LMFDB - Embedded newform 840.2.u.d.629.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [840,2,Mod(629,840)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1, 1, 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("840.629"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	840.u (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,16,0,0,0,0,0,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	no
Analytic conductor:	$6.70743376979$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.2517630976.5
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{6} + 11x^{4} + 4x^{2} + 4 $ x^8 - 2x^6 + 11x^4 + 4*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			629.8
Root			$0.435132 - 0.629640i$ of defining polynomial
Character	$\chi$	$=$	840.629
Dual form			840.2.u.d.629.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.41421 q^{2} +(1.48563 + 0.890446i) q^{3} +2.00000 q^{4} +(-0.615370 - 2.14973i) q^{5} +(2.10100 + 1.25928i) q^{6} -2.64575i q^{7} +2.82843 q^{8} +(1.41421 + 2.64575i) q^{9} +(-0.870264 - 3.04017i) q^{10} +(2.97127 + 1.78089i) q^{12} +0.737669i q^{13} -3.74166i q^{14} +(1.00000 - 3.74166i) q^{15} +4.00000 q^{16} +(2.00000 + 3.74166i) q^{18} -5.43275 q^{19} +(-1.23074 - 4.29945i) q^{20} +(2.35590 - 3.93062i) q^{21} +6.00000 q^{23} +(4.20201 + 2.51856i) q^{24} +(-4.24264 + 2.64575i) q^{25} +1.04322i q^{26} +(-0.254894 + 5.18990i) q^{27} -5.29150i q^{28} +(1.41421 - 5.29150i) q^{30} +5.65685 q^{32} +(-5.68764 + 1.62811i) q^{35} +(2.82843 + 5.29150i) q^{36} -7.68306 q^{38} +(-0.656854 + 1.09591i) q^{39} +(-1.74053 - 6.08034i) q^{40} +(3.33174 - 5.55873i) q^{42} +(4.81738 - 4.66829i) q^{45} +8.48528 q^{46} +(5.94253 + 3.56178i) q^{48} -7.00000 q^{49} +(-6.00000 + 3.74166i) q^{50} +1.47534i q^{52} +(-0.360475 + 7.33962i) q^{54} -7.48331i q^{56} +(-8.07107 - 4.83756i) q^{57} +13.9416i q^{59} +(2.00000 - 7.48331i) q^{60} -10.6543 q^{61} +(7.00000 - 3.74166i) q^{63} +8.00000 q^{64} +(1.58579 - 0.453939i) q^{65} +(8.91380 + 5.34267i) q^{69} +(-8.04354 + 2.30250i) q^{70} +15.8745i q^{71} +(4.00000 + 7.48331i) q^{72} +(-8.65891 + 0.152776i) q^{75} -10.8655 q^{76} +(-0.928932 + 1.54985i) q^{78} -16.9706 q^{79} +(-2.46148 - 8.59890i) q^{80} +(-5.00000 + 7.48331i) q^{81} +13.8368 q^{83} +(4.71179 - 7.86123i) q^{84} +(6.81280 - 6.60195i) q^{90} +1.95169 q^{91} +12.0000 q^{92} +(3.34315 + 11.6789i) q^{95} +(8.40401 + 5.03712i) q^{96} -9.89949 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{4} + 8 q^{15} + 32 q^{16} + 16 q^{18} + 48 q^{23} + 40 q^{39} - 56 q^{49} - 48 q^{50} - 8 q^{57} + 16 q^{60} + 56 q^{63} + 64 q^{64} + 24 q^{65} + 32 q^{72} - 64 q^{78} - 40 q^{81} + 96 q^{92}+ \cdots + 72 q^{95}+O(q^{100}) $ 8 * q + 16 * q^4 + 8 * q^15 + 32 * q^16 + 16 * q^18 + 48 * q^23 + 40 * q^39 - 56 * q^49 - 48 * q^50 - 8 * q^57 + 16 * q^60 + 56 * q^63 + 64 * q^64 + 24 * q^65 + 32 * q^72 - 64 * q^78 - 40 * q^81 + 96 * q^92 + 72 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/840\mathbb{Z}\right)^\times$.

$n$	$241$	$281$	$337$	$421$	$631$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$	$1$

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 $

$2$	1.41421			1.00000
$2$	1.41421			1.00000
$3$	1.48563	+	0.890446i	0.857731	+	0.514099i
$3$	1.48563	+	0.890446i	0.857731	+	0.514099i
$4$	2.00000			1.00000
$5$	−0.615370	−	2.14973i	−0.275202	−	0.961387i
$5$	−0.615370	−	2.14973i	−0.275202	−	0.961387i
$6$	2.10100	+	1.25928i	0.857731	+	0.514099i
$7$		−	2.64575i		−	1.00000i
$7$		−	2.64575i		−	1.00000i
$8$	2.82843			1.00000
$9$	1.41421	+	2.64575i	0.471405	+	0.881917i
$10$	−0.870264	−	3.04017i	−0.275202	−	0.961387i
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	2.97127	+	1.78089i	0.857731	+	0.514099i
$13$			0.737669i			0.204593i	0.994754	+	0.102296i	$0.0326190\pi$
$13$			0.737669i			0.204593i	−0.994754	+	0.102296i	$0.967381\pi$
$14$		−	3.74166i		−	1.00000i
$15$	1.00000	−	3.74166i	0.258199	−	0.966092i
$16$	4.00000			1.00000
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	2.00000	+	3.74166i	0.471405	+	0.881917i
$19$	−5.43275			−1.24636			−0.623179	−	0.782080i	$-0.714159\pi$
$19$	−5.43275			−1.24636			−0.623179	+	0.782080i	$0.714159\pi$
$20$	−1.23074	−	4.29945i	−0.275202	−	0.961387i
$21$	2.35590	−	3.93062i	0.514099	−	0.857731i
$22$		0			0
$23$	6.00000			1.25109			0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000			1.25109			0.625543	+	0.780189i	$0.284877\pi$
$24$	4.20201	+	2.51856i	0.857731	+	0.514099i
$25$	−4.24264	+	2.64575i	−0.848528	+	0.529150i
$26$			1.04322i			0.204593i
$27$	−0.254894	+	5.18990i	−0.0490545	+	0.998796i
$28$		−	5.29150i		−	1.00000i
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$	1.41421	−	5.29150i	0.258199	−	0.966092i
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	5.65685			1.00000
$33$		0			0
$34$		0			0
$35$	−5.68764	+	1.62811i	−0.961387	+	0.275202i
$36$	2.82843	+	5.29150i	0.471405	+	0.881917i
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$	−7.68306			−1.24636
$39$	−0.656854	+	1.09591i	−0.105181	+	0.175485i
$40$	−1.74053	−	6.08034i	−0.275202	−	0.961387i
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$	3.33174	−	5.55873i	0.514099	−	0.857731i
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$	4.81738	−	4.66829i	0.718132	−	0.695907i
$46$	8.48528			1.25109
