LMFDB - Embedded newform 800.2.n.i.543.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [800,2,Mod(543,800)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("800.543"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(800, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 0, 3])) N = Newforms(chi, 2, names="a")

Level:	$N$	$=$	$800 = 2^{5} \cdot 5^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	800.n (of order $4$ , degree $2$ , minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,2,0,0,0,2,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$6.38803216170$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			543.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	800.543
Dual form			800.2.n.i.607.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.00000i) q^{3} +(1.00000 + 1.00000i) q^{7} +1.00000i q^{9} +4.00000i q^{11} +(4.00000 + 4.00000i) q^{13} +(-4.00000 + 4.00000i) q^{17} -4.00000 q^{19} +2.00000 q^{21} +(5.00000 - 5.00000i) q^{23} +(4.00000 + 4.00000i) q^{27} +2.00000i q^{29} -8.00000i q^{31} +(4.00000 + 4.00000i) q^{33} +8.00000 q^{39} -4.00000 q^{41} +(7.00000 - 7.00000i) q^{43} +(3.00000 + 3.00000i) q^{47} -5.00000i q^{49} +8.00000i q^{51} +(4.00000 + 4.00000i) q^{53} +(-4.00000 + 4.00000i) q^{57} -4.00000 q^{59} -8.00000 q^{61} +(-1.00000 + 1.00000i) q^{63} +(-3.00000 - 3.00000i) q^{67} -10.0000i q^{69} -16.0000i q^{71} +(4.00000 + 4.00000i) q^{73} +(-4.00000 + 4.00000i) q^{77} +8.00000 q^{79} +5.00000 q^{81} +(-5.00000 + 5.00000i) q^{83} +(2.00000 + 2.00000i) q^{87} +10.0000i q^{89} +8.00000i q^{91} +(-8.00000 - 8.00000i) q^{93} +(12.0000 - 12.0000i) q^{97} -4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 2 q^{3} + 2 q^{7} + 8 q^{13} - 8 q^{17} - 8 q^{19} + 4 q^{21} + 10 q^{23} + 8 q^{27} + 8 q^{33} + 16 q^{39} - 8 q^{41} + 14 q^{43} + 6 q^{47} + 8 q^{53} - 8 q^{57} - 8 q^{59} - 16 q^{61} - 2 q^{63}+ \cdots - 8 q^{99}+O(q^{100})$ 2 * q + 2 * q^3 + 2 * q^7 + 8 * q^13 - 8 * q^17 - 8 * q^19 + 4 * q^21 + 10 * q^23 + 8 * q^27 + 8 * q^33 + 16 * q^39 - 8 * q^41 + 14 * q^43 + 6 * q^47 + 8 * q^53 - 8 * q^57 - 8 * q^59 - 16 * q^61 - 2 * q^63 - 6 * q^67 + 8 * q^73 - 8 * q^77 + 16 * q^79 + 10 * q^81 - 10 * q^83 + 4 * q^87 - 16 * q^93 + 24 * q^97 - 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$ .

$n$	$101$	$351$	$577$
$\chi(n)$	$1$	$-1$	$e\left(\frac{3}{4}\right)$

Coefficient data

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

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$		0			0
$2$		0			0
$3$	1.00000	−	1.00000i	0.577350	−	0.577350i	−0.356822	−	0.934172i	$-0.616140\pi$
$3$	1.00000	−	1.00000i	0.577350	−	0.577350i	0.934172	+	0.356822i	$0.116140\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	1.00000	+	1.00000i	0.377964	+	0.377964i	0.870367	−	0.492403i	$-0.163881\pi$
$7$	1.00000	+	1.00000i	0.377964	+	0.377964i	−0.492403	+	0.870367i	$0.663881\pi$
$8$		0			0
$9$			1.00000i			0.333333i
$10$		0			0
$11$			4.00000i			1.20605i	0.797724	+	0.603023i	$0.206037\pi$
$11$			4.00000i			1.20605i	−0.797724	+	0.603023i	$0.793963\pi$
$12$		0			0
$13$	4.00000	+	4.00000i	1.10940	+	1.10940i	0.993229	+	0.116171i	$0.0370621\pi$
$13$	4.00000	+	4.00000i	1.10940	+	1.10940i	0.116171	+	0.993229i	$0.462938\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.00000	+	4.00000i	−0.970143	+	0.970143i	−0.999567	−	0.0294245i	$-0.990633\pi$
$17$	−4.00000	+	4.00000i	−0.970143	+	0.970143i	0.0294245	+	0.999567i	$0.490633\pi$
$18$		0			0
$19$	−4.00000			−0.917663			−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000			−0.917663			−0.458831	+	0.888523i	$0.651732\pi$
$20$		0			0
$21$	2.00000			0.436436
$22$		0			0
$23$	5.00000	−	5.00000i	1.04257	−	1.04257i	0.0435195	−	0.999053i	$-0.486143\pi$
$23$	5.00000	−	5.00000i	1.04257	−	1.04257i	0.999053	−	0.0435195i	$-0.0138571\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	4.00000	+	4.00000i	0.769800	+	0.769800i
$28$		0			0
$29$			2.00000i			0.371391i	0.982607	+	0.185695i	$0.0594537\pi$
$29$			2.00000i			0.371391i	−0.982607	+	0.185695i	$0.940546\pi$
$30$		0			0
$31$		−	8.00000i		−	1.43684i	−0.695608	−	0.718421i	$-0.744865\pi$
$31$		−	8.00000i		−	1.43684i	0.695608	−	0.718421i	$-0.255135\pi$
$32$		0			0
$33$	4.00000	+	4.00000i	0.696311	+	0.696311i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$37$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$38$		0			0
$39$	8.00000			1.28103
$40$		0			0
$41$	−4.00000			−0.624695			−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000			−0.624695			−0.312348	+	0.949968i	$0.601115\pi$
$42$		0			0
$43$	7.00000	−	7.00000i	1.06749	−	1.06749i	0.0699387	−	0.997551i	$-0.477720\pi$
$43$	7.00000	−	7.00000i	1.06749	−	1.06749i	0.997551	−	0.0699387i	$-0.0222804\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.00000	+	3.00000i	0.437595	+	0.437595i	0.891202	−	0.453607i	$-0.149863\pi$
$47$	3.00000	+	3.00000i	0.437595	+	0.437595i	−0.453607	+	0.891202i	$0.649863\pi$
$48$		0			0
$49$		−	5.00000i		−	0.714286i
$50$		0			0
$51$			8.00000i			1.12022i
$52$		0			0
$53$	4.00000	+	4.00000i	0.549442	+	0.549442i	0.926279	−	0.376837i	$-0.122988\pi$
$53$	4.00000	+	4.00000i	0.549442	+	0.549442i	−0.376837	+	0.926279i	$0.622988\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−4.00000	+	4.00000i	−0.529813	+	0.529813i
$58$		0			0
$59$	−4.00000			−0.520756			−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000			−0.520756			−0.260378	+	0.965507i	$0.583847\pi$
$60$		0			0
$61$	−8.00000			−1.02430			−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000			−1.02430			−0.512148	+	0.858898i	$0.671150\pi$
$62$		0			0
