LMFDB - Embedded newform 840.2.u.d.629.4

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			629.4
Root			$1.52009 + 1.05050i$ of defining polynomial
Character	$\chi$	$=$	840.629
Dual form			840.2.u.d.629.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.41421 q^{2} +(0.890446 + 1.48563i) q^{3} +2.00000 q^{4} +(2.14973 + 0.615370i) q^{5} +(-1.25928 - 2.10100i) q^{6} -2.64575i q^{7} -2.82843 q^{8} +(-1.41421 + 2.64575i) q^{9} +(-3.04017 - 0.870264i) q^{10} +(1.78089 + 2.97127i) q^{12} -7.17327i q^{13} +3.74166i q^{14} +(1.00000 + 3.74166i) q^{15} +4.00000 q^{16} +(2.00000 - 3.74166i) q^{18} +6.81801 q^{19} +(4.29945 + 1.23074i) q^{20} +(3.93062 - 2.35590i) q^{21} +6.00000 q^{23} +(-2.51856 - 4.20201i) q^{24} +(4.24264 + 2.64575i) q^{25} +10.1445i q^{26} +(-5.18990 + 0.254894i) q^{27} -5.29150i q^{28} +(-1.41421 - 5.29150i) q^{30} -5.65685 q^{32} +(1.62811 - 5.68764i) q^{35} +(-2.82843 + 5.29150i) q^{36} -9.64212 q^{38} +(10.6569 - 6.38741i) q^{39} +(-6.08034 - 1.74053i) q^{40} +(-5.55873 + 3.33174i) q^{42} +(-4.66829 + 4.81738i) q^{45} -8.48528 q^{46} +(3.56178 + 5.94253i) q^{48} -7.00000 q^{49} +(-6.00000 - 3.74166i) q^{50} -14.3465i q^{52} +(7.33962 - 0.360475i) q^{54} +7.48331i q^{56} +(6.07107 + 10.1291i) q^{57} +6.45232i q^{59} +(2.00000 + 7.48331i) q^{60} -11.4230 q^{61} +(7.00000 + 3.74166i) q^{63} +8.00000 q^{64} +(4.41421 - 15.4206i) q^{65} +(5.34267 + 8.91380i) q^{69} +(-2.30250 + 8.04354i) q^{70} +15.8745i q^{71} +(4.00000 - 7.48331i) q^{72} +(-0.152776 + 8.65891i) q^{75} +13.6360 q^{76} +(-15.0711 + 9.03316i) q^{78} +16.9706 q^{79} +(8.59890 + 2.46148i) q^{80} +(-5.00000 - 7.48331i) q^{81} -11.8551 q^{83} +(7.86123 - 4.71179i) q^{84} +(6.60195 - 6.81280i) q^{90} -18.9787 q^{91} +12.0000 q^{92} +(14.6569 + 4.19560i) q^{95} +(-5.03712 - 8.40401i) q^{96} +9.89949 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.890446	+	1.48563i	0.514099	+	0.857731i
$3$	0.890446	+	1.48563i	0.514099	+	0.857731i
$4$	2.00000			1.00000
$5$	2.14973	+	0.615370i	0.961387	+	0.275202i
$5$	2.14973	+	0.615370i	0.961387	+	0.275202i
$6$	−1.25928	−	2.10100i	−0.514099	−	0.857731i
$7$		−	2.64575i		−	1.00000i
$7$		−	2.64575i		−	1.00000i
$8$	−2.82843			−1.00000
$9$	−1.41421	+	2.64575i	−0.471405	+	0.881917i
$10$	−3.04017	−	0.870264i	−0.961387	−	0.275202i
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	1.78089	+	2.97127i	0.514099	+	0.857731i
$13$		−	7.17327i		−	1.98951i	−0.102296	−	0.994754i	$-0.532619\pi$
$13$		−	7.17327i		−	1.98951i	0.102296	−	0.994754i	$-0.467381\pi$
$14$			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
$19$	6.81801			1.56416			0.782080	−	0.623179i	$-0.214159\pi$
$19$	6.81801			1.56416			0.782080	+	0.623179i	$0.214159\pi$
$20$	4.29945	+	1.23074i	0.961387	+	0.275202i
$21$	3.93062	−	2.35590i	0.857731	−	0.514099i
$22$		0			0
$23$	6.00000			1.25109			0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000			1.25109			0.625543	+	0.780189i	$0.284877\pi$
$24$	−2.51856	−	4.20201i	−0.514099	−	0.857731i
$25$	4.24264	+	2.64575i	0.848528	+	0.529150i
$26$			10.1445i			1.98951i
$27$	−5.18990	+	0.254894i	−0.998796	+	0.0490545i
$28$		−	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
$35$	1.62811	−	5.68764i	0.275202	−	0.961387i
$36$	−2.82843	+	5.29150i	−0.471405	+	0.881917i
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$	−9.64212			−1.56416
$39$	10.6569	−	6.38741i	1.70646	−	1.02280i
$40$	−6.08034	−	1.74053i	−0.961387	−	0.275202i
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$	−5.55873	+	3.33174i	−0.857731	+	0.514099i
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$	−4.66829	+	4.81738i	−0.695907	+	0.718132i
$46$	−8.48528			−1.25109
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$	3.56178	+	5.94253i	0.514099	+	0.857731i
$49$	−7.00000			−1.00000
$50$	−6.00000	−	3.74166i	−0.848528	−	0.529150i
$51$		0			0
$52$		−	14.3465i		−	1.98951i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	7.33962	−	0.360475i	0.998796	−	0.0490545i
$55$		0			0
$56$			7.48331i			1.00000i
$57$	6.07107	+	10.1291i	0.804133	+	1.34163i
$58$		0			0
$59$			6.45232i			0.840021i	0.907519	+	0.420010i	$0.137974\pi$
$59$			6.45232i			0.840021i	−0.907519	+	0.420010i	$0.862026\pi$
$60$	2.00000	+	7.48331i	0.258199	+	0.966092i
$61$	−11.4230			−1.46257			−0.731284	−	0.682073i	$-0.761078\pi$
$61$	−11.4230			−1.46257			−0.731284	+	0.682073i	$0.761078\pi$
$62$		0			0
$63$	7.00000	+	3.74166i	0.881917	+	0.471405i
$64$	8.00000			1.00000
$65$	4.41421	−	15.4206i	0.547516	−	1.91269i