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$	5.94253	+	3.56178i	0.857731	+	0.514099i
$49$	−7.00000			−1.00000
$50$	−6.00000	+	3.74166i	−0.848528	+	0.529150i
$51$		0			0
$52$			1.47534i			0.204593i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	−0.360475	+	7.33962i	−0.0490545	+	0.998796i
$55$		0			0
$56$		−	7.48331i		−	1.00000i
$57$	−8.07107	−	4.83756i	−1.06904	−	0.640751i
$58$		0			0
$59$			13.9416i			1.81504i	0.420010	+	0.907519i	$0.362026\pi$
$59$			13.9416i			1.81504i	−0.420010	+	0.907519i	$0.637974\pi$
$60$	2.00000	−	7.48331i	0.258199	−	0.966092i
$61$	−10.6543			−1.36415			−0.682073	−	0.731284i	$-0.738922\pi$
$61$	−10.6543			−1.36415			−0.682073	+	0.731284i	$0.738922\pi$
$62$		0			0
$63$	7.00000	−	3.74166i	0.881917	−	0.471405i
$64$	8.00000			1.00000
$65$	1.58579	−	0.453939i	0.196693	−	0.0563042i
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$		0			0
$69$	8.91380	+	5.34267i	1.07310	+	0.643182i
$70$	−8.04354	+	2.30250i	−0.961387	+	0.275202i
$71$			15.8745i			1.88396i	0.335673	+	0.941979i	$0.391036\pi$
$71$			15.8745i			1.88396i	−0.335673	+	0.941979i	$0.608964\pi$
$72$	4.00000	+	7.48331i	0.471405	+	0.881917i
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$		0			0
$75$	−8.65891	+	0.152776i	−0.999844	+	0.0176411i
$76$	−10.8655			−1.24636
$77$		0			0
$78$	−0.928932	+	1.54985i	−0.105181	+	0.175485i
$79$	−16.9706			−1.90934			−0.954669	−	0.297670i	$-0.903790\pi$
$79$	−16.9706			−1.90934			−0.954669	+	0.297670i	$0.903790\pi$
$80$	−2.46148	−	8.59890i	−0.275202	−	0.961387i
$81$	−5.00000	+	7.48331i	−0.555556	+	0.831479i
$82$		0			0
$83$	13.8368			1.51878			0.759391	−	0.650635i	$-0.225497\pi$
$83$	13.8368			1.51878			0.759391	+	0.650635i	$0.225497\pi$
$84$	4.71179	−	7.86123i	0.514099	−	0.857731i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$	6.81280	−	6.60195i	0.718132	−	0.695907i
$91$	1.95169			0.204593
$92$	12.0000			1.25109
$93$		0			0
$94$		0			0
$95$	3.34315	+	11.6789i	0.343000	+	1.19823i
$96$	8.40401	+	5.03712i	0.857731	+	0.514099i
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$	−9.89949			−1.00000
$99$		0			0
$100$	−8.48528	+	5.29150i	−0.848528	+	0.529150i
$101$		−	6.38589i		−	0.635420i	−0.948188	−	0.317710i	$-0.897086\pi$
$101$		−	6.38589i		−	0.635420i	0.948188	−	0.317710i	$-0.102914\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$			2.08644i			0.204593i
$105$	−9.89949	−	2.64575i	−0.966092	−	0.258199i
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$	−0.509789	+	10.3798i	−0.0490545	+	0.998796i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		−	10.5830i		−	1.00000i
$113$	−14.1421			−1.33038			−0.665190	−	0.746674i	$-0.731650\pi$
$113$	−14.1421			−1.33038			−0.665190	+	0.746674i	$0.731650\pi$
$114$	−11.4142	−	6.84135i	−1.06904	−	0.640751i
$115$	−3.69222	−	12.8984i	−0.344301	−	1.20278i
$116$		0			0
$117$	−1.95169	+	1.04322i	−0.180434	+	0.0964459i
$118$			19.7164i			1.81504i
$119$		0			0
$120$	2.82843	−	10.5830i	0.258199	−	0.966092i
$121$	−11.0000			−1.00000
$122$	−15.0675			−1.36415
$123$		0			0
$124$		0			0
$125$	8.29843	+	7.49240i	0.742234	+	0.670141i
$126$	9.89949	−	5.29150i	0.881917	−	0.471405i
$127$		−	22.4499i		−	1.99211i	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$		−	22.4499i		−	1.99211i	0.0887357	−	0.996055i	$-0.471717\pi$
$128$	11.3137			1.00000
$129$		0			0
$130$	2.24264	−	0.641967i	0.196693	−	0.0563042i
$131$		−	22.5405i		−	1.96937i	−0.174341	−	0.984685i	$-0.555779\pi$
$131$		−	22.5405i		−	1.96937i	0.174341	−	0.984685i	$-0.444221\pi$
$132$		0			0
$133$			14.3737i			1.24636i
$134$		0			0
$135$	11.3137	−	2.64575i	0.973729	−	0.227710i
$136$		0			0
$137$	−18.0000			−1.53784			−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000			−1.53784			−0.768922	+	0.639343i	$0.779207\pi$
$138$	12.6060	+	7.55568i	1.07310	+	0.643182i
$139$	23.2603			1.97292			0.986458	−	0.164012i	$-0.0524434\pi$
$139$	23.2603			1.97292			0.986458	+	0.164012i	$0.0524434\pi$
$140$	−11.3753	+	3.25623i	−0.961387	+	0.275202i
$141$		0			0
$142$			22.4499i			1.88396i
$143$		0			0
$144$	5.65685	+	10.5830i	0.471405	+	0.881917i
$145$		0			0
$146$		0			0
$147$	−10.3994	−	6.23312i	−0.857731	−	0.514099i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$	−12.2455	+	0.216058i	−0.999844	+	0.0176411i
$151$	10.0000			0.813788			0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000			0.813788			0.406894	+	0.913475i	$0.366612\pi$
$152$	−15.3661			−1.24636
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−1.31371	+	2.19181i	−0.105181	+	0.175485i
$157$		−	25.0590i		−	1.99993i	−0.00842626	−	0.999964i	$-0.502682\pi$
$157$		−	25.0590i		−	1.99993i	0.00842626	−	0.999964i	$-0.497318\pi$
$158$	−24.0000			−1.90934
$159$		0			0
$160$	−3.48106	−	12.1607i	−0.275202	−	0.961387i
$161$		−	15.8745i		−	1.25109i
$162$	−7.07107	+	10.5830i	−0.555556	+	0.831479i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$		0			0
$165$		0			0
$166$	19.5681			1.51878
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$	6.66348	−	11.1175i	0.514099	−	0.857731i
$169$	12.4558			0.958142
$170$		0			0
$171$	−7.68306	−	14.3737i	−0.587538	−	1.09918i
$172$		0			0
$173$	23.9813			1.82326			0.911632	−	0.411007i	$-0.134823\pi$
$173$	23.9813			1.82326			0.911632	+	0.411007i	$0.134823\pi$
$174$		0			0
$175$	7.00000	+	11.2250i	0.529150	+	0.848528i
$176$		0			0
$177$	−12.4142	+	20.7121i	−0.933109	+	1.55681i