$63$	−1.00000	+	1.00000i	−0.125988	+	0.125988i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−3.00000	−	3.00000i	−0.366508	−	0.366508i	0.499694	−	0.866202i	$-0.333446\pi$
$67$	−3.00000	−	3.00000i	−0.366508	−	0.366508i	−0.866202	+	0.499694i	$0.833446\pi$
$68$		0			0
$69$		−	10.0000i		−	1.20386i
$70$		0			0
$71$		−	16.0000i		−	1.89885i	−0.313993	−	0.949425i	$-0.601667\pi$
$71$		−	16.0000i		−	1.89885i	0.313993	−	0.949425i	$-0.398333\pi$
$72$		0			0
$73$	4.00000	+	4.00000i	0.468165	+	0.468165i	0.901319	−	0.433155i	$-0.142600\pi$
$73$	4.00000	+	4.00000i	0.468165	+	0.468165i	−0.433155	+	0.901319i	$0.642600\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−4.00000	+	4.00000i	−0.455842	+	0.455842i
$78$		0			0
$79$	8.00000			0.900070			0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000			0.900070			0.450035	+	0.893011i	$0.351411\pi$
$80$		0			0
$81$	5.00000			0.555556
$82$		0			0
$83$	−5.00000	+	5.00000i	−0.548821	+	0.548821i	−0.926100	−	0.377279i	$-0.876860\pi$
$83$	−5.00000	+	5.00000i	−0.548821	+	0.548821i	0.377279	+	0.926100i	$0.376860\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	2.00000	+	2.00000i	0.214423	+	0.214423i
$88$		0			0
$89$			10.0000i			1.06000i	0.847998	+	0.529999i	$0.177808\pi$
$89$			10.0000i			1.06000i	−0.847998	+	0.529999i	$0.822192\pi$
$90$		0			0
$91$			8.00000i			0.838628i
$92$		0			0
$93$	−8.00000	−	8.00000i	−0.829561	−	0.829561i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	12.0000	−	12.0000i	1.21842	−	1.21842i	0.250229	−	0.968187i	$-0.419494\pi$
$97$	12.0000	−	12.0000i	1.21842	−	1.21842i	0.968187	−	0.250229i	$-0.0805058\pi$
$98$		0			0
$99$	−4.00000			−0.402015
$100$		0			0
$101$	6.00000			0.597022			0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000			0.597022			0.298511	+	0.954406i	$0.403510\pi$
$102$		0			0
$103$	3.00000	−	3.00000i	0.295599	−	0.295599i	−0.543688	−	0.839287i	$-0.682973\pi$
$103$	3.00000	−	3.00000i	0.295599	−	0.295599i	0.839287	+	0.543688i	$0.182973\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.00000	+	3.00000i	0.290021	+	0.290021i	0.837088	−	0.547068i	$-0.184256\pi$
$107$	3.00000	+	3.00000i	0.290021	+	0.290021i	−0.547068	+	0.837088i	$0.684256\pi$
$108$		0			0
$109$			16.0000i			1.53252i	0.642529	+	0.766261i	$0.277885\pi$
$109$			16.0000i			1.53252i	−0.642529	+	0.766261i	$0.722115\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−8.00000	−	8.00000i	−0.752577	−	0.752577i	0.222383	−	0.974959i	$-0.428617\pi$
$113$	−8.00000	−	8.00000i	−0.752577	−	0.752577i	−0.974959	+	0.222383i	$0.928617\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−4.00000	+	4.00000i	−0.369800	+	0.369800i
$118$		0			0
$119$	−8.00000			−0.733359
$120$		0			0
$121$	−5.00000			−0.454545
$122$		0			0
$123$	−4.00000	+	4.00000i	−0.360668	+	0.360668i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−15.0000	−	15.0000i	−1.33103	−	1.33103i	−0.904445	−	0.426589i	$-0.859715\pi$
$127$	−15.0000	−	15.0000i	−1.33103	−	1.33103i	−0.426589	−	0.904445i	$-0.640285\pi$
$128$		0			0
$129$		−	14.0000i		−	1.23263i
$130$		0			0
$131$			12.0000i			1.04844i	0.851581	+	0.524222i	$0.175644\pi$
$131$			12.0000i			1.04844i	−0.851581	+	0.524222i	$0.824356\pi$
$132$		0			0
$133$	−4.00000	−	4.00000i	−0.346844	−	0.346844i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	8.00000	−	8.00000i	0.683486	−	0.683486i	−0.277298	−	0.960784i	$-0.589439\pi$
$137$	8.00000	−	8.00000i	0.683486	−	0.683486i	0.960784	+	0.277298i	$0.0894389\pi$
$138$		0			0
$139$	−12.0000			−1.01783			−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000			−1.01783			−0.508913	+	0.860818i	$0.669953\pi$
$140$		0			0
$141$	6.00000			0.505291
$142$		0			0
$143$	−16.0000	+	16.0000i	−1.33799	+	1.33799i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−5.00000	−	5.00000i	−0.412393	−	0.412393i
$148$		0			0
$149$		−	8.00000i		−	0.655386i	−0.944784	−	0.327693i	$-0.893729\pi$
$149$		−	8.00000i		−	0.655386i	0.944784	−	0.327693i	$-0.106271\pi$
$150$		0			0
$151$			8.00000i			0.651031i	0.945537	+	0.325515i	$0.105538\pi$
$151$			8.00000i			0.651031i	−0.945537	+	0.325515i	$0.894462\pi$
$152$		0			0
$153$	−4.00000	−	4.00000i	−0.323381	−	0.323381i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	8.00000	−	8.00000i	0.638470	−	0.638470i	−0.311708	−	0.950178i	$-0.600901\pi$
$157$	8.00000	−	8.00000i	0.638470	−	0.638470i	0.950178	+	0.311708i	$0.100901\pi$
$158$		0			0
$159$	8.00000			0.634441
$160$		0			0
$161$	10.0000			0.788110
$162$		0			0
$163$	−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	$0.416102\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	5.00000	+	5.00000i	0.386912	+	0.386912i	0.873584	−	0.486673i	$-0.161790\pi$
$167$	5.00000	+	5.00000i	0.386912	+	0.386912i	−0.486673	+	0.873584i	$0.661790\pi$
$168$		0			0
$169$			19.0000i			1.46154i
$170$		0			0
$171$		−	4.00000i		−	0.305888i
$172$		0			0
$173$	−8.00000	−	8.00000i	−0.608229	−	0.608229i	0.334254	−	0.942483i	$-0.391516\pi$
$173$	−8.00000	−	8.00000i	−0.608229	−	0.608229i	−0.942483	+	0.334254i	$0.891516\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−4.00000	+	4.00000i	−0.300658	+	0.300658i
$178$		0			0
$179$	12.0000			0.896922			0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000			0.896922			0.448461	+	0.893802i	$0.351972\pi$
$180$		0			0
$181$	10.0000			0.743294			0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000			0.743294			0.371647	+	0.928374i	$0.378793\pi$
$182$		0			0
$183$	−8.00000	+	8.00000i	−0.591377	+	0.591377i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−16.0000	−	16.0000i	−1.17004	−	1.17004i
$188$		0			0