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$		0			0
$69$	5.34267	+	8.91380i	0.643182	+	1.07310i
$70$	−2.30250	+	8.04354i	−0.275202	+	0.961387i
$71$			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
$75$	−0.152776	+	8.65891i	−0.0176411	+	0.999844i
$76$	13.6360			1.56416
$77$		0			0
$78$	−15.0711	+	9.03316i	−1.70646	+	1.02280i
$79$	16.9706			1.90934			0.954669	−	0.297670i	$-0.0962096\pi$
$79$	16.9706			1.90934			0.954669	+	0.297670i	$0.0962096\pi$
$80$	8.59890	+	2.46148i	0.961387	+	0.275202i
$81$	−5.00000	−	7.48331i	−0.555556	−	0.831479i
$82$		0			0
$83$	−11.8551			−1.30127			−0.650635	−	0.759391i	$-0.725497\pi$
$83$	−11.8551			−1.30127			−0.650635	+	0.759391i	$0.725497\pi$
$84$	7.86123	−	4.71179i	0.857731	−	0.514099i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$	6.60195	−	6.81280i	0.695907	−	0.718132i
$91$	−18.9787			−1.98951
$92$	12.0000			1.25109
$93$		0			0
$94$		0			0
$95$	14.6569	+	4.19560i	1.50376	+	0.430459i
$96$	−5.03712	−	8.40401i	−0.514099	−	0.857731i
$97$		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
$101$		−	19.0583i		−	1.89638i	−0.317710	−	0.948188i	$-0.602914\pi$
$101$		−	19.0583i		−	1.89638i	0.317710	−	0.948188i	$-0.397086\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$			20.2891i			1.98951i
$105$	9.89949	−	2.64575i	0.966092	−	0.258199i
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$	−10.3798	+	0.509789i	−0.998796	+	0.0490545i
$109$		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	$-0.268350\pi$
$113$	14.1421			1.33038			0.665190	+	0.746674i	$0.268350\pi$
$114$	−8.58579	−	14.3247i	−0.804133	−	1.34163i
$115$	12.8984	+	3.69222i	1.20278	+	0.344301i
$116$		0			0
$117$	18.9787	+	10.1445i	1.75458	+	0.937863i
$118$		−	9.12496i		−	0.840021i
$119$		0			0
$120$	−2.82843	−	10.5830i	−0.258199	−	0.966092i
$121$	−11.0000			−1.00000
$122$	16.1546			1.46257
$123$		0			0
$124$		0			0
$125$	7.49240	+	8.29843i	0.670141	+	0.742234i
$126$	−9.89949	−	5.29150i	−0.881917	−	0.471405i
$127$			22.4499i			1.99211i	0.0887357	+	0.996055i	$0.471717\pi$
$127$			22.4499i			1.99211i	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	−11.3137			−1.00000
$129$		0			0
$130$	−6.24264	+	21.8080i	−0.547516	+	1.91269i
$131$		−	3.99084i		−	0.348682i	−0.984685	−	0.174341i	$-0.944221\pi$
$131$		−	3.99084i		−	0.348682i	0.984685	−	0.174341i	$-0.0557795\pi$
$132$		0			0
$133$		−	18.0388i		−	1.56416i
$134$		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	$0.779207\pi$
$138$	−7.55568	−	12.6060i	−0.643182	−	1.07310i
$139$	3.86733			0.328023			0.164012	−	0.986458i	$-0.447557\pi$
$139$	3.86733			0.328023			0.164012	+	0.986458i	$0.447557\pi$
$140$	3.25623	−	11.3753i	0.275202	−	0.961387i
$141$		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
$147$	−6.23312	−	10.3994i	−0.514099	−	0.857731i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$	0.216058	−	12.2455i	0.0176411	−	0.999844i
$151$	10.0000			0.813788			0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000			0.813788			0.406894	+	0.913475i	$0.366612\pi$
$152$	−19.2842			−1.56416
$153$		0			0
$154$		0			0
$155$		0			0
$156$	21.3137	−	12.7748i	1.70646	−	1.02280i
$157$			0.211161i			0.0168525i	0.999964	+	0.00842626i	$0.00268219\pi$
$157$			0.211161i			0.0168525i	−0.999964	+	0.00842626i	$0.997318\pi$
$158$	−24.0000			−1.90934
$159$		0			0
$160$	−12.1607	−	3.48106i	−0.961387	−	0.275202i
$161$		−	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
$166$	16.7657			1.30127
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$	−11.1175	+	6.66348i	−0.857731	+	0.514099i
$169$	−38.4558			−2.95814
$170$		0			0
$171$	−9.64212	+	18.0388i	−0.737352	+	1.37946i
$172$		0			0
$173$	−10.8119			−0.822014			−0.411007	−	0.911632i	$-0.634823\pi$
$173$	−10.8119			−0.822014			−0.411007	+	0.911632i	$0.634823\pi$
$174$		0			0
$175$	7.00000	−	11.2250i	0.529150	−	0.848528i
$176$		0			0
$177$	−9.58579	+	5.74544i	−0.720512	+	0.431854i
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	−9.33657	+	9.63475i	−0.695907	+	0.718132i
$181$	−9.94768			−0.739405			−0.369703	−	0.929150i	$-0.620540\pi$
$181$	−9.94768			−0.739405			−0.369703	+	0.929150i	$0.620540\pi$
$182$	26.8399			1.98951
$183$	−10.1716	−	16.9704i	−0.751904	−	1.25449i
$184$	−16.9706			−1.25109
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	0.674387	+	13.7312i	0.0490545	+	0.998796i
$190$	−20.7279	−	5.93347i	−1.50376	−	0.430459i
$191$		−	15.8745i		−	1.14864i	−0.818631	−	0.574320i	$-0.805267\pi$
$191$		−	15.8745i		−	1.14864i	0.818631	−	0.574320i	$-0.194733\pi$