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	9.63475	−	9.33657i	0.718132	−	0.695907i
$181$	−25.0009			−1.85830			−0.929150	−	0.369703i	$-0.879460\pi$
$181$	−25.0009			−1.85830			−0.929150	+	0.369703i	$0.879460\pi$
$182$	2.76011			0.204593
$183$	−15.8284	−	9.48710i	−1.17007	−	0.701307i
$184$	16.9706			1.25109
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	13.7312	+	0.674387i	0.998796	+	0.0490545i
$190$	4.72792	+	16.5165i	0.343000	+	1.19823i
$191$		−	15.8745i		−	1.14864i	−0.818631	−	0.574320i	$-0.805267\pi$
$191$		−	15.8745i		−	1.14864i	0.818631	−	0.574320i	$-0.194733\pi$
$192$	11.8851	+	7.12356i	0.857731	+	0.514099i
$193$			26.4575i			1.90445i	0.305392	+	0.952227i	$0.401213\pi$
$193$			26.4575i			1.90445i	−0.305392	+	0.952227i	$0.598787\pi$
$194$		0			0
$195$	2.76011	+	0.737669i	0.197655	+	0.0528256i
$196$	−14.0000			−1.00000
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	−12.0000	+	7.48331i	−0.848528	+	0.529150i
$201$		0			0
$202$		−	9.03102i		−	0.635420i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	8.48528	+	15.8745i	0.589768	+	1.10335i
$208$			2.95068i			0.204593i
$209$		0			0
$210$	−14.0000	−	3.74166i	−0.966092	−	0.258199i
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$	−14.1354	+	23.5837i	−0.968541	+	1.61593i
$214$		0			0
$215$		0			0
$216$	−0.720950	+	14.6792i	−0.0490545	+	0.998796i
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		−	14.9666i		−	1.00000i
$225$	−13.0000	−	7.48331i	−0.866667	−	0.498888i
$226$	−20.0000			−1.33038
$227$	29.2029			1.93826			0.969132	−	0.246544i	$-0.0792950\pi$
$227$	29.2029			1.93826			0.969132	+	0.246544i	$0.0792950\pi$
$228$	−16.1421	−	9.67513i	−1.06904	−	0.640751i
$229$	−14.5577			−0.962000			−0.481000	−	0.876720i	$-0.659726\pi$
$229$	−14.5577			−0.962000			−0.481000	+	0.876720i	$0.659726\pi$
$230$	−5.22158	−	18.2410i	−0.344301	−	1.20278i
$231$		0			0
$232$		0			0
$233$	−6.00000			−0.393073			−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000			−0.393073			−0.196537	+	0.980497i	$0.562969\pi$
$234$	−2.76011	+	1.47534i	−0.180434	+	0.0964459i
$235$		0			0
$236$			27.8832i			1.81504i
$237$	−25.2120	−	15.1114i	−1.63770	−	0.981588i
$238$		0			0
$239$		−	7.48331i		−	0.484055i	−0.970269	−	0.242028i	$-0.922188\pi$
$239$		−	7.48331i		−	0.484055i	0.970269	−	0.242028i	$-0.0778125\pi$
$240$	4.00000	−	14.9666i	0.258199	−	0.966092i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	−15.5563			−1.00000
$243$	−14.0917	+	6.66524i	−0.903980	+	0.427575i
$244$	−21.3087			−1.36415
$245$	4.30759	+	15.0481i	0.275202	+	0.961387i
$246$		0			0
$247$		−	4.00757i		−	0.254995i
$248$		0			0
$249$	20.5563	+	12.3209i	1.30271	+	0.780804i
$250$	11.7358	+	10.5959i	0.742234	+	0.670141i
$251$			1.16979i			0.0738362i	0.999318	+	0.0369181i	$0.0117541\pi$
$251$			1.16979i			0.0738362i	−0.999318	+	0.0369181i	$0.988246\pi$
$252$	14.0000	−	7.48331i	0.881917	−	0.471405i
$253$		0			0
$254$		−	31.7490i		−	1.99211i
$255$		0			0
$256$	16.0000			1.00000
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$		0			0
$260$	3.17157	−	0.907878i	0.196693	−	0.0563042i
$261$		0			0
$262$		−	31.8771i		−	1.96937i
$263$	28.2843			1.74408			0.872041	−	0.489432i	$-0.162796\pi$
$263$	28.2843			1.74408			0.872041	+	0.489432i	$0.162796\pi$
$264$		0			0
$265$		0			0
$266$			20.3275i			1.24636i
$267$		0			0
$268$		0			0
$269$			19.4108i			1.18350i	0.806122	+	0.591749i	$0.201562\pi$
$269$			19.4108i			1.18350i	−0.806122	+	0.591749i	$0.798438\pi$
$270$	16.0000	−	3.74166i	0.973729	−	0.227710i
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$	2.89949	+	1.73787i	0.175485	+	0.105181i
$274$	−25.4558			−1.53784
$275$		0			0
$276$	17.8276	+	10.6853i	1.07310	+	0.643182i
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$	32.8951			1.97292
$279$		0			0
$280$	−16.0871	+	4.60500i	−0.961387	+	0.275202i
$281$		−	14.9666i		−	0.892834i	−0.894825	−	0.446417i	$-0.852700\pi$
$281$		−	14.9666i		−	0.892834i	0.894825	−	0.446417i	$-0.147300\pi$
$282$		0			0
$283$			29.6640i			1.76334i	0.471863	+	0.881672i	$0.343582\pi$
$283$			29.6640i			1.76334i	−0.471863	+	0.881672i	$0.656418\pi$
$284$			31.7490i			1.88396i
$285$	−5.43275	+	20.3275i	−0.321808	+	1.20410i
$286$		0			0
$287$		0			0
$288$	8.00000	+	14.9666i	0.471405	+	0.881917i
$289$	17.0000			1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−11.6739			−0.681997			−0.340998	−	0.940064i	$-0.610765\pi$
$293$	−11.6739			−0.681997			−0.340998	+	0.940064i	$0.610765\pi$
$294$	−14.7070	−	8.81496i	−0.857731	−	0.514099i
$295$	29.9706	−	8.57922i	1.74495	−	0.499502i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			4.42602i			0.255963i
$300$	−17.3178	+	0.305553i	−0.999844	+	0.0176411i
$301$		0			0
$302$	14.1421			0.813788
$303$	5.68629	−	9.48710i	0.326669	−	0.545020i
$304$	−21.7310			−1.24636
$305$	6.55635	+	22.9039i	0.375415	+	1.31147i
$306$		0			0
$307$			32.6147i			1.86142i	0.365758	+	0.930710i	$0.380810\pi$
$307$			32.6147i			1.86142i	−0.365758	+	0.930710i	$0.619190\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$	−1.85786	+	3.09969i	−0.105181	+	0.175485i
$313$		0			0			−	1.00000i	$-0.5\pi$
$313$		0			0				1.00000i	$0.5\pi$
$314$		−	35.4388i		−	1.99993i
$315$	−12.3511	−	12.7456i	−0.695907	−	0.718132i
$316$	−33.9411			−1.90934
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$		0			0