$189$			8.00000i			0.581914i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$	−4.00000	−	4.00000i	−0.287926	−	0.287926i	0.548333	−	0.836260i	$-0.315262\pi$
$193$	−4.00000	−	4.00000i	−0.287926	−	0.287926i	−0.836260	+	0.548333i	$0.815262\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	4.00000	−	4.00000i	0.284988	−	0.284988i	−0.550106	−	0.835095i	$-0.685413\pi$
$197$	4.00000	−	4.00000i	0.284988	−	0.284988i	0.835095	+	0.550106i	$0.185413\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$	−6.00000			−0.423207
$202$		0			0
$203$	−2.00000	+	2.00000i	−0.140372	+	0.140372i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	5.00000	+	5.00000i	0.347524	+	0.347524i
$208$		0			0
$209$		−	16.0000i		−	1.10674i
$210$		0			0
$211$		−	20.0000i		−	1.37686i	−0.725304	−	0.688428i	$-0.758301\pi$
$211$		−	20.0000i		−	1.37686i	0.725304	−	0.688428i	$-0.241699\pi$
$212$		0			0
$213$	−16.0000	−	16.0000i	−1.09630	−	1.09630i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	8.00000	−	8.00000i	0.543075	−	0.543075i
$218$		0			0
$219$	8.00000			0.540590
$220$		0			0
$221$	−32.0000			−2.15255
$222$		0			0
$223$	9.00000	−	9.00000i	0.602685	−	0.602685i	−0.338340	−	0.941024i	$-0.609865\pi$
$223$	9.00000	−	9.00000i	0.602685	−	0.602685i	0.941024	+	0.338340i	$0.109865\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	7.00000	+	7.00000i	0.464606	+	0.464606i	0.900162	−	0.435556i	$-0.143448\pi$
$227$	7.00000	+	7.00000i	0.464606	+	0.464606i	−0.435556	+	0.900162i	$0.643448\pi$
$228$		0			0
$229$		−	26.0000i		−	1.71813i	−0.511868	−	0.859064i	$-0.671046\pi$
$229$		−	26.0000i		−	1.71813i	0.511868	−	0.859064i	$-0.328954\pi$
$230$		0			0
$231$			8.00000i			0.526361i
$232$		0			0
$233$	−12.0000	−	12.0000i	−0.786146	−	0.786146i	0.194714	−	0.980860i	$-0.437622\pi$
$233$	−12.0000	−	12.0000i	−0.786146	−	0.786146i	−0.980860	+	0.194714i	$0.937622\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	8.00000	−	8.00000i	0.519656	−	0.519656i
$238$		0			0
$239$	−24.0000			−1.55243			−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000			−1.55243			−0.776215	+	0.630468i	$0.782863\pi$
$240$		0			0
$241$	4.00000			0.257663			0.128831	−	0.991667i	$-0.458877\pi$
$241$	4.00000			0.257663			0.128831	+	0.991667i	$0.458877\pi$
$242$		0			0
$243$	−7.00000	+	7.00000i	−0.449050	+	0.449050i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−16.0000	−	16.0000i	−1.01806	−	1.01806i
$248$		0			0
$249$			10.0000i			0.633724i
$250$		0			0
$251$		−	12.0000i		−	0.757433i	−0.925513	−	0.378717i	$-0.876365\pi$
$251$		−	12.0000i		−	0.757433i	0.925513	−	0.378717i	$-0.123635\pi$
$252$		0			0
$253$	20.0000	+	20.0000i	1.25739	+	1.25739i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−16.0000	+	16.0000i	−0.998053	+	0.998053i	−0.999998	−	0.00194553i	$-0.999381\pi$
$257$	−16.0000	+	16.0000i	−0.998053	+	0.998053i	0.00194553	+	0.999998i	$0.499381\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−2.00000			−0.123797
$262$		0			0
$263$	−9.00000	+	9.00000i	−0.554964	+	0.554964i	−0.927869	−	0.372906i	$-0.878362\pi$
$263$	−9.00000	+	9.00000i	−0.554964	+	0.554964i	0.372906	+	0.927869i	$0.378362\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	10.0000	+	10.0000i	0.611990	+	0.611990i
$268$		0			0
$269$			24.0000i			1.46331i	0.681677	+	0.731653i	$0.261251\pi$
$269$			24.0000i			1.46331i	−0.681677	+	0.731653i	$0.738749\pi$
$270$		0			0
$271$			8.00000i			0.485965i	0.970031	+	0.242983i	$0.0781258\pi$
$271$			8.00000i			0.485965i	−0.970031	+	0.242983i	$0.921874\pi$
$272$		0			0
$273$	8.00000	+	8.00000i	0.484182	+	0.484182i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	16.0000	−	16.0000i	0.961347	−	0.961347i	−0.0379334	−	0.999280i	$-0.512077\pi$
$277$	16.0000	−	16.0000i	0.961347	−	0.961347i	0.999280	+	0.0379334i	$0.0120775\pi$
$278$		0			0
$279$	8.00000			0.478947
$280$		0			0
$281$	−20.0000			−1.19310			−0.596550	−	0.802576i	$-0.703462\pi$
$281$	−20.0000			−1.19310			−0.596550	+	0.802576i	$0.703462\pi$
$282$		0			0
$283$	5.00000	−	5.00000i	0.297219	−	0.297219i	−0.542705	−	0.839924i	$-0.682600\pi$
$283$	5.00000	−	5.00000i	0.297219	−	0.297219i	0.839924	+	0.542705i	$0.182600\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−4.00000	−	4.00000i	−0.236113	−	0.236113i
$288$		0			0
$289$		−	15.0000i		−	0.882353i
$290$		0			0
$291$		−	24.0000i		−	1.40690i
$292$		0			0
$293$	8.00000	+	8.00000i	0.467365	+	0.467365i	0.901060	−	0.433695i	$-0.142790\pi$
$293$	8.00000	+	8.00000i	0.467365	+	0.467365i	−0.433695	+	0.901060i	$0.642790\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−16.0000	+	16.0000i	−0.928414	+	0.928414i
$298$		0			0
$299$	40.0000			2.31326
$300$		0			0
$301$	14.0000			0.806947
$302$		0			0
$303$	6.00000	−	6.00000i	0.344691	−	0.344691i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−3.00000	−	3.00000i	−0.171219	−	0.171219i	0.616296	−	0.787515i	$-0.288633\pi$
$307$	−3.00000	−	3.00000i	−0.171219	−	0.171219i	−0.787515	+	0.616296i	$0.788633\pi$
$308$		0			0
$309$		−	6.00000i		−	0.341328i
$310$		0			0
$311$			8.00000i			0.453638i	0.973937	+	0.226819i	$0.0728326\pi$
$311$			8.00000i			0.453638i	−0.973937	+	0.226819i	$0.927167\pi$
$312$		0			0
$313$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$313$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−20.0000	+	20.0000i	−1.12331	+	1.12331i	−0.132072	+	0.991240i	$0.542163\pi$
$317$	−20.0000	+	20.0000i	−1.12331	+	1.12331i	−0.991240	+	0.132072i	$0.957837\pi$
$318$		0			0
$319$	−8.00000			−0.447914
$320$		0			0
$321$	6.00000			0.334887
$322$		0			0
$323$	16.0000	−	16.0000i	0.890264	−	0.890264i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	16.0000	+	16.0000i	0.884802	+	0.884802i