$192$	7.12356	+	11.8851i	0.514099	+	0.857731i
$193$			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
$195$	26.8399	−	7.17327i	1.92205	−	0.513689i
$196$	−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
$202$			26.9526i			1.89638i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−8.48528	+	15.8745i	−0.589768	+	1.10335i
$208$		−	28.6931i		−	1.98951i
$209$		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
$213$	−23.5837	+	14.1354i	−1.61593	+	0.968541i
$214$		0			0
$215$		0			0
$216$	14.6792	−	0.720950i	0.998796	−	0.0490545i
$217$		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
$227$	7.42912			0.493088			0.246544	−	0.969132i	$-0.420705\pi$
$227$	7.42912			0.493088			0.246544	+	0.969132i	$0.420705\pi$
$228$	12.1421	+	20.2581i	0.804133	+	1.34163i
$229$	26.5344			1.75344			0.876720	−	0.481000i	$-0.159726\pi$
$229$	26.5344			1.75344			0.876720	+	0.481000i	$0.159726\pi$
$230$	−18.2410	−	5.22158i	−1.20278	−	0.344301i
$231$		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	$0.562969\pi$
$234$	−26.8399	−	14.3465i	−1.75458	−	0.937863i
$235$		0			0
$236$			12.9046i			0.840021i
$237$	15.1114	+	25.2120i	0.981588	+	1.63770i
$238$		0			0
$239$			7.48331i			0.484055i	0.970269	+	0.242028i	$0.0778125\pi$
$239$			7.48331i			0.484055i	−0.970269	+	0.242028i	$0.922188\pi$
$240$	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
$243$	6.66524	−	14.0917i	0.427575	−	0.903980i
$244$	−22.8460			−1.46257
$245$	−15.0481	−	4.30759i	−0.961387	−	0.275202i
$246$		0			0
$247$		−	48.9075i		−	3.11191i
$248$		0			0
$249$	−10.5563	−	17.6124i	−0.668981	−	1.11614i
$250$	−10.5959	−	11.7358i	−0.670141	−	0.742234i
$251$		−	31.6644i		−	1.99864i	−0.0369181	−	0.999318i	$-0.511754\pi$
$251$		−	31.6644i		−	1.99864i	0.0369181	−	0.999318i	$-0.488246\pi$
$252$	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
$260$	8.82843	−	30.8411i	0.547516	−	1.91269i
$261$		0			0
$262$			5.64391i			0.348682i
$263$	−28.2843			−1.74408			−0.872041	−	0.489432i	$-0.837204\pi$
$263$	−28.2843			−1.74408			−0.872041	+	0.489432i	$0.837204\pi$
$264$		0			0
$265$		0			0
$266$			25.5107i			1.56416i
$267$		0			0
$268$		0			0
$269$		−	26.4428i		−	1.61224i	−0.591749	−	0.806122i	$-0.701562\pi$
$269$		−	26.4428i		−	1.61224i	0.591749	−	0.806122i	$-0.298438\pi$
$270$	16.0000	+	3.74166i	0.973729	+	0.227710i
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$	−16.8995	−	28.1954i	−1.02280	−	1.70646i
$274$	25.4558			1.53784
$275$		0			0
$276$	10.6853	+	17.8276i	0.643182	+	1.07310i
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$	−5.46924			−0.328023
$279$		0			0
$280$	−4.60500	+	16.0871i	−0.275202	+	0.961387i
$281$			14.9666i			0.892834i	0.894825	+	0.446417i	$0.147300\pi$
$281$			14.9666i			0.892834i	−0.894825	+	0.446417i	$0.852700\pi$
$282$		0			0
$283$			15.8759i			0.943725i	0.881672	+	0.471863i	$0.156418\pi$
$283$			15.8759i			0.943725i	−0.881672	+	0.471863i	$0.843582\pi$
$284$			31.7490i			1.88396i
$285$	6.81801	+	25.5107i	0.403864	+	1.51112i
$286$		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
$293$	−32.1826			−1.88013			−0.940064	−	0.340998i	$-0.889235\pi$
$293$	−32.1826			−1.88013			−0.940064	+	0.340998i	$0.889235\pi$
$294$	8.81496	+	14.7070i	0.514099	+	0.857731i
$295$	−3.97056	+	13.8707i	−0.231175	+	0.807585i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	43.0396i		−	2.48905i
$300$	−0.305553	+	17.3178i	−0.0176411	+	0.999844i
$301$		0			0
$302$	−14.1421			−0.813788
$303$	28.3137	−	16.9704i	1.62658	−	0.974925i
$304$	27.2720			1.56416
$305$	−24.5563	−	7.02938i	−1.40609	−	0.402501i
$306$		0			0
$307$		−	12.8172i		−	0.731515i	−0.930710	−	0.365758i	$-0.880810\pi$
$307$		−	12.8172i		−	0.731515i	0.930710	−	0.365758i	$-0.119190\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$	−30.1421	+	18.0663i	−1.70646	+	1.02280i
$313$		0			0			−	1.00000i	$-0.5\pi$
$313$		0			0				1.00000i	$0.5\pi$
$314$		−	0.298627i		−	0.0168525i
$315$	12.7456	+	12.3511i	0.718132	+	0.695907i
$316$	33.9411			1.90934
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$		0			0
$320$	17.1978	+	4.92296i	0.961387	+	0.275202i
$321$		0			0
$322$			22.4499i			1.25109i
$323$		0			0
$324$	−10.0000	−	14.9666i	−0.555556	−	0.831479i
$325$	18.9787	−	30.4336i	1.05275	−	1.68815i
$326$		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			$\pi$
$332$	−23.7103			−1.30127
$333$		0			0
$334$		0			0
$335$		0			0
$336$	15.7225	−	9.42359i	0.857731	−	0.514099i
$337$		−	26.4575i		−	1.44123i	−0.693334	−	0.720616i	$-0.743859\pi$
