0.749403i	$0.269659\pi$
$349$			28.0000i			1.49881i	−0.662114	+	0.749403i	$0.730341\pi$
$350$		0			0
$351$		0			0
$352$	−15.0000	−	15.0000i	−0.799503	−	0.799503i
$353$	−16.0000			−0.851594			−0.425797	−	0.904819i	$-0.640006\pi$
$353$	−16.0000			−0.851594			−0.425797	+	0.904819i	$0.640006\pi$
$354$	6.00000	+	6.00000i	0.318896	+	0.318896i
$355$		0			0
$356$	−6.00000			−0.317999
$357$	24.0000	+	6.00000i	1.27021	+	0.317554i
$358$	−6.00000			−0.317110
$359$		−	30.0000i		−	1.58334i	−0.610949	−	0.791670i	$-0.709212\pi$
$359$		−	30.0000i		−	1.58334i	0.610949	−	0.791670i	$-0.290788\pi$
$360$		0			0
$361$	−17.0000			−0.894737
$362$	−11.0000	−	11.0000i	−0.578147	−	0.578147i
$363$	−7.00000	+	7.00000i	−0.367405	+	0.367405i
$364$		0			0
$365$		0			0
$366$		−	2.00000i		−	0.104542i
$367$	−3.00000	+	3.00000i	−0.156599	+	0.156599i	−0.781058	−	0.624459i	$-0.785320\pi$
$367$	−3.00000	+	3.00000i	−0.156599	+	0.156599i	0.624459	+	0.781058i	$0.285320\pi$
$368$	1.00000	−	1.00000i	0.0521286	−	0.0521286i
$369$	3.00000	+	3.00000i	0.156174	+	0.156174i
$370$		0			0
$371$	−6.00000	−	6.00000i	−0.311504	−	0.311504i
$372$			2.00000i			0.103695i
$373$	24.0000			1.24267			0.621336	−	0.783544i	$-0.286590\pi$
$373$	24.0000			1.24267			0.621336	+	0.783544i	$0.286590\pi$
$374$	−9.00000	−	15.0000i	−0.465379	−	0.775632i
$375$		0			0
$376$			6.00000i			0.309426i
$377$		0			0
$378$	24.0000			1.23443
$379$	−11.0000	−	11.0000i	−0.565032	−	0.565032i	0.365701	−	0.930733i	$-0.380829\pi$
$379$	−11.0000	−	11.0000i	−0.565032	−	0.565032i	−0.930733	+	0.365701i	$0.880829\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	16.0000i		−	0.817562i	−0.912633	−	0.408781i	$-0.865954\pi$
$383$		−	16.0000i		−	0.817562i	0.912633	−	0.408781i	$-0.134046\pi$
$384$	3.00000	−	3.00000i	0.153093	−	0.153093i
$385$		0			0
$386$	−3.00000	−	3.00000i	−0.152696	−	0.152696i
$387$	−12.0000			−0.609994
$388$	−3.00000	−	3.00000i	−0.152302	−	0.152302i
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$	−5.00000	+	3.00000i	−0.252861	+	0.151717i
$392$	33.0000			1.66675
$393$		−	6.00000i		−	0.302660i
$394$	−13.0000	−	13.0000i	−0.654931	−	0.654931i
$395$		0			0
$396$	3.00000	+	3.00000i	0.150756	+	0.150756i
$397$	27.0000	−	27.0000i	1.35509	−	1.35509i	0.475229	−	0.879862i	$-0.342365\pi$
$397$	27.0000	−	27.0000i	1.35509	−	1.35509i	0.879862	−	0.475229i	$-0.157635\pi$
$398$	−1.00000	+	1.00000i	−0.0501255	+	0.0501255i
$399$			36.0000i			1.80225i
$400$		0			0
$401$	9.00000	−	9.00000i	0.449439	−	0.449439i	−0.445729	−	0.895168i	$-0.647056\pi$
$401$	9.00000	−	9.00000i	0.449439	−	0.449439i	0.895168	+	0.445729i	$0.147056\pi$
$402$	6.00000	−	6.00000i	0.299253	−	0.299253i
$403$		0			0
$404$	−6.00000			−0.298511
$405$		0			0
$406$		−	18.0000i		−	0.893325i
$407$	18.0000			0.892227
$408$	15.0000	−	9.00000i	0.742611	−	0.445566i
$409$	30.0000			1.48340			0.741702	−	0.670729i	$-0.234019\pi$
$409$	30.0000			1.48340			0.741702	+	0.670729i	$0.234019\pi$
$410$		0			0
$411$	−4.00000	−	4.00000i	−0.197305	−	0.197305i
$412$	6.00000			0.295599
$413$	−18.0000	−	18.0000i	−0.885722	−	0.885722i
$414$	−1.00000	+	1.00000i	−0.0491473	+	0.0491473i
$415$		0			0
$416$		0			0
$417$			14.0000i			0.685583i
$418$	18.0000	−	18.0000i	0.880409	−	0.880409i
$419$	15.0000	−	15.0000i	0.732798	−	0.732798i	−0.238375	−	0.971173i	$-0.576615\pi$
$419$	15.0000	−	15.0000i	0.732798	−	0.732798i	0.971173	+	0.238375i	$0.0766148\pi$
$420$		0			0
$421$	−6.00000			−0.292422			−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000			−0.292422			−0.146211	+	0.989253i	$0.546708\pi$
$422$	17.0000	+	17.0000i	0.827547	+	0.827547i
$423$		−	2.00000i		−	0.0972433i
$424$	−6.00000			−0.291386
$425$		0			0
$426$	6.00000			0.290701
$427$			6.00000i			0.290360i
$428$	9.00000	+	9.00000i	0.435031	+	0.435031i
$429$		0			0
$430$		0			0
$431$	9.00000	−	9.00000i	0.433515	−	0.433515i	−0.456307	−	0.889822i	$-0.650828\pi$
$431$	9.00000	−	9.00000i	0.433515	−	0.433515i	0.889822	+	0.456307i	$0.150828\pi$
$432$	4.00000	−	4.00000i	0.192450	−	0.192450i
$433$		−	30.0000i		−	1.44171i	−0.693087	−	0.720854i	$-0.743750\pi$
$433$		−	30.0000i		−	1.44171i	0.693087	−	0.720854i	$-0.256250\pi$
$434$			6.00000i			0.288009i
$435$		0			0
$436$	7.00000	−	7.00000i	0.335239	−	0.335239i
$437$	−6.00000	−	6.00000i	−0.287019	−	0.287019i
$438$	6.00000			0.286691
$439$	−19.0000	−	19.0000i	−0.906821	−	0.906821i	0.0891938	−	0.996014i	$-0.471571\pi$
$439$	−19.0000	−	19.0000i	−0.906821	−	0.906821i	−0.996014	+	0.0891938i	$0.971571\pi$