$328$		0			0
$329$			6.00000i			0.330791i
$330$		0			0
$331$		−	12.0000i		−	0.659580i	−0.944054	−	0.329790i	$-0.893022\pi$
$331$		−	12.0000i		−	0.659580i	0.944054	−	0.329790i	$-0.106978\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	16.0000	−	16.0000i	0.871576	−	0.871576i	−0.121069	−	0.992644i	$-0.538632\pi$
$337$	16.0000	−	16.0000i	0.871576	−	0.871576i	0.992644	+	0.121069i	$0.0386321\pi$
$338$		0			0
$339$	−16.0000			−0.869001
$340$		0			0
$341$	32.0000			1.73290
$342$		0			0
$343$	12.0000	−	12.0000i	0.647939	−	0.647939i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1.00000	+	1.00000i	0.0536828	+	0.0536828i	0.733439	−	0.679756i	$-0.237914\pi$
$347$	1.00000	+	1.00000i	0.0536828	+	0.0536828i	−0.679756	+	0.733439i	$0.737914\pi$
$348$		0			0
$349$		−	2.00000i		−	0.107058i	−0.998566	−	0.0535288i	$-0.982953\pi$
$349$		−	2.00000i		−	0.107058i	0.998566	−	0.0535288i	$-0.0170469\pi$
$350$		0			0
$351$			32.0000i			1.70803i
$352$		0			0
$353$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$353$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−8.00000	+	8.00000i	−0.423405	+	0.423405i
$358$		0			0
$359$	−24.0000			−1.26667			−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000			−1.26667			−0.633336	+	0.773877i	$0.718315\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		0			0
$363$	−5.00000	+	5.00000i	−0.262432	+	0.262432i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	23.0000	+	23.0000i	1.20059	+	1.20059i	0.973987	+	0.226603i	$0.0727620\pi$
$367$	23.0000	+	23.0000i	1.20059	+	1.20059i	0.226603	+	0.973987i	$0.427238\pi$
$368$		0			0
$369$		−	4.00000i		−	0.208232i
$370$		0			0
$371$			8.00000i			0.415339i
$372$		0			0
$373$	24.0000	+	24.0000i	1.24267	+	1.24267i	0.958888	+	0.283785i	$0.0915902\pi$
$373$	24.0000	+	24.0000i	1.24267	+	1.24267i	0.283785	+	0.958888i	$0.408410\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−8.00000	+	8.00000i	−0.412021	+	0.412021i
$378$		0			0
$379$	−4.00000			−0.205466			−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000			−0.205466			−0.102733	+	0.994709i	$0.532759\pi$
$380$		0			0
$381$	−30.0000			−1.53695
$382$		0			0
$383$	17.0000	−	17.0000i	0.868659	−	0.868659i	−0.123665	−	0.992324i	$-0.539465\pi$
$383$	17.0000	−	17.0000i	0.868659	−	0.868659i	0.992324	+	0.123665i	$0.0394647\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	7.00000	+	7.00000i	0.355830	+	0.355830i
$388$		0			0
$389$		−	8.00000i		−	0.405616i	−0.979219	−	0.202808i	$-0.934993\pi$
$389$		−	8.00000i		−	0.405616i	0.979219	−	0.202808i	$-0.0650067\pi$
$390$		0			0
$391$			40.0000i			2.02289i
$392$		0			0
$393$	12.0000	+	12.0000i	0.605320	+	0.605320i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	4.00000	−	4.00000i	0.200754	−	0.200754i	−0.599569	−	0.800323i	$-0.704661\pi$
$397$	4.00000	−	4.00000i	0.200754	−	0.200754i	0.800323	+	0.599569i	$0.204661\pi$
$398$		0			0
$399$	−8.00000			−0.400501
$400$		0			0
$401$	30.0000			1.49813			0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000			1.49813			0.749064	+	0.662497i	$0.230503\pi$
$402$		0			0
$403$	32.0000	−	32.0000i	1.59403	−	1.59403i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$			20.0000i			0.988936i	0.869196	+	0.494468i	$0.164637\pi$
$409$			20.0000i			0.988936i	−0.869196	+	0.494468i	$0.835363\pi$
$410$		0			0
$411$		−	16.0000i		−	0.789222i
$412$		0			0
$413$	−4.00000	−	4.00000i	−0.196827	−	0.196827i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−12.0000	+	12.0000i	−0.587643	+	0.587643i
$418$		0			0
$419$	28.0000			1.36789			0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000			1.36789			0.683945	+	0.729534i	$0.260263\pi$
$420$		0			0
$421$	−8.00000			−0.389896			−0.194948	−	0.980814i	$-0.562454\pi$
$421$	−8.00000			−0.389896			−0.194948	+	0.980814i	$0.562454\pi$
$422$		0			0
$423$	−3.00000	+	3.00000i	−0.145865	+	0.145865i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−8.00000	−	8.00000i	−0.387147	−	0.387147i
$428$		0			0
$429$			32.0000i			1.54497i
$430$		0			0
$431$		−	8.00000i		−	0.385346i	−0.981263	−	0.192673i	$-0.938284\pi$
$431$		−	8.00000i		−	0.385346i	0.981263	−	0.192673i	$-0.0617157\pi$
$432$		0			0
$433$	20.0000	+	20.0000i	0.961139	+	0.961139i	0.999273	−	0.0381340i	$-0.0121414\pi$
$433$	20.0000	+	20.0000i	0.961139	+	0.961139i	−0.0381340	+	0.999273i	$0.512141\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−20.0000	+	20.0000i	−0.956730	+	0.956730i
$438$		0			0
$439$	8.00000			0.381819			0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000			0.381819			0.190910	+	0.981608i	$0.438856\pi$
$440$		0			0
$441$	5.00000			0.238095
$442$		0			0
$443$	−13.0000	+	13.0000i	−0.617649	+	0.617649i	−0.944928	−	0.327279i	$-0.893868\pi$
$443$	−13.0000	+	13.0000i	−0.617649	+	0.617649i	0.327279	+	0.944928i	$0.393868\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−8.00000	−	8.00000i	−0.378387	−	0.378387i
$448$		0			0
$449$		−	4.00000i		−	0.188772i	−0.995536	−	0.0943858i	$-0.969911\pi$
$449$		−	4.00000i		−	0.188772i	0.995536	−	0.0943858i	$-0.0300887\pi$
$450$		0			0
$451$		−	16.0000i		−	0.753411i
$452$		0			0
$453$	8.00000	+	8.00000i	0.375873	+	0.375873i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$457$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$458$		0			0
$459$	−32.0000			−1.49363
$460$		0			0
$461$	18.0000			0.838344			0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000			0.838344			0.419172	+	0.907907i	$0.362320\pi$
$462$		0			0
$463$	3.00000	−	3.00000i	0.139422	−	0.139422i	−0.633951	−	0.773373i	$-0.718568\pi$