$337$		−	26.4575i		−	1.44123i	0.693334	−	0.720616i	$-0.256141\pi$
$338$	54.3848			2.95814
$339$	12.5928	+	21.0100i	0.683947	+	1.14111i
$340$		0			0
$341$		0			0
$342$	13.6360	−	25.5107i	0.737352	−	1.37946i
$343$			18.5203i			1.00000i
$344$		0			0
$345$	6.00000	+	22.4499i	0.323029	+	1.20866i
$346$	15.2904			0.822014
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$		0			0
$349$	−37.2197			−1.99233			−0.996163	−	0.0875167i	$-0.972107\pi$
$349$	−37.2197			−1.99233			−0.996163	+	0.0875167i	$0.972107\pi$
$350$	−9.89949	+	15.8745i	−0.529150	+	0.848528i
$351$	1.82843	+	37.2285i	0.0975942	+	1.98711i
$352$		0			0
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$	13.5563	−	8.12528i	0.720512	−	0.431854i
$355$	−9.76869	+	34.1258i	−0.518468	+	1.81121i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			37.4166i			1.97477i	0.158334	+	0.987386i	$0.449388\pi$
$359$			37.4166i			1.97477i	−0.158334	+	0.987386i	$0.550612\pi$
$360$	13.2039	−	13.6256i	0.695907	−	0.718132i
$361$	27.4853			1.44659
$362$	14.0681			0.739405
$363$	−9.79490	−	16.3420i	−0.514099	−	0.857731i
$364$	−37.9574			−1.98951
$365$		0			0
$366$	14.3848	+	23.9998i	0.751904	+	1.25449i
$367$		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
$378$	−0.953728	−	19.4188i	−0.0490545	−	0.998796i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$	29.3137	+	8.39119i	1.50376	+	0.430459i
$381$	−33.3524	+	19.9905i	−1.70869	+	1.02414i
$382$			22.4499i			1.14864i
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$	−10.0742	−	16.8080i	−0.514099	−	0.857731i
$385$		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	$0.5\pi$
$390$	−37.9574	+	10.1445i	−1.92205	+	0.513689i
$391$		0			0
$392$	19.7990			1.00000
$393$	5.92893	−	3.55363i	0.299075	−	0.179257i
$394$		0			0
$395$	36.4821	+	10.4432i	1.83561	+	0.525453i
$396$		0			0
$397$		−	17.6164i		−	0.884144i	−0.896980	−	0.442072i	$-0.854244\pi$
$397$		−	17.6164i		−	0.884144i	0.896980	−	0.442072i	$-0.145756\pi$
$398$		0			0
$399$	26.7990	−	16.0625i	1.34163	−	0.804133i
$400$	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
$404$		−	38.1167i		−	1.89638i
$405$	−6.14362	−	19.1639i	−0.305279	−	0.952263i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$	−16.0280	−	26.7414i	−0.790604	−	1.31906i
$412$		0			0
$413$	17.0712			0.840021
$414$	12.0000	−	22.4499i	0.589768	−	1.10335i
$415$	−25.4853	−	7.29529i	−1.25102	−	0.358112i
$416$			40.5782i			1.98951i
$417$	3.44365	+	5.74544i	0.168636	+	0.281355i
$418$		0			0
$419$			36.5873i			1.78741i	0.448658	+	0.893704i	$0.351902\pi$
$419$			36.5873i			1.78741i	−0.448658	+	0.893704i	$0.648098\pi$
$420$	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
$426$	33.3524	−	19.9905i	1.61593	−	0.968541i
$427$			30.2225i			1.46257i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	37.4166i		−	1.80229i	−0.433515	−	0.901146i	$-0.642727\pi$
$431$		−	37.4166i		−	1.80229i	0.433515	−	0.901146i	$-0.357273\pi$
$432$	−20.7596	+	1.01958i	−0.998796	+	0.0490545i
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	40.9081			1.95690
$438$		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	$0.249646\pi$
$449$			29.9333i			1.41264i	−0.707894	+	0.706319i	$0.750354\pi$
$450$	18.3848	−	10.5830i	0.866667	−	0.498888i
$451$		0			0
$452$	28.2843			1.33038
$453$	8.90446	+	14.8563i	0.418368	+	0.698011i
$454$	−10.5064			−0.493088
$455$	−40.7990	−	11.6789i	−1.91269	−	0.547516i
$456$	−17.1716	−	28.6493i	−0.804133	−	1.34163i
$457$		−	5.29150i		−	0.247526i	−0.992312	−	0.123763i	$-0.960504\pi$
$457$		−	5.29150i		−	0.247526i	0.992312	−	0.123763i	$-0.0394963\pi$
$458$	−37.5253			−1.75344
$459$		0			0
$460$	25.7967	+	7.38443i	1.20278	+	0.344301i
$461$		−	34.4245i		−	1.60331i	−0.597789	−	0.801654i	$-0.703954\pi$
$461$		−	34.4245i		−	1.60331i	0.597789	−	0.801654i	$-0.296046\pi$
$462$		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
$467$	−37.6518			−1.74232			−0.871160	−	0.491000i	$-0.836632\pi$
$467$	−37.6518			−1.74232			−0.871160	+	0.491000i	$0.836632\pi$
$468$	37.9574	+	20.2891i	1.75458	+	0.937863i
$469$		0			0
$470$		0			0
$471$	−0.313708	+	0.188028i	−0.0144549	+	0.00866386i
$472$		−	18.2499i		−	0.840021i
$473$		0			0
$474$	−21.3707	−	35.6552i	−0.981588	−	1.63770i
$475$	28.9264	+	18.0388i	1.32723	+	0.827675i
$476$		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
$483$	23.5837	−	14.1354i	1.07310	−	0.643182i
$484$	−22.0000			−1.00000
$485$		0			0
$486$	−9.42607	+	19.9286i	−0.427575	+	0.903980i
$487$		−	22.4499i		−	1.01730i	−0.860972	−	0.508652i	$-0.830144\pi$