$440$		0			0
$441$	−11.0000			−0.523810
$442$		0			0
$443$	−22.0000			−1.04525			−0.522626	−	0.852562i	$-0.675047\pi$
$443$	−22.0000			−1.04525			−0.522626	+	0.852562i	$0.675047\pi$
$444$			6.00000i			0.284747i
$445$		0			0
$446$		0			0
$447$	−18.0000	−	18.0000i	−0.851371	−	0.851371i
$448$	21.0000	−	21.0000i	0.992157	−	0.992157i
$449$	15.0000	−	15.0000i	0.707894	−	0.707894i	−0.258198	−	0.966092i	$-0.583129\pi$
$449$	15.0000	−	15.0000i	0.707894	−	0.707894i	0.966092	+	0.258198i	$0.0831288\pi$
$450$		0			0
$451$		−	18.0000i		−	0.847587i
$452$	9.00000	−	9.00000i	0.423324	−	0.423324i
$453$	−10.0000	+	10.0000i	−0.469841	+	0.469841i
$454$	−15.0000	−	15.0000i	−0.703985	−	0.703985i
$455$		0			0
$456$	18.0000	+	18.0000i	0.842927	+	0.842927i
$457$			6.00000i			0.280668i	0.990104	+	0.140334i	$0.0448177\pi$
$457$			6.00000i			0.280668i	−0.990104	+	0.140334i	$0.955182\pi$
$458$	−24.0000			−1.12145
$459$	−20.0000	+	12.0000i	−0.933520	+	0.560112i
$460$		0			0
$461$		−	12.0000i		−	0.558896i	−0.960161	−	0.279448i	$-0.909849\pi$
$461$		−	12.0000i		−	0.558896i	0.960161	−	0.279448i	$-0.0901514\pi$
$462$	−18.0000	−	18.0000i	−0.837436	−	0.837436i
$463$	−18.0000			−0.836531			−0.418265	−	0.908325i	$-0.637362\pi$
$463$	−18.0000			−0.836531			−0.418265	+	0.908325i	$0.637362\pi$
$464$	−3.00000	−	3.00000i	−0.139272	−	0.139272i
$465$		0			0
$466$	5.00000	−	5.00000i	0.231621	−	0.231621i
$467$			28.0000i			1.29569i	0.761774	+	0.647843i	$0.224329\pi$
$467$			28.0000i			1.29569i	−0.761774	+	0.647843i	$0.775671\pi$
$468$		0			0
$469$	−18.0000	+	18.0000i	−0.831163	+	0.831163i
$470$		0			0
$471$	−12.0000	−	12.0000i	−0.552931	−	0.552931i
$472$	−18.0000			−0.828517
$473$	36.0000	+	36.0000i	1.65528	+	1.65528i
$474$		−	14.0000i		−	0.643041i
$475$		0			0
$476$	−15.0000	+	9.00000i	−0.687524	+	0.412514i
$477$	2.00000			0.0915737
$478$			24.0000i			1.09773i
$479$	9.00000	+	9.00000i	0.411220	+	0.411220i	0.882164	−	0.470943i	$-0.156086\pi$
$479$	9.00000	+	9.00000i	0.411220	+	0.411220i	−0.470943	+	0.882164i	$0.656086\pi$
$480$		0			0
$481$		0			0
$482$	19.0000	−	19.0000i	0.865426	−	0.865426i
$483$	−6.00000	+	6.00000i	−0.273009	+	0.273009i
$484$		−	7.00000i		−	0.318182i
$485$		0			0
$486$	−7.00000	+	7.00000i	−0.317526	+	0.317526i
$487$	−3.00000	+	3.00000i	−0.135943	+	0.135943i	−0.771804	−	0.635861i	$-0.780645\pi$
$487$	−3.00000	+	3.00000i	−0.135943	+	0.135943i	0.635861	+	0.771804i	$0.280645\pi$
$488$	3.00000	+	3.00000i	0.135804	+	0.135804i
$489$	18.0000			0.813988
$490$		0			0
$491$		−	6.00000i		−	0.270776i	−0.990793	−	0.135388i	$-0.956772\pi$
$491$		−	6.00000i		−	0.270776i	0.990793	−	0.135388i	$-0.0432281\pi$
$492$	6.00000			0.270501
$493$	9.00000	+	15.0000i	0.405340	+	0.675566i
$494$		0			0
$495$		0			0
$496$	1.00000	+	1.00000i	0.0449013	+	0.0449013i
$497$	−18.0000			−0.807410
$498$	−4.00000	−	4.00000i	−0.179244	−	0.179244i
$499$	23.0000	−	23.0000i	1.02962	−	1.02962i	0.0300737	−	0.999548i	$-0.490426\pi$
$499$	23.0000	−	23.0000i	1.02962	−	1.02962i	0.999548	−	0.0300737i	$-0.00957421\pi$
$500$		0			0
$501$			6.00000i			0.268060i
$502$		−	12.0000i		−	0.535586i
$503$	3.00000	−	3.00000i	0.133763	−	0.133763i	−0.637055	−	0.770818i	$-0.719848\pi$
$503$	3.00000	−	3.00000i	0.133763	−	0.133763i	0.770818	+	0.637055i	$0.219848\pi$
$504$	−9.00000	+	9.00000i	−0.400892	+	0.400892i
$505$		0			0
$506$	6.00000			0.266733
$507$	13.0000	+	13.0000i	0.577350	+	0.577350i
$508$		0			0
$509$	−6.00000			−0.265945			−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000			−0.265945			−0.132973	+	0.991120i	$0.542452\pi$
$510$		0			0
$511$	−18.0000			−0.796273
$512$		−	11.0000i		−	0.486136i
$513$	−24.0000	−	24.0000i	−1.05963	−	1.05963i
$514$	−14.0000			−0.617514
$515$		0			0
$516$	−12.0000	+	12.0000i	−0.528271	+	0.528271i
$517$	−6.00000	+	6.00000i	−0.263880	+	0.263880i
$518$			18.0000i			0.790875i
$519$		−	30.0000i		−	1.31685i
$520$		0			0
$521$	9.00000	−	9.00000i	0.394297	−	0.394297i	−0.481919	−	0.876216i	$-0.660060\pi$
$521$	9.00000	−	9.00000i	0.394297	−	0.394297i	0.876216	+	0.481919i	$0.160060\pi$
$522$	3.00000	+	3.00000i	0.131306	+	0.131306i
$523$	−6.00000			−0.262362			−0.131181	−	0.991358i	$-0.541877\pi$
$523$	−6.00000			−0.262362			−0.131181	+	0.991358i	$0.541877\pi$
$524$	3.00000	+	3.00000i	0.131056	+	0.131056i
$525$		0			0
$526$	−16.0000			−0.697633
$527$	−3.00000	−	5.00000i	−0.130682	−	0.217803i
$528$	−6.00000			−0.261116
$529$			21.0000i			0.913043i
$530$		0			0