$463$	3.00000	−	3.00000i	0.139422	−	0.139422i	0.773373	+	0.633951i	$0.218568\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	25.0000	+	25.0000i	1.15686	+	1.15686i	0.985147	+	0.171715i	$0.0549310\pi$
$467$	25.0000	+	25.0000i	1.15686	+	1.15686i	0.171715	+	0.985147i	$0.445069\pi$
$468$		0			0
$469$		−	6.00000i		−	0.277054i
$470$		0			0
$471$		−	16.0000i		−	0.737241i
$472$		0			0
$473$	28.0000	+	28.0000i	1.28744	+	1.28744i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−4.00000	+	4.00000i	−0.183147	+	0.183147i
$478$		0			0
$479$	−16.0000			−0.731059			−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000			−0.731059			−0.365529	+	0.930800i	$0.619112\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	10.0000	−	10.0000i	0.455016	−	0.455016i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−25.0000	−	25.0000i	−1.13286	−	1.13286i	−0.989700	−	0.143158i	$-0.954274\pi$
$487$	−25.0000	−	25.0000i	−1.13286	−	1.13286i	−0.143158	−	0.989700i	$-0.545726\pi$
$488$		0			0
$489$			18.0000i			0.813988i
$490$		0			0
$491$			12.0000i			0.541552i	0.962642	+	0.270776i	$0.0872803\pi$
$491$			12.0000i			0.541552i	−0.962642	+	0.270776i	$0.912720\pi$
$492$		0			0
$493$	−8.00000	−	8.00000i	−0.360302	−	0.360302i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	16.0000	−	16.0000i	0.717698	−	0.717698i
$498$		0			0
$499$	20.0000			0.895323			0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000			0.895323			0.447661	+	0.894203i	$0.352257\pi$
$500$		0			0
$501$	10.0000			0.446767
$502$		0			0
$503$	11.0000	−	11.0000i	0.490466	−	0.490466i	−0.417987	−	0.908453i	$-0.637264\pi$
$503$	11.0000	−	11.0000i	0.490466	−	0.490466i	0.908453	+	0.417987i	$0.137264\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	19.0000	+	19.0000i	0.843820	+	0.843820i
$508$		0			0
$509$		−	30.0000i		−	1.32973i	−0.746965	−	0.664863i	$-0.768490\pi$
$509$		−	30.0000i		−	1.32973i	0.746965	−	0.664863i	$-0.231510\pi$
$510$		0			0
$511$			8.00000i			0.353899i
$512$		0			0
$513$	−16.0000	−	16.0000i	−0.706417	−	0.706417i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−12.0000	+	12.0000i	−0.527759	+	0.527759i
$518$		0			0
$519$	−16.0000			−0.702322
$520$		0			0
$521$	−22.0000			−0.963837			−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000			−0.963837			−0.481919	+	0.876216i	$0.660060\pi$
$522$		0			0
$523$	5.00000	−	5.00000i	0.218635	−	0.218635i	−0.589288	−	0.807923i	$-0.700592\pi$
$523$	5.00000	−	5.00000i	0.218635	−	0.218635i	0.807923	+	0.589288i	$0.200592\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	32.0000	+	32.0000i	1.39394	+	1.39394i
$528$		0			0
$529$		−	27.0000i		−	1.17391i
$530$		0			0
$531$		−	4.00000i		−	0.173585i
$532$		0			0
$533$	−16.0000	−	16.0000i	−0.693037	−	0.693037i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	12.0000	−	12.0000i	0.517838	−	0.517838i
$538$		0			0
$539$	20.0000			0.861461
$540$		0			0
$541$	−30.0000			−1.28980			−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000			−1.28980			−0.644900	+	0.764267i	$0.723101\pi$
$542$		0			0
$543$	10.0000	−	10.0000i	0.429141	−	0.429141i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−17.0000	−	17.0000i	−0.726868	−	0.726868i	0.243127	−	0.969994i	$-0.421827\pi$
$547$	−17.0000	−	17.0000i	−0.726868	−	0.726868i	−0.969994	+	0.243127i	$0.921827\pi$
$548$		0			0
$549$		−	8.00000i		−	0.341432i
$550$		0			0
$551$		−	8.00000i		−	0.340811i
$552$		0			0
$553$	8.00000	+	8.00000i	0.340195	+	0.340195i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$557$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$558$		0			0
$559$	56.0000			2.36855
$560$		0			0
$561$	−32.0000			−1.35104
$562$		0			0
$563$	−15.0000	+	15.0000i	−0.632175	+	0.632175i	−0.948613	−	0.316438i	$-0.897513\pi$
$563$	−15.0000	+	15.0000i	−0.632175	+	0.632175i	0.316438	+	0.948613i	$0.397513\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	5.00000	+	5.00000i	0.209980	+	0.209980i
$568$		0			0
$569$		−	20.0000i		−	0.838444i	−0.907884	−	0.419222i	$-0.862303\pi$
$569$		−	20.0000i		−	0.838444i	0.907884	−	0.419222i	$-0.137697\pi$
$570$		0			0
$571$		−	4.00000i		−	0.167395i	−0.996491	−	0.0836974i	$-0.973327\pi$
$571$		−	4.00000i		−	0.167395i	0.996491	−	0.0836974i	$-0.0266729\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−8.00000	+	8.00000i	−0.333044	+	0.333044i	−0.853741	−	0.520697i	$-0.825672\pi$
$577$	−8.00000	+	8.00000i	−0.333044	+	0.333044i	0.520697	+	0.853741i	$0.325672\pi$
$578$		0			0
$579$	−8.00000			−0.332469
$580$		0			0
$581$	−10.0000			−0.414870
$582$		0			0
$583$	−16.0000	+	16.0000i	−0.662652	+	0.662652i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−25.0000	−	25.0000i	−1.03186	−	1.03186i	−0.999475	−	0.0323850i	$-0.989690\pi$
$587$	−25.0000	−	25.0000i	−1.03186	−	1.03186i	−0.0323850	−	0.999475i	$-0.510310\pi$
$588$		0			0
$589$			32.0000i			1.31854i
$590$		0			0
$591$		−	8.00000i		−	0.329076i
$592$		0			0
$593$	−8.00000	−	8.00000i	−0.328521	−	0.328521i	0.523503	−	0.852024i	$-0.324625\pi$
$593$	−8.00000	−	8.00000i	−0.328521	−	0.328521i	−0.852024	+	0.523503i	$0.824625\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−16.0000			−0.653742			−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000			−0.653742			−0.326871	+	0.945069i	$0.605994\pi$
$600$		0			0
$601$	4.00000			0.163163			0.0815817	−	0.996667i	$-0.474003\pi$
$601$	4.00000			0.163163			0.0815817	+	0.996667i	$0.474003\pi$
$602$		0			0
$603$	3.00000	−	3.00000i	0.122169	−	0.122169i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−5.00000	−	5.00000i	−0.202944	−	0.202944i	0.598316	−	0.801260i	$-0.295837\pi$