$320$	−4.92296	−	17.1978i	−0.275202	−	0.961387i
$321$		0			0
$322$		−	22.4499i		−	1.25109i
$323$		0			0
$324$	−10.0000	+	14.9666i	−0.555556	+	0.831479i
$325$	−1.95169	−	3.12967i	−0.108260	−	0.173603i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$	27.6735			1.51878
$333$		0			0
$334$		0			0
$335$		0			0
$336$	9.42359	−	15.7225i	0.514099	−	0.857731i
$337$		−	26.4575i		−	1.44123i	−0.693334	−	0.720616i	$-0.743859\pi$
$337$		−	26.4575i		−	1.44123i	0.693334	−	0.720616i	$-0.256141\pi$
$338$	17.6152			0.958142
$339$	−21.0100	−	12.5928i	−1.14111	−	0.683947i
$340$		0			0
$341$		0			0
$342$	−10.8655	−	20.3275i	−0.587538	−	1.09918i
$343$			18.5203i			1.00000i
$344$		0			0
$345$	6.00000	−	22.4499i	0.323029	−	1.20866i
$346$	33.9147			1.82326
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$		0			0
$349$	−3.26989			−0.175033			−0.0875167	−	0.996163i	$-0.527893\pi$
$349$	−3.26989			−0.175033			−0.0875167	+	0.996163i	$0.527893\pi$
$350$	9.89949	+	15.8745i	0.529150	+	0.848528i
$351$	−3.82843	−	0.188028i	−0.204346	−	0.0100362i
$352$		0			0
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$	−17.5563	+	29.2913i	−0.933109	+	1.55681i
$355$	34.1258	−	9.76869i	1.81121	−	0.518468i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	37.4166i		−	1.97477i	−0.158334	−	0.987386i	$-0.550612\pi$
$359$		−	37.4166i		−	1.97477i	0.158334	−	0.987386i	$-0.449388\pi$
$360$	13.6256	−	13.2039i	0.718132	−	0.695907i
$361$	10.5147			0.553406
$362$	−35.3566			−1.85830
$363$	−16.3420	−	9.79490i	−0.857731	−	0.514099i
$364$	3.90338			0.204593
$365$		0			0
$366$	−22.3848	−	13.4168i	−1.17007	−	0.701307i
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$	24.0000			1.25109
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$	5.65685	+	18.5203i	0.292119	+	0.956382i
$376$		0			0
$377$		0			0
$378$	19.4188	+	0.953728i	0.998796	+	0.0490545i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$	6.68629	+	23.3578i	0.343000	+	1.19823i
$381$	19.9905	−	33.3524i	1.02414	−	1.70869i
$382$		−	22.4499i		−	1.14864i
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$	16.8080	+	10.0742i	0.857731	+	0.514099i
$385$		0			0
$386$			37.4166i			1.90445i
$387$		0			0
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$	3.90338	+	1.04322i	0.197655	+	0.0528256i
$391$		0			0
$392$	−19.7990			−1.00000
$393$	20.0711	−	33.4869i	1.01245	−	1.68919i
$394$		0			0
$395$	10.4432	+	36.4821i	0.525453	+	1.83561i
$396$		0			0
$397$		−	35.7444i		−	1.79396i	−0.442072	−	0.896980i	$-0.645756\pi$
$397$		−	35.7444i		−	1.79396i	0.442072	−	0.896980i	$-0.354244\pi$
$398$		0			0
$399$	−12.7990	+	21.3540i	−0.640751	+	1.06904i
$400$	−16.9706	+	10.5830i	−0.848528	+	0.529150i
$401$			15.8745i			0.792735i	0.918092	+	0.396368i	$0.129729\pi$
$401$			15.8745i			0.792735i	−0.918092	+	0.396368i	$0.870271\pi$
$402$		0			0
$403$		0			0
$404$		−	12.7718i		−	0.635420i
$405$	19.1639	+	6.14362i	0.952263	+	0.305279i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$	−26.7414	−	16.0280i	−1.31906	−	0.790604i
$412$		0			0
$413$	36.8859			1.81504
$414$	12.0000	+	22.4499i	0.589768	+	1.10335i
$415$	−8.51472	−	29.7452i	−0.417971	−	1.46014i
$416$			4.17289i			0.204593i
$417$	34.5563	+	20.7121i	1.69223	+	1.01427i
$418$		0			0
$419$		−	18.3676i		−	0.897316i	−0.893704	−	0.448658i	$-0.851902\pi$
$419$		−	18.3676i		−	0.897316i	0.893704	−	0.448658i	$-0.148098\pi$
$420$	−19.7990	−	5.29150i	−0.966092	−	0.258199i
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$	−19.9905	+	33.3524i	−0.968541	+	1.61593i
$427$			28.1887i			1.36415i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			37.4166i			1.80229i	0.433515	+	0.901146i	$0.357273\pi$
$431$			37.4166i			1.80229i	−0.433515	+	0.901146i	$0.642727\pi$
$432$	−1.01958	+	20.7596i	−0.0490545	+	0.998796i
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−32.5965			−1.55930
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	−9.89949	−	18.5203i	−0.471405	−	0.881917i
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		−	21.1660i		−	1.00000i
$449$		−	29.9333i		−	1.41264i	−0.707894	−	0.706319i	$-0.750354\pi$
$449$		−	29.9333i		−	1.41264i	0.707894	−	0.706319i	$-0.249646\pi$
$450$	−18.3848	−	10.5830i	−0.866667	−	0.498888i
$451$		0			0
$452$	−28.2843			−1.33038
$453$	14.8563	+	8.90446i	0.698011	+	0.418368i
$454$	41.2991			1.93826
$455$	−1.20101	−	4.19560i	−0.0563042	−	0.196693i
$456$	−22.8284	−	13.6827i	−1.06904	−	0.640751i
$457$		−	5.29150i		−	0.247526i	−0.992312	−	0.123763i	$-0.960504\pi$
$457$		−	5.29150i		−	0.247526i	0.992312	−	0.123763i	$-0.0394963\pi$
$458$	−20.5877			−0.962000
$459$		0			0
$460$	−7.38443	−	25.7967i	−0.344301	−	1.20278i
$461$		−	25.6701i		−	1.19558i	−0.801654	−	0.597789i	$-0.796046\pi$
$461$		−	25.6701i		−	1.19558i	0.801654	−	0.597789i	$-0.203954\pi$
$462$		0			0
$463$			26.4575i			1.22958i	0.788689	+	0.614792i	$0.210760\pi$
$463$			26.4575i			1.22958i	−0.788689	+	0.614792i	$0.789240\pi$
$464$		0			0
$465$		0			0
$466$	−8.48528			−0.393073
$467$	21.2212			0.982000			0.491000	−	0.871160i	$-0.336632\pi$
$467$	21.2212			0.982000			0.491000	+	0.871160i	$0.336632\pi$
$468$	−3.90338	+	2.08644i	−0.180434	+	0.0964459i
$469$		0			0
$470$		0			0
$471$	22.3137	−	37.2285i	1.02816	−	1.71540i
$472$			39.4327i			1.81504i
$473$		0			0
$474$	−35.6552	−	21.3707i	−1.63770	−	0.981588i