$531$	6.00000			0.260378
$532$	−18.0000	−	18.0000i	−0.780399	−	0.780399i
$533$		0			0
$534$	−6.00000	+	6.00000i	−0.259645	+	0.259645i
$535$		0			0
$536$			18.0000i			0.777482i
$537$	6.00000	−	6.00000i	0.258919	−	0.258919i
$538$	−3.00000	+	3.00000i	−0.129339	+	0.129339i
$539$	33.0000	+	33.0000i	1.42141	+	1.42141i
$540$		0			0
$541$	1.00000	+	1.00000i	0.0429934	+	0.0429934i	0.728277	−	0.685283i	$-0.240322\pi$
$541$	1.00000	+	1.00000i	0.0429934	+	0.0429934i	−0.685283	+	0.728277i	$0.740322\pi$
$542$			24.0000i			1.03089i
$543$	22.0000			0.944110
$544$	−5.00000	+	20.0000i	−0.214373	+	0.857493i
$545$		0			0
$546$		0			0
$547$	9.00000	+	9.00000i	0.384812	+	0.384812i	0.872832	−	0.488020i	$-0.162281\pi$
$547$	9.00000	+	9.00000i	0.384812	+	0.384812i	−0.488020	+	0.872832i	$0.662281\pi$
$548$	4.00000			0.170872
$549$	−1.00000	−	1.00000i	−0.0426790	−	0.0426790i
$550$		0			0
$551$	−18.0000	+	18.0000i	−0.766826	+	0.766826i
$552$			6.00000i			0.255377i
$553$			42.0000i			1.78602i
$554$	−21.0000	+	21.0000i	−0.892205	+	0.892205i
$555$		0			0
$556$	−7.00000	−	7.00000i	−0.296866	−	0.296866i
$557$	−32.0000			−1.35588			−0.677942	−	0.735116i	$-0.737128\pi$
$557$	−32.0000			−1.35588			−0.677942	+	0.735116i	$0.737128\pi$
$558$	−1.00000	−	1.00000i	−0.0423334	−	0.0423334i
$559$		0			0
$560$		0			0
$561$	24.0000	+	6.00000i	1.01328	+	0.253320i
$562$	−24.0000			−1.01238
$563$		−	28.0000i		−	1.18006i	−0.807382	−	0.590030i	$-0.799116\pi$
$563$		−	28.0000i		−	1.18006i	0.807382	−	0.590030i	$-0.200884\pi$
$564$	−2.00000	−	2.00000i	−0.0842152	−	0.0842152i
$565$		0			0
$566$	−9.00000	−	9.00000i	−0.378298	−	0.378298i
$567$	−15.0000	+	15.0000i	−0.629941	+	0.629941i
$568$	−9.00000	+	9.00000i	−0.377632	+	0.377632i
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	−23.0000	+	23.0000i	−0.962520	+	0.962520i	−0.999323	−	0.0368025i	$-0.988283\pi$
$571$	−23.0000	+	23.0000i	−0.962520	+	0.962520i	0.0368025	+	0.999323i	$0.488283\pi$
$572$		0			0
$573$		0			0
$574$	18.0000			0.751305
$575$		0			0
$576$			7.00000i			0.291667i
$577$	36.0000			1.49870			0.749350	−	0.662174i	$-0.230366\pi$
$577$	36.0000			1.49870			0.749350	+	0.662174i	$0.230366\pi$
$578$	−8.00000	+	15.0000i	−0.332756	+	0.623918i
$579$	6.00000			0.249351
$580$		0			0
$581$	12.0000	+	12.0000i	0.497844	+	0.497844i
$582$	−6.00000			−0.248708
$583$	−6.00000	−	6.00000i	−0.248495	−	0.248495i
$584$	−9.00000	+	9.00000i	−0.372423	+	0.372423i
$585$		0			0
$586$			4.00000i			0.165238i
$587$		−	4.00000i		−	0.165098i	−0.996587	−	0.0825488i	$-0.973694\pi$
$587$		−	4.00000i		−	0.165098i	0.996587	−	0.0825488i	$-0.0263060\pi$
$588$	−11.0000	+	11.0000i	−0.453632	+	0.453632i
$589$	6.00000	−	6.00000i	0.247226	−	0.247226i
$590$		0			0
$591$	26.0000			1.06950
$592$	3.00000	+	3.00000i	0.123299	+	0.123299i
$593$		−	2.00000i		−	0.0821302i	−0.999156	−	0.0410651i	$-0.986925\pi$
$593$		−	2.00000i		−	0.0821302i	0.999156	−	0.0410651i	$-0.0130751\pi$
$594$	24.0000			0.984732
$595$		0			0
$596$	18.0000			0.737309
$597$		−	2.00000i		−	0.0818546i
$598$		0			0
$599$	−24.0000			−0.980613			−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000			−0.980613			−0.490307	+	0.871550i	$0.663115\pi$
$600$		0			0
$601$	29.0000	−	29.0000i	1.18293	−	1.18293i	0.203954	−	0.978980i	$-0.434621\pi$
$601$	29.0000	−	29.0000i	1.18293	−	1.18293i	0.978980	−	0.203954i	$-0.0653794\pi$
$602$	−36.0000	+	36.0000i	−1.46725	+	1.46725i
$603$		−	6.00000i		−	0.244339i
$604$		−	10.0000i		−	0.406894i
$605$		0			0
$606$	−6.00000	+	6.00000i	−0.243733	+	0.243733i
$607$	33.0000	+	33.0000i	1.33943	+	1.33943i	0.896612	+	0.442816i	$0.146021\pi$
$607$	33.0000	+	33.0000i	1.33943	+	1.33943i	0.442816	+	0.896612i	$0.353979\pi$
$608$	−30.0000			−1.21666
$609$	18.0000	+	18.0000i	0.729397	+	0.729397i
$610$		0			0
$611$		0			0
$612$	1.00000	−	4.00000i	0.0404226	−	0.161690i
$613$	36.0000			1.45403			0.727013	−	0.686624i	$-0.240908\pi$
$613$	36.0000			1.45403			0.727013	+	0.686624i	$0.240908\pi$
$614$		−	18.0000i		−	0.726421i
$615$		0			0
$616$	54.0000			2.17572
$617$	−15.0000	−	15.0000i	−0.603877	−	0.603877i	0.337462	−	0.941339i	$-0.390432\pi$
$617$	−15.0000	−	15.0000i	−0.603877	−	0.603877i	−0.941339	+	0.337462i	$0.890432\pi$
$618$	6.00000	−	6.00000i	0.241355	−	0.241355i
$619$	−13.0000	+	13.0000i	−0.522514	+	0.522514i	−0.918330	−	0.395816i	$-0.870462\pi$
$619$	−13.0000	+	13.0000i	−0.522514	+	0.522514i	0.395816	+	0.918330i	$0.370462\pi$
$620$		0			0
$621$		−	8.00000i		−	0.321029i