$607$	−5.00000	−	5.00000i	−0.202944	−	0.202944i	−0.801260	+	0.598316i	$0.795837\pi$
$608$		0			0
$609$			4.00000i			0.162088i
$610$		0			0
$611$			24.0000i			0.970936i
$612$		0			0
$613$	−12.0000	−	12.0000i	−0.484675	−	0.484675i	0.421946	−	0.906621i	$-0.361347\pi$
$613$	−12.0000	−	12.0000i	−0.484675	−	0.484675i	−0.906621	+	0.421946i	$0.861347\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	4.00000	−	4.00000i	0.161034	−	0.161034i	−0.621991	−	0.783025i	$-0.713676\pi$
$617$	4.00000	−	4.00000i	0.161034	−	0.161034i	0.783025	+	0.621991i	$0.213676\pi$
$618$		0			0
$619$	−12.0000			−0.482321			−0.241160	−	0.970485i	$-0.577528\pi$
$619$	−12.0000			−0.482321			−0.241160	+	0.970485i	$0.577528\pi$
$620$		0			0
$621$	40.0000			1.60514
$622$		0			0
$623$	−10.0000	+	10.0000i	−0.400642	+	0.400642i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−16.0000	−	16.0000i	−0.638978	−	0.638978i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		−	40.0000i		−	1.59237i	−0.605050	−	0.796187i	$-0.706847\pi$
$631$		−	40.0000i		−	1.59237i	0.605050	−	0.796187i	$-0.293153\pi$
$632$		0			0
$633$	−20.0000	−	20.0000i	−0.794929	−	0.794929i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	20.0000	−	20.0000i	0.792429	−	0.792429i
$638$		0			0
$639$	16.0000			0.632950
$640$		0			0
$641$	12.0000			0.473972			0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000			0.473972			0.236986	+	0.971513i	$0.423841\pi$
$642$		0			0
$643$	−11.0000	+	11.0000i	−0.433798	+	0.433798i	−0.889918	−	0.456120i	$-0.849239\pi$
$643$	−11.0000	+	11.0000i	−0.433798	+	0.433798i	0.456120	+	0.889918i	$0.349239\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.00000	−	1.00000i	−0.0393141	−	0.0393141i	0.687176	−	0.726491i	$-0.258850\pi$
$647$	−1.00000	−	1.00000i	−0.0393141	−	0.0393141i	−0.726491	+	0.687176i	$0.758850\pi$
$648$		0			0
$649$		−	16.0000i		−	0.628055i
$650$		0			0
$651$		−	16.0000i		−	0.627089i
$652$		0			0
$653$	20.0000	+	20.0000i	0.782660	+	0.782660i	0.980279	−	0.197619i	$-0.0633207\pi$
$653$	20.0000	+	20.0000i	0.782660	+	0.782660i	−0.197619	+	0.980279i	$0.563321\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−4.00000	+	4.00000i	−0.156055	+	0.156055i
$658$		0			0
$659$	−4.00000			−0.155818			−0.0779089	−	0.996960i	$-0.524824\pi$
$659$	−4.00000			−0.155818			−0.0779089	+	0.996960i	$0.524824\pi$
$660$		0			0
$661$		0			0			−	1.00000i	$-0.5\pi$
$661$		0			0				1.00000i	$0.5\pi$
$662$		0			0
$663$	−32.0000	+	32.0000i	−1.24278	+	1.24278i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	10.0000	+	10.0000i	0.387202	+	0.387202i
$668$		0			0
$669$		−	18.0000i		−	0.695920i
$670$		0			0
$671$		−	32.0000i		−	1.23535i
$672$		0			0
$673$	4.00000	+	4.00000i	0.154189	+	0.154189i	0.779986	−	0.625797i	$-0.215226\pi$
$673$	4.00000	+	4.00000i	0.154189	+	0.154189i	−0.625797	+	0.779986i	$0.715226\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	20.0000	−	20.0000i	0.768662	−	0.768662i	−0.209209	−	0.977871i	$-0.567089\pi$
$677$	20.0000	−	20.0000i	0.768662	−	0.768662i	0.977871	+	0.209209i	$0.0670888\pi$
$678$		0			0
$679$	24.0000			0.921035
$680$		0			0
$681$	14.0000			0.536481
$682$		0			0
$683$	−17.0000	+	17.0000i	−0.650487	+	0.650487i	−0.953110	−	0.302623i	$-0.902138\pi$
$683$	−17.0000	+	17.0000i	−0.650487	+	0.650487i	0.302623	+	0.953110i	$0.402138\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−26.0000	−	26.0000i	−0.991962	−	0.991962i
$688$		0			0
$689$			32.0000i			1.21910i
$690$		0			0
$691$			20.0000i			0.760836i	0.924815	+	0.380418i	$0.124220\pi$
$691$			20.0000i			0.760836i	−0.924815	+	0.380418i	$0.875780\pi$
$692$		0			0
$693$	−4.00000	−	4.00000i	−0.151947	−	0.151947i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	16.0000	−	16.0000i	0.606043	−	0.606043i
$698$		0			0
$699$	−24.0000			−0.907763
$700$		0			0
$701$	−48.0000			−1.81293			−0.906467	−	0.422276i	$-0.861231\pi$
$701$	−48.0000			−1.81293			−0.906467	+	0.422276i	$0.861231\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	6.00000	+	6.00000i	0.225653	+	0.225653i
$708$		0			0
$709$		−	6.00000i		−	0.225335i	−0.993633	−	0.112667i	$-0.964061\pi$
$709$		−	6.00000i		−	0.225335i	0.993633	−	0.112667i	$-0.0359394\pi$
$710$		0			0
$711$			8.00000i			0.300023i
$712$		0			0
$713$	−40.0000	−	40.0000i	−1.49801	−	1.49801i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−24.0000	+	24.0000i	−0.896296	+	0.896296i
$718$		0			0
$719$	−24.0000			−0.895049			−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000			−0.895049			−0.447524	+	0.894272i	$0.647694\pi$
$720$		0			0
$721$	6.00000			0.223452
$722$		0			0
$723$	4.00000	−	4.00000i	0.148762	−	0.148762i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−15.0000	−	15.0000i	−0.556319	−	0.556319i	0.371938	−	0.928257i	$-0.378693\pi$
$727$	−15.0000	−	15.0000i	−0.556319	−	0.556319i	−0.928257	+	0.371938i	$0.878693\pi$
$728$		0			0
$729$			29.0000i			1.07407i
$730$		0			0
$731$			56.0000i			2.07123i
$732$		0			0
$733$	−24.0000	−	24.0000i	−0.886460	−	0.886460i	0.107721	−	0.994181i	$-0.465645\pi$
$733$	−24.0000	−	24.0000i	−0.886460	−	0.886460i	−0.994181	+	0.107721i	$0.965645\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	12.0000	−	12.0000i	0.442026	−	0.442026i
$738$		0			0
$739$	−52.0000			−1.91285			−0.956425	−	0.291977i	$-0.905687\pi$
$739$	−52.0000			−1.91285			−0.956425	+	0.291977i	$0.905687\pi$
$740$		0			0
$741$	−32.0000			−1.17555
$742$		0			0
$743$	1.00000	−	1.00000i	0.0366864	−	0.0366864i	−0.688526	−	0.725212i	$-0.741742\pi$
$743$	1.00000	−	1.00000i	0.0366864	−	0.0366864i	0.725212	+	0.688526i	$0.241742\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−5.00000	−	5.00000i	−0.182940	−	0.182940i