$487$		−	22.4499i		−	1.01730i	0.860972	−	0.508652i	$-0.169856\pi$
$488$	32.3092			1.46257
$489$		0			0
$490$	21.2812	+	6.09185i	0.961387	+	0.275202i
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		0			0
$494$			69.1656i			3.11191i
$495$		0			0
$496$		0			0
$497$	42.0000			1.88396
$498$	14.9289	+	24.9077i	0.668981	+	1.11614i
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$	14.9848	+	16.5969i	0.670141	+	0.742234i
$501$		0			0
$502$			44.7802i			1.99864i
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$	−19.7990	−	10.5830i	−0.881917	−	0.471405i
$505$	11.7279	−	40.9702i	0.521886	−	1.82315i
$506$		0			0
$507$	−34.2428	−	57.1313i	−1.52078	−	2.53729i
$508$			44.8999i			1.99211i
$509$			44.2704i			1.96225i	0.193375	+	0.981125i	$0.438057\pi$
$509$			44.2704i			1.96225i	−0.193375	+	0.981125i	$0.561943\pi$
$510$		0			0
$511$		0			0
$512$	−22.6274			−1.00000
$513$	−35.3848	+	1.73787i	−1.56228	+	0.0767290i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−9.62742	−	16.0625i	−0.422597	−	0.705067i
$520$	−12.4853	+	43.6160i	−0.547516	+	1.91269i
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$			9.33612i			0.408240i	0.978946	+	0.204120i	$0.0654333\pi$
$523$			9.33612i			0.408240i	−0.978946	+	0.204120i	$0.934567\pi$
$524$		−	7.98169i		−	0.348682i
$525$	22.9093	+	0.404208i	0.999844	+	0.0176411i
$526$	40.0000			1.74408
$527$		0			0
$528$		0			0
$529$	13.0000			0.565217
$530$		0			0
$531$	−17.0712	−	9.12496i	−0.740829	−	0.395990i
$532$		−	36.0775i		−	1.56416i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$			37.3957i			1.61224i
$539$		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
$543$	−8.85786	−	14.7786i	−0.380127	−	0.634211i
$544$		0			0
$545$		0			0
$546$	23.8995	+	39.8743i	1.02280	+	1.70646i
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$	−36.0000			−1.53784
$549$	16.1546	−	30.2225i	0.689461	−	1.28986i
$550$		0			0
$551$		0			0
$552$	−15.1114	−	25.2120i	−0.643182	−	1.07310i
$553$		−	44.8999i		−	1.90934i
$554$		0			0
$555$		0			0
$556$	7.73467			0.328023
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$	6.51246	−	22.7506i	0.275202	−	0.961387i
$561$		0			0
$562$		−	21.1660i		−	0.892834i
$563$	26.9665			1.13650			0.568251	−	0.822855i	$-0.307620\pi$
$563$	26.9665			1.13650			0.568251	+	0.822855i	$0.307620\pi$
$564$		0			0
$565$	30.4017	+	8.70264i	1.27901	+	0.366123i
$566$		−	22.4519i		−	0.943725i
$567$	−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	$0.518883\pi$
$570$	−9.64212	−	36.0775i	−0.403864	−	1.51112i
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$	23.5837	−	14.1354i	0.985223	−	0.590514i
$574$		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
$579$	−39.3062	+	23.5590i	−1.63351	+	0.979078i
$580$		0			0
$581$			31.3657i			1.30127i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	34.5563	+	33.4869i	1.42873	+	1.38451i
$586$	45.5131			1.88013
$587$	−48.3372			−1.99509			−0.997545	−	0.0700342i	$-0.977689\pi$
$587$	−48.3372			−1.99509			−0.997545	+	0.0700342i	$0.977689\pi$
$588$	−12.4662	−	20.7989i	−0.514099	−	0.857731i
$589$		0			0
$590$	5.61522	−	19.6162i	0.231175	−	0.807585i
$591$		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
$598$			60.8672i			2.48905i
$599$		−	47.6235i		−	1.94584i	−0.231133	−	0.972922i	$-0.574243\pi$
$599$		−	47.6235i		−	1.94584i	0.231133	−	0.972922i	$-0.425757\pi$
$600$	0.432117	−	24.4911i	0.0176411	−	0.999844i
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$	20.0000			0.813788
$605$	−23.6470	−	6.76906i	−0.961387	−	0.275202i
$606$	−40.0416	+	23.9998i	−1.62658	+	0.974925i
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$	−38.5685			−1.56416
$609$		0			0
$610$	34.7279	+	9.94104i	1.40609	+	0.402501i
$611$		0			0
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$			18.1262i			0.731515i
$615$		0			0
$616$		0			0
$617$	−14.1421			−0.569341			−0.284670	−	0.958625i	$-0.591884\pi$
$617$	−14.1421			−0.569341			−0.284670	+	0.958625i	$0.591884\pi$
$618$		0			0
$619$	−11.2440			−0.451936			−0.225968	−	0.974135i	$-0.572554\pi$
$619$	−11.2440			−0.451936			−0.225968	+	0.974135i	$0.572554\pi$
$620$		0			0
$621$	−31.1394	+	1.52937i	−1.24958	+	0.0613714i
$622$		0			0
$623$		0			0
$624$	42.6274	−	25.5496i	1.70646	−	1.02280i
$625$	11.0000	+	22.4499i	0.440000	+	0.897998i
$626$		0			0
$627$		0			0
$628$			0.422323i			0.0168525i
$629$		0			0
$630$	−18.0250	−	17.4671i	−0.718132	−	0.695907i
$631$	−33.9411			−1.35117			−0.675587	−	0.737280i	$-0.736110\pi$
$631$	−33.9411			−1.35117			−0.675587	+	0.737280i	$0.736110\pi$