$622$	−3.00000	+	3.00000i	−0.120289	+	0.120289i
$623$	18.0000	−	18.0000i	0.721155	−	0.721155i
$624$		0			0
$625$		0			0
$626$	−3.00000	−	3.00000i	−0.119904	−	0.119904i
$627$			36.0000i			1.43770i
$628$	12.0000			0.478852
$629$	−9.00000	−	15.0000i	−0.358854	−	0.598089i
$630$		0			0
$631$		−	34.0000i		−	1.35352i	−0.736204	−	0.676759i	$-0.763384\pi$
$631$		−	34.0000i		−	1.35352i	0.736204	−	0.676759i	$-0.236616\pi$
$632$	21.0000	+	21.0000i	0.835335	+	0.835335i
$633$	−34.0000			−1.35138
$634$	15.0000	+	15.0000i	0.595726	+	0.595726i
$635$		0			0
$636$	2.00000	−	2.00000i	0.0793052	−	0.0793052i
$637$		0			0
$638$		−	18.0000i		−	0.712627i
$639$	3.00000	−	3.00000i	0.118678	−	0.118678i
$640$		0			0
$641$	9.00000	+	9.00000i	0.355479	+	0.355479i	0.862143	−	0.506665i	$-0.169122\pi$
$641$	9.00000	+	9.00000i	0.355479	+	0.355479i	−0.506665	+	0.862143i	$0.669122\pi$
$642$	18.0000			0.710403
$643$	−9.00000	−	9.00000i	−0.354925	−	0.354925i	0.507013	−	0.861938i	$-0.330750\pi$
$643$	−9.00000	−	9.00000i	−0.354925	−	0.354925i	−0.861938	+	0.507013i	$0.830750\pi$
$644$		−	6.00000i		−	0.236433i
$645$		0			0
$646$	−24.0000	−	6.00000i	−0.944267	−	0.236067i
$647$	−46.0000			−1.80845			−0.904223	−	0.427060i	$-0.859549\pi$
$647$	−46.0000			−1.80845			−0.904223	+	0.427060i	$0.859549\pi$
$648$			15.0000i			0.589256i
$649$	−18.0000	−	18.0000i	−0.706562	−	0.706562i
$650$		0			0
$651$	−6.00000	−	6.00000i	−0.235159	−	0.235159i
$652$	−9.00000	+	9.00000i	−0.352467	+	0.352467i
$653$	21.0000	−	21.0000i	0.821794	−	0.821794i	−0.164572	−	0.986365i	$-0.552624\pi$
$653$	21.0000	−	21.0000i	0.821794	−	0.821794i	0.986365	+	0.164572i	$0.0526242\pi$
$654$		−	14.0000i		−	0.547443i
$655$		0			0
$656$	3.00000	−	3.00000i	0.117130	−	0.117130i
$657$	3.00000	−	3.00000i	0.117041	−	0.117041i
$658$	−6.00000	−	6.00000i	−0.233904	−	0.233904i
$659$	−36.0000			−1.40236			−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000			−1.40236			−0.701180	+	0.712984i	$0.747343\pi$
$660$		0			0
$661$			36.0000i			1.40024i	0.714026	+	0.700119i	$0.246870\pi$
$661$			36.0000i			1.40024i	−0.714026	+	0.700119i	$0.753130\pi$
$662$	−18.0000			−0.699590
$663$		0			0
$664$	12.0000			0.465690
$665$		0			0
$666$	−3.00000	−	3.00000i	−0.116248	−	0.116248i
$667$	−6.00000			−0.232321
$668$	−3.00000	−	3.00000i	−0.116073	−	0.116073i
$669$		0			0
$670$		0			0
$671$			6.00000i			0.231627i
$672$			30.0000i			1.15728i
$673$	−15.0000	+	15.0000i	−0.578208	+	0.578208i	−0.934409	−	0.356202i	$-0.884072\pi$
$673$	−15.0000	+	15.0000i	−0.578208	+	0.578208i	0.356202	+	0.934409i	$0.384072\pi$
$674$	15.0000	−	15.0000i	0.577778	−	0.577778i
$675$		0			0
$676$	−13.0000			−0.500000
$677$	−11.0000	−	11.0000i	−0.422764	−	0.422764i	0.463390	−	0.886154i	$-0.346633\pi$
$677$	−11.0000	−	11.0000i	−0.422764	−	0.422764i	−0.886154	+	0.463390i	$0.846633\pi$
$678$		−	18.0000i		−	0.691286i
$679$	18.0000			0.690777
$680$		0			0
$681$	30.0000			1.14960
$682$			6.00000i			0.229752i
$683$	15.0000	+	15.0000i	0.573959	+	0.573959i	0.933232	−	0.359273i	$-0.116975\pi$
$683$	15.0000	+	15.0000i	0.573959	+	0.573959i	−0.359273	+	0.933232i	$0.616975\pi$
$684$	6.00000			0.229416
$685$		0			0
$686$	−12.0000	+	12.0000i	−0.458162	+	0.458162i
$687$	24.0000	−	24.0000i	0.915657	−	0.915657i
$688$			12.0000i			0.457496i
$689$		0			0
$690$		0			0
$691$	−19.0000	+	19.0000i	−0.722794	+	0.722794i	−0.969173	−	0.246379i	$-0.920759\pi$
$691$	−19.0000	+	19.0000i	−0.722794	+	0.722794i	0.246379	+	0.969173i	$0.420759\pi$
$692$	15.0000	+	15.0000i	0.570214	+	0.570214i
$693$	−18.0000			−0.683763
$694$	9.00000	+	9.00000i	0.341635	+	0.341635i
$695$		0			0
$696$	18.0000			0.682288
$697$	−15.0000	+	9.00000i	−0.568166	+	0.340899i
$698$	−28.0000			−1.05982
$699$			10.0000i			0.378235i
$700$		0			0
$701$	−30.0000			−1.13308			−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000			−1.13308			−0.566542	+	0.824033i	$0.691719\pi$
$702$		0			0
$703$	18.0000	−	18.0000i	0.678883	−	0.678883i
$704$	21.0000	−	21.0000i	0.791467	−	0.791467i
$705$		0			0
$706$		−	16.0000i		−	0.602168i
$707$	18.0000	−	18.0000i	0.676960	−	0.676960i
$708$	6.00000	−	6.00000i	0.225494	−	0.225494i
$709$	−5.00000	−	5.00000i	−0.187779	−	0.187779i	0.606956	−	0.794735i	$-0.292390\pi$
$709$	−5.00000	−	5.00000i	−0.187779	−	0.187779i	−0.794735	+	0.606956i	$0.792390\pi$
$710$		0			0
$711$	−7.00000	−	7.00000i	−0.262521	−	0.262521i
$712$		−	18.0000i		−	0.674579i
$713$	2.00000			0.0749006
$714$	−6.00000	+	24.0000i	−0.224544	+	0.898177i