$748$		0			0
$749$			6.00000i			0.219235i
$750$		0			0
$751$			16.0000i			0.583848i	0.956441	+	0.291924i	$0.0942955\pi$
$751$			16.0000i			0.583848i	−0.956441	+	0.291924i	$0.905705\pi$
$752$		0			0
$753$	−12.0000	−	12.0000i	−0.437304	−	0.437304i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	24.0000	−	24.0000i	0.872295	−	0.872295i	−0.120427	−	0.992722i	$-0.538426\pi$
$757$	24.0000	−	24.0000i	0.872295	−	0.872295i	0.992722	+	0.120427i	$0.0384265\pi$
$758$		0			0
$759$	40.0000			1.45191
$760$		0			0
$761$	10.0000			0.362500			0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000			0.362500			0.181250	+	0.983437i	$0.441986\pi$
$762$		0			0
$763$	−16.0000	+	16.0000i	−0.579239	+	0.579239i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−16.0000	−	16.0000i	−0.577727	−	0.577727i
$768$		0			0
$769$		−	2.00000i		−	0.0721218i	−0.999350	−	0.0360609i	$-0.988519\pi$
$769$		−	2.00000i		−	0.0721218i	0.999350	−	0.0360609i	$-0.0114810\pi$
$770$		0			0
$771$			32.0000i			1.15245i
$772$		0			0
$773$	−20.0000	−	20.0000i	−0.719350	−	0.719350i	0.249122	−	0.968472i	$-0.419858\pi$
$773$	−20.0000	−	20.0000i	−0.719350	−	0.719350i	−0.968472	+	0.249122i	$0.919858\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	16.0000			0.573259
$780$		0			0
$781$	64.0000			2.29010
$782$		0			0
$783$	−8.00000	+	8.00000i	−0.285897	+	0.285897i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	11.0000	+	11.0000i	0.392108	+	0.392108i	0.875438	−	0.483330i	$-0.160573\pi$
$787$	11.0000	+	11.0000i	0.392108	+	0.392108i	−0.483330	+	0.875438i	$0.660573\pi$
$788$		0			0
$789$			18.0000i			0.640817i
$790$		0			0
$791$		−	16.0000i		−	0.568895i
$792$		0			0
$793$	−32.0000	−	32.0000i	−1.13635	−	1.13635i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	28.0000	−	28.0000i	0.991811	−	0.991811i	−0.00815585	−	0.999967i	$-0.502596\pi$
$797$	28.0000	−	28.0000i	0.991811	−	0.991811i	0.999967	+	0.00815585i	$0.00259612\pi$
$798$		0			0
$799$	−24.0000			−0.849059
$800$		0			0
$801$	−10.0000			−0.353333
$802$		0			0
$803$	−16.0000	+	16.0000i	−0.564628	+	0.564628i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	24.0000	+	24.0000i	0.844840	+	0.844840i
$808$		0			0
$809$		−	10.0000i		−	0.351581i	−0.984428	−	0.175791i	$-0.943752\pi$
$809$		−	10.0000i		−	0.351581i	0.984428	−	0.175791i	$-0.0562482\pi$
$810$		0			0
$811$			20.0000i			0.702295i	0.936320	+	0.351147i	$0.114208\pi$
$811$			20.0000i			0.702295i	−0.936320	+	0.351147i	$0.885792\pi$
$812$		0			0
$813$	8.00000	+	8.00000i	0.280572	+	0.280572i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−28.0000	+	28.0000i	−0.979596	+	0.979596i
$818$		0			0
$819$	−8.00000			−0.279543
$820$		0			0
$821$	48.0000			1.67521			0.837606	−	0.546275i	$-0.183955\pi$
$821$	48.0000			1.67521			0.837606	+	0.546275i	$0.183955\pi$
$822$		0			0
$823$	13.0000	−	13.0000i	0.453152	−	0.453152i	−0.443248	−	0.896399i	$-0.646174\pi$
$823$	13.0000	−	13.0000i	0.453152	−	0.453152i	0.896399	+	0.443248i	$0.146174\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−23.0000	−	23.0000i	−0.799788	−	0.799788i	0.183274	−	0.983062i	$-0.441331\pi$
$827$	−23.0000	−	23.0000i	−0.799788	−	0.799788i	−0.983062	+	0.183274i	$0.941331\pi$
$828$		0			0
$829$		−	24.0000i		−	0.833554i	−0.909009	−	0.416777i	$-0.863160\pi$
$829$		−	24.0000i		−	0.833554i	0.909009	−	0.416777i	$-0.136840\pi$
$830$		0			0
$831$		−	32.0000i		−	1.11007i
$832$		0			0
$833$	20.0000	+	20.0000i	0.692959	+	0.692959i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	32.0000	−	32.0000i	1.10608	−	1.10608i
$838$		0			0
$839$	24.0000			0.828572			0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000			0.828572			0.414286	+	0.910147i	$0.364031\pi$
$840$		0			0
$841$	25.0000			0.862069
$842$		0			0
$843$	−20.0000	+	20.0000i	−0.688837	+	0.688837i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−5.00000	−	5.00000i	−0.171802	−	0.171802i
$848$		0			0
$849$		−	10.0000i		−	0.343199i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	12.0000	+	12.0000i	0.410872	+	0.410872i	0.882042	−	0.471170i	$-0.156168\pi$
$853$	12.0000	+	12.0000i	0.410872	+	0.410872i	−0.471170	+	0.882042i	$0.656168\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−40.0000	+	40.0000i	−1.36637	+	1.36637i	−0.500826	+	0.865548i	$0.666970\pi$
$857$	−40.0000	+	40.0000i	−1.36637	+	1.36637i	−0.865548	+	0.500826i	$0.833030\pi$
$858$		0			0
$859$	−20.0000			−0.682391			−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000			−0.682391			−0.341196	+	0.939992i	$0.610832\pi$
$860$		0			0
$861$	−8.00000			−0.272639
$862$		0			0
$863$	−35.0000	+	35.0000i	−1.19141	+	1.19141i	−0.214743	+	0.976671i	$0.568892\pi$
$863$	−35.0000	+	35.0000i	−1.19141	+	1.19141i	−0.976671	+	0.214743i	$0.931108\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−15.0000	−	15.0000i	−0.509427	−	0.509427i
$868$		0			0
$869$			32.0000i			1.08553i
$870$		0			0
$871$		−	24.0000i		−	0.813209i
$872$		0			0
$873$	12.0000	+	12.0000i	0.406138	+	0.406138i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−8.00000	+	8.00000i	−0.270141	+	0.270141i	−0.829157	−	0.559016i	$-0.811179\pi$
$877$	−8.00000	+	8.00000i	−0.270141	+	0.270141i	0.559016	+	0.829157i	$0.311179\pi$
$878$		0			0
$879$	16.0000			0.539667
$880$		0			0
$881$	−4.00000			−0.134763			−0.0673817	−	0.997727i	$-0.521465\pi$
$881$	−4.00000			−0.134763			−0.0673817	+	0.997727i	$0.521465\pi$
$882$		0			0
$883$	−23.0000	+	23.0000i	−0.774012	+	0.774012i	−0.978805	−	0.204794i	$-0.934348\pi$