$475$	23.0492	−	14.3737i	1.05757	−	0.659510i
$476$		0			0
$477$		0			0
$478$		−	10.5830i		−	0.484055i
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$	5.65685	−	21.1660i	0.258199	−	0.966092i
$481$		0			0
$482$		0			0
$483$	14.1354	−	23.5837i	0.643182	−	1.07310i
$484$	−22.0000			−1.00000
$485$		0			0
$486$	−19.9286	+	9.42607i	−0.903980	+	0.427575i
$487$			22.4499i			1.01730i	0.860972	+	0.508652i	$0.169856\pi$
$487$			22.4499i			1.01730i	−0.860972	+	0.508652i	$0.830144\pi$
$488$	−30.1350			−1.36415
$489$		0			0
$490$	6.09185	+	21.2812i	0.275202	+	0.961387i
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		0			0
$494$		−	5.66756i		−	0.254995i
$495$		0			0
$496$		0			0
$497$	42.0000			1.88396
$498$	29.0711	+	17.4244i	1.30271	+	0.780804i
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$	16.5969	+	14.9848i	0.742234	+	0.670141i
$501$		0			0
$502$			1.65433i			0.0738362i
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$	19.7990	−	10.5830i	0.881917	−	0.471405i
$505$	−13.7279	+	3.92969i	−0.610885	+	0.174869i
$506$		0			0
$507$	18.5048	+	11.0913i	0.821828	+	0.492580i
$508$		−	44.8999i		−	1.99211i
$509$		−	8.72547i		−	0.386750i	−0.981125	−	0.193375i	$-0.938057\pi$
$509$		−	8.72547i		−	0.386750i	0.981125	−	0.193375i	$-0.0619433\pi$
$510$		0			0
$511$		0			0
$512$	22.6274			1.00000
$513$	1.38478	−	28.1954i	0.0611394	−	1.24486i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	35.6274	+	21.3540i	1.56387	+	0.937338i
$520$	4.48528	−	1.28393i	0.196693	−	0.0563042i
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$		−	44.7754i		−	1.95789i	−0.204120	−	0.978946i	$-0.565433\pi$
$523$		−	44.7754i		−	1.95789i	0.204120	−	0.978946i	$-0.434567\pi$
$524$		−	45.0810i		−	1.96937i
$525$	0.404208	+	22.9093i	0.0176411	+	0.999844i
$526$	40.0000			1.74408
$527$		0			0
$528$		0			0
$529$	13.0000			0.565217
$530$		0			0
$531$	−36.8859	+	19.7164i	−1.60071	+	0.855617i
$532$			28.7474i			1.24636i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$			27.4510i			1.18350i
$539$		0			0
$540$	22.6274	−	5.29150i	0.973729	−	0.227710i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	−37.1421	−	22.2619i	−1.59392	−	0.955350i
$544$		0			0
$545$		0			0
$546$	4.10051	+	2.45772i	0.175485	+	0.105181i
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$	−36.0000			−1.53784
$549$	−15.0675	−	28.1887i	−0.643065	−	1.20306i
$550$		0			0
$551$		0			0
$552$	25.2120	+	15.1114i	1.07310	+	0.643182i
$553$			44.8999i			1.90934i
$554$		0			0
$555$		0			0
$556$	46.5207			1.97292
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$	−22.7506	+	6.51246i	−0.961387	+	0.275202i
$561$		0			0
$562$		−	21.1660i		−	0.892834i
$563$	−39.0488			−1.64571			−0.822855	−	0.568251i	$-0.807620\pi$
$563$	−39.0488			−1.64571			−0.822855	+	0.568251i	$0.807620\pi$
$564$		0			0
$565$	8.70264	+	30.4017i	0.366123	+	1.27901i
$566$			41.9513i			1.76334i
$567$	19.7990	+	13.2288i	0.831479	+	0.555556i
$568$			44.8999i			1.88396i
$569$			47.6235i			1.99648i	0.0592869	+	0.998241i	$0.481117\pi$
$569$			47.6235i			1.99648i	−0.0592869	+	0.998241i	$0.518883\pi$
$570$	−7.68306	+	28.7474i	−0.321808	+	1.20410i
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$	14.1354	−	23.5837i	0.590514	−	0.985223i
$574$		0			0
$575$	−25.4558	+	15.8745i	−1.06158	+	0.662013i
$576$	11.3137	+	21.1660i	0.471405	+	0.881917i
$577$		0			0			−	1.00000i	$-0.5\pi$
$577$		0			0				1.00000i	$0.5\pi$
$578$	24.0416			1.00000
$579$	−23.5590	+	39.3062i	−0.979078	+	1.63351i
$580$		0			0
$581$		−	36.6086i		−	1.51878i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	3.44365	+	3.55363i	0.142377	+	0.146925i
$586$	−16.5094			−0.681997
$587$	3.39359			0.140068			0.0700342	−	0.997545i	$-0.477689\pi$
$587$	3.39359			0.140068			0.0700342	+	0.997545i	$0.477689\pi$
$588$	−20.7989	−	12.4662i	−0.857731	−	0.514099i
$589$		0			0
$590$	42.3848	−	12.1329i	1.74495	−	0.499502i
$591$		0			0
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$			6.25933i			0.255963i
$599$		−	47.6235i		−	1.94584i	−0.231133	−	0.972922i	$-0.574243\pi$
$599$		−	47.6235i		−	1.94584i	0.231133	−	0.972922i	$-0.425757\pi$
$600$	−24.4911	+	0.432117i	−0.999844	+	0.0176411i
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$	20.0000			0.813788
$605$	6.76906	+	23.6470i	0.275202	+	0.961387i
$606$	8.04163	−	13.4168i	0.326669	−	0.545020i
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$	−30.7322			−1.24636
$609$		0			0
$610$	9.27208	+	32.3910i	0.375415	+	1.31147i
$611$		0			0
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$			46.1242i			1.86142i
$615$		0			0
$616$		0			0
$617$	14.1421			0.569341			0.284670	−	0.958625i	$-0.408116\pi$
$617$	14.1421			0.569341			0.284670	+	0.958625i	$0.408116\pi$
$618$		0			0
$619$	48.4724			1.94827			0.974135	−	0.225968i	$-0.0725544\pi$
$619$	48.4724			1.94827			0.974135	+	0.225968i	$0.0725544\pi$
$620$		0			0
$621$	−1.52937	+	31.1394i	−0.0613714	+	1.24958i
$622$		0			0
$623$		0			0
$624$	−2.62742	+	4.38362i	−0.105181	+	0.175485i
$625$	11.0000	−	22.4499i	0.440000	−	0.897998i
$626$		0			0
$627$		0			0
$628$		−	50.1181i		−	1.99993i
$629$		0			0
$630$	−17.4671	−	18.0250i	−0.695907	−	0.718132i
$631$	33.9411			1.35117			0.675587	−	0.737280i	$-0.263890\pi$
$631$	33.9411			1.35117			0.675587	+	0.737280i	$0.263890\pi$
$632$	−48.0000			−1.90934
$633$		0			0
$634$		0			0