$715$		0			0
$716$			6.00000i			0.224231i
$717$	−24.0000	−	24.0000i	−0.896296	−	0.896296i
$718$	30.0000			1.11959
$719$	21.0000	+	21.0000i	0.783168	+	0.783168i	0.980364	−	0.197196i	$-0.0631836\pi$
$719$	21.0000	+	21.0000i	0.783168	+	0.783168i	−0.197196	+	0.980364i	$0.563184\pi$
$720$		0			0
$721$	−18.0000	+	18.0000i	−0.670355	+	0.670355i
$722$		−	17.0000i		−	0.632674i
$723$			38.0000i			1.41324i
$724$	−11.0000	+	11.0000i	−0.408812	+	0.408812i
$725$		0			0
$726$	−7.00000	−	7.00000i	−0.259794	−	0.259794i
$727$	18.0000			0.667583			0.333792	−	0.942647i	$-0.391672\pi$
$727$	18.0000			0.667583			0.333792	+	0.942647i	$0.391672\pi$
$728$		0			0
$729$		−	29.0000i		−	1.07407i
$730$		0			0
$731$	12.0000	−	48.0000i	0.443836	−	1.77534i
$732$	−2.00000			−0.0739221
$733$			42.0000i			1.55131i	0.631160	+	0.775653i	$0.282579\pi$
$733$			42.0000i			1.55131i	−0.631160	+	0.775653i	$0.717421\pi$
$734$	−3.00000	−	3.00000i	−0.110732	−	0.110732i
$735$		0			0
$736$	−5.00000	−	5.00000i	−0.184302	−	0.184302i
$737$	−18.0000	+	18.0000i	−0.663039	+	0.663039i
$738$	−3.00000	+	3.00000i	−0.110432	+	0.110432i
$739$		−	34.0000i		−	1.25071i	−0.780340	−	0.625355i	$-0.784954\pi$
$739$		−	34.0000i		−	1.25071i	0.780340	−	0.625355i	$-0.215046\pi$
$740$		0			0
$741$		0			0
$742$	6.00000	−	6.00000i	0.220267	−	0.220267i
$743$	−9.00000	−	9.00000i	−0.330178	−	0.330178i	0.522476	−	0.852654i	$-0.325008\pi$
$743$	−9.00000	−	9.00000i	−0.330178	−	0.330178i	−0.852654	+	0.522476i	$0.825008\pi$
$744$	−6.00000			−0.219971
$745$		0			0
$746$			24.0000i			0.878702i
$747$	−4.00000			−0.146352
$748$	−15.0000	+	9.00000i	−0.548454	+	0.329073i
$749$	−54.0000			−1.97312
$750$		0			0
$751$	19.0000	+	19.0000i	0.693320	+	0.693320i	0.962961	−	0.269641i	$-0.0869050\pi$
$751$	19.0000	+	19.0000i	0.693320	+	0.693320i	−0.269641	+	0.962961i	$0.586905\pi$
$752$	−2.00000			−0.0729325
$753$	12.0000	+	12.0000i	0.437304	+	0.437304i
$754$		0			0
$755$		0			0
$756$		−	24.0000i		−	0.872872i
$757$			42.0000i			1.52652i	0.646094	+	0.763258i	$0.276401\pi$
$757$			42.0000i			1.52652i	−0.646094	+	0.763258i	$0.723599\pi$
$758$	11.0000	−	11.0000i	0.399538	−	0.399538i
$759$	−6.00000	+	6.00000i	−0.217786	+	0.217786i
$760$		0			0
$761$	−42.0000			−1.52250			−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000			−1.52250			−0.761249	+	0.648459i	$0.775414\pi$
$762$		0			0
$763$			42.0000i			1.52050i
$764$		0			0
$765$		0			0
$766$	16.0000			0.578103
$767$		0			0
$768$	17.0000	+	17.0000i	0.613435	+	0.613435i
$769$	−14.0000			−0.504853			−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000			−0.504853			−0.252426	+	0.967616i	$0.581229\pi$
$770$		0			0
$771$	14.0000	−	14.0000i	0.504198	−	0.504198i
$772$	−3.00000	+	3.00000i	−0.107972	+	0.107972i
$773$		−	26.0000i		−	0.935155i	−0.883952	−	0.467578i	$-0.845127\pi$
$773$		−	26.0000i		−	0.935155i	0.883952	−	0.467578i	$-0.154873\pi$
$774$		−	12.0000i		−	0.431331i
$775$		0			0
$776$	9.00000	−	9.00000i	0.323081	−	0.323081i
$777$	−18.0000	−	18.0000i	−0.645746	−	0.645746i
$778$		0			0
$779$	−18.0000	−	18.0000i	−0.644917	−	0.644917i
$780$		0			0
$781$	−18.0000			−0.644091
$782$	−3.00000	−	5.00000i	−0.107280	−	0.178800i
$783$	−24.0000			−0.857690
$784$			11.0000i			0.392857i
$785$		0			0
$786$	6.00000			0.214013
$787$	−27.0000	−	27.0000i	−0.962446	−	0.962446i	0.0368739	−	0.999320i	$-0.488260\pi$
$787$	−27.0000	−	27.0000i	−0.962446	−	0.962446i	−0.999320	+	0.0368739i	$0.988260\pi$
$788$	−13.0000	+	13.0000i	−0.463106	+	0.463106i
$789$	16.0000	−	16.0000i	0.569615	−	0.569615i
$790$		0			0
$791$			54.0000i			1.92002i
$792$	−9.00000	+	9.00000i	−0.319801	+	0.319801i
$793$		0			0
$794$	27.0000	+	27.0000i	0.958194	+	0.958194i
$795$		0			0
$796$	1.00000	+	1.00000i	0.0354441	+	0.0354441i
$797$		−	34.0000i		−	1.20434i	−0.798367	−	0.602171i	$-0.794303\pi$
$797$		−	34.0000i		−	1.20434i	0.798367	−	0.602171i	$-0.205697\pi$
$798$	−36.0000			−1.27439
$799$	8.00000	+	2.00000i	0.283020	+	0.0707549i
$800$		0			0
$801$			6.00000i			0.212000i
$802$	9.00000	+	9.00000i	0.317801	+	0.317801i
$803$	−18.0000			−0.635206
$804$	−6.00000	−	6.00000i	−0.211604	−	0.211604i
$805$		0			0
$806$		0			0
$807$		−	6.00000i		−	0.211210i
$808$		−	18.0000i		−	0.633238i
$809$	3.00000	−	3.00000i	0.105474	−	0.105474i	−0.652400	−	0.757875i	$-0.726238\pi$
$809$	3.00000	−	3.00000i	0.105474	−	0.105474i	0.757875	+	0.652400i	$0.226238\pi$
$810$		0			0
$811$	−1.00000	−	1.00000i	−0.0351147	−	0.0351147i	0.689331	−	0.724446i	$-0.257904\pi$