$632$	−48.0000			−1.90934
$633$		0			0
$634$		0			0
$635$	−13.8150	+	48.2612i	−0.548232	+	1.91519i
$636$		0			0
$637$			50.2129i			1.98951i
$638$		0			0
$639$	−42.0000	−	22.4499i	−1.66149	−	0.888106i
$640$	−24.3214	−	6.96211i	−0.961387	−	0.275202i
$641$			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
$643$			44.9913i			1.77428i	0.461496	+	0.887142i	$0.347313\pi$
$643$			44.9913i			1.77428i	−0.461496	+	0.887142i	$0.652687\pi$
$644$		−	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
$650$	−26.8399	+	43.0396i	−1.05275	+	1.68815i
$651$		0			0
$652$		0			0
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$	2.45584	−	8.57922i	0.0959578	−	0.335218i
$656$		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
$661$	−50.8557			−1.97806			−0.989030	−	0.147717i	$-0.952807\pi$
$661$	−50.8557			−1.97806			−0.989030	+	0.147717i	$0.952807\pi$
$662$		0			0
$663$		0			0
$664$	33.5314			1.30127
$665$	11.1005	−	38.7784i	0.430459	−	1.50376i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$	−22.2349	+	13.3270i	−0.857731	+	0.514099i
$673$			44.8999i			1.73076i	0.501113	+	0.865382i	$0.332924\pi$
$673$			44.8999i			1.73076i	−0.501113	+	0.865382i	$0.667076\pi$
$674$			37.4166i			1.44123i
$675$	−22.6933	−	12.6498i	−0.873464	−	0.486889i
$676$	−76.9117			−2.95814
$677$	42.8679			1.64755			0.823775	−	0.566918i	$-0.191864\pi$
$677$	42.8679			1.64755			0.823775	+	0.566918i	$0.191864\pi$
$678$	−17.8089	−	29.7127i	−0.683947	−	1.14111i
$679$		0			0
$680$		0			0
$681$	6.61522	+	11.0369i	0.253496	+	0.422937i
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$	−19.2842	+	36.0775i	−0.737352	+	1.37946i
$685$	−38.6951	−	11.0767i	−1.47846	−	0.423217i
$686$		−	26.1916i		−	1.00000i
$687$	23.6274	+	39.4204i	0.901442	+	1.50398i
$688$		0			0
$689$		0			0
$690$	−8.48528	−	31.7490i	−0.323029	−	1.20866i
$691$	40.3494			1.53496			0.767482	−	0.641071i	$-0.221510\pi$
$691$	40.3494			1.53496			0.767482	+	0.641071i	$0.221510\pi$
$692$	−21.6238			−0.822014
$693$		0			0
$694$		0			0
$695$	8.31371	+	2.37984i	0.315357	+	0.0902725i
$696$		0			0
$697$		0			0
$698$	52.6366			1.99233
$699$	−5.34267	−	8.91380i	−0.202078	−	0.337151i
$700$	14.0000	−	22.4499i	0.529150	−	0.848528i
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$	−2.58579	−	52.6491i	−0.0975942	−	1.98711i
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−50.4236			−1.89638
$708$	−19.1716	+	11.4909i	−0.720512	+	0.431854i
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$	13.8150	−	48.2612i	0.518468	−	1.81121i
$711$	−24.0000	+	44.8999i	−0.900070	+	1.68388i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−11.1175	+	6.66348i	−0.415189	+	0.248852i
$718$		−	52.9150i		−	1.97477i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	−18.6731	+	19.2695i	−0.695907	+	0.718132i
$721$		0			0
$722$	−38.8701			−1.44659
$723$		0			0
$724$	−19.8954			−0.739405
$725$		0			0
$726$	13.8521	+	23.1110i	0.514099	+	0.857731i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$	53.6799			1.98951
$729$	26.8701	−	2.64575i	0.995187	−	0.0979908i
$730$		0			0
$731$		0			0
$732$	−20.3431	−	33.9408i	−0.751904	−	1.25449i
$733$		−	53.6940i		−	1.98323i	−0.129220	−	0.991616i	$-0.541247\pi$
$733$		−	53.6940i		−	1.98323i	0.129220	−	0.991616i	$-0.458753\pi$
$734$		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
$741$	72.6586	−	43.5494i	2.66918	−	1.59983i
$742$		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
$747$	16.7657	−	31.3657i	0.613424	−	1.14761i
$748$		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
$753$	47.0416	−	28.1954i	1.71429	−	1.02750i
$754$		0			0
$755$	21.4973	+	6.15370i	0.782365	+	0.223956i
$756$	1.34877	+	27.4624i	0.0490545	+	0.998796i
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$		0			0
$759$		0			0
$760$	−41.4558	−	11.8669i	−1.50376	−	0.430459i
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$	47.1674	−	28.2708i	1.70869	−	1.02414i
$763$		0			0
$764$		−	31.7490i		−	1.14864i
$765$		0			0
$766$		0			0
$767$	46.2843			1.67123
$768$	14.2471	+	23.7701i	0.514099	+	0.857731i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$			52.9150i			1.90445i
$773$	25.9233			0.932395			0.466198	−	0.884681i	$-0.345624\pi$
$773$	25.9233			0.932395			0.466198	+	0.884681i	$0.345624\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$	53.6799	−	14.3465i	1.92205	−	0.513689i
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−28.0000			−1.00000
$785$	−0.129942	+	0.453939i	−0.00463784	+	0.0162018i
$786$	−8.38478	+	5.02559i	−0.299075	+	0.179257i