$635$	−48.2612	+	13.8150i	−1.91519	+	0.548232i
$636$		0			0
$637$		−	5.16368i		−	0.204593i
$638$		0			0
$639$	−42.0000	+	22.4499i	−1.66149	+	0.888106i
$640$	−6.96211	−	24.3214i	−0.275202	−	0.961387i
$641$			15.8745i			0.627005i	0.949587	+	0.313503i	$0.101502\pi$
$641$			15.8745i			0.627005i	−0.949587	+	0.313503i	$0.898498\pi$
$642$		0			0
$643$		−	23.4047i		−	0.922992i	−0.887142	−	0.461496i	$-0.847313\pi$
$643$		−	23.4047i		−	0.922992i	0.887142	−	0.461496i	$-0.152687\pi$
$644$		−	31.7490i		−	1.25109i
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	−14.1421	+	21.1660i	−0.555556	+	0.831479i
$649$		0			0
$650$	−2.76011	−	4.42602i	−0.108260	−	0.173603i
$651$		0			0
$652$		0			0
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$	−48.4558	+	13.8707i	−1.89333	+	0.541974i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	7.59560			0.295434			0.147717	−	0.989030i	$-0.452807\pi$
$661$	7.59560			0.295434			0.147717	+	0.989030i	$0.452807\pi$
$662$		0			0
$663$		0			0
$664$	39.1363			1.51878
$665$	30.8995	−	8.84513i	1.19823	−	0.343000i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$	13.3270	−	22.2349i	0.514099	−	0.857731i
$673$		−	44.8999i		−	1.73076i	−0.501113	−	0.865382i	$-0.667076\pi$
$673$		−	44.8999i		−	1.73076i	0.501113	−	0.865382i	$-0.332924\pi$
$674$		−	37.4166i		−	1.44123i
$675$	−12.6498	−	22.6933i	−0.486889	−	0.873464i
$676$	24.9117			0.958142
$677$	29.5015			1.13384			0.566918	−	0.823775i	$-0.308136\pi$
$677$	29.5015			1.13384			0.566918	+	0.823775i	$0.308136\pi$
$678$	−29.7127	−	17.8089i	−1.14111	−	0.683947i
$679$		0			0
$680$		0			0
$681$	43.3848	+	26.0036i	1.66251	+	0.996459i
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$	−15.3661	−	28.7474i	−0.587538	−	1.09918i
$685$	11.0767	+	38.6951i	0.423217	+	1.47846i
$686$			26.1916i			1.00000i
$687$	−21.6274	−	12.9628i	−0.825137	−	0.494563i
$688$		0			0
$689$		0			0
$690$	8.48528	−	31.7490i	0.323029	−	1.20866i
$691$	33.7035			1.28214			0.641071	−	0.767482i	$-0.278490\pi$
$691$	33.7035			1.28214			0.641071	+	0.767482i	$0.278490\pi$
$692$	47.9626			1.82326
$693$		0			0
$694$		0			0
$695$	−14.3137	−	50.0034i	−0.542950	−	1.89674i
$696$		0			0
$697$		0			0
$698$	−4.62433			−0.175033
$699$	−8.91380	−	5.34267i	−0.337151	−	0.202078i
$700$	14.0000	+	22.4499i	0.529150	+	0.848528i
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$	−5.41421	−	0.265911i	−0.204346	−	0.0100362i
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−16.8955			−0.635420
$708$	−24.8284	+	41.4241i	−0.933109	+	1.55681i
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$	48.2612	−	13.8150i	1.81121	−	0.518468i
$711$	−24.0000	−	44.8999i	−0.900070	−	1.68388i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	6.66348	−	11.1175i	0.248852	−	0.415189i
$718$		−	52.9150i		−	1.97477i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	19.2695	−	18.6731i	0.718132	−	0.695907i
$721$		0			0
$722$	14.8701			0.553406
$723$		0			0
$724$	−50.0017			−1.85830
$725$		0			0
$726$	−23.1110	−	13.8521i	−0.857731	−	0.514099i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$	5.52021			0.204593
$729$	−26.8701	−	2.64575i	−0.995187	−	0.0979908i
$730$		0			0
$731$		0			0
$732$	−31.6569	−	18.9742i	−1.17007	−	0.701307i
$733$		−	6.99700i		−	0.258440i	−0.991616	−	0.129220i	$-0.958753\pi$
$733$		−	6.99700i		−	0.258440i	0.991616	−	0.129220i	$-0.0412474\pi$
$734$		0			0
$735$	−7.00000	+	26.1916i	−0.258199	+	0.966092i
$736$	33.9411			1.25109
$737$		0			0
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$	3.56852	−	5.95378i	0.131093	−	0.218718i
$742$		0			0
$743$	−54.0000			−1.98107			−0.990534	−	0.137268i	$-0.956168\pi$
$743$	−54.0000			−1.98107			−0.990534	+	0.137268i	$0.956168\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	19.5681	+	36.6086i	0.715960	+	1.33944i
$748$		0			0
$749$		0			0
$750$	8.00000	+	26.1916i	0.292119	+	0.956382i
$751$	−50.0000			−1.82453			−0.912263	−	0.409605i	$-0.865667\pi$
$751$	−50.0000			−1.82453			−0.912263	+	0.409605i	$0.865667\pi$
$752$		0			0
$753$	−1.04163	+	1.73787i	−0.0379591	+	0.0633316i
$754$		0			0
$755$	−6.15370	−	21.4973i	−0.223956	−	0.782365i
$756$	27.4624	+	1.34877i	0.998796	+	0.0490545i
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$		0			0
$759$		0			0
$760$	9.45584	+	33.0329i	0.343000	+	1.19823i
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$	28.2708	−	47.1674i	1.02414	−	1.70869i
$763$		0			0
$764$		−	31.7490i		−	1.14864i
$765$		0			0
$766$		0			0
$767$	−10.2843			−0.371344
$768$	23.7701	+	14.2471i	0.857731	+	0.514099i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$			52.9150i			1.90445i
$773$	−49.1933			−1.76936			−0.884681	−	0.466198i	$-0.845624\pi$
$773$	−49.1933			−1.76936			−0.884681	+	0.466198i	$0.845624\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$	5.52021	+	1.47534i	0.197655	+	0.0528256i
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−28.0000			−1.00000
$785$	−53.8701	+	15.4206i	−1.92270	+	0.550384i
$786$	28.3848	−	47.3576i	1.01245	−	1.68919i
$787$			55.4608i			1.97696i	0.151344	+	0.988481i	$0.451640\pi$
$787$			55.4608i			1.97696i	−0.151344	+	0.988481i	$0.548360\pi$
$788$		0			0
$789$	42.0201	+	25.1856i	1.49595	+	0.896631i
$790$	14.7689	+	51.5934i	0.525453	+	1.83561i
$791$			37.4166i			1.33038i
$792$		0			0
$793$		−	7.85937i		−	0.279094i
$794$		−	50.5502i		−	1.79396i