$811$	−1.00000	−	1.00000i	−0.0351147	−	0.0351147i	−0.724446	+	0.689331i	$0.757904\pi$
$812$	−18.0000			−0.631676
$813$	−24.0000	−	24.0000i	−0.841717	−	0.841717i
$814$			18.0000i			0.630900i
$815$		0			0
$816$	3.00000	+	5.00000i	0.105021	+	0.175035i
$817$	72.0000			2.51896
$818$			30.0000i			1.04893i
$819$		0			0
$820$		0			0
$821$	−15.0000	−	15.0000i	−0.523504	−	0.523504i	0.395124	−	0.918628i	$-0.370702\pi$
$821$	−15.0000	−	15.0000i	−0.523504	−	0.523504i	−0.918628	+	0.395124i	$0.870702\pi$
$822$	4.00000	−	4.00000i	0.139516	−	0.139516i
$823$	39.0000	−	39.0000i	1.35945	−	1.35945i	0.484866	−	0.874588i	$-0.338868\pi$
$823$	39.0000	−	39.0000i	1.35945	−	1.35945i	0.874588	−	0.484866i	$-0.161132\pi$
$824$			18.0000i			0.627060i
$825$		0			0
$826$	18.0000	−	18.0000i	0.626300	−	0.626300i
$827$	13.0000	−	13.0000i	0.452054	−	0.452054i	−0.443982	−	0.896036i	$-0.646434\pi$
$827$	13.0000	−	13.0000i	0.452054	−	0.452054i	0.896036	+	0.443982i	$0.146434\pi$
$828$	1.00000	+	1.00000i	0.0347524	+	0.0347524i
$829$	2.00000			0.0694629			0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000			0.0694629			0.0347314	+	0.999397i	$0.488942\pi$
$830$		0			0
$831$		−	42.0000i		−	1.45696i
$832$		0			0
$833$	11.0000	−	44.0000i	0.381127	−	1.52451i
$834$	−14.0000			−0.484780
$835$		0			0
$836$	−18.0000	−	18.0000i	−0.622543	−	0.622543i
$837$	8.00000			0.276520
$838$	15.0000	+	15.0000i	0.518166	+	0.518166i
$839$	3.00000	−	3.00000i	0.103572	−	0.103572i	−0.653422	−	0.756994i	$-0.726667\pi$
$839$	3.00000	−	3.00000i	0.103572	−	0.103572i	0.756994	+	0.653422i	$0.226667\pi$
$840$		0			0
$841$		−	11.0000i		−	0.379310i
$842$		−	6.00000i		−	0.206774i
$843$	24.0000	−	24.0000i	0.826604	−	0.826604i
$844$	17.0000	−	17.0000i	0.585164	−	0.585164i
$845$		0			0
$846$	2.00000			0.0687614
$847$	21.0000	+	21.0000i	0.721569	+	0.721569i
$848$		−	2.00000i		−	0.0686803i
$849$	18.0000			0.617758
$850$		0			0
$851$	6.00000			0.205677
$852$		−	6.00000i		−	0.205557i
$853$	3.00000	+	3.00000i	0.102718	+	0.102718i	0.756598	−	0.653880i	$-0.226860\pi$
$853$	3.00000	+	3.00000i	0.102718	+	0.102718i	−0.653880	+	0.756598i	$0.726860\pi$
$854$	−6.00000			−0.205316
$855$		0			0
$856$	−27.0000	+	27.0000i	−0.922841	+	0.922841i
$857$	11.0000	−	11.0000i	0.375753	−	0.375753i	−0.493814	−	0.869567i	$-0.664398\pi$
$857$	11.0000	−	11.0000i	0.375753	−	0.375753i	0.869567	+	0.493814i	$0.164398\pi$
$858$		0			0
$859$			30.0000i			1.02359i	0.859109	+	0.511793i	$0.171019\pi$
$859$			30.0000i			1.02359i	−0.859109	+	0.511793i	$0.828981\pi$
$860$		0			0
$861$	−18.0000	+	18.0000i	−0.613438	+	0.613438i
$862$	9.00000	+	9.00000i	0.306541	+	0.306541i
$863$	38.0000			1.29354			0.646768	−	0.762687i	$-0.276120\pi$
$863$	38.0000			1.29354			0.646768	+	0.762687i	$0.276120\pi$
$864$	−20.0000	−	20.0000i	−0.680414	−	0.680414i
$865$		0			0
$866$	30.0000			1.01944
$867$	−7.00000	−	23.0000i	−0.237732	−	0.781121i
$868$	6.00000			0.203653
$869$			42.0000i			1.42475i
$870$		0			0
$871$		0			0
$872$	21.0000	+	21.0000i	0.711150	+	0.711150i
$873$	−3.00000	+	3.00000i	−0.101535	+	0.101535i
$874$	6.00000	−	6.00000i	0.202953	−	0.202953i
$875$		0			0
$876$		−	6.00000i		−	0.202721i
$877$	27.0000	−	27.0000i	0.911725	−	0.911725i	−0.0846827	−	0.996408i	$-0.526988\pi$
$877$	27.0000	−	27.0000i	0.911725	−	0.911725i	0.996408	+	0.0846827i	$0.0269877\pi$
$878$	19.0000	−	19.0000i	0.641219	−	0.641219i
$879$	−4.00000	−	4.00000i	−0.134917	−	0.134917i
$880$		0			0
$881$	−3.00000	−	3.00000i	−0.101073	−	0.101073i	0.654762	−	0.755835i	$-0.272769\pi$
$881$	−3.00000	−	3.00000i	−0.101073	−	0.101073i	−0.755835	+	0.654762i	$0.772769\pi$
$882$		−	11.0000i		−	0.370389i
$883$	−6.00000			−0.201916			−0.100958	−	0.994891i	$-0.532191\pi$
$883$	−6.00000			−0.201916			−0.100958	+	0.994891i	$0.532191\pi$
$884$		0			0
$885$		0			0
$886$		−	22.0000i		−	0.739104i
$887$	17.0000	+	17.0000i	0.570804	+	0.570804i	0.932353	−	0.361549i	$-0.117752\pi$
$887$	17.0000	+	17.0000i	0.570804	+	0.570804i	−0.361549	+	0.932353i	$0.617752\pi$
$888$	−18.0000			−0.604040
$889$		0			0
$890$		0			0
$891$	−15.0000	+	15.0000i	−0.502519	+	0.502519i
$892$		0			0
$893$			12.0000i			0.401565i
$894$	18.0000	−	18.0000i	0.602010	−	0.602010i
$895$		0			0
$896$	−9.00000	−	9.00000i	−0.300669	−	0.300669i
$897$		0			0
$898$	15.0000	+	15.0000i	0.500556	+	0.500556i
$899$		−	6.00000i		−	0.200111i
$900$		0			0
$901$	−2.00000	+	8.00000i	−0.0666297	+	0.266519i
$902$	18.0000			0.599334