$883$	−23.0000	+	23.0000i	−0.774012	+	0.774012i	0.204794	+	0.978805i	$0.434348\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−7.00000	−	7.00000i	−0.235037	−	0.235037i	0.579754	−	0.814791i	$-0.303149\pi$
$887$	−7.00000	−	7.00000i	−0.235037	−	0.235037i	−0.814791	+	0.579754i	$0.803149\pi$
$888$		0			0
$889$		−	30.0000i		−	1.00617i
$890$		0			0
$891$			20.0000i			0.670025i
$892$		0			0
$893$	−12.0000	−	12.0000i	−0.401565	−	0.401565i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	40.0000	−	40.0000i	1.33556	−	1.33556i
$898$		0			0
$899$	16.0000			0.533630
$900$		0			0
$901$	−32.0000			−1.06607
$902$		0			0
$903$	14.0000	−	14.0000i	0.465891	−	0.465891i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−3.00000	−	3.00000i	−0.0996134	−	0.0996134i	0.655544	−	0.755157i	$-0.272439\pi$
$907$	−3.00000	−	3.00000i	−0.0996134	−	0.0996134i	−0.755157	+	0.655544i	$0.772439\pi$
$908$		0			0
$909$			6.00000i			0.199007i
$910$		0			0
$911$		−	40.0000i		−	1.32526i	−0.748947	−	0.662630i	$-0.769440\pi$
$911$		−	40.0000i		−	1.32526i	0.748947	−	0.662630i	$-0.230560\pi$
$912$		0			0
$913$	−20.0000	−	20.0000i	−0.661903	−	0.661903i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−12.0000	+	12.0000i	−0.396275	+	0.396275i
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$	−6.00000			−0.197707
$922$		0			0
$923$	64.0000	−	64.0000i	2.10659	−	2.10659i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	3.00000	+	3.00000i	0.0985329	+	0.0985329i
$928$		0			0
$929$			28.0000i			0.918650i	0.888268	+	0.459325i	$0.151909\pi$
$929$			28.0000i			0.918650i	−0.888268	+	0.459325i	$0.848091\pi$
$930$		0			0
$931$			20.0000i			0.655474i
$932$		0			0
$933$	8.00000	+	8.00000i	0.261908	+	0.261908i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	12.0000	−	12.0000i	0.392023	−	0.392023i	−0.483385	−	0.875408i	$-0.660593\pi$
$937$	12.0000	−	12.0000i	0.392023	−	0.392023i	0.875408	+	0.483385i	$0.160593\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	18.0000			0.586783			0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000			0.586783			0.293392	+	0.955992i	$0.405216\pi$
$942$		0			0
$943$	−20.0000	+	20.0000i	−0.651290	+	0.651290i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−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	$0.831337\pi$
$948$		0			0
$949$			32.0000i			1.03876i
$950$		0			0
$951$			40.0000i			1.29709i
$952$		0			0
$953$	24.0000	+	24.0000i	0.777436	+	0.777436i	0.979394	−	0.201958i	$-0.0647304\pi$
$953$	24.0000	+	24.0000i	0.777436	+	0.777436i	−0.201958	+	0.979394i	$0.564730\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−8.00000	+	8.00000i	−0.258603	+	0.258603i
$958$		0			0
$959$	16.0000			0.516667
$960$		0			0
$961$	−33.0000			−1.06452
$962$		0			0
$963$	−3.00000	+	3.00000i	−0.0966736	+	0.0966736i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	27.0000	+	27.0000i	0.868261	+	0.868261i	0.992280	−	0.124018i	$-0.0395782\pi$
$967$	27.0000	+	27.0000i	0.868261	+	0.868261i	−0.124018	+	0.992280i	$0.539578\pi$
$968$		0			0
$969$		−	32.0000i		−	1.02799i
$970$		0			0
$971$			36.0000i			1.15529i	0.816286	+	0.577647i	$0.196029\pi$
$971$			36.0000i			1.15529i	−0.816286	+	0.577647i	$0.803971\pi$
$972$		0			0
$973$	−12.0000	−	12.0000i	−0.384702	−	0.384702i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−12.0000	+	12.0000i	−0.383914	+	0.383914i	−0.872510	−	0.488596i	$-0.837509\pi$
$977$	−12.0000	+	12.0000i	−0.383914	+	0.383914i	0.488596	+	0.872510i	$0.337509\pi$
$978$		0			0
$979$	−40.0000			−1.27841
$980$		0			0
$981$	−16.0000			−0.510841
$982$		0			0
$983$	−1.00000	+	1.00000i	−0.0318950	+	0.0318950i	−0.722874	−	0.690979i	$-0.757180\pi$
$983$	−1.00000	+	1.00000i	−0.0318950	+	0.0318950i	0.690979	+	0.722874i	$0.257180\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	6.00000	+	6.00000i	0.190982	+	0.190982i
$988$		0			0
$989$		−	70.0000i		−	2.22587i
$990$		0			0
$991$		−	16.0000i		−	0.508257i	−0.967170	−	0.254128i	$-0.918211\pi$
$991$		−	16.0000i		−	0.508257i	0.967170	−	0.254128i	$-0.0817886\pi$
$992$		0			0
$993$	−12.0000	−	12.0000i	−0.380808	−	0.380808i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−28.0000	+	28.0000i	−0.886769	+	0.886769i	−0.994211	−	0.107442i	$-0.965734\pi$
$997$	−28.0000	+	28.0000i	−0.886769	+	0.886769i	0.107442	+	0.994211i	$0.465734\pi$
$998$		0			0
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.2.n.i.543.1	yes	2
4.3	odd	2		800.2.n.d.543.1	yes	2
5.2	odd	4		800.2.n.d.607.1	yes	2
5.3	odd	4		800.2.n.g.607.1	yes	2
5.4	even	2		800.2.n.b.543.1	✓	2
8.3	odd	2		1600.2.n.i.1343.1		2
8.5	even	2		1600.2.n.d.1343.1		2
20.3	even	4		800.2.n.b.607.1	yes	2
20.7	even	4	inner	800.2.n.i.607.1	yes	2
20.19	odd	2		800.2.n.g.543.1	yes	2
40.3	even	4		1600.2.n.k.1407.1		2
40.13	odd	4		1600.2.n.f.1407.1		2
40.19	odd	2		1600.2.n.f.1343.1		2
40.27	even	4		1600.2.n.d.1407.1		2
40.29	even	2		1600.2.n.k.1343.1		2
40.37	odd	4		1600.2.n.i.1407.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
800.2.n.b.543.1	✓	2	5.4	even	2
800.2.n.b.607.1	yes	2	20.3	even	4
800.2.n.d.543.1	yes	2	4.3	odd	2
800.2.n.d.607.1	yes	2	5.2	odd	4
800.2.n.g.543.1	yes	2	20.19	odd	2
800.2.n.g.607.1	yes	2	5.3	odd	4
800.2.n.i.543.1	yes	2	1.1	even	1	trivial
800.2.n.i.607.1	yes	2	20.7	even	4	inner
1600.2.n.d.1343.1		2	8.5	even	2
1600.2.n.d.1407.1		2	40.27	even	4
1600.2.n.f.1343.1		2	40.19	odd	2
1600.2.n.f.1407.1		2	40.13	odd	4
1600.2.n.i.1343.1		2	8.3	odd	2
1600.2.n.i.1407.1		2	40.37	odd	4
1600.2.n.k.1343.1		2	40.29	even	2
1600.2.n.k.1407.1		2	40.3	even	4

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	800.2.n.i.543.1

Level	$800$
Weight	$2$
Character	800.543
Analytic conductor	$6.388$
Analytic rank	$0$
Dimension	$2$
Inner twists	$2$