$795$		0			0
$796$		0			0
$797$	−33.8272			−1.19822			−0.599111	−	0.800666i	$-0.704479\pi$
$797$	−33.8272			−1.19822			−0.599111	+	0.800666i	$0.704479\pi$
$798$	−18.1005	+	30.1992i	−0.640751	+	1.06904i
$799$		0			0
$800$	−24.0000	+	14.9666i	−0.848528	+	0.529150i
$801$		0			0
$802$			22.4499i			0.792735i
$803$		0			0
$804$		0			0
$805$	−34.1258	+	9.76869i	−1.20278	+	0.344301i
$806$		0			0
$807$	−17.2843	+	28.8374i	−0.608435	+	1.01512i
$808$		−	18.0620i		−	0.635420i
$809$		−	47.6235i		−	1.67435i	−0.546932	−	0.837177i	$-0.684204\pi$
$809$		−	47.6235i		−	1.67435i	0.546932	−	0.837177i	$-0.315796\pi$
$810$	27.1019	+	8.68840i	0.952263	+	0.305279i
$811$	−55.4345			−1.94657			−0.973284	−	0.229604i	$-0.926257\pi$
$811$	−55.4345			−1.94657			−0.973284	+	0.229604i	$0.926257\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$	2.76011	+	5.16368i	0.0964459	+	0.180434i
$820$		0			0
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$	−37.8181	−	22.6670i	−1.31906	−	0.790604i
$823$			26.4575i			0.922251i	0.887335	+	0.461125i	$0.152554\pi$
$823$			26.4575i			0.922251i	−0.887335	+	0.461125i	$0.847446\pi$
$824$		0			0
$825$		0			0
$826$	52.1646			1.81504
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$	16.9706	+	31.7490i	0.589768	+	1.10335i
$829$	−39.7697			−1.38126			−0.690630	−	0.723208i	$-0.742667\pi$
$829$	−39.7697			−1.38126			−0.690630	+	0.723208i	$0.742667\pi$
$830$	−12.0416	−	42.0661i	−0.417971	−	1.46014i
$831$		0			0
$832$			5.90135i			0.204593i
$833$		0			0
$834$	48.8701	+	29.2913i	1.69223	+	1.01427i
$835$		0			0
$836$		0			0
$837$		0			0
$838$		−	25.9757i		−	0.897316i
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$	−28.0000	−	7.48331i	−0.966092	−	0.258199i
$841$	−29.0000			−1.00000
$842$		0			0
$843$	13.3270	−	22.2349i	0.459005	−	0.765812i
$844$		0			0
$845$	−7.66495	−	26.7766i	−0.263682	−	0.921145i
$846$		0			0
$847$			29.1033i			1.00000i
$848$		0			0
$849$	−26.4142	+	44.0699i	−0.906533	+	1.51247i
$850$		0			0
$851$		0			0
$852$	−28.2708	+	47.1674i	−0.968541	+	1.61593i
$853$			46.4297i			1.58972i	0.606790	+	0.794862i	$0.292457\pi$
$853$			46.4297i			1.58972i	−0.606790	+	0.794862i	$0.707543\pi$
$854$			39.8648i			1.36415i
$855$	−26.1716	+	25.3616i	−0.895049	+	0.867349i
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$	22.8380			0.779223			0.389612	−	0.920979i	$-0.372609\pi$
$859$	22.8380			0.779223			0.389612	+	0.920979i	$0.372609\pi$
$860$		0			0
$861$		0			0
$862$			52.9150i			1.80229i
$863$	−56.5685			−1.92562			−0.962808	−	0.270187i	$-0.912914\pi$
$863$	−56.5685			−1.92562			−0.962808	+	0.270187i	$0.912914\pi$
$864$	−1.44190	+	29.3585i	−0.0490545	+	0.998796i
$865$	−14.7574	−	51.5532i	−0.501765	−	1.75286i
$866$		0			0
$867$	25.2558	+	15.1376i	0.857731	+	0.514099i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$	−46.0984			−1.55930
$875$	19.8230	−	21.9556i	0.670141	−	0.742234i
$876$		0			0
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$		0			0
$879$	−17.3431	−	10.3950i	−0.584970	−	0.350614i
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$	−14.0000	−	26.1916i	−0.471405	−	0.881917i
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$		0			0
$885$	52.1646	+	13.9416i	1.75349	+	0.468641i
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$	−59.3970			−1.99211
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		−	29.9333i		−	1.00000i
$897$	−3.94113	+	6.57544i	−0.131590	+	0.219547i
$898$		−	42.3320i		−	1.41264i
$899$		0			0
$900$	−26.0000	−	14.9666i	−0.866667	−	0.498888i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−40.0000			−1.33038
$905$	15.3848	+	53.7450i	0.511407	+	1.78654i
$906$	21.0100	+	12.5928i	0.698011	+	0.418368i
$907$		0			0			−	1.00000i	$-0.5\pi$
$907$		0			0				1.00000i	$0.5\pi$
$908$	58.4058			1.93826
$909$	16.8955	−	9.03102i	0.560388	−	0.299540i
$910$	−1.69848	−	5.93347i	−0.0563042	−	0.196693i
$911$			52.3832i			1.73553i	0.496972	+	0.867766i	$0.334445\pi$
$911$			52.3832i			1.73553i	−0.496972	+	0.867766i	$0.665555\pi$
$912$	−32.2843	−	19.3503i	−1.06904	−	0.640751i
$913$		0			0
$914$		−	7.48331i		−	0.247526i
$915$	−10.6543	+	39.8648i	−0.352221	+	1.31789i
$916$	−29.1154			−0.962000
$917$	−59.6365			−1.96937
$918$		0			0
$919$	16.9706			0.559807			0.279904	−	0.960028i	$-0.409697\pi$
$919$	16.9706			0.559807			0.279904	+	0.960028i	$0.409697\pi$
$920$	−10.4432	−	36.4821i	−0.344301	−	1.20278i
$921$	−29.0416	+	48.4535i	−0.956954	+	1.59660i
$922$		−	36.3031i		−	1.19558i
$923$	−11.7101			−0.385444
$924$		0			0
$925$		0			0
$926$			37.4166i			1.22958i
$927$		0			0
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	38.0292			1.24636
$932$	−12.0000			−0.393073
$933$		0			0
$934$	30.0113			0.982000
$935$		0			0
$936$	−5.52021	+	2.95068i	−0.180434	+	0.0964459i
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			27.7566i			0.904839i	0.891805	+	0.452419i	$0.149439\pi$
$941$			27.7566i			0.904839i	−0.891805	+	0.452419i	$0.850561\pi$
$942$	31.5563	−	52.6491i	1.02816	−	1.71540i
$943$		0			0
$944$			55.7663i			1.81504i
$945$	−7.00000	−	29.9333i	−0.227710	−	0.973729i
$946$		0			0
$947$		0			0		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$	−50.4241	−	30.2227i	−1.63770	−	0.981588i
$949$		0			0
$950$	32.5965	−	20.3275i	1.05757	−	0.659510i
$951$		0			0
$952$		0			0
$953$	54.0000			1.74923			0.874616	−	0.484817i	$-0.161114\pi$