$903$		−	72.0000i		−	2.39601i
$904$	27.0000	+	27.0000i	0.898007	+	0.898007i
$905$		0			0
$906$	−10.0000	−	10.0000i	−0.332228	−	0.332228i
$907$	−15.0000	+	15.0000i	−0.498067	+	0.498067i	−0.910836	−	0.412769i	$-0.864562\pi$
$907$	−15.0000	+	15.0000i	−0.498067	+	0.498067i	0.412769	+	0.910836i	$0.364562\pi$
$908$	−15.0000	+	15.0000i	−0.497792	+	0.497792i
$909$			6.00000i			0.199007i
$910$		0			0
$911$	21.0000	−	21.0000i	0.695761	−	0.695761i	−0.267732	−	0.963493i	$-0.586274\pi$
$911$	21.0000	−	21.0000i	0.695761	−	0.695761i	0.963493	+	0.267732i	$0.0862743\pi$
$912$	−6.00000	+	6.00000i	−0.198680	+	0.198680i
$913$	12.0000	+	12.0000i	0.397142	+	0.397142i
$914$	−6.00000			−0.198462
$915$		0			0
$916$			24.0000i			0.792982i
$917$	−18.0000			−0.594412
$918$	−12.0000	−	20.0000i	−0.396059	−	0.660098i
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$	18.0000	+	18.0000i	0.593120	+	0.593120i
$922$	12.0000			0.395199
$923$		0			0
$924$	−18.0000	+	18.0000i	−0.592157	+	0.592157i
$925$		0			0
$926$		−	18.0000i		−	0.591517i
$927$		−	6.00000i		−	0.197066i
$928$	−15.0000	+	15.0000i	−0.492399	+	0.492399i
$929$	−21.0000	+	21.0000i	−0.688988	+	0.688988i	−0.962008	−	0.273021i	$-0.911977\pi$
$929$	−21.0000	+	21.0000i	−0.688988	+	0.688988i	0.273021	+	0.962008i	$0.411977\pi$
$930$		0			0
$931$	66.0000			2.16306
$932$	−5.00000	−	5.00000i	−0.163780	−	0.163780i
$933$		−	6.00000i		−	0.196431i
$934$	−28.0000			−0.916188
$935$		0			0
$936$		0			0
$937$			18.0000i			0.588034i	0.955800	+	0.294017i	$0.0949923\pi$
$937$			18.0000i			0.588034i	−0.955800	+	0.294017i	$0.905008\pi$
$938$	−18.0000	−	18.0000i	−0.587721	−	0.587721i
$939$	6.00000			0.195803
$940$		0			0
$941$	−15.0000	+	15.0000i	−0.488986	+	0.488986i	−0.907986	−	0.419000i	$-0.862381\pi$
$941$	−15.0000	+	15.0000i	−0.488986	+	0.488986i	0.419000	+	0.907986i	$0.362381\pi$
$942$	12.0000	−	12.0000i	0.390981	−	0.390981i
$943$		−	6.00000i		−	0.195387i
$944$		−	6.00000i		−	0.195283i
$945$		0			0
$946$	−36.0000	+	36.0000i	−1.17046	+	1.17046i
$947$	−11.0000	−	11.0000i	−0.357452	−	0.357452i	0.505421	−	0.862873i	$-0.331337\pi$
$947$	−11.0000	−	11.0000i	−0.357452	−	0.357452i	−0.862873	+	0.505421i	$0.831337\pi$
$948$	−14.0000			−0.454699
$949$		0			0
$950$		0			0
$951$	−30.0000			−0.972817
$952$	−27.0000	−	45.0000i	−0.875075	−	1.45846i
$953$	−4.00000			−0.129573			−0.0647864	−	0.997899i	$-0.520637\pi$
$953$	−4.00000			−0.129573			−0.0647864	+	0.997899i	$0.520637\pi$
$954$			2.00000i			0.0647524i
$955$		0			0
$956$	24.0000			0.776215
$957$	18.0000	+	18.0000i	0.581857	+	0.581857i
$958$	−9.00000	+	9.00000i	−0.290777	+	0.290777i
$959$	−12.0000	+	12.0000i	−0.387500	+	0.387500i
$960$		0			0
$961$		−	29.0000i		−	0.935484i
$962$		0			0
$963$	9.00000	−	9.00000i	0.290021	−	0.290021i
$964$	−19.0000	−	19.0000i	−0.611949	−	0.611949i
$965$		0			0
$966$	−6.00000	−	6.00000i	−0.193047	−	0.193047i
$967$		−	24.0000i		−	0.771788i	−0.922543	−	0.385894i	$-0.873893\pi$
$967$		−	24.0000i		−	0.771788i	0.922543	−	0.385894i	$-0.126107\pi$
$968$	21.0000			0.674966
$969$	30.0000	−	18.0000i	0.963739	−	0.578243i
$970$		0			0
$971$			42.0000i			1.34784i	0.738802	+	0.673922i	$0.235392\pi$
$971$			42.0000i			1.34784i	−0.738802	+	0.673922i	$0.764608\pi$
$972$	7.00000	+	7.00000i	0.224525	+	0.224525i
$973$	42.0000			1.34646
$974$	−3.00000	−	3.00000i	−0.0961262	−	0.0961262i
$975$		0			0
$976$	−1.00000	+	1.00000i	−0.0320092	+	0.0320092i
$977$		−	46.0000i		−	1.47167i	−0.677161	−	0.735835i	$-0.736790\pi$
$977$		−	46.0000i		−	1.47167i	0.677161	−	0.735835i	$-0.263210\pi$
$978$			18.0000i			0.575577i
$979$	18.0000	−	18.0000i	0.575282	−	0.575282i
$980$		0			0
$981$	−7.00000	−	7.00000i	−0.223493	−	0.223493i
$982$	6.00000			0.191468
$983$	−21.0000	−	21.0000i	−0.669796	−	0.669796i	0.287873	−	0.957669i	$-0.407052\pi$
$983$	−21.0000	−	21.0000i	−0.669796	−	0.669796i	−0.957669	+	0.287873i	$0.907052\pi$
$984$			18.0000i			0.573819i
$985$		0			0
$986$	−15.0000	+	9.00000i	−0.477697	+	0.286618i
$987$	12.0000			0.381964
$988$		0			0
$989$	12.0000	+	12.0000i	0.381578	+	0.381578i
$990$		0			0
$991$	−13.0000	−	13.0000i	−0.412959	−	0.412959i	0.469809	−	0.882768i	$-0.344323\pi$
$991$	−13.0000	−	13.0000i	−0.412959	−	0.412959i	−0.882768	+	0.469809i	$0.844323\pi$
$992$	5.00000	−	5.00000i	0.158750	−	0.158750i
$993$	18.0000	−	18.0000i	0.571213	−	0.571213i
$994$		−	18.0000i		−	0.570925i
$995$		0			0
$996$	−4.00000	+	4.00000i	−0.126745	+	0.126745i
$997$	3.00000	−	3.00000i	0.0950110	−	0.0950110i	−0.658004	−	0.753015i	$-0.728599\pi$