$787$			8.49148i			0.302688i	0.988481	+	0.151344i	$0.0483602\pi$
$787$			8.49148i			0.302688i	−0.988481	+	0.151344i	$0.951640\pi$
$788$		0			0
$789$	−25.1856	−	42.0201i	−0.896631	−	1.49595i
$790$	−51.5934	−	14.7689i	−1.83561	−	0.525453i
$791$		−	37.4166i		−	1.33038i
$792$		0			0
$793$			81.9404i			2.90979i
$794$			24.9134i			0.884144i
$795$		0			0
$796$		0			0
$797$	45.2075			1.60133			0.800666	−	0.599111i	$-0.204479\pi$
$797$	45.2075			1.60133			0.800666	+	0.599111i	$0.204479\pi$
$798$	−37.8995	+	22.7159i	−1.34163	+	0.804133i
$799$		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
$805$	9.76869	−	34.1258i	0.344301	−	1.20278i
$806$		0			0
$807$	39.2843	−	23.5459i	1.38287	−	0.828853i
$808$			53.9051i			1.89638i
$809$		−	47.6235i		−	1.67435i	−0.546932	−	0.837177i	$-0.684204\pi$
$809$		−	47.6235i		−	1.67435i	0.546932	−	0.837177i	$-0.315796\pi$
$810$	8.68840	+	27.1019i	0.305279	+	0.952263i
$811$	−13.0773			−0.459208			−0.229604	−	0.973284i	$-0.573743\pi$
$811$	−13.0773			−0.459208			−0.229604	+	0.973284i	$0.573743\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$	26.8399	−	50.2129i	0.937863	−	1.75458i
$820$		0			0
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$	22.6670	+	37.8181i	0.790604	+	1.31906i
$823$			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
$826$	−24.1424			−0.840021
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$	−16.9706	+	31.7490i	−0.589768	+	1.10335i
$829$	41.6457			1.44642			0.723208	−	0.690630i	$-0.242667\pi$
$829$	41.6457			1.44642			0.723208	+	0.690630i	$0.242667\pi$
$830$	36.0416	+	10.3171i	1.25102	+	0.358112i
$831$		0			0
$832$		−	57.3862i		−	1.98951i
$833$		0			0
$834$	−4.87006	−	8.12528i	−0.168636	−	0.281355i
$835$		0			0
$836$		0			0
$837$		0			0
$838$		−	51.7423i		−	1.78741i
$839$		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
$843$	−22.2349	+	13.3270i	−0.765812	+	0.459005i
$844$		0			0
$845$	−82.6695	−	23.6646i	−2.84392	−	0.814085i
$846$		0			0
$847$			29.1033i			1.00000i
$848$		0			0
$849$	−23.5858	+	14.1366i	−0.809462	+	0.485168i
$850$		0			0
$851$		0			0
$852$	−47.1674	+	28.2708i	−1.61593	+	0.968541i
$853$			35.4440i			1.21358i	0.794862	+	0.606790i	$0.207543\pi$
$853$			35.4440i			1.21358i	−0.794862	+	0.606790i	$0.792457\pi$
$854$		−	42.7410i		−	1.46257i
$855$	−31.8284	+	32.8449i	−1.08851	+	1.12327i
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$	53.9854			1.84196			0.920979	−	0.389612i	$-0.127391\pi$
$859$	53.9854			1.84196			0.920979	+	0.389612i	$0.127391\pi$
$860$		0			0
$861$		0			0
$862$			52.9150i			1.80229i
$863$	56.5685			1.92562			0.962808	−	0.270187i	$-0.0870856\pi$
$863$	56.5685			1.92562			0.962808	+	0.270187i	$0.0870856\pi$
$864$	29.3585	−	1.44190i	0.998796	−	0.0490545i
$865$	−23.2426	−	6.65332i	−0.790273	−	0.226220i
$866$		0			0
$867$	15.1376	+	25.2558i	0.514099	+	0.857731i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$	−57.8527			−1.95690
$875$	21.9556	−	19.8230i	0.742234	−	0.670141i
$876$		0			0
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$		0			0
$879$	−28.6569	−	47.8116i	−0.966572	−	1.61264i
$880$		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
$885$	−24.1424	+	6.45232i	−0.811537	+	0.216892i
$886$		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
$897$	63.9411	−	38.3245i	2.13493	−	1.27962i
$898$		−	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
$905$	−21.3848	−	6.12150i	−0.710854	−	0.203485i
$906$	−12.5928	−	21.0100i	−0.418368	−	0.698011i
$907$		0			0			−	1.00000i	$-0.5\pi$
$907$		0			0				1.00000i	$0.5\pi$
$908$	14.8582			0.493088
$909$	50.4236	+	26.9526i	1.67245	+	0.893960i
$910$	57.6985	+	16.5165i	1.91269	+	0.547516i
$911$		−	52.3832i		−	1.73553i	−0.496972	−	0.867766i	$-0.665555\pi$
$911$		−	52.3832i		−	1.73553i	0.496972	−	0.867766i	$-0.334445\pi$
$912$	24.2843	+	40.5163i	0.804133	+	1.34163i
$913$		0			0
$914$			7.48331i			0.247526i
$915$	−11.4230	−	42.7410i	−0.377633	−	1.41297i
$916$	53.0688			1.75344
$917$	−10.5588			−0.348682
$918$		0			0
$919$	−16.9706			−0.559807			−0.279904	−	0.960028i	$-0.590303\pi$
$919$	−16.9706			−0.559807			−0.279904	+	0.960028i	$0.590303\pi$
$920$	−36.4821	−	10.4432i	−1.20278	−	0.344301i
$921$	19.0416	−	11.4130i	0.627443	−	0.376071i
$922$			48.6835i			1.60331i
$923$	113.872			3.74815
$924$		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
$931$	−47.7261			−1.56416
$932$	−12.0000			−0.393073
$933$		0			0
$934$	53.2477			1.74232
$935$		0			0
$936$	−53.6799	−	28.6931i	−1.75458	−	0.937863i