$953$	54.0000			1.74923			0.874616	+	0.484817i	$0.161114\pi$
$954$		0			0
$955$	−34.1258	+	9.76869i	−1.10429	+	0.316107i
$956$		−	14.9666i		−	0.484055i
$957$		0			0
$958$		0			0
$959$			47.6235i			1.53784i
$960$	8.00000	−	29.9333i	0.258199	−	0.966092i
$961$	31.0000			1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$	56.8764	−	16.2811i	1.83092	−	0.524109i
$966$	19.9905	−	33.3524i	0.643182	−	1.07310i
$967$		−	22.4499i		−	0.721942i	−0.932577	−	0.360971i	$-0.882445\pi$
$967$		−	22.4499i		−	0.721942i	0.932577	−	0.360971i	$-0.117555\pi$
$968$	−31.1127			−1.00000
$969$		0			0
$970$		0			0
$971$		−	44.1643i		−	1.41730i	−0.705560	−	0.708650i	$-0.749305\pi$
$971$		−	44.1643i		−	1.41730i	0.705560	−	0.708650i	$-0.250695\pi$
$972$	−28.1833	+	13.3305i	−0.903980	+	0.427575i
$973$		−	61.5411i		−	1.97292i
$974$			31.7490i			1.01730i
$975$	−0.112698	−	6.38741i	−0.00360924	−	0.204561i
$976$	−42.6173			−1.36415
$977$	18.0000			0.575871			0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000			0.575871			0.287936	+	0.957650i	$0.407031\pi$
$978$		0			0
$979$		0			0
$980$	8.61517	+	30.0962i	0.275202	+	0.961387i
$981$		0			0
$982$		0			0
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		−	8.01514i		−	0.254995i
$989$		0			0
$990$		0			0
$991$	−50.9117			−1.61726			−0.808632	−	0.588315i	$-0.799791\pi$
$991$	−50.9117			−1.61726			−0.808632	+	0.588315i	$0.799791\pi$
$992$		0			0
$993$		0			0
$994$	59.3970			1.88396
$995$		0			0
$996$	41.1127	+	24.6418i	1.30271	+	0.780804i
$997$		−	63.0164i		−	1.99575i	−0.0651544	−	0.997875i	$-0.520754\pi$
$997$		−	63.0164i		−	1.99575i	0.0651544	−	0.997875i	$-0.479246\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	840.2.u.d.629.8	yes	8
3.2	odd	2		840.2.u.c.629.2	yes	8
5.4	even	2		840.2.u.c.629.1	✓	8
7.6	odd	2	inner	840.2.u.d.629.5	yes	8
8.5	even	2	inner	840.2.u.d.629.5	yes	8
15.14	odd	2	inner	840.2.u.d.629.7	yes	8
21.20	even	2		840.2.u.c.629.3	yes	8
24.5	odd	2		840.2.u.c.629.3	yes	8
35.34	odd	2		840.2.u.c.629.4	yes	8
40.29	even	2		840.2.u.c.629.4	yes	8
56.13	odd	2	CM	840.2.u.d.629.8	yes	8
105.104	even	2	inner	840.2.u.d.629.6	yes	8
120.29	odd	2	inner	840.2.u.d.629.6	yes	8
168.125	even	2		840.2.u.c.629.2	yes	8
280.69	odd	2		840.2.u.c.629.1	✓	8
840.629	even	2	inner	840.2.u.d.629.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.u.c.629.1	✓	8	5.4	even	2
840.2.u.c.629.1	✓	8	280.69	odd	2
840.2.u.c.629.2	yes	8	3.2	odd	2
840.2.u.c.629.2	yes	8	168.125	even	2
840.2.u.c.629.3	yes	8	21.20	even	2
840.2.u.c.629.3	yes	8	24.5	odd	2
840.2.u.c.629.4	yes	8	35.34	odd	2
840.2.u.c.629.4	yes	8	40.29	even	2
840.2.u.d.629.5	yes	8	7.6	odd	2	inner
840.2.u.d.629.5	yes	8	8.5	even	2	inner
840.2.u.d.629.6	yes	8	105.104	even	2	inner
840.2.u.d.629.6	yes	8	120.29	odd	2	inner
840.2.u.d.629.7	yes	8	15.14	odd	2	inner
840.2.u.d.629.7	yes	8	840.629	even	2	inner
840.2.u.d.629.8	yes	8	1.1	even	1	trivial
840.2.u.d.629.8	yes	8	56.13	odd	2	CM

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:	\( N \)	\(=\)	\( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	840.u (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +(1.48563 + 0.890446i) q^{3} +2.00000 q^{4} +(-0.615370 - 2.14973i) q^{5} +(2.10100 + 1.25928i) q^{6} -2.64575i q^{7} +2.82843 q^{8} +(1.41421 + 2.64575i) q^{9} +(-0.870264 - 3.04017i) q^{10} +(2.97127 + 1.78089i) q^{12} +0.737669i q^{13} -3.74166i q^{14} +(1.00000 - 3.74166i) q^{15} +4.00000 q^{16} +(2.00000 + 3.74166i) q^{18} -5.43275 q^{19} +(-1.23074 - 4.29945i) q^{20} +(2.35590 - 3.93062i) q^{21} +6.00000 q^{23} +(4.20201 + 2.51856i) q^{24} +(-4.24264 + 2.64575i) q^{25} +1.04322i q^{26} +(-0.254894 + 5.18990i) q^{27} -5.29150i q^{28} +(1.41421 - 5.29150i) q^{30} +5.65685 q^{32} +(-5.68764 + 1.62811i) q^{35} +(2.82843 + 5.29150i) q^{36} -7.68306 q^{38} +(-0.656854 + 1.09591i) q^{39} +(-1.74053 - 6.08034i) q^{40} +(3.33174 - 5.55873i) q^{42} +(4.81738 - 4.66829i) q^{45} +8.48528 q^{46} +(5.94253 + 3.56178i) q^{48} -7.00000 q^{49} +(-6.00000 + 3.74166i) q^{50} +1.47534i q^{52} +(-0.360475 + 7.33962i) q^{54} -7.48331i q^{56} +(-8.07107 - 4.83756i) q^{57} +13.9416i q^{59} +(2.00000 - 7.48331i) q^{60} -10.6543 q^{61} +(7.00000 - 3.74166i) q^{63} +8.00000 q^{64} +(1.58579 - 0.453939i) q^{65} +(8.91380 + 5.34267i) q^{69} +(-8.04354 + 2.30250i) q^{70} +15.8745i q^{71} +(4.00000 + 7.48331i) q^{72} +(-8.65891 + 0.152776i) q^{75} -10.8655 q^{76} +(-0.928932 + 1.54985i) q^{78} -16.9706 q^{79} +(-2.46148 - 8.59890i) q^{80} +(-5.00000 + 7.48331i) q^{81} +13.8368 q^{83} +(4.71179 - 7.86123i) q^{84} +(6.81280 - 6.60195i) q^{90} +1.95169 q^{91} +12.0000 q^{92} +(3.34315 + 11.6789i) q^{95} +(8.40401 + 5.03712i) q^{96} -9.89949 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{4} + 8 q^{15} + 32 q^{16} + 16 q^{18} + 48 q^{23} + 40 q^{39} - 56 q^{49} - 48 q^{50} - 8 q^{57} + 16 q^{60} + 56 q^{63} + 64 q^{64} + 24 q^{65} + 32 q^{72} - 64 q^{78} - 40 q^{81} + 96 q^{92}+ \cdots + 72 q^{95}+O(q^{100}) \) 8 * q + 16 * q^4 + 8 * q^15 + 32 * q^16 + 16 * q^18 + 48 * q^23 + 40 * q^39 - 56 * q^49 - 48 * q^50 - 8 * q^57 + 16 * q^60 + 56 * q^63 + 64 * q^64 + 24 * q^65 + 32 * q^72 - 64 * q^78 - 40 * q^81 + 96 * q^92 + 72 * q^95

Introduction

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Database

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