$997$	3.00000	−	3.00000i	0.0950110	−	0.0950110i	0.753015	+	0.658004i	$0.228599\pi$
$998$	23.0000	+	23.0000i	0.728052	+	0.728052i
$999$	24.0000			0.759326

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	425.2.e.a.251.1		2
5.2	odd	4		85.2.j.a.64.1	yes	2
5.3	odd	4		85.2.j.b.64.1	yes	2
5.4	even	2		425.2.e.b.251.1		2
15.2	even	4		765.2.t.b.64.1		2
15.8	even	4		765.2.t.a.64.1		2
17.2	even	8		7225.2.a.i.1.1		2
17.4	even	4	inner	425.2.e.a.276.1		2
17.15	even	8		7225.2.a.i.1.2		2
85.2	odd	8		1445.2.b.a.579.2		4
85.4	even	4		425.2.e.b.276.1		2
85.19	even	8		7225.2.a.p.1.2		2
85.32	odd	8		1445.2.b.a.579.1		4
85.38	odd	4		85.2.j.a.4.1	✓	2
85.49	even	8		7225.2.a.p.1.1		2
85.53	odd	8		1445.2.b.a.579.3		4
85.72	odd	4		85.2.j.b.4.1	yes	2
85.83	odd	8		1445.2.b.a.579.4		4
255.38	even	4		765.2.t.b.514.1		2
255.242	even	4		765.2.t.a.514.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.j.a.4.1	✓	2	85.38	odd	4
85.2.j.a.64.1	yes	2	5.2	odd	4
85.2.j.b.4.1	yes	2	85.72	odd	4
85.2.j.b.64.1	yes	2	5.3	odd	4
425.2.e.a.251.1		2	1.1	even	1	trivial
425.2.e.a.276.1		2	17.4	even	4	inner
425.2.e.b.251.1		2	5.4	even	2
425.2.e.b.276.1		2	85.4	even	4
765.2.t.a.64.1		2	15.8	even	4
765.2.t.a.514.1		2	255.242	even	4
765.2.t.b.64.1		2	15.2	even	4
765.2.t.b.514.1		2	255.38	even	4
1445.2.b.a.579.1		4	85.32	odd	8
1445.2.b.a.579.2		4	85.2	odd	8
1445.2.b.a.579.3		4	85.53	odd	8
1445.2.b.a.579.4		4	85.83	odd	8
7225.2.a.i.1.1		2	17.2	even	8
7225.2.a.i.1.2		2	17.15	even	8
7225.2.a.p.1.1		2	85.49	even	8
7225.2.a.p.1.2		2	85.19	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 425 = 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	425.e (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +(-1.00000 - 1.00000i) q^{3} +1.00000 q^{4} +(1.00000 - 1.00000i) q^{6} +(-3.00000 + 3.00000i) q^{7} +3.00000i q^{8} -1.00000i q^{9} +(-3.00000 + 3.00000i) q^{11} +(-1.00000 - 1.00000i) q^{12} +(-3.00000 - 3.00000i) q^{14} -1.00000 q^{16} +(4.00000 + 1.00000i) q^{17} +1.00000 q^{18} +6.00000i q^{19} +6.00000 q^{21} +(-3.00000 - 3.00000i) q^{22} +(-1.00000 + 1.00000i) q^{23} +(3.00000 - 3.00000i) q^{24} +(-4.00000 + 4.00000i) q^{27} +(-3.00000 + 3.00000i) q^{28} +(3.00000 + 3.00000i) q^{29} +(-1.00000 - 1.00000i) q^{31} +5.00000i q^{32} +6.00000 q^{33} +(-1.00000 + 4.00000i) q^{34} -1.00000i q^{36} +(-3.00000 - 3.00000i) q^{37} -6.00000 q^{38} +(-3.00000 + 3.00000i) q^{41} +6.00000i q^{42} -12.0000i q^{43} +(-3.00000 + 3.00000i) q^{44} +(-1.00000 - 1.00000i) q^{46} +2.00000 q^{47} +(1.00000 + 1.00000i) q^{48} -11.0000i q^{49} +(-3.00000 - 5.00000i) q^{51} +2.00000i q^{53} +(-4.00000 - 4.00000i) q^{54} +(-9.00000 - 9.00000i) q^{56} +(6.00000 - 6.00000i) q^{57} +(-3.00000 + 3.00000i) q^{58} +6.00000i q^{59} +(1.00000 - 1.00000i) q^{61} +(1.00000 - 1.00000i) q^{62} +(3.00000 + 3.00000i) q^{63} -7.00000 q^{64} +6.00000i q^{66} +6.00000 q^{67} +(4.00000 + 1.00000i) q^{68} +2.00000 q^{69} +(3.00000 + 3.00000i) q^{71} +3.00000 q^{72} +(3.00000 + 3.00000i) q^{73} +(3.00000 - 3.00000i) q^{74} +6.00000i q^{76} -18.0000i q^{77} +(7.00000 - 7.00000i) q^{79} +5.00000 q^{81} +(-3.00000 - 3.00000i) q^{82} -4.00000i q^{83} +6.00000 q^{84} +12.0000 q^{86} -6.00000i q^{87} +(-9.00000 - 9.00000i) q^{88} -6.00000 q^{89} +(-1.00000 + 1.00000i) q^{92} +2.00000i q^{93} +2.00000i q^{94} +(5.00000 - 5.00000i) q^{96} +(-3.00000 - 3.00000i) q^{97} +11.0000 q^{98} +(3.00000 + 3.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{4} + 2 q^{6} - 6 q^{7} - 6 q^{11} - 2 q^{12} - 6 q^{14} - 2 q^{16} + 8 q^{17} + 2 q^{18} + 12 q^{21} - 6 q^{22} - 2 q^{23} + 6 q^{24} - 8 q^{27} - 6 q^{28} + 6 q^{29} - 2 q^{31} + 12 q^{33}+ \cdots + 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^4 + 2 * q^6 - 6 * q^7 - 6 * q^11 - 2 * q^12 - 6 * q^14 - 2 * q^16 + 8 * q^17 + 2 * q^18 + 12 * q^21 - 6 * q^22 - 2 * q^23 + 6 * q^24 - 8 * q^27 - 6 * q^28 + 6 * q^29 - 2 * q^31 + 12 * q^33 - 2 * q^34 - 6 * q^37 - 12 * q^38 - 6 * q^41 - 6 * q^44 - 2 * q^46 + 4 * q^47 + 2 * q^48 - 6 * q^51 - 8 * q^54 - 18 * q^56 + 12 * q^57 - 6 * q^58 + 2 * q^61 + 2 * q^62 + 6 * q^63 - 14 * q^64 + 12 * q^67 + 8 * q^68 + 4 * q^69 + 6 * q^71 + 6 * q^72 + 6 * q^73 + 6 * q^74 + 14 * q^79 + 10 * q^81 - 6 * q^82 + 12 * q^84 + 24 * q^86 - 18 * q^88 - 12 * q^89 - 2 * q^92 + 10 * q^96 - 6 * q^97 + 22 * q^98 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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