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			54.7135i			1.78361i	0.452419	+	0.891805i	$0.350561\pi$
$941$			54.7135i			1.78361i	−0.452419	+	0.891805i	$0.649439\pi$
$942$	0.443651	−	0.265911i	0.0144549	−	0.00866386i
$943$		0			0
$944$			25.8093i			0.840021i
$945$	−7.00000	+	29.9333i	−0.227710	+	0.973729i
$946$		0			0
$947$		0			0		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$	30.2227	+	50.4241i	0.981588	+	1.63770i
$949$		0			0
$950$	−40.9081	−	25.5107i	−1.32723	−	0.827675i
$951$		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
$955$	9.76869	−	34.1258i	0.316107	−	1.10429i
$956$			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
$965$	−16.2811	+	56.8764i	−0.524109	+	1.83092i
$966$	−33.3524	+	19.9905i	−1.07310	+	0.643182i
$967$			22.4499i			0.721942i	0.932577	+	0.360971i	$0.117555\pi$
$967$			22.4499i			0.721942i	−0.932577	+	0.360971i	$0.882445\pi$
$968$	31.1127			1.00000
$969$		0			0
$970$		0			0
$971$			43.9717i			1.41112i	0.708650	+	0.705560i	$0.249305\pi$
$971$			43.9717i			1.41112i	−0.708650	+	0.705560i	$0.750695\pi$
$972$	13.3305	−	28.1833i	0.427575	−	0.903980i
$973$		−	10.2320i		−	0.328023i
$974$			31.7490i			1.01730i
$975$	62.1127	+	1.09591i	1.98920	+	0.0350971i
$976$	−45.6921			−1.46257
$977$	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
$980$	−30.0962	−	8.61517i	−0.961387	−	0.275202i
$981$		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
$988$		−	97.8149i		−	3.11191i
$989$		0			0
$990$		0			0
$991$	50.9117			1.61726			0.808632	−	0.588315i	$-0.200209\pi$
$991$	50.9117			1.61726			0.808632	+	0.588315i	$0.200209\pi$
$992$		0			0
$993$		0			0
$994$	−59.3970			−1.88396
$995$		0			0
$996$	−21.1127	−	35.2248i	−0.668981	−	1.11614i
$997$			4.11454i			0.130309i	0.997875	+	0.0651544i	$0.0207540\pi$
$997$			4.11454i			0.130309i	−0.997875	+	0.0651544i	$0.979246\pi$
$998$		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.629.4	yes	8
3.2	odd	2		840.2.u.c.629.6	yes	8
5.4	even	2		840.2.u.c.629.5	✓	8
7.6	odd	2	inner	840.2.u.d.629.1	yes	8
8.5	even	2	inner	840.2.u.d.629.1	yes	8
15.14	odd	2	inner	840.2.u.d.629.3	yes	8
21.20	even	2		840.2.u.c.629.7	yes	8
24.5	odd	2		840.2.u.c.629.7	yes	8
35.34	odd	2		840.2.u.c.629.8	yes	8
40.29	even	2		840.2.u.c.629.8	yes	8
56.13	odd	2	CM	840.2.u.d.629.4	yes	8
105.104	even	2	inner	840.2.u.d.629.2	yes	8
120.29	odd	2	inner	840.2.u.d.629.2	yes	8
168.125	even	2		840.2.u.c.629.6	yes	8
280.69	odd	2		840.2.u.c.629.5	✓	8
840.629	even	2	inner	840.2.u.d.629.3	yes	8

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

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +(0.890446 + 1.48563i) q^{3} +2.00000 q^{4} +(2.14973 + 0.615370i) q^{5} +(-1.25928 - 2.10100i) q^{6} -2.64575i q^{7} -2.82843 q^{8} +(-1.41421 + 2.64575i) q^{9} +(-3.04017 - 0.870264i) q^{10} +(1.78089 + 2.97127i) q^{12} -7.17327i q^{13} +3.74166i q^{14} +(1.00000 + 3.74166i) q^{15} +4.00000 q^{16} +(2.00000 - 3.74166i) q^{18} +6.81801 q^{19} +(4.29945 + 1.23074i) q^{20} +(3.93062 - 2.35590i) q^{21} +6.00000 q^{23} +(-2.51856 - 4.20201i) q^{24} +(4.24264 + 2.64575i) q^{25} +10.1445i q^{26} +(-5.18990 + 0.254894i) q^{27} -5.29150i q^{28} +(-1.41421 - 5.29150i) q^{30} -5.65685 q^{32} +(1.62811 - 5.68764i) q^{35} +(-2.82843 + 5.29150i) q^{36} -9.64212 q^{38} +(10.6569 - 6.38741i) q^{39} +(-6.08034 - 1.74053i) q^{40} +(-5.55873 + 3.33174i) q^{42} +(-4.66829 + 4.81738i) q^{45} -8.48528 q^{46} +(3.56178 + 5.94253i) q^{48} -7.00000 q^{49} +(-6.00000 - 3.74166i) q^{50} -14.3465i q^{52} +(7.33962 - 0.360475i) q^{54} +7.48331i q^{56} +(6.07107 + 10.1291i) q^{57} +6.45232i q^{59} +(2.00000 + 7.48331i) q^{60} -11.4230 q^{61} +(7.00000 + 3.74166i) q^{63} +8.00000 q^{64} +(4.41421 - 15.4206i) q^{65} +(5.34267 + 8.91380i) q^{69} +(-2.30250 + 8.04354i) q^{70} +15.8745i q^{71} +(4.00000 - 7.48331i) q^{72} +(-0.152776 + 8.65891i) q^{75} +13.6360 q^{76} +(-15.0711 + 9.03316i) q^{78} +16.9706 q^{79} +(8.59890 + 2.46148i) q^{80} +(-5.00000 - 7.48331i) q^{81} -11.8551 q^{83} +(7.86123 - 4.71179i) q^{84} +(6.60195 - 6.81280i) q^{90} -18.9787 q^{91} +12.0000 q^{92} +(14.6569 + 4.19560i) q^{95} +(-5.03712 - 8.40401i) q^{96} +9.89949 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{4} + 8 q^{15} + 32 q^{16} + 16 q^{18} + 48 q^{23} + 40 q^{39} - 56 q^{49} - 48 q^{50} - 8 q^{57} + 16 q^{60} + 56 q^{63} + 64 q^{64} + 24 q^{65} + 32 q^{72} - 64 q^{78} - 40 q^{81} + 96 q^{92}+ \cdots + 72 q^{95}+O(q^{100}) \) 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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more